wpmath0000005_15

Volume_integral.html

  1. f ( x , y , z ) , f(x,y,z),
  2. D f ( x , y , z ) d x d y d z . \iiint\limits_{D}f(x,y,z)\,dx\,dy\,dz.
  3. D f ( r , θ , z ) r d r d θ d z , \iiint\limits_{D}f(r,\theta,z)\,r\,dr\,d\theta\,dz,
  4. θ \theta
  5. ϕ \phi
  6. D f ( r , θ , ϕ ) r 2 sin ϕ d r d θ d ϕ . \iiint\limits_{D}f(r,\theta,\phi)\,r^{2}\sin\phi\,dr\,d\theta\,d\phi.
  7. f ( x , y , z ) = 1 f(x,y,z)=1
  8. 0 1 0 1 0 1 1 d x d y d z = 0 1 0 1 ( 1 - 0 ) d y d z = 0 1 ( 1 - 0 ) d z = 1 - 0 = 1 \int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1}1\,dx\,dy\,dz=\int% \limits_{0}^{1}\int\limits_{0}^{1}(1-0)\,dy\,dz=\int\limits_{0}^{1}(1-0)dz=1-0=1
  9. f : 3 \begin{aligned}\displaystyle f\colon\mathbb{R}^{3}&\displaystyle\to\mathbb{R}% \end{aligned}
  10. ( x , y , z ) (x,y,z)
  11. f = x + y + z f=x+y+z
  12. 0 1 0 1 0 1 ( x + y + z ) d x d y d z = 0 1 0 1 ( 1 2 + y + z ) d y d z = 0 1 ( 1 + z ) d z = 3 2 \int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1}\left(x+y+z\right)\,% dx\,dy\,dz=\int\limits_{0}^{1}\int\limits_{0}^{1}\left(\frac{1}{2}+y+z\right)% \,dy\,dz=\int\limits_{0}^{1}\left(1+z\right)\,dz=\frac{3}{2}

Volume_of_distribution.html

  1. V D = total amount of drug in the body drug blood plasma concentration {V_{D}}=\frac{\mathrm{total\ amount\ of\ drug\ in\ the\ body}}{\mathrm{drug\ % blood\ plasma\ concentration}}
  2. V D = V P + V T ( f u f u t ) {V_{D}}={V_{P}}+{V_{T}}\left(\frac{fu}{fu_{t}}\right)
  3. C 0 C_{0}
  4. V b l o o d V_{blood}
  5. C 0 = D / V b l o o d C_{0}=D/V_{blood}
  6. V D V_{D}
  7. V b l o o d = V_{blood}=
  8. C 0 = C_{0}=
  9. V b l o o d V_{blood}
  10. V D = V_{D}=
  11. V D = V_{D}=
  12. V D = V_{D}=
  13. V D = D / C 0 V_{D}=D/C_{0}
  14. C 0 C_{0}

Von_Mangoldt_function.html

  1. Λ ( n ) Λ(n)
  2. Λ ( n ) = { log p if n = p k for some prime p and integer k 1 , 0 otherwise. \Lambda(n)=\begin{cases}\log p&\,\text{if }n=p^{k}\,\text{ for some prime }p\,% \text{ and integer }k\geq 1,\\ 0&\,\text{otherwise.}\end{cases}
  3. Λ ( n ) Λ(n)
  4. 0 , log 2 , log 3 , log 2 , log 5 , 0 , log 7 , log 2 , log 3 , 0,\log 2,\log 3,\log 2,\log 5,0,\log 7,\log 2,\log 3,
  5. ψ ( x ) ψ(x)
  6. ψ ( x ) = n x Λ ( n ) . \psi(x)=\sum_{n\leq x}\Lambda(n).
  7. ψ ( x ) ψ(x)
  8. log ( n ) = d n Λ ( d ) . \log(n)=\sum_{d\mid n}\Lambda(d).
  9. d d
  10. n n
  11. 0
  12. d 12 Λ ( d ) = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ ( 4 ) + Λ ( 6 ) + Λ ( 12 ) = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ ( 2 2 ) + Λ ( 2 × 3 ) + Λ ( 2 2 × 3 ) = 0 + log ( 2 ) + log ( 3 ) + log ( 2 ) + 0 + 0 = log ( 2 × 3 × 2 ) = log ( 12 ) . \begin{aligned}\displaystyle\sum_{d\mid 12}\Lambda(d)&\displaystyle=\Lambda(1)% +\Lambda(2)+\Lambda(3)+\Lambda(4)+\Lambda(6)+\Lambda(12)\\ &\displaystyle=\Lambda(1)+\Lambda(2)+\Lambda(3)+\Lambda\left(2^{2}\right)+% \Lambda(2\times 3)+\Lambda\left(2^{2}\times 3\right)\\ &\displaystyle=0+\log(2)+\log(3)+\log(2)+0+0\\ &\displaystyle=\log(2\times 3\times 2)\\ &\displaystyle=\log(12).\end{aligned}
  13. Λ ( n ) = - d n μ ( d ) log ( d ) . \Lambda(n)=-\sum_{d\mid n}\mu(d)\log(d)\ .
  14. log ζ ( s ) = n = 2 Λ ( n ) log ( n ) 1 n s , Re ( s ) > 1. \log\zeta(s)=\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^{s}},% \qquad\,\text{Re}(s)>1.
  15. ζ ( s ) ζ ( s ) = - n = 1 Λ ( n ) n s . \frac{\zeta^{\prime}(s)}{\zeta(s)}=-\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{s}}.
  16. F ( s ) = n = 1 f ( n ) n s F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}
  17. f ( n ) f(n)
  18. F ( s ) F ( s ) = - n = 1 f ( n ) Λ ( n ) n s \frac{F^{\prime}(s)}{F(s)}=-\sum_{n=1}^{\infty}\frac{f(n)\Lambda(n)}{n^{s}}
  19. ψ ( x ) = p k x log p = n x Λ ( n ) . \psi(x)=\sum_{p^{k}\leq x}\log p=\sum_{n\leq x}\Lambda(n)\ .
  20. ζ ( s ) ζ ( s ) = - s 1 ψ ( x ) x s + 1 d x \frac{\zeta^{\prime}(s)}{\zeta(s)}=-s\int_{1}^{\infty}\frac{\psi(x)}{x^{s+1}}% \,dx
  21. R e ( s ) > 1 Re(s)>1
  22. F ( y ) = n = 2 ( Λ ( n ) - 1 ) e - n y F(y)=\sum_{n=2}^{\infty}\left(\Lambda(n)-1\right)e^{-ny}
  23. F ( y ) = O ( 1 y ) . F(y)=O\left(\frac{1}{\sqrt{y}}\right).
  24. K > 0 K>0
  25. F ( y ) < - K y , and F ( y ) > K y F(y)<-\frac{K}{\sqrt{y}},\quad\,\text{ and }\quad F(y)>\frac{K}{\sqrt{y}}
  26. n λ ( 1 - n λ ) δ Λ ( n ) = - 1 2 π i c - i c + i Γ ( 1 + δ ) Γ ( s ) Γ ( 1 + δ + s ) ζ ( s ) ζ ( s ) λ s d s = λ 1 + δ + ρ Γ ( 1 + δ ) Γ ( ρ ) Γ ( 1 + δ + ρ ) + n c n λ - n . \begin{aligned}\displaystyle\sum_{n\leq\lambda}\left(1-\frac{n}{\lambda}\right% )^{\delta}\Lambda(n)&\displaystyle=-\frac{1}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)}\frac{\zeta^{\prime% }(s)}{\zeta(s)}\lambda^{s}ds\\ &\displaystyle=\frac{\lambda}{1+\delta}+\sum_{\rho}\frac{\Gamma(1+\delta)% \Gamma(\rho)}{\Gamma(1+\delta+\rho)}+\sum_{n}c_{n}\lambda^{-n}.\end{aligned}
  27. λ λ
  28. δ δ
  29. c > 1 c>1
  30. ρ ρ
  31. n c n λ - n \sum_{n}c_{n}\lambda^{-n}\,
  32. λ > 1 λ>1
  33. - i = 1 n ρ ( i ) -\sum_{i=1}^{\infty}n^{\rho(i)}
  34. ρ ( i ) ρ(i)
  35. i i
  36. x x

Von_Mises_distribution.html

  1. μ \mu
  2. μ \mu
  3. μ \mu
  4. var ( x ) = 1 - I 1 ( κ ) / I 0 ( κ ) \textrm{var}(x)=1-I_{1}(\kappa)/I_{0}(\kappa)
  5. - κ I 1 ( κ ) I 0 ( κ ) + ln [ 2 π I 0 ( κ ) ] -\kappa\frac{I_{1}(\kappa)}{I_{0}(\kappa)}+\ln[2\pi I_{0}(\kappa)]
  6. I | n | ( κ ) I 0 ( κ ) e i n μ \frac{I_{|n|}(\kappa)}{I_{0}(\kappa)}e^{in\mu}
  7. θ \theta
  8. z = e i θ z=e^{i\theta}
  9. f ( x μ , κ ) = e κ cos ( x - μ ) 2 π I 0 ( κ ) f(x\mid\mu,\kappa)=\frac{e^{\kappa\cos(x-\mu)}}{2\pi I_{0}(\kappa)}
  10. f ( x μ , κ ) = 1 2 π ( 1 + 2 I 0 ( κ ) j = 1 I j ( κ ) cos [ j ( x - μ ) ] ) f(x\mid\mu,\kappa)=\frac{1}{2\pi}\left(1+\frac{2}{I_{0}(\kappa)}\sum_{j=1}^{% \infty}I_{j}(\kappa)\cos[j(x-\mu)]\right)
  11. Φ ( x μ , κ ) = f ( t μ , κ ) d t = 1 2 π ( x + 2 I 0 ( κ ) j = 1 I j ( κ ) sin [ j ( x - μ ) ] j ) . \Phi(x\mid\mu,\kappa)=\int f(t\mid\mu,\kappa)\,dt=\frac{1}{2\pi}\left(x+\frac{% 2}{I_{0}(\kappa)}\sum_{j=1}^{\infty}I_{j}(\kappa)\frac{\sin[j(x-\mu)]}{j}% \right).
  12. F ( x μ , κ ) = Φ ( x μ , κ ) - Φ ( x 0 μ , κ ) . F(x\mid\mu,\kappa)=\Phi(x\mid\mu,\kappa)-\Phi(x_{0}\mid\mu,\kappa).\,
  13. m n = z n = Γ z n f ( x | μ , κ ) d x m_{n}=\langle z^{n}\rangle=\int_{\Gamma}z^{n}\,f(x|\mu,\kappa)\,dx
  14. = I | n | ( κ ) I 0 ( κ ) e i n μ =\frac{I_{|n|}(\kappa)}{I_{0}(\kappa)}e^{in\mu}
  15. Γ \Gamma
  16. I n ( κ ) = 1 π 0 π e κ cos ( x ) cos ( n x ) d x . I_{n}(\kappa)=\frac{1}{\pi}\int_{0}^{\pi}e^{\kappa\cos(x)}\cos(nx)\,dx.
  17. m 1 = I 1 ( κ ) I 0 ( κ ) e i μ m_{1}=\frac{I_{1}(\kappa)}{I_{0}(\kappa)}e^{i\mu}
  18. var ( x ) = 1 - E [ cos ( x - μ ) ] = 1 - I 1 ( κ ) I 0 ( κ ) . \textrm{var}(x)=1-E[\cos(x-\mu)]=1-\frac{I_{1}(\kappa)}{I_{0}(\kappa)}.
  19. lim κ f ( x μ , κ ) = 1 σ 2 π exp [ - ( x - μ ) 2 2 σ 2 ] \lim_{\kappa\rightarrow\infty}f(x\mid\mu,\kappa)=\frac{1}{\sigma\sqrt{2\pi}}% \exp\left[\dfrac{-(x-\mu)^{2}}{2\sigma^{2}}\right]
  20. lim κ 0 f ( x μ , κ ) = U ( x ) \lim_{\kappa\rightarrow 0}f(x\mid\mu,\kappa)=\mathrm{U}(x)
  21. z n = e i θ n z_{n}=e^{i\theta_{n}}
  22. z ¯ \overline{z}
  23. z ¯ = 1 N n = 1 N z n \overline{z}=\frac{1}{N}\sum_{n=1}^{N}z_{n}
  24. z ¯ = I 1 ( κ ) I 0 ( κ ) e i μ . \langle\overline{z}\rangle=\frac{I_{1}(\kappa)}{I_{0}(\kappa)}e^{i\mu}.
  25. z ¯ \overline{z}
  26. μ \mu
  27. [ - π , π ) [-\pi,\pi)
  28. ( z ¯ ) (\overline{z})
  29. μ \mu
  30. z n z_{n}
  31. R ¯ 2 \bar{R}^{2}
  32. R ¯ 2 = z ¯ z * ¯ = ( 1 N n = 1 N cos θ n ) 2 + ( 1 N n = 1 N sin θ n ) 2 \bar{R}^{2}=\overline{z}\,\overline{z^{*}}=\left(\frac{1}{N}\sum_{n=1}^{N}\cos% \theta_{n}\right)^{2}+\left(\frac{1}{N}\sum_{n=1}^{N}\sin\theta_{n}\right)^{2}
  33. R ¯ 2 = 1 N + N - 1 N I 1 ( κ ) 2 I 0 ( κ ) 2 . \langle\bar{R}^{2}\rangle=\frac{1}{N}+\frac{N-1}{N}\,\frac{I_{1}(\kappa)^{2}}{% I_{0}(\kappa)^{2}}.
  34. R e 2 = N N - 1 ( R ¯ 2 - 1 N ) R_{e}^{2}=\frac{N}{N-1}\left(\bar{R}^{2}-\frac{1}{N}\right)
  35. I 1 ( κ ) 2 I 0 ( κ ) 2 \frac{I_{1}(\kappa)^{2}}{I_{0}(\kappa)^{2}}\,
  36. R e = I 1 ( κ ) I 0 ( κ ) R_{e}=\frac{I_{1}(\kappa)}{I_{0}(\kappa)}\,
  37. κ \kappa\,
  38. κ \kappa\,
  39. R ¯ = I 1 ( κ ) I 0 ( κ ) \bar{R}=\frac{I_{1}(\kappa)}{I_{0}(\kappa)}\,
  40. κ \kappa\,
  41. κ \kappa\,
  42. z ¯ = R ¯ e i θ ¯ \overline{z}=\bar{R}e^{i\overline{\theta}}
  43. P ( R ¯ , θ ¯ ) d R ¯ d θ ¯ = 1 ( 2 π I 0 ( κ ) ) N Γ n = 1 N ( e κ cos ( θ n - μ ) d θ n ) = e κ N R ¯ cos ( θ ¯ - μ ) I 0 ( κ ) N ( 1 ( 2 π ) N Γ n = 1 N d θ n ) P(\bar{R},\bar{\theta})\,d\bar{R}\,d\bar{\theta}=\frac{1}{(2\pi I_{0}(\kappa))% ^{N}}\int_{\Gamma}\prod_{n=1}^{N}\left(e^{\kappa\cos(\theta_{n}-\mu)}d\theta_{% n}\right)=\frac{e^{\kappa N\bar{R}\cos(\bar{\theta}-\mu)}}{I_{0}(\kappa)^{N}}% \left(\frac{1}{(2\pi)^{N}}\int_{\Gamma}\prod_{n=1}^{N}d\theta_{n}\right)
  44. Γ \Gamma\,
  45. 2 π 2\pi
  46. R ¯ \bar{R}
  47. θ ¯ \bar{\theta}
  48. R ¯ \bar{R}
  49. R ¯ 2 = | z ¯ | 2 = ( 1 N n = 1 N cos ( θ n ) ) 2 + ( 1 N n = 1 N sin ( θ n ) ) 2 \bar{R}^{2}=|\bar{z}|^{2}=\left(\frac{1}{N}\sum_{n=1}^{N}\cos(\theta_{n})% \right)^{2}+\left(\frac{1}{N}\sum_{n=1}^{N}\sin(\theta_{n})\right)^{2}
  50. θ ¯ \overline{\theta}
  51. θ ¯ = Arg ( z ¯ ) . \overline{\theta}=\mathrm{Arg}(\overline{z}).\,
  52. μ \mu
  53. V M ( μ , κ ) VM(\mu,\kappa)
  54. V M ( μ , R ¯ N κ ) VM(\mu,\bar{R}N\kappa)
  55. V M ( μ , R κ ) VM(\mu,R\kappa)
  56. H = - Γ f ( θ ; μ , κ ) ln ( f ( θ ; μ , κ ) ) d θ H=-\int_{\Gamma}f(\theta;\mu,\kappa)\,\ln(f(\theta;\mu,\kappa))\,d\theta\,
  57. Γ \Gamma
  58. 2 π 2\pi
  59. ln ( f ( θ ; μ , κ ) ) = - ln ( 2 π I 0 ( κ ) ) + κ cos ( θ ) \ln(f(\theta;\mu,\kappa))=-\ln(2\pi I_{0}(\kappa))+\kappa\cos(\theta)\,
  60. f ( θ ; μ , κ ) = 1 2 π ( 1 + 2 n = 1 ϕ n cos ( n θ ) ) f(\theta;\mu,\kappa)=\frac{1}{2\pi}\left(1+2\sum_{n=1}^{\infty}\phi_{n}\cos(n% \theta)\right)
  61. ϕ n = I | n | ( κ ) / I 0 ( κ ) \phi_{n}=I_{|n|}(\kappa)/I_{0}(\kappa)
  62. H = ln ( 2 π I 0 ( κ ) ) - κ ϕ 1 = ln ( 2 π I 0 ( κ ) ) - κ I 1 ( κ ) I 0 ( κ ) H=\ln(2\pi I_{0}(\kappa))-\kappa\phi_{1}=\ln(2\pi I_{0}(\kappa))-\kappa\frac{I% _{1}(\kappa)}{I_{0}(\kappa)}
  63. κ = 0 \kappa=0
  64. ln ( 2 π ) \ln(2\pi)

Vortex_shedding.html

  1. St = f D V \mathrm{St}=\frac{fD}{V}
  2. St \mathrm{St}
  3. f f
  4. D D
  5. V V

Vuong's_closeness_test.html

  1. Z = L R N ( β M L , 1 , β M L , 2 ) N ω N Z=\frac{LR_{N}(\beta_{ML,1},\beta_{ML,2})}{\sqrt{N}\omega_{N}}
  2. L R N ( β M L , 1 , β M L , 2 ) = L N 1 - L N 2 - K 1 - K 2 2 log N {LR_{N}(\beta_{ML,1},\beta_{ML,2})}=L^{1}_{N}-L^{2}_{N}-\frac{K_{1}-K_{2}}{2}\log N
  3. ω N \omega_{N}\,
  4. ω N 2 \omega_{N}^{2}
  5. i \ell_{i}\,
  6. i = log f 1 ( y i | x i , β M L , 1 ) f 2 ( y i | x i , β M L , 2 ) . \ell_{i}=\log\frac{f_{1}(y_{i}|x_{i},\beta_{ML,1})}{f_{2}(y_{i}|x_{i},\beta_{% ML,2})}.
  7. 2 L R N ( β M L , 1 , β M L , 2 ) 2LR_{N}(\beta_{ML,1},\beta_{ML,2})\,
  8. M m ( . , λ ) Γ ( b , p ) M_{m}(.,\lambda)\sim\Gamma(b,p)\,
  9. λ = ( λ 1 , λ 2 , , λ m ) , \lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{m}),\,
  10. m = K 1 + K 2 , b = 1 2 λ i λ i 2 m=K_{1}+K_{2},\ b=\frac{1}{2}\frac{\sum\lambda_{i}}{\sum\lambda_{i}^{2}}
  11. p = 1 2 ( λ i ) 2 λ i 2 . p=\frac{1}{2}\frac{{(\sum\lambda i)}^{2}}{\sum\lambda_{i}^{2}}.
  12. λ \lambda

Wahlund_effect.html

  1. P P
  2. p p
  3. q q
  4. p + q = 1 p+q=1
  5. P 1 P_{1}
  6. P 2 P_{2}
  7. P 1 P_{1}
  8. P 2 P_{2}
  9. p 1 p_{1}
  10. p 2 p_{2}
  11. P 1 P_{1}
  12. P 2 P_{2}
  13. q 1 q_{1}
  14. q 2 q_{2}
  15. p 1 p 2 p_{1}\neq p_{2}
  16. H H
  17. H H
  18. = 2 p 1 q 1 + 2 p 2 q 2 2 ={2p_{1}q_{1}+2p_{2}q_{2}\over 2}
  19. = p 1 q 1 + p 2 q 2 ={p_{1}q_{1}+p_{2}q_{2}}
  20. = p 1 ( 1 - p 1 ) + p 2 ( 1 - p 2 ) ={p_{1}(1-p_{1})+p_{2}(1-p_{2})}
  21. 2 p ( 1 - p ) 2p(1-p)
  22. 2 p q 2pq
  23. p 1 = p 2 p_{1}=p_{2}

Wald's_equation.html

  1. N N
  2. N N
  3. E [ X 1 + + X N ] = E [ N ] E [ X 1 ] . \operatorname{E}[X_{1}+\dots+X_{N}]=\operatorname{E}[N]\operatorname{E}[X_{1}]\,.
  4. N N
  5. E [ N ] E [ X ] = 1 + 2 + 3 + 4 + 5 + 6 6 1 + 2 + 3 + 4 + 5 + 6 6 = 441 36 = 12.25 . \operatorname{E}[N]\operatorname{E}[X]=\frac{1+2+3+4+5+6}{6}\cdot\frac{1+2+3+4% +5+6}{6}=\frac{441}{36}=12.25\,.
  6. N N
  7. n n
  8. n = 1 E [ | X n | 1 { N n } ] < . \sum_{n=1}^{\infty}\operatorname{E}\!\bigl[|X_{n}|1_{\{N\geq n\}}\bigr]<\infty.
  9. S N := n = 1 N X n , T N := n = 1 N E [ X n ] S_{N}:=\sum_{n=1}^{N}X_{n},\qquad T_{N}:=\sum_{n=1}^{N}\operatorname{E}[X_{n}]
  10. E [ S N ] = E [ T N ] . \operatorname{E}[S_{N}]=\operatorname{E}[T_{N}].
  11. N N
  12. E [ S N ] = E [ N ] E [ X 1 ] . \operatorname{E}[S_{N}]=\operatorname{E}[N]\,\operatorname{E}[X_{1}].
  13. N N
  14. N N
  15. C C
  16. n n
  17. n = 1 E [ | X n | 1 { N n } ] C n = 1 P ( N n ) , \sum_{n=1}^{\infty}\operatorname{E}\!\bigl[|X_{n}|1_{\{N\geq n\}}\bigr]\leq C% \sum_{n=1}^{\infty}\operatorname{P}(N\geq n),
  18. N N
  19. N N
  20. C C
  21. n n
  22. N N
  23. N N
  24. n n∈ℕ
  25. < s u b > n 1 \mathcal{F}<sub>n–1
  26. n n∈ℕ
  27. S N = n = 1 N X n S_{N}=\sum_{n=1}^{N}X_{n}
  28. N N
  29. N N
  30. Z Z
  31. n n∈ℕ
  32. Z Z
  33. Z Z
  34. Z Z
  35. n n∈ℕ
  36. N = n N=n
  37. + 1 +1
  38. n n∈ℕ
  39. n = 1 n=1
  40. N N
  41. n n∈ℕ
  42. N N
  43. n n∈ℕ
  44. N N
  45. { N n } = { X i = - 2 i for i = 1 , , n - 1 } \{N\geq n\}=\{X_{i}=-2^{i}\,\text{ for }i=1,\ldots,n-1\}
  46. n n∈ℕ
  47. n = 1 E [ | X n | 1 { N n } ] = n = 1 2 n P ( N n ) = n = 1 2 = . \sum_{n=1}^{\infty}\operatorname{E}\!\bigl[|X_{n}|1_{\{N\geq n\}}\bigr]=\sum_{% n=1}^{\infty}2^{n}\,\operatorname{P}(N\geq n)=\sum_{n=1}^{\infty}2=\infty.
  48. M n = i = 1 n ( X i - E [ X i ] ) , n 0 . M_{n}=\sum_{i=1}^{n}(X_{i}-\operatorname{E}[X_{i}]),\quad n\in{\mathbb{N}}_{0}.
  49. n n∈ℕ
  50. E [ S N - T N ] = E [ M 0 ] = 0. \operatorname{E}[S_{N}-T_{N}]=\operatorname{E}[M_{0}]=0.
  51. | T N | = | i = 1 N E [ X i ] | i = 1 N E [ | X i | ] C N , |T_{N}|=\biggl|\sum_{i=1}^{N}\operatorname{E}[X_{i}]\biggr|\leq\sum_{i=1}^{N}% \operatorname{E}[|X_{i}|]\leq CN,
  52. E [ S N ] = E [ T N ] . \operatorname{E}[S_{N}]=\operatorname{E}[T_{N}].
  53. S i = n = 1 i X n , i 0 . S_{i}=\sum_{n=1}^{i}X_{n},\quad i\in{\mathbb{N}}_{0}.
  54. N N
  55. | S N | = i = 1 | S i | 1 { N = i } . |S_{N}|=\sum_{i=1}^{\infty}|S_{i}|\,1_{\{N=i\}}.
  56. E [ | S N | ] = i = 1 E [ | S i | 1 { N = i } ] . \operatorname{E}[|S_{N}|]=\sum_{i=1}^{\infty}\operatorname{E}[|S_{i}|\,1_{\{N=% i\}}].
  57. | S i | n = 1 i | X n | , i . |S_{i}|\leq\sum_{n=1}^{i}|X_{n}|,\quad i\in{\mathbb{N}}.
  58. E [ | S N | ] n = 1 i = n E [ | X n | 1 { N = i } ] n = 1 E [ | X n | 1 { N n } ] , \operatorname{E}[|S_{N}|]\leq\sum_{n=1}^{\infty}\sum_{i=n}^{\infty}% \operatorname{E}[|X_{n}|\,1_{\{N=i\}}]\leq\sum_{n=1}^{\infty}\operatorname{E}[% |X_{n}|\,1_{\{N\geq n\}}],
  59. T i = n = 1 i E [ X n ] , i 0 , T_{i}=\sum_{n=1}^{i}\operatorname{E}[X_{n}],\quad i\in{\mathbb{N}}_{0},
  60. N N
  61. | T N | = i = 1 | T i | 1 { N = i } . |T_{N}|=\sum_{i=1}^{\infty}|T_{i}|\,1_{\{N=i\}}.
  62. E [ | T N | ] = i = 1 | T i | P ( N = i ) . \operatorname{E}[|T_{N}|]=\sum_{i=1}^{\infty}|T_{i}|\operatorname{P}(N=i).
  63. | T i | n = 1 i | E [ X n ] | , i . |T_{i}|\leq\sum_{n=1}^{i}\bigl|\!\operatorname{E}[X_{n}]\bigr|,\quad i\in{% \mathbb{N}}.
  64. E [ | T N | ] n = 1 | E [ X n ] | i = n P ( N = i ) = P ( N n ) , \operatorname{E}[|T_{N}|]\leq\sum_{n=1}^{\infty}\bigl|\!\operatorname{E}[X_{n}% ]\bigr|\underbrace{\sum_{i=n}^{\infty}\operatorname{P}(N=i)}_{=\,\operatorname% {P}(N\geq n)},
  65. | E [ X n ] | P ( N n ) = | E [ X n 1 { N n } ] | E [ | X n | 1 { N n } ] , n . \bigl|\!\operatorname{E}[X_{n}]\bigr|\operatorname{P}(N\geq n)=\bigl|\!% \operatorname{E}[X_{n}1_{\{N\geq n\}}]\bigr|\leq\operatorname{E}[|X_{n}|1_{\{N% \geq n\}}],\quad n\in{\mathbb{N}}.
  66. E [ | T N | ] n = 1 E [ | X n | 1 { N n } ] , \operatorname{E}[|T_{N}|]\leq\sum_{n=1}^{\infty}\operatorname{E}[|X_{n}|1_{\{N% \geq n\}}],
  67. E [ S N ] = i = 1 E [ S i 1 { N = i } ] = i = 1 n = 1 i E [ X n 1 { N = i } ] . \operatorname{E}[S_{N}]=\sum_{i=1}^{\infty}\operatorname{E}[S_{i}1_{\{N=i\}}]=% \sum_{i=1}^{\infty}\sum_{n=1}^{i}\operatorname{E}[X_{n}1_{\{N=i\}}].
  68. E [ S N ] = n = 1 i = n E [ X n 1 { N = i } ] = n = 1 E [ X n 1 { N n } ] , \operatorname{E}[S_{N}]=\sum_{n=1}^{\infty}\sum_{i=n}^{\infty}\operatorname{E}% [X_{n}1_{\{N=i\}}]=\sum_{n=1}^{\infty}\operatorname{E}[X_{n}1_{\{N\geq n\}}],
  69. E [ X n 1 { N n } ] = E [ X n ] P ( N n ) = E [ X n ] i = n P ( N = i ) = i = n E [ E [ X n ] 1 { N = i } ] . \begin{aligned}\displaystyle\operatorname{E}[X_{n}1_{\{N\geq n\}}]&% \displaystyle=\operatorname{E}[X_{n}]\operatorname{P}(N\geq n)\\ &\displaystyle=\operatorname{E}[X_{n}]\sum_{i=n}^{\infty}\operatorname{P}(N=i)% =\sum_{i=n}^{\infty}\operatorname{E}\!\bigl[\operatorname{E}[X_{n}]1_{\{N=i\}}% \bigr].\end{aligned}
  70. E [ S N ] = i = 1 n = 1 i E [ E [ X n ] 1 { N = i } ] = i = 1 E [ T i 1 { N = i } = T N 1 { N = i } ] . \operatorname{E}[S_{N}]=\sum_{i=1}^{\infty}\sum_{n=1}^{i}\operatorname{E}\!% \bigl[\operatorname{E}[X_{n}]1_{\{N=i\}}\bigr]=\sum_{i=1}^{\infty}% \operatorname{E}[\underbrace{T_{i}1_{\{N=i\}}}_{=\,T_{N}1_{\{N=i\}}}].
  71. E [ S N ] = E [ T N i = 1 1 { N = i } = 1 { N 1 } ] = E [ T N ] . \operatorname{E}[S_{N}]=\operatorname{E}\!\biggl[T_{N}\underbrace{\sum_{i=1}^{% \infty}1_{\{N=i\}}}_{=\,1_{\{N\geq 1\}}}\biggr]=\operatorname{E}[T_{N}].
  72. E [ T N ] = E [ n = 1 N E [ X n ] ] = E [ X 1 ] E [ n = 1 N 1 = N ] = E [ N ] E [ X 1 ] . \operatorname{E}[T_{N}]=\operatorname{E}\!\biggl[\sum_{n=1}^{N}\operatorname{E% }[X_{n}]\biggr]=\operatorname{E}[X_{1}]\operatorname{E}\!\biggl[\underbrace{% \sum_{n=1}^{N}1}_{=\,N}\biggr]=\operatorname{E}[N]\operatorname{E}[X_{1}].
  73. 𝐑 < s u p > d \mathbf{R}<sup>d

Wald_test.html

  1. θ \theta
  2. θ \theta
  3. θ \theta
  4. θ \theta
  5. θ \theta
  6. θ ^ \hat{\theta}
  7. θ \theta
  8. θ 0 \theta_{0}
  9. ( θ ^ - θ 0 ) 2 var ( θ ^ ) \frac{(\widehat{\theta}-\theta_{0})^{2}}{\operatorname{var}(\hat{\theta})}
  10. θ ^ - θ 0 se ( θ ^ ) \frac{\widehat{\theta}-\theta_{0}}{\operatorname{se}(\hat{\theta})}
  11. se ( θ ^ ) \operatorname{se}(\widehat{\theta})
  12. 1 I n ( M L E ) \frac{1}{\sqrt{I_{n}(MLE)}}
  13. I n I_{n}
  14. θ ^ n \hat{\theta}_{n}
  15. θ ^ n \hat{\theta}_{n}
  16. n ( θ ^ n - θ ) 𝒟 N ( 0 , V ) \sqrt{n}(\hat{\theta}_{n}-\theta)\xrightarrow{\mathcal{D}}N(0,V)
  17. H 0 : R θ = r H_{0}:R\theta=r
  18. H 1 : R θ r H_{1}:R\theta\neq r
  19. ( R θ ^ n - r ) [ R ( V ^ n / N ) R ] - 1 ( R θ ^ n - r ) 𝒟 \Chi Q 2 (R\hat{\theta}_{n}-r)^{{}^{\prime}}[R(\hat{V}_{n}/N)R^{{}^{\prime}}]^{-1}(R% \hat{\theta}_{n}-r)\quad\xrightarrow{\mathcal{D}}\quad\Chi^{2}_{Q}
  20. V ^ n \hat{V}_{n}
  21. n ( θ ^ n - θ ) 𝒟 N ( 0 , V ) \sqrt{n}(\hat{\theta}_{n}-\theta)\xrightarrow{\mathcal{D}}N(0,V)
  22. R n ( θ ^ n - θ ) = n ( R θ ^ n - r ) 𝒟 N ( 0 , R V R ) R\sqrt{n}(\hat{\theta}_{n}-\theta)=\sqrt{n}(R\hat{\theta}_{n}-r)\xrightarrow{% \mathcal{D}}N(0,RVR^{{}^{\prime}})
  23. n ( R θ ^ n - r ) [ R V R ] - 1 n ( R θ ^ n - r ) 𝒟 \Chi Q 2 \sqrt{n}(R\hat{\theta}_{n}-r)^{{}^{\prime}}[RVR^{{}^{\prime}}]^{-1}\sqrt{n}(R% \hat{\theta}_{n}-r)\xrightarrow{\mathcal{D}}\Chi^{2}_{Q}
  24. ( R θ n - r ) [ R ( V / N ) R ] - 1 ( R θ n - r ) 𝒟 \Chi Q 2 (R\theta_{n}-r)^{{}^{\prime}}[R(V/N)R^{{}^{\prime}}]^{-1}(R\theta_{n}-r)\quad% \xrightarrow{\mathcal{D}}\quad\Chi^{2}_{Q}
  25. V ^ n \hat{V}_{n}
  26. V V
  27. ( R θ n - r ) [ R ( V ^ n / N ) R ] - 1 ( R θ n - r ) 𝒟 \Chi Q 2 (R\theta_{n}-r)^{{}^{\prime}}[R(\hat{V}_{n}/N)R^{{}^{\prime}}]^{-1}(R\theta_{n% }-r)\quad\xrightarrow{\mathcal{D}}\quad\Chi^{2}_{Q}
  28. H 0 : c ( θ ) = 0 H_{0}:c(\theta)=0
  29. H 1 : c ( θ ) 0 H_{1}:c(\theta)\neq 0
  30. c ( θ ^ n ) [ c ( θ ^ n ) ( V ^ n / N ) c ( θ ^ n ) ] - 1 c ( θ ^ n ) 𝒟 \Chi Q 2 c(\hat{\theta}_{n})^{{}^{\prime}}[c^{{}^{\prime}}(\hat{\theta}_{n})(\hat{V}_{n% }/N)c^{{}^{\prime}}(\hat{\theta}_{n})^{{}^{\prime}}]^{-1}c(\hat{\theta}_{n})% \quad\xrightarrow{\mathcal{D}}\quad\Chi^{2}_{Q}
  31. c ( θ ^ n ) c^{{}^{\prime}}(\hat{\theta}_{n})

Walker_circulation.html

  1. x ( n ) , n = 1 , 2 , N x(n),n=1,2,...N
  2. k k
  3. R x x ( k ) = 1 ( N - k ) i = 1 N - k x ( i ) x ( i + k ) . R_{xx}(k)=\frac{1}{(N-k)}\sum_{i=1}^{N-k}x(i)x(i+k).\,
  4. 0
  5. x ( n ) x(n)
  6. x ( n ) x(n)
  7. R x x ( 0 ) = 1 N i = 1 N x ( i ) 2 . R_{xx}(0)=\frac{1}{N}\sum_{i=1}^{N}x(i)^{2}.\,
  8. R x x ( ) , \sqrt{R_{xx}(\infty)},
  9. x ( j ) x(j)
  10. j j
  11. p p
  12. x ( j ) = i = 1 p a i x ( j - i ) + n ( j ) . x(j)=\sum_{i=1}^{p}a_{i}x(j-i)+n(j).\,
  13. n ( j ) n(j)
  14. p p
  15. X t = i = 1 p φ i X t - i + ε t . X_{t}=\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\varepsilon_{t}.\,
  16. X t = i = 1 p φ i X t - i + ε t . X_{t}=\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\varepsilon_{t}.\,
  17. φ i \varphi_{i}
  18. i = 1 , , p i=1,\ldots,p
  19. γ m = i = 1 p φ i γ m - i + σ ε 2 δ m \gamma_{m}=\sum_{i=1}^{p}\varphi_{i}\gamma_{m-i}+\sigma_{\varepsilon}^{2}% \delta_{m}
  20. m = 0 , , p m=0,\ldots,p
  21. p + 1 p+1
  22. γ m \gamma_{m}
  23. X X
  24. σ ε \sigma_{\varepsilon}
  25. δ m \delta_{m}
  26. m = 0 m=0
  27. m > 0 m>0
  28. [ γ 1 γ 2 γ 3 ] = [ γ 0 γ - 1 γ - 2 γ 1 γ 0 γ - 1 γ 2 γ 1 γ 0 ] [ φ 1 φ 2 φ 3 ] \begin{bmatrix}\gamma_{1}\\ \gamma_{2}\\ \gamma_{3}\\ \vdots\\ \end{bmatrix}=\begin{bmatrix}\gamma_{0}&\gamma_{-1}&\gamma_{-2}&\dots\\ \gamma_{1}&\gamma_{0}&\gamma_{-1}&\dots\\ \gamma_{2}&\gamma_{1}&\gamma_{0}&\dots\\ \vdots&\vdots&\vdots&\ddots\\ \end{bmatrix}\begin{bmatrix}\varphi_{1}\\ \varphi_{2}\\ \varphi_{3}\\ \vdots\\ \end{bmatrix}
  29. φ \varphi
  30. m = 0 m=0
  31. γ 0 = i = 1 p φ i γ - i + σ ε 2 \gamma_{0}=\sum_{i=1}^{p}\varphi_{i}\gamma_{-i}+\sigma_{\varepsilon}^{2}
  32. σ ε 2 \sigma_{\varepsilon}^{2}
  33. X t X_{t}
  34. p p

Wallis_product.html

  1. n = 1 ( 2 n 2 n - 1 2 n 2 n + 1 ) = 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8 9 = π 2 \prod_{n=1}^{\infty}\left(\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}\right)=\frac{2}{% 1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6% }{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdots=\frac{\pi}{2}
  2. 0 π sin n x d x \scriptstyle\int_{0}^{\pi}\sin^{n}xdx
  3. sin x x = n = 1 ( 1 - x 2 n 2 π 2 ) \frac{\sin x}{x}=\prod_{n=1}^{\infty}\left(1-\frac{x^{2}}{n^{2}\pi^{2}}\right)
  4. π / 2 {π}/{2}
  5. 2 π \displaystyle\Rightarrow\frac{2}{\pi}
  6. I ( n ) = 0 π sin n x d x I(n)=\int_{0}^{\pi}\sin^{n}xdx
  7. u \displaystyle u
  8. I ( n ) = 0 π sin n x d x = 0 π u d v = u v | x = 0 x = π - 0 π v d u = - sin n - 1 x cos x | x = 0 x = π - 0 π - cos x ( n - 1 ) sin n - 2 x cos x d x = 0 - ( n - 1 ) 0 π - cos 2 x sin n - 2 x d x , n > 1 = ( n - 1 ) 0 π ( 1 - sin 2 x ) sin n - 2 x d x = ( n - 1 ) 0 π sin n - 2 x d x - ( n - 1 ) 0 π sin n x d x = ( n - 1 ) I ( n - 2 ) - ( n - 1 ) I ( n ) = n - 1 n I ( n - 2 ) I ( n ) I ( n - 2 ) = n - 1 n I ( 2 n - 1 ) I ( 2 n + 1 ) = 2 n + 1 2 n \begin{aligned}\displaystyle\Rightarrow I(n)&\displaystyle=\int_{0}^{\pi}\sin^% {n}xdx=\int_{0}^{\pi}udv=uv|_{x=0}^{x=\pi}-\int_{0}^{\pi}vdu\\ &\displaystyle=-\sin^{n-1}x\cos x|_{x=0}^{x=\pi}-\int_{0}^{\pi}-\cos x(n-1)% \sin^{n-2}x\cos xdx\\ &\displaystyle=0-(n-1)\int_{0}^{\pi}-\cos^{2}x\sin^{n-2}xdx,n>1\\ &\displaystyle=(n-1)\int_{0}^{\pi}(1-\sin^{2}x)\sin^{n-2}xdx\\ &\displaystyle=(n-1)\int_{0}^{\pi}\sin^{n-2}xdx-(n-1)\int_{0}^{\pi}\sin^{n}xdx% \\ &\displaystyle=(n-1)I(n-2)-(n-1)I(n)\\ &\displaystyle=\frac{n-1}{n}I(n-2)\\ \displaystyle\Rightarrow\frac{I(n)}{I(n-2)}&\displaystyle=\frac{n-1}{n}\\ \displaystyle\Rightarrow\frac{I(2n-1)}{I(2n+1)}&\displaystyle=\frac{2n+1}{2n}% \end{aligned}
  9. I ( 0 ) \displaystyle I(0)
  10. = 2 n - 1 2 n 2 n - 3 2 n - 2 2 n - 5 2 n - 4 5 6 3 4 1 2 I ( 0 ) = π k = 1 n 2 k - 1 2 k =\frac{2n-1}{2n}\cdot\frac{2n-3}{2n-2}\cdot\frac{2n-5}{2n-4}\cdot\cdots\cdot% \frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}I(0)=\pi\prod_{k=1}^{n}\frac{2k-1}{% 2k}
  11. I ( 2 n + 1 ) = 0 π sin 2 n + 1 x d x = 2 n 2 n + 1 I ( 2 n - 1 ) = 2 n 2 n + 1 2 n - 2 2 n - 1 I ( 2 n - 3 ) I(2n+1)=\int_{0}^{\pi}\sin^{2n+1}xdx=\frac{2n}{2n+1}I(2n-1)=\frac{2n}{2n+1}% \cdot\frac{2n-2}{2n-1}I(2n-3)
  12. = 2 n 2 n + 1 2 n - 2 2 n - 1 2 n - 4 2 n - 3 6 7 4 5 2 3 I ( 1 ) = 2 k = 1 n 2 k 2 k + 1 =\frac{2n}{2n+1}\cdot\frac{2n-2}{2n-1}\cdot\frac{2n-4}{2n-3}\cdot\cdots\cdot% \frac{6}{7}\cdot\frac{4}{5}\cdot\frac{2}{3}I(1)=2\prod_{k=1}^{n}\frac{2k}{2k+1}
  13. sin 2 n + 1 x sin 2 n x sin 2 n - 1 x , 0 x π \sin^{2n+1}x\leq\sin^{2n}x\leq\sin^{2n-1}x,0\leq x\leq\pi
  14. I ( 2 n + 1 ) I ( 2 n ) I ( 2 n - 1 ) \Rightarrow I(2n+1)\leq I(2n)\leq I(2n-1)
  15. 1 I ( 2 n ) I ( 2 n + 1 ) I ( 2 n - 1 ) I ( 2 n + 1 ) = 2 n + 1 2 n \Rightarrow 1\leq\frac{I(2n)}{I(2n+1)}\leq\frac{I(2n-1)}{I(2n+1)}=\frac{2n+1}{% 2n}
  16. lim n I ( 2 n ) I ( 2 n + 1 ) = 1 \Rightarrow\lim_{n\rightarrow\infty}\frac{I(2n)}{I(2n+1)}=1
  17. lim n I ( 2 n ) I ( 2 n + 1 ) = π 2 lim n k = 1 n ( 2 k - 1 2 k 2 k + 1 2 k ) = 1 \lim_{n\rightarrow\infty}\frac{I(2n)}{I(2n+1)}=\frac{\pi}{2}\lim_{n\rightarrow% \infty}\prod_{k=1}^{n}\left(\frac{2k-1}{2k}\cdot\frac{2k+1}{2k}\right)=1
  18. π 2 = k = 1 ( 2 k 2 k - 1 2 k 2 k + 1 ) = 2 1 2 3 4 3 4 5 6 5 6 7 \Rightarrow\frac{\pi}{2}=\prod_{k=1}^{\infty}\left(\frac{2k}{2k-1}\cdot\frac{2% k}{2k+1}\right)=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}% \cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\cdots
  19. n ! = 2 π n ( n e ) n [ 1 + O ( 1 n ) ] n!=\sqrt{2\pi n}{\left(\frac{n}{e}\right)}^{n}\left[1+O\left(\frac{1}{n}\right% )\right]
  20. p k = n = 1 k 2 n 2 n - 1 2 n 2 n + 1 p_{k}=\prod_{n=1}^{k}\frac{2n}{2n-1}\frac{2n}{2n+1}
  21. p k = 1 2 k + 1 n = 1 k ( 2 n ) 4 [ ( 2 n ) ( 2 n - 1 ) ] 2 = 1 2 k + 1 2 4 k ( k ! ) 4 [ ( 2 k ) ! ] 2 \begin{aligned}\displaystyle p_{k}&\displaystyle={1\over{2k+1}}\prod_{n=1}^{k}% \frac{(2n)^{4}}{[(2n)(2n-1)]^{2}}\\ &\displaystyle={1\over{2k+1}}\cdot{{2^{4k}\,(k!)^{4}}\over{[(2k)!]^{2}}}\end{aligned}
  22. π / 2 {π}/{2}
  23. ζ ( s ) \displaystyle\zeta(s)
  24. η ( s ) \displaystyle\eta(s)
  25. η ( 0 ) = - ζ ( 0 ) - ln 2 = - 1 2 n = 1 ( - 1 ) n - 1 [ ln n - ln ( n + 1 ) ] = - 1 2 n = 1 ( - 1 ) n - 1 ln n n + 1 = - 1 2 ( ln 1 2 - ln 2 3 + ln 3 4 - ln 4 5 + ln 5 6 - ) = 1 2 ( ln 2 1 + ln 2 3 + ln 4 3 + ln 4 5 + ln 6 5 + ) = 1 2 ln ( 2 1 2 3 4 3 4 5 ) = 1 2 ln π 2 ζ ( 0 ) = - 1 2 ln ( 2 π ) \begin{aligned}\displaystyle\Rightarrow\eta^{\prime}(0)&\displaystyle=-\zeta^{% \prime}(0)-\ln 2=-\frac{1}{2}\sum_{n=1}^{\infty}(-1)^{n-1}\left[\ln n-\ln(n+1)% \right]\\ &\displaystyle=-\frac{1}{2}\sum_{n=1}^{\infty}(-1)^{n-1}\ln\frac{n}{n+1}\\ &\displaystyle=-\frac{1}{2}\left(\ln\frac{1}{2}-\ln\frac{2}{3}+\ln\frac{3}{4}-% \ln\frac{4}{5}+\ln\frac{5}{6}-\cdots\right)\\ &\displaystyle=\frac{1}{2}\left(\ln\frac{2}{1}+\ln\frac{2}{3}+\ln\frac{4}{3}+% \ln\frac{4}{5}+\ln\frac{6}{5}+\cdots\right)\\ &\displaystyle=\frac{1}{2}\ln\left(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}% \cdot\frac{4}{5}\cdot\cdots\right)=\frac{1}{2}\ln\frac{\pi}{2}\\ \displaystyle\Rightarrow\zeta^{\prime}(0)&\displaystyle=-\frac{1}{2}\ln\left(2% \pi\right)\end{aligned}

Walsh_matrix.html

  1. H ( 2 1 ) = [ 1 1 1 - 1 ] , H(2^{1})=\begin{bmatrix}1&1\\ 1&-1\end{bmatrix},
  2. H ( 2 2 ) = [ 1 1 1 1 1 - 1 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 ] , H(2^{2})=\begin{bmatrix}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\\ \end{bmatrix},
  3. H ( 2 k ) = [ H ( 2 k - 1 ) H ( 2 k - 1 ) H ( 2 k - 1 ) - H ( 2 k - 1 ) ] = H ( 2 ) H ( 2 k - 1 ) , H(2^{k})=\begin{bmatrix}H(2^{k-1})&H(2^{k-1})\\ H(2^{k-1})&-H(2^{k-1})\end{bmatrix}=H(2)\otimes H(2^{k-1}),
  4. \otimes
  5. H ( 4 ) = [ 1 1 1 1 1 - 1 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 ] H(4)=\begin{bmatrix}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\\ \end{bmatrix}
  6. W ( 4 ) = [ 1 1 1 1 1 1 - 1 - 1 1 - 1 - 1 1 1 - 1 1 - 1 ] W(4)=\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&-1&1\\ 1&-1&1&-1\\ \end{bmatrix}
  7. W ( 8 ) = [ 1 1 1 1 1 1 1 1 1 1 1 1 - 1 - 1 - 1 - 1 1 1 - 1 - 1 - 1 - 1 1 1 1 1 - 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 1 - 1 - 1 1 1 - 1 - 1 1 - 1 1 1 - 1 1 - 1 1 - 1 - 1 1 - 1 1 1 - 1 1 - 1 1 - 1 1 - 1 ] W(8)=\begin{bmatrix}1&1&1&1&1&1&1&1\\ 1&1&1&1&-1&-1&-1&-1\\ 1&1&-1&-1&-1&-1&1&1\\ 1&1&-1&-1&1&1&-1&-1\\ 1&-1&-1&1&1&-1&-1&1\\ 1&-1&-1&1&-1&1&1&-1\\ 1&-1&1&-1&-1&1&-1&1\\ 1&-1&1&-1&1&-1&1&-1\\ \end{bmatrix}
  8. W ( 8 ) = [ 1 1 1 1 1 1 1 1 1 1 1 1 - 1 - 1 - 1 - 1 1 1 - 1 - 1 1 1 - 1 - 1 1 1 - 1 - 1 - 1 - 1 1 1 1 - 1 1 - 1 1 - 1 1 - 1 1 - 1 1 - 1 - 1 1 - 1 1 1 - 1 - 1 1 1 - 1 - 1 1 1 - 1 - 1 1 - 1 1 1 - 1 ] W(8)=\begin{bmatrix}1&1&1&1&1&1&1&1\\ 1&1&1&1&-1&-1&-1&-1\\ 1&1&-1&-1&1&1&-1&-1\\ 1&1&-1&-1&-1&-1&1&1\\ 1&-1&1&-1&1&-1&1&-1\\ 1&-1&1&-1&-1&1&-1&1\\ 1&-1&-1&1&1&-1&-1&1\\ 1&-1&-1&1&-1&1&1&-1\\ \end{bmatrix}
  9. W ( 8 ) = [ 1 1 1 1 1 1 1 1 1 - 1 1 - 1 1 - 1 1 - 1 1 1 - 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 1 - 1 - 1 1 1 1 1 1 - 1 - 1 - 1 - 1 1 - 1 1 - 1 - 1 1 - 1 1 1 1 - 1 - 1 - 1 - 1 1 1 1 - 1 - 1 1 - 1 1 1 - 1 ] W(8)=\begin{bmatrix}1&1&1&1&1&1&1&1\\ 1&-1&1&-1&1&-1&1&-1\\ 1&1&-1&-1&1&1&-1&-1\\ 1&-1&-1&1&1&-1&-1&1\\ 1&1&1&1&-1&-1&-1&-1\\ 1&-1&1&-1&-1&1&-1&1\\ 1&1&-1&-1&-1&-1&1&1\\ 1&-1&-1&1&-1&1&1&-1\\ \end{bmatrix}

Washburn's_equation.html

  1. L 2 = γ D t 4 η L^{2}=\frac{\gamma Dt}{4\eta}
  2. t t
  3. η \eta
  4. γ \gamma
  5. L L
  6. D D
  7. D D
  8. l l
  9. d V = π r 2 d l dV=\pi r^{2}dl
  10. δ l δ t = P 8 r 2 η l ( r 4 + 4 ϵ r 3 ) \frac{\delta l}{\delta t}=\frac{\sum P}{8r^{2}\eta l}(r^{4}+4\epsilon r^{3})
  11. P \sum P
  12. P A P_{A}
  13. P h P_{h}
  14. P c P_{c}
  15. η \eta
  16. ϵ \epsilon
  17. r r
  18. P h = h g ρ - l g ρ sin ψ P_{h}=hg\rho-lg\rho\sin\psi
  19. P c = 2 γ r cos ϕ P_{c}=\frac{2\gamma}{r}\cos\phi
  20. ρ \rho
  21. γ \gamma
  22. ψ \psi
  23. ϕ \phi
  24. l l
  25. δ l δ t = [ P A + g ρ ( h - l sin ψ ) + 2 γ r cos ϕ ] ( r 4 + 4 ϵ r 3 ) 8 r 2 η l \frac{\delta l}{\delta t}=\frac{[P_{A}+g\rho(h-l\sin\psi)+\frac{2\gamma}{r}% \cos\phi](r^{4}+4\epsilon r^{3})}{8r^{2}\eta l}
  26. 10 4 [ μ m cm ] [ N m 2 ] 68947.6 [ dynes cm 2 ] = 0.1450 ( 38 ) \frac{10^{4}\left[\mathrm{\frac{\mu m}{cm}}\right]\left[\mathrm{\frac{N}{m^{2}% }}\right]}{68947.6\left[\mathrm{\frac{dynes}{cm^{2}}}\right]}=0.1450(38)
  27. l = [ r cos θ 2 ] 1 2 [ γ η ] 1 2 t 1 2 l=\left[\frac{r\cos\theta}{2}\right]^{\frac{1}{2}}\left[\frac{\gamma}{\eta}% \right]^{\frac{1}{2}}t^{\frac{1}{2}}
  28. [ γ η ] 1 2 \left[\tfrac{\gamma}{\eta}\right]^{\frac{1}{2}}

Water-gas_shift_reaction.html

  1. \rightleftharpoons
  2. log 10 K eq = - 2.4198 + 0.0003855 T + 2180.6 T \log_{10}K_{\mathrm{eq}}=-2.4198+0.0003855T+\frac{2180.6}{T}

Water_content.html

  1. θ = V w V T \theta=\frac{V_{w}}{V_{T}}
  2. V w V_{w}
  3. V T = V s + V v = V s + V w + V a V_{T}=V_{s}+V_{v}=V_{s}+V_{w}+V_{a}
  4. u = m w m t u=\frac{m_{w}}{m_{t}}
  5. m w m_{w}
  6. m t m_{t}
  7. m s m_{s}
  8. m t m_{t}
  9. S w S_{w}
  10. S w = V w V v = V w V T ϕ = θ ϕ S_{w}=\frac{V_{w}}{V_{v}}=\frac{V_{w}}{V_{T}\phi}=\frac{\theta}{\phi}
  11. ϕ = V v / V T \phi=V_{v}/V_{T}
  12. V v V_{v}
  13. Θ \Theta
  14. S e S_{e}
  15. Θ = θ - θ r θ s - θ r \Theta=\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}}
  16. θ \theta
  17. θ r \theta_{r}
  18. d θ / d h d\theta/dh
  19. θ s \theta_{s}
  20. ϕ \phi
  21. θ = m wet - m dry ρ w V b \theta=\frac{m_{\,\text{wet}}-m_{\,\text{dry}}}{\rho_{w}\cdot V_{b}}
  22. m wet m_{\,\text{wet}}
  23. m dry m_{\,\text{dry}}
  24. ρ w \rho_{w}
  25. V b V_{b}
  26. u = m wet - m dry m wet u=\frac{m_{\,\text{wet}}-m_{\,\text{dry}}}{m_{\,\text{wet}}}
  27. u = m wet - m dry m dry u=\frac{m_{\,\text{wet}}-m_{\,\text{dry}}}{m_{\,\text{dry}}}
  28. p = W - D D p=\frac{W-D}{D}
  29. p p
  30. W W
  31. D D

Water_splitting.html

  1. 2 H 2 O H e a t 2 H 2 + O 2 2\,\text{ }H_{2}O\,\text{ }\stackrel{Heat}{\rightleftharpoons}\,\text{ }2\,% \text{ }H_{2}+\,\text{ }O_{2}

Waterline.html

  1. 40 T \tfrac{\triangle}{40T}
  2. \triangle
  3. \triangle

Wave_front_set.html

  1. ϕ C 0 \phi\in C_{0}^{\infty}
  2. ϕ \phi
  3. | ( ϕ f ) ( ξ ) | C N ( 1 + | ξ | ) - N for all ξ Γ |(\phi f)^{\wedge}(\xi)|\leq C_{N}(1+|\xi|)^{-N}\quad\mbox{for all }~{}\ \xi\in\Gamma
  4. ( ϕ f ) (\phi f)^{\wedge}
  5. WF ( f ) = { ( x , ξ ) n × n ξ Σ x ( f ) } {\rm WF}(f)=\{(x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\mid\xi\in\Sigma_{x}% (f)\}
  6. Σ x ( f ) \Sigma_{x}(f)
  7. ξ \xi
  8. ξ \xi
  9. Σ x ( f ) \Sigma_{x}(f)
  10. ( ϕ f ) ( ξ ) < c N ( 1 + | ξ | ) - N forall ξ Γ . (\phi f)^{\wedge}(\xi)<c_{N}(1+|\xi|)^{-N}\quad{\rm forall}\ \xi\in\Gamma.
  11. x , ξ x,\xi
  12. WF ( f ) = { ( x , ξ ) T * ( X ) ξ Σ x ( f ) } {\rm WF}(f)=\{(x,\xi)\in T^{*}(X)\mid\xi\in\Sigma_{x}(f)\}
  13. Σ x ( f ) \Sigma_{x}(f)
  14. ξ \xi
  15. ξ \xi
  16. ξ Σ x ( f ) ξ = 0 or ϕ 𝒟 x , V 𝒱 ξ : ϕ f ^ | V O ( V ) \xi\notin\Sigma_{x}(f)\iff\xi=0\,\text{ or }\exists\phi\in\mathcal{D}_{x},\ % \exists V\in\mathcal{V}_{\xi}:\widehat{\phi f}|_{V}\in O(V)
  17. 𝒟 x \mathcal{D}_{x}
  18. 𝒱 ξ \mathcal{V}_{\xi}
  19. ξ \xi
  20. c V V c\cdot V\subset V
  21. c > 0 c>0
  22. u ^ | V \widehat{u}|_{V}
  23. O : Ω O ( Ω ) O:\Omega\to O(\Omega)
  24. ( 1 + | ξ | ) s v ( ξ ) (1+|\xi|)^{s}v(\xi)
  25. ϕ \phi
  26. C C^{\infty}

Waves_in_plasmas.html

  1. 𝐤 × 𝐄 ~ = ω 𝐁 ~ \mathbf{k}\times\tilde{\mathbf{E}}=\omega\tilde{\mathbf{B}}
  2. B 0 = 0 or k B 0 \vec{B}_{0}=0\ {\rm or}\ \vec{k}\|\vec{B}_{0}
  3. ω 2 = ω p 2 + 3 k 2 v t h 2 \omega^{2}=\omega_{p}^{2}+3k^{2}v_{th}^{2}
  4. k B 0 \vec{k}\perp\vec{B}_{0}
  5. ω 2 = ω p 2 + ω c 2 = ω h 2 \omega^{2}=\omega_{p}^{2}+\omega_{c}^{2}=\omega_{h}^{2}
  6. B 0 = 0 or k B 0 \vec{B}_{0}=0\ {\rm or}\ \vec{k}\|\vec{B}_{0}
  7. ω 2 = k 2 v s 2 = k 2 γ e K T e + γ i K T i M \omega^{2}=k^{2}v_{s}^{2}=k^{2}\frac{\gamma_{e}KT_{e}+\gamma_{i}KT_{i}}{M}
  8. k B 0 \vec{k}\perp\vec{B}_{0}
  9. ω 2 = Ω c 2 + k 2 v s 2 \omega^{2}=\Omega_{c}^{2}+k^{2}v_{s}^{2}
  10. k B 0 \vec{k}\perp\vec{B}_{0}
  11. ω 2 = [ ( Ω c ω c ) - 1 + ω i - 2 ] - 1 \omega^{2}=[(\Omega_{c}\omega_{c})^{-1}+\omega_{i}^{-2}]^{-1}
  12. B 0 = 0 \vec{B}_{0}=0
  13. ω 2 = ω p 2 + k 2 c 2 \omega^{2}=\omega_{p}^{2}+k^{2}c^{2}
  14. k B 0 , E 1 B 0 \vec{k}\perp\vec{B}_{0},\ \vec{E}_{1}\|\vec{B}_{0}
  15. c 2 k 2 ω 2 = 1 - ω p 2 ω 2 \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}}
  16. k B 0 , E 1 B 0 \vec{k}\perp\vec{B}_{0},\ \vec{E}_{1}\perp\vec{B}_{0}
  17. c 2 k 2 ω 2 = 1 - ω p 2 ω 2 ω 2 - ω p 2 ω 2 - ω h 2 \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}}\,\frac{% \omega^{2}-\omega_{p}^{2}}{\omega^{2}-\omega_{h}^{2}}
  18. k B 0 \vec{k}\|\vec{B}_{0}
  19. c 2 k 2 ω 2 = 1 - ω p 2 / ω 2 1 - ( ω c / ω ) \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}/\omega^{2}}{1-(\omega_{c}% /\omega)}
  20. k B 0 \vec{k}\|\vec{B}_{0}
  21. c 2 k 2 ω 2 = 1 - ω p 2 / ω 2 1 + ( ω c / ω ) \frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}/\omega^{2}}{1+(\omega_{c}% /\omega)}
  22. B 0 = 0 \vec{B}_{0}=0
  23. k B 0 \vec{k}\|\vec{B}_{0}
  24. ω 2 = k 2 v A 2 \omega^{2}=k^{2}v_{A}^{2}
  25. k B 0 \vec{k}\perp\vec{B}_{0}
  26. ω 2 k 2 = c 2 v s 2 + v A 2 c 2 + v A 2 \frac{\omega^{2}}{k^{2}}=c^{2}\,\frac{v_{s}^{2}+v_{A}^{2}}{c^{2}+v_{A}^{2}}
  27. ω \omega
  28. k k
  29. c c
  30. ω p \omega_{p}
  31. ω i \omega_{i}
  32. ω c \omega_{c}
  33. Ω c \Omega_{c}
  34. ω h \omega_{h}
  35. v s v_{s}
  36. v A v_{A}

Weak_derivative.html

  1. L 1 ( [ a , b ] ) \mathrm{L}^{1}([a,b])
  2. u u
  3. L 1 ( [ a , b ] ) L^{1}([a,b])
  4. v v
  5. L 1 ( [ a , b ] ) L^{1}([a,b])
  6. u u
  7. a b u ( t ) φ ( t ) d t = - a b v ( t ) φ ( t ) d t \int_{a}^{b}u(t)\varphi^{\prime}(t)dt=-\int_{a}^{b}v(t)\varphi(t)dt
  8. φ \varphi
  9. φ ( a ) = φ ( b ) = 0 \varphi(a)=\varphi(b)=0
  10. n n
  11. u u
  12. v v
  13. L l o c 1 ( U ) L_{loc}^{1}(U)
  14. U n U\subset\mathbb{R}^{n}
  15. α \alpha
  16. v v
  17. α t h \alpha^{th}
  18. u u
  19. U u D α φ = ( - 1 ) | α | U v φ \int_{U}uD^{\alpha}\varphi=(-1)^{|\alpha|}\int_{U}v\varphi
  20. φ C c ( U ) \varphi\in C^{\infty}_{c}(U)
  21. φ \varphi
  22. U U
  23. u u
  24. D α u D^{\alpha}u
  25. v : [ - 1 , 1 ] [ - 1 , 1 ] : t v ( t ) = { 1 , if t > 0 ; 0 , if t = 0 ; - 1 , if t < 0. v\colon[-1,1]\to[-1,1]\colon t\mapsto v(t)=\begin{cases}1,&\mbox{if }~{}t>0;\\ 0,&\mbox{if }~{}t=0;\\ -1,&\mbox{if }~{}t<0.\end{cases}
  26. 1 1_{\mathbb{Q}}
  27. 1 ( t ) φ ( t ) d t = 0. \int 1_{\mathbb{Q}}(t)\varphi(t)dt=0.
  28. v ( t ) = 0 v(t)=0
  29. 1 1_{\mathbb{Q}}
  30. 1 1_{\mathbb{Q}}

Weak_hypercharge.html

  1. Q = T 3 + Y W 2 \qquad Q=T_{3}+{Y_{\rm W}\over 2}
  2. Y W = 2 ( Q - T 3 ) \qquad Y_{\rm W}=2(Q-T_{3})
  3. Y W = Q - T 3 \qquad Y_{\rm W}=Q-T_{3}
  4. X + 2 Y W = 5 ( B - L ) X+2Y_{\rm W}=5(B-L)\,

Weak_isospin.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. 1 / 2 {1}/{2}
  7. Q = T 3 + Y W 2 . Q=T_{3}+\frac{Y_{\mathrm{W}}}{2}.
  8. 1 / 2 {1}/{2}
  9. 1 / 2 {1}/{2}
  10. 1 / 2 {1}/{2}
  11. 1 / 2 {1}/{2}

Weak_topology_(polar_topology).html

  1. ( X , Y , , ) (X,Y,\langle,\rangle)
  2. σ ( X , Y ) \sigma(X,Y)
  3. X X
  4. ( X , σ ( X , Y ) ) Y (X,\sigma(X,Y))^{\prime}\simeq Y
  5. ( X , σ ( X , Y ) ) (X,\sigma(X,Y))
  6. Y Y
  7. y y
  8. Y Y
  9. X X
  10. X X
  11. p y : X p_{y}:X\to\mathbb{R}
  12. p y ( x ) := | x , y | x X p_{y}(x):=|\langle x,y\rangle|\qquad x\in X
  13. X X
  14. X X
  15. X X^{\prime}
  16. σ ( X , X ) \sigma(X,X^{\prime})
  17. X X
  18. σ ( X , X ) \sigma(X^{\prime},X)
  19. X X^{\prime}

Weber_number.html

  1. We = ρ v 2 l σ \mathrm{We}=\frac{\rho\,v^{2}\,l}{\sigma}
  2. ρ \rho
  3. σ \sigma
  4. We * = We 12 \mathrm{We}^{*}=\frac{\mathrm{We}}{12}
  5. We * = E kin E surf \mathrm{We}^{*}=\frac{E_{\mathrm{kin}}}{E_{\mathrm{surf}}}
  6. E kin = π ρ l 3 U 2 12 E_{\mathrm{kin}}=\frac{\pi\rho l^{3}U^{2}}{12}
  7. E surf = π l 2 σ E_{\mathrm{surf}}=\pi l^{2}\sigma

Wedge_(mechanical_device).html

  1. P = F A v A = F B v B , P=F_{A}v_{A}=F_{B}v_{B},\!
  2. F B F A = v A v B . \frac{F_{B}}{F_{A}}=\frac{v_{A}}{v_{B}}.
  3. v B = v A tan α , v_{B}=v_{A}\tan\alpha,\!
  4. M A = F B F A = 1 tan α . MA=\frac{F_{B}}{F_{A}}=\frac{1}{\tan\alpha}.
  5. MechanicalAdvantage = Length Width \rm MechanicalAdvantage={Length\over Width}

Weierstrass_factorization_theorem.html

  1. { c n } \{c_{n}\}
  2. p ( z ) p(z)
  3. p ( z ) = n ( z - c n ) . p(z)=\,\prod_{n}(z-c_{n}).
  4. p ( z ) p(z)
  5. p ( z ) = a n ( z - c n ) , \,p(z)=a\prod_{n}(z-c_{n}),
  6. n ( z - c n ) \,\prod_{n}(z-c_{n})
  7. { c n } \{c_{n}\}
  8. ( z - c n ) (z-c_{n})
  9. n n\to\infty
  10. ( z - c n ) (z-c_{n})
  11. n n\in\mathbb{N}
  12. E n ( z ) = { ( 1 - z ) if n = 0 , ( 1 - z ) exp ( z 1 1 + z 2 2 + + z n n ) otherwise . E_{n}(z)=\begin{cases}(1-z)&\,\text{if }n=0,\\ (1-z)\exp\left(\frac{z^{1}}{1}+\frac{z^{2}}{2}+\cdots+\frac{z^{n}}{n}\right)&% \,\text{otherwise}.\end{cases}
  13. | 1 - E n ( z ) | | z | n + 1 . |1-E_{n}(z)|\leq|z|^{n+1}.
  14. { a n } \{a_{n}\}
  15. | a n | |a_{n}|\to\infty
  16. { p n } \{p_{n}\}
  17. r > 0 r>0
  18. n = 1 ( r / | a n | ) 1 + p n < , \sum_{n=1}^{\infty}\left(r/|a_{n}|\right)^{1+p_{n}}<\infty,
  19. f ( z ) = n = 1 E p n ( z / a n ) f(z)=\prod_{n=1}^{\infty}E_{p_{n}}(z/a_{n})
  20. a n a_{n}
  21. z 0 z_{0}
  22. { a n } \{a_{n}\}
  23. z = z 0 z=z_{0}
  24. { p n } \{p_{n}\}
  25. p n = n p_{n}=n
  26. { a n } \{a_{n}\}
  27. p n = 0 p_{n}=0
  28. f ( z ) = c n ( z - a n ) \,f(z)=c\,{\displaystyle\prod}_{n}(z-a_{n})
  29. { a n } \{a_{n}\}
  30. { p n } \{p_{n}\}
  31. f ( z ) = z m e g ( z ) n = 1 E p n ( z a n ) . f(z)=z^{m}e^{g(z)}\prod_{n=1}^{\infty}E_{p_{n}}\!\!\left(\frac{z}{a_{n}}\right).
  32. sin π z = π z n 0 ( 1 - z n ) e z / n = π z n = 1 ( 1 - z 2 n 2 ) \sin\pi z=\pi z\prod_{n\neq 0}\left(1-\frac{z}{n}\right)e^{z/n}=\pi z\prod_{n=% 1}^{\infty}\left(1-\frac{z^{2}}{n^{2}}\right)
  33. cos π z = q , q odd ( 1 - 2 z q ) e 2 z / q = n = 0 ( 1 - 4 z 2 ( 2 n + 1 ) 2 ) \cos\pi z=\prod_{q\in\mathbb{Z},\,q\;\,\text{odd}}\left(1-\frac{2z}{q}\right)e% ^{2z/q}=\prod_{n=0}^{\infty}\left(1-\frac{4z^{2}}{(2n+1)^{2}}\right)
  34. f ( z ) = z m e g ( z ) n = 1 E p ( z / a n ) f(z)=z^{m}e^{g(z)}\displaystyle\prod_{n=1}^{\infty}E_{p}(z/a_{n})

Weierstrass_functions.html

  1. Λ \Complex \Lambda\subset\Complex
  2. σ ( z ; Λ ) = z w Λ * ( 1 - z w ) e z / w + 1 2 ( z / w ) 2 \sigma(z;\Lambda)=z\prod_{w\in\Lambda^{*}}\left(1-\frac{z}{w}\right)e^{z/w+% \frac{1}{2}(z/w)^{2}}
  3. Λ * \Lambda^{*}
  4. Λ - { 0 } \Lambda-\{0\}
  5. ζ ( z ; Λ ) = σ ( z ; Λ ) σ ( z ; Λ ) = 1 z + w Λ * ( 1 z - w + 1 w + z w 2 ) . \zeta(z;\Lambda)=\frac{\sigma^{\prime}(z;\Lambda)}{\sigma(z;\Lambda)}=\frac{1}% {z}+\sum_{w\in\Lambda^{*}}\left(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}% \right).
  6. ζ ( z ; Λ ) = 1 z - k = 1 𝒢 2 k + 2 ( Λ ) z 2 k + 1 \zeta(z;\Lambda)=\frac{1}{z}-\sum_{k=1}^{\infty}\mathcal{G}_{2k+2}(\Lambda)z^{% 2k+1}
  7. 𝒢 2 k + 2 \mathcal{G}_{2k+2}
  8. - ( z ) -\wp(z)
  9. ( z ) \wp(z)
  10. η ( w ; Λ ) = ζ ( z + w ; Λ ) - ζ ( z ; Λ ) , for any z \Complex \eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda),\mbox{ for any }~{}z\in\Complex
  11. ζ ( z + w ; Λ ) - ζ ( z ; Λ ) \zeta(z+w;\Lambda)-\zeta(z;\Lambda)
  12. ( z ; Λ ) = - ζ ( z ; Λ ) , for any z \Complex \wp(z;\Lambda)=-\zeta^{\prime}(z;\Lambda),\mbox{ for any }~{}z\in\Complex

Weighted_geometric_mean.html

  1. X = { x 1 , x 2 , x n } X=\{x_{1},x_{2}\dots,x_{n}\}
  2. W = { w 1 , w 2 , , w n } W=\{w_{1},w_{2},\dots,w_{n}\}
  3. x ¯ = ( i = 1 n x i w i ) 1 / i = 1 n w i = exp ( i = 1 n w i ln x i i = 1 n w i ) \bar{x}=\left(\prod_{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum_{i=1}^{n}w_{i}}=% \quad\exp\left(\frac{\sum_{i=1}^{n}w_{i}\ln x_{i}}{\sum_{i=1}^{n}w_{i}\quad}\right)

Weil–Châtelet_group.html

  1. Sel ( f ) ( A / K ) = v ker ( H 1 ( G K , ker ( f ) ) H 1 ( G K v , A v [ f ] ) / im ( κ v ) ) \mathrm{Sel}^{(f)}(A/K)=\bigcap_{v}\mathrm{ker}(H^{1}(G_{K},\mathrm{ker}(f))% \rightarrow H^{1}(G_{K_{v}},A_{v}[f])/\mathrm{im}(\kappa_{v}))
  2. κ v \kappa_{v}
  3. B v ( K v ) / f ( A v ( K v ) ) H 1 ( G K v , A v [ f ] ) B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])

Weinberg_angle.html

  1. ( γ Z 0 ) = ( cos θ W sin θ W - sin θ W cos θ W ) ( B 0 W 0 ) \begin{pmatrix}\gamma\\ Z^{0}\end{pmatrix}=\begin{pmatrix}\cos\theta_{W}&\sin\theta_{W}\\ -\sin\theta_{W}&\cos\theta_{W}\end{pmatrix}\begin{pmatrix}B^{0}\\ W^{0}\end{pmatrix}
  2. m Z = m W cos θ W m_{Z}=\frac{m_{W}}{\cos\theta_{W}}
  3. S U ( 2 ) L SU(2)_{L}
  4. U ( 1 ) Y U(1)_{Y}
  5. cos θ W = g g 2 + g 2 \cos\theta_{W}=\frac{g}{\sqrt{g^{2}+g^{\prime 2}}}
  6. sin θ W = g g 2 + g 2 \sin\theta_{W}=\frac{g^{\prime}}{\sqrt{g^{2}+g^{\prime 2}}}
  7. cos θ W = m W m Z \cos\theta_{W}=\frac{m_{W}}{m_{Z}}
  8. M S ¯ \overline{MS}

Wess–Zumino–Witten_model.html

  1. S k ( γ ) = - k 8 π S 2 d 2 x 𝒦 ( γ - 1 μ γ , γ - 1 μ γ ) + 2 π k S W Z ( γ ) . S_{k}(\gamma)=-\,\frac{k}{8\pi}\int_{S^{2}}d^{2}x\,\mathcal{K}(\gamma^{-1}% \partial^{\mu}\gamma\,,\,\gamma^{-1}\partial_{\mu}\gamma)+2\pi k\,S^{\mathrm{W% }Z}(\gamma).
  2. 𝒦 \mathcal{K}
  3. S W Z ( γ ) = - 1 48 π 2 B 3 d 3 y ϵ i j k 𝒦 ( γ - 1 γ y i , [ γ - 1 γ y j , γ - 1 γ y k ] ) S^{\mathrm{W}Z}(\gamma)=-\,\frac{1}{48\pi^{2}}\int_{B^{3}}d^{3}y\,\epsilon^{% ijk}\mathcal{K}\left(\gamma^{-1}\,\frac{\partial\gamma}{\partial y^{i}}\,,\,% \left[\gamma^{-1}\,\frac{\partial\gamma}{\partial y^{j}}\,,\,\gamma^{-1}\,% \frac{\partial\gamma}{\partial y^{k}}\right]\right)
  4. 𝒦 ( e a , [ e b , e c ] ) \mathcal{K}(e_{a},[e_{b},e_{c}])
  5. S W Z ( γ ) = B 3 γ * c . S^{\mathrm{W}Z}(\gamma)=\int_{B^{3}}\gamma^{*}c.
  6. S W Z ( γ ) = S W Z ( γ ) + n , S^{\mathrm{W}Z}(\gamma)=S^{\mathrm{W}Z}(\gamma^{\prime})+n~{},
  7. exp ( i 2 π k S W Z ( γ ) ) = exp ( i 2 π k S W Z ( γ ) ) . \exp\left(i2\pi kS^{\mathrm{W}Z}(\gamma)\right)=\exp\left(i2\pi kS^{\mathrm{W}% Z}(\gamma^{\prime})\right).

Western_Electric_rules.html

  1. x ¯ \bar{x}

Wet-bulb_globe_temperature.html

  1. 𝑊𝐵𝐺𝑇 = 0.7 T w + 0.2 T g + 0.1 T d \mathit{WBGT}=0.7T_{w}+0.2T_{g}+0.1T_{d}
  2. 𝑊𝐵𝐺𝑇 = 0.7 T w + 0.3 T g \mathit{WBGT}=0.7T_{w}+0.3T_{g}

Wetting.html

  1. γ α θ + γ θ β cos θ + γ α β cos α = 0 \gamma_{\alpha\theta}+\gamma_{\theta\beta}\cos{\theta}+\gamma_{\alpha\beta}% \cos{\alpha}\ =0
  2. γ α θ cos θ + γ θ β + γ α β cos β = 0 \gamma_{\alpha\theta}\cos{\theta}+\gamma_{\theta\beta}+\gamma_{\alpha\beta}% \cos{\beta}\ =0
  3. γ α θ cos α + γ θ β cos β + γ α β = 0 \gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{% \alpha\beta}\ =0
  4. α + β + θ = 2 π \alpha+\beta+\theta=2\pi
  5. γ S G = γ S L + γ L G cos θ \gamma_{SG}\ =\gamma_{SL}+\gamma_{LG}\cos{\theta}
  6. θ A \theta_{\mathrm{A}}
  7. θ R \theta_{\mathrm{R}}
  8. θ c \theta_{\mathrm{c}}
  9. θ A \theta_{\mathrm{A}}
  10. θ R \theta_{\mathrm{R}}
  11. θ c = arccos ( r A cos θ A + r R cos θ R r A + r R ) \theta_{\mathrm{c}}=\arccos\left(\frac{r_{\mathrm{A}}\cos{\theta_{\mathrm{A}}}% +r_{\mathrm{R}}\cos{\theta_{\mathrm{R}}}}{r_{\mathrm{A}}+r_{\mathrm{R}}}\right)
  12. r A = ( sin 3 θ A 2 - 3 cos θ A + cos 3 θ A ) 1 / 3 ; r R = ( sin 3 θ R 2 - 3 cos θ R + cos 3 θ R ) 1 / 3 r_{\mathrm{A}}=\left(\frac{\sin^{3}{\theta_{\mathrm{A}}}}{2-3\cos{\theta_{% \mathrm{A}}}+\cos^{3}{\theta_{\mathrm{A}}}}\right)^{1/3}~{};~{}~{}r_{\mathrm{R% }}=\left(\frac{\sin^{3}{\theta_{\mathrm{R}}}}{2-3\cos{\theta_{\mathrm{R}}}+% \cos^{3}{\theta_{\mathrm{R}}}}\right)^{1/3}
  13. S = γ S G - ( γ S L + γ L G ) S\ =\gamma_{SG}-(\gamma_{SL}+\gamma_{LG})
  14. H = θ a - θ r \,\text{H}=\,\theta_{a}-\,\theta_{r}
  15. cos θ * = r cos θ \cos\,{\theta^{*}}=r\cos\,{\theta}
  16. θ * \theta^{*}
  17. cos θ * = r f f cos θ Y + f - 1 \cos\,{\theta^{*}}=r_{f}\,f\,\cos\,{\theta\text{Y}}+f-1
  18. f i f_{i}
  19. γ cos θ * = n = 1 N f i ( γ i,sv - γ i,sl ) \gamma\cos\,{\theta^{*}}=\sum_{n=1}^{N}f_{i}({\gamma\text{i,sv}}-{\gamma\text{% i,sl}})
  20. γ cos θ * = f 1 ( γ 1,sv - γ 1,sl ) - ( 1 - f 1 ) γ \gamma\cos\,{\theta^{*}}=f_{1}({\gamma\text{1,sv}}-{\gamma\text{1,sl}})-(1-f_{% 1}){\gamma}
  21. cos θ C = ϕ - 1 r - ϕ \cos\,{\theta\text{C}}=\frac{\phi-1}{r-\phi}
  22. cos θ * = ϕ cos θ C + ( 1 - ϕ ) \cos\,{\theta^{*}}=\phi\cos\,{\theta_{C}}+(1-{\phi})
  23. r ( t ) = r e [ 1 - exp ( - ( 2 γ L G r e 12 + ρ g 9 r e 10 ) 24 λ V 4 ( t + t 0 ) π 2 η ) ] 1 6 r(t)=r_{e}\left[1-\exp\left(-\left(\frac{2\gamma_{LG}}{r^{12}_{e}}+\frac{\rho g% }{9r^{10}_{e}}\right)\frac{24\lambda V^{4}(t+t_{0})}{\pi^{2}\eta}\right)\right% ]^{\frac{1}{6}}
  24. r ( t ) = [ ( γ L G 96 λ V 4 π 2 η ( t + t 0 ) ) 1 2 + ( λ ( t + t 0 ) η ) 2 3 24 ρ g V 3 8 7 96 1 3 π 4 3 γ L G 1 3 ] 1 6 r(t)=\left[\left(\gamma_{LG}\frac{96\lambda V^{4}}{\pi^{2}\eta}\left(t+t_{0}% \right)\right)^{\tfrac{1}{2}}+\left(\frac{\lambda(t+t_{0})}{\eta}\right)^{% \tfrac{2}{3}}\frac{24\rho gV^{\frac{3}{8}}}{7\cdot 96^{\frac{1}{3}}\pi^{\frac{% 4}{3}}\gamma_{LG}^{\frac{1}{3}}}\right]^{\frac{1}{6}}
  25. σ = γ S + P V + π R 2 ( γ SL - γ SV ) \sigma={\gamma}S+PV+{\pi}\,R^{2}({\gamma\text{SL}}-{\gamma\text{SV}})
  26. cos θ ( t ) = cos θ 0 + ( cos θ - cos θ 0 ) ( 1 - e - t τ ) \cos\,{\theta(t)}=\cos\,{\theta\text{0}}+({\cos\,{\theta_{\infty}}}-\cos\,{% \theta\text{0}})({1-\mathrm{e}^{\frac{-t}{\tau}}})

Weyl_transformation.html

  1. g a b e - 2 ω ( x ) g a b g_{ab}\rightarrow e^{-2\omega(x)}g_{ab}
  2. φ φ e k ω . \varphi\to\varphi e^{k\omega}.
  3. μ ω \partial_{\mu}\omega
  4. B μ = A μ + μ ω . B_{\mu}=A_{\mu}+\partial_{\mu}\omega.
  5. D μ φ μ φ + k B μ φ D_{\mu}\varphi\equiv\partial_{\mu}\varphi+kB_{\mu}\varphi
  6. k - 1 k-1

Wheel_and_axle.html

  1. M A = F B F A = a b . MA=\frac{F_{B}}{F_{A}}=\frac{a}{b}.
  2. I . M . A . = R a d i u s W h e e l R a d i u s A x l e I.M.A.=\frac{Radius_{Wheel}}{Radius_{Axle}}
  3. A . M . A . = R E a c t u a l A.M.A.=\frac{R}{E_{actual}}

Wheel_graph.html

  1. n 2 - 3 n + 3 n^{2}-3n+3
  2. P W n ( x ) = x ( ( x - 2 ) ( n - 1 ) - ( - 1 ) n ( x - 2 ) ) . P_{W_{n}}(x)=x((x-2)^{(n-1)}-(-1)^{n}(x-2)).

Wheeler–DeWitt_equation.html

  1. γ i j \gamma_{ij}
  2. g μ ν d x μ d x ν = ( - N 2 + β k β k ) d t 2 + 2 β k d x k d t + γ i j d x i d x j . g_{\mu\nu}\,\mathrm{d}x^{\mu}\,\mathrm{d}x^{\nu}=(-\,N^{2}+\beta_{k}\beta^{k})% \,\mathrm{d}t^{2}+2\beta_{k}\,\mathrm{d}x^{k}\,\mathrm{d}t+\gamma_{ij}\,% \mathrm{d}x^{i}\,\mathrm{d}x^{j}.
  3. γ i j \gamma_{ij}
  4. π k l \pi^{kl}
  5. = 1 2 γ G i j k l π i j π k l - γ R ( 3 ) = 0 \mathcal{H}=\frac{1}{2\sqrt{\gamma}}G_{ijkl}\pi^{ij}\pi^{kl}-\sqrt{\gamma}\,{}% ^{(3)}\!R=0
  6. γ = det ( γ i j ) \gamma=\det(\gamma_{ij})
  7. G i j k l = ( γ i k γ j l + γ i l γ j k - γ i j γ k l ) G_{ijkl}=(\gamma_{ik}\gamma_{jl}+\gamma_{il}\gamma_{jk}-\gamma_{ij}\gamma_{kl})
  8. ^ = 1 2 γ G ^ i j k l π ^ i j π ^ k l - γ R ^ ( 3 ) . \widehat{\mathcal{H}}=\frac{1}{2\sqrt{\gamma}}\widehat{G}_{ijkl}\widehat{\pi}^% {ij}\widehat{\pi}^{kl}-\sqrt{\gamma}\,{}^{(3)}\!\widehat{R}.
  9. γ ^ i j ( t , x k ) γ i j ( t , x k ) \hat{\gamma}_{ij}(t,x^{k})\to\gamma_{ij}(t,x^{k})
  10. π ^ i j ( t , x k ) - i δ δ γ i j ( t , x k ) . \hat{\pi}^{ij}(t,x^{k})\to-i\frac{\delta}{\delta\gamma_{ij}(t,x^{k})}.
  11. Z = C e - I [ g μ ν , ϕ ] 𝒟 g 𝒟 ϕ Z=\int_{C}\mathrm{e}^{-I[g_{\mu\nu},\phi]}\mathcal{D}{g}\,\mathcal{D}\phi
  12. δ I E H δ N = 0 \frac{\delta I_{EH}}{\delta N}=0
  13. I E H I_{EH}
  14. N N
  15. δ Z δ N = 0 = δ I [ g μ ν , ϕ ] δ N | Σ exp ( - I [ g μ ν , ϕ ] ) 𝒟 g 𝒟 ϕ \frac{\delta Z}{\delta N}=0=\int\left.\frac{\delta I[g_{\mu\nu},\phi]}{\delta N% }\right|_{\Sigma}\exp\left(-I[g_{\mu\nu},\phi]\right)\,\mathcal{D}{g}\,% \mathcal{D}\phi
  16. Σ \Sigma
  17. < g 1 , ϕ 1 | g 2 , ϕ 2 Ψ [ g 1 , ϕ 1 ] Ψ ¯ [ g 2 , ϕ 2 ] e i S [ Ψ ] D [ Ψ ] <g_{1},\phi_{1}|g_{2},\phi_{2}>=\int\Psi[g_{1},\phi_{1}]\overline{\Psi}[g_{2},% \phi_{2}]e^{iS[\Psi]}D[\Psi]
  18. Ψ \Psi
  19. ϕ \phi
  20. Ψ \Psi
  21. H ^ ( x ) \hat{H}(x)
  22. | ψ |\psi\rangle
  23. H ^ \hat{H}
  24. | ψ |\psi\rangle
  25. | ψ |\psi\rangle
  26. H ^ \hat{H}
  27. H ^ | ψ = i / t | ψ \hat{H}|\psi\rangle=i\hbar\partial/\partial t|\psi\rangle
  28. 𝒫 ( x ) | ψ = 0 \vec{\mathcal{P}}(x)\left|\psi\right\rangle=0
  29. t t
  30. ψ e i θ ( r ) ψ \psi\rightarrow e^{i\theta(\vec{r})}\psi
  31. θ ( r ) \theta(\vec{r})

White's_law.html

  1. C = E T , C=ET,

Whitehead_link.html

  1. σ 1 2 σ 2 2 σ 1 - 1 σ 2 - 2 . \sigma^{2}_{1}\sigma^{2}_{2}\sigma^{-1}_{1}\sigma^{-2}_{2}.\,
  2. V ( t ) = t - 3 2 ( - 1 + t - 2 t 2 + t 3 - 2 t 4 + t 5 ) . V(t)=t^{-{3\over 2}}(-1+t-2t^{2}+t^{3}-2t^{4}+t^{5}).
  3. V ( 1 / t ) V(1/t)
  4. V ( 1 / t ) V(1/t)
  5. V ( t ) V(t)

Whitney_umbrella.html

  1. x ( u , v ) = u v x(u,v)=uv
  2. y ( u , v ) = u y(u,v)=u
  3. z ( u , v ) = v 2 z(u,v)=v^{2}
  4. x 2 = y 2 z x^{2}=y^{2}z

Wiedemann–Franz_law.html

  1. κ σ = L T \frac{\kappa}{\sigma}=LT
  2. L = κ σ T = π 2 3 ( k B e ) 2 = 2.44 × 10 - 8 W Ω K - 2 . L=\frac{\kappa}{\sigma T}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=2.% 44\times 10^{-8}\,\mathrm{W\,\Omega\,K^{-2}}.
  3. F ¯ = - e E ¯ = m d v ¯ d t \bar{F}=-e\cdot\bar{E}=m\cdot\frac{\;d\bar{v}}{\;dt}
  4. d v ¯ = - e E ¯ m d t \;d\bar{v}=-\frac{e\cdot\bar{E}}{m}\;dt
  5. d v ¯ d t = - e E ¯ m - 1 τ v \frac{\;d\bar{v}}{\;dt}=-\frac{e\cdot\bar{E}}{m}-\frac{1}{\tau}\cdot v
  6. T 0 T\rightarrow 0
  7. L = κ σ T L=\frac{\kappa}{\sigma T}
  8. T 0 T\rightarrow 0
  9. κ 0 ; L 0 \kappa\rightarrow 0;L\rightarrow 0

Wien_bridge_oscillator.html

  1. f = 1 2 π R C f=\frac{1}{2\pi RC}
  2. ω 2 = 1 R 1 R 2 C 1 C 2 \omega^{2}={1\over R_{1}R_{2}C_{1}C_{2}}
  3. C 1 C 2 = R 4 R 3 - R 2 R 1 {C_{1}\over C_{2}}={R_{4}\over R_{3}}-{R_{2}\over R_{1}}
  4. T = ( R 1 / ( 1 + s C 1 R 1 ) R 1 / ( 1 + s C 1 R 1 ) + R 2 + 1 / ( s C 2 ) - R b R b + R f ) A 0 T=\left(\frac{R_{1}/(1+sC_{1}R_{1})}{R_{1}/(1+sC_{1}R_{1})+R_{2}+1/(sC_{2})}-% \frac{R_{b}}{R_{b}+R_{f}}\right)A_{0}\,
  5. A 0 A_{0}\,
  6. ω = 1 R C f = 1 2 π R C \omega=\frac{1}{RC}\rightarrow f=\frac{1}{2\pi RC}\,
  7. R f R b = 2 A 0 + 3 A 0 - 3 \frac{R_{f}}{R_{b}}=\frac{2A_{0}+3}{A_{0}-3}\,
  8. lim A 0 R f R b = 2 \lim_{A_{0}\rightarrow\infty}\frac{R_{f}}{R_{b}}=2\,
  9. H ( s ) = R 1 / ( 1 + s C 1 R 1 ) R 1 / ( 1 + s C 1 R 1 ) + R 2 + 1 / ( s C 2 ) H(s)=\frac{R_{1}/(1+sC_{1}R_{1})}{R_{1}/(1+sC_{1}R_{1})+R_{2}+1/(sC_{2})}
  10. H ( s ) = s C 2 R 1 ( 1 + s C 1 R 1 ) ( s C 2 R 1 / ( 1 + s C 1 R 1 ) + s C 2 R 2 + 1 ) H(s)=\frac{sC_{2}R_{1}}{(1+sC_{1}R_{1})(sC_{2}R_{1}/(1+sC_{1}R_{1})+sC_{2}R_{2% }+1)}
  11. H ( s ) = s C 2 R 1 s C 2 R 1 + ( 1 + s C 1 R 1 ) ( s C 2 R 2 + 1 ) H(s)=\frac{sC_{2}R_{1}}{sC_{2}R_{1}+(1+sC_{1}R_{1})(sC_{2}R_{2}+1)}
  12. H ( s ) = s C 2 R 1 C 1 C 2 R 1 R 2 s 2 + ( C 2 R 1 + C 2 R 2 + C 1 R 1 ) s + 1 H(s)=\frac{sC_{2}R_{1}}{C_{1}C_{2}R_{1}R_{2}s^{2}+(C_{2}R_{1}+C_{2}R_{2}+C_{1}% R_{1})s+1}
  13. H ( s ) = s C R C 2 R 2 s 2 + 3 C R s + 1 H(s)=\frac{sCR}{C^{2}R^{2}s^{2}+3CRs+1}
  14. H ( s ) = s s 2 + 3 s + 1 H(s)=\frac{s}{s^{2}+3s+1}

Wiener_filter.html

  1. G ( s ) = S x , s ( s ) S x ( s ) e α s . G(s)=\frac{S_{x,s}(s)}{S_{x}(s)}e^{\alpha s}.
  2. S S
  3. g ( t ) g(t)
  4. E ( e 2 ) = R s ( 0 ) - - g ( τ ) R x , s ( τ + α ) d τ , E(e^{2})=R_{s}(0)-\int_{-\infty}^{\infty}{g(\tau)R_{x,s}(\tau+\alpha)\,d\tau},
  5. g ( t ) g(t)
  6. G ( s ) G(s)
  7. G ( s ) = H ( s ) S x + ( s ) , G(s)=\frac{H(s)}{S_{x}^{+}(s)},
  8. H ( s ) H(s)
  9. S x , s ( s ) S x - ( s ) e α s \frac{S_{x,s}(s)}{S_{x}^{-}(s)}e^{\alpha s}
  10. S x + ( s ) S_{x}^{+}(s)
  11. S x ( s ) S_{x}(s)
  12. S x + ( s ) S_{x}^{+}(s)
  13. t 0 t\,\geq\,0
  14. S x - ( s ) S_{x}^{-}(s)
  15. S x ( s ) S_{x}(s)
  16. S x - ( s ) S_{x}^{-}(s)
  17. t < 0 t<0
  18. G ( s ) G(s)
  19. S x ( s ) S_{x}(s)
  20. S x ( s ) = S x + ( s ) S x - ( s ) S_{x}(s)=S_{x}^{+}(s)S_{x}^{-}(s)
  21. S + S^{+}
  22. S - S^{-}
  23. S x , s ( s ) e α s S_{x,s}(s)e^{\alpha s}
  24. S x - ( s ) S_{x}^{-}(s)
  25. H ( s ) H(s)
  26. H ( s ) H(s)
  27. S x + ( s ) S_{x}^{+}(s)
  28. G ( s ) G(s)
  29. { a i } \{a_{i}\}
  30. i = 0 , , N i\,=\,0,\,\ldots,\,N
  31. x [ n ] = i = 0 N a i w [ n - i ] . x[n]=\sum_{i=0}^{N}a_{i}w[n-i].
  32. a i = arg min E { e 2 [ n ] } , a_{i}=\arg\min~{}E\{e^{2}[n]\},
  33. E { } E\{\cdot\}
  34. a i a_{i}
  35. E { e 2 [ n ] } = E { ( x [ n ] - s [ n ] ) 2 } = E { x 2 [ n ] } + E { s 2 [ n ] } - 2 E { x [ n ] s [ n ] } = E { ( i = 0 N a i w [ n - i ] ) 2 } + E { s 2 [ n ] } - 2 E { i = 0 N a i w [ n - i ] s [ n ] } . \begin{array}[]{rcl}E\{e^{2}[n]\}&=&E\{(x[n]-s[n])^{2}\}\\ &=&E\{x^{2}[n]\}+E\{s^{2}[n]\}-2E\{x[n]s[n]\}\\ &=&E\{\big(\sum_{i=0}^{N}a_{i}w[n-i]\big)^{2}\}+E\{s^{2}[n]\}-2E\{\sum_{i=0}^{% N}a_{i}w[n-i]s[n]\}.\end{array}
  36. [ a 0 , , a N ] [a_{0},\,\ldots,\,a_{N}]
  37. a i a_{i}
  38. a i E { e 2 [ n ] } = 2 E { ( j = 0 N a j w [ n - j ] ) w [ n - i ] } - 2 E { s [ n ] w [ n - i ] } i = 0 , , N = 2 j = 0 N E { w [ n - j ] w [ n - i ] } a j - 2 E { w [ n - i ] s [ n ] } . \begin{array}[]{rcl}\frac{\partial}{\partial a_{i}}E\{e^{2}[n]\}&=&2E\{\big(% \sum_{j=0}^{N}a_{j}w[n-j]\big)w[n-i]\}-2E\{s[n]w[n-i]\}\quad i=0,\,\ldots,\,N% \\ &=&2\sum_{j=0}^{N}E\{w[n-j]w[n-i]\}a_{j}-2E\{w[n-i]s[n]\}.\end{array}
  39. R w [ m ] R_{w}[m]
  40. R w s [ m ] R_{ws}[m]
  41. R w [ m ] = \displaystyle R_{w}[m]=
  42. R w s [ - i ] = R s w [ i ] R_{ws}[-i]\,=\,R_{sw}[i]
  43. a i E { e 2 [ n ] } = 2 j = 0 N R w [ j - i ] a j - 2 R s w [ i ] i = 0 , , N . \frac{\partial}{\partial a_{i}}E\{e^{2}[n]\}=2\sum_{j=0}^{N}R_{w}[j-i]a_{j}-2R% _{sw}[i]\quad i=0,\,\ldots,\,N.
  44. j = 0 N R w [ j - i ] a j = R s w [ i ] i = 0 , , N , \sum_{j=0}^{N}R_{w}[j-i]a_{j}=R_{sw}[i]\quad i=0,\,\ldots,\,N,
  45. 𝐓𝐚 = 𝐯 [ R w [ 0 ] R w [ 1 ] R w [ N ] R w [ 1 ] R w [ 0 ] R w [ N - 1 ] R w [ N ] R w [ N - 1 ] R w [ 0 ] ] [ a 0 a 1 a N ] = [ R s w [ 0 ] R s w [ 1 ] R s w [ N ] ] \begin{aligned}&\displaystyle\mathbf{T}\mathbf{a}=\mathbf{v}\\ \displaystyle\Rightarrow&\displaystyle\begin{bmatrix}R_{w}[0]&R_{w}[1]&\cdots&% R_{w}[N]\\ R_{w}[1]&R_{w}[0]&\cdots&R_{w}[N-1]\\ \vdots&\vdots&\ddots&\vdots\\ R_{w}[N]&R_{w}[N-1]&\cdots&R_{w}[0]\end{bmatrix}\begin{bmatrix}a_{0}\\ a_{1}\\ \vdots\\ a_{N}\end{bmatrix}=\begin{bmatrix}R_{sw}[0]\\ R_{sw}[1]\\ \vdots\\ R_{sw}[N]\end{bmatrix}\end{aligned}
  46. R R
  47. 𝐚 = 𝐓 - 1 𝐯 \mathbf{a}\,=\,\mathbf{T}^{-1}\mathbf{v}
  48. 𝐓 \mathbf{T}
  49. 𝐗 \mathbf{X}
  50. 𝐲 \mathbf{y}
  51. s y m b o l β ^ = ( 𝐗 𝐓 𝐗 ) - 1 𝐗 𝐓 s y m b o l y . symbol{\hat{\beta}}=(\mathbf{X}^{\mathbf{T}}\mathbf{X})^{-1}\mathbf{X}^{% \mathbf{T}}symboly.

Wigner_quasiprobability_distribution.html

  1. ψ ( x ) ψ(x)
  2. ħ ħ
  3. ħ ħ
  4. ħ ħ
  5. P ( x , p ) P(x,p)
  6. ψ ψ
  7. x x
  8. p p
  9. x x
  10. ψ ψ
  11. x x
  12. x x
  13. p p
  14. P ( x , p ) = 1 π - φ * ( p + q ) φ ( p - q ) e - 2 i x q / d q P(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}\varphi^{*}(p+q)\varphi(p-q)e^% {-2ixq/\hbar}\,dq
  15. φ φ
  16. ψ ψ
  17. P ( r , p ) = 1 ( 2 π ) 3 ψ * ( r + s / 2 ) ψ ( r - s / 2 ) e i p s d 3 s . P(\vec{r},\vec{p})=\frac{1}{(2\pi)^{3}}\int\psi^{*}(\vec{r}+\hbar\vec{s}/2)% \psi(\vec{r}-\hbar\vec{s}/2)e^{i\vec{p}\cdot\vec{s}}\,d^{3}s.
  18. P ( x , p ) = 1 π - x + y | ρ ^ | x - y e - 2 i p y / d y , P(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}\langle x+y|\hat{\rho}|x-y% \rangle e^{-2ipy/\hbar}\,dy,
  19. ψ ( x ) ψ(x)
  20. g ( x , p ) g(x,p)
  21. G ^ = d x d p P ( x , p ) g ( x , p ) . \langle\hat{G}\rangle=\int\!dx\,dp~{}P(x,p)~{}g(x,p)~{}.
  22. - d p P ( x , p ) = x | ρ ^ | x . \int_{-\infty}^{\infty}dp\,P(x,p)=\langle x|\hat{\rho}|x\rangle.
  23. - d p P ( x , p ) = | ψ ( x ) | 2 \int_{-\infty}^{\infty}dp\,P(x,p)=|\psi(x)|^{2}
  24. - d x P ( x , p ) = p | ρ ^ | p \int_{-\infty}^{\infty}dx\,P(x,p)=\langle p|\hat{\rho}|p\rangle
  25. - d x P ( x , p ) = | φ ( p ) | 2 \int_{-\infty}^{\infty}dx\,P(x,p)=|\varphi(p)|^{2}
  26. - d x - d p P ( x , p ) = T r ( ρ ^ ) \int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dp\,P(x,p)=Tr(\hat{\rho})
  27. ψ ( x ) ψ ( x ) * P ( x , p ) P ( x , - p ) \psi(x)\rightarrow\psi(x)^{*}\Rightarrow P(x,p)\rightarrow P(x,-p)
  28. ψ ( x ) ψ ( - x ) P ( x , p ) P ( - x , - p ) \psi(x)\rightarrow\psi(-x)\Rightarrow P(x,p)\rightarrow P(-x,-p)
  29. ψ ( x ) ψ ( x + y ) P ( x , p ) P ( x + y , p ) \psi(x)\rightarrow\psi(x+y)\Rightarrow P(x,p)\rightarrow P(x+y,p)
  30. P ( x , p ) t = - p m P ( x , p ) x \frac{\partial P(x,p)}{\partial t}=\frac{-p}{m}\frac{\partial P(x,p)}{\partial x}
  31. | ψ | θ | 2 = 2 π - d x - d p P ψ ( x , p ) P θ ( x , p ) |\langle\psi|\theta\rangle|^{2}=2\pi\hbar\int_{-\infty}^{\infty}dx\,\int_{-% \infty}^{\infty}dp\,P_{\psi}(x,p)P_{\theta}(x,p)
  32. g ( x , p ) - d y x - y / 2 | G ^ | x + y / 2 e i p y / , g(x,p)\equiv\int_{-\infty}^{\infty}dy\,\langle x-y/2|\hat{G}|x+y/2\rangle e^{% ipy/\hbar},
  33. ψ | G ^ | ψ = T r ( ρ ^ G ^ ) = - d x - d p P ( x , p ) g ( x , p ) . \langle\psi|\hat{G}|\psi\rangle=Tr(\hat{\rho}\hat{G})=\int_{-\infty}^{\infty}% dx\,\int_{-\infty}^{\infty}dpP(x,p)g(x,p).
  34. - d x - d p P ( x , p ) P θ ( x , p ) 0 , \int_{-\infty}^{\infty}dx\,\int_{-\infty}^{\infty}dp\,P(x,p)P_{\theta}(x,p)% \geq 0~{},
  35. - 2 h P ( x , p ) 2 h . -\frac{2}{h}\leq P(x,p)\leq\frac{2}{h}.
  36. g ( x , p ) = - d s e i p s / x - s 2 | G ^ | x + s 2 . g(x,p)=\int_{-\infty}^{\infty}ds~{}e^{ips/\hbar}\langle x-\frac{s}{2}|\ \hat{G% }\ |x+\frac{s}{2}\rangle.
  37. x | G ^ | y = - d p h e i p ( x - y ) / g ( x + y 2 , p ) , \langle x|\ \hat{G}\ |y\rangle=\int_{-\infty}^{\infty}{dp\over h}~{}e^{ip(x-y)% /\hbar}g\left({x+y\over 2},p\right),
  38. P ( x , p , t ) t = - { { P ( x , p , t ) , H ( x , p ) } } , {\partial P(x,p,t)\over\partial t}=-\{\{P(x,p,t)~{},~{}H(x,p)\}\}~{},
  39. P ( x , p , t ) = P ( ( x - t ( x , p ) , p - t ( x , p ) ) , 0 ) P(x,p,t)=P(\star(x_{-t}(x,p),p_{-t}(x,p)),0)
  40. x t ( x , p ) x_{t}(x,p)
  41. p t ( x , p ) p_{t}(x,p)
  42. x t = 0 ( x , p ) = x x_{t=0}(x,p)=x
  43. p t = 0 ( x , p ) = p p_{t=0}(x,p)=p
  44. \star
  45. \star
  46. ħ ħ
  47. p / ħ p/ħ
  48. k = | k | s i n θ | k | θ k=|k|sinθ≈|k|θ
  49. x x
  50. θ θ
  51. x x
  52. p / ħ p/ħ
  53. ω = 2 π f ω=2πf
  54. f f
  55. f f
  56. t t
  57. x x
  58. p / ħ p/ħ
  59. X X
  60. P P

Wilcoxon_signed-rank_test.html

  1. W W
  2. N N
  3. i = 1 , , N i=1,...,N
  4. x 1 , i x_{1,i}
  5. x 2 , i x_{2,i}
  6. i = 1 , , N i=1,...,N
  7. | x 2 , i - x 1 , i | |x_{2,i}-x_{1,i}|
  8. sgn ( x 2 , i - x 1 , i ) \operatorname{sgn}(x_{2,i}-x_{1,i})
  9. sgn \operatorname{sgn}
  10. | x 2 , i - x 1 , i | = 0 |x_{2,i}-x_{1,i}|=0
  11. N r N_{r}
  12. N r N_{r}
  13. | x 2 , i - x 1 , i | |x_{2,i}-x_{1,i}|
  14. R i R_{i}
  15. W W
  16. W = i = 1 N r [ sgn ( x 2 , i - x 1 , i ) R i ] W=\sum_{i=1}^{N_{r}}[\operatorname{sgn}(x_{2,i}-x_{1,i})\cdot R_{i}]
  17. W W
  18. N r ( N r + 1 ) ( 2 N r + 1 ) 6 \frac{N_{r}(N_{r}+1)(2N_{r}+1)}{6}
  19. W W
  20. H 0 H_{0}
  21. | W | W c r i t i c a l , N r |W|\geq W_{critical,N_{r}}
  22. N r N_{r}
  23. W W
  24. N r 10 N_{r}\geq 10
  25. z = W σ W , σ W = N r ( N r + 1 ) ( 2 N r + 1 ) 6 z=\frac{W}{\sigma_{W}},\sigma_{W}=\sqrt{\frac{N_{r}(N_{r}+1)(2N_{r}+1)}{6}}
  26. | z | > z c r i t i c a l |z|>z_{critical}
  27. H 0 H_{0}
  28. W = i = 1 N [ sgn ( x 2 , i - x 1 , i ) R i ] W=\sum_{i=1}^{N}[\operatorname{sgn}(x_{2,i}-x_{1,i})\cdot R_{i}]
  29. π + = P ( x 2 , i > x 1 , i ) , π - = P ( x 2 , i < x 1 , i ) , π 0 = P ( x 2 , i = x 1 , i ) \pi^{+}=P(x_{2,i}>x_{1,i}),\pi^{-}=P(x_{2,i}<x_{1,i}),\pi^{0}=P(x_{2,i}=x_{1,i})
  30. N 10 {N\geq 10}
  31. Z = 4 W - N ( N + 1 ) 2 N ( N + 1 ) ( 2 N + 1 ) 3 ( π + + π - - ( π + - π - ) 2 ) {Z=\frac{4W-N(N+1)}{\sqrt{\frac{2N(N+1)(2N+1)}{3}(\pi^{+}+\pi^{-}-(\pi^{+}-\pi% ^{-})^{2})}}}
  32. π + = 1 \pi^{+}=1
  33. π - = 1 \pi^{-}=1
  34. x 2 , i - x 1 , i x_{2,i}-x_{1,i}
  35. i i
  36. x 2 , i x_{2,i}
  37. x 1 , i x_{1,i}
  38. sgn \operatorname{sgn}
  39. abs \,\text{abs}
  40. x 2 , i - x 1 , i x_{2,i}-x_{1,i}
  41. i i
  42. x 2 , i x_{2,i}
  43. x 1 , i x_{1,i}
  44. sgn \operatorname{sgn}
  45. abs \,\text{abs}
  46. R i R_{i}
  47. sgn R i \operatorname{sgn}\cdot R_{i}
  48. s g n sgn
  49. abs \,\text{abs}
  50. R i R_{i}
  51. N r = 10 - 1 = 9 , | W | = | 1.5 + 1.5 - 3 - 4 - 5 - 6 + 7 + 8 + 9 | = 9. N_{r}=10-1=9,|W|=|1.5+1.5-3-4-5-6+7+8+9|=9.
  52. | W | < W α = 0.05 , 9 , t w o - s i d e d = 35 fail to reject H 0 . |W|<W_{\alpha=0.05,9,two-sided}=35\therefore\,\text{fail to reject }H_{0}.

Wilkinson_power_divider.html

  1. S 21 , S 31 S_{21},S_{31}
  2. S 11 S_{11}
  3. [ S ] = - j 2 [ 0 1 1 1 0 0 1 0 0 ] [S]=\frac{-j}{\sqrt{2}}\begin{bmatrix}0&1&1\\ 1&0&0\\ 1&0&0\\ \end{bmatrix}
  4. S i j = S j i S_{ij}=S_{ji}
  5. S 11 , S 22 , S 33 = 0 S_{11},S_{22},S_{33}=0
  6. S 23 , S 32 S_{23},S_{32}
  7. S 21 = S 31 S_{21}=S_{31}
  8. S 21 = S 31 = - 3 dB = 10 log 10 ( 1 2 ) S_{21}=S_{31}=-3\,\,\text{dB}=10\log_{10}(\frac{1}{2})
  9. n n

William_Brouncker,_2nd_Viscount_Brouncker.html

  1. π 4 = 1 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + \frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+% \cfrac{7^{2}}{2+\cfrac{9^{2}}{2+\ddots}}}}}}
  2. 1 1 + 1 2 2 = 2 3 = 1 - 1 3 \frac{1}{1+\frac{1^{2}}{2}}=\frac{2}{3}=1-\frac{1}{3}
  3. 1 1 + 1 2 2 + 3 2 2 = 13 15 = 1 - 1 3 + 1 5 . \frac{1}{1+\frac{1^{2}}{2+\frac{3^{2}}{2}}}=\frac{13}{15}=1-\frac{1}{3}+\frac{% 1}{5}.
  4. 4 π = 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + \frac{4}{\pi}=1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\cfrac{9^{2}}{2+\ddots}}}}}

Williams'_p_+_1_algorithm.html

  1. V 0 = 2 , V 1 = A , V j = A V j - 1 - V j - 2 V_{0}=2,V_{1}=A,V_{j}=AV_{j-1}-V_{j-2}
  2. gcd ( N , V M - 2 ) \gcd(N,V_{M}-2)
  3. p - ( D / p ) p-(D/p)
  4. D = A 2 - 4 D=A^{2}-4
  5. ( D / p ) (D/p)
  6. ( D / p ) = - 1 (D/p)=-1
  7. ( D / p ) = + 1 (D/p)=+1
  8. gcd ( N , V M - 2 ) \gcd(N,V_{M}-2)
  9. V M V_{M}
  10. V M - 1 V_{M-1}
  11. V M V_{M}

Wilson_quotient.html

  1. W ( p ) = ( p - 1 ) ! + 1 p W(p)=\frac{(p-1)!+1}{p}

Winsorising.html

  1. { 92 , 19 , 𝟏𝟎𝟏 , 58 , 𝟏𝟎𝟓𝟑 , 91 , 26 , 78 , 10 , 13 , - 𝟒𝟎 , 𝟏𝟎𝟏 , 86 , 85 , 15 , 89 , 89 , 28 , - 𝟓 , 41 } ( N = 20 , m e a n = 101.5 ) \{92,19,\mathbf{101},58,\mathbf{1053},91,26,78,10,13,\mathbf{-40},\mathbf{101}% ,86,85,15,89,89,28,\mathbf{-5},41\}\qquad(N=20,mean=101.5)
  2. { 92 , 19 , 𝟏𝟎𝟏 , 58 , 𝟏𝟎𝟏 , 91 , 26 , 78 , 10 , 13 , - 𝟓 , 𝟏𝟎𝟏 , 86 , 85 , 15 , 89 , 89 , 28 , - 𝟓 , 41 } ( N = 20 , m e a n = 55.65 ) \{92,19,\mathbf{101},58,\mathbf{101},91,26,78,10,13,\mathbf{-5},\mathbf{101},8% 6,85,15,89,89,28,\mathbf{-5},41\}\qquad(N=20,mean=55.65)
  3. { 92 , 19 , 𝟏𝟎𝟏 , 58 , 91 , 26 , 78 , 10 , 13 , 𝟏𝟎𝟏 , 86 , 85 , 15 , 89 , 89 , 28 , - 𝟓 , 41 } ( N = 18 , m e a n = 56.5 ) \{92,19,\mathbf{101},58,\quad 91,26,78,10,13,\quad\mathbf{101},86,85,15,89,89,% 28,\mathbf{-5},41\}\qquad(N=18,mean=56.5)

Wirth–Weber_precedence_relationship.html

  1. ( V t V n ) (V_{t}\cup V_{n})
  2. \gtrdot
  3. \lessdot
  4. = ˙ \dot{=}
  5. G = < V n , V t , S , P Align g t ; G=<V_{n},V_{t},S,P&gt;
  6. X = ˙ Y { A α X Y β P A V n α , β ( V n V t ) * X , Y ( V n V t ) X\dot{=}Y\iff\begin{cases}A\to\alpha XY\beta\in P\\ A\in V_{n}\\ \alpha,\beta\in(V_{n}\cup V_{t})^{*}\\ X,Y\in(V_{n}\cup V_{t})\end{cases}
  7. X Y { A α X B β P B + Y γ A , B V n α , β , γ ( V n V t ) * X , Y ( V n V t ) X\lessdot Y\iff\begin{cases}A\to\alpha XB\beta\in P\\ B\Rightarrow^{+}Y\gamma\\ A,B\in V_{n}\\ \alpha,\beta,\gamma\in(V_{n}\cup V_{t})^{*}\\ X,Y\in(V_{n}\cup V_{t})\end{cases}
  8. X a { A α B Y β P B + γ X Y * a δ A , B V n α , β , γ , δ ( V n V t ) * X , Y ( V n V t ) a V t X\gtrdot a\iff\begin{cases}A\to\alpha BY\beta\in P\\ B\Rightarrow^{+}\gamma X\\ Y\Rightarrow^{*}a\delta\\ A,B\in V_{n}\\ \alpha,\beta,\gamma,\delta\in(V_{n}\cup V_{t})^{*}\\ X,Y\in(V_{n}\cup V_{t})\\ a\in V_{t}\end{cases}
  9. Head + ( X ) = { Y X + Y α } \mathrm{Head}^{+}(X)=\{Y\mid X\Rightarrow^{+}Y\alpha\}
  10. Tail + ( X ) = { Y X + α Y } \mathrm{Tail}^{+}(X)=\{Y\mid X\Rightarrow^{+}\alpha Y\}
  11. Head * ( X ) = ( Head + ( X ) { X } ) V t \mathrm{Head}^{*}(X)=(\mathrm{Head}^{+}(X)\cup\{X\})\cap V_{t}
  12. $\empty$
  13. $\empty$
  14. A α P A\to\alpha\in P
  15. X = ˙ Y X\dot{=}Y
  16. X Head + ( Y ) X\lessdot\mathrm{Head}^{+}(Y)
  17. Tail + ( X ) Head * ( Y ) \mathrm{Tail}^{+}(X)\gtrdot\mathrm{Head}^{*}(Y)
  18. $ H e a d + ( S ) \$\lessdot Head^{+}(S)
  19. Tai l + ( S ) $ \mathrm{Tai}l^{+}(S)\gtrdot\$
  20. \lessdot
  21. \gtrdot
  22. S a S S b | c S\to aSSb|c
  23. $\empty$
  24. $\empty$
  25. $\empty$
  26. $\empty$
  27. $\empty$
  28. $\empty$
  29. S a S S b S\to aSSb
  30. = ˙ \dot{=}
  31. \lessdot
  32. \lessdot
  33. \lessdot
  34. = ˙ \dot{=}
  35. \lessdot
  36. \lessdot
  37. \lessdot
  38. \gtrdot
  39. \gtrdot
  40. \gtrdot
  41. \gtrdot
  42. \gtrdot
  43. = ˙ \dot{=}
  44. \gtrdot
  45. \gtrdot
  46. \gtrdot
  47. S c S\to c
  48. = ˙ \dot{=}
  49. \lessdot
  50. = ˙ \dot{=}
  51. \lessdot
  52. = ˙ \dot{=}
  53. \lessdot
  54. \lessdot
  55. \gtrdot
  56. \gtrdot
  57. \gtrdot
  58. \gtrdot
  59. \gtrdot
  60. \gtrdot
  61. \gtrdot
  62. \gtrdot
  63. \lessdot
  64. \lessdot

Wirtinger's_inequality_for_functions.html

  1. f : f:\mathbb{R}\to\mathbb{R}
  2. 0 2 π f ( x ) d x = 0. \int_{0}^{2\pi}f(x)\,dx=0.
  3. 0 2 π f 2 ( x ) d x 0 2 π f 2 ( x ) d x \int_{0}^{2\pi}f^{\prime 2}(x)\,dx\geq\int_{0}^{2\pi}f^{2}(x)\,dx
  4. π 2 0 a | f | 2 a 2 0 a | f | 2 \pi^{2}\int_{0}^{a}|f|^{2}\leq a^{2}\int_{0}^{a}|f^{\prime}|^{2}
  5. f ( x ) = 1 2 a 0 + n 1 ( a n sin n x π + b n cos n x π ) , f(x)=\frac{1}{2}a_{0}+\sum_{n\geq 1}\left(a_{n}\frac{\sin nx}{\sqrt{\pi}}+b_{n% }\frac{\cos nx}{\sqrt{\pi}}\right),
  6. 0 2 π f 2 ( x ) d x = n = 1 ( a n 2 + b n 2 ) \int_{0}^{2\pi}f^{2}(x)dx=\sum_{n=1}^{\infty}(a_{n}^{2}+b_{n}^{2})
  7. 0 2 π f 2 ( x ) d x = n = 1 n 2 ( a n 2 + b n 2 ) \int_{0}^{2\pi}f^{\prime 2}(x)\,dx=\sum_{n=1}^{\infty}n^{2}(a_{n}^{2}+b_{n}^{2})

Witt_algebra.html

  1. L n = - z n + 1 z L_{n}=-z^{n+1}\frac{\partial}{\partial z}
  2. \mathbb{Z}
  3. [ L m , L n ] = ( m - n ) L m + n . [L_{m},L_{n}]=(m-n)L_{m+n}.

Wobbe_index.html

  1. V C V_{C}
  2. G S G_{S}
  3. I W I_{W}
  4. I W = V C G S . I_{W}=\frac{V_{C}}{\sqrt{G_{S}}}.

Wolfe_conditions.html

  1. min x f ( 𝐱 ) \min_{x}f(\mathbf{x})
  2. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  3. min α f ( 𝐱 k + α 𝐩 k ) \min_{\alpha}f(\mathbf{x}_{k}+\alpha\mathbf{p}_{k})
  4. 𝐱 k \mathbf{x}_{k}
  5. 𝐩 k n \mathbf{p}_{k}\in\mathbb{R}^{n}
  6. α \alpha\in\mathbb{R}
  7. α \alpha
  8. α + \alpha\in\mathbb{R}^{+}
  9. α \alpha
  10. 𝐩 k \mathbf{p}_{k}
  11. ϕ \phi
  12. 𝐩 k \mathbf{p}_{k}
  13. ϕ ( α ) = f ( 𝐱 k + α 𝐩 k ) \phi(\alpha)=f(\mathbf{x}_{k}+\alpha\mathbf{p}_{k})
  14. α k \alpha_{k}
  15. f ( 𝐱 k + α k 𝐩 k ) f ( 𝐱 k ) + c 1 α k 𝐩 k T f ( 𝐱 k ) f(\mathbf{x}_{k}+\alpha_{k}\mathbf{p}_{k})\leq f(\mathbf{x}_{k})+c_{1}\alpha_{% k}\mathbf{p}_{k}^{\mathrm{T}}\nabla f(\mathbf{x}_{k})
  16. 𝐩 k T f ( 𝐱 k + α k 𝐩 k ) c 2 𝐩 k T f ( 𝐱 k ) \mathbf{p}_{k}^{\mathrm{T}}\nabla f(\mathbf{x}_{k}+\alpha_{k}\mathbf{p}_{k})% \geq c_{2}\mathbf{p}_{k}^{\mathrm{T}}\nabla f(\mathbf{x}_{k})
  17. 0 < c 1 < c 2 < 1 0<c_{1}<c_{2}<1
  18. 𝐩 k \mathbf{p}_{k}
  19. 𝐩 k T f ( 𝐱 k ) < 0 \mathbf{p}_{k}^{\mathrm{T}}\nabla f(\mathbf{x}_{k})<0
  20. c 1 c_{1}
  21. c 2 c_{2}
  22. c 1 = 10 - 4 c_{1}=10^{-4}
  23. c 2 = 0.9 c_{2}=0.9
  24. c 2 = 0.1 c_{2}=0.1
  25. α k \alpha_{k}
  26. f f
  27. ϕ \phi
  28. | 𝐩 k T f ( 𝐱 k + α k 𝐩 k ) | c 2 | 𝐩 k T f ( 𝐱 k ) | \big|\mathbf{p}_{k}^{\mathrm{T}}\nabla f(\mathbf{x}_{k}+\alpha_{k}\mathbf{p}_{% k})\big|\leq c_{2}\big|\mathbf{p}_{k}^{\mathrm{T}}\nabla f(\mathbf{x}_{k})\big|
  29. α k \alpha_{k}
  30. ϕ \phi
  31. 𝐱 k + 1 = 𝐱 k + α 𝐩 k \mathbf{x}_{k+1}=\mathbf{x}_{k}+\alpha\mathbf{p}_{k}
  32. 𝐩 k \mathbf{p}_{k}
  33. cos θ k = f ( 𝐱 k ) T 𝐩 k f ( 𝐱 k ) 𝐩 k \cos\theta_{k}=\frac{\nabla f(\mathbf{x}_{k})^{\mathrm{T}}\mathbf{p}_{k}}{\|% \nabla f(\mathbf{x}_{k})\|\|\mathbf{p}_{k}\|}
  34. f ( 𝐱 k ) 0 \nabla f(\mathbf{x}_{k})\rightarrow 0
  35. 𝐩 k = - B k - 1 f ( 𝐱 k ) \mathbf{p}_{k}=-B_{k}^{-1}\nabla f(\mathbf{x}_{k})
  36. B k B_{k}
  37. B k B_{k}
  38. B k + 1 B_{k+1}

Word_metric.html

  1. G G
  2. G G
  3. G G
  4. g g
  5. h h
  6. G G
  7. d ( g , h ) d(g,h)
  8. g - 1 h g^{-1}h
  9. G G
  10. G G
  11. \mathbb{Z}\oplus\mathbb{Z}
  12. \mathbb{Z}\oplus\mathbb{Z}
  13. \mathbb{Z}\oplus\mathbb{Z}
  14. e 1 = < 1 , 0 > e_{1}=<1,0>
  15. e 2 = < 0 , 1 > e_{2}=<0,1>
  16. - e 1 = < - 1 , 0 > -e_{1}=<-1,0>
  17. - e 2 = < 0 , - 1 > -e_{2}=<0,-1>
  18. \mathbb{Z}\oplus\mathbb{Z}
  19. \mathbb{Z}\oplus\mathbb{Z}
  20. e 1 e_{1}
  21. - e 1 -e_{1}
  22. e 2 e_{2}
  23. - e 2 -e_{2}
  24. < 1 , 2 > <1,2>
  25. < - 2 , 4 Align g t ; <-2,4&gt;
  26. v = < i , j > v=<i,j>
  27. w = < k , l > w=<k,l>
  28. \mathbb{Z}\oplus\mathbb{Z}
  29. v v
  30. w w
  31. | i - k | + | j - l | |i-k|+|j-l|
  32. w = s 1 s L w=s_{1}\ldots s_{L}
  33. s 1 , , s L s_{1},\ldots,s_{L}
  34. w w
  35. w = s 1 s L w=s_{1}\ldots s_{L}
  36. w ¯ \bar{w}
  37. w = w=\emptyset
  38. w w
  39. w ¯ \bar{w}
  40. | g - 1 h | |g^{-1}h|
  41. g w ¯ = h g\bar{w}=h
  42. s s
  43. s - 1 s^{-1}
  44. { a , b } \{a,b\}
  45. { a , b , a - 1 , b - 1 } \{a,b,a^{-1},b^{-1}\}
  46. a , a - 1 a,a^{-1}
  47. b , b - 1 b,b^{-1}
  48. g F g\in F
  49. w = b - 1 a w=b^{-1}a
  50. w w
  51. b b
  52. a a
  53. k G k\in G
  54. g G g\in G
  55. k g kg
  56. k g kg
  57. k h kh
  58. | ( k g ) - 1 ( k h ) | = | g - 1 h | |(kg)^{-1}(kh)|=|g^{-1}h|
  59. g g
  60. h h
  61. S S
  62. T T
  63. d S d_{S}
  64. d T d_{T}
  65. K 1 K\geq 1
  66. g , h G g,h\in G
  67. 1 K d T ( g , h ) d S ( g , h ) K d T ( g , h ) \frac{1}{K}\,d_{T}(g,h)\leq d_{S}(g,h)\leq K\,d_{T}(g,h)
  68. d S d_{S}
  69. T T
  70. d T d_{T}
  71. S S

Work_(electrical).html

  1. W = Q a b 𝐄 d 𝐫 = Q a b 𝐅 𝐄 q d 𝐫 = a b 𝐅 𝐄 d 𝐫 W=Q\int_{a}^{b}\mathbf{E}\cdot\,d\mathbf{r}=Q\int_{a}^{b}\frac{\mathbf{F_{E}}}% {q}\cdot\,d\mathbf{r}=\int_{a}^{b}\mathbf{F_{E}}\cdot\,d\mathbf{r}
  2. \cdot
  3. - U 𝐫 = 𝐅 -\frac{\partial U}{\partial\mathbf{r}}=\mathbf{F}
  4. U = - r 0 r 𝐅 d 𝐫 = - r 0 r 1 4 π ε 0 q 1 q 2 𝐫 𝟐 d 𝐫 = q 1 q 2 4 π ε 0 ( 1 r 0 - 1 r ) + c U=-\int_{r_{0}}^{r}\mathbf{F}\cdot\,d\mathbf{r}=-\int_{r_{0}}^{r}\frac{1}{4\pi% \varepsilon_{0}}\frac{q_{1}q_{2}}{\mathbf{r^{2}}}\cdot\,d\mathbf{r}=\frac{q_{1% }q_{2}}{4\pi\varepsilon_{0}}(\frac{1}{r_{0}}-\frac{1}{r})+c
  5. W = - Δ U W=-\Delta U\!
  6. W = q 1 q 2 4 π ε 0 1 r W=\frac{q_{1}q_{2}}{4\pi\varepsilon_{0}}\frac{1}{r}
  7. W = Q ( 𝐄 𝐫 ) = 𝐅 𝐄 𝐫 W=Q(\mathbf{E}\cdot\,\mathbf{r})=\mathbf{F_{E}}\cdot\,\mathbf{r}
  8. P = W t = Q V t P=\frac{\partial W}{\partial t}=\frac{\partial QV}{\partial t}
  9. δ W = 𝐅 𝐯 δ t , \delta W=\mathbf{F}\cdot\mathbf{v}\delta t,
  10. W t = 𝐅 𝐄 𝐯 \frac{\partial W}{\partial t}=\mathbf{F_{E}}\cdot\,\mathbf{v}

Work_hardening.html

  1. τ \tau
  2. ρ \rho_{\perp}
  3. τ = τ 0 + G α b ρ 1 / 2 \tau=\tau_{0}+G\alpha b\rho_{\perp}^{1/2}
  4. τ 0 \tau_{0}
  5. α \alpha
  6. σ = K ϵ p n \sigma=K\epsilon_{p}^{n}\,\!
  7. σ = σ y + K ϵ p n \sigma=\sigma_{y}+K\epsilon_{p}^{n}\,\!
  8. σ = σ y + K ( ϵ 0 + ϵ p ) n \sigma=\sigma_{y}+K(\epsilon_{0}+\epsilon_{p})^{n}\,\!
  9. n = d log ( σ ) d log ( ϵ ) = ϵ σ d σ d ϵ n=\frac{d\log(\sigma)}{d\log(\epsilon)}=\frac{\epsilon}{\sigma}\frac{d\sigma}{% d\epsilon}\,\!
  10. d σ d ϵ = n σ ϵ \frac{d\sigma}{d\epsilon}=n\frac{\sigma}{\epsilon}\,\!

Working_capital.html

  1. W o r k i n g C a p i t a l = C u r r e n t A s s e t s - C u r r e n t L i a b i l i t i e s WorkingCapital=CurrentAssets-CurrentLiabilities

XDH_assumption.html

  1. 𝔾 1 , 𝔾 2 \langle{\mathbb{G}}_{1},{\mathbb{G}}_{2}\rangle
  2. 𝔾 1 {\mathbb{G}}_{1}
  3. 𝔾 2 {\mathbb{G}}_{2}
  4. e ( , ) : 𝔾 1 × 𝔾 2 𝔾 T e(\cdot,\cdot):{\mathbb{G}}_{1}\times{\mathbb{G}}_{2}\rightarrow{\mathbb{G}}_{T}
  5. 𝔾 1 {\mathbb{G}}_{1}
  6. 𝔾 2 {\mathbb{G}}_{2}

Yield_(chemistry).html

  1. percent yield = actual yield theoretical yield × 100 % \mbox{percent yield}~{}=\frac{\mbox{actual yield}~{}}{\mbox{theoretical yield}% ~{}}\times 100\%

Young's_inequality.html

  1. a b a p p + b q q . ab\leq\frac{a^{p}}{p}+\frac{b^{q}}{q}.
  2. log ( t a p + ( 1 - t ) b q ) t log ( a p ) + ( 1 - t ) log ( b q ) = log ( a ) + log ( b ) = log ( a b ) \log(ta^{p}+(1-t)b^{q})\geq t\log(a^{p})+(1-t)\log(b^{q})=\log(a)+\log(b)=\log% (ab)
  3. a b a 2 2 + b 2 2 , ab\leq\frac{a^{2}}{2}+\frac{b^{2}}{2},
  4. a b a 2 2 ε + ε b 2 2 . ab\leq\frac{a^{2}}{2\varepsilon}+\frac{\varepsilon b^{2}}{2}.
  5. 0 ( a - b ) 2 = a 2 + b 2 - 2 a b , 0\leq(a-b)^{2}=a^{2}+b^{2}-2ab,
  6. a = a / ε , b = ε b . a^{\prime}=a/\sqrt{\varepsilon},\,\text{ }b^{\prime}=\sqrt{\varepsilon}b.
  7. a b 0 a f ( x ) d x + 0 b f - 1 ( x ) d x ab\leq\int_{0}^{a}f(x)\,dx+\int_{0}^{b}f^{-1}(x)\,dx
  8. a b f ( a ) + g ( b ) . ab\leq f(a)+g(b).\,
  9. X X
  10. f f^{\star}
  11. X X^{\star}
  12. u , v f ( u ) + f ( v ) . \langle u,v\rangle\leq f^{\star}(u)+f(v).
  13. , : X × X \langle\cdot,\cdot\rangle:X^{\star}\times X\to\mathbb{R}
  14. 1 p + 1 q = 1 r + 1 \frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1
  15. f * g r f p g q . \|f*g\|_{r}\leq\|f\|_{p}\|g\|_{q}.
  16. f p = ( | f ( x ) | p d x ) 1 / p \|f\|_{p}=\Bigl(\int|f(x)|^{p}\,dx\Bigr)^{1/p}
  17. f * g r c p , q f p g q . \|f*g\|_{r}\leq c_{p,q}\|f\|_{p}\|g\|_{q}.

Young_symmetrizer.html

  1. V n V^{\otimes n}
  2. S n S_{n}
  3. V n V^{\otimes n}
  4. P λ P_{\lambda}
  5. Q λ Q_{\lambda}
  6. P λ = { g S n : g preserves each row of λ } P_{\lambda}=\{g\in S_{n}:g\,\text{ preserves each row of }\lambda\}
  7. Q λ = { g S n : g preserves each column of λ } . Q_{\lambda}=\{g\in S_{n}:g\,\text{ preserves each column of }\lambda\}.
  8. S n \mathbb{C}S_{n}
  9. a λ = g P λ e g a_{\lambda}=\sum_{g\in P_{\lambda}}e_{g}
  10. b λ = g Q λ sgn ( g ) e g b_{\lambda}=\sum_{g\in Q_{\lambda}}\operatorname{sgn}(g)e_{g}
  11. e g e_{g}
  12. sgn ( g ) \operatorname{sgn}(g)
  13. c λ := a λ b λ = g P λ , h Q λ sgn ( h ) e g h c_{\lambda}:=a_{\lambda}b_{\lambda}=\sum_{g\in P_{\lambda},h\in Q_{\lambda}}% \operatorname{sgn}(h)e_{gh}
  14. V n = V V V V^{\otimes n}=V\otimes V\otimes\cdots\otimes V
  15. S n End ( V n ) \mathbb{C}S_{n}\rightarrow\,\text{End}(V^{\otimes n})
  16. V n V^{\otimes n}
  17. n = λ 1 + λ 2 + + λ j n=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{j}
  18. a λ a_{\lambda}
  19. Im ( a λ ) := a λ V n Sym λ 1 V Sym λ 2 V Sym λ j V . \,\text{Im}(a_{\lambda}):=a_{\lambda}V^{\otimes n}\cong\,\text{Sym}^{\lambda_{% 1}}\;V\otimes\,\text{Sym}^{\lambda_{2}}\;V\otimes\cdots\otimes\,\text{Sym}^{% \lambda_{j}}\;V.
  20. n = 4 n=4
  21. λ = ( 2 , 2 ) \lambda=(2,2)
  22. { { 1 , 2 } , { 3 , 4 } } \{\{1,2\},\{3,4\}\}
  23. a λ a_{\lambda}
  24. a λ = e id + e ( 1 , 2 ) + e ( 3 , 4 ) + e ( 1 , 2 ) ( 3 , 4 ) a_{\lambda}=e_{\,\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)}
  25. V 4 V^{\otimes 4}
  26. v 1 , 2 , 3 , 4 := v 1 v 2 v 3 v 4 v_{1,2,3,4}:=v_{1}\otimes v_{2}\otimes v_{3}\otimes v_{4}
  27. a λ v 1 , 2 , 3 , 4 = v 1 , 2 , 3 , 4 + v 2 , 1 , 3 , 4 + v 1 , 2 , 4 , 3 + v 2 , 1 , 4 , 3 = ( v 1 v 2 + v 2 v 1 ) ( v 3 v 4 + v 4 v 3 ) . a_{\lambda}v_{1,2,3,4}=v_{1,2,3,4}+v_{2,1,3,4}+v_{1,2,4,3}+v_{2,1,4,3}=(v_{1}% \otimes v_{2}+v_{2}\otimes v_{1})\otimes(v_{3}\otimes v_{4}+v_{4}\otimes v_{3}).
  28. Sym 2 V Sym 2 V \,\text{Sym}^{2}\;V\otimes\,\text{Sym}^{2}\;V
  29. b λ b_{\lambda}
  30. Im ( b λ ) μ 1 V μ 2 V μ k V \,\text{Im}(b_{\lambda})\cong\bigwedge^{\mu_{1}}V\otimes\bigwedge^{\mu_{2}}V% \otimes\cdots\otimes\bigwedge^{\mu_{k}}V
  31. Sym i V \,\text{Sym}^{i}V
  32. j V \bigwedge^{j}V
  33. S n c λ \mathbb{C}S_{n}c_{\lambda}
  34. c λ = a λ b λ c_{\lambda}=a_{\lambda}\cdot b_{\lambda}
  35. S n \mathbb{C}S_{n}
  36. Im ( c λ ) = V λ \,\text{Im}(c_{\lambda})=V_{\lambda}
  37. c λ c_{\lambda}
  38. c λ 2 = α λ c λ c^{2}_{\lambda}=\alpha_{\lambda}c_{\lambda}
  39. α λ \alpha_{\lambda}\in\mathbb{Q}
  40. α λ = n ! / dim V λ \alpha_{\lambda}=n!/\,\text{dim }V_{\lambda}
  41. S n \mathbb{Q}S_{n}
  42. c ( 2 , 1 ) = e 123 + e 213 - e 321 - e 312 c_{(2,1)}=e_{123}+e_{213}-e_{321}-e_{312}
  43. c λ c_{\lambda}
  44. V d V^{\otimes d}

Yukawa_interaction.html

  1. V g ψ ¯ ϕ ψ V\approx g\bar{\psi}\phi\psi
  2. g ψ ¯ i γ 5 ϕ ψ g\bar{\psi}i\gamma^{5}\phi\psi
  3. S [ ϕ , ψ ] = d d x [ meson ( ϕ ) + Dirac ( ψ ) + Yukawa ( ϕ , ψ ) ] S[\phi,\psi]=\int d^{d}x\;\left[\mathcal{L}_{\mathrm{meson}}(\phi)+\mathcal{L}% _{\mathrm{Dirac}}(\psi)+\mathcal{L}_{\mathrm{Yukawa}}(\phi,\psi)\right]
  4. meson ( ϕ ) = 1 2 μ ϕ μ ϕ - V ( ϕ ) . \mathcal{L}_{\mathrm{meson}}(\phi)=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}% \phi-V(\phi).
  5. V ( ϕ ) V(\phi)
  6. V ( ϕ ) = 1 2 μ 2 ϕ 2 V(\phi)=\frac{1}{2}\mu^{2}\phi^{2}
  7. μ \mu
  8. V ( ϕ ) = 1 2 μ 2 ϕ 2 + λ ϕ 4 V(\phi)=\frac{1}{2}\mu^{2}\phi^{2}+\lambda\phi^{4}
  9. Dirac ( ψ ) = ψ ¯ ( i / - m ) ψ \mathcal{L}_{\mathrm{Dirac}}(\psi)=\bar{\psi}(i\partial\!\!\!/-m)\psi
  10. Yukawa ( ϕ , ψ ) = - g ψ ¯ ϕ ψ \mathcal{L}_{\mathrm{Yukawa}}(\phi,\psi)=-g\bar{\psi}\phi\psi
  11. Yukawa ( ϕ , ψ ) = - g ψ ¯ i γ 5 ϕ ψ \mathcal{L}_{\mathrm{Yukawa}}(\phi,\psi)=-g\bar{\psi}i\gamma^{5}\phi\psi
  12. S [ ϕ , ψ ] = d d x [ 1 2 μ ϕ μ ϕ - V ( ϕ ) + ψ ¯ ( i / - m ) ψ - g ψ ¯ ϕ ψ ] . S[\phi,\psi]=\int d^{d}x\left[\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-% V(\phi)+\bar{\psi}(i\partial\!\!\!/-m)\psi-g\bar{\psi}\phi\psi\right].
  13. μ \mu
  14. V ( r ) = - g 2 4 π 1 r e - μ r V(r)=-\frac{g^{2}}{4\pi}\frac{1}{r}e^{-\mu r}
  15. V ( ϕ ) V(\phi)
  16. ϕ = 0 \phi=0
  17. ϕ 0 \phi_{0}
  18. V ( ϕ ) = μ 2 ϕ 2 + λ ϕ 4 V(\phi)=\mu^{2}\phi^{2}+\lambda\phi^{4}
  19. μ \mu
  20. ϕ \phi
  21. ϕ \phi
  22. ϕ ~ = ϕ - ϕ 0 \tilde{\phi}=\phi-\phi_{0}
  23. ϕ 0 \phi_{0}
  24. g ϕ 0 ψ ¯ ψ g\phi_{0}\bar{\psi}\psi
  25. ϕ 0 \phi_{0}
  26. g ϕ 0 g\phi_{0}
  27. ϕ ~ \tilde{\phi}
  28. S [ ϕ , χ ] = d d x [ 1 2 μ ϕ μ ϕ - V ( ϕ ) + χ i σ ¯ χ + i 2 ( m + g ϕ ) χ T σ 2 χ - i 2 ( m + g ϕ ) * χ σ 2 χ * ] S[\phi,\chi]=\int d^{d}x\left[\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-% V(\phi)+\chi^{\dagger}i\bar{\sigma}\cdot\partial\chi+\frac{i}{2}(m+g\phi)\chi^% {T}\sigma^{2}\chi-\frac{i}{2}(m+g\phi)^{*}\chi^{\dagger}\sigma^{2}\chi^{*}\right]

Z_function.html

  1. Z ( t ) = e i θ ( t ) ζ ( 1 2 + i t ) . Z(t)=e^{i\theta(t)}\zeta\left(\frac{1}{2}+it\right).
  2. - 5 < ( t ) < 5 -5<\Re(t)<5
  3. - 40 < ( t ) < 40 -40<\Re(t)<40
  4. Z ( t ) = 2 n 2 < t / 2 π n - 1 / 2 cos ( θ ( t ) - t log n ) + R ( t ) , Z(t)=2\sum_{n^{2}<t/2\pi}n^{-1/2}\cos(\theta(t)-t\log n)+R(t),
  5. Ψ ( z ) = cos 2 π ( z 2 - z - 1 / 16 ) cos 2 π z \Psi(z)=\frac{\cos 2\pi(z^{2}-z-1/16)}{\cos 2\pi z}
  6. u = ( t 2 π ) 1 / 4 u=(\frac{t}{2\pi})^{1/4}
  7. N = u 2 N=\lfloor u^{2}\rfloor
  8. p = u 2 - N p=u^{2}-N
  9. R ( t ) ( - 1 ) N - 1 ( Ψ ( p ) u - 1 - 1 96 π 2 Ψ ( 3 ) ( p ) u - 3 + ) R(t)\sim(-1)^{N-1}\left(\Psi(p)u^{-1}-\frac{1}{96\pi^{2}}\Psi^{(3)}(p)u^{-3}+% \cdots\right)
  10. Q ( a , z ) = Γ ( a , z ) Γ ( a ) = 1 Γ ( a ) z u a - 1 e - u d u Q(a,z)=\frac{\Gamma(a,z)}{\Gamma(a)}=\frac{1}{\Gamma(a)}\int_{z}^{\infty}u^{a-% 1}e^{-u}du
  11. Z ( t ) = 2 ( e i θ ( t ) ( n = 1 Q ( s 2 , π i n 2 ) - π s / 2 e π i s / 4 s Γ ( s 2 ) ) ) Z(t)=2\Re\left(e^{i\theta(t)}\left(\sum_{n=1}^{\infty}Q\left(\frac{s}{2},\pi in% ^{2}\right)-\frac{\pi^{s/2}e^{\pi is/4}}{s\Gamma\left(\frac{s}{2}\right)}% \right)\right)
  12. c 2 π log t 2 π \frac{c}{2\pi}\log\frac{t}{2\pi}
  13. Z ( t ) = Ω ( exp ( 3 4 log t log log t ) ) , Z(t)=\Omega\left(\exp\left(\frac{3}{4}\sqrt{\frac{\log t}{\log\log t}}\right)% \right),
  14. Z ( t ) Z(t)
  15. 1 T 0 T Z ( t ) 2 d t log T \frac{1}{T}\int_{0}^{T}Z(t)^{2}dt\sim\log T
  16. 1 T T 2 T Z ( t ) 2 d t log T \frac{1}{T}\int_{T}^{2T}Z(t)^{2}dt\sim\log T
  17. log t \sqrt{\log t}
  18. 1 T 0 T Z ( t ) 2 d t = log T + ( 2 γ - 2 log ( 2 π ) - 1 ) + O ( T - 15 / 22 ) \frac{1}{T}\int_{0}^{T}Z(t)^{2}dt=\log T+(2\gamma-2\log(2\pi)-1)+O(T^{-15/22})
  19. 1 T 0 T Z ( t ) 4 d t 1 2 π 2 ( log T ) 4 \frac{1}{T}\int_{0}^{T}Z(t)^{4}dt\sim\frac{1}{2\pi^{2}}(\log T)^{4}
  20. 1 2 1 / 4 π log t \frac{1}{2^{1/4}\sqrt{\pi}}\log t
  21. 1 T 0 T Z ( t ) 2 k d t = o ( T ϵ ) \frac{1}{T}\int_{0}^{T}Z(t)^{2k}dt=o(T^{\epsilon})
  22. Z ( t ) = o ( t ϵ ) ; Z(t)=o(t^{\epsilon});
  23. ϵ > 89 570 \epsilon>\frac{89}{570}
  24. Z ( t ) = o ( exp ( 10 log t log log t ) ) , Z(t)=o\left(\exp\left(\frac{10\log t}{\log\log t}\right)\right),

Zacharias_Dase.html

  1. π 4 = arctan 1 2 + arctan 1 5 + arctan 1 8 . \frac{\pi}{4}=\arctan\frac{1}{2}+\arctan\frac{1}{5}+\arctan\frac{1}{8}.

Zeckendorf's_theorem.html

  1. N N
  2. N = i = 0 k F c i , N=\sum_{i=0}^{k}F_{c_{i}},
  3. n n
  4. N N
  5. N N
  6. 100 = 89 + 8 + 3 100=89+8+3
  7. 100 = 89 + 8 + 2 + 1 100=89+8+2+1
  8. 100 = 55 + 34 + 8 + 3 100=55+34+8+3
  9. n n
  10. n n
  11. n = 1 , 2 , 3 n=1, 2, 3
  12. n = 4 n=4
  13. 4 = 3 + 1 4=3+1
  14. n k n≤k
  15. k + 1 k+1
  16. j j
  17. a k a≤k
  18. a a
  19. k + 1 k+1
  20. a a
  21. j j
  22. 𝐒 \mathbf{S}
  23. 𝐓 \mathbf{T}
  24. 𝐒 \mathbf{S}
  25. 𝐓 \mathbf{T}
  26. 𝐒 \mathbf{S}
  27. 𝐓 \mathbf{T}
  28. 𝐒 \mathbf{S}
  29. \cap
  30. 𝐓 \mathbf{T}
  31. 𝐒 = 𝐓 \mathbf{S}=\mathbf{T}
  32. a b a\circ b
  33. a a
  34. b b
  35. a = i = 0 k F c i ( c i 2 ) a=\sum_{i=0}^{k}F_{c_{i}}\;(c_{i}\geq 2)
  36. b = j = 0 l F d j ( d j 2 ) b=\sum_{j=0}^{l}F_{d_{j}}\;(d_{j}\geq 2)
  37. a b = i = 0 k j = 0 l F c i + d j . a\circ b=\sum_{i=0}^{k}\sum_{j=0}^{l}F_{c_{i}+d_{j}}.
  38. F 3 F_{3}
  39. F 4 + F 2 F_{4}+F_{2}
  40. F 1 F_{1}
  41. 2 4 = F 3 + 4 + F 3 + 2 = 13 + 5 = 18. 2\circ 4=F_{3+4}+F_{3+2}=13+5=18.
  42. n n
  43. F n - 2 = F n - F n - 1 , F_{n-2}=F_{n}-F_{n-1},\,
  44. F - n = ( - 1 ) n + 1 F n . F_{-n}=(-1)^{n+1}F_{n}.\,
  45. 11 = F < s u b > 4 + F 6 = ( 3 ) + ( 8 ) −11=F<sub>−4+F_{−6}=(−3)+(−8)

Zeller's_congruence.html

  1. h = ( q + 13 ( m + 1 ) 5 + K + K 4 + J 4 - 2 J ) mod 7 , h=\left(q+\left\lfloor\frac{13(m+1)}{5}\right\rfloor+K+\left\lfloor\frac{K}{4}% \right\rfloor+\left\lfloor\frac{J}{4}\right\rfloor-2J\right)\mod 7,
  2. h = ( q + 13 ( m + 1 ) 5 + K + K 4 + 5 - J ) mod 7 , h=\left(q+\left\lfloor\frac{13(m+1)}{5}\right\rfloor+K+\left\lfloor\frac{K}{4}% \right\rfloor+5-J\right)\mod 7,
  3. y e a r mod 100 year\mod 100
  4. y e a r / 100 \lfloor year/100\rfloor
  5. d = ( ( h + 5 ) mod 7 ) + 1 d=((h+5)\mod 7)+1
  6. 2 J −2J
  7. + 5 J +5J
  8. J −J
  9. + 6 J +6J
  10. h = ( q + 13 ( m + 1 ) 5 + K + K 4 + J 4 + 5 J ) mod 7 , h=\left(q+\left\lfloor\frac{13(m+1)}{5}\right\rfloor+K+\left\lfloor\frac{K}{4}% \right\rfloor+\left\lfloor\frac{J}{4}\right\rfloor+5J\right)\mod 7,
  11. h = ( q + 13 ( m + 1 ) 5 + K + K 4 + 5 + 6 J ) mod 7 , h=\left(q+\left\lfloor\frac{13(m+1)}{5}\right\rfloor+K+\left\lfloor\frac{K}{4}% \right\rfloor+5+6J\right)\mod 7,
  12. Y Y
  13. Y - 1 Y-1
  14. h = ( q + ( m + 1 ) 26 10 + Y + Y 4 + 6 Y 100 + Y 400 ) mod 7 , h=\left(q+\left\lfloor\frac{(m+1)26}{10}\right\rfloor+Y+\left\lfloor\frac{Y}{4% }\right\rfloor+6\left\lfloor\frac{Y}{100}\right\rfloor+\left\lfloor\frac{Y}{40% 0}\right\rfloor\right)\mod 7,
  15. h = ( q + ( m + 1 ) 26 10 + Y + Y 4 + 5 ) mod 7 , h=\left(q+\left\lfloor\frac{(m+1)26}{10}\right\rfloor+Y+\left\lfloor\frac{Y}{4% }\right\rfloor+5\right)\mod 7,
  16. q q
  17. K K
  18. 365 mod 7 = 1 365\mod 7=1
  19. K 4 \left\lfloor\frac{K}{4}\right\rfloor
  20. 36525 mod 7 = 6 36525\mod 7=6
  21. 36524 mod 7 = 5 36524\mod 7=5
  22. J 4 - 2 J \left\lfloor\frac{J}{4}\right\rfloor-2J
  23. J 4 + 5 J \left\lfloor\frac{J}{4}\right\rfloor+5J
  24. 13 ( m + 1 ) 5 \left\lfloor\frac{13(m+1)}{5}\right\rfloor
  25. mod 7 \mod 7

Zernike_polynomials.html

  1. Z n m ( ρ , φ ) = R n m ( ρ ) cos ( m φ ) Z^{m}_{n}(\rho,\varphi)=R^{m}_{n}(\rho)\,\cos(m\,\varphi)\!
  2. Z n - m ( ρ , φ ) = R n m ( ρ ) sin ( m φ ) , Z^{-m}_{n}(\rho,\varphi)=R^{m}_{n}(\rho)\,\sin(m\,\varphi),\!
  3. 0 ρ 1 0\leq\rho\leq 1
  4. | Z n m ( ρ , φ ) | 1 |Z^{m}_{n}(\rho,\varphi)|\leq 1
  5. R n m ( ρ ) = k = 0 n - m 2 ( - 1 ) k ( n - k ) ! k ! ( n + m 2 - k ) ! ( n - m 2 - k ) ! ρ n - 2 k R^{m}_{n}(\rho)=\sum_{k=0}^{\tfrac{n-m}{2}}\frac{(-1)^{k}\,(n-k)!}{k!\left(% \tfrac{n+m}{2}-k\right)!\left(\tfrac{n-m}{2}-k\right)!}\;\rho^{n-2\,k}
  6. R n m ( ρ ) = k = 0 n - m 2 ( - 1 ) k ( n - k k ) ( n - 2 k n - m 2 - k ) ρ n - 2 k R_{n}^{m}(\rho)=\sum_{k=0}^{\tfrac{n-m}{2}}(-1)^{k}{\left({{n-k}\atop{k}}% \right)}{\left({{n-2k}\atop{\tfrac{n-m}{2}-k}}\right)}\rho^{n-2k}
  7. R n m ( ρ ) \displaystyle R_{n}^{m}(\rho)
  8. R n m ( ρ ) + R n - 2 m ( ρ ) = ρ [ R n - 1 | m - 1 | ( ρ ) + R n - 1 m + 1 ( ρ ) ] \begin{aligned}\displaystyle R_{n}^{m}(\rho)+R_{n-2}^{m}(\rho)=\rho\left[R_{n-% 1}^{\left|m-1\right|}(\rho)+R_{n-1}^{m+1}(\rho)\right]\end{aligned}
  9. Z n m Z j Z_{n}^{m}\rightarrow Z_{j}
  10. | |
  11. cos ( m φ ) \cos(m\varphi)
  12. 0 1 ρ 2 n + 2 R n m ( ρ ) 2 n + 2 R n m ( ρ ) d ρ = δ n , n . \int_{0}^{1}\rho\sqrt{2n+2}R_{n}^{m}(\rho)\,\sqrt{2n^{\prime}+2}R_{n^{\prime}}% ^{m}(\rho)\,d\rho=\delta_{n,n^{\prime}}.
  13. 0 2 π cos ( m φ ) cos ( m φ ) d φ = ϵ m π δ | m | , | m | , \int_{0}^{2\pi}\cos(m\varphi)\cos(m^{\prime}\varphi)\,d\varphi=\epsilon_{m}\pi% \delta_{|m|,|m^{\prime}|},
  14. 0 2 π sin ( m φ ) sin ( m φ ) d φ = ( - 1 ) m + m π δ | m | , | m | ; m 0 , \int_{0}^{2\pi}\sin(m\varphi)\sin(m^{\prime}\varphi)\,d\varphi=(-1)^{m+m^{% \prime}}\pi\delta_{|m|,|m^{\prime}|};\quad m\neq 0,
  15. 0 2 π cos ( m φ ) sin ( m φ ) d φ = 0 , \int_{0}^{2\pi}\cos(m\varphi)\sin(m^{\prime}\varphi)\,d\varphi=0,
  16. ϵ m \epsilon_{m}
  17. m = 0 m=0
  18. m 0 m\neq 0
  19. Z n m ( ρ , φ ) Z n m ( ρ , φ ) d 2 r = ϵ m π 2 n + 2 δ n , n δ m , m , \int Z_{n}^{m}(\rho,\varphi)Z_{n^{\prime}}^{m^{\prime}}(\rho,\varphi)\,d^{2}r=% \frac{\epsilon_{m}\pi}{2n+2}\delta_{n,n^{\prime}}\delta_{m,m^{\prime}},
  20. d 2 r = ρ d ρ d φ d^{2}r=\rho\,d\rho\,d\varphi
  21. n - m n-m
  22. n - m n^{\prime}-m^{\prime}
  23. R n m ( 1 ) = 1 , R_{n}^{m}(1)=1,\,
  24. G ( ρ , φ ) G(\rho,\varphi)
  25. G ( ρ , φ ) = m , n [ a m , n Z n m ( ρ , φ ) + b m , n Z n - m ( ρ , φ ) ] , G(\rho,\varphi)=\sum_{m,n}\left[a_{m,n}Z^{m}_{n}(\rho,\varphi)+b_{m,n}Z^{-m}_{% n}(\rho,\varphi)\right],
  26. L 2 L^{2}
  27. F , G := F ( ρ , φ ) G ( ρ , φ ) ρ d ρ d φ . \langle F,G\rangle:=\int F(\rho,\varphi)G(\rho,\varphi)\rho d\rho d\varphi.
  28. a m , n \displaystyle a_{m,n}
  29. Z n m ( ρ , φ ) = ( - 1 ) m Z n m ( ρ , - φ ) . Z_{n}^{m}(\rho,\varphi)=(-1)^{m}Z_{n}^{m}(\rho,-\varphi).
  30. Z n m ( ρ , φ ) = ( - 1 ) m Z n m ( ρ , φ + π ) , Z_{n}^{m}(\rho,\varphi)=(-1)^{m}Z_{n}^{m}(\rho,\varphi+\pi),
  31. ( - 1 ) m (-1)^{m}
  32. ( - 1 ) n (-1)^{n}
  33. n - m n-m
  34. R n m ( ρ ) = ( - 1 ) n R n m ( - ρ ) = ( - 1 ) m R n m ( - ρ ) . R_{n}^{m}(\rho)=(-1)^{n}R_{n}^{m}(-\rho)=(-1)^{m}R_{n}^{m}(-\rho).
  35. 2 π / m 2\pi/m
  36. Z n m ( ρ , φ + 2 π k m ) = Z n m ( ρ , φ ) , k = 0 , ± 1 , ± 2 , . Z_{n}^{m}\left(\rho,\varphi+\tfrac{2\pi k}{m}\right)=Z_{n}^{m}(\rho,\varphi),% \qquad k=0,\pm 1,\pm 2,\cdots.
  37. R 0 0 ( ρ ) = 1 R^{0}_{0}(\rho)=1\,
  38. R 1 1 ( ρ ) = ρ R^{1}_{1}(\rho)=\rho\,
  39. R 2 0 ( ρ ) = 2 ρ 2 - 1 R^{0}_{2}(\rho)=2\rho^{2}-1\,
  40. R 2 2 ( ρ ) = ρ 2 R^{2}_{2}(\rho)=\rho^{2}\,
  41. R 3 1 ( ρ ) = 3 ρ 3 - 2 ρ R^{1}_{3}(\rho)=3\rho^{3}-2\rho\,
  42. R 3 3 ( ρ ) = ρ 3 R^{3}_{3}(\rho)=\rho^{3}\,
  43. R 4 0 ( ρ ) = 6 ρ 4 - 6 ρ 2 + 1 R^{0}_{4}(\rho)=6\rho^{4}-6\rho^{2}+1\,
  44. R 4 2 ( ρ ) = 4 ρ 4 - 3 ρ 2 R^{2}_{4}(\rho)=4\rho^{4}-3\rho^{2}\,
  45. R 4 4 ( ρ ) = ρ 4 R^{4}_{4}(\rho)=\rho^{4}\,
  46. R 5 1 ( ρ ) = 10 ρ 5 - 12 ρ 3 + 3 ρ R^{1}_{5}(\rho)=10\rho^{5}-12\rho^{3}+3\rho\,
  47. R 5 3 ( ρ ) = 5 ρ 5 - 4 ρ 3 R^{3}_{5}(\rho)=5\rho^{5}-4\rho^{3}\,
  48. R 5 5 ( ρ ) = ρ 5 R^{5}_{5}(\rho)=\rho^{5}\,
  49. R 6 0 ( ρ ) = 20 ρ 6 - 30 ρ 4 + 12 ρ 2 - 1 R^{0}_{6}(\rho)=20\rho^{6}-30\rho^{4}+12\rho^{2}-1\,
  50. R 6 2 ( ρ ) = 15 ρ 6 - 20 ρ 4 + 6 ρ 2 R^{2}_{6}(\rho)=15\rho^{6}-20\rho^{4}+6\rho^{2}\,
  51. R 6 4 ( ρ ) = 6 ρ 6 - 5 ρ 4 R^{4}_{6}(\rho)=6\rho^{6}-5\rho^{4}\,
  52. R 6 6 ( ρ ) = ρ 6 . R^{6}_{6}(\rho)=\rho^{6}.\,
  53. j j
  54. 0 2 π 0 1 Z j 2 ρ d ρ d θ = π . \int_{0}^{2\pi}\int_{0}^{1}Z_{j}^{2}\,\rho\,d\rho\,d\theta=\pi.
  55. j j
  56. n n
  57. m m
  58. Z j Z_{j}
  59. 1 1
  60. 2 ρ cos θ 2\rho\cos\theta
  61. 2 ρ sin θ 2\rho\sin\theta
  62. 3 ( 2 ρ 2 - 1 ) \sqrt{3}(2\rho^{2}-1)
  63. 6 ρ 2 sin 2 θ \sqrt{6}\rho^{2}\sin 2\theta
  64. 6 ρ 2 cos 2 θ \sqrt{6}\rho^{2}\cos 2\theta
  65. 8 ( 3 ρ 3 - 2 ρ ) sin θ \sqrt{8}(3\rho^{3}-2\rho)\sin\theta
  66. 8 ( 3 ρ 3 - 2 ρ ) cos θ \sqrt{8}(3\rho^{3}-2\rho)\cos\theta
  67. 8 ρ 3 sin 3 θ \sqrt{8}\rho^{3}\sin 3\theta
  68. 8 ρ 3 cos 3 θ \sqrt{8}\rho^{3}\cos 3\theta
  69. 5 ( 6 ρ 4 - 6 ρ 2 + 1 ) \sqrt{5}(6\rho^{4}-6\rho^{2}+1)
  70. 10 ( 4 ρ 4 - 3 ρ 2 ) cos 2 θ \sqrt{10}(4\rho^{4}-3\rho^{2})\cos 2\theta
  71. 10 ( 4 ρ 4 - 3 ρ 2 ) sin 2 θ \sqrt{10}(4\rho^{4}-3\rho^{2})\sin 2\theta
  72. 10 ρ 4 cos 4 θ \sqrt{10}\rho^{4}\cos 4\theta
  73. 10 ρ 4 sin 4 θ \sqrt{10}\rho^{4}\sin 4\theta
  74. ρ 1 \rho\approx 1
  75. x 1 i x 2 j x D k x_{1}^{i}x_{2}^{j}\cdots x_{D}^{k}
  76. ρ s , s D \rho^{s},s\leq D
  77. D = 3 D=3
  78. ρ s \rho^{s}
  79. R n ( l ) ( ρ ) R_{n}^{(l)}(\rho)
  80. 0 1 ρ D - 1 R n ( l ) ( ρ ) R n ( l ) ( ρ ) d ρ = δ n , n \int_{0}^{1}\rho^{D-1}R_{n}^{(l)}(\rho)R_{n^{\prime}}^{(l)}(\rho)d\rho=\delta_% {n,n^{\prime}}
  81. 2 n + D \sqrt{2n+D}
  82. D = 2 D=2
  83. R n ( l ) ( ρ ) = 2 n + D s = 0 n - l 2 ( - 1 ) s ( n - l 2 s ) ( n - s - 1 + D 2 n - l 2 ) ρ n - 2 s = ( - 1 ) n - l 2 2 n + D s = 0 n - l 2 ( - 1 ) s ( n - l 2 s ) ( s - 1 + n + l + D 2 n - l 2 ) ρ 2 s + l = ( - 1 ) n - l 2 2 n + D ( n + l + D 2 - 1 n - l 2 ) ρ l F 1 2 ( - n - l 2 , n + l + D 2 ; l + D 2 ; ρ 2 ) \begin{aligned}\displaystyle R_{n}^{(l)}(\rho)&\displaystyle=\sqrt{2n+D}\sum_{% s=0}^{\tfrac{n-l}{2}}(-1)^{s}{\tfrac{n-l}{2}\choose s}{n-s-1+\tfrac{D}{2}% \choose\tfrac{n-l}{2}}\rho^{n-2s}\\ &\displaystyle=(-1)^{\tfrac{n-l}{2}}\sqrt{2n+D}\sum_{s=0}^{\tfrac{n-l}{2}}(-1)% ^{s}{\tfrac{n-l}{2}\choose s}{s-1+\tfrac{n+l+D}{2}\choose\tfrac{n-l}{2}}\rho^{% 2s+l}\\ &\displaystyle=(-1)^{\tfrac{n-l}{2}}\sqrt{2n+D}{\tfrac{n+l+D}{2}-1\choose% \tfrac{n-l}{2}}\rho^{l}\ {}_{2}F_{1}\left(-\tfrac{n-l}{2},\tfrac{n+l+D}{2};l+% \tfrac{D}{2};\rho^{2}\right)\end{aligned}
  84. n - l 0 n-l\geq 0

Zirconium_tungstate.html

  1. 3 ¯ \bar{3}

Zitterbewegung.html

  1. 2 m c 2 / 2mc^{2}/\hbar
  2. × 10 2 1 \times 10^{2}1
  3. H ψ ( 𝐱 , t ) = i ψ t ( 𝐱 , t ) H\psi(\mathbf{x},t)=i\hbar\frac{\partial\psi}{\partial t}(\mathbf{x},t)
  4. H H
  5. H = ( α 0 m c 2 + j = 1 3 α j p j c ) H=\left(\alpha_{0}mc^{2}+\sum_{j=1}^{3}\alpha_{j}p_{j}\,c\right)
  6. - i Q t ( t ) = [ H , Q ] . -i\hbar\frac{\partial Q}{\partial t}(t)=\left[H,Q\right]\,.
  7. x k t ( t ) = i [ H , x k ] = c α k \hbar\frac{\partial x_{k}}{\partial t}(t)=i\left[H,x_{k}\right]=\hbar c\alpha_% {k}
  8. α k γ 0 γ k \alpha_{k}\equiv\gamma_{0}\gamma_{k}
  9. α k \alpha_{k}
  10. α k \alpha_{k}
  11. α k ( t ) = e i H t / α k e - i H t / \alpha_{k}(t)=e^{iHt/\hbar}\alpha_{k}e^{-iHt/\hbar}
  12. α k t ( t ) = i [ H , α k ] = 2 ( i γ k m - σ k l p l ) = 2 i ( p k - α k H ) \hbar\frac{\partial\alpha_{k}}{\partial t}(t)=i\left[H,\alpha_{k}\right]=2(i% \gamma_{k}m-\sigma_{kl}p^{l})=2i(p_{k}-\alpha_{k}H)
  13. σ k l i 2 [ γ k , γ l ] \sigma_{kl}\equiv\frac{i}{2}[\gamma_{k},\gamma_{l}]
  14. p k p_{k}
  15. H H
  16. α k ( t ) = ( α k ( 0 ) - c p k H - 1 ) e - 2 i H t / + c p k H - 1 \alpha_{k}(t)=(\alpha_{k}(0)-cp_{k}H^{-1})e^{-2iHt/\hbar}+cp_{k}H^{-1}
  17. x k ( t ) = x k ( 0 ) + c 2 p k H - 1 t + 1 2 i c H - 1 ( α k ( 0 ) - c p k H - 1 ) ( e - 2 i H t / - 1 ) x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{1\over 2}i\hbar cH^{-1}(\alpha_{k}(0)-cp_% {k}H^{-1})(e^{-2iHt/\hbar}-1)
  18. x k ( t ) x_{k}(t)
  19. t t

Znám's_problem.html

  1. { n 1 , , n k } \{n_{1},\ldots,n_{k}\}
  2. ( j i n n j ) + 1 ? \Bigl(\prod_{j\neq i}^{n}n_{j}\Bigr)+1\quad?
  3. 1 x i + 1 x i = y , \sum\frac{1}{x_{i}}+\prod\frac{1}{x_{i}}=y,
  4. 1 x i + 1 x i = 1. \sum\frac{1}{x_{i}}+\prod\frac{1}{x_{i}}=1.

Ł.html

  1. \ \ell

Δ-hyperbolic_space.html

  1. ( x , z ) p min { ( x , y ) p , ( y , z ) p } - δ , (x,z)_{p}\geq\min\big\{(x,y)_{p},(y,z)_{p}\big\}-\delta,