wpmath0000001_19

Radiation_mode.html

  1. β = n 2 ( a ) k 2 - ( l / a ) 2 \beta=\sqrt{n^{2}(a)k^{2}-(l/a)^{2}}

Radiation_pattern.html

  1. G G
  2. r r
  3. W ( θ , Φ ) = G ( θ , Φ ) 4 π r 2 P t \mathrm{W}(\theta,\Phi)=\frac{\mathrm{G}(\theta,\Phi)}{4\pi r^{2}}P_{t}
  4. θ \theta
  5. Φ \Phi
  6. P t P_{t}
  7. G G
  8. P r = A ( θ , Φ ) W P_{r}=\mathrm{A}(\theta,\Phi)W\,
  9. W W
  10. A A
  11. A A
  12. P r = A G 4 π r 2 P t P_{r}=A\frac{G}{4\pi r^{2}}P_{t}
  13. G G
  14. A A
  15. P 1 r = A 1 ( θ , Φ ) G 2 4 π r 2 P 2 t P_{1r}=\mathrm{A_{1}}(\theta,\Phi)\frac{G_{2}}{4\pi r^{2}}P_{2t}
  16. P 2 r = A 2 G 1 ( θ , Φ ) 4 π r 2 P 1 t P_{2r}=A_{2}\frac{\mathrm{G_{1}}(\theta,\Phi)}{4\pi r^{2}}P_{1t}
  17. G 2 G_{2}
  18. A 2 A_{2}
  19. P 1 r P 2 t = P 2 r P 1 t \frac{P_{1r}}{P_{2t}}=\frac{P_{2r}}{P_{1t}}
  20. A 1 ( θ , Φ ) G 1 ( θ , Φ ) = A 2 G 2 \frac{\mathrm{A_{1}}(\theta,\Phi)}{\mathrm{G_{1}}(\theta,\Phi)}=\frac{A_{2}}{G% _{2}}
  21. A 1 ( θ , Φ ) G 1 ( θ , Φ ) = constant \frac{\mathrm{A_{1}}(\theta,\Phi)}{\mathrm{G_{1}}(\theta,\Phi)}=\mathrm{constant}
  22. λ 2 4 π \frac{\lambda^{2}}{4\pi}
  23. λ \lambda
  24. A ( θ , Φ ) = λ 2 G ( θ , Φ ) 4 π \mathrm{A}(\theta,\Phi)=\frac{\lambda^{2}\mathrm{G}(\theta,\Phi)}{4\pi}
  25. P r = λ 2 G r G t ( 4 π r ) 2 P t P_{r}=\frac{\lambda^{2}G_{r}G_{t}}{(4\pi r)^{2}}P_{t}

Radiation_pressure.html

  1. 𝐒 = 𝐄 × 𝐇 \mathbf{S}=\mathbf{E}\times\mathbf{H}
  2. 𝐒 \langle\mathbf{S}\rangle
  3. P a b s o r b = S c = E f c P_{absorb}=\frac{\langle S\rangle}{c}=\frac{E_{f}}{c}
  4. P a b s o r b = E f c cos α P_{absorb}=\frac{E_{f}}{c}\cos\alpha
  5. p = h λ = m c p=\dfrac{h}{\lambda}=mc
  6. E = m c 2 = p c E=mc^{2}=pc
  7. p = E c p=\dfrac{E}{c}
  8. P r e f l e c t = 2 E f c cos 2 α P_{reflect}=\frac{2E_{f}}{c}\cos^{2}\alpha
  9. ε < 1 \varepsilon<1
  10. E f = ε σ T 4 E_{f}=\varepsilon\sigma T^{4}
  11. σ \sigma
  12. T T
  13. ε = ε ( λ ) . \varepsilon=\varepsilon(\lambda).
  14. P e m i s s i o n = E f c = ε σ c T 4 P_{emission}=\frac{E_{f}}{c}=\frac{\varepsilon\sigma}{c}T^{4}
  15. P c o m p r e s s = u 3 = 4 σ 3 c T 4 P_{compress}=\frac{u}{3}=\frac{4\sigma}{3c}T^{4}
  16. u u
  17. T T
  18. σ \sigma
  19. c c
  20. P a b s o r b = W c R 2 cos α P_{absorb}=\frac{W}{cR^{2}}\cos\alpha
  21. P a b s o r b = 4.54 R 2 cos α P_{absorb}=\frac{4.54}{R^{2}}\cos\alpha
  22. P r e f l e c t = 2 W c R 2 cos 2 α P_{reflect}=\frac{2W}{cR^{2}}\cos^{2}\alpha
  23. P r e f l e c t = 9.08 R 2 cos 2 α P_{reflect}=\frac{9.08}{R^{2}}\cos^{2}\alpha

Radio_fix.html

  1. LOP = TH + RB + 180 \mbox{LOP}~{}=\mbox{TH}~{}+\mbox{RB}~{}+180
  2. LOP = 120 + 40 + 180 \mbox{LOP}~{}=120+40+180
  3. LOP = 340 \mbox{LOP}~{}=340
  4. LOP1 = 120 + 320 + 180 \mbox{LOP1}~{}=120+320+180
  5. LOP1 = 260 \mbox{LOP1}~{}=260
  6. LOP2 = 120 + 40 + 180 \mbox{LOP2}~{}=120+40+180
  7. LOP2 = 340 \mbox{LOP2}~{}=340

Radiocarbon_dating.html

  1. n + N 7 14 C 6 14 + p n+\mathrm{{}^{14}_{7}N}\rightarrow\mathrm{{}^{14}_{6}C}+p
  2. C 6 14 N 7 14 + e - + ν ¯ e \mathrm{~{}^{14}_{6}C}\rightarrow\mathrm{~{}^{14}_{7}N}+e^{-}+\bar{\nu}_{e}
  3. N = N 0 e - λ t N=N_{0}e^{-\lambda t}\,
  4. t = 8267 ln ( N 0 / N ) y e a r s = 19035 log ( N 0 / N ) y e a r s t=8267\cdot\ln(N_{0}/N)years=19035\cdot\log(N_{0}/N)years
  5. T 1 2 = 0.693 τ T_{\frac{1}{2}}=0.693\cdot\tau
  6. δ 13 C = ( ( C 13 C 12 ) sample ( C 13 C 12 ) PDB - 1 ) × 1000 o / o o \mathrm{\delta^{13}C}=\Biggl(\mathrm{\frac{\bigl(\frac{{}^{13}C}{{}^{12}C}% \bigr)_{sample}}{\bigl(\frac{{}^{13}C}{{}^{12}C}\bigr)_{PDB}}}-1\Biggr)\times 1% 000\ ^{o}\!/\!_{oo}
  7. A g e = - 8033 l n ( F m ) Age=-8033\cdot ln(Fm)

Radiometry.html

  1. Φ e . \Phi_{\mathrm{e}}.
  2. Φ e , λ = d Φ e d λ , \Phi_{\mathrm{e},\lambda}={\mathrm{d}\Phi_{\mathrm{e}}\over\mathrm{d}\lambda},
  3. d Φ e \mathrm{d}\Phi_{\mathrm{e}}
  4. Φ e , ν = d Φ e d ν , \Phi_{\mathrm{e},\nu}={\mathrm{d}\Phi_{\mathrm{e}}\over\mathrm{d}\nu},
  5. d Φ e \mathrm{d}\Phi_{\mathrm{e}}
  6. λ Φ e , λ = ν Φ e , ν . \lambda\Phi_{\mathrm{e},\lambda}=\nu\Phi_{\mathrm{e},\nu}.
  7. Φ e , λ = c λ 2 Φ e , ν , \Phi_{\mathrm{e},\lambda}={c\over\lambda^{2}}\Phi_{\mathrm{e},\nu},
  8. Φ e , ν = c ν 2 Φ e , λ , \Phi_{\mathrm{e},\nu}={c\over\nu^{2}}\Phi_{\mathrm{e},\lambda},
  9. λ = c ν . \lambda={c\over\nu}.
  10. Φ e = 0 Φ e , λ d λ = 0 Φ e , ν d ν = 0 λ Φ e , λ d ln λ = 0 ν Φ e , ν d ln ν . \Phi_{\mathrm{e}}=\int_{0}^{\infty}\Phi_{\mathrm{e},\lambda}\,\mathrm{d}% \lambda=\int_{0}^{\infty}\Phi_{\mathrm{e},\nu}\,\mathrm{d}\nu=\int_{0}^{\infty% }\lambda\Phi_{\mathrm{e},\lambda}\,\mathrm{d}\ln\lambda=\int_{0}^{\infty}\nu% \Phi_{\mathrm{e},\nu}\,\mathrm{d}\ln\nu.

Radiosity_(computer_graphics).html

  1. B ( x ) d A = E ( x ) d A + ρ ( x ) d A S B ( x ) 1 π r 2 cos θ x cos θ x Vis ( x , x ) d A B(x)\,dA=E(x)\,dA+\rho(x)\,dA\int_{S}B(x^{\prime})\frac{1}{\pi r^{2}}\cos% \theta_{x}\cos\theta_{x^{\prime}}\cdot\mathrm{Vis}(x,x^{\prime})\,\mathrm{d}A^% {\prime}
  2. B i = E i + ρ i j = 1 n F i j B j B_{i}=E_{i}+\rho_{i}\sum_{j=1}^{n}F_{ij}B_{j}
  3. B = ( I - ρ F ) - 1 E B=(I-\rho F)^{-1}E\;
  4. A i B i = A i E i + ρ i j = 1 n A j B j F j i A_{i}B_{i}=A_{i}E_{i}+\rho_{i}\sum_{j=1}^{n}A_{j}B_{j}F_{ji}

Radius_of_convergence.html

  1. f ( z ) = n = 0 c n ( z - a ) n , f(z)=\sum_{n=0}^{\infty}c_{n}(z-a)^{n},
  2. | z - a | < r |z-a|<r\,
  3. | z - a | > r . |z-a|>r.\,
  4. r = sup { | z - z 0 | | n = 0 c n ( z - z 0 ) n converges } r=\sup\left\{|z-z_{0}|\ \left|\ \sum_{n=0}^{\infty}c_{n}(z-z_{0})^{n}\ \,\text% { converges }\right.\right\}
  5. c n c_{n}
  6. C = lim sup n | c n ( z - a ) n | n = lim sup n | c n | n | z - a | C=\limsup_{n\rightarrow\infty}\sqrt[n]{|c_{n}(z-a)^{n}|}=\limsup_{n\rightarrow% \infty}\sqrt[n]{|c_{n}|}|z-a|
  7. r = 1 lim sup n | c n | n r=\frac{1}{\limsup_{n\rightarrow\infty}\sqrt[n]{|c_{n}|}}
  8. r = lim n | c n c n + 1 | . r=\lim_{n\rightarrow\infty}\left|\frac{c_{n}}{c_{n+1}}\right|.
  9. lim n | c n + 1 ( z - a ) n + 1 | | c n ( z - a ) n | < 1. \lim_{n\to\infty}\frac{|c_{n+1}(z-a)^{n+1}|}{|c_{n}(z-a)^{n}|}<1.
  10. | z - a | < 1 lim n | c n + 1 | | c n | = lim n | c n c n + 1 | . |z-a|<\frac{1}{\lim_{n\to\infty}\frac{|c_{n+1}|}{|c_{n}|}}=\lim_{n\to\infty}% \left|\frac{c_{n}}{c_{n+1}}\right|.
  11. c n c_{n}
  12. n n
  13. 1 / r 1/r
  14. { b n } n = 1 \{b_{n}\}_{n=1}^{\infty}
  15. b n 2 = c n + 1 c n - 1 - c n 2 c n c n - 2 - c n - 1 2 . b_{n}^{2}=\frac{c_{n+1}c_{n-1}-c_{n}^{2}}{c_{n}c_{n-2}-c_{n-1}^{2}}.
  16. 1 r = lim n b n \frac{1}{r}=\lim_{n\to\infty}{b_{n}}
  17. b n b_{n}
  18. b n b_{n}
  19. 1 / n 1/n
  20. 1 r = lim n c n c n - 1 \frac{1}{r}=\lim_{n\to\infty}{\frac{c_{n}}{c_{n-1}}}
  21. r r
  22. c n c n - 1 \frac{c_{n}}{c_{n-1}}
  23. 1 / n 1/n
  24. p p
  25. θ \theta
  26. - p + 1 r -\frac{p+1}{r}
  27. c n - 1 b n c n + c n + 1 c n b n 2 \frac{\frac{c_{n-1}b_{n}}{c_{n}}+\frac{c_{n+1}}{c_{n}b_{n}}}{2}
  28. 1 n 2 \frac{1}{n^{2}}
  29. cos θ \cos\theta
  30. f ( z ) = 1 1 + z 2 f(z)=\frac{1}{1+z^{2}}
  31. 1 + z 2 1+z^{2}
  32. n = 0 ( - 1 ) n z 2 n . \sum_{n=0}^{\infty}(-1)^{n}z^{2n}.
  33. arctan ( z ) = z - z 3 3 + z 5 5 - z 7 7 + . \arctan(z)=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\cdots.
  34. z e z - 1 = n = 0 B n n ! z n \frac{z}{e^{z}-1}=\sum_{n=0}^{\infty}\frac{B_{n}}{n!}z^{n}
  35. e z - 1 = 0 e^{z}-1=0\,
  36. e z = e x e i y = e x ( cos ( y ) + i sin ( y ) ) , e^{z}=e^{x}e^{iy}=e^{x}(\cos(y)+i\sin(y)),\,
  37. r r
  38. z z
  39. | z a | = r |z−a|=r
  40. ƒ ( z ) = 1 / ( 1 z ) ƒ(z)=1/(1−z)
  41. z = 0 z=0
  42. n = 0 z n , \sum_{n=0}^{\infty}z^{n},
  43. g ( z ) = l n ( 1 z ) g(z)=−ln(1−z)
  44. z = 0 z=0
  45. n = 1 1 n z n , \sum_{n=1}^{\infty}\frac{1}{n}z^{n},
  46. z = 1 z=1
  47. ƒ ( z ) ƒ(z)
  48. g ( z ) g(z)
  49. n = 1 1 n 2 z n \sum_{n=1}^{\infty}\frac{1}{n^{2}}z^{n}
  50. h h
  51. h ( z ) h(z)
  52. i = 1 a i z i where a i = ( - 1 ) n - 1 2 n n for n = log 2 ( i ) + 1 , the unique integer with 2 n - 1 i < 2 n , \sum_{i=1}^{\infty}a_{i}z^{i}\,\text{ where }a_{i}=\frac{(-1)^{n-1}}{2^{n}n}\,% \text{ for }n=\lfloor\log_{2}(i)\rfloor+1\,\text{, the unique integer with }2^% {n-1}\leq i<2^{n},
  53. f ( x ) = sin x = n = 0 ( - 1 ) n ( 2 n + 1 ) ! x 2 n + 1 = x - x 3 3 ! + x 5 5 ! - for all x f(x)=\sin x=\sum^{\infty}_{n=0}\frac{(-1)^{n}}{(2n+1)!}x^{2n+1}=x-\frac{x^{3}}% {3!}+\frac{x^{5}}{5!}-\cdots\,\text{ for all }x
  54. \scriptstyle\infty

Radius_of_gyration.html

  1. R g 2 = I A R_{\mathrm{g}}^{2}=\frac{I}{A}
  2. R g = I A R_{\mathrm{g}}=\sqrt{\frac{I}{A}}
  3. r g axis r_{\mathrm{g}\,\text{ axis}}
  4. I axis I\text{axis}
  5. r g axis 2 = I axis m r_{\mathrm{g}\,\text{ axis}}^{2}=\frac{I\text{axis}}{m}
  6. r g axis = I axis m r_{\mathrm{g}\,\text{ axis}}=\sqrt{\frac{I\text{axis}}{m}}
  7. I axis I\text{axis}
  8. R g 2 = def 1 N k = 1 N ( 𝐫 k - 𝐫 mean ) 2 R_{\mathrm{g}}^{2}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{N}\sum_{k=1}^{N}\left% (\mathbf{r}_{k}-\mathbf{r}_{\mathrm{mean}}\right)^{2}
  9. 𝐫 mean \mathbf{r}_{\mathrm{mean}}
  10. R g 2 = def 1 2 N 2 i , j ( 𝐫 i - 𝐫 j ) 2 R_{\mathrm{g}}^{2}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{2N^{2}}\sum_{i,j}% \left(\mathbf{r}_{i}-\mathbf{r}_{j}\right)^{2}
  11. R g 2 = def 1 N k = 1 N ( 𝐫 k - 𝐫 mean ) 2 R_{\mathrm{g}}^{2}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{N}\langle\sum_{k=1}^{% N}\left(\mathbf{r}_{k}-\mathbf{r}_{\mathrm{mean}}\right)^{2}\rangle
  12. \langle\ldots\rangle
  13. R g = 1 6 N a R_{\mathrm{g}}=\frac{1}{\sqrt{6}\ }\ \sqrt{N}\ a
  14. a N aN
  15. a a
  16. N N
  17. R g 2 R_{\mathrm{g}}^{2}
  18. R g 2 = def 1 N k = 1 N ( 𝐫 k - 𝐫 mean ) 2 = 1 N k = 1 N [ 𝐫 k 𝐫 k + 𝐫 mean 𝐫 mean - 2 𝐫 k 𝐫 mean ] R_{\mathrm{g}}^{2}\ \stackrel{\mathrm{def}}{=}\ \frac{1}{N}\sum_{k=1}^{N}\left% (\mathbf{r}_{k}-\mathbf{r}_{\mathrm{mean}}\right)^{2}=\frac{1}{N}\sum_{k=1}^{N% }\left[\mathbf{r}_{k}\cdot\mathbf{r}_{k}+\mathbf{r}_{\mathrm{mean}}\cdot% \mathbf{r}_{\mathrm{mean}}-2\mathbf{r}_{k}\cdot\mathbf{r}_{\mathrm{mean}}\right]
  19. 𝐫 mean \mathbf{r}_{\mathrm{mean}}
  20. R g 2 = def - 𝐫 mean 𝐫 mean + 1 N k = 1 N ( 𝐫 k 𝐫 k ) R_{\mathrm{g}}^{2}\ \stackrel{\mathrm{def}}{=}\ -\mathbf{r}_{\mathrm{mean}}% \cdot\mathbf{r}_{\mathrm{mean}}+\frac{1}{N}\sum_{k=1}^{N}\left(\mathbf{r}_{k}% \cdot\mathbf{r}_{k}\right)

Rail_transport.html

  1. R = a + b v + c v 2 \qquad\qquad R=a+bv+cv^{2}

Raman_spectroscopy.html

  1. Δ w = ( 1 λ 0 - 1 λ 1 ) , \Delta w=\left(\frac{1}{\lambda_{0}}-\frac{1}{\lambda_{1}}\right)\ ,
  2. Δ w \Delta w
  3. Δ w ( cm - 1 ) = ( 1 λ 0 ( nm ) - 1 λ 1 ( nm ) ) × ( 10 7 nm ) ( cm ) . \Delta w(\,\text{cm}^{-1})=\left(\frac{1}{\lambda_{0}(\,\text{nm})}-\frac{1}{% \lambda_{1}(\,\text{nm})}\right)\times\frac{(10^{7}\,\text{nm})}{(\,\text{cm})}.

Random_variable.html

  1. X : Ω E X\colon\Omega\to E
  2. Ω \Omega
  3. E E
  4. E = E=
  5. \mathbb{R}
  6. Ω \Omega
  7. E E
  8. X X
  9. X X
  10. X X
  11. X X
  12. Ω = { heads , tails } \Omega=\{\,\text{heads},\,\text{tails}\}
  13. Y Y
  14. Y ( ω ) = { 1 , if ω = heads , 0 , if ω = tails . Y(\omega)=\begin{cases}1,&\,\text{if}\ \ \omega=\,\text{heads},\\ \\ 0,&\,\text{if}\ \ \omega=\,\text{tails}.\end{cases}
  15. f Y f_{Y}
  16. f Y ( y ) = { 1 2 , if y = 1 , 1 2 , if y = 0 , f_{Y}(y)=\begin{cases}\tfrac{1}{2},&\,\text{if }y=1,\\ \\ \tfrac{1}{2},&\,\text{if }y=0,\\ \end{cases}
  17. X ( ( n 1 , n 2 ) ) = n 1 + n 2 X((n_{1},n_{2}))=n_{1}+n_{2}
  18. f X ( S ) = min ( S - 1 , 13 - S ) 36 , for S { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } f_{X}(S)=\tfrac{\min(S-1,13-S)}{36},\,\text{for }S\in\{2,3,4,5,6,7,8,9,10,11,12\}
  19. 1 / 2 {1}/{2}
  20. 1 / 2 {1}/{2}
  21. ( Ω , , P ) (\Omega,\mathcal{F},P)
  22. ( E , ) (E,\mathcal{E})
  23. ( E , ) (E,\mathcal{E})
  24. X : Ω E X\colon\Omega\to E
  25. ( , ) (\mathcal{F},\mathcal{E})
  26. B B\in\mathcal{E}
  27. X - 1 ( B ) X^{-1}(B)\in\mathcal{F}
  28. X - 1 ( B ) = { ω : X ( ω ) B } X^{-1}(B)=\{\omega:X(\omega)\in B\}
  29. B B\in\mathcal{E}
  30. E E
  31. \mathcal{E}
  32. ( E ) \mathcal{B}(E)
  33. E E
  34. ( E , ) (E,\mathcal{E})
  35. E E
  36. E E
  37. \mathbb{R}
  38. ( Ω , , P ) (\Omega,\mathcal{F},P)
  39. X : Ω X\colon\Omega\rightarrow\mathbb{R}
  40. { ω : X ( ω ) r } r . \{\omega:X(\omega)\leq r\}\in\mathcal{F}\qquad\forall r\in\mathbb{R}.
  41. { ( - , r ] : r \R } \{(-\infty,r]:r\in\R\}
  42. { ω : X ( ω ) r } = X - 1 ( ( - , r ] ) \{\omega:X(\omega)\leq r\}=X^{-1}((-\infty,r])
  43. X : Ω X\colon\Omega\to\mathbb{R}
  44. ( Ω , , P ) (\Omega,\mathcal{F},P)
  45. X X
  46. { ω : X ( ω ) = 2 } \{\omega:X(\omega)=2\}\,\!
  47. P ( X = 2 ) P(X=2)\,\!
  48. p X ( 2 ) p_{X}(2)
  49. X X
  50. X X
  51. X X
  52. X X
  53. F X ( x ) = P ( X x ) F_{X}(x)=\operatorname{P}(X\leq x)
  54. p X p_{X}
  55. X X
  56. P P
  57. Ω \Omega
  58. p X p_{X}
  59. \mathbb{R}
  60. Ω \Omega
  61. Ω \Omega
  62. \mathbb{R}
  63. f ( X ) = X f(X)=X
  64. [ X = green ] [X=\,\text{green}]
  65. g : g\colon\mathbb{R}\rightarrow\mathbb{R}
  66. Y Y\,\!
  67. F Y ( y ) = P ( g ( X ) y ) . F_{Y}(y)=\operatorname{P}(g(X)\leq y).
  68. F Y ( y ) = P ( g ( X ) y ) = { P ( X g - 1 ( y ) ) = F X ( g - 1 ( y ) ) , if g - 1 increasing , P ( X g - 1 ( y ) ) = 1 - F X ( g - 1 ( y ) ) , if g - 1 decreasing . F_{Y}(y)=\operatorname{P}(g(X)\leq y)=\begin{cases}\operatorname{P}(X\leq g^{-% 1}(y))=F_{X}(g^{-1}(y)),&\,\text{if }g^{-1}\,\text{ increasing},\\ \\ \operatorname{P}(X\geq g^{-1}(y))=1-F_{X}(g^{-1}(y)),&\,\text{if }g^{-1}\,% \text{ decreasing}.\end{cases}
  69. f Y ( y ) = f X ( g - 1 ( y ) ) | d g - 1 ( y ) d y | . f_{Y}(y)=f_{X}(g^{-1}(y))\left|\frac{dg^{-1}(y)}{dy}\right|.
  70. f Y ( y ) = i f X ( g i - 1 ( y ) ) | d g i - 1 ( y ) d y | f_{Y}(y)=\sum_{i}f_{X}(g_{i}^{-1}(y))\left|\frac{dg_{i}^{-1}(y)}{dy}\right|
  71. X X\!
  72. Ω \Omega\,\!
  73. g : g\colon\mathbb{R}\rightarrow\mathbb{R}
  74. Y = g ( X ) Y=g(X)\,\!
  75. Ω \Omega\,\!
  76. g g
  77. ( Ω , P ) (\Omega,P)\,\!
  78. ( , d F X ) (\mathbb{R},dF_{X})
  79. Y Y\,\!
  80. F Y ( y ) = P ( X 2 y ) . F_{Y}(y)=\operatorname{P}(X^{2}\leq y).
  81. F Y ( y ) = 0 if y < 0. F_{Y}(y)=0\qquad\hbox{if}\quad y<0.
  82. P ( X 2 y ) = P ( | X | y ) = P ( - y X y ) , \operatorname{P}(X^{2}\leq y)=\operatorname{P}(|X|\leq\sqrt{y})=\operatorname{% P}(-\sqrt{y}\leq X\leq\sqrt{y}),
  83. F Y ( y ) = F X ( y ) - F X ( - y ) if y 0. F_{Y}(y)=F_{X}(\sqrt{y})-F_{X}(-\sqrt{y})\qquad\hbox{if}\quad y\geq 0.
  84. X \scriptstyle X
  85. F X ( x ) = P ( X x ) = 1 ( 1 + e - x ) θ F_{X}(x)=P(X\leq x)=\frac{1}{(1+e^{-x})^{\theta}}
  86. θ > 0 \scriptstyle\theta>0
  87. Y = log ( 1 + e - X ) . \scriptstyle Y=\mathrm{log}(1+e^{-X}).
  88. F Y ( y ) = P ( Y y ) = P ( log ( 1 + e - X ) y ) = P ( X > - log ( e y - 1 ) ) . F_{Y}(y)=P(Y\leq y)=P(\mathrm{log}(1+e^{-X})\leq y)=P(X>-\mathrm{log}(e^{y}-1)% ).\,
  89. X , X,
  90. F Y ( y ) = 1 - F X ( - log ( e y - 1 ) ) F_{Y}(y)=1-F_{X}(-\mathrm{log}(e^{y}-1))\,
  91. = 1 - 1 ( 1 + e log ( e y - 1 ) ) θ =1-\frac{1}{(1+e^{\mathrm{log}(e^{y}-1)})^{\theta}}
  92. = 1 - 1 ( 1 + e y - 1 ) θ =1-\frac{1}{(1+e^{y}-1)^{\theta}}
  93. = 1 - e - y θ . =1-e^{-y\theta}.\,
  94. X \scriptstyle X
  95. f X ( x ) = 1 2 π e - x 2 / 2 . f_{X}(x)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}.
  96. Y = X 2 . \scriptstyle Y=X^{2}.
  97. f Y ( y ) = i f X ( g i - 1 ( y ) ) | d g i - 1 ( y ) d y | . f_{Y}(y)=\sum_{i}f_{X}(g_{i}^{-1}(y))\left|\frac{dg_{i}^{-1}(y)}{dy}\right|.
  98. Y \scriptstyle Y
  99. X \scriptstyle X
  100. f Y ( y ) = 2 f X ( g - 1 ( y ) ) | d g - 1 ( y ) d y | . f_{Y}(y)=2f_{X}(g^{-1}(y))\left|\frac{dg^{-1}(y)}{dy}\right|.
  101. x = g - 1 ( y ) = y x=g^{-1}(y)=\sqrt{y}
  102. d g - 1 ( y ) d y = 1 2 y . \frac{dg^{-1}(y)}{dy}=\frac{1}{2\sqrt{y}}.
  103. f Y ( y ) = 2 1 2 π e - y / 2 1 2 y = 1 2 π y e - y / 2 . f_{Y}(y)=2\frac{1}{\sqrt{2\pi}}e^{-y/2}\frac{1}{2\sqrt{y}}=\frac{1}{\sqrt{2\pi y% }}e^{-y/2}.
  104. X = d Y X\stackrel{d}{=}Y
  105. P ( X x ) = P ( Y x ) for all x . \operatorname{P}(X\leq x)=\operatorname{P}(Y\leq x)\quad\hbox{for all}\quad x.
  106. P ( X Y ) = 0. \operatorname{P}(X\neq Y)=0.
  107. d ( X , Y ) = ess sup ω | X ( ω ) - Y ( ω ) | , d_{\infty}(X,Y)=\mathrm{ess}\sup_{\omega}|X(\omega)-Y(\omega)|,
  108. X ( ω ) = Y ( ω ) for all ω . X(\omega)=Y(\omega)\qquad\hbox{for all }\omega.

Range_(statistics).html

  1. F ( t ) = n - g ( x ) [ G ( x + t ) - G ( x ) ] n - 1 d x . F(t)=n\int_{-\infty}^{\infty}g(x)[G(x+t)-G(x)]^{n-1}\,\text{d}x.
  2. n 0 1 x ( G ) [ G n - 1 - ( 1 - G ) n - 1 ] d G n\int_{0}^{1}x(G)[G^{n-1}-(1-G)^{n-1}]\,\text{d}G
  3. - ( 1 - ( 1 - Φ ( x ) ) n - Φ ( x ) n ) d x . \int_{-\infty}^{\infty}(1-(1-\Phi(x))^{n}-\Phi(x)^{n})\,\text{d}x.
  4. F ( t ) = i = 1 n - g i ( x ) j = 1 , j i n [ G j ( x + t ) - G j ( x ) ] d x . F(t)=\sum_{i=1}^{n}\int_{-\infty}^{\infty}g_{i}(x)\prod_{j=1,j\neq i}^{n}[G_{j% }(x+t)-G_{j}(x)]\,\text{d}x.
  5. f ( t ) = { x = 1 N [ g ( x ) ] n t = 0 x = 1 N - t ( [ G ( x + t ) - G ( x - 1 ) ] n - [ G ( x + t ) - G ( x ) ] n - [ G ( x + t - 1 ) - G ( x - 1 ) ] n + [ G ( x + t - 1 ) - G ( x ) ] n ) t = 1 , 2 , 3 , N - 1. f(t)=\begin{cases}\sum_{x=1}^{N}[g(x)]^{n}&t=0\\ \sum_{x=1}^{N-t}\left(\begin{aligned}&\displaystyle[G(x+t)-G(x-1)]^{n}\\ &\displaystyle-[G(x+t)-G(x)]^{n}\\ &\displaystyle-[G(x+t-1)-G(x-1)]^{n}\\ &\displaystyle+[G(x+t-1)-G(x)]^{n}\\ \end{aligned}\right)&t=1,2,3\ldots,N-1.\\ \end{cases}
  6. f ( t ) = { 1 N n - 1 t = 0 x = 1 N - t ( [ t + 1 N ] n - 2 [ t N ] n + [ t - 1 N ] n ) t = 1 , 2 , 3 , N - 1. f(t)=\left\{\begin{array}[]{ll}\frac{1}{N^{n-1}}&t=0\\ \sum_{x=1}^{N-t}\left([\frac{t+1}{N}]^{n}-2[\frac{t}{N}]^{n}+[\frac{t-1}{N}]^{% n}\right)&t=1,2,3\ldots,N-1.\end{array}\right.

Rank_(linear_algebra).html

  1. [ 1 2 1 - 2 - 3 1 3 5 0 ] \begin{bmatrix}1&2&1\\ -2&-3&1\\ 3&5&0\end{bmatrix}
  2. A = [ 1 1 0 2 - 1 - 1 0 - 2 ] A=\begin{bmatrix}1&1&0&2\\ -1&-1&0&-2\end{bmatrix}
  3. A T = [ 1 - 1 1 - 1 0 0 2 - 2 ] A^{T}=\begin{bmatrix}1&-1\\ 1&-1\\ 0&0\\ 2&-2\end{bmatrix}
  4. A = [ 1 2 1 - 2 - 3 1 3 5 0 ] A=\begin{bmatrix}1&2&1\\ -2&-3&1\\ 3&5&0\end{bmatrix}
  5. [ 1 2 1 - 2 - 3 1 3 5 0 ] R 2 2 r 1 + r 2 [ 1 2 1 0 1 3 3 5 0 ] R 3 - 3 r 1 + r 3 [ 1 2 1 0 1 3 0 - 1 - 3 ] R 3 r 2 + r 3 [ 1 2 1 0 1 3 0 0 0 ] R 1 - 2 r 2 + r 1 [ 1 0 - 5 0 1 3 0 0 0 ] \begin{bmatrix}1&2&1\\ -2&-3&1\\ 3&5&0\end{bmatrix}R_{2}\rightarrow 2r_{1}+r_{2}\begin{bmatrix}1&2&1\\ 0&1&3\\ 3&5&0\end{bmatrix}R_{3}\rightarrow-3r_{1}+r_{3}\begin{bmatrix}1&2&1\\ 0&1&3\\ 0&-1&-3\end{bmatrix}R_{3}\rightarrow r_{2}+r_{3}\begin{bmatrix}1&2&1\\ 0&1&3\\ 0&0&0\end{bmatrix}R_{1}\rightarrow-2r_{2}+r_{1}\begin{bmatrix}1&0&-5\\ 0&1&3\\ 0&0&0\end{bmatrix}
  6. x 1 , x 2 , , x r x_{1},x_{2},\ldots,x_{r}
  7. A x 1 , A x 2 , , A x r Ax_{1},Ax_{2},\ldots,Ax_{r}
  8. c 1 , c 2 , , c r c_{1},c_{2},\ldots,c_{r}
  9. 0 = c 1 A x 1 + c 2 A x 2 + + c r A x r = A ( c 1 x 1 + c 2 x 2 + + c r x r ) = A v , 0=c_{1}Ax_{1}+c_{2}Ax_{2}+\cdots+c_{r}Ax_{r}=A(c_{1}x_{1}+c_{2}x_{2}+\cdots+c_% {r}x_{r})=Av,
  10. v = c 1 x 1 + c 2 x 2 + + c r x r v=c_{1}x_{1}+c_{2}x_{2}+\cdots+c_{r}x_{r}
  11. c 1 x 1 + c 2 x 2 + + c r x r = 0. c_{1}x_{1}+c_{2}x_{2}+\cdots+c_{r}x_{r}=0.
  12. x i x_{i}
  13. c 1 = c 2 = = c r = 0 c_{1}=c_{2}=\cdots=c_{r}=0
  14. A x 1 , A x 2 , , A x r Ax_{1},Ax_{2},\ldots,Ax_{r}
  15. A x i Ax_{i}
  16. A x 1 , A x 2 , , A x r Ax_{1},Ax_{2},\ldots,Ax_{r}
  17. c 1 , c 2 , , c k c_{1},c_{2},\dots,c_{k}
  18. A = C R A=CR
  19. c 1 , , c k c_{1},\ldots,c_{k}
  20. c 1 , , c k c_{1},\ldots,c_{k}
  21. m × k m\times k
  22. k × n k\times n
  23. A = C R A=CR
  24. r 1 , , r k r_{1},\ldots,r_{k}
  25. r 1 , , r k r_{1},\ldots,r_{k}
  26. ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) (1)\Leftrightarrow(2)\Leftrightarrow(3)\Leftrightarrow(4)\Leftrightarrow(5)
  27. c 1 , , c k c_{1},\ldots,c_{k}
  28. c 1 , , c k c_{1},\ldots,c_{k}
  29. ( 1 ) ( 5 ) (1)\Leftrightarrow(5)
  30. c r c\cdot r
  31. rank ( A B ) min ( rank A , rank B ) . \operatorname{rank}(AB)\leq\min(\operatorname{rank}\ A,\operatorname{rank}\ B).
  32. rank ( A B ) = rank ( A ) . \operatorname{rank}(AB)=\operatorname{rank}(A).
  33. rank ( C A ) = rank ( A ) . \operatorname{rank}(CA)=\operatorname{rank}(A).
  34. X A Y = [ I r 0 0 0 ] , XAY=\begin{bmatrix}I_{r}&0\\ 0&0\\ \end{bmatrix},
  35. rank ( A ) + rank ( B ) - n rank ( A B ) . \operatorname{rank}(A)+\operatorname{rank}(B)-n\leq\operatorname{rank}(AB).
  36. dim ker ( A B ) dim ker ( A ) + dim ker ( B ) \dim\operatorname{ker}(AB)\leq\dim\operatorname{ker}(A)+\dim\operatorname{ker}% (B)
  37. rank ( A B ) + rank ( B C ) rank ( B ) + rank ( A B C ) . \operatorname{rank}(AB)+\operatorname{rank}(BC)\leq\operatorname{rank}(B)+% \operatorname{rank}(ABC).
  38. C : ker ( A B C ) / ker ( B C ) ker ( A B ) / ker ( B ) C:\operatorname{ker}(ABC)/\operatorname{ker}(BC)\to\operatorname{ker}(AB)/% \operatorname{ker}(B)
  39. rank ( A T A ) = rank ( A A T ) = rank ( A ) = rank ( A T ) \operatorname{rank}(A^{\mathrm{T}}A)=\operatorname{rank}(AA^{\mathrm{T}})=% \operatorname{rank}(A)=\operatorname{rank}(A^{\mathrm{T}})
  40. A T A x = 0 A^{\mathrm{T}}Ax=0
  41. 0 = x T A T A x = | A x | 2 0=x^{\mathrm{T}}A^{\mathrm{T}}Ax=\left|Ax\right|^{2}
  42. rank ( A ) = rank ( A ¯ ) = rank ( A T ) = rank ( A * ) = rank ( A * A ) . \operatorname{rank}(A)=\operatorname{rank}(\overline{A})=\operatorname{rank}(A% ^{\mathrm{T}})=\operatorname{rank}(A^{*})=\operatorname{rank}(A^{*}A).

Raoult's_law.html

  1. p i = p i x i p_{i}=p^{\star}_{i}x_{i}
  2. p i p_{i}
  3. i i
  4. p i p_{i}^{\star}
  5. i i
  6. x i x_{i}
  7. i i
  8. p = p A x A + p B x B + p=p^{\star}_{\rm A}x_{\rm A}+p^{\star}_{\rm B}x_{\rm B}+\cdots
  9. p p
  10. p A p_{A}
  11. p B p_{B}
  12. p = p A x A + p B x B . p=p^{\star}_{\rm A}x_{\rm A}+p^{\star}_{\rm B}x_{\rm B}.
  13. p = p A ( 1 - x B ) + p B x B = p A + ( p B - p A ) x B p=p^{\star}_{\rm A}(1-x_{\rm B})+p^{\star}_{\rm B}x_{\rm B}=p^{\star}_{\rm A}+% (p^{\star}_{\rm B}-p^{\star}_{\rm A})x_{\rm B}
  14. x B x_{B}
  15. μ i = μ i + R T ln x i \mu_{i}=\mu_{i}^{\star}+RT\ln x_{i}\,
  16. μ i \mu_{i}^{\star}
  17. μ i , liq = μ i , vap . \mu_{i,{\rm liq}}=\mu_{i,{\rm vap}}.\,
  18. μ i , liq + R T ln x i = μ i , vap + R T ln < m t p l > f i p \mu_{i,{\rm liq}}^{\star}+RT\ln x_{i}=\mu_{i,{\rm vap}}^{\ominus}+RT\ln\frac{<% }{m}tpl>{{f_{i}}}{{p^{\ominus}}}
  19. f i f_{i}
  20. i i
  21. {}^{\ominus}
  22. i i
  23. μ i , liq = μ i , vap + R T ln f i < m t p l > p \mu_{i,{\rm liq}}^{\star}=\mu_{i,{\rm vap}}^{\ominus}+RT\ln\frac{{f_{i}^{\star% }}}{<}mtpl>{{p^{\ominus}}}
  24. R T ln x i = R T ln < m t p l > f i f i RT\ln x_{i}=RT\ln\frac{<}{m}tpl>{{f_{i}}}{{f_{i}^{\star}}}
  25. f i = x i f i . f_{i}=x_{i}f_{i}^{\star}.
  26. p i = x i p i p_{i}=x_{i}p_{i}^{\star}
  27. Δ mix G = n R T ( x 1 ln x 1 + x 2 ln x 2 ) . \Delta_{\rm mix}G=nRT(x_{1}\ln x_{1}+x_{2}\ln x_{2}).\,
  28. Δ mix H \Delta_{\rm mix}H
  29. x i = y i p total p i , eqm x_{i}=\frac{y_{i}p_{\rm total}}{p_{i,{\rm eqm}}}\,
  30. x i x_{i}
  31. i i
  32. y i y_{i}
  33. ϕ p , i \phi_{p,i}
  34. γ i \gamma_{i}
  35. p i ϕ p , i = p i γ i x i . p_{i}\phi_{p,i}=p_{i}^{\star}\gamma_{i}x_{i}.\,

Rasterisation.html

  1. [ 1 0 0 X 0 1 0 Y 0 0 1 Z 0 0 0 1 ] \begin{bmatrix}1&0&0&X\\ 0&1&0&Y\\ 0&0&1&Z\\ 0&0&0&1\end{bmatrix}
  2. [ X 0 0 0 0 Y 0 0 0 0 Z 0 0 0 0 1 ] \begin{bmatrix}X&0&0&0\\ 0&Y&0&0\\ 0&0&Z&0\\ 0&0&0&1\end{bmatrix}
  3. [ 1 0 0 0 0 cos θ - sin θ 0 0 sin θ cos θ 0 0 0 0 1 ] \begin{bmatrix}1&0&0&0\\ 0&\cos{\theta}&-\sin{\theta}&0\\ 0&\sin{\theta}&\cos{\theta}&0\\ 0&0&0&1\\ \end{bmatrix}
  4. [ cos θ 0 sin θ 0 0 1 0 0 - sin θ 0 cos θ 0 0 0 0 1 ] \begin{bmatrix}\cos{\theta}&0&\sin{\theta}&0\\ 0&1&0&0\\ -\sin{\theta}&0&\cos{\theta}&0\\ 0&0&0&1\end{bmatrix}
  5. [ cos θ - sin θ 0 0 sin θ cos θ 0 0 0 0 1 0 0 0 0 1 ] \begin{bmatrix}\cos{\theta}&-\sin{\theta}&0&0\\ \sin{\theta}&\cos{\theta}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}
  6. [ 1 0 0 0 0 1 0 0 0 0 ( N + F ) / N - F 0 0 1 / N 0 ] \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&(N+F)/N&-F\\ 0&0&1/N&0\end{bmatrix}

Rational_choice_theory.html

  1. A = { a 1 , , a i , , a j } A=\{a_{1},\ldots,a_{i},\ldots,a_{j}\}
  2. A = { V o t e f o r R o g e r , V o t e f o r S a r a , A b s t a i n } A=\{VoteforRoger,VoteforSara,Abstain\}
  3. u ( a i ) > u ( a j ) u\left(a_{i}\right)>u\left(a_{j}\right)
  4. u ( S a r a ) > u ( R o g e r ) > u ( a b s t a i n ) u\left(Sara\right)>u\left(Roger\right)>u\left(abstain\right)

Rational_expectations.html

  1. P = P * + ϵ P=P^{*}+\epsilon
  2. E [ P ] = P * E[P]=P^{*}
  3. P * P^{*}
  4. ϵ \epsilon
  5. P * P^{*}

Rational_root_theorem.html

  1. a n x n + a n - 1 x n - 1 + + a 0 = 0 a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}=0\,\!
  2. P ( p q ) = a n ( p q ) n + a n - 1 ( p q ) n - 1 + + a 1 ( p q ) + a 0 = 0. P\left(\tfrac{p}{q}\right)=a_{n}\left(\tfrac{p}{q}\right)^{n}+a_{n-1}\left(% \tfrac{p}{q}\right)^{n-1}+\cdots+a_{1}\left(\tfrac{p}{q}\right)+a_{0}=0.
  3. p ( a n p n - 1 + a n - 1 q p n - 2 + + a 1 q n - 1 ) = - a 0 q n . \qquad p(a_{n}p^{n-1}+a_{n-1}qp^{n-2}+\cdots+a_{1}q^{n-1})=-a_{0}q^{n}.
  4. q ( a n - 1 p n - 1 + a n - 2 q p n - 2 + + a 0 q n - 1 ) = - a n p n . \qquad q(a_{n-1}p^{n-1}+a_{n-2}qp^{n-2}+\cdots+a_{0}q^{n-1})=-a_{n}p^{n}.
  5. X X ℚXX
  6. X X ℤXX
  7. p / q p/q
  8. X X ℚXX
  9. q x p qx−p
  10. X X ℤXX
  11. q x p qx−p
  12. 3 x 3 - 5 x 2 + 5 x - 2 = 0 3x^{3}-5x^{2}+5x-2=0\,\!
  13. 1 , 2 1 , 3 , \tfrac{1,2}{1,3}\,,
  14. 1 , - 1 , 2 , - 2 , 1 3 , - 1 3 , 2 3 , - 2 3 . 1,-1,2,-2,\frac{1}{3},-\frac{1}{3},\frac{2}{3},-\frac{2}{3}\,.
  15. t = ± 1 1 , 3 t=\pm\tfrac{1}{1,3}
  16. x = 1 + t = 2 , 0 , 4 3 , 2 3 x=1+t=2,0,\frac{4}{3},\frac{2}{3}

Raven_paradox.html

  1. N N
  2. r r
  3. b b
  4. N N
  5. 1 N \tfrac{1}{N}
  6. H i H_{i}
  7. i i
  8. H 1 , H 2 , , H r H_{1},H_{2},...,H_{r}
  9. H 0 H_{0}
  10. r N / average ( r - 1 N , r - 2 N , , 1 N ) = 2 r r - 1 \tfrac{r}{N}\Big/\,\text{average}\left(\tfrac{r-1}{N},\tfrac{r-2}{N},...\ ,% \tfrac{1}{N}\right)\ =\ \tfrac{2r}{r-1}
  11. N - b N / average ( N - b - 1 N , N - b - 2 N , , max ( 0 , N - b - r N ) ) \tfrac{N-b}{N}\Big/\,\text{average}\left(\tfrac{N-b-1}{N},\tfrac{N-b-2}{N},...% \ ,\max(0,\tfrac{N-b-r}{N})\right)
  12. = N - b max ( N - b - r 2 - 1 2 , 1 2 ( N - b - 1 ) ) \ =\ \frac{N-b}{\max\left(N-b-\tfrac{r}{2}-\tfrac{1}{2}\ ,\ \tfrac{1}{2}(N-b-1% )\right)}
  13. r / ( 2 N - 2 b ) r/(2N-2b)
  14. N - b N-b
  15. r r
  16. P ( F a | E ) P(Fa|E)
  17. a a
  18. F F
  19. E E
  20. P ( F a | E ) = n F + λ P ( F a ) n + λ P(Fa|E)\ =\ \frac{n_{F}+\lambda P(Fa)}{n+\lambda}
  21. P ( F a ) P(Fa)
  22. a a
  23. F F
  24. n n
  25. E E
  26. n F n_{F}
  27. F F
  28. λ \lambda
  29. λ \lambda
  30. P ( F a | E ) P(Fa|E)
  31. F F
  32. λ \lambda
  33. n n
  34. P ( F a | E ) P(Fa|E)
  35. P ( F a ) P(Fa)
  36. F F
  37. a a
  38. B a Ba
  39. R a Ra
  40. P ( B a ¯ | H ¯ ) P ( R a | H ¯ ) - P ( B a ¯ | R a H ¯ ) P ( B a | R a H ¯ ) P ( B a ¯ | H ) P ( R a | H ) \frac{P(\overline{Ba}|\overline{H})}{P(Ra|\overline{H})}\ -\ P(\overline{Ba}|% Ra\overline{H})\ \geq\ P(Ba|Ra\overline{H})\frac{P(\overline{Ba}|H)}{P(Ra|H)}
  41. - log P ( B a | R a H ¯ ) -\log P(Ba|Ra\overline{H})
  42. p = P ( B a | R a H ¯ ) p=P(Ba|Ra\overline{H})
  43. 1 / p 1/p
  44. n n
  45. ( 1 / p ) n (1/p)^{n}
  46. n n
  47. H H
  48. H H
  49. H H
  50. E E
  51. H = A a n d B E = B a n d C H=A\ and\ B\ \ \ \ \ \ E=B\ and\ C
  52. A A
  53. B B
  54. C C
  55. P ( A a n d B ) = P ( A ) P ( B ) P(A\ and\ B)=P(A)P(B)
  56. P ( H | E ) > P ( H ) P(H|E)>P(H)
  57. H H
  58. E E
  59. B B
  60. H H
  61. E E
  62. E E
  63. H H
  64. E E
  65. E E
  66. P ( A | E ) = P ( A ) P(A|E)=P(A)
  67. H = ( H o r E ) a n d ( H o r E ¯ ) H=(H\ or\ E)\ and\ (H\ or\ \overline{E})
  68. H H
  69. ( H o r E ) (H\ or\ E)
  70. H H
  71. H H
  72. E E
  73. ( H o r E ¯ ) (H\ or\ \overline{E})
  74. H H
  75. E E
  76. E E
  77. ( H o r E ¯ ) (H\ or\ \overline{E})
  78. E E
  79. H H
  80. E E
  81. ( H o r E ¯ ) (H\ or\ \overline{E})
  82. P ( H | E ) = 1 P(H|E)=1
  83. P ( E ) = 1 P(E)=1
  84. r r
  85. 0 < r < 1 0<r<1
  86. r r
  87. 0 < r < 1 0<r<1
  88. H H
  89. x , R x B x \forall\ x,Rx\ \rightarrow\ Bx
  90. x , B x ¯ R x ¯ \forall\ x,\overline{Bx}\ \rightarrow\ \overline{Rx}
  91. \rightarrow
  92. A A
  93. B B
  94. B B
  95. A ¯ \overline{A}
  96. A A
  97. B B
  98. x x
  99. x x
  100. A A
  101. B B
  102. A A
  103. B B
  104. A A
  105. B B
  106. I I
  107. A A
  108. A A
  109. B B
  110. B ¯ \overline{B}
  111. A ¯ \overline{A}
  112. a a
  113. a a
  114. a a
  115. a a
  116. P ( R a ) = 1 2 P ( B a ) = 1 2 P(Ra)=\frac{1}{2}\ \ \ \ \ \ \ \ P(Ba)=\frac{1}{2}
  117. R a , R b Ra,\ Rb
  118. a a
  119. b b
  120. A A
  121. B B
  122. A A
  123. H 1 H_{1}
  124. H 2 H_{2}
  125. H 1 H_{1}
  126. H 2 H_{2}
  127. x , * R x B x \forall\ x,\ *Rx\rightarrow Bx
  128. x x
  129. H 1 H_{1}
  130. x x
  131. H 2 H_{2}

Ray_tracing_(graphics).html

  1. 𝐜 \mathbf{c}
  2. 𝐫 \mathbf{r}
  3. 𝐱 - 𝐜 2 = r 2 . \left\|\mathbf{x}-\mathbf{c}\right\|^{2}=r^{2}.
  4. 𝐬 \mathbf{s}
  5. 𝐝 \mathbf{d}
  6. 𝐝 \mathbf{d}
  7. 𝐱 = 𝐬 + t 𝐝 , \mathbf{x}=\mathbf{s}+t\mathbf{d},
  8. t t
  9. 𝐱 \mathbf{x}
  10. 𝐬 \mathbf{s}
  11. 𝐜 \mathbf{c}
  12. r r
  13. 𝐬 \mathbf{s}
  14. 𝐝 \mathbf{d}
  15. t t
  16. 𝐱 \mathbf{x}
  17. 𝐬 + t 𝐝 - 𝐜 2 = r 2 . \left\|\mathbf{s}+t\mathbf{d}-\mathbf{c}\right\|^{2}=r^{2}.
  18. 𝐯 = def 𝐬 - 𝐜 \mathbf{v}\ \stackrel{\mathrm{def}}{=}\ \mathbf{s}-\mathbf{c}
  19. 𝐯 + t 𝐝 2 = r 2 \left\|\mathbf{v}+t\mathbf{d}\right\|^{2}=r^{2}
  20. 𝐯 2 + t 2 𝐝 2 + 2 𝐯 t 𝐝 = r 2 \mathbf{v}^{2}+t^{2}\mathbf{d}^{2}+2\mathbf{v}\cdot t\mathbf{d}=r^{2}
  21. ( 𝐝 2 ) t 2 + ( 2 𝐯 𝐝 ) t + ( 𝐯 2 - r 2 ) = 0. (\mathbf{d}^{2})t^{2}+(2\mathbf{v}\cdot\mathbf{d})t+(\mathbf{v}^{2}-r^{2})=0.
  22. t 2 + ( 2 𝐯 𝐝 ) t + ( 𝐯 2 - r 2 ) = 0. t^{2}+(2\mathbf{v}\cdot\mathbf{d})t+(\mathbf{v}^{2}-r^{2})=0.
  23. t = - ( 2 𝐯 𝐝 ) ± ( 2 𝐯 𝐝 ) 2 - 4 ( 𝐯 2 - r 2 ) 2 = - ( 𝐯 𝐝 ) ± ( 𝐯 𝐝 ) 2 - ( 𝐯 2 - r 2 ) . t=\frac{-(2\mathbf{v}\cdot\mathbf{d})\pm\sqrt{(2\mathbf{v}\cdot\mathbf{d})^{2}% -4(\mathbf{v}^{2}-r^{2})}}{2}=-(\mathbf{v}\cdot\mathbf{d})\pm\sqrt{(\mathbf{v}% \cdot\mathbf{d})^{2}-(\mathbf{v}^{2}-r^{2})}.
  24. t t
  25. 𝐬 + t 𝐝 \mathbf{s}+t\mathbf{d}
  26. 𝐬 \mathbf{s}
  27. t t
  28. 𝐧 = 𝐲 - 𝐜 𝐲 - 𝐜 , \mathbf{n}=\frac{\mathbf{y}-\mathbf{c}}{\left\|\mathbf{y}-\mathbf{c}\right\|},
  29. 𝐲 = 𝐬 + t 𝐝 \mathbf{y}=\mathbf{s}+t\mathbf{d}
  30. 𝐝 \mathbf{d}
  31. 𝐧 \mathbf{n}
  32. 𝐫 = 𝐝 - 2 ( 𝐧 𝐝 ) 𝐧 . \mathbf{r}=\mathbf{d}-2(\mathbf{n}\cdot\mathbf{d})\mathbf{n}.
  33. 𝐱 = 𝐲 + u 𝐫 . \mathbf{x}=\mathbf{y}+u\mathbf{r}.\,

Rayleigh_number.html

  1. Ra x = g β ν α ( T s - T ) x 3 = Gr x Pr \mathrm{Ra}_{x}=\frac{g\beta}{\nu\alpha}(T_{s}-T_{\infty})x^{3}=\mathrm{Gr}_{x% }\mathrm{Pr}
  2. T f = T s + T 2 T_{f}=\frac{T_{s}+T_{\infty}}{2}
  3. Ra x * = g β q o ′′ ν α k x 4 \mathrm{Ra}^{*}_{x}=\frac{g\beta q^{\prime\prime}_{o}}{\nu\alpha k}x^{4}
  4. Ra = Δ ρ ρ 0 g K ¯ L α ν = Δ ρ ρ 0 g K ¯ R ν \mathrm{Ra}=\frac{\frac{\Delta\rho}{\rho_{0}}g\bar{K}L}{\alpha\nu}=\frac{\frac% {\Delta\rho}{\rho_{0}}g\bar{K}}{R\nu}
  5. K ¯ \bar{K}
  6. L L
  7. α \alpha
  8. ν \nu
  9. R R
  10. Ra H = g ρ 0 2 β H D 5 η α k \mathrm{Ra}_{H}=\frac{g\rho^{2}_{0}\beta HD^{5}}{\eta\alpha k}
  11. Ra T = ρ 0 g β Δ T s a D 3 c η k \mathrm{Ra}_{T}=\frac{\rho_{0}g\beta\Delta T_{sa}D^{3}c}{\eta k}

Rayleigh_scattering.html

  1. x = 2 π r λ x=\frac{2\pi r}{\lambda}
  2. I = I 0 1 + cos 2 θ 2 R 2 ( 2 π λ ) 4 ( n 2 - 1 n 2 + 2 ) 2 ( d 2 ) 6 I=I_{0}\frac{1+\cos^{2}\theta}{2R^{2}}\left(\frac{2\pi}{\lambda}\right)^{4}% \left(\frac{n^{2}-1}{n^{2}+2}\right)^{2}\left(\frac{d}{2}\right)^{6}
  3. σ s = 2 π 5 3 d 6 λ 4 ( n 2 - 1 n 2 + 2 ) 2 \sigma\text{s}=\frac{2\pi^{5}}{3}\frac{d^{6}}{\lambda^{4}}\left(\frac{n^{2}-1}% {n^{2}+2}\right)^{2}
  4. I = I 0 8 π 4 α 2 λ 4 R 2 ( 1 + cos 2 θ ) . I=I_{0}\frac{8\pi^{4}\alpha^{2}}{\lambda^{4}R^{2}}(1+\cos^{2}\theta).
  5. α scat = 8 π 3 3 λ 4 n 8 p 2 k T f β \alpha\text{scat}=\frac{8\pi^{3}}{3\lambda^{4}}n^{8}p^{2}kT\text{f}\beta

Real_analysis.html

  1. f : I 𝐑 . f\colon I\rightarrow\mathbf{R}.
  2. I = ( a , b ) = { x 𝐑 | a < x < b } , I=(a,b)=\{x\in\mathbf{R}\,|\,a<x<b\},
  3. I = [ a , b ] = { x 𝐑 | a x b } . I=[a,b]=\{x\in\mathbf{R}\,|\,a\leq x\leq b\}.
  4. I I
  5. f : I R f:I\to R
  6. I I
  7. ϵ \epsilon
  8. δ \delta
  9. ( x k , y k ) (x_{k},y_{k})
  10. I I
  11. k | y k - x k | < δ \sum_{k}\left|y_{k}-x_{k}\right|<\delta
  12. k | f ( y k ) - f ( x k ) | < ϵ . \displaystyle\sum_{k}|f(y_{k})-f(x_{k})|<\epsilon.
  13. f ( x ) = f ( a ) + a x f ( t ) d t f(x)=f(a)+\int_{a}^{x}f^{\prime}(t)\,dt
  14. f ( x ) = f ( a ) + a x g ( t ) d t f(x)=f(a)+\int_{a}^{x}g(t)\,dt
  15. n = 1 1 2 n = 1 2 + 1 4 + 1 8 + . \sum_{n=1}^{\infty}\frac{1}{2^{n}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots.
  16. f ( a ) + f ( a ) 1 ! ( x - a ) + f ′′ ( a ) 2 ! ( x - a ) 2 + f ( 3 ) ( a ) 3 ! ( x - a ) 3 + . f(a)+\frac{f^{\prime}(a)}{1!}(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+% \frac{f^{(3)}(a)}{3!}(x-a)^{3}+\cdots.
  17. n = 0 f ( n ) ( a ) n ! ( x - a ) n \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}\,(x-a)^{n}
  18. f ( a ) = lim h 0 f ( a + h ) - f ( a ) h f^{\prime}(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}
  19. a = x 0 t 1 x 1 t 2 x 2 x n - 1 t n x n = b . a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq\cdots\leq x_{n-1}\leq t_{n% }\leq x_{n}=b.\,\!
  20. i = 1 n f ( t i ) Δ i ; \sum_{i=1}^{n}f(t_{i})\Delta_{i};
  21. | S - i = 1 n f ( t i ) Δ i | < ε . \left|S-\sum_{i=1}^{n}f(t_{i})\Delta_{i}\right|<\varepsilon.

Real_line.html

  1. 𝐑 \mathbf{R}
  2. 𝐑 \mathbf{R}
  3. \mathbb{R}
  4. 𝐑 \mathbf{R}
  5. 𝐑 \mathbf{R}
  6. 𝐑 \mathbf{R}
  7. 𝐑 \mathbf{R}
  8. 𝐑 \mathbf{R}
  9. d ( x , y ) = | x y | d(x,y)=|x−y|
  10. n n
  11. n n
  12. n n
  13. p 𝐑 p∈\mathbf{R}
  14. ε > 0 ε>0
  15. ε ε
  16. 𝐑 \mathbf{R}
  17. p p
  18. ( p ε , p + ε ) (p−ε,p+ε)
  19. 𝐑 \mathbf{R}
  20. ( 0 , 1 ) (0,1)
  21. 𝐑 \mathbf{R}
  22. , + −∞,+∞∞
  23. 𝐑 \mathbf{R}
  24. 𝐑 \mathbf{R}
  25. 𝐑 \mathbf{R}
  26. A = R V , A=R\oplus V,
  27. v - v v\mapsto-v

Rectangle.html

  1. 1 4 ( a + c ) ( b + d ) \tfrac{1}{4}(a+c)(b+d)
  2. 1 2 ( a 2 + c 2 ) ( b 2 + d 2 ) . \tfrac{1}{2}\sqrt{(a^{2}+c^{2})(b^{2}+d^{2})}.
  3. \ell
  4. w w
  5. A = w A=\ell w\,
  6. P = 2 + 2 w = 2 ( + w ) P=2\ell+2w=2(\ell+w)\,
  7. d = 2 + w 2 d=\sqrt{\ell^{2}+w^{2}}
  8. = w \ell=w\,
  9. ( A P ) 2 + ( C P ) 2 = ( B P ) 2 + ( D P ) 2 . \displaystyle(AP)^{2}+(CP)^{2}=(BP)^{2}+(DP)^{2}.
  10. 0.5 × Area ( R ) Area ( C ) 2 × Area ( r ) 0.5\,\text{ × Area}(R)\leq\,\text{Area}(C)\leq 2\,\text{ × Area}(r)

Rectifier.html

  1. V rms = V peak 2 V dc = V peak π \begin{aligned}\displaystyle V_{\mathrm{rms}}&\displaystyle=\frac{V_{\mathrm{% peak}}}{2}\\ \displaystyle V_{\mathrm{dc}}&\displaystyle=\frac{V_{\mathrm{peak}}}{\pi}\end{aligned}
  2. V dc = V av \displaystyle V_{\mathrm{dc}}=V_{\mathrm{av}}
  3. V dc = V av = 3 3 V peak π V_{\mathrm{dc}}=V_{\mathrm{av}}=\frac{3{\sqrt{3}}V_{\mathrm{peak}}}{\pi}
  4. V dc = V av = 3 3 V peak π cos α V_{\mathrm{dc}}=V_{\mathrm{av}}=\frac{3{\sqrt{3}}V_{\mathrm{peak}}}{\pi}\cos\alpha
  5. V dc = V av = 3 V LLpeak π cos α V_{\mathrm{dc}}=V_{\mathrm{av}}=\frac{3V_{\mathrm{LLpeak}}}{\pi}\cos\alpha
  6. V dc = V av = 3 V LLpeak π cos ( α + μ ) V_{\mathrm{dc}}=V_{\mathrm{av}}=\frac{3V_{\mathrm{LLpeak}}}{\pi}\cos(\alpha+\mu)
  7. V dc = V av = 3 V LLpeak π cos ( α ) - 6 f L c I d {V_{\mathrm{dc}}=V_{\mathrm{av}}=\frac{3V_{\mathrm{LLpeak}}}{\pi}\cos(\alpha)}% -{6fL_{\mathrm{c}}I_{\mathrm{d}}}
  8. P in = V peak 2 I peak 2 P_{\mathrm{in}}={V_{\mathrm{peak}}\over 2}\cdot{I_{\mathrm{peak}}\over 2}
  9. P out = V peak π I peak π P_{\mathrm{out}}={V_{\mathrm{peak}}\over\pi}\cdot{I_{\mathrm{peak}}\over\pi}
  10. η = P out P in = 4 π 2 = 40.6 % \eta={P_{\mathrm{out}}\over P_{\mathrm{in}}}=\frac{4}{\pi^{2}}=40.6\%
  11. η = P out P in = 8 π 2 = 81.1 % \eta={P_{\mathrm{out}}\over P_{\mathrm{in}}}=\frac{8}{\pi^{2}}=81.1\%

Recursion.html

  1. Fib ( 0 ) = 0 as base case 1, \,\text{Fib}(0)=0\,\text{ as base case 1,}
  2. Fib ( 1 ) = 1 as base case 2, \,\text{Fib}(1)=1\,\text{ as base case 2,}
  3. For all integers n > 1 , Fib ( n ) := Fib ( n - 1 ) + Fib ( n - 2 ) . \,\text{For all integers }n>1,~{}\,\text{ Fib}(n):=\,\text{Fib}(n-1)+\,\text{% Fib}(n-2).
  4. \mathbb{N}
  5. \mathbb{N}
  6. \mathbb{N}
  7. f : X X f:X\rightarrow X
  8. F : X F:\mathbb{N}\rightarrow X
  9. \mathbb{N}
  10. F ( 0 ) = a F(0)=a
  11. F ( n + 1 ) = f ( F ( n ) ) F(n+1)=f(F(n))
  12. F : X F:\mathbb{N}\rightarrow X
  13. G : X G:\mathbb{N}\rightarrow X
  14. F ( 0 ) = a F(0)=a
  15. G ( 0 ) = a G(0)=a
  16. F ( n + 1 ) = f ( F ( n ) ) F(n+1)=f(F(n))
  17. G ( n + 1 ) = f ( G ( n ) ) G(n+1)=f(G(n))
  18. F ( n ) = G ( n ) F(n)=G(n)
  19. F ( 0 ) = a = G ( 0 ) F(0)=a=G(0)
  20. n = 0 n=0
  21. F ( k ) = G ( k ) F(k)=G(k)
  22. k k\in\mathbb{N}
  23. F ( k + 1 ) = f ( F ( k ) ) = f ( G ( k ) ) = G ( k + 1 ) . F(k+1)=f(F(k))=f(G(k))=G(k+1).
  24. F ( n ) = G ( n ) F(n)=G(n)
  25. n n\in\mathbb{N}
  26. ϕ = 1 + ( 1 / ϕ ) = 1 + ( 1 / ( 1 + ( 1 / ( 1 + 1 / ) ) ) ) \phi=1+(1/\phi)=1+(1/(1+(1/(1+1/...))))
  27. n ! = n ( n - 1 ) ! = n ( n - 1 ) 1 n!=n(n-1)!=n(n-1)\cdots 1
  28. f ( n ) = f ( n - 1 ) + f ( n - 2 ) f(n)=f(n-1)+f(n-2)
  29. C 0 = 1 C_{0}=1
  30. C n + 1 = ( 4 n + 2 ) C n / ( n + 2 ) C_{n+1}=(4n+2)C_{n}/(n+2)

Recursively_enumerable_language.html

  1. L * L^{*}
  2. L P L\circ P
  3. L P L\cup P
  4. L P L\cap P

Redshift.html

  1. z z
  2. λ λ
  3. f f
  4. λ f = c λf=c
  5. c c
  6. z z
  7. z z
  8. z = λ obsv - λ emit λ emit z=\frac{\lambda_{\mathrm{obsv}}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit% }}}
  9. z = f emit - f obsv f obsv z=\frac{f_{\mathrm{emit}}-f_{\mathrm{obsv}}}{f_{\mathrm{obsv}}}
  10. 1 + z = λ obsv λ emit 1+z=\frac{\lambda_{\mathrm{obsv}}}{\lambda_{\mathrm{emit}}}
  11. 1 + z = f emit f obsv 1+z=\frac{f_{\mathrm{emit}}}{f_{\mathrm{obsv}}}
  12. z z
  13. z z
  14. z > 0 z>0
  15. z z
  16. 1 + z = γ ( 1 + v c ) 1+z=\gamma\left(1+\frac{v_{\parallel}}{c}\right)
  17. z v c z\approx\frac{v_{\parallel}}{c}
  18. v v
  19. 1 + z = 1 + v c 1 - v c 1+z=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}
  20. 1 + z = 1 1 - v 2 c 2 1+z=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  21. 1 + z = a now a then 1+z=\frac{a_{\mathrm{now}}}{a_{\mathrm{then}}}
  22. 1 + z = g t t ( receiver ) g t t ( source ) 1+z=\sqrt{\frac{g_{tt}(\,\text{receiver})}{g_{tt}(\,\text{source})}}
  23. 1 + z = 1 - 2 G M c 2 r receiver 1 - 2 G M c 2 r source 1+z=\sqrt{\frac{1-\frac{2GM}{c^{2}r_{\,\text{receiver}}}}{1-\frac{2GM}{c^{2}r_% {\,\text{source}}}}}
  24. z > 0 z>0
  25. v v
  26. v c v≪c
  27. z v c z\approx\frac{v}{c}
  28. γ 1 \gamma\approx 1
  29. c c
  30. γ γ
  31. 1 + z = ( 1 + v c ) γ . 1+z=\left(1+\frac{v}{c}\right)\gamma.
  32. θ θ
  33. 1 + z = 1 + v cos ( θ ) / c 1 - v 2 / c 2 1+z=\frac{1+v\cos(\theta)/c}{\sqrt{1-v^{2}/c^{2}}}
  34. θ = 0 ° θ=0°
  35. 1 + z = 1 + v / c 1 - v / c 1+z=\sqrt{\frac{1+v/c}{1-v/c}}
  36. θ = 90 ° θ=90°
  37. 1 + z = 1 1 - v 2 / c 2 1+z=\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  38. × 10 8 \times 10^{8}
  39. d s 2 = 0 = - c 2 d t 2 + a 2 d r 2 1 - k r 2 ds^{2}=0=-c^{2}dt^{2}+\frac{a^{2}dr^{2}}{1-kr^{2}}
  40. d s 2 ds^{2}
  41. d t 2 dt^{2}
  42. d r 2 dr^{2}
  43. c c
  44. a a
  45. k k
  46. r = 0 r=0
  47. t = t n o w t=t_{now}
  48. t = t t h e n t=t_{then}
  49. r = R r=R
  50. c t then t now d t a = R 0 d r 1 - k r 2 . c\int_{t_{\mathrm{then}}}^{t_{\mathrm{now}}}\frac{dt}{a}\;=\int_{R}^{0}\frac{% dr}{\sqrt{1-kr^{2}}}\,.
  51. λ t h e n λ_{then}
  52. t = t then + λ then / c . t=t_{\mathrm{then}}+\lambda_{\mathrm{then}}/c\,.
  53. λ n o w λ_{now}
  54. t = t now + λ now / c . t=t_{\mathrm{now}}+\lambda_{\mathrm{now}}/c\,.
  55. r = R r=R
  56. r = 0 r=0
  57. c t then + λ then / c t now + λ now / c d t a = R 0 d r 1 - k r 2 . c\int_{t_{\mathrm{then}}+\lambda_{\mathrm{then}}/c}^{t_{\mathrm{now}}+\lambda_% {\mathrm{now}}/c}\frac{dt}{a}\;=\int_{R}^{0}\frac{dr}{\sqrt{1-kr^{2}}}\,.
  58. c t then + λ then / c t now + λ now / c d t a = c t then t now d t a c\int_{t_{\mathrm{then}}+\lambda_{\mathrm{then}}/c}^{t_{\mathrm{now}}+\lambda_% {\mathrm{now}}/c}\frac{dt}{a}\;=c\int_{t_{\mathrm{then}}}^{t_{\mathrm{now}}}% \frac{dt}{a}\,
  59. 0 \displaystyle 0
  60. t now t now + λ now / c d t a = t then t then + λ then / c d t a . \int_{t_{\mathrm{now}}}^{t_{\mathrm{now}}+\lambda_{\mathrm{now}}/c}\frac{dt}{a% }\;=\int_{t_{\mathrm{then}}}^{t_{\mathrm{then}}+\lambda_{\mathrm{then}}/c}% \frac{dt}{a}\,.
  61. a = a n o w a=a_{now}
  62. a = a t h e n a=a_{then}
  63. t now + λ now / c a now - t now a now = t then + λ then / c a then - t then a then \frac{t_{\mathrm{now}}+\lambda_{\mathrm{now}}/c}{a_{\mathrm{now}}}-\frac{t_{% \mathrm{now}}}{a_{\mathrm{now}}}\;=\frac{t_{\mathrm{then}}+\lambda_{\mathrm{% then}}/c}{a_{\mathrm{then}}}-\frac{t_{\mathrm{then}}}{a_{\mathrm{then}}}
  64. λ now λ then = a now a then . \frac{\lambda_{\mathrm{now}}}{\lambda_{\mathrm{then}}}=\frac{a_{\mathrm{now}}}% {a_{\mathrm{then}}}\,.
  65. 1 + z = a now a then 1+z=\frac{a_{\mathrm{now}}}{a_{\mathrm{then}}}
  66. z z
  67. Ω 0 = ρ ρ c r i t , \Omega_{0}=\frac{\rho}{\rho_{crit}}\ ,
  68. t ( z ) = 2 3 H 0 Ω 0 1 / 2 ( 1 + z ) 3 / 2 , t(z)=\frac{2}{3H_{0}{\Omega_{0}}^{1/2}(1+z)^{3/2}}\ ,
  69. z z
  70. a ( t ) a(t)
  71. a ( t ) a(t)
  72. a ( t ) a(t)
  73. v > c v>c
  74. v > c v>c
  75. 1 + z = ( 1 + z Doppler ) ( 1 + z expansion ) 1+z=(1+z_{\mathrm{Doppler}})(1+z_{\mathrm{expansion}})
  76. 1 + z = 1 1 - 2 G M r c 2 , 1+z=\frac{1}{\sqrt{1-\frac{2GM}{rc^{2}}}},
  77. G G
  78. M M
  79. r r
  80. c c
  81. z z
  82. δ z = 0.5 δz=0.5
  83. z = 1 z=1
  84. ( 1 + z ) 2 (1+z)^{2}
  85. z = 1089 z=1089
  86. z = 0 z=0
  87. z > 0.1 z>0.1
  88. d H d_{H}
  89. c t L B ct_{LB}
  90. z z
  91. z = 8.6 z=8.6
  92. z = 6.96 z=6.96
  93. z = 7.6 z=7.6
  94. z = 7.0 z=7.0
  95. z = 8.2 z=8.2
  96. z = 7.1 z=7.1
  97. z = 5.2 z=5.2
  98. z = 6.42 z=6.42
  99. z = 1089 z=1089
  100. 20 < z < 100 20<z<100

Reduction.html

  1. G G
  2. B B
  3. H G H\to G
  4. H H
  5. B H B_{H}
  6. B H × H G B_{H}\times_{H}G
  7. B B

Red–black_tree.html

  1. 2 b h ( v ) - 1 2^{bh(v)}-1
  2. 2 b h ( v ) - 1 = 2 0 - 1 = 1 - 1 = 0 2^{bh(v)}-1=2^{0}-1=1-1=0
  3. 2 b h ( v ) - 1 2^{bh(v)}-1
  4. v v^{\prime}
  5. v v^{\prime}
  6. 2 b h ( v ) - 1 2^{bh(v^{\prime})}-1
  7. v v^{\prime}
  8. v v^{\prime}
  9. v v^{\prime}
  10. v v^{\prime}
  11. 2 b h ( v ) - 1 - 1 2^{bh(v^{\prime})-1}-1
  12. v v^{\prime}
  13. 2 b h ( v ) - 1 - 1 + 2 b h ( v ) - 1 - 1 + 1 = 2 b h ( v ) - 1 2^{bh(v^{\prime})-1}-1+2^{bh(v^{\prime})-1}-1+1=2^{bh(v^{\prime})}-1
  14. n 2 h ( root ) 2 - 1 log 2 ( n + 1 ) h ( root ) 2 h ( root ) 2 log 2 ( n + 1 ) . n\geq 2^{{h(\,\text{root})\over 2}}-1\leftrightarrow\;\log_{2}{(n+1)}\geq{h(\,% \text{root})\over 2}\leftrightarrow\;h(\,\text{root})\leq 2\log_{2}{(n+1)}.
  15. O ( l o g l o g n ) O(loglogn)

Reed–Solomon_error_correction.html

  1. t t
  2. t t
  3. t / 2 ⌊t/2⌋
  4. t t
  5. b + 1 b+ 1
  6. b b
  7. t t
  8. k k
  9. n n
  10. k k
  11. n n
  12. n n
  13. k k
  14. n n
  15. k k
  16. k k
  17. n k n−k
  18. n n
  19. k k
  20. k k
  21. k k
  22. n k n−k
  23. p ( x ) p(x)
  24. a 1 , , a n a_{1},\dots,a_{n}
  25. 𝐂 \mathbf{C}
  26. 𝐂 = { ( p ( a 1 ) , p ( a 2 ) , , p ( a n ) ) | p is a polynomial over F of degree < k } . \mathbf{C}=\Big\{\;\big(p(a_{1}),p(a_{2}),\dots,p(a_{n})\big)\;\Big|\;p\,\text% { is a polynomial over }F\,\text{ of degree }<k\;\Big\}\,.
  27. k k
  28. k - 1 k-1
  29. n - ( k - 1 ) = n - k + 1 n-(k-1)=n-k+1
  30. k - 1 k-1
  31. d = n - k + 1 d=n-k+1
  32. δ = d / n = 1 - k / n + 1 / n 1 - R \delta=d/n=1-k/n+1/n\sim 1-R
  33. R = k / n R=k/n
  34. δ + R 1 \delta+R\leq 1
  35. q k q^{k}
  36. a 1 , , a k a_{1},\dots,a_{k}
  37. a 1 , , a n a_{1},\dots,a_{n}
  38. α \alpha
  39. F F
  40. α \alpha
  41. a i = α i a_{i}=\alpha^{i}
  42. i = 1 , , q - 1 i=1,\dots,q-1
  43. F F
  44. 0
  45. n = q - 1 n=q-1
  46. p ( a ) p(a)
  47. F F
  48. p ( α a ) p(\alpha a)
  49. p ( a ) p(a)
  50. x = ( x 1 , , x k ) F k x=(x_{1},\dots,x_{k})\in F^{k}
  51. p x p_{x}
  52. p x ( a ) = i = 1 k x i a i - 1 . p_{x}(a)=\sum_{i=1}^{k}x_{i}a^{i-1}\,.
  53. p p
  54. n n
  55. a 1 , , a n a_{1},\dots,a_{n}
  56. F F
  57. C : F k F n C:F^{k}\to F^{n}
  58. C ( x ) = ( p x ( a 1 ) , , p x ( a n ) ) . C(x)=\big(p_{x}(a_{1}),\dots,p_{x}(a_{n})\big)\,.
  59. C C
  60. C ( x ) = x A C(x)=x\cdot A
  61. ( k × n ) (k\times n)
  62. A A
  63. F F
  64. A = [ 1 1 a 1 a n a 1 2 a n 2 a 1 k - 1 a n k - 1 ] A=\begin{bmatrix}1&\dots&1\\ a_{1}&\dots&a_{n}\\ a_{1}^{2}&\dots&a_{n}^{2}\\ \vdots&\ddots&\vdots\\ a_{1}^{k-1}&\dots&a_{n}^{k-1}\end{bmatrix}
  65. F F
  66. A A
  67. x x
  68. p x p_{x}
  69. p x p_{x}
  70. k k
  71. p x ( a i ) = x i p_{x}(a_{i})=x_{i}
  72. i = 1 , , k i=1,\dots,k
  73. p x p_{x}
  74. x x
  75. a k + 1 , , a n a_{k+1},\dots,a_{n}
  76. C : F k F n C:F^{k}\to F^{n}
  77. C ( x ) = ( p x ( a 1 ) , , p x ( a n ) ) . C(x)=\big(p_{x}(a_{1}),\dots,p_{x}(a_{n})\big)\,.
  78. k k
  79. x x
  80. x x
  81. p x p_{x}
  82. p x p_{x}
  83. p x p_{x}
  84. p x p_{x}
  85. p x p_{x}
  86. s s
  87. n - 1 n-1
  88. n = q - 1 n=q-1
  89. n n
  90. s ( a ) s(a)
  91. p x ( a ) p_{x}(a)
  92. k - 1 k-1
  93. g ( a ) g(a)
  94. n - k n-k
  95. g ( x ) g(x)
  96. α , α 2 , , α n - k \alpha,\alpha^{2},\dots,\alpha^{n-k}
  97. g ( x ) = ( x - α ) ( x - α 2 ) ( x - α n - k ) = g 0 + g 1 x + + g n - k - 1 x n - k - 1 + x n - k . g(x)=(x-\alpha)(x-\alpha^{2})\cdots(x-\alpha^{n-k})=g_{0}+g_{1}x+\cdots+g_{n-k% -1}x^{n-k-1}+x^{n-k}\,.
  98. n = q - 1 n=q-1
  99. s ( a ) = p x ( a ) g ( a ) s(a)=p_{x}(a)\cdot g(a)
  100. 𝐂 \mathbf{C^{\prime}}
  101. n = q - 1 n=q-1
  102. 𝐂 = { ( s 1 , s 2 , , s n ) | s ( a ) = i = 1 n s i a i - 1 is a polynomial that has at least the roots α 1 , α 2 , , α n - k } . \mathbf{C^{\prime}}=\Big\{\;\big(s_{1},s_{2},\dots,s_{n}\big)\;\Big|\;s(a)=% \sum_{i=1}^{n}s_{i}a^{i-1}\,\text{ is a polynomial that has at least the roots% }\alpha^{1},\alpha^{2},\dots,\alpha^{n-k}\;\Big\}\,.
  103. g ( a ) g(a)
  104. α , α 2 , , α n - k \alpha,\alpha^{2},\ldots,\alpha^{n-k}
  105. g ( a ) g(a)
  106. r ( a ) r(a)
  107. r ( a ) = s ( a ) r(a)=s(a)
  108. p x ( a ) = r ( a ) / g ( a ) p_{x}(a)=r(a)/g(a)
  109. p ( a ) p(a)
  110. e ( a ) e(a)
  111. e ( a ) e(a)
  112. g ( a ) g(a)
  113. r ( a ) = p ( a ) g ( a ) + e ( a ) . r(a)=p(a)\cdot g(a)+e(a)\,.
  114. e ( a ) e(a)
  115. r ( a ) r(a)
  116. g ( a ) g(a)
  117. s ( a ) s(a)
  118. r ( a ) r(a)
  119. r ( a ) r(a)
  120. s ( a ) s(a)
  121. s ( x ) = p ( x ) g ( x ) s(x)=p(x)g(x)
  122. s ( x ) s(x)
  123. k k
  124. p ( x ) p(x)
  125. s ( x ) s(x)
  126. s ( x ) s(x)
  127. g ( x ) g(x)
  128. p ( x ) p(x)
  129. s ( x ) s(x)
  130. p ( x ) p(x)
  131. p ( x ) p(x)
  132. x t x^{t}
  133. t = n - k t=n-k
  134. g ( x ) g(x)
  135. t t
  136. s r ( x ) s_{r}(x)
  137. s r ( x ) = p ( x ) x t mod g ( x ) . s_{r}(x)=p(x)\cdot x^{t}\ \bmod\ g(x)\,.
  138. t - 1 t-1
  139. x t - 1 , x t - 2 , , x 1 , x 0 x^{t-1},x^{t-2},\dots,x^{1},x^{0}
  140. p ( x ) x t p(x)\cdot x^{t}
  141. s ( x ) s(x)
  142. k k
  143. p ( x ) p(x)
  144. s ( x ) = p ( x ) x t - s r ( x ) . s(x)=p(x)\cdot x^{t}-s_{r}(x)\,.
  145. s ( x ) s(x)
  146. 𝐂 \mathbf{C^{\prime}}
  147. g ( x ) g(x)
  148. s ( x ) p ( x ) x t - s r ( x ) s r ( x ) - s r ( x ) 0 mod g ( x ) . s(x)\equiv p(x)\cdot x^{t}-s_{r}(x)\equiv s_{r}(x)-s_{r}(x)\equiv 0\mod g(x)\,.
  149. 𝐂 \mathbf{C}
  150. 𝐂 \mathbf{C^{\prime}}
  151. 𝐂 = 𝐂 \mathbf{C}=\mathbf{C^{\prime}}
  152. a 1 , , a n a_{1},\dots,a_{n}
  153. p ( x ) p(x)
  154. q ( x ) q(x)
  155. n n
  156. p ( x ) p(x)
  157. q ( x ) q(x)
  158. q ( x ) q(x)
  159. p ( x ) p(x)
  160. x = α i x=\alpha^{i}
  161. i = 0 , , n - 1 i=0,\dots,n-1
  162. α \alpha
  163. n n
  164. p ( x ) = v 0 + v 1 x + v 2 x 2 + + v n - 1 x n - 1 , p(x)=v_{0}+v_{1}x+v_{2}x^{2}+\cdots+v_{n-1}x^{n-1},
  165. q ( x ) = f 0 + f 1 x + f 2 x 2 + + f n - 1 x n - 1 q(x)=f_{0}+f_{1}x+f_{2}x^{2}+\cdots+f_{n-1}x^{n-1}
  166. p ( x ) p(x)
  167. q ( x ) q(x)
  168. p ( x ) p(x)
  169. q ( x ) q(x)
  170. i = 0 , , n - 1 i=0,\dots,n-1
  171. f i = p ( α i ) f_{i}=p(\alpha^{i})
  172. v i = 1 n q ( α n - i ) \textstyle v_{i}=\frac{1}{n}q(\alpha^{n-i})
  173. ( f 0 , , f n - 1 ) (f_{0},\ldots,f_{n-1})
  174. p ( x ) p(x)
  175. k k
  176. f 0 , , f n - 1 f_{0},\ldots,f_{n-1}
  177. p ( x ) p(x)
  178. v i = 0 v_{i}=0
  179. i = k , , n - 1 i=k,\ldots,n-1
  180. q ( α i ) = 0 q(\alpha^{i})=0
  181. i = 1 , , n - k i=1,\ldots,n-k
  182. q ( α i ) = n v n - i q(\alpha^{i})=nv_{n-i}
  183. ( f 0 , , f n - 1 ) (f_{0},\ldots,f_{n-1})
  184. n - k n-k
  185. ( n - k ) / 2 (n-k)/2
  186. E E
  187. S S
  188. F F
  189. 2 m 2^{m}
  190. m m
  191. n = 2 m - 1 n=2^{m}-1
  192. n = 2 8 - 1 = 255 n=2^{8}-1=255
  193. k k
  194. k < n k<n
  195. k = 223 k=223
  196. n = 255 n=255
  197. ( n , k ) = ( 255 , 223 ) (n,k)=(255,223)
  198. ( n , k ) (n,k)
  199. k k
  200. n n
  201. k k
  202. k k
  203. ( n k ) = n ! ( n - k ) ! k ! \textstyle{\left({{n}\atop{k}}\right)}={n!\over(n-k)!k!}
  204. ( 255 , 249 ) (255,249)
  205. s ( x ) = i = 0 n - 1 c i x i s(x)=\sum_{i=0}^{n-1}c_{i}x^{i}
  206. g ( x ) = j = 1 n - k ( x - α j ) , g(x)=\prod_{j=1}^{n-k}(x-\alpha^{j}),
  207. s ( α i ) = 0 , i = 1 , 2 , , n - k s(\alpha^{i})=0,\ i=1,2,\ldots,n-k
  208. r ( x ) = s ( x ) + e ( x ) r(x)=s(x)+e(x)
  209. e ( x ) = i = 0 n - 1 e i x i e(x)=\sum_{i=0}^{n-1}e_{i}x^{i}
  210. e ( x ) = k = 1 ν e i k x i k e(x)=\sum_{k=1}^{\nu}e_{i_{k}}x^{i_{k}}
  211. S j = r ( α j ) = s ( α j ) + e ( α j ) = 0 + e ( α j ) = e ( α j ) , j = 1 , 2 , , n - k = k = 1 ν e i k ( α j ) i k \begin{aligned}\displaystyle S_{j}&\displaystyle=r(\alpha^{j})=s(\alpha^{j})+e% (\alpha^{j})=0+e(\alpha^{j})=e(\alpha^{j}),\ j=1,2,\ldots,n-k\\ &\displaystyle=\sum_{k=1}^{\nu}e_{i_{k}}\left(\alpha^{j}\right)^{i_{k}}\end{aligned}
  212. ( α j , S j ) (\alpha^{j},S_{j})
  213. S j = e ( α j ) S_{j}=e(\alpha^{j})
  214. X k = α i k , Y k = e i k X_{k}=\alpha^{i_{k}},\ Y_{k}=e_{i_{k}}
  215. S j = k = 1 ν Y k X k j S_{j}=\sum_{k=1}^{\nu}Y_{k}X_{k}^{j}
  216. [ X 1 1 X 2 1 X ν 1 X 1 2 X 2 2 X ν 2 X 1 n - k X 2 n - k X ν n - k ] [ Y 1 Y 2 Y ν ] = [ S 1 S 2 S n - k ] \begin{bmatrix}X_{1}^{1}&X_{2}^{1}&\cdots&X_{\nu}^{1}\\ X_{1}^{2}&X_{2}^{2}&\cdots&X_{\nu}^{2}\\ \vdots&\vdots&&\vdots\\ X_{1}^{n-k}&X_{2}^{n-k}&\cdots&X_{\nu}^{n-k}\\ \end{bmatrix}\begin{bmatrix}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{\nu}\end{bmatrix}=\begin{bmatrix}S_{1}\\ S_{2}\\ \vdots\\ S_{n-k}\end{bmatrix}
  217. Λ ( x ) = k = 1 ν ( 1 - x X k ) = 1 + Λ 1 x 1 + Λ 2 x 2 + + Λ ν x ν \Lambda(x)=\prod_{k=1}^{\nu}(1-xX_{k})=1+\Lambda_{1}x^{1}+\Lambda_{2}x^{2}+% \cdots+\Lambda_{\nu}x^{\nu}
  218. X k - 1 X_{k}^{-1}
  219. Λ ( X k - 1 ) = 0 \Lambda(X_{k}^{-1})=0
  220. Λ ( X k - 1 ) = 1 + Λ 1 X k - 1 + Λ 2 X k - 2 + + Λ ν X k - ν = 0 \Lambda(X_{k}^{-1})=1+\Lambda_{1}X_{k}^{-1}+\Lambda_{2}X_{k}^{-2}+\cdots+% \Lambda_{\nu}X_{k}^{-\nu}=0
  221. Y k X k j + ν Y_{k}X_{k}^{j+\nu}
  222. Y k X k j + ν Λ ( X k - 1 ) = 0. Hence Y k X k j + ν + Λ 1 Y k X k j + ν X k - 1 + Λ 2 Y k X k j + ν X k - 2 + + Λ ν Y k X k j + ν X k - ν = 0 , and so Y k X k j + ν + Λ 1 Y k X k j + ν - 1 + Λ 2 Y k X k j + ν - 2 + + Λ ν Y k X k j = 0 \begin{aligned}&\displaystyle Y_{k}X_{k}^{j+\nu}\Lambda(X_{k}^{-1})=0.\\ \displaystyle\,\text{Hence }&\displaystyle Y_{k}X_{k}^{j+\nu}+\Lambda_{1}Y_{k}% X_{k}^{j+\nu}X_{k}^{-1}+\Lambda_{2}Y_{k}X_{k}^{j+\nu}X_{k}^{-2}+\cdots+\Lambda% _{\nu}Y_{k}X_{k}^{j+\nu}X_{k}^{-\nu}=0,\\ \displaystyle\,\text{and so }&\displaystyle Y_{k}X_{k}^{j+\nu}+\Lambda_{1}Y_{k% }X_{k}^{j+\nu-1}+\Lambda_{2}Y_{k}X_{k}^{j+\nu-2}+\cdots+\Lambda_{\nu}Y_{k}X_{k% }^{j}=0\\ \end{aligned}
  223. k = 1 ν ( Y k X k j + ν + Λ 1 Y k X k j + ν - 1 + Λ 2 Y k X k j + ν - 2 + + Λ ν Y k X k j ) = 0 k = 1 ν ( Y k X k j + ν ) + Λ 1 k = 1 ν ( Y k X k j + ν - 1 ) + Λ 2 k = 1 ν ( Y k X k j + ν - 2 ) + + Λ ν k = 1 ν ( Y k X k j ) = 0 \begin{aligned}&\displaystyle\sum_{k=1}^{\nu}(Y_{k}X_{k}^{j+\nu}+\Lambda_{1}Y_% {k}X_{k}^{j+\nu-1}+\Lambda_{2}Y_{k}X_{k}^{j+\nu-2}+\cdots+\Lambda_{\nu}Y_{k}X_% {k}^{j})=0\\ &\displaystyle\sum_{k=1}^{\nu}(Y_{k}X_{k}^{j+\nu})+\Lambda_{1}\sum_{k=1}^{\nu}% (Y_{k}X_{k}^{j+\nu-1})+\Lambda_{2}\sum_{k=1}^{\nu}(Y_{k}X_{k}^{j+\nu-2})+% \cdots+\Lambda_{\nu}\sum_{k=1}^{\nu}(Y_{k}X_{k}^{j})=0\end{aligned}
  224. S j + ν + Λ 1 S j + ν - 1 + + Λ ν - 1 S j + 1 + Λ ν S j = 0 S_{j+\nu}+\Lambda_{1}S_{j+\nu-1}+\cdots+\Lambda_{\nu-1}S_{j+1}+\Lambda_{\nu}S_% {j}=0\,
  225. S j Λ ν + S j + 1 Λ ν - 1 + + S j + ν - 1 Λ 1 = - S j + ν S_{j}\Lambda_{\nu}+S_{j+1}\Lambda_{\nu-1}+\cdots+S_{j+\nu-1}\Lambda_{1}=-S_{j+\nu}
  226. [ S 1 S 2 S ν S 2 S 3 S ν + 1 S ν S ν + 1 S 2 ν - 1 ] [ Λ ν Λ ν - 1 Λ 1 ] = [ - S ν + 1 - S ν + 2 - S ν + ν ] \begin{bmatrix}S_{1}&S_{2}&\cdots&S_{\nu}\\ S_{2}&S_{3}&\cdots&S_{\nu+1}\\ \vdots&\vdots&&\vdots\\ S_{\nu}&S_{\nu+1}&\cdots&S_{2\nu-1}\end{bmatrix}\begin{bmatrix}\Lambda_{\nu}\\ \Lambda_{\nu-1}\\ \vdots\\ \Lambda_{1}\end{bmatrix}=\begin{bmatrix}-S_{\nu+1}\\ -S_{\nu+2}\\ \vdots\\ -S_{\nu+\nu}\end{bmatrix}
  227. Δ = S i + Λ 1 S i - 1 + + Λ e S i - e \Delta=S_{i}+\Lambda_{1}\ S_{i-1}+\cdots+\Lambda_{e}\ S_{i-e}
  228. G F ( 929 ) GF(929)
  229. α = 3 α=3
  230. t = 4 t=4
  231. g ( x ) = ( x - 3 ) ( x - 3 2 ) ( x - 3 3 ) ( x - 3 4 ) = x 4 + 809 x 3 + 723 x 2 + 568 x + 522 g(x)=(x-3)(x-3^{2})(x-3^{3})(x-3^{4})=x^{4}+809x^{3}+723x^{2}+568x+522
  232. s r ( x ) = p ( x ) x t mod g ( x ) = 547 x 3 + 738 x 2 + 442 x + 455 s_{r}(x)=p(x)\,x^{t}\mod g(x)=547x^{3}+738x^{2}+442x+455
  233. s ( x ) = p ( x ) x t - s r ( x ) = 3 x 6 + 2 x 5 + 1 x 4 + 382 x 3 + 191 x 2 + 487 x + 474 s(x)=p(x)\,x^{t}-s_{r}(x)=3x^{6}+2x^{5}+1x^{4}+382x^{3}+191x^{2}+487x+474
  234. r ( x ) = s ( x ) + e ( x ) = 3 x 6 + 2 x 5 + 123 x 4 + 456 x 3 + 191 x 2 + 487 x + 474 r(x)=s(x)+e(x)=3x^{6}+2x^{5}+123x^{4}+456x^{3}+191x^{2}+487x+474
  235. S 1 = r ( 3 1 ) = 3 3 6 + 2 3 5 + 123 3 4 + 456 3 3 + 191 3 2 + 487 3 + 474 = 732 S_{1}=r(3^{1})=3\cdot 3^{6}+2\cdot 3^{5}+123\cdot 3^{4}+456\cdot 3^{3}+191% \cdot 3^{2}+487\cdot 3+474=732
  236. S 2 = r ( 3 2 ) = 637 , S 3 = r ( 3 3 ) = 762 , S 4 = r ( 3 4 ) = 925 S_{2}=r(3^{2})=637,\;S_{3}=r(3^{3})=762,\;S_{4}=r(3^{4})=925
  237. Ω ( x ) = S ( x ) Λ ( x ) mod x 4 = 546 x + 732 \Omega(x)=S(x)\Lambda(x)\mod x^{4}=546x+732\,
  238. Λ ( x ) = 658 x + 821 \Lambda^{\prime}(x)=658x+821\,
  239. e 1 = - Ω ( x 1 ) / Λ ( x 1 ) = - 649 / 54 = 280 × 843 = 74 e_{1}=-\Omega(x_{1})/\Lambda^{\prime}(x_{1})=-649/54=280\times 843=74\,
  240. e 2 = - Ω ( x 2 ) / Λ ( x 2 ) = 122 e_{2}=-\Omega(x_{2})/\Lambda^{\prime}(x_{2})=122\,
  241. k k
  242. n - k n-k
  243. k k
  244. k - 1 k-1
  245. n - k n-k
  246. G F ( 2 m ) GF(2^{m})

Referential_transparency_(computer_science).html

  1. x - x = 0 x-x=0

Reflectance.html

  1. R = Φ e r Φ e i , R=\frac{\Phi_{\mathrm{e}}^{\mathrm{r}}}{\Phi_{\mathrm{e}}^{\mathrm{i}}},
  2. R ν = Φ e , ν r Φ e , ν i , R_{\nu}=\frac{\Phi_{\mathrm{e},\nu}^{\mathrm{r}}}{\Phi_{\mathrm{e},\nu}^{% \mathrm{i}}},
  3. R λ = Φ e , λ r Φ e , λ i , R_{\lambda}=\frac{\Phi_{\mathrm{e},\lambda}^{\mathrm{r}}}{\Phi_{\mathrm{e},% \lambda}^{\mathrm{i}}},
  4. R Ω = L e , Ω r L e , Ω i , R_{\Omega}=\frac{L_{\mathrm{e},\Omega}^{\mathrm{r}}}{L_{\mathrm{e},\Omega}^{% \mathrm{i}}},
  5. R ν , Ω = L e , Ω , ν r L e , Ω , ν i , R_{\nu,\Omega}=\frac{L_{\mathrm{e},\Omega,\nu}^{\mathrm{r}}}{L_{\mathrm{e},% \Omega,\nu}^{\mathrm{i}}},
  6. R λ , Ω = L e , Ω , λ r L e , Ω , λ i , R_{\lambda,\Omega}=\frac{L_{\mathrm{e},\Omega,\lambda}^{\mathrm{r}}}{L_{% \mathrm{e},\Omega,\lambda}^{\mathrm{i}}},

Reflection_coefficient.html

  1. E - E^{-}
  2. E + E^{+}
  3. Γ \Gamma
  4. Γ = E - E + \Gamma=\frac{E^{-}}{E^{+}}
  5. Z L \scriptstyle Z_{L}\,
  6. Z S \scriptstyle Z_{S}\,
  7. Γ = Z L - Z S Z L + Z S \Gamma={Z_{L}-Z_{S}\over Z_{L}+Z_{S}}
  8. π \pi
  9. S W R SWR
  10. | Γ | = S W R - 1 S W R + 1 |\Gamma|={SWR-1\over SWR+1}

Refraction.html

  1. sin θ 1 sin θ 2 = v 1 v 2 = n 2 n 1 . \frac{\sin\theta_{1}}{\sin\theta_{2}}=\frac{v_{1}}{v_{2}}=\frac{n_{2}}{n_{1}}.

Refractive_index.html

  1. n = c v , n=\frac{\mathrm{c}}{v},
  2. n = c v . n=\frac{\mathrm{c}}{v}.
  3. V = n yellow - 1 n blue - n red . V=\frac{n_{\mathrm{yellow}}-1}{n_{\mathrm{blue}}-n_{\mathrm{red}}}.
  4. n ¯ = n + i κ . \underline{n}=n+i\kappa.
  5. 𝐄 ( z , t ) = Re [ 𝐄 0 e i ( k ¯ z - ω t ) ] = Re [ 𝐄 0 e i ( 2 π ( n + i κ ) z / λ 0 - ω t ) ] = e - 2 π κ z / λ 0 Re [ 𝐄 0 e i ( k z - ω t ) ] . \mathbf{E}(z,t)=\operatorname{Re}\!\left[\mathbf{E}_{0}e^{i(\underline{k}z-% \omega t)}\right]=\operatorname{Re}\!\left[\mathbf{E}_{0}e^{i(2\pi(n+i\kappa)z% /\lambda_{0}-\omega t)}\right]=e^{-2\pi\kappa z/\lambda_{0}}\operatorname{Re}% \!\left[\mathbf{E}_{0}e^{i(kz-\omega t)}\right].
  6. OPL = n d . \,\text{OPL}=nd.
  7. n 1 sin θ 1 = n 2 sin θ 2 . n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}.
  8. n 1 n 2 sin θ 1 > 1 , \frac{n_{1}}{n_{2}}\sin\theta_{1}>1,
  9. θ c = arcsin ( n 2 n 1 ) . \theta_{\mathrm{c}}=\arcsin\!\left(\frac{n_{2}}{n_{1}}\right)\!.
  10. R 0 = | n 1 - n 2 n 1 + n 2 | 2 . R_{0}=\left|\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\right|^{2}\!.
  11. θ B = arctan ( n 2 n 1 ) . \theta_{\mathrm{B}}=\arctan\!\left(\frac{n_{2}}{n_{1}}\right)\!.
  12. 1 f = ( n - 1 ) ( 1 R 1 - 1 R 2 ) , \frac{1}{f}=(n-1)\!\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)\!,
  13. NA = n sin θ . \mathrm{NA}=n\sin\theta.
  14. n = ε r μ r , n=\sqrt{\varepsilon_{\mathrm{r}}\mu_{\mathrm{r}}},
  15. ε ¯ r = ε r + i ε ~ r = n ¯ 2 = ( n + i κ ) 2 , \underline{\varepsilon}_{\mathrm{r}}=\varepsilon_{\mathrm{r}}+i\tilde{% \varepsilon}_{\mathrm{r}}=\underline{n}^{2}=(n+i\kappa)^{2},
  16. ε r = n 2 - κ 2 , \varepsilon_{\mathrm{r}}=n^{2}-\kappa^{2},
  17. ε ~ r = 2 n κ , \tilde{\varepsilon}_{\mathrm{r}}=2n\kappa,
  18. n = | ε ¯ r | + ε r 2 , n=\sqrt{\frac{|\underline{\varepsilon}_{\mathrm{r}}|+\varepsilon_{\mathrm{r}}}% {2}},
  19. κ = | ε ¯ r | - ε r 2 . \kappa=\sqrt{\frac{|\underline{\varepsilon}_{\mathrm{r}}|-\varepsilon_{\mathrm% {r}}}{2}}.
  20. | ε ¯ r | = ε r 2 + ε ~ r 2 |\underline{\varepsilon}_{\mathrm{r}}|=\sqrt{\varepsilon_{\mathrm{r}}^{2}+% \tilde{\varepsilon}_{\mathrm{r}}^{2}}
  21. n g = c v g , n_{\mathrm{g}}=\frac{\mathrm{c}}{v_{\mathrm{g}}},
  22. v g = v - λ d v d λ , v_{\mathrm{g}}=v-\lambda\frac{\mathrm{d}v}{\mathrm{d}\lambda},
  23. n g = n 1 + λ n d n d λ . n_{\mathrm{g}}=\frac{n}{1+\frac{\lambda}{n}\frac{\mathrm{d}n}{\mathrm{d}% \lambda}}.
  24. v g = c ( n - λ 0 d n d λ 0 ) - 1 , v_{\mathrm{g}}=\mathrm{c}\!\left(n-\lambda_{0}\frac{\mathrm{d}n}{\mathrm{d}% \lambda_{0}}\right)^{-1}\!,
  25. n g = n - λ 0 d n d λ 0 , n_{\mathrm{g}}=n-\lambda_{0}\frac{\mathrm{d}n}{\mathrm{d}\lambda_{0}},
  26. p = n E c , p=\frac{nE}{\mathrm{c}},
  27. p = E n c . p=\frac{E}{n\mathrm{c}}.
  28. V = c n + v ( 1 - 1 n 2 ) . V=\frac{\mathrm{c}}{n}+v\!\left(1-\frac{1}{n^{2}}\right)\!.
  29. A = M ρ n 2 - 1 n 2 + 2 , A=\frac{M}{\rho}\frac{n^{2}-1}{n^{2}+2},

Regular_expression.html

  1. ( a b ) * a ( a b ) ( a b ) ( a b ) (a\mid b)^{*}a(a\mid b)(a\mid b)(a\mid b)
  2. ( a b ) * a ( a b ) ( a b ) ( a b ) k - 1 times . (a\mid b)^{*}a\underbrace{(a\mid b)(a\mid b)\cdots(a\mid b)}_{k-1\,\text{ % times}}.\,

Regular_grammar.html

  1. { a i b i : i 0 } \{a^{i}b^{i}:i\geq 0\}

Regular_icosahedron.html

  1. r u = a 2 φ 5 = a 4 10 + 2 5 = a sin 2 π 5 0.9510565163 a r_{u}=\frac{a}{2}\sqrt{\varphi\sqrt{5}}=\frac{a}{4}\sqrt{10+2\sqrt{5}}=a\sin% \frac{2\pi}{5}\approx 0.9510565163\cdot a
  2. r i = φ 2 a 2 3 = 3 12 ( 3 + 5 ) a 0.7557613141 a r_{i}=\frac{\varphi^{2}a}{2\sqrt{3}}=\frac{\sqrt{3}}{12}\left(3+\sqrt{5}\right% )a\approx 0.7557613141\cdot a
  3. r m = a φ 2 = 1 4 ( 1 + 5 ) a = a cos π 5 0.80901699 a r_{m}=\frac{a\varphi}{2}=\frac{1}{4}\left(1+\sqrt{5}\right)a=a\cos\frac{\pi}{5% }\approx 0.80901699\cdot a
  4. A = 5 3 a 2 8.66025404 a 2 , A=5\sqrt{3}a^{2}\approx 8.66025404a^{2},
  5. V = 5 12 ( 3 + 5 ) a 3 2.18169499 a 3 . V=\frac{5}{12}(3+\sqrt{5})a^{3}\approx 2.18169499a^{3}.
  6. F = 20 F=20
  7. f = V / ( 4 π r u 3 / 3 ) = 20 ( 3 + 5 ) ( 2 5 + 10 ) 3 / 2 π 0.6054613829. f=V/(4\pi r_{u}^{3}/3)=\frac{20(3+\surd 5)}{(2\surd 5+10)^{3/2}\pi}\approx 0.6% 054613829.
  8. 1 + 5 2 \tfrac{1+\sqrt{5}}{2}
  9. 5 - 1 2 \tfrac{\sqrt{5}-1}{2}
  10. [ 5 ] \mathbb{Q}[\sqrt{5}]
  11. 3 \mathbb{R}^{3}
  12. A = ( 0 1 1 1 1 1 1 0 1 - 1 - 1 1 1 1 0 1 - 1 - 1 1 - 1 1 0 1 - 1 1 - 1 - 1 1 0 1 1 1 - 1 - 1 1 0 ) . A=\left(\begin{array}[]{crrrrr}0&1&1&1&1&1\\ 1&0&1&-1&-1&1\\ 1&1&0&1&-1&-1\\ 1&-1&1&0&1&-1\\ 1&-1&-1&1&0&1\\ 1&1&-1&-1&1&0\end{array}\right).
  13. - 5 \scriptstyle-\sqrt{5}
  14. 5 \scriptstyle\sqrt{5}
  15. A + 5 I \scriptstyle A+\sqrt{5}I
  16. 6 / ker ( A + 5 I ) \mathbb{R}^{6}/\ker(A+\sqrt{5}I)
  17. 3 \mathbb{R}^{3}
  18. ker ( A + 5 I ) \ker(A+\sqrt{5}I)
  19. A + 5 I \scriptstyle{A+\sqrt{5}I}
  20. π : 6 6 / ker ( A + 5 I ) \pi:\mathbb{R}^{6}\longrightarrow\mathbb{R}^{6}/\ker(A+\sqrt{5}I)
  21. v 1 , , v 6 \mathbb{R}v_{1},\dots,\mathbb{R}v_{6}
  22. 6 \mathbb{R}^{6}
  23. 3 \mathbb{R}^{3}
  24. arccos 1 5 \scriptstyle{\arccos}\tfrac{1}{\sqrt{5}}
  25. 5 \scriptstyle\sqrt{5}

Regular_language.html

  1. { a n b n | n 0 } \{a^{n}b^{n}\,|\;n\geq 0\}
  2. K L K\cup L
  3. K L K\cap L
  4. L ¯ \bar{L}
  5. K - L K-L
  6. K L K\cup L
  7. K L K\circ L
  8. L * L^{*}
  9. K / L K/L
  10. L R L^{R}
  11. { w Σ * | f ( w ) S } \{w\in\Sigma^{*}\,|\,f(w)\in S\}
  12. M Σ * M\subset\Sigma^{*}
  13. { 0 n 1 n : n } \{0^{n}1^{n}:n\in\mathbb{N}\}
  14. u v L v u L uv\in L\Leftrightarrow vu\in L
  15. w L w n L w\in L\Leftrightarrow w^{n}\in L
  16. s L ( n ) s_{L}(n)
  17. n n
  18. L L
  19. S L ( z ) = n 0 s L ( n ) z n . S_{L}(z)=\sum_{n\geq 0}s_{L}(n)z^{n}\ .
  20. L L
  21. n 0 n_{0}
  22. λ 1 , , λ k \lambda_{1},\,\ldots,\,\lambda_{k}
  23. p 1 ( x ) , , p k ( x ) p_{1}(x),\,\ldots,\,p_{k}(x)
  24. n n 0 n\geq n_{0}
  25. s L ( n ) s_{L}(n)
  26. n n
  27. L L
  28. s L ( n ) = p 1 ( n ) λ 1 n + + p k ( n ) λ k n s_{L}(n)=p_{1}(n)\lambda_{1}^{n}+\cdots+p_{k}(n)\lambda_{k}^{n}
  29. L L^{\prime}
  30. L L^{\prime}
  31. 2 n 2n
  32. C n 4 n n 3 / 2 π C_{n}\sim\frac{4^{n}}{n^{3/2}\sqrt{\pi}}
  33. p ( n ) λ n p(n)\lambda^{n}
  34. λ i \lambda_{i}
  35. n n
  36. p ( n ) λ n p(n)\lambda^{n}
  37. 2 , - 2 2,-2
  38. d d
  39. a a
  40. d m + a dm+a
  41. C a m p a λ a m C_{a}m^{p_{a}}\lambda_{a}^{m}
  42. ζ L ( z ) = exp ( n 0 s L ( n ) z n n ) . \zeta_{L}(z)=\exp\left({\sum_{n\geq 0}s_{L}(n)\frac{z^{n}}{n}}\right)\ .

Reinforcement_learning.html

  1. S S
  2. A A
  3. t t
  4. o t o_{t}
  5. r t r_{t}
  6. a t a_{t}
  7. s t + 1 s_{t+1}
  8. r t + 1 r_{t+1}
  9. ( s t , a t , s t + 1 ) (s_{t},a_{t},s_{t+1})
  10. ϵ \epsilon
  11. 1 - ϵ 1-\epsilon
  12. 0 < ϵ < 1 0<\epsilon<1
  13. μ \mu
  14. ρ π \rho^{\pi}
  15. π \pi
  16. ρ π = E [ R | π ] , \rho^{\pi}=E[R|\pi],
  17. R R
  18. R = t = 0 N - 1 r t + 1 , R=\sum_{t=0}^{N-1}r_{t+1},
  19. r t + 1 r_{t+1}
  20. t t
  21. μ \mu
  22. π \pi
  23. N N
  24. R = t = 0 γ t r t + 1 , R=\sum_{t=0}^{\infty}\gamma^{t}r_{t+1},
  25. 0 γ 1 0\leq\gamma\leq 1
  26. π \pi
  27. V π ( s ) = E [ R | s , π ] , V^{\pi}(s)=E[R|s,\pi],
  28. R R
  29. π \pi
  30. s s
  31. V * ( s ) V^{*}(s)
  32. V π ( s ) V^{\pi}(s)
  33. π \pi
  34. V * ( s ) = sup π V π ( s ) . V^{*}(s)=\sup\limits_{\pi}V^{\pi}(s).
  35. ρ π \rho^{\pi}
  36. ρ π = E [ V π ( S ) ] \rho^{\pi}=E[V^{\pi}(S)]
  37. S S
  38. μ \mu
  39. s s
  40. a a
  41. π \pi
  42. ( s , a ) (s,a)
  43. π \pi
  44. Q π ( s , a ) = E [ R | s , a , π ] , Q^{\pi}(s,a)=E[R|s,a,\pi],\,
  45. R R
  46. a a
  47. s s
  48. π \pi
  49. Q Q
  50. Q * Q^{*}
  51. Q k Q_{k}
  52. k = 0 , 1 , 2 , , k=0,1,2,\ldots,
  53. Q * Q^{*}
  54. π \pi
  55. Q π ( s , a ) Q^{\pi}(s,a)
  56. ( s , a ) (s,a)
  57. ( s , a ) (s,a)
  58. ( s , a ) (s,a)
  59. Q Q
  60. Q π Q^{\pi}
  61. Q Q
  62. s s
  63. Q ( s , ) Q(s,\cdot)
  64. ϕ \phi
  65. ( s , a ) (s,a)
  66. ϕ ( s , a ) \phi(s,a)
  67. θ \theta
  68. Q ( s , a ) = i = 1 d θ i ϕ i ( s , a ) Q(s,a)=\sum\limits_{i=1}^{d}\theta_{i}\phi_{i}(s,a)
  69. λ \lambda
  70. ( 0 λ 1 ) (0\leq\lambda\leq 1)
  71. θ \theta
  72. π θ \pi_{\theta}
  73. θ \theta
  74. ρ ( θ ) = ρ π θ . \rho(\theta)=\rho^{\pi_{\theta}}.
  75. θ \theta
  76. ρ \rho

Relational_model.html

  1. r , s , t , r,s,t,\ldots
  2. a , b , c a,b,c
  3. t t
  4. A A
  5. t [ A ] = { ( a , v ) : ( a , v ) t , a A } t[A]=\{(a,v):(a,v)\in t,a\in A\}
  6. ( H , B ) (H,B)
  7. H H
  8. B B
  9. H H
  10. U U
  11. H H
  12. H H
  13. ( H , C ) (H,C)
  14. H H
  15. C ( R ) C(R)
  16. R R
  17. H H
  18. ( H , C ) (H,C)
  19. H H
  20. C C
  21. K K
  22. ( H , B ) (H,B)
  23. K H K\subseteq H
  24. t 1 , t 2 B t_{1},t_{2}\in B
  25. t 1 [ K ] = t 2 [ K ] t_{1}[K]=t_{2}[K]
  26. U U
  27. U U
  28. K K
  29. U U
  30. H H
  31. K H K\subseteq H
  32. K H K\rightarrow H
  33. U U
  34. K K
  35. U U
  36. U U
  37. K K
  38. U U
  39. X Y X\rightarrow Y
  40. X , Y X,Y
  41. X Y X\rightarrow Y
  42. ( H , B ) (H,B)
  43. X , Y H X,Y\subseteq H
  44. \forall
  45. t 1 , t 2 B t_{1},t_{2}\in B
  46. t 1 [ X ] = t 2 [ X ] t 1 [ Y ] = t 2 [ Y ] t_{1}[X]=t_{2}[X]~{}\Rightarrow~{}t_{1}[Y]=t_{2}[Y]
  47. X Y X\rightarrow Y
  48. U U
  49. U U
  50. H H
  51. H H
  52. X Y X\rightarrow Y
  53. H H
  54. Y X H Y\subseteq X\subseteq H
  55. S S
  56. H H
  57. S + S^{+}
  58. S S
  59. Y X H X Y S + Y\subseteq X\subseteq H~{}\Rightarrow~{}X\rightarrow Y\in S^{+}
  60. X Y S + Y Z S + X Z S + X\rightarrow Y\in S^{+}\land Y\rightarrow Z\in S^{+}~{}\Rightarrow~{}X% \rightarrow Z\in S^{+}
  61. X Y S + Z H ( X Z ) ( Y Z ) S + X\rightarrow Y\in S^{+}\land Z\subseteq H~{}\Rightarrow~{}(X\cup Z)\rightarrow% (Y\cup Z)\in S^{+}
  62. H H
  63. S S
  64. H H
  65. X Y S + X\rightarrow Y\in S^{+}
  66. X Y X\rightarrow Y
  67. H H
  68. S S
  69. X X
  70. S S
  71. X + X^{+}
  72. X X
  73. Y Z S Y X + Z X + Y\rightarrow Z\in S\land Y\subseteq X^{+}~{}\Rightarrow~{}Z\subseteq X^{+}
  74. S S
  75. X Y S + X\rightarrow Y\in S^{+}
  76. Y X + Y\subseteq X^{+}
  77. S S
  78. T T
  79. S + = T + S^{+}=T^{+}
  80. U T U\subset T
  81. S + = U + S^{+}=U^{+}
  82. X Y T Y X\rightarrow Y\in T~{}\Rightarrow Y
  83. X Y T Z X Z Y S + X\rightarrow Y\in T\land Z\subset X~{}\Rightarrow~{}Z\rightarrow Y\notin S^{+}

Relative_density.html

  1. 𝑅𝐷 = ρ substance ρ reference \mathit{RD}=\frac{\rho_{\mathrm{substance}}}{\rho_{\mathrm{reference}}}\,
  2. 𝑅𝐷 = ρ gas ρ air M gas M air \mathit{RD}=\frac{\rho_{\mathrm{gas}}}{\rho_{\mathrm{air}}}\approx\frac{M_{% \mathrm{gas}}}{M_{\mathrm{air}}}
  3. 8.15 4 C 20 C 8.15_{4^{\circ}\mathrm{C}}^{20^{\circ}\mathrm{C}}\,
  4. 2.432 0 15 2.432_{0}^{15}
  5. ρ = Mass Volume = Deflection × Spring Constant Gravity Displacement WaterLine × Area Cylinder \rho=\frac{\,\text{Mass}}{\,\text{Volume}}=\frac{\,\text{Deflection}\times% \frac{\,\text{Spring Constant}}{\,\text{Gravity}}}{\,\text{Displacement}_{% \mathrm{WaterLine}}\times\,\text{Area}_{\mathrm{Cylinder}}}\,
  6. R D = ρ object ρ ref = Deflection Obj . Displacement Obj . Deflection Ref . Displacement Ref . = 3 in 20 mm 5 in 34 mm = 3 in × 34 mm 5 in × 20 mm = 1.02 RD=\frac{\rho_{\mathrm{object}}}{\rho_{\mathrm{ref}}}=\frac{\frac{\,\text{% Deflection}_{\mathrm{Obj.}}}{\,\text{Displacement}_{\mathrm{Obj.}}}}{\frac{\,% \text{Deflection}_{\mathrm{Ref.}}}{\,\text{Displacement}_{\mathrm{Ref.}}}}=% \frac{\frac{3\ \mathrm{in}}{20\ \mathrm{mm}}}{\frac{5\ \mathrm{in}}{34\ % \mathrm{mm}}}=\frac{3\ \mathrm{in}\times 34\ \mathrm{mm}}{5\ \mathrm{in}\times 2% 0\ \mathrm{mm}}=1.02\,
  7. R D = W air W air - W water RD=\frac{W_{\mathrm{air}}}{W_{\mathrm{air}}-W_{\mathrm{water}}}\,
  8. m g = ρ ref V g mg=\rho_{\mathrm{ref}}Vg\,
  9. m = ρ ref V m=\rho_{\mathrm{ref}}V\,
  10. m = ρ new ( V - A Δ x ) m=\rho_{\mathrm{new}}(V-A\Delta x)\,
  11. R D new / ref = ρ new ρ ref = V V - A Δ x RD_{\mathrm{new/ref}}=\frac{\rho_{\mathrm{new}}}{\rho_{\mathrm{ref}}}=\frac{V}% {V-A\Delta x}
  12. R D new / ref = 1 1 - A Δ x m ρ ref RD_{\mathrm{new/ref}}=\frac{1}{1-\frac{A\Delta x}{m}\rho_{\mathrm{ref}}}
  13. R D new / ref 1 + A Δ x m ρ ref RD_{\mathrm{new/ref}}\approx 1+\frac{A\Delta x}{m}\rho_{\mathrm{ref}}
  14. F b = g ( m b - ρ a m b ρ b ) F_{b}=g(m_{b}-\rho_{a}{m_{b}\over\rho_{b}})
  15. F w = g ( m b - ρ a m b ρ b + V ρ w - V ρ a ) . F_{w}=g(m_{b}-\rho_{a}{m_{b}\over\rho_{b}}+V\rho_{w}-V\rho_{a}).
  16. F w , n = g V ( ρ w - ρ a ) F_{w,n}=gV(\rho_{w}-\rho_{a})
  17. F s , n = g V ( ρ s - ρ a ) F_{s,n}=gV(\rho_{s}-\rho_{a})
  18. S G A = g V ( ρ s - ρ a ) g V ( ρ w - ρ a ) = ( ρ s - ρ a ) ( ρ w - ρ a ) . SG_{A}={gV(\rho_{s}-\rho_{a})\over gV(\rho_{w}-\rho_{a})}={(\rho_{s}-\rho_{a})% \over(\rho_{w}-\rho_{a})}.
  19. R D A = ρ s ρ w - ρ a ρ w 1 - ρ a ρ w = R D V - ρ a ρ w 1 - ρ a ρ w RD_{A}={{\rho_{s}\over\rho_{w}}-{\rho_{a}\over\rho_{w}}\over 1-{\rho_{a}\over% \rho_{w}}}={RD_{V}-{\rho_{a}\over\rho_{w}}\over 1-{\rho_{a}\over\rho_{w}}}
  20. R D V = R D A - ρ a ρ w ( R D A - 1 ) RD_{V}=RD_{A}-{\rho_{a}\over\rho_{w}}(RD_{A}-1)
  21. R D H 2 O = ρ Material ρ H 2 O = R D , RD_{H_{2}O}=\frac{\rho_{\mathrm{Material}}}{\rho_{\mathrm{H_{2}O}}}\ =RD,

Relative_permittivity.html

  1. κ κ
  2. K K
  3. ε r ( ω ) = ε ( ω ) ε 0 , \varepsilon_{r}(\omega)=\frac{\varepsilon(\omega)}{\varepsilon_{0}},
  4. ε r ( ω ) = ε r ( ω ) + i ε r ′′ ( ω ) . \varepsilon_{r}(\omega)=\varepsilon_{r}^{\prime}(\omega)+i\varepsilon_{r}^{% \prime\prime}(\omega).
  5. 𝐏 \mathbf{P}
  6. 𝐄 \mathbf{E}
  7. ε r = C x C 0 . \varepsilon_{r}=\frac{C_{x}}{C_{0}}.
  8. ε r = ε r + i σ ω ε 0 , \varepsilon_{r}=\varepsilon_{r}^{\prime}+\frac{i\sigma}{\omega\varepsilon_{0}},
  9. ε r = ε r + i σ λ κ , \varepsilon_{r}=\varepsilon_{r}^{\prime}+i\sigma\lambda\kappa,

Rendering_(computer_graphics).html

  1. L o ( x , w ) = L e ( x , w ) + Ω f r ( x , w , w ) L i ( x , w ) ( w n ) d w L_{o}(x,\vec{w})=L_{e}(x,\vec{w})+\int_{\Omega}f_{r}(x,\vec{w}^{\prime},\vec{w% })L_{i}(x,\vec{w}^{\prime})(\vec{w}^{\prime}\cdot\vec{n})\mathrm{d}\vec{w}^{\prime}
  2. f r ( x , w , w ) = d L r ( x , w ) L i ( x , w ) ( w n ) d w f_{r}(x,\vec{w}^{\prime},\vec{w})=\frac{\mathrm{d}L_{r}(x,\vec{w})}{L_{i}(x,% \vec{w}^{\prime})(\vec{w}^{\prime}\cdot\vec{n})\mathrm{d}\vec{w}^{\prime}}

Resistor.html

  1. V = I R . V=I\cdot R.
  2. R eq = R 1 + R 2 + + R n . R_{\mathrm{eq}}=R_{1}+R_{2}+\cdots+R_{n}.
  3. 1 R eq = 1 R 1 + 1 R 2 + + 1 R n . \frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\cdots+\frac{1}{R_{n% }}.
  4. P = I 2 R = I V = V 2 R P=I^{2}R=IV=\frac{V^{2}}{R}
  5. 1 / 8 {1}/{8}
  6. 1 / 4 {1}/{4}
  7. 1 / 2 {1}/{2}

Resonance.html

  1. I ( ω ) ( Γ 2 ) 2 ( ω - Ω ) 2 + ( Γ 2 ) 2 . I(\omega)\propto\frac{\left(\frac{\Gamma}{2}\right)^{2}}{(\omega-\Omega)^{2}+% \left(\frac{\Gamma}{2}\right)^{2}}.
  2. d d\,
  3. 2 d 2d\,
  4. 2 d 2d\,
  5. λ \lambda\,
  6. 2 d = N λ , N { 1 , 2 , 3 , } 2d=N\lambda,\qquad\qquad N\in\{1,2,3,\dots\}
  7. v v\,
  8. f = v / λ f=v/\lambda\,
  9. f = N v 2 d N { 1 , 2 , 3 , } f=\frac{Nv}{2d}\qquad\qquad N\in\{1,2,3,\dots\}
  10. Q / ( 2 π ) Q/(2\pi)
  11. 1 / 2 {1}/{2}
  12. Γ \Gamma
  13. Ω \Omega
  14. M + i Γ M+i\Gamma

Responsivity.html

  1. R = η q h f η λ ( μ m ) 1.23985 ( μ m × W / A ) R=\eta\frac{q}{hf}\approx\eta\frac{\lambda_{(\mu m)}}{1.23985(\mu m\times W/A)}
  2. η \eta
  3. q q
  4. f f
  5. h h
  6. λ \lambda

Return_loss.html

  1. R L ( dB ) = 10 log 10 P i P r RL(\mathrm{dB})=10\log_{10}{P_{\mathrm{i}}\over P_{\mathrm{r}}}
  2. R L ( dB ) = 10 log 10 P i P r RL(\mathrm{dB})=10\log_{10}{P_{\mathrm{i}}\over P_{\mathrm{r}}}
  3. R L ( dB ) = 10 log 10 P r P i RL^{\prime}(\mathrm{dB})=10\log_{10}{P_{\mathrm{r}}\over P_{\mathrm{i}}}
  4. Γ \Gamma
  5. Γ = V r V i \mathit{\Gamma}={V_{\mathrm{r}}\over V_{\mathrm{i}}}
  6. Γ = Z L - Z S Z L + Z S \mathit{\Gamma}={{Z_{\mathrm{L}}-Z_{\mathrm{S}}}\over{Z_{\mathrm{L}}+Z_{% \mathrm{S}}}}
  7. R L ( dB ) = - 20 log 10 | Γ | RL(\mathrm{dB})=-20\log_{10}\left|\mathit{\Gamma}\right|
  8. R L ( dB ) = P i ( dBm ) - P r ( dBm ) RL(\mathrm{dB})=P_{\mathrm{i}}(\mathrm{dBm})-P_{\mathrm{r}}(\mathrm{dBm})\,
  9. ORL ( dB ) = 10 log 10 P i P r \,\text{ORL}(\mathrm{dB})=10\log_{10}{P_{\mathrm{i}}\over P_{\mathrm{r}}}
  10. P r \scriptstyle P_{\mathrm{r}}
  11. P i \scriptstyle P_{\mathrm{i}}

RGB_color_model.html

  1. γ \gamma

Rheology.html

  1. 𝑅𝑒 = ρ u s 2 / L μ u s / L 2 = ρ u s L μ = u s L ν \mathit{Re}={\rho u_{s}^{2}/L\over\mu u_{s}/L^{2}}={\rho u_{s}L\over\mu}={u_{s% }L\over\nu}

Rheumatoid_arthritis.html

  1. D A S 28 = 0.56 × T E N 28 + 0.28 × S W 28 + 0.70 × ln ( E S R ) + 0.014 × S A DAS28=0.56\times\sqrt{TEN28}+0.28\times\sqrt{SW28}+0.70\times\ln(ESR)+0.014% \times SA

Rhombicuboctahedron.html

  1. ( ± 1 , ± 1 , ± ( 1 + 2 ) ) . \left(\pm 1,\pm 1,\pm(1+\sqrt{2})\right).
  2. 2 7 10 - 2 \frac{2}{7}\sqrt{10-\sqrt{2}}
  3. 4 - 2 2 . \sqrt{4-2\sqrt{2}}.
  4. A = ( 18 + 2 3 ) a 2 21.4641016 a 2 A=\left(18+2\sqrt{3}\right)a^{2}\approx 21.4641016a^{2}
  5. V = 1 3 ( 12 + 10 2 ) a 3 8.71404521 a 3 . V=\frac{1}{3}\left(12+10\sqrt{2}\right)a^{3}\approx 8.71404521a^{3}.
  6. η = 4 3 ( 4 2 - 5 ) \eta=\frac{4}{3}\left(4\sqrt{2}-5\right)

Rice's_theorem.html

  1. ϕ : 𝐏 ( 1 ) \phi\colon\mathbb{N}\to\mathbf{P}^{(1)}
  2. 𝐏 ( 1 ) \mathbf{P}^{(1)}
  3. ϕ e := ϕ ( e ) \phi_{e}:=\phi(e)
  4. 𝐏 ( 1 ) \mathbf{P}^{(1)}
  5. F 𝐏 ( 1 ) F\subseteq\mathbf{P}^{(1)}
  6. ϕ e \phi_{e}
  7. ϕ e F \phi_{e}\in F
  8. F 𝐏 ( 1 ) F\subseteq\mathbf{P}^{(1)}
  9. D F D_{F}
  10. ϕ e F \phi_{e}\in F
  11. D F D_{F}
  12. F = F=\emptyset
  13. F = 𝐏 ( 1 ) F=\mathbf{P}^{(1)}
  14. ϕ : 𝐏 ( 1 ) \phi\colon\mathbb{N}\to\mathbf{P}^{(1)}
  15. Q ( x , y ) Q(x,y)
  16. e e
  17. ϕ e ( y ) \phi_{e}(y)
  18. Q ( e , y ) Q(e,y)
  19. f ( x ) f(x)
  20. g ( x ) g(x)
  21. f ( x ) = g ( x ) f(x)=g(x)
  22. f ( x ) f(x)
  23. g ( x ) g(x)
  24. ϕ e \phi_{e}
  25. ϕ e \phi_{e}
  26. Q Q
  27. F F
  28. F 𝐏 ( 1 ) \emptyset\neq F\neq\mathbf{P}^{(1)}
  29. f F f\in F
  30. g F g\notin F
  31. x x
  32. ϕ x F \phi_{x}\in F
  33. Q ( x , y ) Q(x,y)
  34. g ( y ) g(y)
  35. ϕ x F \phi_{x}\in F
  36. f ( y ) f(y)
  37. e e
  38. ϕ e ( y ) \phi_{e}(y)
  39. Q ( e , y ) Q(e,y)
  40. ϕ e F \phi_{e}\in F
  41. ϕ e \phi_{e}
  42. g g
  43. ϕ e F \phi_{e}\notin F
  44. ϕ e F \phi_{e}\notin F
  45. ϕ e \phi_{e}
  46. f f
  47. ϕ e F \phi_{e}\in F
  48. 𝒞 \mathcal{C}
  49. C C
  50. C C
  51. C = C=\emptyset
  52. C = C=\mathbb{N}
  53. \mathbb{N}
  54. i i
  55. i i
  56. i i
  57. W W
  58. S S
  59. e e
  60. ϕ e \phi_{e}
  61. S S
  62. e e
  63. S S
  64. e e
  65. W W
  66. e e
  67. e e
  68. W W
  69. e e
  70. W W
  71. S S\subseteq\mathbb{N}
  72. τ \tau
  73. k < | τ | k<|\tau|
  74. τ \tau
  75. k k
  76. τ \tau
  77. k S k\in S
  78. S = { 1 , 3 , 4 , 7 , } S=\{1,3,4,7,\ldots\}
  79. 01011001 01011001
  80. τ \tau
  81. τ \tau
  82. W W
  83. τ \tau
  84. τ \tau
  85. W W
  86. W W
  87. T 0 T_{0}
  88. T 1 T_{1}
  89. T 0 T 1 T_{0}\cup T_{1}
  90. S S\subseteq\mathbb{N}
  91. S = W e := dom ϕ e := { x : ϕ e ( x ) } S=W_{e}:=\textrm{dom}\,\phi_{e}:=\{x:\phi_{e}(x)\downarrow\}
  92. e e
  93. W e W_{e}
  94. dom ϕ e \textrm{dom}\,\phi_{e}
  95. x x
  96. ϕ e ( x ) \phi_{e}(x)
  97. ϕ e \phi_{e}
  98. { ϕ e : dom ϕ e C } \{\phi_{e}:\textrm{dom}\,\phi_{e}\in C\}
  99. C C
  100. S S\subseteq\mathbb{N}
  101. S S

Riemann_integral.html

  1. S = { ( x , y ) : a x b , 0 < y < f ( x ) } S=\left\{(x,y)\,:\ a\leq x\leq b,0<y<f(x)\right\}
  2. a b f ( x ) d x . \int_{a}^{b}f(x)\,dx.
  3. a = x 0 < x 1 < x 2 < < x n = b a=x_{0}<x_{1}<x_{2}<\cdots<x_{n}=b
  4. [ x i , x i + 1 ] \left[x_{i},x_{i+1}\right]
  5. max ( x i + 1 - x i ) , i [ 0 , n - 1 ] . \max(x_{i+1}-x_{i}),\quad i\in[0,n-1].
  6. P ( x , t ) P(x,t)
  7. t 0 , , t n - 1 t_{0},\dots,t_{n-1}
  8. t i [ x i , x i + 1 ] t_{i}\in[x_{i},x_{i+1}]
  9. P ( x , t ) P(x,t)
  10. Q ( y , s ) Q(y,s)
  11. Q ( y , s ) Q(y,s)
  12. P ( x , t ) P(x,t)
  13. i [ 0 , n ] i\in[0,n]
  14. r ( i ) r(i)
  15. x i = y r ( i ) x_{i}=y_{r(i)}
  16. t i = s j t_{i}=s_{j}
  17. j [ r ( i ) , r ( i + 1 ) ) j\in[r(i),r(i+1))
  18. x 0 , , x n x_{0},\ldots,x_{n}
  19. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  20. i = 0 n - 1 f ( t i ) ( x i + 1 - x i ) . \sum_{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i}).
  21. f ( t i ) f(t_{i})
  22. x i + 1 - x i x_{i+1}-x_{i}
  23. x 0 , , x n x_{0},\ldots,x_{n}
  24. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  25. | i = 0 n - 1 f ( t i ) ( x i + 1 - x i ) - s | < ε . \left|\sum_{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-s\right|<\varepsilon.
  26. y 0 , , y m y_{0},\ldots,y_{m}
  27. r 0 , , r m - 1 r_{0},\ldots,r_{m-1}
  28. x 0 , , x n x_{0},\ldots,x_{n}
  29. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  30. y 0 , , y m y_{0},\ldots,y_{m}
  31. r 0 , , r m - 1 r_{0},\ldots,r_{m-1}
  32. | i = 0 n - 1 f ( t i ) ( x i + 1 - x i ) - s | < ε . \left|\sum_{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-s\right|<\varepsilon.
  33. y 0 , , y m y_{0},\ldots,y_{m}
  34. ε 2 \tfrac{\varepsilon}{2}
  35. r = sup x [ a , b ] | f ( x ) | . r=\sup_{x\in[a,b]}|f(x)|.
  36. δ < min { ε 2 r ( m - 1 ) , ( y 1 - y 0 ) , ( y 2 - y 1 ) , , ( y m - y m - 1 ) } \delta<\min\left\{\frac{\varepsilon}{2r(m-1)},(y_{1}-y_{0}),(y_{2}-y_{1}),% \cdots,(y_{m}-y_{m-1})\right\}
  37. x 0 , , x n x_{0},\ldots,x_{n}
  38. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  39. [ x i , x i + 1 ] [x_{i},x_{i+1}]
  40. [ y j , y j + 1 ] [y_{j},y_{j+1}]
  41. m j < f ( t i ) < M j m_{j}<f(t_{i})<M_{j}
  42. [ y j , y j + 1 ] [y_{j},y_{j+1}]
  43. [ x i , x i + 1 ] [x_{i},x_{i+1}]
  44. [ y j , y j + 1 ] [y_{j},y_{j+1}]
  45. y 0 , , y m y_{0},\ldots,y_{m}
  46. y j < x i < y j + 1 < x i + 1 < y j + 2 . y_{j}<x_{i}<y_{j+1}<x_{i+1}<y_{j+2}.
  47. x 0 , , x n x_{0},\ldots,x_{n}
  48. y j + 1 y_{j+1}
  49. f ( t i ) ( x i - x i + 1 ) f(t_{i})(x_{i}-x_{i+1})
  50. f ( t i ) ( x i - x i + 1 ) = f ( t i ) ( x i - y j + 1 ) + f ( t i ) ( y j + 1 - x i + 1 ) . f(t_{i})(x_{i}-x_{i+1})=f(t_{i})(x_{i}-y_{j+1})+f(t_{i})(y_{j+1}-x_{i+1}).
  51. t i [ x i , x i + 1 ] t_{i}\in[x_{i},x_{i+1}]
  52. m j < f ( t i ) < M j , m_{j}<f(t_{i})<M_{j},
  53. y j + 1 - x i + 1 < δ < ε 2 r ( m - 1 ) , y_{j+1}-x_{i+1}<\delta<\frac{\varepsilon}{2r(m-1)},
  54. f ( t i ) ( y j + 1 - x i + 1 ) < ε 2 ( m - 1 ) . f(t_{i})(y_{j+1}-x_{i+1})<\frac{\varepsilon}{2(m-1)}.
  55. ε 2 \tfrac{\varepsilon}{2}
  56. f : [ 0 , 1 ] 𝐑 f:[0,1]\to\mathbf{R}
  57. I 𝐐 : [ 0 , 1 ] 𝐑 I_{\mathbf{Q}}:[0,1]\to\mathbf{R}
  58. x 0 , , x n x_{0},\ldots,x_{n}
  59. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  60. x i + 1 x_{i+1}
  61. ε n \tfrac{\varepsilon}{n}
  62. ( 0 ε ) n \tfrac{(0\cdot\varepsilon)}{n}
  63. ( 1 ε ) n \tfrac{(1\cdot\varepsilon)}{n}
  64. ε n \tfrac{\varepsilon}{n}
  65. t i - δ 2 t_{i}-\tfrac{\delta}{2}
  66. t i + δ 2 t_{i}+\tfrac{\delta}{2}
  67. [ t i - δ 2 , t i + δ 2 ] . \left[t_{i}-\tfrac{\delta}{2},t_{i}+\tfrac{\delta}{2}\right].
  68. [ t i - δ 2 , x j ] , [ x j , t i + δ 2 ] . \left[t_{i}-\tfrac{\delta}{2},x_{j}\right],\quad\left[x_{j},t_{i}+\tfrac{% \delta}{2}\right].
  69. t i = x i t_{i}=x_{i}
  70. t i = x i + 1 t_{i}=x_{i+1}
  71. n n^{\prime}
  72. [ 0 , 1 n ] , [ 1 n , 2 n ] , , [ n - 1 n , 1 ] . \left[0,\tfrac{1}{n}\right],\left[\tfrac{1}{n},\tfrac{2}{n}\right],\ldots,% \left[\tfrac{n-1}{n},1\right].
  73. 0 2 - 1 I 𝐐 ( x ) d x + 2 - 1 1 I 𝐐 ( x ) d x = 0 1 I 𝐐 ( x ) d x . \int_{0}^{\sqrt{2}-1}\!I_{\mathbf{Q}}(x)\,\mathrm{d}x+\int_{\sqrt{2}-1}^{1}\!I% _{\mathbf{Q}}(x)\,\mathrm{d}x=\int_{0}^{1}\!I_{\mathbf{Q}}(x)\,\mathrm{d}x.
  74. a b ( α f ( x ) + β g ( x ) ) d x = α a b f ( x ) d x + β a b g ( x ) d x . \int_{a}^{b}(\alpha f(x)+\beta g(x))\,dx=\alpha\int_{a}^{b}f(x)\,dx+\beta\int_% {a}^{b}g(x)\,dx.
  75. ε m ( X ε ) ; \varepsilon\cdot m(X_{\varepsilon});
  76. ε ( M - m ) , \varepsilon\cdot(M-m),
  77. ε ( b - a ) \varepsilon\cdot(b-a)
  78. ε ( ( M - m ) + ( b - a ) ) = K ε , \varepsilon\cdot\bigl((M-m)+(b-a)\bigr)=K\varepsilon,
  79. f n {f_{n}}
  80. f n {f_{n}}
  81. a b f d x = a b lim n f n d x = lim n a b f n d x . \int_{a}^{b}f\,dx=\int_{a}^{b}{\lim_{n\to\infty}{f_{n}}\,dx}=\lim_{n\to\infty}% \int_{a}^{b}f_{n}\,dx.
  82. 𝐟 = ( f 1 , , f n ) . \int\mathbf{f}=\left(\int f_{1},\,\dots,\int f_{n}\right).
  83. - f ( x ) d x = lim ( a , b ) ( - , ) a b f ( x ) d x . \int_{-\infty}^{\infty}f(x)\,dx=\lim_{(a,b)\to(-\infty,\infty)}\int_{a}^{b}f(x% )\,dx.
  84. lim a - a a f ( x ) d x \textstyle\lim_{a\to\infty}\int_{-a}^{a}f(x)\,dx
  85. - a a f ( x ) d x = 0 \textstyle\int_{-a}^{a}f(x)\,dx=0
  86. - a 2 a f ( x ) d x \textstyle\int_{-a}^{2a}f(x)\,dx
  87. - 2 a a f ( x ) d x \textstyle\int_{-2a}^{a}f(x)\,dx
  88. lim a - a a f ( x ) d x \textstyle\lim_{a\to\infty}\int_{-a}^{a}f(x)\,dx
  89. - \infty-\infty
  90. lim n a b f n ( x ) d x = a b f ( x ) d x \textstyle\lim_{n\to\infty}\int_{a}^{b}f_{n}(x)\,dx=\int_{a}^{b}f(x)\,dx
  91. - f n d x = 1. \int_{-\infty}^{\infty}f_{n}\,dx=1.
  92. - f d x lim - f n d x . \int_{-\infty}^{\infty}f\,dx\not=\lim\int_{-\infty}^{\infty}f_{n}\,dx.

Riemann_mapping_theorem.html

  1. D = { z 𝐂 : | z | < 1 } . D=\{z\in\mathbf{C}:|z|<1\}.
  2. f ( z ) = ( z - z 0 ) e g ( z ) f(z)=(z-z_{0})e^{g(z)}
  3. u ( z ) = - log | z - z 0 | u(z)=-\log|z-z_{0}|

Riemann_zeta_function.html

  1. ζ ( s ) = n = 1 1 n s \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}
  2. ζ ( s ) = n = 1 n - s = 1 1 s + 1 2 s + 1 3 s + σ = ( s ) > 1. \zeta(s)=\sum_{n=1}^{\infty}n^{-s}=\frac{1}{1^{s}}+\frac{1}{2^{s}}+\frac{1}{3^% {s}}+\cdots\;\;\;\;\;\;\;\sigma=\mathfrak{R}(s)>1.\!
  3. ζ ( s ) = 1 Γ ( s ) 0 x s - 1 e x - 1 d x \zeta(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\mathrm{d}x
  4. lim s 1 ( s - 1 ) ζ ( s ) = 1. \lim_{s\to 1}(s-1)\zeta(s)=1.
  5. ζ ( 2 n ) = ( - 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) ! \zeta(2n)=\frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}
  6. ζ ( - n ) = - B n + 1 n + 1 \zeta(-n)=-\frac{B_{n+1}}{n+1}
  7. ζ ( - 1 ) = - 1 12 \zeta(-1)=-\frac{1}{12}
  8. ζ ( 0 ) = - 1 2 ; \zeta(0)=-\frac{1}{2};\!
  9. ζ ( 1 / 2 ) - 1.4603545 \zeta(1/2)\approx-1.4603545\!
  10. ζ ( 1 ) = 1 + 1 2 + 1 3 + = ; \zeta(1)=1+\frac{1}{2}+\frac{1}{3}+\cdots=\infty;\!
  11. lim ε 0 ζ ( 1 + ε ) + ζ ( 1 - ε ) 2 \lim_{\varepsilon\to 0}\frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2}
  12. γ = 0.5772 \gamma=0.5772\ldots
  13. ζ ( 3 / 2 ) 2.612 ; \zeta(3/2)\approx 2.612;\!
  14. ζ ( 2 ) = 1 + 1 2 2 + 1 3 2 + = π 2 6 1.645 ; \zeta(2)=1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots=\frac{\pi^{2}}{6}\approx 1.6% 45;\!
  15. ζ ( 3 ) = 1 + 1 2 3 + 1 3 3 + 1.202 ; \zeta(3)=1+\frac{1}{2^{3}}+\frac{1}{3^{3}}+\cdots\approx 1.202;\!
  16. ζ ( 4 ) = 1 + 1 2 4 + 1 3 4 + = π 4 90 1.0823 ; \zeta(4)=1+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\cdots=\frac{\pi^{4}}{90}\approx 1.% 0823;\!
  17. n = 1 1 n s = p prime 1 1 - p - s , \sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p\,\text{ prime}}\frac{1}{1-p^{-s}},
  18. p prime 1 1 - p - s = 1 1 - 2 - s 1 1 - 3 - s 1 1 - 5 - s 1 1 - 7 - s 1 1 - 11 - s 1 1 - p - s . \prod_{p\,\text{ prime}}\frac{1}{1-p^{-s}}=\frac{1}{1-2^{-s}}\cdot\frac{1}{1-3% ^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}}% \cdots\frac{1}{1-p^{-s}}\cdots.
  19. p p / ( p - 1 ) \textstyle\prod_{p}p/(p-1)
  20. p ( 1 - 1 p s ) = ( p 1 1 - p - s ) - 1 = 1 ζ ( s ) . \prod_{p}^{\infty}\left(1-\frac{1}{p^{s}}\right)=\left(\prod_{p}^{\infty}\frac% {1}{1-p^{-s}}\right)^{-1}=\frac{1}{\zeta(s)}.
  21. ζ ( s ) = 2 s π s - 1 sin ( π s 2 ) Γ ( 1 - s ) ζ ( 1 - s ) , \zeta(s)=2^{s}\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(% 1-s)\!,
  22. η ( s ) = n = 1 ( - 1 ) n + 1 n s = ( 1 - 2 1 - s ) ζ ( s ) . \eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{s}}=(1-{2^{1-s}})\zeta(s).
  23. ξ ( s ) = 1 2 π - s / 2 s ( s - 1 ) Γ ( s 2 ) ζ ( s ) . \xi(s)=\frac{1}{2}\pi^{-s/2}s(s-1)\Gamma\left(\frac{s}{2}\right)\zeta(s).\!
  24. ξ ( s ) = ξ ( 1 - s ) . \xi(s)=\xi(1-s).\!
  25. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  26. N ( T ) N(T)
  27. N 0 ( T ) N_{0}(T)
  28. ζ ( 1 2 + i t ) \zeta\bigl(\tfrac{1}{2}+it\bigr)
  29. ( 0 , T ] (0,T]
  30. ε > 0 \varepsilon>0
  31. T 0 ( ε ) > 0 T_{0}(\varepsilon)>0
  32. T T 0 ( ε ) T\geq T_{0}(\varepsilon)
  33. H = T 0.25 + ε H=T^{0.25+\varepsilon}
  34. ( T , T + H ] (T,T+H]
  35. ε > 0 \varepsilon>0
  36. T 0 ( ε ) > 0 T_{0}(\varepsilon)>0
  37. c ε > 0 c_{\varepsilon}>0
  38. N 0 ( T + H ) - N 0 ( T ) c ε H N_{0}(T+H)-N_{0}(T)\geq c_{\varepsilon}H
  39. T T 0 ( ε ) T\geq T_{0}(\varepsilon)
  40. H = T 0.5 + ε H=T^{0.5+\varepsilon}
  41. σ 1 - 1 57.54 ( log | t | ) 2 / 3 ( log log | t | ) 1 / 3 . \sigma\geq 1-\frac{1}{57.54(\log{|t|})^{2/3}(\log{\log{|t|}})^{1/3}}.\!
  42. lim n ( γ n + 1 - γ n ) = 0. \lim_{n\rightarrow\infty}\left(\gamma_{n+1}-\gamma_{n}\right)=0.\!
  43. ζ ( s ) = ζ ( s ¯ ) ¯ \zeta(s)=\overline{\zeta(\overline{s})}
  44. 1 ζ ( s ) = n = 1 μ ( n ) n s \frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}\!
  45. F ( T ; H ) F(T;H)
  46. G ( s 0 ; Δ ) G(s_{0};\Delta)
  47. F ( T ; H ) = max | t - T | H | ζ ( 1 2 + i t ) | , G ( s 0 ; Δ ) = max | s - s 0 | Δ | ζ ( s ) | . F(T;H)=\max_{|t-T|\leq H}\bigl|\zeta\bigl(\tfrac{1}{2}+it\bigr)\bigr|,\quad G(% s_{0};\Delta)=\max_{|s-s_{0}|\leq\Delta}|\zeta(s)|.
  48. T T
  49. 0 < H ln ln T 0<H\ll\ln\ln T
  50. s 0 = σ 0 + i T s_{0}=\sigma_{0}+iT
  51. 1 2 σ 0 1 \tfrac{1}{2}\leq\sigma_{0}\leq 1
  52. 0 < Δ < 1 3 0<\Delta<\tfrac{1}{3}
  53. F F
  54. G G
  55. ζ ( s ) \zeta(s)
  56. 0 R e s 1 0\leq Re\ s\leq 1
  57. H ln ln T H\gg\ln\ln T
  58. Δ > c \Delta>c
  59. c c
  60. H H
  61. Δ \Delta
  62. F ( T ; H ) T - c 1 , G ( s 0 ; Δ ) T - c 2 , F(T;H)\geq T^{-c_{1}},\quad G(s_{0};\Delta)\geq T^{-c_{2}},
  63. c 1 , c 2 c_{1},c_{2}
  64. S ( t ) = 1 π arg ζ ( 1 2 + i t ) S(t)=\frac{1}{\pi}\arg{\zeta\bigl(\tfrac{1}{2}+it\bigr)}
  65. arg ζ ( 1 2 + i t ) \arg{\zeta\bigl(\tfrac{1}{2}+it\bigr)}
  66. arg ζ ( s ) \arg\zeta(s)
  67. 2 , 2 + i t 2,2+it
  68. 1 2 + i t \tfrac{1}{2}+it
  69. S ( t ) S(t)
  70. S ( t ) S(t)
  71. S 1 ( t ) = 0 t S ( u ) d u S_{1}(t)=\int_{0}^{t}S(u)du
  72. ( T , T + H ] (T,T+H]
  73. H T 27 / 82 + ε H\geq T^{27/82+\varepsilon}
  74. H ( ln T ) 1 / 3 e - c ln ln T H(\ln T)^{1/3}e^{-c\sqrt{\ln\ln T}}
  75. S ( t ) S(t)
  76. H T 1 / 2 + ε H\geq T^{1/2+\varepsilon}
  77. ζ ( s ) = 1 s - 1 n = 1 ( n ( n + 1 ) s - n - s n s ) \zeta(s)=\frac{1}{s-1}\sum_{n=1}^{\infty}\left(\frac{n}{(n+1)^{s}}-\frac{n-s}{% n^{s}}\right)
  78. s > 0 \Re s>0
  79. ζ ( s ) = 1 s - 1 n = 1 n ( n + 1 ) 2 ( 2 n + 3 + s ( n + 1 ) s + 2 - 2 n - 1 - s n s + 2 ) \zeta(s)=\frac{1}{s-1}\sum_{n=1}^{\infty}\frac{n(n+1)}{2}\left(\frac{2n+3+s}{(% n+1)^{s+2}}-\frac{2n-1-s}{n^{s+2}}\right)
  80. s > - 1 \Re s>-1
  81. s > - k \Re s>-k
  82. k { 1 , 2 , 3 , } k\in\{1,2,3,\dots\}
  83. 0 f ( x ) x s - 1 d x , \int_{0}^{\infty}f(x)x^{s-1}\,dx,
  84. Γ ( s ) ζ ( s ) = 0 x s - 1 e x - 1 d x , \Gamma(s)\zeta(s)=\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\,dx,
  85. 2 sin ( π s ) Γ ( s ) ζ ( s ) = i C ( - x ) s - 1 e x - 1 d x 2\sin(\pi s)\Gamma(s)\zeta(s)=i\oint_{C}\frac{(-x)^{s-1}}{e^{x}-1}\,dx
  86. log ζ ( s ) = s 0 π ( x ) x ( x s - 1 ) d x , \log\zeta(s)=s\int_{0}^{\infty}\frac{\pi(x)}{x(x^{s}-1)}\,dx,
  87. J ( x ) = π ( x 1 / n ) n . J(x)=\sum\frac{\pi(x^{1/n})}{n}.
  88. log ζ ( s ) = s 0 J ( x ) x - s - 1 d x . \log\zeta(s)=s\int_{0}^{\infty}J(x)x^{-s-1}\,dx.
  89. 2 π - s / 2 Γ ( s / 2 ) ζ ( s ) = 0 θ ( i t ) t s / 2 - 1 d t , 2\pi^{-s/2}\Gamma(s/2)\zeta(s)=\int_{0}^{\infty}\theta(it)t^{s/2-1}\,dt,
  90. θ ( τ ) = n = - exp ( π i n 2 τ ) . \theta(\tau)=\sum_{n=-\infty}^{\infty}\exp(\pi in^{2}\tau).
  91. π - s / 2 Γ ( s / 2 ) ζ ( s ) \displaystyle{}\quad\pi^{-s/2}\Gamma(s/2)\zeta(s)
  92. ζ ( s ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( s - 1 ) n . \zeta(s)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}\;(s-1)% ^{n}.
  93. γ n = lim m [ ( k = 1 m ( log k ) n k ) - ( log m ) n + 1 n + 1 ] . \gamma_{n}=\lim_{m\rightarrow\infty}{\left[\left(\sum_{k=1}^{m}\frac{(\log k)^% {n}}{k}\right)-\frac{(\log m)^{n+1}}{n+1}\right]}.
  94. s { 1 } s\in\mathbb{C}\setminus\{1\}
  95. ζ ( s ) = 2 s - 1 s - 1 - 2 s 0 sin ( s arctan t ) ( 1 + t 2 ) s 2 ( e π t + 1 ) d t , \zeta(s)=\frac{2^{s-1}}{s-1}-2^{s}\!\int_{0}^{\infty}\!\!\!\frac{\sin(s\arctan t% )}{(1+t^{2})^{\frac{s}{2}}(\mathrm{e}^{\pi\,t}+1)}\,\mathrm{d}t,
  96. ζ ( s ) = s s - 1 - n = 1 ( ζ ( s + n ) - 1 ) s ( s + 1 ) ( s + n - 1 ) ( n + 1 ) ! . \zeta(s)=\frac{s}{s-1}-\sum_{n=1}^{\infty}\left(\zeta(s+n)-1\right)\frac{s(s+1% )\cdots(s+n-1)}{(n+1)!}.\!
  97. ζ ( s ) = e ( log ( 2 π ) - 1 - γ / 2 ) s 2 ( s - 1 ) Γ ( 1 + s / 2 ) ρ ( 1 - s ρ ) e s / ρ , \zeta(s)=\frac{e^{(\log(2\pi)-1-\gamma/2)s}}{2(s-1)\Gamma(1+s/2)}\prod_{\rho}% \left(1-\frac{s}{\rho}\right)e^{s/\rho},\!
  98. ζ ( s ) = π s / 2 ρ ( 1 - s ρ ) 2 ( s - 1 ) Γ ( 1 + s / 2 ) . \zeta(s)=\pi^{s/2}\frac{\prod_{\rho}\left(1-\frac{s}{\rho}\right)}{2(s-1)% \Gamma(1+s/2)}.\!
  99. π d N d x ( x ) = 1 2 i d d x ( log ( ζ ( 1 / 2 + i x ) ) - log ( ζ ( 1 / 2 - i x ) ) ) - 2 1 + 4 x 2 - n = 0 2 n + 1 / 2 ( 2 n + 1 / 2 ) 2 + x 2 {\pi\frac{dN}{dx}(x)=\frac{1}{2i}\frac{d}{dx}\bigl(\log(\zeta(1/2+ix))-\log(% \zeta(1/2-ix))\bigr)-\frac{2}{1+4x^{2}}-\sum_{n=0}^{\infty}\frac{2n+1/2}{(2n+1% /2)^{2}+x^{2}}}
  100. d N ( x ) d x = ρ δ ( x - ρ ) \frac{dN(x)}{dx}=\sum_{\rho}\delta(x-\rho)
  101. ζ ( s ) = 1 1 - 2 1 - s n = 0 1 2 n + 1 k = 0 n ( n k ) ( - 1 ) k ( k + 1 ) s . \zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\sum_{k=0}^{n}% {n\choose k}\frac{(-1)^{k}}{(k+1)^{s}}.\!
  102. ζ ( s ) = 1 s - 1 n = 0 1 n + 1 k = 0 n ( n k ) ( - 1 ) k ( k + 1 ) s - 1 \zeta(s)=\frac{1}{s-1}\sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{k=0}^{n}{n\choose k% }\frac{(-1)^{k}}{(k+1)^{s-1}}
  103. ζ ( k ) = 2 k 2 k - 1 + r = 2 ( p r - 1 # ) k J k ( p r # ) ( k = 2 , 3 , ) . \zeta(k)=\frac{2^{k}}{2^{k}-1}+\sum_{r=2}^{\infty}\frac{(p_{r-1}\#)^{k}}{J_{k}% (p_{r}\#)}\quad(k=2,3,\dots).
  104. 1 = n = 2 ( ζ ( n ) - 1 ) . 1=\sum_{n=2}^{\infty}(\zeta(n)-1).
  105. n = 1 ( ζ ( 2 n ) - 1 ) = 3 4 \sum_{n=1}^{\infty}(\zeta(2n)-1)=\tfrac{3}{4}
  106. n = 1 ( ζ ( 2 n + 1 ) - 1 ) = 1 4 . \sum_{n=1}^{\infty}(\zeta(2n+1)-1)=\tfrac{1}{4}.
  107. log 2 = n = 1 ζ ( 2 n ) - 1 n . \log 2=\sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{n}.
  108. 1 - γ = n = 2 ζ ( n ) - 1 n 1-\gamma=\sum_{n=2}^{\infty}\frac{\zeta(n)-1}{n}
  109. log π = n = 2 ( 2 ( 3 2 ) n - 3 ) ( ζ ( n ) - 1 ) n . \log\pi=\sum_{n=2}^{\infty}\frac{(2(\tfrac{3}{2})^{n}-3)(\zeta(n)-1)}{n}.
  110. π 4 = n = 2 ζ ( n ) - 1 n ( ( 1 + i ) n - ( 1 + i n ) ) \frac{\pi}{4}=\sum_{n=2}^{\infty}\frac{\zeta(n)-1}{n}\mathfrak{I}((1+i)^{n}-(1% +i^{n}))
  111. \mathfrak{I}
  112. n = 1 ζ ( 2 n ) - 1 2 2 n = 1 6 . \sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{2^{2n}}=\frac{1}{6}.
  113. n = 1 ζ ( 2 n ) - 1 4 2 n = 13 30 - π 8 . \sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{4^{2n}}=\frac{13}{30}-\frac{\pi}{8}.
  114. n = 1 ζ ( 2 n ) - 1 8 2 n = 61 126 - π 16 ( 2 + 1 ) . \sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{8^{2n}}=\frac{61}{126}-\frac{\pi}{16}(% \sqrt{2}+1).
  115. n = 1 ζ ( 2 n ) - 1 a 2 n = 1 2 + 1 1 - a 2 - π cot ( π / a ) 2 a \sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{a^{2n}}=\frac{1}{2}+\frac{1}{1-a^{2}}-% \frac{\pi\cot(\pi/a)}{2a}
  116. | a | > 1 |a|>1
  117. n = 1 ( ζ ( 4 n ) - 1 ) = 7 8 - π 4 ( e 2 π + 1 e 2 π - 1 ) . \sum_{n=1}^{\infty}(\zeta(4n)-1)=\frac{7}{8}-\frac{\pi}{4}\left(\frac{e^{2\pi}% +1}{e^{2\pi}-1}\right).
  118. ζ ( s , q ) = k = 0 1 ( k + q ) s \zeta(s,q)=\sum_{k=0}^{\infty}\frac{1}{(k+q)^{s}}
  119. Li s ( z ) = k = 1 z k k s \mathrm{Li}_{s}(z)=\sum_{k=1}^{\infty}{z^{k}\over k^{s}}\!
  120. Φ ( z , s , q ) = k = 0 z k ( k + q ) s \Phi(z,s,q)=\sum_{k=0}^{\infty}\frac{z^{k}}{(k+q)^{s}}\!
  121. ζ ( s 1 , s 2 , , s n ) = k 1 > k 2 > > k n > 0 k 1 - s 1 k 2 - s 2 k n - s n . \zeta(s_{1},s_{2},\ldots,s_{n})=\sum_{k_{1}>k_{2}>\cdots>k_{n}>0}k_{1}^{-s_{1}% }k_{2}^{-s_{2}}\cdots k_{n}^{-s_{n}}.\!

Riesz_representation_theorem.html

  1. φ x ( y ) = y , x , \varphi_{x}(y)=\left\langle y,x\right\rangle,
  2. , \langle\cdot,\cdot\rangle
  3. Φ \Phi
  4. Φ \Phi
  5. φ \varphi
  6. Φ \Phi
  7. φ \varphi
  8. x = Φ ( x ) \|x\|=\|\Phi(x)\|
  9. Φ \Phi
  10. Φ ( x 1 + x 2 ) = Φ ( x 1 ) + Φ ( x 2 ) \Phi(x_{1}+x_{2})=\Phi(x_{1})+\Phi(x_{2})
  11. Φ ( λ x ) = λ Φ ( x ) \Phi(\lambda x)=\lambda\Phi(x)
  12. Φ ( λ x ) = λ ¯ Φ ( x ) \Phi(\lambda x)=\bar{\lambda}\Phi(x)
  13. λ ¯ \bar{\lambda}
  14. Φ \Phi
  15. φ \varphi
  16. φ \varphi
  17. x = φ ( z ) ¯ z / z 2 x=\overline{\varphi(z)}\cdot z/{\left\|z\right\|}^{2}
  18. Φ \Phi
  19. φ \varphi
  20. | ψ |\psi\rangle
  21. ψ | \langle\psi|

Right_triangle.html

  1. T = 1 2 a b T=\tfrac{1}{2}ab
  2. T = PA PB = ( s - a ) ( s - b ) . T=\,\text{PA}\cdot\,\text{PB}=(s-a)(s-b).
  3. f 2 = d e , \displaystyle f^{2}=de,
  4. b 2 = c e , \displaystyle b^{2}=ce,
  5. a 2 = c d \displaystyle a^{2}=cd
  6. f = a b c . f=\frac{ab}{c}.
  7. 1 a 2 + 1 b 2 = 1 f 2 . \frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{1}{f^{2}}.
  8. a 2 + b 2 = c 2 \displaystyle a^{2}+b^{2}=c^{2}
  9. r = a + b - c 2 = a b a + b + c . r=\frac{a+b-c}{2}=\frac{ab}{a+b+c}.
  10. R = c 2 . R=\frac{c}{2}.
  11. R + r = a + b 2 . R+r=\frac{a+b}{2}.
  12. a = 2 r ( b - r ) b - 2 r . \displaystyle a=\frac{2r(b-r)}{b-2r}.
  13. a b < c a\leq b<c
  14. a 2 + b 2 = c 2 ( Pythagorean theorem ) \displaystyle a^{2}+b^{2}=c^{2}\quad(\,\text{Pythagorean theorem})
  15. ( s - a ) ( s - b ) = s ( s - c ) \displaystyle(s-a)(s-b)=s(s-c)
  16. s = 2 R + r . \displaystyle s=2R+r.
  17. a 2 + b 2 + c 2 = 8 R 2 . \displaystyle a^{2}+b^{2}+c^{2}=8R^{2}.
  18. cos A cos B cos C = 0. \displaystyle\cos{A}\cos{B}\cos{C}=0.
  19. sin 2 A + sin 2 B + sin 2 C = 2. \displaystyle\sin^{2}{A}+\sin^{2}{B}+\sin^{2}{C}=2.
  20. cos 2 A + cos 2 B + cos 2 C = 1. \displaystyle\cos^{2}{A}+\cos^{2}{B}+\cos^{2}{C}=1.
  21. sin 2 A = sin 2 B = 2 sin A sin B . \displaystyle\sin{2A}=\sin{2B}=2\sin{A}\sin{B}.
  22. T = a b 2 \displaystyle T=\frac{ab}{2}
  23. T = r a r b = r r c \displaystyle T=r_{a}r_{b}=rr_{c}
  24. T = r ( 2 R + r ) \displaystyle T=r(2R+r)
  25. T = P A P B , T=PA\cdot PB,
  26. r = s - c = ( a + b - c ) / 2 \displaystyle r=s-c=(a+b-c)/2
  27. r a = s - b = ( a - b + c ) / 2 \displaystyle r_{a}=s-b=(a-b+c)/2
  28. r b = s - a = ( - a + b + c ) / 2 \displaystyle r_{b}=s-a=(-a+b+c)/2
  29. r c = s = ( a + b + c ) / 2 \displaystyle r_{c}=s=(a+b+c)/2
  30. r a + r b + r c + r = a + b + c \displaystyle r_{a}+r_{b}+r_{c}+r=a+b+c
  31. r a 2 + r b 2 + r c 2 + r 2 = a 2 + b 2 + c 2 \displaystyle r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}=a^{2}+b^{2}+c^{2}
  32. r = r a r b r c \displaystyle r=\frac{r_{a}r_{b}}{r_{c}}
  33. h = a b c \displaystyle h=\frac{ab}{c}
  34. m a 2 + m b 2 + m c 2 = 6 R 2 . \displaystyle m_{a}^{2}+m_{b}^{2}+m_{c}^{2}=6R^{2}.
  35. ( c = 2 R ) . \displaystyle(c=2R).
  36. 2 r \sqrt{2}r
  37. sin α = O H , cos α = A H , tan α = O A , sec α = H A , cot α = A O , csc α = H O . \sin\alpha=\frac{O}{H},\,\cos\alpha=\frac{A}{H},\,\tan\alpha=\frac{O}{A},\,% \sec\alpha=\frac{H}{A},\,\cot\alpha=\frac{A}{O},\,\csc\alpha=\frac{H}{O}.
  38. m a 2 + m b 2 = 5 m c 2 = 5 4 c 2 . m_{a}^{2}+m_{b}^{2}=5m_{c}^{2}=\frac{5}{4}c^{2}.
  39. A H = A 2 G 2 = G 2 H 2 = ϕ \frac{A}{H}=\frac{A^{2}}{G^{2}}=\frac{G^{2}}{H^{2}}=\phi\,
  40. a b = ϕ 3 , \frac{a}{b}=\phi^{3},\,
  41. ϕ \phi
  42. 1 + 5 2 . \tfrac{1+\sqrt{5}}{2}.\,
  43. ( 2 - 1 ) . (\sqrt{2}-1).
  44. c 2 2 ( a + b ) c\geq\frac{\sqrt{2}}{2}(a+b)
  45. h c 2 4 ( a + b ) h_{c}\leq\frac{\sqrt{2}}{4}(a+b)
  46. p 2 + q 2 = 5 ( c 3 ) 2 . p^{2}+q^{2}=5\left(\frac{c}{3}\right)^{2}.
  47. 1 c 2 + 1 h 2 = 1 k 2 . \frac{1}{c^{2}}+\frac{1}{h^{2}}=\frac{1}{k^{2}}.
  48. 1 r = - 1 c + 1 h + 1 k . \displaystyle\frac{1}{r}=-{\frac{1}{c}}+\frac{1}{h}+\frac{1}{k}.
  49. a + b + c = r + r a + r b + r c . a+b+c=r+r_{a}+r_{b}+r_{c}.
  50. 4 c 4 + 9 a 2 b 2 = 16 m a 2 m b 2 . 4c^{4}+9a^{2}b^{2}=16m_{a}^{2}m_{b}^{2}.
  51. a - 2 + b - 2 = d - 2 a^{-2}+b^{-2}=d^{-2}

Ring_(mathematics).html

  1. 𝐙 4 = { 0 ¯ , 1 ¯ , 2 ¯ , 3 ¯ } \mathbf{Z}_{4}=\{\overline{0},\overline{1},\overline{2},\overline{3}\}
  2. x ¯ + y ¯ \overline{x}+\overline{y}
  3. 2 ¯ + 3 ¯ = 1 ¯ \overline{2}+\overline{3}=\overline{1}
  4. 3 ¯ + 3 ¯ = 2 ¯ \overline{3}+\overline{3}=\overline{2}
  5. x ¯ y ¯ \overline{x}\cdot\overline{y}
  6. 2 ¯ 3 ¯ = 2 ¯ \overline{2}\cdot\overline{3}=\overline{2}
  7. 3 ¯ 3 ¯ = 1 ¯ \overline{3}\cdot\overline{3}=\overline{1}
  8. x ¯ \overline{x}
  9. x ¯ \overline{x}
  10. - x ¯ \overline{-x}
  11. - 3 ¯ = - 3 ¯ = 1 ¯ . -\overline{3}=\overline{-3}=\overline{1}.
  12. 2 ( ) = { ( a b c d ) | a , b , c , d } . \mathcal{M}_{2}(\mathbb{R})=\left\{\left.\begin{pmatrix}a&b\\ c&d\end{pmatrix}\right|\ a,b,c,d\in\mathbb{R}\right\}.
  13. ( 1 0 0 1 ) \begin{pmatrix}1&0\\ 0&1\end{pmatrix}
  14. A = ( 0 1 1 0 ) A=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  15. B = ( 0 1 0 0 ) B=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}
  16. A B = ( 0 0 0 1 ) AB=\begin{pmatrix}0&0\\ 0&1\end{pmatrix}
  17. B A = ( 1 0 0 0 ) BA=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}
  18. ( f * g ) ( x ) = - f ( y ) g ( x - y ) d y . (f*g)(x)=\int_{-\infty}^{\infty}f(y)g(x-y)dy.
  19. R R
  20. a a
  21. b b
  22. R R
  23. a b = 0 ab=0
  24. a a
  25. a n = 0 a^{n}=0
  26. n > 0 n>0
  27. e e
  28. e 2 = e e^{2}=e
  29. a a
  30. a - 1 a^{-1}
  31. R × R^{\times}
  32. R * R^{*}
  33. U ( R ) U(R)
  34. R × R^{\times}
  35. x y xy
  36. x + y x+y
  37. - x -x
  38. n 1 = 1 + 1 + + 1 n\cdot 1=1+1+\ldots+1
  39. n 1 n\cdot 1
  40. Z ( R ) \operatorname{Z}(R)
  41. x y = y x xy=yx
  42. Z ( R ) \operatorname{Z}(R)
  43. x + y x+y
  44. r x rx
  45. R I RI
  46. r 1 x 1 + + r n x n , r i R , x i I , r_{1}x_{1}+\cdots+r_{n}x_{n},\quad r_{i}\in R,\quad x_{i}\in I,
  47. R I I RI\subseteq I
  48. I R I IR\subseteq I
  49. R E RE
  50. R x Rx
  51. x R xR
  52. R x R RxR
  53. ( x ) (x)
  54. x , y R x,y\in R
  55. x y P xy\in P
  56. x P x\in P
  57. y P y\in P
  58. I , J I,J
  59. I J P IJ\subseteq P
  60. I P I\subseteq P
  61. J P . J\subseteq P.
  62. R , S R,S
  63. R S R\simeq S
  64. u u
  65. R R , x u x u - 1 R\to R,x\mapsto uxu^{-1}
  66. x x p x\mapsto x^{p}
  67. L / K L/K
  68. \mathbb{Z}\to\mathbb{Q}
  69. f : R S f:R\to S
  70. p : R R / I p:R\to R/I
  71. x x + I x\mapsto x+I
  72. f : R S f:R\to S
  73. f ( I ) = 0 f(I)=0
  74. f ¯ : R / I S \overline{f}:R/I\to S
  75. f = f ¯ p f=\overline{f}\circ p
  76. R / ker f R/\operatorname{ker}f
  77. n 0 Γ ( L n ) \oplus_{n\geq 0}\Gamma(L^{\otimes n})
  78. ( 1 , 1 ) (1,1)
  79. R i R_{i}
  80. i I R i \prod_{i\in I}R_{i}
  81. 𝔞 1 , , 𝔞 n \scriptstyle\mathfrak{a}_{1},\cdots,\mathfrak{a}_{n}
  82. 𝔞 i + 𝔞 j = ( 1 ) \mathfrak{a}_{i}+\mathfrak{a}_{j}=(1)
  83. i j i\neq j
  84. R / ( 𝔞 i ) R / 𝔞 i , x ( x mod 𝔞 1 , , x mod 𝔞 n ) R/\left(\cap\mathfrak{a}_{i}\right)\simeq\prod R/\mathfrak{a}_{i},\quad x% \mapsto(x\,\text{ mod }\mathfrak{a}_{1},\ldots,x\,\text{ mod }\mathfrak{a}_{n})
  85. R i , 1 i n R_{i},1\leq i\leq n
  86. R i R = R i R_{i}\to R=\prod R_{i}
  87. 𝔞 i \mathfrak{a}_{i}
  88. 𝔞 i \mathfrak{a}_{i}
  89. 𝔞 i \mathfrak{a}_{i}
  90. R = 𝔞 1 𝔞 n , 𝔞 i 𝔞 j = 0 , i j , 𝔞 i 2 𝔞 i R=\mathfrak{a}_{1}\oplus\cdots\oplus\mathfrak{a}_{n},\quad\mathfrak{a}_{i}% \mathfrak{a}_{j}=0,i\neq j,\quad\mathfrak{a}_{i}^{2}\subseteq\mathfrak{a}_{i}
  91. 1 = e 1 + + e n , e i 𝔞 i . 1=e_{1}+\cdots+e_{n},\quad e_{i}\in\mathfrak{a}_{i}.
  92. 𝔞 i \mathfrak{a}_{i}
  93. e i e_{i}
  94. e i e j = 0 , i j e_{i}e_{j}=0,i\neq j
  95. 𝔞 i = R e i \mathfrak{a}_{i}=Re_{i}
  96. e i e_{i}
  97. R [ t ] = { a n t n + a n - 1 t n - 1 + + a 1 t + a 0 n 0 , a j R } R[t]=\left\{a_{n}t^{n}+a_{n-1}t^{n-1}+\dots+a_{1}t+a_{0}\mid n\geq 0,a_{j}\in R\right\}
  98. R [ t 1 , , t n ] R[t_{1},\ldots,t_{n}]
  99. t 1 , , t n t_{1},\ldots,t_{n}
  100. R [ t i ] R[t_{i}]
  101. R [ t ] R[t]
  102. R [ t ] R[t]
  103. R [ t ] R[t]
  104. R [ t ] R[t]
  105. R S R\subseteq S
  106. R [ t ] S , f f ( x ) R[t]\to S,\quad f\mapsto f(x)
  107. f ( t ) f(t)
  108. f f ( x ) f\mapsto f(x)
  109. R [ x ] R[x]
  110. k [ t 2 , t 3 ] k[t^{2},t^{3}]
  111. k [ x , y ] k [ t ] , f f ( t 2 , t 3 ) . k[x,y]\to k[t],\,f\mapsto f(t^{2},t^{3}).
  112. k [ t ] k[t]
  113. f ( x + h ) f(x+h)
  114. R [ h ] R[h]
  115. f ( x + h ) - f ( x ) f(x+h)-f(x)
  116. ( f ( x + h ) - f ( x ) ) / h (f(x+h)-f(x))/h
  117. f ( x ) f^{\prime}(x)
  118. ϕ : R S \phi:R\to S
  119. ϕ ¯ : R [ t ] S \overline{\phi}:R[t]\to S
  120. ϕ ¯ ( t ) = x \overline{\phi}(t)=x
  121. ϕ ¯ \overline{\phi}
  122. ϕ \phi
  123. R S R\to S
  124. R [ t ] S , f f ¯ R[t]\to S,\quad f\mapsto\overline{f}
  125. f ¯ \overline{f}
  126. R [ t ] R[t]
  127. S [ t ] S[t]
  128. k [ t 1 , , t n ] k[t_{1},\ldots,t_{n}]
  129. k n k^{n}
  130. R [ [ t ] ] R[\![t]\!]
  131. 0 a i t i , a i R \sum_{0}^{\infty}a_{i}t^{i},\quad a_{i}\in R
  132. R [ t ] R[t]
  133. U U
  134. End R ( U ) \operatorname{End}_{R}(U)
  135. End R ( R n ) M n ( R ) \operatorname{End}_{R}(R^{n})\simeq\operatorname{M}_{n}(R)
  136. f : 1 n U 1 n U f:\oplus_{1}^{n}U\to\oplus_{1}^{n}U
  137. f i j f_{ij}
  138. S = End R ( U ) S=\operatorname{End}_{R}(U)
  139. End R ( 1 n U ) M n ( S ) , f ( f i j ) . \operatorname{End}_{R}(\oplus_{1}^{n}U)\to\operatorname{M}_{n}(S),\quad f% \mapsto(f_{ij}).
  140. End R ( U ) \operatorname{End}_{R}(U)
  141. U = i = 1 r U i m i \displaystyle U=\bigoplus_{i=1}^{r}U_{i}^{\oplus m_{i}}
  142. U i U_{i}
  143. End R ( U ) 1 r M m i ( End R ( U i ) ) \operatorname{End}_{R}(U)\simeq\bigoplus_{1}^{r}\operatorname{M}_{m_{i}}(% \operatorname{End}_{R}(U_{i}))
  144. lim R i \underrightarrow{\lim}R_{i}
  145. x y x\sim y
  146. x = y x=y
  147. R [ t 1 , t 2 , ] = lim R [ t 1 , t 2 , , t m ] . R[t_{1},t_{2},\cdots]=\underrightarrow{\lim}R[t_{1},t_{2},\cdots,t_{m}].
  148. 𝐅 ¯ p = lim 𝐅 p m . \overline{\mathbf{F}}_{p}=\underrightarrow{\lim}\mathbf{F}_{p^{m}}.
  149. k ( ( t ) ) = lim t - m k [ [ t ] ] k(\!(t)\!)=\underrightarrow{\lim}t^{-m}k[\![t]\!]
  150. k [ [ t ] ] k[\![t]\!]
  151. lim k [ U ] \underrightarrow{\lim}k[U]
  152. k [ U ] k[U]
  153. R i R_{i}
  154. R j R i , j i R_{j}\to R_{i},j\geq i
  155. R i R i R_{i}\to R_{i}
  156. R k R j R i R_{k}\to R_{j}\to R_{i}
  157. R k R i R_{k}\to R_{i}
  158. k j i k\geq j\geq i
  159. lim R i \underleftarrow{\lim}R_{i}
  160. R i \prod R_{i}
  161. ( x n ) (x_{n})
  162. x j x_{j}
  163. x i x_{i}
  164. R j R i , j i R_{j}\to R_{i},j\geq i
  165. R [ S - 1 ] R[S^{-1}]
  166. R R [ S - 1 ] R\to R[S^{-1}]
  167. R [ S - 1 ] R[S^{-1}]
  168. R [ S - 1 ] R[S^{-1}]
  169. R [ S - 1 ] R[S^{-1}]
  170. R [ f - 1 ] R[f^{-1}]
  171. r / f n , r R , n 0 r/f^{n},\,r\in R,\,n\geq 0
  172. R [ f - 1 ] = R [ t ] / ( t f - 1 ) . R[f^{-1}]=R[t]/(tf-1).
  173. S = R - 𝔭 S=R-\mathfrak{p}
  174. R 𝔭 R_{\mathfrak{p}}
  175. R [ S - 1 ] R[S^{-1}]
  176. R 𝔭 R_{\mathfrak{p}}
  177. 𝔭 R 𝔭 \mathfrak{p}R_{\mathfrak{p}}
  178. 𝔭 \mathfrak{p}
  179. R / 𝔭 R/\mathfrak{p}
  180. R 𝔭 R_{\mathfrak{p}}
  181. k ( 𝔭 ) k(\mathfrak{p})
  182. M [ S - 1 ] = R [ S - 1 ] R M M[S^{-1}]=R[S^{-1}]\otimes_{R}M
  183. 𝔭 𝔭 [ S - 1 ] \mathfrak{p}\mapsto\mathfrak{p}[S^{-1}]
  184. R [ S - 1 ] R[S^{-1}]
  185. R [ S - 1 ] = lim R [ f - 1 ] R[S^{-1}]=\underrightarrow{\lim}R[f^{-1}]
  186. 0 M [ S - 1 ] M [ S - 1 ] M ′′ [ S - 1 ] 0 0\to M^{\prime}[S^{-1}]\to M[S^{-1}]\to M^{\prime\prime}[S^{-1}]\to 0
  187. R [ S - 1 ] R[S^{-1}]
  188. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  189. 0 M 𝔪 M 𝔪 M 𝔪 ′′ 0 0\to M^{\prime}_{\mathfrak{m}}\to M_{\mathfrak{m}}\to M^{\prime\prime}_{% \mathfrak{m}}\to 0
  190. 𝔪 \mathfrak{m}
  191. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  192. R [ S - 1 ] R[S^{-1}]
  193. R [ S - 1 ] R[S^{-1}]
  194. R ^ = lim R / I n \hat{R}=\underleftarrow{\lim}R/I^{n}
  195. R / I n R/I^{n}
  196. R R ^ R\to\hat{R}
  197. x | x | x\mapsto|x|
  198. | n | p = p - v p ( n ) |n|_{p}=p^{-v_{p}(n)}
  199. v p ( n ) v_{p}(n)
  200. | 0 | p = 0 |0|_{p}=0
  201. | m / n | p = | m | p / | n | p |m/n|_{p}=|m|_{p}/|n|_{p}
  202. | x | p 1 |x|_{p}\leq 1
  203. R [ [ t ] ] R[\![t]\!]
  204. R [ t ] R[t]
  205. ( t ) (t)
  206. F R F\to R
  207. E = { x y - y x x , y X } E=\{xy-yx\mid x,y\in X\}
  208. S the free ring generated by the set S S\mapsto\,\text{the free ring generated by the set }S
  209. A R B A\otimes_{R}B
  210. ( x u ) ( y v ) = x y u v (x\otimes u)(y\otimes v)=xy\otimes uv
  211. f : V V f:V\to V
  212. k [ t ] k[t]
  213. q = p 1 e 1 p s e s . q=p_{1}^{e_{1}}...p_{s}^{e_{s}}.
  214. t v = f ( v ) t\cdot v=f(v)
  215. k [ t ] / ( p i k j ) k[t]/(p_{i}^{k_{j}})
  216. p i ( t ) = t - λ i p_{i}(t)=t-\lambda_{i}
  217. p i p_{i}
  218. p i p_{i}
  219. t - λ i t-\lambda_{i}
  220. k [ G ] k[G]
  221. A k F A\otimes_{k}F
  222. F / k F/k
  223. R n R_{n}
  224. A k k n B k k m A\otimes_{k}k_{n}\approx B\otimes_{k}k_{m}
  225. k n k k m k n m k_{n}\otimes_{k}k_{m}\simeq k_{nm}
  226. [ A ] [A]
  227. [ A ] [ B ] = [ A k B ] [A][B]=[A\otimes_{k}B]
  228. Br ( k ) \operatorname{Br}(k)
  229. Br ( k ) \operatorname{Br}(k)
  230. Br ( ) \operatorname{Br}(\mathbb{R})
  231. p \mathbb{Q}_{p}
  232. Br ( k ) = / \operatorname{Br}(k)=\mathbb{Q}/\mathbb{Z}
  233. - k F -\otimes_{k}F
  234. Br ( k ) Br ( F ) \operatorname{Br}(k)\to\operatorname{Br}(F)
  235. Br ( F / k ) \operatorname{Br}(F/k)
  236. [ A ] [A]
  237. A k F A\otimes_{k}F
  238. Br ( F / k ) \operatorname{Br}(F/k)
  239. H 2 ( Gal ( F / k ) , k * ) H^{2}(\operatorname{Gal}(F/k),k^{*})
  240. k ( ( t ) ) k(\!(t)\!)
  241. k [ [ t ] ] k[\![t]\!]
  242. k ( ( G ) ) k(\!(G)\!)
  243. ( f * g ) ( t ) = s G f ( s ) g ( t - s ) (f*g)(t)=\sum_{s\in G}f(s)g(t-s)
  244. + : R × R R +:R\times R\to R\,
  245. : R × R R \cdot:R\times R\to R\,
  246. H * ( X , ) = i = 0 H i ( X , ) , H^{*}(X,\mathbb{Z})=\bigoplus_{i=0}^{\infty}H^{i}(X,\mathbb{Z}),
  247. H i ( X , ) H_{i}(X,\mathbb{Z})
  248. {\mathbb{Z}}
  249. R × R a R R\times R\stackrel{a}{\to}R
  250. R × R m R R\times R\stackrel{m}{\to}R
  251. pt 0 R \operatorname{pt}\stackrel{0}{\to}R
  252. R i R R\stackrel{i}{\to}R
  253. pt 1 R \operatorname{pt}\stackrel{1}{\to}R
  254. h R = Hom ( - , R ) : C op 𝐒𝐞𝐭𝐬 h_{R}=\operatorname{Hom}(-,R):C^{\operatorname{op}}\to\mathbf{Sets}
  255. C op 𝐑𝐢𝐧𝐠𝐬 forgetful 𝐒𝐞𝐭𝐬 C^{\operatorname{op}}\to\mathbf{Rings}\stackrel{\textrm{forgetful}}{% \longrightarrow}\mathbf{Sets}
  256. μ : X X X \mu\colon X\wedge X\to X
  257. S X S\to X

RN.html

  1. 𝐑 n \,\textbf{R}^{n}
  2. n \mathbb{R}^{n}

Robert_Langlands.html

  1. S L ( 2 ) SL(2)
  2. L L
  3. L L
  4. L L
  5. L L
  6. L L
  7. L L
  8. N N
  9. G L ( N ) GL(N)
  10. l l
  11. G L ( 2 ) GL(2)
  12. G L ( 2 ) GL(2)
  13. G L ( 2 ) GL(2)
  14. G L ( 2 ) GL(2)
  15. L L
  16. L L

Roche_limit.html

  1. d d
  2. d = 1.26 R M ( ρ M ρ m ) 1 3 d=1.26\;R_{M}\left(\frac{\rho_{M}}{\rho_{m}}\right)^{\frac{1}{3}}
  3. R M R_{M}
  4. ρ M \rho_{M}
  5. ρ m \rho_{m}
  6. d = 1.26 R m ( M M M m ) 1 3 d=1.26\;R_{m}\left(\frac{M_{M}}{M_{m}}\right)^{\frac{1}{3}}
  7. R m R_{m}
  8. M M M_{M}
  9. M m M_{m}
  10. u u
  11. u u
  12. F G F\text{G}
  13. u u
  14. m m
  15. r r
  16. F G = G m u r 2 F\text{G}=\frac{Gmu}{r^{2}}
  17. F T F\text{T}
  18. u u
  19. R R
  20. M M
  21. d d
  22. F T = 2 G M u r d 3 F\text{T}=\frac{2GMur}{d^{3}}
  23. F T = G M u ( d - r ) 2 - G M u d 2 F\text{T}=\frac{GMu}{(d-r)^{2}}-\frac{GMu}{d^{2}}
  24. F T = G M u d 2 - ( d - r ) 2 d 2 ( d - r ) 2 F\text{T}=GMu\frac{d^{2}-(d-r)^{2}}{d^{2}(d-r)^{2}}
  25. F T = G M u 2 d r - r 2 d 4 - 2 d 3 r + r 2 d 2 F\text{T}=GMu\frac{2dr-r^{2}}{d^{4}-2d^{3}r+r^{2}d^{2}}
  26. r R r\ll R
  27. R < d R<d
  28. r 2 r^{2}
  29. r r
  30. F T = G M u 2 d r d 4 F\text{T}=GMu\frac{2dr}{d^{4}}
  31. F T = 2 G M u r d 3 F\text{T}=\frac{2GMur}{d^{3}}
  32. F G = F T F\text{G}=F\text{T}\;
  33. G m u r 2 = 2 G M u r d 3 \frac{Gmu}{r^{2}}=\frac{2GMur}{d^{3}}
  34. d d
  35. d = r ( 2 M m ) 1 3 d=r\left(2\,\frac{M}{m}\right)^{\frac{1}{3}}
  36. M M
  37. M = 4 π ρ M R 3 3 M=\frac{4\pi\rho_{M}R^{3}}{3}
  38. R R
  39. m = 4 π ρ m r 3 3 m=\frac{4\pi\rho_{m}r^{3}}{3}
  40. r r
  41. 4 π / 3 4\pi/3
  42. d = r ( 2 ρ M R 3 ρ m r 3 ) 1 / 3 d=r\left(\frac{2\rho_{M}R^{3}}{\rho_{m}r^{3}}\right)^{1/3}
  43. d = R ( 2 ρ M ρ m ) 1 3 1.26 R ( ρ M ρ m ) 1 3 d=R\left(2\,\frac{\rho_{M}}{\rho_{m}}\right)^{\frac{1}{3}}\approx 1.26R\left(% \frac{\rho_{M}}{\rho_{m}}\right)^{\frac{1}{3}}
  44. F C = ω 2 u r = G M u r d 3 F_{C}=\omega^{2}ur=\frac{GMur}{d^{3}}
  45. d = R M ( 3 ρ M ρ m ) 1 3 1.442 R M ( ρ M ρ m ) 1 3 d=R_{M}\left(3\;\frac{\rho_{M}}{\rho_{m}}\right)^{\frac{1}{3}}\approx 1.442R_{% M}\left(\frac{\rho_{M}}{\rho_{m}}\right)^{\frac{1}{3}}
  46. d = R m ( 3 M M M m ) 1 3 1.442 R m ( M M M m ) 1 3 d=R_{m}\left(3\;\frac{M_{M}}{M_{m}}\right)^{\frac{1}{3}}\approx 1.442\;R_{m}% \left(\frac{M_{M}}{M_{m}}\right)^{\frac{1}{3}}
  47. m = 4 π ρ m r 3 3 m=\frac{4\pi\rho_{m}r^{3}}{3}
  48. r r
  49. ρ m \rho_{m}
  50. d = ( 9 M M 4 π ρ m ) 1 3 0.8947 ( M M ρ m ) 1 3 d=\left(\frac{9M_{M}}{4\pi\rho_{m}}\right)^{\frac{1}{3}}\approx 0.8947\left(% \frac{M_{M}}{\rho_{m}}\right)^{\frac{1}{3}}
  51. ρ m \rho_{m}
  52. r r
  53. R Roche = 9 M 4 π ρ m 3 R_{\mathrm{Roche}}=\sqrt[3]{\frac{9M}{4\pi\rho_{m}}}
  54. R Hill = R Roche m 3 M 3 R_{\mathrm{Hill}}=R_{\mathrm{Roche}}\sqrt[3]{\frac{m}{3M}}
  55. l = 9 M 4 π ρ m . m 3 M 3 = r l=\sqrt[3]{\frac{9M}{4\pi\rho_{m}}.\frac{m}{3M}}=r
  56. R Roche = R Hill 3 M m 3 = R secondary 3 M m 3 R_{\,\text{Roche}}=R_{\,\text{Hill}}\sqrt[3]{\frac{3M}{m}}=R\text{secondary}% \sqrt[3]{\frac{3M}{m}}
  57. d 2.44 R ( ρ M ρ m ) 1 / 3 d\approx 2.44R\left(\frac{\rho_{M}}{\rho_{m}}\right)^{1/3}
  58. d 2.423 R ( ρ M ρ m ) 1 / 3 ( ( 1 + m 3 M ) + c 3 R ( 1 + m M ) 1 - c / R ) 1 / 3 d\approx 2.423R\left(\frac{\rho_{M}}{\rho_{m}}\right)^{1/3}\left(\frac{(1+% \frac{m}{3M})+\frac{c}{3R}(1+\frac{m}{M})}{1-c/R}\right)^{1/3}
  59. c / R c/R
  60. ρ m \rho_{m}
  61. V V
  62. ω \omega
  63. ω \omega
  64. ω 2 = G M + m d 3 . \omega^{2}=G\,\frac{M+m}{d^{3}}.
  65. ω 2 = G M d 3 . \omega^{2}=G\,\frac{M}{d^{3}}.
  66. m M m\ll M
  67. V T = - 3 G M 2 d 3 Δ d 2 V_{T}=-\frac{3GM}{2d^{3}}\Delta d^{2}\,
  68. V C = - 1 2 ω 2 Δ d 2 = - G M 2 d 3 Δ d 2 V_{C}=-\frac{1}{2}\omega^{2}\Delta d^{2}=-\frac{GM}{2d^{3}}\Delta d^{2}\,
  69. ω \omega
  70. V s = V s 0 + G π ρ m f ( ϵ ) Δ d 2 , V_{s}=V_{s_{0}}+G\pi\rho_{m}\cdot f(\epsilon)\cdot\Delta d^{2},
  71. V s 0 V_{s_{0}}
  72. f ( ϵ ) = 1 - ϵ 2 ϵ 3 [ ( 3 - ϵ 2 ) arsinh ( ϵ 1 - ϵ 2 ) - 3 ϵ ] f(\epsilon)=\frac{1-\epsilon^{2}}{\epsilon^{3}}\cdot\left[\left(3-\epsilon^{2}% \right)\cdot\operatorname{arsinh}\left(\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}% \right)-3\epsilon\right]
  73. 2 G π ρ M R 3 d 3 = G π ρ m f ( ϵ ) \frac{2G\pi\rho_{M}R^{3}}{d^{3}}=G\pi\rho_{m}f(\epsilon)
  74. ϵ max 0.86 \epsilon\text{max}\approx 0{.}86
  75. d 2.423 R ρ M ρ m 3 . d\approx 2{.}423\cdot R\cdot\sqrt[3]{\frac{\rho_{M}}{\rho_{m}}}\,.
  76. d 2.455 R ρ M ρ m 3 . d\approx 2{.}455\cdot R\cdot\sqrt[3]{\frac{\rho_{M}}{\rho_{m}}}\,.

Rocket.html

  1. F n = m ˙ v e F_{n}=\dot{m}\;v_{e}
  2. m ˙ = \dot{m}=\,
  3. v e = v_{e}=\,
  4. v e v_{e}
  5. I = F d t I=\int Fdt
  6. I = F t I=Ft\;
  7. I s p I_{sp}
  8. v e = I s p g 0 v_{e}=I_{sp}\cdot g_{0}
  9. I s p I_{sp}
  10. g 0 g_{0}
  11. I s p I_{sp}
  12. v e v_{e}
  13. Δ v = v e ln m 0 m 1 \Delta v\ =v_{e}\ln\frac{m_{0}}{m_{1}}
  14. m 0 m_{0}
  15. m 1 m_{1}
  16. v e v_{e}
  17. Δ v \Delta v
  18. m 0 m 1 \frac{m_{0}}{m_{1}}
  19. a a
  20. a = F n m a=\frac{F_{n}}{m}
  21. F n F_{n}
  22. η c = 1 2 m ˙ v e 2 η c o m b u s t i o n P c h e m \eta_{c}=\frac{\frac{1}{2}\dot{m}v_{e}^{2}}{\eta_{combustion}P_{chem}}
  23. η c = 100 % \eta_{c}=100\%
  24. η p \eta_{p}
  25. η p \eta_{p}
  26. u u
  27. c c
  28. η p = 2 u c 1 + ( u c ) 2 \eta_{p}=\frac{2\frac{u}{c}}{1+(\frac{u}{c})^{2}}
  29. η \eta
  30. η = η p η c \eta=\eta_{p}\eta_{c}
  31. η c \eta_{c}
  32. η c \eta_{c}

Role-based_access_control.html

  1. P A P × R PA\subseteq P\times R
  2. S A S × R SA\subseteq S\times R
  3. R H R × R RH\subseteq R\times R

Roman_surface.html

  1. x 2 y 2 + y 2 z 2 + z 2 x 2 - r 2 x y z = 0. x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}-r^{2}xyz=0.\,
  2. x 2 + y 2 + z 2 = 1 , x^{2}+y^{2}+z^{2}=1,\,
  3. T ( x , y , z ) = ( y z , z x , x y ) = ( U , V , W ) , T(x,y,z)=(yz,zx,xy)=(U,V,W),\,
  4. U 2 V 2 + V 2 W 2 + W 2 U 2 = z 2 x 2 y 4 + x 2 y 2 z 4 + y 2 z 2 x 4 = ( x 2 + y 2 + z 2 ) ( x 2 y 2 z 2 ) = ( 1 ) ( x 2 y 2 z 2 ) = ( x y ) ( y z ) ( z x ) = U V W , \begin{aligned}\displaystyle U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}&\displaystyle=z^% {2}x^{2}y^{4}+x^{2}y^{2}z^{4}+y^{2}z^{2}x^{4}=(x^{2}+y^{2}+z^{2})(x^{2}y^{2}z^% {2})\\ &\displaystyle=(1)(x^{2}y^{2}z^{2})=(xy)(yz)(zx)=UVW,\end{aligned}
  5. U 2 V 2 + V 2 W 2 + W 2 U 2 - U V W = 0 U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}-UVW=0\,
  6. U 2 V 2 + V 2 W 2 + W 2 U 2 - U V W = 0. U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}-UVW=0.\,
  7. x 2 + y 2 + z 2 = 1 , x^{2}+y^{2}+z^{2}=1,\,
  8. U = x y , V = y z , W = z x , U=xy,V=yz,W=zx,\,
  9. x = W U V , y = U V W , z = V W U . x=\sqrt{\frac{WU}{V}},\ y=\sqrt{\frac{UV}{W}},\ z=\sqrt{\frac{VW}{U}}.\,
  10. U 2 V 2 = 0 U^{2}V^{2}=0\,
  11. U 0 , V = W = 0. U\neq 0,V=W=0.\,
  12. z = 0 , z=0,\,
  13. z 0 , z\neq 0,\,
  14. x = y = 0 , x=y=0,\,
  15. U = 0 , U=0,\,
  16. | U | 1 2 , |U|\leq\frac{1}{2},
  17. x 2 = 1 + 1 - 4 U 2 2 x^{2}=\frac{1+\sqrt{1-4U^{2}}}{2}
  18. y 2 = 1 - 1 - 4 U 2 2 , y^{2}=\frac{1-\sqrt{1-4U^{2}}}{2},
  19. x 2 y 2 = U 2 , x^{2}y^{2}=U^{2},\,
  20. x y = U . xy=U.\,
  21. y z = 0 = V and z x = 0 = W , yz=0=V\,\text{ and }zx=0=W,\,
  22. | U | > 1 2 . |U|>\frac{1}{2}.
  23. x 2 + y 2 = 1 , x^{2}+y^{2}=1,\,
  24. x y 1 2 , xy\leq\frac{1}{2},
  25. | U | > 1 / 2 , V = W = 0 , |U|>1/2,\ V=W=0,
  26. U = x y , V = y z , W = z x . U=xy,\ V=yz,\ W=zx.
  27. | U | > 1 2 |U|>\frac{1}{2}
  28. | V | > 1 2 |V|>\frac{1}{2}
  29. | W | > 1 2 |W|>\frac{1}{2}
  30. ( x y , y z , z x ) = ( 0 , 0 , 0 ) = ( U , V , W ) (xy,yz,zx)=(0,0,0)=(U,V,W)\,
  31. x = r cos θ cos ϕ , x=r\,\cos\theta\,\cos\phi,
  32. y = r cos θ sin ϕ , y=r\,\cos\theta\,\sin\phi,
  33. z = r sin θ . z=r\,\sin\theta.
  34. x = y z = r 2 cos θ sin θ sin ϕ , x^{\prime}=yz=r^{2}\,\cos\theta\,\sin\theta\,\sin\phi,
  35. y = z x = r 2 cos θ sin θ cos ϕ , y^{\prime}=zx=r^{2}\,\cos\theta\,\sin\theta\,\cos\phi,
  36. z = x y = r 2 cos 2 θ cos ϕ sin ϕ , z^{\prime}=xy=r^{2}\,\cos^{2}\theta\,\cos\phi\,\sin\phi,
  37. T : ( x , y , z ) ( y z , z x , x y ) , T:(x,y,z)\rightarrow(yz,zx,xy),
  38. T : ( - x , - y , - z ) ( ( - y ) ( - z ) , ( - z ) ( - x ) , ( - x ) ( - y ) ) = ( y z , z x , x y ) . T:(-x,-y,-z)\rightarrow((-y)(-z),(-z)(-x),(-x)(-y))=(yz,zx,xy).
  39. y z = y z yz={y\over z}

Root_mean_square.html

  1. { x 1 , x 2 , , x n } \{x_{1},x_{2},\dots,x_{n}\}
  2. x rms = 1 n ( x 1 2 + x 2 2 + + x n 2 ) . x_{\mathrm{rms}}=\sqrt{\frac{1}{n}\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}% \right)}.
  3. T 1 t T 2 T_{1}\leq t\leq T_{2}
  4. f rms = 1 T 2 - T 1 T 1 T 2 [ f ( t ) ] 2 d t , f_{\mathrm{rms}}=\sqrt{{1\over{T_{2}-T_{1}}}{\int_{T_{1}}^{T_{2}}{[f(t)]}^{2}% \,dt}},
  5. f rms = lim T 1 T 0 T [ f ( t ) ] 2 d t . f_{\mathrm{rms}}=\lim_{T\rightarrow\infty}\sqrt{{1\over{T}}{\int_{0}^{T}{[f(t)% ]}^{2}\,dt}}.
  6. y = a y=a\,
  7. a a\,
  8. y = a sin ( 2 π f t ) y=a\sin(2\pi ft)\,
  9. a 2 \frac{a}{\sqrt{2}}
  10. y = { a { f t } < 0.5 - a { f t } > 0.5 y=\begin{cases}a&\{ft\}<0.5\\ -a&\{ft\}>0.5\end{cases}
  11. a a\,
  12. y = { a + D C { f t } < 0.5 - a + D C { f t } > 0.5 y=\begin{cases}a+DC&\{ft\}<0.5\\ -a+DC&\{ft\}>0.5\end{cases}
  13. D C 2 + a 2 {\sqrt{DC^{2}+a^{2}}}\,
  14. y = { 0 { f t } < 0.25 a 0.25 < { f t } < 0.5 0 0.5 < { f t } < 0.75 - a { f t } > 0.75 y=\begin{cases}0&\{ft\}<0.25\\ a&0.25<\{ft\}<0.5\\ 0&0.5<\{ft\}<0.75\\ -a&\{ft\}>0.75\end{cases}
  15. a 2 \frac{a}{\sqrt{2}}
  16. y = | 2 a { f t } - a | y=|2a\{ft\}-a\,|
  17. a 3 a\over\sqrt{3}
  18. y = 2 a { f t } - a y=2a\{ft\}-a\,
  19. a 3 a\over\sqrt{3}
  20. y = { a { f t } < D 0 { f t } > D y=\begin{cases}a&\{ft\}<D\\ 0&\{ft\}>D\end{cases}
  21. a D a\sqrt{D}
  22. y = a sin ( t ) - a sin ( t - 2 π 3 ) y=a\sin(t)-a\sin(t-\frac{2\pi}{3})\,
  23. a 3 2 a\sqrt{\frac{3}{2}}
  24. R M S T o t a l = R M S 1 2 + R M S 2 2 + + R M S n 2 RMS_{Total}=\sqrt{{{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots+{RMS_{n}}^{2}}}
  25. R M S T o t a l = R M S D C 2 + R M S A C 2 RMS_{Total}=\sqrt{{{RMS_{DC}}^{2}+{RMS_{AC}}^{2}}}
  26. R M S D C RMS_{DC}
  27. R M S A C RMS_{AC}
  28. P = I 2 R . P=I^{2}R.
  29. P avg P_{\mathrm{avg}}\,\!
  30. = I ( t ) 2 R =\langle I(t)^{2}R\rangle\,\!
  31. \langle\ldots\rangle
  32. = R I ( t ) 2 =R\langle I(t)^{2}\rangle\,\!
  33. = ( I RMS ) 2 R =(I_{\mathrm{RMS}})^{2}R\,\!
  34. P avg = ( V RMS ) 2 R . P_{\mathrm{avg}}={(V_{\mathrm{RMS}})^{2}\over R}.
  35. P avg = V RMS I RMS . P_{\mathrm{avg}}=V_{\mathrm{RMS}}I_{\mathrm{RMS}}.
  36. I RMS = 1 T 2 - T 1 T 1 T 2 ( I p sin ( ω t ) ) 2 d t . I_{\mathrm{RMS}}=\sqrt{{1\over{T_{2}-T_{1}}}{\int_{T_{1}}^{T_{2}}{(I_{\mathrm{% p}}\sin(\omega t)}\,})^{2}dt}.\,\!
  37. I RMS = I p 1 T 2 - T 1 T 1 T 2 sin 2 ( ω t ) d t . I_{\mathrm{RMS}}=I_{\mathrm{p}}\sqrt{{1\over{T_{2}-T_{1}}}{\int_{T_{1}}^{T_{2}% }{\sin^{2}(\omega t)}\,dt}}.
  38. I RMS = I p 1 T 2 - T 1 T 1 T 2 < m t p l > 1 - cos ( 2 ω t ) 2 d t I_{\mathrm{RMS}}=I_{\mathrm{p}}\sqrt{{1\over{T_{2}-T_{1}}}{\int_{T_{1}}^{T_{2}% }<mtpl>{{1-\cos(2\omega t)\over 2}}\,dt}}
  39. I RMS = I p 1 T 2 - T 1 [ t 2 - sin ( 2 ω t ) 4 ω ] T 1 T 2 I_{\mathrm{RMS}}=I_{\mathrm{p}}\sqrt{{1\over{T_{2}-T_{1}}}\left[{{t\over 2}-{% \sin(2\omega t)\over 4\omega}}\right]_{T_{1}}^{T_{2}}}
  40. I RMS = I p 1 T 2 - T 1 [ < m t p l > t 2 ] T 1 T 2 = I p 1 T 2 - T 1 T 2 - T 1 2 = I p 2 . I_{\mathrm{RMS}}=I_{\mathrm{p}}\sqrt{{1\over{T_{2}-T_{1}}}\left[<mtpl>{{t\over 2% }}\right]_{T_{1}}^{T_{2}}}=I_{\mathrm{p}}\sqrt{{1\over{T_{2}-T_{1}}}{{{T_{2}-T% _{1}}\over 2}}}={I_{\mathrm{p}}\over{\sqrt{2}}}.
  41. V RMS = V p 2 . V_{\mathrm{RMS}}={V_{\mathrm{p}}\over{\sqrt{2}}}.
  42. v RMS = 3 R T M {v_{\mathrm{RMS}}}={\sqrt{3RT\over{M}}}
  43. x [ n ] = x ( t = n T ) x[n]=x(t=nT)
  44. T T
  45. n = 1 N x 2 [ n ] = 1 N m = 1 N | X [ m ] | 2 \sum_{n=1}^{N}{{{x}^{2}}[n]}=\frac{1}{N}\sum_{m=1}^{N}\left|X[m]\right|^{2}
  46. X [ m ] = FFT { x [ n ] } X[m]=\mathrm{FFT}\{x[n]\}
  47. RMS { x [ n ] } = 1 N n x 2 [ n ] = 1 N 2 m | X [ m ] | 2 = m | X [ m ] N | 2 . \mathrm{RMS}\{x[n]\}=\sqrt{\frac{1}{N}\sum_{n}{{{x}^{2}}[n]}}=\sqrt{\frac{1}{N% ^{2}}\sum_{m}{{{\left|X[m]\right|}^{2}}}}=\sqrt{\sum_{m}{{{\left|\frac{X[m]}{N% }\right|^{2}}}}}.
  48. x ¯ \bar{x}
  49. σ x \sigma_{x}
  50. x rms 2 = x ¯ 2 + σ x 2 = x 2 ¯ . x_{\mathrm{rms}}^{2}=\bar{x}^{2}+\sigma_{x}^{2}=\overline{x^{2}}.

ROT13.html

  1. ROT ( ROT ( x ) 13 ) 13 = x \mbox{ROT}~{}_{13}(\mbox{ROT}~{}_{13}(x))=x

Roulette.html

  1. p a y o u t = 1 n ( 36 - n ) = 36 n - 1 payout=\frac{1}{n}(36-n)=\frac{36}{n}-1
  2. 37 / 38 {37}/{38}
  3. 1 / 38 {1}/{38}
  4. 1 / 37 {1}/{37}
  5. 36 / 37 {36}/{37}
  6. 36 / 37 {36}/{37}
  7. ( Ω , 2 Ω , ) (\Omega,2^{\Omega},\mathbb{P})
  8. Ω = { 0 , , 36 } \Omega=\{0,\ldots,36\}
  9. ( A ) = | A | 37 \mathbb{P}(A)=\frac{|A|}{37}
  10. A 2 Ω A\in 2^{\Omega}
  11. S S
  12. ( A , r , ξ ) (A,r,\xi)
  13. A A
  14. r + r\in\mathbb{R}_{+}
  15. ξ : Ω \xi:\Omega\to\mathbb{R}
  16. S = ( { ω 0 } , r , ξ ) S=(\{\omega_{0}\},r,\xi)
  17. ω 0 Ω \omega_{0}\in\Omega
  18. ξ \xi
  19. ξ ( ω ) = { - r , ω ω 0 35 r , ω = ω 0 . \xi(\omega)=\begin{cases}-r,&\omega\neq\omega_{0}\\ 35\cdot r,&\omega=\omega_{0}\end{cases}.
  20. M [ ξ ] = 1 37 ω Ω ξ ( ω ) = 1 37 ( ξ ( ω ) + ω ω ξ ( ω ) ) = 1 37 ( 35 r - 36 r ) = - r 37 - 0.027 r . M[\xi]=\frac{1}{37}\sum_{\omega\in\Omega}\xi(\omega)=\frac{1}{37}\left(\xi(% \omega^{\prime})+\sum_{\omega\neq\omega^{\prime}}\xi(\omega)\right)=\frac{1}{3% 7}\left(35\cdot r-36\cdot r\right)=-\frac{r}{37}\approx-0.027r.
  21. ξ ( ω ) = { - r , ω is red - r , ω = 0 r , ω is black , \xi(\omega)=\begin{cases}-r,&\omega\,\text{ is red}\\ -r,&\omega=0\\ r,&\omega\,\text{ is black}\end{cases},
  22. M [ ξ ] = 1 37 ( 18 r - 18 r - r ) = - r 37 M[\xi]=\frac{1}{37}(18\cdot r-18\cdot r-r)=-\frac{r}{37}
  23. - r 37 -\frac{r}{37}
  24. n = 1 M [ ξ n ] = - 1 37 n = 1 r n - . \sum_{n=1}^{\infty}M[\xi_{n}]=-\frac{1}{37}\sum_{n=1}^{\infty}r_{n}\to-\infty.
  25. - 3 38 r - 0.0789 r -\frac{3}{38}r\approx-0.0789r
  26. - r 19 - 0.0526 r -\frac{r}{19}\approx-0.0526r
  27. 1 ( 36 + n ) \frac{1}{(36+n)}
  28. p ( 36 + n ) \frac{p}{(36+n)}
  29. p = 18 p=18
  30. 18 ( 36 + n ) \frac{18}{(36+n)}
  31. 36 p \frac{36}{p}
  32. p = 12 p=12
  33. 36 12 = 3 \frac{36}{12}=3
  34. p ( 36 + n ) × 36 p = 36 ( 36 + n ) \frac{p}{(36+n)}\times\frac{36}{p}=\frac{36}{(36+n)}
  35. n > 0 n>0
  36. n = 1 n=1
  37. 36 37 \frac{36}{37}
  38. 36 37 \frac{36}{37}
  39. n = 2 n=2
  40. 36 38 \frac{36}{38}
  41. 1 - ( 37 / 38 ) 35 1-(37/38)^{35}

RSA_(cryptosystem).html

  1. ( n , e ) (n,e)
  2. d d
  3. M M
  4. M M
  5. m m
  6. c c
  7. c m e ( mod n ) c\equiv m^{e}\;\;(\mathop{{\rm mod}}n)
  8. c c
  9. m m
  10. c c
  11. m m
  12. m m
  13. c c
  14. d d
  15. m c d ( mod n ) m\equiv c^{d}\;\;(\mathop{{\rm mod}}n)
  16. m m
  17. M M
  18. p = 61 p=61
  19. q = 53 q=53
  20. n = 61 × 53 = 3233 n=61\times 53=3233
  21. φ ( 3233 ) = ( 61 - 1 ) ( 53 - 1 ) = 3120 \varphi(3233)=(61-1)(53-1)=3120
  22. e = 17 e=17
  23. d = 2753 d=2753
  24. e × d mod φ ( n ) = 1 e\times d\;\operatorname{mod}\;\varphi(n)=1
  25. 17 × 2753 mod 3120 = 1 17\times 2753\;\operatorname{mod}\;3120=1
  26. c ( m ) = m 17 mod 3233 c(m)=m^{17}\;\operatorname{mod}\;3233
  27. m ( c ) = c 2753 mod 3233 m(c)=c^{2753}\;\operatorname{mod}\;3233
  28. c = 65 17 mod 3233 = 2790 c=65^{17}\;\operatorname{mod}\;3233=2790
  29. m = 2790 2753 mod 3233 = 65 m=2790^{2753}\;\operatorname{mod}\;3233=65
  30. d p \displaystyle d_{p}
  31. m 1 \displaystyle m_{1}
  32. a p - 1 1 ( mod p ) a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p)
  33. e d 1 ( mod ϕ ( p q ) ) . ed\equiv 1\;\;(\mathop{{\rm mod}}\phi(pq)).
  34. ϕ ( p q ) = ( p - 1 ) ( q - 1 ) \phi(pq)=(p-1)(q-1)
  35. e d - 1 = h ( p - 1 ) ( q - 1 ) ed-1=h(p-1)(q-1)
  36. m e d = m ( e d - 1 ) m = m h ( p - 1 ) ( q - 1 ) m = ( m p - 1 ) h ( q - 1 ) m 1 h ( q - 1 ) m m ( mod p ) m^{ed}=m^{(ed-1)}m=m^{h(p-1)(q-1)}m=\left(m^{p-1}\right)^{h(q-1)}m\equiv 1^{h(% q-1)}m\equiv m\;\;(\mathop{{\rm mod}}p)
  37. e d 1 ( mod ϕ ( p q ) ) ed\equiv 1\;\;(\mathop{{\rm mod}}\phi(pq))
  38. ( m e ) d m ( mod p q ) \left(m^{e}\right)^{d}\equiv m\;\;(\mathop{{\rm mod}}pq)
  39. m e d = m 1 + h φ ( n ) = m ( m φ ( n ) ) h m ( 1 ) h m ( mod n ) m^{ed}=m^{1+h\varphi(n)}=m\left(m^{\varphi(n)}\right)^{h}\equiv m(1)^{h}\equiv m% \;\;(\mathop{{\rm mod}}n)
  40. p p
  41. q q
  42. d P = d (mod p - 1 ) d_{P}=d\,\text{ (mod }p-1\,\text{)}
  43. d Q = d (mod q - 1 ) d_{Q}=d\,\text{ (mod }q-1\,\text{)}
  44. q inv = q - 1 (mod p ) q\text{inv}=q^{-1}\,\text{ (mod }p\,\text{)}
  45. m 1 = c d P (mod p ) m_{1}=c^{d_{P}}\,\text{ (mod }p\,\text{)}
  46. m 2 = c d Q (mod q ) m_{2}=c^{d_{Q}}\,\text{ (mod }q\,\text{)}
  47. h = q inv ( m 1 - m 2 ) (mod p ) h=q\text{inv}(m_{1}-m_{2})\,\text{ (mod }p\,\text{)}
  48. m 1 < m 2 m_{1}<m_{2}
  49. q inv ( ( m 1 + q / p p ) - m 2 ) (mod p ) q\text{inv}((m_{1}+\left\lceil q/p\right\rceil p)-m_{2})\,\text{ (mod }p\,% \text{)}
  50. m = m 2 + h q m=m_{2}+hq\,
  51. m m
  52. p p
  53. q q
  54. m m
  55. c = m c=m
  56. p 1 p−1
  57. q 1 q−1
  58. e 1 e−1
  59. m m
  60. m e - 1 (mod p ) = 1 m^{e-1}\,\text{ (mod }p\,\text{)}=1
  61. m e - 1 (mod q ) = 1 m^{e-1}\,\text{ (mod }q\,\text{)}=1

Rubik's_Cube.html

  1. 8 ! × 3 7 × ( 12 ! / 2 ) × 2 11 = 43 , 252 , 003 , 274 , 489 , 856 , 000 {8!\times 3^{7}\times(12!/2)\times 2^{11}}=43,252,003,274,489,856,000
  2. 8 ! × 3 8 × 12 ! × 2 12 = 519 , 024 , 039 , 293 , 878 , 272 , 000. {8!\times 3^{8}\times 12!\times 2^{12}}=519,024,039,293,878,272,000.

Russell's_paradox.html

  1. Let R = { x x x } , then R R R R \,\text{Let }R=\{x\mid x\not\in x\}\,\text{, then }R\in R\iff R\not\in R
  2. \in
  3. y x ( x y P ( x ) ) \exists y\forall x(x\in y\iff P(x))
  4. x x x\notin x
  5. P ( x ) P(x)
  6. y y y y y\in y\iff y\notin y

Rutherford_scattering.html

  1. d 2 u d θ 2 + u = - Z 1 Z 2 e 2 4 π ϵ 0 m v 0 2 b 2 = - κ , \frac{d^{2}u}{d\theta^{2}}+u=-\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}mv_{0}^{2% }b^{2}}=-\kappa,
  2. u = 1 r u={1\over r}
  3. v 0 v_{0}
  4. b b
  5. u = u 0 cos ( θ - θ 0 ) - κ , u=u_{0}\cos(\theta-\theta_{0})-\kappa,
  6. u 0 r sin θ b ( θ π ) . u\to 0\quad r\sin\theta\to b\quad(\theta\to\pi).
  7. θ 0 = π 2 + arctan b κ . \theta_{0}=\frac{\pi}{2}+\arctan b\kappa.
  8. u 0 u\to 0
  9. Θ = 2 θ 0 - π = 2 arctan b κ = 2 arctan Z 1 Z 2 e 2 4 π ϵ 0 m v 0 2 b . \Theta=2\theta_{0}-\pi=2\arctan b\kappa=2\arctan\frac{Z_{1}Z_{2}e^{2}}{4\pi% \epsilon_{0}mv_{0}^{2}b}.
  10. b = Z 1 Z 2 e 2 4 π ϵ 0 m v 0 2 cot Θ 2 . b=\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}mv_{0}^{2}}\cot\frac{\Theta}{2}.
  11. d σ d Ω ( Ω ) d Ω = number of particles scattered into solid angle d Ω per unit time incident intensity {\frac{d\sigma}{d\Omega}(\Omega)d\Omega}={\hbox{number of particles scattered % into solid angle d}\Omega\hbox{ per unit time}\over\hbox{incident intensity}}
  12. E E
  13. b b
  14. Θ \Theta
  15. Θ + d Θ \Theta+d\Theta
  16. b b
  17. b + d b b+db
  18. I I
  19. 2 π I b | d b | = I d σ d Ω d Ω 2\pi Ib\left|db\right|=I\frac{d\sigma}{d\Omega}d\Omega
  20. d Ω = 2 π sin Θ d Θ d\Omega=2\pi\sin{\Theta}d\Theta
  21. d σ d Ω = b sin Θ | d b d Θ | \frac{d\sigma}{d\Omega}=\frac{b}{\sin{\Theta}}\left|\frac{db}{d\Theta}\right|
  22. b ( Θ ) b(\Theta)
  23. d σ d Ω = ( Z 1 Z 2 e 2 8 π ϵ 0 m v 0 2 ) 2 csc 4 ( Θ 2 ) . \frac{d\sigma}{d\Omega}=\left(\frac{Z_{1}Z_{2}e^{2}}{8\pi\epsilon_{0}mv_{0}^{2% }}\right)^{2}\csc^{4}{\left(\frac{\Theta}{2}\right)}.
  24. 1 2 m v 2 = 1 4 π ϵ 0 q 1 q 2 b \frac{1}{2}mv^{2}=\frac{1}{4\pi\epsilon_{0}}\cdot\frac{q_{1}q_{2}}{b}
  25. b = 1 4 π ϵ 0 2 q 1 q 2 m v 2 b=\frac{1}{4\pi\epsilon_{0}}\cdot\frac{2q_{1}q_{2}}{mv^{2}}

SameGame.html

  1. ( n - k ) 2 (n-k)^{2}
  2. n n
  3. k = 1 k=1
  4. 2 2
  5. ( n - 1 ) 2 (n-1)^{2}
  6. n 2 - 3 n + 4 n^{2}-3n+4
  7. n ( n - 1 ) n(n-1)
  8. n ( n - 2 ) n(n-2)
  9. ( n - 2 ) 2 (n-2)^{2}

Sample_(statistics).html

  1. X i X_{i}
  2. X i X_{i}
  3. x i = X i ( ω ) x_{i}=X_{i}(\omega)

Sample_space.html

  1. P ( e v e n t ) = number of outcomes in event number of outcomes in sample space P(event)=\frac{\,\text{number of outcomes in event}}{\,\text{number of % outcomes in sample space}}

Satellite_temperature_measurements.html

  1. T B = W ( S ) T ( 0 ) + 0 T O A W ( Z ) T ( Z ) d Z T_{B}=W(_{S})T(_{0})+\int\limits_{0}^{TOA}W(_{Z})T(_{Z})\,dZ
  2. W ( S ) = e S e - τ ( 0 , ) s e c θ W(_{S})=e_{S}e^{-\tau(0,\infty)sec\theta}
  3. τ = z 1 z 2 κ ( z ) d z \tau=\int\limits_{z1}^{z2}\kappa(z)dz
  4. W ( Z ) = κ ( Z ) s e c θ e - τ ( z , ) s e c θ + κ ( Z ) s e c θ e - τ ( 0 , z ) s e c θ ( 1 - e S ) e - τ ( 0 , ) s e c θ W(_{Z})=\kappa(_{Z})sec\theta e^{-\tau(z,\infty)sec\theta}+\kappa(_{Z})sec% \theta e^{-\tau(0,z)sec\theta}(1-e_{S})e^{-\tau(0,\infty)sec\theta}

Saul_Kripke.html

  1. W , R \langle W,R\rangle
  2. W , R , \langle W,R,\Vdash\rangle
  3. W , R \langle W,R\rangle
  4. \Vdash
  5. w ¬ A w\Vdash\neg A
  6. w A w\nVdash A
  7. w A B w\Vdash A\to B
  8. w A w\nVdash A
  9. w B w\Vdash B
  10. w A w\Vdash\Box A
  11. u ( w R u \forall u\,(w\;R\;u
  12. u A ) u\Vdash A)
  13. w A w\Vdash A
  14. \Vdash
  15. W , R , \langle W,R,\Vdash\rangle
  16. w A w\Vdash A
  17. W , R \langle W,R\rangle
  18. W , R , \langle W,R,\Vdash\rangle
  19. \Vdash
  20. A A \Box A\to A
  21. W , R \langle W,R\rangle
  22. w A w\Vdash\Box A
  23. w A w\Vdash A
  24. u p u\Vdash p
  25. w p w\Vdash\Box p
  26. w p w\Vdash p
  27. \Vdash
  28. ( A A ) A \Box(A\equiv\Box A)\to\Box A
  29. A A \Box A\to\Box\Box A
  30. W , R , \langle W,R,\Vdash\rangle
  31. \Vdash
  32. X R Y X\;R\;Y
  33. A A
  34. A X \Box A\in X
  35. A Y A\in Y
  36. X A X\Vdash A
  37. A X A\in X
  38. { i i I } \{\Box_{i}\mid\,i\in I\}
  39. w i A w\Vdash\Box_{i}A
  40. u ( w R i u u A ) . \forall u\,(w\;R_{i}\;u\Rightarrow u\Vdash A).
  41. W , R , { D i } i I , \langle W,R,\{D_{i}\}_{i\in I},\Vdash\rangle
  42. w i A w\Vdash\Box_{i}A
  43. u D i ( w R u u A ) . \forall u\in D_{i}\,(w\;R\;u\Rightarrow u\Vdash A).
  44. W , , \langle W,\leq,\Vdash\rangle
  45. W , \langle W,\leq\rangle
  46. \Vdash
  47. w u w\leq u
  48. w p w\Vdash p
  49. u p u\Vdash p
  50. w A B w\Vdash A\land B
  51. w A w\Vdash A
  52. w B w\Vdash B
  53. w A B w\Vdash A\lor B
  54. w A w\Vdash A
  55. w B w\Vdash B
  56. w A B w\Vdash A\to B
  57. u w u\geq w
  58. u A u\Vdash A
  59. u B u\Vdash B
  60. w w\Vdash\bot
  61. W , , { M w } w W \langle W,\leq,\{M_{w}\}_{w\in W}\rangle
  62. W , \langle W,\leq\rangle
  63. w A [ e ] w\Vdash A[e]
  64. w P ( t 1 , , t n ) [ e ] w\Vdash P(t_{1},\dots,t_{n})[e]
  65. P ( t 1 [ e ] , , t n [ e ] ) P(t_{1}[e],\dots,t_{n}[e])
  66. w ( A B ) [ e ] w\Vdash(A\land B)[e]
  67. w A [ e ] w\Vdash A[e]
  68. w B [ e ] w\Vdash B[e]
  69. w ( A B ) [ e ] w\Vdash(A\lor B)[e]
  70. w A [ e ] w\Vdash A[e]
  71. w B [ e ] w\Vdash B[e]
  72. w ( A B ) [ e ] w\Vdash(A\to B)[e]
  73. u w u\geq w
  74. u A [ e ] u\Vdash A[e]
  75. u B [ e ] u\Vdash B[e]
  76. w [ e ] w\Vdash\bot[e]
  77. w ( x A ) [ e ] w\Vdash(\exists x\,A)[e]
  78. a M w a\in M_{w}
  79. w A [ e ( x a ) ] w\Vdash A[e(x\to a)]
  80. w ( x A ) [ e ] w\Vdash(\forall x\,A)[e]
  81. u w u\geq w
  82. a M u a\in M_{u}
  83. u A [ e ( x a ) ] u\Vdash A[e(x\to a)]

Sawtooth_wave.html

  1. x ( t ) = t - t = t - floor ( t ) x(t)=t-\lfloor t\rfloor=t-\operatorname{floor}(t)
  2. x ( t ) = 2 ( t a - 1 2 + t a ) x(t)=2\left({t\over a}-\left\lfloor{1\over 2}+{t\over a}\right\rfloor\right)
  3. = 2 ( t a - floor ( 1 2 + t a ) ) =2\left({t\over a}-\operatorname{floor}\left({1\over 2}+{t\over a}\right)\right)
  4. y ( x ) = - 2 a π arctan ( cot ( x π p ) ) y(x)=-\frac{2a}{\pi}\arctan\left(\cot\left(\frac{x\pi}{p}\right)\right)
  5. x reversesawtooth ( t ) = 2 π k = 1 ( - 1 ) k sin ( 2 π k f t ) k x_{\mathrm{reversesawtooth}}(t)=\frac{2}{\pi}\sum_{k=1}^{\infty}{(-1)}^{k}% \frac{\sin(2\pi kft)}{k}
  6. x sawtooth ( t ) = A 2 - A π k = 1 sin ( 2 π k f t ) k x_{\mathrm{sawtooth}}(t)=\frac{A}{2}-\frac{A}{\pi}\sum_{k=1}^{\infty}\frac{% \sin(2\pi kft)}{k}

Scanning_tunneling_microscope.html

  1. - 2 2 m 2 ψ n ( z ) z 2 + U ( z ) ψ n ( z ) = E ψ n ( z ) -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi_{n}(z)}{\partial z^{2}}+U(z)\psi_{% n}(z)=E\psi_{n}(z)
  2. ψ n ( z ) = ψ n ( 0 ) e ± i k z \psi_{n}(z)=\psi_{n}(0)e^{\pm ikz}
  3. k = 2 m ( E - U ( z ) ) k=\frac{\sqrt{2m(E-U(z))}}{\hbar}
  4. κ = 2 m ( U - E ) \kappa=\frac{\sqrt{2m(U-E)}}{\hbar}
  5. - κ -\kappa
  6. P | ψ n ( 0 ) | 2 e - 2 κ W P\propto|\psi_{n}(0)|^{2}e^{-2\kappa W}
  7. I E f - e V E f | ψ n ( 0 ) | 2 e - 2 κ W I\propto\sum_{E_{f}-eV}^{E_{f}}|\psi_{n}(0)|^{2}e^{-2\kappa W}
  8. ρ s ( z , E ) = 1 ϵ E - ϵ E | ψ n ( z ) | 2 \rho_{s}(z,E)=\frac{1}{\epsilon}\sum_{E-\epsilon}^{E}|\psi_{n}(z)|^{2}
  9. I V ρ s ( 0 , E f ) e - 2 κ W I\propto V\rho_{s}(0,E_{f})e^{-2\kappa W}
  10. I V ρ s ( W , E f ) I\propto V\rho_{s}(W,E_{f})
  11. w = 2 π | M | 2 δ ( E ψ - E χ ) w=\frac{2\pi}{\hbar}|M|^{2}\delta(E_{\psi}-E_{\chi})
  12. M = 2 2 m z = z 0 ( χ * ψ z - ψ χ * z ) d S M=\frac{\hbar^{2}}{2m}\int_{z=z_{0}}(\chi*\frac{\partial\psi}{\partial z}-\psi% \frac{\partial\chi*}{\partial z})dS
  13. I = 4 π e - + [ f ( E f - e V + ϵ ) - f ( E f + ϵ ) ] ρ s ( E f - e V + ϵ ) ρ T ( E f + ϵ ) | M | 2 d ϵ I=\frac{4\pi e}{\hbar}\int_{-\infty}^{+\infty}[f(E_{f}-eV+\epsilon)-f(E_{f}+% \epsilon)]\rho_{s}(E_{f}-eV+\epsilon)\rho_{T}(E_{f}+\epsilon)|M|^{2}d\epsilon