wpmath0000011_1

Contourlet.html

  1. 4 / 3 4/3
  2. j j
  3. l j l_{j}
  4. w i d t h width
  5. 2 j 2^{j}
  6. l e n g t h length
  7. 2 j + l j - 2 2^{j+l_{j}-2}
  8. O ( N ) O(N)

Cortisol_O-acetyltransferase.html

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Cost–volume–profit_analysis.html

  1. Total costs = fixed costs + ( unit variable cost × number of units \,\text{Total costs}=\,\text{fixed costs}+(\,\text{unit variable cost}\times\,% \text{number of units}
  2. Total revenue = sales price × number of units \,\text{Total revenue}=\,\text{sales price}\times\,\text{number of units}
  3. TC = TFC + V × X \,\text{TC}=\,\text{TFC}+\,\text{V}\times\,\text{X}
  4. TR = P × X \,\text{TR}=\,\text{P}\times\,\text{X}
  5. TC = TFC + V × X \,\text{TC}=\,\text{TFC}+\,\text{V}\times\,\text{X}
  6. TR \displaystyle\,\text{TR}
  7. PL \displaystyle\,\text{PL}
  8. ( 1 0 - V 1 ) \left(\begin{smallmatrix}1&0\\ -V&1\end{smallmatrix}\right)

Coulomb_gap.html

  1. E i E_{i}
  2. i i
  3. i i
  4. j j
  5. Δ E = E j - E i - e 2 / r i j \Delta E=E_{j}-E_{i}-e^{2}/r_{ij}
  6. E j E_{j}
  7. i i
  8. E f E_{f}
  9. g ( E f ) g(E_{f})
  10. Δ E 0 \Delta E>=0
  11. [ E f - ϵ , E f + ϵ ] . [E_{f}-\epsilon,E_{f}+\epsilon].
  12. N = 2 ϵ g ( E f ) . N=2\epsilon g(E_{f}).
  13. N / 2 N/2
  14. R ( N / V ) - 1 / d R\sim(N/V)^{-1/d}
  15. N N
  16. E j - E i - C e 2 ( ϵ g ( E f ) / V ) 1 / d > 0 E_{j}-E_{i}-Ce^{2}(\epsilon g(E_{f})/V)^{1/d}>0
  17. C C
  18. E j - E i < 2 ϵ E_{j}-E_{i}<2\epsilon
  19. ϵ \epsilon
  20. E f E_{f}
  21. E f E_{f}
  22. ( E - E f ) α (E-E_{f})^{\alpha}
  23. α d - 1 \alpha>=d-1
  24. e - 1 / T 1 / 2 e^{-1/T^{1/2}}
  25. e - 1 / T α e^{-1/T^{\alpha}}

Coupon_collector's_problem.html

  1. Θ ( n log ( n ) ) \Theta(n\log(n))
  2. E ( T ) \displaystyle\operatorname{E}(T)
  3. E ( T ) = n H n = n log n + γ n + 1 2 + o ( 1 ) , as n , \operatorname{E}(T)=n\cdot H_{n}=n\log n+\gamma n+\frac{1}{2}+o(1),\ \ \,\text% {as}\ n\to\infty,
  4. γ 0.5772156649 \gamma\approx 0.5772156649
  5. P ( T c n H n ) 1 c . \operatorname{P}(T\geq c\,nH_{n})\leq\frac{1}{c}.
  6. Var ( T ) \displaystyle\operatorname{Var}(T)
  7. π 2 / 6 \pi^{2}/6
  8. P ( | T - n H n | c n ) π 2 6 c 2 . \operatorname{P}\left(|T-nH_{n}|\geq c\,n\right)\leq\frac{\pi^{2}}{6c^{2}}.
  9. Z i r {Z}_{i}^{r}
  10. i i
  11. r r
  12. P [ Z i r ] = ( 1 - 1 n ) r e - r / n \displaystyle P\left[{Z}_{i}^{r}\right]=\left(1-\frac{1}{n}\right)^{r}\leq e^{% -r/n}
  13. r = β n log n r=\beta n\log n
  14. P [ Z i r ] e ( - β n log n ) / n = n - β P\left[{Z}_{i}^{r}\right]\leq e^{(-\beta n\log n)/n}=n^{-\beta}
  15. P [ T > β n log n ] = P [ i Z i β n log n ] n P [ Z 1 β n log n ] n - β + 1 \begin{aligned}\displaystyle P\left[T>\beta n\log n\right]=P\left[\bigcup_{i}{% Z}_{i}^{\beta n\log n}\right]\leq n\cdot P[{Z}_{1}^{\beta n\log n}]\leq n^{-% \beta+1}\end{aligned}
  16. P ( T < n log n + c n ) e - e - c , as n . \operatorname{P}(T<n\log n+cn)\to e^{-e^{-c}},\ \ \,\text{as}\ n\to\infty.
  17. E ( T m ) = n log n + ( m - 1 ) n log log n + O ( n ) , as n . \operatorname{E}(T_{m})=n\log n+(m-1)n\log\log n+O(n),\ \ \,\text{as}\ n\to\infty.
  18. P ( T k < n log n + ( k - 1 ) n log log n + c n ) e - e - c / ( k - 1 ) ! , as n . \operatorname{P}(T_{k}<n\log n+(k-1)n\log\log n+cn)\to e^{-e^{-c}/(k-1)!},\ \ % \,\text{as}\ n\to\infty.
  19. E ( T ) = 0 ( 1 - i = 1 n ( 1 - e - p i t ) ) d t E(T)=\int_{0}^{\infty}\big(1-\prod_{i=1}^{n}(1-e^{-p_{i}t})\big)dt
  20. n log n + γ n + 1 / 2 n\log n+\gamma n+1/2
  21. 50 log 50 + 50 γ + 1 / 2 195.6011 + 28.8608 + 0.5 224.9619 50\log 50+50\gamma+1/2\approx 195.6011+28.8608+0.5\approx 224.9619

Covering_relation.html

  1. X X
  2. \leq
  3. < <
  4. X X
  5. x < y x<y
  6. x y x\leq y
  7. x y x\neq y
  8. x x
  9. y y
  10. X X
  11. y y
  12. x x
  13. x y x\lessdot y
  14. x < y x<y
  15. z z
  16. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  17. y y
  18. x x
  19. [ x , y ] [x,y]
  20. { x , y } \{x,y\}
  21. x y x\lessdot y
  22. y y
  23. x x
  24. ( x , y ) (x,y)

Creatinase.html

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Creatininase.html

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Creatinine_deaminase.html

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Cribbage_statistics.html

  1. ( 52 4 ) × 48 = 12 , 994 , 800 {52\choose 4}\times 48=12,994,800
  2. ( 52 5 ) × 5 = 12 , 994 , 800 {52\choose 5}\times 5=12,994,800
  3. ( 52 4 ) × ( 48 4 ) × 44 = 2 , 317 , 817 , 502 , 000 {52\choose 4}\times{48\choose 4}\times 44=2,317,817,502,000

Critical_state_soil_mechanics.html

  1. ε s \ \varepsilon_{s}
  2. p \ p^{\prime}
  3. q \ q
  4. σ y \ \sigma_{y}
  5. ν \ \nu
  6. p ε s = q ε s = ν ε s = 0 \ \frac{\partial p^{\prime}}{\partial\varepsilon_{s}}=\frac{\partial q}{% \partial\varepsilon_{s}}=\frac{\partial\nu}{\partial\varepsilon_{s}}=0
  7. ν = 1 + e \ \nu=1+e
  8. p = 1 3 ( σ 1 + σ 2 + σ 3 ) \ p^{\prime}=\frac{1}{3}(\sigma_{1}^{\prime}+\sigma_{2}^{\prime}+\sigma_{3}^{% \prime})
  9. q = ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 1 - σ 3 ) 2 2 \ q=\sqrt{\frac{(\sigma_{1}^{\prime}-\sigma_{2}^{\prime})^{2}+(\sigma_{2}^{% \prime}-\sigma_{3}^{\prime})^{2}+(\sigma_{1}^{\prime}-\sigma_{3}^{\prime})^{2}% }{2}}
  10. σ 2 = σ 3 \ \sigma_{2}^{\prime}=\sigma_{3}^{\prime}
  11. p = 1 3 ( σ 1 + 2 σ 3 ) \ p^{\prime}=\frac{1}{3}(\sigma_{1}^{\prime}+2\sigma_{3}^{\prime})
  12. q = ( σ 1 - σ 3 ) \ q=(\sigma_{1}^{\prime}-\sigma_{3}^{\prime})
  13. ( p , q , v ) \ (p^{\prime},q,v)
  14. q = M p \ q=Mp^{\prime}
  15. ν = Γ - λ ln ( p ) \ \nu=\Gamma-\lambda\ln(p^{\prime})
  16. M \ M
  17. Γ \ \Gamma
  18. λ \ \lambda
  19. q \ q
  20. M \ M
  21. μ \ \mu
  22. p \ p^{\prime}
  23. ν \ \nu
  24. f ( p , q , p c ) = q + M p ln [ p p c ] 0 f(p,q,p_{c})=q+M\,p\,\ln\left[\frac{p}{p_{c}}\right]\leq 0
  25. q q
  26. p p
  27. p c p_{c}
  28. M M
  29. p - q p-q
  30. e e
  31. v v
  32. e = e 0 - λ ln [ p c p c 0 ] e=e_{0}-\lambda\ln\left[\frac{p_{c}}{p_{c0}}\right]
  33. λ \lambda
  34. p c p_{c}
  35. ln [ 1 + e 1 + e 0 ] = ln [ v v 0 ] = - λ ~ ln [ p c p c 0 ] \ln\left[\frac{1+e}{1+e_{0}}\right]=\ln\left[\frac{v}{v_{0}}\right]=-\tilde{% \lambda}\ln\left[\frac{p_{c}}{p_{c0}}\right]
  36. λ ~ \tilde{\lambda}
  37. f ( p , q , p c ) = [ q M ] 2 + p ( p - p c ) 0 f(p,q,p_{c})=\left[\frac{q}{M}\right]^{2}+p\,(p-p_{c})\leq 0
  38. p p
  39. q q
  40. p c p_{c}
  41. M M

Crotonoyl-(acyl-carrier-protein)_hydratase.html

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Cunningham_number.html

  1. b n ± 1 b^{n}\pm 1\,

Curve_resistance_(railroad).html

  1. v 2 R c o s ( θ ) \frac{v^{2}}{R}cos(\theta)

CUSUM.html

  1. θ \theta
  2. θ \theta
  3. x n x_{n}
  4. ω n \omega_{n}
  5. S 0 = 0 S_{0}=0
  6. S n + 1 = max ( 0 , S n + x n - ω n ) S_{n+1}=\max(0,S_{n}+x_{n}-\omega_{n})
  7. ω \omega

Cutinase.html

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Cyanamide_hydratase.html

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Cyanide_hydratase.html

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Cyanidin_3-O-rutinoside_5-O-glucosyltransferase.html

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Cyanoalanine_nitrilase.html

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Cyanohydrin_beta-glucosyltransferase.html

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Cyanuric_acid_amidohydrolase.html

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Cyclohexa-1,5-dienecarbonyl-CoA_hydratase.html

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Cyclohexyl-isocyanide_hydratase.html

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Cyclomaltodextrinase.html

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Cylindrical_harmonics.html

  1. 2 V = 0 \nabla^{2}V=0
  2. V n ( k ) V_{n}(k)
  3. V n ( k ; ρ , φ , z ) = P n ( k , ρ ) Φ n ( φ ) Z ( k , z ) V_{n}(k;\rho,\varphi,z)=P_{n}(k,\rho)\Phi_{n}(\varphi)Z(k,z)\,
  4. ( ρ , φ , z ) (\rho,\varphi,z)
  5. V = P ( ρ ) Φ ( φ ) Z ( z ) V=P(\rho)\,\Phi(\varphi)\,Z(z)
  6. P ¨ P + 1 ρ P ˙ P + 1 ρ 2 Φ ¨ Φ + Z ¨ Z = 0 \frac{\ddot{P}}{P}+\frac{1}{\rho}\,\frac{\dot{P}}{P}+\frac{1}{\rho^{2}}\,\frac% {\ddot{\Phi}}{\Phi}+\frac{\ddot{Z}}{Z}=0
  7. Z ¨ Z = k 2 \frac{\ddot{Z}}{Z}=k^{2}
  8. Z ( k , z ) = cosh ( k z ) or sinh ( k z ) Z(k,z)=\cosh(kz)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sinh(kz)\,
  9. Z ( k , z ) = e k z or e - k z Z(k,z)=e^{kz}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-kz}\,
  10. Z ( k , z ) = cos ( | k | z ) or sin ( | k | z ) Z(k,z)=\cos(|k|z)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sin(|k|z)\,
  11. Z ( k , z ) = e i | k | z or e - i | k | z Z(k,z)=e^{i|k|z}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-i|k|z}\,
  12. k 2 k^{2}
  13. Z ¨ / Z \ddot{Z}/Z
  14. P ¨ P + 1 ρ P ˙ P + 1 ρ 2 Φ ¨ Φ + k 2 = 0 \frac{\ddot{P}}{P}+\frac{1}{\rho}\,\frac{\dot{P}}{P}+\frac{1}{\rho^{2}}\frac{% \ddot{\Phi}}{\Phi}+k^{2}=0
  15. ρ 2 \rho^{2}
  16. Φ ¨ Φ = - n 2 \frac{\ddot{\Phi}}{\Phi}=-n^{2}
  17. ρ 2 P ¨ P + ρ P ˙ P + k 2 ρ 2 = n 2 \rho^{2}\frac{\ddot{P}}{P}+\rho\frac{\dot{P}}{P}+k^{2}\rho^{2}=n^{2}
  18. φ \varphi
  19. Φ ( φ ) \Phi(\varphi)
  20. Φ ( φ ) \Phi(\varphi)
  21. Φ n = cos ( n φ ) or sin ( n φ ) \Phi_{n}=\cos(n\varphi)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sin(n\varphi)
  22. Φ n = e i n φ or e - i n φ \Phi_{n}=e^{in\varphi}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-in\varphi}
  23. ρ \rho
  24. P n ( 0 , ρ ) = ρ n or ρ - n P_{n}(0,\rho)=\rho^{n}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\rho^{-n}\,
  25. P 0 ( 0 , ρ ) = ln ρ or 1 P_{0}(0,\rho)=\ln\rho\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,1\,
  26. P n ( k , ρ ) = J n ( k ρ ) or Y n ( k ρ ) P_{n}(k,\rho)=J_{n}(k\rho)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,Y_{n}(k\rho)\,
  27. J n ( z ) J_{n}(z)
  28. Y n ( z ) Y_{n}(z)
  29. P n ( k , ρ ) = I n ( | k | ρ ) or K n ( | k | ρ ) P_{n}(k,\rho)=I_{n}(|k|\rho)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,K_{n}(|k|\rho)\,
  30. I n ( z ) I_{n}(z)
  31. K n ( z ) K_{n}(z)
  32. V ( ρ , φ , z ) = n d k A n ( k ) P n ( k , ρ ) Φ n ( φ ) Z ( k , z ) V(\rho,\varphi,z)=\sum_{n}\int dk\,\,A_{n}(k)P_{n}(k,\rho)\Phi_{n}(\varphi)Z(k% ,z)\,
  33. A n ( k ) A_{n}(k)
  34. J n ( x ) J_{n}(x)
  35. Φ n ( φ ) \Phi_{n}(\varphi)
  36. Z ( k , z ) Z(k,z)
  37. P n ( k ρ ) P_{n}(k\rho)
  38. J n ( k ρ ) J_{n}(k\rho)
  39. J n J_{n}
  40. Φ n ( φ ) \Phi_{n}(\varphi)
  41. Z ( k , z ) Z(k,z)
  42. ( x ) k (x)_{k}
  43. J n J_{n}
  44. 0 1 J n ( x k ρ ) J n ( x k ρ ) ρ d ρ = 1 2 J n + 1 ( x k ) 2 δ k k \int_{0}^{1}J_{n}(x_{k}\rho)J_{n}(x_{k}^{\prime}\rho)\rho\,d\rho=\frac{1}{2}J_% {n+1}(x_{k})^{2}\delta_{kk^{\prime}}
  45. ( ρ 0 , φ 0 , z 0 ) (\rho_{0},\varphi_{0},z_{0})
  46. z = - L z=-L
  47. z = L z=L
  48. ρ = a \rho=a
  49. q / 4 π ϵ 0 = 1 q/4\pi\epsilon_{0}=1
  50. P n ( k ρ ) P_{n}(k\rho)
  51. J n ( k ρ ) J_{n}(k\rho)
  52. V ( ρ , φ , z ) = n = 0 r = 0 A n r J n ( k n r ρ ) cos ( n ( φ - φ 0 ) ) sinh ( k n r ( L + z ) ) z z 0 V(\rho,\varphi,z)=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\,A_{nr}J_{n}(k_{nr}% \rho)\cos(n(\varphi-\varphi_{0}))\sinh(k_{nr}(L+z))\,\,\,\,\,z\leq z_{0}
  53. k n r a k_{nr}a
  54. J n ( z ) J_{n}(z)
  55. A n r = 4 ( 2 - δ n 0 ) a 2 sinh k n r ( L - z 0 ) sinh 2 k n r L J n ( k n r ρ 0 ) k n r [ J n + 1 ( k n r a ) ] 2 A_{nr}=\frac{4(2-\delta_{n0})}{a^{2}}\,\,\frac{\sinh k_{nr}(L-z_{0})}{\sinh 2k% _{nr}L}\,\,\frac{J_{n}(k_{nr}\rho_{0})}{k_{nr}[J_{n+1}(k_{nr}a)]^{2}}\,
  56. V ( ρ , φ , z ) = n = 0 r = 0 A n r J n ( k n r ρ ) cos ( n ( φ - φ 0 ) ) sinh ( k n r ( L - z ) ) z z 0 V(\rho,\varphi,z)=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\,A_{nr}J_{n}(k_{nr}% \rho)\cos(n(\varphi-\varphi_{0}))\sinh(k_{nr}(L-z))\,\,\,\,\,z\geq z_{0}
  57. A n r = 4 ( 2 - δ n 0 ) a 2 sinh k n r ( L + z 0 ) sinh 2 k n r L J n ( k n r ρ 0 ) k n r [ J n + 1 ( k n r a ) ] 2 . A_{nr}=\frac{4(2-\delta_{n0})}{a^{2}}\,\,\frac{\sinh k_{nr}(L+z_{0})}{\sinh 2k% _{nr}L}\,\,\frac{J_{n}(k_{nr}\rho_{0})}{k_{nr}[J_{n+1}(k_{nr}a)]^{2}}.\,
  58. ρ = a \rho=a
  59. | z | = L |z|=L
  60. z = z 0 z=z_{0}
  61. V ( ρ , φ , z ) = n = 0 r = 0 A n r J n ( k n r ρ ) cos ( n ( φ - φ 0 ) ) e - k n r | z - z 0 | V(\rho,\varphi,z)=\sum_{n=0}^{\infty}\sum_{r=0}^{\infty}\,A_{nr}J_{n}(k_{nr}% \rho)\cos(n(\varphi-\varphi_{0}))e^{-k_{nr}|z-z_{0}|}
  62. A n r = 2 ( 2 - δ n 0 ) a 2 J n ( k n r ρ 0 ) k n r [ J n + 1 ( k n r a ) ] 2 . A_{nr}=\frac{2(2-\delta_{n0})}{a^{2}}\,\,\frac{J_{n}(k_{nr}\rho_{0})}{k_{nr}[J% _{n+1}(k_{nr}a)]^{2}}.\,
  63. V ( ρ , φ , z ) = 1 R = n = 0 0 d k A n ( k ) J n ( k ρ ) cos ( n ( φ - φ 0 ) ) e - k | z - z 0 | V(\rho,\varphi,z)=\frac{1}{R}=\sum_{n=0}^{\infty}\int_{0}^{\infty}dk\,A_{n}(k)% J_{n}(k\rho)\cos(n(\varphi-\varphi_{0}))e^{-k|z-z_{0}|}
  64. A n ( k ) = ( 2 - δ n 0 ) J n ( k ρ 0 ) A_{n}(k)=(2-\delta_{n0})J_{n}(k\rho_{0})\,
  65. R = ( z - z 0 ) 2 + ρ 2 + ρ 0 2 - 2 ρ ρ 0 cos ( φ - φ 0 ) . R=\sqrt{(z-z_{0})^{2}+\rho^{2}+\rho_{0}^{2}-2\rho\rho_{0}\cos(\varphi-\varphi_% {0})}.\,
  66. ρ 0 = z 0 = 0 \rho_{0}=z_{0}=0
  67. V ( ρ , φ , z ) = 1 ρ 2 + z 2 = 0 J 0 ( k ρ ) e - k | z | d k . V(\rho,\varphi,z)=\frac{1}{\sqrt{\rho^{2}+z^{2}}}=\int_{0}^{\infty}J_{0}(k\rho% )e^{-k|z|}\,dk.

Cystathionine_gamma-synthase.html

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Cysteine-conjugate_transaminase.html

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Cysteine-S-conjugate_N-acetyltransferase.html

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Cysteine_synthase.html

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Cysteine_transaminase.html

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Cysteine—tRNA_ligase.html

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Cytidylate_kinase.html

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Cytochrome_c_oxidase_subunit_III.html

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Cytokinin_7-beta-glucosyltransferase.html

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Cytosine_deaminase.html

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D(-)-tartrate_dehydratase.html

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D-4-hydroxyphenylglycine_transaminase.html

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D-alanine_gamma-glutamyltransferase.html

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D-alanine—alanyl-poly(glycerolphosphate)_ligase.html

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D-alanine—D-alanine_ligase.html

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D-alanine—poly(phosphoribitol)_ligase.html

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D-amino-acid_N-acetyltransferase.html

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D-amino-acid_transaminase.html

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D-arabinokinase.html

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D-arabinonolactonase.html

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D-aspartate_ligase.html

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D-benzoylarginine-4-nitroanilide_amidase.html

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D-fuconate_dehydratase.html

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D-glutamate_cyclase.html

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D-glutaminase.html

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D-glutamyltransferase.html

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D-lactate-2-sulfatase.html

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D-methionine—pyruvate_transaminase.html

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D-ribitol-5-phosphate_cytidylyltransferase.html

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D-ribulokinase.html

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D-tryptophan_N-acetyltransferase.html

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D-tryptophan_N-malonyltransferase.html

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Darboux's_formula.html

  1. m = 0 n ( - 1 ) m ( z - a ) m [ ϕ ( n - m ) ( 1 ) f ( m ) ( z ) - ϕ ( n - m ) ( 0 ) f ( m ) ( a ) ] \displaystyle\sum_{m=0}^{n}(-1)^{m}(z-a)^{m}\left[\phi^{(n-m)}(1)f^{(m)}(z)-% \phi^{(n-m)}(0)f^{(m)}(a)\right]

DCTP_deaminase.html

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DCTP_deaminase_(dUMP-forming).html

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DCTP_diphosphatase.html

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De_Longchamps_point.html

  1. A A
  2. B B
  3. C C
  4. a a
  5. b b
  6. c c
  7. Δ \Delta
  8. Δ a \Delta_{a}
  9. Δ b \Delta_{b}
  10. Δ C \Delta_{C}
  11. Δ a \Delta_{a}
  12. A A
  13. a a
  14. A B C ABC
  15. A B C ABC
  16. A A
  17. B B
  18. C C
  19. s - a s-a
  20. s - b s-b
  21. s - c s-c
  22. s s
  23. X X
  24. X X
  25. X X

Deacetyl-(citrate-(pro-3S)-lyase)_S-acetyltransferase.html

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Deacetylcephalosporin-C_acetyltransferase.html

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Deacetylvindoline_O-acetyltransferase.html

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Debtor_days.html

  1. Debtor days = Year end trade debtors Sales × Number of days in financial year \mbox{Debtor days}~{}=\frac{\mbox{Year end trade debtors}~{}}{\mbox{Sales}~{}}% \times{\mbox{Number of days in financial year}~{}}
  2. Debtor days = Average trade debtors Sales × Number of days in financial year \mbox{Debtor days}~{}=\frac{\mbox{Average trade debtors}~{}}{\mbox{Sales}~{}}% \times{\mbox{Number of days in financial year}~{}}
  3. Average trade debtors = Opening trade debtors + Closing trade debtors 2 \mbox{Average trade debtors}~{}=\frac{\mbox{Opening trade debtors}~{}+\mbox{% Closing trade debtors}~{}}{\mbox{2}~{}}

Decylcitrate_synthase.html

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Decylhomocitrate_synthase.html

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Deformation_(meteorology).html

  1. def 𝐕 = A 2 + B 2 \operatorname{def}\mathbf{V}=\sqrt{A^{2}+B^{2}}
  2. A = v x + u y \ A=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}
  3. B = u x - v y \ B=\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}
  4. A \ A
  5. B \ B
  6. A \ A
  7. B \ B

Degasperis–Procesi_equation.html

  1. u t - u x x t + 2 κ u x + 4 u u x = 3 u x u x x + u u x x x \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}
  2. u t - u x x t + 2 κ u x + ( b + 1 ) u u x = b u x u x x + u u x x x , \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx},
  3. κ \kappa
  4. κ > 0 \kappa>0
  5. κ = 0 \kappa=0
  6. u ( x , t ) = i = 1 n m i ( t ) e - | x - x i ( t ) | \displaystyle u(x,t)=\sum_{i=1}^{n}m_{i}(t)e^{-|x-x_{i}(t)|}
  7. m i m_{i}
  8. x i x_{i}
  9. x ˙ i = j = 1 n m j e - | x i - x j | , m ˙ i = 2 m i j = 1 n m j sgn ( x i - x j ) e - | x i - x j | . \dot{x}_{i}=\sum_{j=1}^{n}m_{j}e^{-|x_{i}-x_{j}|},\qquad\dot{m}_{i}=2m_{i}\sum% _{j=1}^{n}m_{j}\,\operatorname{sgn}{(x_{i}-x_{j})}e^{-|x_{i}-x_{j}|}.
  10. κ > 0 \kappa>0
  11. κ \kappa
  12. κ = 0 \kappa=0
  13. t u + x [ u 2 2 + G 2 * 3 u 2 2 ] = 0 , \partial_{t}u+\partial_{x}\left[\frac{u^{2}}{2}+\frac{G}{2}*\frac{3u^{2}}{2}% \right]=0,
  14. G ( x ) = exp ( - | x | ) G(x)=\exp(-|x|)
  15. u 2 u^{2}
  16. u x 2 u_{x}^{2}
  17. H 1 = W 1 , 2 H^{1}=W^{1,2}

Dehydrogluconokinase.html

  1. \rightleftharpoons

Delannoy_number.html

  1. D D
  2. D ( m , n ) D(m,n)
  3. m m
  4. n n
  5. k k
  6. m - k m-k
  7. x x
  8. n - k n-k
  9. y y
  10. ( m , n ) (m,n)
  11. ( m + n - k k , m - k , n - k ) = ( m + n - k m ) ( m k ) {\left({{m+n-k}\atop{k,m-k,n-k}}\right)}={\left({{m+n-k}\atop{m}}\right)}{% \left({{m}\atop{k}}\right)}
  12. D ( m , n ) = k = 0 min ( m , n ) ( m + n - k m ) ( m k ) . D(m,n)=\sum_{k=0}^{\min(m,n)}{\left({{m+n-k}\atop{m}}\right)}{\left({{m}\atop{% k}}\right)}.
  13. D ( m , n ) = k = 0 min ( m , n ) ( m k ) ( n k ) 2 k . D(m,n)=\sum_{k=0}^{\min(m,n)}{\left({{m}\atop{k}}\right)}{\left({{n}\atop{k}}% \right)}2^{k}.
  14. D ( m , n ) = { 1 if m = 0 or n = 0 D ( m - 1 , n ) + D ( m - 1 , n - 1 ) + D ( m , n - 1 ) otherwise D(m,n)=\begin{cases}1&\,\text{if }m=0\,\text{ or }n=0\\ D(m-1,n)+D(m-1,n-1)+D(m,n-1)&\,\text{otherwise}\end{cases}
  15. m , n = 0 D ( m , n ) x m y n = ( 1 - x - y - x y ) - 1 . \sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n}=(1-x-y-xy)^{-1}.
  16. m = n m=n
  17. k n - k k\leftrightarrow n-k
  18. D ( n ) = k = 0 n ( n k ) ( n + k k ) , D(n)=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}{\left({{n+k}\atop{k}}\right)},
  19. D ( n ) = k = 0 n ( n k ) 2 2 k . D(n)=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}^{2}2^{k}.
  20. n D ( n ) = 3 ( 2 n - 1 ) D ( n - 1 ) - ( n - 1 ) D ( n - 2 ) , nD(n)=3(2n-1)D(n-1)-(n-1)D(n-2),
  21. n = 0 D ( n ) x n = ( 1 - 6 x + x 2 ) - 1 / 2 . \sum_{n=0}^{\infty}D(n)x^{n}=(1-6x+x^{2})^{-1/2}.
  22. D ( n ) = c α n n ( 1 + O ( n - 1 ) ) D(n)=\frac{c\,\alpha^{n}}{\sqrt{n}}\,(1+O(n^{-1}))
  23. α = 3 + 2 2 5.828 \alpha=3+2\sqrt{2}\approx 5.828
  24. c = ( 4 π ( 3 2 - 4 ) ) - 1 / 2 0.5727 c=(4\pi(3\sqrt{2}-4))^{-1/2}\approx 0.5727

Delta-v_(physics).html

  1. Δ 𝐯 = 𝐯 1 - 𝐯 0 = t 0 t 1 𝐚 d t \Delta\mathbf{v}=\mathbf{v}_{1}-\mathbf{v}_{0}=\int^{t_{1}}_{t_{0}}\mathbf{a}% \,dt
  2. Δ v = v 1 - v 0 = t 0 t 1 a d t \Delta{v}={v}_{1}-{v}_{0}=\int^{t_{1}}_{t_{0}}{a}\,dt
  3. Δ 𝐩 = m Δ 𝐯 \Delta{\mathbf{p}}=m\Delta{\mathbf{v}}
  4. 𝐩 \mathbf{p}

Denavit–Hartenberg_parameters.html

  1. [ T ] = [ Z 1 ] [ X 1 ] [ Z 2 ] [ X 2 ] [ X n - 1 ] [ Z n ] , [T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\ldots[X_{n-1}][Z_{n}],\!
  2. [ Z i ] = [ cos θ i - sin θ i 0 0 sin θ i cos θ i 0 0 0 0 1 d i 0 0 0 1 ] , [Z_{i}]=\begin{bmatrix}\cos\theta_{i}&-\sin\theta_{i}&0&0\\ \sin\theta_{i}&\cos\theta_{i}&0&0\\ 0&0&1&d_{i}\\ 0&0&0&1\end{bmatrix},
  3. [ X i ] = [ 1 0 0 r i , i + 1 0 cos α i , i + 1 - sin α i , i + 1 0 0 sin α i , i + 1 cos α i , i + 1 0 0 0 0 1 ] , [X_{i}]=\begin{bmatrix}1&0&0&r_{i,i+1}\\ 0&\cos\alpha_{i,i+1}&-\sin\alpha_{i,i+1}&0\\ 0&\sin\alpha_{i,i+1}&\cos\alpha_{i,i+1}&0\\ 0&0&0&1\end{bmatrix},
  4. z z
  5. x x
  6. x n = z n - 1 × z n x_{n}=z_{n-1}\times z_{n}
  7. z z
  8. d d
  9. x n x_{n}
  10. z n - 1 z_{n-1}
  11. z n z_{n}
  12. y y
  13. x x
  14. z z
  15. θ i , d i , a i , α i \theta_{i},d_{i},a_{i},\alpha_{i}
  16. O i - 1 X i - 1 Y i - 1 Z i - 1 O_{i-1}X_{i-1}Y_{i-1}Z_{i-1}
  17. O i X i Y i Z i O_{i}X_{i}Y_{i}Z_{i}
  18. d d\,
  19. z z
  20. θ \theta\,
  21. z z
  22. x x
  23. x x
  24. r r\,
  25. a a
  26. α \alpha
  27. z z
  28. α \alpha\,
  29. z z
  30. z z
  31. x x
  32. x x
  33. x x
  34. x n x_{n}
  35. z n - 1 z_{n-1}
  36. z n z_{n}
  37. x n x_{n}
  38. z n - 1 z_{n-1}
  39. z n z_{n}
  40. n n
  41. x n x_{n}
  42. z n z_{n}
  43. y n y_{n}
  44. x n x_{n}
  45. z n z_{n}
  46. [ Z i ] = Trans Z i ( d i ) Rot Z i ( θ i ) , [Z_{i}]=\operatorname{Trans}_{Z_{i}}(d_{i})\operatorname{Rot}_{Z_{i}}(\theta_{% i}),
  47. [ X i ] = Trans X i ( r i , i + 1 ) Rot X i ( α i , i + 1 ) . [X_{i}]=\operatorname{Trans}_{X_{i}}(r_{i,i+1})\operatorname{Rot}_{X_{i}}(% \alpha_{i,i+1}).
  48. T n n - 1 = Trans z n - 1 ( d n ) Rot z n - 1 ( θ n ) Trans x n ( r n ) Rot x n ( α n ) {}^{n-1}T_{n}=\operatorname{Trans}_{z_{n-1}}(d_{n})\cdot\operatorname{Rot}_{z_% {n-1}}(\theta_{n})\cdot\operatorname{Trans}_{x_{n}}(r_{n})\cdot\operatorname{% Rot}_{x_{n}}(\alpha_{n})
  49. Trans z n - 1 ( d n ) = [ 1 0 0 0 0 1 0 0 0 0 1 d n 0 0 0 1 ] \operatorname{Trans}_{z_{n-1}}(d_{n})=\left[\begin{array}[]{ccc|c}1&0&0&0\\ 0&1&0&0\\ 0&0&1&d_{n}\\ \hline 0&0&0&1\end{array}\right]
  50. Rot z n - 1 ( θ n ) = [ cos θ n - sin θ n 0 0 sin θ n cos θ n 0 0 0 0 1 0 0 0 0 1 ] \operatorname{Rot}_{z_{n-1}}(\theta_{n})=\left[\begin{array}[]{ccc|c}\cos% \theta_{n}&-\sin\theta_{n}&0&0\\ \sin\theta_{n}&\cos\theta_{n}&0&0\\ 0&0&1&0\\ \hline 0&0&0&1\end{array}\right]
  51. Trans x n ( r n ) = [ 1 0 0 r n 0 1 0 0 0 0 1 0 0 0 0 1 ] \operatorname{Trans}_{x_{n}}(r_{n})=\left[\begin{array}[]{ccc|c}1&0&0&r_{n}\\ 0&1&0&0\\ 0&0&1&0\\ \hline 0&0&0&1\end{array}\right]
  52. Rot x n ( α n ) = [ 1 0 0 0 0 cos α n - sin α n 0 0 sin α n cos α n 0 0 0 0 1 ] \operatorname{Rot}_{x_{n}}(\alpha_{n})=\left[\begin{array}[]{ccc|c}1&0&0&0\\ 0&\cos\alpha_{n}&-\sin\alpha_{n}&0\\ 0&\sin\alpha_{n}&\cos\alpha_{n}&0\\ \hline 0&0&0&1\end{array}\right]
  53. n - 1 T n = [ cos θ n - sin θ n cos α n sin θ n sin α n r n cos θ n sin θ n cos θ n cos α n - cos θ n sin α n r n sin θ n 0 sin α n cos α n d n 0 0 0 1 ] = [ R T 0 0 0 1 ] \operatorname{}^{n-1}T_{n}=\left[\begin{array}[]{ccc|c}\cos\theta_{n}&-\sin% \theta_{n}\cos\alpha_{n}&\sin\theta_{n}\sin\alpha_{n}&r_{n}\cos\theta_{n}\\ \sin\theta_{n}&\cos\theta_{n}\cos\alpha_{n}&-\cos\theta_{n}\sin\alpha_{n}&r_{n% }\sin\theta_{n}\\ 0&\sin\alpha_{n}&\cos\alpha_{n}&d_{n}\\ \hline 0&0&0&1\end{array}\right]=\left[\begin{array}[]{ccc|c}&&&\\ &R&&T\\ &&&\\ \hline 0&0&0&1\end{array}\right]
  54. n n
  55. n - 1 n-1
  56. T T
  57. M M
  58. n - 1 T n = M n - 1 , n \operatorname{}^{n-1}T_{n}=M_{n-1,n}
  59. n n
  60. n - 1 n-1
  61. P ( n - 1 ) = M n - 1 , n P ( n ) = [ x n - 1 y n - 1 z n - 1 1 ] = [ X x Y x Z x T x X y Y y Z y T y X z Y z Z z T z 0 0 0 1 ] [ x n y n z n 1 ] P_{(n-1)}=M_{n-1,n}P_{(n)}=\left[\begin{array}[]{c}x_{n-1}\\ y_{n-1}\\ z_{n-1}\\ 1\end{array}\right]=\left[\begin{array}[]{ccc|c}X_{x}&Y_{x}&Z_{x}&T_{x}\\ X_{y}&Y_{y}&Z_{y}&T_{y}\\ X_{z}&Y_{z}&Z_{z}&T_{z}\\ \hline 0&0&0&1\end{array}\right]\left[\begin{array}[]{c}x_{n}\\ y_{n}\\ z_{n}\\ 1\end{array}\right]
  62. 3 × 3 3\times 3
  63. M M
  64. 3 × 1 3\times 1
  65. k k
  66. i i
  67. j j
  68. i i
  69. k k
  70. j j
  71. M i , k = M i , j M j , k M_{i,k}=M_{i,j}M_{j,k}
  72. M - 1 = [ R T - R T T 0 0 0 1 ] M^{-1}=\left[\begin{array}[]{ccc|c}&&&\\ &R^{T}&&-R^{T}T\\ &&&\\ \hline 0&0&0&1\end{array}\right]
  73. R T R^{T}
  74. R R
  75. R i j - 1 = R i j T = R j i R^{-1}_{ij}=R^{T}_{ij}=R_{ji}
  76. i i
  77. j j
  78. k k
  79. W i , j ( k ) = [ 0 - ω z ω y v x ω z 0 - ω x v y - ω y ω x 0 v z 0 0 0 0 ] W_{i,j(k)}=\left[\begin{array}[]{ccc|c}0&-\omega_{z}&\omega_{y}&v_{x}\\ \omega_{z}&0&-\omega_{x}&v_{y}\\ -\omega_{y}&\omega_{x}&0&v_{z}\\ \hline 0&0&0&0\end{array}\right]
  80. ω \omega
  81. j j
  82. i i
  83. k k
  84. v v
  85. j j
  86. i i
  87. j j
  88. i i
  89. H i , j ( k ) = W ˙ i , j ( k ) + W i , j ( k ) 2 H_{i,j(k)}=\dot{W}_{i,j(k)}+W_{i,j(k)}^{2}
  90. i i
  91. j j
  92. P ˙ = W i , j P \dot{P}=W_{i,j}P
  93. P ¨ = H i , j P \ddot{P}=H_{i,j}P
  94. M ˙ i , j = W i , j ( i ) M i , j \dot{M}_{i,j}=W_{i,j(i)}M_{i,j}
  95. M ¨ i , j = H i , j ( i ) M i , j \ddot{M}_{i,j}=H_{i,j(i)}M_{i,j}
  96. W i , k = W i , j + W j , k W_{i,k}=W_{i,j}+W_{j,k}
  97. H i , k = H i , j + H j , k + 2 W i , j W j , k H_{i,k}=H_{i,j}+H_{j,k}+2W_{i,j}W_{j,k}
  98. k k
  99. W ( h ) = M h , k W ( k ) M k , h W_{(h)}=M_{h,k}W_{(k)}M_{k,h}
  100. H ( h ) = M h , k H ( k ) M k , h H_{(h)}=M_{h,k}H_{(k)}M_{k,h}
  101. J J
  102. Γ \Gamma
  103. Φ \Phi
  104. J J
  105. J = [ I x x I x y I x z x g m I x y I y y I y z y g m I x z I y z I z z z g m x g m y g m z g m m ] J=\left[\begin{array}[]{ccc|c}I_{xx}&I_{xy}&I_{xz}&x_{g}m\\ I_{xy}&I_{yy}&I_{yz}&y_{g}m\\ I_{xz}&I_{yz}&I_{zz}&z_{g}m\\ \hline x_{g}m&y_{g}m&z_{g}m&m\end{array}\right]
  106. m m
  107. x g , y g , z g x_{g},\,y_{g},\,z_{g}
  108. I x x , I x y , I_{xx},\,I_{xy},\ldots
  109. I x x = x 2 d m I_{xx}=\int\int x^{2}\,dm
  110. I x y = x y d m I_{xy}=\int\int xy\,dm
  111. I x z = I_{xz}=\cdots
  112. \cdots
  113. Φ \Phi
  114. f f
  115. t t
  116. Φ = [ 0 - t z t y f x t z 0 - t x f y - t y t x 0 f z - f x - f y - f z 0 ] \Phi=\left[\begin{array}[]{ccc|c}0&-t_{z}&t_{y}&f_{x}\\ t_{z}&0&-t_{x}&f_{y}\\ -t_{y}&t_{x}&0&f_{z}\\ \hline-f_{x}&-f_{y}&-f_{z}&0\end{array}\right]
  117. Γ \Gamma
  118. ρ \rho
  119. γ \gamma
  120. Γ = [ 0 - γ z γ y ρ x γ z 0 - γ x ρ y - γ y γ x 0 ρ z - ρ x - ρ y - ρ z 0 ] \Gamma=\left[\begin{array}[]{ccc|c}0&-\gamma_{z}&\gamma_{y}&\rho_{x}\\ \gamma_{z}&0&-\gamma_{x}&\rho_{y}\\ -\gamma_{y}&\gamma_{x}&0&\rho_{z}\\ \hline-\rho_{x}&-\rho_{y}&-\rho_{z}&0\end{array}\right]
  121. k k
  122. k k
  123. h h
  124. J ( h ) = M h , k J ( k ) M h , k T J_{(h)}=M_{h,k}J_{(k)}M_{h,k}^{T}
  125. Γ ( h ) = M h , k Γ ( k ) M h , k T \Gamma_{(h)}=M_{h,k}\Gamma_{(k)}M_{h,k}^{T}
  126. Φ ( h ) = M h , k Φ ( k ) M h , k T \Phi_{(h)}=M_{h,k}\Phi_{(k)}M_{h,k}^{T}
  127. Φ = H J - J H t \Phi=HJ-JH^{t}\,
  128. Γ = W J - J W t \Gamma=WJ-JW^{t}\,
  129. f = m a f=ma
  130. t = J ω ˙ + ω × J ω t=J\dot{\omega}+\omega\times J\omega
  131. O i - 1 O_{i-1}
  132. O i O_{i}
  133. T n n - 1 = Rot x n - 1 ( α n - 1 ) Trans x n - 1 ( a n - 1 ) Rot z n ( θ n ) Trans z n ( d n ) {}^{n-1}T_{n}=\operatorname{Rot}_{x_{n-1}}(\alpha_{n-1})\cdot\operatorname{% Trans}_{x_{n-1}}(a_{n-1})\cdot\operatorname{Rot}_{z_{n}}(\theta_{n})\cdot% \operatorname{Trans}_{z_{n}}(d_{n})
  134. n - 1 T n = [ cos θ n - sin θ n 0 a n - 1 sin θ n cos α n - 1 cos θ n cos α n - 1 - sin α n - 1 - d n sin α n - 1 sin θ n sin α n - 1 cos θ n sin α n - 1 cos α n - 1 d n cos α n - 1 0 0 0 1 ] \operatorname{}^{n-1}T_{n}=\left[\begin{array}[]{ccc|c}\cos\theta_{n}&-\sin% \theta_{n}&0&a_{n-1}\\ \sin\theta_{n}\cos\alpha_{n-1}&\cos\theta_{n}\cos\alpha_{n-1}&-\sin\alpha_{n-1% }&-d_{n}\sin\alpha_{n-1}\\ \sin\theta_{n}\sin\alpha_{n-1}&\cos\theta_{n}\sin\alpha_{n-1}&\cos\alpha_{n-1}% &d_{n}\cos\alpha_{n-1}\\ \hline 0&0&0&1\end{array}\right]
  135. a n a_{n}
  136. α n \alpha_{n}
  137. T n n - 1 {}^{n-1}T_{n}

Deoxyadenosine_kinase.html

  1. \rightleftharpoons

Deoxycytidine_deaminase.html

  1. \rightleftharpoons

Deoxyguanosine_kinase.html

  1. \rightleftharpoons

Deoxylimonate_A-ring-lactonase.html

  1. \rightleftharpoons

Deoxynucleoside_kinase.html

  1. \rightleftharpoons

Deoxynucleotide_3'-phosphatase.html

  1. \rightleftharpoons

Deoxyuridine_phosphorylase.html

  1. \rightleftharpoons

Dephospho-(reductase_kinase)_kinase.html

  1. \rightleftharpoons

Dephospho-CoA_kinase.html

  1. \rightleftharpoons

Dethiobiotin_synthase.html

  1. \rightleftharpoons

Dextransucrase.html

  1. \rightleftharpoons

Dextrin_dextranase.html

  1. \rightleftharpoons

DGTPase.html

  1. \rightleftharpoons

Di-trans,poly-cis-decaprenylcistransferase.html

  1. \rightleftharpoons

Diacylglycerol_cholinephosphotransferase.html

  1. \rightleftharpoons

Diacylglycerol_ethanolaminephosphotransferase.html

  1. \rightleftharpoons

Diacylglycerol—sterol_O-acyltransferase.html

  1. \rightleftharpoons

Diamine_N-acetyltransferase.html

  1. \rightleftharpoons

Diamine_transaminase.html

  1. \rightleftharpoons

Diaminobutyrate_acetyltransferase.html

  1. \rightleftharpoons

Diaminobutyrate—2-oxoglutarate_transaminase.html

  1. \rightleftharpoons

Diaminobutyrate—pyruvate_transaminase.html

  1. \rightleftharpoons

Diaminohydroxyphosphoribosylaminopyrimidine_deaminase.html

  1. \rightleftharpoons

Dicarboxylate—CoA_ligase.html

  1. \rightleftharpoons

Dickson_polynomial.html

  1. D n ( x , α ) = p = 0 n / 2 n n - p ( n - p p ) ( - α ) p x n - 2 p . D_{n}(x,\alpha)=\sum_{p=0}^{\lfloor n/2\rfloor}\frac{n}{n-p}{\left({{n-p}\atop% {p}}\right)}(-\alpha)^{p}x^{n-2p}.
  2. D 0 ( x , α ) = 2 D_{0}(x,\alpha)=2\,
  3. D 1 ( x , α ) = x D_{1}(x,\alpha)=x\,
  4. D 2 ( x , α ) = x 2 - 2 α D_{2}(x,\alpha)=x^{2}-2\alpha\,
  5. D 3 ( x , α ) = x 3 - 3 x α D_{3}(x,\alpha)=x^{3}-3x\alpha\,
  6. D 4 ( x , α ) = x 4 - 4 x 2 α + 2 α 2 . D_{4}(x,\alpha)=x^{4}-4x^{2}\alpha+2\alpha^{2}.\,
  7. E n ( x , α ) = p = 0 n / 2 ( n - p p ) ( - α ) p x n - 2 p . E_{n}(x,\alpha)=\sum_{p=0}^{\lfloor n/2\rfloor}{\left({{n-p}\atop{p}}\right)}(% -\alpha)^{p}x^{n-2p}.
  8. E 0 ( x , α ) = 1 E_{0}(x,\alpha)=1\,
  9. E 1 ( x , α ) = x E_{1}(x,\alpha)=x\,
  10. E 2 ( x , α ) = x 2 - α E_{2}(x,\alpha)=x^{2}-\alpha\,
  11. E 3 ( x , α ) = x 3 - 2 x α E_{3}(x,\alpha)=x^{3}-2x\alpha\,
  12. E 4 ( x , α ) = x 4 - 3 x 2 α + α 2 . E_{4}(x,\alpha)=x^{4}-3x^{2}\alpha+\alpha^{2}.\,
  13. D n ( u + α / u , α ) = u n + ( α / u ) n ; D_{n}(u+\alpha/u,\alpha)=u^{n}+(\alpha/u)^{n}\,;
  14. D m n ( x , α ) = D m ( D n ( x , α ) , α n ) . D_{mn}(x,\alpha)=D_{m}(D_{n}(x,\alpha),\alpha^{n})\,.
  15. D n ( x , α ) = x D n - 1 ( x , α ) - α D n - 2 ( x , α ) D_{n}(x,\alpha)=xD_{n-1}(x,\alpha)-\alpha D_{n-2}(x,\alpha)\,
  16. E n ( x , α ) = x E n - 1 ( x , α ) - α E n - 2 ( x , α ) . E_{n}(x,\alpha)=xE_{n-1}(x,\alpha)-\alpha E_{n-2}(x,\alpha).\,
  17. ( x 2 - 4 α ) y ′′ + x y - n 2 y = 0 (x^{2}-4\alpha)y^{\prime\prime}+xy^{\prime}-n^{2}y=0\,
  18. ( x 2 - 4 α ) y ′′ + 3 x y - n ( n + 2 ) y = 0. (x^{2}-4\alpha)y^{\prime\prime}+3xy^{\prime}-n(n+2)y=0.\,
  19. n D n ( x , α ) z n = 2 - x z 1 - x z + α z 2 \sum_{n}D_{n}(x,\alpha)z^{n}=\frac{2-xz}{1-xz+\alpha z^{2}}\,
  20. n E n ( x , α ) z n = 1 1 - x z + α z 2 . \sum_{n}E_{n}(x,\alpha)z^{n}=\frac{1}{1-xz+\alpha z^{2}}.\,
  21. D n ( 2 x a , a 2 ) = 2 a n T n ( x ) D_{n}(2xa,a^{2})=2a^{n}T_{n}(x)\,
  22. E n ( 2 x a , a 2 ) = a n U n ( x ) . E_{n}(2xa,a^{2})=a^{n}U_{n}(x).\,
  23. D n ( x , 0 ) = x n . D_{n}(x,0)=x^{n}\,.

Differential_geometry_of_surfaces.html

  1. S S
  2. F ( x , y ) = 1 2 k 1 x 2 + 1 2 k 2 y 2 + F(x,y)=\frac{1}{2}k_{1}x^{2}+\frac{1}{2}k_{2}y^{2}+...
  3. K = k 1 k 2 . K=k_{1}\cdot k_{2}.
  4. K m = 1 2 ( k 1 + k 2 ) . K_{m}=\tfrac{1}{2}(k_{1}+k_{2}).
  5. K = R T - S 2 ( 1 + P 2 + Q 2 ) 2 K=\frac{RT-S^{2}}{\left(1+P^{2}+Q^{2}\right)^{2}}
  6. K m = E T + G R - 2 F S ( 1 + P 2 + Q 2 ) 2 K_{m}=\frac{ET+GR-2FS}{\left(1+P^{2}+Q^{2}\right)^{2}}
  7. N ( x , y , z ) = 1 1 + P 2 + Q 2 ( P , Q , - 1 ) . N(x,y,z)=\frac{1}{\sqrt{1+P^{2}+Q^{2}}}(P,Q,-1).
  8. x = φ ( t ) , z = ψ ( t ) x=\varphi(t),\,\,z=\psi(t)
  9. φ ˙ 2 + ψ ˙ 2 = 1. \dot{\varphi}^{2}+\dot{\psi}^{2}=1.
  10. M = { ( φ ( t ) cos θ , φ ( t ) sin θ , ψ ( t ) ) : t ( a , b ) , θ [ 0 , 2 π ) } . M=\{(\varphi(t)\cos\theta,\varphi(t)\sin\theta,\psi(t))\colon t\in(a,b),\theta% \in[0,2\pi)\}.
  11. K = - φ ¨ φ , K m = - ψ ˙ + φ ( ψ ˙ φ ¨ - ψ ¨ φ ˙ ) 2 φ . K=-{\ddot{\varphi}\over\varphi},\,\,K_{m}={-\dot{\psi}+\varphi(\dot{\psi}\ddot% {\varphi}-\ddot{\psi}\dot{\varphi})\over 2\varphi}.
  12. x 2 a + y 2 b + z 2 c = 1. {x^{2}\over a}+{y^{2}\over b}+{z^{2}\over c}=1.
  13. x = a ( a - u ) ( a - v ) ( a - b ) ( a - c ) , y = b ( b - u ) ( b - v ) ( b - a ) ( b - c ) , z = c ( c - u ) ( c - v ) ( c - b ) ( c - a ) . x=\sqrt{a(a-u)(a-v)\over(a-b)(a-c)},\,\,y=\sqrt{b(b-u)(b-v)\over(b-a)(b-c)},\,% \,z=\sqrt{c(c-u)(c-v)\over(c-b)(c-a)}.
  14. K = a b c u 2 v 2 , K m = - ( u + v ) a b c u 3 v 3 . K={abc\over u^{2}v^{2}},\,\,K_{m}=-(u+v)\sqrt{abc\over u^{3}v^{3}}.
  15. u v = 0 , u = 1 , v = 1. u\cdot v=0,\,\,\|u\|=1,\,\,\|v\|=1.
  16. c ( t ) + s u ( t ) c(t)+s\cdot u(t)
  17. a = u t , b = u t v , α = - b / a 2 , β = ( a 2 - b 2 ) / a 2 , a=\|u_{t}\|,\,\,b=u_{t}\cdot v,\,\,\alpha=-b/a^{2},\,\,\beta=(a^{2}-b^{2})/a^{% 2},
  18. K = - β 2 ( ( s - α ) 2 + β 2 ) 2 , K m = - r [ ( s - α ) 2 + β 2 ) ] + β t ( s - α ) + β α t [ ( s - α ) 2 + β 2 ] 3 / 2 . K=-{\beta^{2}\over((s-\alpha)^{2}+\beta^{2})^{2}},\,\,K_{m}=-{r[(s-\alpha)^{2}% +\beta^{2})]+\beta_{t}(s-\alpha)+\beta\alpha_{t}\over[(s-\alpha)^{2}+\beta^{2}% ]^{3/2}}.
  19. ( E ( x , y ) F ( x , y ) F ( x , y ) G ( x , y ) ) \begin{pmatrix}E(x,y)&F(x,y)\\ F(x,y)&G(x,y)\end{pmatrix}
  20. ( e ( x , y ) f ( x , y ) f ( x , y ) g ( x , y ) ) \begin{pmatrix}e(x,y)&f(x,y)\\ f(x,y)&g(x,y)\end{pmatrix}
  21. K = e g - f 2 E G - F 2 K={eg-f^{2}\over EG-F^{2}}
  22. K m = e G + g E - 2 f F E G - F 2 K_{m}={eG+gE-2fF\over EG-F^{2}}
  23. Γ i j k \Gamma_{ij}^{k}
  24. e y - f x = e Γ 12 1 + f ( Γ 12 2 - Γ 11 1 ) - g Γ 11 2 e_{y}-f_{x}=e\Gamma_{12}^{1}+f(\Gamma_{12}^{2}-\Gamma_{11}^{1})-g\Gamma_{11}^{2}
  25. f y - g x = e Γ 22 1 + f ( Γ 22 2 - Γ 12 1 ) - g Γ 12 2 . f_{y}-g_{x}=e\Gamma_{22}^{1}+f(\Gamma_{22}^{2}-\Gamma_{12}^{1})-g\Gamma_{12}^{% 2}.
  26. ( S x v , w ) = ( d f ( v ) , w ) (S_{x}v,w)=(df(v),w)
  27. S = ( E G - F 2 ) - 1 ( e G - f F f G - g F f E - e F g E - f F ) . S=(EG-F^{2})^{-1}\begin{pmatrix}eG-fF&fG-gF\\ fE-eF&gE-fF\end{pmatrix}.
  28. L ( c ) = a b ( E x ˙ 2 + 2 F x ˙ y ˙ + G y ˙ 2 ) 1 / 2 d t L(c)=\int_{a}^{b}(E\dot{x}^{2}+2F\dot{x}\dot{y}+G\dot{y}^{2})^{1/2}\,dt
  29. E ( c ) = a b ( E x ˙ 2 + 2 F x ˙ y ˙ + G y ˙ 2 ) d t . E(c)=\int_{a}^{b}(E\dot{x}^{2}+2F\dot{x}\dot{y}+G\dot{y}^{2})\,dt.
  30. x ¨ + Γ 11 1 x ˙ 2 + 2 Γ 12 1 x ˙ y ˙ + Γ 22 1 y ˙ 2 = 0 \ddot{x}+\Gamma_{11}^{1}\dot{x}^{2}+2\Gamma_{12}^{1}\dot{x}\dot{y}+\Gamma_{22}% ^{1}\dot{y}^{2}=0
  31. y ¨ + Γ 11 2 x ˙ 2 + 2 Γ 12 2 x ˙ y ˙ + Γ 22 2 y ˙ 2 = 0 \ddot{y}+\Gamma_{11}^{2}\dot{x}^{2}+2\Gamma_{12}^{2}\dot{x}\dot{y}+\Gamma_{22}% ^{2}\dot{y}^{2}=0
  32. 1 2 m {1\over 2}\sum_{m}
  33. \partial
  34. \partial
  35. \partial
  36. k g k_{g}
  37. k g = c ¨ ( t ) 𝐧 ( t ) . k_{g}=\ddot{c}(t)\cdot\mathbf{n}(t).
  38. c ˙ ( t ) \dot{c}(t)
  39. c ¨ ( t ) \ddot{c}(t)
  40. K = - 1 2 H [ x ( G x / H ) + y ( E y / H ) ] . K=-{1\over 2H}\left[\partial_{x}(G_{x}/H)+\partial_{y}(E_{y}/H)\right].
  41. φ \varphi
  42. tan φ = H y ˙ / x ˙ . \tan\varphi=H\cdot\dot{y}/\dot{x}.
  43. φ \varphi
  44. φ ˙ = - H x y ˙ . \dot{\varphi}=-H_{x}\cdot\dot{y}.
  45. c ˙ \dot{c}
  46. \mapsto
  47. d s 2 = E d x 2 + 2 F d x d y + G d y 2 ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2}
  48. Δ f = 1 H ( x G H x f - x F H y f - y F H x f + y E H y f ) , \Delta f={1\over H}(\partial_{x}{G\over H}\partial_{x}f-\partial_{x}{F\over H}% \partial_{y}f-\partial_{y}{F\over H}\partial_{x}f+\partial_{y}{E\over H}% \partial_{y}f),
  49. K = - 3 lim r 0 Δ ( log r ) , K=-3\lim_{r\rightarrow 0}\Delta(\log r),
  50. d s 2 = e φ ( d x 2 + d y 2 ) . ds^{2}=e^{\varphi}(dx^{2}+dy^{2}).\,
  51. Δ = e - φ ( 2 x 2 + 2 y 2 ) \Delta=e^{-\varphi}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2% }}{\partial y^{2}}\right)
  52. Δ φ = - 2 K . \Delta\varphi=-2K.\,
  53. Δ K d A = α + β + γ - π . \int_{\Delta}K\,dA=\alpha+\beta+\gamma-\pi.
  54. Δ K d A = Δ K H d r d θ = - 0 α 0 r θ H r r d r d θ = \int_{\Delta}K\,dA=\int_{\Delta}KH\,dr\,d\theta=-\int_{0}^{\alpha}\int_{0}^{r_% {\theta}}H_{rr}\,dr\,d\theta=
  55. = 0 α 1 - H r ( r θ , θ ) d θ = 0 α d θ + π - β γ d φ = =\int_{0}^{\alpha}1-H_{r}(r_{\theta},\theta)\,d\theta=\int_{0}^{\alpha}d\theta% +\int_{\pi-\beta}^{\gamma}d\varphi=
  56. = α + β + γ - π , =\alpha+\beta+\gamma-\pi,
  57. \,
  58. M K d A = 2 π χ ( M ) \int_{M}KdA=2\pi\,\chi(M)
  59. g 2 g\geq 2
  60. c 2 = a 2 + b 2 - 2 a b cos γ . c^{2}=a^{2}+b^{2}-2ab\,\cos\gamma.
  61. cos c = cos a cos b + sin a sin b cos γ . \cos c=\cos a\,\cos b+\sin a\,\sin b\,\cos\gamma.
  62. \cup
  63. π ( x , y , z ) = x + i y 1 - z u + i v . \pi(x,y,z)={x+iy\over 1-z}\equiv u+iv.
  64. d s 2 = 4 ( d u 2 + d v 2 ) ( 1 + u 2 + v 2 ) 2 . ds^{2}={4(du^{2}+dv^{2})\over(1+u^{2}+v^{2})^{2}}.
  65. D = { z : | z | < 1 } D=\{z\,\colon|z|<1\}
  66. d s 2 = 4 ( d x 2 + d y 2 ) ( 1 - x 2 - y 2 ) 2 . ds^{2}={4(dx^{2}+dy^{2})\over(1-x^{2}-y^{2})^{2}}.
  67. d s 2 = 4 ( d r 2 + r 2 d θ 2 ) ( 1 - r 2 ) 2 . ds^{2}={4(dr^{2}+r^{2}\,d\theta^{2})\over(1-r^{2})^{2}}.
  68. \rightarrow
  69. ( γ ) = a b 2 | γ ( t ) | d t 1 - | γ ( t ) | 2 . \ell(\gamma)=\int_{a}^{b}{2|\gamma^{\prime}(t)|\,dt\over 1-|\gamma(t)|^{2}}.
  70. G = { ( α β β ¯ α ¯ ) : α , β 𝐂 , | α | 2 - | β | 2 = 1 } G=\left\{\begin{pmatrix}\alpha&\beta\\ \overline{\beta}&\overline{\alpha}\end{pmatrix}:\alpha,\beta\in\mathbf{C},\,|% \alpha|^{2}-|\beta|^{2}=1\right\}
  71. K = { ( ζ 0 0 ζ ¯ ) : ζ 𝐂 , | ζ | = 1 } . K=\left\{\begin{pmatrix}\zeta&0\\ 0&\overline{\zeta}\end{pmatrix}:\zeta\in\mathbf{C},\,|\zeta|=1\right\}.
  72. d ( z , w ) = 2 tanh - 1 | z - w | / | 1 - w ¯ z | . d(z,w)=2\tanh^{-1}|z-w|/|1-\overline{w}z|.
  73. cosh c = cosh a cosh b - sinh a sinh b cos γ . \cosh c=\cosh a\,\cosh b-\sinh a\,\sinh b\,\cos\gamma.
  74. H = { w = x + i y : y > 0 } H=\{w=x+iy\,\colon\,y>0\}
  75. w = i 1 + z 1 - z , z = w - i w + i . w=i{1+z\over 1-z},\,\,z={w-i\over w+i}.
  76. d s 2 = d x 2 + d y 2 y 2 . ds^{2}={dx^{2}+dy^{2}\over y^{2}}.
  77. M = Γ \ G / K . M=\Gamma\backslash G/K.
  78. K ( x ) = e - 2 u ( K ( x ) - Δ u ) , K^{\prime}(x)=e^{-2u}(K(x)-\Delta u),
  79. Δ u = K e 2 u + K ( x ) . \Delta u=K^{\prime}e^{2u}+K(x).
  80. Δ u = K ( x ) . \Delta u=K(x).
  81. Δ u = - e 2 u + K ( x ) . \Delta u=-e^{2u}+K(x).
  82. Δ u = e 2 u + K ( x ) . \Delta u=e^{2u}+K(x).
  83. u t = 4 π - K ( x , t ) = 4 π - e - 2 u ( K ( x ) - Δ u ) . u_{t}=4\pi-K^{\prime}(x,t)=4\pi-e^{-2u}(K(x)-\Delta u).
  84. f ¨ ( t ) 0. \ddot{f}(t)\geq 0.
  85. r r ¨ + r ˙ 2 1. r\ddot{r}+\dot{r}^{2}\geq 1.
  86. c ˙ ( t ) \dot{c}(t)
  87. 3 \mathbb{R}^{3}
  88. c ˙ ( t ) \dot{c}(t)
  89. θ ˙ ( t ) = - k g ( t ) \dot{\theta}(t)=-k_{g}(t)
  90. δ = π / sup K \delta=\pi/\sqrt{\sup K}
  91. 3 / 2 \sqrt{3}/2
  92. 8 / π \sqrt{8}/\pi
  93. 1 / 2 3 1/2\sqrt{3}

Difructose-anhydride_synthase.html

  1. \rightleftharpoons

Digalactosyldiacylglycerol_synthase.html

  1. \rightleftharpoons

Digital_differential_analyzer.html

  1. y * = y ± Δ y y^{*}=y\pm\sum\Delta y
  2. S * = S ± y * Δ x S^{*}=S\pm y^{*}\sum\Delta x
  3. Δ S = K Δ y Δ x \Delta S=K\int\Delta y\Delta x
  4. K = 1 radix n K={1\over{\,\text{radix}}^{n}}
  5. Δ S = K y Δ x \Delta S=Ky\Delta x

Digital_image_correlation.html

  1. r i j r_{ij}
  2. r i j ( u , v , u x , u y , v x , v y ) = 1 - i j [ F ( x i , y j ) - F ¯ ] [ G ( x i , y j ) - G ¯ ] i j [ F ( x i , y j ) - F ¯ ] 2 i j [ G ( x i , y j ) - G ¯ ] 2 r_{ij}(u,v,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{% \partial v}{\partial x},\frac{\partial v}{\partial y})=1-\frac{\sum_{i}\sum_{j% }[F(x_{i},y_{j})-\bar{F}][G(x_{i}^{\star},y_{j}^{\star})-\bar{G}]}{\sqrt{\sum_% {i}\sum_{j}{[F(x_{i},y_{j})-\bar{F}]^{2}}\sum_{i}\sum_{j}{[G(x_{i}^{\star},y_{% j}^{\star})-\bar{G}]^{2}}}}
  3. F ¯ \bar{F}
  4. G ¯ \bar{G}
  5. x = x + u + u x Δ x + u y Δ y x^{\star}=x+u+\frac{\partial u}{\partial x}\Delta x+\frac{\partial u}{\partial y% }\Delta y
  6. y = y + v + v x Δ x + v y Δ y y^{\star}=y+v+\frac{\partial v}{\partial x}\Delta x+\frac{\partial v}{\partial y% }\Delta y
  7. Δ x \Delta x
  8. Δ y \Delta y
  9. u x , u y , v x , v y \frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial v}{% \partial x},\frac{\partial v}{\partial y}

Diglucosyl_diacylglycerol_synthase.html

  1. \rightleftharpoons

Diguanidinobutanase.html

  1. \rightleftharpoons

Dihydrocoumarin_hydrolase.html

  1. \rightleftharpoons

Dihydrofolate_synthase.html

  1. \rightleftharpoons

Dihydrolipoyllysine-residue_(2-methylpropanoyl)transferase.html

  1. \rightleftharpoons

Dihydrolipoyllysine-residue_succinyltransferase.html

  1. \rightleftharpoons

Dihydropyrimidinase.html

  1. \rightleftharpoons

Dihydrostreptomycin-6-phosphate_3'alpha-kinase.html

  1. \rightleftharpoons

Dihydroxy-acid_dehydratase.html

  1. \rightleftharpoons

Dihydroxyphenylalanine_transaminase.html

  1. \rightleftharpoons

Diiodotyrosine_transaminase.html

  1. \rightleftharpoons

Diisopropyl-fluorophosphatase.html

  1. \rightleftharpoons

Diketene.html

  1. \overrightarrow{\leftarrow}

Dimethylallylcistransferase.html

  1. \rightleftharpoons

Dimethylargininase.html

  1. \rightleftharpoons

Dimethylmaleate_hydratase.html

  1. \rightleftharpoons

Dioxotetrahydropyrimidine_phosphoribosyltransferase.html

  1. \rightleftharpoons

Diphosphate-purine_nucleoside_kinase.html

  1. \rightleftharpoons

Diphosphate—glycerol_phosphotransferase.html

  1. \rightleftharpoons

Diphosphate—serine_phosphotransferase.html

  1. \rightleftharpoons

Diphosphoinositol-pentakisphosphate_kinase.html

  1. \rightleftharpoons

Diphosphoinositol-polyphosphate_diphosphatase.html

  1. \rightleftharpoons

Diphthine—ammonia_ligase.html

  1. \rightleftharpoons

Direct_limit_of_groups.html

  1. GL ( n , A ) GL ( n + 1 , A ) \operatorname{GL}(n,A)\to\operatorname{GL}(n+1,A)
  2. GL ( A ) \operatorname{GL}(A)
  3. GL ( , A ) \operatorname{GL}(\infty,A)
  4. GL ( A ) \operatorname{GL}(A)

Dirichlet's_energy.html

  1. E [ u ] = 1 2 Ω u ( x ) 2 d V , E[u]=\frac{1}{2}\int_{\Omega}\|\nabla u(x)\|^{2}\,\mathrm{d}V,
  2. - Δ u ( x ) = 0 for all x Ω -\Delta u(x)=0\,\text{ for all }x\in\Omega
  3. < V A R > u <VAR>u

Discadenine_synthase.html

  1. \rightleftharpoons

Display_contrast.html

  1. C R = L H L L CR=\frac{L_{H}}{L_{L}}
  2. \infty
  3. C M = L H - L L L H + L L C_{M}=\frac{L_{H}-L_{L}}{L_{H}+L_{L}}
  4. K = L H - L L L H K=\frac{L_{H}-L_{L}}{L_{H}}

Distributed_lag.html

  1. y t = a + w 0 x t + w 1 x t - 1 + w 2 x t - 2 + + error term y_{t}=a+w_{0}x_{t}+w_{1}x_{t-1}+w_{2}x_{t-2}+...+\,\text{error term}
  2. y t = a + w 0 x t + w 1 x t - 1 + w 2 x t - 2 + + w n x t - n + error term , y_{t}=a+w_{0}x_{t}+w_{1}x_{t-1}+w_{2}x_{t-2}+...+w_{n}x_{t-n}+\,\text{error % term},
  3. p p
  4. w i = j = 0 n a j i j w_{i}=\sum_{j=0}^{n}a_{j}i^{j}
  5. i = 0 , , k . i=0,\dots,k.
  6. y t = a + λ y t - 1 + b x t + error term , y_{t}=a+\lambda y_{t-1}+bx_{t}+\,\text{error term},
  7. 0 λ < 1 0\leq\lambda<1
  8. b + λ b + λ 2 b + = b / ( 1 - λ ) . b+\lambda b+\lambda^{2}b+...=b/(1-\lambda).
  9. w i = j = 2 n a j ( i + 1 ) j , w_{i}=\sum_{j=2}^{n}\frac{a_{j}}{(i+1)^{j}},
  10. i = 0 , , . i=0,\dots,\infty.
  11. w i = j = 2 n a j ( 1 / j ) i , w_{i}=\sum_{j=2}^{n}a_{j}(1/j)^{i},
  12. i = 0 , , i=0,\dots,\infty
  13. w i = j = 1 n a j [ j / ( n + 1 ) ] i , w_{i}=\sum_{j=1}^{n}a_{j}[j/(n+1)]^{i},
  14. i = 0 , , . i=0,\dots,\infty.

Disulfoglucosamine-6-sulfatase.html

  1. \rightleftharpoons

Dividend_payout_ratio.html

  1. Dividend payout ratio = Dividends Net Income for the same period \mbox{Dividend payout ratio}~{}=\frac{\mbox{Dividends}~{}}{\mbox{Net Income % for the same period}~{}}
  2. Current Dividend Yield = Most Recent Full-Year Dividend Current Share Price = Dividend payout ratio × Most Recent Full-Year earnings per share Current Share Price \begin{array}[]{lcl}\mbox{Current Dividend Yield}&=&\frac{\mbox{Most Recent % Full-Year Dividend}~{}}{\mbox{Current Share Price}~{}}\\ &=&\frac{\mbox{Dividend payout ratio}~{}\times\mbox{Most Recent Full-Year % earnings per share}~{}}{\mbox{Current Share Price}~{}}\\ \end{array}

Divisia_index.html

  1. p i ( t ) p_{i}(t)
  2. q i ( t ) q_{i}(t)
  3. i p ( 0 ) * q ( 0 ) = P ( 0 ) Q ( 0 ) . \sum_{i}p(0)*q(0)=P(0)Q(0).
  4. i p ( t ) * q ( t ) = P ( t ) Q ( t ) . \sum_{i}p(t)*q(t)=P(t)Q(t).
  5. s j , t * = 1 2 ( s j , t + s j , t - 1 ) s_{j,t}^{*}=\frac{1}{2}(s_{j,t}+s_{j,t-1})

Dmrg_of_Heisenberg_model.html

  1. S = 1 S=1
  2. \mathfrak{H}
  3. { | S , S z } { | 1 , 1 , | 1 , 0 , | 1 , - 1 } \{|S,S_{z}\rangle\}\equiv\{|1,1\rangle,|1,0\rangle,|1,-1\rangle\}
  4. S x S_{x}
  5. S y S_{y}
  6. S z S_{z}
  7. b \mathfrak{H}_{b}
  8. { | w i } \{|w_{i}\rangle\}
  9. i : 1 dim ( b ) i:1\dots\dim(\mathfrak{H}_{b})
  10. O b : b b O_{b}:\mathfrak{H}_{b}\rightarrow\mathfrak{H}_{b}
  11. B \mathfrak{H}_{B}
  12. { | u i } \{|u_{i}\rangle\}
  13. H B H_{B}
  14. S x B S_{x_{B}}
  15. S y B S_{y_{B}}
  16. S z B S_{z_{B}}
  17. l \mathfrak{H}_{l}
  18. { | t i } \{|t_{i}\rangle\}
  19. S x l S_{x_{l}}
  20. S y l S_{y_{l}}
  21. S z l S_{z_{l}}
  22. r \mathfrak{H}_{r}
  23. { | s i } \{|s_{i}\rangle\}
  24. S x r S_{x_{r}}
  25. S y r S_{y_{r}}
  26. S z r S_{z_{r}}
  27. U \mathfrak{H}_{U}
  28. { | r i } \{|r_{i}\rangle\}
  29. H U H_{U}
  30. S x U S_{x_{U}}
  31. S y U S_{y_{U}}
  32. S z U S_{z_{U}}
  33. \mathfrak{H}
  34. S x S_{x}
  35. S y S_{y}
  36. S z S_{z}
  37. H B = H U = 0 H_{B}=H_{U}=0
  38. 3 × 3 3\times 3
  39. 3 × 3 3\times 3
  40. 𝐇 S B = - J < i , j > 𝐒 x i 𝐒 x j + 𝐒 y i 𝐒 y j + 𝐒 z i 𝐒 z j \mathbf{H}_{SB}=-J\sum_{<i,j>}\mathbf{S}_{x_{i}}\mathbf{S}_{x_{j}}+\mathbf{S}_% {y_{i}}\mathbf{S}_{y_{j}}+\mathbf{S}_{z_{i}}\mathbf{S}_{z_{j}}
  41. S B = B l r U \mathfrak{H}_{SB}=\mathfrak{H}_{B}\otimes\mathfrak{H}_{l}\otimes\mathfrak{H}_{% r}\otimes\mathfrak{H}_{U}
  42. { | f = | u | t | s | r } \{|f\rangle=|u\rangle\otimes|t\rangle\otimes|s\rangle\otimes|r\rangle\}
  43. | 1000 0 | f 1 = | u 1 , t 1 , s 1 , r 1 | 100 , 100 , 100 , 100 |1000\dots 0\rangle\equiv|f_{1}\rangle=|u_{1},t_{1},s_{1},r_{1}\rangle\equiv|1% 00,100,100,100\rangle
  44. | 0100 0 | f 2 = | u 1 , t 1 , s 1 , r 2 | 100 , 100 , 100 , 010 |0100\dots 0\rangle\equiv|f_{2}\rangle=|u_{1},t_{1},s_{1},r_{2}\rangle\equiv|1% 00,100,100,010\rangle
  45. J = - 1 J=-1
  46. 𝐇 S B = 𝐇 B + 𝐇 U + < i , j > 𝐒 x i 𝐒 x j + 𝐒 y i 𝐒 y j + 𝐒 z i 𝐒 z j \mathbf{H}_{SB}=\mathbf{H}_{B}+\mathbf{H}_{U}+\sum_{<i,j>}\mathbf{S}_{x_{i}}% \mathbf{S}_{x_{j}}+\mathbf{S}_{y_{i}}\mathbf{S}_{y_{j}}+\mathbf{S}_{z_{i}}% \mathbf{S}_{z_{j}}
  47. ( d * 3 * 3 * d ) × ( d * 3 * 3 * d ) (d*3*3*d)\times(d*3*3*d)
  48. d = dim ( B ) dim ( U ) d=\dim(\mathfrak{H}_{B})\equiv\dim(\mathfrak{H}_{U})
  49. f | 𝐇 B | f u , t , s , r | H B 𝕀 𝕀 𝕀 | u , t , s , r \langle f|\mathbf{H}_{B}|f^{\prime}\rangle\equiv\langle u,t,s,r|H_{B}\otimes% \mathbb{I}\otimes\mathbb{I}\otimes\mathbb{I}|u^{\prime},t^{\prime},s^{\prime},% r^{\prime}\rangle
  50. 𝐒 x B 𝐒 x l = S x B 𝕀 𝕀 S x l 𝕀 𝕀 𝕀 𝕀 = S x B S x l 𝕀 𝕀 \mathbf{S}_{x_{B}}\mathbf{S}_{x_{l}}=S_{x_{B}}\mathbb{I}\otimes\mathbb{I}S_{x_% {l}}\otimes\mathbb{I}\mathbb{I}\otimes\mathbb{I}\mathbb{I}=S_{x_{B}}\otimes S_% {x_{l}}\otimes\mathbb{I}\otimes\mathbb{I}
  51. | Ψ = Ψ i , j , k , w | u i , t j , s k , r w |\Psi\rangle=\sum\Psi_{i,j,k,w}|u_{i},t_{j},s_{k},r_{w}\rangle
  52. | Ψ |\Psi\rangle
  53. ρ \rho
  54. ( d * 3 ) × ( d * 3 ) (d*3)\times(d*3)
  55. ρ i , j ; i , j k , w Ψ i , j , k , w Ψ i , j , k , w \rho_{i,j;i^{\prime},j^{\prime}}\equiv\sum_{k,w}\Psi_{i,j,k,w}\Psi_{i^{\prime}% ,j^{\prime},k,w}
  56. ρ \rho
  57. m × ( d * 3 ) m\times(d*3)
  58. T T
  59. m m
  60. m m
  61. e α e_{\alpha}
  62. ρ \rho
  63. T T
  64. m m
  65. P m α = 1 m e α P_{m}\equiv\sum_{\alpha=1}^{m}e_{\alpha}
  66. 1 - P m 0 1-P_{m}\cong 0
  67. ( d * 3 ) × ( d * 3 ) (d*3)\times(d*3)
  68. H B - l = H B 𝕀 + S x B S x l + S y B S y l + S z B S z l H_{B-l}=H_{B}\otimes\mathbb{I}+S_{x_{B}}\otimes S_{x_{l}}+S_{y_{B}}\otimes S_{% y_{l}}+S_{z_{B}}\otimes S_{z_{l}}
  69. S x B - l = 𝕀 S x l S_{x_{B-l}}=\mathbb{I}\otimes S_{x_{l}}
  70. H r - U = 𝕀 H U + S x r S x U + S y r S y U + S z r S z U H_{r-U}=\mathbb{I}\otimes H_{U}+S_{x_{r}}\otimes S_{x_{U}}+S_{y_{r}}\otimes S_% {y_{U}}+S_{z_{r}}\otimes S_{z_{U}}
  71. S x r - U = S x r 𝕀 S_{x_{r-U}}=S_{x_{r}}\otimes\mathbb{I}
  72. m × m m\times m
  73. T T
  74. H B = T H B - l T S x B = T S x B - l T \begin{matrix}&H_{B}=TH_{B-l}T^{\dagger}&S_{x_{B}}=TS_{x_{B-l}}T^{\dagger}\end% {matrix}

Dodecanoyl-(acyl-carrier-protein)_hydrolase.html

  1. \rightleftharpoons

Dolichol_kinase.html

  1. \rightleftharpoons

Dolichol_O-acyltransferase.html

  1. \rightleftharpoons

Dolichyl-diphosphate—polyphosphate_phosphotransferase.html

  1. \rightleftharpoons

Dolichyl-diphosphooligosaccharide—protein_glycotransferase.html

  1. \rightleftharpoons

Dolichyl-phosphatase.html

  1. \rightleftharpoons

Dolichyl-phosphate-mannose-protein_mannosyltransferase.html

  1. \rightleftharpoons

Dolichyl-phosphate_alpha-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Dolichyl-phosphate_beta-D-mannosyltransferase.html

  1. \rightleftharpoons

Dolichyl-phosphate_beta-glucosyltransferase.html

  1. \rightleftharpoons

Dolichyl-phosphate_D-xylosyltransferase.html

  1. \rightleftharpoons

Dolichyl-xylosyl-phosphate—protein_xylosyltransferase.html

  1. \rightleftharpoons

Dolichyldiphosphatase.html

  1. \rightleftharpoons

Dolichylphosphate-glucose_phosphodiesterase.html

  1. \rightleftharpoons

Dolichylphosphate-mannose_phosphodiesterase.html

  1. \rightleftharpoons

Dominance-based_rough_set_approach.html

  1. S = U , Q , V , f S=\langle U,Q,V,f\rangle
  2. U U\,\!
  3. Q Q\,\!
  4. V = V q q Q V=\bigcup{}_{q\in Q}V_{q}
  5. V q V_{q}\,\!
  6. q q\,\!
  7. f : U × Q V f\colon U\times Q\to V
  8. f ( x , q ) V q f(x,q)\in V_{q}
  9. ( x , q ) U × Q (x,q)\in U\times Q
  10. Q Q\,\!
  11. C C\neq\emptyset
  12. d d\,\!
  13. f ( x , q ) f(x,q)\,\!
  14. x x\,\!
  15. q C q\in C
  16. f ( x , d ) f(x,d)\,\!
  17. q Q q\in Q
  18. q \succeq_{q}
  19. x q y x\succeq_{q}y
  20. x x\,\!
  21. y y\,\!
  22. q q\,\!
  23. q q\,\!
  24. V q V_{q}\subseteq\mathbb{R}
  25. \geq\,\!
  26. x q y f ( x , q ) f ( y , q ) x\succeq_{q}y\iff f(x,q)\geq f(y,q)
  27. V q V_{q}\,\!
  28. T = { 1 , , n } T=\{1,\ldots,n\}\,\!
  29. V d V_{d}\,\!
  30. n n\,\!
  31. V d = T V_{d}=T\,\!
  32. U U\,\!
  33. n n\,\!
  34. 𝐂𝐥 = { C l t , t T } \,\textbf{Cl}=\{Cl_{t},t\in T\}
  35. C l t = { x U : f ( x , d ) = t } Cl_{t}=\{x\in U\colon f(x,d)=t\}
  36. x U x\in U
  37. C l t , t T Cl_{t},t\in T
  38. r , s T r,s\in T
  39. r s r\geq s\,\!
  40. C l r Cl_{r}\,\!
  41. C l s Cl_{s}\,\!
  42. C l t = s t C l s C l t = s t C l s t T Cl^{\geq}_{t}=\bigcup_{s\geq t}Cl_{s}\qquad Cl^{\leq}_{t}=\bigcup_{s\leq t}Cl_% {s}\qquad t\in T
  43. x x\,\!
  44. y y\,\!
  45. P C P\subseteq C
  46. x D p y xD_{p}y\,\!
  47. x x\,\!
  48. y y\,\!
  49. P P\,\!
  50. x q y , q P x\succeq_{q}y,\,\forall q\in P
  51. P C P\subseteq C
  52. D P D_{P}\,\!
  53. P C P\subseteq C
  54. x U x\in U
  55. D P + ( x ) = { y U : y D p x } D_{P}^{+}(x)=\{y\in U\colon yD_{p}x\}
  56. D P - ( x ) = { y U : x D p y } D_{P}^{-}(x)=\{y\in U\colon xD_{p}y\}
  57. x U x\in U
  58. C l t , t T Cl_{t}^{\geq},t\in T
  59. P C P\subseteq C
  60. P ¯ ( C l t ) \underline{P}(Cl_{t}^{\geq})
  61. P ¯ ( C l t ) \overline{P}(Cl_{t}^{\geq})
  62. P ¯ ( C l t ) = { x U : D P + ( x ) C l t } \underline{P}(Cl_{t}^{\geq})=\{x\in U\colon D_{P}^{+}(x)\subseteq Cl_{t}^{\geq}\}
  63. P ¯ ( C l t ) = { x U : D P - ( x ) C l t } \overline{P}(Cl_{t}^{\geq})=\{x\in U\colon D_{P}^{-}(x)\cap Cl_{t}^{\geq}\neq\emptyset\}
  64. C l t , t T Cl_{t}^{\leq},t\in T
  65. P C P\subseteq C
  66. P ¯ ( C l t ) \underline{P}(Cl_{t}^{\leq})
  67. P ¯ ( C l t ) \overline{P}(Cl_{t}^{\leq})
  68. P ¯ ( C l t ) = { x U : D P - ( x ) C l t } \underline{P}(Cl_{t}^{\leq})=\{x\in U\colon D_{P}^{-}(x)\subseteq Cl_{t}^{\leq}\}
  69. P ¯ ( C l t ) = { x U : D P + ( x ) C l t } \overline{P}(Cl_{t}^{\leq})=\{x\in U\colon D_{P}^{+}(x)\cap Cl_{t}^{\leq}\neq\emptyset\}
  70. C l t Cl^{\geq}_{t}
  71. C l t Cl^{\leq}_{t}
  72. x U x\in U
  73. P ¯ ( C l t ) \underline{P}(Cl^{\geq}_{t})
  74. P ¯ ( C l t ) \underline{P}(Cl^{\leq}_{t})
  75. U U\,\!
  76. y U y\in U
  77. x x\,\!
  78. C l t Cl^{\geq}_{t}
  79. C l t Cl^{\leq}_{t}
  80. C l t Cl^{\geq}_{t}
  81. C l t Cl^{\leq}_{t}
  82. x U x\in U
  83. P ¯ ( C l t ) \overline{P}(Cl^{\geq}_{t})
  84. P ¯ ( C l t ) \overline{P}(Cl^{\leq}_{t})
  85. y U y\in U
  86. x x\,\!
  87. C l t Cl^{\geq}_{t}
  88. C l t Cl^{\leq}_{t}
  89. t T t\in T
  90. P C P\subseteq C
  91. P ¯ ( C l t ) C l t P ¯ ( C l t ) \underline{P}(Cl_{t}^{\geq})\subseteq Cl_{t}^{\geq}\subseteq\overline{P}(Cl_{t% }^{\geq})
  92. P ¯ ( C l t ) C l t P ¯ ( C l t ) \underline{P}(Cl_{t}^{\leq})\subseteq Cl_{t}^{\leq}\subseteq\overline{P}(Cl_{t% }^{\leq})
  93. C l t Cl_{t}^{\geq}
  94. C l t Cl_{t}^{\leq}
  95. B n P ( C l t ) = P ¯ ( C l t ) - P ¯ ( C l t ) Bn_{P}(Cl_{t}^{\geq})=\overline{P}(Cl_{t}^{\geq})-\underline{P}(Cl_{t}^{\geq})
  96. B n P ( C l t ) = P ¯ ( C l t ) - P ¯ ( C l t ) Bn_{P}(Cl_{t}^{\leq})=\overline{P}(Cl_{t}^{\leq})-\underline{P}(Cl_{t}^{\leq})
  97. γ P ( 𝐂𝐥 ) = | U - ( ( t T B n P ( C l t ) ) ( t T B n P ( C l t ) ) ) | | U | \gamma_{P}(\,\textbf{Cl})=\frac{\left|U-\left(\left(\bigcup_{t\in T}Bn_{P}(Cl_% {t}^{\geq})\right)\cup\left(\bigcup_{t\in T}Bn_{P}(Cl_{t}^{\leq})\right)\right% )\right|}{|U|}
  98. 𝐂𝐥 \,\textbf{Cl}\,\!
  99. P P\,\!
  100. P C P\subseteq C
  101. γ P ( 𝐂𝐥 ) = γ C ( 𝐂𝐥 ) \gamma_{P}(\mathbf{Cl})=\gamma_{C}(\mathbf{Cl})\,\!
  102. C C\,\!
  103. R E D 𝐂𝐥 ( P ) RED_{\mathbf{Cl}}(P)
  104. f ( x , q 1 ) r 1 f(x,q_{1})\geq r_{1}\,\!
  105. f ( x , q 2 ) r 2 f(x,q_{2})\geq r_{2}\,\!
  106. f ( x , q p ) r p \ldots f(x,q_{p})\geq r_{p}\,\!
  107. x C l t x\in Cl_{t}^{\geq}
  108. f ( x , q 1 ) r 1 f(x,q_{1})\leq r_{1}\,\!
  109. f ( x , q 2 ) r 2 f(x,q_{2})\leq r_{2}\,\!
  110. f ( x , q p ) r p \ldots f(x,q_{p})\leq r_{p}\,\!
  111. x C l t x\in Cl_{t}^{\leq}
  112. x x\,\!
  113. C l t Cl_{t}^{\geq}
  114. x x\,\!
  115. C l t Cl_{t}^{\leq}
  116. f ( x , q 1 ) r 1 f(x,q_{1})\geq r_{1}\,\!
  117. f ( x , q 2 ) r 2 f(x,q_{2})\geq r_{2}\,\!
  118. f ( x , q k ) r k \ldots f(x,q_{k})\geq r_{k}\,\!
  119. f ( x , q k + 1 ) r k + 1 f(x,q_{k+1})\leq r_{k+1}\,\!
  120. f ( x , q k + 2 ) r k + 2 f(x,q_{k+2})\leq r_{k+2}\,\!
  121. f ( x , q p ) r p \ldots f(x,q_{p})\leq r_{p}\,\!
  122. x C l s C l s + 1 C l t x\in Cl_{s}\cup Cl_{s+1}\cup Cl_{t}
  123. q 1 q_{1}
  124. q 2 q_{2}
  125. q 3 q_{3}
  126. d d
  127. x 1 x_{1}
  128. x 2 x_{2}
  129. x 3 x_{3}
  130. x 4 x_{4}
  131. x 5 x_{5}
  132. x 6 x_{6}
  133. x 7 x_{7}
  134. x 8 x_{8}
  135. x 9 x_{9}
  136. x 10 x_{10}
  137. q 1 , q 2 , q 3 q_{1},q_{2},q_{3}\,\!
  138. C l 1 = { b a d } Cl_{1}=\{bad\}
  139. C l 2 = { m e d i u m } Cl_{2}=\{medium\}
  140. C l 3 = { g o o d } Cl_{3}=\{good\}
  141. C l 1 Cl_{1}^{\leq}
  142. C l 2 Cl_{2}^{\leq}
  143. C l 2 Cl_{2}^{\geq}
  144. C l 3 Cl_{3}^{\geq}
  145. x 4 x_{4}\,\!
  146. x 6 x_{6}\,\!
  147. x 4 x_{4}\,\!
  148. x 6 x_{6}\,\!
  149. P ¯ ( C l 1 ) = { x 1 , x 5 } \underline{P}(Cl_{1}^{\leq})=\{x_{1},x_{5}\}
  150. P ¯ ( C l 2 ) = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 8 } = C l 2 \underline{P}(Cl_{2}^{\leq})=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{8}\}=Cl_% {2}^{\leq}
  151. P ¯ ( C l 2 ) = { x 2 , x 3 , x 7 , x 8 , x 9 , x 10 } \underline{P}(Cl_{2}^{\geq})=\{x_{2},x_{3},x_{7},x_{8},x_{9},x_{10}\}
  152. P ¯ ( C l 3 ) = { x 7 , x 9 , x 10 } = C l 3 \underline{P}(Cl_{3}^{\geq})=\{x_{7},x_{9},x_{10}\}=Cl_{3}^{\geq}
  153. C l 1 Cl_{1}^{\leq}
  154. C l 2 Cl_{2}^{\geq}
  155. P ¯ ( C l 1 ) = { x 1 , x 4 , x 5 , x 6 } \overline{P}(Cl_{1}^{\leq})=\{x_{1},x_{4},x_{5},x_{6}\}
  156. P ¯ ( C l 2 ) = { x 2 , x 3 , x 4 , x 6 , x 7 , x 8 , x 9 , x 10 } \overline{P}(Cl_{2}^{\geq})=\{x_{2},x_{3},x_{4},x_{6},x_{7},x_{8},x_{9},x_{10}\}
  157. B n P ( C l 1 ) = B n P ( C l 2 ) = { x 4 , x 6 } Bn_{P}(Cl_{1}^{\leq})=Bn_{P}(Cl_{2}^{\geq})=\{x_{4},x_{6}\}
  158. C l 2 Cl_{2}^{\leq}
  159. C l 3 Cl_{3}^{\geq}
  160. P ¯ ( C l 2 ) = C l 2 \overline{P}(Cl_{2}^{\leq})=Cl_{2}^{\leq}
  161. P ¯ ( C l 3 ) = C l 3 \overline{P}(Cl_{3}^{\geq})=Cl_{3}^{\geq}
  162. B n P ( C l 2 ) = B n P ( C l 3 ) = Bn_{P}(Cl_{2}^{\leq})=Bn_{P}(Cl_{3}^{\geq})=\emptyset
  163. P h y s i c s b a d Physics\leq bad
  164. s t u d e n t b a d student\leq bad
  165. L i t e r a t u r e b a d Literature\leq bad
  166. P h y s i c s m e d i u m Physics\leq medium
  167. M a t h m e d i u m Math\leq medium
  168. s t u d e n t b a d student\leq bad
  169. M a t h b a d Math\leq bad
  170. s t u d e n t m e d i u m student\leq medium
  171. L i t e r a t u r e m e d i u m Literature\leq medium
  172. P h y s i c s m e d i u m Physics\leq medium
  173. s t u d e n t m e d i u m student\leq medium
  174. M a t h m e d i u m Math\leq medium
  175. L i t e r a t u r e b a d Literature\leq bad
  176. s t u d e n t m e d i u m student\leq medium
  177. L i t e r a t u r e g o o d Literature\geq good
  178. M a t h m e d i u m Math\geq medium
  179. s t u d e n t g o o d student\geq good
  180. P h y s i c s g o o d Physics\geq good
  181. M a t h g o o d Math\geq good
  182. s t u d e n t g o o d student\geq good
  183. M a t h g o o d Math\geq good
  184. s t u d e n t m e d i u m student\geq medium
  185. P h y s i c s g o o d Physics\geq good
  186. s t u d e n t m e d i u m student\geq medium
  187. M a t h b a d Math\leq bad
  188. P h y s i c s m e d i u m Physics\geq medium
  189. s t u d e n t = b a d m e d i u m student=bad\lor medium

Dominance_order.html

  1. p q if and only if p 1 + + p k q 1 + + q k for all k 1. p\trianglelefteq q\,\text{ if and only if }p_{1}+\cdots+p_{k}\leq q_{1}+\cdots% +q_{k}\,\text{ for all }k\geq 1.
  2. p q p\trianglelefteq q
  3. q p . q^{\prime}\trianglelefteq p^{\prime}.
  4. p ^ = ( 0 , p 1 , p 1 + p 2 , , p 1 + p 2 + + p n ) . \hat{p}=(0,p_{1},p_{1}+p_{2},\ldots,p_{1}+p_{2}+\cdots+p_{n}).
  5. p i = p ^ i - p ^ i - 1 . p_{i}=\hat{p}_{i}-\hat{p}_{i-1}.
  6. p ^ i p ^ i + 1 ; \hat{p}_{i}\leq\hat{p}_{i+1};
  7. 2 p ^ i p ^ i - 1 + p ^ i + 1 ; 2\hat{p}_{i}\geq\hat{p}_{i-1}+\hat{p}_{i+1};
  8. p ^ 0 = 0 , p ^ n = n . \hat{p}_{0}=0,\hat{p}_{n}=n.
  9. r p , r q r\trianglelefteq p,r\trianglelefteq q
  10. r ^ p ^ , r ^ q ^ . \hat{r}\leq\hat{p},\hat{r}\leq\hat{q}.
  11. min ( p ^ i , q ^ i ) . \operatorname{min}(\hat{p}_{i},\hat{q}_{i}).
  12. p q = ( p q ) . p\lor q=(p^{\prime}\land q^{\prime})^{\prime}.

Double-negation_translation.html

  1. ϕ \phi
  2. ϕ N \phi^{N}
  3. ¬ ¬ ϕ \neg\neg\phi
  4. ( ϕ θ ) N is ϕ N θ N (\phi\wedge\theta)^{N}\mbox{ is }~{}\phi^{N}\wedge\theta^{N}
  5. ( ϕ θ ) N is ¬ ( ¬ ϕ N ¬ θ N ) (\phi\vee\theta)^{N}\mbox{ is }~{}\neg(\neg\phi^{N}\wedge\neg\theta^{N})
  6. ( ϕ θ ) N is ϕ N θ N (\phi\rightarrow\theta)^{N}\mbox{ is }~{}\phi^{N}\rightarrow\theta^{N}
  7. ( ¬ ϕ ) N is ¬ ϕ N (\neg\phi)^{N}\mbox{ is }~{}\neg\phi^{N}
  8. ( x ϕ ) N is x ϕ N (\forall\mbox{ }~{}x\mbox{ }~{}\phi)^{N}\mbox{ is }~{}\forall\mbox{ }~{}x\mbox% { }~{}\phi^{N}
  9. ( x ϕ ) N is ¬ x ¬ ϕ N (\exists\mbox{ }~{}x\mbox{ }~{}\phi)^{N}\mbox{ is }~{}\neg\forall\mbox{ }~{}x% \mbox{ }~{}\neg\phi^{N}

DTDP-4-amino-4,6-dideoxy-D-glucose_transaminase.html

  1. \rightleftharpoons

DTDP-4-amino-4,6-dideoxygalactose_transaminase.html

  1. \rightleftharpoons

DTDP-glucose_4,6-dehydratase.html

  1. \rightleftharpoons

DTMP_kinase.html

  1. \rightleftharpoons

DUTP_diphosphatase.html

  1. \rightleftharpoons

Earnings_growth.html

  1. P = D i = 1 ( 1 + g i 1 + k ) i P=D\cdot\sum_{i=1}^{\infty}\left(\frac{1+g_{i}}{1+k}\right)^{i}
  2. g i g_{i}
  3. i = n i=n
  4. \infty
  5. P = D 1 + g 1 1 + k + D ( 1 + g 2 1 + k ) 2 + + D ( 1 + g n 1 + k ) n + D i = n + 1 ( 1 + g 1 + k ) i P=D\cdot\frac{1+g_{1}}{1+k}+D\cdot(\frac{1+g_{2}}{1+k})^{2}+...+D\cdot(\frac{1% +g_{n}}{1+k})^{n}+D\cdot\sum_{i=n+1}^{\infty}\left(\frac{1+g_{\infty}}{1+k}% \right)^{i}
  6. P = D × 1 + g k - g P=D\times\frac{1+g}{k-g}
  7. g = P l o w b a c k r a t i o × r e t u r n o n e q u i t y g={Plowback\ ratio}\times{return\ on\ equity}

Ecdysone_O-acyltransferase.html

  1. \rightleftharpoons

Ectoine_synthase.html

  1. \rightleftharpoons

Effective_molarity.html

  1. E M k i n e t i c = k i n t r a m o l e c u l a r k i n t e r m o l e c u l a r EM_{kinetic}={k_{intramolecular}\over k_{intermolecular}}
  2. E M t h e r m o = K i n t r a m o l e c u l a r K i n t e r m o l e c u l a r EM_{thermo}={K_{intramolecular}\over K_{intermolecular}}

Eilenberg–Mazur_swindle.html

  1. A = A + ( B + A ) + ( B + A ) + = ( A + B ) + ( A + B ) + = 0 A=A+(B+A)+(B+A)+\cdots=(A+B)+(A+B)+\cdots=0\,

Electronics_and_Radar_Development_Establishment.html

  1. m 2 m^{2}

Elongation_factor_2_kinase.html

  1. \rightleftharpoons

Embedded_pushdown_automaton.html

  1. Γ \,\Gamma
  2. σ i Γ * \,\sigma_{i}\in\Gamma^{*}
  3. j \,j
  4. Υ j = σ j = { σ j , k , σ j , k - 1 , , σ j , 1 } \,\Upsilon_{j}=\ddagger\sigma_{j}=\{\sigma_{j,k},\sigma_{j,k-1},\ldots,\sigma_% {j,1}\}
  5. σ j , k \,\sigma_{j,k}
  6. m \,m
  7. { Υ j } = { σ m , σ m - 1 , , σ 1 } ( Γ + ) * \,\{\Upsilon_{j}\}=\{\ddagger\sigma_{m},\ddagger\sigma_{m-1},\ldots,\ddagger% \sigma_{1}\}\in(\ddagger\Gamma^{+})^{*}
  8. M = ( Q , Σ , Γ , δ , q 0 , Q F , σ 0 ) \,M=(Q,\Sigma,\Gamma,\delta,q_{0},Q_{\textrm{F}},\sigma_{0})
  9. Q \,Q
  10. Σ \,\Sigma
  11. Γ \,\Gamma
  12. q 0 Q \,q_{0}\in Q
  13. Q F Q \,Q_{\textrm{F}}\subseteq Q
  14. σ 0 Γ \,\sigma_{0}\in\Gamma
  15. δ : Q × Σ × Γ S \,\delta:Q\times\Sigma\times\Gamma\rightarrow S
  16. S \,S
  17. Q × ( Γ + ) * × Γ * × ( Γ + ) * \,Q\times(\ddagger\Gamma^{+})^{*}\times\Gamma^{*}\times(\ddagger\Gamma^{+})^{*}
  18. C ( M ) = { q , Υ m Υ 1 , x 1 , x 2 } Q × ( Γ + ) * × Σ * × Σ * \,C(M)=\{q,\Upsilon_{m}\ldots\Upsilon_{1},x_{1},x_{2}\}\in Q\times(\ddagger% \Gamma^{+})^{*}\times\Sigma^{*}\times\Sigma^{*}
  19. q \,q
  20. Υ \,\Upsilon
  21. Υ m \,\Upsilon_{m}
  22. x = x 1 x 2 Σ * \,x=x_{1}x_{2}\in\Sigma^{*}
  23. x 1 \,x_{1}
  24. x 2 \,x_{2}
  25. ϵ Σ \,\epsilon\in\Sigma
  26. L ( M ) = { x | { q 0 , Υ 0 , ϵ , x } M * { q F , Υ m Υ 1 , x , ϵ } } \,L(M)=\left\{x|\{q_{0},\Upsilon_{0},\epsilon,x\}\rightarrow_{M}^{*}\{q_{% \textrm{F}},\Upsilon_{m}\ldots\Upsilon_{1},x,\epsilon\}\right\}
  27. q F Q F \,q_{\textrm{F}}\in Q_{\textrm{F}}
  28. M * \,\rightarrow_{M}^{*}
  29. O ( n 3 2 k - 1 ) O(n^{3\cdot 2^{k-1}})
  30. { a 1 n a 2 k n | n 0 } \{a_{1}^{n}\ldots a_{2^{k}}^{n}|n\geq 0\}
  31. { a 1 n a 2 k + 1 n | n 0 } \{a_{1}^{n}\ldots a_{2^{k+1}}^{n}|n\geq 0\}
  32. { w 2 k - 1 | w { a , b } * } \{w^{2^{k-1}}|w\in\{a,b\}^{*}\}
  33. { w 2 k - 1 + 1 | w { a , b } * } \{w^{2^{k-1}+1}|w\in\{a,b\}^{*}\}

Enamidase.html

  1. \rightleftharpoons

Endoglycosylceramidase.html

  1. \rightleftharpoons

Endopolyphosphatase.html

  1. \rightleftharpoons

Ent-kaurene_synthase.html

  1. \rightleftharpoons

Enveloping_von_Neumann_algebra.html

  1. Φ : π U ( A ) ′′ π ( A ) ′′ \Phi:\pi_{U}(A)^{\prime\prime}\rightarrow\pi(A)^{\prime\prime}

Episodic_tremor_and_slip.html

  1. M w 8.0 M_{w}\geq 8.0
  2. M w M_{w}

Equatorial_electrojet.html

  1. Δ \Delta

Equity_ratio.html

  1. Equity Ratio = Total Shareholder’s Equity Total Assets \mbox{Equity Ratio}~{}=\frac{\mbox{Total Shareholder's Equity}~{}}{\mbox{Total% Assets}~{}}

Erythritol_kinase.html

  1. \rightleftharpoons

Erythronolide_synthase.html

  1. \rightleftharpoons

Eta_meson.html

  1. η 1 = u u ¯ + d d ¯ + s s ¯ 3 \eta_{1}=\mathrm{\tfrac{u\bar{u}+d\bar{d}+s\bar{s}}{\sqrt{3}}}
  2. η 8 = u u ¯ + d d ¯ - 2 s s ¯ 6 \eta_{8}=\mathrm{\tfrac{u\bar{u}+d\bar{d}-2s\bar{s}}{\sqrt{6}}}
  3. ( cos θ P - sin θ P sin θ P cos θ P ) ( η 8 η 1 ) = ( η η ) \left(\begin{array}[]{cc}\cos\theta_{\mathrm{P}}&-\sin\theta_{\mathrm{P}}\\ \sin\theta_{\mathrm{P}}&\cos\theta_{\mathrm{P}}\end{array}\right)\left(\begin{% array}[]{c}\eta_{8}\\ \eta_{1}\end{array}\right)=\left(\begin{array}[]{c}\eta\\ \eta^{\prime}\end{array}\right)
  4. π 0 = u u ¯ - d d ¯ 2 \pi^{0}=\mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt{2}}}

Ethanolamine-phosphate_cytidylyltransferase.html

  1. \rightleftharpoons

Ethanolamine-phosphate_phospho-lyase.html

  1. \rightleftharpoons

Ethanolamine_kinase.html

  1. \rightleftharpoons

Ethyl_iodide.html

  1. 3 C 2 H 5 OH + PI 3 3 C 2 H 5 I + H 3 PO 3 \mathrm{3\ C_{2}H_{5}OH+PI_{3}\rightarrow 3\ C_{2}H_{5}I+H_{3}PO_{3}}

Euclid's_orchard.html

  1. ( m m + n , n m + n , 1 m + n ) . \left(\frac{m}{m+n},\frac{n}{m+n},\frac{1}{m+n}\right).

Euler_force.html

  1. 𝐚 Euler = - d s y m b o l ω d t × 𝐫 , \mathbf{a}_{\mathrm{Euler}}=-\frac{dsymbol\omega}{dt}\times\mathbf{r},
  2. 𝐅 Euler = m 𝐚 Euler = - m d s y m b o l ω d t × 𝐫 . \mathbf{F}_{\mathrm{Euler}}=m\mathbf{a}_{\mathrm{Euler}}=-m\frac{dsymbol\omega% }{dt}\times\mathbf{r}.

Excess_chemical_potential.html

  1. L L
  2. V V
  3. T T
  4. Q ( N , V , T ) = V N Λ d N N ! 0 1 0 1 d s N exp [ - β U ( s N ; L ) ] Q(N,V,T)=\frac{V^{N}}{\Lambda^{dN}N!}\int_{0}^{1}\ldots\int_{0}^{1}ds^{N}\exp[% -\beta U(s^{N};L)]
  5. F ( N , V , T ) = - k B T ln Q = - k B T ln ( V N Λ d N N ! ) - k B T ln d s N exp [ - β U ( s N ; L ) ] = F(N,V,T)=-k_{B}T\ln Q=-k_{B}T\ln\left(\frac{V^{N}}{\Lambda^{dN}N!}\right)-k_{B% }T\ln{\int ds^{N}\exp[-\beta U(s^{N};L)]}=
  6. = F i d ( N , V , T ) + F e x ( N , V , T ) \;=F_{id}(N,V,T)+F_{ex}(N,V,T)
  7. μ a = ( G N a ) P T N \mu_{a}=\left(\frac{\partial G}{\partial N_{a}}\right)_{PTN}
  8. μ = - k B T ln ( Q N + 1 / Q N ) = - k B T ln ( V / Λ d N + 1 ) - k B T ln d s N + 1 exp [ - β U ( s N + 1 ) ] d s N exp [ - β U ( s N ) ] = μ i d ( ρ ) + μ e x \mu=-k_{B}T\ln(Q_{N+1}/Q_{N})=-k_{B}T\ln\left(\frac{V/\Lambda^{d}}{N+1}\right)% -k_{B}T\ln{\frac{\int ds^{N+1}\exp[-\beta U(s^{N+1})]}{\int ds^{N}\exp[-\beta U% (s^{N})]}}=\mu_{id}(\rho)+\mu_{ex}
  9. μ e x \mu_{ex}
  10. Δ U U ( s N + 1 ) - U ( s N ) \Delta U\equiv U(s^{N+1})-U(s^{N})
  11. μ e x = - k B T ln d s N + 1 exp ( - β Δ U ) N \mu_{ex}=-k_{B}T\ln\int ds_{N+1}\langle\exp(-\beta\Delta U)\rangle_{N}

Explained_variation.html

  1. F ( θ ) = d r g ( r ) ln f ( r ; θ ) F(\theta)=\int\textrm{d}r\,g(r)\,\ln f(r;\theta)
  2. g ( r ) g(r)
  3. R R\,
  4. f ( r ; θ ) f(r;\theta)\,
  5. θ Θ i \theta\in\Theta_{i}
  6. i = 0 , 1 i=0,1\,
  7. Θ 0 Θ 1 \Theta_{0}\subset\Theta_{1}
  8. θ i = arg max F θ Θ i ( θ ) \theta_{i}=\mbox{arg max}~{}_{\theta\in\Theta_{i}}F(\theta)
  9. Γ ( θ 1 : θ 0 ) = 2 [ F ( θ 1 ) - F ( θ 0 ) ] \Gamma(\theta_{1}:\theta_{0})=2[F(\theta_{1})-F(\theta_{0})]\,
  10. R = ( X , Y ) R=(X,Y)
  11. f ( y | x ; θ ) f(y|x;\theta)
  12. D ( Y ) = exp [ - 2 F ( θ 0 ) ] D(Y)=\exp[-2F(\theta_{0})]
  13. D ( Y | X ) = exp [ - 2 F ( θ 1 ) ] D(Y|X)=\exp[-2F(\theta_{1})]
  14. ρ C 2 = 1 - D ( Y | X ) / D ( Y ) \rho_{C}^{2}=1-D(Y|X)/D(Y)
  15. μ + Ψ T X \mu+\Psi^{\textrm{T}}X
  16. ρ C 2 \rho_{C}^{2}
  17. R 2 R^{2}
  18. R 2 R^{2}
  19. R 2 R^{2}
  20. R 2 R^{2}
  21. R 2 R^{2}
  22. R 2 R^{2}
  23. R 2 R^{2}

Exponential_dispersion_model.html

  1. f X ( 𝐱 | s y m b o l θ ) = h ( 𝐱 ) exp ( s y m b o l θ 𝐱 - A ( s y m b o l θ ) ) , f_{X}(\mathbf{x}|symbol{\theta})=h(\mathbf{x})\exp(symbol\theta^{\top}\mathbf{% x}-A(symbol\theta))\,\!,
  2. s y m b o l θ symbol\theta
  3. 𝐱 \mathbf{x}
  4. λ \lambda
  5. f X ( 𝐱 | λ , s y m b o l θ ) = h ( λ , 𝐱 ) exp ( λ [ s y m b o l θ 𝐱 - A ( s y m b o l θ ) ] ) . f_{X}(\mathbf{x}|\lambda,symbol{\theta})=h(\lambda,\mathbf{x})\exp(\lambda[% symbol\theta^{\top}\mathbf{x}-A(symbol\theta)])\,\!.
  6. σ 2 = λ - 1 \sigma^{2}=\lambda^{-1}
  7. s y m b o l θ symbol\theta

Extraction_ratio.html

  1. E x t r a c t i o n r a t i o = P a - P v P a Extractionratio=\frac{P_{a}-P_{v}}{P_{a}}

Extraneous_and_missing_solutions.html

  1. x + 2 = 0 x+2=0
  2. 0 = 0 0=0
  3. x ( x + 2 ) = ( 0 ) x x(x+2)=(0)x
  4. x 2 + 2 x = 0 x^{2}+2x=0
  5. ( x + 2 ) ( x + 2 ) = 0 ( x + 2 ) (x+2)(x+2)=0(x+2)
  6. x 2 + 4 x + 4 = 0 x^{2}+4x+4=0
  7. 1 x - 2 = 3 x + 2 - 6 x ( x - 2 ) ( x + 2 ) . \frac{1}{x-2}=\frac{3}{x+2}-\frac{6x}{(x-2)(x+2)}\,.
  8. ( x - 2 ) ( x + 2 ) (x-2)(x+2)
  9. x + 2 = 3 ( x - 2 ) - 6 x . x+2=3(x-2)-6x\,.
  10. 1 - 2 - 2 = 3 - 2 + 2 - 6 ( - 2 ) ( - 2 - 2 ) ( - 2 + 2 ) . \frac{1}{-2-2}=\frac{3}{-2+2}-\frac{6(-2)}{(-2-2)(-2+2)}\,.
  11. 1 - 4 = 3 0 + 12 0 . \frac{1}{-4}=\frac{3}{0}+\frac{12}{0}\,.
  12. 2 x + 4 = 0 2x+4=0
  13. 2 x = - 4 2x=-4
  14. x = - 2 x=-2
  15. x 2 + 2 x = 0 x^{2}+2x=0
  16. x 2 = - 2 x x^{2}=-2x
  17. x = - 2 x=-2
  18. x + 2 = 0 x+2=0
  19. x + 2 x - 2 = 0 \frac{x+2}{x-2}=0
  20. x 2 = 4. x^{2}=4.
  21. x = 2. x=2.
  22. x 2 = 4 . \sqrt{x^{2}}=\sqrt{4}.
  23. | x | = 2. |x|=2.
  24. x = ± 2. x=\pm 2.

F-divergence.html

  1. D f ( P Q ) Ω f ( d P d Q ) d Q . D_{f}(P\parallel Q)\equiv\int_{\Omega}f\left(\frac{dP}{dQ}\right)\,dQ.
  2. D f ( P Q ) = Ω f ( p ( x ) q ( x ) ) q ( x ) d μ ( x ) . D_{f}(P\parallel Q)=\int_{\Omega}f\left(\frac{p(x)}{q(x)}\right)q(x)\,d\mu(x).
  3. t ln t , - ln t t\ln t\,,-\ln t
  4. ( t - 1 ) 2 , 2 ( 1 - t ) (\sqrt{t}-1)^{2},\,2(1-\sqrt{t})
  5. | t - 1 | |t-1|\,
  6. χ 2 \chi^{2}
  7. ( t - 1 ) 2 , t 2 - 1 (t-1)^{2},\,t^{2}-1
  8. { 4 1 - α 2 ( 1 - t ( 1 + α ) / 2 ) , if α ± 1 , t ln t , if α = 1 , - ln t , if α = - 1 \begin{cases}\frac{4}{1-\alpha^{2}}\big(1-t^{(1+\alpha)/2}\big),&\,\text{if}\ % \alpha\neq\pm 1,\\ t\ln t,&\,\text{if}\ \alpha=1,\\ -\ln t,&\,\text{if}\ \alpha=-1\end{cases}

FAD_diphosphatase.html

  1. \rightleftharpoons

Farnesyl-diphosphate_kinase.html

  1. \rightleftharpoons

Farnesyltranstransferase.html

  1. \rightleftharpoons

Fas-activated_serine::threonine_kinase.html

  1. \rightleftharpoons

Fatty-acid_synthase.html

  1. \rightleftharpoons

Fatty-acyl-CoA-transporting_ATPase.html

  1. \rightleftharpoons

Fatty-acyl-CoA_synthase.html

  1. \rightleftharpoons

Fatty-acyl-ethyl-ester_synthase.html

  1. \rightleftharpoons

Favard_operator.html

  1. [ n ( f ) ] ( x ) = n n c π k = - exp ( - n c ( k n - x ) 2 ) f ( k n ) [\mathcal{F}_{n}(f)](x)=\frac{\sqrt{n}}{n\sqrt{c\pi}}\sum_{k=-\infty}^{\infty}% {\exp{\left({\frac{-n}{c}{\left({\frac{k}{n}-x}\right)}^{2}}\right)}f\left(% \frac{k}{n}\right)}
  2. x x\in\mathbb{R}
  3. n n\in\mathbb{N}
  4. c + c\in\mathbb{R^{+}}
  5. [ n ( f ) ] ( x ) = 1 n γ n 2 π k = - exp ( - 1 2 γ n 2 ( k n - x ) 2 ) f ( k n ) [\mathcal{F}_{n}(f)](x)=\frac{1}{n\gamma_{n}\sqrt{2\pi}}\sum_{k=-\infty}^{% \infty}{\exp{\left({\frac{-1}{2\gamma_{n}^{2}}{\left({\frac{k}{n}-x}\right)}^{% 2}}\right)}f\left(\frac{k}{n}\right)}
  6. ( γ n ) n = 1 (\gamma_{n})_{n=1}^{\infty}
  7. γ n 2 = c / ( 2 n ) \gamma_{n}^{2}=c/(2n)

Fe3+-transporting_ATPase.html

  1. \rightleftharpoons

Federal_Bridge_Gross_Weight_Formula.html

  1. w = 500 ( n n - 1 + 12 n + 36 ) w=500\left(\frac{\ell n}{n-1}+12n+36\right)

FENE.html

  1. F i = H R i 1 - ( R i / L m a x ) 2 F_{i}=\frac{HR_{i}}{1-(R_{i}/L_{max})^{2}}

Fermi–Ulam_model.html

  1. x = c o n s t . x=const.
  2. u n + 1 = | u n + U wall ( φ n ) | u_{n+1}=|u_{n}+U_{\mathrm{wall}}(\varphi_{n})|\,
  3. φ n + 1 = φ n + k M u n + 1 ( mod k ) , \varphi_{n+1}=\varphi_{n}+\frac{kM}{u_{n+1}}\;\;(\mathop{{\rm mod}}k),
  4. u n u_{n}
  5. n n
  6. φ n \varphi_{n}
  7. U wall U_{\mathrm{wall}}
  8. M M
  9. ( φ , u ) (\varphi,u)

Ferromagnetic_superconductor.html

  1. H = H 0 + 𝐤 σ E 𝐤 σ γ 𝐤 σ γ 𝐤 σ H=H_{0}+\sum_{\mathbf{k}\sigma}E_{\mathbf{k}\sigma}\gamma_{\mathbf{k}\sigma}^{% \dagger}\gamma_{\mathbf{k}\sigma}
  2. H 0 = 1 2 𝐤 σ ( ξ 𝐤 σ - E 𝐤 σ - Δ 𝐤 σ b 𝐤 σ ) + I N M 2 / 2 H_{0}=\frac{1}{2}\sum_{\mathbf{k}\sigma}(\xi_{\mathbf{k}\sigma}-E_{\mathbf{k}% \sigma}-\Delta_{\mathbf{k}\sigma}^{\dagger}b_{\mathbf{k}\sigma})+INM^{2}/2
  3. E 𝐤 σ = ξ 𝐤 σ 2 + | Δ 𝐤 σ | 2 E_{\mathbf{k}\sigma}=\sqrt{\xi_{\mathbf{k}\sigma}^{2}+|\Delta_{\mathbf{k}% \sigma}|^{2}}

Feruloyl_esterase.html

  1. \rightleftharpoons

Field_arithmetic.html

  1. G = G a l ( 𝐂 / 𝐑 ) = 𝐙 / 2 𝐙 . G=Gal(\mathbf{C}/\mathbf{R})=\mathbf{Z}/2\mathbf{Z}.
  2. 𝐙 ^ = lim 𝐙 / n 𝐙 . \hat{\mathbf{Z}}=\lim_{\longleftarrow}\mathbf{Z}/n\mathbf{Z}.\,

File:2d_multiple_linear_regression.gif.html

  1. Y = β 0 + β 1 X 1 + β 2 X 2 Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}

File:Angle_of_View_F_V_Chambers_1916.png.html

  1. 2 arctan ( d / 2 f ) 2\arctan(d/2f)

File:Direct.png.html

  1. d A dA
  2. r \vec{r}
  3. O O
  4. d Ω d\Omega

File:Energyflow.png.html

  1. d A dA
  2. r \vec{r}
  3. O O
  4. d Ω d\Omega

File:F(r,theta)=r_sin(2_theta).png.html

  1. f ( r , θ ) = r sin ( 2 θ ) f(r,\theta)=r\sin(2\theta)
  2. f ( r , θ ) = ( x , y ) sin ( 2 tan - 1 ( y x ) ) f(r,\theta)=\left\|(x,y)\right\|\sin\left(2\tan^{-1}\left(\frac{y}{x}\right)\right)

File:Part_circle.svg.html

  1. f ( x ) = { 1 - x 2 for x [ 0 , 1 ] , - 1 - x 2 for x [ - 1 , 0 ) . f(x)=\begin{cases}\sqrt{1-x^{2}}&\,\text{for }x\in[0,1],\\ -\sqrt{1-x^{2}}&\,\text{for }x\in[-1,0).\end{cases}

File:Perspective_Transform_Diagram.png.html

  1. s c r e e n x c o o r d i n a t e ( B x ) = m o d e l x c o o r d i n a t e ( A x ) × d i s t a n c e f r o m e y e t o s c r e e n ( B z ) d i s t a n c e f r o m e y e t o p o i n t ( A z ) screen\ x\ coordinate\ (Bx)\ =\ model\ x\ coordinate\ (Ax)\times\frac{distance% \ from\ eye\ to\ screen\ (Bz)}{distance\ from\ eye\ to\ point\ (Az)}
  2. s c r e e n y c o o r d i n a t e ( B y ) = m o d e l y c o o r d i n a t e ( A y ) × d i s t a n c e f r o m e y e t o s c r e e n ( B z ) d i s t a n c e f r o m e y e t o p o i n t ( A z ) screen\ y\ coordinate\ (By)\ =\ model\ y\ coordinate\ (Ay)\times\frac{distance% \ from\ eye\ to\ screen\ (Bz)}{distance\ from\ eye\ to\ point\ (Az)}

File:Semicircles.svg.html

  1. f ( x ) = { 1 - ( x - 1 ) 2 for x [ 0 , 2 ] , - 1 - ( x + 1 ) 2 for x [ - 2 , 0 ] . f(x)=\begin{cases}\sqrt{1-(x-1)^{2}}&\,\text{for }x\in[0,2],\\ -\sqrt{1-(x+1)^{2}}&\,\text{for }x\in[-2,0].\end{cases}

Finite_topological_space.html

  1. \uarr x = { y X : x y } \mathop{\uarr}x=\{y\in X:x\leq y\}
  2. d ( x , y ) = { 0 x y 1 x y d(x,y)=\begin{cases}0&x\equiv y\\ 1&x\not\equiv y\end{cases}
  3. T ( n ) = k = 0 n S ( n , k ) T 0 ( k ) T(n)=\sum_{k=0}^{n}S(n,k)\,T_{0}(k)

Fisher's_inequality.html

  1. b v . b\geq v.\,

FLAME_clustering.html

  1. E ( { s y m b o l p } ) = s y m b o l x \insymbol X s y m b o l p ( x ) - s y m b o l y 𝒩 ( x ) w s y m b o l x y s y m b o l p ( y ) 2 E(\{symbol{p}\})=\sum_{symbol{x}\insymbol{X}}\bigg\|symbol{p(x)}-\sum_{symbol{% y\in\mathcal{N}(x)}}w_{symbol{xy}}symbol{p(y)}\bigg\|^{2}
  2. s y m b o l X symbol{X}
  3. s y m b o l p ( x ) symbol{p(x)}
  4. s y m b o l x symbol{x}
  5. 𝒩 ( x ) \mathcal{N}(x)
  6. s y m b o l x symbol{x}
  7. w s y m b o l x y w_{symbol{xy}}
  8. s y m b o l y 𝒩 ( x ) w s y m b o l x y = 1 \sum_{symbol{y\in\mathcal{N}(x)}}w_{symbol{xy}}=1
  9. p k ( s y m b o l x ) - s y m b o l y 𝒩 ( x ) w s y m b o l x y p k ( s y m b o l y ) = 0 , s y m b o l x s y m b o l X , k = 1 , , M p_{k}(symbol{x})-\sum_{symbol{y\in\mathcal{N}(x)}}w_{symbol{xy}}p_{k}(symbol{y% })=0,\quad\forall{symbol{x}\in symbol{X}},\quad k=1,...,M
  10. M M
  11. s y m b o l p t + 1 ( s y m b o l x ) = s y m b o l y 𝒩 ( x ) w s y m b o l x y s y m b o l p t ( s y m b o l y ) {symbol{p}^{t+1}(symbol{x})}=\sum_{symbol{y\in\mathcal{N}(x)}}w_{symbol{xy}}{% symbol{p}^{t}(symbol{y})}

Flavanone_7-O-beta-glucosyltransferase.html

  1. \rightleftharpoons

Flavone_7-O-beta-glucosyltransferase.html

  1. \rightleftharpoons

Flavone_apiosyltransferase.html

  1. \rightleftharpoons

Flavonol-3-O-beta-glucoside_O-malonyltransferase.html

  1. \rightleftharpoons

Flavonol-3-O-glucoside_glucosyltransferase.html

  1. \rightleftharpoons

Flavonol-3-O-glucoside_L-rhamnosyltransferase.html

  1. \rightleftharpoons

Flavonol-3-O-glycoside_glucosyltransferase.html

  1. \rightleftharpoons

Flavonol-3-O-glycoside_xylosyltransferase.html

  1. \rightleftharpoons

Flavonol-3-O-triglucoside_O-coumaroyltransferase.html

  1. \rightleftharpoons

Flavonol_3-O-glucosyltransferase.html

  1. \rightleftharpoons

Flavonol_7-O-beta-glucosyltransferase.html

  1. \rightleftharpoons

Fluorothreonine_transaldolase.html

  1. \rightleftharpoons

FMN_adenylyltransferase.html

  1. \rightleftharpoons

Formaldehyde_transketolase.html

  1. \rightleftharpoons

Formamidase.html

  1. \rightleftharpoons

Formate_C-acetyltransferase.html

  1. \rightleftharpoons

Formate_kinase.html

  1. \rightleftharpoons

Formate–tetrahydrofolate_ligase.html

  1. \rightleftharpoons

Formate—dihydrofolate_ligase.html

  1. \rightleftharpoons

Formimidoylaspartate_deiminase.html

  1. \rightleftharpoons

Formimidoylglutamase.html

  1. \rightleftharpoons

Formimidoylglutamate_deiminase.html

  1. \rightleftharpoons

Formyl-CoA_hydrolase.html

  1. \rightleftharpoons

Formylaspartate_deformylase.html

  1. \rightleftharpoons

Formylmethanofuran—tetrahydromethanopterin_N-formyltransferase.html

  1. \rightleftharpoons

Formylmethionine_deformylase.html

  1. \rightleftharpoons

Formyltetrahydrofolate_deformylase.html

  1. \rightleftharpoons

Foundations_of_statistics.html

  1. α \alpha

Four_exponentials_conjecture.html

  1. e x 1 y 1 , e x 1 y 2 , e x 2 y 1 , e x 2 y 2 . e^{x_{1}y_{1}},e^{x_{1}y_{2}},e^{x_{2}y_{1}},e^{x_{2}y_{2}}.
  2. λ 11 λ 22 λ 12 λ 21 . \lambda_{11}\lambda_{22}\neq\lambda_{12}\lambda_{21}.\,
  3. M = ( λ 11 λ 12 λ 21 λ 22 ) , M=\begin{pmatrix}\lambda_{11}&\lambda_{12}\\ \lambda_{21}&\lambda_{22}\end{pmatrix},
  4. ( 1 π π π 2 ) \begin{pmatrix}1&\pi\\ \pi&\pi^{2}\end{pmatrix}
  5. e i π , e i π 2 , e i π 2 , e 2 i π . e^{i\pi},e^{i\pi\sqrt{2}},e^{i\pi\sqrt{2}},e^{2i\pi}.
  6. 2 , 3 , 2 t , 3 t . 2,3,2^{t},3^{t}.\,
  7. 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_{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}},
  8. β 0 + i = 1 n β i log α i , \beta_{0}+\sum_{i=1}^{n}\beta_{i}\log\alpha_{i},
  9. e x 1 y , e x 2 y , e γ x 1 / x 2 . e^{x_{1}y},e^{x_{2}y},e^{\gamma x_{1}/x_{2}}.
  10. e x 1 y - β 1 , e x 2 y - β 2 , e ( γ x 1 / x 2 ) - α , e^{x_{1}y-\beta_{1}},e^{x_{2}y-\beta_{2}},e^{(\gamma x_{1}/x_{2})-\alpha},
  11. τ τ
  12. τ τ

Four_factor_formula.html

  1. k = η f p ε k_{\infty}=\eta fp\varepsilon
  2. η \eta
  3. η = ν σ f F σ a F \eta=\frac{\nu\sigma_{f}^{F}}{\sigma_{a}^{F}}
  4. f f
  5. f = Σ a F Σ a f=\frac{\Sigma_{a}^{F}}{\Sigma_{a}}
  6. p p
  7. p exp ( - i = 1 N N i I r , A , i ( ξ ¯ Σ p ) m o d ) p\approx\mathrm{exp}\left(-\frac{\sum\limits_{i=1}^{N}N_{i}I_{r,A,i}}{\left(% \overline{\xi}\Sigma_{p}\right)_{mod}}\right)
  8. ϵ \epsilon
  9. total number of fission neutrons number of fission neutrons from just thermal fissions \frac{\mbox{total number of fission neutrons}~{}}{\mbox{number of fission % neutrons from just thermal fissions}~{}}
  10. ε 1 + 1 - p p u f ν f P F A F f ν t P T A F P T N L \varepsilon\approx 1+\frac{1-p}{p}\frac{u_{f}\nu_{f}P_{FAF}}{f\nu_{t}P_{TAF}P_% {TNL}}
  11. k = number of neutrons in one generation number of neutrons in preceding generation k=\frac{\mbox{number of neutrons in one generation}~{}}{\mbox{number of % neutrons in preceding generation}~{}}
  12. k = k k=k_{\infty}

Frank-Read_Source.html

  1. τ \tau
  2. F = τ b x F=\tau\cdot bx
  3. G b 2 Gb^{2}
  4. F = τ b x = 2 G b 2 F=\tau\cdot bx=2Gb^{2}
  5. τ = 2 G b x \tau=\frac{2Gb}{x}
  6. τ = 2 G b x \tau=\frac{2Gb}{x}

Free_expansion.html

  1. ( P i ) ( V i ) = ( P f ) ( V f ) (P_{i})(V_{i})=(P_{f})(V_{f})
  2. Δ S = d Q r e v T , \Delta S=\int\frac{dQ_{rev}}{T}\,,
  3. Δ S = i f d S = V i V f P d V T = V i V f n R d V V = n R ln V f V i . \Delta S=\int_{i}^{f}\mathrm{d}S=\int_{V_{i}}^{V_{f}}\frac{P\,\mathrm{d}V}{T}=% \int_{V_{i}}^{V_{f}}\frac{nR\,\mathrm{d}V}{V}=nR\ln\frac{V_{f}}{V_{i}}.

Free_surface.html

  1. P r = ρ r ω 2 , P θ = 0 , P z = - ρ g \frac{\partial P}{\partial r}=\rho r\omega^{2},\;\;\;\frac{\partial P}{% \partial\theta}=0,\;\;\;\frac{\partial P}{\partial z}=-\rho g
  2. ρ \rho
  3. r r
  4. ω \omega
  5. g g
  6. ( d P = 0 ) (dP=0)
  7. d P = ρ r ω 2 d r - ρ g d z d z i s o b a r d r = r ω 2 g dP=\rho r\omega^{2}dr-\rho gdz\rightarrow\frac{dz_{isobar}}{dr}=\frac{r\omega^% {2}}{g}
  8. z s = ω 2 2 g r 2 + h c z_{s}=\frac{\omega^{2}}{2g}r^{2}+h_{c}
  9. h c h_{c}
  10. h c = h 0 - ω 2 R 2 4 g h_{c}=h_{0}-\frac{\omega^{2}R^{2}}{4g}
  11. r r
  12. z s = h 0 - ω 2 4 g ( R 2 - 2 r 2 ) z_{s}=h_{0}-\frac{\omega^{2}}{4g}(R^{2}-2r^{2})
  13. D p D t = 0. \ \frac{Dp}{Dt}=0.

Fructose-2,6-bisphosphate_2-phosphatase.html

  1. \rightleftharpoons

Fructose-2,6-bisphosphate_6-phosphatase.html

  1. \rightleftharpoons

Fucokinase.html

  1. \rightleftharpoons

Fucose-1-phosphate_guanylyltransferase.html

  1. \rightleftharpoons

Fucosylgalactoside_3-alpha-galactosyltransferase.html

  1. \rightleftharpoons

Fuhrmann_circle.html

  1. R a 3 - a 2 b - a b 2 + b 3 - a 2 c + 3 a b c - b 2 c - a c 2 + c 3 a b c , R\sqrt{\frac{a^{3}-a^{2}b-ab^{2}+b^{3}-a^{2}c+3abc-b^{2}c-ac^{2}+c^{3}}{abc}},

Fusarinine-C_ornithinesterase.html

  1. \rightleftharpoons

Gabor–Wigner_transform.html

  1. G x ( t , f ) = - e - π ( τ - t ) 2 e - j 2 π f τ x ( τ ) d τ G_{x}(t,f)=\int_{-\infty}^{\infty}e^{-\pi(\tau-t)^{2}}e^{-j2\pi f\tau}x(\tau)% \,d\tau
  2. W x ( t , f ) = - x ( t + τ / 2 ) x * ( t - τ / 2 ) e - j 2 π τ f d τ W_{x}(t,f)=\int_{-\infty}^{\infty}x(t+\tau/2)x^{*}(t-\tau/2)e^{-j2\pi\tau\,f}% \,d\tau
  3. D x ( t , f ) = G x ( t , f ) × W x ( t , f ) D_{x}(t,f)=G_{x}(t,f)\times W_{x}(t,f)
  4. D x ( t , f ) = min { | G x ( t , f ) | 2 , | W x ( t , f ) | } D_{x}(t,f)=\min\left\{|G_{x}(t,f)|^{2},|W_{x}(t,f)|\right\}
  5. D x ( t , f ) = W x ( t , f ) × { | G x ( t , f ) | > 0.25 } D_{x}(t,f)=W_{x}(t,f)\times\{|G_{x}(t,f)|>0.25\}
  6. D x ( t , f ) = G x 2.6 ( t , f ) W x 0.7 ( t , f ) D_{x}(t,f)=G_{x}^{2.6}(t,f)W_{x}^{0.7}(t,f)

Galactarate_dehydratase.html

  1. \rightleftharpoons

Galactarate_O-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Galactinol—raffinose_galactosyltransferase.html

  1. \rightleftharpoons

Galactinol—sucrose_galactosyltransferase.html

  1. \rightleftharpoons

Galactogen_6beta-galactosyltransferase.html

  1. \rightleftharpoons

Galactolipase.html

  1. \rightleftharpoons

Galactolipid_galactosyltransferase.html

  1. \rightleftharpoons

Galactolipid_O-acyltransferase.html

  1. \rightleftharpoons

Galactonate_dehydratase.html

  1. \rightleftharpoons

Galactose-1-phosphate_thymidylyltransferase.html

  1. \rightleftharpoons

Galactoside_2-alpha-L-fucosyltransferase.html

  1. \rightleftharpoons

Galactosylacylglycerol_O-acyltransferase.html

  1. \rightleftharpoons

Galactosyldiacylglycerol_alpha-2,3-sialyltransferase.html

  1. \rightleftharpoons

Galactosylgalactosylglucosylceramidase.html

  1. \rightleftharpoons

Galactosylgalactosylxylosylprotein_3-beta-glucuronosyltransferase.html

  1. \rightleftharpoons

Galactosylxylosylprotein_3-beta-galactosyltransferase.html

  1. \rightleftharpoons

Galacturan_1,4-alpha-galacturonidase.html

  1. \rightleftharpoons

Galacturonokinase.html

  1. \rightleftharpoons

Gallate_1-beta-glucosyltransferase.html

  1. \rightleftharpoons

Galvani_potential.html

  1. μ ¯ j ( 1 ) = μ ¯ j ( 2 ) \overline{\mu}_{j}^{(1)}=\overline{\mu}_{j}^{(2)}
  2. μ ¯ \overline{\mu}
  3. μ ¯ j = μ j + z j F ϕ \overline{\mu}_{j}=\mu_{j}+z_{j}F\phi
  4. ϕ ( 2 ) - ϕ ( 1 ) = μ j ( 1 ) - μ j ( 2 ) z j F \phi^{(2)}-\phi^{(1)}=\frac{\mu_{j}^{(1)}-\mu_{j}^{(2)}}{z_{j}F}
  5. E ( 2 ) - E ( 1 ) = ( ϕ ( 2 ) - ϕ ( S ) - μ j ( 2 ) z j F ) - ( ϕ ( 1 ) - ϕ ( S ) - μ j ( 1 ) z j F ) = ( ϕ ( 2 ) - ϕ ( 1 ) ) - ( μ j ( 2 ) - μ j ( 1 ) z j F ) E^{(2)}-E^{(1)}=\left(\phi^{(2)}-\phi^{(S)}-\frac{\mu_{j}^{(2)}}{z_{j}F}\right% )-\left(\phi^{(1)}-\phi^{(S)}-\frac{\mu_{j}^{(1)}}{z_{j}F}\right)=\left(\phi^{% (2)}-\phi^{(1)}\right)-\left(\frac{\mu_{j}^{(2)}-\mu_{j}^{(1)}}{z_{j}F}\right)

Gamma-glutamyl-gamma-aminobutyrate_hydrolase.html

  1. \rightleftharpoons

Gamma-glutamylcyclotransferase.html

  1. \rightleftharpoons

Gamma-glutamylhistamine_synthase.html

  1. \rightleftharpoons

Gamow_factor.html

  1. P g ( E ) e - E g E 1 / 2 P_{g}(E)\equiv e^{-\frac{E_{g}}{E}^{1/2}}
  2. E g E_{g}
  3. E g 2 m r c 2 ( π α Z a Z b ) 2 E_{g}\equiv 2m_{r}c^{2}(\pi\alpha Z_{a}Z_{b})^{2}
  4. m r m_{r}
  5. m r = m 1 m 2 m 1 + m 2 m_{r}=\frac{m_{1}m_{2}}{m_{1}+m_{2}}
  6. α \alpha
  7. Z a Z_{a}
  8. Z b Z_{b}

Ganglioside_galactosyltransferase.html

  1. \rightleftharpoons

Gauss's_lemma_(Riemannian_geometry).html

  1. exp : T p M M \mathrm{exp}:T_{p}M\to M
  2. p M p\in M
  3. exp p : T p M B ϵ ( 0 ) M , v γ p , v ( 1 ) , \exp_{p}:T_{p}M\supset B_{\epsilon}(0)\longrightarrow M,\quad v\longmapsto% \gamma_{p,v}(1),
  4. γ p , v \gamma_{p,v}
  5. γ ( 0 ) = p \gamma(0)=p
  6. γ p , v ( 0 ) = v T p M \gamma_{p,v}^{\prime}(0)=v\in T_{p}M
  7. ϵ 0 \epsilon_{0}
  8. v B ϵ ( 0 ) T p M v\in B_{\epsilon}(0)\subset T_{p}M
  9. γ p , v \gamma_{p,v}
  10. M M
  11. exp p \exp_{p}
  12. α : I T p M \alpha:I\rightarrow T_{p}M
  13. T p M T_{p}M
  14. α ( 0 ) := 0 \alpha(0):=0
  15. α ( 0 ) := v \alpha^{\prime}(0):=v
  16. T p M n T_{p}M\cong\mathbb{R}^{n}
  17. α ( t ) := v t \alpha(t):=vt
  18. 0
  19. v v
  20. T 0 exp p ( v ) = d d t ( exp p α ( t ) ) | t = 0 = d d t ( exp p ( v t ) ) | t = 0 = d d t ( γ ( 1 , p , v t ) ) | t = 0 = γ ( t , p , v ) | t = 0 = v . T_{0}\exp_{p}(v)=\frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\exp_{p}\circ\alpha(t)% \Bigr)\Big|_{t=0}=\frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\exp_{p}(vt)\Bigr)\Big|_% {t=0}=\frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\gamma(1,p,vt)\Bigr)\Big|_{t=0}=% \gamma^{\prime}(t,p,v)\Big|_{t=0}=v.
  21. T 0 T p M T p M T_{0}T_{p}M\cong T_{p}M
  22. exp p \exp_{p}
  23. exp p \exp_{p}
  24. 0 T p M 0\in T_{p}M
  25. exp p \exp_{p}
  26. p M p\in M
  27. T v T p M T p M n T_{v}T_{p}M\cong T_{p}M\cong\mathbb{R}^{n}
  28. v , w B ϵ ( 0 ) T v T p M T p M v,w\in B_{\epsilon}(0)\subset T_{v}T_{p}M\cong T_{p}M
  29. M q := exp p ( v ) M\ni q:=\exp_{p}(v)
  30. T v exp p ( v ) , T v exp p ( w ) q = v , w p . \langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(w)\rangle_{q}=\langle v,w\rangle_{p}.
  31. p M p\in M
  32. exp p \exp_{p}
  33. v B ϵ ( 0 ) v\in B_{\epsilon}(0)
  34. exp p \exp_{p}
  35. q := exp p ( v ) M q:=\exp_{p}(v)\in M
  36. exp p \exp_{p}
  37. q q
  38. γ \gamma
  39. γ ( 1 , p , v ) = exp p ( v ) \gamma(1,p,v)=\exp_{p}(v)
  40. exp p \exp_{p}
  41. T v exp p : T p M T v T p M T v B ϵ ( 0 ) T exp p ( v ) M . T_{v}\exp_{p}\colon T_{p}M\cong T_{v}T_{p}M\supset T_{v}B_{\epsilon}(0)% \longrightarrow T_{\exp_{p}(v)}M.
  42. T v exp p ( v ) = v T_{v}\exp_{p}(v)=v
  43. α : I T p M \alpha:\mathbb{R}\supset I\rightarrow T_{p}M
  44. α ( 0 ) := v T p M \alpha(0):=v\in T_{p}M
  45. α ( 0 ) := v T v T p M T p M \alpha^{\prime}(0):=v\in T_{v}T_{p}M\cong T_{p}M
  46. T v T p M T p M n T_{v}T_{p}M\cong T_{p}M\cong\mathbb{R}^{n}
  47. α ( t ) := v ( t + 1 ) \alpha(t):=v(t+1)
  48. α ( t ) = v t \alpha(t)=vt
  49. I I
  50. 0
  51. T v exp p ( v ) = d d t ( exp p α ( t ) ) | t = 0 = d d t γ ( t , p , v ) | t = 0 = v . T_{v}\exp_{p}(v)=\frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\exp_{p}\circ\alpha(t)% \Bigr)\Big|_{t=0}=\frac{\mathrm{d}}{\mathrm{d}t}\gamma(t,p,v)\Big|_{t=0}=v.
  52. T v exp p ( v ) , T v exp p ( w ) \langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(w)\rangle
  53. w w
  54. w T w_{T}
  55. v v
  56. w N w_{N}
  57. v v
  58. w T := a v w_{T}:=av
  59. a a\in\mathbb{R}
  60. T v exp p ( v ) , T v exp p ( w ) = T v exp p ( v ) , T v exp p ( w T ) + T v exp p ( v ) , T v exp p ( w N ) \langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(w)\rangle=\langle T_{v}\exp_{p}(v),T_{v% }\exp_{p}(w_{T})\rangle+\langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(w_{N})\rangle
  61. = a T v exp p ( v ) , T v exp p ( v ) + T v exp p ( v ) , T v exp p ( w N ) = v , w T + T v exp p ( v ) , T v exp p ( w N ) . =a\langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(v)\rangle+\langle T_{v}\exp_{p}(v),T_% {v}\exp_{p}(w_{N})\rangle=\langle v,w_{T}\rangle+\langle T_{v}\exp_{p}(v),T_{v% }\exp_{p}(w_{N})\rangle.
  62. T v exp p ( v ) , T v exp p ( w N ) = v , w N = 0. \langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(w_{N})\rangle=\langle v,w_{N}\rangle=0.
  63. T v exp p ( v ) , T v exp p ( w N ) = 0 \langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(w_{N})\rangle=0
  64. α : [ - ϵ , ϵ ] × [ 0 , 1 ] T p M , ( s , t ) t v + t s w N . \alpha\colon[-\epsilon,\epsilon]\times[0,1]\longrightarrow T_{p}M,\qquad(s,t)% \longmapsto tv+tsw_{N}.
  65. α ( 0 , 1 ) = v , α t ( s , t ) = v + s w N , α s ( 0 , t ) = t w N . \alpha(0,1)=v,\qquad\frac{\partial\alpha}{\partial t}(s,t)=v+sw_{N},\qquad% \frac{\partial\alpha}{\partial s}(0,t)=tw_{N}.
  66. f : [ - ϵ , ϵ ] × [ 0 , 1 ] M , ( s , t ) exp p ( t v + t s w N ) , f\colon[-\epsilon,\epsilon]\times[0,1]\longrightarrow M,\qquad(s,t)\longmapsto% \exp_{p}(tv+tsw_{N}),
  67. T v exp p ( v ) = T α ( 0 , 1 ) exp p ( α t ( 0 , 1 ) ) = t ( exp p α ( s , t ) ) | t = 1 , s = 0 = f t ( 0 , 1 ) T_{v}\exp_{p}(v)=T_{\alpha(0,1)}\exp_{p}\left(\frac{\partial\alpha}{\partial t% }(0,1)\right)=\frac{\partial}{\partial t}\Bigl(\exp_{p}\circ\alpha(s,t)\Bigr)% \Big|_{t=1,s=0}=\frac{\partial f}{\partial t}(0,1)
  68. T v exp p ( w N ) = T α ( 0 , 1 ) exp p ( α s ( 0 , 1 ) ) = s ( exp p α ( s , t ) ) | t = 1 , s = 0 = f s ( 0 , 1 ) . T_{v}\exp_{p}(w_{N})=T_{\alpha(0,1)}\exp_{p}\left(\frac{\partial\alpha}{% \partial s}(0,1)\right)=\frac{\partial}{\partial s}\Bigl(\exp_{p}\circ\alpha(s% ,t)\Bigr)\Big|_{t=1,s=0}=\frac{\partial f}{\partial s}(0,1).
  69. T v exp p ( v ) , T v exp p ( w N ) = f t , f s ( 0 , 1 ) . \langle T_{v}\exp_{p}(v),T_{v}\exp_{p}(w_{N})\rangle=\left\langle\frac{% \partial f}{\partial t},\frac{\partial f}{\partial s}\right\rangle(0,1).
  70. t t
  71. f t , f s ( 0 , 1 ) = f t , f s ( 0 , 0 ) = 0 , \left\langle\frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right% \rangle(0,1)=\left\langle\frac{\partial f}{\partial t},\frac{\partial f}{% \partial s}\right\rangle(0,0)=0,
  72. lim t 0 f s ( 0 , t ) = lim t 0 T t v exp p ( t w N ) = 0 \lim_{t\rightarrow 0}\frac{\partial f}{\partial s}(0,t)=\lim_{t\rightarrow 0}T% _{tv}\exp_{p}(tw_{N})=0
  73. t f t , f s = 0 \frac{\partial}{\partial t}\left\langle\frac{\partial f}{\partial t},\frac{% \partial f}{\partial s}\right\rangle=0
  74. t f ( s , t ) t\mapsto f(s,t)
  75. t f t , f s = D t f t = 0 , f s + f t , D t f s = f t , D s f t = 1 2 s f t , f t . \frac{\partial}{\partial t}\left\langle\frac{\partial f}{\partial t},\frac{% \partial f}{\partial s}\right\rangle=\left\langle\underbrace{\frac{D}{\partial t% }\frac{\partial f}{\partial t}}_{=0},\frac{\partial f}{\partial s}\right% \rangle+\left\langle\frac{\partial f}{\partial t},\frac{D}{\partial t}\frac{% \partial f}{\partial s}\right\rangle=\left\langle\frac{\partial f}{\partial t}% ,\frac{D}{\partial s}\frac{\partial f}{\partial t}\right\rangle=\frac{1}{2}% \frac{\partial}{\partial s}\left\langle\frac{\partial f}{\partial t},\frac{% \partial f}{\partial t}\right\rangle.
  76. t f ( s , t ) t\mapsto f(s,t)
  77. t f t , f t t\mapsto\left\langle\frac{\partial f}{\partial t},\frac{\partial f}{\partial t% }\right\rangle
  78. s f t , f t = s v + s w N , v + s w N = 2 v , w N = 0. \frac{\partial}{\partial s}\left\langle\frac{\partial f}{\partial t},\frac{% \partial f}{\partial t}\right\rangle=\frac{\partial}{\partial s}\left\langle v% +sw_{N},v+sw_{N}\right\rangle=2\left\langle v,w_{N}\right\rangle=0.

GDP-glucosidase.html

  1. \rightleftharpoons

GDP-mannose_4,6-dehydratase.html

  1. \rightleftharpoons

Geary's_C.html

  1. C = ( N - 1 ) i j w i j ( X i - X j ) 2 2 W i ( X i - X ¯ ) 2 C=\frac{(N-1)\sum_{i}\sum_{j}w_{ij}(X_{i}-X_{j})^{2}}{2W\sum_{i}(X_{i}-\bar{X}% )^{2}}
  2. N N
  3. i i
  4. j j
  5. X X
  6. X ¯ \bar{X}
  7. X X
  8. w i j w_{ij}
  9. W W
  10. w i j w_{ij}

Generalized_Hebbian_Algorithm.html

  1. Δ w i j = η ( y j x i - y j k = 1 j w i k y k ) \,\Delta w_{ij}~{}=~{}\eta\left(y_{j}x_{i}-y_{j}\sum_{k=1}^{j}w_{ik}y_{k}\right)
  2. i i
  3. j j
  4. x x
  5. y y
  6. η η
  7. d w ( t ) d t = w ( t ) Q - diag [ w ( t ) Q w ( t ) T ] w ( t ) \,\frac{dw(t)}{dt}~{}=~{}w(t)Q-\mathrm{diag}[w(t)Qw(t)^{\mathrm{T}}]w(t)
  8. Δ w ( t ) = - lower [ w ( t ) w ( t ) T ] w ( t ) \,\Delta w(t)~{}=~{}-\mathrm{lower}[w(t)w(t)^{\mathrm{T}}]w(t)
  9. w ( t ) w(t)
  10. d i a g diag
  11. l o w e r lower
  12. Δ w ( t ) = η ( t ) ( 𝐲 ( t ) 𝐱 ( t ) T - LT [ 𝐲 ( t ) 𝐲 ( t ) T ] w ( t ) ) \,\Delta w(t)~{}=~{}\eta(t)\left(\mathbf{y}(t)\mathbf{x}(t)^{\mathrm{T}}-% \mathrm{LT}[\mathbf{y}(t)\mathbf{y}(t)^{\mathrm{T}}]w(t)\right)
  13. L T LT
  14. 𝐲 ( t ) = w ( t ) 𝐱 ( t ) \mathbf{y}(t)=w(t)\mathbf{x}(t)
  15. η η

Generalized_Ozaki_cost_function.html

  1. c ( p , y , t ) = i b i i ( y b y i e b t i t p i + j : j i b i j p i p j y b y e b t t ) . c(p,y,t)=\sum_{i}b_{ii}\left(y^{b_{yi}}e^{b_{ti}t}p_{i}+\sum_{j\,:\,j\neq i}b_% {ij}\sqrt{p_{i}p_{j}}y^{b_{y}}e^{b_{t}t}\right).

Gentamicin_2'-N-acetyltransferase.html

  1. \rightleftharpoons

Gentamicin_3'-N-acetyltransferase.html

  1. \rightleftharpoons

Geometric_flow.html

  1. u t = L u u_{t}=Lu
  2. L u = 0 Lu=0
  3. L u = 0 Lu=0
  4. F ( K ) = K 2 := ( M K 2 ) 1 / 2 F(K)=\|K\|_{2}:=\left(\int_{M}K^{2}\right)^{1/2}
  5. L u = 0 Lu=0
  6. u t = L u u_{t}=Lu

Geranoyl-CoA_carboxylase.html

  1. \rightleftharpoons

Geranylgeranylglycerol-phosphate_geranylgeranyltransferase.html

  1. \rightleftharpoons

Geranyltranstransferase.html

  1. \rightleftharpoons

Germacrene-A_synthase.html

  1. \rightleftharpoons

Gibberellin_beta-D-glucosyltransferase.html

  1. \rightleftharpoons

Glass_batch_calculation.html

  1. N B = ( B T B ) - 1 B T N G N_{B}=(B^{T}\cdot B)^{-1}\cdot B^{T}\cdot N_{G}
  2. 𝐁 = [ 1 0 0 6 6 0 0 0 1.5 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 ] \mathbf{B}=\begin{bmatrix}1&0&0&6&6&0&0\\ 0&1.5&0&1&0&0&1\\ 0&0&1&0&0&1&0\\ 0&0&0&1&1&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&2\end{bmatrix}
  3. 𝐍 𝐆 = [ 1.1151 0.1936 0.1783 0.0490 0.0106 0.0496 0.0431 ] \mathbf{N_{G}}=\begin{bmatrix}1.1151\\ 0.1936\\ 0.1783\\ 0.0490\\ 0.0106\\ 0.0496\\ 0.0431\end{bmatrix}
  4. 𝐍 𝐁 = [ 0.82087 0.08910 0.12870 0.03842 0.01062 0.04962 0.02155 ] \mathbf{N_{B}}=\begin{bmatrix}0.82087\\ 0.08910\\ 0.12870\\ 0.03842\\ 0.01062\\ 0.04962\\ 0.02155\end{bmatrix}
  5. 𝐌 𝐁 = [ 49.321 20.138 12.881 20.150 5.910 9.150 8.217 ] \mathbf{M_{B}}=\begin{bmatrix}49.321\\ 20.138\\ 12.881\\ 20.150\\ 5.910\\ 9.150\\ 8.217\end{bmatrix}
  6. 𝐌 𝐁 ( 𝟏𝟎𝟎 % 𝐧𝐨𝐫𝐦𝐚𝐥𝐢𝐳𝐞𝐝 ) = [ 39.216 16.012 10.242 16.022 4.699 7.276 6.533 ] \mathbf{M_{B}(100\%normalized)}=\begin{bmatrix}39.216\\ 16.012\\ 10.242\\ 16.022\\ 4.699\\ 7.276\\ 6.533\end{bmatrix}

Global_meteoric_water_line.html

  1. δ D = 8.0 δ 18 O + 10 / 00 0 \delta\mathrm{D}=8.0\cdot\delta^{18}\mathrm{O}+10{}^{0\!}\!/\!_{00}

Globoside_alpha-N-acetylgalactosaminyltransferase.html

  1. \rightleftharpoons

Globotriaosylceramide_3-beta-N-acetylgalactosaminyltransferase.html

  1. \rightleftharpoons

Glucarate_dehydratase.html

  1. \rightleftharpoons

Glucarate_O-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Glucarolactone_O-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Glucomannan_4-beta-mannosyltransferase.html

  1. \rightleftharpoons

Gluconate_dehydratase.html

  1. \rightleftharpoons

Gluconokinase.html

  1. \rightleftharpoons

Gluconolactonase.html

  1. \rightleftharpoons

Glucosamine-1-phosphate_N-acetyltransferase.html

  1. \rightleftharpoons

Glucosamine-6-phosphate_deaminase.html

  1. \rightleftharpoons

Glucosamine-phosphate_N-acetyltransferase.html

  1. \rightleftharpoons

Glucosamine_kinase.html

  1. \rightleftharpoons

Glucosamine_N-acetyltransferase.html

  1. \rightleftharpoons

Glucosaminylgalactosylglucosylceramide_beta-galactosyltransferase.html

  1. \rightleftharpoons

Glucose-1-phosphatase.html

  1. \rightleftharpoons

Glucose-1-phosphate_adenylyltransferase.html

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Glucose-1-phosphate_cytidylyltransferase.html

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Glucose-1-phosphate_guanylyltransferase.html

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Glucose-1-phosphate_phosphodismutase.html

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Glucose-1-phosphate_thymidylyltransferase.html

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Glucose-1-phospho-D-mannosylglycoprotein_phosphodiesterase.html

  1. \rightleftharpoons

Glucosylceramidase.html

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Glucosylglycerol-phosphate_synthase.html

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Glucosylglycerol_3-phosphatase.html

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Glucuronate-1-phosphate_uridylyltransferase.html

  1. \rightleftharpoons

Glucuronokinase.html

  1. \rightleftharpoons

Glucuronosyl-disulfoglucosamine_glucuronidase.html

  1. \rightleftharpoons

Glutamate_1-kinase.html

  1. \rightleftharpoons

Glutamate_5-kinase.html

  1. \rightleftharpoons

Glutamate_N-acetyltransferase.html

  1. \rightleftharpoons

Glutamate–cysteine_ligase.html

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Glutamate—ethylamine_ligase.html

  1. \rightleftharpoons

Glutamate—methylamine_ligase.html

  1. \rightleftharpoons

Glutamate—prephenate_aminotransferase.html

  1. \rightleftharpoons

Glutamate—putrescine_ligase.html

  1. \rightleftharpoons

Glutamate—tRNA(Gln)_ligase.html

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Glutamate—tRNA_ligase.html

  1. \rightleftharpoons

Glutamin-(asparagin-)ase.html

  1. \rightleftharpoons

Glutamine_N-acyltransferase.html

  1. \rightleftharpoons

Glutamine_N-phenylacetyltransferase.html

  1. \rightleftharpoons

Glutamine—fructose-6-phosphate_transaminase_(isomerizing).html

  1. \rightleftharpoons

Glutamine—phenylpyruvate_transaminase.html

  1. \rightleftharpoons

Glutamine—pyruvate_transaminase.html

  1. \rightleftharpoons

Glutamine—scyllo-inositol_transaminase.html

  1. \rightleftharpoons

Glutamine—tRNA_ligase.html

  1. \rightleftharpoons

Glutaminyl-peptide_cyclotransferase.html

  1. \rightleftharpoons

Glutaminyl-tRNA_synthase_(glutamine-hydrolysing).html

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Glutarate—CoA_ligase.html

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Glutaryl-7-aminocephalosporanic-acid_acylase.html

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Glutathione_gamma-glutamylcysteinyltransferase.html

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Glutathione_synthase.html

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Glutathione_thiolesterase.html

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Glutathionylspermidine_amidase.html

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Glutathionylspermidine_synthase.html

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Glycerate_kinase.html

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Glycerol-1,2-cyclic-phosphate_2-phosphodiesterase.html

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Glycerol-1-phosphatase.html

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Glycerol-2-phosphatase.html

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Glycerol-3-phosphate-transporting_ATPase.html

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Glycerol-3-phosphate_cytidylyltransferase.html

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Glycerol-3-phosphate_O-acyltransferase.html

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Glycerol-3-phosphate—glucose_phosphotransferase.html

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Glycerol_dehydratase.html

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Glycerone_kinase.html

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Glycerophosphocholine_cholinephosphodiesterase.html

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Glycerophosphocholine_phosphodiesterase.html

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Glycerophosphodiester_phosphodiesterase.html

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Glycerophosphoinositol_glycerophosphodiesterase.html

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Glycerophosphoinositol_inositolphosphodiesterase.html

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Glycerophospholipid_acyltransferase_(CoA-dependent).html

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Glycerophospholipid_arachidonoyl-transferase_(CoA-independent).html

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Glycine_C-acetyltransferase.html

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Glycine_N-acyltransferase.html

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Glycine_N-benzoyltransferase.html

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Glycine_transaminase.html

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Glycine—oxaloacetate_transaminase.html

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Glycine—tRNA_ligase.html

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Glycoprotein-fucosylgalactoside_a-N-acetylgalactosaminyltransferase.html

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Glycoprotein-N-acetylgalactosamine_3-beta-galactosyltransferase.html

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Glycoprotein_2-beta-D-xylosyltransferase.html

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Glycoprotein_3-alpha-L-fucosyltransferase.html

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Glycoprotein_6-alpha-L-fucosyltransferase.html

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Glycoprotein_N-palmitoyltransferase.html

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Glycoprotein_O-fatty-acyltransferase.html

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Glycosaminoglycan_galactosyltransferase.html

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Glycosulfatase.html

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Glycosylceramidase.html

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Glycylpeptide_N-tetradecanoyltransferase.html

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Glycyrrhizinate_beta-glucuronidase.html

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Goldbach–Euler_theorem.html

  1. p 1 p - 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + = 1. \sum_{p}^{\infty}\frac{1}{p-1}={\frac{1}{3}+\frac{1}{7}+\frac{1}{8}+\frac{1}{1% 5}+\frac{1}{24}+\frac{1}{26}+\frac{1}{31}}+\cdots=1.
  2. x = n = 1 1 n \textstyle x=\sum_{n=1}^{\infty}\frac{1}{n}
  3. x = 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 x=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+% \frac{1}{8}\cdots
  4. 1 = 1 2 + 1 4 + 1 8 + 1 16 + \textstyle 1=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots
  5. x - 1 = 1 + 1 3 + 1 5 + 1 6 + 1 7 + 1 9 + 1 10 + 1 11 + x-1=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{9}+\frac{1}{10}% +\frac{1}{11}+\cdots
  6. 1 2 = 1 3 + 1 9 + 1 27 + 1 81 + \textstyle\frac{1}{2}=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\cdots
  7. x - 1 - 1 2 = 1 + 1 5 + 1 6 + 1 7 + 1 10 + 1 11 + 1 12 + x-1-\frac{1}{2}=1+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{10}+\frac{1}{11% }+\frac{1}{12}+\cdots
  8. x - 1 - 1 2 - 1 4 - 1 5 - 1 6 - 1 9 - = 1 x-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}-\frac{1}{9}-\cdots=1
  9. x - 1 = 1 + 1 2 + 1 4 + 1 5 + 1 6 + 1 9 + x-1=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{9}+\cdots
  10. 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + 1=\frac{1}{3}+\frac{1}{7}+\frac{1}{8}+\frac{1}{15}+\frac{1}{24}+\frac{1}{26}+% \frac{1}{31}+\cdots
  11. p 1 p - 1 = m = 2 n = 2 1 m n = 1. \sum_{p}^{\infty}\frac{1}{p-1}=\sum_{m=2}^{\infty}\sum_{n=2}^{\infty}\frac{1}{% m^{n}}=1.

Goldstine_theorem.html

  1. X X
  2. B X B⊂X
  3. B B′′
  4. X X′′
  5. x B x′′∈B′′
  6. n n
  7. X X′
  8. δ > 0 δ>0
  9. x x
  10. ( 1 + δ ) B (1+δ)B
  11. 1 i n 1≤i≤n
  12. [ u ! ! ] x [ u ! ! ] 1 + δ [u^{\prime}!!^{\prime}]x[u^{\prime}!!^{\prime}]≤1+δ
  13. x x
  14. { Φ : X 𝐂 n , x ( φ 1 ( x ) , , φ n ( x ) ) \begin{cases}\Phi:X\to\mathbf{C}^{n},\\ x\mapsto\left(\varphi_{1}(x),\cdots,\varphi_{n}(x)\right)\end{cases}
  15. Y := i ker φ i = ker Φ . Y:=\bigcap_{i}\ker\varphi_{i}=\ker\Phi.
  16. ( x + Y ) ( 1 + δ ) B (x+Y)∩(1+δ)B
  17. d i s t ( x , Y ) 1 + δ dist(x,Y)≥1+δ
  18. φ X φ∈X′
  19. 1 + δ φ ( x ) = x ′′ ( φ ) φ X x ′′ X ′′ 1 , 1+\delta\leq\varphi(x)=x^{\prime\prime}(\varphi)\leq\|\varphi\|_{X^{\prime}}% \left\|x^{\prime\prime}\right\|_{X^{\prime\prime}}\leq 1,

Goodpasture-antigen-binding_protein_kinase.html

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Gorn_address.html

  1. j j
  2. i i
  3. i . j i.j

Görtler_vortices.html

  1. G = U e θ ν ( θ R ) 1 / 2 \mathrm{G}=\frac{U_{e}\theta}{\nu}\left(\frac{\theta}{R}\right)^{1/2}
  2. U e U_{e}
  3. θ \theta
  4. ν \nu
  5. R R

Gradient_copolymers.html

  1. g ( X ) g(X)
  2. ( F 1 ) (F_{1})
  3. ( X ) (X)
  4. g ( X ) = d F 1 ( X ) d X g(X)=\frac{dF_{1}(X)}{dX}
  5. F 1 ( l o c ) ( X ) = 1 N i = 1 N F 1 , i ( X ) F^{(loc)}_{1}(X)=\frac{1}{N}\sum_{i=1}^{N}F_{1,i}(X)
  6. F 1 ( l o c ) ( X ) F^{(loc)}_{1}(X)
  7. X X
  8. N N
  9. F 1 , i ( X ) F_{1,i}(X)
  10. X X
  11. X X

Gradient_method.html

  1. min x n f ( x ) \min_{x\in\mathbb{R}^{n}}\;f(x)

Grain_boundary_strengthening.html

  1. σ G b r , \sigma\propto\dfrac{Gb}{r},
  2. τ felt = τ applied + n dislocation τ dislocation \tau\text{felt}=\tau\text{applied}+n\text{dislocation}\tau\text{dislocation}\,
  3. Δ τ k d x \Delta\tau\propto{k\over{d^{x}}}
  4. σ y = σ 0 + k y d \sigma_{y}=\sigma_{0}+{k_{y}\over\sqrt{d}}

Grubbs'_test_for_outliers.html

  1. G = max i = 1 , , N | Y i - Y ¯ | s G=\frac{\displaystyle\max_{i=1,\ldots,N}\left|Y_{i}-\bar{Y}\right|}{s}
  2. Y ¯ \overline{Y}
  3. G = Y ¯ - Y min s G=\frac{\bar{Y}-Y_{\min}}{s}
  4. G = Y max - Y ¯ s G=\frac{Y_{\max}-\bar{Y}}{s}
  5. G > N - 1 N t α / ( 2 N ) , N - 2 2 N - 2 + t α / ( 2 N ) , N - 2 2 G>\frac{N-1}{\sqrt{N}}\sqrt{\frac{t_{\alpha/(2N),N-2}^{2}}{N-2+t_{\alpha/(2N),% N-2}^{2}}}

Grzegorczyk_hierarchy.html

  1. E 0 ( x , y ) = x + y E_{0}(x,y)=x+y
  2. E 1 ( x ) = x 2 + 2 E_{1}(x)=x^{2}+2
  3. E n ( x ) = E n - 1 x ( 2 ) E_{n}(x)=E^{x}_{n-1}(2)
  4. E n - 1 E_{n-1}
  5. n \mathcal{E}^{n}
  6. p i m ( t 1 , t 2 , , t m ) = t i p_{i}^{m}(t_{1},t_{2},\dots,t_{m})=t_{i}
  7. n \mathcal{E}^{n}
  8. f ( u ¯ ) = h ( g 1 ( u ¯ ) , g 2 ( u ¯ ) , , g m ( u ¯ ) ) f(\bar{u})=h(g_{1}(\bar{u}),g_{2}(\bar{u}),\dots,g_{m}(\bar{u}))
  9. n \mathcal{E}^{n}
  10. f ( t , u ¯ ) j ( t , u ¯ ) f(t,\bar{u})\leq j(t,\bar{u})
  11. u ¯ \bar{u}
  12. f ( 0 , u ¯ ) = g ( u ¯ ) f(0,\bar{u})=g(\bar{u})
  13. f ( t + 1 , u ¯ ) = h ( t , u ¯ , f ( t , u ¯ ) ) f(t+1,\bar{u})=h(t,\bar{u},f(t,\bar{u}))
  14. n \mathcal{E}^{n}
  15. n \mathcal{E}^{n}
  16. B n = { Z , S , ( p i m ) i m , E k : k < n } B_{n}=\{Z,S,(p_{i}^{m})_{i\leq m},E_{k}:k<n\}
  17. 0 1 2 \mathcal{E}^{0}\subseteq\mathcal{E}^{1}\subseteq\mathcal{E}^{2}\subseteq\cdots
  18. B n B_{n}
  19. B 0 B 1 B 2 B_{0}\subseteq B_{1}\subseteq B_{2}\subseteq\cdots
  20. 0 1 2 \mathcal{E}^{0}\subsetneq\mathcal{E}^{1}\subsetneq\mathcal{E}^{2}\subsetneq\cdots
  21. H n H_{n}
  22. n \mathcal{E}^{n}
  23. n - 1 \mathcal{E}^{n-1}
  24. 0 \mathcal{E}^{0}
  25. 1 \mathcal{E}^{1}
  26. 2 \mathcal{E}^{2}
  27. 3 \mathcal{E}^{3}
  28. 4 \mathcal{E}^{4}
  29. U U
  30. T T
  31. 0 \mathcal{E}^{0}
  32. 0 \mathcal{E}^{0}
  33. n \mathcal{E}^{n}
  34. f ( t , u ¯ ) j ( t , u ¯ ) f(t,\bar{u})\leq j(t,\bar{u})
  35. n \mathcal{E}^{n}
  36. ( E k ) k < n (E_{k})_{k<n}
  37. n \mathcal{E}^{n}
  38. n R P \mathcal{E}^{n}\subseteq RP
  39. n n R P \bigcup_{n}{\mathcal{E}^{n}}\subseteq RP
  40. n n = R P \bigcup_{n}{\mathcal{E}^{n}}=RP
  41. 0 , 1 - 0 , 2 - 1 , , n - n - 1 , \mathcal{E}^{0},\mathcal{E}^{1}-\mathcal{E}^{0},\mathcal{E}^{2}-\mathcal{E}^{1% },\dots,\mathcal{E}^{n}-\mathcal{E}^{n-1},\dots
  42. E α E_{\alpha}
  43. E α + 1 ( n ) = E α n ( 2 ) E_{\alpha+1}(n)=E_{\alpha}^{n}(2)
  44. λ m \lambda_{m}
  45. λ \lambda
  46. E λ ( n ) = E λ n ( n ) E_{\lambda}(n)=E_{\lambda_{n}}(n)

GTP_cyclohydrolase_II.html

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GTP_cyclohydrolase_IIa.html

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GTP_diphosphokinase.html

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Guanidinoacetase.html

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Guanidinoacetate_kinase.html

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Guanidinobutyrase.html

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Guanidinodeoxy-scyllo-inositol-4-phosphatase.html

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Guanidinopropionase.html

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Guanine-transporting_ATPase.html

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Guanosine-3',5'-bis(diphosphate)_3'-diphosphatase.html

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Guanosine-5'-triphosphate,3'-diphosphate_diphosphatase.html

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Guanosine-diphosphatase.html

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Guanosine-triphosphate_guanylyltransferase.html

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Guanosine_deaminase.html

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Guanosine_phosphorylase.html

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Guanylate_kinase.html

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Guided_Local_Search.html

  1. f i f_{i}
  2. c i c_{i}
  3. p i p_{i}
  4. i i
  5. I i I_{i}
  6. I i I_{i}
  7. x x
  8. i i
  9. util ( x , i ) \operatorname{util}(x,i)
  10. util ( x , i ) = I i ( x ) c i ( x ) 1 + p i . \operatorname{util}(x,i)=I_{i}(x)\frac{c_{i}(x)}{1+p_{i}}.
  11. g ( x ) = f ( x ) + λ a 1 i m I i ( x ) p i g(x)=f(x)+\lambda a\sum_{1\leq i\leq m}I_{i}(x)p_{i}
  12. a a
  13. a a
  14. a a

Gy's_sampling_theory.html

  1. V = 1 ( i = 1 N q i m i ) 2 i = 1 N q i ( 1 - q i ) m i 2 ( a i - j = 1 N q j a j m j j = 1 N q j m j ) 2 . V=\frac{1}{(\sum_{i=1}^{N}q_{i}m_{i})^{2}}\sum_{i=1}^{N}q_{i}(1-q_{i})m_{i}^{2% }\left(a_{i}-\frac{\sum_{j=1}^{N}q_{j}a_{j}m_{j}}{\sum_{j=1}^{N}q_{j}m_{j}}% \right)^{2}.
  2. V = 1 - q q M batch 2 i = 1 N m i 2 ( a i - a batch ) 2 . V=\frac{1-q}{qM\text{batch}^{2}}\sum_{i=1}^{N}m_{i}^{2}\left(a_{i}-a\text{% batch}\right)^{2}.

Gyrokinetic_ElectroMagnetic.html

  1. δ f \delta f

Gyroscopic_exercise_tool.html

  1. Ω P R groove \mathit{\Omega}_{\mathrm{P}}R_{\mathrm{groove}}
  2. ω r axle \omega r_{\mathrm{axle}}
  3. I ω 2 ( r axle / R groove ) I\omega^{2}\left(r_{\mathrm{axle}}/R_{\mathrm{groove}}\right)
  4. F f = μ k F n F_{\mathrm{f}}=\mu_{\mathrm{k}}F_{\mathrm{n}}
  5. μ k \mu_{\mathrm{k}}
  6. μ k ( r axle / R groove ) \mu_{\mathrm{k}}\left(r_{\mathrm{axle}}/R_{\mathrm{groove}}\right)

Haller_index.html

  1. H I = distance 1 distance 2 \ HI=\frac{\,\text{distance 1}}{\,\text{distance 2}}

Halogen_bond.html

  1. D H - A \mathrm{D\cdots H{-}A}
  2. D X - A \mathrm{D\cdots X{-}A}

Halstead_complexity_measures.html

  1. η 1 \,\eta_{1}
  2. η 2 \,\eta_{2}
  3. N 1 \,N_{1}
  4. N 2 \,N_{2}
  5. η = η 1 + η 2 \eta=\eta_{1}+\eta_{2}\,
  6. N = N 1 + N 2 N=N_{1}+N_{2}\,
  7. N ^ = η 1 log 2 η 1 + η 2 log 2 η 2 \hat{N}=\eta_{1}\log_{2}\eta_{1}+\eta_{2}\log_{2}\eta_{2}
  8. V = N × log 2 η V=N\times\log_{2}\eta
  9. D = η 1 2 × N 2 η 2 D={\eta_{1}\over 2}\times{N_{2}\over\eta_{2}}
  10. E = D × V E=D\times V
  11. T = E 18 T={E\over 18}
  12. B = E 2 3 3000 B={E^{2\over 3}\over 3000}
  13. B = V 3000 B={V\over 3000}
  14. η 1 = 10 \eta_{1}=10
  15. η 2 = 7 \eta_{2}=7
  16. η = 17 \eta=17
  17. N 1 = 16 N_{1}=16
  18. N 2 = 15 N_{2}=15
  19. N = 31 N=31
  20. N ^ = 10 × l o g 2 10 + 7 × l o g 2 7 = 52.9 \hat{N}=10\times log_{2}10+7\times log_{2}7=52.9
  21. V = 31 × l o g 2 17 = 126.7 V=31\times log_{2}17=126.7
  22. D = 10 2 × 15 7 = 10.7 D={10\over 2}\times{15\over 7}=10.7
  23. E = 10.7 × 126.7 = 1 , 355.7 E=10.7\times 126.7=1,355.7
  24. T = 1 , 355.7 18 = 75.3 T={1,355.7\over 18}=75.3
  25. B = 1 , 355.7 2 3 3000 = 0.04 B={1,355.7^{2\over 3}\over 3000}=0.04

Hamaker_constant.html

  1. A = π 2 × C × ρ 1 × ρ 2 A=\pi^{2}\times C\times\rho_{1}\times\rho_{2}
  2. ρ 1 \rho_{1}
  3. ρ 2 \rho_{2}
  4. w ( r ) = - C / r 6 w(r)=-C/r^{6}
  5. 1 / r 6 1/r^{6}

Hamamelose_kinase.html

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Hanner's_inequalities.html

  1. f + g p p + f - g p p ( f p + g p ) p + | f p - g p | p . \|f+g\|_{p}^{p}+\|f-g\|_{p}^{p}\geq\big(\|f\|_{p}+\|g\|_{p}\big)^{p}+\big|\|f% \|_{p}-\|g\|_{p}\big|^{p}.
  2. 2 p ( F p p + G p p ) ( F + G p + F - G p ) p + | F + G p - F - G p | p . 2^{p}\big(\|F\|_{p}^{p}+\|G\|_{p}^{p}\big)\geq\big(\|F+G\|_{p}+\|F-G\|_{p}\big% )^{p}+\big|\|F+G\|_{p}-\|F-G\|_{p}\big|^{p}.

Hardy–Ramanujan_theorem.html

  1. | ω ( n ) - log ( log ( n ) ) | < ψ ( n ) log ( log ( n ) ) |\omega(n)-\log(\log(n))|<\psi(n)\sqrt{\log(\log(n))}
  2. | ω ( n ) - log ( log ( n ) ) | < ( log ( log ( n ) ) ) 1 2 + ε |\omega(n)-\log(\log(n))|<{(\log(\log(n)))}^{\frac{1}{2}+\varepsilon}
  3. n x | ω ( n ) - log log n | 2 x log log x . \sum_{n\leq x}|\omega(n)-\log\log n|^{2}\ll x\log\log x\ .

Harris_affine_region_detector.html

  1. A A
  2. A ( 𝐱 ) = p , q w ( p , q ) [ I x 2 ( 𝐱 ) I x I y ( 𝐱 ) I x I y ( 𝐱 ) I y 2 ( 𝐱 ) ] A(\mathbf{x})=\sum_{p,q}w(p,q)\begin{bmatrix}I_{x}^{2}(\mathbf{x})&I_{x}I_{y}(% \mathbf{x})\\ I_{x}I_{y}(\mathbf{x})&I_{y}^{2}(\mathbf{x})\\ \end{bmatrix}
  3. I x I_{x}
  4. I y I_{y}
  5. x x
  6. y y
  7. 𝐱 \mathbf{x}
  8. p p
  9. q q
  10. I x I_{x}
  11. I y I_{y}
  12. w ( x , y ) w(x,y)
  13. w ( x , y ) = g ( x , y , σ ) = 1 2 π σ 2 e ( - x 2 + y 2 2 σ 2 ) w(x,y)=g(x,y,\sigma)=\frac{1}{2\pi\sigma^{2}}e^{\left(-\frac{x^{2}+y^{2}}{2% \sigma^{2}}\right)}
  14. A A
  15. λ 1 \lambda_{1}
  16. λ 2 \lambda_{2}
  17. A A
  18. A A
  19. R = det ( A ) - α trace 2 ( A ) = λ 1 λ 2 - α ( λ 1 + λ 2 ) 2 R=\det(A)-\alpha\operatorname{trace}^{2}(A)=\lambda_{1}\lambda_{2}-\alpha(% \lambda_{1}+\lambda_{2})^{2}
  20. α \alpha
  21. { x c } = { x c | R ( x c ) > R ( x i ) , x i W ( x c ) } , \displaystyle\{x_{c}\}=\big\{x_{c}|R(x_{c})>R(x_{i}),\forall x_{i}\in W(x_{c})% \big\},
  22. { x c } \{x_{c}\}
  23. R ( x ) R(x)
  24. x x
  25. W ( x c ) W(x_{c})
  26. x c x_{c}
  27. t t h r e s h o l d t_{threshold}
  28. L ( 𝐱 , s ) = G ( s ) I ( 𝐱 ) L(\mathbf{x},s)=G(s)\otimes I(\mathbf{x})
  29. G ( s ) G(s)
  30. s s
  31. m m
  32. D i 1 , i m D_{i_{1},...i_{m}}
  33. s m s^{m}
  34. D i 1 , , i m ( 𝐱 , s ) = s m L i 1 , , i m ( 𝐱 , s ) D_{i_{1},\dots,i_{m}}(\mathbf{x},s)=s^{m}L_{i_{1},\dots,i_{m}}(\mathbf{x},s)
  35. n t h nth
  36. s n = k n s 0 s_{n}=k^{n}s_{0}
  37. M = μ ( 𝐱 , σ I , σ D ) M=\mu(\mathbf{x},\sigma_{\mathit{I}},\sigma_{\mathit{D}})
  38. M = μ ( 𝐱 , σ I , σ D ) = σ D 2 g ( σ I ) [ L x 2 ( 𝐱 , σ D ) L x L y ( 𝐱 , σ D ) L x L y ( 𝐱 , σ D ) L y 2 ( 𝐱 , σ D ) ] M=\mu(\mathbf{x},\sigma_{\mathit{I}},\sigma_{\mathit{D}})=\sigma_{D}^{2}g(% \sigma_{I})\otimes\begin{bmatrix}L_{x}^{2}(\mathbf{x},\sigma_{D})&L_{x}L_{y}(% \mathbf{x},\sigma_{D})\\ L_{x}L_{y}(\mathbf{x},\sigma_{D})&L_{y}^{2}(\mathbf{x},\sigma_{D})\end{bmatrix}
  39. g ( σ I ) g(\sigma_{I})
  40. σ I \sigma_{I}
  41. 𝐱 = ( x , y ) \mathbf{x}=(x,y)
  42. L ( 𝐱 ) L(\mathbf{x})
  43. \mathbf{\otimes}
  44. L x ( 𝐱 , σ D ) L_{x}(\mathbf{x},\sigma_{D})
  45. L y ( 𝐱 , σ D ) L_{y}(\mathbf{x},\sigma_{D})
  46. σ D \sigma_{D}
  47. σ I \sigma_{I}
  48. σ 1 σ n = k 1 σ 0 k n σ 0 {\sigma_{1}\dots\sigma_{n}}={k^{1}\sigma_{0}\dots k^{n}\sigma_{0}}
  49. k = 1.4 k=1.4
  50. σ I \sigma_{I}
  51. σ D = s σ I \sigma_{D}=s\sigma_{I}
  52. s = 0.7 s=0.7
  53. μ ( 𝐱 , σ I , σ D ) \mu(\mathbf{x},\sigma_{\mathit{I}},\sigma_{\mathit{D}})
  54. 𝑐𝑜𝑟𝑛𝑒𝑟𝑛𝑒𝑠𝑠 = det ( μ ( 𝐱 , σ I , σ D ) ) - α trace 2 ( μ ( 𝐱 , σ I , σ D ) ) \mathit{cornerness}=\det(\mu(\mathbf{x},\sigma_{\mathit{I}},\sigma_{\mathit{D}% }))-\alpha\operatorname{trace}^{2}(\mu(\mathbf{x},\sigma_{\mathit{I}},\sigma_{% \mathit{D}}))
  55. 𝐱 \mathbf{x}
  56. σ I \sigma_{I}
  57. k k
  58. k t h kth
  59. σ I ( k + 1 ) \sigma_{I}^{(k+1)}
  60. 1.4 1.4
  61. t [ 0.7 , , 1.4 ] t\in[0.7,\dots,1.4]
  62. σ I ( k + 1 ) = t σ I k \sigma_{I}^{(k+1)}=t\sigma_{I}^{k}
  63. det ( L o G ( 𝐱 , σ I ) ) = σ I 2 det ( L x x ( 𝐱 , σ I ) + L y y ( 𝐱 , σ I ) ) \det(LoG(\mathbf{x},\sigma_{I}))=\sigma_{I}^{2}\det(L_{xx}(\mathbf{x},\sigma_{% I})+L_{yy}(\mathbf{x},\sigma_{I}))
  64. L x x L_{xx}
  65. L y y L_{yy}
  66. σ I 2 \sigma_{I}^{2}
  67. σ I ( k + 1 ) \sigma_{I}^{(k+1)}
  68. 𝐱 ( k + 1 ) \mathbf{x}^{(k+1)}
  69. σ I ( k + 1 ) = = σ I ( k ) \sigma_{I}^{(k+1)}==\sigma_{I}^{(k)}
  70. 𝐱 ( k + 1 ) = = 𝐱 ( k ) \mathbf{x}^{(k+1)}==\mathbf{x}^{(k)}
  71. k + 1 k+1
  72. μ \mathbf{\mu}
  73. μ ( 𝐱 , Σ I , Σ D ) = det ( Σ D ) g ( Σ I ) * ( L ( 𝐱 , Σ D ) L ( 𝐱 , Σ D ) T ) \mu(\mathbf{x},\Sigma_{I},\Sigma_{D})=\det(\Sigma_{D})g(\Sigma_{I})*(\nabla L(% \mathbf{x},\Sigma_{D})\nabla L(\mathbf{x},\Sigma_{D})^{T})
  74. Σ I \Sigma_{I}
  75. Σ D \Sigma_{D}
  76. μ \mu
  77. Σ I \Sigma_{I}
  78. Σ D \Sigma_{D}
  79. σ I \sigma_{I}
  80. σ D \sigma_{D}
  81. 𝐱 L \mathbf{x}_{L}
  82. 𝐱 R = A 𝐱 L \mathbf{x}_{R}=A\mathbf{x}_{L}
  83. 𝐱 R \mathbf{x}_{R}
  84. 𝐱 L \mathbf{x}_{L}
  85. R 2 R^{2}
  86. μ ( 𝐱 L , Σ I , L , Σ D , L ) \displaystyle\mu(\mathbf{x}_{L},\Sigma_{I,L},\Sigma_{D,L})
  87. Σ I , b \Sigma_{I,b}
  88. Σ D , b \Sigma_{D,b}
  89. b b
  90. Σ I , L = σ I M L - 1 \displaystyle\Sigma_{I,L}=\sigma_{I}M_{L}^{-1}
  91. σ I \sigma_{I}
  92. σ D \sigma_{D}
  93. Σ I , R = σ I M R - 1 \displaystyle\Sigma_{I,R}=\sigma_{I}M_{R}^{-1}
  94. M 1 2 M^{\tfrac{1}{2}}
  95. R R
  96. x R x_{R}^{^{\prime}}
  97. x L x_{L}^{^{\prime}}
  98. A = M R - 1 2 R M L 1 2 \displaystyle A=M_{R}^{-\tfrac{1}{2}}RM_{L}^{\tfrac{1}{2}}
  99. M = μ ( 𝐱 , Σ I , Σ D ) M=\mu(\mathbf{x},\Sigma_{I},\Sigma_{D})
  100. 𝒬 = λ min ( M ) λ max ( M ) \mathcal{Q}=\frac{\lambda_{\min}(M)}{\lambda_{\max}(M)}
  101. λ \lambda
  102. [ 0 1 ] [0\dots 1]
  103. 1 1
  104. U U
  105. k k
  106. U ( k ) U^{(k)}
  107. 𝐱 ( k ) \mathbf{x}^{(k)}
  108. σ I ( k ) \sigma_{I}^{(k)}
  109. σ D ( k ) \sigma_{D}^{(k)}
  110. 𝐱 w ( k ) \mathbf{x}_{w}^{(k)}
  111. U ( k ) 𝐱 w ( k ) = 𝐱 ( 𝐤 ) U^{(k)}\mathbf{x}_{w}^{(k)}=\mathbf{x^{(k)}}
  112. U - n o r m a l i z e d U-normalized
  113. σ D ( k ) \sigma_{D}^{(k)}
  114. σ D k = s σ I k \sigma_{D}^{k}=s\sigma_{I}^{k}
  115. s [ 0.5 , 0.75 ] s\in[0.5,0.75]
  116. 𝒬 = λ m i n ( μ ) λ m a x ( μ ) \mathcal{Q}=\frac{\lambda_{min}(\mu)}{\lambda_{max}(\mu)}
  117. σ D ( k ) = argmax σ D = s σ I ( k ) , s [ 0.5 , , 0.75 ] λ min ( μ ( 𝐱 w ( k ) , σ I k , σ D ) ) λ max ( μ ( 𝐱 w ( k ) , σ I k , σ D ) ) \sigma_{D}^{(k)}=\underset{\sigma_{D}=s\sigma_{I}^{(k)},\;s\in[0.5,\dots,0.75]% }{\operatorname{argmax}}\,\frac{\lambda_{\min}(\mu(\mathbf{x}_{w}^{(k)},\sigma% _{I}^{k},\sigma_{D}))}{\lambda_{\max}(\mu(\mathbf{x}_{w}^{(k)},\sigma_{I}^{k},% \sigma_{D}))}
  118. μ ( 𝐱 w ( k ) , σ I k , σ D ) \mu(\mathbf{x}_{w}^{(k)},\sigma_{I}^{k},\sigma_{D})
  119. 𝐱 w ( k ) \mathbf{x}_{w}^{(k)}
  120. 𝑐𝑜𝑟𝑛𝑒𝑟𝑛𝑒𝑠𝑠 \mathit{cornerness}
  121. 𝐱 w ( k - 1 ) \mathbf{x}_{w}^{(k-1)}
  122. 𝐱 w ( k ) = argmax 𝐱 w W ( 𝐱 w ( k - 1 ) ) det ( μ ( 𝐱 w , σ I k , σ D ( k ) ) ) - α trace 2 ( μ ( 𝐱 w , σ I k , σ D ( k ) ) ) \mathbf{x}_{w}^{(k)}=\underset{\mathbf{x}_{w}\in W(\mathbf{x}_{w}^{(k-1)})}{% \operatorname{argmax}}\,\det(\mu(\mathbf{x}_{w},\sigma_{I}^{k},\sigma_{D}^{(k)% }))-\alpha\operatorname{trace}^{2}(\mu(\mathbf{x}_{w},\sigma_{I}^{k},\sigma_{D% }^{(k)}))
  123. μ \mu
  124. W ( 𝐱 w ( k - 1 ) ) W(\mathbf{x}_{w}^{(k-1)})
  125. U U
  126. 𝐱 ( k ) = 𝐱 ( k - 1 ) + U ( k - 1 ) ( 𝐱 w ( k ) - 𝐱 w ( k - 1 ) ) \mathbf{x}^{(k)}=\mathbf{x}^{(k-1)}+U^{(k-1)}\cdot(\mathbf{x}_{w}^{(k)}-% \mathbf{x}_{w}^{(k-1)})
  127. μ i ( k ) = μ - 1 2 ( 𝐱 w ( k ) , σ I ( k ) , σ D ( k ) ) \mu_{i}^{(k)}=\mu^{-\tfrac{1}{2}}(\mathbf{x}_{w}^{(k)},\sigma_{I}^{(k)},\sigma% _{D}^{(k)})
  128. U U
  129. U ( k ) = μ i ( k ) U ( k - 1 ) U^{(k)}=\mu_{i}^{(k)}\cdot U^{(k-1)}
  130. λ m a x ( U ( k ) ) = 1 \lambda_{max}(U^{(k)})=1
  131. U U
  132. U = k μ i ( k ) U ( 0 ) = k ( μ - 1 2 ) ( k ) U ( 0 ) U=\prod_{k}\mu_{i}^{(k)}\cdot U^{(0)}=\prod_{k}(\mu^{-\tfrac{1}{2}})^{(k)}% \cdot U^{(0)}
  133. U - n o r m a l i z a t i o n U-normalization
  134. 𝒬 = λ min ( μ ) λ max ( μ ) \mathcal{Q}=\frac{\lambda_{\min}(\mu)}{\lambda_{\max}(\mu)}
  135. 1 - λ min ( μ i ( k ) ) λ max ( μ i ( k ) ) < ε C 1-\frac{\lambda_{\min}(\mu_{i}^{(k)})}{\lambda_{\max}(\mu_{i}^{(k)})}<% \varepsilon_{C}
  136. ϵ C = 0.05 \epsilon_{C}=0.05
  137. 𝒪 ( n ) \mathcal{O}(n)
  138. n n
  139. U U
  140. 𝒪 ( ( m + k ) p ) \mathcal{O}((m+k)p)
  141. p p
  142. m m
  143. k k
  144. U U
  145. s s
  146. σ D = s σ I , s = c o n s t a n t \sigma_{D}=s\sigma_{I},\;s=constant
  147. U U
  148. 𝐱 \mathbf{x}
  149. σ I \sigma_{I}
  150. λ min ( U ) λ max ( U ) \tfrac{\lambda_{\min}(U)}{\lambda_{\max}(U)}
  151. λ max ( U ) λ min ( U ) > t diverge \tfrac{\lambda_{\max}(U)}{\lambda_{\min}(U)}>t\text{diverge}
  152. t d i v e r g e = 6 t_{diverge}=6
  153. R score = C ( A , B ) min ( n A , n B ) R\text{score}=\frac{C(A,B)}{\min(n_{A},n_{B})}
  154. C ( A , B ) C(A,B)
  155. A A
  156. B B
  157. n B n_{B}
  158. n A n_{A}
  159. n A n_{A}
  160. n B n_{B}
  161. x A = H x B x_{A}=H\cdot x_{B}
  162. H H
  163. 𝐱 𝐚 \mathbf{x_{a}}
  164. 𝐱 𝐛 \mathbf{x_{b}}

Hasse–Davenport_relation.html

  1. Γ ( z ) Γ ( z + 1 k ) Γ ( z + 2 k ) Γ ( z + k - 1 k ) = ( 2 π ) k - 1 2 k 1 / 2 - k z Γ ( k z ) . \Gamma(z)\;\Gamma\left(z+\frac{1}{k}\right)\;\Gamma\left(z+\frac{2}{k}\right)% \cdots\Gamma\left(z+\frac{k-1}{k}\right)=(2\pi)^{\frac{k-1}{2}}\;k^{1/2-kz}\;% \Gamma(kz).\,\!
  2. α \alpha
  3. F s F_{s}
  4. χ \chi
  5. N F s / F ( α ) N_{F_{s}/F}(\alpha)
  6. F s F_{s}
  7. F F
  8. N F s / F ( α ) := α α q α q s - 1 . N_{F_{s}/F}(\alpha):=\alpha\cdot\alpha^{q}\cdots\alpha^{q^{s-1}}.\,
  9. χ \chi^{\prime}
  10. F s F_{s}
  11. χ \chi
  12. χ ( α ) := χ ( N F s / F ( α ) ) \chi^{\prime}(\alpha):=\chi(N_{F_{s}/F}(\alpha))
  13. ψ \psi^{\prime}
  14. F s F_{s}
  15. ψ \psi
  16. ψ ( α ) := ψ ( T r F s / F ( α ) ) \psi^{\prime}(\alpha):=\psi(Tr_{F_{s}/F}(\alpha))
  17. τ ( χ , ψ ) = x F χ ( x ) ψ ( x ) \tau(\chi,\psi)=\sum_{x\in F}\chi(x)\psi(x)
  18. τ ( χ , ψ ) \tau(\chi^{\prime},\psi^{\prime})
  19. F s F_{s}
  20. ( - 1 ) s τ ( χ , ψ ) s = - τ ( χ , ψ ) . (-1)^{s}\cdot\tau(\chi,\psi)^{s}=-\tau(\chi^{\prime},\psi^{\prime}).
  21. a mod m τ ( χ ρ a , ψ ) = - χ - m ( m ) τ ( χ m , ψ ) a mod m τ ( ρ a , ψ ) \prod_{a\bmod m}\tau(\chi\rho^{a},\psi)=-\chi^{-m}(m)\tau(\chi^{m},\psi)\prod_% {a\bmod m}\tau(\rho^{a},\psi)

Hat_matrix.html

  1. 𝐲 ^ = H 𝐲 . \hat{\mathbf{y}}=H\mathbf{y}.
  2. 𝐫 = 𝐲 - 𝐲 ^ = 𝐲 - H 𝐲 = ( I - H ) 𝐲 . \mathbf{r}=\mathbf{y}-\mathbf{\hat{y}}=\mathbf{y}-H\mathbf{y}=(I-H)\mathbf{y}.
  3. h i j = cov [ y ^ i , y j ] / var [ y j ] \displaystyle h_{ij}=\operatorname{cov}[\hat{y}_{i},y_{j}]/\operatorname{var}[% y_{j}]
  4. ( I - H ) Σ ( I - H ) \left(I-H\right)^{\top}\Sigma\left(I-H\right)
  5. 𝐲 = X s y m b o l β + s y m b o l ε , \mathbf{y}=Xsymbol\beta+symbol\varepsilon,
  6. s y m b o l β ^ = ( X X ) - 1 X 𝐲 , \hat{symbol\beta}=\left(X^{\top}X\right)^{-1}X^{\top}\mathbf{y},
  7. 𝐲 ^ = X s y m b o l β ^ = X ( X X ) - 1 X 𝐲 . \hat{\mathbf{y}}=X\hat{symbol\beta}=X\left(X^{\top}X\right)^{-1}X^{\top}% \mathbf{y}.
  8. H = X ( X X ) - 1 X . H=X\left(X^{\top}X\right)^{-1}X^{\top}.
  9. ( X X ) - 1 X \left(X^{\top}X\right)^{-1}X^{\top}
  10. 𝐫 = ( I - H ) 𝐲 , \mathbf{r}=(I-H)\mathbf{y},
  11. 𝐫 = 𝐲 - H 𝐲 X . \mathbf{r}=\mathbf{y}-H\mathbf{y}\perp X.
  12. H 2 = H H^{2}=H
  13. H X = X , HX=X,
  14. ( I - H ) X = 0 (I-H)X=0
  15. ( I - H ) H = H ( I - H ) = 0. (I-H)H=H(I-H)=0.
  16. H 2 = H H^{2}=H
  17. s y m b o l β ^ = ( X Σ - 1 X ) - 1 X Σ - 1 𝐲 , \hat{symbol{\beta}}=\left(X^{\top}\Sigma^{-1}X\right)^{-1}X^{\top}\Sigma^{-1}% \,\mathbf{y},
  18. H = X ( X Σ - 1 X ) - 1 X Σ - 1 , H=X\left(X^{\top}\Sigma^{-1}X\right)^{-1}X^{\top}\Sigma^{-1},\,
  19. C C
  20. C = [ A , B ] C=[A,B]
  21. H ( X ) = X ( X X ) - 1 X H(X)=X\left(X^{\top}X\right)^{-1}X^{\top}
  22. M ( X ) = I - H ( X ) M(X)=I-H(X)
  23. C C
  24. H ( C ) = H ( A ) + H ( M ( A ) B ) H(C)=H(A)+H(M(A)B)
  25. A A
  26. A A
  27. C C
  28. C C

Hatta_number.html

  1. H a 2 = k 2 C A , i C B , b u l k δ L D A δ L C A , i = k 2 C B , b u l k D A ( D A δ L ) 2 = k 2 C B , b u l k D A k L 2 Ha^{2}={{k_{2}C_{A,i}C_{B,bulk}\delta_{L}}\over{\frac{D_{A}}{\delta_{L}}\ C_{A% ,i}}}={{k_{2}C_{B,bulk}D_{A}}\over({\frac{D_{A}}{\delta_{L}}})^{2}}={{k_{2}C_{% B,bulk}D_{A}}\over{{k_{L}}^{2}}}
  2. H a = 2 m + 1 k m , n C A , i m - 1 C B , b u l k n D A k L Ha={{\sqrt{{\frac{2}{{m}+1}}k_{m,n}{C_{A,i}}^{m-1}C_{B,bulk}^{n}{D}_{A}}}\over% {{k}_{L}}}

HD_85512.html

  1. R = L 4 π σ T eff 4 R=\sqrt{\frac{L}{4\pi\sigma T_{\rm eff}^{4}}}
  2. R R
  3. L L
  4. T eff T_{\rm eff}
  5. σ \sigma

Heat_flux.html

  1. ϕ q \overrightarrow{\phi_{q}}
  2. ϕ q \overrightarrow{\phi_{q}}
  3. . E in t - E out t - E accumulated t = 0 \big.\frac{\partial E_{\mathrm{in}}}{\partial t}-\frac{\partial E_{\mathrm{out% }}}{\partial t}-\frac{\partial E_{\mathrm{accumulated}}}{\partial t}=0
  4. . E t \big.\frac{\partial E}{\partial t}
  5. . E in t - E out t = S ϕ q d S \big.\frac{\partial E_{\mathrm{in}}}{\partial t}-\frac{\partial E_{\mathrm{out% }}}{\partial t}=\oint_{S}\overrightarrow{\phi_{q}}\cdot\,\overrightarrow{dS}
  6. ϕ q \overrightarrow{\phi_{q}}
  7. S S

Heat_flux_sensor.html

  1. ϕ q = V sen E sen \phi_{q}=\frac{V_{\mathrm{sen}}}{E_{\mathrm{sen}}}
  2. V sen V_{\mathrm{sen}}
  3. E sen E_{\mathrm{sen}}
  4. R R
  5. C C
  6. R sen R_{\mathrm{sen}}
  7. C sen C_{\mathrm{sen}}
  8. τ sen \tau_{\mathrm{sen}}
  9. τ sen = R sen C sen = d 2 ρ C p λ \tau_{\mathrm{sen}}=R_{\mathrm{sen}}C_{\mathrm{sen}}=\frac{d^{2}\rho C_{p}}{\lambda}
  10. d d
  11. ρ \rho
  12. C p C_{p}
  13. λ \lambda
  14. t > 0 t>0
  15. V sen = E sen ( 1 - e - t τ sen ) V_{\mathrm{sen}}=E_{\mathrm{sen}}\left(1-e^{-\frac{t}{\tau_{\mathrm{sen}}}}\right)
  16. τ sen \tau_{\mathrm{sen}}
  17. R sen C sen R_{\mathrm{sen}}C_{\mathrm{sen}}
  18. d T d t \frac{\mathrm{d}T}{\mathrm{d}t}
  19. t = 0 t=0
  20. τ sen \tau_{\mathrm{sen}}

Helmert_transformation.html

  1. X T = C + μ R X X_{T}=C+\mu RX\,
  2. C C
  3. μ \mu
  4. R R
  5. r x r_{x}
  6. r y r_{y}
  7. r z r_{z}
  8. μ = 1 + s \mu=1+s
  9. c x c_{x}
  10. c y c_{y}
  11. c z c_{z}
  12. [ X Y Z ] B = [ c x c y c z ] + ( 1 + s × 10 - 6 ) [ 1 - r z r y r z 1 - r x - r y r x 1 ] [ X Y Z ] A \begin{bmatrix}X\\ Y\\ Z\end{bmatrix}^{B}=\begin{bmatrix}c_{x}\\ c_{y}\\ c_{z}\end{bmatrix}+(1+s\times 10^{-6})\cdot\begin{bmatrix}1&-r_{z}&r_{y}\\ r_{z}&1&-r_{x}\\ -r_{y}&r_{x}&1\end{bmatrix}\cdot\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}^{A}
  13. X B = c x + ( 1 + s × 10 - 6 ) ( X A - r z Y A + r y Z A ) Y B = c y + ( 1 + s × 10 - 6 ) ( r z X A + Y A - r x Z A ) Z B = c z + ( 1 + s × 10 - 6 ) ( - r y X A + r x Y A + Z A ) . \begin{matrix}X_{B}=c_{x}+(1+s\times 10^{-6})\cdot(X_{A}-r_{z}\cdot Y_{A}+r_{y% }\cdot Z_{A})\\ Y_{B}=c_{y}+(1+s\times 10^{-6})\cdot(r_{z}\cdot X_{A}+Y_{A}-r_{x}\cdot Z_{A})% \\ Z_{B}=c_{z}+(1+s\times 10^{-6})\cdot(-r_{y}\cdot X_{A}+r_{x}\cdot Y_{A}+Z_{A})% .\\ \end{matrix}
  14. c x c_{x}
  15. c y c_{y}
  16. c z c_{z}
  17. ω \omega
  18. ϕ \phi
  19. κ \kappa

Heme-transporting_ATPase.html

  1. \rightleftharpoons

Hessian_affine_region_detector.html

  1. H ( 𝐱 ) = [ L x x ( 𝐱 ) L x y ( 𝐱 ) L x y ( 𝐱 ) L y y ( 𝐱 ) ] H(\mathbf{x})=\begin{bmatrix}L_{xx}(\mathbf{x})&L_{xy}(\mathbf{x})\\ L_{xy}(\mathbf{x})&L_{yy}(\mathbf{x})\\ \end{bmatrix}
  2. L a a ( 𝐱 ) L_{aa}(\mathbf{x})
  3. a a
  4. L a b ( 𝐱 ) L_{ab}(\mathbf{x})
  5. a a
  6. b b
  7. L ( 𝐱 ) = g ( σ I ) I ( 𝐱 ) L(\mathbf{x})=g(\sigma_{I})\otimes I(\mathbf{x})
  8. σ I 2 \sigma_{I}^{2}
  9. D E T = σ I 2 ( L x x L y y ( 𝐱 ) - L x y 2 ( 𝐱 ) ) T R = σ I ( L x x + L y y ) \begin{aligned}\displaystyle DET=\sigma_{I}^{2}(L_{xx}L_{yy}(\mathbf{x})-L_{xy% }^{2}(\mathbf{x}))\\ \displaystyle TR=\sigma_{I}(L_{xx}+L_{yy})\end{aligned}

Heteroglycan_alpha-mannosyltransferase.html

  1. \rightleftharpoons

HGSNAT.html

  1. \rightleftharpoons

Hicks-neutral_technical_change.html

  1. Y = A * F ( K , L ) Y=A*F(K,L)\,
  2. A A

High_Energy_Astronomy_Observatory_3.html

  1. 3 {}^{3}
  2. 2 {}^{2}

Hilbert–Huang_transform.html

  1. X ( t ) - m 1 = h 1 . X(t)-m_{1}=h_{1}.\,
  2. h 1 - m 11 = h 11 . h_{1}-m_{11}=h_{11}.\,
  3. h 1 ( k - 1 ) - m 1 k = h 1 k . h_{1(k-1)}-m_{1k}=h_{1k}.\,
  4. c 1 = h 1 k . c_{1}=h_{1k}.\,
  5. S D k = t = 0 T | h k - 1 ( t ) - h k ( t ) | 2 h k - 1 2 ( t ) . SD_{k}=\sum_{t=0}^{T}\frac{|h_{k-1}(t)-h_{k}(t)|^{2}}{h_{k-1}^{2}(t)}.\,
  6. X ( t ) - c 1 = r 1 . X(t)-c_{1}=r_{1}.\,
  7. r n - 1 - c n = r n . r_{n-1}-c_{n}=r_{n}.\,
  8. X ( t ) = j = 1 n c j + r n . X(t)=\sum_{j=1}^{n}c_{j}+r_{n}.\,
  9. X ( t ) = Real j = 1 n a j ( t ) e i ω j ( t ) d t . X(t)=\,\text{Real}{\sum_{j=1}^{n}a_{j}(t)e^{i\int\omega_{j}(t)dt}}.\,

Hildebrand_solubility_parameter.html

  1. δ = Δ H v - R T V m \delta=\sqrt{\frac{\Delta H_{v}-RT}{V_{m}}}

Hippurate_hydrolase.html

  1. \rightleftharpoons

Histidine_kinase.html

  1. \rightleftharpoons

Histidine_N-acetyltransferase.html

  1. \rightleftharpoons

Histidine_transaminase.html

  1. \rightleftharpoons

Histidine—tRNA_ligase.html

  1. \rightleftharpoons

Histidinol-phosphatase.html

  1. \rightleftharpoons

Histidinol-phosphate_transaminase.html

  1. \rightleftharpoons

Histogram_of_oriented_gradients.html

  1. [ - 1 , 0 , 1 ] and [ - 1 , 0 , 1 ] T . [-1,0,1]\,\text{ and }[-1,0,1]^{T}.\,
  2. v v
  3. v k \|v\|_{k}
  4. k = 1 , 2 k={1,2}
  5. e e
  6. f = v v 2 2 + e 2 f={v\over\sqrt{\|v\|^{2}_{2}+e^{2}}}
  7. f = v ( v 1 + e ) f={v\over(\|v\|_{1}+e)}
  8. f = v ( v 1 + e ) f=\sqrt{v\over(\|v\|_{1}+e)}
  9. 10 - 4 10^{-4}

History_of_quaternions.html

  1. i 2 = j 2 = k 2 = i j k = - 1. i^{2}=j^{2}=k^{2}=ijk=-1.\,
  2. a \mathrm{a}
  3. 𝒜 = cos α + 𝐀 sin α , \mathcal{A}=\cos\alpha+\mathbf{A}\sin\alpha,
  4. 𝐀 \mathbf{A}

History_of_statistics.html

  1. f ( x ) = 1 2 ( 1 - x 2 ) f(x)=\frac{1}{2}\sqrt{(1-x^{2})}
  2. r r

Holo-(acyl-carrier-protein)_synthase.html

  1. \rightleftharpoons

Holo-ACP_synthase.html

  1. \rightleftharpoons

Holographic_algorithm.html

  1. v v
  2. f v . f_{v}.
  3. σ : E { 0 , 1 } v V f v ( σ | E ( v ) ) , ( 1 ) \sum_{\sigma:E\to\{0,1\}}\prod_{v\in V}f_{v}(\sigma|_{E(v)}),~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}(1)
  4. f v f_{v}
  5. v v
  6. u U u\in U
  7. f u f_{u}
  8. v V v\in V
  9. f v f_{v}
  10. Holant ( G , f u , f v ) . \,\text{Holant}(G,f_{u},f_{v}).
  11. f u f_{u}
  12. f u | U | . f_{u}^{\otimes|U|}.
  13. f v | V | . f_{v}^{\otimes|V|}.
  14. f u f_{u}
  15. f v f_{v}
  16. f u | U | f v | V | . f_{u}^{\otimes|U|}f_{v}^{\otimes|V|}.
  17. Holant ( G , f u , f v ) \,\text{Holant}(G,f_{u},f_{v})
  18. Holant ( G , f u T ( deg u ) , ( T - 1 ) ( deg v ) f v ) . \,\text{Holant}(G,f_{u}T^{\otimes(\deg u)},(T^{-1})^{\otimes(\deg v)}f_{v}).
  19. T | E | ( T - 1 ) | E | T^{\otimes|E|}(T^{-1})^{\otimes|E|}
  20. f u | U | f v | V | f_{u}^{\otimes|U|}f_{v}^{\otimes|V|}
  21. f u | U | f v | V | f_{u}^{\otimes|U|}f_{v}^{\otimes|V|}
  22. = f u | U | T | E | ( T - 1 ) | E | f v | V | =f_{u}^{\otimes|U|}T^{\otimes|E|}(T^{-1})^{\otimes|E|}f_{v}^{\otimes|V|}
  23. = ( f u T ( deg u ) ) | U | ( f v ( T - 1 ) ( deg v ) ) | V | . =\left(f_{u}T^{\otimes(\deg u)}\right)^{\otimes|U|}\left(f_{v}(T^{-1})^{% \otimes(\deg v)}\right)^{\otimes|V|}.
  24. Holant ( G , f u , f v ) \,\text{Holant}(G,f_{u},f_{v})
  25. Holant ( G , f u T ( deg u ) , ( T - 1 ) ( deg v ) f v ) \,\text{Holant}(G,f_{u}T^{\otimes(\deg u)},(T^{-1})^{\otimes(\deg v)}f_{v})
  26. Holant ( H , OR 2 , EQUAL 3 ) . \,\text{Holant}(H,\,\text{OR}_{2},\,\text{EQUAL}_{3}).
  27. ( 1 , 0 , 0 , 0 , 0 , 0 , 0 , 1 ) T = [ 1 0 ] 3 + [ 0 1 ] 3 (1,0,0,0,0,0,0,1)^{T}=\begin{bmatrix}1\\ 0\end{bmatrix}^{\otimes 3}+\begin{bmatrix}0\\ 1\end{bmatrix}^{\otimes 3}
  28. [ 0 1 1 0 ] , \begin{bmatrix}0&1\\ 1&0\end{bmatrix},
  29. OR 2 | U | EQUAL 3 | V | \,\text{OR}_{2}^{\otimes|U|}\,\text{EQUAL}_{3}^{\otimes|V|}
  30. = ( 0 , 1 , 1 , 1 ) | U | ( [ 1 0 ] 3 + [ 0 1 ] 3 ) | V | =(0,1,1,1)^{\otimes|U|}\left(\begin{bmatrix}1\\ 0\end{bmatrix}^{\otimes 3}+\begin{bmatrix}0\\ 1\end{bmatrix}^{\otimes 3}\right)^{\otimes|V|}
  31. = ( 0 , 1 , 1 , 1 ) | U | [ 0 1 1 0 ] | E | [ 0 1 1 0 ] | E | ( [ 1 0 ] 3 + [ 0 1 ] 3 ) | V | =(0,1,1,1)^{\otimes|U|}\begin{bmatrix}0&1\\ 1&0\end{bmatrix}^{\otimes|E|}\begin{bmatrix}0&1\\ 1&0\end{bmatrix}^{\otimes|E|}\left(\begin{bmatrix}1\\ 0\end{bmatrix}^{\otimes 3}+\begin{bmatrix}0\\ 1\end{bmatrix}^{\otimes 3}\right)^{\otimes|V|}
  32. = ( ( 0 , 1 , 1 , 1 ) [ 0 1 1 0 ] 2 ) | U | ( ( [ 0 1 1 0 ] [ 1 0 ] ) 3 + ( [ 0 1 1 0 ] [ 0 1 ] ) 3 ) | V | =\left((0,1,1,1)\begin{bmatrix}0&1\\ 1&0\end{bmatrix}^{\otimes 2}\right)^{\otimes|U|}\left(\left(\begin{bmatrix}0&1% \\ 1&0\end{bmatrix}\begin{bmatrix}1\\ 0\end{bmatrix}\right)^{\otimes 3}+\left(\begin{bmatrix}0&1\\ 1&0\end{bmatrix}\begin{bmatrix}0\\ 1\end{bmatrix}\right)^{\otimes 3}\right)^{\otimes|V|}
  33. = ( 1 , 1 , 1 , 0 ) | U | ( [ 0 1 ] 3 + [ 1 0 ] 3 ) | V | =(1,1,1,0)^{\otimes|U|}\left(\begin{bmatrix}0\\ 1\end{bmatrix}^{\otimes 3}+\begin{bmatrix}1\\ 0\end{bmatrix}^{\otimes 3}\right)^{\otimes|V|}
  34. = NAND 2 | U | EQUAL 3 | V | , =\,\text{NAND}_{2}^{\otimes|U|}\,\text{EQUAL}_{3}^{\otimes|V|},
  35. Holant ( H , NAND 2 , EQUAL 3 ) , \,\text{Holant}(H,\,\text{NAND}_{2},\,\text{EQUAL}_{3}),

Homoaconitate_hydratase.html

  1. \rightleftharpoons

Homocitrate_synthase.html

  1. \rightleftharpoons

Homoglutathione_synthase.html

  1. \rightleftharpoons

Homoserine_kinase.html

  1. \rightleftharpoons

Homoserine_O-acetyltransferase.html

  1. \rightleftharpoons

Homoserine_O-succinyltransferase.html

  1. \rightleftharpoons

Homospermidine_synthase_(spermidine-specific).html

  1. \rightleftharpoons

Homothetic_center.html

  1. C 1 C_{1}
  2. C 2 C_{2}
  3. ( x 1 , y 1 ) (x_{1},y_{1})
  4. ( x 2 , y 2 ) (x_{2},y_{2})
  5. r 1 r_{1}
  6. r 2 r_{2}
  7. ( x 0 , y 0 ) , (x_{0},y_{0}),
  8. ( x 0 , y 0 ) = r 2 r 1 + r 2 ( x 1 , y 1 ) + r 1 r 1 + r 2 ( x 2 , y 2 ) . (x_{0},y_{0})=\frac{r_{2}}{r_{1}+r_{2}}(x_{1},y_{1})+\frac{r_{1}}{r_{1}+r_{2}}% (x_{2},y_{2}).
  9. ( x e , y e ) = - r 2 r 1 - r 2 ( x 1 , y 1 ) + r 1 r 1 - r 2 ( x 2 , y 2 ) . (x_{e},y_{e})=\frac{-r_{2}}{r_{1}-r_{2}}(x_{1},y_{1})+\frac{r_{1}}{r_{1}-r_{2}% }(x_{2},y_{2}).
  10. A 1 A 2 A_{1}A_{2}
  11. B 1 B 2 B_{1}B_{2}
  12. E Q E Q = E S E S \frac{EQ}{EQ^{\prime}}=\frac{ES}{ES^{\prime}}
  13. E P E P = E Q E Q ; E P E Q = E Q E P \frac{EP}{EP^{\prime}}=\frac{EQ}{EQ^{\prime}};{EP}\cdot{EQ^{\prime}}={EQ}\cdot% {EP^{\prime}}
  14. H A B B H A B A = r B r A \frac{H_{AB}B}{H_{AB}A}=\frac{r_{B}}{r_{A}}

Hume-Rothery_rules.html

  1. % difference = ( r s o l u t e - r s o l v e n t r s o l v e n t ) × 100 % 15 % . \%\mbox{ difference}~{}=\left(\frac{r_{solute}-r_{solvent}}{r_{solvent}}\right% )\times 100\%\leq 15\%.

Hydrogen-sulfide_S-acetyltransferase.html

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Hydrogenobyrinic_acid_a,c-diamide_synthase_(glutamine-hydrolysing).html

  1. \rightleftharpoons

Hydroperoxide_dehydratase.html

  1. \rightleftharpoons

Hydroquinone_glucosyltransferase.html

  1. \rightleftharpoons

Hydroxyacylglutathione_hydrolase.html

  1. \rightleftharpoons

Hydroxyanthraquinone_glucosyltransferase.html

  1. \rightleftharpoons

Hydroxybutyrate-dimer_hydrolase.html

  1. \rightleftharpoons

Hydroxycinnamate_4-beta-glucosyltransferase.html

  1. \rightleftharpoons

Hydroxydechloroatrazine_ethylaminohydrolase.html

  1. \rightleftharpoons

Hydroxyethylthiazole_kinase.html

  1. \rightleftharpoons

Hydroxyisourate_hydrolase.html

  1. \rightleftharpoons

Hydroxylysine_kinase.html

  1. \rightleftharpoons

Hydroxymandelonitrile_glucosyltransferase.html

  1. \rightleftharpoons

Hydroxymethylglutaryl-CoA_hydrolase.html

  1. \rightleftharpoons

Hydroxymethylglutaryl-CoA_synthase.html

  1. \rightleftharpoons

Hydroxymethylpyrimidine_kinase.html

  1. \rightleftharpoons

Hygromycin-B_kinase.html

  1. \rightleftharpoons

Hyperdeterminant.html

  1. D e t ( A ) = f i j k l f n m o p f q r s t a i n q a j m r a k o s a l p t Det(A)=f^{ijkl}f^{nmop}f^{qrst}a_{inq}a_{jmr}a_{kos}a_{lpt}
  2. f 0011 = f 1100 = f 0110 = f 1001 = - 1 / 2 f^{0011}=f^{1100}=f^{0110}=f^{1001}=-1/2
  3. f 0101 = f 1010 = 1 f^{0101}=f^{1010}=1
  4. f a b c . + f b c a + f c a b + f c b a . + f a c b + f b a c = 0 f^{...abc....}+f^{...bca...}+f^{...cab...}+f^{...cba....}+f^{...acb...}+f^{...% bac...}=0
  5. a 000 = a a_{000}=a
  6. a 001 = a 010 = a 100 = b a_{001}=a_{010}=a_{100}=b
  7. a 100 = a 101 = a 011 = c a_{100}=a_{101}=a_{011}=c
  8. a 111 = d a_{111}=d
  9. a x 3 + 3 b x 2 + 3 c x + d ax^{3}+3bx^{2}+3cx+d
  10. \mathbb{R}
  11. \mathbb{C}
  12. f : V 1 V 2 V r K f:V_{1}\otimes V_{2}\otimes\cdots\otimes V_{r}\to K
  13. f V 1 * V 2 * V r * f\in V^{*}_{1}\otimes V^{*}_{2}\otimes\cdots\otimes V^{*}_{r}
  14. N ( k 2 + + k r , k 2 , , k r ) = ( k 2 + + k r + 1 ) ! k 2 ! k r ! . N(k_{2}+\cdots+k_{r},k_{2},\ldots,k_{r})=\frac{(k_{2}+\cdots+k_{r}+1)!}{k_{2}!% \cdots k_{r}!}.
  15. r = 0 N r z r r ! = e - 2 z ( 1 - z ) 2 . \sum_{r=0}^{\infty}N_{r}\frac{z^{r}}{r!}=\frac{e^{-2z}}{(1-z)^{2}}.
  16. ( k 1 , , k r ) (k_{1},\ldots,k_{r})
  17. S L ( k 1 + 1 ) S L ( k r + 1 ) SL(k_{1}+1)\otimes\cdots\otimes SL(k_{r}+1)
  18. det p {\det}_{p}

Hypernetted-chain_equation.html

  1. ln y ( r 12 ) = ln g ( r 12 ) + β u ( r 12 ) = ρ [ h ( r 13 ) - ln g ( r 13 ) - β u ( r 13 ) ] h ( r 23 ) d 𝐫 𝟑 , \ln y(r_{12})=\ln g(r_{12})+\beta u(r_{12})=\rho\int\left[h(r_{13})-\ln g(r_{1% 3})-\beta u(r_{13})\right]h(r_{23})\,d\mathbf{r_{3}},\,
  2. ρ = N V \rho=\frac{N}{V}
  3. h ( r ) = g ( r ) - 1 h(r)=g(r)-1
  4. g ( r ) g(r)
  5. u ( r ) u(r)
  6. β = 1 k B T \beta=\frac{1}{k_{\rm B}T}
  7. T T
  8. k B k_{\rm B}
  9. c ( r ) = g total ( r ) - g indirect ( r ) c(r)=g_{\rm total}(r)-g_{\rm indirect}(r)\,
  10. g total ( r ) = g ( r ) = exp [ - β w ( r ) ] g_{\rm total}(r)=g(r)=\exp[-\beta w(r)]
  11. w ( r ) w(r)
  12. g indirect ( r ) g_{\rm indirect}(r)
  13. u ( r ) u(r)
  14. g indirect ( r ) = exp { - β [ w ( r ) - u ( r ) ] } g_{\rm indirect}(r)=\exp\{-\beta[w(r)-u(r)]\}
  15. c ( r ) c(r)
  16. c ( r ) = e - β w ( r ) - e - β [ w ( r ) - u ( r ) ] . c(r)=e^{-\beta w(r)}-e^{-\beta[w(r)-u(r)]}.\,
  17. g ( r ) g(r)
  18. y ( r ) = e β u ( r ) g ( r ) ( = g indirect ( r ) ) y(r)=e^{\beta u(r)}g(r)(=g_{\rm indirect}(r))
  19. c ( r ) c(r)
  20. c ( r ) = e - β w ( r ) - 1 + β [ w ( r ) - u ( r ) ] = g ( r ) - 1 - ln y ( r ) = f ( r ) y ( r ) + [ y ( r ) - 1 - ln y ( r ) ] ( HNC ) , c(r)=e^{-\beta w(r)}-1+\beta[w(r)-u(r)]\,=g(r)-1-\ln y(r)\,=f(r)y(r)+[y(r)-1-% \ln y(r)]\,\,(\,\text{HNC}),
  21. f ( r ) = e - β u ( r ) - 1 f(r)=e^{-\beta u(r)}-1
  22. h ( r ) - c ( r ) = g ( r ) - 1 - c ( r ) = ln y ( r ) . h(r)-c(r)=g(r)-1-c(r)=\ln y(r).
  23. h ( r 12 ) - c ( r 12 ) = ρ c ( r 13 ) h ( r 23 ) d 𝐫 3 , h(r_{12})-c(r_{12})=\rho\int c(r_{13})h(r_{23})d\mathbf{r}_{3},
  24. ln y ( r 12 ) = ln g ( r 12 ) + β u ( r 12 ) = ρ [ h ( r 13 ) - ln g ( r 13 ) - β u ( r 13 ) ] h ( r 23 ) d 𝐫 𝟑 . \ln y(r_{12})=\ln g(r_{12})+\beta u(r_{12})=\rho\int\left[h(r_{13})-\ln g(r_{1% 3})-\beta u(r_{13})\right]h(r_{23})\,d\mathbf{r_{3}}.\,

Hypotaurocyamine_kinase.html

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Icosanoyl-CoA_synthase.html

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Image_functors_for_sheaves.html

  1. F G F\leftrightarrows G
  2. 𝒢 f * f * 𝒢 \mathcal{G}\rightarrow f_{*}f^{*}\mathcal{G}
  3. f * f * f^{*}f_{*}\mathcal{F}\rightarrow\mathcal{F}
  4. 𝒢 \mathcal{G}
  5. \mathcal{F}
  6. f : X Z f:X\rightarrow Z
  7. g : Y Z g:Y\rightarrow Z
  8. f ¯ : X × Z Y Y \bar{f}:X\times_{Z}Y\rightarrow Y
  9. g ¯ : X × Z Y X \bar{g}:X\times_{Z}Y\rightarrow X
  10. R f ¯ * R g ¯ ! R f ! R g * R\bar{f}_{*}R\bar{g}^{!}\cong Rf^{!}Rg_{*}
  11. i * i * = i ! i ! i^{*}\leftrightarrows i_{*}=i_{!}\leftrightarrows i^{!}
  12. j ! j ! = j * j * j_{!}\leftrightarrows j^{!}=j^{*}\leftrightarrows j_{*}

Imidazole_N-acetyltransferase.html

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Imidazoleacetate—phosphoribosyldiphosphate_ligase.html

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Imidazoleglycerol-phosphate_dehydratase.html

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Imidazolonepropionase.html

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IMP_cyclohydrolase.html

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Impact_pressure.html

  1. q c q_{c}
  2. Q c Q_{c}
  3. P t P = ( 1 + γ - 1 2 M 2 ) γ γ - 1 \frac{P_{t}}{P}=\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\tfrac{\gamma}{\gamma-% 1}}
  4. P t P_{t}
  5. P P
  6. γ \gamma\;
  7. M M\;
  8. γ \gamma\;
  9. P t = P + q c \;P_{t}=P+q_{c}
  10. q c = P [ ( 1 + 0.2 M 2 ) 7 2 - 1 ] \;q_{c}=P\left[\left(1+0.2M^{2}\right)^{\tfrac{7}{2}}-1\right]
  11. 1 2 γ P M 2 \;\tfrac{1}{2}\gamma PM^{2}
  12. q c = q ( 1 + M 2 4 + M 4 40 + M 6 1600 ) \;q_{c}=q\left(1+\frac{M^{2}}{4}+\frac{M^{4}}{40}+\frac{M^{6}}{1600}...\right)\;
  13. q \;q

Indirect_DNA_damage.html

  1. ( Chromophore ) * + O 2 3 Chromophore + O 2 1 \mathrm{(Chromophore)^{*}+{}^{3}O_{2}\ \xrightarrow{}\ Chromophore+{}^{1}O_{2}}
  2. O 2 1 + intact DNA O 2 3 + damaged DNA \mathrm{{}^{1}O_{2}+intact\ DNA\ \xrightarrow{}\ {}^{3}O_{2}+damaged\ DNA}

Indole-3-acetate_beta-glucosyltransferase.html

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Indoleacetate—lysine_synthetase.html

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Indoleacetylglucose—inositol_O-acyltransferase.html

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Indolylacetyl-myo-inositol_galactosyltransferase.html

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Indolylacetylinositol_arabinosyltransferase.html

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Indoxyl-UDPG_glucosyltransferase.html

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Influence_line.html

  1. F i P F^{P}_{i}
  2. F i Q F^{Q}_{i}
  3. d i P d^{P}_{i}
  4. d i Q d^{Q}_{i}
  5. F i P F^{P}_{i}
  6. F i Q F^{Q}_{i}
  7. F Q F^{Q}
  8. d i Q d^{Q}_{i}
  9. F Q F^{Q}
  10. d 1 Q = - 1 d^{Q}_{1}=-1
  11. - F 1 P + i = 2 n F i P d i Q = F Q × 0 F 1 P = i = 2 n F i P d i Q -F^{P}_{1}+\sum^{n}_{i=2}F^{P}_{i}d^{Q}_{i}=F^{Q}\times 0\iff F^{P}_{1}=\sum^{% n}_{i=2}F^{P}_{i}d^{Q}_{i}

Inorganic_diphosphatase.html

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Inosinate_nucleosidase.html

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Inosine_kinase.html

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Inosine_nucleosidase.html

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Inositol-1,4-bisphosphate_1-phosphatase.html

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Inositol-pentakisphosphate_2-kinase.html

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Inositol-phosphate_phosphatase.html

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Inositol-tetrakisphosphate_1-kinase.html

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Inositol-tetrakisphosphate_5-kinase.html

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Inositol-trisphosphate_3-kinase.html

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Inositol_3-alpha-galactosyltransferase.html

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Inositol_3-kinase.html

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Installment_sales_method.html

  1. G r o s s P r o f i t S a l e s \frac{GrossProfit}{Sales}

Intelligent_Mail_barcode.html

  1. ( 13 2 ) + ( 13 5 ) + ( 13 8 ) + ( 13 11 ) = 78 + 1287 + 1287 + 78 = 2 1365 = 2730 {\left({{13}\atop{2}}\right)}+{\left({{13}\atop{5}}\right)}+{\left({{13}\atop{% 8}}\right)}+{\left({{13}\atop{11}}\right)}=78+1287+1287+78=2\cdot 1365=2730

Interpretation_(logic).html

  1. 𝒲 \mathcal{W}
  2. \triangle
  3. \square
  4. 𝒲 \mathcal{W}
  5. \triangle
  6. \triangle
  7. \square
  8. 𝒲 \mathcal{W}
  9. \triangle
  10. \square
  11. \triangle\square\triangle
  12. 𝒲 \mathcal{W}
  13. \mathcal{I}
  14. \mathcal{I}
  15. ( ϕ x ψ ) x ( ϕ ψ ) (\phi\lor\exists x\psi)\leftrightarrow\exists x(\phi\lor\psi)
  16. [ y ( y = y ) x ( x = x ) ] x [ y ( y = y ) x = x ] [\forall y(y=y)\lor\exists x(x=x)]\equiv\exists x[\forall y(y=y)\lor x=x]
  17. j \mathcal{I}_{j}
  18. i \mathcal{I}_{i}
  19. i j \mathcal{I}_{i}\to\mathcal{I}_{j}
  20. 𝒮 \mathcal{FS^{\prime}}
  21. \blacksquare
  22. \bigstar
  23. \blacklozenge
  24. 𝒮 \mathcal{FS^{\prime}}
  25. 𝒮 \mathcal{FS^{\prime}}
  26. 𝒮 \mathcal{FS^{\prime}}
  27. 𝒮 \mathcal{FS^{\prime}}
  28. \blacksquare
  29. \bigstar
  30. \blacklozenge
  31. \blacksquare
  32. \blacksquare
  33. \blacksquare
  34. \bigstar
  35. \blacksquare
  36. \blacklozenge
  37. \blacksquare
  38. \blacksquare
  39. \blacksquare
  40. \bigstar
  41. \blacksquare
  42. \blacksquare
  43. \blacklozenge
  44. \blacksquare
  45. \blacksquare
  46. \blacksquare
  47. \blacksquare
  48. \bigstar
  49. \blacksquare
  50. \blacksquare
  51. \blacksquare
  52. \blacklozenge
  53. \blacksquare
  54. \blacksquare
  55. \blacksquare
  56. \blacksquare
  57. \blacksquare
  58. \bigstar
  59. \blacksquare
  60. \blacksquare
  61. \blacksquare
  62. \blacklozenge
  63. \blacksquare
  64. \blacksquare
  65. \blacksquare
  66. \blacksquare

Inulosucrase.html

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Inverse_curve.html

  1. X = x x 2 + y 2 , Y = y x 2 + y 2 , X=\frac{x}{x^{2}+y^{2}},\ Y=\frac{y}{x^{2}+y^{2}},
  2. x = X X 2 + Y 2 , y = Y X 2 + Y 2 . x=\frac{X}{X^{2}+Y^{2}},\ y=\frac{Y}{X^{2}+Y^{2}}.
  3. f ( X X 2 + Y 2 , Y X 2 + Y 2 ) = 0. f\left(\frac{X}{X^{2}+Y^{2}},\ \frac{Y}{X^{2}+Y^{2}}\right)=0.
  4. x = x ( t ) , y = y ( t ) x=x(t),\ y=y(t)
  5. X = X ( t ) = x ( t ) x ( t ) 2 + y ( t ) 2 , Y = Y ( t ) = y ( t ) x ( t ) 2 + y ( t ) 2 . X=X(t)=\frac{x(t)}{x(t)^{2}+y(t)^{2}},\ Y=Y(t)=\frac{y(t)}{x(t)^{2}+y(t)^{2}}.
  6. f ( a + k 2 ( X - a ) ( X - a ) 2 + ( Y - b ) 2 , b + k 2 ( Y - b ) ( X - a ) 2 + ( Y - b ) 2 ) = 0. f\left(a+\frac{k^{2}(X-a)}{(X-a)^{2}+(Y-b)^{2}},\ b+\frac{k^{2}(Y-b)}{(X-a)^{2% }+(Y-b)^{2}}\right)=0.
  7. x = x ( t ) , y = y ( t ) x=x(t),\ y=y(t)
  8. X = X ( t ) = a + k 2 ( x ( t ) - a ) ( x ( t ) - a ) 2 + ( y ( t ) - b ) 2 , Y = Y ( t ) = b + k 2 ( y ( t ) - b ) ( x ( t ) - a ) 2 + ( y ( t ) - b ) 2 . X=X(t)=a+\frac{k^{2}(x(t)-a)}{(x(t)-a)^{2}+(y(t)-b)^{2}},\ Y=Y(t)=b+\frac{k^{2% }(y(t)-b)}{(x(t)-a)^{2}+(y(t)-b)^{2}}.
  9. R = 1 r , Θ = θ , R=\frac{1}{r},\ \Theta=\theta,
  10. r = 1 R , θ = Θ . r=\frac{1}{R},\ \theta=\Theta.
  11. ( x 2 + y 2 ) 2 = a 2 ( x 2 - y 2 ) (x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})\,
  12. a 2 ( u 2 - v 2 ) = 1 , a^{2}(u^{2}-v^{2})=1,\,
  13. ( u 2 + v 2 ) n = u n + v n . (u^{2}+v^{2})^{n}=u^{n}+v^{n}.\,
  14. r cos ( θ - θ 0 ) = a r\cos(\theta-\theta_{0})=a
  15. r = a cos ( θ - θ 0 ) r=a\cos(\theta-\theta_{0})
  16. r 2 - 2 r 0 r cos ( θ - θ 0 ) + r 0 2 - a 2 = 0 ( a , r > 0 , a r 0 ) , r^{2}-2r_{0}r\cos(\theta-\theta_{0})+r_{0}^{2}-a^{2}=0\quad(a,\ r>0,\ a\neq r_% {0}),
  17. 1 - 2 r 0 r cos ( θ - θ 0 ) + ( r 0 2 - a 2 ) r 2 = 0 , 1-2r_{0}r\cos(\theta-\theta_{0})+(r_{0}^{2}-a^{2})r^{2}=0,
  18. r 2 - 2 r 0 r 0 2 - a 2 r cos ( θ - θ 0 ) + 1 r 0 2 - a 2 = 0. r^{2}-\frac{2r_{0}}{r_{0}^{2}-a^{2}}r\cos(\theta-\theta_{0})+\frac{1}{r_{0}^{2% }-a^{2}}=0.
  19. A = a | r 0 2 - a 2 | A=\frac{a}{|r_{0}^{2}-a^{2}|}
  20. ( R 0 , Θ 0 ) = ( r 0 r 0 2 - a 2 , θ 0 ) . (R_{0},\ \Theta_{0})=\left(\frac{r_{0}}{r_{0}^{2}-a^{2}},\ \theta_{0}\right).
  21. r 0 2 = a 2 + 1. r_{0}^{2}=a^{2}+1.\,
  22. r 0 2 - a 2 = 1. r_{0}^{2}-a^{2}=1.\,
  23. r = cos θ sin 2 θ . r=\frac{\cos\theta}{\sin^{2}\theta}.
  24. r = sin 2 θ cos θ = sin θ tan θ r=\frac{\sin^{2}\theta}{\cos\theta}=\sin\theta\tan\theta
  25. r = 1 1 + e cos θ , r=\frac{1}{1+e\cos\theta},
  26. r = 1 + e cos θ , r=1+e\cos\theta,\,
  27. x 2 a 2 ± y 2 b 2 = 1. \frac{x^{2}}{a^{2}}\pm\frac{y^{2}}{b^{2}}=1.
  28. ( x - a ) 2 a 2 ± y 2 b 2 = 1 \frac{(x-a)^{2}}{a^{2}}\pm\frac{y^{2}}{b^{2}}=1
  29. x 2 2 a ± a y 2 2 b 2 = x \frac{x^{2}}{2a}\pm\frac{ay^{2}}{2b^{2}}=x
  30. c x 2 + d y 2 = x . cx^{2}+dy^{2}=x.\,
  31. c x 2 ( x 2 + y 2 ) 2 + d y 2 ( x 2 + y 2 ) 2 = x x 2 + y 2 \frac{cx^{2}}{(x^{2}+y^{2})^{2}}+\frac{dy^{2}}{(x^{2}+y^{2})^{2}}=\frac{x}{x^{% 2}+y^{2}}
  32. x ( x 2 + y 2 ) = c x 2 + d y 2 . x(x^{2}+y^{2})=cx^{2}+dy^{2}.\,
  33. c x 2 + d y 2 = 1 cx^{2}+dy^{2}=1\,
  34. ( x 2 + y 2 ) 2 = c x 2 + d y 2 (x^{2}+y^{2})^{2}=cx^{2}+dy^{2}\,

Inverse_exchange-traded_fund.html

  1. x x
  2. A t A_{t}
  3. S t S_{t}
  4. Δ ln ( A t ) = x Δ ln ( S t ) + ( x - x 2 ) σ 2 Δ t 2 \Delta\ln(A_{t})=x\Delta\ln(S_{t})+(x-x^{2})\sigma^{2}\frac{\Delta t}{2}
  5. σ 2 \sigma^{2}
  6. x < 0 x<0
  7. x > 1 x>1
  8. x x

Inverse_mean_curvature_flow.html

  1. R ( t ) R(t)
  2. t t
  3. t t
  4. t t
  5. R ( t ) R^{\prime}(t)
  6. 2 R ( t ) \frac{2}{R(t)}
  7. d R d t = R ( t ) 2 , \frac{dR}{dt}=\frac{R(t)}{2},
  8. R ( t ) = R 0 e t / 2 , R(t)=R_{0}e^{t/2},
  9. R 0 R_{0}
  10. t = 0 t=0

Iron-chelate-transporting_ATPase.html

  1. \rightleftharpoons

Irwin–Hall_distribution.html

  1. X = k = 1 n U k . X=\sum_{k=1}^{n}U_{k}.
  2. f X ( x ; n ) = 1 2 ( n - 1 ) ! k = 0 n ( - 1 ) k ( n k ) ( x - k ) n - 1 sgn ( x - k ) f_{X}(x;n)=\frac{1}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^{k}{n% \choose k}\left(x-k\right)^{n-1}\operatorname{sgn}(x-k)
  3. sgn ( x - k ) = { - 1 x < k 0 x = k 1 x > k . \operatorname{sgn}\left(x-k\right)=\begin{cases}-1&x<k\\ 0&x=k\\ 1&x>k.\end{cases}
  4. f X ( x ; n ) = 1 ( n - 1 ) ! j = 0 n - 1 a j ( k , n ) x j f_{X}(x;n)=\frac{1}{\left(n-1\right)!}\sum_{j=0}^{n-1}a_{j}(k,n)x^{j}
  5. a j ( k , n ) = { 1 k = 0 , j = n - 1 0 k = 0 , j < n - 1 a j ( k - 1 , n ) + ( - 1 ) n + k - j - 1 ( n k ) ( n - 1 j ) k n - j - 1 k > 0 a_{j}(k,n)=\begin{cases}1&k=0,j=n-1\\ 0&k=0,j<n-1\\ a_{j}(k-1,n)+\left(-1\right)^{n+k-j-1}{n\choose k}{{n-1}\choose j}k^{n-j-1}&k>% 0\end{cases}
  6. f X ( x ) = { 1 0 x 1 0 otherwise f_{X}(x)=\begin{cases}1&0\leq x\leq 1\\ 0&\,\text{otherwise}\end{cases}
  7. f X ( x ) = { x 0 x 1 2 - x 1 x 2 f_{X}(x)=\begin{cases}x&0\leq x\leq 1\\ 2-x&1\leq x\leq 2\end{cases}
  8. f X ( x ) = { 1 2 x 2 0 x 1 1 2 ( - 2 x 2 + 6 x - 3 ) 1 x 2 1 2 ( x 2 - 6 x + 9 ) 2 x 3 f_{X}(x)=\begin{cases}\frac{1}{2}x^{2}&0\leq x\leq 1\\ \frac{1}{2}\left(-2x^{2}+6x-3\right)&1\leq x\leq 2\\ \frac{1}{2}\left(x^{2}-6x+9\right)&2\leq x\leq 3\end{cases}
  9. f X ( x ) = { 1 6 x 3 0 x 1 1 6 ( - 3 x 3 + 12 x 2 - 12 x + 4 ) 1 x 2 1 6 ( 3 x 3 - 24 x 2 + 60 x - 44 ) 2 x 3 1 6 ( - x 3 + 12 x 2 - 48 x + 64 ) 3 x 4 f_{X}(x)=\begin{cases}\frac{1}{6}x^{3}&0\leq x\leq 1\\ \frac{1}{6}\left(-3x^{3}+12x^{2}-12x+4\right)&1\leq x\leq 2\\ \frac{1}{6}\left(3x^{3}-24x^{2}+60x-44\right)&2\leq x\leq 3\\ \frac{1}{6}\left(-x^{3}+12x^{2}-48x+64\right)&3\leq x\leq 4\end{cases}
  10. f X ( x ) = { 1 24 x 4 0 x 1 1 24 ( - 4 x 4 + 20 x 3 - 30 x 2 + 20 x - 5 ) 1 x 2 1 24 ( 6 x 4 - 60 x 3 + 210 x 2 - 300 x + 155 ) 2 x 3 1 24 ( - 4 x 4 + 60 x 3 - 330 x 2 + 780 x - 655 ) 3 x 4 1 24 ( x 4 - 20 x 3 + 150 x 2 - 500 x + 625 ) 4 x 5 f_{X}(x)=\begin{cases}\frac{1}{24}x^{4}&0\leq x\leq 1\\ \frac{1}{24}\left(-4x^{4}+20x^{3}-30x^{2}+20x-5\right)&1\leq x\leq 2\\ \frac{1}{24}\left(6x^{4}-60x^{3}+210x^{2}-300x+155\right)&2\leq x\leq 3\\ \frac{1}{24}\left(-4x^{4}+60x^{3}-330x^{2}+780x-655\right)&3\leq x\leq 4\\ \frac{1}{24}\left(x^{4}-20x^{3}+150x^{2}-500x+625\right)&4\leq x\leq 5\end{cases}

Isochron.html

  1. y ( t ) y(t)
  2. d 2 y d t 2 + d y d t = 1 \frac{d^{2}y}{dt^{2}}+\frac{dy}{dt}=1
  3. t = 0 t=0
  4. y ( 0 ) = y 0 y(0)=y_{0}
  5. d y / d t ( 0 ) = y 0 dy/dt(0)=y^{\prime}_{0}
  6. y 0 y_{0}
  7. y 0 y^{\prime}_{0}
  8. y 0 + y 0 = constant y_{0}+y^{\prime}_{0}=\mbox{constant}~{}
  9. y = t + A + B exp ( - t ) y=t+A+B\exp(-t)\,
  10. t t\to\infty
  11. y t + A y\to t+A
  12. A A
  13. B exp ( - t ) B\exp(-t)
  14. A A
  15. t = 0 t=0
  16. y 0 = A + B y_{0}=A+B
  17. y 0 = 1 - B y^{\prime}_{0}=1-B
  18. B B
  19. y 0 + y 0 = 1 + A y_{0}+y^{\prime}_{0}=1+A
  20. A A
  21. d x d t = - x y and d y d t = - y + x 2 - 2 y 2 \frac{dx}{dt}=-xy\,\text{ and }\frac{dy}{dt}=-y+x^{2}-2y^{2}
  22. x = X + X Y + and y = Y + 2 Y 2 + X 2 + x=X+XY+\cdots\,\text{ and }y=Y+2Y^{2}+X^{2}+\cdots
  23. ( X , Y ) (X,Y)
  24. d X d t = - X 3 + and d Y d t = ( - 1 - 2 X 2 + ) Y \frac{dX}{dt}=-X^{3}+\cdots\,\text{ and }\frac{dY}{dt}=(-1-2X^{2}+\cdots)Y
  25. Y Y
  26. d Y / d t = ( negative ) Y dY/dt=(\,\text{negative})Y
  27. X X
  28. X X
  29. X X
  30. ( x 0 , y 0 ) (x_{0},y_{0})
  31. X ( 0 ) X(0)
  32. X 0 X_{0}
  33. X X
  34. Y Y
  35. X X
  36. Y Y
  37. X X
  38. Y Y
  39. ( x , y ) (x,y)
  40. x - X y = X - X 3 x-Xy=X-X^{3}
  41. ( x 0 , y 0 ) (x_{0},y_{0})
  42. X 0 X_{0}
  43. X ( 0 ) = X 0 X(0)=X_{0}

Isocitrate_dehydrogenase_(NADP+)_kinase.html

  1. \rightleftharpoons

Isocitrate_O-dihydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Isoenthalpic–isobaric_ensemble.html

  1. H H\,
  2. P P\,
  3. N p H NpH
  4. N N\,
  5. V V\,
  6. V V\,
  7. P V PV\,
  8. H = E + P V H=E+PV\,

Isoflavone_7-O-glucosyltransferase.html

  1. \rightleftharpoons

Isohexenylglutaconyl-CoA_hydratase.html

  1. \rightleftharpoons

Isoleucine—tRNA_ligase.html

  1. \rightleftharpoons

Isomap.html

  1. K = 1 2 H D 2 H K=\frac{1}{2}HD^{2}H\,
  2. D 2 = D i j 2 := ( D i j ) 2 D^{2}=D^{2}_{ij}:=(D_{ij})^{2}
  3. H = I n - 1 N e N e N T , where e N = [ 1 1 ] T N . H=I_{n}-\frac{1}{N}e_{N}e^{T}_{N},\quad\,\text{where }e_{N}=[1\ \dots\ 1]^{T}% \in\mathbb{R}^{N}.

Isonocardicin_synthase.html

  1. \rightleftharpoons

Isopenicillin_N_N-acyltransferase.html

  1. \rightleftharpoons

Isoprene_synthase.html

  1. \rightleftharpoons

Isovitexin_beta-glucosyltransferase.html

  1. \rightleftharpoons

Itaconyl-CoA_hydratase.html

  1. \rightleftharpoons

Itakura–Saito_distance.html

  1. P ( ω ) P(\omega)
  2. P ^ ( ω ) \hat{P}(\omega)
  3. D I S ( P ( ω ) , P ^ ( ω ) ) = 1 2 π - π π [ P ( ω ) P ^ ( ω ) - log P ( ω ) P ^ ( ω ) - 1 ] d ω D_{IS}(P(\omega),\hat{P}(\omega))=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left[\frac{P% (\omega)}{\hat{P}(\omega)}-\log\frac{P(\omega)}{\hat{P}(\omega)}-1\right]\,d\omega

Jain_philosophy.html

  1. 0 \aleph_{0}

Johnson–Lindenstrauss_lemma.html

  1. Ω ( log ( m ) ε 2 log ( 1 / ε ) ) \Omega\left(\frac{\log(m)}{\varepsilon^{2}\log(1/\varepsilon)}\right)
  2. ( 1 - ε ) u - v 2 f ( u ) - f ( v ) 2 ( 1 + ε ) u - v 2 (1-\varepsilon)\|u-v\|^{2}\leq\|f(u)-f(v)\|^{2}\leq(1+\varepsilon)\|u-v\|^{2}
  3. P ( | A x 2 2 - 1 | > ε ) < δ P(|\|Ax\|_{2}^{2}-1|>\varepsilon)<\delta
  4. x = ( u - v ) / u - v 2 x=(u-v)/\|u-v\|_{2}
  5. δ < 1 / n 2 \delta<1/n^{2}

Juvenile-hormone_esterase.html

  1. \rightleftharpoons

K+-transporting_ATPase.html

  1. \rightleftharpoons

K-approximation_of_k-hitting_set.html

  1. i S : | i | k \forall i\in S:|i|\leq k
  2. j j
  3. p j p_{j}
  4. a T : j S ( a ) p j W ( a ) . \forall a\in T:\sum_{j\in S(a)}p_{j}\leq W(a).\,
  5. Σ j S ( a ) p j = W ( a ) \Sigma_{j\in S(a)}p_{j}=W(a)
  6. p 1 , , p | S | p_{1},\ldots,p_{|S|}
  7. Σ a T * Σ j S ( a ) p j Σ a T * W ( a ) \Sigma_{a\in T^{*}}\Sigma_{j\in S(a)}p_{j}\leq\Sigma_{a\in T^{*}}W(a)
  8. Σ j S p j Σ a T * W ( a ) \Sigma_{j\in S}p_{j}\leq\Sigma_{a\in T^{*}}W(a)
  9. Σ a H Σ j S ( a ) p j = Σ a H W ( a ) \Sigma_{a\in H}\Sigma_{j\in S(a)}p_{j}=\Sigma_{a\in H}W(a)
  10. p j p_{j}
  11. Σ a H W ( a ) k Σ j S p j \Sigma_{a\in H}W(a)\leq k\cdot\Sigma_{j\in S}p_{j}
  12. Σ a H W ( a ) k Σ a T * W ( a ) \Sigma_{a\in H}W(a)\leq k\cdot\Sigma_{a\in T^{*}}W(a)

Kadir–Brady_saliency_detector.html

  1. p log p p\log p\,
  2. { x , R } \{x,R\}
  3. x x
  4. R R
  5. D D
  6. d 1 , , d r {d_{1},\dots,d_{r}}
  7. P D ( d i , x , R ) P_{D}(d_{i},x,R)
  8. d i d_{i}
  9. { x , R } \{x,R\}
  10. R x R_{x}
  11. H D ( x , R ) = - i ( 1 r ) P D ( d i , x , R ) log P D ( d i , x , R ) . H_{D}(x,R)=-\sum_{i\in(1\dots r)}P_{D}(d_{i},x,R)\log P_{D}(d_{i},x,R).
  12. H D ( x , R ) H_{D}(x,R)
  13. x x
  14. R R
  15. H D ( x , R ) H_{D}(x,R)
  16. H D ( x , s ) H_{D}(x,s)
  17. R R
  18. { x i , s i ; i = 1 N } \{x_{i},s_{i};i=1\dots N\}
  19. H D ( x , s ) = - i ( 1 r ) P D ( d i , x , s ) log P D ( d i , x , s ) / 10 H_{D}(x,s)=-\sum_{i\in(1\dots r)}P_{D}(d_{i},x,s)\log P_{D}(d_{i},x,s)/10
  20. s p s_{p}
  21. W D ( x , s ) = i ( 1 r ) | s P D , ( d i , x , s ) | W_{D}(x,s)=\sum_{i\in(1\dots r)}|\frac{\partial}{\partial s}P_{D,}(d_{i},x,s)|
  22. Y D ( x , s p ) Y_{D}(x,s_{p})
  23. H D ( x , s p ) H_{D}(x,s_{p})
  24. W D ( x , s p ) W_{D}(x,s_{p})
  25. s p s_{p}
  26. Y D ( x , s p ) Y_{D}(x,s_{p})
  27. Y D ( x , s p ) Y_{D}(x,s_{p})
  28. x x
  29. R R
  30. W D W_{D}
  31. ϵ \epsilon
  32. μ a \mu_{a}
  33. μ b \mu_{b}
  34. ϵ = 1 - μ a ( A T μ b A ) μ a ( A T μ b A ) \epsilon=1-\frac{\mu_{a}\cap(A^{T}\mu_{b}A)}{\mu_{a}\cup(A^{T}\mu_{b}A)}
  35. μ a ( A T μ b A ) \mu_{a}\cap(A^{T}\mu_{b}A)
  36. μ a ( A T μ b A ) \mu_{a}\cup(A^{T}\mu_{b}A)
  37. μ a \mu_{a}
  38. ϵ \epsilon
  39. ϵ 0 \epsilon_{0}
  40. S i S_{i}
  41. S i = Total number of matches Total number of detected regions = N M i N ( M - 1 ) Si=\frac{\,\text{Total number of matches}}{\,\text{Total number of detected % regions}}=\frac{N_{M}^{i}}{N(M-1)}
  42. S i S_{i}

Kaempferol_3-O-galactosyltransferase.html

  1. \rightleftharpoons

Kanamycin_kinase.html

  1. \rightleftharpoons

Kernelization.html

  1. G G
  2. k k
  3. k k
  4. k > 0 k>0
  5. v v
  6. k k
  7. v v
  8. k k
  9. k k
  10. v v
  11. v v
  12. v v
  13. v v
  14. k 2 k^{2}
  15. k k
  16. k k
  17. k k
  18. k k
  19. k 2 k^{2}
  20. k = 0 k=0
  21. k 2 k^{2}
  22. 2 k 2 2k^{2}
  23. O ( 2 2 k 2 + n + m ) O(2^{2k^{2}}+n+m)
  24. n n
  25. m m
  26. k k
  27. n n
  28. m m
  29. 2 k 2k
  30. 2 k - c log k 2k-c\log k
  31. c c
  32. O ( log k ) O(\log k)
  33. \subseteq
  34. ϵ > 0 \epsilon>0
  35. O ( k 2 - ϵ ) O(k^{2-\epsilon})
  36. ( 2 - ϵ ) k (2-\epsilon)k
  37. ϵ > 0 \epsilon>0
  38. L Σ * × 𝒩 L\subseteq\Sigma^{*}\times\mathcal{N}
  39. L L
  40. ( x , k ) (x,k)
  41. | x | |x|
  42. k k
  43. ( x , k ) (x^{\prime},k^{\prime})
  44. ( x , k ) (x,k)
  45. L L
  46. ( x , k ) (x^{\prime},k^{\prime})
  47. L L
  48. x x^{\prime}
  49. f f
  50. k k
  51. k k^{\prime}
  52. k k
  53. ( x , k ) (x^{\prime},k^{\prime})
  54. x x^{\prime}
  55. L Σ * L\subseteq\Sigma^{*}
  56. κ : Σ * 𝒩 \kappa:\Sigma^{*}\to\mathcal{N}
  57. x x
  58. κ ( x ) \kappa(x)
  59. L L
  60. x x
  61. k k
  62. y y
  63. x x
  64. L L
  65. y y
  66. L L
  67. y y
  68. f f
  69. k k
  70. y y
  71. y y
  72. k k
  73. f f
  74. f = k O ( 1 ) f=k^{O(1)}
  75. L L
  76. f = O ( k ) f={O(k)}
  77. O ( | x | c ) O(|x|^{c})
  78. f ( k ) f(k)
  79. g ( f ( k ) ) + O ( | x | c ) g(f(k))+O(|x|^{c})
  80. g ( n ) g(n)
  81. g ( f ( k ) ) g(f(k))
  82. f ( k ) f(k)
  83. f ( k ) f(k)
  84. I y e s I_{yes}
  85. I n o I_{no}
  86. f ( k ) | x | c f(k)\cdot|x|^{c}
  87. ( x , k ) (x,k)
  88. c c
  89. f ( k ) f(k)
  90. | x | c + 1 |x|^{c+1}
  91. I y e s I_{yes}
  92. I n o I_{no}
  93. | x | c + 1 |x|^{c+1}
  94. ( x , k ) (x,k)
  95. ( x , k ) (x,k)
  96. f ( k ) | x | c > | x | c + 1 f(k)\cdot|x|^{c}>|x|^{c+1}
  97. max { | I y e s | , | I n o | , f ( k ) } \max\{|I_{yes}|,|I_{no}|,f(k)\}
  98. f ( k ) f(k)
  99. 2 k 2k
  100. O ( k 2 ) O(k^{2})
  101. ε > 0 \varepsilon>0
  102. O ( k 2 - ε ) O(k^{2-\varepsilon})
  103. coNP NP/poly \,\text{coNP}\subseteq\,\text{NP/poly}
  104. d d
  105. O ( k d ) O(k^{d})
  106. O ( k d - ε ) O(k^{d-\varepsilon})
  107. coNP NP/poly \,\text{coNP}\subseteq\,\text{NP/poly}
  108. 4 k 2 4k^{2}
  109. O ( k 2 ) O(k^{2})
  110. O ( k 2 - ε ) O(k^{2-\varepsilon})
  111. coNP NP/poly \,\text{coNP}\subseteq\,\text{NP/poly}
  112. k k
  113. k k
  114. k k
  115. coNP NP/poly \,\text{coNP}\subseteq\,\text{NP/poly}
  116. 3 k 3k
  117. 2 k 2k