wpmath0000009_0

(2,3,7)_triangle_group.html

  1. g 2 2 = g 3 3 = ( g 2 g 3 ) 7 = 1. g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=1.
  2. ( 2 - η ) 3 = 7 ( η - 1 ) 2 . (2-\eta)^{3}=7(\eta-1)^{2}.
  3. 𝒬 Hur \mathcal{Q}_{\mathrm{Hur}}
  4. Q Hur Q_{\mathrm{Hur}}
  5. g 2 = 1 η i j g_{2}=\tfrac{1}{\eta}ij
  6. g 3 = 1 2 ( 1 + ( η 2 - 2 ) j + ( 3 - η 2 ) i j ) . g_{3}=\tfrac{1}{2}(1+(\eta^{2}-2)j+(3-\eta^{2})ij).
  7. 1 , g 2 , g 3 , g 2 g 3 1,g_{2},g_{3},g_{2}g_{3}
  8. g 2 2 = g 3 3 = ( g 2 g 3 ) 7 = - 1 , g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=-1,\,
  9. tr ( γ ) = 2 cosh ( γ / 2 ) . \operatorname{tr}(\gamma)=2\cosh(\ell_{\gamma}/2).

1::2_+_1::4_+_1::8_+_1::16_+_⋯.html

  1. 1 2 + 1 4 + 1 8 + 1 16 + = n = 1 ( 1 2 ) n = 1 2 1 - 1 2 = 1. \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=\sum_{n=1}^{\infty}% \left({\frac{1}{2}}\right)^{n}=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1.
  2. 1 2 + 1 4 + 1 8 + 1 16 + \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots
  3. n n
  4. s n = 1 2 + 1 4 + 1 8 + 1 16 + + 1 2 n s_{n}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots+\frac{1}{2^{n}}
  5. n n
  6. 2 s n = 2 2 + 2 4 + 2 8 + 2 16 + + 2 2 n = 1 + 1 2 + 1 4 + 1 8 + + 1 2 n - 1 = 1 + s n - 1 2 n . 2s_{n}=\frac{2}{2}+\frac{2}{4}+\frac{2}{8}+\frac{2}{16}+\cdots+\frac{2}{2^{n}}% =1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^{n-1}}=1+s_{n}-\frac{% 1}{2^{n}}.
  7. s n = 1 - 1 2 n . s_{n}=1-\frac{1}{2^{n}}.
  8. n n
  9. s < s u b > n s<sub>n

1::2_−_1::4_+_1::8_−_1::16_+_⋯.html

  1. 1 2 - 1 4 + 1 8 - 1 16 + = 1 / 2 1 - ( - 1 / 2 ) = 1 3 . \frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)}=% \frac{1}{3}.
  2. 1 - 1 2 - 1 4 + 1 8 - 1 16 + = 1 3 . 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1}{3}.

1::4_+_1::16_+_1::64_+_1::256_+_⋯.html

  1. 1 4 1 - 1 4 = 1 3 . \frac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3}.
  2. 3 ( 1 4 + 1 4 2 + 1 4 3 + 1 4 4 + ) = 1. 3\left(\frac{1}{4}+\frac{1}{4^{2}}+\frac{1}{4^{3}}+\frac{1}{4^{4}}+\cdots% \right)=1.
  3. 3 4 + 3 4 2 + 3 4 3 + 3 4 4 + = 1. \frac{3}{4}+\frac{3}{4^{2}}+\frac{3}{4^{3}}+\frac{3}{4^{4}}+\cdots=1.
  4. A + B + C + D + + Z + 1 3 Z = 4 3 A . A+B+C+D+\cdots+Z+\frac{1}{3}Z=\frac{4}{3}A.
  5. B + C + + Z + B 3 + C 3 + + Z 3 = 4 B 3 + 4 C 3 + + 4 Z 3 = 1 3 ( A + B + + Y ) . \begin{array}[]{rcl}\displaystyle B+C+\cdots+Z+\frac{B}{3}+\frac{C}{3}+\cdots+% \frac{Z}{3}&=&\displaystyle\frac{4B}{3}+\frac{4C}{3}+\cdots+\frac{4Z}{3}\\ &=&\displaystyle\frac{1}{3}(A+B+\cdots+Y).\end{array}
  6. B 3 + C 3 + + Y 3 = 1 3 ( B + C + + Y ) . \frac{B}{3}+\frac{C}{3}+\cdots+\frac{Y}{3}=\frac{1}{3}(B+C+\cdots+Y).
  7. B + C + + Z + Z 3 = 1 3 A B+C+\cdots+Z+\frac{Z}{3}=\frac{1}{3}A
  8. 1 + 1 4 + 1 4 2 + + 1 4 n = 1 - ( 1 4 ) n + 1 1 - 1 4 . 1+\frac{1}{4}+\frac{1}{4^{2}}+\cdots+\frac{1}{4^{n}}=\frac{1-\left(\frac{1}{4}% \right)^{n+1}}{1-\frac{1}{4}}.
  9. lim n 1 - ( 1 4 ) n + 1 1 - 1 4 = 1 1 - 1 4 = 4 3 . \lim_{n\to\infty}\frac{1-\left(\frac{1}{4}\right)^{n+1}}{1-\frac{1}{4}}=\frac{% 1}{1-\frac{1}{4}}=\frac{4}{3}.
  10. 1 + 1 4 + 1 4 2 + 1 4 3 + = 4 3 . 1+\frac{1}{4}+\frac{1}{4^{2}}+\frac{1}{4^{3}}+\cdots=\frac{4}{3}.

10-Formyltetrahydrofolate.html

  1. \rightleftharpoons
  2. \rightleftharpoons

10-simplex.html

  1. ( 1 / 55 , 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , 1 / 3 , ± 1 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15% },\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm 1\right)
  2. ( 1 / 55 , 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , - 2 1 / 3 , 0 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15% },\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)
  3. ( 1 / 55 , 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , - 3 / 2 , 0 , 0 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15% },\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)
  4. ( 1 / 55 , 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , - 2 2 / 5 , 0 , 0 , 0 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15% },\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
  5. ( 1 / 55 , 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , - 5 / 3 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3% },\ 0,\ 0,\ 0,\ 0\right)
  6. ( 1 / 55 , 1 / 45 , 1 / 6 , 1 / 28 , - 12 / 7 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0% ,\ 0,\ 0\right)
  7. ( 1 / 55 , 1 / 45 , 1 / 6 , - 7 / 4 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ 1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
  8. ( 1 / 55 , 1 / 45 , - 4 / 3 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/55},\ \sqrt{1/45},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
  9. ( 1 / 55 , - 3 1 / 5 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/55},\ -3\sqrt{1/5},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
  10. ( - 20 / 11 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(-\sqrt{20/11},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

16-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}
  2. 4 + {}^{+}_{4}
  3. 4 2 {}^{2}_{4}
  4. 4 * {}^{*}_{4}
  5. 4 4 {}^{4}_{4}
  6. 4 2 {}^{2}_{4}
  7. 4 * {}^{*}_{4}
  8. F ~ 4 {\tilde{F}}_{4}
  9. B ~ 4 {\tilde{B}}_{4}
  10. D ~ 4 {\tilde{D}}_{4}

1_+_1_+_1_+_1_+_⋯.html

  1. n = 1 n 0 \sum_{n=1}^{\infty}n^{0}
  2. n = 1 1 n \sum_{n=1}^{\infty}1^{n}
  3. n = 1 1 \sum_{n=1}^{\infty}1
  4. p p
  5. p p
  6. n = 1 1 = + , \sum_{n=1}^{\infty}1=+\infty\,,
  7. s = 0 s=0
  8. ζ ( s ) = n = 1 1 n s = 1 1 - 2 1 - s n = 1 ( - 1 ) n + 1 n s , \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\frac{1}{1-2^{1-s}}\sum_{n=1}^{% \infty}\frac{(-1)^{n+1}}{n^{s}}\,,
  9. 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)\!,
  10. Γ ( 1 ) = 1 \Gamma(1)=1
  11. ζ ( 0 ) = 1 π lim s 0 sin ( π s 2 ) ζ ( 1 - s ) = 1 π lim s 0 ( π s 2 - π 3 s 3 48 + ) ( - 1 s + ) = - 1 2 \zeta(0)=\frac{1}{\pi}\lim_{s\rightarrow 0}\ \sin\left(\frac{\pi s}{2}\right)% \ \zeta(1-s)=\frac{1}{\pi}\lim_{s\rightarrow 0}\ \left(\frac{\pi s}{2}-\frac{% \pi^{3}s^{3}}{48}+...\right)\ \left(-\frac{1}{s}+...\right)=-\frac{1}{2}\!
  12. ζ ( s ) ζ(s)
  13. s = 1 s=1
  14. ζ ( s ) ζ(s)
  15. 1 + 1 + 1 + 1 + · · · = ζ ( 0 ) = < s u p > 1 2 1+1+1+1+···=ζ(0)=−<sup>1⁄_{2}

1_+_2_+_3_+_4_+_⋯.html

  1. k = 1 n k = n ( n + 1 ) 2 \sum_{k=1}^{n}k=\frac{n(n+1)}{2}
  2. 1 + 2 + 3 + 4 + = - 1 12 . 1+2+3+4+\cdots=-\frac{1}{12}.
  3. k = 1 n k = n ( n + 1 ) 2 . \sum_{k=1}^{n}k=\frac{n(n+1)}{2}.
  4. c \displaystyle c
  5. - 3 c = 1 - 2 + 3 - 4 + = 1 ( 1 + 1 ) 2 = 1 4 -3c=1-2+3-4+\cdots=\frac{1}{(1+1)^{2}}=\frac{1}{4}
  6. n = 1 n \sum_{n=1}^{\infty}n
  7. n = 1 n - s \sum_{n=1}^{\infty}n^{-s}
  8. ζ ( s ) \displaystyle\zeta(s)
  9. ( 1 - 2 1 - s ) ζ ( s ) = η ( s ) (1-2^{1-s})\zeta(s)=\eta(s)
  10. - 3 ζ ( - 1 ) = η ( - 1 ) = lim x 1 - ( 1 - 2 x + 3 x 2 - 4 x 3 + ) = lim x 1 - 1 ( 1 + x ) 2 = 1 4 -3\zeta(-1)=\eta(-1)=\lim_{x\to 1^{-}}\left(1-2x+3x^{2}-4x^{3}+\cdots\right)=% \lim_{x\to 1^{-}}\frac{1}{(1+x)^{2}}=\frac{1}{4}
  11. n = 0 N n \sum_{n=0}^{N}n
  12. n = 0 n f ( n / N ) \sum_{n=0}^{\infty}nf(n/N)
  13. k = 1 f ( k ) \sum_{k=1}^{\infty}f(k)
  14. c = - 1 2 f ( 0 ) - k = 1 B 2 k ( 2 k ) ! f ( 2 k - 1 ) ( 0 ) , c=-\frac{1}{2}f(0)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(0),
  15. c = - 1 6 1 2 ! = - 1 12 . c=-\frac{1}{6}\cdot\frac{1}{2!}=-\frac{1}{12}.
  16. D - 2 D-2
  17. D D
  18. ω \omega
  19. n n
  20. n ω / 2 n\hbar\omega/2
  21. - ω ( D - 2 ) / 24 -\hbar\omega(D-2)/24

1_+_2_+_4_+_8_+_⋯.html

  1. f ( x ) = 1 + 2 x + 4 x 2 + 8 x 3 + + 2 n x n + = 1 1 - 2 x f(x)=1+2x+4x^{2}+8x^{3}+\cdots+2^{n}{}x^{n}+\cdots=\frac{1}{1-2x}
  2. 1 + y + y 2 + y 3 + = 1 1 - y 1+y+y^{2}+y^{3}+\cdots=\frac{1}{1-y}
  3. s = 1 + 2 + 4 + 8 + = 1 + 2 ( 1 + 2 + 4 + 8 + ) = 1 + 2 s \begin{array}[]{rcl}s&=&\displaystyle 1+2+4+8+\cdots\\ &=&\displaystyle 1+2(1+2+4+8+\cdots)\\ &=&\displaystyle 1+2s\end{array}
  4. - 1 -1
  5. 1 + 2 + 4 + + 2 n - 1 1+2+4+\ldots+2^{n-1}

1_−_2_+_3_−_4_+_⋯.html

  1. n = 1 m n ( - 1 ) n - 1 . \sum_{n=1}^{m}n(-1)^{n-1}.
  2. 1 - 2 + 3 - 4 + = 1 4 . 1-2+3-4+\cdots=\frac{1}{4}.
  3. 1 / 4 {1}/{4}
  4. \mathbb{Z}
  5. 4 s = ( 1 - 2 + 3 - 4 + ) + ( 1 - 2 + 3 - 4 + ) + ( 1 - 2 + 3 - 4 + ) + ( 1 - 2 + 3 - 4 + ) = ( 1 - 2 + 3 - 4 + ) + 1 + ( - 2 + 3 - 4 + 5 + ) + 1 + ( - 2 + 3 - 4 + 5 + ) + ( 1 - 2 ) + ( 3 - 4 + 5 - 6 ) = ( 1 - 2 + 3 - 4 + ) + 1 + ( - 2 + 3 - 4 + 5 + ) + 1 + ( - 2 + 3 - 4 + 5 + ) - 1 + ( 3 - 4 + 5 - 6 ) = 1 + ( 1 - 2 + 3 - 4 + ) + ( - 2 + 3 - 4 + 5 + ) + ( - 2 + 3 - 4 + 5 + ) + ( 3 - 4 + 5 - 6 ) = 1 + [ ( 1 - 2 - 2 + 3 ) + ( - 2 + 3 + 3 - 4 ) + ( 3 - 4 - 4 + 5 ) + ( - 4 + 5 + 5 - 6 ) + ] = 1 + [ 0 + 0 + 0 + 0 + ] 4 s = 1 \begin{array}[]{rclllll}4s&=&&(1-2+3-4+\cdots)&{}+(1-2+3-4+\cdots)&{}+(1-2+3-4% +\cdots)&{}+(1-2+3-4+\cdots)\\ &=&&(1-2+3-4+\cdots)&{}+1+(-2+3-4+5+\cdots)&{}+1+(-2+3-4+5+\cdots)&{}+(1-2)+(3% -4+5-6\cdots)\\ &=&&(1-2+3-4+\cdots)&{}+1+(-2+3-4+5+\cdots)&{}+1+(-2+3-4+5+\cdots)&{}-1+(3-4+5% -6\cdots)\\ &=&1+&(1-2+3-4+\cdots)&{}+(-2+3-4+5+\cdots)&{}+(-2+3-4+5+\cdots)&{}+(3-4+5-6% \cdots)\\ &=&1+[&(1-2-2+3)&{}+(-2+3+3-4)&{}+(3-4-4+5)&{}+(-4+5+5-6)+\cdots]\\ &=&1+[&0+0+0+0+\cdots]\\ 4s&=&1\end{array}
  6. s = 1 4 s=\frac{1}{4}
  7. 1 / 4 {1}/{4}
  8. 2 s = ( 1 - 2 + 3 - 4 + ) + ( 1 - 2 + 3 - 4 + ) = 1 + ( - 2 + 3 - 4 + ) + 1 - 2 + ( 3 - 4 + 5 ) = 0 + ( - 2 + 3 ) + ( 3 - 4 ) + ( - 4 + 5 ) + 2 s = 1 - 1 + 1 - 1 \begin{array}[]{rcllll}2s&=&&(1-2+3-4+\cdots)&+&(1-2+3-4+\cdots)\\ &=&1+&(-2+3-4+\cdots)&{}+1-2&{}+(3-4+5\cdots)\\ &=&0+&(-2+3)+(3-4)+(-4+5)+\cdots\\ 2s&=&&1-1+1-1\cdots\end{array}
  9. 1 / 4 {1}/{4}
  10. c n = k = 0 n a k b n - k = k = 0 n ( - 1 ) k ( - 1 ) n - k = k = 0 n ( - 1 ) n = ( - 1 ) n ( n + 1 ) . \begin{array}[]{rcl}c_{n}&=&\displaystyle\sum_{k=0}^{n}a_{k}b_{n-k}=\sum_{k=0}% ^{n}(-1)^{k}(-1)^{n-k}\\ &=&\displaystyle\sum_{k=0}^{n}(-1)^{n}=(-1)^{n}(n+1).\end{array}
  11. n = 0 ( - 1 ) n ( n + 1 ) = 1 - 2 + 3 - 4 + . \sum_{n=0}^{\infty}(-1)^{n}(n+1)=1-2+3-4+\cdots.
  12. 1 / 4 {1}/{4}
  13. 2 / 3 {2}/{3}
  14. 3 / 5 {3}/{5}
  15. 4 / 7 {4}/{7}
  16. 1 / 2 {1}/{2}
  17. 1 / 2 {1}/{2}
  18. 1 / 4 {1}/{4}
  19. 1 / 4 {1}/{4}
  20. 1 / 4 {1}/{4}
  21. 1 / 4 {1}/{4}
  22. lim x 1 - n = 1 n ( - x ) n - 1 = lim x 1 - 1 ( 1 + x ) 2 = 1 4 . \lim_{x\rightarrow 1^{-}}\sum_{n=1}^{\infty}n(-x)^{n-1}=\lim_{x\rightarrow 1^{% -}}\frac{1}{(1+x)^{2}}=\frac{1}{4}.
  23. 1 2 a 0 - 1 4 Δ a 0 + 1 8 Δ 2 a 0 - = 1 2 - 1 4 . \frac{1}{2}a_{0}-\frac{1}{4}\Delta a_{0}+\frac{1}{8}\Delta^{2}a_{0}-\cdots=% \frac{1}{2}-\frac{1}{4}.
  24. 1 / 4 {1}/{4}
  25. k = 0 a k = k = 0 ( - 1 ) k ( k + 1 ) , \sum_{k=0}^{\infty}a_{k}=\sum_{k=0}^{\infty}(-1)^{k}(k+1),
  26. a ( x ) = k = 0 ( - 1 ) k ( k + 1 ) x k k ! = e - x ( 1 - x ) . a(x)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(k+1)x^{k}}{k!}=e^{-x}(1-x).
  27. 0 e - x a ( x ) d x = 0 e - 2 x ( 1 - x ) d x = 1 2 - 1 4 . \int_{0}^{\infty}e^{-x}a(x)\,dx=\int_{0}^{\infty}e^{-2x}(1-x)\,dx=\frac{1}{2}-% \frac{1}{4}.
  28. 1 / 4 {1}/{4}
  29. lim δ 0 m = 0 ( - 1 ) m ( m + 1 ) φ ( δ m ) = 1 4 . \lim_{\delta\rightarrow 0}\sum_{m=0}^{\infty}(-1)^{m}(m+1)\varphi(\delta m)=% \frac{1}{4}.
  30. 1 / 8 {1}/{8}
  31. 1 / 16 {1}/{16}
  32. 1 - 2 n + 3 n - = 2 n + 1 - 1 n + 1 B n + 1 1-2^{n}+3^{n}-\cdots=\frac{2^{n+1}-1}{n+1}B_{n+1}
  33. 1 - 2 2 k + 3 2 k - = 0. 1-2^{2k}+3^{2k}-\cdots=0.

1_−_2_+_4_−_8_+_⋯.html

  1. k = 0 n ( - 2 ) k \sum_{k=0}^{n}(-2)^{k}
  2. k = 0 a r k = a 1 - r . \sum_{k=0}^{\infty}ar^{k}=\frac{a}{1-r}.
  3. a 0 2 - Δ a 0 4 + Δ 2 a 0 8 - Δ 3 a 0 16 + = 1 2 - 1 4 + 1 8 - 1 16 + . \frac{a_{0}}{2}-\frac{\Delta a_{0}}{4}+\frac{\Delta^{2}a_{0}}{8}-\frac{\Delta^% {3}a_{0}}{16}+\cdots=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots.

1s_Slater-type_function.html

  1. ψ 1 s ( ζ , 𝐫 - 𝐑 ) = ( ζ 3 π ) 1 2 e - ζ | 𝐫 - 𝐑 | . \psi_{1s}(\zeta,\mathbf{r-R})=\left(\frac{\zeta^{3}}{\pi}\right)^{1\over 2}\,e% ^{-\zeta|\mathbf{r-R}|}.
  2. ζ \zeta
  3. e ( 𝐙 - 1 ) e(\mathbf{Z}-1)
  4. 𝐙 \mathbf{Z}
  5. 𝐇 ^ e = - 2 2 - 𝐙 r \mathbf{\hat{H}}_{e}=-\frac{\nabla^{2}}{2}-\frac{\mathbf{Z}}{r}
  6. 𝐙 \mathbf{Z}
  7. ψ 1 s = ( ζ 3 π ) 0.50 e - ζ r \mathbf{\psi}_{1s}=\left(\frac{\zeta^{3}}{\pi}\right)^{0.50}e^{-\zeta r}
  8. ζ \mathbf{\zeta}
  9. 𝐄 1 s = < ψ 1 s | 𝐇 ^ e | ψ 1 s > < ψ 1 s | ψ 1 s > \mathbf{E}_{1s}=\frac{<\psi_{1s}|\mathbf{\hat{H}}_{e}|\psi_{1s}>}{<\psi_{1s}|% \psi_{1s}>}
  10. < ψ 𝟏 𝐬 | ψ 𝟏 𝐬 1 \mathbf{<\psi_{1s}|\psi_{1s}>}=1
  11. 𝐄 1 s = < ψ 1 s | - 2 2 - 𝐙 r | ψ 1 s Align g t ; \mathbf{E}_{1s}=<\psi_{1s}|\mathbf{-}\frac{\nabla^{2}}{2}-\frac{\mathbf{Z}}{r}% |\psi_{1s}&gt;
  12. 𝐄 1 s = < ψ 1 s | - 2 2 | ψ 1 s > + < ψ 1 s | - 𝐙 r | ψ 1 s Align g t ; \mathbf{E}_{1s}=<\psi_{1s}|\mathbf{-}\frac{\nabla^{2}}{2}|\psi_{1s}>+<\psi_{1s% }|-\frac{\mathbf{Z}}{r}|\psi_{1s}&gt;
  13. 𝐄 1 s = < ψ 1 s | - 1 2 r 2 r ( r 2 r ) | ψ 1 s > + < ψ 1 s | - 𝐙 r | ψ 1 s > \mathbf{E}_{1s}=<\psi_{1s}|\mathbf{-}\frac{1}{2r^{2}}\frac{\partial}{\partial r% }\left(r^{2}\frac{\partial}{\partial r}\right)|\psi_{1s}>+<\psi_{1s}|-\frac{% \mathbf{Z}}{r}|\psi_{1s}>
  14. ψ 1 s = ( ζ 3 π ) 0.50 e - ζ r \mathbf{\psi}_{1s}=\left(\frac{\zeta^{3}}{\pi}\right)^{0.50}e^{-\zeta r}
  15. 𝐄 1 s = < ( ζ 3 π ) 0.50 e - ζ r | - ( ζ 3 π ) 0.50 e - ζ r [ - 2 r ζ + r 2 ζ 2 2 r 2 ] > + < ψ 1 s | - 𝐙 r | ψ 1 s > \mathbf{E}_{1s}=<\left(\frac{\zeta^{3}}{\pi}\right)^{0.50}e^{-\zeta r}|-\left(% \frac{\zeta^{3}}{\pi}\right)^{0.50}e^{-\zeta r}\left[\frac{-2r\zeta+r^{2}\zeta% ^{2}}{2r^{2}}\right]>+<\psi_{1s}|-\frac{\mathbf{Z}}{r}|\psi_{1s}>
  16. 𝐄 1 s = ζ 2 2 - ζ 𝐙 . \mathbf{E}_{1s}=\frac{\zeta^{2}}{2}-\zeta\mathbf{Z}.
  17. ζ \mathbf{\zeta}
  18. ζ \mathbf{\zeta}
  19. d 𝐄 1 s d ζ = ζ - 𝐙 = 0 \frac{d\mathbf{E}_{1s}}{d\zeta}=\zeta-\mathbf{Z}=0
  20. ζ = 𝐙 . \mathbf{\zeta}=\mathbf{Z}.
  21. 𝐙 = 1 \mathbf{Z}=1
  22. ζ = 1 \mathbf{\zeta}=1
  23. 𝐄 1 s = \mathbf{E}_{1s}=
  24. 𝐄 1 s = \mathbf{E}_{1s}=
  25. 𝐄 1 s = \mathbf{E}_{1s}=
  26. 𝐙 = 79 \mathbf{Z}=79
  27. ζ = 79 \mathbf{\zeta}=79
  28. 𝐄 1 s = \mathbf{E}_{1s}=
  29. 𝐄 1 s = \mathbf{E}_{1s}=
  30. 𝐄 1 s = \mathbf{E}_{1s}=
  31. ζ \mathbf{\zeta}
  32. ζ r e l \mathbf{\zeta}_{rel}
  33. ζ r e l = 𝐙 1 - 𝐙 2 / c 2 \mathbf{\zeta}_{rel}=\frac{\mathbf{Z}}{\sqrt{1-\mathbf{Z}^{2}/c^{2}}}
  34. 𝐄 1 s r e l = - ( c 2 + 𝐙 ζ ) + c 4 + 𝐙 2 ζ 2 \mathbf{E}_{1s}^{rel}=-(c^{2}+\mathbf{Z}\zeta)+\sqrt{c^{4}+\mathbf{Z}^{2}\zeta% ^{2}}
  35. 𝐙 \mathbf{Z}
  36. ζ n o n r e l \mathbf{\zeta}_{nonrel}
  37. ζ r e l \mathbf{\zeta}_{rel}
  38. 𝐄 1 s n o n r e l \mathbf{E}_{1s}^{nonrel}
  39. 𝐄 1 s r e l \mathbf{E}_{1s}^{rel}
  40. ζ n o n r e l \mathbf{\zeta}_{nonrel}
  41. 𝐄 1 s r e l \mathbf{E}_{1s}^{rel}
  42. ζ r e l \mathbf{\zeta}_{rel}

2-EXPTIME.html

  1. 2-EXPTIME = k DTIME ( 2 2 n k ) . \mbox{2-EXPTIME}~{}=\bigcup_{k\in\mathbb{N}}\mbox{ DTIME }~{}\left(2^{2^{n^{k}% }}\right).
  2. \subseteq
  3. \subseteq
  4. \subseteq
  5. \subseteq
  6. \subseteq
  7. \subseteq
  8. \subseteq
  9. \subseteq
  10. 2 2 2 n k 2^{2^{2^{n^{k}}}}

2002_(band).html

  1. \infty

24-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}
  2. C ~ 4 {\tilde{C}}_{4}
  3. B ~ 4 {\tilde{B}}_{4}
  4. D ~ 4 {\tilde{D}}_{4}
  5. π 2 16 0.61685. \frac{\pi^{2}}{16}\cong 0.61685.
  6. { ( x i ) 4 : i x i 0 ( mod 2 ) } . \left\{(x_{i})\in\mathbb{Z}^{4}:{\textstyle\sum_{i}}x_{i}\equiv 0\;(\mbox{mod % }~{}2)\right\}.
  7. F ~ 4 {\tilde{F}}_{4}
  8. { 3 , 4 , 3 , 3 } \begin{Bmatrix}3,4,3,3\end{Bmatrix}
  9. { 3 3 , 4 , 3 } \left\{\begin{array}[]{l}3\\ 3,4,3\end{array}\right\}
  10. C ~ 4 {\tilde{C}}_{4}
  11. { 3 , 4 3 , 4 } \left\{\begin{array}[]{l}3,4\\ 3,4\end{array}\right\}
  12. B ~ 4 {\tilde{B}}_{4}
  13. { 3 3 3 , 4 } \left\{\begin{array}[]{l}3\\ 3\\ 3,4\end{array}\right\}
  14. D ~ 4 {\tilde{D}}_{4}
  15. { 3 3 3 3 } \left\{\begin{array}[]{l}3\\ 3\\ 3\\ 3\end{array}\right\}

25-Hydroxyvitamin_D3_1-alpha-hydroxylase.html

  1. \rightleftharpoons

2_×_2_real_matrices.html

  1. q = ( a b c d ) , q=\begin{pmatrix}a&b\\ c&d\end{pmatrix},
  2. q * = ( d - b - c a ) . \quad q^{*}=\begin{pmatrix}d&-b\\ -c&a\end{pmatrix}.
  3. q - 1 = q * / ( a d - b c ) . q^{-1}=q^{*}\,/\,(ad-bc).
  4. ( x y ) ( a b c d ) ( x y ) = ( a x + b y c x + d y ) . \begin{pmatrix}x\\ y\end{pmatrix}\mapsto\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}ax+by\\ cx+dy\end{pmatrix}.
  5. ( a a + b c a b + b d a c + c d b c + d d ) . \begin{pmatrix}aa+bc&ab+bd\\ ac+cd&bc+dd\end{pmatrix}.
  6. ( d u d v ) = ( p r q s ) ( d x d y ) = ( p d x + r d y q d x + s d y ) . \begin{pmatrix}du\\ dv\end{pmatrix}=\begin{pmatrix}p&r\\ q&s\end{pmatrix}\begin{pmatrix}dx\\ dy\end{pmatrix}=\begin{pmatrix}p\,dx+r\,dy\\ q\,dx+s\,dy\end{pmatrix}.
  7. d x d y dx\wedge dy
  8. d u d v = 0 + p s d x d y + q r d y d x + 0 = ( p s - q r ) d x d y = ( det g ) d x d y . \begin{aligned}\displaystyle du\wedge dv&\displaystyle{}=0+ps\ dx\wedge dy+qr% \ dy\wedge dx+0\\ &\displaystyle{}=(ps-qr)\ dx\wedge dy=(\det g)\ dx\wedge dy.\end{aligned}
  9. ρ exp ( a m ) = ρ exp ( a m / 2 ) \sqrt{\rho\ \exp(am)}=\sqrt{\rho}\ \exp(am/2)
  10. z = ( a b c d ) . z=\begin{pmatrix}a&b\\ c&d\end{pmatrix}.
  11. z = x I + n , x = a + d 2 , n = z - x I . z=xI+n,\quad x=\frac{a+d}{2},\quad n=z-xI.
  12. n 2 = p I n^{2}=pI
  13. p = ( a - d ) 2 4 + b c p=\frac{(a-d)^{2}}{4}+bc
  14. q = 1 / - p , m = q n q=1/\sqrt{-p},\quad m=qn
  15. m 2 = - I , z = x I + m - p m^{2}=-I,\quad z=xI+m\sqrt{-p}
  16. z = x I + n z=xI+n
  17. q = 1 / p , m = q n q=1/\sqrt{p},\quad m=qn
  18. m 2 = + I , z = x I + m p m^{2}=+I,\quad z=xI+m\sqrt{p}

3-j_symbol.html

  1. ( j 1 j 2 j 3 m 1 m 2 m 3 ) ( - 1 ) j 1 - j 2 - m 3 2 j 3 + 1 j 1 m 1 j 2 m 2 | j 3 ( - m 3 ) . \begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\equiv\frac{(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt{2j_{% 3}+1}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,(-m_{3})\rangle.
  2. j 1 m 1 j 2 m 2 | j 3 m 3 = ( - 1 ) j 1 - j 2 + m 3 2 j 3 + 1 ( j 1 j 2 j 3 m 1 m 2 - m 3 ) \langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle=(-1)^{j_{1}-j_{2}+m_{3}% }\sqrt{2j_{3}+1}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix}
  3. ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( j 2 j 3 j 1 m 2 m 3 m 1 ) = ( j 3 j 1 j 2 m 3 m 1 m 2 ) . \begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}j_{2}&j_{3}&j_{1}\\ m_{2}&m_{3}&m_{1}\end{pmatrix}=\begin{pmatrix}j_{3}&j_{1}&j_{2}\\ m_{3}&m_{1}&m_{2}\end{pmatrix}.
  4. ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( - 1 ) j 1 + j 2 + j 3 ( j 2 j 1 j 3 m 2 m 1 m 3 ) = ( - 1 ) j 1 + j 2 + j 3 ( j 1 j 3 j 2 m 1 m 3 m 2 ) . \begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{2}&j_% {1}&j_{3}\\ m_{2}&m_{1}&m_{3}\end{pmatrix}=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{1}&j_% {3}&j_{2}\\ m_{1}&m_{3}&m_{2}\end{pmatrix}.
  5. m m
  6. ( j 1 j 2 j 3 - m 1 - m 2 - m 3 ) = ( - 1 ) j 1 + j 2 + j 3 ( j 1 j 2 j 3 m 1 m 2 m 3 ) . \begin{pmatrix}j_{1}&j_{2}&j_{3}\\ -m_{1}&-m_{2}&-m_{3}\end{pmatrix}=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{1}% &j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.
  7. ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( j 1 j 2 + j 3 - m 1 2 j 2 + j 3 + m 1 2 j 3 - j 2 j 2 - j 3 - m 1 2 - m 3 j 2 - j 3 + m 1 2 + m 3 ) . \begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}j_{1}&\frac{j_{2}+j_{3}-m_{1}}{2% }&\frac{j_{2}+j_{3}+m_{1}}{2}\\ j_{3}-j_{2}&\frac{j_{2}-j_{3}-m_{1}}{2}-m_{3}&\frac{j_{2}-j_{3}+m_{1}}{2}+m_{3% }\end{pmatrix}.
  8. ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( - 1 ) j 1 + j 2 + j 3 ( j 2 + j 3 + m 1 2 j 1 + j 3 + m 2 2 j 1 + j 2 + m 3 2 j 1 - j 2 + j 3 - m 1 2 j 2 - j 1 + j 3 - m 2 2 j 3 - j 1 + j 2 - m 3 2 ) . \begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}\frac{j_% {2}+j_{3}+m_{1}}{2}&\frac{j_{1}+j_{3}+m_{2}}{2}&\frac{j_{1}+j_{2}+m_{3}}{2}\\ j_{1}-\frac{j_{2}+j_{3}-m_{1}}{2}&j_{2}-\frac{j_{1}+j_{3}-m_{2}}{2}&j_{3}-% \frac{j_{1}+j_{2}-m_{3}}{2}\end{pmatrix}.
  9. R = - j 1 + j 2 + j 3 j 1 - j 2 + j 3 j 1 + j 2 - j 3 j 1 - m 1 j 2 - m 2 j 3 - m 3 j 1 + m 1 j 2 + m 2 j 3 + m 3 R=\begin{array}[]{|ccc|}\hline-j_{1}+j_{2}+j_{3}&j_{1}-j_{2}+j_{3}&j_{1}+j_{2}% -j_{3}\\ j_{1}-m_{1}&j_{2}-m_{2}&j_{3}-m_{3}\\ j_{1}+m_{1}&j_{2}+m_{2}&j_{3}+m_{3}\\ \hline\end{array}
  10. m 1 + m 2 + m 3 = 0 m_{1}+m_{2}+m_{3}=0\,
  11. j 1 + j 2 + j 3 is an integer (or an even integer if m 1 = m 2 = m 3 = 0 ) j_{1}+j_{2}+j_{3}\,\text{ is an integer}\,\,\text{(or an even integer if}\,m_{% 1}=m_{2}=m_{3}=0)\,
  12. | m i | j i |m_{i}|\leq j_{i}\,
  13. | j 1 - j 2 | j 3 j 1 + j 2 . |j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2}.\,
  14. m 1 = - j 1 j 1 m 2 = - j 2 j 2 m 3 = - j 3 j 3 | j 1 m 1 | j 2 m 2 | j 3 m 3 ( j 1 j 2 j 3 m 1 m 2 m 3 ) , \sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}\sum_{m_{3}=-j_{3}}^{j_{% 3}}|j_{1}m_{1}\rangle|j_{2}m_{2}\rangle|j_{3}m_{3}\rangle\begin{pmatrix}j_{1}&% j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},
  15. ( 2 j + 1 ) m 1 m 2 ( j 1 j 2 j m 1 m 2 m ) ( j 1 j 2 j m 1 m 2 m ) = δ j j δ m m . (2j+1)\sum_{m_{1}m_{2}}\begin{pmatrix}j_{1}&j_{2}&j\\ m_{1}&m_{2}&m\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j^{\prime}\\ m_{1}&m_{2}&m^{\prime}\end{pmatrix}=\delta_{jj^{\prime}}\delta_{mm^{\prime}}.
  16. j m ( 2 j + 1 ) ( j 1 j 2 j m 1 m 2 m ) ( j 1 j 2 j m 1 m 2 m ) = δ m 1 m 1 δ m 2 m 2 . \sum_{jm}(2j+1)\begin{pmatrix}j_{1}&j_{2}&j\\ m_{1}&m_{2}&m\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j\\ m_{1}^{\prime}&m_{2}^{\prime}&m\end{pmatrix}=\delta_{m_{1}m_{1}^{\prime}}% \delta_{m_{2}m_{2}^{\prime}}.
  17. Y l 1 m 1 ( θ , φ ) Y l 2 m 2 ( θ , φ ) Y l 3 m 3 ( θ , φ ) sin θ d θ d φ = ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ( l 1 l 2 l 3 0 0 0 ) ( l 1 l 2 l 3 m 1 m 2 m 3 ) \begin{aligned}&\displaystyle{}\quad\int Y_{l_{1}m_{1}}(\theta,\varphi)Y_{l_{2% }m_{2}}(\theta,\varphi)Y_{l_{3}m_{3}}(\theta,\varphi)\,\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\varphi\\ &\displaystyle=\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi}}\begin{% pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\end{aligned}
  18. l 1 l_{1}
  19. l 2 l_{2}
  20. l 3 l_{3}
  21. d 𝐧 ^ Y j 1 m 1 s 1 ( 𝐧 ^ ) Y j 2 m 2 s 2 ( 𝐧 ^ ) Y j 3 m 3 s 3 ( 𝐧 ^ ) \displaystyle{}\quad\int d{\mathbf{\hat{n}}}\,{}_{s_{1}}Y_{j_{1}m_{1}}({% \mathbf{\hat{n}}})\,{}_{s_{2}}Y_{j_{2}m_{2}}({\mathbf{\hat{n}}})\,{}_{s_{3}}Y_% {j_{3}m_{3}}({\mathbf{\hat{n}}})
  22. - ( l 3 s 3 ) ( l 3 ± s 3 + 1 ) ( l 1 l 2 l 3 s 1 s 2 s 3 ± 1 ) = ( l 1 s 1 ) ( l 1 ± s 1 + 1 ) ( l 1 l 2 l 3 s 1 ± 1 s 2 s 3 ) + ( l 2 s 2 ) ( l 2 ± s 2 + 1 ) ( l 1 l 2 l 3 s 1 s 2 ± 1 s 3 ) \begin{aligned}&\displaystyle{}\quad-\sqrt{(l_{3}\mp s_{3})(l_{3}\pm s_{3}+1)}% \begin{pmatrix}l_{1}&l_{2}&l_{3}\\ s_{1}&s_{2}&s_{3}\pm 1\end{pmatrix}\\ &\displaystyle=\sqrt{(l_{1}\mp s_{1})(l_{1}\pm s_{1}+1)}\begin{pmatrix}l_{1}&l% _{2}&l_{3}\\ s_{1}\pm 1&s_{2}&s_{3}\end{pmatrix}+\sqrt{(l_{2}\mp s_{2})(l_{2}\pm s_{2}+1)}% \begin{pmatrix}l_{1}&l_{2}&l_{3}\\ s_{1}&s_{2}\pm 1&s_{3}\end{pmatrix}\end{aligned}
  23. l 1 l 2 , l 3 l_{1}\ll l_{2},l_{3}
  24. ( l 1 l 2 l 3 m 1 m 2 m 3 ) ( - 1 ) l 3 + m 3 d m 1 , l 3 - l 2 l 1 ( θ ) 2 l 3 + 1 \begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\approx(-1)^{l_{3}+m_{3}}\frac{d^{l_{1}}_{m_{1},% l_{3}-l_{2}}(\theta)}{\sqrt{2l_{3}+1}}
  25. cos ( θ ) = - 2 m 3 / ( 2 l 3 + 1 ) \cos(\theta)=-2m_{3}/(2l_{3}+1)
  26. d m n l d^{l}_{mn}
  27. ( l 1 l 2 l 3 m 1 m 2 m 3 ) ( - 1 ) l 3 + m 3 d m 1 , l 3 - l 2 l 1 ( θ ) l 2 + l 3 + 1 \begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\approx(-1)^{l_{3}+m_{3}}\frac{d^{l_{1}}_{m_{1},% l_{3}-l_{2}}(\theta)}{\sqrt{l_{2}+l_{3}+1}}
  28. cos ( θ ) = ( m 2 - m 3 ) / ( l 2 + l 3 + 1 ) \cos(\theta)=(m_{2}-m_{3})/(l_{2}+l_{3}+1)
  29. m ( - 1 ) j - m ( j j J m - m 0 ) = 2 j + 1 δ J 0 \sum_{m}(-1)^{j-m}\begin{pmatrix}j&j&J\\ m&-m&0\end{pmatrix}=\sqrt{2j+1}~{}\delta_{J0}
  30. 1 2 - 1 1 P l 1 ( x ) P l 2 ( x ) P l ( x ) d x = ( l l 1 l 2 0 0 0 ) 2 \frac{1}{2}\int_{-1}^{1}P_{l_{1}}(x)P_{l_{2}}(x)P_{l}(x)\,dx=\begin{pmatrix}l&% l_{1}&l_{2}\\ 0&0&0\end{pmatrix}^{2}

331_model.html

  1. S U ( 2 ) W × U ( 1 ) Y SU(2)_{W}\times U(1)_{Y}
  2. S U ( 3 ) L × U ( 1 ) X SU(3)_{L}\times U(1)_{X}
  3. S U ( 2 ) W S U ( 3 ) W SU(2)_{W}\subset SU(3)_{W}
  4. Y = β T 8 + I X Y=\beta T_{8}+IX
  5. Q = Y / 2 + T 3 / 2 Q=Y/2+T_{3}/2
  6. S U ( 3 ) C × S U ( 3 ) L × U ( 1 ) X SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}

495_(number).html

  1. ( 12 4 ) {\textstyle\left({{12}\atop{4}}\right)}

5-cubic_honeycomb.html

  1. C ~ 5 {\tilde{C}}_{5}
  2. C ~ 5 {\tilde{C}}_{5}
  3. C ~ 5 {\tilde{C}}_{5}
  4. B ~ 5 {\tilde{B}}_{5}
  5. D ~ 5 {\tilde{D}}_{5}

5-demicubic_honeycomb.html

  1. B ~ 5 {\tilde{B}}_{5}
  2. D ~ 5 {\tilde{D}}_{5}
  3. 5 + {}^{+}_{5}
  4. 5 2 {}^{2}_{5}
  5. 5 * {}^{*}_{5}
  6. 5 4 {}^{4}_{5}
  7. 5 2 {}^{2}_{5}
  8. 5 * {}^{*}_{5}
  9. B ~ 5 {\tilde{B}}_{5}
  10. D ~ 5 {\tilde{D}}_{5}
  11. C ~ 5 {\tilde{C}}_{5}

5-simplex.html

  1. ( 1 / 15 , 1 / 10 , 1 / 6 , 1 / 3 , ± 1 ) \left(\sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm 1\right)
  2. ( 1 / 15 , 1 / 10 , 1 / 6 , - 2 1 / 3 , 0 ) \left(\sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)
  3. ( 1 / 15 , 1 / 10 , - 3 / 2 , 0 , 0 ) \left(\sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)
  4. ( 1 / 15 , - 2 2 / 5 , 0 , 0 , 0 ) \left(\sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
  5. ( - 5 / 3 , 0 , 0 , 0 , 0 ) \left(-\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)

6-cubic_honeycomb.html

  1. C ~ 6 {\tilde{C}}_{6}
  2. B ~ 6 {\tilde{B}}_{6}
  3. C ~ 6 {\tilde{C}}_{6}
  4. C ~ 6 {\tilde{C}}_{6}
  5. B ~ 6 {\tilde{B}}_{6}
  6. D ~ 6 {\tilde{D}}_{6}

6-demicubic_honeycomb.html

  1. B ~ 6 {\tilde{B}}_{6}
  2. D ~ 6 {\tilde{D}}_{6}
  3. 6 + {}^{+}_{6}
  4. 6 2 {}^{2}_{6}
  5. 6 * {}^{*}_{6}
  6. 6 4 {}^{4}_{6}
  7. 6 2 {}^{2}_{6}
  8. B ~ 6 {\tilde{B}}_{6}
  9. D ~ 6 {\tilde{D}}_{6}
  10. C ~ 6 {\tilde{C}}_{6}

6-j_symbol.html

  1. { j 1 j 2 j 3 j 4 j 5 j 6 } = m i ( - 1 ) S ( j 1 j 2 j 3 m 1 m 2 - m 3 ) ( j 1 j 5 j 6 - m 1 m 5 m 6 ) ( j 4 j 5 j 3 m 4 - m 5 m 3 ) ( j 4 j 2 j 6 - m 4 - m 2 - m 6 ) . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\end{Bmatrix}=\sum_{m_{i}}(-1)^{S}\begin{pmatrix}j_{1}&j_{2}&% j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{5}&j_{6}\\ -m_{1}&m_{5}&m_{6}\end{pmatrix}\begin{pmatrix}j_{4}&j_{5}&j_{3}\\ m_{4}&-m_{5}&m_{3}\end{pmatrix}\begin{pmatrix}j_{4}&j_{2}&j_{6}\\ -m_{4}&-m_{2}&-m_{6}\end{pmatrix}.
  2. S = k = 1 6 ( j k - m k ) S=\sum_{k=1}^{6}(j_{k}-m_{k})
  3. { j 1 j 2 j 3 j 4 j 5 j 6 } = ( - 1 ) j 1 + j 2 + j 4 + j 5 W ( j 1 j 2 j 5 j 4 ; j 3 j 6 ) . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{4}+j_{5}}W(j_{1}j_{2}j_{5}% j_{4};j_{3}j_{6}).
  4. { j 1 j 2 j 3 j 4 j 5 j 6 } = { j 2 j 1 j 3 j 5 j 4 j 6 } = { j 1 j 3 j 2 j 4 j 6 j 5 } = { j 3 j 2 j 1 j 6 j 5 j 4 } . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\end{Bmatrix}=\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\ j_{5}&j_{4}&j_{6}\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{3}&j_{2}\\ j_{4}&j_{6}&j_{5}\end{Bmatrix}=\begin{Bmatrix}j_{3}&j_{2}&j_{1}\\ j_{6}&j_{5}&j_{4}\end{Bmatrix}.
  5. { j 1 j 2 j 3 j 4 j 5 j 6 } = { j 4 j 5 j 3 j 1 j 2 j 6 } = { j 1 j 5 j 6 j 4 j 2 j 3 } = { j 4 j 2 j 6 j 1 j 5 j 3 } . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\end{Bmatrix}=\begin{Bmatrix}j_{4}&j_{5}&j_{3}\\ j_{1}&j_{2}&j_{6}\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{5}&j_{6}\\ j_{4}&j_{2}&j_{3}\end{Bmatrix}=\begin{Bmatrix}j_{4}&j_{2}&j_{6}\\ j_{1}&j_{5}&j_{3}\end{Bmatrix}.
  6. { j 1 j 2 j 3 j 4 j 5 j 6 } \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\end{Bmatrix}
  7. j 1 = | j 2 - j 3 | , , j 2 + j 3 j_{1}=|j_{2}-j_{3}|,\ldots,j_{2}+j_{3}
  8. { j 1 j 2 j 3 j 4 j 5 0 } = δ j 2 , j 4 δ j 1 , j 5 ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( - 1 ) j 1 + j 2 + j 3 { j 1 , j 2 , j 3 } . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&0\end{Bmatrix}=\frac{\delta_{j_{2},j_{4}}\delta_{j_{1},j_{5}}}{% \sqrt{(2j_{1}+1)(2j_{2}+1)}}(-1)^{j_{1}+j_{2}+j_{3}}\{j_{1},j_{2},j_{3}\}.
  9. j 3 ( 2 j 3 + 1 ) { j 1 j 2 j 3 j 4 j 5 j 6 } { j 1 j 2 j 3 j 4 j 5 j 6 } = δ j 6 j 6 2 j 6 + 1 { j 1 , j 5 , j 6 } { j 4 , j 2 , j 6 } . \sum_{j_{3}}(2j_{3}+1)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}^{\prime}\end{Bmatrix}=\frac{\delta_{j_{6}j_{6}^{\prime}}}{2j% _{6}+1}\{j_{1},j_{5},j_{6}\}\{j_{4},j_{2},j_{6}\}.
  10. { j 1 j 2 j 3 j 4 j 5 j 6 } 1 12 π | V | cos ( i = 1 6 J i θ i + π 4 ) . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\end{Bmatrix}\sim\frac{1}{\sqrt{12\pi|V|}}\cos{\left(\sum_{i=% 1}^{6}J_{i}\theta_{i}+\frac{\pi}{4}\right)}.
  11. ( V i V j ) V k V i ( V j V k ) (V_{i}\otimes V_{j})\otimes V_{k}\to V_{i}\otimes(V_{j}\otimes V_{k})
  12. H i , j = Hom ( V , V i V j ) H_{i,j}^{\ell}=\operatorname{Hom}(V_{\ell},V_{i}\otimes V_{j})
  13. V i V j = H i , j V V_{i}\otimes V_{j}=\bigoplus_{\ell}H_{i,j}^{\ell}V_{\ell}
  14. ( V i V j ) V k , m H i , j H , k m V m while V i ( V j V k ) m , n H i , n m H j , k n V m (V_{i}\otimes V_{j})\otimes V_{k}\cong\bigoplus_{\ell,m}H_{i,j}^{\ell}\otimes H% _{\ell,k}^{m}\otimes V_{m}\qquad\,\text{while}\qquad V_{i}\otimes(V_{j}\otimes V% _{k})\cong\bigoplus_{m,n}H_{i,n}^{m}\otimes H_{j,k}^{n}\otimes V_{m}
  15. Φ i , j k , m : H i , j H , k m n H i , n m H j , k n \Phi_{i,j}^{k,m}:\bigoplus_{\ell}H_{i,j}^{\ell}\otimes H_{\ell,k}^{m}\to% \bigoplus_{n}H_{i,n}^{m}\otimes H_{j,k}^{n}
  16. { i j k m n } = ( Φ i , j k , m ) , n \begin{Bmatrix}i&j&\ell\\ k&m&n\end{Bmatrix}=(\Phi_{i,j}^{k,m})_{\ell,n}

6-simplex.html

  1. ( 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , 1 / 3 , ± 1 ) \left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm 1\right)
  2. ( 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , - 2 1 / 3 , 0 ) \left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)
  3. ( 1 / 21 , 1 / 15 , 1 / 10 , - 3 / 2 , 0 , 0 ) \left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)
  4. ( 1 / 21 , 1 / 15 , - 2 2 / 5 , 0 , 0 , 0 ) \left(\sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
  5. ( 1 / 21 , - 5 / 3 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)
  6. ( - 12 / 7 , 0 , 0 , 0 , 0 , 0 ) \left(-\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)

6-sphere_coordinates.html

  1. u = x x 2 + y 2 + z 2 , v = y x 2 + y 2 + z 2 , w = z x 2 + y 2 + z 2 . u=\frac{x}{x^{2}+y^{2}+z^{2}},\quad v=\frac{y}{x^{2}+y^{2}+z^{2}},\quad w=% \frac{z}{x^{2}+y^{2}+z^{2}}.
  2. x = u u 2 + v 2 + w 2 , y = v u 2 + v 2 + w 2 , z = w u 2 + v 2 + w 2 . x=\frac{u}{u^{2}+v^{2}+w^{2}},\quad y=\frac{v}{u^{2}+v^{2}+w^{2}},\quad z=% \frac{w}{u^{2}+v^{2}+w^{2}}.
  3. R R

65536_(number).html

  1. 2 16 2^{16}
  2. 2 2 2 2 2^{2^{2^{2}}}
  3. 2 4 {}^{4}2
  4. 2 16 2\uparrow 16
  5. 2 2 2 2 2\uparrow 2\uparrow 2\uparrow 2
  6. 2 4 2\uparrow\uparrow 4
  7. 2 3 2\uparrow\uparrow\uparrow 3

68–95–99.7_rule.html

  1. Pr ( μ - σ x μ + σ ) \displaystyle\Pr(\mu-\;\,\sigma\leq x\leq\mu+\;\,\sigma)
  2. Pr ( μ - 2 σ x μ + 2 σ ) = Φ ( 2 ) - Φ ( - 2 ) 0.9772 - ( 1 - 0.9772 ) 0.9545 \Pr(\mu-2\sigma\leq x\leq\mu+2\sigma)=\Phi(2)-\Phi(-2)\approx 0.9772-(1-0.9772% )\approx 0.9545
  3. erf ( x 2 ) \textstyle\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)
  4. 1 1 - erf ( x 2 ) \textstyle\frac{1}{1-\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)}
  5. 1 1 - erf ( x 2 ) \textstyle\frac{1}{1-\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)}

7-simplex.html

  1. ( 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , 1 / 3 , ± 1 ) \left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ % \sqrt{1/3},\ \pm 1\right)
  2. ( 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , - 2 1 / 3 , 0 ) \left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2% \sqrt{1/3},\ 0\right)
  3. ( 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , - 3 / 2 , 0 , 0 ) \left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,% \ 0\right)
  4. ( 1 / 28 , 1 / 21 , 1 / 15 , - 2 2 / 5 , 0 , 0 , 0 ) \left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
  5. ( 1 / 28 , 1 / 21 , - 5 / 3 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)
  6. ( 1 / 28 , - 12 / 7 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)
  7. ( - 7 / 4 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(-\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

8-simplex.html

  1. ( 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , 1 / 3 , ± 1 ) \left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6}% ,\ \sqrt{1/3},\ \pm 1\right)
  2. ( 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , - 2 1 / 3 , 0 ) \left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6}% ,\ -2\sqrt{1/3},\ 0\right)
  3. ( 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , - 3 / 2 , 0 , 0 ) \left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2% },\ 0,\ 0\right)
  4. ( 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , - 2 2 / 5 , 0 , 0 , 0 ) \left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
  5. ( 1 / 6 , 1 / 28 , 1 / 21 , - 5 / 3 , 0 , 0 , 0 , 0 ) \left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)
  6. ( 1 / 6 , 1 / 28 , - 12 / 7 , 0 , 0 , 0 , 0 , 0 ) \left(1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)
  7. ( 1 / 6 , - 7 / 4 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
  8. ( - 4 / 3 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

9-j_symbol.html

  1. [ ( 2 j 3 + 1 ) ( 2 j 6 + 1 ) ( 2 j 7 + 1 ) ( 2 j 8 + 1 ) ] 1 2 { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 | ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 . [(2j_{3}+1)(2j_{6}+1)(2j_{7}+1)(2j_{8}+1)]^{\frac{1}{2}}\begin{Bmatrix}j_{1}&j% _{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\\ j_{7}&j_{8}&j_{9}\end{Bmatrix}=\langle((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{% 9}|((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}\rangle.
  2. 𝐣 1 \mathbf{j}_{1}
  3. 𝐣 2 \mathbf{j}_{2}
  4. 𝐉 2 \mathbf{J}^{2}
  5. J z J_{z}
  6. 𝐉 = 𝐣 1 + 𝐣 2 \mathbf{J}=\mathbf{j}_{1}+\mathbf{j}_{2}
  7. 𝐣 1 \mathbf{j}_{1}
  8. 𝐣 2 \mathbf{j}_{2}
  9. 𝐣 4 \mathbf{j}_{4}
  10. 𝐣 5 \mathbf{j}_{5}
  11. | ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 m 9 . |((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle.
  12. 𝐣 1 \mathbf{j}_{1}
  13. 𝐣 4 \mathbf{j}_{4}
  14. 𝐣 7 \mathbf{j}_{7}
  15. 𝐣 2 \mathbf{j}_{2}
  16. 𝐣 5 \mathbf{j}_{5}
  17. 𝐣 8 \mathbf{j}_{8}
  18. 𝐣 7 \mathbf{j}_{7}
  19. 𝐣 8 \mathbf{j}_{8}
  20. 𝐣 9 \mathbf{j}_{9}
  21. | ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 m 9 . |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle.
  22. ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( 2 j 4 + 1 ) ( 2 j 5 + 1 ) (2j_{1}+1)(2j_{2}+1)(2j_{4}+1)(2j_{5}+1)
  23. | j 1 m 1 | j 2 m 2 | j 4 m 4 | j 5 m 5 , m 1 = - j 1 , , j 1 ; m 2 = - j 2 , , j 2 ; m 4 = - j 4 , , j 4 ; m 5 = - j 5 , , j 5 . |j_{1}m_{1}\rangle|j_{2}m_{2}\rangle|j_{4}m_{4}\rangle|j_{5}m_{5}\rangle,\;\;m% _{1}=-j_{1},\ldots,j_{1};\;\;m_{2}=-j_{2},\ldots,j_{2};\;\;m_{4}=-j_{4},\ldots% ,j_{4};\;\;m_{5}=-j_{5},\ldots,j_{5}.
  24. m 9 m_{9}
  25. | ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 m 9 = j 3 j 6 | ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 m 9 ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 | ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 . |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle=\sum_{j_{3}}\sum_{j6}|% ((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle\langle((j_{1}j_{2})j_{3% },(j_{4}j_{5})j_{6})j_{9}|((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}\rangle.
  26. 9 - j 9-j
  27. { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = { j 1 j 4 j 7 j 2 j 5 j 8 j 3 j 6 j 9 } = { j 9 j 6 j 3 j 8 j 5 j 2 j 7 j 4 j 1 } . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\\ j_{7}&j_{8}&j_{9}\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{4}&j_{7}\\ j_{2}&j_{5}&j_{8}\\ j_{3}&j_{6}&j_{9}\end{Bmatrix}=\begin{Bmatrix}j_{9}&j_{6}&j_{3}\\ j_{8}&j_{5}&j_{2}\\ j_{7}&j_{4}&j_{1}\end{Bmatrix}.
  28. ( - 1 ) S (-1)^{S}
  29. S = i = 1 9 j i . S=\sum_{i=1}^{9}j_{i}.
  30. { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = ( - 1 ) S { j 4 j 5 j 6 j 1 j 2 j 3 j 7 j 8 j 9 } = ( - 1 ) S { j 2 j 1 j 3 j 5 j 4 j 6 j 8 j 7 j 9 } . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\\ j_{7}&j_{8}&j_{9}\end{Bmatrix}=(-1)^{S}\begin{Bmatrix}j_{4}&j_{5}&j_{6}\\ j_{1}&j_{2}&j_{3}\\ j_{7}&j_{8}&j_{9}\end{Bmatrix}=(-1)^{S}\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\ j_{5}&j_{4}&j_{6}\\ j_{8}&j_{7}&j_{9}\end{Bmatrix}.
  31. { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = x ( - 1 ) 2 x ( 2 x + 1 ) { j 1 j 4 j 7 j 8 j 9 x } { j 2 j 5 j 8 j 4 x j 6 } { j 3 j 6 j 9 x j 1 j 2 } \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\\ j_{7}&j_{8}&j_{9}\end{Bmatrix}=\sum_{x}(-1)^{2x}(2x+1)\begin{Bmatrix}j_{1}&j_{% 4}&j_{7}\\ j_{8}&j_{9}&x\end{Bmatrix}\begin{Bmatrix}j_{2}&j_{5}&j_{8}\\ j_{4}&x&j_{6}\end{Bmatrix}\begin{Bmatrix}j_{3}&j_{6}&j_{9}\\ x&j_{1}&j_{2}\end{Bmatrix}
  32. j 9 = 0 j_{9}=0
  33. { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 0 } = δ j 3 , j 6 δ j 7 , j 8 ( 2 j 3 + 1 ) ( 2 j 7 + 1 ) ( - 1 ) j 2 + j 3 + j 4 + j 7 { j 1 j 2 j 3 j 5 j 4 j 7 } . \begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\\ j_{7}&j_{8}&0\end{Bmatrix}=\frac{\delta_{j_{3},j_{6}}\delta_{j_{7},j_{8}}}{% \sqrt{(2j_{3}+1)(2j_{7}+1)}}(-1)^{j_{2}+j_{3}+j_{4}+j_{7}}\begin{Bmatrix}j_{1}% &j_{2}&j_{3}\\ j_{5}&j_{4}&j_{7}\end{Bmatrix}.
  34. j 7 j 8 ( 2 j 7 + 1 ) ( 2 j 8 + 1 ) { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = δ j 3 j 3 δ j 6 j 6 { j 1 j 2 j 3 } { j 4 j 5 j 6 } { j 3 j 6 j 9 } ( 2 j 3 + 1 ) ( 2 j 6 + 1 ) . \sum_{j_{7}j_{8}}(2j_{7}+1)(2j_{8}+1)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{4}&j_{5}&j_{6}\\ j_{7}&j_{8}&j_{9}\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}^{\prime}\\ j_{4}&j_{5}&j_{6}^{\prime}\\ j_{7}&j_{8}&j_{9}\end{Bmatrix}=\frac{\delta_{j_{3}j_{3}^{\prime}}\delta_{j_{6}% j_{6}^{\prime}}\{j_{1}j_{2}j_{3}\}\{j_{4}j_{5}j_{6}\}\{j_{3}j_{6}j_{9}\}}{(2j_% {3}+1)(2j_{6}+1)}.
  35. { j 1 j 2 j 3 } \{j_{1}j_{2}j_{3}\}
  36. ( j 1 j 2 j 3 ) (j_{1}j_{2}j_{3})
  37. n = 2 n=2
  38. n n
  39. m m
  40. 3 n 3n
  41. 3 n 3n
  42. 2 n 2n

9-simplex.html

  1. ( 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , 1 / 3 , ± 1 ) \left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10% },\ \sqrt{1/6},\ \sqrt{1/3},\ \pm 1\right)
  2. ( 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , 1 / 6 , - 2 1 / 3 , 0 ) \left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10% },\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)
  3. ( 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , 1 / 10 , - 3 / 2 , 0 , 0 ) \left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10% },\ -\sqrt{3/2},\ 0,\ 0\right)
  4. ( 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , 1 / 15 , - 2 2 / 5 , 0 , 0 , 0 ) \left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/% 5},\ 0,\ 0,\ 0\right)
  5. ( 1 / 45 , 1 / 6 , 1 / 28 , 1 / 21 , - 5 / 3 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,% \ 0\right)
  6. ( 1 / 45 , 1 / 6 , 1 / 28 , - 12 / 7 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)
  7. ( 1 / 45 , 1 / 6 , - 7 / 4 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/45},\ 1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
  8. ( 1 / 45 , - 4 / 3 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(\sqrt{1/45},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
  9. ( - 3 1 / 5 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) \left(-3\sqrt{1/5},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

A-weighting.html

  1. R X ( f ) R_{X}(f)
  2. R A ( f ) = 12200 2 f 4 ( f 2 + 20.6 2 ) ( f 2 + 107.7 2 ) ( f 2 + 737.9 2 ) ( f 2 + 12200 2 ) , R_{A}(f)={12200^{2}\cdot f^{4}\over(f^{2}+20.6^{2})\quad\sqrt{(f^{2}+107.7^{2}% )\,(f^{2}+737.9^{2})}\quad(f^{2}+12200^{2})}\ ,
  3. A ( f ) = 2.0 + 20 log 10 ( R A ( f ) ) A(f)=2.0+20\log_{10}\left(R_{A}(f)\right)
  4. R B ( f ) = 12200 2 f 3 ( f 2 + 20.6 2 ) ( f 2 + 158.5 2 ) ( f 2 + 12200 2 ) , R_{B}(f)={12200^{2}\cdot f^{3}\over(f^{2}+20.6^{2})\quad\sqrt{(f^{2}+158.5^{2}% )}\quad(f^{2}+12200^{2})}\ ,
  5. B ( f ) = 0.17 + 20 log 10 ( R B ( f ) ) B(f)=0.17+20\log_{10}\left(R_{B}(f)\right)
  6. R C ( f ) = 12200 2 f 2 ( f 2 + 20.6 2 ) ( f 2 + 12200 2 ) , R_{C}(f)={12200^{2}\cdot f^{2}\over(f^{2}+20.6^{2})\quad(f^{2}+12200^{2})}\ ,
  7. C ( f ) = 0.06 + 20 log 10 ( R C ( f ) ) C(f)=0.06+20\log_{10}\left(R_{C}(f)\right)
  8. - 20 log 10 ( R A ( 1000 ) ) -20\log_{10}\left(R_{A}(1000)\right)
  9. - 20 log 10 ( R B ( 1000 ) ) -20\log_{10}\left(R_{B}(1000)\right)
  10. - 20 log 10 ( R C ( 1000 ) ) -20\log_{10}\left(R_{C}(1000)\right)
  11. R D ( f ) = f 6.8966888496476 10 - 5 h ( f ) ( f 2 + 79919.29 ) ( f 2 + 1345600 ) R_{D}(f)=\frac{f}{6.8966888496476\cdot 10^{-5}}\cdot\sqrt{\frac{h(f)}{(f^{2}+7% 9919.29)\,(f^{2}+1345600)}}
  12. D ( f ) = 20 log 10 ( R D ( f ) ) , D(f)=20\log_{10}\left(R_{D}(f)\right),
  13. h ( f ) = ( 1037918.48 - f 2 ) 2 + 1080768.16 f 2 ( 9837328 - f 2 ) 2 + 11723776 f 2 . h(f)=\frac{(1037918.48-f^{2})^{2}+1080768.16\,f^{2}}{(9837328-f^{2})^{2}+11723% 776\,f^{2}}\ .
  14. H A ( s ) = k A s 4 ( s + 129.4 ) 2 ( s + 676.7 ) ( s + 4636 ) ( s + 76655 ) 2 H_{A}(s)={k_{A}\cdot s^{4}\over(s+129.4)^{2}\quad(s+676.7)\quad(s+4636)\quad(s% +76655)^{2}}
  15. H B ( s ) = k B s 3 ( s + 129.4 ) 2 ( s + 995.9 ) ( s + 76655 ) 2 H_{B}(s)={k_{B}\cdot s^{3}\over(s+129.4)^{2}\quad(s+995.9)\quad(s+76655)^{2}}
  16. H C ( s ) = k C s 2 ( s + 129.4 ) 2 ( s + 76655 ) 2 H_{C}(s)={k_{C}\cdot s^{2}\over(s+129.4)^{2}\quad(s+76655)^{2}}
  17. H D ( s ) = k D s ( s 2 + 6532 s + 4.0975 × 10 7 ) ( s + 1776.3 ) ( s + 7288.5 ) ( s 2 + 21514 s + 3.8836 × 10 8 ) H_{D}(s)={k_{D}\cdot s\cdot(s^{2}+6532s+4.0975\times 10^{7})\over(s+1776.3)% \quad(s+7288.5)\quad(s^{2}+21514s+3.8836\times 10^{8})}

AB_magnitude.html

  1. m AB = - 5 2 log 10 ( f ν 3631 Jy ) , m\text{AB}=-\frac{5}{2}\log_{10}\left(\frac{f_{\nu}}{3631\,\text{ Jy}}\right),
  2. m AB = - 5 2 log 10 ( f ν Jy ) + 8.90. m\text{AB}=-\frac{5}{2}\log_{10}\left(\frac{f_{\nu}}{\,\text{Jy}}\right)+8.90.
  3. m AB = - 5 2 log 10 f ν - 48.600. m\text{AB}=-\frac{5}{2}\log_{10}f_{\nu}-48.600.
  4. m AB = - 5 2 log 10 ( f ν ( h ν ) - 1 e ( ν ) d ν 3631 Jy ( h ν ) - 1 e ( ν ) d ν ) , m\text{AB}=-\frac{5}{2}\log_{10}\left(\frac{\int f_{\nu}(h\nu)^{-1}e(\nu)d\nu}% {\int 3631\,\text{ Jy}(h\nu)^{-1}e(\nu)d\nu}\right),
  5. f ν = λ 2 c f λ , f_{\nu}=\frac{\lambda^{2}}{c}f_{\lambda},
  6. f ν Jy = 3.34 × 10 4 ( λ Å ) 2 f λ erg cm - 2 s - 1 Å - 1 \frac{f_{\nu}}{\,\text{Jy}}=3.34\times 10^{4}\left(\frac{\lambda}{\AA}\right)^% {2}\frac{f_{\lambda}}{\,\text{erg}\,\text{ cm}^{-2}\,\text{ s}^{-1}\AA^{-1}}
  7. λ piv = e ( λ ) d λ e ( λ ) λ - 2 d λ \lambda\text{piv}=\sqrt{\frac{\int e(\lambda)d\lambda}{\int e(\lambda)\lambda^% {-2}d\lambda}}
  8. λ piv = e ( λ ) λ d λ e ( λ ) λ - 1 d λ \lambda\text{piv}=\sqrt{\frac{\int e(\lambda)\lambda d\lambda}{\int e(\lambda)% \lambda^{-1}d\lambda}}

Abel's_binomial_theorem.html

  1. k = 0 m ( m k ) ( w + m - k ) m - k - 1 ( z + k ) k = w - 1 ( z + w + m ) m . \sum_{k=0}^{m}{\left({{m}\atop{k}}\right)}(w+m-k)^{m-k-1}(z+k)^{k}=w^{-1}(z+w+% m)^{m}.
  2. ( 2 0 ) ( w + 2 ) 1 ( z + 0 ) 0 + ( 2 1 ) ( w + 1 ) 0 ( z + 1 ) 1 + ( 2 2 ) ( w + 0 ) - 1 ( z + 2 ) 2 \displaystyle{}\quad{\left({{2}\atop{0}}\right)}(w+2)^{1}(z+0)^{0}+{\left({{2}% \atop{1}}\right)}(w+1)^{0}(z+1)^{1}+{\left({{2}\atop{2}}\right)}(w+0)^{-1}(z+2% )^{2}

Abel's_inequality.html

  1. | k = 1 n a k b k | max k = 1 , , n | B k | ( | a n | + a n - a 1 ) , \left|\sum_{k=1}^{n}a_{k}b_{k}\right|\leq\operatorname{max}_{k=1,\dots,n}|B_{k% }|(|a_{n}|+a_{n}-a_{1}),
  2. | k = 1 n a k b k | max k = 1 , , n | B k | ( | a n | - a n + a 1 ) , \left|\sum_{k=1}^{n}a_{k}b_{k}\right|\leq\operatorname{max}_{k=1,\dots,n}|B_{k% }|(|a_{n}|-a_{n}+a_{1}),
  3. B k = b 1 + + b k . B_{k}=b_{1}+\cdots+b_{k}.
  4. | k = 1 n a k b k | max k = 1 , , n | B k | a 1 , \left|\sum_{k=1}^{n}a_{k}b_{k}\right|\leq\operatorname{max}_{k=1,\dots,n}|B_{k% }|a_{1},
  5. k = 1 n a k b k = a n B n - k = 1 n - 1 B k ( a k + 1 - a k ) . \sum_{k=1}^{n}a_{k}b_{k}=a_{n}B_{n}-\sum_{k=1}^{n-1}B_{k}(a_{k+1}-a_{k}).

Abel's_summation_formula.html

  1. a n a_{n}\,
  2. ϕ ( x ) \phi(x)\,
  3. 𝒞 1 \mathcal{C}^{1}\,
  4. 1 n x a n ϕ ( n ) = A ( x ) ϕ ( x ) - 1 x A ( u ) ϕ ( u ) d u \sum_{1\leq n\leq x}a_{n}\phi(n)=A(x)\phi(x)-\int_{1}^{x}A(u)\phi^{\prime}(u)% \,\mathrm{d}u\,
  5. A ( x ) := 1 n x a n . A(x):=\sum_{1\leq n\leq x}a_{n}\,.
  6. x < n y a n ϕ ( n ) = A ( y ) ϕ ( y ) - A ( x ) ϕ ( x ) - x y A ( u ) ϕ ( u ) d u . \sum_{x<n\leq y}a_{n}\phi(n)=A(y)\phi(y)-A(x)\phi(x)-\int_{x}^{y}A(u)\phi^{% \prime}(u)\,\mathrm{d}u\,.
  7. a n = 1 a_{n}=1\,
  8. ϕ ( x ) = 1 x , \phi(x)=\frac{1}{x}\,,
  9. A ( x ) = x A(x)=\lfloor x\rfloor\,
  10. 1 x 1 n = x x + 1 x u u 2 d u \sum_{1}^{x}\frac{1}{n}=\frac{\lfloor x\rfloor}{x}+\int_{1}^{x}\frac{\lfloor u% \rfloor}{u^{2}}\,\mathrm{d}u
  11. a n = 1 a_{n}=1\,
  12. ϕ ( x ) = 1 x s , \phi(x)=\frac{1}{x^{s}}\,,
  13. A ( x ) = x A(x)=\lfloor x\rfloor\,
  14. 1 1 n s = s 1 u u 1 + s d u . \sum_{1}^{\infty}\frac{1}{n^{s}}=s\int_{1}^{\infty}\frac{\lfloor u\rfloor}{u^{% 1+s}}\mathrm{d}u\,.
  15. ( s ) > 1 . \Re(s)>1\,.
  16. ζ ( s ) \zeta(s)\,
  17. a n = μ ( n ) a_{n}=\mu(n)\,
  18. ϕ ( x ) = 1 x s , \phi(x)=\frac{1}{x^{s}}\,,
  19. A ( x ) = M ( x ) = n x μ ( n ) A(x)=M(x)=\sum_{n\leq x}\mu(n)\,
  20. 1 μ ( n ) n s = s 1 M ( u ) u 1 + s d u . \sum_{1}^{\infty}\frac{\mu(n)}{n^{s}}=s\int_{1}^{\infty}\frac{M(u)}{u^{1+s}}% \mathrm{d}u\,.
  21. ( s ) > 1 . \Re(s)>1\,.

Abel–Jacobi_map.html

  1. H 1 ( C , ) 2 g . H_{1}(C,\mathbb{Z})\cong\mathbb{Z}^{2g}.
  2. γ 1 , , γ 2 g \gamma_{1},\dots,\gamma_{2g}
  3. H 0 ( C , K ) g , H^{0}(C,K)\cong\mathbb{C}^{g},
  4. ω 1 , , ω g \omega_{1},\dots,\omega_{g}
  5. Ω j = ( γ j ω 1 , , γ j ω g ) g . \Omega_{j}=\left(\int_{\gamma_{j}}\omega_{1},\dots,\int_{\gamma_{j}}\omega_{g}% \right)\in\mathbb{C}^{g}.
  6. Ω j \Omega_{j}
  7. Λ \Lambda
  8. g 2 g \mathbb{C}^{g}\cong\mathbb{R}^{2g}
  9. J ( C ) = g / Λ . J(C)=\mathbb{C}^{g}/\Lambda.
  10. p 0 C p_{0}\in C
  11. Λ \Lambda
  12. u : C J ( C ) , u ( p ) = ( p 0 p ω 1 , , p 0 p ω g ) mod Λ . u\colon C\to J(C),u(p)=\left(\int_{p_{0}}^{p}\omega_{1},\dots,\int_{p_{0}}^{p}% \omega_{g}\right)\bmod\Lambda.
  13. p 0 p_{0}
  14. p , p,
  15. C C
  16. H 1 ( C , ) , H_{1}(C,\mathbb{Z}),
  17. Λ . \Lambda.
  18. Λ \Lambda
  19. p 0 p_{0}
  20. M M
  21. π = π 1 ( M ) \pi=\pi_{1}(M)
  22. f : π π a b f:\pi\to\pi^{ab}
  23. t o r = t o r ( π a b ) tor=tor(\pi^{ab})
  24. π a b \pi^{ab}
  25. g : π a b π a b / t o r g:\pi^{ab}\to\pi^{ab}/tor
  26. M M
  27. π a b / t o r \pi^{ab}/tor
  28. 2 g \mathbb{Z}^{2g}
  29. g g
  30. π a b / t o r \pi^{ab}/tor
  31. b \mathbb{Z}^{b}
  32. b b
  33. ϕ = g f : π b \phi=g\circ f:\pi\to\mathbb{Z}^{b}
  34. M ¯ \bar{M}
  35. M M
  36. Ker ( ϕ ) π \mathrm{Ker}(\phi)\subset\pi
  37. E E
  38. 1 1
  39. M M
  40. E * E^{*}
  41. H 1 ( M , ) H_{1}(M,\mathbb{R})
  42. 1 1
  43. x 0 M x_{0}\in M
  44. / = S 1 \mathbb{R}/\mathbb{Z}=S^{1}
  45. M H 1 ( M , ) / H 1 ( M , ) M\to H_{1}(M,\mathbb{R})/H_{1}(M,\mathbb{Z})_{\mathbb{R}}
  46. x x
  47. M ~ \tilde{M}
  48. M M
  49. x x
  50. M M
  51. c c
  52. x 0 x_{0}
  53. c c
  54. h c h h\to\int_{c}h
  55. E E
  56. M ~ E * = H 1 ( M , ) \tilde{M}\to E^{*}=H_{1}(M,\mathbb{R})
  57. A ¯ M : M ¯ E * , c ( h c h ) , \overline{A}_{M}:\overline{M}\to E^{*},\;\;c\mapsto\left(h\mapsto\int_{c}h% \right),
  58. M ¯ \overline{M}
  59. M M
  60. J 1 ( M ) = H 1 ( M , ) / H 1 ( M , ) . J_{1}(M)=H_{1}(M,\mathbb{R})/H_{1}(M,\mathbb{Z})_{\mathbb{R}}.
  61. A M : M J 1 ( M ) , A_{M}:M\to J_{1}(M),
  62. D = i n i p i D=\sum_{i}n_{i}p_{i}
  63. u ( D ) = i n i u ( p i ) u(D)=\sum_{i}n_{i}u(p_{i})
  64. n i n_{i}
  65. u ( D ) = u ( E ) u(D)=u(E)
  66. D D
  67. E . E.

Abhyankar–Moh_theorem.html

  1. L L
  2. 2 \mathbb{C}^{2}
  3. L L
  4. 2 \mathbb{C}^{2}

Absolute_radio-frequency_channel_number.html

  1. ARFCN = f - f b - f o f c \mathrm{ARFCN}=\frac{f-f_{b}-f_{o}}{f_{c}}
  2. ARFCN = f - 300 - 0.0125 0.025 \mathrm{ARFCN}=\frac{f-300-0.0125}{0.025}
  3. f = f c ARFCN + f b + f o f=f_{c}\cdot\mathrm{ARFCN}+f_{b}+f_{o}

Absoluteness.html

  1. Π 2 1 \Pi^{1}_{2}
  2. Σ 2 1 \Sigma^{1}_{2}
  3. Σ 3 1 \Sigma^{1}_{3}
  4. Π 3 1 \Pi^{1}_{3}
  5. Π 2 1 \Pi^{1}_{2}
  6. Σ 3 1 \Sigma^{1}_{3}
  7. Σ 3 1 \Sigma^{1}_{3}
  8. Σ 3 1 \Sigma^{1}_{3}
  9. Π 2 1 \Pi^{1}_{2}

Absorption_cross_section.html

  1. d N d x = - N n σ \frac{dN}{dx}=-Nn\sigma
  2. σ = ( μ / ρ ) m a / N A \sigma=(\mu/\rho)m_{a}/N_{A}
  3. μ / ρ \mu/\rho
  4. m a m_{a}
  5. N A N_{A}
  6. σ = α / N \sigma=\alpha/N
  7. α \alpha
  8. N N

Abstract_family_of_languages.html

  1. L L
  2. Σ \Sigma
  3. L Σ * L\subseteq\Sigma^{*}
  4. ( Σ , Λ ) (\Sigma,\Lambda)
  5. Σ \Sigma
  6. Λ \Lambda
  7. L L
  8. Λ \Lambda
  9. Σ 1 \Sigma_{1}
  10. Σ \Sigma
  11. L L
  12. Σ 1 * \Sigma_{1}^{*}
  13. L L
  14. L L
  15. Λ \Lambda

Abstraction_model_checking.html

  1. η \eta
  2. θ \theta
  3. θ \theta
  4. η \eta
  5. \supset

Academic_grading_in_Singapore.html

  1. 1 + 4 + 6 3 \textstyle\frac{1+4+6}{3}
  2. 11 3 \textstyle\frac{11}{3}

Acid_neutralizing_capacity.html

  1. [ A l 3 + ] = k G [ H + ] 3 [Al^{3+}]=k_{G}[H^{+}]^{3}
  2. [ H + ] + 2 [ C a 2 + ] + [ N a + ] + 3 [ A l 3 + ] + 2 [ A l ( O H ) 2 + ] + [ A l ( O H ) 2 + ] = [ O H - ] + [ C l - ] + 2 [ C O 3 2 - ] + [ H C O 3 - ] + [ R - ] [H^{+}]+2[Ca^{2+}]+[Na^{+}]+3[Al^{3+}]+2[Al(OH)^{2+}]+[Al(OH)_{2}^{+}]=[OH^{-}% ]+[Cl^{-}]+2[CO_{3}^{2-}]+[HCO_{3}^{-}]+[R^{-}]
  3. A N C = + 2 [ C a 2 + ] + [ N a + ] - [ C l - ] ANC=+2[Ca^{2+}]+[Na^{+}]-[Cl^{-}]
  4. A N C = [ O H - ] + 2 [ C O 3 2 - ] + [ H C O 3 - ] + [ R - ] - [ H + ] - 3 [ A l 3 + ] - 2 [ A l ( O H ) 2 + ] - [ A l ( O H ) 2 + ] ANC=[OH^{-}]+2[CO_{3}^{2-}]+[HCO_{3}^{-}]+[R^{-}]-[H^{+}]-3[Al^{3+}]-2[Al(OH)^% {2+}]-[Al(OH)_{2}^{+}]

Acidity_function.html

  1. H 0 = p K a + log c B c BH + H_{0}={\rm p}K_{\rm a}+\log{{c_{\rm B}}\over{c_{\rm BH^{+}}}}
  2. H - = p K a + log c B - c BH H_{-}={\rm p}K_{\rm a}+\log{{c_{\rm B^{-}}}\over{c_{\rm BH}}}

Acid–base_homeostasis.html

  1. H 2 O + CO 2 H 2 CO 3 H + + HCO 3 - \rm H_{2}O+CO_{2}\leftrightarrow H_{2}CO_{3}\leftrightarrow H^{+}+HCO_{3}^{-}

Ackermann_set_theory.html

  1. L A L_{A}
  2. \in
  3. V V
  4. M M
  5. x y x\in y
  6. ( x , y ) \in(x,y)
  7. x y x\in y
  8. x x
  9. y y
  10. V V
  11. L A L_{A}
  12. x y ( z ( z x z y ) x = y ) . \forall x\forall y(\forall z(z\in x\leftrightarrow z\in y)\rightarrow x=y).
  13. F ( y , z 1 , , z n ) F(y,z_{1},\dots,z_{n})
  14. x x
  15. y ( F ( y , z 1 , , z n ) y V ) x y ( y x F ( y , z 1 , , z n ) ) \forall y(F(y,z_{1},\dots,z_{n})\rightarrow y\in V)\rightarrow\exists x\forall y% (y\in x\leftrightarrow F(y,z_{1},\dots,z_{n}))
  16. F ( y , z 1 , , z n ) F(y,z_{1},\dots,z_{n})
  17. V V
  18. x x
  19. z 1 , , z n V z_{1},\dots,z_{n}\in V
  20. y ( F ( y , z 1 , , z n ) y V ) x ( x V y ( y x F ( y , z 1 , , z n ) ) ) . \forall y(F(y,z_{1},\dots,z_{n})\rightarrow y\in V)\rightarrow\exists x(x\in V% \land\forall y(y\in x\leftrightarrow F(y,z_{1},\dots,z_{n}))).
  21. V V
  22. x y y V x V x\in y\land y\in V\rightarrow x\in V
  23. x y y V x V . x\subseteq y\land y\in V\rightarrow x\in V.
  24. x V y ( y x ) y ( y x ¬ z ( z y z x ) ) . x\in V\land\exists y(y\in x)\rightarrow\exists y(y\in x\land\lnot\exists z(z% \in y\land z\in x)).
  25. F F
  26. L = { } L_{\in}=\{\in\}
  27. F F
  28. V V
  29. F F
  30. F V F^{V}
  31. F F
  32. x G ( x , y 1 , y n ) \forall xG(x,y_{1}\dots,y_{n})
  33. x ( x V G ( x , y 1 , y n ) ) \forall x(x\in V\rightarrow G(x,y_{1}\dots,y_{n}))
  34. x G ( x , y 1 , y n ) \exists xG(x,y_{1}\dots,y_{n})
  35. x ( x V G ( x , y 1 , y n ) ) \exists x(x\in V\land G(x,y_{1}\dots,y_{n}))
  36. F F
  37. L L_{\in}
  38. F V F^{V}
  39. F F
  40. F F
  41. L L_{\in}
  42. F F
  43. F V F^{V}

Acoustic_approximation.html

  1. M = 1 M=1

Acoustic_source_localization.html

  1. R x 1 , x 2 ( τ ) = n = - x 1 ( n ) x 2 ( n + τ ) R_{x_{1},x_{2}}(\tau)=\sum_{n=-\infty}^{\infty}x_{1}(n)\ x_{2}(n+\tau)
  2. x 1 x_{1}
  3. x 2 x_{2}
  4. τ \tau
  5. τ true = d spacing c \tau_{\mathrm{true}}=\frac{d_{\mathrm{spacing}}}{c}
  6. c c
  7. Δ t = x sin θ c \Delta t=\frac{x\sin{\theta}}{c}
  8. Δ t \Delta t
  9. x x
  10. θ \theta

Acousto-optic_deflector.html

  1. Δ θ d \Delta\theta_{d}
  2. Δ f \Delta f
  3. ( 12 ) Δ θ d = λ ν Δ f (12)\ \Delta\theta_{d}=\frac{\lambda}{\nu}\Delta f
  4. λ \lambda
  5. ν \nu

Acousto-optics.html

  1. ε \varepsilon
  2. a a
  3. B i B_{i}
  4. a j a_{j}
  5. ( 1 ) Δ B i = p i j a j , (1)\ \Delta B_{i}=p_{ij}a_{j},\,
  6. p i j p_{ij}
  7. i i
  8. j j
  9. a j a_{j}
  10. ( 2 ) n ( z , t ) = n 0 + Δ n cos ( Ω t - K z ) , (2)\ n(z,t)=n_{0}+\Delta n\cos(\Omega t-Kz),\,
  11. n 0 n_{0}
  12. Ω \Omega
  13. K K
  14. Δ n \Delta n
  15. ( 3 ) Δ n = - 1 2 j n 0 3 p z j a j , (3)\ \Delta n=-\frac{1}{2}\sum_{j}n_{0}^{3}p_{zj}a_{j},
  16. θ n \theta_{n}
  17. ( 4 ) Λ sin ( θ m ) = m λ , (4)\ \Lambda\sin(\theta_{m})=m\lambda,\,
  18. λ \lambda
  19. Λ \Lambda
  20. m m
  21. θ 0 \theta_{0}
  22. θ B \theta_{B}
  23. ( 5 ) sin θ B = - λ f 2 n i ν [ 1 + ν 2 λ 2 f 2 ( n i 2 - n d 2 ) ] , (5)\ \sin\theta_{B}=-\frac{\lambda f}{2n_{i}\nu}\left[1+\frac{\nu^{2}}{\lambda% ^{2}f^{2}}\left(n_{i}^{2}-n_{d}^{2}\right)\right],
  24. λ \lambda
  25. f f
  26. v v
  27. n i n_{i}
  28. n d n_{d}
  29. ( 6 ) Q = 2 π λ f 2 n ν 2 , (6)\ Q=\frac{2\pi\lambda\ell f^{2}}{n\nu^{2}},
  30. α \alpha
  31. γ \gamma
  32. φ \varphi
  33. α \alpha_{\ell}
  34. β \beta
  35. \ell
  36. φ \varphi
  37. f i f_{i}
  38. ( 7 ) n φ = n 0 n e n 0 2 cos φ + n e 2 sin 2 φ (7)\ n_{\varphi}=\frac{n_{0}n_{e}}{\sqrt{n_{0}^{2}\cos\varphi+n_{e}^{2}\sin^{2% }\varphi}}
  39. ( 8 ) f i ( φ ) = ν λ [ n φ cos ( φ + α ) ± n 0 2 - n φ 2 ( φ ) sin 2 ( φ + α ) ] (8)\ f_{i}(\varphi)=\frac{\nu}{\lambda}\left[n_{\varphi}\cos(\varphi+\alpha)% \pm\sqrt{n_{0}^{2}-n_{\varphi}^{2}(\varphi)\sin^{2}(\varphi+\alpha)}\right]
  40. n 0 n_{0}
  41. n e n_{e}
  42. v v
  43. ( 9 ) ν ( α ) = ν 110 cos 2 α + ( ν 001 ν 110 ) 2 sin 2 α (9)\ \nu(\alpha)=\nu_{110}\sqrt{\cos^{2}\alpha+\left(\frac{\nu_{001}}{\nu_{110% }}\right)^{2}\sin^{2}\alpha}
  44. v 110 v_{110}
  45. v 001 v_{001}
  46. α 1 \alpha_{1}
  47. φ \varphi
  48. α \alpha
  49. ( 10 ) α = φ + α (10)\ \alpha_{\ell}=\varphi+\alpha
  50. β \beta
  51. ( 11 ) β = arcsin ( λ f 0 n 0 ν sin α + φ ) (11)\ \beta=\arcsin\left(\frac{\lambda f_{0}}{n_{0}\nu}\sin\alpha+\varphi\right)
  52. Δ θ d \Delta\theta_{d}
  53. Δ f \Delta f
  54. ( 12 ) Δ θ d = λ ν Δ f (12)\ \Delta\theta_{d}=\frac{\lambda}{\nu}\Delta f
  55. λ \lambda
  56. ν \nu

Action_learning.html

  1. L = P + Q L=P+Q

Active_contour_model.html

  1. 𝐯 i \mathbf{v}_{i}
  2. i = 0 n - 1 i=0\ldots n-1
  3. E i n t e r n a l E_{internal}
  4. E e x t e r n a l E_{external}
  5. E i m a g e E_{image}
  6. E c o n E_{con}
  7. E s n a k e * = 0 1 E s n a k e ( 𝐯 ( s ) ) d s = 0 1 ( E i n t e r n a l ( 𝐯 ( s ) ) + E i m a g e ( 𝐯 ( s ) ) + E c o n ( 𝐯 ( s ) ) ) d s E_{snake}^{*}=\int\limits_{0}^{1}E_{snake}(\mathbf{v}(s))\,ds=\int\limits_{0}^% {1}(E_{internal}(\mathbf{v}(s))+E_{image}(\mathbf{v}(s))+E_{con}(\mathbf{v}(s)% ))\,ds
  8. E c o n t E_{cont}
  9. E c u r v E_{curv}
  10. E i n t e r n a l = E c o n t + E c u r v E_{internal}=E_{cont}+E_{curv}
  11. E i n t e r n a l = 1 2 ( α ( s ) | 𝐯 s ( s ) | 2 ) + 1 2 ( β ( s ) | 𝐯 s s ( s ) | 2 ) = 1 2 ( α ( s ) d v ¯ d s ( s ) 2 + β ( s ) d 2 v ¯ d s 2 ( s ) 2 ) E_{internal}=\frac{1}{2}(\alpha\,\!(s)\left|\mathbf{v}_{s}(s)\right|^{2})+% \frac{1}{2}(\beta\,\!(s)\left|\mathbf{v}_{ss}(s)\right|^{2})=\frac{1}{2}\bigg(% \alpha\,\!(s)\left\|\frac{d\bar{v}}{ds}(s)\right\|^{2}+\beta\,\!(s)\left\|% \frac{d^{2}\bar{v}}{ds^{2}}(s)\right\|^{2}\bigg)
  12. α ( s ) \alpha(s)
  13. β ( s ) \beta(s)
  14. α ( s ) \alpha(s)
  15. β ( s ) \beta(s)
  16. I ( x , y ) I(x,y)
  17. E i m a g e = w l i n e E l i n e + w e d g e E e d g e + w t e r m E t e r m E_{image}=w_{line}E_{line}+w_{edge}E_{edge}+w_{term}E_{term}
  18. w l i n e w_{line}
  19. w e d g e w_{edge}
  20. w t e r m w_{term}
  21. E l i n e = I ( x , y ) E_{line}=I(x,y)
  22. w l i n e w_{line}
  23. E l i n e = f i l t e r ( I ( x , y ) ) E_{line}=filter(I(x,y))
  24. E e d g e = - | I ( x , y ) | 2 E_{edge}=-\left|\nabla I(x,y)\right|^{2}
  25. E e d g e = - | G σ * 2 I | 2 E_{edge}=-\left|G_{\sigma}*\nabla^{2}I\right|^{2}
  26. G σ G_{\sigma}
  27. σ \sigma
  28. G σ 2 I G_{\sigma}\nabla^{2}I
  29. C ( x , y ) C(x,y)
  30. C ( x , y ) = G σ * I ( x , y ) C(x,y)=G_{\sigma}*I(x,y)
  31. θ = arctan ( C y C x ) \theta=\arctan\Bigg(\frac{C_{y}}{C_{x}}\Bigg)
  32. 𝐧 = ( cos θ , sin θ ) \mathbf{n}=(\cos\theta,\sin\theta)
  33. 𝐧 = ( - sin θ , cos θ ) \mathbf{n}_{\perp}=(-\sin\theta,\cos\theta)
  34. E t e r m = θ n = 2 C / 2 n C / n = C y y C x 2 - 2 C x y C x C y + C x x C y 2 ( 1 + C x 2 + C y 2 ) 3 / 2 E_{term}={\partial\theta\over\partial n_{\perp}}={\partial^{2}C/\partial^{2}n_% {\perp}\over\partial C/\partial n}={{C_{yy}C_{x}^{2}-2C_{xy}C_{x}C_{y}+C_{xx}C% _{y}^{2}}\over(1+C_{x}^{2}+C_{y}^{2})^{3/2}}
  35. E c o n E_{con}
  36. γ \gamma
  37. v ¯ i v ¯ i + F s n a k e ( v ¯ i ) \bar{v}_{i}\leftarrow\bar{v}_{i}+F_{snake}(\bar{v}_{i})
  38. F s n a k e ( v ¯ i ) F_{snake}(\bar{v}_{i})
  39. F s n a k e ( v ¯ i ) = - E s n a k e ( v ¯ i ) = - ( w i n t e r n a l E i n t e r n a l ( v ¯ i ) + w e x t e r n a l E e x t e r n a l ( v ¯ i ) ) F_{snake}(\bar{v}_{i})=-\nabla E_{snake}(\bar{v}_{i})=-\Bigg(w_{internal}% \nabla E_{internal}(\bar{v}_{i})+w_{external}\nabla E_{external}(\bar{v}_{i})\Bigg)
  40. α ( s ) \alpha(s)
  41. β ( s ) \beta(s)
  42. s s
  43. v ¯ i = v ¯ i - γ { w i n t e r n a l [ α 2 v ¯ s 2 ( v ¯ i ) + β 4 v ¯ s 4 ( v ¯ i ) ] + E e x t ( v ¯ i ) } \bar{v}_{i}=\leftarrow\bar{v}_{i}-\gamma\Bigg\{w_{internal}\bigg[\alpha\frac{% \partial^{2}\bar{v}}{\partial s^{2}}(\bar{v}_{i})+\beta\frac{\partial^{4}\bar{% v}}{\partial s^{4}}(\bar{v}_{i})\bigg]+\nabla E_{ext}(\bar{v}_{i})\Bigg\}
  44. τ \tau
  45. E s n a k e * 1 n E s n a k e ( v ¯ i ) E_{snake}^{*}\approx\displaystyle\sum_{1}^{n}E_{snake}(\bar{v}_{i})
  46. F s n a k e * - 1 n E s n a k e ( v ¯ i ) F_{snake}^{*}\approx-\displaystyle\sum_{1}^{n}\nabla E_{snake}(\bar{v}_{i})
  47. F i m a g e = - k E i m a g e E i m a g e F_{image}=-k\frac{\nabla E_{image}}{\|\nabla E_{image}\|}
  48. τ k \tau k
  49. F G V F F_{GVF}
  50. E G V F = μ ( u x 2 + u y 2 + v x 2 + v y 2 ) + | f | 2 | 𝐯 - f | 2 d x d y E_{GVF}=\int\int\mu(u_{x}^{2}+u_{y}^{2}+v_{x}^{2}+v_{y}^{2})+|\nabla f|^{2}|% \mathbf{v}-\nabla f|^{2}dxdy
  51. μ \mu
  52. μ 2 u - ( u - x F e x t ) ( x F e x t ( x , y ) 2 + y F e x t ( x , y ) 2 ) = 0 \mu\nabla^{2}u-\Bigg(u-\frac{\partial}{\partial x}F_{ext}\Bigg)\Bigg(\frac{% \partial}{\partial x}F_{ext}(x,y)^{2}+\frac{\partial}{\partial y}F_{ext}(x,y)^% {2}\Bigg)=0
  53. μ 2 v - ( v - y F e x t ) ( x F e x t ( x , y ) 2 + y F e x t ( x , y ) 2 ) = 0 \mu\nabla^{2}v-\Bigg(v-\frac{\partial}{\partial y}F_{ext}\Bigg)\Bigg(\frac{% \partial}{\partial x}F_{ext}(x,y)^{2}+\frac{\partial}{\partial y}F_{ext}(x,y)^% {2}\Bigg)=0
  54. u i + 1 = u i + μ 2 u i - ( u i - x F e x t ) ( x F e x t ( x , y ) 2 + y F e x t ( x , y ) 2 ) u_{i+1}=u_{i}+\mu\nabla^{2}u_{i}-\Bigg(u_{i}-\frac{\partial}{\partial x}F_{ext% }\Bigg)\Bigg(\frac{\partial}{\partial x}F_{ext}(x,y)^{2}+\frac{\partial}{% \partial y}F_{ext}(x,y)^{2}\Bigg)
  55. v i + 1 = v i + μ 2 v i - ( v i - y F e x t ) ( x F e x t ( x , y ) 2 + y F e x t ( x , y ) 2 ) v_{i+1}=v_{i}+\mu\nabla^{2}v_{i}-\Bigg(v_{i}-\frac{\partial}{\partial y}F_{ext% }\Bigg)\Bigg(\frac{\partial}{\partial x}F_{ext}(x,y)^{2}+\frac{\partial}{% \partial y}F_{ext}(x,y)^{2}\Bigg)
  56. F e x t * = F G V F F_{ext}^{*}=F_{GVF}
  57. μ \mu
  58. μ \mu
  59. F i n f l a t i o n = k 1 n ( s ) F_{inflation}=k_{1}\vec{n}(s)
  60. n ( s ) \vec{n}(s)
  61. v ( s ) v(s)
  62. k 1 k_{1}
  63. k 1 k_{1}
  64. k k
  65. k k
  66. E i m a g e E_{image}
  67. E i m a g e * = E i + α E c E_{image}^{*}=E_{i}+\alpha E_{c}
  68. E i E_{i}
  69. E [ J , B ] = 1 2 D ( I ( x ) - J ( x ) ) 2 d x + λ 1 2 D / B J ( x ) J ( x ) d x + ν 0 1 ( d d s B ( s ) ) 2 d s E[J,B]=\frac{1}{2}\int_{D}(I(\vec{x})-J(\vec{x}))^{2}d\vec{x}+\lambda\frac{1}{% 2}\int_{D/B}\vec{\nabla}J(\vec{x})\cdot\vec{\nabla}J(\vec{x})d\vec{x}+\nu\int_% {0}^{1}\Bigg(\frac{d}{ds}B(s)\Bigg)^{2}ds
  70. J ( x ) J(\vec{x})
  71. I ( x ) I(\vec{x})
  72. D D
  73. B ( s ) B(s)
  74. B ( s ) = n = 1 N p n B n ( s ) B(s)=\sum_{n=1}^{N}\vec{p}_{n}B_{n}(s)
  75. B n ( s ) B_{n}(s)
  76. p n \vec{p}_{n}
  77. λ \lambda\to\infty
  78. E i E_{i}
  79. E c E_{c}
  80. α \alpha
  81. z \vec{z}
  82. z 0 \vec{z}_{0}
  83. Σ \Sigma
  84. E c ( z ) = 1 2 ( z - z 0 ) t Σ * ( z - z 0 ) E_{c}(\vec{z})=\frac{1}{2}(\vec{z}-\vec{z}_{0})^{t}\Sigma^{*}(\vec{z}-\vec{z}_% {0})
  85. Φ \Phi
  86. t Φ = | Φ | D I V ( g ( I ) Φ Φ ) + g ( I ) Φ \frac{\partial}{\partial t}\Phi=|\nabla\Phi|DIV\Bigg(\frac{g(I)}{\nabla\Phi}% \nabla\Phi\Bigg)+\nabla g(I)\cdot\nabla\Phi
  87. g ( I ) g(I)

Acyclic_space.html

  1. H ~ i ( X ) = 0 , i 0. \tilde{H}_{i}(X)=0,\quad\forall i\geq 0.
  2. H ~ i ( G ; 𝐙 ) = 0 \tilde{H}_{i}(G;\mathbf{Z})=0
  3. H 1 ( G ; 𝐙 ) = 0 H_{1}(G;\mathbf{Z})=0
  4. H 1 ( G ; 𝐙 ) = H 2 ( G ; 𝐙 ) = 0 H_{1}(G;\mathbf{Z})=H_{2}(G;\mathbf{Z})=0

Adaptive-additive_algorithm.html

  1. A 0 A_{0}
  2. ϕ n k \phi_{n}^{k}
  3. A 0 e i ϕ n k F F T A n f e i ϕ n f A_{0}e^{i\phi_{n}^{k}}\xrightarrow{FFT}A_{n}^{f}e^{i\phi_{n}^{f}}
  4. I n f I_{n}^{f}
  5. I 0 f I_{0}^{f}
  6. I n f = ( A n f ) 2 , I_{n}^{f}=\left(A_{n}^{f}\right)^{2},
  7. ε = ( I n f ) 2 - ( I 0 ) 2 . \varepsilon=\sqrt{\left(I_{n}^{f}\right)^{2}-\left(I_{0}\right)^{2}}.
  8. ε \varepsilon
  9. A n f A_{n}^{f}
  10. A f A^{f}
  11. A ¯ n f = [ a A f + ( 1 - a ) A n f ] , \bar{A}^{f}_{n}=\left[aA^{f}+(1-a)A_{n}^{f}\right],
  12. A f = I 0 A^{f}=\sqrt{I_{0}}
  13. A ¯ f e i ϕ n f i F F T A ¯ n k e i ϕ n k . \bar{A}^{f}e^{i\phi_{n}^{f}}\xrightarrow{iFFT}\bar{A}_{n}^{k}e^{i\phi_{n}^{k}}.
  14. A ¯ n k \bar{A}_{n}^{k}
  15. ϕ n k \phi^{k}_{n}
  16. A 0 A_{0}
  17. ϕ n k \phi^{k}_{n}
  18. n n + 1 n\to n+1
  19. a = 1 a=1
  20. a = 0 a=0
  21. A ¯ n k = A 0 \bar{A}^{k}_{n}=A_{0}

Adaptive_stepsize.html

  1. y ( t ) = f ( t , y ( t ) ) , y ( a ) = y a y^{\prime}(t)=f(t,y(t)),\qquad y(a)=y_{a}
  2. t = b t=b
  3. f ( t , y ) f(t,y)
  4. t = a t=a
  5. y a = y ( a ) y_{a}=y(a)
  6. y b + ϵ = y ( b ) y_{b}+\epsilon=y(b)
  7. ϵ \epsilon
  8. y ( t ) = f ( t , y ) y^{\prime}(t)=f(t,y)
  9. y ( t + h ) = y ( t ) + h f ( t , y ( t ) ) + h 2 2 f ( η , y ( η ) ) , t η t + h y(t+h)=y(t)+hf(t,y(t))+\frac{h^{2}}{2}f^{\prime}(\eta,y(\eta)),\qquad t\leq% \eta\leq t+h
  10. y n + 1 ( 0 ) = y n + h f ( t n , y n ) y_{n+1}^{(0)}=y_{n}+hf(t_{n},y_{n})
  11. τ n + 1 ( 0 ) = c h 2 \tau_{n+1}^{(0)}=ch^{2}
  12. y n + 1 ( 0 ) + τ n + 1 ( 0 ) = y ( t + h ) y_{n+1}^{(0)}+\tau_{n+1}^{(0)}=y(t+h)
  13. ( 0 ) (0)
  14. c c
  15. f f
  16. ( 1 ) (1)
  17. y n + 1 2 = y n + h 2 f ( t n , y n ) y_{n+\frac{1}{2}}=y_{n}+\frac{h}{2}f(t_{n},y_{n})
  18. y n + 1 ( 1 ) = y n + 1 2 + h 2 f ( t n + 1 2 , y n + 1 2 ) y_{n+1}^{(1)}=y_{n+\frac{1}{2}}+\frac{h}{2}f(t_{n+\frac{1}{2}},y_{n+\frac{1}{2% }})
  19. τ n + 1 ( 1 ) = c ( h 2 ) 2 + c ( h 2 ) 2 = 2 c ( h 2 ) 2 = 1 2 c h 2 = 1 2 τ n + 1 ( 0 ) \tau_{n+1}^{(1)}=c\left(\frac{h}{2}\right)^{2}+c\left(\frac{h}{2}\right)^{2}=2% c\left(\frac{h}{2}\right)^{2}=\frac{1}{2}ch^{2}=\frac{1}{2}\tau_{n+1}^{(0)}
  20. y n + 1 ( 1 ) + τ n + 1 ( 1 ) = y ( t + h ) y_{n+1}^{(1)}+\tau_{n+1}^{(1)}=y(t+h)
  21. c c
  22. [ t , t + h ] [t,t+h]
  23. y ( 3 ) ( t ) y^{(3)}(t)
  24. y n + 1 ( 1 ) - y n + 1 ( 0 ) = τ n + 1 ( 1 ) y_{n+1}^{(1)}-y_{n+1}^{(0)}=\tau_{n+1}^{(1)}
  25. h h
  26. t o l tol
  27. h 0.9 × h × t o l | τ n + 1 ( 1 ) | h\rightarrow 0.9\times h\times\frac{tol}{|\tau_{n+1}^{(1)}|}
  28. 0.9 0.9
  29. 0.9 × t o l 0.9\times tol
  30. | τ n + 1 ( 1 ) | < t o l |\tau_{n+1}^{(1)}|<tol
  31. y n + 1 ( 2 ) = y n + 1 ( 1 ) + τ n + 1 ( 1 ) y_{n+1}^{(2)}=y_{n+1}^{(1)}+\tau_{n+1}^{(1)}
  32. h = b - a h=b-a
  33. a a
  34. b b

Addition-subtraction_chain.html

  1. a 0 = 1 , a_{0}=1,\,
  2. for k > 0 , a k = a i ± a j for some 0 i , j < k . \,\text{for }k>0,\ a_{k}=a_{i}\pm a_{j}\,\text{ for some }0\leq i,j<k.
  3. a L = n a_{L}=n
  4. a 0 = 1 a_{0}=1
  5. a 1 = 2 = 1 + 1 a_{1}=2=1+1
  6. a 2 = 4 = 2 + 2 a_{2}=4=2+2
  7. a 3 = 3 = 4 - 1 a_{3}=3=4-1
  8. a 2 = 3 = 2 + 1 a_{2}=3=2+1
  9. a 0 = 1 , a 1 = 2 = 1 + 1 , a 2 = 4 = 2 + 2 , a 3 = 8 = 4 + 4 , a 4 = 16 = 8 + 8 , a 5 = 32 = 16 + 16 , a 6 = 31 = 32 - 1. a_{0}=1,\ a_{1}=2=1+1,\ a_{2}=4=2+2,\ a_{3}=8=4+4,\ a_{4}=16=8+8,\ a_{5}=32=16% +16,\ a_{6}=31=32-1.
  10. x n x^{n}

Adenosylhomocysteinase.html

  1. \rightleftharpoons

Adjoint_bundle.html

  1. 𝔤 \mathfrak{g}
  2. Ad : G Aut ( 𝔤 ) \sub GL ( 𝔤 ) \mathrm{Ad}:G\to\mathrm{Aut}(\mathfrak{g})\sub\mathrm{GL}(\mathfrak{g})
  3. ad P = P × Ad 𝔤 \mathrm{ad}P=P\times^{\mathrm{Ad}}\mathfrak{g}
  4. 𝔤 P \mathfrak{g}_{P}
  5. 𝔤 \mathfrak{g}
  6. [ p g , x ] = [ p , Ad g - 1 ( x ) ] [p\cdot g,x]=[p,\mathrm{Ad}_{g^{-1}}(x)]
  7. ad P \mathrm{ad}P
  8. ad P \mathrm{ad}P

Admissible_heuristic.html

  1. n n
  2. f ( n ) = g ( n ) + h ( n ) f(n)=g(n)+h(n)
  3. f ( n ) f(n)
  4. g ( n ) g(n)
  5. h ( n ) h(n)
  6. h ( n ) h(n)
  7. f ( n ) f(n)
  8. n n
  9. h h
  10. h ( n ) h(n)
  11. h h
  12. n n
  13. h * ( n ) h^{*}(n)
  14. n n
  15. h ( n ) h(n)
  16. n , h ( n ) h * ( n ) \forall n,h(n)\leq h^{*}(n)
  17. h ( n ) = a l l t i l e s d i s t a n c e ( t i l e , c o r r e c t p o s i t i o n ) h(n)=\sum_{alltiles}distance(tile,correctposition)
  18. h ( n ) = 3 + 1 + 0 + 1 + 2 + 3 + 3 + 4 + 3 + 2 + 4 + 4 + 4 + 1 + 1 = 36 h(n)=3+1+0+1+2+3+3+4+3+2+4+4+4+1+1=36

Admissible_representation.html

  1. ( 𝔤 , K ) (\mathfrak{g},K)
  2. ( 𝔤 , K ) (\mathfrak{g},K)
  3. ( 𝔤 , K ) (\mathfrak{g},K)

Admittance_parameters.html

  1. V n V_{n}\,
  2. I n I_{n}\,
  3. I = Y V I=YV\,
  4. ( I 1 I 2 ) = ( Y 11 Y 12 Y 21 Y 22 ) ( V 1 V 2 ) {I_{1}\choose I_{2}}=\begin{pmatrix}Y_{11}&Y_{12}\\ Y_{21}&Y_{22}\end{pmatrix}{V_{1}\choose V_{2}}
  5. Y 11 = I 1 V 1 | V 2 = 0 Y 12 = I 1 V 2 | V 1 = 0 Y_{11}={I_{1}\over V_{1}}\bigg|_{V_{2}=0}\qquad Y_{12}={I_{1}\over V_{2}}\bigg% |_{V_{1}=0}
  6. Y 21 = I 2 V 1 | V 2 = 0 Y 22 = I 2 V 2 | V 1 = 0 Y_{21}={I_{2}\over V_{1}}\bigg|_{V_{2}=0}\qquad Y_{22}={I_{2}\over V_{2}}\bigg% |_{V_{1}=0}
  7. Y i n = y 11 - y 12 y 21 y 22 + Y L Y_{in}=y_{11}-\frac{y_{12}y_{21}}{y_{22}+Y_{L}}
  8. Y o u t = y 22 - y 12 y 21 y 11 + Y S Y_{out}=y_{22}-\frac{y_{12}y_{21}}{y_{11}+Y_{S}}
  9. Y = y ( 1 N - S ) ( 1 N + S ) - 1 y = y ( 1 N + S ) - 1 ( 1 N - S ) y \begin{aligned}\displaystyle Y&\displaystyle=\sqrt{y}(1_{\!N}-S)(1_{\!N}+S)^{-% 1}\sqrt{y}\\ &\displaystyle=\sqrt{y}(1_{\!N}+S)^{-1}(1_{\!N}-S)\sqrt{y}\\ \end{aligned}
  10. S = ( 1 N - z Y z ) ( 1 N + z Y z ) - 1 = ( 1 N + z Y z ) - 1 ( 1 N - z Y z ) \begin{aligned}\displaystyle S&\displaystyle=(1_{\!N}-\sqrt{z}Y\sqrt{z})(1_{\!% N}+\sqrt{z}Y\sqrt{z})^{-1}\\ &\displaystyle=(1_{\!N}+\sqrt{z}Y\sqrt{z})^{-1}(1_{\!N}-\sqrt{z}Y\sqrt{z})\\ \end{aligned}
  11. 1 N 1_{\!N}
  12. y \sqrt{y}
  13. y = ( y 01 y 02 y 0 N ) \sqrt{y}=\begin{pmatrix}\sqrt{y_{01}}&\\ &\sqrt{y_{02}}\\ &&\ddots\\ &&&\sqrt{y_{0N}}\end{pmatrix}
  14. z = ( y ) - 1 \sqrt{z}=(\sqrt{y})^{-1}
  15. y 01 = y 02 = Y 0 y_{01}=y_{02}=Y_{0}
  16. Y 11 = ( ( 1 - S 11 ) ( 1 + S 22 ) + S 12 S 21 ) Δ S Y 0 Y_{11}={((1-S_{11})(1+S_{22})+S_{12}S_{21})\over\Delta_{S}}Y_{0}\,
  17. Y 12 = - 2 S 12 Δ S Y 0 Y_{12}={-2S_{12}\over\Delta_{S}}Y_{0}\,
  18. Y 21 = - 2 S 21 Δ S Y 0 Y_{21}={-2S_{21}\over\Delta_{S}}Y_{0}\,
  19. Y 22 = ( ( 1 + S 11 ) ( 1 - S 22 ) + S 12 S 21 ) Δ S Y 0 Y_{22}={((1+S_{11})(1-S_{22})+S_{12}S_{21})\over\Delta_{S}}Y_{0}\,
  20. Δ S = ( 1 + S 11 ) ( 1 + S 22 ) - S 12 S 21 \Delta_{S}=(1+S_{11})(1+S_{22})-S_{12}S_{21}\,
  21. S i j S_{ij}
  22. Y i j Y_{ij}
  23. Δ \Delta
  24. S i j S_{ij}
  25. Δ \Delta
  26. Y i j Y_{ij}
  27. S 11 = ( 1 - Z 0 Y 11 ) ( 1 + Z 0 Y 22 ) + Z 0 2 Y 12 Y 21 Δ S_{11}={(1-Z_{0}Y_{11})(1+Z_{0}Y_{22})+Z^{2}_{0}Y_{12}Y_{21}\over\Delta}\,
  28. S 12 = - 2 Z 0 Y 12 Δ S_{12}={-2Z_{0}Y_{12}\over\Delta}\,
  29. S 21 = - 2 Z 0 Y 21 Δ S_{21}={-2Z_{0}Y_{21}\over\Delta}\,
  30. S 22 = ( 1 + Z 0 Y 11 ) ( 1 - Z 0 Y 22 ) + Z 0 2 Y 12 Y 21 Δ S_{22}={(1+Z_{0}Y_{11})(1-Z_{0}Y_{22})+Z^{2}_{0}Y_{12}Y_{21}\over\Delta}\,
  31. Δ = ( 1 + Z 0 Y 11 ) ( 1 + Z 0 Y 22 ) - Z 0 2 Y 12 Y 21 \Delta=(1+Z_{0}Y_{11})(1+Z_{0}Y_{22})-Z^{2}_{0}Y_{12}Y_{21}\,
  32. Z 0 Z_{0}
  33. Y 11 = Z 22 | Z | Y_{11}={Z_{22}\over|Z|}\,
  34. Y 12 = - Z 12 | Z | Y_{12}={-Z_{12}\over|Z|}\,
  35. Y 21 = - Z 21 | Z | Y_{21}={-Z_{21}\over|Z|}\,
  36. Y 22 = Z 11 | Z | Y_{22}={Z_{11}\over|Z|}\,
  37. | Z | = Z 11 Z 22 - Z 12 Z 21 |Z|=Z_{11}Z_{22}-Z_{12}Z_{21}\,
  38. | Z | |Z|
  39. Y = Z - 1 Y=Z^{-1}\,
  40. Z = Y - 1 Z=Y^{-1}\,

Advanced_IRB.html

  1. R = 0.12 * 1 - e - 50 * P D 1 - e - 50 + 0.24 * ( 1 - 1 - e - 50 * P D 1 - e - 50 ) R=0.12*\frac{1-e^{-50*PD}}{1-e^{-50}}+0.24*\left(1-\frac{1-e^{-50*PD}}{1-e^{-5% 0}}\right)
  2. b = ( 0.11852 - 0.05478 * ln ( P D ) ) 2 b=(0.11852-0.05478*\ln(PD))^{2}
  3. K = [ L G D * N ( 1 1 - R * G ( P D ) + R 1 - R * G ( 0.999 ) ) - ( L G D * P D ) ] * 1 + ( M - 2.5 ) b 1 - 1.5 b K=\left[LGD*N\left(\sqrt{\frac{1}{1-R}}*G(PD)+\sqrt{\frac{R}{1-R}}*G(0.999)% \right)-(LGD*PD)\right]*\frac{1+(M-2.5)b}{1-1.5b}
  4. R W A = K * 12.5 * E A D * 1.06 RWA=K*12.5*EAD*1.06
  5. R = 0.12 * 1 - e - 50 * P D 1 - e - 50 + 0.24 * ( 1 - 1 - e - 50 * P D 1 - e - 50 ) - 0.04 * ( 1 - max ( S - 5 , 0 ) 45 ) R=0.12*\frac{1-e^{-50*PD}}{1-e^{-50}}+0.24*\left(1-\frac{1-e^{-50*PD}}{1-e^{-5% 0}}\right)-0.04*(1-\frac{\max(S-5,0)}{45})
  6. R = 0.15 R=0.15
  7. K = L G D * [ N ( 1 1 - R * G ( P D ) + R 1 - R * G ( 0.999 ) ) - P D ] K=LGD*\left[N\left(\sqrt{\frac{1}{1-R}}*G(PD)+\sqrt{\frac{R}{1-R}}*G(0.999)% \right)-PD\right]
  8. R W A = K * 12.5 * E A D RWA=K*12.5*EAD
  9. R = 0.04 R=0.04
  10. K = L G D * [ N ( 1 1 - R * G ( P D ) + R 1 - R * G ( 0.999 ) ) - P D ] K=LGD*\left[N\left(\sqrt{\frac{1}{1-R}}*G(PD)+\sqrt{\frac{R}{1-R}}*G(0.999)% \right)-PD\right]
  11. R W A = K * 12.5 * E A D RWA=K*12.5*EAD

Aerodynamic_drag.html

  1. D p r D_{pr}
  2. D f D_{f}
  3. D p r D_{pr}
  4. D f D_{f}
  5. D v D_{v}
  6. D i D_{i}
  7. D w D_{w}
  8. S = S + S D + S A [ ρ u q + ( p - p ) i - τ x ] n d S = 0 \int_{S=S_{\infty}+S_{D}+S_{A}}\left[\rho\,u\,\vec{q}+\left(p-p_{\infty}\right% )\vec{i}-\vec{\tau}_{x}\right]\cdot\vec{n}dS=0
  9. S = S A Aircraft Surf. + S D + S Far Surf. S=\underbrace{S_{A}}\text{Aircraft Surf.}\;+\;\underbrace{S_{D}+S_{\infty}}% \text{Far Surf.}
  10. S A S_{A}
  11. S D S_{D}
  12. S S_{\infty}
  13. ( V ) (V)
  14. S A [ ρ u q + ( p - p ) i - τ x ] n d S = - S D + S [ ρ u q + ( p - p ) i - τ x ] n d S \int_{S_{A}}\left[\rho\,u\,\vec{q}+\left(p-p_{\infty}\right)\vec{i}-\vec{\tau}% _{x}\right]\,\cdot\,\vec{n}\,dS=-\int_{S_{D}+S_{\infty}}\left[\rho\,u\,\vec{q}% +\left(p-p_{\infty}\right)\vec{i}-\vec{\tau}_{x}\right]\,\cdot\,\vec{n}\,dS
  15. D f + D p r near-field = D i + D w + D v far-field \underbrace{D_{f}+D_{pr}}\text{near-field}=\underbrace{D_{i}+D_{w}+D_{v}}\text% {far-field}

Affine_Hecke_algebra.html

  1. V V
  2. Σ \Sigma
  3. V V
  4. [ W ] \mathbb{C}[W]
  5. W W
  6. Σ \Sigma
  7. H ( Σ , q ) H(\Sigma,q)
  8. q : Σ q:\Sigma\rightarrow\mathbb{C}
  9. q 1 q\equiv 1
  10. H ( Σ , q ) H(\Sigma,q)
  11. [ W ] \mathbb{C}[W]

Affine_manifold.html

  1. n {\mathbb{R}}^{n}
  2. M M\,
  3. ψ i : U i n \psi_{i}\colon U_{i}\to{\mathbb{R}}^{n}
  4. ψ i ψ j - 1 Aff ( n ) \psi_{i}\circ\psi_{j}^{-1}\in{\rm Aff}({\mathbb{R}}^{n})
  5. i , j , i,j\,,
  6. Aff ( n ) {\rm Aff}({\mathbb{R}}^{n})
  7. n {\mathbb{R}}^{n}
  8. M M
  9. G G
  10. M M
  11. M ~ \tilde{M}
  12. n n
  13. D : M ~ n D\colon{\tilde{M}}\to{\mathbb{R}}^{n}
  14. φ : G Aff ( n ) \varphi\colon G\to{\rm Aff}({\mathbb{R}}^{n})
  15. D D
  16. φ \varphi

Age_(model_theory).html

  1. , < \langle\mathbb{Q},<\rangle
  2. , < \langle\mathbb{Q},<\rangle
  3. , < \langle\mathbb{N},<\rangle
  4. f ( 1 ) = 5 f(1)=5
  5. f ( 3 ) = 6 f(3)=6

Air_permeability_specific_surface.html

  1. S = 7 d ρ ( 1 - ϵ ) ϵ 3 π δ P l η Q S=\cfrac{7d}{\rho\,(1-\epsilon\,)}\sqrt{\dfrac{\epsilon\,^{3}\pi\,\delta\,P}{l% \eta\,Q}}
  2. M = π 4 d 2 l ρ ( 1 - ϵ ) M=\tfrac{\pi}{4}\,d^{2}l\rho\,(1-\epsilon\,)

AIR_synthetase_(FGAM_cyclase).html

  1. \rightleftharpoons

Airport_reference_temperature.html

  1. T a + T m - T a 3 . T_{a}+\frac{T_{m}-T_{a}}{3}.

Airway_resistance.html

  1. R A W = Δ P V ˙ R_{AW}=\frac{{\Delta}P}{\dot{V}}
  2. Δ P = P A T M - P A {\Delta P}=P_{ATM}-P_{A}
  3. R A W = P ATM - P A V ˙ R_{AW}=\frac{P_{\mathrm{ATM}}-P_{\mathrm{A}}}{\dot{V}}
  4. R A W R_{AW}
  5. Δ P {\Delta}P
  6. P A T M P_{ATM}
  7. P A P_{A}
  8. V ˙ \dot{V}
  9. V ˙ \dot{V}
  10. Δ P = 8 η l V ˙ π r 4 {\Delta P}=\frac{8\eta l{\dot{V}}}{\pi r^{4}}
  11. Δ P \Delta P
  12. l l
  13. η \eta
  14. V ˙ \dot{V}
  15. r r
  16. V ˙ \dot{V}
  17. R = 8 η l π r 4 R=\frac{8\eta l}{\pi r^{4}}
  18. R e = ρ v d μ Re={{\rho{\mathrm{v}}d}\over\mu}
  19. R e Re
  20. d d
  21. v {\mathrm{v}}
  22. μ {\mu}
  23. ρ {\rho}\,
  24. G A W = 1 R A W G_{AW}=\frac{1}{R_{AW}}
  25. s R A W = R A W V sR_{AW}={R_{AW}}{V}
  26. s R A W = R A W × F R C sR_{AW}={R_{AW}}\times{FRC}
  27. s G A W = G A W V = 1 R A W V = 1 s R A W sG_{AW}=\frac{G_{AW}}{V}=\frac{1}{R_{AW}V}=\frac{1}{sR_{AW}}
  28. s G A W = G A W F R C sG_{AW}=\frac{G_{AW}}{FRC}

Akhmim_wooden_tablets.html

  1. 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + 1 / 32 + 1 / 64 1/2+1/4+1/8+1/16+1/32+1/64
  2. 1 / 4 + 1 / 16 + 1 / 64 1/4+1/16+1/64
  3. 1 / 4 + 1 / 16 + 1 / 64 + ( 1 + 2 / 3 ) r o 1/4+1/16+1/64+(1+2/3)ro
  4. 2 × 7 2\times 7

Albert_algebra.html

  1. x y = 1 2 ( x y + y x ) , x\circ y=\frac{1}{2}(x\cdot y+y\cdot x),
  2. \cdot

Alfred_George_Greenhill.html

  1. T w i s t = C D 2 L × S G 10.9 Twist=\frac{CD^{2}}{L}\times\sqrt{\frac{SG}{10.9}}

Algebraic_character.html

  1. 𝔤 \mathfrak{g}
  2. 𝔥 , \mathfrak{h},
  3. A = [ [ 𝔥 * ] ] A=\mathbb{Z}[[\mathfrak{h}^{*}]]
  4. e μ e^{\mu}
  5. μ 𝔥 * \mu\in\mathfrak{h}^{*}
  6. V V
  7. V V
  8. A A
  9. c h ( V ) = μ dim V μ e μ , ch(V)=\sum_{\mu}\dim V_{\mu}e^{\mu},
  10. V . V.
  11. M λ M_{\lambda}
  12. λ \lambda
  13. c h ( M λ ) = e λ α > 0 ( 1 - e - α ) , ch(M_{\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})},
  14. e μ e ν = e μ + ν e^{\mu}\cdot e^{\nu}=e^{\mu+\nu}
  15. A A

Algebraic_cycle.html

  1. f * ( [ Y ] ) = [ f - 1 ( Y ) ] f^{*}([Y^{\prime}])=[f^{-1}(Y^{\prime})]\,\!
  2. f * ( [ Y ] ) = n [ f ( Y ) ] f_{*}([Y])=n[f(Y)]\,\!
  3. f * : Z k ( X ) Z k ( X ) and f * : Z k ( X ) Z k ( X ) f^{*}\colon Z^{k}(X^{\prime})\to Z^{k}(X)\quad\,\text{and}\quad f_{*}\colon Z_% {k}(X)\to Z_{k}(X^{\prime})\,\!

Algebraic_formula_for_the_variance.html

  1. Var ( X ) = E ( X 2 ) - [ E ( X ) ] 2 . \operatorname{Var}(X)=\operatorname{E}(X^{2})-[\operatorname{E}(X)]^{2}.
  2. σ ^ 2 = 1 N - 1 i = 1 N ( x i - x ¯ ) 2 = N N - 1 ( 1 N ( i = 1 N x i 2 ) - x ¯ 2 ) 1 N - 1 ( ( i = 1 N x i 2 ) - N ( x ¯ ) 2 ) . \hat{\sigma}^{2}=\frac{1}{N-1}\sum_{i=1}^{N}(x_{i}-\bar{x})^{2}=\frac{N}{N-1}% \left(\frac{1}{N}\left(\sum_{i=1}^{N}x_{i}^{2}\right)-\bar{x}^{2}\right)\equiv% \frac{1}{N-1}\left(\left(\sum_{i=1}^{N}x_{i}^{2}\right)-N\left(\bar{x}\right)^% {2}\right).
  3. E ( X ) 2 Var ( X ) \operatorname{E}(X)^{2}\gg\operatorname{Var}(X)
  4. Var ( X ) \displaystyle\operatorname{Var}(X)
  5. Cov ( X i , X j ) = E ( X i X j ) - E ( X i ) E ( X j ) \operatorname{Cov}(X_{i},X_{j})=\operatorname{E}(X_{i}X_{j})-\operatorname{E}(% X_{i})\operatorname{E}(X_{j})
  6. Var ( 𝐗 ) = E ( 𝐗𝐗 ) - E ( 𝐗 ) E ( 𝐗 ) \operatorname{Var}(\mathbf{X})=\operatorname{E}(\mathbf{XX^{\top}})-% \operatorname{E}(\mathbf{X})\operatorname{E}(\mathbf{X})^{\top}
  7. Cov ( 𝐗 , 𝐘 ) = E ( 𝐗𝐘 ) - E ( 𝐗 ) E ( 𝐘 ) \operatorname{Cov}(\,\textbf{X},\,\textbf{Y})=\operatorname{E}(\mathbf{XY^{% \top}})-\operatorname{E}(\mathbf{X})\operatorname{E}(\mathbf{Y})^{\top}
  8. 𝐗 = { X 1 , X 2 , , X n } \mathbf{X}=\{X_{1},X_{2},\ldots,X_{n}\}
  9. 𝐘 = { Y 1 , Y 2 , , Y m } \mathbf{Y}=\{Y_{1},Y_{2},\ldots,Y_{m}\}

Algebraic_fraction.html

  1. 3 x x 2 + 2 x - 3 \frac{3x}{x^{2}+2x-3}
  2. x + 2 x 2 - 3 \frac{\sqrt{x+2}}{x^{2}-3}
  3. 3 x x 2 + 2 x - 3 \frac{3x}{x^{2}+2x-3}
  4. x + 2 x 2 - 3 , \frac{\sqrt{x+2}}{x^{2}-3},
  5. a b \tfrac{a}{b}
  6. f ( x ) g ( x ) \tfrac{f(x)}{g(x)}
  7. deg f ( x ) < deg g ( x ) \deg f(x)<\deg g(x)
  8. 2 x x 2 - 1 \tfrac{2x}{x^{2}-1}
  9. x 3 + x 2 + 1 x 2 - 5 x + 6 \tfrac{x^{3}+x^{2}+1}{x^{2}-5x+6}
  10. x 2 - x + 1 5 x 2 + 3 \tfrac{x^{2}-x+1}{5x^{2}+3}
  11. x 3 + x 2 + 1 x 2 - 5 x + 6 = ( x + 6 ) + 24 x - 35 x 2 - 5 x + 6 , \frac{x^{3}+x^{2}+1}{x^{2}-5x+6}=(x+6)+\frac{24x-35}{x^{2}-5x+6},
  12. 2 x x 2 - 1 = 1 x - 1 + 1 x + 1 . \frac{2x}{x^{2}-1}=\frac{1}{x-1}+\frac{1}{x+1}.
  13. x 1 2 - 1 3 a x 1 3 - x 1 2 . \frac{x^{\tfrac{1}{2}}-\tfrac{1}{3}a}{x^{\tfrac{1}{3}}-x^{\tfrac{1}{2}}}.
  14. x = z 6 x=z^{6}
  15. z 3 - 1 3 a z 2 - z 3 . \frac{z^{3}-\tfrac{1}{3}a}{z^{2}-z^{3}}.

All-electric_range.html

  1. AEREquivalent = ( 1 - G P M C D G P M C S ) d C D \,\text{AER}\text{Equivalent}=\left(1-\frac{GPM_{CD}}{GPM_{CS}}\right)d^{CD}

Allen's_interval_algebra.html

  1. X < Y X\,\mathrel{\mathbf{<}}\,Y
  2. Y > X Y\,\mathrel{\mathbf{>}}\,X
  3. X 𝐦 Y X\,\mathrel{\mathbf{m}}\,Y
  4. Y 𝐦𝐢 X Y\,\mathrel{\mathbf{mi}}\,X
  5. X 𝐨 Y X\,\mathrel{\mathbf{o}}\,Y
  6. Y 𝐨𝐢 X Y\,\mathrel{\mathbf{oi}}\,X
  7. X 𝐬 Y X\,\mathrel{\mathbf{s}}\,Y
  8. Y 𝐬𝐢 X Y\,\mathrel{\mathbf{si}}\,X
  9. X 𝐝 Y X\,\mathrel{\mathbf{d}}\,Y
  10. Y 𝐝𝐢 X Y\,\mathrel{\mathbf{di}}\,X
  11. X 𝐟 Y X\,\mathrel{\mathbf{f}}\,Y
  12. Y 𝐟𝐢 X Y\,\mathrel{\mathbf{fi}}\,X
  13. X = Y X\,\mathrel{\mathbf{=}}\,Y
  14. newspaper { d , s , f } dinner \mbox{newspaper }~{}\mathbf{\{\operatorname{d},\operatorname{s},\operatorname{% f}\}}\mbox{ dinner}~{}
  15. dinner { < , m } bed \mbox{dinner }~{}\mathbf{\{\operatorname{<},\operatorname{m}\}}\mbox{ bed}~{}
  16. X X
  17. Y Y
  18. Y Y
  19. Z Z
  20. X X
  21. Z Z
  22. newspaper { < , m } bed \mbox{newspaper }~{}\mathbf{\{\operatorname{<},\operatorname{m}\}}\mbox{ bed}~{}

Alliinase.html

  1. \rightleftharpoons

Alperin–Brauer–Gorenstein_theorem.html

  1. M 11 M_{11}

Alpha_recursion_theory.html

  1. α \alpha
  2. Σ 1 ( L α ) \Sigma_{1}(L_{\alpha})
  3. α \alpha
  4. α \alpha
  5. α \alpha
  6. α \alpha
  7. Σ 1 \Sigma_{1}
  8. L α L_{\alpha}
  9. α / A \alpha/A
  10. α \alpha
  11. α \alpha
  12. L α L_{\alpha}
  13. α \alpha
  14. α \alpha
  15. H , J , K \langle H,J,K\rangle
  16. R 0 , R 1 R_{0},R_{1}
  17. K A H : J : [ H , J , K R 0 H B J α / B ] , K\subseteq A\leftrightarrow\exists H:\exists J:[\langle H,J,K\rangle\in R_{0}% \wedge H\subseteq B\wedge J\subseteq\alpha/B],
  18. K α / A H : J : [ H , J , K R 1 H B J α / B ] . K\subseteq\alpha/A\leftrightarrow\exists H:\exists J:[\langle H,J,K\rangle\in R% _{1}\wedge H\subseteq B\wedge J\subseteq\alpha/B].
  19. A α B \scriptstyle A\leq_{\alpha}B
  20. \scriptstyle\varnothing
  21. Σ 1 ( L α [ B ] ) \Sigma_{1}(L_{\alpha}[B])
  22. β α : A β L α \forall\beta\in\alpha:A\cap\beta\in L_{\alpha}
  23. α \alpha
  24. α \alpha
  25. α \alpha
  26. B 0 , B 1 B_{0},B_{1}
  27. A = B 0 B 1 B 0 B 1 = A α B i ( i < 2 ) . A=B_{0}\cup B_{1}\wedge B_{0}\cap B_{1}=\varnothing\wedge A\not\leq_{\alpha}B_% {i}(i<2).
  28. A < α C \scriptstyle A<_{\alpha}C
  29. A < α B < α C \scriptstyle A<_{\alpha}B<_{\alpha}C

Alternating_decision_tree.html

  1. T T
  2. T T
  3. T T
  4. 1 1
  5. - 1 -1
  6. 0.657 0.657
  7. ( x 1 , y 1 ) , , ( x m , y m ) (x_{1},y_{1}),\ldots,(x_{m},y_{m})
  8. x i x_{i}
  9. y i y_{i}
  10. w i w_{i}
  11. W + ( c ) W_{+}(c)
  12. c c
  13. W - ( c ) W_{-}(c)
  14. c c
  15. W ( c ) = W + ( c ) + W - ( c ) W(c)=W_{+}(c)+W_{-}(c)
  16. c c
  17. m m
  18. w i = 1 / m w_{i}=1/m
  19. i i
  20. a = 1 / 2 ln W + ( t r u e ) W - ( t r u e ) a=1/2\textrm{ln}\frac{W_{+}(true)}{W_{-}(true)}
  21. R 0 = R_{0}=
  22. a a
  23. 0
  24. 𝒫 = { t r u e } \mathcal{P}=\{true\}
  25. 𝒞 = \mathcal{C}=
  26. j = 1 T j=1\dots T
  27. p 𝒫 , c 𝒞 p\in\mathcal{P},c\in\mathcal{C}
  28. z = 2 ( W + ( p c ) W - ( p c ) + W + ( p ¬ c ) W - ( p ¬ c ) ) + W ( ¬ p ) z=2\left(\sqrt{W_{+}(p\wedge c)W_{-}(p\wedge c)}+\sqrt{W_{+}(p\wedge\neg c)W_{% -}(p\wedge\neg c)}\right)+W(\neg p)
  29. 𝒫 + = p c + p ¬ c \mathcal{P}+=p\wedge c+p\wedge\neg c
  30. a 1 = 1 2 ln W + ( p c ) + 1 W - ( p c ) + 1 a_{1}=\frac{1}{2}\textrm{ln}\frac{W_{+}(p\wedge c)+1}{W_{-}(p\wedge c)+1}
  31. a 2 = 1 2 ln W + ( p ¬ c ) + 1 W - ( p ¬ c ) + 1 a_{2}=\frac{1}{2}\textrm{ln}\frac{W_{+}(p\wedge\neg c)+1}{W_{-}(p\wedge\neg c)% +1}
  32. R j = R_{j}=
  33. p p
  34. c c
  35. a 1 a_{1}
  36. a 2 a_{2}
  37. w i = w i e - y i R j ( x i ) w_{i}=w_{i}e^{-y_{i}R_{j}(x_{i})}
  38. R j R_{j}
  39. 𝒫 \mathcal{P}

Alveolar_gas_equation.html

  1. p A O 2 = F I O 2 ( P A T M - p H 2 O ) - p a C O 2 ( 1 - F I O 2 [ 1 - R E R ] ) R E R p_{A}O_{2}=F_{I}O_{2}(P_{ATM}-pH_{2}O)-\frac{p_{a}CO_{2}(1-F_{I}O_{2}[1-RER])}% {RER}
  2. F I O 2 [ 1 - R E R ] 1 F_{I}O_{2}[1-RER]\ll 1
  3. p A O 2 F I O 2 ( P A T M - p H 2 O ) - p a C O 2 R E R p_{A}O_{2}\approx F_{I}O_{2}(P_{ATM}-pH_{2}O)-\frac{p_{a}CO_{2}}{RER}

Alveolar–arterial_gradient.html

  1. A a G r a d i e n t = P A O 2 - P a O 2 Aa~{}Gradient=P_{A}O_{2}-P_{a}O_{2}
  2. P A O 2 = F i O 2 ( P a t m - P H 2 O ) - P a C O 2 0.8 P_{A}O_{2}=F_{i}O_{2}(P_{atm}-P_{H_{2}O})-\frac{P_{a}CO_{2}}{0.8}
  3. A a G r a d i e n t = ( F i O 2 ( P a t m - P H 2 O ) - P a C O 2 0.8 ) - P a O 2 Aa~{}Gradient=\left(F_{i}O_{2}(P_{atm}-P_{H_{2}O})-\frac{P_{a}CO_{2}}{0.8}% \right)-P_{a}O_{2}
  4. A a G r a d i e n t = ( 150 - 5 4 ( P C O 2 ) ) - P a O 2 Aa~{}Gradient=\left(150-\frac{5}{4}(P_{CO_{2}})\right)-P_{a}O_{2}

Amalgamation_property.html

  1. f f = g g . f^{\prime}\circ f=g^{\prime}\circ g.\,
  2. B * C / A B*C/A
  3. f f = g g f^{\prime}\circ f=g^{\prime}\circ g\,
  4. f [ B ] g [ C ] = ( f f ) [ A ] = ( g g ) [ A ] f^{\prime}[B]\cap g^{\prime}[C]=(f^{\prime}\circ f)[A]=(g^{\prime}\circ g)[A]\,
  5. h [ X ] = { h ( x ) x X } . h[X]=\{h(x)\mid x\in X\}.\,

Ambisonic_decoding.html

  1. P n = W + X cos θ n + Y sin θ n P_{n}=W+X\cos\theta_{n}+Y\sin\theta_{n}
  2. θ n \theta_{n}

Amenable_Banach_algebra.html

  1. a a . x - x . a a\mapsto a.x-x.a
  2. x x
  3. L 1 ( G ) L^{1}(G)
  4. θ \theta
  5. θ ( A ) ¯ \overline{\theta(A)}

American_death_triangle.html

  1. F Anchor = Weight 2 cos ( 1 2 θ Bottom ) Weight × 0.5 + O ( θ Bottom 2 ) F_{\mathrm{Anchor}}=\frac{\mathrm{Weight}}{2\cos(\frac{1}{2}{\theta_{\mathrm{% Bottom}}})}\approx\mathrm{Weight}\times 0.5+O({\theta_{\mathrm{Bottom}}}^{2})
  2. F Anchor = Weight 2 cos ( 45 + 1 4 θ Bottom ) Weight × 0.707 + O ( θ Bottom ) F_{\mathrm{Anchor}}=\frac{\mathrm{Weight}}{2\cos(45^{\circ}+\frac{1}{4}{\theta% _{\mathrm{Bottom}}})}\approx\mathrm{Weight}\times 0.707+O(\theta_{\mathrm{% Bottom}})

Ammonium_perchlorate_composite_propellant.html

  1. A s A_{s}
  2. ρ \rho
  3. b r b_{r}
  4. m ˙ = ρ A s b r \dot{m}=\rho\cdot A_{s}\cdot b_{r}
  5. b r = a p n b_{r}=a\cdot p^{n}

Amoeba_(mathematics).html

  1. Log : ( \ { 0 } ) n n \mathrm{Log}:\left({\mathbb{C}}\backslash\{0\}\right)^{n}\to\mathbb{R}^{n}
  2. z = ( z 1 , z 2 , , z n ) z=(z_{1},z_{2},\dots,z_{n})
  3. n , \mathbb{R}^{n},
  4. Log ( z 1 , z 2 , , z n ) = ( log | z 1 | , log | z 2 | , , log | z n | ) . \mathrm{Log}(z_{1},z_{2},\dots,z_{n})=(\log|z_{1}|,\log|z_{2}|,\dots,\log|z_{n% }|).\,
  5. n n
  6. 𝒜 p {\mathcal{A}}_{p}
  7. 𝒜 p = { Log ( z ) : z ( \ { 0 } ) n , p ( z ) = 0 } . {\mathcal{A}}_{p}=\left\{\mathrm{Log}(z)\,:\,z\in\left({\mathbb{C}}\backslash% \{0\}\right)^{n},p(z)=0\right\}.\,
  8. n \ 𝒜 p \mathbb{R}^{n}\backslash{\mathcal{A}}_{p}
  9. N p : n N_{p}:\mathbb{R}^{n}\to\mathbb{R}
  10. N p ( x ) = 1 ( 2 π i ) n Log - 1 ( x ) log | p ( z ) | d z 1 z 1 d z 2 z 2 d z n z n , N_{p}(x)=\frac{1}{(2\pi i)^{n}}\int_{\mathrm{Log}^{-1}(x)}\log|p(z)|\,\frac{dz% _{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\cdots\wedge\frac{dz_{n}}{z_{n}},
  11. x x
  12. x = ( x 1 , x 2 , , x n ) . x=(x_{1},x_{2},\dots,x_{n}).
  13. N p N_{p}
  14. N p ( x ) = 1 ( 2 π ) n [ 0 , 2 π ] n log | p ( z ) | d θ 1 d θ 2 d θ n , N_{p}(x)=\frac{1}{(2\pi)^{n}}\int_{[0,2\pi]^{n}}\log|p(z)|\,d\theta_{1}\,d% \theta_{2}\cdots d\theta_{n},
  15. z = ( e x 1 + i θ 1 , e x 2 + i θ 2 , , e x n + i θ n ) . z=\left(e^{x_{1}+i\theta_{1}},e^{x_{2}+i\theta_{2}},\dots,e^{x_{n}+i\theta_{n}% }\right).
  16. p ( z ) p(z)
  17. p ( z ) = a z 1 k 1 z 2 k 2 z n k n p(z)=az_{1}^{k_{1}}z_{2}^{k_{2}}\dots z_{n}^{k_{n}}\,
  18. a 0 a\neq 0
  19. N p ( x ) = log | a | + k 1 x 1 + k 2 x 2 + + k n x n . N_{p}(x)=\log|a|+k_{1}x_{1}+k_{2}x_{2}+\cdots+k_{n}x_{n}.\,

Amortizing_loan.html

  1. WAL = i = 1 n P i P t i , \,\text{WAL}=\sum_{i=1}^{n}\frac{P_{i}}{P}t_{i},
  2. P P
  3. P i P_{i}
  4. i i
  5. P i P \frac{P_{i}}{P}
  6. i i
  7. t i t_{i}
  8. i i

Ampere-meter.html

  1. 1 A m = 1 C m s 1~{}\mathrm{A\cdot m}=1~{}\mathrm{C}\cdot\frac{\mathrm{m}}{\mathrm{s}}

AN::FPS-17.html

  1. R = [ P t G 2 λ 2 A ( 4 π ) 3 σ m i n ] 1 4 R=\left[\frac{P_{t}\,G^{2}\,\lambda^{2}\,A}{(4\,\pi)^{3}\,\sigma_{min}}\right]% ^{1\over 4}\,\!
  2. R R\,\!
  3. P t P_{t}\,\!
  4. G G\,\!
  5. λ \lambda\,\!
  6. σ m i n \sigma_{min}\,\!
  7. A A\,\!
  8. λ = c f \lambda=\frac{c}{f}\,\!
  9. c c\,\!
  10. f f\,\!
  11. λ = 3 × 10 8 m / s 192 × 10 6 Hz = 1.56 m \lambda=\frac{3\times 10^{8}\,\mathrm{m/s}}{192\times 10^{6}\,\mathrm{Hz}}=1.5% 6\,\mathrm{m}\,\!
  12. σ m i n = - 130 dBm = 10 - 130 / 10 mW = 10 - 16 W \sigma_{min}=-130\,\mathrm{dBm}=10^{-130/10}\,\mathrm{mW}=10^{-16}\,\mathrm{W}\,\!
  13. R = [ 10 6 W × 5000 2 × ( 1.56 m ) 2 × 1 m 2 12.57 3 × 10 - 16 W ] 1 4 R=\left[\frac{10^{6}\,\mathrm{W}\times 5000^{2}\times\left(1.56\,\mathrm{m}% \right)^{2}\times 1\,\mathrm{m}^{2}}{12.57^{3}\times 10^{-16}\,\mathrm{W}}% \right]^{1\over 4}\,\!

Analemmatic_sundial.html

  1. tan θ = tan ( 15 × t ) sin ϕ \tan\theta=\frac{\tan(15^{\circ}\times t)}{\sin\phi}
  2. Y = W cos ϕ tan δ Y=W\cos\phi\tan\delta\,

Analytic_Fredholm_theorem.html

  1. lim λ λ 0 B ( λ ) - B ( λ 0 ) λ - λ 0 \lim_{\lambda\to\lambda_{0}}\frac{B(\lambda)-B(\lambda_{0})}{\lambda-\lambda_{% 0}}
  2. B ( λ ) ψ = ψ B(\lambda)\psi=\psi

Anderson's_theorem.html

  1. n f ( x ) d x < + . \int_{\mathbb{R}^{n}}f(x)\,\mathrm{d}x<+\infty.
  2. L ( f , t ) = { x n | f ( x ) t } , L(f,t)=\{x\in\mathbb{R}^{n}|f(x)\geq t\},
  3. K f ( x + c y ) d x K f ( x + y ) d x . \int_{K}f(x+cy)\,\mathrm{d}x\geq\int_{K}f(x+y)\,\mathrm{d}x.
  4. Pr ( X K ) Pr ( X + Y K ) \Pr(X\in K)\geq\Pr(X+Y\in K)

Anderson_impurity_model.html

  1. H = σ ϵ f f σ f σ + < j , j > σ t j j c j σ c j σ + j , σ ( V j f σ c j σ + V j * c j σ f σ ) + U f f f f H=\sum_{\sigma}\epsilon_{f}f^{\dagger}_{\sigma}f_{\sigma}+\sum_{<j,j^{\prime}>% \sigma}t_{jj^{\prime}}c^{\dagger}_{j\sigma}c_{j^{\prime}\sigma}+\sum_{j,\sigma% }(V_{j}f^{\dagger}_{\sigma}c_{j\sigma}+V_{j}^{*}c^{\dagger}_{j\sigma}f_{\sigma% })+Uf^{\dagger}_{\uparrow}f_{\uparrow}f^{\dagger}_{\downarrow}f_{\downarrow}
  2. f f
  3. c c
  4. σ \sigma
  5. U U
  6. t j j t_{jj^{\prime}}
  7. j j
  8. j j^{\prime}
  9. V V
  10. f f
  11. H = j σ ϵ f f j σ f j σ + < j , j > σ t j j c j σ c j σ + j , σ ( V j f σ c j σ + V j * c σ f j σ ) + U j f j f j f j f j H=\sum_{j\sigma}\epsilon_{f}f^{\dagger}_{j\sigma}f_{j\sigma}+\sum_{<j,j^{% \prime}>\sigma}t_{jj^{\prime}}c^{\dagger}_{j\sigma}c_{j^{\prime}\sigma}+\sum_{% j,\sigma}(V_{j}f^{\dagger}_{\sigma}c_{j\sigma}+V_{j}^{*}c^{\dagger}_{\sigma}f_% {j\sigma})+U\sum_{j}f^{\dagger}_{j\uparrow}f_{j\uparrow}f^{\dagger}_{j% \downarrow}f_{j\downarrow}
  12. H = i σ ϵ f f i σ f i σ + < j , j > σ t i j j c i j σ c i j σ + i j , σ ( V j f i σ c i j σ + V j * c i j σ f i σ ) + i σ , i σ U 2 n i σ n i σ H=\sum_{i\sigma}\epsilon_{f}f^{\dagger}_{i\sigma}f_{i\sigma}+\sum_{<j,j^{% \prime}>\sigma}t_{ijj^{\prime}}c^{\dagger}_{ij\sigma}c_{ij^{\prime}\sigma}+% \sum_{ij,\sigma}(V_{j}f^{\dagger}_{i\sigma}c_{ij\sigma}+V_{j}^{*}c^{\dagger}_{% ij\sigma}f_{i\sigma})+\sum_{i\sigma,i^{\prime}\sigma^{\prime}}\frac{U}{2}n_{i% \sigma}n_{i^{\prime}\sigma^{\prime}}

Anderson_orthogonality_theorem.html

  1. V ( r ) V(r)
  2. V ( r ) = 0 V(r)=0
  3. V ( r ) 0 V(r)\neq 0

Anomalous_cancellation.html

  1. 64 16 = 64 1 = 4 1 = 4 \frac{64}{16}=\frac{\not 64}{1\not 6}=\frac{4}{1}=4
  2. 26 65 = 2 65 = 2 5 \frac{26}{65}=\frac{2\not 6}{\not 65}=\frac{2}{5}
  3. 19 95 = 1 95 = 1 5 \frac{19}{95}=\frac{1\not 9}{\not 95}=\frac{1}{5}
  4. 98 49 = 98 4 = 8 4 = 2. \frac{98}{49}=\frac{\not 98}{4\not 9}=\frac{8}{4}=2.

Anscombe_transform.html

  1. m m
  2. m m
  3. v v
  4. m = v m=v
  5. A : x 2 x + 3 8 A:x\mapsto 2\sqrt{x+\tfrac{3}{8}}\,
  6. x x
  7. m m
  8. 2 m + 3 / 8 - 1 / ( 4 m ) 2\sqrt{m+3/8}-1/(4\sqrt{m})
  9. m m
  10. x x
  11. m m
  12. y y
  13. A - 1 : y ( y 2 ) 2 - 3 8 A^{-1}:y\mapsto\left(\frac{y}{2}\right)^{2}-\frac{3}{8}
  14. m m
  15. y ( y 2 ) 2 - 1 8 y\mapsto\left(\frac{y}{2}\right)^{2}-\frac{1}{8}
  16. E [ 2 x + 3 8 m ] = 2 x = 0 + ( x + 3 8 m x e - m x ! ) m \operatorname{E}\left[2\sqrt{x+\tfrac{3}{8}}\mid m\right]=2\sum_{x=0}^{+\infty% }\left(\sqrt{x+\tfrac{3}{8}}\cdot\frac{m^{x}e^{-m}}{x!}\right)\mapsto m
  17. y 1 4 y 2 + 1 4 3 2 y - 1 - 11 8 y - 2 + 5 8 3 2 y - 3 - 1 8 . y\mapsto\frac{1}{4}y^{2}+\frac{1}{4}\sqrt{\frac{3}{2}}y^{-1}-\frac{11}{8}y^{-2% }+\frac{5}{8}\sqrt{\frac{3}{2}}y^{-3}-\frac{1}{8}.
  18. A : x x + 1 + x . A:x\mapsto\sqrt{x+1}+\sqrt{x}.\,
  19. A : x 2 x A:x\mapsto 2\sqrt{x}\,

Antiparallel_(mathematics).html

  1. m 1 m_{1}\,
  2. m 2 m_{2}\,
  3. l 1 l_{1}\,
  4. l 2 l_{2}\,
  5. m 1 m_{1}\,
  6. m 2 m_{2}\,
  7. 1 = 2 \angle 1=\angle 2\,
  8. l 1 l_{1}\,
  9. l 2 l_{2}\,
  10. m 1 m_{1}\,
  11. m 2 m_{2}\,
  12. m 1 m_{1}\,
  13. m 2 m_{2}\,
  14. l 1 l_{1}\,
  15. l 2 l_{2}\,
  16. l 1 l_{1}
  17. l 2 l_{2}
  18. A P C \angle APC
  19. m 1 m_{1}\,
  20. m 2 m_{2}\,
  21. l 1 l_{1}\,
  22. l 2 l_{2}\,
  23. m 1 m_{1}\,
  24. m 2 m_{2}\,
  25. 1 = 2 \angle 1=\angle 2\,
  26. l 1 l_{1}\,
  27. l 2 l_{2}\,
  28. A P C \angle APC
  29. m 1 m_{1}\,
  30. m 2 m_{2}\,
  31. l 1 l_{1}\,
  32. l 2 l_{2}\,

Antisymmetrizer.html

  1. 𝒜 \mathcal{A}
  2. 𝒜 \mathcal{A}
  3. 𝒜 \mathcal{A}
  4. Ψ ( 1 , 2 , , N ) with i ( 𝐫 i , σ i ) , \Psi(1,2,\ldots,N)\quad\,\text{with}\quad i\leftrightarrow(\mathbf{r}_{i},% \sigma_{i}),
  5. 3 \mathbb{R}^{3}
  6. Ψ ( 1 , 2 ) - Ψ ( 2 , 1 ) 0 \Psi(1,2)-\Psi(2,1)\neq 0
  7. P ^ i j \hat{P}_{ij}
  8. P ^ i j Ψ ( 1 , 2 , , i , , j , , N ) Ψ ( π ( 1 ) , π ( 2 ) , , π ( i ) , , π ( j ) , , π ( N ) ) Ψ ( 1 , 2 , , j , , i , , N ) = - Ψ ( 1 , 2 , , i , , j , , N ) . \begin{aligned}\displaystyle\hat{P}_{ij}\Psi\big(1,2,\ldots,i,\ldots,j,\ldots,% N\big)&\displaystyle\equiv\Psi\big(\pi(1),\pi(2),\ldots,\pi(i),\ldots,\pi(j),% \ldots,\pi(N)\big)\\ &\displaystyle\equiv\Psi(1,2,\ldots,j,\ldots,i,\ldots,N)\\ &\displaystyle=-\Psi(1,2,\ldots,i,\ldots,j,\ldots,N).\end{aligned}
  9. P ^ i j \hat{P}_{ij}
  10. P ^ Ψ ( 1 , 2 , , N ) Ψ ( π ( 1 ) , π ( 2 ) , , π ( N ) ) = ( - 1 ) π Ψ ( 1 , 2 , , N ) , \hat{P}\Psi\big(1,2,\ldots,N\big)\equiv\Psi\big(\pi(1),\pi(2),\ldots,\pi(N)% \big)=(-1)^{\pi}\Psi(1,2,\ldots,N),
  11. P ^ \hat{P}
  12. 𝒜 1 N ! P S N ( - 1 ) π P ^ . \mathcal{A}\equiv\frac{1}{N!}\sum_{P\in S_{N}}(-1)^{\pi}\hat{P}.
  13. { ( - 1 ) π } \{(-1)^{\pi}\}
  14. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  15. 𝒜 Ψ ( 1 , , N ) = { 0 Ψ ( 1 , , N ) 0. \mathcal{A}\Psi(1,\ldots,N)=\begin{cases}&0\\ &\Psi^{\prime}(1,\dots,N)\neq 0.\end{cases}
  16. P ^ 𝒜 = 𝒜 P ^ = ( - 1 ) π 𝒜 , π S N , \hat{P}\mathcal{A}=\mathcal{A}\hat{P}=(-1)^{\pi}\mathcal{A},\qquad\forall\pi% \in S_{N},
  17. P ^ \hat{P}
  18. P ^ 𝒜 Ψ ( 1 , , N ) P ^ Ψ ( 1 , , N ) = ( - 1 ) π Ψ ( 1 , , N ) , \hat{P}\mathcal{A}\Psi(1,\ldots,N)\equiv\hat{P}\Psi^{\prime}(1,\ldots,N)=(-1)^% {\pi}\Psi^{\prime}(1,\ldots,N),
  19. P ^ \hat{P}
  20. P ^ Ψ = Ψ 𝒜 P ^ Ψ = 𝒜 Ψ - 𝒜 Ψ = 𝒜 Ψ 𝒜 Ψ = 0. \hat{P}\Psi=\Psi\Longrightarrow\mathcal{A}\hat{P}\Psi=\mathcal{A}\Psi% \Longrightarrow-\mathcal{A}\Psi=\mathcal{A}\Psi\Longrightarrow\mathcal{A}\Psi=0.
  21. 𝒜 = 𝒜 . \mathcal{A}^{\dagger}=\mathcal{A}.
  22. H ^ \hat{H}\,
  23. [ 𝒜 , H ^ ] = 0. [\mathcal{A},\hat{H}]=0.
  24. H ^ \hat{H}\,
  25. Ψ ( 1 , 2 , , N ) = ψ n 1 ( 1 ) ψ n 2 ( 2 ) ψ n N ( N ) \Psi(1,2,\ldots,N)=\psi_{n_{1}}(1)\psi_{n_{2}}(2)\cdots\psi_{n_{N}}(N)
  26. N ! 𝒜 Ψ ( 1 , 2 , , N ) = 1 N ! | ψ n 1 ( 1 ) ψ n 1 ( 2 ) ψ n 1 ( N ) ψ n 2 ( 1 ) ψ n 2 ( 2 ) ψ n 2 ( N ) ψ n N ( 1 ) ψ n N ( 2 ) ψ n N ( N ) | \sqrt{N!}\ \mathcal{A}\Psi(1,2,\ldots,N)=\frac{1}{\sqrt{N!}}\begin{vmatrix}% \psi_{n_{1}}(1)&\psi_{n_{1}}(2)&\cdots&\psi_{n_{1}}(N)\\ \psi_{n_{2}}(1)&\psi_{n_{2}}(2)&\cdots&\psi_{n_{2}}(N)\\ \vdots&\vdots&&\vdots\\ \psi_{n_{N}}(1)&\psi_{n_{N}}(2)&\cdots&\psi_{n_{N}}(N)\\ \end{vmatrix}
  27. det ( 𝐁 ) = π S N ( - 1 ) π B 1 , π ( 1 ) B 2 , π ( 2 ) B 3 , π ( 3 ) B N , π ( N ) , \det(\mathbf{B})=\sum_{\pi\in S_{N}}(-1)^{\pi}B_{1,\pi(1)}\cdot B_{2,\pi(2)}% \cdot B_{3,\pi(3)}\cdot\,\cdots\,\cdot B_{N,\pi(N)},
  28. 𝐁 = ( B 1 , 1 B 1 , 2 B 1 , N B 2 , 1 B 2 , 2 B 2 , N B N , 1 B N , 2 B N , N ) . \mathbf{B}=\begin{pmatrix}B_{1,1}&B_{1,2}&\cdots&B_{1,N}\\ B_{2,1}&B_{2,2}&\cdots&B_{2,N}\\ \vdots&\vdots&&\vdots\\ B_{N,1}&B_{N,2}&\cdots&B_{N,N}\\ \end{pmatrix}.
  29. 𝒜 ψ a ( 1 ) ψ b ( 2 ) ψ c ( 3 ) = 1 6 ( ψ a ( 1 ) ψ b ( 2 ) ψ c ( 3 ) + ψ a ( 3 ) ψ b ( 1 ) ψ c ( 2 ) + ψ a ( 2 ) ψ b ( 3 ) ψ c ( 1 ) - ψ a ( 2 ) ψ b ( 1 ) ψ c ( 3 ) - ψ a ( 3 ) ψ b ( 2 ) ψ c ( 1 ) - ψ a ( 1 ) ψ b ( 3 ) ψ c ( 2 ) ) . \begin{aligned}\displaystyle\mathcal{A}\psi_{a}(1)\psi_{b}(2)\psi_{c}(3)=&% \displaystyle\frac{1}{6}\Big(\psi_{a}(1)\psi_{b}(2)\psi_{c}(3)+\psi_{a}(3)\psi% _{b}(1)\psi_{c}(2)+\psi_{a}(2)\psi_{b}(3)\psi_{c}(1)\\ &\displaystyle{}-\psi_{a}(2)\psi_{b}(1)\psi_{c}(3)-\psi_{a}(3)\psi_{b}(2)\psi_% {c}(1)-\psi_{a}(1)\psi_{b}(3)\psi_{c}(2)\Big).\end{aligned}
  30. D 1 6 | ψ a ( 1 ) ψ a ( 2 ) ψ a ( 3 ) ψ b ( 1 ) ψ b ( 2 ) ψ b ( 3 ) ψ c ( 1 ) ψ c ( 2 ) ψ c ( 3 ) | . D\equiv\frac{1}{\sqrt{6}}\begin{vmatrix}\psi_{a}(1)&\psi_{a}(2)&\psi_{a}(3)\\ \psi_{b}(1)&\psi_{b}(2)&\psi_{b}(3)\\ \psi_{c}(1)&\psi_{c}(2)&\psi_{c}(3)\end{vmatrix}.
  31. D = 1 6 ψ a ( 1 ) | ψ b ( 2 ) ψ b ( 3 ) ψ c ( 2 ) ψ c ( 3 ) | - 1 6 ψ a ( 2 ) | ψ b ( 1 ) ψ b ( 3 ) ψ c ( 1 ) ψ c ( 3 ) | + 1 6 ψ a ( 3 ) | ψ b ( 1 ) ψ b ( 2 ) ψ c ( 1 ) ψ c ( 2 ) | , D=\frac{1}{\sqrt{6}}\psi_{a}(1)\begin{vmatrix}\psi_{b}(2)&\psi_{b}(3)\\ \psi_{c}(2)&\psi_{c}(3)\end{vmatrix}-\frac{1}{\sqrt{6}}\psi_{a}(2)\begin{% vmatrix}\psi_{b}(1)&\psi_{b}(3)\\ \psi_{c}(1)&\psi_{c}(3)\end{vmatrix}+\frac{1}{\sqrt{6}}\psi_{a}(3)\begin{% vmatrix}\psi_{b}(1)&\psi_{b}(2)\\ \psi_{c}(1)&\psi_{c}(2)\end{vmatrix},
  32. D = 1 6 ψ a ( 1 ) ( ψ b ( 2 ) ψ c ( 3 ) - ψ b ( 3 ) ψ c ( 2 ) ) - 1 6 ψ a ( 2 ) ( ψ b ( 1 ) ψ c ( 3 ) - ψ b ( 3 ) ψ c ( 1 ) ) + 1 6 ψ a ( 3 ) ( ψ b ( 1 ) ψ c ( 2 ) - ψ b ( 2 ) ψ c ( 1 ) ) . \begin{aligned}\displaystyle D=&\displaystyle\frac{1}{\sqrt{6}}\psi_{a}(1)\Big% (\psi_{b}(2)\psi_{c}(3)-\psi_{b}(3)\psi_{c}(2)\Big)-\frac{1}{\sqrt{6}}\psi_{a}% (2)\Big(\psi_{b}(1)\psi_{c}(3)-\psi_{b}(3)\psi_{c}(1)\Big)\\ &\displaystyle{}+\frac{1}{\sqrt{6}}\psi_{a}(3)\Big(\psi_{b}(1)\psi_{c}(2)-\psi% _{b}(2)\psi_{c}(1)\Big).\end{aligned}
  33. D = 6 𝒜 ψ a ( 1 ) ψ b ( 2 ) ψ c ( 3 ) . D=\sqrt{6}\ \mathcal{A}\psi_{a}(1)\psi_{b}(2)\psi_{c}(3).
  34. Ψ A ( 1 , 2 , , N A ) Ψ B ( N A + 1 , N A + 2 , , N A + N B ) \Psi_{A}(1,2,\dots,N_{A})\Psi_{B}(N_{A}+1,N_{A}+2,\dots,N_{A}+N_{B})
  35. 𝒜 A Ψ A ( 1 , 2 , , N A ) = Ψ A ( 1 , 2 , , N A ) \mathcal{A}^{A}\Psi_{A}(1,2,\dots,N_{A})=\Psi_{A}(1,2,\dots,N_{A})
  36. 𝒜 B Ψ B ( N A + 1 , N A + 2 , , N A + N B ) = Ψ B ( N A + 1 , N A + 2 , , N A + N B ) . \mathcal{A}^{B}\Psi_{B}(N_{A}+1,N_{A}+2,\dots,N_{A}+N_{B})=\Psi_{B}(N_{A}+1,N_% {A}+2,\dots,N_{A}+N_{B}).
  37. 𝒜 A \mathcal{A}^{A}
  38. 𝒜 B \mathcal{A}^{B}
  39. Ψ A ( 1 , 2 , , N A ) \Psi_{A}(1,2,\dots,N_{A})
  40. Ψ B ( N A + 1 , N A + 2 , , N A + N B ) \Psi_{B}(N_{A}+1,N_{A}+2,\dots,N_{A}+N_{B})
  41. 𝒜 A B \mathcal{A}^{AB}
  42. S N A S N B S N A + B B π S N A + B B : π = τ π A π B , π A S N A , π B S N B , S_{N_{A}}\otimes S_{N_{B}}\subset S_{N_{A}+B_{B}}\Longrightarrow\forall\pi\in S% _{N_{A}+B_{B}}:\quad\pi=\tau\pi_{A}\pi_{B},\quad\pi_{A}\in S_{N_{A}},\;\;\pi_{% B}\in S_{N_{B}},
  43. ( - 1 ) π = ( - 1 ) τ ( - 1 ) π A ( - 1 ) π B , (-1)^{\pi}=(-1)^{\tau}(-1)^{\pi_{A}}(-1)^{\pi_{B}},
  44. 𝒜 A B = 𝒜 ~ A B 𝒜 A 𝒜 B with 𝒜 ~ A B = T = 1 C A B ( - 1 ) τ T ^ , C A B = ( N A + N B N A ) . \mathcal{A}^{AB}=\tilde{\mathcal{A}}^{AB}\mathcal{A}^{A}\mathcal{A}^{B}\quad% \hbox{with}\quad\tilde{\mathcal{A}}^{AB}=\sum_{T=1}^{C_{AB}}(-1)^{\tau}\hat{T}% ,\quad C_{AB}={\left({{N_{A}+N_{B}}\atop{N_{A}}}\right)}.
  45. T ^ \hat{T}
  46. 𝒜 ~ A B \tilde{\mathcal{A}}^{AB}
  47. 𝒜 A B Ψ A ( 1 , 2 , , N A ) Ψ B ( N A + 1 , N A + 2 , , N A + N B ) = 𝒜 ~ A B Ψ A ( 1 , 2 , , N A ) Ψ B ( N A + 1 , N A + 2 , , N A + N B ) , \begin{aligned}\displaystyle\mathcal{A}^{AB}\Psi_{A}(1,2,\dots,N_{A})&% \displaystyle\Psi_{B}(N_{A}+1,N_{A}+2,\dots,N_{A}+N_{B})\\ &\displaystyle=\tilde{\mathcal{A}}^{AB}\Psi_{A}(1,2,\dots,N_{A})\Psi_{B}(N_{A}% +1,N_{A}+2,\dots,N_{A}+N_{B}),\end{aligned}
  48. 𝒜 ~ A B \tilde{\mathcal{A}}^{AB}

Aortic_valve_area_calculation.html

  1. Aortic Valve Area (cm 2 ) = LVOT diameter 2 0.78540 LVOT VTI Aortic Valve VTI \,\text{Aortic Valve Area (cm}^{2}\,\text{)}={\,\text{LVOT diameter}^{2}\cdot 0% .78540\cdot\,\text{LVOT VTI}\over\,\text{Aortic Valve VTI}}
  2. Valve Area (cm 2 ) = Cardiac Output ( ml min ) Heart rate ( beats min ) Systolic ejection period (s) 44.3 mean Gradient (mmHg) \,\text{Valve Area (cm}^{2}\,\text{)}=\frac{\,\text{Cardiac Output }(\frac{\,% \text{ml}}{\,\text{min}})}{\,\text{Heart rate }(\frac{\,\text{beats}}{\,\text{% min}})\cdot\,\text{Systolic ejection period (s)}\cdot 44.3\cdot\sqrt{\,\text{% mean Gradient (mmHg)}}}
  3. Aortic Valve Area = 5000 ml min 80 beats min 0.33 s 44.3 50 mmHg 0.6 cm 2 \,\text{Aortic Valve Area}=\frac{5000\frac{\,\text{ml}}{\,\text{min}}}{80\frac% {\,\text{beats}}{\,\text{min}}\cdot 0.33\,\text{s}\cdot 44.3\cdot\sqrt{50\,% \text{mmHg}}}\approx 0.6\,\text{cm}^{2}
  4. heart rate (bpm) systolic ejection period (s) 44.3 1000 \,\text{heart rate (bpm)}\cdot\,\text{systolic ejection period (s)}\cdot 44.3% \approx 1000
  5. Aortic Valve area (cm 2 ) Cardiac Output ( litre min ) Peak to Peak Gradient (mmHg) \,\text{Aortic Valve area (cm}^{2}\,\text{)}\approx\frac{\,\text{Cardiac % Output}(\frac{\,\text{litre}}{\,\text{min}})}{\sqrt{\,\text{Peak to Peak % Gradient (mmHg)}}}
  6. Aortic valve area 3.5 50 0.5 cm 2 \,\text{Aortic valve area}\approx\frac{3.5}{\sqrt{50}}\approx 0.5\ \,\text{cm}% ^{2}
  7. Valve Area (cm 2 ) = (0.83 2 + Q ( m l m i n ) 60 0.35 mean Gradient (dynes/cm2) )0.5 - 0.87 \,\text{Valve Area (cm}^{2}\,\text{)}=\,\text{(0.83}^{2}+\frac{\frac{\,\text{Q% (}\frac{ml}{min})}{\,\text{60}}}{\,\text{0.35 }\cdot\sqrt{\,\text{mean % Gradient (dynes/cm2)}}}\,\text{)}\text{0.5}-\,\text{0.87}
  8. Valve Area (cm 2 ) = (0.83 2 + 13440 ( m l m i n ) 60 0.35 22665 (dynes/cm2) )0.5 - 0.87 1.35 cm 2 \,\text{Valve Area (cm}^{2}\,\text{)}=\,\text{(0.83}^{2}+\frac{\frac{\,\text{1% 3440 (}\frac{ml}{min})}{\,\text{60}}}{\,\text{0.35 }\cdot\sqrt{\,\text{22665 (% dynes/cm2)}}}\,\text{)}\text{0.5}-\,\text{0.87}\approx 1.35\,\text{cm}^{2}

Appell's_equation_of_motion.html

  1. Q r = S α r Q_{r}=\frac{\partial S}{\partial\alpha_{r}}
  2. α r = q ¨ \alpha_{r}=\ddot{q}
  3. d W = r = 1 D Q r d q r dW=\sum_{r=1}^{D}Q_{r}dq_{r}
  4. S = 1 2 k = 1 N m k 𝐚 k 2 , S=\frac{1}{2}\sum_{k=1}^{N}m_{k}\mathbf{a}_{k}^{2}\,,
  5. 𝐚 k = 𝐫 ¨ k = d 2 𝐫 k d t 2 \mathbf{a}_{k}=\ddot{\mathbf{r}}_{k}=\frac{d^{2}\mathbf{r}_{k}}{dt^{2}}
  6. d 𝐫 k = r = 1 D d q r 𝐫 k q r d\mathbf{r}_{k}=\sum_{r=1}^{D}dq_{r}\frac{\partial\mathbf{r}_{k}}{\partial q_{% r}}
  7. 𝐚 k α r = 𝐫 k q r \frac{\partial\mathbf{a}_{k}}{\partial\alpha_{r}}=\frac{\partial\mathbf{r}_{k}% }{\partial q_{r}}
  8. d W = r = 1 D Q r d q r = k = 1 N 𝐅 k d 𝐫 k = k = 1 N m k 𝐚 k d 𝐫 k dW=\sum_{r=1}^{D}Q_{r}dq_{r}=\sum_{k=1}^{N}\mathbf{F}_{k}\cdot d\mathbf{r}_{k}% =\sum_{k=1}^{N}m_{k}\mathbf{a}_{k}\cdot d\mathbf{r}_{k}
  9. 𝐅 k = m k 𝐚 k \mathbf{F}_{k}=m_{k}\mathbf{a}_{k}
  10. d W = r = 1 D Q r d q r = k = 1 N m k 𝐚 k r = 1 D d q r ( 𝐫 k q r ) = r = 1 D d q r k = 1 N m k 𝐚 k ( 𝐫 k q r ) dW=\sum_{r=1}^{D}Q_{r}dq_{r}=\sum_{k=1}^{N}m_{k}\mathbf{a}_{k}\cdot\sum_{r=1}^% {D}dq_{r}\left(\frac{\partial\mathbf{r}_{k}}{\partial q_{r}}\right)=\sum_{r=1}% ^{D}dq_{r}\sum_{k=1}^{N}m_{k}\mathbf{a}_{k}\cdot\left(\frac{\partial\mathbf{r}% _{k}}{\partial q_{r}}\right)
  11. Q r = k = 1 N m k 𝐚 k ( 𝐫 k q r ) = k = 1 N m k 𝐚 k ( 𝐚 k α r ) Q_{r}=\sum_{k=1}^{N}m_{k}\mathbf{a}_{k}\cdot\left(\frac{\partial\mathbf{r}_{k}% }{\partial q_{r}}\right)=\sum_{k=1}^{N}m_{k}\mathbf{a}_{k}\cdot\left(\frac{% \partial\mathbf{a}_{k}}{\partial\alpha_{r}}\right)
  12. S α r = α r 1 2 k = 1 N m k | 𝐚 k | 2 = k = 1 N m k 𝐚 k ( 𝐚 k α r ) \frac{\partial S}{\partial\alpha_{r}}=\frac{\partial}{\partial\alpha_{r}}\frac% {1}{2}\sum_{k=1}^{N}m_{k}\left|\mathbf{a}_{k}\right|^{2}=\sum_{k=1}^{N}m_{k}% \mathbf{a}_{k}\cdot\left(\frac{\partial\mathbf{a}_{k}}{\partial\alpha_{r}}\right)
  13. S α r = Q r \frac{\partial S}{\partial\alpha_{r}}=Q_{r}
  14. s y m b o l ω symbol\omega
  15. s y m b o l α = d s y m b o l ω d t symbol\alpha=\frac{dsymbol\omega}{dt}
  16. δ s y m b o l ϕ \delta symbol\phi
  17. d W = 𝐍 δ s y m b o l ϕ dW=\mathbf{N}\cdot\delta symbol\phi
  18. 𝐯 k = s y m b o l ω × 𝐫 k \mathbf{v}_{k}=symbol\omega\times\mathbf{r}_{k}
  19. 𝐚 k = d 𝐯 k d t = s y m b o l α × 𝐫 k + s y m b o l ω × 𝐯 k \mathbf{a}_{k}=\frac{d\mathbf{v}_{k}}{dt}=symbol\alpha\times\mathbf{r}_{k}+% symbol\omega\times\mathbf{v}_{k}
  20. S = 1 2 k = 1 N m k ( 𝐚 k 𝐚 k ) = 1 2 k = 1 N m k { ( s y m b o l α × 𝐫 k ) 2 + ( s y m b o l ω × 𝐯 k ) 2 + 2 ( s y m b o l α × 𝐫 k ) ( s y m b o l ω × 𝐯 k ) } S=\frac{1}{2}\sum_{k=1}^{N}m_{k}\left(\mathbf{a}_{k}\cdot\mathbf{a}_{k}\right)% =\frac{1}{2}\sum_{k=1}^{N}m_{k}\left\{\left(symbol\alpha\times\mathbf{r}_{k}% \right)^{2}+\left(symbol\omega\times\mathbf{v}_{k}\right)^{2}+2\left(symbol% \alpha\times\mathbf{r}_{k}\right)\cdot\left(symbol\omega\times\mathbf{v}_{k}% \right)\right\}
  21. s y m b o l α symbol\alpha
  22. I x x α x - ( I y y - I z z ) ω y ω z = N x I_{xx}\alpha_{x}-\left(I_{yy}-I_{zz}\right)\omega_{y}\omega_{z}=N_{x}
  23. I y y α y - ( I z z - I x x ) ω z ω x = N y I_{yy}\alpha_{y}-\left(I_{zz}-I_{xx}\right)\omega_{z}\omega_{x}=N_{y}
  24. I z z α z - ( I x x - I y y ) ω x ω y = N z I_{zz}\alpha_{z}-\left(I_{xx}-I_{yy}\right)\omega_{x}\omega_{y}=N_{z}

Approximate_Bayesian_computation.html

  1. θ \theta
  2. D D
  3. D D
  4. θ \theta
  5. p ( θ | D ) = p ( D | θ ) p ( θ ) p ( D ) p(\theta|D)=\frac{p(D|\theta)p(\theta)}{p(D)}
  6. p ( θ | D ) p(\theta|D)
  7. p ( D | θ ) p(D|\theta)
  8. p ( θ ) p(\theta)
  9. p ( D ) p(D)
  10. θ \theta
  11. D D
  12. θ \theta
  13. p ( θ ) p(\theta)
  14. θ \theta
  15. θ \theta
  16. p ( D ) p(D)
  17. p ( D | θ ) p(D|\theta)
  18. p ( θ ) p(\theta)
  19. θ \theta
  20. D ^ \hat{D}
  21. M M
  22. θ \theta
  23. D ^ \hat{D}
  24. D D
  25. D ^ \hat{D}
  26. ϵ 0 \epsilon\geq 0
  27. ρ ( D ^ , D ) ϵ \rho(\hat{D},D)\leq\epsilon
  28. ρ ( D ^ , D ) \rho(\hat{D},D)
  29. D ^ \hat{D}
  30. D D
  31. D ^ = D \hat{D}=D
  32. D ^ \hat{D}
  33. D D
  34. D D
  35. S ( D ) S(D)
  36. D D
  37. ρ ( S ( D ^ ) , S ( D ) ) ϵ \rho(S(\hat{D}),S(D))\leq\epsilon
  38. θ \theta
  39. D D
  40. θ \theta
  41. S ( D ) S(D)
  42. θ \theta
  43. θ \theta
  44. γ \gamma
  45. γ \gamma
  46. ρ ( , ) \rho(\cdot,\cdot)
  47. ϵ = 2 \epsilon=2
  48. θ \theta
  49. θ = 0.25 \theta=0.25
  50. γ = 0.8 \gamma=0.8
  51. ω E = 6 \omega_{E}=6
  52. θ \theta
  53. [ 0 , 1 ] [0,1]
  54. γ \gamma
  55. γ = 0.8 \gamma=0.8
  56. θ i , i = 1 , , n \theta_{i},i=1,\ldots,n
  57. n n
  58. θ i \theta_{i}
  59. ω S , i \omega_{S,i}
  60. ρ ( ω S , i , ω E ) \rho(\omega_{S,i},\omega_{E})
  61. ω S , i , i = 1 , , n \omega_{S,i},i=1,\ldots,n
  62. ρ ( ω S , i , ω E ) = | ω S , i - ω E | \rho(\omega_{S,i},\omega_{E})=|\omega_{S,i}-\omega_{E}|
  63. ϵ \epsilon
  64. θ \theta
  65. ϵ = 0 \epsilon=0
  66. ϵ = 2 \epsilon=2
  67. ϵ = 0 \epsilon=0
  68. θ \theta
  69. θ = 0.25 \theta=0.25
  70. M 1 M_{1}
  71. M 2 M_{2}
  72. B 1 , 2 B_{1,2}
  73. p ( M 1 | D ) p ( M 2 | D ) = p ( D | M 1 ) p ( D | M 2 ) p ( M 1 ) p ( M 2 ) = B 1 , 2 p ( M 1 ) p ( M 2 ) \frac{p(M_{1}|D)}{p(M_{2}|D)}=\frac{p(D|M_{1})}{p(D|M_{2})}\frac{p(M_{1})}{p(M% _{2})}=B_{1,2}\frac{p(M_{1})}{p(M_{2})}
  74. p ( M 1 ) = p ( M 2 ) p(M_{1})=p(M_{2})
  75. ϵ \epsilon
  76. ϵ \epsilon
  77. ϵ \epsilon
  78. p ( θ | ρ ( D ^ , D ) ϵ ) p(\theta|\rho(\hat{D},D)\leq\epsilon)
  79. p ( θ | D ) p(\theta|D)
  80. p ( θ | ρ ( D ^ , D ) ϵ ) p(\theta|\rho(\hat{D},D)\leq\epsilon)
  81. p ( θ | D ) p(\theta|D)
  82. p ( θ | ρ ( D ^ , D ) ϵ ) p(\theta|\rho(\hat{D},D)\leq\epsilon)
  83. p ( θ | D ) p(\theta|D)
  84. ϵ \epsilon
  85. ϵ \epsilon
  86. ϵ \epsilon
  87. ϵ \epsilon
  88. ϵ \epsilon
  89. ϵ \epsilon
  90. ϵ \epsilon
  91. S ( D ) S(D)
  92. B 1 , 2 s B_{1,2}^{s}
  93. B 1 , 2 B_{1,2}
  94. B 1 , 2 s B_{1,2}^{s}
  95. B 1 , 2 = p ( D | M 1 ) p ( D | M 2 ) = p ( D | S ( D ) , M 1 ) p ( D | S ( D ) , M 2 ) p ( S ( D ) | M 1 ) p ( S ( D ) | M 2 ) = p ( D | S ( D ) , M 1 ) p ( D | S ( D ) , M 2 ) B 1 , 2 s B_{1,2}=\frac{p(D|M_{1})}{p(D|M_{2})}=\frac{p(D|S(D),M_{1})}{p(D|S(D),M_{2})}% \frac{p(S(D)|M_{1})}{p(S(D)|M_{2})}=\frac{p(D|S(D),M_{1})}{p(D|S(D),M_{2})}B_{% 1,2}^{s}
  96. S ( D ) S(D)
  97. M 1 M_{1}
  98. M 2 M_{2}
  99. p ( D | S ( D ) , M 1 ) = p ( D | S ( D ) , M 2 ) p(D|S(D),M_{1})=p(D|S(D),M_{2})
  100. B 1 , 2 = B 1 , 2 s B_{1,2}=B_{1,2}^{s}
  101. B 1 , 2 B_{1,2}
  102. B 1 , 2 s B_{1,2}^{s}
  103. M 1 M_{1}
  104. M 2 M_{2}
  105. M M
  106. M 1 M_{1}
  107. M 2 M_{2}
  108. S ( D ) S(D)
  109. D D
  110. S ( D ) S(D)

Archard_equation.html

  1. Q = K W L H Q=\frac{KWL}{H}
  2. W L WL
  3. δ W \,\delta W
  4. a \,a
  5. δ W = P π a 2 \delta W=P\pi{a^{2}}\,\!
  6. δ V \,\delta V
  7. δ V = 2 3 π a 3 \delta V=\frac{2}{3}\pi a^{3}
  8. δ Q \,\delta Q
  9. δ Q = δ V 2 a = π a 2 3 δ W 3 P δ W 3 H \delta Q=\frac{\delta V}{2a}=\frac{\pi a^{2}}{3}\equiv\frac{\delta W}{3P}% \approx\frac{\delta W}{3H}
  10. P H \,P\approx H
  11. Q \,Q

Arden_Buck_equation.html

  1. P s ( T ) = 6.1121 exp ( ( 18.678 - T 234.5 ) ( T 257.14 + T ) ) P_{s}\left(T\right)=6.1121\exp\left(\left(18.678-\frac{T}{234.5}\right)\left(% \frac{T}{257.14+T}\right)\right)

Aristotelian_physics.html

  1. d y d t = y {dy\over dt}=y
  2. y ( 0 ) = 0 y(0)=0
  3. y = 0 y=0

Aronszajn_line.html

  1. 1 \aleph_{1}
  2. ω 1 \omega_{1}\,
  3. ω 1 \omega_{1}\,

Artin–Rees_lemma.html

  1. I n M N = I n - k ( ( I k M ) N ) . I^{n}M\cap N=I^{n-k}((I^{k}M)\cap N).
  2. B I R = n = 0 I n B_{I}R=\textstyle\bigoplus_{n=0}^{\infty}I^{n}
  3. M = M 0 M 1 M 2 M=M_{0}\supset M_{1}\supset M_{2}\supset\cdots
  4. I M n M n + 1 IM_{n}\subset M_{n+1}
  5. I M n = M n + 1 IM_{n}=M_{n+1}
  6. B I M = n = 0 M n B_{I}M=\textstyle\bigoplus_{n=0}^{\infty}M_{n}
  7. B I R B_{I}R
  8. M i M_{i}
  9. B I M B_{I}M
  10. B I R B_{I}R
  11. B I M B_{I}M
  12. k + 1 k+1
  13. M 0 , , M k M_{0},\dots,M_{k}
  14. B I M B_{I}M
  15. j = 0 k M j \textstyle\bigoplus_{j=0}^{k}M_{j}
  16. n k n\geq k
  17. M n M_{n}
  18. f = a i j g i j , a i j I n - j f=\sum a_{ij}g_{ij},\quad a_{ij}\in I^{n-j}
  19. g i j g_{ij}
  20. M j , j k M_{j},j\leq k
  21. f I n - k M k f\in I^{n-k}M_{k}
  22. M n = I n M M_{n}=I^{n}M
  23. M n M_{n}
  24. B I M B_{I}M
  25. B I R B_{I}R
  26. B I R R [ I t ] B_{I}R\simeq R[It]
  27. R [ I t ] R[It]
  28. B I M B_{I}M
  29. B I R B_{I}R
  30. B I N B_{I}N
  31. N n = M n N N_{n}=M_{n}\cap N
  32. n = 1 I n = 0 \textstyle\bigcap_{n=1}^{\infty}I^{n}=0
  33. n k n\geq k
  34. I n N = I n - k ( I k N ) . I^{n}\cap N=I^{n-k}(I^{k}\cap N).
  35. N = I N N=IN
  36. N = 0 N=0

Assortativity.html

  1. r = j k j k ( e j k - q j q k ) σ q 2 r=\frac{\sum_{jk}{jk(e_{jk}-q_{j}q_{k})}}{\sigma_{q}^{2}}
  2. q k q_{k}
  3. p k p_{k}
  4. q k = p k + 1 j 1 p j q_{k}=\frac{p_{k+1}}{\sum_{j\geq 1}p_{j}}
  5. e j k e_{jk}
  6. j k e j k = 1 \sum_{jk}{e_{jk}}=1\,
  7. j e j k = q k \sum_{j}{e_{jk}}=q_{k}\,
  8. r ( in , in ) r(\,\text{in},\,\text{in})
  9. r ( out , out ) r(\,\text{out},\,\text{out})
  10. r ( in , in ) r(\,\text{in},\,\text{in})
  11. r ( in , out ) r(\,\text{in},\,\text{out})
  12. r ( out , in ) r(\,\text{out},\,\text{in})
  13. r ( out , out ) r(\,\text{out},\,\text{out})
  14. ( α , β ) (\alpha,\beta)
  15. ( α , β ) = ( out , in ) (\alpha,\beta)=(\,\text{out},\,\text{in})
  16. E E
  17. 1 , , E 1,\ldots,E
  18. i i
  19. j i α j^{\alpha}_{i}
  20. α \alpha
  21. k i β k^{\beta}_{i}
  22. β \beta
  23. i i
  24. j α ¯ \bar{j^{\alpha}}
  25. k β ¯ \bar{k^{\beta}}
  26. α \alpha
  27. β \beta
  28. r ( α , β ) = i ( j i α - j α ¯ ) ( k i β - k β ¯ ) i ( j i α - j α ¯ ) 2 i ( k i β - k β ¯ ) 2 . r(\alpha,\beta)=\frac{\sum_{i}(j^{\alpha}_{i}-\bar{j^{\alpha}})(k^{\beta}_{i}-% \bar{k^{\beta}})}{\sqrt{\sum_{i}(j^{\alpha}_{i}-\bar{j^{\alpha}})^{2}}\sqrt{% \sum_{i}(k^{\beta}_{i}-\bar{k^{\beta}})^{2}}}.
  29. k n n \langle k_{nn}\rangle
  30. k n n = k k P ( k | k ) \langle k_{nn}\rangle=\sum_{k^{\prime}}{k^{\prime}P(k^{\prime}|k)}
  31. P ( k | k ) P(k^{\prime}|k)
  32. ρ = j ( j + 1 ) ( k ¯ - μ q ) 2 M σ q 2 \rho=\frac{j\ \left(j+1\right)\left(\overline{k}-\ {\mu}_{q}\right)}{2M{\sigma% }^{2}_{q}}
  33. j j
  34. k ¯ \overline{k}
  35. r d r_{d}
  36. ρ d = j o u t 2 ( k ¯ i n - μ q i n ) + j i n 2 ( k ¯ o u t - μ q o u t ) 2 M σ q i n σ q o u t {\rho}_{d}=\ \frac{{j_{out}}^{2}\left({\overline{k}}_{in}-\ {\mu}^{in}_{q}% \right)+\ {j_{in}}^{2}\left({\overline{k}}_{out}-\ {\mu}^{out}_{q}\right)}{2\ % M{\sigma}^{in}_{q}{\sigma}^{out}_{q}}
  37. j o u t j_{out}
  38. j i n j_{in}
  39. k ¯ i n {\overline{k}}_{in}
  40. v v
  41. k ¯ o u t {\overline{k}}_{out}
  42. v v
  43. σ q i n 0 {\sigma}^{in}_{q}\ \neq 0
  44. σ q o u t 0 \ {\ \sigma}^{out}_{q}\ \neq 0
  45. σ q i n {\sigma}^{in}_{q}
  46. σ q o u t {\ \sigma}^{out}_{q}
  47. r d = i = 1 N ρ d r_{d}=\ \sum^{N}_{i=1}{{\rho}_{d}}
  48. r 0 r\to 0
  49. ( log 2 N ) / N (\log^{2}N)/N
  50. N N

Atkinson's_theorem.html

  1. H = Ker ( T ) Ker ( T ) . H=\operatorname{Ker}(T)^{\perp}\oplus\operatorname{Ker}(T).

Atmospheric_tide.html

  1. A = A 0 exp ( - z / 2 H ) A=A_{0}\exp(-z/2H)\,
  2. A 0 A_{0}
  3. z z
  4. H H
  5. u t - 2 Ω sin φ v + 1 a cos φ Φ λ = 0 \frac{\partial u^{\prime}}{\partial t}\,-\,2\Omega\sin\varphi\,v^{\prime}\,+\,% \frac{1}{a\,\cos\varphi}\,\frac{\partial\Phi^{\prime}}{\partial\lambda}=0
  6. v t + 2 Ω sin φ u + 1 a Φ φ = 0 \frac{\partial v^{\prime}}{\partial t}\,+\,2\Omega\sin\varphi\,u^{\prime}\,+\,% \frac{1}{a}\,\frac{\partial\Phi^{\prime}}{\partial\varphi}=0
  7. 2 t z Φ + N 2 w = κ J H \frac{\partial^{2}}{\partial t\partial z}\Phi^{\prime}\,+\,N^{2}w^{\prime}=% \frac{\kappa J^{\prime}}{H}
  8. 1 a cos φ ( u λ + φ ( v cos φ ) ) + 1 ϱ o z ( ϱ o w ) = 0 \frac{1}{a\,\cos\varphi}\,\left(\frac{\partial u^{\prime}}{\partial\lambda}\,+% \,\frac{\partial}{\partial\varphi}(v^{\prime}\,\cos\varphi)\right)\,+\,\frac{1% }{\varrho_{o}}\,\frac{\partial}{\partial z}(\varrho_{o}w^{\prime})=0
  9. u u
  10. v v
  11. w w
  12. Φ \Phi
  13. g ( z , φ ) d z \int g(z,\varphi)\,dz
  14. N 2 N^{2}
  15. Ω \Omega
  16. ϱ o \varrho_{o}
  17. exp ( - z / H ) \propto\exp(-z/H)
  18. z z
  19. λ \lambda
  20. φ \varphi
  21. J J
  22. a a
  23. g g
  24. H H
  25. t t
  26. s s
  27. σ \sigma
  28. s s
  29. σ \sigma
  30. Φ ( φ , λ , z , t ) = Φ ^ ( φ , z ) e i ( s λ - σ t ) \Phi^{\prime}(\varphi,\lambda,z,t)=\hat{\Phi}(\varphi,z)\,e^{i(s\lambda-\sigma t)}
  31. Φ ^ ( φ , z ) = n Θ n ( φ ) G n ( z ) \hat{\Phi}(\varphi,z)=\sum_{n}\Theta_{n}(\varphi)\,G_{n}(z)
  32. L Θ n + ε n Θ n = 0 {L}{\Theta}_{n}+\varepsilon_{n}{\Theta}_{n}=0
  33. L = μ [ ( 1 - μ 2 ) ( η 2 - μ 2 ) μ ] - 1 η 2 - μ 2 [ - s η ( η 2 + μ 2 ) ( η 2 - μ 2 ) + s 2 1 - μ 2 ] {L}=\frac{\partial}{\partial\mu}\left[\frac{(1-\mu^{2})}{(\eta^{2}-\mu^{2})}\,% \frac{\partial}{\partial\mu}\right]-\frac{1}{\eta^{2}-\mu^{2}}\,\left[-\frac{s% }{\eta}\,\frac{(\eta^{2}+\mu^{2})}{(\eta^{2}-\mu^{2})}+\frac{s^{2}}{1-\mu^{2}}\right]
  34. μ = sin φ \mu=\sin\varphi
  35. η = σ / ( 2 Ω ) \eta=\sigma/(2\Omega)
  36. ε n = ( 2 Ω a ) 2 / g h n . \varepsilon_{n}=(2\Omega a)^{2}/gh_{n}.\,
  37. Θ n \Theta_{n}
  38. ε n \varepsilon_{n}
  39. h n h_{n}
  40. 2 G n x 2 + α n 2 G n = F n ( x ) \frac{\partial^{2}G^{\star}_{n}}{\partial x^{2}}\,+\,\alpha_{n}^{2}\,G^{\star}% _{n}=F_{n}(x)
  41. G n ( x ) { e - | α n | x : α n 2 < 0 , evanescent or trapped e i α n x : α n 2 > 0 , propagating e ( κ - 1 2 ) x : h n = H / ( 1 - κ ) , F n ( x ) = 0 x , Lamb waves (free solutions) G^{\star}_{n}(x)\sim\begin{cases}e^{-|\alpha_{n}|x}&\,\text{:}\,\alpha_{n}^{2}% <0,\,\,\text{ evanescent or trapped}\\ e^{i\alpha_{n}x}&\,\text{:}\,\alpha_{n}^{2}>0,\,\,\text{ propagating}\\ e^{\left(\kappa-\frac{1}{2}\right)x}&\,\text{:}\,h_{n}=H/(1-\kappa),F_{n}(x)=0% \,\forall x,\,\,\text{ Lamb waves (free solutions)}\end{cases}
  42. α n 2 = κ H / h n - 1 / 4 \alpha_{n}^{2}=\kappa H/h_{n}-1/4
  43. x = z / H x=z/H
  44. G n = G n ϱ o 1 / 2 N - 1 G^{\star}_{n}=G_{n}\,\varrho_{o}^{1/2}\,N^{-1}
  45. F n ( x ) = - ϱ o - 1 / 2 i σ N x ( ϱ o J n ) . F_{n}(x)=-\frac{\varrho_{o}^{-1/2}}{i\sigma N}\,\frac{\partial}{\partial x}(% \varrho_{o}J_{n}).
  46. h n h_{n}
  47. λ z , n \lambda_{z,n}
  48. α n / H \alpha_{n}/H
  49. λ z , n = 2 π H α n = 2 π H κ H h n - 1 4 . \lambda_{z,n}=\frac{2\pi\,H}{\alpha_{n}}=\frac{2\pi\,H}{\sqrt{\frac{\kappa H}{% h_{n}}-\frac{1}{4}}}.
  50. ( α n 2 > 0 ) (\alpha_{n}^{2}>0)
  51. c g z , n = H σ α n c_{gz,n}=H\frac{\partial\sigma}{\partial\alpha_{n}}
  52. α n > 0 \alpha_{n}>0
  53. ( σ < 0 ) (\sigma<0)
  54. α n < 0 \alpha_{n}<0
  55. ( σ > 0 ) (\sigma>0)
  56. x = z / H x=z/H
  57. K n = s λ + α n x - σ t = 0. K_{n}=s\lambda+\alpha_{n}x-\sigma t=0.
  58. λ \lambda
  59. e z / 2 H \propto e^{z/2H}

Atomic_form_factor.html

  1. ρ ( 𝐫 ) \rho(\mathbf{r})
  2. f ( 𝐐 ) f(\mathbf{Q})
  3. f ( 𝐐 ) = ρ ( 𝐫 ) e i 𝐐 𝐫 d 3 𝐫 f(\mathbf{Q})=\int\rho(\mathbf{r})e^{i\mathbf{Q}\cdot\mathbf{r}}\mathrm{d}^{3}% \mathbf{r}
  4. ρ ( 𝐫 ) \rho(\mathbf{r})
  5. 𝐫 = 0 \mathbf{r}=0
  6. 𝐐 \mathbf{Q}
  7. ρ \rho
  8. 𝐫 \mathbf{r}
  9. f f
  10. 𝐐 \mathbf{Q}
  11. Z Z
  12. ρ ( r ) \rho(r)
  13. ρ ( r ) \rho(r)
  14. ρ ( r ) \rho(r)
  15. b b
  16. Q Q
  17. ρ ( r ) \rho(r)

Atomic_mass_constant.html

  1. 12 {}^{12}
  2. m u = m e A r ( e ) = 2 R h A r ( e ) c α 2 m_{\rm u}=\frac{m_{\rm e}}{A_{\rm r}({\rm e})}=\frac{2R_{\infty}h}{A_{\rm r}({% \rm e})c\alpha^{2}}

Attack_Vector:_Tactical.html

  1. I s p I_{sp}

Attenuation_coefficient.html

  1. μ = - 1 Φ e d Φ e d z , \mu=-\frac{1}{\Phi_{\mathrm{e}}}\frac{\mathrm{d}\Phi_{\mathrm{e}}}{\mathrm{d}z},
  2. μ ν = - 1 Φ e , ν d Φ e , ν d z , \mu_{\nu}=-\frac{1}{\Phi_{\mathrm{e},\nu}}\frac{\mathrm{d}\Phi_{\mathrm{e},\nu% }}{\mathrm{d}z},
  3. μ λ = - 1 Φ e , λ d Φ e , λ d z , \mu_{\lambda}=-\frac{1}{\Phi_{\mathrm{e},\lambda}}\frac{\mathrm{d}\Phi_{% \mathrm{e},\lambda}}{\mathrm{d}z},
  4. μ Ω = - 1 L e , Ω d L e , Ω d z , \mu_{\Omega}=-\frac{1}{L_{\mathrm{e},\Omega}}\frac{\mathrm{d}L_{\mathrm{e},% \Omega}}{\mathrm{d}z},
  5. μ Ω , ν = - 1 L e , Ω , ν d L e , Ω , ν d z , \mu_{\Omega,\nu}=-\frac{1}{L_{\mathrm{e},\Omega,\nu}}\frac{\mathrm{d}L_{% \mathrm{e},\Omega,\nu}}{\mathrm{d}z},
  6. μ Ω , λ = - 1 L e , Ω , λ d L e , Ω , λ d z , \mu_{\Omega,\lambda}=-\frac{1}{L_{\mathrm{e},\Omega,\lambda}}\frac{\mathrm{d}L% _{\mathrm{e},\Omega,\lambda}}{\mathrm{d}z},
  7. μ = μ a + μ s , \mu=\mu_{\mathrm{a}}+\mu_{\mathrm{s}},
  8. μ ν = μ a , ν + μ s , ν , \mu_{\nu}=\mu_{\mathrm{a},\nu}+\mu_{\mathrm{s},\nu},
  9. μ λ = μ a , λ + μ s , λ , \mu_{\lambda}=\mu_{\mathrm{a},\lambda}+\mu_{\mathrm{s},\lambda},
  10. μ Ω = μ a , Ω + μ s , Ω , \mu_{\Omega}=\mu_{\mathrm{a},\Omega}+\mu_{\mathrm{s},\Omega},
  11. μ Ω , ν = μ a , Ω , ν + μ s , Ω , ν , \mu_{\Omega,\nu}=\mu_{\mathrm{a},\Omega,\nu}+\mu_{\mathrm{s},\Omega,\nu},
  12. μ Ω , λ = μ a , Ω , λ + μ s , Ω , λ . \mu_{\Omega,\lambda}=\mu_{\mathrm{a},\Omega,\lambda}+\mu_{\mathrm{s},\Omega,% \lambda}.
  13. μ ρ m , μ a ρ m , μ s ρ m , \frac{\mu}{\rho_{m}},\quad\frac{\mu_{\mathrm{a}}}{\rho_{m}},\quad\frac{\mu_{% \mathrm{s}}}{\rho_{m}},
  14. μ 10 = μ ln 10 . \mu_{10}=\frac{\mu}{\ln 10}.
  15. T = e - 0 μ ( z ) d z = 10 - 0 μ 10 ( z ) d z , T=e^{-\int_{0}^{\ell}\mu(z)\mathrm{d}z}=10^{-\int_{0}^{\ell}\mu_{10}(z)\mathrm% {d}z},
  16. T = e - μ = 10 - μ 10 . T=e^{-\mu\ell}=10^{-\mu_{10}\ell}.
  17. μ ( z ) = i = 1 N μ i ( z ) = i = 1 N σ i n i ( z ) , \mu(z)=\sum_{i=1}^{N}\mu_{i}(z)=\sum_{i=1}^{N}\sigma_{i}n_{i}(z),
  18. μ 10 ( z ) = i = 1 N μ 10 , i ( z ) = i = 1 N ε i c i ( z ) , \mu_{10}(z)=\sum_{i=1}^{N}\mu_{10,i}(z)=\sum_{i=1}^{N}\varepsilon_{i}c_{i}(z),
  19. ε i = N A ln 10 σ i , \varepsilon_{i}=\frac{\mathrm{N_{A}}}{\ln{10}}\,\sigma_{i},
  20. c i = n i N A , c_{i}=\frac{n_{i}}{\mathrm{N_{A}}},

Automorphic_factor.html

  1. ν : Γ × \nu:\Gamma\times\mathbb{H}\to\mathbb{C}
  2. \mathbb{H}
  3. \mathbb{C}
  4. Γ \Gamma
  5. γ Γ \gamma\in\Gamma
  6. γ = [ a b c d ] \gamma=\left[\begin{matrix}a&b\\ c&d\end{matrix}\right]
  7. γ Γ \gamma\in\Gamma
  8. ν ( γ , z ) \nu(\gamma,z)
  9. z z\in\mathbb{H}
  10. z z\in\mathbb{H}
  11. γ Γ \gamma\in\Gamma
  12. | ν ( γ , z ) | = | c z + d | k |\nu(\gamma,z)|=|cz+d|^{k}
  13. z z\in\mathbb{H}
  14. γ , δ Γ \gamma,\delta\in\Gamma
  15. ν ( γ δ , z ) = ν ( γ , δ z ) ν ( δ , z ) \nu(\gamma\delta,z)=\nu(\gamma,\delta z)\nu(\delta,z)
  16. δ z \delta z
  17. z z
  18. δ \delta
  19. - I Γ -I\in\Gamma
  20. z z\in\mathbb{H}
  21. γ Γ \gamma\in\Gamma
  22. ν ( - γ , z ) = ν ( γ , z ) \nu(-\gamma,z)=\nu(\gamma,z)
  23. ν ( γ , z ) = υ ( γ ) ( c z + d ) k \nu(\gamma,z)=\upsilon(\gamma)(cz+d)^{k}
  24. | υ ( γ ) | = 1 |\upsilon(\gamma)|=1
  25. υ : Γ S 1 \upsilon:\Gamma\to S^{1}
  26. υ ( I ) = 1 \upsilon(I)=1
  27. - I Γ -I\in\Gamma
  28. υ ( - I ) = e - i π k \upsilon(-I)=e^{-i\pi k}

Autoregressive_fractionally_integrated_moving_average.html

  1. ( 1 - B ) 2 = 1 - 2 B + B 2 , (1-B)^{2}=1-2B+B^{2}\,,
  2. B 2 X t = X t - 2 , B^{2}X_{t}=X_{t-2}\,,
  3. ( 1 - B ) 2 X t = X t - 2 X t - 1 + X t - 2 . (1-B)^{2}X_{t}=X_{t}-2X_{t-1}+X_{t-2}.
  4. ( 1 - B ) d = k = 0 ( d k ) ( - B ) k = k = 0 a = 0 k - 1 ( d - a ) ( - B ) k k ! = 1 - d B + d ( d - 1 ) 2 ! B 2 - . \begin{aligned}\displaystyle(1-B)^{d}&\displaystyle=\sum_{k=0}^{\infty}\;{d% \choose k}\;(-B)^{k}\\ &\displaystyle=\sum_{k=0}^{\infty}\;\frac{\prod_{a=0}^{k-1}(d-a)\ (-B)^{k}}{k!% }\\ &\displaystyle=1-dB+\frac{d(d-1)}{2!}B^{2}-\cdots\,.\end{aligned}
  5. ( 1 - B ) d X t = ε t , (1-B)^{d}X_{t}=\varepsilon_{t},
  6. X t - d X t - 1 + d ( d - 1 ) 2 ! X t - 2 - = ε t . X_{t}-dX_{t-1}+\frac{d(d-1)}{2!}X_{t-2}-\cdots=\varepsilon_{t}.
  7. ( 1 - i = 1 p ϕ i B i ) ( 1 - B ) d X t = ( 1 + i = 1 q θ i B i ) ε t . \left(1-\sum_{i=1}^{p}\phi_{i}B^{i}\right)\left(1-B\right)^{d}X_{t}=\left(1+% \sum_{i=1}^{q}\theta_{i}B^{i}\right)\varepsilon_{t}\,.

Auxiliary_function.html

  1. | f ( p q ) | 1 q d , \left|f\left(\frac{p}{q}\right)\right|\geq\frac{1}{q^{d}},
  2. | f ( p q ) | c ( α ) | α - p q | . \left|f\left(\frac{p}{q}\right)\right|\leq c(\alpha)\left|\alpha-\frac{p}{q}% \right|.
  3. e x = n = 0 x n n ! . e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}.
  4. R ( x ) = e x - n = 0 N x n n ! R(x)=e^{x}-\sum_{n=0}^{N}\frac{x^{n}}{n!}
  5. R ( x ) = B ( x ) e x - A ( x ) R(x)=B(x)e^{x}-A(x)\,
  6. R k ( x ) = B ( x ) e k x - A k ( x ) . R_{k}(x)=B(x)e^{kx}-A_{k}(x).\,
  7. R = a 0 + a 1 R 1 ( 1 ) + + a m R m ( 1 ) = a 1 A 1 ( 1 ) + + a m A m ( 1 ) . R=a_{0}+a_{1}R_{1}(1)+\cdots+a_{m}R_{m}(1)=a_{1}A_{1}(1)+\cdots+a_{m}A_{m}(1).\,
  8. F ( X , Y ) = P ( X ) + Y * Q ( X ) F(X,Y)=P(X)+Y*Q(X)
  9. i = 0 m + n u i X i = P ( X ) , \sum_{i=0}^{m+n}u_{i}X^{i}=P(X),
  10. i = 0 m + n v i X i = Q ( X ) . \sum_{i=0}^{m+n}v_{i}X^{i}=Q(X).
  11. max 0 i m + n ( | u i | , | v i | ) 2 b 9 ( m + n ) . \max_{0\leq i\leq m+n}{(|u_{i}|,|v_{i}|)}\leq 2b^{9(m+n)}.
  12. m 20 ρ [ K : ] . m\leq 20\rho[K:\mathbb{Q}].
  13. = ( φ i ( ζ j ) ) 1 i , j N \mathcal{M}=\left(\varphi_{i}(\zeta_{j})\right)_{1\leq i,j\leq N}
  14. ( { exp ( j 2 x ) x j 1 - 1 } ( i 1 - 1 ) | x = ( i 2 - 1 ) α ) . \left(\{\exp(j_{2}x)x^{j_{1}-1}\}^{(i_{1}-1)}\Big|_{x=(i_{2}-1)\alpha}\right).
  15. Ω = O ( exp ( ( m + 1 k - 3 2 ) n 8 log n ) ) . \Omega=O\left(\exp\left(\left(\frac{m+1}{k}-\frac{3}{2}\right)n^{8}\log n% \right)\right).

Average_path_length.html

  1. G G
  2. V V
  3. d ( v 1 , v 2 ) d(v_{1},v_{2})
  4. v 1 , v 2 V v_{1},v_{2}\in V
  5. v 1 v_{1}
  6. v 2 v_{2}
  7. d ( v 1 , v 2 ) = 0 d(v_{1},v_{2})=0
  8. v 2 v_{2}
  9. v 1 v_{1}
  10. l G l_{G}
  11. l G = 1 n ( n - 1 ) i j d ( v i , v j ) l_{G}=\frac{1}{n\cdot(n-1)}\cdot\sum_{i\neq j}d(v_{i},v_{j})
  12. n n
  13. G G

Axis–angle_representation.html

  1. ê \mathbf{ê}
  2. θ θ
  3. θ θ
  4. 𝐞 = θ 𝐞 ^ . \mathbf{e}=\theta\mathbf{\hat{e}}~{}.
  5. ( axis , angle ) = ( [ a x a y a z ] , θ ) = ( [ 0 0 1 ] , π 2 ) . (\mathrm{axis},\mathrm{angle})=\left(\begin{bmatrix}a_{x}\\ a_{y}\\ a_{z}\end{bmatrix},\theta\right)=\left(\begin{bmatrix}0\\ 0\\ 1\end{bmatrix},\frac{\pi}{2}\right).
  6. π / 2 {π}/{2}
  7. z z
  8. [ 0 0 π 2 ] . \begin{bmatrix}0\\ 0\\ \frac{\pi}{2}\end{bmatrix}.
  9. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  10. S O ( 3 ) SO(3)
  11. 𝐯 \mathbf{v}
  12. ω \mathbf{ω}
  13. 𝐯 \mathbf{v}
  14. θ θ
  15. 𝐯 rot = ( cos θ ) 𝐯 + ( sin θ ) ( s y m b o l ω × 𝐯 ) + ( 1 - cos θ ) ( s y m b o l ω 𝐯 ) s y m b o l ω . \mathbf{v}_{\mathrm{rot}}=(\cos\theta)\mathbf{v}+(\sin\theta)(symbol{\omega}% \times\mathbf{v})+(1-\cos\theta)(symbol{\omega}\cdot\mathbf{v})symbol{\omega}~% {}.
  16. ω \mathbf{ω}
  17. θ θ
  18. exp : 𝔰 𝔬 ( 3 ) SO ( 3 ) . \exp\colon\mathfrak{so}(3)\to\mathrm{SO}(3)~{}.
  19. 𝔰 𝔬 \mathfrak{so}
  20. R = exp ( θ 𝐊 ) = k = 0 ( θ 𝐊 ) k k ! = I + 𝐊 θ + 1 2 ! ( θ 𝐊 ) 2 + 1 3 ! ( θ 𝐊 ) 3 + R=\exp(\theta\mathbf{K})=\sum_{k=0}^{\infty}\frac{(\theta\mathbf{K})^{k}}{k!}=% I+\mathbf{K}\theta+\frac{1}{2!}(\theta\mathbf{K})^{2}+\frac{1}{3!}(\theta% \mathbf{K})^{3}+\cdots
  21. P ( t ) P(t)
  22. P ( 𝐊 ) P(\mathbf{K})
  23. R = I + ( θ - θ 3 3 ! + θ 5 5 ! - ) 𝐊 + ( θ 2 2 ! - θ 4 4 ! + θ 6 6 ! - ) 𝐊 2 , R=I+\left(\theta-\frac{\theta^{3}}{3!}+\frac{\theta^{5}}{5!}-\cdots\right)% \mathbf{K}+\left(\frac{\theta^{2}}{2!}-\frac{\theta^{4}}{4!}+\frac{\theta^{6}}% {6!}-\cdots\right)\mathbf{K}^{2}~{},
  24. R = I + sin ( θ ) 𝐊 + ( 1 - cos ( θ ) ) 𝐊 2 . R=I+\sin(\theta)\mathbf{K}+(1-\cos(\theta))\mathbf{K}^{2}.
  25. θ = arccos ( trace ( R ) - 1 2 ) \theta=\arccos\left(\frac{\mathrm{trace}(R)-1}{2}\right)
  26. ω = 1 2 sin ( θ ) [ R ( 3 , 2 ) - R ( 2 , 3 ) R ( 1 , 3 ) - R ( 3 , 1 ) R ( 2 , 1 ) - R ( 1 , 2 ) ] . \mathbf{\omega}=\frac{1}{2\sin(\theta)}\begin{bmatrix}R(3,2)-R(2,3)\\ R(1,3)-R(3,1)\\ R(2,1)-R(1,2)\end{bmatrix}~{}.
  27. log R = { 0 if θ = 0 θ 2 sin ( θ ) ( R - R 𝖳 ) if θ 0 and θ ( - π , π ) \log R=\left\{\begin{matrix}0&\mathrm{if}\;\theta=0\\ \frac{\theta}{2\sin(\theta)}(R-R^{\mathsf{T}})&\mathrm{if}\;\theta\neq 0\;% \mathrm{and}\;\theta\in(-\pi,\pi)\end{matrix}\right.
  28. log ( R ) F = 2 | θ | . \|\log(R)\|_{F}=\sqrt{2}|\theta|~{}.
  29. d g ( A , B ) := log ( A 𝖳 B ) F d_{g}(A,B):=\|\log(A^{\mathsf{T}}B)\|_{F}
  30. R I + θ 𝐊 R≈I+θ\mathbf{K}
  31. R = I + 𝐊 sin ( θ ) + 𝐊 2 ( 1 - cos ( θ ) ) R=I+\mathbf{K}\sin(\theta)+\mathbf{K}^{2}(1-\cos(\theta))
  32. R = I + 2 𝐊 2 = I + 2 ( ω ω - I ) = 2 ω ω - I R=I+2\mathbf{K}^{2}=I+2(\mathbf{\omega}\otimes\mathbf{\omega}-I)=2\mathbf{% \omega}\otimes\mathbf{\omega}-I
  33. B := ω ω = 1 2 ( R + I ) , B:=\mathbf{\omega}\otimes\mathbf{\omega}=\frac{1}{2}(R+I)~{},
  34. Q = ( cos ( θ 2 ) , ω sin ( θ 2 ) ) Q=\left(\cos\left(\frac{\theta}{2}\right),\mathbf{\omega}\sin\left(\frac{% \theta}{2}\right)\right)
  35. q = s + 𝐱 q=s+\mathbf{x}
  36. θ = 2 arccos ( s ) \theta=2\,\arccos(s)\,
  37. ω = { x sin ( θ / 2 ) , if θ 0 0 , otherwise \mathbf{\omega}=\left\{\begin{matrix}\frac{x}{\sin(\theta/2)},&\mathrm{if}\;% \theta\neq 0\\ 0,&\mathrm{otherwise}\end{matrix}\right.
  38. θ = 2 atan2 ( 𝐱 , s ) , \theta=2\,\operatorname{atan2}(\|\mathbf{x}\|,s)\,,

Backlash_(engineering).html

  1. b t = t i - t a b_{t}=t_{i}-t_{a}\;
  2. b t b_{t}\;
  3. t i t_{i}\;
  4. t a t_{a}\;
  5. b c = 2 ( Δ c ) tan ϕ b_{c}=2\left(\Delta c\right)\tan\phi
  6. b c b_{c}\;
  7. Δ c \Delta c\;
  8. ϕ \phi\;

Balanced_circuit.html

  1. V + = V in Z in + R 1 Z in + 2 R 1 V_{+}=V_{\mathrm{in}}\frac{Z_{\mathrm{in}}+R_{1}}{Z_{\mathrm{in}}+2R_{1}}
  2. V - = V in R 1 Z in + 2 R 1 V_{-}=V_{\mathrm{in}}\frac{R_{1}}{Z_{\mathrm{in}}+2R_{1}}

Balanced_flow.html

  1. D V D t = - 1 ρ p s - K V \frac{DV}{Dt}=-\frac{1}{\rho}\frac{\partial p}{\partial s}-KV
  2. V 2 R = - 1 ρ p n ± f V \frac{V^{2}}{R}=-\frac{1}{\rho}\frac{\partial p}{\partial n}\pm fV
  3. D V / D t {DV}/{Dt}
  4. - p / s -{\partial p}/{\partial s}
  5. - K V -KV
  6. V 2 / R {V^{2}}/{R}
  7. - p / n -{\partial p}/{\partial n}
  8. ± f V \pm fV
  9. D V / D t = 0 DV/Dt=0
  10. V = - 1 K ρ p s V=-\frac{1}{K\rho}\frac{\partial p}{\partial s}
  11. p / s < 0 {\partial p}/{\partial s}<0
  12. V 2 / R {V^{2}}/{R}
  13. p / n = 0 \partial p/\partial n=0
  14. p / s = 0 \partial p/\partial s=0
  15. V = 1 ρ | 1 f p n | V=\frac{1}{\rho}\left|\frac{1}{f}\frac{\partial p}{\partial n}\right|
  16. V 2 R = - 1 ρ p n \frac{V^{2}}{R}=-\frac{1}{\rho}\frac{\partial p}{\partial n}
  17. V = - R ρ p n V=\sqrt{-\frac{R}{\rho}\frac{\partial p}{\partial n}}
  18. p / s = 0 \partial p/\partial s=0
  19. V 2 R = | f | V \frac{V^{2}}{R}=\left|f\right|V
  20. V = | f | R V=\left|f\right|R
  21. p / s = 0 \partial p/\partial s=0
  22. V = ± f R 2 ± f 2 R 2 4 - R ρ p n V=\pm\frac{fR}{2}\pm\sqrt{\frac{f^{2}R^{2}}{4}-\frac{R}{\rho}\frac{\partial p}% {\partial n}}
  23. V 2 R = 1 ρ | p n | - | f | V \frac{V^{2}}{R}=\frac{1}{\rho}\left|\frac{\partial p}{\partial n}\right|-\left% |f\right|V
  24. V g e o s t r o p h i c V c y c l o n e = 1 + V c y c l o n e V i n e r t i a l > 1 \frac{V_{geostrophic}}{V_{cyclone}}=1+\frac{V_{cyclone}}{V_{inertial}}>1
  25. V c y c l o n e = - V i n e r t i a l 2 + V i n e r t i a l 2 4 + V c y c l o s t r o p h i c 2 V_{cyclone}=-\frac{V_{inertial}}{2}+\sqrt{\frac{V_{inertial}^{2}}{4}+V_{% cyclostrophic}^{2}}
  26. V 2 R = - 1 ρ | p n | + | f | V \frac{V^{2}}{R}=-\frac{1}{\rho}\left|\frac{\partial p}{\partial n}\right|+% \left|f\right|V
  27. V g e o s t r o p h i c V a n t i c y c l o n e = 1 - V a n t i c y c l o n e V i n e r t i a l < 1 \frac{V_{geostrophic}}{V_{anticyclone}}=1-\frac{V_{anticyclone}}{V_{inertial}}<1
  28. V a n t i c y c l o n e = V i n e r t i a l 2 - V i n e r t i a l 2 4 - V c y c l o s t r o p h i c 2 V_{anticyclone}=\frac{V_{inertial}}{2}-\sqrt{\frac{V_{inertial}^{2}}{4}-V_{% cyclostrophic}^{2}}
  29. V i n e r t i a l 2 V c y c l o s t r o p h i c V_{inertial}\geq 2V_{cyclostrophic}
  30. R * = 4 ρ f 2 | p n | R^{*}=\frac{4}{\rho f^{2}}\left|\frac{\partial p}{\partial n}\right|

Balding–Nichols_model.html

  1. I x ( α , β ) I_{x}(\alpha,\beta)\!
  2. p p\!
  3. I 0.5 - 1 ( α , β ) I_{0.5}^{-1}(\alpha,\beta)
  4. F - ( 1 - F ) p 3 F - 1 \frac{F-(1-F)p}{3F-1}
  5. F p ( 1 - p ) Fp(1-p)\!
  6. 2 F ( 1 - 2 p ) ( 1 + F ) F ( 1 - p ) p \frac{2F(1-2p)}{(1+F)\sqrt{F(1-p)p}}
  7. 1 + k = 1 ( r = 0 k - 1 α + r 1 - F F + r ) t k k ! 1+\sum_{k=1}^{\infty}\left(\prod_{r=0}^{k-1}\frac{\alpha+r}{\frac{1-F}{F}+r}% \right)\frac{t^{k}}{k!}
  8. F 1 1 ( α ; α + β ; i t ) {}_{1}F_{1}(\alpha;\alpha+\beta;i\,t)\!
  9. B ( 1 - F F p , 1 - F F ( 1 - p ) ) B\left(\frac{1-F}{F}p,\frac{1-F}{F}(1-p)\right)
  10. { F ( x - 1 ) x f ( x ) + f ( x ) ( F ( - p ) + 3 F x - F + p - x ) = 0 } \left\{F(x-1)xf^{\prime}(x)+f(x)(F(-p)+3Fx-F+p-x)=0\right\}

Ballistic_limit.html

  1. V b = π Γ ρ t σ e D 2 T 4 m [ 1 + 1 + 8 m π Γ 2 ρ t D 2 T ] V_{b}=\frac{\pi\,\Gamma\,\sqrt{\rho_{t}\,\sigma_{e}}\,D^{2}\,T}{4\,m}\left[1+% \sqrt{1+\frac{8\,m}{\pi\,\Gamma^{2}\,\rho_{t}\,D^{2}\,T}}\,\right]
  2. V b V_{b}\,
  3. Γ \Gamma\,
  4. ρ t \rho_{t}\,
  5. σ e \sigma_{e}\,
  6. D D\,
  7. T T\,
  8. m m\,
  9. V 1 = 19.72 [ 7800 d 3 [ ( e h d ) sec θ ] 1.6 W T ] 0.5 V_{1}=19.72\left[\frac{7800d^{3}\left[\left(\frac{e_{h}}{d}\right)\sec\theta% \right]^{1.6}}{W_{T}}\right]^{0.5}
  10. V 1 V_{1}
  11. d d
  12. e h e_{h}
  13. θ \theta
  14. W T W_{T}

Baloot.html

  1. ( 28 4 ) 4 ! ( 32 8 ) \frac{\frac{{\left({{28}\atop{4}}\right)}}{4!}}{{\left({{32}\atop{8}}\right)}}
  2. 16 * ( 27 3 ) 3 ! + 4 * ( 28 4 ) 4 ! ( 32 8 ) \frac{\frac{16*{\left({{27}\atop{3}}\right)}}{3!}+\frac{4*{\left({{28}\atop{4}% }\right)}}{4!}}{{\left({{32}\atop{8}}\right)}}
  3. 20 * ( 28 4 ) 4 ! - 16 * ( 27 3 ) 3 ! ( 32 8 ) \frac{\frac{20*{\left({{28}\atop{4}}\right)}}{4!}-\frac{16*{\left({{27}\atop{3% }}\right)}}{3!}}{{\left({{32}\atop{8}}\right)}}
  4. 24 * ( 29 5 ) 5 ! - 20 * ( 28 4 ) 4 ! ( 32 8 ) \frac{\frac{24*{\left({{29}\atop{5}}\right)}}{5!}-\frac{20*{\left({{28}\atop{4% }}\right)}}{4!}}{{\left({{32}\atop{8}}\right)}}

Banach's_matchbox_problem.html

  1. N N
  2. k k
  3. P [ M = m ] = ( N + m m ) ( 1 2 ) N + 1 + m P[M=m]={\left({{N+m}\atop{m}}\right)}\left(\frac{1}{2}\right)^{N+1+m}
  4. P [ M < N + 1 ] P[M<N+1]
  5. P [ K = k ] = P [ M = N - k | M < N + 1 ] = 2 P [ M = N - k ] = ( 2 N - k N ) ( 1 2 ) 2 N - k P[K=k]=P[M=N-k|M<N+1]=2P[M=N-k]={\left({{2N-k}\atop{N}}\right)}\left(\frac{1}{% 2}\right)^{2N-k}

Banach_*-algebra.html

  1. ( λ x ) * = λ ¯ x * (\lambda x)^{*}=\bar{\lambda}x^{*}
  2. λ ¯ \bar{\lambda}

Bapat–Beg_theorem.html

  1. X 1 , X 2 , , X n X_{1},X_{2},\ldots,X_{n}
  2. F 1 ( x ) , F 2 ( x ) , , F n ( x ) F_{1}(x),F_{2}(x),\ldots,F_{n}(x)
  3. X ( 1 ) , X ( 2 ) , , X ( n ) X_{(1)},X_{(2)},\ldots,X_{(n)}
  4. n 1 , n 2 , , n k n_{1},n_{2},\ldots,n_{k}
  5. n 1 < n 2 < < n k n_{1}<n_{2}<\ldots<n_{k}
  6. x 1 < x 2 < < x k x_{1}<x_{2}<\ldots<x_{k}
  7. F X ( n 1 ) , , X ( n k ) ( x 1 , , x k ) = Pr ( X ( n 1 ) x 1 and X ( n 2 ) x 2 and and X ( n k ) x k ) = i k = n k n i 2 = n 2 i 3 i 1 = n 1 i 2 P i 1 , , i k ( x 1 , , x k ) i 1 ! ( i 2 - i 1 ) ! ( n - i k ) ! , \begin{aligned}\displaystyle F_{X_{(n_{1})},\ldots,X_{(n_{k})}}(x_{1},\ldots,x% _{k})&\displaystyle=\Pr(X_{(n_{1})}\leq x_{1}\and X_{(n_{2})}\leq x_{2}\and% \ldots\and X_{(n_{k})}\leq x_{k})\\ &\displaystyle=\sum_{i_{k}=n_{k}}^{n}\ldots\sum_{i_{2}=n_{2}}^{i_{3}}\,\sum_{i% _{1}=n_{1}}^{i_{2}}\frac{P_{i_{1},\ldots,i_{k}}(x_{1},\ldots,x_{k})}{i_{1}!(i_% {2}-i_{1})!\ldots(n-i_{k})!},\end{aligned}
  8. P i 1 , , i k ( x 1 , , x k ) = P_{i_{1},\ldots,i_{k}}(x_{1},\ldots,x_{k})=
  9. per [ F 1 ( x 1 ) F 1 ( x 1 ) F 1 ( x 2 ) - F 1 ( x 1 ) F 1 ( x 2 ) - F 1 ( x 1 ) 1 - F 1 ( x k ) 1 - F 1 ( x k ) F 2 ( x 1 ) F 2 ( x 1 ) F 2 ( x 2 ) - F 2 ( x 1 ) F 2 ( x 2 ) - F 2 ( x 1 ) 1 - F 2 ( x k ) 1 - F 1 ( x k ) F n ( x 1 ) F n ( x 1 ) i 1 F n ( x 2 ) - F n ( x 1 ) F n ( x 2 ) - F n ( x 1 ) i 2 - i 1 1 - F n ( x k ) 1 - F n ( x k ) n - i k ] \operatorname{per}\begin{bmatrix}F_{1}(x_{1})\ldots F_{1}(x_{1})&F_{1}(x_{2})-% F_{1}(x_{1})\ldots F_{1}(x_{2})-F_{1}(x_{1})&\ldots&1-F_{1}(x_{k})\ldots 1-F_{% 1}(x_{k})\\ F_{2}(x_{1})\ldots F_{2}(x_{1})&F_{2}(x_{2})-F_{2}(x_{1})\ldots F_{2}(x_{2})-F% _{2}(x_{1})&\ldots&1-F_{2}(x_{k})\ldots 1-F_{1}(x_{k})\\ \vdots&\vdots&&\vdots\\ \underbrace{F_{n}(x_{1})\ldots F_{n}(x_{1})}_{i_{1}}&\underbrace{F_{n}(x_{2})-% F_{n}(x_{1})\ldots F_{n}(x_{2})-F_{n}(x_{1})}_{i_{2}-i_{1}}&\ldots&\underbrace% {1-F_{n}(x_{k})\ldots 1-F_{n}(x_{k})}_{n-i_{k}}\end{bmatrix}
  10. X 1 , X 2 , , X n X_{1},X_{2},\ldots,X_{n}
  11. F i = F F_{i}=F
  12. F X ( n 1 ) , , X ( n k ) ( x 1 , , x k ) = i k = n k n i 2 = n 2 i 3 i 1 = n 1 i 2 m ! F ( x 1 ) i 1 i 1 ! ( 1 - F ( x k ) ) m - i k ( m - i k ) ! j = 2 k [ F ( x j ) - F ( x j - 1 ) ] i j - i j - 1 ( i j - i j - 1 ) ! . \begin{aligned}&\displaystyle F_{X_{(n_{1})},\ldots,X_{(n_{k})}}(x_{1},\ldots,% x_{k})\\ &\displaystyle=\sum_{i_{k}=n_{k}}^{n}\cdots\sum_{i_{2}=n_{2}}^{i_{3}}\,\sum_{i% _{1}=n_{1}}^{i_{2}}m!\frac{F(x_{1})^{i_{1}}}{i_{1}!}\frac{(1-F(x_{k}))^{m-i_{k% }}}{(m-i_{k})!}\prod\limits_{j=2}^{k}\frac{\left[F(x_{j})-F(x_{j-1})\right]^{i% _{j}-i_{j-1}}}{(i_{j}-i_{j-1})!}.\end{aligned}

Barabási–Albert_model.html

  1. m 0 m_{0}
  2. m m 0 m\leq m_{0}
  3. p i p_{i}
  4. i i
  5. p i = k i j k j , p_{i}=\frac{k_{i}}{\sum_{j}k_{j}},
  6. k i k_{i}
  7. i i
  8. j j
  9. m 0 = 1 m_{0}=1
  10. P ( k ) k - 3 P(k)\sim k^{-3}\,
  11. H ( k ) k - 6 H(k)\sim k^{-6}\,
  12. m 0 = 1 m_{0}=1
  13. H ( 1 ) | m 0 = 1 = 4 - π H(1)\Big|_{m_{0}=1}=4-\pi\,
  14. ln N ln ln N . \ell\sim\frac{\ln N}{\ln\ln N}.
  15. n k n_{k\ell}
  16. k k
  17. \ell
  18. m = 1 m=1
  19. n k = 4 ( - 1 ) k ( k + 1 ) ( k + ) ( k + + 1 ) ( k + + 2 ) + 12 ( - 1 ) k ( k + - 1 ) ( k + ) ( k + + 1 ) ( k + + 2 ) . n_{k\ell}=\frac{4\left(\ell-1\right)}{k\left(k+1\right)\left(k+\ell\right)% \left(k+\ell+1\right)\left(k+\ell+2\right)}+\frac{12\left(\ell-1\right)}{k% \left(k+\ell-1\right)\left(k+\ell\right)\left(k+\ell+1\right)\left(k+\ell+2% \right)}.
  20. n k = k - 3 - 3 n_{k\ell}=k^{-3}\ell^{-3}
  21. m m
  22. k k
  23. \ell
  24. p ( k , ) = m ( m + 1 ) k ( k + 1 ) ( + 1 ) [ 1 - ( 2 m + 2 m + 1 ) ( k + - 2 m - m ) ( k + + 2 + 1 ) ] . p(k,\ell)=\frac{m(m+1)}{k(k+1)\ell(\ell+1)}\left[1-\frac{{\left({{2m+2}\atop{m% +1}}\right)}{\left({{k+\ell-2m}\atop{\ell-m}}\right)}}{{\left({{k+\ell+2}\atop% {\ell+1}}\right)}}\right].
  25. p ( k ) p(\ell\mid k)
  26. k k
  27. p ( k ) = m ( k + 2 ) k ( + 1 ) [ 1 - ( 2 m + 2 m + 1 ) ( k + - 2 m - m ) ( k + + 2 + 1 ) ] . p(\ell\mid k)=\frac{m(k+2)}{k\ell(\ell+1)}\left[1-\frac{{\left({{2m+2}\atop{m+% 1}}\right)}{\left({{k+\ell-2m}\atop{\ell-m}}\right)}}{{\left({{k+\ell+2}\atop{% \ell+1}}\right)}}\right].
  28. C N - 0.75 . C\sim N^{-0.75}.\,
  29. C ( k ) = k - 1 . C(k)=k^{-1}.\,

Barnes–Hut_simulation.html

  1. s / d s/d

Basal_area.html

  1. B A BA
  2. D B H DBH
  3. B A = π × ( D B H / 2 ) 2 144 BA=\frac{\pi\times(DBH/2)^{2}}{144}
  4. B A = 0.005454 × D B H 2 BA=0.005454\times DBH^{2}
  5. B A = 0.00007854 × D B H 2 BA=0.00007854\times DBH^{2}

Base_runs.html

  1. < m t p l > A * B B + C + D <mtpl>{{A*B\over B+C}}+D

Baumslag–Solitar_group.html

  1. B S ( 1 , 2 ) BS(1,2)
  2. a a
  3. b b
  4. B S ( 1 , 2 ) BS(1,2)
  5. a , b : b a m b - 1 = a n . \left\langle a,b\ :\ ba^{m}b^{-1}=a^{n}\right\rangle.
  6. m m
  7. n n
  8. B S ( m , n ) BS(m,n)
  9. B S ( m , n ) BS(m,n)
  10. B S ( 1 , 1 ) BS(1,1)
  11. B S ( 1 , 1 ) BS(1,−1)
  12. A = ( 1 1 0 1 ) , B = ( n m 0 0 1 ) . A=\begin{pmatrix}1&1\\ 0&1\end{pmatrix},\qquad B=\begin{pmatrix}\frac{n}{m}&0\\ 0&1\end{pmatrix}.
  13. G G
  14. A A
  15. B B
  16. B S ( m , n ) BS(m,n)
  17. a A , b B . a\mapsto A,\qquad b\mapsto B.
  18. B S ( m , n ) BS(m,n)
  19. | m | = 1 |m|=1
  20. | n | = 1 |n|=1
  21. | m | = | n | |m|=|n|

Baum–Connes_conjecture.html

  1. C * C^{*}
  2. μ \mu
  3. μ : R K * Γ ( E Γ ¯ ) K * ( C λ * ( Γ ) ) , \mu\colon RK^{\Gamma}_{*}(\underline{E\Gamma})\to K_{*}(C^{*}_{\lambda}(\Gamma% )),
  4. Γ \Gamma
  5. E Γ ¯ \underline{E\Gamma}
  6. C * C^{*}
  7. Γ \Gamma
  8. B Γ B\Gamma
  9. Γ \Gamma
  10. C * C^{*}
  11. A A
  12. Γ \Gamma
  13. C * C^{*}
  14. μ A , Γ : R K K * Γ ( E Γ ¯ , A ) K * ( A λ Γ ) , \mu_{A,\Gamma}\colon RKK^{\Gamma}_{*}(\underline{E\Gamma},A)\to K_{*}(A\rtimes% _{\lambda}\Gamma),
  15. A = A=\mathbb{C}
  16. Γ \Gamma
  17. \Z \Z
  18. B \Z B\Z
  19. C * C^{*}
  20. R ( K ) R(K)
  21. S O ( n , 1 ) SO(n,1)
  22. S U ( n , 1 ) SU(n,1)
  23. H H
  24. lim n g n ξ \lim_{n\to\infty}g_{n}\xi\to\infty
  25. ξ H \xi\in H
  26. g n g_{n}
  27. lim n g n \lim_{n\to\infty}g_{n}\to\infty
  28. C A T ( 0 ) CAT(0)
  29. S L ( 3 , ) SL(3,\mathbb{R})
  30. S L ( 3 , ) SL(3,\mathbb{C})
  31. S L ( 3 , p ) SL(3,\mathbb{Q}_{p})
  32. S L ( 3 , ) SL(3,\mathbb{R})
  33. k k
  34. k k
  35. k = p k=\mathbb{Q}_{p}
  36. S L 3 ( \Z ) SL_{3}(\Z)

BCJR_algorithm.html

  1. α \alpha
  2. β \beta

Beatty_sequence.html

  1. r r\,
  2. r = r , 2 r , 3 r , = ( n r ) n 1 \mathcal{B}_{r}=\lfloor r\rfloor,\lfloor 2r\rfloor,\lfloor 3r\rfloor,\ldots=(% \lfloor nr\rfloor)_{n\geq 1}
  3. r > 1 , r>1\,,
  4. s = r / ( r - 1 ) s=r/(r-1)\,
  5. 1 r + 1 s = 1 \frac{1}{r}+\frac{1}{s}=1\,
  6. r = ( n r ) n 1 \mathcal{B}_{r}=(\lfloor nr\rfloor)_{n\geq 1}
  7. s = ( n s ) n 1 \mathcal{B}_{s}=(\lfloor ns\rfloor)_{n\geq 1}
  8. r = r + p , 2 r + p , 3 r + p , = ( n r + p ) n 1 \mathcal{B}_{r}=\lfloor r+p\rfloor,\lfloor 2r+p\rfloor,\lfloor 3r+p\rfloor,% \ldots=(\lfloor nr+p\rfloor)_{n\geq 1}
  9. p p\,
  10. p = 1 p=1\,
  11. t = 1 / r t=1/r\,
  12. r = ( n ( r + 1 ) ) n 1 \mathcal{B}_{r}=(\lfloor n(r+1)\rfloor)_{n\geq 1}
  13. t = ( n ( t + 1 ) ) n 1 \mathcal{B}_{t}=(\lfloor n(t+1)\rfloor)_{n\geq 1}
  14. ( n r ) (\lfloor nr\rfloor)
  15. ( n s ) (\lfloor ns\rfloor)
  16. r > 1 , r>1\,,
  17. s > 1 s>1
  18. r \mathcal{B}_{r}
  19. s \mathcal{B}_{s}
  20. r > 1 , r>1\,,
  21. s = r / ( r - 1 ) s=r/(r-1)\,
  22. r \mathcal{B}_{r}
  23. s \mathcal{B}_{s}
  24. j / r = k / s j/r=k/s
  25. r / s = r ( 1 - 1 / r ) = r - 1 , r/s=r(1-1/r)=r-1,
  26. j s / r \lfloor js/r\rfloor
  27. k / s j / r k/s\leq j/r
  28. j / r j/r
  29. j + j s / r j+\lfloor js/r\rfloor
  30. 1 / r + 1 / s = 1 1/r+1/s=1
  31. j + j s / r = j + j ( s - 1 ) = j s . j+\lfloor js/r\rfloor=j+\lfloor j(s-1)\rfloor=\lfloor js\rfloor.
  32. k r \lfloor kr\rfloor
  33. n r \lfloor nr\rfloor
  34. n s \lfloor ns\rfloor
  35. j = k r = m s . j=\left\lfloor{k\cdot r}\right\rfloor=\left\lfloor{m\cdot s}\right\rfloor\,.
  36. j k r < j + 1 and j m s < j + 1. j\leq k\cdot r<j+1\,\text{ and }j\leq m\cdot s<j+1.\,
  37. j < k r < j + 1 and j < m s < j + 1 j<k\cdot r<j+1\,\text{ and }j<m\cdot s<j+1\,
  38. j r < k < j + 1 r and j s < m < j + 1 s . {j\over r}<k<{j+1\over r}\,\text{ and }{j\over s}<m<{j+1\over s}.\,
  39. j < k + m < j + 1 j<k+m<j+1\,
  40. k r < j and j + 1 ( k + 1 ) r and m s < j and j + 1 ( m + 1 ) s . k\cdot r<j\,\text{ and }j+1\leq(k+1)\cdot r\,\text{ and }m\cdot s<j\,\text{ % and }j+1\leq(m+1)\cdot s\,.
  41. k r < j and j + 1 < ( k + 1 ) r and m s < j and j + 1 < ( m + 1 ) s . k\cdot r<j\,\text{ and }j+1<(k+1)\cdot r\,\text{ and }m\cdot s<j\,\text{ and }% j+1<(m+1)\cdot s.\,
  42. k < j r and j + 1 r < k + 1 and m < j s and j + 1 s < m + 1 k<{j\over r}\,\text{ and }{j+1\over r}<k+1\,\text{ and }m<{j\over s}\,\text{ % and }{j+1\over s}<m+1\,
  43. k + m < j and j + 1 < k + m + 2 k+m<j\,\text{ and }j+1<k+m+2\,
  44. k + m < j < k + m + 1 k+m<j<k+m+1\,
  45. m r m\in\mathcal{B}_{r}
  46. 0 1 - 1 r [ m r ] 1 0\leq 1-\frac{1}{r}\leq\left[\frac{m}{r}\right]_{1}
  47. [ x ] 1 [x]_{1}
  48. x mod 1 x\mod 1
  49. x x
  50. [ x ] 1 = x - x [x]_{1}=x-\lfloor x\rfloor
  51. m r m\in\mathcal{B}_{r}
  52. m = ( m r + 1 ) r m=\left\lfloor\left(\left\lfloor\frac{m}{r}\right\rfloor+1\right)r\right\rfloor
  53. m = m r r + [ m r ] 1 r = ( m r + 1 ) r - ( 1 - [ m r ] 1 ) r m=\left\lfloor\frac{m}{r}\right\rfloor r+\left[\frac{m}{r}\right]_{1}r=\left(% \left\lfloor\frac{m}{r}\right\rfloor+1\right)r-\left(1-\left[\frac{m}{r}\right% ]_{1}\right)r
  54. ( m r + 1 ) r = m \left\lfloor\left(\left\lfloor\frac{m}{r}\right\rfloor+1\right)r\right\rfloor=m
  55. ( m r - [ m r ] 1 + 1 ) r = m \left\lfloor\left(\frac{m}{r}-\left[\frac{m}{r}\right]_{1}+1\right)r\right% \rfloor=m
  56. m + ( 1 - [ m r ] 1 ) r = m \left\lfloor m+\left(1-\left[\frac{m}{r}\right]_{1}\right)r\right\rfloor=m
  57. 0 ( 1 - [ m r ] 1 ) r < 1 0\leq\left(1-\left[\frac{m}{r}\right]_{1}\right)r<1
  58. ( n + 1 ) r - n r \lfloor(n+1)r\rfloor-\lfloor nr\rfloor
  59. r r
  60. { r , r + 1 } \{\lfloor r\rfloor,\lfloor r\rfloor+1\}
  61. α 1 , , α n \alpha_{1},\ldots,\alpha_{n}
  62. ( k α i ) k , i 1 (\lfloor k\alpha_{i}\rfloor)_{k,i\geq 1}
  63. n 2. n\leq 2.

Belt_problem.html

  1. φ \varphi
  2. C O + D O + E O + F O + arc C D + arc E F CO+DO+EO+FO+\text{arc}CD+\text{arc}EF\,\!
  3. = 2 r 1 tan ( φ ) + 2 r 2 tan ( φ ) + ( 2 π - 2 φ ) r 1 + ( 2 π - 2 φ ) r 2 =2r_{1}\tan(\varphi)+2r_{2}\tan(\varphi)+(2\pi-2\varphi)r_{1}+(2\pi-2\varphi)r% _{2}\,\!
  4. = 2 ( r 1 + r 2 ) ( tan ( φ ) + π - φ ) =2(r_{1}+r_{2})(\tan(\varphi)+\pi-\varphi)\,\!
  5. φ \varphi
  6. A O B O = A C B E \frac{AO}{BO}=\frac{AC}{BE}\,\!
  7. P - x x = r 1 r 2 \Rightarrow\frac{P-x}{x}=\frac{r_{1}}{r_{2}}\,\!
  8. P x = r 1 + r 2 r 2 \Rightarrow\frac{P}{x}=\frac{r_{1}+r_{2}}{r_{2}}\,\!
  9. x = P r 2 r 1 + r 2 \Rightarrow{x}=\frac{Pr_{2}}{r_{1}+r_{2}}\,\!
  10. cos ( φ ) = r 2 x = r 2 ( P r 2 r 1 + r 2 ) = r 1 + r 2 P \cos(\varphi)=\frac{r_{2}}{x}=\frac{r_{2}}{\left(\dfrac{Pr_{2}}{r_{1}+r_{2}}% \right)}=\frac{r_{1}+r_{2}}{P}\,\!
  11. φ = cos - 1 ( r 1 + r 2 P ) \Rightarrow\varphi=\cos^{-1}\left(\frac{r_{1}+r_{2}}{P}\right)\,\!
  12. 2 P sin ( θ 2 ) + r 1 ( 2 π - θ ) + r 2 θ , 2P\sin\left(\frac{\theta}{2}\right)+r_{1}(2\pi-\theta)+r_{2}{\theta}\,,
  13. θ = 2 cos - 1 ( r 1 - r 2 P ) . \theta=2\cos^{-1}\left(\frac{r_{1}-r_{2}}{P}\right)\,.

Berendsen_thermostat.html

  1. τ \tau
  2. d T d t = T 0 - T τ \frac{dT}{dt}=\frac{T_{0}-T}{\tau}

Berger_code.html

  1. n n
  2. k = log 2 ( n + 1 ) k=\lceil\log_{2}(n+1)\rceil
  3. k k
  4. n = 2 k - 1 n=2^{k}-1

Bergman_metric.html

  1. G n G\subset{\mathbb{C}}^{n}
  2. K ( z , w ) K(z,w)
  3. T z n T_{z}{\mathbb{C}}^{n}
  4. g i j ( z ) := 2 z i z ¯ j log K ( z , z ) , g_{ij}(z):=\frac{\partial^{2}}{\partial z_{i}\,\partial\bar{z}_{j}}\log K(z,z),
  5. z G z\in G
  6. ξ T z n \xi\in T_{z}{\mathbb{C}}^{n}
  7. | ξ | B , z := i , j = 1 n g i j ( z ) ξ i ξ ¯ j . \left|\xi\right|_{B,z}:=\sqrt{\sum_{i,j=1}^{n}g_{ij}(z)\xi_{i}\bar{\xi}_{j}}.
  8. γ : [ 0 , 1 ] n \gamma\colon[0,1]\to{\mathbb{C}}^{n}
  9. ( γ ) = 0 1 | γ t ( t ) | B , γ ( t ) d t . \ell(\gamma)=\int_{0}^{1}\left|\frac{\partial\gamma}{\partial t}(t)\right|_{B,% \gamma(t)}dt.
  10. d G ( p , q ) d_{G}(p,q)
  11. p , q G p,q\in G
  12. d G ( p , q ) := inf { ( γ ) all piecewise C 1 curves γ such that γ ( 0 ) = p and γ ( 1 ) = q } . d_{G}(p,q):=\inf\{\ell(\gamma)\mid\,\text{ all piecewise }C^{1}\,\text{ curves% }\gamma\,\text{ such that }\gamma(0)=p\,\text{ and }\gamma(1)=q\}.
  13. G G^{\prime}
  14. G G^{\prime}
  15. d G ( p , q ) = d G ( f ( p ) , f ( q ) ) d_{G}(p,q)=d_{G^{\prime}}(f(p),f(q))

Bernstein_inequalities_(probability_theory).html

  1. ε \varepsilon
  2. 𝐏 ( | 1 n i = 1 n X i | > ε ) 2 exp ( - n ε 2 2 ( 1 + ε 3 ) ) . \mathbf{P}\left(\left|\frac{1}{n}\sum_{i=1}^{n}X_{i}\right|>\varepsilon\right)% \leq 2\exp\left(-\frac{n\varepsilon^{2}}{2(1+\frac{\varepsilon}{3})}\right).
  3. 𝐏 ( i = 1 n X i > t ) exp ( - 1 2 t 2 𝐄 [ X j 2 ] + 1 3 M t ) . \mathbf{P}\left(\sum_{i=1}^{n}X_{i}>t\right)\leq\exp\left(-\frac{\tfrac{1}{2}t% ^{2}}{\sum\mathbf{E}\left[X_{j}^{2}\right]+\tfrac{1}{3}Mt}\right).
  4. 𝐄 [ | X i k | ] 1 2 𝐄 [ X i 2 ] L k - 2 k ! \mathbf{E}\left[|X_{i}^{k}|\right]\leq\tfrac{1}{2}\mathbf{E}\left[X_{i}^{2}% \right]L^{k-2}k!
  5. 𝐏 ( i = 1 n X i 2 t 𝐄 [ X i 2 ] ) < exp ( - t 2 ) , for 0 < t 1 2 L 𝐄 [ X j 2 ] . \mathbf{P}\left(\sum_{i=1}^{n}X_{i}\geq 2t\sqrt{\sum\mathbf{E}\left[X_{i}^{2}% \right]}\right)<\exp(-t^{2}),\qquad\,\text{for }0<t\leq\tfrac{1}{2L}\sqrt{\sum% \mathbf{E}\left[X_{j}^{2}\right]}.
  6. 𝐄 [ | X i k | ] k ! 4 ! ( L 5 ) k - 4 \mathbf{E}\left[|X_{i}^{k}|\right]\leq\frac{k!}{4!}\left(\frac{L}{5}\right)^{k% -4}
  7. A k = 𝐄 [ X i k ] . A_{k}=\sum\mathbf{E}\left[X_{i}^{k}\right].
  8. 𝐏 ( | j = 1 n X j - A 3 t 2 3 A 2 | 2 A 2 t [ 1 + A 4 t 2 6 A 2 2 ] ) < 2 exp ( - t 2 ) , for 0 < t 5 2 A 2 4 L . \mathbf{P}\left(\left|\sum_{j=1}^{n}X_{j}-\frac{A_{3}t^{2}}{3A_{2}}\right|\geq% \sqrt{2A_{2}}\,t\left[1+\frac{A_{4}t^{2}}{6A_{2}^{2}}\right]\right)<2\exp(-t^{% 2}),\qquad\,\text{for }0<t\leq\frac{5\sqrt{2A_{2}}}{4L}.
  9. 𝐄 [ X i | X 1 , , X i - 1 ] = 0 , 𝐄 [ X i 2 | X 1 , , X i - 1 ] R i 𝐄 [ X i 2 ] , 𝐄 [ X i k | X 1 , , X i - 1 ] 1 2 𝐄 [ X i 2 | X 1 , , X i - 1 ] L k - 2 k ! \begin{aligned}\displaystyle\mathbf{E}\left[X_{i}|X_{1},\dots,X_{i-1}\right]&% \displaystyle=0,\\ \displaystyle\mathbf{E}\left[X_{i}^{2}|X_{1},\dots,X_{i-1}\right]&% \displaystyle\leq R_{i}\mathbf{E}\left[X_{i}^{2}\right],\\ \displaystyle\mathbf{E}\left[X_{i}^{k}|X_{1},\dots,X_{i-1}\right]&% \displaystyle\leq\tfrac{1}{2}\mathbf{E}\left[X_{i}^{2}|X_{1},\dots,X_{i-1}% \right]L^{k-2}k!\end{aligned}
  10. 𝐏 ( i = 1 n X i 2 t i = 1 n R i 𝐄 [ X i 2 ] ) < exp ( - t 2 ) , for 0 < t 1 2 L i = 1 n R i 𝐄 [ X i 2 ] . \mathbf{P}\left(\sum_{i=1}^{n}X_{i}\geq 2t\sqrt{\sum_{i=1}^{n}R_{i}\mathbf{E}% \left[X_{i}^{2}\right]}\right)<\exp(-t^{2}),\qquad\,\text{for }0<t\leq\tfrac{1% }{2L}\sqrt{\sum_{i=1}^{n}R_{i}\mathbf{E}\left[X_{i}^{2}\right]}.
  11. exp ( λ j = 1 n X j ) , \exp\left(\lambda\sum_{j=1}^{n}X_{j}\right),

Bertrand's_box_paradox.html

  1. 1 / 2 {1}/{2}
  2. 2 / 3 {2}/{3}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. P ( s e e g o l d G G ) P ( s e e g o l d G G ) + P ( s e e g o l d S S ) + P ( s e e g o l d G S ) = 1 1 + 0 + 1 / 2 = 2 3 \frac{\mathrm{P}(see\ gold\mid GG)}{\mathrm{P}(see\ gold\mid GG)+\mathrm{P}(% see\ gold\mid SS)+\mathrm{P}(see\ gold\mid GS)}=\frac{1}{1+0+1/2}=\frac{2}{3}
  7. 2 / 3 {2}/{3}
  8. 2 / 3 {2}/{3}
  9. 2 / 3 {2}/{3}
  10. 2 / 3 {2}/{3}
  11. 2 / 3 {2}/{3}
  12. 1 / 2 {1}/{2}
  13. 1 / 2 {1}/{2}
  14. 2 / 3 {2}/{3}
  15. 2 / 3 {2}/{3}
  16. 1 / 6 {1}/{6}
  17. 1 / 3 {1}/{3}
  18. 1 / 3 {1}/{3}
  19. 2 / 3 {2}/{3}
  20. 1 / 2 {1}/{2}
  21. 1 / 3 {1}/{3}
  22. 2 / 3 {2}/{3}
  23. 2 / 3 {2}/{3}
  24. 2 / 3 {2}/{3}
  25. 1 / 2 {1}/{2}
  26. 2 / 3 {2}/{3}
  27. 2 / 3 {2}/{3}
  28. 1 / 3 {1}/{3}
  29. 1 / 3 {1}/{3}
  30. 2 / 3 {2}/{3}
  31. 2 / 3 {2}/{3}
  32. 1 / 3 {1}/{3}
  33. 1 / 2 {1}/{2}
  34. 1 / 3 {1}/{3}
  35. 1 1 / 3 1 / 2 = 2 / 3. \frac{1\cdot 1/3}{1/2}=2/3.
  36. 3 / 4 {3}/{4}
  37. 1 / 2 {1}/{2}
  38. 1 / 2 3 / 4 = 2 / 3. \frac{1/2}{3/4}=2/3.
  39. 2 / 3 {2}/{3}
  40. 2 / 3 {2}/{3}

Bethe_formula.html

  1. - d E d x = 4 π m e c 2 n z 2 β 2 ( e 2 4 π ε 0 ) 2 [ ln ( 2 m e c 2 β 2 I ( 1 - β 2 ) ) - β 2 ] -\frac{dE}{dx}=\frac{4\pi}{m_{e}c^{2}}\cdot\frac{nz^{2}}{\beta^{2}}\cdot\left(% \frac{e^{2}}{4\pi\varepsilon_{0}}\right)^{2}\cdot\left[\ln\left(\frac{2m_{e}c^% {2}\beta^{2}}{I\cdot(1-\beta^{2})}\right)-\beta^{2}\right]
  2. β = v c \beta=\frac{v}{c}
  3. n = N A Z ρ A M u , n=\frac{N_{A}\cdot Z\cdot\rho}{A\cdot M_{u}}\,,

Bethe–Salpeter_equation.html

  1. Γ ( P , p ) = d 4 k ( 2 π ) 4 K ( P , p , k ) S ( k - P 2 ) Γ ( P , k ) S ( k + P 2 ) \Gamma(P,p)=\int\!\frac{d^{4}k}{(2\pi)^{4}}\;K(P,p,k)\,S(k-\tfrac{P}{2})\,% \Gamma(P,k)\,S(k+\tfrac{P}{2})
  2. G = S 1 S 2 + S 1 S 2 K 12 G G=S_{1}\,S_{2}+S_{1}\,S_{2}\,K_{12}\,G
  3. Ω | ϕ 1 ϕ 2 ϕ 3 ϕ 4 | Ω \langle\Omega|\phi_{1}\,\phi_{2}\,\phi_{3}\,\phi_{4}|\Omega\rangle
  4. Ψ = Ω | ϕ 1 ϕ 2 | ψ \Psi=\langle\Omega|\phi_{1}\,\phi_{2}|\psi\rangle
  5. ϕ i \phi_{i}
  6. ψ \psi
  7. G Ψ Ψ ¯ P 2 - M 2 , G\approx\frac{\Psi\;\bar{\Psi}}{P^{2}-M^{2}},
  8. P 2 = M 2 P^{2}=M^{2}
  9. P μ = ( E / c , p ) P_{\mu}=\left(E/c,\vec{p}\right)
  10. P 2 = P μ P μ P^{2}=P_{\mu}\,P^{\mu}
  11. Ψ Ψ ¯ P 2 - M 2 = S 1 S 2 + S 1 S 2 K 12 Ψ Ψ ¯ P 2 - M 2 \frac{\Psi\;\bar{\Psi}}{P^{2}-M^{2}}=S_{1}\,S_{2}+S_{1}\,S_{2}\,K_{12}\frac{% \Psi\;\bar{\Psi}}{P^{2}-M^{2}}
  12. Ψ = S 1 S 2 K 12 Ψ , \Psi=S_{1}\,S_{2}\,K_{12}\Psi,\,
  13. Ψ = S 1 S 2 Γ \Psi=S_{1}\,S_{2}\,\Gamma
  14. Γ = K 12 S 1 S 2 Γ \Gamma=K_{12}\,S_{1}\,S_{2}\,\Gamma
  15. 2 P μ = Γ ¯ ( P μ ( S 1 S 2 ) - S 1 S 2 ( P μ K ) S 1 S 2 ) Γ 2P_{\mu}=\bar{\Gamma}\left(\frac{\partial}{\partial P_{\mu}}\left(S_{1}\otimes S% _{2}\right)-S_{1}\,S_{2}\,\left(\frac{\partial}{\partial P_{\mu}}\,K\right)\,S% _{1}\,S_{2}\right)\Gamma

Beverton–Holt_model.html

  1. n t + 1 = R 0 n t 1 + n t / M . n_{t+1}=\frac{R_{0}n_{t}}{1+n_{t}/M}.
  2. n t = K n 0 n 0 + ( K - n 0 ) R 0 - t . n_{t}=\frac{Kn_{0}}{n_{0}+(K-n_{0})R_{0}^{-t}}.
  3. d N d t = r N ( 1 - N K ) , \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),
  4. N ( t ) = K N ( 0 ) N ( 0 ) + ( K - N ( 0 ) ) e - r t . N(t)=\frac{KN(0)}{N(0)+(K-N(0))e^{-rt}}.

Biangular_coordinates.html

  1. C 1 C_{1}\,\!
  2. C 2 C_{2}\,\!
  3. C 1 C 2 ¯ \overline{C_{1}C_{2}}\,\!
  4. P C 1 C 2 \angle PC_{1}C_{2}\,\!
  5. P C 2 C 1 \angle PC_{2}C_{1}\,\!

Bias_of_an_estimator.html

  1. P θ ( x ) = P ( x θ ) P_{\theta}(x)=P(x\mid\theta)
  2. x x
  3. P θ ( x ) = P ( x θ ) P_{\theta}(x)=P(x\mid\theta)
  4. Bias θ [ θ ^ ] = E θ [ θ ^ ] - θ = E θ [ θ ^ - θ ] , \operatorname{Bias}_{\theta}[\,\hat{\theta}\,]=\operatorname{E}_{\theta}[\,% \hat{\theta}\,]-\theta=\operatorname{E}_{\theta}[\,\hat{\theta}-\theta\,],
  5. E θ \operatorname{E}_{\theta}
  6. P θ ( x ) = P ( x θ ) P_{\theta}(x)=P(x\mid\theta)
  7. x x
  8. P ( x θ ) P(x\mid\theta)
  9. X ¯ = 1 n i = 1 n X i , S 2 = 1 n i = 1 n ( X i - X ¯ ) 2 , \overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i},\qquad S^{2}=\frac{1}{n}\sum_{i=1}% ^{n}\left(X_{i}-\overline{X}\,\right)^{2},
  10. E [ S 2 ] = E [ 1 n i = 1 n ( X i - X ¯ ) 2 ] = E [ 1 n i = 1 n ( ( X i - μ ) - ( X ¯ - μ ) ) 2 ] = E [ 1 n i = 1 n ( X i - μ ) 2 - 2 ( X ¯ - μ ) 1 n i = 1 n ( X i - μ ) + ( X ¯ - μ ) 2 ] = E [ 1 n i = 1 n ( X i - μ ) 2 - ( X ¯ - μ ) 2 ] = σ 2 - E [ ( X ¯ - μ ) 2 ] < σ 2 . \begin{aligned}\displaystyle\operatorname{E}[S^{2}]&\displaystyle=% \operatorname{E}\left[\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)% ^{2}\right]=\operatorname{E}\bigg[\frac{1}{n}\sum_{i=1}^{n}\big((X_{i}-\mu)-(% \overline{X}-\mu)\big)^{2}\bigg]\\ &\displaystyle=\operatorname{E}\bigg[\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\mu)^{2}-% 2(\overline{X}-\mu)\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\mu)+(\overline{X}-\mu)^{2}% \bigg]\\ &\displaystyle=\operatorname{E}\bigg[\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\mu)^{2}-% (\overline{X}-\mu)^{2}\bigg]=\sigma^{2}-\operatorname{E}\left[(\overline{X}-% \mu)^{2}\right]<\sigma^{2}.\end{aligned}
  11. X ¯ \overline{X}
  12. i = 1 n ( X i - X ¯ ) 2 \sum_{i=1}^{n}(X_{i}-\overline{X})^{2}
  13. μ X ¯ \mu\neq\overline{X}
  14. 1 n i = 1 n ( X i - X ¯ ) 2 < 1 n i = 1 n ( X i - μ ) 2 , \frac{1}{n}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}<\frac{1}{n}\sum_{i=1}^{n}(X_% {i}-\mu)^{2},
  15. E [ S 2 ] = E [ 1 n i = 1 n ( X i - X ¯ ) 2 ] < E [ 1 n i = 1 n ( X i - μ ) 2 ] = σ 2 . \begin{aligned}\displaystyle\operatorname{E}[S^{2}]&\displaystyle=% \operatorname{E}\bigg[\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}\bigg]<% \operatorname{E}\bigg[\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\mu)^{2}\bigg]=\sigma^{2% }.\end{aligned}
  16. s 2 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 , s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X}\,)^{2},
  17. E [ ( X ¯ - μ ) 2 ] = 1 n σ 2 . \operatorname{E}\big[(\overline{X}-\mu)^{2}\big]=\frac{1}{n}\sigma^{2}.
  18. P ( X = 0 ) 2 = e - 2 λ \operatorname{P}(X=0)^{2}=e^{-2\lambda}\quad
  19. E ( δ ( X ) ) = x = 0 δ ( x ) λ x e - λ x ! = e - 2 λ , E(\delta(X))=\sum_{x=0}^{\infty}\delta(x)\frac{\lambda^{x}e^{-\lambda}}{x!}=e^% {-2\lambda},
  20. δ ( x ) = ( - 1 ) x . \delta(x)=(-1)^{x}.\,
  21. e - 2 X e^{-2{X}}\quad
  22. e - 4 λ - 2 e λ ( 1 / e 2 - 3 ) + e λ ( 1 / e 4 - 1 ) e^{-4\lambda}-2e^{\lambda(1/e^{2}-3)}+e^{\lambda(1/e^{4}-1)}\,
  23. 1 - e - 4 λ . 1-e^{-4\lambda}.\,
  24. e - 2 λ - e λ ( 1 / e 2 - 1 ) . e^{-2\lambda}-e^{\lambda(1/e^{2}-1)}.\,
  25. MSE ( θ ^ ) = E [ ( θ ^ - θ ) 2 ] . \operatorname{MSE}(\hat{\theta})=\operatorname{E}\big[(\hat{\theta}-\theta)^{2% }\big].
  26. MSE ( θ ^ ) = \displaystyle\operatorname{MSE}(\hat{\theta})=
  27. MSE ( θ ^ ) = trace ( Var ( θ ^ ) ) + Bias ( θ ^ , θ ) 2 \operatorname{MSE}(\hat{\theta})=\operatorname{trace}(\operatorname{Var}(\hat{% \theta}))+\left\|\operatorname{Bias}(\hat{\theta},\theta)\right\|^{2}
  28. trace ( Var ( θ ^ ) ) \operatorname{trace}(\operatorname{Var}(\hat{\theta}))
  29. T 2 = c i = 1 n ( X i - X ¯ ) 2 = c n S 2 T^{2}=c\sum_{i=1}^{n}\left(X_{i}-\overline{X}\,\right)^{2}=cnS^{2}
  30. MSE = \displaystyle\operatorname{MSE}=
  31. E [ n S 2 ] = ( n - 1 ) σ 2 and Var ( n S 2 ) = 2 ( n - 1 ) σ 4 . \operatorname{E}[nS^{2}]=(n-1)\sigma^{2}\,\text{ and }\operatorname{Var}(nS^{2% })=2(n-1)\sigma^{4}.
  32. MSE = ( c ( n - 1 ) - 1 ) 2 σ 4 + 2 c 2 ( n - 1 ) σ 4 \operatorname{MSE}=(c(n-1)-1)^{2}\sigma^{4}+2c^{2}(n-1)\sigma^{4}
  33. p ( θ D , I ) p ( θ I ) p ( D θ , I ) p(\theta\mid D,I)\propto p(\theta\mid I)p(D\mid\theta,I)
  34. ExpectedLoss = E [ ( c n S 2 - σ 2 ) 2 ] = E [ σ 4 ( c n S 2 σ 2 - 1 ) 2 ] \operatorname{ExpectedLoss}=\operatorname{E}\left[\left(cnS^{2}-\sigma^{2}% \right)^{2}\right]=\operatorname{E}\left[\sigma^{4}\left(cn\tfrac{S^{2}}{% \sigma^{2}}-1\right)^{2}\right]
  35. p ( σ 2 ) 1 / σ 2 \scriptstyle{p(\sigma^{2})\;\propto\;1/\sigma^{2}}
  36. p ( S 2 σ 2 S 2 ) = p ( S 2 σ 2 σ 2 ) = g ( S 2 σ 2 ) p\left(\tfrac{S^{2}}{\sigma^{2}}\mid S^{2}\right)=p\left(\tfrac{S^{2}}{\sigma^% {2}}\mid\sigma^{2}\right)=g\left(\tfrac{S^{2}}{\sigma^{2}}\right)
  37. E p ( S 2 σ 2 ) [ σ 4 ( c n S 2 σ 2 - 1 ) 2 ] = σ 4 E p ( S 2 σ 2 ) [ ( c n S 2 σ 2 - 1 ) 2 ] \operatorname{E}_{p(S^{2}\mid\sigma^{2})}\left[\sigma^{4}\left(cn\tfrac{S^{2}}% {\sigma^{2}}-1\right)^{2}\right]=\sigma^{4}\operatorname{E}_{p(S^{2}\mid\sigma% ^{2})}\left[\left(cn\tfrac{S^{2}}{\sigma^{2}}-1\right)^{2}\right]
  38. E p ( σ 2 S 2 ) [ σ 4 ( c n S 2 σ 2 - 1 ) 2 ] σ 4 E p ( σ 2 S 2 ) [ ( c n S 2 σ 2 - 1 ) 2 ] \operatorname{E}_{p(\sigma^{2}\mid S^{2})}\left[\sigma^{4}\left(cn\tfrac{S^{2}% }{\sigma^{2}}-1\right)^{2}\right]\neq\sigma^{4}\operatorname{E}_{p(\sigma^{2}% \mid S^{2})}\left[\left(cn\tfrac{S^{2}}{\sigma^{2}}-1\right)^{2}\right]

Biblical_manuscript.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}

Bicarbonate_buffering_system.html

  1. CO 2 + H 2 O H 2 CO 3 HCO 3 - + H + \rm CO_{2}+H_{2}O\rightleftarrows H_{2}CO_{3}\rightleftarrows HCO_{3}^{-}+H^{+}
  2. p H = p K a H 2 C O 3 + log ( [ H C O 3 - ] [ H 2 C O 3 ] ) pH=pK_{a~{}H_{2}CO_{3}}+\log\left(\frac{[HCO_{3}^{-}]}{[H_{2}CO_{3}]}\right)
  3. [ H 2 C O 3 ] = k H CO 2 × p C O 2 [H_{2}CO_{3}]=k_{\rm H~{}CO_{2}}\,\times pCO_{2}
  4. p H = 6.1 + log ( [ H C O 3 - ] 0.03 × p C O 2 ) pH=6.1+\log\left(\frac{[HCO_{3}^{-}]}{0.03\times pCO_{2}}\right)

Bicycle_and_motorcycle_geometry.html

  1. Trailbicycle = R w cos ( A h ) - O f sin ( A h ) \,\text{Trail}\text{bicycle}=\frac{R_{w}\cos(A_{h})-O_{f}}{\sin(A_{h})}
  2. Trailmotorcycle = R w sin ( A r ) - O f cos ( A r ) \,\text{Trail}\text{motorcycle}=\frac{R_{w}\sin(A_{r})-O_{f}}{\cos(A_{r})}
  3. R w R_{w}
  4. A h A_{h}
  5. A r A_{r}
  6. O f O_{f}
  7. f = b sin ϑ cos ϑ f=b\sin\vartheta\cos\vartheta
  8. f f
  9. b b
  10. ϑ \vartheta

Bidirectional_map.html

  1. ( k e y , v a l u e ) (key,value)
  2. v a l u e value
  3. k e y key
  4. ( a , b ) (a,b)
  5. a a
  6. b b
  7. b b
  8. a a
  9. a a
  10. b b

Bifolium.html

  1. a = 1. a=1.
  2. ( x 2 + y 2 ) 2 = a x 2 y . (x^{2}+y^{2})^{2}=ax^{2}y.\,
  3. ρ = a sin θ cos 2 θ . \rho=a\sin\theta\,\cos^{2}\theta.

Bill_Foster_(politician).html

  1. p e + π 0 p\rightarrow e+\pi^{0}

Binary_octahedral_group.html

  1. Spin ( 3 ) SO ( 3 ) \operatorname{Spin}(3)\to\operatorname{SO}(3)
  2. Spin ( 3 ) Sp ( 1 ) \operatorname{Spin}(3)\cong\operatorname{Sp}(1)
  3. { ± 1 , ± i , ± j , ± k , 1 2 ( ± 1 ± i ± j ± k ) } \{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1\pm i\pm j\pm k)\}
  4. 1 2 ( ± 1 ± 1 i + 0 j + 0 k ) \tfrac{1}{\sqrt{2}}(\pm 1\pm 1i+0j+0k)
  5. 1 { ± 1 } 2 O O 1. 1\to\{\pm 1\}\to 2O\to O\to 1.\,
  6. r , s , t r 2 = s 3 = t 4 = r s t \langle r,s,t\mid r^{2}=s^{3}=t^{4}=rst\rangle
  7. s , t ( s t ) 2 = s 3 = t 4 . \langle s,t\mid(st)^{2}=s^{3}=t^{4}\rangle.
  8. s = 1 2 ( 1 + i + j + k ) t = 1 2 ( 1 + i ) . s=\tfrac{1}{2}(1+i+j+k)\qquad t=\tfrac{1}{\sqrt{2}}(1+i).
  9. Spin ( n ) S O ( n ) . \operatorname{Spin}(n)\to SO(n).

Binary_tetrahedral_group.html

  1. \langle
  2. \rangle
  3. Spin ( 3 ) Sp ( 1 ) \operatorname{Spin}(3)\cong\operatorname{Sp}(1)
  4. { ± 1 , ± i , ± j , ± k , 1 2 ( ± 1 ± i ± j ± k ) } \{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1\pm i\pm j\pm k)\}
  5. 1 { ± 1 } 2 T T 1. 1\to\{\pm 1\}\to 2T\to T\to 1.
  6. T A 4 , T\cong A_{4},
  7. 2 T A 4 ^ . 2T\cong\widehat{A_{4}}.
  8. A 4 A_{4}
  9. S 4 S_{4}
  10. 2 T = Q 3 2T=Q\rtimes\mathbb{Z}_{3}
  11. Q Q
  12. ω = 1 2 ( 1 + i + j + k ) ω=−\frac{1}{2}(1+i+j+k)
  13. Q Q
  14. ω ω
  15. i i
  16. j j
  17. k k
  18. r , s , t r 2 = s 3 = t 3 = r s t \langle r,s,t\mid r^{2}=s^{3}=t^{3}=rst\rangle
  19. s , t ( s t ) 2 = s 3 = t 3 . \langle s,t\mid(st)^{2}=s^{3}=t^{3}\rangle.
  20. s = 1 2 ( 1 + i + j + k ) t = 1 2 ( 1 + i + j - k ) . s=\tfrac{1}{2}(1+i+j+k)\qquad t=\tfrac{1}{2}(1+i+j-k).

Bing_metrization_theorem.html

  1. X X
  2. F F
  3. X X
  4. X X
  5. F F

Binocular_disparity.html

  1. L ( r , c ) R ( r , c - d ) ( L ( r , c ) 2 ) ( R ( r , c - d ) 2 ) \frac{\sum{\sum{L(r,c)\cdot R(r,c-d)}}}{\sqrt{(\sum{\sum{L(r,c)^{2}}})\cdot(% \sum{\sum{R(r,c-d)^{2}}})}}
  2. ( L ( r , c ) - R ( r , c - d ) ) 2 \sum{\sum{(L(r,c)-R(r,c-d))^{2}}}
  3. | L ( r , c ) - R ( r , c - d ) | \sum{\sum{\left|L(r,c)-R(r,c-d)\right|}}

Binomial_approximation.html

  1. x x
  2. α \alpha
  3. ( 1 + x ) α 1 + α x . (1+x)^{\alpha}\approx 1+\alpha x.
  4. x > - 1 x>-1
  5. α \alpha
  6. f ( x ) = ( 1 + x ) α . f(x)=(1+x)^{\alpha}.
  7. f ( x ) = α ( 1 + x ) α - 1 . f^{\prime}(x)=\alpha(1+x)^{\alpha-1}.
  8. f ( 0 ) = α . f^{\prime}(0)=\alpha.
  9. f ( x ) f ( a ) + f ( a ) ( x - a ) . f(x)\approx f(a)+f^{\prime}(a)(x-a).
  10. f ( x ) f ( 0 ) + f ( 0 ) ( x - 0 ) . f(x)\approx f(0)+f^{\prime}(0)(x-0).
  11. ( 1 + x ) α 1 + α x . (1+x)^{\alpha}\approx 1+\alpha x.
  12. M ( p ) = 0 ( 1 + α x ) - γ x p - 1 d x M(p)=\int^{\infty}_{0}(1+\alpha x)^{-\gamma}x^{p-1}dx
  13. y = α x y=\alpha x\,
  14. M ( p ) = α - p 0 ( 1 + y ) - γ y p - 1 d y M(p)=\alpha^{-p}\int^{\infty}_{0}(1+y)^{-\gamma}y^{p-1}dy
  15. y = z / ( 1 - z ) y=z/(1-z)
  16. M ( p ) = α - p 0 1 ( 1 - z ) γ - p - 1 z p - 1 d z M(p)=\alpha^{-p}\int^{1}_{0}(1-z)^{\gamma-p-1}z^{p-1}dz
  17. = α - p B ( γ - p , p ) =\alpha^{-p}B(\gamma-p,p)\,
  18. = α - p Γ ( γ - p ) Γ ( p ) Γ ( γ ) . =\alpha^{-p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}.
  19. ( 1 + α x ) - γ = 1 2 π i c - i c + i ( x α ) - p Γ ( γ - p ) Γ ( p ) Γ ( γ ) d p (1+\alpha x)^{-\gamma}=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}(x\alpha)^{% -p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}dp
  20. | α x | < 1 |\alpha x|<1\,
  21. ( 1 + α x ) - γ = Σ n = 0 ( α x ) n ( - 1 ) n n ! Γ ( γ + n ) Γ ( γ ) (1+\alpha x)^{-\gamma}=\Sigma_{n=0}^{\infty}(\alpha x)^{n}\frac{(-1)^{n}}{n!}% \frac{\Gamma(\gamma+n)}{\Gamma(\gamma)}
  22. = 1 - α x γ + ( 1 / 2 ) ( α x ) 2 ( γ + 1 ) γ - =1-\alpha x\gamma+(1/2)(\alpha x)^{2}(\gamma+1)\gamma-...\,

Binomial_inverse_theorem.html

  1. ( 𝐀 + 𝐔𝐁𝐕 ) - 1 = 𝐀 - 1 - 𝐀 - 1 𝐔𝐁 ( 𝐁 + 𝐁𝐕𝐀 - 1 𝐔𝐁 ) - 1 𝐁𝐕𝐀 - 1 \left(\mathbf{A}+\mathbf{UBV}\right)^{-1}=\mathbf{A}^{-1}-\mathbf{A}^{-1}% \mathbf{UB}\left(\mathbf{B}+\mathbf{BVA}^{-1}\mathbf{UB}\right)^{-1}\mathbf{% BVA}^{-1}
  2. ( 𝐀 + 𝐔𝐁𝐕 ) - 1 = 𝐀 - 1 - 𝐀 - 1 𝐔 ( 𝐁 - 1 + 𝐕𝐀 - 1 𝐔 ) - 1 𝐕𝐀 - 1 . \left(\mathbf{A}+\mathbf{UBV}\right)^{-1}=\mathbf{A}^{-1}-\mathbf{A}^{-1}% \mathbf{U}\left(\mathbf{B}^{-1}+\mathbf{VA}^{-1}\mathbf{U}\right)^{-1}\mathbf{% VA}^{-1}.
  3. ( 𝐀 + 𝐔𝐁𝐕 ) - 1 = 𝐀 - 1 - 𝐀 - 1 𝐔 ( 𝐈 + 𝐁𝐕𝐀 - 1 𝐔 ) - 1 𝐁𝐕𝐀 - 1 . (\mathbf{A+UBV})^{-1}=\mathbf{A}^{-1}-\mathbf{A}^{-1}\mathbf{U}(\mathbf{I+BVA}% ^{-1}\mathbf{U})^{-1}\mathbf{BVA}^{-1}.
  4. ( 𝐀 + 𝐔𝐁𝐕 ) 𝐀 - 1 𝐔𝐁 = 𝐔𝐁 + 𝐔𝐁𝐕𝐀 - 1 𝐔𝐁 = 𝐔 ( 𝐁 + 𝐁𝐕𝐀 - 1 𝐔𝐁 ) . \left(\mathbf{A}+\mathbf{UBV}\right)\mathbf{A}^{-1}\mathbf{UB}=\mathbf{UB}+% \mathbf{UBVA}^{-1}\mathbf{UB}=\mathbf{U}\left(\mathbf{B}+\mathbf{BVA}^{-1}% \mathbf{UB}\right).
  5. ( 𝐀 + 𝐔𝐁𝐕 ) ( 𝐀 - 1 - 𝐀 - 1 𝐔𝐁 ( 𝐁 + 𝐁𝐕𝐀 - 1 𝐔𝐁 ) - 1 𝐁𝐕𝐀 - 1 ) \left(\mathbf{A}+\mathbf{UBV}\right)\left(\mathbf{A}^{-1}-\mathbf{A}^{-1}% \mathbf{UB}\left(\mathbf{B}+\mathbf{BVA}^{-1}\mathbf{UB}\right)^{-1}\mathbf{% BVA}^{-1}\right)
  6. = 𝐈 p + 𝐔𝐁𝐕𝐀 - 1 - 𝐔 ( 𝐁 + 𝐁𝐕𝐀 - 1 𝐔𝐁 ) ( 𝐁 + 𝐁𝐕𝐀 - 1 𝐔𝐁 ) - 1 𝐁𝐕𝐀 - 1 =\mathbf{I}_{p}+\mathbf{UBVA}^{-1}-\mathbf{U}\left(\mathbf{B}+\mathbf{BVA}^{-1% }\mathbf{UB}\right)\left(\mathbf{B}+\mathbf{BVA}^{-1}\mathbf{UB}\right)^{-1}% \mathbf{BVA}^{-1}
  7. = 𝐈 p + 𝐔𝐁𝐕𝐀 - 1 - 𝐔𝐁𝐕𝐀 - 1 = 𝐈 p =\mathbf{I}_{p}+\mathbf{UBVA}^{-1}-\mathbf{UBVA}^{-1}=\mathbf{I}_{p}\!
  8. ( 𝐁 + 𝐁𝐕𝐀 - 1 𝐔𝐁 ) - 1 \left(\mathbf{B}+\mathbf{BVA}^{-1}\mathbf{UB}\right)^{-1}
  9. ( 𝐀 + 𝐔𝐁𝐕 ) - 1 \left(\mathbf{A}+\mathbf{UBV}\right)^{-1}
  10. ( 𝐀 + 𝐁 ) - 1 = 𝐀 - 1 - 𝐀 - 1 𝐁 ( 𝐁 + 𝐁𝐀 - 1 𝐁 ) - 1 𝐁𝐀 - 1 . \left(\mathbf{A}+\mathbf{B}\right)^{-1}=\mathbf{A}^{-1}-\mathbf{A}^{-1}\mathbf% {B}\left(\mathbf{B}+\mathbf{BA}^{-1}\mathbf{B}\right)^{-1}\mathbf{BA}^{-1}.
  11. ( 𝐂𝐃 ) - 1 = 𝐃 - 1 𝐂 - 1 , \left(\mathbf{C}\mathbf{D}\right)^{-1}=\mathbf{D}^{-1}\mathbf{C}^{-1},
  12. ( 𝐀 + 𝐁 ) - 1 = 𝐀 - 1 - 𝐀 - 1 ( 𝐈 + 𝐁𝐀 - 1 ) - 1 𝐁𝐀 - 1 . \left(\mathbf{A}+\mathbf{B}\right)^{-1}=\mathbf{A}^{-1}-\mathbf{A}^{-1}\left(% \mathbf{I}+\mathbf{B}\mathbf{A}^{-1}\right)^{-1}\mathbf{B}\mathbf{A}^{-1}.
  13. ( 𝐀 + 𝐮𝐯 T ) - 1 = 𝐀 - 1 - 𝐀 - 1 𝐮𝐯 T 𝐀 - 1 1 + 𝐯 T 𝐀 - 1 𝐮 . \left(\mathbf{A}+\mathbf{uv}^{\mathrm{T}}\right)^{-1}=\mathbf{A}^{-1}-\frac{% \mathbf{A}^{-1}\mathbf{uv}^{\mathrm{T}}\mathbf{A}^{-1}}{1+\mathbf{v}^{\mathrm{% T}}\mathbf{A}^{-1}\mathbf{u}}.
  14. ( 𝐈 p + 𝐔𝐕 ) - 1 = 𝐈 p - 𝐔 ( 𝐈 q + 𝐕𝐔 ) - 1 𝐕 . \left(\mathbf{I}_{p}+\mathbf{UV}\right)^{-1}=\mathbf{I}_{p}-\mathbf{U}\left(% \mathbf{I}_{q}+\mathbf{VU}\right)^{-1}\mathbf{V}.
  15. ( 𝐈 + 𝐮𝐯 T ) - 1 = 𝐈 - 𝐮𝐯 T 1 + 𝐯 T 𝐮 , \left(\mathbf{I}+\mathbf{uv}^{\mathrm{T}}\right)^{-1}=\mathbf{I}-\frac{\mathbf% {uv}^{\mathrm{T}}}{1+\mathbf{v}^{\mathrm{T}}\mathbf{u}},

Biochemical_systems_theory.html

  1. d X i d t = j μ i j γ j k X k f j k \frac{dX_{i}}{dt}=\sum_{j}\mu_{ij}\cdot\gamma_{j}\prod_{k}X_{k}^{f_{jk}}\,
  2. μ \mu
  3. γ \gamma

Bit_numbering.html

  1. i = 0 N - 1 a i 2 i \sum_{i=0}^{N-1}a_{i}\cdot 2^{i}

Bitcrusher.html

  1. 2 n 2^{n}

Blaschke_selection_theorem.html

  1. { K n } \{K_{n}\}
  2. { K n m } \{K_{n_{m}}\}
  3. K K
  4. K n m K_{n_{m}}
  5. K K

Blind_equalization.html

  1. { h [ n ] } n = - \{h[n]\}_{n=-\infty}^{\infty}
  2. r [ k ] r[k]
  3. s [ k ] s[k]
  4. r [ k ] = n = - h [ n ] s [ k - n ] r[k]=\sum_{n=-\infty}^{\infty}h[n]s[k-n]
  5. r [ k ] r[k]
  6. w [ k ] w[k]
  7. s ^ [ k ] = n = - w [ n ] r [ k - n ] \hat{s}[k]=\sum_{n=-\infty}^{\infty}w[n]r[k-n]
  8. s ^ \hat{s}
  9. s s
  10. s ^ \hat{s}
  11. { s ~ [ n ] , h ~ [ n ] } \{\tilde{s}[n],\tilde{h}[n]\}
  12. { c s ~ [ n + d ] , h ~ [ n - d ] / c } \{c\tilde{s}[n+d],\tilde{h}[n-d]/c\}
  13. r r
  14. c c
  15. d d
  16. s s
  17. h h
  18. n [ k ] n[k]
  19. r [ k ] = n = - h [ n ] s [ k - n ] + n [ k ] r[k]=\sum_{n=-\infty}^{\infty}h[n]s[k-n]+n[k]
  20. r ( t ) r(t)
  21. { h [ n ] } n = - N N \{h[n]\}_{n=-N}^{N}
  22. N N
  23. w n + 1 [ k ] = w n [ k ] + μ e * [ n ] r [ n - k ] , k = - N , N w_{n+1}[k]=w_{n}[k]+\mu\,e^{*}[n]r[n-k],k=-N,...N
  24. e [ n ] = 𝐠 ( s ^ [ n ] ) - s ^ [ n ] e[n]=\mathbf{g}(\hat{s}[n])-\hat{s}[n]
  25. μ \mu
  26. 𝐠 \mathbf{g}

Bloch's_theorem_(complex_variables).html

  1. 0.4332 3 4 + 2 × 10 - 4 B 3 - 1 2 Γ ( 1 3 ) Γ ( 11 12 ) Γ ( 1 4 ) 0.4719 , 0.4332\approx\frac{\sqrt{3}}{4}+2\times 10^{-4}\leq B\leq\sqrt{\frac{\sqrt{3}-% 1}{2}}\cdot\frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})% }\approx 0.4719,

Bloch_oscillations.html

  1. d k d t = - e E \hbar\frac{dk}{dt}=-eE
  2. k ( t ) = k ( 0 ) - e E t k(t)=k(0)-\frac{eE}{\hbar}t
  3. v ( k ) = 1 d d k v(k)=\frac{1}{\hbar}\frac{d\mathcal{E}}{dk}
  4. ( k ) \mathcal{E}(k)
  5. ( k ) = A cos a k \mathcal{E}(k)=A\cos{ak}
  6. v ( k ) = 1 d d k = - A a sin a k v(k)=\frac{1}{\hbar}\frac{d\mathcal{E}}{dk}=-\frac{Aa}{\hbar}\sin{ak}
  7. x x
  8. x ( t ) = v ( k ( t ) ) d t = - A e E cos ( a e E t ) x(t)=\int{v(k(t))}{dt}=-\frac{A}{eE}\cos\left({\frac{aeE}{\hbar}t}\right)
  9. ω B = a e | E | / \omega_{B}=ae|E|/\hbar
  10. ω B \omega_{B}

Bloch_space.html

  1. \mathcal{B}
  2. ( 1 - | z | 2 ) | f ( z ) | (1-|z|^{2})|f^{\prime}(z)|
  3. \mathcal{B}
  4. f = | f ( 0 ) | + sup z 𝐃 ( 1 - | z | 2 ) | f ( z ) | . \|f\|_{\mathcal{B}}=|f(0)|+\sup_{z\in\mathbf{D}}(1-|z|^{2})|f^{\prime}(z)|.

Blom's_scheme.html

  1. n \scriptstyle n
  2. D k , k \scriptstyle D_{k,k}
  3. G F ( p ) \scriptstyle GF(p)
  4. D \scriptstyle D
  5. k \displaystyle k
  6. I Alice , I Bob G F ( p ) I_{\mathrm{Alice}},I_{\mathrm{Bob}}\in GF(p)
  7. I Alice = ( 3 10 11 ) , I Bob = ( 1 3 15 ) I_{\mathrm{Alice}}=\begin{pmatrix}3\\ 10\\ 11\end{pmatrix},I_{\mathrm{Bob}}=\begin{pmatrix}1\\ 3\\ 15\end{pmatrix}
  8. g Alice \displaystyle g_{\mathrm{Alice}}
  9. D D
  10. g Alice \displaystyle g_{\mathrm{Alice}}
  11. I Bob \scriptstyle I_{\mathrm{Bob}}
  12. g Alice \scriptstyle g_{\mathrm{Alice}}
  13. k Alice / Bob = g Alice t I Bob \scriptstyle k_{\mathrm{Alice/Bob}}=g_{\mathrm{Alice}}^{t}I_{\mathrm{Bob}}
  14. t \scriptstyle t
  15. k Alice / Bob = k Alice / Bob t = ( g Alice t I Bob ) t = ( I Alice t D t I Bob ) t = I Bob t D I Alice = k Bob / Alice k_{\mathrm{Alice/Bob}}=k_{\mathrm{Alice/Bob}}^{t}=(g_{\mathrm{Alice}}^{t}I_{% \mathrm{Bob}})^{t}=(I_{\mathrm{Alice}}^{t}D^{t}I_{\mathrm{Bob}})^{t}=I_{% \mathrm{Bob}}^{t}DI_{\mathrm{Alice}}=k_{\mathrm{Bob/Alice}}
  16. k Alice / Bob \displaystyle k_{\mathrm{Alice/Bob}}

Blue_Monday_(date).html

  1. ( C * R * Z Z ) ( ( T t + D ) * S t ) + ( P * P r ) > 400 \frac{(C*R*ZZ)}{((Tt+D)*St)+(P*Pr)}>400
  2. [ W + D - d ] T Q M N a \frac{[W+D-d]T^{Q}}{MN_{a}}

Bodmer_Papyri.html

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

Body_force.html

  1. 𝐅 body = V ρ 𝐠 ( 𝐫 ) d V , \mathbf{F}_{\mathrm{body}}=\int\limits_{V}\rho\mathbf{g}(\mathbf{r})\mathrm{d}% V\,,
  2. 𝐟 ( 𝐫 ) = ρ ( 𝐫 ) 𝐠 ( 𝐫 ) \mathbf{f}(\mathbf{r})=\rho(\mathbf{r})\mathbf{g}(\mathbf{r})
  3. g = 9.81 m s 2 g=9.81\frac{\mathrm{m}}{\mathrm{s}^{2}}

Bogdanov–Takens_bifurcation.html

  1. y 1 = y 2 , y 2 = β 1 + β 2 y 1 + y 1 2 ± y 1 y 2 . \begin{aligned}\displaystyle y_{1}^{\prime}&\displaystyle=y_{2},\\ \displaystyle y_{2}^{\prime}&\displaystyle=\beta_{1}+\beta_{2}y_{1}+y_{1}^{2}% \pm y_{1}y_{2}.\end{aligned}

Boltzmann's_entropy_formula.html

  1. S = k B ln W S=k_{\mathrm{B}}\ln W\!
  2. W W
  3. N N
  4. N i N_{i}
  5. i i
  6. W W
  7. W W
  8. W = N ! / i N i ! W=N!\;/\;\prod_{i}N_{i}!
  9. ! !
  10. W W
  11. S = - k B p i ln p i S=-k_{\mathrm{B}}\sum p_{i}\ln p_{i}
  12. ρ ln ρ \rho\ln\rho
  13. ρ \rho
  14. S B = - N k B i p i ln p i S_{B}=-Nk_{\mathrm{B}}\sum_{i}p_{i}\ln p_{i}\,

Bolza_surface.html

  1. y 2 = x 5 - x y^{2}=x^{5}-x
  2. 2 \mathbb{C}^{2}
  3. π 2 , π 3 , π 8 \tfrac{\pi}{2},\tfrac{\pi}{3},\tfrac{\pi}{8}
  4. s 2 , s 3 , s 8 s_{2},s_{3},s_{8}
  5. s 2 = 2 s 3 = 3 s 8 = 8 1 s_{2}{}^{2}=s_{3}{}^{3}=s_{8}{}^{8}=1
  6. s 2 s 3 = s 8 s_{2}s_{3}=s_{8}
  7. ( 2 ) \mathbb{Q}(\sqrt{2})
  8. i 2 = - 3 , j 2 = 2 , i j = - j i , i^{2}=-3,\;j^{2}=\sqrt{2},\;ij=-ji,

Bombieri_norm.html

  1. \mathbb{R}
  2. \mathbb{C}
  3. α , β N \forall\alpha,\beta\in\mathbb{N}^{N}
  4. X α | X β = 0 \langle X^{\alpha}|X^{\beta}\rangle=0
  5. α β , \alpha\neq\beta,
  6. α N \forall\alpha\in\mathbb{N}^{N}
  7. X α 2 = α ! | α | ! . \|X^{\alpha}\|^{2}=\frac{\alpha!}{|\alpha|!}.
  8. α = ( α 1 , , α N ) N , \alpha=(\alpha_{1},\dots,\alpha_{N})\in\mathbb{N}^{N},
  9. | α | = Σ i = 1 N α i |\alpha|=\Sigma_{i=1}^{N}\alpha_{i}
  10. α ! = Π i = 1 N ( α i ! ) \alpha!=\Pi_{i=1}^{N}(\alpha_{i}!)
  11. X α = Π i = 1 N X i α i . X^{\alpha}=\Pi_{i=1}^{N}X_{i}^{\alpha_{i}}.
  12. P , Q P,Q
  13. d ( P ) d^{\circ}(P)
  14. d ( Q ) d^{\circ}(Q)
  15. N N
  16. d ( P ) ! d ( Q ) ! ( d ( P ) + d ( Q ) ) ! P 2 Q 2 P Q 2 P 2 Q 2 . \frac{d^{\circ}(P)!d^{\circ}(Q)!}{(d^{\circ}(P)+d^{\circ}(Q))!}\|P\|^{2}\,\|Q% \|^{2}\leq\|P\cdot Q\|^{2}\leq\|P\|^{2}\,\|Q\|^{2}.
  17. P , Q P,Q
  18. d d
  19. N N
  20. h h
  21. N \mathbb{R}^{N}
  22. N \mathbb{C}^{N}
  23. P h | Q h = P | Q \langle P\circ h|Q\circ h\rangle=\langle P|Q\rangle
  24. P = Q P=Q
  25. P h = P \|P\circ h\|=\|P\|
  26. P | Q = ( d + N - 1 N - 1 ) S N P ( Z ) Q ( Z ) ¯ d σ ( Z ) \langle P|Q\rangle={d+N-1\choose N-1}\int_{S^{N}}P(Z)\overline{Q(Z)}\,d\sigma(Z)
  27. S N S^{N}
  28. N \mathbb{C}^{N}
  29. d σ ( Z ) d\sigma(Z)
  30. P P
  31. d d
  32. N N
  33. Z N Z\in\mathbb{C}^{N}
  34. | P ( Z ) | P Z E d |P(Z)|\leq\|P\|\,\|Z\|_{E}^{d}
  35. P ( Z ) E d P Z E d \|\nabla P(Z)\|_{E}\leq d\|P\|\,\|Z\|_{E}^{d}
  36. E \|\cdot\|_{E}

Bond_order_potential.html

  1. V i j ( r i j ) = V r e p u l s i v e ( r i j ) + b i j k V a t t r a c t i v e ( r i j ) V_{ij}(r_{ij})=V_{repulsive}(r_{ij})+b_{ijk}V_{attractive}(r_{ij})
  2. r i j r_{ij}
  3. i i
  4. b i j k b_{ijk}
  5. V i j ( r i j ) = V p a i r ( r i j ) - D ρ i V_{ij}(r_{ij})=V_{pair}(r_{ij})-D\sqrt{\rho_{i}}
  6. ρ i \rho_{i}
  7. i i

Bondareva–Shapley_theorem.html

  1. N , v \;\langle N,v\rangle\;
  2. N \;\;N\;
  3. v : 2 N \;v:2^{N}\to\mathbb{R}\;
  4. N N
  5. N N
  6. N , v \;\langle N,v\rangle\;
  7. α : 2 N { } [ 0 , 1 ] \alpha:2^{N}\setminus\{\emptyset\}\to[0,1]
  8. i N : S 2 N : i S α ( S ) = 1 \forall i\in N:\sum_{S\in 2^{N}:\;i\in S}\alpha(S)=1
  9. S 2 N { } α ( S ) v ( S ) v ( N ) . \sum_{S\in 2^{N}\setminus\{\emptyset\}}\alpha(S)v(S)\leq v(N).

Bonnesen's_inequality.html

  1. L L
  2. A A
  3. r r
  4. R R
  5. π 2 ( R - r ) 2 L 2 - 4 π A . \pi^{2}(R-r)^{2}\leq L^{2}-4\pi A.\,
  6. π 2 ( R - r ) 2 \pi^{2}(R-r)^{2}

Bonnet's_theorem.html

  1. v combined = v 1 2 + v 2 2 + + v n 2 v_{\mathrm{combined}}=\sqrt{v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}}
  2. F k = m R v k 2 F_{k}=\frac{m}{R}v_{k}^{2}
  3. k = 1 n F k = m R k = 1 n v k 2 \sum_{k=1}^{n}F_{k}=\frac{m}{R}\sum_{k=1}^{n}v_{k}^{2}
  4. v combined = v 1 2 + v 2 2 + + v n 2 v_{\mathrm{combined}}=\sqrt{v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}}

Boolean_algebra.html

  1. x y \displaystyle x\wedge y
  2. x x
  3. y y
  4. x y x\wedge y
  5. x y x\vee y
  6. x x
  7. ¬ x \neg x
  8. x y \displaystyle x\wedge y
  9. x y = ¬ x y x\rightarrow y=\neg{x}\vee y
  10. x y = ( x y ) ¬ ( x y ) x\oplus y=(x\vee y)\wedge\neg{(x\wedge y)}
  11. x y = ¬ ( x y ) x\equiv y=\neg{(x\oplus y)}
  12. x x
  13. y y
  14. x y x\rightarrow y
  15. x y x\oplus y
  16. x y x\equiv y
  17. Associativity of x ( y z ) = ( x y ) z Associativity of x ( y z ) = ( x y ) z Commutativity of x y = y x Commutativity of x y = y x Distributivity of over x ( y z ) = ( x y ) ( x z ) Identity for x 0 = x Identity for x 1 = x Annihilator for x 0 = 0 \begin{aligned}&\displaystyle\,\text{Associativity of }\vee&\displaystyle x% \vee(y\vee z)&\displaystyle=(x\vee y)\vee z\\ &\displaystyle\,\text{Associativity of }\wedge&\displaystyle x\wedge(y\wedge z% )&\displaystyle=(x\wedge y)\wedge z\\ &\displaystyle\,\text{Commutativity of }\vee&\displaystyle x\vee y&% \displaystyle=y\vee x\\ &\displaystyle\,\text{Commutativity of }\wedge&\displaystyle x\wedge y&% \displaystyle=y\wedge x\\ &\displaystyle\,\text{Distributivity of }\wedge\,\text{ over }\vee&% \displaystyle x\wedge(y\vee z)&\displaystyle=(x\wedge y)\vee(x\wedge z)\\ &\displaystyle\,\text{Identity for }\vee&\displaystyle x\vee 0&\displaystyle=x% \\ &\displaystyle\,\text{Identity for }\wedge&\displaystyle x\wedge 1&% \displaystyle=x\\ &\displaystyle\,\text{Annihilator for }\wedge&\displaystyle x\wedge 0&% \displaystyle=0\\ \end{aligned}
  18. Idempotence of x x = x Idempotence of x x = x Absorption 1 x ( x y ) = x Absorption 2 x ( x y ) = x Distributivity of over x ( y z ) = ( x y ) ( x z ) Annihilator for x 1 = 1 \begin{aligned}&\displaystyle\,\text{Idempotence of }\vee&\displaystyle x\vee x% &\displaystyle=x\\ &\displaystyle\,\text{Idempotence of }\wedge&\displaystyle x\wedge x&% \displaystyle=x\\ &\displaystyle\,\text{Absorption 1}&\displaystyle x\wedge(x\vee y)&% \displaystyle=x\\ &\displaystyle\,\text{Absorption 2}&\displaystyle x\vee(x\wedge y)&% \displaystyle=x\\ &\displaystyle\,\text{Distributivity of }\vee\,\text{ over }\wedge&% \displaystyle x\vee(y\wedge z)&\displaystyle=(x\vee y)\wedge(x\vee z)\\ &\displaystyle\,\text{Annihilator for }\vee&\displaystyle x\vee 1&% \displaystyle=1\end{aligned}
  19. Complementation 1 x ¬ x = 0 Complementation 2 x ¬ x = 1 \begin{aligned}&\displaystyle\,\text{Complementation 1}&\displaystyle x\wedge% \neg x&\displaystyle=0\\ &\displaystyle\,\text{Complementation 2}&\displaystyle x\vee\neg x&% \displaystyle=1\end{aligned}
  20. Double negation ¬ ( ¬ x ) = x \begin{aligned}&\displaystyle\,\text{Double negation}&\displaystyle\neg{(\neg{% x})}&\displaystyle=x\end{aligned}
  21. ( - x ) ( - y ) = x y ( - x ) + ( - y ) = - ( x + y ) \begin{aligned}\displaystyle(-x)(-y)&\displaystyle=xy\\ \displaystyle(-x)+(-y)&\displaystyle=-(x+y)\end{aligned}
  22. De Morgan 1 \displaystyle\,\text{De Morgan 1}
  23. \vdash
  24. \vdash
  25. \vdash
  26. \vdash
  27. x y xy

Boolean_conjunctive_query.html

  1. R 1 ( t 1 ) R n ( t n ) R_{1}(t_{1})\wedge\cdots\wedge R_{n}(t_{n})
  2. R i R_{i}
  3. t i t_{i}
  4. t i t_{i}
  5. R i R_{i}
  6. F a t h e r Father
  7. E m p l o y e d Employed
  8. F a t h e r ( M a r k , x ) E m p l o y e d ( x ) Father(Mark,x)\wedge Employed(x)
  9. x x

Boolean_network.html

  1. A N \langle A\rangle\sim\sqrt{N}
  2. ν N \langle\nu\rangle\sim\sqrt{N}
  3. A > N x x \langle A\rangle>N^{x}\forall x
  4. ν > N x x \langle\nu\rangle>N^{x}\forall x
  5. ν N \langle\nu\rangle\sim N
  6. ν > N x \langle\nu\rangle>N^{x}
  7. x > 1 x>1
  8. ν > N x x \langle\nu\rangle>N^{x}\forall x
  9. A > N x x \langle A\rangle>N^{x}\forall x
  10. ν > N x x \langle\nu\rangle>N^{x}\forall x

Bootstrapping_(finance).html

  1. 100 = 2.25 ( 1 + 4 % / 2 ) 1 + 2.25 ( 1 + 4.3 % / 2 ) 2 + 102.25 ( 1 + Z 3 / 2 ) 3 100={2.25\over(1+4\%/2)^{1}}+{2.25\over(1+4.3\%/2)^{2}}+{102.25\over(1+Z_{3}/2% )^{3}}
  2. Z 3 Z_{3}
  3. 1 = C n Δ 1 d f 1 + C n Δ 2 d f 2 + C n Δ 3 d f 3 + + ( 1 + C n Δ n ) d f n 1=C_{n}\cdot\Delta_{1}\cdot df_{1}+C_{n}\cdot\Delta_{2}\cdot df_{2}+C_{n}\cdot% \Delta_{3}\cdot df_{3}+\cdots+(1+C_{n}\cdot\Delta_{n})\cdot df_{n}
  4. d f n = ( 1 - i = 1 n - 1 C n Δ i d f i ) ( 1 + C n Δ n ) df_{n}={(1-\sum_{i=1}^{n-1}C_{n}\cdot\Delta_{i}\cdot df_{i})\over(1+C_{n}\cdot% \Delta_{n})}
  5. C n C_{n}
  6. Δ i \Delta_{i}
  7. [ i - 1 ; i ] [i-1;i]
  8. d f i df_{i}
  9. d f n df_{n}