wpmath0000011_2

Kievitone_hydratase.html

  1. \rightleftharpoons

Kinetic_chain_length.html

  1. v = R p R i = R p R t v=\frac{R_{p}}{R_{i}}=\frac{R_{p}}{R_{t}}
  2. v = [ M ] 0 - [ M ] [ I ] 0 \ v=\frac{[M]_{0}-[M]}{[I]_{0}}
  3. v = D P n \ v=DP_{n}
  4. v = D P n 2 \ v=\frac{DP_{n}}{2}
  5. v t r = R p R t + R t r \ v_{tr}=\frac{R_{p}}{R_{t}+R_{tr}}

Knaster–Kuratowski_fan.html

  1. C C
  2. p p
  3. ( 1 2 , 1 2 ) 2 (\tfrac{1}{2},\tfrac{1}{2})\in\mathbb{R}^{2}
  4. L ( c ) L(c)
  5. c C c\in C
  6. ( c , 0 ) (c,0)
  7. p p
  8. c C c\in C
  9. X c = { ( x , y ) L ( c ) : y } X_{c}=\{(x,y)\in L(c):y\in\mathbb{Q}\}
  10. C C
  11. X c = { ( x , y ) L ( c ) : y } X_{c}=\{(x,y)\in L(c):y\notin\mathbb{Q}\}
  12. c C X c \bigcup_{c\in C}X_{c}
  13. 2 \mathbb{R}^{2}
  14. p p

Kojibiose_phosphorylase.html

  1. \rightleftharpoons

Korn's_inequality.html

  1. Ω Ω
  2. n n
  3. n 2 n≥ 2
  4. Ω Ω
  5. v H 1 ( Ω ) := ( Ω i = 1 n | v i ( x ) | 2 d x + Ω i , j = 1 n | j v i ( x ) | 2 d x ) 1 / 2 . \|v\|_{H^{1}(\Omega)}:=\left(\int_{\Omega}\sum_{i=1}^{n}|v^{i}(x)|^{2}\,% \mathrm{d}x+\int_{\Omega}\sum_{i,j=1}^{n}|\partial_{j}v^{i}(x)|^{2}\,\mathrm{d% }x\right)^{1/2}.
  6. C 0 C≥ 0
  7. Ω Ω
  8. v H 1 ( Ω ) 2 C Ω i , j = 1 n ( | v i ( x ) | 2 + | ( e i j v ) ( x ) | 2 ) d x \|v\|_{H^{1}(\Omega)}^{2}\leq C\int_{\Omega}\sum_{i,j=1}^{n}\left(|v^{i}(x)|^{% 2}+|(e_{ij}v)(x)|^{2}\right)\,\mathrm{d}x
  9. e e
  10. e i j v = 1 2 ( i v j + j v i ) . e_{ij}v=\frac{1}{2}(\partial_{i}v^{j}+\partial_{j}v^{i}).

Kynurenine—glyoxylate_transaminase.html

  1. \rightleftharpoons

Kynurenine—oxoglutarate_transaminase.html

  1. \rightleftharpoons

L(+)-tartrate_dehydratase.html

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L,L-diaminopimelate_aminotransferase.html

  1. \rightleftharpoons

L-amino-acid_alpha-ligase.html

  1. \rightleftharpoons

L-arabinokinase.html

  1. \rightleftharpoons

L-arabinonate_dehydratase.html

  1. \rightleftharpoons

L-arabinonolactonase.html

  1. \rightleftharpoons

L-fuconate_dehydratase.html

  1. \rightleftharpoons

L-Fuculokinase.html

  1. \rightleftharpoons

L-lysine-lactamase.html

  1. \rightleftharpoons

L-lysine_6-transaminase.html

  1. \rightleftharpoons

L-mimosine_synthase.html

  1. \rightleftharpoons

L-rhamnonate_dehydratase.html

  1. \rightleftharpoons

L-rhamnono-1,4-lactonase.html

  1. \rightleftharpoons

L-xylulokinase.html

  1. \rightleftharpoons

LabelMe.html

  1. I O \mathrm{I}_{\mathrm{O}}\,
  2. I P \mathrm{I}_{\mathrm{P}}\,
  3. S O , P \mathrm{S}_{\mathrm{O},\mathrm{P}}\,
  4. A ( O P ) A ( P ) \frac{\mathrm{A}(\mathrm{O}\cap\mathrm{P})}{\mathrm{A}(\mathrm{P})}\,
  5. I O , P I P \mathrm{I}_{\mathrm{O},\mathrm{P}}\subseteq\mathrm{I}_{\mathrm{P}}\,
  6. S O , P > β \mathrm{S}_{\mathrm{O},\mathrm{P}}>\beta\,
  7. β \beta\,
  8. β = 0.5 \beta=0.5\,
  9. N O , P N P + α \frac{\mathrm{N}_{\mathrm{O},\mathrm{P}}}{\mathrm{N}_{\mathrm{P}}+\alpha}\,
  10. N O , P \mathrm{N}_{\mathrm{O},\mathrm{P}}\,
  11. N P \mathrm{N}_{\mathrm{P}}\,
  12. I O , P \mathrm{I}_{\mathrm{O},\mathrm{P}}\,
  13. I P \mathrm{I}_{\mathrm{P}}\,
  14. α \alpha\,
  15. α = 5 \alpha=5\,

Lactosylceramide_1,3-N-acetyl-beta-D-glucosaminyltransferase.html

  1. \rightleftharpoons

Lactosylceramide_4-alpha-galactosyltransferase.html

  1. \rightleftharpoons

Lactosylceramide_alpha-2,3-sialyltransferase.html

  1. \rightleftharpoons

Lactosylceramide_alpha-2,6-N-sialyltransferase.html

  1. \rightleftharpoons

Lactosylceramide_beta-1,3-galactosyltransferase.html

  1. \rightleftharpoons

Lactoyl-CoA_dehydratase.html

  1. \rightleftharpoons

Lagrange_invariant.html

  1. H = n u ¯ y - n u y ¯ H=n\overline{u}y-nu\overline{y}

Lamb_waves.html

  1. d / λ = 0.6 d/\lambda=0.6
  2. d / λ = 0.3 d/\lambda=0.3
  3. ξ = A x f x ( z ) e i ( ω t - k x ) ( 1 ) \xi=A_{x}f_{x}(z)e^{i(\omega t-kx)}\quad\quad(1)
  4. ζ = A z f z ( z ) e i ( ω t - k x ) ( 2 ) \zeta=A_{z}f_{z}(z)e^{i(\omega t-kx)}\quad\quad(2)
  5. tan ( β d / 2 ) tan ( α d / 2 ) = - 4 α β k 2 ( k 2 - β 2 ) 2 ( 3 ) \frac{\tan(\beta d/2)}{\tan(\alpha d/2)}=-\frac{4\alpha\beta k^{2}}{(k^{2}-% \beta^{2})^{2}}\ \quad\quad\quad\quad(3)
  6. tan ( β d / 2 ) tan ( α d / 2 ) = - ( k 2 - β 2 ) 2 4 α β k 2 ( 4 ) \frac{\tan(\beta d/2)}{\tan(\alpha d/2)}=-\frac{(k^{2}-\beta^{2})^{2}}{4\alpha% \beta k^{2}}\ \quad\quad\quad\quad(4)
  7. α 2 = ω 2 c l 2 - k 2 and β 2 = ω 2 c t 2 - k 2 . \alpha^{2}=\frac{\omega^{2}}{c_{l}^{2}}-k^{2}\quad\quad\,\text{and}\quad\quad% \beta^{2}=\frac{\omega^{2}}{c_{t}^{2}}-k^{2}.
  8. λ \lambda
  9. f d = d c λ , f\cdot d=\frac{d\cdot c}{\lambda},
  10. c = f λ . c=f\lambda.
  11. ω \omega
  12. d / λ d/\lambda
  13. d d
  14. a z a x = π ν ( 1 - ν ) . d λ \frac{a_{z}}{a_{x}}=\frac{\pi\nu}{(1-\nu)}.\frac{d}{\lambda}
  15. ν \nu
  16. d = n λ 2 or f = n c 2 d d=\frac{n\lambda}{2}\quad\quad\,\text{or}\quad\quad f=\frac{nc}{2d}

Laminaribiose_phosphorylase.html

  1. \rightleftharpoons

Landau–Zener_formula.html

  1. Δ E = E 2 ( t ) - E 1 ( t ) α t , \Delta E=E_{2}(t)-E_{1}(t)\equiv\alpha t,\,
  2. E 1 ( t ) \scriptstyle{E_{1}(t)}
  3. E 2 ( t ) \scriptstyle{E_{2}(t)}
  4. t \scriptstyle{t}
  5. α \scriptstyle{\alpha}
  6. 1 / r \scriptstyle{1/r}
  7. v L Z = t | E 2 - E 1 | q | E 2 - E 1 | d q d t , v_{LZ}={\frac{\partial}{\partial t}|E_{2}-E_{1}|\over\frac{\partial}{\partial q% }|E_{2}-E_{1}|}\approx\frac{dq}{dt},
  8. q \scriptstyle{q}
  9. E 1 \scriptstyle{E_{1}}
  10. E 2 \scriptstyle{E_{2}}
  11. v L Z \scriptstyle{v_{LZ}}
  12. P D \scriptstyle{P_{D}}
  13. P D = e - 2 π Γ Γ = a 2 / | t ( E 2 - E 1 ) | = a 2 / | d q d t q ( E 2 - E 1 ) | = a 2 | α | \begin{aligned}\displaystyle P_{D}&\displaystyle=e^{-2\pi\Gamma}\\ \displaystyle\Gamma&\displaystyle={a^{2}/\hbar\over\left|\frac{\partial}{% \partial t}(E_{2}-E_{1})\right|}={a^{2}/\hbar\over\left|\frac{dq}{dt}\frac{% \partial}{\partial q}(E_{2}-E_{1})\right|}\\ &\displaystyle={a^{2}\over\hbar|\alpha|}\end{aligned}
  14. a a
  15. E 1 = E 2 E_{1}=E_{2}

Laplace_operators_in_differential_geometry.html

  1. Δ T = - tr 2 T , \Delta T=-\,\text{tr}\;\nabla^{2}T,
  2. \nabla
  3. X , Y 2 T = - ( X Y T - X Y T ) . \nabla^{2}_{X,Y}T=-\left(\nabla_{X}\nabla_{Y}T-\nabla_{\nabla_{X}Y}T\right).
  4. Δ ϕ = | g | - 1 / 2 μ ( | g | 1 / 2 g μ ν ν ) ϕ \Delta\phi=|g|^{-1/2}\partial_{\mu}\left(|g|^{1/2}g^{\mu\nu}\partial_{\nu}% \right)\phi
  5. ϕ \phi
  6. | g | |g|
  7. g μ ν g^{\mu\nu}
  8. Δ = d δ + δ d = ( d + δ ) 2 , \Delta=\mathrm{d}\delta+\delta\mathrm{d}=(\mathrm{d}+\delta)^{2},\;
  9. \nabla
  10. : Γ ( E ) Γ ( T * M E ) \nabla:\Gamma(E)\rightarrow\Gamma(T^{*}M\otimes E)
  11. Γ ( E ) \Gamma(E)
  12. L 2 L^{2}
  13. \nabla
  14. * : Γ ( T * M E ) Γ ( E ) . \nabla^{*}:\Gamma(T^{*}M\otimes E)\rightarrow\Gamma(E).
  15. Δ = * \Delta=\nabla^{*}\nabla
  16. * = - tr 2 \nabla^{*}\nabla=-\,\text{tr}\,\nabla^{2}
  17. : Γ ( Sym k ( T M ) ) Γ ( Sym k + 1 ( T M ) ) \nabla:\Gamma(\operatorname{Sym}^{k}(TM))\to\Gamma(\operatorname{Sym}^{k+1}(TM))
  18. Δ L = * \Delta_{L}=\nabla^{*}\nabla
  19. * \nabla^{*}
  20. L u = - 4 n - 1 n - 2 Δ u + R u , Lu=-4\frac{n-1}{n-2}\Delta u+Ru,
  21. g ~ = u 4 / ( n - 2 ) g \tilde{g}=u^{4/(n-2)}g\,
  22. R ~ = u - ( n + 2 ) / ( n - 2 ) L u . \tilde{R}=u^{-(n+2)/(n-2)}Lu.\,

Laser_beam_profiler.html

  1. σ 2 ( z ) = σ 0 2 + M 4 ( λ π σ 0 ) 2 ( z - z 0 ) 2 \sigma^{2}(z)=\sigma_{0}^{2}+M^{4}\left(\frac{\lambda}{\pi\sigma_{0}}\right)^{% 2}(z-z_{0})^{2}
  2. σ 2 ( z ) \sigma^{2}(z)
  3. z 0 z_{0}
  4. 2 σ 0 2\sigma_{0}
  5. z 0 z_{0}
  6. σ 0 \sigma_{0}
  7. N \sqrt{N}
  8. N N

Lattice_sieving.html

  1. f ( a , b ) f(a,b)
  2. p [ α ] + ( u + v α ) [ α ] p\mathbb{Z}[\alpha]+(u+v\alpha)\mathbb{Z}[\alpha]
  3. G F ( q ) GF(q)
  4. 𝔮 \mathfrak{q}
  5. 𝐱 , 𝐲 \mathbf{x},\mathbf{y}
  6. 𝔮 \mathfrak{q}
  7. 𝔭 \mathfrak{p}
  8. 𝐱 𝔭 , 𝐲 𝔭 \mathbf{x}_{\mathfrak{p}},\mathbf{y}_{\mathfrak{p}}
  9. 𝔭 𝔮 \mathfrak{pq}
  10. log | 𝔭 | \log|\mathfrak{p}|

Laughlin_wavefunction.html

  1. ν = 1 / n \nu=1/n
  2. n n
  3. ν = 1 / 3 \nu=1/3
  4. ν = 1 / n \nu=1/n
  5. e / n e/n
  6. ψ 0 \psi_{0}
  7. z 1 , z 2 , z 3 , , z N n , N = ψ n , N ( z 1 , z 2 , z 3 , , z N ) = D [ N i > j 1 ( z i - z j ) n ] k = 1 N exp ( - z k 2 ) \langle z_{1},z_{2},z_{3},\ldots,z_{N}\mid n,N\rangle=\psi_{n,N}(z_{1},z_{2},z% _{3},\ldots,z_{N})=D\left[\prod_{N\geqslant i>j\geqslant 1}\left(z_{i}-z_{j}% \right)^{n}\right]\prod^{N}_{k=1}\exp\left(-\mid z_{k}\mid^{2}\right)
  8. z = 1 2 l B ( x + i y ) z={1\over 2\mathit{l}_{B}}\left(x+iy\right)
  9. D = [ 1 2 π ( 2 ) 2 n n ! ( N - 1 ) n + 1 ] N ( N - 1 ) 2 D=\left[{1\over 2\pi\;\left(\sqrt{2}\right)^{2n}\;\sqrt{n!}\;\left(\sqrt{N-1}% \right)^{n+1}}\right]^{N\left(N-1\right)\over 2}
  10. l B = c e B \mathit{l}_{B}=\sqrt{\hbar c\over eB}
  11. x x
  12. y y
  13. \hbar
  14. e e
  15. N N
  16. B B
  17. n n\hbar
  18. V = n , N V n , N , N = 2 \langle V\rangle=\langle n,N\mid V\mid n,N\rangle,\;\;\;N=2
  19. V ( r 12 ) = ( 2 e 2 L B ) 0 k d k k 2 + k B 2 r B 2 M ( l + 1 , 1 , - k 2 4 ) M ( l + 1 , 1 , - k 2 4 ) 𝒥 0 ( k r 12 r B ) V\left(r_{12}\right)=\left({2e^{2}\over L_{B}}\right)\int_{0}^{\infty}{{k\;dk% \;}\over k^{2}+k_{B}^{2}r_{B}^{2}}\;M\left(\mathit{l}+1,1,-{k^{2}\over 4}% \right)\;M\left(\mathit{l}^{\prime}+1,1,-{k^{2}\over 4}\right)\;\mathcal{J}_{0% }\left(k{r_{12}\over r_{B}}\right)
  20. M M
  21. 𝒥 0 \mathcal{J}_{0}
  22. r 12 r_{12}
  23. e e
  24. r B = 2 l B r_{B}=\sqrt{2}\mathit{l}_{B}
  25. L B L_{B}
  26. l \mathit{l}\hbar
  27. l \mathit{l}^{\prime}\hbar
  28. l + l = n \mathit{l}+\mathit{l}^{\prime}=n
  29. k B 2 = 4 π e 2 ω c A L B k_{B}^{2}={4\pi e^{2}\over\hbar\omega_{c}AL_{B}}
  30. ω c \omega_{c}
  31. A A
  32. E = ( 2 e 2 L B ) 0 k d k k 2 + k B 2 r B 2 M ( l + 1 , 1 , - k 2 4 ) M ( l + 1 , 1 , - k 2 4 ) M ( n + 1 , 1 , - k 2 2 ) E=\left({2e^{2}\over L_{B}}\right)\int_{0}^{\infty}{{k\;dk\;}\over k^{2}+k_{B}% ^{2}r_{B}^{2}}\;M\left(\mathit{l}+1,1,-{k^{2}\over 4}\right)\;M\left(\mathit{l% }^{\prime}+1,1,-{k^{2}\over 4}\right)\;M\left(n+1,1,-{k^{2}\over 2}\right)
  33. n {n}
  34. l n = 1 2 ± 1 2 n {\mathit{l}\over n}={1\over 2}\pm{1\over 2n}
  35. k B r B = 0.1 , 1.0 , 10 k_{B}r_{B}=0.1,1.0,10
  36. e 2 L B {e^{2}\over L_{B}}
  37. u 12 = z 1 - z 2 2 u_{12}={z_{1}-z_{2}\over\sqrt{2}}
  38. v 12 = z 1 + z 2 2 v_{12}={z_{1}+z_{2}\over\sqrt{2}}
  39. 1 ( 2 π ) 2 2 2 n n ! d 2 z 1 d 2 z 2 z 1 - z 2 2 n exp [ - 2 ( z 1 2 + z 2 2 ) ] 𝒥 0 ( 2 k z 1 - z 2 ) = {1\over\left(2\pi\right)^{2}\;2^{2n}\;n!}\int d^{2}z_{1}\;d^{2}z_{2}\;\mid z_{% 1}-z_{2}\mid^{2n}\;\exp\left[-2\left(\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}% \right)\right]\;\mathcal{J}_{0}\left(\sqrt{2}\;{k\mid z_{1}-z_{2}\mid}\right)=
  40. 1 ( 2 π ) 2 2 n n ! d 2 u 12 d 2 v 12 u 12 2 n exp [ - 2 ( u 12 2 + v 12 2 ) ] 𝒥 0 ( 2 k u 12 ) = {1\over\left(2\pi\right)^{2}\;2^{n}\;n!}\int d^{2}u_{12}\;d^{2}v_{12}\;\mid u_% {12}\mid^{2n}\;\exp\left[-2\left(\mid u_{12}\mid^{2}+\mid v_{12}\mid^{2}\right% )\right]\;\mathcal{J}_{0}\left({2}k\mid u_{12}\mid\right)=
  41. M ( n + 1 , 1 , - k 2 2 ) . M\left(n+1,1,-{k^{2}\over 2}\right).
  42. l n = 1 3 , 2 5 , 3 7 , etc., {\mathit{l}\over n}={1\over 3},{2\over 5},{3\over 7},\mbox{etc.,}~{}
  43. l n = 2 3 , 3 5 , 4 7 , etc. {\mathit{l}\over n}={2\over 3},{3\over 5},{4\over 7},\mbox{etc.}~{}
  44. n n

Lavandulyl_diphosphate_synthase.html

  1. \rightleftharpoons

Lead-lead_dating.html

  1. ( P 207 b P 204 b ) P = ( P 207 b P 204 b ) I + ( U 235 P 204 b ) P ( e λ 235 t - 1 ) {\left(\frac{{}^{207}Pb}{{}^{204}Pb}\right)_{P}}={\left(\frac{{}^{207}Pb}{{}^{% 204}Pb}\right)_{I}}+{\left(\frac{{}^{235}U}{{}^{204}Pb}\right)_{P}}{\left({e^{% \lambda_{235}t}-1}\right)}
  2. ( P 206 b P 204 b ) P = ( P 206 b P 204 b ) I + ( U 238 P 204 b ) P ( e λ 238 t - 1 ) {\left(\frac{{}^{206}Pb}{{}^{204}Pb}\right)_{P}}={\left(\frac{{}^{206}Pb}{{}^{% 204}Pb}\right)_{I}}+{\left(\frac{{}^{238}U}{{}^{204}Pb}\right)_{P}}{\left({e^{% \lambda_{238}t}-1}\right)}
  3. [ ( P 207 b P 204 b ) P - ( P 207 b P 204 b ) I ( P 206 b P 204 b ) P - ( P 206 b P 204 b ) I ] = ( 1 137.88 ) ( e λ 235 t - 1 e λ 238 t - 1 ) \left[\frac{\left(\frac{{}^{207}Pb}{{}^{204}Pb}\right)_{P}-\left(\frac{{}^{207% }Pb}{{}^{204}Pb}\right)_{I}}{\left(\frac{{}^{206}Pb}{{}^{204}Pb}\right)_{P}-% \left(\frac{{}^{206}Pb}{{}^{204}Pb}\right)_{I}}\right]={\left(\frac{1}{137.88}% \right)}{\left(\frac{e^{\lambda_{235}t}-1}{e^{\lambda_{238}t}-1}\right)}

Leapfrog_integration.html

  1. x ¨ = d 2 x / d t 2 = F ( x ) \ddot{x}=d^{2}x/dt^{2}=F(x)
  2. v ˙ = d v / d t = F ( x ) , x ˙ = d x / d t = v \dot{v}=dv/dt=F(x),\;\dot{x}=dx/dt=v
  3. x ( t ) x(t)
  4. v ( t ) = x ˙ ( t ) v(t)=\dot{x}(t)
  5. Δ t \Delta t
  6. Δ t 2 / ω \Delta t\leq 2/\omega
  7. x i = x i - 1 + v i - 1 / 2 Δ t , a i = F ( x i ) v i + 1 / 2 = v i - 1 / 2 + a i Δ t , \begin{aligned}\displaystyle x_{i}&\displaystyle=x_{i-1}+v_{i-1/2}\,\Delta t,% \\ \displaystyle a_{i}&\displaystyle=F(x_{i})\\ \displaystyle v_{i+1/2}&\displaystyle=v_{i-1/2}+a_{i}\,\Delta t,\end{aligned}
  8. x i x_{i}
  9. i i
  10. v i + 1 / 2 v_{i+1/2\,}
  11. x x
  12. i + 1 / 2 i+1/2\,
  13. a i = F ( x i ) a_{i}=F(x_{i})
  14. x x
  15. i i
  16. Δ t \Delta t
  17. Δ t \Delta t
  18. x i + 1 = x i + v i Δ t + 1 2 a i Δ t 2 , v i + 1 = v i + 1 2 ( a i + a i + 1 ) Δ t . \begin{aligned}\displaystyle x_{i+1}&\displaystyle=x_{i}+v_{i}\,\Delta t+% \tfrac{1}{2}\,a_{i}\,\Delta t^{\,2},\\ \displaystyle v_{i+1}&\displaystyle=v_{i}+\tfrac{1}{2}\,(a_{i}+a_{i+1})\,% \Delta t.\end{aligned}

Leonardo_number.html

  1. L ( n ) = { 1 if n = 0 1 if n = 1 L ( n - 1 ) + L ( n - 2 ) + 1 if n > 1 L(n)=\begin{cases}1&\mbox{if }~{}n=0\\ 1&\mbox{if }~{}n=1\\ L(n-1)+L(n-2)+1&\mbox{if }~{}n>1\\ \end{cases}
  2. L ( n ) = 2 F ( n + 1 ) - 1 , n 0 L(n)=2F(n+1)-1,n\geq 0
  3. L ( n ) = 2 φ n + 1 - ψ n + 1 φ - ψ - 1 = 2 5 ( φ n + 1 - ψ n + 1 ) - 1 = 2 F ( n + 1 ) - 1 L(n)=2\frac{\varphi^{n+1}-\psi^{n+1}}{\varphi-\psi}-1=\frac{2}{\sqrt{5}}\left(% \varphi^{n+1}-\psi^{n+1}\right)-1=2F(n+1)-1
  4. φ = ( 1 + 5 ) / 2 \varphi=\left(1+\sqrt{5}\right)/2
  5. ψ = ( 1 - 5 ) / 2 \psi=\left(1-\sqrt{5}\right)/2
  6. x 2 - x - 1 = 0 x^{2}-x-1=0
  7. 1 , 1 , 3 , 5 , 9 , 15 , 25 , 41 , 67 , 109 , 177 , 287 , 465 , 753 , 1219 , 1973 , 3193 , 5167 , 8361 , 1,\;1,\;3,\;5,\;9,\;15,\;25,\;41,\;67,\;109,\;177,\;287,\;465,\;753,\;1219,\;1% 973,\;3193,\;5167,\;8361,\ldots

Lester's_theorem.html

  1. X 13 , X 14 X_{13},X_{14}
  2. X 5 X_{5}
  3. X 3 X_{3}
  4. H H
  5. G G
  6. S S
  7. F + F_{+}
  8. F - F_{-}
  9. S S
  10. S S
  11. H G HG
  12. K + K_{+}
  13. K - K_{-}
  14. E E
  15. H G HG
  16. K + K - K_{+}K_{-}
  17. H G HG
  18. D D
  19. D E DE
  20. G + G_{+}
  21. G - G_{-}
  22. F + , F - , E , F , G + , G - F_{+},F_{-},E,F,G_{+},G_{-}
  23. S S
  24. F + , F - F_{+},F_{-}
  25. K + , K - K_{+},K_{-}
  26. H , G H,G
  27. P , Q ( P ) , X 13 , X 14 P,Q(P),X_{13},X_{14}

Leucine_N-acetyltransferase.html

  1. \rightleftharpoons

Leucine_transaminase.html

  1. \rightleftharpoons

Leucine—tRNA_ligase.html

  1. \rightleftharpoons

Leucyltransferase.html

  1. \rightleftharpoons

Levansucrase.html

  1. \rightleftharpoons

Lexical_density.html

  1. L d = ( N lex / N ) × 100 L_{d}=(N_{\mathrm{lex}}/N)\times 100
  2. L d L_{d}
  3. N lex N_{\mathrm{lex}}
  4. N N

Limonin-D-ring-lactonase.html

  1. \rightleftharpoons

Limonoid_glucosyltransferase.html

  1. \rightleftharpoons

Linamarin_synthase.html

  1. \rightleftharpoons

Lindemann_mechanism.html

  1. A + M A * + M A+M\rightleftharpoons A^{*}+M
  2. A * P A^{*}\rightarrow P
  3. d [ A * ] d t = k 1 [ A ] [ M ] \frac{d[A^{*}]}{dt}=k_{1}[A][M]
  4. - d [ A * ] d t = k - 1 [ A * ] [ M ] \frac{-d[A^{*}]}{dt}=k_{-1}[A^{*}][M]
  5. - d [ A * ] d t = k 2 [ A * ] \frac{-d[A^{*}]}{dt}=k_{2}[A^{*}]
  6. k 1 [ A ] [ M ] = k - 1 [ A * ] [ M ] + k 2 [ A * ] k_{1}[A][M]=k_{-1}[A^{*}][M]+k_{2}[A^{*}]
  7. [ A * ] [A^{*}]
  8. [ A * ] = k 1 [ A ] [ M ] k - 1 [ M ] + k 2 [A^{*}]=\frac{k_{1}[A][M]}{k_{-1}[M]+k_{2}}
  9. d [ P ] d t = k 2 [ A * ] \frac{d[P]}{dt}=k_{2}[A^{*}]
  10. d [ P ] d t = k 1 k 2 [ A ] [ M ] k - 1 [ M ] + k 2 \frac{d[P]}{dt}=\frac{k_{1}k_{2}[A][M]}{k_{-1}[M]+k_{2}}
  11. k - 1 [ M ] k 2 k_{-1}[M]\ll k_{2}
  12. d [ P ] / d t = k 1 [ A ] [ M ] d[P]/dt=k_{1}[A][M]
  13. k - 1 [ M ] k 2 k_{-1}[M]\gg k_{2}
  14. d [ P ] d t = k 1 k 2 k - 1 [ A ] \frac{d[P]}{dt}=\frac{k_{1}k_{2}}{k_{-1}}[A]
  15. k u n i k_{uni}
  16. d [ P ] d t = k u n i [ A ] , k u n i = 1 [ A ] d [ P ] d t \frac{d[P]}{dt}=k_{uni}[A],\quad k_{uni}=\frac{1}{[A]}\frac{d[P]}{dt}
  17. 1 / k 1/k
  18. 1 / p 1/p
  19. k k
  20. 1 / k 1/k
  21. 1 / p 1/p
  22. R a t e = k 2 [ N 2 O 5 ] * = k 1 k 2 [ N 2 O 5 ] 2 k - 1 [ N 2 O 5 ] + k 2 Rate=k_{2}[N_{2}O_{5}]^{*}=\frac{k_{1}k_{2}[N_{2}O_{5}]^{2}}{k_{-1}[N_{2}O_{5}% ]+k_{2}}
  23. k 2 k - 1 k_{2}\gg k_{-1}
  24. \gg
  25. k - 1 [ N 2 O 5 ] + k 2 k 2 k_{-1}[N_{2}O_{5}]+k_{2}\simeq k_{2}
  26. R a t e = k 1 [ N 2 O 5 ] 2 Rate=k_{1}[N_{2}O_{5}]^{2}
  27. k 2 k - 1 k_{2}\ll k_{-1}
  28. \ll
  29. k - 1 [ N 2 O 5 ] + k 2 k - 1 [ N 2 O 5 ] k_{-1}[N_{2}O_{5}]+k_{2}\simeq k_{-1}[N_{2}O_{5}]
  30. R a t e = k 1 k 2 [ N 2 O 5 ] k - 1 Rate=\frac{k_{1}k_{2}[N_{2}O_{5}]}{k_{-1}}

Lipid-A-disaccharide_synthase.html

  1. \rightleftharpoons

Lipid-phosphate_phosphatase.html

  1. \rightleftharpoons

Lipopolysaccharide-transporting_ATPase.html

  1. \rightleftharpoons

Lipopolysaccharide_3-alpha-galactosyltransferase.html

  1. \rightleftharpoons

Lipopolysaccharide_glucosyltransferase_I.html

  1. \rightleftharpoons

Lipopolysaccharide_glucosyltransferase_II.html

  1. \rightleftharpoons

Lipopolysaccharide_N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Lipopolysaccharide_N-acetylmannosaminouronosyltransferase.html

  1. \rightleftharpoons

Lipoyl(octanoyl)_transferase.html

  1. \rightleftharpoons

List_of_Atlanta_Hawks_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_Dallas_Mavericks_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_forcing_notions.html

  1. \in
  2. \in
  3. \in
  4. \in
  5. \in
  6. \in
  7. Π 1 0 \Pi^{0}_{1}
  8. Π 1 0 \Pi^{0}_{1}
  9. 2 ω 2^{\omega}
  10. 2 < ω 2^{<\omega}
  11. \in
  12. \leq
  13. \leq
  14. \in
  15. \leq
  16. \in
  17. \in
  18. T ω 2 < ω T\subseteq\omega_{2}^{<\omega}
  19. 2 \aleph_{2}
  20. 2 \aleph_{2}
  21. \cup
  22. \in
  23. \in
  24. \in
  25. \in
  26. ω 1 \omega_{1}
  27. P = { σ , C : σ P=\{\langle\sigma,C\rangle\,\colon\sigma
  28. ω 1 } \omega_{1}\}
  29. σ , C σ , C \langle\sigma^{\prime},C^{\prime}\rangle\leq\langle\sigma,C\rangle
  30. σ \sigma^{\prime}
  31. σ \sigma
  32. C C C^{\prime}\subseteq C
  33. σ σ C \sigma^{\prime}\subseteq\sigma\cup C
  34. V [ G ] V[G]
  35. { σ : ( C ) ( σ , C G ) } \bigcup\{\sigma\,\colon(\exists C)(\langle\sigma,C\rangle\in G)\}
  36. 1 \aleph_{1}
  37. ω 1 \omega_{1}
  38. V [ G ] V[G]
  39. G \bigcup G
  40. 1 \aleph_{1}
  41. ω 1 \omega_{1}
  42. p P p\in P
  43. α , β p \langle\alpha,\beta\rangle\in p
  44. α β \alpha\leq\beta
  45. α S \alpha\in S
  46. α , β \langle\alpha,\beta\rangle
  47. γ , δ \langle\gamma,\delta\rangle
  48. β < γ \beta<\gamma
  49. δ < α \delta<\alpha
  50. V [ G ] V[G]
  51. { α : ( β ) ( α , β G ) } \{\alpha\,\colon(\exists\beta)(\langle\alpha,\beta\rangle\in\bigcup G)\}

List_of_gear_nomenclature.html

  1. m n = m t cos β m_{n}=m_{t}\cos\beta\,
  2. d = N P d = p N π spur gears d=\frac{N}{P_{d}}=\frac{pN}{\pi}\qquad\,\text{spur gears}
  3. d = N P n d cos ψ helical gears d=\frac{N}{P_{nd}\cos\psi}\qquad\,\text{helical gears}
  4. N N
  5. p p
  6. P d P_{d}
  7. ψ \psi
  8. d = k m = z p π = z m n cos β d=km=\frac{zp}{\pi}=z\frac{m_{n}}{\cos\beta}
  9. D = N P d = N p π = N P n d cos ψ D=\frac{N}{P_{d}}=\frac{Np}{\pi}=\frac{N}{P_{nd}\cos\psi}

List_of_Houston_Rockets_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_Indiana_Pacers_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_Los_Angeles_Clippers_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_Miami_Heat_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_National_League_slugging_percentage_leaders.html

  1. S L G = ( 1 B ) + ( 2 × 2 B ) + ( 3 × 3 B ) + ( 4 × 𝐻𝑅 ) A B SLG=\frac{(\mathit{1B})+(2\times\mathit{2B})+(3\times\mathit{3B})+(4\times% \mathit{HR})}{AB}

List_of_New_York_Knicks_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_RNA_structure_prediction_software.html

  1. 3 s e q s 60 3\leq seqs\leq 60
  2. 40 \leq 40
  3. 2 s e q s 6 2\leq seqs\leq 6
  4. 50 \leq 50
  5. 2 s e q s 50 2\leq seqs\leq 50

List_of_Washington_Wizards_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

Lithium_cobalt_oxide.html

  1. R 3 ¯ m R\bar{3}m

Log-linear_model.html

  1. exp ( c + i w i f i ( X ) ) , \exp\left(c+\sum_{i}w_{i}f_{i}(X)\right)\,,
  2. f < s u b > i ( X ) f<sub>i(X)

Log-logistic_distribution.html

  1. α > 0 \alpha>0
  2. β > 0 \beta>0
  3. β > 1 \beta>1
  4. β \beta
  5. F ( x ; α , β ) = 1 1 + ( x / α ) - β = ( x / α ) β 1 + ( x / α ) β = x β α β + x β \begin{aligned}\displaystyle F(x;\alpha,\beta)&\displaystyle={1\over 1+(x/% \alpha)^{-\beta}}\\ &\displaystyle={(x/\alpha)^{\beta}\over 1+(x/\alpha)^{\beta}}\\ &\displaystyle={x^{\beta}\over\alpha^{\beta}+x^{\beta}}\end{aligned}
  6. x > 0 x>0
  7. α > 0 \alpha>0
  8. β > 0. \beta>0.
  9. f ( x ; α , β ) = ( β / α ) ( x / α ) β - 1 ( 1 + ( x / α ) β ) 2 f(x;\alpha,\beta)=\frac{(\beta/\alpha)(x/\alpha)^{\beta-1}}{\left(1+(x/\alpha)% ^{\beta}\right)^{2}}
  10. μ , s \mu,s
  11. μ = ln ( α ) \mu=\ln(\alpha)
  12. s = 1 / β s=1/\beta
  13. k k
  14. k < β , k<\beta,
  15. E ( X k ) = α k B ( 1 - k / β , 1 + k / β ) = α k k π / β sin ( k π / β ) \begin{aligned}\displaystyle\operatorname{E}(X^{k})&\displaystyle=\alpha^{k}\,% \operatorname{B}(1-k/\beta,\,1+k/\beta)\\ &\displaystyle=\alpha^{k}\,{k\,\pi/\beta\over\sin(k\,\pi/\beta)}\end{aligned}
  16. b = π / β b=\pi/\beta
  17. E ( X ) = α b / sin b , β > 1 , \operatorname{E}(X)=\alpha b/\sin b,\quad\beta>1,
  18. Var ( X ) = α 2 ( 2 b / sin 2 b - b 2 / sin 2 b ) , β > 2. \operatorname{Var}(X)=\alpha^{2}\left(2b/\sin 2b-b^{2}/\sin^{2}b\right),\quad% \beta>2.
  19. β \beta
  20. α \alpha
  21. F - 1 ( p ; α , β ) = α ( p 1 - p ) 1 / β . F^{-1}(p;\alpha,\beta)=\alpha\left(\frac{p}{1-p}\right)^{1/\beta}.
  22. α \alpha
  23. 3 1 / β α 3^{1/\beta}\alpha
  24. 3 - 1 / β α 3^{-1/\beta}\alpha
  25. α = 1 , \alpha=1,
  26. β \beta
  27. β > 1 , \beta>1,
  28. β \beta
  29. α \alpha
  30. α \alpha
  31. β \beta
  32. log ( α ) \log(\alpha)
  33. S ( t ) = 1 - F ( t ) = [ 1 + ( t / α ) β ] - 1 , S(t)=1-F(t)=[1+(t/\alpha)^{\beta}]^{-1},\,
  34. h ( t ) = f ( t ) S ( t ) = ( β / α ) ( t / α ) β - 1 1 + ( t / α ) β . h(t)=\frac{f(t)}{S(t)}=\frac{(\beta/\alpha)(t/\alpha)^{\beta-1}}{1+(t/\alpha)^% {\beta}}.
  35. 1 / β 1/\beta
  36. X L L ( α , β ) X\sim LL(\alpha,\beta)\,
  37. k X L L ( α , k β ) kX\sim LL(\alpha,k\beta)\,
  38. L L ( α , β ) Dagum ( 1 , α , β ) LL(\alpha,\beta)\sim\textrm{Dagum}(1,\alpha,\beta)\,
  39. L L ( α , β ) SinghMaddala ( 1 , α , β ) LL(\alpha,\beta)\sim\textrm{SinghMaddala}(1,\alpha,\beta)\,
  40. LL ( γ , σ ) β ( 1 , 1 , γ , σ ) \textrm{LL}(\gamma,\sigma)\sim\beta^{{}^{\prime}}(1,1,\gamma,\sigma)\,
  41. α \alpha
  42. β \beta
  43. log ( α ) \log(\alpha)
  44. 1 / β 1/\beta
  45. β \beta
  46. β \beta
  47. L L ( α , β ) L ( α , α / β ) . LL(\alpha,\beta)\to L(\alpha,\alpha/\beta).
  48. β = 1 \beta=1
  49. α \alpha
  50. μ = 0 \mu=0
  51. ξ = 1 \xi=1
  52. α : \alpha:
  53. L L ( α , 1 ) = G P D ( 1 , α , 1 ) . LL(\alpha,1)=GPD(1,\alpha,1).\,

Log-spectral_distance.html

  1. P ( ω ) P\left(\omega\right)
  2. P ^ ( ω ) \hat{P}\left(\omega\right)
  3. D L S = 1 2 π - π π [ 10 log 10 P ( ω ) P ^ ( ω ) ] 2 d ω , D_{LS}=\sqrt{\frac{1}{2\pi}\int_{-\pi}^{\pi}\left[10\log_{10}\frac{P(\omega)}{% \hat{P}(\omega)}\right]^{2}\,d\omega},
  4. P ( ω ) P\left(\omega\right)
  5. P ^ ( ω ) \hat{P}\left(\omega\right)

Log_amplifier.html

  1. V out = K ln V in V ref V_{\mathrm{out}}=K\ln\frac{V_{\mathrm{in}}}{V_{\mathrm{ref}}}
  2. V in V_{\,\text{in}}
  3. V out V_{\,\text{out}}
  4. V out = - V T ln ( V in I S R ) V_{\,\text{out}}=-V_{\,\text{T}}\ln\left(\frac{V_{\,\text{in}}}{I_{\,\text{S}}% \,R}\right)
  5. I S I_{\,\text{S}}
  6. V T V_{\,\text{T}}
  7. V BE = - V out V_{\mathrm{BE}}=-V_{\mathrm{out}}\,\!
  8. I C = I SO ( e V BE / V T - 1 ) I SO e V BE / V T I_{\mathrm{C}}=I_{\mathrm{SO}}(e^{V_{\mathrm{BE}}/V_{\mathrm{T}}}-1)\approx I_% {\mathrm{SO}}e^{V_{\mathrm{BE}}/V_{\mathrm{T}}}
  9. V BE = V T ln I C I SO \Rightarrow V_{\mathrm{BE}}=V_{\mathrm{T}}\ln\frac{I_{\mathrm{C}}}{I_{\mathrm{% SO}}}
  10. I SO I_{\mathrm{SO}}\,
  11. V T V_{\mathrm{T}}\,
  12. I C = V in R 1 I_{\mathrm{C}}=\frac{V_{\mathrm{in}}}{R_{1}}
  13. V out = - V T ln V in I SO R 1 V_{\mathrm{out}}=-V_{\mathrm{T}}\ln\frac{V_{\mathrm{in}}}{I_{\mathrm{SO}}R_{1}}
  14. I SO I_{\mathrm{SO}}\,
  15. V T V_{\mathrm{T}}\,

Logarithmic_decrement.html

  1. δ \delta
  2. δ = 1 n ln x ( t ) x ( t + n T ) , \delta=\frac{1}{n}\ln\frac{x(t)}{x(t+nT)},
  3. ζ = 1 1 + ( 2 π δ ) 2 . \zeta=\frac{1}{\sqrt{1+(\frac{2\pi}{\delta})^{2}}}.
  4. ω d = 2 π T , \omega_{d}=\frac{2\pi}{T},
  5. ω n = ω d 1 - ζ 2 , \omega_{n}=\frac{\omega_{d}}{\sqrt{1-\zeta^{2}}},
  6. ζ = 1 1 + ( 2 π ln ( x 0 / x 1 ) ) 2 , \zeta=\frac{1}{\sqrt{1+(\frac{2\pi}{\ln(x_{0}/x_{1})})^{2}}},
  7. O S = x p - x f x f , OS=\frac{x_{p}-x_{f}}{x_{f}},
  8. ζ = 1 1 + ( π ln O S ) 2 . \zeta=\frac{1}{\sqrt{1+(\frac{\pi}{\ln OS})^{2}}}.

Logarithmic_norm.html

  1. A A
  2. \|\cdot\|
  3. μ \mu
  4. A A
  5. μ ( A ) = lim h 0 + I + h A - 1 h \mu(A)=\lim\limits_{h\rightarrow 0^{+}}\frac{\|I+hA\|-1}{h}
  6. I I
  7. A A
  8. h h
  9. h 0 - h\rightarrow 0^{-}
  10. - μ ( - A ) -\mu(-A)
  11. μ ( A ) \mu(A)
  12. - μ ( - A ) μ ( A ) -\mu(-A)\leq\mu(A)
  13. A \|A\|
  14. A 0 A\neq 0
  15. μ ( A ) \mu(A)
  16. A A
  17. x ˙ = A x . \dot{x}=Ax.
  18. log x \log\|x\|
  19. μ ( A ) \mu(A)
  20. d d t + log x μ ( A ) , \frac{\mathrm{d}}{\mathrm{d}t^{+}}\log\|x\|\leq\mu(A),
  21. d / d t + \mathrm{d}/\mathrm{d}t^{+}
  22. d x d t + μ ( A ) x , \frac{\mathrm{d}\|x\|}{\mathrm{d}t^{+}}\leq\mu(A)\cdot\|x\|,
  23. μ ( A ) \mu(A)
  24. x x
  25. \real x , A x μ ( A ) x 2 \real\langle x,Ax\rangle\leq\mu(A)\cdot\|x\|^{2}
  26. A A
  27. A 2 = sup x 0 A x , A x x , x ; μ ( A ) = sup x 0 \real x , A x x , x \|A\|^{2}=\sup_{x\neq 0}{\frac{\langle Ax,Ax\rangle}{\langle x,x\rangle}}\,;% \qquad\mu(A)=\sup_{x\neq 0}{\frac{\real\langle x,Ax\rangle}{\langle x,x\rangle}}
  28. μ ( z I ) = \real ( z ) \mu(zI)=\real\,(z)
  29. μ ( A ) A \mu(A)\leq\|A\|
  30. μ ( γ A ) = γ μ ( A ) \mu(\gamma A)=\gamma\mu(A)\,
  31. γ > 0 \gamma>0
  32. μ ( A + z I ) = μ ( A ) + \real ( z ) \mu(A+zI)=\mu(A)+\real\,(z)
  33. μ ( A + B ) μ ( A ) + μ ( B ) \mu(A+B)\leq\mu(A)+\mu(B)
  34. α ( A ) μ ( A ) \alpha(A)\leq\mu(A)\,
  35. α ( A ) \alpha(A)
  36. A A
  37. e t A e t μ ( A ) \|\mathrm{e}^{tA}\|\leq\mathrm{e}^{t\mu(A)}\,
  38. t 0 t\geq 0
  39. μ ( A ) < 0 A - 1 - 1 / μ ( A ) \mu(A)<0\,\Rightarrow\,\|A^{-1}\|\leq-1/\mu(A)
  40. a i j a_{ij}
  41. i i
  42. j j
  43. A A
  44. μ 1 ( A ) = sup j ( \real ( a j j ) + i , i j | a i j | ) \mu_{1}(A)=\sup\limits_{j}(\real(a_{jj})+\sum\limits_{i,i\neq j}|a_{ij}|)
  45. μ 2 ( A ) = λ m a x ( A + A T 2 ) \displaystyle\mu_{2}(A)=\lambda_{max}\left(\frac{A+A^{\mathrm{T}}}{2}\right)
  46. μ ( A ) = sup i ( \real ( a i i ) + j , j i | a i j | ) \mu_{\infty}(A)=\sup\limits_{i}(\real(a_{ii})+\sum\limits_{j,j\neq i}|a_{ij}|)
  47. - μ ( - A ) x T A x x T x μ ( A ) , -\mu(-A)\leq{\frac{x^{\mathrm{T}}Ax}{x^{\mathrm{T}}x}}\leq\mu(A),
  48. x 0 x\neq 0
  49. λ k \lambda_{k}
  50. A A
  51. - μ ( - A ) \real λ k μ ( A ) -\mu(-A)\leq\real\,\lambda_{k}\leq\mu(A)
  52. - μ ( - A ) > 0 -\mu(-A)>0
  53. μ ( A ) < 0 \mu(A)<0
  54. A - 1 - 1 μ ( A ) . \|A^{-1}\|\leq-{\frac{1}{\mu(A)}}.
  55. R R
  56. \real ( z ) 0 | R ( z ) | 1 \real\,(z)\leq 0\,\Rightarrow\,|R(z)|\leq 1
  57. μ ( A ) 0 R ( A ) 1. \mu(A)\leq 0\,\Rightarrow\,\|R(A)\|\leq 1.
  58. x ˙ = A x \dot{x}=Ax
  59. x n + 1 = A x n x_{n+1}=Ax_{n}
  60. A A
  61. λ \lambda
  62. | λ | 1 |\lambda|\leq 1
  63. \real λ 0 \real\,\lambda\leq 0
  64. A A
  65. A 1 \|A\|\leq 1
  66. e t A x ( 0 ) \mathrm{e}^{tA}x(0)
  67. e t A 1 \|\mathrm{e}^{tA}\|\leq 1
  68. t 0 t\geq 0
  69. μ ( A ) 0 \mu(A)\leq 0
  70. x \|x\|
  71. x ˙ = A x \dot{x}=Ax
  72. x n + 1 = R ( h A ) x n x_{n+1}=R(hA)\cdot x_{n}
  73. R R
  74. h h
  75. | R ( z ) | 1 |R(z)|\leq 1
  76. \real ( z ) 0 \real\,(z)\leq 0
  77. μ ( A ) 0 \mu(A)\leq 0
  78. R ( h A ) 1 \|R(hA)\|\leq 1
  79. u ( 0 ) = u ( 1 ) = 0 u(0)=u(1)=0
  80. u , v = 0 1 u v d x . \langle u,v\rangle=\int_{0}^{1}uv\,\mathrm{d}x.
  81. u , u ′′ = - u , u - π 2 u 2 , \langle u,u^{\prime\prime}\rangle=-\langle u^{\prime},u^{\prime}\rangle\leq-% \pi^{2}\|u\|^{2},
  82. sin π x \sin\,\pi x
  83. - π 2 -\pi^{2}
  84. u , A u - π 2 u 2 \langle u,Au\rangle\leq-\pi^{2}\|u\|^{2}
  85. A = d 2 / d x 2 A=\mathrm{d}^{2}/\mathrm{d}x^{2}
  86. μ ( d 2 d x 2 ) = - π 2 . \mu({\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}})=-\pi^{2}.
  87. u , A u > 0 \langle u,Au\rangle>0
  88. - d 2 / d x 2 -\mathrm{d}^{2}/\mathrm{d}x^{2}
  89. A A
  90. μ ( - A ) < 0 \mu(-A)<0
  91. - u ′′ = f -u^{\prime\prime}=f
  92. T u = f Tu=f
  93. T T
  94. - μ ( - T ) > 0 -\mu(-T)>0
  95. T T
  96. l ( f ) u - v f ( u ) - f ( v ) L ( f ) u - v , l(f)\cdot\|u-v\|\leq\|f(u)-f(v)\|\leq L(f)\cdot\|u-v\|,
  97. L ( f ) L(f)
  98. f f
  99. l ( f ) l(f)
  100. m ( f ) u - v 2 u - v , f ( u ) - f ( v ) M ( f ) u - v 2 , m(f)\cdot\|u-v\|^{2}\leq\langle u-v,f(u)-f(v)\rangle\leq M(f)\cdot\|u-v\|^{2},
  101. u u
  102. v v
  103. D D
  104. f f
  105. M ( f ) M(f)
  106. f f
  107. l ( f ) l(f)
  108. m ( f ) = - M ( - f ) m(f)=-M(-f)
  109. l ( f ) = L ( f - 1 ) - 1 l(f)=L(f^{-1})^{-1}
  110. L ( f - 1 ) L(f^{-1})
  111. f f
  112. M ( f ) = lim h 0 + L ( I + h f ) - 1 h . M(f)=\lim_{h\rightarrow 0^{+}}{\frac{L(I+hf)-1}{h}}.
  113. f f
  114. D D
  115. L ( f ) = sup x D f ( x ) L(f)=\sup_{x\in D}\|f^{\prime}(x)\|
  116. M ( f ) = sup x D μ ( f ( x ) ) . \displaystyle M(f)=\sup_{x\in D}\mu(f^{\prime}(x)).
  117. f ( x ) f^{\prime}(x)
  118. f f
  119. m ( f ) > 0 m(f)>0
  120. M ( f ) < 0 M(f)<0
  121. L ( f ) < 1 L(f)<1
  122. M ( f ) < 0 L ( f - 1 ) - 1 M ( f ) , M(f)<0\,\Rightarrow\,L(f^{-1})\leq-{\frac{1}{M(f)}},

Lombricine_kinase.html

  1. \rightleftharpoons

Long-chain-alcohol_O-fatty-acyltransferase.html

  1. \rightleftharpoons

Long-chain-enoyl-CoA_hydratase.html

  1. \rightleftharpoons

Long-chain-fatty-acid—(acyl-carrier-protein)_ligase.html

  1. \rightleftharpoons

Long-chain-fatty-acid—luciferin-component_ligase.html

  1. \rightleftharpoons

Long-chain-fatty-acyl-glutamate_deacylase.html

  1. \rightleftharpoons

Longitudinal_static_stability.html

  1. W = L w + L t W=L_{w}+L_{t}
  2. L w L_{w}
  3. L t L_{t}
  4. L w = q S w C L α ( α - α 0 ) L_{w}=qS_{w}\frac{\partial C_{L}}{\partial\alpha}(\alpha-\alpha_{0})
  5. S w S_{w}
  6. C L C_{L}
  7. α \alpha
  8. α 0 \alpha_{0}
  9. q q
  10. q = 1 2 ρ v 2 q=\frac{1}{2}\rho v^{2}
  11. ρ \rho
  12. v v
  13. L t = q S t ( C l α ( α - ϵ α α ) + C l η η ) L_{t}=qS_{t}\left(\frac{\partial C_{l}}{\partial\alpha}\left(\alpha-\frac{% \partial\epsilon}{\partial\alpha}\alpha\right)+\frac{\partial C_{l}}{\partial% \eta}\eta\right)
  14. S t S_{t}\!
  15. C l C_{l}\!
  16. η \eta\!
  17. ϵ \epsilon\!
  18. M = L w x g - ( l t - x g ) L t M=L_{w}x_{g}-(l_{t}-x_{g})L_{t}\!
  19. x g x_{g}\!
  20. l t l_{t}\!
  21. α \alpha
  22. M α = x g L w α - ( l t - x g ) L t α \frac{\partial M}{\partial\alpha}=x_{g}\frac{\partial L_{w}}{\partial\alpha}-(% l_{t}-x_{g})\frac{\partial L_{t}}{\partial\alpha}
  23. M α \frac{\partial M}{\partial\alpha}
  24. M = h ( L w + L t ) M=h(L_{w}+L_{t})\!
  25. M α = h ( L w α + L t α ) \frac{\partial M}{\partial\alpha}=h\left(\frac{\partial L_{w}}{\partial\alpha}% +\frac{\partial L_{t}}{\partial\alpha}\right)
  26. h = x g - l t L t α L w α + L t α h=x_{g}-l_{t}\frac{\frac{\partial L_{t}}{\partial\alpha}}{\frac{\partial L_{w}% }{\partial\alpha}+\frac{\partial L_{t}}{\partial\alpha}}
  27. L L
  28. L w L_{w}
  29. L t L_{t}
  30. h = x g - c ( 1 - ϵ α ) C l α C L α l t S t c S w h=x_{g}-c\left(1-\frac{\partial\epsilon}{\partial\alpha}\right)\frac{\frac{% \partial C_{l}}{\partial\alpha}}{\frac{\partial C_{L}}{\partial\alpha}}\frac{l% _{t}S_{t}}{cS_{w}}
  31. V t = l t S t c S w V_{t}=\frac{l_{t}S_{t}}{cS_{w}}
  32. h = x g - 0.5 c V t h=x_{g}-0.5cV_{t}\!
  33. - h -h
  34. V t = 0 V_{t}=0

Lovastatin_nonaketide_synthase.html

  1. \rightleftharpoons

Low-density-lipoprotein_receptor_kinase.html

  1. \rightleftharpoons

Lucas–Lehmer–Riesel_test.html

  1. u i = u i - 1 2 - 2. u_{i}=u_{i-1}^{2}-2.\,
  2. u 0 = ( 2 + 3 ) k + ( 2 - 3 ) k u_{0}=(2+\sqrt{3})^{k}+(2-\sqrt{3})^{k}
  3. ( P - 2 N ) = 1 and ( P + 2 N ) = - 1 \left(\frac{P-2}{N}\right)=1\quad\,\text{and}\quad\left(\frac{P+2}{N}\right)=-1
  4. u i u_{i}
  5. u i = a 2 i + a - 2 i u_{i}=a^{2^{i}}+a^{-2^{i}}
  6. u i = u i - 1 2 - 2 u_{i}=u_{i-1}^{2}-2
  7. v ( i ) = a i + a - i v(i)=a^{i}+a^{-i}
  8. v ( i ) = α v ( i - 1 ) + β v ( i - 2 ) v(i)=\alpha v(i-1)+\beta v(i-2)

Luteolin-7-O-diglucuronide_4'-O-glucuronosyltransferase.html

  1. \rightleftharpoons

Luteolin_7-O-glucuronosyltransferase.html

  1. \rightleftharpoons

Lysine_N-acetyltransferase.html

  1. \rightleftharpoons

Lysine—pyruvate_6-transaminase.html

  1. \rightleftharpoons

Lysine—tRNA(Pyl)_ligase.html

  1. \rightleftharpoons

Lysine—tRNA_ligase.html

  1. \rightleftharpoons

Lysophospholipase.html

  1. \rightleftharpoons

Lysyltransferase.html

  1. \rightleftharpoons

M7G(5')pppN_diphosphatase.html

  1. \rightleftharpoons

MacCormack_method.html

  1. u t + a u x = 0. \qquad\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0.
  2. u u
  3. n + 1 n+1
  4. u i n + 1 ¯ u_{i}^{\overline{n+1}}
  5. u i n + 1 ¯ = u i n - a Δ t Δ x ( u i + 1 n - u i n ) u_{i}^{\overline{n+1}}=u_{i}^{n}-a\frac{\Delta t}{\Delta x}\left(u_{i+1}^{n}-u% _{i}^{n}\right)
  6. u i n + 1 ¯ u_{i}^{\overline{n+1}}
  7. u i n + 1 = u i n + 1 / 2 - a Δ t 2 Δ x ( u i n + 1 ¯ - u i - 1 n + 1 ¯ ) u_{i}^{n+1}=u_{i}^{n+1/2}-a\frac{\Delta t}{2\Delta x}\left(u_{i}^{\overline{n+% 1}}-u_{i-1}^{\overline{n+1}}\right)
  8. Δ t / 2 \Delta t/2
  9. Δ t \Delta t
  10. u i n + 1 / 2 u_{i}^{n+1/2}
  11. u i n + 1 / 2 = u i n + u i n + 1 ¯ 2 u_{i}^{n+1/2}=\frac{u_{i}^{n}+u_{i}^{\overline{n+1}}}{2}
  12. u i n + 1 = u i n + u i n + 1 ¯ 2 - a Δ t 2 Δ x ( u i n + 1 ¯ - u i - 1 n + 1 ¯ ) u_{i}^{n+1}=\frac{u_{i}^{n}+u_{i}^{\overline{n+1}}}{2}-a\frac{\Delta t}{2% \Delta x}\left(u_{i}^{\overline{n+1}}-u_{i-1}^{\overline{n+1}}\right)

Macrolide_2'-kinase.html

  1. \rightleftharpoons

Madelung_equations.html

  1. t ρ + ( ρ u ) = 0 , \partial_{t}\rho+\nabla\cdot(\rho u)=0,
  2. t u + u u = - 1 m ( 1 ρ H ^ ρ ) = - 1 ρ p Q - 1 m U , \partial_{t}u+u\cdot\nabla u=-\frac{1}{m}\nabla\left(\frac{1}{\sqrt{\rho}}\hat% {H}\sqrt{\rho}\right)=-\frac{1}{\rho}\nabla\cdot p_{Q}-\frac{1}{m}\nabla U,
  3. u u
  4. ρ = m | ψ | 2 \rho=m|\psi|^{2}
  5. Γ m 𝐮 d 𝐥 = 2 π n , n \begin{matrix}\Gamma\doteq\oint{m\mathbf{u}\cdot d\mathbf{l}}=2\pi n\hbar,&n% \in\mathbb{Z}\\ \end{matrix}
  6. μ \mu
  7. H ^ \hat{H}
  8. p Q = - ( / 2 m ) 2 ρ ln ρ p_{Q}=-(\hbar/2m)^{2}\rho\nabla\otimes\nabla\ln\rho
  9. Q Q
  10. p Q p_{Q}
  11. μ = Q + U \mu=Q+U
  12. μ = ( m / ρ ) p Q + U \ \nabla\mu=(m/\rho)\nabla\cdot p_{Q}+\nabla U
  13. ϵ = μ - t r ( p Q ) ( m / ρ ) = - 2 ( ln ρ ) 2 / 8 m + U \ \epsilon=\mu-tr(p_{Q})(m/\rho)=-\hbar^{2}(\nabla\ln\rho)^{2}/8m+U
  14. ϵ = 0 \ \epsilon=0

Magic_circle_(mathematics).html

  1. 40 + 24 + 9 + 56 + 41 + 25 + 8 + 57 = 260 40+24+9+56+41+25+8+57=260
  2. 14 + 51 + 46 + 30 + 3 + 62 + 35 + 19 = 260 14+51+46+30+3+62+35+19=260
  3. 45 + 29 + 4 + 61 + 36 + 20 + 13 + 52 = 260 45+29+4+61+36+20+13+52=260
  4. 37 + 21 + 12 + 53 + 44 + 28 + 5 + 60 = 260 37+21+12+53+44+28+5+60=260
  5. 47 + 31 + 2 + 63 + 34 + 18 + 15 + 50 = 260 47+31+2+63+34+18+15+50=260
  6. 7 + 58 + 39 + 23 + 10 + 55 + 42 + 26 = 260 7+58+39+23+10+55+42+26=260
  7. 38 + 22 + 11 + 54 + 43 + 27 + 6 + 59 = 260 38+22+11+54+43+27+6+59=260
  8. 48 + 32 + 1 + 64 + 33 + 17 + 16 + 49 = 260 48+32+1+64+33+17+16+49=260
  9. 14 + 51 + 62 + 3 + 7 + 58 + 55 + 10 = 260 14+51+62+3+7+58+55+10=260
  10. 49 + 16 + 1 + 64 + 60 + 5 + 12 + 53 = 260 49+16+1+64+60+5+12+53=260
  11. 40 + 57 + 41 + 56 + 50 + 47 + 34 + 63 + 29 + 4 + 13 + 20 + 22 + 11 + 6 + 27 = 2 * 260 = 520 40+57+41+56+50+47+34+63+29+4+13+20+22+11+6+27=2*260=520

Magnesium_chelatase.html

  1. \rightleftharpoons

Magnetic_Prandtl_number.html

  1. Pr m = Re m Re = ν η = viscous diffusion rate magnetic diffusion rate \mathrm{Pr}_{\mathrm{m}}=\frac{\mathrm{Re_{m}}}{\mathrm{Re}}=\frac{\nu}{\eta}=% \frac{\mbox{viscous diffusion rate}~{}}{\mbox{magnetic diffusion rate}~{}}

MAIFI.html

  1. MAIFI = total number of customer interruptions less than the defined time total number of customers served \mbox{MAIFI}~{}=\frac{\mbox{total number of customer interruptions less than % the defined time}~{}}{\mbox{total number of customers served}~{}}

Malate_synthase.html

  1. \rightleftharpoons

Malate—CoA_ligase.html

  1. \rightleftharpoons

Maleate_hydratase.html

  1. \rightleftharpoons

Maleimide_hydrolase.html

  1. \rightleftharpoons

Maltose-6'-phosphate_glucosidase.html

  1. \rightleftharpoons

Maltose-transporting_ATPase.html

  1. \rightleftharpoons

Maltose_O-acetyltransferase.html

  1. \rightleftharpoons

Maltose_phosphorylase.html

  1. \rightleftharpoons

Maltose_synthase.html

  1. \rightleftharpoons

Mandart_inellipse.html

  1. x : y : z = a b + c - a : b a + c - b : c a + b - c x:y:z=\frac{a}{b+c-a}:\frac{b}{a+c-b}:\frac{c}{a+b-c}

Mandelamide_amidase.html

  1. \rightleftharpoons

Manganese-transporting_ATPase.html

  1. \rightleftharpoons

Mannitol-1-phosphatase.html

  1. \rightleftharpoons

Mannokinase.html

  1. \rightleftharpoons

Mannonate_dehydratase.html

  1. \rightleftharpoons

Mannose-1-phosphate_guanylyltransferase.html

  1. \rightleftharpoons

Mannose-1-phosphate_guanylyltransferase_(GDP).html

  1. \rightleftharpoons

Mannosyl-3-phosphoglycerate_phosphatase.html

  1. \rightleftharpoons

Mannosyl-3-phosphoglycerate_synthase.html

  1. \rightleftharpoons

Mannotetraose_2-alpha-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Markov's_principle.html

  1. ( n ( P ( n ) ¬ P ( n ) ) ( ¬ n ¬ P ( n ) ) ) ( n P ( n ) ) . (\forall n(P(n)\vee\neg P(n))\wedge(\neg\forall n\neg P(n)))\rightarrow(% \exists n\;P(n)).
  2. ¬ ¬ n f ( n ) = 0 n f ( n ) = 0 , \neg\neg\exists n\;f(n)=0\rightarrow\exists n\;f(n)=0,
  3. f f
  4. P ( 0 ) , P ( 1 ) , P ( 2 ) , P(0),P(1),P(2),\dots
  5. P P
  6. P P
  7. n P ( n ) \exists n\;P(n)
  8. ¬ ¬ n P ( n ) \neg\neg\exists n\;P(n)
  9. P P
  10. x ( y ¬ ¬ ( 0 < y ) ¬ ¬ ( y < x ) ) 0 < x . \forall x\in\mathbb{R}\ (\forall y\in\mathbb{R}\ \neg\neg(0<y)\vee\neg\neg(y<x% ))\to 0<x.

Mark–Houwink_equation.html

  1. [ η ] [\eta]
  2. M M
  3. [ η ] = K M a [\eta]=KM^{a}
  4. a a
  5. K K
  6. a = 0.5 a=0.5
  7. a = 0.8 a=0.8
  8. 0.5 a 0.8 0.5\leq a\leq 0.8
  9. a 0.8 a\geq 0.8
  10. a = 2.0 a=2.0
  11. K K
  12. a a
  13. K 1 M 1 1 + a 1 = K 2 M 2 1 + a 2 K_{1}M_{1}^{1+a_{1}}=K_{2}M_{2}^{1+a_{2}}

Matrix_polynomial.html

  1. P ( x ) = i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + + a n x n , P(x)=\sum_{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n},
  2. P ( A ) = i = 0 n a i A i = a 0 I + a 1 A + a 2 A 2 + + a n A n , P(A)=\sum_{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots+a_{n}A^{n},
  3. p A ( t ) = det ( t I - A ) p_{A}(t)=\det\left(tI-A\right)
  4. p A ( A ) = 0 p_{A}(A)=0
  5. P ( A ) = Q ( A ) P(A)=Q(A)
  6. P ( j ) ( λ i ) = Q ( j ) ( λ i ) for j = 0 , , n i and i = 1 , , s , P^{(j)}(\lambda_{i})=Q^{(j)}(\lambda_{i})\qquad\,\text{for }j=0,\ldots,n_{i}\,% \text{ and }i=1,\ldots,s,
  7. P ( j ) P^{(j)}
  8. λ 1 , , λ s \lambda_{1},\dots,\lambda_{s}
  9. n 1 , , n s n_{1},\dots,n_{s}
  10. S = I + A + A 2 + + A n S=I+A+A^{2}+\cdots+A^{n}
  11. A S = A + A 2 + A 3 + + A n + 1 AS=A+A^{2}+A^{3}+\cdots+A^{n+1}
  12. ( I - A ) S = S - A S = I - A n + 1 (I-A)S=S-AS=I-A^{n+1}
  13. S = ( I - A ) - 1 ( I - A n + 1 ) S=(I-A)^{-1}(I-A^{n+1})

Maximally_stable_extremal_regions.html

  1. I I
  2. I : D 2 S I:D\subset\mathbb{Z}^{2}\to S
  3. S S
  4. \leq
  5. A D × D A\subset D\times D
  6. Q Q
  7. D D
  8. p , q Q p,q\in Q
  9. p , a 1 , a 2 , . . , a n , q p,a_{1},a_{2},..,a_{n},q
  10. p A a 1 , a i A a i + 1 , a n A q pAa_{1},a_{i}Aa_{i+1},a_{n}Aq
  11. Q = { q D Q : p Q : q A p } \partial Q=\{q\in D\setminus Q:\exists p\in Q:qAp\}
  12. Q \partial Q
  13. Q Q
  14. Q Q
  15. Q Q
  16. Q D Q\subset D
  17. p Q , q Q : I ( p ) > I ( q ) p\in Q,q\in\partial Q:I(p)>I(q)
  18. p Q , q Q : I ( p ) < I ( q ) p\in Q,q\in\partial Q:I(p)<I(q)
  19. Q 1 , . . , Q i - 1 , Q i , Q_{1},..,Q_{i-1},Q_{i},...
  20. Q i Q i + 1 Q_{i}\subset Q_{i+1}
  21. Q i * Q_{i*}
  22. q ( i ) = | Q i + Δ Q i - Δ | / | Q i | q(i)=|Q_{i+\Delta}\setminus Q_{i-\Delta}|/|Q_{i}|
  23. i * i*
  24. | | |\cdot|
  25. Δ S \Delta\in S
  26. Q i + Δ Q_{i+\Delta}
  27. Q i - Δ Q_{i-\Delta}
  28. Q i Q_{i}
  29. I t I_{t}
  30. t t
  31. T : D D T:D\to D
  32. O ( n ) O(n)
  33. n n
  34. O ( n log ( log ( n ) ) ) O(n\,\log(\log(n)))
  35. n n\,
  36. O ( n ) O(n)\,
  37. O ( n log ( log ( n ) ) ) O(n\,\log(\log(n)))
  38. O ( n ) O(n)\,
  39. M A i M_{A}^{i}
  40. A , k A,k
  41. B 1 , , B k B_{1},\dots,B_{k}
  42. M B 1 i , , M B k i M_{B_{1}}^{i},\dots,M_{B_{k}}^{i}
  43. M A i M_{A}^{i}
  44. B 1 , , B k B_{1},\dots,B_{k}

Megaprime.html

  1. 2 13372531 + 1 3 \tfrac{2^{13372531}+1}{3}
  2. 2 13347311 + 1 3 \tfrac{2^{13347311}+1}{3}
  3. 14 1091401 + 1 15 \tfrac{14^{1091401}+1}{15}
  4. 2 4031399 + 1 3 \tfrac{2^{4031399}+1}{3}
  5. 7 1264699 - 1 6 \tfrac{7^{1264699}-1}{6}

Melting-point_depression.html

  1. E = E B ( 1 - d D ) E=E_{B}(1-\frac{d}{D})
  2. T M ( d ) = T M B ( 1 - 4 σ s l H f ρ s d ) T_{M}(d)=T_{MB}(1-\frac{4\sigma\,_{sl}}{H_{f}\rho\,_{s}d})
  3. T M ( d ) = T M B ( 1 - ( c d ) 2 ) T_{M}(d)=T_{MB}(1-(\frac{c}{d})^{2})
  4. T M ( d ) = 4 T M B H f d ( σ s v - σ l v ( ρ s ρ l ) 2 / 3 ) T_{M}(d)=\frac{4T_{MB}}{H_{f}d}\left(\sigma\,_{sv}-\sigma\,_{lv}\left(\frac{% \rho\,_{s}}{\rho\,_{l}}\right)^{2/3}\right)
  5. T M ( d ) = 4 T M B H f d ( σ s v 1 - d 0 d - σ l v ( 1 - ρ s ρ l ) ) T_{M}(d)=\frac{4T_{MB}}{H_{f}d}(\frac{\sigma\,_{sv}}{1-\frac{d_{0}}{d}}-\sigma% \,_{lv}(1-\frac{\rho\,_{s}}{\rho\,_{l}}))
  6. T M ( d ) = 2 T M B H f d ( σ s l - σ l v 3 ( σ s v - σ l v ρ s ρ l ) ) T_{M}(d)=\frac{2T_{MB}}{H_{f}d}(\sigma\,_{sl}-\sigma\,_{lv}3(\sigma\,_{sv}-% \sigma\,_{lv}\frac{\rho\,_{s}}{\rho\,_{l}}))
  7. T M ( d ) = T M B ( 1 - c z d ) T_{M}(d)=T_{MB}(1-\frac{c}{zd})

Metal-induced_gap_states.html

  1. E B = 1 2 [ E V ¯ + E C ¯ ] E_{B}=\frac{1}{2}[\bar{E_{V}}+\bar{E_{C}}]
  2. E V ¯ = E V - 1 3 Δ s o \bar{E_{V}}=E_{V}-\frac{1}{3}\Delta_{so}
  3. Δ s o \Delta_{so}
  4. E V E_{V}
  5. Γ \Gamma
  6. E C ¯ \bar{E_{C}}
  7. e Φ b h e\Phi_{bh}
  8. e V i f eV_{if}
  9. δ q = 0.16 e V | X M - X S C | + 0.035 e V 2 | X M - X S C | 2 \delta q=\frac{0.16}{eV}|X_{M}-X_{SC}|+\frac{0.035}{eV^{2}}|X_{M}-X_{SC}|^{2}
  10. X M X_{M}
  11. X S C X_{SC}
  12. Φ b h = 1 2 [ E C ¯ - E V ¯ ] + δ m = 1 2 [ E C ¯ - E V - Δ s o 3 ] + δ m \Phi_{bh}=\frac{1}{2}[\bar{E_{C}}-\bar{E_{V}}]+\delta_{m}=\frac{1}{2}[\bar{E_{% C}}-E_{V}-\frac{\Delta_{so}}{3}]+\delta_{m}
  13. δ m \delta_{m}
  14. X M X_{M}
  15. Φ b h = 1 1 + α N v s [ Φ M - X M + D J + α N v s ( E g - Φ 0 ) ] \Phi_{bh}=\frac{1}{1+\alpha N_{vs}}[\Phi_{M}-X_{M}+D_{J}+\alpha N_{vs}(E_{g}-% \Phi_{0})]
  16. α \alpha
  17. N v s = N_{vs}=
  18. ϕ M = \phi_{M}=
  19. D J = D_{J}=
  20. E G = E_{G}=
  21. Φ 0 \Phi_{0}
  22. ϕ b h \phi_{bh}
  23. α N v s 1 \alpha N_{vs}>>1
  24. Φ M \Phi_{M}
  25. α N v s 1 \alpha N_{vs}<<1
  26. Φ b h \Phi_{bh}
  27. D J D_{J}
  28. V V
  29. ( V i f - V ) 1 2 (V_{if}-V)^{\frac{1}{2}}

Metal–semiconductor_junction.html

  1. q Φ B ( n ) + q Φ B ( p ) = E g q\Phi_{\rm B}^{(n)}+q\Phi_{\rm B}^{(p)}=E_{\rm g}
  2. q Φ B ( n ) q Φ metal - χ semi q\Phi_{\rm B}^{(n)}\approx q\Phi_{\rm metal}-\chi_{\rm semi}
  3. q Φ B 1 2 E bandgap q\Phi_{\rm B}\approx\frac{1}{2}E_{\rm bandgap}

Methenyltetrahydrofolate_cyclohydrolase.html

  1. \rightleftharpoons

Methenyltetrahydromethanopterin_cyclohydrolase.html

  1. \rightleftharpoons

Methionine—glyoxylate_transaminase.html

  1. \rightleftharpoons

Methionine—tRNA_ligase.html

  1. \rightleftharpoons

Methyl-ONN-azoxymethanol_beta-D-glucosyltransferase.html

  1. \rightleftharpoons

Methylglyoxal_synthase.html

  1. \rightleftharpoons
  2. 3 {}_{3}

Methylguanidinase.html

  1. \rightleftharpoons

Methylphosphothioglycerate_phosphatase.html

  1. \rightleftharpoons

Methylthioadenosine_nucleosidase.html

  1. \rightleftharpoons

Methylthioribulose_1-phosphate_dehydratase.html

  1. \rightleftharpoons

Methylumbelliferyl-acetate_deacetylase.html

  1. \rightleftharpoons

Mg2+-importing_ATPase.html

  1. \rightleftharpoons

Microtubule-severing_ATPase.html

  1. \rightleftharpoons

Mimosinase.html

  1. \rightleftharpoons

Minimal_model_program.html

  1. X X
  2. f : X X f:X\rightarrow X^{\prime}
  3. X X
  4. X X
  5. κ ( X , K X ) = - 1 \kappa(X,K_{X})=-1
  6. X X^{\prime}
  7. X X
  8. f : X Y f:X^{\prime}\rightarrow Y
  9. Y Y
  10. Y Y
  11. - K F -K_{F}
  12. F F
  13. κ ( X , K X ) \kappa(X,K_{X})
  14. X X^{\prime}
  15. X X
  16. K X K_{X^{\prime}}
  17. X X^{\prime}
  18. X X
  19. X X^{\prime}
  20. X X
  21. X X
  22. X X
  23. X X^{\prime}
  24. K X K_{X^{\prime}}
  25. K X C K_{X^{\prime}}\cdot C
  26. n K X nK_{X^{\prime}}
  27. n n
  28. X X
  29. X X
  30. X i X_{i}
  31. K X i K_{X_{i}}
  32. X i X_{i}
  33. X i X_{i}
  34. X X^{\prime}

Minkowski_content.html

  1. ( X , μ , d ) \scriptstyle(X,\,\mu,\,d)
  2. A ε = { x X | d ( x , A ) ε } A_{\varepsilon}=\{x\in X\,|\,d(x,A)\leq\varepsilon\}
  3. M * ( A ) = lim inf ε 0 μ ( A ε ) - μ ( A ) ε , M_{*}(A)=\liminf_{\varepsilon\to 0}\frac{\mu(A_{\varepsilon})-\mu(A)}{% \varepsilon},
  4. M * ( A ) = lim sup ε 0 μ ( A ε ) - μ ( A ) ε . M^{*}(A)=\limsup_{\varepsilon\to 0}\frac{\mu(A_{\varepsilon})-\mu(A)}{% \varepsilon}.
  5. M * m ( A ) = lim inf ε 0 μ ( A ε ) - μ ( A ) α n - m ε n - m M_{*}^{m}(A)=\liminf_{\varepsilon\to 0}\frac{\mu(A_{\varepsilon})-\mu(A)}{% \alpha_{n-m}\varepsilon^{n-m}}
  6. μ \mu
  7. n n
  8. M * m ( A ) = lim sup ε 0 μ ( A ε ) - μ ( A ) α n - m ε n - m . M^{*m}(A)=\limsup_{\varepsilon\to 0}\frac{\mu(A_{\varepsilon})-\mu(A)}{\alpha_% {n-m}\varepsilon^{n-m}}.

Mitochondrial_protein-transporting_ATPase.html

  1. \rightleftharpoons

Mixed_logit.html

  1. β \beta
  2. β \beta
  3. β \beta
  4. U n i = β x n i + ε n i U_{ni}=\beta x_{ni}+\varepsilon_{ni}
  5. ε n i \varepsilon_{ni}
  6. β n \beta_{n}
  7. U n i = β n x n i + ε n i U_{ni}=\beta_{n}x_{ni}+\varepsilon_{ni}
  8. ε n i \varepsilon_{ni}
  9. β n f ( β | θ ) \quad\beta_{n}\sim f(\beta|\theta)
  10. β n \beta_{n}
  11. β n \beta_{n}
  12. β n \beta_{n}
  13. L n i ( β n ) = e β n X n i j e β n X n j L_{ni}(\beta_{n})=\frac{e^{\beta_{n}X_{ni}}}{\sum_{j}e^{\beta_{n}X_{nj}}}
  14. β n \beta_{n}
  15. β n \beta_{n}
  16. P n i = L n i ( β ) f ( β | θ ) d β P_{ni}=\int L_{ni}(\beta)f(\beta|\theta)d\beta
  17. β n \beta_{n}
  18. f ( β | θ ) f(\beta|\theta)
  19. S b S_{b}
  20. E n i x n j m = - x n j m P n i β m L n i ( β ) L n j ( β ) f ( β ) d β = - x n j m β m L n j ( β ) L n i ( β ) P n i f ( β ) d β E_{nix_{nj}^{m}}=-\frac{x_{nj}^{m}}{P_{ni}}\int\beta^{m}L_{ni}(\beta)L_{nj}(% \beta)f(\beta)d\beta=-x_{nj}^{m}\int\beta^{m}L_{nj}(\beta)\frac{L_{ni}(\beta)}% {P_{ni}}f(\beta)d\beta
  21. β \beta
  22. U n i t = β n X n i t + ε n i t U_{nit}=\beta_{n}X_{nit}+\varepsilon_{nit}
  23. ε \varepsilon
  24. ε \varepsilon
  25. ε \varepsilon
  26. β \beta
  27. β ¯ \bar{\beta}
  28. σ 2 \sigma^{2}
  29. U n i t = ( β ¯ + σ η n ) X n i t + ε n i t U_{nit}=(\bar{\beta}+\sigma\eta_{n})X_{nit}+\varepsilon_{nit}
  30. U n i t = β ¯ X n i t + ( σ η n X n i t + ε n i t ) U_{nit}=\bar{\beta}X_{nit}+(\sigma\eta_{n}X_{nit}+\varepsilon_{nit})
  31. U n i t = β ¯ X n i t + e n i t U_{nit}=\bar{\beta}X_{nit}+e_{nit}
  32. e n i t = σ η n X n i t + ε n i t e_{nit}=\sigma\eta_{n}X_{nit}+\varepsilon_{nit}
  33. ε n i t \varepsilon_{nit}
  34. σ η n X n i t \sigma\eta_{n}X_{nit}
  35. i i
  36. j j
  37. C o v ( e n i t , e n j t ) = σ 2 ( X n i t X n j t ) Cov(e_{nit},e_{njt})=\sigma^{2}(X_{nit}X_{njt})
  38. t t
  39. q q
  40. C o v ( e n i t , e n i q ) = σ 2 ( X n i t X n i q ) Cov(e_{nit},e_{niq})=\sigma^{2}(X_{nit}X_{niq})
  41. β n \beta_{n}
  42. L n ( β n ) = t e β n X n i t j e β n X n j t L_{n}(\beta_{n})=\prod_{t}\frac{e^{\beta_{n}X_{nit}}}{\sum_{j}e^{\beta_{n}X_{% njt}}}
  43. ε n i t \varepsilon_{nit}
  44. β \beta
  45. P n i = L n ( β ) f ( β | θ ) d β P_{ni}=\int L_{n}(\beta)f(\beta|\theta)d\beta
  46. f ( β | θ ) f(\beta|\theta)
  47. β r \beta^{r}
  48. r = 1 r=1
  49. L n ( β r ) L_{n}(\beta^{r})
  50. r = 2 , , R r=2,...,R
  51. P ~ n i = r L n ( β r ) R \tilde{P}_{ni}=\frac{\sum_{r}L_{n}(\beta^{r})}{R}

Modified_Wigner_distribution_function.html

  1. C x ( t , f ) = - - W x ( θ , ν ) Π ( t - θ , f - ν ) d θ d ν = [ W x Π ] ( t , f ) C_{x}(t,f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_{x}(\theta,\nu)\Pi(% t-\theta,f-\nu)\,d\theta\,d\nu\quad=[W_{x}\,\ast\,\Pi](t,f)
  2. Π ( t , f ) \Pi\left(t,f\right)
  3. 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
  4. Π ( t , f ) = δ ( 0 , 0 ) ( t , f ) \Pi(t,f)=\delta_{(0,0)}(t,f)
  5. S P x ( t , f ) = | S T x ( t , f ) | 2 = S T x ( t , f ) S T x * ( t , f ) SP_{x}(t,f)=|ST_{x}(t,f)|^{2}=ST_{x}(t,f)\,ST_{x}^{*}(t,f)
  6. S T x ST_{x}
  7. x x
  8. S T x ( t , f ) = - x ( τ ) w * ( t - τ ) e - j 2 π f τ d τ ST_{x}(t,f)=\int_{-\infty}^{\infty}x(\tau)w^{*}(t-\tau)e^{-j2\pi f\tau}\,d\tau
  9. Π ( t , f ) = W h ( t , f ) \Pi(t,f)=W_{h}(t,f)
  10. P W x ( t , f ) = - w ( τ / 2 ) w * ( - τ / 2 ) x ( t + τ / 2 ) x * ( t - τ / 2 ) e - j 2 π τ f d τ PW_{x}(t,f)=\int_{-\infty}^{\infty}w(\tau/2)w^{*}(-\tau/2)x(t+\tau/2)x^{*}(t-% \tau/2)e^{-j2\pi\tau\,f}\,d\tau
  11. Π ( t , f ) = δ 0 ( t ) W h ( t , f ) \Pi(t,f)=\delta_{0}(t)\,W_{h}(t,f)
  12. P W x ( t , f ) = - S T x ( t , f + ν / 2 ) S T x * ( t , f - ν / 2 ) e j 2 π ν t d ν PW_{x}(t,f)=\int_{-\infty}^{\infty}ST_{x}(t,f+\nu/2)ST_{x}^{*}(t,f-\nu/2)e^{j2% \pi\nu\,t}\,d\nu
  13. q q
  14. S P W x ( t , f ) = [ q P W x ( . , f ) ] ( t ) = - q ( t - u ) - w ( τ / 2 ) w * ( - τ / 2 ) x ( u + τ / 2 ) x * ( u - τ / 2 ) e - j 2 π τ f d τ d u SPW_{x}(t,f)=[q\,\ast\,PW_{x}(.,f)](t)=\int_{-\infty}^{\infty}q(t-u)\int_{-% \infty}^{\infty}w(\tau/2)w^{*}(-\tau/2)x(u+\tau/2)x^{*}(u-\tau/2)e^{-j2\pi\tau% \,f}\,d\tau\,du
  15. Π ( t , f ) = q ( t ) W ( f ) \Pi(t,f)=q(t)\,W(f)
  16. W W
  17. w w
  18. S M ( t , f ) = - S T x ( t , f + ν / 2 ) S T x * ( t , f - ν / 2 ) G ( ν ) e j 2 π ν t d ν SM(t,f)=\int_{-\infty}^{\infty}ST_{x}(t,f+\nu/2)ST_{x}^{*}(t,f-\nu/2)G(\nu)e^{% j2\pi\nu\,t}\,d\nu
  19. Π ( t , f ) = g ( t ) W h ( t , f ) \Pi(t,f)=g(t)\,W_{h}(t,f)
  20. g ( t ) g(t)
  21. G ( f ) G(f)
  22. P W x PW_{x}
  23. g ( t ) = 1 g(t)=1
  24. S P x SP_{x}
  25. g ( t ) = δ 0 ( t ) g(t)=\delta_{0}(t)
  26. S M ( t , f ) = - S T ~ x ( t , f + ν ) S T ~ x * ( t , f - ν ) P ( ν ) d ν SM(t,f)=\int_{-\infty}^{\infty}\tilde{ST}_{x}(t,f+\nu)\tilde{ST}_{x}^{*}(t,f-% \nu)P(\nu)\,d\nu
  27. S T ~ x ( t , f ) = - x ( t + τ ) w * ( τ ) e - j 2 π f τ d τ = S T x ( t , f ) e j 2 π f t \tilde{ST}_{x}(t,f)=\int_{-\infty}^{\infty}x(t+\tau)w^{*}(\tau)e^{-j2\pi f\tau% }\,d\tau\quad=ST_{x}(t,f)\,e^{j2\pi ft}
  28. Π ( t , f ) = p ( 2 t ) W h ( t , f ) \Pi(t,f)=p(2t)\,W_{h}(t,f)

Mogensen–Scott_encoding.html

  1. ( ( λ x 1 x n . λ c . c x 1 x n ) v 1 v n ) f ((\lambda x_{1}\ldots x_{n}.\lambda c.c\ x_{1}\ldots x_{n})\ v_{1}\ldots v_{n}% )\ f
  2. x 1 x n x_{1}\ldots x_{n}
  3. v 1 v n v_{1}\ldots v_{n}
  4. f v 1 v n f\ v_{1}\ldots v_{n}
  5. A i A_{i}
  6. ( ( λ x 1 x A 1 . λ c 1 c N . c 1 x 1 x A 1 ) v 1 v A 1 ) ((\lambda x_{1}\ldots x_{A_{1}}.\lambda c_{1}\ldots c_{N}.c_{1}\ x_{1}\ldots x% _{A_{1}})\ v_{1}\ldots v_{A_{1}})
  7. f 1 f N f_{1}\ldots f_{N}
  8. f 1 v 1 v A 1 f_{1}\ v_{1}\ldots v_{A_{1}}
  9. ( ( λ x 1 x A 2 . λ c 1 c N . c 2 x 1 x A 2 ) v 1 v A 2 ) ((\lambda x_{1}\ldots x_{A_{2}}.\lambda c_{1}\ldots c_{N}.c_{2}\ x_{1}\ldots x% _{A_{2}})\ v_{1}\ldots v_{A_{2}})
  10. f 1 f N f_{1}\ldots f_{N}
  11. f 2 v 1 v A 2 f_{2}\ v_{1}\ldots v_{A_{2}}
  12. ( ( λ x 1 x A N . λ c 1 c N . c N x 1 x A N ) v 1 v A N ) ((\lambda x_{1}\ldots x_{A_{N}}.\lambda c_{1}\ldots c_{N}.c_{N}\ x_{1}\ldots x% _{A_{N}})\ v_{1}\ldots v_{A_{N}})
  13. f 1 f N f_{1}\ldots f_{N}
  14. f N v 1 v A 1 f_{N}\ v_{1}\ldots v_{A_{1}}
  15. f 1 f N f_{1}\ldots f_{N}
  16. { c i } i = 1 N \{c_{i}\}_{i=1}^{N}
  17. c i c_{i}
  18. A i A_{i}
  19. c i c_{i}
  20. λ x 1 x A i . λ c 1 c N . c i x 1 x A i \lambda x_{1}\ldots x_{A_{i}}.\lambda c_{1}\ldots c_{N}.c_{i}\ x_{1}\ldots x_{% A_{i}}
  21. mse [ x ] = λ a , b , c . a x mse [ M N ] = λ a , b , c . b mse [ M ] mse [ N ] mse [ λ x . M ] = λ a , b , c . c ( λ x . mse [ M ] ) \begin{array}[]{rcl}\operatorname{mse}[x]&=&\lambda a,b,c.a\ x\\ \ \operatorname{mse}[M\ N]&=&\lambda a,b,c.b\ \operatorname{mse}[M]\ % \operatorname{mse}[N]\\ \ \operatorname{mse}[\lambda x.M]&=&\lambda a,b,c.c\ (\lambda x.\operatorname{% mse}[M])\\ \end{array}
  22. mse [ λ x . f ( x x ) ] \operatorname{mse}[\lambda x.f\ (x\ x)]
  23. λ a , b , c . c ( λ x . mse [ f ( x x ) ] ) \lambda a,b,c.c\ (\lambda x.\operatorname{mse}[f\ (x\ x)])
  24. λ a , b , c . c ( λ x . λ a , b , c . b mse [ f ] mse [ x x ] ) \lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ \operatorname{mse}[f]\ % \operatorname{mse}[x\ x])
  25. λ a , b , c . c ( λ x . λ a , b , c . b ( λ a , b , c . a f ) mse [ x x ] ) \lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ (\lambda a,b,c.a\ f)\ % \operatorname{mse}[x\ x])
  26. λ a , b , c . c ( λ x . λ a , b , c . b ( λ a , b , c . a f ) ( λ a , b , c . b mse [ x ] mse [ x ] ) ) \lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ (\lambda a,b,c.a\ f)\ (\lambda a,% b,c.b\ \operatorname{mse}[x]\ \operatorname{mse}[x]))
  27. λ a , b , c . c ( λ x . λ a , b , c . b ( λ a , b , c . a f ) ( λ a , b , c . b ( λ a , b , c . a x ) ( λ a , b , c . a x ) ) ) \lambda a,b,c.c\ (\lambda x.\lambda a,b,c.b\ (\lambda a,b,c.a\ f)\ (\lambda a,% b,c.b\ (\lambda a,b,c.a\ x)\ (\lambda a,b,c.a\ x)))
  28. c i c_{i}
  29. λ x 1 x A i . λ c 1 c N . c i ( x 1 c 1 c N ) ( x A i c 1 c N ) \lambda x_{1}\ldots x_{A_{i}}.\lambda c_{1}\ldots c_{N}.c_{i}(x_{1}c_{1}\ldots c% _{N})\ldots(x_{A_{i}}c_{1}\ldots c_{N})
  30. λ x 1 x A i . λ c 1 c N . c i x 1 x A i \lambda x_{1}\ldots x_{A_{i}}.\lambda c_{1}\ldots c_{N}.c_{i}x_{1}\ldots x_{A_% {i}}

Molybdate-transporting_ATPase.html

  1. \rightleftharpoons

Monad_transformer.html

  1. M A \mathrm{M}\,A
  2. M ( A ? ) \mathrm{M}\left(A^{?}\right)
  3. A ? A^{?}
  4. return : A M ( A ? ) = a return ( Just a ) \mathrm{return}:A\rightarrow\mathrm{M}\left(A^{?}\right)=a\mapsto\mathrm{% return}(\mathrm{Just}\,a)
  5. bind : M ( A ? ) ( A M ( B ? ) ) M ( B ? ) = m f bind m ( a { return Nothing if a = Nothing f a if a = Just a ) \mathrm{bind}:\mathrm{M}\left(A^{?}\right)\rightarrow\left(A\rightarrow\mathrm% {M}\left(B^{?}\right)\right)\rightarrow\mathrm{M}\left(B^{?}\right)=m\mapsto f% \mapsto\mathrm{bind}\,m\,\left(a\mapsto\begin{cases}\mbox{return Nothing}&% \mbox{if }~{}a=\mathrm{Nothing}\\ f\,a^{\prime}&\mbox{if }~{}a=\mathrm{Just}\,a^{\prime}\end{cases}\right)
  6. lift : M ( A ) M ( A ? ) = m bind m ( a return ( Just a ) ) \mathrm{lift}:\mathrm{M}(A)\rightarrow\mathrm{M}\left(A^{?}\right)=m\mapsto% \mathrm{bind}\,m\,(a\mapsto\mathrm{return}(\mathrm{Just}\,a))
  7. M A \mathrm{M}\,A
  8. M ( A + E ) \mathrm{M}(A+E)
  9. E E
  10. return : A M ( A + E ) = a return ( value a ) \mathrm{return}:A\rightarrow\mathrm{M}(A+E)=a\mapsto\mathrm{return}(\mathrm{% value}\,a)
  11. bind : M ( A + E ) ( A M ( B + E ) ) M ( B + E ) = m f bind m ( a { return err e if a = err e f a if a = value a ) \mathrm{bind}:\mathrm{M}(A+E)\rightarrow(A\rightarrow\mathrm{M}(B+E))% \rightarrow\mathrm{M}(B+E)=m\mapsto f\mapsto\mathrm{bind}\,m\,\left(a\mapsto% \begin{cases}\mbox{return err }~{}e&\mbox{if }~{}a=\mathrm{err}\,e\\ f\,a^{\prime}&\mbox{if }~{}a=\mathrm{value}\,a^{\prime}\end{cases}\right)
  12. lift : M A M ( A + E ) = m bind m ( a return ( value a ) ) \mathrm{lift}:\mathrm{M}\,A\rightarrow\mathrm{M}(A+E)=m\mapsto\mathrm{bind}\,m% \,(a\mapsto\mathrm{return}(\mathrm{value}\,a))
  13. M A \mathrm{M}\,A
  14. E M A E\rightarrow\mathrm{M}\,A
  15. E E
  16. return : A E M A = a e return a \mathrm{return}:A\rightarrow E\rightarrow\mathrm{M}\,A=a\mapsto e\mapsto% \mathrm{return}\,a
  17. bind : ( E M A ) ( A E M B ) E M B = m k e bind ( m e ) ( a k a e ) \mathrm{bind}:(E\rightarrow\mathrm{M}\,A)\rightarrow(A\rightarrow E\rightarrow% \mathrm{M}\,B)\rightarrow E\rightarrow\mathrm{M}\,B=m\mapsto k\mapsto e\mapsto% \mathrm{bind}\,(m\,e)\,(a\mapsto k\,a\,e)
  18. lift : M A E M A = a e a \mathrm{lift}:\mathrm{M}\,A\rightarrow E\rightarrow\mathrm{M}\,A=a\mapsto e\mapsto a
  19. M A \mathrm{M}\,A
  20. S M ( A × S ) S\rightarrow\mathrm{M}(A\times S)
  21. S S
  22. return : A S M ( A × S ) = a s return ( a , s ) \mathrm{return}:A\rightarrow S\rightarrow\mathrm{M}(A\times S)=a\mapsto s% \mapsto\mathrm{return}\,(a,s)
  23. bind : ( S M ( A × S ) ) ( A S M ( B × S ) ) S M ( B × S ) = m k s bind ( m s ) ( ( a , s ) k a s ) \mathrm{bind}:(S\rightarrow\mathrm{M}(A\times S))\rightarrow(A\rightarrow S% \rightarrow\mathrm{M}(B\times S))\rightarrow S\rightarrow\mathrm{M}(B\times S)% =m\mapsto k\mapsto s\mapsto\mathrm{bind}\,(m\,s)\,((a,s^{\prime})\mapsto k\,a% \,s^{\prime})
  24. lift : M A S M ( A × S ) = m s bind m ( a return ( a , s ) ) \mathrm{lift}:\mathrm{M}\,A\rightarrow S\rightarrow\mathrm{M}(A\times S)=m% \mapsto s\mapsto\mathrm{bind}\,m\,(a\mapsto\mathrm{return}\,(a,s))
  25. M A \mathrm{M}\,A
  26. M ( W × A ) \mathrm{M}(W\times A)
  27. W W
  28. * *
  29. ε \varepsilon
  30. return : A M ( W × A ) = a return ( ε , a ) \mathrm{return}:A\rightarrow\mathrm{M}(W\times A)=a\mapsto\mathrm{return}\,(% \varepsilon,a)
  31. bind : M ( W × A ) ( A M ( W × B ) ) M ( W × B ) = m f bind m ( ( w , a ) bind ( f a ) ( ( w , b ) return ( w * w , b ) ) ) \mathrm{bind}:\mathrm{M}(W\times A)\rightarrow(A\rightarrow\mathrm{M}(W\times B% ))\rightarrow\mathrm{M}(W\times B)=m\mapsto f\mapsto\mathrm{bind}\,m\,((w,a)% \mapsto\mathrm{bind}\,(f\,a)\,((w^{\prime},b)\mapsto\mathrm{return}\,(w*w^{% \prime},b)))
  32. lift : M A M ( W × A ) = m bind m ( a return ( ε , a ) ) \mathrm{lift}:\mathrm{M}\,A\rightarrow\mathrm{M}(W\times A)=m\mapsto\mathrm{% bind}\,m\,(a\mapsto\mathrm{return}\,(\varepsilon,a))
  33. M A \mathrm{M}\,A
  34. R R
  35. ( A M R ) M R (A\rightarrow\mathrm{M}\,R)\rightarrow\mathrm{M}\,R
  36. R R
  37. return : A ( A M R ) M R = a k k a \mathrm{return}\colon A\rightarrow\left(A\rightarrow\mathrm{M}\,R\right)% \rightarrow\mathrm{M}\,R=a\mapsto k\mapsto k\,a
  38. bind : ( ( A M R ) M R ) ( A ( B M R ) M R ) ( B M R ) M R \mathrm{bind}\colon\left(\left(A\rightarrow\mathrm{M}\,R\right)\rightarrow% \mathrm{M}\,R\right)\rightarrow\left(A\rightarrow\left(B\rightarrow\mathrm{M}% \,R\right)\rightarrow\mathrm{M}\,R\right)\rightarrow\left(B\rightarrow\mathrm{% M}\,R\right)\rightarrow\mathrm{M}\,R
  39. = c f k c ( a f a k ) =c\mapsto f\mapsto k\mapsto c\,\left(a\mapsto f\,a\,k\right)
  40. lift : M A ( A M R ) M R = bind \mathrm{lift}:\mathrm{M}\,A\rightarrow(A\rightarrow\mathrm{M}\,R)\rightarrow% \mathrm{M}\,R=\mathrm{bind}
  41. S ( A × S ) ? S\rightarrow\left(A\times S\right)^{?}
  42. S ( A ? × S ) S\rightarrow\left(A^{?}\times S\right)

Monogalactosyldiacylglycerol_synthase.html

  1. \rightleftharpoons

Monogenic_system.html

  1. i \mathcal{F}_{i}\,\!
  2. 𝒱 ( q 1 , q 2 , , q N , q ˙ 1 , q ˙ 2 , , q ˙ N , t ) \mathcal{V}(q_{1},\ q_{2},\ \dots,\ q_{N},\ \dot{q}_{1},\ \dot{q}_{2},\ \dots,% \ \dot{q}_{N},\ t)\,\!
  3. i = - 𝒱 q i + d d t ( 𝒱 q i ˙ ) ; \mathcal{F}_{i}=-\frac{\partial\mathcal{V}}{\partial q_{i}}+\frac{d}{dt}\left(% \frac{\partial\mathcal{V}}{\partial\dot{q_{i}}}\right);\,
  4. q i q_{i}\,\!
  5. q i ˙ \dot{q_{i}}\,
  6. t t\,\!
  7. i = - 𝒱 q i \mathcal{F}_{i}=-\frac{\partial\mathcal{V}}{\partial q_{i}}\,

Monomethyl-sulfatase.html

  1. \rightleftharpoons

Monosaccharide-transporting_ATPase.html

  1. \rightleftharpoons

Monosialoganglioside_sialyltransferase.html

  1. \rightleftharpoons

Monoterpenol_beta-glucosyltransferase.html

  1. \rightleftharpoons

Monoterpenol_O-acetyltransferase.html

  1. \rightleftharpoons

Monoterpenyl-diphosphatase.html

  1. \rightleftharpoons

Monte_Carlo_method_for_photon_transport.html

  1. x \displaystyle x
  2. s = - ln ξ μ t s=-\frac{\ln\xi}{\mu_{t}}
  3. ξ \xi
  4. μ t {\mu_{t}}
  5. x \displaystyle x
  6. Δ W = μ a μ t W \Delta W=\frac{\mu_{a}}{\mu_{t}}W
  7. μ a {\mu_{a}}
  8. W W - Δ W W\leftarrow W-\Delta W\,
  9. cos θ = { 1 2 g [ 1 + g 2 - ( 1 - g 2 1 - g + 2 g ξ ) 2 ] if g 0 1 - 2 ξ if g = 0 \cos\theta=\begin{cases}\frac{1}{2g}\left[1+g^{2}-\left(\frac{1-g^{2}}{1-g+2g% \xi}\right)^{2}\right]&\,\text{ if }g\neq 0\\ 1-2\xi&\,\text{ if }g=0\end{cases}
  10. 2 π 2\pi
  11. φ = 2 π ξ \varphi=2\pi\xi\frac{}{}
  12. μ x \displaystyle\mu^{\prime}_{x}
  13. μ z = 1 \displaystyle\mu_{z}=1
  14. μ x \displaystyle\mu^{\prime}_{x}
  15. μ z = - 1 \displaystyle\mu_{z}=-1
  16. μ x \displaystyle\mu^{\prime}_{x}
  17. W = { m W ξ 1 / m 0 ξ > 1 / m W=\begin{cases}mW&\xi\leq 1/m\\ 0&\xi>1/m\end{cases}

Moran's_I.html

  1. I = N i j w i j i j w i j ( X i - X ¯ ) ( X j - X ¯ ) i ( X i - X ¯ ) 2 I=\frac{N}{\sum_{i}\sum_{j}w_{ij}}\frac{\sum_{i}\sum_{j}w_{ij}(X_{i}-\bar{X})(% X_{j}-\bar{X})}{\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. E ( I ) = - 1 N - 1 E(I)=\frac{-1}{N-1}
  10. Var ( I ) = N S 4 - S 3 S 5 ( N - 1 ) ( N - 2 ) ( N - 3 ) ( i j w i j ) 2 - ( E ( I ) ) 2 \operatorname{Var}(I)=\frac{NS_{4}-S_{3}S_{5}}{(N-1)(N-2)(N-3)(\sum_{i}\sum_{j% }w_{ij})^{2}}-(E(I))^{2}
  11. S 1 = 1 2 i j ( w i j + w j i ) 2 S_{1}=\frac{1}{2}\sum_{i}\sum_{j}(w_{ij}+w_{ji})^{2}
  12. S 2 = i ( j w i j + j w j i ) 2 S_{2}=\sum_{i}(\sum_{j}w_{ij}+\sum_{j}w_{ji})^{2}
  13. S 3 = N - 1 i ( x i - x ¯ ) 4 ( N - 1 i ( x i - x ¯ ) 2 ) 2 S_{3}=\frac{N^{-1}\sum_{i}(x_{i}-\bar{x})^{4}}{(N^{-1}\sum_{i}(x_{i}-\bar{x})^% {2})^{2}}
  14. S 4 = ( N 2 - 3 N + 3 ) S 1 - N S 2 + 3 ( i j w i j ) 2 S_{4}=(N^{2}-3N+3)S_{1}-NS_{2}+3(\sum_{i}\sum_{j}w_{ij})^{2}
  15. S 5 = ( N 2 - N ) S 1 - 2 N S 2 + 6 ( i j w i j ) 2 S_{5}=(N^{2}-N)S_{1}-2NS_{2}+6(\sum_{i}\sum_{j}w_{ij})^{2}

Morphological_dictionary.html

  1. Γ \Gamma
  2. A 2 ( L * ) A\subset 2^{(L^{*})}
  3. L = ( ( Σ θ ) × Γ ) ( Σ × ( Γ θ ) ) L=((\Sigma\cup{\theta})\times\Gamma)\cup(\Sigma\times(\Gamma\cup{\theta}))
  4. L * L^{*}
  5. U 2 ( Γ * × Σ * ) U\subset 2^{(\Gamma^{*}\times\Sigma^{*})}
  6. E Σ * E\subset\Sigma^{*}
  7. E = w : ( w , w ) U E={w:(w,w^{\prime})\in U}
  8. τ : E 2 Γ * \tau:E\rightarrow 2^{\Gamma^{*}}
  9. τ ( w ) = w : ( w , w ) U \tau(w)=w^{\prime}:(w,w^{\prime})\in U

Mortgage_yield.html

  1. Mortgage Yield: ri such that P = n = 1 N C ( t ) ( 1 + r i / 1200 ) t - 1 \mbox{Mortgage Yield: ri such that P}~{}=\sum_{n=1}^{N}\frac{C(t)}{(1+ri/1200)% ^{t-1}}
  2. C ( t ) C(t)

Motions_in_the_time-frequency_distribution.html

  1. x ( t - t 0 ) S x ( t - t 0 , f ) e - j 2 π f t 0 x(t-t_{0})\rightarrow S_{x}(t-t_{0},f)e^{-j2\pi ft_{0}}
  2. x ( t - t 0 ) W x ( t - t 0 , f ) x(t-t_{0})\rightarrow W_{x}(t-t_{0},f)\,
  3. e j 2 π f 0 t x ( t ) S x ( t , f - f 0 ) e^{j2\pi f_{0}t}x(t)\rightarrow S_{x}(t,f-f_{0})
  4. e j 2 π f 0 t x ( t ) W x ( t , f - f 0 ) e^{j2\pi f_{0}t}x(t)\rightarrow W_{x}(t,f-f_{0})
  5. 1 | a | x ( t a ) W x ( t a , a f ) \frac{1}{\sqrt{|a|}}x(\frac{t}{a})\rightarrow W_{x}(\frac{t}{a},af)
  6. W x ( a t , f ) \approx W_{x}(at,f)
  7. x ( t ) = e j π a t 2 y ( t ) x(t)=e^{j\pi at^{2}}y(t)\,
  8. S x ( t , f ) S y ( t , f - a t ) S_{x}(t,f)\approx S_{y}(t,f-at)\,
  9. W x ( t , f ) = W y ( t , f - a t ) W_{x}(t,f)=W_{y}(t,f-at)\,
  10. x ( t ) = e j π t 2 a y ( t ) x(t)=e^{j\pi\frac{t^{2}}{a}}y(t)\,
  11. S x ( t , f ) S y ( t - a f , f ) S_{x}(t,f)\approx S_{y}(t-af,f)\,
  12. W x ( t , f ) = W y ( t - a f , f ) W_{x}(t,f)=W_{y}(t-af,f)\,
  13. X ( f ) = F T ( x ( t ) ) X(f)=FT(x(t))\,
  14. | S X ( t , f ) | | S x ( - f , t ) | |S_{X}(t,f)|\approx|S_{x}(-f,t)|\,
  15. G X ( t , f ) = G x ( - f , t ) e - j 2 π f t G_{X}(t,f)=G_{x}(-f,t)e^{-j2\pi ft}\,
  16. W X ( t , f ) = W x ( - f , t ) W_{X}(t,f)=W_{x}(-f,t)\,
  17. X ( f ) = I F T [ x ( t ) ] = - x ( t ) e j 2 π f t d t X(f)=IFT[x(t)]=\int_{-\infty}^{\infty}x(t)e^{j2\pi ft}\,dt
  18. W X ( t , f ) = W x ( f , - t ) W_{X}(t,f)=W_{x}(f,-t)\,
  19. G X ( t , f ) = G x ( f , - t ) e j 2 π t f G_{X}(t,f)=G_{x}(f,-t)e^{j2\pi tf}\,
  20. X ( f ) = x ( - t ) X(f)=x(-t)\,
  21. W X ( t , f ) = W x ( - t , - f ) W_{X}(t,f)=W_{x}(-t,-f)\,
  22. G X ( t , f ) = G x ( - t , - f ) G_{X}(t,f)=G_{x}(-t,-f)\,
  23. | S X ( t , f ) | | S x ( - f , t ) | |S_{X}(t,f)|\approx|S_{x}(-f,t)|\,
  24. G X ( t , f ) = G x ( - f , t ) e - j 2 π f t G_{X}(t,f)=G_{x}(-f,t)e^{-j2\pi ft}\,
  25. W X ( t , f ) = W x ( - f , t ) W_{X}(t,f)=W_{x}(-f,t)\,
  26. S x ( t , f ) S y ( t - 1 3 f , f ) S_{x}(t,f)\approx S_{y}(t-\frac{1}{3}f,f)\,
  27. W x ( t , f ) = W y ( t - 1 3 f , f ) W_{x}(t,f)=W_{y}(t-\frac{1}{3}f,f)\,
  28. x ( t - t 0 ) S x ( t - 2 , f ) e - j 2 π f t 0 x(t-t_{0})\rightarrow S_{x}(t-2,f)e^{-j2\pi ft_{0}}
  29. x ( t - t 0 ) W x ( t - 2 , f ) x(t-t_{0})\rightarrow W_{x}(t-2,f)\,
  30. e j 2 π f 0 t x ( t ) S x ( t , f + 1 ) e^{j2\pi f_{0}t}x(t)\rightarrow S_{x}(t,f+1)
  31. e j 2 π f 0 t x ( t ) W x ( t , f + 1 ) e^{j2\pi f_{0}t}x(t)\rightarrow W_{x}(t,f+1)

MRNA_guanylyltransferase.html

  1. \rightleftharpoons

Mucinaminylserine_mucinaminidase.html

  1. \rightleftharpoons

Multinomial_probit.html

  1. Y i | x 1 , i , , x k , i Categorical ( p i , , p m ) , for i = 1 , , n Y_{i}|x_{1,i},\ldots,x_{k,i}\ \sim\operatorname{Categorical}(p_{i},\ldots,p_{m% }),\,\text{ for }i=1,\dots,n
  2. Pr [ Y i = h | x 1 , i , , x k , i ] = p i , h , for i = 1 , , n , \Pr[Y_{i}=h|x_{1,i},\ldots,x_{k,i}]=p_{i,h},\,\text{ for }i=1,\dots,n,
  3. Y i 1 \displaystyle Y_{i}^{1\ast}
  4. s y m b o l ε 𝒩 ( 0 , s y m b o l Σ ) symbol\varepsilon\sim\mathcal{N}(0,symbol\Sigma)
  5. Y i = { 1 if Y i 1 > Y i 2 , , Y i m 2 if Y i 2 > Y i 1 , Y i 3 , , Y i m m otherwise. Y_{i}=\begin{cases}1&\,\text{if }Y_{i}^{1\ast}>Y_{i}^{2\ast},\ldots,Y_{i}^{m% \ast}\\ 2&\,\text{if }Y_{i}^{2\ast}>Y_{i}^{1\ast},Y_{i}^{3\ast},\ldots,Y_{i}^{m\ast}\\ \ldots&\ldots\\ m&\,\text{otherwise.}\end{cases}
  6. Y i = arg max h = 1 m Y i h Y_{i}=\arg\max_{h=1}^{m}Y_{i}^{h\ast}
  7. \scriptstylesymbol Σ \scriptstylesymbol\Sigma

Multiple_inositol-polyphosphate_phosphatase.html

  1. \rightleftharpoons

Multiple_isomorphous_replacement.html

  1. 𝐅 p h = 𝐅 p + 𝐅 h \mathbf{F}_{ph}=\mathbf{F}_{p}+\mathbf{F}_{h}

Multiple_time_dimensions.html

  1. ( + , , + k , - , , - n ) (\underbrace{+,\cdots,+}_{k},\underbrace{-,\cdots,-}_{n})
  2. ( - , , - k , + , , + n ) (\underbrace{-,\cdots,-}_{k},\underbrace{+,\cdots,+}_{n})
  3. t σ = θ = 1 k ( δ σ θ t θ + c 2 v σ v θ β 2 ( ζ - 1 ) t θ ) - 1 v σ β 2 ζ x k + 1 , t^{\prime}_{\sigma}=\sum_{\theta=1}^{k}\left(\delta_{\sigma\theta}t_{\theta}+% \frac{c^{2}}{v_{\sigma}v_{\theta}}\beta^{2}(\zeta-1)t_{\theta}\right)-\frac{1}% {v_{\sigma}}\beta^{2}\zeta x_{k+1},
  4. x k + 1 = - c 2 β 2 ζ θ = 1 k t θ v θ + ζ x k + 1 , x^{\prime}_{k+1}=-c^{2}\beta^{2}\zeta\sum_{\theta=1}^{k}\frac{t_{\theta}}{v_{% \theta}}+\zeta x_{k+1},
  5. x λ = x λ , x^{\prime}_{\lambda}=x_{\lambda},
  6. 𝐯 1 = ( v 1 , 0 , , 0 n - 1 ) , \mathbf{v}_{1}=(v_{1},\underbrace{0,\cdots,0}_{n-1}),
  7. 𝐯 2 = ( v 2 , 0 , , 0 n - 1 ) , \mathbf{v}_{2}=(v_{2},\underbrace{0,\cdots,0}_{n-1}),
  8. 𝐯 k = ( v k , 0 , , 0 n - 1 ) \mathbf{v}_{k}=(v_{k},\underbrace{0,\cdots,0}_{n-1})
  9. β = 1 μ = 1 k c 2 v μ 2 ; \beta=\frac{1}{\sqrt{\sum_{\mu=1}^{k}\frac{c^{2}}{v^{2}_{\mu}}}};
  10. ζ = 1 1 - β 2 ; \zeta=\frac{1}{\sqrt{1-\beta^{2}}};
  11. d x η d t σ = V σ η \frac{dx_{\eta}}{dt_{\sigma}}=V_{\sigma\eta}
  12. d x η d t σ = V σ η , \frac{dx^{\prime}_{\eta}}{dt^{\prime}_{\sigma}}=V^{\prime}_{\sigma\eta},
  13. V σ ( k + 1 ) = V σ ( k + 1 ) ζ ( 1 - β 2 θ = 1 k c 2 v θ V θ ( k + 1 ) ) 1 + V σ ( k + 1 ) v σ β 2 ( ( ζ - 1 ) θ = 1 k c 2 v θ V θ ( k + 1 ) - ζ ) , V^{\prime}_{\sigma(k+1)}=\frac{V_{\sigma(k+1)}\zeta\left(1-\beta^{2}\sum_{% \theta=1}^{k}\frac{c^{2}}{v_{\theta}V_{\theta(k+1)}}\right)}{1+\frac{V_{\sigma% (k+1)}}{v_{\sigma}}\beta^{2}\left((\zeta-1)\sum_{\theta=1}^{k}\frac{c^{2}}{v_{% \theta}V_{\theta(k+1)}}-\zeta\right)},
  14. V σ λ = V σ λ 1 + V σ ( k + 1 ) v σ β 2 ( ( ζ - 1 ) θ = 1 k c 2 v θ V θ ( k + 1 ) - ζ ) , V^{\prime}_{\sigma\lambda}=\frac{V_{\sigma\lambda}}{1+\frac{V_{\sigma(k+1)}}{v% _{\sigma}}\beta^{2}\left((\zeta-1)\sum_{\theta=1}^{k}\frac{c^{2}}{v_{\theta}V_% {\theta(k+1)}}-\zeta\right)},
  15. Δ T = ( Δ t 1 ) 2 + ( Δ t 2 ) 2 0 \Delta T=\sqrt{(\Delta t_{1})^{2}+(\Delta t_{2})^{2}}\geq 0

Multiple_trace_theory.html

  1. 𝐦 𝟏 = [ m 1 ( 1 ) m 1 ( 2 ) m 1 ( 3 ) m 1 ( L ) ] \mathbf{m_{1}}=\begin{bmatrix}m_{1}(1)\\ m_{1}(2)\\ m_{1}(3)\\ \vdots\\ m_{1}(L)\end{bmatrix}
  2. 𝐌 = [ 𝐦 𝟏 𝐦 𝟐 𝐦 𝟑 𝐦 𝐧 ] = [ m 1 ( 1 ) m 2 ( 1 ) m 3 ( 1 ) m n ( 1 ) m 1 ( 2 ) m 2 ( 2 ) m 3 ( 2 ) m n ( 2 ) m 1 ( L ) m 2 ( L ) m 3 ( L ) m n ( L ) ] \mathbf{M}=\begin{bmatrix}\mathbf{m_{1}}&\mathbf{m_{2}}&\mathbf{m_{3}}&\cdots&% \mathbf{m_{n}}\end{bmatrix}=\begin{bmatrix}m_{1}(1)&m_{2}(1)&m_{3}(1)&\cdots&m% _{n}(1)\\ m_{1}(2)&m_{2}(2)&m_{3}(2)&\cdots&m_{n}(2)\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ m_{1}(L)&m_{2}(L)&m_{3}(L)&\cdots&m_{n}(L)\end{bmatrix}
  3. 𝐦 𝐚𝐛 = [ a ( 1 ) a ( 2 ) a ( k ) b ( 1 ) b ( 2 ) b ( k ) ] = [ 𝐚 𝐛 ] \mathbf{m_{ab}}=\begin{bmatrix}a(1)\\ a(2)\\ \vdots\\ a(k)\\ b(1)\\ b(2)\\ \vdots\\ b(k)\end{bmatrix}=\begin{bmatrix}\mathbf{a}\\ \mathbf{b}\end{bmatrix}
  4. 𝐭 𝐢 + 𝟏 ( 𝐣 ) = 𝐭 𝐢 ( 𝐣 ) + ϵ ( 𝐣 ) \mathbf{t_{i+1}(j)}=\mathbf{t_{i}(j)+\epsilon(j)}
  5. 𝐭 𝐢 + 𝟏 = [ t i ( 1 ) + ϵ ( 1 ) t i ( 2 ) + ϵ ( 2 ) t i ( r ) + ϵ ( r ) ] \mathbf{t_{i+1}}=\begin{bmatrix}t_{i}(1)+\epsilon(1)\\ t_{i}(2)+\epsilon(2)\\ \vdots\\ t_{i}(r)+\epsilon(r)\end{bmatrix}
  6. 𝐩 = [ p ( 1 ) p ( 2 ) p ( L ) ] \mathbf{p}=\begin{bmatrix}p(1)\\ p(2)\\ \vdots\\ p(L)\end{bmatrix}
  7. 𝐩 - 𝐦 𝐢 = j = 1 L ( p ( j ) - m i ( j ) ) 2 \left\|\mathbf{p-m_{i}}\right\|=\sqrt{\sum_{j=1}^{L}(p(j)-m_{i}(j))^{2}}
  8. s i m i l a r i t y ( 𝐩 , 𝐦 𝐢 ) = e - τ 𝐩 - 𝐦 𝐢 similarity(\mathbf{p,m_{i}})=e^{-\tau\left\|\mathbf{p-m_{i}}\right\|}
  9. 𝐒𝐒 ( 𝐩 , 𝐌 ) = i = 1 n e - τ 𝐩 - 𝐦 𝐢 = i = 1 n e - τ j = 1 L ( p ( j ) - m i ( j ) ) 2 \mathbf{SS(p,M)}=\sum_{i=1}^{n}e^{-\tau\left\|\mathbf{p-m_{i}}\right\|}=\sum_{% i=1}^{n}e^{-\tau\sqrt{\sum_{j=1}^{L}(p(j)-m_{i}(j))^{2}}}
  10. P ( r e c o g n i z i n g p ) = P ( 𝐒𝐒 ( 𝐩 , 𝐌 ) > c r i t e r i o n ) P(recognizing~{}p)~{}=~{}P(\mathbf{SS(p,M)}>criterion)
  11. P ( r e c a l l i n g m a b ) = P ( s i m i l a r i t y ( a , m a b ) > c r i t e r i o n ) P(recalling~{}m_{ab})~{}=~{}P(similarity(a,m_{ab})>criterion)
  12. P ( r e c a l l i n g m a b ) = s i m i l a r i t y ( a , m a b ) S S ( a , M ) + e r r o r P(recalling~{}m_{ab})~{}=~{}\frac{similarity(a,m_{ab})}{SS(a,M)+error}

Multiplicative_partition.html

  1. k = t i k=\prod t_{i}
  2. p i t i - 1 , \prod p_{i}^{t_{i}-1},
  3. f ( s ) = n = 1 a n n s = k = 2 1 1 - k - s . f(s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}=\prod_{k=2}^{\infty}\frac{1}{1-k^{% -s}}.
  4. a n n ( exp log n log log log n log log n ) - 2 + o ( 1 ) , a_{n}\leq n\left(\exp\frac{\log n\log\log\log n}{\log\log n}\right)^{-2+o(1)},
  5. a n n ( exp log n log log log n log log n ) - 1 + o ( 1 ) . a_{n}\leq n\left(\exp\frac{\log n\log\log\log n}{\log\log n}\right)^{-1+o(1)}.
  6. a ¯ = exp ( 4 log N 2 e log log N ( 1 + o ( 1 ) ) ) , \bar{a}=\exp\left(\frac{4\sqrt{\log N}}{\sqrt{2e}\log\log N}\bigl(1+o(1)\bigr)% \right),

Mycocerosate_synthase.html

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Myo-inosose-2_dehydratase.html

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Myosin-heavy-chain_kinase.html

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Myrcene_synthase.html

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N,N-dimethylformamidase.html

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N-(long-chain-acyl)ethanolamine_deacylase.html

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N-acetyl-beta-alanine_deacetylase.html

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N-acetyldiaminopimelate_deacetylase.html

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N-acetylgalactosamine_kinase.html

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N-acetylgalactosaminoglycan_deacetylase.html

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N-acetylgalactosaminyl-proteoglycan_3-beta-glucuronosyltransferase.html

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N-acetylglucosamine-1-phosphodiester_alpha-N-acetylglucosaminidase.html

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N-acetylglucosamine-6-phosphate_deacetylase.html

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N-acetylglucosamine_deacetylase.html

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N-acetylglucosamine_kinase.html

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N-acetylglucosaminyl-proteoglycan_4-beta-glucuronosyltransferase.html

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N-acetylglucosaminyldiphosphoundecaprenol_glucosyltransferase.html

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N-acetylglucosaminylphosphatidylinositol_deacetylase.html

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N-acetyllactosaminide_3-alpha-galactosyltransferase.html

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N-acetyllactosaminide_alpha-2,3-sialyltransferase.html

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N-acetyllactosaminide_beta-1,3-N-acetylglucosaminyltransferase.html

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N-acetyllactosaminide_beta-1,6-N-acetylglucosaminyl-transferase.html

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N-acetylneuraminate_4-O-acetyltransferase.html

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N-acetylneuraminate_7-O(or_9-O)-acetyltransferase.html

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N-acetylneuraminate_synthase.html

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N-acyl-D-amino-acid_deacylase.html

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N-acyl-D-aspartate_deacylase.html

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N-acyl-D-glutamate_deacylase.html

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N-acylmannosamine_kinase.html

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N-acylneuraminate-9-phosphatase.html

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N-acylneuraminate-9-phosphate_synthase.html

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N-acylneuraminate_cytidylyltransferase.html

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N-acylsphingosine_galactosyltransferase.html

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N-benzyloxycarbonylglycine_hydrolase.html

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N-carbamoyl-D-amino_acid_hydrolase.html

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N-carbamoyl-L-amino-acid_hydrolase.html

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N-carbamoylputrescine_amidase.html

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N-carbamoylsarcosine_amidase.html

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N-feruloylglycine_deacylase.html

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N-formylglutamate_deformylase.html

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N-formylmethionylaminoacyl-tRNA_deformylase.html

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N-hydroxyarylamine_O-acetyltransferase.html

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N-hydroxythioamide_S-beta-glucosyltransferase.html

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N-isopropylammelide_isopropylaminohydrolase.html

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N-malonylurea_hydrolase.html

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N-methyl-2-oxoglutaramate_hydrolase.html

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N-methyl_nucleosidase.html

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N-methylhydantoinase_(ATP-hydrolysing).html

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N-methylphosphoethanolamine_cytidylyltransferase.html

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N-substituted_formamide_deformylase.html

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N-succinylarginine_dihydrolase.html

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N2-(2-carboxyethyl)arginine_synthase.html

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N6-acetyl-beta-lysine_transaminase.html

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N6-hydroxylysine_O-acetyltransferase.html

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Na+-exporting_ATPase.html

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Na+-transporting_two-sector_ATPase.html

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NAD(+)—dinitrogen-reductase_ADP-D-ribosyltransferase.html

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NAD(+)—diphthamide_ADP-ribosyltransferase.html

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NAD(P)(+)—protein-arginine_ADP-ribosyltransferase.html

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NAD(P)+_nucleosidase.html

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NAD+_diphosphatase.html

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NAD+_nucleosidase.html

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NAD+_synthase.html

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NAD+_synthase_(glutamine-hydrolysing).html

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NADH_kinase.html

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Nalpha-benzyloxycarbonylleucine_hydrolase.html

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Naringenin-chalcone_synthase.html

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Naringenin_8-dimethylallyltransferase.html

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Natural_convection.html

  1. 𝐑𝐚 = Δ ρ g L 3 D μ \,\textbf{Ra}=\frac{\Delta\rho gL^{3}}{D\mu}
  2. Δ ρ \Delta\rho
  3. g g
  4. L L
  5. D D
  6. μ \mu
  7. Δ ρ = ρ 0 β Δ T \Delta\rho=\rho_{0}\beta\Delta T
  8. ρ 0 \rho_{0}
  9. β \beta
  10. Δ T \Delta T
  11. D D
  12. α \alpha
  13. D = α D=\alpha
  14. 𝐑𝐚 = ρ 0 g β Δ T L 3 α μ \,\textbf{Ra}=\frac{\rho_{0}g\beta\Delta TL^{3}}{\alpha\mu}
  15. G r = g β Δ T L 3 ν 2 Gr=\frac{g\beta\Delta TL^{3}}{\nu^{2}}
  16. g Δ ρ L 2 / μ g\Delta\rho L^{2}/\mu
  17. G r = g β Δ C L 3 ν 2 Gr=\frac{g\beta\Delta CL^{3}}{\nu^{2}}
  18. N u = [ N u 0 1 2 + R a 1 6 ( f 4 ( P r ) 300 ) 1 6 ] 2 Nu=\left[Nu_{0}^{\frac{1}{2}}+Ra^{\frac{1}{6}}\left(\frac{f_{4}\left(Pr\right)% }{300}\right)^{\frac{1}{6}}\right]^{2}
  19. f 4 ( P r ) = [ 1 + ( 0.5 P r ) 9 16 ] - 16 9 f_{4}(Pr)=\left[1+\left(\frac{0.5}{Pr}\right)^{\frac{9}{16}}\right]^{\frac{-16% }{9}}
  20. π D / 2 \pi D/2
  21. π \pi
  22. [ g L 3 ( t s - t ) ] / v 2 T [gL^{3}(t_{s}-t_{\infty})]/v^{2}T

NDP-glucose—starch_glucosyltransferase.html

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Neolactotetraosylceramide_alpha-2,3-sialyltransferase.html

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Nickel-transporting_ATPase.html

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Nicotianamine_aminotransferase.html

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Nicotianamine_synthase.html

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Nicotinamidase.html

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Nicotinamide-nucleotide_adenylyltransferase.html

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Nicotinamide-nucleotide_amidase.html

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Nicotinamide_phosphoribosyltransferase.html

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Nicotinate-nucleotide_adenylyltransferase.html

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Nicotinate-nucleotide_diphosphorylase_(carboxylating).html

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Nicotinate-nucleotide—dimethylbenzimidazole_phosphoribosyltransferase.html

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Nicotinate_glucosyltransferase.html

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Nicotinate_phosphoribosyltransferase.html

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Nisnevich_topology.html

  1. u α \coprod u_{\alpha}
  2. X α \coprod X_{\alpha}
  3. G ~ n cd ( X ) \tilde{G}_{n}^{\,\,\text{cd}}(X)
  4. E 2 p , q = H p ( X cd , G ~ q cd ) G q - p ( X ) E^{p,q}_{2}=H^{p}(X\text{cd},\tilde{G}_{q}^{\,\,\text{cd}})\Rightarrow G_{q-p}% (X)
  5. \ell
  6. 𝐙 / 𝐙 \mathbf{Z}/\ell\mathbf{Z}

Nitrate-transporting_ATPase.html

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NMN_nucleosidase.html

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Non-linear_least_squares.html

  1. m m
  2. ( x 1 , y 1 ) , ( x 2 , y 2 ) , , ( x m , y m ) , (x_{1},y_{1}),(x_{2},y_{2}),\dots,(x_{m},y_{m}),
  3. y = f ( x , s y m b o l β ) , y=f(x,symbol\beta),
  4. x x
  5. n n
  6. s y m b o l β = ( β 1 , β 2 , , β n ) , symbol\beta=(\beta_{1},\beta_{2},\dots,\beta_{n}),
  7. m n . m\geq n.
  8. s y m b o l β symbol\beta
  9. S = i = 1 m r i 2 S=\sum_{i=1}^{m}r_{i}^{2}
  10. r i = y i - f ( x i , s y m b o l β ) r_{i}=y_{i}-f(x_{i},symbol\beta)
  11. i = 1 , 2 , , m . i=1,2,\dots,m.
  12. S β j = 2 i r i r i β j = 0 ( j = 1 , , n ) . \frac{\partial S}{\partial\beta_{j}}=2\sum_{i}r_{i}\frac{\partial r_{i}}{% \partial\beta_{j}}=0\quad(j=1,\ldots,n).
  13. r i β j \frac{\partial r_{i}}{\partial\beta_{j}}
  14. β j β j k + 1 = β j k + Δ β j . \beta_{j}\approx\beta_{j}^{k+1}=\beta^{k}_{j}+\Delta\beta_{j}.\,
  15. Δ s y m b o l β \Delta symbol\beta\,
  16. s y m b o l β k symbol\beta^{k}\!
  17. f ( x i , s y m b o l β ) f ( x i , s y m b o l β k ) + j f ( x i , s y m b o l β k ) β j ( β j - β j k ) f ( x i , s y m b o l β k ) + j J i j Δ β j . f(x_{i},symbol\beta)\approx f(x_{i},symbol\beta^{k})+\sum_{j}\frac{\partial f(% x_{i},symbol\beta^{k})}{\partial\beta_{j}}\left(\beta_{j}-\beta^{k}_{j}\right)% \approx f(x_{i},symbol\beta^{k})+\sum_{j}J_{ij}\,\Delta\beta_{j}.
  18. r i β j = - J i j \frac{\partial r_{i}}{\partial\beta_{j}}=-J_{ij}
  19. r i = Δ y i - s = 1 n J i s Δ β s ; Δ y i = y i - f ( x i , s y m b o l β k ) . r_{i}=\Delta y_{i}-\sum_{s=1}^{n}J_{is}\ \Delta\beta_{s};\ \Delta y_{i}=y_{i}-% f(x_{i},symbol\beta^{k}).
  20. - 2 i = 1 m J i j ( Δ y i - s = 1 n J i s Δ β s ) = 0 -2\sum_{i=1}^{m}J_{ij}\left(\Delta y_{i}-\sum_{s=1}^{n}J_{is}\ \Delta\beta_{s}% \right)=0
  21. i = 1 m s = 1 n J i j J i s Δ β s = i = 1 m J i j Δ y i ( j = 1 , , n ) . \sum_{i=1}^{m}\sum_{s=1}^{n}J_{ij}J_{is}\ \Delta\beta_{s}=\sum_{i=1}^{m}J_{ij}% \ \Delta y_{i}\qquad(j=1,\dots,n).\,
  22. ( 𝐉 𝐓 𝐉 ) 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐉 𝐓 𝚫 𝐲 . \mathbf{\left(J^{T}J\right)\Delta symbol\beta=J^{T}\ \Delta y}.
  23. S = i = 1 m W i i r i 2 . S=\sum_{i=1}^{m}W_{ii}r_{i}^{2}.
  24. S = k j r k W k j r j S=\sum_{k}\sum_{j}r_{k}W_{kj}r_{j}\,
  25. ( 𝐉 𝐓 𝐖𝐉 ) 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐉 𝐓 𝐖 𝚫 𝐲 . \mathbf{\left(J^{T}WJ\right)\Delta symbol\beta=J^{T}W\ \Delta y}.
  26. S = i W i i ( y i - j X i j β j ) 2 S=\sum_{i}W_{ii}\left(y_{i}-\sum_{j}X_{ij}\beta_{j}\right)^{2}
  27. 𝐗 𝐓 𝐖𝐗 \mathbf{X^{T}WX}
  28. S i W i i ( y i - j J i j β j ) 2 S\approx\sum_{i}W_{ii}\left(y_{i}-\sum_{j}J_{ij}\beta_{j}\right)^{2}
  29. | S k - S k + 1 S k | < 0.0001. \left|\frac{S^{k}-S^{k+1}}{S^{k}}\right|<0.0001.
  30. | Δ β j β j | < 0.001 , j = 1 , , n . \left|\frac{\Delta\beta_{j}}{\beta_{j}}\right|<0.001,\qquad j=1,\dots,n.
  31. f ( x i , s y m b o l β ) β j δ f ( x i , s y m b o l β ) δ β j \frac{\partial f(x_{i},symbol\beta)}{\partial\beta_{j}}\approx\frac{\delta f(x% _{i},symbol\beta)}{\delta\beta_{j}}
  32. f ( x i , s y m b o l β ) f(x_{i},symbol\beta)\,
  33. β j \beta_{j}\,
  34. β j + δ β j \beta_{j}+\delta\beta_{j}\,
  35. δ β j \delta\beta_{j}\,
  36. f ( x i , s y m b o l β ) = α 1 + ( γ - x i β ) 2 f(x_{i},symbol\beta)=\frac{\alpha}{1+\left(\frac{\gamma-x_{i}}{\beta}\right)^{% 2}}
  37. α \alpha
  38. γ \gamma
  39. β \beta
  40. β ^ \hat{\beta}
  41. - β ^ -\hat{\beta}
  42. α β \alpha\beta
  43. β α \beta\alpha
  44. sin β \sin\beta\,
  45. β ^ + 2 n π \hat{\beta}+2n\pi
  46. f ( x i , s y m b o l β ) = α e β x i f(x_{i},symbol\beta)=\alpha e^{\beta x_{i}}
  47. log f ( x i , s y m b o l β ) = log α + β x i \log f(x_{i},symbol\beta)=\log\alpha+\beta x_{i}
  48. S = i ( log y i - log α - β x i ) 2 . S=\sum_{i}(\log y_{i}-\log\alpha-\beta x_{i})^{2}.\!
  49. V max V_{\max}
  50. K m K_{m}
  51. v = V max [ S ] K m + [ S ] v=\frac{V_{\max}[S]}{K_{m}+[S]}
  52. 1 v = 1 V max + K m V max [ S ] \frac{1}{v}=\frac{1}{V_{\max}}+\frac{K_{m}}{V_{\max}[S]}
  53. 1 v \frac{1}{v}
  54. 1 [ S ] \frac{1}{[S]}
  55. 1 V max \frac{1}{V_{\max}}
  56. K m V max \frac{K_{m}}{V_{\max}}
  57. [ S ] [S]
  58. ( 𝐉 𝐓 𝐖𝐉 ) 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = ( 𝐉 𝐓 𝐖 ) 𝚫 𝐲 \mathbf{\left(J^{T}WJ\right)\Delta symbol\beta=\left(J^{T}W\right)\Delta y}
  59. Δ s y m b o l β \Delta symbol\beta
  60. s y m b o l β k + 1 = s y m b o l β k + Δ s y m b o l β symbol\beta^{k+1}=symbol\beta^{k}+\Delta symbol\beta
  61. 𝚫 β \mathbf{\Delta\beta}
  62. s y m b o l β k + 1 = s y m b o l β k + f Δ s y m b o l β . symbol\beta^{k+1}=symbol\beta^{k}+f\ \Delta symbol\beta.
  63. s y m b o l β k . symbol\beta^{k}.
  64. ( 𝐉 𝐓 𝐖𝐉 + λ 𝐈 ) 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = ( 𝐉 𝐓 𝐖 ) 𝚫 𝐲 \mathbf{\left(J^{T}WJ+\lambda I\right)\Delta symbol\beta=\left(J^{T}W\right)% \Delta y}
  65. λ \lambda
  66. λ \lambda
  67. λ 𝐈 𝐉 𝐓 𝐖𝐉 , 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β 1 / λ 𝐉 𝐓 𝐖 𝚫 𝐲 . \lambda\mathbf{I\gg{}J^{T}WJ},\ \mathbf{\Delta symbol\beta}\approx 1/\lambda% \mathbf{J^{T}W\ \Delta y}.
  68. 𝐉 𝐓 𝐖 𝚫 𝐲 \mathbf{J^{T}W\ \Delta y}
  69. λ \lambda
  70. 1 / trace ( 𝐉 𝐓 𝐖𝐉 ) - 𝟏 1/\mbox{trace}~{}\mathbf{\left(J^{T}WJ\right)^{-1}}
  71. 𝐫 = 𝚫 𝐲 - 𝐉 \Deltasymbol β . \mathbf{r=\Delta y-J\ \Deltasymbol\beta}.
  72. 𝐉 = 𝐐𝐑 \mathbf{J=QR}
  73. m × m m\times m
  74. m × n m\times n
  75. n × n n\times n
  76. 𝐑 n \mathbf{R}_{n}
  77. m - n × n m-n\times n
  78. 𝐑 n \mathbf{R}_{n}
  79. 𝐑 = [ 𝐑 n 𝟎 ] \mathbf{R}=\begin{bmatrix}\mathbf{R}_{n}\\ \mathbf{0}\end{bmatrix}
  80. 𝐐 T \mathbf{Q}^{T}
  81. 𝐐 𝐓 𝐫 = 𝐐 𝐓 𝚫 𝐲 - 𝐑 \Deltasymbol β = [ ( 𝐐 𝐓 𝚫 𝐲 - 𝐑 \Deltasymbol β ) n ( 𝐐 𝐓 𝚫 𝐲 ) m - n ] \mathbf{Q^{T}r=Q^{T}\ \Delta y-R\ \Deltasymbol\beta}=\begin{bmatrix}\mathbf{% \left(Q^{T}\ \Delta y-R\ \Deltasymbol\beta\right)}_{n}\\ \mathbf{\left(Q^{T}\ \Delta y\right)}_{m-n}\end{bmatrix}
  82. S = 𝐫 𝐓 𝐐𝐐 𝐓 𝐫 = 𝐫 𝐓 𝐫 S=\mathbf{r^{T}QQ^{T}r=r^{T}r}
  83. 𝐑 𝐧 \Deltasymbol β = ( 𝐐 𝐓 𝚫 𝐲 ) 𝐧 . \mathbf{R_{n}\ \Deltasymbol\beta=\left(Q^{T}\ \Delta y\right)_{n}}.\,
  84. 𝐉 = 𝐔𝐬𝐲𝐦𝐛𝐨𝐥 𝚺 𝐕 𝐓 \mathbf{J=Usymbol\Sigma V^{T}}\,
  85. 𝐔 \mathbf{U}
  86. s y m b o l Σ symbol\Sigma
  87. 𝐕 \mathbf{V}
  88. 𝐉 𝐓 𝐉 \mathbf{J^{T}J}
  89. 𝐉 \mathbf{J}
  90. 𝐬𝐲𝐦𝐛𝐨𝐥 𝚫 β = 𝐕𝐬𝐲𝐦𝐛𝐨𝐥 𝚺 - 𝟏 ( 𝐔 𝐓 𝐬𝐲𝐦𝐛𝐨𝐥 𝚫 𝐲 ) n . \mathbf{symbol\Delta\beta=Vsymbol\Sigma^{-1}\left(U^{T}\ symbol\Delta y\right)% }_{n}.\,
  91. f ( x i , s y m b o l β ) = f k ( x i , s y m b o l β ) + j J i j Δ β j + 1 2 j k Δ β j Δ β k H j k ( i ) , H j k ( i ) = 2 f ( x i , s y m b o l β ) β j β k . f(x_{i},symbol\beta)=f^{k}(x_{i},symbol\beta)+\sum_{j}J_{ij}\,\Delta\beta_{j}+% \frac{1}{2}\sum_{j}\sum_{k}\Delta\beta_{j}\,\Delta\beta_{k}\,H_{jk_{(i)}},\ H_% {jk_{(i)}}=\frac{\partial^{2}f(x_{i},symbol\beta)}{\partial\beta_{j}\,\partial% \beta_{k}}.

Non-specific_serine::threonine_protein_kinase.html

  1. \rightleftharpoons

Nonhypotenuse_number.html

  1. K ( m 2 + n 2 ) K(m^{2}+n^{2})

Nonlinear_eigenproblem.html

  1. A ( λ ) 𝐱 = 0 , A(\lambda)\mathbf{x}=0,\,
  2. λ \lambda
  3. A ( λ ) A(\lambda)
  4. λ \lambda
  5. B 𝐯 = λ 𝐯 B\mathbf{v}=\lambda\mathbf{v}
  6. A ( λ ) = B - λ I A(\lambda)=B-\lambda I
  7. A ( λ ) 𝐱 = ( A 2 λ 2 + A 1 λ + A 0 ) 𝐱 = 0 , A(\lambda)\mathbf{x}=(A_{2}\lambda^{2}+A_{1}\lambda+A_{0})\mathbf{x}=0,\,
  8. 𝐲 = λ 𝐱 \mathbf{y}=\lambda\mathbf{x}
  9. ( - A 0 0 0 I ) ( 𝐱 𝐲 ) = λ ( A 1 A 2 I 0 ) ( 𝐱 𝐲 ) , \begin{pmatrix}-A_{0}&0\\ 0&I\end{pmatrix}\begin{pmatrix}\mathbf{x}\\ \mathbf{y}\end{pmatrix}=\lambda\begin{pmatrix}A_{1}&A_{2}\\ I&0\end{pmatrix}\begin{pmatrix}\mathbf{x}\\ \mathbf{y}\end{pmatrix},

Nonpolar-amino-acid-transporting_ATPase.html

  1. \rightleftharpoons

Normal_shock_tables.html

  1. γ \gamma
  2. M 1 M_{1}
  3. γ \gamma
  4. M 1 M_{1}
  5. γ \gamma
  6. M 2 M_{2}
  7. M 2 = M 1 2 ( γ - 1 ) + 2 2 γ M 1 2 - ( γ - 1 ) M_{2}=\sqrt{\frac{M_{1}^{2}\left(\gamma-1\right)+2}{2\gamma M_{1}^{2}-\left(% \gamma-1\right)}}
  8. p 2 p_{2}
  9. p 1 p_{1}
  10. p 2 p 1 = 2 γ M 1 2 γ + 1 - γ - 1 γ + 1 \frac{p_{2}}{p_{1}}=\frac{2\gamma M_{1}^{2}}{\gamma+1}-\frac{\gamma-1}{\gamma+1}
  11. ρ 2 \rho_{2}
  12. ρ 1 \rho_{1}
  13. ρ 2 ρ 1 = ( γ + 1 ) M 1 2 ( γ - 1 ) M 1 2 + 2 \frac{\rho_{2}}{\rho_{1}}=\frac{\left(\gamma+1\right)M_{1}^{2}}{\left(\gamma-1% \right)M_{1}^{2}+2}
  14. T 2 T_{2}
  15. T 1 T_{1}
  16. T 2 T 1 = ( 1 + γ - 1 2 M 1 2 ) ( 2 γ γ - 1 M 1 2 - 1 ) M 1 2 ( 2 γ γ - 1 + γ - 1 2 ) \frac{T_{2}}{T_{1}}=\frac{\left(1+\frac{\gamma-1}{2}M_{1}^{2}\right)\left(% \frac{2\gamma}{\gamma-1}M_{1}^{2}-1\right)}{M_{1}^{2}\left(\frac{2\gamma}{% \gamma-1}+\frac{\gamma-1}{2}\right)}
  17. p 01 p_{01}
  18. p 02 p_{02}
  19. p 02 p 01 = ( γ + 1 2 M 1 2 1 + γ - 1 2 M 1 2 ) γ γ - 1 ( 1 2 γ γ + 1 M 1 2 - γ - 1 γ + 1 ) 1 γ - 1 \frac{p_{02}}{p_{01}}=\left(\frac{\frac{\gamma+1}{2}M_{1}^{2}}{1+\frac{\gamma-% 1}{2}M_{1}^{2}}\right)^{\frac{\gamma}{\gamma-1}}\left(\frac{1}{\frac{2\gamma}{% \gamma+1}M_{1}^{2}-\frac{\gamma-1}{\gamma+1}}\right)^{\frac{1}{\gamma-1}}

Nuatigenin_3beta-glucosyltransferase.html

  1. \rightleftharpoons

Nuclear_timescale.html

  1. τ n u c = total mass of fuel available rate of fuel consumption × fraction of star over which fuel is burned = M X L Q × F \tau_{nuc}=\frac{\mbox{total mass of fuel available}~{}}{\mbox{rate of fuel % consumption}~{}}\times\mbox{fraction of star over which fuel is burned}~{}=% \frac{MX}{\frac{L}{Q}}\times F

Nucleoplasmin_ATPase.html

  1. \rightleftharpoons

Nucleoside-diphosphatase.html

  1. \rightleftharpoons

Nucleoside-phosphate_kinase.html

  1. \rightleftharpoons

Nucleoside-triphosphatase.html

  1. \rightleftharpoons

Nucleoside-triphosphate-aldose-1-phosphate_nucleotidyltransferase.html

  1. \rightleftharpoons

Nucleoside-triphosphate_diphosphatase.html

  1. \rightleftharpoons

Nucleoside-triphosphate—adenylate_kinase.html

  1. \rightleftharpoons

Nucleoside_deoxyribosyltransferase.html

  1. \rightleftharpoons

Nucleoside_phosphotransferase.html

  1. \rightleftharpoons

Nucleoside_ribosyltransferase.html

  1. \rightleftharpoons

Nucleotide_diphosphatase.html

  1. \rightleftharpoons

Nucleotide_diphosphokinase.html

  1. \rightleftharpoons

Nullor.html

  1. ( v 1 i 1 ) = ( 0 0 0 0 ) ( v 2 i 2 ) . \begin{pmatrix}v_{1}\\ i_{1}\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}\begin{pmatrix}v_{2}\\ i_{2}\end{pmatrix}\ .
  2. i O U T = v I N R R - i B , i_{OUT}=\frac{v_{IN}}{R_{R}}-i_{B}\ ,

Numerical_range.html

  1. W ( A ) = { 𝐱 * A 𝐱 𝐱 * 𝐱 𝐱 n , x 0 } W(A)=\left\{\frac{\mathbf{x}^{*}A\mathbf{x}}{\mathbf{x}^{*}\mathbf{x}}\mid% \mathbf{x}\in\mathbb{C}^{n},\ x\not=0\right\}
  2. r ( A ) = sup { | λ | : λ W ( A ) } = sup x = 1 | A x , x | . r(A)=\sup\{|\lambda|:\lambda\in W(A)\}=\sup_{\|x\|=1}|\langle Ax,x\rangle|.
  3. W ( α A + β I ) = α W ( A ) + { β } W(\alpha A+\beta I)=\alpha W(A)+\{\beta\}
  4. W ( A ) W(A)
  5. A + A * A+A^{*}
  6. W ( ) W(\cdot)
  7. W ( A + B ) W ( A ) + W ( B ) W(A+B)\subseteq W(A)+W(B)
  8. W ( A ) W(A)
  9. W ( A ) W(A)
  10. W ( A ) W(A)
  11. W ( A ) W(A)
  12. r ( ) r(\cdot)
  13. r ( A n ) r ( A ) n r(A^{n})\leq r(A)^{n}

O-acetylhomoserine_aminocarboxypropyltransferase.html

  1. \rightleftharpoons

O-dihydroxycoumarin_7-O-glucosyltransferase.html

  1. \rightleftharpoons

O-phosphoserine_sulfhydrylase.html

  1. \rightleftharpoons

Object_categorization_from_image_search.html

  1. w w
  2. d \displaystyle d
  3. z \displaystyle z
  4. P ( w | d ) = z = 1 Z P ( w | z ) P ( z | d ) \displaystyle P(w|d)=\sum_{z=1}^{Z}P(w|z)P(z|d)
  5. w \displaystyle w
  6. d \displaystyle d
  7. z \displaystyle z
  8. P ( w | z ) \displaystyle P(w|z)
  9. P ( z | d ) \displaystyle P(z|d)
  10. L = d = 1 D w = 1 W P ( w | d ) n ( w | d ) \displaystyle L=\prod_{d=1}^{D}\prod_{w=1}^{W}P(w|d)^{n(w|d)}
  11. x \displaystyle x
  12. P ( w | d ) = z = 1 Z P ( w , x | z ) P ( z | d ) \displaystyle P(w|d)=\sum_{z=1}^{Z}P(w,x|z)P(z|d)
  13. P ( w , x | z ) \displaystyle P(w,x|z)
  14. P ( d ) \displaystyle P(d)
  15. x \displaystyle x
  16. P ( w , x | d ) = z = 1 Z c = 1 C P ( w , x | c , z ) P ( c ) P ( z | d ) \displaystyle P(w,x|d)=\sum_{z=1}^{Z}\sum_{c=1}^{C}P(w,x|c,z)P(c)P(z|d)
  17. P ( w , x | c , z ) \displaystyle P(w,x|c,z)
  18. P ( d ) \displaystyle P(d)
  19. P ( c ) \displaystyle P(c)
  20. C \displaystyle C
  21. P ( w | z ) \displaystyle P(w|z)
  22. K \displaystyle K
  23. K \displaystyle K
  24. P ( z | c ) \displaystyle P(z|c)
  25. P ( x | z , c ) \displaystyle P(x|z,c)
  26. z \displaystyle z
  27. c \displaystyle c
  28. x \displaystyle x
  29. P ( I | c ) = i j P ( x i | z j , c ) P ( z j | c ) \displaystyle P(I|c)=\prod_{i}\sum_{j}P(x_{i}|z_{j},c)P(z_{j}|c)
  30. P ( I | c f ) P ( I | c b ) > λ A c b - λ R c b λ R c f - λ A c f P ( c b ) P ( c f ) \displaystyle\frac{P(I|c_{f})}{P(I|c_{b})}>\frac{\lambda_{Ac_{b}}-\lambda_{Rc_% {b}}}{\lambda_{Rc_{f}}-\lambda_{Ac_{f}}}\frac{P(c_{b})}{P(c_{f})}
  31. c f \displaystyle c_{f}
  32. c b \displaystyle c_{b}

Objectivity_(frame_invariance).html

  1. O O
  2. ( x 0 , t 0 ) (x_{0},t_{0})
  3. ( x , t ) (x,t)
  4. x x
  5. t t
  6. * *
  7. Q ( t ) Q(t)
  8. x * - x 0 * = Q ( t ) ( x - x 0 ) . \ x^{*}-x_{0}^{*}=Q(t)(x-x_{0}).
  9. C ( t ) C(t)
  10. α \alpha
  11. x x
  12. x * x^{*}
  13. x * = c ( t ) + Q ( t ) x where c ( t ) = x 0 * - Q ( t ) x 0 and α = t * - t = t 0 * - t 0 . \ x^{*}=c(t)+Q(t)x\quad\,\text{where}\quad c(t)=x_{0}^{*}-Q(t)x_{0}\quad\,% \text{and}\quad\alpha=t^{*}-t=t_{0}^{*}-t_{0}.
  14. ( x , t ) (x,t)
  15. ( x * , t * ) (x^{*},t^{*})
  16. O O
  17. x x
  18. y y
  19. O * O^{*}
  20. x * x^{*}
  21. y * y^{*}
  22. O O
  23. u = y - x u=y-x
  24. O * O^{*}
  25. u * = y * - x * = c ( t ) + Q ( t ) y - c ( t ) - Q ( t ) x = Q ( t ) ( y - x ) = Q ( t ) u . \ \begin{aligned}\displaystyle u^{*}&\displaystyle=y^{*}-x^{*}\\ &\displaystyle=c(t)+Q(t)y-c(t)-Q(t)x\\ &\displaystyle=Q(t)(y-x)\\ &\displaystyle=Q(t)u.\end{aligned}
  26. u u
  27. u * = Q ( t ) u , \ u^{*}=Q(t)u,
  28. | u * | = | u | |u^{*}|=|u|
  29. Q ( t ) Q(t)
  30. Q ( t ) T Q ( t ) = I Q(t)^{T}Q(t)=I
  31. I I
  32. x = Q ( t ) T [ x * - c ( t ) ] . \ x=Q(t)^{T}[x^{*}-c(t)].
  33. v ( x , t ) = x ˙ = Q ˙ ( t ) T [ x * - c ( t ) ] + Q ( t ) T [ v * - c ˙ ( t ) ] . \ v(x,t)=\dot{x}=\dot{Q}(t)^{T}[x^{*}-c(t)]+Q(t)^{T}[v^{*}-\dot{c}(t)].
  34. v * ( x * , t ) = Q ( t ) v + c ˙ ( t ) - Q ( t ) Q ˙ ( t ) T [ x * - c ( t ) ] = Q ( t ) v + c ˙ ( t ) + Ω ( t ) [ x * - c ( t ) ] , \ \begin{aligned}\displaystyle v^{*}(x^{*},t)&\displaystyle=Q(t)v+\dot{c}(t)-Q% (t)\dot{Q}(t)^{T}[x^{*}-c(t)]\\ &\displaystyle=Q(t)v+\dot{c}(t)+\Omega(t)[x^{*}-c(t)],\end{aligned}
  35. Ω ( t ) = Q ˙ ( t ) Q ( t ) T = - Ω ( t ) T = - Q ( t ) Q ˙ ( t ) T , \ \Omega(t)=\dot{Q}(t)Q(t)^{T}=-\Omega(t)^{T}=-Q(t)\dot{Q}(t)^{T},
  36. O O
  37. O * O^{*}
  38. c ˙ \dot{c}
  39. Ω [ x * - c ] \Omega[x^{*}-c]
  40. c ˙ + Ω ( x * - c ) = 0 , \ \dot{c}+\Omega(x^{*}-c)=0,
  41. x * = c 0 + Q 0 x where c ˙ 0 = 0 and Q 0 ˙ = 0 , \ x^{*}=c_{0}+Q_{0}x\quad\,\text{where}\quad\dot{c}_{0}=0\quad\,\text{and}% \quad\dot{Q_{0}}=0,
  42. v v
  43. a a
  44. a * = v ˙ * = Q ˙ v + Q a + c ¨ + Ω ˙ ( x * - c ) + Ω ( v - c ˙ ) , \ a^{*}=\dot{v}^{*}=\dot{Q}v+Qa+\ddot{c}+\dot{\Omega}(x^{*}-c)+\Omega(v-\dot{c% }),
  45. a * = Q a + c ¨ + ( Ω ˙ - Ω 2 ) ( x * - c ) + 2 Ω ( v - c ˙ ) . \ a^{*}=Qa+\ddot{c}+(\dot{\Omega}-\Omega^{2})(x^{*}-c)+2\Omega(v-\dot{c}).
  46. n n
  47. u 1 u n u_{1}\otimes\dots\otimes u_{n}
  48. ( u 1 u n ) * = Q u 1 Q u n . \ (u_{1}\otimes\dots\otimes u_{n})^{*}=Qu_{1}\otimes\dots\otimes Qu_{n}.
  49. A = u 1 u 2 A=u_{1}\otimes u_{2}
  50. A * = ( u 1 u 2 ) * = Q u 1 Q u 2 = Q ( u 1 u 2 ) Q T = Q A Q T . \ A^{*}=(u_{1}\otimes u_{2})^{*}=Qu_{1}\otimes Qu_{2}=Q(u_{1}\otimes u_{2})Q^{% T}=QAQ^{T}.
  51. n n
  52. Φ \Phi
  53. n = 0 n=0
  54. Φ * = Φ . \ \Phi^{*}=\Phi.
  55. x x
  56. x * x^{*}
  57. F = x X and F * = x * X , \ F=\frac{\partial x}{\partial X}\qquad\textrm{and}\qquad F^{*}=\frac{\partial x% ^{*}}{\partial X},
  58. X X
  59. F * = x * x x X = Q F . \ F^{*}=\frac{\partial x^{*}}{\partial x}\frac{\partial x}{\partial X}=QF.
  60. F F
  61. X X
  62. t t
  63. σ \sigma
  64. x x
  65. n n
  66. t = σ n t=\sigma n
  67. t * = σ * n * t^{*}=\sigma^{*}n^{*}
  68. t t
  69. n n
  70. t * = σ * n * Q t = σ * Q n Q σ n = σ * Q n σ * = Q σ Q T . \ \begin{array}[]{rrcl}&t^{*}&=&\sigma^{*}n^{*}\\ \Rightarrow&Qt&=&\sigma^{*}Qn\\ \Rightarrow&Q\sigma n&=&\sigma^{*}Qn\\ \Rightarrow&\sigma^{*}&=&Q\sigma Q^{T}.\end{array}
  71. P P
  72. P F T = J σ , \ PF^{T}=J\sigma,
  73. J = det ( F ) J=\det(F)
  74. Q Q
  75. J * = det ( F * ) = det ( Q F ) = det ( Q ) det ( F ) = det ( F ) = J . \ J^{*}=\det(F^{*})=\det(QF)=\det(Q)\det(F)=\det(F)=J.
  76. P * ( F * ) T = J * σ * P * ( Q F ) T = J Q σ Q T P * F T Q T = Q J σ Q T P * F T Q T = Q P F T Q T P * = Q P . \ \begin{array}[]{rrcl}&P^{*}(F^{*})^{T}&=&J^{*}\sigma^{*}\\ \Rightarrow&P^{*}(QF)^{T}&=&JQ\sigma Q^{T}\\ \Rightarrow&P^{*}F^{T}Q^{T}&=&QJ\sigma Q^{T}\\ \Rightarrow&P^{*}F^{T}Q^{T}&=&QPF^{T}Q^{T}\\ \Rightarrow&P^{*}&=&QP.\end{array}
  77. S = F - 1 P S=F^{-1}P
  78. S * = ( F * ) - 1 P * = ( Q F ) - 1 Q P = F - 1 Q - 1 Q P = F - 1 P = S . \ S^{*}=(F^{*})^{-1}P^{*}=(QF)^{-1}QP=F^{-1}Q^{-1}QP=F^{-1}P=S.
  79. σ \sigma
  80. P P
  81. S S
  82. u * = Q u u^{*}=Qu
  83. A * = Q A Q T A^{*}=QAQ^{T}
  84. u ˙ * = Q ˙ u + Q u ˙ and A ˙ * = Q ˙ A Q T + Q A ˙ Q T + Q A Q ˙ T . \ \dot{u}^{*}=\dot{Q}u+Q\dot{u}\quad\,\text{and}\quad\dot{A}^{*}=\dot{Q}AQ^{T}% +Q\dot{A}Q^{T}+QA\dot{Q}^{T}.
  85. l l
  86. l = F ˙ F - 1 = d + w , \ l=\dot{F}F^{-1}=d+w,
  87. d d
  88. w w
  89. l l
  90. d d
  91. w w
  92. l * = F ˙ * ( F * ) - 1 = ( Q ˙ F + Q F ˙ ) ( Q F ) - 1 = ( Q ˙ F + Q F ˙ ) F - 1 Q T = Q ˙ F F - 1 Q T + Q F ˙ F - 1 Q - 1 = Q ˙ Q T + Q l Q - 1 = Ω + Q l Q - 1 . \ \begin{aligned}\displaystyle l^{*}&\displaystyle=\dot{F}^{*}(F^{*})^{-1}\\ &\displaystyle=(\dot{Q}F+Q\dot{F})(QF)^{-1}\\ &\displaystyle=(\dot{Q}F+Q\dot{F})F^{-1}Q^{T}\\ &\displaystyle=\dot{Q}FF^{-1}Q^{T}+Q\dot{F}F^{-1}Q^{-1}\\ &\displaystyle=\dot{Q}Q^{T}+QlQ^{-1}\\ &\displaystyle=\Omega+QlQ^{-1}.\end{aligned}
  93. l = d + w l=d+w
  94. Q ˙ = w * Q - Q w and Q ˙ T = - Q T w * + w Q T \ \dot{Q}=w^{*}Q-Qw\quad\,\text{and}\quad\dot{Q}^{T}=-Q^{T}w^{*}+wQ^{T}
  95. u ˙ * = Q ˙ u + Q u ˙ u ˙ * = ( w * Q - Q w ) u + Q u ˙ u ˙ * = w * u * - Q w u + Q u ˙ ( u ˙ - w u ) * = Q ( u ˙ - w u ) u ¯ * = Q u ¯ , \ \begin{array}[]{rrcl}&\dot{u}^{*}&=&\dot{Q}u+Q\dot{u}\\ \Rightarrow&\dot{u}^{*}&=&(w^{*}Q-Qw)u+Q\dot{u}\\ \Rightarrow&\dot{u}^{*}&=&w^{*}u^{*}-Qwu+Q\dot{u}\\ \Rightarrow&(\dot{u}-wu)^{*}&=&Q(\dot{u}-wu)\\ \Rightarrow&\bar{u}^{*}&=&Q\bar{u},\end{array}
  96. u u
  97. u ¯ = u ˙ - w u , \ \bar{u}=\dot{u}-wu,
  98. A A
  99. ( A ˙ - w A + A w ) * = Q ( A ˙ - w A + A w ) Q T A ¯ * = Q A ¯ Q T . \ \begin{array}[]{rrcl}&(\dot{A}-wA+Aw)^{*}&=&Q(\dot{A}-wA+Aw)Q^{T}\\ \Rightarrow&\bar{A}^{*}&=&Q\bar{A}Q^{T}.\end{array}
  100. A ¯ = A ˙ - w A + A w . \ \bar{A}=\dot{A}-wA+Aw.
  101. σ = G ( F ) . \ \sigma=G(F).
  102. σ * = G ( F * ) \sigma^{*}=G(F^{*})
  103. σ \sigma
  104. F F
  105. σ * = G ( F * ) Q σ Q T = G ( Q F ) Q G ( F ) Q T = G ( Q F ) . \ \begin{array}[]{rrcl}&\sigma^{*}&=&G(F^{*})\\ \Rightarrow&Q\sigma Q^{T}&=&G(QF)\\ \Rightarrow&QG(F)Q^{T}&=&G(QF).\end{array}
  106. G G
  107. σ \sigma
  108. b = F F T b=FF^{T}
  109. σ = h ( b ) . \ \sigma=h(b).
  110. h h
  111. σ * = h ( b * ) Q σ Q T = h ( F * ( F * ) T ) Q h ( b ) Q T = h ( Q F F T Q T ) Q h ( b ) Q T = h ( Q b Q T ) . \ \begin{array}[]{rrcl}&\sigma^{*}&=&h(b^{*})\\ \Rightarrow&Q\sigma Q^{T}&=&h(F^{*}(F^{*})^{T})\\ \Rightarrow&Qh(b)Q^{T}&=&h(QFF^{T}Q^{T})\\ \Rightarrow&Qh(b)Q^{T}&=&h(QbQ^{T}).\end{array}

Occupancy_grid_mapping.html

  1. p ( m z 1 : t , x 1 : t ) p(m\mid z_{1:t},x_{1:t})
  2. m m
  3. z 1 : t z_{1:t}
  4. x 1 : t x_{1:t}
  5. m m
  6. m i m_{i}
  7. p ( m i ) p(m_{i})
  8. p ( m z 1 : t , x 1 : t ) p(m\mid z_{1:t},x_{1:t})
  9. 2 10 , 000 2^{10,000}
  10. p ( m i z 1 : t , x 1 : t ) p(m_{i}\mid z_{1:t},x_{1:t})
  11. m i m_{i}
  12. p ( m z 1 : t , x 1 : t ) = i p ( m i z 1 : t , x 1 : t ) p(m\mid z_{1:t},x_{1:t})=\prod_{i}p(m_{i}\mid z_{1:t},x_{1:t})

Octopamine_dehydratase.html

  1. \rightleftharpoons

Ogden_(hyperelastic_model).html

  1. λ j \,\!\lambda_{j}
  2. j = 1 , 2 , 3 \,\!j=1,2,3
  3. W ( λ 1 , λ 2 , λ 3 ) = p = 1 N μ p α p ( λ 1 α p + λ 2 α p + λ 3 α p - 3 ) W\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)=\sum_{p=1}^{N}\frac{\mu_{p}}% {\alpha_{p}}\left(\lambda_{1}^{\alpha_{p}}+\lambda_{2}^{\alpha_{p}}+\lambda_{3% }^{\alpha_{p}}-3\right)
  4. N N
  5. μ p \,\!\mu_{p}
  6. α p \,\!\alpha_{p}
  7. W ( λ 1 , λ 2 ) = p = 1 N μ p α p ( λ 1 α p + λ 2 α p + λ 1 - α p λ 2 - α p - 3 ) W\left(\lambda_{1},\lambda_{2}\right)=\sum_{p=1}^{N}\frac{\mu_{p}}{\alpha_{p}}% \left(\lambda_{1}^{\alpha_{p}}+\lambda_{2}^{\alpha_{p}}+\lambda_{1}^{-\alpha_{% p}}\lambda_{2}^{-\alpha_{p}}-3\right)
  8. 2 μ = p = 1 N μ p α p . 2\mu=\sum_{p=1}^{N}\mu_{p}\alpha_{p}.
  9. N = 3 N=3
  10. N = 1 N=1
  11. α = 2 \alpha=2
  12. N = 2 N=2
  13. α 1 = 2 \alpha_{1}=2
  14. α 2 = - 2 \alpha_{2}=-2
  15. λ 1 λ 2 λ 3 = 1 \lambda_{1}\lambda_{2}\lambda_{3}=1
  16. σ j = p + λ j W λ j \sigma_{j}=p+\lambda_{j}\frac{\partial W}{\partial\lambda_{j}}
  17. σ j = λ j P j \,\!\sigma_{j}=\lambda_{j}P_{j}
  18. λ = l l 0 \lambda=\frac{l}{l_{0}}
  19. σ j = p + p = 1 N μ p λ j α p \sigma_{j}=p+\sum_{p=1}^{N}\mu_{p}\lambda_{j}^{\alpha_{p}}
  20. p p
  21. σ 2 = σ 3 = 0 \sigma_{2}=\sigma_{3}=0
  22. σ 1 = p = 1 N μ p ( λ α p - λ - 1 2 α p ) \sigma_{1}=\sum_{p=1}^{N}\mu_{p}\left(\lambda^{\alpha_{p}}-\lambda^{-\frac{1}{% 2}\alpha_{p}}\right)
  23. W ( 𝐂 ) = μ 2 ( I 1 C - 3 ) W(\mathbf{C})=\frac{\mu}{2}(I_{1}^{C}-3)
  24. μ \mu
  25. W W
  26. W ( 𝐂 ) = μ 1 2 ( I 1 C - 3 ) - μ 2 2 ( I 2 C - 3 ) W(\mathbf{C})=\frac{\mu_{1}}{2}\left(I_{1}^{C}-3\right)-\frac{\mu_{2}}{2}\left% (I_{2}^{C}-3\right)

Oleate_hydratase.html

  1. \rightleftharpoons

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

  1. \rightleftharpoons

Oligogalacturonide_lyase.html

  1. \rightleftharpoons

Oligopeptide-transporting_ATPase.html

  1. \rightleftharpoons

Oligosaccharide-diphosphodolichol_diphosphatase.html

  1. \rightleftharpoons

Oligosaccharide-transporting_ATPase.html

  1. \rightleftharpoons

Omega-amidase.html

  1. \rightleftharpoons

Omega_equation.html

  1. ω \omega
  2. ω = d p d t \omega=\frac{dp}{dt}
  3. σ H 2 ω + f 2 2 ω p 2 = f p [ 𝐕 g H ( ζ g + f ) ] - H 2 ( 𝐕 g H ϕ p ) \sigma\nabla^{2}_{H}\omega+f^{2}\frac{\partial^{2}\omega}{\partial p^{2}}=f% \frac{\partial}{\partial p}\left[\mathbf{V}_{g}\cdot\nabla_{H}(\zeta_{g}+f)% \right]-\nabla^{2}_{H}\left(\mathbf{V}_{g}\cdot\nabla_{H}\frac{\partial\phi}{% \partial p}\right)
  4. f f
  5. σ \sigma
  6. 𝐕 g \mathbf{V}_{g}
  7. ζ g \zeta_{g}
  8. ϕ \phi
  9. H 2 \nabla^{2}_{H}
  10. H \nabla_{H}
  11. ω \omega
  12. ξ t + V η - f ω p = ( ξ ω p - ω ξ p ) + k ω × V p \frac{\partial\xi}{\partial t}+V\cdot\nabla\eta-f\frac{\partial\omega}{% \partial p}=\left(\xi\frac{\partial\omega}{\partial p}-\omega\frac{\partial\xi% }{\partial p}\right)+k\cdot\nabla\omega\times\frac{\partial V}{\partial p}
  13. ξ \xi
  14. V V
  15. x x
  16. y y
  17. u u
  18. v v
  19. η \eta
  20. f f
  21. ω = d p d t \omega=\frac{dp}{dt}
  22. p p
  23. k k
  24. \nabla
  25. ( ξ ω p - ω ξ p ) \left(\xi\frac{\partial\omega}{\partial p}-\omega\frac{\partial\xi}{\partial p% }\right)
  26. k ω × V p k\cdot\nabla\omega\times\frac{\partial V}{\partial p}
  27. k ( Z p ) p ln θ k\equiv\left(\frac{\partial Z}{\partial p}\right)\frac{\partial}{\partial p}\ln\theta
  28. q q
  29. C p C_{p}
  30. R R
  31. θ \theta
  32. ϕ \phi
  33. ( g Z ) (gZ)
  34. ω \omega
  35. ξ = g f 2 Z \xi=\frac{g}{f}\nabla^{2}Z
  36. k ^ ω × V p = ω y u p - ω x v p \hat{k}\cdot\nabla\omega\times\frac{\partial V}{\partial p}=\frac{\partial% \omega}{\partial y}\frac{\partial u}{\partial p}-\frac{\partial\omega}{% \partial x}\frac{\partial v}{\partial p}
  37. t ( g f 2 Z ) + V η - f ω p = ( ξ ω p - ω ξ p ) + ( ω x v p ) \frac{\partial}{\partial t}\left(\frac{g}{f}\nabla^{2}Z\right)+V\cdot\nabla% \eta-f\frac{\partial\omega}{\partial p}=\left(\xi\frac{\partial\omega}{% \partial p}-\omega\frac{\partial\xi}{\partial p}\right)+\left(\frac{\partial% \omega}{\partial x}\frac{\partial v}{\partial p}\right)
  38. p p
  39. g f t 2 ( Z p ) + p ( V η ) - f 2 ω p 2 - f p ω p = p ( ξ ω p - ω ξ p ) + p ( ω y u p - ω x v p ) \frac{g}{f}\frac{\partial}{\partial t}\nabla^{2}\left(\frac{\partial Z}{% \partial p}\right)+\frac{\partial}{\partial p}(V\cdot\nabla\eta)-f\frac{% \partial^{2}\omega}{\partial p^{2}}-\frac{\partial f}{\partial p}\frac{% \partial\omega}{\partial p}=\frac{\partial}{\partial p}\left(\xi\frac{\partial% \omega}{\partial p}-\omega\frac{\partial\xi}{\partial p}\right)+\frac{\partial% }{\partial p}\left(\frac{\partial\omega}{\partial y}\cdot\frac{\partial u}{% \partial p}-\frac{\partial\omega}{\partial x}\cdot\frac{\partial v}{\partial p% }\right)
  40. 2 \nabla^{2}
  41. 2 ( - Z p ) + 2 V ( - Z p ) - 2 k ω = R C p g 2 q p \nabla^{2}\left(-\frac{\partial Z}{\partial p}\right)+\nabla^{2}V\cdot\nabla% \left(-\frac{\partial Z}{\partial p}\right)-\nabla^{2}k\omega=\frac{R}{C_{p}% \cdot g}\cdot\frac{\nabla^{2}q}{p}
  42. g k = σ gk=\sigma
  43. ω \omega
  44. ω 1 \omega_{1}
  45. ω 2 \omega_{2}
  46. 2 ω 1 + f 2 σ 2 ω 1 p 2 = 1 σ [ p J ( ϕ , η ) + 1 f 2 J ( ϕ , - ϕ p ) ] - f σ p ( ω y u p - ω x v p ) - f σ p ( ξ ω p - ω ξ p ) \nabla^{2}\omega_{1}+\frac{f^{2}}{\sigma}\frac{\partial^{2}\omega_{1}}{% \partial p^{2}}=\frac{1}{\sigma}\left[\frac{\partial}{\partial p}J(\phi,\eta)+% \frac{1}{f}\nabla^{2}J\left(\phi,-\frac{\partial\phi}{\partial p}\right)\right% ]-\frac{f}{\sigma}\frac{\partial}{\partial p}\left(\frac{\partial\omega}{% \partial y}\cdot\frac{\partial u}{\partial p}-\frac{\partial\omega}{\partial x% }\cdot\frac{\partial v}{\partial p}\right)-\frac{f}{\sigma}\frac{\partial}{% \partial p}\left(\xi\frac{\partial\omega}{\partial p}-\omega\frac{\partial\xi}% {\partial p}\right)
  47. 2 ω 2 + f 2 σ 2 ω 2 p 2 = R 2 q C p σ p \nabla^{2}\omega_{2}+\frac{f^{2}}{\sigma}\frac{\partial^{2}\omega_{2}}{% \partial p^{2}}=\frac{R\cdot\nabla^{2}q}{C_{p}\cdot\sigma\cdot p}
  48. ω 1 \omega_{1}
  49. ω 2 \omega_{2}
  50. ω \omega
  51. ω \omega
  52. ω \omega
  53. ω \omega

One-shot_learning.html

  1. I I
  2. O f g O_{fg}
  3. O b g O_{bg}
  4. I t I_{t}
  5. I I
  6. R = p ( O f g | I , I t ) p ( O b g | I , I t ) = p ( I | I t , O f g ) p ( O f g ) p ( I | I t , O b g ) p ( O b g ) , R=\frac{p(O_{fg}|I,I_{t})}{p(O_{bg}|I,I_{t})}=\frac{p(I|I_{t},O_{fg})p(O_{fg})% }{p(I|I_{t},O_{bg})p(O_{bg})},
  7. p ( O f g | I , I t ) p(O_{fg}|I,I_{t})
  8. p ( O b g | I , I t ) p(O_{bg}|I,I_{t})
  9. I I
  10. R R
  11. T T
  12. θ \theta
  13. θ b g \theta_{bg}
  14. I t I_{t}
  15. p ( O f g ) p ( O b g ) \frac{p(O_{fg})}{p(O_{bg})}
  16. θ \theta
  17. θ b g \theta_{bg}
  18. R p ( I | θ , O f g ) p ( θ | I t , O f g ) d θ p ( I | θ b g , O b g ) p ( θ b g | I t , O b g ) d θ b g = p ( I | θ ) p ( θ | I t , O f g ) d θ p ( I | θ b g ) p ( θ b g | I t , O b g ) d θ b g R\propto\frac{\int{p(I|\theta,O_{fg})p(\theta|I_{t},O_{fg})}d\theta}{\int{p(I|% \theta_{bg},O_{bg})p(\theta_{bg}|I_{t},O_{bg})}d\theta_{bg}}=\frac{\int{p(I|% \theta)p(\theta|I_{t},O_{fg})}d\theta}{\int{p(I|\theta_{bg})p(\theta_{bg}|I_{t% },O_{bg})}d\theta_{bg}}
  19. p ( I | θ , O f g ) p(I|\theta,O_{fg})
  20. p ( I | θ , O b g ) p(I|\theta,O_{bg})
  21. p ( I | θ f g ) p(I|\theta_{fg})
  22. p ( I | θ b g ) . p(I|\theta_{bg}).
  23. p ( θ | I t , O f g ) p(\theta|I_{t},O_{fg})
  24. δ ( θ M L ) \delta(\theta^{ML})
  25. I I
  26. I t I_{t}
  27. I I
  28. X i X_{i}
  29. A i A_{i}
  30. X = i = 1 N X i , A = i = 1 N A i X=\sum_{i=1}^{N}X_{i},A=\sum_{i=1}^{N}A_{i}
  31. X t X_{t}
  32. A t A_{t}
  33. R p ( X , A | θ , O f g ) p ( θ | X t , A t , O f g ) d θ p ( X , A | θ b g , O b g ) p ( θ b g | X t , A t , O b g ) d θ b g = p ( X , A | θ ) p ( θ | X t , A t , O f g ) d θ p ( X , A | θ b g ) p ( θ b g | X t , A t , O b g ) d θ b g R\propto\frac{\int{p(X,A|\theta,O_{fg})p(\theta|X_{t},A_{t},O_{fg})}d\theta}{% \int{p(X,A|\theta_{bg},O_{bg})p(\theta_{bg}|X_{t},A_{t},O_{bg})}d\theta_{bg}}=% \frac{\int{p(X,A|\theta)p(\theta|X_{t},A_{t},O_{fg})}d\theta}{\int{p(X,A|% \theta_{bg})p(\theta_{bg}|X_{t},A_{t},O_{bg})}\,d\theta_{bg}}
  34. p ( X , A | θ ) p(X,A|\theta)
  35. p ( X , A | θ b g ) p(X,A|\theta_{bg})
  36. H H
  37. p ( X , A | θ ) = ω = 1 Ω 𝐡 H p ( X , A , 𝐡 , ω | θ ) . p(X,A|\theta)=\sum_{\omega=1}^{\Omega}\sum_{\,\textbf{h}\in H}p(X,A,\,\textbf{% h},\omega|\theta).
  38. ω \omega
  39. ω \omega
  40. X X
  41. p ( X , A , 𝐡 , ω | θ ) p(X,A,\,\textbf{h},\omega|\theta)
  42. p p
  43. θ p , ω A = μ p , ω A , Γ p , ω A \theta_{p,\omega}^{A}={\mu_{p,\omega}^{A},\Gamma_{p,\omega}^{A}}
  44. ω \omega
  45. ω \omega
  46. p ( X , A , 𝐡 , ω | θ ) p(X,A,\,\textbf{h},\omega|\theta)
  47. H H
  48. P ! P!
  49. H H
  50. R R
  51. p ( X , A | θ ) p ( θ | X t , A t , O f g ) d θ \int{p(X,A|\theta)p(\theta|X_{t},A_{t},O_{fg})}d\theta
  52. p ( X , A | θ ) p(X,A|\theta)
  53. p ( θ | X t , A t , O ) p(\theta|X_{t},A_{t},O)
  54. θ \theta
  55. δ \delta
  56. θ * \theta^{*}
  57. p ( X , A | θ * ) p(X,A|\theta^{*})
  58. θ * \theta^{*}
  59. θ * = θ M L \theta^{*}=\theta^{ML}
  60. θ * = θ M A P \theta^{*}=\theta^{MAP}
  61. δ \delta
  62. p ( θ ) p(\theta)
  63. p ( θ | X t , A t , O f g ) p(\theta|X_{t},A_{t},O_{fg})
  64. p ( θ | X t , A t , O f g ) p(\theta|X_{t},A_{t},O_{fg})
  65. p ( X , A | θ ) p(X,A|\theta)
  66. X X
  67. A A
  68. p ( θ | X t , A t , O f g ) p(\theta|X_{t},A_{t},O_{fg})
  69. X t X_{t}
  70. A t A_{t}
  71. θ t \theta_{t}
  72. p ( θ | X t , A t , O f g ) p(\theta|X_{t},A_{t},O_{fg})
  73. Ω = 1 \Omega=1
  74. I = T ( I L ) I=T(I_{L})
  75. E = p = 1 P H ( ν ( p ) ) , E=\sum_{p=1}^{P}H(\nu(p)),
  76. ν ( p ) \nu(p)
  77. H ( ) H()
  78. 1 p P 1\leq p\leq P
  79. I i I_{i}
  80. U i U_{i}
  81. I i I_{i}
  82. I L i I_{L_{i}}
  83. I L i I_{L_{i}}
  84. U i U_{i}
  85. U I U_{I}
  86. I i I_{i}
  87. A A
  88. A U i AU_{i}
  89. U i = A U i U_{i}=AU_{i}
  90. U i ( I ) = I L i U_{i}(I)=I_{L_{i}}
  91. T = U i - 1 T=U_{i}^{-1}
  92. I I
  93. P ( c j | I ) P(c_{j}|I)
  94. T T
  95. T T
  96. c j c_{j}
  97. T T
  98. c j c_{j}
  99. T test = U test - 1 T_{\,\text{test}}=U_{\,\text{test}}^{-1}
  100. B i B_{i}
  101. I t I_{t}
  102. c c
  103. B i B_{i}
  104. I t I_{t}
  105. c c
  106. I I
  107. c c

One-way_analysis_of_variance.html

  1. y i , j = μ j + ε i , j y_{i,j}=\mu_{j}+\varepsilon_{i,j}
  2. y i , j = μ + τ j + ε i , j y_{i,j}=\mu+\tau_{j}+\varepsilon_{i,j}
  3. i = 1 , , I i=1,\ldots,I
  4. j = 1 , , J j=1,\ldots,J
  5. I j I_{j}
  6. I = j I j I=\sum_{j}I_{j}
  7. y i , j y_{i,j}
  8. μ j \mu_{j}
  9. μ \mu
  10. τ j \tau_{j}
  11. τ j = 0 \sum\tau_{j}=0
  12. μ j = μ + τ j \mu_{j}=\mu+\tau_{j}
  13. ε N ( 0 , σ 2 ) \varepsilon\thicksim N(0,\sigma^{2})
  14. ε i , j \varepsilon_{i,j}
  15. j t h j_{th}
  16. y i j y_{ij}
  17. I 1 I_{1}
  18. I 2 I_{2}
  19. I 3 I_{3}
  20. \ldots
  21. I j I_{j}
  22. y 11 y_{11}
  23. y 12 y_{12}
  24. y 13 y_{13}
  25. y 1 j y_{1j}
  26. y 21 y_{21}
  27. y 22 y_{22}
  28. y 23 y_{23}
  29. y 2 j y_{2j}
  30. y 31 y_{31}
  31. y 32 y_{32}
  32. y 33 y_{33}
  33. y 3 j y_{3j}
  34. \vdots
  35. \vdots
  36. i i
  37. y i 1 y_{i1}
  38. y i 2 y_{i2}
  39. y i 3 y_{i3}
  40. \ldots
  41. y i j y_{ij}
  42. I 1 I_{1}
  43. I 2 I_{2}
  44. \ldots
  45. I j I_{j}
  46. \ldots
  47. I J I_{J}
  48. I = I j I=\sum I_{j}
  49. i y i j \sum_{i}y_{ij}
  50. j i y i j \sum_{j}\sum_{i}y_{ij}
  51. i ( y i j ) 2 \sum_{i}(y_{ij})^{2}
  52. j i ( y i j ) 2 \sum_{j}\sum_{i}(y_{ij})^{2}
  53. m 1 m_{1}
  54. \ldots
  55. m j m_{j}
  56. \ldots
  57. m J m_{J}
  58. m m
  59. s 1 2 s_{1}^{2}
  60. \ldots
  61. s j 2 s_{j}^{2}
  62. \ldots
  63. s J 2 s_{J}^{2}
  64. s 2 s^{2}
  65. μ = m \mu=m
  66. μ j = m j \mu_{j}=m_{j}
  67. T r e a t m e n t s I j ( m j - m ) 2 \sum_{Treatments}I_{j}(m_{j}-m)^{2}
  68. j ( i y i j ) 2 I j - ( j i y i j ) 2 I \sum_{j}\frac{(\sum_{i}y_{ij})^{2}}{I_{j}}-\frac{(\sum_{j}\sum_{i}y_{ij})^{2}}% {I}
  69. J - 1 J-1
  70. S S T r e a t m e n t D F T r e a t m e n t \frac{SS_{Treatment}}{DF_{Treatment}}
  71. M S T r e a t m e n t M S E r r o r \frac{MS_{Treatment}}{MS_{Error}}
  72. T r e a t m e n t s ( I j - 1 ) s j 2 \sum_{Treatments}(I_{j}-1)s_{j}^{2}
  73. j i y i j 2 - j ( i y i j ) 2 I j \sum_{j}\sum_{i}y_{ij}^{2}-\sum_{j}\frac{(\sum_{i}y_{ij})^{2}}{I_{j}}
  74. I - J I-J
  75. S S E r r o r D F E r r o r \frac{SS_{Error}}{DF_{Error}}
  76. O b s e r v a t i o n s ( y i j - m ) 2 \sum_{Observations}(y_{ij}-m)^{2}
  77. j i y i j 2 - ( j i y i j ) 2 I \sum_{j}\sum_{i}y_{ij}^{2}-\frac{(\sum_{j}\sum_{i}y_{ij})^{2}}{I}
  78. I - 1 I-1
  79. M S E r r o r MS_{Error}
  80. σ 2 \sigma^{2}
  81. I j I_{j}
  82. i i

Open-circuit_time_constant_method.html

  1. V 1 - V O R 2 = j ω C 2 V O , \frac{V_{1}-V_{O}}{R_{2}}=j\omega C_{2}V_{O}\ ,
  2. V S - V 1 R 1 = j ω C 1 V 1 + V 1 - V O R 2 . \frac{V_{S}-V_{1}}{R_{1}}=j\omega C_{1}V_{1}+\frac{V_{1}-V_{O}}{R_{2}}\ .
  3. V O V S = 1 1 + j ω ( C 2 ( R 1 + R 2 ) + C 1 R 1 ) + ( j ω ) 2 C 1 C 2 R 1 R 2 \frac{V_{O}}{V_{S}}=\frac{1}{1+j\omega\left(C_{2}(R_{1}+R_{2})+C_{1}R_{1}% \right)+(j\omega)^{2}C_{1}C_{2}R_{1}R_{2}}
  4. ( 1 + j ω τ 1 ) ( 1 + j ω τ 2 ) = 1 + j ω ( C 2 ( R 1 + R 2 ) + C 1 R 1 ) + ( j ω ) 2 C 1 C 2 R 1 R 2 \left(1+j\omega{\tau}_{1})(1+j\omega{\tau}_{2}\right)=1+j\omega\left(C_{2}(R_{% 1}+R_{2})+C_{1}R_{1}\right)+(j\omega)^{2}C_{1}C_{2}R_{1}R_{2}
  5. τ 1 + τ 2 = C 2 ( R 1 + R 2 ) + C 1 R 1 , \tau_{1}+\tau_{2}=C_{2}(R_{1}+R_{2})+C_{1}R_{1}\ ,
  6. τ 1 τ 2 = C 1 C 2 R 1 R 2 . \tau_{1}\tau_{2}=C_{1}C_{2}R_{1}R_{2}\ .
  7. τ 1 + τ 2 τ 1 , \tau_{1}+\tau_{2}\approx\tau_{1}\ ,
  8. τ 2 = τ 1 τ 2 τ 1 τ 1 τ 2 τ 1 + τ 2 . \tau_{2}=\frac{\tau_{1}\tau_{2}}{\tau_{1}}\approx\frac{\tau_{1}\tau_{2}}{\tau_% {1}+\tau_{2}}\ .
  9. τ 1 τ 1 ^ = τ 1 + τ 2 = C 2 ( R 1 + R 2 ) + C 1 R 1 , \tau_{1}\approx\hat{\tau_{1}}=\ \tau_{1}+\tau_{2}=C_{2}(R_{1}+R_{2})+C_{1}R_{1% }\ ,
  10. τ 2 τ 2 ^ = τ 1 τ 2 τ 1 + τ 2 = C 1 C 2 R 1 R 2 C 2 ( R 1 + R 2 ) + C 1 R 1 , \tau_{2}\approx\hat{\tau_{2}}=\frac{\tau_{1}\tau_{2}}{\tau_{1}+\tau_{2}}=\frac% {C_{1}C_{2}R_{1}R_{2}}{C_{2}(R_{1}+R_{2})+C_{1}R_{1}}\ ,
  11. f 3 d B = 1 2 π τ 1 ^ , f_{3dB}=\frac{1}{2\pi\hat{\tau_{1}}}\ ,
  12. 1 ( 1 + j ω τ 1 ) ( 1 + j ω τ 2 ) \frac{1}{(1+j\omega\tau_{1})(1+j\omega\tau_{2})}
  13. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  14. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  15. 1 ( 1 + j ω τ 1 ^ ) ( 1 + j ω τ 2 ^ ) . \ \frac{1}{(1+j\omega\hat{\tau_{1}})(1+j\omega\hat{\tau_{2}})}\ .

Open_cluster_family.html

  1. \sim
  2. \sim

Opheline_kinase.html

  1. \rightleftharpoons

Orbit_(control_theory).html

  1. q ˙ = f ( q , u ) {\ }\dot{q}=f(q,u)
  2. 𝒞 \ {\mathcal{C}}^{\infty}
  3. q {\ q}
  4. M \ M
  5. u \ u
  6. U \ U
  7. = { f ( , u ) u U } {\mathcal{F}}=\{f(\cdot,u)\mid u\in U\}
  8. {\mathcal{F}}
  9. f f\in{\mathcal{F}}
  10. t \ t
  11. e t f \ e^{tf}
  12. f \ f
  13. t \ t
  14. q ˙ = f ( q , u ) {\ }\dot{q}=f(q,u)
  15. q 0 M q_{0}\in M
  16. 𝒪 q 0 {\mathcal{O}}_{q_{0}}
  17. M \ M
  18. 𝒪 q 0 = { e t k f k e t k - 1 f k - 1 e t 1 f 1 ( q 0 ) k , t 1 , , t k , f 1 , , f k } . {\mathcal{O}}_{q_{0}}=\{e^{t_{k}f_{k}}\circ e^{t_{k-1}f_{k-1}}\circ\cdots\circ e% ^{t_{1}f_{1}}(q_{0})\mid k\in\mathbb{N},\ t_{1},\dots,t_{k}\in\mathbb{R},\ f_{% 1},\dots,f_{k}\in{\mathcal{F}}\}.
  19. {\mathcal{F}}
  20. f f\in{\mathcal{F}}
  21. - f -f\in{\mathcal{F}}
  22. {\mathcal{F}}
  23. 𝒪 q 0 {\mathcal{O}}_{q_{0}}
  24. M \ M
  25. 𝒪 q 0 {\mathcal{O}}_{q_{0}}
  26. q \ q
  27. T q M \ T_{q}M
  28. P * f ( q ) \ P_{*}f(q)
  29. P * f \ P_{*}f
  30. f \ f
  31. P \ P
  32. f \ f
  33. {\mathcal{F}}
  34. P \ P
  35. M \ M
  36. e t k f k e t 1 f 1 e^{t_{k}f_{k}}\circ\cdots\circ e^{t_{1}f_{1}}
  37. k , t 1 , , t k k\in\mathbb{N},\ t_{1},\dots,t_{k}\in\mathbb{R}
  38. f 1 , , f k f_{1},\dots,f_{k}\in{\mathcal{F}}
  39. {\mathcal{F}}
  40. T q 𝒪 q 0 = Lie q \ T_{q}{\mathcal{O}}_{q_{0}}=\mathrm{Lie}_{q}\,\mathcal{F}
  41. Lie q \mathrm{Lie}_{q}\,\mathcal{F}
  42. q \ q
  43. {\mathcal{F}}
  44. Lie q T q 𝒪 q 0 \mathrm{Lie}_{q}\,\mathcal{F}\subset T_{q}{\mathcal{O}}_{q_{0}}
  45. Lie q = T q M \mathrm{Lie}_{q}\,\mathcal{F}=T_{q}M
  46. q M \ q\in M
  47. M \ M
  48. M \ M

Ordered_dithering.html

  1. 1 5 [ 1 3 4 2 ] \frac{\displaystyle 1}{\displaystyle 5}\begin{bmatrix}1&3\\ 4&2\\ \end{bmatrix}
  2. 1 10 [ 3 7 4 6 1 9 2 8 5 ] \frac{\displaystyle 1}{\displaystyle 10}\begin{bmatrix}3&7&4\\ 6&1&9\\ 2&8&5\\ \end{bmatrix}
  3. 1 17 [ 1 9 3 11 13 5 15 7 4 12 2 10 16 8 14 6 ] \frac{\displaystyle 1}{\displaystyle 17}\begin{bmatrix}1&9&3&11\\ 13&5&15&7\\ 4&12&2&10\\ 16&8&14&6\\ \end{bmatrix}
  4. 1 65 [ 1 49 13 61 4 52 16 64 33 17 45 29 36 20 48 32 9 57 5 53 12 60 8 56 41 25 37 21 44 28 40 24 3 51 15 63 2 50 14 62 35 19 47 31 34 18 46 30 11 59 7 55 10 58 6 54 43 27 39 23 42 26 38 22 ] \frac{\displaystyle 1}{\displaystyle 65}\begin{bmatrix}1&49&13&61&4&52&16&64\\ 33&17&45&29&36&20&48&32\\ 9&57&5&53&12&60&8&56\\ 41&25&37&21&44&28&40&24\\ 3&51&15&63&2&50&14&62\\ 35&19&47&31&34&18&46&30\\ 11&59&7&55&10&58&6&54\\ 43&27&39&23&42&26&38&22\\ \end{bmatrix}

Ordered_weighted_averaging_aggregation_operator.html

  1. n \ n
  2. F : R n R F:R_{n}\rightarrow R
  3. W = [ w 1 , , w n ] \ W=[w_{1},\ldots,w_{n}]
  4. F ( a 1 , , a n ) = j = 1 n w j b j F(a_{1},\ldots,a_{n})=\sum_{j=1}^{n}w_{j}b_{j}
  5. b j b_{j}
  6. a i a_{i}
  7. min ( a 1 , , a n ) F ( a 1 , , a n ) max ( a 1 , , a n ) \min(a_{1},\ldots,a_{n})\leq F(a_{1},\ldots,a_{n})\leq\max(a_{1},\ldots,a_{n})
  8. F ( a 1 , , a n ) F ( g 1 , , g n ) F(a_{1},\ldots,a_{n})\geq F(g_{1},\ldots,g_{n})
  9. a i g i a_{i}\geq g_{i}
  10. i = 1 , 2 , , n \ i=1,2,\ldots,n
  11. F ( a 1 , , a n ) = F ( a s y m b o l π ( 1 ) , , a s y m b o l π ( n ) ) F(a_{1},\ldots,a_{n})=F(a_{s}ymbol{\pi(1)},\ldots,a_{s}ymbol{\pi(n)})
  12. s y m b o l π symbol{\pi}
  13. F ( a 1 , , a n ) = a \ F(a_{1},\ldots,a_{n})=a
  14. a i = a \ a_{i}=a
  15. F ( a 1 , , a n ) = max ( a 1 , , a n ) \ F(a_{1},\ldots,a_{n})=\max(a_{1},\ldots,a_{n})
  16. w 1 = 1 \ w_{1}=1
  17. w j = 0 \ w_{j}=0
  18. j 1 j\neq 1
  19. F ( a 1 , , a n ) = min ( a 1 , , a n ) \ F(a_{1},\ldots,a_{n})=\min(a_{1},\ldots,a_{n})
  20. w n = 1 \ w_{n}=1
  21. w j = 0 \ w_{j}=0
  22. j n j\neq n
  23. A - C ( W ) = 1 n - 1 j = 1 n ( n - j ) w j . A-C(W)=\frac{1}{n-1}\sum_{j=1}^{n}(n-j)w_{j}.
  24. A - C ( W ) [ 0 , 1 ] A-C(W)\in[0,1]
  25. H ( W ) = - j = 1 n w j ln ( w j ) . H(W)=-\sum_{j=1}^{n}w_{j}\ln(w_{j}).
  26. E ( W ) = j = 1 n w j 2 . E(W)=\sum_{j=1}^{n}w_{j}^{2}.
  27. { W i } i = 1 n \left\{{W^{i}}\right\}_{i=1}^{n}
  28. U = [ 0 , 1 ] U=[0,\;\;1]
  29. α [ 0 , 1 ] \alpha\in[0,\;1]
  30. α \alpha
  31. α \alpha
  32. { W α i } i = 1 n \left\{{W_{\alpha}^{i}}\right\}_{i=1}^{n}
  33. α \alpha
  34. { A i } i = 1 n \left\{{A^{i}}\right\}_{i=1}^{n}
  35. Φ α ( A α 1 , , A α n ) = { i = 1 n w i a σ ( i ) i = 1 n w i | w i W α i , a i A α i , i = 1 , , n } \Phi_{\alpha}\left({A_{\alpha}^{1},\ldots,A_{\alpha}^{n}}\right)=\left\{{\frac% {\sum\limits_{i=1}^{n}{w_{i}a_{\sigma(i)}}}{\sum\limits_{i=1}^{n}{w_{i}}}\left% |{w_{i}\in W_{\alpha}^{i},\;a_{i}}\right.\in A_{\alpha}^{i},\;i=1,\ldots,n}\right\}
  36. W α i = { w | μ W i ( w ) α } , A α i = { x | μ A i ( x ) α } W_{\alpha}^{i}=\{w|\mu_{W_{i}}(w)\geq\alpha\},A_{\alpha}^{i}=\{x|\mu_{A_{i}}(x% )\geq\alpha\}
  37. σ : { 1 , , n } { 1 , , n } \sigma:\{\;1,\ldots,n\;\}\to\{\;1,\ldots,n\;\}
  38. a σ ( i ) a σ ( i + 1 ) , i = 1 , , n - 1 a_{\sigma(i)}\geq a_{\sigma(i+1)},\;\forall\;i=1,\ldots,n-1
  39. a σ ( i ) a_{\sigma(i)}
  40. i i
  41. { a 1 , , a n } \left\{{a_{1},\ldots,a_{n}}\right\}
  42. Φ α ( A α 1 , , A α n ) \Phi_{\alpha}\left({A_{\alpha}^{1},\ldots,A_{\alpha}^{n}}\right)
  43. Φ α ( A α 1 , , A α n ) - \Phi_{\alpha}\left({A_{\alpha}^{1},\ldots,A_{\alpha}^{n}}\right)_{-}
  44. Φ α ( A α 1 , , A α n ) + , \Phi_{\alpha}\left({A_{\alpha}^{1},\ldots,A_{\alpha}^{n}}\right)_{+},
  45. A α i = [ A α - i , A α + i ] , W α i = [ W α - i , W α + i ] A_{\alpha}^{i}=[A_{\alpha-}^{i},A_{\alpha+}^{i}],W_{\alpha}^{i}=[W_{\alpha-}^{% i},W_{\alpha+}^{i}]
  46. μ G ( x ) = α : x Φ α ( A α 1 , , A α n ) α α \mu_{G}(x)=\mathop{\vee}\limits_{\alpha:x\in\Phi_{\alpha}\left({A_{\alpha}^{1}% ,\cdots,A_{\alpha}^{n}}\right)_{\alpha}}\alpha
  47. Φ α ( A α 1 , , A α n ) - = min W α - i w i W α + i A α - i a i A α + i i = 1 n w i a σ ( i ) / i = 1 n w i \Phi_{\alpha}\left({A_{\alpha}^{1},\cdots,A_{\alpha}^{n}}\right)_{-}=\mathop{% \min}\limits_{\begin{array}[]{l}W_{\alpha-}^{i}\leq w_{i}\leq W_{\alpha+}^{i}A% _{\alpha-}^{i}\leq a_{i}\leq A_{\alpha+}^{i}\end{array}}\sum\limits_{i=1}^{n}{% w_{i}a_{\sigma(i)}/\sum\limits_{i=1}^{n}{w_{i}}}
  48. Φ α ( A α 1 , , A α n ) + = max W α - i w i W α + i A α - i a i A α + i i = 1 n w i a σ ( i ) / i = 1 n w i \Phi_{\alpha}\left({A_{\alpha}^{1},\cdots,A_{\alpha}^{n}}\right)_{+}=\mathop{% \max}\limits_{\begin{array}[]{l}W_{\alpha-}^{i}\leq w_{i}\leq W_{\alpha+}^{i}A% _{\alpha-}^{i}\leq a_{i}\leq A_{\alpha+}^{i}\end{array}}\sum\limits_{i=1}^{n}{% w_{i}a_{\sigma(i)}/\sum\limits_{i=1}^{n}{w_{i}}}

Ordinal_definable_set.html

  1. S \isin V α 1 S\isin V_{\alpha_{1}}
  2. S S
  3. V α 1 V_{\alpha_{1}}
  4. V α 1 V_{\alpha_{1}}
  5. V α 1 V_{\alpha_{1}}

Ornithine(lysine)_transaminase.html

  1. \rightleftharpoons

Ornithine_N-benzoyltransferase.html

  1. \rightleftharpoons

Orsellinate-depside_hydrolase.html

  1. \rightleftharpoons

Otto_Stolz.html

  1. f ( x + y 2 ) f ( x ) + f ( y ) 2 f\left(\frac{x+y}{2}\right)\leq\frac{f(x)+f(y)}{2}

Overlapping_distribution_method.html

  1. Q 0 Q_{0}
  2. Q 1 Q_{1}
  3. F ( N , V , T ) = - k B T ln Q F(N,V,T)=-k_{B}T\ln Q
  4. Δ F = - k B T ln ( Q 1 / Q 0 ) = - k B T ln ( d s N exp [ - β U 1 ( s N ) ] d s N exp [ - β U 0 ( s N ) ] ) \Delta F=-k_{B}T\ln(Q_{1}/Q_{0})=-k_{B}T\ln(\frac{\int ds^{N}\exp[-\beta U_{1}% (s^{N})]}{\int ds^{N}\exp[-\beta U_{0}(s^{N})]})
  5. Δ U = U 1 ( s N ) - U 0 ( s N ) \Delta U=U_{1}(s^{N})-U_{0}(s^{N})
  6. p 1 ( Δ U ) = d s N exp ( - β U 1 ) δ ( U 1 - U 0 - Δ U ) Q 1 p_{1}(\Delta U)=\frac{\int ds^{N}\exp(-\beta U_{1})\delta(U_{1}-U_{0}-\Delta U% )}{Q_{1}}
  7. p 1 p_{1}
  8. p 1 ( Δ U ) = d s N exp ( - β U 1 ) δ ( U 1 - U 0 - Δ U ) Q 1 = d s N exp [ - β ( U 0 + Δ U ) ] δ ( U 1 - U 0 - Δ U ) Q 1 p_{1}(\Delta U)=\frac{\int ds^{N}\exp(-\beta U_{1})\delta(U_{1}-U_{0}-\Delta U% )}{Q_{1}}=\frac{\int ds^{N}\exp[-\beta(U_{0}+\Delta U)]\delta(U_{1}-U_{0}-% \Delta U)}{Q_{1}}
  9. = Q 0 Q 1 exp ( - β Δ U ) d s N exp ( - β U 0 ) δ ( U 1 - U 0 - Δ U ) Q 0 = Q 0 Q 1 exp ( - β Δ U ) p 0 ( Δ U ) =\frac{Q_{0}}{Q_{1}}\exp(-\beta\Delta U)\frac{\int ds^{N}\exp(-\beta U_{0})% \delta(U_{1}-U_{0}-\Delta U)}{Q_{0}}=\frac{Q_{0}}{Q_{1}}\exp(-\beta\Delta U)p_% {0}(\Delta U)
  10. Δ F = - k B T ln ( Q 1 / Q 0 ) \Delta F=-k_{B}T\ln(Q_{1}/Q_{0})
  11. ln p 1 ( Δ U ) = β ( Δ F - Δ U ) + ln p 0 ( Δ U ) \ln p_{1}(\Delta U)=\beta(\Delta F-\Delta U)+\ln p_{0}(\Delta U)
  12. f 0 ( Δ U ) = ln p 0 ( Δ U ) - β Δ U 2 f 1 ( Δ U ) = ln p 1 ( Δ U ) + β Δ U 2 f_{0}(\Delta U)=\ln p_{0}(\Delta U)-\frac{\beta\Delta U}{2}f_{1}(\Delta U)=\ln p% _{1}(\Delta U)+\frac{\beta\Delta U}{2}
  13. f 1 ( Δ U ) = f 0 ( Δ U ) + β Δ F f_{1}(\Delta U)=f_{0}(\Delta U)+\beta\Delta F
  14. Δ F \Delta F
  15. f 1 f_{1}
  16. f 0 f_{0}

Oxalate—CoA_ligase.html

  1. \rightleftharpoons

Oximinotransferase.html

  1. \rightleftharpoons

PAH_clearance.html

  1. R P F = U P A H P P A H V RPF=\frac{U_{PAH}}{P_{PAH}}V

Palmitoyl(protein)_hydrolase.html

  1. \rightleftharpoons

Palmitoyl-CoA_hydrolase.html

  1. \rightleftharpoons

Pantetheine-phosphate_adenylyltransferase.html

  1. \rightleftharpoons

Pantetheine_hydrolase.html

  1. \rightleftharpoons

Pantetheine_kinase.html

  1. \rightleftharpoons

Pantoate—beta-alanine_ligase.html

  1. \rightleftharpoons

Pantothenase.html

  1. \rightleftharpoons

Paracrystalline.html

  1. ξ \xi

Parker_vector.html

  1. P k = k | G | g G c k ( g ) P_{k}=\frac{k}{|G|}\sum_{g\in G}c_{k}(g)

Penicillin_amidase.html

  1. \rightleftharpoons

Pentalenene_synthase.html

  1. \rightleftharpoons

Pentanamidase.html

  1. \rightleftharpoons

Peptide-transporting_ATPase.html

  1. \rightleftharpoons

Peptide_alpha-N-acetyltransferase.html

  1. \rightleftharpoons

Peptide_deformylase.html

  1. \rightleftharpoons

Peptidyl-glutaminase.html

  1. \rightleftharpoons

Permutation_polynomial.html

  1. x g ( x ) x\mapsto g(x)
  2. g ( x ) = 2 x 2 + x g(x)=2x^{2}+x
  3. g ( 0 ) = 0 ; g ( 1 ) = 3 ; g ( 2 ) = 2 ; g ( 3 ) = 1 g(0)=0;~{}g(1)=3;~{}g(2)=2;~{}g(3)=1
  4. ( 0 1 2 3 0 3 2 1 ) \begin{pmatrix}0&1&2&3\\ 0&3&2&1\end{pmatrix}
  5. g ( x ) = 2 x 2 + x g(x)=2x^{2}+x
  6. g ( 0 ) = 0 ; g ( 1 ) = 3 ; g ( 2 ) = 2 ; g ( 3 ) = 5 ; g ( 4 ) = 4 ; g ( 5 ) = 7 ; g ( 6 ) = 6 ; g ( 7 ) = 1 ; g(0)=0;~{}g(1)=3;~{}g(2)=2;~{}g(3)=5;~{}g(4)=4;~{}g(5)=7;~{}g(6)=6;~{}g(7)=1;
  7. ( 0 1 2 3 4 5 6 7 0 3 2 5 4 7 6 1 ) \begin{pmatrix}0&1&2&3&4&5&6&7\\ 0&3&2&5&4&7&6&1\end{pmatrix}
  8. g ( x ) = a x 2 + b x + c g(x)=ax^{2}+bx+c
  9. a = 0 m o d p a=0~{}mod~{}p
  10. b 0 m o d p b\neq 0~{}mod~{}p
  11. n = p 1 k 1 p 2 k 2 p l k l n=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{l}^{k_{l}}
  12. g ( x ) = a 0 + 0 < i M a i x i g(x)=a_{0}+\sum_{0<i\leq M}a_{i}x^{i}
  13. g p t ( x ) = a 0 , p t + 0 < i M a i , p t x i g_{p_{t}}(x)=a_{0,p_{t}}+\sum_{0<i\leq M}a_{i,p_{t}}x^{i}
  14. Z / p t k t Z Z/p_{t}^{k_{t}}Z
  15. a j , p t a_{j,p_{t}}
  16. a j a_{j}
  17. p t k t p_{t}^{k_{t}}
  18. n = p 1 k 1 p 2 k 2 p l k l n=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{l}^{k_{l}}
  19. a x 2 + b x ax^{2}+bx
  20. a = 0 m o d p 1 a=0~{}mod~{}p_{1}
  21. a 0 m o d p 1 k 1 a\neq 0~{}mod~{}p_{1}^{k_{1}}
  22. a = 0 m o d p i k i a=0~{}mod~{}p_{i}^{k_{i}}
  23. b 0 m o d p i b\neq 0~{}mod~{}p_{i}
  24. a = p 1 p 2 k 2 p l k l a=p_{1}p_{2}^{k_{2}}...p_{l}^{k_{l}}
  25. b = 1 b=1
  26. 6 x 2 + x 6x^{2}+x
  27. ( 0 1 2 3 4 5 6 7 8 0 7 2 9 4 11 6 1 8 ) \begin{pmatrix}0&1&2&3&4&5&6&7&8&...\\ 0&7&2&9&4&11&6&1&8&...\end{pmatrix}
  28. g ( x ) 0 m o d p g^{\prime}(x)\neq 0~{}mod~{}p
  29. g ( x ) = a x 3 + b x g(x)=ax^{3}+bx
  30. y Z / p Z y\in Z/pZ
  31. y 2 - b / a y^{2}\neq-b/a
  32. ( - b / a p ) = - 1. \left(\frac{-b/a}{p}\right)=-1.
  33. L ( x ) = Σ s = 0 r - 1 α s x q s , L(x)=\Sigma_{s=0}^{r-1}\alpha_{s}x^{q^{s}},
  34. det ( α i - j q j ) 0 ( i , j = 0 , 1 , , r - 1 ) . \rm{det}\left(\alpha_{i-j}^{q^{j}}\right)\neq 0\quad(i,j=0,1,\ldots,r-1).
  35. x ( q + m - 1 ) / m + a x x^{(q+m-1)/m}+ax
  36. x r ( x d - a ) ( p n - 1 ) / d x^{r}(x^{d}-a)^{(p^{n}-1)/d}

Peroxisome-assembly_ATPase.html

  1. \rightleftharpoons

Perspective_(geometry).html

  1. A B C a b c ABC\doublebarwedge abc

Pfaffian_function.html

  1. f ( x ) = f ( x ) . f^{\prime}(x)=f(x).
  2. g ( x ) = - g ( x ) 2 . g^{\prime}(x)=-g(x)^{2}.
  3. h ( x ) = e x log x + x - 1 e x = h ( x ) + f ( x ) g ( x ) . h^{\prime}(x)=e^{x}\log x+x^{-1}e^{x}=h(x)+f(x)g(x).
  4. f i x j = P i , j ( s y m b o l x , f 1 ( s y m b o l x ) , , f i ( s y m b o l x ) ) \frac{\partial f_{i}}{\partial x_{j}}=P_{i,j}(symbol{x},f_{1}(symbol{x}),% \ldots,f_{i}(symbol{x}))
  5. f ( s y m b o l x ) = P ( s y m b o l x , f 1 ( s y m b o l x ) , , f r ( s y m b o l x ) ) , f(symbol{x})=P(symbol{x},f_{1}(symbol{x}),\ldots,f_{r}(symbol{x})),\,
  6. f 1 \displaystyle f_{1}^{\prime}
  7. f 1 = P 1 ( x , f 1 , f 2 , f 3 ) f 2 = P 2 ( x , f 1 , f 2 , f 3 ) f 3 = P 3 ( x , f 1 , f 2 , f 3 ) . \begin{aligned}\displaystyle f_{1}^{\prime}&\displaystyle=P_{1}(x,f_{1},f_{2},% f_{3})\\ \displaystyle f_{2}^{\prime}&\displaystyle=P_{2}(x,f_{1},f_{2},f_{3})\\ \displaystyle f_{3}^{\prime}&\displaystyle=P_{3}(x,f_{1},f_{2},f_{3}).\end{aligned}
  8. f 1 ( x ) = f 2 ( x ) f 2 ( x ) = - f 1 ( x ) , \begin{aligned}\displaystyle f_{1}^{\prime}(x)&\displaystyle=f_{2}(x)\\ \displaystyle f_{2}^{\prime}(x)&\displaystyle=-f_{1}(x),\end{aligned}

Phaseollidin_hydratase.html

  1. \rightleftharpoons

Phenol_beta-glucosyltransferase.html

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Phenylacetate—CoA_ligase.html

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Phenylacetyl-CoA_hydrolase.html

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Phenylalanine(histidine)_transaminase.html

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Phenylalanine_adenylyltransferase.html

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Phenylalanine_N-acetyltransferase.html

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Phenylalanine—tRNA_ligase.html

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Phenylglyoxylic_acid.html

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Phloroisovalerophenone_synthase.html

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Phorbol-diester_hydrolase.html

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Phosphate-transporting_ATPase.html

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Phosphate_acetyltransferase.html

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Phosphate_butyryltransferase.html

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Phosphatidate_cytidylyltransferase.html

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Phosphatidate_phosphatase.html

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Phosphatidylcholine_synthase.html

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Phosphatidylcholine—dolichol_O-acyltransferase.html

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Phosphatidylcholine—retinol_O-acyltransferase.html

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Phosphatidylcholine—sterol_O-acyltransferase.html

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Phosphatidylglycerol—membrane-oligosaccharide_glycerophosphotransferase.html

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Phosphatidylglycerophosphatase.html

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Phosphatidylinositol-3,4,5-trisphosphate_3-phosphatase.html

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Phosphatidylinositol-3,4-bisphosphate_4-phosphatase.html

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Phosphatidylinositol-3-phosphatase.html

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Phosphatidylinositol-4,5-bisphosphate_3-kinase.html

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Phosphatidylinositol-4-phosphate_3-kinase.html

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Phosphatidylinositol_deacylase.html

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Phosphatidylinositol_N-acetylglucosaminyltransferase.html

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Phosphatidylserine_decarboxylase.html

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Phospho-N-acetylmuramoyl-pentapeptide-transferase.html

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Phosphoadenylylsulfatase.html

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Phosphoenolpyruvate_phosphatase.html

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Phosphoenolpyruvate—glycerone_phosphotransferase.html

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Phosphoenolpyruvate—protein_phosphotransferase.html

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Phosphoglucan,_water_dikinase.html

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Phosphoglucokinase.html

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Phosphogluconate_dehydratase.html

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Phosphoglycerate_dehydrogenase.html

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Phosphoglycerate_kinase_(GTP).html

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Phosphoglycerate_phosphatase.html

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Phosphoglycerol_geranylgeranyltransferase.html

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Phosphoglycolate_phosphatase.html

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Phosphoinositide_5-phosphatase.html

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Phosphoketolase.html

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Phospholipid-translocating_ATPase.html

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Phospholipid:diacylglycerol_acyltransferase.html

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Phosphomannan_mannosephosphotransferase.html

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Phosphomethylpyrimidine_kinase.html

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Phosphonate-transporting_ATPase.html

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Phosphonopyruvate_decarboxylase.html

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Phosphopantothenate—cysteine_ligase.html

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Phosphopantothenoylcysteine_decarboxylase.html

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Phosphopolyprenol_glucosyltransferase.html

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Phosphoramidate—hexose_phosphotransferase.html

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Phosphoribokinase.html

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Phosphoribosyl-AMP_cyclohydrolase.html

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Phosphoribosyl-ATP_diphosphatase.html

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Phosphoribosylamine—glycine_ligase.html

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Phosphoribosylaminoimidazolesuccinocarboxamide_synthase.html

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Phosphoribosylformylglycinamidine_synthase.html

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Phosphoribulokinase.html

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Phosphoserine_phosphatase.html

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Photo-Carnot_engine.html

  1. U = ε σ T 4 . U=\varepsilon\sigma T^{4}\,.
  2. P = U 3 V = ε σ T 4 3 V . P=\frac{U}{3V}=\frac{\varepsilon\sigma T^{4}}{3V}\,.
  3. d U = d W + d Q dU=dW+dQ
  4. d Q = 0 dQ=0
  5. d U = ε σ d V T 4 + 4 ε σ V T 3 d T dU=\varepsilon\sigma dVT^{4}+4\varepsilon\sigma VT^{3}dT
  6. d W = - P d V = - 1 3 ε σ T 4 d V . dW=-PdV=-\frac{1}{3}\varepsilon\sigma T^{4}dV\,.
  7. 2 3 T 4 d V = 4 V T 3 d T \frac{2}{3}T^{4}dV=4VT^{3}dT
  8. V 1 / 6 T = c o n s t . \frac{V^{1/6}}{T}=const\,.
  9. η = T H - T C T H . \eta=\frac{T_{H}-T_{C}}{T_{H}}\,.

Photochemical_Reflectance_Index.html

  1. P R I = ( ρ 531 - ρ 570 ) ( ρ 531 + ρ 570 ) PRI=\frac{(\rho 531-\rho 570)}{(\rho 531+\rho 570)}
  2. P R I = ( ρ 570 - ρ 531 ) ( ρ 570 + ρ 531 ) PRI=\frac{(\rho 570-\rho 531)}{(\rho 570+\rho 531)}

Photovoltaic_system.html

  1. p = I ( 1 + i ) + t = 0 T I b ( 1 + i ) t t = 0 T E ( 1 - v ) t ( 1 + i ) t p=\frac{I\cdot(1+i)+\sum_{t=0}^{T}\frac{I\cdot b}{(1+i)^{t}}}{\sum_{t=0}^{T}% \frac{E\cdot(1-v)^{t}}{(1+i)^{t}}}

Phthalyl_amidase.html

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Phytanate—CoA_ligase.html

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Piezoelectric_accelerometer.html

  1. F = m a F=ma

Pinene_synthase.html

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Pinosylvin_synthase.html

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Pipe_insulation.html

  1. q = Θ i - Θ a R T q=\frac{\Theta_{i}-\Theta_{a}}{R_{T}}
  2. Θ i \Theta_{i}
  3. Θ a \Theta_{a}
  4. R T R_{T}
  5. R = D x ln ( D e / D i ) λ R=\frac{D_{x}\ln(D_{e}/D_{i})}{\lambda}
  6. D e D_{e}
  7. D i D_{i}
  8. λ \lambda
  9. D x D_{x}
  10. D e D_{e}
  11. D e D_{e}
  12. D i D_{i}
  13. D i D_{i}

Piperidine_N-piperoyltransferase.html

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Pirani_gauge.html

  1. P el = C 1 λ gaz ( T ( t ) - T a ) + C 2 λ fil ( T ( t ) - T a ) + A fil ϵ σ ( T ( t ) 4 - T a 4 ) + c fil m fil d T d t , P_{\,\text{el}}=C_{1}\lambda_{\,\text{gaz}}(T(t)-T_{a})+C_{2}\lambda_{\,\text{% fil}}(T(t)-T_{a})+A_{\,\text{fil}}\epsilon\sigma(T(t)^{4}-T^{4}_{a})+c_{\,% \text{fil}}m_{\,\text{fil}}\frac{\mathrm{d}T}{\mathrm{d}t},
  2. c fil c_{\,\text{fil}}
  3. m fil m_{\,\text{fil}}
  4. C 1 C_{1}
  5. C 2 C_{2}

Pivotal_altitude.html

  1. v v
  2. H = v 2 11.3 H=\frac{v^{2}}{11.3}

Plasmalogen_synthase.html

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Platelet-activating_factor_acetyltransferase.html

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Poisson_sampling.html

  1. π i j = π i × π j . \pi_{ij}=\pi_{i}\times\pi_{j}.\,

Polar-amino-acid-transporting_ATPase.html

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Polar_amplification.html

  1. P A F = Δ T p Δ T ¯ {PAF}={\Delta{T}_{p}\over\Delta\bar{T}}
  2. Δ T p \Delta{T}_{p}
  3. Δ T ¯ \Delta\bar{T}

Pole_splitting.html

  1. v i \ v_{i}
  2. v a \ v_{a}
  3. v i v a = R i R i + R A 1 1 + j ω ( C M + C i ) ( R A / / R i ) , \frac{v_{i}}{v_{a}}=\frac{R_{i}}{R_{i}+R_{A}}\frac{1}{1+j\omega(C_{M}+C_{i})(R% _{A}//R_{i})}\ ,
  4. f 1 \displaystyle f_{1}
  5. τ 1 \tau_{1}
  6. 1 2 π C i ( R A / / R i ) \frac{1}{2\pi C_{i}(R_{A}//R_{i})}
  7. v o \ v_{o}
  8. v o = A v v i . \ v_{o}=A_{v}v_{i}\ .
  9. v v_{\ell}
  10. v i \ v_{i}
  11. v v i = A v R L R L + R o \frac{v_{\ell}}{v_{i}}=A_{v}\frac{R_{L}}{R_{L}+R_{o}}\,\!
  12. \sdot 1 + j ω C C R o / A v 1 + j ω ( C L + C C ) ( R o / / R L ) . \sdot\frac{1+j\omega C_{C}R_{o}/A_{v}}{1+j\omega(C_{L}+C_{C})(R_{o}//R_{L})}\ .
  13. v v a = v v i v i v a \frac{v_{\ell}}{v_{a}}=\frac{v_{\ell}}{v_{i}}\frac{v_{i}}{v_{a}}
  14. = A v R i R i + R A \sdot R L R L + R o =A_{v}\frac{R_{i}}{R_{i}+R_{A}}\sdot\frac{R_{L}}{R_{L}+R_{o}}\,\!
  15. \sdot 1 1 + j ω ( C M + C i ) ( R A / / R i ) \sdot\frac{1}{1+j\omega(C_{M}+C_{i})(R_{A}//R_{i})}\,\!
  16. \sdot 1 + j ω C C R o / A v 1 + j ω ( C L + C C ) ( R o / / R L ) . \sdot\frac{1+j\omega C_{C}R_{o}/A_{v}}{1+j\omega(C_{L}+C_{C})(R_{o}//R_{L})}\ .
  17. C M \displaystyle C_{M}
  18. v v a = A v R i R i + R A R L R L + R o 1 + j ω C C R o / A v D ω , \frac{v_{\ell}}{v_{a}}=A_{v}\frac{R_{i}}{R_{i}+R_{A}}\frac{R_{L}}{R_{L}+R_{o}}% \frac{1+j\omega C_{C}R_{o}/A_{v}}{D_{\omega}}\ ,
  19. D ω D_{\omega}\,\!
  20. = [ 1 + j ω ( C L + C C ) ( R o / / R L ) ] =[1+j\omega(C_{L}+C_{C})(R_{o}//R_{L})]\,\!
  21. \sdot [ 1 + j ω C i ( R A / / R i ) ] \sdot\ [1+j\omega C_{i}(R_{A}//R_{i})]\,\!
  22. + j ω C C ( R A / / R i ) \ +j\omega C_{C}(R_{A}//R_{i})\,\!
  23. \sdot ( 1 - A v R L R L + R O ) \sdot\left(1-A_{v}\frac{R_{L}}{R_{L}+R_{O}}\right)\,\!
  24. + ( j ω ) 2 C C C L ( R A / / R i ) ( R O / / R L ) . \ +(j\omega)^{2}C_{C}C_{L}(R_{A}//R_{i})(R_{O}//R_{L})\ .
  25. D ω = ( 1 + j ω τ 1 ) ( 1 + j ω τ 2 ) \ D_{\omega}=(1+j\omega{\tau}_{1})(1+j\omega{\tau}_{2})
  26. = 1 + j ω ( τ 1 + τ 2 ) ) + ( j ω ) 2 τ 1 τ 2 , =1+j\omega({\tau}_{1}+{\tau}_{2}))+(j\omega)^{2}\tau_{1}\tau_{2}\ ,
  27. τ 1 \tau_{1}
  28. τ 2 \tau_{2}
  29. τ 1 \tau_{1}
  30. τ 1 \tau_{1}
  31. τ 2 \tau_{2}
  32. τ 1 \tau_{1}
  33. τ 2 \tau_{2}
  34. D ω \displaystyle\ D_{\omega}
  35. C M = C C ( 1 - A v R L R L + R o ) , C_{M}=C_{C}\left(1-A_{v}\frac{R_{L}}{R_{L}+R_{o}}\right)\ ,
  36. τ 1 \tau_{1}
  37. τ 1 \tau_{1}
  38. τ 2 \tau_{2}
  39. τ 1 {\tau}_{1}
  40. τ 1 \tau_{1}
  41. τ 2 = τ 1 τ 2 τ 1 τ 1 τ 2 τ 1 + τ 2 . \tau_{2}=\frac{\tau_{1}\tau_{2}}{\tau_{1}}\approx\frac{\tau_{1}\tau_{2}}{\tau_% {1}+\tau_{2}}\ .
  42. τ 1 τ 2 \tau_{1}\tau_{2}
  43. τ 1 \tau_{1}
  44. τ 2 \displaystyle\tau_{2}
  45. τ 2 \tau_{2}
  46. = 20 log 10 ( A v ) log 10 ( f 2 / f 1 ) , =20\frac{\mathrm{log_{10}}(A_{v})}{\mathrm{log_{10}}(f_{2}/f_{1})}\ ,
  47. τ 1 τ 2 A v R i R i + R A \sdot R L R L + R o , \frac{\tau_{1}}{\tau_{2}}\approx A_{v}\frac{R_{i}}{R_{i}+R_{A}}\sdot\frac{R_{L% }}{R_{L}+R_{o}}\ ,
  48. τ 1 τ 2 ( τ 1 + τ 2 ) 2 τ 1 τ 2 A v R i R i + R A \sdot R L R L + R o , \frac{\tau_{1}}{\tau_{2}}\approx\frac{(\tau_{1}+\tau_{2})^{2}}{\tau_{1}\tau_{2% }}\approx A_{v}\frac{R_{i}}{R_{i}+R_{A}}\sdot\frac{R_{L}}{R_{L}+R_{o}}\ ,
  49. [ ( C M + C i ) ( R A / / R i ) + ( C L + C C ) ( R o / / R L ) ] 2 ( C C C L + C L C i + C i C C ) ( R A / / R i ) ( R O / / R L ) \frac{[(C_{M}+C_{i})(R_{A}//R_{i})+(C_{L}+C_{C})(R_{o}//R_{L})]^{2}}{(C_{C}C_{% L}+C_{L}C_{i}+C_{i}C_{C})(R_{A}//R_{i})(R_{O}//R_{L})}\,\!
  50. \color W h i t e \sdot = A v R i R i + R A \sdot R L R L + R o , {\color{White}\sdot}=A_{v}\frac{R_{i}}{R_{i}+R_{A}}\sdot\frac{R_{L}}{R_{L}+R_{% o}}\ ,
  51. τ 1 \tau_{1}

Polo_kinase.html

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Poly(3-hydroxybutyrate)_depolymerase.html

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Poly(glycerol-phosphate)_alpha-glucosyltransferase.html

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Poly(ribitol-phosphate)_beta-glucosyltransferase.html

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Poly(ribitol-phosphate)_N-acetylglucosaminyl-transferase.html

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Polyamine-transporting_ATPase.html

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Polygalacturonate_4-alpha-galacturonosyltransferase.html

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Polyneuridine-aldehyde_esterase.html

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  2. p o l y n e u r i d i n e - a l d e h y d e e s t e r a s e + H 2 O - C H 3 O H \xrightarrow[+H_{2}O\ -CH_{3}OH]{polyneuridine-aldehyde\ esterase}
  3. - C O 2 \xrightarrow[-CO_{2}]{}

Polynomial_and_rational_function_modeling.html

  1. y = a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 y=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}
  2. y = a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 b m x m + b m - 1 x m - 1 + + b 2 x 2 + b 1 x + b 0 y=\frac{a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{2}x^{2}+a_{1}x+a_{0}}{b_{m}x^{m}+b% _{m-1}x^{m-1}+\ldots+b_{2}x^{2}+b_{1}x+b_{0}}
  3. y = A 0 + A 1 x 1 + B 1 x + B 2 x 2 y=\frac{A_{0}+A_{1}x}{1+B_{1}x+B_{2}x^{2}}
  4. y = A 0 + A 1 x + + A p n x p n - B 1 x y - - B p d x p d y y=A_{0}+A_{1}x+\ldots+A_{p_{n}}x^{p_{n}}-B_{1}xy-\ldots-B_{p_{d}}x^{p_{d}}y

Polynomial_Diophantine_equation.html

  1. s a + t b = c sa+tb=c
  2. s ( x 2 + 1 ) + t ( x 3 + 1 ) = 2 x s(x^{2}+1)+t(x^{3}+1)=2x
  3. s = - x 3 - x 2 + x s=-x^{3}-x^{2}+x
  4. t = x 2 + x . t=x^{2}+x.
  5. a b ab
  6. r a b rab
  7. s s
  8. t t
  9. s = s + r b s^{\prime}=s+rb
  10. t = t - r a t^{\prime}=t-ra
  11. ( s + r b ) a + ( t - r a ) b = c . (s+rb)a+(t-ra)b=c.

Polynucleotide_3'-phosphatase.html

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Polynucleotide_5'-hydroxyl-kinase.html

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Polynucleotide_5'-phosphatase.html

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Polynucleotide_adenylyltransferase.html

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Polypeptide_N-acetylgalactosaminyltransferase.html

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Polyphosphate_kinase.html

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Polyphosphate—glucose_phosphotransferase.html

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Polysialic-acid_O-acetyltransferase.html

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Portal:Charles_Dickens.html

  1. 𝕋 𝔼 \mathbb{THE}
  2. 𝔸 𝕃 𝔼 𝕊 \mathbb{CHARLES}
  3. 𝔻 𝕀 𝕂 𝔼 𝕊 \mathbb{DICKENS}
  4. 𝕆 𝕋 𝔸 𝕃 \mathbb{PORTAL}

Post's_lattice.html

  1. π k n ( x 1 , , x n ) = x k , \pi_{k}^{n}(x_{1},\dots,x_{n})=x_{k},
  2. h ( x 1 , , x n ) = f ( g 1 ( x 1 , , x n ) , , g m ( x 1 , , x n ) ) , h(x_{1},\dots,x_{n})=f(g_{1}(x_{1},\dots,x_{n}),\dots,g_{m}(x_{1},\dots,x_{n})),
  3. th k n ( x 1 , , x n ) = { 1 if | { i x i = 1 } | k , 0 otherwise. \mathrm{th}^{n}_{k}(x_{1},\dots,x_{n})=\begin{cases}1&\,\text{if }\bigl|\{i% \mid x_{i}=1\}\bigr|\geq k,\\ 0&\,\text{otherwise.}\end{cases}
  4. maj = th 2 3 = ( x y ) ( x z ) ( y z ) . \mathrm{maj}=\mathrm{th}^{3}_{2}=(x\land y)\lor(x\land z)\lor(y\land z).
  5. ( a 1 , , a n ) ( b 1 , , b n ) a i b i for every i = 1 , , n , (a_{1},\dots,a_{n})\leq(b_{1},\dots,b_{n})\iff a_{i}\leq b_{i}\,\text{ for % every }i=1,\dots,n,
  6. ( a 1 , , a n ) ( b 1 , , b n ) = ( a 1 b 1 , , a n b n ) . (a_{1},\dots,a_{n})\land(b_{1},\dots,b_{n})=(a_{1}\land b_{1},\dots,a_{n}\land b% _{n}).
  7. f ( a 1 , , a i - 1 , c , a i + 1 , , a n ) = f ( a 1 , , d , a i + 1 , ) f ( b 1 , , c , b i + 1 , ) = f ( b 1 , , d , b i + 1 , ) f(a_{1},\dots,a_{i-1},c,a_{i+1},\dots,a_{n})=f(a_{1},\dots,d,a_{i+1},\dots)\ % \Rightarrow\ f(b_{1},\dots,c,b_{i+1},\dots)=f(b_{1},\dots,d,b_{i+1},\dots)
  8. f ( x 1 , , x n ) = i I x i f(x_{1},\dots,x_{n})=\bigwedge_{i\in I}x_{i}
  9. f ( x 1 , , x n ) = i I x i f(x_{1},\dots,x_{n})=\bigvee_{i\in I}x_{i}
  10. 𝐚 1 𝐚 k = 𝟎 f ( 𝐚 1 ) f ( 𝐚 k ) = 0. \mathbf{a}^{1}\land\cdots\land\mathbf{a}^{k}=\mathbf{0}\ \Rightarrow\ f(% \mathbf{a}^{1})\land\cdots\land f(\mathbf{a}^{k})=0.
  11. T 0 = k = 1 T 0 k \mathrm{T}_{0}^{\infty}=\bigcap_{k=1}^{\infty}\mathrm{T}_{0}^{k}
  12. 𝐚 1 𝐚 k = 𝟏 f ( 𝐚 1 ) f ( 𝐚 k ) = 1 , \mathbf{a}^{1}\lor\cdots\lor\mathbf{a}^{k}=\mathbf{1}\ \Rightarrow\ f(\mathbf{% a}^{1})\lor\cdots\lor f(\mathbf{a}^{k})=1,
  13. T 1 = k = 1 T 1 k \mathrm{T}_{1}^{\infty}=\bigcap_{k=1}^{\infty}\mathrm{T}_{1}^{k}
  14. h ( x 1 , , x n + m - 1 ) = f ( x 1 , , x n - 1 , g ( x n , , x n + m - 1 ) ) , h(x_{1},\dots,x_{n+m-1})=f(x_{1},\dots,x_{n-1},g(x_{n},\dots,x_{n+m-1})),

Precision_and_recall.html

  1. precision = | { relevant documents } { retrieved documents } | | { retrieved documents } | \,\text{precision}=\frac{|\{\,\text{relevant documents}\}\cap\{\,\text{% retrieved documents}\}|}{|\{\,\text{retrieved documents}\}|}
  2. recall = | { relevant documents } { retrieved documents } | | { relevant documents } | \,\text{recall}=\frac{|\{\,\text{relevant documents}\}\cap\{\,\text{retrieved % documents}\}|}{|\{\,\text{relevant documents}\}|}
  3. 𝑇𝑃𝑅 = 𝑇𝑃 / P = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑁 ) \mathit{TPR}=\mathit{TP}/P=\mathit{TP}/(\mathit{TP}+\mathit{FN})
  4. 𝑆𝑃𝐶 = 𝑇𝑁 / N = 𝑇𝑁 / ( 𝑇𝑁 + 𝐹𝑃 ) \mathit{SPC}=\mathit{TN}/N=\mathit{TN}/(\mathit{TN}+\mathit{FP})
  5. 𝑃𝑃𝑉 = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑃 ) \mathit{PPV}=\mathit{TP}/(\mathit{TP}+\mathit{FP})
  6. 𝑁𝑃𝑉 = 𝑇𝑁 / ( 𝑇𝑁 + 𝐹𝑁 ) \mathit{NPV}=\mathit{TN}/(\mathit{TN}+\mathit{FN})
  7. 𝐹𝑃𝑅 = 𝐹𝑃 / N = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑁 ) \mathit{FPR}=\mathit{FP}/N=\mathit{FP}/(\mathit{FP}+\mathit{TN})
  8. 𝐹𝑁𝑅 = 𝐹𝑁 / ( 𝐹𝑁 + 𝑇𝑃 ) = 1 - 𝑇𝑃𝑅 \mathit{FNR}=\mathit{FN}/(\mathit{FN}+\mathit{TP})=1-\mathit{TPR}
  9. 𝐹𝐷𝑅 = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑃 ) = 1 - 𝑃𝑃𝑉 \mathit{FDR}=\mathit{FP}/(\mathit{FP}+\mathit{TP})=1-\mathit{PPV}
  10. 𝐴𝐶𝐶 = ( 𝑇𝑃 + 𝑇𝑁 ) / ( P + N ) \mathit{ACC}=(\mathit{TP}+\mathit{TN})/(P+N)
  11. F1 = 2 𝑇𝑃 / ( 2 𝑇𝑃 + 𝐹𝑃 + 𝐹𝑁 ) \mathit{F1}=2\mathit{TP}/(2\mathit{TP}+\mathit{FP}+\mathit{FN})
  12. T P × T N - F P × F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) \frac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}
  13. Precision = t p t p + f p \,\text{Precision}=\frac{tp}{tp+fp}\,
  14. Recall = t p t p + f n \,\text{Recall}=\frac{tp}{tp+fn}\,
  15. True negative rate = t n t n + f p \,\text{True negative rate}=\frac{tn}{tn+fp}\,
  16. Accuracy = t p + t n t p + t n + f p + f n \,\text{Accuracy}=\frac{tp+tn}{tp+tn+fp+fn}\,
  17. F = 2 precision recall precision + recall F=2\cdot\frac{\mathrm{precision}\cdot\mathrm{recall}}{\mathrm{precision}+% \mathrm{recall}}
  18. F 1 F_{1}
  19. F β F_{\beta}
  20. β \beta
  21. F β = ( 1 + β 2 ) precision recall β 2 precision + recall F_{\beta}=(1+\beta^{2})\cdot\frac{\mathrm{precision}\cdot\mathrm{recall}}{% \beta^{2}\cdot\mathrm{precision}+\mathrm{recall}}
  22. F F
  23. F 2 F_{2}
  24. F 0.5 F_{0.5}
  25. F β F_{\beta}
  26. β \beta
  27. E = 1 - 1 α P + 1 - α R E=1-\frac{1}{\frac{\alpha}{P}+\frac{1-\alpha}{R}}
  28. F β = 1 - E F_{\beta}=1-E
  29. α = 1 1 + β 2 \alpha=\frac{1}{1+\beta^{2}}

Predicate_functor_logic.html

  1. { x 1 x n : F x 1 x n } \{x_{1}...x_{n}:Fx_{1}...x_{n}\}
  2. F x 1 x n . Fx_{1}...x_{n}.
  3. I F x 1 x 2 x n F x 1 x 1 x n F x 2 x 2 x n . IFx_{1}x_{2}...x_{n}\leftrightarrow Fx_{1}x_{1}...x_{n}\leftrightarrow Fx_{2}x% _{2}...x_{n}.
  4. ( F x 1 x n and I x 1 y ) F y x 2 x n . (Fx_{1}...x_{n}\and Ix_{1}y)\rightarrow Fyx_{2}...x_{n}.
  5. + F n = d e f { x 0 x 1 x n : F n x 1 x n } . \ +F^{n}=_{def}\{x_{0}x_{1}...x_{n}:F^{n}x_{1}...x_{n}\}.
  6. + F x 1 x n F x 2 x n . +Fx_{1}...x_{n}\leftrightarrow Fx_{2}...x_{n}.
  7. \exist F n = d e f { x 2 x n : \exist x 1 F n x 1 x n } . \exist F^{n}=_{def}\{x_{2}...x_{n}:\exist x_{1}F^{n}x_{1}...x_{n}\}.
  8. F x 1 x n \exist F x 2 x n . Fx_{1}...x_{n}\rightarrow\exist Fx_{2}...x_{n}.
  9. S F n = d e f { x 2 x n : F n x 2 x 2 x n } . \ SF^{n}=_{def}\{x_{2}...x_{n}:F^{n}x_{2}x_{2}...x_{n}\}.
  10. S F n \exist I F n . SF^{n}\leftrightarrow\exist IF^{n}.
  11. × \times
  12. F m × G n F m \exist m G n . F^{m}\times G^{n}\leftrightarrow F^{m}\exist^{m}G^{n}.
  13. F m x 1 x m G n x 1 x n ( F m × G n ) x 1 x m x 1 x n . F^{m}x_{1}...x_{m}G^{n}x_{1}...x_{n}\leftrightarrow(F^{m}\times G^{n})x_{1}...% x_{m}x_{1}...x_{n}.
  14. I n v F n = d e f { x 1 x n : F n x n x 1 x n - 1 } . \ InvF^{n}=_{def}\{x_{1}...x_{n}:F^{n}x_{n}x_{1}...x_{n-1}\}.
  15. I n v F x 1 x n F x n x 1 x n - 1 . InvFx_{1}...x_{n}\leftrightarrow Fx_{n}x_{1}...x_{n-1}.
  16. i n v F n = d e f { x 1 x n : F n x 2 x 1 x n } . \ invF^{n}=_{def}\{x_{1}...x_{n}:F^{n}x_{2}x_{1}...x_{n}\}.
  17. i n v F x 1 x n F x 2 x 1 x n . invFx_{1}...x_{n}\leftrightarrow Fx_{2}x_{1}...x_{n}.
  18. p F n = d e f { x 1 x n : F n x 1 x 3 x n x 2 } . \ pF^{n}=_{def}\{x_{1}...x_{n}:F^{n}x_{1}x_{3}...x_{n}x_{2}\}.
  19. p F x 1 x n I n v i n v F x 1 x 3 x n x 2 . pFx_{1}...x_{n}\leftrightarrow InvinvFx_{1}x_{3}...x_{n}x_{2}.
  20. x 1 x_{1}
  21. ( α and F x 1 x n ) β (\alpha\and Fx_{1}...x_{n})\rightarrow\beta
  22. ( α and \exist F x 2 x n ) β (\alpha\and\exist Fx_{2}...x_{n})\rightarrow\beta
  23. \exist I \exist I
  24. F n p n - 1 F n F^{n}\leftrightarrow p^{n-1}F^{n}
  25. F n + \exist F n F^{n}\rightarrow+\exist F^{n}
  26. F n \exist + F n F^{n}\leftrightarrow\exist+F^{n}
  27. + ¬ F n ¬ + F n +\lnot F^{n}\leftrightarrow\lnot+F^{n}
  28. ¬ \exist F n \exist ¬ F n \lnot\exist F^{n}\rightarrow\exist\lnot F^{n}
  29. p ¬ F n ¬ p F n p\lnot F^{n}\leftrightarrow\lnot pF^{n}
  30. + ( F n G m ) ( + F n + G m ) +(F^{n}G^{m})\leftrightarrow(+F^{n}+G^{m})
  31. p ( F n G m ) ( p F n p G m ) p(F^{n}G^{m})\leftrightarrow(pF^{n}pG^{m})
  32. I F n p n - 2 \exist p + F n IF^{n}\rightarrow p^{n-2}\exist p+F^{n}
  33. α \ \alpha
  34. p α , \ p\alpha,
  35. + α , \ +\alpha,
  36. ¬ \exist ¬ α \lnot\exist\lnot\alpha
  37. \exist x [ α ( x ) γ ( x ) ] ( \exist x α ( x ) \exist x γ ( x ) ) . \exist x[\alpha(x)\gamma(x)]\leftrightarrow(\exist x\alpha(x)\exist x\gamma(x)).
  38. ( F m and G n ) ( F m × G n ) ( F m \exist m G n ) ; m < n . (F^{m}\and G^{n})\leftrightarrow(F^{m}\times G^{n})\leftrightarrow(F^{m}\exist% ^{m}G^{n});m<n.
  39. ( α β ) ¬ ( ¬ α and ¬ β and ) . (\alpha\beta...)\leftrightarrow\lnot(\lnot\alpha\and\lnot\beta\and...).
  40. p ; I n v ; i n v ; ¬ ; I p;Inv;inv;\lnot;I
  41. + F n - 1 ; \ +F^{n-1};
  42. \exist F n + 1 ; \exist F^{n+1};
  43. S F n + 1 \ SF^{n+1}
  44. n \ n
  45. α m β n ; \ \alpha^{m}\beta^{n};
  46. F m × G n F^{m}\times G^{n}
  47. m a x ( m , n ) \ max(m,n)

Preisach_model_of_hysteresis.html

  1. h h
  2. - h -h
  3. R α , β R_{\alpha,\beta}
  4. α \alpha
  5. β \beta
  6. x x
  7. α \alpha
  8. y y
  9. x x
  10. x x
  11. β \beta
  12. x x
  13. x x
  14. y y
  15. x x
  16. α \alpha
  17. R α , β R_{\alpha,\beta}
  18. R α , β R_{\alpha,\beta}
  19. y ( x ) = { 1 if x β 0 if x α k if α < x < β y(x)=\begin{cases}1&\mbox{ if }~{}x\geq\beta\\ 0&\mbox{ if }~{}x\leq\alpha\\ k&\mbox{ if }~{}\alpha<x<\beta\end{cases}
  20. k = 0 k=0
  21. x x
  22. α < x < β \alpha<x<\beta
  23. x α x\leq\alpha
  24. k = 1 k=1
  25. x x
  26. α < x < β \alpha<x<\beta
  27. x β x\geq\beta
  28. y y
  29. x x
  30. α \alpha
  31. β \beta
  32. μ \mu
  33. ( α , β ) (\alpha,\beta)
  34. N N
  35. N N
  36. α β \alpha\beta
  37. α β \alpha\beta
  38. ( α i , β i ) (\alpha_{i},\beta_{i})
  39. R α i , β i R_{\alpha_{i},\beta_{i}}
  40. α < β \alpha<\beta
  41. α = β \alpha=\beta
  42. μ ( α , β ) \mu(\alpha,\beta)
  43. ( α i , β i ) (\alpha_{i},\beta_{i})
  44. μ ( α , β ) = 0 \mu(\alpha,\beta)=0

Prenyl-diphosphatase.html

  1. \rightleftharpoons

Preordered_class.html

  1. × \times

Prephenate_dehydratase.html

  1. \rightleftharpoons

Prismatic_compound_of_antiprisms.html

  1. ( cos k π n , sin k π n , ( - 1 ) k h ) \left(\cos\frac{k\pi}{n},\sin\frac{k\pi}{n},(-1)^{k}h\right)
  2. ( cos k π n , sin k π n , ( - 1 ) k + 1 h ) \left(\cos\frac{k\pi}{n},\sin\frac{k\pi}{n},(-1)^{k+1}h\right)
  3. 2 h 2 = cos π n - cos 2 π n . 2h^{2}=\cos\frac{\pi}{n}-\cos\frac{2\pi}{n}.

Process_management_(computing).html

  1. j = 1 N t j ( processor ) < t i ( execution ) \sum_{j=1}^{N}t_{j\,(\mathrm{processor})}<t_{i\,(\mathrm{execution}\!)}

Proclavaminate_amidinohydrolase.html

  1. \rightleftharpoons

Procollagen_galactosyltransferase.html

  1. \rightleftharpoons

Procollagen_glucosyltransferase.html

  1. \rightleftharpoons

Project_Euler.html

  1. sum 3 or 5 ( n ) \displaystyle\mathrm{sum}_{\text{3 or 5}}(n)
  2. sum k ( n ) \mathrm{sum}_{k}(n)
  3. k k
  4. n n

Proline—tRNA_ligase.html

  1. \rightleftharpoons

Propagule_pressure.html

  1. E ( N l , t ) = 1 - ( 1 - p ) N l , t E(Nl,t)=1-(1-p)Nl,t
  2. E ( N l , t ) = 1 - e 2 < s m a l l > - ( a N l , t ) E(Nl,t)=1-e2<small>-(aNl,t)

Propanediol_dehydratase.html

  1. \rightleftharpoons

Propanoyl-CoA_C-acyltransferase.html

  1. \rightleftharpoons

Proper_velocity.html

  1. η = sinh - 1 w c = tanh - 1 v c = ± cosh - 1 γ \eta=\sinh^{-1}\frac{w}{c}=\tanh^{-1}\frac{v}{c}=\pm\cosh^{-1}\gamma
  2. 𝐮 𝐯 = 𝐮 + 𝐯 + { β 𝐮 1 + β 𝐮 𝐮 𝐯 c 2 + 1 - β 𝐯 β 𝐯 } 𝐮 \mathbf{u}\oplus\mathbf{v}=\mathbf{u}+\mathbf{v}+\left\{{\frac{\beta_{\mathbf{% u}}}{1+\beta_{\mathbf{u}}}}{\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}}+{\frac{1-% \beta_{\mathbf{v}}}{\beta_{\mathbf{v}}}}\right\}\mathbf{u}
  3. β 𝐰 \beta_{\mathbf{w}}
  4. β 𝐰 = 1 1 + | 𝐰 | 2 c 2 \beta_{\mathbf{w}}=\frac{1}{\sqrt{1+\frac{|\mathbf{w}|^{2}}{c^{2}}}}
  5. w AC = ( w AB ) C + w BC = γ AC v AC \vec{w}\text{AC}=\left(\vec{w}\text{AB}\right)\text{C}+\vec{w}\text{BC}=\gamma% \text{AC}\vec{v}\text{AC}
  6. ( w AB ) C ( w AB w BC ( 1 + γ AB ) c 2 + γ BC ) w AB \left(\vec{w}\text{AB}\right)\text{C}\equiv\left(\frac{\vec{w}\text{AB}\cdot% \vec{w}\text{BC}}{(1+\gamma\text{AB})c^{2}}+\gamma\text{BC}\right)\vec{w}\text% {AB}
  7. ( c δ τ ) 2 = ( c δ t ) 2 - ( δ x ) 2 . (c\delta\tau)^{2}=(c\delta t)^{2}-(\delta x)^{2}.\,
  8. γ d t d τ = 1 + ( w c ) 2 = 1 1 - ( v c ) 2 = cosh [ η ] e η + e - η 2 \gamma\equiv\frac{dt}{d\tau}=\sqrt{1+\left(\frac{w}{c}\right)^{2}}=\frac{1}{% \sqrt{1-(\frac{v}{c})^{2}}}=\cosh[\eta]\equiv\frac{e^{\eta}+e^{-\eta}}{2}
  9. w c 1 c d x d τ = v c 1 1 - ( v c ) 2 = sinh [ η ] e η - e - η 2 = ± γ 2 - 1 \frac{w}{c}\equiv\frac{1}{c}\frac{dx}{d\tau}=\frac{v}{c}\frac{1}{\sqrt{1-(% \frac{v}{c})^{2}}}=\sinh[\eta]\equiv\frac{e^{\eta}-e^{-\eta}}{2}=\pm\sqrt{% \gamma^{2}-1}
  10. v c 1 c d x d t = w c 1 1 + ( w c ) 2 = tanh [ η ] e 2 η - 1 e 2 η + 1 = ± 1 - ( 1 γ ) 2 \frac{v}{c}\equiv\frac{1}{c}\frac{dx}{dt}=\frac{w}{c}\frac{1}{\sqrt{1+(\frac{w% }{c})^{2}}}=\tanh[\eta]\equiv\frac{e^{2\eta}-1}{e^{2\eta}+1}=\pm\sqrt{1-\left(% \frac{1}{\gamma}\right)^{2}}
  11. η = sinh - 1 [ w c ] = tanh - 1 [ v c ] = ± cosh - 1 [ γ ] \eta=\sinh^{-1}[\frac{w}{c}]=\tanh^{-1}[\frac{v}{c}]=\pm\cosh^{-1}[\gamma]
  12. η = ln [ w c + ( w c ) 2 + 1 ] = 1 2 ln [ 1 + v c 1 - v c ] = ± ln [ γ + γ 2 - 1 ] \eta=\ln\left[\frac{w}{c}+\sqrt{\left(\frac{w}{c}\right)^{2}+1}\right]=\frac{1% }{2}\ln\left[\frac{1+\frac{v}{c}}{1-\frac{v}{c}}\right]=\pm\ln\left[\gamma+% \sqrt{\gamma^{2}-1}\right]
  13. p A C m 1 = w A C = γ A C v A C = γ A B γ B C ( v A B + v B C ) = γ A B w B C + w A B γ B C \frac{p_{AC}}{m_{1}}=w_{AC}=\gamma_{AC}v_{AC}=\gamma_{AB}\gamma_{BC}\left(v_{% AB}+v_{BC}\right)=\gamma_{AB}w_{BC}+w_{AB}\gamma_{BC}\,
  14. α = Δ w Δ t = c Δ η Δ τ = c 2 Δ γ Δ x \alpha=\frac{\Delta w}{\Delta t}=c\frac{\Delta\eta}{\Delta\tau}=c^{2}\frac{% \Delta\gamma}{\Delta x}
  15. η = sinh - 1 [ w c ] = tanh - 1 [ v c ] = ± cosh - 1 [ γ ] \eta=\sinh^{-1}[\frac{w}{c}]=\tanh^{-1}[\frac{v}{c}]=\pm\cosh^{-1}[\gamma]

Propioin_synthase.html

  1. \rightleftharpoons

Propionate—CoA_ligase.html

  1. \rightleftharpoons

Propionyl-CoA_C2-trimethyltridecanoyltransferase.html

  1. \rightleftharpoons

Proteasome_ATPase.html

  1. \rightleftharpoons

Protein-arginine_deiminase.html

  1. \rightleftharpoons

Protein-glucosylgalactosylhydroxylysine_glucosidase.html

  1. \rightleftharpoons

Protein-glutamate_methylesterase.html

  1. \rightleftharpoons

Protein-glutamine_gamma-glutamyltransferase.html

  1. \rightleftharpoons

Protein-glutamine_glutaminase.html

  1. \rightleftharpoons

Protein-histidine_pros-kinase.html

  1. \rightleftharpoons

Protein-histidine_tele-kinase.html

  1. \rightleftharpoons

Protein-Npi-phosphohistidine-sugar_phosphotransferase.html

  1. \rightleftharpoons

Protein-secreting_ATPase.html

  1. \rightleftharpoons

Protein_geranylgeranyltransferase_type_I.html

  1. \rightleftharpoons

Protein_geranylgeranyltransferase_type_II.html

  1. \rightleftharpoons

Protein_N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Protoaphin-aglucone_dehydratase_(cyclizing).html

  1. \rightleftharpoons

Protocatechuate_decarboxylase.html

  1. \rightleftharpoons

Prunasin_beta-glucosidase.html

  1. \rightleftharpoons

Pseudo-zero_set.html

  1. m + 1 \mathbb{C}^{m+1}

Pseudouridine_kinase.html

  1. \rightleftharpoons

Pseudouridylate_synthase.html

  1. \rightleftharpoons

Pterin_deaminase.html

  1. \rightleftharpoons

Pugh's_closing_lemma.html

  1. f : M M f:M\to M
  2. C 1 C^{1}
  3. M M
  4. x x
  5. f f
  6. g g
  7. f f
  8. C 1 C^{1}
  9. Diff 1 ( M ) \operatorname{Diff}^{1}(M)
  10. x x
  11. g g

Purine-nucleoside_phosphorylase.html

  1. \rightleftharpoons

Purine_nucleosidase.html

  1. \rightleftharpoons

Putrescine_N-hydroxycinnamoyltransferase.html

  1. \rightleftharpoons

Pyrazolylalanine_synthase.html

  1. \rightleftharpoons

Pyridoxal_kinase.html

  1. \rightleftharpoons

Pyridoxal_phosphatase.html

  1. \rightleftharpoons

Pyridoxamine-phosphate_transaminase.html

  1. \rightleftharpoons

Pyridoxamine—oxaloacetate_transaminase.html

  1. \rightleftharpoons

Pyridoxamine—pyruvate_transaminase.html

  1. \rightleftharpoons

Pyridoxine_5'-O-beta-D-glucosyltransferase.html

  1. \rightleftharpoons

Pyridoxine_5'-phosphate_synthase.html

  1. \rightleftharpoons

Pyrimidine-5'-nucleotide_nucleosidase.html

  1. \rightleftharpoons

Pyrimidine-nucleoside_phosphorylase.html

  1. \rightleftharpoons

Pyrithiamine_deaminase.html

  1. \rightleftharpoons

Pyruvate,_phosphate_dikinase.html

  1. \rightleftharpoons

Pyruvate,_water_dikinase.html

  1. \rightleftharpoons

Pythagorean_addition.html

  1. a b = a 2 + b 2 . a\oplus b=\sqrt{a^{2}+b^{2}}.
  2. E = m c 2 p c . E=mc^{2}\oplus pc.
  3. x 1 2 + x 2 2 + + x n 2 = x 1 x 2 x n \sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}=x_{1}\oplus x_{2}\oplus\cdots% \oplus x_{n}
  4. 2 e = e \sqrt{2}e=e
  5. 2 = 1 \sqrt{2}=1

Quadratic_residue_code.html

  1. p p
  2. G F ( l ) GF(l)
  3. p p
  4. l l
  5. p p
  6. l l
  7. p p
  8. f ( x ) = j Q ( x - ζ j ) f(x)=\prod_{j\in Q}(x-\zeta^{j})
  9. Q Q
  10. p p
  11. { 1 , 2 , , p - 1 } \{1,2,\ldots,p-1\}
  12. ζ \zeta
  13. p p
  14. G F ( l ) GF(l)
  15. l l
  16. p p
  17. f f
  18. G F ( l ) GF(l)
  19. ( p + 1 ) / 2 (p+1)/2
  20. ζ \zeta
  21. p p
  22. ζ r \zeta^{r}
  23. r r
  24. p p
  25. g ( x ) = c + j Q x j g(x)=c+\sum_{j\in Q}x^{j}
  26. c G F ( l ) c\in GF(l)
  27. l = 2 l=2
  28. c c
  29. g ( 1 ) = 1 g(1)=1
  30. l l
  31. c = ( 1 + p * ) / 2 c=(1+\sqrt{p^{*}})/2
  32. p * = p p^{*}=p
  33. - p -p
  34. p p
  35. 1 1
  36. 3 3
  37. 4 4
  38. g ( x ) g(x)
  39. F l [ X ] / X p - 1 F_{l}[X]/\langle X^{p}-1\rangle
  40. g ( x ) g(x)
  41. p p
  42. p \sqrt{p}
  43. p 3 p\equiv 3
  44. 4 4
  45. P S L 2 ( p ) PSL_{2}(p)
  46. S L 2 ( p ) SL_{2}(p)
  47. ( 7 , 4 ) (7,4)
  48. G F ( 2 ) GF(2)
  49. ( 23 , 12 ) (23,12)
  50. G F ( 2 ) GF(2)
  51. ( 11 , 6 ) (11,6)
  52. G F ( 3 ) GF(3)