wpmath0000002_21

Vortex.html

  1. ω \vec{\omega}
  2. × u \nabla\times\vec{\mathit{u}}
  3. \nabla
  4. u \vec{\mathit{u}}
  5. ω \vec{\omega}
  6. ω \vec{\omega}
  7. Ω = ( 0 , 0 , Ω ) , r = ( x , y , 0 ) , \vec{\Omega}=(0,0,\Omega),\quad\vec{r}=(x,y,0),
  8. u = Ω × r = ( - Ω y , Ω x , 0 ) , \vec{u}=\vec{\Omega}\times\vec{r}=(-\Omega y,\Omega x,0),
  9. ω = × u = ( 0 , 0 , 2 Ω ) = 2 Ω . \vec{\omega}=\nabla\times\vec{u}=(0,0,2\Omega)=2\vec{\Omega}.
  10. ω \vec{\omega}
  11. Ω = ( 0 , 0 , α r - 2 ) , r = ( x , y , 0 ) , \vec{\Omega}=(0,0,\alpha r^{-2}),\quad\vec{r}=(x,y,0),
  12. u = Ω × r = ( - α y r - 2 , α x r - 2 , 0 ) , \vec{u}=\vec{\Omega}\times\vec{r}=(-\alpha yr^{-2},\alpha xr^{-2},0),
  13. ω = × u = 0. \vec{\omega}=\nabla\times\vec{u}=0.
  14. Γ \Gamma
  15. u θ = Γ / ( 2 π r ) u_{\theta}=\Gamma/(2\pi r)
  16. r u θ = Γ / ( 2 π ) ru_{\theta}=\Gamma/(2\pi)
  17. ω \vec{\omega}
  18. u θ = ( 1 - e - r 2 / ( 4 ν t ) ) Γ / ( 2 π r ) . u_{\theta}=(1-e^{-r^{2}/(4\nu t)})\Gamma/(2\pi r).
  19. ω \vec{\omega}

Vortex_generator.html

  1. V s 0 V_{s0}

Vorticity.html

  1. ω × u , \vec{\omega}\equiv\nabla\times\vec{u}\,,
  2. v v
  3. r r
  4. v v
  5. r r
  6. v v
  7. r r
  8. ω = × v = ( x , y , z ) × ( v x , v y , v z ) = ( v z y - v y z , v x z - v z x , v y x - v x y ) \begin{array}[]{rcl}\vec{\omega}&=&\nabla\times\vec{v}\;=\;\left(\frac{% \partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}% \right)\times(v_{x},v_{y},v_{z})\\ &=&\left(\frac{\partial v_{z}}{\partial y}-\frac{\partial v_{y}}{\partial z},% \;\frac{\partial v_{x}}{\partial z}-\frac{\partial v_{z}}{\partial x},\;\frac{% \partial v_{y}}{\partial x}-\frac{\partial v_{x}}{\partial y}\right)\end{array}
  9. ω = × v = ( x , y , z ) × ( v x , v y , 0 ) = ( v y x - v x y ) z . \vec{\omega}\;=\;\nabla\times\vec{v}\;=\;\left(\tfrac{\partial}{\partial x},% \tfrac{\partial}{\partial y},\tfrac{\partial}{\partial z}\right)\times(v_{x},v% _{y},0)\;=\;\left(\frac{\partial v_{y}}{\partial x}-\frac{\partial v_{x}}{% \partial y}\right)\vec{z}\,.

Vorticity_equation.html

  1. D ω D t \displaystyle\frac{D\vec{\omega}}{Dt}
  2. d ω d t = ( ω ) v + ν 2 ω {d\vec{\omega}\over dt}=(\vec{\omega}\cdot\nabla)\vec{v}+\nu\nabla^{2}\vec{\omega}
  3. ρ t + ( ρ u ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{u})=0
  4. u = - 1 ρ d ρ d t = 1 v d v d t \nabla\cdot\vec{u}=-\frac{1}{\rho}\frac{d\rho}{dt}=\frac{1}{v}\frac{dv}{dt}
  5. d d t ( ω ρ ) = ( ω ρ ) u \frac{d}{dt}\left(\frac{\vec{\omega}}{\rho}\right)=\left(\frac{\vec{\omega}}{% \rho}\right)\cdot\nabla\vec{u}
  6. d ω d t = ( ω ) u \frac{d\vec{\omega}}{dt}=(\vec{\omega}\cdot\nabla)\vec{u}
  7. d u d t = u t + ( u ) u = - 1 ρ p + B + τ ρ \frac{d\vec{u}}{dt}=\frac{\partial\vec{u}}{\partial t}+(\vec{u}\cdot\nabla)% \vec{u}=-\frac{1}{\rho}\nabla p+\vec{B}+\frac{\nabla\cdot\tau}{\rho}
  8. ω = × u \vec{\omega}=\nabla\times\vec{u}
  9. ( u ) u = ( 1 2 u u ) - u × ω (\vec{u}\cdot\nabla)\vec{u}=\nabla(\tfrac{1}{2}\vec{u}\cdot\vec{u})-\vec{u}% \times\vec{\omega}
  10. × ( u × ω ) = - ω ( u ) + ( ω ) u - ( u ) ω \nabla\times(\vec{u}\times\vec{\omega})=-\vec{\omega}(\nabla\cdot\vec{u})+(% \vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega}
  11. × ϕ = 0 \nabla\times\nabla\phi=0
  12. ω = 0 \nabla\cdot\vec{\omega}=0
  13. d ω i d t \displaystyle\frac{d\omega_{i}}{dt}
  14. d η d t = - η h v h - ( ω x v z - ω y u z ) - 1 ρ 2 k ( h p × h ρ ) \frac{d\eta}{dt}=-\eta\nabla_{h}\cdot\vec{v}_{h}-\left(\frac{\partial\omega}{% \partial x}\frac{\partial v}{\partial z}-\frac{\partial\omega}{\partial y}% \frac{\partial u}{\partial z}\right)-\frac{1}{\rho^{2}}\vec{k}\cdot(\nabla_{h}% p\times\nabla_{h}\rho)

W._D._Hamilton.html

  1. C < r × B C<r\times B

W._Edwards_Deming.html

  1. Quality = Results of work efforts Total costs \,\text{Quality}=\frac{\,\text{Results of work efforts}}{\,\text{Total costs}}

Wake.html

  1. F r = U / g L Fr=U/\sqrt{gL}
  2. ω = g k , \omega=\sqrt{gk},
  3. π \pi
  4. v v

Walter_H._Schottky.html

  1. E i n t ( x ) = - q 2 16 π ϵ 0 x E_{int}(x)=-\frac{q^{2}}{16\pi\epsilon_{0}{x}}
  2. d 2 d x 2 Ψ ( x ) = 2 m 2 M ( x ) Ψ ( x ) . \frac{d^{2}}{dx^{2}}\Psi(x)=\frac{2m}{\hbar^{2}}M(x)\Psi(x).
  3. \hbar
  4. M ( x ) = h - e F x - e 2 / 4 π ϵ 0 ϵ r x . M(x)=\;h-eFx-e^{2}/4\pi\epsilon_{0}\epsilon_{r}x\;.

Warrant_(finance).html

  1. P 0 P_{0}
  2. P 0 - ( t = 1 T C ( 1 + r ) t ) - F ( 1 + r ) T . P_{0}-\left(\sum_{t=1}^{T}\frac{C}{(1+r)^{t}}\right)-\frac{F}{(1+r)^{T}}.

Wasting.html

  1. WFH = weight of a given child median weight for a given child of that height × 100 \mathrm{WFH}=\frac{\mbox{weight of a given child}~{}}{\mbox{median weight for % a given child of that height}~{}}\times 100

Water_vapor.html

  1. log 10 ( p ) = \log_{10}\left(p\right)=
  2. - 7.90298 ( 373.16 T - 1 ) + 5.02808 log 10 373.16 T -7.90298\left(\frac{373.16}{T}-1\right)+5.02808\log_{10}\frac{373.16}{T}
  3. - 1.3816 × 10 - 7 ( 10 11.344 ( 1 - T 373.16 ) - 1 ) -1.3816\times 10^{-7}\left(10^{11.344\left(1-\frac{T}{373.16}\right)}-1\right)
  4. + 8.1328 × 10 - 3 ( 10 - 3.49149 ( 373.16 T - 1 ) - 1 ) +8.1328\times 10^{-3}\left(10^{-3.49149\left(\frac{373.16}{T}-1\right)}-1\right)
  5. + log 10 ( 1013.246 ) +\log_{10}\left(1013.246\right)

Wave_function.html

  1. ψ ψ
  2. Ψ Ψ
  3. 3 / 2 {3}/{2}
  4. 2 2
  5. 2 × 1 2×1
  6. 1 / 2 {1}/{2}
  7. ψ ψ
  8. ψ ψ
  9. E = h f E=hf
  10. λ = h / p λ=h/p
  11. λ = h / p λ=h/p
  12. 0
  13. 1 / 2 {1}/{2}
  14. 1 1
  15. 3 / 2 {3}/{2}
  16. 1 1
  17. 2 2
  18. Ψ Ψ
  19. Φ Φ
  20. a Ψ + b Φ aΨ+bΦ
  21. z z
  22. y y
  23. x , y x,y
  24. z z
  25. ( n , l , m ) (n,l,m)
  26. ( n , l , m ) (n,l,m)
  27. P ( Ψ Φ i ) = | ( Ψ , Φ i ) | 2 P(\Psi\rightarrow\Phi_{i})=|(\Psi,\Phi_{i})|^{2}
  28. i i
  29. Ψ Ψ
  30. i i
  31. Ψ = I Ψ = Φ x ( Φ x , Ψ ) d x = Ψ ( x ) Φ x d x = Φ p ( Φ p , Ψ ) d p = Ψ ( p ) Φ p d p , \Psi=I\Psi=\int\Phi_{x}(\Phi_{x},\Psi)dx=\int\Psi(x)\Phi_{x}dx=\int\Phi_{p}(% \Phi_{p},\Psi)dp=\int\Psi(p)\Phi_{p}dp,
  32. Ψ Ψ
  33. x x
  34. ( · , · ) (·,·)
  35. p p
  36. I I
  37. Ψ ( x ) Ψ(x)
  38. Ψ ( p ) Ψ(p)
  39. | Ψ = I | Ψ = | x x | Ψ d x = Ψ ( x ) | x d x = | p p | Ψ d p = Ψ ( p ) | p d p . |\Psi\rangle=I|\Psi\rangle=\int|x\rangle\langle x|\Psi\rangle dx=\int\Psi(x)|x% \rangle dx=\int|p\rangle\langle p|\Psi\rangle dp=\int\Psi(p)|p\rangle dp.
  40. 0
  41. - i -i
  42. 0 , 11 0,11
  43. Ψ ( x , t ) , \Psi(x,t)\,,
  44. x x
  45. t t
  46. x x
  47. t t
  48. | Ψ ( x , t ) | 2 = Ψ ( x , t ) * Ψ ( x , t ) = ρ ( x , t ) , \left|\Psi(x,t)\right|^{2}={\Psi(x,t)}^{*}\Psi(x,t)=\rho(x,t),
  49. x x
  50. x x
  51. a x b a≤x≤b
  52. P a x b ( t ) = a b d x | Ψ ( x , t ) | 2 P_{a\leq x\leq b}(t)=\int\limits_{a}^{b}dx\,|\Psi(x,t)|^{2}
  53. t t
  54. - d x | Ψ ( x , t ) | 2 = 1 , \int\limits_{-\infty}^{\infty}dx\,|\Psi(x,t)|^{2}=1\,,
  55. n = 1 , 2 , n=1,2,...
  56. n a n Ψ n ( x , t ) = a 1 Ψ 1 ( x , t ) + a 2 Ψ 2 ( x , t ) + \sum_{n}a_{n}\Psi_{n}(x,t)=a_{1}\Psi_{1}(x,t)+a_{2}\Psi_{2}(x,t)+\cdots
  57. Ψ Ψ
  58. c c
  59. c Ψ
  60. c c
  61. c Ψ
  62. Φ ( p , t ) \Phi(p,t)
  63. p p
  64. −∞
  65. + +∞
  66. t t
  67. P a p b ( t ) = a b d p | Φ ( p , t ) | 2 , P_{a\leq p\leq b}(t)=\int\limits_{a}^{b}dp\,|\Phi(p,t)|^{2}\,,
  68. - d p | Φ ( p , t ) | 2 = 1 . \int\limits_{-\infty}^{\infty}dp\,\left|\Phi\left(p,t\right)\right|^{2}=1\,.
  69. Φ ( p , t ) = 1 2 π - d x e - i p x / Ψ ( x , t ) Ψ ( x , t ) = 1 2 π - d p e i p x / Φ ( p , t ) \Phi(p,t)=\frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^{\infty}dx\,e^{-ipx/% \hbar}\Psi(x,t)\quad\rightleftharpoons\quad\Psi(x,t)=\frac{1}{\sqrt{2\pi\hbar}% }\int\limits_{-\infty}^{\infty}dp\,e^{ipx/\hbar}\Phi(p,t)
  70. x x
  71. p p
  72. Ψ ( 𝐫 , t ) \Psi(\mathbf{r},t)
  73. 𝐫 \mathbf{r}
  74. t t
  75. Ψ ( 𝐫 , t ) Ψ(\mathbf{r},t)
  76. N N
  77. Ψ ( 𝐫 1 , 𝐫 2 𝐫 N , t ) \Psi(\mathbf{r}_{1},\mathbf{r}_{2}\cdots\mathbf{r}_{N},t)
  78. i i
  79. t t
  80. 3 N + 1 3N+1
  81. Ψ ( 𝐫 a , , 𝐫 b , ) = ± Ψ ( 𝐫 b , , 𝐫 a , ) \Psi\left(\ldots\mathbf{r}_{a},\ldots,\mathbf{r}_{b},\ldots\right)=\pm\Psi% \left(\ldots\mathbf{r}_{b},\ldots,\mathbf{r}_{a},\ldots\right)
  82. + +
  83. N N
  84. Ψ ( 𝐫 a , , 𝐫 b , , 𝐱 1 , 𝐱 2 , ) = ± Ψ ( 𝐫 b , , 𝐫 a , , 𝐱 1 , 𝐱 2 , ) \Psi\left(\ldots\mathbf{r}_{a},\ldots,\mathbf{r}_{b},\ldots,\mathbf{x}_{1},% \mathbf{x}_{2},\ldots\right)=\pm\Psi\left(\ldots\mathbf{r}_{b},\ldots,\mathbf{% r}_{a},\ldots,\mathbf{x}_{1},\mathbf{x}_{2},\ldots\right)
  85. Ψ ( 𝐫 , t , s z ) \Psi(\mathbf{r},t,s_{z})
  86. 𝐫 \mathbf{r}
  87. t t
  88. z z
  89. z z
  90. 𝐫 \mathbf{r}
  91. t t
  92. + 1 / 2 +1/2
  93. 1 / 2 −1/2
  94. s s
  95. s , s 1 , , s + 1 , s s,s−1,...,−s+1,−s
  96. Ψ ( 𝐫 , t ) = [ Ψ ( 𝐫 , t , s ) Ψ ( 𝐫 , t , s - 1 ) Ψ ( 𝐫 , t , - ( s - 1 ) ) Ψ ( 𝐫 , t , - s ) ] \Psi(\mathbf{r},t)=\begin{bmatrix}\Psi(\mathbf{r},t,s)\\ \Psi(\mathbf{r},t,s-1)\\ \vdots\\ \Psi(\mathbf{r},t,-(s-1))\\ \Psi(\mathbf{r},t,-s)\\ \end{bmatrix}
  97. Ψ ( 𝐫 1 , 𝐫 2 𝐫 N , s z 1 , s z 2 s z N , t ) \Psi(\mathbf{r}_{1},\mathbf{r}_{2}\cdots\mathbf{r}_{N},s_{z\,1},s_{z\,2}\cdots s% _{z\,N},t)
  98. N N
  99. Ψ Ψ
  100. ρ ( 𝐫 1 𝐫 N , s z 1 s z N , t ) = | Ψ ( 𝐫 1 𝐫 N , s z 1 s z N , t ) | 2 \rho\left(\mathbf{r}_{1}\cdots\mathbf{r}_{N},s_{z\,1}\cdots s_{z\,N},t\right)=% \left|\Psi\left(\mathbf{r}_{1}\cdots\mathbf{r}_{N},s_{z\,1}\cdots s_{z\,N},t% \right)\right|^{2}
  101. t t
  102. P 𝐫 1 R 1 , s z 1 = m 1 , , 𝐫 N R N , s z N = m N ( t ) = R 1 d 3 𝐫 1 R 2 d 3 𝐫 2 R N d 3 𝐫 N | Ψ ( 𝐫 1 𝐫 N , m 1 m N , t ) | 2 P_{\mathbf{r}_{1}\in R_{1},s_{z\,1}=m_{1},\ldots,\mathbf{r}_{N}\in R_{N},s_{z% \,N}=m_{N}}(t)=\int\limits_{R_{1}}d^{3}\mathbf{r}_{1}\int\limits_{R_{2}}d^{3}% \mathbf{r}_{2}\cdots\int\limits_{R_{N}}d^{3}\mathbf{r}_{N}\left|\Psi\left(% \mathbf{r}_{1}\cdots\mathbf{r}_{N},m_{1}\cdots m_{N},t\right)\right|^{2}
  103. N N
  104. Ψ ( 𝐫 1 , 𝐫 2 , , 𝐫 N , t , s z 1 , s z 2 , , s z N ) = e - i E t / ψ ( 𝐫 1 , 𝐫 2 , , 𝐫 N , s z 1 , s z 2 , , s z N ) , \Psi(\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{N},t,s_{z1},s_{z2},% \ldots,s_{zN})=e^{-iEt/\hbar}\psi(\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf% {r}_{N},s_{z1},s_{z2},\ldots,s_{zN})\,,
  105. E E
  106. Ψ Ψ
  107. ψ ψ
  108. ξ ξ
  109. Ψ ( 𝐫 , t , s z ) = ψ ( 𝐫 , t ) ξ ( s z ) = ϕ ( 𝐫 ) ζ ( s z , t ) . \Psi(\mathbf{r},t,s_{z})=\psi(\mathbf{r},t)\xi(s_{z})=\phi(\mathbf{r})\zeta(s_% {z},t)\,.
  110. Ψ ( 𝐫 , t , s z ) = e - i E t / ψ ( 𝐫 ) ξ ( s z ) , \Psi(\mathbf{r},t,s_{z})=e^{-iEt/\hbar}\psi(\mathbf{r})\xi(s_{z})\,,
  111. E E
  112. Ψ Ψ
  113. N N
  114. Ψ ( 𝐫 , t , s z ) = ψ ( 𝐫 1 , 𝐫 2 , , 𝐫 N , t ) ξ ( s z 1 , s z 2 , , s z N ) = ϕ ( 𝐫 1 , 𝐫 2 , , 𝐫 N ) ζ ( s z 1 , s z 2 , , s z N , t ) . \Psi(\mathbf{r},t,s_{z})=\psi(\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_% {N},t)\xi(s_{z1},s_{z2},\ldots,s_{zN})=\phi(\mathbf{r}_{1},\mathbf{r}_{2},% \ldots,\mathbf{r}_{N})\zeta(s_{z1},s_{z2},\ldots,s_{zN},t)\,.
  115. t t
  116. Ψ 1 , Ψ 2 = - d x Ψ 1 * ( x , t ) Ψ 2 ( x , t ) . \left\langle\Psi_{1},\Psi_{2}\right\rangle=\int\limits_{-\infty}^{\infty}dx\,% \Psi_{1}^{*}(x,t)\Psi_{2}(x,t).
  117. Ψ 1 , Ψ 2 = all space d 3 𝐫 Ψ 1 * ( 𝐫 , t ) Ψ 2 ( 𝐫 , t ) , \langle\Psi_{1},\Psi_{2}\rangle=\int\limits_{\mathrm{all\,space}}d^{3}\mathbf{% r}\,\Psi_{1}^{*}(\mathbf{r},t)\Psi_{2}(\mathbf{r},t)\,,
  118. Ψ 1 , Ψ 2 = all space d 3 𝐫 1 all space d 3 𝐫 2 all space d 3 𝐫 N Ψ 1 * ( 𝐫 1 𝐫 N , t ) Ψ 2 ( 𝐫 1 𝐫 N , t ) \langle\Psi_{1},\Psi_{2}\rangle=\int\limits_{\mathrm{all\,space}}d^{3}\mathbf{% r}_{1}\int\limits_{\mathrm{all\,space}}d^{3}\mathbf{r}_{2}\cdots\int\limits_{% \mathrm{all\,space}}d^{3}\mathbf{r}_{N}\,\Psi_{1}^{*}(\mathbf{r}_{1}\cdots% \mathbf{r}_{N},t)\Psi_{2}(\mathbf{r}_{1}\cdots\mathbf{r}_{N},t)
  119. N N
  120. Ψ 1 , Ψ 2 = all s z all space d 3 𝐫 Ψ 1 * ( 𝐫 , t , s z ) Ψ 2 ( 𝐫 , t , s z ) , \langle\Psi_{1},\Psi_{2}\rangle=\sum_{\mathrm{all\,}s_{z}}\int\limits_{\mathrm% {all\,space}}\,d^{3}\mathbf{r}\Psi^{*}_{1}(\mathbf{r},t,s_{z})\Psi_{2}(\mathbf% {r},t,s_{z})\,,
  121. N N
  122. Ψ 1 , Ψ 2 = s z N s z 2 s z 1 all space d 3 𝐫 1 all space d 3 𝐫 2 all space d 3 𝐫 N Ψ 1 * ( 𝐫 1 𝐫 N , s z 1 s z N , t ) Ψ 2 ( 𝐫 1 𝐫 N , s z 1 s z N , t ) \langle\Psi_{1},\Psi_{2}\rangle=\sum_{s_{z\,N}}\cdots\sum_{s_{z\,2}}\sum_{s_{z% \,1}}\int\limits_{\mathrm{all\,space}}d^{3}\mathbf{r}_{1}\int\limits_{\mathrm{% all\,space}}d^{3}\mathbf{r}_{2}\cdots\int\limits_{\mathrm{all\,space}}d^{3}% \mathbf{r}_{N}\Psi^{*}_{1}\left(\mathbf{r}_{1}\cdots\mathbf{r}_{N},s_{z\,1}% \cdots s_{z\,N},t\right)\Psi_{2}\left(\mathbf{r}_{1}\cdots\mathbf{r}_{N},s_{z% \,1}\cdots s_{z\,N},t\right)
  123. N N
  124. N N
  125. | Ψ 1 , Ψ 2 | 2 = P ( Ψ 2 Ψ 1 ) , \left|\left\langle\Psi_{1},\Psi_{2}\right\rangle\right|^{2}=P\left(\Psi_{2}% \rightarrow\Psi_{1}\right)\,,
  126. Ψ Ψ
  127. Ψ , Ψ = Ψ 2 , \langle\Psi,\Psi\rangle=\|\Psi\|^{2}\,,
  128. || Ψ || ||Ψ||
  129. Ψ Ψ
  130. | Ψ | |Ψ|
  131. Ψ , Ψ = 1 . \left\langle\Psi,\Psi\right\rangle=1\,.
  132. Ψ Ψ
  133. Ψ / || Ψ || Ψ/||Ψ||
  134. Ψ 1 , Ψ 2 = 0 . \left\langle\Psi_{1},\Psi_{2}\right\rangle=0\,.
  135. Ψ m , Ψ n = δ m n , \langle\Psi_{m},\Psi_{n}\rangle=\delta_{mn}\,,
  136. m m
  137. n n
  138. + 1 +1
  139. m = n m=n
  140. 0
  141. m n m≠n
  142. Ψ = n a n ψ n \Psi=\sum_{n}a_{n}\psi_{n}
  143. a n = ψ n , Ψ a_{n}=\langle\psi_{n},\Psi\rangle
  144. Φ 1 , Φ 2 = - d p Φ 1 * ( p , t ) Φ 2 ( p , t ) , \langle\Phi_{1},\Phi_{2}\rangle=\int\limits_{-\infty}^{\infty}dp\,\Phi_{1}^{*}% (p,t)\Phi_{2}(p,t)\,,
  145. Ψ p ( x ) = e i p x / , \Psi_{p}(x)=e^{ipx/\hbar},
  146. p p
  147. { Ψ p ( x , t ) , - p } \{\Psi_{p}(x,t),-\infty\leq p\leq\infty\}
  148. Ψ p , Ψ p = δ ( p - p ) . \langle\Psi_{p},\Psi_{p^{\prime}}\rangle=\delta(p-p^{\prime}).
  149. i i
  150. ψ ψ
  151. N N
  152. n n
  153. | ψ | |ψ|
  154. ψ ψ
  155. | Ψ |\Psi\rangle
  156. | Ψ |Ψ\rangle
  157. 1 1
  158. | ψ |ψ\rangle
  159. | ϕ |ϕ\rangle
  160. a a
  161. b b
  162. | Ψ = a | ψ + b | ϕ |\Psi\rangle=a|\psi\rangle+b|\phi\rangle
  163. Ψ 1 | Ψ 2 , \langle\Psi_{1}|\Psi_{2}\rangle,
  164. Ψ | \langle Ψ|
  165. Ψ | = a * ψ | + b * ϕ | a | ψ + b | ϕ = | Ψ , \langle\Psi|=a^{*}\langle\psi|+b^{*}\langle\phi|\leftrightarrow a|\psi\rangle+% b|\phi\rangle=|\Psi\rangle,
  166. Φ | Ψ = Φ | ( | Ψ ) , \langle\Phi|\Psi\rangle=\langle\Phi|(|\Psi\rangle),
  167. i d d t | Ψ = H ^ | Ψ i\hbar\frac{d}{dt}|\Psi\rangle=\hat{H}|\Psi\rangle
  168. ( a , b , l , m , ) (a,b,…l,m,…)
  169. | a , b , , l , m , . |a,b,\ldots,l,m,\ldots\rangle.
  170. | Ψ |Ψ\rangle
  171. a , b , , l , m , | Ψ , \langle a,b,\ldots,l,m,\ldots|\Psi\rangle,
  172. Ψ ( x ) = x | Ψ . \Psi(x)=\langle x|\Psi\rangle.
  173. i = 1 , 2... n i=1,2...n
  174. ε i | ε j = δ i j . \langle\varepsilon_{i}|\varepsilon_{j}\rangle=\delta_{ij}\,.
  175. | Ψ |Ψ\rangle
  176. | Ψ = i = 1 n c i | ε i = [ c 1 c n ] Ψ | = | Ψ = i = 1 n c i * ε i | = [ c 1 * c n * ] , |\Psi\rangle=\sum_{i=1}^{n}c_{i}|\varepsilon_{i}\rangle=\begin{bmatrix}c_{1}\\ \vdots\\ c_{n}\end{bmatrix}\,\quad\langle\Psi|=|\Psi\rangle^{\dagger}=\sum_{i=1}^{n}c^{% *}_{i}\langle\varepsilon_{i}|=\begin{bmatrix}c_{1}^{*}&\cdots&c_{n}^{*}\end{% bmatrix}\,,
  177. c i = ε i | Ψ c_{i}=\langle\varepsilon_{i}|\Psi\rangle
  178. | Ψ |Ψ\rangle
  179. ε | ε = δ ( ε - ε ) . \langle\varepsilon|\varepsilon^{\prime}\rangle=\delta(\varepsilon-\varepsilon^% {\prime})\,.
  180. ε ε
  181. | ε |ε\rangle
  182. ε ε
  183. ε ε
  184. | Ψ |Ψ\rangle
  185. | Ψ = d ε | ε Ψ ( ε ) , Ψ | = d ε ε | Ψ ( ε ) * , |\Psi\rangle=\int d\varepsilon|\varepsilon\rangle\Psi(\varepsilon)\,,\quad% \langle\Psi|=\int d\varepsilon\langle\varepsilon|{\Psi(\varepsilon)}^{*}\,,
  186. Ψ ( ε ) = ε | Ψ \Psi(\varepsilon)=\langle\varepsilon|\Psi\rangle
  187. ε ε
  188. i = 1 n | ε i ε i | = 1 , d ε | ε ε | = 1 \sum_{i=1}^{n}|\varepsilon_{i}\rangle\langle\varepsilon_{i}|=1\,,\quad\int d% \varepsilon\,|\varepsilon\rangle\langle\varepsilon|=1\,
  189. | Ψ |Ψ\rangle
  190. Ψ 1 | Ψ 2 = ( i z i * ε i | ) ( j c j | ε j ) = i j z i * c j ε i | ε j = i z i * c i . \langle\Psi_{1}|\Psi_{2}\rangle=\left(\sum_{i}z^{*}_{i}\langle\varepsilon_{i}|% \right)\left(\sum_{j}c_{j}|\varepsilon_{j}\rangle\right)=\sum_{ij}z^{*}_{i}c_{% j}\langle\varepsilon_{i}|\varepsilon_{j}\rangle=\sum_{i}z^{*}_{i}c_{i}\,.
  191. Ψ 1 | Ψ 2 = ( d ε Ψ 1 ( ε ) * ε | ) ( d ε Ψ 2 ( ε ) | ε ) = d ε d ε Ψ 1 ( ε ) * Ψ 2 ( ε ) ε | ε = d ε Ψ 1 ( ε ) * Ψ 2 ( ε ) . \langle\Psi_{1}|\Psi_{2}\rangle=\left(\int d\varepsilon^{\prime}{\Psi_{1}(% \varepsilon^{\prime})}^{*}\langle\varepsilon^{\prime}|\right)\left(\int d% \varepsilon\Psi_{2}(\varepsilon)|\varepsilon\rangle\right)=\int d\varepsilon^{% \prime}\int d\varepsilon{\Psi_{1}(\varepsilon^{\prime})}^{*}\Psi_{2}(% \varepsilon)\langle\varepsilon^{\prime}|\varepsilon\rangle=\int d\varepsilon{% \Psi_{1}(\varepsilon)}^{*}\Psi_{2}(\varepsilon)\,.
  192. ε ε
  193. ε ε′
  194. | Ψ |Ψ\rangle
  195. | Ψ |Ψ\rangle
  196. Ψ 2 = Ψ | Ψ = j = 1 n | c j | 2 , Ψ 2 = Ψ | Ψ = d ε | Ψ ( ε ) | 2 \|\Psi\|^{2}=\langle\Psi|\Psi\rangle=\sum_{j=1}^{n}|c_{j}|^{2}\,,\quad\|\Psi\|% ^{2}=\langle\Psi|\Psi\rangle=\int d\varepsilon\,|\Psi(\varepsilon)|^{2}
  197. | Ψ |Ψ\rangle
  198. | Ψ N = 1 Ψ | Ψ |\Psi_{N}\rangle=\frac{1}{\|\Psi\|}|\Psi\rangle
  199. ε q | Ψ N = ε q | 1 Ψ ( i = 1 n c i | ε i ) = c q Ψ , \langle\varepsilon_{q}|\Psi_{N}\rangle=\langle\varepsilon_{q}|\frac{1}{\|\Psi% \|}\left(\sum_{i=1}^{n}c_{i}|\varepsilon_{i}\rangle\right)=\frac{c_{q}}{\|\Psi% \|}\,,
  200. P ( ε q ) = | ε q | Ψ N | 2 = | c q | 2 Ψ 2 , P(\varepsilon_{q})=\left|\langle\varepsilon_{q}|\Psi_{N}\rangle\right|^{2}=% \frac{\left|c_{q}\right|^{2}}{\|\Psi\|^{2}}\,,
  201. | ε |ε′\rangle
  202. ε | Ψ N = ε | ( 1 Ψ d ε | ε Ψ ( ε ) ) = 1 Ψ d ε ε | ε Ψ ( ε ) = 1 Ψ d ε δ ( ε - ε ) Ψ ( ε ) = Ψ ( ε ) Ψ , \langle\varepsilon^{\prime}|\Psi_{N}\rangle=\langle\varepsilon^{\prime}|\left(% \frac{1}{\|\Psi\|}\int d\varepsilon|\varepsilon\rangle\Psi(\varepsilon)\right)% =\frac{1}{\|\Psi\|}\int d\varepsilon\langle\varepsilon^{\prime}|\varepsilon% \rangle\Psi(\varepsilon)=\frac{1}{\|\Psi\|}\int d\varepsilon\delta(\varepsilon% ^{\prime}-\varepsilon)\Psi(\varepsilon)=\frac{\Psi(\varepsilon^{\prime})}{\|% \Psi\|}\,,
  203. ρ ( ε ) = | ε | Ψ N | 2 = | Ψ ( ε ) | 2 Ψ 2 \rho(\varepsilon^{\prime})=\left|\langle\varepsilon^{\prime}|\Psi_{N}\rangle% \right|^{2}=\frac{\left|\Psi(\varepsilon^{\prime})\right|^{2}}{\|\Psi\|^{2}}
  204. ε ε′
  205. ε ε′
  206. a ε b a≤ε′≤b
  207. P a ε b = 1 Ψ 2 a b d ε | Ψ ( ε ) | 2 = 1 Ψ 2 a b d ε | ε | Ψ | 2 , P_{a\leq\varepsilon\leq b}=\frac{1}{\|\Psi\|^{2}}\int_{a}^{b}d\varepsilon^{% \prime}|\Psi(\varepsilon^{\prime})|^{2}=\frac{1}{\|\Psi\|^{2}}\int_{a}^{b}d% \varepsilon^{\prime}|\langle\varepsilon^{\prime}|\Psi\rangle|^{2}\,,
  208. ε ε′
  209. ε = a ε′=a
  210. ε = b ε′=b
  211. | Ψ |Ψ\rangle
  212. ε ε
  213. | Ψ |Ψ\rangle
  214. | Ψ |Ψ\rangle
  215. | Ψ ( t ) = i | ε i ε i | Ψ ( t ) = i c i ( t ) | ε i |\Psi(t)\rangle=\sum_{i}\,|\varepsilon_{i}\rangle\langle\varepsilon_{i}|\Psi(t% )\rangle=\sum_{i}c_{i}(t)|\varepsilon_{i}\rangle
  216. | Ψ ( t ) = d ε | ε ε | Ψ ( t ) = d ε Ψ ( ε , t ) | ε |\Psi(t)\rangle=\int d\varepsilon\,|\varepsilon\rangle\langle\varepsilon|\Psi(% t)\rangle=\int d\varepsilon\,\Psi(\varepsilon,t)|\varepsilon\rangle
  217. | Ψ |Ψ\rangle
  218. | Ψ |Ψ\rangle
  219. | Ψ ( t ) |Ψ(t)\rangle
  220. | Ψ |Ψ\rangle
  221. | Φ |Φ\rangle
  222. | Ψ | Φ |Ψ\rangle⊗|Φ\rangle
  223. | Ψ | Φ |Ψ\rangle|Φ\rangle
  224. | Ψ | Φ |Ψ\rangle|Φ\rangle
  225. | Ψ | Φ | Φ | Ψ |Ψ\rangle|Φ\rangle≠|Φ\rangle|Ψ\rangle
  226. | Ψ |Ψ\rangle
  227. | Φ |Φ\rangle
  228. | Ψ | Φ = ( i c i | ε i ) ( j z j | ε j ) = i , j c i z j | ε i | ε j |\Psi\rangle|\Phi\rangle=\left(\sum_{i}c_{i}|\varepsilon_{i}\rangle\right)% \left(\sum_{j}z_{j}|\varepsilon_{j}\rangle\right)=\sum_{i,j}c_{i}z_{j}|% \varepsilon_{i}\rangle|\varepsilon_{j}\rangle
  229. | A = | Ψ | Φ |A\rangle=|Ψ\rangle|Φ\rangle
  230. | A = i , j A i j | E i j |A\rangle=\sum_{i,j}A_{ij}|E_{ij}\rangle
  231. | Ψ |Ψ\rangle
  232. | x |x\rangle
  233. x x
  234. 1 = - d x | x x | 1=\int\limits_{-\infty}^{\infty}dx\,|x\rangle\langle x|
  235. x | x = δ ( x - x ) \langle x^{\prime}|x\rangle=\delta(x^{\prime}-x)
  236. | Ψ |Ψ\rangle
  237. | Ψ = ( - d x | x x | ) | Ψ = - d x | x x | Ψ = - d x Ψ ( x ) | x |\Psi\rangle=\left(\int\limits_{-\infty}^{\infty}dx\,|x\rangle\langle x|\right% )|\Psi\rangle=\int\limits_{-\infty}^{\infty}dx\,|x\rangle\langle x|\Psi\rangle% =\int\limits_{-\infty}^{\infty}dx\,\Psi(x)|x\rangle
  238. Ψ ( x ) = x | Ψ \Psi(x)=\langle x|\Psi\rangle
  239. Ψ 1 | Ψ 2 = Ψ 1 | ( - d x | x x | ) | Ψ 2 = - d x Ψ 1 | x x | Ψ 2 = - d x Ψ 1 ( x ) * Ψ 2 ( x ) . \langle\Psi_{1}|\Psi_{2}\rangle=\langle\Psi_{1}|\left(\int\limits_{-\infty}^{% \infty}dx\,|x\rangle\langle x|\right)|\Psi_{2}\rangle=\int\limits_{-\infty}^{% \infty}dx\,\langle\Psi_{1}|x\rangle\langle x|\Psi_{2}\rangle=\int\limits_{-% \infty}^{\infty}dx\,\Psi_{1}(x)^{*}\Psi_{2}(x)\,.
  240. R R
  241. R R
  242. | 𝐫 |\mathbf{r}\rangle
  243. | Ψ |Ψ\rangle
  244. | 𝐫 |\mathbf{r}\rangle
  245. | Ψ = all space d 3 𝐫 | 𝐫 𝐫 | Ψ |\Psi\rangle=\int\limits_{\mathrm{all\,space}}d^{3}\mathbf{r}|\mathbf{r}% \rangle\langle\mathbf{r}|\Psi\rangle
  246. 𝐫 | Ψ = Ψ ( 𝐫 ) \langle\mathbf{r}|\Psi\rangle=\Psi(\mathbf{r})
  247. N N
  248. | Ψ |Ψ\rangle
  249. | Ψ = all space d 3 𝐫 N all space d 3 𝐫 2 all space d 3 𝐫 1 | 𝐫 1 , 𝐫 2 , , 𝐫 N 𝐫 1 , 𝐫 2 , , 𝐫 N | Ψ |\Psi\rangle=\int\limits_{\mathrm{all\,space}}d^{3}\mathbf{r}_{N}\cdots\int% \limits_{\mathrm{all\,space}}d^{3}\mathbf{r}_{2}\int\limits_{\mathrm{all\,% space}}d^{3}\mathbf{r}_{1}|\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{N}% \rangle\langle\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{N}|\Psi\rangle
  250. 𝐫 1 , 𝐫 2 , , 𝐫 N | Ψ = Ψ ( 𝐫 1 , 𝐫 2 , , 𝐫 N ) \langle\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{N}|\Psi\rangle=\Psi(% \mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{N})
  251. | 𝐫 |\mathbf{r}\rangle
  252. | Ψ |Ψ\rangle
  253. | Ψ = s z all space d 3 𝐫 | 𝐫 , s z 𝐫 , s z | Ψ |\Psi\rangle=\sum_{s_{z}}\int\limits_{\mathrm{all\,space}}d^{3}\mathbf{r}|% \mathbf{r},s_{z}\rangle\langle\mathbf{r},s_{z}|\Psi\rangle
  254. 𝐫 , s z | Ψ = Ψ ( 𝐫 , s z ) \langle\mathbf{r},s_{z}|\Psi\rangle=\Psi(\mathbf{r},s_{z})
  255. N N
  256. | Ψ = s z 1 , , s z N all space d 3 𝐫 N all space d 3 𝐫 1 | 𝐫 1 , , 𝐫 N , s z 1 , , s z N 𝐫 1 , , 𝐫 N , s z 1 , , s z N | Ψ |\Psi\rangle=\sum_{s_{z\,1},\ldots,s_{z\,N}}\int\limits\limits_{\mathrm{all\,% space}}d^{3}\mathbf{r}_{N}\cdots\int\limits\limits_{\mathrm{all\,space}}d^{3}% \mathbf{r}_{1}\,|\mathbf{r}_{1},\ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N}% \rangle\langle\mathbf{r}_{1},\ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N}|\Psi\rangle
  257. 𝐫 1 , , 𝐫 N , s z 1 , , s z N | Ψ = Ψ ( 𝐫 1 , , 𝐫 N , s z 1 , , s z N ) \langle\mathbf{r}_{1},\ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N}|\Psi% \rangle=\Psi(\mathbf{r}_{1},\ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N})
  258. | Ψ = s z 1 , , s z N R N d 3 𝐫 N R 1 d 3 𝐫 1 Ψ ( 𝐫 1 , , 𝐫 N , s z 1 , , s z N ) | 𝐫 1 , , 𝐫 N , s z 1 , , s z N |\Psi\rangle=\sum_{s_{z\,1},\ldots,s_{z\,N}}\int\limits\limits_{R_{N}}d^{3}% \mathbf{r}_{N}\cdots\int\limits\limits_{R_{1}}d^{3}\mathbf{r}_{1}\,\Psi(% \mathbf{r}_{1},\ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N})|\mathbf{r}_{1},% \ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N}\rangle
  259. 𝐱 1 , , 𝐱 N , m 1 , , m N | 𝐫 1 , , 𝐫 N , s z 1 , , s z N = δ m 1 s z 1 δ m N s z N δ ( 𝐱 1 - 𝐫 1 ) δ ( 𝐱 N - 𝐫 N ) \langle\mathbf{x}_{1},\ldots,\mathbf{x}_{N},m_{1},\ldots,m_{N}|\mathbf{r}_{1},% \ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N}\rangle=\delta_{m_{1}\,s_{z\,1}}% \cdots\delta_{m_{N}\,s_{z\,N}}\delta(\mathbf{x}_{1}-\mathbf{r}_{1})\cdots% \delta(\mathbf{x}_{N}-\mathbf{r}_{N})
  260. Ψ 1 | Ψ 2 = s z 1 , , s z N R N d 3 𝐫 N R 1 d 3 𝐫 1 Ψ 1 ( 𝐫 1 , , 𝐫 N , s z 1 , , s z N ) * Ψ 2 ( 𝐫 1 , , 𝐫 N , s z 1 , , s z N ) . \langle\Psi_{1}|\Psi_{2}\rangle=\sum_{s_{z\,1},\ldots,s_{z\,N}}\int\limits% \limits_{R_{N}}d^{3}\mathbf{r}_{N}\cdots\int\limits\limits_{R_{1}}d^{3}\mathbf% {r}_{1}\Psi_{1}(\mathbf{r}_{1},\ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N})% ^{*}\Psi_{2}(\mathbf{r}_{1},\ldots,\mathbf{r}_{N},s_{z\,1},\ldots,s_{z\,N})\,.
  261. | 𝐩 |\mathbf{p}\rangle
  262. 𝐤 \mathbf{k}
  263. ω ω
  264. Ψ ( 𝐫 , t ) = A e i ( 𝐤 𝐫 - ω t ) . \Psi(\mathbf{r},t)=Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}\,.
  265. x = 0 x=0
  266. x = L x=L
  267. Ψ ( x , t ) \displaystyle\Psi(x,t)
  268. A A
  269. - d x | Ψ | 2 = 1. \int\limits_{-\infty}^{\infty}dx\,|\Psi|^{2}=1.
  270. Ψ Ψ
  271. - 0 d x 0 + 0 L d x A 2 + L d x 0 = 1 , \int\limits_{-\infty}^{0}dx\cdot 0+\int\limits_{0}^{L}dx\,A^{2}+\int\limits_{L% }^{\infty}dx\cdot 0=1,
  272. A = 1 / L A=1/\sqrt{L}
  273. Ψ ( x , t ) = 1 L e i ( k x - ω t ) , 0 x L . \Psi(x,t)=\frac{1}{\sqrt{L}}e^{i(kx-\omega t)},\quad 0\leq x\leq L\,.
  274. V 0 V_{0}
  275. V ( x ) = { V 0 | x | < a 0 otherwise, V(x)=\begin{cases}V_{0}&|x|<a\\ 0&\,\text{otherwise,}\end{cases}
  276. k , κ k,\kappa
  277. ψ ( x ) = { A r exp ( i k x ) + A l exp ( - i k x ) x < - a , B r exp ( κ x ) + B l exp ( - κ x ) | x | a , C r exp ( i k x ) + C l exp ( - i k x ) x > a . \psi(x)=\begin{cases}A_{\mathrm{r}}\exp(ikx)+A_{\mathrm{l}}\exp(-ikx)&x<-a,\\ B_{\mathrm{r}}\exp(\kappa x)+B_{\mathrm{l}}\exp(-\kappa x)&|x|\leq a,\\ C_{\mathrm{r}}\exp(ikx)+C_{\mathrm{l}}\exp(-ikx)&x>a.\end{cases}
  278. x x
  279. A < s u b > r = 1 A<sub>r=1
  280. L < s u p > 2 L<sup>2
  281. L < s u p > 2 L<sup>2

Wave_function_collapse.html

  1. | ψ = i c i | ϕ i . |\psi\rangle=\sum_{i}c_{i}|\phi_{i}\rangle.
  2. | ϕ 1 , | ϕ 2 , | ϕ 3 \scriptstyle{|\phi_{1}\rangle,|\phi_{2}\rangle,|\phi_{3}\rangle\cdots}
  3. ϕ i | ϕ j = δ i j . \langle\phi_{i}|\phi_{j}\rangle=\delta_{ij}.
  4. δ i j \delta_{ij}
  5. | 𝐫 , t = | x , t + | y , t + | z , t , | 𝐩 , t = | p x , t + | p y , t + | p z , t , | E , | s z , | L z , | J z , \scriptstyle{|\mathbf{r},t\rangle=|x,t\rangle+|y,t\rangle+|z,t\rangle,|\mathbf% {p},t\rangle=|p_{x},t\rangle+|p_{y},t\rangle+|p_{z},t\rangle,|E\rangle,|s_{z}% \rangle,|L_{z}\rangle,|J_{z}\rangle,\cdots}
  6. | ϕ 1 , | ϕ 2 , | ϕ 3 \scriptstyle{|\phi_{1}\rangle,|\phi_{2}\rangle,|\phi_{3}\rangle\cdots}
  7. | ϕ i \scriptstyle|\phi_{i}\rangle
  8. ψ | ψ = i | c i | 2 = 1. \langle\psi|\psi\rangle=\sum_{i}|c_{i}|^{2}=1.
  9. { | ϕ i } \{|\phi_{i}\rangle\}
  10. { | ϕ i } \{|\phi_{i}\rangle\}
  11. | ψ |\psi\rangle
  12. | ϕ i |\phi_{i}\rangle
  13. | ψ | ϕ i . |\psi\rangle\rightarrow|\phi_{i}\rangle.
  14. | ϕ k |\phi_{k}\rangle
  15. P k = | c k | 2 P_{k}=|c_{k}|^{2}
  16. c i k c_{i\neq k}
  17. c k = 1 c_{k}=1
  18. Q ^ \hat{Q}
  19. { | ϕ i } \{|\phi_{i}\rangle\}
  20. | ψ |\psi\rangle
  21. Q ^ \hat{Q}
  22. | ϕ i |\phi_{i}\rangle
  23. | ϕ i |\phi_{i}\rangle
  24. | ψ | ϕ i | 2 |\langle\psi|\phi_{i}\rangle|^{2}
  25. | ϕ i |\phi_{i}\rangle
  26. | ψ |\psi\rangle
  27. Q ^ \hat{Q}
  28. c ( q , t ) d q c(q,t)dq
  29. λ i \lambda_{i}

Wavenumber.html

  1. ν ~ = 1 λ \scriptstyle\tilde{\nu}\;=\;\frac{1}{\lambda}
  2. k = 2 π λ \scriptstyle k\;=\;\frac{2\pi}{\lambda}
  3. ν ~ \scriptstyle\tilde{\nu}
  4. ν s c = 1 / λ ν ~ \scriptstyle\frac{\nu_{s}}{c}\;=\;1/{\lambda}\;\equiv\;\tilde{\nu}
  5. k = k 0 ε r = k 0 n k=k_{0}\sqrt{\varepsilon_{r}}=k_{0}n
  6. k k
  7. k = 2 π λ = 2 π ν v p = ω v p k=\frac{2\pi}{\lambda}=\frac{2\pi\nu}{v_{\mathrm{p}}}=\frac{\omega}{v_{\mathrm% {p}}}
  8. ν \nu
  9. λ \lambda
  10. ω = 2 π ν \omega=2\pi\nu
  11. v p v_{\mathrm{p}}
  12. k = E c k=\frac{E}{\hbar c}
  13. k 2 π λ = p = 2 m E k\equiv\frac{2\pi}{\lambda}=\frac{p}{\hbar}=\frac{\sqrt{2mE}}{\hbar}
  14. ν ~ \scriptstyle\tilde{\nu}
  15. ν ~ = ν c = ω 2 π c \tilde{\nu}=\frac{\nu}{c}=\frac{\omega}{2\pi c}
  16. λ vac = 1 ν ~ , \lambda_{\rm vac}=\frac{1}{\tilde{\nu}},
  17. ν ~ = R ( 1 n f 2 - 1 n i 2 ) \tilde{\nu}=R\left(\frac{1}{{n_{f}}^{2}}-\frac{1}{{n_{i}}^{2}}\right)
  18. E = h c ν ~ E=hc\tilde{\nu}
  19. λ = 1 n ν ~ \lambda=\frac{1}{n\tilde{\nu}}
  20. ν ~ \tilde{\nu}
  21. c m < s u p > - 1 cm<sup>-1

Wavenumber–frequency_diagram.html

  1. v p = λ ν v\text{p}=\lambda\nu
  2. v p = ν k v\text{p}=\frac{\nu}{k}
  3. v g = ν k v\text{g}=\frac{\partial\nu}{\partial k}

Weakly_compact_cardinal.html

  1. Π 1 1 \Pi^{1}_{1}

Web_Ontology_Language.html

  1. 𝒮 𝒪 𝒩 ( 𝒟 ) \mathcal{SHOIN}^{\mathcal{(D)}}
  2. 𝒮 𝒪 𝒬 ( 𝒟 ) \mathcal{SROIQ}^{\mathcal{(D)}}
  3. \sqsubseteq
  4. \top
  5. \bot
  6. \sqsubseteq
  7. \sqsubseteq
  8. \sqsubseteq

Weber–Fechner_law.html

  1. d p = k d S S , dp=k\frac{dS}{S},\,\!
  2. p = k ln S + C , p=k\ln{S}+C,\,\!
  3. C C
  4. C C
  5. p = 0 p=0
  6. k ln S 0 k\ln{S_{0}}
  7. C = - k ln S 0 , C=-k\ln{S_{0}},\,\!
  8. S 0 S_{0}
  9. C C
  10. p = k ln S S 0 . p=k\ln{\frac{S}{S_{0}}}.\,\!

Weibull_distribution.html

  1. { 1 - e - ( x / λ ) k x 0 0 x < 0 \begin{cases}1-e^{-(x/\lambda)^{k}}&x\geq 0\\ 0&x<0\end{cases}
  2. λ Γ ( 1 + 1 / k ) \lambda\,\Gamma(1+1/k)\,
  3. λ ( ln ( 2 ) ) 1 / k \lambda(\ln(2))^{1/k}\,
  4. { λ ( k - 1 k ) 1 k k > 1 0 k = 1 \begin{cases}\lambda\left(\frac{k-1}{k}\right)^{\frac{1}{k}}&k>1\\ 0&k=1\end{cases}
  5. λ k - 1 k 1 k \lambda\frac{k-1}{k}^{\frac{1}{k}}\,
  6. k > 1 k>1
  7. λ 2 [ Γ ( 1 + 2 k ) - ( Γ ( 1 + 1 k ) ) 2 ] \lambda^{2}\left[\Gamma\left(1+\frac{2}{k}\right)-\left(\Gamma\left(1+\frac{1}% {k}\right)\right)^{2}\right]\,
  8. Γ ( 1 + 3 / k ) λ 3 - 3 μ σ 2 - μ 3 σ 3 \frac{\Gamma(1+3/k)\lambda^{3}-3\mu\sigma^{2}-\mu^{3}}{\sigma^{3}}
  9. γ ( 1 - 1 / k ) + ln ( λ / k ) + 1 \gamma(1-1/k)+\ln(\lambda/k)+1\,
  10. n = 0 t n λ n n ! Γ ( 1 + n / k ) , k 1 \sum_{n=0}^{\infty}\frac{t^{n}\lambda^{n}}{n!}\Gamma(1+n/k),\ k\geq 1
  11. n = 0 ( i t ) n λ n n ! Γ ( 1 + n / k ) \sum_{n=0}^{\infty}\frac{(it)^{n}\lambda^{n}}{n!}\Gamma(1+n/k)
  12. f ( x ; λ , k ) = { k λ ( x λ ) k - 1 e - ( x / λ ) k x 0 , 0 x < 0 , f(x;\lambda,k)=\begin{cases}\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k% -1}e^{-(x/\lambda)^{k}}&x\geq 0,\\ 0&x<0,\end{cases}
  13. λ = 2 k \lambda=\frac{\sqrt{2}}{k}
  14. F ( x ; k , λ ) = 1 - e - ( x / λ ) k F(x;k,\lambda)=1-e^{-(x/\lambda)^{k}}\,
  15. E [ e t log X ] = λ t Γ ( t k + 1 ) E\left[e^{t\log X}\right]=\lambda^{t}\Gamma\left(\frac{t}{k}+1\right)
  16. Γ Γ
  17. E [ e i t log X ] = λ i t Γ ( i t k + 1 ) . E\left[e^{it\log X}\right]=\lambda^{it}\Gamma\left(\frac{it}{k}+1\right).
  18. m n = λ n Γ ( 1 + n k ) . m_{n}=\lambda^{n}\Gamma\left(1+\frac{n}{k}\right).
  19. E ( X ) = λ Γ ( 1 + 1 k ) \mathrm{E}(X)=\lambda\Gamma\left(1+\frac{1}{k}\right)\,
  20. var ( X ) = λ 2 [ Γ ( 1 + 2 k ) - ( Γ ( 1 + 1 k ) ) 2 ] . \textrm{var}(X)=\lambda^{2}\left[\Gamma\left(1+\frac{2}{k}\right)-\left(\Gamma% \left(1+\frac{1}{k}\right)\right)^{2}\right]\,.
  21. γ 1 = Γ ( 1 + 3 k ) λ 3 - 3 μ σ 2 - μ 3 σ 3 \gamma_{1}=\frac{\Gamma\left(1+\frac{3}{k}\right)\lambda^{3}-3\mu\sigma^{2}-% \mu^{3}}{\sigma^{3}}
  22. μ μ
  23. σ σ
  24. γ 2 = - 6 Γ 1 4 + 12 Γ 1 2 Γ 2 - 3 Γ 2 2 - 4 Γ 1 Γ 3 + Γ 4 [ Γ 2 - Γ 1 2 ] 2 \gamma_{2}=\frac{-6\Gamma_{1}^{4}+12\Gamma_{1}^{2}\Gamma_{2}-3\Gamma_{2}^{2}-4% \Gamma_{1}\Gamma_{3}+\Gamma_{4}}{[\Gamma_{2}-\Gamma_{1}^{2}]^{2}}
  25. Γ i = Γ ( 1 + i / k ) \Gamma_{i}=\Gamma(1+i/k)
  26. γ 2 = λ 4 Γ ( 1 + 4 k ) - 4 γ 1 σ 3 μ - 6 μ 2 σ 2 - μ 4 σ 4 - 3 \gamma_{2}=\frac{\lambda^{4}\Gamma(1+\frac{4}{k})-4\gamma_{1}\sigma^{3}\mu-6% \mu^{2}\sigma^{2}-\mu^{4}}{\sigma^{4}}-3
  27. E [ e t X ] = n = 0 t n λ n n ! Γ ( 1 + n k ) . E\left[e^{tX}\right]=\sum_{n=0}^{\infty}\frac{t^{n}\lambda^{n}}{n!}\Gamma\left% (1+\frac{n}{k}\right).
  28. E [ e t X ] = 0 e t x k λ ( x λ ) k - 1 e - ( x / λ ) k d x . E\left[e^{tX}\right]=\int_{0}^{\infty}e^{tx}\frac{k}{\lambda}\left(\frac{x}{% \lambda}\right)^{k-1}e^{-(x/\lambda)^{k}}\,dx.
  29. E [ e - t X ] = 1 λ k t k p k q / p ( 2 π ) q + p - 2 G p , q q , p ( 1 - k p , 2 - k p , , p - k p 0 q , 1 q , , q - 1 q | p p ( q λ k t k ) q ) E\left[e^{-tX}\right]=\frac{1}{\lambda^{k}\,t^{k}}\,\frac{p^{k}\,\sqrt{q/p}}{(% \sqrt{2\pi})^{q+p-2}}\,G_{p,q}^{\,q,p}\!\left(\left.\begin{matrix}\frac{1-k}{p% },\frac{2-k}{p},\dots,\frac{p-k}{p}\\ \frac{0}{q},\frac{1}{q},\dots,\frac{q-1}{q}\end{matrix}\;\right|\,\frac{p^{p}}% {\left(q\,\lambda^{k}\,t^{k}\right)^{q}}\right)
  30. H ( λ , k ) = γ ( 1 - 1 k ) + ln ( λ k ) + 1 H(\lambda,k)=\gamma\left(1\!-\!\frac{1}{k}\right)+\ln\left(\frac{\lambda}{k}% \right)+1
  31. γ \gamma
  32. λ \lambda
  33. k k
  34. λ ^ k = 1 n i = 1 n x i k \hat{\lambda}^{k}=\frac{1}{n}\sum_{i=1}^{n}x_{i}^{k}
  35. k k
  36. k ^ - 1 = i = 1 n x i k ln x i i = 1 n x i k - 1 n i = 1 n ln x i \hat{k}^{-1}=\frac{\sum_{i=1}^{n}x_{i}^{k}\ln x_{i}}{\sum_{i=1}^{n}x_{i}^{k}}-% \frac{1}{n}\sum_{i=1}^{n}\ln x_{i}
  37. k k
  38. x 1 > x 2 > > x N x_{1}>x_{2}>...>x_{N}
  39. N N
  40. N N
  41. λ \lambda
  42. k k
  43. λ ^ k = 1 N i = 1 N ( x i k - x N k ) \hat{\lambda}^{k}=\frac{1}{N}\sum_{i=1}^{N}(x_{i}^{k}-x_{N}^{k})
  44. k k
  45. k ^ - 1 = i = 1 N ( x i k ln x i - x N k ln x N ) i = 1 N ( x i k - x N k ) - 1 N i = 1 N ln x i \hat{k}^{-1}=\frac{\sum_{i=1}^{N}(x_{i}^{k}\ln x_{i}-x_{N}^{k}\ln x_{N})}{\sum% _{i=1}^{N}(x_{i}^{k}-x_{N}^{k})}-\frac{1}{N}\sum_{i=1}^{N}\ln x_{i}
  46. k k
  47. F ^ ( x ) \hat{F}(x)
  48. ln ( - ln ( 1 - F ^ ( x ) ) ) \ln(-\ln(1-\hat{F}(x)))
  49. ln ( x ) \ln(x)
  50. F ( x ) \displaystyle F(x)
  51. F ^ = i - 0.3 n + 0.4 \hat{F}=\frac{i-0.3}{n+0.4}
  52. i i
  53. n n
  54. k k
  55. λ \lambda
  56. f ( x ; k , λ , θ ) = k λ ( x - θ λ ) k - 1 e - ( x - θ λ ) k f(x;k,\lambda,\theta)={k\over\lambda}\left({x-\theta\over\lambda}\right)^{k-1}% e^{-({x-\theta\over\lambda})^{k}}\,
  57. x θ x\geq\theta
  58. λ > 0 \lambda>0
  59. θ \theta
  60. X = ( W λ ) k X=\left(\frac{W}{\lambda}\right)^{k}
  61. W = λ ( - ln ( U ) ) 1 / k W=\lambda(-\ln(U))^{1/k}\,
  62. - ln ( U ) -\ln(U)
  63. X X
  64. σ = λ / 2 \sigma=\lambda/\sqrt{2}
  65. f Frechet ( x ; k , λ ) = k λ ( x λ ) - 1 - k e - ( x / λ ) - k = * f Weibull ( x ; - k , λ ) . f_{\rm{Frechet}}(x;k,\lambda)=\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^% {-1-k}e^{-(x/\lambda)^{-k}}=*f_{\rm{Weibull}}(x;-k,\lambda).
  66. f ( x ; P 80 , m ) = { 1 - e l n ( 0.2 ) ( x P 80 ) m x 0 , 0 x < 0 , f(x;P_{\rm{80}},m)=\begin{cases}1-e^{ln\left(0.2\right)\left(\frac{x}{P_{\rm{8% 0}}}\right)^{m}}&x\geq 0,\\ 0&x<0,\end{cases}
  67. x x
  68. P 80 P_{\rm{80}}
  69. m m

Weight_(representation_theory).html

  1. χ ( g h ) = χ ( g ) χ ( h ) \chi(gh)=\chi(g)\chi(h)
  2. χ ( a b ) = χ ( a ) χ ( b ) \chi(ab)=\chi(a)\chi(b)
  3. V λ := { v V : ξ 𝔥 , ξ v = λ ( ξ ) v } V_{\lambda}:=\{v\in V:\forall\xi\in\mathfrak{h},\quad\xi\cdot v=\lambda(\xi)v\}
  4. ξ v \xi\cdot v
  5. V = λ 𝔥 * V λ V=\bigoplus_{\lambda\in\mathfrak{h}^{*}}V_{\lambda}
  6. ω 1 , , ω n \omega_{1},\ldots,\omega_{n}
  7. H α 1 , , H α n H_{\alpha_{1}},\ldots,H_{\alpha_{n}}
  8. exp ( t ) = 1 G , λ ( t ) 2 π i 𝐙 \exp(t)=1\in G,\,\,\lambda(t)\in 2\pi i\mathbf{Z}
  9. λ ( H γ ) 0 \lambda(H_{\gamma})\geq 0
  10. v λ V v_{\lambda}\in V

Weight_function.html

  1. w : A + \scriptstyle w\colon A\to{\mathbb{R}}^{+}
  2. A A
  3. w ( a ) := 1 w(a):=1
  4. f : A \scriptstyle f\colon A\to{\mathbb{R}}
  5. f f
  6. A A
  7. a A f ( a ) ; \sum_{a\in A}f(a);
  8. w : A + \scriptstyle w\colon A\to{\mathbb{R}}^{+}
  9. a A f ( a ) w ( a ) . \sum_{a\in A}f(a)w(a).
  10. a B w ( a ) . \sum_{a\in B}w(a).
  11. 1 | A | a A f ( a ) \frac{1}{|A|}\sum_{a\in A}f(a)
  12. a A f ( a ) w ( a ) a A w ( a ) . \frac{\sum_{a\in A}f(a)w(a)}{\sum_{a\in A}w(a)}.
  13. f f
  14. f i f_{i}
  15. σ i 2 \scriptstyle\sigma^{2}_{i}
  16. w i = 1 σ i 2 \scriptstyle w_{i}=\frac{1}{\sigma_{i}^{2}}
  17. σ 2 = 1 / w i \scriptstyle\sigma^{2}=1/\sum w_{i}
  18. w i w_{i}
  19. n n
  20. w 1 , , w n \scriptstyle w_{1},\ldots,w_{n}
  21. \scriptstylesymbol x 1 , , s y m b o l x n \scriptstylesymbol{x}_{1},\ldots,symbol{x}_{n}
  22. i = 1 n w i s y m b o l x i i = 1 n w i , \frac{\sum_{i=1}^{n}w_{i}symbol{x}_{i}}{\sum_{i=1}^{n}w_{i}},
  23. \scriptstylesymbol x i \scriptstylesymbol{x}_{i}
  24. w ( x ) d x w(x)dx
  25. Ω \Omega
  26. n \scriptstyle{\mathbb{R}}^{n}
  27. Ω \Omega
  28. [ a , b ] [a,b]
  29. d x dx
  30. w : Ω \R + \scriptstyle w\colon\Omega\to\R^{+}
  31. w ( x ) w(x)
  32. f : Ω f\colon\Omega\to{\mathbb{R}}
  33. Ω f ( x ) d x \int_{\Omega}f(x)\ dx
  34. Ω f ( x ) w ( x ) d x \int_{\Omega}f(x)w(x)\,dx
  35. f f
  36. w ( x ) d x w(x)dx
  37. Ω \Omega
  38. E w ( x ) d x , \int_{E}w(x)\ dx,
  39. Ω \Omega
  40. 1 vol ( Ω ) Ω f ( x ) d x \frac{1}{\mathrm{vol}(\Omega)}\int_{\Omega}f(x)\ dx
  41. Ω f ( x ) w ( x ) d x Ω w ( x ) d x \frac{\int_{\Omega}f(x)\ w(x)dx}{\int_{\Omega}w(x)\ dx}
  42. f : Ω \scriptstyle f\colon\Omega\to{\mathbb{R}}
  43. g : Ω \scriptstyle g\colon\Omega\to{\mathbb{R}}
  44. f , g := Ω f ( x ) g ( x ) d x \langle f,g\rangle:=\int_{\Omega}f(x)g(x)\ dx
  45. f , g := Ω f ( x ) g ( x ) w ( x ) d x . \langle f,g\rangle:=\int_{\Omega}f(x)g(x)\ w(x)\ dx.

Weighted_average_cost_of_capital.html

  1. WACC = i = 1 N r i M V i i = 1 N M V i \,\text{WACC}=\frac{\sum_{i=1}^{N}r_{i}\cdot MV_{i}}{\sum_{i=1}^{N}MV_{i}}
  2. N N
  3. r i r_{i}
  4. i i
  5. M V i MV_{i}
  6. i i
  7. WACC = D D + E K d + E D + E K e \,\text{WACC}=\frac{D}{D+E}K_{d}+\frac{E}{D+E}K_{e}
  8. M V e MV_{e}
  9. R e R_{e}
  10. M V d MV_{d}
  11. R d R_{d}
  12. t t
  13. WACC = M V e M V d + M V e R e + M V d M V d + M V e R d ( 1 - t ) \,\text{WACC}=\frac{MV_{e}}{MV_{d}+MV_{e}}\cdot R_{e}+\frac{MV_{d}}{MV_{d}+MV_% {e}}\cdot R_{d}\cdot(1-t)

Weil_conjectures.html

  1. \mathfrak{R}
  2. \mathfrak{R}^{\prime}
  3. ′′ \mathfrak{R}^{\prime\prime}
  4. ( ) (\mathfrak{R}\mathfrak{R})
  5. \mathfrak{R}
  6. \mathfrak{R}
  7. \mathfrak{R}
  8. ( ) (\mathfrak{R}\mathfrak{R})
  9. ζ ( X , s ) = exp ( m = 1 N m m ( q - s ) m ) \zeta(X,s)=\exp\left(\sum_{m=1}^{\infty}\frac{N_{m}}{m}(q^{-s})^{m}\right)
  10. i = 0 2 n P i ( q - s ) ( - 1 ) i + 1 = P 1 ( T ) P 2 n - 1 ( T ) P 0 ( T ) P 2 n ( T ) , \textstyle\prod_{i=0}^{2n}P_{i}(q^{-s})^{(-1)^{i+1}}=\frac{P_{1}(T)\cdots P_{2% n-1}(T)}{P_{0}(T)\cdots P_{2n}(T)},
  11. j ( 1 - α i j T ) \textstyle\prod_{j}(1-\alpha_{ij}T)
  12. ζ ( X , n - s ) = ± q n E 2 - E s ζ ( X , s ) \zeta(X,n-s)=\pm q^{\frac{nE}{2}-Es}\zeta(X,s)
  13. ζ ( X , q - n T - 1 ) = ± q n E 2 T E ζ ( X , T ) \zeta(X,q^{-n}T^{-1})=\pm q^{\frac{nE}{2}}T^{E}\zeta(X,T)
  14. ζ ( s ) = P 1 ( T ) P 2 n - 1 ( T ) P 0 ( T ) P 2 ( T ) P 2 n ( T ) \zeta(s)=\frac{P_{1}(T)\cdots P_{2n-1}(T)}{P_{0}(T)P_{2}(T)\cdots P_{2n}(T)}
  15. Z ( X 0 , F 0 , t ) = x | X 0 | det ( 1 - F x * t d e g ( x ) | F 0 ) - 1 Z(X_{0},F_{0},t)=\prod_{x\in|X_{0}|}\det(1-F^{*}_{x}t^{deg(x)}|F_{0})^{-1}
  16. Z ( X 0 , F 0 , t ) = i det ( 1 - F * t | H c i ( F ) ) ( - 1 ) i + 1 Z(X_{0},F_{0},t)=\prod_{i}\det(1-F^{*}t|H^{i}_{c}(F))^{(-1)^{i+1}}
  17. Z ( U , E k , T ) = det ( 1 - F * T | H c 1 ( E k ) ) det ( 1 - F * T | H c 0 ( E k ) ) det ( 1 - F * T | H c 2 ( E k ) ) Z(U,E^{k},T)=\frac{\det(1-F^{*}T|H^{1}_{c}(E^{k}))}{\det(1-F^{*}T|H^{0}_{c}(E^% {k}))\det(1-F^{*}T|H^{2}_{c}(E^{k}))}
  18. Z ( E k , T ) = x 1 Z ( E x k , T ) Z(E^{k},T)=\prod_{x}\frac{1}{Z(E^{k}_{x},T)}
  19. 1 det ( 1 - T d e g ( x ) F x | E k ) = exp ( n > 0 T n n Trace ( F x n | E ) k ) \frac{1}{\det(1-T^{deg(x)}F_{x}|E^{k})}=\exp\left(\sum_{n>0}\frac{T^{n}}{n}\,% \text{Trace}(F_{x}^{n}|E)^{k}\right)
  20. | α k | q k ( d - 1 ) / 2 + 1 |\alpha^{k}|\leq q^{k(d-1)/2+1}
  21. | α | q ( d - 1 ) / 2 . |\alpha|\leq q^{(d-1)/2}.
  22. | α | = q ( d - 1 ) / 2 . |\alpha|=q^{(d-1)/2}.
  23. | α | q d / 2 + 1 / 2 |\alpha|\leq q^{d/2+1/2}
  24. | α k | q k d / 2 + 1 / 2 |\alpha^{k}|\leq q^{kd/2+1/2}
  25. | α | q d / 2 |\alpha|\leq q^{d/2}
  26. | α | = q d / 2 . |\alpha|=q^{d/2}.

Weir.html

  1. Q = C L H n Q=CLH^{n}
  2. P > 2 H m a x , P>2H_{max},
  3. S > 2 H m a x S>2H_{max}
  4. Q = 4.28 C e tan ( θ 2 ) ( H + k ) 5 2 Q=4.28C_{e}\tan{\left(\frac{\theta}{2}\right)}\frac{(H+k)5}{2}
  5. C e C_{e}
  6. C e C_{e}

Weissenberg_number.html

  1. γ ˙ \dot{\gamma}
  2. λ \lambda
  3. Wi = viscous forces elastic forces = μ γ ˙ E ϵ = γ ˙ λ . \,\text{Wi}=\dfrac{\mbox{viscous forces}~{}}{\mbox{elastic forces}~{}}=\frac{% \mu\dot{\gamma}}{E\epsilon}=\dot{\gamma}\lambda.\,

Well-defined.html

  1. A 0 , A 1 A_{0},A_{1}
  2. A = A 0 A 1 A=A_{0}\bigcup A_{1}
  3. f : A { 0 , 1 } f:A\rightarrow\{0,1\}
  4. f ( a ) = 0 f(a)=0
  5. a A 0 a\in A_{0}
  6. f ( a ) = 1 f(a)=1
  7. a A 1 a\in A_{1}
  8. f f
  9. A 0 A 1 = A_{0}\bigcap A_{1}=\emptyset
  10. A 0 := { 2 , 4 } , A 1 := { 3 , 5 } A_{0}:=\{2,4\},A_{1}:=\{3,5\}
  11. mod ( a , 2 ) \operatorname{mod}(a,2)
  12. A 0 A 1 A_{0}\bigcap A_{1}\neq\emptyset
  13. f f
  14. f ( a ) f(a)
  15. a A 0 A 1 a\in A_{0}\bigcap A_{1}
  16. A 0 := { 2 } A_{0}:=\{2\}
  17. A 1 := { 2 } A_{1}:=\{2\}
  18. A 0 A 1 = { 2 } 2 A_{0}\bigcap A_{1}=\{2\}\ni 2
  19. A 0 A 1 = A_{0}\cap A_{1}=\emptyset
  20. f f
  21. f f
  22. A 0 A 1 A_{0}\cap A_{1}\neq\emptyset
  23. a A 0 A 1 a\in A_{0}\cap A_{1}
  24. ( a , 0 ) f (a,0)\in f
  25. ( a , 1 ) f (a,1)\in f
  26. f f
  27. f f
  28. a a
  29. f : \Z / 8 \Z \Z / 4 \Z n ¯ 8 n ¯ 4 , \begin{matrix}f:&\Z/8\Z&\to&\Z/4\Z\\ &\overline{n}_{8}&\mapsto&\overline{n}_{4},\end{matrix}
  30. n \Z , m { 4 , 8 } n\in\Z,m\in\{4,8\}
  31. \Z / m \Z \Z/m\Z
  32. n ¯ m \overline{n}_{m}
  33. n ¯ 4 \overline{n}_{4}
  34. n n ¯ 8 n\in\overline{n}_{8}
  35. n ¯ 8 \overline{n}_{8}
  36. n n mod 8 8 ( n - n ) 2 4 ( n - n ) 4 ( n - n ) n n mod 4. n\equiv n^{\prime}\operatorname{mod}8\;\Leftrightarrow\;8\mid(n-n^{\prime})\;% \Leftrightarrow\;2\cdot 4\mid(n-n^{\prime})\;\Rightarrow\;4\mid(n-n^{\prime})% \;\Leftrightarrow\;n\equiv n^{\prime}\operatorname{mod}4.
  37. [ a ] [ b ] = [ a + b ] [a]\oplus[b]=[a+b]
  38. [ a ] [a]
  39. a + k n a+kn
  40. [ a + k n ] [ b ] = [ ( a + k n ) + b ] = [ ( a + b ) + k n ] = [ a + b ] = [ a ] [ b ] [a+kn]\oplus[b]=[(a+kn)+b]=[(a+b)+kn]=[a+b]=[a]\oplus[b]
  41. [ b ] [b]
  42. a × b × c a\times b\times c
  43. ( a × b ) × c = a × ( b × c ) (a\times b)\times c=a\times(b\times c)
  44. × \times
  45. - -
  46. - -
  47. a - b - c a-b-c
  48. a + ( - b ) + ( - c ) a+(-b)+(-c)
  49. a / b / c a/b/c
  50. / b := * b - 1 /b:=*b^{-1}

Western_text-type.html

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

Weyl_group.html

  1. 𝔰 𝔩 n \mathfrak{sl}_{n}
  2. 𝔥 \mathfrak{h}
  3. 𝔥 \mathfrak{h}
  4. 𝔥 \mathfrak{h}^{\ast}
  5. W ( T , G ) := N ( T ) / Z ( T ) . W(T,G):=N(T)/Z(T).
  6. Z ( T 0 ) = T 0 Z(T_{0})=T_{0}
  7. G = w W B w B G=\bigcup_{w\in W}BwB
  8. [ n ] q ! [n]_{q}!
  9. N = N G ( T ) , N=N_{G}(T),
  10. Out ( N ) H 1 ( W ; T ) Out ( G ) . \operatorname{Out}(N)\cong H^{1}(W;T)\rtimes\operatorname{Out}(G).
  11. ( 𝐙 / 2 ) k (\mathbf{Z}/2)^{k}
  12. H 1 ( W ; T ) H^{1}(W;T)

What_the_Tortoise_Said_to_Achilles.html

  1. P Q , P Q \frac{P\to Q,\;P}{\therefore Q}
  2. p q p q not p or q p implies q \begin{matrix}p\rightarrow q&&&p\Rightarrow q\\ \mbox{not}~{}\ p\ \mbox{or}~{}\ q&&&p\ \mbox{implies}~{}\ q\end{matrix}

Whittaker–Shannon_interpolation_formula.html

  1. x ( t ) = n = - x [ n ] sinc ( t - n T T ) x(t)=\sum_{n=-\infty}^{\infty}x[n]\,{\rm sinc}\left(\frac{t-nT}{T}\right)\,
  2. x ( t ) = ( n = - x [ n ] δ ( t - n T ) ) * sinc ( t T ) . x(t)=\left(\sum_{n=-\infty}^{\infty}x[n]\cdot\delta\left(t-nT\right)\right)*{% \rm sinc}\left(\frac{t}{T}\right).
  3. n \Z , n 0 | x [ n ] n | < . \sum_{n\in\Z,\,n\neq 0}\left|\frac{x[n]}{n}\right|<\infty.
  4. ( x [ n ] ) n \Z \scriptstyle(x[n])_{n\in\Z}
  5. p ( \Z , ) \scriptstyle\ell^{p}(\Z,\mathbb{C})
  6. p ( \Z , ) \scriptstyle\ell^{p}(\Z,\mathbb{C})
  7. p \scriptstyle\ell^{p}

Wick_rotation.html

  1. d s 2 = - ( d t 2 ) + d x 2 + d y 2 + d z 2 ds^{2}=-(dt^{2})+dx^{2}+dy^{2}+dz^{2}
  2. d s 2 = d τ 2 + d x 2 + d y 2 + d z 2 ds^{2}=d\tau^{2}+dx^{2}+dy^{2}+dz^{2}
  3. t = - i τ t=-i\tau
  4. τ \tau
  5. 1 / ( k B T ) 1/(k_{B}T)\,
  6. i t / it/\hbar\,
  7. T T\,
  8. E E\,
  9. exp ( - E / k B T ) \exp(-E/k_{B}T)\,
  10. k B k_{B}\,
  11. Q Q\,
  12. j Q j e - E j / ( k B T ) . \sum_{j}Q_{j}e^{-E_{j}/(k_{B}T)}.\,
  13. t t
  14. H H
  15. E E\,
  16. exp ( - E i t / ) , \exp(-Eit/\hbar),\,
  17. \hbar\,
  18. | ψ = j | j |\psi\rangle=\sum_{j}|j\rangle\,
  19. | Q = j Q j | j |Q\rangle=\sum_{j}Q_{j}|j\rangle\,
  20. Q | e - i H t / | ψ \;\langle Q|e^{-iHt/\hbar}|\psi\rangle
  21. = j Q j e - E j i t / j | j =\sum_{j}Q_{j}e^{-E_{j}it/\hbar}\langle j|j\rangle
  22. = j Q j e - E j i t / . =\sum_{j}Q_{j}e^{-E_{j}it/\hbar}.
  23. n n
  24. n - 1 n-1
  25. n = 2 n=2
  26. y ( x ) y(x)
  27. E = x [ k ( d y ( x ) d x ) 2 + V ( y ( x ) ) ] d x , E=\int_{x}\left[k\left(\frac{dy(x)}{dx}\right)^{2}+V(y(x))\right]dx,
  28. k k
  29. V ( y ( x ) ) V(y(x))
  30. S = t [ m ( d y ( t ) d t ) 2 - V ( y ( t ) ) ] d t S=\int_{t}\left[m\left(\frac{dy(t)}{dt}\right)^{2}-V(y(t))\right]dt
  31. i i
  32. y ( x ) y(x)
  33. y ( i t ) y(it)
  34. k k
  35. m m
  36. i S = t [ m ( d y ( i t ) d t ) 2 + V ( y ( i t ) ) ] d t iS=\int_{t}\left[m\left(\frac{dy(it)}{dt}\right)^{2}+V(y(it))\right]dt
  37. = i t [ m ( d y ( i t ) d i t ) 2 - V ( y ( i t ) ) ] d i t =i\int_{t}\left[m\left(\frac{dy(it)}{dit}\right)^{2}-V(y(it))\right]dit
  38. T T
  39. exp ( i S ) \exp(iS)
  40. π / 2 \scriptstyle\pi/2

Wiedemann_effect.html

  1. α = j h 15 2 G \alpha=j\frac{h_{15}}{2G}
  2. j j
  3. h 15 h_{15}
  4. G G

Wiener_equation.html

  1. 𝐯 = d 𝐱 d t = g ( t ) , \mathbf{v}=\frac{d\mathbf{x}}{dt}=g(t),

Wiener_process.html

  1. α - 1 W α 2 t \alpha^{-1}W_{\alpha^{2}t}
  2. f W t ( x ) = 1 2 π t e - x 2 2 t . f_{W_{t}}(x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^{2}}{2t}}.
  3. E [ W t ] = 0. E[W_{t}]=0.
  4. Var ( W t ) = E [ W t 2 ] - E 2 [ W t ] = E [ W t 2 ] - 0 = E [ W t 2 ] = t . \operatorname{Var}(W_{t})=E\left[W^{2}_{t}\right]-E^{2}[W_{t}]=E\left[W^{2}_{t% }\right]-0=E\left[W^{2}_{t}\right]=t.
  5. cov ( W s , W t ) = min ( s , t ) , \operatorname{cov}(W_{s},W_{t})=\min(s,t),
  6. corr ( W s , W t ) = cov ( W s , W t ) σ W s σ W t = min ( s , t ) s t = min ( s , t ) max ( s , t ) . \operatorname{corr}(W_{s},W_{t})=\frac{\mathrm{cov}(W_{s},W_{t})}{\sigma_{W_{s% }}\sigma_{W_{t}}}=\frac{\min(s,t)}{\sqrt{st}}=\sqrt{\frac{\min(s,t)}{\max(s,t)% }}.
  7. W t = W t - W 0 N ( 0 , t ) . W_{t}=W_{t}-W_{0}\sim N(0,t).
  8. cov ( W t 1 , W t 2 ) = E [ ( W t 1 - E [ W t 1 ] ) ( W t 2 - E [ W t 2 ] ) ] = E [ W t 1 W t 2 ] . \operatorname{cov}(W_{t_{1}},W_{t_{2}})=E\left[(W_{t_{1}}-E[W_{t_{1}}])\cdot(W% _{t_{2}}-E[W_{t_{2}}])\right]=E\left[W_{t_{1}}\cdot W_{t_{2}}\right].
  9. W t 2 = ( W t 2 - W t 1 ) + W t 1 W_{t_{2}}=(W_{t_{2}}-W_{t_{1}})+W_{t_{1}}
  10. E [ W t 1 W t 2 ] = E [ W t 1 ( ( W t 2 - W t 1 ) + W t 1 ) ] = E [ W t 1 ( W t 2 - W t 1 ) ] + E [ W t 1 2 ] . E[W_{t_{1}}\cdot W_{t_{2}}]=E\left[W_{t_{1}}\cdot((W_{t_{2}}-W_{t_{1}})+W_{t_{% 1}})\right]=E\left[W_{t_{1}}\cdot(W_{t_{2}}-W_{t_{1}})\right]+E\left[W_{t_{1}}% ^{2}\right].
  11. E [ W t 1 ( W t 2 - W t 1 ) ] = E [ W t 1 ] E [ W t 2 - W t 1 ] = 0. E\left[W_{t_{1}}\cdot(W_{t_{2}}-W_{t_{1}})\right]=E[W_{t_{1}}]\cdot E[W_{t_{2}% }-W_{t_{1}}]=0.
  12. cov ( W t 1 , W t 2 ) = E [ W t 1 2 ] = t 1 . \operatorname{cov}(W_{t_{1}},W_{t_{2}})=E\left[W_{t_{1}}^{2}\right]=t_{1}.
  13. ξ n \xi_{n}
  14. W t = ξ 0 t + 2 n = 1 ξ n sin π n t π n W_{t}=\xi_{0}t+\sqrt{2}\sum_{n=1}^{\infty}\xi_{n}\frac{\sin\pi nt}{\pi n}
  15. W t = 2 n = 1 ξ n sin ( ( n - 1 2 ) π t ) ( n - 1 2 ) π W_{t}=\sqrt{2}\sum_{n=1}^{\infty}\xi_{n}\frac{\sin\left(\left(n-\frac{1}{2}% \right)\pi t\right)}{\left(n-\frac{1}{2}\right)\pi}
  16. [ 0 , 1 ] [0,1]
  17. c W ( t c ) \sqrt{c}\,W\left(\frac{t}{c}\right)
  18. [ 0 , c ] [0,c]
  19. M t = max 0 s t W s M_{t}=\max_{0\leq s\leq t}W_{s}
  20. f M t , W t ( m , w ) = 2 ( 2 m - w ) t 2 π t e - ( 2 m - w ) 2 2 t , m 0 , w m . f_{M_{t},W_{t}}(m,w)=\frac{2(2m-w)}{t\sqrt{2\pi t}}e^{-\frac{(2m-w)^{2}}{2t}},% \qquad m\geq 0,w\leq m.
  21. f M t f_{M_{t}}
  22. E [ M t ] = 0 m f M t ( m ) d m = 0 m 2 π t e - m 2 2 t d m = 2 t π E[M_{t}]=\int_{0}^{\infty}mf_{M_{t}}(m)\,dm=\int_{0}^{\infty}m\sqrt{\frac{2}{% \pi t}}e^{-\frac{m^{2}}{2t}}\,dm=\sqrt{\frac{2t}{\pi}}
  23. V t = ( 1 / c ) W c t V_{t}=(1/\sqrt{c})W_{ct}
  24. V t = ( 1 / c ) W c t V_{t}=(1/\sqrt{c})W_{ct}
  25. V t = W 1 - W 1 - t V_{t}=W_{1}-W_{1-t}
  26. V t = t W 1 / t V_{t}=tW_{1/t}
  27. ( t + 1 2 2 x 2 ) p ( x , t ) = 0 \left(\frac{\partial}{\partial t}+\frac{1}{2}\frac{\partial^{2}}{\partial x^{2% }}\right)p(x,t)=0
  28. M t = p ( W t , t ) M_{t}=p(W_{t},t)
  29. W t 2 - t W_{t}^{2}-t
  30. M t = p ( W t , t ) - 0 t a ( W s , s ) d s , M_{t}=p(W_{t},t)-\int_{0}^{t}a(W_{s},s)\,\mathrm{d}s,
  31. a ( x , t ) = ( t + 1 2 2 x 2 ) p ( x , t ) . a(x,t)=\left(\frac{\partial}{\partial t}+\frac{1}{2}\frac{\partial^{2}}{% \partial x^{2}}\right)p(x,t).
  32. p ( x , t ) = ( x 2 - t ) 2 , p(x,t)=(x^{2}-t)^{2},
  33. a ( x , t ) = 4 x 2 ; a(x,t)=4x^{2};
  34. ( W t 2 - t ) 2 - 4 0 t W s 2 d s (W_{t}^{2}-t)^{2}-4\int_{0}^{t}W_{s}^{2}\,\mathrm{d}s
  35. W t 2 - t W_{t}^{2}-t
  36. 4 0 t W s 2 d s . 4\int_{0}^{t}W_{s}^{2}\,\mathrm{d}s.
  37. lim s t | w ( s ) - w ( t ) | | s - t | . \lim_{s\to t}\frac{|w(s)-w(t)|}{|s-t|}\to\infty.
  38. lim sup t + | w ( t ) | 2 t log log t = 1 , almost surely . \limsup_{t\to+\infty}\frac{|w(t)|}{\sqrt{2t\log\log t}}=1,\quad\,\text{almost % surely}.
  39. lim sup ε 0 + | w ( ε ) | 2 ε log log ( 1 / ε ) = 1 , almost surely . \limsup_{\varepsilon\to 0+}\frac{|w(\varepsilon)|}{\sqrt{2\varepsilon\log\log(% 1/\varepsilon)}}=1,\qquad\,\text{almost surely}.
  40. lim sup ε 0 + sup 0 s < t 1 , t - s ε | w ( s ) - w ( t ) | 2 ε log ( 1 / ε ) = 1 , almost surely . \limsup_{\varepsilon\to 0+}\sup_{0\leq s<t\leq 1,t-s\leq\varepsilon}\frac{|w(s% )-w(t)|}{\sqrt{2\varepsilon\log(1/\varepsilon)}}=1,\qquad\,\text{almost surely}.
  41. 0 t f ( w ( s ) ) d s = - + f ( x ) L t ( x ) d x \int_{0}^{t}f(w(s))\,\mathrm{d}s=\int_{-\infty}^{+\infty}f(x)L_{t}(x)\,\mathrm% {d}x
  42. X t = μ t + σ W t X_{t}=\mu t+\sigma W_{t}
  43. e μ t - σ 2 t 2 + σ W t . e^{\mu t-\frac{\sigma^{2}t}{2}+\sigma W_{t}}.
  44. X t = e - t W e 2 t X_{t}=e^{-t}W_{e^{2t}}
  45. L x ( t ) = 0 t δ ( x - B t ) d s L^{x}(t)=\int_{0}^{t}\delta(x-B_{t})\,ds
  46. W ( - 1 ) ( t ) := 0 t W ( s ) d s W^{(-1)}(t):=\int_{0}^{t}W(s)ds
  47. t s t\wedge s
  48. W t 2 - t = V A ( t ) W_{t}^{2}-t=V_{A(t)}
  49. A ( t ) = 4 0 t W s 2 d s A(t)=4\int_{0}^{t}W_{s}^{2}\,\mathrm{d}s
  50. M t - M 0 = V A ( t ) M_{t}-M_{0}=V_{A(t)}
  51. M - = lim inf t M t , M^{-}_{\infty}=\liminf_{t\to\infty}M_{t},
  52. M + = lim sup t M t . M^{+}_{\infty}=\limsup_{t\to\infty}M_{t}.
  53. - < M - = M + < + , -\infty<M^{-}_{\infty}=M^{+}_{\infty}<+\infty,
  54. - = M - < M + = + ; -\infty=M^{-}_{\infty}<M^{+}_{\infty}=+\infty;
  55. M - = M + = + , M^{-}_{\infty}=M^{+}_{\infty}=+\infty,
  56. M - < M + < + M^{-}_{\infty}<M^{+}_{\infty}<+\infty
  57. f ( Z t ) - f ( 0 ) f(Z_{t})-f(0)
  58. Z t 2 = ( X t 2 - Y t 2 ) + 2 X t Y t i = U A ( t ) Z_{t}^{2}=(X_{t}^{2}-Y_{t}^{2})+2X_{t}Y_{t}i=U_{A(t)}
  59. A ( t ) = 4 0 t | Z s | 2 d s A(t)=4\int_{0}^{t}|Z_{s}|^{2}\,\mathrm{d}s

Wightman_axioms.html

  1. Ψ , Φ \langle\Psi,\Phi\rangle
  2. Ψ \lVert\Psi\rVert
  3. P ( [ Ψ ] , [ Φ ] ) = | Ψ , Φ | 2 Ψ 2 Φ 2 P([\Psi],[\Phi])=\frac{|\langle\Psi,\Phi\rangle|^{2}}{\lVert\Psi\rVert^{2}% \lVert\Phi\rVert^{2}}
  4. Ψ ( a , L ) , Φ ( a , L ) = Ψ , Φ \left\langle\Psi(a,L),\Phi(a,L)\right\rangle=\left\langle\Psi,\Phi\right\rangle
  5. U ( a , L ) U ( b , M ) = ± U ( ( a , L ) . ( b , M ) ) U(a,L)U(b,M)=\pm U((a,L).(b,M))
  6. π \pi
  7. x = L - 1 ( x - a ) x^{\prime}=L^{-1}(x-a)
  8. P 0 , P j P_{0},P_{j}
  9. P 0 0 P_{0}\geq 0
  10. P 0 2 - P j P j 0. P_{0}^{2}-P_{j}P_{j}\geq 0.
  11. A 1 ( f ) , , A n ( f ) A_{1}(f),\ldots,A_{n}(f)
  12. U ( a , L ) A ( x ) U ( a , L ) = S ( L ) A ( L - 1 ( x - a ) ) . U(a,L)^{\dagger}A(x)U(a,L)=S(L)A(L^{-1}(x-a)).
  13. H i n H^{in}
  14. H o u t H^{out}

Wigner's_classification.html

  1. m P ¯ ² m≡√\overline{P}{²}
  2. m > 0 m>0
  3. m = 0 m=0
  4. m = 0 m=0
  5. i = 1 , 2 i=1,2
  6. ϵ T λ = 1 , 2 \epsilon_{T}^{\lambda=1,2}
  7. ϵ L \epsilon_{L}
  8. ϵ T 2 = - 1 \epsilon_{T}^{2}=-1
  9. ϵ L 2 = + 1 \epsilon_{L}^{2}=+1
  10. Z 0 Z_{0}
  11. ϵ T λ = 1 , 2 , 3 \epsilon_{T}^{\lambda=1,2,3}
  12. ϵ T 2 = - 1 \epsilon_{T}^{2}=-1

Wilhelm_Wien.html

  1. λ max T = constant \lambda_{\mathrm{max}}T=\mathrm{constant}
  2. m = ( 4 / 3 ) E / c 2 m=(4/3)E/c^{2}

Wilkinson_Microwave_Anisotropy_Probe.html

  1. Ω k \Omega_{k}
  2. 1 / 2 {}^{1/2}
  3. n s n_{s}
  4. τ \tau
  5. t 0 t_{0}
  6. k m M p c / · s km{Mpc}/{·s}
  7. H 0 H_{0}
  8. Ω b h 2 \Omega_{b}h^{2}
  9. Ω m h 2 \Omega_{m}h^{2}
  10. τ \tau
  11. n s n_{s}
  12. d n s / d k dn_{s}/dk
  13. σ 8 \sigma_{8}
  14. Ω t o t \Omega_{tot}
  15. t 0 t_{0}
  16. k m / M p c · s {km}/{Mpc·s}
  17. H 0 H_{0}
  18. Ω b h 2 \Omega_{b}h^{2}
  19. Ω m h 2 \Omega_{m}h^{2}
  20. τ \tau
  21. n s n_{s}
  22. σ 8 \sigma_{8}
  23. t 0 t_{0}
  24. k m / M p c · s {km}/{Mpc·s}
  25. H 0 H_{0}
  26. Ω b h 2 \Omega_{b}h^{2}
  27. Ω c h 2 \Omega_{c}h^{2}
  28. Ω Λ \Omega_{\Lambda}
  29. τ \tau
  30. n s n_{s}
  31. d n s / d l n k dn_{s}/dlnk
  32. σ 8 \sigma_{8}
  33. Ω t o t \Omega_{tot}
  34. t 0 t_{0}
  35. k m / M p c · s {km}/{Mpc·s}
  36. H 0 H_{0}
  37. Ω b \Omega_{b}
  38. Ω b h 2 \Omega_{b}h^{2}
  39. Ω c \Omega_{c}
  40. Ω c h 2 \Omega_{c}h^{2}
  41. Ω Λ \Omega_{\Lambda}
  42. σ 8 \sigma_{8}
  43. n s n_{s}
  44. τ \tau
  45. Ω t o t \Omega_{tot}
  46. d n s / d l n k dn_{s}/dlnk
  47. t 0 t_{0}
  48. k m / M p c · s {km}/{Mpc·s}
  49. H 0 H_{0}
  50. Ω b \Omega_{b}
  51. Ω b h 2 \Omega_{b}h^{2}
  52. Ω c \Omega_{c}
  53. Ω c h 2 \Omega_{c}h^{2}
  54. Ω Λ \Omega_{\Lambda}
  55. σ 8 \sigma_{8}
  56. n s n_{s}
  57. τ \tau
  58. - -
  59. Ω tot \Omega_{\rm tot}
  60. d n s / d l n k dn_{s}/dlnk

William_Kingdon_Clifford.html

  1. H C H\otimes C
  2. H D H\otimes D
  3. H N H\otimes N

William_Lawrence_Bragg.html

  1. 3 10 3_{10}
  2. 4 3 4_{3}
  3. 2 1 2_{1}
  4. 3 10 3_{10}
  5. 3 10 3_{10}
  6. 3 10 3_{10}

William_Sealy_Gosset.html

  1. z = t n - 1 z=\frac{t}{\sqrt{n-1}}

Wilson's_theorem.html

  1. ( n - 1 ) ! - 1 ( mod n ) (n-1)!\ \equiv\ -1\;\;(\mathop{{\rm mod}}n)
  2. ( n - 1 ) ! = 1 × 2 × 3 × × ( n - 1 ) (n-1)!=1\times 2\times 3\times\cdots\times(n-1)
  3. n n
  4. ( n - 1 ) ! (n-1)!
  5. ( n - 1 ) ! mod n (n-1)!\ \bmod\ n
  6. 10 ! = [ ( 1 10 ) ] [ ( 2 6 ) ( 3 4 ) ( 5 9 ) ( 7 8 ) ] [ - 1 ] [ 1 1 1 1 ] - 1 ( mod 11 ) . 10!=[(1\cdot 10)]\cdot[(2\cdot 6)(3\cdot 4)(5\cdot 9)(7\cdot 8)]\equiv[-1]% \cdot[1\cdot 1\cdot 1\cdot 1]\equiv-1\;\;(\mathop{{\rm mod}}11).\,
  7. g ( x ) = ( x - 1 ) ( x - 2 ) ( x - ( p - 1 ) ) . g(x)=(x-1)(x-2)\cdots(x-(p-1)).\,
  8. h ( x ) = x p - 1 - 1. h(x)=x^{p-1}-1.\,
  9. f ( x ) = g ( x ) - h ( x ) . f(x)=g(x)-h(x).\,
  10. S p S_{p}
  11. ( p - 1 ) ! (p-1)!
  12. C p C_{p}
  13. S p S_{p}
  14. C p C_{p}
  15. n p = ( p - 2 ) ! n_{p}=(p-2)!
  16. ( p - 2 ) ! 1 ( mod p ) . (p-2)!\equiv 1\;\;(\mathop{{\rm mod}}p).
  17. ( p - 1 ) ! p - 1 - 1 ( mod p ) , (p-1)!\equiv p-1\equiv-1\;\;(\mathop{{\rm mod}}p),
  18. 1 2 ( p - 1 ) - 1 ( mod p ) 1\cdot 2\cdots(p-1)\ \equiv\ -1\ \;\;(\mathop{{\rm mod}}p)
  19. 1 ( p - 1 ) 2 ( p - 2 ) m ( p - m ) 1 ( - 1 ) 2 ( - 2 ) m ( - m ) - 1 ( mod p ) . 1\cdot(p-1)\cdot 2\cdot(p-2)\cdots m\cdot(p-m)\ \equiv\ 1\cdot(-1)\cdot 2\cdot% (-2)\cdots m\cdot(-m)\ \equiv\ -1\;\;(\mathop{{\rm mod}}p).
  20. j = 1 m j 2 ( - 1 ) m + 1 ( mod p ) \prod_{j=1}^{m}\ j^{2}\ \equiv(-1)^{m+1}\;\;(\mathop{{\rm mod}}p)
  21. ( m ! ) 2 ( - 1 ) m + 1 ( mod p ) . (m!)^{2}\equiv(-1)^{m+1}\;\;(\mathop{{\rm mod}}p).
  22. k = 1 gcd ( k , m ) = 1 m k { - 1 ( mod m ) if m = 4 , p α , 2 p α 1 ( mod m ) otherwise \prod_{k=1\atop\gcd(k,m)=1}^{m}\!\!k\ \equiv\begin{cases}-1\;\;(\mathop{{\rm mod% }}m)&\,\text{if }m=4,\;p^{\alpha},\;2p^{\alpha}\\ \;\;\,1\;\;(\mathop{{\rm mod}}m)&\,\text{otherwise}\end{cases}
  23. α \alpha

Wind_chill.html

  1. W C I = ( 10 V - V + 10.5 ) ( 33 - T a ) WCI=(10\sqrt{V}-V+10.5)\cdot(33-T_{\rm a})
  2. W C I WCI
  3. V V
  4. T a T_{\rm a}
  5. T wc = 13.12 + 0.6215 T a - 11.37 V + 0.16 + 0.3965 T a V + 0.16 T_{\rm wc}=13.12+0.6215T_{\rm a}-11.37V^{+0.16}+0.3965T_{\rm a}V^{+0.16}\,\!
  6. T wc T_{\rm wc}\,\!
  7. T a T_{\rm a}\,\!
  8. V V\,\!
  9. T wc = 35.74 + 0.6215 T a - 35.75 V + 0.16 + 0.4275 T a V + 0.16 T_{\rm wc}=35.74+0.6215T_{\rm a}-35.75V^{+0.16}+0.4275T_{\rm a}V^{+0.16}\,\!
  10. T wc T_{\rm wc}\,\!
  11. T a T_{\rm a}\,\!
  12. V V\,\!
  13. A T = T a + 0.33 e - 0.70 w s - 4.00 AT=T_{\rm a}+0.33e-0.70ws-4.00
  14. T a = T_{\rm a}=
  15. e = e=
  16. w s = ws=
  17. e = r h 100 6.105 exp ( 17.27 T a 237.7 + T a ) e=\frac{rh}{100}\cdot 6.105\cdot\exp{\left(\frac{17.27\cdot T_{\rm a}}{237.7+T% _{\rm a}}\right)}
  18. T a = T_{\rm a}=
  19. r h = rh=
  20. exp \exp

Wind_power.html

  1. E = 1 2 m v 2 = 1 2 ( A v t ρ ) v 2 = 1 2 A t ρ v 3 , E=\frac{1}{2}mv^{2}=\frac{1}{2}(Avt\rho)v^{2}=\frac{1}{2}At\rho v^{3},
  2. P = E t = 1 2 A ρ v 3 . P=\frac{E}{t}=\frac{1}{2}A\rho v^{3}.

Wind_shear.html

  1. p 1 p_{1}
  2. p 0 p_{0}
  3. p 1 < p 0 p_{1}<p_{0}
  4. f 𝐯 T = 𝐤 × ( ϕ 1 - ϕ 0 ) f\mathbf{v}_{T}=\mathbf{k}\times\nabla(\phi_{1}-\phi_{0})
  5. ϕ x \phi_{x}
  6. ϕ 1 > ϕ 0 \phi_{1}>\phi_{0}
  7. f f
  8. 𝐤 \mathbf{k}
  9. f f
  10. ( ϕ 1 - ϕ 0 ) \nabla(\phi_{1}-\phi_{0})

Winding_number.html

  1. \cdots
  2. \cdots
  3. x = x ( t ) and y = y ( t ) for 0 t 1. x=x(t)\quad\,\text{and}\quad y=y(t)\qquad\,\text{for }0\leq t\leq 1.
  4. r = r ( t ) and θ = θ ( t ) for 0 t 1. r=r(t)\quad\,\text{and}\quad\theta=\theta(t)\qquad\,\text{for }0\leq t\leq 1.
  5. winding number = θ ( 1 ) - θ ( 0 ) 2 π . \,\text{winding number}=\frac{\theta(1)-\theta(0)}{2\pi}.
  6. d θ = 1 r 2 ( x d y - y d x ) where r 2 = x 2 + y 2 . d\theta=\frac{1}{r^{2}}\left(x\,dy-y\,dx\right)\quad\,\text{where }r^{2}=x^{2}% +y^{2}.
  7. winding number = 1 2 π C x r 2 d y - y r 2 d x . \,\text{winding number}=\frac{1}{2\pi}\oint_{C}\,\frac{x}{r^{2}}\,dy-\frac{y}{% r^{2}}\,dx.
  8. d z = e i θ d r + i r e i θ d θ dz=e^{i\theta}dr+ire^{i\theta}d\theta\!\,
  9. d z z = d r r + i d θ = d [ ln r ] + i d θ . \frac{dz}{z}\;=\;\frac{dr}{r}+i\,d\theta\;=\;d[\ln r]+i\,d\theta.
  10. winding number = 1 2 π i C d z z . \,\text{winding number}=\frac{1}{2\pi i}\oint_{C}\frac{dz}{z}.
  11. 1 2 π i C d z z - a . \frac{1}{2\pi i}\oint_{C}\frac{dz}{z-a}.
  12. S 1 S 1 : s s n S^{1}\to S^{1}:s\mapsto s^{n}

Windlass.html

  1. π ( r - r ) \pi(r-r^{\prime})

Window_function.html

  1. B n o i s e = 1 | H ( f ) | m a x 2 0 | H ( f ) | 2 d f . B_{noise}=\frac{1}{|H(f)|^{2}_{max}}\int_{0}^{\infty}|H(f)|^{2}df.
  2. w [ n ] , 0 n N - 1. w[n],\ 0\leq n\leq N-1.
  3. w [ n ] \scriptstyle w[n]
  4. w [ n ] = w 0 ( n - N - 1 2 ) , 0 n N - 1. w[n]=\ w_{0}\left(n-\frac{N-1}{2}\right),\ 0\leq n\leq N-1.
  5. w 0 ( n - N 2 ) , 0 n N - 1 , \scriptstyle w_{0}\left(n-\frac{N}{2}\right),\ 0\leq n\leq N-1,
  6. w 0 ( n ) \scriptstyle w_{0}(n)
  7. W [ k ] = ( - 1 ) k W 0 [ k ] . \scriptstyle W[k]=(-1)^{k}\cdot W_{0}[k].
  8. w ( n ) = 1. w(n)=1.
  9. w ( n ) = 1 - | n - N - 1 2 L 2 | , w(n)=1-\left|\frac{n-\frac{N-1}{2}}{\frac{L}{2}}\right|,
  10. w ( n ) = { 1 - 6 ( n N / 2 ) 2 ( 1 - | n | N / 2 ) , 0 | n | N 4 2 ( 1 - | n | N / 2 ) 3 , N 4 | n | N 2 w(n)=\left\{\begin{array}[]{ll}1-6\left(\frac{n}{N/2}\right)^{2}\left(1-\frac{% |n|}{N/2}\right),&0\leqslant|n|\leqslant\frac{N}{4}\\ 2\left(1-\frac{|n|}{N/2}\right)^{3},&\frac{N}{4}\leqslant|n|\leqslant\frac{N}{% 2}\\ \end{array}\right.
  11. w ( n ) = 1 - ( n - N - 1 2 N - 1 2 ) 2 w(n)=1-\left(\frac{n-\frac{N-1}{2}}{\frac{N-1}{2}}\right)^{2}
  12. w ( n ) = α - β cos ( 2 π n N - 1 ) w(n)=\alpha-\beta\;\cos\left(\frac{2\pi n}{N-1}\right)\,
  13. w ( n ) = 0.5 ( 1 - cos ( 2 π n N - 1 ) ) w(n)=0.5\;\left(1-\cos\left(\frac{2\pi n}{N-1}\right)\right)
  14. w 0 ( n ) = 0.5 ( 1 + cos ( 2 π n N - 1 ) ) w_{0}(n)=0.5\;\left(1+\cos\left(\frac{2\pi n}{N-1}\right)\right)
  15. w ( n ) = α - β cos ( 2 π n N - 1 ) , w(n)=\alpha-\beta\;\cos\left(\frac{2\pi n}{N-1}\right),
  16. α = 0.54 , β = 1 - α = 0.46 , \alpha=0.54,\;\beta=1-\alpha=0.46,
  17. w 0 ( n ) \displaystyle w_{0}(n)
  18. w ( n ) = k = 0 K a k cos ( 2 π k n N ) w(n)=\sum_{k=0}^{K}a_{k}\;\cos\left(\frac{2\pi kn}{N}\right)
  19. w ( n ) = a 0 - a 1 cos ( 2 π n N - 1 ) + a 2 cos ( 4 π n N - 1 ) w(n)=a_{0}-a_{1}\cos\left(\frac{2\pi n}{N-1}\right)+a_{2}\cos\left(\frac{4\pi n% }{N-1}\right)
  20. a 0 = 1 - α 2 ; a 1 = 1 2 ; a 2 = α 2 a_{0}=\frac{1-\alpha}{2};\quad a_{1}=\frac{1}{2};\quad a_{2}=\frac{\alpha}{2}\,
  21. w ( n ) = a 0 - a 1 cos ( 2 π n N - 1 ) + a 2 cos ( 4 π n N - 1 ) - a 3 cos ( 6 π n N - 1 ) w(n)=a_{0}-a_{1}\cos\left(\frac{2\pi n}{N-1}\right)+a_{2}\cos\left(\frac{4\pi n% }{N-1}\right)-a_{3}\cos\left(\frac{6\pi n}{N-1}\right)
  22. a 0 = 0.355768 ; a 1 = 0.487396 ; a 2 = 0.144232 ; a 3 = 0.012604 a_{0}=0.355768;\quad a_{1}=0.487396;\quad a_{2}=0.144232;\quad a_{3}=0.012604\,
  23. w ( n ) = a 0 - a 1 cos ( 2 π n N - 1 ) + a 2 cos ( 4 π n N - 1 ) - a 3 cos ( 6 π n N - 1 ) w(n)=a_{0}-a_{1}\cos\left(\frac{2\pi n}{N-1}\right)+a_{2}\cos\left(\frac{4\pi n% }{N-1}\right)-a_{3}\cos\left(\frac{6\pi n}{N-1}\right)
  24. a 0 = 0.3635819 ; a 1 = 0.4891775 ; a 2 = 0.1365995 ; a 3 = 0.0106411 a_{0}=0.3635819;\quad a_{1}=0.4891775;\quad a_{2}=0.1365995;\quad a_{3}=0.0106% 411\,
  25. w ( n ) = a 0 - a 1 cos ( 2 π n N - 1 ) + a 2 cos ( 4 π n N - 1 ) - a 3 cos ( 6 π n N - 1 ) w(n)=a_{0}-a_{1}\cos\left(\frac{2\pi n}{N-1}\right)+a_{2}\cos\left(\frac{4\pi n% }{N-1}\right)-a_{3}\cos\left(\frac{6\pi n}{N-1}\right)
  26. a 0 = 0.35875 ; a 1 = 0.48829 ; a 2 = 0.14128 ; a 3 = 0.01168 a_{0}=0.35875;\quad a_{1}=0.48829;\quad a_{2}=0.14128;\quad a_{3}=0.01168\,
  27. w ( n ) = a 0 - a 1 cos ( 2 π n N - 1 ) + a 2 cos ( 4 π n N - 1 ) - a 3 cos ( 6 π n N - 1 ) + a 4 cos ( 8 π n N - 1 ) w(n)=a_{0}-a_{1}\cos\left(\frac{2\pi n}{N-1}\right)+a_{2}\cos\left(\frac{4\pi n% }{N-1}\right)-a_{3}\cos\left(\frac{6\pi n}{N-1}\right)+a_{4}\cos\left(\frac{8% \pi n}{N-1}\right)
  28. a 0 = 1 ; a 1 = 1.93 ; a 2 = 1.29 ; a 3 = 0.388 ; a 4 = 0.028 a_{0}=1;\quad a_{1}=1.93;\quad a_{2}=1.29;\quad a_{3}=0.388;\quad a_{4}=0.028\,
  29. w ( n ) = 1 + l = 1 P a l cos ( l 2 π n N - 1 ) w(n)=1+\sum_{l=1}^{P}a_{l}\cos\left(\frac{l2\pi n}{N-1}\right)
  30. w ( n ) = cos α ( π n N - 1 - π 2 ) w(n)=\cos^{\alpha}\left(\frac{\pi n}{N-1}-\frac{\pi}{2}\right)
  31. w ( n ) = cos ( π n N - 1 - π 2 ) = sin ( π n N - 1 ) w(n)=\cos\left(\frac{\pi n}{N-1}-\frac{\pi}{2}\right)=\sin\left(\frac{\pi n}{N% -1}\right)
  32. w 0 ( n ) w_{0}(n)\,
  33. w ( n ) = e - 1 2 ( n - ( N - 1 ) / 2 σ ( N - 1 ) / 2 ) 2 w(n)=e^{-\frac{1}{2}\left(\frac{n-(N-1)/2}{\sigma(N-1)/2}\right)^{2}}
  34. σ 0.5 \sigma\leq\;0.5\,
  35. w ( n ) = G ( n ) - G ( - 1 2 ) [ G ( n + N ) + G ( n - N ) ] G ( - 1 2 + N ) + G ( - 1 2 - N ) w(n)=G(n)-\frac{G(-\tfrac{1}{2})[G(n+N)+G(n-N)]}{G(-\tfrac{1}{2}+N)+G(-\tfrac{% 1}{2}-N)}
  36. G ( x ) = e - ( x - N - 1 2 2 σ t ) 2 G(x)=e^{-\left(\cfrac{x-\frac{N-1}{2}}{2\sigma_{t}}\right)^{2}}
  37. w ( n , p ) = e - ( n - ( N - 1 ) / 2 σ ( N - 1 ) / 2 ) p w(n,p)=e^{-\left(\frac{n-(N-1)/2}{\sigma(N-1)/2}\right)^{p}}
  38. p p
  39. p = 2 p=2
  40. p p
  41. \infty
  42. p p
  43. w ( n ) = { 1 2 [ 1 + cos ( π ( 2 n α ( N - 1 ) - 1 ) ) ] 0 n α ( N - 1 ) 2 1 α ( N - 1 ) 2 n ( N - 1 ) ( 1 - α 2 ) 1 2 [ 1 + cos ( π ( 2 n α ( N - 1 ) - 2 α + 1 ) ) ] ( N - 1 ) ( 1 - α 2 ) n ( N - 1 ) w(n)=\left\{\begin{matrix}\frac{1}{2}\left[1+\cos\left(\pi\left(\frac{2n}{% \alpha(N-1)}-1\right)\right)\right]&0\leqslant n\leqslant\frac{\alpha(N-1)}{2}% \\ 1&\frac{\alpha(N-1)}{2}\leqslant n\leqslant(N-1)(1-\frac{\alpha}{2})\\ \frac{1}{2}\left[1+\cos\left(\pi\left(\frac{2n}{\alpha(N-1)}-\frac{2}{\alpha}+% 1\right)\right)\right]&(N-1)(1-\frac{\alpha}{2})\leqslant n\leqslant(N-1)\\ \end{matrix}\right.
  44. C C^{\infty}
  45. w ( n ) = { 1 exp ( Z + ) + 1 0 n < ϵ ( N - 1 ) 1 ϵ ( N - 1 ) < n < ( 1 - ϵ ) ( N - 1 ) 1 exp ( Z - ) + 1 ( 1 - ϵ ) ( N - 1 ) < n ( N - 1 ) 0 otherwise w(n)=\left\{\begin{matrix}\frac{1}{\exp(Z_{+})+1}&0\leqslant n<\epsilon(N-1)\\ 1&\epsilon(N-1)<n<(1-\epsilon)(N-1)\\ \frac{1}{\exp(Z_{-})+1}&(1-\epsilon)(N-1)<n\leqslant(N-1)\\ 0&\mbox{otherwise}\\ \end{matrix}\right.
  46. Z ± ( n ; ϵ ) = 2 ϵ [ 1 1 ± 2 n / ( N - 1 ) + 1 1 - 2 ϵ ± 2 n / ( N - 1 ) ] . Z_{\pm}(n;\epsilon)=2\epsilon\left[\frac{1}{1\pm 2n/(N-1)}+\frac{1}{1-2% \epsilon\pm 2n/(N-1)}\right].
  47. w ( n ) = I 0 ( π α 1 - ( 2 n N - 1 - 1 ) 2 ) I 0 ( π α ) w(n)=\frac{I_{0}\left(\pi\alpha\sqrt{1-(\frac{2n}{N-1}-1)^{2}}\right)}{I_{0}(% \pi\alpha)}
  48. 2 1 + α 2 , 2\sqrt{1+\alpha^{2}},
  49. β = def π α . \beta\ \stackrel{\,\text{def}}{=}\ \pi\alpha.
  50. w 0 ( n ) = I 0 ( π α 1 - ( 2 n N - 1 ) 2 ) I 0 ( π α ) w_{0}(n)=\frac{I_{0}\left(\pi\alpha\sqrt{1-(\frac{2n}{N-1})^{2}}\right)}{I_{0}% (\pi\alpha)}
  51. W 0 ( k ) \displaystyle W_{0}(k)
  52. w 0 ( n ) = 1 N k = 0 N - 1 W 0 ( k ) e i 2 π k n / N , - N / 2 n N / 2. w_{0}(n)=\frac{1}{N}\sum_{k=0}^{N-1}W_{0}(k)\cdot e^{i2\pi kn/N},\ -N/2\leq n% \leq N/2.
  53. w ( n ) = w 0 ( n - N - 1 2 ) , w(n)=w_{0}\left(n-\frac{N-1}{2}\right),
  54. w 0 ( n - N - 1 2 ) = 1 N k = 0 N - 1 W 0 ( k ) e i 2 π k ( n - N - 1 2 ) / N = 1 N k = 0 N - 1 [ ( - e i π N ) k W 0 ( k ) ] e i 2 π k n / N , \displaystyle w_{0}\left(n-\frac{N-1}{2}\right)=\frac{1}{N}\sum_{k=0}^{N-1}W_{% 0}(k)\cdot e^{i2\pi k(n-\frac{N-1}{2})/N}=\frac{1}{N}\sum_{k=0}^{N-1}\left[(-e% ^{\frac{i\pi}{N}})^{k}\cdot W_{0}(k)\right]e^{i2\pi kn/N},
  55. ( - e i π N ) k W 0 ( k ) . (-e^{\frac{i\pi}{N}})^{k}\cdot W_{0}(k).
  56. w 0 ( n - N 2 ) , 0 n N - 1 , \scriptstyle w_{0}\left(n-\frac{N}{2}\right),\ 0\leq n\leq N-1,
  57. ( - 1 ) k W 0 ( k ) . (-1)^{k}\cdot W_{0}(k).
  58. ( w 0 ( n - N - 1 2 ) ) \scriptstyle\left(w_{0}\left(n-\frac{N-1}{2}\right)\right)
  59. ( w 0 ( n - N 2 ) ) \scriptstyle\left(w_{0}\left(n-\frac{N}{2}\right)\right)
  60. w ( n ) = 1 N [ C N - 1 μ ( x 0 ) + k = 1 N - 1 2 C N - 1 μ ( x 0 cos k π N ) cos 2 n π k N ] \begin{aligned}\displaystyle w\left(n\right)=\frac{1}{N}\left[C^{\mu}_{N-1}(x_% {0})+\sum_{k=1}^{\frac{N-1}{2}}C^{\mu}_{N-1}\left(x_{0}\cos\frac{k\pi}{N}% \right)\cos\frac{2n\pi k}{N}\right]\end{aligned}
  61. C N μ C^{\mu}_{N}
  62. x 0 x_{0}
  63. μ \mu
  64. μ \mu
  65. μ = 0 \mu=0
  66. μ = 1 \mu=1
  67. w ( n ) = e - | n - N - 1 2 | 1 τ , w(n)=e^{-\left|n-\frac{N-1}{2}\right|\frac{1}{\tau}},
  68. τ = N 2 8.69 D . \tau=\frac{N}{2}\frac{8.69}{D}.
  69. w ( n ) = a 0 - a 1 | n N - 1 - 1 2 | - a 2 cos ( 2 π n N - 1 ) w(n)=a_{0}-a_{1}\left|\frac{n}{N-1}-\frac{1}{2}\right|-a_{2}\cos\left(\frac{2% \pi n}{N-1}\right)
  70. a 0 = 0.62 ; a 1 = 0.48 ; a 2 = 0.38 a_{0}=0.62;\quad a_{1}=0.48;\quad a_{2}=0.38\,
  71. w ( n ) = 1 2 ( 1 - cos ( 2 π n N - 1 ) ) e - α | N - 1 - 2 n | N - 1 w(n)=\frac{1}{2}\left(1-\cos\left(\frac{2\pi n}{N-1}\right)\right)e^{\frac{-% \alpha\left|N-1-2n\right|}{N-1}}\,
  72. w ( n ) = sinc ( 2 n N - 1 - 1 ) w(n)=\mathrm{sinc}\left(\frac{2n}{N-1}-1\right)
  73. w 0 ( n ) = sinc ( 2 n N - 1 ) w_{0}(n)=\mathrm{sinc}\left(\frac{2n}{N-1}\right)\,
  74. W ( m , n ) = w ( m ) w ( n ) \scriptstyle W(m,n)=w(m)w(n)
  75. W ( m , n ) = w ( r ) \scriptstyle W(m,n)=w(r)
  76. r = ( m - M / 2 ) 2 + ( n - N / 2 ) 2 \scriptstyle r=\sqrt{(m-M/2)^{2}+(n-N/2)^{2}}

Woldemar_Voigt.html

  1. v v
  2. x x
  3. x = x - v t x^{\prime}=x-vt
  4. y = y / γ y^{\prime}=y/\gamma
  5. z = z / γ z^{\prime}=z/\gamma
  6. t = t - v x / c 2 t^{\prime}=t-vx/c^{2}
  7. γ = 1 / 1 - v 2 / c 2 \gamma=1/\sqrt{1-v^{2}/c^{2}}
  8. γ \gamma
  9. Δ t Voigt = γ - 2 Δ t = γ - 1 Δ t Lorentz \Delta t_{\mathrm{Voigt}}=\gamma^{-2}\Delta t=\gamma^{-1}\Delta t_{\mathrm{% Lorentz}}
  10. Δ t \Delta t

Woodin_cardinal.html

  1. A V λ A\subseteq V_{\lambda}
  2. λ A \lambda_{A}
  3. A A
  4. λ A \lambda_{A}
  5. < λ <\lambda
  6. A A
  7. λ A \lambda_{A}
  8. j ( λ A ) > α j(\lambda_{A})>\alpha
  9. V α M V_{\alpha}\subseteq M
  10. j ( A ) V α = A V α j(A)\cap V_{\alpha}=A\cap V_{\alpha}
  11. Θ 0 \Theta_{0}
  12. Θ 0 \Theta_{0}
  13. 2 \aleph_{2}
  14. 1 \aleph_{1}
  15. 1 \aleph_{1}
  16. j ( S ) H δ = S H δ . j(S)\cap H_{\delta}=S\cap H_{\delta}.

Work_(physics).html

  1. W = F s . W=Fs.
  2. d E = δ Q - δ W , dE=\delta Q-\delta W,
  3. δ \delta
  4. W = Δ K E . W=\Delta KE.
  5. W = - Δ P E . W=-\Delta PE.
  6. δ W = 𝐅 d 𝐬 = 𝐅 𝐯 d t \delta W=\mathbf{F}\cdot d\mathbf{s}=\mathbf{F}\cdot\mathbf{v}dt
  7. W = t 1 t 2 𝐅 𝐯 d t = t 1 t 2 𝐅 d 𝐬 d t d t = C 𝐅 d 𝐬 , W=\int_{t_{1}}^{t_{2}}\mathbf{F}\cdot\mathbf{v}dt=\int_{t_{1}}^{t_{2}}\mathbf{% F}\cdot{\tfrac{d\mathbf{s}}{dt}}dt=\int_{C}\mathbf{F}\cdot d\mathbf{s},
  8. W = C F d s W=\int_{C}F\,ds
  9. W = C F d s = F C d s = F s W=\int_{C}F\,ds=F\int_{C}ds=Fs
  10. W = C 𝐅 d 𝐬 = F s cos θ . W=\int_{C}\mathbf{F}\cdot d\mathbf{s}=Fs\cos\theta.
  11. F cos θ \scriptstyle F\cos\theta
  12. θ \scriptstyle\theta
  13. δ W = 𝐓 ω δ t , \delta W=\mathbf{T}\cdot\vec{\omega}\delta t,
  14. W = t 1 t 2 𝐓 ω d t . W=\int_{t_{1}}^{t_{2}}\mathbf{T}\cdot\vec{\omega}dt.
  15. ω = ϕ ˙ 𝐒 , \vec{\omega}=\dot{\phi}\mathbf{S},
  16. W = t 1 t 2 𝐓 ω d t = t 1 t 2 𝐓 𝐒 d ϕ d t d t = C 𝐓 𝐒 d ϕ , W=\int_{t_{1}}^{t_{2}}\mathbf{T}\cdot\vec{\omega}dt=\int_{t_{1}}^{t_{2}}% \mathbf{T}\cdot\mathbf{S}\frac{d\phi}{dt}dt=\int_{C}\mathbf{T}\cdot\mathbf{S}d\phi,
  17. 𝐓 = τ 𝐒 , \mathbf{T}=\tau\mathbf{S},
  18. W = t 1 t 2 τ ϕ ˙ d t = τ ( ϕ 2 - ϕ 1 ) . W=\int_{t_{1}}^{t_{2}}\tau\dot{\phi}dt=\tau(\phi_{2}-\phi_{1}).
  19. W = F s = F r ϕ . W=Fs=Fr\phi.
  20. W = F r ϕ = τ ϕ , W=Fr\phi=\tau\phi,
  21. W = C 𝐅 d 𝐱 = t 1 t 2 𝐅 𝐯 d t , W=\int_{C}\mathbf{F}\cdot d\mathbf{x}=\int_{t_{1}}^{t_{2}}\mathbf{F}\cdot% \mathbf{v}dt,
  22. d W d t = P ( t ) = 𝐅 𝐯 . \frac{dW}{dt}=P(t)=\mathbf{F}\cdot\mathbf{v}.
  23. W = C 𝐅 d 𝐱 = 𝐱 ( t 1 ) 𝐱 ( t 2 ) 𝐅 d 𝐱 = U ( 𝐱 ( t 1 ) ) - U ( 𝐱 ( t 2 ) ) . W=\int_{C}\mathbf{F}\cdot\mathrm{d}\mathbf{x}=\int_{\mathbf{x}(t_{1})}^{% \mathbf{x}(t_{2})}\mathbf{F}\cdot\mathrm{d}\mathbf{x}=U(\mathbf{x}(t_{1}))-U(% \mathbf{x}(t_{2})).
  24. W 𝐱 = - U 𝐱 = - ( U x , U y , U z ) = 𝐅 , \frac{\partial W}{\partial\mathbf{x}}=-\frac{\partial U}{\partial\mathbf{x}}=-% \big(\frac{\partial U}{\partial x},\frac{\partial U}{\partial y},\frac{% \partial U}{\partial z}\big)=\mathbf{F},
  25. P ( t ) = - U 𝐱 𝐯 = 𝐅 𝐯 . P(t)=-\frac{\partial U}{\partial\mathbf{x}}\cdot\mathbf{v}=\mathbf{F}\cdot% \mathbf{v}.
  26. W = F g ( y 2 - y 1 ) = F g Δ y = m g Δ y W=F_{g}(y_{2}-y_{1})=F_{g}\Delta y=mg\Delta y
  27. 𝐅 = - G M m r 3 𝐫 , \mathbf{F}=-\frac{GMm}{r^{3}}\mathbf{r},
  28. W = - 𝐫 ( t 1 ) 𝐫 ( t 2 ) G M m r 3 𝐫 d 𝐫 = - t 1 t 2 G M m r 3 𝐫 𝐯 d t . W=-\int^{\mathbf{r}(t_{2})}_{\mathbf{r}(t_{1})}\frac{GMm}{r^{3}}\mathbf{r}% \cdot d\mathbf{r}=-\int^{t_{2}}_{t_{1}}\frac{GMm}{r^{3}}\mathbf{r}\cdot\mathbf% {v}dt.
  29. 𝐫 = r 𝐞 r , 𝐯 = r ˙ 𝐞 r + r θ ˙ 𝐞 t , \mathbf{r}=r\mathbf{e}_{r},\qquad\mathbf{v}=\dot{r}\mathbf{e}_{r}+r\dot{\theta% }\mathbf{e}_{t},
  30. W = - t 1 t 2 G m M r 3 ( r 𝐞 r ) ( r ˙ 𝐞 r + r θ ˙ 𝐞 t ) d t = - t 1 t 2 G m M r 3 r r ˙ d t = G M m r ( t 2 ) - G M m r ( t 1 ) . W=-\int^{t_{2}}_{t_{1}}\frac{GmM}{r^{3}}(r\mathbf{e}_{r})\cdot(\dot{r}\mathbf{% e}_{r}+r\dot{\theta}\mathbf{e}_{t})dt=-\int^{t_{2}}_{t_{1}}\frac{GmM}{r^{3}}r% \dot{r}dt=\frac{GMm}{r(t_{2})}-\frac{GMm}{r(t_{1})}.
  31. d d t r - 1 = - r - 2 r ˙ = - r ˙ r 2 . \frac{d}{dt}r^{-1}=-r^{-2}\dot{r}=-\frac{\dot{r}}{r^{2}}.
  32. U = - G M m r , U=-\frac{GMm}{r},
  33. W = 0 t 𝐅 𝐯 d t = - 0 t k x v x d t = - 1 2 k x 2 . W=\int_{0}^{t}\mathbf{F}\cdot\mathbf{v}dt=-\int_{0}^{t}kxv_{x}dt=-\frac{1}{2}% kx^{2}.
  34. W = a b P d V W=\int_{a}^{b}{P}dV
  35. E k E_{k}
  36. W = Δ E k = 1 2 m v 2 2 - 1 2 m v 1 2 W=\Delta E_{k}=\tfrac{1}{2}mv_{2}^{2}-\tfrac{1}{2}mv_{1}^{2}
  37. v 1 v_{1}
  38. v 2 v_{2}
  39. s = v 2 2 - v 1 2 2 a s=\frac{v_{2}^{2}-v_{1}^{2}}{2a}
  40. v 2 2 = v 1 2 + 2 a s v_{2}^{2}=v_{1}^{2}+2as
  41. W = F s = m a s = m a ( v 2 2 - v 1 2 2 a ) = m v 2 2 2 - m v 1 2 2 = Δ E k W=Fs=mas=ma\left(\frac{v_{2}^{2}-v_{1}^{2}}{2a}\right)=\frac{mv_{2}^{2}}{2}-% \frac{mv_{1}^{2}}{2}=\Delta{E_{k}}
  42. W = F s = m a s = m ( v 2 2 - u 1 2 2 s ) s W=Fs=mas=m\left(\frac{v_{2}^{2}-u_{1}^{2}}{2s}\right)s
  43. W = m ( 0 2 2 - v 1 2 2 s ) s W=m\left(\frac{0_{2}^{2}-v_{1}^{2}}{2s}\right)s
  44. W = - 1 2 m v 2 W=-\frac{1}{2}mv^{2}
  45. W = t 1 t 2 𝐅 𝐯 d t = t 1 t 2 F v d t = t 1 t 2 m a v d t = m t 1 t 2 v d v d t d t = m v 1 v 2 v d v = 1 2 m ( v 2 2 - v 1 2 ) . W=\int_{t_{1}}^{t_{2}}\mathbf{F}\cdot\mathbf{v}dt=\int_{t_{1}}^{t_{2}}F\,vdt=% \int_{t_{1}}^{t_{2}}ma\,vdt=m\int_{t_{1}}^{t_{2}}v\,{dv\over dt}\,dt=m\int_{v_% {1}}^{v_{2}}v\,dv=\tfrac{1}{2}m(v_{2}^{2}-v_{1}^{2}).
  46. W = t 1 t 2 𝐅 𝐯 d t = m t 1 t 2 𝐚 𝐯 d t = m 2 t 1 t 2 d v 2 d t d t = m 2 v 1 2 v 2 2 d v 2 = m v 2 2 2 - m v 1 2 2 = Δ E k W=\int_{t_{1}}^{t_{2}}\mathbf{F}\cdot\mathbf{v}dt=m\int_{t_{1}}^{t_{2}}\mathbf% {a}\cdot\mathbf{v}dt=\frac{m}{2}\int_{t_{1}}^{t_{2}}\frac{dv^{2}}{dt}\,dt=% \frac{m}{2}\int_{v^{2}_{1}}^{v^{2}_{2}}dv^{2}=\frac{mv_{2}^{2}}{2}-\frac{mv_{1% }^{2}}{2}=\Delta{E_{k}}
  47. 𝐚 𝐯 = 1 2 d v 2 d t \textstyle\mathbf{a}\cdot\mathbf{v}=\frac{1}{2}\frac{dv^{2}}{dt}
  48. v 2 = 𝐯 𝐯 \textstyle v^{2}=\mathbf{v}\cdot\mathbf{v}
  49. 𝐚 = d 𝐯 d t \textstyle\mathbf{a}=\frac{d\mathbf{v}}{dt}
  50. d v 2 d t = d ( 𝐯 𝐯 ) d t = d 𝐯 d t 𝐯 + 𝐯 d 𝐯 d t = 2 d 𝐯 d t 𝐯 = 2 𝐚 𝐯 \frac{dv^{2}}{dt}=\frac{d(\mathbf{v}\cdot\mathbf{v})}{dt}=\frac{d\mathbf{v}}{% dt}\cdot\mathbf{v}+\mathbf{v}\cdot\frac{d\mathbf{v}}{dt}=2\frac{d\mathbf{v}}{% dt}\cdot\mathbf{v}=2\mathbf{a}\cdot\mathbf{v}
  51. 𝐅 + 𝐑 = m 𝐗 ¨ , \mathbf{F}+\mathbf{R}=m\ddot{\mathbf{X}},
  52. 𝐅 𝐗 ˙ = m 𝐗 ¨ 𝐗 ˙ , \mathbf{F}\cdot\dot{\mathbf{X}}=m\ddot{\mathbf{X}}\cdot\dot{\mathbf{X}},
  53. t 1 t 2 𝐅 𝐗 ˙ d t = m t 1 t 2 𝐗 ¨ 𝐗 ˙ d t . \int_{t_{1}}^{t_{2}}\mathbf{F}\cdot\dot{\mathbf{X}}dt=m\int_{t_{1}}^{t_{2}}% \ddot{\mathbf{X}}\cdot\dot{\mathbf{X}}dt.
  54. W = t 1 t 2 𝐅 𝐗 ˙ d t = 𝐗 ( t 1 ) 𝐗 ( t 2 ) 𝐅 d 𝐗 . W=\int_{t_{1}}^{t_{2}}\mathbf{F}\cdot\dot{\mathbf{X}}dt=\int_{\mathbf{X}(t_{1}% )}^{\mathbf{X}(t_{2})}\mathbf{F}\cdot d\mathbf{X}.
  55. 1 2 d d t ( 𝐗 ˙ 𝐗 ˙ ) = 𝐗 ¨ 𝐗 ˙ , \frac{1}{2}\frac{d}{dt}(\dot{\mathbf{X}}\cdot\dot{\mathbf{X}})=\ddot{\mathbf{X% }}\cdot\dot{\mathbf{X}},
  56. Δ K = m t 1 t 2 𝐗 ¨ 𝐗 ˙ d t = m 2 t 1 t 2 d d t ( 𝐗 ˙ 𝐗 ˙ ) d t = m 2 𝐗 ˙ 𝐗 ˙ ( t 2 ) - m 2 𝐗 ˙ 𝐗 ˙ ( t 1 ) = 1 2 m Δ 𝐯 𝟐 , \Delta K=m\int_{t_{1}}^{t_{2}}\ddot{\mathbf{X}}\cdot\dot{\mathbf{X}}dt=\frac{m% }{2}\int_{t_{1}}^{t_{2}}\frac{d}{dt}(\dot{\mathbf{X}}\cdot\dot{\mathbf{X}})dt=% \frac{m}{2}\dot{\mathbf{X}}\cdot\dot{\mathbf{X}}(t_{2})-\frac{m}{2}\dot{% \mathbf{X}}\cdot\dot{\mathbf{X}}(t_{1})=\frac{1}{2}m\Delta\mathbf{v^{2}},
  57. K = m 2 𝐗 ˙ 𝐗 ˙ = 1 2 m 𝐯 𝟐 K=\frac{m}{2}\dot{\mathbf{X}}\cdot\dot{\mathbf{X}}=\frac{1}{2}m{\mathbf{v^{2}}}
  58. 𝐗 ˙ = v 𝐓 , and 𝐗 ¨ = v ˙ 𝐓 + v 2 κ 𝐍 . \dot{\mathbf{X}}=v\mathbf{T},\quad\mbox{and}~{}\quad\ddot{\mathbf{X}}=\dot{v}% \mathbf{T}+v^{2}\kappa\mathbf{N}.
  59. v = | 𝐗 ˙ | = 𝐗 ˙ 𝐗 ˙ . v=|\dot{\mathbf{X}}|=\sqrt{\dot{\mathbf{X}}\cdot\dot{\mathbf{X}}}.
  60. Δ K = m t 1 t 2 v ˙ v d t = m 2 t 1 t 2 d d t v 2 d t = m 2 v 2 ( t 2 ) - m 2 v 2 ( t 1 ) , \Delta K=m\int_{t_{1}}^{t_{2}}\dot{v}vdt=\frac{m}{2}\int_{t_{1}}^{t_{2}}\frac{% d}{dt}v^{2}dt=\frac{m}{2}v^{2}(t_{2})-\frac{m}{2}v^{2}(t_{1}),
  61. K = m 2 v 2 = m 2 𝐗 ˙ 𝐗 ˙ . K=\frac{m}{2}v^{2}=\frac{m}{2}\dot{\mathbf{X}}\cdot\dot{\mathbf{X}}.
  62. W = Δ K . W=\Delta K.\!
  63. 𝐅 + 𝐑 = m 𝐗 ¨ . \mathbf{F}+\mathbf{R}=m\ddot{\mathbf{X}}.
  64. F x v = m v ˙ v . F_{x}v=m\dot{v}v.
  65. t 1 t 2 F x v d t = m 2 v 2 ( t 2 ) - m 2 v 2 ( t 1 ) . \int_{t_{1}}^{t_{2}}F_{x}vdt=\frac{m}{2}v^{2}(t_{2})-\frac{m}{2}v^{2}(t_{1}).
  66. F x ( d ( t 2 ) - d ( t 1 ) ) = m 2 v 2 ( t 2 ) - m 2 v 2 ( t 1 ) . F_{x}(d(t_{2})-d(t_{1}))=\frac{m}{2}v^{2}(t_{2})-\frac{m}{2}v^{2}(t_{1}).
  67. k W s = W 2 g v 2 , or v = 2 k s g . kWs=\frac{W}{2g}v^{2},\quad\mbox{or}~{}\quad v=\sqrt{2ksg}.
  68. 𝐅 + 𝐑 = m 𝐗 ¨ . \mathbf{F}+\mathbf{R}=m\ddot{\mathbf{X}}.
  69. W v z = m V ˙ V , Wv_{z}=m\dot{V}V,
  70. t 1 t 2 W v z d t = m 2 V 2 ( t 2 ) - m 2 V 2 ( t 1 ) . \int_{t_{1}}^{t_{2}}Wv_{z}dt=\frac{m}{2}V^{2}(t_{2})-\frac{m}{2}V^{2}(t_{1}).
  71. W Δ z = m 2 V 2 . W\Delta z=\frac{m}{2}V^{2}.
  72. s = Δ z 0.06 = 8.3 V 2 g , or s = 8.3 88 2 32.2 2000 ft . s=\frac{\Delta z}{0.06}=8.3\frac{V^{2}}{g},\quad\mbox{or}~{}\quad s=8.3\frac{8% 8^{2}}{32.2}\approx 2000\mbox{ft}~{}.
  73. 𝐗 i ( t ) = [ A ( t ) ] 𝐱 i + 𝐝 ( t ) i = 1 , , n . \mathbf{X}_{i}(t)=[A(t)]\mathbf{x}_{i}+\mathbf{d}(t)\quad i=1,\ldots,n.
  74. 𝐕 i = ω × ( 𝐗 i - 𝐝 ) + 𝐝 ˙ , \mathbf{V}_{i}=\vec{\omega}\times(\mathbf{X}_{i}-\mathbf{d})+\dot{\mathbf{d}},
  75. [ Ω ] = A ˙ A T , [\Omega]=\dot{A}A^{\mathrm{T}},
  76. δ W = 𝐅 1 𝐕 1 δ t + 𝐅 2 𝐕 2 δ t + + 𝐅 n 𝐕 n δ t \delta W=\mathbf{F}_{1}\cdot\mathbf{V}_{1}\delta t+\mathbf{F}_{2}\cdot\mathbf{% V}_{2}\delta t+\ldots+\mathbf{F}_{n}\cdot\mathbf{V}_{n}\delta t
  77. δ W = i = 1 n 𝐅 i ( ω × ( 𝐗 i - 𝐝 ) + 𝐝 ˙ ) δ t . \delta W=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot(\vec{\omega}\times(\mathbf{X}_{i}-% \mathbf{d})+\dot{\mathbf{d}})\delta t.
  78. δ W = ( i = 1 n 𝐅 i ) 𝐝 ˙ δ t + ( i = 1 n ( 𝐗 i - 𝐝 ) × 𝐅 i ) ω δ t = ( 𝐅 𝐝 ˙ + 𝐓 ω ) δ t , \delta W=(\sum_{i=1}^{n}\mathbf{F}_{i})\cdot\dot{\mathbf{d}}\delta t+(\sum_{i=% 1}^{n}(\mathbf{X}_{i}-\mathbf{d})\times\mathbf{F}_{i})\cdot\vec{\omega}\delta t% =(\mathbf{F}\cdot\dot{\mathbf{d}}+\mathbf{T}\cdot\vec{\omega})\delta t,

Working_mass.html

  1. Δ v = u ln ( m + M M ) \Delta\,v=u\,\ln\left(\frac{m+M}{M}\right)

World_line.html

  1. t t
  2. x x
  3. x a ( τ ) , a = 0 , 1 , 2 , 3 x^{a}(\tau),\;a=0,1,2,3
  4. x 0 x^{0}
  5. τ \tau
  6. τ \tau
  7. x a ( τ ) , a = 0 , 1 , 2 , 3 x^{a}(\tau),\;a=0,1,2,3
  8. τ \tau
  9. p p
  10. τ 0 \tau_{0}
  11. τ 0 + Δ τ \tau_{0}+\Delta\tau
  12. Δ τ 0 \Delta\tau\rightarrow 0
  13. Δ τ \Delta\tau
  14. p p
  15. p p
  16. v \vec{v}
  17. v = ( v 0 , v 1 , v 2 , v 3 ) = ( d x 0 d τ , d x 1 d τ , d x 2 d τ , d x 3 d τ ) \vec{v}=(v^{0},v^{1},v^{2},v^{3})=\left(\frac{dx^{0}}{d\tau}\;,\frac{dx^{1}}{d% \tau}\;,\frac{dx^{2}}{d\tau}\;,\frac{dx^{3}}{d\tau}\right)
  18. p p
  19. τ = τ 0 \tau=\tau_{0}
  20. w ( τ ) \isin R 4 \scriptstyle w(\tau)\isin R^{4}
  21. v = d w d τ \scriptstyle v=\frac{dw}{d\tau}
  22. η ( v , x ) \scriptstyle\eta(v,x)
  23. R 4 R \scriptstyle R^{4}\rightarrow R
  24. x η ( v , x ) . \scriptstyle x\mapsto\eta(v,x).
  25. d u d τ = d w d τ , \scriptstyle\frac{du}{d\tau}=\frac{dw}{d\tau},

Wreath_product.html

  1. K ω Ω A ω K\equiv\prod_{\omega\,\in\,\Omega}A_{\omega}
  2. h ( a ω ) ( a h - 1 ω ) h(a_{\omega})\equiv(a_{h^{-1}\omega})
  3. K ω Ω A ω K\equiv\bigoplus_{\omega\,\in\,\Omega}A_{\omega}
  4. ( ( a ω ) , h ) ( λ , ω ) := ( a h ( ω ) λ , h ω ) ((a_{\omega}),h)\cdot(\lambda,\omega^{\prime}):=(a_{h(\omega^{\prime})}\lambda% ,h\omega^{\prime})
  5. ( ( a ω ) , h ) ( λ ω ) := ( a h - 1 ω λ h - 1 ω ) ((a_{\omega}),h)\cdot(\lambda_{\omega}):=(a_{h^{-1}\omega}\lambda_{h^{-1}% \omega})

Write–read_conflict.html

  1. S = [ T 1 T 2 R ( A ) W ( A ) R ( A ) W ( A ) R ( B ) W ( B ) C o m . R ( B ) W ( B ) C o m . ] S=\begin{bmatrix}T1&T2\\ R(A)&\\ W(A)&\\ &R(A)\\ &W(A)\\ &R(B)\\ &W(B)\\ &Com.\\ R(B)&\\ W(B)&\\ Com.&\end{bmatrix}

Write–write_conflict.html

  1. S = [ T 1 T 2 W ( A ) W ( B ) W ( B ) C o m . W ( A ) C o m . ] S=\begin{bmatrix}T1&T2\\ W(A)&\\ &W(B)\\ W(B)&\\ Com.&\\ &W(A)\\ &Com.\end{bmatrix}

Wronskian.html

  1. f f
  2. g g
  3. W ( f , g ) = f g g f W(f,g)=f g′–g f′
  4. n n
  5. n 1 n–1
  6. I I
  7. I I
  8. W ( f 1 , , f n ) ( x ) = | f 1 ( x ) f 2 ( x ) f n ( x ) f 1 ( x ) f 2 ( x ) f n ( x ) f 1 ( n - 1 ) ( x ) f 2 ( n - 1 ) ( x ) f n ( n - 1 ) ( x ) | , x I . W(f_{1},\ldots,f_{n})(x)=\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots&f_{n}(x)\\ f_{1}^{\prime}(x)&f_{2}^{\prime}(x)&\cdots&f_{n}^{\prime}(x)\\ \vdots&\vdots&\ddots&\vdots\\ f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots&f_{n}^{(n-1)}(x)\end{vmatrix},\qquad x% \in I.
  9. ( n 1 ) (n–1)
  10. f < s u b > i f<sub>i

X-ray_fluorescence.html

  1. λ = h c E \lambda=\frac{hc}{E}
  2. n λ = 2 d sin ( θ ) n\cdot\lambda=2d\cdot\sin(\theta)

X-ray_photoelectron_spectroscopy.html

  1. E binding = E photon - ( E kinetic + ϕ ) E\text{binding}=E\text{photon}-\left(E\text{kinetic}+\phi\right)
  2. ϕ \phi
  3. ϕ \phi
  4. V 1 V_{1}
  5. V 2 V_{2}
  6. V ( r ) = - [ ( V 2 - V 1 ) ( R 2 - R 1 ) ] ( R 1 R 2 ) r + c o n s t . V(r)=-\left[\frac{(V_{2}-V_{1})}{(R_{2}-R_{1})}\right]\cdot\frac{(R_{1}R_{2})}% {r}+const.
  7. | E ( r ) | = - [ ( V 2 - V 1 ) ( R 2 - R 1 ) ] ( R 1 R 2 ) r 2 |E(r)|=-\left[\frac{(V_{2}-V_{1})}{(R_{2}-R_{1})}\right]\cdot\frac{(R_{1}R_{2}% )}{r^{2}}
  8. R 1 R_{1}
  9. R 2 R_{2}
  10. R 0 = ( R 1 + R 2 ) 2 R_{0}=\frac{(R1+R2)}{2}
  11. F E = - e | E ( r ) | F_{E}=-e|E(r)|
  12. F C F_{C}
  13. V ( r ) = ( V 0 R 0 r ) + c o n s t . V(r)=\left(\frac{V_{0}R_{0}}{r}\right)+const.
  14. V 0 = E 0 e V_{0}=\frac{E_{0}}{e}
  15. V 2 - V 1 = V 0 ( R 2 R 1 - R 1 R 2 ) V_{2}-V_{1}=V_{0}\left(\frac{R_{2}}{R_{1}}-\frac{R_{1}}{R_{2}}\right)
  16. E 0 = | e | V 0 E_{0}=|e|V_{0}
  17. E 0 E_{0}
  18. R 0 = ( R 1 + R 2 ) / 2 R_{0}=(R_{1}+R_{2})/2
  19. Δ E = E 0 ( w 2 R 0 + α 2 4 ) \Delta E=E_{0}\left(\frac{w}{2R_{0}}+\frac{\alpha^{2}}{4}\right)
  20. w w
  21. α \alpha
  22. R 0 R_{0}
  23. R 0 R_{0}
  24. E 0 E_{0}
  25. E 0 E_{0}
  26. V 1 V_{1}
  27. V 2 V_{2}
  28. h ν = | E b v | + E k i n h\nu=|E_{b}^{v}|+E_{kin}
  29. h ν h\nu
  30. | E b v | |E_{b}^{v}|
  31. E k i n E_{kin}
  32. | E b v | |E_{b}^{v}|
  33. | E b F | |E_{b}^{F}|
  34. Φ 0 \Phi_{0}
  35. i ψ t = [ 1 2 m ( 𝐩 ^ - e c 𝐀 ^ ) 2 + V ^ ] ψ = H ^ ψ i\hbar\frac{\partial\psi}{\partial t}=\left[\frac{1}{2m}\left(\mathbf{\hat{p}}% -\frac{e}{c}\mathbf{\hat{A}}\right)^{2}+\hat{V}\right]\psi=\hat{H}\psi
  36. ψ \psi
  37. 𝐀 \mathbf{A}
  38. V V
  39. 𝐀 = 0 \nabla\cdot\mathbf{A}=0
  40. [ 𝐩 ^ , 𝐀 ^ ] = 0 [\mathbf{\hat{p}},\mathbf{\hat{A}}]=0
  41. ( 𝐩 ^ - e c 𝐀 ^ ) 2 = p ^ 2 - 2 e c 𝐀 ^ 𝐩 ^ + ( e c ) 2 A ^ 2 \left(\mathbf{\hat{p}}-\frac{e}{c}\mathbf{\hat{A}}\right)^{2}=\hat{p}^{2}-2% \frac{e}{c}\mathbf{\hat{A}}\cdot\mathbf{\hat{p}}+\left(\frac{e}{c}\right)^{2}% \hat{A}^{2}
  42. 𝐀 \nabla\cdot\mathbf{A}
  43. 𝐀 \mathbf{A}
  44. H ^ 0 \hat{H}_{0}
  45. H ^ \hat{H}^{\prime}
  46. H ^ = - e m c 𝐀 ^ 𝐩 ^ \hat{H}^{\prime}=-\frac{e}{mc}\mathbf{\hat{A}}\cdot\mathbf{\hat{p}}
  47. ψ i \psi_{i}
  48. ψ f \psi_{f}
  49. d ω d t 2 π | ψ f | H ^ | ψ i | 2 δ ( E f - E i - h ν ) \frac{d\omega}{dt}\propto\frac{2\pi}{\hbar}|\langle\psi_{f}|\hat{H}^{\prime}|% \psi_{i}\rangle|^{2}\delta(E_{f}-E_{i}-h\nu)
  50. E i E_{i}
  51. E f E_{f}
  52. h ν h\nu
  53. ρ f \rho_{f}
  54. d ω d t 2 π | ψ f | H ^ | ψ i | 2 ρ f = | M f i | 2 ρ f \frac{d\omega}{dt}\propto\frac{2\pi}{\hbar}|\langle\psi_{f}|\hat{H}^{\prime}|% \psi_{i}\rangle|^{2}\rho_{f}=|M_{fi}|^{2}\rho_{f}
  55. τ \tau
  56. exp - t / τ \propto\exp{-t/\tau}
  57. Γ \Gamma
  58. I L ( E ) = I 0 π Γ / 2 ( E - E b ) 2 + ( Γ / 2 ) 2 I_{L}(E)=\frac{I_{0}}{\pi}\frac{\Gamma/2}{(E-E_{b})^{2}+(\Gamma/2)^{2}}
  59. Γ \Gamma
  60. τ \tau
  61. Γ τ \Gamma\tau\geq\hbar
  62. I G ( E ) = I 0 σ 2 exp ( - ( E - E b ) 2 2 σ 2 ) I_{G}(E)=\frac{I_{0}}{\sigma\sqrt{2}}\exp{\left(-\frac{(E-E_{b})^{2}}{2\sigma^% {2}}\right)}
  63. ω s u r f = ω b u l k 2 \omega_{surf}=\frac{\omega_{bulk}}{\sqrt{2}}
  64. ϕ j \phi_{j}
  65. W j = exp ( - Δ k j 2 U j 2 ¯ ) W_{j}=\exp{(-\Delta k_{j}^{2}\bar{U_{j}^{2}})}
  66. Δ k j 2 \Delta k_{j}^{2}
  67. U j 2 ¯ \bar{U_{j}^{2}}
  68. j t h j^{th}
  69. Θ D \Theta_{D}
  70. U j 2 ¯ ( T ) = 9 2 T 2 / m k B Θ D \bar{U_{j}^{2}}(T)=9\hbar^{2}T^{2}/mk_{B}\Theta_{D}

Xanthine_oxidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

XOR_swap_algorithm.html

  1. N N
  2. \oplus
  3. A B = B A A\oplus B=B\oplus A
  4. ( A B ) C = A ( B C ) (A\oplus B)\oplus C=A\oplus(B\oplus C)
  5. A 0 = A A\oplus 0=A
  6. A A
  7. A A
  8. A A = 0 A\oplus A=0
  9. A A
  10. B B
  11. A B A\oplus B
  12. B \ B
  13. A B A\oplus B
  14. ( A B ) B = A ( B B ) (A\oplus B)\oplus B=A\oplus(B\oplus B)
  15. = A 0 =A\oplus 0
  16. = A =A
  17. ( A B ) A = A ( A B ) (A\oplus B)\oplus A=A\oplus(A\oplus B)
  18. = ( A A ) B =(A\oplus A)\oplus B
  19. = 0 B =0\oplus B
  20. = B 0 =B\oplus 0
  21. = B =B
  22. A \ A
  23. ( 1 1 0 1 ) \left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)
  24. ( 1 1 0 1 ) ( x y ) = ( x + y y ) . \begin{pmatrix}1&1\\ 0&1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}x+y\\ y\end{pmatrix}.
  25. ( 1 1 0 1 ) ( 1 0 1 1 ) ( 1 1 0 1 ) = ( 0 1 1 0 ) \begin{pmatrix}1&1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ 1&1\end{pmatrix}\begin{pmatrix}1&1\\ 0&1\end{pmatrix}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  26. 1 + 1 = 0 1+1=0
  27. ( I n I n 0 I n ) . \left(\begin{smallmatrix}I_{n}&I_{n}\\ 0&I_{n}\end{smallmatrix}\right).
  28. / 2 s \mathbb{Z}/2^{s}\mathbb{Z}
  29. s s
  30. ( x + y ) - y = x (x+y)-y=x
  31. ( x + y ) - ( ( x + y ) - y ) = y (x+y)-((x+y)-y)=y
  32. ( / 2 ) s (\mathbb{Z}/2\mathbb{Z})^{s}

Y-Δ_transform.html

  1. R y R_{y}
  2. R R^{\prime}
  3. R ′′ R^{\prime\prime}
  4. R y = R R ′′ R Δ R_{y}=\frac{R^{\prime}R^{\prime\prime}}{\sum R_{\Delta}}
  5. R Δ R_{\Delta}
  6. R 1 = R b R c R a + R b + R c R 2 = R a R c R a + R b + R c R 3 = R a R b R a + R b + R c \begin{aligned}\displaystyle R_{1}&\displaystyle=\frac{R_{b}R_{c}}{R_{a}+R_{b}% +R_{c}}\\ \displaystyle R_{2}&\displaystyle=\frac{R_{a}R_{c}}{R_{a}+R_{b}+R_{c}}\\ \displaystyle R_{3}&\displaystyle=\frac{R_{a}R_{b}}{R_{a}+R_{b}+R_{c}}\end{aligned}
  7. R Δ R_{\Delta}
  8. R Δ = R P R opposite R_{\Delta}=\frac{R_{P}}{R_{\mathrm{opposite}}}
  9. R P = R 1 R 2 + R 2 R 3 + R 3 R 1 R_{P}=R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}
  10. R opposite R_{\mathrm{opposite}}
  11. R Δ R_{\Delta}
  12. R a = R 1 R 2 + R 2 R 3 + R 3 R 1 R 1 R b = R 1 R 2 + R 2 R 3 + R 3 R 1 R 2 R c = R 1 R 2 + R 2 R 3 + R 3 R 1 R 3 \begin{aligned}\displaystyle R_{a}&\displaystyle=\frac{R_{1}R_{2}+R_{2}R_{3}+R% _{3}R_{1}}{R_{1}}\\ \displaystyle R_{b}&\displaystyle=\frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2% }}\\ \displaystyle R_{c}&\displaystyle=\frac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3% }}\end{aligned}
  13. V LL = 3 V LN 30 I LL = 3 I LN - 30 Z Δ / 3 = Z Y S 3 Φ = | S 3 Φ | = 3 V LL I L = 3 V LN I L \begin{aligned}\displaystyle V_{\,\text{LL}}=\sqrt{3}V_{\,\text{LN}}\angle 30% \\ \displaystyle I_{\,\text{LL}}=\sqrt{3}I_{\,\text{LN}}\angle-30\\ \displaystyle Z_{\Delta}/3=Z_{\,\text{Y}}\\ \displaystyle S_{3\Phi}=|S_{3\Phi}|=\sqrt{3}V_{\,\text{LL}}I_{\,\text{L}}=3V_{% \,\text{LN}}I_{\,\text{L}}\\ \end{aligned}
  14. V 1 V_{1}
  15. V 2 V_{2}
  16. V 3 V_{3}
  17. N 1 N_{1}
  18. N 2 N_{2}
  19. N 3 N_{3}
  20. I 1 I_{1}
  21. I 2 I_{2}
  22. I 3 I_{3}
  23. ( I 1 - I 2 ) / 3 (I_{1}-I_{2})/3
  24. - ( I 1 - I 2 ) / 3 -(I_{1}-I_{2})/3
  25. 0
  26. 0
  27. ( I 2 - I 3 ) / 3 (I_{2}-I_{3})/3
  28. - ( I 2 - I 3 ) / 3 -(I_{2}-I_{3})/3
  29. - ( I 3 - I 1 ) / 3 -(I_{3}-I_{1})/3
  30. 0
  31. ( I 3 - I 1 ) / 3 (I_{3}-I_{1})/3
  32. I 1 + I 2 + I 3 = 0 I_{1}+I_{2}+I_{3}=0
  33. R 3 + R 1 = ( R c + R a ) R b R a + R b + R c , R_{3}+R_{1}=\frac{(R_{c}+R_{a})R_{b}}{R_{a}+R_{b}+R_{c}},
  34. R 3 R 1 = R a R c . \frac{R_{3}}{R_{1}}=\frac{R_{a}}{R_{c}}.
  35. R 1 , R 2 , R 3 R_{1},R_{2},R_{3}
  36. R a , R b , R c R_{a},R_{b},R_{c}
  37. { R a , R b , R c } \{R_{a},R_{b},R_{c}\}
  38. { R 1 , R 2 , R 3 } \{R_{1},R_{2},R_{3}\}
  39. R Δ ( N 1 , N 2 ) \displaystyle R_{\Delta}(N_{1},N_{2})
  40. R T R_{T}
  41. { R a , R b , R c } \{R_{a},R_{b},R_{c}\}
  42. R T = R a + R b + R c R_{T}=R_{a}+R_{b}+R_{c}
  43. R Δ ( N 1 , N 2 ) = R c ( R a + R b ) R T R_{\Delta}(N_{1},N_{2})=\frac{R_{c}(R_{a}+R_{b})}{R_{T}}
  44. R Y ( N 1 , N 2 ) = R 1 + R 2 R_{Y}(N_{1},N_{2})=R_{1}+R_{2}
  45. R 1 + R 2 = R c ( R a + R b ) R T R_{1}+R_{2}=\frac{R_{c}(R_{a}+R_{b})}{R_{T}}
  46. R ( N 2 , N 3 ) R(N_{2},N_{3})
  47. R 2 + R 3 = R a ( R b + R c ) R T R_{2}+R_{3}=\frac{R_{a}(R_{b}+R_{c})}{R_{T}}
  48. R ( N 1 , N 3 ) R(N_{1},N_{3})
  49. R 1 + R 3 = R b ( R a + R c ) R T . R_{1}+R_{3}=\frac{R_{b}(R_{a}+R_{c})}{R_{T}}.
  50. { R 1 , R 2 , R 3 } \{R_{1},R_{2},R_{3}\}
  51. R 1 + R 2 + R 1 + R 3 - R 2 - R 3 = R c ( R a + R b ) R T + R b ( R a + R c ) R T - R a ( R b + R c ) R T R_{1}+R_{2}+R_{1}+R_{3}-R_{2}-R_{3}=\frac{R_{c}(R_{a}+R_{b})}{R_{T}}+\frac{R_{% b}(R_{a}+R_{c})}{R_{T}}-\frac{R_{a}(R_{b}+R_{c})}{R_{T}}
  52. 2 R 1 = 2 R b R c R T 2R_{1}=\frac{2R_{b}R_{c}}{R_{T}}
  53. R 1 = R b R c R T . R_{1}=\frac{R_{b}R_{c}}{R_{T}}.
  54. R T = R a + R b + R c R_{T}=R_{a}+R_{b}+R_{c}
  55. R 1 = R b R c R T R_{1}=\frac{R_{b}R_{c}}{R_{T}}
  56. R 2 = R a R c R T R_{2}=\frac{R_{a}R_{c}}{R_{T}}
  57. R 3 = R a R b R T R_{3}=\frac{R_{a}R_{b}}{R_{T}}
  58. R T = R a + R b + R c R_{T}=R_{a}+R_{b}+R_{c}
  59. R 1 = R b R c R T R_{1}=\frac{R_{b}R_{c}}{R_{T}}
  60. R 2 = R a R c R T R_{2}=\frac{R_{a}R_{c}}{R_{T}}
  61. R 3 = R a R b R T . R_{3}=\frac{R_{a}R_{b}}{R_{T}}.
  62. R 1 R 2 = R a R b R c 2 R T 2 R_{1}R_{2}=\frac{R_{a}R_{b}R_{c}^{2}}{R_{T}^{2}}
  63. R 1 R 3 = R a R b 2 R c R T 2 R_{1}R_{3}=\frac{R_{a}R_{b}^{2}R_{c}}{R_{T}^{2}}
  64. R 2 R 3 = R a 2 R b R c R T 2 R_{2}R_{3}=\frac{R_{a}^{2}R_{b}R_{c}}{R_{T}^{2}}
  65. R 1 R 2 + R 1 R 3 + R 2 R 3 = R a R b R c 2 + R a R b 2 R c + R a 2 R b R c R T 2 R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}=\frac{R_{a}R_{b}R_{c}^{2}+R_{a}R_{b}^{2}R_{c}% +R_{a}^{2}R_{b}R_{c}}{R_{T}^{2}}
  66. R a R b R c R_{a}R_{b}R_{c}
  67. R T R_{T}
  68. R T R_{T}
  69. R 1 R 2 + R 1 R 3 + R 2 R 3 = ( R a R b R c ) ( R a + R b + R c ) R T 2 R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}=\frac{(R_{a}R_{b}R_{c})(R_{a}+R_{b}+R_{c})}{R% _{T}^{2}}
  70. R 1 R 2 + R 1 R 3 + R 2 R 3 = R a R b R c R T R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}=\frac{R_{a}R_{b}R_{c}}{R_{T}}
  71. R 1 R 2 + R 1 R 3 + R 2 R 3 R 1 = R a R b R c R T R T R b R c , \frac{R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}{R_{1}}=\frac{R_{a}R_{b}R_{c}}{R_{T}}% \frac{R_{T}}{R_{b}R_{c}},
  72. R 1 R 2 + R 1 R 3 + R 2 R 3 R 1 = R a , \frac{R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}{R_{1}}=R_{a},
  73. R a R_{a}
  74. R 2 R_{2}
  75. R 3 R_{3}

Yagi-Uda_antenna.html

  1. V 1 = Z 11 I 1 + Z 12 I 2 V_{1}=Z_{11}I_{1}+Z_{12}I_{2}
  2. V 2 = Z 21 I 1 + Z 22 I 2 V_{2}=Z_{21}I_{1}+Z_{22}I_{2}
  3. 0 = V 2 = Z 21 I 1 + Z 22 I 2 0=V_{2}=Z_{21}I_{1}+Z_{22}I_{2}
  4. I 2 = - Z 21 Z 22 I 1 I_{2}=-{Z_{21}\over Z_{22}}\,I_{1}
  5. V 1 = Z 11 I 1 + Z 12 I 2 = Z 11 I 1 - Z 12 Z 21 Z 22 I 1 V_{1}=Z_{11}I_{1}+Z_{12}I_{2}=Z_{11}I_{1}-Z_{12}{Z_{21}\over Z_{22}}\,I_{1}
  6. = ( Z 11 - Z 21 2 Z 22 ) I 1 \qquad\qquad=\left(Z_{11}-{Z_{21}^{2}\over Z_{22}}\right)\,I_{1}
  7. Z d p = V 1 / I 1 = Z 11 - Z 21 2 Z 22 Z_{dp}=V_{1}/I_{1}=Z_{11}-{Z_{21}^{2}\over Z_{22}}

Yates's_correction_for_continuity.html

  1. i = 1 N O i = 20 \sum_{i=1}^{N}O_{i}=20\,
  2. χ Yates 2 = i = 1 N ( | O i - E i | - 0.5 ) 2 E i \chi\text{Yates}^{2}=\sum_{i=1}^{N}{(|O_{i}-E_{i}|-0.5)^{2}\over E_{i}}
  3. χ Yates 2 = N ( | a d - b c | - N / 2 ) 2 N S N F N A N B . \chi\text{Yates}^{2}=\frac{N(|ad-bc|-N/2)^{2}}{N_{S}N_{F}N_{A}N_{B}}.
  4. χ Yates 2 = N ( max ( 0 , | a d - b c | - N / 2 ) ) 2 N S N F N A N B . \chi\text{Yates}^{2}=\frac{N(\max(0,|ad-bc|-N/2))^{2}}{N_{S}N_{F}N_{A}N_{B}}.

Yorick_(programming_language).html

  1. j = 1 j = N P i j k l Q m n j \sum_{j=1}^{j=N}{P_{ijkl}Q_{mnj}}

Young's_modulus.html

  1. E tensile stress extensional strain = σ ε = F / A 0 Δ L / L 0 = F L 0 A 0 Δ L E\equiv\frac{\mbox{tensile stress}}{\mbox{extensional strain}}=\frac{\sigma}{% \varepsilon}=\frac{F/A_{0}}{\Delta L/L_{0}}=\frac{FL_{0}}{A_{0}\Delta L}
  2. F = E A 0 Δ L L 0 F=\frac{EA_{0}\Delta L}{L_{0}}
  3. F = ( E A 0 L 0 ) Δ L = k x F=\left(\frac{EA_{0}}{L_{0}}\right)\Delta L=kx\,
  4. k = E A 0 L 0 k=\begin{matrix}\frac{EA_{0}}{L_{0}}\end{matrix}\,
  5. x = Δ L . x=\Delta L.\,
  6. U e = E A 0 Δ L L 0 d Δ L = E A 0 L 0 Δ L d Δ L = E A 0 Δ L 2 2 L 0 U_{e}=\int{\frac{EA_{0}\Delta L}{L_{0}}}\,d\Delta L=\frac{EA_{0}}{L_{0}}\int{% \Delta L}\,d\Delta L=\frac{EA_{0}{\Delta L}^{2}}{2L_{0}}
  7. U e A 0 L 0 = E Δ L 2 2 L 0 2 = 1 2 E ε 2 \frac{U_{e}}{A_{0}L_{0}}=\frac{E{\Delta L}^{2}}{2L_{0}^{2}}=\frac{1}{2}E{% \varepsilon}^{2}
  8. ε = Δ L L 0 \varepsilon=\frac{\Delta L}{L_{0}}
  9. U e = k x d x = 1 2 k x 2 . U_{e}=\int{kx}\,dx=\frac{1}{2}kx^{2}.
  10. E = 2 G ( 1 + ν ) = 3 K ( 1 - 2 ν ) . E=2G(1+\nu)=3K(1-2\nu).\,

YUV.html

  1. W R \displaystyle W_{R}
  2. Y \displaystyle Y^{\prime}
  3. R \displaystyle R
  4. [ Y U V ] \displaystyle\begin{bmatrix}Y^{\prime}\\ U\\ V\end{bmatrix}
  5. W R \displaystyle W_{R}
  6. [ Y U V ] \displaystyle\begin{bmatrix}Y^{\prime}\\ U\\ V\end{bmatrix}
  7. a b a\gg b
  8. [ Y U V ] = [ 66 129 25 - 38 - 74 112 112 - 94 - 18 ] [ R G B ] \begin{bmatrix}Y^{\prime}\\ U\\ V\end{bmatrix}=\begin{bmatrix}66&129&25\\ -38&-74&112\\ 112&-94&-18\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}
  9. Y t = ( Y + 128 ) 8 U t = ( U + 128 ) 8 V t = ( V + 128 ) 8 \begin{array}[]{rcl}Yt^{\prime}&=&(Y^{\prime}+128)\gg 8\\ Ut&=&(U+128)\gg 8\\ Vt&=&(V+128)\gg 8\end{array}
  10. Y u = Y t + 16 U u = U t + 128 V u = V t + 128 \begin{array}[]{rcl}Yu^{\prime}&=&Yt^{\prime}+16\\ Uu&=&Ut+128\\ Vu&=&Vt+128\end{array}
  11. [ Y U V ] = [ 76 150 29 - 43 - 84 127 127 - 106 - 21 ] [ R G B ] \begin{bmatrix}Y^{\prime}\\ U\\ V\end{bmatrix}=\begin{bmatrix}76&150&29\\ -43&-84&127\\ 127&-106&-21\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}
  12. Y t = ( Y + 128 ) 8 U t = ( U + 128 ) 8 V t = ( V + 128 ) 8 \begin{array}[]{rcl}Yt^{\prime}&=&(Y^{\prime}+128)\gg 8\\ Ut&=&(U+128)\gg 8\\ Vt&=&(V+128)\gg 8\end{array}
  13. Y u = Y t U u = U t + 128 V u = V t + 128 \begin{array}[]{rcl}Yu^{\prime}&=&Yt^{\prime}\\ Uu&=&Ut+128\\ Vu&=&Vt+128\end{array}
  14. Y \displaystyle Y^{\prime}
  15. C \displaystyle C
  16. R \displaystyle R
  17. Y \displaystyle Y
  18. C r \displaystyle C_{r}

Z-buffering.html

  1. 𝑛𝑒𝑎𝑟 \mathit{near}
  2. 𝑓𝑎𝑟 \mathit{far}
  3. z z
  4. z z
  5. z z^{\prime}
  6. z = 𝑓𝑎𝑟 + 𝑛𝑒𝑎𝑟 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 + 1 z ( - 2 𝑓𝑎𝑟 𝑛𝑒𝑎𝑟 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) z^{\prime}=\frac{\mathit{far}+\mathit{near}}{\mathit{far}-\mathit{near}}+\frac% {1}{z}\left(\frac{-2\cdot\mathit{far}\cdot\mathit{near}}{\mathit{far}-\mathit{% near}}\right)
  7. z z
  8. z z^{\prime}
  9. z = 2 z - 𝑛𝑒𝑎𝑟 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 - 1 z^{\prime}=2\cdot\frac{{z}-\mathit{near}}{\mathit{far}-\mathit{near}}-1
  10. z z
  11. z z
  12. w w
  13. w w^{\prime}
  14. z z^{\prime}
  15. 𝑛𝑒𝑎𝑟 \mathit{near}
  16. 𝑓𝑎𝑟 \mathit{far}
  17. z 2 = ( z 1 + 1 ) 2 z^{\prime}_{2}=\frac{\left(z^{\prime}_{1}+1\right)}{2}
  18. z = 𝑓𝑎𝑟 + 𝑛𝑒𝑎𝑟 2 ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) + 1 z ( - 𝑓𝑎𝑟 𝑛𝑒𝑎𝑟 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) + 1 2 z^{\prime}=\frac{\mathit{far}+\mathit{near}}{2\cdot\left(\mathit{far}-\mathit{% near}\right)}+\frac{1}{z}\left(\frac{-\mathit{far}\cdot\mathit{near}}{\mathit{% far}-\mathit{near}}\right)+\frac{1}{2}
  19. S = 2 d - 1 S=2^{d}-1
  20. z = f ( z ) = ( 2 d - 1 ) ( 𝑓𝑎𝑟 + 𝑛𝑒𝑎𝑟 2 ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) + 1 z ( - 𝑓𝑎𝑟 𝑛𝑒𝑎𝑟 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) + 1 2 ) z^{\prime}=f\left(z\right)=\left\lfloor\left(2^{d}-1\right)\cdot\left(\frac{% \mathit{far}+\mathit{near}}{2\cdot\left(\mathit{far}-\mathit{near}\right)}+% \frac{1}{z}\left(\frac{-\mathit{far}\cdot\mathit{near}}{\mathit{far}-\mathit{% near}}\right)+\frac{1}{2}\right)\right\rfloor
  21. f ( z ) f\left(z\right)
  22. z = - 𝑓𝑎𝑟 𝑛𝑒𝑎𝑟 z S ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) - f a r = - S f a r 𝑛𝑒𝑎𝑟 z ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) - f a r S z=\frac{-\mathit{far}\cdot\mathit{near}}{\frac{z^{\prime}}{S}\left(\mathit{far% }-\mathit{near}\right)-{far}}=\frac{-\mathit{S}\cdot{far}\cdot\mathit{near}}{z% ^{\prime}\left(\mathit{far}-\mathit{near}\right)-{far}\cdot S}
  23. S = 2 d - 1 S=2^{d}-1
  24. z z
  25. z z^{\prime}
  26. d z d z = - 1 - 1 S f a r 𝑛𝑒𝑎𝑟 ( z ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) - f a r S ) 2 ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) \frac{dz}{dz^{\prime}}=\frac{-1\cdot-1\cdot\mathit{S}\cdot{far}\cdot\mathit{% near}}{\left(z^{\prime}\left(\mathit{far}-\mathit{near}\right)-{far}\cdot S% \right)^{2}}\cdot\left(\mathit{far}-\mathit{near}\right)
  27. z z^{\prime}
  28. f ( z ) f\left(z\right)
  29. d z d z = - 1 - 1 S f a r 𝑛𝑒𝑎𝑟 ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) ( S ( - 𝑓𝑎𝑟 𝑛𝑒𝑎𝑟 z + 𝑓𝑎𝑟 ) - f a r S ) 2 = \frac{dz}{dz^{\prime}}=\frac{-1\cdot-1\cdot\mathit{S}\cdot{far}\cdot\mathit{% near}\cdot\left(\mathit{far}-\mathit{near}\right)}{\left(\mathit{S}\cdot\left(% \frac{-\mathit{far}\cdot\mathit{near}}{z}+\mathit{far}\right)-{far}\cdot S% \right)^{2}}=
  30. ( 𝑓𝑎𝑟 - 𝑛𝑒𝑎𝑟 ) z 2 S 𝑓𝑎𝑟 𝑛𝑒𝑎𝑟 = \frac{\left(\mathit{far}-\mathit{near}\right)\cdot z^{2}}{S\cdot\mathit{far}% \cdot\mathit{near}}=
  31. z 2 S 𝑛𝑒𝑎𝑟 - z 2 S 𝑓𝑎𝑟 = \frac{z^{2}}{S\cdot\mathit{near}}-\frac{z^{2}}{S\cdot\mathit{far}}=
  32. z 2 S 𝑛𝑒𝑎𝑟 \frac{z^{2}}{S\cdot\mathit{near}}
  33. z z^{\prime}
  34. 𝑛𝑒𝑎𝑟 \mathit{near}
  35. 𝑛𝑒𝑎𝑟 / 𝑓𝑎𝑟 \mathit{near}/\mathit{far}
  36. n e a r near
  37. z z^{\prime}
  38. z z
  39. w w
  40. w w
  41. z z^{\prime}
  42. 𝑛𝑒𝑎𝑟 \mathit{near}
  43. 𝑓𝑎𝑟 \mathit{far}

Z-transform.html

  1. X ( z ) = 𝒵 { x [ n ] } = n = - x [ n ] z - n X(z)=\mathcal{Z}\{x[n]\}=\sum_{n=-\infty}^{\infty}x[n]z^{-n}
  2. z = A e j ϕ = A ( cos ϕ + j sin ϕ ) z=Ae^{j\phi}=A(\cos{\phi}+j\sin{\phi})\,
  3. X ( z ) = 𝒵 { x [ n ] } = n = 0 x [ n ] z - n . X(z)=\mathcal{Z}\{x[n]\}=\sum_{n=0}^{\infty}x[n]z^{-n}.
  4. X ( z ) = 𝒵 { x [ n ] } = n x [ n ] z n . X(z)=\mathcal{Z}\{x[n]\}=\sum_{n}x[n]z^{n}.
  5. x [ n ] = 𝒵 - 1 { X ( z ) } = 1 2 π j C X ( z ) z n - 1 d z x[n]=\mathcal{Z}^{-1}\{X(z)\}=\frac{1}{2\pi j}\oint_{C}X(z)z^{n-1}dz
  6. x [ n ] = 1 2 π - π + π X ( e j ω ) e j ω n d ω . x[n]=\frac{1}{2\pi}\int_{-\pi}^{+\pi}X(e^{j\omega})e^{j\omega n}d\omega.
  7. R O C = { z : | n = - x [ n ] z - n | < } ROC=\left\{z:\left|\sum_{n=-\infty}^{\infty}x[n]z^{-n}\right|<\infty\right\}
  8. x [ n ] = { , 0.5 - 3 , 0.5 - 2 , 0.5 - 1 , 1 , 0.5 , 0.5 2 , 0.5 3 , } = { , 2 3 , 2 2 , 2 , 1 , 0.5 , 0.5 2 , 0.5 3 , } . x[n]=\left\{\cdots,0.5^{-3},0.5^{-2},0.5^{-1},1,0.5,0.5^{2},0.5^{3},\cdots% \right\}=\left\{\cdots,2^{3},2^{2},2,1,0.5,0.5^{2},0.5^{3},\cdots\right\}.
  9. n = - x [ n ] z - n . \sum_{n=-\infty}^{\infty}x[n]z^{-n}\to\infty.
  10. x [ n ] = 0.5 n u [ n ] x[n]=0.5^{n}u[n]
  11. x [ n ] = { , 0 , 0 , 0 , 1 , 0.5 , 0.5 2 , 0.5 3 , } . x[n]=\left\{\cdots,0,0,0,1,0.5,0.5^{2},0.5^{3},\cdots\right\}.
  12. n = - x [ n ] z - n = n = 0 0.5 n z - n = n = 0 ( 0.5 z ) n = 1 1 - 0.5 z - 1 . \sum_{n=-\infty}^{\infty}x[n]z^{-n}=\sum_{n=0}^{\infty}0.5^{n}z^{-n}=\sum_{n=0% }^{\infty}\left(\frac{0.5}{z}\right)^{n}=\frac{1}{1-0.5z^{-1}}.
  13. x [ n ] = - ( 0.5 ) n u [ - n - 1 ] x[n]=-(0.5)^{n}u[-n-1]
  14. x [ n ] = { , - ( 0.5 ) - 3 , - ( 0.5 ) - 2 , - ( 0.5 ) - 1 , 0 , 0 , 0 , 0 , } . x[n]=\left\{\cdots,-(0.5)^{-3},-(0.5)^{-2},-(0.5)^{-1},0,0,0,0,\cdots\right\}.
  15. n = - x [ n ] z - n = - n = - - 1 0.5 n z - n = - m = 1 ( z 0.5 ) m = 1 - 1 1 - 0.5 - 1 z = 1 1 - 0.5 z - 1 \sum_{n=-\infty}^{\infty}x[n]z^{-n}=-\sum_{n=-\infty}^{-1}0.5^{n}z^{-n}=-\sum_% {m=1}^{\infty}\left(\frac{z}{0.5}\right)^{m}=1-\frac{1}{1-0.5^{-1}z}=\frac{1}{% 1-0.5z^{-1}}
  16. x [ n ] = 𝒵 - 1 { X ( z ) } x[n]=\mathcal{Z}^{-1}\{X(z)\}
  17. X ( z ) = 𝒵 { x [ n ] } X(z)=\mathcal{Z}\{x[n]\}
  18. r 2 < | z | < r 1 r_{2}<|z|<r_{1}
  19. a 1 x 1 [ n ] + a 2 x 2 [ n ] a_{1}x_{1}[n]+a_{2}x_{2}[n]
  20. a 1 X 1 ( z ) + a 2 X 2 ( z ) a_{1}X_{1}(z)+a_{2}X_{2}(z)
  21. X ( z ) = n = - ( a 1 x 1 ( n ) + a 2 x 2 ( n ) ) z - n = a 1 n = - x 1 ( n ) z - n + a 2 n = - x 2 ( n ) z - n = a 1 X 1 ( z ) + a 2 X 2 ( z ) \begin{aligned}\displaystyle X(z)&\displaystyle=\sum_{n=-\infty}^{\infty}(a_{1% }x_{1}(n)+a_{2}x_{2}(n))z^{-n}\\ &\displaystyle=a_{1}\sum_{n=-\infty}^{\infty}x_{1}(n)z^{-n}+a_{2}\sum_{n=-% \infty}^{\infty}x_{2}(n)z^{-n}\\ &\displaystyle=a_{1}X_{1}(z)+a_{2}X_{2}(z)\end{aligned}
  22. x K [ n ] = { x [ r ] , n = r K 0 , n r K x_{K}[n]=\begin{cases}x[r],&n=rK\\ 0,&n\not=rK\end{cases}
  23. X ( z K ) X(z^{K})
  24. X K ( z ) = n = - x K ( n ) z - n = r = - x ( r ) z - r K = r = - x ( r ) ( z K ) - r = X ( z K ) \begin{aligned}\displaystyle X_{K}(z)&\displaystyle=\sum_{n=-\infty}^{\infty}x% _{K}(n)z^{-n}\\ &\displaystyle=\sum_{r=-\infty}^{\infty}x(r)z^{-rK}\\ &\displaystyle=\sum_{r=-\infty}^{\infty}x(r)(z^{K})^{-r}\\ &\displaystyle=X(z^{K})\end{aligned}
  25. R 1 K R^{\frac{1}{K}}
  26. x [ n K ] x[nK]
  27. 1 K p = 0 K - 1 X ( z 1 K e - i 2 π K p ) \frac{1}{K}\sum_{p=0}^{K-1}X\left(z^{\tfrac{1}{K}}\cdot e^{-i\tfrac{2\pi}{K}p}\right)
  28. x [ n - k ] x[n-k]
  29. z - k X ( z ) z^{-k}X(z)
  30. Z { x [ n - k ] } = n = 0 x [ n - k ] z - n = j = - k x [ j ] z - ( j + k ) j = n - k = j = - k x [ j ] z - j z - k = z - k j = - k x [ j ] z - j = z - k j = 0 x [ j ] z - j x [ β ] = 0 , β < 0 = z - k X ( z ) \begin{aligned}\displaystyle Z\{x[n-k]\}&\displaystyle=\sum_{n=0}^{\infty}x[n-% k]z^{-n}\\ &\displaystyle=\sum_{j=-k}^{\infty}x[j]z^{-(j+k)}&&\displaystyle j=n-k\\ &\displaystyle=\sum_{j=-k}^{\infty}x[j]z^{-j}z^{-k}\\ &\displaystyle=z^{-k}\sum_{j=-k}^{\infty}x[j]z^{-j}\\ &\displaystyle=z^{-k}\sum_{j=0}^{\infty}x[j]z^{-j}&&\displaystyle x[\beta]=0,% \beta<0\\ &\displaystyle=z^{-k}X(z)\end{aligned}
  31. X ( a - 1 z ) X(a^{-1}z)
  32. 𝒵 { a n x [ n ] } = n = - a n x ( n ) z - n = n = - x ( n ) ( a - 1 z ) - n = X ( a - 1 z ) \begin{aligned}\displaystyle\mathcal{Z}\left\{a^{n}x[n]\right\}&\displaystyle=% \sum_{n=-\infty}^{\infty}a^{n}x(n)z^{-n}\\ &\displaystyle=\sum_{n=-\infty}^{\infty}x(n)(a^{-1}z)^{-n}\\ &\displaystyle=X(a^{-1}z)\end{aligned}
  33. | a | r 2 < | z | < | a | r 1 |a|r_{2}<|z|<|a|r_{1}
  34. x [ - n ] x[-n]
  35. X ( z - 1 ) X(z^{-1})
  36. 𝒵 { x ( - n ) } = n = - x ( - n ) z - n = m = - x ( m ) z m = m = - x ( m ) ( z - 1 ) - m = X ( z - 1 ) \begin{aligned}\displaystyle\mathcal{Z}\{x(-n)\}&\displaystyle=\sum_{n=-\infty% }^{\infty}x(-n)z^{-n}\\ &\displaystyle=\sum_{m=-\infty}^{\infty}x(m)z^{m}\\ &\displaystyle=\sum_{m=-\infty}^{\infty}x(m){(z^{-1})}^{-m}\\ &\displaystyle=X(z^{-1})\\ \end{aligned}
  37. 1 r 1 < | z | < 1 r 2 \tfrac{1}{r_{1}}<|z|<\tfrac{1}{r_{2}}
  38. x * [ n ] x^{*}[n]
  39. X * ( z * ) X^{*}(z^{*})
  40. 𝒵 { x * ( n ) } = n = - x * ( n ) z - n = n = - [ x ( n ) ( z * ) - n ] * = [ n = - x ( n ) ( z * ) - n ] * = X * ( z * ) \begin{aligned}\displaystyle\mathcal{Z}\{x^{*}(n)\}&\displaystyle=\sum_{n=-% \infty}^{\infty}x^{*}(n)z^{-n}\\ &\displaystyle=\sum_{n=-\infty}^{\infty}\left[x(n)(z^{*})^{-n}\right]^{*}\\ &\displaystyle=\left[\sum_{n=-\infty}^{\infty}x(n)(z^{*})^{-n}\right]^{*}\\ &\displaystyle=X^{*}(z^{*})\end{aligned}
  41. Re { x [ n ] } \operatorname{Re}\{x[n]\}
  42. 1 2 [ X ( z ) + X * ( z * ) ] \tfrac{1}{2}\left[X(z)+X^{*}(z^{*})\right]
  43. Im { x [ n ] } \operatorname{Im}\{x[n]\}
  44. 1 2 j [ X ( z ) - X * ( z * ) ] \tfrac{1}{2j}\left[X(z)-X^{*}(z^{*})\right]
  45. n x [ n ] nx[n]
  46. - z d X ( z ) d z -z\frac{dX(z)}{dz}
  47. 𝒵 { n x ( n ) } = n = - n x ( n ) z - n = z n = - n x ( n ) z - n - 1 = - z n = - x ( n ) ( - n z - n - 1 ) = - z n = - x ( n ) d d z ( z - n ) = - z d X ( z ) d z \begin{aligned}\displaystyle\mathcal{Z}\{nx(n)\}&\displaystyle=\sum_{n=-\infty% }^{\infty}nx(n)z^{-n}\\ &\displaystyle=z\sum_{n=-\infty}^{\infty}nx(n)z^{-n-1}\\ &\displaystyle=-z\sum_{n=-\infty}^{\infty}x(n)(-nz^{-n-1})\\ &\displaystyle=-z\sum_{n=-\infty}^{\infty}x(n)\frac{d}{dz}(z^{-n})\\ &\displaystyle=-z\frac{dX(z)}{dz}\end{aligned}
  48. x 1 [ n ] * x 2 [ n ] x_{1}[n]*x_{2}[n]
  49. X 1 ( z ) X 2 ( z ) X_{1}(z)X_{2}(z)
  50. 𝒵 { x 1 ( n ) * x 2 ( n ) } = 𝒵 { l = - x 1 ( l ) x 2 ( n - l ) } = n = - [ l = - x 1 ( l ) x 2 ( n - l ) ] z - n = l = - x 1 ( l ) [ n = - x 2 ( n - l ) z - n ] = [ l = - x 1 ( l ) z - l ] [ n = - x 2 ( n ) z - n ] = X 1 ( z ) X 2 ( z ) \begin{aligned}\displaystyle\mathcal{Z}\{x_{1}(n)*x_{2}(n)\}&\displaystyle=% \mathcal{Z}\left\{\sum_{l=-\infty}^{\infty}x_{1}(l)x_{2}(n-l)\right\}\\ &\displaystyle=\sum_{n=-\infty}^{\infty}\left[\sum_{l=-\infty}^{\infty}x_{1}(l% )x_{2}(n-l)\right]z^{-n}\\ &\displaystyle=\sum_{l=-\infty}^{\infty}x_{1}(l)\left[\sum_{n=-\infty}^{\infty% }x_{2}(n-l)z^{-n}\right]\\ &\displaystyle=\left[\sum_{l=-\infty}^{\infty}x_{1}(l)z^{-l}\right]\!\!\left[% \sum_{n=-\infty}^{\infty}x_{2}(n)z^{-n}\right]\\ &\displaystyle=X_{1}(z)X_{2}(z)\end{aligned}
  51. r x 1 , x 2 = x 1 * [ - n ] * x 2 [ n ] r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]
  52. R x 1 , x 2 ( z ) = X 1 * ( 1 z * ) X 2 ( z ) R_{x_{1},x_{2}}(z)=X_{1}^{*}(\tfrac{1}{z^{*}})X_{2}(z)
  53. X 1 ( 1 z * ) X_{1}(\tfrac{1}{z^{*}})
  54. X 2 ( z ) X_{2}(z)
  55. x [ n ] - x [ n - 1 ] x[n]-x[n-1]
  56. ( 1 - z - 1 ) X ( z ) (1-z^{-1})X(z)
  57. k = - n x [ k ] \sum_{k=-\infty}^{n}x[k]
  58. 1 1 - z - 1 X ( z ) \frac{1}{1-z^{-1}}X(z)
  59. n = - k = - n x [ k ] z - n = n = - ( x [ n ] + + x [ - ] ) z - n = X [ z ] ( 1 + z - 1 + z - 2 + ) = X [ z ] j = 0 z - j = X [ z ] 1 1 - z - 1 \begin{aligned}\displaystyle\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{n}x[k]z% ^{-n}&\displaystyle=\sum_{n=-\infty}^{\infty}(x[n]+\cdots+x[-\infty])z^{-n}\\ &\displaystyle=X[z]\left(1+z^{-1}+z^{-2}+\cdots\right)\\ &\displaystyle=X[z]\sum_{j=0}^{\infty}z^{-j}\\ &\displaystyle=X[z]\frac{1}{1-z^{-1}}\end{aligned}
  60. x 1 [ n ] x 2 [ n ] x_{1}[n]x_{2}[n]
  61. 1 j 2 π C X 1 ( v ) X 2 ( z v ) v - 1 d v \frac{1}{j2\pi}\oint_{C}X_{1}(v)X_{2}(\tfrac{z}{v})v^{-1}\mathrm{d}v
  62. r 1 l r 2 l < | z | < r 1 u r 2 u r_{1l}r_{2l}<|z|<r_{1u}r_{2u}
  63. n = - x 1 [ n ] x 2 * [ n ] = 1 j 2 π C X 1 ( v ) X 2 * ( 1 v * ) v - 1 d v \sum_{n=-\infty}^{\infty}x_{1}[n]x^{*}_{2}[n]\quad=\quad\frac{1}{j2\pi}\oint_{% C}X_{1}(v)X^{*}_{2}(\tfrac{1}{v^{*}})v^{-1}\mathrm{d}v
  64. x [ 0 ] = lim z X ( z ) . x[0]=\lim_{z\to\infty}X(z).
  65. x [ ] = lim z 1 ( z - 1 ) X ( z ) . x[\infty]=\lim_{z\to 1}(z-1)X(z).
  66. u : n u [ n ] = { 1 , n 0 0 , n < 0 u:n\mapsto u[n]=\begin{cases}1,&n\geq 0\\ 0,&n<0\end{cases}
  67. δ : n δ [ n ] = { 1 , n = 0 0 , n 0 \delta:n\mapsto\delta[n]=\begin{cases}1,&n=0\\ 0,&n\neq 0\end{cases}
  68. x [ n ] x[n]
  69. X ( z ) X(z)
  70. δ [ n ] \delta[n]
  71. δ [ n - n 0 ] \delta[n-n_{0}]
  72. z - n 0 z^{-n_{0}}
  73. z 0 z\neq 0
  74. u [ n ] u[n]\,
  75. 1 1 - z - 1 \frac{1}{1-z^{-1}}
  76. | z | > 1 |z|>1
  77. e - α n u [ n ] e^{-\alpha n}u[n]
  78. 1 1 - e - α z - 1 1\over 1-e^{-\alpha}z^{-1}
  79. | z | > e - α |z|>e^{-\alpha}\,
  80. - u [ - n - 1 ] -u[-n-1]
  81. 1 1 - z - 1 \frac{1}{1-z^{-1}}
  82. | z | < 1 |z|<1
  83. n u [ n ] nu[n]
  84. z - 1 ( 1 - z - 1 ) 2 \frac{z^{-1}}{(1-z^{-1})^{2}}
  85. | z | > 1 |z|>1
  86. - n u [ - n - 1 ] -nu[-n-1]\,
  87. z - 1 ( 1 - z - 1 ) 2 \frac{z^{-1}}{(1-z^{-1})^{2}}
  88. | z | < 1 |z|<1
  89. n 2 u [ n ] n^{2}u[n]
  90. z - 1 ( 1 + z - 1 ) ( 1 - z - 1 ) 3 \frac{z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}
  91. | z | > 1 |z|>1\,
  92. - n 2 u [ - n - 1 ] -n^{2}u[-n-1]\,
  93. z - 1 ( 1 + z - 1 ) ( 1 - z - 1 ) 3 \frac{z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}
  94. | z | < 1 |z|<1\,
  95. n 3 u [ n ] n^{3}u[n]
  96. z - 1 ( 1 + 4 z - 1 + z - 2 ) ( 1 - z - 1 ) 4 \frac{z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}
  97. | z | > 1 |z|>1\,
  98. - n 3 u [ - n - 1 ] -n^{3}u[-n-1]
  99. z - 1 ( 1 + 4 z - 1 + z - 2 ) ( 1 - z - 1 ) 4 \frac{z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}
  100. | z | < 1 |z|<1\,
  101. a n u [ n ] a^{n}u[n]
  102. 1 1 - a z - 1 \frac{1}{1-az^{-1}}
  103. | z | > | a | |z|>|a|
  104. - a n u [ - n - 1 ] -a^{n}u[-n-1]
  105. 1 1 - a z - 1 \frac{1}{1-az^{-1}}
  106. | z | < | a | |z|<|a|
  107. n a n u [ n ] na^{n}u[n]
  108. a z - 1 ( 1 - a z - 1 ) 2 \frac{az^{-1}}{(1-az^{-1})^{2}}
  109. | z | > | a | |z|>|a|
  110. - n a n u [ - n - 1 ] -na^{n}u[-n-1]
  111. a z - 1 ( 1 - a z - 1 ) 2 \frac{az^{-1}}{(1-az^{-1})^{2}}
  112. | z | < | a | |z|<|a|
  113. n 2 a n u [ n ] n^{2}a^{n}u[n]
  114. a z - 1 ( 1 + a z - 1 ) ( 1 - a z - 1 ) 3 \frac{az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}
  115. | z | > | a | |z|>|a|
  116. - n 2 a n u [ - n - 1 ] -n^{2}a^{n}u[-n-1]
  117. a z - 1 ( 1 + a z - 1 ) ( 1 - a z - 1 ) 3 \frac{az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}
  118. | z | < | a | |z|<|a|
  119. cos ( ω 0 n ) u [ n ] \cos(\omega_{0}n)u[n]
  120. 1 - z - 1 cos ( ω 0 ) 1 - 2 z - 1 cos ( ω 0 ) + z - 2 \frac{1-z^{-1}\cos(\omega_{0})}{1-2z^{-1}\cos(\omega_{0})+z^{-2}}
  121. | z | > 1 |z|>1
  122. sin ( ω 0 n ) u [ n ] \sin(\omega_{0}n)u[n]
  123. z - 1 sin ( ω 0 ) 1 - 2 z - 1 cos ( ω 0 ) + z - 2 \frac{z^{-1}\sin(\omega_{0})}{1-2z^{-1}\cos(\omega_{0})+z^{-2}}
  124. | z | > 1 |z|>1
  125. a n cos ( ω 0 n ) u [ n ] a^{n}\cos(\omega_{0}n)u[n]
  126. 1 - a z - 1 cos ( ω 0 ) 1 - 2 a z - 1 cos ( ω 0 ) + a 2 z - 2 \frac{1-az^{-1}\cos(\omega_{0})}{1-2az^{-1}\cos(\omega_{0})+a^{2}z^{-2}}
  127. | z | > | a | |z|>|a|
  128. a n sin ( ω 0 n ) u [ n ] a^{n}\sin(\omega_{0}n)u[n]
  129. a z - 1 sin ( ω 0 ) 1 - 2 a z - 1 cos ( ω 0 ) + a 2 z - 2 \frac{az^{-1}\sin(\omega_{0})}{1-2az^{-1}\cos(\omega_{0})+a^{2}z^{-2}}
  130. | z | > | a | |z|>|a|
  131. n = - x [ n ] z - n = n = - x [ n ] e - j ω n , \sum_{n=-\infty}^{\infty}x[n]\ z^{-n}=\sum_{n=-\infty}^{\infty}x[n]\ e^{-j% \omega n},
  132. n = - x ( n T ) x [ n ] e - j 2 π f n T DTFT = 1 T k = - X ( f - k / T ) . \underbrace{\sum_{n=-\infty}^{\infty}\overbrace{x(nT)}^{x[n]}\ e^{-j2\pi fnT}}% _{\,\text{DTFT}}=\frac{1}{T}\sum_{k=-\infty}^{\infty}X(f-k/T).
  133. f \scriptstyle f
  134. ω = 2 π f T \scriptstyle\omega=2\pi fT
  135. f = 1 T \scriptstyle f=\frac{1}{T}
  136. f = ω 2 π T , \scriptstyle f=\frac{\omega}{2\pi T},
  137. n = - x [ n ] e - j ω n = 1 T k = - X ( ω 2 π T - k T ) X ( ω - 2 π k 2 π T ) . \sum_{n=-\infty}^{\infty}x[n]\ e^{-j\omega n}=\frac{1}{T}\sum_{k=-\infty}^{% \infty}\underbrace{X\left(\tfrac{\omega}{2\pi T}-\tfrac{k}{T}\right)}_{X\left(% \frac{\omega-2\pi k}{2\pi T}\right)}.
  138. s = 2 T ( z - 1 ) ( z + 1 ) s=\frac{2}{T}\frac{(z-1)}{(z+1)}
  139. H ( s ) H(s)
  140. H ( z ) H(z)
  141. z = 2 + s T 2 - s T z=\frac{2+sT}{2-sT}
  142. . X * ( s ) = X ( z ) | z = e s T \bigg.X^{*}(s)=X(z)\bigg|_{\displaystyle z=e^{sT}}
  143. p = 0 N y [ n - p ] α p = q = 0 M x [ n - q ] β q \sum_{p=0}^{N}y[n-p]\alpha_{p}=\sum_{q=0}^{M}x[n-q]\beta_{q}
  144. y [ n ] = q = 0 M x [ n - q ] β q - p = 1 N y [ n - p ] α p . y[n]=\sum_{q=0}^{M}x[n-q]\beta_{q}-\sum_{p=1}^{N}y[n-p]\alpha_{p}.
  145. Y ( z ) p = 0 N z - p α p = X ( z ) q = 0 M z - q β q Y(z)\sum_{p=0}^{N}z^{-p}\alpha_{p}=X(z)\sum_{q=0}^{M}z^{-q}\beta_{q}
  146. H ( z ) = Y ( z ) X ( z ) = q = 0 M z - q β q p = 0 N z - p α p = β 0 + z - 1 β 1 + z - 2 β 2 + + z - M β M α 0 + z - 1 α 1 + z - 2 α 2 + + z - N α N . H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{q=0}^{M}z^{-q}\beta_{q}}{\sum_{p=0}^{N}z^{-% p}\alpha_{p}}=\frac{\beta_{0}+z^{-1}\beta_{1}+z^{-2}\beta_{2}+\cdots+z^{-M}% \beta_{M}}{\alpha_{0}+z^{-1}\alpha_{1}+z^{-2}\alpha_{2}+\cdots+z^{-N}\alpha_{N% }}.
  147. H ( z ) = ( 1 - q 1 z - 1 ) ( 1 - q 2 z - 1 ) ( 1 - q M z - 1 ) ( 1 - p 1 z - 1 ) ( 1 - p 2 z - 1 ) ( 1 - p N z - 1 ) H(z)=\frac{(1-q_{1}z^{-1})(1-q_{2}z^{-1})\cdots(1-q_{M}z^{-1})}{(1-p_{1}z^{-1}% )(1-p_{2}z^{-1})\cdots(1-p_{N}z^{-1})}
  148. Y ( z ) z \frac{Y(z)}{z}

Zariski_topology.html

  1. 𝔸 n , \mathbb{A}^{n},
  2. 𝔸 n . \mathbb{A}^{n}.
  3. V ( S ) = { x 𝔸 n f ( x ) = 0 , f S } V(S)=\{x\in\mathbb{A}^{n}\mid f(x)=0,\forall f\in S\}
  4. V ( I ) V ( J ) = V ( I J ) ; V(I)\cup V(J)\,=\,V(IJ);
  5. V ( I ) V ( J ) = V ( I + J ) . V(I)\cap V(J)\,=\,V(I+J).
  6. 𝔸 n . \mathbb{A}^{n}.
  7. 𝔸 n . \mathbb{A}^{n}.
  8. A ( X ) = k [ x 1 , , x n ] / I ( X ) A(X)\,=\,k[x_{1},\dots,x_{n}]/I(X)
  9. k [ x 1 , , x n ] k[x_{1},\dots,x_{n}]
  10. 𝔸 n ; \mathbb{A}^{n};
  11. V ( T ) = { x X f ( x ) = 0 , f T } V^{\prime}(T)=\{x\in X\mid f(x)=0,\forall f\in T\}
  12. n \mathbb{P}^{n}
  13. 𝔸 n + 1 \mathbb{A}^{n+1}
  14. k [ x 0 , , x n ] k[x_{0},\dots,x_{n}]
  15. n \mathbb{P}^{n}
  16. V ( S ) = { x n f ( x ) = 0 , f S } . V(S)=\{x\in\mathbb{P}^{n}\mid f(x)=0,\forall f\in S\}.
  17. n . \mathbb{P}^{n}.
  18. 𝔸 1 . \mathbb{A}^{1}.
  19. V ( I ) = { P Spec ( A ) I P } V(I)=\{P\in\operatorname{Spec}\,(A)\mid I\subseteq P\}
  20. e a : ( P Spec ( A ) ) ( a mod P 1 Frac ( A / P ) ) e_{a}\colon\bigl(P\in\operatorname{Spec}(A)\bigr)\mapsto\left(\frac{a\;\bmod P% }{1}\in\operatorname{Frac}(A/P)\right)
  21. e a ( P ) = 0 P V ( a ) e_{a}(P)=0\Leftrightarrow P\in V(a)
  22. 𝔸 1 \mathbb{A}^{1}

Zeeman_effect.html

  1. H = H 0 + V M , H=H_{0}+V_{M},
  2. H 0 H_{0}
  3. V M V_{M}
  4. V M = - μ B , V_{M}=-\vec{\mu}\cdot\vec{B},
  5. μ \vec{\mu}
  6. μ - μ B g J , \vec{\mu}\approx-\frac{\mu_{B}g\vec{J}}{\hbar},
  7. μ B \mu_{B}
  8. J \vec{J}
  9. g g
  10. L \vec{L}
  11. S \vec{S}
  12. μ = - μ B ( g l L + g s S ) , \vec{\mu}=-\frac{\mu_{B}(g_{l}\vec{L}+g_{s}\vec{S})}{\hbar},
  13. g l = 1 g_{l}=1
  14. g s 2.0023192 g_{s}\approx 2.0023192
  15. g J = i ( g l l i + g s s i ) = ( g l L + g s S ) , g\vec{J}=\left\langle\sum_{i}(g_{l}\vec{l_{i}}+g_{s}\vec{s_{i}})\right\rangle=% \left\langle(g_{l}\vec{L}+g_{s}\vec{S})\right\rangle,
  16. L \vec{L}
  17. S \vec{S}
  18. V M V_{M}
  19. V M V_{M}
  20. H 0 H_{0}
  21. H 0 H_{0}
  22. L \scriptstyle\vec{L}
  23. S \scriptstyle\vec{S}
  24. J = L + S \scriptstyle\vec{J}=\vec{L}+\vec{S}
  25. J \scriptstyle\vec{J}
  26. J \scriptstyle\vec{J}
  27. S a v g = ( S J ) J 2 J \vec{S}_{avg}=\frac{(\vec{S}\cdot\vec{J})}{J^{2}}\vec{J}
  28. L a v g = ( L J ) J 2 J . \vec{L}_{avg}=\frac{(\vec{L}\cdot\vec{J})}{J^{2}}\vec{J}.
  29. V M = μ B J ( g L L J J 2 + g S S J J 2 ) B . \langle V_{M}\rangle=\frac{\mu_{B}}{\hbar}\vec{J}\left(g_{L}\frac{\vec{L}\cdot% \vec{J}}{J^{2}}+g_{S}\frac{\vec{S}\cdot\vec{J}}{J^{2}}\right)\cdot\vec{B}.
  30. L = J - S \scriptstyle\vec{L}=\vec{J}-\vec{S}
  31. S J = 1 2 ( J 2 + S 2 - L 2 ) = 2 2 [ j ( j + 1 ) - l ( l + 1 ) + s ( s + 1 ) ] , \vec{S}\cdot\vec{J}=\frac{1}{2}(J^{2}+S^{2}-L^{2})=\frac{\hbar^{2}}{2}[j(j+1)-% l(l+1)+s(s+1)],
  32. S = J - L \scriptstyle\vec{S}=\vec{J}-\vec{L}
  33. L J = 1 2 ( J 2 - S 2 + L 2 ) = 2 2 [ j ( j + 1 ) + l ( l + 1 ) - s ( s + 1 ) ] . \vec{L}\cdot\vec{J}=\frac{1}{2}(J^{2}-S^{2}+L^{2})=\frac{\hbar^{2}}{2}[j(j+1)+% l(l+1)-s(s+1)].
  34. J z = m j \scriptstyle J_{z}=\hbar m_{j}
  35. V M = μ B B m j [ g L j ( j + 1 ) + l ( l + 1 ) - s ( s + 1 ) 2 j ( j + 1 ) + g S j ( j + 1 ) - l ( l + 1 ) + s ( s + 1 ) 2 j ( j + 1 ) ] = μ B B m j [ 1 + ( g S - 1 ) j ( j + 1 ) - l ( l + 1 ) + s ( s + 1 ) 2 j ( j + 1 ) ] , = μ B B m j g j \begin{aligned}\displaystyle V_{M}&\displaystyle=\mu_{B}Bm_{j}\left[g_{L}\frac% {j(j+1)+l(l+1)-s(s+1)}{2j(j+1)}+g_{S}\frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}% \right]\\ &\displaystyle=\mu_{B}Bm_{j}\left[1+(g_{S}-1)\frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+% 1)}\right],\\ &\displaystyle=\mu_{B}Bm_{j}g_{j}\end{aligned}
  36. g L = 1 g_{L}=1
  37. g S 2 g_{S}\approx 2
  38. m j m_{j}
  39. s = 1 / 2 s=1/2
  40. j = l ± s j=l\pm s
  41. g j = 1 ± g S - 1 2 l + 1 g_{j}=1\pm\frac{g_{S}-1}{2l+1}
  42. 2 P 1 / 2 1 S 1 / 2 2P_{1/2}\to 1S_{1/2}
  43. 2 P 3 / 2 1 S 1 / 2 . 2P_{3/2}\to 1S_{1/2}.
  44. m j = 1 / 2 , - 1 / 2 m_{j}=1/2,-1/2
  45. m j = 3 / 2 , 1 / 2 , - 1 / 2 , - 3 / 2 m_{j}=3/2,1/2,-1/2,-3/2
  46. g J = 2 g_{J}=2
  47. 1 S 1 / 2 1S_{1/2}
  48. g J = 2 / 3 g_{J}=2/3
  49. 2 P 1 / 2 2P_{1/2}
  50. g J = 4 / 3 g_{J}=4/3
  51. 2 P 3 / 2 2P_{3/2}
  52. L \vec{L}
  53. S \vec{S}
  54. s = 0 s=0
  55. [ H 0 , S ] = 0 [H_{0},S]=0
  56. L z L_{z}
  57. S z S_{z}
  58. | ψ |\psi\rangle
  59. E z = ψ | H 0 + B z μ B ( L z + g s S z ) | ψ = E 0 + B z μ B ( m l + g s m s ) . E_{z}=\left\langle\psi\left|H_{0}+\frac{B_{z}\mu_{B}}{\hbar}(L_{z}+g_{s}S_{z})% \right|\psi\right\rangle=E_{0}+B_{z}\mu_{B}(m_{l}+g_{s}m_{s}).
  60. m l m_{l}
  61. m s m_{s}
  62. Δ s = 0 , Δ m s = 0 , Δ l = ± 1 , Δ m l = 0 , ± 1 \Delta s=0,\Delta m_{s}=0,\Delta l=\pm 1,\Delta m_{l}=0,\pm 1
  63. Δ m l = 0 , ± 1 \Delta m_{l}=0,\pm 1
  64. Δ E = B μ B Δ m l \Delta E=B\mu_{B}\Delta m_{l}
  65. s 0 s\neq 0
  66. E z + f s = E z + α 2 2 n 3 { 3 4 n - [ l ( l + 1 ) - m l m s l ( l + 1 / 2 ) ( l + 1 ) ] } . E_{z+fs}=E_{z}+\frac{\alpha^{2}}{2n^{3}}\left\{\frac{3}{4n}-\left[\frac{l(l+1)% -m_{l}m_{s}}{l(l+1/2)(l+1)}\right]\right\}.
  67. H = h A I J - μ B H=hA\vec{I}\cdot\vec{J}-\vec{\mu}\cdot\vec{B}
  68. H = h A I J + μ B ( g J J + g I I ) B H=hA\vec{I}\cdot\vec{J}+\mu_{B}(g_{J}\vec{J}+g_{I}\vec{I})\cdot\vec{B}
  69. F | F | = | J + I | F\equiv|\vec{F}|=|\vec{J}+\vec{I}|
  70. I \vec{I}
  71. J J
  72. A A
  73. μ B \mu_{B}
  74. J \hbar\vec{J}
  75. I \hbar\vec{I}
  76. g J g_{J}
  77. g F g_{F}
  78. g J = g L J ( J + 1 ) + L ( L + 1 ) - S ( S + 1 ) 2 J ( J + 1 ) + g S J ( J + 1 ) - L ( L + 1 ) + S ( S + 1 ) 2 J ( J + 1 ) g_{J}=g_{L}\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}+g_{S}\frac{J(J+1)-L(L+1)+S(S+1% )}{2J(J+1)}
  79. g F = g J F ( F + 1 ) + J ( J + 1 ) - I ( I + 1 ) 2 F ( F + 1 ) + g I F ( F + 1 ) - J ( J + 1 ) + I ( I + 1 ) 2 F ( F + 1 ) g_{F}=g_{J}\frac{F(F+1)+J(J+1)-I(I+1)}{2F(F+1)}+g_{I}\frac{F(F+1)-J(J+1)+I(I+1% )}{2F(F+1)}
  80. | F , m f |F,m_{f}\rangle
  81. | I , J , m I , m J |I,J,m_{I},m_{J}\rangle
  82. | m I , m J |m_{I},m_{J}\rangle
  83. I I
  84. J J
  85. | F , m F |F,m_{F}\rangle
  86. | m I , m J |m_{I},m_{J}\rangle
  87. J = 1 / 2 J=1/2
  88. L = 0 L=0
  89. J = 1 / 2 J=1/2
  90. m F = m J + m I m_{F}=m_{J}+m_{I}
  91. J = 1 / 2 J=1/2
  92. m J m_{J}
  93. ± 1 / 2 \pm 1/2
  94. | ± | m J = ± 1 / 2 , m I = m F 1 / 2 |\pm\rangle\equiv|m_{J}=\pm 1/2,m_{I}=m_{F}\mp 1/2\rangle
  95. L L
  96. L ± L x ± i L y L_{\pm}\equiv L_{x}\pm iL_{y}
  97. L ± | L , m L = ( L m L ) ( L ± m L + 1 ) | L , m L ± 1 L_{\pm}|L_{,}m_{L}\rangle=\sqrt{(L\mp m_{L})(L\pm m_{L}+1)}|L,m_{L}\pm 1\rangle
  98. m L m_{L}
  99. - L , , L {-L,\dots...,L}
  100. J ± J_{\pm}
  101. I ± I_{\pm}
  102. H = h A I z J z + h A 2 ( J + I - + J - I + ) + μ B B ( g J J z + g I I Z ) H=hAI_{z}J_{z}+\frac{hA}{2}(J_{+}I_{-}+J_{-}I_{+})+\mu_{B}B(g_{J}J_{z}+g_{I}I_% {Z})
  103. ± | H | ± = - 1 4 A + μ B B g I m F ± 1 2 ( h A m F + μ B B ( g J - g I ) ) \langle\pm|H|\pm\rangle=-\frac{1}{4}A+\mu_{B}Bg_{I}m_{F}\pm\frac{1}{2}(hAm_{F}% +\mu_{B}B(g_{J}-g_{I}))
  104. ± | H | = 1 2 h A ( I + 1 / 2 ) 2 - m F 2 \langle\pm|H|\mp\rangle=\frac{1}{2}hA\sqrt{(I+1/2)^{2}-m_{F}^{2}}
  105. Δ E F = I ± 1 / 2 = - h Δ W 2 ( 2 I + 1 ) + μ B g I m F B ± h Δ W 2 1 + 2 m F x I + 1 / 2 + x 2 \Delta E_{F=I\pm 1/2}=-\frac{h\Delta W}{2(2I+1)}+\mu_{B}g_{I}m_{F}B\pm\frac{h% \Delta W}{2}\sqrt{1+\frac{2m_{F}x}{I+1/2}+x^{2}}
  106. x μ B B ( g J - g I ) h Δ W Δ W = A ( I + 1 2 ) x\equiv\frac{\mu_{B}B(g_{J}-g_{I})}{h\Delta W}\quad\quad\Delta W=A\left(I+% \frac{1}{2}\right)
  107. Δ W \Delta W
  108. B B
  109. x x
  110. m = - ( I + 1 / 2 ) m=-(I+1/2)
  111. + ( 1 - x ) +(1-x)
  112. s s
  113. J = 1 / 2 J=1/2
  114. F F
  115. Δ E F = I ± 1 / 2 \Delta E_{F=I\pm 1/2}
  116. B = 0 B=0
  117. F F
  118. m F m_{F}
  119. | F = I + 1 / 2 , m F = ± F |F=I+1/2,m_{F}=\pm F\rangle

Zener_diode.html

  1. I d i o d e = U I N - U O U T R Ω I_{diode}=\dfrac{U_{IN}-U_{OUT}}{R_{\Omega}}
  2. I D V B < P MAX I_{D}V_{B}<P_{\mathrm{MAX}}

Zenithal_Hourly_Rate.html

  1. Z H R = H R ¯ F r 6.5 - l m sin ( h R ) ZHR=\cfrac{\overline{HR}\cdot F\cdot r^{6.5-lm}}{\sin(hR)}
  2. H R ¯ = N T e f f \overline{HR}=\cfrac{N}{T_{eff}}
  3. F = 1 1 - k F=\cfrac{1}{1-k}
  4. r 6.5 - l m r^{6.5-lm}
  5. sin ( h R ) \sin(hR)

Zermelo–Fraenkel_set_theory.html

  1. x y [ z ( z x z y ) x = y ] . \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow x=y].
  2. z [ z x z y ] w [ x w y w ] . \forall z[z\in x\Leftrightarrow z\in y]\land\forall w[x\in w\Leftrightarrow y% \in w].
  3. x y [ z ( z x z y ) w ( x w y w ) ] , \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow\forall w% (x\in w\Leftrightarrow y\in w)],
  4. x [ a ( a x ) y ( y x ¬ z ( z y z x ) ) ] . \forall x[\exists a(a\in x)\Rightarrow\exists y(y\in x\land\lnot\exists z(z\in y% \land z\in x))].
  5. \mathbb{Z}
  6. x 0 ( mod 2 ) x\equiv 0\;\;(\mathop{{\rm mod}}2)
  7. { x : x 0 ( mod 2 ) } . \{x\in\mathbb{Z}:x\equiv 0\;\;(\mathop{{\rm mod}}2)\}.
  8. ϕ \phi
  9. { x z : ϕ ( x ) } . \{x\in z:\phi(x)\}.
  10. ϕ \phi
  11. ϕ \phi\!
  12. x , z , w 1 , , w n x,z,w_{1},\ldots,w_{n}\!
  13. ϕ \phi\!
  14. z w 1 w 2 w n y x [ x y ( x z ϕ ) ] . \forall z\forall w_{1}\forall w_{2}\ldots\forall w_{n}\exists y\forall x[x\in y% \Leftrightarrow(x\in z\land\phi)].
  15. { x : ϕ ( x ) } . \{x:\phi(x)\}.
  16. \varnothing
  17. ϕ \phi\!
  18. = { u w ( u u ) ¬ ( u u ) } \varnothing=\{u\in w\mid(u\in u)\land\lnot(u\in u)\}
  19. \varnothing
  20. x y z ( x z y z ) . \forall x\forall y\exists z(x\in z\land y\in z).
  21. { { 1 , 2 } , { 2 , 3 } } \{\{1,2\},\{2,3\}\}
  22. { 1 , 2 , 3 } \{1,2,3\}
  23. \mathcal{F}
  24. \mathcal{F}
  25. A Y x [ ( x Y Y ) x A ] . \forall\mathcal{F}\,\exists A\,\forall Y\,\forall x[(x\in Y\land Y\in\mathcal{% F})\Rightarrow x\in A].
  26. ϕ \phi\!
  27. x , y , A , w 1 , , w n x,y,A,w_{1},\ldots,w_{n}\!
  28. B B
  29. ϕ \phi\!
  30. A w 1 w 2 w n [ x ( x A ! y ϕ ) B x ( x A y ( y B ϕ ) ) ] . \forall A\forall w_{1}\forall w_{2}\ldots\forall w_{n}\bigl[\forall x(x\in A% \Rightarrow\exists!y\,\phi)\Rightarrow\exists B\ \forall x\bigl(x\in A% \Rightarrow\exists y(y\in B\land\phi)\bigr)\bigr].
  31. ϕ \phi\!
  32. A A
  33. B B
  34. B B
  35. S ( w ) S(w)\!
  36. w { w } w\cup\{w\}\!
  37. w w\!
  38. { w } \{w\}
  39. x = y = w x=y=w\!
  40. z z\!
  41. { w } \{w\}\!
  42. \varnothing
  43. S ( y ) S(y)\!
  44. X [ X and y ( y X S ( y ) X ) ] . \exists X\left[\varnothing\in X\and\forall y(y\in X\Rightarrow S(y)\in X)% \right].
  45. \mathbb{N}
  46. ( z x ) ( q ( q z q x ) ) . (z\subseteq x)\Leftrightarrow(\forall q(q\in z\Rightarrow q\in x)).
  47. x y z [ z x z y ] . \forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y].
  48. P ( x ) = { z y : z x } P(x)=\{z\in y:z\subseteq x\}
  49. X R ( R well-orders X ) . \forall X\exists R(R\;\mbox{well-orders}~{}\;X).
  50. Y X Y∈X
  51. f ( Y ) Y f(Y)∈Y

Zero-point_energy.html

  1. ϵ = h ν e h ν / k T - 1 \epsilon=\frac{h\nu}{e^{h\nu/kT}-1}
  2. h h
  3. ν \nu
  4. ϵ = h ν 2 + h ν e h ν / k T - 1 \epsilon=\frac{h\nu}{2}+\frac{h\nu}{e^{h\nu/kT}-1}
  5. H ^ = E 0 + 1 2 k ( x ^ - x 0 ) 2 + 1 2 m p ^ 2 \hat{H}=E_{0}+\frac{1}{2}k\left(\hat{x}-x_{0}\right)^{2}+\frac{1}{2m}\hat{p}^{2}
  6. E 0 E_{0}
  7. ( x ^ - x 0 ) 2 p ^ 2 2 , \sqrt{\left\langle\left(\hat{x}-x_{0}\right)^{2}\right\rangle}\sqrt{\left% \langle\hat{p}^{2}\right\rangle}\geq\frac{\hbar}{2},
  8. 1 2 k ( x ^ - x 0 ) 2 1 2 m p ^ 2 ( 4 ) 2 k m . \left\langle\frac{1}{2}k\left(\hat{x}-x_{0}\right)^{2}\right\rangle\left% \langle\frac{1}{2m}\hat{p}^{2}\right\rangle\geq\left(\frac{\hbar}{4}\right)^{2% }\frac{k}{m}.
  9. H ^ E 0 + 2 k m = E 0 + ω 2 \left\langle\hat{H}\right\rangle\geq E_{0}+\frac{\hbar}{2}\sqrt{\frac{k}{m}}=E% _{0}+\frac{\hbar\omega}{2}
  10. ω = k / m \omega=\sqrt{k/m}
  11. E 0 = ω / 2 , E_{0}=\hbar\omega/2,
  12. ν \nu
  13. ω \omega
  14. ω \omega
  15. 2 π ν 2\pi\nu
  16. h h
  17. \hbar
  18. h h
  19. 2 π 2\pi
  20. E = ω / 2 E={\hbar\omega/2}
  21. E = ω / 2 E={\hbar\omega/2}

Zero_of_a_function.html

  1. f ( x ) = 0. f(x)=0.
  2. f ( x ) = x 2 - 5 x + 6 f(x)=x^{2}-5x+6
  3. f ( 2 ) = 2 2 - 5 2 + 6 = 0 and f ( 3 ) = 3 2 - 5 3 + 6 = 0. f(2)=2^{2}-5\cdot 2+6=0\quad\textstyle{\rm{and}}\quad f(3)=3^{2}-5\cdot 3+6=0.
  4. x x
  5. f ( x ) = 0 f(x)=0
  6. f f
  7. f - 1 ( 0 ) f^{-1}(0)
  8. Z ( S ) = { x 𝔸 n f ( x ) = 0 for all f S } . Z(S)=\{x\in\mathbb{A}^{n}\mid f(x)=0\,\text{ for all }f\in S\}.

Zero_set.html

  1. f - 1 ( 0 ) f^{-1}(0)
  2. f ( x ) = 0 f(x)=0
  3. A = f - 1 ( 0 ) . A=f^{-1}(0).\,

Zero_sharp.html

  1. ω 1 \omega_{1}
  2. ω 2 \omega_{2}
  3. ω \omega
  4. G G
  5. ω \omega
  6. ω 2 L \omega_{2}^{L}
  7. ω 2 L \omega_{2}^{L}
  8. ω 1 \omega_{1}
  9. G G
  10. ω 2 \omega_{2}

Zeta_distribution.html

  1. ζ ( s - 1 ) ζ ( s ) for s > 2 \frac{\zeta(s-1)}{\zeta(s)}~{}\textrm{for}~{}s>2
  2. 1 1\,
  3. ζ ( s ) ζ ( s - 2 ) - ζ ( s - 1 ) 2 ζ ( s ) 2 for s > 3 \frac{\zeta(s)\zeta(s-2)-\zeta(s-1)^{2}}{\zeta(s)^{2}}~{}\textrm{for}~{}s>3
  4. k = 1 1 / k s ζ ( s ) log ( k s ζ ( s ) ) . \sum_{k=1}^{\infty}\frac{1/k^{s}}{\zeta(s)}\log(k^{s}\zeta(s)).\,\!
  5. Li s ( e t ) ζ ( s ) \frac{\operatorname{Li}_{s}(e^{t})}{\zeta(s)}
  6. Li s ( e i t ) ζ ( s ) \frac{\operatorname{Li}_{s}(e^{it})}{\zeta(s)}
  7. f s ( k ) = k - s / ζ ( s ) f_{s}(k)=k^{-s}/\zeta(s)\,
  8. k - s k^{-s}
  9. m n = E ( X n ) = 1 ζ ( s ) k = 1 1 k s - n m_{n}=E(X^{n})=\frac{1}{\zeta(s)}\sum_{k=1}^{\infty}\frac{1}{k^{s-n}}
  10. m n = { ζ ( s - n ) / ζ ( s ) for n < s - 1 for n s - 1 m_{n}=\left\{\begin{matrix}\zeta(s-n)/\zeta(s)&\textrm{for}~{}n<s-1\\ \infty&\textrm{for}~{}n\geq s-1\end{matrix}\right.
  11. M ( t ; s ) = E ( e t X ) = 1 ζ ( s ) k = 1 e t k k s . M(t;s)=E(e^{tX})=\frac{1}{\zeta(s)}\sum_{k=1}^{\infty}\frac{e^{tk}}{k^{s}}.
  12. e t < 1 e^{t}<1
  13. M ( t ; s ) = Li s ( e t ) ζ ( s ) for t < 0. M(t;s)=\frac{\operatorname{Li}_{s}(e^{t})}{\zeta(s)}\,\text{ for }t<0.
  14. n = 0 m n t n n ! , \sum_{n=0}^{\infty}\frac{m_{n}t^{n}}{n!},
  15. 1 ζ ( s ) n = 0 , n s - 1 ζ ( s - n ) n ! t n = Li s ( e t ) - Φ ( s , t ) ζ ( s ) \frac{1}{\zeta(s)}\sum_{n=0,n\neq s-1}^{\infty}\frac{\zeta(s-n)}{n!}\,t^{n}=% \frac{\operatorname{Li}_{s}(e^{t})-\Phi(s,t)}{\zeta(s)}
  16. | t | < 2 π \scriptstyle|t|\,<\,2\pi
  17. Φ ( s , t ) \scriptstyle\Phi(s,t)
  18. Φ ( s , t ) = Γ ( 1 - s ) ( - t ) s - 1 for s 1 , 2 , 3 \Phi(s,t)=\Gamma(1-s)(-t)^{s-1}\,\text{ for }s\neq 1,2,3\ldots
  19. Φ ( s , t ) = t s - 1 ( s - 1 ) ! [ H s - ln ( - t ) ] for s = 2 , 3 , 4 \Phi(s,t)=\frac{t^{s-1}}{(s-1)!}\left[H_{s}-\ln(-t)\right]\,\text{ for }s=2,3,4\ldots
  20. Φ ( s , t ) = - ln ( - t ) for s = 1 , \Phi(s,t)=-\ln(-t)\,\text{ for }s=1,\,
  21. lim n N ( A , n ) n \lim_{n\rightarrow\infty}\frac{N(A,n)}{n}
  22. lim s 1 + P ( X A ) \lim_{s\rightarrow 1+}P(X\in A)\,
  23. log ( d + 1 ) - log ( d ) , \log(d+1)-\log(d),\,

Zone_melting.html

  1. d x dx
  2. k O k_{O}
  3. L L
  4. C O C_{O}
  5. C L C_{L}
  6. I I
  7. I O I_{O}
  8. d x dx
  9. d I = ( C O - k O C L ) d x dI=(C_{O}-k_{O}C_{L})\,dx\;
  10. C L = I / L C_{L}=I/L\;
  11. 0 x d x = I O I d I C O - k O I L \int_{0}^{x}dx=\int_{I_{O}}^{I}\frac{dI}{C_{O}-\frac{k_{O}I}{L}}
  12. I O = C O L I_{O}=C_{O}L\;
  13. C S = k O I / L C_{S}=k_{O}I/L\;
  14. C S ( x ) = C O ( 1 - ( 1 - k O ) e - k O x L ) C_{S}(x)=C_{O}\left(1-(1-k_{O})e^{-\frac{k_{O}x}{L}}\right)