wpmath0000014_10

Polymeric_surface.html

  1. V o l I k i o n n e n 0 d V o l I = n e τ n V o l I \textstyle\int\limits_{{V\!ol}_{I}}{k^{ion}}{n_{e}}{n_{0}}\,d{{V\!ol}_{I}}={% \frac{n_{e}}{\tau_{n}}}{V\!ol_{I}}
  2. V o l I {{V\!ol}_{I}}
  3. k i o n k^{ion}
  4. n 0 n_{0}
  5. n e n_{e}
  6. τ n \tau_{n}
  7. s y m b o l γ S G = s y m b o l γ S L + s y m b o l γ L G cos s y m b o l θ c symbol{\gamma}_{SG}=symbol{\gamma}_{SL}+symbol{\gamma}_{LG}~{}{\cos{symbol{% \theta}_{c}}}
  8. s y m b o l γ i j symbol{\gamma}_{ij}
  9. s y m b o l θ c {symbol{\theta}_{c}}

Polyvector_field.html

  1. Λ k T M \Lambda^{k}TM

Pondage.html

  1. 7 × 24 5 × 8 = 4.2 \frac{7\times 24}{5\times 8}=4.2

Porod's_law.html

  1. q 1 nm - 1 q\lesssim 1\,\text{ nm}^{-1}
  2. I ( q ) S q - 4 I(q)\sim Sq^{-4}
  3. lim q I ( q ) S q - ( 6 - d ) \lim_{q\rightarrow\infty}I(q)\propto S^{\prime}q^{-(6-d)}
  4. q q\to\infty
  5. S ( q ) = 4 π 2 q z 2 δ ( q x ) δ ( q y ) . S(\vec{q})=\frac{4\pi^{2}}{q_{z}^{2}}\delta(q_{x})\delta(q_{y}).
  6. S ( q ) = 2 π q 4 . S(q)=\frac{2\pi}{q^{4}}.

Portal:Infrastructure::Economic_analysis.html

  1. r a t i o = P V ( B 2 ) - P V ( B 1 ) P V ( C 2 ) - P V ( C 1 ) \scriptstyle ratio=\frac{PV(B_{2})-PV(B_{1})}{PV(C_{2})-PV(C_{1})}
  2. E A C = N P V A t , r \scriptstyle EAC=\frac{NPV}{A_{t,r}}
  3. F V = P V ( 1 + i ) n \scriptstyle FV=PV\cdot(1+i)^{n}
  4. N P V = n = 0 N C n ( 1 + r ) n = 0 \scriptstyle NPV=\sum_{n=0}^{N}\frac{C_{n}}{(1+r)^{n}}=0
  5. PV = F V ( 1 + i ) n \scriptstyle\mathrm{PV}=\frac{FV}{(1+i)^{n}}\,
  6. R O I = P r o f i t I n v e s t m e n t \scriptstyle ROI=\frac{Profit}{Investment}

Portal:Infrastructure::Selected_article::1.html

  1. Y t = A t * ( N t , K t , G t ) \qquad\qquad Y_{t}=A_{t}*(N_{t},K_{t},G_{t})

Portal:Mathematics::Featured_picture::2011_02.html

  1. f f\,

Portal:Star::Selected_article::19.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Poset_game.html

  1. G ( P ) = min ( { G ( P x ) x P } ) . G(P)=\min\bigl(\mathbb{N}\setminus\{G(P_{x})\mid x\in P\}\bigr).

Positive_harmonic_function.html

  1. f ( r e i θ ) = 0 2 π 1 - r 2 1 - 2 r cos ( θ - φ ) + r 2 d μ ( φ ) . f(re^{i\theta})=\int_{0}^{2\pi}{1-r^{2}\over 1-2r\cos(\theta-\varphi)+r^{2}}\,% d\mu(\varphi).
  2. f n ( z ) = f ( r n z ) . f_{n}(z)=f(r_{n}z).\,
  3. f n ( r e i θ ) = 1 2 π 0 2 π 1 - r 2 1 - 2 r cos ( θ - φ ) + r 2 f n ( φ ) d ϕ = 0 2 π 1 - r 2 1 - 2 r cos ( θ - φ ) + r 2 d μ n ( φ ) f_{n}(re^{i\theta})={1\over 2\pi}\int_{0}^{2\pi}{1-r^{2}\over 1-2r\cos(\theta-% \varphi)+r^{2}}\,f_{n}(\varphi)\,d\phi=\int_{0}^{2\pi}{1-r^{2}\over 1-2r\cos(% \theta-\varphi)+r^{2}}d\mu_{n}(\varphi)
  4. d μ n ( φ ) = 1 2 π f ( r n e i φ ) d φ d\mu_{n}(\varphi)={1\over 2\pi}f(r_{n}e^{i\varphi})\,d\varphi
  5. f ( z ) = 0 2 π 1 + e - i θ z 1 - e - i θ z d μ ( θ ) . f(z)=\int_{0}^{2\pi}{1+e^{-i\theta}z\over 1-e^{-i\theta}z}\,d\mu(\theta).
  6. f ( z ) = 1 + a 1 z + a 2 z 2 + f(z)=1+a_{1}z+a_{2}z^{2}+\cdots
  7. m n a m - n λ m λ n ¯ 0 \sum_{m}\sum_{n}a_{m-n}\lambda_{m}\overline{\lambda_{n}}\geq 0
  8. a 0 = 2 , a - m = a m ¯ a_{0}=2,\,\,\,a_{-m}=\overline{a_{m}}
  9. a n = 2 0 2 π e - i n θ d μ ( θ ) . a_{n}=2\int_{0}^{2\pi}e^{-in\theta}\,d\mu(\theta).
  10. m n a m - n λ m λ n ¯ = 0 2 π | n λ n e - i n θ | 2 d μ ( θ ) 0. \sum_{m}\sum_{n}a_{m-n}\lambda_{m}\overline{\lambda_{n}}=\int_{0}^{2\pi}\left|% \sum_{n}\lambda_{n}e^{-in\theta}\right|^{2}\,d\mu(\theta)\geq 0.
  11. m = 0 n = 0 a m - n λ m λ n ¯ = 2 ( 1 - | z | 2 ) f ( z ) . \sum_{m=0}^{\infty}\sum_{n=0}^{\infty}a_{m-n}\lambda_{m}\overline{\lambda_{n}}% =2(1-|z|^{2})\,\Re\,f(z).

Posterior_predictive_distribution.html

  1. x ~ \tilde{x}
  2. 𝐗 = { x 1 , , x N } \mathbf{X}=\{x_{1},\dots,x_{N}\}
  3. p ( x ~ | 𝐗 , α ) = θ p ( x ~ | θ ) p ( θ | 𝐗 , α ) d θ p(\tilde{x}|\mathbf{X},\alpha)=\int_{\theta}p(\tilde{x}|\theta)\,p(\theta|% \mathbf{X},\alpha)\operatorname{d}\!\theta
  4. θ \theta\,
  5. α \alpha\,
  6. x ~ , θ , α \tilde{x},\theta,\alpha
  7. p ( x ~ | 𝐗 , α ) = 𝔼 θ | 𝐗 , α [ p ( x ~ | θ ) ] p(\tilde{x}|\mathbf{X},\alpha)=\mathbb{E}_{\theta|\mathbf{X},\alpha}\Big[p(% \tilde{x}|\theta)\Big]
  8. x ~ F ( x ~ | θ ) \tilde{x}\sim F(\tilde{x}|\theta)
  9. θ G ( θ | α ) \theta\sim G(\theta|\alpha)
  10. H ( x ~ | α ) H(\tilde{x}|\alpha)
  11. p H ( x ~ | α ) = θ p F ( x ~ | θ ) p G ( θ | α ) d θ p_{H}(\tilde{x}|\alpha)=\int_{\theta}p_{F}(\tilde{x}|\theta)\,p_{G}(\theta|% \alpha)\operatorname{d}\!\theta
  12. G ( θ | α ) G(\theta|\alpha)
  13. G ( θ | α ) G(\theta|\alpha)
  14. p ( θ | 𝐗 , α ) = p G ( θ | α ) , p(\theta|\mathbf{X},\alpha)=p_{G}(\theta|\alpha^{\prime}),
  15. G ( θ | α ) , G(\theta|\alpha),
  16. α \alpha^{\prime}
  17. α . \alpha.
  18. p ( x ~ | 𝐗 , α ) = θ p F ( x ~ | θ ) p ( θ | 𝐗 , α ) d θ = θ p F ( x ~ | θ ) p G ( θ | α ) d θ = p H ( x ~ | α ) \begin{aligned}\displaystyle p(\tilde{x}|\mathbf{X},\alpha)&\displaystyle=\int% _{\theta}p_{F}(\tilde{x}|\theta)\,p(\theta|\mathbf{X},\alpha)\operatorname{d}% \!\theta\\ &\displaystyle=\int_{\theta}p_{F}(\tilde{x}|\theta)\,p_{G}(\theta|\alpha^{% \prime})\operatorname{d}\!\theta\\ &\displaystyle=p_{H}(\tilde{x}|\alpha^{\prime})\end{aligned}
  19. 𝐗 \mathbf{X}
  20. t ( x | μ , ν , σ 2 ) t(x|\mu,\nu,\sigma^{2})
  21. ν , σ 2 \nu^{\prime},{\sigma^{2}}^{\prime}
  22. F ( x | s y m b o l θ ) F(x|symbol{\theta})
  23. s y m b o l θ symbol{\theta}
  24. s y m b o l η = s y m b o l η ( s y m b o l θ ) symbol{\eta}=symbol{\eta}(symbol{\theta})
  25. p F ( x | s y m b o l η ) = h ( x ) g ( s y m b o l η ) e s y m b o l η T 𝐓 ( x ) p_{F}(x|symbol{\eta})=h(x)g(symbol{\eta})e^{symbol{\eta}^{\rm T}\mathbf{T}(x)}
  26. G ( s y m b o l η | s y m b o l χ , ν ) G(symbol{\eta}|symbol{\chi},\nu)
  27. p G ( s y m b o l η | s y m b o l χ , ν ) = f ( s y m b o l χ , ν ) g ( s y m b o l η ) ν e s y m b o l η T s y m b o l χ p_{G}(symbol{\eta}|symbol{\chi},\nu)=f(symbol{\chi},\nu)g(symbol{\eta})^{\nu}e% ^{symbol{\eta}^{\rm T}symbol{\chi}}
  28. H H
  29. F F
  30. G G
  31. p H ( x | s y m b o l χ , ν ) \displaystyle p_{H}(x|symbol{\chi},\nu)
  32. G ( s y m b o l η | s y m b o l χ + 𝐓 ( x ) , ν + 1 ) G(symbol{\eta}|symbol{\chi}+\mathbf{T}(x),\nu+1)
  33. f ( ) f(\dots)\,
  34. s y m b o l θ symbol{\theta}
  35. s y m b o l θ symbol{\theta}
  36. s y m b o l η symbol{\eta}
  37. g ( ) g(\dots)\,
  38. g ( ) g(\dots)\,
  39. F F
  40. G G
  41. h ( x ) h(x)
  42. F F
  43. G G
  44. s y m b o l χ + 𝐓 ( x ) symbol{\chi}+\mathbf{T}(x)
  45. s y m b o l χ + 𝐓 ( x ) χ symbol{\chi}+\mathbf{T}(x){\chi}
  46. f ( ) f(\dots)\,
  47. p ( x ~ | 𝐗 , s y m b o l χ , ν ) = p H ( x ~ | s y m b o l χ + 𝐓 ( 𝐗 ) , ν + N ) \begin{array}[]{lcl}p(\tilde{x}|\mathbf{X},symbol{\chi},\nu)&=&p_{H}\left(% \tilde{x}|symbol{\chi}+\mathbf{T}(\mathbf{X}),\nu+N\right)\end{array}
  48. 𝐓 ( 𝐗 ) = i = 1 N 𝐓 ( x i ) \mathbf{T}(\mathbf{X})=\sum_{i=1}^{N}\mathbf{T}(x_{i})
  49. 𝐓 ( 𝐗 ) = i = 1 N 𝐓 ( x i ) . \mathbf{T}(\mathbf{X})=\sum_{i=1}^{N}\mathbf{T}(x_{i}).
  50. 𝐗 = { x 1 , , x N } \mathbf{X}=\{x_{1},\dots,x_{N}\}
  51. N N
  52. p H ( 𝐗 | s y m b o l χ , ν ) = ( i = 1 N h ( x i ) ) f ( s y m b o l χ , ν ) f ( s y m b o l χ + 𝐓 ( 𝐗 ) , ν + N ) p_{H}(\mathbf{X}|symbol{\chi},\nu)=\left(\prod_{i=1}^{N}h(x_{i})\right)\dfrac{% f(symbol{\chi},\nu)}{f\left(symbol{\chi}+\mathbf{T}(\mathbf{X}),\nu+N\right)}

Potassium_hexafluoronickelate(IV).html

  1. 3 K 2 NiF 6 Δ 2 K 3 NiF 6 + NiF 2 + F 2 \rm\ 3K_{2}NiF_{6}\xrightarrow{\Delta}2K_{3}NiF_{6}+NiF_{2}+F_{2}
  2. K 2 NiF 6 + 5 A s F 5 + XF 5 aHF XF 6 AsF 6 + Ni ( AsF 6 ) 2 + 2 K A s F 6 \rm\ K_{2}NiF_{6}+5AsF_{5}+XF_{5}\xrightarrow{aHF}XF_{6}AsF_{6}+Ni(AsF_{6})_{2% }+2KAsF_{6}

Pólya_urn_model.html

  1. 1 / m 1/m
  2. n n
  3. n n
  4. k k
  5. w + n w w+n_{w}
  6. w + n w w + b + n \frac{w+n_{w}}{w+b+n}
  7. n = 1 , 2 , 3 , n=1,2,3,\dots
  8. α \alpha

Prais–Winsten_estimation.html

  1. y t = α + X t β + ε t , y_{t}=\alpha+X_{t}\beta+\varepsilon_{t},\,
  2. y t y_{t}
  3. β \beta
  4. X t X_{t}
  5. ε t \varepsilon_{t}
  6. ε t = ρ ε t - 1 + e t , | ρ | < 1 \varepsilon_{t}=\rho\varepsilon_{t-1}+e_{t},\ |\rho|<1
  7. e t e_{t}
  8. y t - ρ y t - 1 = α ( 1 - ρ ) + β ( X t - ρ X t - 1 ) + e t . y_{t}-\rho y_{t-1}=\alpha(1-\rho)+\beta(X_{t}-\rho X_{t-1})+e_{t}.\,
  9. 1 - ρ 2 y 1 = α 1 - ρ 2 + ( 1 - ρ 2 X 1 ) β + 1 - ρ 2 ε 1 . \sqrt{1-\rho^{2}}y_{1}=\alpha\sqrt{1-\rho^{2}}+\left(\sqrt{1-\rho^{2}}X_{1}% \right)\beta+\sqrt{1-\rho^{2}}\varepsilon_{1}.\,
  10. cov ( ε t , ε t + h ) = ρ h 1 - ρ 2 , for h = 0 , ± 1 , ± 2 , . \mathrm{cov}(\varepsilon_{t},\varepsilon_{t+h})=\frac{\rho^{h}}{1-\rho^{2}},\,% \text{ for }h=0,\pm 1,\pm 2,\dots\,.
  11. 𝛀 \mathbf{\Omega}
  12. 𝛀 = [ 1 1 - ρ 2 ρ 1 - ρ 2 ρ 2 1 - ρ 2 ρ T - 1 1 - ρ 2 ρ 1 - ρ 2 1 1 - ρ 2 ρ 1 - ρ 2 ρ T - 2 1 - ρ 2 ρ 2 1 - ρ 2 ρ 1 - ρ 2 1 1 - ρ 2 ρ T - 2 1 - ρ 2 ρ T - 1 1 - ρ 2 ρ T - 2 1 - ρ 2 ρ T - 3 1 - ρ 2 1 1 - ρ 2 ] . \mathbf{\Omega}=\begin{bmatrix}\frac{1}{1-\rho^{2}}&\frac{\rho}{1-\rho^{2}}&% \frac{\rho^{2}}{1-\rho^{2}}&\cdots&\frac{\rho^{T-1}}{1-\rho^{2}}\\ \frac{\rho}{1-\rho^{2}}&\frac{1}{1-\rho^{2}}&\frac{\rho}{1-\rho^{2}}&\cdots&% \frac{\rho^{T-2}}{1-\rho^{2}}\\ \frac{\rho^{2}}{1-\rho^{2}}&\frac{\rho}{1-\rho^{2}}&\frac{1}{1-\rho^{2}}&% \cdots&\frac{\rho^{T-2}}{1-\rho^{2}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{\rho^{T-1}}{1-\rho^{2}}&\frac{\rho^{T-2}}{1-\rho^{2}}&\frac{\rho^{T-3}}{% 1-\rho^{2}}&\cdots&\frac{1}{1-\rho^{2}}\end{bmatrix}.
  13. ρ \rho
  14. Θ ^ = ( 𝐙 𝛀 - 1 𝐙 ) - 1 ( 𝐙 𝛀 - 1 𝐘 ) , \hat{\Theta}=(\mathbf{Z}^{\prime}\mathbf{\Omega}^{-1}\mathbf{Z})^{-1}(\mathbf{% Z}^{\prime}\mathbf{\Omega}^{-1}\mathbf{Y}),\,
  15. 𝐙 \mathbf{Z}
  16. 𝐘 \mathbf{Y}
  17. Θ ^ \hat{\Theta}
  18. 𝛀 \mathbf{\Omega}
  19. 𝛀 - 1 = 𝐆 𝐆 \mathbf{\Omega}^{-1}=\mathbf{G}^{\prime}\mathbf{G}
  20. 𝐆 = [ 1 - ρ 2 0 0 0 - ρ 1 0 0 0 - ρ 1 0 0 0 0 1 ] . \mathbf{G}=\begin{bmatrix}\sqrt{1-\rho^{2}}&0&0&\cdots&0\\ -\rho&1&0&\cdots&0\\ 0&-\rho&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\end{bmatrix}.
  21. ρ \rho

Pre-_and_post-test_probability.html

  1. b n = Δ p × r i × ( b i - h i ) - h t b_{n}=\Delta p\times r_{i}\times(b_{i}-h_{i})-h_{t}

Pre-Lie_algebra.html

  1. ( V , ) (V,\triangleleft)
  2. V V
  3. : V V V \triangleleft:V\otimes V\to V
  4. ( x y ) z - x ( y z ) = ( x z ) y - x ( z y ) . (x\triangleleft y)\triangleleft z-x\triangleleft(y\triangleleft z)=(x% \triangleleft z)\triangleleft y-x\triangleleft(z\triangleleft y).
  5. ( x , y , z ) = ( x y ) z - x ( y z ) (x,y,z)=(x\triangleleft y)\triangleleft z-x\triangleleft(y\triangleleft z)
  6. y y
  7. z z
  8. f ( x ) x f(x)\partial_{x}
  9. x f ( x ) x\mapsto f(x)
  10. \triangleleft
  11. f ( x ) g ( x ) = f ( x ) g ( x ) f(x)\triangleleft g(x)=f^{\prime}(x)g(x)
  12. \triangleleft
  13. g ( x ) x g(x)\partial_{x}
  14. f ( x ) x f(x)\partial_{x}
  15. ( g ( x ) x ) ( f ( x ) x ) = g ( x ) x f ( x ) x = g ( x ) f ( x ) x (g(x)\partial_{x})(f(x)\partial_{x})=g(x)\partial_{x}f(x)\partial_{x}=g(x)f^{% \prime}(x)\partial_{x}
  16. ( x y ) z (x\triangleleft y)\triangleleft z
  17. x ( y z ) x\triangleleft(y\triangleleft z)
  18. ( x y ) z - x ( y z ) = ( x y ) z - x y z = x y z x ′′ y z - z y z = x ′′ y z (x\triangleleft y)\triangleleft z-x\triangleleft(y\triangleleft z)=(x^{\prime}% y)^{\prime}z-x^{\prime}y^{\prime}z=x^{\prime}y^{\prime}zx^{\prime\prime}yz-z^{% \prime}y^{\prime}z=x^{\prime\prime}yz
  19. 𝕋 \mathbb{T}
  20. \curvearrowleft
  21. 𝕋 \mathbb{T}
  22. τ 1 \tau_{1}
  23. τ 2 \tau_{2}
  24. τ 1 τ 2 = s Vertices ( τ 1 ) τ 1 s τ 2 \tau_{1}\curvearrowleft\tau_{2}=\sum_{s\in\mathrm{Vertices}(\tau_{1})}\tau_{1}% \circ_{s}\tau_{2}
  25. τ 1 s τ 2 \tau_{1}\circ_{s}\tau_{2}
  26. τ 1 \tau_{1}
  27. τ 2 \tau_{2}
  28. s s
  29. τ 1 \tau_{1}
  30. τ 2 \tau_{2}
  31. ( 𝕋 , ) (\mathbb{T},\curvearrowleft)

Predicative_programming.html

  1. ( P Q ) ( b Q P ) (P\sqsubseteq Q)\equiv(\forall b\cdot Q\Rightarrow P)
  2. x x
  3. y y
  4. z z
  5. x := y + 1 x:=y+1\,\!
  6. x = y + 1 y = y z = z x^{\prime}=y+1\land y^{\prime}=y\land z^{\prime}=z
  7. x x
  8. y y
  9. z z
  10. x x^{\prime}
  11. y y^{\prime}
  12. z z^{\prime}
  13. x > y x := y + 1 x^{\prime}>y\sqsubseteq x:=y+1

Predictable_process.html

  1. ( Ω , , ( n ) n , ) (\Omega,\mathcal{F},(\mathcal{F}_{n})_{n\in\mathbb{N}},\mathbb{P})
  2. ( X n ) n (X_{n})_{n\in\mathbb{N}}
  3. X n + 1 X_{n+1}
  4. n \mathcal{F}_{n}
  5. ( Ω , , ( t ) t 0 , ) (\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})
  6. ( X t ) t 0 (X_{t})_{t\geq 0}
  7. X X
  8. Ω × + \Omega\times\mathbb{R}_{+}

Preference_(economics).html

  1. A A\!
  2. B B\!
  3. B B\!
  4. C C\!
  5. A A\!
  6. C C\!
  7. A B A\succ B
  8. B C , B\succ C,
  9. A C . A\succ C.
  10. \geqslant\!
  11. A B A\geqslant B\!
  12. B A B\geqslant A\!
  13. A = B . A=B.
  14. \sim\!
  15. A B A\succeq B
  16. B A B\succeq A
  17. A B . A\sim B.
  18. \succsim\!
  19. \succsim\!
  20. \succsim\!
  21. u u\!
  22. u u\!
  23. \succsim\!
  24. \succsim\!
  25. A A\!
  26. B B\!
  27. A B A\!\succsim\!B\!
  28. B A B\!\succsim\!A\!
  29. A A A\!\succsim\!A\!
  30. u ( A ) u ( B ) u\left(A\right)\geqslant u(B)
  31. A B A\succsim\!B
  32. \succsim\!
  33. R n R^{n}
  34. \succ\!
  35. \succsim\!
  36. \succ\!

Preference_learning.html

  1. X = { x i } X=\{x_{i}\}\,\!
  2. Y = { y i | i = 1 , 2 , , k } Y=\{y_{i}|i=1,2,\cdots,k\}\,\!
  3. y i x y j y_{i}\succ_{x}y_{j}\,\!
  4. x x\,\!
  5. y i y_{i}\,\!
  6. y j y_{j}\,\!
  7. x x\,\!
  8. y i y_{i}\,\!
  9. j i , y i x y j \forall j\neq i,y_{i}\succ_{x}y_{j}\,\!
  10. x x\,\!
  11. L Y L\subseteq Y\,\!
  12. { y i x y j | y i L , y j Y \ L } \{y_{i}\succ_{x}y_{j}|y_{i}\in L,y_{j}\in Y\backslash L\}\,\!
  13. X X\,\!
  14. Y Y\,\!
  15. y 1 y 2 y k y_{1}\succ y_{2}\succ\cdots\succ y_{k}\,\!
  16. x l x_{l}\,\!
  17. y l y_{l}\,\!
  18. x i x j x_{i}\succ x_{j}\,\!
  19. A B A\succ B\,\!
  20. A A\,\!
  21. B B\,\!
  22. a a\,\!
  23. b b\,\!
  24. a > b a>b\,\!
  25. V ( A , B ) { 0 , 1 } V(A,B)\in\{0,1\}\,\!
  26. ( A , B ) (A,B)\,\!
  27. A B A\succ B\,\!
  28. B A B\succ A\,\!
  29. f : X × Y f:X\times Y\rightarrow\mathbb{R}\,\!
  30. y i x y j f ( x , y i ) > f ( x , y j ) y_{i}\succ_{x}y_{j}\Rightarrow f(x,y_{i})>f(x,y_{j})\,\!
  31. f : X f:X\rightarrow\mathbb{R}\,\!

Preference_ranking_organization_method_for_enrichment_evaluation.html

  1. A = { a 1 , . . , a n } A=\{a_{1},..,a_{n}\}
  2. F = { f 1 , . . , f q } F=\{f_{1},..,f_{q}\}
  3. n × q n\times q
  4. f 1 ( ) f 2 ( ) f j ( ) f q ( ) a 1 f 1 ( a 1 ) f 2 ( a 1 ) f j ( a 1 ) f q ( a 1 ) a 2 f 1 ( a 2 ) f 2 ( a 2 ) f j ( a 2 ) f q ( a 2 ) . a i f 1 ( a i ) f 2 ( a i ) f j ( a i ) f q ( a i ) a n f 1 ( a n ) f 2 ( a n ) f j ( a n ) f q ( a n ) \begin{array}[]{|c|c|c|c|c|c|c|}\hline&f_{1}(\cdot)&f_{2}(\cdot)&\cdots&f_{j}(% \cdot)&\cdots&f_{q}(\cdot)\\ \hline a_{1}&f_{1}(a_{1})&f_{2}(a_{1})&\cdots&f_{j}(a_{1})&\cdots&f_{q}(a_{1})% \\ \hline a_{2}&f_{1}(a_{2})&f_{2}(a_{2})&\cdots&f_{j}(a_{2})&\cdots&f_{q}(a_{2})% \\ \hline\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&.\cdots\\ \hline a_{i}&f_{1}(a_{i})&f_{2}(a_{i})&\cdots&f_{j}(a_{i})&\cdots&f_{q}(a_{i})% \\ \hline\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \hline a_{n}&f_{1}(a_{n})&f_{2}(a_{n})&\cdots&f_{j}(a_{n})&\cdots&f_{q}(a_{n})% \\ \hline\end{array}
  5. d k ( a i , a j ) = f k ( a i ) - f k ( a j ) d_{k}(a_{i},a_{j})=f_{k}(a_{i})-f_{k}(a_{j})
  6. d k ( a i , a j ) d_{k}(a_{i},a_{j})
  7. f k f_{k}
  8. π k ( a i , a j ) = P k [ d k ( a i , a j ) ] \pi_{k}(a_{i},a_{j})=P_{k}[d_{k}(a_{i},a_{j})]
  9. P k : \R [ 0 , 1 ] P_{k}:\R\rightarrow[0,1]
  10. P j ( 0 ) = 0 P_{j}(0)=0
  11. P k ( x ) { 0 , if x q k x - q k p k - q k , if q k < x p k 1 , if x > p k P_{k}(x)\begin{cases}0,&\,\text{if }x\leq q_{k}\\ \frac{x-q_{k}}{p_{k}-q_{k}},&\,\text{if }q_{k}<x\leq p_{k}\\ 1,&\,\text{if }x>p_{k}\end{cases}
  12. q j q_{j}
  13. p j p_{j}
  14. π ( a , b ) = k = 1 q P k ( a , b ) w k \pi(a,b)=\displaystyle\sum_{k=1}^{q}P_{k}(a,b)\cdot w_{k}
  15. w k w_{k}
  16. f k f_{k}
  17. w k 0 w_{k}\geq 0
  18. k = 1 q w k = 1 \sum_{k=1}^{q}w_{k}=1
  19. π ( a i , a j ) 0 \pi(a_{i},a_{j})\geq 0
  20. π ( a i , a j ) + π ( a j , a i ) 1 \pi(a_{i},a_{j})+\pi(a_{j},a_{i})\leq 1
  21. ϕ + ( a ) = 1 n - 1 x A π ( a , x ) \phi^{+}(a)=\frac{1}{n-1}\displaystyle\sum_{x\in A}\pi(a,x)
  22. ϕ - ( a ) = 1 n - 1 x A π ( x , a ) \phi^{-}(a)=\frac{1}{n-1}\displaystyle\sum_{x\in A}\pi(x,a)
  23. ϕ + ( a i ) \phi^{+}(a_{i})
  24. a i a_{i}
  25. ϕ - ( a i ) \phi^{-}(a_{i})
  26. a i a_{i}
  27. a i a_{i}
  28. a j a_{j}
  29. ϕ - ( a i ) ϕ - ( a j ) \phi^{-}(a_{i})\geq\phi^{-}(a_{j})
  30. ϕ - ( a i ) ϕ - ( a j ) \phi^{-}(a_{i})\leq\phi^{-}(a_{j})
  31. ϕ ( a ) = ϕ + ( a ) - ϕ - ( a ) \phi(a)=\phi^{+}(a)-\phi^{-}(a)
  32. ϕ ( a i ) [ - 1 ; 1 ] \phi(a_{i})\in[-1;1]
  33. a i A ϕ ( a i ) = 0 \sum_{a_{i}\in A}\phi(a_{i})=0
  34. ϕ ( a i ) = k = 1 q ϕ k ( a i ) . w k \phi(a_{i})=\displaystyle\sum_{k=1}^{q}\phi_{k}(a_{i}).w_{k}
  35. ϕ k ( a i ) = 1 n - 1 a j A { P k ( a i , a j ) - P k ( a j , a i ) } \phi_{k}(a_{i})=\frac{1}{n-1}\displaystyle\sum_{a_{j}\in A}\{P_{k}(a_{i},a_{j}% )-P_{k}(a_{j},a_{i})\}
  36. ϕ k ( a i ) [ - 1 ; 1 ] \phi_{k}(a_{i})\in[-1;1]
  37. ϕ ( a i ) \phi(a_{i})
  38. a i a_{i}
  39. ϕ ( a i ) = [ ϕ 1 ( a i ) , , ϕ k ( a i ) , ϕ q ( a i ) ] \vec{\phi}(a_{i})=[\phi_{1}(a_{i}),\ldots,\phi_{k}(a_{i}),\phi_{q}(a_{i})]
  40. q q
  41. P j ( d j ) = { 0 if d j 0 1 if d j > 0 P_{j}(d_{j})=\begin{cases}0&\,\text{if }d_{j}\leq 0\\ 1&\,\text{if }d_{j}>0\end{cases}
  42. P j ( d j ) = { 0 if | d j | q j 1 if | d j | > q j \begin{array}[]{cc}P_{j}(d_{j})=\left\{\begin{array}[]{lll}0&\,\text{if}&|d_{j% }|\leq q_{j}\\ \\ 1&\,\text{if}&|d_{j}|>q_{j}\\ \end{array}\right.\end{array}
  43. P j ( d j ) = { | d j | p j if | d j | p j 1 if | d j | > p j \begin{array}[]{cc}P_{j}(d_{j})=\left\{\begin{array}[]{lll}\frac{|d_{j}|}{p_{j% }}&\,\text{if}&|d_{j}|\leq p_{j}\\ \\ 1&\,\text{if}&|d_{j}|>p_{j}\\ \end{array}\right.\end{array}
  44. P j ( d j ) = { 0 if | d j | q j 1 2 if q j < | d j | p j 1 if | d j | > p j \begin{array}[]{cc}P_{j}(d_{j})=\left\{\begin{array}[]{lll}0&\,\text{if}&|d_{j% }|\leq q_{j}\\ \\ \frac{1}{2}&\,\text{if}&q_{j}<|d_{j}|\leq p_{j}\\ \\ 1&\,\text{if}&|d_{j}|>p_{j}\\ \end{array}\right.\end{array}
  45. P j ( d j ) = { 0 if | d j | q j | d j | - q j p j - q j if q j < | d j | p j 1 if | d j | > p j \begin{array}[]{cc}P_{j}(d_{j})=\left\{\begin{array}[]{lll}0&\,\text{if}&|d_{j% }|\leq q_{j}\\ \\ \frac{|d_{j}|-q_{j}}{p_{j}-q_{j}}&\,\text{if}&q_{j}<|d_{j}|\leq p_{j}\\ \\ 1&\,\text{if}&|d_{j}|>p_{j}\\ \end{array}\right.\end{array}
  46. P j ( d j ) = 1 - e - d j 2 2 s j 2 P_{j}(d_{j})=1-e^{-\frac{d_{j}^{2}}{2s_{j}^{2}}}

Presentation_and_access_units.html

  1. C R C_{R}
  2. C B C_{B}

PRESS_statistic.html

  1. PRESS = i = 1 n ( y i - y ^ i , - i ) 2 \operatorname{PRESS}=\sum_{i=1}^{n}(y_{i}-\hat{y}_{i,-i})^{2}

Pressure-volume_loop_analysis_in_cardiology.html

  1. × \times
  2. × \times

Pressure_jump.html

  1. Δ V o = - R T ( ln K P ) T \Delta V^{o}=-RT\left(\frac{\partial\ln K}{\partial P}\right)_{T}

Prime_integer_topology.html

  1. U a ( b ) = { b + n a 𝐙 + | n 𝐙 } U_{a}(b)=\{b+na\in\mathbf{Z}^{+}\,|\,n\in\mathbf{Z}\}
  2. 𝔅 = { U a ( b ) | a , b 𝐙 + , ( a , b ) = 1 } \mathfrak{B}=\{U_{a}(b)\,|\,a,b\in\mathbf{Z}^{+},(a,b)=1\}
  3. 𝔓 = { U p ( b ) | p , b 𝐙 + , p is prime } \mathfrak{P}=\{U_{p}(b)\,|\,p,b\in\mathbf{Z}^{+},p\,\text{ is prime}\}

Principle_of_transformation_groups.html

  1. P ( H | I ) = P ( T | I ) P(H|I)=P(T|I)
  2. P ( H | I ) + P ( T | I ) = 1 2 P ( H | I ) = 1 P ( H | I ) = 0.5 P(H|I)+P(T|I)=1\rightarrow 2P(H|I)=1\rightarrow P(H|I)=0.5
  3. μ \mu
  4. p ( X | μ , I ) = f ( X - μ ) p(X|\mu,I)=f(X-\mu)
  5. p ( X | μ , I ) = f ( X - μ ) p(X|\mu,I)=f(X-\mu)
  6. μ \mu
  7. μ \mu
  8. f ( X - μ ) = f ( [ X + b ] - [ μ + b ] ) = f ( X ( 1 ) - μ ( 1 ) ) f(X-\mu)=f([X+b]-[\mu+b])=f(X^{(1)}-\mu^{(1)})
  9. μ \mu
  10. μ ( 1 ) \mu^{(1)}
  11. f ( μ ) = | μ ( 1 ) μ | f ( μ ( 1 ) ) = f ( μ + b ) f(\mu)=|{\partial\mu^{(1)}\over\partial\mu}|f(\mu^{(1)})=f(\mu+b)
  12. p ( μ | I ) 1 p(\mu|I)\propto 1
  13. σ \sigma
  14. p ( X | σ , I ) = 1 σ f ( X σ ) p(X|\sigma,I)={1\over\sigma}f({X\over\sigma})
  15. σ > 0 \sigma>0
  16. X σ = X a σ a ; a > 0 {X\over\sigma}={Xa\over\sigma a};a>0
  17. p ( X ( 1 ) | σ , I ) = 1 a 1 σ f ( X a σ a ) = 1 σ ( 1 ) f ( X ( 1 ) σ ( 1 ) ) p(X^{(1)}|\sigma,I)={1\over a}{1\over\sigma}f({Xa\over\sigma a})={1\over\sigma% ^{(1)}}f({X^{(1)}\over\sigma^{(1)}})
  18. p ( σ | I ) = 1 a p ( σ ( 1 ) ) = 1 a p ( σ a | I ) p(\sigma|I)={1\over a}p(\sigma^{(1)})={1\over a}p({\sigma\over a}|I)
  19. p ( σ | I ) 1 σ p ( l o g ( σ ) | I ) 1 p(\sigma|I)\propto{1\over\sigma}\rightarrow p(log(\sigma)|I)\propto 1
  20. P ( H | I , N ) = P ( T | I , N ) = P ( S | I , N ) = 1 / 3 P(H|I,N)=P(T|I,N)=P(S|I,N)=1/3
  21. P ( S | smaller coin ) = P ( S | slightly bigger coin ) P(S|\,\text{smaller coin})=P(S|\,\text{slightly bigger coin})
  22. P ( H | smaller coin ) = P ( H | slightly bigger coin ) = P ( T | smaller coin ) = P ( T | slightly bigger coin ) P(H|\,\text{smaller coin})=P(H|\,\text{slightly bigger coin})=P(T|\,\text{% smaller coin})=P(T|\,\text{slightly bigger coin})
  23. f ( M ) = I ( M [ - b , b ] ) 2 b f(M)={I(M\in[-b,b])\over 2b}
  24. b b\rightarrow\infty

Priority_R-tree.html

  1. O ( ( N / B ) 1 - 1 / d + T / B ) I / O s \,O((N/B)^{1-1/d}+T/B)I/Os
  2. ( ( x m i n , y m i n ) , ( x m a x , y m a x ) ) \,((x_{min},y_{min}),(x_{max},y_{max}))
  3. ( x m i n , y m i n , x m a x , y m a x ) \,(x_{min},y_{min},x_{max},y_{max})

Proactive_secret_sharing.html

  1. k k
  2. k k
  3. x i 0 = f 0 ( i ) x_{i}^{0}=f^{0}(i)
  4. i { 1 , , n } i\in\{1,...,n\}
  5. n n
  6. x i 0 x_{i}^{0}
  7. i i
  8. 0
  9. k k
  10. δ i ( z ) = δ i , 1 z 1 + δ i , 2 z 2 + + δ i , k z k \delta_{i}(z)=\delta_{i,1}z^{1}+\delta_{i,2}z^{2}+...+\delta_{i,k}z^{k}
  11. i i
  12. u i , j = δ i ( j ) u_{i,j}=\delta_{i}(j)
  13. x i t + 1 = x i t + u 1 , i t + + u n , i t x_{i}^{t+1}=x_{i}^{t}+u_{1,i}^{t}+...+u_{n,i}^{t}
  14. t t
  15. Z 11 Z_{11}
  16. x = 6 Z 11 x=6\in Z_{11}
  17. Z 11 Z_{11}
  18. f 0 ( x ) = 6 + 2 × x f^{0}(x)=6+2\times x
  19. f 0 ( 0 ) = x = 6 f^{0}(0)=x=6
  20. x 1 0 = f 0 ( 1 ) = 6 + 2 × 1 = 8 x_{1}^{0}=f^{0}(1)=6+2\times 1=8
  21. x 2 0 = f 0 ( 2 ) = 6 + 2 × 2 = 10 x_{2}^{0}=f^{0}(2)=6+2\times 2=10
  22. x 1 0 x_{1}^{0}
  23. x 2 0 x_{2}^{0}
  24. f 0 ( x ) f^{0}(x)
  25. m = ( f 0 ( 2 ) - f 0 ( 1 ) ) / ( 2 - 1 ) = ( x 2 0 - x 1 0 ) / ( 2 - 1 ) = ( 10 - 8 ) / ( 2 - 1 ) = 2 / 1 = 2 m=(f^{0}(2)-f^{0}(1))/(2-1)=(x_{2}^{0}-x_{1}^{0})/(2-1)=(10-8)/(2-1)=2/1=2
  26. b = f 0 ( 1 ) - m = x 1 0 - 2 = 8 - 2 = 6 b=f^{0}(1)-m=x_{1}^{0}-2=8-2=6
  27. f 0 ( x ) = b + m × x = 6 + 2 × x f^{0}(x)=b+m\times x=6+2\times x
  28. f 0 ( 0 ) = 6 + 2 × 0 = 6 = x f^{0}(0)=6+2\times 0=6=x
  29. δ 1 0 ( z ) = δ 1 , 1 0 × z 1 = 2 × z 1 \delta_{1}^{0}(z)=\delta_{1,1}^{0}\times z^{1}=2\times z^{1}
  30. δ 2 0 ( z ) = δ 2 , 1 0 × z 1 = 3 × z 1 \delta_{2}^{0}(z)=\delta_{2,1}^{0}\times z^{1}=3\times z^{1}
  31. u 1 , 1 0 = δ 1 0 ( 1 ) = 2 u_{1,1}^{0}=\delta_{1}^{0}(1)=2
  32. u 1 , 2 0 = δ 1 0 ( 2 ) = 4 u_{1,2}^{0}=\delta_{1}^{0}(2)=4
  33. Z 11 Z_{11}
  34. u 1 , 2 0 u_{1,2}^{0}
  35. u 2 , 1 0 = δ 2 0 ( 1 ) = 3 u_{2,1}^{0}=\delta_{2}^{0}(1)=3
  36. u 2 , 2 0 = δ 2 0 ( 2 ) = 6 u_{2,2}^{0}=\delta_{2}^{0}(2)=6
  37. Z 11 Z_{11}
  38. u 2 , 1 0 u_{2,1}^{0}
  39. x i 1 = x i 0 + u 1 , i 0 + u 2 , i 0 x_{i}^{1}=x_{i}^{0}+u_{1,i}^{0}+u_{2,i}^{0}
  40. x 1 1 = x 1 0 + u 1 , 1 0 + u 2 , 1 0 = 8 + 2 + 3 = 2 Z 11 x_{1}^{1}=x_{1}^{0}+u_{1,1}^{0}+u_{2,1}^{0}=8+2+3=2\in Z_{11}
  41. x 2 1 = x 2 0 + u 1 , 2 0 + u 2 , 2 0 = 10 + 4 + 6 = 9 Z 11 x_{2}^{1}=x_{2}^{0}+u_{1,2}^{0}+u_{2,2}^{0}=10+4+6=9\in Z_{11}
  42. x 1 1 x_{1}^{1}
  43. x 2 1 x_{2}^{1}
  44. f 1 ( x ) f^{1}(x)
  45. f 1 ( x ) f^{1}(x)
  46. m = ( f 1 ( 2 ) - f 1 ( 1 ) ) / ( 2 - 1 ) = ( x 2 1 - x 1 1 ) / ( 2 - 1 ) = ( 9 - 2 ) / ( 2 - 1 ) = 7 / 1 = 7 m=(f^{1}(2)-f^{1}(1))/(2-1)=(x_{2}^{1}-x_{1}{1})/(2-1)=(9-2)/(2-1)=7/1=7
  47. b = f 1 ( 1 ) - m = x 1 1 - 7 = 2 - 7 = - 5 = 6 b=f^{1}(1)-m=x_{1}^{1}-7=2-7=-5=6
  48. f 1 ( x ) = b + m × x = 6 + 7 × x f^{1}(x)=b+m\times x=6+7\times x
  49. f 1 ( 0 ) = 6 + 7 × 0 = 6 = x f^{1}(0)=6+7\times 0=6=x

Probability_box.html

  1. \infty
  2. \infty
  3. \infty

Probable_error.html

  1. γ = 0.6745 × σ . \gamma=0.6745\times\sigma.

Probalign.html

  1. σ ( x , y ) \sigma(x,y)
  2. g ( k ) = α + β k g(k)=\alpha+\beta k
  3. S ( a ) S(a)
  4. S ( a ) = x i - y j a σ ( x i , y j ) + gap cost S(a)=\sum_{x_{i}-y_{j}\in a}\sigma(x_{i},y_{j})+\,\text{gap cost}
  5. e S ( a ) T = e x i - y j a σ ( x i , y j ) + gap cost T = ( x i - y i a e x i - y j a σ ( x i , y j ) T ) e g a p c o s t T e^{\frac{S(a)}{T}}=e^{\frac{\sum_{x_{i}-y_{j}\in a}\sigma(x_{i},y_{j})+\,\text% {gap cost}}{T}}=\left(\prod_{x_{i}-y_{i}\in a}e^{\frac{\sum_{x_{i}-y_{j}\in a}% \sigma(x_{i},y_{j})}{T}}\right)\cdot e^{\frac{gapcost}{T}}
  6. T T
  7. P r [ a | x , y ] = e S ( a ) T Z Pr[a|x,y]=\frac{e^{\frac{S(a)}{T}}}{Z}
  8. Z Z
  9. Z i , j Z_{i,j}
  10. x 0 , x 1 , , x i x_{0},x_{1},...,x_{i}
  11. y 0 , y 1 , , y j y_{0},y_{1},...,y_{j}
  12. Z i , j M : Z^{M}_{i,j}:
  13. Z i , j I : Z^{I}_{i,j}:
  14. ( - , y j ) (-,y_{j})
  15. Z i , j D : Z^{D}_{i,j}:
  16. ( x i , - ) (x_{i},-)
  17. Z i , j = Z i , j M + Z i , j D + Z i , j I Z_{i,j}=Z^{M}_{i,j}+Z^{D}_{i,j}+Z^{I}_{i,j}
  18. Z 0 , j M = Z i , 0 M = 0 Z^{M}_{0,j}=Z^{M}_{i,0}=0
  19. Z 0 , 0 M = 1 Z^{M}_{0,0}=1
  20. Z 0 , j D = 0 Z^{D}_{0,j}=0
  21. Z i , 0 I = 0 Z^{I}_{i,0}=0
  22. x x
  23. y y
  24. Z | x | , | y | Z_{|x|,|y|}
  25. Z i , j M = Z i - 1 , j - 1 e σ ( x i , y j ) T Z^{M}_{i,j}=Z_{i-1,j-1}\cdot e^{\frac{\sigma(x_{i},y_{j})}{T}}
  26. Z i , j D = Z i - 1 , j D e β T + Z i - 1 , j M e g ( 1 ) T + Z i - 1 , j I e g ( 1 ) T Z^{D}_{i,j}=Z^{D}_{i-1,j}\cdot e^{\frac{\beta}{T}}+Z^{M}_{i-1,j}\cdot e^{\frac% {g(1)}{T}}+Z^{I}_{i-1,j}\cdot e^{\frac{g(1)}{T}}
  27. Z i , j I Z^{I}_{i,j}
  28. x i x_{i}
  29. y j y_{j}
  30. P ( x i - y j | x , y ) = Z i - 1 , j - 1 e σ ( x i , y j ) T Z i , j Z | x | , | y | P(x_{i}-y_{j}|x,y)=\frac{Z_{i-1,j-1}\cdot e^{\frac{\sigma(x_{i},y_{j})}{T}}% \cdot Z^{\prime}_{i^{\prime},j^{\prime}}}{Z_{|x|,|y|}}
  31. Z , i , j Z^{\prime},i^{\prime},j^{\prime}
  32. Z Z

Procedural_texture.html

  1. R 3 R^{3}

Product_distribution.html

  1. Z = X Y Z=XY
  2. X X
  3. Y Y
  4. f X f_{X}
  5. f Y f_{Y}
  6. Z = X Y Z=XY
  7. f Z ( z ) = - f X ( x ) f Y ( z / x ) 1 | x | d x . f_{Z}(z)=\int^{\infty}_{-\infty}f_{X}\left(x\right)f_{Y}\left(z/x\right)\frac{% 1}{|x|}\,dx.
  8. 𝒟 \mathcal{D}
  9. { 0 , 1 } n \{0,1\}^{n}
  10. μ 1 , μ 2 , , μ n \mu_{1},\mu_{2},\dots,\mu_{n}
  11. μ i \mu_{i}
  12. x { 0 , 1 } n x\in\{0,1\}^{n}
  13. x 𝒟 x\sim\mathcal{D}
  14. μ i = Pr 𝒟 [ x i = 1 ] \mu_{i}=\operatorname{Pr}_{\mathcal{D}}[x_{i}=1]
  15. μ i = 1 / 2 \mu_{i}=1/2
  16. , \langle\cdot,\cdot\rangle
  17. { 0 , 1 } n \{0,1\}^{n}
  18. f , g 𝒟 = x { 0 , 1 } n 𝒟 ( x ) f ( x ) g ( x ) = 𝔼 𝒟 [ f g ] \langle f,g\rangle_{\mathcal{D}}=\sum_{x\in\{0,1\}^{n}}\mathcal{D}(x)f(x)g(x)=% \mathbb{E}_{\mathcal{D}}[fg]
  19. f 𝒟 = f , f 𝒟 \|f\|_{\mathcal{D}}=\sqrt{\langle f,f\rangle_{\mathcal{D}}}

Product_numerical_range.html

  1. X X
  2. N N
  3. N \mathcal{H}_{N}
  4. Λ ( X ) \mathrm{\Lambda}(X)
  5. λ \lambda
  6. | ψ N {|\psi\rangle}\in\mathcal{H}_{N}
  7. || ψ || = 1 ||\psi||=1
  8. ψ | X | ψ = λ {\langle\psi|}X{|\psi\rangle}=\lambda
  9. N = K M , \mathcal{H}_{N}=\mathcal{H}_{K}\otimes\mathcal{H}_{M},
  10. N = K M N=KM
  11. X X
  12. Λ ( X ) \mathrm{\Lambda}^{\!\otimes}\!\left(X\right)
  13. X X
  14. N \mathcal{H}_{N}
  15. Λ ( X ) = { ψ A ψ B | X | ψ A ψ B : | ψ A K , | ψ B M } , \mathrm{\Lambda}^{\!\otimes}\!\left(X\right)=\left\{{\langle\psi_{A}\otimes% \psi_{B}|}X{|\psi_{A}\otimes\psi_{B}\rangle}:{|\psi_{A}\rangle}\in\mathcal{H}_% {K},{|\psi_{B}\rangle}\in\mathcal{H}_{M}\right\},
  16. | ψ A K {|\psi_{A}\rangle}\in\mathcal{H}_{K}
  17. | ψ B M {|\psi_{B}\rangle}\in\mathcal{H}_{M}
  18. N = K M \mathcal{H}_{N}=\mathcal{H}_{K}\otimes\mathcal{H}_{M}
  19. r ( X ) r^{\otimes}(X)
  20. X X
  21. r ( X ) = max { | z | : z Λ ( X ) } . r^{\otimes}(X)=\max\{|z|:z\in\mathrm{\Lambda}^{\!\otimes}\!\left(X\right)\}.
  22. U ( K ) × U ( M ) U(K)\times U(M)
  23. U ( K M ) U(KM)
  24. A , B 𝕄 n A,B\in\mathbb{M}_{n}
  25. Λ ( A + B ) Λ ( A ) + Λ ( B ) . \mathrm{\Lambda}^{\!\otimes}\!\left(A+B\right)\subset\mathrm{\Lambda}^{\!% \otimes}\!\left(A\right)+\mathrm{\Lambda}^{\!\otimes}\!\left(B\right).
  26. A 𝕄 n A\in\mathbb{M}_{n}
  27. α \alpha\in\mathbb{C}
  28. Λ ( A + α 𝐈 ) = Λ ( A ) + α . \mathrm{\Lambda}^{\!\otimes}\!\left({A+\alpha\mathbf{I}}\right)=\mathrm{% \Lambda}^{\!\otimes}\!\left(A\right)+\alpha.
  29. A 𝕄 n A\in\mathbb{M}_{n}
  30. α \alpha\in\mathbb{C}
  31. Λ ( α A ) = α Λ ( A ) . \mathrm{\Lambda}^{\!\otimes}\!\left({\alpha A}\right)=\alpha\mathrm{\Lambda}^{% \!\otimes}\!\left({A}\right).
  32. A 𝕄 m × n A\in\mathbb{M}_{m\times n}
  33. Λ ( ( U V ) A ( U V ) ) = Λ ( A ) , \mathrm{\Lambda}^{\!\otimes}\!\left({(U\otimes V)A(U\otimes V)^{\dagger}}% \right)=\mathrm{\Lambda}^{\!\otimes}\!\left({A}\right),
  34. U 𝕄 m U\in\mathbb{M}_{m}
  35. V 𝕄 n V\in\mathbb{M}_{n}
  36. A 𝕄 m A\in\mathbb{M}_{m}
  37. B 𝕄 n B\in\mathbb{M}_{n}
  38. Λ ( A B ) = Co ( Λ ( A B ) ) . \mathrm{\Lambda}(A\otimes B)=\mathrm{Co}(\mathrm{\Lambda}^{\!\otimes}\!\left({% A\otimes B}\right)).
  39. e i θ A e^{i\theta}A
  40. θ [ 0 , 2 π ) \theta\in[0,2\pi)
  41. Λ ( A B ) = Λ ( A B ) . \mathrm{\Lambda}(A\otimes B)=\mathrm{\Lambda}^{\!\otimes}\!\left({A\otimes B}% \right).
  42. H ( A ) = 1 2 ( A + A ) H(A)=\frac{1}{2}(A+A^{\dagger})
  43. S ( A ) = 1 2 ( A - A ) S(A)=\frac{1}{2}(A-A^{\dagger})
  44. A 𝕄 n A\in\mathbb{M}_{n}
  45. \mathrm{\Lambda}^{\!\otimes}\! \left( {H(A)} \right) =\mathrm{Re}\ \mathrm{\Lambda}^{\!\otimes}\! \left( {A} \right)
  46. Λ ( S ( A ) ) = i Im Λ ( A ) . \mathrm{\Lambda}^{\!\otimes}\!\left({S(A)}\right)=i\,\mathrm{Im}\ \mathrm{% \Lambda}^{\!\otimes}\!\left({A}\right).
  47. A = ( 1 0 0 0 ) ( 1 0 0 0 ) + i ( 0 0 0 1 ) ( 0 0 0 1 ) . A=\left(\begin{array}[]{cc}1&0\\ 0&0\end{array}\right)\otimes\left(\begin{array}[]{cc}1&0\\ 0&0\end{array}\right)+i\left(\begin{array}[]{cc}0&0\\ 0&1\end{array}\right)\otimes\left(\begin{array}[]{cc}0&0\\ 0&1\end{array}\right).
  48. A A
  49. 0 , 1 , i 0,1,i
  50. 1 Λ ( A ) 1\in\mathrm{\Lambda}^{\!\otimes}\!\left({A}\right)
  51. i Λ ( A ) i\in\mathrm{\Lambda}^{\!\otimes}\!\left({A}\right)
  52. ( 1 + i ) / 2 Λ ( A ) (1+i)/2\not\in\mathrm{\Lambda}^{\!\otimes}\!\left({A}\right)
  53. Λ ( A ) = { x + y i : 0 x , 0 y , x + y 1 } . \mathrm{\Lambda}^{\!\otimes}\!\left({A}\right)=\left\{x+yi:0\leq x,0\leq y,% \sqrt{x}+\sqrt{y}\leq 1\right\}.
  54. A A
  55. A 𝕄 K × M A\in\mathbb{M}_{K\times M}
  56. 1 K M tr A Λ ( A ) . \frac{1}{KM}\;{\mathrm{tr}}A\ \in\ \mathrm{\Lambda}^{\!\otimes}\!\left({A}% \right).

Products_in_algebraic_topology.html

  1. H p ( X ) H q ( Y ) H p + q ( X × Y ) H_{p}(X)\otimes H_{q}(Y)\to H_{p+q}(X\times Y)
  2. : H p ( X ; R ) × H q ( X ; R ) H p - q ( X ; R ) \frown\ :H_{p}(X;R)\times H^{q}(X;R)\rightarrow H_{p-q}(X;R)
  3. : H p ( X ; R ) × H q ( X × Y ; R ) H q - p ( Y ; R ) \frown\ :H_{p}(X;R)\times H^{q}(X\times Y;R)\rightarrow H^{q-p}(Y;R)
  4. H p ( X ) H q ( X ) H p + q ( X ) H^{p}(X)\otimes H^{q}(X)\to H^{p+q}(X)

Projector_augmented_wave_method.html

  1. 𝒯 \mathcal{T}
  2. | Ψ ~ |\tilde{\Psi}\rangle
  3. | Ψ |\Psi\rangle
  4. | Ψ = 𝒯 | Ψ ~ |\Psi\rangle=\mathcal{T}|\tilde{\Psi}\rangle
  5. | Ψ ~ |\tilde{\Psi}\rangle
  6. | Ψ |\Psi\rangle
  7. 𝒯 = 1 + R 𝒯 ^ R \mathcal{T}=1+\sum_{R}\hat{\mathcal{T}}_{R}
  8. 𝒯 ^ R \hat{\mathcal{T}}_{R}
  9. Ω R \Omega_{R}
  10. R R
  11. | Ψ ~ = i | ϕ ~ i c i |\tilde{\Psi}\rangle=\sum_{i}|\tilde{\phi}_{i}\rangle c_{i}
  12. Ω R \Omega_{R}
  13. 𝒯 \mathcal{T}
  14. c i c_{i}
  15. | p i |p_{i}\rangle
  16. c i = p i | Ψ ~ c_{i}=\langle p_{i}|\tilde{\Psi}\rangle
  17. p i | ϕ ~ j = δ i j \langle p_{i}|\tilde{\phi}_{j}\rangle=\delta_{ij}
  18. | ϕ i = 𝒯 | ϕ ~ i |\phi_{i}\rangle=\mathcal{T}|\tilde{\phi}_{i}\rangle
  19. 𝒯 \mathcal{T}
  20. | ϕ i |\phi_{i}\rangle
  21. | ϕ ~ i |\tilde{\phi}_{i}\rangle
  22. | p i |p_{i}\rangle
  23. 𝒯 = 1 + i ( | ϕ i - | ϕ ~ i ) p i | \mathcal{T}=1+\sum_{i}\left(|\phi_{i}\rangle-|\tilde{\phi}_{i}\rangle\right)% \langle p_{i}|
  24. a i = Ψ | A ^ | Ψ a_{i}=\langle\Psi|\hat{A}|\Psi\rangle
  25. | Ψ = 𝒯 | Ψ ~ |\Psi\rangle=\mathcal{T}|\tilde{\Psi}\rangle
  26. a i = Ψ ~ | 𝒯 A ^ 𝒯 | Ψ ~ a_{i}=\langle\tilde{\Psi}|\mathcal{T}^{\dagger}\hat{A}\mathcal{T}|\tilde{\Psi}\rangle
  27. A ~ = 𝒯 A ^ 𝒯 \tilde{A}=\mathcal{T}^{\dagger}\hat{A}\mathcal{T}
  28. A ^ \hat{A}
  29. 𝒯 \mathcal{T}
  30. A ~ = A ^ + i , j | p i ( ϕ i | A ^ | ϕ j - ϕ ~ i | A ^ | ϕ ~ j ) p j | \tilde{A}=\hat{A}+\sum_{i,j}|p_{i}\rangle\left(\langle\phi_{i}|\hat{A}|\phi_{j% }\rangle-\langle\tilde{\phi}_{i}|\hat{A}|\tilde{\phi}_{j}\rangle\right)\langle p% _{j}|
  31. i , j i,j
  32. B ^ \hat{B}
  33. B ^ - i , j | p i ϕ ~ i | B ^ | ϕ ~ j p j | \hat{B}-\sum_{i,j}|p_{i}\rangle\langle\tilde{\phi}_{i}|\hat{B}|\tilde{\phi}_{j% }\rangle\langle p_{j}|

Proof_compression.html

  1. κ \kappa
  2. κ \kappa
  3. η 1 \eta_{1}
  4. η 2 \eta_{2}
  5. η 1 \eta_{1}
  6. η 2 \eta_{2}
  7. η 1 \eta_{1}
  8. η 2 \eta_{2}
  9. η 2 \eta_{2}
  10. η 1 \eta_{1}
  11. κ \kappa
  12. κ \kappa
  13. { a , b , c } \left\{a,b,c\right\}
  14. { η 1 : { a , b , p } , η 2 : { c , ¬ p } } η 1 : a , b , p η 2 : c , ¬ p η 3 : a , b , c p \left\{\eta_{1}:\left\{a,b,p\right\},\eta_{2}:\left\{c,\neg p\right\}\right\}% \quad\frac{\eta_{1}:a,b,p\quad\quad\eta_{2}:c,\neg p}{\eta_{3}:a,b,c}p
  15. η 1 \eta_{1}
  16. η 2 \eta_{2}
  17. η 3 \eta_{3}
  18. p p
  19. p p
  20. ¬ p \neg p
  21. η 3 \eta_{3}
  22. { a , b , c } \left\{a,b,c\right\}
  23. η 3 \eta_{3}
  24. η 1 \eta_{1}
  25. η 2 \eta_{2}
  26. η 1 η 2 η 3 \begin{array}[]{ccc}\eta_{1}&&\eta_{2}\\ &\nwarrow\nearrow\\ &\eta_{3}\end{array}
  27. η 1 \eta_{1}
  28. η 2 \eta_{2}
  29. η 3 \eta_{3}
  30. η 3 \eta_{3}
  31. η 1 \eta_{1}
  32. η 2 \eta_{2}
  33. η 3 \eta_{3}
  34. \bot
  35. η \eta
  36. η \eta
  37. η \eta
  38. η \eta
  39. η \eta
  40. η \eta
  41. κ 1 \kappa_{1}
  42. κ 2 \kappa_{2}
  43. p p
  44. κ 1 p κ 2 \kappa_{1}\odot_{p}\kappa_{2}
  45. κ 1 κ 2 \kappa_{1}\odot\kappa_{2}
  46. { a , b , p } p { c , ¬ p } \left\{a,b,p\right\}\odot_{p}\left\{c,\neg p\right\}
  47. { a , b , p } { c , ¬ p } . \left\{a,b,p\right\}\odot\left\{c,\neg p\right\}.
  48. { a , b , p } η 1 { c , ¬ p } η 2 η 3 \underbrace{\overbrace{\left\{a,b,p\right\}}^{\eta_{1}}\odot\overbrace{\left\{% c,\neg p\right\}}^{\eta_{2}}}_{\eta_{3}}

Proposed_redefinition_of_SI_base_units.html

  1. c 2 = 1 μ 0 ε 0 c^{2}=\frac{1}{\mu_{0}\varepsilon_{0}}
  2. Z 0 = μ 0 ε 0 Z_{0}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}
  3. m ( 𝒦 ) m(\mathcal{K})
  4. m ( 𝒦 ) m(\mathcal{K})
  5. m ( 𝒦 ) m(\mathcal{K})
  6. u r ( m ( 𝒦 ) ) = 4.4 × 10 - 8 u\text{r}(m(\mathcal{K}))=4.4\times 10^{-8}
  7. h h
  8. 8 α c μ 0 K J 2 \frac{8\alpha}{c\mu_{0}K\text{J}^{2}}
  9. K J 2 K\text{J}^{2}
  10. 2 u r ( K J ) = 4.4 × 10 - 8 \sim 2u\text{r}(K\text{J})=4.4\times 10^{-8}
  11. K J K\text{J}
  12. K J K\text{J}
  13. K J K\text{J}
  14. u r ( K J ) = 2.2 × 10 - 8 u\text{r}(K\text{J})=2.2\times 10^{-8}
  15. 2 e h \frac{2e}{h}
  16. R K R\text{K}
  17. c μ 0 2 α \frac{c\mu_{0}}{2\alpha}
  18. α \alpha
  19. u r ( α ) = 3.2 × 10 - 10 u\text{r}(\alpha)=3.2\times 10^{-10}
  20. h e 2 \frac{h}{e^{2}}
  21. e e
  22. 4 α c μ 0 K J \frac{4\alpha}{c\mu_{0}K\text{J}}
  23. K J K\text{J}
  24. u r ( K J ) = 2.2 × 10 - 8 \sim u\text{r}(K\text{J})=2.2\times 10^{-8}
  25. μ 0 \mu_{0}
  26. 2 h α c e 2 \frac{2h\alpha}{ce^{2}}
  27. α \alpha
  28. u r ( α ) = 3.2 × 10 - 10 u\text{r}(\alpha)=3.2\times 10^{-10}
  29. ϵ 0 \epsilon_{0}
  30. 1 c 2 μ 0 \frac{1}{c^{2}\mu_{0}}
  31. e 2 2 h c α \frac{e^{2}}{2hc\alpha}
  32. α \alpha
  33. u r ( α ) = 3.2 × 10 - 10 u\text{r}(\alpha)=3.2\times 10^{-10}
  34. Z 0 Z_{0}
  35. c μ 0 c\mu_{0}
  36. 2 h α e 2 \frac{2h\alpha}{e^{2}}
  37. α \alpha
  38. u r ( α ) = 3.2 × 10 - 10 u\text{r}(\alpha)=3.2\times 10^{-10}
  39. m e m\text{e}
  40. 16 R c 2 α μ 0 K J 2 \frac{16R_{\infty}}{c^{2}\alpha\mu_{0}K\text{J}^{2}}
  41. K J 2 K\text{J}^{2}
  42. 2 u r ( K J ) = 4.4 × 10 - 8 \sim 2u\text{r}(K\text{J})=4.4\times 10^{-8}
  43. 2 h R c α 2 \frac{2hR_{\infty}}{c\alpha^{2}}
  44. α 2 \alpha^{2}
  45. 2 u r ( α ) = 6.4 × 10 - 10 \sim 2u\text{r}(\alpha)=6.4\times 10^{-10}
  46. M ( e ) M(\,\text{e})
  47. A r ( e ) M u A\text{r}(\,\text{e})M\text{u}
  48. A r ( e ) A\text{r}(\,\text{e})
  49. u r ( A r ( e ) ) = 4.0 × 10 - 10 u\text{r}(A\text{r}(\,\text{e}))=4.0\times 10^{-10}
  50. 2 h R N A c α 2 \frac{2hR_{\infty}N\text{A}}{c\alpha^{2}}
  51. α 2 \alpha^{2}
  52. 2 u r ( α ) = 6.4 × 10 - 10 \sim 2u\text{r}(\alpha)=6.4\times 10^{-10}
  53. m u = 1 u m\text{u}=1\,\text{u}
  54. 16 R c 2 α μ 0 K J 2 A r ( e ) \frac{16R_{\infty}}{c^{2}\alpha\mu_{0}K\text{J}^{2}A\text{r}(\,\text{e})}
  55. K J 2 K\text{J}^{2}
  56. 2 u r ( K J ) = 4.4 × 10 - 8 \sim 2u\text{r}(K\text{J})=4.4\times 10^{-8}
  57. 2 h R c α 2 A r ( e ) \frac{2hR_{\infty}}{c\alpha^{2}A\text{r}(\,\text{e})}
  58. α 2 , A r ( e ) \alpha^{2},A\text{r}(\,\text{e})
  59. ( 2 u r ( α ) ) 2 + u r ( A r ( e ) ) 2 7.5 × 10 - 10 \sim\sqrt{(2u\text{r}(\alpha))^{2}+u\text{r}(A\text{r}(\,\text{e}))^{2}}% \approx 7.5\times 10^{-10}
  60. m Da = 1 Da m\text{Da}=1\,\text{Da}
  61. M Da N A \frac{M\text{Da}}{N\text{A}}
  62. M u M\text{u}
  63. 2 h R N A c α 2 A r ( e ) \frac{2hR_{\infty}N\text{A}}{c\alpha^{2}A\text{r}(\,\text{e})}
  64. α 2 , A r ( e ) \alpha^{2},A\text{r}(\,\text{e})
  65. ( 2 u r ( α ) ) 2 + u r ( A r ( e ) ) 2 7.5 × 10 - 10 \sim\sqrt{(2u\text{r}(\alpha))^{2}+u\text{r}(A\text{r}(\,\text{e}))^{2}}% \approx 7.5\times 10^{-10}
  66. M Da M\text{Da}
  67. N A N\text{A}
  68. c 2 α μ 0 K J 2 A r ( e ) M u 16 R \frac{c^{2}\alpha\mu_{0}K\text{J}^{2}A\text{r}(\,\text{e})M\text{u}}{16R_{% \infty}}
  69. K J 2 K\text{J}^{2}
  70. 2 u r ( K J ) = 4.4 × 10 - 8 \sim 2u\text{r}(K\text{J})=4.4\times 10^{-8}
  71. m ( 12 C ) m(^{12}\,\text{C})
  72. 192 R c 2 α μ 0 K J 2 A r ( e ) \frac{192R_{\infty}}{c^{2}\alpha\mu_{0}K\text{J}^{2}A\text{r}(\,\text{e})}
  73. K J 2 K\text{J}^{2}
  74. 2 u r ( K J ) = 4.4 × 10 - 8 \sim 2u\text{r}(K\text{J})=4.4\times 10^{-8}
  75. 24 h R c α 2 A r ( e ) \frac{24hR_{\infty}}{c\alpha^{2}A\text{r}(\,\text{e})}
  76. α 2 , A r ( e ) \alpha^{2},A\text{r}(\,\text{e})
  77. ( 2 u r ( α ) ) 2 + u r ( A r ( e ) ) 2 7.5 × 10 - 10 \sim\sqrt{(2u\text{r}(\alpha))^{2}+u\text{r}(A\text{r}(\,\text{e}))^{2}}% \approx 7.5\times 10^{-10}
  78. M ( 12 C ) M(^{12}\,\text{C})
  79. 24 h R N A c α 2 A r ( e ) \frac{24hR_{\infty}N\text{A}}{c\alpha^{2}A\text{r}(\,\text{e})}
  80. α 2 , A r ( e ) \alpha^{2},A\text{r}(\,\text{e})
  81. ( 2 u r ( α ) ) 2 + u r ( A r ( e ) ) 2 7.5 × 10 - 10 \sim\sqrt{(2u\text{r}(\alpha))^{2}+u\text{r}(A\text{r}(\,\text{e}))^{2}}% \approx 7.5\times 10^{-10}
  82. F F
  83. c α 2 K J A r ( e ) M u 4 R \frac{c\alpha^{2}K\text{J}A\text{r}(\,\text{e})M\text{u}}{4R_{\infty}}
  84. K J K\text{J}
  85. u r ( K J ) = 2.2 × 10 - 8 \sim u\text{r}(K\text{J})=2.2\times 10^{-8}
  86. e N A eN\text{A}
  87. T TPW T\text{TPW}
  88. T TPW T\text{TPW}
  89. T TPW T\text{TPW}
  90. u r ( T TPW ) = 9.1 × 10 - 7 u\text{r}(T\text{TPW})=9.1\times 10^{-7}
  91. R R
  92. R R
  93. R R
  94. u r ( R ) = 9.1 × 10 - 7 u\text{r}(R)=9.1\times 10^{-7}
  95. k N A kN\text{A}
  96. k k
  97. 16 R R c 2 α μ 0 K J 2 A r ( e ) M u \frac{16RR_{\infty}}{c^{2}\alpha\mu_{0}K\text{J}^{2}A\text{r}(\,\text{e})M% \text{u}}
  98. R R
  99. u r ( R ) = 9.1 × 10 - 7 \sim u\text{r}(R)=9.1\times 10^{-7}
  100. σ \sigma
  101. 256 π 5 R 4 R 4 15 c 7 α 7 μ 0 K J 2 A r ( e ) 4 M u 4 \frac{256\pi^{5}R^{4}R_{\infty}^{4}}{15c^{7}\alpha^{7}\mu_{0}K\text{J}^{2}A% \text{r}(\,\text{e})^{4}M\text{u}^{4}}
  102. R 4 R^{4}
  103. 4 u r ( R ) 3.6 × 10 - 6 \sim 4u\text{r}(R)\approx 3.6\times 10^{-6}
  104. 2 π 5 k 4 15 h 3 c 2 \frac{2\pi^{5}k^{4}}{15h^{3}c^{2}}
  105. Θ = k b T ; β = 1 k b T \Theta=k\text{b}T;\beta={1\over k\text{b}T}

Propositional_proof_system.html

  1. P 1 ( x , A ) = { A if P 2 ( A , x ) otherwise P_{1}(\langle x,A\rangle)=\begin{cases}A&\,\text{if }P_{2}(A,x)\\ \top&\,\text{otherwise}\end{cases}
  2. \top
  3. A C 0 {AC}^{0}

Protective_relay.html

  1. T = K × ϕ 1 × ϕ 2 sin θ T=K\times\phi_{1}\times\phi_{2}\sin\theta
  2. K K
  3. ϕ 1 \phi_{1}
  4. ϕ 2 \phi_{2}
  5. θ \theta

Protein_adsorption.html

  1. Δ a d s G = Δ a d s H - T Δ a d s S < 0 \Delta_{ads}G=\Delta_{ads}H-{T}\Delta_{ads}S<0
  2. d n d t = C o ( D π t ) 1 / 2 {dn\over dt}=C_{o}({D\over\pi t})^{1/2}
  3. C t + V ( y ) C x = D 2 C y 2 {\partial C\over\partial t}+V(y){\partial C\over\partial x}=D{\partial^{2}C% \over\partial y^{2}}
  4. V ( y ) = γ y ( 1 - y b ) {V(y)=\gamma y(1-{y\over b})}

Protein_adsorption_in_the_food_industry.html

  1. ϕ \phi
  2. ϕ = ϕ 0 e - x λ D \phi=\phi_{0}e^{\frac{-x}{\lambda_{D}}}
  3. ϕ \phi
  4. Φ 0 \Phi_{0}
  5. λ D \lambda_{D}
  6. Γ = V p C e A p e W a A c R T + V p C e \Gamma=\frac{V_{p}C_{e}}{A_{p}e^{\frac{W_{a}A_{c}}{RT}+V_{p}C_{e}}}
  7. Γ \Gamma
  8. A c A_{c}
  9. V p V_{p}
  10. W a W_{a}
  11. C e C_{e}

Proto-value_functions.html

  1. G = ( V , E ) G=\left(V,E\right)
  2. L L
  3. L = D - A L=D-A
  4. D D
  5. A A
  6. L ϕ λ = λ ϕ λ L\phi_{\lambda}=\lambda\phi_{\lambda}
  7. L L
  8. ϕ λ \phi_{\lambda}
  9. λ \lambda
  10. L normalized = I - D - 1 / 2 A D - 1 / 2 L\text{normalized}=I-D^{-1/2}AD^{-1/2}
  11. P = D - 1 A P=D^{-1}A
  12. G G
  13. S i S_{i}
  14. S j S_{j}
  15. G i , j = { 1 if S i S j 0 otherwise G_{i,j}=\begin{cases}1&\,\text{if }S_{i}\leftrightarrow S_{j}\\ 0&\,\text{otherwise}\end{cases}
  16. G G
  17. Φ G = { V 1 G , , V k G } \Phi_{G}=\left\{V_{1}^{G},\dots,V_{k}^{G}\right\}
  18. V i G V_{i}^{G}
  19. G G
  20. V ^ π \hat{V}^{\pi}
  21. S M G = { s 1 , , s m } S_{M}^{G}=\left\{s_{1},\dots,s_{m}\right\}
  22. K G = ( Φ m G ) T Φ m G . K_{G}=\left(\Phi_{m}^{G}\right)^{T}\Phi_{m}^{G}.
  23. S m G S_{m}^{G}
  24. S G m S_{G}^{m}
  25. K G ( i , j ) = k V i G ( k ) V j G ( k ) K_{G}\left(i,j\right)=\sum_{k}V_{i}^{G}(k)V_{j}^{G}(k)
  26. α = K G - 1 ( Φ M G ) T V ^ π . \alpha=K_{G}^{-1}\left(\Phi_{M}^{G}\right)^{T}\hat{V}^{\pi}.

Proxmap_sort.html

  1. O ( 1 ) O(1)
  2. O ( 1 ) O(1)
  3. O ( 1 ) O(1)
  4. O ( n ) O(n)
  5. O ( n ) O(n)
  6. O ( n 2 ) O(n^{2})
  7. O ( c ) O(c)
  8. O ( n ) O(n)
  9. O ( c 2 ) O(c^{2})
  10. O ( n ) O(n)
  11. O ( 1 ) O(1)

Pseudo-Boolean_function.html

  1. f : 𝐁 n f:\mathbf{B}^{n}\rightarrow\mathbb{R}
  2. f ( s y m b o l x ) = a + i a i x i + i < j a i j x i x j + i < j < k a i j k x i x j x k + f(symbol{x})=a+\sum_{i}a_{i}x_{i}+\sum_{i<j}a_{ij}x_{i}x_{j}+\sum_{i<j<k}a_{% ijk}x_{i}x_{j}x_{k}+\ldots
  3. f f
  4. { - 1 , 1 } n \{-1,1\}^{n}
  5. \mathbb{R}
  6. f f
  7. f ( x ) = I [ n ] f ^ ( I ) i I x i , f(x)=\sum_{I\subseteq[n]}\hat{f}(I)\prod_{i\in I}x_{i},
  8. f ^ ( I ) \hat{f}(I)
  9. f f
  10. [ n ] = { 1 , , n } [n]=\{1,...,n\}
  11. f ( s y m b o l x ) + f ( s y m b o l y ) f ( s y m b o l x s y m b o l y ) + f ( s y m b o l x s y m b o l y ) f(symbol{x})+f(symbol{y})\geq f(symbol{x}\wedge symbol{y})+f(symbol{x}\vee symbol% {y})
  12. - x 1 x 2 x 3 = min z 𝐁 z ( 2 - x 1 - x 2 - x 3 ) \displaystyle-x_{1}x_{2}x_{3}=\min_{z\in\mathbf{B}}z(2-x_{1}-x_{2}-x_{3})
  13. - x 1 x 2 x 3 = min z 𝐁 z ( - x 1 + x 2 + x 3 ) - x 1 x 2 - x 1 x 3 + x 1 . \displaystyle-x_{1}x_{2}x_{3}=\min_{z\in\mathbf{B}}z(-x_{1}+x_{2}+x_{3})-x_{1}% x_{2}-x_{1}x_{3}+x_{1}.
  14. f ( s y m b o l x ) = - 2 x 1 + x 2 - x 3 + 4 x 1 x 2 + 4 x 1 x 3 - 2 x 2 x 3 - 2 x 1 x 2 x 3 . \displaystyle f(symbol{x})=-2x_{1}+x_{2}-x_{3}+4x_{1}x_{2}+4x_{1}x_{3}-2x_{2}x% _{3}-2x_{1}x_{2}x_{3}.
  15. ( 0 , 1 , 1 ) {(0,1,1)}
  16. f f
  17. { - 1 , 1 } n \{-1,1\}^{n}
  18. \mathbb{R}
  19. f ( x ) = I [ n ] f ^ ( I ) i I x i . f(x)=\sum_{I\subseteq[n]}\hat{f}(I)\prod_{i\in I}x_{i}.
  20. f ^ ( I ) \hat{f}(I)
  21. k k
  22. f ( x ) f(x)
  23. k k
  24. O ( k 2 log k ) O(k^{2}\log k)
  25. r r
  26. f f
  27. r ( k - 1 ) r(k-1)

Pseudo-random_number_sampling.html

  1. F ( i ) = j = 1 i f ( j ) . F(i)=\sum_{j=1}^{i}f(j).

Pseudomanifold.html

  1. z 2 = x 2 + y 2 z^{2}=x^{2}+y^{2}
  2. 2 \geq 2

PSI-20.html

  1. I t = i = 1 N Q i , t F i , t f i , t C i , t d t I_{t}=\frac{\sum_{i=1}^{N}Q_{i,t}\,F_{i,t}\,f_{i,t}\,C_{i,t}\,}{d_{t}\,}

PSR_J1719-1438.html

  1. a P sin i = a_{P}\sin i=

Ptolemy's_table_of_chords.html

  1. chord θ = 120 sin ( θ 2 ) = 60 ( 2 sin ( π θ 360 radians ) ) . \mathrm{chord}\ \theta=120\,\sin\left(\frac{\theta^{\circ}}{2}\right)=60\cdot% \left(2\,\sin\left(\frac{\pi\theta}{360}\,\text{ radians}\right)\right).
  2. π \pi
  3. π \pi
  4. 99 + 29 60 + 5 60 2 , 99+\frac{29}{60}+\frac{5}{60^{2}},
  5. chord ( θ + 1 2 ) - chord ( θ ) 1 / 2 . \frac{\mathrm{chord}\left(\theta+\tfrac{1}{2}\right)^{\circ}-\mathrm{chord}% \left(\theta^{\circ}\right)}{1/2}.
  6. arc chord sixtieths 1 2 0 31 25 1 2 50 1 1 2 50 1 2 50 1 1 2 1 34 15 1 2 50 109 97 41 38 0 36 23 109 1 2 97 59 49 0 36 9 110 98 17 54 0 35 56 110 1 2 98 35 52 0 35 42 111 98 53 43 0 35 29 111 1 2 99 11 27 0 35 15 112 99 29 5 0 35 1 112 1 2 99 46 35 0 34 48 113 100 3 59 0 34 34 179 119 59 44 0 0 25 179 1 2 119 59 56 0 0 9 180 120 0 0 0 0 0 \begin{array}[]{|l|rrr|rrr|}\hline\,\text{arc}&\,\text{chord}&&&\,\text{% sixtieths}&&\\ \hline{}\,\,\,\,\,\,\,\,\,\,\tfrac{1}{2}&0&31&25&1&2&50\\ {}\,\,\,\,\,\,\,1&1&2&50&1&2&50\\ {}\,\,\,\,\,\,\,1\tfrac{1}{2}&1&34&15&1&2&50\\ {}\,\,\,\,\,\,\,\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 109&97&41&38&0&36&23\\ 109\tfrac{1}{2}&97&59&49&0&36&9\\ 110&98&17&54&0&35&56\\ 110\tfrac{1}{2}&98&35&52&0&35&42\\ 111&98&53&43&0&35&29\\ 111\tfrac{1}{2}&99&11&27&0&35&15\\ 112&99&29&5&0&35&1\\ 112\tfrac{1}{2}&99&46&35&0&34&48\\ 113&100&3&59&0&34&34\\ {}\,\,\,\,\,\,\,\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 179&119&59&44&0&0&25\\ 179\frac{1}{2}&119&59&56&0&0&9\\ 180&120&0&0&0&0&0\\ \hline\end{array}
  7. α alpha 1 ι iota 10 ϱ rho 100 β beta 2 κ kappa 20 γ gamma 3 λ lambda 30 δ delta 4 μ mu 40 ε epsilon 5 ν nu 50 \stigma stigma ( archaic ) 6 ξ xi 60 ζ zeta 7 \omicron omicron 70 η eta 8 π pi 80 ϑ theta 9 \koppa koppa ( archaic ) 90 \begin{array}[]{|rlr|rlr|rlr|}\hline\alpha&\mathrm{alpha}&1&\iota&\mathrm{iota% }&10&\varrho&\mathrm{rho}&100\\ \beta&\mathrm{beta}&2&\kappa&\mathrm{kappa}&20&&&\\ \gamma&\mathrm{gamma}&3&\lambda&\mathrm{lambda}&30&&&\\ \delta&\mathrm{delta}&4&\mu&\mathrm{mu}&40&&&\\ \varepsilon&\mathrm{epsilon}&5&\nu&\mathrm{nu}&50&&&\\ \stigma&\mathrm{stigma\ (archaic)}&6&\xi&\mathrm{xi}&60&&&\\ \zeta&\mathrm{zeta}&7&\omicron&\mathrm{omicron}&70&&&\\ \eta&\mathrm{eta}&8&\pi&\mathrm{pi}&80&&&\\ \vartheta&\mathrm{theta}&9&\koppa&\mathrm{koppa\ (archaic)}&90&&&\\ \hline\end{array}
  8. 1431 / 2 143{1}/{2}
  9. ϱ μ γ \varrho\mu\gamma\angle^{\prime}
  10. 71 / 2 7{1}/{2}
  11. π ε ϱ ι φ ε ϱ ε ι ω ~ ν ε ν ϑ ε ι ω ~ ν ε ξ η κ \omicron σ τ ω ~ ν α α β β γ γ δ δ ε ε \stigma \stigma ζ ζ λ α κ ε α β ν α λ δ ι ε β ε μ β λ ζ δ γ η κ η γ λ ϑ ν β δ ι α ι \stigma δ μ β μ ε ι δ δ ε μ ε κ ζ \stigma ι \stigma μ ϑ \stigma μ η ι α ζ ι ϑ λ γ ζ ν ν δ α β ν α β ν α β ν α β ν α β μ η α β μ η α β μ η α β μ ζ α β μ ζ α β μ \stigma α β μ ε α β μ δ α β μ γ α β μ β α β μ α \begin{array}[]{ccc}\pi\varepsilon\varrho\iota\varphi\varepsilon\varrho% \varepsilon\iota\tilde{\omega}\nu&\varepsilon\overset{\,\text{'}}{\nu}% \vartheta\varepsilon\iota\tilde{\omega}\nu&\overset{\,\text{`}}{\varepsilon}% \xi\eta\kappa\omicron\sigma\tau\tilde{\omega}\nu\\ \begin{array}[]{|l|}\hline\angle^{\prime}\\ \alpha\\ \alpha\;\angle^{\prime}\\ \hline\beta\\ \beta\;\angle^{\prime}\\ \gamma\\ \hline\gamma\;\angle^{\prime}\\ \delta\\ \delta\;\angle^{\prime}\\ \hline\varepsilon\\ \varepsilon\;\angle^{\prime}\\ \stigma\\ \hline\stigma\;\angle^{\prime}\\ \zeta\\ \zeta\;\angle^{\prime}\\ \hline\end{array}&\begin{array}[]{|r|r|r|}\hline\circ&\lambda\alpha&\kappa% \varepsilon\\ \alpha&\beta&\nu\\ \alpha&\lambda\delta&\iota\varepsilon\\ \hline\beta&\varepsilon&\mu\\ \beta&\lambda\zeta&\delta\\ \gamma&\eta&\kappa\eta\\ \hline\gamma&\lambda\vartheta&\nu\beta\\ \delta&\iota\alpha&\iota\stigma\\ \delta&\mu\beta&\mu\\ \hline\varepsilon&\iota\delta&\delta\\ \varepsilon&\mu\varepsilon&\kappa\zeta\\ \stigma&\iota\stigma&\mu\vartheta\\ \hline\stigma&\mu\eta&\iota\alpha\\ \zeta&\iota\vartheta&\lambda\gamma\\ \zeta&\nu&\nu\delta\\ \hline\end{array}&\begin{array}[]{|r|r|r|r|}\hline\circ&\alpha&\beta&\nu\\ \circ&\alpha&\beta&\nu\\ \circ&\alpha&\beta&\nu\\ \hline\circ&\alpha&\beta&\nu\\ \circ&\alpha&\beta&\mu\eta\\ \circ&\alpha&\beta&\mu\eta\\ \hline\circ&\alpha&\beta&\mu\eta\\ \circ&\alpha&\beta&\mu\zeta\\ \circ&\alpha&\beta&\mu\zeta\\ \hline\circ&\alpha&\beta&\mu\stigma\\ \circ&\alpha&\beta&\mu\varepsilon\\ \circ&\alpha&\beta&\mu\delta\\ \hline\circ&\alpha&\beta&\mu\gamma\\ \circ&\alpha&\beta&\mu\beta\\ \circ&\alpha&\beta&\mu\alpha\\ \hline\end{array}\end{array}
  12. π ε \pi\varepsilon
  13. π \pi
  14. ε \varepsilon
  15. π α \pi\alpha
  16. δ \delta
  17. ι ε \iota\varepsilon
  18. π ε ϱ ι φ ε ϱ ε ι ω ~ ν ε ν ϑ ε ι ω ~ ν ε ξ η κ \omicron σ τ ω ~ ν π δ π ε π ε π \stigma π \stigma π ζ π μ α γ π α δ ι ε π α κ ζ κ β π α ν κ δ π β ι γ ι ϑ π β λ \stigma ϑ μ \stigma κ ε μ \stigma ι δ μ \stigma γ μ ε ν β μ ε μ μ ε κ ϑ \begin{array}[]{ccc}\pi\varepsilon\varrho\iota\varphi\varepsilon\varrho% \varepsilon\iota\tilde{\omega}\nu&\varepsilon\overset{\,\text{'}}{\nu}% \vartheta\varepsilon\iota\tilde{\omega}\nu&\overset{\,\text{`}}{\varepsilon}% \xi\eta\kappa\omicron\sigma\tau\tilde{\omega}\nu\\ \begin{array}[]{|l|}\hline\pi\delta\angle^{\prime}\\ \pi\varepsilon\\ \pi\varepsilon\angle^{\prime}\\ \hline\pi\stigma\\ \pi\stigma\angle^{\prime}\\ \pi\zeta\\ \hline\end{array}&\begin{array}[]{|r|r|r|}\hline\pi&\mu\alpha&\gamma\\ \pi\alpha&\delta&\iota\varepsilon\\ \pi\alpha&\kappa\zeta&\kappa\beta\\ \hline\pi\alpha&\nu&\kappa\delta\\ \pi\beta&\iota\gamma&\iota\vartheta\\ \pi\beta&\lambda\stigma&\vartheta\\ \hline\end{array}&\begin{array}[]{|r|r|r|r|}\hline\circ&\circ&\mu\stigma&% \kappa\varepsilon\\ \circ&\circ&\mu\stigma&\iota\delta\\ \circ&\circ&\mu\stigma&\gamma\\ \hline\circ&\circ&\mu\varepsilon&\nu\beta\\ \circ&\circ&\mu\varepsilon&\mu\\ \circ&\circ&\mu\varepsilon&\kappa\vartheta\\ \hline\end{array}\end{array}

Pulsatile_flow.html

  1. ρ u t = - p x + μ ( 2 u r 2 + 1 r u r ) \rho\frac{\partial u}{\partial t}=-\frac{\partial p}{\partial x}+\mu\left(% \frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}% \right)\,
  2. u x = 0 . \frac{\partial u}{\partial x}=0\,.
  3. p x = n = - N N C n e i n ω t . \frac{\partial p}{\partial x}=\sum^{N}_{n=-N}C_{n}e^{in\omega t}\,.
  4. u ( r , t ) = n = - N N U n e i n ω t . u(r,t)=\sum^{N}_{n=-N}U_{n}e^{in\omega t}\,.
  5. i ρ n ω U n = - C n + μ ( 2 U n r 2 + 1 r U n r ) . i\rho n\omega U_{n}=-C_{n}+\mu\left(\frac{\partial^{2}U_{n}}{\partial r^{2}}+% \frac{1}{r}\frac{\partial U_{n}}{\partial r}\right)\,.
  6. U n ( r ) = A n J 0 ( α r R n 1 / 2 i 3 / 2 ) + B n Y 0 ( α r R n 1 / 2 i 3 / 2 ) + i C n ρ n ω , U_{n}(r)=A_{n}J_{0}\left(\alpha\frac{r}{R}n^{1/2}i^{3/2}\right)+B_{n}Y_{0}% \left(\alpha\frac{r}{R}n^{1/2}i^{3/2}\right)+\frac{iC_{n}}{\rho n\omega}\,,
  7. J 0 ( k r ) J_{0}(kr)
  8. Y 0 ( k r ) Y_{0}(kr)
  9. k k
  10. A n A_{n}
  11. B n B_{n}
  12. α \alpha
  13. A n A_{n}
  14. B n B_{n}
  15. U n / r = 0 \partial U_{n}/\partial r=0
  16. J 0 J_{0}^{\prime}
  17. Y 0 Y_{0}^{\prime}
  18. B n B_{n}
  19. U n ( R ) = 0 = A n J 0 ( α n 1 / 2 i 3 / 2 ) + i C n ρ n ω . U_{n}(R)=0=A_{n}J_{0}\left(\alpha n^{1/2}i^{3/2}\right)+\frac{iC_{n}}{\rho n% \omega}\,.
  20. A n A_{n}
  21. U n ( r ) = i C n ρ n ω [ 1 - J 0 ( α r R n 1 / 2 i 3 / 2 ) J 0 ( α n 1 / 2 i 3 / 2 ) ] , U_{n}(r)=\frac{iC_{n}}{\rho n\omega}\left[1-\frac{J_{0}(\alpha\frac{r}{R}n^{1/% 2}i^{3/2})}{J_{0}(\alpha n^{1/2}i^{3/2})}\right]\,,
  22. u ( r ) = n = - N N i C n ρ n ω [ 1 - J 0 ( α r R n 1 / 2 i 3 / 2 ) J 0 ( α n 1 / 2 i 3 / 2 ) ] e i n ω t . u(r)=\sum^{N}_{n=-N}\frac{iC_{n}}{\rho n\omega}\left[1-\frac{J_{0}(\alpha\frac% {r}{R}n^{1/2}i^{3/2})}{J_{0}(\alpha n^{1/2}i^{3/2})}\right]e^{in\omega t}\,.
  23. α \alpha

Pulse-density_modulation.html

  1. x [ n ] = - A ( - 1 ) a [ n ] x[n]=-A(-1)^{a[n]}
  2. x [ n ] x[n]
  3. y [ n ] y[n]
  4. Y ( z ) = X ( z ) + E ( z ) ( 1 - z - 1 ) Y(z)=X(z)+E(z)\left(1-z^{-1}\right)
  5. Y ( z ) = E ( z ) + [ X ( z ) - Y ( z ) z - 1 ] ( 1 1 - z - 1 ) . Y(z)=E(z)+\left[X(z)-Y(z)z^{-1}\right]\left(\frac{1}{1-z^{-1}}\right).
  6. E ( z ) E(z)
  7. 1 - z - 1 1-z^{-1}
  8. E ( z ) E(z)
  9. Y ( z ) Y(z)
  10. y [ n ] = x [ n ] + e [ n ] - e [ n - 1 ] y[n]=x[n]+e[n]-e[n-1]
  11. y [ n ] y[n]
  12. e [ n ] e[n]
  13. y [ n ] y[n]
  14. y [ n ] = ± 1 y[n]=\pm 1
  15. y [ n ] = { + 1 x [ n ] e [ n - 1 ] - 1 x [ n ] < e [ n - 1 ] y[n]=\begin{cases}+1&x[n]\geq e[n-1]\\ -1&x[n]<e[n-1]\end{cases}
  16. e [ n ] = y [ n ] - x [ n ] + e [ n - 1 ] e[n]=y[n]-x[n]+e[n-1]
  17. y [ n ] y[n]
  18. x [ n ] x[n]

Pulse-Doppler_signal_processing.html

  1. T h r e s h o l d C r i t e r i a { Cell ( n ) > [ Cell ( n - 2 ) + Cell ( n - 1 ) + Cell ( n + 1 ) + Cell ( n + 2 ) ] × Constant Threshold\ Criteria\begin{cases}\mathrm{\begin{aligned}\displaystyle Cell(n)>[% Cell(n-2)+\\ \displaystyle Cell(n-1)+\\ \displaystyle Cell(n+1)+\\ \displaystyle Cell(n+2)]\times\\ \displaystyle Constant\end{aligned}}\end{cases}
  2. + +
  3. - -
  4. P e a k C r i t e r i a { ( Δ Amplitude Δ Frequency ) Cell ( n - 1 ) < 0 ( Δ Amplitude Δ Frequency ) Cell ( n + 1 ) > 0 Peak\ Criteria\begin{cases}\mathrm{\left(\frac{\Delta Amplitude}{\Delta Frequency% }\right)Cell(n-1)<0}\\ \mathrm{\left(\frac{\Delta Amplitude}{\Delta Frequency}\right)Cell(n+1)>0}\end% {cases}
  5. S p e e d C r i t e r i a { ( C × Doppler Frequency 2 × Transmit Frequency ) > Rejection Speed\ Criteria\begin{cases}\mathrm{\left(\frac{C\times Doppler\ Frequency}{2% \times Transmit\ Frequency}\right)>Rejection}\end{cases}
  6. M a i n l o b e C r i t e r i a { Main Lobe > Constant × Side Lobe Mainlobe\ Criteria\begin{cases}\mathrm{Main\ Lobe>Constant\times Side\ Lobe}% \end{cases}
  7. Lock criteria { ( Δ R Δ T ) - ( C × Doppler Frequency 2 × Transmit Frequency ) < Threshold \,\text{Lock criteria}\begin{cases}\mathrm{\left(\frac{\Delta R}{\Delta T}% \right)-\left(\frac{C\times\,\text{Doppler Frequency}}{2\times\,\text{Transmit% Frequency}}\right)<\,\text{Threshold}}\end{cases}

Pulsed_electron_paramagnetic_resonance.html

  1. ω L = - γ B 0 \omega_{L}=-\gamma B_{0}
  2. ω L = ω 0 \omega_{L}=\omega_{0}
  3. ω 1 = - γ B 1 \omega_{1}=-\gamma B_{1}
  4. α = - γ | B 1 | t p \alpha=-\gamma|B_{1}|t_{p}
  5. Δ ω = ω - ω 0 \Delta\omega=\omega-\omega_{0}
  6. ω e f f 2 = ( ω 1 2 + Δ ω 2 ) 1 / 2 \omega_{eff}^{2}=(\omega_{1}^{2}+\Delta\omega^{2})^{1/2}

Purcell_effect.html

  1. F P = 3 4 π 2 ( λ c n ) 3 ( Q V ) , F_{P}=\frac{3}{4\pi^{2}}\left(\frac{\lambda_{c}}{n}\right)^{3}\left(\frac{Q}{V% }\right)\,,
  2. ( λ c / n ) (\lambda_{c}/n)
  3. Q Q
  4. V V
  5. ρ f = 8 π n 3 ν 2 c 3 \rho_{f}=\frac{8\pi n^{3}\nu^{2}}{c^{3}}
  6. ρ c = 1 Δ ν V \rho_{c}=\frac{1}{\Delta\nu V}
  7. Q = ν / Δ ν Q=\nu/\Delta\nu
  8. ρ c / ρ f = c 3 8 π n 3 ν 2 Q ν V = 1 8 π ( λ n ) 3 ( Q V ) \rho_{c}/\rho_{f}=\frac{c^{3}}{8\pi n^{3}\nu^{2}}\frac{Q}{\nu V}=\frac{1}{8\pi% }\left(\frac{\lambda}{n}\right)^{3}\left(\frac{Q}{V}\right)

Pure_bending.html

  1. d M d x = V \frac{dM}{dx}=V

Pure_shear.html

  1. τ \tau
  2. γ \gamma
  3. τ = γ G \tau=\gamma G\,
  4. G G
  5. G = E 2 ( 1 + ν ) G=\frac{E}{2(1+\nu)}
  6. E E
  7. ν \nu
  8. τ = γ E 2 ( 1 + ν ) \tau=\frac{\gamma E}{2(1+\nu)}

Pythagorean_field.html

  1. 0 Tor I W ( F ) W ( F ) W ( F py ) 0\rightarrow\mathrm{Tor}IW(F)\rightarrow W(F)\rightarrow W(F^{\mathrm{py}})

Pytkeev_space.html

  1. π \pi

P–n_diode.html

  1. r D = Δ v D Δ i D | v D = V B I A S , r_{D}=\left.\frac{\Delta v_{D}}{\Delta i_{D}}\right|_{v_{D}=V_{BIAS}}\ ,
  2. p n = p B n B e - φ B / V th pn=p_{B}n_{B}\,e^{-\varphi_{\mathrm{B}}/V_{\mathrm{th}}}
  3. p n = ( p B n B e - φ B / V t h ) e v D / V t h . pn=\left(p_{B}n_{B}\ e^{-\varphi_{B}/V_{th}}\right)\ e^{v_{D}/V_{th}}\ .
  4. i D = I R ( e v D / V th - 1 ) , i_{\mathrm{D}}=I_{\mathrm{R}}\left(e^{v_{\mathrm{D}}/V_{\mathrm{th}}}-1\right),
  5. i D = I R ( e v D / n V th - 1 ) , i_{D}=I_{R}\left(e^{v_{\mathrm{D}}/nV_{\mathrm{th}}}-1\right),
  6. r D = 1 d i D / d v D n V th i D , r_{\mathrm{D}}=\frac{1}{di_{\mathrm{D}}/dv_{\mathrm{D}}}\approx\frac{nV_{% \mathrm{th}}}{i_{\mathrm{D}}},
  7. C J = κ ε 0 A w ( v R ) , C_{J}=\kappa\varepsilon_{0}\frac{A}{w(v_{R})}\ ,
  8. i D = Q D τ T , i_{D}=\frac{Q_{D}}{\tau_{T}}\ ,
  9. C D = d Q D d v D = τ T d i D d v D = i D τ T V t h . C_{D}=\frac{dQ_{D}}{dv_{D}}=\tau_{T}\frac{di_{D}}{dv_{D}}=\frac{i_{D}\tau_{T}}% {V_{th}}\ .
  10. I S = ( j ω ( C J + C D ) + 1 r D + 1 R S ) V O , I_{S}=\left(j\omega(C_{J}+C_{D})+\frac{1}{r_{D}}+\frac{1}{R_{S}}\right)V_{O}\ ,
  11. V O I S = ( R S r D ) 1 + j ω ( C D + C J ) ( R S r D ) , \frac{V_{O}}{I_{S}}=\frac{(R_{S}\mathit{\parallel}r_{D})}{1+j\omega(C_{D}+C_{J% })(R_{S}\mathit{\parallel}r_{D})}\ ,
  12. f C = 1 2 π ( C D + C J ) ( R S r D ) , f_{C}=\frac{1}{2\pi(C_{D}+C_{J})(R_{S}\mathit{\parallel}r_{D})}\ ,
  13. f C = 1 2 π n τ T , f_{C}=\frac{1}{2\pi n\tau_{T}}\ ,
  14. f C = 1 2 π C J ( R S r D ) , f_{C}=\frac{1}{2\pi C_{J}(R_{S}\mathit{\parallel}r_{D})}\ ,

Q-Bessel_polynomials.html

  1. y n ( x ; a ; q ) = 2 ϕ 1 ( q - N - a q n 0 ; q , q x ) y_{n}(x;a;q)=\;_{2}\phi_{1}\left(\begin{matrix}q^{-N}&-aq^{n}\\ 0\end{matrix};q,qx\right)
  2. k = 0 ( a k ( q ; q ) n * q ( k + 1 2 ) * y m * ( q k ; a ; q ) * y n * ( q k ; a ; q ) = ( q ; q ) n * ( - a q n ; q ) a n * q ( n + 1 2 ) 1 + a q 2 n δ m n \sum_{k=0}^{\infty}(\frac{a^{k}}{(q;q)_{n}}*q^{k+1\choose 2}*y_{m}*(q^{k};a;q)% *y_{n}*(q^{k};a;q)=(q;q)_{n}*(-aq^{n};q)_{\infty}\frac{a^{n}*q^{n+1\choose 2}}% {1+aq^{2n}}\delta_{mn}

Q-exponential_distribution.html

  1. ( 2 - q ) λ e q - λ x {(2-q)\lambda e_{q}^{-\lambda x}}
  2. 1 - e q - λ x q where q = 1 2 - q {1-e_{q^{\prime}}^{-\lambda x\over q^{\prime}}}\,\text{ where }q^{\prime}={1% \over{2-q}}
  3. 1 λ ( 3 - 2 q ) for q < 3 2 {1\over\lambda(3-2q)}\,\text{ for }q<{3\over 2}
  4. - q ln q ( 1 2 ) λ where q = 1 2 - q {{-q^{\prime}\,\text{ ln}_{q^{\prime}}({1\over 2})}\over{\lambda}}\,\text{ % where }q^{\prime}={1\over{2-q}}
  5. q - 2 ( 2 q - 3 ) 2 ( 3 q - 4 ) λ 2 for q < 4 3 {{q-2}\over{(2q-3)^{2}(3q-4)\lambda^{2}}}\,\text{ for }q<{4\over 3}
  6. 2 5 - 4 q 3 q - 4 q - 2 for q < 5 4 {2\over{5-4q}}\sqrt{{3q-4}\over{q-2}}\,\text{ for }q<{5\over 4}
  7. 6 - 4 q 3 + 17 q 2 - 20 q + 6 ( q - 2 ) ( 4 q - 5 ) ( 5 q - 6 ) for q < 6 5 6{{-4q^{3}+17q^{2}-20q+6}\over{(q-2)(4q-5)(5q-6)}}\,\text{ for }q<{6\over 5}
  8. q 1 q\rightarrow 1
  9. ( 2 - q ) λ e q ( - λ x ) {(2-q)\lambda e_{q}(-\lambda x)}
  10. e q ( x ) = [ 1 + ( 1 - q ) x ] 1 1 - q e_{q}(x)=[1+(1-q)x]^{1\over 1-q}
  11. μ = 0 , ξ = q - 1 2 - q , σ = 1 λ ( 2 - q ) \mu=0~{},~{}\xi={{q-1}\over{2-q}}~{},~{}\sigma={1\over{\lambda(2-q)}}
  12. α = 2 - q q - 1 , λ Lomax = 1 λ ( q - 1 ) \alpha={{2-q}\over{q-1}}~{},~{}\lambda\text{Lomax}={1\over{\lambda(q-1)}}
  13. If X qExp ( q , λ ) and Y [ Pareto ( x m = 1 λ ( q - 1 ) , α = 2 - q q - 1 ) - x m ] , then X Y \,\text{If }X\sim\mathrm{qExp}(q,\lambda)\,\text{ and }Y\sim\left[\,\text{% Pareto}\left(x_{m}={1\over{\lambda(q-1)}},\alpha={{2-q}\over{q-1}}\right)-x_{m% }\right],\,\text{ then }X\sim Y\,
  14. X = - q ln q ( U ) λ qExp ( q , λ ) X={{-q^{\prime}\,\text{ ln}_{q^{\prime}}(U)}\over\lambda}\sim\mbox{qExp}~{}(q,\lambda)
  15. ln q \,\text{ln}_{q^{\prime}}
  16. q = 1 2 - q q^{\prime}={1\over{2-q}}

Q-Gaussian_distribution.html

  1. q < 1 q<1
  2. β C q e q ( - β x 2 ) {\sqrt{\beta}\over C_{q}}e_{q}({-\beta x^{2}})
  3. 0 for q < 2 0\,\text{ for }q<2
  4. 0
  5. 0
  6. 1 β ( 5 - 3 q ) for q < 5 3 {1\over{\beta(5-3q)}}\,\text{ for }q<{5\over 3}
  7. for 5 3 q < 2 \infty\,\text{ for }{5\over 3}\leq q<2
  8. Undefined for 2 q < 3 \,\text{Undefined for }2\leq q<3
  9. 0 for q < 3 2 0\,\text{ for }q<{3\over 2}
  10. 6 q - 1 7 - 5 q for q < 7 5 6{q-1\over 7-5q}\,\text{ for }q<{7\over 5}
  11. q 1 q\rightarrow 1
  12. 1 < q < 3 1<q<3
  13. f ( x ) = β C q e q ( - β x 2 ) f(x)={\sqrt{\beta}\over C_{q}}e_{q}(-\beta x^{2})
  14. e q ( x ) = [ 1 + ( 1 - q ) x ] 1 1 - q e_{q}(x)=[1+(1-q)x]^{1\over 1-q}
  15. C q C_{q}
  16. C q = 2 π Γ ( 1 1 - q ) ( 3 - q ) 1 - q Γ ( 3 - q 2 ( 1 - q ) ) for - < q < 1 C_{q}={{2\sqrt{\pi}\Gamma\left({1\over 1-q}\right)}\over{(3-q)\sqrt{1-q}\Gamma% \left({3-q\over 2(1-q)}\right)}}\,\text{ for }-\infty<q<1
  17. C q = π for q = 1 C_{q}=\sqrt{\pi}\,\text{ for }q=1\,
  18. C q = π Γ ( 3 - q 2 ( q - 1 ) ) q - 1 Γ ( 1 q - 1 ) for 1 < q < 3. C_{q}={{\sqrt{\pi}\Gamma\left({3-q\over 2(q-1)}\right)}\over{\sqrt{q-1}\Gamma% \left({1\over q-1}\right)}}\,\text{ for }1<q<3.
  19. E ( X ) \operatorname{E}(X)
  20. E ( X 2 ) \operatorname{E}(X^{2})
  21. E ( X 0 ) = 1 \operatorname{E}(X^{0})=1
  22. ν \nu
  23. ν \nu
  24. q and β q\,\text{ and }\beta
  25. ν \nu
  26. ν \nu
  27. q = ν + 3 ν + 1 with β = 1 3 - q q=\frac{\nu+3}{\nu+1}\,\text{ with }\beta=\frac{1}{3-q}
  28. ν = 3 - q q - 1 , but only if β = 1 3 - q . \nu=\frac{3-q}{q-1},\,\text{ but only if }\beta=\frac{1}{3-q}.
  29. β 1 3 - q \beta\neq{1\over{3-q}}
  30. ν < 0 \nu<0
  31. μ \mu
  32. β C q e q ( - β ( x - μ ) 2 ) . {\sqrt{\beta}\over C_{q}}e_{q}({-\beta(x-\mu)^{2}}).
  33. Z 1 = - 2 ln ( U 1 ) cos ( 2 π U 2 ) Z_{1}=\sqrt{-2\ln(U_{1})}\cos(2\pi U_{2})
  34. Z 2 = - 2 ln ( U 1 ) sin ( 2 π U 2 ) Z_{2}=\sqrt{-2\ln(U_{1})}\sin(2\pi U_{2})
  35. β = 1 3 - q \beta={1\over{3-q}}
  36. Z = - 2 ln q ( U 1 ) cos ( 2 π U 2 ) Z=\sqrt{-2\,\text{ ln}_{q^{\prime}}(U_{1})}\,\text{ cos}(2\pi U_{2})
  37. ln q \,\text{ ln}_{q}
  38. q = 1 + q 3 - q q^{\prime}={{1+q}\over{3-q}}
  39. Z = μ + Z β ( 3 - q ) Z^{\prime}=\mu+{Z\over\sqrt{\beta(3-q)}}

Q-Hahn_polynomials.html

  1. Q n ( x ; a , b , N ; q ) = 3 ϕ 2 [ q - n a b q n + 1 x a q q - N ; q , q ] Q_{n}(x;a,b,N;q)=\;_{3}\phi_{2}\left[\begin{matrix}q^{-}n&abq^{n}+1&x\\ aq&q^{-}N\end{matrix};q,q\right]
  2. lim a Q n ( q - x ; a ; p , N | q ) = K n q t m ( q - x ; p , N ; q ) \lim_{a\to\infty}Q_{n}(q^{-}{x};a;p,N|q)=K_{n}^{qtm}(q^{-}{x};p,N;q)
  3. α = q α \alpha=q^{\alpha}
  4. β = q β \beta=q^{\beta}
  5. F 2 3 ( [ - n , α + β + n + 1 , - x ] , [ α + 1 , - N ] , 1 ) {}_{3}F_{2}([-n,\alpha+\beta+n+1,-x],[\alpha+1,-N],1)

Q-Laguerre_polynomials.html

  1. L n ( α ) ( x ; q ) = ( q α + 1 ; q ) n ( q ; q ) n ϕ 1 1 ( q - n ; q α + 1 ; q , - q n + α + 1 x ) \displaystyle L_{n}^{(\alpha)}(x;q)=\frac{(q^{\alpha+1};q)_{n}}{(q;q)_{n}}{}_{% 1}\phi_{1}(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)

Q-system_(geotechnical_engineering).html

  1. Q = R Q D J n × J r J a × J w S R F Q=\frac{RQD}{J_{n}}\times\frac{J_{r}}{J_{a}}\times\frac{J_{w}}{SRF}

Q-Vectors.html

  1. D g u g D t - f 0 v a - β y v g = 0 \frac{D_{g}u_{g}}{Dt}-f_{0}v_{a}-\beta yv_{g}=0
  2. D g v g D t + f 0 u a + β y u g = 0 \frac{D_{g}v_{g}}{Dt}+f_{0}u_{a}+\beta yu_{g}=0
  3. D g T D t - σ p R ω = J c p \frac{D_{g}T}{Dt}-\frac{\sigma p}{R}\omega=\frac{J}{c_{p}}
  4. f 0 u g p = R p T y f_{0}\frac{\partial u_{g}}{\partial p}=\frac{R}{p}\frac{\partial T}{\partial y}
  5. f 0 v g p = - R p T x f_{0}\frac{\partial v_{g}}{\partial p}=-\frac{R}{p}\frac{\partial T}{\partial x}
  6. f 0 f_{0}
  7. R R
  8. β \beta
  9. β = f y \beta=\frac{\partial f}{\partial y}
  10. σ \sigma
  11. c p c_{p}
  12. p p
  13. T T
  14. g g
  15. a a
  16. J J
  17. ω \omega
  18. ω = D p D t \omega=\frac{Dp}{Dt}
  19. - ω -\omega
  20. + w = D z D t +w=\frac{Dz}{Dt}
  21. Q 1 = - R p [ u g x T x + v g x T y ] Q_{1}=-\frac{R}{p}\left[\frac{\partial u_{g}}{\partial x}\frac{\partial T}{% \partial x}+\frac{\partial v_{g}}{\partial x}\frac{\partial T}{\partial y}\right]
  22. Q 2 = - R p [ u g y T x + v g y T y ] Q_{2}=-\frac{R}{p}\left[\frac{\partial u_{g}}{\partial y}\frac{\partial T}{% \partial x}+\frac{\partial v_{g}}{\partial y}\frac{\partial T}{\partial y}\right]
  23. Q 1 = - R p V g x T Q_{1}=-\frac{R}{p}\frac{\partial\vec{V_{g}}}{\partial x}\cdot\vec{\nabla}T
  24. Q 2 = - R p V g y T Q_{2}=-\frac{R}{p}\frac{\partial\vec{V_{g}}}{\partial y}\cdot\vec{\nabla}T
  25. ( σ 2 + f 2 2 p 2 ) ω = - 2 Q + f β v g p - κ p 2 J \left(\sigma\overrightarrow{\nabla^{2}}+f_{\circ}^{2}\frac{\partial^{2}}{% \partial p^{2}}\right)\omega=-2\vec{\nabla}\cdot\vec{Q}+f_{\circ}\beta\frac{% \partial v_{g}}{\partial p}-\frac{\kappa}{p}\overrightarrow{\nabla^{2}}J
  26. - ω - 2 Q -\omega\propto-2\vec{\nabla}\cdot\vec{Q}
  27. - ω -\omega
  28. - ω -\omega
  29. Q \vec{\nabla}\cdot\vec{Q}
  30. Q \vec{Q}
  31. Q \vec{Q}
  32. Φ \Phi
  33. T y < 0 \frac{\partial T}{\partial y}<0

Q_zero.html

  1. ( 1 ) (1)
  2. g o o T and g o o F = x o g o o x o g_{oo}T\and g_{oo}F=\forall x_{o}\centerdot g_{oo}x_{o}
  3. ( 2 α ) (2^{\alpha})
  4. [ x α = y α ] h o α x α = h o α y α [x_{\alpha}=y_{\alpha}]\supset\centerdot\,h_{o\alpha}x_{\alpha}=h_{o\alpha}y_{\alpha}
  5. ( 3 α β ) (3^{\alpha\beta})
  6. f α β = g α β = x β f α β x β = g α β x β f_{\alpha\beta}=g_{\alpha\beta}=\forall x_{\beta}\centerdot f_{\alpha\beta}x_{% \beta}=g_{\alpha\beta}x_{\beta}
  7. ( 4 ) (4)
  8. [ λ x α B β ] A α = S A α x α B β [\lambda{x_{\alpha}}{B}_{\beta}]{A}_{\alpha}={S}^{x_{\alpha}}_{A_{\alpha}}{B}_% {\beta}
  9. ( 5 ) (5)
  10. ι i ( o i ) [ Q o i i y i ] = y i \iota_{i(oi)}[\,\text{Q}_{oii}y_{i}]=y_{i}\,
  11. 𝐀 \mathbf{A}
  12. 𝐁 \mathbf{B}
  13. 𝐂 \mathbf{C}
  14. 𝐱 \mathbf{x}
  15. 𝐲 \mathbf{y}
  16. 𝐒 \mathbf{S}
  17. Q < s u b > ( ( o α ) α ) Q<sub>((oα)α)

QED_vacuum.html

  1. Δ E Δ t 2 , \Delta E\Delta t\geq\frac{\hbar}{2}\ ,
  2. 𝐁 \displaystyle\mathbf{B}
  3. 𝚷 = ε 0 t 𝐀 , \mathbf{\Pi}=\varepsilon_{0}\frac{\partial}{\partial t}\mathbf{A}\ ,
  4. [ Π i ( 𝐫 , t ) , A j ( 𝐫 , t ) ] = - i δ i j δ ( 𝐫 - 𝐫 ) , \left[\Pi_{i}(\mathbf{r},\ t),\ A_{j}(\mathbf{r^{\prime}},\ t)\right]=-i\hbar% \delta_{ij}\delta(\mathbf{r-r^{\prime}})\ ,
  5. [ E ^ k ( s y m b o l r ) , B ^ k ( s y m b o l r ) ] = - ϵ k k m i ε 0 x m δ ( s y m b o l r - r ) , \left[\hat{E}_{k}(symbolr),\ \hat{B}_{k^{\prime}}(symbolr^{\prime})\right]=-% \epsilon_{kk^{\prime}m}\frac{i\hbar}{\varepsilon_{0}}\ \frac{\partial}{% \partial x_{m}}\delta(symbol{r-r^{\prime}})\ ,

QUADPACK.html

  1. x = ( 1 - t ) / t x=(1-t)/t
  2. - + f ( x ) d x = 0 1 d t t 2 ( f ( 1 - t t ) + f ( - 1 - t t ) ) . \int_{-\infty}^{+\infty}f(x)dx=\int_{0}^{1}{dt\over t^{2}}\left(f\left(\frac{1% -t}{t}\right)+f\left(-\frac{1-t}{t}\right)\right)\;.
  3. c o s ( ω x ) f ( x ) cos(ωx)f(x)
  4. s i n ( ω x ) f ( x ) sin(ωx)f(x)
  5. w ( x ) f ( x ) w(x)f(x)
  6. a a
  7. b b
  8. f f
  9. w ( x ) = ( x a ) < s u p > α ( b x ) β l o g k ( x a ) l o g l ( b x ) w(x)=(x–a)<sup>α(b–x)^{β}log^{k}(x–a)log^{l}(b–x)

Quadratic_mean_diameter.html

  1. D i 2 n \sqrt{\frac{\sum{D_{i}}^{2}}{n}}
  2. D i {D_{i}}
  3. B A k * n \sqrt{\frac{BA}{k*n}}

Quadruple_product.html

  1. ( 𝐚 × 𝐛 ) ( 𝐜 × 𝐝 ) , (\mathbf{a\times b})\mathbf{\cdot}(\mathbf{c}\times\mathbf{d})\ ,
  2. ( 𝐚 × 𝐛 ) ( 𝐜 × 𝐝 ) = ( 𝐚 𝐜 ) ( 𝐛 𝐝 ) - ( 𝐚 𝐝 ) ( 𝐛 𝐜 ) . (\mathbf{a\times b})\mathbf{\cdot}(\mathbf{c}\times\mathbf{d})=(\mathbf{a\cdot c% })(\mathbf{b\cdot d})-(\mathbf{a\cdot d})(\mathbf{b\cdot c})\ .
  3. ( 𝐚 × 𝐛 ) ( 𝐜 × 𝐝 ) = | 𝐚 𝐜 𝐚 𝐝 𝐛 𝐜 𝐛 𝐝 | . (\mathbf{a\times b})\mathbf{\cdot}(\mathbf{c}\times\mathbf{d})=\begin{vmatrix}% \mathbf{a\cdot c}&\mathbf{a\cdot d}\\ \mathbf{b\cdot c}&\mathbf{b\cdot d}\end{vmatrix}\ .
  4. ( 𝐚 × 𝐛 ) × ( 𝐜 × 𝐝 ) , (\mathbf{a\times b})\mathbf{\times}(\mathbf{c}\times\mathbf{d})\ ,
  5. ( 𝐚 × 𝐛 ) × ( 𝐜 × 𝐝 ) = [ 𝐚 , 𝐛 , 𝐝 ] 𝐜 - [ 𝐚 , 𝐛 , 𝐜 ] 𝐝 , (\mathbf{a\times b})\mathbf{\times}(\mathbf{c}\times\mathbf{d})=[\mathbf{a,\ b% ,\ d}]\mathbf{c}-[\mathbf{a,\ b,\ c}]\mathbf{d}\ ,
  6. ( 𝐚 × 𝐛 ) × ( 𝐜 × 𝐝 ) = ε i j k a i c j d k b l - ε i j k b i c j d k a l = ε i j k a i b j d k c l - ε i j k a i b j c k d l (\mathbf{a\times b})\mathbf{\times}(\mathbf{c}\times\mathbf{d})=\varepsilon_{% ijk}a^{i}c^{j}d^{k}b^{l}-\varepsilon_{ijk}b^{i}c^{j}d^{k}a^{l}=\varepsilon_{% ijk}a^{i}b^{j}d^{k}c^{l}-\varepsilon_{ijk}a^{i}b^{j}c^{k}d^{l}
  7. [ 𝐚 , 𝐛 , 𝐝 ] = ( 𝐚 × 𝐛 ) 𝐝 = | 𝐚 𝐢 ^ 𝐛 𝐢 ^ 𝐝 𝐢 ^ 𝐚 𝐣 ^ 𝐛 𝐣 ^ 𝐝 𝐣 ^ 𝐚 𝐤 ^ 𝐛 𝐤 ^ 𝐝 𝐤 ^ | = | 𝐚 𝐢 ^ 𝐚 𝐣 ^ 𝐚 𝐤 ^ 𝐛 𝐢 ^ 𝐛 𝐣 ^ 𝐛 𝐤 ^ 𝐝 𝐢 ^ 𝐝 𝐣 ^ 𝐝 𝐤 ^ | , [\mathbf{a,\ b,\ d}]=(\mathbf{a\times b})\mathbf{\cdot d}=\begin{vmatrix}% \mathbf{a\cdot}\hat{\mathbf{i}}&\mathbf{b\cdot}\hat{\mathbf{i}}&\mathbf{d\cdot% }\hat{\mathbf{i}}\\ \mathbf{a\cdot}\hat{\mathbf{j}}&\mathbf{b\cdot}\hat{\mathbf{j}}&\mathbf{d\cdot% }\hat{\mathbf{j}}\\ \mathbf{a\cdot}\hat{\mathbf{k}}&\mathbf{b\cdot}\hat{\mathbf{k}}&\mathbf{d\cdot% }\hat{\mathbf{k}}\end{vmatrix}=\begin{vmatrix}\mathbf{a\cdot}\hat{\mathbf{i}}&% \mathbf{a\cdot}\hat{\mathbf{j}}&\mathbf{a\cdot}\hat{\mathbf{k}}\\ \mathbf{b\cdot}\hat{\mathbf{i}}&\mathbf{b\cdot}\hat{\mathbf{j}}&\mathbf{b\cdot% }\hat{\mathbf{k}}\\ \mathbf{d\cdot}\hat{\mathbf{i}}&\mathbf{d\cdot}\hat{\mathbf{j}}&\mathbf{d\cdot% }\hat{\mathbf{k}}\end{vmatrix}\ ,
  8. 𝐢 ^ , 𝐣 ^ , 𝐤 ^ \hat{\mathbf{i}},\ \hat{\mathbf{j}},\ \hat{\mathbf{k}}
  9. [ 𝐛 , 𝐜 , 𝐝 ] 𝐚 - [ 𝐜 , 𝐝 , 𝐚 ] 𝐛 + [ 𝐝 , 𝐚 , 𝐛 ] 𝐜 - [ 𝐚 , 𝐛 , 𝐜 ] 𝐝 = 0 . [\mathbf{b,\ c,\ d}]\mathbf{a}-[\mathbf{c,\ d,\ a}]\mathbf{b}+[\mathbf{d,\ a,% \ b}]\mathbf{c}-[\mathbf{a,\ b,\ c}]\mathbf{d}=0\ .
  10. ( 𝐚 × 𝐛 ) ( 𝐜 × 𝐝 ) = ( 𝐚 𝐜 ) ( 𝐛 𝐝 ) - ( 𝐚 𝐝 ) ( 𝐛 𝐜 ) , (\mathbf{a\times b})\mathbf{\cdot}(\mathbf{c\times d})=(\mathbf{a\cdot c})(% \mathbf{b\cdot d})-(\mathbf{a\cdot d})(\mathbf{b\cdot c})\ ,
  11. 𝐚 × 𝐛 = a b sin θ a b , \|\mathbf{a\times b}\|=ab\sin\theta_{ab}\ ,
  12. 𝐚 𝐛 = a b cos θ a b , \|\mathbf{a\cdot b}\|=ab\cos\theta_{ab}\ ,
  13. sin θ a b sin θ c d cos x = cos θ a c cos θ b d - cos θ a d cos θ b c , \sin\theta_{ab}\sin\theta_{cd}\cos x=\cos\theta_{ac}\cos\theta_{bd}-\cos\theta% _{ad}\cos\theta_{bc}\ ,

Quantitative_electroencephalography.html

  1. f ( ξ ) = - f ( x ) e - 2 π i x ξ d x f(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}\,dx
  2. X ( a , b ) = 1 a - Ψ ( t - b a ) ¯ x ( t ) d t X(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}\overline{\Psi\left(\frac{t-b}% {a}\right)}x(t)\,dt

Quantitative_susceptibility_mapping.html

  1. δ B \delta B
  2. χ \chi
  3. d d
  4. δ B = d χ \delta B=d\otimes\chi
  5. Δ B = D \Chi \Delta B=D\cdot\Chi
  6. l 1 l_{1}

Quantum_capacity.html

  1. R = 1 - H ( 𝐩 ) R=1-H\left(\mathbf{p}\right)
  2. ρ p I ρ + p X X ρ X + p Y Y ρ Y + p Z Z ρ Z , \rho\mapsto p_{I}\rho+p_{X}X\rho X+p_{Y}Y\rho Y+p_{Z}Z\rho Z,
  3. 𝐩 = ( p I , p X , p Y , p Z ) \mathbf{p}=\left(p_{I},p_{X},p_{Y},p_{Z}\right)
  4. H ( 𝐩 ) H\left(\mathbf{p}\right)
  5. T δ 𝐩 n { a n : | - 1 n log 2 ( Pr { E a n } ) - H ( 𝐩 ) | δ } , T_{\delta}^{\mathbf{p}^{n}}\equiv\left\{a^{n}:\left|-\frac{1}{n}\log_{2}\left(% \Pr\left\{E_{a^{n}}\right\}\right)-H\left(\mathbf{p}\right)\right|\leq\delta% \right\},
  6. a n a^{n}
  7. { I , X , Y , Z } \left\{I,X,Y,Z\right\}
  8. Pr { E a n } \Pr\left\{E_{a^{n}}\right\}
  9. E a n E a 1 E a n E_{a^{n}}\equiv E_{a_{1}}\otimes\cdots\otimes E_{a_{n}}
  10. a n T δ 𝐩 n Pr { E a n } 1 - ϵ , \sum_{a^{n}\in T_{\delta}^{\mathbf{p}^{n}}}\Pr\left\{E_{a^{n}}\right\}\geq 1-\epsilon,
  11. ϵ > 0 \epsilon>0
  12. n n
  13. 𝒮 \mathcal{S}
  14. { E a n : a n T δ 𝐩 n } \{E_{a^{n}}:a^{n}\in T_{\delta}^{\mathbf{p}^{n}}\}
  15. E a n E b n N ( 𝒮 ) \ 𝒮 , E_{a^{n}}^{\dagger}E_{b^{n}}\notin N\left(\mathcal{S}\right)\backslash\mathcal% {S},
  16. E a n E_{a^{n}}
  17. E b n E_{b^{n}}
  18. a n , b n T δ 𝐩 n a^{n},b^{n}\in T_{\delta}^{\mathbf{p}^{n}}
  19. N ( 𝒮 ) N(\mathcal{S})
  20. 𝒮 \mathcal{S}
  21. 𝔼 𝒮 { p e } \displaystyle\mathbb{E}_{\mathcal{S}}\left\{p_{e}\right\}
  22. \mathcal{I}
  23. E a n E_{a^{n}}
  24. 𝒮 \mathcal{S}
  25. = a n T δ 𝐩 n Pr { E a n } Pr 𝒮 { E b n : b n T δ 𝐩 n , b n a n , E a n E b n N ( 𝒮 ) \ 𝒮 } =\sum_{a^{n}\in T_{\delta}^{\mathbf{p}^{n}}}\Pr\left\{E_{a^{n}}\right\}\Pr_{% \mathcal{S}}\left\{\exists E_{b^{n}}:b^{n}\in T_{\delta}^{\mathbf{p}^{n}},\ b^% {n}\neq a^{n},\ E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left(\mathcal{S}\right)% \backslash\mathcal{S}\right\}
  26. a n T δ A n Pr { E a n } Pr 𝒮 { E b n : b n T δ 𝐩 n , b n a n , E a n E b n N ( 𝒮 ) } \leq\sum_{a^{n}\in T_{\delta}^{A^{n}}}\Pr\left\{E_{a^{n}}\right\}\Pr_{\mathcal% {S}}\left\{\exists E_{b^{n}}:b^{n}\in T_{\delta}^{\mathbf{p}^{n}},\ b^{n}\neq a% ^{n},\ E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left(\mathcal{S}\right)\right\}
  27. = a n T δ 𝐩 n Pr { E a n } Pr 𝒮 { b n T δ 𝐩 n , b n a n E a n E b n N ( 𝒮 ) } =\sum_{a^{n}\in T_{\delta}^{\mathbf{p}^{n}}}\Pr\left\{E_{a^{n}}\right\}\Pr_{% \mathcal{S}}\left\{\bigcup\limits_{b^{n}\in T_{\delta}^{\mathbf{p}^{n}},\ b^{n% }\neq a^{n}}E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left(\mathcal{S}\right)\right\}
  28. a n , b n T δ 𝐩 n , b n a n Pr { E a n } Pr 𝒮 { E a n E b n N ( 𝒮 ) } \leq\sum_{a^{n},b^{n}\in T_{\delta}^{\mathbf{p}^{n}},\ b^{n}\neq a^{n}}\Pr% \left\{E_{a^{n}}\right\}\Pr_{\mathcal{S}}\left\{E_{a^{n}}^{\dagger}E_{b^{n}}% \in N\left(\mathcal{S}\right)\right\}
  29. a n , b n T δ 𝐩 n , b n a n Pr { E a n } 2 - ( n - k ) \leq\sum_{a^{n},b^{n}\in T_{\delta}^{\mathbf{p}^{n}},\ b^{n}\neq a^{n}}\Pr% \left\{E_{a^{n}}\right\}2^{-\left(n-k\right)}
  30. 2 2 n [ H ( 𝐩 ) + δ ] 2 - n [ H ( 𝐩 ) + δ ] 2 - ( n - k ) \leq 2^{2n\left[H\left(\mathbf{p}\right)+\delta\right]}2^{-n\left[H\left(% \mathbf{p}\right)+\delta\right]}2^{-\left(n-k\right)}
  31. = 2 - n [ 1 - H ( 𝐩 ) - k / n - 3 δ ] . =2^{-n\left[1-H\left(\mathbf{p}\right)-k/n-3\delta\right]}.
  32. N ( 𝒮 ) N\left(\mathcal{S}\right)
  33. 𝒮 \mathcal{S}
  34. N ( 𝒮 ) N\left(\mathcal{S}\right)
  35. N ( 𝒮 ) \ 𝒮 N ( 𝒮 ) N\left(\mathcal{S}\right)\backslash\mathcal{S}\in N\left(\mathcal{S}\right)
  36. E a n E b n E_{a^{n}}^{\dagger}E_{b^{n}}
  37. Pr 𝒮 { E a n E b n N ( 𝒮 ) } = 2 n + k - 1 2 2 n - 1 2 - ( n - k ) . \Pr_{\mathcal{S}}\left\{E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left(\mathcal{S}% \right)\right\}=\frac{2^{n+k}-1}{2^{2n}-1}\leq 2^{-\left(n-k\right)}.
  38. Z 1 Z_{1}
  39. Z n - k Z_{n-k}
  40. Z ¯ 1 \overline{Z}_{1}
  41. Z ¯ n - k \overline{Z}_{n-k}
  42. 2 n + k - 1 2^{n+k}-1
  43. 2 2 n - 1 2^{2n}-1
  44. a n T δ 𝐩 n : Pr { E a n } 2 - n [ H ( 𝐩 ) + δ ] , \forall a^{n}\in T_{\delta}^{\mathbf{p}^{n}}:\Pr\left\{E_{a^{n}}\right\}\leq 2% ^{-n\left[H\left(\mathbf{p}\right)+\delta\right]},
  45. | T δ 𝐩 n | 2 n [ H ( 𝐩 ) + δ ] . \left|T_{\delta}^{\mathbf{p}^{n}}\right|\leq 2^{n\left[H\left(\mathbf{p}\right% )+\delta\right]}.
  46. k / n = 1 - H ( 𝐩 ) - 4 δ k/n=1-H\left(\mathbf{p}\right)-4\delta

Quantum_depolarizing_channel.html

  1. Δ λ \Delta_{\lambda}
  2. λ \lambda
  3. ρ \rho
  4. Δ λ ( ρ ) = λ ρ + 1 - λ d I \Delta_{\lambda}(\rho)=\lambda\rho+\frac{1-\lambda}{d}I
  5. λ \lambda
  6. - 1 d 2 - 1 λ 1 -\frac{1}{d^{2}-1}\leq\lambda\leq 1
  7. Ψ \Psi
  8. lim n 1 n χ ( Ψ n ) \lim_{n\to\infty}\frac{1}{n}\chi\left(\Psi^{\otimes n}\right)
  9. Ψ \Psi
  10. χ ( Ψ Ψ ) = χ ( Ψ ) + χ ( Ψ ) \chi\left(\Psi\otimes\Psi\right)=\chi\left(\Psi\right)+\chi\left(\Psi\right)
  11. Δ λ \Delta_{\lambda}
  12. Ψ \Psi
  13. χ ( Δ λ Ψ ) = χ ( Δ λ ) + χ ( Ψ ) \chi\left(\Delta_{\lambda}\otimes\Psi\right)=\chi\left(\Delta_{\lambda}\right)% +\chi\left(\Psi\right)
  14. v p v_{p}
  15. v p ( Δ λ Ψ ) = v p ( Δ λ ) v p ( Ψ ) v_{p}\left(\Delta_{\lambda}\otimes\Psi\right)=v_{p}\left(\Delta_{\lambda}% \right)v_{p}\left(\Psi\right)
  16. Δ λ \Delta_{\lambda}
  17. Ψ \Psi
  18. v p ( Δ λ ) v p ( Ψ ) v_{p}(\Delta_{\lambda})v_{p}(\Psi)
  19. Δ λ ( ρ ) = n = 1 2 d 2 ( d + 1 ) c n U n * Φ λ ( n ) ( ρ ) U n \Delta_{\lambda}(\rho)=\sum_{n=1}^{2d^{2}(d+1)}c_{n}U_{n}^{*}\Phi_{\lambda}^{(% n)}(\rho)Un
  20. c n c_{n}
  21. U n U_{n}
  22. Φ λ ( n ) \Phi^{(n)}_{\lambda}
  23. ρ \rho
  24. ( Δ λ Ψ ) ( ρ ) = n = 1 2 d 2 ( d + 1 ) c n ( U n * I ) ( Φ λ ( n ) Ψ ) ( ρ ) ( U n I ) \left(\Delta_{\lambda}\otimes\Psi\right)(\rho)=\sum_{n=1}^{2d^{2}(d+1)}c_{n}% \left(U_{n}^{*}\otimes I\right)\left(\Phi_{\lambda}^{(n)}\otimes\Psi\right)(% \rho)\left(U_{n}\otimes I\right)
  25. ( Φ λ ( n ) Ψ ) ( ρ ) p v p ( Δ λ ) v p ( Ψ ) \|\left(\Phi^{(n)}_{\lambda}\otimes\Psi\right)(\rho)\|_{p}\leq v_{p}(\Delta_{% \lambda})v_{p}(\Psi)

Quantum_dilogarithm.html

  1. ϕ ( x ) ( x ; q ) = n = 0 ( 1 - x q n ) , | q | < 1 \phi(x)\equiv(x;q)_{\infty}=\prod_{n=0}^{\infty}(1-xq^{n}),\quad|q|<1
  2. ϕ ( x ) = E q ( x ) - 1 \phi(x)=E_{q}(x)^{-1}
  3. u , v u,v
  4. u v = q v u uv=qvu
  5. ϕ ( u ) ϕ ( v ) = ϕ ( u + v ) \phi(u)\phi(v)=\phi(u+v)
  6. ϕ ( v ) ϕ ( u ) = ϕ ( u + v - v u ) \phi(v)\phi(u)=\phi(u+v-vu)
  7. ϕ ( v ) ϕ ( u ) = ϕ ( u ) ϕ ( - v u ) ϕ ( v ) \phi(v)\phi(u)=\phi(u)\phi(-vu)\phi(v)
  8. Φ b ( w ) \Phi_{b}(w)
  9. Φ b ( z ) = exp ( 1 4 C e - 2 - 1 z w sinh ( w b ) sinh ( w / b ) d w w ) \Phi_{b}(z)=\exp\left(\frac{1}{4}\int_{C}\frac{e^{-2\sqrt{-1}zw}}{\sinh(wb)% \sinh(w/b)}\frac{dw}{w}\right)
  10. C C
  11. Φ b ( p ^ ) Φ b ( q ^ ) = Φ b ( q ^ ) Φ b ( p ^ + q ^ ) Φ b ( p ^ ) \Phi_{b}(\hat{p})\Phi_{b}(\hat{q})=\Phi_{b}(\hat{q})\Phi_{b}(\hat{p}+\hat{q})% \Phi_{b}(\hat{p})
  12. p ^ \hat{p}
  13. q ^ \hat{q}
  14. [ p ^ , q ^ ] = 1 2 π - 1 . [\hat{p},\hat{q}]=\frac{1}{2\pi\sqrt{-1}}.
  15. Φ b \Phi_{b}
  16. Φ b ( z ) = E e - 2 π i / b 2 ( - e - π i / b 2 + 2 π z / b ) E e 2 π i b 2 ( - e π i b 2 + 2 π z b ) \Phi_{b}(z)=\frac{E_{e^{-2\pi i/b^{2}}}(-e^{-\pi i/b^{2}+2\pi z/b})}{E_{e^{2% \pi ib^{2}}}(-e^{\pi ib^{2}+2\pi zb})}
  17. b 2 > 0 b^{2}>0

Quantum_discord.html

  1. I ( A ; B ) = H ( A ) + H ( B ) - H ( A , B ) I(A;B)=H(A)+H(B)-H(A,B)
  2. J ( A ; B ) = H ( A ) - H ( A | B ) J(A;B)=H(A)-H(A|B)
  3. I ( ρ ) = S ( ρ A ) + S ( ρ B ) - S ( ρ ) I(\rho)=S(\rho_{A})+S(\rho_{B})-S(\rho)
  4. J A ( ρ ) = S ( ρ B ) - S ( ρ B | ρ A ) J_{A}(\rho)=S(\rho_{B})-S(\rho_{B}|\rho_{A})
  5. 𝒟 A ( ρ ) \mathcal{D}_{A}(\rho)
  6. 𝒟 B ( ρ ) \mathcal{D}_{B}(\rho)
  7. 𝒟 A ( ρ ) = I ( ρ ) - max { Π j A } J { Π j A } ( ρ ) = S ( ρ A ) - S ( ρ ) + min { Π j A } S ( ρ B | { Π j A } ) \mathcal{D}_{A}(\rho)=I(\rho)-\max_{\{\Pi_{j}^{A}\}}J_{\{\Pi_{j}^{A}\}}(\rho)=% S(\rho_{A})-S(\rho)+\min_{\{\Pi_{j}^{A}\}}S(\rho_{B|\{\Pi_{j}^{A}\}})

Quantum_finance.html

  1. C 0 N = tr [ ( j = 1 N ρ j ) [ S N - K ] + ] C_{0}^{N}=\mathrm{tr}[(\bigotimes_{j=1}^{N}\rho_{j}){[S_{N}-K]}^{+}]
  2. C 0 N = ( 1 + r ) - N n = 0 N N ! n ! ( N - n ) ! q n ( 1 - q ) N - n [ S 0 ( 1 + b ) n ( 1 + a ) N - n - K ] + C_{0}^{N}=(1+r)^{-N}\sum_{n=0}^{N}\frac{N!}{n!(N-n)!}q^{n}{(1-q)}^{N-n}{[S_{0}% {(1+b)}^{n}{(1+a)}^{N-n}-K]}^{+}
  3. σ = ln ( 1 + x 0 + x 1 2 + x 2 2 + x 3 2 ) 1 / t \sigma=\frac{\ln{(1+x_{0}+\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}})}}{\sqrt{1/t}}
  4. C 0 N = ( 1 + r ) - N n = 0 N ( q n ( 1 - q ) N - n k = 0 N q k ( 1 - q ) N - k ) [ S 0 ( 1 + b ) n ( 1 + a ) N - n - K ] + C_{0}^{N}=(1+r)^{-N}\sum_{n=0}^{N}\left(\frac{q^{n}{(1-q)}^{N-n}}{\sum_{k=0}^{% N}q^{k}{(1-q)}^{N-k}}\right){[S_{0}{(1+b)}^{n}{(1+a)}^{N-n}-K]}^{+}

Quantum_Fourier_transform.html

  1. O ( n 2 ) O(n^{2})
  2. n n
  3. O ( n 2 n ) O(n2^{n})
  4. n n
  5. O ( n 2 ) O(n^{2})
  6. O ( n log n ) O(n\log n)
  7. N \mathbb{C}^{N}
  8. y k = 1 N j = 0 N - 1 x j ω j k y_{k}=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}x_{j}\omega^{jk}
  9. ω = e 2 π i N \omega=e^{\frac{2\pi i}{N}}
  10. i = 0 N - 1 x i | i \sum_{i=0}^{N-1}x_{i}|i\rangle
  11. i = 0 N - 1 y i | i \sum_{i=0}^{N-1}y_{i}|i\rangle
  12. y k = 1 N j = 0 N - 1 x j ω j k . y_{k}=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}x_{j}\omega^{jk}.
  13. | j 1 N k = 0 N - 1 ω j k | k . |j\rangle\mapsto\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\omega^{jk}|k\rangle.
  14. F N F_{N}
  15. F N = 1 N [ 1 1 1 1 1 1 ω ω 2 ω 3 ω N - 1 1 ω 2 ω 4 ω 6 ω 2 ( N - 1 ) 1 ω 3 ω 6 ω 9 ω 3 ( N - 1 ) 1 ω N - 1 ω 2 ( N - 1 ) ω 3 ( N - 1 ) ω ( N - 1 ) ( N - 1 ) ] . F_{N}=\frac{1}{\sqrt{N}}\begin{bmatrix}1&1&1&1&\cdots&1\\ 1&\omega&\omega^{2}&\omega^{3}&\cdots&\omega^{N-1}\\ 1&\omega^{2}&\omega^{4}&\omega^{6}&\cdots&\omega^{2(N-1)}\\ 1&\omega^{3}&\omega^{6}&\omega^{9}&\cdots&\omega^{3(N-1)}\\ \vdots&\vdots&\vdots&\vdots&&\vdots\\ 1&\omega^{N-1}&\omega^{2(N-1)}&\omega^{3(N-1)}&\cdots&\omega^{(N-1)(N-1)}\end{% bmatrix}.
  16. F F = F F = I FF^{\dagger}=F^{\dagger}F=I
  17. F F^{\dagger}
  18. F F
  19. F - 1 = F F^{-1}=F^{\dagger}
  20. | 0 , , | 2 n - 1 . |0\rangle,\ldots,|2^{n}-1\rangle.
  21. | x = | x 1 x 2 x n = | x 1 | x 2 | x n |x\rangle=|x_{1}x_{2}\ldots x_{n}\rangle=|x_{1}\rangle\otimes|x_{2}\rangle% \otimes\cdots\otimes|x_{n}\rangle
  22. \otimes
  23. | x j |x_{j}\rangle
  24. j j
  25. x j x_{j}
  26. x j x_{j}
  27. x x
  28. x = x 1 2 n - 1 + x 2 2 n - 2 + + x n 2 0 . x=x_{1}2^{n-1}+x_{2}2^{n-2}+\cdots+x_{n}2^{0}.\quad
  29. [ 0. x 1 x m ] = k = 1 m x k 2 - k . [0.x_{1}\ldots x_{m}]=\sum_{k=1}^{m}x_{k}2^{-k}.
  30. [ 0. x 1 ] = x 1 2 [0.x_{1}]=\frac{x_{1}}{2}
  31. [ 0. x 1 x 2 ] = x 1 2 + x 2 2 2 . [0.x_{1}x_{2}]=\frac{x_{1}}{2}+\frac{x_{2}}{2^{2}}.
  32. | x 1 x 2 x n 1 N ( | 0 + e 2 π i [ 0. x n ] | 1 ) ( | 0 + e 2 π i [ 0. x n - 1 x n ] | 1 ) ( | 0 + e 2 π i [ 0. x 1 x 2 x n ] | 1 ) , |x_{1}x_{2}\ldots x_{n}\rangle\mapsto\frac{1}{\sqrt{N}}\ \left(|0\rangle+e^{2% \pi i\,[0.x_{n}]}|1\rangle\right)\otimes\left(|0\rangle+e^{2\pi i\,[0.x_{n-1}x% _{n}]}|1\rangle\right)\otimes\cdots\otimes\left(|0\rangle+e^{2\pi i\,[0.x_{1}x% _{2}\ldots x_{n}]}|1\rangle\right),
  33. | 0 |0\rangle
  34. e 2 π i [ 0. x n ] | 1 e^{2\pi i\,[0.x_{n}]}|1\rangle
  35. 1 + 2 + + n = n ( n + 1 ) / 2 = O ( n 2 ) 1+2+\cdots+n=n(n+1)/2=O(n^{2})
  36. | j 1 2 3 k = 0 2 3 - 1 ω j k | k , |j\rangle\mapsto\frac{1}{\sqrt{2^{3}}}\sum_{k=0}^{2^{3}-1}\omega^{jk}|k\rangle,
  37. ω \omega
  38. ω 8 = ( e 2 π i 8 ) 8 = 1 \omega^{8}=\left(e^{\frac{2\pi i}{8}}\right)^{8}=1
  39. N = 2 3 = 8 N=2^{3}=8
  40. F 2 3 = 1 2 3 [ 1 1 1 1 1 1 1 1 1 ω ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 1 ω 2 ω 4 ω 6 ω 8 ω 10 ω 12 ω 14 1 ω 3 ω 6 ω 9 ω 12 ω 15 ω 18 ω 21 1 ω 4 ω 8 ω 12 ω 16 ω 20 ω 24 ω 28 1 ω 5 ω 10 ω 15 ω 20 ω 25 ω 30 ω 35 1 ω 6 ω 12 ω 18 ω 24 ω 30 ω 36 ω 42 1 ω 7 ω 14 ω 21 ω 28 ω 35 ω 42 ω 49 ] = 1 2 3 [ 1 1 1 1 1 1 1 1 1 ω ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 1 ω 2 ω 4 ω 6 1 ω 2 ω 4 ω 6 1 ω 3 ω 6 ω ω 4 ω 7 ω 2 ω 5 1 ω 4 1 ω 4 1 ω 4 1 ω 4 1 ω 5 ω 2 ω 7 ω 4 ω ω 6 ω 3 1 ω 6 ω 4 ω 2 1 ω 6 ω 4 ω 2 1 ω 7 ω 6 ω 5 ω 4 ω 3 ω 2 ω ] . F_{2^{3}}=\frac{1}{\sqrt{2^{3}}}\begin{bmatrix}1&1&1&1&1&1&1&1\\ 1&\omega&\omega^{2}&\omega^{3}&\omega^{4}&\omega^{5}&\omega^{6}&\omega^{7}\\ 1&\omega^{2}&\omega^{4}&\omega^{6}&\omega^{8}&\omega^{10}&\omega^{12}&\omega^{% 14}\\ 1&\omega^{3}&\omega^{6}&\omega^{9}&\omega^{12}&\omega^{15}&\omega^{18}&\omega^% {21}\\ 1&\omega^{4}&\omega^{8}&\omega^{12}&\omega^{16}&\omega^{20}&\omega^{24}&\omega% ^{28}\\ 1&\omega^{5}&\omega^{10}&\omega^{15}&\omega^{20}&\omega^{25}&\omega^{30}&% \omega^{35}\\ 1&\omega^{6}&\omega^{12}&\omega^{18}&\omega^{24}&\omega^{30}&\omega^{36}&% \omega^{42}\\ 1&\omega^{7}&\omega^{14}&\omega^{21}&\omega^{28}&\omega^{35}&\omega^{42}&% \omega^{49}\\ \end{bmatrix}=\frac{1}{\sqrt{2^{3}}}\begin{bmatrix}1&1&1&1&1&1&1&1\\ 1&\omega&\omega^{2}&\omega^{3}&\omega^{4}&\omega^{5}&\omega^{6}&\omega^{7}\\ 1&\omega^{2}&\omega^{4}&\omega^{6}&1&\omega^{2}&\omega^{4}&\omega^{6}\\ 1&\omega^{3}&\omega^{6}&\omega&\omega^{4}&\omega^{7}&\omega^{2}&\omega^{5}\\ 1&\omega^{4}&1&\omega^{4}&1&\omega^{4}&1&\omega^{4}\\ 1&\omega^{5}&\omega^{2}&\omega^{7}&\omega^{4}&\omega&\omega^{6}&\omega^{3}\\ 1&\omega^{6}&\omega^{4}&\omega^{2}&1&\omega^{6}&\omega^{4}&\omega^{2}\\ 1&\omega^{7}&\omega^{6}&\omega^{5}&\omega^{4}&\omega^{3}&\omega^{2}&\omega\\ \end{bmatrix}.
  41. | x 1 , x 2 , x 3 1 2 3 ( | 0 + e 2 π i [ 0. x 3 ] | 1 ) ( | 0 + e 2 π i [ 0. x 2 x 3 ] | 1 ) ( | 0 + e 2 π i [ 0. x 1 x 2 x 3 ] | 1 ) . |x_{1},x_{2},x_{3}\rangle\mapsto\frac{1}{\sqrt{2^{3}}}\ \left(|0\rangle+e^{2% \pi i\,[0.x_{3}]}|1\rangle\right)\otimes\left(|0\rangle+e^{2\pi i\,[0.x_{2}x_{% 3}]}|1\rangle\right)\otimes\left(|0\rangle+e^{2\pi i\,[0.x_{1}x_{2}x_{3}]}|1% \rangle\right).
  42. | x 1 , x 2 , x 3 |x_{1},x_{2},x_{3}\rangle
  43. R θ R_{\theta}
  44. n ( n + 1 ) / 2 n(n+1)/2

Quantum_mechanics_of_time_travel.html

  1. ρ CTC = Tr A [ U ( ρ A ρ CTC ) U ] \rho_{\,\text{CTC}}=\,\text{Tr}_{A}\left[U\left(\rho_{A}\otimes\rho_{\,\text{% CTC}}\right)U^{\dagger}\right]
  2. Tr CTC [ U ( ρ A ρ CTC ) U ] \,\text{Tr}_{\,\text{CTC}}\left[U\left(\rho_{A}\otimes\rho_{\,\text{CTC}}% \right)U^{\dagger}\right]
  3. U = ( 0 1 1 0 ) U=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  4. ρ CTC = ( 1 2 a a 1 2 ) \rho_{\,\text{CTC}}=\begin{pmatrix}\frac{1}{2}&a\\ a&\frac{1}{2}\end{pmatrix}
  5. - 1 / 2 -1/2
  6. 1 / 2 1/2
  7. a = 0 a=0
  8. ( | 0 + | 1 ) / 2 \left(\left|0\right\rangle+\left|1\right\rangle\right)/\sqrt{2}
  9. ( | 0 - | 1 ) / 2 \left(\left|0\right\rangle-\left|1\right\rangle\right)/\sqrt{2}
  10. ρ f = C ρ i C Tr [ C ρ i C ] \rho_{f}=\frac{C\rho_{i}C^{\dagger}}{\,\text{Tr}\left[C\rho_{i}C^{\dagger}% \right]}
  11. C = Tr CTC [ U ] C=\,\text{Tr}_{\,\text{CTC}}\left[U\right]
  12. Tr [ C ρ i C ] = 0 \,\text{Tr}\left[C\rho_{i}C^{\dagger}\right]=0

Quantum_non-equilibrium.html

  1. ρ ( X , t ) \rho(X,t)
  2. \neq
  3. | ψ ( X , t ) | 2 |\psi(X,t)|^{2}
  4. ρ ( X , t ) \rho(X,t)
  5. | ψ ( X , t ) | 2 |\psi(X,t)|^{2}
  6. ρ ( X , t ) = | ψ ( X , t ) | 2 \rho(X,t)=|\psi(X,t)|^{2}
  7. ρ ( X , t ) \rho(X,t)
  8. ψ ( X , t ) \psi(X,t)
  9. ρ ( 𝐱 , t ) = | ψ ( 𝐱 , t ) | 2 \rho(\mathbf{x},t)=|\psi(\mathbf{x},t)|^{2}
  10. ρ \rho
  11. d 3 x d^{3}x
  12. ψ \psi
  13. d 3 x d^{3}x
  14. | ψ ( 𝐱 , t ) | 2 |\psi(\mathbf{x},t)|^{2}
  15. R R
  16. ρ = R 2 \rho=R^{2}\quad
  17. ψ \psi
  18. ρ ( X , t ) = | ψ ( X , t ) | 2 \rho(X,t)=|\psi(X,t)|^{2}
  19. X X
  20. ρ ( X , t ) \rho(X,t)
  21. | ψ ( X , t ) | 2 |\psi(X,t)|^{2}
  22. ρ ( x , y , z , t ) \rho(x,y,z,t)
  23. | ψ ( x , y , z , t ) | 2 |\psi(x,y,z,t)|^{2}

Quantum_oscillations_(experimental_technique).html

  1. ε l = e B m * ( + 1 2 ) \varepsilon_{l}=\frac{eB}{m^{*}}\left(\ell+\frac{1}{2}\right)
  2. \ell
  3. B B
  4. e , m * e,m^{*}
  5. B B

Quantum_pendulum.html

  1. ϕ \phi
  2. T = 1 2 m l 2 ϕ ˙ 2 T=\frac{1}{2}ml^{2}\dot{\phi}^{2}
  3. U = m g l ( 1 - cos ( ϕ ) ) U=mgl(1-\cos(\phi))
  4. H ^ = p ^ 2 2 m l 2 + m g l ( 1 - cos ( ϕ ) ) \hat{H}=\frac{\hat{p}^{2}}{2ml^{2}}+mgl(1-\cos(\phi))
  5. i d Ψ d t = - 2 2 m l 2 d 2 Ψ d ϕ 2 + m g l ( 1 - cos ( ϕ ) ) Ψ i\hbar\frac{d\Psi}{dt}=-\frac{\hbar^{2}}{2ml^{2}}\frac{\mathrm{d}^{2}\Psi}{% \mathrm{d}\phi^{2}}+mgl(1-\cos(\phi))\Psi
  6. η = ϕ + π \eta=\phi+\pi
  7. E ψ = - 2 2 m l 2 d 2 ψ d η 2 + m g l ( 1 + cos ( η ) ) ψ E\psi=-\frac{\hbar^{2}}{2ml^{2}}\frac{\mathrm{d}^{2}\psi}{\mathrm{d}\eta^{2}}+% mgl(1+\cos(\eta))\psi
  8. 0 = d 2 ψ d η 2 + ( 2 m E l 2 2 - 2 m 2 g l 3 2 - 2 m 2 g l 3 2 cos ( η ) ) ψ 0=\frac{\mathrm{d}^{2}\psi}{\mathrm{d}\eta^{2}}+(\frac{2mEl^{2}}{\hbar^{2}}-% \frac{2m^{2}gl^{3}}{\hbar^{2}}-\frac{2m^{2}gl^{3}}{\hbar^{2}}\cos(\eta))\psi
  9. q q
  10. a a
  11. 2 π 2\pi
  12. a n ( q ) , b n ( q ) a_{n}(q),\,b_{n}(q)
  13. C E ( n , q , x ) , S E ( n , q , x ) CE(n,q,x),\,SE(n,q,x)
  14. π \pi
  15. a n ( q ) , b n ( q ) a_{n}(q),\,b_{n}(q)
  16. 0 = d 2 ψ d η 2 + ( 2 m E l 2 2 - 2 m 2 g l 3 2 - 2 m 2 g l 3 2 cos ( η ) ) ψ 0=\frac{\mathrm{d}^{2}\psi}{\mathrm{d}\eta^{2}}+(\frac{2mEl^{2}}{\hbar^{2}}-% \frac{2m^{2}gl^{3}}{\hbar^{2}}-\frac{2m^{2}gl^{3}}{\hbar^{2}}\cos(\eta))\psi
  17. a n ( q ) , b n ( q ) = 2 m E l 2 2 - 2 m 2 g l 3 2 a_{n}(q),\,b_{n}(q)=\frac{2mEl^{2}}{\hbar^{2}}-\frac{2m^{2}gl^{3}}{\hbar^{2}}
  18. E = m g l + 2 a n ( q ) , b n ( q ) 2 m l 2 E=mgl+\frac{\hbar^{2}a_{n}(q),\,b_{n}(q)}{2ml^{2}}
  19. q = m 2 g l 3 2 q=\frac{m^{2}gl^{3}}{\hbar^{2}}
  20. a n ( q ) , b n ( q ) a_{n}(q),\,b_{n}(q)
  21. 2 π 2\pi
  22. C ( a n ( q ) , q , x ) = C E ( n , q , x ) C E ( n , q , 0 ) C\left(a_{n}(q),q,x\right)=\frac{CE(n,q,x)}{CE(n,q,0)}
  23. S ( b n ( q ) , q , x ) = S E ( n , q , x ) S E ( n , q , 0 ) . S\left(b_{n}(q),q,x\right)=\frac{SE(n,q,x)}{SE^{\prime}(n,q,0)}.
  24. C E ( 1 , 1 , x ) CE(1,1,x)

Quantum_phase_estimation_algorithm.html

  1. e i θ e^{i\theta}
  2. | ψ \left|\psi\right\rangle
  3. U | ψ = e i θ | ψ U\left|\psi\right\rangle=e^{i\theta}\left|\psi\right\rangle
  4. θ \theta
  5. | ψ \left|\psi\right\rangle
  6. U U
  7. U U
  8. U U
  9. U 2 n - 1 U^{2^{n-1}}
  10. 1 2 n x | x | ψ \frac{1}{\sqrt{2^{n}}}\sum_{x}\left|x\right\rangle\otimes\left|\psi\right\rangle
  11. U U
  12. 1 2 n x e i x θ | x | ψ \frac{1}{\sqrt{2^{n}}}\sum_{x}e^{ix\theta}\left|x\right\rangle\otimes\left|% \psi\right\rangle
  13. 1 2 n y x e - 2 π i x y / 2 n e i x θ | y | ψ = 1 2 n y e i 2 n θ - 1 e i ( θ - 2 π y / 2 n ) - 1 | y | ψ \frac{1}{2^{n}}\sum_{y}\sum_{x}e^{-2\pi ix\cdot y/2^{n}}e^{ix\theta}\left|y% \right\rangle\otimes\left|\psi\right\rangle=\frac{1}{2^{n}}\sum_{y}\frac{e^{i2% ^{n}\theta}-1}{e^{i\left(\theta-2\pi y/2^{n}\right)}-1}\left|y\right\rangle% \otimes\left|\psi\right\rangle
  14. 2 n 2^{n}
  15. | ψ \left|\psi\right\rangle
  16. U 2 n U^{2^{n}}
  17. n n
  18. U U
  19. U U
  20. U 2 n U^{2^{n}}

Quantum_revival.html

  1. E i E_{i}
  2. ψ i \psi_{i}
  3. H ψ i = E i ψ i H\psi_{i}=E_{i}\psi_{i}
  4. C C
  5. E i = C M i N i E_{i}=C{M_{i}\over N_{i}}
  6. M i = 1 M_{i}=1
  7. N i = i 2 N_{i}=i^{2}
  8. C = - 13.6 e V C=-13.6eV
  9. m a x \mathbb{N}_{max}
  10. Ψ ( t ) = i = 0 m a x a i e - i E i t ψ i \Psi(t)=\sum_{i=0}^{\mathbb{N}_{max}}a_{i}e^{-i{{E_{i}}\over\hbar}t}\psi_{i}
  11. L c m L_{cm}
  12. N i N_{i}
  13. L c d L_{cd}
  14. M i M_{i}
  15. N i N_{i}
  16. L c m / N i {L_{cm}}/N_{i}
  17. M i M_{i}
  18. M i / L c d {M_{i}}/L_{cd}
  19. 2 π M i L c m / ( N i L c d ) 2\pi M_{i}{L_{cm}}/(N_{i}L_{cd})
  20. 2 π 2\pi
  21. Ψ ( t ) = Ψ ( t + T ) \Psi(t)=\Psi(t+T)
  22. T = 2 π L c d C L c m T={2\pi\hbar\over{L_{cd}C}}L_{cm}
  23. m a x \mathbb{N}_{max}
  24. t / 2 π = 100000000 t/2\pi=100000000

Quantum_rotor_model.html

  1. i i
  2. 𝐧 \mathbf{n}
  3. 𝐩 \mathbf{p}
  4. α , β \alpha,\beta
  5. [ n α , p β ] = i δ α β [n_{\alpha},p_{\beta}]=i\delta_{\alpha\beta}
  6. 𝐋 \mathbf{L}
  7. L α = ε α β γ n β p γ L_{\alpha}=\varepsilon_{\alpha\beta\gamma}n_{\beta}p_{\gamma}
  8. H R = J g ¯ 2 i 𝐋 i 2 - J i j 𝐧 i 𝐧 j H_{R}=\frac{J\bar{g}}{2}\sum_{i}\mathbf{L}_{i}^{2}-J\sum_{\langle ij\rangle}% \mathbf{n}_{i}\cdot\mathbf{n}_{j}
  9. J , g ¯ J,\bar{g}
  10. g ¯ \bar{g}
  11. 𝐒 1 i \mathbf{S}_{1i}
  12. 𝐒 2 i \mathbf{S}_{2i}
  13. H d = K i 𝐒 1 i 𝐒 2 i + J i j ( 𝐒 1 i 𝐒 1 j + 𝐒 2 i 𝐒 2 j ) H_{d}=K\sum_{i}\mathbf{S}_{1i}\cdot\mathbf{S}_{2i}+J\sum_{\langle ij\rangle}% \left(\mathbf{S}_{1i}\cdot\mathbf{S}_{1j}+\mathbf{S}_{2i}\cdot\mathbf{S}_{2j}\right)
  14. 𝐋 i = 𝐒 1 i + 𝐒 2 i \mathbf{L}_{i}=\mathbf{S}_{1i}+\mathbf{S}_{2i}

Quantum_spin_liquid.html

  1. Θ c w [ K ] \Theta_{cw}[K]
  2. S = 1 / 2 S=1/2
  3. U ( 1 ) U(1)
  4. f > 100 f>100
  5. f = | Θ c w | T c f=\frac{|\Theta_{cw}|}{T_{c}}
  6. Θ c w \Theta_{cw}
  7. T c T_{c}
  8. J / k B J/k_{B}

Quantum_state.html

  1. { n } \{n\}
  2. | ψ = 1 2 ( | - | ) \left|\psi\right\rangle=\frac{1}{\sqrt{2}}\bigg(\left|\uparrow\downarrow\right% \rangle-\left|\downarrow\uparrow\right\rangle\bigg)
  3. ( α , β ) (\alpha,\beta)
  4. | α | 2 + | β | 2 = 1 , |\alpha|^{2}+|\beta|^{2}=1,
  5. | α | |\alpha|
  6. | β | |\beta|
  7. α \alpha
  8. β \beta
  9. 2 × 2 2\times 2
  10. | Ψ ( t ) = n C n ( t ) | Φ n |\Psi(t)\rangle=\sum_{n}C_{n}(t)|\Phi_{n}\rangle
  11. | |
  12. \rangle
  13. Ψ \Psi
  14. Φ n \Phi_{n}
  15. P n P_{n}
  16. Φ n \Phi_{n}
  17. A σ \langle A\rangle_{\sigma}
  18. | Ψ ( t ) = n C n ( t ) | Φ n |\Psi(t)\rangle=\sum_{n}C_{n}(t)|\Phi_{n}\rangle
  19. | ψ |\psi\rangle
  20. ψ \psi
  21. | ψ |\psi\rangle
  22. ψ | \langle\psi|
  23. | ψ |\psi\rangle
  24. ψ | \langle\psi|
  25. | ψ |\psi\rangle
  26. ψ 1 | ψ 2 \langle\psi_{1}|\psi_{2}\rangle
  27. { - S , - S + 1 , + S - 1 , + S } \{-S,-S+1,\ldots+S-1,+S\}
  28. | ψ ( 𝐫 1 , m 1 ; ; 𝐫 N , m N ) . |\psi(\mathbf{r}_{1},m_{1};\dots;\mathbf{r}_{N},m_{N})\rangle.
  29. { - S ν , - S ν + 1 , + S ν - 1 , + S ν } \{-S_{\nu},-S_{\nu}+1,\ldots+S_{\nu}-1,+S_{\nu}\}
  30. S ν S_{\nu}
  31. S ν = 0 S_{\nu}=0
  32. | k i |{k_{i}}\rangle
  33. | ψ |\psi\rangle
  34. | ψ = i c i | k i |\psi\rangle=\sum_{i}c_{i}|{k_{i}}\rangle
  35. | ψ |\psi\rangle
  36. | k i |{k_{i}}\rangle
  37. c i = k i | ψ c_{i}=\langle{k_{i}}|\psi\rangle
  38. | ψ |\psi\rangle
  39. i | c i | 2 = 1. \sum_{i}\left|c_{i}\right|^{2}=1.
  40. | k i |{k_{i}}\rangle
  41. | ψ |\psi\rangle
  42. | ψ |\psi\rangle
  43. ψ ( 𝐫 ) 𝐫 | ψ . \psi(\mathbf{r})\equiv\langle\mathbf{r}|\psi\rangle.
  44. | ψ |\psi\rangle
  45. | α |\alpha\rangle
  46. | β |\beta\rangle
  47. c α | α + c β | β c_{\alpha}|\alpha\rangle+c_{\beta}|\beta\rangle
  48. c α c_{\alpha}
  49. c β c_{\beta}
  50. | ψ |\psi\rangle
  51. e i θ | ψ e^{i\theta}|\psi\rangle
  52. | ϕ + | ψ |\phi\rangle+|\psi\rangle
  53. | ϕ + e i θ | ψ |\phi\rangle+e^{i\theta}|\psi\rangle
  54. | ϕ + | ψ |\phi\rangle+|\psi\rangle
  55. e i θ ( | ϕ + | ψ ) e^{i\theta}(|\phi\rangle+|\psi\rangle)
  56. H 1 H 2 H_{1}\otimes H_{2}
  57. H 2 H_{2}
  58. H 1 H_{1}
  59. H 2 H_{2}
  60. H H
  61. H K H\otimes K
  62. K K
  63. ρ = s p s | ψ s ψ s | \rho=\sum_{s}p_{s}|\psi_{s}\rangle\langle\psi_{s}|
  64. p s p_{s}
  65. | ψ s . |\psi_{s}\rangle.
  66. A = s p s ψ s | A | ψ s = s i p s a i | α i | ψ s | 2 = tr ( ρ A ) \langle A\rangle=\sum_{s}p_{s}\langle\psi_{s}|A|\psi_{s}\rangle=\sum_{s}\sum_{% i}p_{s}a_{i}|\langle\alpha_{i}|\psi_{s}\rangle|^{2}=\operatorname{tr}(\rho A)
  67. | α i , a i |\alpha_{i}\rangle,\;a_{i}
  68. | ψ s |\psi_{s}\rangle
  69. Tr ( ρ 2 ) = ( Tr ρ ) 2 \operatorname{Tr}(\rho^{2})=(\operatorname{Tr}\rho)^{2}

Quasi-category.html

  1. Λ k [ n ] C \Lambda^{k}[n]\to C
  2. 0 < k < n 0<k<n
  3. Δ [ n ] C \Delta[n]\to C
  4. Δ [ n ] \Delta[n]
  5. Λ k [ n ] \Lambda^{k}[n]
  6. Δ [ 2 ] C \Delta[2]\to C
  7. Λ 1 [ 2 ] C \Lambda^{1}[2]\to C
  8. C Δ [ 2 ] C Λ 1 [ 2 ] C^{\Delta[2]}\to C^{\Lambda^{1}[2]}

Quasi-Frobenius_ring.html

  1. soc ( R R ) R / J \mathrm{soc}(R_{R})\cong R/J
  2. soc ( R R ) R / J \mathrm{soc}(_{R}R)\cong R/J
  3. soc ( R R ) R / J \mathrm{soc}(R_{R})\cong R/J
  4. soc ( R R ) R / J \mathrm{soc}(_{R}R)\cong R/J
  5. soc ( R R ) = soc ( R R ) = R \mathrm{soc}(R_{R})=\mathrm{soc}(_{R}R)=R
  6. n \frac{\mathbb{Z}}{n\mathbb{Z}}

Quasi-geostrophic_equations.html

  1. f o v g = Φ x {f_{o}}{v_{g}}={\partial\Phi\over\partial x}
  2. f o u g = - Φ y {f_{o}}{u_{g}}=-{\partial\Phi\over\partial y}
  3. Φ {\Phi}
  4. ζ g = k ^ × V g {\zeta_{g}}={\hat{k}\cdot\nabla\times\overrightarrow{V_{g}}}
  5. ζ g = v g x - u g y = 1 f o ( 2 Φ x 2 + 2 Φ y 2 ) = 1 f o 2 Φ {\zeta_{g}}={{\partial v_{g}\over\partial x}-{\partial u_{g}\over\partial y}={% 1\over f_{o}}\left({{\partial^{2}\Phi\over\partial x^{2}}+{\partial^{2}\Phi% \over\partial y^{2}}}\right)={1\over f_{o}}{\nabla^{2}\Phi}}
  6. ζ g ( x , y ) {\zeta_{g}(x,y)}
  7. Φ ( x , y ) {\Phi(x,y)}
  8. Φ {\Phi}
  9. ζ g {\zeta_{g}}
  10. x {x}
  11. y {y}
  12. D V D t + f k ^ × V = - Φ {D\overrightarrow{V}\over Dt}+f\hat{k}\times\overrightarrow{V}=-\nabla\Phi
  13. D D t = ( t ) p + ( V ) p + ω p {{D\over Dt}={\left({\partial\over\partial t}\right)_{p}}+{\left({% \overrightarrow{V}\cdot\nabla}\right)_{p}}+{\omega{\partial\over\partial p}}}
  14. ω = D p D t {\omega={Dp\over Dt}}
  15. V {\overrightarrow{V}}
  16. V g {\overrightarrow{V_{g}}}
  17. V a {\overrightarrow{V_{a}}}
  18. V = V g + V a {\overrightarrow{V}=\overrightarrow{V_{g}}+\overrightarrow{V_{a}}}
  19. V g V a {\overrightarrow{V_{g}}\gg\overrightarrow{V_{a}}}
  20. | V a | | V g | O ( Rossby number ) {{|\overrightarrow{V_{a}}|\over|\overrightarrow{V_{g}}|}}\sim O(\,\text{Rossby% number})
  21. f = f o + β y {f=f_{o}+\beta y}
  22. β y f o O ( Rossby number ) {\frac{\beta y}{f_{o}}\sim O(\,\text{Rossby number})}
  23. f o {f_{o}}
  24. f o + β y {f_{o}+\beta y}
  25. f k ^ × V + Φ = ( f o + β y ) k ^ × ( V g + V a ) - f o k ^ × V g = f o k ^ × V a + β y k ^ × V g {f\hat{k}\times\overrightarrow{V}+\nabla\Phi}={(f_{o}+\beta y)\hat{k}\times(% \overrightarrow{V_{g}}+\overrightarrow{V_{a}})-f_{o}\hat{k}\times% \overrightarrow{V_{g}}}={f_{o}\hat{k}\times\overrightarrow{V_{a}}+\beta y\hat{% k}\times\overrightarrow{V_{g}}}
  26. D g V g D t = - f o k ^ × V a - β y k ^ × V g {D_{g}\overrightarrow{V_{g}}\over Dt}={-f_{o}\hat{k}\times\overrightarrow{V_{a% }}-\beta y\hat{k}\times\overrightarrow{V_{g}}}
  27. D g u g D t - f o v a - β y f o v g = 0 {{D_{g}u_{g}\over Dt}-{f_{o}v_{a}}-{\beta yf_{o}v_{g}}=0}
  28. D g v g D t + f o u a + β y f o u g = 0 {{D_{g}v_{g}\over Dt}+{f_{o}u_{a}}+{\beta yf_{o}u_{g}}=0}
  29. ( 8 b ) x - ( 8 a ) y {{\partial(8b)\over\partial x}-{\partial(8a)\over\partial y}}
  30. V = 0 {\nabla\cdot\overrightarrow{V}=0}
  31. D g ζ g D t = f o ( u a x + v a y ) - β v g {{D_{g}\zeta_{g}\over Dt}=f_{o}\left({{\partial u_{a}\over\partial x}+{% \partial v_{a}\over\partial y}}\right)-\beta v_{g}}
  32. f {f}
  33. y {y}
  34. D g f D t = V g f = β v g {{D_{g}f\over Dt}=\overrightarrow{V_{g}}\cdot\nabla f=\beta v_{g}}
  35. ω {\omega}
  36. u a x + v a y + ω p = 0 {{\partial u_{a}\over\partial x}+{\partial v_{a}\over\partial y}+{\partial% \omega\over\partial p}=0}
  37. ζ g t = - V g ( ζ g + f ) + f o ω p {{\partial\zeta_{g}\over\partial t}={-\overrightarrow{V_{g}}\cdot\nabla({\zeta% _{g}+f})}+{f_{o}{\partial\omega\over\partial p}}}
  38. χ = Φ t {\chi={\partial\Phi\over\partial t}}
  39. χ {\chi}
  40. 1 f o 2 χ = - V g ( 1 f o 2 χ + f ) + f o ω p {{1\over f_{o}}{\nabla^{2}\chi}={-\overrightarrow{V_{g}}\cdot\nabla\left({{1% \over f_{o}}{\nabla^{2}\chi}+f}\right)}+{f_{o}{\partial\omega\over\partial p}}}
  41. χ {\chi}
  42. ω {\omega}
  43. ( t + V g ) ( - Φ p ) - σ ω = k J p {{{\left({{\partial\over\partial t}+{\overrightarrow{V_{g}}\cdot\nabla}}\right% )\left({-\partial\Phi\over\partial p}\right)}-\sigma\omega}={kJ\over p}}
  44. σ = - R T o p d log Θ o d p {\sigma={-RT_{o}\over p}{d\log\Theta_{o}\over dp}}
  45. Θ o {\Theta_{o}}
  46. Θ o {\Theta_{o}}
  47. 2.5 × 10 - 6 m P 2 a - 2 s - 2 {2.5\times 10^{-6}m{{}^{2}}Pa^{-2}s^{-2}}
  48. f o σ {f_{o}\over\sigma}
  49. p {p}
  50. χ {\chi}
  51. p ( f o σ χ p ) = - p ( f o σ V g Φ p ) - f o ω p - f o p ( k J σ p ) {{{\partial\over\partial p}\left({{f_{o}\over\sigma}{\partial\chi\over\partial p% }}\right)}=-{{\partial\over\partial p}\left({{f_{o}\over\sigma}{% \overrightarrow{V_{g}}\cdot\nabla}{\partial\Phi\over\partial p}}\right)}-{{f_{% o}}{\partial\omega\over\partial p}}-{{f_{o}}{\partial\over\partial p}\left({kJ% \over\sigma p}\right)}}
  52. J {J}
  53. ω {\omega}
  54. ( 2 + p ( f o 2 σ p ) ) χ = - f o V g ( 1 f o 2 Φ + f ) - p ( - f o 2 σ V g ( Φ p ) ) {{\left({\nabla^{2}+{{\partial\over\partial p}\left({{f_{o}^{2}\over\sigma}{% \partial\over\partial p}}\right)}}\right){\chi}}=-{{f_{o}}{\overrightarrow{V_{% g}}\cdot\nabla}\left({{{1\over f_{o}}{\nabla^{2}\Phi}}+f}\right)}-{{\partial% \over\partial p}\left({{-}{f_{o}^{2}\over\sigma}{\overrightarrow{V_{g}}\cdot% \nabla}\left({\partial\Phi\over\partial p}\right)}\right)}}
  55. - V g p ( f o 2 σ Φ p ) - f o 2 σ V g p Φ p {-{{\overrightarrow{V_{g}}\cdot\nabla}{\partial\over\partial p}\left({{f_{o}^{% 2}\over\sigma}{\partial\Phi\over\partial p}}\right)}-{{f_{o}^{2}\over\sigma}{% \partial\overrightarrow{V_{g}}\over\partial p}{\cdot\nabla}{\partial\Phi\over% \partial p}}}
  56. f o V g p = k ^ × ( Φ p ) {{f_{o}{\partial\overrightarrow{V_{g}}\over\partial p}}={\hat{k}\times\nabla% \left({\partial\Phi\over\partial p}\right)}}
  57. V g p {\partial\overrightarrow{V_{g}}\over\partial p}
  58. ( Φ p ) {\nabla({\partial\Phi\over\partial p})}
  59. f o {f_{o}}
  60. ( t + V g ) q = D g q D t = 0 {{\left({{\partial\over\partial t}+{\overrightarrow{V_{g}}\cdot\nabla}}\right)% q}={D_{g}q\over Dt}=0}
  61. q {q}
  62. q = 1 f o 2 Φ + f + p ( f o σ Φ p ) {q={{{1\over f_{o}}{\nabla^{2}\Phi}}+{f}+{{\partial\over\partial p}\left({{f_{% o}\over\sigma}{\partial\Phi\over\partial p}}\right)}}}
  63. q {q}
  64. Φ {\Phi}
  65. Φ {\Phi}
  66. u g {u_{g}}
  67. v g {v_{g}}
  68. T {T}
  69. q {q}
  70. Φ {\Phi}
  71. ζ g {\zeta_{g}}
  72. V g {\overrightarrow{V_{g}}}
  73. Φ ( x , y , p , t ) {\Phi(x,y,p,t)}
  74. Φ {\Phi}
  75. Φ t {\partial\Phi\over\partial t}

Quasi-linkage_equilibrium.html

  1. X X
  2. Y Y
  3. Z Z
  4. U U
  5. R = X U Y Z R=\frac{XU}{YZ}
  6. R ^ \hat{R}
  7. R ^ \hat{R}

Quasi-relative_interior.html

  1. X X
  2. A X A\subseteq X
  3. qri ( A ) := { x A : cone ¯ ( A - x ) is a linear subspace } \operatorname{qri}(A):=\left\{x\in A:\operatorname{\overline{cone}}(A-x)\,% \text{ is a linear subspace}\right\}\,
  4. cone ¯ ( ) \operatorname{\overline{cone}}(\cdot)
  5. X X
  6. C X C\subset X
  7. qri ( C ) = ri ( C ) \operatorname{qri}(C)=\operatorname{ri}(C)
  8. ri \operatorname{ri}

Quasicircle.html

  1. | z 1 - z 3 | + | z 2 - z 3 | C | z 1 - z 2 | . |z_{1}-z_{3}|+|z_{2}-z_{3}|\leq C|z_{1}-z_{2}|.
  2. | z 3 - z 1 + z 2 2 | C | z 1 - z 2 | . \displaystyle{\left|z_{3}-{z_{1}+z_{2}\over 2}\right|\leq C|z_{1}-z_{2}|.}
  3. C 1 | s - t | | f ( s ) - f ( t ) | C 2 | s - t | C_{1}|s-t|\leq|f(s)-f(t)|\leq C_{2}|s-t|
  4. φ = g - 1 f \varphi=g^{-1}\circ f
  5. R 0 ( z ) = 1 z ¯ \displaystyle{R_{0}(z)={1\over\overline{z}}}
  6. f ( z ) = R 1 f R 0 ( z ) \displaystyle{f(z)=R_{1}fR_{0}(z)}
  7. | z R n ( z ) | c A n |\partial_{z}R^{n}(z)|\geq cA^{n}
  8. μ < 1 \|\mu\|_{\infty}<1
  9. μ ( g ( z ) ) z g ( z ) ¯ z g ( z ) = μ ( z ) \mu(g(z)){\overline{\partial_{z}g(z)}\over\partial_{z}g(z)}=\mu(z)
  10. z ¯ f ( z ) = μ ( z ) z f ( z ) \partial_{\overline{z}}f(z)=\mu(z)\partial_{z}f(z)
  11. α ( g ) = f g f - 1 \alpha(g)=f\circ g\circ f^{-1}
  12. d H ( C ) 1 + k d_{H}(C)\leq 1+k
  13. k = K - 1 K + 1 . k={K-1\over K+1}.
  14. R ( z ) = z 2 + c R(z)=z^{2}+c
  15. 1 < d H ( J c ) < 1 + | c | 2 4 log 2 + o ( | c | 2 ) . 1<d_{H}(J_{c})<1+{|c|^{2}\over 4\log 2}+o(|c|^{2}).
  16. K = 1 + t 1 - t K=\sqrt{1+t\over 1-t}
  17. t = | 1 - 1 - 4 c | , t=|1-\sqrt{1-4c}|,
  18. 1 + 0.36 k 2 d H ( C ) 1 + 37 k 2 . 1+0.36k^{2}\leq d_{H}(C)\leq 1+37k^{2}.
  19. d H ( C ) 1 + k 2 . d_{H}(C)\leq 1+k^{2}.

Quasiregular_map.html

  1. D f ( x ) n K | J f ( x ) | \|Df(x)\|^{n}\leq K|J_{f}(x)|\,
  2. ( r , θ , z ) ( r , 2 θ , z ) . (r,\theta,z)\mapsto(r,2\theta,z).

Quasisymmetric_function.html

  1. x 1 , x 2 , x 3 , x_{1},x_{2},x_{3},\dots
  2. x 1 α 1 x 2 α 2 x k α k x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{k}^{\alpha_{k}}
  3. x i 1 α 1 x i 2 α 2 x i k α k x_{i_{1}}^{\alpha_{1}}x_{i_{2}}^{\alpha_{2}}\cdots x_{i_{k}}^{\alpha_{k}}
  4. i 1 < i 2 < < i k i_{1}<i_{2}<\cdots<i_{k}
  5. ( α 1 , α 2 , , α k ) (\alpha_{1},\alpha_{2},\ldots,\alpha_{k})
  6. S n S_{n}
  7. n n
  8. x 1 , , x n x_{1},\dots,x_{n}
  9. S n S_{n}
  10. p ( x 1 , , x n ) p(x_{1},\dots,x_{n})
  11. ( x i , x i + 1 ) (x_{i},x_{i+1})
  12. S n S_{n}
  13. p ( x 1 , , x n ) p(x_{1},\ldots,x_{n})
  14. ( x i , x i + 1 ) (x_{i},x_{i+1})
  15. x 1 2 x 2 x 3 + x 1 2 x 2 x 4 + x 1 2 x 3 x 4 + x 2 2 x 3 x 4 . x_{1}^{2}x_{2}x_{3}+x_{1}^{2}x_{2}x_{4}+x_{1}^{2}x_{3}x_{4}+x_{2}^{2}x_{3}x_{4% }.\,
  16. x 1 2 x 2 x 3 + x 1 2 x 2 x 4 + x 1 2 x 3 x 4 + x 2 2 x 3 x 4 + x 1 x 2 2 x 3 + x 1 x 2 2 x 4 + x 1 x 3 2 x 4 + x 2 x 3 2 x 4 \displaystyle x_{1}^{2}x_{2}x_{3}+x_{1}^{2}x_{2}x_{4}+x_{1}^{2}x_{3}x_{4}+x_{2% }^{2}x_{3}x_{4}+x_{1}x_{2}^{2}x_{3}+x_{1}x_{2}^{2}x_{4}+x_{1}x_{3}^{2}x_{4}+x_% {2}x_{3}^{2}x_{4}
  17. QSym = n 0 QSym n , \mathrm{QSym}=\bigoplus_{n\geq 0}\mathrm{QSym}_{n},\,
  18. QSym n \mathrm{QSym}_{n}
  19. R R
  20. n n
  21. QSym n \mathrm{QSym}_{n}
  22. { M α } \{M_{\alpha}\}
  23. { F α } \{F_{\alpha}\}
  24. α = ( α 1 , α 2 , , α k ) \alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})
  25. n n
  26. α n \alpha\vDash n
  27. M 0 = 1 M_{0}=1
  28. M α = i 1 < i 2 < < i k x i 1 α 1 x i 2 α 2 x i k α k . M_{\alpha}=\sum_{i_{1}<i_{2}<\cdots<i_{k}}x_{i_{1}}^{\alpha_{1}}x_{i_{2}}^{% \alpha_{2}}\cdots x_{i_{k}}^{\alpha_{k}}.\,
  29. F 0 = 1 F_{0}=1
  30. F α = α β M β , F_{\alpha}=\sum_{\alpha\succeq\beta}M_{\beta},\,
  31. α β \alpha\succeq\beta
  32. α \alpha
  33. β \beta
  34. \succeq
  35. R R
  36. QSym n = span { M α | α n } = span { F α | α n } . \mathrm{QSym}_{n}=\mathrm{span}_{\mathbb{Q}}\{M_{\alpha}|\alpha\vDash n\}=% \mathrm{span}_{\mathbb{Q}}\{F_{\alpha}|\alpha\vDash n\}.\,
  37. Λ = Λ 0 Λ 1 \Lambda=\Lambda_{0}\oplus\Lambda_{1}\oplus\cdots
  38. m 0 = 1 m_{0}=1
  39. m λ = M α , m_{\lambda}=\sum M_{\alpha},
  40. α \alpha
  41. λ \lambda
  42. Λ n = Λ QSym n \Lambda_{n}=\Lambda\cap\mathrm{QSym}_{n}
  43. F ( 1 , 2 ) = M ( 1 , 2 ) + M ( 1 , 1 , 1 ) F_{(1,2)}=M_{(1,2)}+M_{(1,1,1)}
  44. m ( 2 , 1 ) = M ( 2 , 1 ) + M ( 1 , 2 ) . m_{(2,1)}=M_{(2,1)}+M_{(1,2)}.

Quasisymmetric_map.html

  1. d Y ( f ( x ) , f ( y ) ) d Y ( f ( x ) , f ( z ) ) η ( d X ( x , y ) d X ( x , z ) ) . \frac{d_{Y}(f(x),f(y))}{d_{Y}(f(x),f(z))}\leq\eta\left(\frac{d_{X}(x,y)}{d_{X}% (x,z)}\right).
  2. η \eta^{\prime}
  3. η \eta^{\prime}
  4. 1 2 η ( diam B diam A ) diam f ( B ) diam f ( A ) η ( diam B diam A ) . \frac{1}{2\eta(\frac{\,\text{diam }B}{\,\text{diam }A})}\leq\frac{\,\text{diam% }f(B)}{\,\text{diam }f(A)}\leq\eta\left(\frac{\,\text{diam }B}{\,\text{diam }% A}\right).
  5. | f ( x ) - f ( y ) | H | f ( x ) - f ( z ) | whenever | x - y | | x - z | |f(x)-f(y)|\leq H|f(x)-f(z)|\;\;\;\,\text{ whenever }\;\;\;|x-y|\leq|x-z|
  6. f ( x ) - f ( y ) , x - y δ | f ( x ) - f ( y ) | | x - y | . \langle f(x)-f(y),x-y\rangle\geq\delta|f(x)-f(y)|\cdot|x-y|.
  7. f ( x ) = C + 0 x d μ ( t ) . f(x)=C+\int_{0}^{x}\,d\mu(t).
  8. f ( x ) = 1 2 ( x - t | x - t | + t | t | ) d μ ( t ) . f(x)=\frac{1}{2}\int_{\mathbb{R}}\left(\frac{x-t}{|x-t|}+\frac{t}{|t|}\right)d% \mu(t).
  9. | x | > 1 1 | x | d μ ( x ) < \int_{|x|>1}\frac{1}{|x|}\,d\mu(x)<\infty
  10. f ( x ) = 1 2 n ( x - y | x - y | + y | y | ) d μ ( y ) f(x)=\frac{1}{2}\int_{\mathbb{R}^{n}}\left(\frac{x-y}{|x-y|}+\frac{y}{|y|}% \right)\,d\mu(y)

Quasitoric_manifold.html

  1. 2 n 2n
  2. n n
  3. n n
  4. i i
  5. n n
  6. T n T^{n}
  7. T i T_{i}
  8. T 1 × × T n = T n T_{1}\times\ldots\times T_{n}=T^{n}
  9. T n T^{n}
  10. n \mathbb{C}^{n}
  11. X X
  12. M 2 n M^{2n}
  13. Y Y
  14. n \mathbb{C}^{n}
  15. T n T^{n}
  16. T n T^{n}
  17. M 2 n M^{2n}
  18. h ( t x ) = α ( t ) h ( x ) h(tx)=\alpha(t)h(x)
  19. t t
  20. T n T^{n}
  21. x x
  22. X X
  23. h h
  24. X Y X\rightarrow Y
  25. α \alpha
  26. T n T^{n}
  27. P n P^{n}
  28. m m
  29. T n T^{n}
  30. M 2 n M^{2n}
  31. P n P^{n}
  32. T n T^{n}
  33. π : M 2 n P n \pi:M^{2n}\rightarrow P^{n}
  34. l l
  35. l l
  36. P n P^{n}
  37. l = 0 , l=0,
  38. , ...,
  39. n n
  40. M 2 n M^{2n}
  41. T n T^{n}
  42. P n P^{n}
  43. π \pi
  44. F 1 , , F m F_{1},\dots,F_{m}
  45. P n P^{n}
  46. F 1 F n F_{1}\cap\dots\cap F_{n}
  47. v v
  48. P n P^{n}
  49. λ : T m T n \lambda:T^{m}\rightarrow T^{n}
  50. F i 1 F i k F_{i_{1}}\cap\dots\cap F_{i_{k}}
  51. k k
  52. P n P^{n}
  53. λ \lambda
  54. T i 1 × × T i k T_{i_{1}}\times\dots\times T_{i_{k}}
  55. T m T^{m}
  56. T 1 × × T n T_{1}\times\ldots\times T_{n}
  57. v v
  58. λ ( T 1 ) , , λ ( T n ) \lambda(T_{1}),\ldots,\lambda(T_{n})
  59. T n T^{n}
  60. m n \mathbb{Z}^{m}\rightarrow\mathbb{Z}^{n}
  61. n × m n\times m
  62. Λ \Lambda
  63. [ 1 0 0 λ 1 , n + 1 λ 1 , m 0 1 0 λ 2 , n + 1 λ 2 , m 0 0 1 λ n , n + 1 λ n , m ] . \begin{bmatrix}1&0&\dots&0&\lambda_{1,n+1}&\dots&\lambda_{1,m}\\ 0&1&\dots&0&\lambda_{2,n+1}&\dots&\lambda_{2,m}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&1&\lambda_{n,n+1}&\dots&\lambda_{n,m}\end{bmatrix}.
  64. i i
  65. Λ \Lambda
  66. λ i = ( λ 1 , i , , λ n , i ) \lambda_{i}=(\lambda_{1,i},\dots,\lambda_{n,i})
  67. n \mathbb{Z}^{n}
  68. F i 1 F i n F_{i_{1}}\cap\dots\cap F_{i_{n}}
  69. λ i 1 , λ i n \lambda_{i_{1}},\dots\lambda_{i_{n}}
  70. n \mathbb{Z}^{n}
  71. ± 1 \pm 1
  72. F i F_{i}
  73. { ( e 2 π i θ λ 1 , i , , e 2 π i θ λ n , i ) T n } , \{(e^{2\pi i\theta\lambda_{1,i}},\ldots,e^{2\pi i\theta\lambda_{n,i}})\in T^{n% }\},
  74. θ \theta
  75. \mathbb{R}
  76. λ ( T i ) \lambda(T_{i})
  77. λ i \lambda_{i}
  78. Λ \Lambda
  79. ( I n S ) (I_{n}\mid S)
  80. I n I_{n}
  81. S S
  82. n × ( m - n ) n\times(m-n)
  83. K ( λ ) K(\lambda)
  84. Z P n Z_{P^{n}}
  85. K ( λ ) K(\lambda)
  86. Z P n M 2 n Z_{P^{n}}\rightarrow M^{2n}
  87. M 2 n M^{2n}
  88. T n × P n / , T^{n}\times P^{n}/\sim,\,
  89. ( t 1 , p 1 ) (t_{1},p_{1})
  90. ( t 2 , p 2 ) (t_{2},p_{2})
  91. T n × P n T^{n}\times P^{n}
  92. p 1 = p 2 p_{1}=p_{2}
  93. t 1 - 1 t 2 t_{1}^{-1}t_{2}
  94. λ \lambda
  95. T i 1 × × T i k T_{i_{1}}\times\dots\times T_{i_{k}}
  96. F i 1 F i k F_{i_{1}}\cap\dots\cap F_{i_{k}}
  97. P n P^{n}
  98. p 1 p_{1}
  99. 1 k n 1\leq k\leq n
  100. M 2 n M^{2n}
  101. P n P^{n}
  102. n n
  103. P n \mathbb{C}P^{n}
  104. n n
  105. Δ n \Delta^{n}
  106. Δ n \Delta^{n}
  107. n + 1 \mathbb{R}^{n+1}
  108. [ 1 0 0 - 1 0 1 0 - 1 0 0 1 - 1 ] . \begin{bmatrix}1&0&\dots&0&-1\\ 0&1&\dots&0&-1\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\dots&1&-1\end{bmatrix}.
  109. Z Δ n Z_{\Delta^{n}}
  110. ( 2 n + 1 ) (2n+1)
  111. S 2 n + 1 S^{2n+1}
  112. K ( λ ) K(\lambda)
  113. { ( t , , t ) } < T n + 1 \{(t,\dots,t)\}<T^{n+1}
  114. Z Δ n Z_{\Delta^{n}}
  115. K ( λ ) K(\lambda)
  116. P n \mathbb{C}P^{n}
  117. n n
  118. n n
  119. I n I^{n}
  120. 2 n \mathbb{R}^{2n}
  121. ( I n S ) (I_{n}\mid S)
  122. S S
  123. [ 1 0 0 0 0 0 - a ( 1 , 2 ) 1 0 0 0 0 - a ( 1 , i ) - a ( 2 , i ) - a ( i - 1 , i ) 1 0 0 - a ( 1 , n ) - a ( 2 , n ) - a ( i - 1 , n ) - a ( i , n ) - a ( n - 1 , n ) 1 ] , \begin{bmatrix}1&0&\cdots&0&0&\cdots&0&0\\ -a(1,2)&1&\cdots&0&0&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots&&\vdots&\vdots\\ -a(1,i)&-a(2,i)&\cdots&-a(i-1,i)&1&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots&&\vdots&\vdots\\ -a(1,n)&-a(2,n)&\cdots&-a(i-1,n)&-a(i,n)&\cdots&-a(n-1,n)&1\end{bmatrix},
  124. a ( i , j ) a(i,j)
  125. Z I n Z_{I^{n}}
  126. n n
  127. 2 n \mathbb{C}^{2n}
  128. K ( λ ) K(\lambda)
  129. { ( t 1 , t 1 - a ( 1 , 2 ) t 2 , , t 1 - a ( 1 , i ) t i - 1 - a ( i - 1 , i ) t i , , t 1 - a ( 1 , n ) t n - 1 - a ( n - 1 , n ) t n , t 1 - 1 , , t n - 1 ) : t i T , 1 i n } < T 2 n \{(t_{1},t_{1}^{-a(1,2)}t_{2},\dots,t_{1}^{-a(1,i)}\dots t_{i-1}^{-a(i-1,i)}t_% {i},\dots,t_{1}^{-a(1,n)}\dots t_{n-1}^{-a(n-1,n)}t_{n},t_{1}^{-1},\dots,t_{n}% ^{-1}):t_{i}\in T,1\leq i\leq n\}<T^{2n}
  130. Z I n Z_{I^{n}}
  131. K ( λ ) K(\lambda)
  132. n n
  133. a ( i , j ) a(i,j)
  134. ρ i \rho_{i}
  135. M 2 n M^{2n}
  136. Z P n × K ( l ) i M 2 n Z_{P^{n}}\times_{K(l)}\mathbb{C}_{i}\longrightarrow M^{2n}
  137. F i F_{i}
  138. P n P^{n}
  139. 1 i m 1\leq i\leq m
  140. K ( λ ) K(\lambda)
  141. i \mathbb{C}_{i}
  142. K ( λ ) K(\lambda)
  143. i i
  144. T m T^{m}
  145. \mathbb{C}
  146. M 2 n M^{2n}
  147. π - 1 ( F i ) \pi^{-1}(F_{i})
  148. 2 ( n - 1 ) 2(n-1)
  149. M i M_{i}
  150. F i F_{i}
  151. λ \lambda
  152. T i T_{i}
  153. T m T^{m}
  154. ρ i \rho_{i}
  155. M i M_{i}
  156. M i M_{i}
  157. M 2 n M^{2n}
  158. x i x_{i}
  159. H 2 ( M 2 n ; ) H^{2}(M^{2n};\mathbb{Z})
  160. ρ i \rho_{i}
  161. H * ( M 2 n ; ) H^{*}(M^{2n};\mathbb{Z})
  162. x i x_{i}
  163. 1 i m 1\leq i\leq m
  164. P n P^{n}
  165. x i = - λ i , n + 1 x n + 1 - - λ i , m x m , for 1 i n x_{i}=-\lambda_{i,n+1}x_{n+1}-\cdots-\lambda_{i,m}x_{m},\mbox{ for }~{}1\leq i\leq n
  166. x n + 1 , , x m x_{n+1},\dots,x_{m}
  167. H * ( M 2 n ; ) H^{*}(M^{2n};\mathbb{Z})
  168. P 2 P 2 \mathbb{C}P^{2}\sharp\mathbb{C}P^{2}

Quaternary_compound.html

  1. R 3 N + RCl R 4 N + Cl - \mathrm{R_{3}N+RCl\longrightarrow R_{4}N^{+}\ Cl^{-}}

Queap.html

  1. x 1 , x 2 , x 3 , , x k x_{1},x_{2},x_{3},\dots,x_{k}
  2. h v h_{v}
  3. x 0 x_{0}
  4. c v c_{v}
  5. T - T v - { r } T-T_{v}-\{r\}
  6. h v h_{v}
  7. c x 0 c_{x_{0}}
  8. c x 0 c_{x_{0}}
  9. c x 0 c_{x_{0}}
  10. h v h_{v}
  11. x 0 x_{0}
  12. c v c_{v}
  13. x 0 x_{0}
  14. h v h_{v}
  15. c v c_{v}
  16. ϕ ( Q ) = c | L | \phi(Q)=c|L|
  17. Q = ( T , L ) Q=(T,L)
  18. h v h_{v}
  19. c v c_{v}
  20. O ( l g q ( x ) ) O(lgq(x))
  21. a | L | a|L|
  22. h v h_{v}
  23. c v c_{v}
  24. 2 a | L | 2a|L|
  25. x 0 x_{0}
  26. h v h_{v}
  27. c v c_{v}
  28. 2 a | L | + O ( l g q ( x ) ) 2a|L|+O(lgq(x))
  29. c > 2 a c>2a
  30. O ( l g q ( x ) ) O(lgq(x))
  31. O ( l g q ( x ) ) O(lgq(x))

Query_likelihood_model.html

  1. P P
  2. d d
  3. q q
  4. P ( d | q ) = P ( q | d ) P ( d ) P ( q ) P(d|q)=\frac{P(q|d)P(d)}{P(q)}
  5. P ( d | q ) = P ( q | d ) P(d|q)=P(q|d)
  6. P ( q | M d ) = K q t V P ( t | M d ) t f t , q P(q|M_{d})=K_{q}\prod_{t\in V}P(t|M_{d})^{tf_{t,q}}
  7. K q = L q ! / ( t f t 1 , q ! t f t 2 , q ! t f t M , q ! ) K_{q}=L_{q}!/(tf_{t1,q}!tf_{t2,q}!...tf_{tM,q}!)
  8. q q
  9. d d
  10. M d M_{d}
  11. P ( t | M d ) P(t|M_{d})
  12. t t
  13. M d M_{d}
  14. d d
  15. q q
  16. d d
  17. [ 0 , 1 ] [0,1]

Quillen–Lichtenbaum_conjecture.html

  1. E 2 p q = H p étale ( Spec A [ - 1 ] , Z ( - q / 2 ) ) , E_{2}^{pq}=H^{p}\text{étale}(\,\text{Spec }A[\ell^{-1}],Z_{\ell}(-q/2)),
  2. K - p - q A Z K_{-p-q}A\otimes Z_{\ell}

Quintuple_product_identity.html

  1. n 1 ( 1 - s n ) ( 1 - s n t ) ( 1 - s n - 1 t - 1 ) ( 1 - s 2 n - 1 t 2 ) ( 1 - s 2 n - 1 t - 2 ) = n Z s ( 3 n 2 + n ) / 2 ( t 3 n - t - 3 n - 1 ) \prod_{n\geq 1}(1-s^{n})(1-s^{n}t)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^{2})(1-s^{2n-1% }t^{-2})=\sum_{n\in Z}s^{(3n^{2}+n)/2}(t^{3n}-t^{-3n-1})

RA_plot.html

  1. a = { a + ϵ , if a = 0 a , if a > 0 a=\begin{cases}a+\epsilon,&\,\text{if }a=0\\ a,&\,\text{if }a>0\end{cases}
  2. 0 < ϵ < 0.5 0<\epsilon<0.5
  3. R = log 2 ( a / b ) \textstyle{R=\log_{2}(a/b)}
  4. A = 1 2 × ( log 2 a + log 2 b ) \textstyle{A=\frac{1}{2}\times(\log_{2}a+\log_{2}b)}

Rabi_resonance_method.html

  1. d J d t = μ × B \frac{d\vec{J}}{dt}=\vec{\mu}\times\vec{B}
  2. μ = g μ B J = γ J \vec{\mu}=g\frac{\mu_{B}}{\hbar}\vec{J}=\gamma\vec{J}
  3. ω 0 = - γ B 0 \omega_{0}=-\frac{\gamma}{\hbar}B_{0}
  4. d J R d t = d J d t - J × ω \frac{d\vec{J_{R}}}{dt}=\frac{d\vec{J}}{dt}-\vec{J}\times\omega
  5. d J R d t = γ J × ( B 0 + B R ) - J × ω \frac{d\vec{J_{R}}}{dt}=\frac{\gamma}{\hbar}\vec{J}\times(\vec{B_{0}}+\vec{B_{% R}})-\vec{J}\times\omega
  6. γ B 0 = ω \frac{\gamma}{\hbar}B_{0}=\omega
  7. ω R = γ H R \omega_{R}=\frac{\gamma}{\hbar}H_{R}

Racket_features.html

  1. × × \mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}

Radar_horizon.html

  1. D h D_{h}
  2. R e R_{e}
  3. D h = 2 × H × R e D_{h}=\sqrt{2\times H\times R_{e}}
  4. D h D_{h}
  5. R e R_{e}
  6. D h = 2 × H × ( 4 R e 3 ) D_{h}=\sqrt{2\times H\times\left(\frac{4R_{e}}{3}\right)}
  7. T a r g e t H e i g h t > ( T a r g e t R a n g e - 2 × H × R e ) 2 2 × R e Target\ Height>\frac{\left(Target\ Range-\sqrt{2\times H\times Re}\right)^{2}}% {2\times Re}
  8. 1 o 1^{o}

Radar_scalloping.html

  1. V e l o c i t y > ( C 2 × T r a n s m i t F r e q u e n c y × P u l s e W i d t h ) Velocity>\left(\frac{C}{2\times TransmitFrequency\times PulseWidth}\right)
  2. V e l o c i t y > 0.5 × ( C 2 × P e r i o d B e t w e e n P u l s e s × T r a n s m i t F r e q u e n c y ) Velocity>0.5\times\left(\frac{C}{2\times PeriodBetweenPulses\times TransmitFrequency% }\right)
  3. B l i n d V e l o c i t y = ( C × P R F 2 × T r a n s m i t F r e q u e n c y ) Blind\ Velocity=\left(\frac{C\times PRF}{2\times Transmit\ Frequency}\right)
  4. B l i n d R a n g e = ( C 2 × P R F ) Blind\ Range=\left(\frac{C}{2\times PRF}\right)

Rademacher–Menchov_theorem.html

  1. | c ν | 2 log ( ν ) 2 < \sum|c_{\nu}|^{2}\log(\nu)^{2}<\infty

Radial_set.html

  1. X X
  2. A X A\subseteq X
  3. x 0 A x_{0}\in A
  4. x X x\in X
  5. t x > 0 t_{x}>0
  6. t [ 0 , t x ] t\in[0,t_{x}]
  7. x 0 + t x A x_{0}+tx\in A
  8. A A
  9. x 0 A x_{0}\in A
  10. x X t x > 0 t [ 0 , t x ] { x 0 + t x } A . \bigcup_{x\in X}\ \bigcap_{t_{x}>0}\ \bigcup_{t\in[0,t_{x}]}\{x_{0}+tx\}% \subseteq A.
  11. A X A\subseteq X
  12. A X A\subseteq X

Radiant_exposure.html

  1. H e = Q e A = 0 T E e ( t ) d t , H_{\mathrm{e}}=\frac{\partial Q_{\mathrm{e}}}{\partial A}=\int_{0}^{T}E_{% \mathrm{e}}(t)\,\mathrm{d}t,
  2. H e , ν = H e ν , H_{\mathrm{e},\nu}=\frac{\partial H_{\mathrm{e}}}{\partial\nu},
  3. H e , λ = H e λ , H_{\mathrm{e},\lambda}=\frac{\partial H_{\mathrm{e}}}{\partial\lambda},

Radiation_material_science.html

  1. R R
  2. R = N E m i n E m a x T m i n T m a x ϕ ( E i ) σ ( E i , T ) υ ( T ) d T d E i . R=N\int_{E_{min}}^{E_{max}}\int_{T_{min}}^{T_{max}}\phi(E_{i})\,\sigma(E_{i},T% )\,\upsilon(T)\,dT\,dE_{i}.
  3. N N
  4. E m a x E_{max}
  5. E m i n E_{min}
  6. ϕ ( E i ) \phi(E_{i})
  7. T m a x T_{max}
  8. T m i n T_{min}
  9. E i E_{i}
  10. σ ( E i , T ) \sigma(E_{i},T)
  11. E i E_{i}
  12. T T
  13. υ ( T ) \upsilon(T)
  14. σ ( E i , T ) \sigma(E_{i},T)
  15. υ ( T ) \upsilon(T)
  16. σ ( E i , T ) \sigma(E_{i},T)
  17. υ ( T ) \upsilon(T)
  18. E i E_{i}
  19. [ d p a ] \left[dpa\right]
  20. [ M e V ] \left[MeV\right]

Radiation_stress.html

  1. S x x = - h η ( p + ρ u ~ 2 ) d z ¯ - 1 2 ρ g ( h + η ¯ ) 2 , S_{xx}=\overline{\int_{-h}^{\eta}\left(p+\rho\tilde{u}^{2}\right)\;\,\text{d}z% }-\frac{1}{2}\rho g\left(h+\overline{\eta}\right)^{2},
  2. u ~ ( x , z , t ) \tilde{u}(x,z,t)
  3. η ¯ \overline{η}
  4. S x x = ( 2 c g c p - 1 2 ) E , S_{xx}=\left(2\frac{c_{g}}{c_{p}}-\frac{1}{2}\right)E,
  5. E = 1 2 ρ g a 2 = 1 8 ρ g H 2 , E=\frac{1}{2}\rho ga^{2}=\frac{1}{8}\rho gH^{2},
  6. 1 / 16 {1}/{16}
  7. 𝐒 \mathbf{S}
  8. 𝐒 = ( S x x S x y S y x S y y ) . \mathbf{S}=\begin{pmatrix}S_{xx}&S_{xy}\\ S_{yx}&S_{yy}\end{pmatrix}.
  9. S x x = - h η ( p + ρ u ~ 2 ) d z ¯ - 1 2 ρ g ( h + η ¯ ) 2 , S x y = - h η ( ρ u ~ v ~ ) d z ¯ = S y x , S y y = - h η ( p + ρ v ~ 2 ) d z ¯ - 1 2 ρ g ( h + η ¯ ) 2 , \begin{aligned}\displaystyle S_{xx}&\displaystyle=\overline{\int_{-h}^{\eta}% \left(p+\rho\tilde{u}^{2}\right)\;\,\text{d}z}-\frac{1}{2}\rho g\left(h+% \overline{\eta}\right)^{2},\\ \displaystyle S_{xy}&\displaystyle=\overline{\int_{-h}^{\eta}\left(\rho\tilde{% u}\tilde{v}\right)\;\,\text{d}z}=S_{yx},\\ \displaystyle S_{yy}&\displaystyle=\overline{\int_{-h}^{\eta}\left(p+\rho% \tilde{v}^{2}\right)\;\,\text{d}z}-\frac{1}{2}\rho g\left(h+\overline{\eta}% \right)^{2},\end{aligned}
  10. u ~ \tilde{u}
  11. v ~ \tilde{v}
  12. u ~ ( x , y , z , t ) \tilde{u}(x,y,z,t)
  13. S x x = [ k x 2 k 2 c g c p + ( c g c p - 1 2 ) ] E , S x y = ( k x k y k 2 c g c p ) E = S y x , and S y y = [ k y 2 k 2 c g c p + ( c g c p - 1 2 ) ] E , \begin{aligned}\displaystyle S_{xx}&\displaystyle=\left[\frac{k_{x}^{2}}{k^{2}% }\frac{c_{g}}{c_{p}}+\left(\frac{c_{g}}{c_{p}}-\frac{1}{2}\right)\right]E,\\ \displaystyle S_{xy}&\displaystyle=\left(\frac{k_{x}k_{y}}{k^{2}}\frac{c_{g}}{% c_{p}}\right)E=S_{yx},\quad\,\text{and}\\ \displaystyle S_{yy}&\displaystyle=\left[\frac{k_{y}^{2}}{k^{2}}\frac{c_{g}}{c% _{p}}+\left(\frac{c_{g}}{c_{p}}-\frac{1}{2}\right)\right]E,\end{aligned}
  14. s y m b o l M w = s y m b o l k k E ρ c p , symbol{M}_{w}=\frac{symbol{k}}{k}\frac{E}{\rho\,c_{p}},
  15. c p = σ k with σ = ω - s y m b o l k s y m b o l v ¯ , c_{p}=\frac{\sigma}{k}\qquad\,\text{with}\qquad\sigma=\omega-symbol{k}\cdot% \overline{symbol{v}},
  16. 𝐯 ¯ \overline{\mathbf{v}}
  17. 𝐯 ¯ \overline{\mathbf{v}}
  18. s y m b o l M = - h η ρ s y m b o l v d z ¯ = ρ ( h + η ¯ ) s y m b o l v ¯ + s y m b o l M w , symbol{M}=\overline{\int_{-h}^{\eta}\rho\,symbol{v}\;\,\text{d}z}=\rho\,\left(% h+\overline{\eta}\right)\overline{symbol{v}}+symbol{M}_{w},
  19. 𝐮 ¯ \overline{\mathbf{u}}
  20. s y m b o l u ¯ = s y m b o l M ρ ( h + η ¯ ) = s y m b o l v ¯ + s y m b o l M w ρ ( h + η ¯ ) . \overline{symbol{u}}=\frac{symbol{M}}{\rho\,\left(h+\overline{\eta}\right)}=% \overline{symbol{v}}+\frac{symbol{M}_{w}}{\rho\,\left(h+\overline{\eta}\right)}.
  21. η ¯ \overline{η}
  22. t [ ρ ( h + η ¯ ) ] + [ ρ ( h + η ¯ ) s y m b o l u ¯ ] = 0 , \frac{\partial}{\partial t}\left[\rho\left(h+\overline{\eta}\right)\right]+% \nabla\cdot\left[\rho\left(h+\overline{\eta}\right)\overline{symbol{u}}\right]% =0,
  23. 𝐮 ¯ \overline{\mathbf{u}}
  24. t [ ρ ( h + η ¯ ) s y m b o l u ¯ ] + [ ρ ( h + η ¯ ) s y m b o l u ¯ s y m b o l u ¯ + 𝐒 + 1 2 ρ g ( h + η ¯ ) 2 𝐈 ] = ρ g ( h + η ¯ ) h + s y m b o l τ w - s y m b o l τ b , \frac{\partial}{\partial t}\left[\rho\left(h+\overline{\eta}\right)\overline{% symbol{u}}\right]+\nabla\cdot\left[\rho\left(h+\overline{\eta}\right)\overline% {symbol{u}}\otimes\overline{symbol{u}}+\mathbf{S}+\frac{1}{2}\rho g(h+% \overline{\eta})^{2}\,\mathbf{I}\right]=\rho g\left(h+\overline{\eta}\right)% \nabla h+symbol{\tau}_{w}-symbol{\tau}_{b},
  25. 𝐮 ¯ \overline{\mathbf{u}}
  26. 𝐮 ¯ \overline{\mathbf{u}}
  27. 𝐮 ¯ \overline{\mathbf{u}}
  28. t [ ρ ( h + η ¯ ) ] + s y m b o l M = 0 , s y m b o l M t + [ s y m b o l u ¯ s y m b o l M + 𝐒 + 1 2 ρ g ( h + η ¯ ) 2 𝐈 ] = ρ g ( h + η ¯ ) h + s y m b o l τ w - s y m b o l τ b . \begin{aligned}&\displaystyle\frac{\partial}{\partial t}\left[\rho\left(h+% \overline{\eta}\right)\right]+\nabla\cdot symbol{M}=0,\\ &\displaystyle\frac{\partial symbol{M}}{\partial t}+\nabla\cdot\left[\overline% {symbol{u}}\otimes symbol{M}+\mathbf{S}+\frac{1}{2}\rho g(h+\overline{\eta})^{% 2}\,\mathbf{I}\right]=\rho g\left(h+\overline{\eta}\right)\nabla h+symbol{\tau% }_{w}-symbol{\tau}_{b}.\end{aligned}
  29. t [ ρ ( h + η ¯ ) ] + x [ ρ ( h + η ¯ ) u ¯ x ] + y [ ρ ( h + η ¯ ) u ¯ y ] = 0 , \frac{\partial}{\partial t}\left[\rho\left(h+\overline{\eta}\right)\right]+% \frac{\partial}{\partial x}\left[\rho\left(h+\overline{\eta}\right)\overline{u% }_{x}\right]+\frac{\partial}{\partial y}\left[\rho\left(h+\overline{\eta}% \right)\overline{u}_{y}\right]=0,
  30. u ¯ \overline{u}
  31. u ¯ \overline{u}
  32. 𝐮 ¯ \overline{\mathbf{u}}
  33. t [ ρ ( h + η ¯ ) u ¯ x ] \displaystyle\frac{\partial}{\partial t}\left[\rho\left(h+\overline{\eta}% \right)\overline{u}_{x}\right]
  34. E t + [ ( s y m b o l u ¯ + s y m b o l c g ) E ] + 𝐒 : ( s y m b o l u ¯ ) = s y m b o l τ w s y m b o l u ¯ - s y m b o l τ b s y m b o l u ¯ - ε , \frac{\partial E}{\partial t}+\nabla\cdot\left[\left(\overline{symbol{u}}+% symbol{c}_{g}\right)E\right]+\mathbf{S}:\left(\nabla\otimes\overline{symbol{u}% }\right)=symbol{\tau}_{w}\cdot\overline{symbol{u}}-symbol{\tau}_{b}\cdot% \overline{symbol{u}}-\varepsilon,
  35. 𝐒 : ( s y m b o l u ¯ ) \mathbf{S}:\left(\nabla\otimes\overline{symbol{u}}\right)
  36. 𝐮 ¯ \overline{\mathbf{u}}
  37. 𝐮 ¯ \overline{\mathbf{u}}
  38. E t \displaystyle\frac{\partial E}{\partial t}
  39. u ¯ \overline{u}
  40. u ¯ \overline{u}

Radical_extension.html

  1. α n = b \alpha^{n}=b
  2. K = F 0 < F 1 < < F k K=F_{0}<F_{1}<\cdots<F_{k}
  3. F i / F i - 1 F_{i}/F_{i-1}
  4. ω \omega
  5. α \alpha
  6. x n - a x^{n}-a
  7. α ω α \alpha\mapsto\omega\alpha
  8. ϕ \phi
  9. β \beta
  10. α = i = 0 n - 1 ω - i ϕ i ( β ) . \alpha=\sum_{i=0}^{n-1}\omega^{-i}\phi^{i}(\beta).
  11. ϕ ( α ) = ω α \phi(\alpha)=\omega\alpha
  12. α \alpha
  13. α n \alpha^{n}
  14. ± 1 \pm 1
  15. α n K , \alpha^{n}\in K,

Radioactive_displacement_law_of_Fajans_and_Soddy.html

  1. U 92 238 Th 90 234 {}^{238}_{92}\,\text{U}\to{}^{234}_{90}\,\text{Th}
  2. Pb 82 212 Bi 83 212 {}^{212}_{82}\,\text{Pb}\to{}^{212}_{83}\,\text{Bi}
  3. N 7 13 C 6 13 {}^{13}_{7}\,\text{N}\to{}^{13}_{6}\,\text{C}

Radiodrome.html

  1. V t V_{t}
  2. ( A x , A y ) (A_{x}\ ,\ A_{y})
  3. t t
  4. V d V_{d}
  5. ( x ( t ) , y ( t ) ) (x(t)\ ,\ y(t))
  6. x ˙ = V d T x - x ( T x - x ) 2 + ( T y - y ) 2 \dot{x}=V_{d}\ \frac{T_{x}-x}{\sqrt{(T_{x}-x)^{2}+(T_{y}-y)^{2}}}
  7. y ˙ = V d T y - y ( T x - x ) 2 + ( T y - y ) 2 \dot{y}=V_{d}\ \frac{T_{y}-y}{\sqrt{(T_{x}-x)^{2}+(T_{y}-y)^{2}}}
  8. y = f ( x ) y=f(x)
  9. T x - x T_{x}-x
  10. x x
  11. B B
  12. y y^{\prime}
  13. V t V d V_{t}\neq V_{d}
  14. y = C - 1 2 A x ( ( y ( 0 ) + y ( 0 ) 2 + 1 ) ( 1 - x A x ) 1 - V t V d 1 - V t V d - ( 1 - x A x ) 1 + V t V d ( y ( 0 ) + y ( 0 ) 2 + 1 ) ( 1 + V t V d ) ) y=C-\frac{1}{2}\ A_{x}\left(\frac{(y^{\prime}(0)+\sqrt{{y^{\prime}(0)}^{2}+1})% \ (1-\frac{x}{A_{x}})^{1-\frac{V_{t}}{V_{d}}}}{1-\frac{V_{t}}{V_{d}}}-\frac{(1% -\frac{x}{A_{x}})^{1+\frac{V_{t}}{V_{d}}}}{(y^{\prime}(0)+\sqrt{{y^{\prime}(0)% }^{2}+1})\ (1+\frac{V_{t}}{V_{d}})}\right)
  15. C C
  16. V t = V d V_{t}=V_{d}
  17. y = C - 1 2 A x ( ( y ( 0 ) + y ( 0 ) 2 + 1 ) ln ( 1 - x A x ) - ( 1 - x A x ) 2 ( y ( 0 ) + y ( 0 ) 2 + 1 ) 2 ) y=C-\frac{1}{2}A_{x}\ \left(\left(y^{\prime}(0)+\sqrt{{y^{\prime}(0)}^{2}+1}% \right)\ \ln(1-\frac{x}{A_{x}})-\frac{(1-\frac{x}{A_{x}})^{2}}{(y^{\prime}(0)+% \sqrt{{y^{\prime}(0)}^{2}+1})\ 2}\right)
  18. V t < V d V_{t}<V_{d}
  19. lim x A x y ( x ) = C = 1 2 A x ( y ( 0 ) + y ( 0 ) 2 + 1 1 - V t V d - 1 ( y ( 0 ) + y ( 0 ) 2 + 1 ) ( 1 + V t V d ) ) \lim_{x\to A_{x}}y(x)=C=\frac{1}{2}\ A_{x}\left(\frac{y^{\prime}(0)+\sqrt{{y^{% \prime}(0)}^{2}+1}}{1-\frac{V_{t}}{V_{d}}}-\frac{1}{(y^{\prime}(0)+\sqrt{{y^{% \prime}(0)}^{2}+1})\ (1+\frac{V_{t}}{V_{d}})}\right)
  20. V t V d = 1 1.2 \frac{V_{t}}{V_{d}}=\frac{1}{1.2}
  21. ( A x , - 0.6 A x ) (A_{x}\ ,\ -0.6\ A_{x})
  22. y ( 0 ) = - 0.6 y^{\prime}(0)=-0.6
  23. ( A x , 1.21688 A x ) (A_{x}\ ,\ 1.21688\ A_{x})
  24. ( 1.21688 + 0.6 ) A x (1.21688\ +\ 0.6)\ A_{x}
  25. V t V d V_{t}\geq V_{d}
  26. lim x A x y ( x ) = \lim_{x\to A_{x}}y(x)=\infty

Rainville_polynomials.html

  1. e w I 0 ( z w ) = n p n ( z ) w n \displaystyle e^{w}I_{0}(zw)=\sum_{n}p_{n}(z)w^{n}

Raman_cooling.html

  1. | g 1 |g_{1}\rangle
  2. | g 2 |g_{2}\rangle
  3. | e |e\rangle
  4. f 2 - f 1 f_{2}-f_{1}
  5. | g 1 |g_{1}\rangle
  6. | g 2 |g_{2}\rangle
  7. f 2 f_{2}
  8. Δ \Delta
  9. | g 2 |g_{2}\rangle
  10. Δ \Delta
  11. | v | > v m a x |v|>v_{max}
  12. | g 2 |g_{2}\rangle
  13. | v | < v m a x |v|<v_{max}
  14. | g 1 |g_{1}\rangle
  15. | g 2 |g_{2}\rangle
  16. | e |e\rangle
  17. | g 2 |g_{2}\rangle
  18. | g 1 |g_{1}\rangle
  19. | g 2 |g_{2}\rangle
  20. | v | < v m a x |v|<v_{max}
  21. m 1 m\neq 1
  22. σ + \sigma_{+}
  23. π \pi

Ramanujan's_master_theorem.html

  1. f ( x ) f(x)\!
  2. f ( x ) = k = 0 ϕ ( k ) k ! ( - x ) k f(x)=\sum_{k=0}^{\infty}\frac{\phi(k)}{k!}(-x)^{k}\!
  3. f ( x ) f(x)\!
  4. 0 x s - 1 f ( x ) d x = Γ ( s ) ϕ ( - s ) \int_{0}^{\infty}x^{s-1}f(x)\,dx=\Gamma(s)\phi(-s)\!
  5. Γ ( s ) \Gamma(s)\!
  6. 0 x s - 1 ( λ ( 0 ) - x λ ( 1 ) + x 2 λ ( 2 ) - ) d x = π sin ( π s ) λ ( - s ) \int_{0}^{\infty}x^{s-1}({\lambda(0)-x\lambda(1)+x^{2}\lambda(2)-\cdots})\,dx=% \frac{\pi}{\sin(\pi s)}\lambda(-s)
  7. λ ( n ) = ϕ ( n ) Γ ( 1 + n ) \lambda(n)=\frac{\phi(n)}{\Gamma(1+n)}\!
  8. 0 < Re ( s ) < 1 0<\operatorname{Re}(s)<1\!
  9. B k ( x ) B_{k}(x)\!
  10. z e x z e z - 1 = k = 0 B k ( x ) z k k ! \frac{ze^{xz}}{e^{z}-1}=\sum_{k=0}^{\infty}B_{k}(x)\frac{z^{k}}{k!}\!
  11. ζ ( s , a ) = n = 0 1 ( n + a ) s \zeta(s,a)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}\!
  12. ζ ( 1 - n , a ) = - B n ( a ) n \zeta(1-n,a)=-\frac{B_{n}(a)}{n}\!
  13. n 1 n\geq 1\!
  14. 0 x s - 1 ( e - a x 1 - e - x - 1 x ) d x = Γ ( s ) ζ ( s , a ) \int_{0}^{\infty}x^{s-1}\left(\frac{e^{-ax}}{1-e^{-x}}-\frac{1}{x}\right)\,dx=% \Gamma(s)\zeta(s,a)\!
  15. 0 < Re ( s ) < 1 0<\operatorname{Re}(s)<1\!
  16. Γ ( x ) = e - γ x x n = 1 ( 1 + x n ) - 1 e x / n \Gamma(x)=\frac{e^{-\gamma x}}{x}\prod_{n=1}^{\infty}\left(1+\frac{x}{n}\right% )^{-1}e^{x/n}\!
  17. log Γ ( 1 + x ) = - γ x + k = 2 ζ ( k ) k ( - x ) k \log\Gamma(1+x)=-\gamma x+\sum_{k=2}^{\infty}\frac{\zeta(k)}{k}(-x)^{k}\!
  18. ζ ( k ) \zeta(k)\!
  19. 0 x s - 1 γ x + log Γ ( 1 + x ) x 2 d x = π sin ( π s ) ζ ( 2 - s ) 2 - s \int_{0}^{\infty}x^{s-1}\frac{\gamma x+\log\Gamma(1+x)}{x^{2}}\,dx=\frac{\pi}{% \sin(\pi s)}\frac{\zeta(2-s)}{2-s}\!
  20. 0 < R e ( s ) < 1 0<Re(s)<1\!
  21. s = 1 2 s=\frac{1}{2}\!
  22. s = 3 4 s=\frac{3}{4}\!
  23. 0 γ x + log Γ ( 1 + x ) x 5 / 2 d x = 2 π 3 ζ ( 3 2 ) \int_{0}^{\infty}\frac{\gamma x+\log\Gamma(1+x)}{x^{5/2}}\,dx=\frac{2\pi}{3}% \zeta\left(\frac{3}{2}\right)
  24. 0 γ x + log Γ ( 1 + x ) x 9 / 4 d x = 2 4 π 5 ζ ( 5 4 ) \int_{0}^{\infty}\frac{\gamma x+\log\Gamma(1+x)}{x^{9/4}}\,dx=\sqrt{2}\frac{4% \pi}{5}\zeta\left(\frac{5}{4}\right)
  25. F ( a , m ) = 0 d x ( x 4 + 2 a x 2 + 1 ) m + 1 F(a,m)=\int_{0}^{\infty}\frac{dx}{(x^{4}+2ax^{2}+1)^{m+1}}

Randles–Sevcik_equation.html

  1. i p = 0.4463 n F A C ( n F v D R T ) 1 2 i_{p}=0.4463\ nFAC\left(\frac{nFvD}{RT}\right)^{\frac{1}{2}}
  2. i p = 268 , 600 n 3 2 A D 1 2 C v 1 2 i_{p}=268,600\ n^{\frac{3}{2}}AD^{\frac{1}{2}}Cv^{\frac{1}{2}}

Randolph_diagram.html

  1. P Q PQ
  2. \rightarrow
  3. \Rightarrow
  4. \supset
  5. \rightarrow
  6. \leftrightarrow
  7. \equiv
  8. = =
  9. \leftrightarrow
  10. ( Q P ) ( ¬ P Q ) (Q\leftrightarrow P)\lor(\lnot P\land Q)\,
  11. ( (
  12. \leftrightarrow
  13. ) ( )\lor(
  14. \land
  15. ) )
  16. \lor
  17. P Q . P\rightarrow Q.\,
  18. P Q , P Q \frac{P\to Q,P}{\therefore Q}
  19. ( ( P Q ) P ) Q ((P\to Q)\land P)\to Q
  20. ( ( ((
  21. \to
  22. ) )\land
  23. ) )\to
  24. ( (
  25. \land
  26. ) )\to
  27. \to
  28. A B A\cup B
  29. A B A\cap B
  30. A c A^{c}
  31. A B A\smallsetminus B
  32. A Δ B A\Delta B

Random_coordinate_descent.html

  1. F ( x ) = f ( x ) + Ψ ( x ) , F(x)=f(x)+\Psi(x),
  2. Ψ ( x ) = i = 1 n Ψ i ( x ( i ) ) , \Psi(x)=\sum_{i=1}^{n}\Psi_{i}(x^{(i)}),
  3. x R N x\in R^{N}
  4. n n
  5. x = ( x ( 1 ) , , x ( n ) ) x=(x^{(1)},\dots,x^{(n)})
  6. Ψ 1 , , Ψ n \Psi_{1},\dots,\Psi_{n}
  7. x = ( x 1 , x 2 , , x 5 ) R 5 x=(x_{1},x_{2},\dots,x_{5})\in R^{5}
  8. n = 3 n=3
  9. x ( 1 ) = ( x 1 , x 3 ) , x ( 2 ) = ( x 2 , x 5 ) x^{(1)}=(x_{1},x_{3}),x^{(2)}=(x_{2},x_{5})
  10. x ( 3 ) = x 4 x^{(3)}=x_{4}
  11. n = N ; Ψ ( x ) = x 1 = i = 1 n | x i | n=N;\Psi(x)=\|x\|_{1}=\sum_{i=1}^{n}|x_{i}|
  12. N = N 1 + N 2 + + N n ; Ψ ( x ) = i = 1 n x ( i ) 2 N=N_{1}+N_{2}+\dots+N_{n};\Psi(x)=\sum_{i=1}^{n}\|x^{(i)}\|_{2}
  13. x ( i ) R N i x^{(i)}\in R^{N_{i}}
  14. 2 \|\cdot\|_{2}
  15. min x R n f ( x ) , \min_{x\in R^{n}}f(x),
  16. f f
  17. f f
  18. L 1 , L 2 , , L n L_{1},L_{2},\dots,L_{n}
  19. | i f ( x + h e i ) - i f ( x ) | L i | h | , |\nabla_{i}f(x+he_{i})-\nabla_{i}f(x)|\leq L_{i}|h|,
  20. x R n x\in R^{n}
  21. h R h\in R
  22. i \nabla_{i}
  23. x ( i ) x^{(i)}
  24. x 0 R n x_{0}\in R^{n}
  25. x x
  26. i { 1 , 2 , , n } i\in\{1,2,\dots,n\}
  27. x ( i ) = x ( i ) - 1 L i f i ( x ) x^{(i)}=x^{(i)}-\frac{1}{L_{i}}\nabla f_{i}(x)
  28. k 2 n R L ( x 0 ) ϵ log ( f ( x 0 ) - f * ϵ ρ ) k\geq\frac{2nR_{L}(x_{0})}{\epsilon}\log\left(\frac{f(x_{0})-f^{*}}{\epsilon% \rho}\right)
  29. R L ( x ) = max y max x * X * { y - x * L : f ( y ) f ( x ) } R_{L}(x)=\max_{y}\max_{x^{*}\in X^{*}}\{\|y-x^{*}\|_{L}:f(y)\leq f(x)\}
  30. f * f^{*}
  31. f * = min x R n { f ( x ) } f^{*}=\min_{x\in R^{n}}\{f(x)\}
  32. ρ ( 0 , 1 ) \rho\in(0,1)
  33. ϵ > 0 \epsilon>0
  34. P r o b ( f ( x k ) - f * > ϵ ) ρ Prob(f(x_{k})-f^{*}>\epsilon)\leq\rho
  35. x k x_{k}
  36. f ( x ) = 1 2 x T ( 1 0.5 0.5 1 ) x - ( 1.5 1.5 ) x , x 0 = ( 0 0 ) T f(x)=\tfrac{1}{2}x^{T}\left(\begin{array}[]{cc}1&0.5\\ 0.5&1\end{array}\right)x-\left(\begin{array}[]{cc}1.5&1.5\end{array}\right)x,% \quad x_{0}=\left(\begin{array}[]{ cc}0&0\end{array}\right)^{T}
  37. R 5 R^{5}
  38. e 1 = ( 1 , 0 , 0 , 0 , 0 ) T , e 2 = ( 0 , 1 , 0 , 0 , 0 ) T , e 3 = ( 0 , 0 , 1 , 0 , 0 ) T , e 4 = ( 0 , 0 , 0 , 1 , 0 ) T , e 5 = ( 0 , 0 , 0 , 0 , 1 ) T e_{1}=(1,0,0,0,0)^{T},e_{2}=(0,1,0,0,0)^{T},e_{3}=(0,0,1,0,0)^{T},e_{4}=(0,0,0% ,1,0)^{T},e_{5}=(0,0,0,0,1)^{T}

Random_group.html

  1. × \mathbb{Z}\times\mathbb{Z}
  2. a a
  3. b b
  4. a b = b a ab=ba
  5. a b a - 1 b - 1 = 1 aba^{-1}b^{-1}=1
  6. a 1 , a 2 , , a m a_{1},\,a_{2},\,\ldots,\,a_{m}
  7. r 1 = 1 , r 2 = 1 , , r k = 1 r_{1}=1,\,r_{2}=1,\,\ldots,\,r_{k}=1
  8. r j r_{j}
  9. a i a_{i}
  10. a i - 1 a_{i}^{-1}
  11. m m
  12. k k
  13. r j r_{j}
  14. r k r_{k}
  15. G G
  16. G G
  17. F m F_{m}
  18. a 1 , a 2 , , a m a_{1},\,a_{2},\,\ldots,\,a_{m}
  19. R F m R\subset F_{m}
  20. r 1 r 2 , , r k r_{1}\,r_{2},\,\ldots,\,r_{k}
  21. F m F_{m}
  22. G = F m / r 1 , r 2 , , r k . G=F_{m}/\langle r_{1},\,r_{2},\,\ldots,\,r_{k}\rangle.
  23. m 2 m\geq 2
  24. k 1 k\geq 1
  25. \ell
  26. r 1 r 2 , , r k r_{1}\,r_{2},\,\ldots,\,r_{k}
  27. \ell
  28. a i a_{i}
  29. a i - 1 a_{i}^{-1}
  30. \ell
  31. 1 1
  32. \ell\to\infty

Random_modulation.html

  1. x c ( t ) x_{c}(t)
  2. x s ( t ) x_{s}(t)
  3. f [ - B / 2 , B / 2 ] f\in[-B/2,B/2]
  4. f 0 f_{0}
  5. f 0 > B / 2 f_{0}>B/2
  6. x ( t ) x(t)
  7. x ( t ) = x c ( t ) cos ( 2 π f 0 t ) - x s ( t ) sin ( 2 π f 0 t ) = { x ¯ ( t ) e j 2 π f 0 t } , x(t)=x_{c}(t)\cos(2\pi f_{0}t)-x_{s}(t)\sin(2\pi f_{0}t)=\Re\left\{\underline{% x}(t)e^{j2\pi f_{0}t}\right\},
  8. x ¯ ( t ) \underline{x}(t)
  9. x ( t ) x(t)
  10. x ¯ ( t ) = x c ( t ) + j x s ( t ) . \underline{x}(t)=x_{c}(t)+jx_{s}(t).
  11. x c ( t ) x_{c}(t)
  12. x s ( t ) x_{s}(t)
  13. x ( t ) x(t)
  14. x ¯ ( t ) \underline{x}(t)
  15. x c ( t ) x_{c}(t)
  16. x s ( t ) x_{s}(t)
  17. R x c x c ( τ ) = R x s x s ( τ ) and R x c x s ( τ ) = - R x s x c ( τ ) . R_{x_{c}x_{c}}(\tau)=R_{x_{s}x_{s}}(\tau)\qquad\,\text{and }\qquad R_{x_{c}x_{% s}}(\tau)=-R_{x_{s}x_{c}}(\tau).

Random_structure_function.html

  1. X | { d } X|\{d\}
  2. 𝔇 \mathfrak{D}
  3. 𝔇 0 \mathfrak{D}_{0}
  4. 𝔇 \mathfrak{D}
  5. 𝔓 ( { d } ) \mathfrak{P}(\{d\})
  6. { d } \{d\}
  7. 𝔓 ( 𝔇 0 ) ) = 1 | 𝔇 0 | d 𝔇 0 𝔓 ( { d } ) \mathfrak{P}(\mathfrak{D}_{0}))=\frac{1}{|\mathfrak{D}_{0}|}\sum_{d\in% \mathfrak{D}_{0}}\mathfrak{P}(\{d\})
  8. P X | 𝔇 0 ( E ) = 1 | 𝔇 0 | d 𝔇 0 P X | { d } ( E ) P_{X|\mathfrak{D}_{0}}(E)=\frac{1}{|\mathfrak{D}_{0}|}\sum_{d\in\mathfrak{D}_{% 0}}P_{X|\{d\}}(E)
  9. X | 𝔇 0 X|\mathfrak{D}_{0}
  10. X | { d 0 } X|\{d_{0}\}

Range_ambiguity_resolution.html

  1. D i s t a n c e > ( C 2 × P R F ) Distance>\left(\frac{C}{2\times PRF}\right)
  2. A p p a r e n t R a n g e = ( T r u e R a n g e ) M O D ( C 2 × P R F ) Apparent\ Range=(True\ Range)MOD\left(\frac{C}{2\times PRF}\right)
  3. T w o P R F C o m b i n a t i o n { P u l s e S p a c i n g ( A m b i g u o u s R a n g e ) = 1 P R F 1 P R F A - 1 P R F B = ± T r a n s m i t P u l s e W i d t h Two\ PRF\ Combination\begin{cases}Pulse\ Spacing\ (Ambiguous\ Range)=\frac{1}{% PRF}\\ \\ \frac{1}{PRF_{A}}-\frac{1}{PRF_{B}}=\pm\ Transmit\ Pulse\ Width\end{cases}
  4. \Tau \Tau
  5. 1 / P R F 1/PRF
  6. T r a n s m i t t e r C h a r a c t e r i s t i c s { D u t y C y c l e = P R F × T r a n s m i t P u l s e W i d t h P u l s e S p a c i n g = ( C P R F ) \begin{aligned}\displaystyle Transmitter\\ \displaystyle Characteristics\end{aligned}\begin{cases}Duty\ Cycle=PRF\times Transmit% \ Pulse\ Width\\ \\ Pulse\ Spacing=\left(\frac{C}{PRF}\right)\end{cases}
  7. P R F * D u t y C y c l e PRF*DutyCycle
  8. I n s t r u m e n t e d R a n g e { M i n i m u m S a m p l e W i d t h = ( D u t y C y c l e P R F ) M a x i m u m D i s t a n c e = ( P u l s e S p a c i n g S a m p l e W i d t h ) = ( C S a m p l e W i d t h × 2 × P R F 2 ) S a m p l e s P e r T r a n s m i t P u l s e = ( 1 M i n i m u m S a m p l e W i d t h - 1 ) \begin{aligned}\displaystyle Instrumented\\ \displaystyle Range\end{aligned}\begin{cases}Minimum\ Sample\ Width=\left(% \frac{Duty\ Cycle}{PRF}\right)\\ \\ Maximum\ Distance=\left(\frac{Pulse\ Spacing}{Sample\ Width}\right)=\left(% \frac{C}{Sample\ Width\times 2\times PRF^{2}}\right)\\ \\ Samples\ Per\ Transmit\ Pulse=\left(\frac{1}{Minimum\ Sample\ Width}-1\right)% \end{cases}

Range_expansion_index.html

  1. S 1 = j = 1 k n j m j s j S1=\sum_{j=1}^{k}n_{j}m_{j}s_{j}
  2. k k
  3. n j n_{j}
  4. m j m_{j}
  5. s j s_{j}
  6. s j = H i g h [ j ] - H i g h [ j - 2 ] + L o w [ j ] - L o w [ j - 2 ] , s_{j}=High[j]-High[j-2]+Low[j]-Low[j-2],\!
  7. S 2 = j = 1 k | H i g h [ j ] - H i g h [ j - 2 ] | + | L o w [ j ] - L o w [ j - 2 ] | S2=\sum_{j=1}^{k}|High[j]-High[j-2]|+|Low[j]-Low[j-2]|
  8. R E I = S 1 S 2 100 REI=\frac{S1}{S2}100

Rank_3_permutation_group.html

  1. T × T G T 0 wr Z / 2 Z T\times T\leq G\leq T_{0}\operatorname{wr}Z/2Z

Rank_error-correcting_code.html

  1. G F ( q N ) GF(q^{N})
  2. G F ( q N ) GF(q^{N})
  3. G F ( q ) GF(q)
  4. G F ( q N ) GF(q^{N})
  5. X n X^{n}
  6. G F ( q N ) GF\left({q^{N}}\right)
  7. q q
  8. N N
  9. ( u 1 , u 2 , , u N ) \left(u_{1},u_{2},\dots,u_{N}\right)
  10. u i G F ( q ) u_{i}\in GF(q)
  11. G F ( q ) GF\left({q}\right)
  12. x i G F ( q N ) x_{i}\in GF\left({q^{N}}\right)
  13. x i = a 1 i u 1 + a 2 i u 2 + + a N i u N x_{i}=a_{1i}u_{1}+a_{2i}u_{2}+\dots+a_{Ni}u_{N}
  14. x = ( x 1 , x 2 , , x n ) \vec{x}=\left({x_{1},x_{2},\dots,x_{n}}\right)
  15. G F ( q N ) GF\left({q^{N}}\right)
  16. x = a 1 , 1 a 1 , 2 a 1 , n a 2 , 1 a 2 , 2 a 2 , n a N , 1 a N , 2 a N , n \vec{x}=\left\|{\begin{array}[]{*{20}c}a_{1,1}&a_{1,2}&\ldots&a_{1,n}\\ a_{2,1}&a_{2,2}&\ldots&a_{2,n}\\ \ldots&\ldots&\ldots&\ldots\\ a_{N,1}&a_{N,2}&\ldots&a_{N,n}\end{array}}\right\|
  17. x \vec{x}
  18. G F ( q N ) GF\left({q^{N}}\right)
  19. A ( x ) A\left({\vec{x}}\right)
  20. G F ( q ) GF\left({q}\right)
  21. r ( x ; q ) r\left({\vec{x};q}\right)
  22. x \vec{x}
  23. X n = A N n X^{n}=A_{N}^{n}
  24. x r ( x ; q ) \vec{x}\to r\left(\vec{x};q\right)
  25. X n X^{n}
  26. d ( x ; y ) = r ( x - y ; q ) d\left({\vec{x};\vec{y}}\right)=r\left({\vec{x}-\vec{y};q}\right)
  27. { x 1 , x 2 , , x n } \{x_{1},x_{2},\dots,x_{n}\}
  28. X n X^{n}
  29. d = min d ( x i , x j ) d=\min d\left(x_{i},x_{j}\right)
  30. X n X^{n}
  31. d n - k + 1 d\leq n-k+1
  32. [ i ] [i]
  33. x G F ( q N ) x\in GF(q^{N})
  34. x [ i ] = x q i mod N . x^{[i]}=x^{q^{i\mod N}}.\,
  35. g = ( g 1 , g 2 , , g n ) , g i G F ( q N ) , n N \vec{g}=(g_{1},g_{2},\dots,g_{n}),~{}g_{i}\in GF(q^{N}),~{}n\leq N
  36. G F ( q ) GF(q)
  37. G = g 1 g 2 g n g 1 [ m ] g 2 [ m ] g n [ m ] g 1 [ 2 m ] g 2 [ 2 m ] g n [ 2 m ] g 1 [ k m ] g 2 [ k m ] g n [ k m ] , G=\left\|{\begin{array}[]{*{20}c}g_{1}&g_{2}&\dots&g_{n}\\ g_{1}^{[m]}&g_{2}^{[m]}&\dots&g_{n}^{[m]}\\ g_{1}^{[2m]}&g_{2}^{[2m]}&\dots&g_{n}^{[2m]}\\ \dots&\dots&\dots&\dots\\ g_{1}^{[km]}&g_{2}^{[km]}&\dots&g_{n}^{[km]}\end{array}}\right\|,
  38. gcd ( m , N ) = 1 \gcd(m,N)=1

Rank_SIFT.html

  1. { I m , m = 0 , 1 , M } \left\{I_{m},m=0,1,...M\right\}
  2. p p
  3. p p
  4. R ( p ) R(p)
  5. p p
  6. R ( p I 0 ) = m I ( min q I m H m ( p ) - q 2 < ϵ ) , R(p\in I_{0})=\sum_{m}I(\min_{q\in I_{m}}{\lVert H_{m}(p)-q\rVert}_{2}<% \epsilon),
  7. I ( . ) I(.)
  8. H m H_{m}
  9. I 0 I_{0}
  10. I m I_{m}
  11. ϵ \epsilon
  12. x i x_{i}
  13. p i p_{i}
  14. x i x_{i}
  15. p i p_{i}
  16. X f e a t u r e s p a c e = { x 1 , x 2 , } X_{featurespace}=\left\{\vec{x}_{1},\vec{x}_{2},...\right\}
  17. m i n i m i z e : V ( w ) = 1 2 w w s . t . x i a n d x j X f e a t u r e s p a c e , w T ( x i - x j ) 1 i f R ( p i I 0 ) > R ( p j I 0 ) . \begin{array}[]{lcl}minimize:V(\vec{w})={1\over 2}\vec{w}\cdot\vec{w}\\ s.t.\\ \begin{array}[]{lcl}\forall\ \vec{x}_{i}\ and\ \vec{x}_{j}\in X_{featurespace}% ,\\ \vec{w}^{T}(\vec{x}_{i}-\vec{x}_{j})\geqq 1\quad if\ R(p_{i}\in I_{0})>R(p_{j}% \in I_{0}).\end{array}\end{array}
  18. w * \vec{w}^{*}
  19. X = { x 1 , , x N } {\vec{X}}=\left\{x_{1},...,x_{N}\right\}
  20. R = { r 1 , r N } {R}=\left\{r_{1},...r_{N}\right\}
  21. x i x_{i}
  22. X X
  23. r i r_{i}
  24. r i = | { x k : x k x i } | . r_{i}=\left|\left\{x_{k}:x_{k}\geqq x_{i}\right\}\right|.
  25. X \vec{X}
  26. R \vec{R}
  27. R \vec{R}
  28. R \vec{R}^{^{\prime}}
  29. ρ ( R , R ) = 1 - 6 i = 1 N ( r i - r i ) 2 N ( N 2 - 1 ) \rho(\vec{R},\vec{R}^{^{\prime}})=1-{6\sum_{i=1}^{N}(r_{i}-r_{i}^{^{\prime}})^% {2}\over N(N^{2}-1)}
  30. R R
  31. R R^{^{\prime}}
  32. τ ( R , R ) = 2 i = 1 N j = i + 1 N s ( r i - r j , r i - r j ) N ( N - 1 ) , \tau(\vec{R},\vec{R}^{^{\prime}})={2\sum_{i=1}^{N}\sum_{j=i+1}^{N}s(r_{i}-r_{j% },r_{i}^{^{\prime}}-r_{j}^{^{\prime}})\over N(N-1)},
  33. w h e r e s ( a , b ) = { 1 , if s i g n ( a ) = s i g n ( b ) - 1 , o . w . where\quad s(a,b)=\begin{cases}1,&\,\text{if }sign(a)=sign(b)\\ -1,&o.w.\end{cases}

Ranking_SVM.html

  1. \mathbb{C}
  2. C C
  3. c i c_{i}
  4. r r
  5. \mathbb{C}
  6. r r
  7. \mathbb{C}
  8. C C
  9. C C
  10. c i c_{i}
  11. c j c_{j}
  12. r c i < r c j r\ c_{i}<r\ c_{j}
  13. r 1 r_{1}
  14. r 2 r_{2}
  15. \mathbb{C}
  16. r 1 r_{1}
  17. r 2 r_{2}
  18. τ ( r 1 , r 2 ) = P - Q P + Q = 1 - 2 Q P + Q \tau(r_{1},r_{2})={P-Q\over P+Q}=1-{2Q\over P+Q}
  19. P P
  20. r 1 r_{1}
  21. r 2 r_{2}
  22. Q Q
  23. r 1 r_{1}
  24. r 2 r_{2}
  25. P r e l e v a n t Prelevant
  26. P r e t r i e v e d Pretrieved
  27. P r e c i s i o n = | P r e l e v a n t P r e t r i e v e d | | P r e t r i e v e d | ; R e c a l l = | P r e l e v a n t P r e t r i e v e d | | P r e l e v a n t | ; A v e r a g e P r e c i s i o n = 0 1 P r e c ( R e c a l l ) d R e c a l l , \begin{array}[]{lcl}Precision={\left|Prelevant\cap Pretrieved\right|\over\left% |Pretrieved\right|};\\ \\ Recall={\left|Prelevant\cap Pretrieved\right|\over\left|Prelevant\right|};\\ \\ AveragePrecision=\int_{0}^{1}{Prec(R_{ecall})}dR_{ecall},\\ \end{array}
  28. P r e c ( R e c a l l ) Prec(R_{ecall})
  29. R e c a l l Recall
  30. r * r^{*}
  31. r f ( q ) r_{f(q)}
  32. r f ( q ) r_{f(q)}
  33. A v g P r e c ( r f ( q ) ) 1 R [ Q + ( R + 1 2 ) ] - 1 ( i = 1 R i ) 2 AvgPrec(r_{f(q)})\geqq{1\over R}\left[Q+{\left({{R+1}\atop{2}}\right)}\right]^% {-1}(\sum_{i=1}^{R}\sqrt{i})^{2}
  34. Q Q
  35. r * r^{*}
  36. r f ( q ) r_{f(q)}
  37. R R
  38. ( x i , y i ) (\vec{x}_{i},y_{i})
  39. x i \vec{x}_{i}
  40. y i y_{i}
  41. x i \vec{x}_{i}
  42. m i n i m i z e : V ( w , ξ ) = 1 2 w w + C F ξ i σ s . t . σ 0 ; y i ( w x i + b ) 1 - ξ i σ ; w h e r e b i s a s c a l a r ; y i { - 1 , 1 } ; ξ i 0 ; \begin{array}[]{lcl}minimize:V(\vec{w},\vec{\xi})={1\over 2}\vec{w}\cdot\vec{w% }+CF\sum{\xi_{i}^{\sigma}}\\ s.t.\\ \begin{array}[]{lcl}\sigma\geqq 0;\\ \forall y_{i}(\vec{w}\vec{x}_{i}+b)\geqq 1-\xi_{i}^{\sigma};\end{array}\\ where\\ \begin{array}[]{lcl}b\ is\ a\ scalar;\\ \forall y_{i}\in\left\{-1,1\right\};\\ \forall\xi_{i}\geqq 0;\\ \end{array}\end{array}
  43. x i x_{i}
  44. w * = i α i y i x i \vec{w}^{*}=\sum_{i}{\alpha_{i}y_{i}x_{i}}
  45. α i \alpha_{i}
  46. τ P ( f ) \tau_{P(f)}
  47. r * r^{*}
  48. r f ( q ) r_{f(q)}
  49. τ P ( f ) \tau_{P(f)}
  50. r f ( q ) r_{f(q)}
  51. τ P ( f ) \tau_{P(f)}
  52. r f ( q ) r_{f(q)}
  53. L e x p e c t e d = - τ P ( f ) = - τ ( r f ( q ) , r * ) d P r ( q , r * ) L_{expected}=-\tau_{P(f)}=-\int\tau(r_{f(q)},r^{*})dPr(q,r^{*})
  54. P r ( q , r * ) Pr(q,r^{*})
  55. r * r^{*}
  56. q q
  57. L e m p i r i c a l = - τ S ( f ) = - 1 n i = 1 n τ ( r f ( q i ) , r i * ) L_{empirical}=-\tau_{S}(f)=-{1\over n}\sum_{i=1}^{n}{\tau(r_{f(q_{i})},r_{i}^{% *})}
  58. n n
  59. n n
  60. Φ ( q , d ) \Phi(q,d)
  61. c i c_{i}
  62. c j c_{j}
  63. ( c i , c j ) r (c_{i},c_{j})\in r
  64. c i c_{i}
  65. c j c_{j}
  66. r r
  67. w \vec{w}
  68. m i n i m i z e : V ( w , ξ ) = 1 2 w w + C o n s t a n t ξ i , j , k s . t . ξ i , j , k 0 ( c i , c j ) r k * w ( Φ ( q 1 , c i ) - Φ ( q 1 , c j ) ) 1 - ξ i , j , 1 ; w ( Φ ( q n , c i ) - Φ ( q n , c j ) ) 1 - ξ i , j , n ; w h e r e k { 1 , 2 , n } , i , j { 1 , 2 , } . \begin{array}[]{lcl}minimize:V(\vec{w},\vec{\xi})={1\over 2}\vec{w}\cdot\vec{w% }+C_{onstant}\sum{\xi_{i,j,k}}\\ s.t.\\ \begin{array}[]{lcl}\forall\xi_{i,j,k}\geqq 0\\ \forall(c_{i},c_{j})\in r_{k}^{*}\\ \vec{w}(\Phi(q_{1},c_{i})-\Phi(q_{1},c_{j}))\geqq 1-\xi_{i,j,1};\\ ...\\ \vec{w}(\Phi(q_{n},c_{i})-\Phi(q_{n},c_{j}))\geqq 1-\xi_{i,j,n};\\ where\ k\in\left\{1,2,...n\right\},\ i,j\in\left\{1,2,...\right\}.\\ \end{array}\end{array}
  69. w * \vec{w}^{*}
  70. w * = α k , l * Φ ( q k , c i ) \vec{w}^{*}=\sum{\alpha_{k,l}^{*}\Phi(q_{k},c_{i})}
  71. q q
  72. q q
  73. m i n i m i z e : V ( w , ξ ) = 1 2 w w + C o n t a n t ξ i , j , k s . t . ξ i , j , k 0 ( c i , c j ) r k w ( Φ ( q 1 , c i ) - Φ ( q 1 , c j ) ) 1 - ξ i , j , 1 ; w ( Φ ( q n , c i ) - Φ ( q n , c j ) ) 1 - ξ i , j , n ; w h e r e k { 1 , 2 , n } , i , j { 1 , 2 , } . \begin{array}[]{lcl}minimize:V(\vec{w},\vec{\xi})={1\over 2}\vec{w}\cdot\vec{w% }+C_{ontant}\sum{\xi_{i,j,k}}\\ s.t.\\ \begin{array}[]{lcl}\forall\xi_{i,j,k}\geqq 0\\ \forall(c_{i},c_{j})\in r_{k}^{^{\prime}}\\ \vec{w}(\Phi(q_{1},c_{i})-\Phi(q_{1},c_{j}))\geqq 1-\xi_{i,j,1};\\ ...\\ \vec{w}(\Phi(q_{n},c_{i})-\Phi(q_{n},c_{j}))\geqq 1-\xi_{i,j,n};\\ where\ k\in\left\{1,2,...n\right\},\ i,j\in\left\{1,2,...\right\}.\\ \end{array}\end{array}
  74. r r^{\prime}

Rankin–Selberg_method.html

  1. D f ( τ ) g ( τ ) ¯ E ( τ , s ) y k - 2 d x d y \displaystyle\int_{D}f(\tau)\overline{g(\tau)}E(\tau,s)y^{k-2}dxdy

Rapid_shallow_breathing_index.html

  1. R S B I = f V T RSBI=\frac{f}{V_{T}}

Rashba_effect.html

  1. H R = α ( s y m b o l σ × p ) z ^ H_{R}=\alpha(symbol{\sigma}\times{p})\cdot\hat{z}
  2. α \alpha
  3. s y m b o l p symbolp
  4. s y m b o l σ symbol\sigma
  5. s y m b o l k s y m b o l p symbolk\cdot symbolp
  6. α \alpha
  7. H E = - E 0 z H_{E}=-E_{0}z
  8. B = - ( v × E ) / c 2 {B}=-({v}\times{E})/c^{2}
  9. c c
  10. H S O = g μ B 2 c 2 ( v × E ) σ H_{SO}=\frac{g\mu_{B}}{2c^{2}}({v}\times{E})\cdot{\sigma}
  11. - g μ B σ / 2 -g\mu_{B}{\sigma}/2
  12. H R = α ( s y m b o l σ × p ) z ^ H_{R}=\alpha(symbol{\sigma}\times{p})\cdot\hat{z}
  13. α = g μ B E 0 2 m c 2 \alpha=\frac{g\mu_{B}E_{0}}{2mc^{2}}
  14. z ^ \hat{z}
  15. α \alpha
  16. p z p_{z}
  17. p p
  18. Γ \Gamma
  19. H S O = Δ S O L s y m b o l σ H_{SO}=\Delta_{SO}{L}\otimes symbol{\sigma}
  20. H E = E 0 z H_{E}=E_{0}\,z
  21. Δ B G \Delta_{BG}
  22. p z p_{z}
  23. p x p_{x}
  24. p y p_{y}
  25. p z p_{z}
  26. p x p_{x}
  27. p y p_{y}
  28. p z p_{z}
  29. σ \sigma
  30. p x p_{x}
  31. p y p_{y}
  32. σ \sigma^{\prime}
  33. t i j ; σ σ x , y = p z , i ; σ | H | p x , y , j ; σ t_{ij;\sigma\sigma^{\prime}}^{x,y}=\langle p_{z},i;\sigma|H|p_{x,y},j;\sigma^{% \prime}\rangle
  34. H H
  35. H E = 0 H_{E}=0
  36. H E 0 H_{E}\neq 0
  37. t σ σ x , y = E 0 p z , i ; σ | z | p x , y , i + 1 x , y ; σ = t 0 sgn ( 1 x , y ) δ σ σ t_{\sigma\sigma^{\prime}}^{x,y}=E_{0}\langle p_{z},i;\sigma|z|p_{x,y},i+1_{x,y% };\sigma^{\prime}\rangle=t_{0}\,\mathrm{sgn}(1_{x,y})\delta_{\sigma\sigma^{% \prime}}
  38. 1 x , y 1_{x,y}
  39. x , y x,y
  40. δ σ σ \delta_{\sigma\sigma^{\prime}}
  41. | p z , i ; |p_{z},i;\uparrow\rangle
  42. | p x , y , i + 1 x , y ; |p_{x,y},i+1_{x,y};\uparrow\rangle
  43. t 0 t_{0}
  44. | p z , i + 1 x , y ; |p_{z},i+1_{x,y};\downarrow\rangle
  45. Δ S O \Delta_{SO}
  46. Δ B G \Delta_{BG}
  47. α a t 0 Δ S O Δ B G \alpha\approx{a\,t_{0}\,\Delta_{SO}\over\Delta_{BG}}
  48. a a
  49. 3 {}_{3}
  50. 5 {}_{5}
  51. Γ 6 \Gamma_{6}
  52. Γ 6 \Gamma_{6}

Rasiowa–Sikorski_lemma.html

  1. 0 \aleph_{0}
  2. 2 0 2^{\aleph_{0}}

Rational_reciprocity_law.html

  1. ( p | q ) 4 ( q | p ) 4 = ( - 1 ) ( q - 1 ) / 4 ( a B - b A | q ) 2 . (p|q)_{4}(q|p)_{4}=(-1)^{(q-1)/4}(aB-bA|q)_{2}\ .
  2. ( p | q ) 8 = ( q | p ) 8 = ( a B - b A | q ) 4 ( c D - d C | q ) 2 . (p|q)_{8}=(q|p)_{8}=(aB-bA|q)_{4}(cD-dC|q)_{2}\ .

Rational_sequence_topology.html

  1. U n ( x ) := { x k : n k < } { x } . U_{n}(x):=\{x_{k}:n\leq k<\infty\}\cup\{x\}.

Rauzy_fractal.html

  1. s ( 1 ) = 12 , s ( 2 ) = 13 , s ( 3 ) = 1 . s(1)=12,\ s(2)=13,\ s(3)=1\,.
  2. s ( 1 ) = 12 s(1)=12
  3. s ( 2 ) = 13 s(2)=13
  4. s ( 3 ) = 1 s(3)=1
  5. t 0 = 1 t_{0}=1
  6. t 1 = 12 t_{1}=12
  7. t 2 = 1213 t_{2}=1213
  8. t 3 = 1213121 t_{3}=1213121
  9. t 4 = 1213121121312 t_{4}=1213121121312
  10. n > 2 n>2
  11. t n = t n - 1 t n - 2 t n - 3 t_{n}=t_{n-1}t_{n-2}t_{n-3}
  12. R 3 R^{3}
  13. 1 ( 1 , 0 , 0 ) 1\Rightarrow(1,0,0)
  14. 2 ( 1 , 1 , 0 ) 2\Rightarrow(1,1,0)
  15. 1 ( 2 , 1 , 0 ) 1\Rightarrow(2,1,0)
  16. 3 ( 2 , 1 , 1 ) 3\Rightarrow(2,1,1)
  17. 1 ( 3 , 1 , 1 ) 1\Rightarrow(3,1,1)
  18. k k
  19. k 2 k^{2}
  20. k 3 k^{3}
  21. k k
  22. k 3 + k 2 + k - 1 = 0 k^{3}+k^{2}+k-1=0
  23. k = 1 3 ( - 1 - 2 17 + 3 33 3 + 17 + 3 33 3 ) = 0.54368901269207636 \scriptstyle{k=\frac{1}{3}(-1-\frac{2}{\sqrt[3]{17+3\sqrt{33}}}+\sqrt[3]{17+3% \sqrt{33}})=0.54368901269207636}
  24. x 3 - x 2 - x - 1 x^{3}-x^{2}-x-1
  25. β = 1.8392 \beta=1.8392
  26. α \alpha
  27. α ¯ \bar{\alpha}
  28. α α ¯ = 1 / β \alpha\bar{\alpha}=1/\beta
  29. 2 | α | 3 s + | α | 4 s = 1 2|\alpha|^{3s}+|\alpha|^{4s}=1

Rayleigh_sky_model.html

  1. δ = δ m a x sin 2 γ 1 + cos 2 γ \delta=\frac{\delta_{max}\sin^{2}\gamma}{1+\cos^{2}\gamma}
  2. cos γ = sin θ s sin θ cos ψ + cos θ s cos θ {\cos\gamma=\sin\theta_{s}\sin\theta\cos\psi+\cos\theta_{s}\cos\theta}
  3. Q i n = s i n 2 θ + 90 Q_{in}=sin2\theta+90
  4. U i n = c o s 2 θ + 90 U_{in}=cos2\theta+90

Rayleigh–Faber–Krahn_inequality.html

  1. n \mathbb{R}^{n}
  2. n 2 n\geq 2

RD-120.html

  1. 912 , 018 N ( 1 , 125 kg ) ( 9.807 m / s 2 ) = 82.66 \frac{912,018\ \mathrm{N}}{(1,125\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=82.66

Reactor_Experiment_for_Neutrino_Oscillation.html

  1. sin 2 2 θ 13 = 0.113 ± 0.013 ( stat . ) ± 0.019 ( syst . ) \sin^{2}2\theta_{13}=0.113\pm 0.013({\rm stat.})\pm 0.019({\rm syst.})
  2. sin 2 2 θ 13 = 0.100 ± 0.010 ( stat . ) ± 0.015 ( syst . ) \sin^{2}2\theta_{13}=0.100\pm 0.010({\rm stat.})\pm 0.015({\rm syst.})

Real_closed_ring.html

  1. S M S\to M
  2. r : R r c l ( R ) r:R\to rcl(R)
  3. f : R A f:R\to A
  4. g : r c l ( R ) A g:rcl(R)\to A
  5. f = g r f=g\circ r
  6. [ T 1 , , T n ] \mathbb{R}[T_{1},...,T_{n}]
  7. n \mathbb{R}^{n}\to\mathbb{R}
  8. ( 2 ) \mathbb{Q}(\sqrt{2})
  9. \mathbb{R}
  10. a l g \mathbb{R}_{alg}
  11. ( 2 ) \mathbb{Q}(\sqrt{2})
  12. a l g × a l g \mathbb{R}_{alg}\times\mathbb{R}_{alg}
  13. ( 2 ) \mathbb{Q}(\sqrt{2})
  14. P - P P\cap-P

Real_hyperelliptic_curve.html

  1. g 1 g\geq 1
  2. K K
  3. C : y 2 + h ( x ) y = f ( x ) K [ x , y ] C:y^{2}+h(x)y=f(x)\in K[x,y]
  4. h ( x ) , f ( x ) K h(x),f(x)\in K
  5. C : y 2 + h ( x ) y = f ( x ) C:y^{2}+h(x)y=f(x)
  6. h ( x ) K h(x)\in K
  7. f ( x ) K f(x)\in K
  8. ( x , y ) (x,y)
  9. K K
  10. y 2 + h ( x ) y = f ( x ) y^{2}+h(x)y=f(x)
  11. 2 y + h ( x ) = 0 2y+h(x)=0
  12. h ( x ) y = f ( x ) h^{\prime}(x)y=f^{\prime}(x)
  13. K K
  14. C ( K ) = { ( a , b ) K 2 | b 2 + h ( a ) b = f ( a ) } S C(K)=\{(a,b)\in K^{2}|b^{2}+h(a)b=f(a)\}\cup S
  15. S S
  16. 1 \infty_{1}
  17. 2 \infty_{2}
  18. P ( a , b ) C ( K ) P(a,b)\in C(K)
  19. P P
  20. P ¯ = ( a , - b - h ) \overline{P}=(a,-b-h)
  21. C : y 2 = f ( x ) C:y^{2}=f(x)
  22. f ( x ) = x 6 + 3 x 5 - 5 x 4 - 15 x 3 + 4 x 2 + 12 x = x ( x - 1 ) ( x - 2 ) ( x + 1 ) ( x + 2 ) ( x + 3 ) f(x)=x^{6}+3x^{5}-5x^{4}-15x^{3}+4x^{2}+12x=x(x-1)(x-2)(x+1)(x+2)(x+3)\,
  23. R R
  24. deg f ( x ) = 2 g + 2 \deg f(x)=2g+2
  25. f ( x ) f(x)
  26. C C
  27. Y 2 Z 4 = X 6 + 3 X 5 Z - 5 X 4 Z 2 - 15 X 3 Z 3 + 4 X 2 Z 4 + 12 X Z 5 Y^{2}Z^{4}=X^{6}+3X^{5}Z-5X^{4}Z^{2}-15X^{3}Z^{3}+4X^{2}Z^{4}+12XZ^{5}
  28. C C
  29. 1 \infty_{1}
  30. 2 \infty_{2}
  31. C C
  32. D D
  33. C C
  34. P P
  35. C C
  36. D = P C n P P D=\sum_{P\in C}{n_{P}P}
  37. n P \Z n_{P}\in\Z
  38. n p = 0 n_{p}=0
  39. P P
  40. D = P C n P P D=\sum_{P\in C}{n_{P}P}
  41. deg ( D ) = P C n P \deg(D)=\sum_{P\in C}{n_{P}}
  42. D D
  43. K K
  44. D σ = P C n P P σ = D D^{\sigma}=\sum_{P\in C}n_{P}P^{\sigma}=D
  45. K ¯ \overline{K}
  46. K K
  47. D i v ( K ) Div(K)
  48. C C
  49. K K
  50. a P P + b P P = ( a P + b P ) P \sum a_{P}P+\sum b_{P}P=\sum{(a_{P}+b_{P})P}
  51. D i v 0 ( K ) Div^{0}(K)
  52. C C
  53. K K
  54. D i v ( K ) Div(K)
  55. D 1 = 6 P 1 + 4 P 2 D_{1}=6P_{1}+4P_{2}
  56. D 2 = 1 P 1 + 5 P 2 D_{2}=1P_{1}+5P_{2}
  57. D 1 + D 2 = 7 P 1 + 9 P 2 D_{1}+D_{2}=7P_{1}+9P_{2}
  58. D 1 D_{1}
  59. deg ( D 1 ) = 6 + 4 = 10 \deg(D_{1})=6+4=10
  60. D 2 D_{2}
  61. deg ( D 2 ) = 1 + 5 = 6 \deg(D_{2})=1+5=6
  62. deg ( D 1 + D 2 ) = d e g ( D 1 ) + d e g ( D 2 ) = 16. \deg(D_{1}+D_{2})=deg(D_{1})+deg(D_{2})=16.
  63. G K [ C ] G\in K[C]
  64. G G
  65. div ( G ) = P C ord P ( G ) P \mathrm{div}(G)=\sum_{P\in C}{\mathrm{ord}}_{P}(G)P
  66. G G
  67. P P
  68. - ord P ( G ) -{\mathrm{ord}}_{P}(G)
  69. G G
  70. P P
  71. G , H G,H
  72. K [ C ] K[C]
  73. F = G / H F=G/H
  74. div ( F ) = div ( G ) - div ( H ) \mathrm{div}(F)=\mathrm{div}(G)-\mathrm{div}(H)
  75. P ( K ) P(K)
  76. P ( K ) = div ( F ) | F K ( C ) P(K)={\mathrm{div}(F)|F\in K(C)}
  77. C C
  78. K K
  79. J = D i v 0 / P J=Div^{0}/P
  80. J J
  81. C C
  82. K K
  83. J ( K ) J(K)
  84. D ¯ J ( K ) \overline{D}\in J(K)
  85. D D
  86. D i v 0 ( K ) / P ( K ) Div^{0}(K)/P(K)
  87. C C
  88. S = { 1 , 2 } S=\{\infty_{1},\infty_{2}\}
  89. D ¯ \bar{D}
  90. D = i = 1 r P i - r 2 D=\sum_{i=1}^{r}P_{i}-r\infty_{2}
  91. P i C ( 𝔽 ¯ q ) P_{i}\in C(\bar{\mathbb{F}}_{q})
  92. P i 2 P_{i}\not=\infty_{2}
  93. P i P j ¯ P_{i}\not=\bar{P_{j}}
  94. i j i\not=j
  95. D D
  96. D ¯ \bar{D}
  97. D D
  98. r g r\leq g
  99. D D
  100. P i = 1 P_{i}=\infty_{1}
  101. D ¯ \bar{D}
  102. D = D x - d e g ( D x ) 2 + v 1 ( D ) ( 1 - 2 ) D=D_{x}-deg(D_{x})\infty_{2}+v_{1}(D)(\infty_{1}-\infty_{2})
  103. D x D_{x}
  104. 1 \infty_{1}
  105. 2 \infty_{2}
  106. 0 d e g ( D x ) + v 1 ( D ) g 0\leq deg(D_{x})+v_{1}(D)\leq g
  107. D = 1 + 2 D_{\infty}=\infty_{1}+\infty_{2}
  108. K K
  109. 1 \infty_{1}
  110. 2 \infty_{2}
  111. D ¯ \bar{D}
  112. D = D 1 + D D=D_{1}+D_{\infty}
  113. D 1 D_{1}
  114. 1 \infty_{1}
  115. 2 \infty_{2}
  116. d = deg ( D 1 ) d=\deg(D_{1})
  117. d d
  118. D = d 2 ( 1 + 2 ) D_{\infty}=\frac{d}{2}(\infty_{1}+\infty_{2})
  119. d d
  120. D = d + 1 2 1 + d - 1 2 2 D_{\infty}=\frac{d+1}{2}\infty_{1}+\frac{d-1}{2}\infty_{2}
  121. D 1 = 6 P 1 + 4 P 2 D_{1}=6P_{1}+4P_{2}
  122. D 2 = 1 P 1 + 5 P 2 D_{2}=1P_{1}+5P_{2}
  123. D 1 = 6 P 1 + 4 P 2 - 5 D 1 - 5 D 2 D_{1}=6P_{1}+4P_{2}-5D_{\infty_{1}}-5D_{\infty_{2}}
  124. D 2 = 1 P 1 + 5 P 2 - 3 D 1 - 3 D 2 D_{2}=1P_{1}+5P_{2}-3D_{\infty_{1}}-3D_{\infty_{2}}
  125. C C
  126. K K
  127. K K
  128. P = ( a , b ) = P ¯ = ( a , - b - h ( a ) ) P=(a,b)=\overline{P}=(a,-b-h(a))
  129. h ( a ) + 2 b = 0 h(a)+2b=0
  130. P P
  131. D = P - 1 D=P-\infty_{1}
  132. C : y 2 + h ( x ) y = f ( x ) C:y^{2}+h(x)y=f(x)
  133. g g
  134. K K
  135. P = ( a , b ) P=(a,b)
  136. C : y 2 + h ¯ ( x ) y = f ¯ ( x ) C^{\prime}:y^{\prime 2}+\bar{h}(x^{\prime})y^{\prime}=\bar{f}(x^{\prime})
  137. g g
  138. deg ( f ¯ ) = 2 g + 1 \deg(\bar{f})=2g+1
  139. K ( C ) = K ( C ) K(C)=K(C^{\prime})
  140. x = 1 x - a x^{\prime}=\frac{1}{x-a}
  141. y = y + b ( x - a ) g + 1 y^{\prime}=\frac{y+b}{(x-a)^{g+1}}
  142. C : y 2 = f ( x ) C:y^{2}=f(x)
  143. f ( x ) = x 6 + 3 x 5 - 5 x 4 - 15 x 3 + 4 x 2 + 12 x f(x)=x^{6}+3x^{5}-5x^{4}-15x^{3}+4x^{2}+12x
  144. P = ( a , b ) P=(a,b)
  145. h ( a ) h(a)
  146. b = 0 b=0
  147. h ( a ) h(a)
  148. b b
  149. f ( a ) = 0 f(a)=0
  150. f ( a ) = a ( a - 1 ) ( a - 2 ) ( a + 1 ) ( a + 2 ) ( a + 3 ) f(a)=a(a-1)(a-2)(a+1)(a+2)(a+3)
  151. a { 0 , 1 , 2 , - 1 , - 2 , - 3 } a\in\{0,1,2,-1,-2,-3\}
  152. x = a x + 1 x x=\frac{ax^{\prime}+1}{x^{\prime}}
  153. y = y x g + 1 y=\frac{y^{\prime}}{x^{\prime g+1}}
  154. y = y x 3 y=\frac{y^{\prime}}{x^{\prime 3}}
  155. a = 1 a=1
  156. x = x + 1 x x=\frac{x^{\prime}+1}{x^{\prime}}
  157. y = y x 3 y=\frac{y^{\prime}}{x^{\prime 3}}
  158. ( y x 3 ) 2 = x + 1 x ( x + 1 x + 1 ) ( x + 1 x + 2 ) ( x + 1 x + 3 ) ( x + 1 x - 1 ) ( x + 1 x - 2 ) \left(\frac{y^{\prime}}{x^{\prime 3}}\right)^{2}=\frac{x^{\prime}+1}{x^{\prime% }}\left(\frac{x^{\prime}+1}{x^{\prime}}+1\right)\left(\frac{x^{\prime}+1}{x^{% \prime}}+2\right)\left(\frac{x^{\prime}+1}{x^{\prime}}+3\right)\left(\frac{x^{% \prime}+1}{x^{\prime}}-1\right)\left(\frac{x^{\prime}+1}{x^{\prime}}-2\right)
  159. x 6 x^{6}
  160. y 2 = ( x + 1 ) ( 2 x + 1 ) ( 3 x + 1 ) ( 4 x + 1 ) ( 1 ) ( 1 - x ) y^{\prime 2}=(x^{\prime}+1)(2x^{\prime}+1)(3x^{\prime}+1)(4x^{\prime}+1)(1)(1-% x^{\prime})\,
  161. C : y 2 = f ¯ ( x ) C^{\prime}:y^{\prime 2}=\bar{f}(x^{\prime})
  162. f ¯ ( x ) = ( x + 1 ) ( 2 x + 1 ) ( 3 x + 1 ) ( 4 x + 1 ) ( 1 ) ( 1 - x ) = - 24 x 5 - 26 x 4 + 15 x 3 + 25 x 2 + 9 x + 1 \bar{f}(x^{\prime})=(x^{\prime}+1)(2x^{\prime}+1)(3x^{\prime}+1)(4x^{\prime}+1% )(1)(1-x^{\prime})=-24x^{\prime 5}-26x^{\prime 4}+15x^{\prime 3}+25x^{\prime 2% }+9x^{\prime}+1
  163. C C^{\prime}
  164. f ¯ ( x ) \bar{f}(x^{\prime})
  165. 2 g + 1 2g+1