wpmath0000008_6

Integrodifference_equation.html

  1. n t + 1 ( x ) = Ω k ( x , y ) f ( n t ( y ) ) d y , n_{t+1}(x)=\int_{\Omega}k(x,y)\,f(n_{t}(y))\,dy,
  2. { n t } \{n_{t}\}\,
  3. Ω \Omega\,
  4. y Ω y\in\Omega\,
  5. k ( x , y ) k(x,y)\,
  6. Ω \Omega\,
  7. n t n_{t}
  8. { n t } \{n_{t}\}
  9. n t ( x ) n_{t}(x)
  10. x x
  11. t t
  12. f ( n t ( x ) ) f(n_{t}(x))
  13. x x
  14. k ( x , y ) k(x,y)
  15. y y
  16. x x
  17. t t
  18. k ( x - y ) k(x-y)
  19. n t + 1 = - k ( x - y ) R n t ( y ) d y n_{t+1}=\int_{-\infty}^{\infty}k(x-y)Rn_{t}(y)dy
  20. R = d f / d n ( n = 0 ) R=df/dn(n=0)
  21. n t + 1 = f ( 0 ) k * n t n_{t+1}=f^{\prime}(0)k*n_{t}
  22. M ( s ) = - e s x n ( x ) d x M(s)=\int_{-\infty}^{\infty}e^{sx}n(x)dx
  23. c * = min w > 0 [ 1 w ln ( R - k ( s ) e w s d s ) ] c^{*}=\min_{w>0}\left[\frac{1}{w}\ln\left(R\int_{-\infty}^{\infty}k(s)e^{ws}ds% \right)\right]

Inter-rater_reliability.html

  1. r r
  2. ρ \rho
  3. r r
  4. ρ \rho
  5. r r
  6. ρ \rho
  7. x ¯ ± 1.96 s \bar{x}\pm 1.96s
  8. x ¯ ± 2 s \bar{x}\pm 2s
  9. x ¯ ± t 0.05 , n - 1 s 1 + 1 n \bar{x}\pm t_{0.05,n-1}s\sqrt{1+\frac{1}{n}}

Interaction_information.html

  1. { X , Y , Z } \{X,Y,Z\}
  2. I ( X ; Y ; Z ) I(X;Y;Z)
  3. I ( X ; Y ; Z ) = I ( X ; Y | Z ) - I ( X ; Y ) = I ( X ; Z | Y ) - I ( X ; Z ) = I ( Y ; Z | X ) - I ( Y ; Z ) \begin{matrix}I(X;Y;Z)&=&I(X;Y|Z)-I(X;Y)\\ &=&I(X;Z|Y)-I(X;Z)\\ &=&I(Y;Z|X)-I(Y;Z)\end{matrix}
  4. I ( X ; Y ) I(X;Y)
  5. X X
  6. Y Y
  7. I ( X ; Y | Z ) I(X;Y|Z)
  8. X X
  9. Y Y
  10. Z Z
  11. I ( X ; Y | Z ) = H ( X | Z ) + H ( Y | Z ) - H ( X , Y | Z ) = H ( X | Z ) - H ( X | Y , Z ) \begin{aligned}\displaystyle I(X;Y|Z)&\displaystyle=H(X|Z)+H(Y|Z)-H(X,Y|Z)\\ &\displaystyle=H(X|Z)-H(X|Y,Z)\end{aligned}
  12. I ( X ; Y ; Z ) = - [ H ( X ) + H ( Y ) + H ( Z ) ] + [ H ( X , Y ) + H ( X , Z ) + H ( Y , Z ) ] - H ( X , Y , Z ) \begin{aligned}\displaystyle I(X;Y;Z)=&\displaystyle-[H(X)+H(Y)+H(Z)]\\ &\displaystyle+[H(X,Y)+H(X,Z)+H(Y,Z)]\\ &\displaystyle-H(X,Y,Z)\end{aligned}
  13. I ( X ; Y ; Z ) I(X;Y;Z)
  14. { Y , X } \{Y,X\}
  15. Z Z
  16. Z Z
  17. Z Z
  18. { Y , X } \{Y,X\}
  19. I ( X ; Y | Z ) I(X;Y|Z)
  20. { X , Y } \{X,Y\}
  21. Z Z
  22. Z Z
  23. { Y , X } \{Y,X\}
  24. Z Z
  25. { Y , X } \{Y,X\}
  26. - m i n { I ( X ; Y ) , I ( Y ; Z ) , I ( X ; Z ) } I ( X ; Y ; Z ) m i n { I ( X ; Y | Z ) , I ( Y ; Z | X ) , I ( X ; Z | Y ) } -min\ \{I(X;Y),I(Y;Z),I(X;Z)\}\leq I(X;Y;Z)\leq min\ \{I(X;Y|Z),I(Y;Z|X),I(X;Z% |Y)\}
  27. I ( r a i n ; d a r k | c l o u d ) I ( r a i n ; d a r k ) I(rain;dark|cloud)\leq I(rain;dark)
  28. I ( r a i n ; d a r k ; c l o u d ) I(rain;dark;cloud)
  29. I ( X ; Y ; Z ) I(X;Y;Z)
  30. X X
  31. Y Y
  32. Z Z
  33. I ( Y ; Z ) I(Y;Z)
  34. I ( Y ; Z | X ) I(Y;Z|X)
  35. X X
  36. Y Y
  37. Z Z
  38. I ( Y ; Z | X ) > I ( Y ; Z ) I(Y;Z|X)>I(Y;Z)
  39. I ( X ; Y ; Z ) I(X;Y;Z)
  40. X , Y , Z X,Y,Z
  41. I ( X ; Y ; Z ) I(X;Y;Z)
  42. Y Y
  43. X X
  44. Z Z
  45. I ( X ; Y ; Z ) = I ( X ; Y | Z ) - I ( X ; Y ) I(X;Y;Z)=I(X;Y|Z)-I(X;Y)
  46. X X
  47. Y Y
  48. Z Z
  49. ( X ) (X)
  50. ( Y ) (Y)
  51. ( Z ) (Z)
  52. X X
  53. I ( X ; Y ; Z ) I(X;Y;Z)
  54. I ( X ; Y ; Z ) I(X;Y;Z)
  55. \rightarrow
  56. ( n - 1 ) (n-1)
  57. I ( W ; X ; Y ; Z ) = I ( X ; Y ; Z | W ) - I ( X ; Y ; Z ) = I ( X ; Y | Z , W ) - I ( X ; Y | W ) - I ( X ; Y | Z ) + I ( X ; Y ) \begin{aligned}\displaystyle I(W;X;Y;Z)&\displaystyle=I(X;Y;Z|W)-I(X;Y;Z)\\ &\displaystyle=I(X;Y|Z,W)-I(X;Y|W)-I(X;Y|Z)+I(X;Y)\end{aligned}
  58. I ( W ; X ; Y ; Z ) = H ( W ) + H ( X ) + H ( Y ) + H ( Z ) - H ( W , X ) - H ( W , Y ) - H ( W , Z ) - H ( X , Y ) - H ( X , Z ) - H ( Y , Z ) + H ( W , X , Y ) + H ( W , X , Z ) + H ( W , Y , Z ) + H ( X , Y , Z ) - H ( W , X , Y , Z ) \begin{aligned}\displaystyle I(W;X;Y;Z)=&\displaystyle\ H(W)+H(X)+H(Y)+H(Z)\\ &\displaystyle-H(W,X)-H(W,Y)-H(W,Z)-H(X,Y)-H(X,Z)-H(Y,Z)\\ &\displaystyle+H(W,X,Y)+H(W,X,Z)+H(W,Y,Z)+H(X,Y,Z)-H(W,X,Y,Z)\end{aligned}
  59. 𝒱 = { X 1 , X 2 , , X n } \mathcal{V}=\{X_{1},X_{2},\ldots,X_{n}\}
  60. I ( 𝒱 ) - 𝒯 𝒱 ( - 1 ) | 𝒱 | - | 𝒯 | H ( 𝒯 ) I(\mathcal{V})\equiv-\sum_{\mathcal{T}\subseteq\mathcal{V}}(-1)^{\left|% \mathcal{V}\right|-\left|\mathcal{T}\right|}H(\mathcal{T})
  61. 𝒯 𝒱 \mathcal{T}\subseteq\mathcal{V}
  62. | 𝒱 | = n \left|\mathcal{V}\right|=n
  63. { X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , X 8 } \{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7},X_{8}\}
  64. Y 1 = { X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 } Y 2 = { X 4 , X 5 , X 6 , X 7 } Y 3 = { X 5 , X 6 , X 7 , X 8 } \begin{matrix}Y_{1}&=&\{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7}\}\\ Y_{2}&=&\{X_{4},X_{5},X_{6},X_{7}\}\\ Y_{3}&=&\{X_{5},X_{6},X_{7},X_{8}\}\end{matrix}
  65. Y i Y_{i}
  66. { X 5 , X 6 , X 7 } \{X_{5},X_{6},X_{7}\}
  67. I ( Y 1 ; Y 2 ; Y 3 ) I(Y_{1};Y_{2};Y_{3})
  68. - 3 -3
  69. Y 1 = { X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 } Y 2 = { X 4 , X 5 , X 6 , X 7 } Y 3 = { X 5 , X 6 , X 7 , X 8 } Y 4 = { X 7 , X 8 } \begin{matrix}Y_{1}&=&\{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7}\}\\ Y_{2}&=&\{X_{4},X_{5},X_{6},X_{7}\}\\ Y_{3}&=&\{X_{5},X_{6},X_{7},X_{8}\}\\ Y_{4}&=&\{X_{7},X_{8}\}\end{matrix}
  70. Y 4 = { X 7 , X 8 } Y_{4}=\{X_{7},X_{8}\}
  71. Y i Y_{i}
  72. { X 7 } \{X_{7}\}
  73. I ( Y 1 ; Y 2 ; Y 3 ; Y 4 ) I(Y_{1};Y_{2};Y_{3};Y_{4})
  74. - 1 -1
  75. I ( Y 1 ; Y 2 ; Y 3 ; Y 4 ) I(Y_{1};Y_{2};Y_{3};Y_{4})
  76. + 1 +1
  77. I ( Y 1 ; Y 2 ; Y 3 ; Y 4 ) = I ( Y 1 ; Y 2 ; Y 3 | Y 4 ) - I ( Y 1 ; Y 2 ; Y 3 ) = - 2 + 3 = 1 \begin{matrix}I(Y_{1};Y_{2};Y_{3};Y_{4})&=&I(Y_{1};Y_{2};Y_{3}|Y_{4})-I(Y_{1};% Y_{2};Y_{3})\\ &=&-2+3\\ &=&1\end{matrix}

Interactive_skeleton-driven_simulation.html

  1. p : Ω × 3 : ( x , t ) p ( x , t ) p:\Omega\times\mathbb{R}\rightarrow\mathbb{R}^{3}:(x,t)\mapsto p(x,t)
  2. p S : S × 3 p_{S}:S\times\mathbb{R}\rightarrow\mathbb{R}^{3}
  3. r ( x ) = a r a a ( x ) = r a a ( x ) = x r(x)=\sum_{a}r_{a}\emptyset^{a}(x)=r_{a}\emptyset^{a}(x)=x

Interface_conditions_for_electromagnetic_fields.html

  1. 𝐧 12 × ( 𝐄 2 - 𝐄 1 ) = 𝟎 \mathbf{n}_{12}\times(\mathbf{E}_{2}-\mathbf{E}_{1})=\mathbf{0}
  2. 𝐧 12 \mathbf{n}_{12}
  3. ( 𝐃 2 - 𝐃 1 ) 𝐧 12 = ρ s (\mathbf{D}_{2}-\mathbf{D}_{1})\cdot\mathbf{n}_{12}=\rho_{s}
  4. 𝐧 12 \mathbf{n}_{12}
  5. ρ s \rho_{s}
  6. ( 𝐁 2 - 𝐁 1 ) 𝐧 12 = 0 (\mathbf{B}_{2}-\mathbf{B}_{1})\cdot\mathbf{n}_{12}=0
  7. 𝐧 12 \mathbf{n}_{12}
  8. 𝐧 12 × ( 𝐇 2 - 𝐇 1 ) = 𝐣 s \mathbf{n}_{12}\times(\mathbf{H}_{2}-\mathbf{H}_{1})=\mathbf{j}_{s}
  9. 𝐧 12 \mathbf{n}_{12}
  10. 𝐣 s \mathbf{j}_{s}

International_Fisher_effect.html

  1. ( 1 + i $ ) = ( 1 + ρ $ ) × E ( 1 + π $ ) (1+i_{\$})=(1+\rho_{\$})\times E(1+\pi_{\$})
  2. i $ = ρ $ + E ( π $ ) + ρ $ E ( π $ ) ρ $ + E ( π $ ) i_{\$}=\rho_{\$}+E(\pi_{\$})+\rho_{\$}E(\pi_{\$})\approx\rho_{\$}+E(\pi_{\$})
  3. i $ i_{\$}
  4. ρ $ \rho_{\$}
  5. E ( π $ ) E(\pi_{\$})
  6. E ( π $ ) = ( i $ - ρ $ ) ( 1 + ρ $ ) i $ - ρ $ E(\pi_{\$})=\frac{(i_{\$}-\rho_{\$})}{(1+\rho_{\$})}\approx i_{\$}-\rho_{\$}
  7. $ \$
  8. ρ $ = ρ c \rho_{\$}=\rho_{c}
  9. E ( e ) = ( i $ - i c ) ( 1 + i c ) i $ - i c E(e)=\frac{(i_{\$}-i_{c})}{(1+i_{c})}\approx i_{\$}-i_{c}
  10. E ( e ) E(e)
  11. E ( e ) = ( 1 + i $ ) ( 1 + i c ) - 1 E(e)=\frac{(1+i_{\$})}{(1+i_{c})}-1
  12. E ( S t + k ) S t - 1 = ( i $ - i c ) ( 1 + i c ) = E ( e ) \frac{E(S_{t+k})}{S_{t}}-1=\frac{(i_{\$}-i_{c})}{(1+i_{c})}=E(e)
  13. E ( S t + k ) E(S_{t+k})
  14. S t S_{t}
  15. F t , T S t - 1 = ( i $ - i c ) ( 1 + i c ) = E ( e ) \frac{F_{t,T}}{S_{t}}-1=\frac{(i_{\$}-i_{c})}{(1+i_{c})}=E(e)
  16. F t , T F_{t,T}
  17. $ 1.4339 × ( 1 + 5 % ) ( 1 + 7 % ) = $ 1.4071 \$1.4339\times\frac{(1+5\%)}{(1+7\%)}=\$1.4071
  18. E ( e ) = ( 5 % - 7 % ) ( 1 + 7 % ) = - 0.018692 = - 1.87 % E(e)=\frac{(5\%-7\%)}{(1+7\%)}=-0.018692=-1.87\%
  19. E ( e ) = ( 1 + 5 % ) ( 1 + 7 % ) - 1 = - 0.018692 = - 1.87 % E(e)=\frac{(1+5\%)}{(1+7\%)}-1=-0.018692=-1.87\%

Intersection_theorem.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. { A , B , C , a , b , c , P , Q , R , O } \{A,B,C,a,b,c,P,Q,R,O\}
  6. { A B , A C , B C , a b , a c , b c , A a , B b , C c , P Q } \{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}
  7. ( A , A B ) (A,AB)
  8. { ( O , A a ) , ( O , B b ) , ( O , C c ) , ( P , B C ) , ( P , b c ) , ( Q , A C ) , ( Q , a c ) , ( R , A B ) , ( R , a b ) } \{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}
  9. ( R , P Q ) (R,PQ)
  10. R R
  11. [ u o v e r b a r , u P Q ] [u^{\prime}overbar^{\prime},u^{\prime}PQ^{\prime}]
  12. P P
  13. P P
  14. D D
  15. P = 2 D P=\mathbb{P}_{2}D
  16. P P
  17. D D
  18. 2 D \mathbb{P}_{2}D
  19. D D
  20. a , b D , a b = b a \forall a,b\in D,\quad a\cdot b=b\cdot a
  21. 2 D \mathbb{P}_{2}D
  22. D D
  23. 2 \neq 2
  24. a + a = 0 a+a=0

Interspecific_competition.html

  1. d N 1 d t = r 1 N 1 K 1 - N 1 - α 12 N 2 K 1 {dN_{1}\over dt}=r_{1}N_{1}{K_{1}-N_{1}-\alpha_{12}N_{2}\over K_{1}}
  2. d N 2 d t = r 2 N 2 K 2 - N 2 - α 21 N 1 K 2 {dN_{2}\over dt}=r_{2}N_{2}{K_{2}-N_{2}-\alpha_{21}N_{1}\over K_{2}}

Intertemporal_CAPM.html

  1. d X = μ d t + s d Z dX=\mu dt+sdZ
  2. E o { o T U [ C ( t ) , t ] d t + B [ W ( T ) , T ] } E_{o}\left\{\int_{o}^{T}U[C(t),t]dt+B[W(T),T]\right\}
  3. w i w_{i}
  4. W ( t + d t ) = [ W ( t ) - C ( t ) d t ] i = 0 n w i [ 1 + r i ( t + d t ) ] W(t+dt)=[W(t)-C(t)dt]\sum_{i=0}^{n}w_{i}[1+r_{i}(t+dt)]
  5. r i r_{i}
  6. d W = - C ( t ) d t + [ W ( t ) - C ( t ) d t ] w i ( t ) r i ( t + d t ) dW=-C(t)dt+[W(t)-C(t)dt]\sum w_{i}(t)r_{i}(t+dt)
  7. max E o { t = o T - d t t t + d t U [ C ( s ) , s ] d s + B [ W ( T ) , T ] } \max E_{o}\left\{\sum_{t=o}^{T-dt}\int_{t}^{t+dt}U[C(s),s]ds+B[W(T),T]\right\}
  8. t t + d t U [ C ( s ) , s ] d s = U [ C ( t ) , t ] d t + 1 2 U t [ C ( t * ) , t * ] d t 2 U [ C ( t ) , t ] d t \int_{t}^{t+dt}U[C(s),s]ds=U[C(t),t]dt+\frac{1}{2}U_{t}[C(t^{*}),t^{*}]dt^{2}% \approx U[C(t),t]dt
  9. t * t^{*}
  10. r i ( t + d t ) = α i d t + σ i d z i r_{i}(t+dt)=\alpha_{i}dt+\sigma_{i}dz_{i}
  11. E ( r i ) = α i d t ; E ( r i 2 ) = v a r ( r i ) = σ i 2 d t ; c o v ( r i , r j ) = σ i j d t E(r_{i})=\alpha_{i}dt\quad;\quad E(r_{i}^{2})=var(r_{i})=\sigma_{i}^{2}dt\quad% ;\quad cov(r_{i},r_{j})=\sigma_{ij}dt
  12. d W [ W ( t ) w i α i - C ( t ) ] d t + W ( t ) w i σ i d z i dW\approx[W(t)\sum w_{i}\alpha_{i}-C(t)]dt+W(t)\sum w_{i}\sigma_{i}dz_{i}
  13. J ( W , X , t ) = m a x E t { t t + d t U [ C ( s ) , s ] d s + J [ W ( t + d t ) , X ( t + d t ) , t + d t ] } J(W,X,t)=max\;E_{t}\left\{\int_{t}^{t+dt}U[C(s),s]ds+J[W(t+dt),X(t+dt),t+dt]\right\}
  14. d J = J [ W ( t + d t ) , X ( t + d t ) , t + d t ] - J [ W ( t ) , X ( t ) , t + d t ] = J t d t + J W d W + J X d X + 1 2 J X X d X 2 + 1 2 J W W d W 2 + J W X d X d W dJ=J[W(t+dt),X(t+dt),t+dt]-J[W(t),X(t),t+dt]=J_{t}dt+J_{W}dW+J_{X}dX+\frac{1}{% 2}J_{XX}dX^{2}+\frac{1}{2}J_{WW}dW^{2}+J_{WX}dXdW
  15. E t J [ W ( t + d t ) , X ( t + d t ) , t + d t ] = J [ W ( t ) , X ( t ) , t ] + J t d t + J W E [ d W ] + J X E ( d X ) + 1 2 J X X v a r ( d X ) + 1 2 J W W v a r [ d W ] + J W X c o v ( d X , d W ) E_{t}J[W(t+dt),X(t+dt),t+dt]=J[W(t),X(t),t]+J_{t}dt+J_{W}E[dW]+J_{X}E(dX)+% \frac{1}{2}J_{XX}var(dX)+\frac{1}{2}J_{WW}var[dW]+J_{WX}cov(dX,dW)
  16. m a x { U ( C , t ) + J t + J W W [ i = 1 n w i ( α i - r f ) + r f ] - J W C + W 2 2 J W W i = 1 n j = 1 n w i w j σ i j + J X μ + 1 2 J X X s 2 + J W X W i = 1 n w i σ i X } max\left\{U(C,t)+J_{t}+J_{W}W[\sum_{i=1}^{n}w_{i}(\alpha_{i}-r_{f})+r_{f}]-J_{% W}C+\frac{W^{2}}{2}J_{WW}\sum_{i=1}^{n}\sum_{j=1}^{n}w_{i}w_{j}\sigma_{ij}+J_{% X}\mu+\frac{1}{2}J_{XX}s^{2}+J_{WX}W\sum_{i=1}^{n}w_{i}\sigma_{iX}\right\}
  17. r f r_{f}
  18. J W ( α i - r f ) + J W W W j = 1 n w j * σ i j + J W X σ i X = 0 i = 1 , 2 , , n J_{W}(\alpha_{i}-r_{f})+J_{WW}W\sum_{j=1}^{n}w^{*}_{j}\sigma_{ij}+J_{WX}\sigma% _{iX}=0\quad i=1,2,\ldots,n
  19. ( α - r f 1 ) = - J W W J W Ω w * W + - J W X J W c o v r X (\alpha-r_{f}{1})=\frac{-J_{WW}}{J_{W}}\Omega w^{*}W+\frac{-J_{WX}}{J_{W}}cov_% {rX}
  20. α \alpha
  21. Ω \Omega
  22. 1 {1}
  23. c o v r X cov_{rX}
  24. w * = - J W J W W W Ω - 1 ( α - r f 1 ) - J W X J W W W Ω - 1 c o v r X {w^{*}}=\frac{-J_{W}}{J_{WW}W}\Omega^{-1}(\alpha-r_{f}{1})-\frac{J_{WX}}{J_{WW% }W}\Omega^{-1}cov_{rX}
  25. α i = r f + β i m ( α m - r f ) + β i h ( α h - r f ) \alpha_{i}=r_{f}+\beta_{im}(\alpha_{m}-r_{f})+\beta_{ih}(\alpha_{h}-r_{f})
  26. E ( d W ) = - C ( t ) d t + W ( t ) w i ( t ) α i d t E(dW)=-C(t)dt+W(t)\sum w_{i}(t)\alpha_{i}dt
  27. v a r ( d W ) = [ W ( t ) - C ( t ) d t ] 2 v a r [ w i ( t ) r i ( t + d t ) ] = W ( t ) 2 i = 1 i = 1 w i w j σ i j d t var(dW)=[W(t)-C(t)dt]^{2}var[\sum w_{i}(t)r_{i}(t+dt)]=W(t)^{2}\sum_{i=1}\sum_% {i=1}w_{i}w_{j}\sigma_{ij}dt
  28. i = o n w i ( t ) α i = i = 1 n w i ( t ) [ α i - r f ] + r f \sum_{i=o}^{n}w_{i}(t)\alpha_{i}=\sum_{i=1}^{n}w_{i}(t)[\alpha_{i}-r_{f}]+r_{f}

Interval_scheduling.html

  1. k 3 k\geq 3
  2. X = { x 1 , x 2 , , x p } X=\{x_{1},x_{2},...,x_{p}\}
  3. C = { c 1 , c 2 , , c q } C=\{c_{1},c_{2},...,c_{q}\}
  4. x i x_{i}
  5. 50 i - 10 50i-10
  6. x i = f a l s e x_{i}=false
  7. 50 i + 10 50i+10
  8. x i = t r u e x_{i}=true
  9. c j c_{j}
  10. x i x_{i}
  11. 50 i - 12 50i-12
  12. x i x_{i}
  13. 50 i - 8 50i-8
  14. 50 i - 10 50i-10
  15. x i = f a l s e x_{i}=false
  16. x i x_{i}
  17. 50 i + 8 50i+8
  18. 50 i + 10 50i+10
  19. x i = t r u e x_{i}=true
  20. k 3 k\geq 3
  21. x i x_{i}
  22. y i y_{i}
  23. x i y i x_{i}\cup y_{i}
  24. ¬ x i ¬ y i \neg{x_{i}}\cup\neg{y_{i}}
  25. x i x_{i}
  26. y j y_{j}
  27. ¬ x i ¬ y j \neg{x_{i}}\cup\neg{y_{j}}
  28. k 2 k\geq 2

Intrabeam_scattering.html

  1. 1 / γ 4 1/\gamma^{4}
  2. 1 T p = def 1 σ p d σ p d t \frac{1}{T_{p}}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{\sigma_{p}}\frac{d\sigma% _{p}}{dt}
  3. 1 T h = def 1 ϵ h 1 / 2 d ϵ h 1 / 2 d t \frac{1}{T_{h}}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{\epsilon_{h}^{1/2}}\frac% {d\epsilon_{h}^{1/2}}{dt}
  4. 1 T v = def 1 ϵ v 1 / 2 d ϵ v 1 / 2 d t \frac{1}{T_{v}}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{\epsilon_{v}^{1/2}}\frac% {d\epsilon_{v}^{1/2}}{dt}
  5. 1 T i = 4 π A ( log ) 0 d λ λ 1 / 2 [ det ( L + λ I ) ] 1 / 2 { Tr L i Tr ( 1 L + λ I ) - 3 Tr [ L i ( 1 L + λ I ) ] } \frac{1}{T_{i}}=4\pi A(\operatorname{log})\left\langle\int_{0}^{\infty}\,d% \lambda\ \frac{\lambda^{1/2}}{[\operatorname{det}(L+\lambda I)]^{1/2}}\left\{% \operatorname{Tr}L^{i}\operatorname{Tr}\left(\frac{1}{L+\lambda I}\right)-3% \operatorname{Tr}\left[L^{i}\left(\frac{1}{L+\lambda I}\right)\right]\right\}\right\rangle
  6. T p T_{p}
  7. T h T_{h}
  8. T v T_{v}
  9. ( log ) = ln b m i n b m a x = ln 2 θ m i n (\operatorname{log})=\ln\frac{b_{min}}{b_{max}}=\ln\frac{2}{\theta_{min}}
  10. A = r 0 2 c N 64 π 2 β 3 γ 4 ϵ h ϵ v σ s σ p A=\frac{r_{0}^{2}cN}{64\pi^{2}\beta^{3}\gamma^{4}\epsilon_{h}\epsilon_{v}% \sigma_{s}\sigma_{p}}
  11. L = L ( p ) + L ( h ) + L ( v ) L=L^{(p)}+L^{(h)}+L^{(v)}\,
  12. L ( p ) = γ 2 σ p 2 ( 0 0 0 0 1 0 0 0 0 ) L^{(p)}=\frac{\gamma^{2}}{\sigma^{2}_{p}}\begin{pmatrix}0&0&0\\ 0&1&0\\ 0&0&0\end{pmatrix}
  13. L ( h ) = β h ϵ h ( 1 - γ ϕ h 0 - γ ϕ h γ 2 h β h 0 0 0 0 ) L^{(h)}=\frac{\beta_{h}}{\epsilon_{h}}\begin{pmatrix}1&-\gamma\phi_{h}&0\\ -\gamma\phi_{h}&\frac{\gamma^{2}{\mathcal{H}}_{h}}{\beta_{h}}&0\\ 0&0&0\end{pmatrix}
  14. L ( v ) = β v ϵ v ( 0 0 0 0 γ 2 v β v - γ ϕ v 0 - γ ϕ v 1 ) L^{(v)}=\frac{\beta_{v}}{\epsilon_{v}}\begin{pmatrix}0&0&0\\ 0&\frac{\gamma^{2}{\mathcal{H}}_{v}}{\beta_{v}}&-\gamma\phi_{v}\\ 0&-\gamma\phi_{v}&1\end{pmatrix}
  15. h , v = [ η h , v 2 + ( β h , v η h , v - 1 2 β h , v η h ) 2 ] / β h , v {\mathcal{H}}_{h,v}=[\eta^{2}_{h,v}+(\beta_{h,v}\eta^{\prime}_{h,v}-\frac{1}{2% }\beta^{\prime}_{h,v}\eta_{h})^{2}]/\beta_{h,v}
  16. ϕ h , v = η h , v - 1 2 β h , v η h , v / β h , v \phi_{h,v}=\eta^{\prime}_{h,v}-\frac{1}{2}\beta^{\prime}_{h,v}\eta_{h,v}/\beta% _{h,v}
  17. r 0 2 r_{0}^{2}
  18. c c
  19. N N
  20. β \beta
  21. γ \gamma
  22. β h , v \beta_{h,v}
  23. β h , v \beta^{\prime}_{h,v}
  24. η h , v \eta_{h,v}
  25. η h , v \eta^{\prime}_{h,v}
  26. ϵ h , v \epsilon_{h,v}
  27. σ s \sigma_{s}
  28. σ p \sigma_{p}
  29. b m i n b_{min}
  30. b m a x b_{max}
  31. θ m i n \theta_{min}
  32. σ δ γ = σ x = σ y \frac{\sigma_{\delta}}{\gamma}=\sigma_{x^{\prime}}=\sigma_{y^{\prime}}
  33. γ \gamma
  34. γ \gamma
  35. ϵ x β x + ϵ y β y + η s ϵ z β z \frac{\epsilon_{x}}{\beta_{x}}+\frac{\epsilon_{y}}{\beta_{y}}+\eta_{s}\frac{% \epsilon_{z}}{\beta_{z}}
  36. η s = 1 γ 2 - α c \eta_{s}=\frac{1}{\gamma^{2}}-\alpha_{c}
  37. 1 τ 1 , 2 , 3 = 1 ϵ 1 , 2 , 3 d ϵ 1 , 2 , 3 d t \frac{1}{\tau_{1,2,3}}=\frac{1}{\epsilon_{1,2,3}}\frac{d\epsilon_{1,2,3}}{dt}

Intrinsic_viscosity.html

  1. [ η ] \left[\eta\right]
  2. η \eta
  3. [ η ] = lim ϕ 0 η - η 0 η 0 ϕ \left[\eta\right]=\lim_{\phi\rightarrow 0}\frac{\eta-\eta_{0}}{\eta_{0}\phi}
  4. η 0 \eta_{0}
  5. ϕ \phi
  6. [ η ] \left[\eta\right]
  7. 5 2 \frac{5}{2}
  8. ϕ \phi
  9. [ η ] \left[\eta\right]
  10. a a
  11. b b
  12. [ η ] = ( 4 15 ) ( J + K - L ) + ( 2 3 ) L + ( 1 3 ) M + ( 1 15 ) N \left[\eta\right]=\left(\frac{4}{15}\right)(J+K-L)+\left(\frac{2}{3}\right)L+% \left(\frac{1}{3}\right)M+\left(\frac{1}{15}\right)N
  13. M = def 1 a b 4 1 J α M\ \stackrel{\mathrm{def}}{=}\ \frac{1}{ab^{4}}\frac{1}{J_{\alpha}^{\prime}}
  14. K = def M 2 K\ \stackrel{\mathrm{def}}{=}\ \frac{M}{2}
  15. J = def K J α ′′ J β ′′ J\ \stackrel{\mathrm{def}}{=}\ K\frac{J_{\alpha}^{\prime\prime}}{J_{\beta}^{% \prime\prime}}
  16. L = def 2 a b 2 ( a 2 + b 2 ) 1 J β L\ \stackrel{\mathrm{def}}{=}\ \frac{2}{ab^{2}\left(a^{2}+b^{2}\right)}\frac{1% }{J_{\beta}^{\prime}}
  17. N = def 6 a b 2 ( a 2 - b 2 ) a 2 J α + b 2 J β N\ \stackrel{\mathrm{def}}{=}\ \frac{6}{ab^{2}}\frac{\left(a^{2}-b^{2}\right)}% {a^{2}J_{\alpha}+b^{2}J_{\beta}}
  18. J J
  19. J α = 0 d x ( x + b 2 ) ( x + a 2 ) 3 J_{\alpha}=\int_{0}^{\infty}\frac{dx}{\left(x+b^{2}\right)\sqrt{\left(x+a^{2}% \right)^{3}}}
  20. J β = 0 d x ( x + b 2 ) 2 ( x + a 2 ) J_{\beta}=\int_{0}^{\infty}\frac{dx}{\left(x+b^{2}\right)^{2}\sqrt{\left(x+a^{% 2}\right)}}
  21. J α = 0 d x ( x + b 2 ) 3 ( x + a 2 ) J_{\alpha}^{\prime}=\int_{0}^{\infty}\frac{dx}{\left(x+b^{2}\right)^{3}\sqrt{% \left(x+a^{2}\right)}}
  22. J β = 0 d x ( x + b 2 ) 2 ( x + a 2 ) 3 J_{\beta}^{\prime}=\int_{0}^{\infty}\frac{dx}{\left(x+b^{2}\right)^{2}\sqrt{% \left(x+a^{2}\right)^{3}}}
  23. J α ′′ = 0 x d x ( x + b 2 ) 3 ( x + a 2 ) J_{\alpha}^{\prime\prime}=\int_{0}^{\infty}\frac{x\ dx}{\left(x+b^{2}\right)^{% 3}\sqrt{\left(x+a^{2}\right)}}
  24. J β ′′ = 0 x d x ( x + b 2 ) 2 ( x + a 2 ) 3 J_{\beta}^{\prime\prime}=\int_{0}^{\infty}\frac{x\ dx}{\left(x+b^{2}\right)^{2% }\sqrt{\left(x+a^{2}\right)^{3}}}
  25. a a
  26. b b
  27. c c

Introduction_to_entropy.html

  1. δ S = δ q T . {\rm\delta}S=\frac{{\rm\delta}q}{T}.
  2. δ S δ q T . {{\rm\delta}S}\geq{\frac{{\rm\delta}q}{T}}.
  3. S = k B ln Ω , S=k_{B}\ln\Omega,\!
  4. Δ S \Delta S
  5. Δ S = q r e v T \Delta S=\frac{q_{rev}}{T}
  6. Δ S \Delta S
  7. Δ S = S f i n a l - S i n i t i a l \Delta S=S_{final}-S_{initial}
  8. Δ S = S f i n a l - S i n i t i a l = q r e v T \Delta S=S_{final}-S_{initial}=\frac{q_{rev}}{T}
  9. q r e v T = 6008 J 273 K \frac{q_{rev}}{T}=\frac{6008J}{273K}
  10. q r e v T \frac{q_{rev}}{T}
  11. C p C_{p}
  12. d T T \frac{dT}{T}
  13. T i n i t i a l T_{initial}
  14. T f i n a l T_{final}
  15. Δ S = C p ln T f i n a l T i n i t i a l \Delta S=C_{p}\ln\frac{T_{final}}{T_{initial}}

Introduction_to_the_mathematics_of_general_relativity.html

  1. z 1 z^{1}
  2. z 2 z^{2}
  3. y 1 y^{1}
  4. y 2 y^{2}
  5. y 1 = r cos ( x / r ) y^{1}=r\cos(x/r)
  6. y 2 = r sin ( x / r ) y^{2}=r\sin(x/r)
  7. y 1 / x = - sin ( x / r ) \partial y^{1}/\partial x=-\sin(x/r)
  8. y 2 / x = cos ( x / r ) . \partial y^{2}/\partial x=\cos(x/r).
  9. g = cos 2 ( x / r ) + sin 2 ( x / r ) = 1. g=\cos^{2}(x/r)+\sin^{2}(x/r)=1.
  10. d s 2 = g d x 2 = d x 2 . ds^{2}=g\,dx^{2}=dx^{2}.\,
  11. s 2 = Δ r 2 - c 2 Δ t 2 s^{2}=\Delta r^{2}-c^{2}\Delta t^{2}\,
  12. s 2 s^{2}
  13. Δ r 2 \Delta r^{2}\,
  14. ( Δ r ) 2 (\Delta r)^{2}\,
  15. s 2 s^{2}
  16. s s
  17. s 2 s^{2}
  18. c 2 Δ t 2 c^{2}\Delta t^{2}
  19. Δ r 2 \Delta r^{2}
  20. R ρ σ μ ν R^{\rho}{}_{\sigma\mu\nu}
  21. R σ ν R_{\sigma\nu}
  22. 𝐆 \mathbf{G}
  23. 𝐆 = 𝐑 - 1 2 𝐠 R , \mathbf{G}=\mathbf{R}-\frac{1}{2}\mathbf{g}R,
  24. 𝐑 \mathbf{R}
  25. 𝐠 \mathbf{g}
  26. R R
  27. G μ ν = 8 π G c 4 T μ ν , G_{\mu\nu}={8\pi G\over c^{4}}T_{\mu\nu},
  28. G μ ν G_{\mu\nu}
  29. T μ ν T_{\mu\nu}

Invariant_differential_operator.html

  1. n \mathbb{R}^{n}
  2. D D
  3. D f Df
  4. f ( x ) f(x)
  5. f f
  6. x x
  7. G G
  8. D ( g f ) = g ( D f ) . D(g\cdot f)=g\cdot(Df).
  9. ρ : H Aut ( 𝕍 ) \rho:H\rightarrow\mathrm{Aut}(\mathbb{V})
  10. V = G × H 𝕍 where ( g h , v ) ( g , ρ ( h ) v ) g G , h H and v 𝕍 . V=G\times_{H}\mathbb{V}\;\,\text{where}\;(gh,v)\sim(g,\rho(h)v)\;\forall\;g\in G% ,\;h\in H\;\,\text{and}\;v\in\mathbb{V}.
  11. φ Γ ( V ) \varphi\in\Gamma(V)
  12. Γ ( V ) = { φ : G 𝕍 : φ ( g h ) = ρ ( h - 1 ) φ ( g ) g G , h H } . \Gamma(V)=\{\varphi:G\rightarrow\mathbb{V}\;:\;\varphi(gh)=\rho(h^{-1})\varphi% (g)\;\forall\;g\in G,\;h\in H\}.
  13. ( g φ ) ( g ) = φ ( g - 1 g ) . (\ell_{g}\varphi)(g^{\prime})=\varphi(g^{-1}g^{\prime}).
  14. d : Γ ( V ) Γ ( W ) d:\Gamma(V)\rightarrow\Gamma(W)
  15. d ( g φ ) = g ( d φ ) . d(\ell_{g}\varphi)=\ell_{g}(d\varphi).
  16. φ \varphi
  17. Γ ( V ) \Gamma(V)
  18. \nabla
  19. ^ \hat{\nabla}
  20. ω \omega
  21. a ω b = ^ a ω b - Q a b ω c c \nabla_{a}\omega_{b}=\hat{\nabla}_{a}\omega_{b}-Q_{ab}{}^{c}\omega_{c}
  22. Q a b c Q_{ab}{}^{c}
  23. [ ] [\nabla]
  24. Q a b = c Q ( a b ) c Q_{ab}{}^{c}=Q_{(ab)}{}^{c}
  25. [ a ω b ] = ^ [ a ω b ] , \nabla_{[a}\omega_{b]}=\hat{\nabla}_{[a}\omega_{b]},
  26. \nabla
  27. d = j j d x j d=\sum_{j}\partial_{j}\,dx_{j}
  28. d : Ω n ( M ) Ω n + 1 ( M ) d:\Omega^{n}(M)\rightarrow\Omega^{n+1}(M)
  29. X a ( a X b ) - 1 n c X c g a b X^{a}\mapsto\nabla_{(a}X_{b)}-\frac{1}{n}\nabla_{c}X^{c}g_{ab}
  30. g ( x , y ) = x 1 y n + 2 + x n + 2 y 1 + i = 2 n + 1 x i y i g(x,y)=x_{1}y_{n+2}+x_{n+2}y_{1}+\sum_{i=2}^{n+1}x_{i}y_{i}
  31. n + 2 \mathbb{R}^{n+2}
  32. S n S^{n}
  33. S n = { [ x ] n + 1 : g ( x , x ) = 0 } . S^{n}=\{[x]\in\mathbb{RP}_{n+1}\;:\;g(x,x)=0\}.
  34. S n = G / P S^{n}=G/P
  35. G = S O 0 ( n + 1 , 1 ) G=SO_{0}(n+1,1)
  36. n + 2 \mathbb{R}^{n+2}

Invariant_measure.html

  1. μ ( f - 1 ( A ) ) = μ ( A ) . \mu\left(f^{-1}(A)\right)=\mu(A).
  2. μ ( φ t - 1 ( A ) ) = μ ( A ) t T , A Σ . \mu\left(\varphi_{t}^{-1}(A)\right)=\mu(A)\qquad\forall t\in T,A\in\Sigma.
  3. T a ( x ) = x + a . T_{a}(x)=x+a.
  4. T ( x ) = A x + b T(x)=Ax+b
  5. s y m b o l S = { A , B } symbol{\rm S}=\{A,B\}
  6. T = Id T={\rm Id}
  7. μ : s y m b o l S s y m b o l R \mu:symbol{\rm S}\rightarrow symbol{\rm R}

Inverse-Wishart_distribution.html

  1. Γ p \Gamma_{p}
  2. tr \mathrm{tr}
  3. 𝚿 ν - p - 1 \frac{\mathbf{\Psi}}{\nu-p-1}
  4. ν > p + 1 \nu>p+1
  5. 𝚿 ν + p + 1 \frac{\mathbf{\Psi}}{\nu+p+1}
  6. 𝐗 \mathbf{X}
  7. 𝐗 𝒲 - 1 ( 𝚿 , ν ) \mathbf{X}\sim\mathcal{W}^{-1}({\mathbf{\Psi}},\nu)
  8. 𝐗 - 1 \mathbf{X}^{-1}
  9. 𝒲 ( 𝚿 - 1 , ν ) \mathcal{W}({\mathbf{\Psi}}^{-1},\nu)
  10. | 𝚿 | ν 2 2 ν p 2 Γ p ( ν 2 ) | 𝐗 | - ν + p + 1 2 e - 1 2 tr ( 𝚿 𝐗 - 1 ) \frac{\left|{\mathbf{\Psi}}\right|^{\frac{\nu}{2}}}{2^{\frac{\nu p}{2}}\Gamma_% {p}(\frac{\nu}{2})}\left|\mathbf{X}\right|^{-\frac{\nu+p+1}{2}}e^{-\frac{1}{2}% \operatorname{tr}({\mathbf{\Psi}}\mathbf{X}^{-1})}
  11. 𝐗 \mathbf{X}
  12. 𝚿 {\mathbf{\Psi}}
  13. p × p p\times p
  14. 𝐀 𝒲 ( 𝚺 , ν ) {\mathbf{A}}\sim\mathcal{W}({\mathbf{\Sigma}},\nu)
  15. 𝚺 {\mathbf{\Sigma}}
  16. p × p p\times p
  17. 𝐗 = 𝐀 - 1 \mathbf{X}={\mathbf{A}}^{-1}
  18. 𝐗 𝒲 - 1 ( 𝚺 - 1 , ν ) \mathbf{X}\sim\mathcal{W}^{-1}({\mathbf{\Sigma}}^{-1},\nu)
  19. 𝐀 𝒲 - 1 ( 𝚿 , ν ) {\mathbf{A}}\sim\mathcal{W}^{-1}({\mathbf{\Psi}},\nu)
  20. 𝐀 {\mathbf{A}}
  21. 𝚿 {\mathbf{\Psi}}
  22. 𝐀 = [ 𝐀 11 𝐀 12 𝐀 21 𝐀 22 ] , 𝚿 = [ 𝚿 11 𝚿 12 𝚿 21 𝚿 22 ] {\mathbf{A}}=\begin{bmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\end{bmatrix},\;{\mathbf{\Psi}}=\begin{bmatrix}% \mathbf{\Psi}_{11}&\mathbf{\Psi}_{12}\\ \mathbf{\Psi}_{21}&\mathbf{\Psi}_{22}\end{bmatrix}
  23. 𝐀 i j {\mathbf{A}_{ij}}
  24. 𝚿 i j {\mathbf{\Psi}_{ij}}
  25. p i × p j p_{i}\times p_{j}
  26. 𝐀 11 {\mathbf{A}_{11}}
  27. 𝐀 11 - 1 𝐀 12 {\mathbf{A}}_{11}^{-1}{\mathbf{A}}_{12}
  28. 𝐀 22 1 {\mathbf{A}}_{22\cdot 1}
  29. 𝐀 22 1 = 𝐀 22 - 𝐀 21 𝐀 11 - 1 𝐀 12 {\mathbf{A}_{22\cdot 1}}={\mathbf{A}}_{22}-{\mathbf{A}}_{21}{\mathbf{A}}_{11}^% {-1}{\mathbf{A}}_{12}
  30. 𝐀 11 {\mathbf{A}_{11}}
  31. 𝐀 {\mathbf{A}}
  32. 𝐀 11 𝒲 - 1 ( 𝚿 11 , ν - p 2 ) {\mathbf{A}_{11}}\sim\mathcal{W}^{-1}({\mathbf{\Psi}_{11}},\nu-p_{2})
  33. 𝐀 11 - 1 𝐀 12 | 𝐀 22 1 M N p 1 × p 2 ( 𝚿 11 - 1 𝚿 12 , 𝐀 22 1 𝚿 11 - 1 ) {\mathbf{A}}_{11}^{-1}{\mathbf{A}}_{12}|{\mathbf{A}}_{22\cdot 1}\sim MN_{p_{1}% \times p_{2}}({\mathbf{\Psi}}_{11}^{-1}{\mathbf{\Psi}}_{12},{\mathbf{A}}_{22% \cdot 1}\otimes{\mathbf{\Psi}}_{11}^{-1})
  34. M N p × q ( , ) MN_{p\times q}(\cdot,\cdot)
  35. 𝐀 22 1 𝒲 - 1 ( 𝚿 22 1 , ν ) {\mathbf{A}}_{22\cdot 1}\sim\mathcal{W}^{-1}({\mathbf{\Psi}}_{22\cdot 1},\nu)
  36. 𝚿 22 1 = 𝚿 22 - 𝚿 21 𝚿 11 - 1 𝚿 12 {\mathbf{\Psi}_{22\cdot 1}}={\mathbf{\Psi}}_{22}-{\mathbf{\Psi}}_{21}{\mathbf{% \Psi}}_{11}^{-1}{\mathbf{\Psi}}_{12}
  37. 𝚺 {\mathbf{\Sigma}}
  38. p ( 𝚺 ) {p(\mathbf{\Sigma})}
  39. 𝒲 - 1 ( 𝚿 , ν ) \mathcal{W}^{-1}({\mathbf{\Psi}},\nu)
  40. 𝐗 = [ 𝐱 1 , , 𝐱 n ] \mathbf{X}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]
  41. N ( 𝟎 , 𝚺 ) N(\mathbf{0},{\mathbf{\Sigma}})
  42. p ( 𝚺 | 𝐗 ) {p(\mathbf{\Sigma}|\mathbf{X})}
  43. 𝒲 - 1 ( 𝐀 + 𝚿 , n + ν ) \mathcal{W}^{-1}({\mathbf{A}}+{\mathbf{\Psi}},n+\nu)
  44. 𝐀 = 𝐗𝐗 T {\mathbf{A}}=\mathbf{X}\mathbf{X}^{T}
  45. 𝚺 \mathbf{\Sigma}
  46. P ( 𝐗 | 𝚿 , ν ) = P ( 𝐗 | 𝚺 ) P ( 𝚺 | 𝚿 , ν ) d 𝚺 = | 𝚿 | ν 2 Γ p ( ν + n 2 ) π n p 2 | 𝚿 + 𝐀 | ν + n 2 Γ p ( ν 2 ) P(\mathbf{X}|\mathbf{\Psi},\nu)=\int P(\mathbf{X}|\mathbf{\Sigma})P(\mathbf{% \Sigma}|\mathbf{\Psi},\nu)d\mathbf{\Sigma}=\frac{|\mathbf{\Psi}|^{\frac{\nu}{2% }}\Gamma_{p}\left(\frac{\nu+n}{2}\right)}{\pi^{\frac{np}{2}}|\mathbf{\Psi}+% \mathbf{A}|^{\frac{\nu+n}{2}}\Gamma_{p}(\frac{\nu}{2})}
  47. 𝚺 \mathbf{\Sigma}
  48. 𝚿 {\mathbf{\Psi}}
  49. 𝐀 {\mathbf{A}}
  50. E ( 𝐗 ) = 𝚿 ν - p - 1 . E(\mathbf{X})=\frac{\mathbf{\Psi}}{\nu-p-1}.
  51. 𝐗 \mathbf{X}
  52. Var ( x i j ) = ( ν - p + 1 ) ψ i j 2 + ( ν - p - 1 ) ψ i i ψ j j ( ν - p ) ( ν - p - 1 ) 2 ( ν - p - 3 ) \operatorname{Var}(x_{ij})=\frac{(\nu-p+1)\psi_{ij}^{2}+(\nu-p-1)\psi_{ii}\psi% _{jj}}{(\nu-p)(\nu-p-1)^{2}(\nu-p-3)}
  53. i = j i=j
  54. Var ( x i i ) = 2 ψ i i 2 ( ν - p - 1 ) 2 ( ν - p - 3 ) . \operatorname{Var}(x_{ii})=\frac{2\psi_{ii}^{2}}{(\nu-p-1)^{2}(\nu-p-3)}.
  55. 𝐗 \mathbf{X}
  56. Cov ( x i j , x k l ) = 2 ψ i j ψ k l + ( ν - p - 1 ) ( ψ i k ψ j l + ψ i l ψ k j ) ( ν - p ) ( ν - p - 1 ) 2 ( ν - p - 3 ) \operatorname{Cov}(x_{ij},x_{kl})=\frac{2\psi_{ij}\psi_{kl}+(\nu-p-1)(\psi_{ik% }\psi_{jl}+\psi_{il}\psi_{kj})}{(\nu-p)(\nu-p-1)^{2}(\nu-p-3)}
  57. p = 1 p=1
  58. α = ν / 2 \alpha=\nu/2
  59. β = 𝚿 / 2 \beta=\mathbf{\Psi}/2
  60. x = 𝐗 x=\mathbf{X}
  61. p ( x | α , β ) = β α x - α - 1 exp ( - β / x ) Γ 1 ( α ) . p(x|\alpha,\beta)=\frac{\beta^{\alpha}\,x^{-\alpha-1}\exp(-\beta/x)}{\Gamma_{1% }(\alpha)}.
  62. Γ 1 ( ) \Gamma_{1}(\cdot)
  63. 𝒢 𝒲 - 1 \mathcal{GW}^{-1}
  64. p × p p\times p
  65. 𝐗 \mathbf{X}
  66. 𝒢 𝒲 - 1 ( 𝚿 , ν , 𝐒 ) \mathcal{GW}^{-1}(\mathbf{\Psi},\nu,\mathbf{S})
  67. 𝐘 = 𝐗 1 / 2 𝐒 - 1 𝐗 1 / 2 \mathbf{Y}=\mathbf{X}^{1/2}\mathbf{S}^{-1}\mathbf{X}^{1/2}
  68. 𝒲 - 1 ( 𝚿 , ν ) \mathcal{W}^{-1}(\mathbf{\Psi},\nu)
  69. 𝐗 1 / 2 \mathbf{X}^{1/2}
  70. 𝐗 \mathbf{X}
  71. 𝚿 , 𝐒 \mathbf{\Psi},\mathbf{S}
  72. p × p p\times p
  73. ν \nu
  74. 2 p 2p
  75. 𝐒 \mathbf{S}
  76. 𝒢 𝒲 - 1 ( 𝚿 , ν , 𝐒 ) = 𝒲 - 1 ( 𝚿 , ν ) \mathcal{GW}^{-1}(\mathbf{\Psi},\nu,\mathbf{S})=\mathcal{W}^{-1}(\mathbf{\Psi}% ,\nu)

Inverse_Gaussian_distribution.html

  1. Φ ( λ x ( x μ - 1 ) ) \Phi\left(\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}-1\right)\right)
  2. + exp ( 2 λ μ ) Φ ( - λ x ( x μ + 1 ) ) +\exp\left(\frac{2\lambda}{\mu}\right)\Phi\left(-\sqrt{\frac{\lambda}{x}}\left% (\frac{x}{\mu}+1\right)\right)
  3. Φ ( ) \Phi\left(\right)
  4. 𝐄 [ X ] = μ \scriptstyle\mathbf{E}[X]=\mu
  5. 𝐄 [ 1 X ] = 1 μ + 1 λ \scriptstyle\mathbf{E}[\frac{1}{X}]=\frac{1}{\mu}+\frac{1}{\lambda}
  6. μ [ ( 1 + 9 μ 2 4 λ 2 ) 1 2 - 3 μ 2 λ ] \mu\left[\left(1+\frac{9\mu^{2}}{4\lambda^{2}}\right)^{\frac{1}{2}}-\frac{3\mu% }{2\lambda}\right]
  7. 𝐕𝐚𝐫 [ X ] = μ 3 λ \scriptstyle\mathbf{Var}[X]=\frac{\mu^{3}}{\lambda}
  8. 𝐕𝐚𝐫 [ 1 X ] = 1 μ λ + 2 λ 2 \scriptstyle\mathbf{Var}[\frac{1}{X}]=\frac{1}{\mu\lambda}+\frac{2}{\lambda^{2}}
  9. 3 ( μ λ ) 1 / 2 3\left(\frac{\mu}{\lambda}\right)^{1/2}
  10. 15 μ λ \frac{15\mu}{\lambda}
  11. e ( λ μ ) [ 1 - 1 - 2 μ 2 t λ ] e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^{2}t}{\lambda}}% \right]}
  12. e ( λ μ ) [ 1 - 1 - 2 μ 2 i t λ ] e^{\left(\frac{\lambda}{\mu}\right)\left[1-\sqrt{1-\frac{2\mu^{2}\mathrm{i}t}{% \lambda}}\right]}
  13. f ( x ; μ , λ ) = [ λ 2 π x 3 ] 1 / 2 exp - λ ( x - μ ) 2 2 μ 2 x f(x;\mu,\lambda)=\left[\frac{\lambda}{2\pi x^{3}}\right]^{1/2}\exp{\frac{-% \lambda(x-\mu)^{2}}{2\mu^{2}x}}
  14. μ > 0 \mu>0
  15. λ > 0 \lambda>0
  16. X I G ( μ , λ ) . X\sim IG(\mu,\lambda).\,\!
  17. S = i = 1 n X i I G ( μ 0 w i , λ 0 ( w i ) 2 ) . S=\sum_{i=1}^{n}X_{i}\sim IG\left(\mu_{0}\sum w_{i},\lambda_{0}\left(\sum w_{i% }\right)^{2}\right).
  18. Var ( X i ) E ( X i ) = μ 0 2 w i 2 λ 0 w i 2 = μ 0 2 λ 0 \frac{\textrm{Var}(X_{i})}{\textrm{E}(X_{i})}=\frac{\mu_{0}^{2}w_{i}^{2}}{% \lambda_{0}w_{i}^{2}}=\frac{\mu_{0}^{2}}{\lambda_{0}}
  19. X I G ( μ , λ ) t X I G ( t μ , t λ ) . X\sim IG(\mu,\lambda)\,\,\,\,\,\,\Rightarrow\,\,\,\,\,\,tX\sim IG(t\mu,t% \lambda).
  20. { 2 μ 2 x 2 f ( x ) + f ( x ) ( - λ μ 2 + λ x 2 + 3 μ 2 x ) = 0 , f ( 1 ) = λ e - λ ( 1 - μ ) 2 2 μ 2 2 π } \left\{2\mu^{2}x^{2}f^{\prime}(x)+f(x)\left(-\lambda\mu^{2}+\lambda x^{2}+3\mu% ^{2}x\right)=0,f(1)=\frac{\sqrt{\lambda}e^{-\frac{\lambda(1-\mu)^{2}}{2\mu^{2}% }}}{\sqrt{2\pi}}\right\}
  21. X 0 = 0 X_{0}=0\quad
  22. X t = ν t + σ W t X_{t}=\nu t+\sigma W_{t}\quad\quad\quad\quad
  23. ν > 0 \nu>0
  24. α > 0 \alpha>0
  25. T α = inf { 0 < t X t = α } I G ( α ν , α 2 σ 2 ) . T_{\alpha}=\inf\{0<t\mid X_{t}=\alpha\}\sim IG(\tfrac{\alpha}{\nu},\tfrac{% \alpha^{2}}{\sigma^{2}}).\,
  26. f ( x ; 0 , ( α σ ) 2 ) = α σ 2 π x 3 exp ( - α 2 2 x σ 2 ) . f\left(x;0,\left(\frac{\alpha}{\sigma}\right)^{2}\right)=\frac{\alpha}{\sigma% \sqrt{2\pi x^{3}}}\exp\left(-\frac{\alpha^{2}}{2x\sigma^{2}}\right).
  27. c = α 2 σ 2 c=\frac{\alpha^{2}}{\sigma^{2}}
  28. μ = 0 \mu=0
  29. X i I G ( μ , λ w i ) , i = 1 , 2 , , n X_{i}\sim IG(\mu,\lambda w_{i}),\,\,\,\,\,\,i=1,2,\ldots,n
  30. L ( μ , λ ) = ( λ 2 π ) n 2 ( i = 1 n w i X i 3 ) 1 2 exp ( λ μ i = 1 n w i - λ 2 μ 2 i = 1 n w i X i - λ 2 i = 1 n w i 1 X i ) . L(\mu,\lambda)=\left(\frac{\lambda}{2\pi}\right)^{\frac{n}{2}}\left(\prod^{n}_% {i=1}\frac{w_{i}}{X_{i}^{3}}\right)^{\frac{1}{2}}\exp\left(\frac{\lambda}{\mu}% \sum_{i=1}^{n}w_{i}-\frac{\lambda}{2\mu^{2}}\sum_{i=1}^{n}w_{i}X_{i}-\frac{% \lambda}{2}\sum_{i=1}^{n}w_{i}\frac{1}{X_{i}}\right).
  31. μ ^ = i = 1 n w i X i i = 1 n w i , 1 λ ^ = 1 n i = 1 n w i ( 1 X i - 1 μ ^ ) . \hat{\mu}=\frac{\sum_{i=1}^{n}w_{i}X_{i}}{\sum_{i=1}^{n}w_{i}},\,\,\,\,\,\,\,% \,\frac{1}{\hat{\lambda}}=\frac{1}{n}\sum_{i=1}^{n}w_{i}\left(\frac{1}{X_{i}}-% \frac{1}{\hat{\mu}}\right).
  32. μ ^ \hat{\mu}
  33. λ ^ \hat{\lambda}
  34. μ ^ I G ( μ , λ i = 1 n w i ) n λ ^ 1 λ χ n - 1 2 . \hat{\mu}\sim IG\left(\mu,\lambda\sum_{i=1}^{n}w_{i}\right)\,\,\,\,\,\,\,\,% \frac{n}{\hat{\lambda}}\sim\frac{1}{\lambda}\chi^{2}_{n-1}.
  35. ν = N ( 0 , 1 ) . \displaystyle\nu=N(0,1).
  36. y = ν 2 \displaystyle y=\nu^{2}
  37. x = μ + μ 2 y 2 λ - μ 2 λ 4 μ λ y + μ 2 y 2 . x=\mu+\frac{\mu^{2}y}{2\lambda}-\frac{\mu}{2\lambda}\sqrt{4\mu\lambda y+\mu^{2% }y^{2}}.
  38. z = U ( 0 , 1 ) . \displaystyle z=U(0,1).
  39. z μ μ + x z\leq\frac{\mu}{\mu+x}
  40. x \displaystyle x
  41. μ 2 x . \frac{\mu^{2}}{x}.
  42. X IG ( μ , λ ) X\sim\textrm{IG}(\mu,\lambda)\,
  43. k X IG ( k μ , k λ ) kX\sim\textrm{IG}(k\mu,k\lambda)\,
  44. X i IG ( μ , λ ) X_{i}\sim\textrm{IG}(\mu,\lambda)\,
  45. i = 1 n X i IG ( n μ , n 2 λ ) \sum_{i=1}^{n}X_{i}\sim\textrm{IG}(n\mu,n^{2}\lambda)\,
  46. X i IG ( μ , λ ) X_{i}\sim\textrm{IG}(\mu,\lambda)\,
  47. i = 1 , , n i=1,\ldots,n\,
  48. X ¯ IG ( μ , n λ ) \bar{X}\sim\textrm{IG}(\mu,n\lambda)\,
  49. X i IG ( μ i , 2 μ i 2 ) X_{i}\sim\textrm{IG}(\mu_{i},2\mu^{2}_{i})\,
  50. i = 1 n X i IG ( i = 1 n μ i , 2 ( i = 1 n μ i ) 2 ) \sum_{i=1}^{n}X_{i}\sim\textrm{IG}\left(\sum_{i=1}^{n}\mu_{i},2{\left(\sum_{i=% 1}^{n}\mu_{i}\right)}^{2}\right)\,

Inverse_image_functor.html

  1. 𝒢 \mathcal{G}
  2. Y Y
  3. 𝒢 \mathcal{G}
  4. X X
  5. f : X Y f\colon X\to Y
  6. f - 1 𝒢 f^{-1}\mathcal{G}
  7. f - 1 𝒢 ( U ) = 𝒢 ( f ( U ) ) f^{-1}\mathcal{G}(U)=\mathcal{G}(f(U))
  8. U U
  9. X X
  10. f ( U ) f(U)
  11. f - 1 𝒢 f^{-1}\mathcal{G}
  12. U lim V f ( U ) 𝒢 ( V ) . U\mapsto\underrightarrow{\lim}_{V\supseteq f(U)}\mathcal{G}(V).
  13. U U
  14. X X
  15. V V
  16. Y Y
  17. f ( U ) f(U)
  18. f f
  19. y y
  20. Y Y
  21. f - 1 ( ) f^{-1}(\mathcal{F})
  22. \mathcal{F}
  23. f : X Y f\colon X\to Y
  24. 𝒪 Y \mathcal{O}_{Y}
  25. 𝒪 Y \mathcal{O}_{Y}
  26. Y Y
  27. f - 1 f^{-1}
  28. 𝒪 X \mathcal{O}_{X}
  29. 𝒪 Y \mathcal{O}_{Y}
  30. 𝒢 \mathcal{G}
  31. f * 𝒢 := f - 1 𝒢 f - 1 𝒪 Y 𝒪 X f^{*}\mathcal{G}:=f^{-1}\mathcal{G}\otimes_{f^{-1}\mathcal{O}_{Y}}\mathcal{O}_% {X}
  32. f - 1 f^{-1}
  33. f f_{\ast}
  34. x X x\in X
  35. ( f - 1 𝒢 ) x 𝒢 f ( x ) (f^{-1}\mathcal{G})_{x}\cong\mathcal{G}_{f(x)}
  36. f - 1 f^{-1}
  37. f * f^{*}
  38. f * f^{*}
  39. f - 1 f^{-1}
  40. f f_{\ast}
  41. 𝒢 f * f - 1 𝒢 \mathcal{G}\rightarrow f_{*}f^{-1}\mathcal{G}
  42. f - 1 f * f^{-1}f_{*}\mathcal{F}\rightarrow\mathcal{F}
  43. Hom 𝐒𝐡 ( X ) ( f - 1 𝒢 , ) = Hom 𝐒𝐡 ( Y ) ( 𝒢 , f * ) \mathrm{Hom}_{\mathbf{Sh}(X)}(f^{-1}\mathcal{G},\mathcal{F})=\mathrm{Hom}_{% \mathbf{Sh}(Y)}(\mathcal{G},f_{*}\mathcal{F})
  44. i : Z Y i\colon Z\to Y
  45. i * i - 1 𝒢 i_{*}i^{-1}\mathcal{G}
  46. y Y y\in Y
  47. 𝒢 y \mathcal{G}_{y}
  48. y y
  49. Z Z
  50. 0
  51. f - 1 f^{-1}
  52. f * f^{*}

Inversion_transformation.html

  1. t = i t t^{\prime}=it
  2. V μ = O μ ν V ν + P μ V_{\mu}^{\prime}=O_{\mu}^{\nu}V_{\nu}+P_{\mu}\,
  3. O O
  4. P P
  5. r = | x - y | . r=|x-y|.\,
  6. V μ = ( A τ ν V ν + B τ ) ( C τ μ ν V ν + D τ μ ) - 1 . V_{\mu}^{\prime}=\left(A_{\tau}^{\nu}V_{\nu}+B_{\tau}\right)\left(C_{\tau\mu}^% {\nu}V_{\nu}+D_{\tau\mu}\right)^{-1}.
  7. A A T + B C = D D T + C B AA^{T}+BC=DD^{T}+CB\,
  8. D , D,
  9. D D
  10. V μ = ( O μ ν V ν + P τ ) ( δ τ μ + Q τ μ ν V ν ) - 1 . V_{\mu}^{\prime}=\left(O_{\mu}^{\nu}V_{\nu}+P_{\tau}\right)\left(\delta_{\tau% \mu}+Q_{\tau\mu}^{\nu}V_{\nu}\right)^{-1}.\,
  11. Q . Q.
  12. Q = 0. Q=0.
  13. Q = 0 Q=0
  14. O O
  15. ( x - X ) ( y - Y ) ( x - Y ) ( y - X ) . \frac{(x-X)(y-Y)}{(x-Y)(y-X)}.
  16. x x
  17. y y
  18. ( x , X ) (x,X)
  19. ( y , Y ) (y,Y)
  20. ϕ ( x , X ) . \phi(x,X).\,

Invested_capital.html

  1. K = D + E - M K=D+E-M\,

Inviscid_flow.html

  1. ρ ( t + u ) u + p = 0 \rho\left(\frac{\partial}{\partial t}+{u}\cdot\nabla\right){u}+\nabla p=0

Involutory_matrix.html

  1. ( a b c - a ) \begin{pmatrix}a&b\\ c&-a\end{pmatrix}
  2. a 2 + b c = 1. a^{2}+bc=1.
  3. 𝐈 = ( 1 0 0 0 1 0 0 0 1 ) ; 𝐈 - 1 = ( 1 0 0 0 1 0 0 0 1 ) 𝐑 = ( 1 0 0 0 0 1 0 1 0 ) ; 𝐑 - 1 = ( 1 0 0 0 0 1 0 1 0 ) 𝐒 = ( + 1 0 0 0 - 1 0 0 0 - 1 ) ; 𝐒 - 1 = ( + 1 0 0 0 - 1 0 0 0 - 1 ) \begin{array}[]{cc}\mathbf{I}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix};&\mathbf{I}^{-1}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix}\\ \\ \mathbf{R}=\begin{pmatrix}1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix};&\mathbf{R}^{-1}=\begin{pmatrix}1&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}\\ \\ \mathbf{S}=\begin{pmatrix}+1&0&0\\ 0&-1&0\\ 0&0&-1\end{pmatrix};&\mathbf{S}^{-1}=\begin{pmatrix}+1&0&0\\ 0&-1&0\\ 0&0&-1\end{pmatrix}\\ \end{array}

Iron_butterfly_(options_strategy).html

  1. ironfly = Δ ( butterfly strike price ) × ( 1 + r t ) - butterfly \mbox{ironfly}~{}=\Delta(\mbox{butterfly strike price}~{})\times(1+rt)-\mbox{% butterfly}~{}

Irregular_moon.html

  1. r H r_{H}
  2. r H r_{H}
  3. r H r_{H}
  4. N N\,\!
  5. D D\,\!
  6. d N d D D - q \frac{dN}{dD}\sim D^{-q}

Islanding.html

  1. d f d t = R O C O F = Δ P f 2 G H \frac{\mathrm{d}f}{\mathrm{d}t}=ROCOF=\frac{\Delta Pf}{2GH}
  2. f f
  3. t t
  4. Δ P \Delta P
  5. Δ P = P m - P e \Delta P=P_{m}-P_{e}
  6. G G
  7. H H

ISO_metric_screw_thread.html

  1. H = 3 2 × P = cos ( 30 ) × P 0.866 × P H=\frac{\sqrt{3}}{2}\times P=\cos(30^{\circ})\times P\approx 0.866\times P
  2. P = 2 3 × H = H cos ( 30 ) 1.155 × H P=\frac{2}{\sqrt{3}}\times H=\frac{H}{\cos(30^{\circ})}\approx 1.155\times H
  3. D min = D maj - 2 × 5 8 × H = D maj - 5 3 8 × P D maj - 1.082532 × P D p = D maj - 2 × 3 8 × H = D maj - 3 3 8 × P D maj - 0.649519 × P \begin{aligned}\displaystyle D_{\mathrm{min}}&\displaystyle=D_{\mathrm{maj}}-2% \times\frac{5}{8}\times H=D_{\mathrm{maj}}-\frac{5{\sqrt{3}}}{8}\times P% \approx D_{\mathrm{maj}}-1.082532\times P\\ \displaystyle D_{\mathrm{p}}&\displaystyle=D_{\mathrm{maj}}-2\times\frac{3}{8}% \times H=D_{\mathrm{maj}}-\frac{3{\sqrt{3}}}{8}\times P\approx D_{\mathrm{maj}% }-0.649519\times P\end{aligned}

ISO_week_date.html

  1. w e e k ( d a t e ) = o r d i n a l ( d a t e ) - w e e k d a y ( d a t e ) + 10 7 week(date)=\left\lfloor\frac{ordinal(date)-weekday(date)+10}{7}\right\rfloor
  2. i f w e e k < 1 t h e n w e e k = l a s t W e e k ( y e a r - 1 ) if\,week<1\,then\,week=lastWeek(year-1)
  3. i f w e e k > l a s t W e e k ( y e a r ) t h e n w e e k = 1 if\,week>lastWeek(year)\,then\,week=1
  4. o r d i n a l ( d a t e ) = w e e k ( d a t e ) × 7 + w e e k d a y ( d a t e ) - ( w e e k d a y ( y e a r ( d a t e ) , 1 , 4 ) + 3 ) ordinal(date)=week(date)\times 7+weekday(date)-(weekday(year(date),1,4)+3)
  5. i f o r d i n a l < 1 t h e n o r d i n a l = o r d i n a l + d a y s I n Y e a r ( y e a r - 1 ) if\,ordinal<1\,then\,ordinal=ordinal+daysInYear(year-1)
  6. i f o r d i n a l > d a y s I n Y e a r ( y e a r ) t h e n o r d i n a l = o r d i n a l - d a y s I n Y e a r ( y e a r ) if\,ordinal>daysInYear(year)\,then\,ordinal=ordinal-daysInYear(year)

Isodynamic_point.html

  1. S S
  2. S S^{\prime}
  3. S S
  4. S S^{\prime}
  5. A B C ABC
  6. A S B C = B S A C = C S A B AS\cdot BC=BS\cdot AC=CS\cdot AB
  7. S S^{\prime}
  8. A S AS
  9. B S BS
  10. C S CS
  11. B C BC
  12. A C AC
  13. A B AB
  14. S S
  15. S S^{\prime}
  16. A B C ABC
  17. S S SS^{\prime}
  18. S S SS^{\prime}
  19. S S
  20. S S^{\prime}
  21. A B C ABC
  22. A B C ABC
  23. A B C ABC
  24. A B C ABC
  25. { S , S } \{S,S^{\prime}\}
  26. A B C ABC
  27. A B AB
  28. A C AC
  29. B C BC
  30. A B C ABC
  31. A S B = A C B + π / 3 ASB=ACB+\pi/3
  32. A S C = A B C + π / 3 ASC=ABC+\pi/3
  33. B S C = B A C + π / 3 BSC=BAC+\pi/3
  34. A S B = A C B - π / 3 AS^{\prime}B=ACB-\pi/3
  35. A S C = A B C - π / 3 AS^{\prime}C=ABC-\pi/3
  36. B S C = B A C - π / 3 BS^{\prime}C=BAC-\pi/3
  37. S S
  38. A B C ABC
  39. S S
  40. A B C ABC
  41. A B C ABC
  42. A A
  43. A B C ABC
  44. A B AB
  45. A C AC
  46. A A
  47. B C BC
  48. A A^{\prime}
  49. A A
  50. B C BC
  51. B B
  52. C C
  53. A A
  54. B C BC
  55. A ′′ A^{\prime\prime}
  56. B C BC
  57. A A ′′ A^{\prime}A^{\prime\prime}
  58. B B ′′ B^{\prime}B^{\prime\prime}
  59. C C ′′ C^{\prime}C^{\prime\prime}
  60. sin ( A + π / 3 ) : sin ( B + π / 3 ) : sin ( C + π / 3 ) . \sin(A+\pi/3):\sin(B+\pi/3):\sin(C+\pi/3).
  61. - π / 3 -\pi/3
  62. π / 3 \pi/3

Isogonal_trajectory.html

  1. d y d x = f ( x , y ) , \frac{dy}{dx}=f(x,y),
  2. α \alpha
  3. d y d x = f ( x , y ) + tan α 1 - f ( x , y ) tan α . \frac{dy}{dx}=\frac{f(x,y)+\tan\alpha}{1-f(x,y)\tan\alpha}.

Isometry_(Riemannian_geometry).html

  1. ( M , g ) (M,g)
  2. ( M , g ) (M^{\prime},g^{\prime})
  3. f : M M f:M\to M^{\prime}
  4. f f
  5. g = f * g , g=f^{*}g^{\prime},\,
  6. f * g f^{*}g^{\prime}
  7. g g^{\prime}
  8. f f
  9. f * f_{*}
  10. v , w v,w
  11. M M
  12. T M \mathrm{T}M
  13. g ( v , w ) = g ( f * v , f * w ) . g(v,w)=g^{\prime}\left(f_{*}v,f_{*}w\right).\,
  14. f f
  15. g = f * g g=f^{*}g^{\prime}
  16. f f

Isotropic_line.html

  1. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  2. ( a 1 , a 2 , a 3 ) (a_{1},a_{2},a_{3})
  3. a 3 ( x 2 ± i x 1 ) = ( a 2 ± i a 1 ) x 2 . a_{3}(x_{2}\pm ix_{1})=(a_{2}\pm ia_{1})x_{2}.
  4. x 2 = ± i x 1 . x_{2}=\pm ix_{1}.

Iterated_binary_operation.html

  1. , , , \sum,\ \prod,\ \bigcup,
  2. \bigcap
  3. f f
  4. ( a 1 , a 2 , a n ) (a_{1},a_{2}\ldots,a_{n})
  5. f / ( a 1 , a 2 , a n ) f/(a_{1},a_{2}\ldots,a_{n})
  6. F l ( 𝐚 0 , k ) = { a 0 , k = 1 f ( F l ( 𝐚 0 , k - 1 ) , a k - 1 ) , k > 1 . F_{l}(\mathbf{a}_{0,k})=\begin{cases}a_{0},&k=1\\ f(F_{l}(\mathbf{a}_{0,k-1}),a_{k-1}),&k>1\end{cases}.
  7. F r ( 𝐚 0 , k ) = { a 0 , k = 1 f ( a 0 , F r ( 𝐚 1 , k ) ) , k > 1 . F_{r}(\mathbf{a}_{0,k})=\begin{cases}a_{0},&k=1\\ f(a_{0},F_{r}(\mathbf{a}_{1,k})),&k>1\end{cases}.
  8. i = 0 a i \prod_{i=0}^{\infty}a_{i}\,
  9. lim n i = 0 n a i , \lim\limits_{n\rightarrow\infty}\prod_{i=0}^{n}a_{i},

Itō_isometry.html

  1. W : [ 0 , T ] × Ω W:[0,T]\times\Omega\to\mathbb{R}
  2. T > 0 T>0
  3. X : [ 0 , T ] × Ω X:[0,T]\times\Omega\to\mathbb{R}
  4. * W \mathcal{F}_{*}^{W}
  5. 𝔼 [ ( 0 T X t d W t ) 2 ] = 𝔼 [ 0 T X t 2 d t ] , \mathbb{E}\left[\left(\int_{0}^{T}X_{t}\,\mathrm{d}W_{t}\right)^{2}\right]=% \mathbb{E}\left[\int_{0}^{T}X_{t}^{2}\,\mathrm{d}t\right],
  6. 𝔼 \mathbb{E}
  7. γ \gamma
  8. ( X , Y ) L 2 ( W ) := 𝔼 ( 0 T X t d W t 0 T Y t d W t ) = Ω ( 0 T X t d W t 0 T Y t d W t ) d γ ( ω ) (X,Y)_{L^{2}(W)}:=\mathbb{E}\left(\int_{0}^{T}X_{t}\,\mathrm{d}W_{t}\int_{0}^{% T}Y_{t}\,\mathrm{d}W_{t}\right)=\int_{\Omega}\left(\int_{0}^{T}X_{t}\,\mathrm{% d}W_{t}\int_{0}^{T}Y_{t}\,\mathrm{d}W_{t}\right)\,\mathrm{d}\gamma(\omega)
  9. ( A , B ) L 2 ( Ω ) := 𝔼 ( A B ) = Ω A ( ω ) B ( ω ) d γ ( ω ) . (A,B)_{L^{2}(\Omega)}:=\mathbb{E}(AB)=\int_{\Omega}A(\omega)B(\omega)\,\mathrm% {d}\gamma(\omega).

ITU_model_for_indoor_attenuation.html

  1. L = 20 log f + N log d + P f ( n ) - 28 L\;=\;20\;\log f\;+\;N\;\log d\;+\;P_{f}(n)\;-\;28

ITU_terrain_model.html

  1. A = 10 - 20 C N A\;=\;10\;-\;20\;C_{N}
  2. C N = h F 1 C_{N}\;=\;\frac{h}{F_{1}}
  3. h = h L - h O h\;=\;h_{L}\;-\;h_{O}
  4. F 1 = 17.3 d 1 d 2 f d F_{1}\;=\;17.3\;\sqrt{\frac{d_{1}d_{2}}{fd}}
  5. A A\;
  6. C N C_{N}\;
  7. h h\;
  8. h L h_{L}\;
  9. h O h_{O}\;
  10. F 1 F_{1}\;
  11. d 1 d_{1}\;
  12. d 2 d_{2}\;
  13. f f\;
  14. d d\;

J1.html

  1. j 1 j_{1}

J_(disambiguation).html

  1. j j
  2. - 1 \sqrt{-1}

Jacket_matrix.html

  1. A = ( a i j ) A=(a_{ij})
  2. A B = B A = I n \ AB=BA=I_{n}
  3. B = 1 n ( a i j - 1 ) T . \ B={1\over n}(a_{ij}^{-1})^{T}.
  4. u , v { 1 , 2 , , n } : a i u , a i v 0 , i = 1 n a i u - 1 a i v = { n , u = v 0 , u v \forall u,v\in\{1,2,\dots,n\}:~{}a_{iu},a_{iv}\neq 0,~{}~{}~{}~{}\sum_{i=1}^{n% }a_{iu}^{-1}\,a_{iv}=\begin{cases}n,&u=v\\ 0,&u\neq v\end{cases}
  5. 1 4 , 1 2 , \ {1\over 4},{1\over 2},
  6. 2 2 = 4 2^{2}=4
  7. ( 2 2 ) - 1 = 1 4 (2^{2})^{-1}={1\over 4}
  8. 4 * 1 4 = 1 4*{1\over 4}=1
  9. A = [ 1 1 1 1 1 - 2 2 - 1 1 2 - 2 - 1 1 - 1 - 1 1 ] , A=\left[\begin{array}[]{rrrr}1&1&1&1\\ 1&-2&2&-1\\ 1&2&-2&-1\\ 1&-1&-1&1\\ \end{array}\right],
  10. B = 1 4 [ 1 1 1 1 1 - 1 2 1 2 - 1 1 1 2 - 1 2 - 1 1 - 1 - 1 1 ] . B={1\over 4}\left[\begin{array}[]{rrrr}1&1&1&1\\ 1&-{1\over 2}&{1\over 2}&-1\\ 1&{1\over 2}&-{1\over 2}&-1\\ 1&-1&-1&1\\ \end{array}\right].
  11. A = [ a b b a b - c c - b b c - c - b a - b - b a ] , A=\left[\begin{array}[]{rrrr}a&b&b&a\\ b&-c&c&-b\\ b&c&-c&-b\\ a&-b&-b&a\end{array}\right],
  12. B = 1 4 [ 1 a 1 b 1 b 1 a 1 b - 1 c 1 c - 1 b 1 b 1 c - 1 c - 1 b 1 a - 1 b - 1 b 1 a ] , B={1\over 4}\left[\begin{array}[]{rrrr}{1\over a}&{1\over b}&{1\over b}&{1% \over a}\\ {1\over b}&-{1\over c}&{1\over c}&-{1\over b}\\ {1\over b}&{1\over c}&-{1\over c}&-{1\over b}\\ {1\over a}&-{1\over b}&-{1\over b}&{1\over a}\end{array}\right],
  13. 𝐀 𝐣 , \mathbf{A_{j}},
  14. 𝐀 𝐣 = d i a g ( A 1 , A 2 , . . A n ) \mathbf{A_{j}}=diag(A_{1},A_{2},..A_{n})
  15. J 4 = [ I 2 0 0 0 0 c o s θ - s i n θ 0 0 s i n θ c o s θ 0 0 0 0 I 2 ] , J_{4}=\left[\begin{array}[]{rrrr}I_{2}&0&0&0\\ 0&cos\theta&-sin\theta&0\\ 0&sin\theta&cos\theta&0\\ 0&0&0&I_{2}\end{array}\right],
  16. J 4 T J 4 = J 4 J 4 T = I 4 . \ J^{T}_{4}J_{4}=J_{4}J^{T}_{4}=I_{4}.

Jacobi_rotation.html

  1. A Q k T A Q k = A . A\mapsto Q_{k\ell}^{T}AQ_{k\ell}=A^{\prime}.\,\!
  2. [ * * a k k a k a k a * * ] [ * * a k k 0 0 a * * ] . \begin{bmatrix}{*}&&&\cdots&&&*\\ &\ddots&&&&&\\ &&a_{kk}&\cdots&a_{k\ell}&&\\ \vdots&&\vdots&\ddots&\vdots&&\vdots\\ &&a_{\ell k}&\cdots&a_{\ell\ell}&&\\ &&&&&\ddots&\\ {*}&&&\cdots&&&*\end{bmatrix}\to\begin{bmatrix}{*}&&&\cdots&&&*\\ &\ddots&&&&&\\ &&a^{\prime}_{kk}&\cdots&0&&\\ \vdots&&\vdots&\ddots&\vdots&&\vdots\\ &&0&\cdots&a^{\prime}_{\ell\ell}&&\\ &&&&&\ddots&\\ {*}&&&\cdots&&&*\end{bmatrix}.
  3. Q k = [ 1 0 c s - s c 0 1 ] . Q_{k\ell}=\begin{bmatrix}1&&&&&&\\ &\ddots&&&&0&\\ &&c&\cdots&s&&\\ &&\vdots&\ddots&\vdots&&\\ &&-s&\cdots&c&&\\ &0&&&&\ddots&\\ &&&&&&1\end{bmatrix}.
  4. q i j = δ i j + ( δ i k δ j k + δ i δ j ) ( c - 1 ) + ( δ i k δ j - δ i δ j k ) s . q_{ij}=\delta_{ij}+(\delta_{ik}\delta_{jk}+\delta_{i\ell}\delta_{j\ell})(c-1)+% (\delta_{ik}\delta_{j\ell}-\delta_{i\ell}\delta_{jk})s.\,\!
  5. a h k = a k h = c a h k - s a h a^{\prime}_{hk}=a^{\prime}_{kh}=ca_{hk}-sa_{h\ell}\,\!
  6. a h = a h = c a h + s a h k a^{\prime}_{h\ell}=a^{\prime}_{\ell h}=ca_{h\ell}+sa_{hk}\,\!
  7. a k = a k = ( c 2 - s 2 ) a k + s c ( a k k - a ) = 0 a^{\prime}_{k\ell}=a^{\prime}_{\ell k}=(c^{2}-s^{2})a_{k\ell}+sc(a_{kk}-a_{% \ell\ell})=0\,\!
  8. a k k = c 2 a k k + s 2 a - 2 s c a k a^{\prime}_{kk}=c^{2}a_{kk}+s^{2}a_{\ell\ell}-2sca_{k\ell}\,\!
  9. a = s 2 a k k + c 2 a + 2 s c a k . a^{\prime}_{\ell\ell}=s^{2}a_{kk}+c^{2}a_{\ell\ell}+2sca_{k\ell}.\,\!
  10. c 2 - s 2 s c = a - a k k a k . \frac{c^{2}-s^{2}}{sc}=\frac{a_{\ell\ell}-a_{kk}}{a_{k\ell}}.
  11. β = a - a k k 2 a k . \beta=\frac{a_{\ell\ell}-a_{kk}}{2a_{k\ell}}.
  12. t 2 + 2 β t - 1 = 0. t^{2}+2\beta t-1=0.\,\!
  13. t = sgn ( β ) | β | + β 2 + 1 . t=\frac{\operatorname{sgn}(\beta)}{|\beta|+\sqrt{\beta^{2}+1}}.
  14. c = 1 t 2 + 1 c=\frac{1}{\sqrt{t^{2}+1}}\,\!
  15. s = c t s=ct\,\!
  16. ρ = s 1 + c , \rho=\frac{s}{1+c},
  17. a h k = a k h = a h k - s ( a h + ρ a h k ) a^{\prime}_{hk}=a^{\prime}_{kh}=a_{hk}-s(a_{h\ell}+\rho a_{hk})\,\!
  18. a h = a h = a h + s ( a h k - ρ a h ) a^{\prime}_{h\ell}=a^{\prime}_{\ell h}=a_{h\ell}+s(a_{hk}-\rho a_{h\ell})\,\!
  19. a k = a k = 0 a^{\prime}_{k\ell}=a^{\prime}_{\ell k}=0\,\!
  20. a k k = a k k - t a k a^{\prime}_{kk}=a_{kk}-ta_{k\ell}\,\!
  21. a = a + t a k a^{\prime}_{\ell\ell}=a_{\ell\ell}+ta_{k\ell}\,\!

James_Earl_Baumgartner.html

  1. 1 \aleph_{1}
  2. 1 \aleph_{1}
  3. 1 \aleph_{1}
  4. ω 1 ( α ) n 2 \omega_{1}\to(\alpha)^{2}_{n}
  5. α < ω 1 , n < ω \alpha<\omega_{1},n<\omega

Jaro–Winkler_distance.html

  1. d j d_{j}
  2. s 1 s_{1}
  3. s 2 s_{2}
  4. d j = { 0 if m = 0 1 3 ( m | s 1 | + m | s 2 | + m - t m ) otherwise d_{j}=\left\{\begin{array}[]{l l}0&\,\text{if }m=0\\ \frac{1}{3}\left(\frac{m}{|s_{1}|}+\frac{m}{|s_{2}|}+\frac{m-t}{m}\right)&\,% \text{otherwise}\end{array}\right.
  5. m m
  6. t t
  7. s 1 s_{1}
  8. s 2 s_{2}
  9. max ( | s 1 | , | s 2 | ) 2 - 1 \left\lfloor\frac{\max(|s_{1}|,|s_{2}|)}{2}\right\rfloor-1
  10. s 1 s_{1}
  11. s 2 s_{2}
  12. p p
  13. \ell
  14. s 1 s_{1}
  15. s 2 s_{2}
  16. d w d_{w}
  17. d w = d j + ( p ( 1 - d j ) ) d_{w}=d_{j}+(\ell p(1-d_{j}))
  18. d j d_{j}
  19. s 1 s_{1}
  20. s 2 s_{2}
  21. \ell
  22. p p
  23. p p
  24. p = 0.1 p=0.1
  25. d ( x , y ) = 0 x = y d(x,y)=0\rightarrow x=y
  26. p ( 1 - d j ) \ell p(1-d_{j})
  27. b t b_{t}
  28. d w = { d j if d j < b t d j + ( p ( 1 - d j ) ) otherwise d_{w}=\left\{\begin{array}[]{l l}d_{j}&\,\text{if }d_{j}<b_{t}\\ d_{j}+(\ell p(1-d_{j}))&\,\text{otherwise}\end{array}\right.
  29. s 1 s_{1}
  30. s 2 s_{2}
  31. m = 6 m=6
  32. | s 1 | = 6 |s_{1}|=6
  33. | s 2 | = 6 |s_{2}|=6
  34. t = 2 2 = 1 t=\frac{2}{2}=1
  35. d j = 1 3 ( 6 6 + 6 6 + 6 - 1 6 ) = 0.944 d_{j}=\frac{1}{3}\left(\frac{6}{6}+\frac{6}{6}+\frac{6-1}{6}\right)=0.944
  36. p = 0.1 p=0.1
  37. = 3 \ell=3
  38. d w = 0.944 + ( 3 * 0.1 ( 1 - 0.944 ) ) = 0.961 d_{w}=0.944+(3*0.1(1-0.944))=0.961
  39. s 1 s_{1}
  40. s 2 s_{2}
  41. m = 4 m=4
  42. | s 1 | = 6 |s_{1}|=6
  43. | s 2 | = 5 |s_{2}|=5
  44. t = 0 t=0
  45. d j = 1 3 ( 4 6 + 4 5 + 4 - 0 4 ) = 0.822 d_{j}=\frac{1}{3}\left(\frac{4}{6}+\frac{4}{5}+\frac{4-0}{4}\right)=0.822
  46. p = 0.1 p=0.1
  47. = 1 \ell=1
  48. d w = 0.822 + ( 1 * 0.1 ( 1 - 0.822 ) ) = 0.84 d_{w}=0.822+(1*0.1(1-0.822))=0.84
  49. s 1 s_{1}
  50. s 2 s_{2}
  51. m = 4 m=4
  52. | s 1 | = 5 |s_{1}|=5
  53. | s 2 | = 8 |s_{2}|=8
  54. t = 0 t=0
  55. d j = 1 3 ( 4 5 + 4 8 + 4 - 0 4 ) = 0.767 d_{j}=\frac{1}{3}\left(\frac{4}{5}+\frac{4}{8}+\frac{4-0}{4}\right)=0.767
  56. p = 0.1 p=0.1
  57. = 2 \ell=2
  58. d w = 0.767 + ( 2 * 0.1 ( 1 - 0.767 ) ) = 0.814 d_{w}=0.767+(2*0.1(1-0.767))=0.814

Jay_Hambidge.html

  1. δ s = 2.414... \delta_{s}=2.414...

Jaynes–Cummings_model.html

  1. H ^ = H ^ field + H ^ atom + H ^ int \hat{H}=\hat{H}_{\,\text{field}}+\hat{H}_{\,\text{atom}}+\hat{H}_{\,\text{int}}
  2. H ^ field \displaystyle\hat{H}\text{field}
  3. E ^ = a ^ + a ^ \hat{E}=\hat{a}+\hat{a}^{\dagger}
  4. a ^ \hat{a}^{\dagger}
  5. a ^ \hat{a}
  6. ω c \omega_{c}
  7. S ^ = σ ^ + + σ ^ - \hat{S}=\hat{\sigma}_{+}+\hat{\sigma}_{-}
  8. σ ^ + = | e g | \hat{\sigma}_{+}=|e\rangle\langle g|
  9. σ ^ - = | g e | \hat{\sigma}_{-}=|g\rangle\langle e|
  10. σ ^ z = | e e | - | g g | \hat{\sigma}_{z}=|e\rangle\langle e|-|g\rangle\langle g|
  11. ω a \omega_{a}
  12. H 0 = H ^ field + H ^ atom H_{0}=\hat{H}_{\,\text{field}}+\hat{H}_{\,\text{atom}}
  13. H ^ int ( t ) = Ω 2 ( a ^ σ ^ - e - i ( ω c + ω a ) t + a ^ σ ^ + e i ( ω c + ω a ) t + a ^ σ ^ + e i ( - ω c + ω a ) t + a ^ σ ^ - e - i ( - ω c + ω a ) t ) . \hat{H}\text{int}(t)=\frac{\hbar\Omega}{2}\left(\hat{a}\hat{\sigma}_{-}e^{-i(% \omega_{c}+\omega_{a})t}+\hat{a}^{\dagger}\hat{\sigma}_{+}e^{i(\omega_{c}+% \omega_{a})t}+\hat{a}\hat{\sigma}_{+}e^{i(-\omega_{c}+\omega_{a})t}+\hat{a}^{% \dagger}\hat{\sigma}_{-}e^{-i(-\omega_{c}+\omega_{a})t}\right).
  14. ( ω c + ω a ) (\omega_{c}+\omega_{a})
  15. ( ω c - ω a ) (\omega_{c}-\omega_{a})
  16. | ω c - ω a | ω c + ω a |\omega_{c}-\omega_{a}|\ll\omega_{c}+\omega_{a}
  17. H ^ JC = ω c a ^ a ^ + ω a σ ^ z 2 + Ω 2 ( a ^ σ ^ + + a ^ σ ^ - ) . \hat{H}_{\,\text{JC}}=\hbar\omega_{c}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{a}% \frac{\hat{\sigma}_{z}}{2}+\frac{\hbar\Omega}{2}\left(\hat{a}\hat{\sigma}_{+}+% \hat{a}^{\dagger}\hat{\sigma}_{-}\right).
  18. H ^ JC = H ^ I + H ^ I I \hat{H}\text{JC}=\hat{H}_{I}+\hat{H}_{II}
  19. H ^ I = ω c ( a ^ a ^ + σ ^ z 2 ) H ^ I I = δ σ ^ z 2 + Ω 2 ( a ^ σ ^ + + a ^ σ ^ - ) \begin{aligned}\displaystyle\hat{H}_{I}&\displaystyle=\hbar\omega_{c}\left(% \hat{a}^{\dagger}\hat{a}+\frac{\hat{\sigma}_{z}}{2}\right)\\ \displaystyle\hat{H}_{II}&\displaystyle=\hbar\delta\frac{\hat{\sigma}_{z}}{2}+% \frac{\hbar\Omega}{2}\left(\hat{a}\hat{\sigma}_{+}+\hat{a}^{\dagger}\hat{% \sigma}_{-}\right)\end{aligned}
  20. δ = ω a - ω c \delta=\omega_{a}-\omega_{c}
  21. H ^ I \hat{H}_{I}
  22. | n , g , | n , e |n,g\rangle,|n,e\rangle
  23. n n\in\mathbb{N}
  24. | ψ 1 n := | n , e |\psi_{1n}\rangle:=|n,e\rangle
  25. | ψ 2 n := | n + 1 , g |\psi_{2n}\rangle:=|n+1,g\rangle
  26. H ^ I \hat{H}_{I}
  27. n n
  28. H ^ JC \hat{H}_{\,\text{JC}}
  29. span { | ψ 1 n , | ψ 2 n } \,\text{span}\{|\psi_{1n}\rangle,|\psi_{2n}\rangle\}
  30. H ^ JC \hat{H}_{\,\text{JC}}
  31. H i j ( n ) := ψ i n | H ^ JC | ψ j n , {H}^{(n)}_{ij}:=\langle\psi_{in}|\hat{H}_{\,\text{JC}}|\psi_{jn}\rangle,
  32. H ( n ) = ( n ω c + ω a 2 Ω 2 n + 1 Ω 2 n + 1 ( n + 1 ) ω c - ω a 2 ) H^{(n)}=\hbar\begin{pmatrix}n\omega_{c}+\frac{\omega_{a}}{2}&\frac{\Omega}{2}% \sqrt{n+1}\\ \frac{\Omega}{2}\sqrt{n+1}&(n+1)\omega_{c}-\frac{\omega_{a}}{2}\end{pmatrix}
  33. n n
  34. H ( n ) H^{(n)}
  35. E ± ( n ) = ω c ( n + 1 2 ) ± 1 2 Ω n ( δ ) , E_{\pm}(n)=\hbar\omega_{c}\left(n+\frac{1}{2}\right)\pm\frac{1}{2}\hbar\Omega_% {n}(\delta),
  36. Ω n ( δ ) = δ 2 + Ω 2 ( n + 1 ) \Omega_{n}(\delta)=\sqrt{\delta^{2}+\Omega^{2}(n+1)}
  37. | n , ± |n,\pm\rangle~{}
  38. | n , + = cos ( α n 2 ) | ψ 1 n + sin ( α n 2 ) | ψ 2 n |n,+\rangle=\cos\left(\frac{\alpha_{n}}{2}\right)|\psi_{1n}\rangle+\sin\left(% \frac{\alpha_{n}}{2}\right)|\psi_{2n}\rangle
  39. | n , - = - sin ( α n 2 ) | ψ 1 n + cos ( α n 2 ) | ψ 2 n |n,-\rangle=-\sin\left(\frac{\alpha_{n}}{2}\right)|\psi_{1n}\rangle+\cos\left(% \frac{\alpha_{n}}{2}\right)|\psi_{2n}\rangle
  40. α n \alpha_{n}
  41. α n := tan - 1 ( Ω n + 1 δ ) \alpha_{n}:=\tan^{-1}\left(\frac{\Omega\sqrt{n+1}}{\delta}\right)
  42. | ψ field ( 0 ) = n C n | n ~{}|\psi\text{field}(0)\rangle=\sum_{n}{C_{n}|n\rangle}~{}
  43. | ψ tot ( 0 ) = n C n [ cos ( α n 2 ) | n , + - sin ( α n 2 ) | n , - ] . |\psi\text{tot}(0)\rangle=\sum_{n}C_{n}\left[\cos\left(\frac{\alpha_{n}}{2}% \right)|n,+\rangle-\sin\left(\frac{\alpha_{n}}{2}\right)|n,-\rangle\right].
  44. | n , ± ~{}|n,\pm\rangle~{}
  45. t > 0 ~{}t>0~{}
  46. | ψ tot ( t ) = e - i H ^ JC t / | ψ tot ( 0 ) = n C n [ cos ( α n 2 ) | n , + e - i E + ( n ) t / - sin ( α n 2 ) | n , - e - i E - ( n ) t / ] . |\psi\text{tot}(t)\rangle=e^{-i\hat{H}_{\,\text{JC}}t/\hbar}|\psi\text{tot}(0)% \rangle=\sum_{n}C_{n}\left[\cos\left(\frac{\alpha_{n}}{2}\right)|n,+\rangle e^% {-iE_{+}(n)t/\hbar}-\sin\left(\frac{\alpha_{n}}{2}\right)|n,-\rangle e^{-iE_{-% }(n)t/\hbar}\right].
  47. U ^ ( t ) = e - i H ^ JC t / = ( e - i ω c t ( a ^ a ^ + 1 2 ) ( cos t φ ^ + g 2 - i δ / 2 sin t φ ^ + g 2 φ ^ + g 2 ) - i g e - i ω c t ( a ^ a ^ + 1 2 ) sin t φ ^ + g 2 φ ^ + g 2 a ^ - i g e - i ω c t ( a ^ a ^ - 1 2 ) sin t φ ^ φ ^ a ^ e - i ω c t ( a ^ a ^ - 1 2 ) ( cos t φ ^ + i δ / 2 sin t φ ^ φ ^ ) ) \begin{matrix}\begin{aligned}\displaystyle\hat{U}(t)&\displaystyle=e^{-i\hat{H% }_{\,\text{JC}}t/\hbar}\\ &\displaystyle=\begin{pmatrix}e^{-i\omega_{c}t(\hat{a}^{\dagger}\hat{a}+\frac{% 1}{2})}\left(\cos t\sqrt{\hat{\varphi}+g^{2}}-i\delta/2\frac{\sin t\sqrt{\hat{% \varphi}+g^{2}}}{\sqrt{\hat{\varphi}+g^{2}}}\right)&-ige^{-i\omega_{c}t(\hat{a% }^{\dagger}\hat{a}+\frac{1}{2})}\frac{\sin t\sqrt{\hat{\varphi}+g^{2}}}{\sqrt{% \hat{\varphi}+g^{2}}}\,\hat{a}\\ -ige^{-i\omega_{c}t(\hat{a}^{\dagger}\hat{a}-\frac{1}{2})}\frac{\sin t\sqrt{% \hat{\varphi}}}{\sqrt{\hat{\varphi}}}\hat{a}^{\dagger}&e^{-i\omega_{c}t(\hat{a% }^{\dagger}\hat{a}-\frac{1}{2})}\left(\cos t\sqrt{\hat{\varphi}}+i\delta/2% \frac{\sin t\sqrt{\hat{\varphi}}}{\sqrt{\hat{\varphi}}}\right)\end{pmatrix}% \end{aligned}\end{matrix}
  48. φ ^ ~{}\hat{\varphi}~{}
  49. φ ^ = g 2 a ^ a ^ + δ 2 / 4 \hat{\varphi}=g^{2}\hat{a}^{\dagger}\hat{a}+\delta^{2}/4
  50. U ^ ~{}\hat{U}~{}
  51. sin t φ ^ + g 2 φ ^ + g 2 a ^ = a ^ sin t φ ^ φ ^ , \frac{\sin t\,\sqrt{\hat{\varphi}+g^{2}}}{\sqrt{\hat{\varphi}+g^{2}}}\;\hat{a}% =\hat{a}\;\frac{\sin t\,\sqrt{\hat{\varphi}}}{\sqrt{\hat{\varphi}}},
  52. cos t φ ^ + g 2 a ^ = a ^ cos t φ ^ , \cos t\,\sqrt{\hat{\varphi}+g^{2}}\;\hat{a}=\hat{a}\;\cos t\sqrt{\hat{\varphi}},
  53. ρ ^ ( t ) ~{}\hat{\rho}(t)~{}
  54. ρ ^ ( t ) = U ^ ( t ) ρ ^ ( 0 ) U ^ ( t ) \hat{\rho}(t)=\hat{U}^{\dagger}(t)\hat{\rho}(0)\hat{U}(t)
  55. Θ ^ t = Tr [ ρ ^ ( t ) Θ ^ ] \langle\hat{\Theta}\rangle_{t}=\,\text{Tr}[\hat{\rho}(t)\hat{\Theta}]
  56. ρ ^ ( 0 ) ~{}\hat{\rho}(0)~{}
  57. Θ ^ ~{}\hat{\Theta}~{}
  58. a = ( δ / ( 2 g ) ) 2 = 40 a=(\delta/(2g))^{2}=40
  59. δ \delta

Jensen's_formula.html

  1. log | f ( 0 ) | = k = 1 n log ( | a k | r ) + 1 2 π 0 2 π log | f ( r e i θ ) | d θ . \log|f(0)|=\sum_{k=1}^{n}\log\left(\frac{|a_{k}|}{r}\right)+\frac{1}{2\pi}\int% _{0}^{2\pi}\log|f(re^{i\theta})|\,d\theta.
  2. log | f ( 0 ) | = 1 2 π 0 2 π log | f ( r e i θ ) | d θ , \log|f(0)|=\frac{1}{2\pi}\int_{0}^{2\pi}\log|f(re^{i\theta})|\,d\theta,
  3. log | f ( z ) | \log|f(z)|
  4. 1 2 π 0 2 π log | f ( r e i θ ) | d θ - log | f ( 0 ) | = 0 r n ( t ) t d t \frac{1}{2\pi}\int_{0}^{2\pi}\log|f(re^{i\theta})|\;d\theta-\log|f(0)|=\int_{0% }^{r}\frac{n(t)}{t}\;dt
  5. n ( t ) n(t)
  6. f f
  7. t t
  8. f ( z ) = z l g ( z ) h ( z ) , f(z)=z^{l}\frac{g(z)}{h(z)},
  9. a 1 , , a n 𝔻 \ { 0 } a_{1},\ldots,a_{n}\in\mathbb{D}\backslash\{0\}
  10. b 1 , , b m 𝔻 \ { 0 } b_{1},\ldots,b_{m}\in\mathbb{D}\backslash\{0\}
  11. log | g ( 0 ) h ( 0 ) | = log | r m - n a 1 a n b 1 b m | + 1 2 π 0 2 π log | f ( r e i θ ) | d θ . \log\left|\frac{g(0)}{h(0)}\right|=\log\left|r^{m-n}\frac{a_{1}\ldots a_{n}}{b% _{1}\ldots b_{m}}\right|+\frac{1}{2\pi}\int_{0}^{2\pi}\log|f(re^{i\theta})|\,d\theta.
  12. 1 log ( R / r ) log M | f ( z 0 ) | . \frac{1}{\log(R/r)}\log\frac{M}{|f(z_{0})|}.
  13. z 0 = r 0 e i φ 0 z_{0}=r_{0}e^{i\varphi_{0}}
  14. log | f ( z 0 ) | = k = 1 n log | z 0 - a k 1 - a ¯ k z 0 | + 1 2 π 0 2 π P r 0 ( φ 0 - θ ) log | f ( e i θ ) | d θ . \log|f(z_{0})|=\sum_{k=1}^{n}\log\left|\frac{z_{0}-a_{k}}{1-\bar{a}_{k}z_{0}}% \right|+\frac{1}{2\pi}\int_{0}^{2\pi}P_{r_{0}}(\varphi_{0}-\theta)\log|f(e^{i% \theta})|\,d\theta.
  15. P r ( ω ) = n r | n | e i n ω P_{r}(\omega)=\sum_{n\in\mathbb{Z}}r^{|n|}e^{in\omega}
  16. log | f ( z 0 ) | = 1 2 π 0 2 π P r 0 ( φ 0 - θ ) log | f ( e i θ ) | d θ , \log|f(z_{0})|=\frac{1}{2\pi}\int_{0}^{2\pi}P_{r_{0}}(\varphi_{0}-\theta)\log|% f(e^{i\theta})|\,d\theta,
  17. log | f ( z ) | \log|f(z)|

Joseph_F._Traub.html

  1. N N
  2. N N
  3. 2 3 P 2^{3}P

Josephson_energy.html

  1. I s = I c sin ( ϕ ) , I_{s}=I_{c}\sin(\phi),
  2. I s I_{s}\,
  3. I c I_{c}\,
  4. ϕ \phi\,
  5. t = 0 t=0
  6. ϕ = 0 \phi=0
  7. t t
  8. ϕ \phi
  9. U = 0 t I s V d t = Φ 0 2 π 0 t I s d ϕ d t d t = Φ 0 2 π 0 ϕ I c sin ( ϕ ) d ϕ = Φ 0 I c 2 π ( 1 - cos ϕ ) . U=\int_{0}^{t}I_{s}V\,dt=\frac{\Phi_{0}}{2\pi}\int_{0}^{t}I_{s}\frac{d\phi}{dt% }\,dt=\frac{\Phi_{0}}{2\pi}\int_{0}^{\phi}I_{c}\sin(\phi)\,d\phi=\frac{\Phi_{0% }I_{c}}{2\pi}(1-\cos\phi).
  10. E J = Φ 0 I c / 2 π E_{J}={\Phi_{0}I_{c}}/{2\pi}
  11. ( 1 - cos ϕ ) (1-\cos\phi)
  12. ϕ \phi
  13. U ( ϕ ) U(\phi)
  14. U ( ϕ ) U(\phi)
  15. ϕ = 2 π n \phi=2\pi n
  16. n n
  17. ϕ 0 \phi_{0}\,
  18. I 0 = I c sin ϕ 0 I_{0}=I_{c}\sin\phi_{0}\,
  19. I s I_{s}\,
  20. ϕ \phi\,
  21. I 0 I_{0}\,
  22. ϕ 0 \phi_{0}\,
  23. δ I ( t ) I c \delta I(t)\ll I_{c}
  24. ϕ = ϕ 0 + δ ϕ \phi=\phi_{0}+\delta\phi\,
  25. I 0 + δ I = I c sin ( ϕ 0 + δ ϕ ) I_{0}+\delta I=I_{c}\sin(\phi_{0}+\delta\phi)\,
  26. δ ϕ \delta\phi\,
  27. δ I = I c cos ( ϕ 0 ) δ ϕ \delta I=I_{c}\cos(\phi_{0})\delta\phi\,
  28. V = Φ 0 2 π ϕ ˙ = Φ 0 2 π ( ϕ 0 ˙ = 0 + δ ϕ ˙ ) = Φ 0 2 π δ I ˙ I c cos ( ϕ 0 ) . V=\frac{\Phi_{0}}{2\pi}\dot{\phi}=\frac{\Phi_{0}}{2\pi}(\underbrace{\dot{\phi_% {0}}}_{=0}+\dot{\delta\phi})=\frac{\Phi_{0}}{2\pi}\frac{\dot{\delta I}}{I_{c}% \cos(\phi_{0})}.
  29. V = L I t V=L\frac{\partial I}{\partial t}
  30. L J ( ϕ 0 ) = Φ 0 2 π I c cos ( ϕ 0 ) = L J ( 0 ) cos ( ϕ 0 ) . L_{J}(\phi_{0})=\frac{\Phi_{0}}{2\pi I_{c}\cos(\phi_{0})}=\frac{L_{J}(0)}{\cos% (\phi_{0})}.
  31. ( ϕ 0 ) (\phi_{0})\,
  32. L J ( 0 ) L_{J}(0)\,
  33. I c I_{c}\,
  34. cos ( ϕ 0 ) 0 \cos(\phi_{0})<=0\,
  35. δ U ( ϕ 0 ) = U ( ϕ ) - U ( ϕ 0 ) = E J ( cos ( ϕ 0 ) - cos ( ϕ 0 + δ ϕ ) \delta U(\phi_{0})=U(\phi)-U(\phi_{0})=E_{J}(\cos(\phi_{0})-\cos(\phi_{0}+% \delta\phi)\,
  36. δ ϕ \delta\phi\,
  37. E J sin ( ϕ 0 ) δ ϕ = E J sin ( ϕ 0 ) I c cos ϕ 0 δ I \approx E_{J}\sin(\phi_{0})\delta\phi=\frac{E_{J}\sin(\phi_{0})}{I_{c}\cos\phi% _{0}}\delta I
  38. d E L = L I δ I dE_{L}=LI\delta I\,
  39. L L\,

JPEG_XR.html

  1. V = B - R V=B-R\,
  2. U = G - R - V 2 U=G-R-\left\lceil\frac{V}{2}\right\rceil
  3. Y = G - U 2 Y=G-\left\lceil\frac{U}{2}\right\rceil

Juliette_Peirce.html

  1. \downarrow

Jung's_theorem.html

  1. K n K\subset\mathbb{R}^{n}
  2. d = max p , q K p - q 2 d=\max\nolimits_{p,q\in K}\|p-q\|_{2}
  3. r d n 2 ( n + 1 ) r\leq d\sqrt{\frac{n}{2(n+1)}}
  4. r d 3 . r\leq\frac{d}{\sqrt{3}}.
  5. r = d 3 . r=\frac{d}{\sqrt{3}}.

K-medoids.html

  1. l 2 l_{2}
  2. l 1 l_{1}
  3. k k
  4. n n
  5. m m
  6. o o
  7. m m
  8. o o
  9. c < s u b > 1 c<sub>1
  10. c < s u b > 2 c<sub>2
  11. ( 2 , 6 ) (2,6)
  12. ( 3 , 8 ) (3,8)
  13. ( 4 , 7 ) (4,7)
  14. 20 20
  15. cost ( x , c ) = i = 1 d | x i - c i | \mbox{cost}~{}(x,c)=\sum_{i=1}^{d}|x_{i}-c_{i}|
  16. 2 2
  17. total cost = { cost ( ( 3 , 4 ) , ( 2 , 6 ) ) + cost ( ( 3 , 4 ) , ( 3 , 8 ) ) + cost ( ( 3 , 4 ) , ( 4 , 7 ) ) } + { cost ( ( 7 , 4 ) , ( 6 , 2 ) ) + cost ( ( 7 , 4 ) , ( 6 , 4 ) ) + cost ( ( 7 , 4 ) , ( 7 , 3 ) ) + cost ( ( 7 , 4 ) , ( 8 , 5 ) ) + cost ( ( 7 , 4 ) , ( 7 , 6 ) ) } = ( 3 + 4 + 4 ) + ( 3 + 1 + 1 + 2 + 2 ) = 20 \begin{aligned}\displaystyle\mbox{total cost}&\displaystyle=\{\mbox{cost}~{}((% 3,4),(2,6))+\mbox{cost}~{}((3,4),(3,8))+\mbox{cost}~{}((3,4),(4,7))\}\\ &\displaystyle~{}+\{\mbox{cost}~{}((7,4),(6,2))+\mbox{cost}~{}((7,4),(6,4))+% \mbox{cost}~{}((7,4),(7,3))\\ &\displaystyle~{}+\mbox{cost}~{}((7,4),(8,5))+\mbox{cost}~{}((7,4),(7,6))\}\\ &\displaystyle=(3+4+4)+(3+1+1+2+2)\\ &\displaystyle=20\\ \end{aligned}
  18. O O′
  19. O = ( 7 , 3 ) O′=(7,3)
  20. O ( 7 , 3 ) O′(7,3)
  21. c 1 c1
  22. O O′
  23. i i
  24. c < s u b > 1 c<sub>1
  25. i i
  26. O O′
  27. X < s u b > i X<sub>i
  28. total cost = 3 + 4 + 4 + 2 + 2 + 1 + 3 + 3 = 22 \begin{aligned}\displaystyle\mbox{total cost}&\displaystyle=3+4+4+2+2+1+3+3\\ &\displaystyle=22\\ \end{aligned}
  29. O O′
  30. S = current total cost - past total cost = 22 - 20 = 2 > 0. \begin{aligned}\displaystyle S&\displaystyle=\mbox{current total cost}~{}-% \mbox{past total cost}\\ &\displaystyle=22-20\\ &\displaystyle=2>0.\end{aligned}
  31. O O′
  32. < v a r > k <var>k
  33. L 1 L_{1}

K-set_(geometry).html

  1. O ( n d / 2 k d / 2 ) O(n^{\lfloor d/2\rfloor}k^{\lceil d/2\rceil})

K-vertex-connected_graph.html

  1. ( s , t ) (s,t)
  2. ( s , t ) (s,t)
  3. s s
  4. t t
  5. k k
  6. k k
  7. s s
  8. t t

K_(disambiguation).html

  1. K n K_{n}
  2. K m , n K_{m,n}
  3. 𝕂 \mathbb{K}
  4. B {}_{B}
  5. κ \kappa

K_correction.html

  1. K K
  2. M = m - 5 ( log 10 D L - 1 ) - K C o r r M=m-5(\log_{10}{D_{L}}-1)-K_{Corr}\!\,

Kalman–Yakubovich–Popov_lemma.html

  1. γ > 0 \gamma>0
  2. ( A , B ) (A,B)
  3. A T P + P A = - Q Q T A^{T}P+PA=-QQ^{T}\,
  4. P B - C = γ Q PB-C=\sqrt{\gamma}Q\,
  5. γ + 2 R e [ C T ( j ω I - A ) - 1 B ] 0 \gamma+2Re[C^{T}(j\omega I-A)^{-1}B]\geq 0
  6. { x : x T P x = 0 } \{x:x^{T}Px=0\}
  7. ( A , B ) (A,B)
  8. A \R n × n , B \R n × m , M = M T \R ( n + m ) × ( n + m ) A\in\R^{n\times n},B\in\R^{n\times m},M=M^{T}\in\R^{(n+m)\times(n+m)}
  9. det ( j ω I - A ) 0 \det(j\omega I-A)\neq 0
  10. ω \R \omega\in\R
  11. ( A , B ) (A,B)
  12. ω \R { } \omega\in\R\cup\{\infty\}
  13. [ ( j ω I - A ) - 1 B I ] * M [ ( j ω I - A ) - 1 B I ] 0 \left[\begin{matrix}(j\omega I-A)^{-1}B\\ I\end{matrix}\right]^{*}M\left[\begin{matrix}(j\omega I-A)^{-1}B\\ I\end{matrix}\right]\leq 0
  14. P \R n × n P\in\R^{n\times n}
  15. P = P T P=P^{T}
  16. M + [ A T P + P A P B B T P 0 ] 0. M+\left[\begin{matrix}A^{T}P+PA&PB\\ B^{T}P&0\end{matrix}\right]\leq 0.
  17. ( A , B ) (A,B)

Karatsuba_algorithm.html

  1. n log 2 3 n 1.585 n^{\log_{2}3}\approx n^{1.585}
  2. n log 2 3 n^{\log_{2}3}
  3. n 2 n^{2}\,\!
  4. Θ ( n 2 ) \Theta(n^{2})\,\!
  5. Ω ( n 2 ) \Omega(n^{2})\,\!
  6. Ω ( n 2 ) \Omega(n^{2})\,\!
  7. Θ ( n log 2 3 ) \Theta(n^{\log_{2}3})
  8. x x
  9. y y
  10. x x
  11. y y
  12. x x
  13. y y
  14. n n
  15. B B
  16. m m
  17. n n
  18. x = x 1 B m + x 0 x=x_{1}B^{m}+x_{0}
  19. y = y 1 B m + y 0 y=y_{1}B^{m}+y_{0}
  20. x 0 x_{0}
  21. y 0 y_{0}
  22. B m B^{m}
  23. x y = ( x 1 B m + x 0 ) ( y 1 B m + y 0 ) xy=(x_{1}B^{m}+x_{0})(y_{1}B^{m}+y_{0})
  24. x y = z 2 B 2 m + z 1 B m + z 0 xy=z_{2}B^{2m}+z_{1}B^{m}+z_{0}
  25. z 2 = x 1 y 1 z_{2}=x_{1}y_{1}
  26. z 1 = x 1 y 0 + x 0 y 1 z_{1}=x_{1}y_{0}+x_{0}y_{1}
  27. z 0 = x 0 y 0 z_{0}=x_{0}y_{0}
  28. x y xy
  29. z 0 z_{0}
  30. z 2 z_{2}
  31. z 1 = ( x 1 + x 0 ) ( y 1 + y 0 ) - z 2 - z 0 z_{1}=(x_{1}+x_{0})(y_{1}+y_{0})-z_{2}-z_{0}
  32. z 1 = x 1 y 0 + x 0 y 1 z_{1}=x_{1}y_{0}+x_{0}y_{1}
  33. z 1 = ( x 1 + x 0 ) ( y 1 + y 0 ) - x 1 y 1 - x 0 y 0 z_{1}=(x_{1}+x_{0})(y_{1}+y_{0})-x_{1}y_{1}-x_{0}y_{0}
  34. x y = ( b 2 + b ) x 1 y 1 - b ( x 1 - x 0 ) ( y 1 - y 0 ) + ( b + 1 ) x 0 y 0 xy=(b^{2}+b)x_{1}y_{1}-b(x_{1}-x_{0})(y_{1}-y_{0})+(b+1)x_{0}y_{0}
  35. b = B m b=B^{m}
  36. 3 log 2 n 3 n log 2 3 3^{\lceil\log_{2}n\rceil}\leq 3n^{\log_{2}3}\,\!
  37. T ( n ) = 3 t ( n / 2 ) + c n + d T(n)=3t(\lceil n/2\rceil)+cn+d
  38. T ( n ) = Θ ( n log 2 3 ) T(n)=\Theta(n^{\log_{2}3})\,\!

Karplus_equation.html

  1. J ( ϕ ) = A cos 2 ϕ + B cos ϕ + C J(\phi)=A\cos^{2}\phi+B\cos\,\phi+C
  2. ϕ \phi

Kasner_metric.html

  1. D > 3 D>3
  2. D > 3 D>3
  3. d s 2 = - d t 2 + j = 1 D - 1 t 2 p j [ d x j ] 2 \,\text{d}s^{2}=-\,\text{d}t^{2}+\sum_{j=1}^{D-1}t^{2p_{j}}[\,\text{d}x^{j}]^{2}
  4. D - 1 D-1
  5. p j p_{j}
  6. p j p_{j}
  7. Δ x j \Delta x^{j}
  8. t p j Δ x j t^{p_{j}}\Delta x^{j}
  9. j = 1 D - 1 p j = 1 , \sum_{j=1}^{D-1}p_{j}=1,
  10. j = 1 D - 1 p j 2 = 1. \sum_{j=1}^{D-1}p_{j}^{2}=1.
  11. p j p_{j}
  12. D D
  13. D - 3 D-3
  14. S D - 3 S^{D-3}
  15. t t
  16. - g \sqrt{-g}
  17. - g = t p 1 + p 2 + + p D - 1 = t \sqrt{-g}=t^{p_{1}+p_{2}+\cdots+p_{D-1}}=t
  18. t 0 t\to 0
  19. t t
  20. p j = 1 / ( D - 1 ) p_{j}=1/(D-1)
  21. j = 1 D - 1 p j 2 = 1 D - 1 1. \sum_{j=1}^{D-1}p_{j}^{2}=\frac{1}{D-1}\neq 1.
  22. p j = 1 p_{j}=1
  23. t t
  24. t t
  25. p j = 1 p_{j}=1
  26. t = t cosh x j t^{\prime}=t\cosh x_{j}
  27. x j = t sinh x j x_{j}^{\prime}=t\sinh x_{j}

Katětov–Tong_insertion_theorem.html

  1. X X
  2. g , h : X g,h\colon X\to\mathbb{R}
  3. g h g\leq h
  4. f : X f\colon X\to\mathbb{R}
  5. g f h . g\leq f\leq h.

Kautz_graph.html

  1. K M N + 1 K_{M}^{N+1}
  2. M M
  3. N + 1 N+1
  4. ( M + 1 ) M N (M+1)M^{N}
  5. s 0 s N s_{0}\cdots s_{N}
  6. N + 1 N+1
  7. s i s_{i}
  8. A A
  9. M + 1 M+1
  10. s i s i + 1 s_{i}\neq s_{i+1}
  11. K M N + 1 K_{M}^{N+1}
  12. ( M + 1 ) M N + 1 (M+1)M^{N+1}
  13. { ( s 0 s 1 s N , s 1 s 2 s N s N + 1 ) | s i A s i s i + 1 } \{(s_{0}s_{1}\cdots s_{N},s_{1}s_{2}\cdots s_{N}s_{N+1})|\;s_{i}\in A\;s_{i}% \neq s_{i+1}\}\,
  14. K M N + 1 K_{M}^{N+1}
  15. s 0 s 1 s N + 1 s_{0}s_{1}\cdots s_{N+1}
  16. K M N + 1 K_{M}^{N+1}
  17. K M N + 2 K_{M}^{N+2}
  18. M M
  19. V = ( M + 1 ) M N V=(M+1)M^{N}
  20. V V
  21. M M
  22. K M N K_{M}^{N}
  23. K M N + 1 K_{M}^{N+1}
  24. K M N + 1 K_{M}^{N+1}
  25. K M N K_{M}^{N}
  26. k k
  27. k k
  28. x x
  29. y y

KeeLoq.html

  1. F ( a , b , c , d , e ) = d e a c a e b c b e c d d e a d e a c e a b d a b c F(a,b,c,d,e)=d\oplus e\oplus ac\oplus ae\oplus bc\oplus be\oplus cd\oplus de% \oplus ade\oplus ace\oplus abd\oplus abc

Kelvin's_circulation_theorem.html

  1. D Γ D t = 0 \frac{\mathrm{D}\Gamma}{\mathrm{D}t}=0
  2. Γ \Gamma
  3. C ( t ) C(t)
  4. Γ \Gamma
  5. C ( t ) C(t)
  6. Γ ( t ) = C s y m b o l u s y m b o l d s \Gamma(t)=\oint_{C}symbol{u}\cdot symbol{ds}
  7. D s y m b o l u D t = - 1 ρ s y m b o l p + s y m b o l Φ \frac{\mathrm{D}symbol{u}}{\mathrm{D}t}=-\frac{1}{\rho}symbol{\nabla}p+symbol{% \nabla}\Phi
  8. ρ = ρ ( p ) \rho=\rho(p)
  9. D Γ D t = C D s y m b o l u D t s y m b o l d s + C s y m b o l u D s y m b o l d s D t . \frac{\mathrm{D}\Gamma}{\mathrm{D}t}=\oint_{C}\frac{\mathrm{D}symbol{u}}{% \mathrm{D}t}\cdot symbol{\mathrm{d}s}+\oint_{C}symbol{u}\cdot\frac{\mathrm{D}% symbol{\mathrm{d}s}}{\mathrm{D}t}.
  10. C D s y m b o l u D t s y m b o l d s = A s y m b o l × ( - 1 ρ s y m b o l p + s y m b o l Φ ) s y m b o l n d S = A 1 ρ 2 ( s y m b o l ρ × s y m b o l p ) s y m b o l n d S = 0. \oint_{C}\frac{\mathrm{D}symbol{u}}{\mathrm{D}t}\cdot symbol{ds}=\int_{A}% symbol{\nabla}\times\left(-\frac{1}{\rho}symbol{\nabla}p+symbol{\nabla}\Phi% \right)\cdot symbol{n}\,\mathrm{d}S=\int_{A}\frac{1}{\rho^{2}}\left(symbol{% \nabla}\rho\times symbol{\nabla}p\right)\cdot symbol{n}\,\mathrm{d}S=0.
  11. s y m b o l ρ × s y m b o l p = 0 symbol{\nabla}\rho\times symbol{\nabla}p=0
  12. D s y m b o l d s D t = ( s y m b o l d s s y m b o l ) s y m b o l u . \frac{\mathrm{D}symbol{\mathrm{d}s}}{\mathrm{D}t}=\left(symbol{\mathrm{d}s}% \cdot symbol{\nabla}\right)symbol{u}.
  13. C s y m b o l u D s y m b o l d s D t = C s y m b o l u [ ( s y m b o l d s s y m b o l ) s y m b o l u ] = 1 2 C s y m b o l ( | s y m b o l u | 2 ) s y m b o l d s = 0. \oint_{C}symbol{u}\cdot\frac{\mathrm{D}symbol{\mathrm{d}s}}{\mathrm{D}t}=\oint% _{C}symbol{u}\cdot\left[\left(symbol{\mathrm{d}s}\cdot symbol{\nabla}\right)% symbol{u}\right]=\frac{1}{2}\oint_{C}symbol{\nabla}\left(|symbol{u}|^{2}\right% )\cdot symbol{\mathrm{d}s}=0.
  14. D Γ D t = 0. \frac{\mathrm{D}\Gamma}{\mathrm{D}t}=0.
  15. s y m b o l Ω symbol{\Omega}
  16. Γ ( t ) = C ( s y m b o l u + s y m b o l Ω × s y m b o l r ) s y m b o l d s \Gamma(t)=\oint_{C}(symbol{u}+symbol{\Omega}\times symbol{r})\cdot symbol{ds}
  17. s y m b o l r symbol{r}
  18. Γ ( t ) = A s y m b o l × ( s y m b o l u + s y m b o l Ω × s y m b o l r ) s y m b o l n d S = A ( s y m b o l × s y m b o l u + 2 s y m b o l Ω ) s y m b o l n d S \Gamma(t)=\int_{A}symbol{\nabla}\times(symbol{u}+symbol{\Omega}\times symbol{r% })\cdot symbol{n}\,\mathrm{d}S=\int_{A}(symbol{\nabla}\times symbol{u}+2symbol% {\Omega})\cdot symbol{n}\,\mathrm{d}S

Kelvin_equation.html

  1. r r
  2. ln p p 0 = 2 γ V m r R T \ln{p\over p_{0}}={2\gamma V_{\rm{m}}\over rRT}
  3. p p
  4. p 0 p_{0}
  5. γ \gamma
  6. V m V_{\rm{m}}
  7. R R
  8. r r
  9. T T
  10. p > p 0 p>p_{0}
  11. p < p 0 p<p_{0}
  12. r r
  13. p p
  14. T T
  15. p 0 p_{0}
  16. p / p 0 p/p_{0}
  17. γ \gamma
  18. V m V_{\rm{m}}
  19. r r
  20. p p
  21. r r
  22. p 0 p_{0}

Kelvin_transform.html

  1. x * = R 2 | x | 2 x . x^{*}=\frac{R^{2}}{|x|^{2}}x.
  2. \infty
  3. \infty
  4. f * ( x * ) = | x | n - 2 R 2 n - 4 f ( x ) = 1 | x * | n - 2 f ( x ) = 1 | x * | n - 2 f ( R 2 | x * | 2 x * ) . f^{*}(x^{*})=\frac{|x|^{n-2}}{R^{2n-4}}f(x)=\frac{1}{|x^{*}|^{n-2}}f(x)=\frac{% 1}{|x^{*}|^{n-2}}f\left(\frac{R^{2}}{|x^{*}|^{2}}x^{*}\right).
  5. Δ u * ( x * ) = R 4 | x * | n + 2 ( Δ u ) ( R 2 | x * | 2 x * ) . \Delta u^{*}(x^{*})=\frac{R^{4}}{|x^{*}|^{n+2}}(\Delta u)\left(\frac{R^{2}}{|x% ^{*}|^{2}}x^{*}\right).

Kendall's_W.html

  1. R i = j = 1 m r i , j , R_{i}=\sum_{j=1}^{m}r_{i,j},
  2. R ¯ = 1 n i = 1 n R i . \bar{R}=\frac{1}{n}\sum_{i=1}^{n}R_{i}.
  3. S = i = 1 n ( R i - R ¯ ) 2 , S=\sum_{i=1}^{n}(R_{i}-\bar{R})^{2},
  4. W = 12 S m 2 ( n 3 - n ) . W=\frac{12S}{m^{2}(n^{3}-n)}.
  5. T j = i = 1 g j ( t i 3 - t i ) , T_{j}=\sum_{i=1}^{g_{j}}(t_{i}^{3}-t_{i}),
  6. W = 12 i = 1 n ( R i 2 ) - 3 m 2 n ( n + 1 ) 2 m 2 n ( n 2 - 1 ) - m j = 1 m ( T j ) , W=\frac{12\sum_{i=1}^{n}(R_{i}^{2})-3m^{2}n(n+1)^{2}}{m^{2}n(n^{2}-1)-m\sum_{j% =1}^{m}(T_{j})},
  7. j = 1 m ( T j ) \sum_{j=1}^{m}(T_{j})

Kendall_rank_correlation_coefficient.html

  1. τ = ( number of concordant pairs ) - ( number of discordant pairs ) 1 2 n ( n - 1 ) . \tau=\frac{(\,\text{number of concordant pairs})-(\,\text{number of discordant% pairs})}{\frac{1}{2}n(n-1)}.
  2. 2 ( 2 n + 5 ) 9 n ( n - 1 ) \frac{2(2n+5)}{9n(n-1)}
  3. τ A = n c - n d n 0 \tau_{A}=\frac{n_{c}-n_{d}}{n_{0}}
  4. τ B = n c - n d ( n 0 - n 1 ) ( n 0 - n 2 ) \tau_{B}=\frac{n_{c}-n_{d}}{\sqrt{(n_{0}-n_{1})(n_{0}-n_{2})}}
  5. n 0 \displaystyle n_{0}
  6. τ \tau
  7. τ A \tau_{A}
  8. z A z_{A}
  9. z A = 3 ( n c - n d ) n ( n - 1 ) ( 2 n + 5 ) / 2 z_{A}={3(n_{c}-n_{d})\over\sqrt{n(n-1)(2n+5)/2}}
  10. z A z_{A}
  11. - | z A | -|z_{A}|
  12. z A z_{A}
  13. z B z_{B}
  14. τ B \tau_{B}
  15. z B = n c - n d v z_{B}={n_{c}-n_{d}\over\sqrt{v}}
  16. v = ( v 0 - v t - v u ) / 18 + v 1 + v 2 v 0 = n ( n - 1 ) ( 2 n + 5 ) v t = i t i ( t i - 1 ) ( 2 t i + 5 ) v u = j u j ( u j - 1 ) ( 2 u j + 5 ) v 1 = i t i ( t i - 1 ) j u j ( u j - 1 ) / ( 2 n ( n - 1 ) ) v 2 = i t i ( t i - 1 ) ( t i - 2 ) j u j ( u j - 1 ) ( u j - 2 ) / ( 9 n ( n - 1 ) ( n - 2 ) ) \begin{array}[]{ccl}v&=&(v_{0}-v_{t}-v_{u})/18+v_{1}+v_{2}\\ v_{0}&=&n(n-1)(2n+5)\\ v_{t}&=&\sum_{i}t_{i}(t_{i}-1)(2t_{i}+5)\\ v_{u}&=&\sum_{j}u_{j}(u_{j}-1)(2u_{j}+5)\\ v_{1}&=&\sum_{i}t_{i}(t_{i}-1)\sum_{j}u_{j}(u_{j}-1)/(2n(n-1))\\ v_{2}&=&\sum_{i}t_{i}(t_{i}-1)(t_{i}-2)\sum_{j}u_{j}(u_{j}-1)(u_{j}-2)/(9n(n-1% )(n-2))\end{array}
  17. n c - n d n_{c}-n_{d}
  18. O ( n 2 ) O(n^{2})
  19. O ( n log n ) O(n\cdot\log{n})
  20. x x
  21. x x
  22. y y
  23. y y
  24. y y
  25. O ( n log n ) O(n\log n)
  26. S ( y ) S(y)
  27. y i y_{i}
  28. τ \tau
  29. n c - n d = n 0 - n 1 - n 2 + n 3 - 2 S ( y ) , n_{c}-n_{d}=n_{0}-n_{1}-n_{2}+n_{3}-2S(y),
  30. n 3 n_{3}
  31. n 1 n_{1}
  32. n 2 n_{2}
  33. x x
  34. y y
  35. y y
  36. y left y_{\mathrm{left}}
  37. y right y_{\mathrm{right}}
  38. S ( y ) = S ( y left ) + S ( y right ) + M ( Y left , Y right ) S(y)=S(y_{\mathrm{left}})+S(y_{\mathrm{right}})+M(Y_{\mathrm{left}},Y_{\mathrm% {right}})
  39. Y left Y_{\mathrm{left}}
  40. Y right Y_{\mathrm{right}}
  41. y left y_{\mathrm{left}}
  42. y right y_{\mathrm{right}}
  43. M ( , ) M(\cdot,\cdot)
  44. M ( , ) M(\cdot,\cdot)
  45. y y
  46. t i t_{i}
  47. u j u_{j}
  48. τ B \tau_{B}
  49. O ( n log n ) O(n\cdot\log{n})
  50. O ( n log n ) O(n\cdot\log{n})

Kendall_tau_distance.html

  1. L 1 L1
  2. L 2 L2
  3. K ( τ 1 , τ 2 ) = | { ( i , j ) : i < j , ( τ 1 ( i ) < τ 1 ( j ) τ 2 ( i ) > τ 2 ( j ) ) ( τ 1 ( i ) > τ 1 ( j ) τ 2 ( i ) < τ 2 ( j ) ) } | . K(\tau_{1},\tau_{2})=|\{(i,j):i<j,(\tau_{1}(i)<\tau_{1}(j)\wedge\tau_{2}(i)>% \tau_{2}(j))\vee(\tau_{1}(i)>\tau_{1}(j)\wedge\tau_{2}(i)<\tau_{2}(j))\}|.
  4. τ 1 ( i ) \tau_{1}(i)
  5. τ 2 ( i ) \tau_{2}(i)
  6. L 1 L1
  7. L 2 L2
  8. K ( τ 1 , τ 2 ) K(\tau_{1},\tau_{2})
  9. n ( n - 1 ) / 2 n(n-1)/2
  10. n n
  11. n ( n - 1 ) / 2 n(n-1)/2
  12. K ( τ 1 , τ 2 ) = { i , j } P K ¯ i , j ( τ 1 , τ 2 ) K(\tau_{1},\tau_{2})=\begin{matrix}\sum_{\{i,j\}\in P}\bar{K}_{i,j}(\tau_{1},% \tau_{2})\end{matrix}
  13. τ 1 \tau_{1}
  14. τ 2 \tau_{2}
  15. K ¯ i , j ( τ 1 , τ 2 ) \bar{K}_{i,j}(\tau_{1},\tau_{2})
  16. τ 1 \tau_{1}
  17. τ 2 \tau_{2}
  18. K ¯ i , j ( τ 1 , τ 2 ) \bar{K}_{i,j}(\tau_{1},\tau_{2})
  19. τ 1 \tau_{1}
  20. τ 2 . \tau_{2}.
  21. K ( L 1 , L 2 ) K(L1,L2)
  22. K ( τ 1 , τ 2 ) K(\tau_{1},\tau_{2})
  23. τ 1 \tau_{1}
  24. τ 2 \tau_{2}
  25. L 1 L1
  26. L 2 L2
  27. 4 5 ( 5 - 1 ) / 2 = 0.4. \frac{4}{5(5-1)/2}=0.4.

Kenneth_Kunen.html

  1. 2 \aleph_{2}
  2. 1 \aleph_{1}
  3. κ \kappa
  4. 2 κ > κ + 2^{\kappa}>\kappa^{+}
  5. κ \kappa
  6. κ \kappa
  7. V V V\to V
  8. P ( ω ) / F i n P(\omega)/Fin
  9. ω 2 \omega_{2}
  10. 2 \aleph_{2}

Kent_distribution.html

  1. κ \kappa\,
  2. S 2 S^{2}\,
  3. 3 \mathbb{R}^{3}
  4. f ( 𝐱 ) f(\mathbf{x})\,
  5. f ( 𝐱 ) = 1 c ( κ , β ) exp { \kappasymbol γ 1 𝐱 + β [ ( s y m b o l γ 2 𝐱 ) 2 - ( s y m b o l γ 3 𝐱 ) 2 ] } f(\mathbf{x})=\frac{1}{\textrm{c}(\kappa,\beta)}\exp\{\kappasymbol{\gamma}_{1}% \cdot\mathbf{x}+\beta[(symbol{\gamma}_{2}\cdot\mathbf{x})^{2}-(symbol{\gamma}_% {3}\cdot\mathbf{x})^{2}]\}
  6. 𝐱 \mathbf{x}\,
  7. c ( κ , β ) \textrm{c}(\kappa,\beta)\,
  8. c ( κ , β ) = 2 π j = 0 Γ ( j + 1 2 ) Γ ( j + 1 ) β 2 j ( 1 2 κ ) - 2 j - 1 2 I 2 j + 1 2 ( κ ) c(\kappa,\beta)=2\pi\sum_{j=0}^{\infty}\frac{\Gamma(j+\frac{1}{2})}{\Gamma(j+1% )}\beta^{2j}\left(\frac{1}{2}\kappa\right)^{-2j-\frac{1}{2}}I_{2j+\frac{1}{2}}% (\kappa)
  9. I v ( κ ) I_{v}(\kappa)
  10. c ( 0 , 0 ) = 4 π c(0,0)=4\pi
  11. c ( κ , 0 ) = 4 π κ - 1 sinh ( κ ) c(\kappa,0)=4\pi\kappa^{-1}\sinh(\kappa)
  12. κ \kappa\,
  13. κ > 0 \kappa>0\,
  14. β \beta\,
  15. 0 2 β < κ 0\leq 2\beta<\kappa
  16. κ \kappa\,
  17. β \beta\,
  18. γ 1 \gamma_{1}\,
  19. γ 2 , γ 3 \gamma_{2},\gamma_{3}\,
  20. ( γ 1 , γ 2 , γ 3 ) (\gamma_{1},\gamma_{2},\gamma_{3})\,
  21. x x
  22. S p - 1 S^{p-1}
  23. p \mathbb{R}^{p}
  24. p p
  25. exp { κ s y m b o l γ 1 𝐱 + j = 2 p β j ( s y m b o l γ j 𝐱 ) 2 } \exp\{\kappa symbol{\gamma}_{1}\cdot\mathbf{x}+\sum_{j=2}^{p}\beta_{j}(symbol{% \gamma}_{j}\cdot\mathbf{x})^{2}\}
  26. j = 2 p β j = 0 \sum_{j=2}^{p}\beta_{j}=0
  27. 0 2 | β j | < κ 0\leq 2|\beta_{j}|<\kappa
  28. { s y m b o l γ j j = 1 p } \{symbol{\gamma}_{j}\mid j=1\ldots p\}
  29. p > 3 p>3

Kepler–Bouwkamp_constant.html

  1. k = 3 cos ( π k ) = 0.1149420448 . \prod_{k=3}^{\infty}\cos\left(\frac{\pi}{k}\right)=0.1149420448\dots.
  2. k = 3 , 5 , 7 , 11 , 13 , 17 , cos ( π k ) = 0.312832 \prod_{k=3,5,7,11,13,17,\ldots}\cos\left(\frac{\pi}{k}\right)=0.312832\ldots

Kernel_principal_component_analysis.html

  1. 1 N i = 1 N 𝐱 i = 𝟎 \frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_{i}=\mathbf{0}
  2. C = 1 N i = 1 N 𝐱 i 𝐱 i C=\frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_{i}\mathbf{x}_{i}^{\top}
  3. λ 𝐯 = C 𝐯 \lambda\mathbf{v}=C\mathbf{v}
  4. λ 𝐱 i 𝐯 = 𝐱 i C 𝐯 i [ 1 , N ] \lambda\mathbf{x}_{i}^{\top}\mathbf{v}=\mathbf{x}_{i}^{\top}C\mathbf{v}\quad% \forall i\in[1,N]
  5. d < N d<N
  6. d N d\geq N
  7. 𝐱 i \mathbf{x}_{i}
  8. Φ ( 𝐱 i ) \Phi(\mathbf{x}_{i})
  9. Φ : d N \Phi:\mathbb{R}^{d}\to\mathbb{R}^{N}
  10. Φ \Phi
  11. Φ \Phi
  12. Φ \Phi
  13. Φ \Phi
  14. K = k ( 𝐱 , 𝐲 ) = ( Φ ( 𝐱 ) , Φ ( 𝐲 ) ) = Φ ( 𝐱 ) T Φ ( 𝐲 ) K=k(\mathbf{x},\mathbf{y})=(\Phi(\mathbf{x}),\Phi(\mathbf{y}))=\Phi(\mathbf{x}% )^{T}\Phi(\mathbf{y})
  15. Φ ( 𝐱 ) \Phi(\mathbf{x})
  16. Φ ( 𝐱 ) \Phi(\mathbf{x})
  17. V V
  18. 𝐕 k T Φ ( 𝐱 ) = ( i = 1 N 𝐚 𝐢 k Φ ( 𝐱 𝐢 ) ) T Φ ( 𝐱 ) {\mathbf{V}^{k}}^{T}\Phi(\mathbf{x})=\left(\sum_{i=1}^{N}\mathbf{a_{i}}^{k}% \Phi(\mathbf{x_{i}})\right)^{T}\Phi(\mathbf{x})
  19. Φ ( 𝐱 𝐢 ) T Φ ( 𝐱 ) \Phi(\mathbf{x_{i}})^{T}\Phi(\mathbf{x})
  20. K K
  21. 𝐚 𝐢 k \mathbf{a_{i}}^{k}
  22. N λ 𝐚 = K 𝐚 N\lambda\mathbf{a}=K\mathbf{a}
  23. λ \lambda
  24. 𝐚 \mathbf{a}
  25. 𝐚 k \mathbf{a}^{k}
  26. 1 = ( 𝐚 k ) T 𝐚 k 1=(\mathbf{a}^{k})^{T}\mathbf{a}^{k}
  27. x x
  28. K K^{\prime}
  29. K = K - 𝟏 𝐍 K - K 𝟏 𝐍 + 𝟏 𝐍 K 𝟏 𝐍 K^{\prime}=K-\mathbf{1_{N}}K-K\mathbf{1_{N}}+\mathbf{1_{N}}K\mathbf{1_{N}}
  30. 𝟏 𝐍 \mathbf{1_{N}}
  31. 1 / N 1/N
  32. K K^{\prime}
  33. k ( s y m b o l x , s y m b o l y ) = ( s y m b o l x T s y m b o l y + 1 ) 2 k(symbol{x},symbol{y})=(symbol{x}^{\mathrm{T}}symbol{y}+1)^{2}
  34. k ( s y m b o l x , s y m b o l y ) = ( s y m b o l x T s y m b o l y + 1 ) 2 k(symbol{x},symbol{y})=(symbol{x}^{\mathrm{T}}symbol{y}+1)^{2}
  35. k ( s y m b o l x , s y m b o l y ) = e - || s y m b o l x - s y m b o l y || 2 2 σ 2 , k(symbol{x},symbol{y})=e^{\frac{-||symbol{x}-symbol{y}||^{2}}{2\sigma^{2}}},

Kharitonov's_theorem.html

  1. p ( s ) = a 0 + a 1 s 1 + a 2 s 2 + + a n s n p(s)=a_{0}+a_{1}s^{1}+a_{2}s^{2}+...+a_{n}s^{n}
  2. a i R a_{i}\in R
  3. l i a i u i . l_{i}\leq a_{i}\leq u_{i}.
  4. 0 [ l n , u n ] 0\notin[l_{n},u_{n}]
  5. k 1 ( s ) = l 0 + l 1 s 1 + u 2 s 2 + u 3 s 3 + l 4 s 4 + l 5 s 5 + k_{1}(s)=l_{0}+l_{1}s^{1}+u_{2}s^{2}+u_{3}s^{3}+l_{4}s^{4}+l_{5}s^{5}+\cdots\,
  6. k 2 ( s ) = u 0 + u 1 s 1 + l 2 s 2 + l 3 s 3 + u 4 s 4 + u 5 s 5 + k_{2}(s)=u_{0}+u_{1}s^{1}+l_{2}s^{2}+l_{3}s^{3}+u_{4}s^{4}+u_{5}s^{5}+\cdots\,
  7. k 3 ( s ) = l 0 + u 1 s 1 + u 2 s 2 + l 3 s 3 + l 4 s 4 + u 5 s 5 + k_{3}(s)=l_{0}+u_{1}s^{1}+u_{2}s^{2}+l_{3}s^{3}+l_{4}s^{4}+u_{5}s^{5}+\cdots\,
  8. k 4 ( s ) = u 0 + l 1 s 1 + l 2 s 2 + u 3 s 3 + u 4 s 4 + l 5 s 5 + k_{4}(s)=u_{0}+l_{1}s^{1}+l_{2}s^{2}+u_{3}s^{3}+u_{4}s^{4}+l_{5}s^{5}+\cdots\,

Killing_tensor.html

  1. K K
  2. ( α K β γ ) = 0 \nabla_{(\alpha}K_{\beta\gamma)}=0\,

Kinetic_inductance.html

  1. L K L_{K}
  2. τ \tau
  3. ω \omega
  4. σ ( ω ) = σ 1 - i σ 2 {\sigma(\omega)=\sigma_{1}-i\sigma_{2}}
  5. σ = n e 2 τ m ( 1 + i ω τ ) = n e 2 τ m ( 1 + ω 2 τ 2 ) - i n e 2 ω τ 2 m ( 1 + ω 2 τ 2 ) \sigma=\frac{ne^{2}\tau}{m(1+i\omega\tau)}=\frac{ne^{2}\tau}{m(1+\omega^{2}% \tau^{2})}-i\frac{ne^{2}\omega\tau^{2}}{m(1+\omega^{2}\tau^{2})}
  6. m m
  7. n n
  8. 10 - 14 \approx 10^{-14}
  9. τ {\tau\rightarrow\infty}
  10. 1 2 ( 2 m v 2 ) ( n s l A ) = 1 2 L K I 2 \frac{1}{2}(2mv^{2})(n_{s}lA)=\frac{1}{2}L_{K}I^{2}
  11. m m
  12. 2 m 2m
  13. v v
  14. n s n_{s}
  15. l l
  16. A A
  17. I I
  18. I = 2 e v n s A I=2evn_{s}A
  19. e e
  20. L K = ( m 2 n s e 2 ) ( l A ) L_{K}=\left(\frac{m}{2n_{s}e^{2}}\right)\left(\frac{l}{A}\right)
  21. 2 m 2m
  22. m m
  23. 2 e 2e
  24. e e
  25. n s n_{s}
  26. n n
  27. L K = ( m n e 2 ) ( l A ) L_{K}=\left(\frac{m}{ne^{2}}\right)\left(\frac{l}{A}\right)
  28. n n
  29. β \beta

Kinetic_Monte_Carlo.html

  1. t = 0 t=0
  2. r i r_{i}
  3. R i = j = 1 i r j R_{i}=\sum_{j=1}^{i}r_{j}
  4. i = 1 , , N i=1,\ldots,N
  5. u ( 0 , 1 ] u\in(0,1]
  6. R i - 1 < u R N R i R_{i-1}<uR_{N}\leq R_{i}
  7. u ( 0 , 1 ] u^{\prime}\in(0,1]
  8. t = t + Δ t t=t+\Delta t
  9. Δ t = R N - 1 ln ( 1 / u ) \Delta t=R_{N}^{-1}\ln(1/u^{\prime})
  10. r i r_{i}
  11. ln ( 1 / u ) \ln(1/u^{\prime})
  12. Δ t = R N - 1 \Delta t=R_{N}^{-1}
  13. r i r_{i}
  14. r i ( t ) r_{i}(t)
  15. 0 Δ t R i ( t ) d t = ln ( 1 / u ) \int_{0}^{\Delta t}R_{i}(t^{\prime})dt^{\prime}=\ln(1/u^{\prime})
  16. R i - 1 ( Δ t ) < u R N ( Δ t ) R i ( Δ t ) R_{i-1}(\Delta t)<uR_{N}(\Delta t)\leq R_{i}(\Delta t)
  17. Δ t i \Delta t_{i}
  18. 0 Δ t i r i ( t ) d t = ln ( 1 / u i ) \int_{0}^{\Delta t_{i}}r_{i}(t^{\prime})dt^{\prime}=\ln(1/u_{i})
  19. u i ( 0 , 1 ] u_{i}\in(0,1]
  20. N N
  21. r i r_{i}
  22. w w

Kinetic_resolution.html

  1. Δ G \Delta G^{\ddagger}
  2. Δ G \Delta G^{\ddagger}
  3. Δ Δ G \Delta\Delta G^{\ddagger}
  4. s = k R k S = e Δ Δ G / R T s=\frac{k_{R}}{k_{S}}=e^{\Delta\Delta G^{\ddagger}/RT}
  5. [ S ] = ( 1 + e e ) ( 1 - c ) 2 [S]=\frac{(1+ee)(1-c)}{2}
  6. [ R ] = ( 1 - e e ) ( 1 - c ) 2 [R]=\frac{(1-ee)(1-c)}{2}
  7. S 0 = R 0 = 1 2 S_{0}=R_{0}=\frac{1}{2}
  8. d [ S ] d t = - k S [ S ] [ B * ] log [ S ] = - k S [ B * ] t + log S 0 \frac{d[S]}{dt}=-k_{S}[S][B^{*}]\implies\log[S]=-k_{S}[B^{*}]t+\log S_{0}
  9. s = k R k S = log [ R ] - log R 0 log [ S ] - log S 0 = log [ ( 1 - c ) ( 1 - e e ) ] + log 1 2 - log R 0 log [ ( 1 - c ) ( 1 - e e ) ] + log 1 2 - log S 0 = log [ ( 1 - c ) ( 1 - e e ) ] log [ ( 1 - c ) ( 1 + e e ) ] s=\frac{k_{R}}{k_{S}}=\frac{\log[R]-\log R_{0}}{\log[S]-\log S_{0}}=\frac{\log% [(1-c)(1-ee)]+\log\frac{1}{2}-\log R_{0}}{\log[(1-c)(1-ee)]+\log\frac{1}{2}-% \log S_{0}}=\frac{\log[(1-c)(1-ee)]}{\log[(1-c)(1+ee)]}
  10. e e ′′ = [ R ] - [ S ] [ R ] + [ S ] = e e ( 1 - c ) c e e = e e ′′ c 1 - c ee^{\prime\prime}=\frac{[R^{\prime}]-[S^{\prime}]}{[R^{\prime}]+[S^{\prime}]}=% \frac{ee(1-c)}{c}\implies ee=ee^{\prime\prime}\frac{c}{1-c}
  11. 1 - e e = 1 - c - c e e ′′ 1 - c 1-ee=\frac{1-c-cee^{\prime\prime}}{1-c}
  12. 1 + e e = 1 - c + c e e ′′ 1 - c 1+ee=\frac{1-c+cee^{\prime\prime}}{1-c}
  13. ( 1 - c ) ( 1 - e e ) = 1 - c ( 1 + e e ′′ ) (1-c)(1-ee)=1-c(1+ee^{\prime\prime})
  14. ( 1 - c ) ( 1 - e e ) = 1 - c ( 1 - e e ′′ ) (1-c)(1-ee)=1-c(1-ee^{\prime\prime})
  15. s = log [ 1 - c ( 1 - e e ′′ ) ] log [ 1 - c ( 1 + e e ′′ ) ] s=\frac{\log[1-c(1-ee^{\prime\prime})]}{\log[1-c(1+ee^{\prime\prime})]}
  16. d [ S ] d t = - k S [ S ] S = 1 2 e - k S t \frac{d[S]}{dt}=-k_{S}[S]\implies S=\frac{1}{2}e^{-k_{S}t}
  17. e e = [ S ] - [ R ] [ S ] + [ R ] = e - k S t - e - k R t e - k S t + e - k R t ee=\frac{[S]-[R]}{[S]+[R]}=\frac{e^{-k_{S}t}-e^{-k_{R}t}}{e^{-k_{S}t}+e^{-k_{R% }t}}
  18. c = 1 - ( [ S ] - [ R ] ) = 1 - e - k S t + e - k R t 2 c=1-\big([S]-[R]\big)=1-\frac{e^{-k_{S}t}+e^{-k_{R}t}}{2}
  19. e e ′′ = 1 - c c e e = e - t + e - s t 2 - e - t - e - s t ( e - t - e - s t e - t + e - s t ) ee^{\prime\prime}=\frac{1-c}{c}ee=\frac{e^{-t}+e^{-st}}{2-e^{-t}-e^{-st}}\left% (\frac{e^{-t}-e^{-st}}{e^{-t}+e^{-st}}\right)
  20. Δ G r a c < Δ G R < Δ G S \Delta G_{rac}<\Delta G^{\ddagger}_{R}<\Delta G^{\ddagger}_{S}

Kirkwood_approximation.html

  1. P ( x 1 , x 2 , , x n ) P(x_{1},x_{2},\ldots,x_{n})
  2. P ( x 1 , x 2 , , x n ) = 𝒯 n - 1 𝒱 p ( 𝒯 n - 1 ) 𝒯 n - 2 𝒱 p ( 𝒯 n - 2 ) 𝒯 1 𝒱 p ( 𝒯 1 ) P^{\prime}(x_{1},x_{2},\ldots,x_{n})=\frac{\frac{\frac{\prod_{\mathcal{T}_{n-1% }\subseteq\mathcal{V}}p(\mathcal{T}_{n-1})}{\prod_{\mathcal{T}_{n-2}\subseteq% \mathcal{V}}p(\mathcal{T}_{n-2})}}{\vdots}}{\prod_{\mathcal{T}_{1}\subseteq% \mathcal{V}}p(\mathcal{T}_{1})}
  3. 𝒯 i 𝒱 p ( 𝒯 i ) \prod_{\mathcal{T}_{i}\subseteq\mathcal{V}}p(\mathcal{T}_{i})
  4. 𝒱 \scriptstyle\mathcal{V}
  5. P ( x 1 , x 2 , x 3 ) = p ( x 1 , x 2 ) p ( x 2 , x 3 ) p ( x 1 , x 3 ) p ( x 1 ) p ( x 2 ) p ( x 3 ) P^{\prime}(x_{1},x_{2},x_{3})=\frac{p(x_{1},x_{2})p(x_{2},x_{3})p(x_{1},x_{3})% }{p(x_{1})p(x_{2})p(x_{3})}

Kirsch_equations.html

  1. σ r r = σ 2 ( 1 - a 2 r 2 ) + σ 2 ( 1 + 3 a 4 r 4 - 4 a 2 r 2 ) cos 2 θ \sigma_{rr}=\frac{\sigma}{2}\left(1-\frac{a^{2}}{r^{2}}\right)+\frac{\sigma}{2% }\left(1+3\frac{a^{4}}{r^{4}}-4\frac{a^{2}}{r^{2}}\right)\cos 2\theta
  2. σ θ θ = σ 2 ( 1 + a 2 r 2 ) - σ 2 ( 1 + 3 a 4 r 4 ) cos 2 θ \sigma_{\theta\theta}=\frac{\sigma}{2}\left(1+\frac{a^{2}}{r^{2}}\right)-\frac% {\sigma}{2}\left(1+3\frac{a^{4}}{r^{4}}\right)\cos 2\theta
  3. σ r θ = - σ 2 ( 1 - 3 a 4 r 4 + 2 a 2 r 2 ) sin 2 θ \sigma_{r\theta}=-\frac{\sigma}{2}\left(1-3\frac{a^{4}}{r^{4}}+2\frac{a^{2}}{r% ^{2}}\right)\sin 2\theta

Knudsen_diffusion.html

  1. D A A * = λ u 3 = λ 3 8 k B T π M A {D_{AA*}}={{\lambda u}\over{3}}={{\lambda}\over{3}}\sqrt{{8k_{B}T}\over{\pi M_% {A}}}
  2. D j = 8 r a 3 R T 2 π M j {D_{j}}={8r_{a}\over{3}}{\sqrt{{RT}\over{2}\pi M_{j}}}
  3. d p o r e d_{pore}
  4. D K A D_{KA}
  5. D K A = d p o r e 3 u = d p o r e 3 8 R g T π M A {D_{KA}}={d_{pore}\over{3}}u={{d_{pore}\over{3}}}\sqrt{{8R_{g}T}\over{\pi M_{A% }}}
  6. R g R_{g}
  7. M A M_{A}
  8. D K A D_{KA}
  9. D A B D_{AB}
  10. D A e D_{Ae}
  11. 1 D A e = 1 - α y a D A B + 1 D K A \frac{1}{{{D}_{Ae}}}=\frac{1-\alpha{{y}_{a}}}{{{D}_{AB}}}+\frac{1}{{{D}_{KA}}}
  12. α = 1 + N B N A \alpha=1+\frac{{{N}_{B}}}{{{N}_{A}}}
  13. N A N_{A}
  14. N B N_{B}
  15. y A y_{A}
  16. 1 D A e = 1 D A B + 1 D K A \frac{1}{{{D}_{Ae}}}=\frac{1}{{{D}_{AB}}}+\frac{1}{{{D}_{KA}}}
  17. D S D_{S}

Koide_formula.html

  1. Q = m e + m μ + m τ ( m e + m μ + m τ ) 2 2 3 . Q=\frac{m_{e}+m_{\mu}+m_{\tau}}{(\sqrt{m_{e}}+\sqrt{m_{\mu}}+\sqrt{m_{\tau}})^% {2}}\approx\frac{2}{3}.
  2. ( m e , m μ , m τ ) (\sqrt{m_{e}},\sqrt{m_{\mu}},\sqrt{m_{\tau}})
  3. ( 1 , 1 , 1 ) . (1,1,1).
  4. 1 / 3 {1}/{3}
  5. Q = m B + m T ( m B + m T ) 2 3 4 . Q=\frac{m_{B}+m_{T}}{(\sqrt{m_{B}}+\sqrt{m_{T}})^{2}}\approx\frac{3}{4}.
  6. Δ Q = Q m τ Δ m τ \scriptstyle{\Delta Q=\frac{\partial Q}{\partial m_{\tau}}\Delta m_{\tau}}

Kolmogorov_backward_equations_(diffusion).html

  1. p t ( x ) p_{t}(x)
  2. s > t s>t
  3. p t ( x ) p_{t}(x)
  4. p t ( x ) p_{t}(x)
  5. u s ( x ) u_{s}(x)
  6. u s ( x ) = 1 B u_{s}(x)=1_{B}
  7. t , ( t < s ) t,(t<s)
  8. u s ( x ) u_{s}(x)
  9. x ( t ) x(t)
  10. d x ( t ) = μ ( x ( t ) , t ) d t + σ ( x ( t ) , t ) d W ( t ) dx(t)=\mu(x(t),t)\,dt+\sigma(x(t),t)\,dW(t)
  11. - t p ( x , t ) = μ ( x , t ) x p ( x , t ) + 1 2 σ 2 ( x , t ) 2 x 2 p ( x , t ) -\frac{\partial}{\partial t}p(x,t)=\mu(x,t)\frac{\partial}{\partial x}p(x,t)+% \frac{1}{2}\sigma^{2}(x,t)\frac{\partial^{2}}{\partial x^{2}}p(x,t)
  12. t s t\leq s
  13. p ( x , s ) = u s ( x ) p(x,s)=u_{s}(x)
  14. p ( x , t ) p(x,t)
  15. u s ( x ) u_{s}(x)
  16. P ( X s B X t = x ) = E [ u s ( x ) X t = x ] P(X_{s}\in B\mid X_{t}=x)=E[u_{s}(x)\mid X_{t}=x]
  17. s p ( x , s ) = - x [ μ ( x , s ) p ( x , s ) ] + 1 2 2 x 2 [ σ 2 ( x , s ) p ( x , s ) ] \frac{\partial}{\partial s}p(x,s)=-\frac{\partial}{\partial x}[\mu(x,s)p(x,s)]% +\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}[\sigma^{2}(x,s)p(x,s)]
  18. s t s\geq t
  19. p ( x , t ) = p t ( x ) p(x,t)=p_{t}(x)

Kolmogorov_continuity_theorem.html

  1. X : [ 0 , + ) × Ω n X:[0,+\infty)\times\Omega\to\mathbb{R}^{n}
  2. T > 0 T>0
  3. α , β , K \alpha,\beta,K
  4. 𝔼 [ | X t - X s | α ] K | t - s | 1 + β \mathbb{E}\left[|X_{t}-X_{s}|^{\alpha}\right]\leq K|t-s|^{1+\beta}
  5. 0 s , t T 0\leq s,t\leq T
  6. X X
  7. X ~ : [ 0 , + ) × Ω n \tilde{X}:[0,+\infty)\times\Omega\to\mathbb{R}^{n}
  8. X ~ \tilde{X}
  9. t 0 t\geq 0
  10. ( X t = X ~ t ) = 1. \mathbb{P}(X_{t}=\tilde{X}_{t})=1.
  11. X ~ \tilde{X}
  12. γ \gamma
  13. 0 < γ < β α 0<\gamma<\tfrac{\beta}{\alpha}
  14. n \mathbb{R}^{n}
  15. α = 4 \alpha=4
  16. β = 1 \beta=1
  17. K = n ( n + 2 ) K=n(n+2)

Kolmogorov_extension_theorem.html

  1. T T
  2. n n\in\mathbb{N}
  3. k k\in\mathbb{N}
  4. t 1 , , t k T t_{1},\dots,t_{k}\in T
  5. ν t 1 t k \nu_{t_{1}\dots t_{k}}
  6. ( n ) k (\mathbb{R}^{n})^{k}
  7. π \pi
  8. { 1 , , k } \{1,\dots,k\}
  9. F i n F_{i}\subseteq\mathbb{R}^{n}
  10. ν t π ( 1 ) t π ( k ) ( F π ( 1 ) × × F π ( k ) ) = ν t 1 t k ( F 1 × × F k ) ; \nu_{t_{\pi(1)}\dots t_{\pi(k)}}\left(F_{\pi(1)}\times\dots\times F_{\pi(k)}% \right)=\nu_{t_{1}\dots t_{k}}\left(F_{1}\times\dots\times F_{k}\right);
  11. F i n F_{i}\subseteq\mathbb{R}^{n}
  12. m m\in\mathbb{N}
  13. ν t 1 t k ( F 1 × × F k ) = ν t 1 t k t k + 1 , , t k + m ( F 1 × × F k × n × × n ) . \nu_{t_{1}\dots t_{k}}\left(F_{1}\times\dots\times F_{k}\right)=\nu_{t_{1}% \dots t_{k}t_{k+1},\dots,t_{k+m}}\left(F_{1}\times\dots\times F_{k}\times% \mathbb{R}^{n}\times\dots\times\mathbb{R}^{n}\right).
  14. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  15. X : T × Ω n X:T\times\Omega\to\mathbb{R}^{n}
  16. ν t 1 t k ( F 1 × × F k ) = ( X t 1 F 1 , , X t k F k ) \nu_{t_{1}\dots t_{k}}\left(F_{1}\times\dots\times F_{k}\right)=\mathbb{P}% \left(X_{t_{1}}\in F_{1},\dots,X_{t_{k}}\in F_{k}\right)
  17. t i T t_{i}\in T
  18. k k\in\mathbb{N}
  19. F i n F_{i}\subseteq\mathbb{R}^{n}
  20. X X
  21. ν t 1 t k \nu_{t_{1}\dots t_{k}}
  22. t 1 t k t_{1}\dots t_{k}
  23. Ω = ( n ) T \Omega=(\mathbb{R}^{n})^{T}
  24. X X
  25. X : ( t , Y ) Y t X\colon(t,Y)\mapsto Y_{t}
  26. ν \nu
  27. ( n ) T (\mathbb{R}^{n})^{T}
  28. ν t 1 t k \nu_{t_{1}\dots t_{k}}
  29. t 1 t k t_{1}\dots t_{k}
  30. T T
  31. ν \nu
  32. ( n ) T (\mathbb{R}^{n})^{T}
  33. X X
  34. ( X 1 > 0 , X 2 < 0 ) \mathbb{P}(X_{1}>0,X_{2}<0)
  35. ν 1 , 2 ( + × - ) \nu_{1,2}(\mathbb{R}_{+}\times\mathbb{R}_{-})
  36. ν 2 , 1 ( - × + ) \nu_{2,1}(\mathbb{R}_{-}\times\mathbb{R}_{+})
  37. ν 1 , 2 ( + × - ) = ν 2 , 1 ( - × + ) \nu_{1,2}(\mathbb{R}_{+}\times\mathbb{R}_{-})=\nu_{2,1}(\mathbb{R}_{-}\times% \mathbb{R}_{+})
  38. t i t_{i}
  39. F i F_{i}
  40. ( X 1 > 0 ) = ( X 1 > 0 , X 2 ) \mathbb{P}(X_{1}>0)=\mathbb{P}(X_{1}>0,X_{2}\in\mathbb{R})
  41. n \mathbb{R}^{n}
  42. n \mathbb{R}^{n}
  43. T T
  44. { ( Ω t , t ) } t T \{(\Omega_{t},\mathcal{F}_{t})\}_{t\in T}
  45. t T t\in T
  46. τ t \tau_{t}
  47. Ω t \Omega_{t}
  48. J T J\subset T
  49. Ω J := t J Ω t \Omega_{J}:=\prod_{t\in J}\Omega_{t}
  50. I J T I\subset J\subset T
  51. π I J : Ω J Ω I \pi_{I\leftarrow J}:\Omega_{J}\to\Omega_{I}
  52. ω ω | I \omega\mapsto\omega|_{I}
  53. F T F\subset T
  54. μ F \mu_{F}
  55. Ω F \Omega_{F}
  56. τ t \tau_{t}
  57. Ω F \Omega_{F}
  58. { μ F } \{\mu_{F}\}
  59. F G T F\subset G\subset T
  60. μ F = ( π F G ) * μ G \mu_{F}=(\pi_{F\leftarrow G})_{*}\mu_{G}
  61. ( π F G ) * μ G (\pi_{F\leftarrow G})_{*}\mu_{G}
  62. μ G \mu_{G}
  63. π F G \pi_{F\leftarrow G}
  64. μ \mu
  65. Ω T \Omega_{T}
  66. μ F = ( π F T ) * μ \mu_{F}=(\pi_{F\leftarrow T})_{*}\mu
  67. F T F\subset T
  68. μ F , μ \mu_{F},\mu
  69. μ \mu
  70. Ω t = n \Omega_{t}=\mathbb{R}^{n}
  71. t T t\in T
  72. μ { t 1 , , t k } = ν t 1 t k \mu_{\{t_{1},...,t_{k}\}}=\nu_{t_{1}\dots t_{k}}
  73. t 1 , , t k T t_{1},...,t_{k}\in T
  74. ( π t ) t T (\pi_{t})_{t\in T}
  75. Ω = ( n ) T \Omega=(\mathbb{R}^{n})^{T}
  76. P = μ P=\mu
  77. ν t 1 t k \nu_{t_{1}\dots t_{k}}
  78. p , q p,q

Komar_mass.html

  1. a r ^ = m r 2 1 - 2 m r c 2 a^{\hat{r}}=\frac{m}{r^{2}\sqrt{1-\frac{2m}{rc^{2}}}}
  2. 4 π m 1 - 2 m r c 2 \frac{4\pi m}{\sqrt{1-\frac{2m}{rc^{2}}}}
  3. g t t \sqrt{g_{tt}}
  4. d s 2 = - g t t d t 2 + quadratic form ( d x d y d z ) ds^{2}=-g_{tt}\,dt^{2}+\mathrm{quadratic\ form}(dx\,dy\,dz)\,
  5. a b = u u b = u c c u b a^{b}=\nabla_{u}u^{b}=u^{c}\nabla_{c}u^{b}
  6. a b = u c c u b a_{b}=u^{c}\nabla_{c}u_{b}
  7. a norm = N b a b a_{\mathrm{norm}}=N^{b}a_{b}\,
  8. N b a b = ( g t t r c 2 ) / ( 2 g t t g r r ) = m r 2 1 - 2 m r c 2 N^{b}a_{b}=\left(\frac{\partial g_{tt}}{\partial r}c^{2}\right)/\left(2g_{tt}% \sqrt{g_{rr}}\right)=\frac{m}{r^{2}\sqrt{1-\frac{2m}{rc^{2}}}}
  9. a inf = g t t a a\mathrm{inf}=\sqrt{g_{tt}}a\,
  10. N b a inf b = m / r 2 N^{b}a\mathrm{inf}_{b}=m/r^{2}\,
  11. a b = b Z 1 Z 1 = ln g t t a_{b}=\nabla_{b}Z_{1}\qquad Z_{1}=\ln{gtt}
  12. a inf b = b Z 2 Z 2 = g t t a\mathrm{inf}_{b}=\nabla_{b}Z_{2}\qquad Z_{2}=\sqrt{g_{tt}}
  13. m = - 1 4 π A N b a inf b d A m=-\frac{1}{4\pi}\int_{A}N^{b}a\mathrm{inf}_{b}dA
  14. - a inf -\nabla\cdot a\mathrm{inf}
  15. F a b = a inf a u b - a inf b u a F_{ab}=a\,\mathrm{inf}_{a}\,u_{b}-a\,\mathrm{inf}_{b}\,u_{a}\,
  16. a inf a = F a b u b a\mathrm{inf}_{a}=F_{ab}u^{b}\,
  17. a F a b u b \nabla^{a}F_{ab}u^{b}
  18. - u b a F a b = g t t R 00 u a u b = g t t R a b u a u b -u^{b}\nabla^{a}F_{ab}=\sqrt{g_{tt}}R_{00}u^{a}u^{b}=\sqrt{g_{tt}}R_{ab}u^{a}u% ^{b}
  19. m = g t t 4 π V R a b u a u b m=\frac{\sqrt{g_{tt}}}{4\pi}\int_{V}R_{ab}u^{a}u^{b}
  20. G u = v R u - v 1 2 R I u = v 8 π T u v G^{u}{}_{v}=R^{u}{}_{v}-\frac{1}{2}RI^{u}{}_{v}=8\pi T^{u}{}_{v}
  21. m = V g t t ( 2 T a b - T g a b ) u a u b d V m=\int_{V}\sqrt{g_{tt}}\left(2T_{ab}-Tg_{ab}\right)u^{a}u^{b}dV
  22. m = V ( 2 T a b - T g a b ) u a ξ b d V m=\int_{V}\left(2T_{ab}-Tg_{ab}\right)u^{a}\xi^{b}dV
  23. ξ b \xi^{b}
  24. ξ a ξ a = - 1 \xi^{a}\xi_{a}=-1
  25. ξ b \xi^{b}\,
  26. g t t u b \sqrt{g_{tt}}u^{b}\,
  27. g a b g_{ab}\,
  28. ξ a = ( 1 , 0 , 0 , 0 ) \xi^{a}=\left(1,0,0,0\right)
  29. ξ a \xi^{a}\,
  30. u a u^{a}\,
  31. ξ a = K u a \xi^{a}=Ku^{a}\,
  32. m = V ( 2 T 00 - T g 00 ) K d V m=\int_{V}\left(2T_{00}-Tg_{00}\right)KdV
  33. u a u^{a}
  34. ξ b \xi^{b}
  35. - ξ a ξ a \sqrt{-\xi^{a}\xi_{a}}
  36. ξ a \xi^{a}
  37. g t t \sqrt{g_{tt}}
  38. g a b = η a b g_{ab}=\eta_{ab}\,
  39. g 00 = - 1 , T = - T 00 + T 11 + T 22 + T 33 g_{00}=-1,T=-T_{00}+T_{11}+T_{22}+T_{33}\,
  40. m = V - ξ a ξ a ( T 00 + T 11 + T 22 + T 33 ) d V m=\int_{V}\sqrt{-\xi^{a}\xi_{a}}\left(T_{00}+T_{11}+T_{22}+T_{33}\right)dV
  41. K = g t t = - ξ a ξ a K=\sqrt{g_{tt}}=\sqrt{-\xi^{a}\xi_{a}}
  42. m = - 1 8 π S ϵ a b c d c ξ d m=-\frac{1}{8\pi}\int_{S}\epsilon_{abcd}\nabla^{c}\xi^{d}
  43. ϵ a b c d \epsilon_{abcd}\,
  44. ξ d \xi^{d}
  45. ξ a ξ a = - 1 \xi^{a}\xi_{a}=-1
  46. g a b g_{ab}\,
  47. ξ a = ( 1 , 0 , 0 , 0 ) \xi^{a}=\left(1,0,0,0\right)

Kopp's_law.html

  1. C = i = 1 N ( C i f i ) C=\sum_{i=1}^{N}(C_{i}\cdot f_{i})

Koszul_algebra.html

  1. R R
  2. k k
  3. k k
  4. R ( - i ) b i R ( - 2 ) b 2 R ( - 1 ) b 1 R k 0. \cdots\rightarrow R(-i)^{b_{i}}\rightarrow\cdots\rightarrow R(-2)^{b_{2}}% \rightarrow R(-1)^{b_{1}}\rightarrow R\rightarrow k\rightarrow 0.
  5. R = k [ x , y ] / ( x y ) R=k[x,y]/(xy)

König's_theorem_(graph_theory).html

  1. G = ( V , E ) G=(V,E)
  2. V V
  3. L L
  4. R R
  5. M M
  6. G G
  7. M M
  8. M M
  9. | M | |M|
  10. U U
  11. L L
  12. Z Z
  13. U U
  14. U U
  15. K = ( L Z ) ( R Z ) . K=(L\setminus Z)\cup(R\cap Z).
  16. e e
  17. E E
  18. K K
  19. K K
  20. e e
  21. e e
  22. L Z L\setminus Z
  23. e e
  24. e e
  25. K K
  26. K K
  27. L Z L\setminus Z
  28. R Z R\cap Z
  29. K K
  30. K K
  31. M M

Kramers_theorem.html

  1. T : t - t . T:t\mapsto-t.
  2. [ H , T ] = 0 , [H,T]=0,
  3. | n |n\rangle
  4. T | n T|n\rangle

Kramers–Heisenberg_formula.html

  1. d 2 σ d Ω k d ( ω k ) = ω k ω k | f | | n f | T | n n | T | i E i - E n + ω k + i Γ n 2 | 2 δ ( E i - E f + ω k - ω k ) \frac{d^{2}\sigma}{d\Omega_{k^{\prime}}d(\hbar\omega_{k}^{\prime})}=\frac{% \omega_{k}^{\prime}}{\omega_{k}}\sum_{|f\rangle}\left|\sum_{|n\rangle}\frac{% \langle f|T^{\dagger}|n\rangle\langle n|T|i\rangle}{E_{i}-E_{n}+\hbar\omega_{k% }+i\frac{\Gamma_{n}}{2}}\right|^{2}\delta(E_{i}-E_{f}+\hbar\omega_{k}-\hbar% \omega_{k}^{\prime})
  2. ω k \hbar\omega_{k}^{\prime}
  3. d Ω k d\Omega_{k^{\prime}}
  4. k k^{\prime}
  5. ω k \hbar\omega_{k}
  6. | i , | n , | f |i\rangle,|n\rangle,|f\rangle
  7. E i , E n , E f E_{i},E_{n},E_{f}
  8. T T
  9. Γ n \Gamma_{n}

Kronecker_limit_formula.html

  1. E ( τ , s ) = π s - 1 + 2 π ( γ - log ( 2 ) - log ( y | η ( τ ) | 2 ) ) + O ( s - 1 ) , E(\tau,s)={\pi\over s-1}+2\pi(\gamma-\log(2)-\log(\sqrt{y}|\eta(\tau)|^{2}))+O% (s-1),
  2. E ( τ , s ) = ( m , n ) ( 0 , 0 ) y s | m τ + n | 2 s E(\tau,s)=\sum_{(m,n)\neq(0,0)}{y^{s}\over|m\tau+n|^{2s}}
  3. η ( τ ) = q 1 / 24 n 1 ( 1 - q n ) \eta(\tau)=q^{1/24}\prod_{n\geq 1}(1-q^{n})
  4. E u , v ( τ , 1 ) = - 2 π log | f ( u - v τ ; τ ) q v 2 / 2 | , E_{u,v}(\tau,1)=-2\pi\log|f(u-v\tau;\tau)q^{v^{2}/2}|,
  5. E u , v ( τ , s ) = ( m , n ) ( 0 , 0 ) e 2 π i ( m u + n v ) y s | m τ + n | 2 s E_{u,v}(\tau,s)=\sum_{(m,n)\neq(0,0)}e^{2\pi i(mu+nv)}{y^{s}\over|m\tau+n|^{2s}}
  6. f ( z , τ ) = q 1 / 12 ( p 1 / 2 - p - 1 / 2 ) n 1 ( 1 - q n p ) ( 1 - q n / p ) . f(z,\tau)=q^{1/12}(p^{1/2}-p^{-1/2})\prod_{n\geq 1}(1-q^{n}p)(1-q^{n}/p).

Krylov–Bogolyubov_theorem.html

  1. μ ( F - 1 ( A ) ) = μ ( A ) . \mu\left(F^{-1}(A)\right)=\mu(A).
  2. F * ( μ ) = μ . F_{*}(\mu)=\mu.
  3. P t , t 0 , P_{t},t\geq 0,
  4. Pr [ X t A | X 0 = x ] = P t ( x , A ) . \Pr[X_{t}\in A|X_{0}=x]=P_{t}(x,A).
  5. x X x\in X
  6. ( P t ) ( μ ) = μ for all t > 0. (P_{t})_{\ast}(\mu)=\mu\mbox{ for all }~{}t>0.

KSE_BRIndex30.html

  1. Index Level = ( P i Q i ) D i v i s o r × 5000 \,\text{Index Level}={\sum\left({P_{i}}\cdot{Q_{i}}\right)\over Divisor}{% \times 5000}

Kummer_sum.html

  1. Σ χ ( r ) e ( r / p ) = G ( χ ) \Sigma\chi(r)e(r/p)=G(\chi)
  2. K ( n , p ) = x = 1 p e ( n x 3 / p ) K(n,p)=\sum_{x=1}^{p}e(nx^{3}/p)

Kurepa_tree.html

  1. 2 \aleph_{2}
  2. ω 2 \omega_{2}
  3. 2 1 2^{\aleph_{1}}
  4. 1 \aleph_{1}
  5. 1 \aleph_{1}

L-lactate_dehydrogenase_(cytochrome).html

  1. \rightleftharpoons

L_pad.html

  1. R b = Z 2 1 - Z 2 / Z 1 R_{b}=\frac{Z_{2}}{\sqrt{1-Z_{2}/Z_{1}}}\,
  2. R a = Z 1 Z 2 R b R_{a}=\frac{Z_{1}Z_{2}}{R_{b}}\,
  3. L o s s = 10 log P p a d + P l o a d P l o a d = - 20 log 2 Z 1 / Z 2 1 + ( R a + Z 1 ) ( R b + Z 2 ) / ( R b Z 2 ) Loss=10\log{\frac{P_{pad}+P_{load}}{P_{load}}}=-20\log{\frac{2\sqrt{Z_{1}/Z_{2% }}}{1+(R_{a}+Z_{1})(R_{b}+Z_{2})/(R_{b}Z_{2})}}\,
  4. P l o a d P_{load}\,
  5. P p a d P_{pad}\,

Lagrangian_and_Eulerian_specification_of_the_flow_field.html

  1. 𝐮 ( 𝐱 ( t ) , t ) \mathbf{u}\left(\mathbf{x}(t),t\right)
  2. 𝐔 ( 𝐱 0 , t ) \mathbf{U}\left(\mathbf{x}_{0},t\right)
  3. 𝐮 ( 𝐔 ( 𝐱 0 , t ) , t ) = 𝐔 t ( 𝐱 0 , t ) \mathbf{u}\left(\mathbf{U}(\mathbf{x}_{0},t),t\right)=\frac{\partial\mathbf{U}% }{\partial t}\left(\mathbf{x}_{0},t\right)
  4. D 𝐅 D t = 𝐅 t + ( 𝐮 ) 𝐅 \frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t}=\frac{\partial\mathbf{F}}{\partial t}% +(\mathbf{u}\cdot\nabla)\mathbf{F}

Lagrangian_relaxation.html

  1. x n x\in\mathbb{R}^{n}
  2. A m , n A\in\mathbb{R}^{m,n}
  3. c T x c^{T}x
  4. A x b Ax\leq b
  5. A A
  6. A 1 m 1 , n A_{1}\in\mathbb{R}^{m_{1},n}
  7. A 2 m 2 , n A_{2}\in\mathbb{R}^{m_{2},n}
  8. m 1 + m 2 = m m_{1}+m_{2}=m
  9. c T x c^{T}x
  10. A 1 x b 1 A_{1}x\leq b_{1}
  11. A 2 x b 2 A_{2}x\leq b_{2}
  12. c T x + λ T ( b 2 - A 2 x ) c^{T}x+\lambda^{T}(b_{2}-A_{2}x)
  13. A 1 x b 1 A_{1}x\leq b_{1}
  14. λ = ( λ 1 , , λ m 2 ) \lambda=(\lambda_{1},\ldots,\lambda_{m_{2}})
  15. λ ~ \tilde{\lambda}
  16. x ^ \hat{x}
  17. x ¯ \bar{x}
  18. c T x ^ c T x ^ + λ ~ T ( b 2 - A 2 x ^ ) c T x ¯ + λ ~ T ( b 2 - A 2 x ¯ ) c^{T}\hat{x}\leq c^{T}\hat{x}+\tilde{\lambda}^{T}(b_{2}-A_{2}\hat{x})\leq c^{T% }\bar{x}+\tilde{\lambda}^{T}(b_{2}-A_{2}\bar{x})
  19. x ^ \hat{x}
  20. x ¯ \bar{x}
  21. P ( λ ) P(\lambda)
  22. λ 0 \lambda\geq 0
  23. P ( λ ) P(\lambda)
  24. c T x + λ T ( b 2 - A 2 x ) c^{T}x+\lambda^{T}(b_{2}-A_{2}x)
  25. A 1 x b 1 A_{1}x\leq b_{1}
  26. λ \lambda
  27. P P
  28. P P
  29. x ¯ \bar{x}
  30. P P
  31. λ \lambda

Lambda_baryon.html

  1. 1 / 2 {1}/{2}

Lambert_(unit).html

  1. 1 π \frac{1}{\pi}
  2. 10 4 π \frac{10^{4}}{\pi}

Lamplighter_group.html

  1. - / 2 , \bigoplus_{-\infty}^{\infty}\mathbb{Z}/2\mathbb{Z},
  2. a , t a 2 , [ t m a t - m , t n a t - n ] , m , n \langle a,t\mid a^{2},[t^{m}at^{-m},t^{n}at^{-n}],m,n\in\mathbb{Z}\rangle
  3. a , t ( a t n a t - n ) 2 , n \langle a,t\mid(at^{n}at^{-n})^{2},n\in\mathbb{Z}\rangle

Landau's_problems.html

  1. 2 n = p + q 2n=p+q
  2. 2 p 2\sqrt{p}
  3. x 1 / 6 x^{1/6}
  4. 2 p \sqrt{2p}
  5. x p n 2 x p n + 1 - p n > x 1 / 2 p n + 1 - p n x 2 / 3 . \sum_{\stackrel{p_{n+1}-p_{n}>x^{1/2}}{x\leq p_{n}\leq 2x}}p_{n+1}-p_{n}\ll x^% {2/3}.
  6. n 3 n^{3}
  7. ( n + 1 ) 3 (n+1)^{3}
  8. x 2 + y 4 x^{2}+y^{4}
  9. n 2 + 1 n^{2}+1

Landau_quantization.html

  1. q q
  2. S S
  3. x - y x-y
  4. 𝐁 = ( 0 0 B ) \mathbf{B}=\begin{pmatrix}0\\ 0\\ B\end{pmatrix}
  5. z z
  6. H ^ = 1 2 m ( 𝐩 ^ - q 𝐀 ^ / c ) 2 . \hat{H}=\frac{1}{2m}(\hat{\mathbf{p}}-q\hat{\mathbf{A}}/c)^{2}.
  7. 𝐁 = × 𝐀 ^ . \mathbf{B}=\mathbf{\nabla}\times\hat{\mathbf{A}}.\,
  8. 𝐀 ^ = ( 0 B x 0 ) . \hat{\mathbf{A}}=\begin{pmatrix}0\\ Bx\\ 0\end{pmatrix}.
  9. B B
  10. x x
  11. H ^ = p ^ x 2 2 m + 1 2 m ( p ^ y - q B x ^ c ) 2 . \hat{H}=\frac{\hat{p}_{x}^{2}}{2m}+\frac{1}{2m}\left(\hat{p}_{y}-\frac{qB\hat{% x}}{c}\right)^{2}.
  12. p ^ y \hat{p}_{y}
  13. p ^ y \hat{p}_{y}
  14. H ^ = p ^ x 2 2 m + 1 2 m ω c 2 ( x ^ - k y m ω c ) 2 . \hat{H}=\frac{\hat{p}_{x}^{2}}{2m}+\frac{1}{2}m\omega_{c}^{2}\left(\hat{x}-% \frac{\hbar k_{y}}{m\omega_{c}}\right)^{2}.
  15. E n = ω c ( n + 1 2 ) , n 0 . E_{n}=\hbar\omega_{c}\left(n+\frac{1}{2}\right),\quad n\geq 0~{}.
  16. p ^ y \hat{p}_{y}
  17. y y
  18. | ϕ n |\phi_{n}\rangle
  19. x x
  20. x x
  21. Ψ ( x , y ) = e i k y y ϕ n ( x - x 0 ) . \Psi(x,y)=e^{ik_{y}y}\phi_{n}(x-x_{0})~{}.
  22. n n
  23. n n
  24. k y = 2 π N L y k_{y}=\frac{2\pi N}{L_{y}}
  25. N N
  26. N N
  27. N N
  28. 0 N < m ω c L x L y 2 π . 0\leq N<\frac{m\omega_{c}L_{x}L_{y}}{2\pi\hbar}.
  29. q = Z e q=Ze
  30. N N
  31. Z B L x L y ( h c / e ) = Z Φ Φ 0 , \frac{ZBL_{x}L_{y}}{(hc/e)}=Z\frac{\Phi}{\Phi_{0}},
  32. Φ = B A Φ=BA
  33. S S
  34. D D
  35. D = Z ( 2 S + 1 ) Φ Φ 0 . D=Z(2S+1)\frac{\Phi}{\Phi_{0}}~{}.
  36. x x
  37. Z Z
  38. S S
  39. D D
  40. x x
  41. x x
  42. y y
  43. z z
  44. z z
  45. E E
  46. x x
  47. y y
  48. 𝐀 ^ = 1 2 ( - B y B x 0 ) \hat{\mathbf{A}}=\frac{1}{2}\begin{pmatrix}-By\\ Bx\\ 0\end{pmatrix}
  49. H ^ = 1 2 [ ( - i x - y 2 ) 2 + ( - i y + x 2 ) 2 ] \hat{H}=\frac{1}{2}\left[\left(-i\frac{\partial}{\partial x}-\frac{y}{2}\right% )^{2}+\left(-i\frac{\partial}{\partial y}+\frac{x}{2}\right)^{2}\right]
  50. q , c , , 𝐁 q,c,\hbar,\mathbf{B}
  51. m m
  52. a ^ = 1 2 [ ( x 2 + x ) - i ( y 2 + y ) ] \hat{a}=\frac{1}{\sqrt{2}}\left[\left(\frac{x}{2}+\frac{\partial}{\partial x}% \right)-i\left(\frac{y}{2}+\frac{\partial}{\partial y}\right)\right]
  53. a ^ = 1 2 [ ( x 2 - x ) + i ( y 2 - y ) ] \hat{a}^{\dagger}=\frac{1}{\sqrt{2}}\left[\left(\frac{x}{2}-\frac{\partial}{% \partial x}\right)+i\left(\frac{y}{2}-\frac{\partial}{\partial y}\right)\right]
  54. b ^ = 1 2 [ ( x 2 + x ) + i ( y 2 + y ) ] \hat{b}=\frac{1}{\sqrt{2}}\left[\left(\frac{x}{2}+\frac{\partial}{\partial x}% \right)+i\left(\frac{y}{2}+\frac{\partial}{\partial y}\right)\right]
  55. b ^ = 1 2 [ ( x 2 - x ) - i ( y 2 - y ) ] \hat{b}^{\dagger}=\frac{1}{\sqrt{2}}\left[\left(\frac{x}{2}-\frac{\partial}{% \partial x}\right)-i\left(\frac{y}{2}-\frac{\partial}{\partial y}\right)\right]
  56. [ a ^ , a ^ ] = [ b ^ , b ^ ] = 1 [\hat{a},\hat{a}^{\dagger}]=[\hat{b},\hat{b}^{\dagger}]=1
  57. H ^ = a ^ a ^ + 1 2 \hat{H}=\hat{a}^{\dagger}\hat{a}+\frac{1}{2}
  58. n n
  59. a ^ a ^ \hat{a}^{\dagger}\hat{a}
  60. L ^ z = - i θ = - ( b ^ b ^ - a ^ a ^ ) \hat{L}_{z}=-i\hbar\frac{\partial}{\partial\theta}=-\hbar(\hat{b}^{\dagger}% \hat{b}-\hat{a}^{\dagger}\hat{a})
  61. [ H ^ , L ^ z ] = 0 [\hat{H},\hat{L}_{z}]=0
  62. H ^ \hat{H}
  63. L ^ z \hat{L}_{z}
  64. L ^ z \hat{L}_{z}
  65. - m -m\hbar
  66. m - n m\geq-n
  67. n n
  68. b ^ \hat{b}^{\dagger}
  69. m m
  70. n n
  71. a ^ \hat{a}^{\dagger}
  72. n n
  73. m m
  74. H ^ | n , m = E n | n , m \hat{H}|n,m\rangle=E_{n}|n,m\rangle
  75. E n = ( n + 1 2 ) E_{n}=\left(n+\frac{1}{2}\right)
  76. | n , m = ( b ^ ) m + n ( m + n ) ! ( a ^ ) n n ! | 0 , 0 |n,m\rangle=\frac{(\hat{b}^{\dagger})^{m+n}}{\sqrt{(m+n)!}}\frac{(\hat{a}^{% \dagger})^{n}}{\sqrt{n!}}|0,0\rangle
  77. m m
  78. ψ n , m ( x , y ) = ( w - w ¯ 4 ) n w n + m e - | w | 2 / 4 \psi_{n,m}(x,y)=\left(\frac{\partial}{\partial w}-\frac{\bar{w}}{4}\right)^{n}% w^{n+m}e^{-|w|^{2}/4}
  79. w = x + i y w=x+iy
  80. n = 0 n=0
  81. ψ ( x , y ) = f ( w ) e - | w | 2 / 4 \psi(x,y)=f(w)e^{-|w|^{2}/4}
  82. A A = A + λ ( x ) \vec{A}\to\vec{A}^{\prime}=\vec{A}+\vec{\nabla}\lambda(\vec{x})
  83. π ^ = 𝐩 ^ - q 𝐀 ^ / c \hat{\pi}=\hat{\mathbf{p}}-q\hat{\mathbf{A}}/c
  84. 𝐩 ^ \hat{\mathbf{p}}
  85. π ^ \langle\hat{\pi}\rangle
  86. x ^ \langle\hat{x}\rangle
  87. 𝐩 ^ \langle\hat{\mathbf{p}}\rangle
  88. | α |\alpha\rangle
  89. | α |\alpha^{\prime}\rangle
  90. x ^ \langle\hat{x}\rangle
  91. π ^ \langle\hat{\pi}\rangle
  92. α | x ^ | α = α | x ^ | α \langle\alpha|\hat{x}|\alpha\rangle=\langle\alpha^{\prime}|\hat{x}|\alpha^{% \prime}\rangle
  93. α | π ^ | α = α | π ^ | α \langle\alpha|\hat{\pi}|\alpha\rangle=\langle\alpha^{\prime}|\hat{\pi^{\prime}% }|\alpha^{\prime}\rangle
  94. α | α = α | α \langle\alpha|\alpha\rangle=\langle\alpha^{\prime}|\alpha^{\prime}\rangle
  95. 𝒢 \mathcal{G}
  96. | α = 𝒢 | α |\alpha^{\prime}\rangle=\mathcal{G}|\alpha\rangle
  97. 𝒢 x ^ 𝒢 = x ^ \mathcal{G}^{\dagger}\hat{x}\mathcal{G}=\hat{x}
  98. 𝒢 ( p ^ - e A ^ c - e λ ( x ) c ) 𝒢 = p ^ - e A ^ c \mathcal{G}^{\dagger}\left(\hat{p}-\frac{e\hat{A}}{c}-\frac{e\vec{\nabla}% \lambda(x)}{c}\right)\mathcal{G}=\hat{p}-\frac{e\hat{A}}{c}
  99. 𝒢 𝒢 = 1 \mathcal{G}^{\dagger}\mathcal{G}=1
  100. 𝒢 = exp ( i e λ ( x ) c ) \mathcal{G}=\exp\left(\frac{ie\lambda(\vec{x})}{\hbar c}\right)

Langer_correction.html

  1. - 2 2 m d 2 R ( r ) d r 2 + [ E - V eff ( r ) ] R ( r ) = 0 -\frac{\hbar^{2}}{2m}\frac{d^{2}R(r)}{dr^{2}}+[E-V_{\textrm{eff}}(r)]R(r)=0
  2. V eff ( r ) = V ( r ) - ( + 1 ) 2 2 m r 2 V_{\textrm{eff}}(r)=V(r)-\frac{\ell(\ell+1)\hbar^{2}}{2mr^{2}}
  3. ( + 1 ) ( + 1 2 ) 2 \ell(\ell+1)\rightarrow\left(\ell+\frac{1}{2}\right)^{2}

Langevin_dynamics.html

  1. N N
  2. M M
  3. X = X ( t ) X=X(t)
  4. M X ¨ = - U ( X ) - γ M X ˙ + 2 γ k B T M R ( t ) , M\ddot{X}=-\nabla U(X)-\gamma M\dot{X}+\sqrt{2\gamma k_{B}TM}R(t)\,,
  5. U ( X ) U(X)
  6. \nabla
  7. - U ( X ) -\nabla U(X)
  8. X ˙ \dot{X}
  9. X ¨ \ddot{X}
  10. R ( t ) R(t)
  11. R ( t ) = 0 \left\langle R(t)\right\rangle=0
  12. R ( t ) R ( t ) = δ ( t - t ) \left\langle R(t)R(t^{\prime})\right\rangle=\delta(t-t^{\prime})
  13. δ \delta
  14. γ \gamma
  15. γ \gamma

Langley_extrapolation.html

  1. τ \tau
  2. I / I 0 = e - m τ , I/I_{0}=e^{-m\tau},\,
  3. τ \tau
  4. τ \tau
  5. τ \tau
  6. τ \tau
  7. V s V_{s}
  8. V d V_{d}
  9. τ \tau
  10. τ = ln I - ln I 0 m \tau=\frac{\ln I-\ln I_{0}}{m}
  11. I 0 I_{0}

Langton's_loops.html

  1. A 121 \left\lfloor\frac{A}{121}\right\rfloor

Latimer–MacDuffee_theorem.html

  1. f f
  2. n n
  3. \mathbb{Z}
  4. n × n n\times n
  5. f f
  6. [ x ] / ( f ( x ) ) . \mathbb{Z}[x]/(f(x)).\,
  7. f ( x ) f(x)

Lattice_Boltzmann_methods.html

  1. f i t ( x , t + δ t ) = f i ( x , t ) + 1 τ f ( f i e q - f i ) f_{i}^{t}(\vec{x},t+\delta_{t})=f_{i}(\vec{x},t)+\frac{1}{\tau_{f}}(f_{i}^{eq}% -f_{i})\,\!
  2. f i ( x + e i δ t , t + δ t ) = f i t ( x , t + δ t ) f_{i}(\vec{x}+\vec{e}_{i}\delta_{t},t+\delta_{t})=f_{i}^{t}(\vec{x},t+\delta_{% t})\,\!
  3. δ x \delta_{x}\,\!
  4. L L\,\!
  5. N N\,\!
  6. δ x = L / N \delta_{x}=L/N\,\!
  7. δ t = δ x C s \delta_{t}=\frac{\delta_{x}}{C_{s}}\,\!
  8. C s C_{s}
  9. 2 n d 2^{nd}\,\!
  10. 1 s t 1^{st}\,\!
  11. 2 n d 2^{nd}\,\!
  12. f i ( x + e i δ t , t + δ t ) = f i ( x , t ) + 1 τ f ( f i e q - f i ) f_{i}(\vec{x}+\vec{e}_{i}\delta_{t},t+\delta_{t})=f_{i}(\vec{x},t)+\frac{1}{% \tau_{f}}(f_{i}^{eq}-f_{i})
  13. f i ( x , t ) f_{i}(\vec{x},t)
  14. f i f_{i}\,\!
  15. f i t + e i f i + ( 1 2 e i e i : f i + e i f i t + 1 2 2 f i t 2 ) = 1 τ ( f i e q - f i ) \frac{\partial f_{i}}{\partial t}+\vec{e}_{i}\cdot\nabla f_{i}+\left(\frac{1}{% 2}\vec{e}_{i}\vec{e}_{i}:\nabla\nabla f_{i}+\vec{e}_{i}\cdot\nabla\frac{% \partial f_{i}}{\partial t}+\frac{1}{2}\frac{\partial^{2}f_{i}}{\partial t^{2}% }\right)=\frac{1}{\tau}(f_{i}^{eq}-f_{i})
  16. K K\,\!
  17. f i = f i e q + K f i n e q f_{i}=f_{i}^{eq}+Kf_{i}^{neq}\,\!
  18. f i n e q = f i ( 1 ) + K f i ( 2 ) + O ( K 2 ) f_{i}^{neq}=f_{i}^{(1)}+Kf_{i}^{(2)}+O(K^{2})
  19. ρ = i f i e q \rho=\sum_{i}f_{i}^{eq}
  20. ρ u = i f i e q e i \rho\vec{u}=\sum_{i}f_{i}^{eq}\vec{e}_{i}
  21. 0 = i f i ( k ) s . t . k = 1 , 2 0=\sum_{i}f_{i}^{(k)}\qquad s.t.\ k=1,2\,\!
  22. 0 = i f i ( k ) e i 0=\sum_{i}f_{i}^{(k)}\vec{e}_{i}
  23. t = K t 1 + K 2 t 2 s . t . t 2 ( diffusive time-scale ) t 1 ( convective time-scale ) \frac{\partial}{\partial t}=K\frac{\partial}{\partial t_{1}}+K^{2}\frac{% \partial}{\partial t_{2}}\qquad s.t.\ t_{2}(\,\text{diffusive time-scale})\ll t% _{1}(\,\text{convective time-scale})
  24. x = K x 1 \frac{\partial}{\partial x}=K\frac{\partial}{\partial x_{1}}
  25. K K\,\!
  26. K 0 K^{0}\,\!
  27. f i e q t 1 + e i 1 f i e q = - f i ( 1 ) τ \frac{\partial f_{i}^{eq}}{\partial t_{1}}+\vec{e}_{i}\nabla_{1}f_{i}^{eq}=-% \frac{f_{i}^{(1)}}{\tau}
  28. K 1 K^{1}\,\!
  29. f i ( 1 ) t 1 + f i e q t 2 + e i f i ( 1 ) + 1 2 e i e i : f i e q + e i f i e q t 1 + 1 2 2 f i e q t 1 2 = - f i ( 2 ) τ \frac{\partial f_{i}^{(1)}}{\partial t_{1}}+\frac{\partial f_{i}^{eq}}{% \partial t_{2}}+\vec{e}_{i}\nabla f_{i}^{(1)}+\frac{1}{2}\vec{e}_{i}\vec{e}_{i% }:\nabla\nabla f_{i}^{eq}+\vec{e}_{i}\cdot\nabla\frac{\partial f_{i}^{eq}}{% \partial t_{1}}+\frac{1}{2}\frac{\partial^{2}f_{i}^{eq}}{\partial t_{1}^{2}}=-% \frac{f_{i}^{(2)}}{\tau}
  30. f i ( e q ) t 2 + ( 1 - 1 2 τ ) [ f i ( 1 ) t 1 + e i 1 f i ( 1 ) ] = - f i ( 2 ) τ \frac{\partial f_{i}^{(eq)}}{\partial t_{2}}+\left(1-\frac{1}{2\tau}\right)% \left[\frac{\partial f_{i}^{(1)}}{\partial t_{1}}+\vec{e}_{i}\nabla_{1}f_{i}^{% (1)}\right]=-\frac{f_{i}^{(2)}}{\tau}
  31. ρ t + ρ u = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot\rho\vec{u}=0
  32. ρ u t + Π = 0 \frac{\partial\rho\vec{u}}{\partial t}+\nabla\cdot\Pi=0
  33. Π \Pi\,\!
  34. Π x y = i e i x e i y [ f i e q + ( 1 - 1 2 τ ) f i ( 1 ) ] \Pi_{xy}=\sum_{i}\vec{e}_{ix}\vec{e}_{iy}\left[f_{i}^{eq}+\left(1-\frac{1}{2% \tau}\right)f_{i}^{(1)}\right]\,\!
  35. e i x e i y \vec{e}_{ix}\vec{e}_{iy}
  36. e i \vec{e}_{i}
  37. ( x e i x ) 2 = x y e i x e i y \textstyle\left(\sum_{x}\vec{e}_{ix}\right)^{2}=\sum_{x}\sum_{y}\vec{e}_{ix}% \vec{e}_{iy}
  38. f i e q = ω i ρ ( 1 + e i . u c s 2 + ( e i . u ) 2 2 c s 4 - u 2 2 c s 2 ) f_{i}^{eq}=\omega_{i}\rho\left(1+\frac{\vec{e}_{i}.\vec{u}}{c_{s}^{2}}+\frac{(% \vec{e}_{i}.\vec{u})^{2}}{2c_{s}^{4}}-\frac{\vec{u}^{2}}{2c_{s}^{2}}\right)
  39. Π x y ( 0 ) = i e i x e i y f i e q = p δ x y + ρ u x u y \Pi_{xy}^{(0)}=\sum_{i}\vec{e}_{ix}\vec{e}_{iy}f_{i}^{eq}=p\delta_{xy}+\rho u_% {x}u_{y}\,\!
  40. Π x y ( 1 ) = ( 1 - 1 2 τ ) i e i x e i y f i ( 1 ) = ν ( x ( ρ u y ) + y ( ρ u x ) ) \Pi_{xy}^{(1)}=\left(1-\frac{1}{2\tau}\right)\sum_{i}\vec{e}_{ix}\vec{e}_{iy}f% _{i}^{(1)}=\nu\left(\nabla_{x}\left(\rho\vec{u}_{y}\right)+\nabla_{y}\left(% \rho\vec{u}_{x}\right)\right)\,\!
  41. ρ ( u x t + y u x u y ) = - x p + ν y ( x ( ρ u y ) + y ( ρ u x ) ) \rho\left(\frac{\partial\vec{u}_{x}}{\partial t}+\nabla_{y}\cdot\vec{u}_{x}% \vec{u}_{y}\right)=-\nabla_{x}p+\nu\nabla_{y}\cdot\left(\nabla_{x}\left(\rho% \vec{u}_{y}\right)+\nabla_{y}\left(\rho\vec{u}_{x}\right)\right)\,\!
  42. f ( x , e i , t ) f(\vec{x},\vec{e}_{i},t)\,\!
  43. g ( x , e i , t ) g(\vec{x},\vec{e}_{i},t)\,\!
  44. t f + ( e ) f + F v f = Ω ( f ) \partial_{t}f+(\vec{e}\cdot\nabla)f+F\partial_{v}f=\Omega(f)
  45. t g + ( e ) g + G v f = Ω ( g ) \partial_{t}g+(\vec{e}\cdot\nabla)g+G\partial_{v}f=\Omega(g)
  46. g ( x , e i , t ) g(\vec{x},\vec{e}_{i},t)\,\!
  47. f ( x , e i , t ) f(\vec{x},\vec{e}_{i},t)\,\!
  48. g ( x , e i , t ) = ( e - u ) 2 2 f ( x , e i , t ) g(\vec{x},\vec{e}_{i},t)=\frac{(\vec{e}-\vec{u})^{2}}{2}f(\vec{x},\vec{e}_{i},t)
  49. F F\,\!
  50. Ω \Omega\,\!
  51. e \vec{e}
  52. ξ \vec{\xi}
  53. F F\,\!
  54. G G\,\!
  55. G G\,\!
  56. F = G ( e - u ) R T f e q F=\frac{\vec{G}\cdot(\vec{e}-\vec{u})}{RT}f^{eq}
  57. G = β g 0 ( T - T a v g ) k \vec{G}=\beta g_{0}(T-T_{avg})\vec{k}
  58. ρ \rho\,\!
  59. u \vec{u}\,\!
  60. T T\,\!
  61. ρ = f d e \rho=\int\ f\,d\vec{e}
  62. ρ u = e f d e \rho\vec{u}=\int\ \vec{e}f\,d\vec{e}
  63. ρ D R T 2 = ρ ϵ = g d e \frac{\rho DRT}{2}=\rho\epsilon=\int\ g\,d\vec{e}
  64. e i = ( e i x , e i y ) \vec{e}_{i}=(\vec{e}_{ix},\vec{e}_{iy})
  65. e i = c × { ( 0 , 0 ) i = 0 ( 1 , 0 ) , ( 0 , 1 ) , ( - 1 , 0 ) , ( 0 , - 1 ) i = 1 , 2 , 3 , 4 ( 1 , 1 ) , ( - 1 , 1 ) , ( - 1 , - 1 ) , ( 1 , - 1 ) i = 5 , 6 , 7 , 8 \vec{e}_{i}=c\times\begin{cases}(0,0)&i=0\\ (1,0),(0,1),(-1,0),(0,-1)&i=1,2,3,4\\ (1,1),(-1,1),(-1,-1),(1,-1)&i=5,6,7,8\\ \end{cases}
  66. e i = c × { ( 0 , 0 , 0 ) i = 0 ( \plusmn 1 , 0 , 0 ) , ( 0 , \plusmn 1 , 0 ) , ( 0 , 0 , \plusmn 1 ) i = 1 , 2 , , 5 , 6 ( \plusmn 1 , \plusmn 1 , \plusmn 1 ) i = 7 , 8 , , 13 , 14 \vec{e}_{i}=c\times\begin{cases}(0,0,0)&i=0\\ (\plusmn 1,0,0),(0,\plusmn 1,0),(0,0,\plusmn 1)&i=1,2,...,5,6\\ (\plusmn 1,\plusmn 1,\plusmn 1)&i=7,8,...,13,14\\ \end{cases}
  67. e i = c × { ( 0 , 0 , 0 ) i = 0 ( \plusmn 1 , 0 , 0 ) , ( 0 , \plusmn 1 , 0 ) , ( 0 , 0 , \plusmn 1 ) i = 1 , 2 , , 5 , 6 ( \plusmn 1 , \plusmn 1 , 0 ) , ( \plusmn 1 , 0 , \plusmn 1 ) , ( 0 , \plusmn 1 , \plusmn 1 ) i = 7 , 8 , , 17 , 18 \vec{e}_{i}=c\times\begin{cases}(0,0,0)&i=0\\ (\plusmn 1,0,0),(0,\plusmn 1,0),(0,0,\plusmn 1)&i=1,2,...,5,6\\ (\plusmn 1,\plusmn 1,0),(\plusmn 1,0,\plusmn 1),(0,\plusmn 1,\plusmn 1)&i=7,8,% ...,17,18\\ \end{cases}
  68. f i ( x + e i δ t , t + δ t ) - f i ( x , t ) + F i = Ω ( f ) f_{i}(\vec{x}+\vec{e}_{i}\delta_{t},t+\delta_{t})-f_{i}(\vec{x},t)+F_{i}=% \Omega(f)
  69. g i ( x + e i δ t , t + δ t ) - g i ( x , t ) + G i = Ω ( g ) g_{i}(\vec{x}+\vec{e}_{i}\delta_{t},t+\delta_{t})-g_{i}(\vec{x},t)+G_{i}=% \Omega(g)
  70. Ω ( f ) = 1 τ f ( f i e q - f i ) \Omega(f)=\frac{1}{\tau_{f}}(f_{i}^{eq}-f_{i})
  71. Ω ( g ) = 1 τ g ( g i e q - g i ) \Omega(g)=\frac{1}{\tau_{g}}(g_{i}^{eq}-g_{i})
  72. f i e q f_{i}^{eq}
  73. f e q = ρ ( 2 π R T ) D / 2 e - ( e - u ) 2 2 R T f^{eq}=\frac{\rho}{(2\pi RT)^{D/2}}e^{-\frac{(\vec{e}-\vec{u})^{2}}{2RT}}
  74. = ρ ( 2 π R T ) D / 2 e - ( e ) 2 2 R T e e u R T - u 2 2 R T =\frac{\rho}{(2\pi RT)^{D/2}}e^{-\frac{(\vec{e})^{2}}{2RT}}e^{\frac{\vec{e}% \vec{u}}{RT}-\frac{\vec{u}^{2}}{2RT}}
  75. = ρ ( 2 π R T ) D / 2 e - ( e ) 2 2 R T ( 1 + e u R T + ( e u ) 2 2 ( R T ) 2 - u 2 2 R T + ) =\frac{\rho}{(2\pi RT)^{D/2}}e^{-\frac{(\vec{e})^{2}}{2RT}}(1+\frac{\vec{e}% \vec{u}}{RT}+\frac{(\vec{e}\vec{u})^{2}}{2(RT)^{2}}-\frac{\vec{u}^{2}}{2RT}+...)
  76. c = 3 R T c=\sqrt{3RT}\,\!
  77. f i e q = ω i ρ ( 1 + 3 e i u c 2 + 9 ( e i u ) 2 2 c 4 - 3 ( u ) 2 2 c 2 ) f_{i}^{eq}=\omega_{i}\rho\left(1+\frac{3\vec{e}_{i}\vec{u}}{c^{2}}+\frac{9(% \vec{e}_{i}\vec{u})^{2}}{2c^{4}}-\frac{3(\vec{u})^{2}}{2c^{2}}\right)
  78. g e q = ρ ( e - u ) 2 2 ( 2 π R T ) D / 2 e - ( e - u ) 2 2 R T g^{eq}=\frac{\rho(\vec{e}-\vec{u})^{2}}{2(2\pi RT)^{D/2}}e^{-\frac{(\vec{e}-% \vec{u})^{2}}{2RT}}
  79. ω i = { 4 / 9 i = 0 1 / 9 i = 1 , 2 , 3 , 4 1 / 36 i = 5 , 6 , 7 , 8 \omega_{i}=\begin{cases}4/9&i=0\\ 1/9&i=1,2,3,4\\ 1/36&i=5,6,7,8\\ \end{cases}
  80. ω i = { 1 / 3 i = 0 1 / 18 i = 1 , 2 , , 5 , 6 1 / 36 i = 7 , 8 , , 17 , 18 \omega_{i}=\begin{cases}1/3&i=0\\ 1/18&i=1,2,...,5,6\\ 1/36&i=7,8,...,17,18\\ \end{cases}
  81. Ψ \Psi\,\!
  82. σ j \sigma_{j}\,\!
  83. f i σ ( x + e i δ t , t + δ t ) - f i σ ( x , t ) + F i = 1 τ f σ ( f i σ , e q ( ρ σ , v σ ) - f i σ ) f_{i}^{\sigma}(\vec{x}+\vec{e}_{i}\delta_{t},t+\delta_{t})-f_{i}^{\sigma}(\vec% {x},t)+F_{i}=\frac{1}{\tau_{f}^{\sigma}}(f_{i}^{\sigma,eq}(\rho^{\sigma},v^{% \sigma})-f_{i}^{\sigma})
  84. τ f σ j \tau_{f}^{\sigma_{j}}\,\!
  85. ν f σ j \nu_{f}^{\sigma_{j}}\,\!
  86. ν f σ j = ( τ f σ j - 0.5 ) c s 2 δ t \nu_{f}^{\sigma_{j}}=(\tau_{f}^{\sigma_{j}}-0.5)c_{s}^{2}\delta_{t}
  87. f i f_{i}\,\!
  88. ρ = σ i f i \rho=\sum_{\sigma}\sum_{i}f_{i}\,\!
  89. ρ ϵ = i g i \rho\epsilon=\sum_{i}g_{i}\,\!
  90. ρ σ = i f i σ \rho^{\sigma}=\sum_{i}f_{i}^{\sigma}\,\!
  91. u \vec{u^{\prime}}\,\!
  92. u = ( σ ρ σ u σ τ f σ ) / ( σ ρ σ τ f σ ) \vec{u^{\prime}}=\left(\sum_{\sigma}\frac{\rho^{\sigma}\vec{u^{\sigma}}}{\tau_% {f}^{\sigma}}\right)/\left(\sum_{\sigma}\frac{\rho^{\sigma}}{\tau_{f}^{\sigma}% }\right)
  93. ρ σ u σ = i f i σ e i . \rho^{\sigma}\vec{u^{\sigma}}=\sum_{i}f_{i}^{\sigma}\vec{e}_{i}.
  94. v σ = u + τ f σ ρ σ F σ v^{\sigma}=\vec{u^{\prime}}+\frac{\tau_{f}^{\sigma}}{\rho^{\sigma}}\vec{F}^{\sigma}
  95. v σ v^{\sigma}\,\!
  96. F σ \vec{F}^{\sigma}\,\!
  97. F σ \vec{F}^{\sigma}\,\!
  98. F σ = - ψ σ ( x ) σ j H σ σ j ( x , x ) i ψ σ j ( x + e i ) e i \vec{F}^{\sigma}=-\psi^{\sigma}(\vec{x})\sum_{\sigma_{j}}H^{\sigma\sigma_{j}}(% \vec{x},\vec{x}^{\prime})\sum_{i}\psi^{\sigma_{j}}(\vec{x}+\vec{e}_{i})\vec{e}% _{i}\,\!
  99. ψ ( x ) \psi(\vec{x})\,\!
  100. H ( x , x ) H(\vec{x},\vec{x}^{\prime})\,\!
  101. x \vec{x}^{\prime}\,\!
  102. H ( x , x ) = H ( x , x ) H(\vec{x},\vec{x}^{\prime})=H(\vec{x}^{\prime},\vec{x})\,\!
  103. H ( x , x ) > 0 H(\vec{x},\vec{x}^{\prime})>0\,\!
  104. H σ σ j ( x , x ) = { h σ σ j | x - x | c 0 | x - x | > c H^{\sigma\sigma_{j}}(\vec{x},\vec{x}^{\prime})=\begin{cases}h^{\sigma\sigma_{j% }}&\left|\vec{x}-\vec{x}^{\prime}\right|\leq c\\ 0&\left|\vec{x}-\vec{x}^{\prime}\right|>c\\ \end{cases}
  105. H σ σ j ( x , x ) = { h σ σ j | x - x | = c h σ σ j / 2 | x - x | = 2 c 0 otherwise H^{\sigma\sigma_{j}}(\vec{x},\vec{x}^{\prime})=\begin{cases}h^{\sigma\sigma_{j% }}&\left|\vec{x}-\vec{x}^{\prime}\right|=c\\ h^{\sigma\sigma_{j}}/2&\left|\vec{x}-\vec{x}^{\prime}\right|=\sqrt{2c}\\ 0&\,\text{otherwise}\\ \end{cases}
  106. ψ ( x ) = ψ ( ρ σ ) = ρ 0 σ [ 1 - e ( - ρ σ / ρ 0 σ ) ] \psi(\vec{x})=\psi(\rho^{\sigma})=\rho_{0}^{\sigma}\left[1-e^{(-\rho^{\sigma}/% \rho_{0}^{\sigma})}\right]\,\!
  107. p = c s 2 ρ + c 0 h [ ψ ( x ) ] 2 p=c_{s}^{2}\rho+c_{0}h[\psi(\vec{x})]^{2}\,\!
  108. ρ 0 σ \rho_{0}^{\sigma}\,\!
  109. h σ σ j h^{\sigma\sigma_{j}}\,\!
  110. ( P / ρ ) T = ( 2 P / ρ 2 ) T = 0 (\partial P/\partial{\rho})_{T}=(\partial^{2}P/\partial{\rho^{2}})_{T}=0\,\!
  111. p = p c p=p_{c}\,\!
  112. c 0 c_{0}\,\!
  113. ρ θ + ρ u u = i f i e i e i . \rho\theta+\rho uu=\sum_{i}f_{i}\vec{e}_{i}\vec{e}_{i}.

Law_(stochastic_processes).html

  1. X t : Ω S : ω X ( t , ω ) X_{t}:\Omega\to S:\omega\mapsto X(t,\omega)
  2. ( Φ X ( ω ) ) ( t ) := X t ( ω ) . \left(\Phi_{X}(\omega)\right)(t):=X_{t}(\omega).
  3. X := ( Φ X ) * ( 𝐏 ) = 𝐏 Φ X - 1 \mathcal{L}_{X}:=\left(\Phi_{X}\right)_{*}(\mathbf{P})=\mathbf{P}\circ\Phi_{X}% ^{-1}

Lax_pair.html

  1. L ( t ) , P ( t ) L(t),P(t)
  2. d L d t = [ P , L ] \frac{dL}{dt}=[P,L]
  3. [ P , L ] = P L - L P [P,L]=PL-LP
  4. P P
  5. L L
  6. L L
  7. t t
  8. t t
  9. L ( t ) L(t)
  10. L ( t ) = U ( t , s ) L ( s ) U ( t , s ) - 1 L(t)=U(t,s)L(s)U(t,s)^{-1}
  11. U ( t , s ) U(t,s)
  12. d d t U ( t , s ) = P ( t ) U ( t , s ) , U ( s , s ) = I , \frac{d}{dt}U(t,s)=P(t)U(t,s),\qquad U(s,s)=I,
  13. λ ( t ) = λ ( 0 ) \lambda(t)=\lambda(0)
  14. ψ t = P ψ . \frac{\partial\psi}{\partial t}=P\psi.
  15. x \|x\|\to\infty
  16. L ( 0 ) L(0)
  17. λ \lambda
  18. ψ ( 0 , x ) \psi(0,x)
  19. P P
  20. ψ \psi
  21. ψ t ( t , x ) = P ψ ( t , x ) \frac{\partial\psi}{\partial t}(t,x)=P\psi(t,x)
  22. ψ ( 0 , x ) \psi(0,x)
  23. ψ \psi
  24. L ( t ) L(t)
  25. u ( t , x ) u(t,x)
  26. u t = 6 u u x - u x x x . u_{t}=6uu_{x}-u_{xxx}.\,
  27. L t = [ P , L ] L_{t}=[P,L]\,
  28. L = - x 2 + u L=-\partial_{x}^{2}+u\,
  29. P = - 4 x 3 + 3 ( u x + x u ) P=-4\partial_{x}^{3}+3(u\partial_{x}+\partial_{x}u)\,

LCS35.html

  1. w = 2 2 t ( mod n ) w=2^{2^{t}}\;\;(\mathop{{\rm mod}}n)

Lebesgue_differentiation_theorem.html

  1. f 𝟏 A f\cdot\mathbf{1}_{A}
  2. 𝟏 A \mathbf{1}_{A}
  3. A f d λ , \int_{A}f\ \mathrm{d}\lambda,
  4. lim B x 1 | B | B f d λ , \lim_{B\rightarrow x}\frac{1}{|B|}\int_{B}f\,\mathrm{d}\lambda,
  5. | 1 | B | B f ( y ) d λ ( y ) - f ( x ) | = | 1 | B | B f ( y ) - f ( x ) d λ ( y ) | 1 | B | B | f ( y ) - f ( x ) | d λ ( y ) . \left|\frac{1}{|B|}\int_{B}f(y)\,\mathrm{d}\lambda(y)-f(x)\right|=\left|\frac{% 1}{|B|}\int_{B}f(y)-f(x)\,\mathrm{d}\lambda(y)\right|\leq\frac{1}{|B|}\int_{B}% |f(y)-f(x)|\,\mathrm{d}\lambda(y).
  6. 𝒱 \mathcal{V}
  7. | U | c | B | |U|\geq c\,|B|
  8. 𝒱 \mathcal{V}
  9. f ( x ) = lim U x , U 𝒱 1 | U | U f d λ . f(x)=\lim_{U\rightarrow x,\,U\in\mathcal{V}}\frac{1}{|U|}\int_{U}f\,\mathrm{d}\lambda.
  10. 𝒱 \mathcal{V}
  11. 𝒱 \mathcal{V}
  12. F ( x ) = - x f ( t ) d t F(x)=\int_{-\infty}^{x}f(t)\,\mathrm{d}t
  13. F ( x ) = f ( x ) . F^{\prime}(x)=f(x).
  14. E α = { x 𝐑 n : lim sup | B | 0 , x B 1 | B | B | f ( y ) - f ( x ) | d y > 2 α } E_{\alpha}=\Bigl\{x\in\mathbf{R}^{n}:\limsup_{|B|\rightarrow 0,\,x\in B}\frac{% 1}{|B|}\int_{B}|f(y)-f(x)|\,\mathrm{d}y>2\alpha\Bigr\}
  15. f - g L 1 = 𝐑 n | f ( x ) - g ( x ) | d x < ε . \|f-g\|_{L^{1}}=\int_{\mathbf{R}^{n}}|f(x)-g(x)|\,\mathrm{d}x<\varepsilon.
  16. 1 | B | B f ( y ) d y - f ( x ) = ( 1 | B | B ( f ( y ) - g ( y ) ) d y ) + ( 1 | B | B g ( y ) d y - g ( x ) ) + ( g ( x ) - f ( x ) ) . \frac{1}{|B|}\int_{B}f(y)\,\mathrm{d}y-f(x)=\Bigl(\frac{1}{|B|}\int_{B}\bigl(f% (y)-g(y)\bigr)\,\mathrm{d}y\Bigr)+\Bigl(\frac{1}{|B|}\int_{B}g(y)\,\mathrm{d}y% -g(x)\Bigr)+\bigl(g(x)-f(x)\bigr).
  17. ( f - g ) * ( x ) (f-g)^{*}(x)
  18. 1 | B | B | f ( y ) - g ( y ) | d y sup r > 0 1 | B r ( x ) | B r ( x ) | f ( y ) - g ( y ) | d y = ( f - g ) * ( x ) . \frac{1}{|B|}\int_{B}|f(y)-g(y)|\,\mathrm{d}y\leq\sup_{r>0}\frac{1}{|B_{r}(x)|% }\int_{B_{r}(x)}|f(y)-g(y)|\,\mathrm{d}y=(f-g)^{*}(x).
  19. | { x : ( f - g ) * ( x ) > α } | A n α f - g L 1 < A n α ε , \Bigl|\left\{x:(f-g)^{*}(x)>\alpha\right\}\Bigr|\leq\frac{A_{n}}{\alpha}\,\|f-% g\|_{L^{1}}<\frac{A_{n}}{\alpha}\,\varepsilon,
  20. | { x : | f ( x ) - g ( x ) | > α } | 1 α f - g L 1 < 1 α ε \Bigl|\left\{x:|f(x)-g(x)|>\alpha\right\}\Bigr|\leq\frac{1}{\alpha}\,\|f-g\|_{% L^{1}}<\frac{1}{\alpha}\,\varepsilon
  21. | E α | A n + 1 α ε . |E_{\alpha}|\leq\frac{A_{n}+1}{\alpha}\,\varepsilon.

Lebesgue_spine.html

  1. \R n \R^{n}
  2. n 3 , n\geq 3,
  3. S = { ( x 1 , x 2 , , x n ) \R n : x n > 0 , x 1 2 + x 2 2 + + x n - 1 2 exp ( - 1 / x n 2 ) } . S=\{(x_{1},x_{2},\dots,x_{n})\in\R^{n}:x_{n}>0,x_{1}^{2}+x_{2}^{2}+\cdots+x_{n% -1}^{2}\leq\exp(-1/x_{n}^{2})\}.
  4. \R n \R^{n}
  5. S S
  6. S S
  7. \R n \R^{n}
  8. \R 2 \R^{2}

LED_circuit.html

  1. resistance ( R ) = power supply voltage ( V s ) - LED voltage drop ( V f ) LED current ( I ) , \mbox{resistance}~{}(R)=\frac{\mbox{power supply voltage}~{}(V_{s})-\mbox{LED % voltage drop}~{}(V_{f})}{\mbox{LED current}~{}(I)},
  2. V f I Ω {V_{f}\over I}\;\Omega
  3. V s = V r + V f = R I + V f V_{s}=V_{r}+V_{f}=RI+V_{f}
  4. R I = V s - V f RI=V_{s}-V_{f}\;
  5. R = V s - V f I R={V_{s}-V_{f}\over I}

Leeway.html

  1. L α L_{\alpha}

Leftover_hash_lemma.html

  1. X \scriptstyle X
  2. n \scriptstyle n
  3. t < n \scriptstyle t\;<\;n
  4. n - t \scriptstyle n\,-\,t
  5. n - t \scriptstyle n\,-\,t
  6. H ( X ) \scriptstyle H_{\infty}(X)
  7. X \scriptstyle X
  8. X \scriptstyle X
  9. X \scriptstyle X
  10. X \scriptstyle X
  11. 𝒳 \scriptstyle\mathcal{X}
  12. m > 0 \scriptstyle m\;>\;0
  13. h : 𝒮 × 𝒳 { 0 , 1 } m \scriptstyle h:\;\mathcal{S}\,\times\,\mathcal{X}\;\rightarrow\;\{0,\,1\}^{m}
  14. m H ( X ) - 2 log ( 1 ε ) m\leq H_{\infty}(X)-2\log\left(\frac{1}{\varepsilon}\right)
  15. S \scriptstyle S
  16. 𝒮 \scriptstyle\mathcal{S}
  17. X \scriptstyle X
  18. δ [ ( h ( S , X ) , S ) , ( U , S ) ] ε \delta[(h(S,X),S),(U,S)]\leq\varepsilon
  19. U \scriptstyle U
  20. { 0 , 1 } m \scriptstyle\{0,\,1\}^{m}
  21. S \scriptstyle S
  22. H ( X ) = - log max x Pr [ X = x ] \scriptstyle H_{\infty}(X)\;=\;-\log\max_{x}\Pr[X=x]
  23. X \scriptstyle X
  24. X \scriptstyle X
  25. max x Pr [ X = x ] \scriptstyle\max_{x}\Pr[X=x]
  26. X \scriptstyle X
  27. X \scriptstyle X
  28. δ ( X , Y ) = 1 2 v | Pr [ X = v ] - Pr [ Y = v ] | \scriptstyle\delta(X,\,Y)\;=\;\frac{1}{2}\sum_{v}\left|\Pr[X=v]\,-\,\Pr[Y=v]\right|
  29. X \scriptstyle X
  30. Y \scriptstyle Y

Legendre_rational_functions.html

  1. R n ( x ) = 2 x + 1 L n ( x - 1 x + 1 ) R_{n}(x)=\frac{\sqrt{2}}{x+1}\,L_{n}\left(\frac{x-1}{x+1}\right)
  2. L n ( x ) L_{n}(x)
  3. ( x + 1 ) x ( x x ( ( x + 1 ) v ( x ) ) ) + λ v ( x ) = 0 (x+1)\partial_{x}(x\partial_{x}((x+1)v(x)))+\lambda v(x)=0
  4. λ n = n ( n + 1 ) \lambda_{n}=n(n+1)\,
  5. R n + 1 ( x ) = 2 n + 1 n + 1 x - 1 x + 1 R n ( x ) - n n + 1 R n - 1 ( x ) for n 1 R_{n+1}(x)=\frac{2n+1}{n+1}\,\frac{x-1}{x+1}\,R_{n}(x)-\frac{n}{n+1}\,R_{n-1}(% x)\quad\mathrm{for\,n\geq 1}
  6. 2 ( 2 n + 1 ) R n ( x ) = ( x + 1 ) 2 ( x R n + 1 ( x ) - x R n - 1 ( x ) ) + ( x + 1 ) ( R n + 1 ( x ) - R n - 1 ( x ) ) 2(2n+1)R_{n}(x)=(x+1)^{2}(\partial_{x}R_{n+1}(x)-\partial_{x}R_{n-1}(x))+(x+1)% (R_{n+1}(x)-R_{n-1}(x))
  7. lim x ( x + 1 ) R n ( x ) = 2 \lim_{x\rightarrow\infty}(x+1)R_{n}(x)=\sqrt{2}
  8. lim x x x ( ( x + 1 ) R n ( x ) ) = 0 \lim_{x\rightarrow\infty}x\partial_{x}((x+1)R_{n}(x))=0
  9. 0 R m ( x ) R n ( x ) d x = 2 2 n + 1 δ n m \int_{0}^{\infty}R_{m}(x)\,R_{n}(x)\,dx=\frac{2}{2n+1}\delta_{nm}
  10. δ n m \delta_{nm}
  11. R 0 ( x ) = 1 R_{0}(x)=1\,
  12. R 1 ( x ) = x - 1 x + 1 R_{1}(x)=\frac{x-1}{x+1}\,
  13. R 2 ( x ) = x 2 - 4 x + 1 ( x + 1 ) 2 R_{2}(x)=\frac{x^{2}-4x+1}{(x+1)^{2}}\,
  14. R 3 ( x ) = x 3 - 9 x 2 + 9 x - 1 ( x + 1 ) 3 R_{3}(x)=\frac{x^{3}-9x^{2}+9x-1}{(x+1)^{3}}\,
  15. R 4 ( x ) = x 4 - 16 x 3 + 36 x 2 - 16 x + 1 ( x + 1 ) 4 R_{4}(x)=\frac{x^{4}-16x^{3}+36x^{2}-16x+1}{(x+1)^{4}}\,

Lehmer_matrix.html

  1. A i j = { i / j , j i j / i , j < i . A_{ij}=\begin{cases}i/j,&j\geq i\\ j/i,&j<i.\end{cases}
  2. A i j = min ( i , j ) max ( i , j ) . A_{ij}=\frac{\mbox{min}~{}(i,j)}{\mbox{max}~{}(i,j)}.
  3. A 2 = ( 1 1 / 2 1 / 2 1 ) ; A 2 - 1 = ( 4 / 3 - 2 / 3 - 2 / 3 \color B r o w n 𝟒 / 𝟑 ) ; A 3 = ( 1 1 / 2 1 / 3 1 / 2 1 2 / 3 1 / 3 2 / 3 1 ) ; A 3 - 1 = ( 4 / 3 - 2 / 3 - 2 / 3 32 / 15 - 6 / 5 - 6 / 5 \color B r o w n 𝟗 / 𝟓 ) ; A 4 = ( 1 1 / 2 1 / 3 1 / 4 1 / 2 1 2 / 3 1 / 2 1 / 3 2 / 3 1 3 / 4 1 / 4 1 / 2 3 / 4 1 ) ; A 4 - 1 = ( 4 / 3 - 2 / 3 - 2 / 3 32 / 15 - 6 / 5 - 6 / 5 108 / 35 - 12 / 7 - 12 / 7 \color B r o w n 𝟏𝟔 / 𝟕 ) . \begin{array}[]{lllll}A_{2}=\begin{pmatrix}1&1/2\\ 1/2&1\end{pmatrix};&A_{2}^{-1}=\begin{pmatrix}4/3&-2/3\\ -2/3&{\color{Brown}{\mathbf{4/3}}}\end{pmatrix};\\ \\ A_{3}=\begin{pmatrix}1&1/2&1/3\\ 1/2&1&2/3\\ 1/3&2/3&1\end{pmatrix};&A_{3}^{-1}=\begin{pmatrix}4/3&-2/3&\\ -2/3&32/15&-6/5\\ &-6/5&{\color{Brown}{\mathbf{9/5}}}\end{pmatrix};\\ \\ A_{4}=\begin{pmatrix}1&1/2&1/3&1/4\\ 1/2&1&2/3&1/2\\ 1/3&2/3&1&3/4\\ 1/4&1/2&3/4&1\end{pmatrix};&A_{4}^{-1}=\begin{pmatrix}4/3&-2/3&&\\ -2/3&32/15&-6/5&\\ &-6/5&108/35&-12/7\\ &&-12/7&{\color{Brown}{\mathbf{16/7}}}\end{pmatrix}.\\ \end{array}

Lehmer_mean.html

  1. x x
  2. L p ( x ) = k = 1 n x k p k = 1 n x k p - 1 . L_{p}(x)=\frac{\sum_{k=1}^{n}x_{k}^{p}}{\sum_{k=1}^{n}x_{k}^{p-1}}.
  3. w w
  4. L p , w ( x ) = k = 1 n w k x k p k = 1 n w k x k p - 1 . L_{p,w}(x)=\frac{\sum_{k=1}^{n}w_{k}\cdot x_{k}^{p}}{\sum_{k=1}^{n}w_{k}\cdot x% _{k}^{p-1}}.
  5. p L p ( x ) p\mapsto L_{p}(x)
  6. p L p ( x ) = j = 1 n k = j + 1 n ( x j - x k ) ( ln x j - ln x k ) ( x j x k ) p - 1 ( k = 1 n x k p - 1 ) 2 , \frac{\partial}{\partial p}L_{p}(x)=\frac{\sum_{j=1}^{n}\sum_{k=j+1}^{n}(x_{j}% -x_{k})\cdot(\ln x_{j}-\ln x_{k})\cdot(x_{j}\cdot x_{k})^{p-1}}{\left(\sum_{k=% 1}^{n}x_{k}^{p-1}\right)^{2}},
  7. p q L p ( x ) L q ( x ) p\leq q\Rightarrow L_{p}(x)\leq L_{q}(x)
  8. lim p - L p ( x ) \lim_{p\to-\infty}L_{p}(x)
  9. x x
  10. L 0 ( x ) L_{0}(x)
  11. L 1 2 ( ( x 0 , x 1 ) ) L_{\frac{1}{2}}\left((x_{0},x_{1})\right)
  12. x 0 x_{0}
  13. x 1 x_{1}
  14. L 1 ( x ) L_{1}(x)
  15. L 2 ( x ) L_{2}(x)
  16. lim p L p ( x ) \lim_{p\to\infty}L_{p}(x)
  17. x x
  18. x 1 , , x k x_{1},\dots,x_{k}
  19. L p ( x ) = x 1 k + ( x k + 1 x 1 ) p + + ( x n x 1 ) p k + ( x k + 1 x 1 ) p - 1 + + ( x n x 1 ) p - 1 L_{p}(x)=x_{1}\cdot\frac{k+\left(\frac{x_{k+1}}{x_{1}}\right)^{p}+\cdots+\left% (\frac{x_{n}}{x_{1}}\right)^{p}}{k+\left(\frac{x_{k+1}}{x_{1}}\right)^{p-1}+% \cdots+\left(\frac{x_{n}}{x_{1}}\right)^{p-1}}
  20. p p
  21. p p
  22. p p
  23. p p

Leibniz_algebra.html

  1. [ [ a , b ] , c ] = [ a , [ b , c ] ] + [ [ a , c ] , b ] . [[a,b],c]=[a,[b,c]]+[[a,c],b].\,
  2. [ a 1 a n , x ] = a 1 a n x for a 1 , , a n , x V . [a_{1}\otimes\cdots\otimes a_{n},x]=a_{1}\otimes\cdots a_{n}\otimes x\quad\,% \text{for }a_{1},\ldots,a_{n},x\in V.
  3. ( a b ) c = a ( b c ) + a ( c b ) . (a\circ b)\circ c=a\circ(b\circ c)+a\circ(c\circ b).

Leibniz_formula_for_determinants.html

  1. A = ( a i j ) i , j = 1 , , n A=(a_{ij})_{i,j=1,\dots,n}
  2. det ( A ) = σ S n sgn ( σ ) i = 1 n a σ ( i ) , i \det(A)=\sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}a_{% \sigma(i),i}
  3. det ( A ) = ϵ i 1 i n a 1 i 1 a n i n , \det(A)=\epsilon^{i_{1}\cdots i_{n}}{a}_{1i_{1}}\cdots{a}_{ni_{n}},
  4. Ω ( n ! n ) \Omega(n!\cdot n)
  5. A = L U A=LU
  6. det A = ( det L ) ( det U ) \det A=(\det L)(\det U)
  7. F : M n ( 𝕂 ) 𝕂 F:M_{n}(\mathbb{K})\rightarrow\mathbb{K}
  8. F ( I ) = 1 F(I)=1
  9. F F
  10. A = ( a i j ) i = 1 , , n j = 1 , , n A=(a_{i}^{j})_{i=1,\dots,n}^{j=1,\dots,n}
  11. n × n n\times n
  12. A j A^{j}
  13. j j
  14. A A
  15. A j = ( a i j ) i = 1 , , n A^{j}=(a_{i}^{j})_{i=1,\dots,n}
  16. A = ( A 1 , , A n ) . A=\left(A^{1},\dots,A^{n}\right).
  17. E k E^{k}
  18. k k
  19. A j A^{j}
  20. E k E^{k}
  21. A j = k = 1 n a k j E k A^{j}=\sum_{k=1}^{n}a_{k}^{j}E^{k}
  22. F F
  23. F ( A ) = F ( k 1 = 1 n a k 1 1 E k 1 , , k n = 1 n a k n n E k n ) = k 1 , , k n = 1 n ( i = 1 n a k i i ) F ( E k 1 , , E k n ) . \begin{aligned}\displaystyle F(A)&\displaystyle=F\left(\sum_{k_{1}=1}^{n}a_{k_% {1}}^{1}E^{k_{1}},\dots,\sum_{k_{n}=1}^{n}a_{k_{n}}^{n}E^{k_{n}}\right)\\ &\displaystyle=\sum_{k_{1},\dots,k_{n}=1}^{n}\left(\prod_{i=1}^{n}a_{k_{i}}^{i% }\right)F\left(E^{k_{1}},\dots,E^{k_{n}}\right).\end{aligned}
  24. F ( A ) = σ S n ( i = 1 n a σ ( i ) i ) F ( E σ ( 1 ) , , E σ ( n ) ) . F(A)=\sum_{\sigma\in S_{n}}\left(\prod_{i=1}^{n}a_{\sigma(i)}^{i}\right)F(E^{% \sigma(1)},\dots,E^{\sigma(n)}).
  25. E E
  26. sgn ( σ ) \operatorname{sgn}(\sigma)
  27. F ( A ) \displaystyle F(A)
  28. F ( I ) F(I)
  29. 1 1
  30. F ( I ) = 1 F\left(I\right)=1
  31. F ( A 1 , , c A j , ) \displaystyle F(A^{1},\dots,cA^{j},\dots)
  32. F ( , A j 1 , , A j 2 , ) \displaystyle F(\dots,A^{j_{1}},\dots,A^{j_{2}},\dots)
  33. σ S n \sigma\in S_{n}
  34. σ \sigma^{\prime}
  35. σ \sigma
  36. j 1 j_{1}
  37. j 2 j_{2}
  38. F ( A ) = σ S n , σ ( j 1 ) < σ ( j 2 ) [ sgn ( σ ) ( i = 1 , i j 1 , i j 2 n a σ ( i ) i ) a σ ( j 1 ) j 1 a σ ( j 2 ) j 2 + sgn ( σ ) ( i = 1 , i j 1 , i j 2 n a σ ( i ) i ) a σ ( j 1 ) j 1 a σ ( j 2 ) j 2 ] = σ S n , σ ( j 1 ) < σ ( j 2 ) [ sgn ( σ ) ( i = 1 , i j 1 , i j 2 n a σ ( i ) i ) a σ ( j 1 ) j 1 a σ ( j 2 ) j 2 - sgn ( σ ) ( i = 1 , i j 1 , i j 2 n a σ ( i ) i ) a σ ( j 2 ) j 1 a σ ( j 1 ) j 2 ] = σ S n , σ ( j 1 ) < σ ( j 2 ) sgn ( σ ) ( i = 1 , i j 1 , i j 2 n a σ ( i ) i ) ( a σ ( j 1 ) j 1 a σ ( j 2 ) j 2 - a σ ( j 1 ) j 2 a σ ( j 2 ) j 1 ) \begin{aligned}\displaystyle F(A)&\displaystyle=\sum_{\sigma\in S_{n},\sigma(j% _{1})<\sigma(j_{2})}\left[\operatorname{sgn}(\sigma)\left(\prod_{i=1,i\neq j_{% 1},i\neq j_{2}}^{n}a_{\sigma(i)}^{i}\right)a_{\sigma(j_{1})}^{j_{1}}a_{\sigma(% j_{2})}^{j_{2}}+\operatorname{sgn}(\sigma^{\prime})\left(\prod_{i=1,i\neq j_{1% },i\neq j_{2}}^{n}a_{\sigma^{\prime}(i)}^{i}\right)a_{\sigma^{\prime}(j_{1})}^% {j_{1}}a_{\sigma^{\prime}(j_{2})}^{j_{2}}\right]\\ &\displaystyle=\sum_{\sigma\in S_{n},\sigma(j_{1})<\sigma(j_{2})}\left[% \operatorname{sgn}(\sigma)\left(\prod_{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{% \sigma(i)}^{i}\right)a_{\sigma(j_{1})}^{j_{1}}a_{\sigma(j_{2})}^{j_{2}}-% \operatorname{sgn}(\sigma)\left(\prod_{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{% \sigma(i)}^{i}\right)a_{\sigma(j_{2})}^{j_{1}}a_{\sigma(j_{1})}^{j_{2}}\right]% \\ &\displaystyle=\sum_{\sigma\in S_{n},\sigma(j_{1})<\sigma(j_{2})}\operatorname% {sgn}(\sigma)\left(\prod_{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{\sigma(i)}^{i}% \right)\left(a_{\sigma(j_{1})}^{j_{1}}a_{\sigma(j_{2})}^{j_{2}}-a_{\sigma(j_{1% })}^{j_{2}}a_{\sigma(j_{2})}^{j_{{}_{1}}}\right)\\ \\ \end{aligned}
  39. A j 1 = A j 2 A^{j_{1}}=A^{j_{2}}
  40. F ( , A j 1 , , A j 2 , ) = 0 F(\dots,A^{j_{1}},\dots,A^{j_{2}},\dots)=0
  41. F ( I ) = 1 F(I)=1
  42. F ( I ) \displaystyle F(I)
  43. F ( I ) = 1 F(I)=1
  44. det : M n ( 𝕂 ) 𝕂 \det:M_{n}(\mathbb{K})\rightarrow\mathbb{K}

Lemniscate.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  2. ( x 2 + y 2 ) 2 - c x 2 - d y 2 (x^{2}+y^{2})^{2}-cx^{2}-dy^{2}
  3. ( x 2 + y 2 ) 2 = 2 a 2 ( x 2 - y 2 ) (x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})
  4. d = - c d=-c
  5. y 2 = x 2 ( a 2 - x 2 ) y^{2}=x^{2}(a^{2}-x^{2})
  6. y 2 ( y 2 - a 2 ) = x 2 ( x 2 - b 2 ) y^{2}(y^{2}-a^{2})=x^{2}(x^{2}-b^{2})
  7. ( x 2 + y 2 ) ( x 2 + y 2 - d 2 ) 2 + 4 a 2 y 2 ( x 2 + y 2 - b 2 ) = 0 (x^{2}+y^{2})(x^{2}+y^{2}-d^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0

Lemniscate_of_Gerono.html

  1. \infty
  2. x 4 - x 2 + y 2 = 0. x^{4}-x^{2}+y^{2}=0.
  3. x = t 2 - 1 t 2 + 1 , y = 2 t ( t 2 - 1 ) ( t 2 + 1 ) 2 . x=\frac{t^{2}-1}{t^{2}+1},\ y=\frac{2t(t^{2}-1)}{(t^{2}+1)^{2}}.
  4. x = cos φ , y = sin φ cos φ = sin ( 2 φ ) / 2 x=\cos\varphi,\ y=\sin\varphi\,\cos\varphi=\sin(2\varphi)/2
  5. ( x 2 - y 2 ) 3 + 8 y 4 + 20 x 2 y 2 - x 4 - 16 y 2 = 0. (x^{2}-y^{2})^{3}+8y^{4}+20x^{2}y^{2}-x^{4}-16y^{2}=0.

Lenoir_cycle.html

  1. Q 2 1 = m c v ( T 2 - T 1 ) {}_{1}Q_{2}=mc_{v}\left({T_{2}-T_{1}}\right)
  2. W 2 1 = 1 2 p d V = 0 {}_{1}W_{2}=\int\limits_{1}^{2}{pdV}=0
  3. c v = R < m t p l > γ - 1 c_{v}=\frac{R}{<}mtpl>{{\gamma-1}}
  4. p 2 V 2 = R T 2 p_{2}V_{2}=RT_{2}
  5. < m t p l > T 2 T 3 = ( p 2 p 3 ) γ - 1 γ = ( V 3 V 2 ) γ - 1 \frac{<}{m}tpl>{{T_{2}}}{{T_{3}}}=\left({\frac{{p_{2}}}{{p_{3}}}}\right)^{{% \textstyle{{\gamma-1}\over\gamma}}}=\left({\frac{{V_{3}}}{{V_{2}}}}\right)^{% \gamma-1}
  6. p 3 = p 1 p_{3}=p_{1}
  7. W 3 2 = m c v ( T 2 - T 3 ) {}_{2}W_{3}=mc_{v}\left({T_{2}-T_{3}}\right)
  8. Q 3 2 = 0 {}_{2}Q_{3}=0
  9. Q 1 3 - 3 W 1 = U 1 - U 3 {}_{3}Q_{1}-_{3}W_{1}=U_{1}-U_{3}
  10. W 1 3 = 3 1 p d V = p 1 ( V 1 - V 3 ) {}_{3}W_{1}=\int\limits_{3}^{1}{pdV}=p_{1}\left({V_{1}-V_{3}}\right)
  11. Q 1 3 = ( U 1 + p 1 V 1 ) - ( U 3 + p 3 V 3 ) = H 1 - H 3 {}_{3}Q_{1}=\left({U_{1}+p_{1}V_{1}}\right)-\left({U_{3}+p_{3}V_{3}}\right)=H_% {1}-H_{3}
  12. Q 1 3 = m c p ( T 1 - T 3 ) {}_{3}Q_{1}=mc_{p}\left({T_{1}-T_{3}}\right)
  13. c p = < m t p l > γ R γ - 1 c_{p}=\frac{<}{m}tpl>{{\gamma R}}{{\gamma-1}}
  14. η t h = W 3 2 + W 1 3 Q 2 1 \eta_{th}=\frac{{{}_{2}W_{3}+{}_{3}W_{1}}}{{{}_{1}Q_{2}}}

Leopold_Gegenbauer.html

  1. P a , r ( n ) := d | n ; d 1 / r d a = : n a ρ - a , r ( n ) P_{a,r}(n):=\sum_{d|n;d^{1/r}\in\mathbb{N}}d^{a}=:n^{a}\rho_{-a,r}(n)

Lerche–Newberger_sum_rule.html

  1. γ ( 0 , 1 ] \scriptstyle\gamma\in(0,1]
  2. n = - ( - 1 ) n J α - γ n ( z ) J β + γ n ( z ) n + μ = π sin μ π J α + γ μ ( z ) J β - γ μ ( z ) . \sum_{n=-\infty}^{\infty}\frac{(-1)^{n}J_{\alpha-\gamma n}(z)J_{\beta+\gamma n% }(z)}{n+\mu}=\frac{\pi}{\sin\mu\pi}J_{\alpha+\gamma\mu}(z)J_{\beta-\gamma\mu}(% z).

Level_set_(data_structures).html

  1. O ( n d ) O(n^{d})
  2. n n
  3. d d
  4. O ( n 2 ) O(n^{2})
  5. O ( n 3 ) O(n^{3})
  6. O ( n 3 ) . O(n^{3}).
  7. O ( n 3 ) O(n^{3})
  8. O ( n 2 ) O(n^{2})
  9. O ( n 3 ) O(n^{3})
  10. n 3 n^{3}
  11. m 3 m^{3}
  12. ( n / m ) 3 (n/m)^{3}
  13. O ( ( n m ) 3 + m 3 n 2 ) O\left((nm)3+m^{3}n^{2}\right)
  14. O ( n 2 ) , O(n^{2}),
  15. O ( log n ) . O(\log\,n).
  16. O ( log r ) O(\log r)
  17. O ( 1 ) O(1)

Levinson's_inequality.html

  1. a > 0 a>0
  2. f f
  3. ( 0 , 2 a ) (0,2a)
  4. f ′′′ ( x ) 0 f^{\prime\prime\prime}(x)\geq 0
  5. x ( 0 , 2 a ) x\in(0,2a)
  6. 0 < x i a 0<x_{i}\leq a
  7. i = 1 , , n i=1,\ldots,n
  8. 0 < p 0<p
  9. i = 1 n p i f ( x i ) i = 1 n p i - f ( i = 1 n p i x i i = 1 n p i ) i = 1 n p i f ( 2 a - x i ) i = 1 n p i - f ( i = 1 n p i ( 2 a - x i ) i = 1 n p i ) . \frac{\sum_{i=1}^{n}p_{i}f(x_{i})}{\sum_{i=1}^{n}p_{i}}-f\left(\frac{\sum_{i=1% }^{n}p_{i}x_{i}}{\sum_{i=1}^{n}p_{i}}\right)\leq\frac{\sum_{i=1}^{n}p_{i}f(2a-% x_{i})}{\sum_{i=1}^{n}p_{i}}-f\left(\frac{\sum_{i=1}^{n}p_{i}(2a-x_{i})}{\sum_% {i=1}^{n}p_{i}}\right).
  10. p i = 1 , a = 1 2 , p_{i}=1,\ a=\frac{1}{2},
  11. f ( x ) = log x . f(x)=\log x.\,

Lewy's_example.html

  1. F ( t , z ) F(t,z)
  2. u z ¯ - i z u t = F ( t , z ) \frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t}=F(t,z)
  3. F F
  4. F F
  5. u ( t , z ) u(t,z)
  6. u z ¯ - i z u t = φ ( t ) \frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t}=\varphi^{% \prime}(t)
  7. u x + i x u y = F ( x , y ) \frac{\partial u}{\partial x}+ix\frac{\partial u}{\partial y}=F(x,y)
  8. ¯ b \scriptstyle\bar{\partial}_{b}
  9. ¯ b \scriptstyle\bar{\partial}_{b}

Lévy's_modulus_of_continuity_theorem.html

  1. B : [ 0 , 1 ] × Ω B:[0,1]\times\Omega\to\mathbb{R}
  2. lim h 0 sup 0 t 1 - h | B t + h - B t | 2 h log ( 1 / h ) = 1. \lim_{h\to 0}\sup_{0\leq t\leq 1-h}\frac{|B_{t+h}-B_{t}|}{\sqrt{2h\log(1/h)}}=1.
  3. ω B ( δ ) = 2 δ log ( 1 / δ ) \omega_{B}(\delta)=\sqrt{2\delta\log(1/\delta)}
  4. δ > 0 \delta>0

Lévy_metric.html

  1. F , G : [ 0 , 1 ] F,G:\mathbb{R}\to[0,1]
  2. L ( F , G ) := inf { ε > 0 | F ( x - ε ) - ε G ( x ) F ( x + ε ) + ε for all x } . L(F,G):=\inf\{\varepsilon>0|F(x-\varepsilon)-\varepsilon\leq G(x)\leq F(x+% \varepsilon)+\varepsilon\mathrm{\,for\,all\,}x\in\mathbb{R}\}.

Lévy–Prokhorov_metric.html

  1. ( M , d ) (M,d)
  2. ( M ) \mathcal{B}(M)
  3. 𝒫 ( M ) \mathcal{P}(M)
  4. ( M , ( M ) ) (M,\mathcal{B}(M))
  5. A M A\subseteq M
  6. A A
  7. A ε := { p M | q A , d ( p , q ) < ε } = p A B ε ( p ) . A^{\varepsilon}:=\{p\in M~{}|~{}\exists q\in A,\ d(p,q)<\varepsilon\}=\bigcup_% {p\in A}B_{\varepsilon}(p).
  8. B ε ( p ) B_{\varepsilon}(p)
  9. ε \varepsilon
  10. p p
  11. π : 𝒫 ( M ) 2 [ 0 , + ) \pi:\mathcal{P}(M)^{2}\to[0,+\infty)
  12. μ \mu
  13. ν \nu
  14. π ( μ , ν ) := inf { ε > 0 | μ ( A ) ν ( A ε ) + ε and ν ( A ) μ ( A ε ) + ε for all A ( M ) } . \pi(\mu,\nu):=\inf\left\{\varepsilon>0~{}|~{}\mu(A)\leq\nu(A^{\varepsilon})+% \varepsilon\ \,\text{and}\ \nu(A)\leq\mu(A^{\varepsilon})+\varepsilon\ \,\text% {for all}\ A\in\mathcal{B}(M)\right\}.
  15. π ( μ , ν ) 1 \pi(\mu,\nu)\leq 1
  16. A A
  17. ( A ¯ ) ε = A ε (\bar{A})^{\varepsilon}=A^{\varepsilon}
  18. M M
  19. ( M , d ) (M,d)
  20. π \pi
  21. ( 𝒫 ( M ) , π ) \left(\mathcal{P}(M),\pi\right)
  22. ( M , d ) (M,d)
  23. ( 𝒫 ( M ) , π ) \left(\mathcal{P}(M),\pi\right)
  24. ( M , d ) (M,d)
  25. 𝒫 ( M ) \mathcal{P}(M)
  26. ( M , d ) (M,d)
  27. ( 𝒫 ( M ) , π ) \left(\mathcal{P}(M),\pi\right)
  28. ( M , d ) (M,d)
  29. 𝒦 𝒫 ( M ) \mathcal{K}\subseteq\mathcal{P}(M)
  30. π \pi
  31. π \pi

LIBOR_market_model.html

  1. n n
  2. L j L_{j}
  3. j = 1 , , n j=1,\ldots,n
  4. T j T_{j}
  5. Q T j + 1 Q_{T_{j+1}}
  6. d L j ( t ) = σ j ( t ) L j ( t ) d W Q T j + 1 ( t ) . dL_{j}(t)=\sigma_{j}(t)L_{j}(t)dW^{Q_{T_{j+1}}}(t)\,\text{.}
  7. L j L_{j}
  8. [ T j , T j + 1 ] [T_{j},T_{j+1}]
  9. T T
  10. d W Q T j ( t ) = { d W Q T p ( t ) - k = j + 1 p δ L k ( t ) 1 + δ L k ( t ) σ k ( t ) d t j < p d W Q T p ( t ) j = p d W Q T p ( t ) + k = p j - 1 δ L k ( t ) 1 + δ L k ( t ) σ k ( t ) d t j > p dW^{Q_{T_{j}}}(t)=\begin{cases}dW^{Q_{T_{p}}}(t)-\sum\limits_{k=j+1}^{p}\frac{% \delta L_{k}(t)}{1+\delta L_{k}(t)}{\sigma}_{k}(t)dt\qquad j<p\\ dW^{Q_{T_{p}}}(t)\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad j=p\\ dW^{Q_{T_{p}}}(t)+\sum\limits_{k=p}^{j-1}\frac{\delta L_{k}(t)}{1+\delta L_{k}% (t)}{\sigma}_{k}(t)dt\qquad\quad j>p\\ \end{cases}
  11. d L j ( t ) = { L j ( t ) σ j ( t ) d W Q T p ( t ) - L j ( t ) k = j + 1 p δ L k ( t ) 1 + δ L k ( t ) σ j ( t ) σ k ( t ) ρ j k d t j < p L j ( t ) σ j ( t ) d W Q T p ( t ) j = p L j ( t ) σ j ( t ) d W Q T p ( t ) + L j ( t ) k = p j - 1 δ L k ( t ) 1 + δ L k ( t ) σ j ( t ) σ k ( t ) ρ j k d t j > p dL_{j}(t)=\begin{cases}L_{j}(t){\sigma}_{j}(t)dW^{Q_{T_{p}}}(t)-L_{j}(t)\sum% \limits_{k=j+1}^{p}\frac{\delta L_{k}(t)}{1+\delta L_{k}(t)}{\sigma}_{j}(t){% \sigma}_{k}(t){\rho}_{jk}dt\qquad j<p\\ L_{j}(t){\sigma}_{j}(t)dW^{Q_{T_{p}}}(t)\qquad\qquad\qquad\qquad\qquad\qquad% \qquad\qquad\quad\quad j=p\\ L_{j}(t){\sigma}_{j}(t)dW^{Q_{T_{p}}}(t)+L_{j}(t)\sum\limits_{k=p}^{j-1}\frac{% \delta L_{k}(t)}{1+\delta L_{k}(t)}{\sigma}_{j}(t){\sigma}_{k}(t){\rho}_{jk}dt% \quad\qquad j>p\\ \end{cases}

Lie_coalgebra.html

  1. d : E E E d\colon E\to E\wedge E
  2. d ( a b ) = ( d a ) b + ( - 1 ) deg a a ( d b ) d(a\wedge b)=(da)\wedge b+(-1)^{\operatorname{deg}a}a\wedge(db)
  3. d : E + 1 E . d\colon\bigwedge^{\bullet}E\rightarrow\bigwedge^{\bullet+1}E.
  4. ( * E , d ) (\bigwedge^{*}E,d)
  5. E d E E d 3 E d E\ \rightarrow^{\!\!\!\!\!\!d}\ E\wedge E\ \rightarrow^{\!\!\!\!\!\!d}\ % \bigwedge^{3}E\rightarrow^{\!\!\!\!\!\!d}\ \dots
  6. C ( M ) C^{\infty}(M)
  7. d ( f g ) = ( d f ) g + f ( d g ) f ( d g ) d(fg)=(df)g+f(dg)\neq f(dg)
  8. Ω 1 Ω 2 \Omega^{1}\to\Omega^{2}
  9. C ( M ) Ω 1 ( M ) C^{\infty}(M)\to\Omega^{1}(M)
  10. [ , ] : 𝔤 × 𝔤 𝔤 [\cdot,\cdot]\colon\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}
  11. [ , ] : 𝔤 𝔤 𝔤 [\cdot,\cdot]\colon\mathfrak{g}\wedge\mathfrak{g}\to\mathfrak{g}
  12. d : E E E d\colon E\to E\otimes E
  13. τ d = - d \tau\circ d=-d
  14. τ \tau
  15. E E E E E\otimes E\to E\otimes E
  16. ( d id ) d = ( id d ) d + ( id τ ) ( d id ) d \left(d\otimes\mathrm{id}\right)\circ d=\left(\mathrm{id}\otimes d\right)\circ d% +\left(\mathrm{id}\otimes\tau\right)\circ\left(d\otimes\mathrm{id}\right)\circ d
  17. d : E E E d\colon E\to E\otimes E
  18. d : E E E d\colon E\to E\wedge E
  19. 𝔤 \mathfrak{g}
  20. [ , ] * : 𝔤 * ( 𝔤 𝔤 ) * 𝔤 * 𝔤 * [\cdot,\cdot]^{*}\colon\mathfrak{g}^{*}\to(\mathfrak{g}\wedge\mathfrak{g})^{*}% \cong\mathfrak{g}^{*}\wedge\mathfrak{g}^{*}
  21. \cong
  22. d 2 α ( x y z ) = 1 3 d 2 α ( x y z + y z x + z x y ) = 1 3 ( d α ( [ x , y ] z ) + d α ( [ y , z ] x ) + d α ( [ z , x ] y ) ) , \begin{aligned}\displaystyle d^{2}\alpha(x\wedge y\wedge z)&\displaystyle=% \frac{1}{3}d^{2}\alpha(x\wedge y\wedge z+y\wedge z\wedge x+z\wedge x\wedge y)% \\ &\displaystyle=\frac{1}{3}\left(d\alpha([x,y]\wedge z)+d\alpha([y,z]\wedge x)+% d\alpha([z,x]\wedge y)\right),\end{aligned}
  23. d 2 α ( x y z ) = 1 3 ( α ( [ [ x , y ] , z ] ) + α ( [ [ y , z ] , x ] ) + α ( [ [ z , x ] , y ] ) ) . d^{2}\alpha(x\wedge y\wedge z)=\frac{1}{3}\left(\alpha([[x,y],z])+\alpha([[y,z% ],x])+\alpha([[z,x],y])\right).
  24. α ( [ [ x , y ] , z ] + [ [ y , z ] , x ] + [ [ z , x ] , y ] ) = 0 \alpha([[x,y],z]+[[y,z],x]+[[z,x],y])=0

Lifted_condensation_level.html

  1. h L C L = T - T d Γ d - Γ d e w = 125 ( T - T d ) h_{LCL}=\frac{T-T_{d}}{\Gamma_{d}-\Gamma_{dew}}=125(T-T_{d})
  2. h L C L = ( 20 + T 5 ) ( 100 - R H ) h_{LCL}=(20+\frac{T}{5})(100-RH)

Limit_set.html

  1. X X
  2. f : X X f:X\rightarrow X
  3. ω \omega
  4. x X x\in X
  5. ω ( x , f ) \omega(x,f)
  6. { f n ( x ) } n \{f^{n}(x)\}_{n\in\mathbb{N}}
  7. f f
  8. y ω ( x , f ) y\in\omega(x,f)
  9. { n k } k \{n_{k}\}_{k\in\mathbb{N}}
  10. f n k ( x ) y f^{n_{k}}(x)\rightarrow y
  11. k k\rightarrow\infty
  12. ω ( x , f ) = n { f k ( x ) : k > n } ¯ , \omega(x,f)=\bigcap_{n\in\mathbb{N}}\overline{\{f^{k}(x):k>n\}},
  13. S ¯ \overline{S}
  14. S S
  15. ω ( x , f ) = n = 1 k = n { f k ( x ) } ¯ . \omega(x,f)=\bigcap_{n=1}^{\infty}\overline{\bigcup_{k=n}^{\infty}\{f^{k}(x)\}}.
  16. f f
  17. α \alpha
  18. α ( x , f ) = ω ( x , f - 1 ) \alpha(x,f)=\omega(x,f^{-1})
  19. f f
  20. X X
  21. φ : × X X \varphi:\mathbb{R}\times X\to X
  22. ( t n ) n (t_{n})_{n\in\mathbb{N}}
  23. lim n t n = \lim_{n\to\infty}t_{n}=\infty
  24. lim n φ ( t n , x ) = y \lim_{n\to\infty}\varphi(t_{n},x)=y
  25. ( t n ) n (t_{n})_{n\in\mathbb{N}}
  26. lim n t n = - \lim_{n\to\infty}t_{n}=-\infty
  27. lim n φ ( t n , x ) = y \lim_{n\to\infty}\varphi(t_{n},x)=y
  28. lim ω γ := s { φ ( x , t ) : t > s } ¯ \lim_{\omega}\gamma:=\bigcap_{s\in\mathbb{R}}\overline{\{\varphi(x,t):t>s\}}
  29. lim α γ := s { φ ( x , t ) : t < s } ¯ . \lim_{\alpha}\gamma:=\bigcap_{s\in\mathbb{R}}\overline{\{\varphi(x,t):t<s\}}.
  30. x 0 x_{0}
  31. x 0 x_{0}
  32. x 0 x_{0}
  33. x 0 x_{0}

Limited-memory_BFGS.html

  1. d k = - H k g k d_{k}=-H_{k}g_{k}\,\!
  2. x k x_{k}\,\!
  3. k k\,\!
  4. g k f ( x k ) g_{k}\equiv\nabla f(x_{k})
  5. f f\,\!
  6. m m
  7. s k = x k + 1 - x k s_{k}=x_{k+1}-x_{k}\,\!
  8. y k = g k + 1 - g k y_{k}=g_{k+1}-g_{k}\,\!
  9. ρ k = 1 y k T s k \rho_{k}=\frac{1}{y^{\rm T}_{k}s_{k}}
  10. H k 0 H^{0}_{k}\,\!
  11. k k\,\!
  12. q = g k q=g_{k}\,\!
  13. i = k - 1 , k - 2 , , k - m i=k-1,k-2,\ldots,k-m
  14. α i = ρ i s i T q \alpha_{i}=\rho_{i}s^{\rm T}_{i}q\,\!
  15. q = q - α i y i q=q-\alpha_{i}y_{i}\,\!
  16. H k = y k - 1 T s k - 1 / y k - 1 T y k - 1 H_{k}=y^{\rm T}_{k-1}s_{k-1}/y^{\rm T}_{k-1}y_{k-1}
  17. z = H k q z=H_{k}q
  18. i = k - m , k - m + 1 , , k - 1 i=k-m,k-m+1,\ldots,k-1
  19. β i = ρ i y i T z \beta_{i}=\rho_{i}y^{\rm T}_{i}z\,\!
  20. z = z + s i ( α i - β i ) z=z+s_{i}(\alpha_{i}-\beta_{i})\,\!
  21. H k g k = z H_{k}g_{k}=z\,\!
  22. H k 0 H^{0}_{k}
  23. H k 0 H^{0}_{k}\,\!
  24. z z\,\!
  25. B k B_{k}\,\!
  26. 2 \ell_{2}
  27. l i x i u i l_{i}\leq x_{i}\leq u_{i}
  28. 1 \ell_{1}
  29. f ( x ) = g ( x ) + C x 1 f(\vec{x})=g(\vec{x})+C\|\vec{x}\|_{1}
  30. g g
  31. x 1 \|\vec{x}\|_{1}

Lindelöf's_theorem.html

  1. Ω = { z | x 1 Re ( z ) x 2 and Im ( z ) y 0 } . \Omega=\{z\in\mathbb{C}|x_{1}\leq\mathrm{Re}(z)\leq x_{2}\,\text{ and }\mathrm% {Im}(z)\geq y_{0}\}\subsetneq\mathbb{C}.\,
  2. | f ( z ) | M for all z Ω |f(z)|\leq M\,\text{ for all }z\in\partial\Omega\,
  3. | f ( x + i y ) | y A B for all x + i y Ω . \frac{|f(x+iy)|}{y^{A}}\leq B\,\text{ for all }x+iy\in\Omega.\,
  4. | f ( z ) | M for all z Ω . |f(z)|\leq M\,\text{ for all }z\in\Omega.\,
  5. ξ = σ + i τ \xi=\sigma+i\tau
  6. Ω \Omega
  7. λ > - y 0 \lambda>-y_{0}
  8. N > A N>A
  9. y 1 > τ y_{1}>\tau
  10. B y 1 A ( y 1 + λ ) N M ( y 0 + λ ) N \frac{By_{1}^{A}}{(y_{1}+\lambda)^{N}}\leq\frac{M}{(y_{0}+\lambda)^{N}}
  11. g ( z ) = f ( z ) ( z + i λ ) N g(z)=\frac{f(z)}{(z+i\lambda)^{N}}
  12. { z | x 1 Re ( z ) x 2 and y 0 Im ( z ) y 1 } \{z\in\mathbb{C}|x_{1}\leq\mathrm{Re}(z)\leq x_{2}\,\text{ and }y_{0}\leq% \mathrm{Im}(z)\leq y_{1}\}
  13. | g ( ξ ) | M ( y 0 + λ ) N |g(\xi)|\leq\frac{M}{(y_{0}+\lambda)^{N}}
  14. | f ( ξ ) | M ( | ξ + λ | y 0 + λ ) N |f(\xi)|\leq M\left(\frac{|\xi+\lambda|}{y_{0}+\lambda}\right)^{N}
  15. λ + \lambda\rightarrow+\infty
  16. | f ( ξ ) | M |f(\xi)|\leq M

Lindley's_paradox.html

  1. x \textstyle x
  2. H 0 \textstyle H_{0}
  3. H 1 \textstyle H_{1}
  4. π \textstyle\pi
  5. x \textstyle x
  6. x \textstyle x
  7. H 0 \textstyle H_{0}
  8. H 0 \textstyle H_{0}
  9. H 0 \textstyle H_{0}
  10. x \textstyle x
  11. H 0 \textstyle H_{0}
  12. x \textstyle x
  13. H 1 \textstyle H_{1}
  14. H 0 \textstyle H_{0}
  15. H 1 \textstyle H_{1}
  16. x \textstyle x
  17. θ \textstyle\theta
  18. θ \textstyle\theta
  19. H 0 : θ = 0.5 \textstyle H_{0}:\theta=0.5
  20. H 1 : θ 0.5 \textstyle H_{1}:\theta\neq 0.5
  21. H 0 \textstyle H_{0}
  22. x \textstyle x
  23. H 0 \textstyle H_{0}
  24. X N ( μ , σ 2 ) \textstyle X\sim N(\mu,\sigma^{2})
  25. μ = n p = n θ = 98 , 451 × 0.5 = 49 , 225.5 \textstyle\mu=np=n\theta=98,451\times 0.5=49,225.5
  26. σ 2 = n θ ( 1 - θ ) = 98 , 451 × 0.5 × 0.5 = 24 , 612.75 \textstyle\sigma^{2}=n\theta(1-\theta)=98,451\times 0.5\times 0.5=24,612.75
  27. P ( X x μ = 49222.5 ) = x = 49581 98451 1 2 π σ 2 e - ( u - μ σ ) 2 / 2 d u = x = 49581 98451 1 2 π ( 24 , 612.75 ) e - ( u - 49225.5 ) 2 24612.75 / 2 d u 0.0117. \begin{aligned}\displaystyle P(X\geq x\mid\mu=49222.5)=\int_{x=49581}^{98451}% \frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-(\frac{u-\mu}{\sigma})^{2}/2}du\\ \displaystyle=\int_{x=49581}^{98451}\frac{1}{\sqrt{2\pi(24,612.75)}}e^{-\frac{% (u-49225.5)^{2}}{24612.75}/2}du\approx 0.0117.\end{aligned}
  28. x 0.4964 \textstyle x\approx 0.4964
  29. p 2 × 0.0117 = 0.0235 \textstyle p\approx 2\times 0.0117=0.0235
  30. H 0 \textstyle H_{0}
  31. π ( H 0 ) = π ( H 1 ) = 0.5 \textstyle\pi(H_{0})=\pi(H_{1})=0.5
  32. θ \textstyle\theta
  33. H 1 H_{1}
  34. H 0 \textstyle H_{0}
  35. P ( H 0 k ) = P ( k H 0 ) π ( H 0 ) P ( k H 0 ) π ( H 0 ) + P ( k H 1 ) π ( H 1 ) . P(H_{0}\mid k)=\frac{P(k\mid H_{0})\pi(H_{0})}{P(k\mid H_{0})\pi(H_{0})+P(k% \mid H_{1})\pi(H_{1})}.
  36. k = 49 , 581 \textstyle k=49,581
  37. n = 98 , 451 \textstyle n=98,451
  38. P ( k H 0 ) = ( n k ) ( 0.5 ) k ( 1 - 0.5 ) n - k 1.95 × 10 - 4 P ( k H 1 ) = 0 1 ( n k ) u k ( 1 - u ) n - k d u = ( n k ) B ( k + 1 , n - k + 1 ) 1.02 × 10 - 5 \begin{aligned}\displaystyle P(k\mid H_{0})&\displaystyle={n\choose k}(0.5)^{k% }(1-0.5)^{n-k}\approx 1.95\times 10^{-4}\\ \displaystyle P(k\mid H_{1})&\displaystyle=\int_{0}^{1}{n\choose k}u^{k}(1-u)^% {n-k}du={n\choose k}\mathrm{B}(k+1,n-k+1)\approx 1.02\times 10^{-5}\end{aligned}
  39. B ( a , b ) \textstyle\mathrm{B}(a,b)
  40. P ( H 0 k ) 0.95 P(\textstyle H_{0}\mid k)\approx 0.95
  41. H 0 \textstyle H_{0}
  42. H 1 \textstyle H_{1}
  43. H 0 \textstyle H_{0}
  44. H 1 \textstyle H_{1}
  45. H 0 \textstyle H_{0}
  46. H 1 \textstyle H_{1}
  47. θ \textstyle\theta
  48. [ 0 , 1 ] \textstyle[0,1]
  49. H 0 \textstyle H_{0}
  50. θ 0.500 \textstyle\theta\approx 0.500
  51. H 1 \textstyle H_{1}
  52. θ \textstyle\theta
  53. θ \textstyle\theta
  54. H 1 \textstyle H_{1}
  55. H 0 \textstyle H_{0}
  56. H 0 \textstyle H_{0}
  57. H 1 \textstyle H_{1}
  58. θ 0.504 \textstyle\theta\approx 0.504
  59. θ 0.500 \textstyle\theta\approx 0.500
  60. θ \textstyle\theta
  61. θ \textstyle\theta
  62. \approx
  63. θ = 0.5 \theta=0.5
  64. H 0 H_{0}
  65. x 0.5036 x\approx 0.5036
  66. x x
  67. 2.28 σ 2.28\sigma
  68. H 0 H_{0}
  69. θ = 0.5 \textstyle\theta=0.5
  70. θ \textstyle\theta
  71. H 0 \textstyle H_{0}
  72. H 1 \textstyle H_{1}
  73. H 2 : θ = x \textstyle H_{2}:\theta=x
  74. θ \textstyle\theta
  75. H 0 \textstyle H_{0}
  76. H 2 \textstyle H_{2}
  77. P ( θ x , n ) \textstyle P(\theta\mid x,n)
  78. θ \textstyle\theta
  79. π ( θ [ 0 , 1 ] ) = 1 \textstyle\pi(\theta\in[0,1])=1
  80. P ( θ k , n ) = B ( k + 1 , n - k + 1 ) . P(\theta\mid k,n)=\mathrm{B}(k+1,n-k+1).
  81. P ( θ > 0.5 k , n ) P(\theta>0.5\mid k,n)
  82. 0.5 1 B ( 49582 , 48871 ) 0.983. \int_{0.5}^{1}\mathrm{B}(49582,48871)\approx 0.983.

Lindström_quantifier.html

  1. ϕ A , x , a ¯ = { x A : A ϕ [ x , a ¯ ] } \phi^{A,x,\bar{a}}=\{x\in A\colon A\models\phi[x,\bar{a}]\}
  2. a ¯ \bar{a}
  3. ϕ A , x , a ¯ \phi^{A,x,\bar{a}}
  4. x ¯ \bar{x}
  5. ϕ A , x ¯ , a ¯ \phi^{A,\bar{x},\bar{a}}
  6. Q A Q_{A}
  7. A x ϕ [ x , a ¯ ] ϕ A , x , a ¯ A A\models\forall x\phi[x,\bar{a}]\iff\phi^{A,x,\bar{a}}\in\forall_{A}
  8. A x ϕ [ x , a ¯ ] ϕ A , x , a ¯ A , A\models\exists x\phi[x,\bar{a}]\iff\phi^{A,x,\bar{a}}\in\exists_{A},
  9. A \forall_{A}
  10. A \exists_{A}
  11. Q A x 1 x 2 y 1 z 1 z 2 z 3 ( ϕ ( x 1 x 2 ) , ψ ( y 1 ) , θ ( z 1 z 2 z 3 ) ) Q_{A}x_{1}x_{2}y_{1}z_{1}z_{2}z_{3}(\phi(x_{1}x_{2}),\psi(y_{1}),\theta(z_{1}z% _{2}z_{3}))
  12. A Q A x 1 x 2 y 1 z 1 z 2 z 3 ( ϕ , ψ , θ ) [ a ] ( ϕ A , x 1 x 2 , a ¯ , ψ A , y 1 , a ¯ , θ A , z 1 z 2 z 3 , a ¯ ) Q A A\models Q_{A}x_{1}x_{2}y_{1}z_{1}z_{2}z_{3}(\phi,\psi,\theta)[a]\iff(\phi^{A,% x_{1}x_{2},\bar{a}},\psi^{A,y_{1},\bar{a}},\theta^{A,z_{1}z_{2}z_{3},\bar{a}})% \in Q_{A}
  13. ϕ A , x ¯ , a ¯ = { ( x 1 , , x n ) A n : A ϕ [ x ¯ , a ¯ ] } \phi^{A,\bar{x},\bar{a}}=\{(x_{1},\dots,x_{n})\in A^{n}\colon A\models\phi[% \bar{x},\bar{a}]\}
  14. x ¯ \bar{x}
  15. Q x y ϕ ( x ) ψ ( y ) Qxy\phi(x)\psi(y)
  16. Q x y ϕ ( x , y ) Qxy\phi(x,y)

Line_chart.html

  1. t t
  2. v v
  3. v ( t ) v(t)
  4. v v
  5. t t

Linear-quadratic-Gaussian_control.html

  1. 𝐱 ˙ ( t ) = A ( t ) 𝐱 ( t ) + B ( t ) 𝐮 ( t ) + 𝐯 ( t ) , \dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)+B(t)\mathbf{u}(t)+\mathbf{v}(t),
  2. 𝐲 ( t ) = C ( t ) 𝐱 ( t ) + 𝐰 ( t ) , \mathbf{y}(t)=C(t)\mathbf{x}(t)+\mathbf{w}(t),
  3. 𝐱 {\mathbf{x}}
  4. 𝐮 {\mathbf{u}}
  5. 𝐲 {\mathbf{y}}
  6. 𝐯 ( t ) \mathbf{v}(t)
  7. 𝐰 ( t ) \mathbf{w}(t)
  8. 𝐮 ( t ) {\mathbf{u}}(t)
  9. t {\mathbf{}}t
  10. 𝐲 ( t ) , 0 t < t {\mathbf{y}}(t^{\prime}),0\leq t^{\prime}<t
  11. J = E ( 𝐱 T ( T ) F 𝐱 ( T ) + 0 T 𝐱 T ( t ) Q ( t ) 𝐱 ( t ) + 𝐮 T ( t ) R ( t ) 𝐮 ( t ) d t ) , J=E\left({\mathbf{x}^{\mathrm{T}}}(T)F{\mathbf{x}}(T)+\int_{0}^{T}{\mathbf{x}^% {\mathrm{T}}}(t)Q(t){\mathbf{x}}(t)+{\mathbf{u}^{\mathrm{T}}}(t)R(t){\mathbf{u% }}(t)\,dt\right),
  12. F 0 , Q ( t ) 0 , R ( t ) > 0 , F\geq 0,\quad Q(t)\geq 0,\quad R(t)>0,
  13. E {\mathbf{}}E
  14. T {\mathbf{}}T
  15. 𝐱 T ( T ) F 𝐱 ( T ) {\mathbf{x}}^{\mathrm{T}}(T)F{\mathbf{x}}(T)
  16. J / T {\mathbf{}}J/T
  17. 𝐱 ^ ˙ ( t ) = A ( t ) 𝐱 ^ ( t ) + B ( t ) 𝐮 ( t ) + K ( t ) ( 𝐲 ( t ) - C ( t ) 𝐱 ^ ( t ) ) , 𝐱 ^ ( 0 ) = E ( 𝐱 ( 0 ) ) , \dot{\hat{\mathbf{x}}}(t)=A(t)\hat{\mathbf{x}}(t)+B(t){\mathbf{u}}(t)+K(t)% \left({\mathbf{y}}(t)-C(t)\hat{\mathbf{x}}(t)\right),\hat{\mathbf{x}}(0)=E% \left({\mathbf{x}}(0)\right),
  18. 𝐮 ( t ) = - L ( t ) 𝐱 ^ ( t ) . {\mathbf{u}}(t)=-L(t)\hat{\mathbf{x}}(t).
  19. K ( t ) {\mathbf{}}K(t)
  20. t {\mathbf{}}t
  21. 𝐱 ^ ( t ) \hat{\mathbf{x}}(t)
  22. 𝐱 ( t ) {\mathbf{x}}(t)
  23. K ( t ) {\mathbf{}}K(t)
  24. A ( t ) , C ( t ) {\mathbf{}}A(t),C(t)
  25. V ( t ) , W ( t ) \mathbf{}V(t),W(t)
  26. 𝐯 ( t ) \mathbf{v}(t)
  27. 𝐰 ( t ) \mathbf{w}(t)
  28. E ( 𝐱 ( 0 ) 𝐱 T ( 0 ) ) E\left({\mathbf{x}}(0){\mathbf{x}}^{\mathrm{T}}(0)\right)
  29. P ˙ ( t ) = A ( t ) P ( t ) + P ( t ) A T ( t ) - P ( t ) C T ( t ) W - 1 ( t ) C ( t ) P ( t ) + V ( t ) , \dot{P}(t)=A(t)P(t)+P(t)A^{\mathrm{T}}(t)-P(t)C^{\mathrm{T}}(t){\mathbf{}}W^{-% 1}(t)C(t)P(t)+V(t),
  30. P ( 0 ) = E ( 𝐱 ( 0 ) 𝐱 T ( 0 ) ) . P(0)=E\left({\mathbf{x}}(0){\mathbf{x}}^{\mathrm{T}}(0)\right).
  31. P ( t ) , 0 t T P(t),0\leq t\leq T
  32. K ( t ) = P ( t ) C T ( t ) W - 1 ( t ) . {\mathbf{}}K(t)=P(t)C^{\mathrm{T}}(t)W^{-1}(t).
  33. L ( t ) {\mathbf{}}L(t)
  34. A ( t ) , B ( t ) , Q ( t ) , R ( t ) {\mathbf{}}A(t),B(t),Q(t),R(t)
  35. F {\mathbf{}}F
  36. - S ˙ ( t ) = A T ( t ) S ( t ) + S ( t ) A ( t ) - S ( t ) B ( t ) R - 1 ( t ) B T ( t ) S ( t ) + Q ( t ) , -\dot{S}(t)=A^{\mathrm{T}}(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B^{\mathrm{T}}(t)S% (t)+Q(t),
  37. S ( T ) = F . {\mathbf{}}S(T)=F.
  38. S ( t ) , 0 t T {\mathbf{}}S(t),0\leq t\leq T
  39. L ( t ) = R - 1 ( t ) B T ( t ) S ( t ) . {\mathbf{}}L(t)=R^{-1}(t)B^{\mathrm{T}}(t)S(t).
  40. A ( t ) , B ( t ) , C ( t ) , Q ( t ) , R ( t ) {\mathbf{}}A(t),B(t),C(t),Q(t),R(t)
  41. V ( t ) \mathbf{}V(t)
  42. W ( t ) \mathbf{}W(t)
  43. t {\mathbf{}}t
  44. T {\mathbf{}}T
  45. 𝐱 i + 1 = A i 𝐱 i + B i 𝐮 i + 𝐯 i , {\mathbf{x}}_{i+1}=A_{i}\mathbf{x}_{i}+B_{i}\mathbf{u}_{i}+\mathbf{v}_{i},
  46. 𝐲 i = C i 𝐱 i + 𝐰 i . \mathbf{y}_{i}=C_{i}\mathbf{x}_{i}+\mathbf{w}_{i}.
  47. i \mathbf{}i
  48. 𝐯 i , 𝐰 i \mathbf{v}_{i},\mathbf{w}_{i}
  49. V i , W i \mathbf{}V_{i},W_{i}
  50. J = E ( 𝐱 N T F 𝐱 N + i = 0 N - 1 𝐱 i T Q i 𝐱 i + 𝐮 i T R i 𝐮 i ) , J=E\left({\mathbf{x}}^{\mathrm{T}}_{N}F{\mathbf{x}}_{N}+\sum_{i=0}^{N-1}% \mathbf{x}_{i}^{\mathrm{T}}Q_{i}\mathbf{x}_{i}+\mathbf{u}_{i}^{\mathrm{T}}R_{i% }\mathbf{u}_{i}\right),
  51. F 0 , Q i 0 , R i > 0. F\geq 0,Q_{i}\geq 0,R_{i}>0.\,
  52. 𝐱 ^ i + 1 = A i 𝐱 ^ i + B i 𝐮 i + K i ( 𝐲 i - C i 𝐱 ^ i ) , 𝐱 ^ 0 = E ( 𝐱 0 ) \hat{\mathbf{x}}_{i+1}=A_{i}\hat{\mathbf{x}}_{i}+B_{i}{\mathbf{u}}_{i}+K_{i}% \left({\mathbf{y}}_{i}-C_{i}{\hat{\mathbf{x}}}_{i}\right),\hat{\mathbf{x}}_{0}% =E({\mathbf{x}}_{0})
  53. 𝐮 i = - L i 𝐱 ^ i . \mathbf{u}_{i}=-L_{i}\hat{\mathbf{x}}_{i}.\,
  54. K i = P i C i T ( C i P i C i T + W i ) - 1 , {\mathbf{}}K_{i}=P_{i}C^{\mathrm{T}}_{i}(C_{i}P_{i}C^{\mathrm{T}}_{i}+W_{i})^{% -1},
  55. P i {\mathbf{}}P_{i}
  56. P i + 1 = A i ( P i - P i C i T ( C i P i C i T + W i ) - 1 C i P i ) A i T + V i , P 0 = E ( 𝐱 0 - 𝐱 ^ 0 ) ( 𝐱 0 - 𝐱 ^ 0 ) T . P_{i+1}=A_{i}\left(P_{i}-P_{i}C^{\mathrm{T}}_{i}\left(C_{i}P_{i}C^{\mathrm{T}}% _{i}+W_{i}\right)^{-1}C_{i}P_{i}\right)A^{\mathrm{T}}_{i}+V_{i},P_{0}=E\left({% \mathbf{x}}_{0}-\hat{\mathbf{x}}_{0}\right)\left({\mathbf{x}}_{0}-\hat{\mathbf% {x}}_{0}\right)^{\mathrm{T}}.
  57. L i = ( B i T S i + 1 B i + R i ) - 1 B i T S i + 1 A i {\mathbf{}}L_{i}=(B^{\mathrm{T}}_{i}S_{i+1}B_{i}+R_{i})^{-1}B^{\mathrm{T}}_{i}% S_{i+1}A_{i}
  58. S i {\mathbf{}}S_{i}
  59. S i = A i T ( S i + 1 - S i + 1 B i ( B i T S i + 1 B i + R i ) - 1 B i T S i + 1 ) A i + Q i , S N = F . S_{i}=A^{\mathrm{T}}_{i}\left(S_{i+1}-S_{i+1}B_{i}\left(B^{\mathrm{T}}_{i}S_{i% +1}B_{i}+R_{i}\right)^{-1}B^{\mathrm{T}}_{i}S_{i+1}\right)A_{i}+Q_{i},S_{N}=F.
  60. N {\mathbf{}}N
  61. J {\mathbf{}}J
  62. J / N {\mathbf{}}J/N