wpmath0000016_2

Chirp_spectrum.html

  1. s ( t ) = a ( t ) . e x p [ j ( ω 0 . t + θ ( t ) ) ] s(t)=a(t).exp[j(\omega_{0}.t+\theta(t))]
  2. S ( ω ) = - s ( t ) . e x p ( - j ω t ) . d t = - a ( t ) . e x p [ j ( ω 0 t + θ ( t ) ) ] . e x p ( - j ω t ) . d t S(\omega)=\int_{-\infty}^{\infty}s(t).exp(-j\omega t).dt=\int_{-\infty}^{% \infty}a(t).exp[j(\omega_{0}t+\theta(t))].exp(-j\omega t).dt
  3. S ( ω ) = - a ( t ) . e x p [ j { ( ω 0 - ω ) . t + θ ( t ) } ] . d t S(\omega)=\int_{-\infty}^{\infty}a(t).exp[j\left\{(\omega_{0}-\omega).t+\theta% (t)\right\}].dt
  4. s ( t ) = 1. e x p [ j ( ω 0 . t + Δ Ω 2. T . t 2 ) ] w h e r e Δ Ω = 2 π . Δ F a n d - T 2 < t < T 2 s(t)=1.exp[j(\omega_{0}.t+\frac{\Delta\Omega}{2.T}.t^{2})]\qquad where\quad% \Delta\Omega=2\pi.\Delta F\qquad and\quad\frac{-T}{2}<t<\frac{T}{2}
  5. S ( ω ) = - T / 2 T / 2 e x p [ j ( ω 0 . t + Δ Ω 2. T . t 2 ) ] . e x p ( - j ω . t ) . d t = - T / 2 T / 2 e x p [ j { ( ω 0 - ω ) . t + Δ Ω 2. T . t 2 } ] . d t S(\omega)=\int_{-T/2}^{T/2}exp\left[j(\omega_{0}.t+\frac{\Delta\Omega}{2.T}.t^% {2})\right].exp(-j\omega.t).dt=\int_{-T/2}^{T/2}exp\left[j\left\{(\omega_{0}-% \omega).t+\frac{\Delta\Omega}{2.T}.t^{2}\right\}\right].dt
  6. C ( X ) = 0 X c o s π . y 2 2 . d y a n d S ( X ) = 0 X s i n π . y 2 2 . d y C(X)=\int_{0}^{X}cos\frac{\pi.y^{2}}{2}.dy\qquad and\qquad S(X)=\int_{0}^{X}% sin\frac{\pi.y^{2}}{2}.dy
  7. S ( ω ) = ( π . T 2. Δ Ω ) . e x p [ - j ( ( ω - ω 0 ) 2 . T 2. Δ Ω ) ] . [ C ( X 1 ) + j . S ( X 1 ) + C ( X 2 ) + j . S ( X 2 ) ] S(\omega)=\sqrt{\left(\frac{\pi.T}{2.\Delta\Omega}\right)}.exp\left[-j\left((% \omega-\omega_{0})^{2}.\frac{T}{2.\Delta\Omega}\right)\right].[C(X_{1})+j.S(X_% {1})+C(X_{2})+j.S(X_{2})]
  8. X 1 X_{1}
  9. X 2 X_{2}
  10. X 1 = Δ Ω 2 + ( ω - ω 0 ) π . Ω T a n d X 2 = Δ Ω 2 + ( ω + ω 0 ) π . Ω T \quad X_{1}=\frac{\frac{\Delta\Omega}{2}+(\omega-\omega_{0})}{\sqrt{\frac{\pi.% \Omega}{T}}}\quad and\quad X_{2}=\frac{\frac{\Delta\Omega}{2}+(\omega+\omega_{% 0})}{\sqrt{\frac{\pi.\Omega}{T}}}
  11. | S ( ω ) | = π . T Δ Ω . [ ( C ( X 1 ) + C ( X 2 ) ) 2 + ( S ( X 1 ) + S ( X 2 ) ) 2 ] 1 2 |S(\omega)|=\sqrt{\frac{\pi.T}{\Delta\Omega}}.\left[\left(C(X_{1})+C(X_{2})% \right)^{2}+\left(S(X_{1})+S(X_{2})\right)^{2}\right]^{\frac{1}{2}}
  12. - Φ 1 ( ω ) = ( ω - ω 0 ) 2 . T 2. Δ Ω \quad-\Phi_{1}(\omega)=(\omega-\omega_{0})^{2}.\frac{T}{2.\Delta\Omega}
  13. Φ 2 ( ω ) = a r c t a n [ S ( X 1 ) + S ( X 2 ) C ( X 1 ) + C ( X 2 ) ] \quad\Phi_{2}(\omega)=arctan\left[\frac{S(X_{1})+S(X_{2})}{C(X_{1})+C(X_{2})}\right]
  14. [ S ( X 1 ) + S ( X 2 ) C ( X 1 ) + C ( X 2 ) ] \left[\frac{S(X_{1})+S(X_{2})}{C(X_{1})+C(X_{2})}\right]
  15. n = 2. ( ω - ω 0 ) Δ Ω n=2.\frac{(\omega-\omega_{0})}{\Delta\Omega}
  16. X 1 = T . Δ F 2 . ( 1 + n ) X_{1}=\frac{\sqrt{T.\Delta F}}{\sqrt{2}}.(1+n)
  17. X 2 = T . Δ F 2 . ( 1 - n ) X_{2}=\frac{\sqrt{T.\Delta F}}{\sqrt{2}}.(1-n)
  18. ω 0 ± Δ Ω / 2 \omega_{0}\pm\Delta\Omega/2
  19. ω 0 ± Δ Ω / 2 \omega_{0}\pm\Delta\Omega/2
  20. d d t [ ( ω 0 - ω ) t + θ ( t ) ] = 0 ] o r ( ω - ω 0 ) - θ ( t ) = 0 \frac{d}{dt}[(\omega_{0}-\omega)t+\theta(t)]=0]\qquad or\qquad(\omega-\omega_{% 0})-\theta^{\prime}(t)=0
  21. S ( ω ) a ( t s ) t s - δ t s + δ e x p [ - j { ( ω s - ω 0 ) . t - θ ( t s ) - θ ′′ ( t s ) 2 . ( t - t s ) 2 } ] . d t S(\omega)\approxeq a(t_{s})\int_{t_{s}-\delta}^{t_{s}+\delta}exp\left[-j\left% \{(\omega_{s}-\omega_{0}).t-\theta(t_{s})-\frac{\theta^{\prime\prime}(t_{s})}{% 2}.(t-t_{s})^{2}\right\}\right].dt
  22. S ( ω ) 2 π . a ( t s ) | θ ′′ ( t ) | . e x p [ j { ( ω 0 - ω s ) t + θ ( t s ) + π 4 } ] S(\omega)\approxeq\sqrt{2\pi}.\frac{a(t_{s})}{\sqrt{|\theta^{\prime\prime}(t)|% }}.exp\left[j\left\{(\omega_{0}-\omega_{s})t+\theta(t_{s})+\frac{\pi}{4}\right% \}\right]
  23. | S ( ω t ) | 2 2 π . a 2 ( t ) | θ ′′ ( t ) | |S(\omega_{t})|^{2}\approxeq 2\pi.\frac{a^{2}(t)}{|\theta^{\prime\prime}(t)|}
  24. s ( t ) = 1 2 π - | S ( ω ) | . e x p [ j ( Φ ( ω ) + ω . t ) ] . d ω s(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}|S(\omega)|.exp[j(\Phi(\omega)+% \omega.t)].d\omega
  25. Φ ( ω ) = - t \Phi^{\prime}(\omega)=-t
  26. a 2 ( t ω ) 1 2 π . | S ( ω ) | 2 | Φ ′′ ( ω ) | a^{2}(t_{\omega})\approxeq\frac{1}{2\pi}.\frac{|S(\omega)|^{2}}{|\Phi^{\prime% \prime}(\omega)|}
  27. a ( t ) . e x p [ j θ ( t ) ] 1 2 π . - S ( ω ) | . e x p [ j { Φ ( ω ) + ω . t } ] . d ω a(t).exp[j\theta(t)]\approxeq\frac{1}{2\pi}.\int_{-\infty}^{\infty}S(\omega)|.% exp[j\{\Phi(\omega)+\omega.t\}].d\omega
  28. | S ( ω ) | . e x p [ j Φ ( ω ) ] - a ( t ) . e x p [ - j { ω t - θ ( t ) } ] d t |S(\omega)|.exp[j\Phi(\omega)]\approxeq\int_{-\infty}^{\infty}a(t).exp[-j\{% \omega t-\theta(t)\}]dt
  29. Φ ′′ ( ω ) = ± 1 2 π . | S ( ω ) | 2 \Phi^{\prime\prime}(\omega)=\pm\frac{1}{2\pi}.|S(\omega)|^{2}
  30. D ( ω ) = - Φ ( ω ) = - 0 Φ ′′ ( ω ) . d ω + K D(\omega)=-\Phi^{\prime}(\omega)=-\int_{0}^{\infty}\Phi^{\prime\prime}(\omega)% .d\omega+K
  31. D ( ω ) = ± 1 2 π . 0 | S ( ω ) | 2 . d ω + K D(\omega)=\pm\frac{1}{2\pi}.\int_{0}^{\infty}|S(\omega)|^{2}.d\omega+K
  32. | S ( ω ) | 2 = A f o r | ω | < ω m a x 2 |S(\omega)|^{2}=A\qquad for\qquad|\omega|<\frac{\omega_{max}}{2}
  33. D ( ω ) = 1 2 π . A . d ω + K = A 2 π . ω + K D(\omega)=\frac{1}{2\pi}.\int A.d\omega+K=\frac{A}{2\pi}.\omega+K
  34. D ( ω ) = T . [ 1 2 + ω ω m a x ] D(\omega)=T.\left[\frac{1}{2}+\frac{\omega}{\omega_{max}}\right]
  35. | S ( ω ) | 2 = A n . c o s n ( π ω ω m a x ) w h e r e | ω | < ω m a x 2 a n d n i s a n i n t e g e r |S(\omega)|^{2}=A_{n}.cos^{n}\left(\frac{\pi\omega}{\omega_{max}}\right)\qquad where% \qquad|\omega|<\frac{\omega_{max}}{2}\qquad and\ n\ is\ an\ integer
  36. | S ( ω ) | 2 = A . [ 0.54 + 0.46. c o s ( 2 π ω ω m a x ) ] = A . [ 0.08 + 0.92. c o s 2 ( π ω ω m a x ) ] |S(\omega)|^{2}=A.\left[0.54+0.46.cos\left(\frac{2\pi\omega}{\omega_{max}}% \right)\right]=A.\left[0.08+0.92.cos^{2}\left(\frac{\pi\omega}{\omega_{max}}% \right)\right]
  37. D H ( ω ) = T . [ 1 2 + ω ω m a x + 1.7037 4 π . s i n ( 2 π ω ω m a x ) ] D_{H}(\omega)=T.\left[\frac{1}{2}+\frac{\omega}{\omega_{max}}+\frac{1.7037}{4% \pi}.sin\left(\frac{2\pi\omega}{\omega_{max}}\right)\right]
  38. t T = 1 2 + ω ω m a x + 1.7037 4 π . s i n ( 2 π ω ω m a x ) \frac{t}{T}=\frac{1}{2}+\frac{\omega}{\omega_{max}}+\frac{1.7037}{4\pi}.sin% \left(\frac{2\pi\omega}{\omega_{max}}\right)
  39. S s ( ω ) ) = W ( s i n ω W / 2 ) ω W / 2 . [ n = - s n . e x p ( - j n ω W ) ] Ss(\omega))=\frac{W(sin\omega W/2)}{\omega W/2}.\left[\sum_{n=-\infty}^{\infty% }s_{n}.exp(-jn\omega W)\right]
  40. ω m = 2 π m N W f o r m = 0 , 1 , 2 , N - 1 \omega_{m}=\frac{2\pi m}{NW}\qquad for\qquad m=0,1,2,...N-1
  41. S s m = S s ( j 2 π m N W ) = n = 0 N - 1 s n . e x p ( - j ( 2 π m n N ) Ss_{m}\quad=Ss\left(j\frac{2\pi m}{NW}\right)\quad=\quad\sum_{n=0}^{N-1}s_{n}.% exp(-j\left(\frac{2\pi mn}{N}\right)
  42. Δ f p = 0.75 Δ F a n d δ = 1 / Δ F \Delta f_{p}=0.75\Delta F\qquad and\qquad\delta=1/\Delta F
  43. Δ f p = 0.73 Δ F a n d δ = 0.86 / Δ F \Delta f_{p}=0.73\Delta F\qquad and\qquad\delta=0.86/\Delta F
  44. | U ( ω ) | 2 = 0.42323 - 0.49755 c o s ( 2 π ω ω m a x ) + 0.07922 c o s ( 4 π ω ω m a x ) |U(\omega)|^{2}=0.42323-0.49755cos\left(\frac{2\pi\omega}{\omega_{max}}\right)% +0.07922cos\left(\frac{4\pi\omega}{\omega_{max}}\right)
  45. t T = 1 2 + ω ω m a x + 0.1871 s i n ( 2 π ω ω m a x ) + 0.014895 s i n ( 4 π ω ω m a x ) \frac{t}{T}=\frac{1}{2}+\frac{\omega}{\omega_{max}}+0.1871sin\left(\frac{2\pi% \omega}{\omega_{max}}\right)+0.014895sin\left(\frac{4\pi\omega}{\omega_{max}}\right)

Cholate—CoA_ligase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Chomsky–Schützenberger_enumeration_theorem.html

  1. \mathbb{N}
  2. f = f ( x ) = k = 0 a k x k = a 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 + f=f(x)=\sum_{k=0}^{\infty}a_{k}x^{k}=a_{0}+a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+\cdots
  3. a k a_{k}
  4. \mathbb{N}
  5. f f
  6. g g
  7. a n a_{n}
  8. b n b_{n}
  9. f ( x ) g ( x ) = k = 0 ( i = 0 k a i b k - i ) x k . f(x)\cdot g(x)=\sum_{k=0}^{\infty}\left(\sum_{i=0}^{k}a_{i}b_{k-i}\right)x^{k}.
  10. f 2 = f ( x ) f ( x ) f^{2}=f(x)\cdot f(x)
  11. f 3 = f ( x ) f ( x ) f ( x ) f^{3}=f(x)\cdot f(x)\cdot f(x)
  12. f ( x ) f(x)
  13. ( x ) \mathbb{Q}(x)
  14. p 0 ( x ) , p 1 ( x ) , p 2 ( x ) , , p n ( x ) p_{0}(x),p_{1}(x),p_{2}(x),\ldots,p_{n}(x)
  15. p 0 ( x ) + p 1 ( x ) f + p 2 ( x ) f 2 + + p n ( x ) f n = 0. p_{0}(x)+p_{1}(x)\cdot f+p_{2}(x)\cdot f^{2}+\cdots+p_{n}(x)\cdot f^{n}=0.
  16. L L
  17. a k := | L Σ k | a_{k}:=|L\ \cap\Sigma^{k}|
  18. k k
  19. L L
  20. G ( x ) = k = 0 a k x k G(x)=\sum_{k=0}^{\infty}a_{k}x^{k}
  21. \mathbb{N}
  22. ( x ) \mathbb{Q}(x)
  23. a n a_{n}
  24. a n O ( 2 + ϵ ) n a_{n}\in O(2+\epsilon)^{n}
  25. a n O ( 2 - ϵ ) n a_{n}\notin O(2-\epsilon)^{n}
  26. ϵ > 0 \epsilon>0
  27. L G L_{G}
  28. { a , b } \{a,b\}
  29. a n 1 b a n 2 b a n p b a^{n_{1}}ba^{n_{2}}b\cdots a^{n_{p}}b
  30. p 1 p\geq 1
  31. n i > 0 n_{i}>0
  32. i { 1 , 2 , , p } i\in\{1,2,\ldots,p\}
  33. n j j n_{j}\neq j
  34. j { 1 , 2 , , p } j\in\{1,2,\ldots,p\}
  35. L G L_{G}
  36. L G L_{G}
  37. g k g_{k}
  38. k k
  39. L G L_{G}
  40. G ( x ) = k = 0 g k x k = 1 - x 1 - 2 x - 1 x k 1 x k ( k + 1 ) / 2 - 1 G(x)=\sum_{k=0}^{\infty}g_{k}x^{k}=\frac{1-x}{1-2x}-\frac{1}{x}\sum_{k\geq 1}x% ^{k(k+1)/2-1}
  41. ( x ) \mathbb{Q}(x)
  42. L G L_{G}

Chomsky–Schützenberger_representation_theorem.html

  1. h h
  2. Γ \Gamma
  3. Σ \Sigma
  4. Γ \Gamma
  5. h ( x y ) = h ( x ) h ( y ) h(xy)=h(x)h(y)
  6. x x
  7. y y
  8. h : Γ * Σ * h:\Gamma^{*}\to\Sigma^{*}
  9. T T ¯ T\cup\overline{T}
  10. T T
  11. T ¯ \overline{T}
  12. T T ¯ T\cup\overline{T}
  13. D T D_{T}
  14. D T = { w ( T T ¯ ) * w is a correctly nested sequence of parentheses } D_{T}=\{\,w\in(T\cup\overline{T})^{*}\mid w\,\text{ is a correctly nested % sequence of parentheses}\,\}
  15. T T ¯ T\cup\overline{T}
  16. Σ \Sigma
  17. T T ¯ T\cup\overline{T}
  18. R R
  19. T T ¯ T\cup\overline{T}
  20. h : ( T T ¯ ) * Σ * h:(T\cup\overline{T})^{*}\to\Sigma^{*}
  21. L = h ( D T R ) L=h(D_{T}\cap R)

Christ–Kiselev_maximal_inequality.html

  1. ( M , μ ) (M,\mu)
  2. { A α } α \{A_{\alpha}\}_{\alpha\in\mathbb{R}}
  3. A α M A_{\alpha}\nearrow M
  4. α A α = \bigcap_{\alpha\in\mathbb{R}}A_{\alpha}=\emptyset
  5. μ ( A β A α ) < \mu(A_{\beta}\setminus A_{\alpha})<\infty
  6. β > α \beta>\alpha
  7. lim ε 0 + μ ( A α + ε A α ) = lim ε 0 + μ ( A α A α + ε ) = 0 \lim_{\varepsilon\to 0^{+}}\mu(A_{\alpha+\varepsilon}\setminus A_{\alpha})=% \lim_{\varepsilon\to 0^{+}}\mu(A_{\alpha}\setminus A_{\alpha+\varepsilon})=0
  8. = M \mathbb{R}=M
  9. μ \mu
  10. A α := { { | x | α } , α > 0 , , α 0. A_{\alpha}:=\begin{cases}\{|x|\leq\alpha\},&\alpha>0,\\ \emptyset,&\alpha\leq 0.\end{cases}
  11. 1 p < q 1\leq p<q\leq\infty
  12. T : L p ( M , μ ) L q ( N , ν ) T:L^{p}(M,\mu)\to L^{q}(N,\nu)
  13. σ - \sigma-
  14. ( M , μ ) , ( N , ν ) (M,\mu),(N,\nu)
  15. T * f := sup α | T ( f χ α ) | , T^{*}f:=\sup_{\alpha}|T(f\chi_{\alpha})|,
  16. χ α := χ A α \chi_{\alpha}:=\chi_{A_{\alpha}}
  17. T * : L p ( M , μ ) L q ( N , ν ) T^{*}:L^{p}(M,\mu)\to L^{q}(N,\nu)
  18. T * f q 2 - ( p - 1 - q - 1 ) ( 1 - 2 - ( p - 1 - q - 1 ) ) - 1 T f p . \|T^{*}f\|_{q}\leq 2^{-(p^{-1}-q^{-1})}(1-2^{-(p^{-1}-q^{-1})})^{-1}\|T\|\|f\|% _{p}.
  19. 1 p < q 1\leq p<q\leq\infty
  20. W : p ( ) L q ( N , ν ) W:\ell^{p}(\mathbb{Z})\to L^{q}(N,\nu)
  21. σ - \sigma-
  22. ( M , μ ) , ( N , ν ) (M,\mu),(N,\nu)
  23. a p ( ) a\in\ell^{p}(\mathbb{Z})
  24. ( χ n a ) := { a k , | k | n 0 , otherwise . (\chi_{n}a):=\begin{cases}a_{k},&|k|\leq n\\ 0,&\,\text{otherwise}.\end{cases}
  25. sup n 0 | W ( χ n a ) | = : W * ( a ) \sup_{n\in\mathbb{Z}^{\geq 0}}|W(\chi_{n}a)|=:W^{*}(a)
  26. W * : p ( ) L q ( N , ν ) W^{*}:\ell^{p}(\mathbb{Z})\to L^{q}(N,\nu)
  27. A α = { [ - α , α ] , α > 0 , α 0 A_{\alpha}=\begin{cases}[-\alpha,\alpha],&\alpha>0\\ \emptyset,&\alpha\leq 0\end{cases}
  28. T : L p ( , d x ) L q ( N , ν ) T:L^{p}(\mathbb{R},dx)\to L^{q}(N,\nu)

Chvátal–Sankoff_constants.html

  1. γ k \gamma_{k}
  2. γ 2 \gamma_{2}
  3. λ n , k \lambda_{n,k}
  4. λ n , k \lambda_{n,k}
  5. γ k \gamma_{k}
  6. E [ λ n , k ] E[\lambda_{n,k}]
  7. E [ λ m + n , k ] E [ λ m , k ] + E [ λ n , k ] E[\lambda_{m+n,k}]\geq E[\lambda_{m,k}]+E[\lambda_{n,k}]
  8. γ k = lim n E [ λ n , k ] n \gamma_{k}=\lim_{n\to\infty}\frac{E[\lambda_{n,k}]}{n}
  9. E [ λ n , k ] / n E[\lambda_{n,k}]/n
  10. γ k \gamma_{k}
  11. γ k \gamma_{k}
  12. E [ λ n , k ] E[\lambda_{n,k}]
  13. γ k \gamma_{k}
  14. E [ λ n , k ] E[\lambda_{n,k}]
  15. γ 2 0.788071 \gamma_{2}\geq 0.788071
  16. γ k \gamma_{k}
  17. γ 2 0.826280 \gamma_{2}\leq 0.826280
  18. γ 2 = 2 / ( 1 + 2 ) \gamma_{2}=2/(1+\sqrt{2})
  19. γ 2 \gamma_{2}
  20. γ k \gamma_{k}
  21. lim k γ k k = 2. \lim_{k\to\infty}\gamma_{k}\sqrt{k}=2.

Circle-throw_vibrating_machine.html

  1. E o = 100 m a s s f l o w r a t e o f s o l i d s c o a r s e r t h a n s c r e e n s i z e i n f e e d s t r e a m m a s s f l o w r a t e o f s o l i d s i n t h e o v e r s i z e s t r e a m E_{o}=100{{mass\ flow\ rate\ of\ solids\ coarser\ than\ screen\ size\ in\ feed% \ stream}\over{mass\ flow\ rate\ of\ solids\ in\ the\ oversize\ stream}}
  2. E o = 100 [ F ( 1 - f x ) O ] = 100 ( 1 - o x ) E_{o}=100{\left[\frac{F(1-f_{x})}{O}\right]}=100(1-o_{x})
  3. E u = 100 m a s s f l o w r a t e o f s o l i d s i n t h e u n d e r s i z e s t r e a m m a s s f l o w r a t e o f s o l i d s f i n e r t h a n s c r e e n s i z e i n f e e d s t r e a m E_{u}=100{{mass\ flow\ rate\ of\ solids\ in\ the\ undersize\ stream}\over{mass% \ flow\ rate\ of\ solids\ finer\ than\ screen\ size\ in\ feed\ stream}}
  4. E u = 100 ( U F f x ) = 100 [ f x - o x f x ( 1 - o x ) ] = E o - ( 1 - f x ) E o f x E_{u}=100{\left(\frac{U}{Ff_{x}}\right)}=100{\left[\frac{f_{x}-o_{x}}{f_{x}(1-% o_{x})}\right]}=\frac{E_{o}-(1-f_{x})}{E_{o}f_{x}}
  5. F r = 1 z × G g × r × ( π × n 30 ) 2 F_{r}={\frac{1}{z}\times\frac{G}{g}\times r\times(\pi\times\frac{n}{30})^{2}}
  6. P = F z × F r P=F_{z}\times F_{r}
  7. C = P × F L F n C=P\times\frac{F_{L}}{F_{n}}
  8. D e f l e c t i o n = F o r c e S t i f f n e s s = F s t a t i c 2 k Deflection=\frac{Force}{Stiffness}=\frac{F_{static}}{2k}
  9. A m p l i t u b e = M F ( F d y n a m i c 2 k ) Amplitube=MF\left(\frac{F_{dynamic}}{2k}\right)
  10. M F = 1 [ 1 - ( f d f n ) 2 ] 2 + [ 2 ζ ( f d f n ) 2 ] 1 2 MF=\frac{1}{\left[1-\left(\frac{f_{d}}{f_{n}}\right)^{2}\right]^{2}+\left[2% \zeta\left(\frac{f_{d}}{f_{n}}\right)^{2}\right]^{\frac{1}{2}}}
  11. M F = 1 [ 1 - ( f d f n ) 2 ] 2 MF=\frac{1}{\left[1-\left(\frac{f_{d}}{f_{n}}\right)^{2}\right]^{2}}
  12. W = 400 F D T × ( b u l k d e n s i t y ) W=\frac{400F}{DT\times(bulk\ density)}
  13. T = - 120 + 10 × ( i n c l i n a t i o n a n g l e ) T=-120+10\times(inclination\ angle)
  14. D * = [ 2 + 0.2 × ( b u l k d e n s i t y ) ] × X s D^{*}=[2+0.2\times(bulk\ density)]\times X_{s}
  15. L e n g t h = A W Length=\frac{A}{W}
  16. A = 15.5 F W A=15.5\sqrt{\frac{F}{W}}

Circle_Hough_Transform.html

  1. ( x - a ) 2 + ( y - b ) 2 = r 2 ( 1 ) \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}\ \ \ \ \ (1)

Circle_of_equal_altitude.html

  1. s i n H o = s i n B * s i n D e c + c o s B * c o s D e c * c o s L H A sinHo=sinB*sinDec+cosB*cosDec*cosLHA\,

Circumcenter_of_mass.html

  1. P P
  2. V 1 , V 2 , , V n V_{1},V_{2},\ldots,V_{n}
  3. O O
  4. P P
  5. O V i V i + 1 OV_{i}V_{i+1}
  6. i i
  7. n n
  8. C i C_{i}
  9. P P
  10. O O
  11. V i = ( x i , y i ) V_{i}=(x_{i},y_{i})
  12. P P
  13. A A
  14. C C M ( P ) CCM(P)
  15. P P
  16. C C M ( P ) = 1 4 A ( i = 0 n - 1 - y i y i + 1 2 + y i 2 y i + 1 + x i 2 y i + 1 - x i + 1 2 y i , i = 0 n - 1 - x i + 1 y i 2 + x i y i + 1 2 + x i x i + 1 2 - x i 2 x i + 1 ) . CCM(P)=\frac{1}{4A}(\sum_{i=0}^{n-1}-y_{i}y_{i+1}^{2}+y_{i}^{2}y_{i+1}+x_{i}^{% 2}y_{i+1}-x_{i+1}^{2}y_{i},\sum_{i=0}^{n-1}-x_{i+1}y_{i}^{2}+x_{i}y_{i+1}^{2}+% x_{i}x_{i+1}^{2}-x_{i}^{2}x_{i+1}).

Cis-abienol_synthase.html

  1. \rightleftharpoons

Cis-muuroladiene_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Class_(knowledge_representation).html

  1. : :

Class_logic.html

  1. y : ( y { x A ( x ) } A ( y ) ) \forall y\colon(y\in\{x\mid A(x)\}\iff A(y))
  2. A = B x : ( x A x B ) A=B\iff\forall x\colon(x\in A\iff x\in B)
  3. { x A ( x ) } B y : ( y = { x A ( x ) } and y B ) \{x\mid A(x)\}\in B\iff\exists y\colon(y=\{x\mid A(x)\}\and y\in B)

Classical_nucleation_theory.html

  1. R = N S Z j exp ( - Δ G * k B T ) R\ =\ N_{S}Zj\exp\left(\frac{-\Delta G^{*}}{k_{B}T}\right)
  2. Δ G * \Delta G^{*}
  3. N S N_{S}
  4. j j
  5. Z Z
  6. N S exp ( - Δ G * / k B T ) N_{S}\exp\left(-\Delta G^{*}/k_{B}T\right)
  7. exp [ - Δ G * / k T ] \exp[-\Delta G^{*}/kT]
  8. Δ G * \Delta G^{*}
  9. Z j Zj
  10. j j
  11. Z Z
  12. Z Z
  13. Δ G * = 275 k B T \Delta G^{*}=275k_{B}T
  14. Δ G * \Delta G^{*}
  15. Δ G * \Delta G^{*}
  16. r * r*
  17. Δ G ( r ) \Delta G(r)
  18. Δ G \Delta G
  19. Δ G = 4 3 π r 3 Δ g + 4 π r 2 σ \Delta G=\frac{4}{3}\pi r^{3}\Delta g+4\pi r^{2}\sigma
  20. r r
  21. Δ g \Delta g
  22. Δ g \Delta g
  23. Δ g \Delta g
  24. σ \sigma
  25. r r
  26. Δ G ( r ) > 0 \Delta G(r)>0
  27. r 2 r^{2}
  28. r 3 r^{3}
  29. r 3 r^{3}
  30. r r
  31. r 2 r^{2}
  32. r r
  33. r 2 r^{2}
  34. r r
  35. r 3 r^{3}
  36. r r
  37. Δ G \Delta G
  38. d G d r = 0 \frac{dG}{dr}=0
  39. r * = - 2 σ Δ g r^{*}=-\frac{2\sigma}{\Delta g}
  40. Δ G * \Delta G^{*}
  41. Δ G * = 16 π σ 3 3 ( Δ g ) 2 \Delta G^{*}=\frac{16\pi\sigma^{3}}{3(\Delta g)^{2}}
  42. R R
  43. 4 π r 2 4\pi r^{2}

Clebsch–Gordan_coefficients_for_SU(3).html

  1. ( G , * ) ) (G,*))
  2. G G
  3. x x
  4. y y
  5. G G
  6. x * y x*y
  7. G G
  8. x , y G , x * y x,y\in G,x*y
  9. G . G.
  10. x x
  11. y y
  12. z z
  13. G G
  14. ( x * y ) * z (x*y)*z
  15. x * ( y * z ) x*(y*z)
  16. G G
  17. ( x * y ) * z = x * ( y * z ) (x*y)*z=x*(y*z)
  18. x , y x,y
  19. z G z\in G
  20. e e
  21. G G
  22. G G
  23. e e
  24. x * e = e * x = x x*e=e*x=x
  25. x G x\in G
  26. x x
  27. G G
  28. y y
  29. G G
  30. x x
  31. y y
  32. e e
  33. x G x\in G
  34. y G y\in G
  35. x * y = y * x = e x*y=y*x=e
  36. x G x\in G
  37. x , y G \forall x,y\in G
  38. x * y = y * x x*y=y*x
  39. H ^ = T ^ H ^ T ^ - 1 = H ^ , \hat{H^{\prime}}=\hat{T}\hat{H}\hat{T}^{-1}=\hat{H},
  40. T ^ H ^ = H ^ T ^ , \hat{T}\hat{H}=\hat{H}\hat{T},
  41. [ T ^ , H ^ ] = 0 = [ H ^ , T ^ ] [\hat{T},\hat{H}]=0=[\hat{H},\hat{T}]
  42. T ^ \hat{T}
  43. T ^ = 𝕀 \hat{T}=\mathbb{I}
  44. S U ( 3 ) SU(3)
  45. S U ( 3 ) SU(3)
  46. S U ( 3 ) SU(3)
  47. U ( 3 ) U(3)
  48. U ( 3 ) U(3)
  49. U ( 1 ) S U ( 3 ) U(1)⊗SU(3)
  50. S U ( 3 ) SU(3)
  51. U U
  52. U = e i H U=e^{iH}\,
  53. S U ( 3 ) SU(3)
  54. U = e i a k λ k U=e^{i\sum{a_{k}\lambda_{k}}}
  55. λ k \lambda_{k}
  56. S U ( 3 ) SU(3)
  57. λ k \lambda_{k}
  58. det ( e A ) = e tr A \det(e^{A})=e^{\operatorname{tr}A}
  59. λ 1 = ( 0 1 0 1 0 0 0 0 0 ) λ 2 = ( 0 - i 0 i 0 0 0 0 0 ) λ 3 = ( 1 0 0 0 - 1 0 0 0 0 ) λ 4 = ( 0 0 1 0 0 0 1 0 0 ) λ 5 = ( 0 0 - i 0 0 0 i 0 0 ) λ 6 = ( 0 0 0 0 0 1 0 1 0 ) λ 7 = ( 0 0 0 0 0 - i 0 i 0 ) λ 8 = 1 3 ( 1 0 0 0 1 0 0 0 - 2 ) . \begin{array}[]{ccc}\lambda_{1}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}&\lambda_{2}=\begin{pmatrix}0&-i&0\\ i&0&0\\ 0&0&0\end{pmatrix}&\lambda_{3}=\begin{pmatrix}1&0&0\\ 0&-1&0\\ 0&0&0\end{pmatrix}\\ \\ \lambda_{4}=\begin{pmatrix}0&0&1\\ 0&0&0\\ 1&0&0\end{pmatrix}&\lambda_{5}=\begin{pmatrix}0&0&-i\\ 0&0&0\\ i&0&0\end{pmatrix}\\ \\ \lambda_{6}=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}&\lambda_{7}=\begin{pmatrix}0&0&0\\ 0&0&-i\\ 0&i&0\end{pmatrix}&\lambda_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&-2\end{pmatrix}.\end{array}
  60. S U ( 3 ) SU(3)
  61. tr ( λ j λ k ) = 2 δ i j . \operatorname{tr}(\lambda_{j}\lambda_{k})=2\delta_{ij}.
  62. λ k \lambda_{k}
  63. [ λ j , λ k ] = 2 i f j k l λ k , [\lambda_{j},\lambda_{k}]=2if_{jkl}\lambda_{k}~{},
  64. f j k l f_{jkl}
  65. f j k l f_{jkl}
  66. ϵ j k l \epsilon_{jkl}
  67. S U ( 2 ) SU(2)
  68. { λ j , λ k } = 4 3 δ j k + 2 d j k l λ l \{\lambda_{j},\lambda_{k}\}=\frac{4}{3}\delta_{jk}+2d_{jkl}\lambda_{l}
  69. d j k l d_{jkl}
  70. F i ^ = 1 2 λ i \hat{F_{i}}=\frac{1}{2}\lambda_{i}
  71. S U ( 3 ) SU(3)
  72. I ± ^ = F 1 ^ ± i F 2 ^ \hat{I_{\pm}}=\hat{F_{1}}\pm i\hat{F_{2}}
  73. I 3 ^ = F 3 ^ \hat{I_{3}}=\hat{F_{3}}
  74. V ± ^ = F 4 ^ ± i F 5 ^ \hat{V_{\pm}}=\hat{F_{4}}\pm i\hat{F_{5}}
  75. U ± ^ = F 6 ^ ± i F 7 ^ \hat{U_{\pm}}=\hat{F_{6}}\pm i\hat{F_{7}}
  76. Y ^ = 2 3 F 8 ^ . \hat{Y}=\frac{2}{\sqrt{3}}\hat{F_{8}}~{}.
  77. S U ( 3 ) SU(3)
  78. [ Y ^ , I ^ 3 ] = 0 , [\hat{Y},\hat{I}_{3}]=0,
  79. [ Y ^ , I ^ ± ] = 0 , [\hat{Y},\hat{I}_{\pm}]=0,
  80. [ Y ^ , U ^ ± ] = ± U ± ^ , [\hat{Y},\hat{U}_{\pm}]=\pm\hat{U_{\pm}},
  81. [ Y ^ , V ^ ± ] = ± V ± ^ , [\hat{Y},\hat{V}_{\pm}]=\pm\hat{V_{\pm}},
  82. [ I ^ 3 , I ^ ± ] = ± I ± ^ , [\hat{I}_{3},\hat{I}_{\pm}]=\pm\hat{I_{\pm}},
  83. [ I ^ 3 , U ^ ± ] = 1 2 U ± ^ , [\hat{I}_{3},\hat{U}_{\pm}]=\mp\frac{1}{2}\hat{U_{\pm}},
  84. [ I ^ 3 , V ^ ± ] = ± 1 2 V ± ^ , [\hat{I}_{3},\hat{V}_{\pm}]=\pm\frac{1}{2}\hat{V_{\pm}},
  85. [ I ^ + , I ^ - ] = 2 I ^ 3 , [\hat{I}_{+},\hat{I}_{-}]=2\hat{I}_{3},
  86. [ U ^ + , U ^ - ] = 3 2 Y ^ - I ^ 3 , [\hat{U}_{+},\hat{U}_{-}]=\frac{3}{2}\hat{Y}-\hat{I}_{3},
  87. [ V ^ + , V ^ - ] = 3 2 Y ^ + I ^ 3 , [\hat{V}_{+},\hat{V}_{-}]=\frac{3}{2}\hat{Y}+\hat{I}_{3},
  88. [ I ^ + , V ^ - ] = - U ^ - , [\hat{I}_{+},\hat{V}_{-}]=-\hat{U}_{-},
  89. [ I ^ + , U ^ + ] = V ^ + , [\hat{I}_{+},\hat{U}_{+}]=\hat{V}_{+},
  90. [ U ^ + , V ^ - ] = I ^ - , [\hat{U}_{+},\hat{V}_{-}]=\hat{I}_{-},
  91. [ I ^ + , V ^ + ] = 0 , [\hat{I}_{+},\hat{V}_{+}]=0,
  92. [ I ^ + , U ^ - ] = 0 , [\hat{I}_{+},\hat{U}_{-}]=0,
  93. [ U ^ + , V ^ + ] = 0. [\hat{U}_{+},\hat{V}_{+}]=0.
  94. S U ( 3 ) SU(3)
  95. I 3 - Y I_{3}-Y
  96. I ^ 3 \hat{I}_{3}
  97. Y ^ \hat{Y}
  98. S U ( 3 ) SU(3)
  99. S U ( 2 ) SU(2)
  100. S U ( 3 ) SU(3)
  101. C 1 ^ = k F k ^ F k ^ \hat{C_{1}}=\sum_{k}\hat{F_{k}}\hat{F_{k}}
  102. C 2 ^ = j k l d j k l F j ^ F k ^ F l ^ . \hat{C_{2}}=\sum_{jkl}d_{jkl}\hat{F_{j}}\hat{F_{k}}\hat{F_{l}}~{}.
  103. S U ( 3 ) SU(3)
  104. D D
  105. D ( p , q ) D(p,q)
  106. ( p 2 + q 3 + 3 p + 3 q + p q ) / 3 (p^{2}+q^{3}+3p+3q+pq)/3
  107. ( p - q ) ( 3 + p + 2 q ) ( 3 + q + 2 p ) / 18. (p-q)(3+p+2q)(3+q+2p)/18.
  108. p p
  109. q q
  110. p = q p=q
  111. D D
  112. D ( p , q ) D(p,q)
  113. p p
  114. q q
  115. p p
  116. q q
  117. d ( p , q ) = 1 2 ( p + 1 ) ( q + 1 ) ( p + q + 2 ) . d(p,q)=\frac{1}{2}(p+1)(q+1)(p+q+2).
  118. S U ( 3 ) SU(3)
  119. S U ( 2 ) SU(2)
  120. I I
  121. [ I ^ 3 , Y ^ ] = 0 [\hat{I}_{3},\hat{Y}]=0
  122. I ^ 3 \hat{I}_{3}
  123. Y ^ \hat{Y}
  124. | t , y |t,y\rangle
  125. I ^ 3 | t , y = t | t , y \hat{I}_{3}|t,y\rangle=t|t,y\rangle
  126. Y ^ | t , y = y | t , y \hat{Y}|t,y\rangle=y|t,y\rangle
  127. U ^ 0 | t , y = ( 3 4 y - 1 2 t ) | t , y \hat{U}_{0}|t,y\rangle=(\frac{3}{4}y-\frac{1}{2}t)|t,y\rangle
  128. V ^ 0 | t , y = ( 3 4 y + 1 2 t ) | t , y \hat{V}_{0}|t,y\rangle=(\frac{3}{4}y+\frac{1}{2}t)|t,y\rangle
  129. I ^ ± | t , y = α | t ± 1 2 , y \hat{I}_{\pm}|t,y\rangle=\alpha|t\pm\frac{1}{2},y\rangle
  130. U ^ ± | t , y = β | t ± 1 2 , y ± 1 \hat{U}_{\pm}|t,y\rangle=\beta|t\pm\frac{1}{2},y\pm 1\rangle
  131. V ^ ± | t , y = γ | t 1 2 , y ± 1 \hat{V}_{\pm}|t,y\rangle=\gamma|t\mp\frac{1}{2},y\pm 1\rangle
  132. U ^ 0 1 2 [ U ^ + , U ^ - ] = 3 4 Y ^ - 1 2 I ^ 3 \hat{U}_{0}\equiv\frac{1}{2}[\hat{U}_{+},\hat{U}_{-}]=\frac{3}{4}\hat{Y}-\frac% {1}{2}\hat{I}_{3}
  133. V ^ 0 1 2 [ V ^ + , V ^ - ] = 3 4 Y ^ + 1 2 I ^ 3 . \hat{V}_{0}\equiv\frac{1}{2}[\hat{V}_{+},\hat{V}_{-}]=\frac{3}{4}\hat{Y}+\frac% {1}{2}\hat{I}_{3}.
  134. I ^ ± , U ^ ± \hat{I}_{\pm},\hat{U}_{\pm}
  135. V ^ ± \hat{V}_{\pm}
  136. D ( p 1 , q 1 ) D(p_{1},q_{1})
  137. D ( p 2 , q 2 ) D(p_{2},q_{2})
  138. D ( p 1 , q 1 ) D ( p 2 , q 2 ) = P , Q σ ( P , Q ) D ( P , Q ) , D(p_{1},q_{1})\otimes D(p_{2},q_{2})=\sum_{P,Q}\oplus\sigma(P,Q)D(P,Q)~{},
  139. σ ( P , Q ) σ(P,Q)
  140. D ( 1 , 1 ) D ( 1 , 1 ) = D ( 2 , 2 ) D ( 3 , 0 ) D ( 1 , 1 ) D ( 1 , 1 ) D ( 0 , 3 ) D ( 0 , 0 ) , D(1,1)\otimes D(1,1)=D(2,2)\oplus D(3,0)\oplus D(1,1)\oplus D(1,1)\oplus D(0,3% )\oplus D(0,0)~{},
  141. D ( P , Q ) D(P,Q)
  142. σ ( P , Q ) \sigma(P,Q)
  143. D ( p 1 , q 1 ) D(p_{1},q_{1})
  144. D ( p 2 , q 2 ) D(p_{2},q_{2})
  145. D ( p , q ) D(p,q)
  146. { C ^ 1 , C ^ 2 , I ^ 3 , I ^ 2 , Y ^ } , \{\hat{C}_{1},\hat{C}_{2},\hat{I}_{3},\hat{I}^{2},\hat{Y}\}~{},
  147. I 2 I 1 2 + I 3 2 + I 3 2 I^{2}\equiv{I_{1}}^{2}+{I_{3}}^{2}+{I_{3}}^{2}
  148. { C ^ 1 ( 1 ) , C ^ 2 ( 1 ) , I ^ 3 ( 1 ) , I ^ 2 ( 1 ) , Y ^ ( 1 ) , C ^ 1 ( 2 ) , C ^ 2 ( 2 ) , I ^ 3 ( 2 ) , I ^ 2 ( 2 ) , Y ^ ( 2 ) } , \{\hat{C}_{1}(1),\hat{C}_{2}(1),\hat{I}_{3}(1),\hat{I}^{2}(1),\hat{Y}(1),\hat{% C}_{1}(2),\hat{C}_{2}(2),\hat{I}_{3}(2),\hat{I}^{2}(2),\hat{Y}(2)\},
  149. ^ 1 = C ^ 1 ( 1 ) + C ^ 1 ( 2 ) \hat{\mathbb{C}}_{1}=\hat{C}_{1}(1)+\hat{C}_{1}(2)
  150. ^ 2 = C ^ 2 ( 1 ) + C ^ 2 ( 2 ) \hat{\mathbb{C}}_{2}=\hat{C}_{2}(1)+\hat{C}_{2}(2)
  151. 𝕀 ^ 2 = I ^ 2 ( 1 ) + I ^ 2 ( 2 ) \hat{\mathbb{I}}^{2}=\hat{I}^{2}(1)+\hat{I}^{2}(2)
  152. 𝕐 ^ = Y ^ ( 1 ) + Y ^ ( 2 ) \hat{\mathbb{Y}}=\hat{Y}(1)+\hat{Y}(2)
  153. 𝕀 ^ 3 = I ^ 3 ( 1 ) + I ^ 3 ( 2 ) . \hat{\mathbb{I}}_{3}=\hat{I}_{3}(1)+\hat{I}_{3}(2).
  154. { ^ 1 , ^ 2 , 𝕐 ^ , 𝕀 ^ 3 , 𝕀 ^ 2 , C ^ 1 ( 1 ) , C ^ 1 ( 2 ) , C ^ 2 ( 1 ) , C ^ 2 ( 2 ) } . \{\hat{\mathbb{C}}_{1},\hat{\mathbb{C}}_{2},\hat{\mathbb{Y}},\hat{\mathbb{I}}_% {3},\hat{\mathbb{I}}^{2},\hat{C}_{1}(1),\hat{C}_{1}(2),\hat{C}_{2}(1),\hat{C}_% {2}(2)\}.
  155. Γ \Gamma
  156. D ( P , Q ) D(P,Q)
  157. P P
  158. Q Q
  159. C ^ 1 ( 1 ) \hat{C}_{1}(1)
  160. c 1 1 {c^{1}}_{1}
  161. C ^ 1 ( 2 ) \hat{C}_{1}(2)
  162. c 1 2 {c^{1}}_{2}
  163. C ^ 2 ( 1 ) \hat{C}_{2}(1)
  164. c 2 1 {c^{2}}_{1}
  165. C ^ 2 ( 2 ) \hat{C}_{2}(2)
  166. c 2 2 {c^{2}}_{2}
  167. 𝕀 ^ 2 \hat{\mathbb{I}}^{2}
  168. i 2 {i^{2}}
  169. 𝕀 ^ 3 \hat{\mathbb{I}}_{3}
  170. i z {i^{z}}
  171. 𝕐 ^ \hat{\mathbb{Y}}
  172. y {y}
  173. Γ ^ \hat{\Gamma}
  174. γ \gamma
  175. ^ 1 \hat{\mathbb{C}}_{1}
  176. c 1 {c^{1}}
  177. ^ 2 \hat{\mathbb{C}}_{2}
  178. c 1 {c^{1}}
  179. Y ^ 1 \hat{Y}_{1}
  180. y 1 {y}_{1}
  181. Y ^ 2 \hat{Y}_{2}
  182. y 2 {y}_{2}
  183. I ^ 2 ( 1 ) \hat{I}^{2}(1)
  184. i 2 1 {i^{2}}_{1}
  185. I ^ 2 ( 2 ) \hat{I}^{2}(2)
  186. i 2 2 {i^{2}}_{2}
  187. I ^ 3 ( 1 ) \hat{I}_{3}(1)
  188. i z 1 {i^{z}}_{1}
  189. I ^ 3 ( 2 ) \hat{I}_{3}(2)
  190. i z 2 {i^{z}}_{2}
  191. | c 1 1 , c 1 2 , c 2 1 , c 2 2 , y 1 , y 2 , i 2 1 , i 2 2 , i z 1 , i z 2 |{c^{1}}_{1},{c^{1}}_{2},{c^{2}}_{1},{c^{2}}_{2},y_{1},y_{2},{i^{2}}_{1},{i^{2% }}_{2},{i^{z}}_{1},{i^{z}}_{2}\rangle
  192. | c 1 1 , c 1 2 , c 2 1 , c 2 2 , y , γ , i 2 , i z , c 1 , c 2 |{c^{1}}_{1},{c^{1}}_{2},{c^{2}}_{1},{c^{2}}_{2},y,\gamma,{i^{2}},{i^{z}},c^{1% },c^{2}\rangle
  193. c 1 1 , c 1 2 , c 2 1 , c 2 2 {c^{1}}_{1},{c^{1}}_{2},{c^{2}}_{1},{c^{2}}_{2}
  194. | y 1 , y 2 , i 2 1 , i 2 2 , i z 1 , i z 2 |y_{1},y_{2},{i^{2}}_{1},{i^{2}}_{2},{i^{z}}_{1},{i^{z}}_{2}\rangle
  195. | y , γ , i 2 , i z , c 1 , c 2 , |y,\gamma,{i^{2}},{i^{z}},c^{1},c^{2}\rangle~{},
  196. D ( p 2 , q 2 ) D(p_{2},q_{2})
  197. ϕ μ 1 ν 1 , ϕ μ 2 ν 2 {\phi^{\mu_{1}}}_{\nu_{1}},{\phi^{\mu_{2}}}_{\nu_{2}}
  198. ν 1 , ν 2 \nu_{1},\nu_{2}
  199. ( i 2 1 , i z 1 , y 1 ) ({i^{2}}_{1},{i^{z}}_{1},y_{1})
  200. ( i 2 2 , i z 2 , y 2 ) ({i^{2}}_{2},{i^{z}}_{2},y_{2})
  201. ψ ( μ 1 μ 2 γ ν ) = ν 1 , ν 2 ( μ 1 μ 2 γ ν 1 ν 2 ν ) ϕ μ 1 ν 1 ϕ μ 2 ν 2 \psi\begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ &&\nu\end{pmatrix}=\sum_{\nu_{1},\nu_{2}}\begin{pmatrix}\mu_{1}&\mu_{2}&\gamma% \\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}{\phi^{\mu_{1}}}_{\nu_{1}}{\phi^{\mu_{2}}}_{% \nu_{2}}
  202. ( μ 1 μ 2 γ ν 1 ν 2 ν ) \begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}
  203. ϕ μ 1 ν 1 ϕ μ 2 ν 2 = μ , ν , γ ( μ 1 μ 2 γ ν 1 ν 2 ν ) ψ ( μ 1 μ 2 γ ν ) . {\phi^{\mu_{1}}}_{\nu_{1}}{\phi^{\mu_{2}}}_{\nu_{2}}=\sum_{\mu,\nu,\gamma}% \begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}\psi\begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ &&\nu\end{pmatrix}.
  204. ν 1 , ν 2 ( μ 1 μ 2 γ ν 1 ν 2 ν ) ( μ 1 μ 2 γ ν 1 ν 2 ν ) = δ ν ν δ γ , γ \sum_{\nu_{1},\nu_{2}}\begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}\begin{pmatrix}\mu_{1}&\mu_{2}&\gamma^{\prime}% \\ \nu_{1}&\nu_{2}&\nu^{\prime}\end{pmatrix}=\delta_{\nu\nu^{\prime}}\delta_{% \gamma,\gamma^{\prime}}
  205. μ ν γ ( μ 1 μ 2 γ ν 1 ν 2 ν ) ( μ 1 μ 2 γ ν 1 ν 2 ν ) = δ ν 1 ν 1 δ ν 2 , ν 2 \sum_{\mu\nu\gamma}\begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}\begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}^{\prime}&\nu_{2}^{\prime}&\nu\end{pmatrix}=\delta_{\nu_{1}\nu_{1}^{% \prime}}\delta_{\nu_{2},\nu_{2}^{\prime}}
  206. μ γ {{\mu}_{\gamma}}
  207. μ 1 μ 2 {{\mu}_{1}\otimes{\mu}_{2}}
  208. μ 2 μ 1 {{\mu}_{2}\otimes{\mu}_{1}}
  209. ( μ 1 μ 2 γ ν 1 ν 2 ν ) = ξ 1 ( μ 2 μ 1 γ ν 2 ν 1 ν ) \begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}=\xi_{1}\begin{pmatrix}\mu_{2}&\mu_{1}&\gamma% \\ \nu_{2}&\nu_{1}&\nu\end{pmatrix}
  210. ξ 1 = ξ 1 ( μ 1 , μ 2 , γ ) = ± 1 \xi_{1}=\xi_{1}(\mu_{1},\mu_{2},\gamma)=\pm 1
  211. ( μ 1 μ 2 γ ν 1 ν 2 ν ) = ξ 2 ( μ 1 * μ 2 * γ * ν 1 ν 2 ν ) \begin{pmatrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}=\xi_{2}\begin{pmatrix}{\mu_{1}}^{*}&{\mu_{2}}% ^{*}&{\gamma}^{*}\\ \nu_{1}&\nu_{2}&\nu\end{pmatrix}
  212. ξ 2 = ξ 2 ( μ 1 , μ 2 , γ ) = ± 1 \xi_{2}=\xi_{2}(\mu_{1},\mu_{2},\gamma)=\pm 1
  213. H ^ = - 1 2 2 + 1 2 ( x 2 + y 2 + z 2 ) , \hat{H}=-\frac{1}{2}\nabla^{2}+\frac{1}{2}(x^{2}+y^{2}+z^{2}),
  214. ħ ħ
  215. m m
  216. V = x 2 + y 2 + z 2 V=x^{2}+y^{2}+z^{2}
  217. S O ( 3 ) SO(3)
  218. S U ( 3 ) SU(3)
  219. S U ( 3 ) SU(3)
  220. A ^ i j = 1 2 p i p j + ω 2 r i r j \hat{A}_{ij}=\frac{1}{2}p_{i}p_{j}+\omega^{2}r_{i}r_{j}
  221. [ A ^ i j , H ^ ] = 0. [\hat{A}_{ij},\hat{H}]=0.
  222. j A ^ i j L ^ j = i L ^ i A ^ i j = 0 \sum_{j}\hat{A}_{ij}\hat{L}_{j}=\sum_{i}\hat{L}_{i}\hat{A}_{ij}=0
  223. j A ^ i j A ^ j k = H ^ A ^ i k + 1 4 ω 2 { L ^ i L ^ k - δ i k L ^ 2 + 2 [ L ^ i , L ^ k ] - 2 2 δ i k } \sum_{j}\hat{A}_{ij}\hat{A}_{jk}=\hat{H}\hat{A}_{ik}+\frac{1}{4}\omega^{2}\{% \hat{L}_{i}\hat{L}_{k}-\delta_{ik}\hat{L}^{2}+2[\hat{L}_{i},\hat{L}_{k}]-2% \hbar^{2}\delta_{ik}\}
  224. T r [ A ^ i j ] = i A ^ i i = H ^ . Tr[\hat{A}_{ij}]=\sum_{i}{\hat{A}_{ii}}=\hat{H}.
  225. A ^ i j , \hat{A}_{ij},
  226. A ^ 0 = ω - 1 ( 2 A ^ 33 - A ^ 11 - A ^ 22 ) \hat{A}_{0}=\omega^{-1}(2\hat{A}_{33}-\hat{A}_{11}-\hat{A}_{22})
  227. A ^ ± = ω - 1 ( A ^ 13 ± i A ^ 2 3 ) \hat{A}_{\pm}=\mp\omega^{-1}(\hat{A}_{13}\pm i\hat{A}_{2}3)
  228. A ^ ± = ω - 1 ( A ^ 11 - A ^ 22 ± 2 i A ^ 22 ) \hat{A^{\prime}}_{\pm}=\omega^{-1}(\hat{A}_{11}-\hat{A}_{22}\pm 2i\hat{A}_{22})
  229. L ^ 3 = L ^ 3 \hat{L}_{3}=\hat{L}_{3}
  230. L ^ ± = ( L ^ 1 ± i L ^ 2 ) . \hat{L}_{\pm}=(\hat{L}_{1}\pm i\hat{L}_{2}).
  231. [ L ^ 3 , A ^ 0 ] = [ A ^ 0 , A ^ ± ] = [ A ^ ± , A ^ ± ] = [ L ^ ± , A ^ ± ] [\hat{L}_{3},\hat{A}_{0}]=[\hat{A}_{0},\hat{A^{\prime}}_{\pm}]=[\hat{A}_{\pm},% \hat{A^{\prime}}_{\pm}]=[\hat{L}_{\pm},\hat{A^{\prime}}_{\pm}]
  232. [ L ^ ± , L ^ ] = - 4 [ A ^ ± , A ^ ] = 1 2 [ A ^ ± , A ^ ] = ± 2 L ^ 3 [\hat{L}_{\pm},\hat{L}_{\mp}]=-4[\hat{A}_{\pm},\hat{A}_{\mp}]=\frac{1}{2}[\hat% {A^{\prime}}_{\pm},\hat{A^{\prime}}_{\mp}]=\pm 2\hbar\hat{L}_{3}
  233. [ L ^ ± , A ^ ] = A ^ 0 [\hat{L}_{\pm},\hat{A}_{\mp}]=\hbar\hat{A}_{0}
  234. ± [ L ^ 3 , L ^ ± ] = - 2 3 [ A ^ 0 , A ^ ± ] = [ A ^ , A ^ ± ] = L ^ ± \pm[\hat{L}_{3},\hat{L}_{\pm}]=-\frac{2}{3}[\hat{A}_{0},\hat{A}_{\pm}]=[\hat{A% }_{\mp},\hat{A^{\prime}}_{\pm}]=\hbar\hat{L}_{\pm}
  235. ± [ L ^ 3 , A ^ ± ] = - 1 6 [ A ^ 0 , L ^ ± ] = 1 4 [ L ^ , A ^ ± ] = A ^ ± \pm[\hat{L}_{3},\hat{A}_{\pm}]=-\frac{1}{6}[\hat{A}_{0},\hat{L}_{\pm}]=\frac{1% }{4}[\hat{L}_{\mp},\hat{A^{\prime}}_{\pm}]=\hbar\hat{A}_{\pm}
  236. ± [ L ^ 3 , A ^ ± ] = 2 [ L ^ ± , A ^ ± ] = 2 A ^ ± . \pm[\hat{L}_{3},\hat{A^{\prime}}_{\pm}]=2[\hat{L}_{\pm},\hat{A}_{\pm}]=2\hbar% \hat{A^{\prime}}_{\pm}.
  237. { A ^ i j ; i j } \{\hat{A}_{ij};i\neq j\}
  238. S U ( 3 ) SU(3)
  239. S U ( 3 ) SU(3)
  240. a i ^ = X i ^ + i P i ^ \hat{a_{i}}=\hat{X_{i}}+i\hat{P_{i}}~{}~{}
  241. a i ^ = X i ^ + i P i ^ ~{}~{}\hat{a_{i}}^{\dagger}=\hat{X_{i}}+i\hat{P_{i}}
  242. a i ^ \hat{a_{i}}
  243. a i ^ \hat{a_{i}}^{\dagger}
  244. a i ^ a j ^ \hat{a_{i}}\hat{a_{j}}^{\dagger}
  245. i , j = 1 , 2 , 3 i,j=1,2,3
  246. a i ^ a j ^ \hat{a_{i}}^{\dagger}\hat{a_{j}}
  247. [ a i ^ , a j ^ ] = 2 i δ i j , [\hat{a_{i}},\hat{a_{j}}^{\dagger}]=2i\delta_{ij},
  248. [ a i ^ , a j ^ ] = [ a i ^ , a j ^ ] = 0 , [\hat{a_{i}},\hat{a_{j}}]=[\hat{a_{i}}^{\dagger},\hat{a_{j}}^{\dagger}]=0,
  249. N i ^ = a i ^ a i ^ \hat{N_{i}}=\hat{a_{i}}\hat{a_{i}}^{\dagger}
  250. H ^ = ω [ 3 2 + N 1 ^ + N 2 ^ + N 3 ^ ] \hat{H}=\omega\left[\frac{3}{2}+\hat{N_{1}}+\hat{N_{2}}+\hat{N_{3}}\right]
  251. a i ^ \hat{a_{i}}
  252. a j ^ \hat{a_{j}}
  253. N i N_{i}
  254. N j N_{j}
  255. N = N i N=\sum{N_{i}}
  256. = 1 𝕀 + 𝕀 2 . \mathbb{H}=\mathbb{H}_{1}\otimes\mathbb{I}+\mathbb{I}\otimes\mathbb{H}_{2}.
  257. 1 \mathbb{H}_{1}
  258. 2 \mathbb{H}_{2}
  259. S U ( 3 ) SU(3)
  260. 𝐒 𝐧 \mathbf{S_{n}}
  261. 𝐒 𝐧 \mathbf{S_{n}}
  262. ψ 1 , 2 \psi_{1,2}
  263. 𝐒 𝟏𝟐 = 𝐈 + 𝐏 𝟏𝟐 \mathbf{S_{12}}=\mathbf{I}+\mathbf{P_{12}}
  264. 𝐀 𝟏𝟐 = 𝐈 - 𝐏 𝟏𝟐 \mathbf{A_{12}}=\mathbf{I}-\mathbf{P_{12}}
  265. 𝐏 𝟏𝟐 \mathbf{P_{12}}
  266. 𝐏 𝟏𝟐 𝐏 𝟏𝟐 = 𝐈 \mathbf{P_{12}P_{12}}=\mathbf{I}
  267. 𝐏 𝟏𝟐 𝐒 𝟏𝟐 = 𝐏 𝟏𝟐 + 𝐈 = 𝐒 𝟏𝟐 \mathbf{P_{12}S_{12}}=\mathbf{P_{12}}+\mathbf{I}=\mathbf{S_{12}}
  268. 𝐏 𝟏𝟐 𝐀 𝟏𝟐 = 𝐏 𝟏𝟐 - 𝐈 = - 𝐀 𝟏𝟐 \mathbf{P_{12}A_{12}}=\mathbf{P_{12}}-\mathbf{I}=-\mathbf{A_{12}}
  269. 𝐏 𝟏𝟐 𝐒 𝟏𝟐 ψ 12 = + 𝐒 𝟏𝟐 ψ 12 \mathbf{P_{12}}\mathbf{S_{12}}\psi_{12}=+\mathbf{S_{12}}\psi_{12}
  270. 𝐏 𝟏𝟐 𝐒 𝟏𝟐 ψ 12 = - 𝐀 𝟏𝟐 ψ 12 . \mathbf{P_{12}}\mathbf{S_{12}}\psi_{12}=-\mathbf{A_{12}}\psi_{12}.
  271. 𝐒 𝟏𝟐 \mathbf{S_{12}}
  272. 𝐀 𝟏𝟐 \mathbf{A_{12}}
  273. ψ \psi
  274. ( ) (\Box)
  275. N p N_{p}
  276. N p N_{p}\Box
  277. \Box
  278. D ( p , q ) D(p,q)
  279. p + q p+q
  280. q q
  281. D ( p 1 , q 1 ) D ( p 2 , q 2 ) = P , Q D ( P , Q ) D(p_{1},q_{1})\otimes D(p_{2},q_{2})=\sum_{P,Q}\oplus D(P,Q)
  282. D ( 1 , 0 ) D ( 1 , 1 ) = D ( 2 , 1 ) D ( 0 , 2 ) D ( 1 , 0 ) D(1,0)\otimes D(1,1)=D(2,1)\oplus D(0,2)\oplus D(1,0)

Clement_W._H._Lam.html

  1. A 2 = d I + λ J A^{2}=dI+\lambda J

Clifford_module_bundle.html

  1. S ( M ) = F Spin ( M ) × σ V S(M)=F_{\mathrm{Spin}}(M)\times_{\sigma}V\,
  2. C ( T * M ) S ( M ) S ( M ) C\ell(T^{*}M)\otimes S(M)\to S(M)

Clohessy-Wiltshire_equations.html

  1. x ¨ = 3 n 2 x + 2 n y ˙ \ddot{x}=3n^{2}x+2n\dot{y}
  2. y ¨ = - 2 n x ˙ \ddot{y}=-2n\dot{x}
  3. z ¨ = - n 2 z \ddot{z}=-n^{2}z
  4. n = μ a 3 n=\sqrt{\frac{\mu}{a^{3}}}
  5. μ = 3.986 E 14 m 3 s 2 \mu=3.986E14\frac{m^{3}}{s^{2}}
  6. a = 6378137 m + 415000 m = 6793137 m a=6378137m+415000m=6793137m
  7. n = 0.0011 s - 1 n=0.0011s^{-1}

Closed_subgroup_theorem.html

  1. H H
  2. G G
  3. H H
  4. G G
  5. 𝐠 \mathbf{g}
  6. H G H⊂G
  7. 𝐠 \mathbf{g}
  8. e x p ( x ) exp(x)
  9. H H
  10. e x p ( 𝐡 ) exp(\mathbf{h})
  11. U 𝐡 U⊂\mathbf{h}
  12. l o g log
  13. e x p ( U ) exp(U)
  14. X X
  15. U U
  16. l o g ( e x p ( X ) ) = X log(exp(X))=X
  17. x x
  18. e x p ( U ) exp(U)
  19. e x p ( l o g ( x ) ) = x exp(log(x))=x
  20. h e x p ( U ) hexp(U)
  21. h h
  22. H H
  23. U 𝐡 U⊂\mathbf{h}
  24. H H
  25. h h
  26. e x p ( U ) exp(U)
  27. l o g ( h ) log(h)
  28. 𝐡 \mathbf{h}
  29. H H
  30. h h
  31. g e x p ( U ) gexp(U)
  32. 𝐡 \mathbf{h}
  33. g g
  34. H H
  35. 𝐡 \mathbf{h}
  36. 𝐡 \mathbf{h}
  37. H H
  38. ( U , φ ) (U,φ)
  39. M M
  40. N M N⊂M
  41. n n
  42. N N
  43. p N p∈N
  44. N N
  45. M M
  46. N N
  47. H H
  48. G G
  49. G = 𝕋 2 = { ( e 2 π i θ 0 0 e 2 π i ϕ ) | θ , ϕ } , G=\mathbb{T}^{2}=\left\{\left.\left(\begin{matrix}e^{2\pi i\theta}&0\\ 0&e^{2\pi i\phi}\end{matrix}\right)\right|\theta,\phi\in\mathbb{R}\right\},
  50. H = { ( e 2 π i θ 0 0 e 2 π i a θ ) | θ } with Lie algebra 𝔥 = { ( i θ 0 0 i a θ ) | θ } , H=\left\{\left.\left(\begin{matrix}e^{2\pi i\theta}&0\\ 0&e^{2\pi ia\theta}\end{matrix}\right)\right|\theta\in\mathbb{R}\right\}\,% \text{with Lie algebra }\mathfrak{h}=\left\{\left.\left(\begin{matrix}i\theta&% 0\\ 0&ia\theta\end{matrix}\right)\right|\theta\in\mathbb{R}\right\},
  51. a a
  52. H H
  53. G G
  54. H H
  55. H H
  56. H H
  57. H H
  58. U U
  59. 𝐡 \mathbf{h}
  60. U U
  61. U 𝐡 U⊂\mathbf{h}
  62. V V
  63. H H
  64. 𝐡 \mathbf{h}
  65. H G H⊂G
  66. G / H G/H
  67. π : G G / H π:G→G/H
  68. G / H G/H
  69. G G
  70. X X
  71. x X x∈X
  72. X X
  73. G L ( n , ) GL(n,ℝ)
  74. G L ( n , ) GL(n,ℂ)
  75. H G H⊂G
  76. G L ( F , n ) GL(F,n)
  77. F = , F=ℝ,ℂ
  78. U G U⊂G
  79. H U H∩U
  80. U U
  81. A A
  82. B B
  83. H H
  84. 𝐡 𝐠 \mathbf{h}⊂\mathbf{g}
  85. X 𝐠 𝐡 , X X , 𝐡 𝐡 X∈\mathbf{g}\ \mathbf{h},XX,\mathbf{h}∈\mathbf{h}
  86. Γ ( 𝐡 ) Γ(\mathbf{h})
  87. G G
  88. X 𝐠 X∈\mathbf{g}
  89. X X
  90. X X
  91. 𝐡 𝐠 \mathbf{h}⊂\mathbf{g}
  92. 𝐤 \mathbf{k}
  93. 𝐡 \mathbf{h}
  94. Γ ( 𝐡 ) Γ(\mathbf{h})
  95. G G
  96. H G H⊂G
  97. H H
  98. G = G L ( n , ) G=GL(n,ℝ)
  99. G G
  100. G G
  101. H G H⊂G
  102. G G
  103. G L ( n , ) GL(n,ℝ)
  104. H H
  105. G L ( n , ) GL(n,ℝ)
  106. G L ( n , ) GL(n,ℝ)
  107. G G L ( n , ) G⊂GL(n,ℝ)
  108. H H
  109. G L ( n , ) GL(n,ℝ)
  110. U U
  111. 0
  112. V V
  113. I I
  114. G L ( n , ) GL(n,ℝ)
  115. e x p : U e x p ( U ) exp:U→exp(U)
  116. H V H∩V
  117. I I
  118. H H
  119. 𝐠 \mathbf{g}
  120. ( X , Y ) T r ( X Y ) (X,Y)→Tr(XY)
  121. 𝐡 \mathbf{h}
  122. H H
  123. 𝐡 \mathbf{h}
  124. 𝐠 \mathbf{g}
  125. 𝐠 = 𝐬 𝐡 \mathbf{g}=\mathbf{s}⊕\mathbf{h}
  126. X 𝐠 X∈\mathbf{g}
  127. X = S + H X=S+H
  128. S 𝐬 , H 𝐡 S∈\mathbf{s},H∈\mathbf{h}
  129. Φ : 𝐠 G L ( n , ) Φ:\mathbf{g}→GL(n,ℝ)
  130. Φ ( S , H ) = e t S e t H = ( I + t S + O ( t 2 ) ) ( I + t H + O ( t 2 ) ) = I + t S + t H + O ( t 2 ) , \Phi(S,H)=e^{tS}e^{tH}=(I+tS+O(t^{2}))(I+tH+O(t^{2}))=I+tS+tH+O(t^{2}),
  131. 0
  132. S + H S+H
  133. Φ Φ
  134. Φ Φ
  135. U U
  136. V V
  137. Β Β
  138. 0 𝐠 0∈\mathbf{g}
  139. i i
  140. 0
  141. Β Β
  142. 𝐬 \mathbf{s}
  143. 𝐬 \mathbf{s}
  144. Y 𝐬 Y∈\mathbf{s}
  145. i i
  146. t = τ t=τ
  147. ( e S i ) m i = e m i S i = e m i S i S i S i e τ Y . (e^{S_{i}})^{m_{i}}=e^{m_{i}S_{i}}=e^{m_{i}\|S_{i}\|\frac{S_{i}}{\|S_{i}\|}}% \rightarrow e^{\tau Y}.
  148. H H
  149. H H
  150. i i
  151. H H
  152. i i
  153. e < s u p > t Y H , t e<sup>tY∈H,∀t
  154. U 𝐠 U⊂\mathbf{g}
  155. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  156. h h
  157. h U hU
  158. h U hU
  159. h h
  160. U U
  161. m m
  162. B < s u b > 1 U 1 B<sub>1⊂U_{1}

Closest_point_method.html

  1. 𝒮 , c p ( x ) \mathcal{S},cp(x)
  2. 𝒮 \mathcal{S}
  3. x x
  4. 𝒮 \mathcal{S}
  5. d \mathbb{R}^{d}
  6. u : 𝒮 u:\mathcal{S}\rightarrow\mathbb{R}
  7. Ω \Omega
  8. 𝒮 \mathcal{S}
  9. v : Ω v:\Omega\rightarrow\mathbb{R}
  10. v ( x ) = u ( c p ( x ) ) v(x)=u(cp(x))
  11. 3 \mathbb{R}^{3}
  12. 3 \mathbb{R}^{3}
  13. Ω c x : x - c p ( x ) 2 λ \Omega_{c}{x:\|x-cp(x)\|_{2}\leq\lambda}
  14. λ \lambda
  15. u S ( θ , t ) = sin ( θ ) u_{S}(\theta,t)=\sin(\theta)
  16. u S ( θ , t ) = exp ( - t ) sin ( θ ) u_{S}(\theta,t)=\exp(-t)\sin(\theta)
  17. Δ t = 0.1 Δ x 2 \Delta t=0.1\Delta x^{2}
  18. u u

Clumping_factor.html

  1. C ( ρ ) = ρ 2 / ρ 2 \operatorname{C}(\rho)=\langle\rho^{2}\rangle/\langle\rho\rangle^{2}
  2. C ( ρ ) = Var ( ρ ) / E ( ρ ) 2 - 1. \operatorname{C}(\rho)=\operatorname{Var}(\rho)/\operatorname{E}(\rho)^{2}-1.
  3. ρ t r u e = ρ o b s C 1 / 2 \rho_{true}=\frac{\rho_{obs}}{\operatorname{C}^{1/2}}

Cobordism_ring.html

  1. Ω * S O = 0 Ω n S O \Omega^{SO}_{*}=\oplus_{0}^{\infty}\Omega^{SO}_{n}
  2. Ω n S O \Omega^{SO}_{n}
  3. Ω * O \Omega^{O}_{*}
  4. Ω * B \Omega^{B}_{*}
  5. Ω n O = π n ( M O ) \Omega^{O}_{n}=\pi_{n}(MO)

COBYLA.html

  1. x 𝒮 \vec{x}\in\mathcal{S}
  2. 𝒮 n \mathcal{S}\subseteq\mathbb{R}^{n}
  3. f ( x ) f(\vec{x})
  4. f f

Cobyrinate_a,c-diamide_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Cocycle_category.html

  1. X 𝑓 Z 𝑔 Y X\overset{f}{\leftarrow}Z\overset{g}{\rightarrow}Y
  2. H ( X , Y ) H(X,Y)
  3. π 0 H ( X , Y ) [ X , Y ] , ( f , g ) g f - 1 \pi_{0}H(X,Y)\to[X,Y],\quad(f,g)\mapsto g\circ f^{-1}

Coefficient_of_colligation.html

  1. Y = a d - b c a d + b c . Y=\frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}.
  2. Y = O R - 1 O R + 1 Y=\frac{\sqrt{OR}-1}{\sqrt{OR}+1}

Coenzyme_F420-0:L-glutamate_ligase.html

  1. \rightleftharpoons

Coenzyme_F420-1:gamma-L-glutamate_ligase.html

  1. \rightleftharpoons

Coenzyme_gamma-F420-2:alpha-L-glutamate_ligase.html

  1. \rightleftharpoons

Coexistence_theory.html

  1. r i ^ = b i ( k i - k ^ + A ) \hat{r_{i}}=b_{i}(k_{i}-\hat{k}+A)
  2. r i ^ \hat{r_{i}}
  3. b i b_{i}
  4. k i - k ^ k_{i}-\hat{k}
  5. A A

Cohen's_h.html

  1. ϕ = 2 arcsin p \phi=2\arcsin\sqrt{p}
  2. p 1 p_{1}
  3. p 2 p_{2}
  4. h = ϕ 1 - ϕ 2 h=\phi_{1}-\phi_{2}
  5. h = | ϕ 1 - ϕ 2 | h=\left|\phi_{1}-\phi_{2}\right|

Coherent_turbulent_structure.html

  1. Ω c \Omega_{c}
  2. - u c ν c -u_{c}\nu_{c}
  3. - u r ν r -\langle u_{r}\nu_{r}\rangle
  4. - u v ¯ -\overline{u^{\prime}v^{\prime}}

Cohn-Vossen's_inequality.html

  1. S K d A 2 π χ ( S ) , \iint_{S}K\,dA\leq 2\pi\chi(S),
  2. S K d A = 2 π χ ( S ) - S k g d s \iint_{S}K\,dA=2\pi\chi(S)-\int_{\partial S}k_{g}\,ds
  3. k g k_{g}

Cohomological_descent.html

  1. a * : D + ( S ) D + ( X ) a^{*}:D^{+}(S)\to D^{+}(X)
  2. id D + ( S ) R a * a * \operatorname{id}_{D^{+}(S)}\to Ra_{*}\circ a^{*}

Coins_in_a_fountain.html

  1. f ( n , k ) f(n,k)
  2. n k C n , k x n y k \sum_{n}\sum_{k}C_{n,k}x^{n}y^{k}
  3. k C n , k x n y k \sum_{k}C_{n,k}x^{n}y^{k}
  4. f ( n , k ) f(n,k)
  5. g ( n , k ) g(n,k)
  6. F ( x , y ) \displaystyle F(x,y)

Cokurtosis.html

  1. K ( X , X , X , Y ) = E [ ( X - E [ X ] ) 3 ( Y - E [ Y ] ) ] σ X 3 σ Y , K(X,X,X,Y)={\operatorname{E}{\big[(X-\operatorname{E}[X])^{3}(Y-\operatorname{% E}[Y])\big]}\over\sigma_{X}^{3}\sigma_{Y}},
  2. K ( X , X , Y , Y ) = E [ ( X - E [ X ] ) 2 ( Y - E [ Y ] ) 2 ] σ X 2 σ Y 2 , K(X,X,Y,Y)={\operatorname{E}{\big[(X-\operatorname{E}[X])^{2}(Y-\operatorname{% E}[Y])^{2}\big]}\over\sigma_{X}^{2}\sigma_{Y}^{2}},
  3. K ( X , Y , Y , Y ) = E [ ( X - E [ X ] ) ( Y - E [ Y ] ) 3 ] σ X σ Y 3 , K(X,Y,Y,Y)={\operatorname{E}{\big[(X-\operatorname{E}[X])(Y-\operatorname{E}[Y% ])^{3}\big]}\over\sigma_{X}\sigma_{Y}^{3}},
  4. σ X \sigma_{X}
  5. K ( X , X , X , X ) = E [ ( X - E [ X ] ) 4 ] σ X 4 = kurtosis [ X ] , K(X,X,X,X)={\operatorname{E}{\big[(X-\operatorname{E}[X])^{4}\big]}\over\sigma% _{X}^{4}}={\operatorname{kurtosis}\big[X\big]},
  6. K X + Y = 1 σ X + Y 4 [ \displaystyle K_{X+Y}={1\over\sigma_{X+Y}^{4}}\big[
  7. K X K_{X}
  8. σ X \sigma_{X}
  9. K X + Y 0 K_{X+Y}\neq 0
  10. K X = 0 K_{X}=0
  11. K Y = 0 K_{Y}=0

Combinant.html

  1. G X ( t ) = M X ( log ( 1 + t ) ) G_{X}(t)=M_{X}(\log(1+t))
  2. G X ( t ) := E [ ( 1 + t ) X ] , t , G_{X}(t):=E\left[(1+t)^{X}\right],\quad t\in\mathbb{R},
  3. c n = 1 n ! n t n log ( G ( t ) ) | t = - 1 c_{n}=\frac{1}{n!}\frac{\partial^{n}}{\partial t^{n}}\log(G(t))\bigg|_{t=-1}

Combining_rules.html

  1. σ i j = σ i i + σ j j 2 \sigma_{ij}=\frac{\sigma_{ii}+\sigma_{jj}}{2}
  2. ϵ i j = ϵ i i ϵ j j \epsilon_{ij}=\sqrt{\epsilon_{ii}\epsilon_{jj}}
  3. r i j 0 = ( ( r i 0 ) 6 + ( r j 0 ) 6 2 ) 1 / 6 r_{ij}^{0}=\left(\frac{(r_{i}^{0})^{6}+(r_{j}^{0})^{6}}{2}\right)^{1/6}
  4. ϵ i j = 2 ϵ i ϵ j ( ( r i 0 ) 3 ( r j 0 ) 3 ( r i 0 ) 6 + ( r j 0 ) 6 ) \epsilon_{ij}=2\sqrt{\epsilon_{i}\cdot\epsilon_{j}}\left(\frac{(r_{i}^{0})^{3}% \cdot(r_{j}^{0})^{3}}{(r_{i}^{0})^{6}+(r_{j}^{0})^{6}}\right)
  5. ϵ i j = 2 ϵ i ϵ j ϵ i + ϵ j \epsilon_{ij}=\frac{2\epsilon_{i}\epsilon_{j}}{\epsilon_{i}+\epsilon_{j}}
  6. ϵ i j σ i j 6 = ( ϵ i i σ i i 6 ϵ j j σ j j 6 ) 1 / 2 \epsilon_{ij}\sigma_{ij}^{6}=\left(\epsilon_{ii}\sigma_{ii}^{6}\epsilon_{jj}% \sigma_{jj}^{6}\right)^{1/2}
  7. ϵ i j σ i j 12 = [ ( ϵ i i σ i i 12 ) 1 / 13 + ( ϵ j j σ j j 12 ) 1 / 13 2 ] 13 \epsilon_{ij}\sigma_{ij}^{12}=\left[\frac{(\epsilon_{ii}\sigma_{ii}^{12})^{1/1% 3}+(\epsilon_{jj}\sigma_{jj}^{12})^{1/13}}{2}\right]^{13}
  8. σ i j = σ i i σ j j \sigma_{ij}=\sqrt{\sigma_{ii}\sigma_{jj}}
  9. ϵ 12 = 2 ϵ 11 ϵ 22 ϵ 11 + ϵ 22 \epsilon_{12}=\frac{2\epsilon_{11}\epsilon_{22}}{\epsilon_{11}+\epsilon_{22}}
  10. α 12 = 1 2 ( α 11 + α 22 ) \alpha_{12}=\frac{1}{2}(\alpha_{11}+\alpha_{22})
  11. [ ϵ 12 α 12 e α 12 ( α 12 - 6 ) σ 12 ] 2 σ 12 / α 12 = [ ϵ 11 α 11 e α 11 ( α 11 - 6 ) σ 11 ] σ 11 / α 11 [ ϵ 22 α 22 e α 22 ( α 22 - 6 ) σ 22 ] σ 22 / α 22 \left[\frac{\epsilon_{12}\alpha_{12}e^{\alpha_{12}}}{(\alpha_{12}-6)\sigma_{12% }}\right]^{2\sigma_{12}/\alpha_{12}}=\left[\frac{\epsilon_{11}\alpha_{11}e^{% \alpha_{11}}}{(\alpha_{11}-6)\sigma_{11}}\right]^{\sigma_{11}/\alpha_{11}}% \left[\frac{\epsilon_{22}\alpha_{22}e^{\alpha_{22}}}{(\alpha_{22}-6)\sigma_{22% }}\right]^{\sigma_{22}/\alpha_{22}}
  12. σ 12 α 12 = 1 2 ( σ 11 α 11 + σ 22 α 22 ) \frac{\sigma_{12}}{\alpha_{12}}=\frac{1}{2}\left(\frac{\sigma_{11}}{\alpha_{11% }}+\frac{\sigma_{22}}{\alpha_{22}}\right)
  13. ϵ 12 α 12 σ 12 6 ( α 12 - 6 ) = [ ϵ 11 α 11 σ 11 6 ( α 11 - 6 ) ϵ 22 α 22 σ 22 6 ( α 22 - 6 ) ] 1 2 \frac{\epsilon_{12}\alpha_{12}\sigma_{12}^{6}}{(\alpha_{12}-6)}=\left[\frac{% \epsilon_{11}\alpha_{11}\sigma_{11}^{6}}{(\alpha_{11}-6)}\frac{\epsilon_{22}% \alpha_{22}\sigma_{22}^{6}}{(\alpha_{22}-6)}\right]^{\frac{1}{2}}

Commutative_ring_spectrum.html

  1. E E_{\infty}
  2. \mathbb{Q}
  3. \mathbb{Q}
  4. E E_{\infty}
  5. E E_{\infty}
  6. E E_{\infty}

Commutator_collecting_process.html

  1. g = c 1 n 1 c 2 n 2 c k n k c g=c_{1}^{n_{1}}c_{2}^{n_{2}}\cdots c_{k}^{n_{k}}c

Commutator_subspace.html

  1. = =
  2. { 1 1 + n k = 0 n ( μ ( k , A ) - μ ( k , B ) ) } n = 0 j \left\{\frac{1}{1+n}\sum_{k=0}^{n}\left(\mu(k,A)-\mu(k,B)\right)\right\}_{n=0}% ^{\infty}\in j
  3. { 1 1 + n k = 0 n ( λ ( k , A ) - λ ( k , B ) ) } n = 0 j \left\{\frac{1}{1+n}\sum_{k=0}^{n}\left(\lambda(k,A)-\lambda(k,B)\right)\right% \}_{n=0}^{\infty}\in j
  4. = =
  5. = =
  6. = =
  7. { a 1 + a 2 + + a n n } n = 1 c 00 \left\{\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right\}_{n=1}^{\infty}\in c_{00}
  8. a 1 + a 2 + + a N = 0 a_{1}+a_{2}+\cdots+a_{N}=0
  9. = =
  10. { a 1 + a 2 + + a n n } n = 1 1 \left\{\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right\}_{n=1}^{\infty}\in\ell_{1}
  11. a n = 1 n log 2 ( n ) , n 2. a_{n}=\frac{1}{n\log^{2}(n)},\quad n\geq 2.
  12. a 1 = - n = 2 a n . a_{1}=-\sum_{n=2}^{\infty}a_{n}.
  13. { a 1 + a 2 + + a n n } n = 1 1 , \left\{\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right\}_{n=1}^{\infty}\in\ell_{1,\infty}
  14. { a 1 + a 2 + + a n } n = 1 = O ( 1 ) \left\{a_{1}+a_{2}+\cdots+a_{n}\right\}_{n=1}^{\infty}=O(1)
  15. = =

Compact_finite_difference.html

  1. H i + 1 = m = - ( a m f i + m + b m f i + m ) = 0. ( 1 ) H_{i+1}=\sum_{m=-\ell}^{\ell}(a_{m}f_{i+m}+b_{m}f^{\prime}_{i+m})=0.\qquad(1)

Comparator_applications.html

  1. V out = { V S + if V 1 > V 2 , V S - if V 1 < V 2 , 0 if V 1 = V 2 , V_{\,\text{out}}=\begin{cases}V_{\,\text{S}+}&\,\text{if }V_{1}>V_{2},\\ V_{\,\text{S}-}&\,\text{if }V_{1}<V_{2},\\ 0&\,\text{if }V_{1}=V_{2},\end{cases}
  2. V S + V_{\,\text{S}+}
  3. V S - V_{\,\text{S}-}
  4. V a = R 1 R 1 + R 5 V i n + R 5 R 1 + R 5 V T H R o u t V_{a}={R_{1}\over{R_{1}+R_{5}}}V_{in}+{R_{5}\over{R_{1}+R_{5}}}V_{THRout}
  5. f i = Δ t 2 f_{i}=\frac{\Delta t}{2}
  6. Δ t \Delta t
  7. R 1 {R_{1}}
  8. ± R 1 R 2 V sat \pm\frac{R_{1}}{R_{2}}V_{\,\text{sat}}
  9. V sat V_{\,\text{sat}}
  10. ± R 1 R 1 + R 2 V sat \pm\frac{R_{1}}{R_{1}+R_{2}}V_{\,\text{sat}}

Competitive_Regret.html

  1. p p
  2. 𝒳 \mathcal{X}
  3. X X
  4. q q
  5. max p 𝒫 r n ( q , p ) . \max_{p\in\mathcal{P}}r_{n}(q,p).
  6. 𝒫 \mathcal{P}
  7. r n ( q , p ) = 𝔼 ( D ( p | | q ( X ) ) ) . r_{n}(q,p)=\mathbb{E}(D(p||q(X))).
  8. D ( p | | q ) D(p||q)
  9. p p
  10. q q
  11. p p
  12. p p
  13. \mathbb{P}
  14. P P
  15. p P p\in P
  16. r n ( P ) = min q max p P r n ( q , p ) . r_{n}(P)=\min_{q}\max_{p\in P}r_{n}(q,p).
  17. r n ( q , 𝒫 ) = max P ( r n ( q , P ) - r n ( P ) ) . r_{n}^{\mathbb{P}}(q,\mathcal{P})=\max_{P\in\mathbb{P}}(r_{n}(q,P)-r_{n}(P)).
  18. p p
  19. r n n a t ( p ) = min q 𝒬 n a t r n ( q , p ) , r_{n}^{nat}(p)=\min_{q\in\mathcal{Q}_{nat}}r_{n}(q,p),
  20. max p 𝒫 ( r n ( q , p ) - r n n a t ( p ) ) . \max_{p\in\mathcal{P}}(r_{n}(q,p)-r_{n}^{nat}(p)).
  21. q q
  22. r n σ ( q , Δ k ) r n n a t ( q , Δ k ) 𝒪 ~ ( min ( 1 n , k n ) ) . r_{n}^{\mathbb{P}_{\sigma}}(q,\Delta_{k})\leq r^{nat}_{n}(q,\Delta_{k})\leq% \tilde{\mathcal{O}}(\min(\frac{1}{\sqrt{n}},\frac{k}{n})).
  23. Δ k \Delta_{k}
  24. σ \mathbb{P}_{\sigma}
  25. Δ k \Delta_{k}
  26. p p
  27. p p^{\prime}
  28. p p^{\prime}
  29. p p

Complex-oriented_cohomology_theory.html

  1. E 2 ( 𝐏 ) E 2 ( 𝐏 1 ) E^{2}(\mathbb{C}\mathbf{P}^{\infty})\to E^{2}(\mathbb{C}\mathbf{P}^{1})
  2. E 2 ( 𝐏 ) E^{2}(\mathbb{C}\mathbf{P}^{\infty})
  3. E ~ 2 ( 𝐏 1 ) \widetilde{E}^{2}(\mathbb{C}\mathbf{P}^{1})
  4. π 3 E = π 5 E = \pi_{3}E=\pi_{5}E=\cdots
  5. H 2 ( 𝐏 ; R ) H 2 ( 𝐏 1 ; R ) \operatorname{H}^{2}(\mathbb{C}\mathbf{P}^{\infty};R)\simeq\operatorname{H}^{2% }(\mathbb{C}\mathbf{P}^{1};R)
  6. π 3 K = π 5 K = = 0 \pi_{3}K=\pi_{5}K=\cdots=0
  7. 𝐏 × 𝐏 𝐏 , ( [ x ] , [ y ] ) [ x y ] \mathbb{C}\mathbf{P}^{\infty}\times\mathbb{C}\mathbf{P}^{\infty}\to\mathbb{C}% \mathbf{P}^{\infty},([x],[y])\mapsto[xy]
  8. [ x ] [x]
  9. [ t ] \mathbb{C}[t]
  10. 𝐏 \mathbb{C}\mathbf{P}^{\infty}
  11. E * ( 𝐏 ) = lim E * ( 𝐏 n ) = lim R [ t ] / ( t n + 1 ) = R [ [ t ] ] , R = π * E = π 2 n E E^{*}(\mathbb{C}\mathbf{P}^{\infty})=\underleftarrow{\lim}E^{*}(\mathbb{C}% \mathbf{P}^{n})=\underleftarrow{\lim}R[t]/(t^{n+1})=R[\![t]\!],\quad R=\pi_{*}% E=\oplus\pi_{2n}E
  12. f = m * ( t ) f=m^{*}(t)
  13. E * ( 𝐏 × 𝐏 ) = lim E * ( 𝐏 n × 𝐏 m ) = lim R [ x , y ] / ( x n + 1 , y m + 1 ) = R [ [ x , y ] ] E^{*}(\mathbb{C}\mathbf{P}^{\infty}\times\mathbb{C}\mathbf{P}^{\infty})=% \underleftarrow{\lim}E^{*}(\mathbb{C}\mathbf{P}^{n}\times\mathbb{C}\mathbf{P}^% {m})=\underleftarrow{\lim}R[x,y]/(x^{n+1},y^{m+1})=R[\![x,y]\!]

Composite_Higgs_models.html

  1. Adj [ G ] = Adj [ H ] + R [ Π ] \rm{Adj}[G]={\rm Adj}[H]+{\rm R}[\Pi]
  2. S O ( 5 ) S U ( 2 ) L × S U ( 2 ) R G B = ( 2 , 2 ) \frac{SO(5)}{SU(2)_{L}\times SU(2)_{R}}\rightarrow GB=(2,2)
  3. M = g f M=gf
  4. ξ = v 2 f 2 \xi=\frac{v^{2}}{f^{2}}
  5. h V V h V V S M 1 - ξ 2 \frac{h_{VV}}{h_{VV}^{SM}}\approx 1-\frac{\xi}{2}
  6. m h 2 3 2 π 2 M 2 f 2 v 2 m_{h}^{2}\sim\frac{3}{2\pi^{2}}\frac{M^{2}}{f^{2}}v^{2}
  7. | S M = cos α | E l > + sin α | C o |SM\rangle=\cos{\alpha}|El>+\sin{\alpha}|Co\rangle
  8. m f v sin α L Y sin α R \frac{m_{f}}{v}\approx\sin\alpha_{L}\cdot Y\cdot\sin\alpha_{R}

Composition_drift.html

  1. F 1 = F_{1}=
  2. f 1 = f_{1}=
  3. F 1 F_{1}
  4. f 1 f_{1}
  5. ( f 1 ) a = 1 - r 2 2 - r 1 - r 2 (f_{1})_{a}=\frac{1-r_{2}}{2-r_{1}-r_{2}}\,
  6. F 1 = f 1 F_{1}=f_{1}
  7. f 1 f_{1}
  8. f 1 f_{1}

Compressible_duct_flow.html

  1. ρ \rho
  2. ρ V = G A \rho V=\frac{G}{A}
  3. d p p + d V V = 0 \frac{dp}{p}+\frac{dV}{V}=0
  4. ρ A - ( p + d p ) A - τ w Π D d x = m ˙ ( V + d V - V ) \rho A-(p+dp)A-\tau_{w}\Pi Ddx=\dot{m}(V+dV-V)
  5. d p + 4 τ w d x D + ρ V d V = 0 dp+\frac{4\tau_{w}dx}{D}+\rho VdV=0
  6. τ w \tau_{w}
  7. τ w Π D d x \tau_{w}\Pi Ddx
  8. h + 1 2 V 2 + h o = c p T o = c p T + 1 2 V 2 h+\frac{1}{2}V^{2}+h_{o}=c_{p}T_{o}=c_{p}T+\frac{1}{2}V^{2}
  9. c p d T + V d V = 0 c_{p}dT+VdV=0
  10. p = ρ R T p=\rho RT
  11. d p p = d ρ ρ + d T T \frac{dp}{p}=\frac{d\rho}{\rho}+\frac{dT}{T}
  12. τ w = 1 8 f ρ V 2 \tau_{w}=\frac{1}{8}f\rho V^{2}
  13. a 2 = k p ρ a^{2}=\frac{kp}{\rho}
  14. τ w = 1 8 f k p M a 2 \tau_{w}=\frac{1}{8}fkpMa^{2}
  15. V 2 = M a 2 k R T V^{2}=Ma^{2}kRT
  16. 2 d V V = 2 d M a M a + d T T \frac{2dV}{V}=\frac{2dMa}{Ma}+\frac{dT}{T}
  17. d p p = - k M a 2 1 + ( k - 1 ) M a 2 2 ( 1 - M a 2 ) f d x D \frac{dp}{p}=-kMa^{2}\frac{1+(k-1)Ma^{2}}{2(1-Ma^{2})}f\frac{dx}{D}
  18. d ρ ρ = - k M a 2 2 ( 1 - M a 2 ) f d x D = - d V V \frac{d\rho}{\rho}=-\frac{kMa^{2}}{2(1-Ma^{2})}f\frac{dx}{D}=-\frac{dV}{V}
  19. d p o p o = d ρ o ρ o = - 1 2 k M a 2 f d x D \frac{dp_{o}}{p_{o}}=\frac{d\rho_{o}}{\rho_{o}}=-\frac{1}{2}kMa^{2}f\frac{dx}{D}
  20. d T T = k ( k - 1 ) M a 4 2 ( 1 - M a 2 ) f d x D \frac{dT}{T}=\frac{k(k-1)Ma^{4}}{2(1-Ma^{2})}f\frac{dx}{D}
  21. d M a 2 M a 2 = k M a 2 1 + 1 2 k ( k - 1 ) M a 2 ( 1 - M a 2 ) f d x D \frac{dMa^{2}}{Ma^{2}}=kMa^{2}\frac{1+\frac{1}{2}k(k-1)Ma^{2}}{(1-Ma^{2})}f% \frac{dx}{D}
  22. 1 - M a 2 1-Ma^{2}
  23. d p o p o \frac{dp_{o}}{p_{o}}
  24. ρ \rho
  25. P o , ρ o P_{o},\rho_{o}
  26. p o p_{o}
  27. ρ o \rho_{o}
  28. p o * p^{*}_{o}
  29. ρ o * \rho^{*}_{o}
  30. 0 L * f d x D = M a 2 1.0 1 - M a 2 k M a 4 [ 1 + 1 2 ( k - 1 ) M a 2 ] d M a 2 \int_{0}^{L^{*}}{f\frac{dx}{D}}=\int_{Ma^{2}}^{1.0}{\frac{1-Ma^{2}}{kMa^{4}[1+% \frac{1}{2}(k-1)Ma^{2}]}dMa^{2}}
  31. f ¯ L * D = 1 - M a 2 k M a 2 + k + 1 k ln ( k + 1 ) M a 2 2 + ( k - 1 ) M a 2 \frac{\bar{f}L^{*}}{D}=\frac{1-Ma^{2}}{kMa^{2}}+\frac{k+1}{k}\ln{\frac{(k+1)Ma% ^{2}}{2+(k-1)Ma^{2}}}

Computational_Fluid_Dynamics_for_Phase_Change_Materials.html

  1. ρ t + ( ρ 𝐮 ) = S m {\partial\rho\over\partial t}+\nabla\cdot(\rho\mathbf{u})=S_{m}
  2. ( ρ H ) t + x j ( ρ * u j * c p * T ) = x j ( λ T x j ) + S E \begin{aligned}\displaystyle{\partial(\rho{H})\over\partial t}+{\partial\over% \partial x_{j}}{(\rho*u_{j}*c_{p}*{T})}={\partial\over\partial x_{j}}(\lambda% \cdot{\partial{T}\over\partial x_{j}})+{S_{E}}\end{aligned}
  3. ρ ( u i t + u j u i x j ) = - p x i + μ 2 u i x j x j + f i \rho\left(\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{% \partial x_{j}}\right)=-\frac{\partial p}{\partial x_{i}}+\mu\frac{\partial^{2% }u_{i}}{\partial x_{j}\partial x_{j}}+f_{i}
  4. s y m b o l 𝖳 \nabla\cdot symbol{\mathsf{T}}
  5. μ 2 𝐯 \mu\nabla^{2}\mathbf{v}
  6. 2 \nabla^{2}
  7. - υ i υ j ¯ = 2 ν t S i j - 2 3 K δ i j -\overline{\upsilon_{i}^{\prime}\upsilon_{j}^{\prime}}=2\nu_{t}S_{ij}-\frac{2}% {3}K\delta_{ij}
  8. S i j S_{ij}
  9. ν t \nu_{t}
  10. K = 1 2 υ i υ i ¯ K=\frac{1}{2}\overline{\upsilon_{i}^{\prime}\upsilon_{i}^{\prime}}
  11. δ i j \delta_{ij}

Computational_methods_for_free_surface_flow.html

  1. [ ( v - v [(v-v
  2. b b
  3. . n ] .n]
  4. f s fs
  5. = 0 , =0,
  6. f s fs
  7. = 0 =0
  8. ( n . T ) (n.T)
  9. l l
  10. . n + .n+
  11. K = - ( n . T ) K=-(n.T)
  12. g g
  13. . n , .n,
  14. ( n . T ) (n.T)
  15. l l
  16. . t - .t-
  17. σ t \frac{∂σ}{∂t}
  18. = ( n . T ) =(n.T)
  19. g g
  20. . t , .t,
  21. ( n . T ) (n.T)
  22. l l
  23. . s - .s-
  24. σ s \frac{∂σ}{∂s}
  25. = ( n . T ) =(n.T)
  26. g g
  27. . s , .s,
  28. K = K=
  29. + +
  30. 1 R < s u b > s \frac{1}{R<sub>s}
  31. f f
  32. d l dl
  33. Z = H ( x , y , t ) Z=H(x,y,t)
  34. H t \frac{∂H}{∂t}
  35. H x \frac{∂H}{∂x}
  36. H y \frac{∂H}{∂y}
  37. u u
  38. i i
  39. * *
  40. c t \frac{∂c}{∂t}

Concentration_polarization.html

  1. J 1 m J_{1}^{m}
  2. J 1 s J_{1}^{s}
  3. J 1 s = J 1 m J_{1}^{s}=J_{1}^{m}
  4. J 1 m J_{1}^{m}
  5. J 1 s J_{1}^{s}
  6. c 1 c_{1}^{\prime}
  7. c 1 ′′ c_{1}^{\prime\prime}
  8. J 1 s = J 1 m J_{1}^{s}=J_{1}^{m}
  9. c 1 c_{1}^{\prime}
  10. c 1 c_{1}^{\prime}

Concurrent_average_memory_access_time.html

  1. T M e m C y c l e T_{MemCycle}
  2. C M e m A c c C_{MemAcc}
  3. C - A M A T = T M e m C y c l e C M e m A c c C\mbox{-}~{}AMAT=\frac{T_{MemCycle}}{C_{MemAcc}}
  4. T M e m C y c l e T_{MemCycle}
  5. C M e m A c c C_{MemAcc}
  6. T M e m C y c l e T_{MemCycle}
  7. T M e m C y c l e T_{MemCycle}
  8. C - A M A T = 1 A P C = T M e m C y c l e C M e m A c c C\mbox{-}~{}AMAT=\frac{1}{APC}=\frac{T_{MemCycle}}{C_{MemAcc}}
  9. C - A M A T = H C H + p M R p A M P C M C\mbox{-}~{}AMAT=\frac{H}{C_{H}}+pMR\cdot\frac{pAMP}{C_{M}}
  10. H {}_{\,\text{H}}
  11. M {}_{\,\text{M}}
  12. H {}_{\,\text{H}}
  13. M {}_{\,\text{M}}
  14. C - A M A T 1 = H 1 C H 1 + p M R 1 η 1 C - A M A T 2 C\mbox{-}~{}AMAT_{1}=\frac{H_{1}}{C_{H_{1}}}+pMR_{1}\cdot\eta_{1}\cdot C\mbox{% -}~{}AMAT_{2}
  15. C - A M A T 1 = H 1 C H 1 + p M R 1 p A M P 1 C M 1 C\mbox{-}~{}AMAT_{1}=\frac{H_{1}}{C_{H_{1}}}+pMR_{1}\cdot\frac{pAMP_{1}}{C_{M_% {1}}}
  16. C - A M A T 2 = H 2 C H 2 + p M R 2 p A M P 2 C M 2 C\mbox{-}~{}AMAT_{2}=\frac{H_{2}}{C_{H_{2}}}+pMR_{2}\cdot\frac{pAMP_{2}}{C_{M_% {2}}}
  17. η 1 = p A M P 1 A M P 1 C m 1 C M 1 \eta_{1}=\frac{pAMP_{1}}{AMP_{1}}\cdot\frac{C_{m_{1}}}{C_{M_{1}}}
  18. p M R 1 η 1 pMR_{1}\cdot\eta_{1}
  19. 1 {}_{\,\text{1}}
  20. C m C_{m}
  21. 2 {}_{\,\text{2}}
  22. C m C_{m}
  23. 1 {}_{\,\text{1}}
  24. C M C_{M}

Conditional_probability_table.html

  1. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  2. K K
  3. x 1 x_{1}
  4. P ( x 1 x 2 , x 3 ) P(x_{1}\mid x_{2},x_{3})
  5. M M
  6. x 1 , x 2 , , x M x_{1},x_{2},\ldots,x_{M}
  7. K K
  8. P ( x j x i ) = T i j P(x_{j}\mid x_{i})=T_{ij}
  9. j T i j \sum_{j}T_{ij}

Conductor_of_an_elliptic_curve.html

  1. δ = dim Z / l Z Hom Z l [ G ] ( P , M ) . \delta=\dim_{Z/lZ}\text{Hom}_{Z_{l}[G]}(P,M).
  2. f 𝐩 = ν 𝐩 ( Δ ) + 1 - n , f_{\mathbf{p}}=\nu_{\mathbf{p}}(\Delta)+1-n\ ,
  3. f ( E ) = 𝐩 𝐩 f 𝐩 . f(E)=\prod_{\mathbf{p}}\mathbf{p}^{f_{\mathbf{p}}}\ .

Conflict-free_replicated_data_type.html

  1. i \sum\nolimits_{i}
  2. i \forall i\in
  3. \leq
  4. i \forall i\in
  5. i \sum\nolimits_{i}
  6. i \sum\nolimits_{i}
  7. i \forall i\in
  8. \leq
  9. and \and
  10. i \forall i\in
  11. \leq
  12. i \forall i\in
  13. i \forall i\in
  14. \emptyset
  15. \cup
  16. \in
  17. \subseteq
  18. \cup
  19. \emptyset
  20. \emptyset
  21. \in
  22. and \and
  23. \notin
  24. \cup
  25. \cup
  26. \subseteq
  27. $\or$
  28. \subseteq
  29. \cup
  30. \cup

Congruence-permutable_algebra.html

  1. α , β \alpha,\beta
  2. α β = β α \alpha\circ\beta=\beta\circ\alpha
  3. M ( x , x , y ) = y = M ( y , x , x ) . M(x,x,y)=y=M(y,x,x).

Conical_plate_centrifuge.html

  1. t d t_{d}
  2. t d = v s η d Q f Φ f t_{d}={v_{s}\eta_{d}\over Q_{f}\Phi_{f}}
  3. v s v_{s}
  4. η d \eta_{d}
  5. Q f Q_{f}
  6. Φ f \Phi_{f}
  7. t d t_{d}

Connective_spectrum.html

  1. π k \pi_{k}

Consistent_and_inconsistent_equations.html

  1. x + y + z = 3 , x+y+z=3,
  2. x + y + 2 z = 4 x+y+2z=4
  3. x 2 + y 2 + z 2 = 10 , x^{2}+y^{2}+z^{2}=10,
  4. x 2 + y 2 = 5 x^{2}+y^{2}=5
  5. z = ± 5 . z=\pm\sqrt{5}.
  6. x + y + z = 3 , x+y+z=3,
  7. x + y + z = 4 x+y+z=4
  8. x 2 + y 2 + z 2 = 10 , x^{2}+y^{2}+z^{2}=10,
  9. x 2 + y 2 + z 2 = 12 x^{2}+y^{2}+z^{2}=12
  10. x + y = 3 , x+y=3,
  11. x + 2 y = 5 x+2y=5
  12. x + y = 1 , x+y=1,
  13. x 2 + y 2 = 1 x^{2}+y^{2}=1
  14. x 3 + y 3 + z 3 = 10 , x^{3}+y^{3}+z^{3}=10,
  15. x 3 + 2 y 3 + z 3 = 12 , x^{3}+2y^{3}+z^{3}=12,
  16. 3 x 3 + 5 y 3 + 3 z 3 = 34 3x^{3}+5y^{3}+3z^{3}=34
  17. x + y = 3 , x+y=3,
  18. 4 x + 4 y = 10 4x+4y=10
  19. x 3 + y 3 + z 3 = 10 , x^{3}+y^{3}+z^{3}=10,
  20. x 3 + 2 y 3 + z 3 = 12 , x^{3}+2y^{3}+z^{3}=12,
  21. 3 x 3 + 5 y 3 + 3 z 3 = 32 3x^{3}+5y^{3}+3z^{3}=32
  22. x + y = 3 , x+y=3,
  23. x + 2 y = 7 , x+2y=7,
  24. 4 x + 6 y = 20 4x+6y=20
  25. x + 2 y = 7 , x+2y=7,
  26. 3 x + 6 y = 21 , 3x+6y=21,
  27. 7 x + 14 y = 49 7x+14y=49
  28. x 2 - 1 = 0 , x^{2}-1=0,
  29. y 2 - 1 = 0 , y^{2}-1=0,
  30. ( x - 1 ) ( y - 1 ) = 0 (x-1)(y-1)=0
  31. x + y = 3 , x+y=3,
  32. x + 2 y = 7 , x+2y=7,
  33. 4 x + 6 y = 21 4x+6y=21
  34. x 2 + y 2 = 1 , x^{2}+y^{2}=1,
  35. x 2 + 2 y 2 = 2 , x^{2}+2y^{2}=2,
  36. 2 x 2 + 3 y 2 = 4 2x^{2}+3y^{2}=4

Construction_Irreducible_Markov_Chain_in_Ising_Model.html

  1. z Z N 1 × . . × N d z\in Z^{N_{1}\times..\times N_{d}}
  2. z = z + - z - z=z^{+}-z^{-}
  3. z + z^{+}
  4. z - z^{-}
  5. Z ~ Z N 1 × . . × N d \widetilde{Z}\subset Z^{N_{1}\times..\times N_{d}}
  6. z w i d e t i l d e Z z\in widetilde{Z}
  7. T 1 ( z + ) = T 1 ( z - ) T_{1}(z^{+})=T_{1}(z^{-})
  8. T 2 ( z + ) = T 2 ( z - ) T_{2}(z^{+})=T_{2}(z^{-})
  9. a , b Z > 0 a,b\in Z_{>0}
  10. x , y S ( a , b ) x,y\in S(a,b)
  11. z 1 , . . , z k Z ~ z_{1},..,z_{k}\in\widetilde{Z}
  12. y = x + i = 1 k z i y=x+\sum_{i=1}^{k}z_{i}
  13. x + i = 1 l z i S ( a , b ) x+\sum_{i=1}^{l}z_{i}\in S(a,b)
  14. w i d e t i l d e Z widetilde{Z}
  15. I := k e r ( ψ * ϕ ) I:=ker({\psi}*{\phi})
  16. S ( a , b ) S(a,b)
  17. z Z N 1 × . . × N d z\in Z^{N_{1}\times..\times N_{d}}
  18. e i e_{i}
  19. z Z N 1 × . . × N d z\in Z^{N_{1}\times..\times N_{d}}
  20. y , y S ( a , b ) y^{\prime},y\in S(a,b)
  21. S ( a , b ) S(a,b)
  22. y y
  23. y y^{\prime}
  24. S ( a , b ) S(a,b)
  25. z Z z\in Z
  26. z 1 , , z k Z z_{1},...,z_{k}\in Z
  27. y = y + i = 1 k z i y^{\prime}=y+\sum_{i=1}^{k}z_{i}
  28. y + i = 1 l z i S ( a , b ) y+\sum_{i=1}^{l}z_{i}\in S(a,b)
  29. y S ( a , b ) y\in S(a,b)
  30. z Z z\in Z
  31. y = y + z y^{\prime}=y+z
  32. y y^{\prime}
  33. y S ( a , b ) y^{\prime}\in S(a,b)
  34. S ( a , b ) S^{\star}(a,b)
  35. S ( a , b ) S(a,b)
  36. a , b N a,b\in N
  37. y S ( a , b ) y\in S(a,b)
  38. y S ( a , b ) y^{\star}S(a,b)
  39. z 1 , , z k Z z_{1},...,z_{k}\in Z
  40. y = y + i = 1 k z i y^{\star}=y+\sum_{i=1}^{k}z_{i}
  41. y + i = 1 l z i S ( a , b ) y+\sum_{i=1}^{l}z_{i}\in S(a,b)
  42. S ( a , b ) S^{\star}(a,b)
  43. S ( a , b ) S(a,b)
  44. S ( a , b ) S(a,b)
  45. S ( a , b ) S^{\star}(a,b)

Continuous-time_random_walk.html

  1. X ( t ) X(t)
  2. X ( t ) = X 0 + i = 1 N ( t ) Δ X i , X(t)=X_{0}+\sum_{i=1}^{N(t)}\Delta X_{i},
  3. Δ X i \Delta X_{i}
  4. Ω \Omega
  5. N ( t ) N(t)
  6. ( 0 , t ) (0,t)
  7. X X
  8. t t
  9. P ( X , t ) = n = 0 P ( n , t ) P n ( X ) . P(X,t)=\sum_{n=0}^{\infty}P(n,t)P_{n}(X).
  10. P n ( X ) P_{n}(X)
  11. X X
  12. n n
  13. P ( n , t ) P(n,t)
  14. n n
  15. t t
  16. N ( t ) N(t)
  17. ψ ( τ ) \psi(\tau)
  18. ψ ~ ( s ) = 0 d τ e - τ s ψ ( τ ) . \tilde{\psi}(s)=\int_{0}^{\infty}d\tau e^{-\tau s}\psi(\tau).
  19. f ( Δ X ) f(\Delta X)
  20. f ^ ( k ) = Ω d ( Δ X ) e i k Δ X f ( Δ X ) . \hat{f}(k)=\int_{\Omega}d(\Delta X)e^{ik\Delta X}f(\Delta X).
  21. P ( X , t ) P(X,t)
  22. P ~ ^ ( k , s ) = 1 - ψ ~ ( s ) s 1 1 - ψ ~ ( s ) f ^ ( k ) . \hat{\tilde{P}}(k,s)=\frac{1-\tilde{\psi}(s)}{s}\frac{1}{1-\tilde{\psi}(s)\hat% {f}(k)}.

Continuum_percolation_theory.html

  1. { x i } \{x_{i}\}
  2. Φ \Phi
  3. λ \lambda
  4. x i Φ x_{i}\in\Phi
  5. D i D_{i}
  6. x i x_{i}
  7. D i D_{i}
  8. R i R_{i}
  9. { x i } \{x_{i}\}
  10. { D i } \{D_{i}\}
  11. i D i \cup_{i}D_{i}
  12. S i S_{i}
  13. x i x_{i}
  14. S i S_{i}
  15. { x i } \{x_{i}\}
  16. S i S_{i}
  17. 𝐑 < s u p > 2 \mathbf{R}<sup>2

Contour_boxplot.html

  1. S s B ( S 1 , , S j ) k = 1 j S k S k = 1 j S k . S\in sB(S_{1},\ldots,S_{j})\Longleftrightarrow\bigcap_{k=1}^{j}S_{k}\subset S% \subset\bigcup_{k=1}^{j}S_{k}.\,
  2. s B D J ( S ) = j = 2 J P [ S s B ( S 1 , , S j ] . sBD_{J}(S)=\sum_{j=2}^{J}P\left[S\in sB(S_{1},\ldots,S_{j}\right].

Convective_Boundary_Layer.html

  1. c ¯ t = - w c ¯ z \frac{\partial\bar{c}}{\partial t}=-\frac{\partial\overline{w^{\prime}c^{% \prime}}}{\partial z}
  2. c ¯ \bar{c}
  3. c c
  4. q q
  5. θ \theta
  6. u u
  7. v v
  8. w c ¯ \overline{w^{\prime}c^{\prime}}
  9. c c
  10. K c K_{c}
  11. c z \frac{\partial c}{\partial z}
  12. w c ¯ = - K c c z \overline{w^{\prime}c^{\prime}}=-K_{c}\frac{\partial c}{\partial z}
  13. K c K_{c}
  14. c c
  15. l c l_{c}
  16. c c
  17. K c K_{c}
  18. K c = l c 2 | u ¯ z | K_{c}=l_{c}^{2}\left|\frac{\partial\bar{u}}{\partial z}\right|
  19. z z
  20. c c
  21. l c l_{c}
  22. l c = κ z l_{c}=\kappa z
  23. κ \kappa
  24. K c K_{c}
  25. K c = S l c e 1/2 K_{c}=Sl_{c}e^{\,\text{1/2}}
  26. S S
  27. e e
  28. S S
  29. l c l_{c}
  30. c c
  31. w c ¯ = - K c ( c z - γ c ) \overline{w^{\prime}c^{\prime}}=-K_{c}(\frac{\partial c}{\partial z}-\gamma_{c})
  32. γ c \gamma_{c}
  33. γ c = b ( w c ) s ¯ w s \gamma_{c}=b\frac{\overline{(w^{\prime}c^{\prime})_{s}}}{w_{s}}
  34. ( w c ) s ¯ \overline{(w^{\prime}c^{\prime})_{s}}
  35. c c
  36. b b
  37. w s w_{s}
  38. K c = κ z w s ( 1 - z h ) 2 K_{c}=\kappa zw_{s}(1-\frac{z}{h})^{2}
  39. κ \kappa
  40. z z
  41. h h
  42. w c ¯ = ( w c ) s ¯ ( 1 - z / h 1 ) + ( w c ) l ¯ ( z / h 1 ) \overline{w^{\prime}c^{\prime}}=\overline{(w^{\prime}c^{\prime})_{s}}(1-z/h_{1% })+\overline{(w^{\prime}c^{\prime})_{l}}(z/h_{1})
  43. h 1 h_{1}
  44. ( w c ) l ¯ \overline{(w^{\prime}c^{\prime})_{l}}
  45. ( w c ) s ¯ \overline{(w^{\prime}c^{\prime})_{s}}
  46. ( w c ) s ¯ = K s c s z \overline{(w^{\prime}c^{\prime})_{s}}=K_{s}\frac{\partial c_{s}}{\partial z}
  47. ( w c ) l ¯ = K l c l z \overline{(w^{\prime}c^{\prime})_{l}}=K_{l}\frac{\partial c_{l}}{\partial z}
  48. K s K_{s}
  49. K l K_{l}
  50. K s = w * h 1 ( 1 - z / h 1 ) g s K_{s}=\frac{w_{*}h_{1}(1-z/h_{1})}{g_{s}}
  51. K l = w * z g l K_{l}=\frac{w_{*}z}{g_{l}}
  52. w * w_{*}
  53. w * = ( g w s θ ¯ h 1 T ) 1/3 w_{*}=(g\overline{w_{s}\theta}h_{1}T)\text{1/3}
  54. g b g_{b}
  55. z / h 1 z/h_{1}
  56. g l g_{l}
  57. g t g_{t}
  58. g l g_{l}

Convenient_vector_space.html

  1. C [ u s u , u p = 21 e ] C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  2. c [ u s u , u p = 21 e ] c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  3. c [ u s u , u p = 21 e ] c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  4. E E
  5. c : E c:ℝ→E
  6. C [ u s u , u p = 21 e ] C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  7. C [ u s u , u p = 21 e ] ( , E ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](ℝ,E)
  8. E E
  9. C [ u s u , u p = 21 e ] ( , E ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](ℝ,E)
  10. E E
  11. C C
  12. E [ u s u , u b = , u B ] E E[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}B^{\prime}]→E
  13. B B
  14. E E
  15. E [ u s u , u b = , u B ] E[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}B^{\prime}]
  16. B B
  17. = =
  18. x [ u s u , u b = , u n ] x x[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime}]→x
  19. λ [ u s u , u b = , u n ] ( x [ u s u , u b = , u n ] x ) λ[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime}](x[u^{% \prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime}]−x)
  20. c [ u s u , u p = 21 e ] c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  21. E E
  22. c [ u s u , u p = 21 e ] E c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]E
  23. D D
  24. c [ u s u , u p = 21 e ] ( D × D ) ( c [ u s u , u p = 21 e ] D ) × ( c [ u s u , u p = 21 e ] D ) c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](D×D)≠(c[u^{\prime}su^{% \prime},u^{\prime}p=\u{2}21e^{\prime}]D)×(c[u^{\prime}su^{\prime},u^{\prime}p=% \u{2}21e^{\prime}]D)
  25. E E
  26. c [ u s u , u p = 21 e ] E c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]E
  27. E E
  28. c [ u s u , u p = 21 e ] E = E c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]E=E
  29. E E
  30. c [ u s u , u p = 21 e ] c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  31. c C [ u s u , u p = 21 e ] ( , E ) c∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](ℝ,E)
  32. [ u s u , u p = 1 , u b = 0 ] c ( t ) d t ∫[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}]c(t)dt
  33. E E
  34. E E
  35. C [ u s u , u p = 21 e ] C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  36. C [ u s u , u p = 21 e ] C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  37. c : E c:ℝ→E
  38. λ o c : t λ ( c ( t ) ) λoc:t→λ(c(t))
  39. C [ u s u , u p = 21 e ] ( , ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](ℝ,ℝ)
  40. λ E [ u s u , u p = 217 ] λ∈E[u^{\prime}su^{\prime},u^{\prime}p=\u{2}217^{\prime}]
  41. E [ u s u , u p = 217 ] E[u^{\prime}su^{\prime},u^{\prime}p=\u{2}217^{\prime}]
  42. E E
  43. λ E λ∈E′
  44. λ V λ∈V
  45. V V
  46. E E′
  47. E E
  48. t [ u s u , u b = n m ] ( x [ u s u , u b - n ] x [ u s u , u b = m ] ) 0 t[u^{\prime}su^{\prime},u^{\prime}b=nm^{\prime}](x[u^{\prime}su^{\prime},u^{% \prime}b-n^{\prime}]−x[u^{\prime}su^{\prime},u^{\prime}b=m^{\prime}])→0
  49. t [ u s u , u b = n m ] t[u^{\prime}su^{\prime},u^{\prime}b=nm^{\prime}]→∞
  50. 𝐑 \mathbf{R}
  51. E E
  52. B B
  53. E [ u s u , u b = , u B ] E[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}B^{\prime}]
  54. f : E f:ℝ→E
  55. L i p [ u s u , u p = , u k ] Lip[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime}]
  56. f f
  57. L i p [ u s u , u p = , u k ] Lip[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime}]
  58. k > 1 k>1
  59. f : E f:ℝ→E
  60. C [ u s u , u p = 21 e ] C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  61. f f
  62. f : E f:ℝ→E
  63. L i p [ u s u , u p = , u k ] Lip[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime}]
  64. k k
  65. 𝐑 \mathbf{R}
  66. E E
  67. F F
  68. U E U⊆E
  69. c [ u s u , u p = 21 e ] c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  70. f : U F f:U→F
  71. C [ u s u , u p = 21 e ] C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  72. f o c C [ u s u , u p = 21 e ] ( , F ) foc∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](ℝ,F)
  73. c C [ u s u , u p = 21 e ] ( , U ) c∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](ℝ,U)
  74. [ u s u , u p = 2 ] ℝ[u^{\prime}su^{\prime},u^{\prime}p=2^{\prime}]
  75. f : E U F f:E⊇U→F
  76. d f : U × E F df:U×E→F
  77. d f : U L ( E , F ) df:U→L(E,F)
  78. L ( E , F ) L(E,F)
  79. C [ u s u , u p = 21 e ] ( U , F ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](U,F)
  80. U F U→F
  81. C [ u s u , u p = 21 e ] ( , ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](ℝ,ℝ)
  82. C ( U , F ) c C ( , U ) , F * C ( , ) , f ( f c ) c , . C^{\infty}(U,F)\to\prod_{c\in C^{\infty}(\mathbb{R},U),\ell\in F^{*}}C^{\infty% }(\mathbb{R},\mathbb{R}),\quad f\mapsto(\ell\circ f\circ c)_{c,\ell}\,.
  83. c [ u s u , u p = 21 e ] c[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  84. V F V⊆F
  85. C ( U , C ( V , G ) ) C ( U × V , G ) , f g , f ( u ) ( v ) = g ( u , v ) . C^{\infty}(U,C^{\infty}(V,G))\cong C^{\infty}(U\times V,G),\qquad f\mapsto g,% \qquad f(u)(v)=g(u,v).
  86. f : E C [ u s u , u p = 21 e ] ( V , G ) f:E→C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](V,G)
  87. e v [ u s u , u b = , u v ] o f : V G ev[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}v^{\prime}]of:V→G
  88. v V v∈V
  89. ev : C ( E , F ) × E F , ev ( f , x ) = f ( x ) \displaystyle\operatorname{ev}:C^{\infty}(E,F)\times E\to F,\quad\,\text{ev}(f% ,x)=f(x)
  90. L i p [ u s u , u p = , u k ] Lip[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime}]
  91. C [ u s u , u p = , u k , u , 3 b 1 ] C[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}k^{\prime},u^{\prime},% \u{0}3b1^{\prime}]
  92. k k
  93. M M
  94. N N
  95. M M
  96. g ¯ \bar{g}
  97. N N
  98. g ¯ \bar{g}
  99. C [ u s u , u p = 21 e ] ( M , N ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](M,N)
  100. M N M→N
  101. f C [ u s u , u p = 21 e ] ( M , N ) f∈C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](M,N)
  102. u f : C ( M , N ) U f = { g : ( f , g ) ( M ) V N × N } U ~ f Γ ( f * T N ) , u_{f}:C^{\infty}(M,N)\supset U_{f}=\{g:(f,g)(M)\subset V^{N\times N}\}\to% \tilde{U}_{f}\subset\Gamma(f^{*}TN),
  103. u f ( g ) = ( π N , exp g ¯ ) - 1 ( f , g ) , u f ( g ) ( x ) = ( exp f ( x ) g ¯ ) - 1 ( g ( x ) ) , u_{f}(g)=(\pi_{N},\exp^{\bar{g}})^{-1}\circ(f,g),\quad u_{f}(g)(x)=(\exp^{\bar% {g}}_{f(x)})^{-1}(g(x)),
  104. ( u f ) - 1 ( s ) = exp f g ¯ s , ( u f ) - 1 ( s ) ( x ) = exp f ( x ) g ¯ ( s ( x ) ) . (u_{f})^{-1}(s)=\exp^{\bar{g}}_{f}\circ s,\qquad\quad(u_{f})^{-1}(s)(x)=\exp^{% \bar{g}}_{f(x)}(s(x)).
  105. f [ u s u , u p = 217 ] T N f[u^{\prime}su^{\prime},u^{\prime}p=\u{2}217^{\prime}]TN
  106. C ( , Γ ( M ; f * T N ) ) = Γ ( × M ; pr 2 * f * T N ) . C^{\infty}(\mathbb{R},\Gamma(M;f^{*}TN))=\Gamma(\mathbb{R}\times M;% \operatorname{pr_{2}}^{*}f^{*}TN).
  107. C [ u s u , u p = 21 e ] C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}]
  108. U ~ f 1 s ( π N , exp g ¯ ) s ( π N , exp g ¯ ) ( f 2 , exp f 1 g ¯ s ) . \tilde{U}_{f_{1}}\ni s\mapsto(\pi_{N},\exp^{\bar{g}})\circ s\mapsto(\pi_{N},% \exp^{\bar{g}})\circ(f_{2},\exp^{\bar{g}}_{f_{1}}\circ s).
  109. C [ u s u , u p = 21 e ] ( M , N ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](M,N)
  110. C ( , C ( M , N ) ) C ( × M , N ) . C^{\infty}(\mathbb{R},C^{\infty}(M,N))\cong C^{\infty}(\mathbb{R}\times M,N).
  111. C ( P , M ) × C ( M , N ) C ( P , N ) , ( f , g ) g f , C^{\infty}(P,M)\times C^{\infty}(M,N)\to C^{\infty}(P,N),\qquad(f,g)\mapsto g% \circ f,
  112. π C ( M , N ) = C ( M , π N ) : T C ( M , N ) = C ( M , T N ) C ( M , N ) . \pi_{C^{\infty}(M,N)}=C^{\infty}(M,\pi_{N}):TC^{\infty}(M,N)=C^{\infty}(M,TN)% \to C^{\infty}(M,N).
  113. G G
  114. 𝔤 = T e G \mathfrak{g}=T_{e}G
  115. μ : G × G G , μ ( x , y ) = x . y = μ x ( y ) = μ y ( x ) , ν : G G , ν ( x ) = x - 1 . \mu:G\times G\to G,\quad\mu(x,y)=x.y=\mu_{x}(y)=\mu^{y}(x),\qquad\nu:G\to G,% \nu(x)=x^{-1}.
  116. G G
  117. X C ( , 𝔤 ) X\in C^{\infty}(\mathbb{R},\mathfrak{g})
  118. g C ( , G ) g\in C^{\infty}(\mathbb{R},G)
  119. X X
  120. g g
  121. g ( 0 ) g(0)
  122. g ( 0 ) = e , t g ( t ) = T e ( μ g ( t ) ) X ( t ) = X ( t ) . g ( t ) . g(0)=e,\qquad\partial_{t}g(t)=T_{e}(\mu^{g(t)})X(t)=X(t).g(t).
  123. g g
  124. X X
  125. evol G r ( X ) = g ( 1 ) , Evol G r ( X ) ( t ) := g ( t ) = evol G r ( t X ) . \operatorname{evol}^{r}_{G}(X)=g(1),\quad\operatorname{Evol}^{r}_{G}(X)(t):=g(% t)=\operatorname{evol}^{r}_{G}(tX).
  126. evol G r : C ( , 𝔤 ) G . \operatorname{evol}^{r}_{G}:C^{\infty}(\mathbb{R},\mathfrak{g})\to G.
  127. X X
  128. e v o l [ u s u , u p = r , u b = , u G ] ( X ) = e x p [ u s u , u p = , u G ] ( X ) evol[u^{\prime}su^{\prime},u^{\prime}p=r^{\prime},u^{\prime}b=^{\prime},u^{% \prime}G^{\prime}](X)=exp[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{% \prime}G^{\prime}](X)
  129. M M
  130. D i f f ( M ) Diff(M)
  131. 𝔛 ( M ) \mathfrak{X}(M)
  132. M M
  133. D i f f ( M ) Diff(M)
  134. C [ u s u , u p = 21 e ] ( M , M ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](M,M)
  135. t f ( t , ) t→f(t,  )
  136. D i f f ( M ) Diff(M)
  137. f ( t , ) [ u s u , u p = 2121 ] f(t,  )[u^{\prime}su^{\prime},u^{\prime}p=\u{2}2121^{\prime}]
  138. f ( t , f ( t , ) - 1 ( x ) ) = x f(t,f(t,\quad)^{-1}(x))=x
  139. ( t . x ) f ( t , ) [ u s u , u p = 2121 ] ( x ) (t.x)↦f(t,  )[u^{\prime}su^{\prime},u^{\prime}p=\u{2}2121^{\prime}](x)
  140. X ( t , x ) X(t,x)
  141. M M
  142. C ( , 𝔛 ( M ) ) C^{\infty}(\mathbb{R},\mathfrak{X}(M))
  143. F l Fl
  144. t × X \partial_{t}\times X
  145. × M ℝ×M
  146. Fl s ( t , x ) = ( t + s , Evol ( X ) ( t , x ) ) \operatorname{Fl}_{s}(t,x)=(t+s,\operatorname{Evol}(X)(t,x))
  147. t Evol ( X ) ( t , x ) = X ( t , Evol ( X ) ( t , x ) ) . \partial_{t}\operatorname{Evol}(X)(t,x)=X(t,\operatorname{Evol}(X)(t,x)).
  148. X ( s , t , x ) C ( 2 , 𝔛 ( M ) ) X(s,t,x)\in C^{\infty}(\mathbb{R}^{2},\mathfrak{X}(M))
  149. s s
  150. e v o l [ u s u , u p = r , u b = D i f f ( , u M ] evol[u^{\prime}su^{\prime},u^{\prime}p=r^{\prime},u^{\prime}b=Diff(^{\prime},u% ^{\prime}M^{\prime}]
  151. M M
  152. N N
  153. M M
  154. E m b ( M , N ) Emb(M,N)
  155. M M
  156. N N
  157. C [ u s u , u p = 21 e ] ( M , N ) C[u^{\prime}su^{\prime},u^{\prime}p=\u{2}21e^{\prime}](M,N)
  158. D i f f ( M ) Diff(M)
  159. E m b ( M , N ) Emb(M,N)
  160. E m b ( M , N ) E m b ( M , N ) / D i f f ( M ) Emb(M,N)→Emb(M,N)/Diff(M)
  161. D i f f ( M ) Diff(M)
  162. g ¯ \bar{g}
  163. N N
  164. f E m b ( M , N ) f∈Emb(M,N)
  165. f ( M ) f(M)
  166. N N
  167. T N TN
  168. f ( M ) f(M)
  169. f ( M ) f(M)
  170. f ( M ) f(M)
  171. T N | f ( M ) = Nor ( f ( M ) ) T f ( M ) TN|_{f(M)}=\operatorname{Nor}(f(M))\oplus Tf(M)
  172. p f ( M ) : Nor ( f ( M ) ) W f ( M ) f ( M ) . p_{f(M)}:\operatorname{Nor}(f(M))\supset W_{f(M)}\to f(M).
  173. g : M N g:M→N
  174. C [ u s u , u p = 1 ] C[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime}]
  175. f f
  176. ϕ ( g ) := f - 1 p f ( M ) g Diff ( M ) and g ϕ ( g ) - 1 Γ ( f * W f ( M ) ) Γ ( f * Nor ( f ( M ) ) ) . \phi(g):=f^{-1}\circ\,p_{f(M)}\circ\,g\in\operatorname{Diff}(M)\quad\,\text{% and}\quad g\circ\,\phi(g)^{-1}\in\Gamma(f^{*}W_{f(M)})\subset\Gamma(f^{*}% \operatorname{Nor}(f(M))).

Convex_position.html

  1. ( ( 2 n - 2 n - 1 ) / n ! ) 2 . \left({\left({{2n-2}\atop{n-1}}\right)}/n!\right)^{2}.

Convolutional_neural_network.html

  1. f ( x ) = m a x ( 0 , x ) f(x)=max(0,x)
  2. f ( x ) = t a n h ( x ) f(x)=tanh(x)
  3. f ( x ) = | t a n h ( x ) | f(x)=|tanh(x)|
  4. f ( x ) = ( 1 + e - x ) - 1 f(x)=(1+e^{-x})^{-1}

Copal-8-ol_diphosphate_hydratase.html

  1. \rightleftharpoons

Corank.html

  1. m × n m\times n
  2. m - r m-r
  3. r r
  4. n n
  5. r r
  6. n - r n-r

Coshc_function.html

  1. Coshc ( z ) = cosh ( z ) z \operatorname{Coshc}(z)=\frac{\cosh(z)}{z}
  2. w ( z ) z - 2 d d z w ( z ) - z d 2 d z 2 w ( z ) = 0 w(z)z-2\frac{d}{dz}w(z)-z\frac{d^{2}}{dz^{2}}w(z)=0
  3. Im ( cosh ( x + i y ) x + i y ) \operatorname{Im}\left(\frac{\cosh(x+iy)}{x+iy}\right)
  4. Re ( cosh ( x + i y ) x + i y ) \operatorname{Re}\left(\frac{\cosh(x+iy)}{x+iy}\right)
  5. | cosh ( x + i y ) x + i y | \left|\frac{\cosh(x+iy)}{x+iy}\right|
  6. 1 - cosh ( z ) ) 2 z - cosh ( z ) z 2 \frac{1-\cosh(z))^{2}}{z}-\frac{\cosh(z)}{z^{2}}
  7. - Re ( - 1 - ( cosh ( x + i y ) ) 2 x + i y + cosh ( x + i y ) ( x + i y ) 2 ) -\operatorname{Re}\left(-\frac{1-(\cosh(x+iy))^{2}}{x+iy}+\frac{\cosh(x+iy)}{(% x+iy)^{2}}\right)
  8. - Im ( - 1 - ( cosh ( x + i y ) ) 2 x + i y + cosh ( x + i y ) ( x + i y ) 2 ) -\operatorname{Im}\left(-\frac{1-(\cosh(x+iy))^{2}}{x+iy}+\frac{\cosh(x+iy)}{(% x+iy)^{2}}\right)
  9. | - 1 - ( cosh ( x + i y ) ) 2 x + i y + cosh ( x + i y ) ( x + i y ) 2 | \left|-\frac{1-(\cosh(x+iy))^{2}}{x+iy}+\frac{\cosh(x+iy)}{(x+iy)^{2}}\right|
  10. Coshc ( z ) = ( i z + 1 / 2 π ) M ( 1 , 2 , i π - 2 z ) e ( i / 2 ) π - z z \operatorname{Coshc}(z)=\frac{(iz+1/2\,\pi){\rm M}(1,2,i\pi-2z)}{{\rm e}^{(i/2% )\pi-z}z}
  11. Coshc ( z ) = 1 2 ( 2 i z + π ) HeunB ( 2 , 0 , 0 , 0 , 2 1 / 2 i π - z ) e 1 / 2 i π - z z \operatorname{Coshc}(z)=\frac{1}{2}\,\frac{(2\,iz+\pi)\operatorname{HeunB}% \left(2,0,0,0,\sqrt{2}\sqrt{1/2\,i\pi-z}\right)}{{\rm e}^{1/2\,i\pi-z}z}
  12. Coshc ( z ) = - i ( 2 i z + π ) 𝐖 hittakerM ( 0 , 1 / 2 , i π - 2 z ) ( 4 i z + 2 π ) z \operatorname{Coshc}(z)=\frac{-i(2\,iz+\pi){{\rm\mathbf{W}hittakerM}(0,\,1/2,% \,i\pi-2z)}}{(4iz+2\pi)z}
  13. Coshc z ( z - 1 + 1 2 z + 1 24 z 3 + 1 720 z 5 + 1 40320 z 7 + 1 3628800 z 9 + 1 479001600 z 11 + 1 87178291200 z 13 + O ( z 15 ) ) \operatorname{Coshc}z\approx\left(z^{-1}+\frac{1}{2}z+\frac{1}{24}z^{3}+\frac{% 1}{720}z^{5}+\frac{1}{40320}z^{7}+\frac{1}{3628800}z^{9}+\frac{1}{479001600}z^% {11}+\frac{1}{87178291200}z^{13}+O(z^{15})\right)
  14. 𝐶𝑜𝑠ℎ𝑐 ( z ) = 23594700729600 + 11275015752000 z 2 + 727718024880 z 4 + 13853547000 z 6 + 80737373 z 8 147173 z 9 - 39328920 z 7 + 5772800880 z 5 - 522334612800 z 3 + 23594700729600 z {\it Coshc}\left(z\right)={\frac{23594700729600+11275015752000\,{z}^{2}+727718% 024880\,{z}^{4}+13853547000\,{z}^{6}+80737373\,{z}^{8}}{147173\,{z}^{9}-393289% 20\,{z}^{7}+5772800880\,{z}^{5}-522334612800\,{z}^{3}+23594700729600\,z}}

Cosheaf.html

  1. U i U_{i}
  2. lim F ( U i ) F ( U ) \underrightarrow{\lim}F(U_{i})\to F(U)
  3. F ( U V ) F(U\cup V)
  4. F ( U V ) F ( U ) F(U\cap V)\to F(U)
  5. F ( U V ) F ( V ) F(U\cap V)\to F(V)
  6. U C * ( U ; A ) U\mapsto C_{*}(U;A)
  7. U f - 1 ( U ) U\mapsto f^{-1}(U)

Coskewness.html

  1. S ( X , X , Y ) = E [ ( X - E [ X ] ) 2 ( Y - E [ Y ] ) ] σ X 2 σ Y S(X,X,Y)=\frac{\operatorname{E}\left[(X-\operatorname{E}[X])^{2}(Y-% \operatorname{E}[Y])\right]}{\sigma_{X}^{2}\sigma_{Y}}
  2. S ( X , Y , Y ) = E [ ( X - E [ X ] ) ( Y - E [ Y ] ) 2 ] σ X σ Y 2 S(X,Y,Y)=\frac{\operatorname{E}\left[(X-\operatorname{E}[X])(Y-\operatorname{E% }[Y])^{2}\right]}{\sigma_{X}\sigma_{Y}^{2}}
  3. σ X \sigma_{X}
  4. S ( X , X , X ) = E [ ( X - E [ X ] ) 3 ] σ X 3 = skewness [ X ] , S(X,X,X)=\frac{\operatorname{E}\left[(X-\operatorname{E}[X])^{3}\right]}{% \sigma_{X}^{3}}={\operatorname{skewness}[X]},
  5. S X + Y = 1 σ X + Y 3 [ σ X 3 S X + 3 σ X 2 σ Y S ( X , X , Y ) + 3 σ X σ Y 2 S ( X , Y , Y ) + σ Y 3 S Y ] , S_{X+Y}={1\over\sigma_{X+Y}^{3}}{\left[\sigma_{X}^{3}S_{X}+3\sigma_{X}^{2}% \sigma_{Y}S(X,X,Y)+3\sigma_{X}\sigma_{Y}^{2}S(X,Y,Y)+\sigma_{Y}^{3}S_{Y}\right% ]},
  6. σ X \sigma_{X}

Cost-loss_model.html

  1. C < p L C<pL
  2. C / L < p C/L<p
  3. C > p L C>pL
  4. C / L > p C/L>p
  5. C = p L C=pL
  6. C / L C/L
  7. C > L C>L

Cost_of_transport.html

  1. C O T E m g d = P m g v COT\triangleq\frac{E}{mgd}=\frac{P}{mgv}
  2. E E
  3. m m
  4. d d
  5. g g
  6. P P
  7. v v

Cotangent_sheaf.html

  1. 𝒪 X \mathcal{O}_{X}
  2. 𝒪 X \mathcal{O}_{X}
  3. Hom 𝒪 X ( Ω X / S , F ) = Der S ( 𝒪 X , F ) \operatorname{Hom}_{\mathcal{O}_{X}}(\Omega_{X/S},F)=\operatorname{Der}_{S}(% \mathcal{O}_{X},F)
  4. d : 𝒪 X Ω X / S d:\mathcal{O}_{X}\to\Omega_{X/S}
  5. D : 𝒪 X F D:\mathcal{O}_{X}\to F
  6. D = α d D=\alpha\circ d
  7. α : Ω X / S F \alpha:\Omega_{X/S}\to F
  8. Ω X / S \Omega_{X/S}
  9. Θ X \Theta_{X}
  10. f * Ω S / T Ω X / T Ω X / S 0. f^{*}\Omega_{S/T}\to\Omega_{X/T}\to\Omega_{X/S}\to 0.
  11. I / I 2 Ω X / S 𝒪 Z Ω Z / S 0. I/I^{2}\to\Omega_{X/S}\otimes\mathcal{O}_{Z}\to\Omega_{Z/S}\to 0.
  12. f : X S f:X\to S
  13. Ω X / S = Δ * ( I / I 2 ) \Omega_{X/S}=\Delta^{*}(I/I^{2})
  14. 𝐏 R n \mathbf{P}^{n}_{R}
  15. 0 Ω 𝐏 R n / R 𝒪 𝐏 R n ( - 1 ) ( n + 1 ) 𝒪 𝐏 R n 0. 0\to\Omega_{\mathbf{P}^{n}_{R}/R}\to\mathcal{O}_{\mathbf{P}^{n}_{R}}(-1)^{% \oplus(n+1)}\to\mathcal{O}_{\mathbf{P}^{n}_{R}}\to 0.
  16. 𝐒𝐩𝐞𝐜 ( Sym ( E ˇ ) ) \mathbf{Spec}(\operatorname{Sym}(\check{E}))
  17. Bun G ( X ) \operatorname{Bun}_{G}(X)
  18. Θ X = def o m 𝒪 X ( Ω X , 𝒪 X ) = 𝒟 e r ( 𝒪 X ) . \Theta_{X}\overset{\mathrm{def}}{=}\mathcal{H}om_{\mathcal{O}_{X}}(\Omega_{X},% \mathcal{O}_{X})=\mathcal{D}er(\mathcal{O}_{X}).

Cotriple_homology.html

  1. E = - R N E=-\otimes_{R}N
  2. F U FU
  3. E ( F U * M ) E(FU_{*}M)
  4. Tor n R ( M , N ) \operatorname{Tor}^{R}_{n}(M,N)
  5. R lim n G L n ( R ) R\mapsto\underrightarrow{\lim}_{n}GL_{n}(R)
  6. F U FU
  7. K n ( R ) = π n - 2 G L ( F U * R ) , n 3 K_{n}(R)=\pi_{n-2}GL(FU_{*}R),\,n\geq 3

Count-distinct_problem.html

  1. x 1 , x 2 , , x s x_{1},x_{2},\ldots,x_{s}
  2. m m
  3. n n
  4. n = | { x 1 , x 2 , , x s } | n=|\left\{{x_{1},x_{2},\ldots,x_{s}}\right\}|
  5. { e 1 , e 2 , , e n } \left\{{e_{1},e_{2},\ldots,e_{n}}\right\}
  6. n ^ \widehat{n}
  7. n n
  8. m m
  9. m n m\ll n
  10. a , b , a , c , d , b , d a,b,a,c,d,b,d
  11. n = | { a , b , c , d } | = 4 n=|\left\{{a,b,c,d}\right\}|=4
  12. c c
  13. c 0 c\leftarrow 0
  14. D D
  15. x i x_{i}
  16. x i x_{i}
  17. D D
  18. x i D x_{i}\notin D
  19. x i x_{i}
  20. D D
  21. c c
  22. c c + 1 c\leftarrow c+1
  23. x i D x_{i}\in D
  24. n = c n=c
  25. D D
  26. x i x_{i}
  27. n n
  28. e j e_{j}
  29. h ( e j ) h(e_{j})
  30. e j e_{j}
  31. h ( e j ) U ( 0 , 1 ) h(e_{j})\sim U(0,1)
  32. h ( e 1 ) , h ( e 2 ) , , h ( e n ) h(e_{1}),h(e_{2}),\ldots,h(e_{n})
  33. 1 / ( n + 1 ) 1/(n+1)
  34. h ( e j ) h(e_{j})
  35. e j e_{j}
  36. m m
  37. m 1 m\geq 1
  38. x 1 , x 2 , , x s x_{1},x_{2},\ldots,x_{s}
  39. m m
  40. n n
  41. n = | { x 1 , x 2 , , x s } | n=|\left\{{x_{1},x_{2},\ldots,x_{s}}\right\}|
  42. { e 1 , e 2 , , e n } \left\{{e_{1},e_{2},\ldots,e_{n}}\right\}
  43. w j w_{j}
  44. e j e_{j}
  45. w ^ \widehat{w}
  46. w = j = 1 n w j w=\sum_{j=1}^{n}w_{j}
  47. m m
  48. m n m\ll n
  49. a ( 3 ) , b ( 4 ) , a ( 3 ) , c ( 2 ) , d ( 3 ) , b ( 4 ) , d ( 3 ) a(3),b(4),a(3),c(2),d(3),b(4),d(3)
  50. e 1 = a , e 2 = b , e 3 = c , e 4 = d e_{1}=a,e_{2}=b,e_{3}=c,e_{4}=d
  51. w 1 = 3 , w 2 = 4 , w 3 = 2 , w 4 = 3 w_{1}=3,w_{2}=4,w_{3}=2,w_{4}=3
  52. w j = 12 \sum{w_{j}}=12
  53. x 1 , x 2 , , x s x_{1},x_{2},\ldots,x_{s}
  54. n n
  55. e 1 , e 2 , , e n e_{1},e_{2},\ldots,e_{n}
  56. w j w_{j}
  57. e j e_{j}
  58. j = 1 n w j \sum_{j=1}^{n}{w_{j}}
  59. x 1 , x 2 , , x s x_{1},x_{2},\ldots,x_{s}

Coupling_coefficient_of_resonators.html

  1. k k
  2. k = | f o - f e | / f 0 , k=|f_{o}-f_{e}|/f_{0},
  3. f e , f_{e},
  4. f o f_{o}
  5. f 0 = f e f o . f_{0}=\sqrt{f_{e}f_{o}}.
  6. f 0 . f_{0}.
  7. k k
  8. k = K 12 x 1 x 2 k=\frac{K_{12}}{\sqrt{x_{1}x_{2}}}
  9. k = J 12 b 1 b 2 k=\frac{J_{12}}{\sqrt{b_{1}b_{2}}}
  10. K 12 , K_{12},
  11. J 12 J_{12}
  12. x 1 , x_{1},
  13. x 2 x_{2}
  14. f 0 , f_{0},
  15. b 1 , b_{1},
  16. b 2 b_{2}
  17. k L = L m L 1 L 2 k_{L}=\frac{L_{m}}{\sqrt{L_{1}L_{2}}}
  18. k C = C m ( C 1 + C m ) ( C 2 + C m ) . k_{C}=\frac{C_{m}}{\sqrt{(C_{1}+C_{m})(C_{2}+C_{m})}}.
  19. L 1 , L_{1},
  20. C 1 C_{1}
  21. L 2 , L_{2},
  22. C 2 C_{2}
  23. L m , L_{m},
  24. C m C_{m}
  25. k = f o 2 - f e 2 f o 2 + f e 2 . k=\frac{f_{o}^{2}-f_{e}^{2}}{f_{o}^{2}+f_{e}^{2}}.
  26. k k
  27. f o < f e . f_{o}<f_{e}.
  28. k L k_{L}
  29. L m > 0 L_{m}>0
  30. L m < 0. L_{m}<0.
  31. k C k_{C}
  32. k C = - C m ( C 1 + C m ) ( C 2 + C m ) . k_{C}=\frac{-C_{m}}{\sqrt{(C_{1}+C_{m})(C_{2}+C_{m})}}.
  33. k L k_{L}
  34. k C . k_{C}.
  35. k L k_{L}
  36. k C k_{C}
  37. k = k L + k C 1 + k L k C . k=\frac{k_{L}+k_{C}}{1+k_{L}k_{C}}.
  38. k k
  39. k ( f ) k(f)
  40. k p k_{p}
  41. p p
  42. k ( f ) k(f)
  43. k ( f ) | f = f p = k p . k(f)|_{f=f_{p}}=k_{p}.
  44. k ( f ) k(f)
  45. f z f_{z}
  46. k ( f ) | f = f z = 0. k(f)|_{f=f_{z}}=0.
  47. L m > 0. L_{m}>0.
  48. k ( f ) k(f)
  49. f z = 1 2 π L m ( L 1 L 2 - L m 2 ) C m f_{z}=\frac{1}{2\pi}\sqrt{\frac{L_{m}}{(L_{1}L_{2}-L_{m}^{2})C_{m}}}
  50. k ( f ) k(f)
  51. k L ( f ) k_{L}(f)
  52. k C ( f ) k_{C}(f)
  53. k L ( f ) = W ˙ 12 L ( f ) [ W ¯ 11 L ( f ) + W ¯ 11 C ( f ) ] [ W ¯ 22 L ( f ) + W ¯ 22 C ( f ) ] , k_{L}(f)=\frac{\dot{W}_{12L}(f)}{\sqrt{[\bar{W}_{11L}(f)+\bar{W}_{11C}(f)][% \bar{W}_{22L}(f)+\bar{W}_{22C}(f)]}},
  54. k C ( f ) = W ˙ 12 C ( f ) [ W ¯ 11 L ( f ) + W ¯ 11 C ( f ) ] [ W ¯ 22 L ( f ) + W ¯ 22 C ( f ) ] . k_{C}(f)=\frac{\dot{W}_{12C}(f)}{\sqrt{[\bar{W}_{11L}(f)+\bar{W}_{11C}(f)][% \bar{W}_{22L}(f)+\bar{W}_{22C}(f)]}}.
  55. W W
  56. W W
  57. L L
  58. C C
  59. | U 1 | 2 , |U_{1}|^{2},
  60. | U 1 | | U 2 | |U_{1}||U_{2}|
  61. | U 2 | 2 |U_{2}|^{2}
  62. U 1 U_{1}
  63. U 2 U_{2}
  64. k L ( f ) = L m L 1 L 2 2 ( 1 + f 1 - 2 f 2 ) ( 1 + f 2 - 2 f 2 ) , k_{L}(f)=\frac{L_{m}}{\sqrt{L_{1}L_{2}}}\frac{2}{\sqrt{(1+f_{1}^{-2}f^{2})(1+f% _{2}^{-2}f^{2})}},
  65. k C ( f ) = - C m ( C 1 + C m ) ( C 2 + C m ) 2 ( 1 + f 1 2 f - 2 ) ( 1 + f 2 2 f - 2 ) k_{C}(f)=\frac{-C_{m}}{\sqrt{(C_{1}+C_{m})(C_{2}+C_{m})}}\frac{2}{\sqrt{(1+f_{% 1}^{2}f^{-2})(1+f_{2}^{2}f^{-2})}}
  66. f 1 , f_{1},
  67. f 2 f_{2}
  68. f = f 1 = f 2 f=f_{1}=f_{2}
  69. k L k_{L}
  70. k C k_{C}
  71. k ( f ) k(f)
  72. f z f_{z}
  73. f 0 . f_{0}.
  74. k i , i + 1 k_{i,i+1}
  75. i i
  76. i + 1 i+1
  77. k i , i + 1 = f 2 - f 1 f 1 f 2 g i g i + 1 , k_{i,i+1}=\frac{f_{2}-f_{1}}{\sqrt{f_{1}f_{2}g_{i}g_{i+1}}},
  78. g i g_{i}
  79. ( i = 0 , 1 , 2... n ) (i=0,1,2...n)
  80. n n
  81. f 1 , f_{1},
  82. f 2 f_{2}
  83. g i g_{i}
  84. g 0 = 1 , g_{0}=1,
  85. g 1 = 2 a 1 / γ , g_{1}=2a_{1}/\gamma,
  86. g i = 4 a i - 1 a i b i - 1 g i - 1 , ( i = 2 , 3 , n ) , g_{i}=\frac{4a_{i-1}a_{i}}{b_{i-1}g_{i-1}},(i=2,3,...n),
  87. g n + 1 = 1 , g_{n+1}=1,
  88. n n
  89. g n + 1 = coth 2 ( β / 4 ) , g_{n+1}=\mathrm{coth}^{2}(\beta/4),
  90. n n
  91. β = 2 a r t a n h 10 - Δ L / 10 , \beta=2\mathrm{artanh}\sqrt{10^{-\Delta L/10}},
  92. γ = sh ( β 2 n ) , \gamma=\mathrm{sh}(\frac{\beta}{2n}),
  93. a i = sin ( 2 i - 1 ) π 2 n , a_{i}=\mathrm{sin}\frac{(2i-1)\pi}{2n},
  94. b i = γ 2 + sin 2 ( i π n ) , ( i = 1 , 2 , n ) , b_{i}=\gamma^{2}+\mathrm{sin}^{2}(\frac{i\pi}{n}),(i=1,2,...n),
  95. Δ L \Delta L
  96. k i , i + 1 k_{i,i+1}
  97. f 0 f_{0}
  98. d k i , i + 1 / d f | f = f 0 . dk_{i,i+1}/df|_{f=f_{0}}.
  99. k i , i + 1 k_{i,i+1}
  100. k i , i + 1 . k_{i,i+1}.
  101. k i , i + 1 k_{i,i+1}
  102. k i , i + 1 k_{i,i+1}
  103. k i , i + 1 k_{i,i+1}
  104. 𝐌 \mathbf{M}
  105. n × n n\times n
  106. M i j M_{ij}
  107. k i j . k_{ij}.
  108. M i i M_{ii}
  109. M i i M_{ii}
  110. 𝐌 \mathbf{M}
  111. 𝐌 \mathbf{M}

Cours_d'Analyse.html

  1. 1 4 , 1 3 , 1 6 , 1 5 , 1 8 , 1 7 , \frac{1}{4},\frac{1}{3},\frac{1}{6},\frac{1}{5},\frac{1}{8},\frac{1}{7},\ldots
  2. α \alpha
  3. α \alpha
  4. α , α 2 , α 3 , \alpha,\alpha^{2},\alpha^{3},\ldots
  5. k α n k\alpha^{n}\quad{}
  6. k α n ( 1 ± ε ) {}\quad k\alpha^{n}(1\pm\varepsilon)
  7. α \alpha
  8. f ( x + α ) - f ( x ) f(x+\alpha)-f(x)
  9. f ( x + α ) - f ( x ) f(x+\alpha)-f(x)
  10. α \alpha
  11. u 0 , u 1 , u 2 , , u n , u n + 1 , u_{0},u_{1},u_{2},\ldots,u_{n},u_{n+1},\ldots

Covariance_mapping.html

  1. X n ( E ) X_{n}(E)
  2. n n
  3. E E
  4. X n ( E ) X_{n}(E)
  5. n n
  6. E E
  7. E i E_{i}
  8. 𝐗 = 𝐗 n = [ X n ( E 1 ) , X n ( E 2 ) , X n ( E 3 ) , up to the last sample ] . \mathbf{X}=\mathbf{X}_{n}=[X_{n}(E_{1}),X_{n}(E_{2}),X_{n}(E_{3}),\ \ldots\,% \text{ up to the last sample}].
  9. N N
  10. 𝐗 = 1 N n = 1 N 𝐗 n . \langle\mathbf{X}\rangle=\frac{1}{N}\sum^{N}_{n=1}\mathbf{X}_{n}.
  11. 𝐜𝐨𝐯 ( 𝐘 , 𝐗 ) = 𝐘𝐗 - 𝐘 𝐗 , \mathbf{cov}(\mathbf{Y},\mathbf{X})=\langle\mathbf{YX}\rangle-\langle\mathbf{Y% }\rangle\langle\mathbf{X}\rangle,
  12. 𝐘 \mathbf{Y}
  13. 𝐗 \mathbf{X}
  14. 𝐗 \langle\mathbf{X}\rangle
  15. 𝐘 \langle\mathbf{Y}\rangle
  16. X n ( E i ) X_{n}(E_{i})
  17. E x = E y E_{x}=E_{y}
  18. I I
  19. 𝐩𝐜𝐨𝐯 ( 𝐘 , 𝐗 ; I ) = 𝐜𝐨𝐯 ( 𝐘 , 𝐗 ) - 𝐜𝐨𝐯 ( 𝐘 , I ) 𝐜𝐨𝐯 ( I , 𝐗 ) / cov ( I , I ) , \mathbf{pcov}(\mathbf{Y},\mathbf{X};I)=\mathbf{cov}(\mathbf{Y},\mathbf{X})-% \mathbf{cov}(\mathbf{Y},I)\mathbf{cov}(I,\mathbf{X})/\operatorname{cov}(I,I),
  20. cov ( I , I ) \operatorname{cov}(I,I)
  21. 𝐘𝐗 \langle\mathbf{YX}\rangle
  22. 𝐘 𝐗 \langle\mathbf{Y}\rangle\langle\mathbf{X}\rangle

Cramer's_theorem_(algebraic_curves).html

  1. x n , x n - 1 y 1 , , y n , x^{n},\,x^{n-1}y^{1},\,\dots,\,y^{n},
  2. x n - 1 , x n - 2 y 1 , , y n - 1 , x^{n-1},\,x^{n-2}y^{1},\,\dots,\,y^{n-1},
  3. x x
  4. y , y,
  5. x 3 - x = 0 x^{3}-x=0
  6. y 3 - y = 0. y^{3}-y=0.
  7. ( x - a ) 2 + ( y - b ) 2 = r 2 (x-a)^{2}+(y-b)^{2}=r^{2}
  8. x 2 - 2 a x + y 2 - 2 b y = k , x^{2}-2ax+y^{2}-2by=k,
  9. k = r 2 - a 2 - b 2 . k=r^{2}-a^{2}-b^{2}.

Cramer–Castillon_problem.html

  1. Z Z
  2. A , B , C A,B,C
  3. Z Z
  4. A , B , C A,B,C
  5. Z Z
  6. Z Z
  7. A , B , C A,B,C
  8. n n

Crenel_function.html

  1. P ( x ) = { 1 , x [ - Δ / 2 , Δ / 2 ] , 0 , x [ - Δ / 2 , Δ / 2 ] , P(x)=\begin{cases}1,&x\in[-\Delta/2,\Delta/2],\\ 0,&x\notin[-\Delta/2,\Delta/2],\end{cases}
  2. P m ( Δ , x ) = exp ( 2 π i m x ) sin ( π m Δ ) π m . P_{m}(\Delta,x)=\frac{\exp(2\pi imx)\sin(\pi m\Delta)}{\pi m}.

Critical_point_(network_science).html

  1. < k Align g t ; <k&gt;
  2. < k 1 <k>=1
  3. e e
  4. N N
  5. < k e N <k>=\frac{e}{N}
  6. < k 0 <k>=0
  7. < k > < 1 <k><1
  8. < k N - 1 <k>=N-1
  9. < k 1 <k>>1
  10. < k 1 <k>=1
  11. < k 2 <k>=2

Cross-serial_dependencies.html

  1. L = De Jan s a ¨ it das mer (d’chind) (em m Hans) s n huus h a ¨ nd wele (laa) (h m a ¨ lfe) aastriiche. n L=\,\text{De Jan}\,\text{ s}\ddot{\mathrm{a}}\,\text{it}\,\text{ das}\,\text{ % mer}\,\text{ (d'chind)}{}^{m}\,\text{ (em}\,\text{ Hans)}{}^{n}\,\text{ s}\,% \text{ huus}\,\text{ h}\ddot{\mathrm{a}}\,\text{nd}\,\text{ wele}\,\text{ (laa% )}{}^{m}\,\text{ (h}\ddot{\mathrm{a}}\,\text{lfe)}{}^{n}\,\text{ aastriiche.}
  2. L L
  3. x a m b n y c m d n z xa^{m}b^{n}yc^{m}d^{n}z
  4. x x
  5. y y
  6. z z
  7. L = a m b n c m d n L^{\prime}=a^{m}b^{n}c^{m}d^{n}
  8. L L^{\prime}

Crosswind_kite_power.html

  1. P = 2 27 ρ a A C L ( C L C D ) 2 V 3 P={2\over 27}\rho_{a}AC_{L}({C_{L}\over C_{D}})^{2}V^{3}
  2. P = 2 27 ρ a A C L G 2 V 3 P={2\over 27}\rho_{a}AC_{L}G^{2}V^{3}

Crowther_Criterion.html

  1. m = π D d m=\pi\frac{D}{d}

CryptoNote.html

  1. S = 64 n + 32 S=64n+32
  2. n n
  3. n n
  4. n n

CTAIDI.html

  1. CTAIDI = U i N i N i o \mbox{CTAIDI}~{}=\frac{\sum{U_{i}N_{i}}}{N_{io}}
  2. N i N_{i}
  3. U i U_{i}
  4. i i
  5. N i o N_{io}
  6. i i
  7. CTAIDI = sum of durations of customer interruptions number of distinct customers interrupted \mbox{CTAIDI}~{}=\frac{\mbox{sum of durations of customer interruptions}~{}}{% \mbox{number of distinct customers interrupted}~{}}
  8. N i o N i = SAIDI CTAIDI = SAIFI CAIFI \frac{N_{io}}{N_{i}}=\frac{\mbox{SAIDI}~{}}{\mbox{CTAIDI}~{}}=\frac{\mbox{% SAIFI}~{}}{\mbox{CAIFI}~{}}

Cubebol_synthase.html

  1. \rightleftharpoons

Cubic_mean.html

  1. x ¯ cubic \bar{x}_{\mathrm{cubic}}
  2. n n
  3. x i x_{i}\in\mathbb{R}
  4. x ¯ cubic = 1 n i = 1 n x i 3 3 = x 1 3 + x 2 3 + + x n 3 n 3 . \bar{x}_{\mathrm{cubic}}=\sqrt[3]{\frac{1}{n}\sum_{i=1}^{n}{x_{i}^{3}}}=\sqrt[% 3]{{x_{1}^{3}+x_{2}^{3}+\cdots+x_{n}^{3}}\over n}.
  5. p = 3 p=3

Cucurbitadienol_synthase.html

  1. \rightleftharpoons

Curtright_field.html

  1. T α β μ T_{\alpha\beta\mu}
  2. T α β μ = T [ α β ] μ T_{\alpha\beta\mu}=T_{[\alpha\beta]\mu}
  3. T [ α β μ ] = 0 T_{[\alpha\beta\mu]}=0
  4. T α β μ T_{\alpha\beta\mu}
  5. F α β γ μ = 3 [ γ T α β ] μ . F_{\alpha\beta\gamma\mu}=3\partial_{[\gamma}T_{\alpha\beta]\mu}.
  6. F α β = η γ μ F α β γ μ F_{\alpha\beta}=\eta^{\gamma\mu}F_{\alpha\beta\gamma\mu}
  7. η γ μ \eta^{\gamma\mu}
  8. T α β μ T_{\alpha\beta\mu}
  9. S = - 1 6 d D x ( F α β γ μ F α β γ μ - 3 F α β F α β ) . S=\frac{-1}{6}\int d^{D}x(F_{\alpha\beta\gamma\mu}F^{\alpha\beta\gamma\mu}-3F_% {\alpha\beta}F^{\alpha\beta}).
  10. δ T α β μ = [ α S β ] μ + [ α A β ] μ - μ A α β \delta T_{\alpha\beta\mu}=\partial_{[\alpha}S_{\beta]\mu}+\partial_{[\alpha}A_% {\beta]\mu}-\partial_{\mu}A_{\alpha\beta}

Cut-insertion_theorem.html

  1. S S
  2. U U
  3. X u X_{u}
  4. W r W_{r}
  5. W p W_{p}
  6. S S
  7. W r = W p W_{r}=W_{p}
  8. W r ¯ = W p ¯ \bar{W_{r}}=\bar{W_{p}}
  9. W p W_{p}
  10. X p X_{p}
  11. A U W p | S = 0 A\equiv\frac{U}{W_{p}}|_{S=0}\!\,
  12. β W r U | S = 0 \beta\equiv\frac{W_{r}}{U}|_{S=0}\!\,
  13. X i W p W p ¯ | S = 0 X_{i}\equiv\frac{W_{p}}{\bar{W_{p}}}|_{S=0}\!\,
  14. γ U S | W p = 0 \gamma\equiv\frac{U}{S}|_{W_{p}=0}\!\,
  15. α W r S | W p = 0 \alpha\equiv\frac{W_{r}}{S}|_{W_{p}=0}\!\,
  16. ρ W p ¯ S | W p = 0 \rho\equiv\frac{\bar{W_{p}}}{S}|_{W_{p}=0}\!\,
  17. W r = α S + β A W p W_{r}=\alpha S+\beta AW_{p}
  18. W p ¯ = ρ S + W p X i \bar{W_{p}}=\rho S+\frac{W_{p}}{X_{i}}
  19. W p = α 1 - β A S W_{p}=\frac{\alpha}{1-\beta A}S
  20. W r ¯ = W r X p \bar{W_{r}}=\frac{W_{r}}{X_{p}}
  21. W p ¯ = ( 1 X i + ρ α ( 1 - β A ) ) W r \bar{W_{p}}=\left(\frac{1}{X_{i}}+\frac{\rho}{\alpha}(1-\beta A)\right)W_{r}
  22. 1 X p = 1 X i + ρ α ( 1 - β A ) \frac{1}{X_{p}}=\frac{1}{X_{i}}+\frac{\rho}{\alpha}(1-\beta A)
  23. γ \gamma
  24. A A
  25. U = γ S + A W p U=\gamma S+AW_{p}
  26. A f U S A_{f}\equiv\frac{U}{S}
  27. A f = α A 1 - β A + γ A_{f}=\frac{\alpha A}{1-\beta A}+\gamma
  28. α \alpha
  29. γ \gamma
  30. ρ \rho
  31. α \alpha
  32. γ \gamma
  33. ρ \rho
  34. α = 1 \alpha=1
  35. ρ = 0 \rho=0
  36. γ = 0 \gamma=0
  37. A f = A 1 - β A A_{f}=\frac{A}{1-\beta A}
  38. S S
  39. Z Z
  40. X p X_{p}
  41. S S
  42. S S
  43. V s = S V_{s}=S
  44. I s = S ¯ I_{s}=\bar{S}
  45. Z p = X p Z_{p}=X_{p}
  46. Z = V s I s = V s I r = Z p V s V r = Z p V s V p = Z p 1 - β A α Z=\frac{V_{s}}{I_{s}}=\frac{V_{s}}{I_{r}}=Z_{p}\frac{V_{s}}{V_{r}}=Z_{p}\frac{% V_{s}}{V_{p}}=Z_{p}\frac{1-\beta A}{\alpha}
  47. α = V r V s | V p = 0 = Z p Z p + Z b \alpha=\frac{V_{r}}{V_{s}}|_{V_{p}=0}=\frac{Z_{p}}{Z_{p}+Z_{b}}
  48. Z b Z_{b}
  49. Z p Z_{p}
  50. Z Z
  51. Z = ( Z p + Z b ) ( 1 - β A ) Z=\left(Z_{p}+Z_{b}\right)\left(1-\beta A\right)
  52. S S
  53. X p X_{p}
  54. I s = S I_{s}=S
  55. V s = S ¯ V_{s}=\bar{S}
  56. Y p = X p Y_{p}=X_{p}
  57. Y Y
  58. Y = I s V s = I s V r = Y p I s I r = Y p I s I p = Y p 1 - β A α Y=\frac{I_{s}}{V_{s}}=\frac{I_{s}}{V_{r}}=Y_{p}\frac{I_{s}}{I_{r}}=Y_{p}\frac{% I_{s}}{I_{p}}=Y_{p}\frac{1-\beta A}{\alpha}
  59. α = I r I s | I p = 0 = Y p Y p + Y b \alpha=\frac{I_{r}}{I_{s}}|_{I_{p}=0}=\frac{Y_{p}}{Y_{p}+Y_{b}}
  60. Y b Y_{b}
  61. Y p Y_{p}
  62. Y Y
  63. Y = ( Y p + Y b ) ( 1 - β A ) Y=\left(Y_{p}+Y_{b}\right)\left(1-\beta A\right)
  64. W p W_{p}
  65. X p X_{p}
  66. X p X_{p}
  67. W p ¯ \bar{W_{p}}
  68. X p X_{p}
  69. X X

Cutthroat_flume.html

  1. Q = K H n Q=KH^{n}
  2. H a = 2 L / 9 H_{a}=2L/9
  3. H b = 5 L / 9 H_{b}=5L/9

Cyanophycin_synthase_(L-arginine-adding).html

  1. \rightleftharpoons

Cyanophycin_synthase_(L-aspartate-adding).html

  1. \rightleftharpoons

Cyclic_sieving.html

  1. π \pi
  2. [ n k ] q \left[{n\atop k}\right]_{q}
  3. [ n k ] q = i = 1 n ( 1 + q + q 2 + + q i - 1 ) ( i = 1 k ( 1 + q + q 2 + + q i - 1 ) ) ( i = 1 n - k ( 1 + q + q 2 + + q i - 1 ) ) . \left[{n\atop k}\right]_{q}=\frac{\prod_{i=1}^{n}(1+q+q^{2}+\cdots+q^{i-1})}{% \left(\prod_{i=1}^{k}(1+q+q^{2}+\cdots+q^{i-1})\right)\cdot\left(\prod_{i=1}^{% n-k}(1+q+q^{2}+\cdots+q^{i-1})\right)}.
  4. ( n k ) {\left({{n}\atop{k}}\right)}
  5. { 1 , 3 } { 2 , 4 } { 1 , 3 } \{1,3\}\to\{2,4\}\to\{1,3\}
  6. { 1 , 2 } { 2 , 3 } { 3 , 4 } { 1 , 4 } { 1 , 2 } \{1,2\}\to\{2,3\}\to\{3,4\}\to\{1,4\}\to\{1,2\}
  7. [ 4 2 ] q = 1 + q + 2 q 2 + q 3 + q 4 ; \left[{4\atop 2}\right]_{q}=1+q+2q^{2}+q^{3}+q^{4};

Cyclic_subspace.html

  1. T : V V T:V\rightarrow V
  2. V V
  3. v v
  4. V V
  5. T T
  6. V V
  7. v v
  8. W W
  9. V V
  10. { v , T ( v ) , T 2 ( v ) , , T r ( v ) , } \{v,T(v),T^{2}(v),\ldots,T^{r}(v),\ldots\}
  11. Z ( v ; T ) Z(v;T)
  12. V = Z ( v ; T ) V=Z(v;T)
  13. v v
  14. T T
  15. T : V V T:V\rightarrow V
  16. F F
  17. v v
  18. V V
  19. g ( T ) v g(T)v
  20. g ( x ) g(x)
  21. F [ x ] F[x]
  22. x x
  23. F F
  24. T T
  25. v v
  26. V V
  27. T T
  28. V V
  29. T T
  30. V V
  31. I I
  32. I I
  33. Z ( v ; T ) Z(v;T)
  34. v v
  35. T T
  36. V V
  37. T T
  38. V V
  39. [ 0 1 0 0 ] \begin{bmatrix}0&1\\ 0&0\end{bmatrix}
  40. V V
  41. v = [ 0 1 ] v=\begin{bmatrix}0\\ 1\end{bmatrix}
  42. T v = [ 1 0 ] , T 2 v = 0 , , T r v = 0 , Tv=\begin{bmatrix}1\\ 0\end{bmatrix},\quad T^{2}v=0,\ldots,T^{r}v=0,\ldots
  43. { v , T ( v ) , T 2 ( v ) , , T r ( v ) , } = { [ 0 1 ] , [ 1 0 ] } \{v,T(v),T^{2}(v),\ldots,T^{r}(v),\ldots\}=\left\{\begin{bmatrix}0\\ 1\end{bmatrix},\begin{bmatrix}1\\ 0\end{bmatrix}\right\}
  44. Z ( v ; T ) = V Z(v;T)=V
  45. v v
  46. T T
  47. T : V V T:V\rightarrow V
  48. n n
  49. V V
  50. F F
  51. v v
  52. T T
  53. B = { v 1 = v , v 2 = T v , v 3 = T 2 v , v n = T n - 1 v } B=\{v_{1}=v,v_{2}=Tv,v_{3}=T^{2}v,\ldots v_{n}=T^{n-1}v\}
  54. V V
  55. T T
  56. p ( x ) = c 0 + c 1 x + c 2 x 2 + + c n - 1 x n - 1 + x n p(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots+c_{n-1}x^{n-1}+x^{n}
  57. T v 1 \displaystyle Tv_{1}
  58. B B
  59. T T
  60. [ 0 0 0 0 - c 0 1 0 0 0 - c 1 0 1 0 0 - c 2 0 0 0 1 - c n - 1 ] \begin{bmatrix}0&0&0&\cdots&0&-c_{0}\\ 1&0&0&\ldots&0&-c_{1}\\ 0&1&0&\ldots&0&-c_{2}\\ \vdots&&&&&\\ 0&0&0&\ldots&1&-c_{n-1}\end{bmatrix}
  61. p ( x ) p(x)

Cyclone_Waste_Heat_Engine.html

  1. E f f i c i e n c y = power out power in = 10 k W 146.5 k W = 6.8 % Efficiency={\,\text{power out}\over\,\text{power in}}={10kW\over 146.5kW}=6.8\%

Cyclotomic_fast_Fourier_transform.html

  1. G F ( 2 m ) GF(2^{m})
  2. { f i } 0 N - 1 \{f_{i}\}_{0}^{N-1}
  3. F j = i = 0 N - 1 f i α i j , 0 j N - 1 , F_{j}=\sum_{i=0}^{N-1}f_{i}\alpha^{ij},0\leq j\leq N-1,
  4. α \alpha
  5. { f i } 0 N - 1 \{f_{i}\}_{0}^{N-1}
  6. f ( x ) = f 0 + f 1 x + f 2 x 2 + + f N - 1 x N - 1 = 0 N - 1 f i x i , f(x)=f_{0}+f_{1}x+f_{2}x^{2}+\cdots+f_{N-1}x^{N-1}=\sum_{0}^{N-1}f_{i}x^{i},
  7. F j F_{j}
  8. f ( α j ) f(\alpha^{j})
  9. 𝐅 = [ F 0 F 1 F N - 1 ] = [ α 0 α 0 α 0 α 0 α 1 α N - 1 α 0 α N - 1 α ( N - 1 ) ( N - 1 ) ] [ f 0 f 1 f N - 1 ] = 𝐟 . \mathbf{F}=\left[\begin{matrix}F_{0}\\ F_{1}\\ \vdots\\ F_{N-1}\end{matrix}\right]=\left[\begin{matrix}\alpha^{0}&\alpha^{0}&\cdots&% \alpha^{0}\\ \alpha^{0}&\alpha^{1}&\cdots&\alpha^{N-1}\\ \vdots&\vdots&\ddots&\vdots\\ \alpha^{0}&\alpha^{N-1}&\cdots&\alpha^{(N-1)(N-1)}\end{matrix}\right]\left[% \begin{matrix}f_{0}\\ f_{1}\\ \vdots\\ f_{N-1}\end{matrix}\right]=\mathcal{F}\mathbf{f}.
  10. O ( N 2 ) O(N^{2})
  11. L ( x ) = i l i x p i , l i GF ( p m ) . L(x)=\sum_{i}l_{i}x^{p^{i}},l_{i}\in\mathrm{GF}(p^{m}).
  12. L ( x ) L(x)
  13. L ( x 1 + x 2 ) = L ( x 1 ) + L ( x 2 ) L(x_{1}+x_{2})=L(x_{1})+L(x_{2})
  14. x 1 , x 2 GF ( p m ) , x_{1},x_{2}\in\mathrm{GF}(p^{m}),
  15. ( x 1 + x 2 ) p = x 1 p + x 2 p . (x_{1}+x_{2})^{p}=x_{1}^{p}+x_{2}^{p}.
  16. p p
  17. N N
  18. N N
  19. p m - 1 p^{m}-1
  20. GF ( p m ) \mathrm{GF}(p^{m})
  21. { 0 , 1 , 2 , , N - 1 } \{0,1,2,\ldots,N-1\}
  22. l + 1 l+1
  23. N N
  24. { 0 } , \{0\},
  25. { k 1 , p k 1 , p 2 k 1 , , p m 1 - 1 k 1 } , \{k_{1},pk_{1},p^{2}k_{1},\ldots,p^{m_{1}-1}k_{1}\},
  26. , \ldots,
  27. { k l , p k l , p 2 k l , , p m l - 1 k l } , \{k_{l},pk_{l},p^{2}k_{l},\ldots,p^{m_{l}-1}k_{l}\},
  28. k i = p m i k i ( mod N ) k_{i}=p^{m_{i}}k_{i}\;\;(\mathop{{\rm mod}}N)
  29. f ( x ) = i = 0 l L i ( x k i ) , L i ( y ) = t = 0 m i - 1 y p t f p t k i mod N . f(x)=\sum_{i=0}^{l}L_{i}(x^{k_{i}}),\quad L_{i}(y)=\sum_{t=0}^{m_{i}-1}y^{p^{t% }}f_{p^{t}k_{i}\bmod{N}}.
  30. F j F_{j}
  31. F j = f ( α j ) = i = 0 l L i ( α j k i ) F_{j}=f(\alpha^{j})=\sum_{i=0}^{l}L_{i}(\alpha^{jk_{i}})
  32. α j k i GF ( p m i ) \alpha^{jk_{i}}\in\mathrm{GF}(p^{m_{i}})
  33. { β i , 0 , β i , 1 , , β i , m i - 1 } \{\beta_{i,0},\beta_{i,1},\ldots,\beta_{i,m_{i}-1}\}
  34. α j k i = s = 0 m i - 1 a i j s β i , s \alpha^{jk_{i}}=\sum_{s=0}^{m_{i}-1}a_{ijs}\beta_{i,s}
  35. a i j s GF ( p ) a_{ijs}\in\mathrm{GF}(p)
  36. L i ( x ) L_{i}(x)
  37. F j = i = 0 l s = 0 m i - 1 a i j s ( t = 0 m i - 1 β i , s p t f p t k i mod N ) F_{j}=\sum_{i=0}^{l}\sum_{s=0}^{m_{i}-1}a_{ijs}\left(\sum_{t=0}^{m_{i}-1}\beta% _{i,s}^{p^{t}}f_{p^{t}k_{i}\bmod{N}}\right)
  38. 𝐅 = 𝐀𝐋 𝚷 𝐟 \mathbf{F}=\mathbf{AL\Pi f}
  39. 𝐀 \mathbf{A}
  40. N × N N\times N
  41. a i j s a_{ijs}
  42. 𝐋 \mathbf{L}
  43. 𝚷 \mathbf{\Pi}
  44. 𝐟 \mathbf{f}
  45. { γ i p 0 , γ i p 1 , , γ i p m i - 1 } \{\gamma_{i}^{p^{0}},\gamma_{i}^{p^{1}},\cdots,\gamma_{i}^{p^{m_{i}-1}}\}
  46. GF ( p m i ) \mathrm{GF}(p^{m_{i}})
  47. 𝐋 \mathbf{L}
  48. 𝐋 i = [ γ i p 0 γ i p 1 γ i p m i - 1 γ i p 1 γ i p 2 γ i p 0 γ i p m i - 1 γ i p 0 γ i p m i - 2 ] \mathbf{L}_{i}=\begin{bmatrix}\gamma_{i}^{p^{0}}&\gamma_{i}^{p^{1}}&\cdots&% \gamma_{i}^{p^{m_{i}-1}}\\ \gamma_{i}^{p^{1}}&\gamma_{i}^{p^{2}}&\cdots&\gamma_{i}^{p^{0}}\\ \vdots&\vdots&\ddots&\vdots\\ \gamma_{i}^{p^{m_{i}-1}}&\gamma_{i}^{p^{0}}&\cdots&\gamma_{i}^{p^{m_{i}-2}}\\ \end{bmatrix}
  49. 𝐀 \mathbf{A}
  50. A \mathrm{A}
  51. L Π f \mathrm{L\Pi f}
  52. O ( n ( log 2 n ) log 2 3 2 ) O(n(\log_{2}n)^{\log_{2}\frac{3}{2}})
  53. O ( n 2 / ( log 2 n ) log 2 8 3 ) O(n^{2}/(\log_{2}n)^{\log_{2}\frac{8}{3}})

Cyclotruncated_7-simplex_honeycomb.html

  1. A ~ 7 {\tilde{A}}_{7}

Cyclotruncated_8-simplex_honeycomb.html

  1. A ~ 8 {\tilde{A}}_{8}

D-alanine—(R)-lactate_ligase.html

  1. \rightleftharpoons

D-alanine—D-serine_ligase.html

  1. \rightleftharpoons

D-lactate_dehydratase.html

  1. \rightleftharpoons

D-sedoheptulose_7-phosphate_isomerase.html

  1. \rightleftharpoons

Dahlander_pole_changing_motor.html

  1. n s = 120 × f p n_{s}={120\times{f}\over{p}}

Dammaradiene_synthase.html

  1. \rightleftharpoons

Dammarenediol_II_synthase.html

  1. \rightleftharpoons

Dan_Willard.html

  1. Θ ( n log n ) \Theta(n\log n)
  2. n n
  3. 1 1
  4. N N
  5. O ( n ( 1 + log N log n ) ) O(n(1+\tfrac{\log N}{\log n}))
  6. N N
  7. N N
  8. O ( n log n log log n ) O(n\tfrac{\log n}{\log\log n})

Darcy_number.html

  1. Da = K d 2 \mathrm{Da}=\frac{K}{d^{2}}

Dark_Ages_Radio_Explorer.html

  1. 5.9 × 10 - 6 5.9\times 10^{-6}

Datafly_algorithm.html

  1. \leftarrow
  2. \leftarrow
  3. \leftarrow
  4. \not\in
  5. \leftarrow
  6. \leftarrow
  7. \leftarrow
  8. \leftarrow
  9. \leftarrow
  10. \leftarrow
  11. \leftarrow
  12. \leftarrow
  13. \leftarrow
  14. \leftarrow

Davenport_chained_rotations.html

  1. ( z - x - z , x - y - x , y - z - y , z - y - z , x - z - x , y - x - y ) (z-x-z,x-y-x,y-z-y,z-y-z,x-z-x,y-x-y)
  2. ( x - y - z , y - z - x , z - x - y , x - z - y , z - y - x , y - x - z ) (x-y-z,y-z-x,z-x-y,x-z-y,z-y-x,y-x-z)
  3. R x ( ϕ ) = Roll ( ϕ ) = [ 1 0 0 0 cos ϕ - sin ϕ 0 sin ϕ cos ϕ ] R_{x}(\phi)=\mathrm{Roll}(\phi)=\begin{bmatrix}1&0&0\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi&\cos\phi\end{bmatrix}
  4. R y ( θ ) = Pitch ( θ ) = [ cos θ 0 - sin θ 0 1 0 sin θ 0 cos θ ] \begin{aligned}\\ \displaystyle R_{y}(\theta)=\mathrm{Pitch}(\theta)=\begin{bmatrix}\cos\theta&0% &-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\end{bmatrix}\end{aligned}
  5. R z ( ψ ) = Yaw ( ψ ) = [ cos ψ - sin ψ 0 sin ψ cos ψ 0 0 0 1 ] \begin{aligned}\\ \displaystyle R_{z}(\psi)=\mathrm{Yaw}(\psi)=\begin{bmatrix}\cos\psi&-\sin\psi% &0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix}\end{aligned}
  6. R Z ( ϕ ) = Roll 1 ( ϕ ) = [ cos ϕ - sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ] R_{Z}(\phi)=\mathrm{Roll}_{1}(\phi)=\begin{bmatrix}\cos\phi&-\sin\phi&0\\ \sin\phi&\cos\phi&0\\ 0&0&1\end{bmatrix}
  7. R y ( θ ) = Pitch ( θ ) = [ cos θ 0 - sin θ 0 1 0 sin θ 0 cos θ ] R_{y}(\theta)=\mathrm{Pitch}(\theta)=\begin{bmatrix}\cos\theta&0&-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\end{bmatrix}
  8. R Z ( ψ ) = Roll 2 ( ψ ) = [ cos ψ - sin ψ 0 sin ψ cos ψ 0 0 0 1 ] R_{Z}(\psi)=\mathrm{Roll}_{2}(\psi)=\begin{bmatrix}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix}
  9. R = X ( α ) Y ( β ) Z ( γ ) R=X(\alpha)Y(\beta)Z(\gamma)
  10. R = Z ( γ ) Y ( β ) X ( α ) R=Z(\gamma)Y(\beta)X(\alpha)

David_A._Cox.html

  1. x 2 + n y 2 x^{2}+n\cdot y^{2}

De-sparsified_lasso.html

  1. Y = X β 0 + ϵ Y=X\beta^{0}+\epsilon
  2. n × p n\times p
  3. X = : [ X 1 , , X p ] X=:[X_{1},...,X_{p}]
  4. n × p n\times p
  5. X j X_{j}
  6. ϵ N n ( 0 , σ ϵ 2 I ) \epsilon\sim N_{n}(0,\sigma^{2}_{\epsilon}I)
  7. X X
  8. p × 1 p\times 1
  9. β 0 \beta^{0}
  10. β ^ n ( λ ) = a r g m i n β p 1 2 n Y - X β 2 2 + λ β 1 \hat{\beta}^{n}(\lambda)=\underset{\beta\in\mathbb{R}^{p}}{argmin}\ \frac{1}{2% n}\left\|Y-X\beta\right\|^{2}_{2}+\lambda\left\|\beta\right\|_{1}
  11. β ^ n ( λ , M ) = β ^ n ( λ ) + 1 n M X T ( Y - X β ^ n ( λ ) ) \hat{\beta}^{n}(\lambda,M)=\hat{\beta}^{n}(\lambda)+\frac{1}{n}MX^{T}(Y-X\hat{% \beta}^{n}(\lambda))
  12. M R ( p × p ) M\in R^{(}p\times p)
  13. M M
  14. l 1 l_{1}
  15. 1 × p 1\times p
  16. x i χ R p x_{i}\in\chi\subset R^{p}
  17. y i Y R y_{i}\in Y\subset R
  18. i = 1 , , n i=1,...,n
  19. ρ β ( y , x ) = ρ ( y , x β ) ( β R p ) \rho_{\beta}(y,x)=\rho(y,x\beta)(\beta\in R^{p})
  20. β R p \beta\in R^{p}
  21. l 1 l_{1}
  22. β ^ = a r g m i n 𝛽 ( P n ρ β + λ β 1 ) \hat{\beta}=\underset{\beta}{argmin}(P_{n}\rho_{\beta}+\lambda\left\|\beta% \right\|_{1})
  23. Σ ^ \hat{\Sigma}
  24. l 1 l_{1}
  25. γ j ^ := a r g m i n γ R p - 1 ( Σ ^ j , j - 2 Σ ^ j , / j γ + γ T Σ ^ / j , / j γ + 2 λ j γ 1 \hat{\gamma_{j}}:=\underset{\gamma\in R^{p-1}}{argmin}(\hat{\Sigma}_{j,j}-2% \hat{\Sigma}_{j,/j}\gamma+\gamma^{T}\hat{\Sigma}_{/j,/j}\gamma+2\lambda_{j}% \left\|\gamma\right\|_{1}
  26. Σ ^ j , / j \hat{\Sigma}_{j,/j}
  27. j j
  28. Σ ^ \hat{\Sigma}
  29. ( j , j ) (j,j)
  30. Σ ^ / j , / j \hat{\Sigma}_{/j,/j}
  31. j j
  32. j j

Deductive_lambda_calculus.html

  1. λ v . y \lambda v.y
  2. f x f\ x
  3. a b a\cdot b
  4. let x in y \operatorname{let}x\operatorname{in}y
  5. m = n m=n
  6. m = β n m=_{\beta}n
  7. eta - reduct [ X ] = X \operatorname{eta-reduct}[X]=X
  8. beta - reduct [ X ] = X \operatorname{beta-reduct}[X]=X
  9. x FV ( f ) eta - reduct [ λ x . ( f x ) ] = f x\not\in\operatorname{FV}(f)\to\operatorname{eta-reduct}[\lambda x.(f\ x)]=f
  10. eta - reduct [ X ] = X \operatorname{eta-reduct}[X]=X
  11. λ x . ( f x ) = f \lambda x.(f\ x)=f
  12. f x = y f\ x=y
  13. f x = y f = λ x . y f\ x=y\iff f=\lambda x.y
  14. λ x . y \lambda x.y
  15. f x = y f\ x=y
  16. beta - reduct [ ( λ x . b ) z ] = b [ x := z ] \operatorname{beta-reduct}[(\lambda x.b)\ z]=b[x:=z]
  17. beta - reduct [ X ] = X \operatorname{beta-reduct}[X]=X
  18. ( λ x . b ) z = b [ x := z ] (\lambda x.b)\ z=b[x:=z]
  19. x : f x = y \forall x:f\ x=y
  20. f z f\ z
  21. f z = y [ x := z ] f\ z=y[x:=z]
  22. ( λ x . y ) z = y [ x := z ] (\lambda x.y)\ z=y[x:=z]
  23. f x = y f = λ x . y f\ x=y\iff f=\lambda x.y
  24. λ x . y \lambda x.y
  25. f x = y f\ x=y
  26. \sqrt{}
  27. y = x 2 x = y y=x^{2}\iff x=\sqrt{y}
  28. y = x 2 y=x^{2}
  29. ( y = x 2 and x 0 ) x = y (y=x^{2}\and x>=0)\iff x=\sqrt{y}
  30. f f
  31. f x = y f\ x=y
  32. { f : f x = y } \{f:f\ x=y\}
  33. ( f x = y f = λ x . y ) (f\ x=y\iff f=\lambda x.y)
  34. | { f : f = y } | = 1 ( f x = y x f = λ x . y x ) |\{f:f=y\}|=1\equiv(f\ x=y\ x\iff f=\lambda x.y\ x)
  35. f = g ( x f x = g x ) f=g\iff(\forall xf\ x=g\ x)
  36. | { f : x f x = y x } | = 1 ( f x = y x f = λ x . y x ) |\{f:\forall xf\ x=y\ x\}|=1\equiv(f\ x=y\ x\iff f=\lambda x.y\ x)
  37. | { f : x f x = y x } | = 1 |\{f:\forall xf\ x=y\ x\}|=1
  38. x x
  39. c c
  40. | { f : c } | = 1 f x | { z : z = f x and c } | = 1 |\{f:c\}|=1\equiv\exists f\forall x|\{z:z=f\ x\and c\}|=1
  41. f x | { z : z = f x and w f w = y w } | = 1 \exists f\forall x|\{z:z=f\ x\and\forall w\ f\ w=y\ w\}|=1
  42. f x f\ x
  43. x = w x=w
  44. ( f x | { z : z = f x and f x = y x } | = 1 ) ( f x = y x f = λ x . y x ) (\exists f\forall x|\{z:z=f\ x\and f\ x=y\ x\}|=1)\equiv(f\ x=y\ x\iff f=% \lambda x.y\ x)
  45. f x | { z : z = f x and f x = y x } | = 1 \exists f\forall x|\{z:z=f\ x\and f\ x=y\ x\}|=1
  46. x x
  47. f f
  48. f = y f=y
  49. x x
  50. f f
  51. f x ( a x = a f ) | { z : z = f x and f x = y x } | = 1 \exists f\forall x(\exists a\ x=a\ f)\to|\{z:z=f\ x\and f\ x=y\ x\}|=1
  52. f x = y x f\ x=y\ x
  53. f ( a f ) = y ( a f ) f\ (a\ f)=y\ (a\ f)
  54. f ( a f ) f\ (a\ f)
  55. y y
  56. f ( a f ) f\ (a\ f)
  57. g g
  58. f ( a f ) = g ( f ( a f ) ) f\ (a\ f)=g\ (f\ (a\ f))
  59. z = f ( a f ) z=f\ (a\ f)
  60. z = g z z=g\ z
  61. y y
  62. g g
  63. y ( a f ) = g ( f ( a f ) ) y\ (a\ f)=g\ (f\ (a\ f))
  64. y y
  65. f ( a f ) f\ (a\ f)
  66. a f a\ f
  67. f f
  68. f f
  69. a f a\ f
  70. b b
  71. m b ( a m ) = m \forall m\ b\ (a\ m)=m
  72. y y
  73. g g
  74. n y n = g ( ( b n ) n ) \forall n\ y\ n=g\ ((b\ n)\ n)
  75. ( g ( a b ( n y n = g ( ( b n ) n ) ) and ( m b ( a m ) = m ) ) | { z : z = g z } | = 1 ) (\exists g\ (\exists a\exists b\ (\forall n\ y\ n=g\ ((b\ n)\ n))\and(\forall m% \ b\ (a\ m)=m))\to|\{z:z=g\ z\}|=1)
  76. ( f x = y x f = λ x . y x ) \equiv(f\ x=y\ x\iff f=\lambda x.y\ x)
  77. g z = ¬ z g\ z=\neg z
  78. z = ¬ z z=\neg z
  79. a a
  80. b b
  81. y n = ¬ ( n n ) y\ n=\neg(n\ n)
  82. | { z : z = ¬ z } | = 0 |\{z:z=\neg z\}|=0
  83. f x = ¬ ( x x ) f = λ x . ¬ ( x x ) f\ x=\neg(x\ x)\iff f=\lambda x.\neg(x\ x)
  84. f = λ x . ¬ ( x x ) f=\lambda x.\neg(x\ x)
  85. f f f\ f
  86. = ( λ x . ¬ ( x x ) ) ( λ x . ¬ ( x x ) ) =(\lambda x.\neg(x\ x))(\lambda x.\neg(x\ x))
  87. = ¬ ( ( λ x . ¬ ( x x ) ) ( λ x . ¬ ( x x ) ) ) =\neg((\lambda x.\neg(x\ x))(\lambda x.\neg(x\ x)))
  88. = ¬ ( f f ) =\neg(f\ f)
  89. f f f\ f
  90. f x f\ x
  91. g n = 4 n g\ n=\frac{4}{n}
  92. z = 4 z z=\frac{4}{z}
  93. z 2 = 4 z^{2}=4
  94. a a
  95. b b
  96. y n = 4 n n y\ n=\frac{4}{n\ n}
  97. | { z : z = 4 z } | = 2 |\{z:z=\frac{4}{z}\}|=2
  98. f x = 4 x x f = λ x . 4 x x f\ x=\frac{4}{x\ x}\iff f=\lambda x.\frac{4}{x\ x}
  99. f = λ x . 4 x x f=\lambda x.\frac{4}{x\ x}
  100. f f f\ f
  101. = ( λ x . 4 x x ) ( λ x . 4 x x ) =(\lambda x.\frac{4}{x\ x})(\lambda x.\frac{4}{x\ x})
  102. = 4 ( λ x . 4 x x ) ( λ x . 4 x x ) =\frac{4}{(\lambda x.\frac{4}{x\ x})(\lambda x.\frac{4}{x\ x})}
  103. = 4 f f =\frac{4}{f\ f}
  104. ( f f ) 2 = 4 (f\ f)^{2}=4
  105. ( f f ) (f\ f)
  106. ( f f ) 2 = 4 ( f f ) = ( λ x . 4 x x ) ( λ x . 4 x x ) (f\ f)^{2}=4\implies(f\ f)=(\lambda x.\frac{4}{x\ x})\ (\lambda x.\frac{4}{x\ % x})
  107. ( f f ) (f\ f)
  108. true ( λ x . 4 x x ) ( λ x . 4 x x ) = 2 \operatorname{true}\implies(\lambda x.\frac{4}{x\ x})\ (\lambda x.\frac{4}{x\ % x})=2
  109. ( f f ) (f\ f)
  110. true ( λ x . 4 x x ) ( λ x . 4 x x ) = - 2 \operatorname{true}\implies(\lambda x.\frac{4}{x\ x})\ (\lambda x.\frac{4}{x\ % x})=-2
  111. ( λ x . 4 x x ) ( λ x . 4 x x ) = 2 = - 2 (\lambda x.\frac{4}{x\ x})\ (\lambda x.\frac{4}{x\ x})=2=-2
  112. f = g ( x f x = g x ) f=g\iff(\forall xf\ x=g\ x)
  113. 2 * ( r + s ) = 2 * r + 2 * s 2*(r+s)=2*r+2*s
  114. λ r . λ s .2 * ( r + s ) = λ r . λ s .2 * r + 2 * s \lambda r.\lambda s.2*(r+s)=\lambda r.\lambda s.2*r+2*s
  115. λ r . λ s . mult 2 ( plus r s ) \lambda r.\lambda s.\operatorname{mult}\ 2\ (\operatorname{plus}\ r\ s)
  116. λ r . λ s . ( λ m . λ n . λ f . m ( n f ) ) ( λ f . λ x . f ( f x ) ) ( ( λ m . λ n . λ f . λ x . m f ( n f x ) ) r s ) \lambda r.\lambda s.(\lambda m.\lambda n.\lambda f.m\ (n\ f))\ (\lambda f.% \lambda x.f\ (f\ x))\ ((\lambda m.\lambda n.\lambda f.\lambda x.m\ f\ (n\ f\ x% ))\ r\ s)
  117. λ r . λ s . λ f . λ x . r f ( s f ( r f ( s f x ) ) ) \lambda r.\lambda s.\lambda f.\lambda x.r\ f\ (s\ f\ (r\ f\ (s\ f\ x)))
  118. λ r . λ s . plus ( mult 2 r ) ( mult 2 s ) \lambda r.\lambda s.\operatorname{plus}\ (\operatorname{mult}\ 2\ r)\ (% \operatorname{mult}\ 2\ s)
  119. λ r . λ s . ( λ m . λ n . λ f . λ x . m f ( n f x ) ) ( ( λ m . λ n . λ f . m ( n f ) ) ( λ f . λ x . f ( f x ) ) r ) ( ( λ m . λ n . λ f . m ( n f ) ) ( λ f . λ x . f ( f x ) ) s ) \lambda r.\lambda s.(\lambda m.\lambda n.\lambda f.\lambda x.m\ f\ (n\ f\ x))% \ ((\lambda m.\lambda n.\lambda f.m\ (n\ f))\ (\lambda f.\lambda x.f\ (f\ x))% \ r)\ ((\lambda m.\lambda n.\lambda f.m\ (n\ f))\ (\lambda f.\lambda x.f\ (f\ % x))\ s)
  120. λ r . λ s . λ f . λ x . r f ( r f ( s f ( s f x ) ) ) \lambda r.\lambda s.\lambda f.\lambda x.r\ f\ (r\ f\ (s\ f\ (s\ f\ x)))
  121. λ r . λ s . λ f . λ x . r f ( s f ( r f ( s f x ) ) ) \lambda r.\lambda s.\lambda f.\lambda x.r\ f\ (s\ f\ (r\ f\ (s\ f\ x)))
  122. λ r . λ s . λ f . λ x . r f ( r f ( s f ( s f x ) ) ) \lambda r.\lambda s.\lambda f.\lambda x.r\ f\ (r\ f\ (s\ f\ (s\ f\ x)))
  123. f K ( x : x D f x R ) f\in K\iff(\forall x:x\in D\implies f\ x\in R)
  124. f F ( x : x F f x F ) f\in F\iff(\forall x:x\in F\implies f\ x\in F)
  125. g g F g\ g\not\in F
  126. x g g x F x\neq g\implies g\ x\in F
  127. g F ( x : x F g x F ) g\in F\iff(\forall x:x\in F\implies g\ x\in F)
  128. ( g F g g F ) \implies(g\in F\implies g\ g\in F)
  129. g g F \implies g\ g\in F
  130. g g F g\ g\not\in F
  131. g F g\not\in F
  132. ( x : x F g x F ) g F (\forall x:x\in F\implies g\ x\in F)\implies g\in F
  133. ( x : x F g x g x F ) g F (\forall x:x\in F\implies g\neq x\implies g\ x\in F)\implies g\in F
  134. ( x : x F true ) g F (\forall x:x\in F\implies\operatorname{true})\implies g\in F
  135. true g F \operatorname{true}\implies g\in F
  136. g F g\in F
  137. g F g\not\in F
  138. x = ¬ x x=\neg x
  139. not 1 = λ p . λ a . λ b . p b a \operatorname{not}_{1}=\lambda p.\lambda a.\lambda b.p\ b\ a
  140. ( λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ) ( λ p . λ a . λ b . p b a ) (\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x)))(\lambda p.\lambda a.% \lambda b.p\ b\ a)
  141. ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) (\lambda x.(\lambda p.\lambda a.\lambda b.p\ b\ a)\ (x\ x))\ (\lambda x.(% \lambda p.\lambda a.\lambda b.p\ b\ a)\ (x\ x))
  142. ( λ p . λ a . λ b . p b a ) ( ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ) (\lambda p.\lambda a.\lambda b.p\ b\ a)\ ((\lambda x.(\lambda p.\lambda a.% \lambda b.p\ b\ a)\ (x\ x))\ (\lambda x.(\lambda p.\lambda a.\lambda b.p\ b\ a% )\ (x\ x)))
  143. λ a . λ b . ( ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ) b a \lambda a.\lambda b.((\lambda x.(\lambda p.\lambda a.\lambda b.p\ b\ a)\ (x\ x% ))\ (\lambda x.(\lambda p.\lambda a.\lambda b.p\ b\ a)\ (x\ x)))\ b\ a
  144. λ a . λ b . ( λ a . λ b . ( ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ) b a ) b a \lambda a.\lambda b.(\lambda a.\lambda b.((\lambda x.(\lambda p.\lambda a.% \lambda b.p\ b\ a)\ (x\ x))\ (\lambda x.(\lambda p.\lambda a.\lambda b.p\ b\ a% )\ (x\ x)))\ b\ a)\ b\ a
  145. ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) ( λ x . ( λ p . λ a . λ b . p b a ) ( x x ) ) (\lambda x.(\lambda p.\lambda a.\lambda b.p\ b\ a)\ (x\ x))\ (\lambda x.(% \lambda p.\lambda a.\lambda b.p\ b\ a)\ (x\ x))
  146. x 2 = n x = n x f x = n x and Y f = x x^{2}=n\Rightarrow x=\frac{n}{x}\Rightarrow f\ x=\frac{n}{x}\and Y\ f=x
  147. Y ( divide n ) Y(\operatorname{divide}n)

Degree-preserving_randomization.html

  1. N N
  2. E E
  3. Q * E Q*E
  4. Q Q
  5. E 2 * l n ( 1 / ϵ ) \frac{E}{2}*ln(1/\epsilon)
  6. 10 - 6 10^{-6}
  7. 10 - 7 10^{-7}
  8. π \pi
  9. N N
  10. E 2 * l n ( 1 / ϵ ) \frac{E}{2}*ln(1/\epsilon)

Deligne–Mumford_stack.html

  1. F F × S F F\to F\times_{S}F
  2. F ( B ) F(B)

Delta-amyrin_synthase.html

  1. \rightleftharpoons

Delta-guaiene_synthase.html

  1. \rightleftharpoons

Delta6-protoilludene_synthase.html

  1. \rightleftharpoons

Denaturation_mapping.html

  1. I ( s ) = I d s p d s ( s ) + I s s * [ 1 - p d s ( s ) ] I(s)=I_{ds}p_{ds}(s)+I_{ss}*[1-p_{ds}(s)]
  2. I d s I_{ds}
  3. I s s I_{ss}
  4. p d s p_{ds}

Denjoy–Carleman–Ahlfors_theorem.html

  1. exp ( z ) \exp(z)
  2. 0 / exp ( z ) 0/\exp(z)
  3. exp ( z ) \exp(z)
  4. sin ( z ) / z , \sin(z)/z,
  5. Si ( z ) = 0 z sin ζ ζ d ζ \,\text{Si}(z)=\int_{0}^{z}\frac{\sin\zeta}{\zeta}\,d\zeta
  6. a sin ( z 2 ) / z + b sin ( z 2 ) / z 2 a\sin(z^{2})/z+b\sin(z^{2})/z^{2}
  7. f ( z ) = 0 z sin ( ζ ρ ) ζ ρ d ζ , f(z)=\int_{0}^{z}\frac{\sin(\zeta^{\rho})}{\zeta^{\rho}}d\zeta,

Denoising_Algorithm_based_on_Relevance_network_Topology.html

  1. γ i j = 1 2 log 1 + c i j 1 - c i j \gamma_{ij}=\frac{1}{2}\log{\frac{1+c_{ij}}{1-c_{ij}}}

Depletion_force.html

  1. σ \sigma
  2. V ( h ) = { 0 if h σ if h < σ V(h)=\left\{\begin{matrix}0&\mbox{if}~{}\quad h\geq\sigma\\ \infty&\mbox{if}~{}\quad h<\sigma\end{matrix}\right.
  3. h h
  4. D D
  5. d d
  6. V ( h ) = { 0 if h ( D + d 2 ) if h < ( D + d 2 ) V(h)=\left\{\begin{matrix}0&\mbox{if}~{}\quad h\geq\Big(\frac{D+d}{2}\Big)\\ \infty&\mbox{if}~{}\quad h<\Big(\frac{D+d}{2}\Big)\end{matrix}\right.
  7. h h
  8. d D d\ll D
  9. V E = π ( D + d ) 3 6 V_{\mathrm{E}}=\frac{\pi\big(D+d\big)^{3}}{6}
  10. D D
  11. d d
  12. V E V^{\prime}_{\mathrm{E}}
  13. V E = V E - 2 π l 2 3 [ 3 ( D + d ) 2 - l ] V^{\prime}_{\mathrm{E}}=V_{\mathrm{E}}-\frac{2\pi l^{2}}{3}\bigg[\frac{3\left(% D+d\right)}{2}-l\bigg]
  14. l = ( D + d ) / 2 - h / 2 l=(D+d)/2-h/2
  15. V A V_{\mathrm{A}}
  16. V A = { V - V E if h D + d V - V E if h < D + d V_{\mathrm{A}}=\left\{\begin{matrix}V-V_{\mathrm{E}}&\mbox{if}~{}\quad h\geq D% +d\\ V-V^{\prime}_{\mathrm{E}}&\mbox{if}~{}\quad h<D+d\end{matrix}\right.
  17. A = - T S A=-TS
  18. A A
  19. S S
  20. T T
  21. A = - k B T ln Q A=-k_{\mathrm{B}}T\ln Q
  22. Q Q
  23. Q Q
  24. Q = V A N N ! Λ 3 N Q=\frac{V_{\mathrm{A}}^{N}}{N!\Lambda^{3N}}
  25. V A V_{\mathrm{A}}
  26. N N
  27. Λ \Lambda
  28. Q Q
  29. A = - k B T ln ( V A N N ! Λ 3 N ) A=-k_{\mathrm{B}}T\ln\bigg(\frac{V_{\mathrm{A}}^{N}}{N!\Lambda^{3N}}\bigg)
  30. \mathcal{F}
  31. = - ( A h ) T \mathcal{F}=-\bigg(\frac{\partial A}{\partial h}\bigg)_{T}
  32. p 0 = ρ k B T p_{0}=\rho k_{\mathrm{B}}T
  33. p 0 p_{0}
  34. ρ \rho
  35. k B k_{\mathrm{B}}
  36. ρ ( r ) \rho(r)
  37. Δ G \Delta G
  38. Δ G ( r ) = Π Δ V e x c l u s i o n \Delta G(r)=\Pi\Delta V_{exclusion}
  39. Π \Pi
  40. Δ V e x c l u s i o n \Delta V_{exclusion}
  41. a a
  42. d d
  43. p = k B T N ( ln Q a ) p=k_{\mathrm{B}}TN\left(\frac{\partial\ln Q}{\partial a}\right)
  44. p p
  45. N N
  46. a < d a<d
  47. l l
  48. l 2 A l^{2}\ll A
  49. p p
  50. p = - A p o { ( 1 - f ) - a ( f a ) } p=-Ap_{o}\Bigg\{(1-f)-a\left(\frac{\partial f}{\partial a}\right)\Bigg\}
  51. 1 - f 1-f
  52. f a \frac{\partial f}{\partial a}
  53. p p
  54. r \langle r\rangle
  55. D D
  56. d d
  57. h h
  58. ( D + d ) / 2 (D+d)/2
  59. R / r R/r
  60. R 1 R_{1}
  61. R 2 R_{2}
  62. Z Z
  63. h + R 1 + R 2 h+R_{1}+R_{2}
  64. h h
  65. R 1 R_{1}
  66. R 2 R_{2}
  67. F F
  68. z z
  69. F ( h ) 2 π ( R 1 R 2 R 1 + R 2 ) W ( h ) F(h)\approx 2\pi\left(\frac{R_{1}R_{2}}{R_{1}+R_{2}}\right)W(h)
  70. W ( h ) = h f ( z ) d z W(h)=\textstyle\int_{h}^{\infty}f(z)dz
  71. f ( z ) f(z)
  72. z z
  73. ϵ \epsilon
  74. γ ( ρ , ) = 2 γ ( ρ ) \gamma(\rho,\infty)=2\gamma(\rho)
  75. V ( R ) V(R)
  76. ρ ( R ) \rho(R)
  77. Ω ( [ ρ ( R ) ] ; μ , T ) \Omega\left(\big[\rho(R)\big];\mu,T\right)
  78. Ω ( [ ρ ( R ) ] ; μ , T ) = A ( [ ρ ( R ) ] ; T ) - d 3 R [ μ - V ( R ) ] ρ ( R ) , \Omega\left(\big[\rho(R)\big];\mu,T\right)=A\left(\big[\rho(R)\big];T\right)-% \int d^{3}R\big[\mu-V(R)\big]\rho(R),
  79. μ \mu
  80. T T
  81. A [ ρ ] A[\rho]

Derivations_of_the_Lorentz_transformations.html

  1. c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 - ( z 2 - z 1 ) 2 = 0. c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=0.
  2. d s 2 = c 2 d t 2 - d x 2 - d y 2 - d z 2 , ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2},
  3. K K
  4. d s 2 = a d s 2 . ds^{2}=ads^{\prime 2}.
  5. a a
  6. K K
  7. d s 2 = a ( V 1 ) d s 1 2 , d s 2 = a ( V 2 ) d s 2 2 , d s 1 2 = a ( V 12 ) d s 2 2 . ds^{2}=a(V_{1})ds_{1}^{2},\quad ds^{2}=a(V_{2})ds_{2}^{2},\quad ds_{1}^{2}=a(V% _{12})ds_{2}^{2}.
  8. a ( V 2 ) a ( V 1 ) = a ( V 12 ) . \frac{a(V_{2})}{a(V_{1})}=a(V_{12}).
  9. a ( V ) a(V)
  10. d s 2 = d s 2 ds^{2}=ds^{\prime 2}
  11. c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 - ( z 2 - z 1 ) 2 = c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 - ( z 2 - z 1 ) 2 . \begin{aligned}&\displaystyle\,\,\,\,\,\,\,\,c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{% 1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}\\ &\displaystyle=c^{2}(t_{2}^{\prime}-t_{1}^{\prime})^{2}-(x_{2}^{\prime}-x_{1}^% {\prime})^{2}-(y_{2}^{\prime}-y_{1}^{\prime})^{2}-(z_{2}^{\prime}-z_{1}^{% \prime})^{2}.\end{aligned}
  12. F F
  13. x x
  14. V V
  15. c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 = c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 . c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}=c^{2}(t_{2}^{\prime}-t_{1}^{\prime})^% {2}-(x_{2}^{\prime}-x_{1}^{\prime})^{2}.
  16. c 2 t 2 - x 2 = c 2 t 2 - x 2 c^{2}t^{2}-x^{2}=c^{2}t^{\prime 2}-x^{\prime 2}
  17. 1. cosh 2 Ψ - sinh 2 Ψ = 1 , 2. sinh Ψ = tanh Ψ 1 - tanh 2 Ψ , 3. cosh Ψ = 1 1 - tanh 2 Ψ , \,\text{1. }\cosh^{2}\Psi-\sinh^{2}\Psi=1,\quad\,\text{2. }\sinh\Psi=\frac{% \tanh\Psi}{\sqrt{1-\tanh^{2}\Psi}},\quad\,\text{3. }\cosh\Psi=\frac{1}{\sqrt{1% -\tanh^{2}\Psi}},
  18. c 2 t 2 - x 2 = c 2 t 2 - x 2 . c^{2}t^{2}-x^{2}=c^{2}t^{\prime 2}-x^{\prime 2}.
  19. Ψ Ψ
  20. x = c t sinh Ψ , c t = c t cosh Ψ . x=ct^{\prime}\sinh\Psi,\quad ct=ct^{\prime}\cosh\Psi.
  21. x c t = tanh Ψ = v c sinh Ψ = v c 1 - v 2 c 2 , cosh Ψ = 1 1 - v 2 c 2 , \frac{x}{ct}=\tanh\Psi=\frac{v}{c}\Rightarrow\quad\sinh\Psi=\frac{\frac{v}{c}}% {\sqrt{1-{\frac{v^{2}}{c^{2}}}}},\quad\cosh\Psi=\frac{1}{\sqrt{1-{\frac{v^{2}}% {c^{2}}}}},
  22. x = v t x=vt
  23. x = x + v t 1 - v 2 c 2 , t = t + v c 2 x 1 - v 2 c 2 , x=\frac{x^{\prime}+vt^{\prime}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\quad t=\frac{t^% {\prime}+\frac{v}{c^{2}}x^{\prime}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},
  24. r \displaystyle r
  25. x 2 + y 2 + z 2 = r 2 . x^{2}+y^{2}+z^{2}=r^{2}.
  26. x 2 + y 2 + z 2 = c 2 t 2 . x^{2}+y^{2}+z^{2}=c^{2}t^{2}.
  27. x 2 + y 2 + z 2 = r 2 , x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=r^{\prime 2},
  28. x 2 + y 2 + z 2 = c 2 t 2 . x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=c^{2}t^{\prime 2}.
  29. y = y z = z . \begin{aligned}\displaystyle y^{\prime}&\displaystyle=y\\ \displaystyle z^{\prime}&\displaystyle=z.\end{aligned}
  30. x = γ x + σ t . x^{\prime}=\gamma x+\sigma t.
  31. x = 0 x = v t , \begin{aligned}\displaystyle x^{\prime}&\displaystyle=0\\ \displaystyle x&\displaystyle=vt,\end{aligned}
  32. 0 = γ v t + σ t 0=\gamma vt+\sigma t
  33. σ = - γ v . \sigma=-\gamma v.
  34. x = γ ( x - v t ) x^{\prime}=\gamma\left(x-vt\right)
  35. x = γ ( x + v t ) . x=\gamma\left(x^{\prime}+vt^{\prime}\right).
  36. x = γ [ γ ( x - v t ) + v t ] x=\gamma\left[\gamma\left(x-vt\right)+vt^{\prime}\right]
  37. t = γ t + ( 1 - γ 2 ) x γ v . t^{\prime}=\gamma t+\frac{\left(1-{\gamma^{2}}\right)x}{\gamma v}.
  38. x 2 + y 2 + z 2 = c 2 t 2 , x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=c^{2}t^{\prime 2},
  39. γ 2 ( x - v t ) 2 + y 2 + z 2 = c 2 [ γ t + ( 1 - γ 2 ) x γ v ] 2 {\gamma^{2}}\left(x-vt\right)^{2}+y^{2}+z^{2}=c^{2}\left[\gamma t+\frac{\left(% 1-{\gamma^{2}}\right)x}{\gamma v}\right]^{2}
  40. γ 2 x 2 + γ 2 v 2 t 2 - 2 γ 2 v t x + y 2 + z 2 = c 2 γ 2 t 2 + ( 1 - γ 2 ) 2 c 2 x 2 γ 2 v 2 + 2 ( 1 - γ 2 ) t x c 2 v \gamma^{2}x^{2}+\gamma^{2}v^{2}t^{2}-2\gamma^{2}vtx+y^{2}+z^{2}=c^{2}{\gamma^{% 2}}t^{2}+\frac{\left(1-{\gamma^{2}}\right)^{2}c^{2}x^{2}}{{\gamma^{2}}v^{2}}+2% \frac{\left(1-{\gamma^{2}}\right)txc^{2}}{v}
  41. [ γ 2 - ( 1 - γ 2 ) 2 c 2 γ 2 v 2 ] x 2 - 2 γ 2 v t x + y 2 + z 2 = ( c 2 γ 2 - v 2 γ 2 ) t 2 + 2 [ 1 - γ 2 ] t x c 2 v \left[{\gamma^{2}}-\frac{\left(1-{\gamma^{2}}\right)^{2}c^{2}}{{\gamma^{2}}v^{% 2}}\right]x^{2}-2{\gamma^{2}}vtx+y^{2}+z^{2}=\left(c^{2}{\gamma^{2}}-v^{2}{% \gamma^{2}}\right)t^{2}+2\frac{\left[1-{\gamma^{2}}\right]txc^{2}}{v}
  42. [ γ 2 - ( 1 - γ 2 ) 2 c 2 γ 2 v 2 ] x 2 - [ 2 γ 2 v + 2 ( 1 - γ 2 ) c 2 v ] t x + y 2 + z 2 = [ c 2 γ 2 - v 2 γ 2 ] t 2 \left[{\gamma^{2}}-\frac{\left(1-{\gamma^{2}}\right)^{2}c^{2}}{{\gamma^{2}}v^{% 2}}\right]x^{2}-\left[2{\gamma^{2}}v+2\frac{\left(1-{\gamma^{2}}\right)c^{2}}{% v}\right]tx+y^{2}+z^{2}=\left[c^{2}{\gamma^{2}}-v^{2}{\gamma^{2}}\right]t^{2}
  43. c 2 γ 2 - v 2 γ 2 = c 2 c^{2}{\gamma^{2}}-v^{2}{\gamma^{2}}=c^{2}
  44. γ 2 = 1 1 - v 2 c 2 {\gamma^{2}}=\frac{1}{1-\frac{v^{2}}{c^{2}}}
  45. γ = 1 1 - v 2 c 2 {\gamma}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  46. x = γ ( x - v t ) t = γ ( t - v x c 2 ) y = y z = z \begin{aligned}\displaystyle x^{\prime}&\displaystyle=\gamma\left(x-vt\right)% \\ \displaystyle t^{\prime}&\displaystyle=\gamma\left(t-\frac{vx}{c^{2}}\right)\\ \displaystyle y^{\prime}&\displaystyle=y\\ \displaystyle z^{\prime}&\displaystyle=z\end{aligned}
  47. λ ( δ x 2 + δ y 2 + δ z 2 - c 2 δ t 2 ) \lambda\left(\delta x^{2}+\delta y^{2}+\delta z^{2}-c^{2}\delta t^{2}\right)
  48. λ = 1 \lambda=1
  49. v c v\ll c
  50. x = γ x + b t x^{\prime}=\gamma x+bt\,
  51. t = A x + B t . t^{\prime}=Ax+Bt.\,
  52. x = γ ( x - v t ) . x^{\prime}=\gamma(x-vt)\,.
  53. x = γ ( x - ( - v ) t ) , x=\gamma\left(x^{\prime}-(-v)t^{\prime}\right),
  54. x = γ ( x + v t ) . x=\gamma\left(x^{\prime}+vt^{\prime}\right).
  55. x = γ ( 1 - v / c ) x , x^{\prime}=\gamma\left(1-v/c\right)x,
  56. x = γ ( 1 + v / c ) x . x=\gamma\left(1+v/c\right)x^{\prime}.
  57. x x = γ 2 ( 1 - v 2 / c 2 ) x x . xx^{\prime}=\gamma^{2}\left(1-v^{2}/c^{2}\right)xx^{\prime}.
  58. γ = 1 1 - v 2 c 2 , \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},
  59. { x = c t x = c t . \begin{cases}x^{\prime}=ct^{\prime}\\ x=ct.\end{cases}
  60. x = γ ( x - v t ) , x^{\prime}=\gamma(x-vt),\,
  61. c t = γ ( c t - v c x ) , ct^{\prime}=\gamma\left(ct-\frac{v}{c}x\right),
  62. t = γ ( t - v c 2 x ) , t^{\prime}=\gamma\left(t-\frac{v}{c^{2}}x\right),
  63. A = - γ v / c 2 , A=-\gamma v/c^{2},\,
  64. B = γ . B=\gamma.\,
  65. λ λ
  66. μ μ
  67. { x - c t = λ ( x - c t ) x + c t = μ ( x + c t ) \begin{cases}x^{\prime}-ct^{\prime}=\lambda\left(x-ct\right)\\ x^{\prime}+ct^{\prime}=\mu\left(x+ct\right)\end{cases}
  68. x = c t x=ct
  69. x = c t x′=ct′
  70. { γ = ( λ + μ ) / 2 b = ( λ - μ ) / 2 , \begin{cases}\gamma=\left(\lambda+\mu\right)/2\\ b=\left(\lambda-\mu\right)/2,\end{cases}
  71. { x = γ x - b c t c t = γ c t - b x . \begin{cases}x^{\prime}=\gamma x-bct\\ ct^{\prime}=\gamma ct-bx.\end{cases}
  72. x = 0 x′=0
  73. x = v t x=vt
  74. v = b c / γ v=bc/γ
  75. { x = γ ( x - v t ) t = γ ( t - v c 2 x ) \begin{cases}x^{\prime}=\gamma\left(x-vt\right)\\ t^{\prime}=\gamma\left(t-\frac{v}{c^{2}}x\right)\end{cases}
  76. γ γ
  77. ( c t ) 2 - ( x 2 + y 2 + z 2 ) = ( c t ) 2 - ( x 2 + y 2 + z 2 ) = 0 (ct)^{2}-(x^{2}+y^{2}+z^{2})=(ct^{\prime})^{2}-(x^{\prime 2}+y^{\prime 2}+z^{% \prime 2})=0
  78. ( c t ) 2 - x 2 = ( c t ) 2 - x 2 (ct)^{2}-x^{2}=(ct^{\prime})^{2}-x^{\prime 2}
  79. x \displaystyle x^{\prime}
  80. ( c t ) 2 - x 2 = [ ( C x ) 2 + ( D c t ) 2 + 2 C D c x t ] - [ ( A x ) 2 + ( B c t ) 2 + 2 A B c x t ] (ct)^{2}-x^{2}=[(Cx)^{2}+(Dct)^{2}+2CDcxt]-[(Ax)^{2}+(Bct)^{2}+2ABcxt]
  81. - 1 = C 2 - A 2 \displaystyle-1=C^{2}-A^{2}
  82. cosh 2 ϕ - sinh 2 ϕ = 1 \cosh^{2}\phi-\sinh^{2}\phi=1
  83. A = D = cosh ϕ , C = B = - sinh ϕ A=D=\cosh\phi\,,\quad C=B=-\sinh\phi
  84. x \displaystyle x^{\prime}
  85. 0 = cosh ϕ v t - sinh ϕ c t tanh ϕ = v c = β 0=\cosh\phi vt-\sinh\phi ct\,\Rightarrow\,\tanh\phi=\frac{v}{c}=\beta
  86. cosh ϕ = γ , sinh ϕ = β γ . \cosh\phi=\gamma,\,\quad\sinh\phi=\beta\gamma\,.
  87. [ t x ] = [ γ δ β α ] [ t x ] , \begin{bmatrix}t^{\prime}\\ x^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&\delta\\ \beta&\alpha\end{bmatrix}\begin{bmatrix}t\\ x\end{bmatrix},
  88. [ t 0 ] = [ γ δ β α ] [ t v t ] , \begin{bmatrix}t^{\prime}\\ 0\end{bmatrix}=\begin{bmatrix}\gamma&\delta\\ \beta&\alpha\end{bmatrix}\begin{bmatrix}t\\ vt\end{bmatrix},
  89. β = - v α \beta=-v\alpha\,
  90. [ t - v t ] = [ γ δ β α ] [ t 0 ] , \begin{bmatrix}t^{\prime}\\ -vt^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&\delta\\ \beta&\alpha\end{bmatrix}\begin{bmatrix}t\\ 0\end{bmatrix},
  91. β = - v γ \beta=-v\gamma\,
  92. [ t x ] = [ γ δ - v γ γ ] [ t x ] . \begin{bmatrix}t^{\prime}\\ x^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&\delta\\ -v\gamma&\gamma\end{bmatrix}\begin{bmatrix}t\\ x\end{bmatrix}.
  93. [ t x ] = 1 γ 2 + v δ γ [ γ - δ v γ γ ] [ t x ] . \begin{bmatrix}t\\ x\end{bmatrix}=\frac{1}{\gamma^{2}+v\delta\gamma}\begin{bmatrix}\gamma&-\delta% \\ v\gamma&\gamma\end{bmatrix}\begin{bmatrix}t^{\prime}\\ x^{\prime}\end{bmatrix}.
  94. [ t x ] = [ γ ( - v ) δ ( - v ) v γ ( - v ) γ ( - v ) ] [ t x ] , \begin{bmatrix}t\\ x\end{bmatrix}=\begin{bmatrix}\gamma(-v)&\delta(-v)\\ v\gamma(-v)&\gamma(-v)\end{bmatrix}\begin{bmatrix}t^{\prime}\\ x^{\prime}\end{bmatrix},
  95. γ 2 + v δ γ = 1. \gamma^{2}+v\delta\gamma=1.\,
  96. [ t ′′ x ′′ ] \displaystyle\begin{bmatrix}t^{\prime\prime}\\ x^{\prime\prime}\end{bmatrix}
  97. γ ( v ) γ ( v ) - v δ ( v ) γ ( v ) = - v γ ( v ) δ ( v ) + γ ( v ) γ ( v ) \gamma(v^{\prime})\gamma(v)-v\delta(v^{\prime})\gamma(v)=-v^{\prime}\gamma(v^{% \prime})\delta(v)+\gamma(v^{\prime})\gamma(v)\,
  98. v δ ( v ) γ ( v ) = v γ ( v ) δ ( v ) v\delta(v^{\prime})\gamma(v)=v^{\prime}\gamma(v^{\prime})\delta(v)\,
  99. δ ( v ) v γ ( v ) = δ ( v ) v γ ( v ) . \frac{\delta(v)}{v\gamma(v)}=\frac{\delta(v^{\prime})}{v^{\prime}\gamma(v^{% \prime})}.\,
  100. γ 2 + v δ γ = 1 \gamma^{2}+v\delta\gamma=1
  101. 1 = γ 2 + v δ γ = γ 2 ( 1 + κ v 2 ) 1=\gamma^{2}+v\delta\gamma=\gamma^{2}(1+\kappa v^{2})\,
  102. γ = 1 / 1 + κ v 2 \gamma=1/\sqrt{1+\kappa v^{2}}
  103. [ t x ] = 1 1 + κ v 2 [ 1 κ v - v 1 ] [ t x ] . \begin{bmatrix}t^{\prime}\\ x^{\prime}\end{bmatrix}=\frac{1}{\sqrt{1+\kappa v^{2}}}\begin{bmatrix}1&\kappa v% \\ -v&1\end{bmatrix}\begin{bmatrix}t\\ x\end{bmatrix}.
  104. X , Y , Z , T X,Y,Z,T
  105. x , y , z , t x,y,z,t
  106. a ( v ) a(v)
  107. b ( v ) b(v)
  108. d ( v ) d(v)
  109. ε ( v ) \varepsilon(v)
  110. t \displaystyle t
  111. ε ( v ) \varepsilon(v)
  112. - v / c 2 -v/c^{2}
  113. b ( v ) b(v)
  114. d ( v ) d(v)
  115. a ( v ) a(v)
  116. b ( v ) b(v)
  117. a ( v ) a(v)
  118. 1 / a ( v ) = b ( v ) = γ 1/a(v)=b(v)=\gamma
  119. d ( v ) = 1 d(v)=1

Derivative_of_the_exponential_map.html

  1. 𝐠 \mathbf{g}
  2. G G
  3. G G
  4. G G
  5. e x p : 𝐠 G exp:\mathbf{g}→G
  6. d d t e x p ( X ( t ) ) : T 𝐠 T G \frac{d}{dt}exp(X(t)):T\mathbf{g}→TG
  7. X ( t ) X(t)
  8. d e x p : T 𝐠 T G dexp:T\mathbf{g}→TG
  9. d e x p dexp
  10. e x p ( X ) exp(X)
  11. e x p exp
  12. X = X ( t ) X=X(t)
  13. X ´ ( t ) = d X ( t X´(t)=dX\frac{(}{t}
  14. t t
  15. e x p exp
  16. G G
  17. G G
  18. e x p exp
  19. e x p exp
  20. G = G L ( n , ) G=GL(n,ℂ)
  21. G L ( n , ) GL(n,ℝ)
  22. d e x p dexp
  23. e x p exp
  24. X X
  25. d exp X Y = d d t e Z ( t ) | t = 0 , Z ( 0 ) = X , Z ( 0 ) = Y d\exp_{X}Y=\left.\frac{d}{dt}e^{Z(t)}\right|_{t=0},Z(0)=X,Z^{\prime}(0)=Y
  26. Z ( t ) = X + t Y Z(t)=X+tY
  27. d exp X Y = e X 1 - e - ad X ad X Y d\exp_{X}Y=e^{X}\frac{1-e^{-\mathrm{ad}_{X}}}{\mathrm{ad}_{X}}Y
  28. 𝐠 \mathbf{g}
  29. e x p exp
  30. s s
  31. Γ ( s , t ) = e - s X ( t ) t e s X ( t ) \Gamma(s,t)=e^{-sX(t)}\frac{\partial}{\partial t}e^{sX(t)}
  32. Γ Γ
  33. s s
  34. A d Ad
  35. A G , X 𝐠 A∈G,X∈\mathbf{g}
  36. A d Ad
  37. a d ad
  38. Γ s = e - s X ( - X ) t e s X ( t ) + e - s X t [ X ( t ) e s X ( t ) ] = e - s X d X d t e s X . \frac{\partial\Gamma}{\partial s}=e^{-sX}(-X)\frac{\partial}{\partial t}e^{sX(% t)}+e^{-sX}\frac{\partial}{\partial t}[X(t)e^{sX(t)}]=e^{-sX}\frac{dX}{dt}e^{% sX}.
  39. Γ s = Ad e - s X X = e - ad s X X , \frac{\partial\Gamma}{\partial s}=\mathrm{Ad}_{e^{-sX}}X^{\prime}=e^{-\mathrm{% ad}_{sX}}X^{\prime},
  40. Γ ( 1 , t ) = e - X ( t ) t e X ( t ) = 0 1 Γ s d s = 0 1 e - ad s X X d s . \Gamma(1,t)=e^{-X(t)}\frac{\partial}{\partial t}e^{X(t)}=\int_{0}^{1}\frac{% \partial\Gamma}{\partial s}ds=\int_{0}^{1}e^{-\mathrm{ad}_{sX}}X^{\prime}ds.
  41. Γ ( 1 , t ) = 0 1 k = 0 ( - 1 ) k s k k ! ( ad X ) k d X d t d s = k = 0 ( - 1 ) k ( k + 1 ) ! ( ad X ) k d X d t = 1 - e - ad X ad X d X d t , \Gamma(1,t)=\int_{0}^{1}\sum_{k=0}^{\infty}\frac{(-1)^{k}s^{k}}{k!}(\mathrm{ad% }_{X})^{k}\frac{dX}{dt}ds=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)!}(\mathrm{% ad}_{X})^{k}\frac{dX}{dt}=\frac{1-e^{-\mathrm{ad}_{X}}}{\mathrm{ad}_{X}}\frac{% dX}{dt},
  42. d d t e X ( t ) = lim N d d t ( 1 + X ( t ) N ) N = lim N k = 1 N ( 1 + X ( t ) N ) N - k 1 N d X ( t ) d t ( 1 + X ( t ) N ) k - 1 , \begin{aligned}\displaystyle\frac{d}{dt}e^{X(t)}&\displaystyle=\lim_{N\to% \infty}\frac{d}{dt}\left(1+\frac{X(t)}{N}\right)^{N}\\ &\displaystyle=\lim_{N\to\infty}\sum_{k=1}^{N}\left(1+\frac{X(t)}{N}\right)^{N% -k}\frac{1}{N}\frac{dX(t)}{dt}\left(1+\frac{X(t)}{N}\right)^{k-1}~{}~{}~{},% \end{aligned}
  43. X ( t ) X(t)
  44. X ´ ( t ) X´(t)
  45. N N
  46. Δ k = 1 N Δk=\frac{1}{N}
  47. N N
  48. Δ k d k d s , k s Δk→dk≡ds,k→s
  49. Σ Σ→∫
  50. d d t e X ( t ) = 0 1 e ( 1 - s ) X X e s X d s = e X 0 1 Ad e - s X X d s = e X 0 1 e - ad s X d s X = e X 1 - e - a d X a d X d X d t . \begin{aligned}\displaystyle\frac{d}{dt}e^{X(t)}&\displaystyle=\int_{0}^{1}e^{% (1-s)X}X^{\prime}e^{sX}ds\\ &\displaystyle=e^{X}\int_{0}^{1}\mathrm{Ad}_{e^{-sX}}X^{\prime}ds\\ &\displaystyle=e^{X}\int_{0}^{1}e^{-\mathrm{ad}_{sX}}dsX^{\prime}\\ &\displaystyle=e^{X}\frac{1-e^{-ad_{X}}}{ad_{X}}\frac{dX}{dt}.\end{aligned}
  51. d d t exp ( C ( t ) ) = exp ( C ) ϕ ( - ad ( C ) ) C , \frac{d}{dt}\mathrm{exp}(C(t))=\mathrm{exp}(C)\phi(-\mathrm{ad}(C))C~{}^{% \prime},
  52. ϕ ( z ) = e z - 1 z = 1 + 1 2 ! z + 1 3 ! z 2 + , \phi(z)=\frac{e^{z}-1}{z}=1+\frac{1}{2!}z+\frac{1}{3!}z^{2}+\cdots,
  53. d d t exp ( C ( t ) ) = exp ( C ) 1 - e - ad C ad C d C ( t ) d t . \frac{d}{dt}\mathrm{exp}(C(t))=\mathrm{exp}(C)\frac{1-e^{-\mathrm{ad}_{C}}}{% \mathrm{ad}_{C}}\frac{dC(t)}{dt}.
  54. e x p exp
  55. e x p exp
  56. f f
  57. f f
  58. x x
  59. 1 - e ad X ad X \frac{1-e^{\mathrm{ad_{X}}}}{\mathrm{ad}_{X}}
  60. g g
  61. g ( U ) g(U)
  62. U U
  63. g ( U ) g(U)
  64. U U
  65. g ( U ) = 1 e x p ( U ) g(U)=1−exp(−\frac{U}{)}
  66. 1 - e - λ i j λ i j , \frac{1-e^{-\lambda_{ij}}}{\lambda_{ij}},
  67. d e x p dexp
  68. λ i j k 2 π i , k = ± 1 , ± 2 , . \lambda_{ij}\neq k2\pi i,k=\pm 1,\pm 2,\ldots.
  69. X X
  70. X X
  71. V V
  72. X X
  73. X e i = λ i e i + , Xe_{i}=\lambda_{i}e_{i}+\cdots,
  74. n > i n>i
  75. ad X E i j = ( λ i - λ j ) E i j + λ i j E i j + , \mathrm{ad}_{X}E_{ij}=(\lambda_{i}-\lambda_{j})E_{ij}+\cdots\equiv\lambda_{ij}% E_{ij}+\cdots,
  76. e x p exp
  77. X X
  78. X X
  79. λ i - λ j k 2 π i , k = ± 1 , ± 2 , , 1 i , j n = dim V . \lambda_{i}-\lambda_{j}\neq k2\pi i,\quad k=\pm 1,\pm 2,\ldots,\quad 1\leq i,j% \leq n=\mathrm{dim}V.
  80. e x p exp
  81. 0 𝐠 0∈\mathbf{g}
  82. e x p exp
  83. 0 𝐠 0∈\mathbf{g}
  84. 𝐠 \mathbf{g}
  85. e G e∈G
  86. ξ ξ
  87. Z ( t ) Z(t)
  88. e Z ( t ) = e X e t Y , e^{Z(t)}=e^{X}e^{tY},
  89. Z ( 1 ) = l o g ( e x p X e x p Y ) Z(1)=log(expXexpY)
  90. exp ( - Z ( t ) ) d d t exp ( Z ( t ) ) = 1 - e - ad Z ad Z Z ( t ) . \exp(-Z(t))\frac{d}{dt}\mathrm{exp}(Z(t))=\frac{1-e^{-\mathrm{ad}_{Z}}}{% \mathrm{ad}_{Z}}Z^{\prime}(t).
  91. Y = 1 - e - ad Z ad Z Z ( t ) , Y=\frac{1-e^{-\mathrm{ad}_{Z}}}{\mathrm{ad}_{Z}}Z^{\prime}(t),
  92. Z ( t ) = ad Z 1 - e - ad Z Y ψ ( e ad Z ) Y , ψ ( w ) = w log w w - 1 = 1 + m = 1 ( - 1 ) m + 1 m ( m + 1 ) ( w - 1 ) m , || w || < 1. Z^{\prime}(t)=\frac{\mathrm{ad}_{Z}}{1-e^{-\mathrm{ad}_{Z}}}Y\equiv\psi(e^{% \mathrm{ad}_{Z}})Y,\quad\psi(w)=\frac{w\log w}{w-1}=1+\sum_{m=1}^{\infty}\frac% {(-1)^{m+1}}{m(m+1)}(w-1)^{m},||w||<1.
  93. A d Ad
  94. a d ad
  95. e ad Z = e ad X e t ad Y e^{\mathrm{ad}_{Z}}=e^{\mathrm{ad}_{X}}e^{t\mathrm{ad}_{Y}}
  96. Z ( t ) = ψ ( e ad X e t ad Y ) Y . Z^{\prime}(t)=\psi(e^{\mathrm{ad}_{X}}e^{t\mathrm{ad}_{Y}})Y.
  97. Z ( 1 ) = log ( exp X exp Y ) = X + ( 0 1 ψ ( e ad X e t ad Y ) d t ) Y , Z(1)=\log(\exp X\exp Y)=X+\left(\int^{1}_{0}\psi\left(e^{\operatorname{ad}_{X}% }~{}e^{t\,\,\text{ad}_{Y}}\right)\,dt\right)\,Y,
  98. Z ( 1 ) Z(1)
  99. ψ ψ
  100. X + Y X+Y
  101. X X
  102. Y Y
  103. e Z ( t ) = e t X e t Y , e^{Z(t)}=e^{tX}e^{tY},
  104. e - Z ( t ) d e Z ( t ) d t = e - t ad Y X + Y , e^{-Z(t)}\frac{de^{Z(t)}}{dt}=e^{-t\mathrm{ad}_{Y}}X+Y~{},
  105. Z = ad Z 1 - e - ad Z ( e - t ad Y X + Y ) = ad Z e ad Z - 1 ( X + e t ad X Y ) . Z^{\prime}=\frac{\mathrm{ad}_{Z}}{1-e^{-\mathrm{ad}_{Z}}}~{}(e^{-t\mathrm{ad}_% {Y}}X+Y)=\frac{\mathrm{ad}_{Z}}{e^{\mathrm{ad}_{Z}}-1}~{}(X+e^{t\mathrm{ad}_{X% }}Y)~{}.
  106. ad Z = log ( exp ( ad Z ) ) = log ( 1 + ( exp ( ad Z ) - 1 ) ) = n = 1 ( - ) n + 1 n ( exp ( ad Z ) - 1 ) n , || ad Z || < log 2 , \begin{aligned}\displaystyle\mathrm{ad_{Z}}&\displaystyle=\mathrm{log}(\mathrm% {exp(\mathrm{ad}_{Z})})=\mathrm{log}(1+(\mathrm{exp(\mathrm{ad}_{Z})-1)})\\ &\displaystyle=\sum\limits^{\infty}_{n=1}\frac{(-)^{n+1}}{n}(\exp(\mathrm{ad}_% {Z})-1)^{n}~{},\quad||\mathrm{ad}_{Z}||<\log 2~{}~{},\end{aligned}
  107. Z = n = 1 ( - ) n - 1 n ( e ad Z - 1 ) n - 1 ( X + e t ad X Y ) , Z^{\prime}=\sum\limits^{\infty}_{n=1}\frac{(-)^{n-1}}{n}(e^{\mathrm{ad}_{Z}}-1% )^{n-1}~{}(X+e^{t\mathrm{ad}_{X}}Y)~{},
  108. Z ( 1 ) = 0 1 d t d Z ( t ) d t = n = 1 ( - ) n - 1 n 0 1 d t ( e t ad X e t ad Y - 1 ) n - 1 ( X + e t ad X Y ) . Z(1)=\int^{1}_{0}dt~{}\frac{dZ(t)}{dt}=\sum\limits^{\infty}_{n=1}\frac{(-)^{n-% 1}}{n}\int^{1}_{0}dt~{}(e^{t\mathrm{ad}_{X}}e^{t\mathrm{ad}_{Y}}-1)^{n-1}~{}(X% +e^{t\mathrm{ad}_{X}}Y)~{}.
  109. Z Z
  110. X , Y X,Y
  111. k k
  112. k = n 1 k=n−1
  113. d Z d t = k = 0 ( - 1 ) k k + 1 { ( e ad t X e ad t Y - 1 ) k X + ( e ad t X e ad t Y - 1 ) k e ad t X Y } \frac{dZ}{dt}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+1}\left\{(e^{\mathrm{ad}_{% tX}}e^{\mathrm{ad}_{tY}}-1)^{k}X+(e^{\mathrm{ad}_{tX}}e^{\mathrm{ad}_{tY}}-1)^% {k}e^{\mathrm{ad}_{tX}}Y\right\}
  114. l o g log
  115. e x p exp
  116. log ( A ) = k = 1 ( - 1 ) k + 1 k ( A - I ) k , and e X = k = 0 X k k ! \log(A)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}{(A-I)}^{k},\quad\,\text{and}% \quad e^{X}=\sum_{k=0}^{\infty}\frac{X^{k}}{k!}
  117. log ( e X e Y ) = k = 1 ( - 1 ) k + 1 k ( e X e Y - I ) k = k = 1 ( - 1 ) k + 1 k ( i = 0 X i i ! j = 0 X j j ! - I ) ) k = k = 1 ( - 1 ) k + 1 k ( i , j 0 , i + j > 1 X i Y j i ! j ! ) k . \log(e^{X}e^{Y})=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}{(e^{X}e^{Y}-I)}^{k}=% \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\left({\sum_{i=0}^{\infty}\frac{X^{i}}{% i!}\sum_{j=0}^{\infty}\frac{X^{j}}{j!}-I)}\right)^{k}=\sum_{k=1}^{\infty}\frac% {(-1)^{k+1}}{k}\left(\sum_{i,j\geq 0,i+j>1}^{\infty}\frac{X^{i}Y^{j}}{i!j!}% \right)^{k}.
  118. 2 k 2k
  119. d Z d t = k = 0 ( - 1 ) k k + 1 s S k , i k + 1 0 t i 1 + j 1 + + i k + j k ad X i 1 ad Y j 1 ad X i k ad Y j k i 1 ! j 1 ! i k ! j k ! X + t i 1 + j 1 + + i k + j k + i k + 1 ad X i 1 ad Y j 1 ad X i k ad Y j k X i k + 1 i 1 ! j 1 ! i k ! j k ! i k + 1 ! Y , i r , j r 0 , i r + j r > 0 , 1 r k , \begin{aligned}\displaystyle\frac{dZ}{dt}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k% +1}\sum_{s\in S_{k},i_{k+1}\geq 0}&\displaystyle t^{i_{1}+j_{1}+\cdots+i_{k}+j% _{k}}\frac{{\mathrm{ad}_{X}}^{i_{1}}{\mathrm{ad}_{Y}}^{j_{1}}\cdots{\mathrm{ad% }_{X}}^{i_{k}}{\mathrm{ad}_{Y}}^{j_{k}}}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!}X\\ \displaystyle+&\displaystyle t^{i_{1}+j_{1}+\cdots+i_{k}+j_{k}+i_{k+1}}\frac{{% \mathrm{ad}_{X}}^{i_{1}}{\mathrm{ad}_{Y}}^{j_{1}}\cdots{\mathrm{ad}_{X}}^{i_{k% }}{\mathrm{ad}_{Y}}^{j_{k}}X^{i_{k+1}}}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!i_{k+1}% !}Y,\quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r\leq k,\end{aligned}
  120. d Z d t = k = 0 ( - 1 ) k k + 1 s S k , i k + 1 0 t i 1 + j 1 + + i k + j k [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) X ] i 1 ! j 1 ! i k ! j k ! + t i 1 + j 1 + + i k + j k + i k + 1 [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) X ( i k + 1 ) Y ] i 1 ! j 1 ! i k ! j k ! i k + 1 ! , i r , j r 0 , i r + j r > 0 , 1 r k . \begin{aligned}\displaystyle\frac{dZ}{dt}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k% +1}\sum_{s\in S_{k},i_{k+1}\geq 0}&\displaystyle t^{i_{1}+j_{1}+\cdots+i_{k}+j% _{k}}\frac{[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X]}{i_{1}!j_{1}% !\cdots i_{k}!j_{k}!}\\ \displaystyle+&\displaystyle t^{i_{1}+j_{1}+\cdots+i_{k}+j_{k}+i_{k+1}}\frac{[% X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X^{(i_{k+1})}Y]}{i_{1}!j_{1% }!\cdots i_{k}!j_{k}!i_{k+1}!},\quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,% \quad 1\leq r\leq k\end{aligned}.
  121. Z = Z ( 1 ) = d Z d t d t Z=Z(1)=∫\frac{dZ}{dt}dt
  122. Z ( 0 ) = 0 Z(0)=0
  123. Z = k = 0 ( - 1 ) k k + 1 s S k , i k + 1 0 1 i 1 + j 1 + + i k + j k + 1 [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) X ] i 1 ! j 1 ! i k ! j k ! + 1 i 1 + j 1 + + i k + j k + i k + 1 + 1 [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) X ( i k + 1 ) Y ] i 1 ! j 1 ! i k ! j k ! i k + 1 ! , i r , j r 0 , i r + j r > 0 , 1 r k . \begin{aligned}\displaystyle Z=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+1}\sum_{s% \in S_{k},i_{k+1}\geq 0}&\displaystyle\frac{1}{i_{1}+j_{1}+\cdots+i_{k}+j_{k}+% 1}\frac{[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X]}{i_{1}!j_{1}!% \cdots i_{k}!j_{k}!}\\ \displaystyle+&\displaystyle\frac{1}{i_{1}+j_{1}+\cdots+i_{k}+j_{k}+i_{k+1}+1}% \frac{[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X^{(i_{k+1})}Y]}{i_{% 1}!j_{1}!\cdots i_{k}!j_{k}!i_{k+1}!},\quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r% }>0,\quad 1\leq r\leq k\end{aligned}.
  124. Z = k = 0 ( - 1 ) k k + 1 s S k , i k + 1 0 1 i 1 + j 1 + + i k + j k + ( i k + 1 = 1 ) + ( j k + 1 = 0 ) [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) X ( i k + 1 = 1 ) Y ( j k + 1 = 0 ) ] i 1 ! j 1 ! i k ! j k ! ( i k + 1 = 1 ) ! ( j k + 1 = 0 ) ! + 1 i 1 + j 1 + + i k + j k + i k + 1 + ( j k + 1 = 1 ) [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) X ( i k + 1 ) Y ( j k + 1 = 1 ) ] i 1 ! j 1 ! i k ! j k ! i k + 1 ! ( j k + 1 = 1 ) ! , i r , j r 0 , i r + j r > 0 , 1 r k . \begin{aligned}\displaystyle Z=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+1}\sum_{s% \in S_{k},i_{k+1}\geq 0}&\displaystyle\frac{1}{i_{1}+j_{1}+\cdots+i_{k}+j_{k}+% (i_{k+1}=1)+(j_{k+1}=0)}\frac{[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{% k})}X^{(i_{k+1}=1)}Y^{(j_{k+1}=0)}]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!(i_{k+1}=1% )!(j_{k+1}=0)!}\\ \displaystyle+&\displaystyle\frac{1}{i_{1}+j_{1}+\cdots+i_{k}+j_{k}+i_{k+1}+(j% _{k+1}=1)}\frac{[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(i_{k})}Y^{(j_{k})}X^{(i_{k+1% })}Y^{(j_{k+1}=1)}]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!i_{k+1}!(j_{k+1}=1)!},% \quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r\leq k\end{aligned}.
  125. Z = k = 0 ( - 1 ) k k + 1 s S k + 1 1 i 1 + j 1 + + i k + j k + i k + 1 + j k + 1 [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) X ( i k + 1 ) Y ( j k + 1 ) ] i 1 ! j 1 ! i k ! j k ! i k + 1 ! j k + 1 ! , i r , j r 0 , i r + j r > 0 , 1 r k + 1 , Z=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+1}\sum_{s\in S_{k+1}}\frac{1}{i_{1}+j_{% 1}+\cdots+i_{k}+j_{k}+i_{k+1}+j_{k+1}}\frac{[X^{(i_{1})}Y^{(j_{1})}\cdots X^{(% i_{k})}Y^{(j_{k})}X^{(i_{k+1})}Y^{(j_{k+1})}]}{i_{1}!j_{1}!\cdots i_{k}!j_{k}!% i_{k+1}!j_{k+1}!},\quad i_{r},j_{r}\geq 0,\quad i_{r}+j_{r}>0,\quad 1\leq r% \leq k+1,
  126. T T , T = 0 TT,T=0
  127. T T
  128. 0
  129. 1 1
  130. 1 1
  131. = =
  132. k k 1 k→k−1
  133. X i 1 Y j 1 X i k Y j k X^{i_{1}}Y^{j_{1}}\cdots X^{i_{k}}Y^{j_{k}}
  134. X i 1 Y j 1 X i k Y j k = [ X ( i 1 ) Y ( j 1 ) X ( i k ) Y ( j k ) ] i 1 + j 1 + + i k + j k . X^{i_{1}}Y^{j_{1}}\cdots X^{i_{k}}Y^{j_{k}}=\frac{[X^{(i_{1})}Y^{(j_{1})}% \cdots X^{(i_{k})}Y^{(j_{k})}]}{i_{1}+j_{1}+\cdots+i_{k}+j_{k}}.
  135. A d Ad
  136. a d ad
  137. a d = d A d ad=dAd
  138. τ ( l o g z ) ϕ ( - l o g z ) = 1 \tau(logz)\phi(-logz)=1
  139. τ ( w ) = w 1 - e - w . \tau(w)=\frac{w}{1-e^{-w}}.
  140. τ τ
  141. ( - 1 ) k b k , (-1)^{k}b_{k},
  142. b < s u b > k b<sub>k
  143. U U
  144. U < s u p > k U<sup>k
  145. λ λ
  146. | I m λ | < π |Imλ|<π
  147. G G
  148. G L ( n , ) GL(n,ℂ)
  149. G L ( n , ) GL(n,ℝ)
  150. e x p exp