wpmath0000003_13

Proper_time.html

  1. η μ ν = ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) , \eta_{\mu\nu}=\left(\begin{matrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{matrix}\right),
  2. ( x 0 , x 1 , x 2 , x 3 ) = ( c t , x , y , z ) (x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)
  3. d s 2 = c 2 d t 2 - d x 2 - d y 2 - d z 2 = η μ ν d x μ d x ν , ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu},
  4. d s 2 = c 2 d τ 2 - d x τ 2 - d y τ 2 - d z τ 2 = η μ ν d x τ μ d x τ ν = c 2 d τ 2 , ds^{2}=c^{2}d\tau^{2}-dx_{\tau}^{2}-dy_{\tau}^{2}-dz_{\tau}^{2}=\eta_{\mu\nu}% dx_{\tau}^{\mu}dx_{\tau}^{\nu}=c^{2}d\tau^{2},
  5. d s = c d τ , ds=cd\tau,
  6. d τ = d s c . d\tau=\frac{ds}{c}.
  7. τ τ
  8. P P
  9. d s = 0 ds=0
  10. v ( t ) v(t)
  11. t t
  12. x ( t ) x(t)
  13. y ( t ) y(t)
  14. z ( t ) z(t)
  15. t , x , y , z t,x,y,z
  16. Δ τ = ( d t d λ ) 2 - 1 c 2 [ ( d x d λ ) 2 + ( d y d λ ) 2 + ( d z d λ ) 2 ] d λ . \Delta\tau=\int\sqrt{\left(\frac{dt}{d\lambda}\right)^{2}-\frac{1}{c^{2}}\left% [\left(\frac{dx}{d\lambda}\right)^{2}+\left(\frac{dy}{d\lambda}\right)^{2}+% \left(\frac{dz}{d\lambda}\right)^{2}\right]}\,d\lambda.
  17. Δ τ = ( Δ t ) 2 - ( Δ x ) 2 c 2 - ( Δ y ) 2 c 2 - ( Δ z ) 2 c 2 , \Delta\tau=\sqrt{\left(\Delta t\right)^{2}-\frac{\left(\Delta x\right)^{2}}{c^% {2}}-\frac{\left(\Delta y\right)^{2}}{c^{2}}-\frac{\left(\Delta z\right)^{2}}{% c^{2}}},
  18. Δ τ Δτ
  19. P P
  20. Δ τ = P d τ = P 1 c g μ ν d x μ d x ν . \Delta\tau=\int_{P}\,d\tau=\int_{P}\frac{1}{c}\sqrt{g_{\mu\nu}\;dx^{\mu}\;dx^{% \nu}}.
  21. Δ τ = P d τ = P 1 c g 00 d x 0 . \Delta\tau=\int_{P}d\tau=\int_{P}\frac{1}{c}\sqrt{g_{00}}dx^{0}.
  22. x = y = z = 0 x=y=z=0
  23. Δ τ = ( 10 years ) 2 = 10 years \Delta\tau=\sqrt{(10\,\text{ years})^{2}}=10\,\text{ years}
  24. Δ τ = ( 5 years ) 2 - ( 4.33 years ) 2 = 6.25 years 2 = 6.25 years = 2.5 years . \Delta\tau=\sqrt{(5\;\mathrm{years})^{2}-(4.33\;\mathrm{years})^{2}}=\sqrt{6.2% 5\;\mathrm{years}^{2}}=\sqrt{6.25\;}\mathrm{years}=2.5\;\mathrm{years}.
  25. Δ T \Delta T
  26. Δ τ = Δ T 2 - ( v x Δ T / c ) 2 - ( v y Δ T / c ) 2 - ( v z Δ T / c ) 2 = Δ T 1 - v 2 / c 2 , \Delta\tau=\sqrt{\Delta T^{2}-(v_{x}\Delta T/c)^{2}-(v_{y}\Delta T/c)^{2}-(v_{% z}\Delta T/c)^{2}}=\Delta T\sqrt{1-v^{2}/c^{2}},
  27. d τ d\tau
  28. ω \omega
  29. ( T , r cos ( ω T ) , r sin ( ω T ) , 0 ) (T,\;\,r\cos(\omega T),\;\,r\sin(\omega T),\;\,0)
  30. T T
  31. ω \omega
  32. d x = - r ω sin ( ω T ) d T dx=-r\omega\sin(\omega T)\;dT
  33. d y = r ω cos ( ω T ) d T dy=r\omega\cos(\omega T)\;dT
  34. d τ = d T 2 - ( r ω / c ) 2 sin 2 ( ω T ) d T 2 - ( r ω / c ) 2 cos 2 ( ω T ) d T 2 = d T 1 - ( r ω c ) 2 . d\tau=\sqrt{dT^{2}-(r\omega/c)^{2}\sin^{2}(\omega T)\;dT^{2}-(r\omega/c)^{2}% \cos^{2}(\omega T)\;dT^{2}}=dT\sqrt{1-\left(\frac{r\omega}{c}\right)^{2}}.
  35. T 1 T_{1}
  36. T 2 T_{2}
  37. T 1 T 2 d τ = ( T 2 - T 1 ) 1 - ( r ω c ) 2 . \int_{T_{1}}^{T_{2}}d\tau=(T_{2}-T_{1})\sqrt{1-\left(\frac{r\omega}{c}\right)^% {2}}.
  38. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}
  39. θ = arctan ( y x ) - ω t . \theta=\arctan\left(\frac{y}{x}\right)-\omega t.
  40. d τ = [ 1 - ( r ω c ) 2 ] d t 2 - d r 2 c 2 - r 2 d θ 2 c 2 - d z 2 c 2 - 2 r 2 ω d t d θ c 2 . d\tau=\sqrt{\left[1-\left(\frac{r\omega}{c}\right)^{2}\right]dt^{2}-\frac{dr^{% 2}}{c^{2}}-\frac{r^{2}\,d\theta^{2}}{c^{2}}-\frac{dz^{2}}{c^{2}}-2\frac{r^{2}% \omega\,dt\,d\theta}{c^{2}}}.
  41. d τ = d t 1 - ( r ω c ) 2 , d\tau=dt\sqrt{1-\left(\frac{r\omega}{c}\right)^{2}},
  42. d τ = [ 1 - ( R ω c ) 2 ] d t 2 - ( R ω c ) 2 d t 2 + 2 ( R ω c ) 2 d t 2 = d t . d\tau=\sqrt{\left[1-\left(\frac{R\omega}{c}\right)^{2}\right]dt^{2}-\left(% \frac{R\omega}{c}\right)^{2}\,dt^{2}+2\left(\frac{R\omega}{c}\right)^{2}\,dt^{% 2}}=dt.
  43. d τ = ( 1 - 2 m r ) d t 2 - 1 c 2 ( 1 - 2 m r ) - 1 d r 2 - r 2 c 2 d ϕ 2 - r 2 c 2 sin 2 ( ϕ ) d θ 2 , d\tau=\sqrt{\left(1-\frac{2m}{r}\right)dt^{2}-\frac{1}{c^{2}}\left(1-\frac{2m}% {r}\right)^{-1}dr^{2}-\frac{r^{2}}{c^{2}}d\phi^{2}-\frac{r^{2}}{c^{2}}\sin^{2}% (\phi)\,d\theta^{2}},
  44. d r = d θ = d ϕ = 0 dr=d\theta=d\phi=0
  45. d τ = d t 1 - 2 m / r d\tau=dt\,\sqrt{1-2m/r}
  46. r = 6 , 356 , 752 r=6,356,752
  47. d τ = ( 1 - 1.3908 × 10 - 9 ) d t 2 = ( 1 - 6.9540 × 10 - 10 ) d t . d\tau=\sqrt{\left(1-1.3908\times 10^{-9}\right)\;dt^{2}}=\left(1-6.9540\times 1% 0^{-10}\right)\,dt.
  48. d θ / d t \ d\theta/dt
  49. d θ = 7.2923 × 10 - 5 d t d\theta=7.2923\times 10^{-5}\,dt
  50. d τ = ( 1 - 1.3908 × 10 - 9 ) d t 2 - 2.4069 × 10 - 12 d t 2 = ( 1 - 6.9660 × 10 - 10 ) d t . d\tau=\sqrt{\left(1-1.3908\times 10^{-9}\right)dt^{2}-2.4069\times 10^{-12}\,% dt^{2}}=\left(1-6.9660\times 10^{-10}\right)\,dt.

Prothrombin_time.html

  1. INR = ( PTtest PTnormal ) ISI \,\text{INR}=\left(\frac{\,\text{PT}\text{test}}{\,\text{PT}\text{normal}}% \right)\text{ISI}

Protonation.html

  1. H 2 SO 4 + H 2 O H 3 O + + HSO 4 - \mathrm{H_{2}SO_{4}\ +\ H_{2}O\ \rightleftharpoons\ H_{3}O^{+}\ +\ HSO_{4}^{-}}
  2. ( CH 3 ) 2 C \mathrm{(CH_{3})_{2}C}
  3. CH 2 + HBF 4 ( CH 3 ) 3 C + + BF 4 - \mathrm{CH_{2}\ +\ HBF_{4}\ \rightleftharpoons\ (CH_{3})_{3}C^{+}\ +\ BF_{4}^{% -}}
  4. NH 3 ( g ) + HCl ( g ) NH 4 Cl ( s ) \mathrm{NH_{3(g)}\ +\ HCl_{(g)}\ \rightarrow\ NH_{4}Cl_{(s)}}

Pseudo-spectral_method.html

  1. i t ψ ( x , t ) = [ - 2 x 2 + V ( x ) ] ψ ( x , t ) , ψ ( t 0 ) = ψ 0 i\frac{\partial}{\partial t}\psi(x,t)=\Bigl[-\frac{\partial^{2}}{\partial x^{2% }}+V(x)\Bigr]\psi(x,t),\qquad\qquad\psi(t_{0})=\psi_{0}
  2. ψ ( x + 2 π , t ) = ψ ( x , t ) \psi(x+2\pi,t)=\psi(x,t)
  3. V ( x ) V(x)
  4. ψ \psi
  5. ψ ( x , t ) = 1 2 π n c n ( t ) e 2 π i n x . \psi(x,t)=\frac{1}{\sqrt{2\pi}}\sum_{n}c_{n}(t)e^{2\pi inx}.
  6. i d d t c n ( t ) = ( 2 π n ) 2 c n + k V n k c k , i\frac{d}{dt}c_{n}(t)=(2\pi n)^{2}c_{n}+\sum_{k}V_{nk}c_{k},
  7. V n k V_{nk}
  8. V n k = 1 2 π 0 2 π V ( x ) e 2 π i ( k - n ) x d x . V_{nk}=\frac{1}{2\pi}\int_{0}^{2\pi}V(x)\ e^{2\pi i(k-n)x}dx.
  9. N N
  10. c n ( t ) c_{n}(t)
  11. V ( x ) V(x)
  12. V ( x ) V(x)
  13. N 2 N^{2}
  14. V n k V_{nk}
  15. c n ( t ) c_{n}(t)
  16. ψ \psi
  17. x j = 2 π j / N x_{j}=2\pi j/N
  18. ψ ( x i , t ) = V ( x i ) ψ ( x i , t ) \psi^{\prime}(x_{i},t)=V(x_{i})\psi(x_{i},t)
  19. c n ( t ) c^{\prime}_{n}(t)
  20. k V n k c k ( t ) \sum_{k}V_{nk}c_{k}(t)
  21. O ( N ln N ) O(N\ln N)
  22. V ( x ) V(x)
  23. V ( x ) V(x)
  24. f ( x ) f(x)
  25. f ( x ) , f ~ ( x ) = V ( x ) f ( x ) f(x),\tilde{f}(x)=V(x)f(x)
  26. V , f V,f
  27. f , f ~ f,\tilde{f}
  28. { ϕ n } n = 0 , , N \{\phi_{n}\}_{n=0,\ldots,N}
  29. f ( x ) = n = 0 N c n ϕ n ( x ) f(x)=\sum_{n=0}^{N}c_{n}\phi_{n}(x)
  30. f ~ ( x ) = n = 0 N c ~ n ϕ n ( x ) \tilde{f}(x)=\sum_{n=0}^{N}\tilde{c}_{n}\phi_{n}(x)
  31. ϕ n , ϕ m = δ n m \langle\phi_{n},\phi_{m}\rangle=\delta_{nm}
  32. f , g = a b f ( x ) g ( x ) ¯ d x \langle f,g\rangle=\int_{a}^{b}f(x)\overline{g(x)}dx
  33. a , b a,b
  34. c n = f , ϕ n c_{n}=\langle f,\phi_{n}\rangle
  35. c ~ n = f ~ , ϕ n \tilde{c}_{n}=\langle\tilde{f},\phi_{n}\rangle
  36. c ~ n = m = 0 N V n m c m \tilde{c}_{n}=\sum_{m=0}^{N}V_{nm}c_{m}
  37. V n m = V ϕ m , ϕ n V_{nm}=\langle V\phi_{m},\phi_{n}\rangle
  38. ϕ n \phi_{n}
  39. { ϕ n } \{\phi_{n}\}
  40. N + 1 N+1
  41. N + 1 N+1
  42. ϕ n , ϕ m = i = 0 N w i ϕ n ( x i ) ϕ m ( x i ) ¯ n , m = 0 , , N \langle\phi_{n},\phi_{m}\rangle=\sum_{i=0}^{N}w_{i}\phi_{n}(x_{i})\overline{% \phi_{m}(x_{i})}\qquad\qquad n,m=0,\ldots,N
  43. x i , w i x_{i},w_{i}
  44. N N
  45. f ( x ) , f ~ ( x ) f(x),\tilde{f}(x)
  46. f ( x i ) = n = 0 N c n ϕ n ( x i ) f(x_{i})=\sum_{n=0}^{N}c_{n}\phi_{n}(x_{i})
  47. c n = f , ϕ n = n = 0 N w i f ( x i ) ϕ n ( x i ) ¯ c_{n}=\langle f,\phi_{n}\rangle=\sum_{n=0}^{N}w_{i}f(x_{i})\overline{\phi_{n}(% x_{i})}
  48. V ( x ) V(x)
  49. f ~ ( x i ) = V ( x i ) f ( x i ) . \tilde{f}(x_{i})=V(x_{i})f(x_{i}).
  50. c ~ n \tilde{c}_{n}
  51. c ~ n = f ~ , ϕ n = i w i f ~ ( x i ) ϕ n ( x i ) ¯ = i w i V ( x i ) f ( x i ) ϕ n ( x i ) ¯ \tilde{c}_{n}=\langle\tilde{f},\phi_{n}\rangle=\sum_{i}w_{i}\tilde{f}(x_{i})% \overline{\phi_{n}(x_{i})}=\sum_{i}w_{i}V(x_{i})f(x_{i})\overline{\phi_{n}(x_{% i})}
  52. c ~ n = V f , ϕ n \tilde{c}_{n}=\langle Vf,\phi_{n}\rangle
  53. V f , ϕ n i w i V ( x i ) f ( x i ) ϕ n ( x i ) ¯ . \langle Vf,\phi_{n}\rangle\approx\sum_{i}w_{i}V(x_{i})f(x_{i})\overline{\phi_{% n}(x_{i})}.
  54. V f Vf
  55. [ 0 , L ] [0,L]
  56. ϕ n ( x ) = 1 L e - ı k n x \phi_{n}(x)=\frac{1}{\sqrt{L}}e^{-\imath k_{n}x}
  57. k n = ( - 1 ) n n / 2 2 π / L k_{n}=(-1)^{n}\lceil n/2\rceil 2\pi/L
  58. \lceil\rceil
  59. n max = N n_{\,\text{max}}=N
  60. x i = i Δ x x_{i}=i\Delta x
  61. Δ x = L / ( N + 1 ) \Delta x=L/(N+1)
  62. w i = Δ x w_{i}=\Delta x
  63. ϕ a + ϕ b = ϕ c \phi_{a}+\phi_{b}=\phi_{c}
  64. c a + b c\leq a+b
  65. f ( x ) , V ( x ) f(x),V(x)
  66. N f , N V N_{f},N_{V}
  67. N f + N V N_{f}+N_{V}
  68. N ln N N\ln N
  69. w i w_{i}
  70. x i x_{i}
  71. a b w ( x ) p ( x ) d x = i = 0 N w i p ( x i ) \int_{a}^{b}w(x)p(x)dx=\sum_{i=0}^{N}w_{i}p(x_{i})
  72. p ( x ) p(x)
  73. 2 N + 1 2N+1
  74. w ( x ) w(x)
  75. a , b a,b
  76. ϕ n ( x ) = w ( x ) P n ( x ) \phi_{n}(x)=\sqrt{w(x)}P_{n}(x)
  77. P n P_{n}
  78. n n
  79. a b w ( x ) P n ( x ) P m ( x ) d x = δ m n . \int_{a}^{b}w(x)P_{n}(x)P_{m}(x)dx=\delta_{mn}.
  80. ϕ n \phi_{n}
  81. f , g = a b f ( x ) g ( x ) ¯ d x \langle f,g\rangle=\int_{a}^{b}f(x)\overline{g(x)}dx
  82. f f
  83. N f N_{f}
  84. V V
  85. N V N_{V}
  86. N f + N V N_{f}+N_{V}

Pseudoscalar.html

  1. e 1 e_{1}
  2. e 2 e_{2}
  3. e 1 e 2 = e 12 . e_{1}e_{2}=e_{12}.

Psi_(letter).html

  1. ϕ | ψ \langle\phi|\psi\rangle
  2. ψ ( m ) ( z ) = d m d z m Γ ( z ) Γ ( z ) \psi^{(m)}(z)=\frac{d^{m}}{dz^{m}}\frac{\Gamma^{\prime}(z)}{\Gamma(z)}
  3. Γ ( z ) \Gamma(z)
  4. ψ P ( x ) \psi_{P}(x)\,\!
  5. Y Y\,\!
  6. P P\,\!

PSL(2,7).html

  1. 1 A 1 2 A 21 4 A 42 3 A 56 7 A 24 7 B 24 χ 1 1 1 1 1 1 1 χ 2 3 - 1 1 0 σ σ ¯ χ 3 3 - 1 1 0 σ ¯ σ χ 4 6 2 0 0 - 1 - 1 χ 5 7 - 1 - 1 1 0 0 χ 6 8 0 0 - 1 1 1 , \begin{array}[]{r|cccccc}&1A_{1}&2A_{21}&4A_{42}&3A_{56}&7A_{24}&7B_{24}\\ \hline\chi_{1}&1&1&1&1&1&1\\ \chi_{2}&3&-1&1&0&\sigma&\bar{\sigma}\\ \chi_{3}&3&-1&1&0&\bar{\sigma}&\sigma\\ \chi_{4}&6&2&0&0&-1&-1\\ \chi_{5}&7&-1&-1&1&0&0\\ \chi_{6}&8&0&0&-1&1&1\\ \end{array},
  2. σ = - 1 + i 7 2 . \sigma=\frac{-1+i\sqrt{7}}{2}.
  3. For γ = ( a b c d ) PSL ( 2 , 7 ) and x 𝐏 1 ( 7 ) , γ x = a x + b c x + d \mbox{For }~{}\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mbox{PSL}~{}(2,7)\mbox{ and }~{}x\in\mathbf{P}^{1}(7),\ % \gamma\cdot x=\frac{ax+b}{cx+d}
  4. For γ = ( a b c d e f g h i ) PSL ( 3 , 2 ) and 𝐱 = ( x y z ) 𝐏 2 ( 2 ) , γ 𝐱 = ( a x + b y + c z d x + e y + f z g x + h y + i z ) \mbox{For }~{}\gamma=\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}\in\mbox{PSL}~{}(3,2)\mbox{ and }~{}\mathbf{x}=\begin{% pmatrix}x\\ y\\ z\end{pmatrix}\in\mathbf{P}^{2}(2),\ \gamma\ \cdot\ \mathbf{x}=\begin{pmatrix}% ax+by+cz\\ dx+ey+fz\\ gx+hy+iz\end{pmatrix}

Pullback_(differential_geometry).html

  1. φ * s = s φ \varphi^{*}s=s\circ\varphi
  2. F : W × W × × W F:W\times W\times\cdots\times W\rightarrow\mathbb{R}
  3. ( Φ * F ) ( v 1 , v 2 , , v s ) = F ( Φ ( v 1 ) , Φ ( v 2 ) , , Φ ( v s ) ) , (\Phi^{*}F)(v_{1},v_{2},\ldots,v_{s})=F(\Phi(v_{1}),\Phi(v_{2}),\ldots,\Phi(v_% {s})),
  4. Φ : V W , Φ * : W * V * . \Phi\colon V\rightarrow W,\qquad\Phi^{*}\colon W^{*}\rightarrow V^{*}.
  5. W W W W\otimes W\otimes\cdots\otimes W
  6. V V V V\otimes V\otimes\cdots\otimes V
  7. W W W W\otimes W\otimes\cdots\otimes W
  8. Φ * ( v 1 v 2 v r ) = Φ ( v 1 ) Φ ( v 2 ) Φ ( v r ) . \Phi_{*}(v_{1}\otimes v_{2}\otimes\cdots\otimes v_{r})=\Phi(v_{1})\otimes\Phi(% v_{2})\otimes\cdots\otimes\Phi(v_{r}).
  9. ( φ * α ) x ( X ) = α φ ( x ) ( d φ x ( X ) ) (\varphi^{*}\alpha)_{x}(X)=\alpha_{\varphi(x)}(\mathrm{d}\varphi_{x}(X))
  10. F : T y N × × T y N \R . F\colon T_{y}N\times\cdots\times T_{y}N\to\R.
  11. ( φ * S ) x ( X 1 , , X s ) = S φ ( x ) ( d φ x ( X 1 ) , d φ x ( X s ) ) (\varphi^{*}S)_{x}(X_{1},\ldots,X_{s})=S_{\varphi(x)}(\mathrm{d}\varphi_{x}(X_% {1}),\ldots\mathrm{d}\varphi_{x}(X_{s}))
  12. ( φ * α ) x ( X 1 , , X k ) = α φ ( x ) ( d φ x ( X 1 ) , , d φ x ( X k ) ) (\varphi^{*}\alpha)_{x}(X_{1},\ldots,X_{k})=\alpha_{\varphi(x)}(\mathrm{d}% \varphi_{x}(X_{1}),\ldots,\mathrm{d}\varphi_{x}(X_{k}))
  13. φ * ( α β ) = φ * α φ * β . \varphi^{*}(\alpha\wedge\beta)=\varphi^{*}\alpha\wedge\varphi^{*}\beta.
  14. φ * ( d α ) = d ( φ * α ) . \varphi^{*}(\mathrm{d}\alpha)=\mathrm{d}(\varphi^{*}\alpha).
  15. Φ = d φ x G L ( T x M , T φ ( x ) N ) \Phi=\mathrm{d}\varphi_{x}\in GL(T_{x}M,T_{\varphi(x)}N)
  16. Φ - 1 = ( d φ x ) - 1 G L ( T φ ( x ) N , T x M ) . \Phi^{-1}=({\mathrm{d}\varphi_{x}})^{-1}\in GL(T_{\varphi(x)}N,T_{x}M).
  17. \nabla
  18. E E
  19. N N
  20. φ \varphi
  21. M M
  22. N N
  23. φ * \varphi^{*}\nabla
  24. φ \varphi
  25. M M
  26. ( φ * ) X ( φ * s ) = φ * ( d φ ( X ) s ) . (\varphi^{*}\nabla)_{X}(\varphi^{*}s)=\varphi^{*}(\nabla_{\mathrm{d}\varphi(X)% }s).

Purely_functional.html

  1. 2 k log 2 n 2k\log_{2}n
  2. k k

Pursuit_curve.html

  1. L ( t ) = F ( t ) + x F ( t ) . L(t)=F(t)+xF^{\prime}(t).\,

Pushout_(category_theory).html

  1. X Z Y X\leftarrow Z\rightarrow Y
  2. P = X Y / P=X\coprod Y\Bigg/\sim
  3. X f Y X\cup_{f}Y
  4. X Y X\vee Y
  5. A C B = { i I ( a i , b i ) | a i A , b i B } / ( f ( c ) a , b ) - ( a , g ( c ) b ) | a A , b B , c C A\otimes_{C}B=\left\{\sum_{i\in I}(a_{i},b_{i})\;\big|\;a_{i}\in A,b_{i}\in B% \right\}\Bigg/\bigg\langle(f(c)a,b)-(a,g(c)b)\;\big|\;a\in A,b\in B,c\in C\bigg\rangle
  6. g : A A C B g^{\prime}:A\rightarrow A\otimes_{C}B
  7. f : B A C B f^{\prime}:B\rightarrow A\otimes_{C}B
  8. f g = g f f^{\prime}\circ g=g^{\prime}\circ f
  9. π 1 ( D , * ) π 1 ( A , * ) \pi_{1}(D,*)\to\pi_{1}(A,*)
  10. π 1 ( D , * ) π 1 ( B , * ) . \pi_{1}(D,*)\to\pi_{1}(B,*).

Pyramidal_number.html

  1. P n r = 3 n 2 + n 3 ( r - 2 ) - n ( r - 5 ) 6 , P_{n}^{r}=\frac{3n^{2}+n^{3}(r-2)-n(r-5)}{6},
  2. P n r = n ( n + 1 ) [ n ( r - 2 ) - ( r - 5 ) ] ( 2 ) ( 3 ) = [ n ( n + 1 ) 2 ] [ n ( r - 2 ) - ( r - 5 ) 3 ] = T n [ n ( r - 2 ) - ( r - 5 ) 3 ] . \begin{aligned}\displaystyle P_{n}^{r}=\frac{n(n+1)[n(r-2)-(r-5)]}{(2)(3)}=% \left[\frac{n(n+1)}{2}\right]\left[\frac{n(r-2)-(r-5)}{3}\right]=T_{n}\ \left[% \frac{n(r-2)-(r-5)}{3}\right]\end{aligned}.

P–n_junction.html

  1. V bi V_{\rm bi}
  2. C A ( x ) C_{A}(x)
  3. C D ( x ) C_{D}(x)
  4. N 0 ( x ) N_{0}(x)
  5. P 0 ( x ) P_{0}(x)
  6. - d 2 V d x 2 = ρ ε = q ε [ ( N 0 - P 0 ) + ( C D - C A ) ] -\frac{\mathrm{d}^{2}V}{\mathrm{d}x^{2}}=\frac{\rho}{\varepsilon}=\frac{q}{% \varepsilon}\left[(N_{0}-P_{0})+(C_{D}-C_{A})\right]
  7. V V
  8. ρ \rho
  9. ε \varepsilon
  10. q q
  11. d p d_{p}
  12. d n d_{n}
  13. d p C A = d n C D d_{p}C_{A}=d_{n}C_{D}
  14. D D
  15. Δ V \Delta V
  16. Δ V = D q ε [ ( N 0 - P 0 ) + ( C D - C A ) ] d x d x \Delta V=\int_{D}\int\frac{q}{\varepsilon}\left[(N_{0}-P_{0})+(C_{D}-C_{A})% \right]\,\mathrm{d}x\,\mathrm{d}x
  17. = C A C D C A + C D 2 q ε ( d p + d n ) 2 =\frac{C_{A}C_{D}}{C_{A}+C_{D}}\frac{2q}{\varepsilon}(d_{p}+d_{n})^{2}
  18. P 0 = N 0 = 0 P_{0}=N_{0}=0
  19. d d
  20. d = 2 ε q C A + C D C A C D Δ V d=\sqrt{\frac{2\varepsilon}{q}\frac{C_{A}+C_{D}}{C_{A}C_{D}}\Delta V}
  21. Δ V \Delta V
  22. Δ V 0 + Δ V ext \Delta V_{0}+\Delta V\text{ext}
  23. Δ V 0 \Delta V_{0}
  24. P 0 N 0 {{P}_{0}}{{N}_{0}}
  25. Δ V 0 = k T q ln ( C A C D P 0 N 0 ) \Delta{{V}_{0}}=\frac{kT}{q}\ln\left(\frac{{{C}_{A}}{{C}_{D}}}{{{P}_{0}}{{N}_{% 0}}}\right)
  26. 𝐉 F \mathbf{J}_{F}
  27. n n
  28. 𝐉 D - q n \mathbf{J}_{D}\propto-q\nabla n
  29. 𝐉 R \mathbf{J}_{R}

Q_(disambiguation).html

  1. \mathbb{Q}

QR_algorithm.html

  1. A k + 1 = R k Q k = Q k - 1 Q k R k Q k = Q k - 1 A k Q k = Q k T A k Q k , A_{k+1}=R_{k}Q_{k}=Q_{k}^{-1}Q_{k}R_{k}Q_{k}=Q_{k}^{-1}A_{k}Q_{k}=Q_{k}^{T}A_{% k}Q_{k},
  2. 10 3 n 3 + O ( n 2 ) \begin{matrix}\frac{10}{3}\end{matrix}n^{3}+O(n^{2})
  3. 6 n 2 + O ( n ) 6n^{2}+O(n)
  4. 4 3 n 3 + O ( n 2 ) \begin{matrix}\frac{4}{3}\end{matrix}n^{3}+O(n^{2})
  5. O ( n ) O(n)
  6. A 0 = Q A Q T A_{0}=QAQ^{T}
  7. A k A_{k}
  8. p ( A k ) p(A_{k})
  9. p ( A k ) e 1 p(A_{k})e_{1}
  10. p ( A k ) p(A_{k})
  11. r r
  12. p ( x ) = ( x - λ ) ( x - λ ¯ ) p(x)=(x-\lambda)(x-\bar{\lambda})
  13. λ \lambda
  14. λ ¯ \bar{\lambda}
  15. 2 × 2 2\times 2
  16. A k A_{k}
  17. r + 1 r+1
  18. A k A_{k}
  19. A k A_{k}

Quadratic_Gauss_sum.html

  1. g ( a ; p ) = n = 0 p - 1 e 2 π i a n 2 / p = n = 0 p - 1 ζ p a n 2 , ζ p = e 2 π i / p . g(a;p)=\sum_{n=0}^{p-1}e^{2{\pi}ian^{2}/p}=\sum_{n=0}^{p-1}\zeta_{p}^{an^{2}},% \quad\zeta_{p}=e^{2{\pi}i/p}.
  2. G ( a , χ ) = n = 1 p - 1 ( n p ) e 2 π i a n / p . G(a,\chi)=\sum_{n=1}^{p-1}\left(\frac{n}{p}\right)e^{2{\pi}ian/p}.
  3. χ ( n ) = ( n p ) \chi(n)=\left(\frac{n}{p}\right)
  4. g ( a ; p ) = ( a p ) g ( 1 ; p ) . g(a;p)=\left(\frac{a}{p}\right)g(1;p).
  5. g ( 1 ; p ) = n = 0 p - 1 e 2 π i n 2 / p = { p p 1 mod 4 i p p 3 mod 4 . g(1;p)=\sum_{n=0}^{p-1}e^{2{\pi}in^{2}/p}=\begin{cases}\sqrt{p}&p\equiv 1\mod 4% \\ i\sqrt{p}&p\equiv 3\mod 4\end{cases}.
  6. g ( a ; p ) 2 = ( - 1 p ) p g(a;p)^{2}=\left(\frac{-1}{p}\right)p
  7. G ( a , b , c ) = n = 0 c - 1 e ( a n 2 + b n c ) , G(a,b,c)=\sum_{n=0}^{c-1}e\left(\frac{an^{2}+bn}{c}\right),
  8. G ( a , c ) = G ( a , 0 , c ) G(a,c)=G(a,0,c)
  9. G ( a , b , c ) = gcd ( a , c ) G ( a gcd ( a , c ) , b gcd ( a , c ) , c gcd ( a , c ) ) G(a,b,c)=\gcd(a,c)\cdot G\left(\frac{a}{\gcd(a,c)},\frac{b}{\gcd(a,c)},\frac{c% }{\gcd(a,c)}\right)
  10. a c 0 ac\neq 0
  11. n = 0 | c | - 1 e π i ( a n 2 + b n ) / c = | c / a | 1 / 2 e π i ( | a c | - b 2 ) / ( 4 a c ) n = 0 | a | - 1 e - π i ( c n 2 + b n ) / a . \sum_{n=0}^{|c|-1}e^{\pi i(an^{2}+bn)/c}=|c/a|^{1/2}e^{\pi i(|ac|-b^{2})/(4ac)% }\sum_{n=0}^{|a|-1}e^{-\pi i(cn^{2}+bn)/a}.
  12. ε m = { 1 m 1 mod 4 i m 3 mod 4 \varepsilon_{m}=\begin{cases}1&m\equiv 1\mod 4\\ i&m\equiv 3\mod 4\end{cases}
  13. G ( a , c ) = G ( a , 0 , c ) = { 0 c 2 mod 4 ε c c ( a c ) c odd ( 1 + i ) ε a - 1 c ( c a ) a odd , 4 c . G(a,c)=G(a,0,c)=\begin{cases}0&c\equiv 2\mod 4\\ \varepsilon_{c}\sqrt{c}\left(\frac{a}{c}\right)&c\ \,\text{odd}\\ (1+i)\varepsilon_{a}^{-1}\sqrt{c}\left(\frac{c}{a}\right)&a\ \,\text{odd},4% \mid c.\end{cases}
  14. ( a c ) \left(\frac{a}{c}\right)
  15. G ( a , b , c ) = ε c c ( a c ) e - 2 π i ψ ( a ) b 2 / c G(a,b,c)=\varepsilon_{c}\sqrt{c}\cdot\left(\frac{a}{c}\right)e^{-2\pi i\psi(a)% b^{2}/c}
  16. ψ ( a ) \psi(a)
  17. 4 ψ ( a ) a 1 mod c 4\psi(a)a\equiv 1\ \,\text{mod}\ c
  18. G ( a , b , 2 n ) = 0 G(a,b,2^{n})=0
  19. a n 2 + b n an^{2}+bn
  20. 0 n < c - 1 0\leq n<c-1
  21. a n 2 + b n + q = 0 an^{2}+bn+q=0
  22. / 2 n \mathbb{Z}/2^{n}\mathbb{Z}
  23. a n 2 + b n an^{2}+bn
  24. G ( a , b , 2 n ) = 0 G(a,b,2^{n})=0
  25. G ( a , 0 , c ) = n = 0 c - 1 ( n c ) e 2 π i a n / c . G(a,0,c)=\sum_{n=0}^{c-1}\left(\frac{n}{c}\right)e^{2\pi ian/c}.

Quadratic_integral.html

  1. d x a + b x + c x 2 . \int\frac{dx}{a+bx+cx^{2}}.
  2. d x a + b x + c x 2 = 1 c d x ( x + b 2 c ) 2 + ( a c - b 2 4 c 2 ) . \int\frac{dx}{a+bx+cx^{2}}=\frac{1}{c}\int\frac{dx}{\left(x+\frac{b}{2c}\right% )^{2}+\left(\frac{a}{c}-\frac{b^{2}}{4c^{2}}\right)}.
  3. u = x + b 2 c u=x+\frac{b}{2c}
  4. - A 2 = a c - b 2 4 c 2 = 1 4 c 2 ( 4 a c - b 2 ) . -A^{2}=\frac{a}{c}-\frac{b^{2}}{4c^{2}}=\frac{1}{4c^{2}}\left(4ac-b^{2}\right).
  5. d x a + b x + c x 2 = 1 c d u u 2 - A 2 = 1 c d u ( u + A ) ( u - A ) . \int\frac{dx}{a+bx+cx^{2}}=\frac{1}{c}\int\frac{du}{u^{2}-A^{2}}=\frac{1}{c}% \int\frac{du}{(u+A)(u-A)}.
  6. 1 ( u + A ) ( u - A ) = 1 2 A ( 1 u - A - 1 u + A ) \frac{1}{(u+A)(u-A)}=\frac{1}{2A}\left(\frac{1}{u-A}-\frac{1}{u+A}\right)
  7. 1 c d u ( u + A ) ( u - A ) = 1 2 A c ln ( u - A u + A ) + constant . \frac{1}{c}\int\frac{du}{(u+A)(u-A)}=\frac{1}{2Ac}\ln\left(\frac{u-A}{u+A}% \right)+\,\text{constant}.
  8. d x a + b x + c x 2 = 1 q ln ( 2 c x + b - q 2 c x + b + q ) + constant, where q = b 2 - 4 a c . \int\frac{dx}{a+bx+cx^{2}}=\frac{1}{\sqrt{q}}\ln\left(\frac{2cx+b-\sqrt{q}}{2% cx+b+\sqrt{q}}\right)+\,\text{constant, where }q=b^{2}-4ac.
  9. d x a + b x + c x 2 = 1 c d x ( x + b 2 c ) 2 + ( a c - b 2 4 c 2 ) . \int\frac{dx}{a+bx+cx^{2}}=\frac{1}{c}\int\frac{dx}{\left(x+\frac{b}{2c}\right% )^{2}+\left(\frac{a}{c}-\frac{b^{2}}{4c^{2}}\right)}.
  10. 1 c d u u 2 + A 2 = 1 c A d u / A ( u / A ) 2 + 1 = 1 c A d w w 2 + 1 = 1 c A arctan ( w ) + constant = 1 c A arctan ( u A ) + constant = 1 c a c - b 2 4 c 2 arctan ( x + b 2 c a c - b 2 4 c 2 ) + constant = 2 4 a c - b 2 arctan ( 2 c x + b 4 a c - b 2 ) + constant . \begin{aligned}&\displaystyle{}\qquad\frac{1}{c}\int\frac{du}{u^{2}+A^{2}}\\ &\displaystyle=\frac{1}{cA}\int\frac{du/A}{(u/A)^{2}+1}\\ &\displaystyle=\frac{1}{cA}\int\frac{dw}{w^{2}+1}\\ &\displaystyle=\frac{1}{cA}\arctan(w)+\mathrm{constant}\\ &\displaystyle=\frac{1}{cA}\arctan\left(\frac{u}{A}\right)+\,\text{constant}\\ &\displaystyle=\frac{1}{c\sqrt{\frac{a}{c}-\frac{b^{2}}{4c^{2}}}}\arctan\left(% \frac{x+\frac{b}{2c}}{\sqrt{\frac{a}{c}-\frac{b^{2}}{4c^{2}}}}\right)+\,\text{% constant}\\ &\displaystyle=\frac{2}{\sqrt{4ac-b^{2}\,}}\arctan\left(\frac{2cx+b}{\sqrt{4ac% -b^{2}}}\right)+\,\text{constant}.\end{aligned}

Quadratic_sieve.html

  1. e ( 1 + o ( 1 ) ) ln n ln ln n = L n [ 1 / 2 , 1 ] e^{(1+o(1))\sqrt{\ln n\ln\ln n}}=L_{n}\left[1/2,1\right]
  2. ( a + b ) ( a - b ) = 0 ( mod n ) (a+b)(a-b)=0\;\;(\mathop{{\rm mod}}n)
  3. y ( x ) = ( n + x ) 2 - n (where x is a small integer) y(x)=\left(\left\lceil\sqrt{n}\right\rceil+x\right)^{2}-n\hbox{ (where }x\hbox% { is a small integer)}
  4. y ( x ) 2 x n y(x)\approx 2x\left\lceil\sqrt{n}\right\rceil
  5. 21 2 7 1 11 ( mod 91 ) {21^{2}\equiv 7^{1}\cdot 11\;\;(\mathop{{\rm mod}}91)}
  6. 29 2 2 1 11 ( mod 91 ) {29^{2}\equiv 2^{1}\cdot 11\;\;(\mathop{{\rm mod}}91)}
  7. ( 21 29 ) 2 2 1 7 1 11 2 ( mod 91 ) {(21\cdot 29)^{2}\equiv 2^{1}\cdot 7^{1}\cdot 11^{2}\;\;(\mathop{{\rm mod}}91)}
  8. ( 58 21 29 ) 2 2 1 7 1 ( mod 91 ) (58\cdot 21\cdot 29)^{2}\equiv 2^{1}\cdot 7^{1}\;\;(\mathop{{\rm mod}}91)
  9. 14 2 2 1 7 1 ( mod 91 ) 14^{2}\equiv 2^{1}\cdot 7^{1}\;\;(\mathop{{\rm mod}}91)
  10. f ( x ) = x 2 - n f(x)=x^{2}-n
  11. f ( x + k p ) = ( x + k p ) 2 - n f(x+kp)=(x+kp)^{2}-n
  12. f ( x + k p ) = x 2 + 2 x k p + ( k p ) 2 - n f(x+kp)=x^{2}+2xkp+(kp)^{2}-n
  13. f ( x + k p ) = f ( x ) + 2 x k p + ( k p ) 2 f ( x ) ( mod p ) f(x+kp)=f(x)+2xkp+(kp)^{2}\equiv f(x)\;\;(\mathop{{\rm mod}}p)
  14. V X V_{X}
  15. Y ( X ) = ( X + N ) 2 - N = ( X + 124 ) 2 - 15347 Y(X)=(X+\lceil\sqrt{N}\rceil)^{2}-N=(X+124)^{2}-15347
  16. { 2 , 17 , 23 , 29 } \{2,17,23,29\}
  17. Y ( X ) ( X + N ) 2 - N 0 ( mod p ) Y(X)\equiv(X+\lceil\sqrt{N}\rceil)^{2}-N\equiv 0\;\;(\mathop{{\rm mod}}p)
  18. p = 2 p=2
  19. ( X + 124 ) 2 - 15347 0 ( mod 2 ) (X+124)^{2}-15347\equiv 0\;\;(\mathop{{\rm mod}}2)
  20. X 15347 - 124 1 ( mod 2 ) X\equiv\sqrt{15347}-124\equiv 1\;\;(\mathop{{\rm mod}}2)
  21. V = [ 29 139 529 391 1037 647 17191 ] V=\begin{bmatrix}29&139&529&391&1037&647&\cdots&17191\end{bmatrix}
  22. { 17 , 23 , 29 } \{17,23,29\}
  23. X 15347 - 124 ( mod p ) X\equiv\sqrt{15347}-124\;\;(\mathop{{\rm mod}}p)
  24. X \displaystyle X
  25. X a ( mod p ) X\equiv a\;\;(\mathop{{\rm mod}}p)
  26. V x V_{x}
  27. V = [ 1 139 23 1 61 647 17191 ] V=\begin{bmatrix}1&139&23&1&61&647&\cdots&17191\end{bmatrix}
  28. V 0 V_{0}
  29. V 3 V_{3}
  30. V 71 V_{71}
  31. Y Z 2 ( mod N ) Y\equiv Z^{2}\;\;(\mathop{{\rm mod}}N)
  32. 29 \displaystyle 29
  33. ( mod 2 ) \;\;(\mathop{{\rm mod}}2)
  34. S [ 0 0 0 1 1 1 1 0 1 1 1 1 ] [ 0 0 0 0 ] ( mod 2 ) S\cdot\begin{bmatrix}0&0&0&1\\ 1&1&1&0\\ 1&1&1&1\end{bmatrix}\equiv\begin{bmatrix}0&0&0&0\end{bmatrix}\;\;(\mathop{{\rm mod% }}2)
  35. S = [ 1 1 1 ] S=\begin{bmatrix}1&1&1\end{bmatrix}
  36. 29 782 22678 = 22678 2 29\cdot 782\cdot 22678=22678^{2}
  37. 124 2 127 2 195 2 = 3070860 2 124^{2}\cdot 127^{2}\cdot 195^{2}=3070860^{2}
  38. 22678 2 3070860 2 ( mod 15347 ) 22678^{2}\equiv 3070860^{2}\;\;(\mathop{{\rm mod}}15347)
  39. y ( x ) = ( A x + B ) 2 - n A , B y(x)=(Ax+B)^{2}-n\qquad A,B\in\mathbb{Z}
  40. B 2 - n B^{2}-n
  41. B 2 - n = A C B^{2}-n=AC
  42. y ( x ) = A ( A x 2 + 2 B x + C ) y(x)=A\cdot(Ax^{2}+2Bx+C)
  43. ( A x 2 + 2 B x + C ) (Ax^{2}+2Bx+C)
  44. 2 803 - 2 402 + 1 2^{803}-2^{402}+1
  45. 2 1606 + 1 2^{1606}+1

Quadrupole_magnet.html

  1. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) , \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}),
  2. B y = K x B x = K y B_{y}=K\cdot x\qquad B_{x}=K\cdot y
  3. K K
  4. T / m \mathrm{T}/\mathrm{m}
  5. K K

Quantization_(signal_processing).html

  1. x x
  2. Δ \Delta
  3. Q ( x ) = sgn ( x ) Δ | x | Δ + 1 2 Q(x)=\operatorname{sgn}(x)\cdot\Delta\cdot\left\lfloor\frac{\left|x\right|}{% \Delta}+\frac{1}{2}\right\rfloor
  4. sgn \operatorname{sgn}
  5. Δ \Delta
  6. Δ = 1 \Delta=1
  7. Δ \Delta
  8. Δ 2 / 12 \Delta^{2}/12
  9. 10 log 10 ( 1 4 ) = - 6 dB . \scriptstyle 10\cdot\log_{10}\left(\tfrac{1}{4}\right)\ =\ -6\ \mathrm{dB}.
  10. k k
  11. k k
  12. y k y_{k}
  13. k = sgn ( x ) | x | Δ + 1 2 k=\operatorname{sgn}(x)\cdot\left\lfloor\frac{\left|x\right|}{\Delta}+\frac{1}% {2}\right\rfloor
  14. y k = k Δ y_{k}=k\cdot\Delta
  15. Q ( x ) = Δ ( x Δ + 1 2 ) Q(x)=\Delta\cdot\left(\left\lfloor\frac{x}{\Delta}\right\rfloor+\frac{1}{2}\right)
  16. k = x Δ k=\left\lfloor\frac{x}{\Delta}\right\rfloor
  17. y k = Δ ( k + 1 2 ) y_{k}=\Delta\cdot\left(k+\tfrac{1}{2}\right)
  18. Δ \Delta
  19. Δ \Delta
  20. 1 12 LSB 0.289 LSB \scriptstyle{\frac{1}{\sqrt{12}}}\mathrm{LSB}\ \approx\ 0.289\,\mathrm{LSB}
  21. 1 2 LSB \scriptstyle{\frac{1}{2}}\mathrm{LSB}
  22. 1 3 LSB \scriptstyle{\frac{1}{\sqrt{3}}}\mathrm{LSB}
  23. 10 log 10 ( 4 ) = 6.02 \scriptstyle 10\cdot\log_{10}(4)\ =\ 6.02
  24. SQNR = 20 log 10 ( 2 Q ) 6.02 Q dB \mathrm{SQNR}=20\log_{10}(2^{Q})\approx 6.02\cdot Q\ \mathrm{dB}\,\!
  25. SQNR 1.761 + 6.02 Q dB \mathrm{SQNR}\approx 1.761+6.02\cdot Q\ \mathrm{dB}\,\!
  26. N = ( δ v ) 2 12 W \mathrm{N}=\frac{(\delta\mathrm{v})^{2}}{12}\mathrm{W}\,\!
  27. δ v \delta\mathrm{v}
  28. M M
  29. { I k } k = 1 M \{I_{k}\}_{k=1}^{M}
  30. M - 1 M-1
  31. { b k } k = 1 M - 1 \{b_{k}\}_{k=1}^{M-1}
  32. I k = [ b k - 1 , b k ) I_{k}=[b_{k-1}~{},~{}b_{k})
  33. k = 1 , 2 , , M k=1,2,\ldots,M
  34. b 0 = - b_{0}=-\infty
  35. b M = b_{M}=\infty
  36. x x
  37. I k I_{k}
  38. k k
  39. I k I_{k}
  40. y k y_{k}
  41. x I k y = y k x\in I_{k}\Rightarrow y=y_{k}
  42. y = Q ( x ) y=Q(x)
  43. k k
  44. c k c_{k}
  45. length ( c k ) \mathrm{length}(c_{k})
  46. M M
  47. { b k } k = 1 M - 1 \{b_{k}\}_{k=1}^{M-1}
  48. { c k } k = 1 M \{c_{k}\}_{k=1}^{M}
  49. { y k } k = 1 M \{y_{k}\}_{k=1}^{M}
  50. R R
  51. D D
  52. S S
  53. X X
  54. f ( x ) f(x)
  55. p k p_{k}
  56. I k I_{k}
  57. p k = P [ x I k ] = b k - 1 b k f ( x ) d x p_{k}=P[x\in I_{k}]=\int_{b_{k-1}}^{b_{k}}f(x)dx
  58. R R
  59. R = k = 1 M p k length ( c k ) = k = 1 M length ( c k ) b k - 1 b k f ( x ) d x R=\sum_{k=1}^{M}p_{k}\cdot\mathrm{length}(c_{k})=\sum_{k=1}^{M}\mathrm{length}% (c_{k})\int_{b_{k-1}}^{b_{k}}f(x)dx
  60. D = E [ ( x - Q ( x ) ) 2 ] = - ( x - Q ( x ) ) 2 f ( x ) d x = k = 1 M b k - 1 b k ( x - y k ) 2 f ( x ) d x D=E[(x-Q(x))^{2}]=\int_{-\infty}^{\infty}(x-Q(x))^{2}f(x)dx=\sum_{k=1}^{M}\int% _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx
  61. R R
  62. { b k } k = 1 M - 1 \{b_{k}\}_{k=1}^{M-1}
  63. { length ( c k ) } k = 1 M \{\mathrm{length}(c_{k})\}_{k=1}^{M}
  64. D D
  65. { b k } k = 1 M - 1 \{b_{k}\}_{k=1}^{M-1}
  66. { y k } k = 1 M \{y_{k}\}_{k=1}^{M}
  67. D D max D\leq D_{\max}
  68. R R
  69. R R max R\leq R_{\max}
  70. D D
  71. min { D + λ R } \min\left\{D+\lambda\cdot R\right\}
  72. λ \lambda
  73. { y k } k = 1 M \{y_{k}\}_{k=1}^{M}
  74. y k y_{k}
  75. d k d_{k}
  76. D = k = 1 M d k D=\sum_{k=1}^{M}d_{k}
  77. d k = b k - 1 b k ( x - y k ) 2 f ( x ) d x d_{k}=\int_{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx
  78. { b k } k = 1 M - 1 \{b_{k}\}_{k=1}^{M-1}
  79. y k y_{k}
  80. D D
  81. { y k * } k = 1 M \{y^{*}_{k}\}_{k=1}^{M}
  82. y k y_{k}
  83. I k I_{k}
  84. y k * = 1 p k b k - 1 b k x f ( x ) d x y^{*}_{k}=\frac{1}{p_{k}}\int_{b_{k-1}}^{b_{k}}xf(x)dx
  85. { k } k = 1 M \{k\}_{k=1}^{M}
  86. length ( c k ) - log 2 ( p k ) \mathrm{length}(c_{k})\approx-\log_{2}\left(p_{k}\right)
  87. R = k = 1 M - p k log 2 ( p k ) R=\sum_{k=1}^{M}-p_{k}\cdot\log_{2}\left(p_{k}\right)
  88. { p k } k = 1 M \{p_{k}\}_{k=1}^{M}
  89. M M
  90. M M
  91. M M
  92. λ \lambda
  93. D D
  94. M M
  95. R = log 2 M R=\lceil\log_{2}M\rceil
  96. M = M=
  97. R R
  98. M M
  99. X X
  100. f ( x ) f(x)
  101. M M
  102. { b k } k = 1 M - 1 \{b_{k}\}_{k=1}^{M-1}
  103. { y k } k = 1 M \{y_{k}\}_{k=1}^{M}
  104. D = E [ ( x - Q ( x ) ) 2 ] = - ( x - Q ( x ) ) 2 f ( x ) d x = k = 1 M b k - 1 b k ( x - y k ) 2 f ( x ) d x = k = 1 M d k D=E[(x-Q(x))^{2}]=\int_{-\infty}^{\infty}(x-Q(x))^{2}f(x)dx=\sum_{k=1}^{M}\int% _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx=\sum_{k=1}^{M}d_{k}
  105. D / b k = 0 {\partial D/\partial b_{k}}=0
  106. D / y k = 0 {\partial D/\partial y_{k}}=0
  107. D b k = 0 b k = y k + y k + 1 2 {\partial D\over\partial b_{k}}=0\Rightarrow b_{k}={y_{k}+y_{k+1}\over 2}
  108. D y k = 0 y k = b k - 1 b k x f ( x ) d x b k - 1 b k f ( x ) d x = 1 p k b k - 1 b k x f ( x ) d x {\partial D\over\partial y_{k}}=0\Rightarrow y_{k}={\int_{b_{k-1}}^{b_{k}}xf(x% )dx\over\int_{b_{k-1}}^{b_{k}}f(x)dx}=\frac{1}{p_{k}}\int_{b_{k-1}}^{b_{k}}xf(% x)dx
  109. [ y 1 - Δ / 2 , y M + Δ / 2 ) [y_{1}-\Delta/2,~{}y_{M}+\Delta/2)
  110. f ( x ) = 1 2 X m a x f(x)=\frac{1}{2X_{max}}
  111. x [ - X m a x , X m a x ] x\in[-X_{max},X_{max}]
  112. Δ = 2 X m a x M \Delta=\frac{2X_{max}}{M}
  113. SQNR = 10 log 10 σ x 2 σ q 2 = 10 log 10 ( M Δ ) 2 / 12 Δ 2 / 12 = 10 log 10 M 2 = 20 log 10 M {\rm SQNR}=10\log_{10}{\frac{\sigma_{x}^{2}}{\sigma_{q}^{2}}}=10\log_{10}{% \frac{(M\Delta)^{2}/12}{\Delta^{2}/12}}=10\log_{10}M^{2}=20\log_{10}M
  114. N N
  115. M = 2 N M=2^{N}
  116. SQNR = 20 log 10 2 N = N ( 20 log 10 2 ) = N 6.0206 dB {\rm SQNR}=20\log_{10}{2^{N}}=N\cdot(20\log_{10}2)=N\cdot 6.0206\,\rm{dB}
  117. N N
  118. M M
  119. N N
  120. M M
  121. Δ 2 / 12 \Delta^{2}/12

Quantum_algorithm.html

  1. | f ~ ( z i ) | 1 \left|\tilde{f}\left(z_{i}\right)\right|\geqslant 1
  2. | f ~ ( z i ) | 2 \left|\tilde{f}\left(z_{i}\right)\right|\geqslant 2
  3. O ( N ) O(\sqrt{N})
  4. N N
  5. ϵ \epsilon
  6. Θ ( 1 ϵ N k ) \Theta\left(\frac{1}{\epsilon}\sqrt{\frac{N}{k}}\right)
  7. k k
  8. k k^{\prime}
  9. k k
  10. | k - k | ϵ k |k-k^{\prime}|\leq\epsilon k
  11. Θ ( N 2 / 3 ) \Theta(N^{2/3})
  12. c = log 2 ( 1 + 33 ) / 4 0.754 c=\log_{2}(1+\sqrt{33})/4\approx 0.754
  13. Θ ( k 2 ) \Theta(k^{2})
  14. Θ ( k ) \Theta(k)
  15. Ω ( k 2 / 3 ) \Omega(k^{2/3})
  16. O ( k 2 / 3 log k ) O(k^{2/3}\log k)

Quantum_dot.html

  1. a b * = ε r ( m μ ) a b a^{*}_{b}=\varepsilon_{r}\left(\frac{m}{\mu}\right)a_{b}
  2. E confinement \displaystyle E_{\textrm{confinement}}
  3. μ ( N ) = E ( N ) - E ( N - 1 ) \mu(N)=E(N)-E(N-1)
  4. 1 C Δ V Δ Q {1\over C}\equiv{\Delta\,V\over\Delta\,Q}
  5. Δ V = Δ μ e = μ ( N + Δ N ) - μ ( N ) e \Delta\,V={\Delta\,\mu\,\over e}={\mu(N+\Delta\,N)-\mu(N)\over e}
  6. Δ N = 1 \Delta\,N=1
  7. Δ Q = e \Delta\,Q=e
  8. C ( N ) = e 2 μ ( N + 1 ) - μ ( N ) = e 2 I ( N ) - A ( N ) C(N)={e^{2}\over\mu(N+1)-\mu(N)}={e^{2}\over I(N)-A(N)}

Quantum_group.html

  1. k 0 = 1 k_{0}=1
  2. k λ k μ = k λ + μ k_{\lambda}k_{\mu}=k_{\lambda+\mu}
  3. k λ e i k λ - 1 = q ( λ , α i ) e i k_{\lambda}e_{i}k_{\lambda}^{-1}=q^{(\lambda,\alpha_{i})}e_{i}
  4. k λ f i k λ - 1 = q - ( λ , α i ) f i k_{\lambda}f_{i}k_{\lambda}^{-1}=q^{-(\lambda,\alpha_{i})}f_{i}
  5. [ e i , f j ] = δ i j k i - k i - 1 q i - q i - 1 [e_{i},f_{j}]=\delta_{ij}\frac{k_{i}-k_{i}^{-1}}{q_{i}-q_{i}^{-1}}
  6. n = 0 1 - a i j ( - 1 ) n [ 1 - a i j ] q i ! [ 1 - a i j - n ] q i ! [ n ] q i ! e i n e j e i 1 - a i j - n = 0 , \sum_{n=0}^{1-a_{ij}}(-1)^{n}\frac{[1-a_{ij}]_{q_{i}}!}{[1-a_{ij}-n]_{q_{i}}![% n]_{q_{i}}!}e_{i}^{n}e_{j}e_{i}^{1-a_{ij}-n}=0,
  7. n = 0 1 - a i j ( - 1 ) n [ 1 - a i j ] q i ! [ 1 - a i j - n ] q i ! [ n ] q i ! f i n f j f i 1 - a i j - n = 0. \sum_{n=0}^{1-a_{ij}}(-1)^{n}\frac{[1-a_{ij}]_{q_{i}}!}{[1-a_{ij}-n]_{q_{i}}![% n]_{q_{i}}!}f_{i}^{n}f_{j}f_{i}^{1-a_{ij}-n}=0.
  8. k i = k α i , q i = q 1 2 ( α i , α i ) , [ 0 ] q i ! = 1 , [ n ] q i ! = m = 1 n [ m ] q i k_{i}=k_{\alpha_{i}},q_{i}=q^{\frac{1}{2}(\alpha_{i},\alpha_{i})},[0]_{q_{i}}!% =1,[n]_{q_{i}}!=\prod_{m=1}^{n}[m]_{q_{i}}
  9. [ m ] q i = q i m - q i - m q i - q i - 1 . [m]_{q_{i}}=\frac{q_{i}^{m}-q_{i}^{-m}}{q_{i}-q_{i}^{-1}}.
  10. k λ - k - λ q - q - 1 t λ \frac{k_{\lambda}-k_{-\lambda}}{q-q^{-1}}\to t_{\lambda}
  11. Δ 1 ( k λ ) = k λ k λ \Delta_{1}(k_{\lambda})=k_{\lambda}\otimes k_{\lambda}
  12. Δ 1 ( e i ) = 1 e i + e i k i \Delta_{1}(e_{i})=1\otimes e_{i}+e_{i}\otimes k_{i}
  13. Δ 1 ( f i ) = k i - 1 f i + f i 1 \Delta_{1}(f_{i})=k_{i}^{-1}\otimes f_{i}+f_{i}\otimes 1
  14. Δ 2 ( k λ ) = k λ k λ \Delta_{2}(k_{\lambda})=k_{\lambda}\otimes k_{\lambda}
  15. Δ 2 ( e i ) = k i - 1 e i + e i 1 \Delta_{2}(e_{i})=k_{i}^{-1}\otimes e_{i}+e_{i}\otimes 1
  16. Δ 2 ( f i ) = 1 f i + f i k i \Delta_{2}(f_{i})=1\otimes f_{i}+f_{i}\otimes k_{i}
  17. Δ 3 ( k λ ) = k λ k λ \Delta_{3}(k_{\lambda})=k_{\lambda}\otimes k_{\lambda}
  18. Δ 3 ( e i ) = k i - 1 2 e i + e i k i 1 2 \Delta_{3}(e_{i})=k_{i}^{-\frac{1}{2}}\otimes e_{i}+e_{i}\otimes k_{i}^{\frac{% 1}{2}}
  19. Δ 3 ( f i ) = k i - 1 2 f i + f i k i 1 2 \Delta_{3}(f_{i})=k_{i}^{-\frac{1}{2}}\otimes f_{i}+f_{i}\otimes k_{i}^{\frac{% 1}{2}}
  20. S 1 ( k λ ) = k - λ , S 1 ( e i ) = - e i k i - 1 , S 1 ( f i ) = - k i f i S_{1}(k_{\lambda})=k_{-\lambda},\ S_{1}(e_{i})=-e_{i}k_{i}^{-1},\ S_{1}(f_{i})% =-k_{i}f_{i}
  21. S 2 ( k λ ) = k - λ , S 2 ( e i ) = - k i e i , S 2 ( f i ) = - f i k i - 1 S_{2}(k_{\lambda})=k_{-\lambda},\ S_{2}(e_{i})=-k_{i}e_{i},\ S_{2}(f_{i})=-f_{% i}k_{i}^{-1}
  22. S 3 ( k λ ) = k - λ , S 3 ( e i ) = - q i e i , S 3 ( f i ) = - q i - 1 f i . S_{3}(k_{\lambda})=k_{-\lambda},\ S_{3}(e_{i})=-q_{i}e_{i},\ S_{3}(f_{i})=-q_{% i}^{-1}f_{i}.
  23. Ad x y = ( x ) x ( 1 ) y S ( x ( 2 ) ) , {\mathrm{Ad}}_{x}\cdot y=\sum_{(x)}x_{(1)}yS(x_{(2)}),
  24. Δ ( x ) = ( x ) x ( 1 ) x ( 2 ) \Delta(x)=\sum_{(x)}x_{(1)}\otimes x_{(2)}
  25. d 0 = 1 d_{0}=1
  26. d λ d μ = d λ + μ d_{\lambda}d_{\mu}=d_{\lambda+\mu}
  27. e i k . v = f i k . v = 0 e_{i}^{k}.v=f_{i}^{k}.v=0
  28. d λ = c λ q ( λ , ν ) d_{\lambda}=c_{\lambda}q^{(\lambda,\nu)}
  29. c 0 = 1 , c_{0}=1,\,
  30. c λ c μ = c λ + μ c_{\lambda}c_{\mu}=c_{\lambda+\mu}
  31. c 2 α i = 1 c_{2\alpha_{i}}=1
  32. k λ . v = q ( λ , ν ) v k_{\lambda}.v=q^{(\lambda,\nu)}v
  33. 2 ( μ , α i ) / ( α i , α i ) 2(\mu,\alpha_{i})/(\alpha_{i},\alpha_{i})
  34. k λ . v = c λ q ( λ , ν ) v k_{\lambda}.v=c_{\lambda}q^{(\lambda,\nu)}v
  35. c 0 = 1 c_{0}=1
  36. c λ c μ = c λ + μ c_{\lambda}c_{\mu}=c_{\lambda+\mu}
  37. c 2 α i = 1 c_{2\alpha_{i}}=1
  38. k λ . ( v w ) = k λ . v k λ . w k_{\lambda}.(v\otimes w)=k_{\lambda}.v\otimes k_{\lambda}.w
  39. e i . ( v w ) = k i . v e i . w + e i . v w e_{i}.(v\otimes w)=k_{i}.v\otimes e_{i}.w+e_{i}.v\otimes w
  40. f i . ( v w ) = v f i . w + f i . v k i - 1 . w f_{i}.(v\otimes w)=v\otimes f_{i}.w+f_{i}.v\otimes k_{i}^{-1}.w
  41. k λ . v 0 = q ( λ , ν ) v 0 k_{\lambda}.v_{0}=q^{(\lambda,\nu)}v_{0}
  42. e i . v 0 = 0 e_{i}.v_{0}=0
  43. q η j t λ j t μ j q^{\eta\sum_{j}t_{\lambda_{j}}\otimes t_{\mu_{j}}}
  44. q η j t λ j t μ j . ( v w ) = q η ( α , β ) v w q^{\eta\sum_{j}t_{\lambda_{j}}\otimes t_{\mu_{j}}}.(v\otimes w)=q^{\eta(\alpha% ,\beta)}v\otimes w
  45. ( 𝔅 ( V ) k [ 𝐙 n ] 𝔅 ( V * ) ) σ \left(\mathfrak{B}(V)\otimes k[\mathbf{Z}^{n}]\otimes\mathfrak{B}(V^{*})\right% )^{\sigma}
  46. 𝔅 ( V ) \mathfrak{B}(V)
  47. g ( u i j ( g ) ) i , j g\mapsto(u_{ij}(g))_{i,j}
  48. Δ ( u i j ) = k u i k u k j . \Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj}.
  49. 1 = k u 1 k κ ( u k 1 ) = k κ ( u 1 k ) u k 1 . 1=\sum_{k}u_{1k}\kappa(u_{k1})=\sum_{k}\kappa(u_{1k})u_{k1}.
  50. u = ( u i j ) i , j = 1 , , n u=(u_{ij})_{i,j=1,\dots,n}
  51. Δ ( u i j ) = k u i k u k j \Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj}
  52. k κ ( u i k ) u k j = k u i k κ ( u k j ) = δ i j I , \sum_{k}\kappa(u_{ik})u_{kj}=\sum_{k}u_{ik}\kappa(u_{kj})=\delta_{ij}I,
  53. v = ( v i j ) i , j = 1 , , n v=(v_{ij})_{i,j=1,\dots,n}
  54. Δ ( v i j ) = k = 1 n v i k v k j \Delta(v_{ij})=\sum_{k=1}^{n}v_{ik}\otimes v_{kj}
  55. γ γ * = γ * γ , \gamma\gamma^{*}=\gamma^{*}\gamma,
  56. α γ = μ γ α , \alpha\gamma=\mu\gamma\alpha,
  57. α γ * = μ γ * α , \alpha\gamma^{*}=\mu\gamma^{*}\alpha,
  58. α α * + μ γ * γ = α * α + μ - 1 γ * γ = I , \alpha\alpha^{*}+\mu\gamma^{*}\gamma=\alpha^{*}\alpha+\mu^{-1}\gamma^{*}\gamma% =I,
  59. u = ( α γ - γ * α * ) , u=\left(\begin{matrix}\alpha&\gamma\\ -\gamma^{*}&\alpha^{*}\end{matrix}\right),
  60. v = ( α μ γ - 1 μ γ * α * ) . v=\left(\begin{matrix}\alpha&\sqrt{\mu}\gamma\\ -\frac{1}{\sqrt{\mu}}\gamma^{*}&\alpha^{*}\end{matrix}\right).
  61. β β * = β * β , \beta\beta^{*}=\beta^{*}\beta,
  62. α β = μ β α , \alpha\beta=\mu\beta\alpha,
  63. α β * = μ β * α , \alpha\beta^{*}=\mu\beta^{*}\alpha,
  64. α α * + μ 2 β * β = α * α + β * β = I , \alpha\alpha^{*}+\mu^{2}\beta^{*}\beta=\alpha^{*}\alpha+\beta^{*}\beta=I,
  65. w = ( α μ β - β * α * ) , w=\left(\begin{matrix}\alpha&\mu\beta\\ -\beta^{*}&\alpha^{*}\end{matrix}\right),
  66. γ = μ β \gamma=\sqrt{\mu}\beta
  67. [ p , K ] = h K ( K - 1 ) [p,K]=hK(K-1)
  68. Δ p = p K + 1 p \Delta p=p\otimes K+1\otimes p
  69. Δ K = K K \Delta K=K\otimes K

Quasi-arithmetic_mean.html

  1. f f
  2. I I
  3. x 1 , x 2 I x_{1},x_{2}\in I
  4. M f ( x 1 , x 2 ) = f - 1 ( f ( x 1 ) + f ( x 2 ) 2 ) . M_{f}(x_{1},x_{2})=f^{-1}\left(\frac{f(x_{1})+f(x_{2})}{2}\right).
  5. n n
  6. x 1 , , x n I x_{1},\dots,x_{n}\in I
  7. M f ( x 1 , , x n ) = f - 1 ( f ( x 1 ) + + f ( x n ) n ) . M_{f}(x_{1},\dots,x_{n})=f^{-1}\left(\frac{f(x_{1})+\cdots+f(x_{n})}{n}\right).
  8. f - 1 f^{-1}
  9. f f
  10. f ( x 1 ) + f ( x 2 ) 2 \frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}
  11. f - 1 f^{-1}
  12. x x
  13. x x
  14. I I
  15. f ( x ) = x f(x)=x
  16. x a x + b x\mapsto a\cdot x+b
  17. a a
  18. I I
  19. f ( x ) = log ( x ) f(x)=\log(x)
  20. I I
  21. f ( x ) = 1 x f(x)=\frac{1}{x}
  22. I I
  23. f ( x ) = x p f(x)=x^{p}
  24. p p
  25. I I
  26. f ( x ) = exp ( x ) f(x)=\exp(x)
  27. M f ( x 1 , , x n ) = L S E ( x 1 , , x n ) - log ( n ) M_{f}(x_{1},\dots,x_{n})=LSE(x_{1},\dots,x_{n})-\log(n)
  28. M f ( x 1 , , x n k ) = M f ( M f ( x 1 , , x k ) , M f ( x k + 1 , , x 2 k ) , , M f ( x ( n - 1 ) k + 1 , , x n k ) ) M_{f}(x_{1},\dots,x_{n\cdot k})=M_{f}(M_{f}(x_{1},\dots,x_{k}),M_{f}(x_{k+1},% \dots,x_{2\cdot k}),\dots,M_{f}(x_{(n-1)\cdot k+1},\dots,x_{n\cdot k}))
  29. m = M f ( x 1 , , x k ) m=M_{f}(x_{1},\dots,x_{k})
  30. M f ( x 1 , , x k , x k + 1 , , x n ) = M f ( m , , m k times , x k + 1 , , x n ) M_{f}(x_{1},\dots,x_{k},x_{k+1},\dots,x_{n})=M_{f}(\underbrace{m,\dots,m}_{k\,% \text{ times}},x_{k+1},\dots,x_{n})
  31. f f
  32. a b 0 ( ( t g ( t ) = a + b f ( t ) ) x M f ( x ) = M g ( x ) \forall a\ \forall b\neq 0((\forall t\ g(t)=a+b\cdot f(t))\Rightarrow\forall x% \ M_{f}(x)=M_{g}(x)
  33. f f
  34. M f M_{f}
  35. M M
  36. M ( M ( x , y ) , M ( z , w ) ) = M ( M ( x , z ) , M ( y , w ) ) M(M(x,y),M(z,w))=M(M(x,z),M(y,w))
  37. M ( x , M ( y , z ) ) = M ( M ( x , y ) , M ( x , z ) ) M(x,M(y,z))=M(M(x,y),M(x,z))
  38. M M
  39. M ( M ( x , M ( x , y ) ) , M ( y , M ( x , y ) ) ) = M ( x , y ) M\big(M(x,M(x,y)),M(y,M(x,y))\big)=M(x,y)
  40. M M
  41. f f
  42. C C
  43. M f , C x = C x f - 1 ( f ( x 1 C x ) + + f ( x n C x ) n ) M_{f,C}x=Cx\cdot f^{-1}\left(\frac{f\left(\frac{x_{1}}{Cx}\right)+\cdots+f% \left(\frac{x_{n}}{Cx}\right)}{n}\right)

Quasi-Monte_Carlo_method.html

  1. [ 0 , 1 ] s f ( u ) d u 1 N i = 1 N f ( x i ) . \int_{[0,1]^{s}}f(u)\,{\rm d}u\approx\frac{1}{N}\,\sum_{i=1}^{N}f(x_{i}).
  2. ϵ = | [ 0 , 1 ] s f ( u ) d u - 1 N i = 1 N f ( x i ) | \epsilon=\left|\int_{[0,1]^{s}}f(u)\,{\rm d}u-\frac{1}{N}\,\sum_{i=1}^{N}f(x_{% i})\right|
  3. | ϵ | V ( f ) D N |\epsilon|\leq V(f)D_{N}
  4. D N = sup Q [ 0 , 1 ] s | number of points in Q N - volume ( Q ) | D_{N}=\sup_{Q\subset[0,1]^{s}}\left|\frac{\mbox{number of points in }~{}Q}{N}-% \mbox{volume}~{}(Q)\right|
  5. | ϵ | V ( f ) D N |\epsilon|\leq V(f)D_{N}
  6. O ( ( log N ) s N ) O\left(\frac{(\log N)^{s}}{N}\right)
  7. O ( 1 N ) O\left(\frac{1}{\sqrt{N}}\right)
  8. O ( ( log N ) s N ) O\left(\frac{(\log N)^{s}}{N}\right)
  9. O ( 1 N ) O\left(\frac{1}{\sqrt{N}}\right)
  10. s s
  11. N N
  12. V ( f ) = V(f)=\infty
  13. D N * D_{N}^{*}
  14. V ( f ) V(f)
  15. y j = x j + U ( mod 1 ) y_{j}=x_{j}+U\;\;(\mathop{{\rm mod}}1)
  16. ( y j ) (y_{j})
  17. ( x j ) (x_{j})

Quasistatic_process.html

  1. W 1 - 2 = P d V = P ( V 2 - V 1 ) W_{1-2}=\int PdV=P(V_{2}-V_{1})
  2. W 1 - 2 = P d V = 0 W_{1-2}=\int PdV=0
  3. W 1 - 2 = P d V , W_{1-2}=\int PdV,
  4. P V = P 1 V 1 = C \quad PV=P_{1}V_{1}=C
  5. W 1 - 2 = P 1 V 1 ln V 2 V 1 W_{1-2}=P_{1}V_{1}\ln\frac{V_{2}}{V_{1}}
  6. W 1 - 2 = P 1 V 1 - P 2 V 2 n - 1 W_{1-2}=\frac{P_{1}V_{1}-P_{2}V_{2}}{n-1}

Quine–McCluskey_algorithm.html

  1. f ( A , B , C , D ) = m ( 4 , 8 , 10 , 11 , 12 , 15 ) + d ( 9 , 14 ) . f(A,B,C,D)=\sum m(4,8,10,11,12,15)+d(9,14).\,
  2. f A , B , C , D = A B C D + A B C D + A B C D + A B C D + A B C D + A B C D , f_{A,B,C,D}=A^{\prime}BC^{\prime}D^{\prime}+AB^{\prime}C^{\prime}D^{\prime}+AB% ^{\prime}CD^{\prime}+AB^{\prime}CD+ABC^{\prime}D^{\prime}+ABCD,
  3. f A , B , C , D = B C D + A B + A C f_{A,B,C,D}=BC^{\prime}D^{\prime}+AB^{\prime}+AC
  4. f A , B , C , D = A B C D + A B C D + A B C D + A B C D + A B C D + A B C D + A B C D + A B C D . f_{A,B,C,D}=A^{\prime}BC^{\prime}D^{\prime}+AB^{\prime}C^{\prime}D^{\prime}+AB% ^{\prime}C^{\prime}D+AB^{\prime}CD^{\prime}+AB^{\prime}CD+ABC^{\prime}D^{% \prime}+ABCD^{\prime}+ABCD.

Quiver_(mathematics).html

  1. Γ = ( V , E , s , t ) \Gamma=(V,E,s,t)
  2. Γ = ( V , E , s , t ) \Gamma^{\prime}=(V^{\prime},E^{\prime},s^{\prime},t^{\prime})
  3. m = ( m v , m e ) m=(m_{v},m_{e})
  4. m v : V V m_{v}:V\to V^{\prime}
  5. m e : E E m_{e}:E\to E^{\prime}
  6. m v s = s m e m_{v}\circ s=s^{\prime}\circ m_{e}
  7. m v t = t m e m_{v}\circ t=t^{\prime}\circ m_{e}
  8. E s t V E\;\begin{matrix}s\\ \rightrightarrows\\ t\end{matrix}\;V
  9. e i e_{i}
  10. e E := v E 1 v e_{E}:=\sum_{v\in E}1_{v}
  11. f ( x ) : V ( x ) V ( x ) f(x):V(x)\rightarrow V^{\prime}(x)
  12. V ( a ) f ( x ) = f ( y ) V ( a ) V^{\prime}(a)f(x)=f(y)V(a)
  13. V W V\oplus W
  14. ( V W ) ( x ) = V ( x ) W ( x ) (V\oplus W)(x)=V(x)\oplus W(x)
  15. ( V W ) ( a ) (V\oplus W)(a)
  16. A n A_{n}
  17. D n D_{n}
  18. E 6 E_{6}
  19. E 7 E_{7}
  20. E 8 E_{8}

R-parity.html

  1. 2 \mathbb{Z}_{2}
  2. P R = ( - 1 ) 3 B + L + 2 s , P_{\mathrm{R}}=(-1)^{3B+L+2s},
  3. P R = ( - 1 ) 3 ( B - L ) + 2 s , P_{\mathrm{R}}=(-1)^{3(B-L)+2s},
  4. d 2 θ λ 1 U c D c D c \int d^{2}\theta\;\lambda_{1}\;U^{c}D^{c}D^{c}
  5. d 2 θ λ 2 Q D c L \int d^{2}\theta\;\lambda_{2}\;QD^{c}L
  6. G F G_{F}
  7. d 2 θ λ 3 L E c L \int d^{2}\theta\;\lambda_{3}\;LE^{c}L
  8. d 2 θ κ L H u \int d^{2}\theta\;\kappa\;LH_{u}
  9. 𝒪 ( 1 ) \mathcal{O}(1)
  10. U ( 1 ) B - L U(1)_{B-L}
  11. U ( 1 ) B - L U(1)_{B-L}
  12. U ( 1 ) B - L U(1)_{B-L}

Ra_(disambiguation).html

  1. R a R_{a}
  2. R a R_{a}

Rabi_cycle.html

  1. | c b ( t ) | 2 sin 2 ( ω t / 2 ) |c_{b}(t)|^{2}\propto\sin^{2}(\omega t/2)
  2. ω \omega
  3. | ψ |\psi\rangle
  4. | ψ = ( c 1 c 2 ) = c 1 ( 1 0 ) + c 2 ( 0 1 ) ; |\psi\rangle=\begin{pmatrix}c_{1}\\ c_{2}\end{pmatrix}=c_{1}\begin{pmatrix}1\\ 0\end{pmatrix}+c_{2}\begin{pmatrix}0\\ 1\end{pmatrix};
  5. c 1 c_{1}
  6. c 2 c_{2}
  7. c 1 c_{1}
  8. c 2 c_{2}
  9. | c 1 | 2 + | c 2 | 2 = 1 {|c_{1}|}^{2}+{|c_{2}|}^{2}=1
  10. | 1 = ( 1 0 ) |1\rangle=\begin{pmatrix}1\\ 0\end{pmatrix}
  11. | 2 = ( 0 1 ) |2\rangle=\begin{pmatrix}0\\ 1\end{pmatrix}
  12. × \times
  13. | 1 |1\rangle
  14. | 1 |1\rangle
  15. | 1 |1\rangle
  16. | 1 |1\rangle
  17. | 1 |1\rangle
  18. 𝐇 = ( a 0 + a 3 a 1 - i a 2 a 1 + i a 2 a 0 - a 3 ) \mathbf{H}=\begin{pmatrix}a_{0}+a_{3}&a_{1}-ia_{2}\\ a_{1}+ia_{2}&a_{0}-a_{3}\end{pmatrix}
  19. a 0 , a 1 , a 2 a_{0},a_{1},a_{2}
  20. a 3 a_{3}
  21. 𝐇 = a 0 σ 0 + a 1 σ 1 + a 2 σ 2 + a 3 σ 3 ; \mathbf{H}=a_{0}\cdot\sigma_{0}+a_{1}\cdot\sigma_{1}+a_{2}\cdot\sigma_{2}+a_{3% }\cdot\sigma_{3};
  22. σ 0 \sigma_{0}
  23. × \times
  24. σ k ( k = 1 , 2 , 3 ) \sigma_{k}(k=1,2,3)
  25. a 0 , a 1 , a 2 a_{0},a_{1},a_{2}
  26. a 3 a_{3}
  27. 𝐁 = B 𝐳 ^ \mathbf{B}=B\mathbf{\hat{z}}
  28. H = - s y m b o l μ 𝐁 = - γ 𝐒 𝐁 = - γ S z B H=-symbol{\mu}\cdot\mathbf{B}=-\gamma\mathbf{S}\cdot\mathbf{B}=-\gamma\ S_{z}B
  29. S z = 2 σ 3 = 2 [ 1 0 0 - 1 ] S_{z}=\frac{\hbar}{2}\sigma_{3}=\frac{\hbar}{2}\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  30. μ \mu
  31. γ \gamma
  32. s y m b o l σ symbol{\sigma}
  33. σ 3 \sigma_{3}
  34. | 1 |1\rangle
  35. | 2 |2\rangle
  36. | ψ |\psi\rangle
  37. | ϕ |\phi\rangle
  38. | ϕ | ψ | 2 {|\langle\phi|\psi\rangle|}^{2}
  39. t = 0 t=0
  40. | + X |+X\rangle
  41. σ 1 \sigma_{1}
  42. | ψ ( 0 ) = 1 2 ( 1 1 ) |\psi(0)\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\end{pmatrix}
  43. | ψ ( 0 ) = 1 2 ( 1 0 ) + 1 2 ( 0 1 ) |\psi(0)\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 0\end{pmatrix}+\frac{1}{\sqrt{2}}\begin{pmatrix}0\\ 1\end{pmatrix}
  44. | ψ ( t ) = e - i E t | ψ ( 0 ) |\psi(t)\rangle=e^{\frac{-iEt}{\hbar}}|\psi(0)\rangle
  45. | ψ ( t ) = e - i E + t 1 2 | 1 + e - i E - t 1 2 | 2 |\psi(t)\rangle=e^{\frac{-iE_{+}t}{\hbar}}\frac{1}{\sqrt{2}}|1\rangle+e^{\frac% {-iE_{-}t}{\hbar}}\frac{1}{\sqrt{2}}|2\rangle
  46. | + X | ψ ( t ) | 2 = | 1 2 ( 1 | + 2 | ) ( e - i E + t 1 2 | 1 + e - i E - t 1 2 | 2 ) | 2 = cos 2 ( ω t 2 ) {|\langle\ +X|\psi(t)\rangle|}^{2}={|\frac{1}{\sqrt{2}}(\langle\ 1|+\langle\ 2% |)(e^{\frac{-iE_{+}t}{\hbar}}\frac{1}{\sqrt{2}}|1\rangle+e^{\frac{-iE_{-}t}{% \hbar}}\frac{1}{\sqrt{2}}|2\rangle)|}^{2}={\cos}^{2}(\frac{\omega t}{2})
  47. ω \omega
  48. ω = E - - E + = γ B \omega=\frac{E_{-}-E_{+}}{\hbar}=\gamma B
  49. E - E + E_{-}\geq E_{+}
  50. 2 \frac{\hbar}{2}
  51. 1 2 \frac{1}{2}
  52. E + = E - E_{+}=E_{-}
  53. H = - γ S z B s i n ( ω t ) H=-\gamma\ S_{z}Bsin(\omega t)
  54. + 2 \frac{+\hbar}{2}
  55. | + Y | ψ ( t ) | 2 = cos 2 ( γ B 2 ω cos ω t ) {|\langle\ +Y|\psi(t)\rangle|}^{2}={\cos}^{2}(\frac{\gamma B}{2\omega}\cos% \omega t)
  56. | + Y |+Y\rangle
  57. P 1 P_{1}
  58. P 2 P_{2}
  59. | 1 |1\rangle
  60. | 2 |2\rangle
  61. P 1 P_{1}
  62. P 2 P_{2}
  63. P 1 P_{1}
  64. | E + |E_{+}\rangle
  65. | E - |E_{-}\rangle
  66. 𝐇 = E 0 σ 0 + W 1 σ 1 + W 2 σ 2 + Δ σ 3 = ( E 0 + Δ W 1 - i W 2 W 1 + i W 2 E 0 - Δ ) \mathbf{H}=E_{0}\cdot\sigma_{0}+W_{1}\cdot\sigma_{1}+W_{2}\cdot\sigma_{2}+% \Delta\cdot\sigma_{3}=\begin{pmatrix}E_{0}+\Delta&W_{1}-iW_{2}\\ W_{1}+iW_{2}&E_{0}-\Delta\end{pmatrix}
  67. λ + = E + = E 0 + Δ 2 + W 1 2 + W 2 2 = E 0 + Δ 2 + | W | 2 \lambda_{+}=E_{+}=E_{0}+\sqrt{{\Delta}^{2}+{W_{1}}^{2}+{W_{2}}^{2}}=E_{0}+% \sqrt{{\Delta}^{2}+{\left|W\right|}^{2}}
  68. λ - = E - = E 0 - Δ 2 + W 1 2 + W 2 2 = E 0 - Δ 2 + | W | 2 \lambda_{-}=E_{-}=E_{0}-\sqrt{{\Delta}^{2}+{W_{1}}^{2}+{W_{2}}^{2}}=E_{0}-% \sqrt{{\Delta}^{2}+{\left|W\right|}^{2}}
  69. 𝐖 = W 1 + ı W 2 \mathbf{W}=W_{1}+\imath W_{2}
  70. | W | 2 = W 1 2 + W 2 2 = W W * {\left|W\right|}^{2}={W_{1}}^{2}+{W_{2}}^{2}=WW^{*}
  71. 𝐖 = | W | e - ı ϕ \mathbf{W}={\left|W\right|}e^{-\imath\phi}
  72. E + E_{+}
  73. ( E 0 + Δ W 1 - i W 2 W 1 + i W 2 E 0 - Δ ) ( a b ) = E + ( a b ) \begin{pmatrix}E_{0}+\Delta&W_{1}-iW_{2}\\ W_{1}+iW_{2}&E_{0}-\Delta\end{pmatrix}\begin{pmatrix}a\\ b\\ \end{pmatrix}=E_{+}\begin{pmatrix}a\\ b\\ \end{pmatrix}
  74. b = - a ( E 0 + Δ - E + ) W 1 - ı W 2 b=-\frac{a(E_{0}+\Delta-E_{+})}{W_{1}-\imath W_{2}}
  75. | a | 2 + | b | 2 = 1 {\left|a\right|}^{2}+{\left|b\right|}^{2}=1
  76. | a | 2 + | a | 2 ( Δ | W | - Δ 2 + | W | 2 | W | ) 2 = 1 {\left|a\right|}^{2}+{\left|a\right|}^{2}(\frac{\Delta}{\left|W\right|}-\frac{% \sqrt{{\Delta}^{2}+{\left|W\right|}^{2}}}{\left|W\right|})^{2}=1
  77. sin θ = | W | Δ 2 + | W | 2 \sin\theta=\frac{\left|W\right|}{\sqrt{{\Delta}^{2}+{\left|W\right|}^{2}}}
  78. cos θ = Δ Δ 2 + | W | 2 \cos\theta=\frac{\Delta}{\sqrt{{\Delta}^{2}+{\left|W\right|}^{2}}}
  79. tan θ = | W | Δ \tan\theta=\frac{\left|W\right|}{\Delta}
  80. | a | 2 + | a | 2 ( 1 - cos θ ) 2 sin 2 θ = 1 {\left|a\right|}^{2}+{\left|a\right|}^{2}\frac{({1-\cos\theta})^{2}}{\sin^{2}% \theta}=1
  81. | a | 2 = cos 2 θ / 2 {\left|a\right|}^{2}=\cos^{2}\theta/2
  82. ϕ \phi
  83. a = exp ( ı ϕ / 2 ) cos θ / 2 a=\exp(\imath\phi/2)\cos\theta/2
  84. b = exp ( - ı ϕ / 2 ) sin θ / 2 b=\exp(-\imath\phi/2)\sin\theta/2
  85. E + E_{+}
  86. | E + = e ı ϕ / 2 ( cos θ / 2 e - ı ϕ sin θ / 2 ) |E_{+}\rangle=e^{\imath\phi/2}\begin{pmatrix}\cos\theta/2\\ e^{-\imath\phi}\sin\theta/2\end{pmatrix}
  87. | E + = ( cos θ / 2 e - ı ϕ sin θ / 2 ) = cos ( θ / 2 ) | 1 + e - ı ϕ sin ( θ / 2 ) | 2 |E_{+}\rangle=\begin{pmatrix}\cos\theta/2\\ e^{-\imath\phi}\sin\theta/2\end{pmatrix}=\cos(\theta/2)|1\rangle+e^{-\imath% \phi}\sin(\theta/2)|2\rangle
  88. E - E_{-}
  89. | E - = - e ı ϕ sin ( θ / 2 ) | 1 + cos ( θ / 2 ) | 2 |E_{-}\rangle=-e^{\imath\phi}\sin(\theta/2)|1\rangle+\cos(\theta/2)|2\rangle
  90. | 1 = cos ( θ / 2 ) | E + - sin ( θ / 2 ) e - ı ϕ | E - |1\rangle=\cos(\theta/2)|E_{+}\rangle-\sin(\theta/2)e^{-\imath\phi}|E_{-}\rangle
  91. | 2 = e ı ϕ sin ( θ / 2 ) | E + + cos ( θ / 2 ) | E - |2\rangle=e^{\imath\phi}\sin(\theta/2)|E_{+}\rangle+\cos(\theta/2)|E_{-}\rangle
  92. | 1 |1\rangle
  93. | ψ ( 0 ) = | 1 = cos ( θ / 2 ) | E + - sin ( θ / 2 ) e - ı ϕ | E - |\psi(0)\rangle=|1\rangle=\cos(\theta/2)|E_{+}\rangle-\sin(\theta/2)e^{-\imath% \phi}|E_{-}\rangle
  94. | ψ ( t ) = e - i E t | ψ ( 0 ) = cos ( θ / 2 ) e - ı E + t | E + - sin ( θ / 2 ) e - ı ϕ e - ı E - t | E - |\psi(t)\rangle=e^{\frac{-iEt}{\hbar}}|\psi(0)\rangle=\cos(\theta/2)e^{\frac{-% \imath E_{+}t}{\hbar}}|E_{+}\rangle-\sin(\theta/2)e^{-\imath\phi}e^{\frac{-% \imath E_{-}t}{\hbar}}|E_{-}\rangle
  95. | 1 |1\rangle
  96. | 2 |2\rangle
  97. | 2 |2\rangle
  98. | 2 | ψ ( t ) | = e - ı ϕ sin ( θ / 2 ) cos ( θ / 2 ) ( e - ı E + t - e - ı E - t ) |\langle\ 2|\psi(t)\rangle|=e^{-\imath\phi}\sin(\theta/2)\cos(\theta/2)(e^{% \frac{-\imath E_{+}t}{\hbar}}-e^{\frac{-\imath E_{-}t}{\hbar}})
  99. | ψ ( t ) |\psi(t)\rangle
  100. | 2 |2\rangle
  101. P 1 2 ( t ) = | 2 | ψ ( t ) | 2 = e + ı ϕ sin ( θ / 2 ) cos ( θ / 2 ) ( e + ı E + t - e + ı E t ) e - ı ϕ sin ( θ / 2 ) cos ( θ / 2 ) ( e - ı E + t - e i ı E t ) = sin 2 θ 4 ( 2 - 2 cos ( ( E + - E - ) t 2 ) ) P_{1}\to_{2}(t)={|\langle\ 2|\psi(t)\rangle|}^{2}=e^{+\imath\phi}\sin(\theta/2% )\cos(\theta/2)(e^{\frac{+\imath E_{+}t}{\hbar}}-e^{\frac{+\imath E_{t}}{\hbar% }})e^{-\imath\phi}\sin(\theta/2)\cos(\theta/2)(e^{\frac{-\imath E_{+}t}{\hbar}% }-e^{\frac{i\imath E_{t}}{\hbar}})=\frac{\sin^{2}\theta}{4}(2-2\cos(\frac{(E_{% +}-E_{-})t}{2\hbar}))
  102. P 1 2 ( t ) = sin 2 ( θ ) sin 2 ( ( E + - E - ) t 2 ) = | W | 2 Δ 2 + | W | 2 sin 2 ( ( E + - E - ) t 2 ) P_{1}\to_{2}(t)=\sin^{2}(\theta)\sin^{2}(\frac{(E_{+}-E_{-})t}{2\hbar})=\frac{% {\left|W\right|}^{2}}{{\Delta}^{2}+{\left|W\right|}^{2}}\sin^{2}(\frac{(E_{+}-% E_{-})t}{2\hbar})
  103. | 2 |2\rangle
  104. | 1 |1\rangle
  105. ω = E + - E - 2 = Δ 2 + | W | 2 \omega=\frac{E_{+}-E_{-}}{2\hbar}=\frac{\sqrt{{\Delta}^{2}+{\left|W\right|}^{2% }}}{\hbar}
  106. | 1 |1\rangle
  107. | + X | ψ ( t ) | 2 = 1 - sin 2 ( θ ) sin 2 ( ( E + - E - ) t 2 ) {|\langle\ +X|\psi(t)\rangle|}^{2}=1-\sin^{2}(\theta)\sin^{2}(\frac{(E_{+}-E_{% -})t}{2\hbar})
  108. 1 2 \frac{1}{2}
  109. s y m b o l μ symbol{\mu}
  110. s y m b o l B = B 0 z ^ + B 1 ( cos ω t x ^ - sin ω t y ^ ) symbol{B}=B_{0}\hat{z}+B_{1}(\cos\omega t\hat{x}-\sin\omega t\hat{y})
  111. γ \gamma
  112. s y m b o l μ = 2 γ s y m b o l σ symbol{\mu}=\frac{\hbar}{2}\gamma symbol{\sigma}
  113. 𝐇 = - s y m b o l μ 𝐁 = - 2 ω 0 σ z - 2 ω 1 ( σ x cos ω t - σ y sin ω t ) \mathbf{H}=-symbol{\mu}\cdot\mathbf{B}=-\frac{\hbar}{2}\omega_{0}\sigma_{z}-% \frac{\hbar}{2}\omega_{1}(\sigma_{x}\cos\omega t-\sigma_{y}\sin\omega t)
  114. ω 0 = γ B 0 \omega_{0}=\gamma B_{0}
  115. ω 1 = γ B 1 \omega_{1}=\gamma B_{1}
  116. | 1 |1\rangle
  117. t = 0 t=0
  118. t t
  119. | 2 |2\rangle
  120. P 1 2 ( t ) = ( ω 1 Ω ) 2 sin 2 ( Ω t 2 ) P_{1\to 2}(t)=\left(\frac{\omega_{1}}{\Omega}\right)^{2}\sin^{2}\left(\frac{% \Omega t}{2}\right)
  121. Ω = ( ω - ω 0 ) 2 + ω 1 2 \Omega=\sqrt{(\omega-\omega_{0})^{2}+\omega_{1}^{2}}
  122. | 1 |1\rangle
  123. | 2 |2\rangle
  124. ω = ω 0 \omega=\omega_{0}
  125. P 1 2 ( t ) = sin 2 ( ω t 2 ) P_{1\to 2}(t)=\sin^{2}\left(\frac{\omega t}{2}\right)
  126. | 1 |1\rangle
  127. | 2 |2\rangle
  128. t t
  129. ω 1 t 2 = π 2 \frac{\omega_{1}t}{2}=\frac{\pi}{2}
  130. t = π ω 1 t=\frac{\pi}{\omega_{1}}
  131. π \pi
  132. π ω 1 \frac{\pi}{\omega_{1}}
  133. | 1 |1\rangle
  134. | 2 |2\rangle
  135. t = π 2 ω 1 t=\frac{\pi}{2\omega_{1}}
  136. π 2 \frac{\pi}{2}
  137. | 1 | 1 + | 2 2 |1\rangle\to\frac{|1\rangle+|2\rangle}{\sqrt{2}}
  138. ω 0 \hbar\omega_{0}
  139. ω \omega
  140. ω 1 \omega_{1}
  141. d \vec{d}
  142. E \vec{E}
  143. ω 1 d E \omega_{1}\propto\vec{d}\cdot\vec{E}\hbar

Rabin_cryptosystem.html

  1. p q 3 ( mod 4 ) p\equiv q\equiv 3\;\;(\mathop{{\rm mod}}4)
  2. n = p q n=p\cdot q
  3. p = 7 p=7
  4. q = 11 q=11
  5. n = 77 n=77
  6. P = { 0 , , n - 1 } P=\{0,\dots,n-1\}
  7. m P m\in P
  8. c c
  9. c = m 2 mod n c=m^{2}\,\bmod\,n
  10. P = { 0 , , 76 } P=\{0,\dots,76\}
  11. m = 20 m=20
  12. c = m 2 mod n = 400 mod 77 = 15 c=m^{2}\,\bmod\,n=400\,\bmod\,77=15
  13. m { 13 , 20 , 57 , 64 } m\in\{13,20,57,64\}
  14. m { 0 , , n - 1 } m\in\{0,\dots,n-1\}
  15. m 2 c ( mod n ) m^{2}\equiv c\;\;(\mathop{{\rm mod}}n)
  16. n = p q n=p\cdot q
  17. n n
  18. m p = c mod p m_{p}=\sqrt{c}\,\bmod\,p
  19. m q = c mod q m_{q}=\sqrt{c}\,\bmod\,q
  20. m p = 1 m_{p}=1
  21. m q = 9 m_{q}=9
  22. y p y_{p}
  23. y q y_{q}
  24. y p p + y q q = 1 y_{p}\cdot p+y_{q}\cdot q=1
  25. y p = - 3 y_{p}=-3
  26. y q = 2 y_{q}=2
  27. + r +r
  28. - r -r
  29. + s +s
  30. - s -s
  31. c + n / n c+n\mathbb{Z}\in\mathbb{Z}/n\mathbb{Z}
  32. / n \mathbb{Z}/n\mathbb{Z}
  33. { 0 , , n - 1 } \{0,\dots,n-1\}
  34. r = ( y p p m q + y q q m p ) mod n - r = n - r s = ( y p p m q - y q q m p ) mod n - s = n - s \begin{matrix}r&=&(y_{p}\cdot p\cdot m_{q}+y_{q}\cdot q\cdot m_{p})\,\bmod\,n% \\ -r&=&n-r\\ s&=&(y_{p}\cdot p\cdot m_{q}-y_{q}\cdot q\cdot m_{p})\,\bmod\,n\\ -s&=&n-s\end{matrix}
  35. mod n \mod\,n
  36. m { 64 , 𝟐𝟎 , 13 , 57 } m\in\{64,\mathbf{20},13,57\}
  37. r r
  38. s s
  39. n n
  40. gcd ( | r - s | , n ) = p \gcd(|r-s|,n)=p
  41. gcd ( | r - s | , n ) = q \gcd(|r-s|,n)=q
  42. gcd \gcd
  43. n n
  44. r r
  45. s s
  46. 57 57
  47. 13 13
  48. r r
  49. s s
  50. gcd ( 57 - 13 , 77 ) = gcd ( 44 , 77 ) = 11 = q \gcd(57-13,77)=\gcd(44,77)=11=q
  51. p q 3 ( mod 4 ) p\equiv q\equiv 3\;\;(\mathop{{\rm mod}}4)
  52. m p = c 1 4 ( p + 1 ) mod p m_{p}=c^{\frac{1}{4}(p+1)}\,\bmod\,p
  53. m q = c 1 4 ( q + 1 ) mod q m_{q}=c^{\frac{1}{4}(q+1)}\,\bmod\,q
  54. p 3 ( mod 4 ) p\equiv 3\!\!\!\;\;(\mathop{{\rm mod}}4)
  55. m p 2 c 1 2 ( p + 1 ) c c 1 2 ( p - 1 ) c ( c p ) ( mod p ) , m_{p}^{2}\equiv c^{\frac{1}{2}(p+1)}\equiv c\cdot c^{\frac{1}{2}(p-1)}\equiv c% \cdot\left({c\over p}\right)\;\;(\mathop{{\rm mod}}p),
  56. ( c p ) \left({c\over p}\right)
  57. c m 2 ( mod p q ) c\equiv m^{2}\;\;(\mathop{{\rm mod}}pq)
  58. c m 2 ( mod p ) c\equiv m^{2}\;\;(\mathop{{\rm mod}}p)
  59. ( c p ) = 1 \left({c\over p}\right)=1
  60. m p 2 c ( mod p ) . m_{p}^{2}\equiv c\;\;(\mathop{{\rm mod}}p).
  61. p 3 ( mod 4 ) p\equiv 3\;\;(\mathop{{\rm mod}}4)
  62. mod p \mod p
  63. mod q \mod q
  64. n = 77 n=77

Radar_gun.html

  1. v = Δ f f c 2 {v}=\frac{\Delta f}{f}\frac{c}{2}\,
  2. Δ f = 2 f v c {\Delta f}=\frac{2{f}{v}}{c}\,

Radiance.html

  1. L e , Ω = 2 Φ e Ω A cos θ , L_{\mathrm{e},\Omega}=\frac{\partial^{2}\Phi_{\mathrm{e}}}{\partial\Omega\,% \partial A\cos\theta},
  2. L e , Ω , ν = L e , Ω ν , L_{\mathrm{e},\Omega,\nu}=\frac{\partial L_{\mathrm{e},\Omega}}{\partial\nu},
  3. L e , Ω , λ = L e , Ω λ , L_{\mathrm{e},\Omega,\lambda}=\frac{\partial L_{\mathrm{e},\Omega}}{\partial% \lambda},
  4. L e , Ω = n 2 Φ e G , L_{\mathrm{e},\Omega}=n^{2}\frac{\partial\Phi_{\mathrm{e}}}{\partial G},
  5. L e , Ω * = L e , Ω n 2 L_{\mathrm{e},\Omega}^{*}=\frac{L_{\mathrm{e},\Omega}}{n^{2}}

Radiation_resistance.html

  1. P = I 2 R P=I^{2}R\,
  2. I I
  3. P P
  4. R = P I 2 R=\frac{P}{I^{2}}\,

Radiative_forcing.html

  1. Δ T s = λ Δ F \Delta T_{s}=~{}\lambda~{}\Delta F
  2. Δ F = 5.35 × ln C C 0 W m - 2 \Delta F=5.35\times\ln{C\over C_{0}}~{}\mathrm{W}~{}\mathrm{m}^{-2}\,

Radical_of_a_ring.html

  1. 𝒵 ( R R ) \mathcal{Z}(_{R}R)\,
  2. N / 𝒵 ( R R ) = 𝒵 ( R / 𝒵 ( R R ) R / 𝒵 ( R R ) ) N/\mathcal{Z}(_{R}R)=\mathcal{Z}(_{R/\mathcal{Z}(_{R}R)}R/\mathcal{Z}(_{R}R))\,
  3. 𝒵 2 ( R R ) \mathcal{Z}_{2}(_{R}R)

Radius_of_curvature_(applications).html

  1. γ : n \gamma:\mathbb{R}\rightarrow\mathbb{R}^{n}
  2. n \mathbb{R}^{n}
  3. ρ : \rho:\mathbb{R}\rightarrow\mathbb{R}
  4. ρ = | γ | 3 | γ | 2 | γ ′′ | 2 - ( γ γ ′′ ) 2 \rho=\frac{|\gamma^{\prime}|^{3}}{\sqrt{|\gamma^{\prime}|^{2}\;|\gamma^{\prime% \prime}|^{2}-(\gamma^{\prime}\cdot\gamma^{\prime\prime})^{2}}}
  5. \mathbb{R}
  6. \mathbb{R}
  7. γ ( t ) = ( t , f ( t ) ) \gamma(t)=(t,f(t))
  8. ρ ( t ) = | 1 + f 2 ( t ) | 3 / 2 | f ( t ) | . \rho(t)=\frac{|1+f^{\prime 2}(t)|^{3/2}}{|f^{\prime}{}^{\prime}(t)|}.
  9. γ \gamma
  10. t t
  11. ρ \rho
  12. γ \gamma
  13. t t
  14. γ ( t ) \gamma(t)
  15. γ ( t ) \gamma^{\prime}(t)
  16. γ ′′ ( t ) \gamma^{\prime\prime}(t)
  17. | γ 2 ( t ) | |\gamma^{\prime 2}(t)|\,\!
  18. | γ ′′ 2 ( t ) | |\gamma^{\prime\prime 2}(t)|\,\!
  19. γ ( t ) γ ′′ ( t ) \gamma^{\prime}(t)\cdot\gamma^{\prime\prime}(t)
  20. n \mathbb{R}^{n}
  21. g ( u ) = A cos ( h ( u ) ) + B sin ( h ( u ) ) + C g(u)=A\cos(h(u))+B\sin(h(u))+C\,\!
  22. C n C\in\mathbb{R}^{n}
  23. A , B n A,B\in\mathbb{R}^{n}
  24. ρ \rho
  25. A A = B B = ρ 2 A\cdot A=B\cdot B=\rho^{2}
  26. A B = 0 A\cdot B=0
  27. h : h:\mathbb{R}\rightarrow\mathbb{R}
  28. | g | 2 = ρ 2 ( h ) 2 g g ′′ = ρ 2 h h ′′ | g ′′ | 2 = ρ 2 ( ( h ) 4 + ( h ′′ ) 2 ) \begin{array}[]{lll}|g^{\prime}|^{2}&=&\rho^{2}(h^{\prime})^{2}\\ g^{\prime}\cdot g^{\prime\prime}&=&\rho^{2}h^{\prime}h^{\prime\prime}\\ |g^{\prime\prime}|^{2}&=&\rho^{2}\left((h^{\prime})^{4}+(h^{\prime\prime})^{2}% \right)\end{array}
  29. γ \gamma
  30. | γ 2 ( t ) | = ρ 2 h 2 ( t ) γ ( t ) γ ′′ ( t ) = ρ 2 h ( t ) h ′′ ( t ) | γ ′′ 2 ( t ) | = ρ 2 ( h 4 ( t ) + h ′′ 2 ( t ) ) \begin{array}[]{lll}|\gamma^{\prime 2}(t)|&=&\rho^{2}h^{\prime 2}(t)\\ \gamma^{\prime}(t)\cdot\gamma^{\prime\prime}(t)&=&\rho^{2}h^{\prime}(t)h^{% \prime\prime}(t)\\ |\gamma^{\prime\prime 2}(t)|&=&\rho^{2}(h^{\prime 4}(t)+h^{\prime\prime 2}(t))% \end{array}
  31. ρ \rho
  32. h ( t ) h^{\prime}(t)
  33. h ′′ ( t ) h^{\prime\prime}(t)
  34. ρ \rho
  35. ρ ( t ) = | γ 3 ( t ) | | γ 2 ( t ) | | γ ′′ 2 ( t ) | - ( γ ( t ) γ ′′ ( t ) ) 2 \rho(t)=\frac{|\gamma^{\prime 3}(t)|}{\sqrt{|\gamma^{\prime 2}(t)|\;|\gamma^{% \prime\prime 2}(t)|-(\gamma^{\prime}(t)\cdot\gamma^{\prime\prime}(t))^{2}}}
  36. ρ = | γ | 3 | γ | 2 | γ ′′ | 2 - ( γ γ ′′ ) 2 \rho=\frac{|\gamma^{\prime}|^{3}}{\sqrt{|\gamma^{\prime}|^{2}\;|\gamma^{\prime% \prime}|^{2}-(\gamma^{\prime}\cdot\gamma^{\prime\prime})^{2}}}

Radon_measure.html

  1. 𝒦 ( X ) = C C ( X ) \mathcal{K}(X)=C_{C}(X)
  2. 𝒦 ( X ) \mathcal{K}(X)
  3. 𝒦 ( X , K ) \mathcal{K}(X,K)
  4. 𝒦 ( X , K ) \mathcal{K}(X,K)
  5. 𝒦 ( X ) \mathcal{K}(X)
  6. 𝒦 ( X , K ) \mathcal{K}(X,K)
  7. X , X,
  8. I : f f d m I:f\mapsto\int f\,dm
  9. 𝒦 ( X ) \mathcal{K}(X)
  10. | I ( f ) | M K sup x X | f ( x ) | . |I(f)|\leq M_{K}\sup_{x\in X}|f(x)|.
  11. 𝒦 ( X ) \mathcal{K}(X)
  12. 𝒦 ( X ) \mathcal{K}(X)
  13. 𝒦 ( X ) \mathcal{K}(X)
  14. 𝒦 ( X ) \mathcal{K}(X)
  15. 𝒦 ( X ) \mathcal{K}(X)
  16. n \mathbb{R}^{n}
  17. { [ a , b ) : 0 a < b 1 } \{[a,b):0\leq a<b\leq 1\}
  18. [ 0 , 1 ] [0,1]
  19. ( 0 , 1 ) κ (0,1)^{\kappa}
  20. κ \kappa
  21. M ( B ) = inf { m ( V ) V is an open set with B V X } . M(B)=\inf\{m(V)\mid V\,\text{ is an open set with }B\subseteq V\subseteq X\}.
  22. + ( X ) \mathcal{M}_{+}(X)
  23. X X
  24. m 1 , m 2 + ( X ) m_{1},m_{2}\in\mathcal{M}_{+}(X)
  25. ρ ( m 1 , m 2 ) := sup { X f ( x ) d ( m 1 - m 2 ) ( x ) | continuous f : X [ - 1 , 1 ] } . \rho(m_{1},m_{2}):=\sup\left\{\left.\int_{X}f(x)\,d(m_{1}-m_{2})(x)\ \right|% \mathrm{continuous\,}f:X\to[-1,1]\subset\mathbb{R}\right\}.
  26. X X
  27. 𝒫 ( X ) := { m + ( X ) m ( X ) = 1 } , \mathcal{P}(X):=\{m\in\mathcal{M}_{+}(X)\mid m(X)=1\},
  28. X X
  29. 𝒫 ( X ) \mathcal{P}(X)
  30. ρ ( m n , m ) 0 m n m , \rho(m_{n},m)\to 0\Rightarrow m_{n}\rightharpoonup m,

Radon_transform.html

  1. R f ( L ) = L f ( 𝐱 ) | d 𝐱 | . Rf(L)=\int_{L}f(\mathbf{x})\,|d\mathbf{x}|.
  2. ( x ( z ) , y ( z ) ) = ( ( z sin α + s cos α ) , ( - z cos α + s sin α ) ) (x(z),y(z))=\Big((z\sin\alpha+s\cos\alpha),(-z\cos\alpha+s\sin\alpha)\Big)\,
  3. α \alpha
  4. R f ( α , s ) = - f ( x ( z ) , y ( z ) ) d z = - f ( ( z sin α + s cos α ) , ( - z cos α + s sin α ) ) d z \begin{aligned}\displaystyle Rf(\alpha,s)&\displaystyle=\int_{-\infty}^{\infty% }f(x(z),y(z))\,dz\\ &\displaystyle=\int_{-\infty}^{\infty}f\big((z\sin\alpha+s\cos\alpha),(-z\cos% \alpha+s\sin\alpha)\big)\,dz\end{aligned}
  5. R f ( ξ ) = ξ f ( 𝐱 ) d σ ( 𝐱 ) Rf(\xi)=\int_{\xi}f(\mathbf{x})\,d\sigma(\mathbf{x})
  6. 𝐱 α = s \mathbf{x}\cdot\alpha=s
  7. R f ( α , s ) = 𝐱 α = s f ( 𝐱 ) d σ ( 𝐱 ) . Rf(\alpha,s)=\int_{\mathbf{x}\cdot\alpha=s}f(\mathbf{x})\,d\sigma(\mathbf{x}).
  8. f ^ ( ω ) = - f ( x ) e - 2 π i x ω d x . \hat{f}(\omega)=\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\omega}\,dx.
  9. 𝐱 = ( x , y ) \mathbf{x}=(x,y)
  10. f ^ ( 𝐰 ) = - - f ( 𝐱 ) e - 2 π i 𝐱 𝐰 d x d y . \hat{f}(\mathbf{w})=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{% \infty}f(\mathbf{x})e^{-2\pi i\mathbf{x}\cdot\mathbf{w}}\,dx\,dy.
  11. R α [ f ] ( s ) = R [ f ] ( α , s ) R_{\alpha}[f](s)=R[f](\alpha,s)
  12. s s
  13. R α [ f ] ^ ( σ ) = f ^ ( σ 𝐧 ( α ) ) \widehat{R_{\alpha}[f]}(\sigma)=\hat{f}(\sigma\mathbf{n}(\alpha))
  14. 𝐧 ( α ) = ( cos α , sin α ) . \mathbf{n}(\alpha)=(\cos\alpha,\sin\alpha).
  15. f ^ ( r α ) = - R f ( α , s ) e - 2 π i s r d s . \hat{f}(r\alpha)=\int_{-\infty}^{\infty}Rf(\alpha,s)e^{-2\pi isr}\,ds.
  16. f ^ ( r α ) = - d s 𝐱 α = s e - 2 π i r ( 𝐱 α ) d m ( 𝐱 ) . \hat{f}(r\alpha)=\int_{-\infty}^{\infty}ds\int_{\mathbf{x}\cdot\alpha=s}e^{-2% \pi ir(\mathbf{x}\cdot\alpha)}dm(\mathbf{x}).
  17. R * g ( x ) = x ξ g ( ξ ) d μ ( ξ ) . R^{*}g(x)=\int_{x\in\xi}g(\xi)\,d\mu(\xi).
  18. { ξ | x ξ } \{\xi|x\in\xi\}
  19. R * g ( x ) = 1 2 π α = 0 2 π g ( α , 𝐧 ( α ) 𝐱 ) d α . R^{*}g(x)=\frac{1}{2\pi}\int_{\alpha=0}^{2\pi}g(\alpha,\mathbf{n}(\alpha)\cdot% \mathbf{x})\,d\alpha.
  20. Δ = 2 x 1 2 + + 2 x n 2 . \Delta=\frac{\partial^{2}}{\partial x_{1}^{2}}+\cdots+\frac{\partial^{2}}{% \partial x_{n}^{2}}.
  21. L f ( α , s ) 2 s 2 f ( α , s ) Lf(\alpha,s)\equiv\frac{\partial^{2}}{\partial s^{2}}f(\alpha,s)
  22. R ( Δ f ) = L ( R f ) , R * ( L g ) = Δ ( R * g ) . R(\Delta f)=L(Rf),\quad R^{*}(Lg)=\Delta(R^{*}g).
  23. c n f = ( - Δ ) ( n - 1 ) / 2 R * R f c_{n}f=(-\Delta)^{(n-1)/2}R^{*}Rf\,
  24. c n = ( 4 π ) ( n - 1 ) / 2 Γ ( n / 2 ) Γ ( 1 / 2 ) . c_{n}=(4\pi)^{(n-1)/2}\frac{\Gamma(n/2)}{\Gamma(1/2)}.
  25. [ ( - Δ ) ( n - 1 ) / 2 ϕ ] ( ξ ) = | 2 π ξ | n - 1 ϕ ( ξ ) . \mathcal{F}\left[(-\Delta)^{(n-1)/2}\phi\right](\xi)=|2\pi\xi|^{n-1}\mathcal{F% }\phi(\xi).
  26. c n f = { R * d n - 1 d s n - 1 R f n odd R * H s d n - 1 d s n - 1 R f n even c_{n}f=\begin{cases}R^{*}\frac{d^{n-1}}{ds^{n-1}}Rf&n\rm{\ odd}\\ R^{*}H_{s}\frac{d^{n-1}}{ds^{n-1}}Rf&n\rm{\ even}\end{cases}
  27. f = 1 2 R * H s d d s R f . f=\frac{1}{2}R^{*}H_{s}\frac{d}{ds}Rf.
  28. s s
  29. f ( x ) = 1 2 ( 2 π ) 1 - n ( - 1 ) ( n - 1 ) / 2 S n - 1 n - 1 s n - 1 R f ( α , α x ) d α f(x)=\frac{1}{2}(2\pi)^{1-n}(-1)^{(n-1)/2}\int_{S^{n-1}}\frac{\partial^{n-1}}{% \partial s^{n-1}}Rf(\alpha,\alpha\cdot x)\,d\alpha
  30. f ( x ) = ( 2 π ) - n ( - 1 ) n / 2 - 1 q S n - 1 n - 1 s n - 1 R f ( α , α x + q ) d α d q f(x)=(2\pi)^{-n}(-1)^{n/2}\int_{-\infty}^{\infty}\frac{1}{q}\int_{S^{n-1}}% \frac{\partial^{n-1}}{\partial s^{n-1}}Rf(\alpha,\alpha\cdot x+q)\,d\alpha\,dq
  31. c n g = ( - L ) ( n - 1 ) / 2 R ( R * g ) . c_{n}g=(-L)^{(n-1)/2}R(R^{*}g).\,

Radon–Nikodym_theorem.html

  1. ( X , Σ ) (X,\Sigma)
  2. ν \nu
  3. ( X , Σ ) (X,\Sigma)
  4. μ μ
  5. ( X , Σ ) (X,\Sigma)
  6. f : X [ 0 , ) f:X\rightarrow[0,\infty)
  7. A X A\subset X
  8. ν ( A ) = A f d μ \nu(A)=\int_{A}f\,d\mu
  9. f f
  10. d ν d μ \frac{d\nu}{d\mu}
  11. Y Y
  12. Y Y
  13. Y Y
  14. f f
  15. μ μ
  16. g g
  17. f = g f=g
  18. μ μ
  19. f f
  20. d ν d μ \scriptstyle\frac{d\nu}{d\mu}
  21. μ μ
  22. μ μ
  23. μ μ
  24. g g
  25. X X
  26. A A
  27. ν ( A ) = A g d μ . \nu(A)=\int_{A}g\,d\mu.
  28. d ( ν + μ ) d λ = d ν d λ + d μ d λ λ -almost everywhere . \frac{d(\nu+\mu)}{d\lambda}=\frac{d\nu}{d\lambda}+\frac{d\mu}{d\lambda}\quad% \lambda\,\text{-almost everywhere}.
  29. d ν d λ = d ν d μ d μ d λ λ -almost everywhere . \frac{d\nu}{d\lambda}=\frac{d\nu}{d\mu}\frac{d\mu}{d\lambda}\quad\lambda\,% \text{-almost everywhere}.
  30. d μ d ν = ( d ν d μ ) - 1 ν -almost everywhere . \frac{d\mu}{d\nu}=\left(\frac{d\nu}{d\mu}\right)^{-1}\quad\nu\,\text{-almost % everywhere}.
  31. g g
  32. X g d μ = X g d μ d λ d λ . \int_{X}g\,d\mu=\int_{X}g\frac{d\mu}{d\lambda}\,d\lambda.
  33. d | ν | d μ = | d ν d μ | . {d|\nu|\over d\mu}=\left|{d\nu\over d\mu}\right|.
  34. X X
  35. D KL ( μ ν ) = X log ( d μ d ν ) d μ . D_{\mathrm{KL}}(\mu\|\nu)=\int_{X}\log\left(\frac{d\mu}{d\nu}\right)\;d\mu.
  36. D α ( μ ν ) = 1 α - 1 log ( X ( d μ d ν ) α - 1 d μ ) . D_{\alpha}(\mu\|\nu)=\frac{1}{\alpha-1}\log\left(\int_{X}\left(\frac{d\mu}{d% \nu}\right)^{\alpha-1}\;d\mu\right).
  37. μ μ
  38. A A
  39. A A
  40. A A
  41. μ μ
  42. σ σ
  43. ν ν
  44. ν ν
  45. μ μ
  46. A A
  47. μ ( A ) = 0 μ(A)=0
  48. A A
  49. ν ( A ) ν(A)
  50. f f
  51. ν ( A ) = A f d μ \nu(A)=\int_{A}f\,d\mu
  52. A A
  53. 0 = f ( a ) 0=f(a)
  54. a a
  55. f f
  56. ν ν
  57. μ μ
  58. ν ν
  59. f f
  60. f d μ d ν f dμ≤dν
  61. μ μ
  62. ν ν
  63. σ σ
  64. μ μ
  65. ν ν
  66. F F
  67. f : X 0 , ) ) f:X→0,∞))
  68. A Σ : A f d μ ν ( A ) \forall A\in\Sigma:\qquad\int_{A}f\,d\mu\leq\nu(A)
  69. F F≠∅
  70. A A
  71. A 1 \displaystyle A_{1}
  72. A max { f 1 , f 2 } d μ = A 1 f 1 d μ + A 2 f 2 d μ ν ( A 1 ) + ν ( A 2 ) = ν ( A ) , \int_{A}\max\{f_{1},f_{2}\}\,d\mu=\int_{A_{1}}f_{1}\,d\mu+\int_{A_{2}}f_{2}\,d% \mu\leq\nu(A_{1})+\nu(A_{2})=\nu(A),
  73. F F
  74. lim n X f n d μ = sup f F X f d μ . \lim_{n\to\infty}\int_{X}f_{n}\,d\mu=\sup_{f\in F}\int_{X}f\,d\mu.
  75. n n
  76. g g
  77. g ( x ) := lim n f n ( x ) . g(x):=\lim_{n\to\infty}f_{n}(x).
  78. A g d μ = lim n A f n d μ ν ( A ) \int_{A}g\,d\mu=\lim_{n\to\infty}\int_{A}f_{n}\,d\mu\leq\nu(A)
  79. A Σ A∈Σ
  80. g F g∈F
  81. g g
  82. X g d μ = sup f F X f d μ . \int_{X}g\,d\mu=\sup_{f\in F}\int_{X}f\,d\mu.
  83. g F g∈F
  84. ν 0 ( A ) := ν ( A ) - A g d μ \nu_{0}(A):=\nu(A)-\int_{A}g\,d\mu
  85. Σ Σ
  86. μ μ
  87. ε > 0 ε>0
  88. A Σ A∈Σ
  89. ν ( A ) = A g d μ + ν 0 ( A ) A g d μ + ν 0 ( A P ) A g d μ + ε μ ( A P ) = A ( g + ε 1 P ) d μ . \begin{aligned}\displaystyle\nu(A)&\displaystyle=\int_{A}g\,d\mu+\nu_{0}(A)\\ &\displaystyle\geq\int_{A}g\,d\mu+\nu_{0}(A\cap P)\\ &\displaystyle\geq\int_{A}g\,d\mu+\varepsilon\mu(A\cap P)\\ &\displaystyle=\int_{A}(g+\varepsilon 1_{P})\,d\mu.\end{aligned}
  90. μ ( P ) > 0 μ(P)>0
  91. μ ( P ) = 0 μ(P)=0
  92. ν ν
  93. μ μ
  94. ν 0 ( X ) - ε μ ( X ) = ( ν 0 - ε μ ) ( N ) 0 , \nu_{0}(X)-\varepsilon\mu(X)=(\nu_{0}-\varepsilon\mu)(N)\leq 0,
  95. X ( g + ε 1 P ) d μ ν ( X ) < + , \int_{X}(g+\varepsilon 1_{P})\,d\mu\leq\nu(X)<+\infty,
  96. X ( g + ε 1 P ) d μ > X g d μ = sup f F X f d μ . \int_{X}(g+\varepsilon 1_{P})\,d\mu>\int_{X}g\,d\mu=\sup_{f\in F}\int_{X}f\,d\mu.
  97. g g
  98. μ μ
  99. μ μ
  100. f f
  101. f ( x ) = { g ( x ) if g ( x ) < 0 otherwise, f(x)=\begin{cases}g(x)&\,\text{if }g(x)<\infty\\ 0&\,\text{otherwise,}\end{cases}
  102. f f
  103. f , g : X 0 , ) ) f,g:X→0,∞))
  104. ν ( A ) = A f d μ = A g d μ \nu(A)=\int_{A}f\,d\mu=\int_{A}g\,d\mu
  105. A A
  106. g f g−f
  107. μ μ
  108. A ( g - f ) d μ = 0. \int_{A}(g-f)\,d\mu=0.
  109. μ μ
  110. f = g f=g
  111. μ μ
  112. σ σ
  113. μ μ
  114. ν ν
  115. σ σ
  116. X X
  117. Σ Σ
  118. μ μ
  119. ν ν
  120. n n
  121. Σ Σ
  122. ν ( A ) = A f n d μ \nu(A)=\int_{A}f_{n}\,d\mu
  123. Σ Σ
  124. A A
  125. B < s u b > n B<sub>n

Raman_scattering.html

  1. x x
  2. y y
  3. z z
  4. 3 N - 3 - 2 = 3 N - 5 3N-3-2=3N-5
  5. 3 N - 6 3N-6
  6. E E
  7. E = h ν E=h\nu
  8. h h
  9. ν \nu
  10. α / Q 0 \partial\alpha/\partial Q\neq 0
  11. α / Q 0 \partial\alpha/\partial Q\neq 0
  12. χ ( 3 ) \chi^{(3)}
  13. x x
  14. λ λ
  15. λ λ
  16. Θ = 2 π x ( 1 / λ 1 / λ ) ) Θ=2πx(1/λ−1/λ))
  17. Θ = π Θ=π
  18. x x

Ramanujan–Soldner_constant.html

  1. li ( x ) = 0 x d t ln t , \mathrm{li}(x)=\int_{0}^{x}\frac{dt}{\ln t},
  2. li ( x ) = li ( x ) - li ( μ ) \mathrm{li}(x)\;=\;\mathrm{li}(x)-\mathrm{li}(\mu)
  3. 0 x d t ln t = 0 x d t ln t - 0 μ d t ln t \int_{0}^{x}\frac{dt}{\ln t}=\int_{0}^{x}\frac{dt}{\ln t}-\int_{0}^{\mu}\frac{% dt}{\ln t}
  4. li ( x ) = μ x d t ln t , \mathrm{li}(x)=\int_{\mu}^{x}\frac{dt}{\ln t},
  5. li ( x ) = Ei ( ln x ) , \mathrm{li}(x)\;=\;\mathrm{Ei}(\ln{x}),

Ramification_(mathematics).html

  1. \to
  2. \to
  3. B / A B/A
  4. Tr : B A \mathrm{Tr}:B\to A
  5. f : X Y f:X\to Y
  6. Ω X / Y \Omega_{X/Y}
  7. f f
  8. f ( Supp Ω X / Y ) f\left(\mathrm{Supp}\Omega_{X/Y}\right)
  9. f f
  10. Ω X / Y = 0 \Omega_{X/Y}=0
  11. f f
  12. f f
  13. f f

Ramification_group.html

  1. G G
  2. L / K L/K
  3. w , 𝒪 L , 𝔭 w,\mathcal{O}_{L},\mathfrak{p}
  4. L L
  5. 𝒪 L = 𝒪 K [ α ] \mathcal{O}_{L}=\mathcal{O}_{K}[\alpha]
  6. α L \alpha\in L
  7. O K O_{K}
  8. K K
  9. i - 1 i\geq-1
  10. G i G_{i}
  11. s G s\in G
  12. s s
  13. 𝒪 L / 𝔭 i + 1 . \mathcal{O}_{L}/\mathfrak{p}^{i+1}.
  14. w ( s ( x ) - x ) i + 1 w(s(x)-x)\geq i+1
  15. x 𝒪 L x\in\mathcal{O}_{L}
  16. w ( s ( α ) - α ) i + 1. w(s(\alpha)-\alpha)\geq i+1.
  17. G i G_{i}
  18. i i
  19. G - 1 = G G 0 G 1 { * } . G_{-1}=G\supset G_{0}\supset G_{1}\supset\dots\{*\}.
  20. G i G_{i}
  21. i i
  22. G 0 G_{0}
  23. G G
  24. G 1 G_{1}
  25. G G
  26. G 0 / G 1 G_{0}/G_{1}
  27. G G
  28. G i G_{i}
  29. G / G 0 = Gal ( l / k ) , G/G_{0}=\operatorname{Gal}(l/k),
  30. l , k l,k
  31. L , K L,K
  32. G 0 = 1 L / K G_{0}=1\Leftrightarrow L/K
  33. G 1 = 1 L / K G_{1}=1\Leftrightarrow L/K
  34. G i = ( G 0 ) i G_{i}=(G_{0})_{i}
  35. i 0 i\geq 0
  36. i G ( s ) = w ( s ( α ) - α ) , s G i_{G}(s)=w(s(\alpha)-\alpha),s\in G
  37. i G i_{G}
  38. α \alpha
  39. G i G_{i}
  40. i G i_{G}
  41. i G i_{G}
  42. s , t G s,t\in G
  43. i G ( s ) i + 1 s G i . i_{G}(s)\geq i+1\Leftrightarrow s\in G_{i}.
  44. i G ( t s t - 1 ) = i G ( s ) . i_{G}(tst^{-1})=i_{G}(s).
  45. i G ( s t ) min { i G ( s ) , i G ( t ) } . i_{G}(st)\geq\min\{i_{G}(s),i_{G}(t)\}.
  46. π \pi
  47. L L
  48. s s ( π ) / π s\mapsto s(\pi)/\pi
  49. G i / G i + 1 U L , i / U L , i + 1 , i 0 G_{i}/G_{i+1}\to U_{L,i}/U_{L,i+1},i\geq 0
  50. U L , 0 = 𝒪 L × , U L , i = 1 + 𝔭 i U_{L,0}=\mathcal{O}_{L}^{\times},U_{L,i}=1+\mathfrak{p}^{i}
  51. G 0 / G 1 G_{0}/G_{1}
  52. p p
  53. G i / G i + 1 G_{i}/G_{i+1}
  54. p p
  55. G 1 G_{1}
  56. G G
  57. 𝔇 L / K \mathfrak{D}_{L/K}
  58. L / K L/K
  59. w ( 𝔇 L / K ) = s 1 i G ( s ) = 0 ( | G i | - 1 ) . w(\mathfrak{D}_{L/K})=\sum_{s\neq 1}i_{G}(s)=\sum_{0}^{\infty}(|G_{i}|-1).
  60. H H
  61. G G
  62. σ G \sigma\in G
  63. i G / H ( σ ) = 1 e L / K s σ i G ( s ) i_{G/H}(\sigma)={1\over e_{L/K}}\sum_{s\mapsto\sigma}i_{G}(s)
  64. F / K F/K
  65. H H
  66. v F ( 𝔇 F / K ) = 1 e L / F s H i G ( s ) . v_{F}(\mathfrak{D}_{F/K})={1\over e_{L/F}}\sum_{s\not\in H}i_{G}(s).
  67. s G i , t G j , i , j 1 s\in G_{i},t\in G_{j},i,j\geq 1
  68. s t s - 1 t - 1 G i + j + 1 sts^{-1}t^{-1}\in G_{i+j+1}
  69. gr ( G 1 ) = i 1 G i / G i + 1 \operatorname{gr}(G_{1})=\sum_{i\geq 1}G_{i}/G_{i+1}
  70. 2 + 2 \sqrt{2+\sqrt{2}\ }
  71. 2 - 2 \sqrt{2-\sqrt{2}\ }
  72. π \pi
  73. 2 \sqrt{2}
  74. π \pi
  75. π \pi
  76. π \pi
  77. 4 - 2 2 \sqrt{4-2\sqrt{2}\ }
  78. π \pi
  79. C 4 C_{4}
  80. G 0 G_{0}
  81. G 1 G_{1}
  82. G 2 G_{2}
  83. C 4 C_{4}
  84. G 3 G_{3}
  85. G 4 G_{4}
  86. w ( 𝔇 K / Q ) w(\mathfrak{D}_{K/Q})
  87. 𝔇 K / Q \mathfrak{D}_{K/Q}
  88. π \pi
  89. u u
  90. - 1 \geq-1
  91. G u G_{u}
  92. G i G_{i}
  93. u \geq u
  94. s G u i G ( s ) u + 1. s\in G_{u}\Leftrightarrow i_{G}(s)\geq u+1.
  95. ϕ \phi
  96. ϕ ( u ) = 0 u d t ( G 0 : G t ) \phi(u)=\int_{0}^{u}{dt\over(G_{0}:G_{t})}
  97. ( G 0 : G t ) (G_{0}:G_{t})
  98. ( G - 1 : G 0 ) - 1 (G_{-1}:G_{0})^{-1}
  99. t = - 1 t=-1
  100. 1 1
  101. - 1 < t 0 -1<t\leq 0
  102. ϕ ( u ) = u \phi(u)=u
  103. - 1 u 0 -1\leq u\leq 0
  104. ϕ \phi
  105. ψ \psi
  106. [ - 1 , ) [-1,\infty)
  107. G v = G ψ ( v ) G^{v}=G_{\psi(v)}
  108. G v G^{v}
  109. G ϕ ( u ) = G u G^{\phi(u)}=G_{u}
  110. G - 1 = G , G 0 = G 0 G^{-1}=G,G^{0}=G_{0}
  111. H H
  112. G G
  113. ( G / H ) v = G v H / H (G/H)^{v}=G^{v}H/H
  114. v v
  115. G u H / H = ( G / H ) v G_{u}H/H=(G/H)_{v}
  116. v = ϕ L / F ( u ) v=\phi_{L/F}(u)
  117. L / F L/F
  118. H H
  119. G u H / H = ( G / H ) u G^{u}H/H=(G/H)^{u}
  120. G G
  121. G v G^{v}
  122. G i = G i + 1 G_{i}=G_{i+1}
  123. ϕ ( i ) \phi(i)
  124. G n ( L / K ) G^{n}(L/K)
  125. G ( L / K ) ab K * / N L / K ( L * ) G(L/K)^{\mathrm{ab}}\leftrightarrow K^{*}/N_{L/K}(L^{*})
  126. U K n / ( U K n N L / K ( L * ) ) . U^{n}_{K}/(U^{n}_{K}\cap N_{L/K}(L^{*}))\ .
  127. G / G 0 G/G_{0}
  128. U L , 0 / U L , 1 l × U_{L,0}/U_{L,1}\simeq l^{\times}
  129. U L , i / U L , i + 1 l + U_{L,i}/U_{L,i+1}\approx l^{+}

Random_coil.html

  1. \scriptstyle\ell
  2. N × \scriptstyle N\,\times\,\ell
  3. P ( r ) = 4 π r 2 ( 2 / 3 π r 2 ) 3 / 2 e - 3 r 2 2 r 2 P(r)=\frac{4\pi r^{2}}{(2/3\;\pi\langle r^{2}\rangle)^{3/2}}\;e^{-\,\frac{3r^{% 2}}{2\langle r^{2}\rangle}}
  4. r 2 \scriptstyle\sqrt{\langle r^{2}\rangle}
  5. \scriptstyle\ell

Random_Fibonacci_sequence.html

  1. f n = { f n - 1 + f n - 2 , with probability 1/2 ; f n - 1 - f n - 2 , with probability 1/2 . f_{n}=\begin{cases}f_{n-1}+f_{n-2},&\,\text{ with probability 1/2};\\ f_{n-1}-f_{n-2},&\,\text{ with probability 1/2}.\end{cases}
  2. 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , . 1,1,2,3,5,8,13,21,34,55,\ldots.
  3. 1 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1 , . 1,1,0,1,1,0,1,1,0,1,\ldots.
  4. 1 , 1 , 2 , 3 , 1 , - 2 , - 3 , - 5 , - 2 , - 3 , for the signs + , + , + , - , - , + , - , - , . 1,1,2,3,1,-2,-3,-5,-2,-3,\ldots\,\text{ for the signs }+,+,+,-,-,+,-,-,\ldots.
  5. ( f n - 1 f n ) = ( 0 1 ± 1 1 ) ( f n - 2 f n - 1 ) , {f_{n-1}\choose f_{n}}=\begin{pmatrix}0&1\\ \pm 1&1\end{pmatrix}{f_{n-2}\choose f_{n-1}},
  6. ( f n - 1 f n ) = M n M n - 1 M 3 ( f 1 f 2 ) , {f_{n-1}\choose f_{n}}=M_{n}M_{n-1}\ldots M_{3}{f_{1}\choose f_{2}},
  7. A = ( 0 1 1 1 ) , B = ( 0 1 - 1 1 ) . A=\begin{pmatrix}0&1\\ 1&1\end{pmatrix},\quad B=\begin{pmatrix}0&1\\ -1&1\end{pmatrix}.
  8. φ = ( 1 + 5 ) / 2 , \varphi=(1+\sqrt{5})/2,
  9. F n = φ n - ( - 1 / φ ) n 5 . F_{n}={{\varphi^{n}-(-1/\varphi)^{n}}\over{\sqrt{5}}}.
  10. | f n | n 1.13198824 as n . \sqrt[n]{|f_{n}|}\to 1.13198824\dots\,\text{ as }n\to\infty.
  11. f n = f n - 1 ± β f n - 2 f_{n}=f_{n-1}\pm\beta f_{n-2}

Random_field.html

  1. ( Ω , , P ) (\Omega,\mathcal{F},P)
  2. { F t : t T } \{F_{t}:t\in T\}
  3. F t F_{t}
  4. P ( X i = x i | X j = x j , i j ) = P ( X i = x i | i ) , P(X_{i}=x_{i}|X_{j}=x_{j},i\neq j)=P(X_{i}=x_{i}|\partial_{i}),\,
  5. i \partial_{i}
  6. P ( X i = x i | i ) = P ( ω ) ω P ( ω ) , P(X_{i}=x_{i}|\partial_{i})=\frac{P(\omega)}{\sum_{\omega^{\prime}}P(\omega^{% \prime})},

Random_graph.html

  1. N = ( n 2 ) N={\textstyle\left({{n}\atop{2}}\right)}
  2. ( N M ) {\textstyle\left({{N}\atop{M}}\right)}
  3. 1 / ( N M ) 1/{\textstyle\left({{N}\atop{M}}\right)}
  4. G ~ n \tilde{G}_{n}
  5. a 1 , , a n , b 1 , , b m V a_{1},\ldots,a_{n},b_{1},\ldots,b_{m}\in V
  6. a 1 , , a n a_{1},\ldots,a_{n}
  7. b 1 , , b m b_{1},\ldots,b_{m}
  8. p i , j p_{i,j}
  9. e i , j e_{i,j}
  10. k \langle k\rangle
  11. p c = 1 k p_{c}=\tfrac{1}{\langle k\rangle}
  12. V n ( 2 ) = { i j : 1 j n , i j } V ( 2 ) , i = 1 , , n . V_{n}^{(2)}=\left\{ij\ :\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i% =1,\cdots,n.

Randomized_algorithm.html

  1. Θ ( 1 ) \Theta(1)
  2. Pr [ find a ] = 1 - ( 1 / 2 ) k \Pr[\mathrm{find~{}a}]=1-(1/2)^{k}
  3. Θ ( 1 ) \Theta(1)
  4. O ( k n 2 ) O(kn^{2})
  5. 1 - ( 1 2 ) k 1-(\frac{1}{2})^{k}
  6. A B C A\cdot B\neq C
  7. P r [ A B r = C r ] 1 / 2 Pr[A\cdot B\cdot r=C\cdot r]\leq 1/2
  8. A B C A\cdot B\neq C
  9. D = A B - C 0 D=A\cdot B-C\neq 0
  10. d 11 0 d_{11}\neq 0
  11. P r [ A B r = C r ] = P r [ ( A B - C ) r = 0 ] = P r [ D r = 0 ] Pr[A\cdot B\cdot r=C\cdot r]=Pr[(A\cdot B-C)\cdot r=0]=Pr[D\cdot r=0]
  12. D r = 0 D\cdot r=0
  13. D r D\cdot r
  14. j = 1 n d 1 j r j = 0 \sum_{j=1}^{n}d_{1j}r_{j}=0
  15. d 11 0 d_{11}\neq 0
  16. r 1 r_{1}
  17. r 1 = - j = 2 n d 1 j r j d 11 r_{1}=\frac{-\sum_{j=2}^{n}d_{1j}r_{j}}{d_{11}}
  18. r j r_{j}
  19. r 1 r_{1}
  20. r 1 { 0 , 1 } r_{1}\in\{0,1\}
  21. P r [ A B r = C r ] 1 / 2 Pr[ABr=Cr]\leq 1/2
  22. C = A B C=A\cdot B
  23. C A B C\neq A\cdot B
  24. 1 - ( 1 2 ) k 1-(\frac{1}{2})^{k}
  25. O ( n ) O(n)
  26. i = 1 m Pr ( C i C ) = i = 1 m ( 1 - Pr ( C i = C ) ) . \prod_{i=1}^{m}\Pr(C_{i}\neq C)=\prod_{i=1}^{m}(1-\Pr(C_{i}=C)).
  27. 1 - k | E ( G j ) | 1-\frac{k}{|E(G_{j})|}
  28. ( n - j ) k 2 \frac{(n-j)k}{2}
  29. 1 - k | E ( G j ) | 1 - 2 n - j = n - j - 2 n - j 1-\frac{k}{|E(G_{j})|}\geq 1-\frac{2}{n-j}=\frac{n-j-2}{n-j}
  30. P r [ C i = C ] ( n - 2 n ) ( n - 3 n - 1 ) ( n - 4 n - 2 ) ( 3 5 ) ( 2 4 ) ( 1 3 ) . Pr[C_{i}=C]\geq\left(\frac{n-2}{n}\right)\left(\frac{n-3}{n-1}\right)\left(% \frac{n-4}{n-2}\right)\ldots\left(\frac{3}{5}\right)\left(\frac{2}{4}\right)% \left(\frac{1}{3}\right).
  31. Pr [ C i = C ] 2 n ( n - 1 ) \Pr[C_{i}=C]\geq\frac{2}{n(n-1)}
  32. 1 - ( 1 - 2 n ( n - 1 ) ) m 1-\left(1-\frac{2}{n(n-1)}\right)^{m}
  33. m = n ( n - 1 ) 2 ln n m=\frac{n(n-1)}{2}\ln n
  34. 1 - 1 n 1-\frac{1}{n}
  35. 1 - 1 n 1-\frac{1}{n}
  36. O ( m n ) = O ( n 3 log n ) O(mn)=O(n^{3}\log n)
  37. log n \log n
  38. Θ ( n ) \Theta(n)

RANDU.html

  1. V j + 1 = 65539 V j mod 2 31 V_{j+1}=65539\cdot V_{j}\,\bmod\,2^{31}\,
  2. V 0 \scriptstyle V_{0}
  3. V j \scriptstyle V_{j}
  4. X j \scriptstyle X_{j}
  5. X j = V j 2 31 X_{j}=\frac{V_{j}}{2^{31}}
  6. 65539 65539
  7. ( i.e., 2 16 + 3 ) (\,\text{i.e., }2^{16}+3)
  8. x k + 2 = ( 2 16 + 3 ) x k + 1 = ( 2 16 + 3 ) 2 x k x_{k+2}=(2^{16}+3)x_{k+1}=(2^{16}+3)^{2}x_{k}\,
  9. x k + 2 = ( 2 32 + 6 2 16 + 9 ) x k = [ 6 ( 2 16 + 3 ) - 9 ] x k x_{k+2}=(2^{32}+6\cdot 2^{16}+9)x_{k}=[6\cdot(2^{16}+3)-9]x_{k}\,
  10. x k + 2 = 6 x k + 1 - 9 x k x_{k+2}=6x_{k+1}-9x_{k}\,
  11. V 0 = 1 \scriptstyle V_{0}=1

Rankine–Hugoniot_conditions.html

  1. ρ 1 u s = ρ 2 ( u s - u 2 ) \displaystyle\rho_{1}\,u_{s}=\rho_{2}(u_{s}-u_{2})
  2. e 2 - e 1 = 1 2 ( p 2 + p 1 ) ( v 1 - v 2 ) e_{2}-e_{1}=\tfrac{1}{2}\,(p_{2}+p_{1})\,(v_{1}-v_{2})
  3. ( 1 ) ρ t = - x ( ρ u ) (\;1)\quad\quad\frac{\partial\rho}{\partial t}\;\;=-\frac{\partial}{\partial x% }\left(\rho u\right)
  4. ( 2 ) t ( ρ u ) = - x ( ρ u 2 + p ) (\;2)\quad\quad\frac{\partial}{\partial t}(\rho u)\,=-\frac{\partial}{\partial x% }\left(\rho u^{2}+p\right)
  5. ( 3 ) t ( E t ) = - x [ u ( E t + p ) ] , (\;3)\quad\quad\frac{\partial}{\partial t}(E^{t})=-\frac{\partial}{\partial x}% \left[u\left(E^{t}+p\right)\right],
  6. ρ = \rho=\,
  7. u = u=\,
  8. e = e=\,
  9. p = p=\,
  10. t = t=\,
  11. x = x=\,
  12. E t = ρ e + ρ 1 2 u 2 , E^{t}=\rho e+\rho\frac{1}{2}u^{2},
  13. ( 4 ) p = ( γ - 1 ) ρ e , (\;4)\quad\quad p=\left(\gamma-1\right)\rho e,
  14. γ \gamma
  15. c p / c v c_{p}/c_{v}
  16. ( 5 ) p ρ γ = constant . (\;5)\quad\quad\frac{p}{\rho^{\gamma}}=\,\text{constant}.
  17. γ \gamma\,
  18. γ \gamma\,
  19. w w
  20. ( 6 ) d d t x 1 x 2 w d x = - f ( w ) | x 1 x 2 (\;6)\quad\quad\frac{d}{dt}\int_{x_{1}}^{x_{2}}w\,dx=-\left.f\left(w\right)% \right|_{x_{1}}^{x_{2}}
  21. x 1 x_{1}
  22. x 2 x_{2}
  23. x 1 < x 2 x_{1}<x_{2}
  24. ( 6 ) w t + x f ( w ) = 0 (\;6^{\prime})\quad\quad\frac{\partial w}{\partial t}+\frac{\partial}{\partial x% }f\left(w\right)=0
  25. x = x s ( t ) x=x_{s}(t)
  26. x 1 < x s ( t ) x_{1}<x_{s}(t)
  27. x s ( t ) < x 2 x_{s}(t)<x_{2}
  28. ( 7 ) d d t [ ( x 1 x s ( t ) w d x + x s ( t ) x 2 w d x ) ] = - x 1 x 2 x f ( w ) d x (\;7)\quad\quad\frac{d}{dt}\left[\left(\int_{x_{1}}^{x_{s}(t)}w\,dx+\int_{x_{s% }(t)}^{x_{2}}w\,dx\right)\right]=-\int_{x_{1}}^{x_{2}}\frac{\partial}{\partial x% }f\left(w\right)\,dx
  29. ( 8 ) w 1 d x s d t - w 2 d x s d t + x 1 x s ( t ) w t d x + x s ( t ) x 2 w t d x = - f ( w ) | x 1 x 2 (\;8)\quad\quad\therefore w_{1}\frac{dx_{s}}{dt}-w_{2}\frac{dx_{s}}{dt}+\int_{% x_{1}}^{x_{s}(t)}w_{t}\,dx+\int_{x_{s}(t)}^{x_{2}}w_{t}\,dx=-\left.f\left(w% \right)\right|_{x_{1}}^{x_{2}}
  30. w 1 = lim ϵ 0 + w ( x s - ϵ ) w_{1}=\lim_{\epsilon\rightarrow 0^{+}}w(x_{s}-\epsilon)
  31. w 2 = lim ϵ 0 + w ( x s + ϵ ) w_{2}=\lim_{\epsilon\rightarrow 0^{+}}w(x_{s}+\epsilon)
  32. d x 1 / d t = 0 {\displaystyle dx_{1}/dt=0}
  33. d x 2 / d t = 0 {\displaystyle dx_{2}/dt=0}
  34. x 1 x s ( t ) - ϵ x_{1}\rightarrow x_{s}(t)-\epsilon
  35. x 2 x s ( t ) + ϵ x_{2}\rightarrow x_{s}(t)+\epsilon
  36. x 1 x s ( t ) - ϵ w t d x 0 \int_{x_{1}}^{x_{s}(t)-\epsilon}w_{t}\,dx\rightarrow 0
  37. x s ( t ) + ϵ x 2 w t d x 0 \int_{x_{s}(t)+\epsilon}^{x_{2}}w_{t}\,dx\rightarrow 0
  38. ( 9 ) u s ( w 1 - w 2 ) = f ( w 1 ) - f ( w 2 ) , (\;9)\quad\quad u_{s}\left(w_{1}-w_{2}\right)=f\left(w_{1}\right)-f\left(w_{2}% \right),
  39. u s = d x s ( t ) / d t u_{s}=dx_{s}(t)/dt\,
  40. ( 10 ) u s = f ( w 1 ) - f ( w 2 ) w 1 - w 2 . (10)\quad\quad u_{s}=\frac{f\left(w_{1}\right)-f\left(w_{2}\right)}{w_{1}-w_{2% }}.
  41. ( 11 ) f ( w 1 ) > u s > f ( w 2 ) , (11)\quad\quad f^{\prime}\left(w_{1}\right)>u_{s}>f^{\prime}\left(w_{2}\right),
  42. f ( w 1 ) f^{\prime}\left(w_{1}\right)
  43. f ( w 2 ) f^{\prime}\left(w_{2}\right)
  44. [ ρ , ρ u , ρ E ] T \left[\rho,\rho u,\rho E\right]^{T}
  45. ( 12 ) u s ( ρ 2 - ρ 1 ) = ρ 2 u 2 - ρ 1 u 1 (12)\quad\quad\quad\quad\;u_{s}\left(\rho_{2}-\rho_{1}\right)=\rho_{2}u_{2}-% \rho_{1}u_{1}
  46. ( 13 ) u s ( ρ 2 u 2 - ρ 1 u 1 ) = ( ρ 2 u 2 2 + p 2 ) - ( ρ 1 u 1 2 + p 1 ) (13)\quad\quad\;u_{s}\left(\rho_{2}u_{2}-\rho_{1}u_{1}\right)=\left(\rho_{2}u_% {2}^{2}+p_{2}\right)-\left(\rho_{1}u_{1}^{2}+p_{1}\right)
  47. ( 14 ) u s ( ρ 2 E 2 - ρ 1 E 1 ) = [ ρ 2 u 2 ( e 2 + 1 2 u 2 2 + p 2 / ρ 2 ) ] - [ ρ 1 u 1 ( e 1 + 1 2 u 1 2 + p 1 / ρ 1 ) ] . (14)\quad\quad u_{s}\left(\rho_{2}E_{2}-\rho_{1}E_{1}\right)=\left[\rho_{2}u_{% 2}\left(e_{2}+\frac{1}{2}u_{2}^{2}+p_{2}/\rho_{2}\right)\right]-\left[\rho_{1}% u_{1}\left(e_{1}+\frac{1}{2}u_{1}^{2}+p_{1}/\rho_{1}\right)\right].
  48. ρ \rho
  49. u u
  50. T T
  51. u s u_{s}
  52. u s := u s - u 1 u_{s}^{\prime}:=u_{s}-u_{1}
  53. u 1 := 0 u^{\prime}_{1}:=0
  54. u 2 := u 2 - u 1 u^{\prime}_{2}:=u_{2}-u_{1}
  55. u 1 u_{1}
  56. u 2 u^{\prime}_{2}
  57. ( 15 ) u s = u 1 + c 1 1 + γ + 1 2 γ ( p 2 p 1 - 1 ) , (15)\quad\quad u_{s}=u_{1}+c_{1}\sqrt{1+\frac{\gamma+1}{2\gamma}\left(\frac{p_% {2}}{p_{1}}-1\right)},
  58. c 1 = γ p 1 / ρ 1 c_{1}=\sqrt{\gamma p_{1}/\rho_{1}}
  59. u s = 0 u_{s}=0
  60. ( 16 ) ρ 1 u 1 = ρ 2 u 2 (16)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\rho_{1}u_{1}=\rho_{% 2}u_{2}
  61. ( 17 ) ρ 1 u 1 2 + p 1 = ρ 2 u 2 2 + p 2 (17)\quad\quad\quad\quad\quad\quad\quad\quad\;\;\rho_{1}u_{1}^{2}+p_{1}=\rho_{% 2}u_{2}^{2}+p_{2}
  62. ( 18 ) ρ 1 u 1 ( e 1 + 1 2 u 1 2 + p 1 / ρ 1 ) = ρ 2 u 2 ( e 2 + 1 2 u 2 2 + p 2 / ρ 2 ) . (18)\quad\quad\rho_{1}u_{1}\left(e_{1}+\frac{1}{2}u_{1}^{2}+p_{1}/\rho_{1}% \right)=\rho_{2}u_{2}\left(e_{2}+\frac{1}{2}u_{2}^{2}+p_{2}/\rho_{2}\right).
  63. ( 19 ) e 1 + 1 2 u 1 2 + p 1 / ρ 1 = e 2 + 1 2 u 2 2 + p 2 / ρ 2 , (19)\quad\quad e_{1}+\frac{1}{2}u_{1}^{2}+p_{1}/\rho_{1}=e_{2}+\frac{1}{2}u_{2% }^{2}+p_{2}/\rho_{2},
  64. u 1 u_{1}
  65. u 2 u_{2}
  66. ( 20 ) 2 ( h 2 - h 1 ) = ( p 2 - p 1 ) ( 1 ρ 1 + 1 ρ 2 ) , (20)\quad\quad 2\left(h_{2}-h_{1}\right)=\left(p_{2}-p_{1}\right)\cdot\left(% \frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}\right),
  67. h = p ρ + e h=\frac{p}{\rho}+e
  68. e e
  69. ( 21 ) ρ 2 ρ 1 = p 2 p 1 ( γ + 1 ) + ( γ - 1 ) ( γ + 1 ) + p 2 p 1 ( γ - 1 ) = u 1 u 2 (21)\quad\quad\frac{\rho_{2}}{\rho_{1}}=\frac{\frac{p_{2}}{p_{1}}(\gamma+1)+(% \gamma-1)}{(\gamma+1)+\frac{p_{2}}{p_{1}}(\gamma-1)}=\frac{u_{1}}{u_{2}}
  70. ( 22 ) p 2 p 1 = ρ 2 ρ 1 ( γ + 1 ) - ( γ - 1 ) ( γ + 1 ) - ρ 2 ρ 1 ( γ - 1 ) . (22)\quad\quad\frac{p_{2}}{p_{1}}=\frac{\frac{\rho_{2}}{\rho_{1}}(\gamma+1)-(% \gamma-1)}{(\gamma+1)-\frac{\rho_{2}}{\rho_{1}}(\gamma-1)}.
  71. ρ 2 / ρ 1 < ( γ + 1 ) / ( γ - 1 ) \rho_{2}/\rho_{1}<(\gamma+1)/(\gamma-1)\,
  72. ρ 2 / ρ 1 \rho_{2}/\rho_{1}
  73. ( γ = 5 / 3 ) (\gamma=5/3)
  74. ( γ = 1.4 ) (\gamma=1.4)
  75. γ air 1.4 \gamma_{\mathrm{air}}\simeq 1.4
  76. ( 23 ) u s = c 0 + s u p = c 0 + s u 2 (23)\qquad u_{s}=c_{0}+s\,u_{p}=c_{0}+s\,u_{2}
  77. ( 24 ) p 2 - p 1 = c 0 2 ρ 1 ρ 2 ( ρ 2 - ρ 1 ) [ ρ 2 - s ( ρ 2 - ρ 1 ) ] 2 = c 0 2 ( v 1 - v 2 ) [ v 1 - s ( v 1 - v 2 ) ] 2 . (24)\qquad p_{2}-p_{1}=\frac{c_{0}^{2}\,\rho_{1}\,\rho_{2}\,(\rho_{2}-\rho_{1}% )}{[\rho_{2}-s(\rho_{2}-\rho_{1})]^{2}}=\frac{c_{0}^{2}\,(v_{1}-v_{2})}{[v_{1}% -s(v_{1}-v_{2})]^{2}}\,.
  78. ( 25 ) p 2 - p 1 = u s 2 ( ρ 1 - ρ 1 2 ρ 2 ) (25)\qquad p_{2}-p_{1}=u_{s}^{2}\left(\rho_{1}-\frac{\rho_{1}^{2}}{\rho_{2}}% \right)\,
  79. ( 6 ) \scriptstyle(6)
  80. ( 6 ) \scriptstyle(6^{\prime})
  81. [ x 1 ; x 2 ] \scriptstyle[x_{1};x_{2}]
  82. ( 6 ) \scriptstyle(6^{\prime})

Rank–nullity_theorem.html

  1. rk ( A ) + nul ( A ) = n . \operatorname{rk}(A)+\operatorname{nul}(A)=n.
  2. dim ( im ( T ) ) + dim ( ker ( T ) ) = dim ( V ) , \operatorname{dim}(\operatorname{im}(T))+\operatorname{dim}(\operatorname{ker}% (T))=\operatorname{dim}(V),
  3. rk ( T ) + nul ( T ) = dim ( V ) . \operatorname{rk}(T)+\operatorname{nul}(T)=\operatorname{dim}(V).
  4. { 𝐮 1 , , 𝐮 m } \{\mathbf{u}_{1},\ldots,\mathbf{u}_{m}\}
  5. { 𝐮 1 , , 𝐮 m , 𝐰 1 , , 𝐰 n } \{\mathbf{u}_{1},\ldots,\mathbf{u}_{m},\mathbf{w}_{1},\ldots,\mathbf{w}_{n}\}
  6. { T 𝐰 1 , , T 𝐰 n } \{T\mathbf{w}_{1},\ldots,T\mathbf{w}_{n}\}
  7. 𝐯 = a 1 𝐮 1 + + a m 𝐮 m + b 1 𝐰 1 + + b n 𝐰 n \mathbf{v}=a_{1}\mathbf{u}_{1}+\cdots+a_{m}\mathbf{u}_{m}+b_{1}\mathbf{w}_{1}+% \cdots+b_{n}\mathbf{w}_{n}
  8. T 𝐯 = a 1 T 𝐮 1 + + a m T 𝐮 m + b 1 T 𝐰 1 + + b n T 𝐰 n \Rightarrow T\mathbf{v}=a_{1}T\mathbf{u}_{1}+\cdots+a_{m}T\mathbf{u}_{m}+b_{1}% T\mathbf{w}_{1}+\cdots+b_{n}T\mathbf{w}_{n}
  9. T 𝐯 = b 1 T 𝐰 1 + + b n T 𝐰 n T 𝐮 i = 0 \Rightarrow T\mathbf{v}=b_{1}T\mathbf{w}_{1}+\cdots+b_{n}T\mathbf{w}_{n}\;\;% \because T\mathbf{u}_{i}=0
  10. { T 𝐰 1 , , T 𝐰 n } \{T\mathbf{w}_{1},\ldots,T\mathbf{w}_{n}\}
  11. { T 𝐰 1 , , T 𝐰 n } \{T\mathbf{w}_{1},\ldots,T\mathbf{w}_{n}\}
  12. c 1 T 𝐰 1 + + c n T 𝐰 n = 0 T { c 1 𝐰 1 + + c n 𝐰 n } = 0 c_{1}T\mathbf{w}_{1}+\cdots+c_{n}T\mathbf{w}_{n}=0\Leftrightarrow T\{c_{1}% \mathbf{w}_{1}+\cdots+c_{n}\mathbf{w}_{n}\}=0
  13. c 1 𝐰 1 + + c n 𝐰 n ker T \therefore c_{1}\mathbf{w}_{1}+\cdots+c_{n}\mathbf{w}_{n}\in\operatorname{ker}\;T
  14. c 1 𝐰 1 + + c n 𝐰 n = d 1 𝐮 1 + + d m 𝐮 m c_{1}\mathbf{w}_{1}+\cdots+c_{n}\mathbf{w}_{n}=d_{1}\mathbf{u}_{1}+\cdots+d_{m% }\mathbf{u}_{m}
  15. { 𝐮 1 , , 𝐮 m , 𝐰 1 , , 𝐰 n } \{\mathbf{u}_{1},\ldots,\mathbf{u}_{m},\mathbf{w}_{1},\ldots,\mathbf{w}_{n}\}
  16. { T 𝐰 1 , , T 𝐰 n } \{T\mathbf{w}_{1},\ldots,T\mathbf{w}_{n}\}
  17. 𝐗 = ( - 𝐁 𝐈 n - r ) \displaystyle\mathbf{X}=\begin{pmatrix}-\mathbf{B}\\ \mathbf{I}_{n-r}\end{pmatrix}
  18. 𝐈 n - r \mathbf{I}_{n-r}
  19. 𝐀𝐗 = [ 𝐀 1 : 𝐀 1 𝐁 ] ( - 𝐁 𝐈 n - r ) = - 𝐀 1 𝐁 + 𝐀 1 𝐁 = 𝐎 . \mathbf{A}\mathbf{X}=[\mathbf{A}_{1}:\mathbf{A}_{1}\mathbf{B}]\begin{pmatrix}-% \mathbf{B}\\ \mathbf{I}_{n-r}\end{pmatrix}=-\mathbf{A}_{1}\mathbf{B}+\mathbf{A}_{1}\mathbf{% B}=\mathbf{O}\;.
  20. 𝐗𝐮 = 𝟎 ( - 𝐁 𝐈 n - r ) 𝐮 = 𝟎 ( - 𝐁𝐮 𝐮 ) = ( 𝟎 𝟎 ) 𝐮 = 0 . \mathbf{X}\mathbf{u}=\mathbf{0}\Rightarrow\begin{pmatrix}-\mathbf{B}\\ \mathbf{I}_{n-r}\end{pmatrix}\mathbf{u}=\mathbf{0}\Rightarrow\begin{pmatrix}-% \mathbf{B}\mathbf{u}\\ \mathbf{u}\end{pmatrix}=\begin{pmatrix}\mathbf{0}\\ \mathbf{0}\end{pmatrix}\Rightarrow\mathbf{u}=\mathbf{0}\;.
  21. 𝐮 = ( 𝐮 1 𝐮 2 ) \displaystyle\mathbf{u}=\begin{pmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{pmatrix}
  22. 𝐀𝐮 = 𝟎 [ 𝐀 1 : 𝐀 1 𝐁 ] ( 𝐮 1 𝐮 2 ) = 𝟎 𝐀 1 ( 𝐮 1 + 𝐁𝐮 2 ) = 𝟎 𝐮 1 + 𝐁𝐮 2 = 𝟎 𝐮 1 = - 𝐁𝐮 2 \mathbf{A}\mathbf{u}=\mathbf{0}\Rightarrow[\mathbf{A}_{1}:\mathbf{A}_{1}% \mathbf{B}]\begin{pmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{pmatrix}=\mathbf{0}\Rightarrow\mathbf{A}_{1}(\mathbf{u}_{1}% +\mathbf{B}\mathbf{u}_{2})=\mathbf{0}\Rightarrow\mathbf{u}_{1}+\mathbf{B}% \mathbf{u}_{2}=\mathbf{0}\Rightarrow\mathbf{u}_{1}=-\mathbf{B}\mathbf{u}_{2}
  23. 𝐮 = ( 𝐮 1 𝐮 2 ) = ( - 𝐁 𝐈 n - r ) 𝐮 2 = 𝐗𝐮 2 . \Rightarrow\mathbf{u}=\begin{pmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{pmatrix}=\begin{pmatrix}-\mathbf{B}\\ \mathbf{I}_{n-r}\end{pmatrix}\mathbf{u}_{2}=\mathbf{X}\mathbf{u}_{2}.
  24. 0 ker T I d V 𝑇 im T 0 0\rightarrow\ker T~{}\overset{Id}{\rightarrow}~{}V~{}\overset{T}{\rightarrow}~% {}\operatorname{im}T\rightarrow 0
  25. i = 1 r ( - 1 ) i dim ( V i ) = 0. \sum_{i=1}^{r}(-1)^{i}\dim(V_{i})=0.

Rao–Blackwell_theorem.html

  1. E ( ( δ 1 ( X ) - θ ) 2 ) E ( ( δ ( X ) - θ ) 2 ) . \operatorname{E}((\delta_{1}(X)-\theta)^{2})\leq\operatorname{E}((\delta(X)-% \theta)^{2}).\,\!
  2. 0 Var ( Y ) = E ( ( Y - E ( Y ) ) 2 ) = E ( Y 2 ) - ( E ( Y ) ) 2 . 0\leq\operatorname{Var}(Y)=\operatorname{E}((Y-\operatorname{E}(Y))^{2})=% \operatorname{E}(Y^{2})-(\operatorname{E}(Y))^{2}.\,\!
  3. E ( L ( δ 1 ( X ) ) ) E ( L ( δ ( X ) ) ) \operatorname{E}(L(\delta_{1}(X)))\leq\operatorname{E}(L(\delta(X)))\,\!
  4. δ 0 = { 1 if X 1 = 0 , 0 otherwise, \delta_{0}=\left\{\begin{matrix}1&\,\text{if}\ X_{1}=0,\\ 0&\,\text{otherwise,}\end{matrix}\right.
  5. S n = i = 1 n X i = X 1 + + X n S_{n}=\sum_{i=1}^{n}X_{i}=X_{1}+\cdots+X_{n}\,\!
  6. δ 1 = E ( δ 0 S n = s n ) . \delta_{1}=\operatorname{E}(\delta_{0}\mid S_{n}=s_{n}).
  7. δ 1 = E ( 𝟏 { X 1 = 0 } | i = 1 n X i = s n ) = P ( X 1 = 0 | i = 1 n X i = s n ) = P ( X 1 = 0 , i = 2 n X i = s n ) × P ( i = 1 n X i = s n ) - 1 = e - λ ( ( n - 1 ) λ ) s n e - ( n - 1 ) λ s n ! × ( ( n λ ) s n e - n λ s n ! ) - 1 = ( ( n - 1 ) λ ) s n e - n λ s n ! × s n ! ( n λ ) s n e - n λ = ( 1 - 1 n ) s n \begin{aligned}\displaystyle\delta_{1}&\displaystyle=\operatorname{E}\left(% \mathbf{1}_{\{X_{1}=0\}}\Bigg|\sum_{i=1}^{n}X_{i}=s_{n}\right)\\ &\displaystyle=P\left(X_{1}=0\Bigg|\sum_{i=1}^{n}X_{i}=s_{n}\right)\\ &\displaystyle=P\left(X_{1}=0,\sum_{i=2}^{n}X_{i}=s_{n}\right)\times P\left(% \sum_{i=1}^{n}X_{i}=s_{n}\right)^{-1}\\ &\displaystyle=e^{-\lambda}\frac{\left((n-1)\lambda\right)^{s_{n}}e^{-(n-1)% \lambda}}{s_{n}!}\times\left(\frac{(n\lambda)^{s_{n}}e^{-n\lambda}}{s_{n}!}% \right)^{-1}\\ &\displaystyle=\frac{\left((n-1)\lambda\right)^{s_{n}}e^{-n\lambda}}{s_{n}!}% \times\frac{s_{n}!}{(n\lambda)^{s_{n}}e^{-n\lambda}}\\ &\displaystyle=\left(1-\frac{1}{n}\right)^{s_{n}}\end{aligned}
  8. ( 1 - 1 n ) n λ e - λ . \left(1-{1\over n}\right)^{n\lambda}\approx e^{-\lambda}.

Rate_(mathematics).html

  1. Average rate of change = f ( a + h ) - f ( a ) h Instantaneous rate of change = lim h 0 f ( a + h ) - f ( a ) h \begin{aligned}\displaystyle\mbox{Average rate of change}&\displaystyle=\frac{% f(a+h)-f(a)}{h}\\ \displaystyle\mbox{Instantaneous rate of change}&\displaystyle=\lim_{h\to 0}% \frac{f(a+h)-f(a)}{h}\end{aligned}

Ratio_test.html

  1. n = 1 a n , \sum_{n=1}^{\infty}a_{n},
  2. n n
  3. n = 1 ( 1 2 ) n = 1 2 + 1 4 + 1 8 + \sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=\frac{1}{2}+\frac{1}{4}+\frac{% 1}{8}+\cdots
  4. a n + 1 / a n = ( 1 / 2 ) n + 1 / ( 1 / 2 ) n a_{n+1}/a_{n}=(1/2)^{n+1}/(1/2)^{n}
  5. 1 - 1 2 m . 1-\frac{1}{2^{m}}.
  6. n = 1 2 n = 2 + 4 + 8 + \sum_{n=1}^{\infty}2^{n}=2+4+8+\cdots
  7. a n + 1 / a n a_{n+1}/a_{n}
  8. 2 m + 1 - 2 , 2^{m+1}-2,
  9. n = 1 m r n = r r - 1 ( r m - 1 ) . \sum_{n=1}^{m}r^{n}=\frac{r}{r-1}(r^{m}-1).
  10. n = 1 n + 1 n ( 1 2 ) n = 2 1 1 2 + 3 2 1 4 + 4 3 1 8 + \sum_{n=1}^{\infty}\frac{n+1}{n}\left(\frac{1}{2}\right)^{n}=\frac{2}{1}\cdot% \frac{1}{2}+\frac{3}{2}\cdot\frac{1}{4}+\frac{4}{3}\cdot\frac{1}{8}+\cdots
  11. ( n + 1 n ( 1 2 ) n ) / ( n n - 1 ( 1 2 ) n - 1 ) = n 2 - 1 2 n 2 = 1 2 - 1 2 n 2 . \left(\frac{n+1}{n}\left(\frac{1}{2}\right)^{n}\right)/\left(\frac{n}{n-1}% \left(\frac{1}{2}\right)^{n-1}\right)=\frac{n^{2}-1}{2n^{2}}=\frac{1}{2}-\frac% {1}{2n^{2}}.
  12. L = lim n | a n + 1 a n | . L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|.
  13. R = lim sup | a n + 1 a n | R=\lim\sup\left|\frac{a_{n+1}}{a_{n}}\right|
  14. r = lim inf | a n + 1 a n | r=\lim\inf\left|\frac{a_{n+1}}{a_{n}}\right|
  15. | a n + 1 a n | 1 \left|\frac{a_{n+1}}{a_{n}}\right|\geq 1
  16. | a n | |a_{n}|
  17. L = lim n | a n + 1 a n | = lim n | n + 1 e n + 1 n e n | = 1 e < 1. L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim_{n\to\infty}\left|% \frac{\frac{n+1}{e^{n+1}}}{\frac{n}{e^{n}}}\right|=\frac{1}{e}<1.
  18. n = 1 e n n . \sum_{n=1}^{\infty}\frac{e^{n}}{n}.
  19. L = lim n | a n + 1 a n | = lim n | e n + 1 n + 1 e n n | = e > 1. L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim_{n\to\infty}\left|% \frac{\frac{e^{n+1}}{n+1}}{\frac{e^{n}}{n}}\right|=e>1.
  20. n = 1 1 , \sum_{n=1}^{\infty}1,
  21. n = 1 1 n 2 , \sum_{n=1}^{\infty}\frac{1}{n^{2}},
  22. n = 1 ( - 1 ) n 1 n . \sum_{n=1}^{\infty}(-1)^{n}\frac{1}{n}.
  23. | a n + 1 a n | \left|\frac{a_{n+1}}{a_{n}}\right|
  24. n 2 ( n + 1 ) 2 \frac{n^{2}}{(n+1)^{2}}
  25. n n + 1 \frac{n}{n+1}
  26. lim n | a n + 1 a n | = 1 \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|=1
  27. n = 1 1 \sum_{n=1}^{\infty}1
  28. | a n + 1 a n | = 1 \left|\frac{a_{n+1}}{a_{n}}\right|=1
  29. L = lim n | a n + 1 a n | < 1 L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|<1
  30. r = L + 1 2 r=\frac{L+1}{2}
  31. | a n + 1 | < r | a n | |a_{n+1}|<r|a_{n}|
  32. | a n + i | < r i | a n | |a_{n+i}|<r^{i}|a_{n}|
  33. i = N + 1 | a i | = i = 1 | a N + i | < i = 1 r i | a N + 1 | = | a N + 1 | i = 1 r i = | a N + 1 | r 1 - r < . \sum_{i=N+1}^{\infty}|a_{i}|=\sum_{i=1}^{\infty}\left|a_{N+i}\right|<\sum_{i=1% }^{\infty}r^{i}|a_{N+1}|=|a_{N+1}|\sum_{i=1}^{\infty}r^{i}=|a_{N+1}|\frac{r}{1% -r}<\infty.
  34. | a n + 1 | > | a n | |a_{n+1}|>|a_{n}|
  35. | a n + 1 a n | 1. \left|\frac{a_{n+1}}{a_{n}}\right|\geq 1.
  36. lim n | a n a n + 1 | = 1 , \lim_{n\to\infty}\left|\frac{a_{n}}{a_{n+1}}\right|=1,
  37. lim n n ( | a n a n + 1 | - 1 ) = R , \lim_{n\to\infty}n\left(\left|\frac{a_{n}}{a_{n+1}}\right|-1\right)=R,
  38. R > p > 1 R>p>1
  39. lim n n ( | a n a n + 1 | - 1 ) = R > p = lim n n ( ( 1 + 1 n ) p - 1 ) \lim_{n\to\infty}n\left(\left|\frac{a_{n}}{a_{n+1}}\right|-1\right)=R>p=\lim_{% n\to\infty}n\left(\left(1+\frac{1}{n}\right)^{p}-1\right)
  40. n ( | a n a n + 1 | - 1 ) > n ( ( 1 + 1 n ) p - 1 ) for large n \Rightarrow\quad n\left(\left|\frac{a_{n}}{a_{n+1}}\right|-1\right)>n\left(% \left(1+\frac{1}{n}\right)^{p}-1\right)\quad\,\text{ for large }n
  41. | a n a n + 1 | > ( n + 1 ) p n p \Rightarrow\quad\left|\frac{a_{n}}{a_{n+1}}\right|>\frac{(n+1)^{p}}{n^{p}}
  42. n - p \sum n^{-p}
  43. p > 1 p>1
  44. a n \sum{{a}_{n}}
  45. R > 1 R>1
  46. R < 1 R<1
  47. | a n a n + 1 | = 1 + 1 n + ρ n n ln n \left|\frac{a_{n}}{a_{n+1}}\right|=1+\frac{1}{n}+\frac{\rho_{n}}{n\ln n}
  48. | a n a n + 1 | = 1 + h n + C n n r \left|\frac{a_{n}}{a_{n+1}}\right|=1+\frac{h}{n}+\frac{C_{n}}{n^{r}}
  49. ρ = lim n ( ζ n a n a n + 1 - ζ n + 1 ) . \rho=\lim_{n\to\infty}\left(\zeta_{n}\frac{a_{n}}{a_{n+1}}-\zeta_{n+1}\right).
  50. ρ > 0 \rho>0
  51. 0 < δ < ρ 0<\delta<\rho
  52. N 0 N_{0}
  53. n > N 0 n>N_{0}
  54. δ ζ n a n a n + 1 - ζ n + 1 . \delta\leq\zeta_{n}\frac{a_{n}}{a_{n+1}}-\zeta_{n+1}.
  55. a n + 1 > 0 a_{n+1}>0
  56. n > N 0 n>N_{0}
  57. 0 δ a n + 1 ζ n a n - ζ n + 1 a n + 1 . 0\leq\delta a_{n+1}\leq\zeta_{n}a_{n}-\zeta_{n+1}a_{n+1}.
  58. ζ n + 1 a n + 1 ζ n a n \zeta_{n+1}a_{n+1}\leq\zeta_{n}a_{n}
  59. n N 0 n\geq N_{0}
  60. N 0 N_{0}
  61. ζ n a n > 0 \zeta_{n}a_{n}>0
  62. lim n ζ n a n = L \lim_{n\to\infty}\zeta_{n}a_{n}=L
  63. n = 1 ( ζ n a n - ζ n + 1 a n + 1 ) \sum_{n=1}^{\infty}\Big(\zeta_{n}a_{n}-\zeta_{n+1}a_{n+1}\Big)
  64. n > N 0 n>N_{0}
  65. δ a n + 1 ζ n a n - ζ n + 1 a n + 1 \delta a_{n+1}\leq\zeta_{n}a_{n}-\zeta_{n+1}a_{n+1}
  66. n = 1 δ a n + 1 \sum_{n=1}^{\infty}\delta a_{n+1}
  67. ρ < 0 \rho<0
  68. ζ n a n \zeta_{n}a_{n}
  69. n > N n>N
  70. ϵ > 0 \epsilon>0
  71. ζ n a n > ϵ \zeta_{n}a_{n}>\epsilon
  72. n > N n>N
  73. n a n = a n ζ n / ζ n \sum_{n}a_{n}=\sum a_{n}\zeta_{n}/\zeta_{n}
  74. n ϵ / ζ n \sum_{n}\epsilon/\zeta_{n}

Rational_function.html

  1. f ( x ) f(x)
  2. f ( x ) = P ( x ) Q ( x ) f(x)=\frac{P(x)}{Q(x)}
  3. P P\,
  4. Q Q\,
  5. x x\,
  6. Q Q\,
  7. f f\,
  8. x x\,
  9. Q ( x ) Q(x)\,
  10. P \textstyle P
  11. Q \textstyle Q
  12. R \textstyle R
  13. P = P 1 R \textstyle P=P_{1}R
  14. Q = Q 1 R \textstyle Q=Q_{1}R
  15. f 1 ( x ) = P 1 ( x ) Q 1 ( x ) , f_{1}(x)=\frac{P_{1}(x)}{Q_{1}(x)},
  16. f ( x ) f(x)
  17. f ( x ) f(x)
  18. f ( x ) . f(x).
  19. f ( x ) f(x)
  20. f 1 ( x ) f_{1}(x)
  21. f ( x ) f(x)
  22. f 1 ( x ) . f_{1}(x).
  23. P ( x ) Q ( x ) \frac{P(x)}{Q(x)}
  24. P 1 ( x ) Q 1 ( x ) \frac{P_{1}(x)}{Q_{1}(x)}
  25. f ( x ) = x 3 - 2 x 2 ( x 2 - 5 ) f(x)=\frac{x^{3}-2x}{2(x^{2}-5)}
  26. x 2 = 5 x = ± 5 x^{2}=5\Leftrightarrow x=\pm\sqrt{5}
  27. x 2 \frac{x}{2}
  28. f ( x ) = x 2 + 2 x 2 + 1 f(x)=\frac{x^{2}+2}{x^{2}+1}
  29. - 1 -1
  30. f ( i ) = i 2 + 2 i 2 + 1 = - 1 + 2 - 1 + 1 = 1 0 f(i)=\frac{i^{2}+2}{i^{2}+1}=\frac{-1+2}{-1+1}=\frac{1}{0}
  31. f ( x ) = P ( x ) f(x)=P(x)
  32. Q ( x ) = 1 Q(x)=1
  33. f ( x ) = sin ( x ) f(x)=\sin(x)
  34. f ( x ) = x x f(x)=\frac{x}{x}
  35. 1 x 2 - x + 2 = k = 0 a k x k . \frac{1}{x^{2}-x+2}=\sum_{k=0}^{\infty}a_{k}x^{k}.
  36. 1 = ( x 2 - x + 2 ) k = 0 a k x k 1=(x^{2}-x+2)\sum_{k=0}^{\infty}a_{k}x^{k}
  37. 1 = k = 0 a k x k + 2 - k = 0 a k x k + 1 + 2 k = 0 a k x k . 1=\sum_{k=0}^{\infty}a_{k}x^{k+2}-\sum_{k=0}^{\infty}a_{k}x^{k+1}+2\sum_{k=0}^% {\infty}a_{k}x^{k}.
  38. 1 = k = 2 a k - 2 x k - k = 1 a k - 1 x k + 2 k = 0 a k x k . 1=\sum_{k=2}^{\infty}a_{k-2}x^{k}-\sum_{k=1}^{\infty}a_{k-1}x^{k}+2\sum_{k=0}^% {\infty}a_{k}x^{k}.
  39. 1 = 2 a 0 + ( 2 a 1 - a 0 ) x + k = 2 ( a k - 2 - a k - 1 + 2 a k ) x k . 1=2a_{0}+(2a_{1}-a_{0})x+\sum_{k=2}^{\infty}(a_{k-2}-a_{k-1}+2a_{k})x^{k}.
  40. a 0 = 1 2 . a_{0}=\frac{1}{2}.
  41. a 1 = 1 4 a_{1}=\frac{1}{4}
  42. a k = 1 2 ( a k - 1 - a k - 2 ) f o r k 2. a_{k}=\frac{1}{2}(a_{k-1}-a_{k-2})\quad for\ k\geq 2.
  43. f ( z ) = P ( z ) Q ( z ) f(z)=\frac{P(z)}{Q(z)}
  44. f ( z ) = w f(z)=w\,

Ray_Solomonoff.html

  1. T i / P i T_{i}/P_{i}
  2. T i T_{i}
  3. P i P_{i}

Rayleigh_quotient_iteration.html

  1. μ 0 \mu_{0}
  2. A A
  3. b 0 b_{0}
  4. b i + 1 b_{i+1}
  5. b i + 1 = ( A - μ i I ) - 1 b i || ( A - μ i I ) - 1 b i || , b_{i+1}=\frac{(A-\mu_{i}I)^{-1}b_{i}}{||(A-\mu_{i}I)^{-1}b_{i}||},
  6. I I
  7. μ i = b i * A b i b i * b i . \mu_{i}=\frac{b^{*}_{i}Ab_{i}}{b^{*}_{i}b_{i}}.
  8. A = [ 1 2 3 1 2 1 3 2 1 ] A=\left[\begin{matrix}1&2&3\\ 1&2&1\\ 3&2&1\\ \end{matrix}\right]
  9. λ 1 = 3 + 5 \lambda_{1}=3+\sqrt{5}
  10. λ 2 = 3 - 5 \lambda_{2}=3-\sqrt{5}
  11. λ 3 = - 2 \lambda_{3}=-2
  12. v 1 = [ 1 φ - 1 1 ] v_{1}=\left[\begin{matrix}1\\ \varphi-1\\ 1\\ \end{matrix}\right]
  13. v 2 = [ 1 - φ 1 ] v_{2}=\left[\begin{matrix}1\\ -\varphi\\ 1\\ \end{matrix}\right]
  14. v 3 = [ 1 0 1 ] v_{3}=\left[\begin{matrix}1\\ 0\\ 1\\ \end{matrix}\right]
  15. φ = 1 + 5 2 \textstyle\varphi=\frac{1+\sqrt{5}}{2}
  16. λ 1 5.2361 \lambda_{1}\approx 5.2361
  17. v 1 [ 1 0.6180 1 ] . v_{1}\approx\left[\begin{matrix}1\\ 0.6180\\ 1\\ \end{matrix}\right].
  18. b 0 = [ 1 1 1 ] , μ 0 = 200 b_{0}=\left[\begin{matrix}1\\ 1\\ 1\\ \end{matrix}\right],~{}\mu_{0}=200
  19. b 1 [ - 0.57927 - 0.57348 - 0.57927 ] , μ 1 5.3355 b_{1}\approx\left[\begin{matrix}-0.57927\\ -0.57348\\ -0.57927\\ \end{matrix}\right],~{}\mu_{1}\approx 5.3355
  20. b 2 [ 0.64676 0.40422 0.64676 ] , μ 2 5.2418 b_{2}\approx\left[\begin{matrix}0.64676\\ 0.40422\\ 0.64676\\ \end{matrix}\right],~{}\mu_{2}\approx 5.2418
  21. b 3 [ - 0.64793 - 0.40045 - 0.64793 ] , μ 3 5.2361 b_{3}\approx\left[\begin{matrix}-0.64793\\ -0.40045\\ -0.64793\\ \end{matrix}\right],~{}\mu_{3}\approx 5.2361

RC_time_constant.html

  1. τ = R C \tau=RC
  2. 1 - e - 1 1-e^{-1}
  3. V ( t ) = V 0 ( 1 - e - t / τ ) V(t)=V_{0}(1-e^{-t/\tau})
  4. V ( t ) = V 0 ( e - t / τ ) V(t)=V_{0}(e^{-t/\tau})
  5. τ \tau
  6. τ = R C = 1 2 π f c \tau=RC=\frac{1}{2\pi f_{c}}
  7. f c = 1 2 π R C = 1 2 π τ f_{c}=\frac{1}{2\pi RC}=\frac{1}{2\pi\tau}
  8. t r 1.4 τ 0.22 f c t_{r}\approx 1.4\tau\approx\frac{0.22}{f_{c}}
  9. t r 2.2 τ 0.35 f c t_{r}\approx 2.2\tau\approx\frac{0.35}{f_{c}}
  10. τ \tau
  11. τ \tau

Reaction_mechanism.html

  1. r = k [ N O 2 ] 2 r=k[NO_{2}]^{2}
  2. r = k [ N O 2 ] 2 r=k[NO_{2}]^{2}

Reactivity_(chemistry).html

  1. R a t e = k * [ A ] Rate=k*[A]
  2. R a t e = k * [ A ] n * [ B ] m Rate=k*[A]^{n}*[B]^{m}

Real_projective_plane.html

  1. k ( x , y ) = ( 1 + x 2 + y 2 ) 1 / 2 . ( x , y ) k(x,y)=(1+x^{2}+y^{2})^{1/2}.(x,y)
  2. X ( u , v ) = r ( 1 + cos v ) cos u , X(u,v)=r\,(1+\cos v)\,\cos u,
  3. Y ( u , v ) = r ( 1 + cos v ) sin u , Y(u,v)=r\,(1+\cos v)\,\sin u,
  4. Z ( u , v ) = - tanh ( u - π ) r sin v , Z(u,v)=-\hbox{tanh}\left(u-\pi\right)\,r\,\sin v,
  5. X ( u , v ) = r v cos 2 u , X(u,v)=r\,v\,\cos 2u,
  6. Y ( u , v ) = r v sin 2 u , Y(u,v)=r\,v\,\sin 2u,
  7. Z ( u , v ) = r v cos u , Z(u,v)=r\,v\,\cos u,
  8. ( r cos 2 u , r sin 2 u , r cos u ) (r\,\cos 2u,r\,\sin 2u,r\,\cos u)
  9. ( r cos 2 u , r sin 2 u , - r cos u ) (r\,\cos 2u,r\,\sin 2u,-r\,\cos u)

Real_tree.html

  1. 𝐑 \mathbf{R}

Receptor_(biochemistry).html

  1. [ Ligand ] [ Receptor ] K d [ Ligand-receptor complex ] \left[\mathrm{Ligand}\right]\cdot\left[\mathrm{Receptor}\right]\;\;\overset{K_% {d}}{\rightleftharpoons}\;\;\left[\,\text{Ligand-receptor complex}\right]

Recurrence_plot.html

  1. x ( i ) x ( j ) , \vec{x}(i)\approx\vec{x}(j),\,
  2. i i
  3. j j
  4. x \vec{x}
  5. ( i , j ) (i,j)
  6. x ( i ) = x ( j ) \vec{x}(i)=\vec{x}(j)
  7. T T
  8. T T
  9. x ( i ) \vec{x}(i)
  10. i τ i\tau
  11. R ( i , j ) = { 1 if x ( i ) - x ( j ) ε 0 otherwise , R(i,j)=\begin{cases}1&\,\text{if}\quad\|\vec{x}(i)-\vec{x}(j)\|\leq\varepsilon% \\ 0&\,\text{otherwise},\end{cases}
  12. ( i , j ) (i,j)
  13. R ( i , j ) = 1 R(i,j)=1
  14. y y
  15. 𝐂𝐑 ( i , j ) = Θ ( ε - x ( i ) - y ( j ) ) , x ( i ) , y ( i ) m , i = 1 , , N x , j = 1 , , N y . \mathbf{CR}(i,j)=\Theta(\varepsilon-\|\vec{x}(i)-\vec{y}(j)\|),\quad\vec{x}(i)% ,\,\vec{y}(i)\in\mathbb{R}^{m},\quad i=1,\dots,N_{x},\ j=1,\dots,N_{y}.
  16. x \vec{x}
  17. y \vec{y}
  18. 𝐉𝐑 ( i , j ) = Θ ( ε x - x ( i ) - x ( j ) ) Θ ( ε y - y ( i ) - y ( j ) ) , x ( i ) m , y ( i ) n , i , j = 1 , , N x , y . \mathbf{JR}(i,j)=\Theta(\varepsilon_{x}-\|\vec{x}(i)-\vec{x}(j)\|)\cdot\Theta(% \varepsilon_{y}-\|\vec{y}(i)-\vec{y}(j)\|),\quad\vec{x}(i)\in\mathbb{R}^{m},% \quad\vec{y}(i)\in\mathbb{R}^{n},\quad i,j=1,\dots,N_{x,y}.

Recursively_enumerable_set.html

  1. \mathcal{E}
  2. f ( x ) = { 1 if x S undefined/does not halt if x S f(x)=\left\{\begin{matrix}1&\mbox{if}~{}\ x\in S\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ x\notin S\end{matrix}\right.
  3. x S a , b , c , d , e , f , g , h , i ( p ( x , a , b , c , d , e , f , g , h , i ) = 0 ) . x\in S\Leftrightarrow\exists a,b,c,d,e,f,g,h,i\ (p(x,a,b,c,d,e,f,g,h,i)=0).
  4. ϕ \phi
  5. { i , x ϕ i ( x ) } \{\langle i,x\rangle\mid\phi_{i}(x)\downarrow\}
  6. i , x \langle i,x\rangle
  7. ϕ i ( x ) \phi_{i}(x)\downarrow
  8. ϕ i ( x ) \phi_{i}(x)
  9. ϕ \phi
  10. { x , y , z ϕ x ( y ) = z } \{\left\langle x,y,z\right\rangle\mid\phi_{x}(y)=z\}
  11. x , f ( x ) \langle x,f(x)\rangle
  12. Σ 1 0 \Sigma^{0}_{1}
  13. T T
  14. T \mathbb{N}\setminus T
  15. Π 1 0 \Pi^{0}_{1}

Reduced_ring.html

  1. 𝔭 i \mathfrak{p}_{i}
  2. D 𝔭 i : D\subset\cup\mathfrak{p}_{i}:
  3. 𝔭 i \mathfrak{p}_{i}
  4. 𝔭 i \mathfrak{p}_{i}
  5. 𝔭 j \mathfrak{p}_{j}
  6. 𝔭 i \mathfrak{p}_{i}
  7. 𝔭 i \mathfrak{p}_{i}
  8. D 𝔭 i : D\supset\mathfrak{p}_{i}:
  9. S = { x y | x R - D , y R - 𝔭 } S=\{xy|x\in R-D,y\in R-\mathfrak{p}\}
  10. R R [ S - 1 ] R\to R[S^{-1}]
  11. 𝔮 \mathfrak{q}
  12. 𝔮 \mathfrak{q}
  13. 𝔭 \mathfrak{p}
  14. 𝔮 = 𝔭 \mathfrak{q}=\mathfrak{p}

Reference_dose.html

  1. R f D ( m g / k g / d a y ) = N O E L ( m g / k g / d a y ) U f i n t e r * U f i n t r a * U f o t h e r RfD(mg/kg/day)={NOEL(mg/kg/day)\over Uf_{inter}*Uf_{intra}*Uf_{other}}

Reference_ellipsoid.html

  1. f \displaystyle f
  2. f f\,\!

Referential_integrity.html

  1. R R
  2. S S
  3. R [ A 1 , , A n ] S [ B 1 , , B n ] R[A_{1},...,A_{n}]\subseteq S[B_{1},...,B_{n}]
  4. A i A_{i}
  5. B i B_{i}
  6. R R
  7. S S
  8. A 1 , , A n A_{1},...,A_{n}
  9. R R
  10. B 1 , , B n B_{1},...,B_{n}
  11. S S

Reflection_high-energy_electron_diffraction.html

  1. k 0 = 2 π λ k_{0}=\frac{2\pi}{\lambda}
  2. | k 0 | = | k i | |k_{0}|=|k_{i}|
  3. G = k i - k 0 G=k_{i}-k_{0}

Reflexive_operator_algebra.html

  1. { ( a b 0 a ) : a , b } . \left\{\begin{pmatrix}a&b\\ 0&a\end{pmatrix}\ :\ a,b\in\mathbb{C}\right\}.
  2. { ( a b 0 c ) : a , b , c } \left\{\begin{pmatrix}a&b\\ 0&c\end{pmatrix}\ :\ a,b,c\in\mathbb{C}\right\}
  3. { ( a b 0 0 a 0 0 0 a ) : a , b } \left\{\begin{pmatrix}a&b&0\\ 0&a&0\\ 0&0&a\end{pmatrix}\ :\ a,b\in\mathbb{C}\right\}
  4. T = ( 0 1 0 0 0 0 0 0 0 ) T=\begin{pmatrix}0&1&0\\ 0&0&0\\ 0&0&0\end{pmatrix}
  5. 𝒜 \mathcal{A}
  6. β ( T , 𝒜 ) = sup { P T P : P is a projection and P 𝒜 P = ( 0 ) } \beta(T,\mathcal{A})=\sup\{\|P^{\perp}TP\|\ :\ P\mbox{ is a projection and }~{% }P^{\perp}\mathcal{A}P=(0)\}
  7. 𝒜 \mathcal{A}
  8. 𝒜 \mathcal{A}
  9. β ( T , 𝒜 ) = 0 implies that T is in 𝒜 \beta(T,\mathcal{A})=0\mbox{ implies that }~{}T\mbox{ is in }~{}\mathcal{A}
  10. β ( T , 𝒜 ) dist ( T , 𝒜 ) \beta(T,\mathcal{A})\leq\mbox{dist}~{}(T,\mathcal{A})
  11. dist ( T , 𝒜 ) \mbox{dist}~{}(T,\mathcal{A})
  12. 𝒜 \mathcal{A}
  13. dist ( T , 𝒜 ) K β ( T , 𝒜 ) \mbox{dist}~{}(T,\mathcal{A})\leq K\beta(T,\mathcal{A})
  14. 𝒜 \mathcal{A}

Regenerative_circuit.html

  1. bandwidth = frequency / Q \mathrm{bandwidth}=\mathrm{frequency}/Q
  2. Q = Z / R Q=Z/R
  3. Q reg = Z / ( R - R neg ) Q_{\mathrm{reg}}=Z/(R-R_{\mathrm{neg}})
  4. M = Q reg / Q = R / ( R - R neg ) M=Q_{\mathrm{reg}}/Q=R/(R-R_{\mathrm{neg}})
  5. Q = F / f = 4000 Q=F/f=4000
  6. Q = 100 Q=100
  7. M = 40 M=40

Register_allocation.html

  1. O ( n 2.5 ) O(n^{2.5})
  2. n n

Register_machine.html

  1. r 0 r n r_{0}\ldots r_{n}
  2. I 1 I m I_{1}\ldots I_{m}

Regular.html

  1. T 3 T_{3}

Regular_cardinal.html

  1. κ \kappa
  2. κ \kappa
  3. κ \kappa
  4. α \alpha
  5. α \alpha
  6. ω \omega
  7. ω \omega
  8. ω \omega
  9. ω \omega
  10. 0 \aleph_{0}
  11. ω \omega
  12. ω + 1 \omega+1
  13. ω \omega
  14. ω + ω \omega+\omega
  15. ω \omega
  16. ω \omega
  17. ω + 1 \omega+1
  18. ω + 2 \omega+2
  19. ω + 3 \omega+3
  20. ω \omega
  21. ω + ω \omega+\omega
  22. ω + ω \omega+\omega
  23. ω + ω \omega+\omega
  24. 1 \aleph_{1}
  25. 0 \aleph_{0}
  26. 1 \aleph_{1}
  27. 1 \aleph_{1}
  28. ω \aleph_{\omega}
  29. 0 \aleph_{0}
  30. 1 \aleph_{1}
  31. 2 \aleph_{2}
  32. 3 \aleph_{3}
  33. ω ω \omega_{\omega}
  34. ω \omega
  35. ω 1 \omega_{1}
  36. ω 2 \omega_{2}
  37. ω 3 \omega_{3}
  38. ω \omega
  39. ω ω \omega_{\omega}
  40. ω \aleph_{\omega}
  41. ω \aleph_{\omega}
  42. ω + 1 \omega+1
  43. ω \aleph_{\omega}
  44. ω \omega
  45. 0 , 0 , 0 , \aleph_{0},\aleph_{\aleph_{0}},\aleph_{\aleph_{\aleph_{0}}},...
  46. 1 \aleph_{1}
  47. ω 1 \omega_{1}
  48. 0 \aleph_{0}

Regular_local_ring.html

  1. 𝒪 X , x \mathcal{O}_{X,x}
  2. A A
  3. 𝔪 \mathfrak{m}
  4. 𝔪 = ( a 1 , , a n ) \mathfrak{m}=(a_{1},\ldots,a_{n})
  5. n n
  6. A A
  7. dim A = n \mbox{dim }~{}A=n\,
  8. a 1 , , a n a_{1},\ldots,a_{n}
  9. k = A / 𝔪 k=A/\mathfrak{m}
  10. A A
  11. A A
  12. dim k 𝔪 / 𝔪 2 = dim A \dim_{k}\mathfrak{m}/\mathfrak{m}^{2}=\dim A\,
  13. gl dim A := sup { pd M | M is an A -module } \mbox{gl dim }~{}A:=\sup\{\mbox{pd }~{}M\mbox{ }~{}|\mbox{ }~{}M\mbox{ is an }% ~{}A\mbox{-module}~{}\}
  14. A A
  15. A A
  16. A A
  17. gl dim A < \mbox{gl dim }~{}A<\infty\,
  18. gl dim A = dim A \mbox{gl dim }~{}A=\dim A
  19. ( A , 𝔪 ) (A,\mathfrak{m})
  20. A k [ [ x 1 , , x d ] ] A\cong k[[x_{1},\ldots,x_{d}]]
  21. k = A / 𝔪 k=A/\mathfrak{m}
  22. d = dim A d=\dim A

Regular_polygon.html

  1. A = 1 4 n s 2 cot π n A=\tfrac{1}{4}ns^{2}\cot\frac{\pi}{n}
  2. ( n - 2 ) × 180 n (n-2)\times\frac{180^{\circ}}{n}
  3. ( n - 2 ) × 180 \left(n-2\right)\times 180^{\circ}
  4. ( 1 - 2 n ) × 180 \left(1-\frac{2}{n}\right)\times 180
  5. ( n - 2 ) × 180 n (n-2)\times\frac{180}{n}
  6. ( n - 2 ) π n \frac{(n-2)\pi}{n}
  7. ( n - 2 ) 2 n \frac{(n-2)}{2n}
  8. 360 n \tfrac{360}{n}
  9. n ( n - 3 ) 2 \tfrac{n(n-3)}{2}
  10. R = s 2 sin π n = a cos π n R=\frac{s}{2\sin{\frac{\pi}{n}}}=\frac{a}{\cos{\frac{\pi}{n}}}
  11. A = 1 2 n s a = 1 2 p a = 1 4 n s 2 cot π n = n a 2 tan π n = 1 2 n R 2 sin 2 π n A=\tfrac{1}{2}nsa=\tfrac{1}{2}pa=\tfrac{1}{4}ns^{2}\cot{\tfrac{\pi}{n}}=na^{2}% \tan{\tfrac{\pi}{n}}=\tfrac{1}{2}nR^{2}\sin{\tfrac{2\pi}{n}}
  12. n 4 cot π n \tfrac{n}{4}\cot{\tfrac{\pi}{n}}
  13. n 2 sin 2 π n \tfrac{n}{2}\sin{\tfrac{2\pi}{n}}
  14. n 2 π sin 2 π n \tfrac{n}{2\pi}\sin{\tfrac{2\pi}{n}}
  15. n tan π n n\tan{\tfrac{\pi}{n}}
  16. n π tan π n \tfrac{n}{\pi}\tan{\tfrac{\pi}{n}}
  17. [ u r a d i c a l , u 3 ] / 4 [u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]/4
  18. 3 [ u r a d i c a l , u 3 ] / 4 3[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]/4
  19. 3 [ u r a d i c a l , u 3 ] 3[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]
  20. 1 / 4 [ u r a d i c a l , u 25 + 10 , " [ u r a d i c a l , u 5 ] " ] 1/4[u^{\prime}radical^{\prime},u^{\prime}25+10^{\prime},"[u^{\prime}radical^{% \prime},u^{\prime}5^{\prime}]"]
  21. 5 / 4 [ u r a d i c a l , u ( 5 + , " [ u r a d i c a l , u 5 ] " , u ) / 2 ] 5/4[u^{\prime}radical^{\prime},u^{\prime}(5+^{\prime},"[u^{\prime}radical^{% \prime},u^{\prime}5^{\prime}]",u^{\prime})/2^{\prime}]
  22. 5 [ u r a d i c a l , u 5 - 2 , " [ u r a d i c a l , u 5 ] " ] 5[u^{\prime}radical^{\prime},u^{\prime}5-2^{\prime},"[u^{\prime}radical^{% \prime},u^{\prime}5^{\prime}]"]
  23. 3 [ u r a d i c a l , u 3 ] / 2 3[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]/2
  24. 3 [ u r a d i c a l , u 3 ] / 2 3[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]/2
  25. 2 [ u r a d i c a l , u 3 ] 2[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]
  26. 2 + 2 [ u r a d i c a l , u 2 ] 2+2[u^{\prime}radical^{\prime},u^{\prime}2^{\prime}]
  27. 2 [ u r a d i c a l , u 2 ] 2[u^{\prime}radical^{\prime},u^{\prime}2^{\prime}]
  28. 8 ( [ u r a d i c a l , u 2 ] - 1 ) 8([u^{\prime}radical^{\prime},u^{\prime}2^{\prime}]-1)
  29. 5 / 2 [ u r a d i c a l , u 5 + 2 , " [ u r a d i c a l , u 5 ] " ] 5/2[u^{\prime}radical^{\prime},u^{\prime}5+2^{\prime},"[u^{\prime}radical^{% \prime},u^{\prime}5^{\prime}]"]
  30. 5 / 2 [ u r a d i c a l , u ( 5 - , " [ u r a d i c a l , u 5 ] " , u ) / 2 ] 5/2[u^{\prime}radical^{\prime},u^{\prime}(5-^{\prime},"[u^{\prime}radical^{% \prime},u^{\prime}5^{\prime}]",u^{\prime})/2^{\prime}]
  31. 2 [ u r a d i c a l , u 25 - 10 , " [ u r a d i c a l , u 5 ] " ] 2[u^{\prime}radical^{\prime},u^{\prime}25-10^{\prime},"[u^{\prime}radical^{% \prime},u^{\prime}5^{\prime}]"]
  32. 6 + 3 [ u r a d i c a l , u 3 ] 6+3[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]
  33. 12 ( 2 - [ u r a d i c a l , u 3 ] ) 12(2-[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}])
  34. 4 ( 1 + [ u r a d i c a l , u 2 ] + [ u r a d i c a l , u 2 ( 2 + , " [ u r a d i c a l , u 2 ] " , u ) ] ) 4(1+[u^{\prime}radical^{\prime},u^{\prime}2^{\prime}]+[u^{\prime}radical^{% \prime},u^{\prime}2(2+^{\prime},"[u^{\prime}radical^{\prime},u^{\prime}2^{% \prime}]",u^{\prime})^{\prime}])
  35. 4 [ u r a d i c a l , u 2 - , " [ u r a d i c a l , u 2 ] " ] 4[u^{\prime}radical^{\prime},u^{\prime}2-^{\prime},"[u^{\prime}radical^{\prime% },u^{\prime}2^{\prime}]"]
  36. 16 ( 1 + [ u r a d i c a l , u 2 ] ) ( [ u r a d i c a l , u 2 ( 2 - , " [ u r a d i c a l , u 2 ] " , u ) ] - 1 ) 16(1+[u^{\prime}radical^{\prime},u^{\prime}2^{\prime}])([u^{\prime}radical^{% \prime},u^{\prime}2(2-^{\prime},"[u^{\prime}radical^{\prime},u^{\prime}2^{% \prime}]",u^{\prime})^{\prime}]-1)
  37. 5 ( 1 + [ u r a d i c a l , u 5 ] + [ u r a d i c a l , u 5 + 2 , " [ u r a d i c a l , u 5 ] " ] ) 5(1+[u^{\prime}radical^{\prime},u^{\prime}5^{\prime}]+[u^{\prime}radical^{% \prime},u^{\prime}5+2^{\prime},"[u^{\prime}radical^{\prime},u^{\prime}5^{% \prime}]"])
  38. 5 / 2 ( [ u r a d i c a l , u 5 ] - 1 ) 5/2([u^{\prime}radical^{\prime},u^{\prime}5^{\prime}]-1)
  39. 20 ( 1 + [ u r a d i c a l , u 5 ] - [ u r a d i c a l , u 5 + 2 , " [ u r a d i c a l , u 5 ] " ] ) 20(1+[u^{\prime}radical^{\prime},u^{\prime}5^{\prime}]-[u^{\prime}radical^{% \prime},u^{\prime}5+2^{\prime},"[u^{\prime}radical^{\prime},u^{\prime}5^{% \prime}]"])
  40. 180 ( p - 2 q ) p \frac{180(p-2q)}{p}

Regular_polyhedron.html

  1. N 2 = 4 n 2 m + 2 n - m n N_{2}=\frac{4n}{2m+2n-mn}

Regular_polytope.html

  1. p 1 = a 1 x + b 1 y + + h 1 z 0 , p_{1}=a_{1}x+b_{1}y+\cdots+h_{1}z\geq 0,
  2. p 2 > 0 , , p n > 0 p_{2}>0,\ldots,p_{n}>0
  3. x 2 + y 2 + + z 2 < 1 x^{2}+y^{2}+\cdots+z^{2}<1

Regular_prime.html

  1. E n = B n ( 4 n - 2 n ) n E_{n}=\frac{B_{n}(4^{n}-2^{n})}{n}

Regular_representation.html

  1. λ ( g ) : h g h , for all h G . \lambda(g):h\mapsto gh,\,\text{ for all }h\in G.
  2. ρ ( g ) : h h g - 1 , for all h G . \rho(g):h\mapsto hg^{-1},\,\text{ for all }h\in G.
  3. ( λ ( g ) f ) ( x ) = f ( g - 1 x ) (\lambda(g)f)(x)=f(g^{-1}x)
  4. ( ρ ( g ) f ) ( x ) = f ( x g ) . (\rho(g)f)(x)=f(xg).

Regular_sequence.html

  1. 0 R ( d d ) R ( d 1 ) R R / ( r 1 , , r d ) 0 0\rightarrow R^{{\left({{d}\atop{d}}\right)}}\rightarrow\cdots\rightarrow R^{{% \left({{d}\atop{1}}\right)}}\rightarrow R\rightarrow R/(r_{1},\ldots,r_{d})\rightarrow 0
  2. j 0 I j / I j + 1 \oplus_{j\geq 0}I^{j}/I^{j+1}

Rejection_sampling.html

  1. m \mathbb{R}^{m}
  2. ( x , y ) (x,y)
  3. x x
  4. y y
  5. x 2 + y 2 1 x^{2}+y^{2}\leq 1
  6. f ( x ) f(x)
  7. g ( x ) g(x)
  8. f ( x ) < M g ( x ) f(x)<Mg(x)
  9. M > 1 M>1
  10. f ( x ) / g ( x ) f(x)/g(x)
  11. f ( x ) f(x)
  12. f ( x ) f(x)
  13. M g ( x ) Mg(x)
  14. M g ( x ) Mg(x)
  15. f ( x ) f(x)
  16. P r ( u < f ( x ) / M g ( x ) ) Pr(u<f(x)/Mg(x))
  17. = E [ f ( x ) / M g ( x ) ] = =E[f(x)/Mg(x)]=
  18. ( f ( x ) / M g ( x ) ) g ( x ) d x = ( 1 / M ) \int(f(x)/Mg(x))g(x)dx=(1/M)
  19. f ( x ) d x = 1 / M \int f(x)dx=1/M
  20. M M
  21. M M
  22. f ( x ) / g ( x ) f(x)/g(x)
  23. f ( x ) = g ( x ) f(x)=g(x)
  24. x x
  25. g ( x ) g(x)
  26. u u
  27. U ( 0 , 1 ) U(0,1)
  28. u < f ( x ) / M g ( x ) u<f(x)/Mg(x)
  29. x x
  30. f ( x ) f(x)
  31. x x
  32. ( x , v = u M g ( x ) ) (x,v=u\cdot Mg(x))
  33. M g ( x ) Mg(x)
  34. u < f ( x ) / M g ( x ) u<f(x)/Mg(x)
  35. ( x , v ) (x,v)
  36. f ( x ) f(x)
  37. f ( x ) . f(x).
  38. f ( x ) f(x)
  39. h ( x ) = log g ( x ) h\left(x\right)=\mathrm{log}\;g\left(x\right)
  40. g ( x ) g\left(x\right)
  41. f ( x ) f\left(x\right)
  42. log f ( x ) \mathrm{log}\;f\left(x\right)
  43. f ( x ) f\left(x\right)
  44. h ( x ) h\left(x\right)
  45. f ( x ) f\left(x\right)
  46. h ( x ) h\left(x\right)
  47. f ( x ) f\left(x\right)
  48. g l ( x ) g_{l}\left(x\right)
  49. h l ( x ) h_{l}\left(x\right)

Relative_strength_index.html

  1. U = closenow - closeprevious U=\,\text{close}\text{now}-\,\text{close}\text{previous}
  2. D = 0 D=0
  3. U = 0 U=0
  4. D = closeprevious - closenow D=\,\text{close}\text{previous}-\,\text{close}\text{now}
  5. R S = SMMA ( U , n ) SMMA ( D , n ) RS=\frac{\,\text{SMMA}(U,n)}{\,\text{SMMA}(D,n)}
  6. R S I = 100 - 100 1 + R S RSI=100-{100\over{1+RS}}
  7. R S = SMA ( U , n ) SMA ( D , n ) RS=\frac{\,\text{SMA}(U,n)}{\,\text{SMA}(D,n)}

Relativistic_Doppler_effect.html

  1. v v\,
  2. v v\,
  3. λ = c / f s \lambda=c/f_{s}\,
  4. λ \lambda\,
  5. f s f_{s}\,
  6. c c\,
  7. c c\,
  8. v v
  9. λ + v t = c t \lambda+vt=ct
  10. t = λ c - v = c ( c - v ) f s = 1 ( 1 - β ) f s , t=\frac{\lambda}{c-v}=\frac{c}{(c-v)f_{s}}=\frac{1}{(1-\beta)f_{s}},
  11. β = v / c \beta=v/c\,
  12. t o = t γ , t_{o}=\frac{t}{\gamma},
  13. γ = 1 1 - β 2 \gamma=\frac{1}{\sqrt{1-\beta^{2}}}
  14. f o = 1 t o = γ ( 1 - β ) f s = 1 - β 1 + β f s . f_{o}=\frac{1}{t_{o}}=\gamma(1-\beta)f_{s}=\sqrt{\frac{1-\beta}{1+\beta}}\,f_{% s}.
  15. f s f o = 1 + β 1 - β \frac{f_{s}}{f_{o}}=\sqrt{\frac{1+\beta}{1-\beta}}
  16. λ o λ s = f s f o = 1 + β 1 - β , \frac{\lambda_{o}}{\lambda_{s}}=\frac{f_{s}}{f_{o}}=\sqrt{\frac{1+\beta}{1-% \beta}},
  17. z = λ o - λ s λ s = f s - f o f o z=\frac{\lambda_{o}-\lambda_{s}}{\lambda_{s}}=\frac{f_{s}-f_{o}}{f_{o}}
  18. z = 1 + β 1 - β - 1. z=\sqrt{\frac{1+\beta}{1-\beta}}-1.
  19. v c v\ll c
  20. z β = v c , z\simeq\beta=\frac{v}{c},
  21. S S
  22. S S^{\prime}
  23. S S
  24. S S^{\prime}
  25. t = t = 0 t=t^{\prime}=0
  26. t t
  27. S S
  28. t t^{\prime}
  29. S S^{\prime}
  30. S S^{\prime}
  31. S S
  32. v v
  33. x x
  34. = =
  35. γ ( x + β c t ) \gamma(x^{\prime}+\beta ct^{\prime})
  36. y y
  37. = =
  38. y y^{\prime}
  39. z z
  40. = =
  41. z z^{\prime}
  42. c t ct
  43. = =
  44. γ ( c t + β x ) \gamma(ct^{\prime}+\beta x^{\prime})
  45. d x d t \frac{dx}{dt}
  46. = =
  47. d x d t + v 1 + v c 2 d x d t , \frac{\frac{dx^{\prime}}{dt^{\prime}}+v}{1+\frac{v}{c^{2}}\frac{dx^{\prime}}{% dt^{\prime}}},
  48. β = v / c \beta=v/c
  49. γ = ( 1 - β 2 ) - 1 / 2 \gamma=(1-\beta^{2})^{-1/2}
  50. c c
  51. S S^{\prime}
  52. O O^{\prime}
  53. S S^{\prime}
  54. t 1 = 0 t_{1}^{\prime}=0
  55. t 2 = 1 / f t_{2}^{\prime}=1/f^{\prime}
  56. f f^{\prime}
  57. S S^{\prime}
  58. S S
  59. S S^{\prime}
  60. S S
  61. x 1 = 0 x_{1}^{\prime}=0
  62. t 1 = 0 t_{1}^{\prime}=0
  63. x 1 = 0 x_{1}=0
  64. t 1 = 0 t_{1}=0
  65. x 2 = 0 x_{2}^{\prime}=0
  66. t 2 = 1 f t_{2}^{\prime}=\frac{1}{f^{\prime}}
  67. x 2 = γ v f x_{2}=\gamma\frac{v}{f^{\prime}}
  68. t 2 = γ 1 f t_{2}=\gamma\frac{1}{f^{\prime}}
  69. S S
  70. t 2 - t 1 t_{2}-t_{1}
  71. S S
  72. x 2 x 1 x_{2}\neq x_{1}
  73. x 2 x_{2}
  74. x 1 x_{1}
  75. x 2 - x 1 x_{2}-x_{1}
  76. S S
  77. - u -u^{\prime}
  78. S S^{\prime}
  79. S S
  80. O O
  81. O O
  82. S S
  83. - u = - u + v 1 + ( - u ) v c 2 , -u=\frac{-u^{\prime}+v}{1+(-u^{\prime})\frac{v}{c^{2}}},
  84. S S
  85. τ \tau
  86. = t 2 - t 1 + ( γ v f ) ( u - v 1 - v u / c 2 ) - 1 =t_{2}-t_{1}+\left(\gamma\frac{v}{f^{\prime}}\right)\left(\frac{u^{\prime}-v}{% 1-vu^{\prime}/c^{2}}\right)^{-1}
  87. = γ f + γ f v u - v ( 1 - v u / c 2 ) . =\frac{\gamma}{f^{\prime}}+\frac{\gamma}{f^{\prime}}\frac{v}{u^{\prime}-v}% \left(1-vu^{\prime}/c^{2}\right).
  88. τ \tau
  89. 1 / f 1/f
  90. f f^{\prime}
  91. f = γ ( 1 - v u ) f . f=\gamma\left(1-\frac{v}{u^{\prime}}\right)f^{\prime}.
  92. v c v\ll c
  93. c c\rightarrow\infty
  94. γ 1 \gamma\rightarrow 1
  95. f = ( 1 - v u ) f . f=\left(1-\frac{v}{u^{\prime}}\right)f^{\prime}.
  96. u = c u^{\prime}=c
  97. f = γ ( 1 - v c ) f = γ ( 1 - β ) f = f 1 - β 1 + β f=\gamma\left(1-\frac{v}{c}\right)f^{\prime}=\gamma\left(1-\beta\right)f^{% \prime}=f^{\prime}\sqrt{\frac{1-\beta}{1+\beta}}
  98. λ = λ 1 + β 1 - β , \lambda=\lambda^{\prime}\sqrt{\frac{1+\beta}{1-\beta}},
  99. λ \lambda^{\prime}
  100. O O^{\prime}
  101. S S^{\prime}
  102. c c\rightarrow\infty
  103. f = f f=f^{\prime}
  104. 1 γ = 1 - v 2 / c 2 . \frac{1}{\gamma}=\sqrt{1-v^{2}/c^{2}\,}.
  105. γ = 1 1 - v 2 / c 2 . \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}\,}}.
  106. f o = f s γ ( 1 + v cos θ o c ) f_{o}=\frac{f_{s}}{\gamma\left(1+\frac{v\cos\theta_{o}}{c}\right)}
  107. f o = f s γ f_{o}=\frac{f_{s}}{\gamma}\,
  108. v v\,
  109. θ o \theta_{o}\,
  110. f o = f s γ ( 1 + v c cos θ o ) . f_{o}=\frac{f_{s}}{\gamma\left(1+\frac{v}{c}\cos\theta_{o}\right)}.
  111. θ o = 90 \theta_{o}=90^{\circ}\,
  112. cos θ o = 0 \cos\theta_{o}=0\,
  113. f o = f s γ . f_{o}=\frac{f_{s}}{\gamma}.\,
  114. θ o \theta_{o}\,
  115. θ s \theta_{s}\,
  116. cos θ o \cos\theta_{o}\,
  117. cos θ s \cos\theta_{s}\,
  118. cos θ o = cos θ s - v c 1 - v c cos θ s . \cos\theta_{o}=\frac{\cos\theta_{s}-\frac{v}{c}}{1-\frac{v}{c}\cos\theta_{s}}\,.
  119. f o = γ ( 1 - v cos θ s c ) f s . f_{o}=\gamma\left(1-\frac{v\cos\theta_{s}}{c}\right)f_{s}.
  120. cos θ s = 0 \cos\theta_{s}=0\,
  121. f o = γ f s . f_{o}=\gamma f_{s}.\,
  122. Δ f f - v cos θ c . \frac{\Delta f}{f}\simeq-\frac{v\cos\theta}{c}.
  123. ν 3 / ( e h ν / k T - 1 ) \nu^{3}/(e^{h\nu/kT}-1)
  124. f o f s = 1 - v o c c o s ( θ c o ) 1 - v s c c o s ( θ c s ) 1 - ( v s / c ) 2 1 - ( v o / c ) 2 \frac{f_{o}}{f_{s}}=\frac{1-\frac{\|\vec{v_{o}}\|}{\|\vec{c}\|}cos(\theta_{co}% )}{1-\frac{\|\vec{v_{s}}\|}{\|\vec{c}\|}cos(\theta_{cs})}\sqrt{\frac{1-(v_{s}/% c)^{2}}{1-(v_{o}/c)^{2}}}
  125. v s \vec{v_{s}}
  126. v o \vec{v_{o}}
  127. c \vec{c}
  128. θ c s \theta_{cs}
  129. θ c o \theta_{co}
  130. c \vec{c}
  131. v s \vec{v_{s}}
  132. θ c s = 0 \theta_{cs}=0^{\circ}
  133. f o f_{o}
  134. f s f_{s}
  135. c \vec{c}
  136. v s \vec{v_{s}}
  137. θ c s = 180 \theta_{cs}=180^{\circ}
  138. f o f_{o}
  139. f s f_{s}

Relativistic_kill_vehicle.html

  1. 1 2 m v 2 \begin{matrix}\frac{1}{2}\end{matrix}mv^{2}
  2. E k = γ m c 2 - m c 2 E_{k}=\gamma mc^{2}-mc^{2}\,
  3. γ \gamma
  4. γ = 1 1 - v 2 c 2 \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  5. E k = m c 2 ( 1 1 - v 2 c 2 - 1 ) E_{k}=mc^{2}\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)

Relativistic_rocket.html

  1. γ \gamma
  2. Δ v \Delta v
  3. I s p I_{sp}
  4. m 0 m_{0}
  5. m 1 m_{1}
  6. I s p = v e I_{sp}=v_{e}
  7. I s p = v e 1 - v e 2 c 2 = γ e v e , I_{sp}=\frac{v_{e}}{\sqrt{1-\frac{v_{e}^{2}}{c^{2}}}}=\gamma_{e}\ v_{e},
  8. γ \gamma
  9. I s p I_{sp}
  10. η \eta
  11. η = 1 - 1 - I s p 2 c 2 = 1 - 1 γ s p . \eta=1-\sqrt{1-\frac{I_{sp}^{2}}{c^{2}}}=1-\frac{1}{\gamma_{sp}}.
  12. I s p = c 2 η - η 2 . I_{sp}=c\cdot\sqrt{2\eta-\eta^{2}}.
  13. η \eta
  14. I s p / c I_{sp}/c
  15. I s p I_{sp}
  16. Δ v = I s p ln m 0 m 1 \Delta v=I_{sp}\ln\frac{m_{0}}{m_{1}}
  17. a a
  18. t t
  19. t = I s p a ln m 0 m 1 t=\frac{I_{sp}}{a}\ln\frac{m_{0}}{m_{1}}
  20. a a
  21. t t
  22. m 0 m 1 = exp [ a t I s p ] \frac{m_{0}}{m_{1}}=\exp\left[\frac{at}{I_{sp}}\right]
  23. Δ v \Delta v
  24. m 0 m 1 = [ 1 + Δ v c 1 - Δ v c ] c 2 I s p \frac{m_{0}}{m_{1}}=\left[\frac{1+{\frac{\Delta v}{c}}}{1-{\frac{\Delta v}{c}}% }\right]^{\frac{c}{2I_{sp}}}
  25. Δ v c = tanh [ a t c ] \frac{\Delta v}{c}=\tanh\left[\frac{at}{c}\right]
  26. tanh x = e 2 x - 1 e 2 x + 1 \tanh x=\frac{e^{2x}-1}{e^{2x}+1}
  27. m 0 m 1 = exp [ a t I s p ] \frac{m_{0}}{m_{1}}=\exp\left[\frac{at}{I_{sp}}\right]
  28. Δ v \Delta v
  29. t t^{\prime}
  30. Δ v = a t 1 + ( a t ) 2 c 2 \Delta v=\frac{a\cdot t^{\prime}}{\sqrt{1+\frac{(a\cdot t^{\prime})^{2}}{c^{2}% }}}
  31. t = c a sinh ( a t c ) t^{\prime}=\frac{c}{a}\sinh\left(\frac{a\cdot t}{c}\right)
  32. Δ v \Delta v
  33. Δ v = c tanh ( I s p c ln m 0 m 1 ) \Delta v=c\cdot\tanh\left(\frac{I_{sp}}{c}\ln\frac{m_{0}}{m_{1}}\right)
  34. Δ r = I s p c ln m 0 m 1 \Delta r=\frac{I_{sp}}{c}\ln\frac{m_{0}}{m_{1}}
  35. Δ v \Delta v
  36. β \beta
  37. γ \gamma
  38. d v s h i p dv_{ship}
  39. d v s h i p dv_{ship}
  40. m s h i p c 2 = d m e c 2 1 - v e 2 c 2 + m s h i p c 2 - d m f u e l c 2 m_{ship}\ c^{2}=\frac{dm_{e}\ c^{2}}{\sqrt{1-\frac{v_{e}^{2}}{c^{2}}}}+m_{ship% }\ c^{2}-dm_{fuel}\ c^{2}
  41. d m f u e l c 2 = d m e c 2 1 - v e 2 c 2 dm_{fuel}\ c^{2}=\frac{dm_{e}\ c^{2}}{\sqrt{1-\frac{v_{e}^{2}}{c^{2}}}}
  42. η \eta
  43. d m e = ( 1 - η ) d m f u e l dm_{e}=(1-\eta)\ dm_{fuel}
  44. d m f u e l dm_{fuel}
  45. d m e dm_{e}
  46. η \eta
  47. c 2 c^{2}
  48. d m e dm_{e}
  49. 1 1 - η = 1 1 - v e 2 c 2 \frac{1}{1-\eta}=\frac{1}{\sqrt{1-\frac{v_{e}^{2}}{c^{2}}}}
  50. v e v_{e}
  51. v e = c 2 η - η 2 v_{e}=c\ \sqrt{2\eta-\eta^{2}}
  52. d p e = d m e v e 1 - v e 2 c 2 dp_{e}=\frac{dm_{e}\ v_{e}}{\sqrt{1-\frac{v_{e}^{2}}{c^{2}}}}
  53. d p e = d m f u e l ( 1 - η ) v e 1 - v e 2 c 2 dp_{e}=\frac{dm_{fuel}\ (1-\eta)\ v_{e}}{\sqrt{1-\frac{v_{e}^{2}}{c^{2}}}}
  54. I s p I_{sp}
  55. I s p = d p e d m f u e l = ( 1 - η ) v e 1 - v e 2 c 2 I_{sp}=\frac{dp_{e}}{dm_{fuel}}=\frac{(1-\eta)\ v_{e}}{\sqrt{1-\frac{v_{e}^{2}% }{c^{2}}}}
  56. I s p = c 2 η - η 2 I_{sp}=c\ \sqrt{2\eta-\eta^{2}}

Relativistic_wave_equations.html

  1. ψ ψ
  2. Ψ Ψ
  3. i t ψ = H ^ ψ i\hbar\frac{\partial}{\partial t}\psi=\hat{H}\psi
  4. ħ ħ
  5. 1 2 \frac{1}{2}
  6. - 2 2 ψ t 2 + ( c ) 2 2 ψ = ( m c 2 ) 2 ψ , -\hbar^{2}\frac{\partial^{2}\psi}{\partial t^{2}}+(\hbar c)^{2}\nabla^{2}\psi=% (mc^{2})^{2}\psi\,,
  7. 1 2 \frac{1}{2}
  8. ( E c - s y m b o l α 𝐩 - β m c ) ( E c + s y m b o l α 𝐩 + β m c ) ψ = 0 , \left(\frac{E}{c}-symbol{\alpha}\cdot\mathbf{p}-\beta mc\right)\left(\frac{E}{% c}+symbol{\alpha}\cdot\mathbf{p}+\beta mc\right)\psi=0\,,
  9. α \mathbf{α}
  10. β β
  11. 1 2 \frac{1}{2}
  12. 1 2 \frac{1}{2}
  13. ( E c + s y m b o l α 𝐩 - β m c ) ψ = 0 , \left(\frac{E}{c}+symbol{\alpha}\cdot\mathbf{p}-\beta mc\right)\psi=0\,,
  14. ψ ψ
  15. α \mathbf{α}
  16. β β
  17. A A
  18. B B
  19. n + ½ n+½
  20. n n
  21. p γ α ˙ A ϵ 1 ϵ 2 ϵ n α ˙ β ˙ 1 β ˙ 2 β ˙ n = m c B γ ϵ 1 ϵ 2 ϵ n β ˙ 1 β ˙ 2 β ˙ n p_{\gamma\dot{\alpha}}A_{\epsilon_{1}\epsilon_{2}\cdots\epsilon_{n}}^{\dot{% \alpha}\dot{\beta}_{1}\dot{\beta}_{2}\cdots\dot{\beta}_{n}}=mcB_{\gamma% \epsilon_{1}\epsilon_{2}\cdots\epsilon_{n}}^{\dot{\beta}_{1}\dot{\beta}_{2}% \cdots\dot{\beta}_{n}}
  22. p γ α ˙ B γ ϵ 1 ϵ 2 ϵ n β ˙ 1 β ˙ 2 β ˙ n = m c A ϵ 1 ϵ 2 ϵ n α ˙ β ˙ 1 β ˙ 2 β ˙ n p^{\gamma\dot{\alpha}}B_{\gamma\epsilon_{1}\epsilon_{2}\cdots\epsilon_{n}}^{% \dot{\beta}_{1}\dot{\beta}_{2}\cdots\dot{\beta}_{n}}=mcA_{\epsilon_{1}\epsilon% _{2}\cdots\epsilon_{n}}^{\dot{\alpha}\dot{\beta}_{1}\dot{\beta}_{2}\cdots\dot{% \beta}_{n}}
  23. p p
  24. n = 0 n=0
  25. A A
  26. B B
  27. A A
  28. B B
  29. A A
  30. B B
  31. 3 / 2 {3}/{2}
  32. n + ½ n+½
  33. n n
  34. ψ ψ
  35. μ = 0 , 1 , 2 , 3 μ=0,1,2,3
  36. 2 × 2 2×2
  37. σ 0 = ( 1 0 0 1 ) \sigma^{0}=\begin{pmatrix}1&0\\ 0&1\\ \end{pmatrix}
  38. σ μ μ σ 0 0 + σ 1 1 + σ 2 2 + σ 3 3 \sigma^{\mu}\partial_{\mu}\equiv\sigma^{0}\partial_{0}+\sigma^{1}\partial_{1}+% \sigma^{2}\partial_{2}+\sigma^{3}\partial_{3}
  39. 2 × 2 2×2
  40. μ = 0 , 1 , 2 , 3 μ=0,1,2,3
  41. 4 × 4 4×4
  42. i γ μ μ + m c i ( γ 0 0 + γ 1 1 + γ 2 2 + γ 3 3 ) + m c ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) i\hbar\gamma^{\mu}\partial_{\mu}+mc\equiv i\hbar(\gamma^{0}\partial_{0}+\gamma% ^{1}\partial_{1}+\gamma^{2}\partial_{2}+\gamma^{3}\partial_{3})+mc\begin{% pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  43. 4 × 4 4×4
  44. m c mc
  45. 2 × 2 2×2
  46. 4 × 4 4×4
  47. ( μ + i m c ) ( μ - i m c ) ψ = 0 (\hbar\partial_{\mu}+imc)(\hbar\partial^{\mu}-imc)\psi=0
  48. σ μ μ ψ = 0 \sigma^{\mu}\partial_{\mu}\psi=0
  49. ( i / - m c ) ψ = 0 \left(i\hbar\partial\!\!\!/-mc\right)\psi=0
  50. [ ( γ 1 ) μ ( p 1 - A ~ 1 ) μ + m 1 + S ~ 1 ] Ψ = 0 , [(\gamma_{1})_{\mu}(p_{1}-\tilde{A}_{1})^{\mu}+m_{1}+\tilde{S}_{1}]\Psi=0,
  51. [ ( γ 2 ) μ ( p 2 - A ~ 2 ) μ + m 2 + S ~ 2 ] Ψ = 0. [(\gamma_{2})_{\mu}(p_{2}-\tilde{A}_{2})^{\mu}+m_{2}+\tilde{S}_{2}]\Psi=0.
  52. i / ψ - m c ψ c = 0 i\hbar\partial\!\!\!/\psi-mc\psi_{c}=0
  53. i Ψ t = ( i H ^ D ( i ) + i > j 1 r i j - i > j B ^ i j ) Ψ i\hbar\frac{\partial\Psi}{\partial t}=\left(\sum_{i}\hat{H}_{D}(i)+\sum_{i>j}% \frac{1}{r_{ij}}-\sum_{i>j}\hat{B}_{ij}\right)\Psi
  54. μ μ A ν = e ψ ¯ γ ν ψ \partial_{\mu}\partial^{\mu}A^{\nu}=e\overline{\psi}\gamma^{\nu}\psi
  55. μ ( μ A ν - ν A μ ) + ( m c ) 2 A ν = 0 \partial_{\mu}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})+\left(\frac{mc}{% \hbar}\right)^{2}A^{\nu}=0
  56. ϵ μ ν ρ σ γ 5 γ ν ρ ψ σ + m ψ μ = 0 \epsilon^{\mu\nu\rho\sigma}\gamma^{5}\gamma_{\nu}\partial_{\rho}\psi_{\sigma}+% m\psi^{\mu}=0
  57. ( - i γ μ μ + m c ) α 1 α 1 ψ α 1 α 2 α 3 α 2 s = 0 (-i\hbar\gamma^{\mu}\partial_{\mu}+mc)_{\alpha_{1}\alpha_{1}^{\prime}}\psi_{% \alpha^{\prime}_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{2s}}=0
  58. ( - i γ μ μ + m c ) α 2 α 2 ψ α 1 α 2 α 3 α 2 s = 0 (-i\hbar\gamma^{\mu}\partial_{\mu}+mc)_{\alpha_{2}\alpha_{2}^{\prime}}\psi_{% \alpha_{1}\alpha^{\prime}_{2}\alpha_{3}\cdots\alpha_{2s}}=0
  59. \qquad\vdots
  60. ( - i γ μ μ + m c ) α 2 s α 2 s ψ α 1 α 2 α 3 α 2 s = 0 (-i\hbar\gamma^{\mu}\partial_{\mu}+mc)_{\alpha_{2s}\alpha^{\prime}_{2s}}\psi_{% \alpha_{1}\alpha_{2}\alpha_{3}\cdots\alpha^{\prime}_{2s}}=0
  61. ψ ψ
  62. ( i β a a - m c ) ψ = 0 (i\hbar\beta^{a}\partial_{a}-mc)\psi=0
  63. R μ ν - 1 2 g μ ν R + g μ ν Λ = 8 π G c 4 T μ ν R_{\mu\nu}-{1\over 2}g_{\mu\nu}\,R+g_{\mu\nu}\Lambda={8\pi G\over c^{4}}T_{\mu\nu}

Remainder.html

  1. a ( x ) = b ( x ) q ( x ) + r ( x ) a(x)=b(x)q(x)+r(x)
  2. deg ( r ( x ) ) < deg ( b ( x ) ) , \deg(r(x))<\deg(b(x)),

Renal_clearance_ratio.html

  1. c l e a r a n c e r a t i o o f X = C x C i n clearance\ ratio\ of\ X=\frac{C_{x}}{C_{in}}

Renal_function.html

  1. G F R = Urine Concentration × Urine Flow Plasma Concentration GFR=\frac{\mbox{Urine Concentration}~{}\times\mbox{Urine Flow}~{}}{\mbox{% Plasma Concentration}~{}}
  2. d Q d t = K f × ( P G - P B - Π G + Π B ) {\operatorname{d}Q\over\operatorname{d}t}=K_{f}\times(P_{G}-P_{B}-\Pi_{G}+\Pi_% {B})
  3. d Q d t {\operatorname{d}Q\over\operatorname{d}t}
  4. K f K_{f}
  5. P G P_{G}
  6. P B P_{B}
  7. Π G \Pi_{G}
  8. Π B \Pi_{B}
  9. K f K_{f}
  10. K f = GFR Net Filt. Pressure = GFR ( P G - P B - Π G + Π B ) K_{f}=\frac{\textrm{GFR}}{\textrm{Net\ Filt.\ Pressure}}=\frac{\textrm{GFR}}{(% P_{G}-P_{B}-\Pi_{G}+\Pi_{B})}
  11. P a - P G = R a × Q a P_{a}-P_{G}=R_{a}\times Q_{a}
  12. P G - P e = R e × Q e P_{G}-P_{e}=R_{e}\times Q_{e}
  13. P a P_{a}
  14. P e P_{e}
  15. R a R_{a}
  16. R e R_{e}
  17. Q a Q_{a}
  18. Q e Q_{e}
  19. P B - P d = R d × ( Q a - Q e ) P_{B}-P_{d}=R_{d}\times(Q_{a}-Q_{e})
  20. P d P_{d}
  21. R d R_{d}
  22. Π G = R T c \Pi_{G}=RTc
  23. C C r = U C r × V ˙ P C r C_{Cr}=\frac{U_{Cr}\times\dot{V}}{P_{Cr}}
  24. C C r = 1.25 m g / m L × 60 m L 60 m i n 0.01 m g / m L = 1.25 m g / m L × 1 m L / m i n 0.01 m g / m L = 1.25 m g / m i n 0.01 m g / m L = 125 m L / m i n C_{Cr}=\frac{1.25mg/mL\times\frac{60mL}{60min}}{0.01mg/mL}=\frac{{1.25mg/mL}% \times{1mL/min}}{0.01mg/mL}=\frac{1.25mg/min}{0.01mg/mL}={125mL/min}
  25. C C r = U C r × 24-hour volume P C r × 24 × 60 m i n s C_{Cr}=\frac{U_{Cr}\ \times\ \mbox{24-hour volume}~{}}{P_{Cr}\ \times\ 24% \times 60mins}
  26. C C r - c o r r e c t e d = C C r × 1.73 B S A C_{Cr-corrected}=\frac{{C_{Cr}}\ \times\ {1.73}}{BSA}
  27. e C C r = (140 - Age) × Mass (in kilograms) × [ 0.85 i f F e m a l e ] 72 × Serum Creatinine (in mg/dL) eC_{Cr}=\frac{\mbox{(140 - Age)}~{}\ \times\ \mbox{Mass (in kilograms)}~{}\ % \times\ [{0.85\ if\ Female}]}{\mbox{72}~{}\ \times\ \mbox{Serum Creatinine (in% mg/dL)}~{}}
  28. e C C r = (140 - Age) × Mass (in kilograms) × C o n s t a n t Serum Creatinine (in μ mol/L) eC_{Cr}=\frac{\mbox{(140 - Age)}~{}\ \times\ \mbox{Mass (in kilograms)}~{}\ % \times\ {Constant}}{\mbox{Serum Creatinine (in }~{}\mu\mbox{mol/L)}~{}}
  29. eGFR = 32788 × Serum Creatinine × - 1.154 Age × - 0.203 [ 1.210 i f B l a c k ] × [ 0.742 i f F e m a l e ] \mbox{eGFR}~{}=\mbox{32788}~{}\ \times\ \mbox{Serum Creatinine}~{}^{-1.154}\ % \times\ \mbox{Age}~{}^{-0.203}\ \times\ {[1.210\ if\ Black]}\ \times\ {[0.742% \ if\ Female]}
  30. eGFR = 186 × Serum Creatinine × - 1.154 Age × - 0.203 [ 1.210 i f B l a c k ] × [ 0.742 i f F e m a l e ] \mbox{eGFR}~{}=\mbox{186}~{}\ \times\ \mbox{Serum Creatinine}~{}^{-1.154}\ % \times\ \mbox{Age}~{}^{-0.203}\ \times\ {[1.210\ if\ Black]}\ \times\ {[0.742% \ if\ Female]}
  31. eGFR = 170 × Serum Creatinine × - 0.999 Age × - 0.176 [ 0.762 i f F e m a l e ] × [ 1.180 i f B l a c k ] × BUN × - 0.170 Albumin + 0.318 \mbox{eGFR}~{}=\mbox{170}~{}\ \times\ \mbox{Serum Creatinine}~{}^{-0.999}\ % \times\ \mbox{Age}~{}^{-0.176}\ \times\ {[0.762\ if\ Female]}\ \times\ {[1.180% \ if\ Black]}\ \times\ \mbox{BUN}~{}^{-0.170}\ \times\ \mbox{Albumin}~{}^{+0.3% 18}
  32. eGFR = 141 × min(SCr/k,1) × a max(SCr/k,1) × - 1.209 0.993 × A g e [ 1.018 i f F e m a l e ] × [ 1.159 i f B l a c k ] \mbox{eGFR}~{}=\mbox{141}~{}\ \times\ \mbox{min(SCr/k,1)}~{}^{a}\ \times\ % \mbox{max(SCr/k,1)}~{}^{-1.209}\ \times\ \mbox{0.993}~{}^{Age}\ \times\ {[1.01% 8\ if\ Female]}\ \times\ {[1.159\ if\ Black]}
  33. eGFR = 166 × (SCr/0.7) × - 1.209 0.993 A g e \mbox{eGFR}~{}=\mbox{166}~{}\ \times\ \mbox{(SCr/0.7)}~{}^{-1.209}\ \times\ % \mbox{0.993}~{}^{Age}
  34. eGFR = 163 × (SCr/0.9) × - 1.209 0.993 A g e \mbox{eGFR}~{}=\mbox{163}~{}\ \times\ \mbox{(SCr/0.9)}~{}^{-1.209}\ \times\ % \mbox{0.993}~{}^{Age}
  35. eGFR = 144 × (SCr/0.7) × - 1.209 0.993 A g e \mbox{eGFR}~{}=\mbox{144}~{}\ \times\ \mbox{(SCr/0.7)}~{}^{-1.209}\ \times\ % \mbox{0.993}~{}^{Age}
  36. eGFR = 141 × (SCr/0.9) × - 1.209 0.993 A g e \mbox{eGFR}~{}=\mbox{141}~{}\ \times\ \mbox{(SCr/0.9)}~{}^{-1.209}\ \times\ % \mbox{0.993}~{}^{Age}
  37. eGFR = exp ( 1.911 + 5.249 / S e r u m C r e a t i n i n e - 2.114 / S e r u m C r e a t i n i n e 2 - 0.00686 × Age - [ 0.205 i f F e m a l e ] ) \mbox{eGFR}~{}=\mbox{exp}~{}{(1.911+5.249/{Serum\ Creatinine}-2.114/{Serum\ % Creatinine}^{2}-0.00686\ \times\ \mbox{Age}~{}-{[0.205\ if\ Female]})}
  38. eGFR = k × H e i g h t S e r u m C r e a t i n i n e \mbox{eGFR}~{}=\frac{{k}\times{Height}}{Serum\ Creatinine}

Renal_physiology.html

  1. V ˙ - U o s m P o s m V ˙ \dot{V}-\frac{U_{osm}}{P_{osm}}\dot{V}
  2. = C H 2 O =C_{H_{2}O}

Representativeness_heuristic.html

  1. P ( H | D ) = P ( D | H ) P ( H ) P ( D ) . P(H|D)=\frac{P(D|H)\,P(H)}{P(D)}.
  2. P ( H | D ) = P ( D | H ) P(H|D)=P(D|H)
  3. P ( c o n s c i e n t i o u s | n e u r o t i c ) = P ( n e u r o t i c | c o n s c i e n t i o u s ) P(conscientious|neurotic)=P(neurotic|conscientious)

Residual_value.html

  1. Residual value = 10 % × ( 20 , 000 ) = 2 , 000 \,\text{Residual value}=10\%\times(20{,}000)=2{,}000

Resistance_force.html

  1. R × D R = E × D E R\times D_{R}=E\times D_{E}
  2. W o r k o u t p u t = R × D R Work\,output=R\times D_{R}

Resonator.html

  1. d d\,
  2. 2 d 2d\,
  3. 2 d 2d\,
  4. λ \lambda\,
  5. 2 d = N λ , N { 1 , 2 , 3 , } 2d=N\lambda,\qquad\qquad N\in\{1,2,3,\dots\}
  6. c c\,
  7. f = c / λ f=c/\lambda\,
  8. f = N c 2 d N { 1 , 2 , 3 , } f=\frac{Nc}{2d}\qquad\qquad N\in\{1,2,3,\dots\}

Resultant_force.html

  1. F 1 {\scriptstyle\vec{F}_{1}}
  2. F 2 \scriptstyle\vec{F}_{2}
  3. F 1 \scriptstyle\vec{F}_{1}
  4. F R \scriptstyle\vec{F}_{R}
  5. F 2 \scriptstyle\vec{F}_{2}
  6. F R \scriptstyle\vec{F}_{R}
  7. τ = F d \scriptstyle\tau=Fd
  8. d \scriptstyle d
  9. 𝐅 = i = 1 n 𝐅 i , \mathbf{F}=\sum_{i=1}^{n}\mathbf{F}_{i},
  10. 𝐓 = i = 1 n ( 𝐑 i - 𝐑 ) × 𝐅 i . \mathbf{T}=\sum_{i=1}^{n}(\mathbf{R}_{i}-\mathbf{R})\times\mathbf{F}_{i}.
  11. 𝐓 = i = 1 n ( 𝐑 i - ( 𝐑 + k 𝐅 ) ) × 𝐅 i . \mathbf{T}=\sum_{i=1}^{n}(\mathbf{R}_{i}-(\mathbf{R}+k\mathbf{F}))\times% \mathbf{F}_{i}.
  12. 𝐓 = i = 1 n ( 𝐑 i - 𝐑 ) × 𝐅 i - i = 1 n k 𝐅 × 𝐅 i = i = 1 n ( 𝐑 i - 𝐑 ) × 𝐅 i , \mathbf{T}=\sum_{i=1}^{n}(\mathbf{R}_{i}-\mathbf{R})\times\mathbf{F}_{i}-\sum_% {i=1}^{n}k\mathbf{F}\times\mathbf{F}_{i}=\sum_{i=1}^{n}(\mathbf{R}_{i}-\mathbf% {R})\times\mathbf{F}_{i},
  13. i = 1 n k 𝐅 × 𝐅 i = k 𝐅 × ( i = 1 n 𝐅 i ) = 0 , \sum_{i=1}^{n}k\mathbf{F}\times\mathbf{F}_{i}=k\mathbf{F}\times(\sum_{i=1}^{n}% \mathbf{F}_{i})=0,
  14. 𝐑 × 𝐅 = i = 1 n 𝐑 i × 𝐅 i , \mathbf{R}\times\mathbf{F}=\sum_{i=1}^{n}\mathbf{R}_{i}\times\mathbf{F}_{i},
  15. 𝐅 ( i = 1 n 𝐑 i × 𝐅 i ) = 0. \mathbf{F}\cdot(\sum_{i=1}^{n}\mathbf{R}_{i}\times\mathbf{F}_{i})=0.
  16. 𝖶 = i = 1 n 𝖶 i = i = 1 n ( 𝐅 i , 𝐑 i × 𝐅 i ) . \mathsf{W}=\sum_{i=1}^{n}\mathsf{W}_{i}=\sum_{i=1}^{n}(\mathbf{F}_{i},\mathbf{% R}_{i}\times\mathbf{F}_{i}).

Reuleaux_triangle.html

  1. 1 2 ( π - 3 ) s 2 0.70477 s 2 , \frac{1}{2}(\pi-\sqrt{3})s^{2}\approx 0.70477s^{2},
  2. π s 2 / 4 0.78540 s 2 \pi s^{2}/4\approx 0.78540s^{2}
  3. π s \pi s
  4. ( 1 - 1 3 ) s 0.42265 s \displaystyle\left(1-\frac{1}{\sqrt{3}}\right)s\approx 0.42265s
  5. s 3 0.57735 s \displaystyle\frac{s}{\sqrt{3}}\approx 0.57735s
  6. 2 ( π - 3 ) 15 + 7 - 12 0.922888 , \frac{2(\pi-\sqrt{3})}{\sqrt{15}+\sqrt{7}-\sqrt{12}}\approx 0.922888,
  7. w - r d 3 , w-r\leq\frac{d}{\sqrt{3}},

Reversal_potential.html

  1. I i o n = g i o n ( V m - E i o n ) {I_{ion}}={g_{ion}}({V_{m}}-{E_{ion}})\,
  2. V m - E i o n {V_{m}}-{E_{ion}}
  3. E P C = g A C h ( V m - E r e v ) EPC={g_{ACh}}({V_{m}}-{E_{rev}})\,

Reverse_mathematics.html

  1. Γ 0 \Gamma_{0}

Reversible_reaction.html

  1. a A + b B c C + d D aA+bB\rightleftharpoons cC+dD

Rewriting.html

  1. ¬ ¬ A A \neg\neg A\to A
  2. ¬ ( A B ) ¬ A ¬ B \neg(A\land B)\to\neg A\lor\neg B
  3. ¬ ( A B ) ¬ A ¬ B \neg(A\lor B)\to\neg A\land\neg B
  4. ( A B ) C ( A C ) ( B C ) (A\land B)\lor C\to(A\lor C)\land(B\lor C)
  5. A ( B C ) ( A B ) ( A C ) A\lor(B\land C)\to(A\lor B)\land(A\lor C)
  6. \to
  7. * \stackrel{*}{\rightarrow}
  8. = \rightarrow\cup=
  9. * \stackrel{*}{\rightarrow}
  10. \rightarrow
  11. \rightarrow
  12. \leftrightarrow
  13. - 1 \rightarrow\cup\rightarrow^{-1}
  14. \rightarrow
  15. * \stackrel{*}{\leftrightarrow}
  16. = \leftrightarrow\cup=
  17. * \stackrel{*}{\leftrightarrow}
  18. \rightarrow
  19. \rightarrow
  20. x y x\rightarrow y
  21. x * y x\stackrel{*}{\rightarrow}y
  22. x x\downarrow
  23. c = a = b c=a\downarrow=b\downarrow
  24. x * z * y x\stackrel{*}{\rightarrow}z\stackrel{*}{\leftarrow}y
  25. * * \stackrel{*}{\rightarrow}\circ\stackrel{*}{\leftarrow}
  26. \circ
  27. \downarrow
  28. x , y x\mathbin{\downarrow}y
  29. * \stackrel{*}{\leftrightarrow}
  30. * \stackrel{*}{\leftrightarrow}
  31. x * y x\stackrel{*}{\leftrightarrow}y
  32. x , y x\mathbin{\downarrow}y
  33. ( A , ) (A,\rightarrow)
  34. x * w * y x\stackrel{*}{\leftarrow}w\stackrel{*}{\rightarrow}y
  35. x , y x\mathbin{\downarrow}y
  36. x w y x\leftarrow w\rightarrow y
  37. x , y x\mathbin{\downarrow}y
  38. x * y x\stackrel{*}{\leftrightarrow}y
  39. x * y x\stackrel{*}{\rightarrow}y
  40. x * y x\stackrel{*}{\leftrightarrow}y
  41. a b , b a , a c , b d a\rightarrow b,\;b\rightarrow a,\;a\rightarrow c,\;b\rightarrow d
  42. c c
  43. d d
  44. x 0 x 1 x 2 x_{0}\rightarrow x_{1}\rightarrow x_{2}\rightarrow\cdots
  45. a b a b a\rightarrow b\rightarrow a\rightarrow b\rightarrow\cdots
  46. R R
  47. ( Σ , R ) (\Sigma,R)
  48. Σ \Sigma
  49. R R
  50. R \rightarrow_{R}
  51. R R
  52. Σ * \Sigma^{*}
  53. s s
  54. t t
  55. Σ * \Sigma^{*}
  56. s R t s\rightarrow_{R}t
  57. x x
  58. y y
  59. u u
  60. v v
  61. Σ * \Sigma^{*}
  62. s = x u y s=xuy
  63. t = x v y t=xvy
  64. u R v uRv
  65. R \rightarrow_{R}
  66. Σ * \Sigma^{*}
  67. ( Σ * , R ) (\Sigma^{*},\rightarrow_{R})
  68. R R
  69. R \rightarrow_{R}
  70. R R
  71. * R \stackrel{*}{\rightarrow}_{R}
  72. x * R y x\stackrel{*}{\rightarrow}_{R}y
  73. u x v * R u y v uxv\stackrel{*}{\rightarrow}_{R}uyv
  74. x x
  75. y y
  76. u u
  77. v v
  78. Σ * \Sigma^{*}
  79. R \rightarrow_{R}
  80. * R \stackrel{*}{\leftrightarrow}_{R}
  81. * R \stackrel{*}{\leftrightarrow}_{R}
  82. R R
  83. R R
  84. * R \stackrel{*}{\rightarrow}_{R}
  85. * R \stackrel{*}{\leftrightarrow}_{R}
  86. * R \stackrel{*}{\leftrightarrow}_{R}
  87. R = Σ * / * R \mathcal{M}_{R}=\Sigma^{*}/\stackrel{*}{\leftrightarrow}_{R}
  88. Σ * \Sigma^{*}
  89. \mathcal{M}
  90. R \mathcal{M}_{R}
  91. ( Σ , R ) (\Sigma,R)
  92. \mathcal{M}
  93. ( Σ , R ) (\Sigma,R)
  94. ( ) (\vee)
  95. ( ) (\wedge)
  96. ( ¬ ) (\neg)
  97. l r l\longrightarrow r
  98. l l
  99. r r
  100. R R
  101. l r l\longrightarrow r
  102. s s
  103. l l
  104. s s
  105. s | p = l σ s\mid_{p}=l\sigma
  106. p p
  107. s s
  108. σ \sigma
  109. t t
  110. t = s [ r σ ] p t=s[r\sigma]_{p}
  111. s s
  112. t t
  113. R R
  114. s R t s\longrightarrow_{R}t
  115. s R t s\stackrel{R}{\longrightarrow}t
  116. t 1 t_{1}
  117. t n t_{n}
  118. t 1 R t 2 R R t n t_{1}\longrightarrow_{R}t_{2}\longrightarrow_{R}\ldots\longrightarrow_{R}t_{n}
  119. t 1 t_{1}
  120. t n t_{n}
  121. t 1 R + t n t_{1}\longrightarrow_{R}^{+}t_{n}
  122. R + \longrightarrow_{R}^{+}
  123. R \longrightarrow_{R}
  124. R * \longrightarrow_{R}^{*}
  125. R \longrightarrow_{R}
  126. s R * t s\longrightarrow_{R}^{*}t
  127. s = t s=t
  128. s R + t s\longrightarrow_{R}^{+}t
  129. R R
  130. R \longrightarrow_{R}
  131. x * ( y * z ) ( x * y ) * z x*(y*z)\longrightarrow(x*y)*z
  132. * *
  133. a * ( ( a + 1 ) * ( a + 2 ) ) 1 * ( 2 * 3 ) \frac{a*((a+1)*(a+2))}{1*(2*3)}
  134. { x a , y a + 1 , z a + 2 } \{x\mapsto a,\;y\mapsto a+1,\;z\mapsto a+2\}
  135. ( a * ( a + 1 ) ) * ( a + 2 ) (a*(a+1))*(a+2)
  136. ( a * ( a + 1 ) ) * ( a + 2 ) 1 * ( 2 * 3 ) \frac{(a*(a+1))*(a+2)}{1*(2*3)}
  137. * *
  138. a * ( ( a + 1 ) * ( a + 2 ) ) 1 * ( 2 * 3 ) \frac{a*((a+1)*(a+2))}{1*(2*3)}
  139. a * ( ( a + 1 ) * ( a + 2 ) ) ( 1 * 2 ) * 3 \frac{a*((a+1)*(a+2))}{(1*2)*3}
  140. f ( 0 , 1 , x ) f ( x , x , x ) f(0,1,x)\rightarrow f(x,x,x)
  141. g ( x , y ) x g(x,y)\rightarrow x
  142. g ( x , y ) y g(x,y)\rightarrow y
  143. f ( g ( 0 , 1 ) , g ( 0 , 1 ) , g ( 0 , 1 ) ) f ( 0 , g ( 0 , 1 ) , g ( 0 , 1 ) ) f ( 0 , 1 , g ( 0 , 1 ) ) f ( g ( 0 , 1 ) , g ( 0 , 1 ) , g ( 0 , 1 ) ) f(g(0,1),g(0,1),g(0,1))\rightarrow f(0,g(0,1),g(0,1))\rightarrow f(0,1,g(0,1))% \rightarrow f(g(0,1),g(0,1),g(0,1))\rightarrow\ldots
  144. R 1 R_{1}
  145. R 2 R_{2}
  146. R 1 R_{1}
  147. R 2 R_{2}
  148. R 1 R_{1}
  149. R 2 R_{2}
  150. s | p s\mid_{p}
  151. s s
  152. p p
  153. l σ l\sigma
  154. σ \sigma
  155. l l
  156. s [ r σ ] p s[r\sigma]_{p}
  157. p p
  158. s s
  159. r σ r\sigma
  160. x * ( y * z ) x*(y*z)
  161. a * ( ( a + 1 ) * ( a + 2 ) ) a*((a+1)*(a+2))

Reynolds-averaged_Navier–Stokes_equations.html

  1. X ¯ \bar{X}
  2. x x
  3. X ¯ = lim T 1 T t 0 t 0 + T x d t . \bar{X}=\lim_{T\to\infty}\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x\,dt.
  4. X ¯ \bar{X}
  5. t 0 t_{0}
  6. T T
  7. t 0 t_{0}
  8. T T
  9. T T
  10. ρ u ¯ j u ¯ i x j = ρ f ¯ i + x j [ - p ¯ δ i j + μ ( u ¯ i x j + u ¯ j x i ) - ρ u i u j ¯ ] . \rho\bar{u}_{j}\frac{\partial\bar{u}_{i}}{\partial x_{j}}=\rho\bar{f}_{i}+% \frac{\partial}{\partial x_{j}}\left[-\bar{p}\delta_{ij}+\mu\left(\frac{% \partial\bar{u}_{i}}{\partial x_{j}}+\frac{\partial\bar{u}_{j}}{\partial x_{i}% }\right)-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right].
  11. ( - ρ u i u j ¯ ) \left(-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right)
  12. . ¯ \overline{.}
  13. u u
  14. u ¯ \overline{u}
  15. u u^{\prime}
  16. ( u ¯ = 0 ) (\bar{u^{\prime}}=0)
  17. u ( s y m b o l x , t ) = u ¯ ( s y m b o l x ) + u ( s y m b o l x , t ) u(symbol{x},t)=\bar{u}(symbol{x})+u^{\prime}(symbol{x},t)\,
  18. s y m b o l x = ( x , y , z ) symbol{x}=(x,y,z)
  19. U U
  20. u ¯ \bar{u}
  21. u u^{\prime}
  22. u u
  23. u , u ¯ , and u u,\bar{u},\mbox{ and }~{}u^{\prime}
  24. u i x i = 0 \frac{\partial u_{i}}{\partial x_{i}}=0
  25. u i t + u j u i x j = f i - 1 ρ p x i + ν 2 u i x j x j \frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}=f% _{i}-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\nu\frac{\partial^{2}u_{i% }}{\partial x_{j}\partial x_{j}}
  26. f i f_{i}
  27. ( u i ¯ + u i ) x i = 0 \frac{\partial\left(\bar{u_{i}}+u_{i}^{\prime}\right)}{\partial x_{i}}=0
  28. ( u i ¯ + u i ) t + ( u j ¯ + u j ) ( u i ¯ + u i ) x j = ( f i ¯ + f i ) - 1 ρ ( p ¯ + p ) x i + ν 2 ( u i ¯ + u i ) x j x j . \frac{\partial\left(\bar{u_{i}}+u_{i}^{\prime}\right)}{\partial t}+\left(\bar{% u_{j}}+u_{j}^{\prime}\right)\frac{\partial\left(\bar{u_{i}}+u_{i}^{\prime}% \right)}{\partial x_{j}}=\left(\bar{f_{i}}+f_{i}^{\prime}\right)-\frac{1}{\rho% }\frac{\partial\left(\bar{p}+p^{\prime}\right)}{\partial x_{i}}+\nu\frac{% \partial^{2}\left(\bar{u_{i}}+u_{i}^{\prime}\right)}{\partial x_{j}\partial x_% {j}}.
  29. ( u i ¯ + u i ) x i ¯ = 0 \overline{\frac{\partial\left(\bar{u_{i}}+u_{i}^{\prime}\right)}{\partial x_{i% }}}=0
  30. ( u i ¯ + u i ) t ¯ + ( u j ¯ + u j ) ( u i ¯ + u i ) x j ¯ = ( f i ¯ + f i ) ¯ - 1 ρ ( p ¯ + p ) x i ¯ + ν 2 ( u i ¯ + u i ) x j x j ¯ . \overline{\frac{\partial\left(\bar{u_{i}}+u_{i}^{\prime}\right)}{\partial t}}+% \overline{\left(\bar{u_{j}}+u_{j}^{\prime}\right)\frac{\partial\left(\bar{u_{i% }}+u_{i}^{\prime}\right)}{\partial x_{j}}}=\overline{\left(\bar{f_{i}}+f_{i}^{% \prime}\right)}-\frac{1}{\rho}\overline{\frac{\partial\left(\bar{p}+p^{\prime}% \right)}{\partial x_{i}}}+\nu\overline{\frac{\partial^{2}\left(\bar{u_{i}}+u_{% i}^{\prime}\right)}{\partial x_{j}\partial x_{j}}}.
  31. u i u i ¯ \overline{u_{i}u_{i}}
  32. u i u i ¯ = ( u i ¯ + u i ) ( u i ¯ + u i ) ¯ = u i ¯ u i ¯ + u i ¯ u i + u i u i ¯ + u i u i ¯ = u i ¯ u i ¯ + u i u i ¯ \overline{u_{i}u_{i}}=\overline{\left(\bar{u_{i}}+u_{i}^{\prime}\right)\left(% \bar{u_{i}}+u_{i}^{\prime}\right)}=\overline{\bar{u_{i}}\bar{u_{i}}+\bar{u_{i}% }u_{i}^{\prime}+u_{i}^{\prime}\bar{u_{i}}+u_{i}^{\prime}u_{i}^{\prime}}=\bar{u% _{i}}\bar{u_{i}}+\overline{u_{i}^{\prime}u_{i}^{\prime}}
  33. u i ¯ x i = 0 \frac{\partial\bar{u_{i}}}{\partial x_{i}}=0
  34. u i ¯ t + u j ¯ u i ¯ x j + u j u i x j ¯ = f i ¯ - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j . \frac{\partial\bar{u_{i}}}{\partial t}+\bar{u_{j}}\frac{\partial\bar{u_{i}}}{% \partial x_{j}}+\overline{u_{j}^{\prime}\frac{\partial u_{i}^{\prime}}{% \partial x_{j}}}=\bar{f_{i}}-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{% i}}+\nu\frac{\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}.
  35. u i x i = u i ¯ x i + u i x i = 0 \frac{\partial u_{i}}{\partial x_{i}}=\frac{\partial\bar{u_{i}}}{\partial x_{i% }}+\frac{\partial u_{i}^{\prime}}{\partial x_{i}}=0
  36. u i ¯ t + u j ¯ u i ¯ x j = f i ¯ - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j - u i u j ¯ x j . \frac{\partial\bar{u_{i}}}{\partial t}+\bar{u_{j}}\frac{\partial\bar{u_{i}}}{% \partial x_{j}}=\bar{f_{i}}-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i% }}+\nu\frac{\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}-\frac{% \partial\overline{u_{i}^{\prime}u_{j}^{\prime}}}{\partial x_{j}}.
  37. ρ u i ¯ t + ρ u j ¯ u i ¯ x j = ρ f i ¯ + x j [ - p ¯ δ i j + 2 μ S i j ¯ - ρ u i u j ¯ ] \rho\frac{\partial\bar{u_{i}}}{\partial t}+\rho\bar{u_{j}}\frac{\partial\bar{u% _{i}}}{\partial x_{j}}=\rho\bar{f_{i}}+\frac{\partial}{\partial x_{j}}\left[-% \bar{p}\delta_{ij}+2\mu\bar{S_{ij}}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}% }\right]
  38. S i j ¯ = 1 2 ( u i ¯ x j + u j ¯ x i ) \bar{S_{ij}}=\frac{1}{2}\left(\frac{\partial\bar{u_{i}}}{\partial x_{j}}+\frac% {\partial\bar{u_{j}}}{\partial x_{i}}\right)
  39. ρ u j ¯ u i ¯ x j = ρ f i ¯ + x j [ - p ¯ δ i j + 2 μ S i j ¯ - ρ u i u j ¯ ] . \rho\bar{u_{j}}\frac{\partial\bar{u_{i}}}{\partial x_{j}}=\rho\bar{f_{i}}+% \frac{\partial}{\partial x_{j}}\left[-\bar{p}\delta_{ij}+2\mu\bar{S_{ij}}-\rho% \overline{u_{i}^{\prime}u_{j}^{\prime}}\right].

Rg_chromaticity.html

  1. ( r , g , b ) (r,g,b)
  2. r , g , b r,g,b
  3. r = R R + G + B r=\frac{R}{R+G+B}
  4. g = G R + G + B g=\frac{G}{R+G+B}
  5. b = B R + G + B b=\frac{B}{R+G+B}
  6. r + g + b = 1 r+g+b=1
  7. R = r G g R=\frac{rG}{g}
  8. G = G G=G
  9. B = ( 1 - r - g ) G g B=\frac{(1-r-g)G}{g}
  10. f c ( x ) = m b ( x ) ω s ( λ , x ) ρ c ( λ ) d λ f^{c}(x)=m^{b}(x)\int_{\omega}s(\lambda,x)\rho^{c}(\lambda)d\lambda
  11. r = m b ( x ) k R m b ( x ) ( k R + k G + k B ) = k R k R + k G + k B r=\frac{m^{b}(x)k_{R}}{m^{b}(x)(k_{R}+k_{G}+k_{B})}=\frac{k_{R}}{k_{R}+k_{G}+k% _{B}}
  12. g = m b ( x ) k G m b ( x ) ( k R + k G + k B ) = k G k R + k G + k B g=\frac{m^{b}(x)k_{G}}{m^{b}(x)(k_{R}+k_{G}+k_{B})}=\frac{k_{G}}{k_{R}+k_{G}+k% _{B}}
  13. b = m b ( x ) k B m b ( x ) ( k R + k G + k B ) = k B k R + k G + k B b=\frac{m^{b}(x)k_{B}}{m^{b}(x)(k_{R}+k_{G}+k_{B})}=\frac{k_{B}}{k_{R}+k_{G}+k% _{B}}
  14. k c = ω s ( λ , x ) ρ c ( λ ) d λ k_{c}=\int_{\omega}s(\lambda,x)\rho^{c}(\lambda)d\lambda
  15. c { R , G , B } c\in{\{R,G,B\}}
  16. m b ( x ) m^{b}(x)
  17. k c k_{c}
  18. k c k_{c}
  19. ρ c ( λ ) \rho^{c}(\lambda)
  20. s ( λ , x ) s(\lambda,x)
  21. ρ \rho
  22. s s
  23. λ R = 700.0 n m , λ G = 546.1 n m , λ B = 435.8 n m \lambda_{R}=700.0nm,\lambda_{G}=546.1nm,\lambda_{B}=435.8nm
  24. [ F λ ] [F_{\lambda}]
  25. R [ R ] , G [ G ] R[R],G[G]
  26. B [ B ] B[B]
  27. R R
  28. [ R ] , [ G ] [R],[G]
  29. [ B ] [B]
  30. [ F ] [F]
  31. [ R ] , [ G ] [R],[G]
  32. [ B ] [B]
  33. [ F λ ] + R [ R ] = G [ G ] + B [ B ] [F_{\lambda}]+R[R]=G[G]+B[B]
  34. [ F λ ] = - R [ R ] + G [ G ] + B [ B ] [F_{\lambda}]=-R[R]+G[G]+B[B]
  35. [ R ] [R]
  36. r ¯ \overline{r}

Rho.html

  1. A A
  2. ρ ( A ) \rho(A)
  3. π \pi
  4. ρ π \rho^{\pi}
  5. ρ \rho
  6. ϱ \varrho

Rhombic_dodecahedron.html

  1. r i = 6 3 a 0.8164965809 a , r_{i}=\frac{\sqrt{6}}{3}a\approx 0.8164965809a,
  2. r m = 2 2 3 a 0.94280904158 a , r_{m}=\frac{2\sqrt{2}}{3}a\approx 0.94280904158a,
  3. r o = 2 3 3 a 1.154700538 a , r_{o}=\frac{2\sqrt{3}}{3}a\approx 1.154700538a,
  4. A = 8 2 a 2 11.3137085 a 2 A=8\sqrt{2}a^{2}\approx 11.3137085a^{2}
  5. V = 16 9 3 a 3 3.07920144 a 3 V=\frac{16}{9}\sqrt{3}a^{3}\approx 3.07920144a^{3}

Richardson_number.html

  1. Ri = buoyancy term flow gradient term = g ρ ρ ( u ) 2 \mathrm{Ri}=\frac{\,\text{buoyancy term}}{\,\text{flow gradient term}}=\frac{g% }{\rho}\frac{\nabla\rho}{(\nabla u)^{2}}
  2. Ri = g ρ ρ ( u ) 2 \mathrm{Ri}=\frac{g^{\prime}}{\rho}\frac{\nabla\rho}{(\nabla u)^{2}}
  3. Ri = g β ( T hot - T ref ) L V 2 \mathrm{Ri}=\frac{g\beta(T\text{hot}-T\text{ref})L}{V^{2}}
  4. β \beta
  5. Ri = Gr Re 2 . \mathrm{Ri}=\frac{\mathrm{Gr}}{\mathrm{Re}^{2}}.
  6. Ri = N 2 / ( d u / d z ) 2 \mathrm{Ri}=N^{2}/(\mathrm{d}u/\mathrm{d}z)^{2}

Riemann–Hurwitz_formula.html

  1. 2 - 2 g 2-2g\,
  2. π : S S \pi:S^{\prime}\to S\,
  3. χ ( S ) = N χ ( S ) . \chi(S^{\prime})=N\cdot\chi(S).\,
  4. χ ( S ) = N χ ( S ) - P S ( e P - 1 ) \chi(S^{\prime})=N\cdot\chi(S)-\sum_{P\in S^{\prime}}(e_{P}-1)
  5. \wp
  6. 0 = 2 2 - Σ 1 0=2\cdot 2-\Sigma\ 1
  7. 2 = n 2 - ( n - 1 ) - ( e - 1 ) 2=n\cdot 2-(n-1)-(e_{\infty}-1)
  8. χ ( S ) = N χ ( S ) \chi(S^{\prime})=N\cdot\chi(S)\,
  9. χ \chi\,