wpmath0000012_8

Maxwell_construction.html

  1. P P
  2. V V
  3. v R 3 - ( 8 T R p R + 1 ) v R 2 3 + 9 v R 3 p R - 1 p R = 0 v_{R}^{3}-\left(\frac{8T_{R}}{p_{R}}+1\right)\frac{v_{R}^{2}}{3}+\frac{9v_{R}}% {3p_{R}}-\frac{1}{p_{R}}=0
  4. v r 3 + a 1 v r 2 + a 2 v r + a 3 = 0 v_{r}^{3}+a_{1}v_{r}^{2}+a_{2}v_{r}+a_{3}=0
  5. a 1 ( T R , p R ) = - 8 T R p R + 1 3 a_{1}(T_{R},p_{R})=-\frac{\frac{8T_{R}}{p_{R}}+1}{3}
  6. a 2 ( p R ) = 3 p R a_{2}(p_{R})=\frac{3}{p_{R}}
  7. a 3 ( p R ) = - 1 p R a_{3}(p_{R})=-\frac{1}{p_{R}}
  8. η = a 2 - a 1 2 3 \eta=a_{2}-\frac{a_{1}^{2}}{3}
  9. q = - ( + 2 27 a 1 3 - a 1 a 2 3 + a 3 ) q=-\left(+\frac{2}{27}a_{1}^{3}-\frac{a_{1}a_{2}}{3}+a_{3}\right)
  10. z 1 = 4 η - 3 c o s [ 1 3 c o s - 1 ( - 3 q 2 η - 3 η ) ] z_{1}=\sqrt{\frac{4\eta}{-3}}cos\left[\frac{1}{3}cos^{-1}\left(-\frac{3q}{2% \eta}\sqrt{\frac{-3}{\eta}}\right)\right]
  11. v R 1 = 4 η - 3 c o s [ 1 3 c o s - 1 ( 3 q 4 η - 3 η ) ] - a 1 3 v_{R_{1}}=\sqrt{\frac{4\eta}{-3}}cos\left[\frac{1}{3}cos^{-1}\left(\frac{3q}{4% \eta}\sqrt{\frac{-3}{\eta}}\right)\right]-\frac{a_{1}}{3}
  12. v R 2 = - z 1 + z 1 2 - 4 ( η + z 1 2 ) 2 - a 1 3 v_{R_{2}}=\frac{-z_{1}+\sqrt{z_{1}^{2}-4(\eta+z_{1}^{2})}}{2}-\frac{a_{1}}{3}
  13. v R 3 = - z 1 + z 1 2 + 4 ( η + z 1 2 ) 2 - a 1 3 v_{R_{3}}=\frac{-z_{1}+\sqrt{z_{1}^{2}+4(\eta+z_{1}^{2})}}{2}-\frac{a_{1}}{3}
  14. 8 T R 3 n ( ( 3 v R s m a l l ( p R , T R ) - 1 ) ( 3 v R l a r g e ( p R , T R ) - 1 ) + 1 v R s m a l l ( p R , T R ) 3 - 1 v R l a r g e ( p R , T R ) 3 = 0 \frac{8T_{R}}{3}\ell n\left(\frac{(3v_{R_{small}}(p_{R},T_{R})-1)}{(3v_{R_{% large}}(p_{R},T_{R})-1}\right)+\frac{1}{v_{R_{small}}(p_{R},T_{R})^{3}}-\frac{% 1}{v{R_{large}}(p_{R},T_{R})^{3}}=0
  15. P P
  16. V V

McLaughlin_sporadic_group.html

  1. × 10 8 \times 10^{8}
  2. T 2 A ( τ ) T_{2A}(\tau)
  3. T 4 A ( τ ) T_{4A}(\tau)

Mean_and_predicted_response.html

  1. y i = α + β x i + ϵ i y_{i}=\alpha+\beta x_{i}+\epsilon_{i}\,
  2. y i y_{i}
  3. x i x_{i}
  4. α \alpha
  5. β \beta
  6. y ^ d = α ^ + β ^ x d , \hat{y}_{d}=\hat{\alpha}+\hat{\beta}x_{d},
  7. y d = α + β x d + ϵ d y_{d}=\alpha+\beta x_{d}+\epsilon_{d}\,
  8. α ^ \hat{\alpha}
  9. β ^ \hat{\beta}
  10. E ( y | x d ) = y ^ d E(y|x_{d})=\hat{y}_{d}\!
  11. Var ( α ^ + β ^ x d ) = Var ( α ^ ) + ( Var β ^ ) x d 2 + 2 x d Cov ( α ^ , β ^ ) . \,\text{Var}\left(\hat{\alpha}+\hat{\beta}x_{d}\right)=\,\text{Var}\left(\hat{% \alpha}\right)+\left(\,\text{Var}\hat{\beta}\right)x_{d}^{2}+2x_{d}\,\text{Cov% }\left(\hat{\alpha},\hat{\beta}\right).
  12. Var ( α ^ + β ^ x d ) = σ 2 ( 1 m + ( x d - x ¯ ) 2 ( x i - x ¯ ) 2 ) . \,\text{Var}\left(\hat{\alpha}+\hat{\beta}x_{d}\right)=\sigma^{2}\left(\frac{1% }{m}+\frac{\left(x_{d}-\bar{x}\right)^{2}}{\sum(x_{i}-\bar{x})^{2}}\right).
  13. ( x i - x ¯ ) 2 = x i 2 - 1 m ( x i ) 2 . \sum(x_{i}-\bar{x})^{2}=\sum x_{i}^{2}-\frac{1}{m}\left(\sum x_{i}\right)^{2}.
  14. Var ( y d - [ α ^ + β ^ x d ] ) = Var ( y d ) + Var ( α ^ + β ^ x d ) . \,\text{Var}\left(y_{d}-\left[\hat{\alpha}+\hat{\beta}x_{d}\right]\right)=\,% \text{Var}\left(y_{d}\right)+\,\text{Var}\left(\hat{\alpha}+\hat{\beta}x_{d}% \right).
  15. Var ( y d ) = σ 2 \,\text{Var}\left(y_{d}\right)=\sigma^{2}
  16. Var ( y d - [ α ^ + β ^ x d ] ) = σ 2 + σ 2 ( 1 m + ( x d - x ¯ ) 2 ( x i - x ¯ ) 2 ) = σ 2 ( 1 + 1 m + ( x d - x ¯ ) 2 ( x i - x ¯ ) 2 ) . \,\text{Var}\left(y_{d}-\left[\hat{\alpha}+\hat{\beta}x_{d}\right]\right)=% \sigma^{2}+\sigma^{2}\left(\frac{1}{m}+\frac{\left(x_{d}-\bar{x}\right)^{2}}{% \sum(x_{i}-\bar{x})^{2}}\right)=\sigma^{2}\left(1+\frac{1}{m}+\frac{\left(x_{d% }-\bar{x}\right)^{2}}{\sum(x_{i}-\bar{x})^{2}}\right).
  17. 100 ( 1 - α ) % 100(1-\alpha)\%
  18. y d ± t α 2 , m - n - 1 Var y_{d}\pm t_{\frac{\alpha}{2},m-n-1}\sqrt{\text{Var}}
  19. y y
  20. y y
  21. α ^ \hat{\alpha}
  22. β ^ \hat{\beta}
  23. α + β x d \alpha+\beta x_{d}
  24. y i = j = 1 j = n X i j β j + ϵ i y_{i}=\sum_{j=1}^{j=n}X_{ij}\beta_{j}+\epsilon_{i}\,
  25. y d = j = 1 j = n X d j β ^ j y_{d}=\sum_{j=1}^{j=n}X_{dj}\hat{\beta}_{j}
  26. Var ( j = 1 j = n X d j β ^ j ) = i = 1 i = n j = 1 j = n X d i M i j X d j , \,\text{Var}\left(\sum_{j=1}^{j=n}X_{dj}\hat{\beta}_{j}\right)=\sum_{i=1}^{i=n% }\sum_{j=1}^{j=n}X_{di}M_{ij}X_{dj},
  27. 𝐌 = σ 2 ( 𝐗 𝐓 𝐗 ) - 1 \mathbf{M}=\sigma^{2}\left(\mathbf{X^{T}X}\right)^{-1}

Mean_annual_increment.html

  1. M A I = Y ( t ) / t MAI=Y(t)/t

Mean_integrated_squared_error.html

  1. E f n - f 2 2 = E ( f n ( x ) - f ( x ) ) 2 d x \operatorname{E}\|f_{n}-f\|_{2}^{2}=\operatorname{E}\int(f_{n}(x)-f(x))^{2}\,dx

Mean_kinetic_temperature.html

  1. T K = Δ H R - ln ( t 1 e ( - Δ H R T 1 ) + t 2 e ( - Δ H R T 2 ) + + t n e ( - Δ H R T n ) t 1 + t 2 + + t n ) T_{K}=\cfrac{\frac{\Delta H}{R}}{-\ln\left(\frac{{t_{1}}e^{\left(}\frac{-% \Delta H}{RT_{1}}\right)+{t_{2}}e^{\left(}\frac{-\Delta H}{RT_{2}}\right)+% \cdots+{t_{n}}e^{\left(}\frac{-\Delta H}{RT_{n}}\right)}{{t_{1}}+{t_{2}}+% \cdots+{t_{n}}}\right)}
  2. T K T_{K}\,\!
  3. Δ H \Delta H\,\!
  4. R R\,\!
  5. T 1 T_{1}\,\!
  6. T n T_{n}\,\!
  7. t 1 t_{1}\,\!
  8. t n t_{n}\,\!
  9. t 1 t_{1}\,\!
  10. t 2 t_{2}\,\!
  11. \cdots
  12. t n t_{n}\,\!
  13. T K = Δ H R - ln ( e ( - Δ H R T 1 ) + e ( - Δ H R T 2 ) + + e ( - Δ H R T n ) n ) T_{K}=\cfrac{\frac{\Delta H}{R}}{-\ln\left(\frac{e^{\left(}\frac{-\Delta H}{RT% _{1}}\right)+e^{\left(}\frac{-\Delta H}{RT_{2}}\right)+\cdots+e^{\left(}\frac{% -\Delta H}{RT_{n}}\right)}{n}\right)}
  14. n n\,\!

Mean_square_weighted_deviation.html

  1. x i \ {x_{i}}
  2. w i \ {w_{i}}
  3. σ x i \sigma_{x_{i}}
  4. x ¯ = i = 1 N x i N \overline{x}=\frac{\sum_{i=1}^{N}x_{i}}{N}
  5. σ 2 = i = 1 N ( x i - x ¯ ) 2 N and s 2 = N N - 1 σ 2 = N N 2 - N i = 1 N ( x i - x ¯ ) 2 . \sigma^{2}=\frac{\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}}{N}{\rm\ \ and\ \ }s^{% 2}=\frac{N}{N-1}\cdot\sigma^{2}=\frac{N}{N^{2}-N}\cdot\sum_{i=1}^{N}(x_{i}-% \overline{x})^{2}.
  6. x ¯ * = i = 1 N w i x i i = 1 N w i \overline{x}^{\,*}=\frac{\sum_{i=1}^{N}w_{i}x_{i}}{\sum_{i=1}^{N}w_{i}}
  7. σ 2 = i = 1 N w i ( x i - x ¯ * ) 2 i = 1 N w i \sigma^{2}=\frac{\sum_{i=1}^{N}w_{i}(x_{i}-\overline{x}^{\,*})^{2}}{\sum_{i=1}% ^{N}w_{i}}
  8. σ 2 = i = 1 N w i x i 2 i = 1 N w i - ( i = 1 N w i x i ) 2 ( i = 1 N w i ) 2 \sigma^{2}=\frac{\sum_{i=1}^{N}w_{i}x_{i}^{2}\cdot\sum_{i=1}^{N}w_{i}-(\sum_{i% =1}^{N}w_{i}x_{i})^{2}}{(\sum_{i=1}^{N}w_{i})^{2}}
  9. s 2 = i = 1 N w i ( i = 1 N w i ) 2 - i = 1 N w i 2 . i = 1 N w i ( x i - x ¯ * ) 2 s^{2}=\frac{\sum_{i=1}^{N}w_{i}}{{(\sum_{i=1}^{N}w_{i}})^{2}-{\sum_{i=1}^{N}w_% {i}^{2}}}\ .\ {\sum_{i=1}^{N}w_{i}(x_{i}-\overline{x}^{\,*})^{2}}
  10. s 2 = i = 1 N w i x i 2 i = 1 N w i - ( i = 1 N w i x i ) 2 ( i = 1 N w i ) 2 - i = 1 N w i 2 s^{2}=\frac{\sum_{i=1}^{N}w_{i}x_{i}^{2}\cdot\sum_{i=1}^{N}w_{i}-(\sum_{i=1}^{% N}w_{i}x_{i})^{2}}{(\sum_{i=1}^{N}w_{i})^{2}-\sum_{i=1}^{N}w_{i}^{2}}
  11. = u 1 N - 1 . i = 1 N ( x i - x ¯ ) 2 σ x i 2 {}_{u}=\frac{1}{N-1}\ .\ \sum_{i=1}^{N}\frac{(x_{i}-\overline{x})^{2}}{\sigma_% {x_{i}}^{2}}
  12. = w i = 1 N w i ( i = 1 N w i ) 2 - i = 1 N w i 2 . i = 1 N w i . ( x i - x ¯ * ) 2 ( σ x i ) 2 {}_{w}=\frac{\sum_{i=1}^{N}w_{i}}{(\sum_{i=1}^{N}w_{i})^{2}-\sum_{i=1}^{N}w_{i% }^{2}}\ .\ \sum_{i=1}^{N}\frac{w_{i}.(x_{i}-\overline{x}^{\,*})^{2}}{(\sigma_{% x_{i}})^{2}}

Measurement_of_a_Circle.html

  1. 3 10 71 3\tfrac{10}{71}
  2. 3 1 7 3\tfrac{1}{7}
  3. 1351 780 > 3 > 265 153 . \tfrac{1351}{780}>\sqrt{3}>\tfrac{265}{153}\,.

Measures_of_conditioned_emotional_response.html

  1. S R = D / ( D + B ) SR=D/(D+B)

Mechanics_of_planar_particle_motion.html

  1. 𝐫 ( s ) = [ x ( s ) , y ( s ) ] . \mathbf{r}(s)=\left[x(s),\ y(s)\right]\ .
  2. d 𝐫 ( s ) = [ d x ( s ) , d y ( s ) ] = [ x ( s ) , y ( s ) ] d s , d\mathbf{r}(s)=\left[dx(s),\ dy(s)\right]=\left[x^{\prime}(s),\ y^{\prime}(s)% \right]ds\ ,
  3. [ x ( s ) 2 + y ( s ) 2 ] = 1 . \left[x^{\prime}(s)^{2}+y^{\prime}(s)^{2}\right]=1\ .
  4. 𝐮 t ( s ) = [ x ( s ) , y ( s ) ] , \mathbf{u}_{t}(s)=\left[x^{\prime}(s),\ y^{\prime}(s)\right]\ ,
  5. 𝐮 n ( s ) = [ y ( s ) , - x ( s ) ] , \mathbf{u}_{n}(s)=\left[y^{\prime}(s),\ -x^{\prime}(s)\right]\ ,
  6. sin θ = y ( s ) x ( s ) 2 + y ( s ) 2 = y ( s ) ; \sin\theta=\frac{y^{\prime}(s)}{\sqrt{x^{\prime}(s)^{2}+y^{\prime}(s)^{2}}}=y^% {\prime}(s)\ ;
  7. cos θ = x ( s ) x ( s ) 2 + y ( s ) 2 = x ( s ) . \cos\theta=\frac{x^{\prime}(s)}{\sqrt{x^{\prime}(s)^{2}+y^{\prime}(s)^{2}}}=x^% {\prime}(s)\ .
  8. 1 ρ = d θ d s . \frac{1}{\rho}=\frac{d\theta}{ds}\ .
  9. d sin θ d s = cos θ d θ d s = 1 ρ cos θ \frac{d\sin\theta}{ds}=\cos\theta\frac{d\theta}{ds}=\frac{1}{\rho}\cos\theta
  10. = 1 ρ x ( s ) . =\frac{1}{\rho}x^{\prime}(s)\ .
  11. d sin θ d s = d d s y ( s ) x ( s ) 2 + y ( s ) 2 \frac{d\sin\theta}{ds}=\frac{d}{ds}\frac{y^{\prime}(s)}{\sqrt{x^{\prime}(s)^{2% }+y^{\prime}(s)^{2}}}
  12. = y ′′ ( s ) x ( s ) 2 - y ( s ) x ( s ) x ′′ ( s ) ( x ( s ) 2 + y ( s ) 2 ) 3 / 2 , =\frac{y^{\prime\prime}(s)x^{\prime}(s)^{2}-y^{\prime}(s)x^{\prime}(s)x^{% \prime\prime}(s)}{\left(x^{\prime}(s)^{2}+y^{\prime}(s)^{2}\right)^{3/2}}\ ,
  13. d θ d s = 1 ρ = y ′′ ( s ) x ( s ) - y ( s ) x ′′ ( s ) \frac{d\theta}{ds}=\frac{1}{\rho}=y^{\prime\prime}(s)x^{\prime}(s)-y^{\prime}(% s)x^{\prime\prime}(s)
  14. = y ′′ ( s ) x ( s ) = - x ′′ ( s ) y ( s ) , =\frac{y^{\prime\prime}(s)}{x^{\prime}(s)}=-\frac{x^{\prime\prime}(s)}{y^{% \prime}(s)}\ ,
  15. x ( s ) x ′′ ( s ) + y ( s ) y ′′ ( s ) = 0 . x^{\prime}(s)x^{\prime\prime}(s)+y^{\prime}(s)y^{\prime\prime}(s)=0\ .
  16. 𝐚 ( s ) = d d t 𝐯 ( s ) \mathbf{a}(s)=\frac{d}{dt}\mathbf{v}(s)
  17. = d d t [ d s d t ( x ( s ) , y ( s ) ) ] =\frac{d}{dt}\left[\frac{ds}{dt}\left(x^{\prime}(s),\ y^{\prime}(s)\right)\right]
  18. = ( d 2 s d t 2 ) 𝐮 t ( s ) + ( d s d t ) 2 ( x ′′ ( s ) , y ′′ ( s ) ) =\left(\frac{d^{2}s}{dt^{2}}\right)\mathbf{u}_{t}(s)+\left(\frac{ds}{dt}\right% )^{2}\left(x^{\prime\prime}(s),\ y^{\prime\prime}(s)\right)
  19. = ( d 2 s d t 2 ) 𝐮 t ( s ) - ( d s d t ) 2 1 ρ 𝐮 n ( s ) , =\left(\frac{d^{2}s}{dt^{2}}\right)\mathbf{u}_{t}(s)-\left(\frac{ds}{dt}\right% )^{2}\frac{1}{\rho}\mathbf{u}_{n}(s)\ ,
  20. s y m b o l a = d s y m b o l v d t = d 2 𝐫 d t 2 = ( r ¨ - r θ ˙ 2 ) s y m b o l r ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ , symbol{a}=\frac{dsymbol{v}}{dt}=\frac{d^{2}\mathbf{r}}{dt^{2}}=(\ddot{r}-r\dot% {\theta}^{2})\hat{symbol{r}}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\hat{symbol% \theta}\ ,
  21. T T
  22. U U
  23. L = T - U . L=T-U.\quad
  24. L ( s y m b o l q , s y m b o l q ˙ , t ) = T - U L(symbol{q},\ symbol{\dot{q}},\ t)=T-U
  25. q i ˙ \dot{q_{i}}
  26. L / q i ˙ \partial L/\partial\dot{q_{i}}
  27. L / q i \partial L/\partial q_{i}
  28. d d t L q i ˙ - L q i = 0 \frac{d}{dt}\frac{\partial L}{\partial\dot{q_{i}}}-\frac{\partial L}{\partial q% _{i}}=0
  29. p r = m r ˙ , p θ = m r 2 θ ˙ , p_{r}=m\dot{r}\ ,\ p_{\theta}=mr^{2}\dot{\theta}\ ,
  30. d d t p r = Q r + m r θ ˙ 2 , \frac{d}{dt}p_{r}=Q_{r}+mr{\dot{\theta}}^{2}\ ,
  31. m r θ ˙ 2 . mr{\dot{\theta}}^{2}\ .
  32. j = 1 n M i j ( s y m b o l q ) q ¨ j + j , k = 1 n Γ i j k q ˙ j q ˙ k + V q i = Υ i ; i = 1 , , n , \sum_{j=1}^{n}\ M_{ij}(symbolq)\ddot{q}_{j}+\sum_{j,k=1}^{n}\Gamma_{ijk}\dot{q% }_{j}\dot{q}_{k}+\frac{\partial V}{\partial q_{i}}=\Upsilon_{i}\ ;i=1,...,n\ ,
  33. Υ i \Upsilon_{i}
  34. 𝐫 \mathbf{r}
  35. 𝐫 = ( r cos θ , r sin θ ) , \mathbf{r}=(r\cos\theta,\ r\sin\theta)\ ,
  36. 𝐫 \mathbf{r}
  37. s y m b o l r ^ = 𝐫 r = ( cos θ , sin θ ) \hat{symbol{r}}=\frac{\partial\mathbf{r}}{\partial r}=(\cos\theta,\ \sin\theta)
  38. 𝐫 \mathbf{r}
  39. s y m b o l θ ^ = 2 𝐫 r θ = ( - sin θ , cos θ ) . \hat{symbol\theta}=\frac{\partial^{2}{\mathbf{r}}}{\partial r\,\partial\theta}% =(-\sin\theta\ ,\cos\theta)\ .
  40. d d t s y m b o l r ^ = ( - sin θ , cos θ ) d θ d t = d θ d t s y m b o l θ ^ , \frac{d}{dt}\hat{symbol{r}}=(-\sin\theta,\ \cos\theta)\frac{d\theta}{dt}=\frac% {d\theta}{dt}\hat{symbol\theta},
  41. d d t s y m b o l θ ^ = ( - cos θ , - sin θ ) d θ d t = - d θ d t s y m b o l r ^ . \frac{d}{dt}\hat{symbol{\theta}}=(-\cos\theta,\ -\sin\theta)\frac{d\theta}{dt}% =-\frac{d\theta}{dt}\hat{symbolr}.
  42. s y m b o l v = d 𝐫 d t = r ˙ s y m b o l r ^ + r θ ˙ s y m b o l θ ^ , symbol{v}=\frac{d\mathbf{r}}{dt}=\dot{r}\hat{symbol{r}}+r\dot{\theta}\hat{% symbol\theta},
  43. s y m b o l a = d s y m b o l v d t = d 2 𝐫 d t 2 = ( r ¨ - r θ ˙ 2 ) s y m b o l r ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ , symbol{a}=\frac{dsymbol{v}}{dt}=\frac{d^{2}\mathbf{r}}{dt^{2}}=(\ddot{r}-r\dot% {\theta}^{2})\hat{symbol{r}}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\hat{symbol% \theta}\ ,
  44. s y m b o l a symbol{a}
  45. s y m b o l F = m s y m b o l a = m ( r ¨ - r θ ˙ 2 ) s y m b o l r ^ + m ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ , symbol{F}=msymbol{a}=m(\ddot{r}-r\dot{\theta}^{2})\hat{symbol{r}}+m(r\ddot{% \theta}+2\dot{r}\dot{\theta})\hat{symbol\theta}\ ,
  46. s y m b o l F + m r θ ˙ 2 𝐫 ^ - m 2 r ˙ θ ˙ s y m b o l θ ^ = m s y m b o l a ~ = m r ¨ s y m b o l r ^ + m r θ ¨ s y m b o l θ ^ , symbol{F}+mr\dot{\theta}^{2}\hat{\mathbf{r}}-m2\dot{r}\dot{\theta}\hat{symbol% \theta}=m\tilde{symbol{a}}=m\ddot{r}\hat{symbol{r}}+mr\ddot{\theta}\hat{symbol% \theta}\ ,
  47. s y m b o l a ~ = r ¨ s y m b o l r ^ + r θ ¨ s y m b o l θ ^ , \tilde{symbol{a}}=\ddot{r}\hat{symbol{r}}+r\ddot{\theta}\hat{symbol\theta}\ ,
  48. θ ˙ = 0 \dot{\theta}=0
  49. θ ˙ 0 . \dot{\theta}^{\prime}\neq 0\ .
  50. r θ ˙ 2 r\dot{\theta}^{2}
  51. s y m b o l a = ( r ¨ - r θ ˙ 2 ) s y m b o l r ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ = 0 , symbola=\left(\ddot{r}-r{\dot{\theta}}^{2}\right)\hat{symbolr}+\left(r\ddot{% \theta}+2\dot{r}\dot{\theta}\right)\hat{symbol\theta}=0\ ,
  52. θ ˙ = 0 , θ ¨ = 0 \dot{\theta}=0,\ \ddot{\theta}=0
  53. r ¨ = 0 \ddot{r}=0
  54. r θ ˙ 2 r\dot{\theta}^{2}
  55. s y m b o l a = ( r ¨ - r θ ˙ ) 2 s y m b o l r ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ symbola^{\prime}=\left(\ddot{r}^{\prime}-r^{\prime}\dot{\theta}^{\prime}{}^{2}% \right)\hat{symbolr}^{\prime}+\left(r^{\prime}\ddot{\theta}^{\prime}+2\dot{r}^% {\prime}\dot{\theta}^{\prime}\right)\hat{symbol\theta}^{\prime}
  56. r ¨ = r θ ˙ . 2 \ddot{r}^{\prime}=r^{\prime}\dot{\theta}^{\prime}{}^{2}\ .
  57. r r^{\prime}
  58. θ ˙ \dot{\theta}^{\prime}
  59. s y m b o l a symbola^{\prime}
  60. r ¨ \ddot{r}^{\prime}
  61. r θ ˙ 2 r\dot{\theta}^{2}
  62. r ¨ \ddot{r}
  63. s y m b o l F θ ˙ = - s y m b o l ω × ( s y m b o l ω × r ) , symbol{F_{\dot{\theta}}}=-symbol{\omega\times}\left(symbol{\omega\times r}% \right)\ ,
  64. s y m b o l ω = θ ˙ s y m b o l k ^ , symbol\omega=\dot{\theta}symbol{\hat{k}}\ ,
  65. s y m b o l k ^ symbol{\hat{k}}
  66. θ ˙ \dot{\theta}
  67. r θ ˙ 2 r\dot{\theta}^{2}
  68. r θ ˙ 2 r\dot{\theta}^{2}
  69. r θ ˙ 2 r\dot{\theta}^{2}
  70. r ¨ \ddot{r}
  71. r θ ˙ 2 r\dot{\theta}^{2}
  72. r θ ˙ 2 r\dot{\theta}^{2}
  73. r ¨ - r θ ˙ 2 \ddot{r}-r\dot{\theta}^{2}
  74. θ = θ - Ω t . \theta^{\prime}=\theta-\Omega t\ .
  75. θ ˙ = θ ˙ - Ω . \dot{\theta}^{\prime}=\dot{\theta}-\Omega\ .
  76. d 2 𝐫 d t 2 = [ r ¨ - r ( θ ˙ + Ω ) 2 ] 𝐫 ^ + [ r θ ¨ + 2 r ˙ ( θ ˙ + Ω ) ] s y m b o l θ ^ \frac{d^{2}\mathbf{r}}{dt^{2}}=\left[\ddot{r}-r\left(\dot{\theta}^{\prime}+% \Omega\right)^{2}\right]\hat{\mathbf{r}}+\left[r\ddot{\theta}^{\prime}+2\dot{r% }\left(\dot{\theta}^{\prime}+\Omega\right)\right]\hat{symbol\theta}
  77. = ( r ¨ - r θ ˙ 2 ) 𝐫 ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ - ( 2 r Ω θ ˙ + r Ω 2 ) 𝐫 ^ + ( 2 r ˙ Ω ) s y m b o l θ ^ . =(\ddot{r}-r\dot{\theta}^{\prime 2})\hat{\mathbf{r}}+(r\ddot{\theta}^{\prime}+% 2\dot{r}\dot{\theta}^{\prime})\hat{symbol\theta}-\left(2r\Omega\dot{\theta}^{% \prime}+r\Omega^{2}\right)\hat{\mathbf{r}}+\left(2\dot{r}\Omega\right)\hat{% symbol\theta}\ .
  78. - ( 2 r Ω θ ˙ + r Ω 2 ) 𝐫 ^ + ( 2 r ˙ Ω ) s y m b o l θ ^ -\left(2r\Omega\dot{\theta}^{\prime}+r\Omega^{2}\right)\hat{\mathbf{r}}+\left(% 2\dot{r}\Omega\right)\hat{symbol\theta}
  79. 2 r Ω θ ˙ 2r\Omega\dot{\theta}^{\prime}
  80. r θ ˙ r\dot{\theta}^{\prime}
  81. - ( 2 r ˙ Ω ) s y m b o l θ ^ -\left(2\dot{r}\Omega\right)\hat{symbol\theta}
  82. s y m b o l F + m r θ ˙ 2 𝐫 ^ - m 2 r ˙ θ ˙ s y m b o l θ ^ + m ( 2 r Ω θ ˙ + r Ω 2 ) 𝐫 ^ - m ( 2 r ˙ Ω ) s y m b o l θ ^ symbol{F}+mr\dot{\theta}^{\prime 2}\hat{\mathbf{r}}-m2\dot{r}\dot{\theta}^{% \prime}\hat{symbol\theta}+m\left(2r\Omega\dot{\theta}^{\prime}+r\Omega^{2}% \right)\hat{\mathbf{r}}-m\left(2\dot{r}\Omega\right)\hat{symbol\theta}
  83. = m r ¨ 𝐫 ^ + m r θ ¨ s y m b o l θ ^ =m\ddot{r}\hat{\mathbf{r}}+mr\ddot{\theta}^{\prime}\ \hat{symbol\theta}
  84. = m s y m b o l a ~ =m\tilde{symbol{a}}
  85. s y m b o l a ~ \tilde{symbol{a}}
  86. ( m r θ ˙ 2 𝐫 ^ - m 2 r ˙ θ ˙ s y m b o l θ ^ ) (mr\dot{\theta}^{\prime 2}\hat{\mathbf{r}}-m2\dot{r}\dot{\theta}^{\prime}\hat{% symbol\theta})
  87. x = x ( q 1 , q 2 , q 3 ) ; x=x(q_{1},\ q_{2},\ q_{3})\ ;
  88. q 1 = q 1 ( x , y , z ) , \ q_{1}=q_{1}(x,\ y,\ z)\ ,
  89. d s 2 = k = 1 d ( h k ) 2 ( d q k ) 2 , ds^{2}=\sum_{k=1}^{d}\left(h_{k}\right)^{2}\left(dq_{k}\right)^{2}\ ,
  90. s y m b o l r = k = 1 d q k s y m b o l e k symbol{r}=\sum_{k=1}^{d}q_{k}\ symbol{e_{k}}\,
  91. s y m b o l v = k = 1 d v k s y m b o l e k symbol{v}=\sum_{k=1}^{d}v_{k}\ symbol{e_{k}}\,
  92. = d d t s y m b o l r = k = 1 d q ˙ k s y m b o l e k + k = 1 d q k s y m b o l e k ˙ =\frac{d}{dt}symbol{r}=\sum_{k=1}^{d}\dot{q}_{k}\ symbol{e_{k}}+\sum_{k=1}^{d}% q_{k}\ \dot{symbol{e_{k}}}\,
  93. q 2 s y m b o l e 1 = - s y m b o l e 2 1 h 2 h 1 q 2 - s y m b o l e 3 1 h 3 h 1 q 3 , \frac{\partial}{\partial q_{2}}symbol{e_{1}}=-symbol{e}_{2}\frac{1}{h_{2}}% \frac{\partial h_{1}}{\partial q_{2}}-symbol{e}_{3}\frac{1}{h_{3}}\frac{% \partial h_{1}}{\partial q_{3}}\ ,
  94. s y m b o l e j q k = n = 1 d Γ n k j s y m b o l e n , \frac{\partial symbol{e_{j}}}{\partial q_{k}}=\sum_{n=1}^{d}{\Gamma^{n}}_{kj}% symbol{e_{n}}\ ,
  95. Γ i i i = { i i i } = 1 h i h i q i ; {\Gamma^{i}}_{ii}=\begin{Bmatrix}\,i\\ i\,\,i\end{Bmatrix}=\frac{1}{h_{i}}\frac{\partial h_{i}}{\partial q_{i}}\!\ ;
  96. Γ i i j = { i i j } = 1 h i h i q j = { i j i } ; {\Gamma^{i}}_{ij}=\ \begin{Bmatrix}\,i\\ i\,\,j\end{Bmatrix}=\frac{1}{h_{i}}\frac{\partial h_{i}}{\partial q_{j}}=% \begin{Bmatrix}\,i\\ j\,\,i\end{Bmatrix}\!\ ;
  97. Γ j i i = { j i i } = - h i h j 2 h i q j , {\Gamma^{j}}_{ii}=\begin{Bmatrix}\,j\\ i\,\,i\end{Bmatrix}=-\frac{h_{i}}{{h_{j}}^{2}}\frac{\partial h_{i}}{\partial q% _{j}}\ ,
  98. s y m b o l e j ˙ = k = 1 d q k s y m b o l e j q ˙ k \dot{symbol{e_{j}}}=\sum_{k=1}^{d}\frac{\partial}{\partial q_{k}}symbol{e_{j}}% \dot{q}_{k}
  99. = k = 1 d i = 1 d Γ k i j q ˙ i s y m b o l e k , =\sum_{k=1}^{d}\sum_{i=1}^{d}{\Gamma^{k}}_{ij}\dot{q}_{i}symbol{e_{k}}\ ,
  100. s y m b o l v = d d t s y m b o l r = k = 1 d q ˙ k s y m b o l e k + k = 1 d q k s y m b o l e k ˙ symbol{v}=\frac{d}{dt}symbol{r}=\sum_{k=1}^{d}\dot{q}_{k}\ symbol{e_{k}}+\sum_% {k=1}^{d}q_{k}\ \dot{symbol{e_{k}}}
  101. = k = 1 d q ˙ k s y m b o l e k + j = 1 d q j s y m b o l e j ˙ , =\sum_{k=1}^{d}\dot{q}_{k}\ symbol{e_{k}}+\sum_{j=1}^{d}q_{j}\ \dot{symbol{e_{% j}}},
  102. = k = 1 d q ˙ k s y m b o l e k + k = 1 d j = 1 d i = 1 d q j Γ k i j s y m b o l e k q ˙ i =\sum_{k=1}^{d}\dot{q}_{k}\ symbol{e_{k}}+\sum_{k=1}^{d}\sum_{j=1}^{d}\sum_{i=% 1}^{d}q_{j}\ {\Gamma^{k}}_{ij}symbol{e_{k}}\dot{q}_{i}
  103. = k = 1 d ( q ˙ k + j = 1 d i = 1 d q j Γ k i j q ˙ i ) s y m b o l e k , =\sum_{k=1}^{d}\left(\dot{q}_{k}\ +\sum_{j=1}^{d}\sum_{i=1}^{d}q_{j}\ {\Gamma^% {k}}_{ij}\dot{q}_{i}\right)symbol{e_{k}}\ ,
  104. s y m b o l a = d d t s y m b o l v = k = 1 d v ˙ k s y m b o l e k + k = 1 d v k s y m b o l e k ˙ . symbol{a}=\frac{d}{dt}symbol{v}=\sum_{k=1}^{d}\dot{v}_{k}\ symbol{e_{k}}+\sum_% {k=1}^{d}v_{k}\ \dot{symbol{e_{k}}}\ .
  105. = k = 1 d ( v ˙ k + j = 1 d i = 1 d v j Γ k i j q ˙ i ) s y m b o l e k . =\sum_{k=1}^{d}\left(\dot{v}_{k}\ +\sum_{j=1}^{d}\sum_{i=1}^{d}v_{j}{\Gamma^{k% }}_{ij}\dot{q}_{i}\right)symbol{e_{k}}\ .
  106. s y m b o l F = m s y m b o l a = m k = 1 d ( v ˙ k + j = 1 d i = 1 d v j Γ k i j q ˙ i ) s y m b o l e k , symbol{F}=msymbol{a}=m\sum_{k=1}^{d}\left(\dot{v}_{k}\ +\sum_{j=1}^{d}\sum_{i=% 1}^{d}v_{j}{\Gamma^{k}}_{ij}\dot{q}_{i}\right)symbol{e_{k}}\ ,
  107. s y m b o l F - m j = 1 d i = 1 d v j Γ k i j q ˙ i s y m b o l e k = m s y m b o l a ~ , symbol{F}-m\sum_{j=1}^{d}\sum_{i=1}^{d}v_{j}{\Gamma^{k}}_{ij}\dot{q}_{i}symbol% {e_{k}}=m\tilde{symbol{a}}\ ,
  108. s y m b o l a ~ \tilde{symbol{a}}
  109. s y m b o l a ~ = k = 1 d v ˙ k s y m b o l e k . \tilde{symbol{a}}=\sum_{k=1}^{d}\dot{v}_{k}symbol{e_{k}}\ .

Median_algebra.html

  1. x , y , z \langle x,y,z\rangle
  2. x , y , y = y \langle x,y,y\rangle=y
  3. x , y , z = z , x , y \langle x,y,z\rangle=\langle z,x,y\rangle
  4. x , y , z = x , z , y \langle x,y,z\rangle=\langle x,z,y\rangle
  5. x , w , y , w , z = x , w , y , w , z \langle\langle x,w,y\rangle,w,z\rangle=\langle x,w,\langle y,w,z\rangle\rangle
  6. x , y , y = y \langle x,y,y\rangle=y
  7. u , v , u , w , x = u , x , w , u , v \langle u,v,\langle u,w,x\rangle\rangle=\langle u,x,\langle w,u,v\rangle\rangle
  8. x , y , z = ( x y ) ( y z ) ( z x ) \langle x,y,z\rangle=(x\vee y)\wedge(y\vee z)\wedge(z\vee x)

Median_graph.html

  1. ( x 11 x 12 ) ( x 21 x 22 ) ( x n 1 x n 2 ) . (x_{11}\lor x_{12})\land(x_{21}\lor x_{22})\land\cdots\land(x_{n1}\lor x_{n2})% \land\cdots.

MediaWiki:Gadget-charinsert-core.js.html

  1. + +

Megagon.html

  1. A = 250000 a 2 cot π 1000000 . A=250000a^{2}\cot\frac{\pi}{1000000}.
  2. 2000000 sin π 1000000 , 2000000\sin\frac{\pi}{1000000},

Meixner_polynomials.html

  1. M n ( x , β , γ ) = k = 0 n ( - 1 ) k ( n k ) ( x k ) k ! ( x - β ) n - k γ - k M_{n}(x,\beta,\gamma)=\sum_{k=0}^{n}(-1)^{k}{n\choose k}{x\choose k}k!(x-\beta% )_{n-k}\gamma^{-k}

Menachem_Magidor.html

  1. ω \aleph_{\omega}
  2. 2 ω = ω + 2 2^{\aleph_{\omega}}=\aleph_{\omega+2}
  3. ω \aleph_{\omega}
  4. ω \aleph_{\omega}
  5. L L

Menzerath's_law.html

  1. y = a x - b e - c x y=a\cdot x^{-b}\cdot e^{-cx}
  2. y y
  3. x x
  4. a a
  5. b b
  6. c c
  7. x x
  8. y y
  9. y y
  10. x x

Meridian_arc.html

  1. φ \varphi
  2. a a
  3. b b
  4. f f
  5. e e
  6. n n
  7. f \displaystyle f
  8. M ( φ ) = a ( 1 - e 2 ) ( 1 - e 2 sin 2 φ ) 3 / 2 , M(\varphi)=\frac{a(1-e^{2})}{\bigl(1-e^{2}\sin^{2}\varphi\bigr)^{3/2}},
  9. d m = M ( φ ) d φ dm=M(\varphi)\,d\varphi
  10. φ \varphi
  11. φ \varphi
  12. m ( φ ) = 0 φ M ( φ ) d φ = a ( 1 - e 2 ) 0 φ ( 1 - e 2 sin 2 φ ) - 3 / 2 d φ . \begin{aligned}\displaystyle m(\varphi)&\displaystyle=\int_{0}^{\varphi}M(% \varphi)\,d\varphi=a(1-e^{2})\int_{0}^{\varphi}\bigl(1-e^{2}\sin^{2}\varphi% \bigr)^{-3/2}\,d\varphi.\end{aligned}
  13. m ( φ ) = b 0 β 1 + e 2 sin 2 β d β , m(\varphi)=b\int_{0}^{\beta}\sqrt{1+e^{\prime 2}\sin^{2}\beta}\,d\beta,
  14. tan β = ( 1 - f ) tan φ \tan\beta=(1-f)\tan\varphi
  15. e 2 = e 2 / ( 1 - e 2 ) e^{\prime 2}=e^{2}/(1-e^{2})
  16. m p = m ( π / 2 ) . m_{p}=m(\pi/2).\,
  17. [ - π / 2 , π / 2 ] [-\pi/2,\pi/2]
  18. φ \varphi
  19. β \beta
  20. μ \mu
  21. m ( φ ) = a ( 1 - e 2 ) Π ( φ , e 2 , e ) . m(\varphi)=a\big(1-e^{2}\big)\,\Pi(\varphi,e^{2},e).
  22. m ( φ ) = a ( E ( φ , e ) - e 2 sin φ cos φ 1 - e 2 sin 2 φ ) = a ( E ( φ , e ) + d 2 d φ 2 E ( φ , e ) ) = b E ( β , i e ) . \begin{aligned}\displaystyle m(\varphi)&\displaystyle=a\biggl(E(\varphi,e)-% \frac{e^{2}\sin\varphi\cos\varphi}{\sqrt{1-e^{2}\sin^{2}\varphi}}\biggr)\\ &\displaystyle=a\biggl(E(\varphi,e)+\frac{d^{2}}{d\varphi^{2}}E(\varphi,e)% \biggr)\\ &\displaystyle=bE(\beta,ie^{\prime}).\end{aligned}
  23. m p = a E ( e ) = b E ( i e ) . m_{p}=aE(e)=bE(ie^{\prime}).
  24. m ( φ ) = b 2 a ( D 0 φ + D 2 sin 2 φ + D 4 sin 4 φ + D 6 sin 6 φ + D 8 sin 8 φ + ) , m(\varphi)=\frac{b^{2}}{a}\bigl(D_{0}\varphi+D_{2}\sin 2\varphi+D_{4}\sin 4% \varphi+D_{6}\sin 6\varphi+D_{8}\sin 8\varphi+\cdots\bigr),
  25. D 0 = 1 + 3 4 e 2 + 45 64 e 4 + 175 256 e 6 + 11025 16384 e 8 + , D 2 = - 3 8 e 2 - 15 32 e 4 - 525 1024 e 6 - 2205 4096 e 8 - , D 4 = 15 256 e 4 + 105 1024 e 6 + 2205 16384 e 8 + , D 6 = - 35 3072 e 6 - 105 4096 e 8 - , D 8 = 315 131072 e 8 + . \begin{aligned}\displaystyle D_{0}&\displaystyle=\textstyle 1+\frac{3}{4}e^{2}% +\frac{45}{64}e^{4}+\frac{175}{256}e^{6}+\frac{11025}{16384}e^{8}+\cdots,\\ \displaystyle D_{2}&\displaystyle=\textstyle-\frac{3}{8}e^{2}-\frac{15}{32}e^{% 4}-\frac{525}{1024}e^{6}-\frac{2205}{4096}e^{8}-\cdots,\\ \displaystyle D_{4}&\displaystyle=\textstyle\frac{15}{256}e^{4}+\frac{105}{102% 4}e^{6}+\frac{2205}{16384}e^{8}+\cdots,\\ \displaystyle D_{6}&\displaystyle=\textstyle-\frac{35}{3072}e^{6}-\frac{105}{4% 096}e^{8}-\cdots,\\ \displaystyle D_{8}&\displaystyle=\textstyle\frac{315}{131072}e^{8}+\cdots.% \end{aligned}
  26. n n
  27. e 2 = 4 n ( 1 + n ) 2 . e^{2}=\frac{4n}{(1+n)^{2}}.
  28. m ( φ ) = a + b 2 ( H 0 φ + H 2 sin 2 φ + H 4 sin 4 φ + H 6 sin 6 φ + H 8 sin 8 φ + ) , m(\varphi)=\frac{a+b}{2}\bigl(H_{0}\varphi+H_{2}\sin 2\varphi+H_{4}\sin 4% \varphi+H_{6}\sin 6\varphi+H_{8}\sin 8\varphi+\cdots\bigr),
  29. H 0 = 1 + 1 4 n 2 + 1 64 n 4 + , H 2 = - 3 2 n + 3 16 n 3 + , H 6 = - 35 48 n 3 + , H 4 = 15 16 n 2 - 15 64 n 4 - , H 8 = 315 512 n 4 - . \begin{aligned}\displaystyle H_{0}&\displaystyle=\textstyle 1+\frac{1}{4}n^{2}% +\frac{1}{64}n^{4}+\cdots,\\ \displaystyle H_{2}&\displaystyle=\textstyle-\frac{3}{2}n+\frac{3}{16}n^{3}+% \cdots,&\displaystyle H_{6}&\displaystyle=\textstyle-\frac{35}{48}n^{3}+\cdots% ,\\ \displaystyle H_{4}&\displaystyle=\textstyle\frac{15}{16}n^{2}-\frac{15}{64}n^% {4}-\cdots,&\displaystyle H_{8}&\displaystyle=\textstyle\frac{315}{512}n^{4}-% \cdots.\end{aligned}
  30. n n
  31. a a
  32. b b
  33. 1 2 ( a + b ) \frac{1}{2}(a+b)
  34. H 2 k H_{2k}
  35. a a
  36. b b
  37. 1 2 ( a + b ) = a / ( 1 + n ) = a ( 1 - n + n 2 - n 3 + n 4 - ) \frac{1}{2}(a+b)=a/(1+n)=a(1-n+n^{2}-n^{3}+n^{4}-\cdots)
  38. n n
  39. β \beta
  40. m ( φ ) = a + b 2 ( B 0 β + B 2 sin 2 β + B 4 sin 4 β + B 6 sin 6 β + B 8 sin 8 β + ) , m(\varphi)=\frac{a+b}{2}\bigl(B_{0}\beta+B_{2}\sin 2\beta+B_{4}\sin 4\beta+B_{% 6}\sin 6\beta+B_{8}\sin 8\beta+\cdots\bigr),
  41. B 0 = 1 + 1 4 n 2 + 1 64 n 4 + = H 0 , B 2 = - 1 2 n + 1 16 n 3 + , B 6 = - 1 48 n 3 + , B 4 = - 1 16 n 2 + 1 64 n 4 + , B 8 = - 5 512 n 4 + . \begin{aligned}\displaystyle B_{0}&\displaystyle=\textstyle 1+\frac{1}{4}n^{2}% +\frac{1}{64}n^{4}+\cdots=H_{0},\\ \displaystyle B_{2}&\displaystyle=\textstyle-\frac{1}{2}n+\frac{1}{16}n^{3}+% \cdots,&\displaystyle B_{6}&\displaystyle=\textstyle-\frac{1}{48}n^{3}+\cdots,% \\ \displaystyle B_{4}&\displaystyle=\textstyle-\frac{1}{16}n^{2}+\frac{1}{64}n^{% 4}+\cdots,&\displaystyle B_{8}&\displaystyle=\textstyle-\frac{5}{512}n^{4}+% \cdots.\end{aligned}
  42. m ( φ ) = a + b 2 ( B 0 φ - B 2 sin 2 φ + B 4 sin 4 φ - B 6 sin 6 φ + B 8 sin 8 φ - - 2 n sin 2 φ 1 + 2 n cos 2 φ + n 2 ) . \begin{aligned}\displaystyle m(\varphi)&\displaystyle=\frac{a+b}{2}\biggl(B_{0% }\varphi-B_{2}\sin 2\varphi+B_{4}\sin 4\varphi-B_{6}\sin 6\varphi+B_{8}\sin 8% \varphi-\cdots\\ &\displaystyle\qquad-\frac{2n\sin 2\varphi}{\sqrt{1+2n\cos 2\varphi+n^{2}}}% \biggr).\end{aligned}
  43. B 2 k = { c 0 , if k = 0 , c k / k , if k > 0 , B_{2k}=\begin{cases}c_{0},&\,\text{if }k=0,\\ c_{k}/k,&\,\text{if }k>0,\end{cases}
  44. c k = j = 0 ( 2 j - 3 ) ! ! ( 2 j + 2 k - 3 ) ! ! ( 2 j ) ! ! ( 2 j + 2 k ) ! ! n k + 2 j c_{k}=\sum_{j=0}^{\infty}\frac{(2j-3)!!\,(2j+2k-3)!!}{(2j)!!\,(2j+2k)!!}n^{k+2j}
  45. k ! ! k!!
  46. ( - 1 ) ! ! = 1 (-1)!!=1
  47. ( - 3 ) ! ! = - 1 (-3)!!=-1
  48. H 2 k = ( - 1 ) k ( 1 - 2 k ) ( 1 + 2 k ) B 2 k . H_{2k}=(-1)^{k}(1-2k)(1+2k)B_{2k}.
  49. ( 1 - 2 k ) ( 1 + 2 k ) (1-2k)(1+2k)
  50. φ \varphi
  51. β \beta
  52. m p = π ( a + b ) 4 c 0 = π ( a + b ) 4 j = 0 ( ( 2 j - 3 ) ! ! ( 2 j ) ! ! ) 2 n 2 j , m_{p}=\frac{\pi(a+b)}{4}c_{0}=\frac{\pi(a+b)}{4}\sum_{j=0}^{\infty}\biggl(% \frac{(2j-3)!!}{(2j)!!}\biggr)^{2}n^{2j},
  53. m ( φ 1 ) - m ( φ 2 ) m(\varphi_{1})-m(\varphi_{2})
  54. m ( φ ) = ( 111 132.95255 φ ( ) - 16 038.509 sin 2 φ + 16.833 sin 4 φ - 0.022 sin 6 φ + 0.00003 sin 8 φ ) metres = ( 111 132.95255 β ( ) - 5 346.170 sin 2 β - 1.122 sin 4 β - 0.001 sin 6 β - 0.5 × 10 - 6 sin 8 β ) metres \begin{aligned}\displaystyle m(\varphi)&\displaystyle=\bigl(111\,132.95255\,% \varphi^{(\circ)}-16\,038.509\,\sin 2\varphi+16.833\,\sin 4\varphi-0.022\,\sin 6% \varphi+0.00003\,\sin 8\varphi\bigr)\,\mathrm{metres}\\ &\displaystyle=\bigl(111\,132.95255\,\beta^{(\circ)}-5\,346.170\,\sin 2\beta-1% .122\,\sin 4\beta-0.001\,\sin 6\beta-0.5\times 10^{-6}\,\sin 8\beta\bigr)\,% \mathrm{metres}\end{aligned}
  55. φ ( ) = φ / 1 \varphi^{(\circ)}=\varphi/1^{\circ}
  56. φ \varphi
  57. β ( ) \beta^{(\circ)}
  58. m p = π ( a + b ) 4 c 0 = 10 001 965.729 metres . m_{p}=\frac{\pi(a+b)}{4}c_{0}=10\,001\,965.729\,\mathrm{metres}.
  59. 4 m p = 2 π ( a + b ) c 0 4m_{p}=2\pi(a+b)c_{0}
  60. 1 2 ( a + b ) c 0 \frac{1}{2}(a+b)c_{0}
  61. 6 367 449.146 m 6\,367\,449.146\,\mathrm{m}
  62. φ 1 \varphi_{1}
  63. φ 2 \varphi_{2}
  64. m ( φ 1 ) - m ( φ 2 ) m(\varphi_{1})-m(\varphi_{2})
  65. Δ m \Delta m
  66. φ \varphi
  67. Δ m = ( 111 133 - 560 cos 2 φ ) metres . \Delta m=(111\,133-560\cos 2\varphi)\,\mathrm{metres}.
  68. m m
  69. φ \varphi
  70. φ i + 1 = φ i - m ( φ i ) - m M ( φ i ) , \varphi_{i+1}=\varphi_{i}-\frac{m(\varphi_{i})-m}{M(\varphi_{i})},
  71. φ 0 = μ \varphi_{0}=\mu
  72. μ = π 2 m m p \mu=\frac{\pi}{2}\frac{m}{m_{p}}
  73. m ( φ ) m(\varphi)
  74. M ( φ ) M(\varphi)
  75. φ = μ + H 2 sin 2 μ + H 4 sin 4 μ + H 6 sin 6 μ + H 8 sin 8 μ + \varphi=\mu+H^{\prime}_{2}\sin 2\mu+H^{\prime}_{4}\sin 4\mu+H^{\prime}_{6}\sin 6% \mu+H^{\prime}_{8}\sin 8\mu+\cdots
  76. H 2 = 3 2 n - 27 32 n 3 + , H 6 = 151 96 n 3 + , H 4 = 21 16 n 2 - 55 32 n 4 + , H 8 = 1097 512 n 4 + . \begin{aligned}\displaystyle H^{\prime}_{2}&\displaystyle=\textstyle\frac{3}{2% }n-\frac{27}{32}n^{3}+\cdots,&\displaystyle H^{\prime}_{6}&\displaystyle=% \textstyle\frac{151}{96}n^{3}+\cdots,\\ \displaystyle H^{\prime}_{4}&\displaystyle=\textstyle\frac{21}{16}n^{2}-\frac{% 55}{32}n^{4}+\cdots,&\displaystyle H^{\prime}_{8}&\displaystyle=\textstyle% \frac{1097}{512}n^{4}+\cdots.\end{aligned}
  77. m m
  78. β \beta
  79. β = μ + B 2 sin 2 μ + B 4 sin 4 μ + B 6 sin 6 μ + B 8 sin 8 μ + , \beta=\mu+B^{\prime}_{2}\sin 2\mu+B^{\prime}_{4}\sin 4\mu+B^{\prime}_{6}\sin 6% \mu+B^{\prime}_{8}\sin 8\mu+\cdots,
  80. B 2 = 1 2 n - 9 32 n 3 + , B 6 = 29 96 n 3 - , B 4 = 5 16 n 2 - 37 96 n 4 + , B 8 = 539 1536 n 4 - . \begin{aligned}\displaystyle B^{\prime}_{2}&\displaystyle=\textstyle\frac{1}{2% }n-\frac{9}{32}n^{3}+\cdots,&\displaystyle B^{\prime}_{6}&\displaystyle=% \textstyle\frac{29}{96}n^{3}-\cdots,\\ \displaystyle B^{\prime}_{4}&\displaystyle=\textstyle\frac{5}{16}n^{2}-\frac{3% 7}{96}n^{4}+\cdots,&\displaystyle B^{\prime}_{8}&\displaystyle=\textstyle\frac% {539}{1536}n^{4}-\cdots.\end{aligned}
  81. m m
  82. β \beta
  83. m m
  84. s s
  85. β \beta
  86. σ \sigma
  87. ϵ \epsilon
  88. n n
  89. τ \tau
  90. μ \mu

Merkle_signature_scheme.html

  1. p u b pub
  2. N = 2 n N=2^{n}
  3. p u b pub
  4. X i X_{i}
  5. Y i Y_{i}
  6. 2 n 2^{n}
  7. X i X_{i}
  8. 1 i 2 n 1\leq i\leq 2^{n}
  9. h i = X i = H ( Y i ) h_{i}=X_{i}=H(Y_{i})
  10. h i h_{i}
  11. a i , j a_{i,j}
  12. i i
  13. j j
  14. i = 0 i=0
  15. i = n i=n
  16. a i , 0 a_{i,0}
  17. i i
  18. h i h_{i}
  19. h i = a 0 , i h_{i}=a_{0,i}
  20. a 1 , 0 = H ( a 0 , 0 | | a 0 , 1 ) a_{1,0}=H(a_{0,0}||a_{0,1})
  21. a 2 , 0 = H ( a 1 , 0 | | a 1 , 1 ) a_{2,0}=H(a_{1,0}||a_{1,1})
  22. 2 n 2^{n}
  23. 2 n + 1 - 1 2^{n+1}-1
  24. a n , 0 a_{n,0}
  25. p u b pub
  26. M M
  27. M M
  28. s i g sig^{\prime}
  29. ( X i , Y i , ) (X_{i},Y_{i},)
  30. s i g sig^{\prime}
  31. X i X_{i}
  32. a 0 , i = H ( X i ) a_{0,i}=H(X_{i})
  33. a 0 , i a_{0,i}
  34. A A
  35. A A
  36. n + 1 n+1
  37. A 0 , A n A_{0},...A_{n}
  38. A 0 = a 0 , i A_{0}=a_{0,i}
  39. A n = a n , 0 = p u b A_{n}=a_{n,0}=pub
  40. A A
  41. A 1 , , A n A_{1},...,A_{n}
  42. A i A_{i}
  43. A i + 1 A_{i+1}
  44. A i + 1 A_{i+1}
  45. A A
  46. A i + 1 A_{i+1}
  47. A i A_{i}
  48. a u t h i auth_{i}
  49. A i + 1 = H ( A i | | a u t h i ) A_{i+1}=H(A_{i}||auth_{i})
  50. n n
  51. a u t h 0 , , a u t h n - 1 auth_{0},...,auth_{n-1}
  52. A A
  53. a u t h 0 , , a u t h n - 1 auth_{0},...,auth_{n-1}
  54. s i g sig^{\prime}
  55. M M
  56. s i g = ( s i g || a u t h 0 || a u t h 1 || || a u t h n - 1 ) sig=(sig^{\prime}||auth_{0}||auth_{1}||...||auth_{n-1})
  57. p u b pub
  58. M M
  59. s i g = ( s i g || a u t h 0 || a u t h 1 || || a u t h n - 1 ) sig=(sig^{\prime}||auth_{0}||auth_{1}||...||auth_{n-1})
  60. s i g sig^{\prime}
  61. M M
  62. s i g sig^{\prime}
  63. M M
  64. A 0 = H ( X i ) A_{0}=H(X_{i})
  65. j = 1 , . . , n - 1 j=1,..,n-1
  66. A j A_{j}
  67. A A
  68. A j = H ( A j - 1 | | a u t h j - 1 ) A_{j}=H(A_{j-1}||auth_{j-1})
  69. A n A_{n}
  70. p u b pub

Methanation.html

  1. CO + 3 H 2 CH 4 + H 2 O \mathrm{\,CO+3\,H_{2}\rightarrow CH_{4}+\,H_{2}O}

Meyer's_law.html

  1. P = k d n P\,=\,kd^{n}
  2. P = k 1 d 1 n 1 = k 2 d 2 n 2 = k 3 d 3 n 3 = P=k_{1}d_{1}^{n_{1}}=k_{2}d_{2}^{n_{2}}=k_{3}d_{3}^{n_{3}}=...
  3. P = 1.854 k d n - 2 P\,=\,1.854kd^{n-2}

Ménage_problem.html

  1. M n = 2 n ! k = 0 n ( - 1 ) k 2 n 2 n - k ( 2 n - k k ) ( n - k ) ! . M_{n}=2\cdot n!\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k}{2n-k\choose k}(n-k)!.
  2. A n = k = 0 n ( - 1 ) k 2 n 2 n - k ( 2 n - k k ) ( n - k ) ! A_{n}=\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k}{2n-k\choose k}(n-k)!
  3. A n = n A n - 1 + n n - 2 A n - 2 + 4 ( - 1 ) n - 1 n - 2 A_{n}=nA_{n-1}+\frac{n}{n-2}A_{n-2}+\frac{4(-1)^{n-1}}{n-2}
  4. A n = n A n - 1 + 2 A n - 2 - ( n - 4 ) A n - 3 - A n - 4 , \displaystyle A_{n}=nA_{n-1}+2A_{n-2}-(n-4)A_{n-3}-A_{n-4},

MHV_amplitudes.html

  1. 𝒜 ( 1 + n + ) = 0 , \mathcal{A}(1^{+}\cdots n^{+})=0,
  2. 𝒜 ( 1 + i - n + ) = 0. \mathcal{A}(1^{+}\cdots i^{-}\cdots n^{+})=0.
  3. 𝒜 ( 1 + i - j - n + ) = i ( - g ) n - 2 i j 4 1 2 2 3 ( n - 1 ) n n 1 \mathcal{A}(1^{+}\cdots i^{-}\cdots j^{-}\cdots n^{+})=i(-g)^{n-2}\frac{% \langle i\;j\rangle^{4}}{\langle 1\;2\rangle\langle 2\;3\rangle\cdots\langle(n% -1)\;n\rangle\langle n\;1\rangle}
  4. ( + + - ) (++-)
  5. ( + + + + ) (++++)
  6. L [ A ] = L + - [ A ] + L - + + [ A ] + L - - + + [ A ] . L[A]=L^{+-}[A]+L^{-++}[A]+L^{--++}[A].
  7. L + - [ A ] + L + + - [ A ] = L + - [ B ] . L^{+-}[A]+L^{++-}[A]=L^{+-}[B].
  8. L [ B ] = L + - [ B ] + L - - + [ B ] + L - - + + [ B ] + L - - + + + [ B ] + L[B]=L^{+-}[B]+L^{--+}[B]+L^{--++}[B]+L^{--+++}[B]+\cdots

Michael_J._Hopkins.html

  1. A A_{\infty}
  2. E E_{\infty}

Michell_solution.html

  1. r , θ r,\theta\,
  2. θ \theta\,
  3. φ = A 0 r 2 + B 0 r 2 ln ( r ) + C 0 ln ( r ) + D 0 θ + ( A 1 r + B 1 r - 1 + B 1 r θ + C 1 r 3 + D 1 r ln ( r ) ) cos θ + ( E 1 r + F 1 r - 1 + F 1 r θ + G 1 r 3 + H 1 r ln ( r ) ) sin θ + n = 2 ( A n r n + B n r - n + C n r n + 2 + D n r - n + 2 ) cos ( n θ ) + n = 2 ( E n r n + F n r - n + G n r n + 2 + H n r - n + 2 ) sin ( n θ ) \begin{aligned}\displaystyle\varphi&\displaystyle=A_{0}~{}r^{2}+B_{0}~{}r^{2}~% {}\ln(r)+C_{0}~{}\ln(r)+D_{0}~{}\theta\\ &\displaystyle+\left(A_{1}~{}r+B_{1}~{}r^{-1}+B_{1}^{{}^{\prime}}~{}r~{}\theta% +C_{1}~{}r^{3}+D_{1}~{}r~{}\ln(r)\right)\cos\theta\\ &\displaystyle+\left(E_{1}~{}r+F_{1}~{}r^{-1}+F_{1}^{{}^{\prime}}~{}r~{}\theta% +G_{1}~{}r^{3}+H_{1}~{}r~{}\ln(r)\right)\sin\theta\\ &\displaystyle+\sum_{n=2}^{\infty}\left(A_{n}~{}r^{n}+B_{n}~{}r^{-n}+C_{n}~{}r% ^{n+2}+D_{n}~{}r^{-n+2}\right)\cos(n\theta)\\ &\displaystyle+\sum_{n=2}^{\infty}\left(E_{n}~{}r^{n}+F_{n}~{}r^{-n}+G_{n}~{}r% ^{n+2}+H_{n}~{}r^{-n+2}\right)\sin(n\theta)\end{aligned}
  4. A 1 r cos θ A_{1}~{}r~{}\cos\theta\,
  5. E 1 r sin θ E_{1}~{}r~{}\sin\theta\,
  6. φ \varphi
  7. σ r r \sigma_{rr}\,
  8. σ r θ \sigma_{r\theta}\,
  9. σ θ θ \sigma_{\theta\theta}\,
  10. r 2 r^{2}\,
  11. 2 2
  12. 0
  13. 2 2
  14. r 2 ln r r^{2}~{}\ln r
  15. 2 ln r + 1 2~{}\ln r+1
  16. 0
  17. 2 ln r + 3 2~{}\ln r+3
  18. ln r \ln r\,
  19. r - 2 r^{-2}\,
  20. 0
  21. - r - 2 -r^{-2}\,
  22. θ \theta\,
  23. 0
  24. r - 2 r^{-2}\,
  25. 0
  26. r 3 cos θ r^{3}~{}\cos\theta\,
  27. 2 r cos θ 2~{}r~{}\cos\theta\,
  28. 2 r sin θ 2~{}r~{}\sin\theta\,
  29. 6 r cos θ 6~{}r~{}\cos\theta\,
  30. r θ cos θ r\theta~{}\cos\theta\,
  31. 2 r - 1 sin θ 2~{}r^{-1}~{}\sin\theta\,
  32. 0
  33. 0
  34. r ln r cos θ r~{}\ln r~{}\cos\theta\,
  35. r - 1 cos θ r^{-1}~{}\cos\theta\,
  36. r - 1 sin θ r^{-1}~{}\sin\theta\,
  37. r - 1 cos θ r^{-1}~{}\cos\theta\,
  38. r - 1 cos θ r^{-1}~{}\cos\theta\,
  39. - 2 r - 3 cos θ -2~{}r^{-3}~{}\cos\theta\,
  40. - 2 r - 3 sin θ -2~{}r^{-3}~{}\sin\theta\,
  41. 2 r - 3 cos θ 2~{}r^{-3}~{}\cos\theta\,
  42. r 3 sin θ r^{3}~{}\sin\theta\,
  43. 2 r sin θ 2~{}r~{}\sin\theta\,
  44. - 2 r cos θ -2~{}r~{}\cos\theta\,
  45. 6 r sin θ 6~{}r~{}\sin\theta\,
  46. r θ sin θ r\theta~{}\sin\theta\,
  47. 2 r - 1 cos θ 2~{}r^{-1}~{}\cos\theta\,
  48. 0
  49. 0
  50. r ln r sin θ r~{}\ln r~{}\sin\theta\,
  51. r - 1 sin θ r^{-1}~{}\sin\theta\,
  52. - r - 1 cos θ -r^{-1}~{}\cos\theta\,
  53. r - 1 sin θ r^{-1}~{}\sin\theta\,
  54. r - 1 sin θ r^{-1}~{}\sin\theta\,
  55. - 2 r - 3 sin θ -2~{}r^{-3}~{}\sin\theta\,
  56. 2 r - 3 cos θ 2~{}r^{-3}~{}\cos\theta\,
  57. 2 r - 3 sin θ 2~{}r^{-3}~{}\sin\theta\,
  58. r n + 2 cos ( n θ ) r^{n+2}~{}\cos(n\theta)\,
  59. - ( n + 1 ) ( n - 2 ) r n cos ( n θ ) -(n+1)(n-2)~{}r^{n}~{}\cos(n\theta)\,
  60. n ( n + 1 ) r n sin ( n θ ) n(n+1)~{}r^{n}~{}\sin(n\theta)\,
  61. ( n + 1 ) ( n + 2 ) r n cos ( n θ (n+1)(n+2)~{}r^{n}~{}\cos(n\theta\,
  62. r - n + 2 cos ( n θ ) r^{-n+2}~{}\cos(n\theta)\,
  63. - ( n + 2 ) ( n - 1 ) r - n cos ( n θ ) -(n+2)(n-1)~{}r^{-n}~{}\cos(n\theta)\,
  64. - n ( n - 1 ) r - n sin ( n θ ) -n(n-1)~{}r^{-n}~{}\sin(n\theta)\,
  65. ( n - 1 ) ( n - 2 ) r - n cos ( n θ ) (n-1)(n-2)~{}r^{-n}~{}\cos(n\theta)
  66. r n cos ( n θ ) r^{n}~{}\cos(n\theta)\,
  67. - n ( n - 1 ) r n - 2 cos ( n θ ) -n(n-1)~{}r^{n-2}~{}\cos(n\theta)\,
  68. n ( n - 1 ) r n - 2 sin ( n θ ) n(n-1)~{}r^{n-2}~{}\sin(n\theta)\,
  69. n ( n - 1 ) r n - 2 cos ( n θ ) n(n-1)~{}r^{n-2}~{}\cos(n\theta)\,
  70. r - n cos ( n θ ) r^{-n}~{}\cos(n\theta)\,
  71. - n ( n + 1 ) r - n - 2 cos ( n θ ) -n(n+1)~{}r^{-n-2}~{}\cos(n\theta)\,
  72. - n ( n + 1 ) r - n - 2 sin ( n θ ) -n(n+1)~{}r^{-n-2}~{}\sin(n\theta)\,
  73. n ( n + 1 ) r - n - 2 cos ( n θ ) n(n+1)~{}r^{-n-2}~{}\cos(n\theta)\,
  74. r n + 2 sin ( n θ ) r^{n+2}~{}\sin(n\theta)\,
  75. - ( n + 1 ) ( n - 2 ) r n sin ( n θ ) -(n+1)(n-2)~{}r^{n}~{}\sin(n\theta)\,
  76. - n ( n + 1 ) r n cos ( n θ ) -n(n+1)~{}r^{n}~{}\cos(n\theta)\,
  77. ( n + 1 ) ( n + 2 ) r n sin ( n θ (n+1)(n+2)~{}r^{n}~{}\sin(n\theta\,
  78. r - n + 2 sin ( n θ ) r^{-n+2}~{}\sin(n\theta)\,
  79. - ( n + 2 ) ( n - 1 ) r - n sin ( n θ ) -(n+2)(n-1)~{}r^{-n}~{}\sin(n\theta)\,
  80. n ( n - 1 ) r - n cos ( n θ ) n(n-1)~{}r^{-n}~{}\cos(n\theta)\,
  81. ( n - 1 ) ( n - 2 ) r - n sin ( n θ ) (n-1)(n-2)~{}r^{-n}~{}\sin(n\theta)
  82. r n sin ( n θ ) r^{n}~{}\sin(n\theta)\,
  83. - n ( n - 1 ) r n - 2 sin ( n θ ) -n(n-1)~{}r^{n-2}~{}\sin(n\theta)\,
  84. - n ( n - 1 ) r n - 2 cos ( n θ ) -n(n-1)~{}r^{n-2}~{}\cos(n\theta)\,
  85. n ( n - 1 ) r n - 2 sin ( n θ ) n(n-1)~{}r^{n-2}~{}\sin(n\theta)\,
  86. r - n sin ( n θ ) r^{-n}~{}\sin(n\theta)\,
  87. - n ( n + 1 ) r - n - 2 sin ( n θ ) -n(n+1)~{}r^{-n-2}~{}\sin(n\theta)\,
  88. n ( n + 1 ) r - n - 2 cos ( n θ ) n(n+1)~{}r^{-n-2}~{}\cos(n\theta)\,
  89. n ( n + 1 ) r - n - 2 sin ( n θ ) n(n+1)~{}r^{-n-2}~{}\sin(n\theta)\,
  90. ( u r , u θ ) (u_{r},u_{\theta})
  91. κ = { 3 - 4 ν for plane strain 3 - ν 1 + ν for plane stress \kappa=\begin{cases}3-4~{}\nu&\rm{for~{}plane~{}strain}\\ \cfrac{3-\nu}{1+\nu}&\rm{for~{}plane~{}stress}\\ \end{cases}
  92. ν \nu
  93. μ \mu
  94. φ \varphi
  95. 2 μ u r 2~{}\mu~{}u_{r}\,
  96. 2 μ u θ 2~{}\mu~{}u_{\theta}\,
  97. r 2 r^{2}\,
  98. ( κ - 1 ) r (\kappa-1)~{}r
  99. 0
  100. r 2 ln r r^{2}~{}\ln r
  101. ( κ - 1 ) r ln r - r (\kappa-1)~{}r~{}\ln r-r
  102. ( κ + 1 ) r θ (\kappa+1)~{}r~{}\theta
  103. ln r \ln r\,
  104. - r - 1 -r^{-1}\,
  105. 0
  106. θ \theta\,
  107. 0
  108. - r - 1 -r^{-1}\,
  109. r 3 cos θ r^{3}~{}\cos\theta\,
  110. ( κ - 2 ) r 2 cos θ (\kappa-2)~{}r^{2}~{}\cos\theta\,
  111. ( κ + 2 ) r 2 sin θ (\kappa+2)~{}r^{2}~{}\sin\theta\,
  112. r θ cos θ r\theta~{}\cos\theta\,
  113. 1 2 [ ( κ - 1 ) θ cos θ + { 1 - ( κ + 1 ) ln r } sin θ ] \frac{1}{2}[(\kappa-1)\theta~{}\cos\theta+\{1-(\kappa+1)\ln r\}~{}\sin\theta]\,
  114. - 1 2 [ ( κ - 1 ) θ sin θ + { 1 + ( κ + 1 ) ln r } cos θ ] -\frac{1}{2}[(\kappa-1)\theta~{}\sin\theta+\{1+(\kappa+1)\ln r\}~{}\cos\theta]\,
  115. r ln r cos θ r~{}\ln r~{}\cos\theta\,
  116. 1 2 [ ( κ + 1 ) θ sin θ - { 1 - ( κ - 1 ) ln r } cos θ ] \frac{1}{2}[(\kappa+1)\theta~{}\sin\theta-\{1-(\kappa-1)\ln r\}~{}\cos\theta]\,
  117. 1 2 [ ( κ + 1 ) θ cos θ - { 1 + ( κ - 1 ) ln r } sin θ ] \frac{1}{2}[(\kappa+1)\theta~{}\cos\theta-\{1+(\kappa-1)\ln r\}~{}\sin\theta]\,
  118. r - 1 cos θ r^{-1}~{}\cos\theta\,
  119. r - 2 cos θ r^{-2}~{}\cos\theta\,
  120. r - 2 sin θ r^{-2}~{}\sin\theta\,
  121. r 3 sin θ r^{3}~{}\sin\theta\,
  122. ( κ - 2 ) r 2 sin θ (\kappa-2)~{}r^{2}~{}\sin\theta\,
  123. - ( κ + 2 ) r 2 cos θ -(\kappa+2)~{}r^{2}~{}\cos\theta\,
  124. r θ sin θ r\theta~{}\sin\theta\,
  125. 1 2 [ ( κ - 1 ) θ sin θ - { 1 - ( κ + 1 ) ln r } cos θ ] \frac{1}{2}[(\kappa-1)\theta~{}\sin\theta-\{1-(\kappa+1)\ln r\}~{}\cos\theta]\,
  126. 1 2 [ ( κ - 1 ) θ cos θ - { 1 + ( κ + 1 ) ln r } sin θ ] \frac{1}{2}[(\kappa-1)\theta~{}\cos\theta-\{1+(\kappa+1)\ln r\}~{}\sin\theta]\,
  127. r ln r sin θ r~{}\ln r~{}\sin\theta\,
  128. - 1 2 [ ( κ + 1 ) θ cos θ + { 1 - ( κ - 1 ) ln r } sin θ ] -\frac{1}{2}[(\kappa+1)\theta~{}\cos\theta+\{1-(\kappa-1)\ln r\}~{}\sin\theta]\,
  129. 1 2 [ ( κ + 1 ) θ sin θ + { 1 + ( κ - 1 ) ln r } cos θ ] \frac{1}{2}[(\kappa+1)\theta~{}\sin\theta+\{1+(\kappa-1)\ln r\}~{}\cos\theta]\,
  130. r - 1 sin θ r^{-1}~{}\sin\theta\,
  131. r - 2 sin θ r^{-2}~{}\sin\theta\,
  132. - r - 2 cos θ -r^{-2}~{}\cos\theta\,
  133. r n + 2 cos ( n θ ) r^{n+2}~{}\cos(n\theta)\,
  134. ( κ - n - 1 ) r n + 1 cos ( n θ ) (\kappa-n-1)~{}r^{n+1}~{}\cos(n\theta)\,
  135. ( κ + n + 1 ) r n + 1 sin ( n θ ) (\kappa+n+1)~{}r^{n+1}~{}\sin(n\theta)\,
  136. r - n + 2 cos ( n θ ) r^{-n+2}~{}\cos(n\theta)\,
  137. ( κ + n - 1 ) r - n + 1 cos ( n θ ) (\kappa+n-1)~{}r^{-n+1}~{}\cos(n\theta)\,
  138. - ( κ - n + 1 ) r - n + 1 sin ( n θ ) -(\kappa-n+1)~{}r^{-n+1}~{}\sin(n\theta)\,
  139. r n cos ( n θ ) r^{n}~{}\cos(n\theta)\,
  140. - n r n - 1 cos ( n θ ) -n~{}r^{n-1}~{}\cos(n\theta)\,
  141. n r n - 1 sin ( n θ ) n~{}r^{n-1}~{}\sin(n\theta)\,
  142. r - n cos ( n θ ) r^{-n}~{}\cos(n\theta)\,
  143. n r - n - 1 cos ( n θ ) n~{}r^{-n-1}~{}\cos(n\theta)\,
  144. n ( r - n - 1 sin ( n θ ) n(~{}r^{-n-1}~{}\sin(n\theta)\,
  145. r n + 2 sin ( n θ ) r^{n+2}~{}\sin(n\theta)\,
  146. ( κ - n - 1 ) r n + 1 sin ( n θ ) (\kappa-n-1)~{}r^{n+1}~{}\sin(n\theta)\,
  147. - ( κ + n + 1 ) r n + 1 cos ( n θ ) -(\kappa+n+1)~{}r^{n+1}~{}\cos(n\theta)\,
  148. r - n + 2 sin ( n θ ) r^{-n+2}~{}\sin(n\theta)\,
  149. ( κ + n - 1 ) r - n + 1 sin ( n θ ) (\kappa+n-1)~{}r^{-n+1}~{}\sin(n\theta)\,
  150. ( κ - n + 1 ) r - n + 1 cos ( n θ ) (\kappa-n+1)~{}r^{-n+1}~{}\cos(n\theta)\,
  151. r n sin ( n θ ) r^{n}~{}\sin(n\theta)\,
  152. - n r n - 1 sin ( n θ ) -n~{}r^{n-1}~{}\sin(n\theta)\,
  153. - n r n - 1 cos ( n θ ) -n~{}r^{n-1}~{}\cos(n\theta)\,
  154. r - n sin ( n θ ) r^{-n}~{}\sin(n\theta)\,
  155. n r - n - 1 sin ( n θ ) n~{}r^{-n-1}~{}\sin(n\theta)\,
  156. - n r - n - 1 cos ( n θ ) -n~{}r^{-n-1}~{}\cos(n\theta)\,
  157. u r = A cos θ + B sin θ u θ = - A sin θ + B cos θ + C r \begin{aligned}\displaystyle u_{r}&\displaystyle=A~{}\cos\theta+B~{}\sin\theta% \\ \displaystyle u_{\theta}&\displaystyle=-A~{}\sin\theta+B~{}\cos\theta+C~{}r\\ \end{aligned}

Microactuator.html

  1. W = F Δ r W=\overrightarrow{F}\cdot\Delta\overrightarrow{r}

Microdispensing.html

  1. We = ρ v 2 D σ \mathrm{We}=\frac{\rho v^{2}D}{\sigma}
  2. ρ \rho
  3. v v
  4. D D
  5. σ \sigma
  6. Q = Δ P π r 4 8 μ L Q=\frac{\Delta P\pi r^{4}}{8\mu L}
  7. Q Q
  8. Δ P \Delta P
  9. r r
  10. μ \mu
  11. L L
  12. R e < R e c r i t Re<Re_{crit}
  13. R e = ρ v D μ Re=\frac{\rho vD}{\mu}
  14. μ \mu
  15. 1800 R e c r i t 2400 1800\lessapprox Re_{crit}\lessapprox 2400

Microplasma.html

  1. V b = < m t p l > B ( p d ) ln ( p d ) + ln ( A / ln ( 1 + 1 γ ) ) V_{b}=\frac{<}{m}tpl>{{B(pd)}}{{\ln(pd)+\ln(A/\ln(1+\frac{1}{\gamma}))}}
  2. A A
  3. B B
  4. γ \gamma

Mild-slope_equation.html

  1. ζ ( x , y , t ) = { η ( x , y ) e - i ω t } \zeta(x,y,t)=\Re\left\{\eta(x,y)\,\,\text{e}^{-i\omega t}\right\}
  2. h ( x , y ) h(x,y)
  3. ( c p c g η ) + k 2 c p c g η = 0 , \nabla\cdot\left(c_{p}\,c_{g}\,\nabla\eta\right)\,+\,k^{2}\,c_{p}\,c_{g}\,\eta% \,=\,0,
  4. η ( x , y ) \eta(x,y)
  5. ζ ( x , y , t ) ; \zeta(x,y,t);
  6. ( x , y ) (x,y)
  7. ω \omega
  8. i i
  9. { } \Re\{\cdot\}
  10. \nabla
  11. \nabla\cdot
  12. k k
  13. c p c_{p}
  14. c g c_{g}
  15. ω 2 = g k tanh ( k h ) , c p = ω k and c g = 1 2 c p [ 1 + k h 1 - tanh 2 ( k h ) tanh ( k h ) ] \begin{aligned}\displaystyle\omega^{2}&\displaystyle=\,g\,k\,\tanh\,(kh),\\ \displaystyle c_{p}&\displaystyle=\,\frac{\omega}{k}\quad\,\text{and}\\ \displaystyle c_{g}&\displaystyle=\,\frac{1}{2}\,c_{p}\,\left[1\,+\,kh\,\frac{% 1-\tanh^{2}(kh)}{\tanh\,(kh)}\right]\end{aligned}
  16. g g
  17. tanh \tanh
  18. ω \omega
  19. k k
  20. h h
  21. ψ = η c p c g , \psi\,=\,\eta\,\sqrt{c_{p}\,c_{g}},
  22. Δ ψ + k c 2 ψ = 0 with k c 2 = k 2 - Δ ( c p c g ) c p c g , \Delta\psi\,+\,k_{c}^{2}\,\psi\,=\,0\qquad\,\text{with}\qquad k_{c}^{2}\,=\,k^% {2}\,-\,\frac{\Delta\left(\sqrt{c_{p}\,c_{g}}\right)}{\sqrt{c_{p}\,c_{g}}},
  23. Δ \Delta
  24. η ( x , y ) \eta(x,y)
  25. η ( x , y ) = a ( x , y ) e i θ ( x , y ) , \eta(x,y)\,=\,a(x,y)\,\,\text{e}^{i\,\theta(x,y)},
  26. a = | η | a=|\eta|\,
  27. η \eta\,
  28. θ = arg { η } \theta=\arg\{\eta\}\,
  29. η . \eta.\,
  30. θ \nabla\theta
  31. κ y x - κ x y = 0 with κ x = θ x and κ y = θ y , κ 2 = k 2 + ( c p c g a ) c p c g a with κ = κ x 2 + κ y 2 and ( s y m b o l v g E ) = 0 with E = 1 2 ρ g a 2 and s y m b o l v g = c g s y m b o l κ k , \begin{aligned}\displaystyle\frac{\partial\kappa_{y}}{\partial{x}}\,-\,\frac{% \partial\kappa_{x}}{\partial{y}}\,=\,0&\displaystyle\,\text{ with }\kappa_{x}% \,=\,\frac{\partial\theta}{\partial{x}}\,\text{ and }\kappa_{y}\,=\,\frac{% \partial\theta}{\partial{y}},\\ \displaystyle\kappa^{2}\,=\,k^{2}\,+\,\frac{\nabla\cdot\left(c_{p}\,c_{g}\,% \nabla a\right)}{c_{p}\,c_{g}\,a}&\displaystyle\,\text{ with }\kappa\,=\,\sqrt% {\kappa_{x}^{2}\,+\,\kappa_{y}^{2}}\quad\,\text{ and}\\ \displaystyle\nabla\cdot\left(symbol{v}_{g}\,E\right)\,=\,0&\displaystyle\,% \text{ with }E\,=\,\frac{1}{2}\,\rho\,g\,a^{2}\quad\,\text{and}\quad symbol{v}% _{g}\,=\,c_{g}\,\frac{symbol{\kappa}}{k},\end{aligned}
  32. E E
  33. s y m b o l κ symbol{\kappa}
  34. ( κ x , κ y ) , (\kappa_{x},\kappa_{y}),\,
  35. s y m b o l v g symbol{v}_{g}
  36. ρ \rho
  37. g g
  38. E E
  39. s y m b o l κ symbol{\kappa}
  40. | s y m b o l v g | |symbol{v}_{g}|
  41. c g . c_{g}.
  42. s y m b o l κ symbol{\kappa}
  43. θ \theta
  44. ( c p c g a ) k 2 c p c g a , \nabla\cdot(c_{p}\,c_{g}\,\nabla a)\ll k^{2}\,c_{p}\,c_{g}\,a,
  45. a a
  46. θ \theta
  47. a a
  48. s y m b o l κ symbol{\kappa}
  49. k k
  50. η = a exp ( i θ ) \eta=a\mbox{ }~{}\exp(i\theta)
  51. exp ( i θ ) \exp(i\theta)
  52. c p c g ( Δ a + 2 i a θ - a θ θ + i a Δ θ ) + ( c p c g ) ( a + i a θ ) + k 2 c p c g a = 0. c_{p}\,c_{g}\,\left(\Delta a\,+\,2i\,\nabla a\cdot\nabla\theta\,-\,a\,\nabla% \theta\cdot\nabla\theta\,+\,i\,a\,\Delta\theta\right)\,+\,\nabla\left(c_{p}\,c% _{g}\right)\cdot\left(\nabla a\,+\,i\,a\,\nabla\theta\right)\,+\,k^{2}\,c_{p}% \,c_{g}\,a\,=\,0.
  53. c p c g Δ a - c p c g a θ θ + ( c p c g ) a + k 2 c p c g a = 0 and \displaystyle c_{p}\,c_{g}\,\Delta a\,-\,c_{p}\,c_{g}\,a\,\nabla\theta\cdot% \nabla\theta\,+\,\nabla\left(c_{p}\,c_{g}\right)\cdot\nabla a\,+\,k^{2}\,c_{p}% \,c_{g}\,a\,=\,0\quad\,\text{and}
  54. s y m b o l κ symbol{\kappa}
  55. s y m b o l κ = θ symbol{\kappa}\,=\,\nabla\theta
  56. κ = | s y m b o l κ | . \kappa=|symbol{\kappa}|.
  57. s y m b o l κ symbol{\kappa}
  58. × s y m b o l κ = 0. \nabla\,\times\,symbol{\kappa}\,=\,0.
  59. a a
  60. κ 2 = k 2 + ( c p c g ) c p c g a a + Δ a a and \displaystyle\kappa^{2}\,=\,k^{2}\,+\,\frac{\nabla(c_{p}\,c_{g})}{c_{p}\,c_{g}% }\cdot\frac{\nabla a}{a}\,+\,\frac{\Delta a}{a}\quad\,\text{and}
  61. κ \kappa\,
  62. ( s y m b o l κ c p c g a 2 ) = 0 , \nabla\cdot\left(symbol{\kappa}\,c_{p}\,c_{g}\,a^{2}\right)\,=\,0,
  63. c p = ω / k c_{p}=\omega/k
  64. ω \omega
  65. ρ \rho
  66. z = ζ ( x , y , t ) z=\zeta(x,y,t)
  67. z = - h ( x , y ) , z=-h(x,y),
  68. δ = 0 \delta\mathcal{L}=0
  69. = t 0 t 1 L d x d y d t , \mathcal{L}=\int_{t_{0}}^{t_{1}}\iint L\;\,\text{d}x\;\,\text{d}y\;\,\text{d}t,
  70. L L
  71. L = - ρ { - h ( x , y ) ζ ( x , y , t ) [ Φ t + 1 2 ( ( Φ x ) 2 + ( Φ y ) 2 + ( Φ z ) 2 ) ] d z + 1 2 g ( ζ 2 - h 2 ) } , L=-\rho\,\left\{\int_{-h(x,y)}^{\zeta(x,y,t)}\left[\frac{\partial\Phi}{% \partial t}+\,\frac{1}{2}\left(\left(\frac{\partial\Phi}{\partial x}\right)^{2% }+\left(\frac{\partial\Phi}{\partial y}\right)^{2}+\left(\frac{\partial\Phi}{% \partial z}\right)^{2}\right)\right]\;\,\text{d}z\;+\,\frac{1}{2}\,g\,(\zeta^{% 2}\,-\,h^{2})\right\},
  72. Φ ( x , y , z , t ) \Phi(x,y,z,t)
  73. Φ / x , \partial\Phi/\partial{x},
  74. Φ / y \partial\Phi/\partial{y}
  75. Φ / z \partial\Phi/\partial{z}
  76. x x
  77. y y
  78. z z
  79. ( Φ , ζ ) \mathcal{L}(\Phi,\zeta)
  80. Φ ( x , y , z , t ) \Phi(x,y,z,t)
  81. ζ ( x , y , t ) \zeta(x,y,t)
  82. Φ \Phi
  83. z = ζ ( x , y , t ) z=\zeta(x,y,t)
  84. z = - h ( x , y ) . z=-h(x,y).
  85. L L
  86. z = - h z=-h
  87. z = 0 , z=0,
  88. z = 0 z=0
  89. z = ζ z=\zeta
  90. z = 0 , z=0,
  91. Φ \Phi
  92. ζ , \zeta,
  93. L 0 L_{0}
  94. L 0 = - ρ { ζ [ Φ t ] z = 0 + - h 0 1 2 [ ( Φ x ) 2 + ( Φ y ) 2 + ( Φ z ) 2 ] d z + 1 2 g ζ 2 } . L_{0}=-\rho\,\left\{\zeta\,\left[\frac{\partial\Phi}{\partial t}\right]_{z=0}% \,+\,\int_{-h}^{0}\frac{1}{2}\left[\left(\frac{\partial\Phi}{\partial x}\right% )^{2}+\left(\frac{\partial\Phi}{\partial y}\right)^{2}+\left(\frac{\partial% \Phi}{\partial z}\right)^{2}\right]\;\,\text{d}z\;+\,\frac{1}{2}\,g\,\zeta^{2}% \,\right\}.
  95. Φ / t \partial\Phi/\partial{t}
  96. h 2 h^{2}
  97. ( x , y ) (x,y)
  98. Φ \Phi
  99. z z
  100. Φ : \Phi:
  101. Φ ( x , y , z , t ) = f ( z ; x , y ) φ ( x , y , t ) \Phi(x,y,z,t)=f(z;x,y)\,\varphi(x,y,t)
  102. f ( 0 ; x , y ) = 1 f(0;x,y)=1
  103. z = 0. z=0.
  104. φ ( x , y , t ) \varphi(x,y,t)
  105. z = 0. z=0.
  106. f f
  107. ( x , y ) (x,y)
  108. f f
  109. ( Φ x Φ y Φ z ) ( f φ x f φ y f z φ ) . \begin{pmatrix}\displaystyle\frac{\partial\Phi}{\partial{x}}\\ \displaystyle\frac{\partial\Phi}{\partial{y}}\\ \displaystyle\frac{\partial\Phi}{\partial{z}}\end{pmatrix}\,\approx\,\begin{% pmatrix}\displaystyle f\,\frac{\partial\varphi}{\partial{x}}\\ \displaystyle f\,\frac{\partial\varphi}{\partial{y}}\\ \displaystyle\frac{\partial{f}}{\partial{z}}\,\varphi\end{pmatrix}.
  110. L 0 = - ρ { ζ φ t + 1 2 F [ ( φ x ) 2 + ( φ y ) 2 ] + 1 2 G φ 2 + 1 2 g ζ 2 } , L_{0}=-\rho\,\left\{\zeta\,\frac{\partial\varphi}{\partial t}\,+\,\frac{1}{2}% \,F\,\left[\left(\frac{\partial\varphi}{\partial{x}}\right)^{2}\,+\,\left(% \frac{\partial\varphi}{\partial{y}}\right)^{2}\right]\,+\,\frac{1}{2}\,G\,% \varphi^{2}\,+\,\frac{1}{2}\,g\,\zeta^{2}\,\right\},
  111. F = - h 0 f 2 d z F\,=\,\int_{-h}^{0}f^{2}\;\,\text{d}z
  112. G = - h 0 ( d f d z ) 2 d z . G\,=\,\int_{-h}^{0}\left(\frac{\,\text{d}f}{\,\text{d}z}\right)^{2}\;\,\text{d% }z.
  113. L 0 L_{0}
  114. ξ ( x , y , t ) \xi(x,y,t)
  115. φ \varphi
  116. ζ : \zeta:
  117. L 0 ξ - t ( L 0 ( ξ / t ) ) - x ( L 0 ( ξ / x ) ) - y ( L 0 ( ξ / y ) ) = 0. \frac{\partial{L_{0}}}{\partial\xi}-\frac{\partial}{\partial{t}}\left(\frac{% \partial{L_{0}}}{\partial(\partial\xi/\partial{t})}\right)-\frac{\partial}{% \partial{x}}\left(\frac{\partial{L_{0}}}{\partial(\partial\xi/\partial{x})}% \right)-\frac{\partial}{\partial{y}}\left(\frac{\partial{L_{0}}}{\partial(% \partial\xi/\partial{y})}\right)=0.
  118. ξ \xi
  119. φ \varphi
  120. ζ . \zeta.
  121. ζ t + ( F φ ) - G φ = 0 and φ t + g ζ = 0 , \begin{aligned}\displaystyle\frac{\partial\zeta}{\partial t}&\displaystyle+% \nabla\cdot\left(F\,\nabla\varphi\right)\,-\,G\,\varphi\,=\,0\quad\,\text{and}% \\ \displaystyle\frac{\partial\varphi}{\partial t}&\displaystyle+\,g\,\zeta\,=\,0% ,\end{aligned}
  122. f f
  123. F F
  124. G . G.
  125. f ( z ) f(z)
  126. h . h.
  127. f = cosh ( k ( z + h ) ) cosh ( k h ) , f=\frac{\cosh\,\bigl(k\,(z+h)\bigr)}{\cosh\,(kh)},
  128. k ( x , y ) k(x,y)
  129. x x
  130. y y
  131. h ( x , y ) h(x,y)
  132. ω 0 2 = g k tanh ( k h ) . \omega_{0}^{2}\,=\,g\,k\,\tanh\,(kh).
  133. ω 0 \omega_{0}
  134. F F
  135. G G
  136. F = h 0 f 2 d z = 1 g c p c g and G = h 0 ( f z ) 2 d z = 1 g ( ω 0 2 - k 2 c p c g ) . \begin{aligned}\displaystyle F&\displaystyle=\int_{h}^{0}f^{2}\;\,\text{d}z=% \frac{1}{g}\,c_{p}\,c_{g}\quad\,\text{and}\\ \displaystyle G&\displaystyle=\int_{h}^{0}\left(\frac{\partial{f}}{\partial{z}% }\right)^{2}\;\,\text{d}z=\frac{1}{g}\left(\omega_{0}^{2}\,-\,k^{2}\,c_{p}\,c_% {g}\right).\end{aligned}
  137. ζ ( x , y , t ) \zeta(x,y,t)
  138. ϕ ( x , y , t ) : \phi(x,y,t):
  139. g ζ t + ( c p c g φ ) + ( k 2 c p c g - ω 0 2 ) φ = 0 , φ t + g ζ = 0 , with ω 0 2 = g k tanh ( k h ) . \begin{aligned}\displaystyle g\,\frac{\partial\zeta}{\partial{t}}&% \displaystyle+\nabla\cdot\left(c_{p}\,c_{g}\,\nabla\varphi\right)+\left(k^{2}% \,c_{p}\,c_{g}\,-\,\omega_{0}^{2}\right)\,\varphi=0,\\ \displaystyle\frac{\partial\varphi}{\partial{t}}&\displaystyle+g\zeta=0,\quad% \,\text{with}\quad\omega_{0}^{2}\,=\,g\,k\,\tanh\,(kh).\end{aligned}
  140. φ \varphi
  141. ζ \zeta
  142. - 2 ζ t 2 + ( c p c g ζ ) + ( k 2 c p c g - ω 0 2 ) ζ = 0 , -\frac{\partial^{2}\zeta}{\partial{t^{2}}}+\nabla\cdot\left(c_{p}\,c_{g}\,% \nabla\zeta\right)+\left(k^{2}\,c_{p}\,c_{g}\,-\,\omega_{0}^{2}\right)\,\zeta=0,
  143. ζ \zeta
  144. φ . \varphi.
  145. ω 0 . \omega_{0}.
  146. η ( x , y ) \eta(x,y)
  147. ω : \omega:
  148. ζ ( x , y , t ) = { η ( x , y ) e - i ω t } , \zeta(x,y,t)\,=\,\Re\left\{\eta(x,y)\;\,\text{e}^{-i\,\omega\,t}\right\},
  149. ω \omega
  150. ω 0 \omega_{0}
  151. ω = ω 0 . \omega=\omega_{0}.
  152. ( c p c g η ) + k 2 c p c g η = 0. \nabla\cdot\left(c_{p}\,c_{g}\,\nabla\eta\right)\,+\,k^{2}\,c_{p}\,c_{g}\,\eta% \,=\,0.

Miles_per_gallon_gasoline_equivalent.html

  1. M P G e = t o t a l m i l e s d r i v e n [ t o t a l e n e r g y o f a l l f u e l s c o n s u m e d e n e r g y o f o n e g a l l o n o f g a s o l i n e ] = ( t o t a l m i l e s d r i v e n ) × ( e n e r g y o f o n e g a l l o n o f g a s o l i n e ) t o t a l e n e r g y o f a l l f u e l s c o n s u m e d MPGe=\frac{total~{}miles~{}driven}{\left[\frac{total~{}energy~{}of~{}all~{}% fuels~{}consumed}{energy~{}of~{}one~{}gallon~{}of~{}gasoline}\right]}=\frac{(% total\ miles\ driven)\times(energy\ of\ one\ gallon\ of\ gasoline)}{total~{}% energy~{}of~{}all~{}fuels~{}consumed}
  2. M P G e = E G E M * E E = 33 , 705 E M MPGe=\frac{E_{G}}{E_{M}*E_{E}}=\frac{33,705}{E_{M}}
  3. M P G e MPGe
  4. E G = E_{G}=
  5. E M = E_{M}=
  6. E E = E_{E}=
  7. M P G e = E G E M * E E = 32 , 600 E M MPGe=\frac{E_{G}}{E_{M}*E_{E}}=\frac{32,600}{E_{M}}
  8. M P G e = M P k g H 2 × G G E MPGe=MPkg_{H_{2}}\times{GGE}
  9. M P G e = 72 m i k g H 2 × 1.012 k g H 2 g a l l o n g a s o l i n e = 72.8 M P G e MPGe=72\frac{mi}{kg_{H_{2}}}\times{1.012\frac{kg_{H_{2}}}{gallon_{gasoline}}}=% 72.8MPGe

Milnor_number.html

  1. f : ( n , 0 ) ( , 0 ) . f:(\mathbb{C}^{n},0)\to(\mathbb{C},0)\ .
  2. z 1 , , z n z_{1},\ldots,z_{n}
  3. f ( z 1 , , z n ) . f(z_{1},\ldots,z_{n}).
  4. z := ( z 1 , , z n ) . z:=(z_{1},\ldots,z_{n}).
  5. z 0 n z_{0}\in\mathbb{C}^{n}
  6. f / z 1 , , f / z n \partial f/\partial z_{1},\ldots,\partial f/\partial z_{n}
  7. z = z 0 z=z_{0}
  8. z 0 n z_{0}\in\mathbb{C}^{n}
  9. U n U\subset\mathbb{C}^{n}
  10. z 0 z_{0}
  11. z 0 z_{0}
  12. z 0 n z_{0}\in\mathbb{C}^{n}
  13. z 0 z_{0}
  14. z 0 z_{0}
  15. det ( 2 f z i z j ) 1 i j n z = z 0 = 0. \det\left(\frac{\partial^{2}f}{\partial z_{i}\partial z_{j}}\right)_{1\leq i% \leq j\leq n}^{z=z_{0}}=0.
  16. 𝒪 \mathcal{O}
  17. ( n , 0 ) ( , 0 ) (\mathbb{C}^{n},0)\to(\mathbb{C},0)
  18. J f J_{f}
  19. J f := f z i : 1 i n . J_{f}:=\left\langle\frac{\partial f}{\partial z_{i}}:1\leq i\leq n\right\rangle.
  20. 𝒜 f := 𝒪 / J f . \mathcal{A}_{f}:=\mathcal{O}/J_{f}.
  21. μ ( f ) = dim 𝒜 f . \mu(f)=\dim_{\mathbb{C}}\mathcal{A}_{f}\ .
  22. μ ( f ) \mu(f)
  23. n \mathbb{C}^{n}
  24. f ( x , y ) = x 2 + y 2 f(x,y)=x^{2}+y^{2}
  25. 2 x , 2 y = x , y \langle 2x,2y\rangle=\langle x,y\rangle
  26. 𝒜 f = 𝒪 / x , y = 1 . \mathcal{A}_{f}=\mathcal{O}/\langle x,y\rangle=\langle 1\rangle.
  27. h 𝒪 h\in\mathcal{O}
  28. h ( x , y ) = k + x h 1 ( x , y ) + y h 2 ( x , y ) h(x,y)=k+xh_{1}(x,y)+yh_{2}(x,y)
  29. h 1 h_{1}
  30. h 2 h_{2}
  31. 𝒪 \mathcal{O}
  32. h 1 h_{1}
  33. h 2 h_{2}
  34. 𝒜 f = 1 \mathcal{A}_{f}=\langle 1\rangle
  35. f ( x , y ) = x 3 + x y 2 f(x,y)=x^{3}+xy^{2}
  36. 𝒜 f = 𝒪 / 3 x 2 + y 2 , x y = 1 , x , y , x 2 . \mathcal{A}_{f}=\mathcal{O}/\langle 3x^{2}+y^{2},xy\rangle=\langle 1,x,y,x^{2}\rangle.
  37. μ ( f ) = 4 \mu(f)=4
  38. f ( x , y ) = x 2 y 2 + y 3 f(x,y)=x^{2}y^{2}+y^{3}
  39. μ ( f ) = . \mu(f)=\infty.
  40. g 1 , , g μ g_{1},\ldots,g_{\mu}
  41. F : ( n × μ , 0 ) ( , 0 ) , F:(\mathbb{C}^{n}\times\mathbb{C}^{\mu},0)\to(\mathbb{C},0),
  42. F ( z , a ) := f ( z ) + a 1 g 1 ( z ) + + a μ g μ ( z ) , F(z,a):=f(z)+a_{1}g_{1}(z)+\cdots+a_{\mu}g_{\mu}(z),
  43. ( a 1 , , a μ ) μ (a_{1},\dots,a_{\mu})\in\mathbb{C}^{\mu}
  44. f , g : ( n , 0 ) ( , 0 ) f,g:(\mathbb{C}^{n},0)\to(\mathbb{C},0)
  45. ϕ : ( n , 0 ) ( n , 0 ) \phi:(\mathbb{C}^{n},0)\to(\mathbb{C}^{n},0)
  46. ψ : ( , 0 ) ( , 0 ) \psi:(\mathbb{C},0)\to(\mathbb{C},0)
  47. f ϕ = ψ g f\circ\phi=\psi\circ g
  48. f ( x , y ) = x 3 + y 3 f(x,y)=x^{3}+y^{3}
  49. g ( x , y ) = x 2 + y 5 g(x,y)=x^{2}+y^{5}
  50. μ ( f ) = μ ( g ) = 4 \mu(f)=\mu(g)=4

Milnor–Thurston_kneading_theory.html

  1. D ( t ) = 1 + D 1 t + D 2 t 2 + D(t)=1+D_{1}t+D_{2}t^{2}+\cdots\,

Minimal_prime_ideal.html

  1. R / I R/I
  2. I \sqrt{I}

Minimax_estimator.html

  1. θ Θ \theta\in\Theta
  2. x 𝒳 , x\in\mathcal{X},
  3. δ M \delta^{M}\,\!
  4. θ \theta\,\!
  5. δ M \delta^{M}\,\!
  6. θ Θ \theta\in\Theta
  7. x 𝒳 x\in\mathcal{X}
  8. P ( x | θ ) P(x|\theta)\,\!
  9. δ ( x ) \delta(x)\,\!
  10. θ \theta\,\!
  11. R ( θ , δ ) R(\theta,\delta)\,\!
  12. L ( θ , δ ) L(\theta,\delta)\,\!
  13. P ( x | θ ) P(x|\theta)\,\!
  14. L ( θ , δ ) = θ - δ 2 L(\theta,\delta)=\|\theta-\delta\|^{2}\,\!
  15. θ \theta\,\!
  16. θ \theta\,\!
  17. δ M : 𝒳 Θ \delta^{M}:\mathcal{X}\rightarrow\Theta\,\!
  18. R ( θ , δ ) R(\theta,\delta)\,\!
  19. sup θ Θ R ( θ , δ M ) = inf δ sup θ Θ R ( θ , δ ) . \sup_{\theta\in\Theta}R(\theta,\delta^{M})=\inf_{\delta}\sup_{\theta\in\Theta}% R(\theta,\delta).\,
  20. θ \theta\,\!
  21. δ π \delta_{\pi}\,\!
  22. π \pi\,\!
  23. r π = R ( θ , δ π ) d π ( θ ) r_{\pi}=\int R(\theta,\delta_{\pi})\,d\pi(\theta)\,
  24. π \pi\,\!
  25. π \pi^{\prime}\,\!
  26. r π r π r_{\pi}\geq r_{\pi^{\prime}}\,
  27. r π = sup θ R ( θ , δ π ) , r_{\pi}=\sup_{\theta}R(\theta,\delta_{\pi}),\,
  28. δ π \delta_{\pi}\,\!
  29. δ π \delta_{\pi}\,\!
  30. π \pi\,\!
  31. x B ( n , θ ) x\sim B(n,\theta)\,\!
  32. θ Beta ( n / 2 , n / 2 ) \theta\sim\,\text{Beta}(\sqrt{n}/2,\sqrt{n}/2)\,
  33. δ M = x + 0.5 n n + n , \delta^{M}=\frac{x+0.5\sqrt{n}}{n+\sqrt{n}},\,
  34. r = 1 4 ( 1 + n ) 2 r=\frac{1}{4(1+\sqrt{n})^{2}}\,
  35. π n {\pi}_{n}\,\!
  36. π \pi^{\prime}\,\!
  37. lim n r π n r π . \lim_{n\rightarrow\infty}r_{\pi_{n}}\geq r_{\pi^{\prime}}.\,
  38. π n \pi_{n}\,\!
  39. δ \delta\,\!
  40. sup θ R ( θ , δ ) = lim n r π n \sup_{\theta}R(\theta,\delta)=\lim_{n\rightarrow\infty}r_{\pi_{n}}\,\!
  41. δ \delta\,\!
  42. π n {\pi}_{n}\,\!
  43. π n U [ - n , n ] \pi_{n}\sim U[-n,n]\,\!
  44. π n N ( 0 , n σ 2 ) \pi_{n}\sim N(0,n\sigma^{2})\,\!
  45. p p\,\!
  46. x N ( θ , I p σ 2 ) x\sim N(\theta,I_{p}\sigma^{2})\,\!
  47. θ \theta\,\!
  48. δ M L = x \delta_{ML}=x\,\!
  49. R ( θ , δ M L ) = E δ M L - θ 2 = 1 p E ( x i - θ i ) 2 = p σ 2 . R(\theta,\delta_{ML})=E{\|\delta_{ML}-\theta\|^{2}}=\sum\limits_{1}^{p}E{(x_{i% }-\theta_{i})^{2}}=p\sigma^{2}.\,
  50. π n N ( 0 , n σ 2 ) \pi_{n}\sim N(0,n\sigma^{2})\,\!
  51. p > 2 p>2\,\!
  52. p > 2 p>2\,\!
  53. p σ 2 p\sigma^{2}\,\!
  54. θ \|\theta\|\rightarrow\infty\,\!
  55. θ \|\theta\|\,\!
  56. x N ( θ , I n σ 2 ) x\sim N(\theta,I_{n}\sigma^{2})\,\!
  57. θ 2 M \|\theta\|^{2}\leq M\,\!
  58. M n M\leq n\,\!
  59. δ M = n J n + 1 ( n x ) x J n ( n x ) , \delta^{M}=\frac{nJ_{n+1}(n\|x\|)}{\|x\|J_{n}(n\|x\|)},\,
  60. J n ( t ) J_{n}(t)\,\!
  61. δ \delta^{\prime}
  62. c c
  63. sup θ Θ R ( θ , δ ) c inf δ sup θ Θ R ( θ , δ ) . \sup_{\theta\in\Theta}R(\theta,\delta^{\prime})\leq c\inf_{\delta}\sup_{\theta% \in\Theta}R(\theta,\delta).
  64. Θ \Theta

Minimum_distance_estimation.html

  1. X 1 , , X n \displaystyle X_{1},\ldots,X_{n}
  2. F ( x ; θ ) : θ Θ F(x;\theta)\colon\theta\in\Theta
  3. Θ k ( k 1 ) \Theta\subseteq\mathbb{R}^{k}(k\geq 1)
  4. F n ( x ) \displaystyle F_{n}(x)
  5. θ ^ \hat{\theta}
  6. θ \displaystyle\theta
  7. F ( x ; θ ^ ) F(x;\hat{\theta})
  8. F ( x ; θ ) \displaystyle F(x;\theta)
  9. d [ , ] d[\cdot,\cdot]
  10. d \displaystyle d
  11. θ ^ Θ \hat{\theta}\in\Theta
  12. d [ F ( x ; θ ^ ) , F n ( x ) ] = inf { d [ F ( x ; θ ) , F n ( x ) ] ; θ Θ } d[F(x;\hat{\theta}),F_{n}(x)]=\inf\{d[F(x;\theta),F_{n}(x)];\theta\in\Theta\}
  13. θ ^ \hat{\theta}
  14. θ \displaystyle\theta

Minimum_mass.html

  1. M true = M min sin i M\text{true}=\frac{M_{\min}}{\sin i}\,

Minimum_polynomial_extrapolation.html

  1. x k + 1 = f ( x k ) . x_{k+1}=f(x_{k}).
  2. x 1 , x 2 , , x k x_{1},x_{2},...,x_{k}
  3. n \mathbb{R}^{n}
  4. n × ( k - 1 ) n\times(k-1)
  5. U = ( x 2 - x 1 , x 3 - x 2 , , x k - x k - 1 ) U=(x_{2}-x_{1},x_{3}-x_{2},...,x_{k}-x_{k-1})
  6. k - 1 k-1
  7. c = - U + ( x k + 1 - x k ) c=-U^{+}(x_{k+1}-x_{k})
  8. U + U^{+}
  9. U U
  10. c c
  11. s = X c i = 1 k c i , s={Xc\over\sum_{i=1}^{k}c_{i}},
  12. X = ( x 2 , x 3 , , x k + 1 ) X=(x_{2},x_{3},...,x_{k+1})
  13. k k

Minimum_railway_curve_radius.html

  1. tan θ = v 2 g r \tan\theta=\frac{v^{2}}{gr}
  2. r = v 2 g tan θ r=\frac{v^{2}}{g\tan\theta}
  3. tan θ sin θ = h a + h b G \tan\theta\approx\sin\theta=\frac{h_{a}+h_{b}}{G}
  4. r = v 2 g h a + h b G = G v 2 g ( h a + h b ) r=\frac{v^{2}}{g\frac{h_{a}+h_{b}}{G}}=\frac{Gv^{2}}{g(h_{a}+h_{b})}
  5. R = m g \plusmn m v 2 r R=mg\plusmn\frac{mv^{2}}{r}

Minkowski_distance.html

  1. X = ( x 1 , x 2 , , x n ) and Y = ( y 1 , y 2 , , y n ) n X=(x_{1},x_{2},\ldots,x_{n})\,\text{ and }Y=(y_{1},y_{2},\ldots,y_{n})\in% \mathbb{R}^{n}
  2. ( i = 1 n | x i - y i | p ) 1 / p \left(\sum_{i=1}^{n}|x_{i}-y_{i}|^{p}\right)^{1/p}
  3. p 1 p\geq 1
  4. p < 1 p<1
  5. 2 1 / p > 2 2^{1/p}>2
  6. p < 1 p<1
  7. lim p ( i = 1 n | x i - y i | p ) 1 p = max i = 1 n | x i - y i | . \lim_{p\to\infty}{\left(\sum_{i=1}^{n}|x_{i}-y_{i}|^{p}\right)^{\frac{1}{p}}}=% \max_{i=1}^{n}|x_{i}-y_{i}|.\,
  8. lim p - ( i = 1 n | x i - y i | p ) 1 p = min i = 1 n | x i - y i | . \lim_{p\to-\infty}{\left(\sum_{i=1}^{n}|x_{i}-y_{i}|^{p}\right)^{\frac{1}{p}}}% =\min_{i=1}^{n}|x_{i}-y_{i}|.\,

Minnaert_function.html

  1. α \alpha
  2. φ \varphi
  3. λ \lambda
  4. RADF = I F = π A M μ 0 k μ k - 1 \,\text{RADF}=\frac{I}{F}=\pi~{}A_{M}~{}\mu_{0}^{k}~{}\mu^{k-1}
  5. A M A_{M}
  6. k k
  7. I I
  8. ( α , φ , λ ) (\alpha,\varphi,\lambda)
  9. π F \pi F
  10. μ 0 = cos φ cos ( α - λ ) ; μ = cos φ cos λ . \mu_{0}=\cos\varphi~{}\cos(\alpha-\lambda)~{};~{}~{}\mu=\cos\varphi~{}\cos% \lambda~{}.
  11. k k

Minnaert_resonance.html

  1. f = 1 2 π a ( 3 γ p A ρ ) 1 / 2 f=\cfrac{1}{2\pi a}\left(\cfrac{3\gamma~{}p_{A}}{\rho}\right)^{1/2}
  2. a a
  3. γ \gamma
  4. p A p_{A}
  5. ρ \rho
  6. a a
  7. ρ \rho
  8. ( p A = 100 kPa , ρ = 1000 kg / m 3 ) (p_{A}=100~{}{\rm kPa},~{}\rho=1000~{}{\rm kg/m^{3}})
  9. f a 3.26 m / s fa\approx 3.26~{}\rm m/s
  10. f f~{}

Minuscule_104.html

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

Minuscule_1739.html

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

Minuscule_181.html

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

Minuscule_33.html

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

Minuscule_6.html

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

Minuscule_700.html

  1. 𝔓 \mathfrak{P}

Minuscule_81.html

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

Minuscule_88.html

  1. 𝔓 \mathfrak{P}

Miquel's_theorem.html

  1. x = a ( - a 2 d a d a + b 2 d a d b + c 2 d a d c ) x=a\left(-a^{2}d_{a}d_{a}^{\,{}^{\prime}}+b^{2}d_{a}d_{b}+c^{2}d_{a}^{\,{}^{% \prime}}d_{c}^{\,{}^{\prime}}\right)
  2. y = b ( a 2 d a d b - b 2 d b d b + c 2 d b d c ) y=b\left(a^{2}d_{a}^{\,{}^{\prime}}d_{b}^{\,{}^{\prime}}-b^{2}d_{b}d_{b}^{\,{}% ^{\prime}}+c^{2}d_{b}d_{c}\right)
  3. z = c ( a 2 d a d c + b 2 d b d c - c 2 d c d c ) , z=c\left(a^{2}d_{a}d_{c}+b^{2}d_{b}^{\,{}^{\prime}}d_{c}^{\,{}^{\prime}}-c^{2}% d_{c}d_{c}^{\,{}^{\prime}}\right),
  4. ( c o s α : c o s β : c o s γ ) (cosα:cosβ:cosγ)

Mismatch_loss.html

  1. M L dB = 10 log 10 ( P i P i - P r ) ML_{\mathrm{dB}}=10\log_{10}\bigg(\frac{P_{i}}{P_{i}-P_{r}}\bigg)\,
  2. P r = P i - P d P_{r}=P_{i}-P_{d}\,
  3. P i P_{i}
  4. P r P_{r}
  5. P d P_{d}
  6. P d P i = 1 - ρ 2 \frac{P_{d}}{P_{i}}=1-\rho^{2}
  7. ρ \rho
  8. M L dB = - 10 log 10 ( 1 - ρ 2 ) ML_{\mathrm{dB}}=-10\log_{10}\bigg(1-\rho^{2}\bigg)\,
  9. M L dB = - 10 log 10 ( 1 - ( V S W R - 1 V S W R + 1 ) 2 ) ML_{\mathrm{dB}}=-10\log_{10}\bigg(1-\bigg(\frac{VSWR-1}{VSWR+1}\bigg)^{2}% \bigg)\,
  10. M E dB = 20 log 10 ( 1 - ρ 1 ρ 2 e - j 2 θ ) ME_{\mathrm{dB}}=20\log_{10}\bigg(1-\rho_{1}\rho_{2}\,e^{-j2\theta}\bigg)\,
  11. θ \theta

Mixing_length_model.html

  1. ξ \ \xi^{\prime}
  2. T \ T
  3. T \ T^{\prime}
  4. T \ T^{\prime}
  5. ξ \ \xi^{\prime}
  6. T = T ¯ + T T=\overline{T}+T^{\prime}
  7. T ¯ \ \overline{T}
  8. T \ T^{\prime}
  9. T \ T^{\prime}
  10. T = - ξ T ¯ z . \ T^{\prime}=-\xi^{\prime}\frac{\partial\overline{T}}{\partial z}.
  11. u \ u^{\prime}
  12. v \ v^{\prime}
  13. w \ w^{\prime}
  14. u = - ξ u ¯ z , v = - ξ v ¯ z , w = - ξ w ¯ z . \ u^{\prime}=-\xi^{\prime}\frac{\partial\overline{u}}{\partial z},\qquad\ v^{% \prime}=-\xi^{\prime}\frac{\partial\overline{v}}{\partial z},\qquad\ w^{\prime% }=-\xi^{\prime}\frac{\partial\overline{w}}{\partial z}.
  15. w \ w^{\prime}
  16. u w ¯ = ξ 2 ¯ | w ¯ z | u ¯ z \ \overline{u^{\prime}w^{\prime}}=\overline{\xi^{\prime}{}^{2}}\left|\frac{% \partial\overline{w}}{\partial z}\right|\frac{\partial\overline{u}}{\partial z}
  17. K m = ξ 2 ¯ | w ¯ z | \ K_{m}=\overline{\xi^{\prime 2}}\left|\frac{\partial\overline{w}}{\partial z}\right|
  18. K m \ K_{m}
  19. ξ \ \xi^{\prime}

MKER.html

  1. M L M_{\mathrm{L}}

Mm'-type_filter.html

  1. ω c = 1 L C . \omega_{c}=\frac{1}{\sqrt{LC}}.
  2. ω = ω c 1 - ( m m ) 2 . \omega_{\infty}=\frac{\omega_{c}}{\sqrt{1-(mm^{\prime})^{2}}}.
  3. Z i T m m = 1 - ω 2 ( 1 - ( 1 - m 2 ) ω 2 ) 1 - ( ω / ω ) 2 Z_{iTmm^{\prime}}=\frac{\sqrt{1-\omega^{2}}\left(1-(1-m^{2})\omega^{2}\right)}% {1-\left(\omega/\omega_{\infty}\right)^{2}}
  4. Z i T = 1 - ω 2 Z_{iT}=\sqrt{1-\omega^{2}}
  5. Z i T m = 1 - ω 2 1 - ( ω / ω ) 2 Z_{iTm}=\frac{\sqrt{1-\omega^{2}}}{1-\left(\omega/\omega_{\infty}\right)^{2}}
  6. Z i Π m m = 1 - ( ω / ω ) 2 1 - ω 2 ( 1 - ( 1 - m 2 ) ω 2 ) Z_{i\Pi mm^{\prime}}=\frac{1-\left(\omega/\omega_{\infty}\right)^{2}}{\sqrt{1-% \omega^{2}}\left(1-(1-m^{2})\omega^{2}\right)}
  7. Z i Π = 1 1 - ω 2 Z_{i\Pi}=\frac{1}{\sqrt{1-\omega^{2}}}
  8. Z i Π m = 1 - ( ω / ω ) 2 1 - ω 2 . Z_{i\Pi m}=\frac{1-\left(\omega/\omega_{\infty}\right)^{2}}{\sqrt{1-\omega^{2}% }}.
  9. ω \omega_{\infty}
  10. m = 0.7230 m=0.7230\,
  11. m = 0.4134 m^{\prime}=0.4134\,
  12. γ = α + i β \gamma=\alpha+i\beta\,

Modal_algebra.html

  1. A , , , - , 0 , 1 , \langle A,\land,\lor,-,0,1,\Box\rangle
  2. A , , , - , 0 , 1 \langle A,\land,\lor,-,0,1\rangle
  3. \Box
  4. 1 = 1 \Box 1=1
  5. ( x y ) = x y \Box(x\land y)=\Box x\land\Box y

Modal_matrix.html

  1. M M
  2. A A
  3. A A
  4. M M
  5. D = M - 1 A M , D=M^{-1}AM,
  6. D D
  7. A A
  8. D D
  9. D D
  10. A A
  11. M M
  12. A = ( 3 2 0 2 0 0 1 0 2 ) A=\begin{pmatrix}3&2&0\\ 2&0&0\\ 1&0&2\end{pmatrix}
  13. λ 1 = - 1 , b 1 = ( - 3 , 6 , 1 ) , \lambda_{1}=-1,\quad\,b_{1}=\left(-3,6,1\right),
  14. λ 2 = 2 , b 2 = ( 0 , 0 , 1 ) , \lambda_{2}=2,\qquad b_{2}=\left(0,0,1\right),
  15. λ 3 = 4 , b 3 = ( 2 , 1 , 1 ) . \lambda_{3}=4,\qquad b_{3}=\left(2,1,1\right).
  16. D D
  17. A A
  18. D = ( - 1 0 0 0 2 0 0 0 4 ) . D=\begin{pmatrix}-1&0&0\\ 0&2&0\\ 0&0&4\end{pmatrix}.
  19. M M
  20. D = M - 1 A M , D=M^{-1}AM,
  21. M = ( - 3 0 2 6 0 1 1 1 1 ) . M=\begin{pmatrix}-3&0&2\\ 6&0&1\\ 1&1&1\end{pmatrix}.
  22. M M
  23. D D
  24. M M
  25. D D
  26. A A
  27. M M
  28. A A
  29. A A
  30. M M
  31. M M
  32. M M
  33. M M
  34. J J
  35. M - 1 M^{-1}
  36. J = M - 1 A M . J=M^{-1}AM.
  37. A = ( - 1 0 - 1 1 1 3 0 0 1 0 0 0 0 0 2 1 2 - 1 - 1 - 6 0 - 2 0 - 1 2 1 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 - 1 - 1 0 1 2 4 1 ) A=\begin{pmatrix}-1&0&-1&1&1&3&0\\ 0&1&0&0&0&0&0\\ 2&1&2&-1&-1&-6&0\\ -2&0&-1&2&1&3&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ -1&-1&0&1&2&4&1\end{pmatrix}
  38. λ 1 = 1 \lambda_{1}=1
  39. μ 1 = 7 \mu_{1}=7
  40. A A
  41. { x 3 , x 2 , x 1 } \left\{x_{3},x_{2},x_{1}\right\}
  42. { y 2 , y 1 } \left\{y_{2},y_{1}\right\}
  43. { z 1 } \left\{z_{1}\right\}
  44. { w 1 } \left\{w_{1}\right\}
  45. J J
  46. A A
  47. M = ( z 1 w 1 x 1 x 2 x 3 y 1 y 2 ) = ( 0 1 - 1 0 0 - 2 1 0 3 0 0 1 0 0 - 1 1 1 1 0 2 0 - 2 0 - 1 0 0 - 2 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 - 1 0 - 1 0 ) , M=\begin{pmatrix}z_{1}&w_{1}&x_{1}&x_{2}&x_{3}&y_{1}&y_{2}\end{pmatrix}=\begin% {pmatrix}0&1&-1&0&0&-2&1\\ 0&3&0&0&1&0&0\\ -1&1&1&1&0&2&0\\ -2&0&-1&0&0&-2&0\\ 1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&0&-1&0&-1&0\end{pmatrix},
  48. J = ( 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 ) , J=\begin{pmatrix}1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&1&0&0&0\\ 0&0&0&1&1&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&1\\ 0&0&0&0&0&0&1\end{pmatrix},
  49. M M
  50. A A
  51. M M
  52. A A
  53. A M = M J AM=MJ
  54. M M
  55. J J
  56. M M
  57. J J

Modem.html

  1. bandwidth × log 2 ( 1 + P u / P n ) \,\text{bandwidth}\times\log_{2}(1+P_{u}/P_{n})
  2. P u / P n P_{u}/P_{n}
  3. P u / P n = 1000 P_{u}/P_{n}=1000

Moderation_(statistics).html

  1. Y = b 0 + b 1 x 1 + b 2 x 2 + b 3 ( x 1 × x 2 ) + ε Y=b_{0}+b_{1}x_{1}+b_{2}x_{2}+b_{3}(x_{1}\times x_{2})+\varepsilon\,
  2. x 1 x 2 x_{1}x_{2}
  3. Y = b 0 + b 1 A + b 2 B + b 3 A * B + ε Y=b_{0}+b_{1}A+b_{2}B+b_{3}A*B+\varepsilon
  4. Y = b 0 + b 1 A + b 2 B + b 3 C + b 4 A * B + b 5 A * C + b 6 B * C + b 7 A * B * C + ε . Y=b_{0}+b_{1}A+b_{2}B+b_{3}C+b_{4}A*B+b_{5}A*C+b_{6}B*C+b_{7}A*B*C+\varepsilon.

Modes_of_convergence_(annotated_index).html

  1. \Rightarrow
  2. \Rightarrow
  3. \Rightarrow
  4. \Rightarrow
  5. | b k | \sum|b_{k}|
  6. \Rightarrow
  7. \Rightarrow
  8. \Rightarrow
  9. \Rightarrow
  10. ⇏ \not\Rightarrow
  11. \equiv
  12. \Rightarrow
  13. \Rightarrow
  14. \Rightarrow
  15. \Rightarrow
  16. X X
  17. \equiv
  18. \Rightarrow
  19. \Rightarrow
  20. \Rightarrow
  21. \Rightarrow
  22. \Rightarrow
  23. \Rightarrow
  24. Σ | g k | \Sigma|g_{k}|
  25. Σ g k \Sigma g_{k}
  26. Σ | g k | \Sigma|g_{k}|
  27. Σ | g k | \Sigma|g_{k}|
  28. Σ || g k || u \Sigma||g_{k}||_{u}
  29. \Rightarrow
  30. \Rightarrow
  31. \Rightarrow
  32. \Rightarrow
  33. \Rightarrow
  34. \Rightarrow
  35. \Rightarrow
  36. \equiv
  37. \equiv

Modified_nodal_analysis.html

  1. G G
  2. G = 1 / R G=1/R
  3. R R
  4. I R = G V R I_{R}=GV_{R}
  5. I C = C d V C d t I_{C}=C\frac{dV_{C}}{dt}
  6. e 1 e_{1}
  7. e 2 e_{2}
  8. i V s i_{V_{s}}
  9. i R i_{R}
  10. i C i_{C}
  11. i V s + i R = 0 i_{V_{s}}+i_{R}=0
  12. - i R + i C = 0 -i_{R}+i_{C}=0
  13. V s = e 1 V_{s}=e_{1}
  14. V R = e 1 - e 2 V_{R}=e_{1}-e_{2}
  15. V C = e 2 , V_{C}=e_{2},
  16. G ( e 1 - e 2 ) + i V S = 0 G(e_{1}-e_{2})+i_{V_{S}}=0
  17. C d e 2 d t + G ( e 2 - e 1 ) = 0 C\frac{de_{2}}{dt}+G(e_{2}-e_{1})=0
  18. e 1 = V s e_{1}=V_{s}
  19. 𝐱 = ( e 1 e 2 i V S ) T \mathbf{x}=\begin{pmatrix}e_{1}&e_{2}&i_{V_{S}}\end{pmatrix}^{T}
  20. E x ( t ) + A x ( t ) = f , Ex^{\prime}(t)+Ax(t)=f,
  21. A = ( G - G 1 - G G 0 1 0 0 ) A=\begin{pmatrix}G&-G&1\\ -G&G&0\\ 1&0&0\end{pmatrix}
  22. E = ( 0 0 0 0 C 0 0 0 0 ) E=\begin{pmatrix}0&0&0\\ 0&C&0\\ 0&0&0\end{pmatrix}
  23. f = ( 0 0 V s ) T f=\begin{pmatrix}0&0&V_{s}\end{pmatrix}^{T}
  24. E E

Modularity_(networks).html

  1. n n
  2. m m
  3. s s
  4. v v
  5. s v = 1 s_{v}=1
  6. v v
  7. s v = - 1 s_{v}=-1
  8. A A
  9. A v w = 0 A_{vw}=0
  10. v v
  11. w w
  12. A v w = 1 A_{vw}=1
  13. A v w = A w v A_{vw}=A_{wv}
  14. l n = v k v = 2 m l_{n}=\sum_{v}k_{v}=2m
  15. Expectation of full edges between v and w = (Full edges between v and w ) / (Total number of rewiring possibilities) \,\text{Expectation of full edges between }v\,\text{ and }w={\,\text{ (Full % edges between }v\,\text{ and }w\,\text{)}}/{\,\text{(Total number of rewiring % possibilities)}}
  16. Q = 1 2 m v w [ A v w - k v * k w 2 m ] s v s w + 1 2 Q=\frac{1}{2m}\sum_{vw}\left[A_{vw}-\frac{k_{v}*k_{w}}{2m}\right]\frac{s_{v}s_% {w}+1}{2}
  17. Q = v w [ A v w 2 m - k v * k w ( 2 m ) ( 2 m ) ] δ ( c v , c w ) = i = 1 c ( e i i - a i 2 ) Q=\sum_{vw}\left[\frac{A_{vw}}{2m}-\frac{k_{v}*k_{w}}{(2m)(2m)}\right]\delta(c% _{v},c_{w})=\sum_{i=1}^{c}(e_{ii}-a_{i}^{2})
  18. e i j = v w A v w 2 m 1 v c i 1 w c j e_{ij}=\sum_{vw}\frac{A_{vw}}{2m}1_{v\in c_{i}}1_{w\in c_{j}}
  19. a i = k i 2 m = j e i j a_{i}=\frac{k_{i}}{2m}=\sum_{j}e_{ij}
  20. δ ( c v , c w ) = r S v r S w r \delta(c_{v},c_{w})=\sum_{r}S_{vr}S_{wr}
  21. Q = 1 2 m v w r [ A v w - k v k w 2 m ] S v r S w r = 1 2 m Tr ( 𝐒 T 𝐁𝐒 ) , Q=\frac{1}{2m}\sum_{vw}\sum_{r}\left[A_{vw}-\frac{k_{v}k_{w}}{2m}\right]S_{vr}% S_{wr}=\frac{1}{2m}\mathrm{Tr}(\mathbf{S}^{\mathrm{T}}\mathbf{BS}),
  22. B v w = A v w - k v k w 2 m . B_{vw}=A_{vw}-\frac{k_{v}k_{w}}{2m}.
  23. Q = 1 2 m v w B v w s v s w = 1 2 m 𝐬 T 𝐁𝐬 , Q={1\over 2m}\sum_{vw}B_{vw}s_{v}s_{w}={1\over 2m}\mathbf{s}^{\mathrm{T}}% \mathbf{Bs},

Modulus_(algebraic_number_theory).html

  1. 𝐦 = 𝐩 𝐩 ν ( 𝐩 ) , ν ( 𝐩 ) 0 \mathbf{m}=\prod_{\mathbf{p}}\mathbf{p}^{\nu(\mathbf{p})},\,\,\nu(\mathbf{p})\geq 0
  2. a b ( mod 𝐩 ν ) ord 𝐩 ( a b - 1 ) ν a\equiv^{\ast}\!b\,(\mathrm{mod}\,\mathbf{p}^{\nu})\Leftrightarrow\mathrm{ord}% _{\mathbf{p}}\left(\frac{a}{b}-1\right)\geq\nu
  3. a b ( mod 𝐩 ) a b > 0 a\equiv^{\ast}\!b\,(\mathrm{mod}\,\mathbf{p})\Leftrightarrow\frac{a}{b}>0
  4. K 𝐦 , 1 = { a K × : a 1 ( mod 𝐦 ) } . K_{\mathbf{m},1}=\left\{a\in K^{\times}:a\equiv^{\ast}\!1\,(\mathrm{mod}\,% \mathbf{m})\right\}.

Molecular_conductance.html

  1. G = I / V G=I/V

Momentum_compaction.html

  1. α p = d L / L d p / p = p L d L d p = 1 L D x ( s ) ρ ( s ) d s \alpha_{p}=\frac{\mathrm{d}L/L}{\mathrm{d}p/p}=\frac{p}{L}\frac{\mathrm{d}L}{% \mathrm{d}p}=\frac{1}{L}\oint\frac{D_{x}(s)}{\rho(s)}\mathrm{d}s
  2. η \eta
  3. D x D_{x}
  4. ρ \rho
  5. α p = 1 γ 2 - η \alpha_{p}=\frac{1}{\gamma^{2}}-\eta
  6. γ \gamma

Monad_(non-standard_analysis).html

  1. monad ( x ) = { y * x - y is infinitesimal } . \,\text{monad}(x)=\{y\in\mathbb{R}^{*}\mid x-y\,\text{ is infinitesimal}\}.

Monge_cone.html

  1. F ( x , y , u , u x , u y ) = 0 ( 1 ) F(x,y,u,u_{x},u_{y})=0\qquad\qquad(1)
  2. F u x F_{u_{x}}
  3. F u y F_{u_{y}}
  4. z 0 = u ( x 0 , y 0 ) . ( 2 ) z_{0}=u(x_{0},y_{0}).\qquad\qquad(2)
  5. z = u ( x , y ) z=u(x,y)\,
  6. a d x + b d y + c d z F ( x , y , z , - a / c , - b / c ) . a\,dx+b\,dy+c\,dz\mapsto F(x,y,z,-a/c,-b/c).
  7. F ( x 1 , , x n , u , u x 1 , , u x n ) = 0. F\left(x_{1},\dots,x_{n},u,\frac{\partial u}{\partial x_{1}},\dots,\frac{% \partial u}{\partial x_{n}}\right)=0.
  8. ( x 1 0 , , x n 0 , z 0 ) (x_{1}^{0},\dots,x_{n}^{0},z^{0})
  9. u ( x 1 0 , , x n 0 ) = z 0 u(x_{1}^{0},\dots,x_{n}^{0})=z^{0}
  10. | u | 2 = 1 , |\nabla u|^{2}=1,
  11. F ( x , y , u , u x , u y ) = u x 2 + u y 2 - 1. F(x,y,u,u_{x},u_{y})=u_{x}^{2}+u_{y}^{2}-1.
  12. a 2 + b 2 - c 2 = 0. a^{2}+b^{2}-c^{2}=0.

Monge_equation.html

  1. F ( u , x 1 , x 2 , , x n , u x 1 , , u x n ) = 0 F\left(u,x_{1},x_{2},\dots,x_{n},\frac{\partial u}{\partial x_{1}},\dots,\frac% {\partial u}{\partial x_{n}}\right)=0
  2. i 0 + + i n = k P i 0 i n ( x 0 , x 1 , , x k ) d x 0 i 0 d x 1 i 1 d x n i n = 0 \sum_{i_{0}+\cdots+i_{n}=k}P_{i_{0}\dots i_{n}}(x_{0},x_{1},\dots,x_{k})\,dx_{% 0}^{i_{0}}\,dx_{1}^{i_{1}}\cdots dx_{n}^{i_{n}}=0

Monogenic_field.html

  1. K = 𝐐 ( d ) K=\mathbf{Q}(\sqrt{d})
  2. d d
  3. O K = 𝐙 [ a ] O_{K}=\mathbf{Z}[a]
  4. a = ( 1 + d ) / 2 a=(1+\sqrt{d})/2
  5. a = d a=\sqrt{d}
  6. K = 𝐐 ( ζ ) K=\mathbf{Q}(\zeta)
  7. ζ \zeta
  8. O K = 𝐙 [ ζ ] . O_{K}=\mathbf{Z}[\zeta].
  9. 𝐐 ( ζ ) + = 𝐐 ( ζ + ζ - 1 ) \mathbf{Q}(\zeta)^{+}=\mathbf{Q}(\zeta+\zeta^{-1})
  10. 𝐙 [ ζ + ζ - 1 ] . \mathbf{Z}[\zeta+\zeta^{-1}].
  11. X 3 - X 2 - 2 X - 8 X^{3}-X^{2}-2X-8

Monogenic_semigroup.html

  1. a \langle a\rangle
  2. a \langle a\rangle
  3. a \langle a\rangle
  4. a \langle a\rangle
  5. a \langle a\rangle
  6. a \langle a\rangle
  7. a \langle a\rangle
  8. a \langle a\rangle
  9. a \langle a\rangle
  10. a \langle a\rangle

Monopsony.html

  1. w w
  2. L L
  3. w ( L ) w(L)
  4. w ( L ) L w(L)\cdot L
  5. R R
  6. L L
  7. R ( L ) R(L)
  8. L L
  9. P P
  10. P ( L ) = R ( L ) - w ( L ) L P(L)=R(L)-w(L)\cdot L\,\!
  11. P ( L ) = 0 P^{\prime}(L)=0
  12. 0 = R ( L ) - w ( L ) L - w ( L ) 0=R^{\prime}(L)-w^{\prime}(L)\cdot L-w(L)
  13. w ( L ) w^{\prime}(L)
  14. w ( L ) , w(L),
  15. R ( L ) = w ( L ) L + w ( L ) . R^{\prime}(L)=w^{\prime}(L)\cdot L+w(L).
  16. R ( L ) R^{\prime}(L)
  17. w ( L ) w(L)
  18. w ( L ) L w^{\prime}(L)L\,\!
  19. e = R ( w ) - w w e=\frac{R^{\prime}(w)-w}{w}\,\!
  20. e e
  21. e e
  22. w = { w m i n , if w m i n w ( L ) w ( L ) , if w m i n w ( L ) w=\begin{cases}w_{min},&\mbox{if }~{}w_{min}\geq\;w(L)\\ w(L),&\mbox{if }~{}w_{min}\leq\;w(L)\end{cases}\,\!
  23. w ( L ) w(L)
  24. w m i n w_{min}
  25. w ( L ) = w m i n w(L)=w_{min}\,\!

Montessori_sensorial_materials.html

  1. ( a + b ) 3 (a+b)^{3}
  2. a 3 + 3 a 2 b + 3 a b 2 + b 3 a^{3}+3a^{2}b+3ab^{2}+b^{3}
  3. ( a + b + c ) 3 = a 3 + 3 a 2 b + 3 a 2 c + b 3 + 3 a b 2 + 3 b 2 c + c 3 + 3 a c 2 + 3 b c 2 + 6 a b c (a+b+c)^{3}=a^{3}+3a^{2}b+3a^{2}c+b^{3}+3ab^{2}+3b^{2}c+c^{3}+3ac^{2}+3bc^{2}+% 6abc

Moody_chart.html

  1. Δ P \Delta P
  2. h f h_{\mathrm{f}}
  3. h f = f l d V 2 2 g ; {h_{\mathrm{f}}=f\frac{l}{d}\frac{V^{2}}{2\,g}};
  4. h f = 4 f l d V 2 2 g , {h_{\mathrm{f}}=4f\frac{l}{d}\frac{V^{2}}{2\,g}},
  5. Δ P = ρ g h f \Delta P=\rho\,g\,h_{\mathrm{f}}
  6. Δ P = f ρ V 2 2 l d , \Delta P=f\frac{\rho V^{2}}{2}\frac{l}{d},
  7. ρ \rho
  8. V V
  9. f f
  10. l l
  11. d d
  12. ϵ / d \epsilon/d
  13. 64 / R e 64/Re
  14. f f
  15. 1 f = - 2.0 log 10 ( ϵ / d 3.7 + 2.51 R e f ) , turbulent flow . {1\over\sqrt{\mathit{f}}}=-2.0\log_{10}\left(\frac{\epsilon/d}{3.7}+{\frac{2.5% 1}{Re\sqrt{\mathit{f}}}}\right),\,\text{turbulent flow}.

Moore_matrix.html

  1. M = [ α 1 α 1 q α 1 q n - 1 α 2 α 2 q α 2 q n - 1 α 3 α 3 q α 3 q n - 1 α m α m q α m q n - 1 ] M=\begin{bmatrix}\alpha_{1}&\alpha_{1}^{q}&\dots&\alpha_{1}^{q^{n-1}}\\ \alpha_{2}&\alpha_{2}^{q}&\dots&\alpha_{2}^{q^{n-1}}\\ \alpha_{3}&\alpha_{3}^{q}&\dots&\alpha_{3}^{q^{n-1}}\\ \vdots&\vdots&\ddots&\vdots\\ \alpha_{m}&\alpha_{m}^{q}&\dots&\alpha_{m}^{q^{n-1}}\\ \end{bmatrix}
  2. M i , j = α i q j - 1 M_{i,j}=\alpha_{i}^{q^{j-1}}
  3. det ( V ) = 𝐜 ( c 1 α 1 + + c n α n ) , \det(V)=\prod_{\mathbf{c}}\left(c_{1}\alpha_{1}+\cdots+c_{n}\alpha_{n}\right),
  4. det ( V ) = 1 i n c 1 , , c i - 1 ( c 1 α 1 + + c i - 1 α i - 1 + α i ) . \det(V)=\prod_{1\leq i\leq n}\prod_{c_{1},\dots,c_{i-1}}\left(c_{1}\alpha_{1}+% \cdots+c_{i-1}\alpha_{i-1}+\alpha_{i}\right).

Morphological_gradient.html

  1. f : E R f:E\mapsto R
  2. b ( x ) b(x)
  3. b ( x ) = { 0 , | x | 1 , - , otherwise b(x)=\left\{\begin{array}[]{ll}0,&|x|\leq 1,\\ -\infty,&\mbox{otherwise}\end{array}\right.
  4. G ( f ) = f b - f b G(f)=f\oplus b-f\ominus b
  5. \oplus
  6. \ominus
  7. G i ( f ) = f - f b G_{i}(f)=f-f\ominus b
  8. G e ( f ) = f b - f G_{e}(f)=f\oplus b-f
  9. G i + G e = G G_{i}+G_{e}=G
  10. b ( 0 ) 0 b(0)\geq 0

Morphological_skeleton.html

  1. X 2 X\subset\mathbb{R}^{2}
  2. S ( X ) = ρ > 0 μ > 0 [ ( X ρ B ) - ( X ρ B ) μ B ¯ ] S(X)=\bigcup_{\rho>0}\bigcap_{\mu>0}\left[(X\ominus\rho B)-(X\ominus\rho B)% \circ\mu\overline{B}\right]
  3. \ominus
  4. \circ
  5. ρ B \rho B
  6. ρ \rho
  7. B ¯ \overline{B}
  8. B B
  9. { n B } \{nB\}
  10. n = 0 , 1 , n=0,1,\ldots
  11. n B = B B n times nB=\underbrace{B\oplus\cdots\oplus B}_{n\mbox{ times}~{}}
  12. 0 B = { o } 0B=\{o\}
  13. X 2 X\subset\mathbb{Z}^{2}
  14. { S n ( X ) } \{S_{n}(X)\}
  15. n = 0 , 1 , , N n=0,1,\ldots,N
  16. S n ( X ) = ( X n B ) - ( X n B ) B S_{n}(X)=(X\ominus nB)-(X\ominus nB)\circ B
  17. { S n ( X ) } \{S_{n}(X)\}
  18. X = n ( S n ( X ) n B ) X=\bigcup_{n}(S_{n}(X)\oplus nB)
  19. n m ( S n ( X ) n B ) = X m B \bigcup_{n\geq m}(S_{n}(X)\oplus nB)=X\circ mB
  20. n B z nB_{z}
  21. n B nB
  22. n B z = { x E | x - z n B } nB_{z}=\{x\in E|x-z\in nB\}
  23. n B z nB_{z}
  24. n B z A nB_{z}\in A
  25. n B z m B y nB_{z}\subseteq mB_{y}
  26. m B y A mB_{y}\not\subseteq A
  27. S n ( X ) S_{n}(X)

Motor_cognition.html

  1. 1 2 \tfrac{1}{2}
  2. 1 2 \tfrac{1}{2}
  3. 1 2 \tfrac{1}{2}

Mott_Criterion.html

  1. n - 1 / 3 < C a 0 * , n^{-1/3}<Ca_{0}^{*},
  2. n ~{}n
  3. a 0 * a_{0}^{*}
  4. C C

Mott_transition.html

  1. 1 2 k B T \tfrac{1}{2}k_{B}T
  2. a H a_{H}
  3. N 1 / 3 a H 0.2 N^{1/3}a_{H}\simeq 0.2

Mountain_climbing_problem.html

  1. f f
  2. g g
  3. [ 0 , 1 ] [0,1]
  4. [ 0 , 1 ] [0,1]
  5. f ( 0 ) = g ( 0 ) = 0 f(0)=g(0)=0
  6. f ( 1 ) = g ( 1 ) = 1 f(1)=g(1)=1
  7. s s
  8. t t
  9. [ 0 , 1 ] [0,1]
  10. [ 0 , 1 ] [0,1]
  11. s ( 0 ) = t ( 0 ) = 0 s(0)=t(0)=0
  12. s ( 1 ) = t ( 1 ) = 1 s(1)=t(1)=1
  13. f s = g t f\circ s\,=\,g\circ t
  14. \circ
  15. f f
  16. g g
  17. G G
  18. G G
  19. G G
  20. G G
  21. ( 0 , 0 ) (0,0)
  22. ( 1 , 1 ) (1,1)
  23. G G
  24. ( 0 , 0 ) (0,0)
  25. ( 0 , 0 ) (0,0)
  26. ( 1 , 1 ) (1,1)
  27. G G
  28. ( 0 , 0 ) (0,0)
  29. ( 1 , 1 ) (1,1)
  30. G G

Moving-average_model.html

  1. X t = μ + ε t + θ 1 ε t - 1 + + θ q ε t - q X_{t}=\mu+\varepsilon_{t}+\theta_{1}\varepsilon_{t-1}+\cdots+\theta_{q}% \varepsilon_{t-q}\,
  2. X t = μ + ( 1 + θ 1 B + + θ q B q ) ε t . X_{t}=\mu+(1+\theta_{1}B+\cdots+\theta_{q}B^{q})\varepsilon_{t}.
  3. ε t - 1 \varepsilon_{t-1}
  4. X t X_{t}
  5. ε t - 1 \varepsilon_{t-1}
  6. X t X_{t}
  7. X t - 1 X_{t-1}
  8. X t - 1 X_{t-1}
  9. X t X_{t}
  10. ε t - 1 \varepsilon_{t-1}
  11. X t X_{t}
  12. X X
  13. X X
  14. ε t \varepsilon_{t}
  15. X t X_{t}
  16. X t + 1 X_{t+1}
  17. X t + 2 X_{t+2}

Moving_target_indication.html

  1. D o p p l e r = 180 , 000 o / s = 720 ( 75 × 10 9 3 × 10 8 ) = 720 ( V e l o c i t y × T r a n s m i t F r e q u e n c y C ) Doppler=180,000^{o}/s=720\left(\frac{75\times 10^{9}}{3\times 10^{8}}\right)=7% 20\left(\frac{Velocity\times Transmit\ Frequency}{C}\right)
  2. 360 o 360^{o}
  3. v p v_{p}
  4. R m a x R_{max}
  5. E L EL
  6. A Z AZ
  7. M D V = λ 2 ( 4 v p B ( sin ( A Z ) sin ( E L ) ) 2 + ( cos ( A Z ) cos ( E L ) ) 2 ) MDV=\frac{\lambda}{2}\left(\frac{4v_{p}}{B}\sqrt{(\sin(AZ)\sin(EL))^{2}+(\cos(% AZ)\cos(EL))^{2}}\right)

Möbius_configuration.html

  1. ( 0 , 0 , 0 ) , (0,0,0),
  2. ( 0 , 0 , 1 ) , (0,0,1),
  3. ( 0 , 1 , 0 ) (0,1,0)
  4. ( 1 , 0 , 0 ) (1,0,0)
  5. ( 0 , - γ , γ ) , (0,-\gamma,\gamma),
  6. ( γ , 0 , - γ ) , (\gamma,0,-\gamma),
  7. ( - γ , γ , 0 ) (-\gamma,\gamma,0)
  8. ( λ , λ , λ ) , (\lambda,\lambda,\lambda),
  9. γ = 1 2 \gamma=\frac{1}{\sqrt{2}}
  10. λ = 1 3 \lambda=\frac{1}{3}
  11. 8 4 8_{4}
  12. 8 4 8_{4}
  13. 8 4 8_{4}
  14. S 4 S_{4}
  15. A 0 , B 0 , C 0 , D 0 A_{0},B_{0},C_{0},D_{0}
  16. A 1 , B 1 , C 1 , D 1 A_{1},B_{1},C_{1},D_{1}
  17. A i , B j , C k , D l A_{i},B_{j},C_{k},D_{l}
  18. i + j + k + l i+j+k+l
  19. 8 4 8_{4}
  20. 8 4 8_{4}
  21. 8 4 8_{4}

Möbius_energy.html

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

Muckenhoupt_weights.html

  1. ω ω
  2. f f
  3. M ( f ) M(f)
  4. M ( f ) ( x ) = sup r > 0 1 r n B r ( x ) | f | , M(f)(x)=\sup_{r>0}\frac{1}{r^{n}}\int_{B_{r}(x)}|f|,
  5. r r
  6. x x
  7. | M ( f ) ( x ) | p ω ( x ) d x C | f | p ω ( x ) d x , \int|M(f)(x)|^{p}\,\omega(x)dx\leq C\int|f|^{p}\,\omega(x)\,dx,
  8. C C
  9. p p
  10. ω ω
  11. C C
  12. B B
  13. ( 1 | B | B ω ( x ) d x ) ( 1 | B | B ω ( x ) - q p d x ) p q C < , \left(\frac{1}{|B|}\int_{B}\omega(x)\,dx\right)\left(\frac{1}{|B|}\int_{B}% \omega(x)^{-\frac{q}{p}}\,dx\right)^{\frac{p}{q}}\leq C<\infty,
  14. | B | |B|
  15. B B
  16. q q
  17. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  18. C C
  19. 1 | B | B ω ( x ) d x C ω ( x ) , \frac{1}{|B|}\int_{B}\omega(x)\,dx\leq C\omega(x),
  20. x B x∈B
  21. B B
  22. ω ω
  23. | M ( f ) ( x ) | p ω ( x ) d x C | f | p ω ( x ) d x , \int|M(f)(x)|^{p}\,\omega(x)\,dx\leq C\int|f|^{p}\,\omega(x)\,dx,
  24. C C
  25. p p
  26. A A
  27. c c
  28. f f
  29. B B
  30. ( f B ) p c ω ( B ) B f ( x ) p ω ( x ) d x , (f_{B})^{p}\leq\frac{c}{\omega(B)}\int_{B}f(x)^{p}\,\omega(x)\,dx,
  31. f B = 1 | B | B f , ω ( B ) = B ω ( x ) d x . f_{B}=\frac{1}{|B|}\int_{B}f,\qquad\omega(B)=\int_{B}\omega(x)\,dx.
  32. sup B 1 | B | B e φ - φ B d x < \sup_{B}\frac{1}{|B|}\int_{B}e^{\varphi-\varphi_{B}}dx<\infty
  33. sup B 1 | B | B e - φ - φ B p - 1 d x < . \sup_{B}\frac{1}{|B|}\int_{B}e^{-\frac{\varphi-\varphi_{B}}{p-1}}dx<\infty.
  34. E B E⊂B
  35. | E | γ | B | |E|≤γ|B|
  36. ω ( E ) δ ω ( B ) ω(E)≤δω(B)
  37. c c
  38. ω ω
  39. B B
  40. 1 | B | B ω q ( c | B | B ω ) q . \frac{1}{|B|}\int_{B}\omega^{q}\leq\left(\frac{c}{|B|}\int_{B}\omega\right)^{q}.
  41. ω ω
  42. l o g ( w ) B M O log(w)∈BMO
  43. l o g ( w ) log(w)
  44. f B M O f∈BMO
  45. δ > 0 δ>0
  46. p 1 p≥1
  47. δ > 0 δ>0
  48. l o g | x | B M O −log|x|∈BMO
  49. e - log | x | = 1 e log | x | = 1 | x | e^{-\log|x|}=\frac{1}{e^{\log|x|}}=\frac{1}{|x|}
  50. A 1 A p A , 1 p . A_{1}\subseteq A_{p}\subseteq A_{\infty},\qquad 1\leq p\leq\infty.
  51. A = p < A p . A_{\infty}=\bigcup_{p<\infty}A_{p}.
  52. w d x wdx
  53. B B
  54. 2 B 2B
  55. w ( 2 B ) C w ( B ) w(2B)≤Cw(B)
  56. C > 1 C>1
  57. w w
  58. δ > 1 δ>1
  59. δ > 0 δ>0
  60. w 1 , w 2 A 1 w_{1},w_{2}\in A_{1}
  61. w = w 1 w 2 - δ w=w_{1}w_{2}^{-\delta}
  62. T T
  63. f C c : T ( f ) L 2 C f L 2 . \forall f\in C^{\infty}_{c}:\qquad\|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}}.
  64. T T
  65. K K
  66. f , g f,g
  67. g ( x ) T ( f ) ( x ) d x = g ( x ) K ( x - y ) f ( y ) d y d x . \int g(x)T(f)(x)\,dx=\iint g(x)K(x-y)f(y)\,dy\,dx.
  68. K K
  69. x 0 , | α | 1 : | α K | C | x | - n - α . \forall x\neq 0,\forall|\alpha|\leq 1:\qquad\left|\partial^{\alpha}K\right|% \leq C|x|^{-n-\alpha}.
  70. | T ( f ) ( x ) | p ω ( x ) d x C | f ( x ) | p ω ( x ) d x , \int|T(f)(x)|^{p}\,\omega(x)\,dx\leq C\int|f(x)|^{p}\,\omega(x)\,dx,
  71. f f
  72. K K
  73. | K ( x ) | a | x | - n |K(x)|\geq a|x|^{-n}
  74. x = t u ˙ 0 x=t\dot{u}_{0}
  75. | T ( f ) ( x ) | p ω ( x ) d x C | f ( x ) | p ω ( x ) d x , \int|T(f)(x)|^{p}\,\omega(x)\,dx\leq C\int|f(x)|^{p}\,\omega(x)\,dx,
  76. K > 1 K>1
  77. K K
  78. f W l o c 1 , 2 ( 𝐑 n ) , and D f ( x ) n J ( f , x ) K , f\in W^{1,2}_{loc}(\mathbf{R}^{n}),\quad\,\text{ and }\quad\frac{\|Df(x)\|^{n}% }{J(f,x)}\leq K,
  79. D f ( x ) Df(x)
  80. f f
  81. x x
  82. J ( f , x ) = d e t ( D f ( x ) ) J(f,x)=det(Df(x))
  83. K K
  84. f : 𝐑 < s u p > n 𝐑 n f:\mathbf{R}<sup>n→\mathbf{R}^{n}

Mucoadhesion.html

  1. cos ( θ ) = γ m g - γ b m γ b g {{{\cos(\theta)\;=\;\frac{\gamma_{mg}\;-\gamma_{bm}\;}{\gamma_{bg}}}}}
  2. γ m g \gamma_{mg}
  3. γ b m \gamma_{bm}
  4. γ b g \gamma_{bg}
  5. θ \theta
  6. l = ( t × D b ) 1 / 2 {{l={(t\times D_{b}})^{1/2}}}
  7. t t
  8. D b D_{b}

Multi-jackbolt_tensioner.html

  1. M A = T T j = n D A d MA=\frac{T}{T_{j}}=\frac{nDA}{d}

Multi-junction_solar_cell.html

  1. n L 2 = n A l I n P 1 / 2 n L 1 n_{L2}=n_{AlInP}^{1/2}\cdot n_{L1}
  2. l d e p l = 2 ϵ ( ϕ 0 - V ) q N A + N D N A N D l_{depl}=\sqrt{\frac{2\epsilon(\phi_{0}-V)}{q}\frac{N_{A}+N_{D}}{N_{A}N_{D}}}
  3. J P J_{P}
  4. J = J S ( exp ( q V k T ) - 1 ) J=J_{S}\left(\exp\left(\frac{qV}{kT}\right)-1\right)
  5. Q E i ( λ ) = J S C i ( λ ) q ϕ i ( λ ) J S C i = 0 λ 2 q ϕ i ( λ ) Q E i ( λ ) d λ QE_{i}(\lambda)=\frac{J_{SCi}(\lambda)}{q\phi_{i}(\lambda)}\Rightarrow J_{SCi}% =\int_{0}^{\lambda 2}q\phi_{i}(\lambda)QE_{i}(\lambda)\,d\lambda
  6. Q E i ( λ ) QE_{i}(\lambda)
  7. α ( λ ) \alpha(\lambda)
  8. Q E i ( λ ) = 1 - e - α ( λ ) d i QE_{i}(\lambda)=1-e^{-\alpha(\lambda)d_{i}}
  9. e - α ( λ ) d i e^{-\alpha(\lambda)d_{i}}
  10. V = i = 1 3 V i V=\sum_{i=1}^{3}V_{i}
  11. V O C = i = 1 3 V O C i V_{OC}=\sum_{i=1}^{3}V_{OCi}
  12. V O C i V_{OCi}
  13. J i = J 0 i ( e q V i k T - 1 ) - J S C i V O C i k T q ln ( J S C i J 0 i ) J_{i}=J_{0i}\left(e^{\frac{qV_{i}}{kT}}-1\right)-J_{SCi}\Rightarrow V_{OCi}% \approx\frac{kT}{q}\ln(\frac{J_{SCi}}{J_{0i}})
  14. d n p h d h v = E E p h \frac{dn_{ph}}{dhv}\,=\frac{E}{E_{ph}}\,
  15. n p h ( E g ) = E g d n p h d h v d h v = i = E g ( h v i + 1 - h v i ) 1 2 ( d n p h d h v ( h v i + 1 ) + d n p h d h v ( h v i ) ) n_{ph}(E_{g})=\int_{E_{g}}^{\infty}\frac{dn_{ph}}{dhv}\,dhv=\sum_{i=E_{g}}^{% \infty}(hv_{i+1}-hv_{i})\frac{1}{2}(\frac{dn_{ph}}{dhv}(hv_{i+1})+\frac{dn_{ph% }}{dhv}(hv_{i}))\,
  16. J r a d = A e x p ( e V - E g k T ) J_{rad}=Aexp(\frac{eV-E_{g}}{kT})\,
  17. A = 2 π e x p ( n 2 + 1 ) E g 2 k T h 3 c 2 A=\frac{2\pi\,exp(n^{2}+1)E_{g}^{2}kT}{h^{3}c^{2}}\,
  18. J t h = A e x p ( - E g k T ) J_{th}=Aexp(\frac{-E_{g}}{kT})\,
  19. J = e n p h - A e x p ( e V - E g k T ) J=en_{ph}-Aexp(\frac{eV-E_{g}}{kT})\,
  20. e V O C = E g - k T l n ( A e n p h ) eV_{OC}=E_{g}-kTln(\frac{A}{en_{ph}})\,
  21. d J V d V = 0 \frac{dJV}{dV}\,=0
  22. e V m = e V O C - k T l n ( 1 + e V m k T ) eV_{m}=eV_{OC}-kTln(1+\frac{eV_{m}}{kT})\,
  23. J m = e n p h 1 + k T / e V m J_{m}=\frac{en_{ph}}{1+kT/eV_{m}}\,
  24. W m = J m V m n p h = e V m 1 + k T / e V m = e V m - k T W_{m}=\frac{J_{m}V_{m}}{n_{ph}}\,=\frac{eV_{m}}{1+kT/eV_{m}}\,=eV_{m}-kT
  25. W m = E g - k T [ l n ( A e n p h ) + l n ( 1 + e V m k T ) + 1 ] W_{m}=E_{g}-kT[ln(\frac{A}{en_{ph}})+ln(1+\frac{eV_{m}}{kT})+1]\,

Multibrot_set.html

  1. z z d + c . z\mapsto z^{d}+c.\,
  2. d = 2 d=2\,
  3. \mapsto
  4. \mapsto
  5. \mapsto
  6. \mapsto
  7. \mapsto
  8. \mapsto
  9. \mapsto
  10. \mapsto
  11. \mapsto
  12. \mapsto
  13. \mapsto
  14. \mapsto
  15. λ = 1 N ln | 𝐳 | \lambda=\frac{1}{N}\ln|\mathbf{z}|
  16. \mapsto

Multicritical_point.html

  1. T c T_{c}
  2. T c T_{c}
  3. T c T_{c}
  4. P P
  5. T c T_{c}
  6. P P
  7. T , P T,P
  8. 1 1
  9. K K
  10. T c T_{c}
  11. P , K P,K
  12. T , P , K T,P,K
  13. 2 2
  14. d > 1 d>1
  15. d > 2 d>2
  16. d - 1 d-1
  17. d - 2 d-2
  18. 2 2
  19. 1 1
  20. P P
  21. T T
  22. ϕ 4 \phi^{4}
  23. ( ϕ ) 2 \left(\nabla\phi\right)^{2}

Multidimensional_Chebyshev's_inequality.html

  1. μ = 𝔼 [ X ] \mu=\mathbb{E}\left[X\right]
  2. V = 𝔼 [ ( X - μ ) ( X - μ ) T ] . V=\mathbb{E}\left[\left(X-\mu\right)\left(X-\mu\right)^{T}\right].\,
  3. V V
  4. t > 0 t>0
  5. Pr ( ( X - μ ) T V - 1 ( X - μ ) > t ) N t 2 \mathrm{Pr}\left(\sqrt{\left(X-\mu\right)^{T}\,V^{-1}\,\left(X-\mu\right)}>t% \right)\leq\frac{N}{t^{2}}
  6. V V
  7. V - 1 V^{-1}
  8. y = ( X - μ ) T V - 1 ( X - μ ) . y=\left(X-\mu\right)^{T}\,V^{-1}\,\left(X-\mu\right).
  9. y y
  10. Pr ( ( X - μ ) T V - 1 ( X - μ ) > t ) = Pr ( y > t ) = Pr ( y > t 2 ) 𝔼 [ y ] t 2 . \begin{array}[]{lll}\mathrm{Pr}\left(\sqrt{\left(X-\mu\right)^{T}\,V^{-1}\,% \left(X-\mu\right)}>t\right)&=\mathrm{Pr}\left(\sqrt{y}>t\right)\\ &=\mathrm{Pr}\left(y>t^{2}\right)\\ &\leq\frac{\mathbb{E}[y]}{t^{2}}.\end{array}
  11. 𝔼 [ y ] = 𝔼 [ ( X - μ ) T V - 1 ( X - μ ) ] = 𝔼 [ trace ( V - 1 ( X - μ ) ( X - μ ) T ) ] = trace ( V - 1 V ) = N . \begin{array}[]{lll}\mathbb{E}[y]&=\mathbb{E}[\left(X-\mu\right)^{T}\,V^{-1}\,% \left(X-\mu\right)]\\ &=\mathbb{E}[\mathrm{trace}(V^{-1}\,\left(X-\mu\right)\,\left(X-\mu\right)^{T}% )]\\ &=\mathrm{trace}(V^{-1}V)=N\end{array}.

Multidimensional_panel_data.html

  1. X i s t h , i = 1 , , N , s = 1 , , S , t = 1 , , T , h = 1 , , H , X_{isth},\;i=1,\dots,N,\;s=1,\dots,S,\;t=1,\dots,T,\;h=1,\dots,H,\,
  2. y i s t h = α + X s i t h β + u s i t h . y_{isth}=\alpha+X_{sith}\beta+u_{sith}.\,

Multielectrode_array.html

  1. V p a d = V o v e r l a p × A o v e r l a p A e l e c t r o d e V_{pad}=V_{overlap}\times\frac{A_{overlap}}{A_{electrode}}

Multinomial_test.html

  1. N N
  2. k k
  3. 𝐱 = ( x 1 , x 2 , , x k ) \mathbf{x}=(x_{1},x_{2},\dots,x_{k})
  4. i = 1 k x i = N \textstyle\sum_{i=1}^{k}x_{i}=N
  5. H 0 : π = ( π 1 , π 2 , , π k ) H_{0}:\mathbf{\pi}=(\pi_{1},\pi_{2},\dots,\pi_{k})
  6. i = 1 k π i = 1 \textstyle\sum_{i=1}^{k}\pi_{i}=1
  7. 𝐱 \mathbf{x}
  8. Pr ( 𝐱 ) 𝟎 = N ! i = 1 k π i x i x i ! . \Pr(\mathbf{x)_{0}}=N!\prod_{i=1}^{k}\frac{\pi_{i}^{x_{i}}}{x_{i}!}.
  9. Pr ( 𝐬𝐢𝐠 ) = y : P r ( 𝐲 ) P r ( 𝐱 ) 𝟎 Pr ( 𝐲 ) \Pr(\mathbf{sig})=\sum_{y:Pr(\mathbf{y})\leq Pr(\mathbf{x)_{0}}}\Pr(\mathbf{y})
  10. k k
  11. N N
  12. π i \pi_{i}
  13. p i = x i / N p_{i}=x_{i}/N
  14. 𝐱 \mathbf{x}
  15. Pr ( 𝐱 ) 𝐀 = N ! i = 1 k p i x i x i ! . \Pr(\mathbf{x)_{A}}=N!\prod_{i=1}^{k}\frac{p_{i}^{x_{i}}}{x_{i}!}.
  16. - 2 -2
  17. - 2 ln ( L R ) = - 2 i = 1 k x i ln ( π i / p i ) . -2\ln(LR)=\textstyle-2\sum_{i=1}^{k}x_{i}\ln(\pi_{i}/p_{i}).
  18. N N
  19. - 2 ln ( L R ) -2\ln(LR)
  20. k - 1 k-1
  21. - 2 ln ( L R ) -2\ln(LR)
  22. N - 1 N^{-1}
  23. N - 2 N^{-2}
  24. q 1 = 1 + i = 1 k π i - 1 - 1 6 N ( k - 1 ) . q_{1}=1+\frac{\sum_{i=1}^{k}\pi_{i}^{-1}-1}{6N(k-1)}.
  25. π i \pi_{i}
  26. 1 / k 1/k
  27. q 1 = 1 + k + 1 6 N . q_{1}=1+\frac{k+1}{6N}.
  28. N - 3 N^{-3}
  29. π i \pi_{i}
  30. q 2 = 1 + k + 1 6 N + k 2 6 N 2 . q_{2}=1+\frac{k+1}{6N}+\frac{k^{2}}{6N^{2}}.
  31. χ 2 = i = 1 k ( x i - E i ) 2 E i \chi^{2}=\sum_{i=1}^{k}{(x_{i}-E_{i})^{2}\over E_{i}}
  32. E i = N π i E_{i}=N\pi_{i}
  33. i i
  34. k - 1 k-1
  35. - 2 ln ( L R ) -2\ln(LR)
  36. - 2 ln ( L R ) -2\ln(LR)

Multipartite_entanglement.html

  1. m > 2 m>2
  2. m \;m
  3. m \;m
  4. m \;m
  5. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  6. m \;m
  7. A 1 , , A m \;A_{1},\ldots,A_{m}
  8. A 1 A m = A 1 A m \;\mathcal{H}_{A_{1}\ldots A_{m}}=\mathcal{H}_{A_{1}}\otimes\ldots\otimes% \mathcal{H}_{A_{m}}
  9. ϱ A 1 A m = i = 1 k p i ϱ A 1 i ϱ A m i . \;\varrho_{A_{1}\ldots A_{m}}=\sum_{i=1}^{k}p_{i}\varrho_{A_{1}}^{i}\otimes% \ldots\otimes\varrho_{A_{m}}^{i}.
  10. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  11. m \;m
  12. m \;m
  13. i Ω i 1 Ω i n \;\sum_{i}\Omega_{i}^{1}\otimes\ldots\otimes\Omega_{i}^{n}
  14. ϱ A 1 A m i Ω i 1 Ω i n ϱ A 1 A m ( Ω i 1 Ω i n ) T r [ i Ω i 1 Ω i n ϱ A 1 A m ( Ω i 1 Ω i n ) ] . \;\varrho_{A_{1}\ldots A_{m}}\to\frac{\sum_{i}\Omega_{i}^{1}\otimes\ldots% \otimes\Omega_{i}^{n}\varrho_{A_{1}\ldots A_{m}}(\Omega_{i}^{1}\otimes\ldots% \otimes\Omega_{i}^{n})^{\dagger}}{Tr[\sum_{i}\Omega_{i}^{1}\otimes\ldots% \otimes\Omega_{i}^{n}\varrho_{A_{1}\ldots A_{m}}(\Omega_{i}^{1}\otimes\ldots% \otimes\Omega_{i}^{n})^{\dagger}]}.
  15. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  16. m \;m
  17. A 1 , , A m \;A_{1},\ldots,A_{m}
  18. { I 1 , , I k } \;\{I_{1},\ldots,I_{k}\}
  19. I i \;I_{i}
  20. I = { 1 , , m } , j = 1 k I j = I \;I=\{1,\ldots,m\},\cup_{j=1}^{k}I_{j}=I
  21. ϱ A 1 A m = i = 1 N p i ϱ 1 i ϱ k i . \;\varrho_{A_{1}\ldots A_{m}}=\sum_{i=1}^{N}p_{i}\varrho_{1}^{i}\otimes\ldots% \otimes\varrho_{k}^{i}.
  22. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  23. 1 \;1
  24. ( m - 1 ) \;(m-1)
  25. { I 1 = { k } , I 2 = { 1 , , k - 1 , k + 1 , , m } } , 1 k m \;\big\{I_{1}=\{k\},I_{2}=\{1,\ldots,k-1,k+1,\ldots,m\}\big\},1\leq k\leq m
  26. m \;m
  27. s \;s
  28. { I 1 , , I k } \;\{I_{1},\ldots,I_{k}\}
  29. I k \;I_{k}
  30. N s \;N\leq s
  31. | Ψ A 1 A m |\Psi_{A_{1}\ldots A_{m}}\rangle
  32. m \;m
  33. A 1 , , A m \;A_{1},\ldots,A_{m}
  34. m \;m
  35. | Ψ A 1 A m = | ψ A 1 | ψ A m . \;|\Psi_{A_{1}\ldots A_{m}}\rangle=|\psi_{A_{1}}\rangle\otimes\ldots\otimes|% \psi_{A_{m}}\rangle.
  36. | Ψ A 1 A m = i = 1 m i n { d A 1 , , d A m } a i | e A 1 i | e A m i \;|\Psi_{A_{1}\ldots A_{m}}\rangle=\sum_{i=1}^{min\{d_{A_{1}},\ldots,d_{A_{m}}% \}}a_{i}|e_{A_{1}}^{i}\rangle\otimes\ldots\otimes|e_{A_{m}}^{i}\rangle
  37. m \;m
  38. 2 2 2\otimes 2
  39. 2 3 2\otimes 3
  40. Λ A 2 A m : ( A 2 A m ) ( A 1 ) \;\Lambda_{A_{2}\ldots A_{m}}:\mathcal{B}(\mathcal{H}_{A_{2}\ldots A_{m}})\to% \mathcal{B}(\mathcal{H}_{A_{1}})
  41. ( I A 1 Λ A 2 A m ) [ ϱ A 1 A m ] 0 , \;(I_{A_{1}}\otimes\Lambda_{A_{2}\ldots A_{m}})[\varrho_{A_{1}\ldots A_{m}}]% \geq 0,
  42. I A 1 \;I_{A_{1}}
  43. A 1 \;\mathcal{H}_{A_{1}}
  44. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  45. Λ A 2 A m : ( A 2 A m ) ( A 1 ) \;\Lambda_{A_{2}\ldots A_{m}}:\mathcal{B}(\mathcal{H}_{A_{2}\ldots A_{m}})\to% \mathcal{B}(\mathcal{H}_{A_{1}})
  46. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  47. T r ( W ϱ A 1 A m ) 0 \;Tr(W\varrho_{A_{1}\ldots A_{m}})\geq 0
  48. W W
  49. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  50. W \;W
  51. T r ( W ϱ A 1 A m ) < 0 \;Tr(W\varrho_{A_{1}\ldots A_{m}})<0
  52. m \;m
  53. m \;m
  54. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  55. { | ϕ A 1 , , | ϕ A m } \;\{|\phi_{A_{1}}\rangle,\ldots,|\phi_{A_{m}}\rangle\}
  56. ϱ A 1 A m T A k 1 A k l \;\varrho_{A_{1}\ldots A_{m}}^{T_{A_{k_{1}}\ldots A_{k_{l}}}}
  57. { A k 1 A k l } { A 1 A m } \;\{A_{k_{1}}\ldots A_{k_{l}}\}\subset\{A_{1}\ldots A_{m}\}
  58. k 1 , , k l \;k_{1},\ldots,k_{l}
  59. ϱ A 1 A m \;\varrho_{A_{1}\ldots A_{m}}
  60. ϱ A 1 A m T A k 1 A k l \;\varrho_{A_{1}\ldots A_{m}}^{T_{A_{k_{1}}\ldots A_{k_{l}}}}
  61. ϱ A 1 A m \varrho_{A_{1}\ldots A_{m}}
  62. [ R π ( ϱ A 1 A m ) ] i 1 j 1 , i 2 j 2 , , i n j n ϱ π ( i 1 j 1 , i 2 j 2 , , i n j n ) \;[R_{\pi}(\varrho_{A_{1}\ldots A_{m}})]_{i_{1}j_{1},i_{2}j_{2},\ldots,i_{n}j_% {n}}\equiv\varrho_{\pi(i_{1}j_{1},i_{2}j_{2},\ldots,i_{n}j_{n})}
  63. π \;\pi
  64. | | R π ( ϱ A 1 A m ) ] | | T r 1 \;||R_{\pi}(\varrho_{A_{1}\ldots A_{m}})]||_{Tr}\leq 1
  65. E P R A B E P R C D EPR_{AB}\otimes EPR_{CD}
  66. k k
  67. I ( A 1 : : A N ) = S ( A 1 ) + + S ( A N ) - S ( A 1 A N ) I(A_{1}:\ldots:A_{N})=S(A_{1})+\ldots+S(A_{N})-S(A_{1}\ldots A_{N})
  68. τ ( A : B : C ) = τ ( A : B C ) - τ ( A B ) - τ ( A C ) , \;\tau(A:B:C)=\tau(A:BC)-\tau(AB)-\tau(AC),
  69. 2 \;2
  70. log r \;\log r
  71. r \;r
  72. E P R EPR
  73. E g = 1 - Λ k [ ψ ] , \;E_{g}=1-\Lambda^{k}[\psi],
  74. Λ k [ ψ ] = s u p ϕ S k | ψ | ϕ | 2 \;\Lambda^{k}[\psi]=sup_{\phi\in S_{k}}|\langle\psi|\phi\rangle|^{2}
  75. S k \;S_{k}
  76. k \;k

Multiple-try_Metropolis.html

  1. Q ( x ; x t ) = 𝒩 ( x t ; σ 2 I ) Q(x^{\prime};x^{t})=\mathcal{N}(x^{t};\sigma^{2}I)\,
  2. π ( x ) \pi(x)\,
  3. Q ( x ; x t ) = 𝒩 ( x t ; 𝚺 ) Q(x^{\prime};x^{t})=\mathcal{N}(x^{t};\mathbf{\Sigma})
  4. 𝚺 \mathbf{\Sigma}
  5. σ 2 \sigma^{2}\,
  6. σ 2 \sigma^{2}\,
  7. Q ( x ; x t ) = 𝒩 ( x t ; I ) Q(x^{\prime};x^{t})=\mathcal{N}(x^{t};I)\,
  8. N N\,
  9. N \sqrt{N}\,
  10. σ 2 \sigma^{2}\,
  11. 𝚺 \mathbf{\Sigma}
  12. Q ( x ; x t ) = 𝒩 ( x t ; I ) Q(x^{\prime};x^{t})=\mathcal{N}(x^{t};I)\,
  13. x 2 = - x 2 1 2 π e - x 2 2 = 1 \langle x^{2}\rangle=\int_{-\infty}^{\infty}x^{2}\frac{1}{\sqrt{2\pi}}e^{-% \frac{x^{2}}{2}}=1
  14. N N\,
  15. P n ( r ) P_{n}(r)\,
  16. P n ( r ) r n - 1 e - r 2 / 2 P_{n}(r)\propto r^{n-1}e^{-r^{2}/2}
  17. r = N - 1 r=\sqrt{N-1}\,
  18. N \approx\sqrt{N}\,
  19. N N\,
  20. σ 2 \sigma^{2}\,
  21. σ 2 σ 2 / N \sigma^{2}\rightarrow\sigma^{2}/N
  22. σ \sigma\,
  23. σ / N \sigma/\sqrt{N}
  24. σ \sigma\,
  25. Q ( 𝐱 , 𝐲 ) Q(\mathbf{x},\mathbf{y})
  26. Q ( 𝐱 , 𝐲 ) > 0 Q(\mathbf{x},\mathbf{y})>0
  27. Q ( 𝐲 , 𝐱 ) > 0 Q(\mathbf{y},\mathbf{x})>0
  28. π ( 𝐱 ) \pi(\mathbf{x})
  29. w ( 𝐱 , 𝐲 ) = π ( 𝐱 ) Q ( 𝐱 , 𝐲 ) λ ( 𝐱 , 𝐲 ) w(\mathbf{x},\mathbf{y})=\pi(\mathbf{x})Q(\mathbf{x},\mathbf{y})\lambda(% \mathbf{x},\mathbf{y})
  30. λ ( 𝐱 , 𝐲 ) \lambda(\mathbf{x},\mathbf{y})
  31. 𝐱 \mathbf{x}
  32. 𝐲 \mathbf{y}
  33. 𝐱 \mathbf{x}
  34. 𝐲 1 , , 𝐲 k \mathbf{y}_{1},\ldots,\mathbf{y}_{k}
  35. Q ( 𝐱 , . ) Q(\mathbf{x},.)
  36. w ( 𝐲 j , 𝐱 ) w(\mathbf{y}_{j},\mathbf{x})
  37. 𝐲 \mathbf{y}
  38. 𝐲 i \mathbf{y}_{i}
  39. 𝐱 1 , , 𝐱 k - 1 \mathbf{x}_{1},\ldots,\mathbf{x}_{k-1}
  40. Q ( 𝐲 , . ) Q(\mathbf{y},.)
  41. 𝐱 k = 𝐱 \mathbf{x}_{k}=\mathbf{x}
  42. 𝐲 \mathbf{y}
  43. r = min ( 1 , w ( 𝐲 1 , 𝐱 ) + + w ( 𝐲 k , 𝐱 ) w ( 𝐱 1 , 𝐲 ) + + w ( 𝐱 k , 𝐲 ) ) r=\,\text{min}\left(1,\frac{w(\mathbf{y}_{1},\mathbf{x})+\ldots+w(\mathbf{y}_{% k},\mathbf{x})}{w(\mathbf{x}_{1},\mathbf{y})+\ldots+w(\mathbf{x}_{k},\mathbf{y% })}\right)
  44. π ( 𝐱 ) \pi(\mathbf{x})
  45. Q ( 𝐱 , 𝐲 ) Q(\mathbf{x},\mathbf{y})
  46. λ ( 𝐱 , 𝐲 ) = 1 Q ( 𝐱 , 𝐲 ) \lambda(\mathbf{x},\mathbf{y})=\frac{1}{Q(\mathbf{x},\mathbf{y})}
  47. w ( 𝐱 , 𝐲 ) = π ( 𝐱 ) w(\mathbf{x},\mathbf{y})=\pi(\mathbf{x})
  48. 2 k - 1 2k-1

Multiple_zeta_function.html

  1. ζ ( s 1 , , s k ) = n 1 > n 2 > > n k > 0 1 n 1 s 1 n k s k = n 1 > n 2 > > n k > 0 i = 1 k 1 n i s i , \zeta(s_{1},\ldots,s_{k})=\sum_{n_{1}>n_{2}>\cdots>n_{k}>0}\ \frac{1}{n_{1}^{s% _{1}}\cdots n_{k}^{s_{k}}}=\sum_{n_{1}>n_{2}>\cdots>n_{k}>0}\ \prod_{i=1}^{k}% \frac{1}{n_{i}^{s_{i}}},\!
  2. ζ ( 2 , 1 , 2 , 1 , 3 ) = ζ ( { 2 , 1 } 2 , 3 ) \zeta(2,1,2,1,3)=\zeta(\{2,1\}^{2},3)
  3. ζ ( s , t ) = n > m 1 1 n s m t = n = 2 1 n s m = 1 n - 1 1 m t = n = 1 1 ( n + 1 ) s m = 1 n 1 m t \zeta(s,t)=\sum_{n>m\geq 1}\ \frac{1}{n^{s}m^{t}}=\sum_{n=2}^{\infty}\frac{1}{% n^{s}}\sum_{m=1}^{n-1}\frac{1}{m^{t}}=\sum_{n=1}^{\infty}\frac{1}{(n+1)^{s}}% \sum_{m=1}^{n}\frac{1}{m^{t}}
  4. ζ ( s , t ) = n = 1 H n , t ( n + 1 ) s \zeta(s,t)=\sum_{n=1}^{\infty}\frac{H_{n,t}}{(n+1)^{s}}
  5. H n , t H_{n,t}
  6. n = 1 H n ( n + 1 ) 2 = ζ ( 2 , 1 ) = ζ ( 3 ) = n = 1 1 n 3 , \sum_{n=1}^{\infty}\frac{H_{n}}{(n+1)^{2}}=\zeta(2,1)=\zeta(3)=\sum_{n=1}^{% \infty}\frac{1}{n^{3}},\!
  7. ζ ( s , t ) = ζ ( s ) ζ ( t ) + 1 2 [ ( s + t s ) - 1 ] ζ ( s + t ) - r = 1 N - 1 [ ( 2 r s - 1 ) + ( 2 r t - 1 ) ] ζ ( 2 r + 1 ) ζ ( s + t - 1 - 2 r ) \zeta(s,t)=\zeta(s)\zeta(t)+\tfrac{1}{2}\Big[{\textstyle\left({{s+t}\atop{s}}% \right)}-1\Big]\zeta(s+t)-\sum_{r=1}^{N-1}\Big[{\textstyle\left({{2r}\atop{s-1% }}\right)}+{\textstyle\left({{2r}\atop{t-1}}\right)}\Big]\zeta(2r+1)\zeta(s+t-% 1-2r)
  8. 3 4 ζ ( 4 ) \tfrac{3}{4}\zeta(4)
  9. 3 ζ ( 2 ) ζ ( 3 ) - 11 2 ζ ( 5 ) 3\zeta(2)\zeta(3)-\tfrac{11}{2}\zeta(5)
  10. ( ζ ( 3 ) ) 2 - 4 3 ζ ( 6 ) \left(\zeta(3)\right)^{2}-\tfrac{4}{3}\zeta(6)
  11. 5 ζ ( 2 ) ζ ( 5 ) + 2 ζ ( 3 ) ζ ( 4 ) - 11 ζ ( 7 ) 5\zeta(2)\zeta(5)+2\zeta(3)\zeta(4)-11\zeta(7)
  12. 9 2 ζ ( 5 ) - 2 ζ ( 2 ) ζ ( 3 ) \tfrac{9}{2}\zeta(5)-2\zeta(2)\zeta(3)
  13. 1 2 ( ( ζ ( 3 ) ) 2 - ζ ( 6 ) ) \tfrac{1}{2}\left(\left(\zeta(3)\right)^{2}-\zeta(6)\right)
  14. 17 ζ ( 7 ) - 10 ζ ( 2 ) ζ ( 5 ) 17\zeta(7)-10\zeta(2)\zeta(5)
  15. 5 ζ ( 3 ) ζ ( 5 ) - 147 24 ζ ( 8 ) - 5 2 ζ ( 6 , 2 ) 5\zeta(3)\zeta(5)-\tfrac{147}{24}\zeta(8)-\tfrac{5}{2}\zeta(6,2)
  16. 25 12 ζ ( 6 ) - ( ζ ( 3 ) ) 2 \tfrac{25}{12}\zeta(6)-\left(\zeta(3)\right)^{2}
  17. 10 ζ ( 2 ) ζ ( 5 ) + ζ ( 3 ) ζ ( 4 ) - 18 ζ ( 7 ) 10\zeta(2)\zeta(5)+\zeta(3)\zeta(4)-18\zeta(7)
  18. 1 2 ( ( ζ ( 4 ) ) 2 - ζ ( 8 ) ) \tfrac{1}{2}\left(\left(\zeta(4)\right)^{2}-\zeta(8)\right)
  19. s + t = 2 p + 2 s+t=2p+2
  20. p / 3 p/3
  21. ζ ( a ) \zeta(a)
  22. ζ ( a , b , c ) = n > j > i 1 1 n a j b i c = n = 1 1 ( n + 2 ) a j = 1 n 1 ( j + 1 ) b i = 1 j 1 ( i ) c = n = 1 1 ( n + 2 ) a j = 1 n H i , c ( j + 1 ) b \zeta(a,b,c)=\sum_{n>j>i\geq 1}\ \frac{1}{n^{a}j^{b}i^{c}}=\sum_{n=1}^{\infty}% \frac{1}{(n+2)^{a}}\sum_{j=1}^{n}\frac{1}{(j+1)^{b}}\sum_{i=1}^{j}\frac{1}{(i)% ^{c}}=\sum_{n=1}^{\infty}\frac{1}{(n+2)^{a}}\sum_{j=1}^{n}\frac{H_{i,c}}{(j+1)% ^{b}}
  23. ζ ( a , b ) + ζ ( b , a ) = ζ ( a ) ζ ( b ) - ζ ( a + b ) \zeta(a,b)+\zeta(b,a)=\zeta(a)\zeta(b)-\zeta(a+b)
  24. a , b > 1 a,b>1
  25. ζ ( a , b , c ) + ζ ( a , c , b ) + ζ ( b , a , c ) + ζ ( b , c , a ) + ζ ( c , a , b ) + ζ ( c , b , a ) = ζ ( a ) ζ ( b ) ζ ( c ) + 2 ζ ( a + b + c ) - ζ ( a ) ζ ( b + c ) - ζ ( b ) ζ ( a + c ) - ζ ( c ) ζ ( a + b ) \zeta(a,b,c)+\zeta(a,c,b)+\zeta(b,a,c)+\zeta(b,c,a)+\zeta(c,a,b)+\zeta(c,b,a)=% \zeta(a)\zeta(b)\zeta(c)+2\zeta(a+b+c)-\zeta(a)\zeta(b+c)-\zeta(b)\zeta(a+c)-% \zeta(c)\zeta(a+b)
  26. a , b , c > 1 a,b,c>1
  27. S ( i 1 , i 2 , , i k ) = n 1 n 2 n k 1 1 n 1 i 1 n 2 i 2 n k i k S(i_{1},i_{2},\cdots,i_{k})=\sum_{n_{1}\geq n_{2}\geq\cdots n_{k}\geq 1}\frac{% 1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}
  28. Π = { P 1 , P 2 , , P l } \Pi=\{P_{1},P_{2},\dots,P_{l}\}
  29. { 1 , 2 , , k } \{1,2,\dots,k\}
  30. c ( Π ) = ( | P 1 | - 1 ) ! ( | P 2 | - 1 ) ! ( | P l | - 1 ) ! c(\Pi)=(\left|P_{1}\right|-1)!(\left|P_{2}\right|-1)!\cdots(\left|P_{l}\right|% -1)!
  31. Π \Pi
  32. i = { i 1 , , i k } i=\{i_{1},...,i_{k}\}
  33. s = 1 l ζ ( j P s i j ) \prod_{s=1}^{l}\zeta(\sum_{j\in P_{s}}i_{j})
  34. ζ \zeta
  35. S S
  36. S ( i 1 , i 2 ) = ζ ( i 1 , i 2 ) + ζ ( i 1 + i 2 ) S(i_{1},i_{2})=\zeta(i_{1},i_{2})+\zeta(i_{1}+i_{2})
  37. S ( i 1 , i 2 , i 3 ) = ζ ( i 1 , i 2 , i 3 ) + ζ ( i 1 + i 2 , i 3 ) + ζ ( i 1 , i 2 + i 3 ) + ζ ( i 1 + i 2 + i 3 ) S(i_{1},i_{2},i_{3})=\zeta(i_{1},i_{2},i_{3})+\zeta(i_{1}+i_{2},i_{3})+\zeta(i% _{1},i_{2}+i_{3})+\zeta(i_{1}+i_{2}+i_{3})
  38. i 1 , , i k > 1 , i_{1},\cdots,i_{k}>1,
  39. σ k S ( i σ ( 1 ) , , i σ ( k ) ) = partitions Π of { 1 , , k } c ( Π ) ζ ( i , Π ) \sum_{{\sigma\in\sum_{k}}}S(i_{\sigma(1)},\dots,i_{\sigma(k)})=\sum_{\,\text{% partitions }\Pi\,\text{ of }\{1,\dots,k\}}c(\Pi)\zeta(i,\Pi)
  40. i j i_{j}
  41. σ n 1 n 2 n k 1 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) n i k σ ( k ) \sum_{\sigma}\sum_{n_{1}\geq n_{2}\geq\cdots\geq n_{k}\geq 1}\frac{1}{{n^{i_{1% }}}_{\sigma(1)}{n^{i_{2}}}_{\sigma(2)}\cdots{n^{i_{k}}}_{\sigma(k)}}
  42. k \sum_{k}
  43. n = ( 1 , , k ) n=(1,\cdots,k)
  44. n = ( n 1 , , n k ) n=(n_{1},\cdots,n_{k})
  45. k ( n ) \sum_{k}(n)
  46. Λ \Lambda
  47. ( 1 , 2 , , k ) (1,2,\cdots,k)
  48. Λ \Lambda
  49. i j i\sim j
  50. n i = n j n_{i}=n_{j}
  51. k ( n ) = { σ k : σ ( i ) i } \sum_{k}(n)=\{\sigma\in\sum_{k}:\sigma(i)\sim\forall i\}
  52. 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) n i k σ ( k ) \frac{1}{{n^{i_{1}}}_{\sigma(1)}{n^{i_{2}}}_{\sigma(2)}\cdots{n^{i_{k}}}_{% \sigma(k)}}
  53. σ k S ( i σ ( 1 ) , , i σ ( k ) ) = partitions Π of { 1 , , k } c ( Π ) ζ ( i , Π ) \sum_{{\sigma\in\sum_{k}}}S(i_{\sigma(1)},\dots,i_{\sigma(k)})=\sum_{\,\text{% partitions }\Pi\,\text{ of }\{1,\dots,k\}}c(\Pi)\zeta(i,\Pi)
  54. | k ( n ) | \left|\sum_{k}(n)\right|
  55. Π \Pi
  56. Λ \Lambda
  57. \succeq
  58. 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) n i k σ ( k ) \frac{1}{{n^{i_{1}}}_{\sigma(1)}{n^{i_{2}}}_{\sigma(2)}\cdots{n^{i_{k}}}_{% \sigma(k)}}
  59. Π Λ ( Π ) \sum_{\Pi\succeq\Lambda}(\Pi)
  60. | k ( n ) | = Π Λ c ( Π ) \left|\sum_{k}(n)\right|=\sum_{\Pi\succeq\Lambda}c(\Pi)
  61. n = { n 1 , , n k } n=\{n_{1},\cdots,n_{k}\}
  62. Λ \Lambda
  63. c ( Π ) c(\Pi)
  64. Π \Pi
  65. k ( n ) \sum_{k}(n)
  66. Λ \Lambda
  67. k = 3 k=3
  68. σ 3 S ( i σ ( 1 ) , i σ ( 2 ) , i σ ( 3 ) ) = ζ ( i 1 ) ζ ( i 2 ) ζ ( i 3 ) + ζ ( i 1 + i 2 ) ζ ( i 3 ) + ζ ( i 1 ) ζ ( i 2 + i 3 ) + ζ ( i 1 + i 3 ) ζ ( i 2 ) + 2 ζ ( i 1 + i 2 + i 3 ) \sum_{{\sigma\in\sum_{3}}}S(i_{\sigma(1)},i_{\sigma(2)},i_{\sigma(3)})=\zeta(i% _{1})\zeta(i_{2})\zeta(i_{3})+\zeta(i_{1}+i_{2})\zeta(i_{3})+\zeta(i_{1})\zeta% (i_{2}+i_{3})+\zeta(i_{1}+i_{3})\zeta(i_{2})+2\zeta(i_{1}+i_{2}+i_{3})
  69. i 1 , i 2 , i 3 > 1 i_{1},i_{2},i_{3}>1
  70. ζ ( i 1 , i 2 , , i k ) = n 1 > n 2 > n k 1 1 n 1 i 1 n 2 i 2 n k i k \zeta(i_{1},i_{2},\cdots,i_{k})=\sum_{n_{1}>n_{2}>\cdots n_{k}\geq 1}\frac{1}{% n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}
  71. ζ s \zeta^{\prime}s
  72. Π = { P 1 , , P l } \Pi=\{P_{1},\cdots,P_{l}\}
  73. { 1 , 2 , k } \{1,2\cdots,k\}
  74. c ~ ( Π ) = ( - 1 ) k - l c ( Π ) \tilde{c}(\Pi)=(-1)^{k-l}c(\Pi)
  75. i 1 , , i k > 1 i_{1},\cdots,i_{k}>1
  76. σ k ζ ( i σ ( 1 ) , , i σ ( k ) ) = partitions Π of { 1 , , k } c ~ ( Π ) ζ ( i , Π ) \sum_{{\sigma\in\sum_{k}}}\zeta(i_{\sigma(1)},\dots,i_{\sigma(k)})=\sum_{\,% \text{partitions }\Pi\,\text{ of }\{1,\dots,k\}}\tilde{c}(\Pi)\zeta(i,\Pi)
  77. σ n 1 > n 2 > > n k 1 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) n i k σ ( k ) \sum_{\sigma}\sum_{n_{1}>n_{2}>\cdots>n_{k}\geq 1}\frac{1}{{n^{i_{1}}}_{\sigma% (1)}{n^{i_{2}}}_{\sigma(2)}\cdots{n^{i_{k}}}_{\sigma(k)}}
  78. 1 n 1 i 1 n 2 i 2 n k i k \frac{1}{n^{i_{1}}_{1}n^{i_{2}}_{2}\cdots n^{i_{k}}_{k}}
  79. n i n_{i}
  80. Π Λ c ~ ( Π ) = { 1 , if | Λ | = k 0 , otherwise . \sum_{\Pi\succeq\Lambda}\tilde{c}(\Pi)=\begin{cases}1,\,\text{ if }\left|% \Lambda\right|=k\\ 0,\,\text{ otherwise }.\end{cases}
  81. c ~ ( Π ) \tilde{c}(\Pi)
  82. Π \Pi
  83. k ( n ) \sum_{k}(n)
  84. Λ \Lambda
  85. { { 1 } , { 2 } , , { k } } \{\{1\},\{2\},\cdots,\{k\}\}
  86. i 1 + + i k = n , i 1 > 1 ζ ( i 1 , , i k ) = ζ ( n ) \sum_{i_{1}+\cdots+i_{k}=n,i_{1}>1}\zeta(i_{1},\cdots,i_{k})=\zeta(n)
  87. i 1 , , i k i_{1},\cdots,i_{k}
  88. i 1 > 1 i_{1}>1
  89. i 1 + + i k = n , i 1 > 1 S ( i 1 , , i k ) = ( n - 1 k - 1 ) ζ ( n ) \sum_{i_{1}+\cdots+i_{k}=n,i_{1}>1}S(i_{1},\cdots,i_{k})={n-1\choose k-1}\zeta% (n)
  90. k = 2 k=2
  91. ζ ( n - 1 , 1 ) + ζ ( n - 2 , 2 ) + + ζ ( 2 , n - 2 ) = ζ ( n ) \zeta(n-1,1)+\zeta(n-2,2)+\cdots+\zeta(2,n-2)=\zeta(n)
  92. ζ s \zeta^{\prime}s
  93. S s S^{\prime}s
  94. 2 S ( n - 1 , 1 ) = ( n + 1 ) ζ ( n ) - k = 2 n - 2 ζ ( k ) ζ ( n - k ) . 2S(n-1,1)=(n+1)\zeta(n)-\sum_{k=2}^{n-2}\zeta(k)\zeta(n-k).
  95. τ \tau
  96. \Im
  97. \Tau \Tau
  98. Σ : \Tau \Sigma:\Im\rightarrow\Tau
  99. \Im
  100. \Tau n \Tau_{n}
  101. T a u Tau
  102. n n
  103. R n R_{n}
  104. C n C_{n}
  105. \Tau n \Tau_{n}
  106. R n ( a 1 , a 2 , , a l ) = ( n + 1 - a l , n + 1 - a l - 1 , , n + 1 - a 1 ) R_{n}(a_{1},a_{2},\cdots,a_{l})=(n+1-a_{l},n+1-a_{l-1},\cdots,n+1-a_{1})
  107. C n ( a 1 , , a l ) C_{n}(a_{1},\cdots,a_{l})
  108. { a 1 , , a l } \{a_{1},\cdots,a_{l}\}
  109. { 1 , 2 , , n } \{1,2,\cdots,n\}
  110. τ \tau
  111. τ ( I ) = Σ - 1 R n C n Σ ( I ) = Σ - 1 C n R n Σ ( I ) \tau(I)=\Sigma^{-1}R_{n}C_{n}\Sigma(I)=\Sigma^{-1}C_{n}R_{n}\Sigma(I)
  112. I = ( i 1 , i 2 , , i k ) I=(i_{1},i_{2},\cdots,i_{k})\in\Im
  113. i 1 + + i k = n i_{1}+\cdots+i_{k}=n
  114. τ ( 3 , 4 , 1 ) = Σ - 1 C 8 R 8 ( 3 , 7 , 8 ) = Σ - 1 ( 3 , 4 , 5 , 7 , 8 ) = ( 3 , 1 , 1 , 2 , 1 ) . \tau(3,4,1)=\Sigma^{-1}C_{8}R_{8}(3,7,8)=\Sigma^{-1}(3,4,5,7,8)=(3,1,1,2,1).
  115. ( i 1 , , i k ) (i_{1},\cdots,i_{k})
  116. τ ( i 1 , , i k ) \tau(i_{1},\cdots,i_{k})
  117. τ \tau
  118. ( h 1 , , h n - k ) (h_{1},\cdots,h_{n-k})
  119. ( i 1 , , i k ) (i_{1},\cdots,i_{k})
  120. ζ ( h 1 , , h n - k ) = ζ ( i 1 , , i k ) \zeta(h_{1},\cdots,h_{n-k})=\zeta(i_{1},\cdots,i_{k})
  121. s 1 > 1 s 1 + + s k = n ζ ( s 1 , , s k ) = ζ ( n ) \sum_{\stackrel{s_{1}+\cdots+s_{k}=n}{s_{1}>1}}\zeta(s_{1},\ldots,s_{k})=\zeta% (n)
  122. ζ ( 6 , 1 ) + ζ ( 5 , 2 ) + ζ ( 4 , 3 ) + ζ ( 3 , 4 ) + ζ ( 2 , 5 ) = ζ ( 7 ) \zeta(6,1)+\zeta(5,2)+\zeta(4,3)+\zeta(3,4)+\zeta(2,5)=\zeta(7)
  123. n = 1 H n ( b ) ( - 1 ) ( n + 1 ) ( n + 1 ) a = ζ ( a ¯ , b ) \sum_{n=1}^{\infty}\frac{H_{n}^{(b)}(-1)^{(n+1)}}{(n+1)^{a}}=\zeta(\bar{a},b)
  124. H n ( b ) = + 1 + 1 2 b + 1 3 b + H_{n}^{(b)}=+1+\frac{1}{2^{b}}+\frac{1}{3^{b}}+\cdots
  125. n = 1 H ¯ n ( b ) ( n + 1 ) a = ζ ( a , b ¯ ) \sum_{n=1}^{\infty}\frac{\bar{H}_{n}^{(b)}}{(n+1)^{a}}=\zeta(a,\bar{b})
  126. H ¯ n ( b ) = - 1 + 1 2 b - 1 3 b + \bar{H}_{n}^{(b)}=-1+\frac{1}{2^{b}}-\frac{1}{3^{b}}+\cdots
  127. n = 1 H ¯ n ( b ) ( - 1 ) ( n + 1 ) ( n + 1 ) a = ζ ( a ¯ , b ¯ ) \sum_{n=1}^{\infty}\frac{\bar{H}_{n}^{(b)}(-1)^{(n+1)}}{(n+1)^{a}}=\zeta(\bar{% a},\bar{b})
  128. n = 1 ( - 1 ) n ( n + 2 ) a n = 1 H ¯ n ( c ) ( - 1 ) ( n + 1 ) ( n + 1 ) b = ζ ( a ¯ , b ¯ , c ¯ ) \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(n+2)^{a}}\sum_{n=1}^{\infty}\frac{\bar{H}_% {n}^{(c)}(-1)^{(n+1)}}{(n+1)^{b}}=\zeta(\bar{a},\bar{b},\bar{c})
  129. H ¯ n ( c ) = - 1 + 1 2 c - 1 3 c + \bar{H}_{n}^{(c)}=-1+\frac{1}{2^{c}}-\frac{1}{3^{c}}+\cdots
  130. n = 1 ( - 1 ) n ( n + 2 ) a n = 1 H n ( c ) ( n + 1 ) b = ζ ( a ¯ , b , c ) \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(n+2)^{a}}\sum_{n=1}^{\infty}\frac{H_{n}^{(% c)}}{(n+1)^{b}}=\zeta(\bar{a},b,c)
  131. H n ( c ) = + 1 + 1 2 c + 1 3 c + H_{n}^{(c)}=+1+\frac{1}{2^{c}}+\frac{1}{3^{c}}+\cdots
  132. n = 1 1 ( n + 2 ) a n = 1 H n ( c ) ( - 1 ) ( n + 1 ) ( n + 1 ) b = ζ ( a , b ¯ , c ) \sum_{n=1}^{\infty}\frac{1}{(n+2)^{a}}\sum_{n=1}^{\infty}\frac{H_{n}^{(c)}(-1)% ^{(n+1)}}{(n+1)^{b}}=\zeta(a,\bar{b},c)
  133. n = 1 1 ( n + 2 ) a n = 1 H ¯ n ( c ) ( n + 1 ) b = ζ ( a , b , c ¯ ) \sum_{n=1}^{\infty}\frac{1}{(n+2)^{a}}\sum_{n=1}^{\infty}\frac{\bar{H}_{n}^{(c% )}}{(n+1)^{b}}=\zeta(a,b,\bar{c})
  134. ϕ ( s ) = 1 - 2 ( s - 1 ) 2 ( s - 1 ) ζ ( s ) \phi(s)=\frac{1-2^{(s-1)}}{2^{(s-1)}}\zeta(s)
  135. s > 1 s>1
  136. ϕ ( 1 ) = - ln 2 \phi(1)=-\ln 2
  137. ζ ( a , b ) + ζ ( b , a ) = ζ ( a ) ζ ( b ) - ζ ( a + b ) \zeta(a,b)+\zeta(b,a)=\zeta(a)\zeta(b)-\zeta(a+b)
  138. ζ ( a , b ¯ ) + ζ ( b ¯ , a ) = ζ ( a ) ϕ ( b ) - ϕ ( a + b ) \zeta(a,\bar{b})+\zeta(\bar{b},a)=\zeta(a)\phi(b)-\phi(a+b)
  139. ζ ( a ¯ , b ) + ζ ( b , a ¯ ) = ζ ( b ) ϕ ( a ) - ϕ ( a + b ) \zeta(\bar{a},b)+\zeta(b,\bar{a})=\zeta(b)\phi(a)-\phi(a+b)
  140. ζ ( a ¯ , b ¯ ) + ζ ( b ¯ , a ¯ ) = ϕ ( a ) ϕ ( b ) - ζ ( a + b ) \zeta(\bar{a},\bar{b})+\zeta(\bar{b},\bar{a})=\phi(a)\phi(b)-\zeta(a+b)
  141. a = b a=b
  142. ζ ( a ¯ , a ¯ ) = 1 2 [ ϕ 2 ( a ) - ζ ( 2 a ) ] \zeta(\bar{a},\bar{a})=\tfrac{1}{2}\Big[\phi^{2}(a)-\zeta(2a)\Big]
  143. ζ ( a , b ) + ζ ( a , b ¯ ) + ζ ( a ¯ , b ) + ζ ( a ¯ , b ¯ ) = ζ ( a , b ) 2 ( a + b - 2 ) \zeta(a,b)+\zeta(a,\bar{b})+\zeta(\bar{a},b)+\zeta(\bar{a},\bar{b})=\frac{% \zeta(a,b)}{2^{(a+b-2)}}
  144. a > 1 a>1
  145. ζ ( a , b , c ) + ζ ( a , b , c ¯ ) + ζ ( a , b ¯ , c ) + ζ ( a ¯ , b , c ) + ζ ( a , b ¯ , c ¯ ) + ζ ( a ¯ , b , c ¯ ) + ζ ( a ¯ , b ¯ , c ) + ζ ( a ¯ , b ¯ , c ¯ ) = ζ ( a , b , c ) 2 ( a + b + c - 3 ) \zeta(a,b,c)+\zeta(a,b,\bar{c})+\zeta(a,\bar{b},c)+\zeta(\bar{a},b,c)+\zeta(a,% \bar{b},\bar{c})+\zeta(\bar{a},b,\bar{c})+\zeta(\bar{a},\bar{b},c)+\zeta(\bar{% a},\bar{b},\bar{c})=\frac{\zeta(a,b,c)}{2^{(a+b+c-3)}}
  146. a > 1 a>1
  147. ζ ( a , b ) + ζ ( a ¯ , b ¯ ) = s > 0 ( a + b - s - 1 ) ! [ Z a ( a + b - s , s ) ( a - s ) ! ( b - 1 ) ! + Z b ( a + b - s , s ) ( b - s ) ! ( a - 1 ) ! ] \zeta(a,b)+\zeta(\bar{a},\bar{b})=\sum_{s>0}(a+b-s-1)!\Big[\frac{Z_{a}(a+b-s,s% )}{(a-s)!(b-1)!}+\frac{Z_{b}(a+b-s,s)}{(b-s)!(a-1)!}\Big]
  148. Z a ( s , t ) = ζ ( s , t ) + ζ ( s ¯ , t ) - [ ζ ( s , t ) + ζ ( s + t ) ] 2 ( s - 1 ) Z_{a}(s,t)=\zeta(s,t)+\zeta(\bar{s},t)-\frac{\Big[\zeta(s,t)+\zeta(s+t)\Big]}{% 2^{(s-1)}}
  149. Z b ( s , t ) = ζ ( s , t ) 2 ( s - 1 ) Z_{b}(s,t)=\frac{\zeta(s,t)}{2^{(s-1)}}
  150. s s
  151. > 1 >1
  152. 0 \geqslant 0
  153. a , b , , k a,b,\dots,k
  154. n = 2 ζ ( n , k ) = ζ ( k + 1 ) \sum_{n=2}^{\infty}\zeta(n,k)=\zeta(k+1)
  155. n = 2 ζ ( n , a , b , , k ) = ζ ( a + 1 , b , , k ) \sum_{n=2}^{\infty}\zeta(n,a,b,\dots,k)=\zeta(a+1,b,\dots,k)
  156. n = 2 ζ ( n , k ¯ ) = - ϕ ( k + 1 ) \sum_{n=2}^{\infty}\zeta(n,\bar{k})=-\phi(k+1)
  157. n = 2 ζ ( n , a ¯ , b ) = ζ ( a + 1 ¯ , b ) \sum_{n=2}^{\infty}\zeta(n,\bar{a},b)=\zeta(\overline{a+1},b)
  158. n = 2 ζ ( n , a , b ¯ ) = ζ ( a + 1 , b ¯ ) \sum_{n=2}^{\infty}\zeta(n,a,\bar{b})=\zeta(a+1,\bar{b})
  159. n = 2 ζ ( n , a ¯ , b ¯ ) = ζ ( a + 1 ¯ , b ¯ ) \sum_{n=2}^{\infty}\zeta(n,\bar{a},\bar{b})=\zeta(\overline{a+1},\bar{b})
  160. lim k ζ ( n , k ) = ζ ( n ) - 1 \lim_{k\to\infty}\zeta(n,k)=\zeta(n)-1
  161. 1 - ζ ( 2 ) + ζ ( 3 ) - ζ ( 4 ) + = | 1 2 | 1-\zeta(2)+\zeta(3)-\zeta(4)+\cdots=|\frac{1}{2}|
  162. ζ ( a , a ) = 1 2 [ ( ζ ( a ) ) 2 - ζ ( 2 a ) ] \zeta(a,a)=\tfrac{1}{2}\Big[(\zeta(a))^{2}-\zeta(2a)\Big]
  163. ζ ( a , a , a ) = 1 6 ( ζ ( a ) ) 3 + 1 3 ζ ( 3 a ) - 1 2 ζ ( a ) ζ ( 2 a ) \zeta(a,a,a)=\tfrac{1}{6}(\zeta(a))^{3}+\tfrac{1}{3}\zeta(3a)-\tfrac{1}{2}% \zeta(a)\zeta(2a)
  164. ζ M T , r ( s 1 , , s r ; s r + 1 ) = m 1 , , m r > 0 1 m 1 s 1 m r s r ( m 1 + + m r ) s r + 1 \zeta_{MT,r}(s_{1},\dots,s_{r};s_{r+1})=\sum_{m_{1},\dots,m_{r}>0}\frac{1}{m_{% 1}^{s_{1}}\cdots m_{r}^{s_{r}}(m_{1}+\dots+m_{r})^{s_{r+1}}}

Multiplicative_calculus.html

  1. f ( x ) = lim h 0 f ( x + h ) - f ( x ) h f^{\prime}(x)=\lim_{h\to 0}{f(x+h)-f(x)\over{h}}
  2. f * ( x ) = lim h 0 ( f ( x + h ) f ( x ) ) 1 h f^{*}(x)=\lim_{h\to 0}{\left({f(x+h)\over{f(x)}}\right)^{1\over{h}}}
  3. f * ( x ) = e f ( x ) f ( x ) f^{*}(x)=e^{f^{\prime}(x)\over f(x)}
  4. f * ( x ) = lim h 0 ( f ( ( 1 + h ) x ) f ( x ) ) 1 h = lim k 1 ( f ( k x ) f ( x ) ) 1 ln ( k ) f^{*}(x)=\lim_{h\to 0}{\left({f((1+h)x)\over{f(x)}}\right)^{1\over{h}}}=\lim_{% k\to 1}{\left({f(kx)\over{f(x)}}\right)^{1\over{\ln(k)}}}
  5. f * ( x ) = e x f ( x ) f ( x ) . f^{*}(x)=e^{xf^{\prime}(x)\over f(x)}.
  6. [ f ( x + h ) - f ( x ) ] h [f(x+h)-f(x)]\over h
  7. [ f ( x + h ) f ( x ) ] 1 / h ? \left[{f(x+h)\over f(x)}\right]^{1/h}\,\text{?}
  8. f ( x ) f^{\prime}(x)
  9. f ( x ) = [ exp ( ln [ f ( x ) ] ) ] f^{\star}(x)=\left[\exp(\ln[f(x)])\right]^{\prime}
  10. f ( x ) = [ exp ( ln [ f ( x ) ] ) ] f^{\star}(x)=\left[\exp(\ln[f(x)])\right]^{\prime}
  11. f ( x ) = exp [ ( ln f ) ( x ) ] f^{\star}(x)=\exp[(\ln\circ f)^{\prime}(x)]

Multiplicative_character.html

  1. χ 1 , χ 2 , , χ n \chi_{1},\chi_{2},\ldots,\chi_{n}
  2. a 1 χ 1 + a 2 χ 2 + + a n χ n = 0 a_{1}\chi_{1}+a_{2}\chi_{2}+\cdots+a_{n}\chi_{n}=0
  3. a 1 = a 2 = = a n = 0 a_{1}=a_{2}=\cdots=a_{n}=0
  4. G := { ( a b 0 1 ) | a > 0 , b 𝐑 } . G:=\left\{\left.\begin{pmatrix}a&b\\ 0&1\end{pmatrix}\ \right|\ a>0,\ b\in\mathbf{R}\right\}.
  5. f u ( ( a b 0 1 ) ) = a u , f_{u}\left(\begin{pmatrix}a&b\\ 0&1\end{pmatrix}\right)=a^{u},

Multiplicity_(mathematics).html

  1. R = K [ X ] / f , R=K[X]/\langle f\rangle,
  2. f ( X ) = i = 1 k ( X - α i ) m i f(X)=\prod_{i=1}^{k}(X-\alpha_{i})^{m_{i}}
  3. X - α i \langle X-\alpha_{i}\rangle
  4. K [ X ] / ( X - α ) m i . K[X]/\langle(X-\alpha)^{m_{i}}\rangle.
  5. m i m_{i}

Multivariate_adaptive_regression_splines.html

  1. y ^ = - 37 + 5.1 x \hat{y}=-37+5.1x
  2. y ^ \hat{y}
  3. y ^ \hat{y}
  4. y ^ \hat{y}
  5. y ^ = \displaystyle\hat{y}=
  6. y ^ \hat{y}
  7. max \max
  8. max ( a , b ) \max(a,b)
  9. a a
  10. a > b a>b
  11. b b
  12. ozone = \displaystyle\mathrm{ozone}=
  13. wind \mathrm{wind}
  14. vis \mathrm{vis}
  15. ozone \mathrm{ozone}
  16. wind \mathrm{wind}
  17. vis \mathrm{vis}
  18. f ^ ( x ) = i = 1 k c i B i ( x ) \hat{f}(x)=\sum_{i=1}^{k}c_{i}B_{i}(x)
  19. B i ( x ) B_{i}(x)
  20. c i c_{i}
  21. B i ( x ) B_{i}(x)
  22. max ( 0 , x - c o n s t ) \max(0,x-const)
  23. max ( 0 , c o n s t - x ) \max(0,const-x)
  24. max ( 0 , x - c ) \max(0,x-c)
  25. max ( 0 , c - x ) \max(0,c-x)
  26. c c
  27. 6.1 max ( 0 , x - 13 ) - 3.1 max ( 0 , 13 - x ) 6.1\max(0,x-13)-3.1\max(0,13-x)
  28. max \max
  29. [ ± ( x i - c ) ] + [\pm(x_{i}-c)]_{+}
  30. [ ] + [\cdot]_{+}
  31. y ^ \hat{y}

Multivariate_cryptography.html

  1. G F n GF^{n}
  2. G F m GF^{m}
  3. G F n GF^{n}
  4. G F n GF^{n}
  5. G F m GF^{m}
  6. G F m GF^{m}
  7. x = x 1 , , x n x=\langle x_{1},...,x_{n}\rangle
  8. x x
  9. M S M_{S}
  10. v s v_{s}
  11. n n
  12. M S M_{S}
  13. n n
  14. n n
  15. S = M S * x + v S S=M_{S}*x+v_{S}
  16. T = M T * y + v T T=M_{T}*y^{\prime}+v_{T}
  17. y 1 = x 1 x 2 + x 1 x 4 + x 3 x 4 y_{1}=x_{1}x_{2}+x_{1}x_{4}+x_{3}x_{4}
  18. y 2 = x 1 x 3 + x 2 x 4 y_{2}=x_{1}x_{3}+x_{2}x_{4}
  19. y 3 = x 1 x 4 + x 2 x 3 + x 2 x 4 + x 3 x 4 y_{3}=x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4}
  20. y 4 = x 1 x 2 + x 1 x 3 + x 1 x 4 + x 2 x 3 + x 2 x 4 + x 3 x 4 y_{4}=x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4}

Murray_loop_bridge.html

  1. R x R g + R y = R B 1 R B 2 \frac{R_{x}}{R_{g}+R_{y}}=\frac{R_{B1}}{R_{B2}}
  2. R x = ( R g + R y ) R B 1 R B 2 R_{x}=(R_{g}+R_{y})\cdot\frac{R_{B1}}{R_{B2}}
  3. L x = 2 L R B 1 R B 1 + R B 2 L_{x}=2\cdot L\cdot\frac{R_{B1}}{R_{B1}+R_{B2}}

Music_without_sound.html

  1. 10 29 10^{29}

Mutual_coherence_(linear_algebra).html

  1. a 1 , , a m d a_{1},\ldots,a_{m}\in{\mathbb{C}}^{d}
  2. a i H a i = 1. a_{i}^{H}a_{i}=1.
  3. M = max 1 i j m | a i H a j | . M=\max_{1\leq i\neq j\leq m}\left|a_{i}^{H}a_{j}\right|.
  4. M m - d d ( m - 1 ) M\geq\sqrt{\frac{m-d}{d(m-1)}}

Mutually_unbiased_bases.html

  1. { | e 1 , , | e d } \{|e_{1}\rangle,\dots,|e_{d}\rangle\}
  2. { | f 1 , , | f d } \{|f_{1}\rangle,\dots,|f_{d}\rangle\}
  3. | e j |e_{j}\rangle
  4. | f k |f_{k}\rangle
  5. | e j | f k | 2 = 1 d , j , k { 1 , , d } . |\langle e_{j}|f_{k}\rangle|^{2}=\frac{1}{d},\quad\forall j,k\in\{1,\dots,d\}.
  6. 𝔐 ( d ) \mathfrak{M}(d)
  7. 𝔐 ( d ) \mathfrak{M}(d)
  8. d = p 1 n 1 p 2 n 2 p k n k d=p_{1}^{n_{1}}p_{2}^{n_{2}}...p_{k}^{n_{k}}
  9. p 1 n 1 < p 2 n 2 < < p k n k p_{1}^{n_{1}}<p_{2}^{n_{2}}<...<p_{k}^{n_{k}}
  10. p 1 n 1 + 1 𝔐 ( d ) d + 1. p_{1}^{n_{1}}+1\leq\mathfrak{M}(d)\leq d+1.
  11. d = p 1 n 1 d=p_{1}^{n_{1}}
  12. 𝔐 ( d ) = d + 1. \mathfrak{M}(d)=d+1.
  13. M 0 = { | 0 , | 1 } M_{0}=\left\{|0\rangle,|1\rangle\right\}
  14. M 1 = { | 0 + | 1 2 , | 0 - | 1 2 } M_{1}=\left\{\frac{|0\rangle+|1\rangle}{\sqrt{2}},\frac{|0\rangle-|1\rangle}{% \sqrt{2}}\right\}
  15. M 2 = { | 0 + i | 1 2 , | 0 - i | 1 2 } M_{2}=\left\{\frac{|0\rangle+i|1\rangle}{\sqrt{2}},\frac{|0\rangle-i|1\rangle}% {\sqrt{2}}\right\}
  16. σ x , σ z \sigma_{x},\sigma_{z}
  17. σ x σ z \sigma_{x}\sigma_{z}
  18. M 0 = { ( 1 , 0 , 0 , 0 ) , ( 0 , 1 , 0 , 0 ) , ( 0 , 0 , 1 , 0 ) , ( 0 , 0 , 0 , 1 ) } M_{0}=\left\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\right\}
  19. M 1 = { 1 2 ( 1 , 1 , 1 , 1 ) , 1 2 ( 1 , 1 , - 1 , - 1 ) , 1 2 ( 1 , - 1 , - 1 , 1 ) , 1 2 ( 1 , - 1 , 1 , - 1 ) } M_{1}=\left\{\frac{1}{2}(1,1,1,1),\frac{1}{2}(1,1,-1,-1),\frac{1}{2}(1,-1,-1,1% ),\frac{1}{2}(1,-1,1,-1)\right\}
  20. M 2 = { 1 2 ( 1 , - 1 , - i , - i ) , 1 2 ( 1 , - 1 , i , i ) , 1 2 ( 1 , 1 , i , - i ) , 1 2 ( 1 , 1 , - i , i ) } M_{2}=\left\{\frac{1}{2}(1,-1,-i,-i),\frac{1}{2}(1,-1,i,i),\frac{1}{2}(1,1,i,-% i),\frac{1}{2}(1,1,-i,i)\right\}
  21. M 3 = { 1 2 ( 1 , - i , - i , - 1 ) , 1 2 ( 1 , - i , i , 1 ) , 1 2 ( 1 , i , i , - 1 ) , 1 2 ( 1 , i , - i , 1 ) } M_{3}=\left\{\frac{1}{2}(1,-i,-i,-1),\frac{1}{2}(1,-i,i,1),\frac{1}{2}(1,i,i,-% 1),\frac{1}{2}(1,i,-i,1)\right\}
  22. M 4 = { 1 2 ( 1 , - i , - 1 , - i ) , 1 2 ( 1 , - i , 1 , i ) , 1 2 ( 1 , i , - 1 , i ) , 1 2 ( 1 , i , 1 , - i ) } M_{4}=\left\{\frac{1}{2}(1,-i,-1,-i),\frac{1}{2}(1,-i,1,i),\frac{1}{2}(1,i,-1,% i),\frac{1}{2}(1,i,1,-i)\right\}
  23. X ^ \hat{X}
  24. Z ^ \hat{Z}
  25. Z ^ X ^ = ω X ^ Z ^ \hat{Z}\hat{X}=\omega\hat{X}\hat{Z}
  26. ω \omega
  27. ω \omega
  28. ω e 2 π i d \omega\equiv e^{\frac{2\pi i}{d}}
  29. X ^ \hat{X}
  30. Z ^ \hat{Z}
  31. Z ^ \hat{Z}
  32. F a b = ω a b , 0 a , b N - 1 F_{ab}=\omega^{ab},0\leq a,b\leq N-1
  33. X ^ \hat{X}
  34. Z ^ \hat{Z}
  35. X ^ = k = 0 d - 1 | k + 1 k | \hat{X}=\sum_{k=0}^{d-1}|k+1\rangle\langle k|
  36. Z ^ = k = 0 d - 1 ω k | k k | \hat{Z}=\sum_{k=0}^{d-1}\omega^{k}|k\rangle\langle k|
  37. { | k | 0 j d - 1 } \{|k\rangle|0\leq j\leq d-1\}
  38. ω = e 2 π i d \omega=e^{\frac{2\pi i}{d}}
  39. X ^ , Z ^ , X ^ Z ^ , X ^ Z ^ 2 X ^ Z ^ d - 1 \hat{X},\hat{Z},\hat{X}\hat{Z},\hat{X}\hat{Z}^{2}...\hat{X}\hat{Z}^{d-1}
  40. d = p r d=p^{r}
  41. 𝔽 d \mathbb{F}_{d}
  42. { | a | a 𝔽 d } \{|a\rangle|a\in\mathbb{F}_{d}\}
  43. X a ^ \hat{X_{a}}
  44. Z b ^ \hat{Z_{b}}
  45. X a ^ = c 𝔽 d | c + a c | \hat{X_{a}}=\sum_{c\in\mathbb{F}_{d}}|c+a\rangle\langle c|
  46. Z b ^ = c 𝔽 d χ ( b c ) | c c | \hat{Z_{b}}=\sum_{c\in\mathbb{F}_{d}}\chi(bc)|c\rangle\langle c|
  47. χ ( θ ) = exp [ 2 π i p ( θ + θ p + θ p 2 + + θ p r - 1 ) ] , \chi(\theta)=\exp\left[\frac{2\pi i}{p}\left(\theta+\theta^{p}+\theta^{p^{2}}+% \cdots+\theta^{p^{r-1}}\right)\right],
  48. χ ( ) \chi(\cdot)
  49. 𝔽 d \mathbb{F}_{d}
  50. { Z s ^ | s 𝔽 d } \{\hat{Z_{s}}|s\in\mathbb{F}_{d}\}
  51. { X s ^ Z s r ^ | s 𝔽 d } \{\hat{X_{s}}\hat{Z_{sr}}|s\in\mathbb{F}_{d}\}
  52. r 𝔽 d r\in\mathbb{F}_{d}
  53. U = 1 d [ 1 1 1 e i ϕ 10 e i ϕ 11 e i ϕ 12 e i ϕ 20 e i ϕ 21 e i ϕ 22 ] U=\frac{1}{\sqrt{d}}\begin{bmatrix}1&1&1\\ e^{i\phi_{10}}&e^{i\phi_{11}}&e^{i\phi_{12}}\\ e^{i\phi_{20}}&e^{i\phi_{21}}&e^{i\phi_{22}}\end{bmatrix}
  54. H 4 ( ϕ ) = 1 2 [ 1 1 1 1 1 e i ϕ - 1 - e i ϕ 1 - 1 1 - 1 1 - e i ϕ - 1 e i ϕ ] H_{4}(\phi)=\frac{1}{2}\begin{bmatrix}1&1&1&1\\ 1&e^{i\phi}&-1&-e^{i\phi}\\ 1&-1&1&-1\\ 1&-e^{i\phi}&-1&e^{i\phi}\end{bmatrix}
  55. 𝔐 ( 6 ) = 3 \mathfrak{M}(6)=3
  56. B 1 = { | a i i = 1 d } B_{1}=\{|a_{i}\rangle_{i=1}^{d}\}
  57. B 2 = { | b j j = 1 d } B_{2}=\{|b_{j}\rangle_{j=1}^{d}\}
  58. H B 1 + H B 2 - 2 log c . H_{B_{1}}+H_{B_{2}}\geq-2\log c.
  59. c = max | a j | b k | c=\max|\langle a_{j}|b_{k}\rangle|
  60. H B 1 H_{B_{1}}
  61. H B 2 H_{B_{2}}
  62. B 1 B_{1}
  63. B 2 B_{2}
  64. H B 1 + H B 2 log ( d ) H_{B_{1}}+H_{B_{2}}\geq\log(d)
  65. B 1 B_{1}
  66. B 2 B_{2}
  67. log ( d ) \log(d)
  68. k = 1 d + 1 H B k d + 1 2 log ( d + 1 2 ) \sum_{k=1}^{d+1}H_{B_{k}}\geq\frac{d+1}{2}\log\left(\frac{d+1}{2}\right)
  69. d + 1 d+1
  70. | ψ s b |\psi_{s}^{b}\rangle
  71. | ψ s b |\psi_{s^{\prime}}^{b^{\prime}}\rangle
  72. | ψ s b | ψ s b | 2 = k > 0 , s , s |\langle\psi_{s}^{b}|\psi_{s^{\prime}}^{b^{\prime}}\rangle|^{2}=k>0,s,s^{% \prime}\in\mathbb{R}
  73. | q , q |q\rangle,q\in\mathbb{R}
  74. | p , p |p\rangle,p\in\mathbb{R}
  75. | q | p | 2 = 1 2 π |\langle q|p\rangle|^{2}=\frac{1}{2\pi\hbar}
  76. | q |q\rangle
  77. | p |p\rangle
  78. x ^ \hat{x}
  79. - i x -i\frac{\partial}{\partial x}
  80. α x ^ - i β x \alpha\hat{x}-i\beta\frac{\partial}{\partial x}
  81. exp ( i ( a x 2 + b x ) ) \exp(i(ax^{2}+bx))\,
  82. α + 2 β a = 0 \alpha+2\beta a=0
  83. b β b\beta
  84. α \alpha
  85. β \beta
  86. cos θ \cos\theta
  87. sin θ \sin\theta
  88. β \beta
  89. | x θ | x | 2 = 1 2 π | sin θ | , |\langle x_{\theta}|x\rangle|^{2}=\frac{1}{2\pi|\sin\theta|},
  90. | x |x\rangle
  91. | x θ |x_{\theta}\rangle
  92. x ^ \hat{x}
  93. cos θ x ^ - i sin θ x \cos\theta\hat{x}-i\sin\theta\frac{\partial}{\partial x}

M–sigma_relation.html

  1. M 10 8 M 3.1 ( σ 200 km s - 1 ) 4 . \frac{M}{10^{8}M_{\odot}}\approx 3.1\left(\frac{\sigma}{200~{}{\rm km}~{}{\rm s% }^{-1}}\right)^{4}.
  2. M M_{\odot}
  3. M 10 8 M 1.9 ( σ 200 km s - 1 ) 5.1 . \frac{M}{10^{8}M_{\odot}}\approx 1.9\left(\frac{\sigma}{200~{}{\rm km}~{}{\rm s% }^{-1}}\right)^{5.1}.

N!_conjecture.html

  1. P λ P_{\lambda}\,
  2. H μ H_{\mu}\,

N-dimensional_sequential_move_puzzle.html

  1. = 12 ! 8 ! 2 2 12 2 3 8 3 10 20 =\frac{12!\cdot 8!}{2}\cdot\frac{2^{12}}{2}\cdot\frac{3^{8}}{3}\sim 10^{20}
  2. P = V + E + F + C P=V+E+F+C\,\!
  3. = 24 ! 32 ! 2 16 ! 2 2 23 ( 3 ! ) 31 3 ( 4 ! 2 ) 15 4 =\frac{24!\cdot 32!}{2}\cdot\frac{16!}{2}\cdot 2^{23}\cdot(3!)^{31}\cdot 3% \cdot{\left(\frac{4!}{2}\right)}^{15}\cdot 4
  4. 10 120 \sim 10^{120}\,\!
  5. = 15 ! 2 ( 4 ! 2 ) 14 4 {}=\frac{15!}{2}\cdot{\left(\frac{4!}{2}\right)}^{14}\cdot 4
  6. 10 28 {}\sim 10^{28}\,\!
  7. = 15 ! 2 ( 4 ! 2 ) 14 4 64 ! 2 3 63 96 ! 2 2 ( 4 ! ) 24 2 95 64 ! 2 2 ( 8 ! ) 8 =\frac{15!}{2}\cdot\left(\frac{4!}{2}\right)^{14}\cdot 4\cdot\frac{64!}{2}% \cdot 3^{63}\cdot\frac{96!\cdot 2}{2\cdot(4!)^{24}}\cdot\frac{2^{95}\cdot 64!% \cdot 2}{2\cdot(8!)^{8}}
  8. 10 334 \sim 10^{334}\,\!
  9. = 48 ! ( 6 ! ) 8 96 ! ( 12 ! ) 8 64 ! ( 8 ! ) 8 24 ! 32 ! 2 ( 3 ! ) 31 2 23 64 ! 2 =\frac{48!}{(6!)^{8}}\cdot\frac{96!}{(12!)^{8}}\cdot\frac{64!}{(8!)^{8}}\cdot% \frac{24!\cdot 32!}{2}\cdot(3!)^{31}\cdot 2^{23}\cdot\frac{64!}{2}\cdot
  10. 3 63 16 ! ( 4 ! 2 ) 15 4 96 ! ( 4 ! ) 24 2 95 96 ! ( 4 ! ) 24 2 95 3^{63}\cdot 16!\cdot\left(\frac{4!}{2}\right)^{15}\cdot 4\cdot\frac{96!}{(4!)^% {24}}\cdot 2^{95}\cdot\frac{96!}{(4!)^{24}}\cdot 2^{95}
  11. 10 701 \sim 10^{701}\,\!
  12. = 32 ! 2 60 32 80 ! 2 24 80 2 40 ! 80 ! 2 6 80 2 2 40 2 =\frac{32!}{2}\cdot 60^{32}\cdot\frac{80!}{2}\cdot\frac{24^{80}}{2}\cdot\frac{% 40!\cdot 80!}{2}\cdot\frac{6^{80}}{2}\cdot\frac{2^{40}}{2}
  13. 10 561 \sim 10^{561}\,\!
  14. = 31 ! 2 60 31 =\frac{31!}{2}\cdot 60^{31}
  15. 10 89 \sim 10^{89}\,\!
  16. = 31 ! 2 60 31 160 ! 2 12 160 3 320 ! 24 80 6 320 2 320 ! 8 ! 40 2 320 2 160 ! 16 ! 10 =\frac{31!}{2}\cdot 60^{31}\cdot\frac{160!}{2}\cdot\frac{12^{160}}{3}\cdot% \frac{320!}{24^{80}}\cdot\frac{6^{320}}{2}\cdot\frac{320!}{8!^{40}}\cdot\frac{% 2^{320}}{2}\cdot\frac{160!}{16!^{10}}
  17. 10 2075 \sim 10^{2075}\,\!
  18. = 32 ! 2 60 32 80 ! 2 24 80 2 160 ! 2 12 160 3 40 ! 80 ! 2 6 80 2 2 40 2 320 ! 24 80 6 320 2 320 ! 24 80 6 320 2 240 ! ( 6 ! ) 40 2 240 2 320 ! ( 8 ! ) 40 2 320 2 480 ! ( 12 ! ) 40 2 480 2 80 ! ( 8 ! ) 10 160 ! ( 16 ) 10 240 ! ( 24 ) 10 320 ! ( 32 ) 10 =\frac{32!}{2}\cdot 60^{32}\cdot\frac{80!}{2}\cdot\frac{24^{80}}{2}\cdot\frac{% 160!}{2}\cdot\frac{12^{160}}{3}\cdot\frac{40!\cdot 80!}{2}\cdot\frac{6^{80}}{2% }\cdot\frac{2^{40}}{2}\cdot\frac{320!}{24^{80}}\cdot\frac{6^{320}}{2}\cdot% \frac{320!}{24^{80}}\cdot\frac{6^{320}}{2}\cdot\frac{240!}{(6!)^{40}}\cdot% \frac{2^{240}}{2}\cdot\frac{320!}{(8!)^{40}}\cdot\frac{2^{320}}{2}\cdot\frac{4% 80!}{(12!)^{40}}\cdot\frac{2^{480}}{2}\cdot\frac{80!}{(8!)^{10}}\cdot\frac{160% !}{(16)^{10}}\cdot\frac{240!}{(24)^{10}}\cdot\frac{320!}{(32)^{10}}
  19. 10 5267 \sim 10^{5267}\,\!
  20. = 600 ! 2 1200 ! 2 720 ! 2 2 720 2 6 1200 2 12 600 3 =\frac{600!}{2}\cdot\frac{1200!}{2}\cdot\frac{720!}{2}\cdot\frac{2^{720}}{2}% \cdot\frac{6^{1200}}{2}\cdot\frac{12^{600}}{3}
  21. 10 8126 \sim 10^{8126}\,
  22. = 4 ! = 24 =4!\,\!=24

N0.html

  1. 0 \mathbb{N}_{0}

NAIRU.html

  1. U * U*
  2. U U
  3. U < U * U<U*
  4. U > U * U>U*
  5. U = U * U=U*

Nasalance.html

  1. A n ( A m + A n ) \dfrac{A_{n}}{(A_{m}+A_{n})}

Natural_pseudodistance.html

  1. ( M , φ : M ) (M,\varphi:M\to\mathbb{R})
  2. ( N , ψ : N ) (N,\psi:N\to\mathbb{R})
  3. inf h φ - ψ h \inf_{h}\|\varphi-\psi\circ h\|_{\infty}
  4. h h
  5. M M
  6. N N
  7. \|\cdot\|_{\infty}
  8. M M
  9. N N
  10. \infty
  11. M M
  12. N N
  13. C 1 C^{1}
  14. φ , ψ \varphi,\psi
  15. C 1 C^{1}
  16. M M
  17. N N
  18. φ \varphi
  19. m \mathbb{R}^{m}
  20. k k
  21. M M
  22. N N
  23. k k
  24. 1 1
  25. 2 2
  26. 3 3
  27. M M
  28. N N
  29. k k
  30. 1 1
  31. 2 2
  32. h ¯ \bar{h}
  33. φ - ψ h ¯ = inf h φ - ψ h \|\varphi-\psi\circ\bar{h}\|_{\infty}=\inf_{h}\|\varphi-\psi\circ h\|_{\infty}
  34. k k
  35. 1 1

Necklace_splitting_problem.html

  1. \ldots
  2. D 1 D 2 = D_{1}\cap D_{2}=\varnothing
  3. i D 1 i\in D_{1}
  4. i D 2 i\in D_{2}

Neptune.html

  1. × 10 2 6 \times 10^{2}6
  2. × 10 2 4 \times 10^{2}4
  3. M N e p t u n e M E a r t h = 1.02 × 10 26 5.97 × 10 24 = 17.09 \begin{smallmatrix}\frac{M_{Neptune}}{M_{Earth}}\ =\ \frac{1.02\times 10^{26}}% {5.97\times 10^{24}}\ =\ 17.09\end{smallmatrix}
  4. × 10 2 5 \times 10^{2}5
  5. M U r a n u s M E a r t h = 8.68 × 10 25 5.97 × 10 24 = 14.54 \begin{smallmatrix}\frac{M_{Uranus}}{M_{Earth}}\ =\ \frac{8.68\times 10^{25}}{% 5.97\times 10^{24}}\ =\ 14.54\end{smallmatrix}
  6. × 10 2 7 \times 10^{2}7
  7. M J u p i t e r M N e p t u n e = 1.90 × 10 27 1.02 × 10 26 = 18.63 \begin{smallmatrix}\frac{M_{Jupiter}}{M_{Neptune}}\ =\ \frac{1.90\times 10^{27% }}{1.02\times 10^{26}}\ =\ 18.63\end{smallmatrix}
  8. × 10 2 2 \times 10^{2}2
  9. × 10 1 9 \times 10^{1}9
  10. r a r p = 2 1 - e - 1 = 2 / 0.2488 - 1 = 7.039. \begin{smallmatrix}\frac{r_{a}}{r_{p}}=\frac{2}{1-e}-1=2/0.2488-1=7.039.\end{smallmatrix}

Net_tonnage.html

  1. K 2 = 0.2 + 0.02 × log 10 ( V c ) K_{2}=0.2+0.02\times\log_{10}(V_{c})
  2. N T = K 2 × V c × ( 4 d 3 D ) 2 NT=K_{2}\times V_{c}\times(\tfrac{4d}{3D})^{2}
  3. ( 4 d 3 D ) 2 (\tfrac{4d}{3D})^{2}
  4. K 2 × V c × ( 4 d 3 D ) 2 K_{2}\times V_{c}\times(\tfrac{4d}{3D})^{2}
  5. K 3 = 1.25 × ( G T + 10000 ) 10000 K_{3}=\frac{1.25\times(GT+10000)}{10000}
  6. N T = K 2 × V c × ( 4 d 3 D ) 2 + K 3 × ( N 1 + N 2 10 ) NT=K_{2}\times V_{c}\times(\tfrac{4d}{3D})^{2}+K_{3}\times(N_{1}+\frac{N_{2}}{% 10})
  7. ( 4 d 3 D ) 2 (\tfrac{4d}{3D})^{2}
  8. K 2 × V c × ( 4 d 3 D ) 2 K_{2}\times V_{c}\times(\tfrac{4d}{3D})^{2}

Network_science.html

  1. D D
  2. E E
  3. ( N 2 ) {\textstyle\left({{N}\atop{2}}\right)}
  4. D = 2 E N ( N - 1 ) . D=\frac{2E}{N(N-1)}.
  5. D = T N ( N - 1 ) . D=\frac{T}{N(N-1)}.
  6. T T
  7. N N
  8. E E
  9. N - 1 N-1
  10. E m a x E_{max}
  11. k k
  12. < k 2 E N <k>=\tfrac{2E}{N}
  13. < k p ( N - 1 ) <k>=p(N-1)
  14. p p
  15. i i
  16. C i = 2 e i k i ( k i - 1 ) , C_{i}={2e_{i}\over k_{i}{(k_{i}-1)}}\,,
  17. k i k_{i}
  18. i i
  19. e i e_{i}
  20. ( k 2 ) = k ( k - 1 ) 2 . {{\left({{k}\atop{2}}\right)}}={{k(k-1)}\over 2}\,.
  21. P ( deg ( v ) = k ) = ( n - 1 k ) p k ( 1 - p ) n - 1 - k P(\operatorname{deg}(v)=k)={n-1\choose k}p^{k}(1-p)^{n-1-k}
  22. 0
  23. < k > <k>
  24. p p
  25. p E = p N < k Align g t ; / 2 pE=pN<k&gt;/2
  26. m m
  27. p i = k i j k j , p_{i}=\frac{k_{i}}{\sum_{j}k_{j}},
  28. P ( k ) k - 3 P\left(k\right)\sim k^{-3}\,
  29. S = β ( 1 / N ) S=\beta(1/N)
  30. β β
  31. Δ S = β × S 1 N Δ t \Delta S=\beta\times S{1\over N}\Delta t
  32. Δ I = μ I Δ t \Delta I=\mu I\Delta t
  33. μ \mu
  34. 1 τ {1\over\tau}
  35. I I
  36. Δ t \Delta t
  37. R 0 R_{0}
  38. R 0 = β τ = β μ R_{0}=\beta\tau={\beta\over\mu}
  39. x i x_{i}
  40. j j
  41. i i
  42. j j
  43. j j
  44. x i = j i 1 N j x j ( k ) x_{i}=\sum_{j\rightarrow i}{1\over N_{j}}x_{j}^{(k)}
  45. α \alpha
  46. 1 - α 1-\alpha
  47. R ( p ) = α N + ( 1 - α ) j i 1 N j x j ( k ) R{(p)}={\alpha\over N}+(1-\alpha)\sum_{j\rightarrow i}{1\over N_{j}}x_{j}^{(k)}
  48. R ( A ) = R B B ( o u t l i n k s ) + + R n n ( o u t l i n k s ) R(A)=\sum{R_{B}\over B_{(outlinks)}}+...+{R_{n}\over n_{(outlinks)}}
  49. S ( t ) S(t)
  50. I ( t ) I(t)
  51. R ( t ) R(t)
  52. S ( t ) S(t)
  53. I ( t ) I(t)
  54. R ( t ) R(t)
  55. \color b l u e 𝒮 {\color{blue}{\mathcal{S}\rightarrow\mathcal{I}\rightarrow\mathcal{R}}}
  56. N = S ( t ) + I ( t ) + R ( t ) N=S(t)+I(t)+R(t)
  57. d S d t = - β S I \frac{dS}{dt}=-\beta SI
  58. d I d t = β S I - γ I \frac{dI}{dt}=\beta SI-\gamma I
  59. d R d t = γ I \frac{dR}{dt}=\gamma I
  60. β \beta
  61. β N \beta N
  62. S / N S/N
  63. β N ( S / N ) \beta N(S/N)
  64. β N ( S / N ) I = β S I \beta N(S/N)I=\beta SI
  65. γ \gamma
  66. 1 / γ 1/\gamma

Neumann–Neumann_methods.html

  1. - Δ u = f , u | Ω = 0 -\Delta u=f,\qquad u|_{\partial\Omega}=0
  2. u 1 = u 2 , n u 1 = n u 2 u_{1}=u_{2},\qquad\partial_{n}u_{1}=\partial_{n}u_{2}
  3. - Δ u i ( k ) = f i , u i ( k ) | Ω = 0 , u i ( k ) | Γ = λ ( k ) -\Delta u_{i}^{(k)}=f_{i},\qquad u_{i}^{(k)}|_{\partial\Omega}=0,\quad u^{(k)}% _{i}|_{\Gamma}=\lambda^{(k)}
  4. - Δ ψ i ( k ) = 0 , ψ i ( k ) | Ω = 0 , n ψ i ( k ) = n u 1 ( k ) - n u 2 ( k ) . -\Delta\psi_{i}^{(k)}=0,\qquad\psi_{i}^{(k)}|_{\partial\Omega}=0,\quad\partial% _{n}\psi_{i}^{(k)}=\partial_{n}u_{1}^{(k)}-\partial_{n}u_{2}^{(k)}.
  5. λ ( k + 1 ) = λ ( k ) - ω ( θ 1 ψ 1 ( k ) | Γ - θ 2 ψ 2 ( k ) | Γ ) \lambda^{(k+1)}=\lambda^{(k)}-\omega(\theta_{1}\psi_{1}^{(k)}|_{\Gamma}-\theta% _{2}\psi_{2}^{(k)}|_{\Gamma})

Neural_modeling_fields.html

  1. X ( n ) = { X d ( n ) } , d = 1.. D . \vec{X}(n)=\{X_{d}(n)\},d=1..D.
  2. M m ( S m , n ) , m = 1.. M . \vec{M}_{m}(\vec{S}_{m},n),m=1..M.
  3. S m = { S m a } , a = 1.. A . \vec{S}_{m}=\{S_{m}^{a}\},a=1..A.
  4. L ( { X ( n ) } , { M m ( S m , n ) } ) = n = 1 N l ( X ( n ) ) . L(\{\vec{X}(n)\},\{\vec{M}_{m}(\vec{S}_{m},n)\})=\prod_{n=1}^{N}{l(\vec{X}(n))}.
  5. L ( { X ( n ) } , { M m ( S m , n ) } ) = n = 1 N m = 1 M r ( m ) l ( X ( n ) | m ) . L(\{\vec{X}(n)\},\{\vec{M}_{m}(\vec{S}_{m},n)\})=\prod_{n=1}^{N}{\sum_{m=1}^{M% }{r(m)l(\vec{X}(n)|m)}}.
  6. f ( m | n ) = r ( m ) l ( X ( n | m ) ) m = 1 M r ( m ) l ( X ( n | m ) ) f(m|n)=\frac{r(m)l(\vec{X}(n|m))}{\sum_{m^{\prime}=1}^{M}{r(m^{\prime})l(\vec{% X}(n|m^{\prime}))}}
  7. d S m d t = n = 1 N f ( m | n ) ln l ( n | m ) M m M m S m \frac{d\vec{S}_{m}}{dt}=\sum_{n=1}^{N}{f(m|n)\frac{\partial{\ln l(n|m)}}{% \partial{\vec{M}_{m}}}\frac{\partial{\vec{M}_{m}}}{\partial{\vec{S}_{m}}}}
  8. d f ( m | n ) d t = f ( m | n ) m = 1 M [ δ m m - f ( m | n ) ] ln l ( n | m ) M m M m S m d S m d t \frac{df(m|n)}{dt}=f(m|n)\sum_{m^{\prime}=1}^{M}{[\delta_{mm^{\prime}}-f(m^{% \prime}|n)]\frac{\partial{\ln l(n|m^{\prime})}}{\partial{\vec{M}_{m^{\prime}}}% }}\frac{\partial{\vec{M}_{m^{\prime}}}}{\partial{\vec{S}_{m^{\prime}}}}\frac{d% \vec{S}_{m^{\prime}}}{dt}

Neutron_electric_dipole_moment.html

  1. h ν = 2 μ n B ± 2 d n E h\nu=2\mu\text{n}B\pm 2d\text{n}E
  2. d n = h Δ ν 4 E d\text{n}=\frac{h\,\Delta\nu}{4E}

Nevanlinna–Pick_interpolation.html

  1. n n
  2. λ 1 , , λ n \lambda_{1},\ldots,\lambda_{n}
  3. 𝔻 \mathbb{D}
  4. n n
  5. z 1 , , z n z_{1},\ldots,z_{n}
  6. 𝔻 \mathbb{D}
  7. φ \varphi
  8. i i
  9. φ ( λ i ) = z i \varphi(\lambda_{i})=z_{i}
  10. | φ ( λ ) | 1 \left|\varphi(\lambda)\right|\leq 1
  11. λ 𝔻 \lambda\in\mathbb{D}
  12. n n
  13. f : 𝔻 𝔻 f:\mathbb{D}\to\mathbb{D}
  14. λ 1 , λ 2 𝔻 \lambda_{1},\lambda_{2}\in\mathbb{D}
  15. | f ( λ 1 ) - f ( λ 2 ) 1 - f ( λ 2 ) ¯ f ( λ 1 ) | | λ 1 - λ 2 1 - λ 2 ¯ λ 1 | . \left|\frac{f(\lambda_{1})-f(\lambda_{2})}{1-\overline{f(\lambda_{2})}f(% \lambda_{1})}\right|\leq\left|\frac{\lambda_{1}-\lambda_{2}}{1-\overline{% \lambda_{2}}\lambda_{1}}\right|.
  16. f ( λ i ) = z i f(\lambda_{i})=z_{i}
  17. [ 1 - | z 1 | 2 1 - | λ 1 | 2 1 - z 1 ¯ z 2 1 - λ 1 ¯ λ 2 1 - z 2 ¯ z 1 1 - λ 2 ¯ λ 1 1 - | z 2 | 2 1 - | λ 2 | 2 ] 0 , \begin{bmatrix}\frac{1-|z_{1}|^{2}}{1-|\lambda_{1}|^{2}}&\frac{1-\overline{z_{% 1}}z_{2}}{1-\overline{\lambda_{1}}\lambda_{2}}\\ \frac{1-\overline{z_{2}}z_{1}}{1-\overline{\lambda_{2}}\lambda_{1}}&\frac{1-|z% _{2}|^{2}}{1-|\lambda_{2}|^{2}}\end{bmatrix}\geq 0,
  18. λ 1 , λ 2 , z 1 , z 2 𝔻 \lambda_{1},\lambda_{2},z_{1},z_{2}\in\mathbb{D}
  19. φ : 𝔻 𝔻 \varphi:\mathbb{D}\to\mathbb{D}
  20. φ ( λ 1 ) = z 1 \varphi(\lambda_{1})=z_{1}
  21. φ ( λ 2 ) = z 2 \varphi(\lambda_{2})=z_{2}
  22. ( 1 - z j ¯ z i 1 - λ j ¯ λ i ) i , j = 1 , 2 0. \left(\frac{1-\overline{z_{j}}z_{i}}{1-\overline{\lambda_{j}}\lambda_{i}}% \right)_{i,j=1,2}\geq 0.
  23. λ 1 , , λ n , z 1 , , z n 𝔻 \lambda_{1},\ldots,\lambda_{n},z_{1},\ldots,z_{n}\in\mathbb{D}
  24. φ : 𝔻 𝔻 ¯ \varphi:\mathbb{D}\to\overline{\mathbb{D}}
  25. φ ( λ i ) = z i \varphi(\lambda_{i})=z_{i}
  26. ( 1 - z j ¯ z i 1 - λ j ¯ λ i ) i , j = 1 n \left(\frac{1-\overline{z_{j}}z_{i}}{1-\overline{\lambda_{j}}\lambda_{i}}% \right)_{i,j=1}^{n}
  27. φ \varphi
  28. φ \varphi
  29. K ( a , b ) = ( 1 - b a ¯ ) - 1 . K(a,b)=\left(1-b\bar{a}\right)^{-1}.\,
  30. ( ( 1 - w i w j ¯ ) K ( z j , z i ) ) i , j = 1 N . \left((1-w_{i}\overline{w_{j}})K(z_{j},z_{i})\right)_{i,j=1}^{N}.\,
  31. f : R 𝔻 f:R\to\mathbb{D}
  32. ( ( 1 - w i w j ¯ ) K λ ( z j , z i ) ) i , j = 1 N \left((1-w_{i}\overline{w_{j}})K_{\lambda}(z_{j},z_{i})\right)_{i,j=1}^{N}\,

News_analytics.html

  1. S S
  2. X X
  3. Y Y
  4. + 20 +20
  5. S X - S Y S_{X}-S_{Y}
  6. 20 20
  7. X X
  8. Y Y
  9. X X
  10. Y Y
  11. S X - S Y S_{X}-S_{Y}
  12. 0
  13. X X
  14. Y Y
  15. X X
  16. 70 70
  17. 100 100
  18. X X
  19. X X
  20. X X
  21. 70 70
  22. 100 100
  23. X X
  24. X X
  25. 60 60
  26. X X
  27. Z Z
  28. 70 70
  29. 100 100
  30. Z Z
  31. Z Z
  32. 60 60
  33. Z Z
  34. X X
  35. X X

Newton–Cartan_theory.html

  1. Δ U = 4 π G ρ \Delta U=4\pi G\rho\,
  2. U U
  3. G G
  4. ρ \rho
  5. U U
  6. m t x ¨ = - m g U m_{t}\ddot{\vec{x}}=-m_{g}\nabla U
  7. m t m_{t}
  8. m g m_{g}
  9. m t = m g m_{t}=m_{g}
  10. x ¨ = - U \ddot{\vec{x}}=-\nabla U
  11. d 2 x λ d s 2 + Γ μ ν λ d x μ d s d x ν d s = 0 \frac{d^{2}x^{\lambda}}{ds^{2}}+\Gamma_{\mu\nu}^{\lambda}\frac{dx^{\mu}}{ds}% \frac{dx^{\nu}}{ds}=0
  12. U U
  13. Γ μ ν λ = γ λ ρ U , ρ Ψ μ Ψ ν \Gamma_{\mu\nu}^{\lambda}=\gamma^{\lambda\rho}U_{,\rho}\Psi_{\mu}\Psi_{\nu}
  14. Ψ μ = δ μ 0 \Psi_{\mu}=\delta_{\mu}^{0}
  15. γ μ ν = δ A μ δ B ν δ A B \gamma^{\mu\nu}=\delta^{\mu}_{A}\delta^{\nu}_{B}\delta^{AB}
  16. A , B = 1 , 2 , 3 A,B=1,2,3
  17. Ψ μ \Psi_{\mu}
  18. γ μ ν \gamma^{\mu\nu}
  19. R κ μ ν λ = 2 γ λ σ U , σ [ μ Ψ ν ] Ψ κ R^{\lambda}_{\kappa\mu\nu}=2\gamma^{\lambda\sigma}U_{,\sigma[\mu}\Psi_{\nu]}% \Psi_{\kappa}
  20. A [ μ ν ] = 1 2 ! [ A μ ν - A ν μ ] A_{[\mu\nu]}=\frac{1}{2!}[A_{\mu\nu}-A_{\nu\mu}]
  21. A μ ν A_{\mu\nu}
  22. R κ ν = Δ U Ψ κ Ψ ν R_{\kappa\nu}=\Delta U\Psi_{\kappa}\Psi_{\nu}\,
  23. R μ ν = 4 π G ρ Ψ μ Ψ ν R_{\mu\nu}=4\pi G\rho\Psi_{\mu}\Psi_{\nu}\,

Newton–Pepys_problem.html

  1. P ( A ) = 1 - ( 5 6 ) 6 = 31031 46656 0.6651 , P(A)=1-\left(\frac{5}{6}\right)^{6}=\frac{31031}{46656}\approx 0.6651\,,
  2. P ( B ) = 1 - x = 0 1 ( 12 x ) ( 1 6 ) x ( 5 6 ) 12 - x = 1346704211 2176782336 0.6187 , P(B)=1-\sum_{x=0}^{1}{\left({{12}\atop{x}}\right)}\left(\frac{1}{6}\right)^{x}% \left(\frac{5}{6}\right)^{12-x}=\frac{1346704211}{2176782336}\approx 0.6187\,,
  3. P ( C ) = 1 - x = 0 2 ( 18 x ) ( 1 6 ) x ( 5 6 ) 18 - x = 15166600495229 25389989167104 0.5973 . P(C)=1-\sum_{x=0}^{2}{\left({{18}\atop{x}}\right)}\left(\frac{1}{6}\right)^{x}% \left(\frac{5}{6}\right)^{18-x}=\frac{15166600495229}{25389989167104}\approx 0% .5973\,.
  4. P ( N ) = 1 - x = 0 n - 1 ( 6 n x ) ( 1 6 ) x ( 5 6 ) 6 n - x . P(N)=1-\sum_{x=0}^{n-1}{\left({{6n}\atop{x}}\right)}\left(\frac{1}{6}\right)^{% x}\left(\frac{5}{6}\right)^{6n-x}\,.
  5. P ( r k ; n , p ) P(r\geq k;n,p)
  6. ν 1 , ν 2 \nu_{1},\nu_{2}
  7. ν 1 ν 2 \nu_{1}\leq\nu_{2}
  8. P ( r ν 1 k ; ν 1 n , p ) P(r\geq\nu_{1}k;\nu_{1}n,p)
  9. P ( r ν 2 k ; ν 2 n , p ) P(r\geq\nu_{2}k;\nu_{2}n,p)
  10. P ( r 1 ; 6 , 1 / 6 ) P ( r 2 ; 12 , 1 / 6 ) P ( r 3 ; 18 , 1 / 6 ) P(r\geq 1;6,1/6)\geq P(r\geq 2;12,1/6)\geq P(r\geq 3;18,1/6)
  11. k 1 , k 2 , n k_{1},k_{2},n
  12. k 1 < k 2 k_{1}<k_{2}
  13. P ( r k 1 ; k 1 n , 1 n ) > P ( r k 2 ; k 2 n , 1 n ) P(r\geq k_{1};k_{1}n,\frac{1}{n})>P(r\geq k_{2};k_{2}n,\frac{1}{n})
  14. k , n 1 , n 2 k,n_{1},n_{2}
  15. n 1 < n 2 n_{1}<n_{2}
  16. P ( r k ; k n 1 , 1 n 1 ) > P ( r k ; k n 2 , 1 n 2 ) P(r\geq k;kn_{1},\frac{1}{n_{1}})>P(r\geq k;kn_{2},\frac{1}{n_{2}})
  17. ν 1 , ν 2 , n , k \nu_{1},\nu_{2},n,k
  18. ν 1 ν 2 , k n , p [ 0 , 1 ] \nu_{1}\leq\nu_{2},k\leq n,p\in[0,1]
  19. P ( r = ν 1 k ; ν 1 n , p ) P ( r = ν 2 k ; ν 2 n , p ) P(r=\nu_{1}k;\nu_{1}n,p)\geq P(r=\nu_{2}k;\nu_{2}n,p)

Niederreiter_cryptosystem.html

  1. 1 9 ! \frac{1}{9!}

Nielsen–Schreier_theorem.html

  1. G G
  2. G G
  3. S S
  4. G G
  5. G G
  6. S S
  7. S S
  8. G G
  9. n n
  10. H H
  11. e e
  12. H H
  13. 1 + e ( n - 1 ) . 1+e(n-1)\ .
  14. G G
  15. a a
  16. b b
  17. E E
  18. E E
  19. p = a a p=aa
  20. q = a b q=ab
  21. r = a b < s u p > 1 r=ab<sup>−1

Nine-point_hyperbola.html

  1. z z * = a 2 . zz^{*}=a^{2}.
  2. t 1 , t 2 , t 3 . t_{1},t_{2},t_{3}.
  3. s = t 1 + t 2 + t 3 . s=t_{1}+t_{2}+t_{3}.
  4. ( z - s / 2 ) ( z * - s * / 2 ) = a 2 4 . (z-s/2)(z^{*}-s^{*}/2)=\frac{a^{2}}{4}.

NNPDF.html

  1. χ 2 \chi^{2}
  2. Q 0 2 Q^{2}_{0}
  3. Q 2 Q^{2}
  4. N r e p N_{rep}
  5. s s
  6. s ¯ \bar{s}

Noisy_channel_model.html

  1. Σ \Sigma
  2. Σ * \Sigma^{*}
  3. Σ \Sigma
  4. D D
  5. Σ * \Sigma^{*}
  6. D Σ * D\subseteq\Sigma^{*}
  7. Γ w s = Pr ( s | w ) \Gamma_{ws}=\Pr(s|w)
  8. w D w\in D
  9. s Σ * s\in\Sigma^{*}
  10. Σ = { a , b , c , , y , z , A , B , , Z , } \Sigma=\{a,b,c,...,y,z,A,B,...,Z,...\}
  11. D Σ * D\subseteq\Sigma^{*}
  12. Γ \Gamma
  13. Pr ( s | w ) \Pr(s|w)
  14. w D w\in D
  15. s Σ * s\in\Sigma^{*}
  16. s s
  17. w w
  18. σ : Σ * D \sigma:\Sigma^{*}\to D

Non-dictatorship.html

  1. P i P_{i}

Non-photochemical_quenching.html

  1. F m F_{m}

Nonequilibrium_partition_identity.html

  1. exp [ - Σ ¯ t t ] = 1 , t \left\langle{\exp[-\overline{\Sigma}_{t}\;t]}\right\rangle=1,\quad\forall t

Nonstandard_finite_difference_scheme.html

  1. ( t 2 - v 2 x 2 ) Ψ ( x , t ) = 0. (\partial_{t}^{2}-v^{2}\partial_{x}^{2})\Psi(x,t)=0.
  2. f ( x ) f ( x + Δ x / 2 ) - f ( x - Δ x / 2 ) Δ x . f^{\prime}(x)\approx\frac{f(x+\Delta x/2)-f(x-\Delta x/2)}{\Delta x}.
  3. f ( x ) f^{\prime}(x)
  4. f ′′ ( x ) f^{\prime\prime}(x)
  5. f ′′ ( x ) d x 2 f ( x ) Δ x 2 , f^{\prime\prime}(x)\approx\frac{\,\text{d}_{x}^{2}f(x)}{\Delta x^{2}},
  6. d x f ( x ) = f ( x + Δ x / 2 ) - f ( x - Δ x / 2 ) \,\text{d}_{x}f(x)=f(x+\Delta x/2)-f(x-\Delta x/2)
  7. d x f 2 ( x ) = f ( x + Δ x ) + f ( x - Δ x ) - 2 f ( x ) \,\text{d}_{x}{}^{2}f(x)=f(x+\Delta x)+f(x-\Delta x)-2f(x)
  8. d x \,\text{d}_{x}
  9. f ( x ) f(x)
  10. [ d t 2 - ( v Δ t / Δ x ) 2 d x 2 ] Ψ ( x , t ) = 0. \left[\,\text{d}_{t}^{2}-(v\Delta t/\Delta x)^{2}\,\text{d}_{x}^{2}\right]\Psi% (x,t)=0.
  11. ϕ ( x , t ) = e i ( k x - ω t ) \phi(x,t)=e^{i(kx-\omega t)}
  12. ω / k = v \omega/k=v
  13. [ d t 2 - ( v Δ t / Δ x ) 2 d x 2 ] ϕ ( x , t ) = ϵ . \left[\,\text{d}_{t}^{2}-(v\Delta t/\Delta x)^{2}\,\text{d}_{x}^{2}\right]\phi% (x,t)=\epsilon.
  14. ϵ 0 \epsilon\neq 0
  15. v Δ t / Δ x v\Delta t/\Delta x
  16. ϵ = 0 \epsilon=0
  17. u = sin ( ω Δ t / 2 ) sin ( k Δ x / 2 ) . u=\frac{\sin(\omega\Delta t/2)}{\sin(k\Delta x/2)}.
  18. [ d t 2 - ( u Δ t / Δ x ) 2 d x 2 ] Ψ ( x , t ) = 0. \left[\,\text{d}_{t}^{2}-(u\Delta t/\Delta x)^{2}\,\text{d}_{x}^{2}\right]\Psi% (x,t)=0.

Noriko_Yui.html

  1. X X
  2. \mathbb{Q}
  3. L L
  4. X X
  5. L L

Normal-inverse-gamma_distribution.html

  1. x | σ 2 , μ , λ N ( μ , σ 2 / λ ) x|\sigma^{2},\mu,\lambda\sim\mathrm{N}(\mu,\sigma^{2}/\lambda)\,\!
  2. μ \mu
  3. σ 2 / λ \sigma^{2}/\lambda
  4. σ 2 | α , β Γ - 1 ( α , β ) \sigma^{2}|\alpha,\beta\sim\Gamma^{-1}(\alpha,\beta)\!
  5. ( x , σ 2 ) (x,\sigma^{2})
  6. ( x , σ 2 ) N- Γ - 1 ( μ , λ , α , β ) . (x,\sigma^{2})\sim\,\text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta)\!.
  7. NIG \,\text{NIG}
  8. N- Γ - 1 . \,\text{N-}\Gamma^{-1}.
  9. 𝐱 | σ 2 , s y m b o l μ , 𝐕 - 1 N ( s y m b o l μ , σ 2 𝐕 ) \mathbf{x}|\sigma^{2},symbol{\mu},\mathbf{V}^{-1}\sim\mathrm{N}(symbol{\mu},% \sigma^{2}\mathbf{V})\,\!
  10. σ 2 \sigma^{2}
  11. 𝐱 \mathbf{x}
  12. k × 1 k\times 1
  13. s y m b o l μ symbol{\mu}
  14. σ 2 𝐕 \sigma^{2}\mathbf{V}
  15. σ 2 | α , β Γ - 1 ( α , β ) \sigma^{2}|\alpha,\beta\sim\Gamma^{-1}(\alpha,\beta)\!
  16. f ( x , σ 2 | μ , λ , α , β ) = λ σ 2 π β α Γ ( α ) ( 1 σ 2 ) α + 1 exp ( - 2 β + λ ( x - μ ) 2 2 σ 2 ) f(x,\sigma^{2}|\mu,\lambda,\alpha,\beta)=\frac{\sqrt{\lambda}}{\sigma\sqrt{2% \pi}}\,\frac{\beta^{\alpha}}{\Gamma(\alpha)}\,\left(\frac{1}{\sigma^{2}}\right% )^{\alpha+1}\exp\left(-\frac{2\beta+\lambda(x-\mu)^{2}}{2\sigma^{2}}\right)
  17. 𝐱 \mathbf{x}
  18. k × 1 k\times 1
  19. f ( 𝐱 , σ 2 | μ , 𝐕 - 1 , α , β ) = | 𝐕 | - 1 / 2 ( 2 π ) - k / 2 β α Γ ( α ) ( 1 σ 2 ) k / 2 + α + 1 exp ( - 2 β + ( 𝐱 - s y m b o l μ ) 𝐕 - 1 ( 𝐱 - s y m b o l μ ) 2 σ 2 ) . f(\mathbf{x},\sigma^{2}|\mu,\mathbf{V}^{-1},\alpha,\beta)=|\mathbf{V}|^{-1/2}{% (2\pi)^{-k/2}}\,\frac{\beta^{\alpha}}{\Gamma(\alpha)}\,\left(\frac{1}{\sigma^{% 2}}\right)^{k/2+\alpha+1}\exp\left(-\frac{2\beta+(\mathbf{x}-symbol{\mu})^{% \prime}\mathbf{V}^{-1}(\mathbf{x}-symbol{\mu})}{2\sigma^{2}}\right).
  20. | 𝐕 | |\mathbf{V}|
  21. k × k k\times k
  22. 𝐕 \mathbf{V}
  23. k = 1 k=1
  24. 𝐱 , 𝐕 , s y m b o l μ \mathbf{x},\mathbf{V},symbol{\mu}
  25. γ = 1 / λ \gamma=1/\lambda
  26. f ( x , σ 2 | μ , γ , α , β ) = 1 σ 2 π γ β α Γ ( α ) ( 1 σ 2 ) α + 1 exp ( - 2 γ β + ( x - μ ) 2 2 γ σ 2 ) f(x,\sigma^{2}|\mu,\gamma,\alpha,\beta)=\frac{1}{\sigma\sqrt{2\pi\gamma}}\,% \frac{\beta^{\alpha}}{\Gamma(\alpha)}\,\left(\frac{1}{\sigma^{2}}\right)^{% \alpha+1}\exp\left(-\frac{2\gamma\beta+(x-\mu)^{2}}{2\gamma\sigma^{2}}\right)
  27. 𝐕 \mathbf{V}
  28. 𝐕 - 1 \mathbf{V}^{-1}
  29. F ( x , σ 2 | μ , λ , α , β ) = e - β σ 2 ( β σ 2 ) α ( erf ( λ ( x - μ ) 2 σ ) + 1 ) 2 σ 2 Γ ( α ) F(x,\sigma^{2}|\mu,\lambda,\alpha,\beta)=\frac{e^{-\frac{\beta}{\sigma^{2}}}% \left(\frac{\beta}{\sigma^{2}}\right)^{\alpha}\left(\,\text{erf}\left(\frac{% \sqrt{\lambda}(x-\mu)}{\sqrt{2}\sigma}\right)+1\right)}{2\sigma^{2}\Gamma(% \alpha)}
  30. { σ 2 f ( x ) + λ f ( x ) ( x - μ ) = 0 , f ( 0 ) = λ β α ( 1 σ 2 ) α + 1 e - 2 β - λ μ 2 2 σ 2 2 π σ Γ ( α ) } \left\{\begin{array}[]{l}\sigma^{2}f^{\prime}(x)+\lambda f(x)(x-\mu)=0,\\ f(0)=\frac{\sqrt{\lambda}\beta^{\alpha}\left(\frac{1}{\sigma^{2}}\right)^{% \alpha+1}e^{\frac{-2\beta-\lambda\mu^{2}}{2\sigma^{2}}}}{\sqrt{2\pi}\sigma% \Gamma(\alpha)}\end{array}\right\}
  31. ( x , σ 2 ) N- Γ - 1 ( μ , λ , α , β ) . (x,\sigma^{2})\sim\,\text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta)\!.
  32. σ 2 \sigma^{2}
  33. σ 2 Γ - 1 ( α , β ) \sigma^{2}\sim\Gamma^{-1}(\alpha,\beta)\!
  34. α λ β ( x - μ ) \sqrt{\frac{\alpha\lambda}{\beta}}(x-\mu)
  35. 2 α 2\alpha
  36. 𝐱 \mathbf{x}
  37. 𝐱 t 2 α ( s y m b o l μ , β α 𝐕 ) \mathbf{x}\sim t_{2\alpha}(symbol{\mu},\frac{\beta}{\alpha}\mathbf{V})\!
  38. σ 2 \sigma^{2}
  39. α \alpha
  40. β \beta
  41. x x
  42. μ \mu
  43. σ 2 / λ \sigma^{2}/\lambda
  44. σ 2 𝐕 \sigma^{2}\mathbf{V}
  45. σ 2 \sigma^{2}

Normal_invariant.html

  1. \geq
  2. n n
  3. f : M X f\colon M\to X
  4. ξ \xi
  5. ν M \nu_{M}
  6. M M
  7. ξ \xi
  8. M M
  9. f f
  10. X X
  11. f * ( [ M ] ) = [ X ] H n ( X ) f_{*}([M])=[X]\in H_{n}(X)
  12. X X
  13. n n
  14. X X
  15. f : M X f\colon M\to X
  16. n n
  17. M M
  18. ξ \xi
  19. X X
  20. τ M ε k \tau_{M}\oplus\varepsilon^{k}
  21. M M
  22. ξ \xi
  23. M M
  24. f f
  25. X X
  26. f * ( [ M ] ) = [ X ] H n ( X ) f_{*}([M])=[X]\in H_{n}(X)
  27. M X M\to X
  28. M M
  29. X X
  30. M X M\to X
  31. X X
  32. f : M X f\colon M\to X
  33. M M
  34. X X
  35. K * ( M ) = k e r ( f * : H * ( M ) H * ( X ) ) K_{*}(M)=ker(f_{*}\colon H_{*}(M)\to H_{*}(X))
  36. H * ( M ) = K * ( M ) H * ( X ) H_{*}(M)=K_{*}(M)\oplus H_{*}(X)
  37. f f
  38. Z [ π 1 ( X ) ] Z[\pi_{1}(X)]
  39. f f
  40. α π p + 1 ( f ) \alpha\in\pi_{p+1}(f)
  41. f f
  42. ϕ : S p M \phi:S^{p}\to M
  43. f ϕ : S p X f\circ\phi:S^{p}\to X
  44. p p
  45. X X
  46. S p X S^{p}\to X
  47. π p ( B O , B O k ) = 0 \pi_{p}(BO,BO_{k})=0
  48. k > p k>p
  49. 𝒩 ( X ) \mathcal{N}(X)
  50. X X
  51. n n
  52. X X
  53. f i : M i X f_{i}\colon M_{i}\rightarrow X
  54. i = 0 , 1 i=0,1
  55. h : M 0 M 1 h\colon M_{0}\rightarrow M_{1}
  56. f 1 h f 0 f_{1}\circ h\simeq f_{0}
  57. X X
  58. ( f , b ) : M X (f,b)\colon M\rightarrow X
  59. f i : M i X f_{i}\colon M_{i}\rightarrow X
  60. i = 0 , 1 i=0,1
  61. ( F , B ) : ( W , M , M ) ( M × I , M × 0 , M × 1 ) (F,B)\colon(W,M,M^{\prime})\to(M\times I,M\times 0,M\times 1)
  62. F 0 = f 0 \partial F_{0}=f_{0}
  63. F 1 = f 1 \partial F_{1}=f_{1}
  64. 𝒩 ( X ) \mathcal{N}(X)\neq\emptyset
  65. f 0 = f 1 f_{0}=f_{1}
  66. 𝒩 ( X ) \mathcal{N}(X)
  67. 𝒩 ( X ) \mathcal{N}(X)
  68. 𝒮 ( X ) \mathcal{S}(X)
  69. 𝒩 ( X ) \mathcal{N}(X)
  70. 𝒩 ( X ) \mathcal{N}(X)
  71. X X
  72. 𝒩 ( X ) \mathcal{N}(X)\neq\emptyset
  73. B G BG
  74. B O BO
  75. J : B O B G J\colon BO\rightarrow BG
  76. O G O\hookrightarrow G
  77. B O B G B ( G / O ) BO\rightarrow BG\rightarrow B(G/O)
  78. ν X : X B G \nu_{X}\colon X\rightarrow BG
  79. ν X \nu_{X}
  80. ν ~ X : X B O \tilde{\nu}_{X}\colon X\rightarrow BO
  81. X B G B ( G / O ) X\rightarrow BG\rightarrow B(G/O)
  82. B ( G / O ) B(G/O)
  83. X X
  84. 𝒩 ( X ) [ X , G / O ] . \mathcal{N}(X)\cong[X,G/O].
  85. 𝒩 ( X ) \mathcal{N}(X)
  86. G / O G/O
  87. 𝒩 ( X ) [ X , G / O ] \mathcal{N}(X)\cong[X,G/O]
  88. 𝒩 P L ( X ) [ X , G / P L ] \mathcal{N}^{PL}(X)\cong[X,G/PL]
  89. 𝒩 T O P ( X ) [ X , G / T O P ] . \mathcal{N}^{TOP}(X)\cong[X,G/TOP].
  90. G / O G/O
  91. G / P L G/PL
  92. G / T O P G/TOP
  93. G / P L G/PL
  94. G / T O P G/TOP
  95. C A T = P L CAT=PL
  96. T O P TOP