wpmath0000010_2

Kovats_retention_index.html

  1. log ( t r ) \log(t_{r}^{\prime})
  2. I = 100 × [ n + log ( t r ( unknown ) ) - log ( t r ( n ) ) log ( t r ( N ) ) - log ( t r ( n ) ) ] I=100\times\left[n+\frac{\log(t_{r(\,\text{unknown})}^{\prime})-\log(t_{r(n)}^% {\prime})}{\log(t_{r(N)}^{\prime})-\log(t_{r(n)}^{\prime})}\right]
  3. I = I=
  4. n = n=
  5. N = N=
  6. t r = t_{r}^{\prime}=
  7. I = 100 × [ n + t r ( unknown ) - t r ( n ) t r ( N ) - t r ( n ) ] I=100\times\left[n+\frac{t_{r(\,\text{unknown})}-t_{r(n)}}{t_{r(N)}-t_{r(n)}}\right]
  8. I = I=
  9. n = n=
  10. N = N=
  11. t r = t_{r}=

Kozeny–Carman_equation.html

  1. Δ p L = 180 μ Φ s 2 D p 2 ( 1 - ϵ ) 2 ϵ 3 v s \frac{\Delta p}{L}=\frac{180\mu}{\Phi_{\mathrm{s}}^{2}D_{\mathrm{p}}^{2}}\frac% {(1-\epsilon)^{2}}{\epsilon^{3}}v_{s}
  2. Δ p \Delta p
  3. L L
  4. v s v_{s}
  5. μ \mu
  6. ϵ \epsilon
  7. Φ s \Phi_{\mathrm{s}}
  8. D p D_{\mathrm{p}}

Kōmura's_theorem.html

  1. Φ ( t ) = 0 t φ ( s ) d s , \Phi(t)=\int_{0}^{t}\varphi(s)\,\mathrm{d}s,
  2. φ ( t ) = φ ( 0 ) + 0 t φ ( s ) d s . \varphi(t)=\varphi(0)+\int_{0}^{t}\varphi^{\prime}(s)\,\mathrm{d}s.

Krener's_theorem.html

  1. q ˙ = f ( q , u ) {\ }\dot{q}=f(q,u)
  2. q {\ q}
  3. M \ M
  4. u \ u
  5. U \ U
  6. = { f ( , u ) u U } {\mathcal{F}}=\{f(\cdot,u)\mid u\in U\}
  7. Lie \ \mathrm{Lie}\,\mathcal{F}
  8. {\mathcal{F}}
  9. q M \ q\in M
  10. Lie q = { g ( q ) g Lie } \ \mathrm{Lie}_{q}\,\mathcal{F}=\{g(q)\mid g\in\mathrm{Lie}\,\mathcal{F}\}
  11. T q M \ T_{q}M
  12. q \ q
  13. q \ q
  14. Lie q \mathrm{Lie}_{q}\,\mathcal{F}
  15. T q M \ T_{q}M
  16. q \ q
  17. q \ q
  18. \ \mathcal{F}
  19. Lie q = T q M \ \mathrm{Lie}_{q}\,\mathcal{F}=T_{q}M
  20. q \ q
  21. q \ q
  22. q M \ q\in M
  23. M \ M
  24. q \ q
  25. M \ M

Kuratowski_convergence.html

  1. d ( x , A ) = inf { d ( x , a ) | a A } d(x,A)=\inf\{d(x,a)|a\in A\}
  2. Li n A n = { x X | lim sup n d ( x , A n ) = 0 } \mathop{\mathrm{Li}}_{n\to\infty}A_{n}=\left\{x\in X\left|\limsup_{n\to\infty}% d(x,A_{n})=0\right.\right\}
  3. = { x X | for all open neighbourhoods U of x , U A n for large enough n } ; =\left\{x\in X\left|\begin{matrix}\mbox{for all open neighbourhoods }~{}U\mbox% { of }~{}x,\\ U\cap A_{n}\neq\emptyset\mbox{ for large enough }~{}n\end{matrix}\right.\right\};
  4. Ls n A n = { x X | lim inf n d ( x , A n ) = 0 } \mathop{\mathrm{Ls}}_{n\to\infty}A_{n}=\left\{x\in X\left|\liminf_{n\to\infty}% d(x,A_{n})=0\right.\right\}
  5. = { x X | for all open neighbourhoods U of x , U A n for infinitely many n } . =\left\{x\in X\left|\begin{matrix}\mbox{for all open neighbourhoods }~{}U\mbox% { of }~{}x,\\ U\cap A_{n}\neq\emptyset\mbox{ for infinitely many }~{}n\end{matrix}\right.% \right\}.
  6. Li n A n Ls n A n . \mathop{\mathrm{Li}}_{n\to\infty}A_{n}\subseteq\mathop{\mathrm{Ls}}_{n\to% \infty}A_{n}.
  7. A n = { x 𝐑 | sin ( n x ) = 0 } . A_{n}=\big\{x\in\mathbf{R}\big|\sin(nx)=0\big\}.

Kynurenate-7,8-dihydrodiol_dehydrogenase.html

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Kynurenine_3-monooxygenase.html

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Kynurenine_7,8-hydroxylase.html

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L-2-amino-4-chloropent-4-enoate_dehydrochlorinase.html

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L-3-cyanoalanine_synthase.html

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L-amino-acid_dehydrogenase.html

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L-amino-acid_oxidase.html

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L-aminoadipate-semialdehyde_dehydrogenase.html

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L-arabinitol_2-dehydrogenase.html

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L-arabinitol_4-dehydrogenase.html

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L-arabinose_1-dehydrogenase.html

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L-arabinose_isomerase.html

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L-ascorbate_oxidase.html

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L-ascorbate_peroxidase.html

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L-ascorbate—cytochrome-b5_reductase.html

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L-aspartate_oxidase.html

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L-cysteate_sulfo-lyase.html

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L-dopachrome_isomerase.html

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L-erythro-3,5-diaminohexanoate_dehydrogenase.html

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L-fucose_isomerase.html

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L-fuculose-phosphate_aldolase.html

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L-galactonolactone_oxidase.html

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L-glutamate_oxidase.html

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L-glycol_dehydrogenase.html

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L-gulonate_3-dehydrogenase.html

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L-iditol_2-dehydrogenase.html

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L-idonate_5-dehydrogenase.html

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L-lysine_6-monooxygenase_(NADPH).html

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L-lysine_6-oxidase.html

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L-lysine_oxidase.html

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L-methionine_(R)-S-oxide_reductase.html

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L-methionine_(S)-S-oxide_reductase.html

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L-pipecolate_dehydrogenase.html

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L-pipecolate_oxidase.html

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L-rhamnose_1-dehydrogenase.html

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L-rhamnose_isomerase.html

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L-ribulose-5-phosphate_3-epimerase.html

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L-ribulose-5-phosphate_4-epimerase.html

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L-serine_ammonia-lyase.html

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L-seryl-tRNASec_selenium_transferase.html

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L-shell.html

  1. L = 2 L=2
  2. r = L cos 2 λ r=L\cos^{2}\lambda
  3. r r
  4. λ \lambda
  5. L L

L-sorbose_oxidase.html

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L-threonate_3-dehydrogenase.html

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L-threonine_3-dehydrogenase.html

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L-xylose_1-dehydrogenase.html

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Lactaldehyde_dehydrogenase.html

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Lactaldehyde_reductase.html

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Lactaldehyde_reductase_(NADPH).html

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Lactate_2-monooxygenase.html

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Lactate_aldolase.html

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Lactate—malate_transhydrogenase.html

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Lactoylglutathione_lyase.html

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Lagrange_bracket.html

  1. [ u , v ] p , q = i = 1 n ( q i u p i v - p i u q i v ) . [u,v]_{p,q}=\sum_{i=1}^{n}\left(\frac{\partial q_{i}}{\partial u}\frac{% \partial p_{i}}{\partial v}-\frac{\partial p_{i}}{\partial u}\frac{\partial q_% {i}}{\partial v}\right).
  2. Q = Q ( q , p ) , P = P ( q , p ) Q=Q(q,p),P=P(q,p)
  3. [ u , v ] q , p = [ u , v ] Q , P [u,v]_{q,p}=[u,v]_{Q,P}
  4. [ u i , u j ] p , q , 1 i , j 2 n [u_{i},u_{j}]_{p,q},\quad 1\leq i,j\leq 2n
  5. { u i , u j } , 1 i , j 2 n \{u_{i},u_{j}\},\quad 1\leq i,j\leq 2n
  6. [ Q i , Q j ] p , q = 0 , [ P i , P j ] p , q = 0 , [ Q i , P j ] p , q = - [ P j , Q i ] p , q = δ i j . [Q_{i},Q_{j}]_{p,q}=0,\quad[P_{i},P_{j}]_{p,q}=0,\quad[Q_{i},P_{j}]_{p,q}=-[P_% {j},Q_{i}]_{p,q}=\delta_{ij}.

Laman_graph.html

  1. ( k , l ) (k,l)
  2. n n
  3. k n - l kn-l
  4. ( k , l ) (k,l)
  5. ( k , l ) (k,l)
  6. k n - l kn-l

Lambda-mu_calculus.html

  1. ( λ x . u ) v c u [ v / x ] (\lambda x.u)v\;\triangleright_{c}\;u[v/x]
  2. ( μ β . u ) v c μ β . u [ [ β ] ( w v ) / [ β ] w ] (\mu\beta.u)v\;\triangleright_{c}\;\mu\beta.u\left[[\beta](wv)/[\beta]w\right]
  3. [ α ] μ β . u c u [ α / β ] [\alpha]\mu\beta.u\;\triangleright_{c}\;u[\alpha/\beta]
  4. μ α . [ α ] u c u \mu\alpha.[\alpha]u\;\triangleright_{c}\;u

Langmuir_circulation.html

  1. ν T \nu_{T}
  2. τ \tau
  3. u * u_{*}
  4. La = ν T 3 k 6 σ a 2 u * 2 k 4 or ν T 3 β 6 u * 2 S 0 β 3 \mathrm{La}=\sqrt{\frac{\nu^{3}_{T}k^{6}}{\sigma a^{2}u^{2}_{*}k^{4}}}\ % \mathrm{or}\ \sqrt{\frac{\nu_{T}^{3}\beta^{6}}{u^{2}_{*}S_{0}\beta^{3}}}
  5. a a
  6. σ \sigma
  7. k k
  8. β \beta
  9. S 0 S_{0}
  10. u i t + u j j u i = - 2 ϵ i j k Ω j ( u k s + u k ) - i ( P ρ 0 + 1 2 u j s u j s + u j s u j ) + ϵ i j k u j s ϵ k l m l u m + g i ρ ρ 0 + j ν j u i \frac{\partial u_{i}}{\partial t}+u_{j}\nabla_{j}u_{i}=\begin{array}[]{l}-2% \epsilon_{ijk}\Omega_{j}\left(u^{s}_{k}+u_{k}\right)-\nabla_{i}\left(\frac{P}{% \rho_{0}}+\frac{1}{2}u^{s}_{j}u^{s}_{j}+u^{s}_{j}u_{j}\right)\\ +\epsilon_{ijk}u^{s}_{j}\epsilon_{klm}\nabla_{l}u_{m}+g_{i}\frac{\rho}{\rho_{0% }}+\nabla_{j}\nu\nabla_{j}u_{i}\end{array}
  11. i u i = 0 \nabla_{i}u_{i}=0
  12. ρ t + u j j ρ = i κ i ρ \frac{\partial\rho}{\partial t}+u_{j}\nabla_{j}\rho=\nabla_{i}\kappa\nabla_{i}\rho
  13. u i u_{i}
  14. Ω \Omega
  15. u i s u^{s}_{i}
  16. P P
  17. g i g_{i}
  18. ρ \rho
  19. ρ 0 \rho_{0}
  20. ν \nu
  21. κ \kappa

Laplace_limit.html

  1. E = M + sin ( M ) ε + 1 2 sin ( 2 M ) ε 2 + ( 3 8 sin ( 3 M ) - 1 8 sin ( M ) ) ε 3 + E=M+\sin(M)\,\varepsilon+\tfrac{1}{2}\sin(2M)\,\varepsilon^{2}+\left(\tfrac{3}% {8}\sin(3M)-\tfrac{1}{8}\sin(M)\right)\,\varepsilon^{3}+\cdots

Laplace_principle_(large_deviations_theory).html

  1. A e - φ ( x ) d x < + . \int_{A}e^{-\varphi(x)}\,\mathrm{d}x<+\infty.
  2. lim θ + 1 θ log A e - θ φ ( x ) d x = - ess inf x A φ ( x ) , \lim_{\theta\to+\infty}\frac{1}{\theta}\log\int_{A}e^{-\theta\varphi(x)}\,% \mathrm{d}x=-\mathop{\mathrm{ess\,inf}}_{x\in A}\varphi(x),
  3. A e - θ φ ( x ) d x exp ( - θ ess inf x A φ ( x ) ) . \int_{A}e^{-\theta\varphi(x)}\,\mathrm{d}x\approx\exp\left(-\theta\mathop{% \mathrm{ess\,inf}}_{x\in A}\varphi(x)\right).
  4. 𝐏 θ ( A ) = ( A e - θ φ ( x ) d x ) / ( 𝐑 d e - θ φ ( y ) d y ) \mathbf{P}_{\theta}(A)=\left(\int_{A}e^{-\theta\varphi(x)}\,\mathrm{d}x\right)% \Big/\left(\int_{\mathbf{R}^{d}}e^{-\theta\varphi(y)}\,\mathrm{d}y\right)
  5. lim ε 0 ε log 𝐏 [ ε X A ] = - ess inf x A x 2 2 \lim_{\varepsilon\downarrow 0}\varepsilon\log\mathbf{P}\big[\sqrt{\varepsilon}% X\in A\big]=-\mathop{\mathrm{ess\,inf}}_{x\in A}\frac{x^{2}}{2}

Lathosterol_oxidase.html

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Latia-luciferin_monooxygenase_(demethylating).html

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Lead_(engineering).html

  1. Lead angle = arctan ( l π d m ) \mbox{Lead angle}~{}=\arctan\left(\frac{l}{\pi d_{m}}\right)

Least-squares_spectral_analysis.html

  1. ϕ 𝐀 x \phi\approx\,\textbf{A}x
  2. x = ( 𝐀 T 𝐀 ) - 1 𝐀 T ϕ . x=(\,\textbf{A}^{\mathrm{T}}\,\textbf{A})^{-1}\,\textbf{A}^{\mathrm{T}}\phi.
  3. x = 𝐀 T ϕ x=\,\textbf{A}^{\mathrm{T}}\phi
  4. tan 2 ω τ = j sin 2 ω t j j cos 2 ω t j . \tan{2\omega\tau}=\frac{\sum_{j}\sin 2\omega t_{j}}{\sum_{j}\cos 2\omega t_{j}}.
  5. P x ( ω ) = 1 2 ( [ j X j cos ω ( t j - τ ) ] 2 j cos 2 ω ( t j - τ ) + [ j X j sin ω ( t j - τ ) ] 2 j sin 2 ω ( t j - τ ) ) P_{x}(\omega)=\frac{1}{2}\left(\frac{\left[\sum_{j}X_{j}\cos\omega(t_{j}-\tau)% \right]^{2}}{\sum_{j}\cos^{2}\omega(t_{j}-\tau)}+\frac{\left[\sum_{j}X_{j}\sin% \omega(t_{j}-\tau)\right]^{2}}{\sum_{j}\sin^{2}\omega(t_{j}-\tau)}\right)
  6. ϕ ( t ) A sin ω t + B cos ω t . \phi(t)\approx A\sin\omega t+B\cos\omega t.

Leghemoglobin_reductase.html

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Lehmer's_conjecture.html

  1. μ > 1 \mu>1
  2. P ( x ) [ x ] P(x)\in\mathbb{Z}[x]
  3. ( P ( x ) ) \mathcal{M}(P(x))
  4. P ( x ) P(x)
  5. μ \mu
  6. P ( x ) P(x)
  7. x x
  8. ( P ( x ) ) = 1 \mathcal{M}(P(x))=1
  9. P ( x ) P(x)
  10. P ( x ) P(x)
  11. \mathbb{C}
  12. P ( x ) = a 0 ( x - α 1 ) ( x - α 2 ) ( x - α D ) , P(x)=a_{0}(x-\alpha_{1})(x-\alpha_{2})\cdots(x-\alpha_{D}),
  13. ( P ( x ) ) = | a 0 | i = 1 D max ( 1 , | α i | ) . \mathcal{M}(P(x))=|a_{0}|\prod_{i=1}^{D}\max(1,|\alpha_{i}|).
  14. P ( x ) = x 10 + x 9 - x 7 - x 6 - x 5 - x 4 - x 3 + x + 1 , P(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1\,,
  15. ( P ( x ) ) = 1.176280818 . \mathcal{M}(P(x))=1.176280818\dots\ .
  16. μ = 1.176280818 \mu=1.176280818\dots
  17. P ( x ) = a 0 ( x - α 1 ) ( x - α 2 ) ( x - α D ) P(x)=a_{0}(x-\alpha_{1})(x-\alpha_{2})\cdots(x-\alpha_{D})
  18. ( P ( x ) ) = | a 0 | i = 1 D max ( 1 , | α i | ) . \mathcal{M}(P(x))=|a_{0}|\prod_{i=1}^{D}\max(1,|\alpha_{i}|).
  19. m ( P ) = log ( ( P ( x ) ) m(P)=\log(\mathcal{M}(P(x))
  20. P P
  21. ( P ) \mathcal{M}(P)
  22. m ( P ) m(P)
  23. m ( P ) 0 m(P)\geq 0
  24. m ( P ) = 0 m(P)=0
  25. P P
  26. x x
  27. x n x^{n}
  28. n n
  29. m ( P ) = 0 m(P)=0
  30. Δ n = Res ( P ( x ) , x n - 1 ) = i = 1 D ( α i n - 1 ) \Delta_{n}=\,\text{Res}(P(x),x^{n}-1)=\prod^{D}_{i=1}(\alpha_{i}^{n}-1)
  31. P P
  32. P P
  33. lim | Δ n | 1 / n = ( P ) \lim|\Delta_{n}|^{1/n}=\mathcal{M}(P)
  34. P P
  35. c > 0 c>0
  36. m ( P ) > c m(P)>c
  37. P P
  38. c > 0 c>0
  39. P P
  40. 0 < m ( P ) < c 0<m(P)<c
  41. P ( x ) [ x ] P(x)\in\mathbb{Z}[x]
  42. D D
  43. x D P ( x - 1 ) P ( x ) x^{D}P(x^{-1})\neq P(x)
  44. C > 1 C>1
  45. ( P ( x ) ) = 1 \mathcal{M}(P(x))=1
  46. log ( P ( x ) ) C D log D . \log\mathcal{M}(P(x))\geq\frac{C}{D\log D}.
  47. log ( P ( x ) ) C ( log log D log D ) 3 . \log\mathcal{M}(P(x))\geq C\left(\frac{\log\log D}{\log D}\right)^{3}.
  48. E / K E/K
  49. K K
  50. h ^ E : E ( K ¯ ) \hat{h}_{E}:E(\bar{K})\to\mathbb{R}
  51. ( deg P ) - 1 log ( P ( x ) ) (\deg P)^{-1}\log\mathcal{M}(P(x))
  52. h ^ E ( Q ) = 0 \hat{h}_{E}(Q)=0
  53. Q Q
  54. E ( K ¯ ) E(\bar{K})
  55. C ( E / K ) > 0 C(E/K)>0
  56. h ^ E ( Q ) C ( E / K ) D \hat{h}_{E}(Q)\geq\frac{C(E/K)}{D}
  57. Q E ( K ¯ ) Q\in E(\bar{K})
  58. D = [ K ( Q ) : K ] D=[K(Q):K]
  59. h ^ E ( Q ) C ( E / K ) D ( log log D log D ) 3 , \hat{h}_{E}(Q)\geq\frac{C(E/K)}{D}\left(\frac{\log\log D}{\log D}\right)^{3},
  60. h ^ E ( Q ) C ( E / K ) D 3 ( log D ) 2 , \hat{h}_{E}(Q)\geq\frac{C(E/K)}{D^{3}(\log D)^{2}},
  61. h ^ E ( Q ) C ( E / K ) D 2 ( log D ) 2 , \hat{h}_{E}(Q)\geq\frac{C(E/K)}{D^{2}(\log D)^{2}},
  62. M ( P ) M ( x 3 - x - 1 ) 1.3247 M(P)\geq M(x^{3}-x-1)\approx 1.3247
  63. M ( P ) M ( x 2 - x - 1 ) 1.618. M(P)\geq M(x^{2}-x-1)\approx 1.618.

Lemaître_coordinates.html

  1. d s 2 = ( 1 - r g r ) d t 2 - d r 2 1 - r g r - r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , ds^{2}=\left(1-{r_{g}\over r}\right)dt^{2}-{dr^{2}\over 1-{r_{g}\over r}}-r^{2% }\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)\;,
  2. d s 2 ds^{2}
  3. r g = 2 M r_{g}=2M
  4. M M
  5. t , r , θ , ϕ t,r,\theta,\phi
  6. c c
  7. G G
  8. r = r g r=r_{g}
  9. { t , r } \{t,r\}
  10. { τ , ρ } , \{\tau,\rho\},
  11. { d τ = d t + r g r 1 ( 1 - r g r ) d r d ρ = d t + r r g 1 ( 1 - r g r ) d r \begin{cases}d\tau=dt+\sqrt{\frac{r_{g}}{r}}\frac{1}{(1-\frac{r_{g}}{r})}dr\\ d\rho=dt+\sqrt{\frac{r}{r_{g}}}\frac{1}{(1-\frac{r_{g}}{r})}dr\end{cases}
  12. d s 2 = d τ 2 - r g r d ρ 2 - r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}=d\tau^{2}-\frac{r_{g}}{r}d\rho^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta d% \phi^{2})
  13. r = [ 3 2 ( ρ - τ ) ] 2 / 3 r g 1 / 3 . r=\left[\frac{3}{2}(\rho-\tau)\right]^{2/3}r_{g}^{1/3}\;.
  14. 3 2 ( ρ - τ ) = r g \frac{3}{2}(\rho-\tau)=r_{g}
  15. ρ - τ = 0 \rho-\tau=0
  16. d r = ( ± 1 - r g r ) d τ , dr=\left(\pm 1-\sqrt{r_{g}\over r}\right)d\tau,
  17. d r < 0 dr<0

Leucine_2,3-aminomutase.html

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Leucine_dehydrogenase.html

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Leucoanthocyanidin_reductase.html

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Leucocyanidin_oxygenase.html

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Leukotriene-B4_20-monooxygenase.html

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Leukotriene-C4_synthase.html

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Leukotriene-E4_20-monooxygenase.html

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Licodione_2'-O-methyltransferase.html

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Licodione_synthase.html

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Lignin_peroxidase.html

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Lignostilbene_alphabeta-dioxygenase.html

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Limonene-1,2-epoxide_hydrolase.html

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Linalool_8-monooxygenase.html

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Linear_space_(geometry).html

  1. ( v , k , 1 ) (v,k,1)

Linear_sweep_voltammetry.html

  1. A + e - = A - , E s = 0.00 V A+e^{-}=A^{-},E_{s}=0.00V

Linoleate_11-lipoxygenase.html

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Linoleate_diol_synthase.html

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Linoleate_isomerase.html

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Linoleoyl-CoA_desaturase.html

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Liouville's_equation.html

  1. f f
  2. f 2 ( d x 2 + d y 2 ) f^{2}(\mathrm{d}x^{2}+\mathrm{d}y^{2})
  3. K K
  4. Δ 0 log f = - K f 2 , \Delta_{0}\log f=-Kf^{2},
  5. Δ 0 \Delta_{0}
  6. Δ 0 = 2 x 2 + 2 y 2 = 4 z z ¯ \Delta_{0}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^% {2}}=4\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}
  7. x , y x,y
  8. f f
  9. f 2 f^{2}
  10. f f
  11. log f = u \log\,f=u
  12. Δ 0 u = - K e 2 u . \Delta_{0}u=-Ke^{2u}.
  13. 2 log f = u 2\log\,f=u
  14. Δ 0 u = - 2 K e u 2 u z z ¯ = - K 2 e u . \Delta_{0}u=-2Ke^{u}\quad\Longleftrightarrow\quad\frac{\partial^{2}u}{{% \partial z}{\partial\bar{z}}}=-\frac{K}{2}e^{u}.
  15. Δ LB = 1 f 2 Δ 0 \Delta_{\mathrm{LB}}=\frac{1}{f^{2}}\Delta_{0}
  16. Δ LB log f = - K . \Delta_{\mathrm{LB}}\log\;f=-K.
  17. Ω \Omega
  18. u ( z , z ¯ ) = 1 2 ln ( 4 | d f ( z ) / d z | 2 ( 1 + K | f ( z ) | 2 ) 2 ) u(z,\bar{z})=\frac{1}{2}\ln\left(4\frac{\left|{\mathrm{d}f(z)}/{\mathrm{d}z}% \right|^{2}}{(1+K\left|f(z)\right|^{2})^{2}}\right)
  19. f ( z ) f(z)
  20. d f ( z ) / d z 0 {\mathrm{d}f(z)}/{\mathrm{d}z}\neq 0
  21. z Ω z\in\Omega
  22. f ( z ) f(z)
  23. Ω \Omega
  24. d l 2 = g ( z , z ¯ ) d z d z ¯ \mathrm{d}l^{2}=g(z,{\bar{z}}){\mathrm{d}z}{\mathrm{d}\bar{z}}
  25. K K
  26. K > 0 K>0
  27. K < n o w i k i 0 K<nowiki>=0
  28. 2 f x 2 + 2 f y 2 = e f \frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}=e^{f}

Lipoyl_synthase.html

  1. \rightleftharpoons

List_of_Chicago_Bulls_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_Lakers_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_price_index_formulas.html

  1. p 0 p_{0}
  2. p t p_{t}
  3. q 0 q_{0}
  4. q t q_{t}
  5. p t / p 0 p_{t}/p_{0}
  6. P L = ( p n q 0 ) ( p 0 q 0 ) P_{L}=\frac{\sum(p_{n}\cdot q_{0})}{\sum(p_{0}\cdot q_{0})}
  7. q 0 q_{0}
  8. P P = ( p n q n ) ( p 0 q n ) P_{P}=\frac{\sum(p_{n}\cdot q_{n})}{\sum(p_{0}\cdot q_{n})}
  9. q n q_{n}
  10. P C = 1 n ( p t p 0 ) P_{C}=\frac{1}{n}\cdot\sum\left(\frac{p_{t}}{p_{0}}\right)
  11. P D = 1 n ( p t ) 1 n ( p 0 ) = ( p t ) ( p 0 ) P_{D}=\frac{\frac{1}{n}\cdot\sum(p_{t})}{\frac{1}{n}\cdot\sum(p_{0})}=\frac{% \sum(p_{t})}{\sum(p_{0})}
  12. P J = ( p t p 0 ) 1 / n P_{J}=\prod\left(\frac{p_{t}}{p_{0}}\right)^{1/n}
  13. P H R = 1 1 n ( p 0 p t ) P_{HR}=\frac{1}{\frac{1}{n}\cdot\sum\left(\frac{p_{0}}{p_{t}}\right)}
  14. P C S W D = P C P H R P_{CSWD}=\sqrt{P_{C}\cdot P_{HR}}
  15. P R H = ( n p 0 ) ( n p t ) P_{RH}=\frac{\sum\left(\frac{n}{p_{0}}\right)}{\sum\left(\frac{n}{p_{t}}\right)}
  16. P M E = [ p c , t n 1 2 ( q c , t 0 + q c , t n ) ] [ p c , t 0 1 2 ( q c , t 0 + q c , t n ) ] = [ p c , t n ( q c , t 0 + q c , t n ) ] [ p c , t 0 ( q c , t 0 + q c , t n ) ] P_{ME}=\frac{\sum[p_{c,t_{n}}\cdot\frac{1}{2}\cdot(q_{c,t_{0}}+q_{c,t_{n}})]}{% \sum[p_{c,t_{0}}\cdot\frac{1}{2}\cdot(q_{c,t_{0}}+q_{c,t_{n}})]}=\frac{\sum[p_% {c,t_{n}}\cdot(q_{c,t_{0}}+q_{c,t_{n}})]}{\sum[p_{c,t_{0}}\cdot(q_{c,t_{0}}+q_% {c,t_{n}})]}
  17. P F = P P P L P_{F}=\sqrt{P_{P}\cdot P_{L}}
  18. P t = i = 1 n ( p i t p i 0 ) 1 2 [ p i 0 q i 0 i = 1 m ( p i 0 q i 0 ) + p i t q i t i = 1 m ( p i t q i t ) ] P_{t}=\prod_{i=1}^{n}\left(\frac{p_{it}}{p_{i0}}\right)^{\frac{1}{2}\left[% \frac{p_{i0}q_{i0}}{\sum_{i=1}^{m}\left(p_{i0}q_{i0}\right)}+\frac{p_{it}q_{it% }}{\sum_{i=1}^{m}\left(p_{it}q_{it}\right)}\right]}
  19. P W = ( p t q 0 q t ) ( p 0 q 0 q t ) P_{W}=\frac{\sum\left(p_{t}\cdot\sqrt{q_{0}\cdot q_{t}}\right)}{\sum\left(p_{0% }\cdot\sqrt{q_{0}\cdot q_{t}}\right)}

List_of_Runge–Kutta_methods.html

  1. d y d t = f ( t , y ) \frac{dy}{dt}=f(t,y)\,
  2. y n + 1 = y n + h i = 1 s b i k i y_{n+1}=y_{n}+h\sum_{i=1}^{s}b_{i}k_{i}\,
  3. k i = f ( t n + c i h , y n + h j = 1 s a i j k j ) . k_{i}=f\left(t_{n}+c_{i}h,y_{n}+h\sum_{j=1}^{s}a_{ij}k_{j}\right).
  4. c 1 a 11 a 12 a 1 s c 2 a 21 a 22 a 2 s c s a s 1 a s 2 a s s b 1 b 2 b s \begin{array}[]{c|cccc}c_{1}&a_{11}&a_{12}&\dots&a_{1s}\\ c_{2}&a_{21}&a_{22}&\dots&a_{2s}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ c_{s}&a_{s1}&a_{s2}&\dots&a_{ss}\\ \hline&b_{1}&b_{2}&\dots&b_{s}\\ \end{array}
  5. [ a i j ] [a_{ij}]
  6. 0 0 1 \begin{array}[]{c|c}0&0\\ \hline&1\\ \end{array}
  7. 0 0 0 1 / 2 1 / 2 0 0 1 \begin{array}[]{c|cc}0&0&0\\ 1/2&1/2&0\\ \hline&0&1\\ \end{array}
  8. 0 0 0 1 1 0 1 / 2 1 / 2 \begin{array}[]{c|cc}0&0&0\\ 1&1&0\\ \hline&1/2&1/2\\ \end{array}
  9. 0 0 0 2 / 3 2 / 3 0 1 / 4 3 / 4 \begin{array}[]{c|cc}0&0&0\\ 2/3&2/3&0\\ \hline&1/4&3/4\\ \end{array}
  10. 0 0 0 x x 0 1 - 1 2 x 1 2 x \begin{array}[]{c|ccc}0&0&0\\ x&x&0\\ \hline&1-\frac{1}{2x}&\frac{1}{2x}\\ \end{array}
  11. 0 0 0 0 1 / 2 1 / 2 0 0 1 - 1 2 0 1 / 6 2 / 3 1 / 6 \begin{array}[]{c|ccc}0&0&0&0\\ 1/2&1/2&0&0\\ 1&-1&2&0\\ \hline&1/6&2/3&1/6\\ \end{array}
  12. 0 0 0 0 0 1 / 2 1 / 2 0 0 0 1 / 2 0 1 / 2 0 0 1 0 0 1 0 1 / 6 1 / 3 1 / 3 1 / 6 \begin{array}[]{c|cccc}0&0&0&0&0\\ 1/2&1/2&0&0&0\\ 1/2&0&1/2&0&0\\ 1&0&0&1&0\\ \hline&1/6&1/3&1/3&1/6\\ \end{array}
  13. 0 0 0 0 0 1 / 3 1 / 3 0 0 0 2 / 3 - 1 / 3 1 0 0 1 1 - 1 1 0 1 / 8 3 / 8 3 / 8 1 / 8 \begin{array}[]{c|cccc}0&0&0&0&0\\ 1/3&1/3&0&0&0\\ 2/3&-1/3&1&0&0\\ 1&1&-1&1&0\\ \hline&1/8&3/8&3/8&1/8\\ \end{array}
  14. y n + 1 * = y n + h i = 1 s b i * k i , y^{*}_{n+1}=y_{n}+h\sum_{i=1}^{s}b^{*}_{i}k_{i},
  15. k i k_{i}
  16. e n + 1 = y n + 1 - y n + 1 * = h i = 1 s ( b i - b i * ) k i , e_{n+1}=y_{n+1}-y^{*}_{n+1}=h\sum_{i=1}^{s}(b_{i}-b^{*}_{i})k_{i},
  17. b i * b^{*}_{i}
  18. c 1 a 11 a 12 a 1 s c 2 a 21 a 22 a 2 s c s a s 1 a s 2 a s s b 1 b 2 b s b 1 * b 2 * b s * \begin{array}[]{c|cccc}c_{1}&a_{11}&a_{12}&\dots&a_{1s}\\ c_{2}&a_{21}&a_{22}&\dots&a_{2s}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ c_{s}&a_{s1}&a_{s2}&\dots&a_{ss}\\ \hline&b_{1}&b_{2}&\dots&b_{s}\\ &b_{1}^{*}&b_{2}^{*}&\dots&b_{s}^{*}\\ \end{array}
  19. 0 1 1 1 / 2 1 / 2 1 0 \begin{array}[]{c|cc}0&\\ 1&1\\ \hline&1/2&1/2\\ &1&0\end{array}
  20. 1 1 1 \begin{array}[]{c|c}1&1\\ \hline&1\\ \end{array}
  21. 1 / 2 1 / 2 1 \begin{array}[]{c|c}1/2&1/2\\ \hline&1\end{array}
  22. 1 2 - 3 6 1 4 1 4 - 3 6 1 2 + 3 6 1 4 + 3 6 1 4 1 2 1 2 \begin{array}[]{c|cc}\frac{1}{2}-\frac{\sqrt{3}}{6}&\frac{1}{4}&\frac{1}{4}-% \frac{\sqrt{3}}{6}\\ \frac{1}{2}+\frac{\sqrt{3}}{6}&\frac{1}{4}+\frac{\sqrt{3}}{6}&\frac{1}{4}\\ \hline&\frac{1}{2}&\frac{1}{2}\\ \end{array}
  23. 1 2 - 15 10 5 36 2 9 - 15 15 5 36 - 15 30 1 2 5 36 + 15 24 2 9 5 36 - 15 24 1 2 + 15 10 5 36 + 15 30 2 9 + 15 15 5 36 5 18 4 9 5 18 \begin{array}[]{c|ccc}\frac{1}{2}-\frac{\sqrt{15}}{10}&\frac{5}{36}&\frac{2}{9% }-\frac{\sqrt{15}}{15}&\frac{5}{36}-\frac{\sqrt{15}}{30}\\ \frac{1}{2}&\frac{5}{36}+\frac{\sqrt{15}}{24}&\frac{2}{9}&\frac{5}{36}-\frac{% \sqrt{15}}{24}\\ \frac{1}{2}+\frac{\sqrt{15}}{10}&\frac{5}{36}+\frac{\sqrt{15}}{30}&\frac{2}{9}% +\frac{\sqrt{15}}{15}&\frac{5}{36}\\ \hline&\frac{5}{18}&\frac{4}{9}&\frac{5}{18}\\ \end{array}
  24. 0 0 0 1 1 / 2 1 / 2 1 / 2 1 / 2 \begin{array}[]{c|cc}0&0&0\\ 1&1/2&1/2\\ \hline&1/2&1/2\\ \end{array}
  25. 0 0 0 0 1 / 2 5 / 24 1 / 3 - 1 / 24 1 1 / 6 2 / 3 1 / 6 1 / 6 2 / 3 1 / 6 \begin{array}[]{c|ccc}0&0&0&0\\ 1/2&5/24&1/3&-1/24\\ 1&1/6&2/3&1/6\\ \hline&1/6&2/3&1/6\\ \end{array}
  26. 0 1 / 2 0 1 1 / 2 0 1 / 2 1 / 2 \begin{array}[]{c|cc}0&1/2&0\\ 1&1/2&0\\ \hline&1/2&1/2\\ \end{array}
  27. 0 1 / 6 - 1 / 6 0 1 / 2 1 / 6 1 / 3 0 1 1 / 6 5 / 6 0 1 / 6 2 / 3 1 / 6 \begin{array}[]{c|ccc}0&1/6&-1/6&0\\ 1/2&1/6&1/3&0\\ 1&1/6&5/6&0\\ \hline&1/6&2/3&1/6\\ \end{array}
  28. 0 1 / 2 - 1 / 2 1 1 / 2 1 / 2 1 / 2 1 / 2 \begin{array}[]{c|cc}0&1/2&-1/2\\ 1&1/2&1/2\\ \hline&1/2&1/2\\ \end{array}
  29. 0 1 / 6 - 1 / 3 1 / 6 1 / 2 1 / 6 5 / 12 - 1 / 12 1 1 / 6 2 / 3 1 / 6 1 / 6 2 / 3 1 / 6 \begin{array}[]{c|ccc}0&1/6&-1/3&1/6\\ 1/2&1/6&5/12&-1/12\\ 1&1/6&2/3&1/6\\ \hline&1/6&2/3&1/6\\ \end{array}
  30. 0 1 / 4 - 1 / 4 2 / 3 1 / 4 5 / 12 1 / 4 3 / 4 \begin{array}[]{c|cc}0&1/4&-1/4\\ 2/3&1/4&5/12\\ \hline&1/4&3/4\\ \end{array}
  31. 0 1 9 - 1 - 6 18 - 1 + 6 18 3 5 - 6 10 1 9 11 45 + 7 6 360 11 45 - 43 6 360 3 5 + 6 10 1 9 11 45 + 43 6 360 11 45 - 7 6 360 1 9 4 9 + 6 36 4 9 - 6 36 \begin{array}[]{c|ccc}0&\frac{1}{9}&\frac{-1-\sqrt{6}}{18}&\frac{-1+\sqrt{6}}{% 18}\\ \frac{3}{5}-\frac{\sqrt{6}}{10}&\frac{1}{9}&\frac{11}{45}+\frac{7\sqrt{6}}{360% }&\frac{11}{45}-\frac{43\sqrt{6}}{360}\\ \frac{3}{5}+\frac{\sqrt{6}}{10}&\frac{1}{9}&\frac{11}{45}+\frac{43\sqrt{6}}{36% 0}&\frac{11}{45}-\frac{7\sqrt{6}}{360}\\ \hline&\frac{1}{9}&\frac{4}{9}+\frac{\sqrt{6}}{36}&\frac{4}{9}-\frac{\sqrt{6}}% {36}\\ \end{array}
  32. P s ( 2 x - 1 ) - P s - 1 ( 2 x - 1 ) = 0 , P_{s}(2x-1)-P_{s-1}(2x-1)=0,
  33. P s P_{s}
  34. 1 / 3 5 / 12 - 1 / 12 1 3 / 4 1 / 4 3 / 4 1 / 4 \begin{array}[]{c|cc}1/3&5/12&-1/12\\ 1&3/4&1/4\\ \hline&3/4&1/4\\ \end{array}
  35. 2 5 - 6 10 11 45 - 7 6 360 37 225 - 169 6 1800 - 2 225 + 6 75 2 5 + 6 10 37 225 + 169 6 1800 11 45 + 7 6 360 - 2 225 - 6 75 1 4 9 - 6 36 4 9 + 6 36 1 9 4 9 - 6 36 4 9 + 6 36 1 9 \begin{array}[]{c|ccc}\frac{2}{5}-\frac{\sqrt{6}}{10}&\frac{11}{45}-\frac{7% \sqrt{6}}{360}&\frac{37}{225}-\frac{169\sqrt{6}}{1800}&-\frac{2}{225}+\frac{% \sqrt{6}}{75}\\ \frac{2}{5}+\frac{\sqrt{6}}{10}&\frac{37}{225}+\frac{169\sqrt{6}}{1800}&\frac{% 11}{45}+\frac{7\sqrt{6}}{360}&-\frac{2}{225}-\frac{\sqrt{6}}{75}\\ 1&\frac{4}{9}-\frac{\sqrt{6}}{36}&\frac{4}{9}+\frac{\sqrt{6}}{36}&\frac{1}{9}% \\ \hline&\frac{4}{9}-\frac{\sqrt{6}}{36}&\frac{4}{9}+\frac{\sqrt{6}}{36}&\frac{1% }{9}\\ \end{array}

Lithium_burning.html

  1. M J M_{J}

Lithocholate_6beta-hydroxylase.html

  1. \rightleftharpoons

Load_line_(electronics).html

  1. V D = V D D - I R V_{D}=V_{DD}-IR\,

Local_flatness.html

  1. x N , x\in N,
  2. U M U\subset M
  3. ( U , U N ) (U,U\cap N)
  4. ( n , d ) (\mathbb{R}^{n},\mathbb{R}^{d})
  5. d \mathbb{R}^{d}
  6. n \mathbb{R}^{n}
  7. U R n U\to R^{n}
  8. U N U\cap N
  9. d \mathbb{R}^{d}
  10. U M U\subset M
  11. ( U , U N ) (U,U\cap N)
  12. ( + n , d ) (\mathbb{R}^{n}_{+},\mathbb{R}^{d})
  13. + n \mathbb{R}^{n}_{+}
  14. d \mathbb{R}^{d}
  15. + n = { y n : y n 0 } \mathbb{R}^{n}_{+}=\{y\in\mathbb{R}^{n}\colon y_{n}\geq 0\}
  16. d = { y n : y d + 1 = = y n = 0 } \mathbb{R}^{d}=\{y\in\mathbb{R}^{n}\colon y_{d+1}=\cdots=y_{n}=0\}
  17. χ : N M \chi\colon N\to M
  18. χ ( U ) \chi(U)

Loganate_O-methyltransferase.html

  1. \rightleftharpoons

Lommel_function.html

  1. z 2 d 2 y d z 2 + z d y d z + ( z 2 - ν 2 ) y = z μ + 1 . z^{2}\frac{d^{2}y}{dz^{2}}+z\frac{dy}{dz}+(z^{2}-\nu^{2})y=z^{\mu+1}.
  2. s μ , ν ( z ) = 1 2 π [ Y ν ( z ) 0 z z μ J ν ( z ) d z - J ν ( z ) 0 z z μ Y ν ( z ) d z ] s_{\mu,\nu}(z)=\frac{1}{2}\pi\left[Y_{\nu}(z)\int_{0}^{z}z^{\mu}J_{\nu}(z)\,dz% -J_{\nu}(z)\int_{0}^{z}z^{\mu}Y_{\nu}(z)\,dz\right]
  3. S μ , ν ( z ) = s μ , ν ( z ) - 2 μ - 1 Γ ( 1 + μ + ν 2 ) π Γ ( ν - μ 2 ) ( J ν ( z ) - cos ( π ( μ - ν ) / 2 ) Y ν ( z ) ) \displaystyle S_{\mu,\nu}(z)=s_{\mu,\nu}(z)-\frac{2^{\mu-1}\Gamma(\frac{1+\mu+% \nu}{2})}{\pi\Gamma(\frac{\nu-\mu}{2})}\left(J_{\nu}(z)-\cos(\pi(\mu-\nu)/2)Y_% {\nu}(z)\right)

Long-chain-acyl-CoA_dehydrogenase.html

  1. \rightleftharpoons

Long-chain-alcohol_dehydrogenase.html

  1. \rightleftharpoons

Long-chain-alcohol_oxidase.html

  1. \rightleftharpoons

Long-chain-fatty-acyl-CoA_reductase.html

  1. \rightleftharpoons

Low-energy_ion_scattering.html

  1. E 0 = 1 2 m x v 0 2 E_{0}=\tfrac{1}{2}m_{x}v_{0}^{2}\,\!
  2. E 1 = 1 2 m x v 1 2 E_{1}=\tfrac{1}{2}m_{x}v_{1}^{2}\,\!
  3. E 2 = 1 2 m y v 2 2 E_{2}=\tfrac{1}{2}m_{y}v_{2}^{2}\,\!
  4. E 0 = E 1 + E 2 E_{0}=E_{1}+E_{2}\,\!
  5. 1 2 m x v 0 2 = 1 2 m x v 1 2 + 1 2 m y v 2 2 \tfrac{1}{2}m_{x}v_{0}^{2}=\tfrac{1}{2}m_{x}v_{1}^{2}+\tfrac{1}{2}m_{y}v_{2}^{% 2}\,\!
  6. m x v 0 = m x v 1 cos θ 1 + m y v 2 cos θ 2 m_{x}v_{0}=m_{x}v_{1}\cos\theta_{1}+m_{y}v_{2}\cos\theta_{2}\,\!
  7. E 1 = E 0 ( m x cos θ 1 ± m y 2 - m x 2 sin 2 θ 1 m x + m y ) 2 E_{1}=E_{0}\left(\frac{m_{x}\cos\theta_{1}\pm\sqrt{m_{y}^{2}-m_{x}^{2}\sin^{2}% \theta_{1}}}{m_{x}+m_{y}}\right)^{2}
  8. E 2 = E 0 ( 4 m x m y c o s 2 ( θ 1 ) ( m x + m y ) 2 ) E_{2}=E_{0}\left(\frac{4m_{x}m_{y}cos^{2}(\theta_{1})}{(m_{x}+m_{y})^{2}}\right)
  9. V ( r ) = Z 1 Z 2 e 2 r ϕ ( r ) ( 1 ) V(r)=\frac{Z_{1}Z_{2}e^{2}}{r}\phi(r)\qquad(1)
  10. Z 1 Z_{1}\,\!
  11. Z 2 Z_{2}\,\!
  12. e e\,\!
  13. r r\,\!
  14. ϕ ( r ) \phi(r)\,\!
  15. ϕ ( r ) \phi(r)\,\!
  16. d σ d Ω \tfrac{d\sigma}{d\Omega}
  17. d σ d Ω = ( Z 1 Z 2 e 2 4 E 0 ) 2 1 sin 4 ( θ 2 ) ( 2 ) \frac{d\sigma}{d\Omega}=\left(\frac{Z_{1}Z_{2}e^{2}}{4E_{0}}\right)^{2}\frac{1% }{\sin^{4}\left(\frac{\theta}{2}\right)}\qquad(2)
  18. d σ d\sigma\,\!
  19. d Ω d\Omega\,\!
  20. ϕ ( r ) \phi(r)\,\!
  21. α c r i t . \alpha_{crit}.\,\!
  22. r = 2 Z 1 Z 2 e 2 L E 0 r=2\sqrt{\tfrac{Z_{1}Z_{2}e^{2}L}{E_{0}}}
  23. α c r i t \alpha_{crit}\,\!
  24. α c r i t \alpha_{crit}\,\!
  25. λ = h m v \lambda=\tfrac{h}{mv}
  26. α \alpha\,\!
  27. r = d sin α c r i t r=d\sin\alpha_{crit}\,\!
  28. L = d cos α c r i t L=d\cos\alpha_{crit}\,\!
  29. α 0 \alpha_{0}\,\!
  30. α 1 \alpha_{1}\,\!
  31. α 2 \alpha_{2}\,\!

Lower_flammable_limit.html

  1. L F L m i x LFL_{mix}
  2. L F L i LFL_{i}
  3. i i
  4. L F L m i x = 1 x i L F L i LFL_{mix}=\frac{1}{\sum\frac{x_{i}}{LFL_{i}}}
  5. L F L m i x LFL_{mix}
  6. L F L i LFL_{i}
  7. i i
  8. x i x_{i}
  9. i i

Luminosity_function_(astronomy).html

  1. n ( x ) d x = ϕ * x α e - x d x , n(x)\ \mathrm{d}x=\phi^{*}x^{\alpha}\mathrm{e}^{-x}\mathrm{d}x,
  2. x = L / L * x=L/L^{*}
  3. L * L^{*}
  4. ϕ * \,\!\phi^{*}
  5. α = - 1.25 , ϕ * = 1.2 × 10 - 2 h 3 Mpc - 3 \alpha=-1.25,\ \phi^{*}=1.2\times 10^{-2}h^{3}\mathrm{Mpc}^{-3}
  6. n ( M ) d M = 0.4 ln 10 ϕ * [ 10 0.4 ( M * - M ) ] α + 1 exp [ - 10 0.4 ( M * - M ) ] d M . n(M)\ \mathrm{d}M=0.4\ \ln 10\ \phi^{*}[10^{0.4(M^{*}-M)}]^{\alpha+1}\exp[-10^% {0.4(M^{*}-M)}]\ \mathrm{d}M.
  7. α + 1 \alpha+1
  8. α = - 1 \alpha=-1
  9. a b x α e - x d x = Γ ( α + 1 , a ) - Γ ( α + 1 , b ) \int_{a}^{b}x^{\alpha}e^{-x}\mathrm{d}x=\Gamma(\alpha+1,a)-\Gamma(\alpha+1,b)

Lundquist_number.html

  1. S S
  2. S = μ 0 L V A η , S=\frac{\mu_{0}LV_{A}}{\eta},
  3. L L
  4. μ 0 \mu_{0}
  5. V A = B μ 0 n 0 m i V_{A}=\frac{B}{\sqrt{\mu_{0}n_{0}m_{i}}}
  6. B B
  7. n 0 n_{0}
  8. m i m_{i}
  9. η \eta
  10. 10 2 - 10 8 10^{2}-10^{8}
  11. 10 20 10^{20}

Lune_(mathematics).html

  1. L = A - A B L=A-A\cap B

Luteolin_O-methyltransferase.html

  1. \rightleftharpoons

Lysine_2-monooxygenase.html

  1. \rightleftharpoons

Lysine_carbamoyltransferase.html

  1. \rightleftharpoons

Lysine_dehydrogenase.html

  1. \rightleftharpoons

Lysolecithin_acylmutase.html

  1. \rightleftharpoons

M::M::1_queue.html

  1. Q = ( - λ λ μ - ( μ + λ ) λ μ - ( μ + λ ) λ μ - ( μ + λ ) λ ) Q=\begin{pmatrix}-\lambda&\lambda\\ \mu&-(\mu+\lambda)&\lambda\\ &\mu&-(\mu+\lambda)&\lambda\\ &&\mu&-(\mu+\lambda)&\lambda&\\ &&&&\ddots\end{pmatrix}
  2. p k ( t ) = e - ( λ + μ ) t [ ρ k - i 2 I k - i ( a t ) + ρ k - i - 1 2 I k + i + 1 ( a t ) + ( 1 - ρ ) ρ k j = k + i + 2 ρ - j / 2 I j ( a t ) ] p_{k}(t)=e^{-(\lambda+\mu)t}\left[\rho^{\frac{k-i}{2}}I_{k-i}(at)+\rho^{\frac{% k-i-1}{2}}I_{k+i+1}(at)+(1-\rho)\rho^{k}\sum_{j=k+i+2}^{\infty}\rho^{-j/2}I_{j% }(at)\right]
  3. ρ = λ / μ \rho=\lambda/\mu
  4. a = 2 λ μ a=2\sqrt{\lambda\mu}
  5. π i = ( 1 - ρ ) ρ i . \pi_{i}=(1-\rho)\rho^{i}.\,
  6. f ( t ) = { 1 t ρ e - ( λ + μ ) t I 1 ( 2 t λ μ ) t > 0 0 otherwise f(t)=\begin{cases}\frac{1}{t\sqrt{\rho}}e^{-(\lambda+\mu)t}I_{1}(2t\sqrt{% \lambda\mu})&t>0\\ 0&\,\text{otherwise}\end{cases}
  7. 𝔼 ( e - θ F ) = 1 2 λ ( λ + μ + θ - ( λ + μ + θ ) 2 - 4 λ μ ) \mathbb{E}(e^{-\theta F})=\frac{1}{2\lambda}(\lambda+\mu+\theta-\sqrt{(\lambda% +\mu+\theta)^{2}-4\lambda\mu})
  8. 1 + λ μ μ 2 ( 1 - λ μ ) 3 . \frac{1+\frac{\lambda}{\mu}}{\mu^{2}(1-\frac{\lambda}{\mu})^{3}}.
  9. f ( t ) = { ( μ - λ ) e - ( μ - λ ) t t > 0 0 otherwise. f(t)=\begin{cases}(\mu-\lambda)e^{-(\mu-\lambda)t}&t>0\\ 0&\,\text{otherwise.}\end{cases}
  10. W ( s | x ) = ( 1 - ρ ) ( 1 - ρ r 2 ) e - [ λ ( 1 - r ) + s ] x ( 1 - ρ r 2 ) - ρ ( 1 - r ) 2 e - ( μ / r - λ r ) x W^{\ast}(s|x)=\frac{(1-\rho)(1-\rho r^{2})e^{-[\lambda(1-r)+s]x}}{(1-\rho r^{2% })-\rho(1-r)^{2}e^{-(\mu/r-\lambda r)x}}
  11. λ r 2 - ( λ + μ + s ) r + μ = 0. \lambda r^{2}-(\lambda+\mu+s)r+\mu=0.

Macbeath_surface.html

  1. 2 \langle 2\rangle

Macrocin_O-methyltransferase.html

  1. \rightleftharpoons

Magic_angle_(EELS).html

  1. θ M \theta_{M}
  2. θ E \theta_{E}
  3. A A
  4. C C
  5. α \alpha
  6. A ( α ) = 1 2 0 α 2 d x x ( x + θ E 2 ( 1 - β 2 ) ) 2 A(\alpha)=\frac{1}{2}\int_{0}^{\alpha^{2}}dx\frac{x}{(x+\theta_{E}^{2}{(1-% \beta^{2}))}^{2}}
  7. C ( α ) = θ E 2 ( 1 - β 2 ) 2 0 α 2 d x 1 ( x + θ E 2 ( 1 - β 2 ) ) 2 C(\alpha)=\theta_{E}^{2}{(1-\beta^{2})}^{2}\int_{0}^{\alpha^{2}}dx\frac{1}{{(x% +\theta_{E}^{2}(1-\beta^{2}))}^{2}}
  8. β \beta
  9. β \beta
  10. α \alpha
  11. θ M 2 θ E \theta_{M}\approx 2\theta_{E}

Magic_number_(oil).html

  1. Q × P > S Q\times P>S

MagmaFS.html

  1. l n = k l n b n b a s n s a l_{n}=kl_{n}\cdot\frac{b_{n}}{b_{a}}\cdot\frac{s_{n}}{s_{a}}
  2. k l n kl_{n}
  3. b n b_{n}
  4. b a b_{a}
  5. s n s_{n}
  6. s a s_{a}

Magnesium_protoporphyrin_IX_methyltransferase.html

  1. \rightleftharpoons

Magnetic_anisotropy.html

  1. \scriptstylesymbol μ \scriptstylesymbol{\mu}
  2. V \scriptstyle V
  3. 𝐌 = s y m b o l μ / V = M s ( α , β , γ ) \scriptstyle\mathbf{M}=symbol{\mu}/V=M_{s}\left(\alpha,\beta,\gamma\right)
  4. M s \scriptstyle M_{s}
  5. α , β , γ \scriptstyle\alpha,\beta,\gamma
  6. α 2 + β 2 + γ 2 = 1 \scriptstyle\alpha^{2}+\beta^{2}+\gamma^{2}=1
  7. z z
  8. E = K V ( 1 - γ 2 ) = K V sin 2 θ , E=KV\left(1-\gamma^{2}\right)=KV\sin^{2}\theta,
  9. V \scriptstyle V
  10. K \scriptstyle K
  11. θ \scriptstyle\theta
  12. 𝒩 \scriptstyle\mathcal{N}
  13. K \scriptstyle K
  14. E = K a V α 2 + K b V β 2 . \displaystyle E=K_{a}V\alpha^{2}+K_{b}V\beta^{2}.
  15. K a > K b > 0 , \scriptstyle K_{a}>K_{b}>0,
  16. z z
  17. y y
  18. x x
  19. E = K V ( α 2 β 2 + β 2 γ 2 + γ 2 α 2 ) . E=KV\left(\alpha^{2}\beta^{2}+\beta^{2}\gamma^{2}+\gamma^{2}\alpha^{2}\right).
  20. K > 0 , \scriptstyle K>0,
  21. x , y , x,y,
  22. z z
  23. K < 0 , \scriptstyle K<0,
  24. x = ± y = ± z x=\pm y=\pm z

Magnetic_anomaly.html

  1. × 10 - 3 \times 10^{-3}
  2. 𝐌 = 𝐌 i + 𝐌 r . \mathbf{M}=\mathbf{M}\text{i}+\mathbf{M}\text{r}.
  3. χ χ
  4. 𝐌 i = χ 𝐇 . \mathbf{M}\text{i}=\chi\mathbf{H}.
  5. Q = M < s u b > r / M i Q=M<sub>r/M_{i}

Malate_dehydrogenase_(decarboxylating).html

  1. \rightleftharpoons

Malate_dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Malate_dehydrogenase_(oxaloacetate-decarboxylating).html

  1. \rightleftharpoons

Malate_dehydrogenase_(oxaloacetate-decarboxylating)_(NADP+).html

  1. \rightleftharpoons

Malate_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Malate_oxidase.html

  1. \rightleftharpoons

Maleate_isomerase.html

  1. \rightleftharpoons

Maleylacetate_reductase.html

  1. \rightleftharpoons

Maleylacetoacetate_isomerase.html

  1. \rightleftharpoons

Maleylpyruvate_isomerase.html

  1. \rightleftharpoons

Malgrange_preparation_theorem.html

  1. f ( 0 , 0 ) = 0 , f t ( 0 , 0 ) = 0 , , k - 1 f t k - 1 ( 0 , 0 ) = 0 , k f t k ( 0 , 0 ) 0. f(0,0)=0,{\partial f\over\partial t}(0,0)=0,\dots,{\partial^{k-1}f\over% \partial t^{k-1}}(0,0)=0,{\partial^{k}f\over\partial t^{k}}(0,0)\neq 0.
  2. f ( t , x ) = c ( t , x ) ( t k + a k - 1 ( x ) t k - 1 + + a 0 ( x ) ) f(t,x)=c(t,x)\left(t^{k}+a_{k-1}(x)t^{k-1}+\cdots+a_{0}(x)\right)
  3. g = q f + r g=qf+r
  4. r ( x ) = 0 j < k t j r j ( x ) r(x)=\sum_{0\leq j<k}t^{j}r_{j}(x)

Malgrange–Ehrenpreis_theorem.html

  1. P ( x 1 , , x ) u ( 𝐱 ) = δ ( 𝐱 ) , P\left(\frac{\partial}{\partial x_{1}},\cdots,\frac{\partial}{\partial x_{\ell% }}\right)u(\mathbf{x})=\delta(\mathbf{x}),
  2. P ( x 1 , , x ) u ( 𝐱 ) = f ( 𝐱 ) P\left(\frac{\partial}{\partial x_{1}},\cdots,\frac{\partial}{\partial x_{\ell% }}\right)u(\mathbf{x})=f(\mathbf{x})
  3. E = 1 P m ( 2 η ) ¯ j = 0 m a j e λ j η x ξ - 1 ( P ( i ξ + λ j η ) ¯ P ( i ξ + λ j η ) ) E=\frac{1}{\overline{P_{m}(2\eta)}}\sum_{j=0}^{m}a_{j}e^{\lambda_{j}\eta x}% \mathcal{F}^{-1}_{\xi}\left(\frac{\overline{P(i\xi+\lambda_{j}\eta)}}{P(i\xi+% \lambda_{j}\eta)}\right)
  4. a j = k = 0 , k j m ( λ j - λ k ) - 1 . a_{j}=\prod_{k=0,k\neq j}^{m}(\lambda_{j}-\lambda_{k})^{-1}.

Malliavin's_absolute_continuity_lemma.html

  1. | 𝐑 n D φ ( y ) ( x ) d μ ( y ) | C ( x ) φ \left|\int_{\mathbf{R}^{n}}\mathrm{D}\varphi(y)(x)\,\mathrm{d}\mu(y)\right|% \leq C(x)\|\varphi\|_{\infty}

Malonate-semialdehyde_dehydrogenase.html

  1. \rightleftharpoons

Malonate-semialdehyde_dehydrogenase_(acetylating).html

  1. \rightleftharpoons

Malonate_CoA-transferase.html

  1. \rightleftharpoons

Maltose_alpha-D-glucosyltransferase.html

  1. \rightleftharpoons

Maltose_epimerase.html

  1. \rightleftharpoons

Malyl-CoA_lyase.html

  1. \rightleftharpoons

Mandelate_4-monooxygenase.html

  1. \rightleftharpoons

Mandelonitrile_lyase.html

  1. \rightleftharpoons

Maneuvering_speed.html

  1. V s n V_{s}\sqrt{n}
  2. V s n V_{s}\sqrt{n}
  3. V s n V_{s}\sqrt{n}
  4. V A W 2 W 1 \scriptstyle V_{A}\sqrt{W_{2}\over W_{1}}

Manganese_peroxidase.html

  1. \rightleftharpoons

Mannitol-1-phosphate_5-dehydrogenase.html

  1. \rightleftharpoons

Mannitol_2-dehydrogenase.html

  1. \rightleftharpoons

Mannitol_2-dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Mannitol_dehydrogenase.html

  1. \rightleftharpoons

Mannitol_dehydrogenase_(cytochrome).html

  1. \rightleftharpoons

Mannose-6-phosphate_6-reductase.html

  1. \rightleftharpoons

Mannose_isomerase.html

  1. \rightleftharpoons

Mannuronate_reductase.html

  1. \rightleftharpoons

Marginal_model.html

  1. Y i j Y_{ij}
  2. Y i j = β 0 j + R i j Y_{ij}=\beta_{0j}+R_{ij}
  3. R i j R_{ij}
  4. v a r ( R i j ) = σ 2 var(R_{ij})=\sigma^{2}
  5. β 0 j = γ 00 + U 0 j \beta_{0j}=\gamma_{00}+U_{0j}
  6. U 0 j U_{0j}
  7. v a r ( U 0 j ) = τ 0 2 var(U_{0j})=\tau_{0}^{2}
  8. Y i j N ( γ 00 , ( τ 0 2 + σ 2 ) ) Y_{ij}\sim N(\gamma_{00},(\tau_{0}^{2}+\sigma^{2}))

Mass_(mass_spectrometry).html

  1. K e n d r i c k m a s s ( F ) = ( o b s e r v e d m a s s ) × n o m i n a l m a s s F e x a c t m a s s F Kendrick~{}mass~{}(F)=(observed~{}mass)\times\frac{nominal~{}mass~{}F}{exact~{% }mass~{}F}~{}

Mass_attenuation_coefficient.html

  1. μ ρ m , \frac{\mu}{\rho_{m}},
  2. μ a ρ m , μ s ρ m , \frac{\mu_{\mathrm{a}}}{\rho_{m}},\quad\frac{\mu_{\mathrm{s}}}{\rho_{m}},
  3. μ = ( μ / ρ 1 ) ρ 1 + ( μ / ρ 2 ) ρ 2 + , \mu=(\mu/\rho_{1})\rho_{1}+(\mu/\rho_{2})\rho_{2}+\ldots,
  4. μ = ( μ / ρ 1 ) ρ 1 + ( μ / ρ 2 ) ρ 2 + , \mu=(\mu/\rho_{1})\rho_{1}+(\mu/\rho_{2})\rho_{2}+\ldots,

Math_Girl.html

  1. lim t 0 sin t t = 1 \lim_{t\to 0}\frac{\sin t}{t}=1

Matrix_function.html

  1. f f
  2. f ( x ) = f ( 0 ) + f ( 0 ) x + f ′′ ( 0 ) x 2 2 ! + f(x)=f(0)+f^{\prime}(0)\cdot x+f^{\prime\prime}(0)\cdot\frac{x^{2}}{2!}+\cdots
  3. x x
  4. | x | < r |x|<r
  5. A < r \|A\|<r
  6. \|\cdot\|
  7. A B A B \|AB\|\leq\|A\|\cdot\|B\|
  8. A = P D P - 1 A=P\cdot D\cdot P^{-1}
  9. f ( A ) f(A)
  10. f ( A ) = P [ f ( d 1 ) 0 0 f ( d n ) ] P - 1 , f(A)=P\begin{bmatrix}f(d_{1})&\dots&0\\ \vdots&\ddots&\vdots\\ 0&\dots&f(d_{n})\end{bmatrix}P^{-1},
  11. d 1 , , d n d_{1},\dots,d_{n}
  12. A = P J P - 1 A=P\cdot J\cdot P^{-1}
  13. f ( [ λ 1 0 0 0 λ 1 0 0 λ 1 0 0 λ ] ) = [ f ( λ ) 0 ! f ( λ ) 1 ! f ′′ ( λ ) 2 ! f ( n ) ( λ ) n ! 0 f ( λ ) 0 ! f ( λ ) 1 ! f ( n - 1 ) ( λ ) ( n - 1 ) ! 0 0 f ( λ ) 0 ! f ( λ ) 1 ! 0 0 f ( λ ) 0 ! ] . f\left(\begin{bmatrix}\lambda&1&0&\ldots&0\\ 0&\lambda&1&\vdots&\vdots\\ 0&0&\ddots&\ddots&\vdots\\ \vdots&\ldots&\ddots&\lambda&1\\ 0&\ldots&\ldots&0&\lambda\end{bmatrix}\right)=\begin{bmatrix}\frac{f(\lambda)}% {0!}&\frac{f^{\prime}(\lambda)}{1!}&\frac{f^{\prime\prime}(\lambda)}{2!}&% \ldots&\frac{f^{(n)}(\lambda)}{n!}\\ 0&\frac{f(\lambda)}{0!}&\frac{f^{\prime}(\lambda)}{1!}&\vdots&\frac{f^{(n-1)}(% \lambda)}{(n-1)!}\\ 0&0&\ddots&\ddots&\vdots\\ \vdots&\ldots&\ddots&\frac{f(\lambda)}{0!}&\frac{f^{\prime}(\lambda)}{1!}\\ 0&\ldots&\ldots&0&\frac{f(\lambda)}{0!}\end{bmatrix}.
  14. f ( a ) g ( a ) f(a)\leq g(a)
  15. A A
  16. f ( A ) g ( A ) f(A)\preceq g(A)
  17. X Y Y - X X\preceq Y\Leftrightarrow Y-X
  18. f ( x ) = 1 2 π i C f ( z ) z - x d z f(x)=\frac{1}{2\pi i}\oint_{C}{\frac{f(z)}{z-x}}\,\mathrm{d}z
  19. A \scriptstyle\|A\|
  20. \scriptstyle\|\cdot\|
  21. f ( A ) f(A)
  22. f ( A ) = 1 2 π i C f ( z ) ( z I - A ) - 1 d z . f(A)=\frac{1}{2\pi i}\oint_{C}{f(z)(zI-A)^{-1}}\,\mathrm{d}z.
  23. x x
  24. A ( η ) = A + η B A(\eta)=A+\eta B
  25. η = 0 \eta=0
  26. [ A , B ] = 0 [A,B]=0
  27. f ( x ) = x 3 f(x)=x^{3}
  28. f ( A + η B ) = ( A + η B ) 3 = A 3 + η ( A 2 B + A B A + B A 2 ) + η 2 ( A B 2 + B A B + B 2 A ) + η 3 B 3 f(A+\eta B)=(A+\eta B)^{3}=A^{3}+\eta(A^{2}B+ABA+BA^{2})+\eta^{2}(AB^{2}+BAB+B% ^{2}A)+\eta^{3}B^{3}
  29. f ( a + η b ) f(a+\eta b)
  30. f ( a + η b ) = f ( a ) + f ( a ) η b 1 ! + f ′′ ( a ) ( η b ) 2 2 ! + f ′′′ ( a ) ( η b ) 3 3 ! = a 3 + 3 a 2 ( η b ) + 3 a ( η b ) 2 + ( η b ) 3 A 3 + 3 A 2 ( η B ) + 3 A ( η B ) 2 + ( η B ) 3 f(a+\eta b)=f(a)+f^{\prime}(a)\frac{\eta b}{1!}+f^{\prime\prime}(a)\frac{(\eta b% )^{2}}{2!}+f^{\prime\prime\prime}(a)\frac{(\eta b)^{3}}{3!}=a^{3}+3a^{2}(\eta b% )+3a(\eta b)^{2}+(\eta b)^{3}\to A^{3}+3A^{2}(\eta B)+3A(\eta B)^{2}+(\eta B)^% {3}
  31. [ A , B ] = 0 [A,B]=0
  32. f ( x ) = 1 x f(x)=\frac{1}{x}
  33. A - 1 A^{-1}
  34. f ( A + η B ) = f ( 𝕀 + η A - 1 B ) f ( A ) f(A+\eta B)=f(\mathbb{I}+\eta A^{-1}B)f(A)
  35. f ( 𝕀 + η A - 1 B ) = 𝕀 - η A - 1 B + ( - η A - 1 B ) 2 + = n = 0 ( - η A - 1 B ) n f(\mathbb{I}+\eta A^{-1}B)=\mathbb{I}-\eta A^{-1}B+(-\eta A^{-1}B)^{2}+\ldots=% \sum_{n=0}^{\infty}(-\eta A^{-1}B)^{n}
  36. η A - 1 B \|\eta A^{-1}B\|
  37. X Y Y - X X\preceq Y\Leftrightarrow Y-X
  38. X Y Y - X X\prec Y\Leftrightarrow Y-X
  39. f f
  40. 0 A H f ( A ) f ( H ) 0\prec A\preceq H\Rightarrow f(A)\preceq f(H)
  41. A , H A,H
  42. f f
  43. τ f ( A ) + ( 1 - τ ) f ( H ) f ( τ A + ( 1 - τ ) H ) \tau f(A)+(1-\tau)f(H)\preceq f\left(\tau A+(1-\tau)H\right)
  44. A , H A,H
  45. τ [ 0 , 1 ] \tau\in[0,1]
  46. \preceq
  47. \succeq

Mauchly's_sphericity_test.html

  1. H 0 : σ Tx A - Tx B 2 = σ Tx A - Tx C 2 = σ Tx B - Tx C 2 H_{0}:\sigma_{\,\text{Tx A}-\,\text{Tx B}}^{2}=\sigma_{\,\text{Tx A}-\,\text{% Tx C}}^{2}=\sigma_{\,\text{Tx B}-\,\text{Tx C}}^{2}
  2. H 1 : σ Tx A - Tx B 2 σ Tx A - Tx C 2 σ Tx B - Tx C 2 H_{1}:\sigma_{\,\text{Tx A}-\,\text{Tx B}}^{2}\neq\sigma_{\,\text{Tx A}-\,% \text{Tx C}}^{2}\neq\sigma_{\,\text{Tx B}-\,\text{Tx C}}^{2}
  3. α \alpha
  4. α \alpha
  5. α \alpha
  6. α \alpha

Mazur_manifold.html

  1. S 1 × D 3 S^{1}\times D^{3}
  2. S 4 S^{4}
  3. Σ ( 2 , 5 , 7 ) \Sigma(2,5,7)
  4. Σ ( 3 , 4 , 5 ) \Sigma(3,4,5)
  5. Σ ( 2 , 3 , 13 ) \Sigma(2,3,13)
  6. n 5 n\geq 5
  7. n 6 n\geq 6
  8. n n
  9. D n D^{n}
  10. D 5 D^{5}
  11. S 4 S^{4}
  12. S 4 S^{4}
  13. D 4 D^{4}
  14. M M
  15. S 1 × D 3 S^{1}\times D^{3}
  16. S 4 S^{4}
  17. M × [ 0 , 1 ] M\times[0,1]
  18. S 1 × D 4 S^{1}\times D^{4}
  19. S 1 × S 3 S^{1}\times S^{3}
  20. S 1 × D 4 S^{1}\times D^{4}
  21. D 5 D^{5}
  22. D 5 D^{5}
  23. S 4 S^{4}
  24. M × [ 0 , 1 ] M\times[0,1]
  25. M M

McClellan_oscillator.html

  1. O s c i l l a t o r = ( 19 day EMA of advances minus declines ) - ( 39 day EMA of advances minus declines ) Oscillator=(\,\text{19 day EMA of advances minus declines})-(\,\text{39 day % EMA of advances minus declines})

McCumber_relation.html

  1. σ a ( ω ) \sigma_{\rm a}(\omega)
  2. σ e ( ω ) \sigma_{\rm e}(\omega)
  3. ω \omega
  4. T ~{}T~{}
  5. σ e ( ω ) σ a ( ω ) exp ( ω k B T ) = ( N 1 N 2 ) T = exp ( ω z k B T ) \frac{\sigma_{\rm e}(\omega)}{\sigma_{\rm a}(\omega)}\exp\!\left(\frac{\hbar% \omega}{k_{\rm B}T}\right)=\left(\frac{N_{1}}{N_{2}}\right)_{T}=\exp\!\left(% \frac{\hbar\omega_{\rm z}}{k_{\rm B}T}\right)
  6. ( N 1 N 2 ) T \left(\frac{N_{1}}{N_{2}}\right)_{T}
  7. ω z \omega_{\rm z}
  8. \hbar
  9. k B k_{\rm B}
  10. ω ~{}\omega~{}
  11. σ a ( ω ) \sigma_{\rm a}(\omega)
  12. σ e ( ω ) \sigma_{\rm e}(\omega)
  13. ω ~{}\omega~{}
  14. G ( ω ) = N 2 σ e ( ω ) - N 1 σ a ( ω ) ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(\omega)=N_{2}\sigma_{\rm e}(% \omega)-N_{1}\sigma_{\rm a}(\omega)
  15. σ a \sigma_{\rm a}
  16. σ e \sigma_{\rm e}
  17. N 1 N_{1}
  18. N 2 N_{2}
  19. v ( ω ) ~{}v(\omega)~{}
  20. n 2 σ e ( ω ) v ( ω ) D ( ω ) ~{}n_{2}\sigma_{\rm e}(\omega)v(\omega)D(\omega)~{}
  21. n 1 σ a ( ω ) v ( ω ) D ( ω ) ~{}n_{1}\sigma_{\rm a}(\omega)v(\omega)D(\omega)~{}
  22. a ( ω ) n 2 a(\omega)n_{2}
  23. n 2 σ e ( ω ) v ( ω ) D ( ω ) + n 2 a ( ω ) = n 1 σ a ( ω ) v ( ω ) D ( ω ) ( balance ) ~{}~{}~{}n_{2}\sigma_{\rm e}(\omega)v(\omega)D(\omega)+n_{2}a(\omega)=n_{1}% \sigma_{\rm a}(\omega)v(\omega)D(\omega)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}{\rm(balance)}
  24. D ( ω ) = a ( ω ) σ e ( ω ) v ( ω ) n 1 n 2 σ a ( ω ) σ e ( ω ) - 1 ( D1 ) ~{}~{}~{}D(\omega)=\frac{\frac{a(\omega)}{\sigma_{\rm e}(\omega)v(\omega)}}{% \frac{n_{1}}{n_{2}}\frac{\sigma_{\rm a}(\omega)}{\sigma_{\rm e}(\omega)}-1}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm(D1)}
  25. D ( ω ) = 1 π 2 ω 2 c 3 exp ( ω k B T ) - 1 ( D2 ) ~{}~{}~{}D(\omega)~{}=~{}\frac{\frac{1}{\pi^{2}}\frac{\omega^{2}}{c^{3}}}{\exp% \!\left(\frac{\hbar\omega}{k_{\rm B}T}\right)-1}~{}~{}~{}~{}~{}{\rm(D2)}
  26. ω ~{}\omega~{}
  27. σ a ( ω ) = σ e ( ω ) ~{}\sigma_{\rm a}(\omega)=\sigma_{\rm e}(\omega)~{}
  28. n 1 / n 2 = exp ( ω k B T ) ~{}n_{1}/n_{2}=\exp\!\left(\frac{\hbar\omega}{k_{\rm B}T}\right)
  29. a ( ω ) a(\omega)
  30. σ e ( ω ) ~{}\sigma_{\rm e}(\omega)~{}
  31. a ( ω ) d ω d t ~{}a(\omega){\rm d}\omega{\rm d}t~{}
  32. ( ω , ω + d ω ) ~{}(\omega,\omega+{\rm d}\omega)~{}
  33. ( t , t + d t ) ~{}(t,t+{\rm d}t)~{}
  34. t ~{}t~{}
  35. s ~{}s~{}
  36. j j
  37. a s , j ( ω ) ~{}a_{s,j}(\omega)~{}
  38. a s , j ( ω ) = σ s , j ( ω ) ω 2 v ( ω ) π 2 c 3 . comparison1 partial ~{}~{}~{}a_{s,j}(\omega)=\sigma_{s,j}(\omega)\frac{\omega^{2}v(\omega)}{\pi^{2% }c^{3}}~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm comparison1% }~{}~{}{\rm partial}
  39. q s ~{}q_{s}~{}
  40. n s , j ~{}n_{s,j}~{}
  41. a ~{}a~{}
  42. σ e ~{}\sigma_{\rm e}~{}
  43. a ( ω ) σ e ( ω ) = ω 2 v ( ω ) π 2 c 3 ( comparison ) ( av ) \frac{a(\omega)}{\sigma_{\rm e}(\omega)}=\frac{\omega^{2}v(\omega)}{\pi^{2}c^{% 3}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\rm comparison)(av)
  44. n 1 n 2 σ a ( ω ) σ e ( ω ) = exp ( ω k B T ) . ( n1n2 ) ( mc1 ) \frac{n_{1}}{n_{2}}\frac{\sigma_{\rm a}(\omega)}{\sigma_{\rm e}(\omega)}=\exp% \!\left(\frac{\hbar\omega}{k_{\rm B}T}\right)~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}{% \rm(n1n2)(mc1)}
  45. ω Z \omega_{Z}
  46. ( n 2 n 1 ) T = exp ( ω Z k B T ) , ~{}\left(\frac{n_{2}}{n_{1}}\right)_{\!T}=\exp\!\left(\frac{\hbar\omega_{\rm Z% }}{k_{\rm B}T}\right)~{}~{}~{}~{},~{}~{}~{}
  47. T ~{}T~{}
  48. ω Z = k B T ln ( n 1 n 2 ) T . ( oz ) \omega_{\rm Z}=\frac{k_{\rm B}T}{\hbar}\ln\left(\frac{n_{1}}{n_{2}}\right)_{T}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}.~{}~{}{(\rm oz)}
  49. λ = 2 π c ω \lambda=\frac{2\pi c}{\omega}

Mean_curvature_flow.html

  1. z = S ( x , y ) z=S(x,y)
  2. S t = 2 D H ( x , y ) 1 + ( S x ) 2 + ( S y ) 2 \frac{\partial S}{\partial t}=2D\ H(x,y)\sqrt{1+\left(\frac{\partial S}{% \partial x}\right)^{2}+\left(\frac{\partial S}{\partial y}\right)^{2}}
  3. D D
  4. H ( x , y ) = 1 2 ( 1 + ( S x ) 2 ) 2 S y 2 - 2 S x S y 2 S x y + ( 1 + ( S y ) 2 ) 2 S x 2 ( 1 + ( S x ) 2 + ( S y ) 2 ) 3 / 2 . \begin{aligned}\displaystyle H(x,y)&\displaystyle=\frac{1}{2}\frac{\left(1+% \left(\frac{\partial S}{\partial x}\right)^{2}\right)\frac{\partial^{2}S}{% \partial y^{2}}-2\frac{\partial S}{\partial x}\frac{\partial S}{\partial y}% \frac{\partial^{2}S}{\partial x\partial y}+\left(1+\left(\frac{\partial S}{% \partial y}\right)^{2}\right)\frac{\partial^{2}S}{\partial x^{2}}}{\left(1+% \left(\frac{\partial S}{\partial x}\right)^{2}+\left(\frac{\partial S}{% \partial y}\right)^{2}\right)^{3/2}}.\end{aligned}
  5. | S x | 1 |\frac{\partial S}{\partial x}|\ll 1
  6. | S y | 1 |\frac{\partial S}{\partial y}|\ll 1
  7. S t = D 2 S \frac{\partial S}{\partial t}=D\ \nabla^{2}S

Mechanostat.html

  1. E = Δ l l E=\frac{\Delta l}{l}

Melilotate_3-monooxygenase.html

  1. \rightleftharpoons

Memristor.html

  1. f ( Φ m ( t ) , q ( t ) ) = 0 f(\mathrm{\Phi}_{\mathrm{m}}(t),q(t))=0
  2. M ( q ) = d Φ m d q M(q)=\frac{\mathrm{d}\Phi_{m}}{\mathrm{d}q}
  3. M ( q ( t ) ) = d Φ m / d t d q / d t = V ( t ) I ( t ) M(q(t))=\cfrac{\mathrm{d}\Phi_{m}/\mathrm{d}t}{\mathrm{d}q/\mathrm{d}t}=\frac{% V(t)}{I(t)}
  4. V ( t ) = M ( q ( t ) ) I ( t ) V(t)=\ M(q(t))I(t)
  5. P ( t ) = I ( t ) V ( t ) = I 2 ( t ) M ( q ( t ) ) P(t)=\ I(t)V(t)=\ I^{2}(t)M(q(t))
  6. M ( q ( t ) ) = R OFF ( 1 - μ v R ON D 2 q ( t ) ) M(q(t))=R_{\mathrm{OFF}}\cdot\left(1-\frac{\mu_{v}R_{\mathrm{ON}}}{D^{2}}q(t)\right)
  7. E switch = V 2 T off T on d t M ( q ( t ) ) = V 2 Q off Q on d q I ( q ) M ( q ) = V 2 Q off Q on d q V ( q ) = V Δ Q E_{\mathrm{switch}}=\ V^{2}\int_{T_{\mathrm{off}}}^{T_{\mathrm{on}}}\frac{% \mathrm{d}t}{M(q(t))}=\ V^{2}\int_{Q_{\mathrm{off}}}^{Q_{\mathrm{on}}}\frac{% \mathrm{d}q}{I(q)M(q)}=\ V^{2}\int_{Q_{\mathrm{off}}}^{Q_{\mathrm{on}}}\frac{% \mathrm{d}q}{V(q)}=\ V\Delta Q
  8. y ( t ) = g ( 𝐱 , u , t ) u ( t ) , 𝐱 ˙ = f ( 𝐱 , u , t ) \begin{aligned}\displaystyle y(t)&\displaystyle=g(\,\textbf{x},u,t)u(t),\\ \displaystyle\dot{\,\textbf{x}}&\displaystyle=f(\,\textbf{x},u,t)\end{aligned}
  9. y ( t ) = g 0 ( 𝐱 , u ) u ( t ) + g 1 ( 𝐱 , u ) d 2 u d t 2 + g 2 ( 𝐱 , u ) d 4 u d t 4 + + g m ( 𝐱 , u ) d 2 m u d t 2 m , 𝐱 ˙ = f ( 𝐱 , u ) \begin{aligned}\displaystyle y(t)&\displaystyle=g_{0}(\,\textbf{x},u)u(t)+g_{1% }(\,\textbf{x},u){\operatorname{d}^{2}u\over\operatorname{d}t^{2}}+g_{2}(\,% \textbf{x},u){\operatorname{d}^{4}u\over\operatorname{d}t^{4}}+\ldots+g_{m}(\,% \textbf{x},u){\operatorname{d}^{2m}u\over\operatorname{d}t^{2m}},\\ \displaystyle\dot{\,\textbf{x}}&\displaystyle=f(\,\textbf{x},u)\end{aligned}
  10. y ( t ) = g 0 ( 𝐱 , u ) ( u ( t ) - a ) , 𝐱 ˙ = f ( 𝐱 , u ) \begin{aligned}\displaystyle y(t)&\displaystyle=g_{0}(\,\textbf{x},u)(u(t)-a),% \\ \displaystyle\dot{\,\textbf{x}}&\displaystyle=f(\,\textbf{x},u)\end{aligned}

Meso-tartrate_dehydrogenase.html

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METAP2.html

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Methanethiol_oxidase.html

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Methanol_dehydrogenase.html

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Methanol—5-hydroxybenzimidazolylcobamide_Co-methyltransferase.html

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Methanosarcina-phenazine_hydrogenase.html

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Methionine_decarboxylase.html

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Methionine_gamma-lyase.html

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Methionine_racemase.html

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Methionine_S-methyltransferase.html

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Methionyl-tRNA_formyltransferase.html

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Methomyl.html

  1. H 3 C- C | | O -OH+HS-C H 3 30–50 C ,- H 2 O H 3 C- C | | O -S-C H 3 {{\,\text{H}}_{\,\text{3}}}\,\text{C-}\underset{\underset{\,\text{O}}{\mathop{% \,\text{ }\!\!|\!\!\,\text{ }\!\!|\!\!\,\text{ }}}\,}{\mathop{\,\text{C}}}\,\,% \text{-OH+HS-C}{{\,\text{H}}_{\,\text{3}}}\xrightarrow{\,\text{30--50 }\!\!~{}% \!\!\,\text{ }\!\!{}^{\circ}\!\!\,\text{ C}\,\text{,-}{{\,\text{H}}_{2}}O}{{\,% \text{H}}_{\,\text{3}}}\,\text{C-}\underset{\underset{\,\text{O}}{\mathop{\,% \text{ }\!\!|\!\!\,\text{ }\!\!|\!\!\,\text{ }}}\,}{\mathop{\,\text{C}}}\,\,% \text{-S-C}{{\,\text{H}}_{\,\text{3}}}
  2. H 3 C- C | | O -S-C H 3 + H 2 NOH - H 2 O HO-N= C | C H 3 -S-C H 3 {{\,\text{H}}_{\,\text{3}}}\,\text{C-}\underset{\underset{\,\text{O}}{\mathop{% \,\text{ }\!\!|\!\!\,\text{ }\!\!|\!\!\,\text{ }}}\,}{\mathop{\,\text{C}}}\,\,% \text{-S-C}{{\,\text{H}}_{\,\text{3}}}\,\text{+}{{\,\text{H}}_{\,\text{2}}}\,% \text{NOH}\xrightarrow{-{{H}_{2}}O}\,\text{HO-N=}\underset{\,\text{C}{{\,\text% {H}}_{\,\text{3}}}}{\mathop{\underset{\,\text{ }\!\!|\!\!\,\text{ }}{\mathop{% \,\text{C}}}\,}}\,\,\text{-S-C}{{\,\text{H}}_{\,\text{3}}}
  3. H 3 C-N=C=O+HO-N= C | C H 3 -S-C H 3 C H 2 C l 2 ,30–50 C Methomyl {{\,\text{H}}_{\,\text{3}}}\,\text{C-N=C=O+HO-N=}\underset{\,\text{C}{{\,\text% {H}}_{\,\text{3}}}}{\mathop{\underset{\,\text{ }\!\!|\!\!\,\text{ }}{\mathop{% \,\text{C}}}\,}}\,\,\text{-S-C}{{\,\text{H}}_{\,\text{3}}}\xrightarrow{\,\text% {C}{{\,\text{H}}_{\,\text{2}}}\,\text{C}{{\,\text{l}}_{\,\text{2}}}\,\text{,30% --50 }\!\!~{}\!\!\,\text{ }\!\!{}^{\circ}\!\!\,\text{ C}}\,\text{Methomyl}

Methylamine—glutamate_N-methyltransferase.html

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Methylarsonate_reductase.html

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Methylaspartate_ammonia-lyase.html

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Methylaspartate_mutase.html

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Methylated-DNA—(protein)-cysteine_S-methyltransferase.html

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Methylene-fatty-acyl-phospholipid_synthase.html

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Methylenetetrahydrofolate_dehydrogenase_(NAD+).html

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Methylenetetrahydrofolate_dehydrogenase_(NADP+).html

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Methylenetetrahydrofolate_reductase_(ferredoxin).html

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Methylenetetrahydrofolate—tRNA-(uracil-5-)-methyltransferase.html

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Methylenetetrahydromethanopterin_dehydrogenase.html

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Methylglutamate_dehydrogenase.html

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Methylglyoxal_reductase_(NADH-dependent).html

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Methylglyoxal_reductase_(NADPH-dependent).html

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Methylisocitrate_lyase.html

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Methylitaconate_Delta-isomerase.html

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Methylmalonate-semialdehyde_dehydrogenase_(acylating).html

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Methylmalonyl-CoA_carboxytransferase.html

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Methylmalonyl-CoA_decarboxylase.html

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Methylquercetagetin_6-O-methyltransferase.html

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Methyltetrahydroprotoberberine_14-monooxygenase.html

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Metric_space_aimed_at_its_subspace.html

  1. ( Y , d ) (Y,d)
  2. X X
  3. Y Y
  4. ( X , d | X ) (X,d|_{X})
  5. X X
  6. Y Y
  7. X X
  8. ( Y , d ) (Y,d)
  9. Y Y
  10. X X
  11. y , z y,z
  12. Y Y
  13. ϵ > 0 \epsilon>0
  14. p p
  15. X X
  16. | d ( p , y ) - d ( p , z ) | > d ( y , z ) - ϵ . |d(p,y)-d(p,z)|>d(y,z)-\epsilon.
  17. Met ( X ) \,\text{Met}(X)
  18. X X
  19. Aim ( X ) := { f Met ( X ) : f ( p ) + f ( q ) d ( p , q ) for all p , q X } . \,\text{Aim}(X):=\{f\in\operatorname{Met}(X):f(p)+f(q)\geq d(p,q)\,\text{ for % all }p,q\in X\}.
  20. d ( f , g ) := sup x X | f ( x ) - g ( x ) | < d(f,g):=\sup_{x\in X}|f(x)-g(x)|<\infty
  21. f , g Aim ( X ) f,g\in\,\text{Aim}(X)
  22. Aim ( X ) \,\text{Aim}(X)
  23. δ X : x d x \delta_{X}\colon x\mapsto d_{x}
  24. d x ( p ) := d ( x , p ) d_{x}(p):=d(x,p)\,
  25. X X
  26. Aim ( X ) \operatorname{Aim}(X)
  27. X X
  28. C ( X ) C(X)
  29. Aim ( X ) \operatorname{Aim}(X)
  30. δ X ( X ) \delta_{X}(X)
  31. i : X Y i\colon X\to Y
  32. j : Y Aim ( X ) j\colon Y\to\operatorname{Aim}(X)
  33. j i = δ X j\circ i=\delta_{X}
  34. ( j ( y ) ) ( x ) := d ( x , y ) (j(y))(x):=d(x,y)\,
  35. x X x\in X\,
  36. y Y y\in Y\,
  37. j : Y Aim ( X ) j\colon Y\to\operatorname{Aim}(X)

Mevaldate_reductase.html

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Mevaldate_reductase_(NADPH).html

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Microsomal_epoxide_hydrolase.html

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MIMO.html

  1. N t N r \scriptstyle N_{t}N_{r}
  2. N t \scriptstyle N_{t}
  3. N r \scriptstyle N_{r}
  4. 𝐲 = 𝐇𝐱 + 𝐧 \mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}
  5. 𝐲 \scriptstyle\mathbf{y}
  6. 𝐱 \scriptstyle\mathbf{x}
  7. 𝐇 \scriptstyle\mathbf{H}
  8. 𝐧 \scriptstyle\mathbf{n}
  9. C perfect - CSI = E [ max 𝐐 ; tr ( 𝐐 ) 1 log 2 det ( 𝐈 + ρ 𝐇𝐐𝐇 H ) ] = E [ log 2 det ( 𝐈 + ρ 𝐃𝐒𝐃 ) ] C_{\mathrm{perfect-CSI}}=E\left[\max_{\mathbf{Q};\,\mbox{tr}~{}(\mathbf{Q})% \leq 1}\log_{2}\det\left(\mathbf{I}+\rho\mathbf{H}\mathbf{Q}\mathbf{H}^{H}% \right)\right]=E\left[\log_{2}\det\left(\mathbf{I}+\rho\mathbf{D}\mathbf{S}% \mathbf{D}\right)\right]
  10. ( ) H \scriptstyle()^{H}
  11. ρ \scriptstyle\rho
  12. 𝐐 = 𝐕𝐒𝐕 H \scriptstyle\mathbf{Q}=\mathbf{VSV}^{H}
  13. 𝐔𝐃𝐕 H = 𝐇 \scriptstyle\mathbf{UDV}^{H}\,=\,\mathbf{H}
  14. 𝐒 = diag ( s 1 , , s min ( N t , N r ) , 0 , , 0 ) \scriptstyle\mathbf{S}=\textrm{diag}(s_{1},\ldots,s_{\min(N_{t},N_{r})},0,% \ldots,0)
  15. s i = ( μ - 1 ρ d i 2 ) + , for i = 1 , , min ( N t , N r ) , s_{i}=\left(\mu-\frac{1}{\rho d_{i}^{2}}\right)^{+},\quad\textrm{for}\,\,i=1,% \ldots,\min(N_{t},N_{r}),
  16. d 1 , , d min ( N t , N r ) \scriptstyle d_{1},\ldots,d_{\min(N_{t},N_{r})}
  17. 𝐃 \scriptstyle\mathbf{D}
  18. ( ) + \scriptstyle(\cdot)^{+}
  19. μ \mu
  20. s 1 + + s min ( N t , N r ) = N t \scriptstyle s_{1}+\ldots+s_{\min(N_{t},N_{r})}=N_{t}
  21. 𝐐 \scriptstyle\mathbf{Q}
  22. C statistical - CSI = max 𝐐 E [ log 2 det ( 𝐈 + ρ 𝐇𝐐𝐇 H ) ] . C_{\mathrm{statistical-CSI}}=\max_{\mathbf{Q}}E\left[\log_{2}\det\left(\mathbf% {I}+\rho\mathbf{H}\mathbf{Q}\mathbf{H}^{H}\right)\right].
  23. 𝐐 \scriptstyle\mathbf{Q}
  24. 𝐐 = 1 / N t 𝐈 \scriptstyle\mathbf{Q}=1/N_{t}\mathbf{I}
  25. C no - CSI = E [ log 2 det ( 𝐈 + ρ N t 𝐇𝐇 H ) ] . C_{\mathrm{no-CSI}}=E\left[\log_{2}\det\left(\mathbf{I}+\frac{\rho}{N_{t}}% \mathbf{H}\mathbf{H}^{H}\right)\right].
  26. min ( N t , N r ) \scriptstyle\min(N_{t},N_{r})

Modified_Dietz_method.html

  1. R M D i e t z = G a i n A v e r a g e C a p i t a l = E M V - B M V - F B M V + i = 1 n W i × F i R_{MDietz}=\cfrac{Gain}{AverageCapital}=\cfrac{EMV-BMV-F}{BMV+\sum_{i=1}^{n}W_% {i}\times F_{i}}
  2. E M V EMV
  3. B M V BMV
  4. F F
  5. i = 1 n W i × F i = \sum_{i=1}^{n}W_{i}\times{F_{i}}=
  6. F i F_{i}
  7. W i W_{i}
  8. W i W_{i}
  9. F i F_{i}
  10. W i W_{i}
  11. W i = C D - D i C D W_{i}=\frac{CD-D_{i}}{CD}
  12. C D CD
  13. D i D_{i}
  14. F i F_{i}
  15. W i = C D - D i + 1 C D W_{i}=\frac{CD-D_{i}+1}{CD}
  16. R R
  17. E M V = B M V × ( 1 + R ) + i = 1 n F i × ( 1 + R × T - t i T ) EMV=BMV\times(1+R)+\sum_{i=1}^{n}F_{i}\times(1+R\times\frac{T-t_{i}}{T})
  18. E M V EMV
  19. B M V BMV
  20. T T
  21. t i t_{i}
  22. i i
  23. R R
  24. E M V = B M V × ( 1 + R ) + i = 1 n F i × ( 1 + R ) T - t i T EMV=BMV\times(1+R)+\sum_{i=1}^{n}F_{i}\times(1+R)^{\frac{T-t_{i}}{T}}
  25. G a i n O r L o s s = E M V - B M V - F = 300 - 100 - 50 = 150 U S D GainOrLoss=EMV-BMV-F=300-100-50=150USD
  26. A v e r a g e C a p i t a l = B M V + W e i g h t × F l o w = 100 + 0.5 × 50 = 125 U S D AverageCapital=BMV+\sum Weight\times Flow=100+0.5\times 50=125USD
  27. G a i n O r L o s s / A v e r a g e C a p i t a l = 150 / 125 = 120 % GainOrLoss/AverageCapital=150/125=120\%
  28. 300 = 100 × ( 1 + 125 % ) + 50 × ( 1 + 125 % ) 2 - 1 2 = 225 + 50 × 150 % = 225 + 75 300=100\times(1+125\%)+50\times(1+125\%)^{\frac{2-1}{2}}=225+50\times 150\%=22% 5+75
  29. R 1 R_{1}
  30. R 2 R_{2}
  31. W 1 × R 1 + W 2 × R 2 W_{1}\times R_{1}+W_{2}\times R_{2}
  32. W i = A v e r a g e C a p i t a l i A v e r a g e C a p i t a l 1 + A v e r a g e C a p i t a l 2 W_{i}=\frac{AverageCapital_{i}}{AverageCapital_{1}+AverageCapital_{2}}

Modulus_and_characteristic_of_convexity.html

  1. δ ( ε ) = inf { 1 - x + y 2 : x , y S , x - y ε } , \delta(\varepsilon)=\inf\left\{1-\left\|\frac{x+y}{2}\right\|\,:\,x,y\in S,\|x% -y\|\geq\varepsilon\right\},
  2. ε 0 = sup { ε : δ ( ε ) = 0 } . \varepsilon_{0}=\sup\{\varepsilon\,:\,\delta(\varepsilon)=0\}.
  3. δ ( ε / 2 ) δ 1 ( ε ) δ ( ε ) , ε [ 0 , 2 ] . \delta(\varepsilon/2)\leq\delta_{1}(\varepsilon)\leq\delta(\varepsilon),\quad% \varepsilon\in[0,2].
  4. δ ( ε ) c ε q , ε [ 0 , 2 ] . \delta(\varepsilon)\geq c\,\varepsilon^{q},\quad\varepsilon\in[0,2].

Molar_mass_constant.html

  1. M u = M ( C 12 ) A r ( C 12 ) = 1 g / mol M_{\rm u}={{M({}^{12}{\rm C})}\over{A_{\rm r}({}^{12}{\rm C})}}={1\ {\rm g/mol}}
  2. m ( C 12 ) = 12 M u N A m({}^{12}{\rm C})=\frac{12M_{\rm u}}{N_{\rm A}}
  3. × 10 8 \times 10^{−}8

Moment_distribution_method.html

  1. k k
  2. j j
  3. D j k = E k I k L k i = 1 i = n E i I i L i D_{jk}=\frac{\frac{E_{k}I_{k}}{L_{k}}}{\sum_{i=1}^{i=n}\frac{E_{i}I_{i}}{L_{i}}}
  4. M A M_{A}
  5. θ A \theta_{A}
  6. M B M_{B}
  7. M B M_{B}
  8. M A M_{A}
  9. C A B = M B M A C_{AB}=\frac{M_{B}}{M_{A}}
  10. E I EI
  11. M A = 4 E I L θ A + 2 E I L θ B = 4 E I L θ A M_{A}=4\frac{EI}{L}\theta_{A}+2\frac{EI}{L}\theta_{B}=4\frac{EI}{L}\theta_{A}
  12. M B = 2 E I L θ A + 4 E I L θ B = 2 E I L θ A M_{B}=2\frac{EI}{L}\theta_{A}+4\frac{EI}{L}\theta_{B}=2\frac{EI}{L}\theta_{A}
  13. C A B = M B M A = 1 2 C_{AB}=\frac{M_{B}}{M_{A}}=\frac{1}{2}
  14. L = 10 m L=10\ m
  15. P = 10 k N P=10\ kN
  16. a = 3 m a=3\ m
  17. q = 1 k N / m q=1\ kN/m
  18. P = 10 k N P=10\ kN
  19. M A B f = - P b 2 a L 2 = - 10 × 7 2 × 3 10 2 = - 14.700 k N m M_{AB}^{f}=-\frac{Pb^{2}a}{L^{2}}=-\frac{10\times 7^{2}\times 3}{10^{2}}=-14.7% 00\ kN\cdot m
  20. M B A f = P a 2 b L 2 = 10 × 3 2 × 7 10 2 = + 6.300 k N m M_{BA}^{f}=\frac{Pa^{2}b}{L^{2}}=\frac{10\times 3^{2}\times 7}{10^{2}}=+6.300% \ kN\cdot m
  21. M B C f = - q L 2 12 = - 1 × 10 2 12 = - 8.333 k N m M_{BC}^{f}=-\frac{qL^{2}}{12}=-\frac{1\times 10^{2}}{12}=-8.333\ kN\cdot m
  22. M C B f = q L 2 12 = 1 × 10 2 12 = + 8.333 k N m M_{CB}^{f}=\frac{qL^{2}}{12}=\frac{1\times 10^{2}}{12}=+8.333\ kN\cdot m
  23. M C D f = - P L 8 = - 10 × 10 8 = - 12.500 k N m M_{CD}^{f}=-\frac{PL}{8}=-\frac{10\times 10}{8}=-12.500\ kN\cdot m
  24. M D C f = P L 8 = 10 × 10 8 = + 12.500 k N m M_{DC}^{f}=\frac{PL}{8}=\frac{10\times 10}{8}=+12.500\ kN\cdot m
  25. 3 E I L \frac{3EI}{L}
  26. 4 × 2 E I L \frac{4\times 2EI}{L}
  27. 4 E I L \frac{4EI}{L}
  28. D B A = 3 E I L 3 E I L + 4 × 2 E I L = 3 10 3 10 + 8 10 = 3 11 = 0. ( 27 ) D_{BA}=\frac{\frac{3EI}{L}}{\frac{3EI}{L}+\frac{4\times 2EI}{L}}=\frac{\frac{3% }{10}}{\frac{3}{10}+\frac{8}{10}}=\frac{3}{11}=0.(27)
  29. D B C = 4 × 2 E I L 3 E I L + 4 × 2 E I L = 8 10 3 10 + 8 10 = 8 11 = 0. ( 72 ) D_{BC}=\frac{\frac{4\times 2EI}{L}}{\frac{3EI}{L}+\frac{4\times 2EI}{L}}=\frac% {\frac{8}{10}}{\frac{3}{10}+\frac{8}{10}}=\frac{8}{11}=0.(72)
  30. D C B = 4 × 2 E I L 4 × 2 E I L + 4 E I L = 8 10 8 10 + 4 10 = 8 12 = 0. ( 67 ) D_{CB}=\frac{\frac{4\times 2EI}{L}}{\frac{4\times 2EI}{L}+\frac{4EI}{L}}=\frac% {\frac{8}{10}}{\frac{8}{10}+\frac{4}{10}}=\frac{8}{12}=0.(67)
  31. D C D = 4 E I L 4 × 2 E I L + 4 E I L = 4 10 8 10 + 4 10 = 4 12 = 0. ( 33 ) D_{CD}=\frac{\frac{4EI}{L}}{\frac{4\times 2EI}{L}+\frac{4EI}{L}}=\frac{\frac{4% }{10}}{\frac{8}{10}+\frac{4}{10}}=\frac{4}{12}=0.(33)
  32. D A B = 1 D_{AB}=1
  33. D D C = 0 D_{DC}=0
  34. 1 2 \frac{1}{2}
  35. M A B f = 14.700 kN m M_{AB}^{f}=14.700\mathrm{\,kN\,m}
  36. M B A f M_{BA}^{f}
  37. M B C f M_{BC}^{f}
  38. D B A = 0.2727 D_{BA}=0.2727
  39. D B C = 0.7273 D_{BC}=0.7273
  40. M B C = 3.867 kN m M_{BC}=3.867\mathrm{\,kN\,m}
  41. M C B f M_{CB}^{f}
  42. M C D f M_{CD}^{f}
  43. M A = 0 k N m M_{A}=0\ kN\cdot m
  44. M B = - 11.569 k N m M_{B}=-11.569\ kN\cdot m
  45. M C = 10.186 k N m M_{C}=10.186\ kN\cdot m
  46. M D = - 13.657 k N m M_{D}=-13.657\ kN\cdot m
  47. M A = 0 k N m M_{A}=0\ kN\cdot m
  48. M B = - 11.569 k N m M_{B}=-11.569\ kN\cdot m
  49. M C = - 10.186 k N m M_{C}=-10.186\ kN\cdot m
  50. M D = - 13.657 k N m M_{D}=-13.657\ kN\cdot m
  51. [ K ] { d } = { - f } \left[K\right]\left\{d\right\}=\left\{-f\right\}
  52. [ K ] = [ 3 E I L + 4 2 E I L 2 2 E I L 2 2 E I L 4 2 E I L + 4 E I L ] \left[K\right]=\begin{bmatrix}3\frac{EI}{L}+4\frac{2EI}{L}&2\frac{2EI}{L}\\ 2\frac{2EI}{L}&4\frac{2EI}{L}+4\frac{EI}{L}\end{bmatrix}
  53. { f } T = { - P a b ( L + a ) 2 L 2 + q L 2 12 , - q L 2 12 + P L 8 } \left\{f\right\}^{T}=\left\{-P\frac{ab(L+a)}{2L^{2}}+q\frac{L^{2}}{12},-q\frac% {L^{2}}{12}+P\frac{L}{8}\right\}
  54. { d } \left\{d\right\}
  55. { d } T = { 6.9368 ; - 5.7845 } \left\{d\right\}^{T}=\left\{6.9368;-5.7845\right\}
  56. M B A = 3 E I L d 1 - P a b ( L + a ) 2 L 2 = - 11.569 M_{BA}=3\frac{EI}{L}d_{1}-P\frac{ab(L+a)}{2L^{2}}=-11.569
  57. M B C = - 4 2 E I L d 1 - 2 2 E I L d 2 - q L 2 12 = - 11.569 M_{BC}=-4\frac{2EI}{L}d_{1}-2\frac{2EI}{L}d_{2}-q\frac{L^{2}}{12}=-11.569
  58. M C B = 2 2 E I L d 1 + 4 2 E I L d 2 - q L 2 12 = - 10.186 M_{CB}=2\frac{2EI}{L}d_{1}+4\frac{2EI}{L}d_{2}-q\frac{L^{2}}{12}=-10.186
  59. M C D = - 4 E I L d 2 - P L 8 = - 10.186 M_{CD}=-4\frac{EI}{L}d_{2}-P\frac{L}{8}=-10.186

Monodehydroascorbate_reductase_(NADH).html

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Monoprenyl_isoflavone_epoxidase.html

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Moreau's_theorem.html

  1. J α = ( id + α A ) - 1 ; J_{\alpha}=(\mathrm{id}+\alpha A)^{-1};
  2. A α = 1 α ( id - J α ) . A_{\alpha}=\frac{1}{\alpha}(\mathrm{id}-J_{\alpha}).
  3. φ α ( x ) = inf y H 1 2 α y - x 2 + φ ( y ) . \varphi_{\alpha}(x)=\inf_{y\in H}\frac{1}{2\alpha}\|y-x\|^{2}+\varphi(y).
  4. φ α ( x ) = α 2 A α x 2 + φ ( J α ( x ) ) \varphi_{\alpha}(x)=\frac{\alpha}{2}\|A_{\alpha}x\|^{2}+\varphi(J_{\alpha}(x))

Morphine_6-dehydrogenase.html

  1. \rightleftharpoons

Mott_scattering.html

  1. A = I r i g h t - I l e f t I r i g h t + I l e f t A=\frac{I^{right}-I^{left}}{I^{right}+I^{left}}

Möbius–Kantor_configuration.html

  1. 3 × 3 \mathbb{Z}_{3}\times\mathbb{Z}_{3}
  2. ( i , 0 ) (i,0)
  3. ( i , i ) (i,i)
  4. ( i , 2 i ) (i,2i)
  5. ( 0 , i ) (0,i)

Möbius–Kantor_graph.html

  1. 3 × 3 \mathbb{Z}_{3}\times\mathbb{Z}_{3}
  2. ( i , 0 ) (i,0)
  3. ( i , i ) (i,i)
  4. ( i , 2 i ) (i,2i)
  5. ( 0 , i ) (0,i)
  6. G ( 4 , 1 ) G(4,1)
  7. G ( 5 , 2 ) G(5,2)
  8. G ( 10 , 2 ) G(10,2)
  9. G ( 10 , 3 ) G(10,3)
  10. G ( 12 , 5 ) G(12,5)
  11. ( x - 3 ) ( x - 1 ) 3 ( x + 1 ) 3 ( x + 3 ) ( x 2 - 3 ) 4 . (x-3)(x-1)^{3}(x+1)^{3}(x+3)(x^{2}-3)^{4}.

MRNA_(2'-O-methyladenosine-N6-)-methyltransferase.html

  1. \rightleftharpoons

MRNA_(guanine-N7-)-methyltransferase.html

  1. \rightleftharpoons

MRNA_(nucleoside-2'-O-)-methyltransferase.html

  1. \rightleftharpoons

Mu_problem.html

  1. X M p l H u H d {X\over M_{pl}}H_{u}H_{d}
  2. F X M p l H u H d {\langle F_{X}\rangle\over M_{pl}}H_{u}H_{d}
  3. μ = F X M p l \mu={\langle F_{X}\rangle\over M_{pl}}
  4. F X M p l {\langle F_{X}\rangle\over M_{pl}}

Muconolactone_Delta-isomerase.html

  1. \rightleftharpoons

Multiple-instance_learning.html

  1. X = { X 1 , . . , X N } X=\{X_{1},..,X_{N}\}
  2. X i X_{i}

Multiple_of_the_median.html

  1. M o M ( P a t i e n t ) = R e s u l t ( P a t i e n t ) M e d i a n ( P a t i e n t P o p u l a t i o n ) MoM(Patient)=\frac{Result(Patient)}{Median(PatientPopulation)}

Multiplicative_cascade.html

  1. { p 1 , p 2 , p 3 , p 4 } = { 1 , 1 , 1 , 0 } \{p_{1},p_{2},p_{3},p_{4}\}=\{1,1,1,0\}
  2. { p 1 , p 2 , p 3 , p 4 } = { 1 , 0.75 , 0.75 , 0.5 } \{p_{1},p_{2},p_{3},p_{4}\}=\{1,0.75,0.75,0.5\}
  3. { p 1 , p 2 , p 3 , p 4 } = { 1 , 0.5 , 0.5 , 0.25 } \{p_{1},p_{2},p_{3},p_{4}\}=\{1,0.5,0.5,0.25\}
  4. { p 1 , p 2 , p 3 , p 4 } \{p_{1},p_{2},p_{3},p_{4}\}
  5. p i [ 0 , 1 ] p_{i}\in[0,1]
  6. N N\rightarrow\infty
  7. D q = log 2 ( f 1 q + f 2 q + f 3 q + f 4 q ) 1 - q , D_{q}=\frac{\log_{2}\left(f^{q}_{1}+f^{q}_{2}+f^{q}_{3}+f^{q}_{4}\right)}{1-q},
  8. f i = p i i p i . f_{i}=\frac{p_{i}}{\sum_{i}p_{i}}.

Multiplier_(economics).html

  1. C t = C 0 + c Y t - 1 C_{t}=C_{0}+cY_{t-1}
  2. 1 / ( 1 - c ( 1 - t ) + m ) 1/(1-c(1-t)+m)
  3. I t = I 0 + I ( r ) + b ( C t - C t - 1 ) I_{t}=I_{0}+I(r)+b(C_{t}-C_{t-1})
  4. I t = I 0 + b ( C t - C t - 1 ) I_{t}=I_{0}+b(C_{t}-C_{t-1})
  5. Y t d = C t + I t = C 0 + I 0 + c Y t - 1 + b ( C t - C t - 1 ) Ytd=C_{t}+I_{t}=C_{0}+I_{0}+cY_{t-1}+b(C_{t}-C_{t-1})
  6. Y t = Y t d Y_{t}=Ytd
  7. Y t = C 0 + I 0 + c Y t - 1 + b ( C t - C t - 1 ) Y_{t}=C_{0}+I_{0}+cY_{t-1}+b(C_{t}-C_{t-1})
  8. C t C_{t}
  9. C t - 1 C_{t-1}
  10. C t = C 0 + c Y t - 1 C_{t}=C_{0}+cY_{t-1}
  11. C t - 1 = C 0 + c Y t - 2 C_{t-1}=C_{0}+cY_{t-2}
  12. Y t = C 0 + I 0 + c Y t - 1 + b ( C 0 + c Y t - 1 - C 0 - c Y t - 2 ) Y_{t}=C_{0}+I_{0}+cY_{t-1}+b(C_{0}+cY_{t-1}-C_{0}-cY_{t-2})
  13. Y t - ( 1 + b ) c Y t - 1 + b c Y t - 2 = ( C 0 + I 0 ) Y_{t}-(1+b)cY_{t-1}+bcY_{t-2}=(C_{0}+I_{0})
  14. Y p Y_{p}
  15. Y t = Y t - 1 = Y t - 2 = Y p Y_{t}=Y_{t-1}=Y_{t-2}=Y_{p}
  16. ( 1 - c - b c + b c ) Y p = ( C 0 + I 0 ) (1-c-bc+bc)Y_{p}=(C_{0}+I_{0})
  17. Y p = ( C 0 + I 0 ) / ( 1 - c ) Y_{p}=(C_{0}+I_{0})/(1-c)
  18. Y c Y_{c}
  19. Y c = A 1 r 1 t + A 2 r 2 t Y_{c}=A_{1}r_{1}t+A_{2}r_{2}t
  20. A 1 A_{1}
  21. A 2 A_{2}
  22. r 1 r_{1}
  23. r 2 r_{2}
  24. r 2 - ( 1 + b ) c r + b c = 0 r^{2}-(1+b)cr+bc=0
  25. Y = Y c + Y p Y=Y_{c}+Y_{p}

Multiplier_algebra.html

  1. l a ( x ) = a x , r a ( x ) = x a . l_{a}(x)=\|ax\|,\;r_{a}(x)=\|xa\|.
  2. C b ( X ) C ( Y ) C_{b}(X)\simeq C(Y)

Multipliers_and_centralizers_(Banach_spaces).html

  1. p T = a T ( p ) p for all p Ext ( X ) , p\circ T=a_{T}(p)p\mbox{ for all }~{}p\in\mathrm{Ext}(X),
  2. a T ( p ) a_{T}(p)
  3. a S = a T ¯ , a_{S}=\overline{a_{T}},

Mycothiol-dependent_formaldehyde_dehydrogenase.html

  1. \rightleftharpoons

Mycothione_reductase.html

  1. \rightleftharpoons

Myricetin_O-methyltransferase.html

  1. \rightleftharpoons

Myristoyl-CoA_11-(E)_desaturase.html

  1. \rightleftharpoons

Myristoyl-CoA_11-(Z)_desaturase.html

  1. \rightleftharpoons

N-acetyl-gamma-glutamyl-phosphate_reductase.html

  1. \rightleftharpoons

N-acetylhexosamine_1-dehydrogenase.html

  1. \rightleftharpoons

N-acetylneuraminate_lyase.html

  1. \rightleftharpoons

N-acetylornithine_carbamoyltransferase.html

  1. \rightleftharpoons

N-acylglucosamine-6-phosphate_2-epimerase.html

  1. \rightleftharpoons

N-acylglucosamine_2-epimerase.html

  1. \rightleftharpoons

N-acylhexosamine_oxidase.html

  1. \rightleftharpoons

N-acylmannosamine_1-dehydrogenase.html

  1. \rightleftharpoons

N-benzoyl-4-hydroxyanthranilate_4-O-methyltransferase.html

  1. \rightleftharpoons

N-hydroxy-2-acetamidofluorene_reductase.html

  1. \rightleftharpoons

N-methyl-L-amino-acid_oxidase.html

  1. \rightleftharpoons

N-methylalanine_dehydrogenase.html

  1. \rightleftharpoons

N-methylcoclaurine_3'-monooxygenase.html

  1. \rightleftharpoons

N-sulfoglucosamine_sulfohydrolase.html

  1. \rightleftharpoons

N5-(carboxyethyl)ornithine_synthase.html

  1. \rightleftharpoons

N6-methyl-lysine_oxidase.html

  1. \rightleftharpoons

NAD(P)+_transhydrogenase_(Re::Si-specific).html

  1. \rightleftharpoons

NAD(P)+_transhydrogenase_(Si-specific).html

  1. \rightleftharpoons

NAD(P)H_dehydrogenase_(quinone).html

  1. \rightleftharpoons

NAD+_kinase.html

  1. \rightleftharpoons

NADH_dehydrogenase_(quinone).html

  1. \rightleftharpoons

NADH_peroxidase.html

  1. \rightleftharpoons
  2. \rightarrow
  3. \rightarrow
  4. \rightarrow
  5. \rightarrow
  6. \rightarrow
  7. \rightarrow
  8. \rightarrow

NADPH:quinone_reductase.html

  1. \rightleftharpoons

NADPH_dehydrogenase.html

  1. \rightleftharpoons

NADPH_dehydrogenase_(quinone).html

  1. \rightleftharpoons

NADPH_peroxidase.html

  1. \rightleftharpoons

NADPH—cytochrome-c2_reductase.html

  1. \rightleftharpoons

NADPH—hemoprotein_reductase.html

  1. \rightleftharpoons

Naphthalene_1,2-dioxygenase.html

  1. \rightleftharpoons

Nash_functions.html

  1. x 1 + x 2 x\mapsto\sqrt{1+x^{2}}
  2. 𝒩 \mathcal{N}
  3. \mathcal{I}
  4. 𝒩 \mathcal{N}
  5. \mathcal{I}
  6. { U i } \{U_{i}\}
  7. | U i \mathcal{I}|_{U_{i}}
  8. U i U_{i}
  9. \mathcal{I}
  10. H 0 ( M , 𝒩 ) H 0 ( M , 𝒩 / ) H^{0}(M,\mathcal{N})\to H^{0}(M,\mathcal{N}/\mathcal{I})
  11. H 1 ( M , 𝒩 ) 0 , if dim ( M ) > 0 , H^{1}(M,\mathcal{N})\neq 0,\ \,\text{if}\ \dim(M)>0,

Nasik_magic_hypercube.html

  1. S = m ( m n + 1 ) 2 S=\frac{m(m^{n}+1)}{2}

Neher–McGrath.html

  1. Δ T D {\Delta}TD
  2. Δ T D {\Delta}TD
  3. I = T c - T a R d c ( 1 + Y C ) R c a I=\sqrt{Tc-Ta\over Rdc(1+YC)Rca}

Neil_J._Gunther.html

  1. C ( N ) = N 1 + α ( ( N - 1 ) + β N ( N - 1 ) ) C(N)=\frac{N}{1+\alpha((N-1)+\beta N(N-1))}
  2. α \alpha
  3. β \beta
  4. β \beta
  5. β = 0 \beta=0

Neoxanthin_synthase.html

  1. \rightleftharpoons

Neves_(video_game).html

  1. 1 / 2 \scriptstyle{1/2}
  2. 1 / 2 \scriptstyle{1/\sqrt{2}}
  3. 1 / 4 \scriptstyle{1/4}
  4. 1 / 4 \scriptstyle{1/4}
  5. 1 / 2 \scriptstyle{1/2}
  6. 1 / 2 2 \scriptstyle{1/2\sqrt{2}}
  7. 1 / 2 2 \scriptstyle{1/2\sqrt{2}}
  8. 1 / 2 2 \scriptstyle{1/2\sqrt{2}}
  9. 1 / 2 \scriptstyle{1/\sqrt{2}}
  10. 1 / 2 \scriptstyle{1/2}
  11. 1 / 2 \scriptstyle{1/\sqrt{2}}
  12. 1 / 2 2 \scriptstyle{1/2\sqrt{2}}
  13. 3 / 2 2 \scriptstyle{3/2\sqrt{2}}
  14. 1 / 2 \scriptstyle{1/2}
  15. 1 / 2 \scriptstyle{1/2}
  16. 1 / 2 \scriptstyle{1/2}
  17. 1 / 2 \scriptstyle{1/2}
  18. 1 / 2 2 \scriptstyle{1/2\sqrt{2}}
  19. 1 / 2 2 \scriptstyle{1/2\sqrt{2}}

Nicotinamide_N-methyltransferase.html

  1. \rightleftharpoons

Nicotinate_dehydrogenase.html

  1. \rightleftharpoons

Nicotinate_N-methyltransferase.html

  1. \rightleftharpoons

Nicotine_dehydrogenase.html

  1. \rightleftharpoons

Nitric-oxide_reductase.html

  1. \rightleftharpoons

Nitric_oxide_dioxygenase.html

  1. \rightleftharpoons

Nitrite_reductase_(NAD(P)H).html

  1. \rightleftharpoons

Nitroalkane_oxidase.html

  1. \rightleftharpoons

Nitrogenase_(flavodoxin).html

  1. \rightleftharpoons

Nitroquinoline-N-oxide_reductase.html

  1. \rightleftharpoons

Nitrous-oxide_reductase.html

  1. \rightleftharpoons

Nocardicin-A_epimerase.html

  1. \rightleftharpoons

Node_(autonomous_system).html

  1. t t\rightarrow\infty
  2. t - t\rightarrow-\infty

Non-Hausdorff_manifold.html

  1. ( x , a ) ( x , b ) if x 0. (x,a)\sim(x,b)\,\text{ if }x\neq 0.\;
  2. ( x , a ) ( x , b ) if x < 0. (x,a)\sim(x,b)\,\text{ if }x<0.\;
  3. x a , x b x_{a},x_{b}

Non-random_two-liquid_model.html

  1. γ i \gamma_{i}
  2. x i x_{i}
  3. U i i U_{ii}
  4. U i j U_{ij}
  5. { ln γ 1 = x 2 2 [ τ 21 ( G 21 x 1 + x 2 G 21 ) 2 + τ 12 G 12 ( x 2 + x 1 G 12 ) 2 ] ln γ 2 = x 1 2 [ τ 12 ( G 12 x 2 + x 1 G 12 ) 2 + τ 21 G 21 ( x 1 + x 2 G 21 ) 2 ] \left\{\begin{matrix}\ln\ \gamma_{1}=x^{2}_{2}\left[\tau_{21}\left(\frac{G_{21% }}{x_{1}+x_{2}G_{21}}\right)^{2}+\frac{\tau_{12}G_{12}}{(x_{2}+x_{1}G_{12})^{2% }}\right]\\ \\ \ln\ \gamma_{2}=x^{2}_{1}\left[\tau_{12}\left(\frac{G_{12}}{x_{2}+x_{1}G_{12}}% \right)^{2}+\frac{\tau_{21}G_{21}}{(x_{1}+x_{2}G_{21})^{2}}\right]\end{matrix}\right.
  6. { ln G 12 = - α 12 τ 12 ln G 21 = - α 21 τ 21 \left\{\begin{matrix}\ln\ G_{12}=-\alpha_{12}\ \tau_{12}\\ \ln\ G_{21}=-\alpha_{21}\ \tau_{21}\end{matrix}\right.
  7. τ 12 \tau_{12}
  8. τ 21 \tau_{21}
  9. Δ g 12 \Delta g_{12}
  10. Δ g 21 \Delta g_{21}
  11. { τ 12 = Δ g 12 R T = U 12 - U 22 R T τ 21 = Δ g 21 R T = U 21 - U 11 R T \left\{\begin{matrix}\tau_{12}=\frac{\Delta g_{12}}{RT}=\frac{U_{12}-U_{22}}{% RT}\\ \tau_{21}=\frac{\Delta g_{21}}{RT}=\frac{U_{21}-U_{11}}{RT}\end{matrix}\right.
  12. Δ g i j \Delta g_{ij}
  13. Δ g j i \Delta g_{ji}
  14. α 12 \alpha_{12}
  15. α 21 \alpha_{21}
  16. α 12 \alpha_{12}
  17. α 21 \alpha_{21}
  18. α 12 = 0 \alpha_{12}=0
  19. { ln γ 1 = x 2 2 [ τ 21 + τ 12 ] = A x 2 2 ln γ 2 = x 1 2 [ τ 12 + τ 21 ] = A x 1 2 \left\{\begin{matrix}\ln\ \gamma_{1}=x^{2}_{2}\left[\tau_{21}+\tau_{12}\right]% =Ax^{2}_{2}\\ \ln\ \gamma_{2}=x^{2}_{1}\left[\tau_{12}+\tau_{21}\right]=Ax^{2}_{1}\end{% matrix}\right.
  20. α 12 \alpha_{12}
  21. α 12 = - 1 \alpha_{12}=-1
  22. α 12 \alpha_{12}
  23. { ln γ 1 = [ τ 21 + τ 12 exp ( - α 12 τ 12 ) ] ln γ 2 = [ τ 12 + τ 21 exp ( - α 12 τ 21 ) ] \left\{\begin{matrix}\ln\ \gamma_{1}^{\infty}=\left[\tau_{21}+\tau_{12}\exp{(-% \alpha_{12}\ \tau_{12})}\right]\\ \ln\ \gamma_{2}^{\infty}=\left[\tau_{12}+\tau_{21}\exp{(-\alpha_{12}\ \tau_{21% })}\right]\end{matrix}\right.
  24. α 12 = 0 \alpha_{12}=0
  25. ln ( γ i ) \ln(\gamma_{i})
  26. i i
  27. n n
  28. ln ( γ i ) = j = 1 n x j τ j i G j i k = 1 n x k G k i + j = 1 n x j G i j k = 1 n x k G k j ( τ i j - m = 1 n x m τ m j G m j k = 1 n x k G k j ) \ln(\gamma_{i})=\frac{\displaystyle\sum_{j=1}^{n}{x_{j}\tau_{ji}G_{ji}}}{% \displaystyle\sum_{k=1}^{n}{x_{k}G_{ki}}}+\sum_{j=1}^{n}{\frac{x_{j}G_{ij}}{% \displaystyle\sum_{k=1}^{n}{x_{k}G_{kj}}}}{\left({\tau_{ij}-\frac{% \displaystyle\sum_{m=1}^{n}{x_{m}\tau_{mj}G_{mj}}}{\displaystyle\sum_{k=1}^{n}% {x_{k}G_{kj}}}}\right)}
  29. G i j = exp ( - α i j τ i j ) G_{ij}=\exp\left({-\alpha_{ij}\tau_{ij}}\right)
  30. α i j = α i j 0 + α i j 1 T \alpha_{ij}=\alpha_{ij_{0}}+\alpha_{ij_{1}}T
  31. τ i , j = A i j + B i j T + C i j T 2 + D i j ln ( T ) + E i j T F i j \tau_{i,j}=A_{ij}+\frac{B_{ij}}{T}+\frac{C_{ij}}{T^{2}}+D_{ij}\ln{\left({T}% \right)}+E_{ij}T^{F_{ij}}
  32. α i j \alpha_{ij}
  33. τ i j \tau_{ij}
  34. τ i j = f ( T ) = a i j + b i j T + c i j ln T + d i j T \tau_{ij}=f(T)=a_{ij}+\frac{b_{ij}}{T}+c_{ij}\ \ln\ T+d_{ij}T
  35. Δ g i j = f ( T ) = a i j + b i j T + c i j T 2 \Delta g_{ij}=f(T)=a_{ij}+b_{ij}\cdot T+c_{ij}T^{2}

Nonnegative_rank_(linear_algebra).html

  1. rank ( A ) + = min { q j = 1 q R j = A , rank R 1 = = rank R q = 1 , R 1 , , R q 0 } , \mbox{rank}~{}_{+}(A)=\min\{q\mid\sum_{j=1}^{q}R_{j}=A,\;\mbox{rank}~{}\,R_{1}% =\dots=\mbox{rank}~{}\,R_{q}=1,\;R_{1},\dots,R_{q}\geq 0\},
  2. m × n m\times n
  3. r a n k + ( A ) rank_{+}(A)
  4. rank ( A ) rank ( A ) + min ( m , n ) , \mbox{rank}~{}\,(A)\leq\mbox{rank}~{}_{+}(A)\leq\min(m,n),
  5. r a n k ( A ) rank(A)
  6. 𝐀 = [ 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 ] , \mathbf{A}=\begin{bmatrix}1&1&0&0\\ 1&0&1&0\\ 0&1&0&1\\ 0&0&1&1\end{bmatrix},
  7. r a n k + ( A ) = r a n k ( A ) {{rank}_{+}}(A)=rank(A)

Normalisation_by_evaluation.html

  1. α t = 𝐒𝐘𝐍 t τ 1 τ 2 v = 𝐋𝐀𝐌 ( λ S . τ 2 ( 𝐚𝐩𝐩 ( v , τ 1 S ) ) ) τ 1 × τ 2 v = 𝐏𝐀𝐈𝐑 ( τ 1 ( 𝐟𝐬𝐭 v ) , τ 2 ( 𝐬𝐧𝐝 v ) ) α ( 𝐒𝐘𝐍 t ) = t τ 1 τ 2 ( 𝐋𝐀𝐌 S ) = 𝐥𝐚𝐦 ( x , τ 2 ( S ( τ 1 ( 𝐯𝐚𝐫 x ) ) ) ) where x is fresh τ 1 × τ 2 ( 𝐏𝐀𝐈𝐑 ( S , T ) ) = 𝐩𝐚𝐢𝐫 ( τ 1 S , τ 2 T ) \begin{aligned}\displaystyle\uparrow_{\alpha}t&\displaystyle=\mathbf{SYN}\ t\\ \displaystyle\uparrow_{\tau_{1}\to\tau_{2}}v&\displaystyle=\mathbf{LAM}(% \lambda S.\ \uparrow_{\tau_{2}}(\mathbf{app}\ (v,\downarrow^{\tau_{1}}S)))\\ \displaystyle\uparrow_{\tau_{1}\times\tau_{2}}v&\displaystyle=\mathbf{PAIR}(% \uparrow_{\tau_{1}}(\mathbf{fst}\ v),\uparrow_{\tau_{2}}(\mathbf{snd}\ v))\\ \displaystyle\downarrow^{\alpha}(\mathbf{SYN}\ t)&\displaystyle=t\\ \displaystyle\downarrow^{\tau_{1}\to\tau_{2}}(\mathbf{LAM}\ S)&\displaystyle=% \mathbf{lam}\ (x,\downarrow^{\tau_{2}}(S\ (\uparrow_{\tau_{1}}(\mathbf{var}\ x% ))))\,\text{ where }x\,\text{ is fresh}\\ \displaystyle\downarrow^{\tau_{1}\times\tau_{2}}(\mathbf{PAIR}\ (S,T))&% \displaystyle=\mathbf{pair}\ (\downarrow^{\tau_{1}}S,\downarrow^{\tau_{2}}T)% \end{aligned}
  2. 𝐯𝐚𝐫 x Γ = Γ ( x ) 𝐥𝐚𝐦 ( x , s ) Γ = 𝐋𝐀𝐌 ( λ S . s Γ , x S ) 𝐚𝐩𝐩 ( s , t ) Γ = S ( t Γ ) where s Γ = 𝐋𝐀𝐌 S 𝐩𝐚𝐢𝐫 ( s , t ) Γ = 𝐏𝐀𝐈𝐑 ( s Γ , t Γ ) 𝐟𝐬𝐭 s Γ = S where s Γ = 𝐏𝐀𝐈𝐑 ( S , T ) 𝐬𝐧𝐝 t Γ = T where t Γ = 𝐏𝐀𝐈𝐑 ( S , T ) \begin{aligned}\displaystyle\|\mathbf{var}\ x\|_{\Gamma}&\displaystyle=\Gamma(% x)\\ \displaystyle\|\mathbf{lam}\ (x,s)\|_{\Gamma}&\displaystyle=\mathbf{LAM}\ (% \lambda S.\ \|s\|_{\Gamma,x\mapsto S})\\ \displaystyle\|\mathbf{app}\ (s,t)\|_{\Gamma}&\displaystyle=S\ (\|t\|_{\Gamma}% )\,\text{ where }\|s\|_{\Gamma}=\mathbf{LAM}\ S\\ \displaystyle\|\mathbf{pair}\ (s,t)\|_{\Gamma}&\displaystyle=\mathbf{PAIR}\ (% \|s\|_{\Gamma},\|t\|_{\Gamma})\\ \displaystyle\|\mathbf{fst}\ s\|_{\Gamma}&\displaystyle=S\,\text{ where }\|s\|% _{\Gamma}=\mathbf{PAIR}\ (S,T)\\ \displaystyle\|\mathbf{snd}\ t\|_{\Gamma}&\displaystyle=T\,\text{ where }\|t\|% _{\Gamma}=\mathbf{PAIR}\ (S,T)\end{aligned}

O-6-methylguanine-DNA_methyltransferase.html

  1. MGMT \mathrm{\ \xrightarrow{MGMT}}

O-demethylpuromycin_O-methyltransferase.html

  1. \rightleftharpoons

O-pyrocatechuate_decarboxylase.html

  1. \rightleftharpoons

Oberth_effect.html

  1. W W
  2. F \overrightarrow{F}
  3. s \overrightarrow{s}
  4. W = F s W=\overrightarrow{F}\cdot\overrightarrow{s}
  5. F s = || F || || s || \overrightarrow{F}\cdot\overrightarrow{s}=||F||\cdot||s||
  6. Δ E k \Delta E_{k}
  7. Δ E k = F s \Delta E_{k}=F\cdot s
  8. d E k d t = F d s d t \frac{\operatorname{d}E_{k}}{\operatorname{d}t}=F\cdot\frac{\operatorname{d}s}% {\operatorname{d}t}
  9. d E k d t = F v \frac{\operatorname{d}E_{k}}{\operatorname{d}t}=F\cdot v
  10. v v
  11. m m
  12. e k e_{k}
  13. d e k d t = F m v = a v \frac{\operatorname{d}e_{k}}{\operatorname{d}t}=\frac{F}{m}\cdot v=a\cdot v
  14. a a
  15. Δ v \Delta v
  16. e k \displaystyle e_{k}
  17. V = V esc + Δ v V=V\text{esc}+\Delta v
  18. 1 2 V esc 2 \frac{1}{2}V\text{esc}^{2}
  19. Δ v V esc + 1 2 Δ v 2 \Delta vV\text{esc}+\frac{1}{2}\Delta v^{2}
  20. 1 2 Δ v 2 \frac{1}{2}\Delta v^{2}
  21. Δ v V esc \Delta vV\text{esc}
  22. 1 + 2 V esc Δ v \sqrt{1+\frac{2V\text{esc}}{\Delta v}}
  23. v Δ v + 1 2 ( Δ v ) 2 v\Delta v+\frac{1}{2}(\Delta v)^{2}

Ocean_heat_content.html

  1. H = ρ c p h 2 h 1 T ( z ) d z H=\rho c_{p}\int_{h2}^{h1}T(z)dz
  2. ρ \rho
  3. c p c_{p}
  4. T ( z ) T(z)

Octadecanal_decarbonylase.html

  1. \rightleftharpoons

Octanol_dehydrogenase.html

  1. \rightleftharpoons

One-way_quantum_computer.html

  1. { | 0 , | 1 } \{|0\rangle,|1\rangle\}
  2. | 0 ± e i θ | 1 |0\rangle\pm e^{i\theta}|1\rangle
  3. 3 2 \tfrac{3}{2}

Ono's_inequality.html

  1. 27 ( b 2 + c 2 - a 2 ) 2 ( c 2 + a 2 - b 2 ) 2 ( a 2 + b 2 - c 2 ) 2 ( 4 A ) 6 . 27(b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2}\leq(4A% )^{6}.
  2. a = 2 , b = 3 , c = 4 , A = 3 15 / 4. a=2,\,\,b=3,\,\,c=4,\,\,A=3\sqrt{15}/4.
  3. 1 , 1 , 1 1,1,1
  4. 3 / 4. \sqrt{3}/4.

OpenFOAM.html

  1. ρ 𝐔 t + ϕ 𝐔 - μ 𝐔 = - p \frac{\partial\rho\mathbf{U}}{\partial t}+\nabla\cdot\phi\mathbf{U}-\nabla% \cdot\mu\nabla\mathbf{U}=-\nabla p

Operator_K-theory.html

  1. K 0 K_{0}
  2. K 0 K_{0}
  3. K 1 K_{1}
  4. M n ( ) M_{n}(\mathbb{C})
  5. K C ( X ) K\otimes C(X)
  6. K K
  7. K 0 K_{0}
  8. K C ( X ) K\otimes C(X)
  9. K 0 K_{0}
  10. K n ( A ) = K 0 ( S n ( A ) ) K_{n}(A)=K_{0}(S^{n}(A))
  11. S A = C 0 ( 0 , 1 ) A . SA=C_{0}(0,1)\otimes A.
  12. K n + 2 ( A ) K_{n+2}(A)
  13. K n ( A ) K_{n}(A)
  14. K 0 K_{0}
  15. K 1 K_{1}

Opine_dehydrogenase.html

  1. \rightleftharpoons

Oplophorus-luciferin_2-monooxygenase.html

  1. \rightarrow

Optical_downconverter.html

  1. ω 1 \omega_{1}
  2. ω 2 \omega_{2}
  3. ω 3 = ω 1 - ω 2 \omega_{3}=\omega_{1}-\omega_{2}
  4. ω 1 = ω 2 \omega_{1}=\omega_{2}
  5. E ν = ω , E_{\nu}=\hbar\omega,
  6. ω 3 = ω 1 - ω 2 \omega_{3}=\omega_{1}-\omega_{2}

Optical_modulation_amplitude.html

  1. OMA = P 1 - P 0 \,\text{OMA}=P_{1}-P_{0}\,
  2. P av = ( P 1 + P 0 ) / 2 P_{\,\text{av}}=(P_{1}+P_{0})/2
  3. r e = P 1 / P 0 r_{e}=P_{1}/P_{0}
  4. OMA = 2 P av r e - 1 r e + 1 \,\text{OMA}=2P_{\,\text{av}}\frac{r_{e}-1}{r_{e}+1}
  5. OMA 2 P a v \,\text{OMA}\approx 2P_{av}

Orcinol_2-monooxygenase.html

  1. \rightleftharpoons

Ornithine_cyclodeaminase.html

  1. \rightleftharpoons

Ornithine_racemase.html

  1. \rightleftharpoons

Orotate_reductase_(NADH).html

  1. \rightleftharpoons

Orotate_reductase_(NADPH).html

  1. \rightleftharpoons

Orsellinate_decarboxylase.html

  1. \rightleftharpoons

Orthogonal_Procrustes_problem.html

  1. A A
  2. B B
  3. R R
  4. A A
  5. B B
  6. R = arg min Ω A Ω - B F subject to Ω T Ω = I , R=\arg\min_{\Omega}\|A\Omega-B\|_{F}\quad\mathrm{subject\ to}\quad\Omega^{T}% \Omega=I,
  7. F \|\cdot\|_{F}
  8. M = A T B M=A^{T}B
  9. R R
  10. M = U Σ V T M=U\Sigma V^{T}\,\!
  11. R = U V T . R=UV^{T}.\,\!
  12. R \displaystyle R
  13. M = U Σ V T M=U\Sigma V^{T}
  14. R = U Σ V T , R=U\Sigma^{\prime}V^{T},\,\!
  15. Σ \Sigma^{\prime}\,\!
  16. Σ \Sigma\,\!
  17. sign ( det ( U V T ) ) \operatorname{sign}(\det(UV^{T}))

Orthostochastic_matrix.html

  1. B i j = O i j 2 for i , j = 1 , , n . B_{ij}=O_{ij}^{2}\,\text{ for }i,j=1,\dots,n.\,
  2. B = [ a 1 - a 1 - a a ] B=\begin{bmatrix}a&1-a\\ 1-a&a\end{bmatrix}
  3. O = [ cos ϕ sin ϕ - sin ϕ cos ϕ ] , O=\begin{bmatrix}\cos\phi&\sin\phi\\ -\sin\phi&\cos\phi\end{bmatrix},
  4. cos 2 ϕ = a , \cos^{2}\phi=a,
  5. B i j = O i j 2 . B_{ij}=O_{ij}^{2}.

Oscillating_U-tube.html

  1. ρ = A τ 2 - B \rho=A\cdot\tau^{2}-B

Oscillator_strength.html

  1. f 12 f_{12}
  2. | 1 |1\rangle
  3. | 2 |2\rangle
  4. f 12 = 2 3 m e 2 ( E 2 - E 1 ) α = x , y , z | 1 m 1 | R α | 2 m 2 | 2 , f_{12}=\frac{2}{3}\frac{m_{e}}{\hbar^{2}}(E_{2}-E_{1})\sum_{\alpha=x,y,z}|% \langle 1m_{1}|R_{\alpha}|2m_{2}\rangle|^{2},
  5. m e m_{e}
  6. \hbar
  7. | n , n = |n\rangle,n=
  8. m n m_{n}
  9. E n E_{n}
  10. R x R_{x}
  11. r i , x r_{i,x}
  12. N N
  13. R α = i = 1 N r i , α . R_{\alpha}=\sum_{i=1}^{N}r_{i,\alpha}.
  14. | n m n |nm_{n}\rangle
  15. s y m b o l p symbol{p}
  16. H = 1 2 m s y m b o l p 2 + V ( s y m b o l r ) H=\frac{1}{2m}symbol{p}^{2}+V(symbol{r})
  17. [ H , x ] [H,x]
  18. H H
  19. x n k = - i / m E n - E k ( p x ) n k . x_{nk}=-\frac{i\hbar/m}{E_{n}-E_{k}}(p_{x})_{nk}.
  20. [ p x , x ] [p_{x},x]
  21. x x
  22. n | [ p x , x ] | n = 2 i m k n | n | p x | k | 2 E n - E k . \langle n|[p_{x},x]|n\rangle=\frac{2i\hbar}{m}\sum_{k\neq n}\frac{|\langle n|p% _{x}|k\rangle|^{2}}{E_{n}-E_{k}}.
  23. [ p x , x ] = - i [p_{x},x]=-i\hbar
  24. k n f n k = 1 , f n k = 2 m | n | p x | k | 2 E n - E k , \sum_{k\neq n}f_{nk}=1,\,\,\,\,\,f_{nk}=\frac{2}{m}\frac{|\langle n|p_{x}|k% \rangle|^{2}}{E_{n}-E_{k}},
  25. f n k f_{nk}
  26. n n
  27. k k
  28. k = n k=n
  29. n | p x | n = 0 \langle n|p_{x}|n\rangle=0
  30. H H
  31. E n ( s y m b o l p ) E_{n}(symbol{p})
  32. s y m b o l p symbol{p}
  33. E n ( s y m b o l p ) = s y m b o l p 2 / 2 m * E_{n}(symbol{p})=symbol{p}^{2}/2m^{*}
  34. m * m^{*}
  35. 2 m k n | n | p x | k | 2 E k - E n + m m * = 1. \frac{2}{m}\sum_{k\neq n}\frac{|\langle n|p_{x}|k\rangle|^{2}}{E_{k}-E_{n}}+% \frac{m}{m^{*}}=1.
  36. k n k\neq n
  37. m / m * m/m^{*}
  38. m m
  39. m * m^{*}
  40. n n

Ostrowski–Hadamard_gap_theorem.html

  1. p j + 1 p j > λ . \frac{p_{j+1}}{p_{j}}>\lambda.
  2. f ( z ) = j 𝐍 α j z p j f(z)=\sum_{j\in\mathbf{N}}\alpha_{j}z^{p_{j}}

Other_dimensions_of_the_Discworld.html

  1. p \ell^{p}
  2. Books = Knowledge = Power = Mass × Distance 2 Time 3 . \,\text{Books}=\,\text{Knowledge}=\,\text{Power}=\frac{\,\text{Mass}\times\,% \text{Distance}^{2}}{\,\text{Time}^{3}}.

Oxalate_CoA-transferase.html

  1. \rightleftharpoons

Oxalate_decarboxylase.html

  1. \rightleftharpoons

Oxalate_oxidase.html

  1. \rightleftharpoons

Oxaloacetase.html

  1. \rightleftharpoons

Oxaloacetate_tautomerase.html

  1. \rightleftharpoons

Oxaloglycolate_reductase_(decarboxylating).html

  1. \rightleftharpoons

Oxalomalate_lyase.html

  1. \rightleftharpoons

Oxalyl-CoA_decarboxylase.html

  1. \rightleftharpoons

Oxamate_carbamoyltransferase.html

  1. \rightleftharpoons

Oxidizable_carbon_ratio_dating.html

  1. OCRDATE = OCR × Depth × Mean temperature × Mean rainfall Mean texture × p H × % C × 14.4888 \,\text{OCR}\text{DATE}=\frac{\,\text{OCR}\times\,\text{Depth}\times\,\text{% Mean temperature}\times\,\text{Mean rainfall}}{\,\text{Mean texture}\times% \sqrt{pH}\times\sqrt{\%C}\times 14.4888}

Oxoglutarate_dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Ozsváth–Schücking_metric.html

  1. { x 0 , x 1 , x 2 , x 3 } \{x^{0},x^{1},x^{2},x^{3}\}
  2. e ( 0 ) = 1 2 + ( x 3 ) 2 ( x 3 0 - 1 + 2 ) e_{(0)}=\frac{1}{\sqrt{2+(x^{3})^{2}}}\left(x^{3}\partial_{0}-\partial_{1}+% \partial_{2}\right)
  3. e ( 1 ) = 1 4 + 2 ( x 3 ) 2 [ ( x 3 - 2 + ( x 3 ) 2 ) 0 + ( 1 + ( x 3 ) 2 - x 3 2 + ( x 3 ) 2 ) 1 + 2 ] e_{(1)}=\frac{1}{\sqrt{4+2(x^{3})^{2}}}\left[\left(x^{3}-\sqrt{2+(x^{3})^{2}}% \right)\partial_{0}+\left(1+(x^{3})^{2}-x^{3}\sqrt{2+(x^{3})^{2}}\right)% \partial_{1}+\partial_{2}\right]
  4. e ( 2 ) = 1 4 + 2 ( x 3 ) 2 [ ( x 3 + 2 + ( x 3 ) 2 ) 0 + ( 1 + ( x 3 ) 2 + x 3 2 + ( x 3 ) 2 ) 1 + 2 ] e_{(2)}=\frac{1}{\sqrt{4+2(x^{3})^{2}}}\left[\left(x^{3}+\sqrt{2+(x^{3})^{2}}% \right)\partial_{0}+\left(1+(x^{3})^{2}+x^{3}\sqrt{2+(x^{3})^{2}}\right)% \partial_{1}+\partial_{2}\right]
  5. e ( 3 ) = 3 e_{(3)}=\partial_{3}
  6. d τ 2 = - ( d x 0 ) 2 + 4 ( x 3 ) ( d x 0 ) ( d x 2 ) - 2 ( d x 1 ) ( d x 2 ) - 2 ( x 3 ) 2 ( d x 2 ) 2 - ( d x 3 ) 2 . {d\tau}^{2}=-(dx^{0})^{2}+4(x^{3})(dx^{0})(dx^{2})-2(dx^{1})(dx^{2})-2(x^{3})^% {2}(dx^{2})^{2}-(dx^{3})^{2}.
  7. R 0202 = - 1 , R_{0202}=-1,
  8. d τ 2 = [ ( x 2 - y 2 ) cos ( 2 u ) + 2 x y sin ( 2 u ) ] d u 2 - 2 d u d v - d x 2 - d y 2 d\tau^{2}=[(x^{2}-y^{2})\cos(2u)+2xy\sin(2u)]du^{2}-2dudv-dx^{2}-dy^{2}

P-benzoquinone_reductase_(NADPH).html

  1. \rightleftharpoons

Packing_dimension.html

  1. P 0 s ( S ) = lim δ 0 sup { i I diam ( B i ) s | { B i } i I is a countable collection of pairwise disjoint closed balls with diameters δ and centres in S } . P_{0}^{s}(S)=\lim_{\delta\downarrow 0}\sup\left\{\left.\sum_{i\in I}\mathrm{% diam}(B_{i})^{s}\right|\begin{matrix}\{B_{i}\}_{i\in I}\,\text{ is a countable% collection}\\ \,\text{of pairwise disjoint closed balls with}\\ \,\text{diameters }\leq\delta\,\text{ and centres in }S\end{matrix}\right\}.
  2. P s ( S ) = inf { j J P 0 s ( S j ) | S j J S j , J countable } , P^{s}(S)=\inf\left\{\left.\sum_{j\in J}P_{0}^{s}(S_{j})\right|S\subseteq% \bigcup_{j\in J}S_{j},J\,\text{ countable}\right\},
  3. dim P ( S ) \displaystyle\dim_{\mathrm{P}}(S)
  4. ( a n ) (a_{n})
  5. a 0 = 1 a_{0}=1
  6. 0 < a n + 1 < a n / 2 0<a_{n+1}<a_{n}/2
  7. E 0 E 1 E 2 E_{0}\supset E_{1}\supset E_{2}\supset\cdots
  8. E 0 = [ 0 , 1 ] E_{0}=[0,1]
  9. E n E_{n}
  10. a n a_{n}
  11. a n - 2 a n + 1 a_{n}-2a_{n+1}
  12. a n + 1 a_{n+1}
  13. E n + 1 E_{n+1}
  14. K = n E n K=\bigcap_{n}E_{n}
  15. K K
  16. K K
  17. a n = 3 - n a_{n}=3^{-n}
  18. K K
  19. dim H ( K ) \displaystyle\dim_{\mathrm{H}}(K)
  20. 0 d 1 d 2 1 0\leq d_{1}\leq d_{2}\leq 1
  21. ( a n ) (a_{n})
  22. K K
  23. d 1 d_{1}
  24. d 2 d_{2}
  25. P 0 h ( S ) = lim δ 0 sup { i I h ( diam ( B i ) ) | { B i } i I is a countable collection of pairwise disjoint balls with diameters δ and centres in S } P_{0}^{h}(S)=\lim_{\delta\downarrow 0}\sup\left\{\left.\sum_{i\in I}h\big(% \mathrm{diam}(B_{i})\big)\right|\begin{matrix}\{B_{i}\}_{i\in I}\,\text{ is a % countable collection}\\ \,\text{of pairwise disjoint balls with}\\ \,\text{diameters }\leq\delta\,\text{ and centres in }S\end{matrix}\right\}
  26. P h ( S ) = inf { j J P 0 h ( S j ) | S j J S j , J countable } . P^{h}(S)=\inf\left\{\left.\sum_{j\in J}P_{0}^{h}(S_{j})\right|S\subseteq% \bigcup_{j\in J}S_{j},J\,\text{ countable}\right\}.
  27. dim P ( S ) = dim ¯ MB ( S ) . \dim_{\mathrm{P}}(S)=\overline{\dim}_{\mathrm{MB}}(S).

Padua_points.html

  1. [ - 1 , 1 ] × [ - 1 , 1 ] 2 \scriptstyle[-1,1]\times[-1,1]\subset\mathbb{R}^{2}
  2. n n
  3. s s
  4. Pad n s = { ξ = ( ξ 1 , ξ 2 ) } = { γ s ( k π n ( n + 1 ) ) , k = 0 , , n ( n + 1 ) } . \,\text{Pad}_{n}^{s}=\{\mathbf{\xi}=(\xi_{1},\xi_{2})\}=\left\{\gamma_{s}\left% (\frac{k\pi}{n(n+1)}\right),k=0,\ldots,n(n+1)\right\}.
  5. [ - 1 , 1 ] 2 [-1,1]^{2}
  6. Pad n s \scriptstyle\,\text{Pad}_{n}^{s}
  7. | Pad n s | = N = ( n + 1 ) ( n + 2 ) 2 \scriptstyle|\,\text{Pad}_{n}^{s}|=N=\frac{(n+1)(n+2)}{2}
  8. [ - 1 , 1 ] 2 [-1,1]^{2}
  9. 2 n - 1 2n-1
  10. [ 0 , 2 π ] [0,2\pi]
  11. γ 1 ( t ) = [ - cos ( ( n + 1 ) t ) , - cos ( n t ) ] , t [ 0 , π ] . \gamma_{1}(t)=[-\cos((n+1)t),-\cos(nt)],\quad t\in[0,\pi].
  12. Pad n 1 = { ξ = ( μ j , η k ) , 0 j n ; 1 k n 2 + 1 + δ j } , \,\text{Pad}_{n}^{1}=\{\mathbf{\xi}=(\mu_{j},\eta_{k}),0\leq j\leq n;1\leq k% \leq\lfloor\frac{n}{2}\rfloor+1+\delta_{j}\},
  13. δ j = 0 \delta_{j}=0
  14. n n
  15. j j
  16. δ j = 1 \delta_{j}=1
  17. n n
  18. k k
  19. μ j = cos ( j π n ) , η k = { cos ( ( 2 k - 2 ) π n + 1 ) j odd cos ( ( 2 k - 1 ) π n + 1 ) j even. \mu_{j}=\cos\left(\frac{j\pi}{n}\right),\eta_{k}=\begin{cases}\cos\left(\frac{% (2k-2)\pi}{n+1}\right)&j\mbox{ odd}\\ \cos\left(\frac{(2k-1)\pi}{n+1}\right)&j\mbox{ even.}\end{cases}
  20. n n
  21. n n
  22. γ 2 ( t ) = [ - cos ( n t ) , - cos ( ( n + 1 ) t ) ] , t [ 0 , π ] , \gamma_{2}(t)=[-\cos(nt),-\cos((n+1)t)],\quad t\in[0,\pi],
  23. n n
  24. n n
  25. γ 3 ( t ) = [ cos ( ( n + 1 ) t ) , cos ( n t ) ] , t [ 0 , π ] , \gamma_{3}(t)=[\cos((n+1)t),\cos(nt)],\quad t\in[0,\pi],
  26. n n
  27. n n
  28. γ 4 ( t ) = [ cos ( n t ) , cos ( ( n + 1 ) t ) ] , t [ 0 , π ] , \gamma_{4}(t)=[\cos(nt),\cos((n+1)t)],\quad t\in[0,\pi],
  29. n n
  30. n n
  31. K n ( 𝐱 , 𝐲 ) \scriptstyle K_{n}(\mathbf{x},\mathbf{y})
  32. 𝐱 = ( x 1 , x 2 ) \scriptstyle\mathbf{x}=(x_{1},x_{2})
  33. 𝐲 = ( y 1 , y 2 ) \scriptstyle\mathbf{y}=(y_{1},y_{2})
  34. Π n 2 ( [ - 1 , 1 ] 2 ) \scriptstyle\Pi_{n}^{2}([-1,1]^{2})
  35. f , g = 1 π 2 [ - 1 , 1 ] 2 f ( x 1 , x 2 ) g ( x 1 , x 2 ) d x 1 1 - x 1 2 d x 2 1 - x 2 2 \langle f,g\rangle=\frac{1}{\pi^{2}}\int_{[-1,1]^{2}}f(x_{1},x_{2})g(x_{1},x_{% 2})\frac{dx_{1}}{\sqrt{1-x_{1}^{2}}}\frac{dx_{2}}{\sqrt{1-x_{2}^{2}}}
  36. K n ( 𝐱 , 𝐲 ) = k = 0 n j = 0 k T ^ j ( x 1 ) T ^ k - j ( x 2 ) T ^ j ( y 1 ) T ^ k - j ( y 2 ) K_{n}(\mathbf{x},\mathbf{y})=\sum_{k=0}^{n}\sum_{j=0}^{k}\hat{T}_{j}(x_{1})% \hat{T}_{k-j}(x_{2})\hat{T}_{j}(y_{1})\hat{T}_{k-j}(y_{2})
  37. T ^ j \scriptstyle\hat{T}_{j}
  38. j j
  39. T ^ 0 = T 0 \scriptstyle\hat{T}_{0}=T_{0}
  40. T ^ p = 2 T p \scriptstyle\hat{T}_{p}=\sqrt{2}T_{p}
  41. T p ( ) = cos ( p arccos ( ) ) \scriptstyle T_{p}(\cdot)=\cos(p\arccos(\cdot))
  42. p p
  43. Pad n s = { ξ = ( ξ 1 , ξ 2 ) } \scriptstyle\,\text{Pad}_{n}^{s}=\{\mathbf{\xi}=(\xi_{1},\xi_{2})\}
  44. s = { 1 , 2 , 3 , 4 } s=\{1,2,3,4\}
  45. n n
  46. f : [ - 1 , 1 ] 2 2 \scriptstyle f\colon[-1,1]^{2}\to\mathbb{R}^{2}
  47. 𝐱 [ - 1 , 1 ] 2 \scriptstyle\mathbf{x}\in[-1,1]^{2}
  48. n s f ( 𝐱 ) = ξ Pad n s f ( ξ ) L ξ s ( 𝐱 ) \mathcal{L}_{n}^{s}f(\mathbf{x})=\sum_{\mathbf{\xi}\in\,\text{Pad}_{n}^{s}}f(% \mathbf{\xi})L^{s}_{\mathbf{\xi}}(\mathbf{x})
  49. L ξ s ( 𝐱 ) \scriptstyle L^{s}_{\mathbf{\xi}}(\mathbf{x})
  50. L ξ s ( 𝐱 ) = w ξ ( K n ( ξ , 𝐱 ) - T n ( ξ i ) T n ( x i ) ) , s = 1 , 2 , 3 , 4 , i = 2 - ( s mod 2 ) . L^{s}_{\mathbf{\xi}}(\mathbf{x})=w_{\mathbf{\xi}}(K_{n}(\mathbf{\xi},\mathbf{x% })-T_{n}(\xi_{i})T_{n}(x_{i})),\quad s=1,2,3,4,\quad i=2-(s\mod 2).
  51. w ξ \scriptstyle w_{\mathbf{\xi}}
  52. w ξ = 1 n ( n + 1 ) { 1 2 if ξ is a vertex point 1 if ξ is an edge point 2 if ξ is an interior point. w_{\mathbf{\xi}}=\frac{1}{n(n+1)}\cdot\begin{cases}\frac{1}{2}\,\text{ if }% \mathbf{\xi}\,\text{ is a vertex point}\\ 1\,\text{ if }\mathbf{\xi}\,\text{ is an edge point}\\ 2\,\text{ if }\mathbf{\xi}\,\text{ is an interior point.}\end{cases}

Panjer_recursion.html

  1. S = i = 1 N X i S=\sum_{i=1}^{N}X_{i}\,
  2. N N\,
  3. X i X_{i}\,
  4. S = i = 1 N X i S=\sum_{i=1}^{N}X_{i}\,
  5. N N\,
  6. X i X_{i}\,
  7. X i X_{i}\,
  8. N N\,
  9. X i X_{i}\,
  10. h 0 h\mathbb{N}_{0}\,
  11. h > 0 h>0\,
  12. f k = P [ X i = h k ] . f_{k}=P[X_{i}=hk].\,
  13. X i X_{i}\,
  14. P [ N = k ] = p k = ( a + b k ) p k - 1 , k 1. P[N=k]=p_{k}=(a+\frac{b}{k})\cdot p_{k-1},~{}~{}k\geq 1.\,
  15. a + b 0 a+b\geq 0\,
  16. p 0 p_{0}\,
  17. k = 0 p k = 1. \sum_{k=0}^{\infty}p_{k}=1.\,
  18. W N ( x ) W_{N}(x)\,
  19. n n
  20. g k = P [ S = h k ] g_{k}=P[S=hk]\,
  21. g 0 = W N ( f 0 ) g_{0}=W_{N}(f_{0})\,
  22. g 0 = p 0 exp ( f 0 b ) if a = 0 , g_{0}=p_{0}\cdot\exp(f_{0}b)\,\text{ if }a=0,\,
  23. g 0 = p 0 ( 1 - f 0 a ) 1 + b / a for a 0 , g_{0}=\frac{p_{0}}{(1-f_{0}a)^{1+b/a}}\,\text{ for }a\neq 0,\,
  24. g k = 1 1 - f 0 a j = 1 k ( a + b j k ) f j g k - j . g_{k}=\frac{1}{1-f_{0}a}\sum_{j=1}^{k}\left(a+\frac{b\cdot j}{k}\right)\cdot f% _{j}\cdot g_{k-j}.\,
  25. S = i = 1 N X i \scriptstyle S\,=\,\sum_{i=1}^{N}X_{i}
  26. N NegBin ( 3.5 , 0.3 ) \scriptstyle N\,\sim\,\,\text{NegBin}(3.5,0.3)\,
  27. X Frechet ( 1.7 , 1 ) \scriptstyle X\,\sim\,\,\text{Frechet}(1.7,1)

Pantoate_4-dehydrogenase.html

  1. \rightleftharpoons

Pantothenoylcysteine_decarboxylase.html

  1. \rightleftharpoons

Papyrus_104.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}

Papyrus_3.html

  1. 𝔓 \mathfrak{P}

Papyrus_4.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Papyrus_42.html

  1. 𝔓 \mathfrak{P}

Papyrus_75.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}

Paradoxes_of_material_implication.html

  1. ( ¬ p p ) q (\neg p\land p)\to q
  2. p ( q p ) p\to(q\to p)
  3. ¬ p ( p q ) \neg p\to(p\to q)
  4. p ( q ¬ q ) p\to(q\lor\neg q)
  5. ( p q ) ( q r ) (p\to q)\lor(q\to r)
  6. ¬ ( p q ) ( p ¬ q ) \neg(p\to q)\to(p\land\neg q)
  7. ( p q ) p (p\land q)\to p
  8. ( p q ) ( r s ) ( p s ) ( r q ) (p\to q)\land(r\to s)\ \vdash\ (p\to s)\lor(r\to q)
  9. ( p q ) r ( p r ) ( q r ) (p\land q)\to r\ \vdash\ (p\to r)\lor(q\to r)

Partial_current.html

  1. I = Σ I a , i + Σ I c , i I=\Sigma I_{a,i}+\Sigma I_{c,i}

Particle_size.html

  1. D = 2 3 V 4 π 3 D=2\sqrt[3]{\frac{3V}{4\pi}}
  2. D D
  3. V V
  4. D = 2 3 W 4 π d g 3 D=2\sqrt[3]{\frac{3W}{4\pi dg}}
  5. D D
  6. W W
  7. d d
  8. g g
  9. D = 2 A 4 π 2 D=2\sqrt[2]{\frac{A}{4\pi}}
  10. D D
  11. A A

Penman–Monteith_equation.html

  1. Δ \Delta
  2. c p c_{p}
  3. δ q \delta_{q}
  4. λ v E = Δ ( R n - G ) + ρ a c p ( δ e ) g a Δ + γ ( 1 + g a / g s ) Energy flux rate E T o = Δ ( R n - G ) + ρ a c p ( δ e ) g a ( Δ + γ ( 1 + g a / g s ) ) L v Volume flux rate \overset{\,\text{Energy flux rate}}{\lambda_{v}E=\frac{\Delta(R_{n}-G)+\rho_{a% }c_{p}\left(\delta e\right)g_{a}}{\Delta+\gamma\left(1+g_{a}/g_{s}\right)}}~{}% \iff~{}\overset{\,\text{Volume flux rate}}{ET_{o}=\frac{\Delta(R_{n}-G)+\rho_{% a}c_{p}\left(\delta e\right)g_{a}}{\left(\Delta+\gamma\left(1+g_{a}/g_{s}% \right)\right)L_{v}}}
  5. g a = 1 r a & g s = 1 r s = 1 r c g_{a}=\tfrac{1}{r_{a}}~{}~{}\And~{}~{}g_{s}=\tfrac{1}{r_{s}}=\tfrac{1}{r_{c}}
  6. g s g_{s}
  7. g s g_{s}
  8. δ q \delta q
  9. Δ \Delta
  10. α \alpha
  11. α \alpha
  12. α \alpha
  13. α \alpha

Peptide-aspartate_beta-dioxygenase.html

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Peptide-methionine_(R)-S-oxide_reductase.html

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Peptide-tryptophan_2,3-dioxygenase.html

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Peptidylamidoglycolate_lyase.html

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Peptidylglycine_monooxygenase.html

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Peridynamics.html

  1. ρ ( x ) u ¨ ( x , t ) = R f ( u ( x , t ) - u ( x , t ) , x - x , x ) d V x + b ( x , t ) \rho(x)\ddot{u}(x,t)=\int_{R}f(u(x^{\prime},t)-u(x,t),x^{\prime}-x,x)dV_{x^{% \prime}}+b(x,t)
  2. x x
  3. R R
  4. t t
  5. u u
  6. ρ \rho
  7. x x^{\prime}
  8. f f
  9. x x^{\prime}
  10. x x
  11. x x^{\prime}
  12. x x
  13. f f
  14. f f
  15. x x
  16. x x^{\prime}
  17. f f
  18. x x^{\prime}
  19. x x
  20. u = u ( x , t ) u=u(x,t)
  21. u = u ( x , t ) u^{\prime}=u(x^{\prime},t)
  22. f f
  23. f ( u - u , x - x , x ) = - f ( u - u , x - x , x ) \displaystyle f(u-u^{\prime},x-x^{\prime},x^{\prime})=-f(u^{\prime}-u,x^{% \prime}-x,x)
  24. x , x , u , u x,x^{\prime},u,u^{\prime}
  25. x x
  26. x x^{\prime}
  27. x x^{\prime}
  28. x x
  29. f f
  30. x x
  31. x x^{\prime}
  32. ( ( x + u ) - ( x + u ) ) × f ( u - u , x - x , x ) = 0. \displaystyle((x^{\prime}+u^{\prime})-(x+u))\times f(u^{\prime}-u,x^{\prime}-x% ,x)=0.
  33. | f | |f|
  34. e e
  35. e = | ( x + u ) - ( x + u ) | - | x - x | . \displaystyle e=|(x^{\prime}+u^{\prime})-(x+u)|-|x^{\prime}-x|.

Perillyl-alcohol_dehydrogenase.html

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Petromyzonol_sulfotransferase.html

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Phenol_2-monooxygenase.html

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Phenol_O-methyltransferase.html

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Phenylacetaldehyde_dehydrogenase.html

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Phenylacetaldoxime_dehydratase.html

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Phenylacetone_monooxygenase.html

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Phenylacetyl-CoA_dehydrogenase.html

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Phenylalanine_2-monooxygenase.html

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Phenylalanine_ammonia-lyase.html

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Phenylalanine_decarboxylase.html

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Phenylalanine_dehydrogenase.html

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Phenylglyoxylate_dehydrogenase_(acylating).html

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Phenylpyruvate_decarboxylase.html

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Phenylpyruvate_tautomerase.html

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Phenylserine_aldolase.html

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Phloretin_hydrolase.html

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Phloroglucinol_reductase.html

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Phosphatidyl-N-methylethanolamine_N-methyltransferase.html

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Phosphatidylcholine_12-monooxygenase.html

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Phosphatidylcholine_desaturase.html

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Phosphatidylinositol_diacylglycerol-lyase.html

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Phosphoacetylglucosamine_mutase.html

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Phosphoadenylyl-sulfate_reductase_(thioredoxin).html

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Phosphoamidase.html

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Phosphoenolpyruvate_mutase.html

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Phosphoethanolamine_N-methyltransferase.html

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Phosphoglucomutase_(glucose-cofactor).html

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Phosphogluconate_2-dehydrogenase.html

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Phosphogluconate_dehydrogenase_(decarboxylating).html

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Phosphoglucosamine_mutase.html

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Phospholipid-hydroperoxide_glutathione_peroxidase.html

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Phosphomannomutase.html

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Phosphonate_dehydrogenase.html

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Phosphonoacetaldehyde_hydrolase.html

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Phosphonoacetate_hydrolase.html

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Phosphonopyruvate_hydrolase.html

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Phosphopentomutase.html

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Phosphoribosylaminoimidazolecarboxamide_formyltransferase.html

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Phosphoribosylanthranilate_isomerase.html

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Phosphosulfolactate_synthase.html

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Phthalate_4,5-cis-dihydrodiol_dehydrogenase.html

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Phthalate_4,5-dioxygenase.html

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Phycocyanobilin:ferredoxin_oxidoreductase.html

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Phycoerythrobilin:ferredoxin_oxidoreductase.html

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Phylloquinone_monooxygenase_(2,3-epoxidizing).html

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Phytanoyl-CoA_dioxygenase.html

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Phytochromobilin:ferredoxin_oxidoreductase.html

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Pimeloyl-CoA_dehydrogenase.html

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Planarity.html

  1. L L
  2. ( L 2 ) = L ( L - 1 ) 2 {\textstyle\left({{L}\atop{2}}\right)}=\tfrac{L(L-1)}{2}
  3. L - 1 L-1
  4. L ( L - 2 ) L(L-2)
  5. L - 2 L-2
  6. L = 4 L=4
  7. L ( L - 1 ) / 2 = 6 L(L-1)/2=6
  8. L ( L - 2 ) = 8 L(L-2)=8
  9. L L
  10. L L
  11. 2 L - 1 2L-1
  12. O ( L 2 ) O(L^{2})
  13. x x
  14. O ( L 2 log L ) O(L^{2}\log L)
  15. n ϵ n^{\epsilon}
  16. n n
  17. ϵ \epsilon

Plane_wave_expansion_method.html

  1. 1 ϵ r = m = - + K m ϵ r e - i G . r \frac{1}{\epsilon_{r}}=\sum_{m=-\infty}^{+\infty}K_{m}^{\epsilon_{r}}e^{-i\vec% {G}.\vec{r}}
  2. E ( ω , r ) = n = - + K n E y e - i G . r e - i k r E(\omega,\vec{r})=\sum_{n=-\infty}^{+\infty}K_{n}^{E_{y}}e^{-i\vec{G}.\vec{r}}% e^{-i\vec{k}\vec{r}}
  3. G \vec{G}
  4. ± N m a x \pm Nmax
  5. 1 ϵ ( r ) × × E ( r , ω ) = ( ω c ) 2 E ( r , ω ) \frac{1}{\epsilon(\vec{r})}\nabla\times\nabla\times E(\vec{r},\omega)=\left(% \frac{\omega}{c}\right)^{2}E(\vec{r},\omega)
  6. 1 ϵ r = m = - + K m ϵ r e - i 2 π m a z \frac{1}{\epsilon_{r}}=\sum_{m=-\infty}^{+\infty}K_{m}^{\epsilon_{r}}e^{-i% \frac{2\pi m}{a}z}
  7. E ( ω , r ) = n = - + K n E y e - i 2 π n a z e - i k r E(\omega,\vec{r})=\sum_{n=-\infty}^{+\infty}K_{n}^{E_{y}}e^{-i\frac{2\pi n}{a}% z}e^{-i\vec{k}\vec{r}}
  8. n ( 2 π n a + k z ) ( 2 π m a + k z ) K m - n ϵ r K n E y = ω 2 c 2 K m E y \sum_{n}{\left(\frac{2\pi n}{a}+k_{z}\right)\left(\frac{2\pi m}{a}+k_{z}\right% )K_{m-n}^{\epsilon_{r}}K_{n}^{E_{y}}}=\frac{\omega^{2}}{c^{2}}K_{m}^{E_{y}}

Plasmanylethanolamine_desaturase.html

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Pollaczek–Khinchine_formula.html

  1. L = ρ + ρ 2 + λ 2 Var ( S ) 2 ( 1 - ρ ) L=\rho+\frac{\rho^{2}+\lambda^{2}\operatorname{Var}(S)}{2(1-\rho)}
  2. λ \lambda
  3. 1 / μ 1/\mu
  4. ρ = λ / μ \rho=\lambda/\mu
  5. ρ < 1 \rho<1
  6. λ a \lambda_{a}
  7. λ s \lambda_{s}
  8. W = W + μ - 1 W=W^{\prime}+\mu^{-1}
  9. W W^{\prime}
  10. μ \mu
  11. L = λ W L=\lambda W
  12. λ \lambda
  13. W = ρ + λ μ Var ( S ) 2 ( μ - λ ) + μ - 1 . W=\frac{\rho+\lambda\mu\,\text{Var}(S)}{2(\mu-\lambda)}+\mu^{-1}.
  14. W = L λ - μ - 1 = ρ + λ μ Var ( S ) 2 ( μ - λ ) . W^{\prime}=\frac{L}{\lambda}-\mu^{-1}=\frac{\rho+\lambda\mu\,\text{Var}(S)}{2(% \mu-\lambda)}.
  15. π ( z ) = ( 1 - z ) ( 1 - ρ ) g ( λ ( 1 - z ) ) g ( λ ( 1 - z ) ) - z \pi(z)=\frac{(1-z)(1-\rho)g(\lambda(1-z))}{g(\lambda(1-z))-z}
  16. W ( s ) = ( 1 - ρ ) s g ( s ) s - λ ( 1 - g ( s ) ) W^{\ast}(s)=\frac{(1-\rho)sg(s)}{s-\lambda(1-g(s))}

Pollard's_kangaroo_algorithm.html

  1. G G
  2. n n
  3. α \alpha
  4. x x
  5. β \beta
  6. α \alpha
  7. x Z n x\in Z_{n}
  8. α x = β \alpha^{x}=\beta
  9. x x
  10. { a , , b } Z n \{a,\ldots,b\}\subset Z_{n}
  11. a = 0 a=0
  12. b = n - 1 b=n-1
  13. S S
  14. f : G S f:G\rightarrow S
  15. N N
  16. { x 0 , x 1 , , x N } \{x_{0},x_{1},\ldots,x_{N}\}
  17. x 0 = α b x_{0}=\alpha^{b}\,
  18. x i + 1 = x i α f ( x i ) for i = 0 , 1 , , N - 1 x_{i+1}=x_{i}\alpha^{f(x_{i})}\mbox{ for }~{}i=0,1,\ldots,N-1
  19. d = i = 0 N - 1 f ( x i ) d=\sum_{i=0}^{N-1}f(x_{i})
  20. x N = x 0 α d = α b + d . x_{N}=x_{0}\alpha^{d}=\alpha^{b+d}\,.
  21. { y 0 , y 1 , } \{y_{0},y_{1},\ldots\}
  22. y 0 = β y_{0}=\beta\,
  23. y i + 1 = y i α f ( y i ) for i = 0 , 1 , , N - 1 y_{i+1}=y_{i}\alpha^{f(y_{i})}\mbox{ for }~{}i=0,1,\ldots,N-1
  24. { d 0 , d 1 , } \{d_{0},d_{1},\ldots\}
  25. d n = i = 0 n - 1 f ( y i ) d_{n}=\sum_{i=0}^{n-1}f(y_{i})
  26. y i = y 0 α d i = β α d i for i = 0 , 1 , , N - 1 y_{i}=y_{0}\alpha^{d_{i}}=\beta\alpha^{d_{i}}\mbox{ for }~{}i=0,1,\ldots,N-1
  27. { y i } \{y_{i}\}
  28. { d i } \{d_{i}\}
  29. y j = x N y_{j}=x_{N}
  30. j j
  31. { x i } \{x_{i}\}
  32. { y j } \{y_{j}\}
  33. x N = y j α b + d = β α d j β = α b + d - d j ( mod n ) x b + d - d j ( mod n ) x_{N}=y_{j}\Rightarrow\alpha^{b+d}=\beta\alpha^{d_{j}}\Rightarrow\beta=\alpha^% {b+d-d_{j}}\;\;(\mathop{{\rm mod}}n)\Rightarrow x\equiv b+d-d_{j}\;\;(\mathop{% {\rm mod}}n)
  34. d i > b - a + d d_{i}>b-a+d
  35. x x
  36. S S
  37. f f
  38. O ( b - a ) {\scriptstyle O(\sqrt{b-a})}
  39. O ( b - a ) = O ( 2 1 2 log ( b - a ) ) {\scriptstyle O(\sqrt{b-a})=O(2^{\frac{1}{2}\log(b-a)})}
  40. λ \lambda
  41. { x i } \{x_{i}\}
  42. { y i } \{y_{i}\}

Polyamine_oxidase.html

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Polyenoic_fatty_acid_isomerase.html

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Polysaccharide_O-methyltransferase.html

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Polyvinyl-alcohol_dehydrogenase_(acceptor).html

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Polyvinyl-alcohol_oxidase.html

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Porous_set.html

  1. B ( y , α r ) B ( x , r ) E . B(y,\alpha r)\subseteq B(x,r)\setminus E.
  2. B ( y , s ) B ( x , r ) E . B(y,s)\subseteq B(x,r)\setminus E.

Portal:Chemistry::Useful_equations_and_links.html

  1. P 1 V 1 T 1 = P 2 V 2 T 2 . \frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}.
  2. P V = n R T PV=nRT\,
  3. P V = N k T PV=NkT\,
  4. Rate 1 Rate 2 = M 2 M 1 {\mbox{Rate}~{}_{1}\over\mbox{Rate}~{}_{2}}=\sqrt{M_{2}\over M_{1}}
  5. P = N m v r m s 2 3 V P={Nmv_{rms}^{2}\over 3V}
  6. P = 1 3 ρ v r m s 2 P={1\over 3}\rho\ v_{rms}^{2}
  7. v r m s 2 = 3 R T molar mass v_{rms}^{2}=\frac{3RT}{\mbox{molar mass}~{}}
  8. P t o t a l = i = 1 n p i P_{total}=\sum_{i=1}^{n}{p_{i}}
  9. P t o t a l = p 1 + p 2 + + p n P_{total}=p_{1}+p_{2}+\cdots+p_{n}
  10. p 1 , p 2 , p n p_{1},\ p_{2},\ p_{n}
  11. P i = P t o t a l m i \ P_{i}=P_{total}m_{i}
  12. m i = m_{i}\ =
  13. ( p + n 2 a V 2 ) ( V - n b ) = n R T \left(p+\frac{n^{2}a}{V^{2}}\right)\left(V-nb\right)=nRT
  14. a = N A 2 a a=N_{\mathrm{A}}^{2}a^{\prime}
  15. b = N A b \,b=N_{\mathrm{A}}b^{\prime}
  16. R = N A k \,R=N_{\mathrm{A}}k
  17. l o g P o c t / w a t = l o g ( [ s o l u t e ] o c t a n o l [ s o l u t e ] w a t e r u n - i o n i z e d ) log\ P_{oct/wat}=log\Bigg(\frac{\big[solute\big]_{octanol}}{\big[solute\big]_{% water}^{un-ionized}}\Bigg)
  18. d N d t = - λ N . \frac{dN}{dt}=-\lambda N.
  19. N = N 0 e - λ t N=N_{0}e^{-\lambda t}\,
  20. N 0 N_{0}
  21. t = 0 t=0
  22. N N
  23. t t
  24. λ = {\lambda}=
  25. t a v g t_{avg}\,
  26. 1 λ \frac{1}{\lambda}
  27. t 1 2 t_{\frac{1}{2}}\,
  28. t a v g ln 2 t_{avg}\cdot\ln 2
  29. t ( B P ) = 1 λ ln N N 0 t(BP)=\frac{1}{\lambda}{\ln\frac{N}{N_{0}}}
  30. t ( B P ) = - 1 λ ln N N 0 t(BP)=-\frac{1}{\lambda}{\ln\frac{N}{N_{0}}}
  31. A B B C A C A\sim B\wedge B\sim C\Rightarrow A\sim C
  32. d U = δ Q - δ W \mathrm{d}U=\delta Q-\delta W\,
  33. δ Q T 0 \oint\frac{\delta Q}{T}\geq 0
  34. T 0 , S C T\rightarrow 0,S\rightarrow C
  35. d W = - P d V dW=-PdV\,
  36. W = - V i V f P d V W=-\int_{V_{i}}^{V_{f}}P\,dV
  37. d S = δ Q T dS=\frac{\delta Q}{T}\!
  38. δ Q \delta Q
  39. δ Q \delta Q
  40. d Q dQ
  41. H = E + p V , H=E+pV,\,
  42. H = U + p V , H=U+pV,\,
  43. Δ U = Q + W + W \Delta U=Q+W+W^{\prime}\,
  44. d U = δ Q - d W = T d S - p d V \mathrm{d}U=\delta Q-dW=T\mathrm{d}S-p\mathrm{d}V\,
  45. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S\,
  46. Δ G = - R T ln K \Delta G^{\circ}=-RT\ln K\,
  47. Δ G = Δ G + R T ln Q \Delta G=\Delta G^{\circ}+RT\ln Q\,
  48. Δ G = - n F Δ E \Delta G=-nF\Delta E\,
  49. n F Δ E = R T ln K nF\Delta E^{\circ}=RT\ln K\,
  50. n F Δ E = n F Δ E - R T ln Q nF\Delta E=nF\Delta E^{\circ}-RT\ln Q\,\,
  51. Δ E = Δ E - R T n F ln Q \Delta E=\Delta E^{\circ}-\frac{RT}{nF}\ln Q\,\,
  52. K e q = e - Δ G R T K_{eq}=e^{-\frac{\Delta G^{\circ}}{RT}}
  53. Δ G = - R T ( ln K e q ) = - 2.303 R T ( log K e q ) \Delta G^{\circ}=-RT(\ln K_{eq})=-2.303RT(\log K_{eq})
  54. A U - T S A\equiv U-TS\,
  55. d U = δ Q - δ W {\rm d}U=\delta Q-\delta W\,
  56. U U
  57. δ Q \delta Q
  58. δ W = p d V \delta W=p{\rm d}V
  59. δ Q = T d S \delta Q=T{\rm d}S
  60. A A
  61. d A = d U - ( T d S + S d T ) {\rm d}A={\rm d}U-(T{\rm d}S+S{\rm d}T)\,
  62. = ( T d S - p d V ) - T d S - S d T =(T{\rm d}S-p\,{\rm d}V)-T{\rm d}S-S{\rm d}T\,
  63. = - p d V - S d T =-p\,{\rm d}V-S{\rm d}T\,
  64. ( T V ) S = - ( p S ) V = 2 U S V \left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial p}{% \partial S}\right)_{V}\qquad=\frac{\partial^{2}U}{\partial S\partial V}
  65. ( T p ) S = + ( V S ) p = 2 H S p \left(\frac{\partial T}{\partial p}\right)_{S}=+\left(\frac{\partial V}{% \partial S}\right)_{p}\qquad=\frac{\partial^{2}H}{\partial S\partial p}
  66. ( S V ) T = + ( p T ) V = - 2 A T V \left(\frac{\partial S}{\partial V}\right)_{T}=+\left(\frac{\partial p}{% \partial T}\right)_{V}\qquad=-\frac{\partial^{2}A}{\partial T\partial V}
  67. ( S p ) T = - ( V T ) p = 2 G T P \left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{% \partial T}\right)_{p}\qquad=\frac{\partial^{2}G}{\partial T\partial P}
  68. U ( S , V ) U(S,V)\,
  69. H ( S , p ) H(S,p)\,
  70. A ( T , V ) A(T,V)\,
  71. G ( T , p ) G(T,p)\,
  72. N i N = g i e - E i / k B T Z ( T ) {{N_{i}}\over{N}}={{g_{i}e^{-E_{i}/k_{B}T}}\over{Z(T)}}
  73. k B k_{B}
  74. g i g_{i}
  75. E i E_{i}
  76. N = i N i N=\sum_{i}N_{i}\,
  77. Z ( T ) = i g i e - E i / k B T . Z(T)=\sum_{i}g_{i}e^{-E_{i}/k_{B}T}.
  78. - d [ A ] d t = k -\frac{d[A]}{dt}=k
  79. - d [ A ] d t = k [ A ] -\frac{d[A]}{dt}=k[A]
  80. - d [ A ] d t = k [ A ] 2 -\frac{d[A]}{dt}=k[A]^{2}
  81. - d [ A ] d t = k [ A ] n -\frac{d[A]}{dt}=k[A]^{n}
  82. [ A ] = [ A ] 0 - k t \ [A]=[A]_{0}-kt
  83. [ A ] = [ A ] 0 e - k t \ [A]=[A]_{0}e^{-kt}
  84. 1 [ A ] = 1 [ A ] 0 + k t \frac{1}{[A]}=\frac{1}{[A]_{0}}+kt
  85. 1 [ A ] n - 1 = 1 [ A ] 0 n - 1 + ( n - 1 ) k t \frac{1}{[A]^{n-1}}=\frac{1}{{[A]_{0}}^{n-1}}+(n-1)kt
  86. k \ k
  87. M s \frac{M}{s}
  88. 1 s \frac{1}{s}
  89. 1 M s \frac{1}{M\cdot s}
  90. 1 M n - 1 s \frac{1}{M^{n-1}\cdot s}
  91. k \ k
  92. [ A ] vs. t [A]\ \mbox{vs.}~{}\ t
  93. ln ( [ A ] ) vs. t \ln([A])\ \mbox{vs.}~{}\ t
  94. 1 [ A ] vs. t \frac{1}{[A]}\ \mbox{vs.}~{}\ t
  95. 1 [ A ] n - 1 vs. t \frac{1}{[A]^{n-1}}\ \mbox{vs.}~{}\ t
  96. t 1 / 2 = [ A ] 0 2 k t_{1/2}=\frac{[A]_{0}}{2k}
  97. t 1 / 2 = ln ( 2 ) k t_{1/2}=\frac{\ln(2)}{k}
  98. t 1 / 2 = 1 [ A ] 0 k t_{1/2}=\frac{1}{[A]_{0}k}
  99. t 1 / 2 = 2 n - 1 - 1 ( n - 1 ) k [ A 0 ] n - 1 t_{1/2}=\frac{2^{n-1}-1}{(n-1)k[A_{0}]^{n-1}}
  100. t 1 / 2 t_{1/2}
  101. t 1 / 2 = ln ( 2 ) λ t_{1/2}=\frac{\ln(2)}{\lambda}
  102. ln ( 2 ) \ln(2)
  103. t 1 / 2 = ln ( 2 ) τ t_{1/2}=\ln(2)\cdot\tau
  104. k = A e - E a / R T k=Ae^{{-E_{a}}/{RT}}
  105. H = A + B u + C u H=A+\frac{B}{u}+C\cdot u
  106. 1 λ vac = R H ( 1 n 1 2 - 1 n 2 2 ) \frac{1}{\lambda_{\mathrm{vac}}}=R_{\mathrm{H}}\left(\frac{1}{n_{1}^{2}}-\frac% {1}{n_{2}^{2}}\right)
  107. λ vac \lambda_{\mathrm{vac}}
  108. R H R_{\mathrm{H}}
  109. n 1 n_{1}
  110. n 2 n_{2}
  111. n 1 < n 2 n_{1}<n_{2}
  112. n 1 n_{1}
  113. n 2 n_{2}
  114. n 1 n_{1}
  115. n 2 n_{2}
  116. 2 2\rightarrow\infty
  117. 3 3\rightarrow\infty
  118. 4 4\rightarrow\infty
  119. 5 5\rightarrow\infty
  120. 6 6\rightarrow\infty
  121. 7 7\rightarrow\infty
  122. ψ n x , n y , n z = 8 L x L y L z sin ( n x π x L x ) sin ( n y π y L y ) sin ( n z π z L z ) ( 22 ) \psi_{n_{x},n_{y},n_{z}}=\sqrt{\frac{8}{L_{x}L_{y}L_{z}}}\sin\left(\frac{n_{x}% \pi x}{L_{x}}\right)\sin\left(\frac{n_{y}\pi y}{L_{y}}\right)\sin\left(\frac{n% _{z}\pi z}{L_{z}}\right)\quad(22)
  123. E n x , n y , n z = 2 π 2 2 m [ ( n x L x ) 2 + ( n y L y ) 2 + ( n z L z ) 2 ] ( 23 ) E_{n_{x},n_{y},n_{z}}=\frac{\hbar^{2}\pi^{2}}{2m}\left[\left(\frac{n_{x}}{L_{x% }}\right)^{2}+\left(\frac{n_{y}}{L_{y}}\right)^{2}+\left(\frac{n_{z}}{L_{z}}% \right)^{2}\right]\quad(23)
  124. n i = 1 , 2 , 3 , n_{i}=1,2,3,\ldots

Portal:Mathematics::Featured_article::2007_43.html

  1. φ \varphi
  2. a + b a = a b = φ . \frac{a+b}{a}=\frac{a}{b}=\varphi\,.

Portal:Mathematics::Featured_picture::2008_01.html

  1. f f

Portal:Systems_science::Article::4.html

  1. r r
  2. t t

Post_canonical_system.html

  1. g 10 $ 11 g 11 $ 12 g 12 $ 1 m 1 g 1 m 1 g 20 $ 21 g 21 $ 22 g 22 $ 2 m 2 g 2 m 2 g k 0 $ k 1 g k 1 $ k 2 g k 2 $ k m k g k m k h 0 $ 1 h 1 $ 2 h 2 $ n h n \begin{matrix}g_{10}\ \$_{11}\ g_{11}\ \$_{12}\ g_{12}\ \dots\ \$_{1m_{1}}\ g_% {1m_{1}}\\ g_{20}\ \$_{21}\ g_{21}\ \$_{22}\ g_{22}\ \dots\ \$_{2m_{2}}\ g_{2m_{2}}\\ \dots\ \dots\ \dots\ \dots\ \dots\ \dots\ \dots\ \dots\\ g_{k0}\ \$_{k1}\ g_{k1}\ \$_{k2}\ g_{k2}\ \dots\ \$_{km_{k}}\ g_{km_{k}}\\ \\ \downarrow\\ \\ h_{0}\ \$^{\prime}_{1}\ h_{1}\ \$^{\prime}_{2}\ h_{2}\ \dots\ \$^{\prime}_{n}% \ h_{n}\\ \end{matrix}
  2. g 0 $ 1 g 1 $ 2 g 2 $ m g m h 0 $ 1 h 1 $ 2 h 2 $ n h n g_{0}\ \$_{1}\ g_{1}\ \$_{2}\ g_{2}\ \dots\ \$_{m}\ g_{m}\ \rightarrow\ h_{0}% \ \$^{\prime}_{1}\ h_{1}\ \$^{\prime}_{2}\ h_{2}\ \dots\ \$^{\prime}_{n}\ h_{n}
  3. g $ $ h g\$\ \rightarrow\ \$h
  4. P 1 g P 2 P 1 h P 2 P_{1}gP_{2}\ \rightarrow\ P_{1}hP_{2}

Pre-measure.html

  1. μ 0 ( ) = 0 \mu_{0}(\emptyset)=0
  2. μ 0 ( n = 1 A n ) = n = 1 μ 0 ( A n ) \mu_{0}\left(\bigcup_{n=1}^{\infty}A_{n}\right)=\sum_{n=1}^{\infty}\mu_{0}(A_{% n})
  3. μ * ( S ) = inf { i = 1 μ 0 ( A i ) | A i R , S i = 1 A i } \mu^{*}(S)=\inf\left\{\left.\sum_{i=1}^{\infty}\mu_{0}(A_{i})\right|A_{i}\in R% ,S\subseteq\bigcup_{i=1}^{\infty}A_{i}\right\}
  4. μ ( A ) = μ 0 ( A ) \mu(A)=\mu_{0}(A)
  5. A R A\in R
  6. + +\infty

Precorrin-2_C20-methyltransferase.html

  1. \rightleftharpoons

Precorrin-2_dehydrogenase.html

  1. \rightleftharpoons

Precorrin-3B_C17-methyltransferase.html

  1. \rightleftharpoons

Precorrin-3B_synthase.html

  1. \rightleftharpoons

Precorrin-4_C11-methyltransferase.html

  1. \rightleftharpoons

Precorrin-6A_reductase.html

  1. \rightleftharpoons

Precorrin-6A_synthase_(deacetylating).html

  1. \rightleftharpoons

Precorrin-6Y_C5,15-methyltransferase_(decarboxylating).html

  1. \rightleftharpoons

Precorrin-8X_methylmutase.html

  1. \rightleftharpoons

Predominant_chord.html

  1. 3 6 {}^{6}_{3}
  2. 3 4 {}^{4}_{3}
  3. 5 6 {}^{6}_{5}
  4. 4 6 {}^{6}_{4}
  5. 4 6 {}^{6}_{4}
  6. 6 {}^{6}

Prehomogeneous_vector_space.html

  1. ( G × S L ( n ) , V 𝔽 n ) (G\times SL(n),V\otimes\mathbb{F}^{n})
  2. ( G × S L ( m - n ) , V * 𝔽 m - n ) (G\times SL(m-n),V^{*}\otimes\mathbb{F}^{m-n})
  3. ( G × S L ( n ) , V 𝔽 n ) (G\times SL(n),V\otimes\mathbb{F}^{n})
  4. ( G × S L ( n ) , V 𝔽 n ) (G\times SL(n),V\otimes\mathbb{F}^{n})
  5. ( G × G L ( n ) , V 𝔽 n ) (G\times GL(n),V\otimes\mathbb{F}^{n})
  6. G S L ( m , ) G\subseteq SL(m,\mathbb{C})
  7. m \mathbb{C}^{m}
  8. G G
  9. S L ( m , ) SL(m,\mathbb{C})
  10. m \mathbb{C}^{m}
  11. S L ( m , ) SL(m,\mathbb{C})
  12. Λ 2 m \Lambda^{2}\mathbb{C}^{m}
  13. S p ( m , ) Sp(m,\mathbb{C})
  14. S L ( m , ) SL(m,\mathbb{C})
  15. S 2 m S^{2}\mathbb{C}^{m}
  16. S O ( m , ) SO(m,\mathbb{C})
  17. S O ( m , ) SO(m,\mathbb{C})
  18. m \mathbb{C}^{m}
  19. S O ( n , ) × S O ( m - n , ) SO(n,\mathbb{C})\times SO(m-n,\mathbb{C})
  20. S p ( 2 m , ) Sp(2m,\mathbb{C})
  21. 2 m \mathbb{C}^{2m}
  22. S p ( n , ) × S p ( 2 m - n , ) Sp(n,\mathbb{C})\times Sp(2m-n,\mathbb{C})
  23. S p i n ( 10 , ) on 16 Spin(10,\mathbb{C})\quad\mathrm{on}\quad\mathbb{C}^{16}
  24. S p ( 2 m , ) × S O ( 3 , ) on 2 m 3 Sp(2m,\mathbb{C})\times SO(3,\mathbb{C})\quad\mathrm{on}\quad\mathbb{C}^{2m}% \otimes\mathbb{C}^{3}
  25. S L ( 2 , ) SL(2,\mathbb{C})
  26. S 3 2 S^{3}\mathbb{C}^{2}
  27. S L ( 6 , ) SL(6,\mathbb{C})
  28. Λ 3 6 \Lambda^{3}\mathbb{C}^{6}
  29. 𝔰 𝔩 ( 3 , ) × 𝔰 𝔩 ( 3 , ) \mathfrak{sl}(3,\mathbb{C})\times\mathfrak{sl}(3,\mathbb{C})
  30. S L ( 7 , ) SL(7,\mathbb{C})
  31. Λ 3 7 \Lambda^{3}\mathbb{C}^{7}
  32. 𝔤 2 \mathfrak{g}_{2}^{\mathbb{C}}
  33. S L ( 8 , ) SL(8,\mathbb{C})
  34. Λ 3 8 \Lambda^{3}\mathbb{C}^{8}
  35. 𝔰 𝔩 ( 3 , ) \mathfrak{sl}(3,\mathbb{C})
  36. S L ( 3 , ) SL(3,\mathbb{C})
  37. S 2 3 S^{2}\mathbb{C}^{3}
  38. S L ( 5 , ) SL(5,\mathbb{C})
  39. Λ 2 3 \Lambda^{2}\mathbb{C}^{3}
  40. 𝔰 𝔩 ( 2 , ) , 0 \mathfrak{sl}(2,\mathbb{C}),0
  41. S L ( 6 , ) SL(6,\mathbb{C})
  42. Λ 2 3 \Lambda^{2}\mathbb{C}^{3}
  43. 𝔰 𝔩 ( 2 , ) × 𝔰 𝔩 ( 2 , ) × 𝔰 𝔩 ( 2 , ) \mathfrak{sl}(2,\mathbb{C})\times\mathfrak{sl}(2,\mathbb{C})\times\mathfrak{sl% }(2,\mathbb{C})
  44. S L ( 3 , ) × S L ( 3 , ) SL(3,\mathbb{C})\times SL(3,\mathbb{C})
  45. 3 3 \mathbb{C}^{3}\otimes\mathbb{C}^{3}
  46. 𝔤 𝔩 ( 1 , ) × 𝔤 𝔩 ( 1 , ) \mathfrak{gl}(1,\mathbb{C})\times\mathfrak{gl}(1,\mathbb{C})
  47. S p ( 6 , ) Sp(6,\mathbb{C})
  48. Λ 0 3 6 \Lambda^{3}_{0}\mathbb{C}^{6}
  49. 𝔰 𝔩 ( 3 , ) \mathfrak{sl}(3,\mathbb{C})
  50. S p i n ( 7 , ) Spin(7,\mathbb{C})
  51. 8 \mathbb{C}^{8}
  52. 𝔤 2 , 𝔰 𝔩 ( 3 , ) × 𝔰 𝔬 ( 2 , ) , 𝔰 𝔩 ( 2 , ) × 𝔰 𝔬 ( 3 , ) \mathfrak{g}_{2}^{\mathbb{C}},\mathfrak{sl}(3,\mathbb{C})\times\mathfrak{so}(2% ,\mathbb{C}),\mathfrak{sl}(2,\mathbb{C})\times\mathfrak{so}(3,\mathbb{C})
  53. S p i n ( 9 , ) Spin(9,\mathbb{C})
  54. 16 \mathbb{C}^{16}
  55. 𝔰 𝔭 𝔦 𝔫 ( 7 , ) \mathfrak{spin}(7,\mathbb{C})
  56. S p i n ( 10 , ) Spin(10,\mathbb{C})
  57. 16 \mathbb{C}^{16}
  58. 𝔤 2 × 𝔰 𝔩 ( 2 , ) , 𝔰 𝔩 ( 2 , ) × 𝔰 𝔬 ( 3 , ) \mathfrak{g}_{2}^{\mathbb{C}}\times\mathfrak{sl}(2,\mathbb{C}),\mathfrak{sl}(2% ,\mathbb{C})\times\mathfrak{so}(3,\mathbb{C})
  59. S p i n ( 11 , ) Spin(11,\mathbb{C})
  60. 32 \mathbb{C}^{32}
  61. 𝔰 𝔩 ( 5 , ) \mathfrak{sl}(5,\mathbb{C})
  62. S p i n ( 12 , ) Spin(12,\mathbb{C})
  63. 32 \mathbb{C}^{32}
  64. 𝔰 𝔩 ( 6 , ) \mathfrak{sl}(6,\mathbb{C})
  65. S p i n ( 14 , ) Spin(14,\mathbb{C})
  66. 64 \mathbb{C}^{64}
  67. 𝔤 2 × 𝔤 2 \mathfrak{g}_{2}^{\mathbb{C}}\times\mathfrak{g}_{2}^{\mathbb{C}}
  68. G 2 G_{2}^{\mathbb{C}}
  69. 7 \mathbb{C}^{7}
  70. 𝔰 𝔩 ( 3 , ) , 𝔤 𝔩 ( 2 , ) \mathfrak{sl}(3,\mathbb{C}),\mathfrak{gl}(2,\mathbb{C})
  71. E 6 E_{6}^{\mathbb{C}}
  72. 27 \mathbb{C}^{27}
  73. 𝔣 4 , 𝔰 𝔬 ( 8 , ) \mathfrak{f}_{4}^{\mathbb{C}},\mathfrak{so}(8,\mathbb{C})
  74. E 7 E_{7}^{\mathbb{C}}
  75. 56 \mathbb{C}^{56}
  76. 𝔢 6 \mathfrak{e}_{6}^{\mathbb{C}}
  77. Λ 0 3 6 14 \Lambda^{3}_{0}\mathbb{C}^{6}\cong\mathbb{C}^{14}
  78. 𝔭 \mathfrak{p}^{\perp}
  79. V := 𝔭 / [ 𝔭 , 𝔭 ] V:=\mathfrak{p}^{\perp}/[\mathfrak{p}^{\perp},\mathfrak{p}^{\perp}]
  80. Sp ( 6 , ) \mathrm{Sp}(6,\mathbb{C})
  81. SL ( 3 , ) \mathrm{SL}(3,\mathbb{C})
  82. S 2 3 S^{2}\mathbb{C}^{3}
  83. 𝔰 𝔬 ( 3 , ) \mathfrak{so}(3,\mathbb{C})
  84. J 3 ( ) J_{3}(\mathbb{R})
  85. SL ( 6 , ) \mathrm{SL}(6,\mathbb{C})
  86. SL ( 3 , ) × S L ( 3 , ) \mathrm{SL}(3,\mathbb{C})\times SL(3,\mathbb{C})
  87. 3 3 \mathbb{C}^{3}\otimes\mathbb{C}^{3}
  88. 𝔰 𝔩 ( 3 , ) \mathfrak{sl}(3,\mathbb{C})
  89. J 3 ( ) J_{3}(\mathbb{C})
  90. SO ( 12 , ) \mathrm{SO}(12,\mathbb{C})
  91. SL ( 6 , ) \mathrm{SL}(6,\mathbb{C})
  92. Λ 2 6 \Lambda^{2}\mathbb{C}^{6}
  93. 𝔰 𝔭 ( 6 , ) \mathfrak{sp}(6,\mathbb{C})
  94. J 3 ( ) J_{3}(\mathbb{H})
  95. E 7 E_{7}^{\mathbb{C}}
  96. E 6 E_{6}^{\mathbb{C}}
  97. 27 \mathbb{C}^{27}
  98. 𝔣 4 \mathfrak{f}_{4}^{\mathbb{C}}
  99. J 3 ( 𝕆 ) J_{3}(\mathbb{O})
  100. Sp ( 2 n , ) \mathrm{Sp}(2n,\mathbb{C})
  101. SL ( n , ) \mathrm{SL}(n,\mathbb{C})\,
  102. S 2 n S^{2}\mathbb{C}^{n}
  103. 𝔰 𝔬 ( n , ) \mathfrak{so}(n,\mathbb{C})
  104. J n ( ) J_{n}(\mathbb{R})
  105. SL ( 2 n , ) \mathrm{SL}(2n,\mathbb{C})
  106. SL ( n , ) × SL ( n , ) \mathrm{SL}(n,\mathbb{C})\times\mathrm{SL}(n,\mathbb{C})
  107. n n \mathbb{C}^{n}\otimes\mathbb{C}^{n}
  108. 𝔰 𝔩 ( n , ) \mathfrak{sl}(n,\mathbb{C})
  109. J n ( ) J_{n}(\mathbb{C})
  110. SO ( 4 n , ) \mathrm{SO}(4n,\mathbb{C})
  111. SL ( 2 n , ) \mathrm{SL}(2n,\mathbb{C})
  112. Λ 2 2 n \Lambda^{2}\mathbb{C}^{2n}
  113. 𝔰 𝔭 ( 2 n , ) \mathfrak{sp}(2n,\mathbb{C})
  114. J n ( ) J_{n}(\mathbb{H})
  115. SO ( m + 2 , ) \mathrm{SO}(m+2,\mathbb{C})
  116. SO ( m , ) \mathrm{SO}(m,\mathbb{C})
  117. m \mathbb{C}^{m}
  118. 𝔰 𝔬 ( m - 1 , ) \mathfrak{so}(m-1,\mathbb{C})
  119. J ( m - 1 ) J(m-1)\,
  120. J 2 ( ) , J 2 ( ) , J 2 ( ) , J 2 ( 𝕆 ) J_{2}(\mathbb{R}),J_{2}(\mathbb{C}),J_{2}(\mathbb{H}),J_{2}(\mathbb{O})

Prenylcysteine_oxidase.html

  1. \rightleftharpoons

Prephenate_dehydrogenase.html

  1. \rightleftharpoons

Prephenate_dehydrogenase_(NADP+).html

  1. \rightleftharpoons

PreQ1_synthase.html

  1. \rightleftharpoons

Presentation_complex.html

  1. G = x , y | x y x - 1 y - 1 . G=\langle x,y|xyx^{-1}y^{-1}\rangle.

Procollagen-proline_3-dioxygenase.html

  1. \rightleftharpoons

Production_(computer_science).html

  1. P P
  2. N N
  3. Σ \Sigma
  4. N N
  5. S N S\in N
  6. u v u\to v
  7. u u
  8. v v
  9. u u
  10. v v
  11. ϵ \epsilon
  12. λ \lambda
  13. ( N Σ ) * N ( N Σ ) * ( N Σ ) * (N\cup\Sigma)^{*}N(N\cup\Sigma)^{*}\to(N\cup\Sigma)^{*}
  14. * {}^{*}
  15. \cup
  16. N ( N Σ ) * N\to(N\cup\Sigma)^{*}
  17. a a
  18. b b
  19. S S
  20. S a S b S\rightarrow aSb
  21. S b a S\rightarrow ba
  22. S S
  23. S S
  24. a S b aSb
  25. a S b aSb
  26. S S
  27. a S b aSb
  28. a a S b b aaSbb
  29. a a
  30. b b
  31. S S
  32. b a ba
  33. a a b a b b aababb
  34. S a S b a a S b b a a b a b b S\Rightarrow aSb\Rightarrow aaSbb\Rightarrow aababb
  35. { b a , a b a b , a a b a b b , a a a b a b b b , } \{ba,abab,aababb,aaababbb,\ldots\}

Progesterone_11alpha-monooxygenase.html

  1. \rightleftharpoons

Progesterone_5alpha-reductase.html

  1. \rightleftharpoons

Progesterone_monooxygenase.html

  1. \rightleftharpoons

Programmable_metallization_cell.html

  1. E = - V d E=-\frac{V}{d}

Prolate_spheroidal_wave_function.html

  1. Q T Q_{T}
  2. x = Q T x x=Q_{T}x
  3. [ - T / 2 ; T / 2 ] [-T/2;T/2]
  4. P W P_{W}
  5. x = P W x x=P_{W}x
  6. [ - W ; W ] [-W;W]
  7. Q T P W Q T Q_{T}P_{W}Q_{T}
  8. n = 1 , 2 , n=1,2,\ldots
  9. ψ n \psi_{n}
  10. Q T P W Q T ψ n = λ n ψ n , \ Q_{T}P_{W}Q_{T}\psi_{n}=\lambda_{n}\psi_{n},
  11. { λ n } n \{\lambda_{n}\}_{n}
  12. { ψ n } n \{\psi_{n}\}_{n}
  13. Δ Φ + k 2 Φ = 0 \Delta\Phi+k^{2}\Phi=0
  14. ( ξ , η , ϕ ) (\xi,\eta,\phi)
  15. x = ( d / 2 ) ξ η , \ x=(d/2)\xi\eta,
  16. y = ( d / 2 ) ( ξ 2 - 1 ) ( 1 - η 2 ) cos ϕ , \ y=(d/2)\sqrt{(\xi^{2}-1)(1-\eta^{2})}\cos\phi,
  17. z = ( d / 2 ) ( ξ 2 - 1 ) ( 1 - η 2 ) sin ϕ , \ z=(d/2)\sqrt{(\xi^{2}-1)(1-\eta^{2})}\sin\phi,
  18. ξ 1 \ \xi>=1
  19. | η | 1 |\eta|<=1
  20. Φ ( ξ , η , ϕ ) \Phi(\xi,\eta,\phi)
  21. R m n ( c , ξ ) R_{mn}(c,\xi)
  22. S m n ( c , η ) S_{mn}(c,\eta)
  23. e i m ϕ e^{im\phi}
  24. c = k d / 2 c=kd/2
  25. d d
  26. R m n ( c , ξ ) R_{mn}(c,\xi)
  27. ( ξ 2 - 1 ) d 2 R m n ( c , ξ ) d ξ 2 + 2 ξ d R m n ( c , ξ ) d ξ - ( λ m n ( c ) - c 2 ξ 2 + m 2 ξ 2 - 1 ) R m n ( c , ξ ) = 0 \ (\xi^{2}-1)\frac{d^{2}R_{mn}(c,\xi)}{d\xi^{2}}+2\xi\frac{dR_{mn}(c,\xi)}{d% \xi}-\left(\lambda_{mn}(c)-c^{2}\xi^{2}+\frac{m^{2}}{\xi^{2}-1}\right){R_{mn}(% c,\xi)}=0
  28. λ m n ( c ) \lambda_{mn}(c)
  29. R m n ( c , ξ ) {R_{mn}(c,\xi)}
  30. | ξ | 1 + |\xi|\to 1_{+}
  31. ( η 2 - 1 ) d 2 S m n ( c , η ) d η 2 + 2 η d S m n ( c , η ) d η - ( λ m n ( c ) - c 2 η 2 + m 2 η 2 - 1 ) S m n ( c , η ) = 0 \ (\eta^{2}-1)\frac{d^{2}S_{mn}(c,\eta)}{d\eta^{2}}+2\eta\frac{dS_{mn}(c,\eta)% }{d\eta}-\left(\lambda_{mn}(c)-c^{2}\eta^{2}+\frac{m^{2}}{\eta^{2}-1}\right){S% _{mn}(c,\eta)}=0
  32. ξ 1 \xi>=1
  33. | η | 1 |\eta|<=1
  34. c = 0 c=0
  35. c 0 c\neq 0
  36. S m n ( c , η ) = ( 1 - η 2 ) m / 2 Y m n ( c , η ) S_{mn}(c,\eta)=(1-\eta^{2})^{m/2}Y_{mn}(c,\eta)
  37. Y m n ( c , η ) Y_{mn}(c,\eta)
  38. ( 1 - η 2 ) d 2 Y m n ( c , η ) d η 2 - 2 ( m + 1 ) η d Y m n ( c , η ) d η + ( c 2 η 2 + m ( m + 1 ) - λ m n ( c ) ) Y m n ( c , η ) = 0 , \ (1-\eta^{2})\frac{d^{2}Y_{mn}(c,\eta)}{d\eta^{2}}-2(m+1)\eta\frac{dY_{mn}(c,% \eta)}{d\eta}+\left(c^{2}\eta^{2}+m(m+1)-\lambda_{mn}(c)\right){Y_{mn}(c,\eta)% }=0,

Proline_3-hydroxylase.html

  1. \rightleftharpoons

Proline_dehydrogenase.html

  1. \rightleftharpoons

Proline_racemase.html

  1. \rightleftharpoons

Proof_that_π_is_irrational.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. tan ( x ) = x 1 - x 2 3 - x 2 5 - x 2 7 - . \tan(x)=\cfrac{x}{1-\cfrac{x^{2}}{3-\cfrac{x^{2}}{5-\cfrac{x^{2}}{7-{}\ddots}}% }}.
  5. π \pi
  6. π \pi
  7. π \pi
  8. π \pi
  9. π \pi
  10. A 0 ( x ) = sin ( x ) ; A_{0}(x)=\sin(x);\,
  11. ( n + ) : A n + 1 ( x ) = 0 x y A n ( y ) d y ; (\forall n\in\mathbb{Z}_{+}):A_{n+1}(x)=\int_{0}^{x}yA_{n}(y)\,dy;
  12. U 0 ( x ) = sin ( x ) x ; U_{0}(x)=\frac{\sin(x)}{x};
  13. ( n + ) : U n + 1 ( x ) = - U n ( x ) x . (\forall n\in\mathbb{Z}_{+}):U_{n+1}(x)=-\frac{U_{n}^{\prime}(x)}{x}.
  14. ( n + ) : A n ( x ) = x 2 n + 1 ( 2 n + 1 ) ! ! - x 2 n + 3 2 × ( 2 n + 3 ) ! ! + x 2 n + 5 2 × 4 × ( 2 n + 5 ) ! ! (\forall n\in\mathbb{Z}_{+}):A_{n}(x)=\frac{x^{2n+1}}{(2n+1)!!}-\frac{x^{2n+3}% }{2\times(2n+3)!!}+\frac{x^{2n+5}}{2\times 4\times(2n+5)!!}\mp\cdots
  15. ( n + ) : U n ( x ) = 1 ( 2 n + 1 ) ! ! - x 2 2 × ( 2 n + 3 ) ! ! + x 4 2 × 4 × ( 2 n + 5 ) ! ! (\forall n\in\mathbb{Z}_{+}):U_{n}(x)=\frac{1}{(2n+1)!!}-\frac{x^{2}}{2\times(% 2n+3)!!}+\frac{x^{4}}{2\times 4\times(2n+5)!!}\mp\cdots
  16. U n ( x ) = A n ( x ) x 2 n + 1 . U_{n}(x)=\frac{A_{n}(x)}{x^{2n+1}}.
  17. A n + 1 ( x ) x 2 n + 3 = U n + 1 ( x ) = - U n ( x ) x = - 1 x d d x ( A n ( x ) x 2 n + 1 ) , \frac{A_{n+1}(x)}{x^{2n+3}}=U_{n+1}(x)=-\frac{U_{n}^{\prime}(x)}{x}=-\frac{1}{% x}\frac{d}{dx}\left(\frac{A_{n}(x)}{x^{2n+1}}\right),
  18. A n + 1 ( x ) = ( 2 n + 1 ) A n ( x ) - x A n ( x ) = ( 2 n + 1 ) A n ( x ) - x 2 A n - 1 ( x ) . A_{n+1}(x)=(2n+1)A_{n}(x)-xA_{n}^{\prime}(x)=(2n+1)A_{n}(x)-x^{2}A_{n-1}(x).\,
  19. π \pi
  20. π \pi
  21. A n ( x ) = x 2 n + 1 2 n n ! 0 1 ( 1 - z 2 ) n cos ( x z ) d z . A_{n}(x)=\frac{x^{2n+1}}{2^{n}n!}\int_{0}^{1}(1-z^{2})^{n}\cos(xz)\,dz.
  22. 1 2 n n ! 0 1 ( 1 - z 2 ) n cos ( x z ) d z = A n ( x ) x 2 n + 1 = U n ( x ) . \frac{1}{2^{n}n!}\int_{0}^{1}(1-z^{2})^{n}\cos(xz)\,dz=\frac{A_{n}(x)}{x^{2n+1% }}=U_{n}(x).
  23. 0 1 cos ( x z ) d z = sin ( x ) x = U 0 ( x ) \int_{0}^{1}\cos(xz)\,dz=\frac{\sin(x)}{x}=U_{0}(x)
  24. 1 2 n n ! 0 1 ( 1 - z 2 ) n cos ( x z ) d z = U n ( x ) , \frac{1}{2^{n}n!}\int_{0}^{1}(1-z^{2})^{n}\cos(xz)\,dz=U_{n}(x),
  25. 1 2 n + 1 ( n + 1 ) ! 0 1 ( 1 - z 2 ) n + 1 cos ( x z ) d z = 1 2 n + 1 ( n + 1 ) ! ( ( 1 - z 2 ) n + 1 sin ( x z ) x | z = 0 z = 1 = 0 + 0 1 2 ( n + 1 ) ( 1 - z 2 ) n z sin ( x z ) x d z ) = 1 x 1 2 n n ! 0 1 ( 1 - z 2 ) n z sin ( x z ) d z = - 1 x d d x ( 1 2 n n ! 0 1 ( 1 - z 2 ) n cos ( x z ) d z ) = - U n ( x ) x = U n + 1 ( x ) . \begin{aligned}&\displaystyle{}\quad\frac{1}{2^{n+1}(n+1)!}\int_{0}^{1}(1-z^{2% })^{n+1}\cos(xz)\,dz\\ &\displaystyle=\frac{1}{2^{n+1}(n+1)!}\Biggl(\overbrace{\left.(1-z^{2})^{n+1}% \frac{\sin(xz)}{x}\right|_{z=0}^{z=1}}^{=\,0}+\int_{0}^{1}2(n+1)(1-z^{2})^{n}z% \frac{\sin(xz)}{x}\,dz\Biggr)\\ &\displaystyle=\frac{1}{x}\cdot\frac{1}{2^{n}n!}\int_{0}^{1}(1-z^{2})^{n}z\sin% (xz)\,dz\\ &\displaystyle=-\frac{1}{x}\cdot\frac{d}{dx}\left(\frac{1}{2^{n}n!}\int_{0}^{1% }(1-z^{2})^{n}\cos(xz)\,dz\right)\\ &\displaystyle=-\frac{U_{n}^{\prime}(x)}{x}=U_{n+1}(x).\end{aligned}
  26. π \pi
  27. π \pi
  28. N = q n 2 A n ( π 2 ) = q n 2 ( p q ) n + 1 2 2 n n ! 0 1 ( 1 - z 2 ) cos ( π 2 z ) d z . \begin{aligned}\displaystyle N&\displaystyle=q^{\left\lfloor\frac{n}{2}\right% \rfloor}A_{n}\left(\frac{\pi}{2}\right)\\ &\displaystyle=q^{\left\lfloor\frac{n}{2}\right\rfloor}\frac{\left(\frac{p}{q}% \right)^{n+\frac{1}{2}}}{2^{n}n!}\int_{0}^{1}(1-z^{2})\cos\left(\frac{\pi}{2}z% \right)\,dz.\end{aligned}
  29. lim n q n 2 ( p q ) n + 1 2 2 n n ! = 0. \lim_{n\in\mathbb{N}}q^{\left\lfloor\frac{n}{2}\right\rfloor}\frac{\left(\frac% {p}{q}\right)^{n+\frac{1}{2}}}{2^{n}n!}=0.
  30. I n ( x ) = - 1 1 ( 1 - z 2 ) n cos ( x z ) d z , I_{n}(x)=\int_{-1}^{1}(1-z^{2})^{n}\cos(xz)\,dz,
  31. ( n { 1 } ) : x 2 I n ( x ) = 2 n ( 2 n - 1 ) I n - 1 ( x ) - 4 n ( n - 1 ) I n - 2 ( x ) . (\forall n\in\mathbb{N}\setminus\{1\}):x^{2}I_{n}(x)=2n(2n-1)I_{n-1}(x)-4n(n-1% )I_{n-2}(x).
  32. J n ( x ) = x 2 n + 1 I n ( x ) , J_{n}(x)=x^{2n+1}I_{n}(x),\,
  33. J n ( x ) = 2 n ( 2 n - 1 ) J n - 1 ( x ) - 4 n ( n - 1 ) x 2 J n - 2 ( x ) . J_{n}(x)=2n(2n-1)J_{n-1}(x)-4n(n-1)x^{2}J_{n-2}(x).
  34. J n ( x ) = x 2 n + 1 I n ( x ) = n ! ( P n ( x ) sin ( x ) + Q n ( x ) cos ( x ) ) , J_{n}(x)=x^{2n+1}I_{n}(x)=n!\bigl(P_{n}(x)\sin(x)+Q_{n}(x)\cos(x)\bigr),\,
  35. π \pi
  36. π \pi
  37. π \pi
  38. b 2 n + 1 n ! I n ( π 2 ) = P n ( π 2 ) a 2 n + 1 . \frac{b^{2n+1}}{n!}I_{n}\left(\frac{\pi}{2}\right)=P_{n}\left(\frac{\pi}{2}% \right)a^{2n+1}.
  39. π \pi
  40. 0 < b 2 n + 1 I n ( π 2 ) n ! < 1 , 0<\frac{b^{2n+1}I_{n}\left(\frac{\pi}{2}\right)}{n!}<1,
  41. π \pi
  42. J n ( x ) = x 2 n + 1 - 1 1 ( 1 - z 2 ) n cos ( x z ) d z = 2 x 2 n + 1 0 1 ( 1 - z 2 ) n cos ( x z ) d z = 2 n + 1 n ! A n ( x ) . \begin{aligned}\displaystyle J_{n}(x)&\displaystyle=x^{2n+1}\int_{-1}^{1}(1-z^% {2})^{n}\cos(xz)\,dz\\ &\displaystyle=2x^{2n+1}\int_{0}^{1}(1-z^{2})^{n}\cos(xz)\,dz\\ &\displaystyle=2^{n+1}n!A_{n}(x).\end{aligned}
  43. π \pi
  44. π \pi
  45. f ( x ) = x n ( a - b x ) n n ! f(x)=\frac{x^{n}(a-bx)^{n}}{n!}
  46. F ( x ) = f ( x ) - f ′′ ( x ) + f ( 4 ) ( x ) + + ( - 1 ) j f ( 2 j ) ( x ) + + ( - 1 ) n f ( 2 n ) ( x ) F(x)=f(x)-f^{\prime\prime}(x)+f^{(4)}(x)+\cdots+(-1)^{j}f^{(2j)}(x)+\cdots+(-1% )^{n}f^{(2n)}(x)
  47. π \pi
  48. π \pi
  49. π \pi
  50. 0 π f ( x ) sin ( x ) d x = F ( 0 ) + F ( π ) \int_{0}^{\pi}f(x)\sin(x)\,dx=F(0)+F(\pi)
  51. F ′′ + F = f . F^{\prime\prime}+F=f.\,
  52. ( F sin - F cos ) = f sin (F^{\prime}\cdot\sin-F\cdot\cos)^{\prime}=f\cdot\sin\!
  53. 0 π f ( x ) sin ( x ) d x = ( F ( x ) sin x - F ( x ) cos x ) | 0 π . \int_{0}^{\pi}f(x)\sin(x)\,dx=\bigl(F^{\prime}(x)\sin x-F(x)\cos x\bigr)\Big|_% {0}^{\pi}.\!
  54. π \pi
  55. 0 π f ( x ) sin ( x ) d x π ( π a ) n n ! \int_{0}^{\pi}f(x)\sin(x)\,dx\leq\pi\frac{(\pi a)^{n}}{n!}
  56. 0 π f ( x ) sin ( x ) d x = j = 0 n ( - 1 ) j ( f ( 2 j ) ( π ) + f ( 2 j ) ( 0 ) ) + ( - 1 ) n + 1 0 π f ( 2 n + 2 ) ( x ) sin ( x ) d x , \begin{aligned}\displaystyle\int_{0}^{\pi}f(x)\sin(x)\,dx&\displaystyle=\sum_{% j=0}^{n}(-1)^{j}\bigl(f^{(2j)}(\pi)+f^{(2j)}(0)\bigr)\\ &\displaystyle\qquad+(-1)^{n+1}\int_{0}^{\pi}f^{(2n+2)}(x)\sin(x)\,dx,\end{aligned}
  57. J n ( x ) = x 2 n + 1 - 1 1 ( 1 - z 2 ) n cos ( x z ) d z = - 1 1 ( x 2 - ( x z ) 2 ) n x cos ( x z ) d z . \begin{aligned}\displaystyle J_{n}(x)&\displaystyle=x^{2n+1}\int_{-1}^{1}(1-z^% {2})^{n}\cos(xz)\,dz\\ &\displaystyle=\int_{-1}^{1}\bigl(x^{2}-(xz)^{2}\bigr)^{n}x\cos(xz)\,dz.\end{aligned}
  58. - x x ( x 2 - y 2 ) n cos ( y ) d y . \int_{-x}^{x}(x^{2}-y^{2})^{n}\cos(y)\,dy.
  59. J n ( π 2 ) = - π / 2 π / 2 ( π 2 4 - y 2 ) n cos ( y ) d y = 0 π ( π 2 4 - ( y - π 2 ) 2 ) n cos ( y - π 2 ) d y = 0 π y n ( π - y ) n sin ( y ) d y = n ! b n 0 π f ( x ) sin ( x ) d x . \begin{aligned}\displaystyle J_{n}\left(\frac{\pi}{2}\right)&\displaystyle=% \int_{-\pi/2}^{\pi/2}\left(\frac{\pi^{2}}{4}-y^{2}\right)^{n}\cos(y)\,dy\\ &\displaystyle=\int_{0}^{\pi}\left(\frac{\pi^{2}}{4}-\left(y-\frac{\pi}{2}% \right)^{2}\right)^{n}\cos\left(y-\frac{\pi}{2}\right)\,dy\\ &\displaystyle=\int_{0}^{\pi}y^{n}(\pi-y)^{n}\sin(y)\,dy\\ &\displaystyle=\frac{n!}{b^{n}}\int_{0}^{\pi}f(x)\sin(x)\,dx.\end{aligned}
  60. F = f - f ( 2 ) + f ( 4 ) , F=f-f^{(2)}+f^{(4)}\mp\cdots,
  61. f ( x ) sin ( x ) d x = F ( x ) sin ( x ) - F ( x ) cos ( x ) , \int f(x)\sin(x)\,dx=F^{\prime}(x)\sin(x)-F(x)\cos(x),
  62. 0 π f ( x ) sin ( x ) d x = F ( π ) + F ( 0 ) . \int_{0}^{\pi}f(x)\sin(x)\,dx=F(\pi)+F(0).
  63. A n ( b ) = b n 0 π x n ( π - x ) n n ! sin ( x ) d x . A_{n}(b)=b^{n}\int_{0}^{\pi}\frac{x^{n}(\pi-x)^{n}}{n!}\sin(x)\,dx.
  64. π \pi
  65. π \pi
  66. A n ( b ) π b n 1 n ! ( π 2 ) 2 n = π ( b π 2 / 4 ) n n ! . A_{n}(b)\leq\pi b^{n}\frac{1}{n!}\left(\frac{\pi}{2}\right)^{2n}=\pi\frac{(b% \pi^{2}/4)^{n}}{n!}.
  67. π \pi
  68. π \pi
  69. f ( x ) = x n ( a - b x ) n n ! , f(x)=\frac{x^{n}(a-bx)^{n}}{n!},
  70. A n ( b ) \displaystyle A_{n}(b)
  71. π \pi
  72. f k ( x ) = 1 - x 2 k + x 4 2 ! k ( k + 1 ) - x 6 3 ! k ( k + 1 ) ( k + 2 ) + ( k { 0 , - 1 , - 2 , } ) . \begin{aligned}\displaystyle f_{k}(x)&\displaystyle=1-\frac{x^{2}}{k}+\frac{x^% {4}}{2!k(k+1)}-\frac{x^{6}}{3!k(k+1)(k+2)}+\cdots\\ &\displaystyle{}\quad(k\notin\{0,-1,-2,\ldots\}).\end{aligned}
  73. f 1 / 2 ( x ) = cos ( 2 x ) and f 3 / 2 ( x ) = sin ( 2 x ) 2 x . f_{1/2}(x)=\cos(2x)\,\text{ and }f_{3/2}(x)=\frac{\sin(2x)}{2x}.
  74. ( x ) : x 2 k ( k + 1 ) f k + 2 ( x ) = f k + 1 ( x ) - f k ( x ) . (\forall x\in\mathbb{R}):\frac{x^{2}}{k(k+1)}f_{k+2}(x)=f_{k+1}(x)-f_{k}(x).
  75. lim k + f k ( x ) = 1. \lim_{k\to+\infty}f_{k}(x)=1.
  76. | f k ( x ) - 1 | n = 1 C k n = C 1 / k 1 - 1 / k = C k - 1 . \bigl|f_{k}(x)-1\bigr|\leqslant\sum_{n=1}^{\infty}\frac{C}{k^{n}}=C\frac{1/k}{% 1-1/k}=\frac{C}{k-1}.
  77. ( k { 0 , - 1 , - 2 , } ) : f k ( x ) 0 and f k + 1 ( x ) f k ( x ) . (\forall k\in\mathbb{Q}\setminus\{0,-1,-2,\ldots\}):f_{k}(x)\neq 0\,\text{ and% }\frac{f_{k+1}(x)}{f_{k}(x)}\notin\mathbb{Q}.
  78. g n = { f k ( x ) if n = 0 c n k ( k + 1 ) ( k + n - 1 ) f k + n ( x ) otherwise. g_{n}=\begin{cases}f_{k}(x)&\,\text{ if }n=0\\ \frac{c^{n}}{k(k+1)\cdots(k+n-1)}f_{k+n}(x)&\,\text{ otherwise.}\end{cases}
  79. g 0 = f k ( x ) = a y y and g 1 = c k f k + 1 ( x ) = b c k y y . g_{0}=f_{k}(x)=ay\in\mathbb{Z}y\,\text{ and }g_{1}=\frac{c}{k}f_{k+1}(x)=\frac% {bc}{k}y\in\mathbb{Z}y.
  80. g n + 2 = c n + 2 x 2 k ( k + 1 ) ( k + n - 1 ) x 2 ( k + n ) ( k + n + 1 ) f k + n + 2 ( x ) = c n + 2 x 2 k ( k + 1 ) ( k + n - 1 ) f k + n + 1 ( x ) - c n + 2 x 2 k ( k + 1 ) ( k + n - 1 ) f k + n ( x ) = c ( k + n ) x 2 g n + 1 - c 2 x 2 g n = ( c k x 2 + c x 2 n ) g n + 1 - c 2 x 2 g n , \begin{aligned}\displaystyle g_{n+2}&\displaystyle=\frac{c^{n+2}}{x^{2}k(k+1)% \cdots(k+n-1)}\cdot\frac{x^{2}}{(k+n)(k+n+1)}f_{k+n+2}(x)\\ &\displaystyle=\frac{c^{n+2}}{x^{2}k(k+1)\cdots(k+n-1)}f_{k+n+1}(x)-\frac{c^{n% +2}}{x^{2}k(k+1)\cdots(k+n-1)}f_{k+n}(x)\\ &\displaystyle=\frac{c(k+n)}{x^{2}}g_{n+1}-\frac{c^{2}}{x^{2}}g_{n}\\ &\displaystyle=\left(\frac{ck}{x^{2}}+\frac{c}{x^{2}}n\right)g_{n+1}-\frac{c^{% 2}}{x^{2}}g_{n},\end{aligned}
  81. π \pi
  82. π \pi
  83. π \pi
  84. π \pi
  85. tan x = sin x cos x = x f 3 / 2 ( x / 2 ) f 1 / 2 ( x / 2 ) , \tan x=\frac{\sin x}{\cos x}=x\frac{f_{3/2}(x/2)}{f_{1/2}(x/2)},
  86. ( k { 0 , - 1 , - 2 , } ) : x J k ( x ) J k - 1 ( x ) . (\forall k\in\mathbb{Q}\setminus\{0,-1,-2,\ldots\}):\frac{xJ_{k}(x)}{J_{k-1}(x% )}\notin\mathbb{Q}.
  87. π \pi

Propanediol-phosphate_dehydrogenase.html

  1. \rightleftharpoons

Propionate_CoA-transferase.html

  1. \rightleftharpoons

Prostaglandin-A1_Delta-isomerase.html

  1. \rightleftharpoons

Prostaglandin-D_synthase.html

  1. \rightleftharpoons

Prostaglandin-E2_9-reductase.html

  1. \rightleftharpoons

Prostaglandin-F_synthase.html

  1. \rightleftharpoons

Protein-disulfide_reductase.html

  1. \rightleftharpoons

Protein-disulfide_reductase_(glutathione).html

  1. \rightleftharpoons

Protein-glutamate_O-methyltransferase.html

  1. \rightleftharpoons

Protein-histidine_N-methyltransferase.html

  1. \rightleftharpoons

Protein-S-isoprenylcysteine_O-methyltransferase.html

  1. \rightleftharpoons

Protein-serine_epimerase.html

  1. \rightleftharpoons

Protocatechuate_3,4-dioxygenase.html

  1. \rightleftharpoons

Protocatechuate_4,5-dioxygenase.html

  1. \rightleftharpoons

Protochlorophyllide_reductase.html

  1. \rightleftharpoons

Protopine_6-monooxygenase.html

  1. \rightleftharpoons

Pseudoelementary_class.html

  1. ˘ \breve{\ }
  2. ˘ \breve{\ }
  3. ˘ \breve{\ }

Pseudoforest.html

  1. n k = 1 n ( - 1 ) k - 1 k n 1 + + n k = n n ! n 1 ! n k ! ( ( n 1 2 ) + + ( n k 2 ) n ) . n\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k}\sum_{n_{1}+\cdots+n_{k}=n}\frac{n!}{n_{1}!% \cdots n_{k}!}{\left({{{\left({{n_{1}}\atop{2}}\right)}+\cdots+{\left({{n_{k}}% \atop{2}}\right)}}\atop{n}}\right)}.

Psoralen_synthase.html

  1. \rightleftharpoons

Psychometric_Entrance_Test.html

  1. ( 3 4 ) 168 1 9.77 × 10 20 \left(\tfrac{3}{4}\right)^{168}\approx\tfrac{1}{9.77\times 10^{20}}
  2. P = ( 2 S - 575 ) 2 - 625 5000 ; S = 575 + 625 ( 8 P + 1 ) 2 P=\frac{{(2S-575)^{2}}-625}{5000};\quad S=\frac{575+\sqrt{625(8P+1)}}{2}

Psychosine_sulfotransferase.html

  1. \rightleftharpoons

Psychrometric_constant.html

  1. γ \gamma
  2. γ = ( c p ) a i r * P λ v * M W r a t i o \gamma=\frac{\left(c_{p}\right)_{air}*P}{\lambda_{v}*MW_{ratio}}
  3. γ = \gamma=
  4. λ v = \lambda_{v}=
  5. c p = c_{p}=
  6. M W r a t i o = MW_{ratio}=
  7. λ v \lambda_{v}
  8. M W r a t i o MW_{ratio}
  9. γ \gamma
  10. ( c p ) H 2 O \left(c_{p}\right)_{H_{2}O}
  11. ( c p ) a i r \left(c_{p}\right)_{air}
  12. e s = e [ T w e t ] e_{s}=e\left[T_{wet}\right]
  13. e a = e s - γ * ( T d r y - T w e t ) e_{a}=e_{s}-\gamma*\left(T_{dry}-T_{wet}\right)

Pteridine_oxidase.html

  1. \rightleftharpoons

Pteridine_reductase.html

  1. \rightleftharpoons

Pterocarpin_synthase.html

  1. \rightleftharpoons

Pulse_tube_refrigerator.html

  1. Q ˙ L \dot{Q}_{L}
  2. ξ \xi
  3. Q ˙ L \dot{Q}_{L}
  4. ξ = Q ˙ L / P \xi=\dot{Q}_{L}/P
  5. ξ \xi
  6. ξ C = T L T H - T L \xi_{C}=\frac{T_{L}}{T_{H}-T_{L}}
  7. ξ P T R = T L T H \xi_{PTR}=\frac{T_{L}}{T_{H}}

Purine_imidazole-ring_cyclase.html

  1. \rightleftharpoons

Putnam–Norden–Rayleigh_curve.html

  1. t o t_{o}
  2. t m i n t_{min}
  3. E a = m ( t d 4 t a 4 ) E_{a}=m\left(\frac{{t_{d}}^{4}}{{t_{a}}^{4}}\right)
  4. E a E_{a}
  5. t d t_{d}
  6. t a t_{a}

Putrescine_carbamoyltransferase.html

  1. \rightleftharpoons

Putrescine_N-methyltransferase.html

  1. \rightleftharpoons

Putrescine_oxidase.html

  1. \rightleftharpoons

Pyranose_oxidase.html

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Pyridine_N-methyltransferase.html

  1. \rightleftharpoons

Pyridoxal_4-dehydrogenase.html

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Pyridoxal_oxidase.html

  1. \rightleftharpoons

Pyridoxine_4-dehydrogenase.html

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Pyridoxine_4-oxidase.html

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Pyridoxine_5-dehydrogenase.html

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Pyrimidine-deoxynucleoside_1'-dioxygenase.html

  1. \rightleftharpoons

Pyrimidine-deoxynucleoside_2'-dioxygenase.html

  1. \rightleftharpoons

Pyrimidodiazepine_synthase.html

  1. \rightleftharpoons

Pyrogallol_1,2-oxygenase.html

  1. \rightleftharpoons

Pyrogallol_hydroxytransferase.html

  1. \rightleftharpoons

Pyrroline-2-carboxylate_reductase.html

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Pyrroline-5-carboxylate_reductase.html

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Pyrroloquinoline-quinone_synthase.html

  1. \rightleftharpoons

Pyruvate_dehydrogenase_(acetyl-transferring).html

  1. \rightleftharpoons

Pyruvate_dehydrogenase_(cytochrome).html

  1. \rightleftharpoons

Pyruvate_oxidase.html

  1. \rightleftharpoons

Pyruvate_oxidase_(CoA-acetylating).html

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Pyruvate_synthase.html

  1. \rightleftharpoons

Q-function.html

  1. ϕ ( x ) \phi(x)
  2. x = y - μ σ x=\frac{y-\mu}{\sigma}
  3. Q ( x ) = 1 2 π x exp ( - u 2 2 ) d u . Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp\left(-\frac{u^{2}}{2}\right)\,du.
  4. Q ( x ) = 1 - Q ( - x ) = 1 - Φ ( x ) , Q(x)=1-Q(-x)=1-\Phi(x)\,\!,
  5. Φ ( x ) \Phi(x)
  6. Q ( x ) = 1 2 ( 2 π x / 2 exp ( - t 2 ) d t ) = 1 2 - 1 2 erf ( x 2 ) -or- = 1 2 erfc ( x 2 ) . \begin{aligned}\displaystyle Q(x)&\displaystyle=\frac{1}{2}\left(\frac{2}{% \sqrt{\pi}}\int_{x/\sqrt{2}}^{\infty}\exp\left(-t^{2}\right)\,dt\right)\\ &\displaystyle=\frac{1}{2}-\frac{1}{2}\operatorname{erf}\left(\frac{x}{\sqrt{2% }}\right)~{}~{}\,\text{ -or-}\\ &\displaystyle=\frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right).% \end{aligned}
  7. Q ( x ) = 1 π 0 π 2 exp ( - x 2 2 sin 2 θ ) d θ . Q(x)=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\exp\left(-\frac{x^{2}}{2\sin^{2}% \theta}\right)d\theta.
  8. ( x 1 + x 2 ) ϕ ( x ) < Q ( x ) < ϕ ( x ) x , x > 0 , \left(\frac{x}{1+x^{2}}\right)\phi(x)<Q(x)<\frac{\phi(x)}{x},\qquad x>0,
  9. Q ( x ) = x ϕ ( u ) d u < x u x ϕ ( u ) d u = x 2 2 e - v x 2 π d v = - . e - v x 2 π | x 2 2 = ϕ ( x ) x . Q(x)=\int_{x}^{\infty}\phi(u)\,du<\int_{x}^{\infty}\frac{u}{x}\phi(u)\,du=\int% _{\frac{x^{2}}{2}}^{\infty}\frac{e^{-v}}{x\sqrt{2\pi}}\,dv=-\biggl.\frac{e^{-v% }}{x\sqrt{2\pi}}\biggr|_{\frac{x^{2}}{2}}^{\infty}=\frac{\phi(x)}{x}.
  10. ϕ ( u ) = - u ϕ ( u ) \phi^{\prime}(u)=-u\phi(u)
  11. ( 1 + 1 x 2 ) Q ( x ) = x ( 1 + 1 x 2 ) ϕ ( u ) d u > x ( 1 + 1 u 2 ) ϕ ( u ) d u = - . ϕ ( u ) u | x = ϕ ( x ) x . \left(1+\frac{1}{x^{2}}\right)Q(x)=\int_{x}^{\infty}\left(1+\frac{1}{x^{2}}% \right)\phi(u)\,du>\int_{x}^{\infty}\left(1+\frac{1}{u^{2}}\right)\phi(u)\,du=% -\biggl.\frac{\phi(u)}{u}\biggr|_{x}^{\infty}=\frac{\phi(x)}{x}.
  12. Q ( x ) e - x 2 2 , x > 0 Q(x)\leq e^{-\frac{x^{2}}{2}},\qquad x>0
  13. Q ( x ) 1 4 e - x 2 + 1 4 e - x 2 2 1 2 e - x 2 2 , x > 0 Q(x)\leq\tfrac{1}{4}e^{-x^{2}}+\tfrac{1}{4}e^{-\frac{x^{2}}{2}}\leq\tfrac{1}{2% }e^{-\frac{x^{2}}{2}},\qquad x>0
  14. Q ( x ) 1 12 e - x 2 2 + 1 4 e - 2 3 x 2 , x > 0 Q(x)\approx\frac{1}{12}e^{-\frac{x^{2}}{2}}+\frac{1}{4}e^{-\frac{2}{3}x^{2}},% \qquad x>0
  15. Q ( x ) ( 1 - e - 1.4 x ) e - x 2 2 1.135 2 π x , x > 0 Q(x)\approx\frac{\left(1-e^{-1.4x}\right)e^{-\frac{x^{2}}{2}}}{1.135\sqrt{2\pi% }x},x>0
  16. Q - 1 ( x ) = 2 erf - 1 ( 1 - 2 x ) Q^{-1}(x)=\sqrt{2}\ \mathrm{erf}^{-1}(1-2x)

Quadratic_eigenvalue_problem.html

  1. λ \lambda\,
  2. y y\,
  3. x x\,
  4. Q ( λ ) x = 0 and y Q ( λ ) = 0 , Q(\lambda)x=0\,\text{ and }y^{\ast}Q(\lambda)=0,\,
  5. Q ( λ ) = λ 2 A 2 + λ A 1 + A 0 Q(\lambda)=\lambda^{2}A_{2}+\lambda A_{1}+A_{0}\,
  6. A 2 , A 1 , A 0 n × n A_{2},\,A_{1},A_{0}\in\mathbb{C}^{n\times n}
  7. A 2 0 A_{2}\,\neq 0
  8. 2 n 2n\,
  9. Q ( λ ) Q(\lambda)
  10. Q ( λ ) Q(\lambda)\,
  11. Q ( λ ) = λ 2 M + λ C + K Q(\lambda)=\lambda^{2}M+\lambda C+K\,
  12. M M\,
  13. C C\,
  14. K K\,
  15. A x = λ x Ax=\lambda x
  16. A x = λ B x Ax=\lambda Bx
  17. A - λ B A-\lambda B
  18. L ( λ ) = λ [ M 0 0 I n ] + [ C K - I n 0 ] , L(\lambda)=\lambda\begin{bmatrix}M&0\\ 0&I_{n}\end{bmatrix}+\begin{bmatrix}C&K\\ -I_{n}&0\end{bmatrix},
  19. I n I_{n}
  20. n n
  21. n n
  22. z = [ λ x x ] . z=\begin{bmatrix}\lambda x\\ x\end{bmatrix}.
  23. L ( λ ) z = 0 L(\lambda)z=0
  24. λ \lambda
  25. z z
  26. n n
  27. z z
  28. x x
  29. Q ( λ ) Q(\lambda)

Quantum_nondemolition_measurement.html

  1. A A
  2. 𝒮 \mathcal{S}
  3. H 𝒮 H_{\mathcal{S}}
  4. 𝒮 \mathcal{S}
  5. \mathcal{R}
  6. 𝒮 \mathcal{S}
  7. H 𝒮 H_{\mathcal{RS}}
  8. 𝒮 {\mathcal{S}}
  9. H 𝒮 H_{\mathcal{S}}
  10. A A
  11. 𝒮 \mathcal{S}
  12. \mathcal{R}
  13. | ψ i | A i 𝒮 | R i |\psi\rangle\approx\sum_{i}|A_{i}\rangle_{\mathcal{S}}|R_{i}\rangle_{\mathcal{% R}}
  14. | A i 𝒮 |A_{i}\rangle_{\mathcal{S}}
  15. A A
  16. | R i |R_{i}\rangle_{\mathcal{R}}
  17. A ( t n ) = e - i t H 𝒮 A e + i t H 𝒮 . A(t_{n})=e^{-itH_{\mathcal{S}}}Ae^{+itH_{\mathcal{S}}}.
  18. A A
  19. [ A ( t n ) , A ( t m ) ] = 0 [A(t_{n}),A(t_{m})]=0
  20. t n t_{n}
  21. t m t_{m}
  22. t n t_{n}
  23. t m t_{m}
  24. A A
  25. A A
  26. A A
  27. d d t A ( t ) = - i [ H 𝒮 , A ] = 0 , \frac{\mathrm{d}}{\mathrm{d}t}A(t)=-\frac{i}{\hbar}[H_{\mathcal{S}},A]=0,
  28. A A

Quantum_spin_Hall_effect.html

  1. S z S_{z}
  2. σ x y s p i n = 2 \sigma_{xy}^{spin}=2
  3. e 4 π \frac{e}{4\pi}
  4. Z 2 Z_{2}
  5. S z S_{z}
  6. G x x = 2 e 2 h G_{xx}=2\frac{e^{2}}{h}
  7. S z S_{z}

Quasi-isometry.html

  1. f f
  2. ( M 1 , d 1 ) (M_{1},d_{1})
  3. ( M 2 , d 2 ) (M_{2},d_{2})
  4. f f
  5. ( M 1 , d 1 ) (M_{1},d_{1})
  6. ( M 2 , d 2 ) (M_{2},d_{2})
  7. A 1 A\geq 1
  8. B 0 B\geq 0
  9. C 0 C\geq 0
  10. x x
  11. y y
  12. M 1 M_{1}
  13. B B
  14. A A
  15. x , y M 1 : 1 A d 1 ( x , y ) - B d 2 ( f ( x ) , f ( y ) ) A d 1 ( x , y ) + B . \forall x,y\in M_{1}:\frac{1}{A}\;d_{1}(x,y)-B\leq d_{2}(f(x),f(y))\leq A\;d_{% 1}(x,y)+B.
  16. M 2 M_{2}
  17. C C
  18. z M 2 : x M 1 : d 2 ( z , f ( x ) ) C . \forall z\in M_{2}:\exists x\in M_{1}:d_{2}(z,f(x))\leq C.
  19. ( M 1 , d 1 ) (M_{1},d_{1})
  20. ( M 2 , d 2 ) (M_{2},d_{2})
  21. f f
  22. ( M 1 , d 1 ) (M_{1},d_{1})
  23. ( M 2 , d 2 ) (M_{2},d_{2})
  24. 2 \sqrt{2}
  25. f : n n f:\mathbb{Z}^{n}\mapsto\mathbb{R}^{n}
  26. n n
  27. n / 4 \sqrt{n/4}
  28. n / 4 \sqrt{n/4}
  29. 2 n / 4 2\sqrt{n/4}
  30. f : M 1 M 2 f:M_{1}\mapsto M_{2}
  31. g : M 2 M 1 g:M_{2}\mapsto M_{1}
  32. g ( x ) g(x)
  33. y y
  34. f f
  35. C C
  36. x x
  37. g ( x ) g(x)
  38. f - 1 ( y ) f^{-1}(y)
  39. k 0 k_{0}
  40. # ( n ) n k 0 \#(n)\sim n^{k_{0}}
  41. # ( n ) \#(n)
  42. \mathbb{N}
  43. ( X , d n , p n ) (X,\frac{d}{n},p_{n})
  44. ( p n ) n (p_{n})_{n}\,
  45. C o n e ω ( X , d , ( p n ) n ) Cone_{\omega}(X,d,(p_{n})_{n})\,
  46. C o n e ω ( X , d ) Cone_{\omega}(X,d)\,
  47. C o n e ω ( X ) Cone_{\omega}(X)\,

Quasi-Monte_Carlo_methods_in_finance.html

  1. ϵ \epsilon
  2. d d
  3. ϵ \epsilon
  4. ϵ - d \epsilon^{-d}
  5. ϵ \epsilon
  6. ϵ \epsilon
  7. d d
  8. φ MC ( f ) = 1 n i = 1 n f ( x i ) , \varphi^{\mathop{\rm MC}}(f)=\frac{1}{n}\sum_{i=1}^{n}f(x_{i}),
  9. x i x_{i}
  10. n - 1 / 2 n^{-1/2}
  11. ϵ \epsilon
  12. ϵ - 2 \epsilon^{-2}
  13. 2 d 2^{d}
  14. φ QMC ( f ) = 1 n i = 1 n f ( x i ) , \varphi^{\mathop{\rm QMC}}(f)=\frac{1}{n}\sum_{i=1}^{n}f(x_{i}),
  15. x i x_{i}
  16. ( log n ) d n , \frac{(\log n)^{d}}{n},
  17. n n
  18. n - 1 / 2 n^{-1/2}
  19. d d
  20. d d
  21. ( log n ) d (\log n)^{d}
  22. d = 360 d=360
  23. log n = 2 \log n=2
  24. 2 360 2^{360}
  25. d > 12 d>12
  26. d < 10 d<10
  27. 10 - 2 10^{-2}
  28. 10 - 2 10^{-2}
  29. ϵ - p \epsilon^{-p}
  30. p p
  31. d d
  32. d d
  33. n - 1 n^{-1}
  34. n n
  35. ( 1 2 π ) d / 2 d cos ( x ) e - x 2 d x , \left(\frac{1}{2\pi}\right)^{d/2}\int_{\mathbb{R}^{d}}\cos(\|x\|)e^{-\|x\|^{2}% }\,dx,
  36. \|\cdot\|
  37. d = 25 d=25
  38. 10 - 2 10^{-2}
  39. d d
  40. d = 100 d=100
  41. c n - 1 , c\cdot n^{-1},
  42. c < 110 c<110
  43. n n
  44. f f
  45. n - 1 / 2 n^{-1/2}
  46. d d
  47. log n / n . \sqrt{\log n}/n.
  48. n - 1 / 2 n^{-1/2}
  49. n - 1 + p ( log n ) - 1 / 2 , n^{-1+p(\log n)^{-1/2}},
  50. p 0 p\geq 0

Quasivariety.html

  1. s 1 t 1 s n t n s t s_{1}\approx t_{1}\land\ldots\land s_{n}\approx t_{n}\rightarrow s\approx t
  2. s , s 1 , , s n , t , t 1 , , t n s,s_{1},\ldots,s_{n},t,t_{1},\ldots,t_{n}

Quercetin-3,3'-bissulfate_7-sulfotransferase.html

  1. \rightleftharpoons

Quercetin-3-sulfate_3'-sulfotransferase.html

  1. \rightleftharpoons

Quercetin-3-sulfate_4'-sulfotransferase.html

  1. \rightleftharpoons

Quercetin_2,3-dioxygenase.html

  1. \rightleftharpoons

Quercetin_3-O-methyltransferase.html

  1. \rightleftharpoons

Questin_monooxygenase.html

  1. \rightleftharpoons

Queue_automaton.html

  1. M = ( Q , Σ , Γ , $ , s , δ ) M=(Q,\Sigma,\Gamma,\$,s,\delta)
  2. Q \,Q
  3. Σ Γ \,\Sigma\subset\Gamma
  4. Γ \,\Gamma
  5. $ Γ - Σ \,\$\in\Gamma-\Sigma
  6. s Q \,s\in Q
  7. δ : Q × Γ Q × Γ * \,\delta:Q\times\Gamma\rightarrow Q\times\Gamma^{*}
  8. ( q , γ ) Q × Γ * \,(q,\gamma)\in Q\times\Gamma^{*}
  9. Γ * \,\Gamma^{*}
  10. Γ \,\Gamma
  11. x \,x
  12. ( s , x $ ) \,(s,x\$)
  13. M 1 \rightarrow_{M}^{1}
  14. ( p , A α ) M 1 ( q , α γ ) \,(p,A\alpha)\rightarrow_{M}^{1}(q,\alpha\gamma)
  15. A A
  16. α \alpha
  17. α Γ * \alpha\in\Gamma^{*}
  18. ( q , γ ) = δ ( p , A ) (q,\gamma)=\delta(p,A)
  19. x Σ * \,x\in\Sigma^{*}
  20. ϵ \,\epsilon
  21. ( s , x $ ) M * ( q , ϵ ) . \,(s,x\$)\rightarrow_{M}^{*}(q,\epsilon).

Quinaldate_4-oxidoreductase.html

  1. \rightleftharpoons

Quinate_dehydrogenase.html

  1. \rightleftharpoons

Quinate_dehydrogenase_(pyrroloquinoline-quinone).html

  1. \rightleftharpoons

Quinine_3-monooxygenase.html

  1. \rightleftharpoons

Quinoline-4-carboxylate_2-oxidoreductase.html

  1. \rightleftharpoons

Quinoline_2-oxidoreductase.html

  1. \rightleftharpoons

Quinoprotein_glucose_dehydrogenase.html

  1. \rightleftharpoons

Rankine_theory.html

  1. K a = cos β - ( cos 2 β - cos 2 ϕ ) 1 / 2 cos β + ( cos 2 β - cos 2 ϕ ) 1 / 2 K_{a}=\frac{\cos\beta-\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}{\cos\beta% +\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}
  2. K p = cos β + ( cos 2 β - cos 2 ϕ ) 1 / 2 cos β - ( cos 2 β - cos 2 ϕ ) 1 / 2 K_{p}=\frac{\cos\beta+\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}{\cos\beta% -\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}
  3. K a = tan 2 ( 45 - ϕ 2 ) K_{a}=\tan^{2}\left(45-\frac{\phi}{2}\right)
  4. K p = tan 2 ( 45 + ϕ 2 ) K_{p}=\tan^{2}\left(45+\frac{\phi}{2}\right)
  5. P a = K a w h P_{a}=K_{a}wh
  6. K a = cos β - ( cos 2 β - cos 2 ϕ ) 1 / 2 cos β + ( cos 2 β - cos 2 ϕ ) 1 / 2 K_{a}=\frac{\cos\beta-\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}{\cos\beta% +\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}
  7. P p = K p w h P_{p}=K_{p}wh
  8. K p = 1 K a K_{p}=\frac{1}{K_{a}}

Rational_homotopy_theory.html

  1. V = n > 0 V n , V=\oplus_{n>0}V^{n},\,
  2. [ a ] , [ a ] , [ b ] \langle[a],[a],[b]\rangle

Read-only_Turing_machine.html

  1. M = ( Q , Σ , Γ , , _ , δ , s , t , r ) M=(Q,\Sigma,\Gamma,\vdash,\_,\delta,s,t,r)
  2. Q Q
  3. Σ \Sigma
  4. Γ \Gamma
  5. Γ - Σ \vdash\in\Gamma-\Sigma
  6. _ Γ - Σ \_\in\Gamma-\Sigma
  7. δ : Q × Γ Q × Γ × { L , R } \delta:Q\times\Gamma\rightarrow Q\times\Gamma\times\{L,R\}
  8. s Q s\in Q
  9. t Q t\in Q
  10. r Q , r t r\in Q,~{}r\neq t
  11. q q
  12. a a
  13. δ ( q , a ) = ( q 2 , a 2 , d ) \delta(q,a)=(q_{2},a_{2},d)
  14. a a
  15. a 2 a_{2}
  16. q 2 q_{2}
  17. d d
  18. a = a 2 a=a_{2}

Regular_matroid.html

  1. M M
  2. F F
  3. M M
  4. F F
  5. U 4 2 U{}^{2}_{4}
  6. U 4 2 U{}^{2}_{4}

Regulated_rewriting.html

  1. M G MG
  2. G = ( N , T , M , S ) G=(N,T,M,S)
  3. N N
  4. T T
  5. N N
  6. M = m 1 , m 2 , , m n M={m_{1},m_{2},...,m_{n}}
  7. m i = [ p i 1 , , p i k ( i ) ] m_{i}=[p_{i_{1}},...,p_{i_{k(i)}}]
  8. k ( i ) 1 k(i)\geq 1
  9. 1 i n 1\leq i\leq n
  10. p i j 1 j k ( i ) p_{i_{j}}1\leq j\leq k(i)
  11. p i j = ( L , R ) p_{i_{j}}=(L,R)
  12. L ( N T ) * N ( N T ) * , R ( N T ) * L\in(N\cup T)^{*}N(N\cup T)^{*},R\in(N\cup T)^{*}
  13. L R L\rightarrow R
  14. m i = [ L i 1 R i 1 , , L i k ( i ) R i k ( i ) ] m_{i}=[L_{i_{1}}\rightarrow R_{i_{1}},...,L_{i_{k(i)}}\rightarrow R_{i_{k(i)}}]
  15. M G = ( N , T , M , S ) MG=(N,T,M,S)
  16. P P
  17. M G MG
  18. M G MG
  19. i = 0 , 1 , 2 , 3 i=0,1,2,3
  20. λ \lambda
  21. G = ( N , T , P , S ) G=(N,T,P,S)
  22. L ( G ) = { a n b n c n : n 1 } L(G)=\{a^{n}b^{n}c^{n}:n\geq 1\}
  23. C F M G CFMG
  24. G = ( N , T , M , S ) G=(N,T,M,S)
  25. N = { S , A , B , C } N=\{S,A,B,C\}
  26. T = { a , b , c } T=\{a,b,c\}
  27. M : M:
  28. [ S a b c ] \left[S\rightarrow abc\right]
  29. [ S a A b B c C ] \left[S\rightarrow aAbBcC\right]
  30. [ A a A , B b B , C c C ] \left[A\rightarrow aA,B\rightarrow bB,C\rightarrow cC\right]
  31. [ A a , B b , C c ] \left[A\rightarrow a,B\rightarrow b,C\rightarrow c\right]
  32. ( G , v ) (G,v)
  33. G = ( N , T , P , S ) G=(N,T,P,S)
  34. v : 2 P v:\mathbb{N}\rightarrow 2^{P}
  35. ( G , s ) (G,s)
  36. G = ( N , T , P , S ) G=(N,T,P,S)
  37. s , f : P 2 P s,f:P\rightarrow 2^{P}
  38. G W R C L GWRCL
  39. ( G , e ) (G,e)
  40. G = ( N , T , P , S ) G=(N,T,P,S)
  41. e e
  42. G = ( N , T , P , S ) G=(N,T,P,S)
  43. N = { S , A , B , C } N=\{S,A,B,C\}
  44. T = { a , b , c } T=\{a,b,c\}
  45. P = { p 0 , p 1 , p 2 , p 3 , p 4 , p 5 , p 6 } P=\{p_{0},p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}\}
  46. p 0 = S A B C p_{0}=S\rightarrow ABC
  47. p 1 = A a A p_{1}=A\rightarrow aA
  48. p 2 = B b B p_{2}=B\rightarrow bB
  49. p 3 = C c C p_{3}=C\rightarrow cC
  50. p 4 = A a p_{4}=A\rightarrow a
  51. p 5 = B b p_{5}=B\rightarrow b
  52. p 6 = C c p_{6}=C\rightarrow c
  53. L ( G ) = { a * b * c * } L(G)=\{a^{*}b^{*}c^{*}\}
  54. P P
  55. P P
  56. e = p 0 ( p 1 p 2 p 3 ) * ( p 4 p 5 p 6 ) e=p_{0}(p_{1}p_{2}p_{3})^{*}(p_{4}p_{5}p_{6})
  57. G G
  58. e e
  59. ( G , e ) = ( G , p 0 ( p 1 p 2 p 3 ) * ( p 4 p 5 p 6 ) ) (G,e)=(G,p_{0}(p_{1}p_{2}p_{3})^{*}(p_{4}p_{5}p_{6}))
  60. L ( G ) = { a n b n c n : n 1 } L(G)=\{a^{n}b^{n}c^{n}:n\geq 1\}

Relative_dimension.html

  1. V Q V\to Q
  2. V W V\to W
  3. W * V * W^{*}\to V^{*}

Relative_volatility.html

  1. α \alpha
  2. α = ( y i / x i ) ( y j / x j ) = K i / K j \alpha=\frac{(y_{i}/x_{i})}{(y_{j}/x_{j})}=K_{i}/K_{j}
  3. α \alpha
  4. i i
  5. j j
  6. y i y_{i}
  7. i i
  8. x i x_{i}
  9. i i
  10. y j y_{j}
  11. j j
  12. x j x_{j}
  13. j j
  14. ( y / x ) (y/x)
  15. K K
  16. K K
  17. y / x y/x
  18. K K
  19. α \alpha
  20. K K
  21. K K
  22. α \alpha
  23. α \alpha
  24. α \alpha
  25. α = ( y L K / x L K ) ( y H K / x H K ) = K L K / K H K \alpha=\frac{(y_{LK}/x_{LK})}{(y_{HK}/x_{HK})}=K_{LK}/K_{HK}
  26. K K
  27. K K

Renilla-luciferin_2-monooxygenase.html

  1. \rightleftharpoons

Renilla-luciferin_sulfotransferase.html

  1. \rightleftharpoons

Repeating_decimal.html

  1. 3227 555 \frac{3227}{555}
  2. 0.333 0.333…
  3. 3.14159 3.14159…
  4. 0. 1 ˙ 0.\dot{1}
  5. 0. 3 ˙ 0.\dot{3}
  6. 0. 6 ˙ 0.\dot{6}
  7. 0. 8 ˙ 1 ˙ 0.\dot{8}\dot{1}
  8. 0.58 3 ˙ 0.58\dot{3}
  9. 0. 0 ˙ 1234567 9 ˙ 0.\dot{0}1234567\dot{9}
  10. 3. 1 ˙ 4285 7 ˙ 3.\dot{1}4285\dot{7}
  11. 1 / 7 {1}/{7}
  12. 10 p - 1 - 1 p . \frac{10^{p-1}-1}{p}.
  13. 10 11 - 1 - 1 11 = 909090909 \frac{10^{11-1}-1}{11}=909090909
  14. λ ( n ) \lambda(n)
  15. ϕ ( n ) \phi(n)
  16. ϕ \phi
  17. ϕ ( n ) \phi(n)
  18. λ ( p ) = p - 1 \lambda(p)=p-1
  19. 1 p 2 \tfrac{1}{p^{2}}
  20. a m 1 ( mod n ) a^{m}\equiv 1\;\;(\mathop{{\rm mod}}n)
  21. 1 p 2 \tfrac{1}{p^{2}}
  22. 1 p \tfrac{1}{p}
  23. 1 p 2 \tfrac{1}{p^{2}}
  24. 1 p \tfrac{1}{p}
  25. 1 p k \tfrac{1}{p^{k}}
  26. 1 p q \tfrac{1}{p\ q}
  27. 1 p q \tfrac{1}{p\ q}
  28. 1 p q \tfrac{1}{p\ q}
  29. 1 p \tfrac{1}{p}
  30. 1 q \tfrac{1}{q}
  31. 1 p k q r m \frac{1}{p^{k}q^{\ell}r^{m}\cdots}
  32. LCM ( T p k , T q , T r m , ) \mathrm{LCM}(T_{p^{k}},T_{q^{\ell}},T_{r^{m}},\ldots)
  33. T p k , T q , T r m T_{p^{k}},\ T_{q^{\ell}},\ T_{r^{m}}
  34. 1 p k , 1 q , 1 r m , \frac{1}{p^{k}},\ \frac{1}{q^{\ell}},\ \frac{1}{r^{m}},
  35. 1 2 a 5 b p k q , \frac{1}{2^{a}5^{b}p^{k}q^{\ell}\cdots}\,,
  36. 5 a - b 10 a p k q , \frac{5^{a-b}}{10^{a}p^{k}q^{\ell}\cdots}\,,
  37. 2 b - a 10 b p k q , \frac{2^{b-a}}{10^{b}p^{k}q^{\ell}\cdots}\,,
  38. 1 10 a p k q , \frac{1}{10^{a}p^{k}q^{\ell}\cdots}\,,
  39. 1 p k q \frac{1}{p^{k}q^{\ell}\cdots}
  40. x \displaystyle x
  41. x \displaystyle x
  42. x \displaystyle x
  43. 7.48181818 \displaystyle 7.48181818\ldots
  44. n = 1 1 10 n = 1 10 + 1 100 + 1 1000 + = 0. 1 ¯ \sum_{n=1}^{\infty}\frac{1}{10^{n}}={1\over 10}+{1\over 100}+{1\over 1000}+% \cdots=0.\overline{1}
  45. a 1 - r = 1 10 1 - 1 10 = 1 9 = 0. 1 ¯ \ \frac{a}{1-r}=\frac{\frac{1}{10}}{1-\frac{1}{10}}=\frac{1}{9}=0.\overline{1}
  46. k = 2 a 5 b n k=2^{a}5^{b}n
  47. 10 r 1 ( mod n ) 10^{r}\equiv 1\;\;(\mathop{{\rm mod}}n)
  48. 1 k k \frac{1}{kk^{\prime}}
  49. 1 k \frac{1}{k}
  50. 1 k \frac{1}{k^{\prime}}
  51. period ( 1 p ) = period ( 1 p 2 ) = = period ( 1 p m ) \,\text{period}(\tfrac{1}{p})=\,\text{period}(\tfrac{1}{p^{2}})=\cdots=\,\text% {period}(\tfrac{1}{p^{m}})
  52. period ( 1 p m ) period ( 1 p m + 1 ) \,\text{period}(\tfrac{1}{p^{m}})\neq\,\text{period}(\tfrac{1}{p^{m+1}})
  53. c 0 c\geq 0
  54. period ( 1 p m + c ) = p c period ( 1 p ) \,\text{period}(\tfrac{1}{p^{m+c}})=p^{c}\cdot\,\text{period}(\tfrac{1}{p})
  55. 1 / p 1/p
  56. a ( i ) = 2 i mod p mod 2 a(i)=2^{i}~{}\bmod p~{}\bmod 2