wpmath0000007_10

Pitchfork_bifurcation.html

  1. d x d t = r x - x 3 . \frac{dx}{dt}=rx-x^{3}.
  2. r r
  3. x = 0 x=0
  4. r > 0 r>0
  5. x = 0 x=0
  6. x = ± r x=\pm\sqrt{r}
  7. d x d t = r x + x 3 . \frac{dx}{dt}=rx+x^{3}.
  8. r < 0 r<0
  9. x = 0 x=0
  10. x = ± - r x=\pm\sqrt{-r}
  11. r > 0 r>0
  12. x = 0 x=0
  13. x ˙ = f ( x , r ) \dot{x}=f(x,r)\,
  14. f ( x , r ) f(x,r)
  15. r r\in\mathbb{R}
  16. - f ( x , r ) = f ( - x , r ) -f(x,r)=f(-x,r)\,\,
  17. f x ( 0 , r o ) = 0 , 2 f x 2 ( 0 , r o ) = 0 , 3 f x 3 ( 0 , r o ) 0 , f r ( 0 , r o ) = 0 , 2 f r x ( 0 , r o ) 0. \begin{array}[]{lll}\displaystyle\frac{\partial f}{\partial x}(0,r_{o})=0,&% \displaystyle\frac{\partial^{2}f}{\partial x^{2}}(0,r_{o})=0,&\displaystyle% \frac{\partial^{3}f}{\partial x^{3}}(0,r_{o})\neq 0,\\ \displaystyle\frac{\partial f}{\partial r}(0,r_{o})=0,&\displaystyle\frac{% \partial^{2}f}{\partial r\partial x}(0,r_{o})\neq 0.\end{array}
  18. ( x , r ) = ( 0 , r o ) (x,r)=(0,r_{o})
  19. 3 f x 3 ( 0 , r o ) { < 0 , supercritical > 0 , subcritical \frac{\partial^{3}f}{\partial x^{3}}(0,r_{o})\left\{\begin{matrix}<0,&\mathrm{% supercritical}\\ >0,&\mathrm{subcritical}\end{matrix}\right.\,\,

Pitching_moment.html

  1. M c / 4 M_{c/4}
  2. M c / 4 = M indicated + 𝐱 × ( D indicated , L indicated ) M_{c/4}=M\text{indicated}+\mathbf{x}\times(D\text{indicated},L\text{indicated})
  3. C m C_{m}
  4. C m = M q S c C_{m}=\frac{M}{qSc}
  5. C m C_{m}

Planar_ternary_ring.html

  1. ( R , T ) (R,T)
  2. R R
  3. T : R 3 R T\colon R^{3}\to R\,
  4. ( R , T ) (R,T)
  5. R R
  6. T : R 3 R T\colon R^{3}\to R\,
  7. T ( a , 0 , b ) = T ( 0 , a , b ) = b , a , b R T(a,0,b)=T(0,a,b)=b,\quad\forall a,b\in R
  8. T ( 1 , a , 0 ) = T ( a , 1 , 0 ) = a , a R T(1,a,0)=T(a,1,0)=a,\quad\forall a\in R
  9. a , b , c , d R , a c \forall a,b,c,d\in R,a\neq c
  10. x R x\in R
  11. T ( x , a , b ) = T ( x , c , d ) T(x,a,b)=T(x,c,d)\,
  12. a , b , c R \forall a,b,c\in R
  13. x R x\in R
  14. T ( a , b , x ) = c T(a,b,x)=c\,
  15. a , b , c , d R , a c \forall a,b,c,d\in R,a\neq c
  16. T ( a , x , y ) = b , T ( c , x , y ) = d T(a,x,y)=b,T(c,x,y)=d\,
  17. ( x , y ) R 2 (x,y)\in R^{2}
  18. R R
  19. R 2 R^{2}
  20. T T
  21. a b = T ( a , 1 , b ) a\oplus b=T(a,1,b)
  22. ( R , ) (R,\oplus)
  23. a b = T ( a , b , 0 ) a\otimes b=T(a,b,0)
  24. R 0 = R { 0 } R_{0}=R\setminus\{0\}\,
  25. ( R 0 , ) (R_{0},\otimes)
  26. ( R , T ) (R,T)
  27. T ( a , b , c ) = ( a b ) c , a , b , c R T(a,b,c)=(a\otimes b)\oplus c,\quad\forall a,b,c\in R
  28. ( R , T ) (R,T)
  29. \infty
  30. R R
  31. P = { ( a , b ) | a , b R } { ( a ) | a R } { ( ) } P=\{(a,b)|a,b\in R\}\cup\{(a)|a\in R\}\cup\{(\infty)\}
  32. L = { [ a , b ] | a , b R } { [ a ] | a R } { [ ] } L=\{[a,b]|a,b\in R\}\cup\{[a]|a\in R\}\cup\{[\infty]\}
  33. a , b , c , d R \forall a,b,c,d\in R
  34. I I
  35. ( ( a , b ) , [ c , d ] ) I T ( a , c , d ) = b ((a,b),[c,d])\in I\Longleftrightarrow T(a,c,d)=b
  36. ( ( a , b ) , [ c ] ) I a = c ((a,b),[c])\in I\Longleftrightarrow a=c
  37. ( ( a , b ) , [ ] ) I ((a,b),[\infty])\notin I
  38. ( ( a ) , [ c , d ] ) I a = c ((a),[c,d])\in I\Longleftrightarrow a=c
  39. ( ( a ) , [ c ] ) I ((a),[c])\notin I
  40. ( ( a ) , [ ] ) I ((a),[\infty])\in I
  41. ( ( ( ) , [ c , d ] ) I (((\infty),[c,d])\notin I
  42. ( ( ) , [ a ] ) I ((\infty),[a])\in I
  43. ( ( ) , [ ] ) I ((\infty),[\infty])\in I
  44. \infty
  45. x - x a + x b x\longrightarrow-x\otimes a+x\otimes b
  46. x a x - b x x\longrightarrow a\otimes x-b\otimes x
  47. a b a\neq b
  48. ( x + y ) z = x z + y z (x+y)\otimes z=x\otimes z+y\otimes z
  49. x ( y + z ) = x y + x z . x\otimes(y+z)=x\otimes y+x\otimes z.
  50. a b = T ( 1 , a , b ) a\oplus b=T(1,a,b)

Plasma_effect.html

  1. f ( x , y ) f(x,y)
  2. c c
  3. ( x , y ) (x,y)
  4. sin ( f ( x , y ) ω c ) \sin(f(x,y)\cdot\omega_{c})

Plasma_parameters.html

  1. μ = m i / m p \mu=m_{i}/m_{p}
  2. Z Z
  3. k k
  4. K K
  5. ln Λ \ln\Lambda
  6. ω c e = e B / m e c = 1.76 × 10 7 B rad/s \omega_{ce}=eB/m_{e}c=1.76\times 10^{7}B\mbox{rad/s}~{}\,
  7. ω c i = Z e B / m i c = 9.58 × 10 3 Z μ - 1 B rad/s \omega_{ci}=ZeB/m_{i}c=9.58\times 10^{3}Z\mu^{-1}B\mbox{rad/s}~{}\,
  8. ω p e = ( 4 π n e e 2 / m e ) 1 / 2 = 5.64 × 10 4 n e 1 / 2 rad/s \omega_{pe}=(4\pi n_{e}e^{2}/m_{e})^{1/2}=5.64\times 10^{4}n_{e}^{1/2}\mbox{% rad/s}~{}
  9. ω p i = ( 4 π n i Z 2 e 2 / m i ) 1 / 2 = 1.32 × 10 3 Z μ - 1 / 2 n i 1 / 2 rad/s \omega_{pi}=(4\pi n_{i}Z^{2}e^{2}/m_{i})^{1/2}=1.32\times 10^{3}Z\mu^{-1/2}n_{% i}^{1/2}\mbox{rad/s}~{}
  10. ν T e = ( e K E / m e ) 1 / 2 = 7.26 × 10 8 K 1 / 2 E 1 / 2 s - 1 \nu_{Te}=(eKE/m_{e})^{1/2}=7.26\times 10^{8}K^{1/2}E^{1/2}\mbox{s}~{}^{-1}\,
  11. ν T i = ( Z e K E / m i ) 1 / 2 = 1.69 × 10 7 Z 1 / 2 K 1 / 2 E 1 / 2 μ - 1 / 2 s - 1 \nu_{Ti}=(ZeKE/m_{i})^{1/2}=1.69\times 10^{7}Z^{1/2}K^{1/2}E^{1/2}\mu^{-1/2}% \mbox{s}~{}^{-1}\,
  12. ν e = 2.91 × 10 - 6 n e ln Λ T e - 3 / 2 s - 1 \nu_{e}=2.91\times 10^{-6}n_{e}\,\ln\Lambda\,T_{e}^{-3/2}\mbox{s}~{}^{-1}
  13. ν i = 4.80 × 10 - 8 Z 4 μ - 1 / 2 n i ln Λ T i - 3 / 2 s - 1 \nu_{i}=4.80\times 10^{-8}Z^{4}\mu^{-1/2}n_{i}\,\ln\Lambda\,T_{i}^{-3/2}\mbox{% s}~{}^{-1}
  14. ν e , i = N σ e , i v ¯ = N 0 σ ( v ) e , i f ( v ) v d v \nu_{e,i}=N\overline{\sigma_{e,i}v}=N\int\limits_{0}^{\infty}\sigma(v)_{e,i}f(% v)vdv
  15. σ ( v ) e , i \sigma(v)_{e,i}
  16. f ( v ) f(v)
  17. N N
  18. Λ e = h 2 2 π m e k T e = 6.919 × 10 - 8 T e - 1 / 2 cm \Lambda_{e}=\sqrt{\frac{h^{2}}{2\pi m_{e}kT_{e}}}=6.919\times 10^{-8}\,T_{e}^{% -1/2}\,\mbox{cm}~{}
  19. e 2 / k T = 1.44 × 10 - 7 T - 1 cm e^{2}/kT=1.44\times 10^{-7}\,T^{-1}\,\mbox{cm}~{}
  20. r e = v T e / ω c e = 2.38 T e 1 / 2 B - 1 cm r_{e}=v_{Te}/\omega_{ce}=2.38\,T_{e}^{1/2}B^{-1}\,\mbox{cm}~{}
  21. r i = v T i / ω c i = 1.02 × 10 2 μ 1 / 2 Z - 1 T i 1 / 2 B - 1 cm r_{i}=v_{Ti}/\omega_{ci}=1.02\times 10^{2}\,\mu^{1/2}Z^{-1}T_{i}^{1/2}B^{-1}\,% \mbox{cm}~{}
  22. c / ω p e = 5.31 × 10 5 n e - 1 / 2 cm c/\omega_{pe}=5.31\times 10^{5}\,n_{e}^{-1/2}\,\mbox{cm}~{}
  23. λ D = ( k T / 4 π n e 2 ) 1 / 2 = 7.43 × 10 2 T 1 / 2 n - 1 / 2 cm \lambda_{D}=(kT/4\pi ne^{2})^{1/2}=7.43\times 10^{2}\,T^{1/2}n^{-1/2}\,\mbox{% cm}~{}
  24. d i = c / ω p i d_{i}=c/\omega_{pi}
  25. λ e , i = v e , i ¯ ν e , i \lambda_{e,i}=\frac{\overline{v_{e,i}}}{\nu_{e,i}}
  26. v e , i ¯ \overline{v_{e,i}}
  27. ν e , i \nu_{e,i}
  28. v T e = ( k T e / m e ) 1 / 2 = 4.19 × 10 7 T e 1 / 2 cm/s v_{Te}=(kT_{e}/m_{e})^{1/2}=4.19\times 10^{7}\,T_{e}^{1/2}\,\mbox{cm/s}~{}
  29. v T i = ( k T i / m i ) 1 / 2 = 9.79 × 10 5 μ - 1 / 2 T i 1 / 2 cm/s v_{Ti}=(kT_{i}/m_{i})^{1/2}=9.79\times 10^{5}\,\mu^{-1/2}T_{i}^{1/2}\,\mbox{cm% /s}~{}
  30. c s = ( γ Z k T e / m i ) 1 / 2 = 9.79 × 10 5 ( γ Z T e / μ ) 1 / 2 cm/s c_{s}=(\gamma ZkT_{e}/m_{i})^{1/2}=9.79\times 10^{5}\,(\gamma ZT_{e}/\mu)^{1/2% }\,\mbox{cm/s}~{}
  31. γ = 1 + 2 / n \gamma=1+2/n
  32. n n
  33. v A = B / ( 4 π n i m i ) 1 / 2 = 2.18 × 10 11 μ - 1 / 2 n i - 1 / 2 B cm/s v_{A}=B/(4\pi n_{i}m_{i})^{1/2}=2.18\times 10^{11}\,\mu^{-1/2}n_{i}^{-1/2}B\,% \mbox{cm/s}~{}
  34. ( m e / m p ) 1 / 2 = 2.33 × 10 - 2 = 1 / 42.9 (m_{e}/m_{p})^{1/2}=2.33\times 10^{-2}=1/42.9\,
  35. ( 4 π / 3 ) n λ D 3 = 1.72 × 10 9 T 3 / 2 n - 1 / 2 (4\pi/3)n\lambda_{D}^{3}=1.72\times 10^{9}\,T^{3/2}n^{-1/2}
  36. v A / c = 7.28 μ - 1 / 2 n i - 1 / 2 B v_{A}/c=7.28\,\mu^{-1/2}n_{i}^{-1/2}B
  37. ω p e / ω c e = 3.21 × 10 - 3 n e 1 / 2 B - 1 \omega_{pe}/\omega_{ce}=3.21\times 10^{-3}\,n_{e}^{1/2}B^{-1}
  38. ω p i / ω c i = 0.137 μ 1 / 2 n i 1 / 2 B - 1 \omega_{pi}/\omega_{ci}=0.137\,\mu^{1/2}n_{i}^{1/2}B^{-1}
  39. β = 8 π n k T / B 2 = 4.03 × 10 - 11 n T B - 2 \beta=8\pi nkT/B^{2}=4.03\times 10^{-11}\,nTB^{-2}
  40. B 2 / 8 π n i m i c 2 = 26.5 μ - 1 n i - 1 B 2 B^{2}/8\pi n_{i}m_{i}c^{2}=26.5\,\mu^{-1}n_{i}^{-1}B^{2}
  41. ln Λ 13.6 \ln\Lambda\simeq 13.6
  42. ln Λ 6.8 \ln\Lambda\simeq 6.8

Plastic_bending.html

  1. M r M_{r}
  2. M r M_{r}

Plastic_moment.html

  1. M p = ( b d 2 / 4 ) σ y M_{p}=(bd^{2}/4)\sigma_{y}
  2. Z P Z_{P}
  3. M p = σ Z p M_{p}=\sigma Z_{p}

Plotkin_bound.html

  1. { 0 , 1 } \{0,1\}
  2. 𝔽 2 n \mathbb{F}_{2}^{n}
  3. 𝔽 2 \mathbb{F}_{2}
  4. d d
  5. C C
  6. d = min x , y C , x y d ( x , y ) d=\min_{x,y\in C,x\neq y}d(x,y)
  7. d ( x , y ) d(x,y)
  8. x x
  9. y y
  10. A 2 ( n , d ) A_{2}(n,d)
  11. n n
  12. d d
  13. d d
  14. 2 d > n 2d>n
  15. A 2 ( n , d ) 2 d 2 d - n . A_{2}(n,d)\leq 2\left\lfloor\frac{d}{2d-n}\right\rfloor.
  16. d d
  17. 2 d + 1 > n 2d+1>n
  18. A 2 ( n , d ) 2 d + 1 2 d + 1 - n . A_{2}(n,d)\leq 2\left\lfloor\frac{d+1}{2d+1-n}\right\rfloor.
  19. d d
  20. A 2 ( 2 d , d ) 4 d . A_{2}(2d,d)\leq 4d.
  21. d d
  22. A 2 ( 2 d + 1 , d ) 4 d + 4 A_{2}(2d+1,d)\leq 4d+4
  23. \left\lfloor~{}\right\rfloor
  24. d ( x , y ) d(x,y)
  25. x x
  26. y y
  27. M M
  28. C C
  29. M M
  30. A 2 ( n , d ) A_{2}(n,d)
  31. ( x , y ) C 2 , x y d ( x , y ) \sum_{(x,y)\in C^{2},x\neq y}d(x,y)
  32. M M
  33. x x
  34. M - 1 M-1
  35. y y
  36. d ( x , y ) d d(x,y)\geq d
  37. x x
  38. y y
  39. x y x\neq y
  40. ( x , y ) C 2 , x y d ( x , y ) M ( M - 1 ) d . \sum_{(x,y)\in C^{2},x\neq y}d(x,y)\geq M(M-1)d.
  41. A A
  42. M × n M\times n
  43. C C
  44. s i s_{i}
  45. i i
  46. A A
  47. i i
  48. M - s i M-s_{i}
  49. 2 2
  50. d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)
  51. x , y C d ( x , y ) \sum_{x,y\in C}d(x,y)
  52. x , y C d ( x , y ) = i = 1 n 2 s i ( M - s i ) . \sum_{x,y\in C}d(x,y)=\sum_{i=1}^{n}2s_{i}(M-s_{i}).
  53. M M
  54. s i = M / 2 s_{i}=M/2
  55. i i
  56. x , y C d ( x , y ) 1 2 n M 2 . \sum_{x,y\in C}d(x,y)\leq\frac{1}{2}nM^{2}.
  57. x , y C d ( x , y ) \sum_{x,y\in C}d(x,y)
  58. M ( M - 1 ) d 1 2 n M 2 M(M-1)d\leq\frac{1}{2}nM^{2}
  59. 2 d > n 2d>n
  60. M 2 d 2 d - n . M\leq\frac{2d}{2d-n}.
  61. M M
  62. M 2 d 2 d - n . M\leq 2\left\lfloor\frac{d}{2d-n}\right\rfloor.
  63. M M
  64. i = 1 n 2 s i ( M - s i ) \sum_{i=1}^{n}2s_{i}(M-s_{i})
  65. s i = M ± 1 2 s_{i}=\frac{M\pm 1}{2}
  66. x , y C d ( x , y ) 1 2 n ( M 2 - 1 ) . \sum_{x,y\in C}d(x,y)\leq\frac{1}{2}n(M^{2}-1).
  67. x , y C d ( x , y ) \sum_{x,y\in C}d(x,y)
  68. M ( M - 1 ) d 1 2 n ( M 2 - 1 ) M(M-1)d\leq\frac{1}{2}n(M^{2}-1)
  69. 2 d > n 2d>n
  70. M 2 d 2 d - n - 1. M\leq\frac{2d}{2d-n}-1.
  71. M M
  72. M 2 d 2 d - n - 1 = 2 d 2 d - n - 1 2 d 2 d - n . M\leq\left\lfloor\frac{2d}{2d-n}-1\right\rfloor=\left\lfloor\frac{2d}{2d-n}% \right\rfloor-1\leq 2\left\lfloor\frac{d}{2d-n}\right\rfloor.

PLS_(complexity).html

  1. L L
  2. D L D_{L}
  3. Σ \Sigma
  4. x x
  5. F L ( x ) F_{L}(x)
  6. s F L ( x ) s\in F_{L}(x)
  7. c L ( s , x ) c_{L}(s,x)
  8. N ( s , x ) F L ( x ) N(s,x)\subseteq F_{L}(x)
  9. A L A_{L}
  10. A L ( x ) F L ( x ) A_{L}(x)\in F_{L}(x)
  11. B L B_{L}
  12. c L ( s , x ) c_{L}(s,x)
  13. C L C_{L}
  14. s F L ( x ) s\in F_{L}(x)
  15. s N ( s , x ) s^{\prime}\in N(s,x)
  16. c L ( s , x ) < c L ( s , x ) c_{L}(s^{\prime},x)<c_{L}(s,x)
  17. C L C_{L}
  18. D L D_{L}
  19. s , s F L ( x ) s,s^{\prime}\in F_{L}(x)
  20. s N ( s , x ) s^{\prime}\in N(s,x)
  21. x x
  22. L L
  23. c L ( s , x ) c_{L}(s,x)
  24. s F L ( x ) s\in F_{L}(x)
  25. c L ( s , x ) c L ( s , x ) c_{L}(s^{\prime},x)\geq c_{L}(s,x)
  26. s N ( s , x ) s^{\prime}\in N(s,x)
  27. A L A_{L}
  28. s s
  29. C L C_{L}
  30. s N ( s , x ) s^{\prime}\in N(s,x)
  31. s s
  32. s s^{\prime}
  33. s s
  34. L L

Pneumatic_cylinder.html

  1. F = A σ F=A\sigma
  2. F F
  3. A A
  4. σ \sigma
  5. F r = P A e F_{r}=PA_{e}
  6. F r F_{r}
  7. P P
  8. A e A_{e}
  9. A e A_{e}
  10. F r = P ( π r 2 ) F_{r}=P(\pi r^{2})
  11. F r F_{r}
  12. r r
  13. π \pi
  14. F r = P ( π r 1 2 - π r 2 2 ) = P π ( r 1 2 - r 2 2 ) F_{r}=P(\pi r_{1}^{2}-\pi r_{2}^{2})=P\pi(r_{1}^{2}-r_{2}^{2})
  15. F r F_{r}
  16. r 1 r_{1}
  17. r 2 r_{2}
  18. π \pi

Poincaré_inequality.html

  1. u L p ( Ω ) C u L p ( Ω ) , \|u\|_{L^{p}(\Omega)}\leq C\|\nabla u\|_{L^{p}(\Omega)},
  2. u - u Ω L p ( Ω ) C u L p ( Ω ) , \|u-u_{\Omega}\|_{L^{p}(\Omega)}\leq C\|\nabla u\|_{L^{p}(\Omega)},
  3. u Ω = 1 | Ω | Ω u ( y ) d y u_{\Omega}=\frac{1}{|\Omega|}\int_{\Omega}u(y)\,\mathrm{d}y
  4. 1 q , p < 1\leq q,p<\infty
  5. λ 1 \lambda\geq 1
  6. μ ( B ) - 1 / q u - u B L q ( B ) C rad ( B ) μ ( B ) - 1 / p u L p ( λ B ) . \mu(B)^{-1/q}\|u-u_{B}\|_{L^{q}(B)}\leq C\,\text{rad}(B)\mu(B)^{-1/p}\|\nabla u% \|_{L^{p}(\lambda B)}.
  7. | u | |\nabla u|
  8. [ u ] H 1 / 2 ( 𝐓 2 ) 2 = k 𝐙 2 | k | | u ^ ( k ) | 2 < + : [u]_{H^{1/2}(\mathbf{T}^{2})}^{2}=\sum_{k\in\mathbf{Z}^{2}}|k|\big|\hat{u}(k)% \big|^{2}<+\infty:
  9. 𝐓 2 | u ( x ) | 2 d x C ( 1 + 1 cap ( E × { 0 } ) ) [ u ] H 1 / 2 ( 𝐓 2 ) 2 , \int_{\mathbf{T}^{2}}|u(x)|^{2}\,\mathrm{d}x\leq C\left(1+\frac{1}{\mathrm{cap% }(E\times\{0\})}\right)[u]_{H^{1/2}(\mathbf{T}^{2})}^{2},
  10. d 2 / π 2 \scriptstyle{d^{2}/\pi^{2}}
  11. Ω \Omega
  12. W 0 1 , 2 ( Ω ) W^{1,2}_{0}(\Omega)
  13. u W 0 1 , 2 ( Ω ) u\in W^{1,2}_{0}(\Omega)
  14. || u || L 2 2 λ 1 - 1 || u || L 2 2 \displaystyle||u||_{L^{2}}^{2}\leq\lambda_{1}^{-1}||\nabla u||_{L^{2}}^{2}

Poinsot's_spirals.html

  1. r = a csch ( n θ ) r=a\ \operatorname{csch}(n\theta)
  2. r = a sech ( n θ ) r=a\ \operatorname{sech}(n\theta)

Point-biserial_correlation_coefficient.html

  1. r p b = M 1 - M 0 s n n 1 n 0 n 2 , r_{pb}=\frac{M_{1}-M_{0}}{s_{n}}\sqrt{\frac{n_{1}n_{0}}{n^{2}}},
  2. s n = 1 n i = 1 n ( X i - X ¯ ) 2 , s_{n}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}\,,
  3. r p b = M 1 - M 0 s n - 1 n 1 n 0 n ( n - 1 ) , r_{pb}=\frac{M_{1}-M_{0}}{s_{n-1}}\sqrt{\frac{n_{1}n_{0}}{n(n-1)}},
  4. s n - 1 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 . s_{n-1}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}.
  5. r p b = M 1 - M 0 s n n 1 n 0 n 2 = M 1 - M 0 s n - 1 n 1 n 0 n ( n - 1 ) . r_{pb}=\frac{M_{1}-M_{0}}{s_{n}}\sqrt{\frac{n_{1}n_{0}}{n^{2}}}=\frac{M_{1}-M_% {0}}{s_{n-1}}\sqrt{\frac{n_{1}n_{0}}{n(n-1)}}.
  6. ( M 1 - M 0 ) 2 i = 1 n ( X i - X ¯ ) 2 ( n 1 n 0 n ) . \frac{(M_{1}-M_{0})^{2}}{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}\left(\frac{n_% {1}n_{0}}{n}\right)\,.
  7. r p b n 1 + n 0 - 2 1 - r p b 2 r_{pb}\sqrt{\frac{n_{1}+n_{0}-2}{1-r_{pb}^{2}}}
  8. r b = M 1 - M 0 s n n 1 n 0 n 2 u , r_{b}=\frac{M_{1}-M_{0}}{s_{n}}\frac{n_{1}n_{0}}{n^{2}u},
  9. r u p b = M 1 - M 0 - 1 n 2 s n 2 n 1 n 0 - 2 ( M 1 - M 0 ) + 1 . r_{upb}=\frac{M_{1}-M_{0}-1}{\sqrt{\frac{n^{2}s_{n}^{2}}{n_{1}n_{0}}-2(M_{1}-M% _{0})+1}}.
  10. r r b = 2 M 1 - M 0 n 1 + n 0 , r_{rb}=2\frac{M_{1}-M_{0}}{n_{1}+n_{0}},
  11. ( 1 + r r b ) n 1 n 0 2 (1+r_{rb})\frac{n_{1}n_{0}}{2}
  12. ( 1 - r r b ) n 1 n 0 2 (1-r_{rb})\frac{n_{1}n_{0}}{2}

Pointwise.html

  1. f ( x ) f(x)
  2. f . f.
  3. ( f + g ) ( x ) \displaystyle(f+g)(x)
  4. f , g : X R f,g:X\to R
  5. A A
  6. R R
  7. X X
  8. A A
  9. K n K^{n}
  10. n n
  11. K K
  12. i i
  13. v v
  14. v i v_{i}
  15. ( u + v ) i = u i + v i (u+v)_{i}=u_{i}+v_{i}
  16. v v
  17. f : n K f:n\to K
  18. f ( i ) = v i f(i)=v_{i}
  19. { f n } n = 1 \{f_{n}\}_{n=1}^{\infty}
  20. f n : X Y f_{n}:X\longrightarrow Y
  21. f f
  22. x x
  23. X X
  24. lim n f n ( x ) = f ( x ) . \lim_{n\rightarrow\infty}f_{n}(x)=f(x).

Pointwise_product.html

  1. : Y × Y Y \cdot:Y\times Y\longrightarrow Y
  2. y z = y z y\cdot z=yz
  3. ( f g ) ( x ) = f ( x ) g ( x ) (f\cdot g)(x)=f(x)\cdot g(x)
  4. ( f g ) ( x ) = f ( x ) g ( x ) = 2 x ( x + 1 ) = 2 x 2 + 2 x (fg)(x)=f(x)g(x)=2x(x+1)=2x^{2}+2x\,
  5. { f * g } = { f } { g } \mathcal{F}\{f*g\}=\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}
  6. ( r f ) ( x ) = r f ( x ) (rf)(x)=rf(x)\,

Poisson_ring.html

  1. [ , ] [\cdot,\cdot]
  2. [ f , g ] = - [ g , f ] [f,g]=-[g,f]
  3. [ f + g , h ] = [ f , h ] + [ g , h ] [f+g,h]=[f,h]+[g,h]
  4. [ f g , h ] = f [ g , h ] + [ f , h ] g [fg,h]=f[g,h]+[f,h]g
  5. [ f , [ g , h ] ] + [ g , [ h , f ] ] + [ h , [ f , g ] ] = 0 [f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0
  6. f , g , h f,g,h
  7. [ s f , g ] = s [ f , g ] [sf,g]=s[f,g]
  8. a d g ad_{g}
  9. a d g ( f ) = [ f , g ] ad_{g}(f)=[f,g]
  10. { a d g | g A } \{ad_{g}|g\in A\}
  11. [ , ] [\cdot,\cdot]

Poisson–Lie_group.html

  1. μ : G × G G \mu:G\times G\to G
  2. μ ( g 1 , g 2 ) = g 1 g 2 \mu(g_{1},g_{2})=g_{1}g_{2}
  3. { f 1 , f 2 } ( g g ) = { f 1 L g , f 2 L g } ( g ) + { f 1 R g , f 2 R g } ( g ) \{f_{1},f_{2}\}(gg^{\prime})=\{f_{1}\circ L_{g},f_{2}\circ L_{g}\}(g^{\prime})% +\{f_{1}\circ R_{g^{\prime}},f_{2}\circ R_{g^{\prime}}\}(g)
  4. 𝒫 \mathcal{P}
  5. 𝒫 ( g g ) = L g ( 𝒫 ( g ) ) + R g ( 𝒫 ( g ) ) \mathcal{P}(gg^{\prime})=L_{g\ast}(\mathcal{P}(g^{\prime}))+R_{g^{\prime}\ast}% (\mathcal{P}(g))
  6. { f , g } ( e ) = 0 \{f,g\}(e)=0
  7. 𝒫 ( e ) = 0 \mathcal{P}(e)=0
  8. ϕ : G H \phi:G\to H
  9. ι : G G \iota:G\to G
  10. ι ( g ) = g - 1 \iota(g)=g^{-1}
  11. { f 1 ι , f 2 ι } = - { f 1 , f 2 } ι \{f_{1}\circ\iota,f_{2}\circ\iota\}=-\{f_{1},f_{2}\}\circ\iota
  12. f 1 , f 2 f_{1},f_{2}

Polar_moment_of_inertia.html

  1. I = A x 2 d A I=\int_{A}x^{2}\,\mathrm{d}A
  2. I = A r 2 d A I=\int_{A}r^{2}\,\mathrm{d}A
  3. I = m r 2 d m I=\int_{m}r^{2}\mathrm{d}m
  4. I z = A ρ 2 d A I_{z}=\int_{A}\rho^{2}\,dA
  5. I z = 0 2 π 0 r ρ 2 ρ d ρ d ϕ = π r 4 2 I_{z}=\int_{0}^{2\pi}\int_{0}^{r}\rho^{2}\rho\,d\rho\,d\phi=\frac{\pi r^{4}}{2}
  6. I z = I x + I y I_{z}=I_{x}+I_{y}
  7. τ = T r J z \tau=\frac{Tr}{J_{z}}
  8. T T
  9. r r
  10. J z {J_{z}}
  11. T max = τ max J z r T_{\max}=\frac{{\tau}_{\max}J_{z}}{r}
  12. ω = 2 π f \omega=2\pi f
  13. P = T ω P=T\omega
  14. T max = τ max J z r T_{\max}=\frac{{\tau}_{\max}J_{z}}{r}
  15. r = 2 T max π τ max 3 r=\sqrt[3]{\frac{2T_{\max}}{\pi{\tau}_{\max}}}
  16. J z = π r 4 2 J_{z}=\frac{\pi r^{4}}{2}
  17. J z = π d 4 32 J_{z}=\frac{\pi d^{4}}{32}

Polar_space.html

  1. n - 1 2 \left\lfloor\frac{n-1}{2}\right\rfloor
  2. n + 1 2 \left\lfloor\frac{n+1}{2}\right\rfloor

Polarimeter.html

  1. [ α ] λ T [\alpha]_{\lambda}^{T}
  2. [ α ] λ T = 100 × α l × c [\alpha]_{\lambda}^{T}=\frac{100\times\alpha}{l\times c}

Polarography.html

  1. I d = k . n . F . D 1 / 2 . m r 2 / 3 . t 1 / 6 . c \,I_{d}=k.n.F.D^{1/2}.m_{r}^{2/3}.t^{1/6}.c

Pole–zero_plot.html

  1. H ( s ) = B ( s ) A ( s ) = m = 0 M b m s m s N + n = 0 N - 1 a n s n = b 0 + b 1 s + b 2 s 2 + + b M s M a 0 + a 1 s + a 2 s 2 + + a ( N - 1 ) s ( N - 1 ) + s N H(s)=\frac{B(s)}{A(s)}={\displaystyle\sum_{m=0}^{M}{b_{m}s^{m}}\over s^{N}+% \displaystyle\sum_{n=0}^{N-1}{a_{n}s^{n}}}=\frac{b_{0}+b_{1}s+b_{2}s^{2}+% \cdots+b_{M}s^{M}}{a_{0}+a_{1}s+a_{2}s^{2}+\cdots+a_{(N-1)}s^{(N-1)}+s^{N}}
  2. B B
  3. A A
  4. s s
  5. M M
  6. b m b_{m}
  7. N N
  8. a n a_{n}
  9. M N M\leq N
  10. s = { β m | m 1 , M } s=\{\beta_{m}|m\in 1,\ldots M\}
  11. B ( s ) | s = β m = 0 B(s)|_{s=\beta_{m}}=0
  12. s = { α n | n 1 , N } s=\{\alpha_{n}|n\in 1,\ldots N\}
  13. A ( s ) | s = α n = 0 A(s)|_{s=\alpha_{n}}=0
  14. H ( s ) = 25 s 2 + 6 s + 25 H(s)={25\over s^{2}+6s+25}
  15. s = α 1 = - 3 + 4 j s=\alpha_{1}=-3+4j
  16. s = α 2 = - 3 - 4 j s=\alpha_{2}=-3-4j
  17. H ( z ) = P ( z ) Q ( z ) = m = 0 M b m z - m 1 + n = 1 N a n z - n = b 0 + b 1 z - 1 + b 2 z - 2 + b M z - M 1 + a 1 z - 1 + a 2 z - 2 + a N z - N H(z)=\frac{P(z)}{Q(z)}=\frac{\displaystyle\sum_{m=0}^{M}{b_{m}z^{-m}}}{1+% \displaystyle\sum_{n=1}^{N}{a_{n}z^{-n}}}=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}% \cdots+b_{M}z^{-M}}{1+a_{1}z^{-1}+a_{2}z^{-2}\cdots+a_{N}z^{-N}}
  18. M M
  19. b m b_{m}
  20. N N
  21. a n a_{n}
  22. z = β m z=\beta_{m}
  23. P ( z ) | z = β m = 0 P(z)|_{z=\beta_{m}}=0
  24. z = α n z=\alpha_{n}
  25. Q ( z ) | z = α n = 0 Q(z)|_{z=\alpha_{n}}=0
  26. P ( z ) P(z)
  27. Q ( z ) Q(z)
  28. H ( z ) = z 1 + 2 z 2 + 1 4 H(z)=\frac{z^{1}+2}{z^{2}+\frac{1}{4}}
  29. z = - 2 z=-2
  30. z = ± j 2 z=\pm\frac{j}{2}

Polignac's_conjecture.html

  1. π n ( x ) \pi_{n}(x)
  2. π n ( x ) 2 C n x ( ln x ) 2 2 C n 2 x d t ( ln t ) 2 \pi_{n}(x)\sim 2C_{n}\frac{x}{(\ln x)^{2}}\sim 2C_{n}\int_{2}^{x}{dt\over(\ln t% )^{2}}
  3. \sim
  4. C 2 = p 3 p ( p - 2 ) ( p - 1 ) 2 0.660161815846869573927812110014 C_{2}=\prod_{p\geq 3}\frac{p(p-2)}{(p-1)^{2}}\approx 0.66016181584686957392781% 2110014\dots
  5. C n = C 2 q | n q - 1 q - 2 . C_{n}=C_{2}\prod_{q|n}\frac{q-1}{q-2}.
  6. q - 1 q - 2 \tfrac{q-1}{q-2}
  7. 2 q \tfrac{2}{q}
  8. 1 q \tfrac{1}{q}
  9. q - 1 q \tfrac{q-1}{q}
  10. q - 2 q \tfrac{q-2}{q}
  11. q - 1 q - 2 \tfrac{q-1}{q-2}

Polite_number.html

  1. f ( n ) = n + log 2 ( n + log 2 n ) . f(n)=n+\left\lfloor\log_{2}\left(n+\log_{2}n\right)\right\rfloor.
  2. 90 = 2 × 3 2 × 5 1 90=2\times 3^{2}\times 5^{1}
  3. ( 2 + 1 ) × ( 1 + 1 ) - 1 = 5 (2+1)\times(1+1)-1=5
  4. x = i = x y - y - 1 2 x y + y - 1 2 i . x=\sum_{i=\frac{x}{y}-\frac{y-1}{2}}^{\frac{x}{y}+\frac{y-1}{2}}i.
  5. T n = n ( n + 1 ) 2 = 1 + 2 + + n . T_{n}=\frac{n(n+1)}{2}=1+2+\cdots+n.\,\!
  6. i + ( i + 1 ) + ( i + 2 ) + + j = T j - T i - 1 . i+(i+1)+(i+2)+\cdots+j=T_{j}-T_{i-1}.\,\!

Poly-Bernoulli_number.html

  1. B n ( k ) B_{n}^{(k)}
  2. L i k ( 1 - e - x ) 1 - e - x = n = 0 B n ( k ) x n n ! {Li_{k}(1-e^{-x})\over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}
  3. B n ( 1 ) B_{n}^{(1)}
  4. L i k ( 1 - ( a b ) - x ) b x - a - x c x t = n = 0 B n ( k ) ( t ; a , b , c ) x n n ! {Li_{k}(1-(ab)^{-x})\over b^{x}-a^{-x}}c^{xt}=\sum_{n=0}^{\infty}B_{n}^{(k)}(t% ;a,b,c){x^{n}\over n!}
  5. B n ( - k ) = m = 0 n ( - 1 ) m + n m ! S ( n , m ) ( m + 1 ) k , B_{n}^{(-k)}=\sum_{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},
  6. B n ( - k ) = j = 0 min ( n , k ) ( j ! ) 2 S ( n + 1 , j + 1 ) S ( k + 1 , j + 1 ) , B_{n}^{(-k)}=\sum_{j=0}^{\min(n,k)}(j!)^{2}S(n+1,j+1)S(k+1,j+1),
  7. S ( n , k ) S(n,k)
  8. n n
  9. k k
  10. n n
  11. k k
  12. B n ( - p ) 2 n ( mod p ) , B_{n}^{(-p)}\equiv 2^{n}\;\;(\mathop{{\rm mod}}p),
  13. B x ( - n ) + B y ( - n ) = B z ( - n ) B_{x}^{(-n)}+B_{y}^{(-n)}=B_{z}^{(-n)}

Polyconic_projection.html

  1. x = cot ( φ ) sin ( ( λ - λ 0 ) sin ( φ ) ) x=\cot(\varphi)\sin((\lambda-\lambda_{0})\sin(\varphi))\,
  2. y = φ - φ 0 + cot ( φ ) ( 1 - cos ( ( λ - λ 0 ) sin ( φ ) ) ) y=\varphi-\varphi_{0}+\cot(\varphi)(1-\cos((\lambda-\lambda_{0})\sin(\varphi)))\,
  3. λ \lambda
  4. φ \varphi
  5. λ 0 \lambda_{0}
  6. φ 0 \varphi_{0}
  7. λ 0 \lambda_{0}
  8. φ = 0 \varphi=0
  9. x = λ - λ 0 x=\lambda-\lambda_{0}
  10. y = 0 y=0

Polymer_brush.html

  1. ϕ ( z , ρ ) = n z \phi(z,\rho)=\frac{\partial n}{\partial z}
  2. n ( z , ρ ) = 2 N π arcsin ( z ρ ) n(z,\rho)=\frac{2N}{\pi}\arcsin\left(\frac{z}{\rho}\right)
  3. ρ \rho
  4. N N
  5. ϵ ( ρ ) \epsilon(\rho)
  6. ϕ ( z ) = z n ( z , ρ ) z ϵ ( ρ ) d ρ \phi(z)=\int_{z}^{\infty}\frac{\partial n(z,\rho)}{\partial z}\,\epsilon(\rho)% \,{\rm d}\rho
  7. H H
  8. ϵ dry ( ρ , H ) = ρ / H N a 1 - ρ 2 / H 2 \epsilon_{\rm dry}(\rho,H)=\frac{\rho/H}{Na\sqrt{1-\rho^{2}/H^{2}}}
  9. a a
  10. n ( z , ρ ) n(z,\rho)
  11. U = 0 ϵ ( ρ ) d ρ 0 N d n k T 2 N a 2 ( z ( n , ρ ) n ) 2 U=\int_{0}^{\infty}\epsilon(\rho)\,{\rm d}\rho\,\int_{0}^{N}\,{\rm d}n\,\frac{% kT}{2Na^{2}}\left(\frac{\partial z(n,\rho)}{\partial n}\right)^{2}
  12. ϵ ( ρ ) \epsilon(\rho)
  13. ϕ ( z ) \phi(z)
  14. ϵ ( ρ ) = ρ - d ϕ ( H ) d H ϵ dry ( ρ , H ) \epsilon(\rho)=\int_{\rho}^{\infty}-\frac{{\rm d}\phi(H)}{{\rm d}H}\epsilon_{% \rm dry}(\rho,H)
  15. F el k T = π 2 24 N 2 a 5 0 { - z 3 d ϕ ( z ) d z } d z \frac{F_{\rm el}}{kT}=\frac{\pi^{2}}{24N^{2}a^{5}}\int_{0}^{\infty}\left\{-z^{% 3}\frac{{\rm d}\phi(z)}{{\rm d}z}\right\}{\rm d}z

Polynomial_expansion.html

  1. ( x + y ) 2 = x 2 + 2 x y + y 2 (x+y)^{2}=x^{2}+2xy+y^{2}
  2. ( x + y ) ( x - y ) = x 2 - y 2 (x+y)(x-y)=x^{2}-y^{2}
  3. ( a + b + c + d ) ( x + y + z ) = a x + a y + a z + b x + b y + b z + c x + c y + c z + d x + d y + d z (a+b+c+d)(x+y+z)=ax+ay+az+bx+by+bz+cx+cy+cz+dx+dy+dz
  4. 1 + x ( - 3 + x ( 4 + x ( 0 + x ( - 12 + x 2 ) ) ) ) = 1 - 3 x + 4 x 2 - 12 x 4 + 2 x 5 1+x(-3+x(4+x(0+x(-12+x\cdot 2))))=1-3x+4x^{2}-12x^{4}+2x^{5}
  5. ( x + 2 ) ( 2 x - 5 ) (x+2)(2x-5)\,
  6. 2 x 2 - 5 x + 4 x - 10 = 2 x 2 - x - 10. 2x^{2}-5x+4x-10=2x^{2}-x-10.
  7. ( x + y ) n (x+y)^{n}
  8. ( x + y ) 6 (x+y)^{6}
  9. \color r e d 1 x 6 + \color r e d 6 x 5 y + \color r e d 15 x 4 y 2 + \color r e d 20 x 3 y 3 + \color r e d 15 x 2 y 4 + \color r e d 6 x y 5 + \color r e d 1 y 6 {\color{red}1}x^{6}+{\color{red}6}x^{5}y+{\color{red}{15}}x^{4}y^{2}+{\color{% red}{20}}x^{3}y^{3}+{\color{red}{15}}x^{2}y^{4}+{\color{red}{6}}xy^{5}+{\color% {red}1}y^{6}\,

Polynomial_transformations.html

  1. P ( x ) = a 0 x n + a 1 x n - 1 + + a n P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n}
  2. α 1 , , α n \alpha_{1},\ldots,\alpha_{n}
  3. c c
  4. α 1 + c , , α n + c \alpha_{1}+c,\ldots,\alpha_{n}+c
  5. Q ( y ) = P ( y - c ) = a 0 ( y - c ) n + a 1 ( y - c ) n - 1 + + a n . Q(y)=P(y-c)=a_{0}(y-c)^{n}+a_{1}(y-c)^{n-1}+\cdots+a_{n}.
  6. P P
  7. c = p q c=\frac{p}{q}
  8. Q Q
  9. Q Q
  10. c = a 1 a 0 . c=\frac{a_{1}}{a_{0}}.
  11. Q Q
  12. P ( x ) = a 0 x n + a 1 x n - 1 + + a n P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n}
  13. P P
  14. Q ( y ) = y n P ( 1 y ) = a n y n + a n - 1 y n - 1 + + a 0 . Q(y)=y^{n}P\left(\frac{1}{y}\right)=a_{n}y^{n}+a_{n-1}y^{n-1}+\cdots+a_{0}.
  15. P ( x ) = a 0 x n + a 1 x n - 1 + + a n P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n}
  16. c c
  17. c c
  18. P P
  19. Q ( y ) = c n P ( y c ) = a 0 y n + a 1 c y n - 1 + + a n c n . Q(y)=c^{n}P\left(\frac{y}{c}\right)=a_{0}y^{n}+a_{1}cy^{n-1}+\cdots+a_{n}c^{n}.
  20. c c
  21. P P
  22. Q Q
  23. c = a 0 c=a_{0}
  24. Q Q
  25. c c
  26. Q c \frac{Q}{c}
  27. c c
  28. P P
  29. a 1 a 0 \frac{a_{1}}{a_{0}}
  30. n 1 n−1
  31. f ( x ) = g ( x ) h ( x ) f(x)=\frac{g(x)}{h(x)}
  32. g g
  33. h h
  34. P P
  35. f f
  36. Q Q
  37. f f
  38. P P
  39. Q Q
  40. y y
  41. x x
  42. P , g P,g
  43. h h
  44. P ( x ) = 0 y h ( x ) - g ( x ) = 0 . \begin{aligned}\displaystyle P(x)&\displaystyle=0\\ \displaystyle y\,h(x)-g(x)&\displaystyle=0\,.\end{aligned}
  45. Res x ( y h ( x ) - g ( x ) , P ( x ) ) \operatorname{Res}_{x}(y\,h(x)-g(x),P(x))
  46. P P
  47. Q Q
  48. α \alpha
  49. P P
  50. L L
  51. α \alpha
  52. f ( α ) f(\alpha)
  53. L L
  54. Q Q
  55. f ( α ) f(\alpha)
  56. L L
  57. Q Q

Polynormal_subgroup.html

  1. H H
  2. G G
  3. g G g\in G
  4. K = H < g > K=H^{<g>}
  5. H H < g > H^{H^{<g>}}

Polytropic_process.html

  1. p v n = C pv^{\,n}=C
  2. \infty
  3. \infty
  4. K = δ Q / δ W = c o n s t a n t K=\delta Q/\delta W=constant
  5. P 1 v 1 n = P 2 v 2 n = = C P_{1}{v_{1}^{\,n}}=P_{2}v_{2}^{\,n}=...=C
  6. δ q - δ w = d u \delta q-\delta w=du
  7. K = δ q δ w K=\frac{\delta q}{\delta w}
  8. δ q = K δ w \delta q=K\delta w
  9. δ w = P d v \delta w=Pdv
  10. δ w \delta w
  11. δ q \delta q
  12. ( K - 1 ) P d v = c v d T (K-1)Pdv=c_{\,v}dT
  13. P d v + v d P = Z R d T Pdv+vdP=ZRdT
  14. - v d P P d v = ( 1 - γ ) K + γ -{vdP\over Pdv}=(1-\gamma)K+\gamma
  15. P v n = C Pv^{\,n}=C
  16. ln P + n ln v = C \ln P+n\ln v=C
  17. n = - v d P P d v n=-{vdP\over Pdv}
  18. n = ( 1 - γ ) K + γ n=(1-\gamma)K+\gamma
  19. ( 1 - γ ) (1-\gamma)
  20. δ q / δ w δq/δw
  21. C n C_{n}
  22. C n = C V γ - n 1 - n C_{n}=C_{V}{\gamma-n\over 1-n}
  23. n n
  24. n < 0 n<0
  25. n = 0 n=0
  26. p V 0 = p pV^{0}=p
  27. n = 1 n=1
  28. p V = N R T pV=NRT
  29. 1 < n < γ 1<n<\gamma
  30. n = γ n=\gamma
  31. γ = \gamma=
  32. C p C V \frac{C_{p}}{C_{V}}
  33. γ < n < \gamma<n<\infty
  34. n = n=\infty
  35. 1 < γ < 2 1<\gamma<2
  36. γ = C p C V = C V + R C V = 1 + R C V = C p C p - R \gamma=\frac{C_{p}}{C_{V}}=\frac{C_{V}+R}{C_{V}}=1+\frac{R}{C_{V}}=\frac{C_{p}% }{C_{p}-R}
  37. γ \gamma
  38. γ \gamma
  39. Γ \Gamma
  40. Γ \Gamma
  41. n n
  42. n = 1 Γ - 1 . n=\frac{1}{\Gamma-1}.

Pore_water_pressure.html

  1. p s = g w h w p_{s}=g_{w}h_{w}
  2. g w = 9.81 k N / m 3 g_{w}=9.81kN/m^{3}
  3. p g = - g w d w p_{g}=-g_{w}d_{w}
  4. g w = 9.81 k N / m 3 g_{w}=9.81kN/m^{3}
  5. p u = - g w ( z - d w ) p_{u}=-g_{w}(z-d_{w})
  6. Ψ m \Psi_{m}

Porosimetry.html

  1. P L - P G = 4 σ cos θ D P P_{L}-P_{G}=\frac{4\sigma\cos\theta}{D_{P}}
  2. P L P_{L}
  3. P G P_{G}
  4. σ \sigma
  5. θ \theta
  6. D P D_{P}
  7. D P = 1470 kPa μ m P L D_{P}=\frac{1470\ \,\text{kPa}\cdot\mu\,\text{m}}{P_{L}}

Portal:Bible::Selected_person::March,_2006.html

  1. \infty

Position_operator.html

  1. | ψ | 2 = ψ * ψ |\psi|^{2}=\psi^{*}\psi
  2. x x
  3. x = - + x | ψ | 2 d x = - + ψ * x ψ d x \langle x\rangle=\int_{-\infty}^{+\infty}x|\psi|^{2}dx=\int_{-\infty}^{+\infty% }\psi^{*}x\psi dx
  4. x ^ \hat{x}
  5. ( x ^ ψ ) ( x ) = x ψ ( x ) (\hat{x}\psi)(x)=x\psi(x)
  6. ψ \psi
  7. x 0 x_{0}
  8. x ^ ψ ( x ) = x ψ ( x ) = x 0 ψ ( x ) \hat{x}\psi(x)=x\psi(x)=x_{0}\psi(x)
  9. x ^ \hat{x}
  10. x x
  11. x x
  12. x 0 x_{0}
  13. ψ \psi
  14. x = x 0 x=x_{0}
  15. ψ ( x ) = δ ( x - x 0 ) \psi(x)=\delta(x-x_{0})
  16. x 0 x_{0}
  17. ψ ( r , t ) \psi({r},t)
  18. r = r | ψ | 2 d 3 r \langle{r}\rangle=\int{r}|\psi|^{2}d^{3}{r}
  19. r ^ ψ = r ψ {\hat{r}}\psi={r}\psi
  20. x ^ = i d d p \hat{x}=i\hbar\frac{d}{dp}
  21. Q ( ψ ) ( x ) = x ψ ( x ) Q(\psi)(x)=x\psi(x)
  22. D ( Q ) = { ψ L 2 ( 𝐑 ) | Q ψ L 2 ( 𝐑 ) } . D(Q)=\{\psi\in L^{2}({\mathbf{R}})\,|\,Q\psi\in L^{2}({\mathbf{R}})\}.
  23. Q = λ d Ω Q ( λ ) . Q=\int\lambda d\Omega_{Q}(\lambda).
  24. χ B \chi_{B}
  25. Ω Q ( B ) ψ = χ B ψ , \Omega_{Q}(B)\psi=\chi_{B}\psi,
  26. | Ω Q ( B ) ψ | 2 = | χ B ψ | 2 = B | ψ | 2 d μ , |\Omega_{Q}(B)\psi|^{2}=|\chi_{B}\psi|^{2}=\int_{B}|\psi|^{2}d\mu,
  27. Ω Q ( B ) ψ Ω Q ( B ) ψ \frac{\Omega_{Q}(B)\psi}{\|\Omega_{Q}(B)\psi\|}
  28. ( 1 - χ B ) ψ ( 1 - χ B ) ψ \frac{(1-\chi_{B})\psi}{\|(1-\chi_{B})\psi\|}
  29. \|\cdots\|

Position_weight_matrix.html

  1. M k , j = 1 N i = 1 N I ( X i , j = k ) , M_{k,j}=\frac{1}{N}\sum_{i=1}^{N}I(X_{i,j}=k),
  2. \in
  3. \in
  4. M = A C G T [ 3 6 1 0 0 6 7 2 1 2 2 1 0 0 2 1 1 2 1 1 7 10 0 1 1 5 1 4 1 1 0 10 1 1 2 6 ] . M=\begin{matrix}A\\ C\\ G\\ T\end{matrix}\begin{bmatrix}3&6&1&0&0&6&7&2&1\\ 2&2&1&0&0&2&1&1&2\\ 1&1&7&10&0&1&1&5&1\\ 4&1&1&0&10&1&1&2&6\end{bmatrix}.
  5. M = A C G T [ 0.3 0.6 0.1 0.0 0.0 0.6 0.7 0.2 0.1 0.2 0.2 0.1 0.0 0.0 0.2 0.1 0.1 0.2 0.1 0.1 0.7 1.0 0.0 0.1 0.1 0.5 0.1 0.4 0.1 0.1 0.0 1.0 0.1 0.1 0.2 0.6 ] . M=\begin{matrix}A\\ C\\ G\\ T\end{matrix}\begin{bmatrix}0.3&0.6&0.1&0.0&0.0&0.6&0.7&0.2&0.1\\ 0.2&0.2&0.1&0.0&0.0&0.2&0.1&0.1&0.2\\ 0.1&0.1&0.7&1.0&0.0&0.1&0.1&0.5&0.1\\ 0.4&0.1&0.1&0.0&1.0&0.1&0.1&0.2&0.6\end{bmatrix}.
  6. p ( S | M ) = 0.1 × 0.6 × 0.7 × 1.0 × 1.0 × 0.6 × 0.7 × 0.2 × 0.2 = 0.0007056. p(S|M)=0.1\times 0.6\times 0.7\times 1.0\times 1.0\times 0.6\times 0.7\times 0% .2\times 0.2=0.0007056.
  7. b b
  8. M k , j = log ( M k , j / b k ) . M_{k,j}=\mathrm{log}\;(M_{k,j}/b_{k}).
  9. M k , j M_{k,j}
  10. b k = 1 / | k | b_{k}=1/|k|
  11. M = A C G T [ 0.18 0.87 - 0.91 - - 0.87 1.02 - 0.22 - 0.91 - 0.22 - 0.22 - 0.91 - - - 0.22 - 0.91 - 0.91 - 0.22 - 0.91 - 0.91 1.02 1.38 - - 0.91 - 0.91 0.69 - 0.91 0.47 - 0.91 - 0.91 - 1.38 - 0.91 - 0.91 - 0.22 0.87 ] . M=\begin{matrix}A\\ C\\ G\\ T\end{matrix}\begin{bmatrix}0.18&0.87&-0.91&-\infty&-\infty&0.87&1.02&-0.22&-0% .91\\ -0.22&-0.22&-0.91&-\infty&-\infty&-0.22&-0.91&-0.91&-0.22\\ -0.91&-0.91&1.02&1.38&-\infty&-0.91&-0.91&0.69&-0.91\\ 0.47&-0.91&-0.91&-\infty&1.38&-0.91&-0.91&-0.22&0.87\end{bmatrix}.
  12. - -\infty
  13. - log ( p i , j ) -\log(p_{i,j})
  14. - p i , j log ( p i , j ) -p_{i,j}\cdot\log(p_{i,j})
  15. - i , j p i , j log ( p i , j ) \textstyle-\sum_{i,j}p_{i,j}\cdot\log(p_{i,j})
  16. - i , j p i , j log ( p i , j / p b ) \textstyle-\sum_{i,j}p_{i,j}\cdot\log(p_{i,j}/p_{b})
  17. p b p_{b}

Positive_and_negative_parts.html

  1. f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if f ( x ) > 0 0 otherwise. f^{+}(x)=\max(f(x),0)=\begin{cases}f(x)&\mbox{ if }~{}f(x)>0\\ 0&\mbox{ otherwise.}\end{cases}
  2. f + f^{+}
  3. f f
  4. f + f^{+}
  5. f - ( x ) = - min ( f ( x ) , 0 ) = { - f ( x ) if f ( x ) < 0 0 otherwise. f^{-}(x)=-\min(f(x),0)=\begin{cases}-f(x)&\mbox{ if }~{}f(x)<0\\ 0&\mbox{ otherwise.}\end{cases}
  6. f = f + - f - . f=f^{+}-f^{-}.\,
  7. | f | = f + + f - |f|=f^{+}+f^{-}\,
  8. f + = | f | + f 2 f^{+}=\frac{|f|+f}{2}\,
  9. f - = | f | - f 2 . f^{-}=\frac{|f|-f}{2}.\,
  10. f + = [ f > 0 ] f f^{+}=[f>0]f\,
  11. f - = - [ f < 0 ] f . f^{-}=-[f<0]f.\,
  12. f = 1 V - 1 2 , f=1_{V}-{1\over 2},

Positive_and_negative_sets.html

  1. n = 1 A n \bigcup_{n=1}^{\infty}A_{n}

Post-Newtonian_expansion.html

  1. h α β = g α β - η α β . h_{\alpha\beta}=g_{\alpha\beta}-\eta_{\alpha\beta}\,.
  2. h α β η β γ h_{\alpha\beta}\eta^{\beta\gamma}\,
  3. g μ ν η μ ν - η μ α h α β η β ν + η μ α h α β η β γ h γ δ η δ ν . g^{\mu\nu}\approx\eta^{\mu\nu}-\eta^{\mu\alpha}h_{\alpha\beta}\eta^{\beta\nu}+% \eta^{\mu\alpha}h_{\alpha\beta}\eta^{\beta\gamma}h_{\gamma\delta}\eta^{\delta% \nu}\,.
  4. - g 1 + 1 2 h α β η β α + 1 8 h α β η β α h γ δ η δ γ - 1 4 h α β η β γ h γ δ η δ α . \sqrt{-g}\approx 1+\tfrac{1}{2}h_{\alpha\beta}\eta^{\beta\alpha}+\tfrac{1}{8}h% _{\alpha\beta}\eta^{\beta\alpha}h_{\gamma\delta}\eta^{\delta\gamma}-\tfrac{1}{% 4}h_{\alpha\beta}\eta^{\beta\gamma}h_{\gamma\delta}\eta^{\delta\alpha}\,.

Post_hoc_analysis.html

  1. α \alpha\,\!
  2. α \alpha\,\!
  3. p i p_{i}
  4. H i H_{i}
  5. H i H_{i}
  6. p i α m p_{i}\leq\frac{\alpha}{m}
  7. m m
  8. m m
  9. P ( 1 ) P ( m ) P_{(1)}\ldots P_{(m)}
  10. H ( 1 ) H ( m ) H_{(1)}\ldots H_{(m)}
  11. R R
  12. k k
  13. P ( k ) > α m + 1 - k P_{(k)}>\frac{\alpha}{m+1-k}
  14. H ( 1 ) H ( R - 1 ) H_{(1)}\ldots H_{(R-1)}
  15. R = 1 R=1
  16. α \alpha
  17. H 0 H_{0}
  18. E α \operatorname{E}\alpha
  19. Y A - Y B S E \frac{Y_{A}-Y_{B}}{SE}
  20. Y A Y_{A}
  21. Y B Y_{B}
  22. S E SE
  23. H 1 , , H m H_{1},\ldots,H_{m}
  24. P ( 1 ) , , P ( m ) P_{(1)},\ldots,P_{(m)}
  25. α \alpha
  26. m m

PostBQP.html

  1. | x |x\rangle
  2. | α x | p |\alpha_{x}|^{p}
  3. | α x | 2 |\alpha_{x}|^{2}
  4. ( 2 / 3 ) 3 + 3 ( 1 / 3 ) ( 2 / 3 ) 2 = 20 / 27 (2/3)^{3}+3(1/3)(2/3)^{2}=20/27
  5. L 1 L 2 L1\cap L2
  6. Ψ \Psi
  7. Ψ \Psi
  8. Ψ \Psi
  9. 2 B × 2 B 2^{B}\times 2^{B}
  10. Ψ = A G A G - 1 A 2 A 1 | x \Psi=A^{G}A^{G-1}\cdots A^{2}A^{1}|x\rangle
  11. π 1 := Pr [ P = 1 , Q = 1 ] = ω S 1 Ψ ω 2 \pi_{1}:=\,\text{Pr}[P=1,Q=1]=\sum_{\omega\in S_{1}}\Psi^{2}_{\omega}
  12. π 0 := Pr [ P = 1 , Q = 0 ] = ω S 0 Ψ ω 2 . \pi_{0}:=\,\text{Pr}[P=1,Q=0]=\sum_{\omega\in S_{0}}\Psi^{2}_{\omega}.
  13. π 1 2 π 0 \pi_{1}\geq 2\pi_{0}
  14. π 0 2 π 1 \pi_{0}\geq 2\pi_{1}
  15. π 1 \pi_{1}
  16. π 0 \pi_{0}
  17. Ψ ω = α 1 , , α G A ω , α G G A α G , α G - 1 G - 1 A α 3 , α 2 2 A α 2 , α 1 1 x α 1 \Psi_{\omega}=\sum_{\alpha_{1},\ldots,\alpha_{G}}A^{G}_{\omega,\alpha_{G}}A^{G% -1}_{\alpha_{G},\alpha_{G-1}}\cdots A^{2}_{\alpha_{3},\alpha_{2}}A^{1}_{\alpha% _{2},\alpha_{1}}x_{\alpha_{1}}
  18. α i \alpha_{i}
  19. π 1 \pi_{1}
  20. π 0 \pi_{0}
  21. 1 2 ( 1 + π 1 - π 0 ) \frac{1}{2}(1+\pi_{1}-\pi_{0})
  22. x L x\in L
  23. 1 2 ( 1 + π 1 - π 0 ) > 1 / 2 \frac{1}{2}(1+\pi_{1}-\pi_{0})>1/2
  24. x L x\not\in L
  25. 1 2 ( 1 + π 1 - π 0 ) < 1 / 2 \frac{1}{2}(1+\pi_{1}-\pi_{0})<1/2
  26. 2 f ( n ) 2^{f(n)}
  27. π 1 2 3 ( π 0 + π 1 ) \pi_{1}\geq\frac{2}{3}(\pi_{0}+\pi_{1})
  28. π 0 2 3 ( π 0 + π 1 ) \pi_{0}\geq\frac{2}{3}(\pi_{0}+\pi_{1})
  29. 2 f ( n ) 2^{f(n)}
  30. π \pi
  31. π 1 > 1 2 ( π 0 + π 1 ) \pi_{1}>\frac{1}{2}(\pi_{0}+\pi_{1})
  32. π 0 > 1 2 ( π 0 + π 1 ) \pi_{0}>\frac{1}{2}(\pi_{0}+\pi_{1})
  33. ( 1 + 2 - f ( n ) 2 B ) G - 1 < 1 6 2 - n c , (1+2^{-f(n)}2^{B})^{G}-1<\frac{1}{6}2^{-n^{c}},
  34. α \alpha
  35. { α i } i = 1 G \{\alpha_{i}\}_{i=1}^{G}
  36. Π ( A , ω , α , x ) := A ω , α G G A α G , α G - 1 G - 1 A α 3 , α 2 2 A α 2 , α 1 1 x α 1 \Pi(A,\omega,\alpha,x):=A^{G}_{\omega,\alpha_{G}}A^{G-1}_{\alpha_{G},\alpha_{G% -1}}\cdots A^{2}_{\alpha_{3},\alpha_{2}}A^{1}_{\alpha_{2},\alpha_{1}}x_{\alpha% _{1}}
  37. π 1 - π 0 = ω S 1 α , α Π ( A , ω , α , x ) Π ( A , ω , α , x ) - ω S 0 α , α Π ( A , ω , α , x ) Π ( A , ω , α , x ) . \pi_{1}-\pi_{0}=\sum_{\omega\in S_{1}}\sum_{\alpha,\alpha^{\prime}}\Pi(A,% \omega,\alpha,x)\Pi(A,\omega,\alpha^{\prime},x)-\sum_{\omega\in S_{0}}\sum_{% \alpha,\alpha^{\prime}}\Pi(A,\omega,\alpha,x)\Pi(A,\omega,\alpha^{\prime},x).
  38. ω \omega
  39. ω S 0 S 1 \omega\not\in S_{0}\cup S_{1}
  40. α , α \alpha,\alpha^{\prime}
  41. X = Π ( A , ω , α , x ) Π ( A , ω , α , x ) X=\Pi(A,\omega,\alpha,x)\Pi(A,\omega,\alpha^{\prime},x)
  42. 2 2 f ( n ) G ( n ) 2^{2f(n)G(n)}
  43. - 1 X 1 -1\leq X\leq 1
  44. ω S 1 \omega\in S_{1}
  45. 1 + X 2 \frac{1+X}{2}
  46. 1 - X 2 \frac{1-X}{2}
  47. ω S 0 \omega\in S_{0}
  48. 1 - X 2 \frac{1-X}{2}
  49. 1 + X 2 \frac{1+X}{2}
  50. 1 2 + ( π 1 - π 0 ) / ( 2 1 + B ( n ) + 2 B ( n ) G ( n ) ) , \frac{1}{2}+(\pi_{1}-\pi_{0})/(2^{1+B(n)+2B(n)G(n)}),
  51. x L # { r { 0 , 1 } T f ( x , r ) = 1 } 2 T - 1 x\in L\Leftrightarrow\#\{r\in\{0,1\}^{T}\mid f(x,r)=1\}\geq 2^{T-1}
  52. s := # { r { 0 , 1 } T f ( x , r ) = 1 } s:=\#\{r\in\{0,1\}^{T}\mid f(x,r)=1\}
  53. 2 T - s 2^{T}-s
  54. s { 0 , 2 T / 2 , 2 T } s\not\in\{0,2^{T}/2,2^{T}\}
  55. x { 0 , 1 } T | x | f ( x ) \sum_{x\in\{0,1\}^{T}}|x\rangle|f(x)\rangle
  56. | ψ := ( 2 T - s ) | 0 + s | 1 . |\psi\rangle:=(2^{T}-s)|0\rangle+s|1\rangle.
  57. H | ψ = ( 2 T | 0 + ( 2 T - 2 s ) | 1 ) / 2 H|\psi\rangle=(2^{T}|0\rangle+(2^{T}-2s)|1\rangle)/\sqrt{2}
  58. α , β \alpha,\beta
  59. α 2 + β 2 = 1 \alpha^{2}+\beta^{2}=1
  60. α | 0 | ψ + β | 1 | H ψ \alpha|0\rangle|\psi\rangle+\beta|1\rangle|H\psi\rangle
  61. β / α \beta/\alpha
  62. | ϕ β / α := α s | 0 + β 2 ( 2 T - 2 s ) | 1 |\phi_{\beta/\alpha}\rangle:=\alpha s|0\rangle+\frac{\beta}{\sqrt{2}}(2^{T}-2s% )|1\rangle
  63. s > 2 T - 1 s>2^{T-1}
  64. x L x\in L
  65. ϕ β / α \phi_{\beta/\alpha}
  66. Q a c c := ( - | 1 , | 0 ) Q_{acc}:=(-|1\rangle,|0\rangle)
  67. s < 2 T - 1 s<2^{T-1}
  68. x L x\not\in L
  69. ϕ β / α \phi_{\beta/\alpha}
  70. Q r e j := ( | 0 , | 1 ) Q_{rej}:=(|0\rangle,|1\rangle)
  71. β / α \beta/\alpha
  72. ( 0 , ) (0,\infty)
  73. | ϕ β / α |\phi_{\beta/\alpha}\rangle
  74. | + = ( | 1 + | 0 ) / 2 |+\rangle=(|1\rangle+|0\rangle)/\sqrt{2}
  75. Q r e j Q_{rej}
  76. | - |-\rangle
  77. | + |+\rangle
  78. Q a c c Q_{acc}
  79. { | + , | - } \{|+\rangle,|-\rangle\}
  80. | + |+\rangle
  81. x L x\not\in L
  82. β / α = r * := 2 s / ( 2 T - 2 s ) \beta/\alpha=r^{*}:=\sqrt{2}s/(2^{T}-2s)
  83. | ϕ β / α |\phi_{\beta/\alpha}\rangle
  84. { | + , | - } \{|+\rangle,|-\rangle\}
  85. | + |+\rangle
  86. β / α \beta/\alpha
  87. 2 - T < r * < 2 T 2^{-T}<r*<2^{T}
  88. β / α \beta/\alpha
  89. 2 i 2^{i}
  90. - T i T -T\leq i\leq T
  91. | ϕ 2 i |\phi_{2^{i}}\rangle
  92. { | + , | - } \{|+\rangle,|-\rangle\}
  93. | + |+\rangle
  94. ( 3 + 2 2 ) / 6 0.971. (3+2\sqrt{2})/6\approx 0.971.
  95. ( 3 + 2 2 ) / 6 (3+2\sqrt{2})/6
  96. - T i T -T\leq i\leq T
  97. | ϕ 2 i |\phi_{2^{i}}\rangle
  98. { | + , | - } \{|+\rangle,|-\rangle\}
  99. C log T C\log T
  100. | + |+\rangle
  101. | ϕ 2 i |\phi_{2^{i}}\rangle
  102. 1 / 2 O ( T ) 1/2^{O(T)}
  103. 1 / 2 O ( T 2 log T ) 1/2^{O(T^{2}\log T)}

POVM.html

  1. { F i } \{F_{i}\}
  2. \mathcal{H}
  3. i = 1 n F i = I H . \sum_{i=1}^{n}F_{i}=\operatorname{I}_{H}.
  4. { E i } \{E_{i}\}
  5. { | ϕ i } \{\left|\phi_{i}\right\rangle\}
  6. i = 1 N E i = I H , E i E j = δ i j E i , E i = | ϕ i ϕ i | . \sum_{i=1}^{N}E_{i}=\operatorname{I}_{H},\quad E_{i}E_{j}=\delta_{ij}E_{i},{% \quad}E_{i}=\left|\phi_{i}\right\rangle\left\langle\phi_{i}\right|.
  7. \in
  8. E F ( E ) ξ ξ E\mapsto\langle F(E)\xi\mid\xi\rangle
  9. F i F_{i}
  10. P ( i ) = tr ( ρ F i ) , P(i)={\rm tr}(\rho F_{i}),\;
  11. ρ \rho
  12. H A H_{A}
  13. H A H A H_{A}\oplus H^{\perp}_{A}
  14. { π ^ i } \{\hat{\pi}_{i}\}
  15. π ^ i \hat{\pi}_{i}
  16. P ( i ) = tr ( ρ π ^ i ) = tr ( ρ π ^ A π ^ i π ^ A ) , P(i)={\rm tr}(\rho\hat{\pi}_{i})={\rm tr}(\rho\hat{\pi}_{A}\hat{\pi}_{i}\hat{% \pi}_{A}),\;
  17. π ^ A \hat{\pi}_{A}
  18. H A H A H_{A}\oplus H^{\perp}_{A}
  19. H A H_{A}
  20. H A H_{A}
  21. F i = π ^ A π ^ i π ^ A F_{i}=\hat{\pi}_{A}\hat{\pi}_{i}\hat{\pi}_{A}
  22. | 0 B |0\rangle_{B}
  23. | ϕ A | ψ B |\phi\rangle_{A}\otimes|\psi\rangle_{B}
  24. ψ | 0 B = 0 \langle\psi|0\rangle_{B}=0
  25. | 0 B |0\rangle_{B}
  26. | ψ A | 0 B i M i | ψ A | i B , |\psi\rangle_{A}|0\rangle_{B}\rightarrow\sum_{i}M_{i}|\psi\rangle_{A}|i\rangle% _{B},
  27. { | i B } \{|i\rangle_{B}\}
  28. F i = M i M i F_{i}=M_{i}^{\dagger}M_{i}
  29. M i M_{i}
  30. ρ = M i ρ M i tr ( M i ρ M i ) \rho^{\prime}={M_{i}\rho M_{i}^{\dagger}\over{\rm tr}(M_{i}\rho M_{i}^{\dagger% })}
  31. M i M_{i}
  32. ρ \rho^{\prime}
  33. ρ ′′ = M i ρ M i tr ( M i ρ M i ) = M i M i ρ M i M i tr ( M i M i ρ M i M i ) \rho^{\prime\prime}={M_{i}\rho^{\prime}M_{i}^{\dagger}\over{\rm tr}(M_{i}\rho^% {\prime}M_{i}^{\dagger})}={M_{i}M_{i}\rho M_{i}^{\dagger}M_{i}^{\dagger}\over{% \rm tr}(M_{i}M_{i}\rho M_{i}^{\dagger}M_{i}^{\dagger})}
  34. ρ \rho^{\prime}
  35. M i 2 = M i , M_{i}^{2}=M_{i},
  36. ρ ^ retr [ n ] = Π ^ n Tr { Π ^ n } . \hat{\rho}_{\mathrm{retr}}^{[n]}=\frac{\hat{\Pi}_{n}}{\mathrm{Tr}\{\hat{\Pi}_{% n}\}}.
  37. Pr ( m | n ) = Tr { ρ ^ retr [ n ] Θ ^ m } , \mathrm{Pr}\left(m|n\right)=\mathrm{Tr}\{\hat{\rho}_{\mathrm{retr}}^{[n]}\hat{% \Theta}_{m}\},
  38. Θ ^ m \hat{\Theta}_{m}
  39. ρ ^ m \hat{\rho}_{m}
  40. π n \pi_{n}
  41. π n = Tr [ ( ρ ^ retr [ n ] ) 2 ] . \pi_{n}=\mathrm{Tr}\left[\left(\hat{\rho}_{\mathrm{retr}}^{[n]}\right)^{2}% \right].
  42. | ψ n ( π n = 1 ) |\psi_{n}\rangle(\pi_{n}=1)
  43. Π ^ n = η n | ψ n ψ n | , \hat{\Pi}_{n}=\eta_{n}|\psi_{n}\rangle\langle\psi_{n}|,
  44. η n = Tr { Π n } \eta_{n}=\mathrm{Tr}\{\Pi_{n}\}
  45. | ψ n |\psi_{n}\rangle
  46. Pr ( n | ψ n ) = η n \mathrm{Pr}\left(n|\psi_{n}\right)=\eta_{n}
  47. | ψ |\psi\rangle
  48. | ψ T |\psi^{T}\rangle
  49. A ^ = a | ψ T ψ T | + b | ψ ψ | , \hat{A}=a|\psi^{T}\rangle\langle\psi^{T}|+b|\psi\rangle\langle\psi|,
  50. | ψ T |\psi^{T}\rangle
  51. | ψ |\psi\rangle
  52. | ψ |\psi\rangle
  53. | ϕ |\phi\rangle
  54. | ϕ | ψ | = cos ( θ ) , |\langle\phi|\psi\rangle|=\operatorname{cos}(\theta),
  55. θ > 0 \theta>0
  56. π ^ ψ T = | ψ T ψ T | , \hat{\pi}_{\psi^{T}}=|\psi^{T}\rangle\langle\psi^{T}|,
  57. π ^ ϕ T = | ϕ T ϕ T | , \hat{\pi}_{\phi^{T}}=|\phi^{T}\rangle\langle\phi^{T}|,
  58. π ^ ψ T \hat{\pi}_{\psi^{T}}
  59. | ϕ |\phi\rangle
  60. π ^ ϕ T \hat{\pi}_{\phi^{T}}
  61. | ψ |\psi\rangle
  62. P proj = 1 - | ϕ | ψ | 2 2 . P_{\mathrm{proj}}=\frac{1-|\langle\phi|\psi\rangle|^{2}}{2}.
  63. P POVM = 1 - | ϕ | ψ | . P_{\mathrm{POVM}}=1-|\langle\phi|\psi\rangle|.
  64. F ^ ψ = 1 - | ϕ ϕ | 1 + | ϕ | ψ | \hat{F}_{\psi}=\frac{1-|\phi\rangle\langle\phi|}{1+|\langle\phi|\psi\rangle|}
  65. F ^ ϕ = 1 - | ψ ψ | 1 + | ϕ | ψ | \hat{F}_{\phi}=\frac{1-|\psi\rangle\langle\psi|}{1+|\langle\phi|\psi\rangle|}
  66. F ^ inconcl . = 1 - F ^ ψ - F ^ ϕ , \hat{F}_{\mathrm{inconcl.}}=1-\hat{F}_{\psi}-\hat{F}_{\phi},
  67. F ^ i \hat{F}_{i}
  68. | ψ |\psi\rangle
  69. | ϕ |\phi\rangle
  70. ( 1 - 1 / 2 ) / 2 = 25 % (1-1/2)/2=25\%
  71. 1 - 1 / 2 = 29.3 % 1-1/\sqrt{2}=29.3\%

Pólya_conjecture.html

  1. L ( n ) = k = 1 n λ ( k ) 0 L(n)=\sum_{k=1}^{n}\lambda(k)\leq 0

Pólya_enumeration_theorem.html

  1. | Y | = t \left|Y\right|=t
  2. X Y X\to Y
  3. | Y X / G | = 1 | G | g G t c ( g ) |Y^{X}/G|=\frac{1}{|G|}\sum_{g\in G}t^{c(g)}
  4. | Y | = t \left|Y\right|=t
  5. f ( x ) = f 0 + f 1 x + f 2 x 2 + f(x)=f_{0}+f_{1}x+f_{2}x^{2}+\cdots
  6. Z G ( t 1 , t 2 , , t n ) = 1 | G | g G t 1 j 1 ( g ) t 2 j 2 ( g ) t n j n ( g ) . Z_{G}(t_{1},t_{2},\ldots,t_{n})=\frac{1}{|G|}\sum_{g\in G}t_{1}^{j_{1}(g)}t_{2% }^{j_{2}(g)}\cdots t_{n}^{j_{n}(g)}.
  7. F ( x ) = Z G ( f ( x ) , f ( x 2 ) , f ( x 3 ) , , f ( x n ) ) F(x)=Z_{G}(f(x),f(x^{2}),f(x^{3}),\ldots,f(x^{n}))
  8. F ( a , b , ) = Z G ( f ( a , b , ) , f ( a 2 , b 2 , ) , f ( a 3 , b 3 , ) , , f ( a n , b n , ) ) . F(a,b,\ldots)=Z_{G}(f(a,b,\ldots),f(a^{2},b^{2},\ldots),f(a^{3},b^{3},\ldots),% \ldots,f(a^{n},b^{n},\ldots)).
  9. | Y X / G | = Z G ( t , t , , t ) . |Y^{X}/G|=Z_{G}(t,t,\ldots,t).
  10. ( m 2 ) {\left({{m}\atop{2}}\right)}
  11. f ( t ) = 1 + t f(t)=1+t
  12. G = S m G=S_{m}
  13. S ( m 2 ) S_{{\left({{m}\atop{2}}\right)}}
  14. Z G ( t 1 , t 2 , t 3 ) = t 1 3 + 3 t 1 t 2 + 2 t 3 6 . Z_{G}(t_{1},t_{2},t_{3})=\frac{t_{1}^{3}+3t_{1}t_{2}+2t_{3}}{6}.
  15. F ( t ) = Z G ( t + 1 , t 2 + 1 , t 3 + 1 ) = ( t + 1 ) 3 + 3 ( t + 1 ) ( t 2 + 1 ) + 2 ( t 3 + 1 ) 6 , F(t)=Z_{G}(t+1,t^{2}+1,t^{3}+1)=\frac{(t+1)^{3}+3(t+1)(t^{2}+1)+2(t^{3}+1)}{6},
  16. F ( t ) = t 3 + t 2 + t + 1. F(t)=t^{3}+t^{2}+t+1.
  17. Z G ( t 1 , t 2 , t 3 , t 4 ) = t 1 6 + 9 t 1 2 t 2 2 + 8 t 3 2 + 6 t 2 t 4 24 . Z_{G}(t_{1},t_{2},t_{3},t_{4})=\frac{t_{1}^{6}+9t_{1}^{2}t_{2}^{2}+8t_{3}^{2}+% 6t_{2}t_{4}}{24}.
  18. F ( t ) = Z G ( t + 1 , t 2 + 1 , t 3 + 1 , t 4 + 1 ) = ( t + 1 ) 6 + 9 ( t + 1 ) 2 ( t 2 + 1 ) 2 + 8 ( t 3 + 1 ) 2 + 6 ( t 2 + 1 ) ( t 4 + 1 ) 24 F(t)=Z_{G}(t+1,t^{2}+1,t^{3}+1,t^{4}+1)=\frac{(t+1)^{6}+9(t+1)^{2}(t^{2}+1)^{2% }+8(t^{3}+1)^{2}+6(t^{2}+1)(t^{4}+1)}{24}\,
  19. F ( t ) = t 6 + t 5 + 2 t 4 + 3 t 3 + 2 t 2 + t + 1. F(t)=t^{6}+t^{5}+2t^{4}+3t^{3}+2t^{2}+t+1.
  20. Z S 3 ( t 1 , t 2 , t 3 ) = t 1 3 + 3 t 1 t 2 + 2 t 3 6 . Z_{S_{3}}(t_{1},t_{2},t_{3})=\frac{t_{1}^{3}+3t_{1}t_{2}+2t_{3}}{6}.
  21. T ( x ) = 1 + x T ( x ) 3 + 3 T ( x ) T ( x 2 ) + 2 T ( x 3 ) 6 . T(x)=1+x\frac{T(x)^{3}+3T(x)T(x^{2})+2T(x^{3})}{6}.
  22. t 0 = 1 t_{0}=1
  23. t n + 1 = 1 6 ( a + b + c = n t a t b t c + 3 a + 2 b = n t a t b + 2 3 a = n t a ) t_{n+1}=\frac{1}{6}\left(\sum_{a+b+c=n}t_{a}t_{b}t_{c}+3\sum_{a+2b=n}t_{a}t_{b% }+2\sum_{3a=n}t_{a}\right)
  24. t n t_{n}
  25. Z C ( t 1 , t 2 , t 3 , t 4 ) = t 1 6 + 6 t 1 2 t 4 + 3 t 1 2 t 2 2 + 8 t 3 2 + 6 t 2 3 24 . Z_{C}(t_{1},t_{2},t_{3},t_{4})=\frac{t_{1}^{6}+6t_{1}^{2}t_{4}+3t_{1}^{2}t_{2}% ^{2}+8t_{3}^{2}+6t_{2}^{3}}{24}.
  26. Z C ( t , t , t , t ) = t 6 + 3 t 4 + 12 t 3 + 8 t 2 24 Z_{C}(t,t,t,t)=\frac{t^{6}+3t^{4}+12t^{3}+8t^{2}}{24}
  27. Y X Y^{X}
  28. x 1 , x 2 , x_{1},x_{2},\ldots
  29. Y Y
  30. ω \omega
  31. x ω x^{\omega}
  32. ω \omega
  33. 1 | G | g G | ( Y X ) ω , g | \frac{1}{|G|}\sum_{g\in G}|(Y^{X})_{\omega,g}|
  34. ( Y X ) ω , g (Y^{X})_{\omega,g}
  35. ω \omega
  36. F ( x 1 , x 2 , ) = 1 | G | g G , ω x ω | ( Y X ) ω , g | . F(x_{1},x_{2},\ldots)=\frac{1}{|G|}\sum_{g\in G,\omega}x^{\omega}|(Y^{X})_{% \omega,g}|.
  37. j 1 , j 2 , , j n j_{1},j_{2},\ldots,j_{n}
  38. t 1 j 1 ( g ) t 2 j 2 ( g ) t n j n ( g ) t_{1}^{j_{1}(g)}t_{2}^{j_{2}(g)}\cdots t_{n}^{j_{n}(g)}
  39. Y X Y^{X}
  40. f ( x 1 | q | , x 2 | q | , x 3 | q | , ) . f(x_{1}^{|q|},x_{2}^{|q|},x_{3}^{|q|},\ldots).
  41. ω x ω | ( Y X ) ω , g | = q g f ( x 1 | q | , x 2 | q | , x 3 | q | , ) , \sum_{\omega}x^{\omega}|(Y^{X})_{\omega,g}|=\prod_{q\in g}f(x_{1}^{|q|},x_{2}^% {|q|},x_{3}^{|q|},\ldots),
  42. f ( x 1 , x 2 , ) j 1 ( g ) f ( x 1 2 , x 2 2 , ) j 2 ( g ) f ( x 1 n , x 2 n , ) j n ( g ) . f(x_{1},x_{2},\ldots)^{j_{1}(g)}f(x_{1}^{2},x_{2}^{2},\ldots)^{j_{2}(g)}\cdots f% (x_{1}^{n},x_{2}^{n},\ldots)^{j_{n}(g)}.
  43. ω x ω | ( Y X ) ω , g | \sum_{\omega}x^{\omega}|(Y^{X})_{\omega,g}|\,

Predicate_abstraction.html

  1. A ( t ) \Box A(t)
  2. ( λ x . A ( x ) ) ( t ) (\lambda x.\Box A(x))(t)

Premelting.html

  1. T m T_{m}
  2. T T
  3. T m T_{m}
  4. γ 1 - 2 \gamma_{1-2}
  5. γ 1 - 2 > γ 1 - l + γ l - 2 \gamma_{1-2}>\gamma_{1-l}+\gamma_{l-2}
  6. G ( T , P , d ) = ( n l μ l ( T , P ) ) d + γ t o t a l ( d ) G\left(T,P,d\right)=\left(n_{l}\mu_{l}\left(T,P\right)\right)d+\gamma_{total}% \left(d\right)
  7. T T
  8. P P
  9. d d
  10. N N
  11. n l n_{l}
  12. μ l \mu_{l}
  13. γ t o t a l = γ 1 - l + γ l - 2 \gamma_{total}=\gamma_{1-l}+\gamma_{l-2}
  14. d d
  15. N N
  16. d d
  17. μ ( T , P , d ) = m u l ( T , P ) + Δ γ n l f ( d ) d = μ 1 \mu\left(T,P,d\right)=mu_{l}\left(T,P\right)+\frac{\Delta\gamma}{n_{l}}\frac{% \partial f\left(d\right)}{\partial d}=\mu_{1}
  18. γ t o t a l = Δ γ 1 - l f ( d ) + γ 1 - 2 \gamma_{total}=\Delta\gamma_{1-l}\cdot f\left(d\right)+\gamma_{1-2}
  19. μ 1 \mu_{1}
  20. μ l \mu_{l}
  21. Δ μ = μ 1 - μ l \Delta\mu=\mu_{1}-\mu_{l}
  22. ( T m , P m ) \left(T_{m},P_{m}\right)
  23. f ( d ) = 1 - σ 2 / d 2 f\left(d\right)=1-\sigma^{2}/d^{2}
  24. d σ d>>\sigma
  25. ( d = - 2 σ 2 Δ γ n l q m t ) 1 / 3 \left(d=-\frac{2\sigma^{2}\Delta\gamma}{n_{l}q_{m}t}\right)^{1/3}
  26. f d e - d / d 0 \frac{\partial f}{\partial d}~{}e^{-d/d_{0}}
  27. d | l n | t | | d\propto\left|ln\left|t\right|\right|
  28. σ \sigma
  29. q m q_{m}
  30. t = T - T m T m t=\frac{T-T_{m}}{T_{m}}
  31. γ t o t a l ( d ) \gamma_{total}\left(d\right)
  32. γ t o t a l ( d ) \gamma_{total}\left(d\right)
  33. η \eta
  34. t = 0 t=0
  35. T T m T\leq T_{m}
  36. η 0 { c o n s t . , a < a m | t | 1 / 4 , a = a m | t | 1 / 2 , a > a m \eta_{0}\propto\begin{cases}const.&\,\text{, }a<\sqrt{a_{m}}\\ \left|t\right|^{1/4}&\,\text{, }a=\sqrt{a_{m}}\\ \left|t\right|^{1/2}&\,\text{, }a>\sqrt{a_{m}}\end{cases}
  37. η 0 \eta_{0}
  38. 1 / a 1/a
  39. a m a_{m}
  40. d | l n | t | | d\propto\left|ln\left|t\right|\right|
  41. δ d d = 2 γ 1 - l 3 p l q m t r \frac{\delta d}{d}=\frac{2\gamma_{1-l}}{3p_{l}q_{m}tr}
  42. μ = 0.6 \mu=0.6
  43. μ m \mu m

Presheaf_(category_theory).html

  1. C C
  2. F : C op 𝐒𝐞𝐭 F\colon C^{\mathrm{op}}\to\mathbf{Set}
  3. C C
  4. C ^ = 𝐒𝐞𝐭 C op \widehat{C}=\mathbf{Set}^{C^{\mathrm{op}}}
  5. C ^ \widehat{C}
  6. F : C op 𝐕 F\colon C^{\mathrm{op}}\to\mathbf{V}
  7. 𝐕 \mathbf{V}
  8. C = Δ C=\Delta
  9. C C
  10. C ^ = 𝐒𝐞𝐭 C op \widehat{C}=\mathbf{Set}^{C^{\mathrm{op}}}
  11. P P
  12. P P
  13. C ^ = 𝐒𝐞𝐭 C op \widehat{C}=\mathbf{Set}^{C^{\mathrm{op}}}
  14. C C
  15. f : X Y f:X\to Y
  16. C ^ \widehat{C}
  17. f * : Sub C ^ ( Y ) Sub C ^ ( X ) f^{*}:\mathrm{Sub}_{\widehat{C}}(Y)\to\mathrm{Sub}_{\widehat{C}}(X)
  18. f \forall_{f}
  19. f \exists_{f}
  20. C C
  21. C ^ \widehat{C}
  22. Y c \mathrm{Y}_{c}
  23. A A
  24. C C
  25. C ( - , A ) C(-,A)
  26. C ^ \widehat{C}
  27. C C

Primality_certificate.html

  1. 4 log 2 n - 4 4\log_{2}n-4
  2. 4 log 2 p i - 4 4\log_{2}p_{i}-4
  3. 1 + i = 1 k ( 4 log 2 p i - 4 ) = - 4 k + 4 log 2 p 1 p k 4 log 2 p - 4 , 1+\sum_{i=1}^{k}(4\log_{2}p_{i}-4)=-4k+4\log_{2}p_{1}\cdots p_{k}\leq 4\log_{2% }p-4,
  4. n \mathbb{Z}_{n}
  5. q > n 1 / 2 + 1 + 2 n 1 / 4 q>n^{1/2}+1+2n^{1/4}
  6. n \mathbb{Z}_{n}
  7. n \mathbb{Z}_{n}

Primary_cyclic_group.html

  1. C p m C_{p^{m}}\!
  2. G = 1 i n C p i m i G=\bigoplus_{1\leq i\leq n}C_{{p_{i}}^{m_{i}}}\;

Principal_protected_note.html

  1. P P N T = 100 + A ( S T - K ) + PPN_{T}=100+A(S_{T}-K)^{+}
  2. A A
  3. S T S_{T}
  4. K K

Principle_of_minimum_energy.html

  1. U ( S , X 1 , X 2 , ) U(S,X_{1},X_{2},\dots)
  2. X i X_{i}
  3. S ( U , X 1 , X 2 , ) S(U,X_{1},X_{2},...)
  4. X i X_{i}
  5. ( S X ) U = 0 \left(\frac{\partial S}{\partial X}\right)_{U}=0
  6. ( 2 S X 2 ) U < 0 \left(\frac{\partial^{2}S}{\partial X^{2}}\right)_{U}<0
  7. ( U X ) S = - ( S X ) U ( S U ) X = - T ( S X ) U = 0 \left(\frac{\partial U}{\partial X}\right)_{S}=-\,\frac{\left(\frac{\partial S% }{\partial X}\right)_{U}}{\left(\frac{\partial S}{\partial U}\right)_{X}}=-T% \left(\frac{\partial S}{\partial X}\right)_{U}=0
  8. ( 2 U X 2 ) S = - T ( 2 S X 2 ) U \left(\frac{\partial^{2}U}{\partial X^{2}}\right)_{S}=-T\left(\frac{\partial^{% 2}S}{\partial X^{2}}\right)_{U}
  9. U = T S - P A x + μ N + m g x U=TS-PAx+\mu N+mgx\,
  10. d U = T d S - P A d x + m g d x dU=T\,dS-PA\,dx+mg\,dx
  11. 0 = - P A + m g 0=-PA+mg\,
  12. U ( S , V , { N j } ) = T S - P V + j μ j N j U(S,V,\{N_{j}\})=TS-PV+\sum_{j}\mu_{j}N_{j}\,
  13. ( S , V , { N j } ) (S,V,\{N_{j}\})
  14. A ( T , V , { N j } ) = U - T S A(T,V,\{N_{j}\})=U-TS\,
  15. A ( T , V , { N j } ) = max S ( U ( S , V , { N j } ) - T S ) A(T,V,\{N_{j}\})=\mathrm{max}_{S}(U(S,V,\{N_{j}\})-TS)\,
  16. T = ( U S ) V , { N j } T=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_{j}\}}
  17. U o ( S o ) = min x ( U ( S o , x ) ) U_{o}(S_{o})=\mathrm{min}_{x}(U(S_{o},x))\,
  18. U o U_{o}
  19. S o S_{o}
  20. N 1 N_{1}
  21. N 2 N_{2}
  22. N A N_{A}
  23. N A = N 1 + 2 N 2 N_{A}=N_{1}+2N_{2}\,
  24. N 1 N_{1}
  25. N 2 N_{2}
  26. x = N 1 / N 2 x=N_{1}/N_{2}
  27. U o U_{o}
  28. S o S_{o}
  29. A ( T , x ) = max S ( U ( S , x ) - T S ) A(T,x)=\mathrm{max}_{S}(U(S,x)-TS)\,
  30. A o ( T o ) = max S o ( U o ( S o ) - T o S o ) A_{o}(T_{o})=\mathrm{max}_{S_{o}}(U_{o}(S_{o})-T_{o}S_{o})
  31. T o T_{o}
  32. U o U_{o}
  33. A o = max S o ( min x ( U ( S o , x ) ) - T o S o ) A_{o}=\mathrm{max}_{S_{o}}(\mathrm{min}_{x}(U(S_{o},x))-T_{o}S_{o})
  34. A o = min x ( max S o ( U ( S o , x ) - T o S o ) ) = min x ( A o ( T o , x ) ) A_{o}=\mathrm{min}_{x}(\mathrm{max}_{S_{o}}(U(S_{o},x)-T_{o}S_{o}))=\mathrm{% min}_{x}(A_{o}(T_{o},x))

Probabilistic_logic.html

  1. W W
  2. V V
  3. V V
  4. V = { } V=\{\}
  5. V = W V=W

Probability_current.html

  1. j = 2 m i ( Ψ * Ψ x - Ψ Ψ * x ) , j=\frac{\hbar}{2mi}\left(\Psi^{*}\frac{\partial\Psi}{\partial x}-\Psi\frac{% \partial\Psi^{*}}{\partial x}\right),
  2. 𝐣 = 2 m i ( Ψ * Ψ - Ψ Ψ * ) , \mathbf{j}=\frac{\hbar}{2mi}\left(\Psi^{*}\mathbf{\nabla}\Psi-\Psi\mathbf{% \nabla}\Psi^{*}\right)\,,
  3. 𝐩 ^ = i = - i \mathbf{\hat{p}}=\frac{\hbar}{i}\nabla=-i\hbar\nabla
  4. 𝐣 = 1 2 m ( Ψ * 𝐩 ^ Ψ - Ψ 𝐩 ^ Ψ * ) . \mathbf{j}=\frac{1}{2m}\left(\Psi^{*}\mathbf{\hat{p}}\Psi-\Psi\mathbf{\hat{p}}% \Psi^{*}\right)\,.
  5. 𝐣 = 1 2 m [ ( Ψ * 𝐩 ^ Ψ - Ψ 𝐩 ^ Ψ * ) - 2 q 𝐀 | Ψ | 2 ] \mathbf{j}=\frac{1}{2m}\left[\left(\Psi^{*}\mathbf{\hat{p}}\Psi-\Psi\mathbf{% \hat{p}}\Psi^{*}\right)-2q\mathbf{A}|\Psi|^{2}\right]\,\!
  6. 𝐣 = 1 2 m [ ( Ψ * 𝐩 ^ Ψ - Ψ 𝐩 ^ Ψ * ) - 2 q c 𝐀 | Ψ | 2 ] \mathbf{j}=\frac{1}{2m}\left[\left(\Psi^{*}\mathbf{\hat{p}}\Psi-\Psi\mathbf{% \hat{p}}\Psi^{*}\right)-2\frac{q}{c}\mathbf{A}|\Psi|^{2}\right]\,\!
  7. 𝐣 = 1 2 m [ ( Ψ * 𝐩 ^ Ψ - Ψ 𝐩 ^ Ψ * ) - 2 q 𝐀 | Ψ | 2 ] + μ S s × ( Ψ * 𝐒 Ψ ) \mathbf{j}=\frac{1}{2m}\left[\left(\Psi^{*}\mathbf{\hat{p}}\Psi-\Psi\mathbf{% \hat{p}}\Psi^{*}\right)-2q\mathbf{A}|\Psi|^{2}\right]+\frac{\mu_{S}}{s}\nabla% \times(\Psi^{*}\mathbf{S}\Psi)\,\!
  8. 𝐣 = 1 2 m [ ( Ψ * 𝐩 ^ Ψ - Ψ 𝐩 ^ Ψ * ) - 2 q c 𝐀 | Ψ | 2 ] + μ S c s × ( Ψ * 𝐒 Ψ ) \mathbf{j}=\frac{1}{2m}\left[\left(\Psi^{*}\mathbf{\hat{p}}\Psi-\Psi\mathbf{% \hat{p}}\Psi^{*}\right)-2\frac{q}{c}\mathbf{A}|\Psi|^{2}\right]+\frac{\mu_{S}c% }{s}\nabla\times(\Psi^{*}\mathbf{S}\Psi)\,\!
  9. Ψ = R e i S / \Psi=Re^{iS/\hbar}
  10. ρ = Ψ * Ψ = R 2 \rho=\Psi^{*}\Psi=R^{2}
  11. 𝐣 \displaystyle\mathbf{j}
  12. = 2 m i [ i R 2 S + i R 2 S ] =\frac{\hbar}{2mi}\left[\frac{i}{\hbar}R^{2}\mathbf{\nabla}S+\frac{i}{\hbar}R^% {2}\mathbf{\nabla}S\right]
  13. 𝐣 = ρ S m \mathbf{j}=\rho\frac{\mathbf{\nabla}S}{m}
  14. 𝐣 = ρ 𝐯 \mathbf{j}=\rho\mathbf{v}
  15. 𝐩 = S \mathbf{p}=\nabla S
  16. ρ t + 𝐣 = 0 \frac{\partial\rho}{\partial t}+\mathbf{\nabla}\cdot\mathbf{j}=0
  17. ρ \rho\,
  18. ρ ( 𝐫 , t ) = | Ψ | 2 = Ψ * ( 𝐫 , t ) Ψ ( 𝐫 , t ) \rho(\mathbf{r},t)=|\Psi|^{2}=\Psi^{*}(\mathbf{r},t)\Psi(\mathbf{r},t)\,
  19. V ( | Ψ | 2 t ) d V + V ( 𝐣 ) d V = 0 \int_{V}\left(\frac{\partial|\Psi|^{2}}{\partial t}\right)\mathrm{d}V+\int_{V}% \left(\mathbf{\nabla}\cdot\mathbf{j}\right)\mathrm{d}V=0
  20. T + R = 1 , T+R=1\,,
  21. T = | 𝐣 trans | | 𝐣 inc | , R = | 𝐣 ref | | 𝐣 inc | , T=\frac{|\mathbf{j}_{\mathrm{trans}}|}{|\mathbf{j}_{\mathrm{inc}}|}\,,\quad R=% \frac{|\mathbf{j}_{\mathrm{ref}}|}{|\mathbf{j}_{\mathrm{inc}}|}\,,
  22. 𝐣 trans + 𝐣 ref = 𝐣 inc . \mathbf{j}_{\mathrm{trans}}+\mathbf{j}_{\mathrm{ref}}=\mathbf{j}_{\mathrm{inc}% }\,.
  23. T = | 𝐣 trans 𝐧 𝐣 inc 𝐧 | , R = | 𝐣 ref 𝐧 𝐣 inc 𝐧 | , T=\left|\frac{\mathbf{j}_{\mathrm{trans}}\cdot\mathbf{n}}{\mathbf{j}_{\mathrm{% inc}}\cdot\mathbf{n}}\right|\,,\qquad R=\left|\frac{\mathbf{j}_{\mathrm{ref}}% \cdot\mathbf{n}}{\mathbf{j}_{\mathrm{inc}}\cdot\mathbf{n}}\right|\,,
  24. Ψ ( 𝐫 , t ) = A e i ( 𝐤 𝐫 - ω t ) \Psi(\mathbf{r},t)=\,Ae^{i(\mathbf{k}\cdot{\mathbf{r}}-\omega t)}
  25. ρ ( 𝐫 , t ) = | A | 2 | Ψ | 2 t = 0 \rho(\mathbf{r},t)=|A|^{2}\rightarrow\frac{\partial|\Psi|^{2}}{\partial t}=0
  26. 𝐣 ( 𝐫 , t ) = | A | 2 𝐤 m = ρ 𝐩 m = ρ 𝐯 \mathbf{j}\left(\mathbf{r},t\right)=\left|A\right|^{2}{\hbar\mathbf{k}\over m}% =\rho\frac{\mathbf{p}}{m}=\rho\mathbf{v}
  27. 0 < x < L 0<x<L\,\!
  28. Ψ n = 2 L sin ( n π L x ) \Psi_{n}=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}x\right)
  29. j n = 2 m i ( Ψ n * Ψ n x - Ψ n Ψ n * x ) = 0 j_{n}=\frac{\hbar}{2mi}\left(\Psi_{n}^{*}\frac{\partial\Psi_{n}}{\partial x}-% \Psi_{n}\frac{\partial\Psi_{n}^{*}}{\partial x}\right)=0
  30. Ψ n = Ψ n * \Psi_{n}=\Psi_{n}^{*}

Procedural_parameter.html

  1. \leftarrow
  2. \leq
  3. \leftarrow
  4. \leftarrow
  5. \leftarrow
  6. \leftarrow
  7. \leftarrow
  8. \leftarrow
  9. \leftarrow
  10. \leftarrow
  11. \leftarrow
  12. \leftarrow
  13. \leftarrow
  14. \leftarrow
  15. \neq
  16. \neq
  17. = =
  18. \neq
  19. \neq
  20. \leftarrow
  21. \leftarrow
  22. = =
  23. \leftarrow
  24. \leftarrow
  25. \leftarrow
  26. \leftarrow
  27. = =
  28. \leftarrow
  29. \leftarrow
  30. \leftarrow
  31. \leftarrow
  32. = =
  33. \leftarrow
  34. \leftarrow
  35. \leftarrow
  36. \leftarrow
  37. a b \textstyle\int_{a}^{b}
  38. \leftarrow
  39. \leftarrow
  40. \leftarrow
  41. \leftarrow
  42. \leftarrow
  43. \leftarrow
  44. D g ( x , y ) d x d y = 0 R z ( 0 2 π g ( 𝑥𝑐 + z cos t , 𝑦𝑐 + z sin t ) d t ) d z \int\!\int_{D}g(x,y)\,\mathrm{d}x\,\mathrm{d}y=\int_{0}^{R}z\left(\int_{0}^{2% \pi}g(\mathit{xc}+z\cos t,\mathit{yc}+z\sin t)\,\mathrm{d}t\right)\,\mathrm{d}z
  45. \leftarrow
  46. \leftarrow
  47. \leftarrow

Process_function.html

  1. δ X \delta X
  2. d Y dY
  3. d Y dY
  4. δ X \delta X
  5. λ \lambda
  6. Y = λ X Y=\lambda X
  7. λ \lambda
  8. d Y = λ δ X dY=\lambda\delta X
  9. λ = 1 / p \lambda=1/p
  10. d V = δ W / p dV=\delta W/p
  11. λ = 1 / T \lambda=1/T
  12. d S = δ Q / T dS=\delta Q/T

Procrustes_analysis.html

  1. k k
  2. ( ( x 1 , y 1 ) , ( x 2 , y 2 ) , , ( x k , y k ) ) ((x_{1},y_{1}),(x_{2},y_{2}),\dots,(x_{k},y_{k}))\,
  3. ( x ¯ , y ¯ ) (\bar{x},\bar{y})
  4. x ¯ = x 1 + x 2 + + x k k , y ¯ = y 1 + y 2 + + y k k . \bar{x}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k},\quad\bar{y}=\frac{y_{1}+y_{2}+% \cdots+y_{k}}{k}.
  5. ( x , y ) ( x - x ¯ , y - y ¯ ) (x,y)\to(x-\bar{x},y-\bar{y})
  6. ( x 1 - x ¯ , y 1 - y ¯ ) , (x_{1}-\bar{x},y_{1}-\bar{y}),\dots
  7. s = ( x 1 - x ¯ ) 2 + ( y 1 - y ¯ ) 2 + k s=\sqrt{{(x_{1}-\bar{x})^{2}+(y_{1}-\bar{y})^{2}+\cdots}\over k}
  8. ( ( x 1 - x ¯ ) / s , ( y 1 - y ¯ ) / s ) ((x_{1}-\bar{x})/s,(y_{1}-\bar{y})/s)
  9. ( ( x 1 , y 1 ) , ) ((x_{1},y_{1}),\ldots)
  10. ( ( w 1 , z 1 ) , ) ((w_{1},z_{1}),\ldots)
  11. θ \theta\,\!
  12. θ \theta\,\!
  13. ( u 1 , v 1 ) = ( cos θ w 1 - sin θ z 1 , sin θ w 1 + cos θ z 1 ) (u_{1},v_{1})=(\cos\theta\,w_{1}-\sin\theta\,z_{1},\,\sin\theta\,w_{1}+\cos% \theta\,z_{1})\,\!
  14. ( u 1 - x 1 ) 2 + ( v 1 - y 1 ) 2 + (u_{1}-x_{1})^{2}+(v_{1}-y_{1})^{2}+\cdots
  15. θ \theta
  16. θ \theta
  17. θ = tan - 1 ( i = 1 k ( w i y i - z i x i ) i = 1 k ( w i x i + z i y i ) ) . \theta=\tan^{-1}\left(\frac{\sum_{i=1}^{k}(w_{i}y_{i}-z_{i}x_{i})}{\sum_{i=1}^% {k}(w_{i}x_{i}+z_{i}y_{i})}\right).
  18. d = ( u 1 - x 1 ) 2 + ( v 1 - y 1 ) 2 + . d=\sqrt{(u_{1}-x_{1})^{2}+(v_{1}-y_{1})^{2}+\cdots}.

Profunctor.html

  1. ϕ \,\phi
  2. C C
  3. D D
  4. ϕ : C D \phi\colon C\nrightarrow D
  5. ϕ : D op × C 𝐒𝐞𝐭 \phi\colon D^{\mathrm{op}}\times C\to\mathbf{Set}
  6. D op D^{\mathrm{op}}
  7. D D
  8. 𝐒𝐞𝐭 \mathbf{Set}
  9. f : d d , g : c c f\colon d\to d^{\prime},g\colon c\to c^{\prime}
  10. D , C D,C
  11. x ϕ ( d , c ) x\in\phi(d^{\prime},c)
  12. x f ϕ ( d , c ) , g x ϕ ( d , c ) xf\in\phi(d,c),gx\in\phi(d^{\prime},c^{\prime})
  13. 𝐂𝐚𝐭 \mathbf{Cat}
  14. ϕ \phi
  15. ϕ ^ : C D ^ \hat{\phi}\colon C\to\hat{D}
  16. D ^ \hat{D}
  17. Set D op \mathrm{Set}^{D^{\mathrm{op}}}
  18. D D
  19. C C
  20. D D
  21. D C D\nrightarrow C
  22. ψ ϕ \psi\phi
  23. ϕ : C D \phi\colon C\nrightarrow D
  24. ψ : D E \psi\colon D\nrightarrow E
  25. ψ ϕ = Lan Y D ( ψ ^ ) ϕ ^ \psi\phi=\mathrm{Lan}_{Y_{D}}(\hat{\psi})\circ\hat{\phi}
  26. Lan Y D ( ψ ^ ) \mathrm{Lan}_{Y_{D}}(\hat{\psi})
  27. ψ ^ \hat{\psi}
  28. Y D : D D ^ Y_{D}\colon D\to\hat{D}
  29. D D
  30. d d
  31. D D
  32. D ( - , d ) : D op Set D(-,d)\colon D^{\mathrm{op}}\to\mathrm{Set}
  33. ( ψ ϕ ) ( e , c ) = ( d D ψ ( e , d ) × ϕ ( d , c ) ) / (\psi\phi)(e,c)=\left(\coprod_{d\in D}\psi(e,d)\times\phi(d,c)\right)\Bigg/\sim
  34. \sim
  35. ( y , x ) ( y , x ) (y^{\prime},x^{\prime})\sim(y,x)
  36. v v
  37. D D
  38. y = v y ψ ( e , d ) y^{\prime}=vy\in\psi(e,d^{\prime})
  39. x v = x ϕ ( d , c ) x^{\prime}v=x\in\phi(d,c)
  40. F : C D F\colon C\to D
  41. ϕ F : C D \phi_{F}\colon C\nrightarrow D
  42. ϕ F = Y D F \phi_{F}=Y_{D}\circ F
  43. ϕ F \phi_{F}
  44. ϕ : C D \phi\colon C\nrightarrow D
  45. ϕ ^ : C D ^ \hat{\phi}\colon C\to\hat{D}
  46. D D
  47. F : C D F\colon C\to D
  48. ϕ ^ = Y D F \hat{\phi}=Y_{D}\circ F

Programming_Computable_Functions.html

  1. Γ t : 𝐧𝐚𝐭 , Γ s 0 : σ , Γ s 1 : σ Γ 𝐢𝐟 ( t , s 0 , s 1 ) : σ \frac{\Gamma\;\vdash\;t\;:\,\textbf{nat},\quad\quad\Gamma\;\vdash\;s_{0}\;:% \sigma,\quad\quad\Gamma\;\vdash\;s_{1}\;:\sigma}{\Gamma\;\vdash\;\,\textbf{if}% (t,s_{0},s_{1})\;:\sigma}
  2. [ [ 𝐧𝐚𝐭 ] ] := [\![\,\textbf{nat}]\!]:=\mathbb{N}_{\bot}
  3. [ [ σ τ ] ] [\![\sigma\to\tau\,]\!]
  4. [ [ σ ] ] [\![\sigma]\!]\,
  5. [ [ τ ] ] [\![\tau]\!]\,
  6. x 1 : σ 1 , , x n : σ n x_{1}:\sigma_{1},\;\dots,\;x_{n}:\sigma_{n}
  7. [ [ σ 1 ] ] × × [ [ σ n ] ] [\![\sigma_{1}]\!]\times\;\dots\;\times[\![\sigma_{n}]\!]
  8. Γ x : σ \Gamma\;\vdash\;x\;:\;\sigma
  9. [ [ Γ ] ] [ [ σ ] ] [\![\Gamma]\!]\;\to\;[\![\sigma]\!]

Projection_(mathematics).html

  1. i X Y i \prod_{i\in X}Y_{i}

Projection_(set_theory).html

  1. proj j \mathrm{proj}_{j}\!
  2. x = ( x 1 , , x j , , x k ) \vec{x}=(x_{1},\ \ldots,\ x_{j},\ \ldots,\ x_{k})
  3. ( X 1 × × X j × × X k ) (X_{1}\times\cdots\times X_{j}\times\cdots\times X_{k})
  4. proj j ( x ) = x j \mathrm{proj}_{j}(\vec{x})=x_{j}

Projective_cone.html

  1. \cap
  2. | R A | |RA|
  3. q r + 1 q^{r+1}
  4. | A | |A|
  5. q r + 1 - 1 q - 1 \frac{q^{r+1}-1}{q-1}

Projective_orthogonal_group.html

  1. - I S O ( 2 k ) -I\in SO(2k)
  2. - I S O ( 2 k + 1 ) -I\not\in SO(2k+1)
  3. B k = 𝔰 𝔬 2 k + 1 , D k = 𝔰 𝔬 2 k . B_{k}=\mathfrak{so}_{2k+1},D_{k}=\mathfrak{so}_{2k}.
  4. O ( 2 k + 1 ) = S O ( 2 k + 1 ) × { ± I } , O(2k+1)=SO(2k+1)\times\{\pm I\},
  5. O ( 2 k ) S O ( 2 k ) × { ± I } O(2k)\neq SO(2k)\times\{\pm I\}
  6. 𝐑𝐏 2 k = 𝐏 ( 𝐑 2 k + 1 ) , \mathbf{RP}^{2k}=\mathbf{P}(\mathbf{R}^{2k+1}),
  7. 𝐑𝐏 2 k - 1 = 𝐏 ( 𝐑 2 k ) \mathbf{RP}^{2k-1}=\mathbf{P}(\mathbf{R}^{2k})
  8. S O ( 2 k + 1 ) P S O ( 2 k + 1 ) = P O ( 2 k + 1 ) , SO(2k+1)\cong PSO(2k+1)=PO(2k+1),
  9. - I -I
  10. - I -I
  11. π 1 \pi_{1}
  12. π 0 ( P S O ) 1 \pi_{0}(PSO)\cong 1
  13. π 0 ( P O ( 2 k ) ) 𝐙 / 2 , π 0 ( P O ( 2 k + 1 ) ) 1. \pi_{0}\big(PO(2k)\big)\cong\mathbf{Z}/2,\pi_{0}\big(PO(2k+1)\big)\cong 1.
  14. π 1 ( P S O ( n ) ) = π 1 ( P O ( n ) ) = Z ( S p i n ( n ) ) . \pi_{1}\big(PSO(n)\big)=\pi_{1}\big(PO(n)\big)=Z\big(Spin(n)\big).
  15. k 1 k\geq 1
  16. π 1 ( P S O ( 4 k ) ) = 𝐙 / 2 𝐙 / 2 , \pi_{1}\big(PSO(4k)\big)=\mathbf{Z}/2\oplus\mathbf{Z}/2,
  17. π 1 ( P S O ( 4 k + 2 ) ) = 𝐙 / 4 , \pi_{1}\big(PSO(4k+2)\big)=\mathbf{Z}/4,
  18. π 1 ( P S O ( 2 k + 1 ) ) = π 1 ( S O ( 2 k + 1 ) ) = 𝐙 / 2 , \pi_{1}\big(PSO(2k+1)\big)=\pi_{1}\big(SO(2k+1)\big)=\mathbf{Z}/2,
  19. π 1 ( P S O ( 1 ) ) = 1 , \pi_{1}\big(PSO(1)\big)=1,
  20. π 1 ( P S O ( 2 ) ) = 𝐙 , \pi_{1}\big(PSO(2)\big)=\mathbf{Z},
  21. 𝐙 / 4 , \mathbf{Z}/4,
  22. 4 k + 2. 4k+2.
  23. I = - I , I=-I,
  24. ( - 1 ) D (-1)^{D}
  25. O P O O\to PO
  26. S O P S O SO\to PSO

Projective_unitary_group.html

  1. e i θ e^{i\theta}
  2. Z ( SU ( n ) ) = SU ( n ) Z ( U ( n ) ) 𝐙 / n Z(\mbox{SU}~{}(n))=\mbox{SU}~{}(n)\cap Z(\mbox{U}~{}(n))\cong\mathbf{Z}/n
  3. PSU ( n ) = SU ( n ) / Z ( SU ( n ) ) PU ( n ) = U ( n ) / Z ( U ( n ) ) \mbox{PSU}~{}(n)=\mbox{SU}~{}(n)/Z(\mbox{SU}~{}(n))\to\mbox{PU}~{}(n)=\mbox{U}% ~{}(n)/Z(\mbox{U}~{}(n))
  4. SU ( 2 ) Spin ( 3 ) Sp ( 1 ) \mbox{SU}~{}(2)\cong\mbox{Spin}~{}(3)\cong\mbox{Sp}~{}(1)
  5. PU ( 2 ) = PSU ( 2 ) = SU ( 2 ) / ( 𝐙 / 2 ) Spin ( 3 ) / ( 𝐙 / 2 ) = SO ( 3 ) \mbox{PU}~{}(2)=\mbox{PSU}~{}(2)=\mbox{SU}~{}(2)/(\mathbf{Z}/2)\cong\mbox{Spin% }~{}(3)/(\mathbf{Z}/2)=\mbox{SO}~{}(3)
  6. 𝐅 q 2 \mathbf{F}_{q^{2}}
  7. U ( n , q 2 ) U\left(n,q^{2}\right)
  8. U ( n , q 2 ) U(n,q^{2})
  9. 𝐅 q 2 \mathbf{F}_{q^{2}}
  10. G L ( n , q 2 ) GL(n,q^{2})
  11. q 2 - 1 q^{2}-1
  12. U ( n , q 2 ) U(n,q^{2})
  13. c I V cI_{V}
  14. c q + 1 = 1 c^{q+1}=1
  15. gcd ( n , q + 1 ) \gcd(n,q+1)
  16. P U ( n , q 2 ) PU(n,q^{2})
  17. P S U ( n , q 2 ) PSU(n,q^{2})
  18. n 2 n\geq 2
  19. ( n , q 2 ) { ( 2 , 2 2 ) , ( 2 , 3 2 ) , ( 3 , 2 2 ) } (n,q^{2})\notin\{(2,2^{2}),(2,3^{2}),(3,2^{2})\}
  20. S U ( n , q 2 ) SU(n,q^{2})
  21. P S U ( n , q 2 ) PSU(n,q^{2})
  22. \mathcal{H}
  23. u 1 2 u 1 2 1 2 u u - 1 u u - 1 1 2 1 2 ( u U ( H ) ) u\oplus 1_{\ell^{2}}\sim u\oplus 1_{\ell^{2}}\oplus 1_{\ell^{2}}\oplus\cdots% \sim u\oplus u^{-1}\oplus u\oplus u^{-1}\oplus\cdots\sim 1_{\ell^{2}}\oplus 1_% {\ell^{2}}\oplus\cdots(u\in{\rm U}(H))
  24. \mathcal{H}
  25. \mathcal{H}
  26. \mathcal{H}
  27. \mathcal{H}
  28. \mathcal{H}
  29. π n ( X ) = π n + 1 ( B X ) \pi_{n}(X)=\pi_{n+1}(BX)
  30. π 1 ( U ( 1 ) ) = 𝐙 , \pi_{1}(U(1))=\mathbf{Z},
  31. π k 1 ( U ( 1 ) ) = 0 \pi_{k\neq 1}(U(1))=0
  32. \mathcal{H}
  33. π 2 ( P U ( ) ) = 𝐙 , \pi_{2}(PU(\mathcal{H}))=\mathbf{Z},
  34. π k 2 ( P U ( ) ) = 0 \pi_{k\neq 2}(PU(\mathcal{H}))=0
  35. \mathcal{H}
  36. \mathcal{H}
  37. \mathcal{H}
  38. \mathcal{H}
  39. \mathcal{H}
  40. \mathcal{H}
  41. \mathcal{H}
  42. \mathcal{H}
  43. \mathcal{H}
  44. \mathcal{H}
  45. \mathcal{H}
  46. \mathcal{H}
  47. \mathcal{H}

Prolate_spheroidal_coordinates.html

  1. ( μ , ν , ϕ ) (\mu,\nu,\phi)
  2. x = a sinh μ sin ν cos ϕ x=a\ \sinh\mu\ \sin\nu\ \cos\phi
  3. y = a sinh μ sin ν sin ϕ y=a\ \sinh\mu\ \sin\nu\ \sin\phi
  4. z = a cosh μ cos ν z=a\ \cosh\mu\ \cos\nu
  5. μ \mu
  6. ν [ 0 , π ] \nu\in[0,\pi]
  7. ϕ \phi
  8. [ 0 , 2 π ] [0,2\pi]
  9. z 2 a 2 cosh 2 μ + x 2 + y 2 a 2 sinh 2 μ = cos 2 ν + sin 2 ν = 1 \frac{z^{2}}{a^{2}\cosh^{2}\mu}+\frac{x^{2}+y^{2}}{a^{2}\sinh^{2}\mu}=\cos^{2}% \nu+\sin^{2}\nu=1
  10. μ \mu
  11. z 2 a 2 cos 2 ν - x 2 + y 2 a 2 sin 2 ν = cosh 2 μ - sinh 2 μ = 1 \frac{z^{2}}{a^{2}\cos^{2}\nu}-\frac{x^{2}+y^{2}}{a^{2}\sin^{2}\nu}=\cosh^{2}% \mu-\sinh^{2}\mu=1
  12. ν \nu
  13. ( μ , ν ) (\mu,\nu)
  14. h μ = h ν = a sinh 2 μ + sin 2 ν h_{\mu}=h_{\nu}=a\sqrt{\sinh^{2}\mu+\sin^{2}\nu}
  15. h ϕ = a sinh μ sin ν h_{\phi}=a\sinh\mu\ \sin\nu
  16. d V = a 3 sinh μ sin ν ( sinh 2 μ + sin 2 ν ) d μ d ν d ϕ dV=a^{3}\sinh\mu\ \sin\nu\ \left(\sinh^{2}\mu+\sin^{2}\nu\right)d\mu d\nu d\phi
  17. 2 Φ = 1 a 2 ( sinh 2 μ + sin 2 ν ) [ 2 Φ μ 2 + 2 Φ ν 2 + coth μ Φ μ + cot ν Φ ν ] + 1 a 2 sinh 2 μ sin 2 ν 2 Φ ϕ 2 \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)}\left[\frac% {\partial^{2}\Phi}{\partial\mu^{2}}+\frac{\partial^{2}\Phi}{\partial\nu^{2}}+% \coth\mu\frac{\partial\Phi}{\partial\mu}+\cot\nu\frac{\partial\Phi}{\partial% \nu}\right]+\frac{1}{a^{2}\sinh^{2}\mu\sin^{2}\nu}\frac{\partial^{2}\Phi}{% \partial\phi^{2}}
  18. 𝐅 \nabla\cdot\mathbf{F}
  19. × 𝐅 \nabla\times\mathbf{F}
  20. ( μ , ν , ϕ ) (\mu,\nu,\phi)
  21. ( σ , τ , ϕ ) (\sigma,\tau,\phi)
  22. σ = cosh μ \sigma=\cosh\mu
  23. τ = cos ν \tau=\cos\nu
  24. σ \sigma
  25. τ \tau
  26. τ \tau
  27. σ \sigma
  28. σ \sigma
  29. τ \tau
  30. F 1 F_{1}
  31. F 2 F_{2}
  32. d 1 + d 2 d_{1}+d_{2}
  33. 2 a σ 2a\sigma
  34. d 1 - d 2 d_{1}-d_{2}
  35. 2 a τ 2a\tau
  36. F 1 F_{1}
  37. a ( σ + τ ) a(\sigma+\tau)
  38. F 2 F_{2}
  39. a ( σ - τ ) a(\sigma-\tau)
  40. F 1 F_{1}
  41. F 2 F_{2}
  42. z = - a z=-a
  43. z = + a z=+a
  44. σ \sigma
  45. τ \tau
  46. ϕ \phi
  47. σ = 1 2 a ( x 2 + y 2 + ( z + a ) 2 + x 2 + y 2 + ( z - a ) 2 ) \sigma=\frac{1}{2a}\left(\sqrt{x^{2}+y^{2}+(z+a)^{2}}+\sqrt{x^{2}+y^{2}+(z-a)^% {2}}\right)
  48. τ = 1 2 a ( x 2 + y 2 + ( z + a ) 2 - x 2 + y 2 + ( z - a ) 2 ) \tau=\frac{1}{2a}\left(\sqrt{x^{2}+y^{2}+(z+a)^{2}}-\sqrt{x^{2}+y^{2}+(z-a)^{2% }}\right)
  49. ϕ = arctan ( y x ) \phi=\arctan\left(\frac{y}{x}\right)
  50. x = a ( σ 2 - 1 ) ( 1 - τ 2 ) cos ϕ x=a\sqrt{\left(\sigma^{2}-1\right)\left(1-\tau^{2}\right)}\cos\phi
  51. y = a ( σ 2 - 1 ) ( 1 - τ 2 ) sin ϕ y=a\sqrt{\left(\sigma^{2}-1\right)\left(1-\tau^{2}\right)}\sin\phi
  52. z = a σ τ z=a\ \sigma\ \tau
  53. ( σ , τ , ϕ ) (\sigma,\tau,\phi)
  54. h σ = a σ 2 - τ 2 σ 2 - 1 h_{\sigma}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{\sigma^{2}-1}}
  55. h τ = a σ 2 - τ 2 1 - τ 2 h_{\tau}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{1-\tau^{2}}}
  56. h ϕ = a ( σ 2 - 1 ) ( 1 - τ 2 ) h_{\phi}=a\sqrt{\left(\sigma^{2}-1\right)\left(1-\tau^{2}\right)}
  57. d V = a 3 ( σ 2 - τ 2 ) d σ d τ d ϕ dV=a^{3}\left(\sigma^{2}-\tau^{2}\right)d\sigma d\tau d\phi
  58. 2 Φ = 1 a 2 ( σ 2 - τ 2 ) { σ [ ( σ 2 - 1 ) Φ σ ] + τ [ ( 1 - τ 2 ) Φ τ ] } + 1 a 2 ( σ 2 - 1 ) ( 1 - τ 2 ) 2 Φ ϕ 2 \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sigma^{2}-\tau^{2}\right)}\left\{\frac{% \partial}{\partial\sigma}\left[\left(\sigma^{2}-1\right)\frac{\partial\Phi}{% \partial\sigma}\right]+\frac{\partial}{\partial\tau}\left[\left(1-\tau^{2}% \right)\frac{\partial\Phi}{\partial\tau}\right]\right\}+\frac{1}{a^{2}\left(% \sigma^{2}-1\right)\left(1-\tau^{2}\right)}\frac{\partial^{2}\Phi}{\partial% \phi^{2}}
  59. 𝐅 \nabla\cdot\mathbf{F}
  60. × 𝐅 \nabla\times\mathbf{F}
  61. ( σ , τ ) (\sigma,\tau)

Proof_of_impossibility.html

  1. π π
  2. 3 , 5 , , \sqrt{3},\sqrt{5},...,
  3. x n + y n = z n x^{n}+y^{n}=z^{n}
  4. n > 2 n>2
  5. 0 \aleph_{0}

Proportional_reasoning.html

  1. a b = c d \frac{a}{b}=\frac{c}{d}
  2. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  3. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}

Propositional_directed_acyclic_graph.html

  1. \top
  2. \bot
  3. \bigtriangleup
  4. \bigtriangledown
  5. \Diamond
  6. \bigtriangleup
  7. \bigtriangledown
  8. \Diamond
  9. \top
  10. \bot
  11. x x
  12. x = 1 x=1
  13. x = 1 x=1
  14. \bigtriangleup
  15. \bigtriangledown
  16. \Diamond
  17. f ( x 1 , x 2 , x 3 ) = - x 1 * - x 2 * - x 3 + x 1 * x 2 + x 2 * x 3 f(x1,x2,x3)=-x1*-x2*-x3+x1*x2+x2*x3

Propulsive_efficiency.html

  1. η \eta
  2. η = η c η p \eta=\eta_{c}\eta_{p}
  3. η c \eta_{c}
  4. η p \eta_{p}
  5. d W = d Q c - ( - d Q h ) dW\ =\ dQ_{c}\ -\ (-dQ_{h})
  6. d W = - P d V dW=-PdV
  7. d Q h = T h d S h dQ_{h}=T_{h}dS_{h}
  8. ( - d Q h ) (-dQ_{h})
  9. d Q c = T c d S c dQ_{c}=T_{c}dS_{c}
  10. η c = - d W - d Q h = - d Q h - d Q c - d Q h = 1 - d Q c - d Q h \eta_{c}=\frac{-dW}{-dQ_{h}}=\frac{-dQ_{h}-dQ_{c}}{-dQ_{h}}=1-\frac{dQ_{c}}{-% dQ_{h}}
  11. d S c = - d S h dS_{c}=-dS_{h}
  12. η cmax = 1 - T c d S c - T h d S h = 1 - T c T h \eta\text{cmax}=1-\frac{T_{c}dS_{c}}{-T_{h}dS_{h}}=1-\frac{T_{c}}{T_{h}}
  13. T h T_{h}
  14. T c T_{c}
  15. d S c dS_{c}
  16. d S h dS_{h}
  17. η p = 2 1 + c v \eta_{p}=\frac{2}{1+\frac{c}{v}}
  18. η p \eta_{p}
  19. η c \eta_{c}
  20. η p \eta_{p}
  21. η p = 2 v c 1 + ( v c ) 2 \eta_{p}=\frac{2\frac{v}{c}}{1+(\frac{v}{c})^{2}}
  22. 550 P e = η c H h J 3600 , 550P_{e}=\frac{\eta_{c}HhJ}{3600},
  23. P e P_{e}
  24. η c = 14 C p . \eta_{c}=\frac{14}{C_{p}}.
  25. η p \eta_{p}
  26. η = 12 C p . \eta=\frac{12}{C_{p}}.

Protein_contact_map.html

  1. i i
  2. j j
  3. i j ij

Protein_pKa_calculations.html

  1. p K a HH ( pH ) = pH - Δ G prot ( pH ) RT ln 10 ] \mathrm{p}K_{\mathrm{a}}^{\mathrm{HH}}(\mathrm{pH})=\mathrm{pH}-\frac{\Delta G% ^{\mathrm{prot}}(\mathrm{pH})}{\mathrm{RT}\ln 10}]
  2. Δ G prot ( pH ) = RT ln 10 ( pH - p K a HH ) \Delta G^{\mathrm{prot}}(\mathrm{pH})=\mathrm{RT}\ln 10(\mathrm{pH}-\mathrm{p}% K_{\mathrm{a}}^{\mathrm{HH}})
  3. Δ G prot ( pH ) = - RT ln [ < x > 1 - < x > ] \Delta G^{\mathrm{prot}}(\mathrm{pH})=-\mathrm{RT}\ln\left[\frac{<x>}{1-<x>}\right]
  4. < < v a r > x < / v a r > Align g t ; <<var>x</var>&gt;

Proth's_theorem.html

  1. a ( p - 1 ) / 2 - 1 mod p , a^{(p-1)/2}\equiv-1\mod{p},
  2. ( a p ) = - 1 \left(\frac{a}{p}\right)=-1

Proth_number.html

  1. k 2 n + 1 k\cdot 2^{n}+1
  2. k k
  3. n n
  4. 2 n > k 2^{n}>k
  5. p p
  6. a a
  7. a p - 1 2 - 1 ( mod p ) a^{\frac{p-1}{2}}\equiv-1\ \;\;(\mathop{{\rm mod}}p)
  8. 19249 2 13018586 + 1 19249\cdot 2^{13018586}+1

Proton_nuclear_magnetic_resonance.html

  1. B 0 B_{0}
  2. B 0 B_{0}

Prüfer_group.html

  1. 𝐙 ( p ) = { exp ( 2 π i m / p n ) m 𝐙 + , n 𝐙 + } . \mathbf{Z}(p^{\infty})=\{\exp(2\pi im/p^{n})\mid m\in\mathbf{Z}^{+},\,n\in% \mathbf{Z}^{+}\}.\;
  2. 𝐙 ( p ) = 𝐙 [ 1 / p ] / 𝐙 \mathbf{Z}(p^{\infty})=\mathbf{Z}[1/p]/\mathbf{Z}
  3. 𝐙 ( p ) = 𝐐 p / 𝐙 p \mathbf{Z}(p^{\infty})=\mathbf{Q}_{p}/\mathbf{Z}_{p}
  4. 𝐙 ( p ) = g 1 , g 2 , g 3 , g 1 p = 1 , g 2 p = g 1 , g 3 p = g 2 , . \mathbf{Z}(p^{\infty})=\langle\,g_{1},g_{2},g_{3},\ldots\mid g_{1}^{p}=1,g_{2}% ^{p}=g_{1},g_{3}^{p}=g_{2},\dots\,\rangle.
  5. 0 ( 1 p 𝐙 ) / 𝐙 ( 1 p 2 𝐙 ) / 𝐙 ( 1 p 3 𝐙 ) / 𝐙 𝐙 ( p ) 0\subset\left({1\over p}\mathbf{Z}\right)/\mathbf{Z}\subset\left({1\over p^{2}% }\mathbf{Z}\right)/\mathbf{Z}\subset\left({1\over p^{3}}\mathbf{Z}\right)/% \mathbf{Z}\subset\cdots\subset\mathbf{Z}(p^{\infty})
  6. ( 1 p n 𝐙 ) / 𝐙 \left({1\over p^{n}}\mathbf{Z}\right)/\mathbf{Z}

Pseudo-arc.html

  1. 𝒞 = { C 1 , C 2 , , C n } \mathcal{C}=\{C_{1},C_{2},\ldots,C_{n}\}
  2. C i C j C_{i}\cap C_{j}\neq\emptyset
  3. | i - j | 1. |i-j|\leq 1.
  4. 𝒞 \mathcal{C}
  5. 𝒟 \mathcal{D}
  6. 𝒟 \mathcal{D}
  7. 𝒞 \mathcal{C}
  8. D i C m D_{i}\cap C_{m}\neq\emptyset
  9. D j C n D_{j}\cap C_{n}\neq\emptyset
  10. m < n - 2 m<n-2
  11. k k
  12. \ell
  13. i < k < < j i<k<\ell<j
  14. i > k > > j i>k>\ell>j
  15. D k C n - 1 D_{k}\subseteq C_{n-1}
  16. D C m + 1 . D_{\ell}\subseteq C_{m+1}.
  17. 𝒟 \mathcal{D}
  18. 𝒞 . \mathcal{C}.
  19. C * C^{*}
  20. C * = S C S . C^{*}=\bigcup_{S\in C}S.
  21. { 𝒞 i } i \left\{\mathcal{C}^{i}\right\}_{i\in\mathbb{N}}
  22. 𝒞 i \mathcal{C}^{i}
  23. 𝒞 i \mathcal{C}^{i}
  24. 1 / 2 i 1/2^{i}
  25. 𝒞 i + 1 \mathcal{C}^{i+1}
  26. 𝒞 i \mathcal{C}^{i}
  27. 𝒞 i + 1 \mathcal{C}^{i+1}
  28. 𝒞 i \mathcal{C}^{i}
  29. P = i ( 𝒞 i ) * . P=\bigcap_{i\in\mathbb{N}}\left(\mathcal{C}^{i}\right)^{*}.

Pseudo-ring.html

  1. a ( b + c ) + a 0 = a b + a c a(b+c)+a0=ab+ac
  2. ( b + c ) a + 0 a = b a + c a (b+c)a+0a=ba+ca

Pseudo_algebraically_closed_field.html

  1. K K
  2. V V
  3. K K
  4. K K
  5. f K [ T 1 , T 2 , , T r , X ] f\in K[T_{1},T_{2},\cdots,T_{r},X]
  6. f X 0 \frac{\partial f}{\partial X}\not=0
  7. g K [ T 1 , T 2 , , T r ] g\in K[T_{1},T_{2},\cdots,T_{r}]
  8. ( 𝐚 , b ) K r + 1 (\,\textbf{a},b)\in K^{r+1}
  9. f ( 𝐚 , b ) = 0 f(\,\textbf{a},b)=0
  10. g ( 𝐚 ) 0 g(\,\textbf{a})\not=0
  11. f K [ T , X ] f\in K[T,X]
  12. K K
  13. R R
  14. K K
  15. K K
  16. h : R K h:R\to K
  17. h ( a ) = a h(a)=a
  18. a K a\in K
  19. G G
  20. K K
  21. K K
  22. e e
  23. e e
  24. ( σ 1 , , σ e ) G e (\sigma_{1},...,\sigma_{e})\in G^{e}

Pseudocircle.html

  1. { { a , b , c , d } , { a , b , c } , { a , b , d } , { a , b } , { a } , { b } , } \left\{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\emptyset\right\}
  2. a < c , b < c , a < d , b < d a<c,b<c,a<d,b<d
  3. f ( x , y ) = { a x < 0 b x > 0 c ( x , y ) = ( 0 , 1 ) d ( x , y ) = ( 0 , - 1 ) f(x,y)=\begin{cases}a\quad x<0\\ b\quad x>0\\ c\quad(x,y)=(0,1)\\ d\quad(x,y)=(0,-1)\end{cases}

Pseudoconvex_function.html

  1. f ( x ) ( y - x ) 0 \nabla f(x)\cdot(y-x)\geq 0
  2. f ( y ) f ( x ) f(y)\geq f(x)
  3. f = ( f x 1 , , f x n ) . \nabla f=\left(\frac{\partial f}{\partial x_{1}},\dots,\frac{\partial f}{% \partial x_{n}}\right).
  4. f ( x * ) = 0. \nabla f(x^{*})=0.
  5. f + ( x , u ) = lim sup h 0 + f ( x + h u ) - f ( x ) h f^{+}(x,u)=\limsup_{h\to 0^{+}}\frac{f(x+hu)-f(x)}{h}
  6. x * , y - x 0 , \langle x^{*},y-x\rangle\geq 0\,,

Pseudoconvexity.html

  1. G n G\subset{\mathbb{C}}^{n}
  2. G G
  3. φ \varphi
  4. G G
  5. { z G φ ( z ) < x } \{z\in G\mid\varphi(z)<x\}
  6. G G
  7. x . x.
  8. G G
  9. G G
  10. C 2 C^{2}
  11. C 2 C^{2}
  12. G G
  13. ρ : n \rho:\mathbb{C}^{n}\to\mathbb{R}
  14. C 2 C^{2}
  15. G = { ρ < 0 } G=\{\rho<0\}
  16. G = { ρ = 0 } \partial G=\{\rho=0\}
  17. G G
  18. p G p\in\partial G
  19. w w
  20. ρ ( p ) w = i = 1 n ρ ( p ) z j w j = 0 \nabla\rho(p)w=\sum_{i=1}^{n}\frac{\partial\rho(p)}{\partial z_{j}}w_{j}=0
  21. i , j = 1 n 2 ρ ( p ) z i z j ¯ w i w j ¯ 0. \sum_{i,j=1}^{n}\frac{\partial^{2}\rho(p)}{\partial z_{i}\partial\bar{z_{j}}}w% _{i}\bar{w_{j}}\geq 0.
  22. G G
  23. C 2 C^{2}
  24. G G
  25. G k G G_{k}\subset G
  26. C C^{\infty}
  27. G G
  28. G = k = 1 G k . G=\bigcup_{k=1}^{\infty}G_{k}.
  29. φ \varphi

Pseudolikelihood.html

  1. X = X 1 , X 2 , , X n X=X_{1},X_{2},...,X_{n}
  2. E E
  3. { X i , X j } E \{X_{i},X_{j}\}\notin E
  4. X i X_{i}
  5. X j X_{j}
  6. X i X_{i}
  7. X = x = ( x 1 , x 2 , x n ) X=x=(x_{1},x_{2},...x_{n})
  8. Pr ( X = x ) = i Pr ( X i = x i | X j = x j for all j for which { X i , X j } E ) . \Pr(X=x)=\prod_{i}\Pr(X_{i}=x_{i}|X_{j}=x_{j}\ \mathrm{for\ all\ }j\ \mathrm{% for\ which}\ \{X_{i},X_{j}\}\in E).
  9. X X
  10. x x
  11. X = x X=x
  12. X i X_{i}
  13. X X
  14. x i x_{i}
  15. x x
  16. Pr ( X = x ) \Pr(X=x)
  17. X X
  18. x x
  19. Pr ( X = x ) \Pr(X=x)
  20. log Pr ( X = x ) = i log Pr ( X i = x i | X j = x j for all { X i , X j } E ) . \log\Pr(X=x)=\sum_{i}\log\Pr(X_{i}=x_{i}|X_{j}=x_{j}\ \mathrm{for\ all}\ \{X_{% i},X_{j}\}\in E).
  21. X i X_{i}

Pseudospectrum.html

  1. Λ ϵ ( A ) = { λ x n { 0 } , E n × n : ( A + E ) x = λ x , E ϵ } . \Lambda_{\epsilon}(A)=\{\lambda\in\mathbb{C}\mid\exists x\in\mathbb{C}^{n}% \setminus\{0\},\exists E\in\mathbb{C}^{n\times n}\colon(A+E)x=\lambda x,\|E\|% \leq\epsilon\}.

Pulsar_kick.html

  1. 10 15 10^{15}
  2. 10 16 10^{16}

Purely_inseparable_extension.html

  1. E F E\supseteq F
  2. α E F \alpha\in E\setminus F
  3. α \alpha
  4. F F F\supseteq F
  5. E F E\supseteq F
  6. α E \alpha\in E
  7. n 0 n\geq 0
  8. α p n F \alpha^{p^{n}}\in F
  9. X p n - a X^{p^{n}}-a
  10. n 0 n\geq 0
  11. a F a\in F
  12. E = F [ α ] E=F[\alpha]
  13. α p n F \alpha^{p^{n}}\in F
  14. n 0 n\geq 0
  15. x p n F x^{p^{n}}\in F
  16. n 0 n\geq 0
  17. α \alpha
  18. E F E\supseteq F
  19. a F a\in F
  20. α \alpha
  21. f ( α ) = 0 f(\alpha)=0
  22. α p = a \alpha^{p}=a
  23. F [ α ] F F[\alpha]\supseteq F
  24. E = F [ α ] E=F[\alpha]
  25. E F E\supseteq F
  26. E F E\supseteq F
  27. F K E F\subseteq K\subseteq E
  28. F K E F\subseteq K\subseteq E
  29. F K F\subseteq K
  30. K E K\subseteq E
  31. E F E\supseteq F
  32. α E F \alpha\in E\setminus F
  33. α \alpha
  34. E F E\supseteq F
  35. E F E\supseteq F
  36. K = Fix ( Gal ( E / F ) ) K=\mbox{Fix}~{}(\mbox{Gal}~{}(E/F))

Purification_theorem.html

  1. P r ( a i a * ) = 1 2 + 3 / A + A 2 A = A 4 A 2 + 6 A + 1 2 . \textstyle Pr(a_{i}\leq a^{*})=\frac{\frac{1}{2+3/A}+A}{2A}=\frac{A}{4A^{2}+6A% }+\frac{1}{2}.

Purkinje_images.html

  1. Intensity (brightness) = ( n - n ) 2 / ( n + n ) 2 , \,\text{Intensity (brightness)}=(n^{\prime}-n)^{2}/(n^{\prime}+n)^{2},
  2. n n
  3. n n^{\prime}

Push–relabel_maximum_flow_algorithm.html

  1. G = ( V , E ) G=(V,E)
  2. s s
  3. t t
  4. c ( u , v ) 0 c(u,v)\geq 0
  5. ( u , v ) E (u,v)\in E
  6. ( u , v ) E (u,v)\notin E
  7. c ( u , v ) = 0 c(u,v)=0
  8. G G
  9. f : V × V \ f:V\times V\rightarrow\mathbb{R}
  10. f ( u , v ) c ( u , v ) , u , v V \ f(u,v)\leq c(u,v),\quad\forall u,v\in V
  11. f ( u , v ) = - f ( v , u ) , u , v V \ f(u,v)=-f(v,u),\quad\forall u,v\in V
  12. v V f ( u , v ) = 0 , u V - { s , t } \ \sum_{v\in V}f(u,v)=0,\quad\forall u\in V-\{s,t\}
  13. u V f ( u , v ) 0 , v V - { s } \ \sum\limits_{u\in V}f(u,v)\geq 0,\quad\forall v\in V-\{s\}
  14. e ( v ) = { u V f ( u , v ) , v V - { s } , v = s e(v)=\begin{cases}\sum\limits_{u\in V}f(u,v),\quad\forall v\in V-\{s\}\\ \infty,\quad v=s\end{cases}
  15. v v
  16. e ( v ) > 0 e(v)>0
  17. v V - { s , t } v\in V-\{s,t\}
  18. ( u , v ) V × V (u,v)\in V\times V
  19. c f ( u , v ) = c ( u , v ) - f ( u , v ) c_{f}(u,v)=c(u,v)-f(u,v)
  20. G G
  21. f f
  22. G f ( V , E f ) G_{f}(V,E_{f})
  23. E f = { ( u , v ) | u , v V and c f ( u , v ) > 0 } E_{f}=\{(u,v)|u,v\in V\and c_{f}(u,v)>0\}
  24. G f G_{f}
  25. e ( t ) e(t)
  26. T T
  27. G f G_{f}
  28. S = V \ T S=V\backslash T
  29. ( S , T ) (S,T)
  30. h ( v ) , v V h(v),v\in V
  31. h ( u ) h ( v ) + 1 , ( u , v ) E f h(u)\leq h(v)+1,\quad\forall(u,v)\in E_{f}
  32. h ( s ) = | V | h(s)=|V|
  33. h ( t ) = 0 h(t)=0
  34. h ( u ) h(u)
  35. G f G_{f}
  36. h ( u ) - | V | h(u)-|V|
  37. G f G_{f}
  38. | V | - 1 |V|-1
  39. ( u , v ) E f (u,v)\in E_{f}
  40. h ( u ) = h ( v ) + 1 h(u)=h(v)+1
  41. G ~ f ( V , E ~ f ) \tilde{G}_{f}(V,\tilde{E}_{f})
  42. E ~ f = { ( u , v ) | ( u , v ) E f and h ( u ) = h ( v ) + 1 } \tilde{E}_{f}=\{(u,v)|(u,v)\in E_{f}\and h(u)=h(v)+1\}
  43. min { e ( u ) , c f ( u , v ) } \min\{e(u),c_{f}(u,v)\}

Pyramid_(geometry).html

  1. V = 1 3 b h \scriptstyle{V=}\tfrac{1}{3}\scriptstyle{bh}
  2. 1 - y h \scriptstyle{1-}\tfrac{y}{h}
  3. h - y h \scriptstyle\tfrac{h-y}{h}
  4. ( h - y ) 2 h 2 \tfrac{(h-y)^{2}}{h^{2}}
  5. b h 2 \tfrac{b}{h^{2}}
  6. ( h - y ) 2 \scriptstyle(h-y)^{2}
  7. b h 2 0 h ( h - y ) 2 d y = - b 3 h 2 ( h - y ) 3 | 0 h = 1 3 b h . \frac{b}{h^{2}}\int_{0}^{h}(h-y)^{2}\,dy=\frac{-b}{3h^{2}}(h-y)^{3}\bigg|_{0}^% {h}=\tfrac{1}{3}bh.
  8. V = 1 3 b h \scriptstyle{V=}\tfrac{1}{3}\scriptstyle{bh}
  9. V = n 12 h s 2 cot π n . V=\frac{n}{12}hs^{2}\cot\frac{\pi}{n}.
  10. A = B + P L 2 \scriptstyle{A=B+}\tfrac{PL}{2}
  11. L = h 2 + r 2 \scriptstyle{L=}\sqrt{\scriptstyle{h^{2}+r^{2}}}

Pyramidal_alkene.html

  1. ϕ \phi
  2. c o s ϕ = - c o s ( R - C - C ) / [ c o s ( 1 / 2 ( R - C - R ) ) ] \ cos\phi\ =-cos(R-C-C)/[cos(1/2(R-C-R))]
  3. ψ \psi

Pythagorean_means.html

  1. A M ( x 1 , , x n ) = 1 n ( x 1 + + x n ) AM(x_{1},\ldots,x_{n})=\frac{1}{n}(x_{1}+\cdots+x_{n})
  2. G M ( x 1 , , x n ) = x 1 x n n GM(x_{1},\ldots,x_{n})=\sqrt[n]{x_{1}\cdots x_{n}}
  3. H M ( x 1 , , x n ) = n 1 x 1 + + 1 x n HM(x_{1},\ldots,x_{n})=\frac{n}{\frac{1}{x_{1}}+\cdots+\frac{1}{x_{n}}}
  4. M ( x , x , , x ) = x M(x,x,\ldots,x)=x
  5. M ( b x 1 , , b x n ) = b M ( x 1 , , x n ) M(bx_{1},\ldots,bx_{n})=bM(x_{1},\ldots,x_{n})
  6. M ( , x i , , x j , ) = M ( , x j , , x i , ) M(\ldots,x_{i},\ldots,x_{j},\ldots)=M(\ldots,x_{j},\ldots,x_{i},\ldots)
  7. i i
  8. j j
  9. min ( x 1 , , x n ) M ( x 1 , , x n ) max ( x 1 , , x n ) \min(x_{1},\ldots,x_{n})\leq M(x_{1},\ldots,x_{n})\leq\max(x_{1},\ldots,x_{n})
  10. H M ( 1 / x 1 1 / x n ) = 1 / A M ( x 1 x n ) HM(1/x_{1}\ldots 1/x_{n})=1/AM(x_{1}\ldots x_{n})
  11. x i x_{i}
  12. min H M G M A M max \min\leq HM\leq GM\leq AM\leq\max
  13. x i x_{i}
  14. A M m a x AM\leq max
  15. min \min
  16. max \max

P²-irreducible.html

  1. P 2 \mathbb{R}P^{2}

Q-derivative.html

  1. ( d d x ) q f ( x ) = f ( q x ) - f ( x ) q x - x . \left(\frac{d}{dx}\right)_{q}f(x)=\frac{f(qx)-f(x)}{qx-x}.
  2. D q f ( x ) D_{q}f(x)
  3. D q = 1 x q d d ( ln x ) - 1 q - 1 , D_{q}=\frac{1}{x}~{}\frac{q^{d~{}~{}~{}\over d(\ln x)}-1}{q-1}~{},
  4. D q ( f ( x ) + g ( x ) ) = D q f ( x ) + D q g ( x ) . \displaystyle D_{q}(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)~{}.
  5. D q ( f ( x ) g ( x ) ) = g ( x ) D q f ( x ) + f ( q x ) D q g ( x ) = g ( q x ) D q f ( x ) + f ( x ) D q g ( x ) . \displaystyle D_{q}(f(x)g(x))=g(x)D_{q}f(x)+f(qx)D_{q}g(x)=g(qx)D_{q}f(x)+f(x)% D_{q}g(x).
  6. D q ( f ( x ) / g ( x ) ) = g ( x ) D q f ( x ) - f ( x ) D q g ( x ) g ( q x ) g ( x ) , g ( x ) g ( q x ) 0. \displaystyle D_{q}(f(x)/g(x))=\frac{g(x)D_{q}f(x)-f(x)D_{q}g(x)}{g(qx)g(x)},% \quad g(x)g(qx)\neq 0.
  7. g ( x ) = c x k g(x)=cx^{k}
  8. D q f ( g ( x ) ) = D q k ( f ) ( g ( x ) ) D q ( g ) ( x ) . \displaystyle D_{q}f(g(x))=D_{q^{k}}(f)(g(x))D_{q}(g)(x).
  9. ( d d z ) q z n = 1 - q n 1 - q z n - 1 = [ n ] q z n - 1 \left(\frac{d}{dz}\right)_{q}z^{n}=\frac{1-q^{n}}{1-q}z^{n-1}=[n]_{q}z^{n-1}
  10. [ n ] q [n]_{q}
  11. lim q 1 [ n ] q = n \lim_{q\to 1}[n]_{q}=n
  12. ( D q n f ) ( 0 ) = f ( n ) ( 0 ) n ! ( q ; q ) n ( 1 - q ) n = f ( n ) ( 0 ) n ! [ n ] q ! (D^{n}_{q}f)(0)=\frac{f^{(n)}(0)}{n!}\frac{(q;q)_{n}}{(1-q)^{n}}=\frac{f^{(n)}% (0)}{n!}[n]_{q}!
  13. ( q ; q ) n (q;q)_{n}
  14. [ n ] q ! [n]_{q}!
  15. f ( x ) f(x)
  16. D q ( f ( x ) ) D_{q}(f(x))
  17. D q ( f ( x ) ) = k = 0 ( q - 1 ) k ( k + 1 ) ! x k f ( k + 1 ) ( x ) . \displaystyle D_{q}(f(x))=\sum_{k=0}^{\infty}\frac{(q-1)^{k}}{(k+1)!}x^{k}f^{(% k+1)}(x).
  18. f ( z ) = n = 0 f ( n ) ( 0 ) z n n ! = n = 0 ( D q n f ) ( 0 ) z n [ n ] q ! f(z)=\sum_{n=0}^{\infty}f^{(n)}(0)\,\frac{z^{n}}{n!}=\sum_{n=0}^{\infty}(D^{n}% _{q}f)(0)\,\frac{z^{n}}{[n]_{q}!}

Q-difference_polynomial.html

  1. ( d d z ) q p n ( z ) = p n ( q z ) - p n ( z ) q z - z = q n - 1 q - 1 p n - 1 ( z ) = [ n ] q p n - 1 ( z ) \left(\frac{d}{dz}\right)_{q}p_{n}(z)=\frac{p_{n}(qz)-p_{n}(z)}{qz-z}=\frac{q^% {n}-1}{q-1}p_{n-1}(z)=[n]_{q}p_{n-1}(z)
  2. q 1 q\to 1
  3. d d z p n ( z ) = n p n - 1 ( z ) . \frac{d}{dz}p_{n}(z)=np_{n-1}(z).
  4. A ( w ) e q ( z w ) = n = 0 p n ( z ) [ n ] q ! w n A(w)e_{q}(zw)=\sum_{n=0}^{\infty}\frac{p_{n}(z)}{[n]_{q}!}w^{n}
  5. e q ( t ) e_{q}(t)
  6. e q ( t ) = n = 0 t n [ n ] q ! = n = 0 t n ( 1 - q ) n ( q ; q ) n . e_{q}(t)=\sum_{n=0}^{\infty}\frac{t^{n}}{[n]_{q}!}=\sum_{n=0}^{\infty}\frac{t^% {n}(1-q)^{n}}{(q;q)_{n}}.
  7. [ n ] q ! [n]_{q}!
  8. ( q ; q ) n = ( 1 - q n ) ( 1 - q n - 1 ) ( 1 - q ) (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots(1-q)
  9. A ( w ) A(w)
  10. A ( w ) = n = 0 a n w n with a 0 0. A(w)=\sum_{n=0}^{\infty}a_{n}w^{n}\mbox{ with }~{}a_{0}\neq 0.
  11. A ( w ) A(w)

Q-exponential.html

  1. e q ( z ) e_{q}(z)
  2. e q ( z ) = n = 0 z n [ n ] q ! = n = 0 z n ( 1 - q ) n ( q ; q ) n = n = 0 z n ( 1 - q ) n ( 1 - q n ) ( 1 - q n - 1 ) ( 1 - q ) e_{q}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{[n]_{q}!}=\sum_{n=0}^{\infty}\frac{z^% {n}(1-q)^{n}}{(q;q)_{n}}=\sum_{n=0}^{\infty}z^{n}\frac{(1-q)^{n}}{(1-q^{n})(1-% q^{n-1})\cdots(1-q)}
  3. [ n ] q ! [n]_{q}!
  4. ( q ; q ) n = ( 1 - q n ) ( 1 - q n - 1 ) ( 1 - q ) (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots(1-q)
  5. ( d d z ) q e q ( z ) = e q ( z ) \left(\frac{d}{dz}\right)_{q}e_{q}(z)=e_{q}(z)
  6. ( d d z ) q z n = z n - 1 1 - q n 1 - q = [ n ] q z n - 1 . \left(\frac{d}{dz}\right)_{q}z^{n}=z^{n-1}\frac{1-q^{n}}{1-q}=[n]_{q}z^{n-1}.
  7. [ n ] q [n]_{q}
  8. q > 1 q>1
  9. e q ( z ) e_{q}(z)
  10. q < 1 q<1
  11. e q ( z ) e_{q}(z)
  12. | z | < 1 / ( 1 - q ) |z|<1/(1-q)
  13. e q ( z ) e 1 / q ( - z ) = 1 ~{}e_{q}(z)~{}e_{1/q}(-z)=1
  14. q < 1 q<1
  15. e q ( z ) = E q ( z ( 1 - q ) ) . e_{q}(z)=E_{q}(z(1-q)).
  16. E q ( t ) E_{q}(t)
  17. E q ( z ) = 1 ϕ 0 ( 0 ; q , z ) = n = 0 1 1 - q n z . E_{q}(z)=\;_{1}\phi_{0}(0;q,z)=\prod_{n=0}^{\infty}\frac{1}{1-q^{n}z}~{}.

Q10_(temperature_coefficient).html

  1. Q 10 = ( R 2 R 1 ) 10 / ( T 2 - T 1 ) Q_{10}=\left(\frac{R_{2}}{R_{1}}\right)^{10/(T_{2}-T_{1})}

Q_(number_format).html

  1. [ - ( 2 m ) , 2 m - 2 - n ] [-(2^{m}),2^{m}-2^{-n}]
  2. 2 - n 2^{-n}
  3. [ 0 , 2 m - 2 - n ] [0,2^{m}-2^{-n}]
  4. 2 - n 2^{-n}
  5. N 1 N_{1}
  6. N 2 N_{2}
  7. N 1 d + N 2 d = N 1 + N 2 d N 1 d - N 2 d = N 1 - N 2 d ( N 1 d × N 2 d ) × d = N 1 × N 2 d ( N 1 d / N 2 d ) / d = N 1 / N 2 d \begin{aligned}\displaystyle\frac{N_{1}}{d}+\frac{N_{2}}{d}&\displaystyle=% \frac{N_{1}+N_{2}}{d}\\ \displaystyle\frac{N_{1}}{d}-\frac{N_{2}}{d}&\displaystyle=\frac{N_{1}-N_{2}}{% d}\\ \displaystyle\left(\frac{N_{1}}{d}\times\frac{N_{2}}{d}\right)\times d&% \displaystyle=\frac{N_{1}\times N_{2}}{d}\\ \displaystyle\left(\frac{N_{1}}{d}/\frac{N_{2}}{d}\right)/d&\displaystyle=% \frac{N_{1}/N_{2}}{d}\end{aligned}

Quadratic_form_(statistics).html

  1. ϵ \epsilon
  2. n n
  3. Λ \Lambda
  4. n n
  5. ϵ T Λ ϵ \epsilon^{T}\Lambda\epsilon
  6. ϵ \epsilon
  7. E [ ϵ T Λ ϵ ] = tr [ Λ Σ ] + μ T Λ μ \operatorname{E}\left[\epsilon^{T}\Lambda\epsilon\right]=\operatorname{tr}% \left[\Lambda\Sigma\right]+\mu^{T}\Lambda\mu
  8. μ \mu
  9. Σ \Sigma
  10. ϵ \epsilon
  11. μ \mu
  12. Σ \Sigma
  13. ϵ \epsilon
  14. E [ ϵ T Λ ϵ ] = tr ( E [ ϵ T Λ ϵ ] ) \operatorname{E}\left[\epsilon^{T}\Lambda\epsilon\right]=\operatorname{tr}(% \operatorname{E}[\epsilon^{T}\Lambda\epsilon])
  15. E \operatorname{E}
  16. tr \operatorname{tr}
  17. E tr = tr E \operatorname{E}\circ\operatorname{tr}=\operatorname{tr}\circ\operatorname{E}
  18. tr ( E [ ϵ T Λ ϵ ] ) = E [ tr ( ϵ T Λ ϵ ) ] , \operatorname{tr}(\operatorname{E}\left[\epsilon^{T}\Lambda\epsilon\right])=% \operatorname{E}[\operatorname{tr}(\epsilon^{T}\Lambda\epsilon)],
  19. E [ tr ( ϵ T Λ ϵ ) ] = E [ tr ( Λ ϵ ϵ T ) ] = tr ( Λ ( Σ + μ μ T ) ) = tr ( Λ Σ ) + μ T Λ μ . \operatorname{E}[\operatorname{tr}(\epsilon^{T}\Lambda\epsilon)]=\operatorname% {E}[\operatorname{tr}(\Lambda\epsilon\epsilon^{T})]=\operatorname{tr}(\Lambda(% \Sigma+\mu\mu^{T}))=\operatorname{tr}(\Lambda\Sigma)+\mu^{T}\Lambda\mu.
  20. ϵ \epsilon
  21. ϵ \epsilon
  22. Λ \Lambda
  23. var [ ϵ T Λ ϵ ] = 2 tr [ Λ Σ Λ Σ ] + 4 μ T Λ Σ Λ μ \operatorname{var}\left[\epsilon^{T}\Lambda\epsilon\right]=2\operatorname{tr}% \left[\Lambda\Sigma\Lambda\Sigma\right]+4\mu^{T}\Lambda\Sigma\Lambda\mu
  24. ϵ \epsilon
  25. Λ 1 \Lambda_{1}
  26. Λ 2 \Lambda_{2}
  27. cov [ ϵ T Λ 1 ϵ , ϵ T Λ 2 ϵ ] = 2 tr [ Λ 1 Σ Λ 2 Σ ] + 4 μ T Λ 1 Σ Λ 2 μ \operatorname{cov}\left[\epsilon^{T}\Lambda_{1}\epsilon,\epsilon^{T}\Lambda_{2% }\epsilon\right]=2\operatorname{tr}\left[\Lambda_{1}\Sigma\Lambda_{2}\Sigma% \right]+4\mu^{T}\Lambda_{1}\Sigma\Lambda_{2}\mu
  28. Λ \Lambda
  29. Λ \Lambda
  30. ϵ T Λ T ϵ = ϵ T Λ ϵ \epsilon^{T}\Lambda^{T}\epsilon=\epsilon^{T}\Lambda\epsilon
  31. ϵ T Λ ~ ϵ = ϵ T ( Λ + Λ T ) ϵ / 2 \epsilon^{T}\tilde{\Lambda}\epsilon=\epsilon^{T}\left(\Lambda+\Lambda^{T}% \right)\epsilon/2
  32. Λ ~ = ( Λ + Λ T ) / 2 \tilde{\Lambda}=\left(\Lambda+\Lambda^{T}\right)/2
  33. Λ \Lambda
  34. Λ ~ \tilde{\Lambda}
  35. y y
  36. H H
  37. y y
  38. RSS = y T ( I - H ) T ( I - H ) y . \textrm{RSS}=y^{T}\left(I-H\right)^{T}\left(I-H\right)y.
  39. H H
  40. σ 2 I \sigma^{2}I
  41. RSS / σ 2 \textrm{RSS}/\sigma^{2}
  42. k k
  43. λ \lambda
  44. k = tr [ ( I - H ) T ( I - H ) ] k=\operatorname{tr}\left[\left(I-H\right)^{T}\left(I-H\right)\right]
  45. λ = μ T ( I - H ) T ( I - H ) μ / 2 \lambda=\mu^{T}\left(I-H\right)^{T}\left(I-H\right)\mu/2
  46. H y Hy
  47. μ \mu
  48. λ \lambda
  49. RSS / σ 2 \textrm{RSS}/\sigma^{2}

Quadrature_filter.html

  1. q ( t ) q(t)
  2. f ( t ) f(t)
  3. q ( t ) = f a ( t ) = ( δ ( t ) + i π t ) * f ( t ) q(t)=f_{a}(t)=\left(\delta(t)+{i\over\pi t}\right)*f(t)
  4. q ( t ) q(t)
  5. s ( t ) s(t)
  6. h ( t ) = ( q * s ) ( t ) = ( δ ( t ) + i π t ) * f ( t ) * s ( t ) h(t)=(q*s)(t)=\left(\delta(t)+{i\over\pi t}\right)*f(t)*s(t)
  7. h ( t ) h(t)
  8. ( f * s ) ( t ) (f*s)(t)
  9. q q
  10. q q
  11. f ( t ) f(t)
  12. 1 π t {1\over\pi t}
  13. s ( t ) s(t)
  14. q ( t ) q(t)
  15. q ( t ) q(t)
  16. h ( t ) h(t)
  17. f * s f*s
  18. s s
  19. f f
  20. f f
  21. f ( t ) f(t)
  22. ω \omega
  23. ω \omega
  24. h ( t ) = ( s * q ) ( t ) = 1 2 π - S ( u ) Q ( u ) e i u t d u = 1 2 π - A π [ δ ( u + ω ) + δ ( u - ω ) ] Q ( u ) e i u t d u = h(t)=(s*q)(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}S(u)Q(u)e^{iut}du=\frac{1}{% 2\pi}\int_{-\infty}^{\infty}A\pi[\delta(u+\omega)+\delta(u-\omega)]Q(u)e^{iut}du=
  25. = A 2 0 δ ( u - ω ) Q ( u ) e i u t d u = A 2 Q ( ω ) e i ω t =\frac{A}{2}\int_{0}^{\infty}\delta(u-\omega)Q(u)e^{iut}du=\frac{A}{2}Q(\omega% )e^{i\omega t}
  26. | h ( t ) | = A 2 | Q ( ω ) | |h(t)|=\frac{A}{2}|Q(\omega)|
  27. ω \omega

Quadric_(projective_geometry).html

  1. K K
  2. 𝒱 ( K ) \mathcal{V}(K)
  3. K K
  4. ρ \rho
  5. 𝒱 ( K ) \mathcal{V}(K)
  6. K K
  7. ρ ( λ x ) = λ 2 ρ ( x ) \rho(\lambda\vec{x})=\lambda^{2}\rho(\vec{x})
  8. λ K \lambda\in K
  9. x 𝒱 ( K ) \vec{x}\in\mathcal{V}(K)
  10. f ( x , y ) := ρ ( x + y ) - ρ ( x ) - ρ ( y ) f(\vec{x},\vec{y}):=\rho(\vec{x}+\vec{y})-\rho(\vec{x})-\rho(\vec{y})
  11. f f
  12. char K 2 \operatorname{char}K\neq 2
  13. f ( x , x ) = 2 ρ ( x ) f(\vec{x},\vec{x})=2\rho(\vec{x})
  14. f f
  15. ρ \rho
  16. char K = 2 \operatorname{char}K=2
  17. f ( x , x ) = 0 f(\vec{x},\vec{x})=0
  18. f f
  19. 𝒱 ( K ) = K n \mathcal{V}(K)=K^{n}
  20. x = i = 1 n x i e i \vec{x}=\sum_{i=1}^{n}x_{i}\vec{e}_{i}
  21. { e 1 , , e n } \{\vec{e}_{1},\ldots,\vec{e}_{n}\}
  22. 𝒱 ( K ) \mathcal{V}(K)
  23. ρ \rho
  24. ρ ( x ) = 1 = i k n a i k x i x k with a i k := f ( e i , e k ) for i k and a i k := ρ ( e i ) for i = k \rho(\vec{x})=\sum_{1=i\leq k}^{n}a_{ik}x_{i}x_{k}\,\text{ with }a_{ik}:=f(% \vec{e}_{i},\vec{e}_{k})\,\text{ for }i\neq k\,\text{ and }a_{ik}:=\rho(\vec{e% }_{i})\,\text{ for }i=k
  25. f ( x , y ) = 1 = i k n a i k ( x i y k + x k y i ) f(\vec{x},\vec{y})=\sum_{1=i\leq k}^{n}a_{ik}(x_{i}y_{k}+x_{k}y_{i})
  26. n = 3 , ρ ( x ) = x 1 x 2 - x 3 2 , f ( x , y ) = x 1 y 2 + x 2 y 1 - 2 x 3 y 3 . n=3,\ \rho(\vec{x})=x_{1}x_{2}-x^{2}_{3},\ f(\vec{x},\vec{y})=x_{1}y_{2}+x_{2}% y_{1}-2x_{3}y_{3}.
  27. K K
  28. 2 n 𝒩 2\leq n\in\mathcal{N}
  29. 𝔓 n ( K ) = ( 𝒫 , 𝒢 , ) \mathfrak{P}_{n}(K)=({\mathcal{P}},{\mathcal{G}},\in)
  30. n n
  31. K K
  32. 𝒫 = { x 0 x V n + 1 ( K ) } , {\mathcal{P}}=\{\langle\vec{x}\rangle\mid\vec{0}\neq\vec{x}\in V_{n+1}(K)\},
  33. V n + 1 ( K ) V_{n+1}(K)
  34. ( n + 1 ) (n+1)
  35. K K
  36. x \langle\vec{x}\rangle
  37. x \vec{x}
  38. 𝒢 = { { x 𝒫 x U } U 2-dimensional subspace of V n + 1 ( K ) } , {\mathcal{G}}=\{\{\langle\vec{x}\rangle\in{\mathcal{P}}\mid\vec{x}\in U\}\mid U% \,\text{ 2-dimensional subspace of }V_{n+1}(K)\},
  39. ρ \rho
  40. V n + 1 ( K ) V_{n+1}(K)
  41. x 𝒫 \langle\vec{x}\rangle\in{\mathcal{P}}
  42. ρ ( x ) = 0 \rho(\vec{x})=0
  43. 𝒬 = { x 𝒫 ρ ( x ) = 0 } \mathcal{Q}=\{\langle\vec{x}\rangle\in{\mathcal{P}}\mid\rho(\vec{x})=0\}
  44. ρ \rho
  45. ρ \rho
  46. P = p 𝒫 P=\langle\vec{p}\rangle\in{\mathcal{P}}
  47. P := { x 𝒫 f ( p , x ) = 0 } P^{\perp}:=\{\langle\vec{x}\rangle\in{\mathcal{P}}\mid f(\vec{p},\vec{x})=0\}
  48. P P
  49. ρ \rho
  50. P P^{\perp}
  51. 𝒫 {\mathcal{P}}
  52. 𝒬 \mathcal{Q}\neq\emptyset
  53. ρ ( x ) = x 1 x 2 - x 3 2 \rho(\vec{x})=x_{1}x_{2}-x^{2}_{3}
  54. 𝔓 2 ( K ) \mathfrak{P}_{2}(K)
  55. 𝒬 \mathcal{Q}
  56. g g
  57. P n ( K ) P_{n}(K)
  58. g 𝒬 = g\cap\mathcal{Q}=\emptyset
  59. g g
  60. g 𝒬 g\subset\mathcal{Q}
  61. g g
  62. | g 𝒬 | = 1 |g\cap\mathcal{Q}|=1
  63. g g
  64. | g 𝒬 | = 2 |g\cap\mathcal{Q}|=2
  65. g g
  66. g g
  67. P 𝒬 P\in\mathcal{Q}
  68. g P g\subset P^{\perp}
  69. := { P 𝒫 P = 𝒫 } \mathcal{R}:=\{P\in{\mathcal{P}}\mid P^{\perp}=\mathcal{P}\}
  70. \mathcal{R}
  71. 𝒬 \mathcal{Q}
  72. 𝒮 := 𝒬 \mathcal{S}:=\mathcal{R}\cap\mathcal{Q}
  73. 𝒮 \mathcal{S}
  74. ρ \rho
  75. 𝒬 \mathcal{Q}
  76. char K 2 \operatorname{char}K\neq 2
  77. = 𝒮 \mathcal{R}=\mathcal{S}
  78. 𝒮 = \mathcal{S}=\emptyset
  79. char K = 2 \operatorname{char}K=2
  80. = 𝒮 \emptyset=\mathcal{S}\neq\mathcal{R}
  81. P 𝒫 ( 𝒬 ) P\in{\mathcal{P}}\setminus(\mathcal{Q}\cup{\mathcal{R}})
  82. σ P \sigma_{P}
  83. P P
  84. σ P ( 𝒬 ) = 𝒬 \sigma_{P}(\mathcal{Q})=\mathcal{Q}
  85. P 𝒫 ( 𝒬 ) P\in{\mathcal{P}}\setminus(\mathcal{Q}\cup{\mathcal{R}})
  86. P P^{\perp}
  87. φ : x x - f ( p , x ) ρ ( p ) p \varphi:\vec{x}\rightarrow\vec{x}-\frac{f(\vec{p},\vec{x})}{\rho(\vec{p})}\vec% {p}
  88. P P^{\perp}
  89. P P
  90. 𝒬 \mathcal{Q}
  91. char K 2 \operatorname{char}K\neq 2
  92. φ \varphi
  93. φ : x x - 2 f ( p , x ) f ( p , p ) p \varphi:\vec{x}\rightarrow\vec{x}-2\frac{f(\vec{p},\vec{x})}{f(\vec{p},\vec{p}% )}\vec{p}
  94. φ ( p ) = - p \varphi(\vec{p})=-\vec{p}
  95. φ ( x ) = x \varphi(\vec{x})=\vec{x}
  96. x P \langle\vec{x}\rangle\in P^{\perp}
  97. σ P \sigma_{P}
  98. {\mathcal{R}}
  99. σ P \sigma_{P}
  100. Π ( 𝒬 ) \Pi(\mathcal{Q})
  101. 𝔓 n ( K ) \mathfrak{P}_{n}(K)
  102. 𝒬 \mathcal{Q}
  103. Π ( 𝒬 ) \Pi(\mathcal{Q})
  104. 𝒬 \mathcal{Q}\setminus{\mathcal{R}}
  105. 𝒰 \mathcal{U}
  106. 𝔓 n ( K ) \mathfrak{P}_{n}(K)
  107. ρ \rho
  108. 𝒰 𝒬 \mathcal{U}\subset\mathcal{Q}
  109. ρ \rho
  110. m m
  111. m m
  112. ρ \rho
  113. 𝒬 \mathcal{Q}
  114. i := m + 1 i:=m+1
  115. 𝒬 \mathcal{Q}
  116. i i
  117. 𝒬 \mathcal{Q}
  118. 𝔓 n ( K ) \mathfrak{P}_{n}(K)
  119. i n + 1 2 i\leq\frac{n+1}{2}
  120. 𝒬 \mathcal{Q}
  121. 𝔓 n ( K ) , n 2 \mathfrak{P}_{n}(K),n\geq 2
  122. i i
  123. i = 1 i=1
  124. 𝒬 \mathcal{Q}
  125. n = 2 n=2
  126. i = 2 i=2
  127. 𝒬 \mathcal{Q}
  128. 𝒬 \mathcal{Q}
  129. 𝔓 2 ( K ) \mathfrak{P}_{2}(K)
  130. ρ ( x ) = x 1 x 2 - x 3 2 \rho(\vec{x})=x_{1}x_{2}-x^{2}_{3}
  131. q ( ξ ) = ξ 2 + a 0 ξ + b 0 q(\xi)=\xi^{2}+a_{0}\xi+b_{0}
  132. K K
  133. ρ ( x ) = x 1 2 + a 0 x 1 x 2 + b 0 x 2 2 - x 3 x 4 \rho(\vec{x})=x^{2}_{1}+a_{0}x_{1}x_{2}+b_{0}x^{2}_{2}-x_{3}x_{4}
  134. 𝒬 \mathcal{Q}
  135. 𝔓 3 ( K ) \mathfrak{P}_{3}(K)
  136. 𝔓 3 ( K ) \mathfrak{P}_{3}(K)
  137. ρ ( x ) = x 1 x 2 + x 3 x 4 \rho(\vec{x})=x_{1}x_{2}+x_{3}x_{4}
  138. K K
  139. x 2 + a x + b = 0 , a , b K x^{2}+ax+b=0,\ a,b\in K

Quantum_inequalities.html

  1. G μ ν = κ T μ ν G_{\mu\nu}=\kappa T_{\mu\nu}
  2. G μ ν G_{\mu\nu}
  3. T μ ν T_{\mu\nu}
  4. κ \kappa

Quantum_point_contact.html

  1. V V
  2. I = G V I=GV
  3. G G
  4. N N
  5. e 2 / h e^{2}/h
  6. G = N G Q G=NG_{Q}
  7. G Q = 2 e 2 / h G_{Q}=2e^{2}/h
  8. e e
  9. h h
  10. N N
  11. G Q G_{Q}
  12. N N
  13. N N
  14. 0.7 G Q 0.7G_{Q}

Quantum_reflection.html

  1. y y
  2. z z
  3. x x
  4. x x
  5. λ ( x ) = h 2 m ( E - V ( x ) ) \lambda\left(x\right)=\frac{h}{\sqrt{2m\left(E-V\left(x\right)\right)}}
  6. m m
  7. E ~{}E~{}
  8. V ( x ) ~{}V(x)~{}
  9. | d λ ( x ) d x | 1 \left|\frac{d\lambda\left(x\right)}{dx}\right|\sim 1
  10. V ( x ) V(x)
  11. V ( x ) ~{}V(x)~{}
  12. θ ~{}\theta~{}
  13. r ~{}r~{}
  14. v = 2 g h ~{}v=\sqrt{2gh}~{}
  15. g ~{}g~{}
  16. h ~{}h~{}
  17. k = sin ( θ ) m v ~{}k=\sin(\theta)\frac{mv}{\hbar}~{}
  18. m ~{}m~{}
  19. \hbar
  20. v = 1 t h ~{}v=\frac{1}{t\!~{}h}~{}
  21. t ~{}t~{}
  22. v v
  23. r = 1 ( 1 + k w ) 4 r=\frac{1}{(1+kw)^{4}}
  24. m m
  25. k = 2 π / λ k=2\pi/\lambda
  26. E = ( k ) 2 2 m E=\frac{(\hbar k)^{2}}{2m}
  27. V ( x ) V(x)
  28. | x t | |x_{t}|
  29. E = V ( x ) E=V(x)
  30. k | x t | < 1 k|x_{t}|<1
  31. r r
  32. r = 1 ( 1 + k | x t | ) 4 r=\frac{1}{(1+k|x_{t}|)^{4}}
  33. L L
  34. L L

Quarter-comma_meantone.html

  1. ( 3 / 2 ) 2 2 = 9 / 4 2 = 9 8 , {(3/2)^{2}\over 2}={9/4\over 2}={9\over 8},
  2. ( 3 / 2 ) 4 2 2 = 81 / 16 4 = 81 64 5 4 = 5 16 4 16 = 80 64 . {(3/2)^{4}\over 2^{2}}={81/16\over 4}={81\over 64}\approx{5\over 4}={5\cdot 16% \over 4\cdot 16}={80\over 64}.
  3. ( 3 2 ) 4 = 81 16 = 80 16 81 80 = 5 81 80 . \left({3\over 2}\right)^{4}={81\over 16}={80\over 16}\cdot{81\over 80}=5\cdot{% 81\over 80}.
  4. 4 1 5 4 = 5. {4\over 1}\cdot{5\over 4}=5.
  5. x 4 = 5 , x^{4}=5,
  6. x = 5 4 = 5 1 / 4 , x=\sqrt[4]{5}=5^{1/4},
  7. x 2 2 = 5 2 , {x^{2}\over 2}={\sqrt{5}\over 2},
  8. 2 3 x 5 = 8 5 5 / 4 . {{2^{3}\over x^{5}}={8\over{5^{5/4}}}}.
  9. 5 1 / 4 1.495349 5^{1/4}\approx 1.495349
  10. 1200 log 2 5 1 / 4 cents 696.578 cents , 1200\log_{2}5^{1/4}\ \hbox{cents}\approx 696.578\ \hbox{cents},
  11. 3 2 = 1.5 {3\over 2}=1.5
  12. 1200 log 2 3 2 cents 701.955 cents . 1200\log_{2}{3\over 2}\ \hbox{cents}\approx 701.955\ \hbox{cents}.
  13. 701.955 - 696.578 5.377 21.506 4 cents . \approx 701.955-696.578\approx 5.377\approx{21.506\over 4}\ \hbox{cents}.
  14. 5 : 2 \sqrt{5}:2
  15. 8 : 5 5 4 8:5^{5\over 4}
  16. x - 6 2 4 = 16 5 25 x^{-6}\cdot 2^{4}=\frac{16\sqrt{5}}{25}
  17. x - 5 2 3 = 8 5 x 25 x^{-5}\cdot 2^{3}=\frac{8\sqrt{5}x}{25}
  18. x - 4 2 3 = 8 5 x^{-4}\cdot 2^{3}=\frac{8}{5}
  19. x - 3 2 2 = 4 x 5 x^{-3}\cdot 2^{2}=\frac{4x}{5}
  20. x - 2 2 2 = 4 5 5 x^{-2}\cdot 2^{2}=\frac{4\sqrt{5}}{5}
  21. x - 1 2 1 = 2 5 x 5 x^{-1}\cdot 2^{1}=\frac{2\sqrt{5}x}{5}
  22. x 0 2 0 = 1 x^{0}\cdot 2^{0}=1
  23. x 1 2 0 = x x^{1}\cdot 2^{0}=x
  24. x 2 2 - 1 = 5 2 x^{2}\cdot 2^{-1}=\frac{\sqrt{5}}{2}
  25. x 3 2 - 1 = 5 x 2 x^{3}\cdot 2^{-1}=\frac{\sqrt{5}x}{2}
  26. x 4 2 - 2 = 5 4 x^{4}\cdot 2^{-2}=\frac{5}{4}
  27. x 5 2 - 2 = 5 x 4 x^{5}\cdot 2^{-2}=\frac{5x}{4}
  28. x 6 2 - 3 = 5 5 8 x^{6}\cdot 2^{-3}=\frac{5\sqrt{5}}{8}
  29. x = 5 4 = 5 1 / 4 x=\sqrt[4]{5}=5^{1/4}
  30. ( 81 / 80 ) 1 / 4 (81/80)^{1/4}
  31. S = 8 : 5 5 / 4 , S={8:5^{5/4}},
  32. T = 5 : 2. T=\sqrt{5}:2.
  33. S X = T . S\cdot X=T.
  34. X = T S = 5 / 2 8 / 5 5 / 4 = 5 1 / 2 5 5 / 4 8 2 = 5 7 / 4 16 . X={T\over S}={\sqrt{5}/2\over 8/5^{5/4}}={5^{1/2}\cdot 5^{5/4}\over 8\cdot 2}=% {5^{7/4}\over 16}.
  35. S ( or S 1 ) = 8 5 5 / 4 117.1 cents S\ (\hbox{or }S_{1})={8\over{5^{5/4}}}\approx 117.1\ \hbox{cents}
  36. X ( or S 2 ) = 5 7 / 4 16 76.0 cents X\ (\hbox{or }S_{2})={5^{7/4}\over 16}\approx 76.0\ \hbox{cents}
  37. S E = 2 12 = 100 cents . S_{E}=\sqrt[12]{2}=100\ \hbox{cents}.
  38. S 4 X 3 1.4953 696.6 cents S^{4}\cdot X^{3}\approx 1.4953\approx 696.6\ \hbox{cents}
  39. S 5 X 2 1.5312 737.6 cents S^{5}\cdot X^{2}\approx 1.5312\approx 737.6\ \hbox{cents}
  40. S 2 X 2 = 1.25 386.3 cents S^{2}\cdot X^{2}=1.25\approx 386.3\ \hbox{cents}
  41. S 3 X = 1.28 427.4 cents S^{3}\cdot X=1.28\approx 427.4\ \hbox{cents}
  42. S 2 X 1.1963 310.3 cents S^{2}\cdot X\approx 1.1963\approx 310.3\ \hbox{cents}
  43. S X 2 1.1682 269.2 cents S\cdot X^{2}\approx 1.1682\approx 269.2\ \hbox{cents}
  44. S = 8 : 5 5 / 4 , S={8:5^{5/4}},
  45. X = 5 7 / 4 : 16 , X={5^{7/4}:16},
  46. T = 5 : 2 , T=\sqrt{5}:2,
  47. P = 5 1 / 4 . P={5^{1/4}}.
  48. T = S X = 8 5 5 / 4 5 7 / 4 16 = 5 2 / 4 2 = 5 2 . T=S\cdot X={8\over{5^{5/4}}}\cdot{5^{7/4}\over 16}={5^{2/4}\over 2}={\sqrt{5}% \over 2}.
  49. P = T 3 S = 5 3 / 2 2 3 8 5 5 / 4 = 5 6 / 4 - 5 / 4 = 5 1 / 4 , P=T^{3}\cdot S={5^{3/2}\over 2^{3}}\cdot{8\over 5^{5/4}}=5^{6/4-5/4}=5^{1/4},
  50. P = T 3 S = S 4 X 3 . P=T^{3}\cdot S=S^{4}\cdot X^{3}.

Quarter_cubic_honeycomb.html

  1. 3 ¯ \overline{3}
  2. A ~ 3 {\tilde{A}}_{3}
  3. A ~ 3 {\tilde{A}}_{3}
  4. A ~ 3 {\tilde{A}}_{3}
  5. 4 ¯ \overline{4}
  6. 3 ¯ \overline{3}
  7. A ~ 3 {\tilde{A}}_{3}

Quartic_plane_curve.html

  1. A x 4 + B y 4 + C x 3 y + D x 2 y 2 + E x y 3 + F x 3 + G y 3 + H x 2 y + I x y 2 + J x 2 + K y 2 + L x y + M x + N y + P = 0. Ax^{4}+By^{4}+Cx^{3}y+Dx^{2}y^{2}+Exy^{3}+Fx^{3}+Gy^{3}+Hx^{2}y+Ixy^{2}+Jx^{2}% +Ky^{2}+Lxy+Mx+Ny+P=0.
  2. 14 \mathbb{RP}^{14}
  3. ( y 2 - x 2 ) ( x - 1 ) ( 2 x - 3 ) = 4 ( x 2 + y 2 - 2 x ) 2 . \ (y^{2}-x^{2})(x-1)(2x-3)=4(x^{2}+y^{2}-2x)^{2}.
  4. x 4 + x 2 y 2 + y 4 = x ( x 2 + y 2 ) x^{4}+x^{2}y^{2}+y^{4}=x(x^{2}+y^{2})\,
  5. ( x 2 - a 2 ) ( x - a ) 2 + ( y 2 - a 2 ) 2 = 0 (x^{2}-a^{2})(x-a)^{2}+(y^{2}-a^{2})^{2}=0\,
  6. x 4 = x 2 y - y 3 . x^{4}=x^{2}y-y^{3}.\,
  7. x 2 y 2 - b 2 x 2 - a 2 y 2 = 0 x^{2}y^{2}-b^{2}x^{2}-a^{2}y^{2}=0\,
  8. x = - t 2 - 2 t + 5 t 2 - 2 t - 3 , y = t 2 - 2 t + 5 2 t - 2 x=-\frac{t^{2}-2t+5}{t^{2}-2t-3},y=\frac{t^{2}-2t+5}{2t-2}
  9. x 4 + 2 x 2 y 2 + y 4 - x 3 + 3 x y 2 = 0 x^{4}+2x^{2}y^{2}+y^{4}-x^{3}+3xy^{2}=0\,
  10. y = ± - 2 x 2 - 3 x ± 16 x 3 - 9 x 2 2 . y=\pm\sqrt{\frac{-2x^{2}-3x\pm\sqrt{16x^{3}-9x^{2}}}{2}}.
  11. x = cos ( 3 t ) cos t , y = cos ( 3 t ) sin t x=\cos(3t)\cos t,y=\cos(3t)\sin t\,
  12. r = cos ( 3 φ ) r=\cos(3\varphi)\,

Quasi-bialgebra.html

  1. Φ \Phi
  2. 𝒜 = ( 𝒜 , Δ , ε , Φ , l , r ) \mathcal{B_{A}}=(\mathcal{A},\Delta,\varepsilon,\Phi,l,r)
  3. 𝒜 \mathcal{A}
  4. 𝔽 \mathbb{F}
  5. Δ : 𝒜 𝒜 𝒜 \Delta:\mathcal{A}\rightarrow\mathcal{A\otimes A}
  6. ε : 𝒜 𝔽 \varepsilon:\mathcal{A}\rightarrow\mathbb{F}
  7. Φ 𝒜 𝒜 𝒜 \Phi\in\mathcal{A\otimes A\otimes A}
  8. r , l A r,l\in A
  9. ( i d Δ ) Δ ( a ) = Φ [ ( Δ i d ) Δ ( a ) ] Φ - 1 , a 𝒜 (id\otimes\Delta)\circ\Delta(a)=\Phi[(\Delta\otimes id)\circ\Delta(a)]\Phi^{-1% },\quad\forall a\in\mathcal{A}
  10. [ ( i d i d Δ ) ( Φ ) ] [ ( Δ i d i d ) ( Φ ) ] = ( 1 Φ ) [ ( i d Δ i d ) ( Φ ) ] ( Φ 1 ) [(id\otimes id\otimes\Delta)(\Phi)]\ [(\Delta\otimes id\otimes id)(\Phi)]=(1% \otimes\Phi)\ [(id\otimes\Delta\otimes id)(\Phi)]\ (\Phi\otimes 1)
  11. ( ε i d ) ( Δ a ) = l - 1 a l , ( i d ε ) Δ = r - 1 a r , a 𝒜 (\varepsilon\otimes id)(\Delta a)=l^{-1}al,\qquad(id\otimes\varepsilon)\circ% \Delta=r^{-1}ar,\quad\forall a\in\mathcal{A}
  12. ( i d ε i d ) ( Φ ) = 1 1. (id\otimes\varepsilon\otimes id)(\Phi)=1\otimes 1.
  13. Δ \Delta
  14. ϵ \epsilon
  15. r r
  16. l l
  17. Φ \Phi
  18. 𝒜 - M o d \mathcal{A}-Mod
  19. l = r = 1 l=r=1
  20. l = r = 1 l=r=1
  21. Φ = 1 1 1 \Phi=1\otimes 1\otimes 1
  22. 𝒜 - M o d \mathcal{A}-Mod
  23. ( 𝒜 , Δ , ϵ , Φ , l , r ) (\mathcal{A},\Delta,\epsilon,\Phi,l,r)
  24. R 𝒜 𝒜 R\in\mathcal{A\otimes A}
  25. ( Δ o p ) ( a ) = R Δ ( a ) R - 1 (\Delta^{op})(a)=R\Delta(a)R^{-1}
  26. ( i d Δ ) ( R ) = ( Φ 231 ) - 1 R 13 Φ 213 R 12 ( Φ 213 ) - 1 (id\otimes\Delta)(R)=(\Phi_{231})^{-1}R_{13}\Phi_{213}R_{12}(\Phi_{213})^{-1}
  27. ( Δ i d ) ( R ) = ( Φ 321 ) R 13 ( Φ 213 ) - 1 R 23 Φ 123 (\Delta\otimes id)(R)=(\Phi_{321})R_{13}(\Phi_{213})^{-1}R_{23}\Phi_{123}
  28. a 1 a k 𝒜 k a_{1}\otimes...\otimes a_{k}\in\mathcal{A}^{\otimes k}
  29. a i 1 i 2 i n a_{i_{1}i_{2}...i_{n}}
  30. a j a_{j}
  31. i j i_{j}
  32. 𝒜 k \mathcal{A}^{\otimes k}
  33. R 12 Φ 321 R 13 ( Φ 132 ) - 1 R 23 Φ 123 = Φ 321 R 23 ( Φ 231 ) - 1 R 13 Φ 213 R 12 R_{12}\Phi_{321}R_{13}(\Phi_{132})^{-1}R_{23}\Phi_{123}=\Phi_{321}R_{23}(\Phi_% {231})^{-1}R_{13}\Phi_{213}R_{12}
  34. r = l = 1 r=l=1
  35. 𝒜 \mathcal{B_{A}}
  36. F 𝒜 𝒜 F\in\mathcal{A\otimes A}
  37. ( ε i d ) F = ( i d ε ) F = 1 (\varepsilon\otimes id)F=(id\otimes\varepsilon)F=1
  38. Δ ( a ) = F Δ ( a ) F - 1 , a 𝒜 \Delta^{\prime}(a)=F\Delta(a)F^{-1},\quad\forall a\in\mathcal{A}
  39. Φ = ( 1 F ) ( ( i d Δ ) F ) Φ ( ( Δ i d ) F - 1 ) ( F - 1 1 ) . \Phi^{\prime}=(1\otimes F)\ ((id\otimes\Delta)F)\ \Phi\ ((\Delta\otimes id)F^{% -1})\ (F^{-1}\otimes 1).
  40. ( 𝒜 , Δ , ε , Φ ) (\mathcal{A},\Delta^{\prime},\varepsilon,\Phi^{\prime})
  41. 𝒜 \mathcal{B_{A}}
  42. ( 𝒜 , Δ , ε , Φ ) (\mathcal{A},\Delta,\varepsilon,\Phi)
  43. R R
  44. ( 𝒜 , Δ , ε , Φ ) (\mathcal{A},\Delta^{\prime},\varepsilon,\Phi^{\prime})
  45. F 21 R F - 1 F_{21}RF^{-1}
  46. F 1 F_{1}
  47. F 2 F_{2}
  48. F 2 F 1 F_{2}F_{1}
  49. F F
  50. F - 1 F^{-1}
  51. 𝒜 \mathcal{B_{A}}
  52. 𝒜 \mathcal{B_{A^{\prime}}}
  53. 𝒜 \mathcal{B^{\prime}_{A^{\prime}}}
  54. 𝒜 \mathcal{B_{A^{\prime}}}
  55. F F
  56. α : 𝒜 𝒜 \alpha:\mathcal{B_{A}}\to\mathcal{B^{\prime}_{A^{\prime}}}
  57. ( α * , i d , ϕ 2 F ) (\alpha^{*},id,\phi_{2}^{F})
  58. 𝒜 - m o d \mathcal{A^{\prime}}-mod
  59. 𝒜 - m o d \mathcal{A}-mod
  60. ϕ 2 F ( v w ) = F - 1 ( v w ) \phi_{2}^{F}(v\otimes w)=F^{-1}(v\otimes w)
  61. α \alpha

Quasi-finite_field.html

  1. ϕ : 𝐙 ^ Gal ( K s / K ) , \phi:\hat{\mathbf{Z}}\to\operatorname{Gal}(K_{s}/K),
  2. 𝐙 ^ \widehat{\mathbf{Z}}
  3. K n = 𝐂 ( ( T 1 / n ) ) K_{n}=\mathbf{C}((T^{1/n}))
  4. F n ( T 1 / n ) = e 2 π i / n T 1 / n . F_{n}(T^{1/n})=e^{2\pi i/n}T^{1/n}.

Quasi-Hopf_algebra.html

  1. 𝒜 = ( 𝒜 , Δ , ε , Φ ) \mathcal{B_{A}}=(\mathcal{A},\Delta,\varepsilon,\Phi)
  2. α , β 𝒜 \alpha,\beta\in\mathcal{A}
  3. 𝒜 \mathcal{A}
  4. i S ( b i ) α c i = ε ( a ) α \sum_{i}S(b_{i})\alpha c_{i}=\varepsilon(a)\alpha
  5. i b i β S ( c i ) = ε ( a ) β \sum_{i}b_{i}\beta S(c_{i})=\varepsilon(a)\beta
  6. a 𝒜 a\in\mathcal{A}
  7. Δ ( a ) = i b i c i \Delta(a)=\sum_{i}b_{i}\otimes c_{i}
  8. i X i β S ( Y i ) α Z i = 𝕀 , \sum_{i}X_{i}\beta S(Y_{i})\alpha Z_{i}=\mathbb{I},
  9. j S ( P j ) α Q j β S ( R j ) = 𝕀 . \sum_{j}S(P_{j})\alpha Q_{j}\beta S(R_{j})=\mathbb{I}.
  10. Φ \Phi
  11. Φ - 1 \Phi^{-1}
  12. Φ = i X i Y i Z i \Phi=\sum_{i}X_{i}\otimes Y_{i}\otimes Z_{i}
  13. Φ - 1 = j P j Q j R j . \Phi^{-1}=\sum_{j}P_{j}\otimes Q_{j}\otimes R_{j}.

Quasi-isomorphism.html

  1. H n ( A ) H n ( B ) ( respectively, H n ( A ) H n ( B ) ) H_{n}(A_{\bullet})\to H_{n}(B_{\bullet})\ (\,\text{respectively, }H^{n}(A^{% \bullet})\to H^{n}(B^{\bullet}))

Quasi-phase-matching.html

  1. P 3 = 4 d A 1 A 2 e i ( k 1 + k 2 ) z P_{3}=4dA_{1}A_{2}e^{i(k_{1}+k_{2})z}
  2. d d
  3. i i
  4. P 3 = - 4 d A 1 A 2 e i ( k 1 + k 2 ) z = 4 d A 1 A 2 e i ( ( k 1 + k 2 ) z + π ) P_{3}=-4dA_{1}A_{2}e^{i(k_{1}+k_{2})z}=4dA_{1}A_{2}e^{i((k_{1}+k_{2})z+\pi)}
  5. A 2 z = A 1 χ e i Δ k z , \frac{\partial A_{2}}{\partial z}=A_{1}\chi e^{i\Delta kz},
  6. A 2 A_{2}
  7. A 1 A_{1}
  8. Δ k \Delta k
  9. χ \chi
  10. χ \chi
  11. n t h n^{th}
  12. χ \chi
  13. χ = χ 0 ( - 1 ) n \chi=\chi_{0}(-1)^{n}
  14. n n
  15. A 2 A_{2}
  16. A 2 = A 1 χ 0 n = 0 N - 1 ( - 1 ) n Λ n Λ ( n + 1 ) e i Δ k z z A_{2}=A_{1}\chi_{0}\sum^{N-1}_{n=0}(-1)^{n}\int^{\Lambda(n+1)}_{\Lambda n}e^{i% \Delta kz}\partial z
  17. A 2 = - i A 1 χ 0 Δ k n = 0 N - 1 ( - 1 ) n ( e i Δ k Λ ( n + 1 ) - e i Δ k Λ n ) A_{2}=-\frac{iA_{1}\chi_{0}}{\Delta k}\sum^{N-1}_{n=0}(-1)^{n}(e^{i\Delta k% \Lambda(n+1)}-e^{i\Delta k\Lambda n})
  18. A 2 = - i A 1 χ 0 e i Δ k Λ - 1 Δ k n = 0 N - 1 ( - 1 ) n e i Δ k Λ n A_{2}=-iA_{1}\chi_{0}\frac{e^{i\Delta k\Lambda}-1}{\Delta k}\sum^{N-1}_{n=0}(-% 1)^{n}e^{i\Delta k\Lambda n}
  19. s = n = 0 N - 1 ( - 1 ) n e i Δ k Λ n = 1 - e i Δ k Λ + e i 2 Δ k Λ n - e i 3 Δ k Λ + + ( - 1 ) N e i Δ k Λ ( N - 2 ) - ( - 1 ) N e i Δ k Λ ( N - 1 ) . s=\sum^{N-1}_{n=0}(-1)^{n}e^{i\Delta k\Lambda n}=1-e^{i\Delta k\Lambda}+e^{i2% \Delta k\Lambda n}-e^{i3\Delta k\Lambda}+...+(-1)^{N}e^{i\Delta k\Lambda(N-2)}% -(-1)^{N}e^{i\Delta k\Lambda(N-1)}.
  20. e i Δ k Λ e^{i\Delta k\Lambda}
  21. s e i Δ k Λ = e i Δ k Λ - e i 2 Δ k Λ n + e i 3 Δ k Λ + + ( - 1 ) N e i Δ k Λ ( N - 1 ) - ( - 1 ) N e i Δ k Λ N . se^{i\Delta k\Lambda}=e^{i\Delta k\Lambda}-e^{i2\Delta k\Lambda n}+e^{i3\Delta k% \Lambda}+...+(-1)^{N}e^{i\Delta k\Lambda(N-1)}-(-1)^{N}e^{i\Delta k\Lambda N}.
  22. s ( 1 + e i Δ k Λ ) = 1 - ( - 1 ) N e i Δ k Λ N . s(1+e^{i\Delta k\Lambda})=1-(-1)^{N}e^{i\Delta k\Lambda N}.
  23. s s
  24. s = 1 - ( - 1 ) N e i Δ k Λ N 1 + e i Δ k Λ , s=\frac{1-(-1)^{N}e^{i\Delta k\Lambda N}}{1+e^{i\Delta k\Lambda}},
  25. A 2 = - i A 1 χ 0 ( e i Δ k Λ - 1 Δ k ) ( 1 - ( - 1 ) N e i Δ k Λ N e i Δ k Λ + 1 ) . A_{2}=-iA_{1}\chi_{0}\left(\frac{e^{i\Delta k\Lambda}-1}{\Delta k}\right)\left% (\frac{1-(-1)^{N}e^{i\Delta k\Lambda N}}{e^{i\Delta k\Lambda}+1}\right).
  26. I 2 = A 2 A 2 * = A 1 2 χ 0 2 Λ 2 sinc ( Δ k Λ / 2 ) 2 ( 1 - ( - 1 ) N cos ( Δ k Λ N ) 1 + cos ( Δ k Λ ) ) . I_{2}=A_{2}A_{2}^{*}=A_{1}^{2}\chi_{0}^{2}\Lambda^{2}\mbox{sinc}~{}^{2}(\Delta k% \Lambda/2)\left(\frac{1-(-1)^{N}\cos(\Delta k\Lambda N)}{1+\cos(\Delta k% \Lambda)}\right).
  27. Λ = π Δ k \Lambda=\frac{\pi}{\Delta k}
  28. Δ k Λ π \Delta k\Lambda\rightarrow\pi
  29. lim Δ k Λ π 1 - ( - 1 ) N cos ( Δ k Λ N ) 1 + cos ( Δ k Λ ) = N 2 \lim_{\Delta k\Lambda\to\pi}\frac{1-(-1)^{N}\cos(\Delta k\Lambda N)}{1+\cos(% \Delta k\Lambda)}=N^{2}
  30. I 2 = 4 A 1 2 χ 0 2 L 2 π 2 . I_{2}=\frac{4A_{1}^{2}\chi_{0}^{2}L^{2}}{\pi^{2}}.
  31. Λ = m π Δ k \Lambda=\frac{m\pi}{\Delta k}
  32. m = 1 , 3 , 5 , m=1,3,5,...
  33. I 2 = A 2 A 2 * = A 1 2 χ 0 2 Λ 2 sinc ( m Δ k Λ / 2 ) 2 ( 1 - ( - 1 ) N cos ( m Δ k Λ N ) 1 + cos ( m Δ k Λ ) ) . I_{2}=A_{2}A_{2}^{*}=A_{1}^{2}\chi_{0}^{2}\Lambda^{2}\mbox{sinc}~{}^{2}(m% \Delta k\Lambda/2)\left(\frac{1-(-1)^{N}\cos(m\Delta k\Lambda N)}{1+\cos(m% \Delta k\Lambda)}\right).
  34. Λ - > m π Δ k \Lambda->\frac{m\pi}{\Delta k}
  35. I 2 = 4 A 1 2 χ 0 2 L 2 m 2 π 2 . I_{2}=\frac{4A_{1}^{2}\chi_{0}^{2}L^{2}}{m^{2}\pi^{2}}.
  36. Λ \Lambda
  37. m m
  38. m 2 m^{2}
  39. Δ k = k 3 - k 1 - k 2 \Delta k=k_{3}-k_{1}-k_{2}
  40. k 1 , k 2 , and k 3 k_{1},k_{2},\mbox{and }~{}k_{3}
  41. k i = 2 π n ( λ i ) λ i k_{i}=\frac{2\pi n(\lambda_{i})}{\lambda_{i}}
  42. Δ k \Delta k
  43. Λ = π Δ k \Lambda=\frac{\pi}{\Delta k}

Quasi-triangular_Quasi-Hopf_algebra.html

  1. 𝒜 = ( 𝒜 , R , Δ , ε , Φ ) \mathcal{H_{A}}=(\mathcal{A},R,\Delta,\varepsilon,\Phi)
  2. 𝒜 = ( 𝒜 , Δ , ε , Φ ) \mathcal{B_{A}}=(\mathcal{A},\Delta,\varepsilon,\Phi)
  3. R 𝒜 𝒜 R\in\mathcal{A\otimes A}
  4. R Δ ( a ) = σ Δ ( a ) R , a 𝒜 R\Delta(a)=\sigma\circ\Delta(a)R,a\in\mathcal{A}
  5. σ : 𝒜 𝒜 𝒜 𝒜 \sigma:\mathcal{A\otimes A}\rightarrow\mathcal{A\otimes A}
  6. x y y x x\otimes y\rightarrow y\otimes x
  7. σ \sigma
  8. ( Δ i d ) R = Φ 321 R 13 Φ 132 - 1 R 23 Φ 123 (\Delta\otimes id)R=\Phi_{321}R_{13}\Phi_{132}^{-1}R_{23}\Phi_{123}
  9. ( i d Δ ) R = Φ 231 - 1 R 13 Φ 213 R 12 Φ 123 - 1 (id\otimes\Delta)R=\Phi_{231}^{-1}R_{13}\Phi_{213}R_{12}\Phi_{123}^{-1}
  10. Φ a b c = x a x b x c \Phi_{abc}=x_{a}\otimes x_{b}\otimes x_{c}
  11. Φ 123 = Φ = x 1 x 2 x 3 𝒜 𝒜 𝒜 \Phi_{123}=\Phi=x_{1}\otimes x_{2}\otimes x_{3}\in\mathcal{A\otimes A\otimes A}
  12. R 21 R 12 = 1 R_{21}R_{12}=1
  13. 𝒜 \mathcal{H_{A}}
  14. F 𝒜 𝒜 F\in\mathcal{A\otimes A}
  15. Φ = 1 \Phi=1

Quasifield.html

  1. ( Q , + , ) (Q,+,\cdot)
  2. \cdot
  3. ( Q , + , ) (Q,+,\cdot)
  4. \cdot\,
  5. ( Q , + ) (Q,+)\,
  6. ( Q 0 , ) (Q_{0},\cdot)
  7. Q 0 = Q { 0 } Q_{0}=Q\setminus\{0\}\,
  8. a ( b + c ) = a b + a c a , b , c Q a\cdot(b+c)=a\cdot b+a\cdot c\quad\forall a,b,c\in Q
  9. a x = b x + c a\cdot x=b\cdot x+c
  10. a , b , c Q , a b \forall a,b,c\in Q,a\neq b
  11. ( Q , + ) (Q,+)
  12. ( Q 0 , ) (Q_{0},\cdot)
  13. a ( b c ) = ( a b ) c a , b Q a\cdot(b\cdot c)=(a\cdot b)\cdot c\quad\forall a,b\in Q
  14. ( a + b ) c = ( a c ) + ( b c ) a , b Q (a+b)\cdot c=(a\cdot c)+(b\cdot c)\quad\forall a,b\in Q
  15. \cdot
  16. ( K , + , ) (K,+,\cdot)
  17. v l = v l v Q , l K v\otimes l=v\cdot l\quad\forall v\in Q,l\in K
  18. Q Q
  19. T : Q × Q × Q Q \scriptstyle T\colon Q\times Q\times Q\to Q\,
  20. T ( a , b , c ) = a b + c a , b , c Q T(a,b,c)=a\cdot b+c\quad\forall a,b,c\in Q
  21. ( Q , T ) (Q,T)
  22. ( Q , T ) (Q,T)

Quasiprobability_distribution.html

  1. ρ ^ \hat{\rho}
  2. a ^ \hat{a}
  3. a ^ | α \displaystyle\hat{a}|\alpha\rangle
  4. β | α = e - 1 2 ( | β | 2 + | α | 2 - 2 β * α ) δ ( α - β ) . \langle\beta|\alpha\rangle=e^{-{1\over 2}(|\beta|^{2}+|\alpha|^{2}-2\beta^{*}% \alpha)}\neq\delta(\alpha-\beta).
  5. d 2 α = r d r d θ d^{2}\alpha=r\,drd\theta
  6. | α α | d 2 α \displaystyle\int|\alpha\rangle\langle\alpha|\,d^{2}\alpha
  7. | ψ = 1 π | α α | ψ d 2 α . |\psi\rangle=\frac{1}{\pi}\int|\alpha\rangle\langle\alpha|\psi\rangle\,d^{2}\alpha.
  8. ρ ^ = f ( α , α * ) | α α | d 2 α \hat{\rho}=\int f(\alpha,\alpha^{*})|{\alpha}\rangle\langle{\alpha}|\,d^{2}\alpha
  9. f ( α , α * ) d 2 α = tr ( ρ ^ ) = 1 \int f(\alpha,\alpha^{*})\,d^{2}\alpha=\mathrm{tr}(\hat{\rho})=1
  10. g Ω ( a ^ , a ^ ) g_{\Omega}(\hat{a},\hat{a}^{\dagger})
  11. g Ω ( a ^ , a ^ ) = f ( α , α * ) g Ω ( α , α * ) d α d α * \langle g_{\Omega}(\hat{a},\hat{a}^{\dagger})\rangle=\int f(\alpha,\alpha^{*})% g_{\Omega}(\alpha,\alpha^{*})\,d\alpha d\alpha^{*}
  12. Ω Ω
  13. P P
  14. P P
  15. P P
  16. χ W ( 𝐳 , 𝐳 * ) = tr ( ρ e i 𝐳 𝐚 ^ + i 𝐳 * 𝐚 ^ ) \chi_{W}(\mathbf{z},\mathbf{z}^{*})=\operatorname{tr}(\rho e^{i\mathbf{z}\cdot% \widehat{\mathbf{a}}+i\mathbf{z}^{*}\cdot\widehat{\mathbf{a}}^{\dagger}})
  17. χ P ( 𝐳 , 𝐳 * ) = tr ( ρ e i 𝐳 𝐚 ^ e i 𝐳 * 𝐚 ^ ) \chi_{P}(\mathbf{z},\mathbf{z}^{*})=\operatorname{tr}(\rho e^{i\mathbf{z}\cdot% \widehat{\mathbf{a}}}e^{i\mathbf{z}^{*}\cdot\widehat{\mathbf{a}}^{\dagger}})
  18. χ Q ( 𝐳 , 𝐳 * ) = tr ( ρ e i 𝐳 * 𝐚 ^ e i 𝐳 𝐚 ^ ) \chi_{Q}(\mathbf{z},\mathbf{z}^{*})=\operatorname{tr}(\rho e^{i\mathbf{z}^{*}% \cdot\widehat{\mathbf{a}}^{\dagger}}e^{i\mathbf{z}\cdot\widehat{\mathbf{a}}})
  19. 𝐚 ^ \widehat{\mathbf{a}}
  20. 𝐚 ^ \widehat{\mathbf{a}}^{\dagger}
  21. χ P \chi_{P}\,
  22. a ^ j m a ^ k n = m + n ( i z j * ) m ( i z k ) n χ P ( 𝐳 , 𝐳 * ) | 𝐳 = 𝐳 * = 0 \langle\widehat{a}_{j}^{\dagger m}\widehat{a}_{k}^{n}\rangle=\frac{\partial^{m% +n}}{\partial(iz_{j}^{*})^{m}\partial(iz_{k})^{n}}\chi_{P}(\mathbf{z},\mathbf{% z}^{*})\Big|_{\mathbf{z}=\mathbf{z}^{*}=0}
  23. { W | P | Q } ( α , α * ) = 1 π 2 N χ { W | P | Q } ( 𝐳 , 𝐳 * ) e - i 𝐳 * α * e - i 𝐳 α d 2 N 𝐳 . \{W|P|Q\}(\mathbf{\alpha},\mathbf{\alpha}^{*})=\frac{1}{\pi^{2N}}\int\chi_{\{W% |P|Q\}}(\mathbf{z},\mathbf{z}^{*})e^{-i\mathbf{z}^{*}\cdot\mathbf{\alpha}^{*}}% e^{-i\mathbf{z}\cdot\mathbf{\alpha}}\,d^{2N}\mathbf{z}.
  24. α j \alpha_{j}\,
  25. α k * \alpha^{*}_{k}
  26. 𝐚 ^ j m 𝐚 ^ k n = P ( α , α * ) α j n α k * m d 2 N α \langle\widehat{\mathbf{a}}_{j}^{\dagger m}\widehat{\mathbf{a}}_{k}^{n}\rangle% =\int P(\mathbf{\alpha},\mathbf{\alpha}^{*})\alpha_{j}^{n}\alpha_{k}^{*m}\,d^{% 2N}\mathbf{\alpha}
  27. 𝐚 ^ j m 𝐚 ^ k n = Q ( α , α * ) α j m α k * n d 2 N α \langle\widehat{\mathbf{a}}_{j}^{m}\widehat{\mathbf{a}}_{k}^{\dagger n}\rangle% =\int Q(\mathbf{\alpha},\mathbf{\alpha}^{*})\alpha_{j}^{m}\alpha_{k}^{*n}\,d^{% 2N}\mathbf{\alpha}
  28. ( 𝐚 ^ j m 𝐚 ^ k n ) S = W ( α , α * ) α j m α k * n d 2 N α \langle(\widehat{\mathbf{a}}_{j}^{\dagger m}\widehat{\mathbf{a}}_{k}^{n})_{S}% \rangle=\int W(\mathbf{\alpha},\mathbf{\alpha}^{*})\alpha_{j}^{m}\alpha_{k}^{*% n}\,d^{2N}\mathbf{\alpha}
  29. ( ) S (\ldots)_{S}
  30. W ( α , α * ) = 2 π P ( β , β * ) e - 2 | α - β | 2 d 2 β W(\alpha,\alpha^{*})=\frac{2}{\pi}\int P(\beta,\beta^{*})e^{-2|\alpha-\beta|^{% 2}}\,d^{2}\beta
  31. Q ( α , α * ) = 2 π W ( β , β * ) e - 2 | α - β | 2 d 2 β Q(\alpha,\alpha^{*})=\frac{2}{\pi}\int W(\beta,\beta^{*})e^{-2|\alpha-\beta|^{% 2}}\,d^{2}\beta
  32. Q ( α , α * ) = 1 π P ( β , β * ) e - | α - β | 2 d 2 β . Q(\alpha,\alpha^{*})=\frac{1}{\pi}\int P(\beta,\beta^{*})e^{-|\alpha-\beta|^{2% }}\,d^{2}\beta.
  33. ρ ρ
  34. ρ ˙ \dot{\rho}
  35. a ^ j \widehat{a}_{j}\,
  36. ρ ρ
  37. tr ( a ^ j ρ e i 𝐳 𝐚 ^ e i 𝐳 * 𝐚 ^ ) = ( i z j ) χ P ( 𝐳 , 𝐳 * ) . \operatorname{tr}(\widehat{a}_{j}\rho e^{i\mathbf{z}\cdot\widehat{\mathbf{a}}}% e^{i\mathbf{z}^{*}\cdot\widehat{\mathbf{a}}^{\dagger}})=\frac{\partial}{% \partial(iz_{j})}\chi_{P}(\mathbf{z},\mathbf{z}^{*}).
  38. 𝐳 \mathbf{z}\,
  39. a ^ j ρ α j P ( α , α * ) . \widehat{a}_{j}\rho\rightarrow\alpha_{j}P(\mathbf{\alpha},\mathbf{\alpha}^{*}).
  40. a ^ j ρ ( α j + κ α j * ) { W | P | Q } ( α , α * ) \widehat{a}_{j}\rho\rightarrow\left(\alpha_{j}+\kappa\frac{\partial}{\partial% \alpha_{j}^{*}}\right)\{W|P|Q\}(\mathbf{\alpha},\mathbf{\alpha}^{*})
  41. ρ a ^ j ( α j * + κ α j ) { W | P | Q } ( α , α * ) \rho\widehat{a}^{\dagger}_{j}\rightarrow\left(\alpha_{j}^{*}+\kappa\frac{% \partial}{\partial\alpha_{j}}\right)\{W|P|Q\}(\mathbf{\alpha},\mathbf{\alpha}^% {*})
  42. a ^ j ρ ( α j * - ( 1 - κ ) α j ) { W | P | Q } ( α , α * ) \widehat{a}^{\dagger}_{j}\rho\rightarrow\left(\alpha_{j}^{*}-(1-\kappa)\frac{% \partial}{\partial\alpha_{j}}\right)\{W|P|Q\}(\mathbf{\alpha},\mathbf{\alpha}^% {*})
  43. ρ a ^ j ( α j - ( 1 - κ ) α j * ) { W | P | Q } ( α , α * ) \rho\widehat{a}_{j}\rightarrow\left(\alpha_{j}-(1-\kappa)\frac{\partial}{% \partial\alpha_{j}^{*}}\right)\{W|P|Q\}(\mathbf{\alpha},\mathbf{\alpha}^{*})
  44. κ = 0 , 1 / 2 κ=0,1/2
  45. | α 0 |\alpha_{0}\rangle
  46. P ( α , α * ) = δ 2 ( α - α 0 ) . P(\alpha,\alpha^{*})=\delta^{2}(\alpha-\alpha_{0}).
  47. W ( α , α * ) = 2 π δ 2 ( β - α 0 ) e - 2 | α - β | 2 d 2 β = 2 π e - 2 | α - α 0 | 2 W(\alpha,\alpha^{*})=\frac{2}{\pi}\int\delta^{2}(\beta-\alpha_{0})e^{-2|\alpha% -\beta|^{2}}\,d^{2}\beta=\frac{2}{\pi}e^{-2|\alpha-\alpha_{0}|^{2}}
  48. Q ( α , α * ) = 1 π δ 2 ( β - α 0 ) e - | α - β | 2 d 2 β = 1 π e - | α - α 0 | 2 . Q(\alpha,\alpha^{*})=\frac{1}{\pi}\int\delta^{2}(\beta-\alpha_{0})e^{-|\alpha-% \beta|^{2}}\,d^{2}\beta=\frac{1}{\pi}e^{-|\alpha-\alpha_{0}|^{2}}.
  49. Q ( α , α * ) = 1 π α | ρ ^ | α = 1 π | α 0 | α | 2 = 1 π e - | α - α 0 | 2 Q(\alpha,\alpha^{*})=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle=\frac% {1}{\pi}|\langle\alpha_{0}|\alpha\rangle|^{2}=\frac{1}{\pi}e^{-|\alpha-\alpha_% {0}|^{2}}
  50. | n |n\rangle
  51. P ( α , α * ) = e | α | 2 n ! 2 n α * n α n δ 2 ( α ) . P(\alpha,\alpha^{*})=\frac{e^{|\alpha|^{2}}}{n!}\frac{\partial^{2n}}{\partial% \alpha^{*n}\partial\alpha^{n}}\delta^{2}(\alpha).
  52. W ( α , α * ) = ( - 1 ) n 2 π e - 2 | α | 2 L n ( 4 | α | 2 ) , W(\alpha,\alpha^{*})=(-1)^{n}\frac{2}{\pi}e^{-2|\alpha|^{2}}L_{n}\left(4|% \alpha|^{2}\right)~{},
  53. Q ( α , α * ) = 1 π α | ρ ^ | α = 1 π | n | α | 2 = 1 π n ! | 0 | a ^ n | α | 2 = | α | 2 n π n ! e - | α | 2 Q(\alpha,\alpha^{*})=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle=\frac% {1}{\pi}|\langle n|\alpha\rangle|^{2}=\frac{1}{\pi n!}|\langle 0|\hat{a}^{n}|% \alpha\rangle|^{2}=\frac{|\alpha|^{2n}}{\pi n!}e^{-|\alpha|^{2}}
  54. d ρ ^ d t = i ω 0 [ ρ ^ , a ^ a ^ ] + γ 2 ( 2 a ^ ρ ^ a ^ - a ^ a ^ ρ ^ - ρ a ^ a ^ ) + γ n ( a ^ ρ ^ a ^ + a ^ ρ ^ a ^ - a ^ a ^ ρ ^ - ρ ^ a ^ a ^ ) . \frac{d\hat{\rho}}{dt}=i\omega_{0}[\hat{\rho},\hat{a}^{\dagger}\hat{a}]+\frac{% \gamma}{2}(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{% \rho}-\rho\hat{a}^{\dagger}\hat{a})+\gamma\langle n\rangle(\hat{a}\hat{\rho}% \hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{\rho}\hat{a}-\hat{a}^{\dagger}\hat{a}% \hat{\rho}-\hat{\rho}\hat{a}\hat{a}^{\dagger}).
  55. t { W | P | Q } ( α , α * , t ) = [ ( γ + i ω 0 ) α α + ( γ - i ω 0 ) α * α * + γ 2 ( n + κ ) 2 α α * ] { W | P | Q } ( α , α * , t ) \frac{\partial}{\partial t}\{W|P|Q\}(\alpha,\alpha^{*},t)=\left[(\gamma+i% \omega_{0})\frac{\partial}{\partial\alpha}\alpha+(\gamma-i\omega_{0})\frac{% \partial}{\partial\alpha^{*}}\alpha^{*}+\frac{\gamma}{2}(\langle n\rangle+% \kappa)\frac{\partial^{2}}{\partial\alpha\partial\alpha^{*}}\right]\{W|P|Q\}(% \alpha,\alpha^{*},t)
  56. | α 0 |\alpha_{0}\rangle
  57. { W | P | Q } ( α , α * , t ) = 1 π [ κ + n ( 1 - e - 2 γ t ) ] exp ( - | α - α 0 e - ( γ + i ω 0 ) t | 2 κ + n ( 1 - e - 2 γ t ) ) \{W|P|Q\}(\alpha,\alpha^{*},t)=\frac{1}{\pi\left[\kappa+\langle n\rangle\left(% 1-e^{-2\gamma t}\right)\right]}\exp{\left(-\frac{\left|\alpha-\alpha_{0}e^{-(% \gamma+i\omega_{0})t}\right|^{2}}{\kappa+\langle n\rangle\left(1-e^{-2\gamma t% }\right)}\right)}

Quasitriangular_Hopf_algebra.html

  1. H H H\otimes H
  2. R Δ ( x ) = ( T Δ ) ( x ) R R\ \Delta(x)=(T\circ\Delta)(x)\ R
  3. x H x\in H
  4. Δ \Delta
  5. T : H H H H T:H\otimes H\to H\otimes H
  6. T ( x y ) = y x T(x\otimes y)=y\otimes x
  7. ( Δ 1 ) ( R ) = R 13 R 23 (\Delta\otimes 1)(R)=R_{13}\ R_{23}
  8. ( 1 Δ ) ( R ) = R 13 R 12 (1\otimes\Delta)(R)=R_{13}\ R_{12}
  9. R 12 = ϕ 12 ( R ) R_{12}=\phi_{12}(R)
  10. R 13 = ϕ 13 ( R ) R_{13}=\phi_{13}(R)
  11. R 23 = ϕ 23 ( R ) R_{23}=\phi_{23}(R)
  12. ϕ 12 : H H H H H \phi_{12}:H\otimes H\to H\otimes H\otimes H
  13. ϕ 13 : H H H H H \phi_{13}:H\otimes H\to H\otimes H\otimes H
  14. ϕ 23 : H H H H H \phi_{23}:H\otimes H\to H\otimes H\otimes H
  15. ϕ 12 ( a b ) = a b 1 , \phi_{12}(a\otimes b)=a\otimes b\otimes 1,
  16. ϕ 13 ( a b ) = a 1 b , \phi_{13}(a\otimes b)=a\otimes 1\otimes b,
  17. ϕ 23 ( a b ) = 1 a b . \phi_{23}(a\otimes b)=1\otimes a\otimes b.
  18. ( ϵ 1 ) R = ( 1 ϵ ) R = 1 H (\epsilon\otimes 1)R=(1\otimes\epsilon)R=1\in H
  19. R - 1 = ( S 1 ) ( R ) R^{-1}=(S\otimes 1)(R)
  20. R = ( 1 S ) ( R - 1 ) R=(1\otimes S)(R^{-1})
  21. ( S S ) ( R ) = R (S\otimes S)(R)=R
  22. S 2 ( x ) = u x u - 1 S^{2}(x)=uxu^{-1}
  23. u := m ( S 1 ) R 21 u:=m(S\otimes 1)R^{21}
  24. F = i f i f i 𝒜 𝒜 F=\sum_{i}f^{i}\otimes f_{i}\in\mathcal{A\otimes A}
  25. ( ε i d ) F = ( i d ε ) F = 1 (\varepsilon\otimes id)F=(id\otimes\varepsilon)F=1
  26. ( F 1 ) ( Δ i d ) F = ( 1 F ) ( i d Δ ) F (F\otimes 1)\circ(\Delta\otimes id)F=(1\otimes F)\circ(id\otimes\Delta)F
  27. u = i f i S ( f i ) u=\sum_{i}f^{i}S(f_{i})
  28. S ( a ) = u S ( a ) u - 1 S^{\prime}(a)=uS(a)u^{-1}

Query_throughput.html

  1. 𝑄𝑡ℎ𝐷 = S × 17 × 3600 T s × 𝑆𝐹 \mathit{QthD}=\frac{S\times 17\times 3600}{T_{s}}\times\mathit{SF}

Quicksort.html

  1. O ( n 2 ) O(n^{2})
  2. l o lo
  3. h i hi
  4. A A

Quotient_category.html

  1. f 1 , f 2 : X Y f_{1},f_{2}:X\to Y\,
  2. g 1 , g 2 : Y Z g_{1},g_{2}:Y\to Z\,
  3. Hom 𝒞 / ( X , Y ) = Hom 𝒞 ( X , Y ) / R X , Y . \mathrm{Hom}_{\mathcal{C}/\mathcal{R}}(X,Y)=\mathrm{Hom}_{\mathcal{C}}(X,Y)/R_% {X,Y}.
  4. lim Hom A ( X , Y / Y ) \underrightarrow{\lim}\mathrm{Hom}_{A}(X^{\prime},Y/Y^{\prime})
  5. X X X^{\prime}\subseteq X
  6. Y Y Y^{\prime}\subseteq Y
  7. X / X , Y B X/X^{\prime},Y^{\prime}\in B
  8. Q : A A / B Q\colon A\to A/B
  9. F : A C F\colon A\to C
  10. b B b\in B
  11. F ¯ : A / B C \overline{F}\colon A/B\to C
  12. F = F ¯ Q F=\overline{F}\circ Q

Q–Q_plot.html

  1. N ( i ) = G ( U ( i ) ) N(i)=G(U(i))
  2. m ( i ) = { 1 - m ( n ) i = 1 i - 0.3175 n + 0.365 i = 2 , 3 , , n - 1 0.5 1 / n i = n . m(i)=\begin{cases}1-m(n)&i=1\\ \\ \dfrac{i-0.3175}{n+0.365}&i=2,3,\ldots,n-1\\ \\ 0.5^{1/n}&i=n.\end{cases}

Radar_altimeter.html

  1. H 2 + r p 2 = ( H + l p ) 2 = H 2 + l p 2 + 2 H l p H^{2}+r_{p}^{2}=(H+l_{p})^{2}=H^{2}+l_{p}^{2}+2Hl_{p}
  2. r p = ( 2 H l p ) 1 2 = ( H c t p ) 1 2 r_{p}=(2Hl_{p})^{\frac{1}{2}}=(Hct_{p})^{\frac{1}{2}}
  3. P ( t ) = { 0 t < t o π r 2 ( t ) t o < t < t o + t p π [ r 2 ( t ) - r 2 ( t - t p ) ] t > t o + t p P(t)=\begin{cases}0&t<t_{o}\\ \pi r^{2}(t)&t_{o}<t<t_{o}+t_{p}\\ \pi[r^{2}(t)-r^{2}(t-t_{p})]&t>t_{o}+t_{p}\\ \end{cases}
  4. P ( t ) = { 0 t < t o t - t o t p t o < t < t o + t p 1 t > t o + t p P(t)=\begin{cases}0&t<t_{o}\\ \frac{t-t_{o}}{t_{p}}&t_{o}<t<t_{o}+t_{p}\\ \ 1&t>t_{o}+t_{p}\\ \end{cases}
  5. P ( t ) = { 0 t < t o 2 W r ( t ) t o < t < t o + t p 2 W [ r ( t ) - r ( t - t p ) ] t > t o + t p P(t)=\begin{cases}0&t<t_{o}\\ \ 2Wr(t)&t_{o}<t<t_{o}+t_{p}\\ \ 2W[r(t)-r(t-t_{p})]&t>t_{o}+t_{p}\\ \end{cases}
  6. t = t - t o t^{\prime}=t-t_{o}
  7. P ( t ) = { 0 t < 0 ( t t p ) 1 2 0 < t < t p ( t t p ) 1 2 - ( t t - t p ) 1 2 t > t p P(t^{\prime})=\begin{cases}0&t^{\prime}<0\\ \ (\frac{t^{\prime}}{t_{p}})^{\frac{1}{2}}&0<t^{\prime}<t_{p}\\ \ (\frac{t^{\prime}}{t_{p}})^{\frac{1}{2}}-(\frac{t^{\prime}}{t^{\prime}-t_{p}% })^{\frac{1}{2}}&t^{\prime}>t_{p}\\ \end{cases}

Radial_distribution_function.html

  1. g ( r ) g(r)
  2. ρ = N / V \rho=N/V
  3. r r
  4. ρ g ( r ) \rho g(r)
  5. r r
  6. g ( r ) I = 4 π r 2 ρ d r g(r)_{I}=4\pi r^{2}\rho dr
  7. ρ \rho
  8. N N
  9. V V
  10. ρ = N / V \rho=N/V
  11. T T
  12. β = 1 k T \textstyle\beta=\frac{1}{kT}
  13. 𝐫 i \mathbf{r}_{i}
  14. i = 1 , , N \textstyle i=1,\,\ldots,\,N
  15. U N ( 𝐫 1 , 𝐫 N ) \textstyle U_{N}(\mathbf{r}_{1}\,\ldots,\,\mathbf{r}_{N})
  16. ( N , V , T ) (N,V,T)
  17. Z N = e - β U N d 𝐫 1 d 𝐫 N \textstyle Z_{N}=\int\cdots\int\mathrm{e}^{-\beta U_{N}}\mathrm{d}\mathbf{r}_{% 1}\cdots\mathrm{d}\mathbf{r}_{N}
  18. d 𝐫 1 \textstyle\mathrm{d}\mathbf{r}_{1}
  19. d 𝐫 2 \textstyle\mathrm{d}\mathbf{r}_{2}
  20. P ( N ) P^{(N)}
  21. n < N n<N
  22. 𝐫 1 , 𝐫 n \textstyle\mathbf{r}_{1}\,\ldots,\,\mathbf{r}_{n}
  23. N - n N-n
  24. 𝐫 n + 1 , 𝐫 N \mathbf{r}_{n+1}\,\ldots,\,\mathbf{r}_{N}
  25. P ( n ) ( 𝐫 1 , , 𝐫 n ) = 1 Z N e - β U N d 𝐫 n + 1 d 𝐫 N P^{(n)}(\mathbf{r}_{1},\ldots,\mathbf{r}_{n})=\frac{1}{Z_{N}}\int\cdots\int% \mathrm{e}^{-\beta U_{N}}\,\mathrm{d}\mathbf{r}_{n+1}\cdots\mathrm{d}\mathbf{r% }_{N}\,
  26. n n
  27. 𝐫 1 , 𝐫 n \textstyle\mathbf{r}_{1}\,\ldots,\,\mathbf{r}_{n}
  28. n n
  29. n = 1 n=1
  30. 𝐫 1 \textstyle\mathbf{r}_{1}
  31. 1 V ρ ( 1 ) ( 𝐫 1 ) d 𝐫 1 = ρ ( 1 ) = N V = ρ \frac{1}{V}\int\rho^{(1)}(\mathbf{r}_{1})\,\mathrm{d}\mathbf{r}_{1}=\rho^{(1)}% =\frac{N}{V}=\rho\,
  32. g ( n ) g^{(n)}
  33. g ( n ) g^{(n)}
  34. ρ ( n ) \rho^{(n)}
  35. ρ n \rho^{n}
  36. g ( n ) g^{(n)}
  37. g ( r ) g(r)
  38. g ( 2 ) ( 𝐫 1 , 𝐫 2 ) g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})
  39. g ( 2 ) ( 𝐫 1 , 𝐫 2 ) g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})
  40. 𝐫 12 = 𝐫 2 - 𝐫 1 \mathbf{r}_{12}=\mathbf{r}_{2}-\mathbf{r}_{1}
  41. g ( 𝐫 ) g ( 2 ) ( 𝐫 12 ) \textstyle g(\mathbf{r})\equiv g^{(2)}(\mathbf{r}_{12})
  42. ρ g ( 𝐫 ) dr = dn ( 𝐫 ) \textstyle\rho g(\mathbf{r})\mathrm{d}\rm{r}=\mathrm{d}n(\mathbf{r})
  43. N - 1 N-1
  44. d 𝐫 \textstyle\mathrm{d}\mathbf{r}
  45. 𝐫 \textstyle\mathbf{r}
  46. d n ( 𝐫 ) = i 0 δ ( 𝐫 - 𝐫 i ) d 𝐫 \textstyle\mathrm{d}n(\mathbf{r})=\langle\sum_{i\neq 0}\delta(\mathbf{r}-% \mathbf{r}_{i})\rangle\mathrm{d}\mathbf{r}
  47. \textstyle\langle\cdot\rangle
  48. g ( 𝐫 ) = 1 ρ i 0 δ ( 𝐫 - 𝐫 i ) = V N - 1 N δ ( 𝐫 - 𝐫 1 ) g(\mathbf{r})=\frac{1}{\rho}\langle\sum_{i\neq 0}\delta(\mathbf{r}-\mathbf{r}_% {i})\rangle=V\frac{N-1}{N}\left\langle\delta(\mathbf{r}-\mathbf{r}_{1})\right\rangle
  49. 1 , , N - 1 \textstyle 1,\,\ldots,\,N-1
  50. g ( 𝐫 ) g(\mathbf{r})
  51. S ( 𝐪 ) S(\mathbf{q})
  52. S ( 𝐪 ) = 1 / N i j e - i 𝐪 ( 𝐫 i - 𝐫 j ) \textstyle S(\mathbf{q})=1/N\langle\sum_{ij}\mathrm{e}^{-i\mathbf{q}(\mathbf{r% }_{i}-\mathbf{r}_{j})}\rangle
  53. S ( 𝐪 ) \displaystyle S(\mathbf{q})
  54. S ( 𝐪 ) = 1 + ρ V d 𝐫 e - i 𝐪𝐫 g ( 𝐫 ) S(\mathbf{q})=1+\rho\int_{V}\mathrm{d}\mathbf{r}\,\mathrm{e}^{-i\mathbf{q}% \mathbf{r}}g(\mathbf{r})
  55. g ( 𝐫 ) g(\mathbf{r})
  56. lim r g ( 𝐫 ) = 1 \textstyle\lim_{r\rightarrow\infty}g(\mathbf{r})=1
  57. V d 𝐫 g ( 𝐫 ) \textstyle\int_{V}\mathrm{d}\mathbf{r}g(\mathbf{r})
  58. V V
  59. S ( 𝐪 ) = S ( 𝐪 ) - ρ δ ( 𝐪 ) = 1 + ρ V d 𝐫 e - i 𝐪𝐫 [ g ( 𝐫 ) - 1 ] S^{\prime}(\mathbf{q})=S(\mathbf{q})-\rho\delta(\mathbf{q})=1+\rho\int_{V}% \mathrm{d}\mathbf{r}\,\mathrm{e}^{-i\mathbf{q}\mathbf{r}}[g(\mathbf{r})-1]
  60. S ( 𝐪 ) S ( 𝐪 ) S(\mathbf{q})\equiv S^{\prime}(\mathbf{q})
  61. q = 0 q=0
  62. χ T \textstyle\chi_{T}
  63. w ( 2 ) ( r ) w^{(2)}(r)
  64. U N = i > j = 1 N u ( | 𝐫 i - 𝐫 j | ) \textstyle U_{N}=\sum_{i>j=1}^{N}u(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|)
  65. P E = N 2 4 π ρ 0 r 2 u ( r ) g ( r ) d r PE=\frac{N}{2}4\pi\rho\int^{\infty}_{0}r^{2}u(r)g(r)dr
  66. ρ \rho
  67. P = ρ k B T - 2 3 π ρ 2 0 d r d u ( r ) d r r 3 g ( r ) P=\rho k_{B}T-\frac{2}{3}\pi\rho^{2}\int_{0}^{\infty}dr\frac{du(r)}{dr}r^{3}g(r)
  68. T T
  69. g ( r ) g(r)
  70. g ( r ) g(r)
  71. u ( r ) u(r)
  72. u ( r ) u(r)
  73. r r
  74. g ( r ) = 1 g(r)=1
  75. ρ \rho
  76. r r
  77. u ( r ) u(r)
  78. ρ \rho
  79. u ( r ) u(r)
  80. u ( r ) u(r)
  81. 𝐫 \mathbf{r}
  82. g ( r ) g(r)
  83. y ( r ) y(r)
  84. g ( r ) g(r)
  85. S ( q ) S(q)
  86. S ( q ) S(q)
  87. g ( r ) g(r)
  88. g ( r ) g(r)
  89. g ( k ) \textstyle g^{(k)}
  90. k > 2 \textstyle k>2

Radian_per_second.html

  1. ν = ω / 2 π \nu=\omega/{2\pi}

Radiation-dominated_era.html

  1. a ( t ) t 1 / 2 . a(t)\propto t^{1/2}.\,

Radius_of_curvature_(optics).html

  1. z ( r ) = r 2 R ( 1 + 1 - ( 1 + K ) r 2 R 2 ) + α 1 r 2 + α 2 r 4 + α 3 r 6 + , z(r)=\frac{r^{2}}{R\left(1+\sqrt{1-(1+K)\frac{r^{2}}{R^{2}}}\right)}+\alpha_{1% }r^{2}+\alpha_{2}r^{4}+\alpha_{3}r^{6}+\cdots,
  2. z ( r ) z(r)
  3. r r
  4. α 1 \alpha_{1}
  5. α 2 \alpha_{2}
  6. R R
  7. K K
  8. r = 0 r=0
  9. α i \alpha_{i}
  10. R R
  11. K K

Ramsey_sentence.html

  1. [ ( T 1 T 2 T n ) + ( C 1 C 2 C m ) ] [(T_{1}\land T_{2}\land\cdots\land T_{n})+(C_{1}\land C_{2}\land\cdots\land C_% {m})]
  2. \exists
  3. x 1 x n T C ( x 1 x n , o 1 o m ) \exists x_{1}\ldots\exists x_{n}TC(x_{1}\ldots x_{n},o_{1}\ldots o_{m})
  4. 1 i m 1\leq i\leq m
  5. F T + A T T C F_{T}+A_{T}\Leftrightarrow TC
  6. [ T C O R T C O ] [TC\models O\Leftrightarrow^{R}TC\models O]

Ramsey–Cass–Koopmans_model.html

  1. F ( K , A L ) F(K,AL)
  2. K K
  3. L L
  4. A A
  5. n n
  6. g g
  7. k ˙ = f ( k ) - c - ( n + g + δ ) k \dot{k}=f(k)-c-(n+g+\delta)k
  8. k k
  9. k ˙ \dot{k}
  10. d k d t \tfrac{dk}{dt}
  11. c c
  12. f ( k ) f(k)
  13. δ \delta\,
  14. k ˙ = 0 \dot{k}=0
  15. c c
  16. s = 1 - c s=1-c
  17. I = s Y = ( 1 - c ) Y I=sY=(1-c)Y
  18. I I
  19. Y Y
  20. s s
  21. r = ρ - % d M U * c ˙ r=\rho\ -\%dMU*\dot{c}\,
  22. u ( c ) = c 1 - θ - 1 1 - θ u(c)=\frac{c^{1-\theta}-1}{1-\theta}\,
  23. % d M U = d 2 u d c 2 d u d c = - θ c \%dMU=\frac{\frac{d^{2}u}{dc^{2}}}{\frac{du}{dc}}=-\frac{\theta}{c}
  24. c ˙ c = r - ρ θ \frac{\dot{c}}{c}=\frac{r-\rho}{\theta}\,
  25. y = k α y=k^{\alpha}\,
  26. R = α k α - 1 R=\alpha k^{\alpha-1}\,
  27. r = R - δ = α k α - 1 - δ r=R-\delta=\alpha k^{\alpha-1}-\delta\,
  28. k ˙ \dot{k}
  29. c ˙ \dot{c}

Rand_index.html

  1. n n
  2. S = { o 1 , , o n } S=\{o_{1},\ldots,o_{n}\}
  3. S S
  4. X = { X 1 , , X r } X=\{X_{1},\ldots,X_{r}\}
  5. Y = { Y 1 , , Y s } Y=\{Y_{1},\ldots,Y_{s}\}
  6. a a
  7. S S
  8. X X
  9. Y Y
  10. b b
  11. S S
  12. X X
  13. Y Y
  14. c c
  15. S S
  16. X X
  17. Y Y
  18. d d
  19. S S
  20. X X
  21. Y Y
  22. R R
  23. R = a + b a + b + c + d = a + b < m t p l > ( n 2 ) R=\frac{a+b}{a+b+c+d}=\frac{a+b}{<}mtpl>{{n\choose 2}}
  24. a + b a+b
  25. X X
  26. Y Y
  27. c + d c+d
  28. X X
  29. Y Y
  30. a = | S * | a=|S^{*}|
  31. S * = { ( o i , o j ) | o i , o j X k , o i , o j Y l } S^{*}=\{(o_{i},o_{j})|o_{i},o_{j}\in X_{k},o_{i},o_{j}\in Y_{l}\}
  32. b = | S * | b=|S^{*}|
  33. S * = { ( o i , o j ) | o i X k 1 , o j X k 2 , o i Y l 1 , o j Y l 2 } S^{*}=\{(o_{i},o_{j})|o_{i}\in X_{k_{1}},o_{j}\in X_{k_{2}},o_{i}\in Y_{l_{1}}% ,o_{j}\in Y_{l_{2}}\}
  34. c = | S * | c=|S^{*}|
  35. S * = { ( o i , o j ) | o i , o j X k , o i Y l 1 , o j Y l 2 } S^{*}=\{(o_{i},o_{j})|o_{i},o_{j}\in X_{k},o_{i}\in Y_{l_{1}},o_{j}\in Y_{l_{2% }}\}
  36. d = | S * | d=|S^{*}|
  37. S * = { ( o i , o j ) | o i X k 1 , o j X k 2 , o i , o j Y l } S^{*}=\{(o_{i},o_{j})|o_{i}\in X_{k_{1}},o_{j}\in X_{k_{2}},o_{i},o_{j}\in Y_{% l}\}
  38. 1 i , j n , i j , 1 k , k 1 , k 2 r , k 1 k 2 , 1 l , l 1 , l 2 s , l 1 l 2 1\leq i,j\leq n,i\neq j,1\leq k,k_{1},k_{2}\leq r,k_{1}\neq k_{2},1\leq l,l_{1% },l_{2}\leq s,l_{1}\neq l_{2}
  39. S S
  40. n n
  41. X = { X 1 , X 2 , , X r } X=\{X_{1},X_{2},\ldots,X_{r}\}
  42. Y = { Y 1 , Y 2 , , Y s } Y=\{Y_{1},Y_{2},\ldots,Y_{s}\}
  43. X X
  44. Y Y
  45. [ n i j ] \left[n_{ij}\right]
  46. n i j n_{ij}
  47. X i X_{i}
  48. Y j Y_{j}
  49. n i j = | X i Y j | n_{ij}=|X_{i}\cap Y_{j}|
  50. Y 1 Y_{1}
  51. Y 2 Y_{2}
  52. \ldots
  53. Y s Y_{s}
  54. X 1 X_{1}
  55. n 11 n_{11}
  56. n 12 n_{12}
  57. \ldots
  58. n 1 s n_{1s}
  59. a 1 a_{1}
  60. X 2 X_{2}
  61. n 21 n_{21}
  62. n 22 n_{22}
  63. \ldots
  64. n 2 s n_{2s}
  65. a 2 a_{2}
  66. \vdots
  67. \vdots
  68. \vdots
  69. \ddots
  70. \vdots
  71. \vdots
  72. X r X_{r}
  73. n r 1 n_{r1}
  74. n r 2 n_{r2}
  75. \ldots
  76. n r s n_{rs}
  77. a r a_{r}
  78. b 1 b_{1}
  79. b 2 b_{2}
  80. \ldots
  81. b s b_{s}
  82. A d j u s t e d I n d e x = I n d e x - E x p e c t e d I n d e x M a x I n d e x - E x p e c t e d I n d e x AdjustedIndex=\frac{Index-ExpectedIndex}{MaxIndex-ExpectedIndex}
  83. A R I = i j ( n i j 2 ) - [ i ( a i 2 ) j ( b j 2 ) ] / ( n 2 ) 1 2 [ i ( a i 2 ) + j ( b j 2 ) ] - [ i ( a i 2 ) j ( b j 2 ) ] / ( n 2 ) ARI=\frac{\sum_{ij}{\left({{n_{ij}}\atop{2}}\right)}-[\sum_{i}{\left({{a_{i}}% \atop{2}}\right)}\sum_{j}{\left({{b_{j}}\atop{2}}\right)}]/{\left({{n}\atop{2}% }\right)}}{\frac{1}{2}[\sum_{i}{\left({{a_{i}}\atop{2}}\right)}+\sum_{j}{\left% ({{b_{j}}\atop{2}}\right)}]-[\sum_{i}{\left({{a_{i}}\atop{2}}\right)}\sum_{j}{% \left({{b_{j}}\atop{2}}\right)}]/{\left({{n}\atop{2}}\right)}}
  84. n i j , a i , b j n_{ij},a_{i},b_{j}

Random_permutation_statistics.html

  1. 𝒫 \scriptstyle\mathcal{P}
  2. 𝒵 \scriptstyle\mathcal{Z}
  3. 𝔓 ( ( 𝒵 ) ) = 𝒫 . \mathfrak{P}(\mathfrak{C}(\mathcal{Z}))=\mathcal{P}.
  4. exp log 1 1 - z = 1 1 - z \exp\log\frac{1}{1-z}=\frac{1}{1-z}
  5. n 0 n ! n ! z n = 1 1 - z . \sum_{n\geq 0}\frac{n!}{n!}z^{n}=\frac{1}{1-z}.
  6. 𝔓 \scriptstyle\mathfrak{P}
  7. 𝔓 2 \scriptstyle\mathfrak{P}_{2}
  8. ( 𝒵 ) \scriptstyle\mathfrak{C}(\mathcal{Z})
  9. log 1 1 - z = k 1 z k k \log\frac{1}{1-z}=\sum_{k\geq 1}\frac{z^{k}}{k}
  10. b ( k ) : \scriptstyle b(k):\mathbb{N}\rightarrow\mathbb{R}
  11. b ( σ ) = c σ b ( c ) b(\sigma)=\sum_{c\in\sigma}b(c)
  12. σ \sigma
  13. g ( z , u ) = 1 + n 1 ( σ S n u b ( σ ) ) z n n ! = exp k 1 u b ( k ) z k k g(z,u)=1+\sum_{n\geq 1}\left(\sum_{\sigma\in S_{n}}u^{b(\sigma)}\right)\frac{z% ^{n}}{n!}=\exp\sum_{k\geq 1}u^{b(k)}\frac{z^{k}}{k}
  14. u g ( z , u ) | u = 1 = 1 1 - z k 1 b ( k ) z k k = n 1 ( σ S n b ( σ ) ) z n n ! \frac{\partial}{\partial u}g(z,u)\Bigg|_{u=1}=\frac{1}{1-z}\sum_{k\geq 1}b(k)% \frac{z^{k}}{k}=\sum_{n\geq 1}\left(\sum_{\sigma\in S_{n}}b(\sigma)\right)% \frac{z^{n}}{n!}
  15. g ( z ) = exp ( z + 1 2 z 2 ) . g(z)=\exp\left(z+\frac{1}{2}z^{2}\right).
  16. I ( n ) I(n)
  17. I ( n ) = n ! [ z n ] g ( z ) = n ! a + 2 b = n 1 a ! 2 b b ! = n ! b = 0 n / 2 1 ( n - 2 b ) ! 2 b b ! . I(n)=n![z^{n}]g(z)=n!\sum_{a+2b=n}\frac{1}{a!\;2^{b}\;b!}=n!\sum_{b=0}^{% \lfloor n/2\rfloor}\frac{1}{(n-2b)!\;2^{b}\;b!}.
  18. g ( z ) = exp ( d m z d d ) . g(z)=\exp\left(\sum_{d\mid m}\frac{z^{d}}{d}\right).
  19. n ! [ z n ] g ( z ) = n ! a + p b = n 1 a ! p b b ! = n ! b = 0 n / p 1 ( n - p b ) ! p b b ! . n![z^{n}]g(z)=n!\sum_{a+pb=n}\frac{1}{a!\;p^{b}\;b!}=n!\sum_{b=0}^{\lfloor n/p% \rfloor}\frac{1}{(n-pb)!\;p^{b}\;b!}.
  20. 𝒬 \mathcal{Q}
  21. 𝒬 = 𝔓 ( d k d ( 𝒵 ) ) . \mathcal{Q}=\mathfrak{P}\left(\sum_{d\mid k}\mathfrak{C}_{d}(\mathcal{Z})% \right).
  22. Q k ( z ) = exp ( d k z d d ) . Q_{k}(z)=\exp\left(\sum_{d\mid k}\frac{z^{d}}{d}\right).
  23. p n , d p_{n,d}
  24. q n , k q_{n,k}
  25. d | k p n , d = q n , k . \sum_{d|k}p_{n,d}=q_{n,k}.
  26. d | k q n , d × μ ( k / d ) = p n , k . \sum_{d|k}q_{n,d}\times\mu(k/d)=p_{n,k}.
  27. Q ( z ) = d k μ ( k / d ) × Q d ( z ) = d k μ ( k / d ) exp ( m d z m m ) . Q(z)=\sum_{d\mid k}\mu(k/d)\times Q_{d}(z)=\sum_{d\mid k}\mu(k/d)\exp\left(% \sum_{m\mid d}\frac{z^{m}}{m}\right).
  28. n ! [ z n ] Q ( z ) . n![z^{n}]Q(z).
  29. Q ( z ) = e z - e z + 1 / 2 z 2 - e z + 1 / 3 z 3 + e z + 1 / 2 z 2 + 1 / 3 z 3 + 1 / 6 z 6 Q(z)={\rm e}^{z}-{\rm e}^{z+1/2\,z^{2}}-{\rm e}^{z+1/3\,z^{3}}+{\rm e}^{z+1/2% \,z^{2}+1/3\,z^{3}+1/6\,z^{6}}
  30. 20 , 240 , 1470 , 10640 , 83160 , 584640 , 4496030 , 42658440 , 371762820 , 3594871280 , 20,240,1470,10640,83160,584640,4496030,42658440,371762820,3594871280,\ldots
  31. Q ( z ) = - e z + 1 / 2 z 2 + 1 / 4 z 4 + e z + 1 / 2 z 2 + 1 / 4 z 4 + 1 / 8 z 8 Q(z)=-{\rm e}^{z+1/2\,z^{2}+1/4\,z^{4}}+{\rm e}^{z+1/2\,z^{2}+1/4\,z^{4}+1/8\,% z^{8}}
  32. 5040 , 45360 , 453600 , 3326400 , 39916800 , 363242880 , 3874590720 , 34767532800 , 5040,45360,453600,3326400,39916800,363242880,3874590720,34767532800,\ldots
  33. Q ( z ) = e z + 1 / 2 z 2 - e z + 1 / 2 z 2 + 1 / 4 z 4 - e z + 1 / 2 z 2 + 1 / 3 z 3 + 1 / 6 z 6 + e z + 1 / 2 z 2 + 1 / 3 z 3 + 1 / 4 z 4 + 1 / 6 z 6 + 1 / 12 z 12 Q(z)={\rm e}^{z+1/2\,z^{2}}-{\rm e}^{z+1/2\,z^{2}+1/4\,z^{4}}-{\rm e}^{z+1/2\,% z^{2}+1/3\,{z}^{3}+1/6\,z^{6}}+{\rm e}^{z+1/2\,z^{2}+1/3\,z^{3}+1/4\,z^{4}+1/6% \,z^{6}+1/12\,z^{12}}
  34. 420 , 3360 , 30240 , 403200 , 4019400 , 80166240 , 965284320 , 12173441280 , 162850287600 , 420,3360,30240,403200,4019400,80166240,965284320,12173441280,162850287600,\ldots
  35. exp ( - z + k 1 z k k ) = e - z 1 - z . \exp\left(-z+\sum_{k\geq 1}\frac{z^{k}}{k}\right)=\frac{e^{-z}}{1-z}.
  36. 1 / ( 1 - z ) 1/(1-z)
  37. D ( n ) D(n)
  38. D ( n ) = n ! k = 0 n ( - 1 ) k k ! n ! e . D(n)=n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}\;\approx\;\frac{n!}{e}.
  39. n ! / e n!/e
  40. 1 / e . 1/e.
  41. A p A_{p}
  42. 1 p n \begin{matrix}1\leq p\leq n\end{matrix}
  43. | p A p | = p | A p | - p < q | A p A q | + p < q < r | A p A q A r | - ± | A p A s | . \left|\bigcup_{p}A_{p}\right|=\sum_{p}\left|A_{p}\right|\;-\;\sum_{p<q}\left|A% _{p}\cap A_{q}\right|\;+\;\sum_{p<q<r}\left|A_{p}\cap A_{q}\cap A_{r}\right|\;% -\;\cdots\;\pm\;\left|A_{p}\cap\;\cdots\;\cap A_{s}\right|.
  44. | A p | = ( n - 1 ) ! , | A p A q | = ( n - 2 ) ! , | A p A q A r | = ( n - 3 ) ! , \left|A_{p}\right|=(n-1)!\;,\;\;\left|A_{p}\cap A_{q}\right|=(n-2)!\;,\;\;% \left|A_{p}\cap A_{q}\cap A_{r}\right|=(n-3)!\;,\;\ldots
  45. n ! - ( n 1 ) ( n - 1 ) ! + ( n 2 ) ( n - 2 ) ! - ( n 3 ) ( n - 3 ) ! + ± ( n n ) ( n - n ) ! n!\;\;-\;\;{n\choose 1}(n-1)!\;\;+\;\;{n\choose 2}(n-2)!\;\;-\;\;{n\choose 3}(% n-3)!\;\;+\;\;\cdots\;\;\pm\;\;{n\choose n}(n-n)!
  46. n ! ( 1 - 1 1 ! + 1 2 ! - 1 3 ! + ± 1 n ! ) = n ! k = 0 n ( - 1 ) k k ! n!\left(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots\pm\frac{1}{n!}\right)=% n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}
  47. D ( n , m ) D(n,m)
  48. [ n ] [n]
  49. k = 1 k=1
  50. g ( z , u ) g(z,u)
  51. g ( z , u ) = exp ( - z + u z + k 1 z k k ) = e - z 1 - z e u z . g(z,u)=\exp\left(-z+uz+\sum_{k\geq 1}\frac{z^{k}}{k}\right)=\frac{e^{-z}}{1-z}% e^{uz}.
  52. [ u m ] g ( z , u ) = e - z 1 - z z m m ! [u^{m}]g(z,u)=\frac{e^{-z}}{1-z}\frac{z^{m}}{m!}
  53. D ( n , m ) = n ! [ z n ] [ u m ] g ( z , u ) = n ! m ! [ z n - m ] e - z 1 - z = n ! m ! k = 0 n - m ( - 1 ) k k ! . D(n,m)=n![z^{n}][u^{m}]g(z,u)=\frac{n!}{m!}[z^{n-m}]\frac{e^{-z}}{1-z}=\frac{n% !}{m!}\sum_{k=0}^{n-m}\frac{(-1)^{k}}{k!}.
  54. D ( n , m ) = ( n m ) D ( n - m , 0 ) and D ( n , m ) n ! e - 1 m ! D(n,m)={n\choose m}D(n-m,0)\;\;\mbox{ and }~{}\;\;\frac{D(n,m)}{n!}\approx% \frac{e^{-1}}{m!}
  55. D 0 ( n ) D_{0}(n)
  56. D 1 ( n ) D_{1}(n)
  57. g ( z , u ) = exp ( - u z + u log 1 1 - z ) = exp ( - u z ) ( 1 1 - z ) u . g(z,u)=\exp\left(-uz+u\log\frac{1}{1-z}\right)=\exp(-uz)\left(\frac{1}{1-z}% \right)^{u}.
  58. q ( z ) q(z)
  59. D 0 ( n ) D_{0}(n)
  60. q ( z ) = 1 2 × g ( z , - 1 ) + 1 2 × g ( z , 1 ) = 1 2 exp ( - z ) 1 1 - z + 1 2 exp ( z ) ( 1 - z ) . q(z)=\frac{1}{2}\times g(z,-1)+\frac{1}{2}\times g(z,1)=\frac{1}{2}\exp(-z)% \frac{1}{1-z}+\frac{1}{2}\exp(z)(1-z).
  61. D 0 ( n ) = n ! [ z n ] q ( z ) = 1 2 n ! k = 0 n ( - 1 ) k k ! + 1 2 n ! 1 n ! - 1 2 n ! 1 ( n - 1 ) ! D_{0}(n)=n![z^{n}]q(z)=\frac{1}{2}n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}+\frac{1}% {2}n!\frac{1}{n!}-\frac{1}{2}n!\frac{1}{(n-1)!}
  62. 1 2 n ! k = 0 n ( - 1 ) k k ! + 1 2 ( 1 - n ) 1 2 e n ! + 1 2 ( 1 - n ) . \frac{1}{2}n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}+\frac{1}{2}(1-n)\sim\frac{1}{2e% }n!+\frac{1}{2}(1-n).
  63. D 0 ( n ) D_{0}(n)
  64. D ( n ) D(n)
  65. D 1 ( n ) = 1 2 n ! k = 0 n ( - 1 ) k k ! - 1 2 ( 1 - n ) . D_{1}(n)=\frac{1}{2}n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}-\frac{1}{2}(1-n).
  66. D 0 ( n ) D_{0}(n)
  67. D 1 ( n ) D_{1}(n)
  68. n - 1. n-1.
  69. ( < m t p l > ( 99 49 ) ( 100 50 ) ) 100 = 1 2 100 , \left(\frac{<}{m}tpl>{{99\choose 49}}{{100\choose 50}}\right)^{100}=\frac{1}{2% ^{100}},
  70. 2 n 2n
  71. n n
  72. n n
  73. g ( z , u ) = exp ( z + z 2 2 + z 3 3 + + u z n + 1 n + 1 + u z n + 2 n + 2 + ) g(z,u)=\exp\left(z+\frac{z^{2}}{2}+\frac{z^{3}}{3}+\cdots+u\frac{z^{n+1}}{n+1}% +u\frac{z^{n+2}}{n+2}+\cdots\right)
  74. 1 1 - z exp ( ( u - 1 ) ( z n + 1 n + 1 + z n + 2 n + 2 + ) ) , \frac{1}{1-z}\exp\left((u-1)\left(\frac{z^{n+1}}{n+1}+\frac{z^{n+2}}{n+2}+% \cdots\right)\right),
  75. [ z 2 n ] [ u ] g ( z , u ) , [z^{2n}][u]g(z,u),\,
  76. n n
  77. 2 ( n + 1 ) > 2 n 2(n+1)>2n
  78. [ z 2 n ] [ u ] g ( z , u ) = [ z 2 n ] [ u ] 1 1 - z ( 1 + ( u - 1 ) ( z n + 1 n + 1 + z n + 2 n + 2 + ) ) , [z^{2n}][u]g(z,u)=[z^{2n}][u]\frac{1}{1-z}\left(1+(u-1)\left(\frac{z^{n+1}}{n+% 1}+\frac{z^{n+2}}{n+2}+\cdots\right)\right),
  79. [ z 2 n ] [ u ] g ( z , u ) = [ z 2 n ] 1 1 - z ( z n + 1 n + 1 + z n + 2 n + 2 + ) = k = n + 1 2 n 1 k = H 2 n - H n . [z^{2n}][u]g(z,u)=[z^{2n}]\frac{1}{1-z}\left(\frac{z^{n+1}}{n+1}+\frac{z^{n+2}% }{n+2}+\cdots\right)=\sum_{k=n+1}^{2n}\frac{1}{k}=H_{2n}-H_{n}.
  80. H 2 n - H n log 2 - 1 4 n + 1 16 n 2 - 1 128 n 4 + 1 256 n 6 - 17 4096 n 8 + , H_{2n}-H_{n}\sim\log 2-\frac{1}{4n}+\frac{1}{16n^{2}}-\frac{1}{128n^{4}}+\frac% {1}{256n^{6}}-\frac{17}{4096n^{8}}+\cdots,
  81. [ z 2 n ] [ u ] g ( z , u ) < log 2 and 1 - [ z 2 n ] [ u ] g ( z , u ) > 1 - log 2 = 0.30685281 , [z^{2n}][u]g(z,u)<\log 2\quad\mbox{and}~{}\quad 1-[z^{2n}][u]g(z,u)>1-\log 2=0% .30685281,
  82. 2 n 2n
  83. n n
  84. p k = Pr [ there is a cycle of length k ] , p_{k}=\Pr[\mbox{there is a cycle of length }~{}k],
  85. Pr [ there is a cycle of length > n ] = k = n + 1 2 n p k . \Pr[\mbox{there is a cycle of length}~{}>n]=\sum_{k=n+1}^{2n}p_{k}.
  86. k > n k>n
  87. k k
  88. ( 2 n k ) k ! k ( 2 n - k ) ! . {{2n}\choose k}\cdot\frac{k!}{k}\cdot(2n-k)!.
  89. ( 2 n k ) {{2n}\choose k}
  90. k k
  91. k ! k \frac{k!}{k}
  92. k k
  93. ( 2 n - k ) ! (2n-k)!
  94. p k = ( 2 n k ) k ! k ( 2 n - k ) ! ( 2 n ) ! = 1 k . p_{k}=\frac{{{2n}\choose k}\cdot\frac{k!}{k}\cdot(2n-k)!}{(2n)!}=\frac{1}{k}.
  95. Pr [ there is a cycle of length > n ] = k = n + 1 2 n 1 k = H 2 n - H n . \Pr[\mbox{there is a cycle of length}~{}>n]=\sum_{k=n+1}^{2n}\frac{1}{k}=H_{2n% }-H_{n}.
  96. 𝒬 \mathcal{Q}
  97. 𝒬 = 𝔓 ( q 1 = q ( 𝒵 ) × p = 1 q ( q p ) 𝒰 p ) . \mathcal{Q}=\mathfrak{P}\left(\sum_{q\geq 1}\mathfrak{C}_{=q}(\mathcal{Z})% \times\sum_{p=1}^{q}{q\choose p}\mathcal{U}^{p}\right).
  98. G ( z , u ) = exp ( q 1 z q q p = 1 q ( q p ) u p ) . G(z,u)=\exp\left(\sum_{q\geq 1}\frac{z^{q}}{q}\sum_{p=1}^{q}{q\choose p}u^{p}% \right).
  99. exp ( q 1 z q q ( u + 1 ) q - q 1 z q q ) \exp\left(\sum_{q\geq 1}\frac{z^{q}}{q}(u+1)^{q}-\sum_{q\geq 1}\frac{z^{q}}{q}\right)
  100. exp ( log 1 1 - ( u + 1 ) z - log 1 1 - z ) = 1 - z 1 - ( u + 1 ) z . \exp\left(\log\frac{1}{1-(u+1)z}-\log\frac{1}{1-z}\right)=\frac{1-z}{1-(u+1)z}.
  101. ( 1 - z ) q 0 ( u + 1 ) q z q . (1-z)\sum_{q\geq 0}(u+1)^{q}z^{q}.
  102. [ z n ] G ( z , u ) = ( u + 1 ) n - ( u + 1 ) n - 1 [z^{n}]G(z,u)=(u+1)^{n}-(u+1)^{n-1}
  103. [ u k ] [ z n ] G ( z , u ) = ( n k ) - ( n - 1 k ) . [u^{k}][z^{n}]G(z,u)={n\choose k}-{n-1\choose k}.
  104. ( n k ) {n\choose k}
  105. 1 - ( n - 1 ) ! k ! ( n - 1 - k ) ! k ! ( n - k ) ! n ! = 1 - n - k n = k n . 1-\frac{(n-1)!}{k!(n-1-k)!}\frac{k!(n-k)!}{n!}=1-\frac{n-k}{n}=\frac{k}{n}.
  106. G ( z , u ) G(z,u)
  107. G = S m G=S_{m}
  108. 𝔓 m ( ( 𝒵 ) ) \mathfrak{P}_{m}(\mathfrak{C}(\mathcal{Z}))
  109. g m ( z ) = 1 | S m | ( log 1 1 - z ) m = 1 m ! ( log 1 1 - z ) m . g_{m}(z)=\frac{1}{|S_{m}|}\left(\log\frac{1}{1-z}\right)^{m}=\frac{1}{m!}\left% (\log\frac{1}{1-z}\right)^{m}.
  110. ( - 1 ) n + m n ! [ z n ] g m ( z ) = s ( n , m ) (-1)^{n+m}n!\;[z^{n}]g_{m}(z)=s(n,m)
  111. g m ( z ) g_{m}(z)
  112. n ! [ z n ] g m ( z ) = [ n m ] . n![z^{n}]g_{m}(z)=\left[\begin{matrix}n\\ m\end{matrix}\right].
  113. s n ( w ) = m = 0 n s ( n , m ) w m . s_{n}(w)=\sum_{m=0}^{n}s(n,m)w^{m}.
  114. g m ( z ) = n m ( - 1 ) n + m n ! s ( n , m ) z n g_{m}(z)=\sum_{n\geq m}\frac{(-1)^{n+m}}{n!}s(n,m)z^{n}
  115. ( - 1 ) m g m ( z ) w m = n m ( - 1 ) n n ! s ( n , m ) w m z n . (-1)^{m}g_{m}(z)w^{m}=\sum_{n\geq m}\frac{(-1)^{n}}{n!}s(n,m)w^{m}z^{n}.
  116. m 0 ( - 1 ) m g m ( z ) w m = m 0 n m ( - 1 ) n n ! s ( n , m ) w m z n = n 0 ( - 1 ) n n ! z n m = 0 n s ( n , m ) w m . \sum_{m\geq 0}(-1)^{m}g_{m}(z)w^{m}=\sum_{m\geq 0}\sum_{n\geq m}\frac{(-1)^{n}% }{n!}s(n,m)w^{m}z^{n}=\sum_{n\geq 0}\frac{(-1)^{n}}{n!}z^{n}\sum_{m=0}^{n}s(n,% m)w^{m}.
  117. g m ( z ) g_{m}(z)
  118. s n ( w ) s_{n}(w)
  119. ( 1 - z ) w = n 0 ( w n ) ( - 1 ) n z n = n 0 ( - 1 ) n n ! s n ( w ) z n . (1-z)^{w}=\sum_{n\geq 0}{w\choose n}(-1)^{n}z^{n}=\sum_{n\geq 0}\frac{(-1)^{n}% }{n!}s_{n}(w)z^{n}.
  120. z n z^{n}
  121. s n ( w ) = w ( w - 1 ) ( w - 2 ) ( w - ( n - 1 ) ) = ( w ) n , s_{n}(w)=w\;(w-1)\;(w-2)\;\cdots\;(w-(n-1))=(w)_{n},
  122. u g ( z , u ) | u = 1 = 1 1 - z k 1 b ( k ) z k k = 1 1 - z z m m \frac{\partial}{\partial u}g(z,u)\Bigg|_{u=1}=\frac{1}{1-z}\sum_{k\geq 1}b(k)% \frac{z^{k}}{k}=\frac{1}{1-z}\frac{z^{m}}{m}
  123. 1 m z m + 1 m z m + 1 + 1 m z m + 2 + \frac{1}{m}z^{m}\;+\;\frac{1}{m}z^{m+1}\;+\;\frac{1}{m}z^{m+2}\;+\;\cdots
  124. 1 1 - z k = 1 m z k k and [ z n ] 1 1 - z k = 1 m z k k = H m for n m \frac{1}{1-z}\sum_{k=1}^{m}\frac{z^{k}}{k}\mbox{ and }~{}[z^{n}]\frac{1}{1-z}% \sum_{k=1}^{m}\frac{z^{k}}{k}=H_{m}\mbox{ for }~{}n\geq m
  125. g ( z , u ) g(z,u)
  126. g ( z , u ) = exp ( - z + u z + log 1 1 - z ) = 1 1 - z exp ( - z + u z ) . g(z,u)=\exp\left(-z+uz+\log\frac{1}{1-z}\right)=\frac{1}{1-z}\exp(-z+uz).
  127. E ( X m ) = E ( k = 0 m { m k } ( X ) k ) = k = 0 m { m k } E ( ( X ) k ) , E(X^{m})=E\left(\sum_{k=0}^{m}\left\{\begin{matrix}m\\ k\end{matrix}\right\}(X)_{k}\right)=\sum_{k=0}^{m}\left\{\begin{matrix}m\\ k\end{matrix}\right\}E((X)_{k}),
  128. ( X ) k (X)_{k}
  129. g ( z , u ) g(z,u)
  130. E ( ( X ) k ) = [ z n ] ( d d u ) k g ( z , u ) | u = 1 = [ z n ] z k 1 - z exp ( - z + u z ) | u = 1 = [ z n ] z k 1 - z , E((X)_{k})=[z^{n}]\left(\frac{d}{du}\right)^{k}g(z,u)\Bigg|_{u=1}=[z^{n}]\frac% {z^{k}}{1-z}\exp(-z+uz)\Bigg|_{u=1}=[z^{n}]\frac{z^{k}}{1-z},
  131. k > n k>n
  132. k n k<=n
  133. E ( X m ) = k = 0 n { m k } . E(X^{m})=\sum_{k=0}^{n}\left\{\begin{matrix}m\\ k\end{matrix}\right\}.
  134. σ \sigma
  135. k k
  136. k k
  137. E [ F k ] E[F_{k}]
  138. d d
  139. k k
  140. d d
  141. d d
  142. k . k.
  143. u d . u^{d}.
  144. E [ F 6 ] . E[F_{6}].
  145. g ( z , u ) = exp ( u z - z + u 2 z 2 2 - z 2 2 + u 3 z 3 3 - z 3 3 + u 6 z 6 6 - z 6 6 + log 1 1 - z ) g(z,u)=\exp\left(uz-z+u^{2}\frac{z^{2}}{2}-\frac{z^{2}}{2}+u^{3}\frac{z^{3}}{3% }-\frac{z^{3}}{3}+u^{6}\frac{z^{6}}{6}-\frac{z^{6}}{6}+\log\frac{1}{1-z}\right)
  146. 1 1 - z exp ( u z - z + u 2 z 2 2 - z 2 2 + u 3 z 3 3 - z 3 3 + u 6 z 6 6 - z 6 6 ) . \frac{1}{1-z}\exp\left(uz-z+u^{2}\frac{z^{2}}{2}-\frac{z^{2}}{2}+u^{3}\frac{z^% {3}}{3}-\frac{z^{3}}{3}+u^{6}\frac{z^{6}}{6}-\frac{z^{6}}{6}\right).
  147. u g ( z , u ) | u = 1 = z + z 2 + z 3 + z 6 1 - z exp ( u z - z + u 2 z 2 2 - z 2 2 + u 3 z 3 3 - z 3 3 + u 6 z 6 6 - z 6 6 ) | u = 1 \left.\frac{\partial}{\partial u}g(z,u)\right|_{u=1}=\left.\frac{z+z^{2}+z^{3}% +z^{6}}{1-z}\exp\left(uz-z+u^{2}\frac{z^{2}}{2}-\frac{z^{2}}{2}+u^{3}\frac{z^{% 3}}{3}-\frac{z^{3}}{3}+u^{6}\frac{z^{6}}{6}-\frac{z^{6}}{6}\right)\right|_{u=1}
  148. z + z 2 + z 3 + z 6 1 - z . \frac{z+z^{2}+z^{3}+z^{6}}{1-z}.
  149. E [ F 6 ] = 4 E[F_{6}]=4
  150. n 6 n\geq 6
  151. g ( z , u ) = exp ( d k ( u d z d d - z d d ) + log 1 1 - z ) = 1 1 - z exp ( d k ( u d z d d - z d d ) ) . g(z,u)=\exp\left(\sum_{d\mid k}\left(u^{d}\frac{z^{d}}{d}-\frac{z^{d}}{d}% \right)+\log\frac{1}{1-z}\right)=\frac{1}{1-z}\exp\left(\sum_{d\mid k}\left(u^% {d}\frac{z^{d}}{d}-\frac{z^{d}}{d}\right)\right).
  152. u g ( z , u ) | u = 1 = d k z d 1 - z exp ( d k ( u d z d d - z d d ) ) | u = 1 = d k z d 1 - z . \left.\frac{\partial}{\partial u}g(z,u)\right|_{u=1}=\left.\frac{\sum_{d\mid k% }z^{d}}{1-z}\exp\left(\sum_{d\mid k}\left(u^{d}\frac{z^{d}}{d}-\frac{z^{d}}{d}% \right)\right)\right|_{u=1}=\frac{\sum_{d\mid k}z^{d}}{1-z}.
  153. E [ F k ] E[F_{k}]
  154. τ ( k ) \tau(k)
  155. k k
  156. n k . n\geq k.
  157. 1 1
  158. n = 1 n=1
  159. n n
  160. k k
  161. k k
  162. g ( z , u ) g(z,u)
  163. b ( k ) b(k)
  164. b ( k ) b(k)
  165. g ( z , u ) g(z,u)
  166. g ( z , u ) = ( 1 1 - z ) u g(z,u)=\left(\frac{1}{1-z}\right)^{u}
  167. u g ( z , u ) | u = 1 = 1 1 - z k 1 b ( k ) z k k = 1 1 - z k 1 z k k = 1 1 - z log 1 1 - z . \frac{\partial}{\partial u}g(z,u)\Bigg|_{u=1}=\frac{1}{1-z}\sum_{k\geq 1}b(k)% \frac{z^{k}}{k}=\frac{1}{1-z}\sum_{k\geq 1}\frac{z^{k}}{k}=\frac{1}{1-z}\log% \frac{1}{1-z}.
  168. H n H_{n}
  169. log n \log n
  170. g ( z , u ) g(z,u)
  171. 𝒫 \mathcal{P}
  172. n / 2 n/2
  173. u u
  174. g ( z , u ) = exp ( u k > n 2 z k k + k = 1 n 2 z k k ) . g(z,u)=\exp\left(u\sum_{k>\lfloor\frac{n}{2}\rfloor}^{\infty}\frac{z^{k}}{k}+% \sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\frac{z^{k}}{k}\right).
  175. n 2 \frac{n}{2}
  176. n ! [ u z n ] g ( z , u ) = n ! [ z n ] exp ( k = 1 n 2 z k k ) k > n 2 z k k n![uz^{n}]g(z,u)=n![z^{n}]\exp\left(\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}% \frac{z^{k}}{k}\right)\sum_{k>\lfloor\frac{n}{2}\rfloor}^{\infty}\frac{z^{k}}{k}
  177. n ! [ z n ] exp ( log 1 1 - z - k > n 2 z k k ) k > n 2 z k k n![z^{n}]\exp\left(\log\frac{1}{1-z}-\sum_{k>\lfloor\frac{n}{2}\rfloor}^{% \infty}\frac{z^{k}}{k}\right)\sum_{k>\lfloor\frac{n}{2}\rfloor}^{\infty}\frac{% z^{k}}{k}
  178. n ! [ z n ] 1 1 - z exp ( - k > n 2 z k k ) k > n 2 z k k = n ! [ z n ] 1 1 - z m = 0 ( - 1 ) m m ! ( k > n 2 z k k ) m + 1 n![z^{n}]\frac{1}{1-z}\exp\left(-\sum_{k>\lfloor\frac{n}{2}\rfloor}^{\infty}% \frac{z^{k}}{k}\right)\sum_{k>\lfloor\frac{n}{2}\rfloor}^{\infty}\frac{z^{k}}{% k}=n![z^{n}]\frac{1}{1-z}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!}\left(\sum_{k>% \lfloor\frac{n}{2}\rfloor}^{\infty}\frac{z^{k}}{k}\right)^{m+1}
  179. z z
  180. m + 1 m+1
  181. n 2 \lfloor\frac{n}{2}\rfloor
  182. m > 0 m>0
  183. [ z n ] . [z^{n}].
  184. n ! [ z n ] 1 1 - z k > n 2 z k k = n ! k = n 2 + 1 n 1 k . n![z^{n}]\frac{1}{1-z}\sum_{k>\lfloor\frac{n}{2}\rfloor}^{\infty}\frac{z^{k}}{% k}=n!\sum_{k=\lfloor\frac{n}{2}\rfloor+1}^{n}\frac{1}{k}.
  185. k = 1 n 1 k - k = 1 n 2 1 k = k = 1 n 1 k - 2 k = 1 n 2 1 2 k = k = 1 k even n ( 1 - 2 ) 1 k + k = 1 k odd n 1 k \sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\frac{1}{k}=% \sum_{k=1}^{n}\frac{1}{k}-2\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\frac{1}{2k}=% \sum_{k=1\atop k\;\,\text{even}}^{n}(1-2)\frac{1}{k}+\sum_{k=1\atop k\;\,\text% {odd}}^{n}\frac{1}{k}
  186. n ! k = 1 n ( - 1 ) k + 1 k n ! log 2. n!\sum_{k=1}^{n}\frac{(-1)^{k+1}}{k}\sim n!\log 2.
  187. ( 1 2 34 ) (1\;2\;34)\,
  188. ( 1 2 ) ( 2 3 ) ( 3 4 ) (1\;2)\;(2\;3)\;(3\;4)
  189. b ( k ) b(k)
  190. k - 1 k-1
  191. g ( z , u ) = ( 1 1 - u z ) 1 / u g(z,u)=\left(\frac{1}{1-uz}\right)^{1/u}
  192. u g ( z , u ) | u = 1 = 1 1 - z k 1 ( k - 1 ) z k k = z ( 1 - z ) 2 - 1 1 - z log 1 1 - z . \frac{\partial}{\partial u}g(z,u)\Bigg|_{u=1}=\frac{1}{1-z}\sum_{k\geq 1}(k-1)% \frac{z^{k}}{k}=\frac{z}{(1-z)^{2}}-\frac{1}{1-z}\log\frac{1}{1-z}.
  193. T ( n ) T(n)
  194. T ( n ) = n - H n . T(n)=n-H_{n}.\,
  195. log n \log n\,
  196. g ( z , u ) g(z,u)
  197. ( - 1 ) m n ! [ z n ] [ u m ] g ( z , u ) = [ n n - m ] (-1)^{m}n!\;[z^{n}][u^{m}]g(z,u)=\left[\begin{matrix}n\\ n-m\end{matrix}\right]
  198. ( - 1 ) n + m n ! [ z n ] [ u m ] g ( z , u ) | u = 1 / u | z = u z = [ n m ] (-1)^{n+m}n!\;[z^{n}][u^{m}]g(z,u)|_{u=1/u}|_{z=uz}=\left[\begin{matrix}n\\ m\end{matrix}\right]
  199. [ u m ] g ( z , u ) | u = 1 / u | z = u z = [ u m ] ( 1 1 - z ) u = 1 m ! ( log 1 1 - z ) m , [u^{m}]g(z,u)|_{u=1/u}|_{z=uz}=[u^{m}]\left(\frac{1}{1-z}\right)^{u}=\frac{1}{% m!}\left(\log\frac{1}{1-z}\right)^{m},
  200. σ \sigma
  201. b ( k ) b(k)
  202. k 2 k^{2}
  203. u g ( z , u ) | u = 1 = 1 1 - z k 1 k 2 z k k = 1 1 - z z ( 1 - z ) 2 = z ( 1 - z ) 3 . \frac{\partial}{\partial u}g(z,u)\Bigg|_{u=1}=\frac{1}{1-z}\sum_{k\geq 1}k^{2}% \frac{z^{k}}{k}=\frac{1}{1-z}\frac{z}{(1-z)^{2}}=\frac{z}{(1-z)^{3}}.
  204. 1 n [ z n ] z ( 1 - z ) 3 = 1 n 1 2 n ( n + 1 ) = 1 2 ( n + 1 ) . \frac{1}{n}[z^{n}]\frac{z}{(1-z)^{3}}=\frac{1}{n}\frac{1}{2}n(n+1)=\frac{1}{2}% (n+1).
  205. [ n ] [n]
  206. b ( k ) b(k)
  207. m m
  208. m = k m=k
  209. u g ( z , u ) | u = 1 = 1 1 - z k 1 b ( k ) z k k = 1 1 - z m z m m = z m 1 - z . \frac{\partial}{\partial u}g(z,u)\Bigg|_{u=1}=\frac{1}{1-z}\sum_{k\geq 1}b(k)% \frac{z^{k}}{k}=\frac{1}{1-z}\;m\;\frac{z^{m}}{m}=\frac{z^{m}}{1-z}.
  210. 1 n [ z n ] z m 1 - z = { 1 n , if n m 0 , otherwise. \frac{1}{n}[z^{n}]\frac{z^{m}}{1-z}=\begin{cases}\frac{1}{n},&\mbox{if }~{}n% \geq m\\ 0,&\mbox{otherwise.}\end{cases}
  211. ( k m ) \begin{matrix}{k\choose m}\end{matrix}
  212. ( k m ) \begin{matrix}{k\choose m}\end{matrix}
  213. ( k m ) = 0 \begin{matrix}{k\choose m}=0\end{matrix}
  214. u g ( z , u ) | u = 1 = 1 1 - z k m ( k m ) z k k = 1 1 - z 1 m z m ( 1 - z ) m = 1 m z m ( 1 - z ) m + 1 . \frac{\partial}{\partial u}g(z,u)\Bigg|_{u=1}=\frac{1}{1-z}\sum_{k\geq m}{k% \choose m}\frac{z^{k}}{k}=\frac{1}{1-z}\frac{1}{m}\frac{z^{m}}{(1-z)^{m}}=% \frac{1}{m}\frac{z^{m}}{(1-z)^{m+1}}.
  215. ( n m ) - 1 [ z n ] 1 m z m ( 1 - z ) m + 1 = ( n m ) - 1 1 m [ z n - m ] 1 ( 1 - z ) m + 1 {n\choose m}^{-1}[z^{n}]\frac{1}{m}\frac{z^{m}}{(1-z)^{m+1}}={n\choose m}^{-1}% \frac{1}{m}[z^{n-m}]\frac{1}{(1-z)^{m+1}}
  216. 1 m ( n m ) - 1 ( ( n - m ) + m m ) = 1 m . \frac{1}{m}{n\choose m}^{-1}{(n-m)\;+\;m\choose m}=\frac{1}{m}.
  217. 𝔓 ( odd ( 𝒵 ) ) 𝔓 even ( even ( 𝒵 ) ) . \mathfrak{P}(\mathfrak{C}_{\operatorname{odd}}(\mathcal{Z}))\mathfrak{P}_{% \operatorname{even}}(\mathfrak{C}_{\operatorname{even}}(\mathcal{Z})).
  218. exp ( 1 2 log 1 + z 1 - z ) cosh ( 1 2 log 1 1 - z 2 ) \exp\left(\frac{1}{2}\log\frac{1+z}{1-z}\right)\cosh\left(\frac{1}{2}\log\frac% {1}{1-z^{2}}\right)
  219. 1 2 exp ( 1 2 ( log 1 + z 1 - z + log 1 1 - z 2 ) ) + 1 2 exp ( 1 2 ( log 1 + z 1 - z - log 1 1 - z 2 ) ) . \frac{1}{2}\exp\left(\frac{1}{2}\left(\log\frac{1+z}{1-z}+\log\frac{1}{1-z^{2}% }\right)\right)+\frac{1}{2}\exp\left(\frac{1}{2}\left(\log\frac{1+z}{1-z}-\log% \frac{1}{1-z^{2}}\right)\right).
  220. 1 2 exp ( 1 2 log 1 ( 1 - z ) 2 ) + 1 2 exp ( 1 2 log ( 1 + z ) 2 ) \frac{1}{2}\exp\left(\frac{1}{2}\log\frac{1}{(1-z)^{2}}\right)+\frac{1}{2}\exp% \left(\frac{1}{2}\log(1+z)^{2}\right)
  221. 1 2 1 1 - z + 1 2 ( 1 + z ) = 1 + z + 1 2 z 2 1 - z . \frac{1}{2}\frac{1}{1-z}+\frac{1}{2}(1+z)=1+z+\frac{1}{2}\frac{z^{2}}{1-z}.
  222. n 2 n\geq 2
  223. n ! / 2 n!/2\,
  224. ( 1 8 9 11 13 ) (1\;8\;9\;11\;13)
  225. ( 1 9 13 8 11 ) (1\;9\;13\;8\;11)
  226. ( 5 13 6 9 ) (5\;13\;6\;9)
  227. ( 5 6 ) ( 9 13 ) (5\;6)\;(9\;13)
  228. 𝔓 ( odd ( 𝒵 ) ) 𝔓 even ( 2 ( 𝒵 ) ) 𝔓 even ( 4 ( 𝒵 ) ) 𝔓 even ( 6 ( 𝒵 ) ) \mathfrak{P}(\mathfrak{C}_{\operatorname{odd}}(\mathcal{Z}))\mathfrak{P}_{% \operatorname{even}}(\mathfrak{C}_{2}(\mathcal{Z}))\mathfrak{P}_{\operatorname% {even}}(\mathfrak{C}_{4}(\mathcal{Z}))\mathfrak{P}_{\operatorname{even}}(% \mathfrak{C}_{6}(\mathcal{Z}))\cdots
  229. exp ( 1 2 log 1 + z 1 - z ) m 1 cosh z 2 m 2 m = 1 + z 1 - z m 1 cosh z 2 m 2 m . \exp\left(\frac{1}{2}\log\frac{1+z}{1-z}\right)\prod_{m\geq 1}\cosh\frac{z^{2m% }}{2m}=\sqrt{\frac{1+z}{1-z}}\prod_{m\geq 1}\cosh\frac{z^{2m}}{2m}.
  230. 𝔓 ( odd ( 𝒵 ) ) ( 𝔓 3 ( 2 ( 𝒵 ) ) + 2 ( 𝒵 ) 4 ( 𝒵 ) + 6 ( 𝒵 ) ) \mathfrak{P}(\mathfrak{C}_{\operatorname{odd}}(\mathcal{Z}))\left(\mathfrak{P}% _{3}(\mathfrak{C}_{2}(\mathcal{Z}))+\mathfrak{C}_{2}(\mathcal{Z})\mathfrak{C}_% {4}(\mathcal{Z})+\mathfrak{C}_{6}(\mathcal{Z})\right)
  231. 1 + z 1 - z ( 1 6 ( z 2 2 ) 3 + z 2 2 z 4 4 + z 6 6 ) = 5 16 z 6 1 + z 1 - z . \sqrt{\frac{1+z}{1-z}}\left(\frac{1}{6}\left(\frac{z^{2}}{2}\right)^{3}+\frac{% z^{2}}{2}\frac{z^{4}}{4}+\frac{z^{6}}{6}\right)=\frac{5}{16}z^{6}\sqrt{\frac{1% +z}{1-z}}.
  232. 0 , 0 , 0 , 0 , 0 , 225 , 1575 , 6300 , 56700 , 425250 , 4677750 , 46777500 , 608107500 , 0,0,0,0,0,225,1575,6300,56700,425250,4677750,46777500,608107500,\ldots
  233. 𝔓 ( odd ( 𝒵 ) ) ( 𝔓 1 ( 2 ( 𝒵 ) ) + 𝔓 1 ( 4 ( 𝒵 ) ) + 𝔓 1 ( 6 ( 𝒵 ) ) + ) \mathfrak{P}(\mathfrak{C}_{\operatorname{odd}}(\mathcal{Z}))\left(\mathfrak{P}% _{\geq 1}(\mathfrak{C}_{2}(\mathcal{Z}))+\mathfrak{P}_{\geq 1}(\mathfrak{C}_{4% }(\mathcal{Z}))+\mathfrak{P}_{\geq 1}(\mathfrak{C}_{6}(\mathcal{Z}))+\cdots\right)
  234. 1 + z 1 - z ( exp ( z 2 2 ) - 1 + exp ( z 4 4 ) - 1 + exp ( z 6 6 ) - 1 + ) . \sqrt{\frac{1+z}{1-z}}\left(\exp\left(\frac{z^{2}}{2}\right)-1\,+\,\exp\left(% \frac{z^{4}}{4}\right)-1\,+\,\exp\left(\frac{z^{6}}{6}\right)-1\,+\,\cdots% \right).
  235. 0 , 1 , 3 , 15 , 75 , 405 , 2835 , 22155 , 199395 , 1828575 , 0,1,3,15,75,405,2835,22155,199395,1828575,\ldots
  236. 𝔓 ( odd ( 𝒵 ) ) 𝔓 ( 2 ( 𝒵 ) + 4 ( 𝒵 ) ) \mathfrak{P}(\mathfrak{C}_{\operatorname{odd}}(\mathcal{Z}))\mathfrak{P}(% \mathfrak{C}_{2}(\mathcal{Z})+\mathfrak{C}_{4}(\mathcal{Z}))
  237. 1 + z 1 - z exp ( z 2 2 + z 4 4 ) . \sqrt{\frac{1+z}{1-z}}\exp\left(\frac{z^{2}}{2}+\frac{z^{4}}{4}\right).
  238. 1 , 2 , 6 , 24 , 120 , 600 , 4200 , 28560 , 257040 , 2207520 , 24282720 , 258128640 , 1,2,6,24,120,600,4200,28560,257040,2207520,24282720,258128640,\ldots
  239. 𝒵 \mathcal{Z}
  240. g ( z ) = h ( z ) 1 + z 1 - z , g(z)=h(z)\sqrt{\frac{1+z}{1-z}},
  241. h ( z ) h(z)
  242. 1 1 + z g ( z ) = h ( z ) 1 1 - z 2 \frac{1}{1+z}\;g(z)=h(z)\;\frac{1}{\sqrt{1-z^{2}}}
  243. 1 1 + z g ( z ) = 1 1 - z g ( - z ) or ( 1 - z ) g ( z ) = ( 1 + z ) g ( - z ) . \frac{1}{1+z}\;g(z)=\frac{1}{1-z}\;g(-z)\quad\mbox{ or }~{}\quad(1-z)\;g(z)=(1% +z)\;g(-z).
  244. g n = [ z n ] g ( z ) g_{n}=[z^{n}]g(z)\,
  245. g 2 m + 1 ( 2 m + 1 ) ! - g 2 m ( 2 m ) ! = - g 2 m + 1 ( 2 m + 1 ) ! + g 2 m ( 2 m ) ! or 2 g 2 m + 1 ( 2 m + 1 ) ! = 2 g 2 m ( 2 m ) ! \frac{g_{2m+1}}{(2m+1)!}-\frac{g_{2m}}{(2m)!}=-\frac{g_{2m+1}}{(2m+1)!}+\frac{% g_{2m}}{(2m)!}\quad\mbox{ or }~{}\quad 2\frac{g_{2m+1}}{(2m+1)!}=2\frac{g_{2m}% }{(2m)!}
  246. g 2 m + 1 = ( 2 m + 1 ) g 2 m . g_{2m+1}=(2m+1)g_{2m}\,.
  247. π S n σ ( π ) ν ( π ) + 1 = ( - 1 ) n + 1 n n + 1 , \sum_{\pi\in S_{n}}\frac{\sigma(\pi)}{\nu(\pi)+1}=(-1)^{n+1}\frac{n}{n+1},
  248. n ! n!
  249. [ n ] [n]
  250. σ ( π ) \sigma(\pi)
  251. π \pi
  252. σ ( π ) = 1 \sigma(\pi)=1
  253. π \pi
  254. σ ( π ) = - 1 \sigma(\pi)=-1
  255. π \pi
  256. ν ( π ) \nu(\pi)
  257. π \pi
  258. π \pi
  259. σ ( π ) = c π ( - 1 ) | c | - 1 , \sigma(\pi)=\prod_{c\in\pi}(-1)^{|c|-1},
  260. π \pi
  261. 𝔓 ( - 𝒵 + 𝒱 𝒵 + 1 ( 𝒵 ) + 𝒰 2 ( 𝒵 ) + 𝒰 2 3 ( 𝒵 ) + 𝒰 3 4 ( 𝒵 ) + ) \mathfrak{P}(-\mathcal{Z}+\mathcal{V}\mathcal{Z}+\mathfrak{C}_{1}(\mathcal{Z})% +\mathcal{U}\mathfrak{C}_{2}(\mathcal{Z})+\mathcal{U}^{2}\mathfrak{C}_{3}(% \mathcal{Z})+\mathcal{U}^{3}\mathfrak{C}_{4}(\mathcal{Z})+\cdots)
  262. 𝒰 \mathcal{U}
  263. 𝒱 \mathcal{V}
  264. g ( z , u , v ) = exp ( - z + v z + k 1 u k - 1 z k k ) g(z,u,v)=\exp\left(-z+vz+\sum_{k\geq 1}u^{k-1}\frac{z^{k}}{k}\right)
  265. exp ( - z + v z + 1 u log 1 1 - u z ) = exp ( - z + v z ) ( 1 1 - u z ) 1 / u . \exp\left(-z+vz+\frac{1}{u}\log\frac{1}{1-uz}\right)=\exp(-z+vz)\left(\frac{1}% {1-uz}\right)^{1/u}.
  266. n ! [ z n ] g ( z , - 1 , v ) = n ! [ z n ] exp ( - z + v z ) ( 1 + z ) = π S n σ ( π ) v ν ( π ) n![z^{n}]g(z,-1,v)=n![z^{n}]\exp(-z+vz)(1+z)=\sum_{\pi\in S_{n}}\sigma(\pi)v^{% \nu(\pi)}
  267. n ! [ z n ] 0 1 g ( z , - 1 , v ) d v = π S n σ ( π ) ν ( π ) + 1 . n![z^{n}]\int_{0}^{1}g(z,-1,v)dv=\sum_{\pi\in S_{n}}\frac{\sigma(\pi)}{\nu(\pi% )+1}.
  268. 0 1 g ( z , - 1 , v ) d v = exp ( - z ) ( 1 + z ) ( 1 z exp ( z ) - 1 z ) \int_{0}^{1}g(z,-1,v)dv=\exp(-z)(1+z)\left(\frac{1}{z}\exp(z)-\frac{1}{z}\right)
  269. ( 1 z + 1 ) ( 1 - exp ( - z ) ) = 1 z + 1 - exp ( - z ) - 1 z exp ( - z ) . \left(\frac{1}{z}+1\right)\left(1-\exp(-z)\right)=\frac{1}{z}+1-\exp(-z)-\frac% {1}{z}\exp(-z).
  270. 1 / z 1/z
  271. n n
  272. n ! [ z n ] ( - exp ( - z ) - 1 z exp ( - z ) ) = n ! ( - ( - 1 ) n 1 n ! - ( - 1 ) n + 1 1 ( n + 1 ) ! ) n![z^{n}]\left(-\exp(-z)-\frac{1}{z}\exp(-z)\right)=n!\left(-(-1)^{n}\frac{1}{% n!}-(-1)^{n+1}\frac{1}{(n+1)!}\right)
  273. ( - 1 ) n + 1 ( 1 - 1 n + 1 ) = ( - 1 ) n + 1 n n + 1 , (-1)^{n+1}\left(1-\frac{1}{n+1}\right)=(-1)^{n+1}\frac{n}{n+1},
  274. g ( z , u , v ) g(z,u,v)
  275. n × n n\times n
  276. d ( n ) = det ( A n ) = | a b b b b a b b b b a b b b b a | . d(n)=\det(A_{n})=\begin{vmatrix}a&&b&&b&&\cdots&&b\\ b&&a&&b&&\cdots&&b\\ b&&b&&a&&\cdots&&b\\ \vdots&&\vdots&&\vdots&&\ddots&&\vdots\\ b&&b&&b&&\cdots&&a\end{vmatrix}.
  277. a , b 0 a,b\neq 0
  278. det ( A ) = π S n σ ( π ) i = 1 n A i , π ( i ) . \det(A)=\sum_{\pi\in S_{n}}\sigma(\pi)\prod_{i=1}^{n}A_{i,\pi(i)}.
  279. π \pi
  280. a f b n - f a^{f}b^{n-f}\,
  281. π \pi
  282. d ( n ) = b n n ! [ z n ] g ( z , - 1 , a b ) = b n n ! [ z n ] exp ( a - b b z ) ( 1 + z ) d(n)=b^{n}n![z^{n}]g\left(z,-1,\frac{a}{b}\right)=b^{n}n![z^{n}]\exp\left(% \frac{a-b}{b}z\right)(1+z)
  283. b n ( a - b b ) n + b n n ( a - b b ) n - 1 = ( a - b ) n + n b ( a - b ) n - 1 b^{n}\left(\frac{a-b}{b}\right)^{n}+b^{n}n\left(\frac{a-b}{b}\right)^{n-1}=(a-% b)^{n}+nb(a-b)^{n-1}
  284. d ( n ) = ( a + ( n - 1 ) b ) ( a - b ) n - 1 . d(n)=(a+(n-1)b)(a-b)^{n-1}\,.
  285. ( - 1 ) n ( n - 2 ) ! (-1)^{n}(n-2)!
  286. σ ( π ) \sigma(\pi)
  287. π \pi
  288. σ ( π ) = c π ( - 1 ) | c | - 1 \sigma(\pi)=\prod_{c\in\pi}(-1)^{|c|-1}
  289. π \pi
  290. 𝒬 \mathcal{Q}
  291. 𝒬 = 𝔓 ( 𝒱 1 ( 𝒵 ) + 𝒰 𝒱 2 ( 𝒵 ) ) + 𝒰 2 𝒱 3 ( 𝒵 ) + 𝒰 3 𝒱 4 ( 𝒵 ) + 𝒰 4 𝒱 5 ( 𝒵 ) + ) \mathcal{Q}=\mathfrak{P}(\mathcal{V}\mathfrak{C}_{1}(\mathcal{Z})+\mathcal{U}% \mathcal{V}\mathfrak{C}_{2}(\mathcal{Z}))+\mathcal{U}^{2}\mathcal{V}\mathfrak{% C}_{3}(\mathcal{Z})+\mathcal{U}^{3}\mathcal{V}\mathfrak{C}_{4}(\mathcal{Z})+% \mathcal{U}^{4}\mathcal{V}\mathfrak{C}_{5}(\mathcal{Z})+\cdots)
  292. 𝒰 \mathcal{U}
  293. 𝒱 \mathcal{V}
  294. Q ( z , u , v ) = exp ( v z 1 + v u z 2 2 + v u 2 z 3 3 + v u 3 z 4 4 + v u 4 z 5 5 + ) . Q(z,u,v)=\exp\left(v\frac{z}{1}+vu\frac{z^{2}}{2}+vu^{2}\frac{z^{3}}{3}+vu^{3}% \frac{z^{4}}{4}+vu^{4}\frac{z^{5}}{5}+\cdots\right).
  295. Q ( z , u , v ) = exp ( v u ( z u 1 + z 2 u 2 2 + z 3 u 3 3 + z 4 u 4 4 + z 5 u 5 5 + ) ) Q(z,u,v)=\exp\left(\frac{v}{u}\left(\frac{zu}{1}+\frac{z^{2}u^{2}}{2}+\frac{z^% {3}u^{3}}{3}+\frac{z^{4}u^{4}}{4}+\frac{z^{5}u^{5}}{5}+\cdots\right)\right)
  296. exp ( v u log 1 1 - u z ) = ( 1 1 - u z ) v u . \exp\left(\frac{v}{u}\log\frac{1}{1-uz}\right)=\left(\frac{1}{1-uz}\right)^{% \frac{v}{u}}.
  297. Q 1 ( z , v ) Q_{1}(z,v)
  298. Q 2 ( z , v ) Q_{2}(z,v)
  299. Q 1 ( z , v ) = 1 2 Q ( z , + 1 , v ) + 1 2 Q ( z , - 1 , v ) = 1 2 ( 1 1 - z ) v + 1 2 ( 1 1 + z ) - v Q_{1}(z,v)=\frac{1}{2}Q(z,+1,v)+\frac{1}{2}Q(z,-1,v)=\frac{1}{2}\left(\frac{1}% {1-z}\right)^{v}+\frac{1}{2}\left(\frac{1}{1+z}\right)^{-v}
  300. Q 2 ( z , v ) = 1 2 Q ( z , + 1 , v ) - 1 2 Q ( z , - 1 , v ) = 1 2 ( 1 1 - z ) v - 1 2 ( 1 1 + z ) - v . Q_{2}(z,v)=\frac{1}{2}Q(z,+1,v)-\frac{1}{2}Q(z,-1,v)=\frac{1}{2}\left(\frac{1}% {1-z}\right)^{v}-\frac{1}{2}\left(\frac{1}{1+z}\right)^{-v}.
  301. G ( z , v ) = d d v ( Q 1 ( z , v ) - Q 2 ( z , v ) ) | v = 1 G(z,v)=\left.\frac{d}{dv}(Q_{1}(z,v)-Q_{2}(z,v))\right|_{v=1}
  302. d d v ( 1 1 + z ) - v | v = 1 = - log 1 1 + z ( 1 1 + z ) - v | v = 1 = - ( 1 + z ) log 1 1 + z . \left.\frac{d}{dv}\left(\frac{1}{1+z}\right)^{-v}\right|_{v=1}=-\left.\log% \frac{1}{1+z}\left(\frac{1}{1+z}\right)^{-v}\right|_{v=1}=-(1+z)\log\frac{1}{1% +z}.
  303. - n ! [ z n ] ( 1 + z ) log 1 1 + z = - n ! ( ( - 1 ) n n + ( - 1 ) n - 1 n - 1 ) -n![z^{n}](1+z)\log\frac{1}{1+z}=-n!\left(\frac{(-1)^{n}}{n}+\frac{(-1)^{n-1}}% {n-1}\right)
  304. - n ! ( - 1 ) n - 1 ( - 1 n + 1 n - 1 ) = n ! ( - 1 ) n n - ( n - 1 ) n ( n - 1 ) -n!(-1)^{n-1}\left(-\frac{1}{n}+\frac{1}{n-1}\right)=n!(-1)^{n}\frac{n-(n-1)}{% n(n-1)}
  305. n ! ( - 1 ) n 1 n ( n - 1 ) = ( - 1 ) n ( n - 2 ) ! n!(-1)^{n}\frac{1}{n(n-1)}=(-1)^{n}(n-2)!