wpmath0000001_23

Wave_equation.html

  1. u u
  2. 2 u t 2 = c 2 2 u {\partial^{2}u\over\partial t^{2}}=c^{2}\nabla^{2}u
  3. 2 u t 2 = c 2 2 u x 2 {\partial^{2}u\over\partial t^{2}}=c^{2}{\partial^{2}u\over\partial x^{2}}
  4. F 𝑁𝑒𝑤𝑡𝑜𝑛 = m a ( t ) = m 2 t 2 u ( x + h , t ) F_{\mathit{Newton}}=m\cdot a(t)=m\cdot{{\partial^{2}\over\partial t^{2}}u(x+h,% t)}
  5. F 𝐻𝑜𝑜𝑘𝑒 = F x + 2 h - F x = k [ u ( x + 2 h , t ) - u ( x + h , t ) ] - k [ u ( x + h , t ) - u ( x , t ) ] F_{\mathit{Hooke}}=F_{x+2h}-F_{x}=k\left[{u(x+2h,t)-u(x+h,t)}\right]-k[u(x+h,t% )-u(x,t)]
  6. m 2 t 2 u ( x + h , t ) = k [ u ( x + 2 h , t ) - u ( x + h , t ) - u ( x + h , t ) + u ( x , t ) ] m{\partial^{2}\over\partial t^{2}}u(x+h,t)=k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]
  7. 2 t 2 u ( x + h , t ) = K L 2 M u ( x + 2 h , t ) - 2 u ( x + h , t ) + u ( x , t ) h 2 {\partial^{2}\over\partial t^{2}}u(x+h,t)={KL^{2}\over M}{u(x+2h,t)-2u(x+h,t)+% u(x,t)\over h^{2}}
  8. 2 u ( x , t ) t 2 = K L 2 M 2 u ( x , t ) x 2 {\partial^{2}u(x,t)\over\partial t^{2}}={KL^{2}\over M}{\partial^{2}u(x,t)% \over\partial x^{2}}
  9. K = E A L K={EA\over L}
  10. 2 u ( x , t ) t 2 = E A L M 2 u ( x , t ) x 2 {\partial^{2}u(x,t)\over\partial t^{2}}={EAL\over M}{\partial^{2}u(x,t)\over% \partial x^{2}}
  11. A L M = 1 ρ {AL\over M}={1\over\rho}
  12. ρ \rho
  13. 2 u ( x , t ) t 2 = E ρ 2 u ( x , t ) x 2 {\partial^{2}u(x,t)\over\partial t^{2}}={E\over\rho}{\partial^{2}u(x,t)\over% \partial x^{2}}
  14. E ρ \sqrt{E\over\rho}
  15. ξ = x - c t ; η = x + c t \xi=x-ct\quad;\quad\eta=x+ct
  16. 2 u ξ η = 0 \frac{\partial^{2}u}{\partial\xi\partial\eta}=0
  17. u ( ξ , η ) = F ( ξ ) + G ( η ) u(\xi,\eta)=F(\xi)+G(\eta)
  18. u ( x , t ) = F ( x - c t ) + G ( x + c t ) u(x,t)=F(x-ct)+G(x+ct)
  19. [ t - c x ] [ t + c x ] u = 0 \left[\frac{\partial}{\partial t}-c\frac{\partial}{\partial x}\right]\left[% \frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\right]u=0
  20. either u t - c u x = 0 or u t + c u x = 0 \mbox{either}~{}\qquad\frac{\partial u}{\partial t}-c\frac{\partial u}{% \partial x}=0\qquad\mbox{or}~{}\qquad\frac{\partial u}{\partial t}+c\frac{% \partial u}{\partial x}=0
  21. u ( x , 0 ) = f ( x ) u(x,0)=f(x)\,
  22. u t ( x , 0 ) = g ( x ) u_{t}(x,0)=g(x)\,
  23. u ( x , t ) = f ( x - c t ) + f ( x + c t ) 2 + 1 2 c x - c t x + c t g ( s ) d s u(x,t)=\frac{f(x-ct)+f(x+ct)}{2}+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)ds
  24. ω \omega
  25. e - i ω t e^{-i\omega t}
  26. x x
  27. u ω ( x , t ) = e - i ω t f ( x ) , u_{\omega}(x,t)=e^{-i\omega t}f(x),
  28. f ( x ) f(x)
  29. 2 u ω t 2 = 2 t 2 ( e - i ω t f ( x ) ) = - ω 2 e - i ω t f ( x ) = c 2 2 x 2 ( e - i ω t f ( x ) ) , \frac{\partial^{2}u_{\omega}}{\partial t^{2}}=\frac{\partial^{2}}{\partial t^{% 2}}\left(e^{-i\omega t}f(x)\right)=-\omega^{2}e^{-i\omega t}f(x)=c^{2}\frac{% \partial^{2}}{\partial x^{2}}\left(e^{-i\omega t}f(x)\right),
  30. d 2 d x 2 f ( x ) = - ( ω c ) 2 f ( x ) , \frac{d^{2}}{dx^{2}}f(x)=-\left(\frac{\omega}{c}\right)^{2}f(x),
  31. f ( x ) f(x)
  32. f ( x ) = A e ± i k x f(x)=Ae^{\pm ikx}
  33. k = ω / c k=\omega/c
  34. u ω ( x , t ) = e - i ω t ( A e - i k x + B e i k x ) = A e - i ( k x + ω t ) + B e i ( k x - ω t ) , u_{\omega}(x,t)=e^{-i\omega t}\left(Ae^{-ikx}+Be^{ikx}\right)=Ae^{-i(kx+\omega t% )}+Be^{i(kx-\omega t)},
  35. A , B A,B
  36. e - i ω t e^{-i\omega t}
  37. u ( x , t ) = - s ( ω ) u ω ( x , t ) d ω u(x,t)=\int_{-\infty}^{\infty}s(\omega)u_{\omega}(x,t)d\omega
  38. u ( x , t ) = - s + ( ω ) e - i ( k x + ω t ) d ω + - s - ( ω ) e i ( k x - ω t ) d ω = - s + ( ω ) e - i k ( x + c t ) d ω + - s - ( ω ) e i k ( x - c t ) d ω = F ( x - c t ) + G ( x + c t ) \begin{aligned}\displaystyle u(x,t)&\displaystyle=\int_{-\infty}^{\infty}s_{+}% (\omega)e^{-i(kx+\omega t)}d\omega+\int_{-\infty}^{\infty}s_{-}(\omega)e^{i(kx% -\omega t)}d\omega\\ &\displaystyle=\int_{-\infty}^{\infty}s_{+}(\omega)e^{-ik(x+ct)}d\omega+\int_{% -\infty}^{\infty}s_{-}(\omega)e^{ik(x-ct)}d\omega\\ &\displaystyle=F(x-ct)+G(x+ct)\end{aligned}
  39. s ± ( ω ) s_{\pm}(\omega)
  40. u ( x , t ) u(x,t)
  41. ω \omega
  42. Ψ ( r , t ) = - Ψ ( r , ω ) e - i ω t d ω . \Psi(\vec{r},t)=\int_{-\infty}^{\infty}\Psi(\vec{r},\omega)e^{-i\omega t}d\omega.
  43. ( 2 + ω 2 c 2 ) Ψ ( r , ω ) = 0 \left(\nabla^{2}+\frac{\omega^{2}}{c^{2}}\right)\Psi(\vec{r},\omega)=0
  44. [ d 2 d r 2 + 2 r d d r + k 2 - l ( l + 1 ) r 2 ] f l m ( r ) = 0 \left[\frac{d^{2}}{dr^{2}}+\frac{2}{r}\frac{d}{dr}+k^{2}-\frac{l(l+1)}{r^{2}}% \right]f_{lm}(r)=0
  45. k ω c k\equiv\frac{\omega}{c}
  46. Ψ ( r , t ) = l m [ A l m ( 1 ) h l m ( 1 ) ( k r ) + A l m ( 2 ) h l m ( 2 ) ( k r ) ] ( r ) Y l m ( θ , ϕ ) , \Psi(\vec{r},t)=\sum_{lm}\left[A_{lm}^{(1)}h_{lm}^{(1)}(kr)+A_{lm}^{(2)}h_{lm}% ^{(2)}(kr)\right](r)Y_{lm}(\theta,\phi),
  47. h l m ( 1 ) ( k r ) h_{lm}^{(1)}(kr)
  48. h l m ( 2 ) ( k r ) h_{lm}^{(2)}(kr)
  49. l = 0 l=0
  50. Ψ ( r , t ) u ( r , t ) \Psi(\vec{r},t)\rightarrow u(r,t)
  51. ( 2 - 1 c 2 2 t 2 ) Ψ ( r , t ) = 0 ( 2 r 2 + 2 r r - 1 c 2 2 t 2 ) u ( r , t ) = 0 \left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\right)\Psi% (\vec{r},t)=0\rightarrow\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r}% \frac{\partial}{\partial r}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}% \right)u(r,t)=0
  52. 2 ( r u ) t 2 - c 2 2 ( r u ) r 2 = 0 ; \frac{\partial^{2}(ru)}{\partial t^{2}}-c^{2}\frac{\partial^{2}(ru)}{\partial r% ^{2}}=0;\,
  53. r u ru
  54. u ( r , t ) = 1 r F ( r - c t ) + 1 r G ( r + c t ) , u(r,t)=\frac{1}{r}F(r-ct)+\frac{1}{r}G(r+ct),
  55. ω \omega
  56. r u ( r , t ) ru(r,t)
  57. r u ( r , t ) = A e i ( ω t ± k r ) ru(r,t)=Ae^{i(\omega t\pm kr)}
  58. u ( r , t ) = A r e i ( ω t ± k r ) u(r,t)=\frac{A}{r}e^{i\left(\omega t\pm kr\right)}
  59. I = | u ( r , t ) | 2 = | A | 2 r 2 I=|u(r,t)|^{2}=\frac{|A|^{2}}{r^{2}}
  60. 1 / r 2 1/r^{2}
  61. r 2 = ( x - ξ ) 2 + ( y - η ) 2 + ( z - ζ ) 2 . r^{2}=(x-\xi)^{2}+(y-\eta)^{2}+(z-\zeta)^{2}.
  62. u ( t , x , y , z ) = 1 4 π c φ ( ξ , η , ζ ) δ ( r - c t ) r d ξ d η d ζ ; u(t,x,y,z)=\frac{1}{4\pi c}\iiint\varphi(\xi,\eta,\zeta)\frac{\delta(r-ct)}{r}% d\xi\,d\eta\,d\zeta;\,
  63. u ( t , x , y , z ) = t 4 π S φ ( x + c t α , y + c t β , z + c t γ ) d ω , u(t,x,y,z)=\frac{t}{4\pi}\iint_{S}\varphi(x+ct\alpha,y+ct\beta,z+ct\gamma)d% \omega,\,
  64. u ( t , x , y , z ) = t M c t [ ϕ ] . u(t,x,y,z)=tM_{ct}[\phi].
  65. u ( 0 , x , y , z ) = 0 , u t ( 0 , x , y , z ) = ϕ ( x , y , z ) . u(0,x,y,z)=0,\quad u_{t}(0,x,y,z)=\phi(x,y,z).
  66. v ( t , x , y , z ) = t ( t M c t [ ψ ] ) , v(t,x,y,z)=\frac{\partial}{\partial t}\left(tM_{ct}[\psi]\right),
  67. v ( 0 , x , y , z ) = ψ ( x , y , z ) , v t ( 0 , x , y , z ) = 0. v(0,x,y,z)=\psi(x,y,z),\quad v_{t}(0,x,y,z)=0.
  68. u t t = c 2 ( u x x + u y y ) . u_{tt}=c^{2}\left(u_{xx}+u_{yy}\right).\,
  69. u ( 0 , x , y ) = 0 , u t ( 0 , x , y ) = ϕ ( x , y ) , u(0,x,y)=0,\quad u_{t}(0,x,y)=\phi(x,y),\,
  70. u ( t , x , y ) = t M c t [ ϕ ] = t 4 π S ϕ ( x + c t α , y + c t β ) d ω , u(t,x,y)=tM_{ct}[\phi]=\frac{t}{4\pi}\iint_{S}\phi(x+ct\alpha,\,y+ct\beta)d% \omega,\,
  71. u ( t , x , y ) = 1 2 π c D ϕ ( x + ξ , y + η ) ( c t ) 2 - ξ 2 - η 2 d ξ d η . u(t,x,y)=\frac{1}{2\pi c}\iint_{D}\frac{\phi(x+\xi,y+\eta)}{\sqrt{(ct)^{2}-\xi% ^{2}-\eta^{2}}}d\xi\,d\eta.\,
  72. ( x - ξ ) 2 + ( y - η ) 2 = c 2 t 2 , (x-\xi)^{2}+(y-\eta)^{2}=c^{2}t^{2},\,
  73. γ n = 1 3 5 . . ( n - 2 ) \gamma_{n}=1\cdot 3\cdot 5\cdot..\cdot(n-2)
  74. u ( x , t ) = 1 γ n [ t ( 1 t t ) n - 3 2 ( t n - 2 B t ( x ) average g d S ) + ( 1 t t ) n - 3 2 ( t n - 2 B t ( x ) average h d S ) ] u(x,t)=\frac{1}{\gamma_{n}}\left[\partial_{t}\left(\frac{1}{t}\partial_{t}% \right)^{\frac{n-3}{2}}\left(t^{n-2}\int^{\,\text{average}}_{\partial B_{t}(x)% }gdS\right)+\left(\frac{1}{t}\partial_{t}\right)^{\frac{n-3}{2}}\left(t^{n-2}% \int^{\,\text{average}}_{\partial B_{t}(x)}hdS\right)\right]
  75. lim ( x , t ) ( x 0 , 0 ) u ( x , t ) = g ( x 0 ) lim ( x , t ) ( x 0 , 0 ) u t ( x , t ) = h ( x 0 ) \begin{aligned}\displaystyle\lim_{(x,t)\to(x^{0},0)}u(x,t)&\displaystyle=g(x^{% 0})\\ \displaystyle\lim_{(x,t)\to(x^{0},0)}u_{t}(x,t)&\displaystyle=h(x^{0})\end{aligned}
  76. γ n = 2 4 . . n \gamma_{n}=2\cdot 4\cdot..\cdot n
  77. u ( x , t ) = 1 γ n [ t ( 1 t t ) n - 2 2 ( t n B t ( x ) average g ( t 2 - | y - x | 2 ) 1 2 d y ) + ( 1 t t ) n - 2 2 ( t n B t ( x ) average h ( t 2 - | y - x | 2 ) 1 2 d y ) ] u(x,t)=\frac{1}{\gamma_{n}}\left[\partial_{t}\left(\frac{1}{t}\partial_{t}% \right)^{\frac{n-2}{2}}\left(t^{n}\int^{\,\text{average}}_{B_{t}(x)}\frac{g}{(% t^{2}-|y-x|^{2})^{\frac{1}{2}}}dy\right)+\left(\frac{1}{t}\partial_{t}\right)^% {\frac{n-2}{2}}\left(t^{n}\int^{\,\text{average}}_{B_{t}(x)}\frac{h}{(t^{2}-|y% -x|^{2})^{\frac{1}{2}}}dy\right)\right]
  78. lim ( x , t ) ( x 0 , 0 ) u ( x , t ) = g ( x 0 ) lim ( x , t ) ( x 0 , 0 ) u t ( x , t ) = h ( x 0 ) \begin{aligned}\displaystyle\lim_{(x,t)\to(x^{0},0)}u(x,t)&\displaystyle=g(x^{% 0})\\ \displaystyle\lim_{(x,t)\to(x^{0},0)}u_{t}(x,t)&\displaystyle=h(x^{0})\end{aligned}
  79. u x ( t , L ) + b u ( t , L ) = 0 , u_{x}(t,L)+bu(t,L)=0,\,
  80. u ( t , x ) = T ( t ) v ( x ) . u(t,x)=T(t)v(x).\,
  81. T ′′ c 2 T = v ′′ v = - λ . \frac{T^{\prime\prime}}{c^{2}T}=\frac{v^{\prime\prime}}{v}=-\lambda.\,
  82. v ′′ + λ v = 0 , v^{\prime\prime}+\lambda v=0,\,
  83. - v ( 0 ) + a v ( 0 ) = 0 , v ( L ) + b v ( L ) = 0. -v^{\prime}(0)+av(0)=0,\quad v^{\prime}(L)+bv(L)=0.\,
  84. u i + 1 - u i Δ x f \frac{u_{i+1}-u_{i}}{\Delta x}\ f
  85. u i - 1 - u i Δ x f \frac{u_{i-1}-u_{i}}{\Delta x}\ f
  86. u ¨ i = ( f m Δ x ) ( u i + 1 + u i - 1 - 2 u i ) \ddot{u}_{i}=\left(\frac{f}{m\ \Delta x}\right)\left(u_{i+1}+u_{i-1}\ -\ 2u_{i% }\right)
  87. ρ = m Δ x \rho=\frac{m}{\Delta x}
  88. u ¨ i = ( f ρ Δ x 2 ) ( u i + 1 + u i - 1 - 2 u i ) \ddot{u}_{i}=\left(\frac{f}{\rho\ {\Delta x}^{2}}\right)\left(u_{i+1}+u_{i-1}% \ -\ 2u_{i}\right)
  89. u ¨ i \ddot{u}_{i}
  90. 2 u t 2 \partial^{2}u\over\partial t^{2}
  91. u i + 1 + u i - 1 - 2 u i Δ x 2 2 u x 2 \frac{u_{i+1}+u_{i-1}\ -\ 2u_{i}}{{\Delta x}^{2}}\rightarrow\frac{\partial^{2}% u}{\partial x^{2}}
  92. u ( 0 , t ) = u ( L , t ) = 0 u(0,t)=u(L,t)=0
  93. u ¨ 1 = ( c Δ x ) 2 ( u 2 - 2 u 1 ) \ddot{u}_{1}={\left(\frac{c}{\Delta x}\right)}^{2}\left(u_{2}\ -\ 2u_{1}\right)
  94. u ¨ n = ( c Δ x ) 2 ( u n - 1 - 2 u n ) \ddot{u}_{n}={\left(\frac{c}{\Delta x}\right)}^{2}\left(u_{n-1}\ -\ 2u_{n}\right)
  95. c = f ρ c=\sqrt{\frac{f}{\rho}}
  96. L c k 0.05 k = 0 , , 5 \frac{L}{c}\ k\ 0.05\ \ k=0,\cdots,5
  97. u ˙ i = 0 \dot{u}_{i}=0
  98. L c 0.25 \frac{L}{c}\ 0.25
  99. c = f ρ c=\sqrt{\frac{f}{\rho}}
  100. L c k 0.05 k = 6 , , 11 \frac{L}{c}\ k\ 0.05\ \ k=6,\cdots,11
  101. c = f ρ c=\sqrt{\frac{f}{\rho}}
  102. L c k 0.05 k = 12 , , 17 \frac{L}{c}\ k\ 0.05\ \ k=12,\cdots,17
  103. L c k 0.05 k = 18 , , 23 \frac{L}{c}\ k\ 0.05\ \ k=18,\cdots,23
  104. L c k 0.05 k = 18 , , 20 \frac{L}{c}\ k\ 0.05\ \ k=18,\cdots,20
  105. L c k 0.05 k = 21 , , 23 \frac{L}{c}\ k\ 0.05\ \ k=21,\cdots,23
  106. L c k 0.05 k = 24 , , 29 \frac{L}{c}\ k\ 0.05\ \ k=24,\cdots,29
  107. L c k 0.05 k = 30 , , 35 \frac{L}{c}\ k\ 0.05\ \ k=30,\cdots,35
  108. u n + a u = 0 , \frac{\partial u}{\partial n}+au=0,\,
  109. u ( 0 , x ) = f ( x ) , u t ( 0 , x ) = g ( x ) , u(0,x)=f(x),\quad u_{t}(0,x)=g(x),\,
  110. v + λ v = 0 , \nabla\cdot\nabla v+\lambda v=0,\,
  111. v n + a v = 0 , \frac{\partial v}{\partial n}+av=0,\,
  112. c 2 u x x ( x , t ) - u t t ( x , t ) = s ( x , t ) c^{2}u_{xx}(x,t)-u_{tt}(x,t)=s(x,t)\,
  113. u ( x , 0 ) = f ( x ) u(x,0)=f(x)\,
  114. u t ( x , 0 ) = g ( x ) u_{t}(x,0)=g(x)\,
  115. R C ( c 2 u x x ( x , t ) - u t t ( x , t ) ) d x d t = R C s ( x , t ) d x d t . \iint\limits_{R_{C}}\left(c^{2}u_{xx}(x,t)-u_{tt}(x,t)\right)dxdt=\iint\limits% _{R_{C}}s(x,t)dxdt.
  116. L 0 + L 1 + L 2 ( - c 2 u x ( x , t ) d t - u t ( x , t ) d x ) = R C s ( x , t ) d x d t . \int_{L_{0}+L_{1}+L_{2}}\left(-c^{2}u_{x}(x,t)dt-u_{t}(x,t)dx\right)=\iint% \limits_{R_{C}}s(x,t)dxdt.
  117. x i - c t i x i + c t i - u t ( x , 0 ) d x = - x i - c t i x i + c t i g ( x ) d x . \int^{x_{i}+ct_{i}}_{x_{i}-ct_{i}}-u_{t}(x,0)dx=-\int^{x_{i}+ct_{i}}_{x_{i}-ct% _{i}}g(x)dx.
  118. L 1 ( - c 2 u x ( x , t ) d t - u t ( x , t ) d x ) \displaystyle\int_{L_{1}}\left(-c^{2}u_{x}(x,t)dt-u_{t}(x,t)dx\right)
  119. L 2 ( - c 2 u x ( x , t ) d t - u t ( x , t ) d x ) \displaystyle\int_{L_{2}}\left(-c^{2}u_{x}(x,t)dt-u_{t}(x,t)dx\right)
  120. R C s ( x , t ) d x d t \displaystyle\iint_{R_{C}}s(x,t)dxdt
  121. u ( x i , t i ) = f ( x i + c t i ) + f ( x i - c t i ) 2 + 1 2 c x i - c t i x i + c t i g ( x ) d x + 1 2 c 0 t i x i - c ( t i - t ) x i + c ( t i - t ) s ( x , t ) d x d t . u(x_{i},t_{i})=\frac{f(x_{i}+ct_{i})+f(x_{i}-ct_{i})}{2}+\frac{1}{2c}\int^{x_{% i}+ct_{i}}_{x_{i}-ct_{i}}g(x)dx+\frac{1}{2c}\int^{t_{i}}_{0}\int^{x_{i}+c\left% (t_{i}-t\right)}_{x_{i}-c\left(t_{i}-t\right)}s(x,t)dxdt.
  122. ρ u ¨ = f + ( λ + 2 μ ) ( u ) - μ × ( × u ) \rho{\ddot{{u}}}={f}+(\lambda+2\mu)\nabla(\nabla\cdot{u})-\mu\nabla\times(% \nabla\times{u})
  123. ω = ω ( k ) , \omega=\omega({k}),
  124. v p = ω ( k ) k . v_{\mathrm{p}}=\frac{\omega(k)}{k}.
  125. u ( 0 , x ) = u 0 ( 1 - ( x - x 1 x 1 ) 2 ) u(0,x)=u_{0}\ \left(1-\left(\frac{x-x_{1}}{x_{1}}\right)^{2}\right)
  126. 0 x x 2 0\leq x\leq x_{2}
  127. u ( 0 , x ) = u 0 ( x - x 3 x 1 ) 2 u(0,x)=u_{0}\ \left({\frac{x-x_{3}}{x_{1}}}\right)^{2}
  128. x 2 x x 3 x_{2}\leq x\leq x_{3}
  129. u ( 0 , x ) = 0 u(0,x)=0
  130. x 3 x L x_{3}\leq x\leq L
  131. x 1 = 1 10 L , x 2 = x 1 + 1 2 x 1 , x 3 = x 2 + 1 2 x 1 x_{1}=\frac{1}{10}\ L\ ,\ x_{2}=x_{1}+\sqrt{\frac{1}{2}}\ x_{1}\ ,\ x_{3}=x_{2% }+\sqrt{\frac{1}{2}}\ x_{1}

Wave_impedance.html

  1. Z = E 0 - ( x ) H 0 - ( x ) Z={E_{0}^{-}(x)\over H_{0}^{-}(x)}
  2. E 0 - ( x ) E_{0}^{-}(x)
  3. H 0 - ( x ) H_{0}^{-}(x)
  4. Z = j ω μ σ + j ω ε Z=\sqrt{j\omega\mu\over\sigma+j\omega\varepsilon}
  5. Z = μ ε . Z=\sqrt{\mu\over\varepsilon}.
  6. Z 0 = μ 0 ε 0 Z_{0}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}
  7. c 0 = 1 μ 0 ε 0 = 299 , 792 , 458 c_{0}=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}=299,792,458
  8. c 0 c_{0}
  9. Z 0 = μ 0 c 0 = 4 π × 10 - 7 H / m × 299 , 792 , 458 m / s = 376.730313 Ω Z_{0}=\mu_{0}c_{0}=4\pi\times 10^{-7}H/m\times 299,792,458m/s=376.730313\Omega
  10. μ = μ 0 = 4 π × 10 - 7 \mu=\mu_{0}=4\pi\times 10^{-7}
  11. ε = ε r × 8.854 × 10 - 12 \varepsilon=\varepsilon_{r}\times 8.854\times 10^{-12}
  12. Z = μ ε = μ 0 ε 0 ε r = Z 0 ε r 377 ε r Ω Z=\sqrt{\mu\over\varepsilon}=\sqrt{\mu_{0}\over\varepsilon_{0}\varepsilon_{r}}% ={Z_{0}\over\sqrt{\varepsilon}_{r}}\approx{377\over\sqrt{\varepsilon_{r}}}\,\Omega
  13. f f
  14. Z = Z 0 1 - ( f c f ) 2 (TE modes) , Z=\frac{Z_{0}}{\sqrt{1-\left(\frac{f_{c}}{f}\right)^{2}}}\qquad\mbox{(TE modes% )}~{},
  15. Z = Z 0 1 - ( f c f ) 2 (TM modes) Z=Z_{0}\sqrt{1-\left(\frac{f_{c}}{f}\right)^{2}}\qquad\mbox{(TM modes)}~{}

Waveguide.html

  1. Ω \Omega
  2. Γ = Z 2 / Z 1 - 1 Z 2 / Z 1 + 1 \Gamma=\frac{Z_{2}/Z_{1}-1}{Z_{2}/Z_{1}+1}
  3. Γ \Gamma
  4. Z 1 Z_{1}
  5. Z 2 Z_{2}
  6. V S W R = | V | m a x | V | m i n = 1 + | Γ | 1 - | Γ | VSWR=\frac{|V|_{max}}{|V|_{min}}=\frac{1+|\Gamma|}{1-|\Gamma|}
  7. | V | m i n , m a x \left|V\right|_{min,max}

Wavelength.html

  1. λ = v f , \lambda=\frac{v}{f}\,\,,
  2. y ( x , t ) = A cos ( 2 π ( x λ - f t ) ) = A cos ( 2 π λ ( x - v t ) ) y(x,\ t)=A\cos\left(2\pi\left(\frac{x}{\lambda}-ft\right)\right)=A\cos\left(% \frac{2\pi}{\lambda}(x-vt)\right)
  3. y ( x , t ) = A cos ( k x - ω t ) = A cos ( k ( x - v t ) ) y(x,\ t)=A\cos\left(kx-\omega t\right)=A\cos\left(k(x-vt)\right)
  4. k = 2 π λ = 2 π f v = ω v , k=\frac{2\pi}{\lambda}=\frac{2\pi f}{v}=\frac{\omega}{v},
  5. λ = 2 π k = 2 π v ω = v f . \lambda=\frac{2\pi}{k}=\frac{2\pi v}{\omega}=\frac{v}{f}.
  6. A e i ( k x - ω t ) . Ae^{i\left(kx-\omega t\right)}.
  7. v = c n ( λ 0 ) , v=\frac{c}{n(\lambda_{0})},
  8. λ = λ 0 n ( λ 0 ) . \lambda=\frac{\lambda_{0}}{n(\lambda_{0})}.
  9. d sin θ = m λ , d\sin\theta=m\lambda\ ,
  10. d sin θ = ( m + 1 / 2 ) λ . d\sin\theta=(m+1/2)\lambda\ .
  11. I q = I 1 sin 2 ( q π g sin α λ ) / sin 2 ( π g sin α λ ) , I_{q}=I_{1}\sin^{2}\left(\frac{q\pi g\sin\alpha}{\lambda}\right)/\sin^{2}\left% (\frac{\pi g\sin\alpha}{\lambda}\right)\ ,
  12. S ( u ) = sinc 2 ( u ) = ( sin π u π u ) 2 ; S(u)=\mathrm{sinc}^{2}(u)=\left(\frac{\sin\pi u}{\pi u}\right)^{2}\ ;
  13. u = x L λ R , u=\frac{xL}{\lambda R}\ ,
  14. r A i r y = 1.22 λ 2 N A , r_{Airy}=1.22\frac{\lambda}{2\mathrm{NA}}\ ,
  15. NA = n sin θ \mathrm{NA}=n\sin\theta\;
  16. δ = 1.22 λ D , \delta=1.22\frac{\lambda}{D}\ ,

Wavelet.html

  1. ψ ( t ) = 2 sinc ( 2 t ) - sinc ( t ) = sin ( 2 π t ) - sin ( π t ) π t \psi(t)=2\,\operatorname{sinc}(2t)-\,\operatorname{sinc}(t)=\frac{\sin(2\pi t)% -\sin(\pi t)}{\pi t}
  2. ψ a , b ( t ) = 1 a ψ ( t - b a ) , \psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right),
  3. x a ( t ) = \R W T ψ { x } ( a , b ) ψ a , b ( t ) d b x_{a}(t)=\int_{\R}WT_{\psi}\{x\}(a,b)\cdot\psi_{a,b}(t)\,db
  4. W T ψ { x } ( a , b ) = x , ψ a , b = \R x ( t ) ψ a , b ( t ) d t . WT_{\psi}\{x\}(a,b)=\langle x,\psi_{a,b}\rangle=\int_{\R}x(t){\psi_{a,b}(t)}\,dt.
  5. ψ m , n ( t ) = a - m / 2 ψ ( a - m t - n b ) . \psi_{m,n}(t)=a^{-m/2}\psi(a^{-m}t-nb).\,
  6. x ( t ) = m \Z n \Z x , ψ m , n ψ m , n ( t ) x(t)=\sum_{m\in\Z}\sum_{n\in\Z}\langle x,\,\psi_{m,n}\rangle\cdot\psi_{m,n}(t)
  7. { ψ m , n : m , n \Z } \{\psi_{m,n}:m,n\in\Z\}
  8. V m = span ( ϕ m , n : n \Z ) , where ϕ m , n ( t ) = 2 - m / 2 ϕ ( 2 - m t - n ) V_{m}=\operatorname{span}(\phi_{m,n}:n\in\Z),\,\text{ where }\phi_{m,n}(t)=2^{% -m/2}\phi(2^{-m}t-n)
  9. W m = span ( ψ m , n : n \Z ) , where ψ m , n ( t ) = 2 - m / 2 ψ ( 2 - m t - n ) . W_{m}=\operatorname{span}(\psi_{m,n}:n\in\Z),\,\text{ where }\psi_{m,n}(t)=2^{% -m/2}\psi(2^{-m}t-n).
  10. V i V_{i}
  11. W i W_{i}
  12. { 0 } V 1 V 0 V - 1 V - 2 L 2 ( \R ) \{0\}\subset\dots\subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset% \dots\subset L^{2}(\R)
  13. , W 1 , W 0 , W - 1 , \dots,W_{1},W_{0},W_{-1},\dots\dots
  14. V m W m = V m - 1 . V_{m}\oplus W_{m}=V_{m-1}.
  15. V 0 W 0 = V - 1 V_{0}\oplus W_{0}=V_{-1}
  16. h = { h n } n \Z h=\{h_{n}\}_{n\in\Z}
  17. g = { g n } n \Z g=\{g_{n}\}_{n\in\Z}
  18. g n = ϕ 0 , 0 , ϕ - 1 , n g_{n}=\langle\phi_{0,0},\,\phi_{-1,n}\rangle
  19. ϕ ( t ) = 2 n \Z g n ϕ ( 2 t - n ) , \phi(t)=\sqrt{2}\sum_{n\in\Z}g_{n}\phi(2t-n),
  20. h n = ψ 0 , 0 , ϕ - 1 , n h_{n}=\langle\psi_{0,0},\,\phi_{-1,n}\rangle
  21. ψ ( t ) = 2 n \Z h n ϕ ( 2 t - n ) . \psi(t)=\sqrt{2}\sum_{n\in\Z}h_{n}\phi(2t-n).
  22. L 2 = V j 0 W j 0 W j 0 - 1 W j 0 - 2 W j 0 - 3 L^{2}=V_{j_{0}}\oplus W_{j_{0}}\oplus W_{j_{0}-1}\oplus W_{j_{0}-2}\oplus W_{j% _{0}-3}\oplus\dots
  23. S L 2 S\in L^{2}
  24. S = k c j 0 , k ϕ j 0 , k + j j 0 k d j , k ψ j , k S=\sum_{k}c_{j_{0},k}\phi_{j_{0},k}+\sum_{j\leq j_{0}}\sum_{k}d_{j,k}\psi_{j,k}
  25. c j 0 , k = S , ϕ j 0 , k c_{j_{0},k}=\langle S,\phi_{j_{0},k}\rangle
  26. d j , k = S , ψ j , k d_{j,k}=\langle S,\psi_{j,k}\rangle
  27. L 1 ( \R ) L 2 ( \R ) . L^{1}(\R)\cap L^{2}(\R).
  28. - | ψ ( t ) | d t < \int_{-\infty}^{\infty}|\psi(t)|\,dt<\infty
  29. - | ψ ( t ) | 2 d t < . \int_{-\infty}^{\infty}|\psi(t)|^{2}\,dt<\infty.
  30. - ψ ( t ) d t = 0 \int_{-\infty}^{\infty}\psi(t)\,dt=0
  31. - | ψ ( t ) | 2 d t = 1 \int_{-\infty}^{\infty}|\psi(t)|^{2}\,dt=1
  32. ψ a , b ( t ) = 1 a ψ ( t - b a ) . \psi_{a,b}(t)={1\over\sqrt{a}}\psi\left({t-b\over a}\right).
  33. 1 a - φ a 1 , b 1 ( t ) φ ( t - b a ) d t \frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}\varphi_{a1,b1}(t)\varphi\left(\frac{% t-b}{a}\right)\,dt
  34. Ψ ( t ) \Psi(t)
  35. ψ ( t ) = e - 2 π i t \psi(t)=e^{-2\pi it}
  36. ψ ( t ) = g ( t - u ) e - 2 π i t \psi(t)=g(t-u)e^{-2\pi it}
  37. g ( t - u ) g(t-u)
  38. rect ( t - u Δ t ) \rm{rect}\left(\frac{t-u}{\Delta_{t}}\right)
  39. Δ t \Delta_{t}
  40. E = - | ψ ( t ) | 2 d t E=\int_{-\infty}^{\infty}|\psi(t)|^{2}\,dt
  41. 1 2 π - | ψ ( ω ) ^ | 2 d ω \frac{1}{2\pi}\int_{-\infty}^{\infty}|\hat{\psi(\omega)}|^{2}\,d\omega
  42. σ t 2 = 1 E | t - u | 2 | ψ ( t ) | 2 d t \sigma_{t}^{2}=\frac{1}{E}\int|t-u|^{2}|\psi(t)|^{2}\,dt
  43. ξ \xi
  44. σ ω 2 = 1 2 π E | ω - ξ | 2 | ψ ^ ( ω ) | 2 d ω \sigma_{\omega}^{2}=\frac{1}{2\pi E}\int|\omega-\xi|^{2}|\hat{\psi}(\omega)|^{% 2}\,d\omega
  45. σ t 2 σ ω 2 1 / 4 \sigma_{t}^{2}\sigma_{\omega}^{2}\geq 1/4
  46. Δ t \Delta_{t}
  47. sinc ( Δ t ω ) \rm{sinc(\Delta_{t}\omega)}
  48. Δ t \Delta_{t}\rightarrow\infty
  49. x ( t ) x(t)
  50. x = s + v x=s+v
  51. v 𝒩 ( 0 , σ 2 I ) v\ \sim\ \mathcal{N}(0,\,\sigma^{2}I)
  52. y = W T x = W T s + W T v = p + z y=W^{T}x=W^{T}s+W^{T}v=p+z
  53. z 𝒩 ( 0 , σ 2 I ) z\ \sim\ \ \mathcal{N}(0,\,\sigma^{2}I)
  54. p a 𝒩 ( 0 , σ 1 2 ) + ( 1 - a ) 𝒩 ( 0 , σ 2 2 ) p\ \sim\ a\mathcal{N}(0,\,\sigma_{1}^{2})+(1-a)\mathcal{N}(0,\,\sigma_{2}^{2})
  55. σ 1 2 \sigma_{1}^{2}
  56. σ 2 2 \sigma_{2}^{2}
  57. p ~ = E ( p / y ) = τ ( y ) y \tilde{p}=E(p/y)=\tau(y)y
  58. τ ( y ) \tau(y)
  59. σ 1 2 \sigma_{1}^{2}
  60. σ 2 2 \sigma_{2}^{2}
  61. s ~ = W p ~ \tilde{s}=W\tilde{p}

Waveplate.html

  1. Γ = 2 π Δ n L λ 0 , \Gamma=\frac{2\pi\,\Delta n\,L}{\lambda_{0}},
  2. 𝐩 ^ \mathbf{\hat{p}}
  3. 𝐩 ^ \mathbf{\hat{p}}
  4. 𝐟 ^ \mathbf{\hat{f}}
  5. 𝐟 ^ \mathbf{\hat{f}}
  6. 𝐄 e i ( k z - ω t ) = E 𝐩 ^ e i ( k z - ω t ) = E ( cos θ 𝐟 ^ + sin θ 𝐬 ^ ) e i ( k z - ω t ) , \mathbf{E}\,\mathrm{e}^{i(kz-\omega t)}=E\,\mathbf{\hat{p}}\,\mathrm{e}^{i(kz-% \omega t)}=E(\cos\theta\,\mathbf{\hat{f}}+\sin\theta\,\mathbf{\hat{s}})\mathrm% {e}^{i(kz-\omega t)},
  7. 𝐬 ^ \mathbf{\hat{s}}
  8. E ( cos θ 𝐟 ^ - sin θ 𝐬 ^ ) e i ( k z - ω t ) = E [ cos ( - θ ) 𝐟 ^ + sin ( - θ ) 𝐬 ^ ] e i ( k z - ω t ) . E(\cos\theta\,\mathbf{\hat{f}}-\sin\theta\,\mathbf{\hat{s}})\mathrm{e}^{i(kz-% \omega t)}=E[\cos(-\theta)\mathbf{\hat{f}}+\sin(-\theta)\mathbf{\hat{s}}]% \mathrm{e}^{i(kz-\omega t)}.
  9. 𝐩 ^ \mathbf{\hat{p}}^{\prime}
  10. 𝐩 ^ \mathbf{\hat{p}}^{\prime}
  11. 𝐟 ^ \mathbf{\hat{f}}
  12. 𝐟 ^ \mathbf{\hat{f}}
  13. 𝐳 ^ \mathbf{\hat{z}}
  14. ( E f 𝐟 ^ + E s 𝐬 ^ ) e i ( k z - ω t ) , (E_{f}\mathbf{\hat{f}}+E_{s}\mathbf{\hat{s}})\mathrm{e}^{i(kz-\omega t)},
  15. ( E f 𝐟 ^ + i E s 𝐬 ^ ) e i ( k z - ω t ) . (E_{f}\mathbf{\hat{f}}+iE_{s}\mathbf{\hat{s}})\mathrm{e}^{i(kz-\omega t)}.
  16. E ( 𝐟 ^ + i 𝐬 ^ ) e i ( k z - ω t ) , E(\mathbf{\hat{f}}+i\mathbf{\hat{s}})\mathrm{e}^{i(kz-\omega t)},

Wave–particle_duality.html

  1. E = h f E=hf\,
  2. λ = h p \lambda=\frac{h}{p}
  3. E c \tfrac{E}{c}
  4. c f \tfrac{c}{f}
  5. Δ x Δ p 2 \Delta x\Delta p\geq\frac{\hbar}{2}
  6. Δ \Delta
  7. \hbar
  8. π \pi

Weak_interaction.html

  1. e - e^{-}\,
  2. - 1 / 2 -1/2\,
  3. μ - \mu^{-}\,
  4. - 1 / 2 -1/2\,
  5. τ - \tau^{-}\,
  6. - 1 / 2 -1/2\,
  7. ν e \nu_{e}\,
  8. + 1 / 2 +1/2\,
  9. ν μ \nu_{\mu}\,
  10. + 1 / 2 +1/2\,
  11. ν τ \nu_{\tau}\,
  12. + 1 / 2 +1/2\,
  13. u u\,
  14. + 1 / 2 +1/2\,
  15. c c\,
  16. + 1 / 2 +1/2\,
  17. t t\,
  18. + 1 / 2 +1/2\,
  19. d d\,
  20. - 1 / 2 -1/2\,
  21. s s\,
  22. - 1 / 2 -1/2\,
  23. b b\,
  24. - 1 / 2 -1/2\,
  25. 1 / 2 {1}/{2}
  26. 1 / 2 {1}/{2}
  27. 1 / 2 {1}/{2}
  28. 1 / 2 {1}/{2}
  29. 1 / 2 {1}/{2}
  30. 1 / 2 {1}/{2}
  31. Y W = 2 ( Q - T 3 ) \qquad Y_{W}=2(Q-T_{3})
  32. μ - + W + ν μ \mu^{-}+W^{+}\to\nu_{\mu}
  33. 1 / 3 {1}/{3}
  34. 2 / 3 {2}/{3}
  35. d \displaystyle d
  36. W - \displaystyle W^{-}
  37. d u + e - + ν ¯ e d\to u+e^{-}+\bar{\nu}_{e}~{}
  38. e - e - + Z 0 e^{-}\to e^{-}+Z^{0}
  39. Z 0 b + b ¯ Z^{0}\to b+\bar{b}

Weak_topology.html

  1. | | |\cdot|
  2. x f = def | f ( x ) | \|x\|_{f}\overset{\,\text{def}}{=}|f(x)|
  3. ϕ ( x n ) ϕ ( x ) \phi(x_{n})\to\phi(x)
  4. x n w x x_{n}\overset{\mathrm{w}}{\longrightarrow}x
  5. x n x . x_{n}\rightharpoonup x.
  6. x T x x\mapsto T_{x}
  7. T x ( ϕ ) = ϕ ( x ) . T_{x}(\phi)=\phi(x).
  8. ϕ λ ( x ) ϕ ( x ) \phi_{\lambda}(x)\rightarrow\phi(x)
  9. ϕ n ( x ) ϕ ( x ) \phi_{n}(x)\to\phi(x)
  10. ϕ n w * ϕ \phi_{n}\overset{w^{*}}{\rightarrow}\phi
  11. 𝐑 n | ψ k - ψ | 2 d μ 0 \int_{\mathbf{R}^{n}}|\psi_{k}-\psi|^{2}\,{\rm d}\mu\,\to 0\,
  12. 𝐑 n ψ ¯ k f d μ 𝐑 n ψ ¯ f d μ \int_{\mathbf{R}^{n}}\bar{\psi}_{k}f\,\mathrm{d}\mu\to\int_{\mathbf{R}^{n}}% \bar{\psi}f\,\mathrm{d}\mu
  13. ψ k ( x ) = 2 / π sin ( k x ) \psi_{k}(x)=\sqrt{2/\pi}\sin(kx)
  14. f f ( x ) Y . f\mapsto\|f(x)\|_{Y}.
  15. p q , x : f q ( f ( x ) ) , p_{q,x}:f\mapsto q(f(x)),

Weakly_interacting_massive_particles.html

  1. σ v 3 × 10 - 26 cm 3 s - 1 \langle\sigma v\rangle\simeq 3\times 10^{-26}\mathrm{cm}^{3}\;\mathrm{s}^{-1}

Web_crawler.html

  1. F p ( t ) = { 1 if p is equal to the local copy at time t 0 otherwise F_{p}(t)=\begin{cases}1&{\rm if}~{}p~{}{\rm~{}is~{}equal~{}to~{}the~{}local~{}% copy~{}at~{}time}~{}t\\ 0&{\rm otherwise}\end{cases}
  2. A p ( t ) = { 0 if p is not modified at time t t - modification time of p otherwise A_{p}(t)=\begin{cases}0&{\rm if}~{}p~{}{\rm~{}is~{}not~{}modified~{}at~{}time}% ~{}t\\ t-{\rm modification~{}time~{}of}~{}p&{\rm otherwise}\end{cases}

Weighted_arithmetic_mean.html

  1. x ¯ = 4300 50 = 86. \bar{x}=\frac{4300}{50}=86.
  2. x ¯ = ( 20 × 80 ) + ( 30 × 90 ) 20 + 30 = 86. \bar{x}=\frac{(20\times 80)+(30\times 90)}{20+30}=86.
  3. 20 20 + 30 = 0.4 \frac{20}{20+30}=0.4\,
  4. 30 20 + 30 = 0.6 \frac{30}{20+30}=0.6\,
  5. x ¯ = ( 0.4 × 80 ) + ( 0.6 × 90 ) = 86. \bar{x}=(0.4\times 80)+(0.6\times 90)=86.
  6. { x 1 , x 2 , , x n } , \{x_{1},x_{2},\dots,x_{n}\},
  7. x ¯ = i = 1 n w i x i i = 1 n w i , \bar{x}=\frac{\sum_{i=1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}},
  8. x ¯ = w 1 x 1 + w 2 x 2 + + w n x n w 1 + w 2 + + w n . \bar{x}=\frac{w_{1}x_{1}+w_{2}x_{2}+\cdots+w_{n}x_{n}}{w_{1}+w_{2}+\cdots+w_{n% }}.
  9. 1 1
  10. i = 1 n w i = 1 \sum_{i=1}^{n}{w_{i}}=1
  11. x ¯ = i = 1 n w i x i \bar{x}=\sum_{i=1}^{n}{w_{i}x_{i}}
  12. w i = w i j = 1 n w j w_{i}^{\prime}=\frac{w_{i}}{\sum_{j=1}^{n}{w_{j}}}
  13. x ¯ = i = 1 n w i x i = i = 1 n w i j = 1 n w j x i = i = 1 n w i x i j = 1 n w j = i = 1 n w i x i i = 1 n w i . \begin{aligned}\displaystyle\bar{x}&\displaystyle=\sum_{i=1}^{n}w^{\prime}_{i}% x_{i}=\sum_{i=1}^{n}\frac{w_{i}}{\sum_{j=1}^{n}w_{j}}x_{i}=\frac{\sum_{i=1}^{n% }w_{i}x_{i}}{\sum_{j=1}^{n}w_{j}}\\ &\displaystyle=\frac{\sum_{i=1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}}.\end{aligned}
  14. 1 n i = 1 n x i \frac{1}{n}\sum_{i=1}^{n}{x_{i}}
  15. w i = w w_{i}=w
  16. w i = 1 n . w_{i}^{\prime}=\frac{1}{n}.
  17. X ¯ \bar{X}
  18. E ( X i ) = μ i ¯ , E(X_{i})=\bar{\mu_{i}},
  19. E ( X ¯ ) = i = 1 n w i μ i . E(\bar{X})=\sum_{i=1}^{n}{w_{i}\mu_{i}}.
  20. μ i = μ \mu_{i}=\mu
  21. E ( X ¯ ) = μ . E(\bar{X})=\mu.\,
  22. σ i 2 \sigma^{2}_{i}
  23. σ X ¯ 2 = i = 1 n w i 2 σ i 2 . \sigma^{2}_{\bar{X}}=\sum_{i=1}^{n}{w_{i}^{2}\sigma^{2}_{i}}.
  24. σ i 2 = σ 0 2 \sigma^{2}_{i}=\sigma^{2}_{0}
  25. σ X ¯ 2 = σ 0 2 i = 1 n w i 2 , \sigma^{2}_{\bar{X}}=\sigma^{2}_{0}\sum_{i=1}^{n}{w_{i}^{2}},
  26. 1 / n i = 1 n w i 2 1 1/n\leq\sum_{i=1}^{n}{w_{i}^{2}}\leq 1
  27. σ X ¯ = σ / n \sigma_{\bar{X}}=\sigma/\sqrt{n}
  28. w i w_{i}
  29. w i = w i i = 1 n w i w_{i}^{\prime}=\frac{w_{i}}{\sum_{i=1}^{n}{w_{i}}}
  30. x i x_{i}\,\!
  31. σ i 2 \sigma_{i}^{2}\,
  32. w i = 1 σ i 2 . w_{i}=\frac{1}{\sigma_{i}^{2}}.
  33. x ¯ = i = 1 n ( x i σ i - 2 ) i = 1 n σ i - 2 , \bar{x}=\frac{\sum_{i=1}^{n}\left(x_{i}\sigma_{i}^{-2}\right)}{\sum_{i=1}^{n}% \sigma_{i}^{-2}},
  34. σ x ¯ 2 = 1 i = 1 n σ i - 2 , \sigma_{\bar{x}}^{2}=\frac{1}{\sum_{i=1}^{n}\sigma_{i}^{-2}},
  35. σ x ¯ 2 = σ 0 2 / n \sigma_{\bar{x}}^{2}=\sigma_{0}^{2}/n
  36. σ i = σ 0 \sigma_{i}=\sigma_{0}
  37. x ¯ = σ x ¯ 2 i = 1 n x i / σ i 2 . \bar{x}=\sigma_{\bar{x}}^{2}\sum_{i=1}^{n}x_{i}/\sigma_{i}^{2}.
  38. χ 2 \chi^{2}
  39. σ x ¯ 2 σ x ¯ 2 χ ν 2 \sigma_{\bar{x}}^{2}\rightarrow\sigma_{\bar{x}}^{2}\chi^{2}_{\nu}\,
  40. χ ν 2 \chi^{2}_{\nu}
  41. χ 2 \chi^{2}
  42. σ x ¯ 2 = 1 i = 1 n σ i - 2 × 1 ( n - 1 ) i = 1 n ( x i - x ¯ ) 2 σ i 2 ; \sigma_{\bar{x}}^{2}=\frac{1}{\sum_{i=1}^{n}\sigma_{i}^{-2}}\times\frac{1}{(n-% 1)}\sum_{i=1}^{n}\frac{(x_{i}-\bar{x})^{2}}{\sigma_{i}^{2}};
  43. σ i = σ 0 \sigma_{i}=\sigma_{0}
  44. σ x ¯ 2 \sigma_{\bar{x}}^{2}
  45. σ x ¯ 2 = σ 2 / n \sigma_{\bar{x}}^{2}=\sigma^{2}/n
  46. σ 2 = i = 1 n ( x i - x ¯ ) 2 / ( n - 1 ) \sigma^{2}=\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}/(n-1)
  47. μ * \mu^{*}
  48. σ ^ 2 \displaystyle\hat{\sigma}^{2}
  49. V 1 = i = 1 n w i V_{1}=\sum_{i=1}^{n}w_{i}
  50. σ ^ weighted 2 \hat{\sigma}^{2}_{\mathrm{weighted}}
  51. σ 2 \sigma^{2}
  52. s 2 \displaystyle s^{2}
  53. { 2 , 2 , 4 , 5 , 5 , 5 } \{2,2,4,5,5,5\}
  54. { 2 , 4 , 5 } \{2,4,5\}
  55. { 2 , 1 , 3 } \{2,1,3\}
  56. E [ σ ^ 2 ] = i = 1 N E [ ( x i - μ ) 2 ] N = E [ ( X - E [ X ] ) 2 ] - 1 N E [ ( X - E [ X ] ) 2 ] = ( N - 1 N ) σ actual 2 E [ σ ^ weighted 2 ] = i = 1 N w i E [ ( x i - μ * ) 2 ] V 1 = E [ ( X - E [ X ] ) 2 ] - V 2 V 1 2 E [ ( X - E [ X ] ) 2 ] = ( 1 - V 2 V 1 2 ) σ actual 2 \begin{aligned}\displaystyle\mathrm{E}\left[\hat{\sigma}^{2}\right]&% \displaystyle=\frac{\sum_{i=1}^{N}{\mathrm{E}\left[\left(x_{i}-\mu\right)^{2}% \right]}}{N}\\ &\displaystyle=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)^{2}% \right]-\frac{1}{N}\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)^{2}% \right]\\ &\displaystyle=\left(\frac{N-1}{N}\right)\sigma_{\mathrm{actual}}^{2}\\ \displaystyle\mathrm{E}\left[\hat{\sigma}^{2}_{\mathrm{weighted}}\right]&% \displaystyle=\frac{\sum_{i=1}^{N}w_{i}\mathrm{E}\left[\left(x_{i}-\mu^{*}% \right)^{2}\right]}{V_{1}}\\ &\displaystyle=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)^{2}% \right]-\frac{V_{2}}{V_{1}^{2}}\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right% ]\right)^{2}\right]\\ &\displaystyle=\left(1-\frac{V_{2}}{V_{1}^{2}}\right)\sigma_{\mathrm{actual}}^% {2}\end{aligned}
  57. V 2 = i = 1 n w i 2 V_{2}=\sum_{i=1}^{n}w_{i}^{2}
  58. ( 1 - V 2 V 1 2 ) \left(1-\frac{V_{2}}{V_{1}^{2}}\right)
  59. ( N - 1 N ) \left(\frac{N-1}{N}\right)
  60. 1 - ( V 2 / V 1 2 ) 1-\left(V_{2}/V_{1}^{2}\right)
  61. s 2 \displaystyle s^{2}
  62. E [ s 2 ] = σ actual 2 \mathrm{E}\left[s^{2}\right]=\sigma_{\mathrm{actual}}^{2}
  63. 𝐱 i \textstyle\,\textbf{x}_{i}
  64. w i 0 \textstyle w_{i}\geq 0
  65. i = 1 N w i = 1. \sum_{i=1}^{N}w_{i}=1.
  66. w i = w i i = 1 N w i w_{i}^{\prime}=\frac{w_{i}}{\sum_{i=1}^{N}w_{i}}
  67. μ * \textstyle\mathbf{\mu^{*}}
  68. μ * = i = 1 N w i 𝐱 i . \mathbf{\mu^{*}}=\sum_{i=1}^{N}w_{i}\mathbf{x}_{i}.
  69. μ * = i = 1 N w i 𝐱 i i = 1 N w i . \mathbf{\mu^{*}}=\frac{\sum_{i=1}^{N}w_{i}\mathbf{x}_{i}}{\sum_{i=1}^{N}w_{i}}.
  70. 𝚺 \textstyle\mathbf{\Sigma}
  71. Σ = i = 1 N w i ( i = 1 N w i ) 2 - i = 1 N w i 2 i = 1 N w i ( 𝐱 i - μ * ) T ( 𝐱 i - μ * ) . \Sigma=\frac{\sum_{i=1}^{N}w_{i}}{\left(\sum_{i=1}^{N}w_{i}\right)^{2}-\sum_{i% =1}^{N}w_{i}^{2}}\sum_{i=1}^{N}w_{i}\left(\mathbf{x}_{i}-\mu^{*}\right)^{T}% \left(\mathbf{x}_{i}-\mu^{*}\right).
  72. w i / V 1 = 1 / N \textstyle w_{i}/V_{1}=1/N
  73. σ 2 \sigma^{2}
  74. Σ \Sigma
  75. W i = Σ i - 1 . \,\text{W}_{i}=\Sigma_{i}^{-1}.
  76. 𝐱 ¯ = Σ 𝐱 ¯ ( i = 1 n W i 𝐱 i ) , \bar{\mathbf{x}}=\Sigma_{\bar{\mathbf{x}}}\left(\sum_{i=1}^{n}\,\text{W}_{i}% \mathbf{x}_{i}\right),
  77. Σ 𝐱 ¯ = ( i = 1 n W i ) - 1 , \Sigma_{\bar{\mathbf{x}}}=\left(\sum_{i=1}^{n}\,\text{W}_{i}\right)^{-1},
  78. 𝐱 1 := [ 10 ] , Σ 1 := [ 1 0 0 100 ] \mathbf{x}_{1}:=[10]^{\top},\qquad\Sigma_{1}:=\begin{bmatrix}1&0\\ 0&100\end{bmatrix}
  79. 𝐱 2 := [ 01 ] , Σ 2 := [ 100 0 0 1 ] \mathbf{x}_{2}:=[01]^{\top},\qquad\Sigma_{2}:=\begin{bmatrix}100&0\\ 0&1\end{bmatrix}
  80. 𝐱 ¯ = ( Σ 1 - 1 + Σ 2 - 1 ) - 1 ( Σ 1 - 1 𝐱 1 + Σ 2 - 1 𝐱 2 ) \bar{\mathbf{x}}=\left(\Sigma_{1}^{-1}+\Sigma_{2}^{-1}\right)^{-1}\left(\Sigma% _{1}^{-1}\mathbf{x}_{1}+\Sigma_{2}^{-1}\mathbf{x}_{2}\right)
  81. = [ 0.9901 0 0 0.9901 ] [ 1 1 ] = [ 0.9901 0.9901 ] =\begin{bmatrix}0.9901&0\\ 0&0.9901\end{bmatrix}\begin{bmatrix}1\\ 1\end{bmatrix}=\begin{bmatrix}0.9901\\ 0.9901\end{bmatrix}
  82. 𝐗 = [ x 1 , , x n ] \mathbf{X}=[x_{1},\dots,x_{n}]
  83. 𝐂 \mathbf{C}
  84. x i x_{i}
  85. x ¯ \bar{x}
  86. 𝐖 \mathbf{W}
  87. n n
  88. σ x ¯ 2 = ( 𝐖 T 𝐂 - 1 𝐖 ) - 1 , \sigma^{2}_{\bar{x}}=(\mathbf{W}^{T}\mathbf{C}^{-1}\mathbf{W})^{-1},
  89. x ¯ = σ x ¯ 2 ( 𝐖 T 𝐂 - 1 𝐗 ) . \bar{x}=\sigma^{2}_{\bar{x}}(\mathbf{W}^{T}\mathbf{C}^{-1}\mathbf{X}).
  90. x x
  91. y y
  92. n n
  93. t i t_{i}
  94. y y
  95. t i t_{i}
  96. x i x_{i}
  97. z z
  98. m m
  99. z k = i = 1 m w i x k + 1 - i . z_{k}=\sum_{i=1}^{m}w_{i}x_{k+1-i}.
  100. 0 < Δ < 1 0<\Delta<1
  101. w = 1 - Δ w=1-\Delta
  102. m m
  103. w i = w i - 1 V 1 , w_{i}=\frac{w^{i-1}}{V_{1}},
  104. V 1 V_{1}
  105. V 1 V_{1}
  106. V 1 = i = 1 m w i - 1 = 1 - w m 1 - w , V_{1}=\sum_{i=1}^{m}{w^{i-1}}=\frac{1-w^{m}}{1-w},
  107. V 1 = 1 / ( 1 - w ) V_{1}=1/(1-w)
  108. m m
  109. w w
  110. ( 1 - w ) - 1 (1-w)^{-1}
  111. e - 1 ( 1 - w ) = 0.39 ( 1 - w ) {e^{-1}}(1-w)=0.39(1-w)
  112. e - 1 e^{-1}
  113. 1 - e - 1 = 0.61 {1-e^{-1}}=0.61
  114. n n
  115. e - n ( 1 - w ) \leq{e^{-n(1-w)}}
  116. n n
  117. w w

Welding.html

  1. Q = ( V × I × 60 S × 1000 ) × 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 Q=\left(\frac{V\times I\times 60}{S\times 1000}\right)\times\mathit{Efficiency}

Wheatstone_bridge.html

  1. R x \scriptstyle R_{x}
  2. R 1 \scriptstyle R_{1}
  3. R 2 \scriptstyle R_{2}
  4. R 3 \scriptstyle R_{3}
  5. R 2 \scriptstyle R_{2}
  6. ( R 2 / R 1 ) \scriptstyle(R_{2}/R_{1})
  7. ( R x / R 3 ) \scriptstyle(R_{x}/R_{3})
  8. V g \scriptstyle V_{g}
  9. R 2 \scriptstyle R_{2}
  10. R 2 \scriptstyle R_{2}
  11. R 1 \scriptstyle R_{1}
  12. R 2 \scriptstyle R_{2}
  13. R 3 \scriptstyle R_{3}
  14. R x \scriptstyle R_{x}
  15. R x \scriptstyle R_{x}
  16. R 2 R 1 = R x R 3 R x = R 2 R 1 R 3 \begin{aligned}\displaystyle\frac{R_{2}}{R_{1}}&\displaystyle=\frac{R_{x}}{R_{% 3}}\\ \displaystyle\Rightarrow R_{x}&\displaystyle=\frac{R_{2}}{R_{1}}\cdot R_{3}% \end{aligned}
  17. R 1 \scriptstyle R_{1}
  18. R 2 \scriptstyle R_{2}
  19. R 3 \scriptstyle R_{3}
  20. R 2 \scriptstyle R_{2}
  21. R x \scriptstyle R_{x}
  22. I 3 - I x + I G \displaystyle I_{3}-I_{x}+I_{G}
  23. ( I 3 R 3 ) - ( I G R G ) - ( I 1 R 1 ) \displaystyle(I_{3}\cdot R_{3})-(I_{G}\cdot R_{G})-(I_{1}\cdot R_{1})
  24. I 3 R 3 \displaystyle I_{3}\cdot R_{3}
  25. R x = R 2 I 2 I 3 R 3 R 1 I 1 I x R_{x}={{R_{2}\cdot I_{2}\cdot I_{3}\cdot R_{3}}\over{R_{1}\cdot I_{1}\cdot I_{% x}}}
  26. R x = R 3 R 2 R 1 R_{x}={{R_{3}\cdot R_{2}}\over{R_{1}}}
  27. V G = ( R 2 R 1 + R 2 - R x R x + R 3 ) V s V_{G}=\left({{R_{2}}\over{R_{1}+R_{2}}}-{{R_{x}}\over{R_{x}+R_{3}}}\right)V_{s}
  28. V < s u b > G V<sub>G

Where_Mathematics_Comes_From.html

  1. A = { { } , { , { } } } A=\{\,\{\emptyset\},\,\{\emptyset,\{\emptyset\}\}\,\}
  2. $\empty$
  3. \cup
  4. { { a } , { a , b } } \{\,\{a\},\{a,b\}\,\}
  5. x 2 = 2 x^{2}=2
  6. 2 \sqrt{2}

Whistle.html

  1. W m = π ρ 0 c 0 Q 2 ^ S t 2 U 4 L 2 {{W}_{m}}=\frac{\pi{{\rho}_{0}}}{{{c}_{0}}}\widehat{{{Q}^{2}}}S{{t}^{2}}{{U}^{% 4}}{{L}^{2}}
  2. W d = 3 π ρ 0 c 0 3 F ^ 2 S t 2 U 6 L 2 {{W}_{d}}=\frac{3\pi\rho{}_{0}}{c_{0}^{3}}{{\widehat{F}}^{2}}S{{t}^{2}}{{U}^{6% }}{{L}^{2}}
  3. S t = f L 1 U , S t a = f L 2 c 0 St=\frac{f{{L}_{1}}}{U},S{{t}_{a}}=\frac{f{{L}_{2}}}{{{c}_{0}}}
  4. 2 π 2\pi
  5. M = U c 0 M=\frac{U}{{{c}_{0}}}
  6. Re = U L υ \operatorname{Re}=\frac{UL}{\upsilon}
  7. R o = U f 0 L \operatorname{R}o=\frac{U}{{{f}_{0}}L}
  8. F ^ = F ρ 0 U 2 L 2 \widehat{F}=\frac{F}{{{\rho}_{0}}{{U}^{2}}{{L}^{2}}}
  9. Q ^ = Q U L 2 \widehat{Q}=\frac{Q}{U{{L}^{2}}}
  10. U 4 {{U}^{4}}
  11. S t = f n n L U {{S}_{t}}=\frac{{{f}_{n}}nL}{U}
  12. λ = 4 L \lambda=4L
  13. λ d = 5.8 + 2.5 { h d - ( 1 + 0.0041 ( P - 0.9 ) 2 ) } S t = f d c 0 0.17 f h U \begin{aligned}&\displaystyle\frac{\lambda}{d}=5.8+2.5\left\{\frac{h}{d}-\left% (1+0.0041{{\left(P-0.9\right)}^{2}}\right)\right\}\\ &\displaystyle{{S}_{t}}=\frac{fd}{{{c}_{0}}}\approx 0.17\approx\frac{fh}{U}\\ \end{aligned}
  14. c 0 {{c}_{0}}
  15. W h = ρ 2 π d 2 4 ( 2 π f a ) 2 c 0 = A ρ f 2 d 2 h 2 c 0 = A ρ c 0 ( f d c 0 ) 2 c 0 4 a 2 W h = A ρ 0 c 0 S t 2 U 4 L 2 \begin{aligned}&\displaystyle{{W}_{h}}=\frac{\rho}{2}\frac{\pi{{d}^{2}}}{4}{{% \left(2\pi fa\right)}^{2}}{{c}_{0}}=A\rho{{f}^{2}}{{d}^{2}}{{h}^{2}}{{c}_{0}}=% A\frac{\rho}{{{c}_{0}}}{{\left(\frac{fd}{{{c}_{0}}}\right)}^{2}}c_{0}^{4}{{a}^% {2}}\\ &\displaystyle{{W}_{h}}=A\frac{{{\rho}_{0}}}{{{c}_{0}}}S{{t}^{2}}{{U}^{4}}{{L}% ^{2}}\\ \end{aligned}
  16. c 0 {{c}_{0}}
  17. c 0 {{c}_{0}}
  18. F = F d + F d + F l = C d ρ 0 2 ( U + u + v ) 2 d w W d = 3 π ρ 0 c 0 3 S 2 U 6 d w C d 2 ( u U ) 2 ¯ W l = 3 π ρ 0 c 0 3 S 2 U 6 d w C d 2 ( v U ) 2 ¯ \begin{aligned}&\displaystyle F={{F}_{d}}+F_{d}^{{}^{\prime}}+F_{l}^{{}^{% \prime}}=\frac{{{C}_{d}}{{\rho}_{0}}}{2}{{\left(U+u^{\prime}+v^{\prime}\right)% }^{2}}dw\\ &\displaystyle{{W}_{d}}=\frac{3\pi{{\rho}_{0}}}{c_{0}^{3}}{{S}^{2}}{{U}^{6}}% dwC_{d}^{2}\overline{{{\left(\frac{{{u}^{{}^{\prime}}}}{U}\right)}^{2}}}\\ &\displaystyle{{W}_{l}}=\frac{3\pi{{\rho}_{0}}}{c_{0}^{3}}{{S}^{2}}{{U}^{6}}% dwC_{d}^{2}\overline{{{\left(\frac{{{v}^{{}^{\prime}}}}{U}\right)}^{2}}}\\ \end{aligned}
  19. U 6 {{U}^{6}}
  20. R Ω R\Omega
  21. S t = f h U St=\frac{fh}{U}
  22. U 2 {{U}^{2}}
  23. U 4.5 {{U}^{4.5}}
  24. U 6.0 {{U}^{6.0}}
  25. ( f d ) 2 {{(fd)}^{2}}
  26. S t n = f h U = [ 4 n + 1 8 - h 60 d ] S{{t}_{n}}=\frac{fh}{U}=\left[\frac{4n+1}{8}-\frac{h}{60d}\right]
  27. h > d 10 {}^{h}\!\!\diagup\!\!{}_{d}\;>10
  28. U 2 {{U}^{2}}
  29. S t n = f n L U = n - β U ( 1 c 0 + 1 u v ) S{{t}_{n}}=\frac{{{f}_{n}}L}{U}=\frac{n-\beta}{U\left(\frac{1}{{{c}_{0}}}+% \frac{1}{{{u}_{v}}}\right)}
  30. c 0 {{c}_{0}}
  31. U f D \frac{U}{fD}
  32. U f D \frac{U}{fD}
  33. c 0 {{c}_{0}}
  34. cot ( k L ) = A c ( L o + δ e + δ i ) A 0 k L k L = 2 π f L c 0 = 2 π S t \begin{aligned}&\displaystyle\cot\left(kL\right)=\frac{{{A}_{c}}\left({{L}_{o}% }+{{\delta}_{e}}+{{\delta}_{i}}\right)}{{{A}_{0}}}kL\\ &\displaystyle kL=2\pi\frac{fL}{{{c}_{0}}}=2\pi{{S}_{t}}\\ \end{aligned}
  35. 2 π 2\pi
  36. c 0 {{c}_{0}}
  37. S t = f D U S t a = f L 1 c 0 = ( 2 n + 1 ) 4 L 1 = L ( 1 + β D L ) \begin{aligned}&\displaystyle St=\frac{fD}{U}\\ &\displaystyle S{{t}_{a}}=\frac{f{{L}_{1}}}{{{c}_{0}}}=\frac{\left(2n+1\right)% }{4}\\ &\displaystyle{{L}_{1}}=L\left(1+\beta\frac{D}{L}\right)\\ \end{aligned}
  38. λ = L 2 \lambda={}^{L}\!\!\diagup\!\!{}_{2}\;
  39. λ 4 {}^{\lambda}\!\!\diagup\!\!{}_{4}\;
  40. c 0 {{c}_{0}}
  41. S = t c f L 1 c 0 , S t e = f h U \displaystyle S{}_{tc}=\frac{f{{L}_{1}}}{{{c}_{0}}},{{S}_{te}}=\frac{fh}{U}
  42. L 1 = L + δ 1 + δ 2 {{L}_{1}}=L+{{\delta}_{1}}+{{\delta}_{2}}
  43. c 0 {{c}_{0}}

White_dwarf.html

  1. 1 × 10 < s u p > 9 1 × 10<sup>9
  2. G M R −GM∕R
  3. G G
  4. R R
  5. p p
  6. m m
  7. N N
  8. p p
  9. Δ p Δp
  10. Δ p Δ x ΔpΔx
  11. Δ x Δx
  12. n n
  13. N M NM
  14. n n
  15. E k N ( Δ p ) 2 2 m N 2 n 2 / 3 2 m M 2 / 3 N 5 / 3 2 2 m R 2 . E_{k}\approx\frac{N(\Delta p)^{2}}{2m}\approx\frac{N\hbar^{2}n^{2/3}}{2m}% \approx\frac{M^{2/3}N^{5/3}\hbar^{2}}{2mR^{2}}.
  16. | E g | G M R = E k M 2 / 3 N 5 / 3 2 2 m R 2 . |E_{g}|\approx\frac{GM}{R}=E_{k}\approx\frac{M^{2/3}N^{5/3}\hbar^{2}}{2mR^{2}}.
  17. R R
  18. R N 5 / 3 2 2 m G M 1 / 3 . R\approx\frac{N^{5/3}\hbar^{2}}{2mGM^{1/3}}.
  19. N N
  20. R M - 1 / 3 R\sim M^{-1/3}
  21. c c
  22. p c pc
  23. E k relativistic M 1 / 3 N 4 / 3 c R . E_{k\ {\rm relativistic}}\approx\frac{M^{1/3}N^{4/3}\hbar c}{R}.
  24. R R
  25. M M
  26. M limit N 2 ( c G ) 3 / 2 . M_{\rm limit}\approx N^{2}\left(\frac{\hbar c}{G}\right)^{3/2}.
  27. c c
  28. M M
  29. M < s u b > l i m i t M<sub>limit

White_noise.html

  1. σ 2 \sigma^{2}
  2. σ 2 \sigma^{2}
  3. σ 2 \sigma^{2}
  4. μ \mu
  5. μ \mu
  6. μ n \mu\sqrt{n}
  7. w w
  8. t t
  9. w ( t ) w(t)
  10. t t
  11. t t
  12. w ( t 1 ) w(t_{1})
  13. w ( t 2 ) w(t_{2})
  14. t 1 t_{1}
  15. t 2 t_{2}
  16. w ( t 1 ) w(t_{1})
  17. w ( t 2 ) w(t_{2})
  18. w w
  19. n \mathbb{R}^{n}
  20. w w
  21. w w
  22. w ( t ) w(t)
  23. σ 2 \sigma^{2}
  24. E ( w ( t 1 ) w ( t 2 ) ) \mathrm{E}(w(t_{1})\cdot w(t_{2}))
  25. t 1 t_{1}
  26. t 2 t_{2}
  27. σ 2 \sigma^{2}
  28. W [ a , a + r ] = a a + r w ( t ) d t W_{[a,a+r]}=\int_{a}^{a+r}w(t)\,dt
  29. r r
  30. w w
  31. w ( t ) w(t)
  32. | w ( t ) | 2 |w(t)|^{2}
  33. [ a , a + r ] [a,a+r]
  34. r r
  35. w ( t ) w(t)
  36. E ( w ( t 1 ) w ( t 2 ) ) \mathrm{E}(w(t_{1})\cdot w(t_{2}))
  37. t 1 = t 2 t_{1}=t_{2}
  38. R ( t 1 , t 2 ) \mathrm{R}(t_{1},t_{2})
  39. N δ ( t 1 - t 2 ) N\delta(t_{1}-t_{2})
  40. N N
  41. δ \delta
  42. W I W_{I}
  43. w ( t ) w(t)
  44. I = [ a , b ] I=[a,b]
  45. ( b - a ) σ 2 (b-a)\sigma^{2}
  46. E ( W I W J ) \mathrm{E}(W_{I}\cdot W_{J})
  47. W I W_{I}
  48. W J W_{J}
  49. r σ 2 r\sigma^{2}
  50. r r
  51. I J I\cap J
  52. I , J I,J

Wien's_displacement_law.html

  1. λ max = b T \lambda\text{max}=\frac{b}{T}
  2. d ν d\nu
  3. ν max \nu_{\max}
  4. ν max = α h k T ( 5.879 × 10 10 Hz / K ) T \nu_{\max}={\alpha\over h}kT\approx(5.879\times 10^{10}\ \mathrm{Hz/K})\cdot T
  5. u λ ( λ , T ) = 2 h c 2 λ 5 1 e h c / λ k T - 1 . u_{\lambda}(\lambda,T)={2hc^{2}\over\lambda^{5}}{1\over e^{hc/\lambda kT}-1}.
  6. u λ = 2 h c 2 ( h c k T λ 7 e h c / λ k T ( e h c / λ k T - 1 ) 2 - 1 λ 6 5 e h c / λ k T - 1 ) = 0 , {\partial u\over\partial\lambda}=2hc^{2}\left({hc\over kT\lambda^{7}}{e^{hc/% \lambda kT}\over\left(e^{hc/\lambda kT}-1\right)^{2}}-{1\over\lambda^{6}}{5% \over e^{hc/\lambda kT}-1}\right)=0,
  7. h c λ k T e h c / λ k T e h c / λ k T - 1 - 5 = 0. {hc\over\lambda kT}{e^{hc/\lambda kT}\over e^{hc/\lambda kT}-1}-5=0.
  8. x h c λ k T , x\equiv{hc\over\lambda kT},
  9. x e x e x - 1 - 5 = 0. {xe^{x}\over e^{x}-1}-5=0.
  10. λ max = h c x 1 k T = 2.89776829 × 10 6 nm K T . \lambda_{\max}={hc\over x}{1\over kT}={2.89776829\times 10^{6}\ \mathrm{nm}% \cdot K\over T}.

Willard_Van_Orman_Quine.html

  1. x F x x F x \forall x\,Fx\rightarrow\exists x\,Fx

William_Rowan_Hamilton.html

  1. i 2 = j 2 = k 2 = i j k = - 1 \displaystyle i^{2}=j^{2}=k^{2}=ijk=-1
  2. \mathbb{H}

Word_problem_for_groups.html

  1. Σ \Sigma
  2. Σ \Sigma
  3. a , b , c , d , e , p , q , r , t , k | p 10 a = a p , p a c q r = r p c a q , r a = a r , p 10 b = b p , p 2 a d q 2 r = r p 2 d a q 2 , r b = b r , p 10 c = c p , p 3 b c q 3 r = r p 3 c b q 3 , r c = c r , p 10 d = d p , p 4 b d q 4 r = r p 4 d b q 4 , r d = d r , p 10 e = e p , p 5 c e q 5 r = r p 5 e c a q 5 , r e = e r , a q 10 = q a , p 6 d e q 6 r = r p 6 e d b q 6 , p t = t p , b q 10 = q b , p 7 c d c q 7 r = r p 7 c d c e q 7 , q t = t q , c q 10 = q c , p 8 c a 3 q 8 r = r p 8 a 3 q 8 , d q 10 = q d , p 9 d a 3 q 9 r = r p 9 a 3 q 9 , e q 10 = q e , a - 3 t a 3 k = k a - 3 t a 3 \begin{array}[]{lllll}\langle&a,b,c,d,e,p,q,r,t,k&|&&\\ &p^{10}a=ap,&pacqr=rpcaq,&ra=ar,&\\ &p^{10}b=bp,&p^{2}adq^{2}r=rp^{2}daq^{2},&rb=br,&\\ &p^{10}c=cp,&p^{3}bcq^{3}r=rp^{3}cbq^{3},&rc=cr,&\\ &p^{10}d=dp,&p^{4}bdq^{4}r=rp^{4}dbq^{4},&rd=dr,&\\ &p^{10}e=ep,&p^{5}ceq^{5}r=rp^{5}ecaq^{5},&re=er,&\\ &aq^{10}=qa,&p^{6}deq^{6}r=rp^{6}edbq^{6},&pt=tp,&\\ &bq^{10}=qb,&p^{7}cdcq^{7}r=rp^{7}cdceq^{7},&qt=tq,&\\ &cq^{10}=qc,&p^{8}ca^{3}q^{8}r=rp^{8}a^{3}q^{8},&&\\ &dq^{10}=qd,&p^{9}da^{3}q^{9}r=rp^{9}a^{3}q^{9},&&\\ &eq^{10}=qe,&a^{-3}ta^{3}k=ka^{-3}ta^{3}&&\rangle\end{array}
  4. S = { u , v : u and v are words in X and u = v in G } S=\{\langle u,v\rangle:u\mbox{ and }~{}v\mbox{ are words in }~{}X\mbox{ and }~% {}u=v\mbox{ in }~{}G\ \}
  5. f P ( u , v ) = { 0 if u , v S undefined/does not halt if u , v S f_{P}(\langle u,v\rangle)=\left\{\begin{matrix}0&\mbox{if}~{}\ \langle u,v% \rangle\in S\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ \langle u,v\rangle\notin S\end{% matrix}\right.
  6. g ( u , v ) = { 0 if u , v S undefined/does not halt if u , v S g(\langle u,v\rangle)=\left\{\begin{matrix}0&\mbox{if}~{}\ \langle u,v\rangle% \notin S\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ \langle u,v\rangle\in S\end{% matrix}\right.
  7. h ( x ) = { 0 if x 1 in G undefined/does not halt if x = 1 in G h(x)=\left\{\begin{matrix}0&\mbox{if}~{}\ x\neq 1\ \mbox{in}~{}\ G\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ x=1\ \mbox{in}~{}\ G\end{matrix}\right.
  8. h ( x ) = { 0 if x 1 in G undefined/does not halt if x = 1 in G h(x)=\left\{\begin{matrix}0&\mbox{if}~{}\ x\neq 1\ \mbox{in}~{}\ G\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ x=1\ \mbox{in}~{}\ G\end{matrix}\right.
  9. f ( P , w ) = { 0 if w 1 in G undefined/does not halt if w = 1 in G f(P,w)=\left\{\begin{matrix}0&\mbox{if}~{}\ w\neq 1\ \mbox{in}~{}\ G\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ w=1\ \mbox{in}~{}\ G\end{matrix}\right.
  10. If w 1 in H , h n ( w ) 1 in G for some h n \mbox{If}~{}\ w\neq 1\ \mbox{in}~{}\ H,\ h_{n}(w)\neq 1\ \mbox{in}~{}\ G\ % \mbox{for some}~{}\ h_{n}
  11. If w = 1 in H , h n ( w ) = 1 in G for all h n \mbox{If}~{}\ w=1\ \mbox{in}~{}\ H,\ h_{n}(w)=1\ \mbox{in}~{}\ G\ \mbox{for % all}~{}\ h_{n}
  12. f ( w ) = { 0 if w 1 in H undefined/does not halt if w = 1 in H . f(w)=\left\{\begin{matrix}0&\mbox{if}~{}\ w\neq 1\ \mbox{in}~{}\ H\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ w=1\ \mbox{in}~{}\ H.\end{matrix}\right.
  13. f ( P , w ) = { 0 if w 1 in H undefined/does not halt if w = 1 in H . f(P,w)=\left\{\begin{matrix}0&\mbox{if}~{}\ w\neq 1\ \mbox{in}~{}\ H\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ w=1\ \mbox{in}~{}\ H.\end{matrix}\right.
  14. S w = X | R { w } . S_{w}=\langle X|R\cup\{w\}\rangle.
  15. f X | R { w } f_{\langle X|R\cup\{w\}\rangle}
  16. f X | R { w } ( x ) = { 0 if x = 1 in S w undefined/does not halt if x 1 in S w . f_{\langle X|R\cup\{w\}\rangle}(x)=\left\{\begin{matrix}0&\mbox{if}~{}\ x=1\ % \mbox{in}~{}\ S_{w}\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ x\neq 1\ \mbox{in}~{}\ S_{w}.\end% {matrix}\right.
  17. g ( w , x ) = f X | R { w } ( x ) . g(w,x)=f_{\langle X|R\cup\{w\}\rangle}(x).
  18. h ( w ) = { 0 if a = 1 in S w undefined/does not halt if a 1 in S w . h(w)=\left\{\begin{matrix}0&\mbox{if}~{}\ a=1\ \mbox{in}~{}\ S_{w}\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ a\neq 1\ \mbox{in}~{}\ S_{w}.\end% {matrix}\right.
  19. h ( w ) = { 0 if w 1 in S undefined/does not halt if w = 1 in S . h(w)=\left\{\begin{matrix}0&\mbox{if}~{}\ w\neq 1\ \mbox{in}~{}\ S\\ \mbox{undefined/does not halt}&\mbox{if}~{}\ w=1\ \mbox{in}~{}\ S.\end{matrix}\right.

Work_function.html

  1. W W
  2. W = - e ϕ - E F , W=-e\phi-E_{\rm F},
  3. e −e
  4. ϕ ϕ
  5. e ϕ −eϕ
  6. ϕ ϕ
  7. ϕ ϕ
  8. ϕ = V - W e \phi=V-\frac{W}{e}
  9. ϕ ϕ
  10. W W
  11. E barrier = W e E_{\rm barrier}=W_{\rm e}
  12. J e = - A e T e 2 e - E barrier / k T e J_{\rm e}=-A_{\rm e}T_{\rm e}^{2}e^{-E_{\rm barrier}/kT_{\rm e}}
  13. E barrier = W c - e ( Δ V ce - Δ V S ) E_{\rm barrier}=W_{\rm c}-e(\Delta V_{\rm ce}-\Delta V_{\rm S})
  14. J c = A T e 2 e - E barrier / k T e J_{\rm c}=AT_{\rm e}^{2}e^{-E_{\rm barrier}/kT_{\rm e}}
  15. ω \hbar\omega
  16. ω = W e \hbar\omega=W_{\rm e}
  17. e Δ V sp = W s - W p , when ϕ is flat . e\Delta V_{\rm sp}=W_{\rm s}-W_{\rm p},\quad\,\text{when}~{}\phi~{}\,\text{is % flat}.
  18. W = E EA + E C - E F W=E_{\rm EA}+E_{\rm C}-E_{\rm F}

World_file.html

  1. A 2 + D 2 \sqrt{A^{2}+D^{2}}
  2. B 2 + E 2 \sqrt{B^{2}+E^{2}}
  3. [ x y ] = [ A B C D E F ] [ x y 1 ] \begin{bmatrix}x\prime\\ y\prime\end{bmatrix}=\begin{bmatrix}A&B&C\\ D&E&F\end{bmatrix}\begin{bmatrix}x\\ y\\ 1\end{bmatrix}
  4. x \displaystyle x^{\prime}
  5. x = ( E x - B y + B F - E C ) / ( A E - D B ) \displaystyle x=(Ex^{\prime}-By^{\prime}+BF-EC)/(AE-DB)

Wormhole.html

  1. d s 2 = - c 2 d t 2 + d l 2 + ( k 2 + l 2 ) ( d θ 2 + sin 2 θ d ϕ 2 ) . ds^{2}=-c^{2}dt^{2}+dl^{2}+(k^{2}+l^{2})(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).
  2. d s 2 = - c 2 ( 1 - 2 G M r c 2 ) d t 2 + d r 2 1 - 2 G M r c 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . ds^{2}=-c^{2}\left(1-\frac{2GM}{rc^{2}}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2GM% }{rc^{2}}}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).

Wright_brothers.html

  1. L = k S V 2 C L L=k\;S\;V^{2}\;C_{L}
  2. 203 / 4 20{3}/{4}

X-ray_crystallography.html

  1. 2 d sin θ = n λ 2d\sin\theta=n\lambda
  2. θ \theta
  3. I o = I e ( q 4 m 2 c 4 ) 1 + cos 2 2 θ 2 = I e 7.94.10 - 26 1 + cos 2 2 θ 2 = I e f I_{o}=I_{e}\left(\frac{q^{4}}{m^{2}c^{4}}\right)\frac{1+\cos^{2}2\theta}{2}=I_% {e}7.94.10^{-26}\frac{1+\cos^{2}2\theta}{2}=I_{e}f
  4. R = all reflections | F o - F c | all reflections | F o | R=\frac{\sum_{\mathrm{all\ reflections}}\left|F_{o}-F_{c}\right|}{\sum_{% \mathrm{all\ reflections}}\left|F_{o}\right|}
  5. f ( 𝐫 ) = 1 ( 2 π ) 3 F ( 𝐪 ) e i 𝐪 𝐫 d 𝐪 f(\mathbf{r})=\frac{1}{\left(2\pi\right)^{3}}\int F(\mathbf{q})e^{\mathrm{i}% \mathbf{q}\cdot\mathbf{r}}\mathrm{d}\mathbf{q}
  6. π \pi
  7. F ( 𝐪 ) = f ( 𝐫 ) e - i 𝐪 𝐫 d 𝐫 F(\mathbf{q})=\int f(\mathbf{r})\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\mathbf{% r}}\mathrm{d}\mathbf{r}
  8. F ( 𝐪 ) = | F ( 𝐪 ) | e i ϕ ( 𝐪 ) F(\mathbf{q})=\left|F(\mathbf{q})\right|\mathrm{e}^{\mathrm{i}\phi(\mathbf{q})}
  9. 2 d sin θ = n λ 2d\sin\theta=n\lambda\,
  10. A e i 𝐤 in 𝐫 A\mathrm{e}^{\mathrm{i}\mathbf{k}_{\mathrm{in}}\cdot\mathbf{r}}
  11. amplitude of scattered wave = A e i 𝐤 𝐫 S f ( 𝐫 ) d V \mathrm{amplitude\ of\ scattered\ wave}=A\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot% \mathbf{r}}Sf(\mathbf{r})\mathrm{d}V
  12. e i 𝐤 o u t ( 𝐫 screen - 𝐫 ) e^{i\mathbf{k}_{out}\cdot\left(\mathbf{r}_{\mathrm{screen}}-\mathbf{r}\right)}
  13. A S d 𝐫 f ( 𝐫 ) e i 𝐤 i n 𝐫 e i 𝐤 o u t ( 𝐫 screen - 𝐫 ) = A S e i 𝐤 o u t 𝐫 screen d 𝐫 f ( 𝐫 ) e i ( 𝐤 i n - 𝐤 o u t ) 𝐫 AS\int\mathrm{d}\mathbf{r}f(\mathbf{r})\mathrm{e}^{\mathrm{i}\mathbf{k}_{in}% \cdot\mathbf{r}}e^{i\mathbf{k}_{out}\cdot\left(\mathbf{r}_{\mathrm{screen}}-% \mathbf{r}\right)}=ASe^{i\mathbf{k}_{out}\cdot\mathbf{r}_{\mathrm{screen}}}% \int\mathrm{d}\mathbf{r}f(\mathbf{r})\mathrm{e}^{\mathrm{i}\left(\mathbf{k}_{% in}-\mathbf{k}_{out}\right)\cdot\mathbf{r}}
  14. A S e i 𝐤 o u t 𝐫 screen d 𝐫 f ( 𝐫 ) e - i 𝐪 𝐫 = A S e i 𝐤 o u t 𝐫 screen F ( 𝐪 ) AS\mathrm{e}^{\mathrm{i}\mathbf{k}_{out}\cdot\mathbf{r}_{\mathrm{screen}}}\int d% \mathbf{r}f(\mathbf{r})\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\mathbf{r}}=AS% \mathrm{e}^{\mathrm{i}\mathbf{k}_{out}\cdot\mathbf{r}_{\mathrm{screen}}}F(% \mathbf{q})
  15. A 2 S 2 | F ( 𝐪 ) | 2 A^{2}S^{2}\left|F(\mathbf{q})\right|^{2}
  16. F ( - 𝐪 ) = | F ( - 𝐪 ) | e i ϕ ( - 𝐪 ) = F * ( 𝐪 ) = | F ( 𝐪 ) | e - i ϕ ( 𝐪 ) F(-\mathbf{q})=\left|F(-\mathbf{q})\right|\mathrm{e}^{\mathrm{i}\phi(-\mathbf{% q})}=F^{*}(\mathbf{q})=\left|F(\mathbf{q})\right|\mathrm{e}^{-\mathrm{i}\phi(% \mathbf{q})}
  17. f ( 𝐫 ) = d 𝐪 ( 2 π ) 3 F ( 𝐪 ) e i 𝐪 𝐫 = d 𝐪 ( 2 π ) 3 | F ( 𝐪 ) | e i ϕ ( 𝐪 ) e i 𝐪 𝐫 f(\mathbf{r})=\int\frac{d\mathbf{q}}{\left(2\pi\right)^{3}}F(\mathbf{q})% \mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}}=\int\frac{d\mathbf{q}}{\left(% 2\pi\right)^{3}}\left|F(\mathbf{q})\right|\mathrm{e}^{\mathrm{i}\phi(\mathbf{q% })}\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}}
  18. f ( 𝐫 ) = d 𝐪 ( 2 π ) 3 | F ( 𝐪 ) | e i ( ϕ + 𝐪 𝐫 ) = d 𝐪 ( 2 π ) 3 | F ( 𝐪 ) | cos ( ϕ + 𝐪 𝐫 ) + i d 𝐪 ( 2 π ) 3 | F ( 𝐪 ) | sin ( ϕ + 𝐪 𝐫 ) = I cos + i I sin f(\mathbf{r})=\int\frac{d\mathbf{q}}{\left(2\pi\right)^{3}}\left|F(\mathbf{q})% \right|\mathrm{e}^{\mathrm{i}\left(\phi+\mathbf{q}\cdot\mathbf{r}\right)}=\int% \frac{d\mathbf{q}}{\left(2\pi\right)^{3}}\left|F(\mathbf{q})\right|\cos\left(% \phi+\mathbf{q}\cdot\mathbf{r}\right)+i\int\frac{d\mathbf{q}}{\left(2\pi\right% )^{3}}\left|F(\mathbf{q})\right|\sin\left(\phi+\mathbf{q}\cdot\mathbf{r}\right% )=I_{\mathrm{cos}}+iI_{\mathrm{sin}}
  19. I sin = d 𝐪 ( 2 π ) 3 | F ( 𝐪 ) | sin ( ϕ + 𝐪 𝐫 ) = d 𝐪 ( 2 π ) 3 | F ( - 𝐪 ) | sin ( - ϕ - 𝐪 𝐫 ) = - I sin I_{\mathrm{sin}}=\int\frac{d\mathbf{q}}{\left(2\pi\right)^{3}}\left|F(\mathbf{% q})\right|\sin\left(\phi+\mathbf{q}\cdot\mathbf{r}\right)=\int\frac{d\mathbf{q% }}{\left(2\pi\right)^{3}}\left|F(\mathbf{-q})\right|\sin\left(-\phi-\mathbf{q}% \cdot\mathbf{r}\right)=-I_{\mathrm{sin}}
  20. c ( 𝐫 ) = d 𝐱 f ( 𝐱 ) f ( 𝐱 + 𝐫 ) = d 𝐪 ( 2 π ) 3 C ( 𝐪 ) e i 𝐪 𝐫 c(\mathbf{r})=\int d\mathbf{x}f(\mathbf{x})f(\mathbf{x}+\mathbf{r})=\int\frac{% d\mathbf{q}}{\left(2\pi\right)^{3}}C(\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{r}}
  21. C ( 𝐪 ) = | F ( 𝐪 ) | 2 C(\mathbf{q})=\left|F(\mathbf{q})\right|^{2}

X.html

  1. x x\!

X86.html

  1. { C S : D S : S S : E S : } [ { B X B P } ] + [ { S I D I } ] + [ displacement ] \begin{Bmatrix}CS:\\ DS:\\ SS:\\ ES:\end{Bmatrix}\begin{bmatrix}\begin{Bmatrix}BX\\ BP\end{Bmatrix}\end{bmatrix}+\begin{bmatrix}\begin{Bmatrix}SI\\ DI\end{Bmatrix}\end{bmatrix}+\rm[displacement]
  2. { C S : D S : S S : E S : F S : G S : } [ { E A X E B X E C X E D X E S P E B P E S I E D I } ] + [ { E A X E B X E C X E D X E B P E S I E D I } * { 1 2 4 8 } ] + [ displacement ] \begin{Bmatrix}CS:\\ DS:\\ SS:\\ ES:\\ FS:\\ GS:\end{Bmatrix}\begin{bmatrix}\begin{Bmatrix}EAX\\ EBX\\ ECX\\ EDX\\ ESP\\ EBP\\ ESI\\ EDI\end{Bmatrix}\end{bmatrix}+\begin{bmatrix}\begin{Bmatrix}EAX\\ EBX\\ ECX\\ EDX\\ EBP\\ ESI\\ EDI\end{Bmatrix}*\begin{Bmatrix}1\\ 2\\ 4\\ 8\end{Bmatrix}\end{bmatrix}+\rm[displacement]
  3. { { F S : G S : } [ general register ] + [ general register * { 1 2 4 8 } ] R I P } + [ displacement ] \begin{Bmatrix}\begin{Bmatrix}FS:\\ GS:\end{Bmatrix}\begin{bmatrix}{\rm general\;register}\end{bmatrix}+\begin{% bmatrix}{\rm general\;register}*\begin{Bmatrix}1\\ 2\\ 4\\ 8\end{Bmatrix}\end{bmatrix}\\ \\ RIP\end{Bmatrix}+\rm[displacement]

Yoneda_lemma.html

  1. h A = Hom ( A , - ) . h^{A}=\mathrm{Hom}(A,-).
  2. f - f\circ-
  3. f Hom ( A , f ) = [ [ Hom ( A , X ) g f g Hom ( A , Y ) ] ] f\longmapsto\mathrm{Hom}(A,f)=[\![\mathrm{Hom}(A,X)\ni g\mapsto f\circ g\in% \mathrm{Hom}(A,Y)]\!]
  4. Nat ( h A , F ) F ( A ) . \mathrm{Nat}(h^{A},F)\cong F(A).
  5. u = Φ A ( id A ) u=\Phi_{A}(\mathrm{id}_{A})
  6. h A = Hom ( - , A ) , h_{A}=\mathrm{Hom}(-,A),
  7. Nat ( h A , G ) G ( A ) . \mathrm{Nat}(h_{A},G)\cong G(A).
  8. Φ A ( id A ) = u \Phi_{A}(\mathrm{id}_{A})=u
  9. Φ X ( f ) = ( F f ) u . \Phi_{X}(f)=(Ff)u.\,
  10. Nat ( h A , h B ) Hom ( B , A ) . \mathrm{Nat}(h^{A},h^{B})\cong\mathrm{Hom}(B,A).
  11. h - : 𝒞 op 𝐒𝐞𝐭 𝒞 . h^{-}\colon\mathcal{C}^{\,\text{op}}\to\mathbf{Set}^{\mathcal{C}}.
  12. Nat ( h A , h B ) Hom ( A , B ) . \mathrm{Nat}(h_{A},h_{B})\cong\mathrm{Hom}(A,B).
  13. h - : 𝒞 𝐒𝐞𝐭 𝒞 op . h_{-}\colon\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\mathrm{op}}}.

Zeno's_paradoxes.html

  1. { , 1 16 , 1 8 , 1 4 , 1 2 , 1 } \left\{\cdots,\frac{1}{16},\frac{1}{8},\frac{1}{4},\frac{1}{2},1\right\}

Zero-sum_game.html

  1. M M
  2. M i , j M_{i,j}
  3. i i
  4. j j
  5. M M
  6. u u
  7. i u i \sum_{i}u_{i}
  8. u u
  9. M u Mu
  10. u u
  11. M u Mu
  12. u u
  13. u u
  14. M M

Zero_divisor.html

  1. a a
  2. R R
  3. x x
  4. a x = 0 ax=0
  5. R R
  6. R R
  7. x x
  8. a x ax
  9. a a
  10. y y
  11. y a = 0 ya=0
  12. a a
  13. x x
  14. a x = 0 ax=0
  15. y y
  16. y a = 0 ya=0
  17. / 4 \mathbb{Z}/4\mathbb{Z}
  18. 2 ¯ \overline{2}
  19. \mathbb{Z}
  20. e 1 e\neq 1
  21. e ( 1 - e ) = 0 = ( 1 - e ) e e(1-e)=0=(1-e)e
  22. 2 × 2 2\times 2
  23. ( 1 1 2 2 ) ( 1 1 - 1 - 1 ) = ( - 2 1 - 2 1 ) ( 1 1 2 2 ) = ( 0 0 0 0 ) , \begin{pmatrix}1&1\\ 2&2\end{pmatrix}\begin{pmatrix}1&1\\ -1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\ -2&1\end{pmatrix}\begin{pmatrix}1&1\\ 2&2\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix},
  24. ( 1 0 0 0 ) ( 0 0 0 1 ) = ( 0 0 0 1 ) ( 1 0 0 0 ) = ( 0 0 0 0 ) \begin{pmatrix}1&0\\ 0&0\end{pmatrix}\begin{pmatrix}0&0\\ 0&1\end{pmatrix}=\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ 0&0\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}
  25. ( x y 0 z ) \begin{pmatrix}x&y\\ 0&z\end{pmatrix}
  26. x , z x,z\in\mathbb{Z}
  27. y / 2 y\in\mathbb{Z}/2\mathbb{Z}
  28. ( x y 0 z ) ( a b 0 c ) = ( x a x b + y c 0 z c ) \begin{pmatrix}x&y\\ 0&z\end{pmatrix}\begin{pmatrix}a&b\\ 0&c\end{pmatrix}=\begin{pmatrix}xa&xb+yc\\ 0&zc\end{pmatrix}
  29. ( a b 0 c ) ( x y 0 z ) = ( x a y a + z b 0 z c ) \begin{pmatrix}a&b\\ 0&c\end{pmatrix}\begin{pmatrix}x&y\\ 0&z\end{pmatrix}=\begin{pmatrix}xa&ya+zb\\ 0&zc\end{pmatrix}
  30. x 0 y x\neq 0\neq y
  31. ( x y 0 z ) \begin{pmatrix}x&y\\ 0&z\end{pmatrix}
  32. x x
  33. ( x y 0 z ) ( 0 1 0 0 ) = ( 0 x 0 0 ) \begin{pmatrix}x&y\\ 0&z\end{pmatrix}\begin{pmatrix}0&1\\ 0&0\end{pmatrix}=\begin{pmatrix}0&x\\ 0&0\end{pmatrix}
  34. z z
  35. x , z x,z
  36. 0
  37. S S
  38. ( a 1 , a 2 , a 3 , ) (a1,a2,a3,...)
  39. S S
  40. S S
  41. End ( S ) \mathrm{End}(S)
  42. S S
  43. R ( a 1 , a 2 , a 3 , ) = ( 0 , a 1 , a 2 , ) R(a1,a2,a3,...)=(0,a1,a2,...)
  44. L ( a 1 , a 2 , a 3 , ) = ( a 2 , a 3 , a 4 , ) L(a1,a2,a3,...)=(a2,a3,a4,...)
  45. P ( a 1 , a 2 , a 3 , ) = ( a 1 , 0 , 0 , ) P(a1,a2,a3,...)=(a1,0,0,...)
  46. L P LP
  47. P R PR
  48. L L
  49. R R
  50. S S
  51. S S
  52. L L
  53. R R
  54. L R LR
  55. R L RL
  56. R L P = 0 = P R L RLP=0=PRL
  57. L R = 1 LR=1
  58. n n
  59. n n
  60. n n
  61. n n
  62. a a
  63. a x = 0 ax=0
  64. x x
  65. a = 0 a=0
  66. R R
  67. 0 · 1 = 0 0·1=0
  68. 1 · 0 = 0 1·0=0
  69. R R
  70. 0 = 1 0=1
  71. R R
  72. R R
  73. R R
  74. R R
  75. R R
  76. M M
  77. R R
  78. a a
  79. R R
  80. a a
  81. M M
  82. a a
  83. M a M M\stackrel{a}{\to}M
  84. a a
  85. M M
  86. M M
  87. R R
  88. M M
  89. M M
  90. M M
  91. R R
  92. a x ax
  93. a y ay
  94. x x
  95. y y
  96. a a
  97. x x
  98. y y

Zhang_Heng.html

  1. 9 16 \tfrac{9}{16}
  2. 1 16 \tfrac{1}{16}
  3. 9 16 \tfrac{9}{16}
  4. 1 16 \tfrac{1}{16}
  5. 5 8 \tfrac{5}{8}
  6. 730 232 \tfrac{730}{232}
  7. 355 113 \tfrac{355}{113}

Zipf's_law.html

  1. H k , s H N , s \frac{H_{k,s}}{H_{N,s}}
  2. H N , s - 1 H N , s \frac{H_{N,s-1}}{H_{N,s}}
  3. 1 1\,
  4. s H N , s k = 1 N ln ( k ) k s + ln ( H N , s ) \frac{s}{H_{N,s}}\sum_{k=1}^{N}\frac{\ln(k)}{k^{s}}+\ln(H_{N,s})
  5. 1 H N , s n = 1 N e n t n s \frac{1}{H_{N,s}}\sum_{n=1}^{N}\frac{e^{nt}}{n^{s}}
  6. 1 H N , s n = 1 N e i n t n s \frac{1}{H_{N,s}}\sum_{n=1}^{N}\frac{e^{int}}{n^{s}}
  7. n t h n^{th}
  8. 1 n 1.07 \frac{1}{n^{1.07}}
  9. f ( k ; s , N ) = 1 / k s n = 1 N ( 1 / n s ) . f(k;s,N)=\frac{1/k^{s}}{\sum_{n=1}^{N}(1/n^{s})}.
  10. p ( f ) = α f - 1 - 1 / s . p(f)=\alpha f^{-1-1/s}.
  11. f ( k ; s , N ) = 1 k s H N , s f(k;s,N)=\frac{1}{k^{s}H_{N,s}}
  12. n = 1 1 n = . \sum_{n=1}^{\infty}\frac{1}{n}=\infty.\!
  13. ζ ( s ) = n = 1 1 n s < . \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}<\infty.\!
  14. f ( k ; N , q , s ) = [ constant ] ( k + q ) s . f(k;N,q,s)=\frac{[\mbox{constant}~{}]}{(k+q)^{s}}.\,
  15. f ( k ; ρ ) [ constant ] k ρ + 1 f(k;\rho)\approx\frac{[\mbox{constant}~{}]}{k^{\rho+1}}
  16. n n
  17. P ( n ) = P(n)=
  18. log 10 ( n + 1 ) - log 10 ( n ) \log_{10}(n+1)-\log_{10}(n)
  19. log ( P ( n ) / P ( n - 1 ) ) log ( n / ( n - 1 ) ) \tfrac{\log(P(n)/P(n-1))}{\log(n/(n-1))}

Zorn's_lemma.html

  1. \Q \Q
  2. a b = 0 ab=0
  3. a , b a,b
  4. \Q / A \Q/A
  5. A A
  6. A A

Μ-law_algorithm.html

  1. x x
  2. F ( x ) = sgn ( x ) ln ( 1 + μ | x | ) ln ( 1 + μ ) - 1 x 1 F(x)=\operatorname{sgn}(x)\frac{\ln(1+\mu|x|)}{\ln(1+\mu)}~{}~{}~{}~{}-1\leq x\leq 1
  3. μ = 255 μ=255
  4. F - 1 ( y ) = sgn ( y ) ( 1 / μ ) ( ( 1 + μ ) | y | - 1 ) - 1 y 1 F^{-1}(y)=\operatorname{sgn}(y)(1/\mu)((1+\mu)^{|y|}-1)~{}~{}~{}~{}-1\leq y\leq 1

Μ-recursive_function.html

  1. n n\,
  2. k k\,
  3. f ( x 1 , , x k ) = n f(x_{1},\ldots,x_{k})=n\,
  4. S ( x ) = def f ( x ) = x + 1 S(x)\stackrel{\mathrm{def}}{=}f(x)=x+1\,
  5. P i k P_{i}^{k}
  6. I i k I_{i}^{k}
  7. i , k i,k\,
  8. 1 i k 1\leq i\leq k
  9. P i k = def f ( x 1 , , x k ) = x i P_{i}^{k}\stackrel{\mathrm{def}}{=}f(x_{1},\ldots,x_{k})=x_{i}
  10. \circ\,
  11. h ( x 1 , , x m ) h(x_{1},\ldots,x_{m})\,
  12. g 1 ( x 1 , , x k ) , , g m ( x 1 , , x k ) g_{1}(x_{1},\ldots,x_{k}),\ldots,g_{m}(x_{1},\ldots,x_{k})
  13. h ( g 1 , , g m ) = def f ( x 1 , , x k ) = h ( g 1 ( x 1 , , x k ) , , g m ( x 1 , , x k ) ) h\circ(g_{1},\ldots,g_{m})\stackrel{\mathrm{def}}{=}f(x_{1},\ldots,x_{k})=h(g_% {1}(x_{1},\ldots,x_{k}),\ldots,g_{m}(x_{1},\ldots,x_{k}))\,
  14. ρ \rho\,
  15. g ( x 1 , , x k ) g(x_{1},\ldots,x_{k})\,
  16. h ( y , z , x 1 , , x k ) h(y,z,x_{1},\ldots,x_{k})\,
  17. ρ ( g , h ) = def f ( y , x 1 , , x k ) where f ( 0 , x 1 , , x k ) = g ( x 1 , , x k ) f ( y + 1 , x 1 , , x k ) = h ( y , f ( y , x 1 , , x k ) , x 1 , , x k ) \begin{aligned}\displaystyle\rho(g,h)&\displaystyle\stackrel{\mathrm{def}}{=}f% (y,x_{1},\ldots,x_{k})\quad{\rm where}\\ \displaystyle f(0,x_{1},\ldots,x_{k})&\displaystyle=g(x_{1},\ldots,x_{k})\\ \displaystyle f(y+1,x_{1},\ldots,x_{k})&\displaystyle=h(y,f(y,x_{1},\ldots,x_{% k}),x_{1},\ldots,x_{k})\end{aligned}
  18. μ \mu\,
  19. f ( y , x 1 , , x k ) f(y,x_{1},\ldots,x_{k})\,
  20. μ ( f ) ( x 1 , , x k ) = z def f ( z , x 1 , , x k ) = 0 and f ( i , x 1 , , x k ) > 0 for i = 0 , , z - 1. \begin{aligned}\displaystyle\mu(f)(x_{1},\ldots,x_{k})=z\stackrel{\mathrm{def}% }{\iff}\ f(z,x_{1},\ldots,x_{k})&\displaystyle=0\quad\,\text{and}\\ \displaystyle f(i,x_{1},\ldots,x_{k})&\displaystyle>0\quad\,\text{for}\ i=0,% \ldots,z-1.\end{aligned}
  21. \simeq
  22. f ( x 1 , , x k ) g ( x 1 , , x l ) f(x_{1},\ldots,x_{k})\simeq g(x_{1},\ldots,x_{l})
  23. U ( y ) U(y)\!
  24. T ( y , e , x 1 , , x k ) T(y,e,x_{1},\ldots,x_{k})\!
  25. f ( x 1 , , x k ) f(x_{1},\ldots,x_{k})\!
  26. f ( x 1 , , x k ) U ( μ y T ( y , e , x 1 , , x k ) ) f(x_{1},\ldots,x_{k})\simeq U(\mu y\,T(y,e,x_{1},\ldots,x_{k}))

Genetic_algorithm.html

  1. \frac{1}{N}\sum_{1\leq i\leq N}~f\left(\xi^i_n\right)~\approx_{N\uparrow\infty}~\sum_{x\in S}~f(x)~\pi_{\beta_n}(x)\quad\mbox{and the unbiased approximation}~\quad