wpmath0000001_20

Schrödinger's_cat.html

  1. | ψ = 1 2 ( | 00 0 + | 11 1 ) . |\psi\rangle=\frac{1}{\sqrt{2}}\bigg(|00\ldots 0\rangle+|11\ldots 1\rangle% \bigg).

Schrödinger_equation.html

  1. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}
  2. i i
  3. ħ ħ
  4. 2 π 2\pi
  5. t t
  6. Ψ Ψ
  7. Ĥ Ĥ
  8. μ μ
  9. V V
  10. Ψ Ψ
  11. Ψ Ψ
  12. Ψ Ψ
  13. Ψ Ψ
  14. E E
  15. Ψ Ψ
  16. p p
  17. λ λ
  18. k k
  19. p = h λ = k p=\frac{h}{\lambda}=\hbar k
  20. h h
  21. L L
  22. L = n h 2 π = n . L=n{h\over 2\pi}=n\hbar.
  23. n λ = 2 π r . n\lambda=2\pi r.\,
  24. r r
  25. i t Ψ ( 𝐫 , t ) = - 2 2 m 2 Ψ ( 𝐫 , t ) + V ( 𝐫 ) Ψ ( 𝐫 , t ) . i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^{2}}{2m}% \nabla^{2}\Psi(\mathbf{r},\,t)+V(\mathbf{r})\Psi(\mathbf{r},\,t).
  26. ( E + e 2 r ) 2 ψ ( x ) = - 2 ψ ( x ) + m 2 ψ ( x ) . \left(E+{e^{2}\over r}\right)^{2}\psi(x)=-\nabla^{2}\psi(x)+m^{2}\psi(x).
  27. Ψ ( x , t ) Ψ(x,t)
  28. V V
  29. Ψ Ψ
  30. Ψ Ψ
  31. ψ ψ
  32. ψ ψ
  33. ψ ψ
  34. E E
  35. T T
  36. V V
  37. H H
  38. E = T + V = H E=T+V=H\,\!
  39. x x
  40. m m
  41. p p
  42. V V
  43. t t
  44. E = p 2 2 m + V ( x , t ) = H . E=\frac{p^{2}}{2m}+V(x,t)=H.
  45. 𝐫 \mathbf{r}
  46. 𝐩 \mathbf{p}
  47. E = 𝐩 𝐩 2 m + V ( 𝐫 , t ) = H E=\frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H
  48. V V
  49. E = n = 1 N 𝐩 n 𝐩 n 2 m n + V ( 𝐫 1 , 𝐫 2 𝐫 N , t ) = H E=\sum_{n=1}^{N}\frac{\mathbf{p}_{n}\cdot\mathbf{p}_{n}}{2m_{n}}+V(\mathbf{r}_% {1},\mathbf{r}_{2}\cdots\mathbf{r}_{N},t)=H\,\!
  50. Ψ ( 𝐫 , t ) = A e i ( 𝐤 𝐫 - ω t ) \Psi(\mathbf{r},t)=Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}\,\!
  51. A A
  52. 𝐤 \mathbf{k}
  53. ω ω
  54. 𝐤 \mathbf{k}
  55. Ψ ( 𝐫 , t ) = n = 1 A n e i ( 𝐤 n 𝐫 - ω n t ) \Psi(\mathbf{r},t)=\sum_{n=1}^{\infty}A_{n}e^{i(\mathbf{k}_{n}\cdot\mathbf{r}-% \omega_{n}t)}\,\!
  56. 𝐤 \mathbf{k}
  57. Ψ ( 𝐫 , t ) = 1 ( 2 π ) 3 Φ ( 𝐤 ) e i ( 𝐤 𝐫 - ω t ) d 3 𝐤 \Psi(\mathbf{r},t)=\frac{1}{(\sqrt{2\pi})^{3}}\int\Phi(\mathbf{k})e^{i(\mathbf% {k}\cdot\mathbf{r}-\omega t)}d^{3}\mathbf{k}\,\!
  58. 𝐤 \mathbf{k}
  59. 𝐤 \mathbf{k}
  60. Φ ( 𝐤 ) Φ(\mathbf{k})
  61. E E
  62. ν ν
  63. ω = 2 π ν ω=2\pi ν
  64. E = h ν = ω E=h\nu=\hbar\omega\,\!
  65. p p
  66. λ λ
  67. k = 2 π / λ k=2\pi/λ
  68. p = h λ = k , p=\frac{h}{\lambda}=\hbar k\;,
  69. λ λ
  70. 𝐤 \mathbf{k}
  71. 𝐩 = 𝐤 , | 𝐤 | = 2 π λ . \mathbf{p}=\hbar\mathbf{k}\,,\quad|\mathbf{k}|=\frac{2\pi}{\lambda}\,.
  72. ħ = 1 ħ=1
  73. Ψ = A e i ( 𝐤 𝐫 - ω t ) = A e i ( 𝐩 𝐫 - E t ) / \Psi=Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}=Ae^{i(\mathbf{p}\cdot\mathbf{r% }-Et)/\hbar}
  74. Ψ = i 𝐩 A e i ( 𝐩 𝐫 - E t ) / = i 𝐩 Ψ \nabla\Psi=\dfrac{i}{\hbar}\mathbf{p}Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar% }=\dfrac{i}{\hbar}\mathbf{p}\Psi
  75. Ψ t = - i E A e i ( 𝐩 𝐫 - E t ) / = - i E Ψ \dfrac{\partial\Psi}{\partial t}=-\dfrac{iE}{\hbar}Ae^{i(\mathbf{p}\cdot% \mathbf{r}-Et)/\hbar}=-\dfrac{iE}{\hbar}\Psi
  76. E ^ Ψ = i t Ψ = E Ψ \hat{E}\Psi=i\hbar\dfrac{\partial}{\partial t}\Psi=E\Psi
  77. E E
  78. 𝐩 ^ Ψ = - i Ψ = 𝐩 Ψ \hat{\mathbf{p}}\Psi=-i\hbar\nabla\Psi=\mathbf{p}\Psi
  79. 𝐩 \mathbf{p}
  80. $^$
  81. V V
  82. E = 𝐩 𝐩 2 m + V E ^ = 𝐩 ^ 𝐩 ^ 2 m + V E=\dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}+V\quad\rightarrow\quad\hat{E}=\dfrac{% \hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m}+V
  83. Ψ Ψ
  84. i Ψ t = - 2 2 m 2 Ψ + V Ψ i\hbar\dfrac{\partial\Psi}{\partial t}=-\dfrac{\hbar^{2}}{2m}\nabla^{2}\Psi+V\Psi
  85. T T
  86. 𝐩 \mathbf{p}
  87. | 𝐤 | |\mathbf{k}|
  88. λ λ
  89. 𝐩 𝐩 𝐤 𝐤 T 1 λ 2 \mathbf{p}\cdot\mathbf{p}\propto\mathbf{k}\cdot\mathbf{k}\propto T\propto% \dfrac{1}{\lambda^{2}}
  90. T ^ Ψ = - 2 2 m Ψ 2 Ψ . \hat{T}\Psi=\frac{-\hbar^{2}}{2m}\nabla\cdot\nabla\Psi\,\propto\,\nabla^{2}% \Psi\,.
  91. 𝐫 \mathbf{r}
  92. 𝐤 \mathbf{k}
  93. 𝐤 \mathbf{k}
  94. 𝐫 \mathbf{r}
  95. 𝐤 \mathbf{k}
  96. ħ ħ
  97. ħ ħ
  98. ħ ħ
  99. ħ 0 ħ→0
  100. σ ( x ) σ ( p x ) 2 σ ( x ) σ ( p x ) 0 \sigma(x)\sigma(p_{x})\geqslant\frac{\hbar}{2}\quad\rightarrow\quad\sigma(x)% \sigma(p_{x})\geqslant 0\,\!
  101. σ σ
  102. x x
  103. y y
  104. z z
  105. i t Ψ ( 𝐫 , t ) = H ^ Ψ ( 𝐫 , t ) i\hbar\frac{\partial}{\partial t}\Psi\left(\mathbf{r},t\right)=\hat{H}\Psi% \left(\mathbf{r},t\right)\,\!
  106. t S ( q i , t ) = H ( q i , S q i , t ) \frac{\partial}{\partial t}S(q_{i},t)=H\left(q_{i},\frac{\partial S}{\partial q% _{i}},t\right)\,\!
  107. S S
  108. H H
  109. i = 1 , 2 , 3 i=1,2,3
  110. Ψ = ρ ( 𝐫 , t ) e i S ( 𝐫 , t ) / \Psi=\sqrt{\rho(\mathbf{r},t)}e^{iS(\mathbf{r},t)/\hbar}\,\!
  111. ρ ρ
  112. ħ 0 ħ→0
  113. N N
  114. Ψ ( space coords , t ) = ψ ( space coords ) τ ( t ) . \Psi(\,\text{space coords},t)=\psi(\,\text{space coords})\tau(t)\,.
  115. ψ ( s p a c e c o o r d s ) ψ(spacecoords)
  116. τ ( t ) τ(t)
  117. ψ ψ
  118. Ψ ( space coords , t ) = ψ ( space coords ) e - i E t / . \Psi(\,\text{space coords},t)=\psi(\,\text{space coords})e^{-i{Et/\hbar}}\,.
  119. Ê = i ħ t Ê=iħ∂\frac{∂}{t}
  120. E E
  121. H ^ ψ = E ψ \hat{H}\psi=E\psi
  122. H ^ = p ^ 2 2 m + V ( x ) , p ^ = - i d d x \hat{H}=\frac{\hat{p}^{2}}{2m}+V(x)\,,\quad\hat{p}=-i\hbar\frac{d}{dx}
  123. - 2 2 m d 2 d x 2 ψ ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) -\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi(x)+V(x)\psi(x)=E\psi(x)
  124. Ψ ( x , t ) = ψ ( x ) e - i E t / . \Psi(x,t)=\psi(x)e^{-iEt/\hbar}\,.
  125. N N
  126. H ^ = n = 1 N p ^ n 2 2 m n + V ( x 1 , x 2 , x N ) , p ^ n = - i x n \hat{H}=\sum_{n=1}^{N}\frac{\hat{p}_{n}^{2}}{2m_{n}}+V(x_{1},x_{2},\cdots x_{N% })\,,\quad\hat{p}_{n}=-i\hbar\frac{\partial}{\partial x_{n}}
  127. n n
  128. - 2 2 n = 1 N 1 m n 2 x n 2 ψ ( x 1 , x 2 , x N ) + V ( x 1 , x 2 , x N ) ψ ( x 1 , x 2 , x N ) = E ψ ( x 1 , x 2 , x N ) . -\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\frac{\partial^{2}}{\partial x% _{n}^{2}}\psi(x_{1},x_{2},\cdots x_{N})+V(x_{1},x_{2},\cdots x_{N})\psi(x_{1},% x_{2},\cdots x_{N})=E\psi(x_{1},x_{2},\cdots x_{N})\,.
  129. Ψ ( x 1 , x 2 , x N , t ) = e - i E t / ψ ( x 1 , x 2 x N ) \Psi(x_{1},x_{2},\cdots x_{N},t)=e^{-iEt/\hbar}\psi(x_{1},x_{2}\cdots x_{N})
  130. V ( x 1 , x 2 , x N ) = n = 1 N V ( x n ) . V(x_{1},x_{2},\cdots x_{N})=\sum_{n=1}^{N}V(x_{n})\,.
  131. Ψ ( x 1 , x 2 , x N , t ) = e - i E t / n = 1 N ψ ( x n ) , \Psi(x_{1},x_{2},\cdots x_{N},t)=e^{-i{Et/\hbar}}\prod_{n=1}^{N}\psi(x_{n})\,,
  132. V = 0 V=0
  133. - E ψ = 2 2 m d 2 ψ d x 2 -E\psi=\frac{\hbar^{2}}{2m}{d^{2}\psi\over dx^{2}}\,
  134. E > 0 E>0
  135. ψ E ( x ) = C 1 e i 2 m E / 2 x + C 2 e - i 2 m E / 2 x \psi_{E}(x)=C_{1}e^{i\sqrt{2mE/\hbar^{2}}\,x}+C_{2}e^{-i\sqrt{2mE/\hbar^{2}}\,% x}\,
  136. ψ - | E | ( x ) = C 1 e 2 m | E | / 2 x + C 2 e - 2 m | E | / 2 x . \psi_{-|E|}(x)=C_{1}e^{\sqrt{2m|E|/\hbar^{2}}\,x}+C_{2}e^{-\sqrt{2m|E|/\hbar^{% 2}}\,x}.\,
  137. E ψ = - 2 2 m d 2 d x 2 ψ + 1 2 m ω 2 x 2 ψ E\psi=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi+\frac{1}{2}m\omega^{2}x^{2}\psi
  138. ψ n ( x ) = 1 2 n n ! ( m ω π ) 1 / 4 e - m ω x 2 2 H n ( m ω x ) \psi_{n}(x)=\sqrt{\frac{1}{2^{n}\,n!}}\cdot\left(\frac{m\omega}{\pi\hbar}% \right)^{1/4}\cdot e^{-\frac{m\omega x^{2}}{2\hbar}}\cdot H_{n}\left(\sqrt{% \frac{m\omega}{\hbar}}x\right)
  139. n = 0 , 1 , 2 , n=0,1,2,...
  140. H ^ = 𝐩 ^ 𝐩 ^ 2 m + V ( 𝐫 ) , 𝐩 ^ = - i \hat{H}=\frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m}+V(\mathbf{r})\,,\quad% \hat{\mathbf{p}}=-i\hbar\nabla
  141. - 2 2 m 2 ψ ( 𝐫 ) + V ( 𝐫 ) ψ ( 𝐫 ) = E ψ ( 𝐫 ) -\frac{\hbar^{2}}{2m}\nabla^{2}\psi(\mathbf{r})+V(\mathbf{r})\psi(\mathbf{r})=% E\psi(\mathbf{r})
  142. Ψ ( 𝐫 , t ) = ψ ( 𝐫 ) e - i E t / \Psi(\mathbf{r},t)=\psi(\mathbf{r})e^{-iEt/\hbar}
  143. 𝐫 = ( x , y , z ) \mathbf{r}=(x,y,z)
  144. 𝐫 = ( r , θ , φ ) \mathbf{r}=(r,θ,φ)
  145. N N
  146. H ^ = n = 1 N 𝐩 ^ n 𝐩 ^ n 2 m n + V ( 𝐫 1 , 𝐫 2 , 𝐫 N ) , 𝐩 ^ n = - i n \hat{H}=\sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_{n}\cdot\hat{\mathbf{p}}_{n}}{2m_% {n}}+V(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N})\,,\quad\hat{\mathbf% {p}}_{n}=-i\hbar\nabla_{n}
  147. n n
  148. n n
  149. n = 𝐞 x x n + 𝐞 y y n + 𝐞 z z n , n 2 = n n = 2 x n 2 + 2 y n 2 + 2 z n 2 \nabla_{n}=\mathbf{e}_{x}\frac{\partial}{\partial x_{n}}+\mathbf{e}_{y}\frac{% \partial}{\partial y_{n}}+\mathbf{e}_{z}\frac{\partial}{\partial z_{n}}\,,% \quad\nabla_{n}^{2}=\nabla_{n}\cdot\nabla_{n}=\frac{\partial^{2}}{{\partial x_% {n}}^{2}}+\frac{\partial^{2}}{{\partial y_{n}}^{2}}+\frac{\partial^{2}}{{% \partial z_{n}}^{2}}
  150. - 2 2 n = 1 N 1 m n n 2 Ψ ( 𝐫 1 , 𝐫 2 , 𝐫 N ) + V ( 𝐫 1 , 𝐫 2 , 𝐫 N ) Ψ ( 𝐫 1 , 𝐫 2 , 𝐫 N ) = E Ψ ( 𝐫 1 , 𝐫 2 , 𝐫 N ) -\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\nabla_{n}^{2}\Psi(\mathbf{r}% _{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N})+V(\mathbf{r}_{1},\mathbf{r}_{2},% \cdots\mathbf{r}_{N})\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N})=% E\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N})
  151. Ψ ( 𝐫 1 , 𝐫 2 𝐫 N , t ) = e - i E t / ψ ( 𝐫 1 , 𝐫 2 𝐫 N ) \Psi(\mathbf{r}_{1},\mathbf{r}_{2}\cdots\mathbf{r}_{N},t)=e^{-iEt/\hbar}\psi(% \mathbf{r}_{1},\mathbf{r}_{2}\cdots\mathbf{r}_{N})
  152. V ( 𝐫 1 , 𝐫 2 , 𝐫 N ) = n = 1 N V ( 𝐫 n ) V(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N})=\sum_{n=1}^{N}V(\mathbf{% r}_{n})
  153. Ψ ( 𝐫 1 , 𝐫 2 𝐫 N , t ) = e - i E t / n = 1 N ψ ( 𝐫 n ) . \Psi(\mathbf{r}_{1},\mathbf{r}_{2}\cdots\mathbf{r}_{N},t)=e^{-i{Et/\hbar}}% \prod_{n=1}^{N}\psi(\mathbf{r}_{n})\,.
  154. E ψ = - 2 2 μ 2 ψ - e 2 4 π ε 0 r ψ E\psi=-\frac{\hbar^{2}}{2\mu}\nabla^{2}\psi-\frac{e^{2}}{4\pi\varepsilon_{0}r}\psi
  155. e e
  156. 𝐫 \mathbf{r}
  157. r = | 𝐫 | r=|\mathbf{r}|
  158. μ = m e m p m e + m p \mu=\frac{m_{e}m_{p}}{m_{e}+m_{p}}
  159. ψ ( r , θ , ϕ ) = R ( r ) Y m ( θ , ϕ ) = R ( r ) Θ ( θ ) Φ ( ϕ ) \psi(r,\theta,\phi)=R(r)Y_{\ell}^{m}(\theta,\phi)=R(r)\Theta(\theta)\Phi(\phi)
  160. R R
  161. Y m ( θ , ϕ ) \scriptstyle Y_{\ell}^{m}(\theta,\phi)\,
  162. m m
  163. ψ n m ( r , θ , ϕ ) = ( 2 n a 0 ) 3 ( n - - 1 ) ! 2 n [ ( n + ) ! ] e - r / n a 0 ( 2 r n a 0 ) L n - - 1 2 + 1 ( 2 r n a 0 ) Y m ( θ , ϕ ) \psi_{n\ell m}(r,\theta,\phi)=\sqrt{{\left(\frac{2}{na_{0}}\right)}^{3}\frac{(% n-\ell-1)!}{2n[(n+\ell)!]}}e^{-r/na_{0}}\left(\frac{2r}{na_{0}}\right)^{\ell}L% _{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_{0}}\right)\cdot Y_{\ell}^{m}(\theta,\phi)
  164. a 0 = 4 π ε 0 2 m e e 2 a_{0}=\frac{4\pi\varepsilon_{0}\hbar^{2}}{m_{e}e^{2}}
  165. L n - - 1 2 + 1 ( ) L_{n-\ell-1}^{2\ell+1}(\cdots)
  166. n 1 n−ℓ−1
  167. n , , m n,ℓ,m
  168. n \displaystyle n
  169. Z = 2 Z=2
  170. Z = 1 Z=1
  171. Z = 3 Z=3
  172. E ψ = - 2 [ 1 2 μ ( 1 2 + 2 2 ) + 1 M 1 2 ] ψ + e 2 4 π ε 0 [ 1 r 12 - Z ( 1 r 1 + 1 r 2 ) ] ψ E\psi=-\hbar^{2}\left[\frac{1}{2\mu}\left(\nabla_{1}^{2}+\nabla_{2}^{2}\right)% +\frac{1}{M}\nabla_{1}\cdot\nabla_{2}\right]\psi+\frac{e^{2}}{4\pi\varepsilon_% {0}}\left[\frac{1}{r_{12}}-Z\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\right]\psi
  173. | 𝐫 12 | = | 𝐫 2 - 𝐫 1 | |\mathbf{r}_{12}|=|\mathbf{r}_{2}-\mathbf{r}_{1}|\,\!
  174. μ μ
  175. M M
  176. μ = m e M m e + M \mu=\frac{m_{e}M}{m_{e}+M}\,\!
  177. Z Z
  178. 1 M 1 2 \frac{1}{M}\nabla_{1}\cdot\nabla_{2}\,\!
  179. ψ = ψ ( 𝐫 1 , 𝐫 2 ) . \psi=\psi(\mathbf{r}_{1},\mathbf{r}_{2}).
  180. i t Ψ = H ^ Ψ . i\hbar\frac{\partial}{\partial t}\Psi=\hat{H}\Psi.
  181. H ^ = p ^ 2 2 m + V ( x , t ) , p ^ = - i x \hat{H}=\frac{\hat{p}^{2}}{2m}+V(x,t)\,,\quad\hat{p}=-i\hbar\frac{\partial}{% \partial x}
  182. i t Ψ ( x , t ) = - 2 2 m 2 x 2 Ψ ( x , t ) + V ( x , t ) Ψ ( x , t ) i\hbar\frac{\partial}{\partial t}\Psi(x,t)=-\frac{\hbar^{2}}{2m}\frac{\partial% ^{2}}{\partial x^{2}}\Psi(x,t)+V(x,t)\Psi(x,t)
  183. N N
  184. H ^ = n = 1 N p ^ n 2 2 m n + V ( x 1 , x 2 , x N , t ) , p ^ n = - i x n \hat{H}=\sum_{n=1}^{N}\frac{\hat{p}_{n}^{2}}{2m_{n}}+V(x_{1},x_{2},\cdots x_{N% },t)\,,\quad\hat{p}_{n}=-i\hbar\frac{\partial}{\partial x_{n}}
  185. n n
  186. i t Ψ ( x 1 , x 2 x N , t ) = - 2 2 n = 1 N 1 m n 2 x n 2 Ψ ( x 1 , x 2 x N , t ) + V ( x 1 , x 2 x N , t ) Ψ ( x 1 , x 2 x N , t ) . i\hbar\frac{\partial}{\partial t}\Psi(x_{1},x_{2}\cdots x_{N},t)=-\frac{\hbar^% {2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\frac{\partial^{2}}{\partial x_{n}^{2}}% \Psi(x_{1},x_{2}\cdots x_{N},t)+V(x_{1},x_{2}\cdots x_{N},t)\Psi(x_{1},x_{2}% \cdots x_{N},t)\,.
  187. H ^ = 𝐩 ^ 𝐩 ^ 2 m + V ( 𝐫 , t ) , 𝐩 ^ = - i \hat{H}=\frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m}+V(\mathbf{r},t)\,,% \quad\hat{\mathbf{p}}=-i\hbar\nabla
  188. i t Ψ ( 𝐫 , t ) = - 2 2 m 2 Ψ ( 𝐫 , t ) + V ( 𝐫 , t ) Ψ ( 𝐫 , t ) i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t)=-\frac{\hbar^{2}}{2m}% \nabla^{2}\Psi(\mathbf{r},t)+V(\mathbf{r},t)\Psi(\mathbf{r},t)
  189. N N
  190. H ^ = n = 1 N 𝐩 ^ n 𝐩 ^ n 2 m n + V ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) , 𝐩 ^ n = - i n \hat{H}=\sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_{n}\cdot\hat{\mathbf{p}}_{n}}{2m_% {n}}+V(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N},t)\,,\quad\hat{% \mathbf{p}}_{n}=-i\hbar\nabla_{n}
  191. n n
  192. i t Ψ ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) = - 2 2 n = 1 N 1 m n n 2 Ψ ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) + V ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) Ψ ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\cdots% \mathbf{r}_{N},t)=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\nabla_{n}^% {2}\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N},t)+V(\mathbf{r}_{1}% ,\mathbf{r}_{2},\cdots\mathbf{r}_{N},t)\Psi(\mathbf{r}_{1},\mathbf{r}_{2},% \cdots\mathbf{r}_{N},t)
  193. ψ = a ψ 1 + b ψ 2 \displaystyle\psi=a\psi_{1}+b\psi_{2}
  194. a a
  195. b b
  196. Ψ ( x , t ) Ψ(x,t)
  197. e - i E n t / , e^{{-iE_{n}t}/\hbar},
  198. Ψ ( x , t ) = n A n ψ E n ( x ) e - i E n t / . \displaystyle\Psi(x,t)=\sum\limits_{n}A_{n}\psi_{E_{n}}(x)e^{{-iE_{n}t}/\hbar}.
  199. Ψ = n A n ψ n \displaystyle\Psi=\sum\limits_{n}A_{n}\psi_{n}
  200. n | A n | 2 = 1. \displaystyle\sum\limits_{n}|A_{n}|^{2}=1.
  201. - | Ψ ( x ) | 2 d x = - Ψ ( x ) Ψ * ( x ) d x = 1. \displaystyle\int\limits_{-\infty}^{\infty}|\Psi(x)|^{2}\,dx=\int\limits_{-% \infty}^{\infty}\Psi(x)\Psi^{*}(x)\,dx=1.
  202. E E
  203. H ^ ( a ψ 1 + b ψ 2 ) = a H ^ ψ 1 + b H ^ ψ 2 = E ( a ψ 1 + b ψ 2 ) . \hat{H}(a\psi_{1}+b\psi_{2})=a\hat{H}\psi_{1}+b\hat{H}\psi_{2}=E(a\psi_{1}+b% \psi_{2}).
  204. ψ ψ
  205. ψ * ψ*
  206. ψ ψ
  207. Ψ ( x , t ) Ψ(x,t)
  208. Ψ ( x , t ) Ψ(x,–t)
  209. x x
  210. y y
  211. z z
  212. t t
  213. i Ψ t = - 2 2 m ( 2 Ψ x 2 + 2 Ψ y 2 + 2 Ψ z 2 ) + V ( x , y , z , t ) Ψ . i\hbar{\partial\Psi\over\partial t}=-{\hbar^{2}\over 2m}\left({\partial^{2}% \Psi\over\partial x^{2}}+{\partial^{2}\Psi\over\partial y^{2}}+{\partial^{2}% \Psi\over\partial z^{2}}\right)+V(x,y,z,t)\Psi.\,\!
  214. t = 0 t=0
  215. Ψ ( x , y , z , 0 ) \Psi(x,y,z,0)\,\!
  216. Ψ ( x b , y b , z b , t ) x Ψ ( x b , y b , z b , t ) y Ψ ( x b , y b , z b , t ) z Ψ ( x b , y b , z b , t ) \begin{aligned}&\displaystyle\Psi(x_{b},y_{b},z_{b},t)\\ &\displaystyle\frac{\partial}{\partial x}\Psi(x_{b},y_{b},z_{b},t)\quad\frac{% \partial}{\partial y}\Psi(x_{b},y_{b},z_{b},t)\quad\frac{\partial}{\partial z}% \Psi(x_{b},y_{b},z_{b},t)\end{aligned}\,\!
  217. b b
  218. t ρ ( 𝐫 , t ) + 𝐣 = 0 , {\partial\over\partial t}\rho\left(\mathbf{r},t\right)+\nabla\cdot\mathbf{j}=0,
  219. ρ = | Ψ | 2 = Ψ * ( 𝐫 , t ) Ψ ( 𝐫 , t ) \rho=|\Psi|^{2}=\Psi^{*}(\mathbf{r},t)\Psi(\mathbf{r},t)\,\!
  220. * *
  221. 𝐣 = 1 2 m ( Ψ * 𝐩 ^ Ψ - Ψ 𝐩 ^ Ψ * ) \mathbf{j}={1\over 2m}\left(\Psi^{*}\hat{\mathbf{p}}\Psi-\Psi\hat{\mathbf{p}}% \Psi^{*}\right)\,\!
  222. Â Â
  223. ψ ψ
  224. ψ | A ^ | ψ \langle\psi|\hat{A}|\psi\rangle
  225. ψ ψ
  226. Ĥ Ĥ
  227. ψ | H ^ | ψ = ψ * ( 𝐫 ) [ - 2 2 m 2 ψ ( 𝐫 ) + V ( 𝐫 ) ψ ( 𝐫 ) ] d 3 𝐫 = [ 2 2 m | ψ | 2 + V ( 𝐫 ) | ψ | 2 ] d 3 𝐫 = H ^ \langle\psi|\hat{H}|\psi\rangle=\int\psi^{*}(\mathbf{r})\left[-\frac{\hbar^{2}% }{2m}\nabla^{2}\psi(\mathbf{r})+V(\mathbf{r})\psi(\mathbf{r})\right]d^{3}% \mathbf{r}=\int\left[\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}+V(\mathbf{r})|\psi|^% {2}\right]d^{3}\mathbf{r}=\langle\hat{H}\rangle
  228. ψ ψ
  229. V ( x ) V(x)
  230. V ( x ) V(x)
  231. E E
  232. τ = i t τ=it
  233. τ X ( 𝐫 , τ ) = 2 m 2 X ( 𝐫 , τ ) , X ( 𝐫 , τ ) = Ψ ( 𝐫 , τ / i ) {\partial\over\partial\tau}X(\mathbf{r},\tau)=\frac{\hbar}{2m}\nabla^{2}X(% \mathbf{r},\tau)\,,\quad X(\mathbf{r},\tau)=\Psi(\mathbf{r},\tau/i)
  234. ħ 2 m ħ\frac{2}{m}
  235. E 2 = ( p c ) 2 + ( m 0 c 2 ) 2 , E^{2}=(pc)^{2}+(m_{0}c^{2})^{2}\,,
  236. m m
  237. q q
  238. φ φ
  239. 𝐀 \mathbf{A}
  240. H ^ Dirac = γ 0 [ c s y m b o l γ ( 𝐩 ^ - q 𝐀 ) + m c 2 + γ 0 q ϕ ] , \hat{H}_{\,\text{Dirac}}=\gamma^{0}\left[csymbol{\gamma}\cdot\left(\hat{% \mathbf{p}}-q\mathbf{A}\right)+mc^{2}+\gamma^{0}q\phi\right]\,,
  241. γ = ( γ < s u p > 1 , γ 2 , γ 3 ) \mathbf{γ}=(γ<sup>1,γ^{2},γ^{3})

Scientific_notation.html

  1. b {}^{b}
  2. 10 \scriptstyle\sqrt{10}
  3. x 0 = a 0 × 10 b 0 x_{0}=a_{0}\times 10^{b_{0}}
  4. x 1 = a 1 × 10 b 1 x_{1}=a_{1}\times 10^{b_{1}}
  5. x 0 x 1 = a 0 a 1 × 10 b 0 + b 1 x_{0}x_{1}=a_{0}a_{1}\times 10^{b_{0}+b_{1}}
  6. x 0 x 1 = a 0 a 1 × 10 b 0 - b 1 \frac{x_{0}}{x_{1}}=\frac{a_{0}}{a_{1}}\times 10^{b_{0}-b_{1}}
  7. 5.67 × 10 - 5 × 2.34 × 10 2 13.3 × 10 - 3 = 1.33 × 10 - 2 5.67\times 10^{-5}\times 2.34\times 10^{2}\approx 13.3\times 10^{-3}=1.33% \times 10^{-2}
  8. 2.34 × 10 2 5.67 × 10 - 5 0.413 × 10 7 = 4.13 × 10 6 \frac{2.34\times 10^{2}}{5.67\times 10^{-5}}\approx 0.413\times 10^{7}=4.13% \times 10^{6}
  9. x 1 = c × 10 b 0 x_{1}=c\times 10^{b_{0}}
  10. x 0 ± x 1 = ( a 0 ± c ) × 10 b 0 x_{0}\pm x_{1}=(a_{0}\pm c)\times 10^{b_{0}}
  11. 2.34 × 10 - 5 + 5.67 × 10 - 6 = 2.34 × 10 - 5 + 0.567 × 10 - 5 2.907 × 10 - 5 2.34\times 10^{-5}+5.67\times 10^{-6}=2.34\times 10^{-5}+0.567\times 10^{-5}% \approx 2.907\times 10^{-5}

Scrambler.html

  1. 1 + x - 14 + x - 15 1+x^{-14}+x^{-15}
  2. 1 + x - 18 + x - 23 1+x^{-18}+x^{-23}

Sedenion.html

  1. 𝕊 \mathbb{S}
  2. 𝕊 \mathbb{S}
  3. x n x^{n}
  4. x = x 0 e 0 + x 1 e 1 + x 2 e 2 + + x 14 e 14 + x 15 e 15 , x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\ldots+x_{14}e_{14}+x_{15}e_{15},\,
  5. × \mathbf{×}
  6. e 0 e i = e i e 0 = e i , e_{0}e_{i}=e_{i}e_{0}=e_{i},
  7. e i e i = - e 0 for i 0 , e_{i}e_{i}=-e_{0}\,\,\,\text{for}\,\,i\neq 0,
  8. e i e j = - e j e i for i j and i , j 0. e_{i}e_{j}=-e_{j}e_{i}\,\,\,\text{for}\,\,i\neq j\,\,\,\text{and}\,\,i,j\neq 0.
  9. e i ( e j e k ) = - ( e i e j ) e k for i j , i , j 0 and e i e j \plusmn e k . e_{i}(e_{j}e_{k})=-(e_{i}e_{j})e_{k}\,\,\,\text{for}\,\,i\neq j,\,\,i,j\neq 0% \,\,\,\text{and}\,\,e_{i}e_{j}\neq\plusmn e_{k}.

Sediment.html

  1. 𝐑𝐨𝐮𝐬𝐞 = Settling velocity Upwards velocity from lift and drag = w s κ u * \,\textbf{Rouse}=\frac{\,\text{Settling velocity}}{\,\text{Upwards velocity % from lift and drag}}=\frac{w_{s}}{\kappa u_{*}}
  2. w s w_{s}
  3. κ \kappa
  4. u * u_{*}

Self-similarity.html

  1. { f s : s S } \{f_{s}:s\in S\}
  2. X = s S f s ( X ) X=\bigcup_{s\in S}f_{s}(X)
  3. X Y X\subset Y
  4. { f s : s S } \{f_{s}:s\in S\}
  5. 𝔏 = ( X , S , { f s : s S } ) \mathfrak{L}=(X,S,\{f_{s}:s\in S\})

Semi-continuity.html

  1. f ( x ) = x f(x)=\lfloor x\rfloor
  2. f ( x ) = x f(x)=\lceil x\rceil
  3. f ( x ) = { 1 if x < 1 , 2 if x = 1 , 1 / 2 if x > 1 , f(x)=\begin{cases}1&\mbox{if }~{}x<1,\\ 2&\mbox{if }~{}x=1,\\ 1/2&\mbox{if }~{}x>1,\end{cases}
  4. f ( x ) = { sin ( 1 / x ) if x 0 , 1 if x = 0 , f(x)=\begin{cases}\sin(1/x)&\mbox{if }~{}x\neq 0,\\ 1&\mbox{if }~{}x=0,\end{cases}
  5. X = n X=\mathbb{R}^{n}
  6. Γ = C ( [ 0 , 1 ] , X ) \Gamma=C([0,1],X)
  7. X X
  8. d Γ ( α , β ) = sup t d X ( α ( t ) , β ( t ) ) d_{\Gamma}(\alpha,\beta)=\sup_{t}\ d_{X}(\alpha(t),\beta(t))
  9. L : Γ [ 0 , + ] L:\Gamma\to[0,+\infty]
  10. α \alpha
  11. L ( α ) L(\alpha)
  12. ( X , μ ) (X,\mu)
  13. L + ( X , μ ) L^{+}(X,\mu)
  14. μ \mu
  15. L + ( X , μ ) L^{+}(X,\mu)
  16. [ - , + ] [-\infty,+\infty]
  17. lim sup x x 0 f ( x ) f ( x 0 ) \limsup_{x\to x_{0}}f(x)\leq f(x_{0})
  18. lim inf x x 0 f ( x ) f ( x 0 ) \liminf_{x\to x_{0}}f(x)\geq f(x_{0})
  19. f ( x ) = sup i I f i ( x ) , x X . f(x)=\sup_{i\in I}f_{i}(x),\qquad x\in X.

Semicolon.html

  1. f ( x 1 , x 2 , ; a 1 , a 2 , ) f(x_{1},x_{2},\dots;a_{1},a_{2},\dots)

Semidirect_product.html

  1. G = N H , G=N\rtimes H,
  2. G G
  3. N N
  4. H H
  5. A u t ( N ) Aut(N)
  6. N N
  7. φ : H A u t ( N ) φ:H→Aut(N)
  8. h h
  9. H H
  10. n n
  11. N N
  12. N N
  13. N N
  14. H H
  15. φ φ
  16. G G
  17. N N
  18. H H
  19. φ \varphi
  20. H A u t ( N ) H→Aut(N)
  21. N φ H N\rtimes_{\varphi}H
  22. N N
  23. H H
  24. φ φ
  25. N φ H N\rtimes_{\varphi}H
  26. N × H N×H
  27. N φ H N\rtimes_{\varphi}H
  28. φ \varphi
  29. * : ( N × H ) × ( N × H ) N φ H *\colon(N\times H)\times(N\times H)\to N\rtimes_{\varphi}H
  30. ( n 1 , h 1 ) * ( n 2 , h 2 ) = ( n 1 φ h 1 ( n 2 ) , h 1 h 2 ) (n_{1},h_{1})*(n_{2},h_{2})=(n_{1}\varphi_{h_{1}}(n_{2}),h_{1}h_{2})
  31. n < s u b > 1 , n 2 n<sub>1,n_{2}
  32. ( n , h ) (n,h)
  33. H H
  34. G G
  35. N N
  36. H H
  37. g g
  38. G G
  39. g = n h g=nh
  40. n n
  41. N N
  42. h h
  43. H H
  44. φ : H A u t ( N ) φ:H→Aut(N)
  45. φ h ( n ) = h n h - 1 \varphi_{h}(n)=hnh^{-1}
  46. n N , h H n∈N,h∈H
  47. G G
  48. N φ H N\rtimes_{\varphi}H
  49. n h nh
  50. ( n , h ) (n,h)
  51. G G
  52. ( n 1 h 1 ) ( n 2 h 2 ) = n 1 h 1 n 2 h 1 - 1 h 1 h 2 = ( n 1 φ h 1 ( n 2 ) ) ( h 1 h 2 ) (n_{1}h_{1})(n_{2}h_{2})=n_{1}h_{1}n_{2}h_{1}^{-1}h_{1}h_{2}=(n_{1}\varphi_{h_% {1}}(n_{2}))(h_{1}h_{2})
  53. N φ H N\rtimes_{\varphi}H
  54. φ φ
  55. H H
  56. N N
  57. N φ H N\rtimes_{\varphi}H
  58. N × H N\times H
  59. G G
  60. N N
  61. H H
  62. 1 N β G α H 1 1\longrightarrow N\longrightarrow^{\!\!\!\!\!\!\!\!\!\beta}\ \,G% \longrightarrow^{\!\!\!\!\!\!\!\!\!\alpha}\ \,H\longrightarrow 1
  63. γ : H G γ:H→G
  64. H H
  65. φ : H A u t ( N ) φ:H→Aut(N)
  66. φ h ( n ) = β - 1 ( γ ( h ) β ( n ) γ ( h - 1 ) ) . \varphi_{h}(n)=\beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).
  67. a , b a 2 = e , b n = e , a b a - 1 = b - 1 . \langle a,\;b\mid a^{2}=e,\;b^{n}=e,\;aba^{-1}=b^{-1}\rangle.
  68. C m C_{m}\;
  69. a a\;
  70. C n C_{n}\;
  71. b b\;
  72. a b a - 1 = b k aba^{-1}=b^{k}\;
  73. k k\;
  74. n n\;
  75. a , b a m = e , b n = e , a b a - 1 = b k . \langle a,\;b\mid a^{m}=e,\;b^{n}=e,\;aba^{-1}=b^{k}\;\rangle.
  76. r r\;
  77. m m\;
  78. a r a^{r}\;
  79. C m C_{m}\;
  80. a r b a - r = b k r a^{r}ba^{-r}=b^{k^{r}}\;
  81. a , b a m = e , b n = e , a b a - 1 = b k r \langle a,\;b\mid a^{m}=e,\;b^{n}=e,\;aba^{-1}=b^{k^{r}}\;\rangle
  82. a , b a b a - 1 = b - 1 \langle a,\;b\mid aba^{-1}=b^{-1}\;\rangle
  83. \mathbb{Z}
  84. \mathbb{Z}
  85. φ : Aut ( ) \varphi:\mathbb{Z}\to\mathrm{Aut}(\mathbb{Z})
  86. φ ( h ) ( n ) = ( - 1 ) h n \varphi(h)(n)=(-1)^{h}n
  87. φ : O ( 2 ) Aut ( 2 ) \varphi:O(2)\to\mathrm{Aut}(\mathbb{R}^{2})
  88. φ ( h ) ( n ) = h n \varphi(h)(n)=hn
  89. 𝕂 \mathbb{K}
  90. Γ L ( V ) \operatorname{\Gamma L}(V)
  91. GL ( V ) \operatorname{GL}(V)
  92. Γ L ( V ) \operatorname{\Gamma L}(V)
  93. 𝕂 \mathbb{K}
  94. G G
  95. X X
  96. π 1 ( X ) \pi_{1}(X)
  97. π 1 ( X ) G \pi_{1}(X)\rtimes G
  98. X / G X/G
  99. N H N\rtimes H
  100. H N H\ltimes N
  101. ϕ : H Aut ( N ) \phi:H\rightarrow\operatorname{Aut}(N)
  102. N ϕ H N\rtimes_{\phi}H
  103. N H N\rtimes H
  104. \triangleleft
  105. × \times

Semigroup.html

  1. S S
  2. \cdot
  3. : S × S S \cdot:S\times S\rightarrow S
  4. a , b , c S a,b,c\in S
  5. ( a b ) c = a ( b c ) (a\cdot b)\cdot c=a\cdot(b\cdot c)
  6. e S e\notin S
  7. S S
  8. e s = s e = s e\cdot s=s\cdot e=s
  9. s S { e } s\in S\cup\{e\}
  10. S S
  11. S 1 S^{1}
  12. f : S 0 S 1 f:S_{0}\to S_{1}
  13. f f
  14. S 0 S_{0}
  15. e 0 e_{0}
  16. f ( e 0 ) f(e_{0})
  17. f f
  18. S 1 S_{1}
  19. e 1 e_{1}
  20. e 1 e_{1}
  21. f f
  22. f ( e 0 ) = e 1 f(e_{0})=e_{1}
  23. f f
  24. f f
  25. \sim
  26. S × S \sim\;\subseteq S\times S
  27. x y x\sim y\,
  28. u v u\sim v\,
  29. x u y v xu\sim yv\,
  30. x , y , u , v x,y,u,v
  31. \sim
  32. [ a ] = { x S | x a } [a]_{\sim}=\{x\in S|\;x\sim a\}
  33. \circ
  34. [ u ] [ v ] = [ u v ] [u]_{\sim}\circ[v]_{\sim}=[uv]_{\sim}
  35. \sim
  36. \sim
  37. \circ
  38. S / S/\sim
  39. x [ x ] x\mapsto[x]_{\sim}
  40. [ 1 ] [1]_{\sim}
  41. ( L , ) (L,\leq)
  42. a , b L a,b\in L
  43. a b a\wedge b
  44. \wedge
  45. L L
  46. a a = a a\wedge a=a
  47. f : S L f:S\to L
  48. S a = f - 1 { a } S_{a}=f^{-1}\{a\}
  49. S S
  50. L L
  51. S a S b S a b S_{a}S_{b}\subseteq S_{a\wedge b}
  52. f f
  53. L L
  54. S S
  55. \sim
  56. x y x\sim y
  57. f ( x ) = f ( y ) f(x)=f(y)
  58. S S
  59. \sim
  60. S S
  61. L L
  62. f f
  63. S S
  64. L L
  65. S S
  66. S a S_{a}
  67. x , y x,y
  68. z z
  69. n > 0 n>0
  70. x n = y z x^{n}=yz
  71. L L
  72. f ( x ) f ( y ) f(x)\leq f(y)
  73. x n = y z x^{n}=yz
  74. z z
  75. n > 0 n>0
  76. { t u ( t , x ) = x 2 u ( t , x ) , x ( 0 , 1 ) , t > 0 ; u ( t , x ) = 0 , x { 0 , 1 } , t > 0 ; u ( t , x ) = u 0 ( x ) , x ( 0 , 1 ) , t = 0. \begin{cases}\partial_{t}u(t,x)=\partial_{x}^{2}u(t,x),&x\in(0,1),t>0;\\ u(t,x)=0,&x\in\{0,1\},t>0;\\ u(t,x)=u_{0}(x),&x\in(0,1),t=0.\end{cases}
  77. D ( A ) = { u H 2 ( ( 0 , 1 ) ; 𝐑 ) | u ( 0 ) = u ( 1 ) = 0 } , D(A)=\big\{u\in H^{2}((0,1);\mathbf{R})\big|u(0)=u(1)=0\big\},
  78. { u ˙ ( t ) = A u ( t ) ; u ( 0 ) = u 0 . \begin{cases}\dot{u}(t)=Au(t);\\ u(0)=u_{0}.\end{cases}

Semivariance.html

  1. γ ^ ( h ) = 1 2 1 n ( h ) i = 1 n ( h ) ( z ( x i + h ) - z ( x i ) ) 2 \hat{\gamma}(h)=\frac{1}{2}\cdot\frac{1}{n(h)}\sum_{i=1}^{n(h)}(z(x_{i}+h)-z(x% _{i}))^{2}
  2. z ( x i + h ) - z ( x i ) z(x_{i}+h)-z(x_{i})
  3. 2 γ ^ ( h ) 2\hat{\gamma}(h)
  4. γ ^ ( h ) \hat{\gamma}(h)

Separable_space.html

  1. { x n } n = 1 \{x_{n}\}_{n=1}^{\infty}
  2. ( r 1 , , r n ) n (r_{1},\ldots,r_{n})\in\mathbb{R}^{n}
  3. r i r_{i}
  4. n \mathbb{R}^{n}
  5. n n
  6. n n
  7. { U n } \scriptstyle\{U_{n}\}
  8. x n U n \scriptstyle x_{n}\in U_{n}
  9. U n \scriptstyle U_{n}
  10. 𝔠 \mathfrak{c}
  11. X X
  12. 2 𝔠 2^{\mathfrak{c}}
  13. 𝔠 \mathfrak{c}
  14. Y X Y\subseteq X
  15. z X z\in X
  16. z Y ¯ z\in\overline{Y}
  17. \mathcal{B}
  18. Y Y
  19. z z
  20. S ( Y ) S(Y)
  21. 2 2 | Y | 2^{2^{|Y|}}
  22. S ( Y ) X S(Y)\rightarrow X
  23. Y ¯ = X . \overline{Y}=X.
  24. X X
  25. κ \kappa
  26. X X
  27. 2 2 κ 2^{2^{\kappa}}
  28. 2 κ 2^{\kappa}
  29. \mathbb{R}^{\mathbb{R}}
  30. 2 𝔠 2^{\mathfrak{c}}
  31. κ \kappa
  32. 2 κ 2^{\kappa}
  33. κ \kappa
  34. κ \kappa
  35. C ( K ) C(K)
  36. K K\subseteq\mathbb{R}
  37. \mathbb{R}
  38. L p ( X , μ ) L^{p}\left(X,\mu\right)
  39. X , , μ \left\langle X,\mathcal{M},\mu\right\rangle
  40. 1 p < 1\leq p<\infty
  41. n n
  42. C ( [ 0 , 1 ] ) C([0,1])
  43. [ 0 , 1 ] [0,1]
  44. [ x ] \mathbb{Q}[x]
  45. C ( [ 0 , 1 ] ) C([0,1])
  46. C ( [ 0 , 1 ] ) C([0,1])
  47. 2 \ell^{2}
  48. 𝕊 \mathbb{S}
  49. ω 1 \omega_{1}
  50. \ell^{\infty}
  51. L L^{\infty}
  52. 𝒞 ( X , ) \mathcal{C}(X,\mathbb{R})
  53. α α
  54. C ( 0 , 11 < s u p > α , 𝐑 ) C(0,11<sup>α,\mathbf{R})

Septuagint.html

  1. 𝔊 \mathfrak{G}

Sequence.html

  1. a 1 1st element a 2 2nd element a 3 3rd element a n - 1 (n-1)th element a n nth element a n + 1 (n+1)th element \begin{aligned}\displaystyle a_{1}&\displaystyle\leftrightarrow&\displaystyle% \,\text{ 1st element}\\ \displaystyle a_{2}&\displaystyle\leftrightarrow&\displaystyle\,\text{ 2nd % element }\\ \displaystyle a_{3}&\displaystyle\leftrightarrow&\displaystyle\,\text{ 3rd % element }\\ \displaystyle\vdots&&\displaystyle\vdots\\ \displaystyle a_{n-1}&\displaystyle\leftrightarrow&\displaystyle\,\text{ (n-1)% th element}\\ \displaystyle a_{n}&\displaystyle\leftrightarrow&\displaystyle\,\text{ nth % element}\\ \displaystyle a_{n+1}&\displaystyle\leftrightarrow&\displaystyle\,\text{ (n+1)% th element}\\ \displaystyle\vdots&&\displaystyle\vdots\end{aligned}
  2. ( a 1 , a 2 , , a 10 ) , a k = k 2 . (a_{1},a_{2},...,a_{10}),\qquad a_{k}=k^{2}.
  3. ( a k ) k = 1 10 , a k = k 2 . (a_{k})_{k=1}^{10},\qquad a_{k}=k^{2}.
  4. ( a k ) k = 1 , a k = k 2 . (a_{k})_{k=1}^{\infty},\qquad a_{k}=k^{2}.
  5. ( a k ) k = 0 = ( a 0 , a 1 , a 2 , ) . (a_{k})_{k=0}^{\infty}=(a_{0},a_{1},a_{2},...).
  6. ( 1 , 9 , 25 , ) (1,9,25,...)
  7. ( a 1 , a 3 , a 5 , ) , a k = k 2 (a_{1},a_{3},a_{5},...),\qquad a_{k}=k^{2}
  8. ( a 2 k - 1 ) k = 1 , a k = k 2 (a_{2k-1})_{k=1}^{\infty},\qquad a_{k}=k^{2}
  9. ( a k ) k = 1 , a k = ( 2 k - 1 ) 2 (a_{k})_{k=1}^{\infty},\qquad a_{k}=(2k-1)^{2}
  10. ( ( 2 k - 1 ) 2 ) k = 1 ((2k-1)^{2})_{k=1}^{\infty}
  11. ( a k ) k 𝐍 (a_{k})_{k\in\mathbf{N}}
  12. a n = a n - 1 + a n - 2 a_{n}=a_{n-1}+a_{n-2}
  13. a 0 = 0 a_{0}=0
  14. a 1 = 1 a_{1}=1
  15. a 0 = 0 a_{0}=0
  16. a n = a n - 1 - n a_{n}=a_{n-1}-n
  17. a n = a n - 1 + n a_{n}=a_{n-1}+n
  18. a n + 1 = f ( a n ) a_{n+1}=f(a_{n})
  19. ( 2 n ) n = - (2n)_{n=-\infty}^{\infty}
  20. ( a n ) n = 1 (a_{n})_{n=1}^{\infty}
  21. lim n a n = L . \lim_{n\to\infty}a_{n}=L.
  22. lim n a n = . \lim_{n\to\infty}a_{n}=\infty.
  23. lim n a n = - \lim_{n\to\infty}a_{n}=-\infty
  24. ( a n ) n = 1 (a_{n})_{n=1}^{\infty}
  25. ϵ > 0 , N 𝐍 s.t. n N , | a n - L | < ϵ . \forall\epsilon>0,\exists N\in\mathbf{N}\,\text{ s.t. }\forall n\geq N,|a_{n}-% L|<\epsilon.
  26. z * z \sqrt{z^{*}z}
  27. lim n ( a n ± b n ) = lim n a n ± lim n b n \lim_{n\to\infty}(a_{n}\pm b_{n})=\lim_{n\to\infty}a_{n}\pm\lim_{n\to\infty}b_% {n}
  28. lim n c a n = c lim n a n \lim_{n\to\infty}ca_{n}=c\lim_{n\to\infty}a_{n}
  29. lim n ( a n b n ) = ( lim n a n ) ( lim n b n ) \lim_{n\to\infty}(a_{n}b_{n})=(\lim_{n\to\infty}a_{n})(\lim_{n\to\infty}b_{n})
  30. lim n a n b n = lim n a n lim n b n \lim_{n\to\infty}\frac{a_{n}}{b_{n}}=\frac{\lim_{n\to\infty}a_{n}}{\lim_{n\to% \infty}b_{n}}
  31. lim n b n 0 \lim_{n\to\infty}b_{n}\neq 0
  32. lim n a n p = [ lim n a n ] p \lim_{n\to\infty}a_{n}^{p}=\left[\lim_{n\to\infty}a_{n}\right]^{p}
  33. lim n a n lim n b n \lim_{n\to\infty}a_{n}\leq\lim_{n\to\infty}b_{n}
  34. a n c n b n a_{n}\leq c_{n}\leq b_{n}
  35. lim n a n = lim n b n = L \lim_{n\to\infty}a_{n}=\lim_{n\to\infty}b_{n}=L
  36. lim n c n = L \lim_{n\to\infty}c_{n}=L
  37. S 1 \displaystyle S_{1}
  38. S N = n = 1 N a n . S_{N}=\sum_{n=1}^{N}a_{n}.
  39. lim N S N = n = 1 a n . \lim_{N\to\infty}S_{N}=\sum_{n=1}^{\infty}a_{n}.
  40. { X i } \{X_{i}\}
  41. X := i I X i , X:=\prod_{i\in I}X_{i},
  42. { x i } \{x_{i}\}
  43. x i x_{i}
  44. X i X_{i}
  45. ( x 1 , x 2 , x 3 , ) or ( x 0 , x 1 , x 2 , ) (x_{1},x_{2},x_{3},\dots)\,\text{ or }(x_{0},x_{1},x_{2},\dots)\,
  46. G 0 f 1 G 1 f 2 G 2 f 3 f n G n G_{0}\;\xrightarrow{f_{1}}\;G_{1}\;\xrightarrow{f_{2}}\;G_{2}\;\xrightarrow{f_% {3}}\;\cdots\;\xrightarrow{f_{n}}\;G_{n}
  47. im ( f k ) = ker ( f k + 1 ) \mathrm{im}(f_{k})=\mathrm{ker}(f_{k+1})

Series_(mathematics).html

  1. n = 1 1 2 n = 1 2 + 1 4 + 1 8 + . \sum_{n=1}^{\infty}\frac{1}{2^{n}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots.
  2. { a n } \{a_{n}\}
  3. n = 0 a n = a 0 + a 1 + a 2 + \sum_{n=0}^{\infty}a_{n}=a_{0}+a_{1}+a_{2}+\cdots
  4. { S k } \{S_{k}\}
  5. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  6. k k
  7. { a n } \{a_{n}\}
  8. a 0 a_{0}
  9. a k a_{k}
  10. S k = n = 0 k a n = a 0 + a 1 + + a k . S_{k}=\sum_{n=0}^{k}a_{n}=a_{0}+a_{1}+\cdots+a_{k}.
  11. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  12. L L
  13. { S k } \{S_{k}\}
  14. L L
  15. L = n = 0 a n L = lim k S k . L=\sum_{n=0}^{\infty}a_{n}\Leftrightarrow L=\lim_{k\rightarrow\infty}S_{k}.
  16. a : I G a:I\mapsto G
  17. a a
  18. a ( x ) G a(x)\in G
  19. x I x\in I
  20. x I a ( x ) . \sum_{x\in I}a(x).
  21. I = I=\mathbb{N}
  22. a : G a:\mathbb{N}\mapsto G
  23. a ( n ) = a n a(n)=a_{n}
  24. n \sum_{n\in\mathbb{N}}
  25. n = 0 \sum_{n=0}^{\infty}
  26. n = 0 a n = a 0 + a 1 + a 2 + . \sum_{n=0}^{\infty}a_{n}=a_{0}+a_{1}+a_{2}+\cdots.
  27. G G
  28. { S k } G \{S_{k}\}\subset G
  29. { a n } G \{a_{n}\}\subset G
  30. k k
  31. a 0 , a 1 , , a k a_{0},a_{1},\cdots,a_{k}
  32. S k = n = 0 k a n = a 0 + a 1 + + a k . S_{k}=\sum_{n=0}^{k}a_{n}=a_{0}+a_{1}+\cdots+a_{k}.
  33. G G
  34. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  35. L G L\in G
  36. { S k } \{S_{k}\}
  37. L L
  38. L = n = 0 a n L = lim k S k . L=\sum_{n=0}^{\infty}a_{n}\Leftrightarrow L=\lim_{k\rightarrow\infty}S_{k}.
  39. n = 0 a n = lim N S N = lim N n = 0 N a n . \sum_{n=0}^{\infty}a_{n}=\lim_{N\to\infty}S_{N}=\lim_{N\to\infty}\sum_{n=0}^{N% }a_{n}.
  40. 1 + 1 2 + 1 4 + 1 8 + + 1 2 n + . 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^{n}}+\cdots.
  41. S / 2 = 1 + 1 2 + 1 4 + 1 8 + 2 = 1 2 + 1 4 + 1 8 + 1 16 + . S/2=\frac{1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots}{2}=\frac{1}{2}+\frac{1% }{4}+\frac{1}{8}+\frac{1}{16}+\cdots.
  42. S - S / 2 = 1 S = 2. S-S/2=1\Rightarrow S=2.\,\!
  43. x = 0.111 x=0.111\dots\,
  44. n = 1 1 10 n . \sum_{n=1}^{\infty}\frac{1}{10^{n}}.
  45. 1 + 1 2 + 1 4 + 1 8 + 1 16 + = n = 0 1 2 n . 1+{1\over 2}+{1\over 4}+{1\over 8}+{1\over 16}+\cdots=\sum_{n=0}^{\infty}{1% \over 2^{n}}.
  46. n = 0 z n \sum_{n=0}^{\infty}z^{n}
  47. | z | . |z|.
  48. 3 + 5 2 + 7 4 + 9 8 + 11 16 + = n = 0 ( 3 + 2 n ) 2 n . 3+{5\over 2}+{7\over 4}+{9\over 8}+{11\over 16}+\cdots=\sum_{n=0}^{\infty}{(3+% 2n)\over 2^{n}}.
  49. 1 + 1 2 + 1 3 + 1 4 + 1 5 + = n = 1 1 n . 1+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+\cdots=\sum_{n=1}^{\infty}{1% \over n}.
  50. 1 - 1 2 + 1 3 - 1 4 + 1 5 - = n = 1 ( - 1 ) n + 1 1 n = ln ( 2 ) . 1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-\cdots=\sum_{n=1}^{\infty}(-1)^{% n+1}{1\over n}=\ln(2).
  51. n = 1 1 n r \sum_{n=1}^{\infty}\frac{1}{n^{r}}
  52. n = 1 ( b n - b n + 1 ) \sum_{n=1}^{\infty}(b_{n}-b_{n+1})
  53. 0 x 1 d t = x . \int_{0}^{x}1\,dt=x.
  54. n 1 1 n 2 \sum_{n\geq 1}\frac{1}{n^{2}}
  55. 1 n 2 1 n - 1 - 1 n , n 2 , \frac{1}{n^{2}}\leq\frac{1}{n-1}-\frac{1}{n},\quad n\geq 2,
  56. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  57. n = 0 | a n | \sum_{n=0}^{\infty}\left|a_{n}\right|
  58. n = 1 ( - 1 ) n + 1 n = 1 - 1 2 + 1 3 - 1 4 + 1 5 - \sum\limits_{n=1}^{\infty}{(-1)^{n+1}\over n}=1-{1\over 2}+{1\over 3}-{1\over 4% }+{1\over 5}-\cdots
  59. a n = λ n b n \sum a_{n}=\sum\lambda_{n}b_{n}
  60. sup N | n = 0 N b n | < , | λ n + 1 - λ n | < and λ n B n converges, \sup_{N}\Bigl|\sum_{n=0}^{N}b_{n}\Bigr|<\infty,\ \ \sum|\lambda_{n+1}-\lambda_% {n}|<\infty\ \,\text{and}\ \lambda_{n}B_{n}\ \,\text{converges,}
  61. n = 2 sin ( n x ) ln n \sum_{n=2}^{\infty}\frac{\sin(nx)}{\ln n}
  62. ( λ n - λ n + 1 ) B n . \sum(\lambda_{n}-\lambda_{n+1})\,B_{n}.
  63. n = 0 f n ( x ) \sum_{n=0}^{\infty}f_{n}(x)
  64. s N ( x ) = n = 0 N f n ( x ) s_{N}(x)=\sum_{n=0}^{N}f_{n}(x)
  65. | s N ( x ) - f ( x ) | |s_{N}(x)-f(x)|
  66. E | s N ( x ) - f ( x ) | 2 d x 0 \int_{E}\left|s_{N}(x)-f(x)\right|^{2}\,dx\to 0
  67. n = 0 a n ( x - c ) n . \sum_{n=0}^{\infty}a_{n}(x-c)^{n}.
  68. n = 0 x n n ! \sum_{n=0}^{\infty}\frac{x^{n}}{n!}
  69. e x e^{x}
  70. n = - a n x n . \sum_{n=-\infty}^{\infty}a_{n}x^{n}.
  71. n = 1 a n n s , \sum_{n=1}^{\infty}{a_{n}\over n^{s}},
  72. ζ ( s ) = n = 1 1 n s . \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.
  73. 𝐂 { 1 } \mathbf{C}\setminus\{1\}
  74. 1 2 A 0 + n = 1 ( A n cos n x + B n sin n x ) . \tfrac{1}{2}A_{0}+\sum_{n=1}^{\infty}\left(A_{n}\cos nx+B_{n}\sin nx\right).
  75. 1 + α β 1 γ x + α ( α + 1 ) β ( β + 1 ) 1 2 γ ( γ + 1 ) x 2 + 1+\frac{\alpha\beta}{1\cdot\gamma}x+\frac{\alpha(\alpha+1)\beta(\beta+1)}{1% \cdot 2\cdot\gamma(\gamma+1)}x^{2}+\cdots
  76. 1 + m 1 ! x + m ( m - 1 ) 2 ! x 2 + 1+\frac{m}{1!}x+\frac{m(m-1)}{2!}x^{2}+\cdots
  77. m m
  78. x x
  79. F ( x ) = 1 n + 2 n + + ( x - 1 ) n . F(x)=1^{n}+2^{n}+\cdots+(x-1)^{n}.\,
  80. x - n = 0 N x n 0 \biggl\|x-\sum_{n=0}^{N}x_{n}\biggr\|\to 0
  81. i I a i = sup { i A a i | A finite, A I } [ 0 , + ] . \sum_{i\in I}a_{i}=\sup\Bigl\{\sum_{i\in A}a_{i}\,\big|A\,\text{ finite, }A% \subset I\Bigr\}\in[0,+\infty].
  82. A n = { i I : a i > 1 / n } \scriptstyle A_{n}=\{i\in I\,:\,a_{i}>1/n\}
  83. 1 n card ( A n ) i A n a i i I a i < . \frac{1}{n}\,\textrm{card}(A_{n})\leq\sum_{i\in A_{n}}a_{i}\leq\sum_{i\in I}a_% {i}<\infty.
  84. i I a i = k = 0 + a i k , \sum_{i\in I}a_{i}=\sum_{k=0}^{+\infty}a_{i_{k}},
  85. S = i I a i = lim { i A a i | A F } S=\sum_{i\in I}a_{i}=\lim\Bigl\{\sum_{i\in A}a_{i}\,\big|A\in F\Bigr\}
  86. S - i A a i V , A A 0 . S-\sum_{i\in A}a_{i}\in V,\quad A\supset A_{0}.
  87. i A 1 a i - i A 2 a i W , A 1 , A 2 A 0 . \sum_{i\in A_{1}}a_{i}-\sum_{i\in A_{2}}a_{i}\in W,\quad A_{1},A_{2}\supset A_% {0}.
  88. n = 0 a n = n 𝐍 a n . \sum_{n=0}^{\infty}a_{n}=\sum_{n\in\mathbf{N}}a_{n}.
  89. n = 0 a σ ( n ) = n = 0 a n . \sum_{n=0}^{\infty}a_{\sigma(n)}=\sum_{n=0}^{\infty}a_{n}.
  90. n = 0 ε n a n \sum_{n=0}^{\infty}\varepsilon_{n}a_{n}
  91. n 𝐍 a n < + . \sum_{n\in\mathbf{N}}\|a_{n}\|<+\infty.
  92. β < α + 1 a β = a α + β < α a β \sum_{\beta<\alpha+1}a_{\beta}=a_{\alpha}+\sum_{\beta<\alpha}a_{\beta}\,\!
  93. β < α a β = lim γ α β < γ a β \sum_{\beta<\alpha}a_{\beta}=\lim_{\gamma\to\alpha}\sum_{\beta<\gamma}a_{\beta}

Set_(game).html

  1. < m t p l > ( 81 2 ) 3 = 81 × 80 2 × 3 = 1080 \frac{<}{m}tpl>{{81\choose 2}}{3}=\frac{81\times 80}{2\times 3}=1080
  2. d d
  3. 4 - d 4-d
  4. ( 4 d ) 2 d 80 \frac{{4\choose d}2^{d}}{80}
  5. ( 81 12 ) = 81 ! 12 ! 69 ! = 70724320184700 7.07 × 10 13 {81\choose 12}=\frac{81!}{12!69!}=70724320184700\approx 7.07\times 10^{13}
  6. 1 - 78 79 × ( 77 78 ) 78 = 0.64 = 64 % 1-\frac{78}{79}\times(\frac{77}{78})^{78}=0.64=64\%
  7. ( 12 3 ) 1 79 2.78 {12\choose 3}\cdot\frac{1}{79}\approx 2.78
  8. ( 15 3 ) 1 79 5.76 {15\choose 3}\cdot\frac{1}{79}\approx 5.76

Set_(mathematics).html

  1. A Δ B = ( A B ) ( B A ) . A\,\Delta\,B=(A\setminus B)\cup(B\setminus A).
  2. | A 1 A 2 A 3 A n | = ( | A 1 | + | A 2 | + | A 3 | + | A n | ) - ( | A 1 A 2 | + | A 1 A 3 | + | A n - 1 A n | ) + + ( - 1 ) n - 1 ( | A 1 A 2 A 3 A n | ) \begin{aligned}\displaystyle\left|A_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n% }\right|=&\displaystyle\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}% \right|+\ldots\left|A_{n}\right|\right)-\\ &\displaystyle\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+% \ldots\left|A_{n-1}\cap A_{n}\right|\right)+\\ &\displaystyle\ldots+\\ &\displaystyle\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap% \ldots\cap A_{n}\right|\right)\end{aligned}

SHA-1.html

  1. S H A ( m e s s a g e | | k e y ) SHA(message||key)
  2. S H A ( k e y | | m e s s a g e ) SHA(key||message)
  3. S H A d ( m e s s a g e ) = S H A ( S H A ( 0 b | | m e s s a g e ) ) SHA_{d}(message)=SHA(SHA(0^{b}||message))
  4. 0 b 0^{b}

Shannon–Fano_coding.html

  1. - log P ( x ) {-\log}P(x)
  2. 1 2 n \textstyle\frac{1}{2^{n}}
  3. 2 bits ( 15 + 7 + 6 ) + 3 bits ( 6 + 5 ) 39 symbols 2.28 bits per symbol. \frac{2\,\,\text{bits}\cdot(15+7+6)+3\,\,\text{bits}\cdot(6+5)}{39\,\,\text{% symbols}}\approx 2.28\,\,\text{bits per symbol.}
  4. 1 bit 15 + 3 bits ( 7 + 6 + 6 + 5 ) 39 symbols 2.23 bits per symbol. \frac{1\,\,\text{bit}\cdot 15+3\,\,\text{bits}\cdot(7+6+6+5)}{39\,\,\text{% symbols}}\approx 2.23\,\,\text{bits per symbol.}

Sharp-P.html

  1. ( 1 - ϵ ) P ( a ) x ( 1 + ϵ ) P ( a ) (1-\epsilon)P(a)\leq x\leq(1+\epsilon)P(a)

Sheffer_stroke.html

  1. A B ~{}A\uparrow B
  2. \uparrow
  3. P Q P\uparrow Q
  4. \Leftrightarrow
  5. ¬ ( P and Q ) \neg(P\and Q)
  6. \Leftrightarrow
  7. ¬ \neg
  8. \uparrow
  9. ¬ P \neg P
  10. \Leftrightarrow
  11. P P
  12. \uparrow
  13. P P
  14. \Leftrightarrow
  15. \uparrow
  16. P Q P\rightarrow Q
  17. \Leftrightarrow
  18. P ~{}P
  19. \uparrow
  20. ( Q Q ) (Q\uparrow Q)
  21. \Leftrightarrow
  22. P ~{}P
  23. \uparrow
  24. ( P Q ) (P\uparrow Q)
  25. \Leftrightarrow
  26. \uparrow
  27. \Leftrightarrow
  28. \uparrow
  29. P and Q P\and Q
  30. \Leftrightarrow
  31. ( P Q ) (P\uparrow Q)
  32. \uparrow
  33. ( P Q ) (P\uparrow Q)
  34. \Leftrightarrow
  35. \uparrow
  36. P Q PQ
  37. \Leftrightarrow
  38. ( P P ) (P\uparrow P)
  39. \uparrow
  40. ( Q Q ) (Q\uparrow Q)
  41. \Leftrightarrow
  42. \uparrow
  43. \vdash
  44. \equiv

Ship.html

  1. knots 1.34 × L ft \mbox{knots}~{}\approx 1.34\times\sqrt{L\mbox{ft}~{}}
  2. knots 2.5 × L m \mbox{knots}~{}\approx 2.5\times\sqrt{L\mbox{m}~{}}

Shor's_algorithm.html

  1. N N
  2. d d
  3. 1 1
  4. N N
  5. N N
  6. N N
  7. N N
  8. 2 2
  9. N N
  10. N N
  11. k k
  12. N N
  13. k log 2 ( N ) k\leq\log_{2}(N)
  14. N = M k N=M^{k}
  15. M M
  16. k > 1 k>1
  17. N N
  18. 1 1
  19. 1 1
  20. N N
  21. 1 1
  22. - 1 -1
  23. b b
  24. 1 1
  25. - 1 -1
  26. b b
  27. N N
  28. b b
  29. a a
  30. a a
  31. N = 15 , a = 7 , r = 4 N=15,a=7,r=4
  32. g c d ( 7 2 ± 1 , 15 ) = g c d ( 49 ± 1 , 15 ) gcd(7^{2}\pm 1,15)=gcd(49\pm 1,15)
  33. g c d ( 48 , 15 ) = 3 gcd(48,15)=3
  34. g c d ( 50 , 15 ) = 5 gcd(50,15)=5
  35. N 2 Q < 2 N 2 N^{2}\leq Q<2N^{2}
  36. Q / r > N Q/r>N
  37. G G
  38. ϕ ( N ) \phi(N)
  39. a r 1 mod N . a^{r}\equiv 1\ \mbox{mod}~{}\ N.\,
  40. b a r / 2 ( mod N ) b\equiv a^{r/2}\;\;(\mathop{{\rm mod}}N)
  41. N N
  42. r r
  43. a a
  44. N N
  45. a r / 2 1 ( mod N ) a^{r/2}\not\equiv 1\;\;(\mathop{{\rm mod}}N)
  46. a a
  47. r / 2 r/2
  48. a r / 2 - 1 ( mod N ) a^{r/2}\equiv-1\;\;(\mathop{{\rm mod}}N)
  49. a a
  50. a a
  51. r r
  52. G G
  53. b a r / 2 1 , - 1 ( mod N ) b\equiv a^{r/2}\not\equiv 1,-1\;\;(\mathop{{\rm mod}}N)
  54. b b
  55. N N
  56. - 1 -1
  57. N N
  58. d = gcd ( b - 1 , N ) d=\operatorname{gcd}(b-1,N)
  59. N N
  60. d 1 , N d\neq 1,N
  61. d = N d=N
  62. N N
  63. b - 1 b-1
  64. b 1 ( mod N ) b\equiv 1\;\;(\mathop{{\rm mod}}N)
  65. b b
  66. d = gcd ( b - 1 , N ) = 1 d=\operatorname{gcd}(b-1,N)=1
  67. u , v u,v
  68. ( b - 1 ) u + N v = 1 (b-1)u+Nv=1
  69. b + 1 b+1
  70. ( b 2 - 1 ) u + N ( b + 1 ) v = b + 1 (b^{2}-1)u+N(b+1)v=b+1
  71. N N
  72. b 2 - 1 a r - 1 ( mod N ) b^{2}-1\equiv a^{r}-1\;\;(\mathop{{\rm mod}}N)
  73. N N
  74. b + 1 b+1
  75. b - 1 ( mod N ) b\equiv-1\;\;(\mathop{{\rm mod}}N)
  76. b b
  77. d d
  78. N N
  79. log N \log N
  80. q ( q - 1 ) / 2 = O ( ( log Q ) 2 ) q(q-1)/2=O((\log Q)^{2})
  81. e - 2 π i b y r / Q = 1 e^{-2\pi ibyr/Q}=1\,
  82. 1 / r 2 1/r^{2}
  83. f ( x 0 ) f(x_{0})
  84. n 3 n^{3}
  85. n n
  86. p p
  87. g g
  88. 1 < g < p - 1 1<g<p-1
  89. x = g r ( mod p ) x=g^{r}\;\;(\mathop{{\rm mod}}p)
  90. r = log g x ( mod p ) r=\log_{g}x\;\;(\mathop{{\rm mod}}p)
  91. ( p ) × × ( p ) × \left(\mathbb{Z}_{p}\right)^{\times}\times\left(\mathbb{Z}_{p}\right)^{\times}
  92. f ( a , b ) = g a x - b ( mod p ) . f(a,b)=g^{a}x^{-b}\;\;(\mathop{{\rm mod}}p).

Shortest_path_problem.html

  1. P = ( v 1 , v 2 , , v n ) V × V × × V P=(v_{1},v_{2},\ldots,v_{n})\in V\times V\times\ldots\times V
  2. v i v_{i}
  3. v i + 1 v_{i+1}
  4. 1 i < n 1\leq i<n
  5. P P
  6. n n
  7. v 1 v_{1}
  8. v n v_{n}
  9. v i v_{i}
  10. e i , j e_{i,j}
  11. v i v_{i}
  12. v j v_{j}
  13. f : E f:E\rightarrow\mathbb{R}
  14. G G
  15. v v
  16. v v^{\prime}
  17. P = ( v 1 , v 2 , , v n ) P=(v_{1},v_{2},\ldots,v_{n})
  18. v 1 = v v_{1}=v
  19. v n = v v_{n}=v^{\prime}
  20. n n
  21. i = 1 n - 1 f ( e i , i + 1 ) . \sum_{i=1}^{n-1}f(e_{i,i+1}).
  22. f : E { 1 } f:E\rightarrow\{1\}
  23. Θ ( E + V ) Θ(E+V)
  24. l o g ¯ L \overline{log}{L}
  25. v v
  26. v v
  27. i j A w i j x i j \sum_{ij\in A}w_{ij}x_{ij}
  28. x 0 x\geq 0
  29. j x i j - j x j i = { 1 , if i = s ; - 1 , if i = t ; 0 , otherwise. \sum_{j}x_{ij}-\sum_{j}x_{ji}=\begin{cases}1,&\,\text{if }i=s;\\ -1,&\,\text{if }i=t;\\ 0,&\,\text{ otherwise.}\end{cases}
  30. x i j x_{ij}
  31. w i j = w i j - y j + y i w^{\prime}_{ij}=w_{ij}-y_{j}+y_{i}

Shot_noise.html

  1. SNR = N N = N . \mathrm{SNR}=\frac{N}{\sqrt{N}}=\sqrt{N}.\,
  2. N \sqrt{N}
  3. ω \omega
  4. S ( ω ) = 2 e | I | , S(\omega)=2e|I|\ ,
  5. e e
  6. I I
  7. T n T_{n}
  8. n n
  9. I = e 2 π V n T n , I=\frac{e^{2}}{\pi\hbar}V\sum_{n}T_{n}\ ,
  10. V V
  11. S = 2 e 3 π | V | n T n , S=\frac{2e^{3}}{\pi\hbar}|V|\sum_{n}T_{n}\ ,
  12. S P S_{P}
  13. S = 2 e 3 π | V | n T n ( 1 - T n ) . S=\frac{2e^{3}}{\pi\hbar}|V|\sum_{n}T_{n}(1-T_{n})\ .
  14. F = S / S P F=S/S_{P}
  15. T n = 1 T_{n}=1
  16. T n = 0 T_{n}=0
  17. σ i = 2 q I Δ f \sigma_{i}=\sqrt{2\,q\,I\,\Delta f}
  18. σ i = 0.18 nA . \sigma_{i}=0.18\,\mathrm{nA}\;.
  19. σ v = σ i R \sigma_{v}=\sigma_{i}\,R
  20. P = 1 2 q I Δ f R . P={\frac{1}{2}}\,q\,I\,\Delta fR.
  21. ( Δ I ) 2 = def ( I - I ) 2 I . (\Delta I)^{2}\ \stackrel{\mathrm{def}}{=}\ \langle\left(I-\langle I\rangle% \right)^{2}\rangle\propto I.

Shotgun_sequencing.html

  1. N × L / G N\times L/G

Shuffling.html

  1. 1 , 2 , 3 , 4 , 5 , 6 , , 2 n \scriptstyle 1,2,3,4,5,6,\dots,2n
  2. 2 n , 2 n - 2 , 2 n - 4 , , 4 , 2 , 1 , 3 , , 2 n - 3 , 2 n - 1 \scriptstyle 2n,2n-2,2n-4,\dots,4,2,1,3,\dots,2n-3,2n-1
  3. × 10 6 7 \times 10^{6}7

Sidereal_time.html

  1. d R A dRA
  2. d ψ d\psi
  3. ϵ \epsilon
  4. d R A = d ψ * c o s ( ϵ ) dRA=d\psi*cos(\epsilon)
  5. 365.242 190 402 366.242 190 402 \tfrac{365.242\ 190\ 402}{366.242\ 190\ 402}
  6. number of sidereal days per orbital period = 1 + number of solar days per orbital period \,\text{number of sidereal days per orbital period}=1+\,\text{number of solar % days per orbital period}
  7. length of solar day = length of sidereal day 1 - length of sidereal day orbital period . \,\text{length of solar day}=\frac{\,\text{length of sidereal day}}{1-\tfrac{% \,\text{length of sidereal day}}{\,\text{orbital period}}}.
  8. number of sidereal days per orbital period = - 1 + number of solar days per orbital period \,\text{number of sidereal days per orbital period}=-1+\,\text{number of solar% days per orbital period}
  9. length of solar day = length of sidereal day 1 + length of sidereal day orbital period . \,\text{length of solar day}=\frac{\,\text{length of sidereal day}}{1+\tfrac{% \,\text{length of sidereal day}}{\,\text{orbital period}}}.

Sierpinski_carpet.html

  1. 8 / 9 {8}/{9}
  2. 8 / 9 {8}/{9}
  3. C 1 , C 2 , C 3 , C_{1},C_{2},C_{3},\dots
  4. C i C_{i}
  5. i i\to\infty
  6. C i C_{i}
  7. C j C_{j}
  8. i j i\neq j
  9. C i C_{i}
  10. i i
  11. C i C_{i}

Sierpinski_triangle.html

  1. d a d_{a}
  2. d a d_{a}
  3. d b d_{b}
  4. d c d_{c}
  5. d a d_{a}
  6. d b d_{b}
  7. d c d_{c}
  8. L 2 3 L^{2}\sqrt{3}
  9. L L 2 L\rightarrow{L\over 2}
  10. 4 ( ( L 2 ) 2 3 ) = 4 L 2 4 3 = L 2 3 . 4\left(\left({L\over 2}\right)^{2}\sqrt{3}\right)=4{{L^{2}}\over 4}\sqrt{3}=L^% {2}\sqrt{3}.
  11. ln 4 ln 2 = 2 \textstyle\frac{\ln 4}{\ln 2}=2

Sigma-algebra.html

  1. lim sup n A n = n = 1 m = n A m . \limsup_{n\to\infty}A_{n}=\bigcap_{n=1}^{\infty}\bigcup_{m=n}^{\infty}A_{m}.
  2. lim inf n A n = n = 1 m = n A m . \liminf_{n\to\infty}A_{n}=\bigcup_{n=1}^{\infty}\bigcap_{m=n}^{\infty}A_{m}.
  3. lim inf n A n = lim sup n A n \liminf_{n\to\infty}A_{n}=\limsup_{n\to\infty}A_{n}
  4. lim n A n \lim_{n\to\infty}A_{n}
  5. Ω = { H , T } = { ( x 1 , x 2 , x 3 , ) : x i { H , T } , i 1 } \Omega=\{H,T\}^{\infty}=\{(x_{1},x_{2},x_{3},\dots):x_{i}\in\{H,T\},i\geq 1\}
  6. 𝒢 n = { A × { H , T } : A { H , T } n } \mathcal{G}_{n}=\{A\times\{H,T\}^{\infty}:A\subset\{H,T\}^{n}\}
  7. 𝒢 1 𝒢 2 𝒢 3 𝒢 \mathcal{G}_{1}\subset\mathcal{G}_{2}\subset\mathcal{G}_{3}\subset\cdots% \subset\mathcal{G}_{\infty}
  8. 𝒢 \mathcal{G}_{\infty}
  9. X A X\ A
  10. ( X , Σ ) (X,Σ)
  11. ( X A ) = A F ( d x ) \mathbb{P}(X\in A)=\int_{A}\,F(dx)
  12. \mathbb{P}
  13. { Σ α : α 𝒜 } \textstyle\{\Sigma_{\alpha}:\alpha\in\mathcal{A}\}
  14. α 𝒜 Σ α . \bigwedge_{\alpha\in\mathcal{A}}\Sigma_{\alpha}.
  15. α 𝒜 Σ α = σ ( α 𝒜 Σ α ) . \bigvee_{\alpha\in\mathcal{A}}\Sigma_{\alpha}=\sigma\left(\bigcup_{\alpha\in% \mathcal{A}}\Sigma_{\alpha}\right).
  16. 𝒫 = { i = 1 n A i : A i Σ α i , α i 𝒜 , n 1 } . \mathcal{P}=\left\{\bigcap_{i=1}^{n}A_{i}:A_{i}\in\Sigma_{\alpha_{i}},\alpha_{% i}\in\mathcal{A},\ n\geq 1\right\}.
  17. Σ α 𝒫 \Sigma_{\alpha}\subset\mathcal{P}
  18. α 𝒜 Σ α 𝒫 . \bigcup_{\alpha\in\mathcal{A}}\Sigma_{\alpha}\subset\mathcal{P}.
  19. σ ( α 𝒜 Σ α ) σ ( 𝒫 ) \sigma\left(\bigcup_{\alpha\in\mathcal{A}}\Sigma_{\alpha}\right)\subset\sigma(% \mathcal{P})
  20. 𝒫 σ ( α 𝒜 Σ α ) \mathcal{P}\subset\sigma\left(\bigcup_{\alpha\in\mathcal{A}}\Sigma_{\alpha}\right)
  21. σ ( 𝒫 ) σ ( α 𝒜 Σ α ) . \sigma(\mathcal{P})\subset\sigma\left(\bigcup_{\alpha\in\mathcal{A}}\Sigma_{% \alpha}\right).
  22. ( X , Σ ) (X,Σ)
  23. ( X , ) \scriptstyle(X,\,\mathcal{F})
  24. ( X , 𝔉 ) \scriptstyle(X,\,\mathfrak{F})
  25. τ \tau
  26. σ \sigma
  27. τ \mathcal{F}_{\tau}
  28. τ \tau
  29. τ \tau
  30. τ \mathcal{F}_{\tau}
  31. σ ( f ) = { f - 1 ( S ) | S B } . \sigma(f)=\{f^{-1}(S)\,|\,S\in B\}.
  32. σ ( f ) = σ ( { f - 1 ( ( a 1 , b 1 ] × × ( a n , b n ] ) : a i , b i } ) . \sigma(f)=\sigma\left(\{f^{-1}((a_{1},b_{1}]\times\cdots\times(a_{n},b_{n}]):a% _{i},b_{i}\in\mathbb{R}\}\right).
  33. ( X 1 , Σ 1 ) (X_{1},\Sigma_{1})
  34. ( X 2 , Σ 2 ) (X_{2},\Sigma_{2})
  35. X 1 × X 2 X_{1}\times X_{2}
  36. Σ 1 × Σ 2 = σ ( { B 1 × B 2 : B 1 Σ 1 , B 2 Σ 2 } ) . \Sigma_{1}\times\Sigma_{2}=\sigma(\{B_{1}\times B_{2}:B_{1}\in\Sigma_{1},B_{2}% \in\Sigma_{2}\}).
  37. { B 1 × B 2 : B 1 Σ 1 , B 2 Σ 2 } \{B_{1}\times B_{2}:B_{1}\in\Sigma_{1},B_{2}\in\Sigma_{2}\}
  38. ( n ) = σ ( { ( - , b 1 ] × × ( - , b n ] : b i } ) = σ ( { ( a 1 , b 1 ] × × ( a n , b n ] : a i , b i } ) . \mathcal{B}(\mathbb{R}^{n})=\sigma\left(\left\{(-\infty,b_{1}]\times\cdots% \times(-\infty,b_{n}]:b_{i}\in\mathbb{R}\right\}\right)=\sigma\left(\left\{(a_% {1},b_{1}]\times\cdots\times(a_{n},b_{n}]:a_{i},b_{i}\in\mathbb{R}\right\}% \right).
  39. X 𝕋 = { f : f ( t ) , t 𝕋 } X\subset\mathbb{R}^{\mathbb{T}}=\{f:f(t)\in\mathbb{R},\ t\in\mathbb{T}\}
  40. ( ) \mathcal{B}(\mathbb{R})
  41. X X
  42. C t 1 , , t n ( B 1 , , B n ) = { f X : f ( t i ) B i , 1 i n } . C_{t_{1},\dots,t_{n}}(B_{1},\dots,B_{n})=\{f\in X:f(t_{i})\in B_{i},1\leq i% \leq n\}.
  43. { C t 1 , , t n ( B 1 , , B n ) : B i ( ) , 1 i n } \{C_{t_{1},\dots,t_{n}}(B_{1},\dots,B_{n}):B_{i}\in\mathcal{B}(\mathbb{R}),1% \leq i\leq n\}
  44. Σ t 1 , , t n \textstyle\Sigma_{t_{1},\dots,t_{n}}
  45. X = n = 1 t i 𝕋 , i n Σ t 1 , , t n \mathcal{F}_{X}=\bigcup_{n=1}^{\infty}\bigcup_{t_{i}\in\mathbb{T},i\leq n}% \Sigma_{t_{1},\dots,t_{n}}
  46. X X
  47. 𝕋 \mathbb{R}^{\mathbb{T}}
  48. X X
  49. 𝕋 \mathbb{T}
  50. X X
  51. C n ( B 1 , , B n ) = ( B 1 × × B n × ) X = { ( x 1 , x 2 , , x n , x n + 1 , ) X : x i B i , 1 i n } , C_{n}(B_{1},\dots,B_{n})=(B_{1}\times\cdots\times B_{n}\times\mathbb{R}^{% \infty})\cap X=\{(x_{1},x_{2},\dots,x_{n},x_{n+1},\dots)\in X:x_{i}\in B_{i},1% \leq i\leq n\},
  52. Σ n = σ ( { C n ( B 1 , , B n ) : B i ( ) , 1 i n } ) \Sigma_{n}=\sigma(\{C_{n}(B_{1},\dots,B_{n}):B_{i}\in\mathcal{B}(\mathbb{R}),1% \leq i\leq n\})
  53. ( Ω , Σ , ) (\Omega,\Sigma,\mathbb{P})
  54. Y : Ω n \textstyle Y:\Omega\to\mathbb{R}^{n}
  55. Y Y
  56. Y Y
  57. σ ( Y ) = { Y - 1 ( A ) : A ( n ) } . \sigma(Y)=\{Y^{-1}(A):A\in\mathcal{B}(\mathbb{R}^{n})\}.
  58. ( Ω , Σ , ) (\Omega,\Sigma,\mathbb{P})
  59. 𝕋 \mathbb{R}^{\mathbb{T}}
  60. 𝕋 \mathbb{T}
  61. Y : Ω X 𝕋 \textstyle Y:\Omega\to X\subset\mathbb{R}^{\mathbb{T}}
  62. σ ( X ) \sigma(\mathcal{F}_{X})
  63. X X
  64. Y Y
  65. Y Y
  66. σ ( Y ) = { Y - 1 ( A ) : A σ ( X ) } = σ ( { Y - 1 ( A ) : A X } ) , \sigma(Y)=\left\{Y^{-1}(A):A\in\sigma(\mathcal{F}_{X})\right\}=\sigma(\{Y^{-1}% (A):A\in\mathcal{F}_{X}\}),

Signal-to-noise_ratio.html

  1. SNR = P signal P noise , \mathrm{SNR}=\frac{P_{\mathrm{signal}}}{P_{\mathrm{noise}}},
  2. SNR = σ signal 2 σ noise 2 . \mathrm{SNR}=\frac{\sigma^{2}_{\mathrm{signal}}}{\sigma^{2}_{\mathrm{noise}}}.
  3. SNR = P signal P noise = ( A signal A noise ) 2 , \mathrm{SNR}=\frac{P_{\mathrm{signal}}}{P_{\mathrm{noise}}}=\left(\frac{A_{% \mathrm{signal}}}{A_{\mathrm{noise}}}\right)^{2},
  4. P signal , dB = 10 log 10 ( P signal ) P_{\mathrm{signal,dB}}=10\log_{10}\left(P_{\mathrm{signal}}\right)
  5. P noise , dB = 10 log 10 ( P noise ) . P_{\mathrm{noise,dB}}=10\log_{10}\left(P_{\mathrm{noise}}\right).
  6. SNR dB = 10 log 10 ( SNR ) . \mathrm{SNR_{dB}}=10\log_{10}\left(\mathrm{SNR}\right).
  7. SNR dB = 10 log 10 ( P signal P noise ) . \mathrm{SNR_{dB}}=10\log_{10}\left(\frac{P_{\mathrm{signal}}}{P_{\mathrm{noise% }}}\right).
  8. 10 log 10 ( P signal P noise ) = 10 log 10 ( P signal ) - 10 log 10 ( P noise ) . 10\log_{10}\left(\frac{P_{\mathrm{signal}}}{P_{\mathrm{noise}}}\right)=10\log_% {10}\left(P_{\mathrm{signal}}\right)-10\log_{10}\left(P_{\mathrm{noise}}\right).
  9. SNR dB = P signal , dB - P noise , dB . \mathrm{SNR_{dB}}={P_{\mathrm{signal,dB}}-P_{\mathrm{noise,dB}}}.
  10. SNR dB = 10 log 10 [ ( A signal A noise ) 2 ] = 20 log 10 ( A signal A noise ) = 2 ( P signal , dB - P noise , dB ) . \mathrm{SNR_{dB}}=10\log_{10}\left[\left(\frac{A_{\mathrm{signal}}}{A_{\mathrm% {noise}}}\right)^{2}\right]=20\log_{10}\left(\frac{A_{\mathrm{signal}}}{A_{% \mathrm{noise}}}\right)=2\left({P_{\mathrm{signal,dB}}-P_{\mathrm{noise,dB}}}% \right).
  11. P = V rms I rms \mathrm{P}=V_{\mathrm{rms}}I_{\mathrm{rms}}
  12. P = V rms 2 R = I rms 2 R \mathrm{P}=\frac{V_{\mathrm{rms}}^{2}}{R}=I_{\mathrm{rms}}^{2}R
  13. R = 1 Ω R=1\Omega
  14. P = V rms 2 = A 2 2 \mathrm{P}=V_{\mathrm{rms}}^{2}=\frac{A^{2}}{2}
  15. SNR = μ σ \mathrm{SNR}=\frac{\mu}{\sigma}
  16. μ \mu
  17. σ \sigma
  18. ( SNR ) C , AM = A C 2 ( 1 + k a 2 P ) 2 W N 0 \mathrm{(SNR)_{C,AM}}=\frac{A_{C}^{2}(1+k_{a}^{2}P)}{2WN_{0}}
  19. k a k_{a}
  20. ( SNR ) O , AM = A c 2 k a 2 P 2 W N 0 \mathrm{(SNR)_{O,AM}}=\frac{A_{c}^{2}k_{a}^{2}P}{2WN_{0}}
  21. ( SNR ) C , FM = A c 2 2 W N 0 \mathrm{(SNR)_{C,FM}}=\frac{A_{c}^{2}}{2WN_{0}}
  22. ( SNR ) O , FM = A c 2 k f 2 P 2 N 0 W 3 \mathrm{(SNR)_{O,FM}}=\frac{A_{c}^{2}k_{f}^{2}P}{2N_{0}W^{3}}
  23. DR dB = SNR dB = 20 log 10 ( 2 n ) 6.02 n \mathrm{DR_{dB}}=\mathrm{SNR_{dB}}=20\log_{10}(2^{n})\approx 6.02\cdot n
  24. SNR dB 20 log 10 ( 2 n 3 / 2 ) 6.02 n + 1.761 \mathrm{SNR_{dB}}\approx 20\log_{10}(2^{n}\sqrt{3/2})\approx 6.02\cdot n+1.761
  25. DR dB = 6.02 2 m \mathrm{DR_{dB}}=6.02\cdot 2^{m}
  26. SNR dB = 6.02 ( n - m ) \mathrm{SNR_{dB}}=6.02\cdot(n-m)
  27. σ \sigma
  28. σ \sigma

Siméon_Denis_Poisson.html

  1. 2 ϕ = - 4 π ρ \nabla^{2}\phi=-4\pi\rho\;
  2. 2 ϕ = 0 . \nabla^{2}\phi=0\;.
  3. 2 ϕ = ρ ( x , y , z ) . \nabla^{2}\phi=\rho(x,y,z)\;.
  4. 2 Ψ = 2 Ψ x 2 + 2 Ψ y 2 + 2 Ψ z 2 = - ρ e ε ε 0 . \nabla^{2}\Psi={\partial^{2}\Psi\over\partial x^{2}}+{\partial^{2}\Psi\over% \partial y^{2}}+{\partial^{2}\Psi\over\partial z^{2}}=-{\rho_{e}\over% \varepsilon\varepsilon_{0}}\;.
  5. 2 Ψ = n 0 e ε ε 0 ( e e Ψ ( x , y , z ) / k B T - e - e Ψ ( x , y , z ) / k B T ) , \nabla^{2}\Psi={n_{0}e\over\varepsilon\varepsilon_{0}}\left(e^{e\Psi(x,y,z)/k_% {B}T}-e^{-e\Psi(x,y,z)/k_{B}T}\right),\;
  6. 1 r 2 d d r ( r 2 d Ψ d r ) = n 0 e ε ε 0 ( e e Ψ ( r ) / k B T - e - e Ψ ( r ) / k B T ) {1\over r^{2}}{d\over dr}\left(r^{2}{d\Psi\over dr}\right)={n_{0}e\over% \varepsilon\varepsilon_{0}}\left(e^{e\Psi(r)/k_{B}T}-e^{-e\Psi(r)/k_{B}T}% \right)\;
  7. ϕ i k = ρ ( x , y , z , c t ) . \sqrt{\phi}_{ik}=\rho(x,y,z,ct)\;.
  8. ϕ M = - 1 4 π ρ ( x , y , z ) d v r \phi_{M}=-{1\over 4\pi}\int{\rho(x,y,z)\,dv\over r}\;
  9. ϕ ( ξ η ) = 1 4 π 0 2 π R 2 - ρ 2 R 2 + ρ 2 - 2 R ρ cos ( ψ - χ ) ϕ ( χ ) d χ \phi(\xi\eta)={1\over 4\pi}\int_{0}^{2\pi}{R^{2}-\rho^{2}\over R^{2}+\rho^{2}-% 2R\rho\cos(\psi-\chi)}\phi(\chi)\,d\chi\;
  10. ξ = ρ cos ψ , \xi=\rho\cos\psi,\;
  11. η = ρ sin ψ , \quad\eta=\rho\sin\psi,\;
  12. G ( x , y , z ; ξ , η , ζ ) = 1 r - R r 1 ρ , G(x,y,z;\xi,\eta,\zeta)={1\over r}-{R\over r_{1}\rho}\;,
  13. ρ = ξ 2 + η 2 + ζ 2 \rho=\sqrt{\xi^{2}+\eta^{2}+\zeta^{2}}
  14. ϕ ( ξ , η , ζ ) = 1 4 π S R 2 - ρ 2 R r 3 ϕ d s . \phi(\xi,\eta,\zeta)={1\over 4\pi}\iint_{S}{R^{2}-\rho^{2}\over Rr^{3}}\phi\,% ds\;.
  15. p i = T q i / t , p_{i}={\partial T\over{\partial q_{i}/\partial t}},

Similarity_(geometry).html

  1. A B C \triangle ABC
  2. A B C \triangle A^{\prime}B^{\prime}C^{\prime}
  3. B A C \angle BAC
  4. B A C \angle B^{\prime}A^{\prime}C^{\prime}
  5. A B C \angle ABC
  6. A B C \angle A^{\prime}B^{\prime}C^{\prime}
  7. A C B \angle ACB
  8. A C B \angle A^{\prime}C^{\prime}B^{\prime}
  9. A B A B = B C B C = A C A C {AB\over A^{\prime}B^{\prime}}={BC\over B^{\prime}C^{\prime}}={AC\over A^{% \prime}C^{\prime}}
  10. A B A B = B C B C {AB\over A^{\prime}B^{\prime}}={BC\over B^{\prime}C^{\prime}}
  11. A B C \angle ABC
  12. A B C \angle A^{\prime}B^{\prime}C^{\prime}
  13. A B C \triangle ABC
  14. A B C \triangle A^{\prime}B^{\prime}C^{\prime}
  15. A B C A B C \triangle ABC\sim\triangle A^{\prime}B^{\prime}C^{\prime}\,
  16. A B C \triangle ABC
  17. D E ¯ \overline{DE}
  18. A B C D E F \triangle ABC\sim\triangle DEF
  19. d ( f ( x ) , f ( y ) ) = r d ( x , y ) , d(f(x),f(y))=rd(x,y),\,
  20. f ( z ) = a z + b f(z)=az+b
  21. f ( z ) = a z ¯ + b f(z)=a\overline{z}+b
  22. d ( f ( x ) , f ( y ) ) = r d ( x , y ) . d(f(x),f(y))=rd(x,y).\,\,
  23. lim d ( f ( x ) , f ( y ) ) d ( x , y ) = r . \lim\frac{d(f(x),f(y))}{d(x,y)}=r.
  24. { f s } s S \{f_{s}\}_{s\in S}
  25. 0 r s < 1 0\leq r_{s}<1
  26. s S f s ( K ) = K . \bigcup_{s\in S}f_{s}(K)=K.\,
  27. μ D \mu^{D}
  28. s S ( r s ) D = 1 \sum_{s\in S}(r_{s})^{D}=1\,
  29. f s ( K ) f_{s}(K)
  30. μ D ( f s 1 f s 2 f s n ( K ) ) = ( r s 1 r s 2 r s n ) D . \mu^{D}(f_{s_{1}}\circ f_{s_{2}}\circ\cdots\circ f_{s_{n}}(K))=(r_{s_{1}}\cdot r% _{s_{2}}\cdots r_{s_{n}})^{D}.\,
  31. ( a , b ) , S ( a , b ) 0 \forall(a,b),S(a,b)\geq 0
  32. S ( a , b ) S ( a , a ) S(a,b)\leq S(a,a)
  33. ( a , b ) , S ( a , b ) = S ( a , a ) a = b \forall(a,b),S(a,b)=S(a,a)\Leftrightarrow a=b
  34. ( a , b ) S ( a , b ) = S ( b , a ) \forall(a,b)\ S(a,b)=S(b,a)
  35. ( a , b ) S ( a , b ) < \forall(a,b)\ S(a,b)<\infty
  36. { 2 i , 3 2 i } \{2^{i},3\cdot 2^{i}\}
  37. i i

Simple_group.html

  1. A A_{\infty}
  2. A n A_{n}
  3. A n A n + 1 A_{n}\to A_{n+1}
  4. PSL n ( F ) \mathrm{PSL}_{n}(F)
  5. F F
  6. n 3 n\geq 3
  7. n 5 n\geq 5

Simple_harmonic_motion.html

  1. 𝐅 = - k 𝐱 , \mathbf{F}=-k\mathbf{x},
  2. F n e t = m d 2 x d t 2 = - k x , F_{net}=m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=-kx,
  3. d 2 x d t 2 = - ( k m ) x , \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=-\left(\frac{k}{m}\right)x,
  4. x ( t ) = c 1 cos ( ω t ) + c 2 sin ( ω t ) = A cos ( ω t - φ ) , x(t)=c_{1}\cos\left(\omega t\right)+c_{2}\sin\left(\omega t\right)=A\cos\left(% \omega t-\varphi\right),
  5. ω = k m , \omega=\sqrt{\frac{k}{m}},
  6. A = c 1 2 + c 2 2 , A=\sqrt{{c_{1}}^{2}+{c_{2}}^{2}},
  7. tan φ = ( c 2 c 1 ) , \tan\varphi=\left(\frac{c_{2}}{c_{1}}\right),
  8. v ( t ) = d x d t = - A ω sin ( ω t - φ ) , v(t)=\frac{\mathrm{d}x}{\mathrm{d}t}=-A\omega\sin(\omega t-\varphi),
  9. ω A 2 - x 2 {\omega}\sqrt{A^{2}-x^{2}}
  10. = w A =wA
  11. a ( t ) = d 2 x d t 2 = - A ω 2 cos ( ω t - φ ) . a(t)=\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=-A\omega^{2}\cos(\omega t-\varphi).
  12. A ω 2 A\omega^{2}
  13. a ( x ) = - ω 2 x . a(x)=-\omega^{2}x.\!
  14. ω 2 = ( k m ) \omega^{2}=\left(\frac{k}{m}\right)\!
  15. f = 1 2 π k m , f=\frac{1}{2\pi}\sqrt{\frac{k}{m}},
  16. T = 2 π m k . T=2\pi\sqrt{\frac{m}{k}}.
  17. K ( t ) = 1 2 m v 2 ( t ) = 1 2 m ω 2 A 2 sin 2 ( ω t - φ ) = 1 2 k A 2 sin 2 ( ω t - φ ) , K(t)=\frac{1}{2}mv^{2}(t)=\frac{1}{2}m\omega^{2}A^{2}\sin^{2}(\omega t-\varphi% )=\frac{1}{2}kA^{2}\sin^{2}(\omega t-\varphi),
  18. U ( t ) = 1 2 k x 2 ( t ) = 1 2 k A 2 cos 2 ( ω t - φ ) . U(t)=\frac{1}{2}kx^{2}(t)=\frac{1}{2}kA^{2}\cos^{2}(\omega t-\varphi).
  19. E = K + U = 1 2 k A 2 . E=K+U=\frac{1}{2}kA^{2}.
  20. T = 2 π m k T=2\pi{\sqrt{\frac{m}{k}}}
  21. T = 2 π g T=2\pi\sqrt{\frac{\ell}{g}}
  22. m g sin ( θ ) = I α , mg\ell\sin(\theta)=I\alpha,
  23. I I
  24. - m g θ = I α -mg\ell\theta=I\alpha

Simple_machine.html

  1. F in F\text{in}\,
  2. F out F\text{out}\,
  3. MA = F out / F in \mathrm{MA}=F\text{out}/F\text{in}\,
  4. P out P\text{out}\,
  5. P in P\text{in}\,
  6. P out = P in P\text{out}=P\text{in}\!
  7. v out v\text{out}\,
  8. P out = F out v out P\text{out}=F\text{out}v\text{out}\!
  9. v in v\text{in}\,
  10. P in = F in v in P\text{in}=F\text{in}v\text{in}\!
  11. F out v out = F in v in F\text{out}v\text{out}=F\text{in}v\text{in}\,
  12. v out v in = d out d in {v\text{out}\over v\text{in}}={d\text{out}\over d\text{in}}\,
  13. MA > 1 \mathrm{MA}>1\,
  14. d out d\text{out}\,
  15. d in d\text{in}\,
  16. MA < 1 \mathrm{MA}<1\,
  17. P fric P\text{fric}\,
  18. P in = P out + P fric P\text{in}=P\text{out}+P\text{fric}\,
  19. η \eta\,
  20. η P out P in \eta\equiv{P\text{out}\over P\text{in}}\,
  21. P out = η P in P\text{out}=\eta P\text{in}\,
  22. F out v out = η F in v in F\text{out}v\text{out}=\eta F\text{in}v\text{in}\,
  23. MA compound = F outN F in1 \mathrm{MA}\text{compound}={F\text{outN}\over F\text{in1}}\,
  24. F out1 = F in2 , F out2 = F in3 , F outK = F inK+1 F\text{out1}=F\text{in2},\;F\text{out2}=F\text{in3},\ldots\;F\text{outK}=F% \text{inK+1}
  25. MA compound = F out1 F in1 F out2 F in2 F out3 F in3 F outN F inN \mathrm{MA}\text{compound}={F\text{out1}\over F\text{in1}}{F\text{out2}\over F% \text{in2}}{F\text{out3}\over F\text{in3}}\ldots{F\text{outN}\over F\text{inN}}\,
  26. MA compound = MA 1 MA 2 MA N \mathrm{MA}\text{compound}=\mathrm{MA}_{1}\mathrm{MA}_{2}\ldots\mathrm{MA}% \text{N}\,
  27. η compound = η 1 η 2 η N . \eta\text{compound}=\eta_{1}\eta_{2}\ldots\;\eta\text{N}.\,
  28. η F o u t / F i n d i n / d o u t < 0.50 \eta\equiv\frac{F_{out}/F_{in}}{d_{in}/d_{out}}<0.50\,
  29. W 1,2 W\text{1,2}\,
  30. W load W\text{load}\,
  31. W fric W\text{fric}\,
  32. W 1,2 = W load + W fric ( 1 ) W\text{1,2}=W\text{load}+W\text{fric}\qquad\qquad(1)\,
  33. η = W load / W 1,2 < 1 / 2 \eta=W\text{load}/W\text{1,2}<1/2\,
  34. 2 W load < W 1,2 2W\text{load}<W\text{1,2}\,
  35. 2 W load < W load + W fric 2W\text{load}<W\text{load}+W\text{fric}\,
  36. W load < W fric W\text{load}<W\text{fric}\,
  37. W fric W\text{fric}\,
  38. W load = W 2,1 + W fric W\text{load}=W\text{2,1}+W\text{fric}\,
  39. W 2,1 = W load - W fric < 0 W\text{2,1}=W\text{load}-W\text{fric}<0\,

Simple_module.html

  1. 0 N M M / N 0. 0\to N\to M\to M/N\to 0.
  2. M 2 M 1 M . \cdots\subset M_{2}\subset M_{1}\subset M.

Simplex.html

  1. u 0 , , u k n u_{0},\dots,u_{k}\in\mathbb{R}^{n}
  2. u 1 - u 0 , , u k - u 0 u_{1}-u_{0},\dots,u_{k}-u_{0}
  3. C = { θ 0 u 0 + + θ k u k | θ i 0 , 0 i k , i = 0 k θ i = 1 } C=\{\theta_{0}u_{0}+\dots+\theta_{k}u_{k}|\theta_{i}\geq 0,0\leq i\leq k,\sum_% {i=0}^{k}\theta_{i}=1\}
  4. ( n + 1 m + 1 ) {\textstyle\left({{n+1}\atop{m+1}}\right)}
  5. Δ n = { ( t 0 , , t n ) n + 1 i = 0 n t i = 1 and t i 0 for all i } \Delta^{n}=\left\{(t_{0},\cdots,t_{n})\in\mathbb{R}^{n+1}\mid\sum_{i=0}^{n}{t_% {i}}=1\mbox{ and }~{}t_{i}\geq 0\mbox{ for all }~{}i\right\}
  6. \vdots
  7. ( t 0 , , t n ) i = 0 n t i v i (t_{0},\cdots,t_{n})\mapsto\sum_{i=0}^{n}t_{i}v_{i}
  8. ( n - 1 ) (n-1)
  9. ( t 1 , , t n ) i = 1 n t i v i (t_{1},\cdots,t_{n})\mapsto\sum_{i=1}^{n}t_{i}v_{i}
  10. Δ n - 1 P . \Delta^{n-1}\twoheadrightarrow P.
  11. s 0 \displaystyle s_{0}
  12. Δ * n = { ( s 1 , , s n ) n 0 = s 0 s 1 s 2 s n s n + 1 = 1 } . \Delta_{*}^{n}=\left\{(s_{1},\cdots,s_{n})\in\mathbb{R}^{n}\mid 0=s_{0}\leq s_% {1}\leq s_{2}\leq\dots\leq s_{n}\leq s_{n+1}=1\right\}.
  13. n \mathbb{R}^{n}
  14. n + 1 \mathbb{R}^{n+1}
  15. t i = 0 , t_{i}=0,
  16. s i = s i + 1 , s_{i}=s_{i+1},
  17. n ! n!
  18. 1 / n ! 1/n!
  19. 1 , x , x 2 / 2 , x 3 / 3 ! , , x n / n ! 1,x,x^{2}/2,x^{3}/3!,\dots,x^{n}/n!
  20. ( p i ) i \,(p_{i})_{i}
  21. ( t i ) i \left(t_{i}\right)_{i}
  22. t i = max { p i + Δ , 0 } , t_{i}=\max\{p_{i}+\Delta\,,0\},
  23. Δ \Delta
  24. i max { p i + Δ , 0 } = 1. \sum_{i}\max\{p_{i}+\Delta\,,0\}=1.
  25. Δ \Delta
  26. p i p_{i}
  27. O ( n log n ) O(n\log n)
  28. O ( n ) O(n)
  29. 1 \ell_{1}
  30. Δ c n = { ( t 1 , , t n ) n i = 1 n t i 1 and t i 0 for all i } . \Delta_{c}^{n}=\left\{(t_{1},\cdots,t_{n})\in\mathbb{R}^{n}\mid\sum_{i=1}^{n}{% t_{i}}\leq 1\mbox{ and }~{}t_{i}\geq 0\mbox{ for all }~{}i\right\}.
  31. arccos ( - 1 n ) \arccos\left(\tfrac{-1}{n}\right)
  32. - 1 / n -1/n
  33. ( x 0 y 0 z 0 ) , ( x 1 y 1 z 1 ) , ( x 2 y 2 z 2 ) , ( x 3 y 3 z 3 ) \begin{pmatrix}x_{0}\\ y_{0}\\ z_{0}\end{pmatrix},\begin{pmatrix}x_{1}\\ y_{1}\\ z_{1}\end{pmatrix},\begin{pmatrix}x_{2}\\ y_{2}\\ z_{2}\end{pmatrix},\begin{pmatrix}x_{3}\\ y_{3}\\ z_{3}\end{pmatrix}
  34. ( 1 0 0 ) , ( x 1 y 1 z 1 ) , ( x 2 y 2 z 2 ) , ( x 3 y 3 z 3 ) \begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}x_{1}\\ y_{1}\\ z_{1}\end{pmatrix},\begin{pmatrix}x_{2}\\ y_{2}\\ z_{2}\end{pmatrix},\begin{pmatrix}x_{3}\\ y_{3}\\ z_{3}\end{pmatrix}
  35. 1 / 3 {1}/{3}
  36. ( 1 0 0 ) , ( - 1 3 y 1 z 1 ) , ( - 1 3 y 2 z 2 ) , ( - 1 3 y 3 z 3 ) \begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ y_{1}\\ z_{1}\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ y_{2}\\ z_{2}\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ y_{3}\\ z_{3}\end{pmatrix}
  37. ( 1 0 0 ) , ( - 1 3 8 3 0 ) , ( - 1 3 y 2 z 2 ) , ( - 1 3 y 3 z 3 ) \begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ \frac{\sqrt{8}}{3}\\ 0\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ y_{2}\\ z_{2}\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ y_{3}\\ z_{3}\end{pmatrix}
  38. ( 1 0 0 ) , ( - 1 3 8 3 0 ) , ( - 1 3 - 2 3 z 2 ) , ( - 1 3 - 2 3 z 3 ) \begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ \frac{\sqrt{8}}{3}\\ 0\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ -\frac{\sqrt{2}}{3}\\ z_{2}\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ -\frac{\sqrt{2}}{3}\\ z_{3}\end{pmatrix}
  39. ( 1 0 0 ) , ( - 1 3 8 3 0 ) , ( - 1 3 - 2 3 2 3 ) , ( - 1 3 - 2 3 - 2 3 ) \begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ \frac{\sqrt{8}}{3}\\ 0\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ -\frac{\sqrt{2}}{3}\\ \sqrt{\frac{2}{3}}\end{pmatrix},\begin{pmatrix}-\frac{1}{3}\\ -\frac{\sqrt{2}}{3}\\ -\sqrt{\frac{2}{3}}\end{pmatrix}
  40. | 1 n ! det ( v 1 - v 0 , v 2 - v 0 , , v n - v 0 ) | \left|{1\over n!}\det\begin{pmatrix}v_{1}-v_{0},&v_{2}-v_{0},&\dots,&v_{n}-v_{% 0}\end{pmatrix}\right|
  41. 1 ( n + 1 ) ! {1\over(n+1)!}
  42. n + 1 n ! 2 n {\frac{\sqrt{n+1}}{n!\sqrt{2^{n}}}}
  43. x = 1 / 2 x=1/\sqrt{2}
  44. d x / n + 1 dx/\sqrt{n+1}\,
  45. ( d x / ( n + 1 ) , , d x / ( n + 1 ) ) (dx/(n+1),\dots,dx/(n+1))
  46. k = 1 n | A k | 2 = | A 0 | 2 \sum_{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}
  47. A 1 A n A_{1}\ldots A_{n}
  48. A 0 A_{0}
  49. ( n + 1 ) (n+1)
  50. σ = [ v 0 , v 1 , v 2 , , v n ] \sigma=[v_{0},v_{1},v_{2},...,v_{n}]
  51. v j v_{j}
  52. σ \partial\sigma
  53. σ = j = 0 n ( - 1 ) j [ v 0 , , v j - 1 , v j + 1 , , v n ] \partial\sigma=\sum_{j=0}^{n}(-1)^{j}[v_{0},...,v_{j-1},v_{j+1},...,v_{n}]
  54. 2 σ = ( j = 0 n ( - 1 ) j [ v 0 , , v j - 1 , v j + 1 , , v n ] ) = 0. \partial^{2}\sigma=\partial(~{}\sum_{j=0}^{n}(-1)^{j}[v_{0},...,v_{j-1},v_{j+1% },...,v_{n}]~{})=0.
  55. 2 ρ = 0 \partial^{2}\rho=0
  56. f : n M f\colon\mathbb{R}^{n}\rightarrow M
  57. f ( i a i σ i ) = i a i f ( σ i ) f(\sum\nolimits_{i}a_{i}\sigma_{i})=\sum\nolimits_{i}a_{i}f(\sigma_{i})
  58. a i a_{i}
  59. \partial
  60. f ( ρ ) = f ( ρ ) \partial f(\rho)=f(\partial\rho)
  61. f : σ X f:\sigma\rightarrow X
  62. Δ n := { x 𝔸 n + 1 | i = 1 n + 1 x i - 1 = 0 } \Delta^{n}:=\{x\in\mathbb{A}^{n+1}|\sum_{i=1}^{n+1}x_{i}-1=0\}
  63. Δ n ( R ) = S p e c ( R [ Δ n ] ) \Delta_{n}(R)=Spec(R[\Delta^{n}])
  64. R [ Δ n ] := R [ x 1 , , x n + 1 ] / ( x i - 1 ) R[\Delta^{n}]:=R[x_{1},...,x_{n+1}]/(\sum x_{i}-1)
  65. R R
  66. R [ Δ n ] R[\Delta^{n}]
  67. R [ Δ ] R[\Delta^{\bullet}]

Simpson's_paradox.html

  1. S L ( 1 ) = 0 % S_{L}(1)=0\%
  2. S B ( 1 ) = 14.2 % S_{B}(1)=14.2\%
  3. S L ( 2 ) = 71.4 % S_{L}(2)=71.4\%
  4. S B ( 2 ) = 100 % S_{B}(2)=100\%
  5. S L = 5 10 S_{L}=\begin{matrix}\frac{5}{10}\end{matrix}
  6. S B = 4 10 S_{B}=\begin{matrix}\frac{4}{10}\end{matrix}
  7. S L > S B S_{L}>S_{B}\,
  8. S B ( 1 ) > S L ( 1 ) S_{B}(1)>S_{L}(1)
  9. S B ( 2 ) > S L ( 2 ) S_{B}(2)>S_{L}(2)
  10. S B S_{B}
  11. S L S_{L}
  12. 3 10 \begin{matrix}\frac{3}{10}\end{matrix}
  13. 7 10 \begin{matrix}\frac{7}{10}\end{matrix}
  14. S L = 3 10 S L ( 1 ) + 7 10 S L ( 2 ) S_{L}=\begin{matrix}\frac{3}{10}\end{matrix}S_{L}(1)+\begin{matrix}\frac{7}{10% }\end{matrix}S_{L}(2)
  15. S B = 7 10 S B ( 1 ) + 3 10 S B ( 2 ) S_{B}=\begin{matrix}\frac{7}{10}\end{matrix}S_{B}(1)+\begin{matrix}\frac{3}{10% }\end{matrix}S_{B}(2)
  16. p / q p/q
  17. A = ( q , p ) \overrightarrow{A}=(q,p)
  18. p / q p/q
  19. p 1 / q 1 p_{1}/q_{1}
  20. p 2 / q 2 p_{2}/q_{2}
  21. ( q 1 , p 1 ) (q_{1},p_{1})
  22. ( q 2 , p 2 ) (q_{2},p_{2})
  23. ( q 1 + q 2 , p 1 + p 2 ) (q_{1}+q_{2},p_{1}+p_{2})
  24. p 1 + p 2 q 1 + q 2 \frac{p_{1}+p_{2}}{q_{1}+q_{2}}
  25. b 1 \overrightarrow{b_{1}}
  26. r 1 \overrightarrow{r_{1}}
  27. b 2 \overrightarrow{b_{2}}
  28. r 2 \overrightarrow{r_{2}}
  29. b 1 + b 2 \overrightarrow{b_{1}}+\overrightarrow{b_{2}}
  30. r 1 + r 2 \overrightarrow{r_{1}}+\overrightarrow{r_{2}}

Simula.html

  1. \sum

SINAD.html

  1. SINAD = P signal + P noise + P distortion P noise + P distortion \mathrm{SINAD}=\frac{P_{\mathrm{signal}}+P_{\mathrm{noise}}+P_{\mathrm{% distortion}}}{P_{\mathrm{noise}}+P_{\mathrm{distortion}}}
  2. P P

Single-mode_optical_fiber.html

  1. 2.405 g + 2 g 2.405\sqrt{\frac{g+2}{g}}

Single-sideband_modulation.html

  1. s s s b ( t ) = s ( t ) cos ( 2 π f 0 t ) - s ^ ( t ) sin ( 2 π f 0 t ) , s_{ssb}(t)=s(t)\cdot\cos(2\pi f_{0}t)-\widehat{s}(t)\cdot\sin(2\pi f_{0}t),\,
  2. s ( t ) s(t)\,
  3. s ^ ( t ) \widehat{s}(t)\,
  4. f 0 f_{0}\,
  5. s ( t ) = 1 2 ( s ( t ) + j s ^ ( t ) ) s a ( t ) + 1 2 ( s ( t ) - j s ^ ( t ) ) s a * ( t ) , s(t)=\tfrac{1}{2}\underbrace{(s(t)+j\cdot\widehat{s}(t))}_{s_{a}(t)}+\tfrac{1}% {2}\underbrace{(s(t)-j\cdot\widehat{s}(t))}_{s_{a}^{*}(t)},
  6. j j
  7. s a ( t ) s_{a}(t)
  8. s ( t ) , s(t),
  9. s a * ( t ) s_{a}^{*}(t)
  10. s ( t ) s(t)
  11. 1 2 S a ( f ) = { S ( f ) , for f > 0 , 0 , for f < 0 , \tfrac{1}{2}S_{\mathrm{a}}(f)=\begin{cases}S(f),&\,\text{for}\ f>0,\\ 0,&\,\text{for}\ f<0,\end{cases}
  12. S a ( f ) S_{\mathrm{a}}(f)
  13. S ( f ) S(f)
  14. s a ( t ) s_{a}(t)
  15. s ( t ) . s(t).
  16. S a ( f - f 0 ) S_{\mathrm{a}}(f-f_{0})
  17. S ( f ) . S(f).
  18. s s s b ( t ) : s_{ssb}(t):
  19. - 1 { S a ( f - f 0 ) } = s a ( t ) e j 2 π f 0 t = s s s b ( t ) + j s ^ s s b ( t ) . \mathcal{F}^{-1}\{S_{\mathrm{a}}(f-f_{0})\}=s_{a}(t)\cdot e^{j2\pi f_{0}t}=s_{% ssb}(t)+j\cdot\widehat{s}_{ssb}(t).\,
  20. e j 2 π f 0 t , e^{j2\pi f_{0}t},\,
  21. s s s b ( t ) \displaystyle s_{ssb}(t)
  22. s s s b ( t ) s_{ssb}(t)
  23. s ( t ) s(t)
  24. cos ( 2 π f 0 t ) , \cos(2\pi f_{0}t),
  25. 2 f 0 2f_{0}
  26. s ( t ) s(t)
  27. s ^ ( t ) \widehat{s}(t)
  28. s ( t ) s(t)
  29. s a * ( t ) , s_{a}^{*}(t),
  30. S ( f ) . S(f).
  31. f 0 f_{0}\,
  32. S ( f - f 0 ) S(f-f_{0})
  33. s a * ( t ) e j 2 π f 0 t s_{a}^{*}(t)\cdot e^{j2\pi f_{0}t}
  34. s a * ( t ) e j 2 π f 0 t = s l s b ( t ) + j s ^ l s b ( t ) s_{a}^{*}(t)\cdot e^{j2\pi f_{0}t}=s_{lsb}(t)+j\cdot\widehat{s}_{lsb}(t)\,
  35. s l s b ( t ) = R e { s a * ( t ) e j 2 π f 0 t } = s ( t ) cos ( 2 π f 0 t ) + s ^ ( t ) sin ( 2 π f 0 t ) . \begin{aligned}\displaystyle s_{lsb}(t)&\displaystyle=Re\big\{s_{a}^{*}(t)% \cdot e^{j2\pi f_{0}t}\big\}\\ &\displaystyle=s(t)\cdot\cos(2\pi f_{0}t)+\widehat{s}(t)\cdot\sin(2\pi f_{0}t)% .\end{aligned}
  36. 2 s ( t ) cos ( 2 π f 0 t ) , 2s(t)\cdot\cos(2\pi f_{0}t),\,
  37. exp ( j ω t ) \exp(j\omega t)
  38. F i f F_{if}\,
  39. F b F_{b}\,
  40. c o s ( 2 π F b f o t ) cos(2\pi\cdot F_{bfo}\cdot t)\,
  41. ( F i f + F b f o ) (F_{if}+F_{bfo})\,
  42. | F i f - F b f o | |F_{if}-F_{bfo}|\,
  43. F b f o F_{bfo}\,
  44. | F i f - F b f o | = F b |F_{if}-F_{bfo}|=F_{b}\,
  45. ( F i f + F b f o ) (F_{if}+F_{bfo})\,
  46. F b f o F_{bfo}\,
  47. F b f o F_{bfo}\,
  48. F b F_{b}\,

Single-stage-to-orbit.html

  1. Δ v = I sp g 0 ln ( M R ) \Delta v=I\text{sp}\cdot g_{0}\ln(MR)
  2. Δ v \Delta v
  3. I sp I\text{sp}
  4. g 0 g_{0}
  5. M R MR
  6. ln \ln
  7. ( m i ) \left(m_{i}\right)
  8. ( m f ) \left(m_{f}\right)
  9. M R = m i m f = m p + m s + m pl m s + m pl MR=\frac{m_{i}}{m_{f}}=\frac{m_{p}+m_{s}+m\text{pl}}{m_{s}+m\text{pl}}
  10. m i m_{i}
  11. ( G L O W ) \left(GLOW\right)
  12. m f m_{f}
  13. m s m_{s}
  14. m p m_{p}
  15. m pl m\text{pl}
  16. ζ \zeta
  17. ζ = m p m i = m i - m f m i = 1 - m f m i = 1 - 1 M R = M R - 1 M R \zeta=\frac{m_{p}}{m_{i}}=\frac{m_{i}-m_{f}}{m_{i}}=1-\frac{m_{f}}{m_{i}}=1-% \frac{1}{MR}=\frac{MR-1}{MR}
  18. λ \lambda
  19. λ = m s m p + m s = m s m i - m pl = m s m i 1 - m pl m i \lambda=\frac{m_{s}}{m_{p}+m_{s}}=\frac{m_{s}}{m_{i}-m\text{pl}}=\frac{\frac{m% _{s}}{m_{i}}}{1-\frac{m\text{pl}}{m_{i}}}
  20. ( m s m i ) \left(\frac{m_{s}}{m_{i}}\right)
  21. m s m i = λ ( 1 - m pl m i ) \frac{m_{s}}{m_{i}}=\lambda\left(1-\frac{m\text{pl}}{m_{i}}\right)
  22. ( m pl m i ) \left(\frac{m\text{pl}}{m_{i}}\right)
  23. 1 = m pl m i + m p m i + m s m i = m pl m i + ζ + m s m i 1=\frac{m\text{pl}}{m_{i}}+\frac{m_{p}}{m_{i}}+\frac{m_{s}}{m_{i}}=\frac{m% \text{pl}}{m_{i}}+\zeta+\frac{m_{s}}{m_{i}}
  24. m s m i = 1 - ζ - m pl m i \frac{m_{s}}{m_{i}}=1-\zeta-\frac{m\text{pl}}{m_{i}}
  25. m i = G L O W = m pl 1 - ( ζ 1 - λ ) m_{i}=GLOW=\frac{m\text{pl}}{1-\left(\frac{\zeta}{1-\lambda}\right)}
  26. ( Δ v , m pl ) \left(\Delta v,m\text{pl}\right)
  27. ( I sp ) \left(I\text{sp}\right)
  28. λ max = 1 - ζ = 1 M R \lambda\text{max}=1-\zeta=\frac{1}{MR}

Single_transferable_vote.html

  1. votes needed to win = ( valid votes cast seats to fill + 1 ) + 1 \mbox{votes needed to win}~{}=\left({{\rm\mbox{valid votes cast}~{}}\over{\rm% \mbox{seats to fill}~{}}+1}\right)+1
  2. ( 20 votes cast 3 seats to fill + 1 ) + 1 = 6 votes required \left({\mbox{20 votes cast}~{}\over{\mbox{3 seats to fill}~{}+1}}\right)+1=% \mbox{6 votes required}~{}

Singularity_(mathematics).html

  1. f ( x ) = 1 x f(x)=\frac{1}{x}
  2. { ( x , y ) : | x | = | y | } \{(x,y):|x|=|y|\}
  3. f ( x ) f(x)
  4. x x
  5. c c
  6. f ( c - ) f(c^{-})
  7. f ( c + ) f(c^{+})
  8. f ( c - ) = lim x c f ( x ) f(c^{-})=\lim_{x\to c}f(x)
  9. x < c x<c
  10. f ( c + ) = lim x c f ( x ) f(c^{+})=\lim_{x\to c}f(x)
  11. x > c x>c
  12. f ( c - ) f(c^{-})
  13. f ( x ) f(x)
  14. x x
  15. c c
  16. f ( c + ) f(c^{+})
  17. f ( x ) f(x)
  18. x x
  19. c c
  20. x = c x=c
  21. g ( x ) = sin ( 1 x ) g(x)=\sin\left(\frac{1}{x}\right)
  22. x x
  23. c = 0 c=0
  24. g ( x ) g(x)
  25. c c
  26. f ( c - ) = f ( c ) = f ( c + ) f(c^{-})=f(c)=f(c^{+})
  27. f ( c - ) f(c^{-})
  28. f ( c + ) f(c^{+})
  29. f ( c - ) f ( c + ) f(c^{-})\neq f(c^{+})
  30. f ( c ) f(c)
  31. x x
  32. f ( c ) f(c)
  33. f ( c - ) f ( c + ) f(c^{-})\neq f(c^{+})
  34. f ( c ) f(c)
  35. f ( c - ) = f ( c + ) f(c^{-})=f(c^{+})
  36. f ( c ) f(c)
  37. x = c x=c
  38. f ( c - ) f(c^{-})
  39. f ( c + ) f(c^{+})
  40. f ( c - ) f(c^{-})
  41. f ( c + ) f(c^{+})
  42. ± \pm\infty
  43. z \sqrt{z}
  44. log ( z ) \log(z)
  45. z = 0 z=0
  46. z = z=\infty
  47. log ( z ) \log(z)
  48. x - α , x^{-\alpha},
  49. x - 1 . x^{-1}.
  50. ( t 0 - t ) - α (t_{0}-t)^{-\alpha}
  51. - t -t
  52. t 0 t_{0}
  53. y 2 x 3 = 0 y^{2}−x^{3}=0
  54. x = y = 0 x=y=0
  55. x x
  56. x x

Sirius.html

  1. [ F e H ] = 0.5 \begin{smallmatrix}[\frac{Fe}{H}]=0.5\end{smallmatrix}

Skewness.html

  1. ( μ - ν ) / σ , (\mu-\nu)/\sigma,
  2. γ 1 = E [ ( X - μ σ ) 3 ] = μ 3 σ 3 = E [ ( X - μ ) 3 ] ( E [ ( X - μ ) 2 ] ) 3 / 2 = κ 3 κ 2 3 / 2 , \gamma_{1}=\operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^{3}\right]=% \frac{\mu_{3}}{\sigma^{3}}=\frac{\operatorname{E}\left[(X-\mu)^{3}\right]}{\ % \ \ (\operatorname{E}\left[(X-\mu)^{2}\right])^{3/2}}=\frac{\kappa_{3}}{\kappa% _{2}^{3/2}},
  3. γ 1 \displaystyle\gamma_{1}
  4. Pr [ X > x ] = x - 3 for x > 1 , Pr [ X < 1 ] = 0 \Pr\left[X>x\right]=x^{-3}\mbox{ for }~{}x>1,\ \Pr[X<1]=0
  5. Pr [ X < x ] = ( 1 - x ) - 3 / 2 for negative x and Pr [ X > x ] = ( 1 + x ) - 3 / 2 for positive x . \Pr[X<x]=(1-x)^{-3}/2\mbox{ for negative }~{}x\mbox{ and }~{}\Pr[X>x]=(1+x)^{-% 3}/2\mbox{ for positive }~{}x.
  6. Pr [ X > x ] = x - 2 for x > 1 , Pr [ X < 1 ] = 0 \Pr\left[X>x\right]=x^{-2}\mbox{ for }~{}x>1,\ \Pr[X<1]=0
  7. Skew [ Y ] = Skew [ X ] / n \mbox{Skew}~{}[Y]=\mbox{Skew}~{}[X]/\sqrt{n}
  8. b 1 = m 3 s 3 = 1 n i = 1 n ( x i - x ¯ ) 3 [ 1 n - 1 i = 1 n ( x i - x ¯ ) 2 ] 3 / 2 , b_{1}=\frac{m_{3}}{s^{3}}=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})% ^{3}}{\left[\tfrac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\right]^{3/2}}\ ,
  9. x ¯ \scriptstyle\overline{x}
  10. G 1 = k 3 k 2 3 / 2 = n 2 ( n - 1 ) ( n - 2 ) m 3 s 3 , G_{1}=\frac{k_{3}}{k_{2}^{3/2}}=\frac{n^{2}}{(n-1)(n-2)}\;\frac{m_{3}}{s^{3}},
  11. k 3 k_{3}
  12. k 2 = s 2 k_{2}=s^{2}
  13. b 1 b_{1}
  14. G 1 G_{1}
  15. γ 1 \gamma_{1}
  16. G 1 G_{1}
  17. b 1 b_{1}
  18. G 1 G_{1}
  19. var ( G 1 ) = 6 n ( n - 1 ) ( n - 2 ) ( n + 1 ) ( n + 3 ) . \operatorname{var}(G_{1})=\frac{6n(n-1)}{(n-2)(n+1)(n+3)}.
  20. b 1 b_{1}
  21. var ( b 1 ) < var ( m 3 m 2 3 / 2 ) < var ( G 1 ) , \operatorname{var}(b_{1})<\operatorname{var}\left(\frac{m_{3}}{m_{2}^{3/2}}% \right)<\operatorname{var}(G_{1}),
  22. G 1 G_{1}
  23. γ ( u ) = F - 1 ( u ) + F - 1 ( 1 - u ) - 2 F - 1 ( 1 / 2 ) F - 1 ( u ) - F - 1 ( 1 - u ) \gamma(u)=\frac{F^{-1}(u)+F^{-1}(1-u)-2F^{-1}(1/2)}{F^{-1}(u)-F^{-1}(1-u)}
  24. \|\cdot\|
  25. dSkew ( X ) := 1 - E X - X E X + X if Pr ( X = 0 ) 1 \operatorname{dSkew}(X):=1-\frac{\operatorname{E}\|X-X^{\prime}\|}{% \operatorname{E}\|X+X^{\prime}\|}\,\text{ if }\Pr(X=0)\neq 1
  26. dSkew n ( X ) := 1 - i , j x i - x j i , j x i + x j . \operatorname{dSkew}_{n}(X):=1-\frac{\sum_{i,j}\|x_{i}-x_{j}\|}{\sum_{i,j}\|x_% {i}+x_{j}\|}.
  27. skew ( X ) = ( μ - ν ) E ( | X - ν | ) \mathrm{skew}(X)=\frac{(\mu-\nu)}{E(|X-\nu|)}
  28. h ( x i , x j ) = ( x i - x m ) - ( x m - x j ) x i - x j h(x_{i},x_{j})=\frac{(x_{i}-x_{m})-(x_{m}-x_{j})}{x_{i}-x_{j}}
  29. ( x i , x j ) (x_{i},x_{j})
  30. x i x m x j x_{i}\geq x_{m}\geq x_{j}
  31. x m x_{m}
  32. { x 1 , x 2 , , x n } \{x_{1},x_{2},\ldots,x_{n}\}

Skyscraper.html

  1. Simple price of floor area (currency/ m 2 ) = price of land area (currency) total floor area ( m 2 ) \,\text{Simple price of floor area (currency/}\mathrm{m}^{2}\,\text{)}=\frac{% \,\text{price of land area (currency)}}{\,\text{total floor area (}\mathrm{m}^% {2}\,\text{)}}

Slide_rule.html

  1. log ( x y ) = log ( x ) + log ( y ) \log(xy)=\log(x)+\log(y)
  2. log ( x / y ) = log ( x ) - log ( y ) \log(x/y)=\log(x)-\log(y)
  3. log ( x ) \log(x)
  4. x x
  5. y y
  6. log ( y ) \log(y)
  7. log ( x ) + log ( y ) \log(x)+\log(y)
  8. log ( x ) + log ( y ) = log ( x y ) \log(x)+\log(y)=\log(xy)
  9. x y xy
  10. x x
  11. y y
  12. e x e^{x}
  13. x 2 x^{2}
  14. x y x^{y}
  15. x y / 2 x^{y/2}
  16. k sin x k\sin x
  17. x + y = ( x y + 1 ) y . x+y=\left(\frac{x}{y}+1\right)y.
  18. x - y = ( x y - 1 ) y . x-y=\left(\frac{x}{y}-1\right)y.

Slope.html

  1. m > 0 m>0
  2. m < 0 m<0
  3. m = y 2 - y 1 x 2 - x 1 . m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}.
  4. m = tan ( θ ) m=\tan(\theta)\!
  5. m = Δ y Δ x = vertical change horizontal change = rise run . m=\frac{\Delta y}{\Delta x}=\frac{\,\text{vertical}\,\,\text{change}}{\,\text{% horizontal}\,\,\text{change}}=\frac{\,\text{rise}}{\,\text{run}}.
  6. m = y 2 - y 1 x 2 - x 1 . m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}.
  7. m = Δ y Δ x = y 2 - y 1 x 2 - x 1 = 8 - 2 13 - 1 = 6 12 = 1 2 m=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-2}{13-1}=% \frac{6}{12}=\frac{1}{2}
  8. m = 21 - 15 3 - 4 = 6 - 1 = - 6. m=\frac{21-15}{3-4}=\frac{6}{-1}=-6.
  9. y = m x + b y=mx+b\,
  10. y - y 1 = m ( x - x 1 ) . y-y_{1}=m(x-x_{1}).\!
  11. a x + b y + c = 0 ax+by+c=0\,
  12. - a b -\frac{a}{b}\;
  13. m = tan ( θ ) m=\tan(\theta)
  14. θ = arctan ( m ) \theta=\arctan(m)
  15. ( 20 - 8 ) ( 3 - 2 ) = 12. \frac{(20-8)}{(3-2)}\;=12.\,
  16. y - 8 = 12 ( x - 2 ) = 12 x - 24 y-8=12(x-2)=12x-24\,
  17. y = 12 x - 16. y=12x-16.\,
  18. θ = arctan ( 12 ) 85.2 . \theta=\arctan(12)\approx 85.2^{\circ}\,.
  19. angle = arctan ( slope 100 % ) \,\text{angle}=\arctan\left(\frac{\,\text{slope}}{100\%}\right)
  20. slope = 100 % tan ( angle ) , \mbox{slope}~{}=100\%\cdot\tan(\mbox{angle}~{}),\,
  21. m = Δ y Δ x m=\frac{\Delta y}{\Delta x}
  22. d y d x = lim Δ x 0 Δ y Δ x \frac{dy}{dx}=\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}

Slugging_percentage.html

  1. S L G = ( 1 B ) + ( 2 × 2 B ) + ( 3 × 3 B ) + ( 4 × 𝐻𝑅 ) A B SLG=\frac{(\mathit{1B})+(2\times\mathit{2B})+(3\times\mathit{3B})+(4\times% \mathit{HR})}{AB}
  2. [ 0 , 4 ] \left[0,4\right]
  3. RC = ( Hits + Walks ) ( Total Bases ) At Bats + Walks \,\text{RC}=\frac{(\,\text{Hits}+\,\text{Walks})(\,\text{Total Bases})}{\,% \text{At Bats}+\,\text{Walks}}

Smalltalk.html

  1. f f
  2. f ( x ) = x + 1 f(x)=x+1
  3. λ x \lambda x
  4. x + 1 x+1
  5. f ( 3 ) = 3 + 1 f(3)=3+1
  6. ( λ x : x + 1 ) 3 β 4 (\lambda x:x+1)3_{\beta}\rightarrow 4

Snell's_law.html

  1. sin θ 1 sin θ 2 = v 1 v 2 = λ 1 λ 2 = n 2 n 1 \frac{\sin\theta_{1}}{\sin\theta_{2}}=\frac{v_{1}}{v_{2}}=\frac{\lambda_{1}}{% \lambda_{2}}=\frac{n_{2}}{n_{1}}
  2. θ \theta
  3. v v
  4. λ \lambda
  5. n n
  6. n 1 n_{1}
  7. n 2 n_{2}
  8. sin θ 1 sin θ 2 = v 1 v 2 = λ 1 λ 2 \frac{\sin\theta_{1}}{\sin\theta_{2}}=\frac{v_{1}}{v_{2}}=\frac{\lambda_{1}}{% \lambda_{2}}
  9. k \vec{k}
  10. ( k x , k y , 0 ) (k_{x},k_{y},0)
  11. z , x z,x
  12. k x Region 1 = k x Region 2 k_{x\,\text{Region}_{1}}=k_{x\,\text{Region}_{2}}
  13. k x Region 1 = k x Region 2 k_{x\,\text{Region}_{1}}=k_{x\,\text{Region}_{2}}\,
  14. n 1 k 0 sin θ 1 = n 2 k 0 sin θ 2 n_{1}k_{0}\sin\theta_{1}=n_{2}k_{0}\sin\theta_{2}\,
  15. n 1 sin θ 1 = n 2 sin θ 2 n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}\,
  16. k 0 = 2 π λ 0 = ω c k_{0}=\frac{2\pi}{\lambda_{0}}=\frac{\omega}{c}
  17. θ 1 \theta_{1}
  18. θ 2 \theta_{2}
  19. cos θ 1 = - 𝐧 𝐥 \cos\theta_{1}=-\mathbf{n}\cdot\mathbf{l}
  20. cos θ 1 \cos\theta_{1}
  21. n 1 n_{1}
  22. cos θ 1 \cos\theta_{1}
  23. 𝐯 reflect = 𝐥 + 2 cos θ 1 𝐧 \mathbf{v}_{\mathrm{reflect}}=\mathbf{l}+2\cos\theta_{1}\mathbf{n}
  24. sin θ 2 = ( n 1 n 2 ) sin θ 1 = ( n 1 n 2 ) 1 - ( cos θ 1 ) 2 \sin\theta_{2}=\left(\frac{n_{1}}{n_{2}}\right)\sin\theta_{1}=\left(\frac{n_{1% }}{n_{2}}\right)\sqrt{1-\left(\cos\theta_{1}\right)^{2}}
  25. cos θ 2 = 1 - sin θ 2 2 = 1 - ( n 1 n 2 ) 2 ( 1 - ( cos θ 1 ) 2 ) \cos\theta_{2}=\sqrt{1-\sin\theta_{2}^{2}}=\sqrt{1-\left(\frac{n_{1}}{n_{2}}% \right)^{2}\left(1-\left(\cos\theta_{1}\right)^{2}\right)}
  26. 𝐯 refract = ( n 1 n 2 ) 𝐥 + ( n 1 n 2 cos θ 1 - cos θ 2 ) 𝐧 \mathbf{v}_{\mathrm{refract}}=\left(\frac{n_{1}}{n_{2}}\right)\mathbf{l}+\left% (\frac{n_{1}}{n_{2}}\cos\theta_{1}-\cos\theta_{2}\right)\mathbf{n}
  27. r = n 1 / n 2 r=n_{1}/n_{2}
  28. c = - 𝐧 𝐥 c=-\mathbf{n}\cdot\mathbf{l}
  29. 𝐯 refract = r 𝐥 + ( r c - 1 - r 2 ( 1 - c 2 ) ) 𝐧 \mathbf{v}_{\mathrm{refract}}=r\mathbf{l}+\left(rc-\sqrt{1-r^{2}\left(1-c^{2}% \right)}\right)\mathbf{n}
  30. 𝐥 = { 0.707107 , - 0.707107 } , 𝐧 = { 0 , 1 } , r = n 1 n 2 = 0.9 \mathbf{l}=\{0.707107,-0.707107\},~{}\mathbf{n}=\{0,1\},~{}r=\frac{n_{1}}{n_{2% }}=0.9
  31. c = cos θ 1 = 0.707107 , 1 - r 2 ( 1 - c 2 ) = cos θ 2 = 0.771362 c=\cos\theta_{1}=0.707107,~{}\sqrt{1-r^{2}\left(1-c^{2}\right)}=\cos\theta_{2}% =0.771362
  32. 𝐯 reflect = { 0.707107 , 0.707107 } , 𝐯 refract = { 0.636396 , - 0.771362 } \mathbf{v}_{\mathrm{reflect}}=\{0.707107,0.707107\},~{}\mathbf{v}_{\mathrm{% refract}}=\{0.636396,-0.771362\}
  33. cos θ 2 \cos\theta_{2}
  34. n 2 < n 1 n_{2}<n_{1}
  35. sin θ 2 = n 1 n 2 sin θ 1 = 1.333 1 sin ( 50 ) = 1.333 0.766 = 1.021 , \sin\theta_{2}=\frac{n_{1}}{n_{2}}\sin\theta_{1}=\frac{1.333}{1}\cdot\sin\left% (50^{\circ}\right)=1.333\cdot 0.766=1.021,
  36. θ crit = arcsin ( n 2 n 1 sin θ 2 ) = arcsin n 2 n 1 = 48.6 . \theta\text{crit}=\arcsin\left(\frac{n_{2}}{n_{1}}\sin\theta_{2}\right)=% \arcsin\frac{n_{2}}{n_{1}}=48.6^{\circ}.

Software_bug.html

  1. P ( p , s ) = B - k p S P(p,s)=B-\lceil kpS\rceil

Software_engineering.html

  1. T = k * ( S L O C ) ( 1 + x ) T=k*(SLOC)^{(1+x)}

Soil.html

  1. q = - k T \overrightarrow{q}=-k{\nabla}T
  2. q \overrightarrow{q}
  3. . k . \big.k\big.
  4. . T . \big.\nabla T\big.
  5. k k
  6. k k
  7. k k
  8. q x = - k d T d x q_{x}=-k\frac{dT}{dx}

Solid-fuel_rocket.html

  1. A s A_{s}
  2. b r b_{r}
  3. m ˙ = ρ A s b r \dot{m}=\rho\cdot A_{s}\cdot b_{r}

Solubility.html

  1. ( ln N i P ) T = - V i , a q - V i , c r R T \left(\frac{\partial\ln N_{i}}{\partial P}\right)_{T}=-\frac{V_{i,aq}-V_{i,cr}% }{RT}
  2. p = k H c p=k_{\rm H}\,c
  3. l o g S ( V m ) = Δ G s o l v a t i o n - 2.303 R T logS(V_{m})=\frac{\Delta G_{solvation}}{-2.303RT}

Solubility_equilibrium.html

  1. log ( * K A ) = log ( * K A 0 ) + γ A m 3.454 R T \log(^{*}K_{A})=\log(^{*}K_{A\to 0})+\frac{\gamma A_{m}}{3.454RT}
  2. K A * {}^{*}K_{A}
  3. K A 0 * {}^{*}K_{A\to 0}
  4. ( ln N i P ) T = - V i , a q - V i , c r R T \left(\frac{\partial\ln N_{i}}{\partial P}\right)_{T}=-\frac{V_{i,aq}-V_{i,cr}% }{RT}
  5. C 12 H 22 O 11 ( s ) C 12 H 22 O 11 ( aq ) \mathrm{{C}_{12}{H}_{22}{O}_{11}(s)}\rightleftharpoons\mathrm{{C}_{12}{H}_{22}% {O}_{11}(aq)}
  6. K = { C 12 H 22 O 11 ( a q ) } { C 12 H 22 O 11 ( s ) } K^{\ominus}=\frac{\left\{\mathrm{{C}_{12}{H}_{22}{O}_{11}}(aq)\right\}}{\left% \{\mathrm{{C}_{12}{H}_{22}{O}_{11}}(s)\right\}}
  7. K = { C 12 H 22 O 11 ( a q ) } K^{\ominus}=\left\{\mathrm{{C}_{12}{H}_{22}{O}_{11}}(aq)\right\}
  8. K s = [ C 12 H 22 O 11 ( a q ) ] K_{s}=\left[\mathrm{{C}_{12}{H}_{22}{O}_{11}}(aq)\right]\,
  9. CaSO 4 ( s ) Ca ( a q ) 2 + + SO ( a q ) 4 2 - \mathrm{CaSO}_{4}(s)\rightleftharpoons\mbox{Ca}~{}^{2+}(aq)+\mbox{SO}~{}_{4}^{% 2-}(aq)\,
  10. K = { Ca ( a q ) 2 + } { SO ( a q ) 4 2 - } { CaSO ( s ) 4 } = { Ca ( a q ) 2 + } { SO ( a q ) 4 2 - } K^{\ominus}=\frac{\left\{\mbox{Ca}~{}^{2+}(aq)\right\}\left\{\mbox{SO}~{}_{4}^% {2-}(aq)\right\}}{\left\{\mbox{CaSO}~{}_{4}(s)\right\}}=\left\{\mbox{Ca}~{}^{2% +}(aq)\right\}\left\{\mbox{SO}~{}_{4}^{2-}(aq)\right\}
  11. K sp = [ Ca ( a q ) 2 + ] [ SO ( a q ) 4 2 - ] . K_{\mathrm{sp}}=\left[\mbox{Ca}~{}^{2+}(aq)\right]\left[\mbox{SO}~{}_{4}^{2-}(% aq)\right].\,
  12. [ A ] = K sp ( q / p ) q p + q [A]=\sqrt[p+q]{K_{\mathrm{sp}}\over{(q/p)^{q}}}
  13. S = [ A ] p = [ B ] q = K sp ( q / p ) q p p + q p + q = K sp q q p p p + q S={[A]\over p}={[B]\over q}=\sqrt[p+q]{K_{\mathrm{sp}}\over{(q/p)^{q}}p^{p+q}}% =\sqrt[p+q]{K_{\mathrm{sp}}\over{q^{q}}p^{p}}
  14. S = K s p S=\sqrt{K_{sp}}
  15. S = K s p 4 3 S=\sqrt[3]{K_{sp}\over 4}
  16. S = K s p 108 5 S=\sqrt[5]{K_{sp}\over 108}
  17. K s p \sqrt{K_{sp}}

Sonoluminescence.html

  1. R R ¨ + 3 2 R ˙ 2 = 1 ρ ( p g - P 0 - P ( t ) - 4 μ R ˙ R - 2 γ R ) R\ddot{R}+\frac{3}{2}\dot{R}^{2}=\frac{1}{\rho}\left(p_{g}-P_{0}-P(t)-4\mu% \frac{\dot{R}}{R}-\frac{2\gamma}{R}\right)

Sophie_Germain.html

  1. N 2 ( 4 z x 4 + 4 z x 2 y 2 + 4 z y 4 ) + 2 z t 2 = 0 N^{2}\left(\frac{\partial^{4}z}{\partial x^{4}}+\frac{\partial^{4}z}{\partial x% ^{2}\partial y^{2}}+\frac{\partial^{4}z}{\partial y^{4}}\right)+\frac{\partial% ^{2}z}{\partial t^{2}}=0
  2. y y
  3. z z
  4. 4 ( x p - 1 ) x - 1 = y 2 ± p z 2 \textstyle\frac{4(x^{p}-1)}{x-1}=y^{2}\pm pz^{2}
  5. n n
  6. θ θ
  7. θ = k n + 1 θ =kn+ 1
  8. n n
  9. θ θ
  10. x x
  11. y y
  12. n n
  13. p p
  14. 2 p + 1 2p+ 1
  15. k 1 + k 2 2 \frac{k_{1}+k_{2}}{2}
  16. x 4 + 4 y 4 = ( ( x + y ) 2 + y 2 ) ( ( x - y ) 2 + y 2 ) = ( x 2 + 2 x y + 2 y 2 ) ( x 2 - 2 x y + 2 y 2 ) . x^{4}+4y^{4}=((x+y)^{2}+y^{2})((x-y)^{2}+y^{2})=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2% y^{2}).\,

Sorting_algorithm.html

  1. log 2 n \log^{2}n
  2. ( log n ) 2 (\log n)^{2}
  3. n n
  4. Ω ( n log n ) \Omega(n\log n)
  5. n + 2 k n+2^{k}
  6. n + 2 k n+2^{k}
  7. 2 k 2^{k}
  8. n + k n+k
  9. n 2 k n^{2}\cdot k
  10. n k n\cdot k
  11. n + r n+r
  12. n + r n+r
  13. n + r n+r
  14. n + r n+r
  15. n + r n+r
  16. n + r n+r
  17. n k d n\cdot\frac{k}{d}
  18. n k d n\cdot\frac{k}{d}
  19. n + 2 d n+2^{d}
  20. n k d n\cdot\frac{k}{d}
  21. n k d n\cdot\frac{k}{d}
  22. n + 2 d n+2^{d}
  23. n k d n\cdot\frac{k}{d}
  24. n k d n\cdot\frac{k}{d}
  25. 2 d 2^{d}
  26. k d \frac{k}{d}
  27. n k d n\cdot\frac{k}{d}
  28. n ( k s + d ) n\cdot\left({\frac{k}{s}+d}\right)
  29. k d 2 d \frac{k}{d}\cdot 2^{d}
  30. O ( n log log n ) O\bigl(n\sqrt{\log\log n}\bigr)

Space_elevator.html

  1. g r = - G M / r 2 gr=-G\cdot M/r^{2}
  2. a = ω 2 r a=\omega^{2}\cdot r
  3. g = - G M / r 2 + ω 2 r g=-G\cdot M/r^{2}+\omega^{2}\cdot r
  4. r 1 = ( G M / ω 2 ) 1 / 3 r_{1}=(G\cdot M/\omega^{2})^{1/3}
  5. σ d S = g ρ S d r \sigma\cdot dS=g\cdot\rho\cdot S\cdot dr
  6. d S / S = g ρ / σ d r dS/S=g\cdot\rho/\sigma\cdot dr
  7. d S / S = ρ / σ ( G M / r 2 - ω 2 r ) d r dS/S=\rho/\sigma\cdot(G\cdot M/r^{2}-\omega^{2}\cdot r)\cdot dr
  8. Δ [ ln ( S ) ] = r 0 r 1 - ρ / σ Δ [ G M / r + ω 2 r 2 / 2 ] r 0 r 1 \Delta\left[\ln(S)\right]{}_{r_{0}}^{r_{1}}=-\rho/\sigma\cdot\Delta\left[G% \cdot M/r+\omega^{2}\cdot r^{2}/2\right]{}_{r_{0}}^{r_{1}}
  9. Δ [ ln ( S ) ] = ρ / σ g 0 r 0 ( 1 + x / 2 - 3 / 2 x 1 / 3 ) \Delta\left[\ln(S)\right]=\rho/\sigma\cdot g_{0}\cdot r_{0}\cdot(1+x/2-3/2% \cdot x^{1/3})
  10. S 0 = S 1 . e ρ / σ g 0 r 0 ( 1 + x / 2 - 3 / 2 x 1 / 3 ) S_{0}=S_{1}.e^{\rho/\sigma\cdot g_{0}\cdot r_{0}\cdot(1+x/2-3/2\cdot x^{1/3})}
  11. x = ω 2 r 0 / g 0 x=\omega^{2}\cdot r_{0}/g_{0}
  12. Δ [ ln ( S ) ] = ρ / σ g 0 r 0 ( 1 + x / 2 - 3 / 2 x 1 / 3 ) \Delta\left[\ln(S)\right]=\rho/\sigma\cdot g_{0}\cdot r_{0}\cdot(1+x/2-3/2% \cdot x^{1/3})
  13. x = w 2 r 0 / g 0 . x=w^{2}\cdot r_{0}/g_{0}.
  14. σ = ρ r 0 g 0 . \sigma=\rho\cdot r_{0}\cdot g_{0}.

Space_suit.html

  1. W = V i V f P d V W=\int_{V_{i}}^{V_{f}}\,P\,dV

Spacecraft_propulsion.html

  1. I s p I_{sp}
  2. v e v_{e}
  3. I s p g n = v e I_{sp}g_{\mathrm{n}}=v_{e}
  4. Δ v \Delta v
  5. Δ v \Delta v
  6. v e v_{e}
  7. Δ v = v e ln ( M + P P ) . \Delta v=v_{e}\ln\left(\frac{M+P}{P}\right).
  8. v e v_{e}
  9. v e = I s p g o v_{e}=I_{sp}g_{o}
  10. I s p I_{sp}
  11. g o g_{o}
  12. Δ v \Delta v
  13. I s p I_{sp}
  14. M = P ( e Δ v / v e - 1 ) . M=P\left(e^{\Delta v/v_{e}}-1\right).
  15. Δ v \Delta v
  16. Δ v \Delta v
  17. 1 2 m ˙ v e 2 \frac{1}{2}\dot{m}v_{e}^{2}
  18. I s p I_{sp}
  19. I s p I_{sp}
  20. v e v_{e}
  21. v e v_{e}
  22. v e v_{e}
  23. m 1 m_{1}
  24. m 0 - m 1 m_{0}-m_{1}
  25. 1 2 ( m 0 - m 1 ) v e 2 \frac{1}{2}(m_{0}-m_{1})v\text{e}^{2}
  26. v e v_{e}
  27. d ( 1 2 v 2 ) = v d v = v v e d m / m = 1 2 ( v e 2 - ( v - v e ) 2 + v 2 ) d m / m d\left(\frac{1}{2}v^{2}\right)=vdv=vv\text{e}dm/m=\frac{1}{2}\left(v\text{e}^{% 2}-(v-v\text{e})^{2}+v^{2}\right)dm/m
  28. Δ ϵ = v d ( Δ v ) \Delta\epsilon=\int v\,d(\Delta v)
  29. ϵ \epsilon
  30. Δ v \Delta v
  31. v v
  32. v v
  33. v e 2 / 2 \scriptstyle{v\text{e}^{2}/2}
  34. v e v\text{e}
  35. E E
  36. E = 1 2 m 1 ( e Δ v / v e - 1 ) v e 2 E=\frac{1}{2}m_{1}\left(e^{\Delta v\ /v\text{e}}-1\right)v\text{e}^{2}
  37. Δ v v e \Delta v\ll v_{e}
  38. E 1 2 m 1 v e Δ v E\approx\frac{1}{2}m_{1}v\text{e}\Delta v
  39. Δ v \Delta v
  40. v e = 0.6275 Δ v v\text{e}=0.6275\Delta v
  41. E = 0.772 m 1 ( Δ v ) 2 E=0.772m_{1}(\Delta v)^{2}
  42. Δ v \Delta v
  43. Δ v \Delta v
  44. Δ v \Delta v
  45. Δ v \Delta v
  46. Δ v \Delta v
  47. Δ v \Delta v
  48. g g
  49. P = 1 2 m a v e = 1 2 F v e P=\frac{1}{2}mav\text{e}=\frac{1}{2}Fv\text{e}
  50. F F
  51. a a
  52. a a
  53. v e v\text{e}
  54. v e v\text{e}
  55. Δ v v e \Delta v\ll v\text{e}
  56. t m v e Δ v 2 P t\approx\frac{mv\text{e}\Delta v}{2P}
  57. Δ v \Delta v
  58. v e v\text{e}
  59. Δ v \Delta v
  60. v e v\text{e}
  61. v e v\text{e}
  62. P F = 1 2 m ˙ v 2 m ˙ v = 1 2 v \frac{P}{F}=\frac{\frac{1}{2}{\dot{m}v^{2}}}{\dot{m}v}=\frac{1}{2}v
  63. F = P 1 2 v = 2 P v F=\frac{P}{\frac{1}{2}v}=\frac{2P}{v}
  64. Δ v \Delta v

Spacetime.html

  1. ( x , y , z , t ) (x,y,z,t)
  2. ( x 0 , x 1 , x 2 , x 3 ) = ( c t , x , y , z ) (x_{0},x_{1},x_{2},x_{3})=(ct,x,y,z)
  3. ( x 1 , x 2 , x 3 , x 4 ) = ( x , y , z , i c t ) (x_{1},x_{2},x_{3},x_{4})=(x,y,z,ict)
  4. c c
  5. s 2 = Δ r 2 - c 2 Δ t 2 s^{2}=\Delta r^{2}-c^{2}\Delta t^{2}\,
  6. s 2 s^{2}
  7. c 2 Δ t 2 c^{2}\Delta t^{2}
  8. Δ r 2 \Delta r^{2}
  9. c 2 Δ t 2 > Δ r 2 s 2 < 0 \begin{aligned}\\ \displaystyle c^{2}\Delta t^{2}&\displaystyle>\Delta r^{2}\\ \displaystyle s^{2}&\displaystyle<0\\ \end{aligned}
  10. s 2 < 0 s^{2}<0
  11. Δ τ \Delta\tau
  12. Δ τ = Δ t 2 - Δ r 2 c 2 \Delta\tau=\sqrt{\Delta t^{2}-\frac{\Delta r^{2}}{c^{2}}}
  13. c 2 Δ t 2 = Δ r 2 s 2 = 0 \begin{aligned}\displaystyle c^{2}\Delta t^{2}&\displaystyle=\Delta r^{2}\\ \displaystyle s^{2}&\displaystyle=0\\ \end{aligned}
  14. s 2 = 0 s^{2}=0
  15. c c
  16. c 2 Δ t 2 < Δ r 2 s 2 > 0 \begin{aligned}\\ \displaystyle c^{2}\Delta t^{2}&\displaystyle<\Delta r^{2}\\ \displaystyle s^{2}&\displaystyle>0\\ \end{aligned}
  17. s 2 > 0 s^{2}>0
  18. Δ σ \Delta\sigma
  19. Δ σ = s 2 = Δ r 2 - c 2 Δ t 2 \Delta\sigma=\sqrt{s^{2}}=\sqrt{\Delta r^{2}-c^{2}\Delta t^{2}}
  20. ( cosh ϕ sinh ϕ sinh ϕ cosh ϕ ) , \begin{pmatrix}\cosh\phi&\sinh\phi\\ \sinh\phi&\cosh\phi\end{pmatrix},
  21. e j ϕ = cosh ϕ + j sinh ϕ . e^{j\phi}=\cosh\phi\ +j\ \sinh\phi.
  22. ( M , g ) (M,g)
  23. g g
  24. ( 3 , 1 ) (3,1)
  25. ( x , y , z , t ) (x,y,z,t)
  26. c c
  27. p p
  28. p p
  29. p p
  30. p p
  31. q q
  32. p p
  33. ( x , y , z , t ) (x,y,z,t)
  34. η \eta
  35. η a b = diag ( 1 , - 1 , - 1 , - 1 ) \eta_{ab}\,=\operatorname{diag}(1,-1,-1,-1)
  36. | s < s u p > 2 | \sqrt{|s<sup>2|}
  37. s 2 = g α β Δ x α Δ x β s^{2}=g_{\alpha\beta}\Delta x^{\alpha}\Delta x^{\beta}

Special_relativity.html

  1. t \displaystyle t^{\prime}
  2. γ = 1 1 - v 2 c 2 \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  3. Δ x = x 2 - x 1 , Δ x = x 2 - x 1 , Δ t = t 2 - t 1 , Δ t = t 2 - t 1 , \begin{array}[]{ll}\Delta x^{\prime}=x^{\prime}_{2}-x^{\prime}_{1}\ ,&\Delta x% =x_{2}-x_{1}\ ,\\ \Delta t^{\prime}=t^{\prime}_{2}-t^{\prime}_{1}\ ,&\Delta t=t_{2}-t_{1}\ ,\\ \end{array}
  4. Δ x = γ ( Δ x - v Δ t ) , Δ x = γ ( Δ x + v Δ t ) , Δ t = γ ( Δ t - v Δ x c 2 ) , Δ t = γ ( Δ t + v Δ x c 2 ) . \begin{array}[]{ll}\Delta x^{\prime}=\gamma\ (\Delta x-v\,\Delta t)\ ,&\Delta x% =\gamma\ (\Delta x^{\prime}+v\,\Delta t^{\prime})\ ,\\ \Delta t^{\prime}=\gamma\ \left(\Delta t-\dfrac{v\,\Delta x}{c^{2}}\right)\ ,&% \Delta t=\gamma\ \left(\Delta t^{\prime}+\dfrac{v\,\Delta x^{\prime}}{c^{2}}% \right)\ .\\ \end{array}
  5. Δ t = γ ( Δ t - v Δ x c 2 ) \Delta t^{\prime}=\gamma\left(\Delta t-\frac{v\,\Delta x}{c^{2}}\right)
  6. Δ t = γ Δ t \Delta t^{\prime}=\gamma\,\Delta t
  7. Δ x = 0 . \Delta x=0\ .
  8. Δ x = Δ x γ \Delta x^{\prime}=\frac{\Delta x}{\gamma}
  9. Δ t = 0 . \Delta t^{\prime}=0\ .
  10. u = d x d t = γ ( d x - v d t ) γ ( d t - v d x / c 2 ) = ( d x / d t ) - v 1 - ( v / c 2 ) ( d x / d t ) = u - v 1 - u v / c 2 . u^{\prime}=\frac{dx^{\prime}}{dt^{\prime}}=\frac{\gamma\ (dx-vdt)}{\gamma\ (dt% -vdx/c^{2})}=\frac{(dx/dt)-v}{1-(v/c^{2})(dx/dt)}=\frac{u-v}{1-uv/c^{2}}\ .
  11. u = d x d t = γ ( d x + v d t ) γ ( d t + v d x / c 2 ) = ( d x / d t ) + v 1 + ( v / c 2 ) ( d x / d t ) = u + v 1 + u v / c 2 . u=\frac{dx}{dt}=\frac{\gamma\ (dx^{\prime}+vdt^{\prime})}{\gamma\ (dt^{\prime}% +vdx^{\prime}/c^{2})}=\frac{(dx^{\prime}/dt^{\prime})+v}{1+(v/c^{2})(dx^{% \prime}/dt^{\prime})}=\frac{u^{\prime}+v}{1+u^{\prime}v/c^{2}}\ .
  12. u u - v . u^{\prime}\approx u-v\ .
  13. p = m γ v p=m\gamma v
  14. v v
  15. d s 2 = d 𝐱 d 𝐱 = d x 1 2 + d x 2 2 + d x 3 2 , ds^{2}=d\mathbf{x}\cdot d\mathbf{x}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2},
  16. d s 2 = - d X 0 2 + d X 1 2 + d X 2 2 + d X 3 2 , ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2},
  17. d s 2 = d x 1 2 + d x 2 2 - c 2 d t 2 , ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2},
  18. d s 2 = 0 = d x 1 2 + d x 2 2 - c 2 d t 2 ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}
  19. d x 1 2 + d x 2 2 = c 2 d t 2 , dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2},
  20. d s 2 = 0 = d x 1 2 + d x 2 2 + d x 3 2 - c 2 d t 2 ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}
  21. d x 1 2 + d x 2 2 + d x 3 2 = c 2 d t 2 . dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}.
  22. d = x 1 2 + x 2 2 + x 3 2 d=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}
  23. ( c t x y z ) = ( γ - β γ 0 0 - β γ γ 0 0 0 0 1 0 0 0 0 1 ) ( c t x y z ) = ( γ c t - γ β x γ x - β γ c t y z ) . \begin{pmatrix}ct^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}=\begin{pmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}ct\\ x\\ y\\ z\end{pmatrix}=\begin{pmatrix}\gamma ct-\gamma\beta x\\ \gamma x-\beta\gamma ct\\ y\\ z\end{pmatrix}.
  24. X ν = ( X 0 , X 1 , X 2 , X 3 ) = ( c t , x , y , z ) = ( c t , 𝐱 ) . X^{\nu}=(X^{0},X^{1},X^{2},X^{3})=(ct,x,y,z)=(ct,\mathbf{x}).
  25. X μ = Λ μ X ν ν X^{\mu^{\prime}}=\Lambda^{\mu^{\prime}}{}_{\nu}X^{\nu}
  26. Λ μ ν \Lambda^{\mu^{\prime}}{}_{\nu}
  27. T ν T^{\nu}
  28. T μ = Λ μ T ν ν T^{\mu^{\prime}}=\Lambda^{\mu^{\prime}}{}_{\nu}T^{\nu}
  29. U μ = d X μ d τ = γ ( v ) ( c , v x , v y , v z ) = γ ( v ) ( c , 𝐯 ) . U^{\mu}=\frac{dX^{\mu}}{d\tau}=\gamma(v)(c,v_{x},v_{y},v_{z})=\gamma(v)(c,% \mathbf{v}).
  30. γ ( v ) = 1 1 - ( v / c ) 2 , v 2 = v x 2 + v y 2 + v z 2 . \gamma(v)=\frac{1}{\sqrt{1-(v/c)^{2}}}\,,\quad v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}% ^{2}\,.
  31. E = γ ( v ) m c 2 E=\gamma(v)mc^{2}
  32. 𝐩 = γ ( v ) m 𝐯 \mathbf{p}=\gamma(v)m\mathbf{v}
  33. P μ = m U μ = m γ ( v ) ( c , v x , v y , v z ) = ( E / c , p x , p y , p z ) = ( E / c , 𝐩 ) . P^{\mu}=mU^{\mu}=m\gamma(v)(c,v_{x},v_{y},v_{z})=(E/c,p_{x},p_{y},p_{z})=(E/c,% \mathbf{p}).
  34. A μ = d U μ d τ . A^{\mu}=\frac{dU^{\mu}}{d\tau}\,.
  35. ( 1 c ϕ t ϕ x ϕ y ϕ z ) = ( 1 c ϕ t ϕ x ϕ y ϕ z ) ( γ - β γ 0 0 - β γ γ 0 0 0 0 1 0 0 0 0 1 ) . \begin{pmatrix}\frac{1}{c}\frac{\partial\phi}{\partial t^{\prime}}&\frac{% \partial\phi}{\partial x^{\prime}}&\frac{\partial\phi}{\partial y^{\prime}}&% \frac{\partial\phi}{\partial z^{\prime}}\end{pmatrix}=\begin{pmatrix}\frac{1}{% c}\frac{\partial\phi}{\partial t}&\frac{\partial\phi}{\partial x}&\frac{% \partial\phi}{\partial y}&\frac{\partial\phi}{\partial z}\end{pmatrix}\begin{% pmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\,.
  36. ( μ ϕ ) = Λ μ ( ν ϕ ) ν , μ x μ . (\partial_{\mu^{\prime}}\phi)=\Lambda_{\mu^{\prime}}{}^{\nu}(\partial_{\nu}% \phi)\,,\quad\partial_{\mu}\equiv\frac{\partial}{\partial x^{\mu}}\,.
  37. Λ μ T μ ν = T ν \Lambda_{\mu^{\prime}}{}^{\nu}T^{\mu^{\prime}}=T^{\nu}
  38. Λ μ ν \Lambda_{\mu^{\prime}}{}^{\nu}
  39. Λ μ ν \Lambda^{\mu^{\prime}}{}_{\nu}
  40. T θ ι κ α β ζ = Λ α Λ β μ ν Λ ζ Λ θ ρ Λ ι σ υ Λ κ T σ υ ϕ μ ν ρ ϕ T^{\alpha^{\prime}\beta^{\prime}\cdots\zeta^{\prime}}_{\theta^{\prime}\iota^{% \prime}\cdots\kappa^{\prime}}=\Lambda^{\alpha^{\prime}}{}_{\mu}\Lambda^{\beta^% {\prime}}{}_{\nu}\cdots\Lambda^{\zeta^{\prime}}{}_{\rho}\Lambda_{\theta^{% \prime}}{}^{\sigma}\Lambda_{\iota^{\prime}}{}^{\upsilon}\cdots\Lambda_{\kappa^% {\prime}}{}^{\phi}T^{\mu\nu\cdots\rho}_{\sigma\upsilon\cdots\phi}
  41. Λ χ ψ \Lambda_{\chi^{\prime}}{}^{\psi}
  42. Λ χ ψ \Lambda^{\chi^{\prime}}{}_{\psi}
  43. η α β = ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \eta_{\alpha\beta}=\begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  44. η α β \eta^{\alpha\beta}
  45. η α β = η μ ν Λ μ Λ ν α β \eta_{\alpha\beta}=\eta_{\mu^{\prime}\nu^{\prime}}\Lambda^{\mu^{\prime}}{}_{% \alpha}\Lambda^{\nu^{\prime}}{}_{\beta}\!
  46. T α S α = T α η α β S β = T α η α β S β = invariant scalar T^{\alpha}S_{\alpha}=T^{\alpha}\eta_{\alpha\beta}S^{\beta}=T_{\alpha}\eta^{% \alpha\beta}S_{\beta}=\,\text{invariant scalar}
  47. | 𝐓 | = T α T α |\mathbf{T}|=\sqrt{T^{\alpha}T_{\alpha}}
  48. T α , α T α T β β , α T α T β β T γ γ = α invariant scalars , T^{\alpha}{}_{\alpha}\,,T^{\alpha}{}_{\beta}T^{\beta}{}_{\alpha}\,,T^{\alpha}{% }_{\beta}T^{\beta}{}_{\gamma}T^{\gamma}{}_{\alpha}=\,\text{invariant scalars}\,,
  49. d X μ = Λ μ d ν X ν dX^{\mu^{\prime}}=\Lambda^{\mu^{\prime}}{}_{\nu}dX^{\nu}
  50. d 𝐗 2 = d X μ d X μ = η μ ν d X μ d X ν = - ( c d t ) 2 + ( d x ) 2 + ( d y ) 2 + ( d z ) 2 d\mathbf{X}^{2}=dX^{\mu}\,dX_{\mu}=\eta_{\mu\nu}\,dX^{\mu}\,dX^{\nu}=-(cdt)^{2% }+(dx)^{2}+(dy)^{2}+(dz)^{2}\,
  51. 𝐔 2 = η ν μ U ν U μ = - c 2 , {\mathbf{U}}^{2}=\eta_{\nu\mu}U^{\nu}U^{\mu}=-c^{2}\,,
  52. 2 η μ ν A μ U ν = 0. 2\eta_{\mu\nu}A^{\mu}U^{\nu}=0.
  53. 𝐏 2 = η μ ν P μ P ν = - ( E / c ) 2 + p 2 . \mathbf{P}^{2}=\eta^{\mu\nu}P_{\mu}P_{\nu}=-(E/c)^{2}+p^{2}.
  54. 𝐏 2 = - ( E rest / c ) 2 = - ( m c ) 2 . \mathbf{P}^{2}=-(E_{\mathrm{rest}}/c)^{2}=-(mc)^{2}.
  55. E rest = m c 2 . E_{\mathrm{rest}}=mc^{2}.
  56. F ν = d P ν d τ = m A ν F_{\nu}=\frac{dP_{\nu}}{d\tau}=mA_{\nu}

Specific_detectivity.html

  1. D * = A f N E P D^{*}=\frac{\sqrt{Af}}{NEP}
  2. A A
  3. f f
  4. c m H z / W cm\cdot\sqrt{Hz}/W
  5. \mathfrak{R}
  6. A / W A/W
  7. V / W V/W
  8. S n S_{n}
  9. A / H z 1 / 2 A/Hz^{1/2}
  10. V / H z 1 / 2 V/Hz^{1/2}
  11. N E P = S n NEP=\frac{S_{n}}{\mathfrak{R}}
  12. D * = A S n D^{*}=\frac{\mathfrak{R}\cdot\sqrt{A}}{S_{n}}
  13. D * = q λ η h c [ 4 k T R 0 A + 2 q 2 η Φ b ] - 1 / 2 D^{*}=\frac{q\lambda\eta}{hc}\left[\frac{4kT}{R_{0}A}+2q^{2}\eta\Phi_{b}\right% ]^{-1/2}
  14. λ \lambda
  15. R 0 A R_{0}A
  16. η \eta
  17. Φ b \Phi_{b}
  18. Δ f \Delta f
  19. t c t_{c}
  20. Δ f = 1 2 t c \Delta f=\frac{1}{2t_{c}}
  21. N N
  22. S i g n a l r m s = 1 N ( i N S i g n a l i 2 ) Signal_{rms}=\sqrt{\frac{1}{N}\big(\sum_{i}^{N}Signal_{i}^{2}\big)}
  23. N o i s e r m s = σ 2 = 1 N i N ( S i g n a l i - S i g n a l a v g ) 2 Noise_{rms}=\sigma^{2}=\sqrt{\frac{1}{N}\sum_{i}^{N}(Signal_{i}-Signal_{avg})^% {2}}
  24. H H
  25. A d A_{d}
  26. R = S i g n a l r m s H G = S i g n a l d H d A d d Ω B B R=\frac{Signal_{rms}}{HG}=\frac{Signal}{\int dHdA_{d}d\Omega_{BB}}
  27. R R
  28. H H
  29. G G
  30. A d A_{d}
  31. Ω B B \Omega_{BB}
  32. N E P = N o i s e r m s R = N o i s e r m s S i g n a l r m s H G NEP=\frac{Noise_{rms}}{R}=\frac{Noise_{rms}}{Signal_{rms}}HG
  33. D * = Δ f A d N E P = Δ f A d H G S i g n a l r m s N o i s e r m s D^{*}=\frac{\sqrt{\Delta fA_{d}}}{NEP}=\frac{\sqrt{\Delta fA_{d}}}{HG}\frac{% Signal_{rms}}{Noise_{rms}}

Specific_impulse.html

  1. F thrust = I sp m ˙ g 0 , F\text{thrust}=I\text{sp}\cdot\dot{m}\cdot g_{0},
  2. F thrust F\text{thrust}
  3. I sp I\text{sp}
  4. m ˙ \dot{m}
  5. g 0 g_{0}
  6. F thrust = I sp m ˙ . F\text{thrust}=I\text{sp}\cdot\dot{m}.
  7. I sp = v e g 0 , I_{\rm sp}=\frac{v\text{e}}{g_{0}},
  8. v e v\text{e}
  9. v e = g 0 I sp , v\text{e}=g_{0}I\text{sp},
  10. I sp I\text{sp}
  11. v e v\text{e}
  12. g 0 g_{0}
  13. I sp I\text{sp}
  14. F thrust = v e m ˙ , F\text{thrust}=v\text{e}\cdot\dot{m},
  15. m ˙ \dot{m}
  16. I s p I_{sp}
  17. d m d t v e 2 2 \frac{dm}{dt}\frac{v_{e}^{2}}{2}
  18. d m d t v e \frac{dm}{dt}v_{e}
  19. v e 2 \frac{v_{e}}{2}
  20. v e = 0.6275 Δ v v\text{e}=0.6275\Delta v
  21. I s p I_{sp}

Speckle_pattern.html

  1. P ( I ) d I = 1 μ exp ( - I / μ ) P(I)dI=\frac{1}{\mu}\exp(-I/\mu)
  2. μ \mu

Spectroscopy.html

  1. ( E ) (E)
  2. ( ν ) (\nu)
  3. E = h ν E=h\nu
  4. h h

Spectrum_of_a_ring.html

  1. V I V_{I}
  2. { V I : I is an ideal of R } . \{V_{I}\colon I\,\text{ is an ideal of }R\}.
  3. { D f : f R } \{D_{f}:f\in R\}
  4. f : S p e c A Y f\colon{Spec}\ A\to Y
  5. U Y U\subseteq Y
  6. f - 1 ( U ) Spec A ( U ) f^{-1}(U)\cong\mathrm{Spec}\ A(U)
  7. U V U\subseteq V
  8. f - 1 ( U ) f - 1 ( V ) f^{-1}(U)\to f^{-1}(V)
  9. A ( V ) A ( U ) . A(V)\to A(U).
  10. R = K [ x 1 , , x n ] R=K[x_{1},\dots,x_{n}]
  11. R = K [ V ] . R=K[V].
  12. x i x_{i}
  13. R / I , R/I,
  14. ( x 1 - a 1 ) , ( x 2 - a 2 ) , , ( x n - a n ) (x_{1}-a_{1}),(x_{2}-a_{2}),\ldots,(x_{n}-a_{n})
  15. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  16. K [ V ] K[V]
  17. V * , V^{*},
  18. x i x_{i}
  19. a i a_{i}
  20. K n K^{n}
  21. K n K K^{n}\to K
  22. K n K . K^{n}\to K.
  23. R = K [ x 1 , , x n ] , R=K[x_{1},\dots,x_{n}],
  24. K [ T ] / ( T - 1 ) K [ T ] / ( T - 1 ) K[T]/(T-1)\oplus K[T]/(T-1)
  25. K [ T ] / ( T - 0 ) K [ T ] / ( T - 0 ) , K[T]/(T-0)\oplus K[T]/(T-0),
  26. K [ T ] / T 2 , K[T]/T^{2},

Speed.html

  1. v = d t , v=\frac{d}{t},
  2. v v
  3. d d
  4. t t
  5. v v
  6. s y m b o l v symbol{v}
  7. s y m b o l r symbol{r}
  8. v = | s y m b o l v | = | s y m b o l r ˙ | = | d s y m b o l r d t | . v=\left|symbolv\right|=\left|\dot{symbolr}\right|=\left|\frac{dsymbolr}{dt}% \right|\,.
  9. s s
  10. t t
  11. s s
  12. v = d s d t . v=\frac{ds}{dt}.
  13. v = s / t v=s/t
  14. d = s y m b o l v ¯ t . d=symbol{\bar{v}}t\,.
  15. π \pi
  16. v r ω , v\propto\!\,r\omega\,,
  17. v = r ω . v=r\omega\,.

Sphere.html

  1. r r
  2. r r
  3. A = 4 π r 2 . A=4\pi r^{2}.
  4. δ V A ( r ) δ r . \delta V\approx A(r)\cdot\delta r.\,
  5. V A ( r ) δ r . V\approx\sum A(r)\cdot\delta r.
  6. V = 0 r A ( r ) d r . V=\int_{0}^{r}A(r)\,dr.
  7. 4 3 π r 3 = 0 r A ( r ) d r . \frac{4}{3}\pi r^{3}=\int_{0}^{r}A(r)\,dr.
  8. 4 π r 2 = A ( r ) . \!4\pi r^{2}=A(r).
  9. A = 4 π r 2 \!A=4\pi r^{2}
  10. d A = r 2 sin θ d θ d ϕ . dA=r^{2}\sin\theta\,d\theta\,d\phi.
  11. d S = r r 2 - i k x i 2 Π i k d x i , k dS=\frac{r}{\sqrt{r^{2}-\sum_{i\neq k}x_{i}^{2}}}\Pi_{i\neq k}dx_{i},\;\forall k
  12. A = 0 2 π 0 π r 2 sin θ d θ d ϕ = 4 π r 2 . A=\int_{0}^{2\pi}\int_{0}^{\pi}r^{2}\sin\theta\,d\theta\,d\phi=4\pi r^{2}.
  13. V = 4 3 π r 3 \!V=\frac{4}{3}\pi r^{3}
  14. δ V π y 2 δ x . \!\delta V\approx\pi y^{2}\cdot\delta x.
  15. V π y 2 δ x . \!V\approx\sum\pi y^{2}\cdot\delta x.
  16. V = - r r π y 2 d x . \!V=\int_{-r}^{r}\pi y^{2}dx.
  17. y 2 = r 2 - x 2 . \!y^{2}=r^{2}-x^{2}.
  18. V = - r r π ( r 2 - x 2 ) d x . \!V=\int_{-r}^{r}\pi(r^{2}-x^{2})dx.
  19. V = π [ r 2 x - x 3 3 ] - r r = π ( r 3 - r 3 3 ) - π ( - r 3 + r 3 3 ) = 4 3 π r 3 . \!V=\pi\left[r^{2}x-\frac{x^{3}}{3}\right]_{-r}^{r}=\pi\left(r^{3}-\frac{r^{3}% }{3}\right)-\pi\left(-r^{3}+\frac{r^{3}}{3}\right)=\frac{4}{3}\pi r^{3}.
  20. V = 4 3 π r 3 . \!V=\frac{4}{3}\pi r^{3}.
  21. d V = r 2 sin θ d r d θ d φ \mathrm{d}V=r^{2}\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\varphi
  22. V = 0 2 π 0 π 0 r r 2 sin θ d r d θ d φ = 4 3 π r 3 V=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{r}r^{2}\sin\theta\,\mathrm{d}r\,% \mathrm{d}\theta\,\mathrm{d}\varphi=\frac{4}{3}\pi r^{3}
  23. π / 6 0.5236 \pi/6\approx 0.5236
  24. ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 = r 2 . \,(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.
  25. x = x 0 + r cos θ sin φ \,x=x_{0}+r\cos\theta\;\sin\varphi
  26. y = y 0 + r sin θ sin φ ( 0 θ 2 π and 0 φ π ) \,y=y_{0}+r\sin\theta\;\sin\varphi\qquad(0\leq\theta\leq 2\pi\mbox{ and }~{}0% \leq\varphi\leq\pi)\,
  27. z = z 0 + r cos φ \,z=z_{0}+r\cos\varphi\,
  28. x d x + y d y + z d z = 0. \,x\,dx+y\,dy+z\,dz=0.
  29. S S A = A V ρ = 3 r ρ , SSA=\frac{A}{V\rho}=\frac{3}{r\rho},
  30. ρ \rho
  31. S n S^{n}
  32. S 0 S^{0}
  33. S 1 S^{1}
  34. S 2 S^{2}
  35. S 3 S^{3}
  36. 2 π n / 2 Γ ( n / 2 ) \frac{2\pi^{n/2}}{\Gamma(n/2)}
  37. { ( 2 π ) n / 2 r n - 1 2 4 ( n - 2 ) , if n is even ; 2 ( 2 π ) ( n - 1 ) / 2 r n - 1 1 3 ( n - 2 ) , if n is odd . \begin{cases}\displaystyle\frac{(2\pi)^{n/2}\,r^{n-1}}{2\cdot 4\cdots(n-2)},&% \,\text{if }n\,\text{ is even};\\ \\ \displaystyle\frac{2(2\pi)^{(n-1)/2}\,r^{n-1}}{1\cdot 3\cdots(n-2)},&\,\text{% if }n\,\text{ is odd}.\end{cases}
  38. r n {r\over n}
  39. { ( 2 π ) n / 2 r n 2 4 n , if n is even ; 2 ( 2 π ) ( n - 1 ) / 2 r n 1 3 n , if n is odd . \begin{cases}\displaystyle\frac{(2\pi)^{n/2}\,r^{n}}{2\cdot 4\cdots n},&\,% \text{if }n\,\text{ is even};\\ \\ \displaystyle\frac{2(2\pi)^{(n-1)/2}\,r^{n}}{1\cdot 3\cdots n},&\,\text{if }n% \,\text{ is odd}.\end{cases}

Spherical_coordinate_system.html

  1. ( Z , X , Y ) (Z,\ X,\ Y)
  2. ( r , θ inc , ϕ az,right ) \ (r,\ \theta\text{inc},\ \phi\text{az,right})
  3. ( U , S , E ) \ (U,\ S,\ E)
  4. ( r , ϕ az,right , θ el ) \ (r,\ \phi\text{az,right},\ \theta\text{el})
  5. ( U , E , N ) \ (U,\ E,\ N)
  6. ( r , θ el , ϕ az,right ) \ (r,\ \theta\text{el},\ \phi\text{az,right})
  7. ( U , N , E ) \ (U,\ N,\ E)
  8. E E
  9. N N
  10. U U
  11. S S
  12. E E
  13. ( U , S , E ) (U,\ S,\ E)
  14. r = x 2 + y 2 + z 2 r=\sqrt{x^{2}+y^{2}+z^{2}}
  15. θ = arccos ( z x 2 + y 2 + z 2 ) \theta=\operatorname{arccos}\left(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)
  16. φ = arctan ( y x ) \varphi=\operatorname{arctan}\left(\frac{y}{x}\right)
  17. φ = a r c t a n ( y / x ) φ=arctan(y/x)
  18. x = r sin θ cos φ x=r\,\sin\theta\,\cos\varphi
  19. y = r sin θ sin φ y=r\,\sin\theta\,\sin\varphi
  20. z = r cos θ z=r\,\cos\theta
  21. r = ρ 2 + z 2 r=\sqrt{\rho^{2}+z^{2}}
  22. θ = arctan ( ρ z ) = arccos ( z ρ 2 + z 2 ) \theta=\operatorname{arctan}\left(\frac{\rho}{z}\right)=\operatorname{arccos}% \left(\frac{z}{\sqrt{\rho^{2}+z^{2}}}\right)
  23. φ = φ \varphi=\varphi\quad
  24. ρ = r sin θ \rho=r\sin\theta\,
  25. φ = φ \varphi=\varphi\,
  26. z = r cos θ z=r\cos\theta\,
  27. ( r , θ , φ ) (r,\theta,\varphi)
  28. ( r + d r , θ + d θ , φ + d φ ) (r+\mathrm{d}r,\,\theta+\mathrm{d}\theta,\,\varphi+\mathrm{d}\varphi)
  29. d 𝐫 = d r s y m b o l r ^ + r d θ s y m b o l θ ^ + r sin θ d φ 𝐬𝐲𝐦𝐛𝐨𝐥 φ ^ . \mathrm{d}\mathbf{r}=\mathrm{d}r\,\hat{symbolr}+r\,\mathrm{d}\theta\,\hat{% symbol\theta}+r\sin{\theta}\,\mathrm{d}\varphi\,\mathbf{\hat{symbol\varphi}}.
  30. s y m b o l r ^ = sin θ cos φ s y m b o l x ^ + sin θ sin φ s y m b o l y ^ + cos θ s y m b o l z ^ \hat{symbolr}=\sin\theta\cos\varphi\,\hat{symbolx}+\sin\theta\sin\varphi\,\hat% {symboly}+\cos\theta\,\hat{symbolz}
  31. s y m b o l θ ^ = cos θ cos φ s y m b o l x ^ + cos θ sin φ s y m b o l y ^ - sin θ s y m b o l z ^ \hat{symbol\theta}=\cos\theta\cos\varphi\,\hat{symbolx}+\cos\theta\sin\varphi% \,\hat{symboly}-\sin\theta\,\hat{symbolz}
  32. s y m b o l φ ^ = - sin φ s y m b o l x ^ + cos φ s y m b o l y ^ \hat{symbol\varphi}=-\sin\varphi\,\hat{symbolx}+\cos\varphi\,\hat{symboly}
  33. r r
  34. θ θ
  35. φ φ
  36. s y m b o l x ^ \hat{symbolx}
  37. s y m b o l y ^ \hat{symboly}
  38. s y m b o l z ^ \hat{symbolz}
  39. θ \theta
  40. θ + d θ \theta+\mathrm{d}\theta
  41. φ \varphi
  42. φ + d φ \varphi+\mathrm{d}\varphi
  43. r r
  44. d S r = r 2 sin θ d θ d φ . \mathrm{d}S_{r}=r^{2}\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi.
  45. d Ω = d S r r 2 = sin θ d θ d φ . \mathrm{d}\Omega=\frac{\mathrm{d}S_{r}}{r^{2}}=\sin\theta\,\mathrm{d}\theta\,% \mathrm{d}\varphi.
  46. θ \theta
  47. d S θ = r sin θ d φ d r . \mathrm{d}S_{\theta}=r\sin\theta\,\mathrm{d}\varphi\,\mathrm{d}r.
  48. φ \varphi
  49. d S φ = r d r d θ . \mathrm{d}S_{\varphi}=r\,\mathrm{d}r\,\mathrm{d}\theta.
  50. r r
  51. r + d r r+\mathrm{d}r
  52. θ \theta
  53. θ + d θ \theta+\mathrm{d}\theta
  54. φ \varphi
  55. φ + d φ \varphi+\mathrm{d}\varphi
  56. d V = r 2 sin θ d r d θ d φ . \mathrm{d}V=r^{2}\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\varphi.
  57. f ( r , θ , φ ) f(r,\theta,\varphi)
  58. φ = 0 2 π θ = 0 π r = 0 f ( r , θ , φ ) r 2 sin θ d r d θ d φ . \int_{\varphi=0}^{2\pi}\int_{\theta=0}^{\pi}\int_{r=0}^{\infty}f(r,\theta,% \varphi)r^{2}\sin\theta\,\mathrm{d}r\ \mathrm{d}\theta\ \mathrm{d}\varphi.
  59. f = \displaystyle\nabla f=
  60. 𝐫 = r 𝐫 ^ . \mathbf{r}=r\mathbf{\hat{r}}.
  61. 𝐯 = r ˙ 𝐫 ^ + r θ ˙ s y m b o l θ ^ + r φ ˙ sin θ 𝐬𝐲𝐦𝐛𝐨𝐥 φ ^ , \mathbf{v}=\dot{r}\mathbf{\hat{r}}+r\,\dot{\theta}\,\hat{symbol\theta}+r\,\dot% {\varphi}\,\sin\theta\mathbf{\hat{symbol\varphi}},
  62. 𝐚 \displaystyle\mathbf{a}
  63. θ = π 2 \theta=\tfrac{\pi}{2}

Spinor.html

  1. V V
  2. V V
  3. 2 dim V / 2 2^{\lfloor\dim V/2\rfloor}
  4. n n
  5. 2 ν 2^{\nu}
  6. n = 2 ν + 1 n=2\nu+1
  7. 2 ν 2\nu
  8. e i e j = { + 1 i = j , i ( 1 p ) - 1 i = j , i ( p + 1 n ) - e j e i i j . e_{i}e_{j}=\Bigg\{\begin{matrix}+1&i=j,\,i\in(1\ldots p)\\ -1&i=j,\,i\in(p+1\ldots n)\\ -e_{j}e_{i}&i\not=j.\end{matrix}
  9. ( a + b σ 1 σ 2 ) * = a + b σ 2 σ 1 (a+b\sigma_{1}\sigma_{2})^{*}=a+b\sigma_{2}\sigma_{1}\,
  10. ( a + b σ 1 σ 2 ) * = a + b σ 2 σ 1 = a - b σ 1 σ 2 (a+b\sigma_{1}\sigma_{2})^{*}=a+b\sigma_{2}\sigma_{1}=a-b\sigma_{1}\sigma_{2}\,
  11. γ ( u ) = γ u γ * \gamma(u)=\gamma u\gamma^{*}\,
  12. γ ( ϕ ) = γ ϕ \gamma(\phi)=\gamma\phi
  13. γ ( u ) = γ u γ * = γ 2 u \gamma(u)=\gamma u\gamma^{*}=\gamma^{2}u\,
  14. γ = 1 2 ( 1 - σ 1 σ 2 ) \gamma=\tfrac{1}{\sqrt{2}}(1-\sigma_{1}\sigma_{2})\,
  15. 1 2 ( 1 - σ 1 σ 2 ) { a 1 σ 1 + a 2 σ 2 } ( 1 - σ 2 σ 1 ) = a 1 σ 2 - a 2 σ 1 \tfrac{1}{2}(1-\sigma_{1}\sigma_{2})\,\{a_{1}\sigma_{1}+a_{2}\sigma_{2}\}\,(1-% \sigma_{2}\sigma_{1})=a_{1}\sigma_{2}-a_{2}\sigma_{1}\,
  16. 1 2 ( 1 - σ 1 σ 2 ) { a 1 + a 2 σ 1 σ 2 } = a 1 + a 2 2 + - a 1 + a 2 2 σ 1 σ 2 \tfrac{1}{\sqrt{2}}(1-\sigma_{1}\sigma_{2})\,\{a_{1}+a_{2}\sigma_{1}\sigma_{2}% \}=\frac{a_{1}+a_{2}}{\sqrt{2}}+\frac{-a_{1}+a_{2}}{\sqrt{2}}\sigma_{1}\sigma_% {2}
  17. ( - σ 1 σ 2 ) { a 1 σ 1 + a 2 σ 2 } ( - σ 2 σ 1 ) = - a 1 σ 1 - a 2 σ 2 (-\sigma_{1}\sigma_{2})\,\{a_{1}\sigma_{1}+a_{2}\sigma_{2}\}\,(-\sigma_{2}% \sigma_{1})=-a_{1}\sigma_{1}-a_{2}\sigma_{2}\,
  18. ( - σ 1 σ 2 ) { a 1 + a 2 σ 1 σ 2 } = a 2 - a 1 σ 1 σ 2 (-\sigma_{1}\sigma_{2})\,\{a_{1}+a_{2}\sigma_{1}\sigma_{2}\}=a_{2}-a_{1}\sigma% _{1}\sigma_{2}
  19. ( - 1 ) { a 1 σ 1 + a 2 σ 2 } ( - 1 ) = a 1 σ 1 + a 2 σ 2 (-1)\,\{a_{1}\sigma_{1}+a_{2}\sigma_{2}\}\,(-1)=a_{1}\sigma_{1}+a_{2}\sigma_{2}\,
  20. u = ρ ( 1 / 2 ) u ρ ( 1 / 2 ) = ρ u , u^{\prime}=\rho^{(1/2)}u\rho^{(1/2)}=\rho u,
  21. u = γ u γ * , u^{\prime}=\gamma\,u\,\gamma^{*},
  22. γ = cos ( θ / 2 ) - { a 1 σ 2 σ 3 + a 2 σ 3 σ 1 + a 3 σ 1 σ 2 } sin ( θ / 2 ) = cos ( θ / 2 ) - i { a 1 σ 1 + a 2 σ 2 + a 3 σ 3 } sin ( θ / 2 ) = cos ( θ / 2 ) - i v sin ( θ / 2 ) } \left.\begin{array}[]{rcl}\gamma&=&\cos(\theta/2)-\{a_{1}\sigma_{2}\sigma_{3}+% a_{2}\sigma_{3}\sigma_{1}+a_{3}\sigma_{1}\sigma_{2}\}\sin(\theta/2)\\ &=&\cos(\theta/2)-i\{a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{3}\sigma_{3}\}\sin(% \theta/2)\\ &=&\cos(\theta/2)-iv\sin(\theta/2)\end{array}\right\}
  23. ( cos ( θ / 2 ) - i σ 3 sin ( θ / 2 ) ) σ 3 ( cos ( θ / 2 ) + i σ 3 sin ( θ / 2 ) ) = ( cos 2 ( θ / 2 ) + sin 2 ( θ / 2 ) ) σ 3 = σ 3 . (\cos(\theta/2)-i\sigma_{3}\sin(\theta/2))\,\sigma_{3}\,(\cos(\theta/2)+i% \sigma_{3}\sin(\theta/2))=(\cos^{2}(\theta/2)+\sin^{2}(\theta/2))\,\sigma_{3}=% \sigma_{3}.
  24. 𝐢 = - σ 2 σ 3 = - i σ 1 𝐣 = - σ 3 σ 1 = - i σ 2 𝐤 = - σ 1 σ 2 = - i σ 3 . \begin{matrix}\mathbf{i}=-\sigma_{2}\sigma_{3}=-i\sigma_{1}\\ \mathbf{j}=-\sigma_{3}\sigma_{1}=-i\sigma_{2}\\ \mathbf{k}=-\sigma_{1}\sigma_{2}=-i\sigma_{3}.\end{matrix}
  25. C ( V , g ) Cℓ(V,g)
  26. C ( V , g ) ω Cℓ(V,g)ω
  27. C ( V , g ) Cℓ(V,g)
  28. c : x ω c x ω c:xω→cxω
  29. ω ω
  30. V V
  31. W W
  32. V V
  33. ( V , g ) (V,g)
  34. n n
  35. V V
  36. V V
  37. g g
  38. W W
  39. V V
  40. n = 2 k n= 2k
  41. W W
  42. n = 2 k + 1 n= 2k+1
  43. U U
  44. W W
  45. k k
  46. U U
  47. u u
  48. ω = w 1 w 2 w k . \omega=w^{\prime}_{1}w^{\prime}_{2}\cdots w^{\prime}_{k}.
  49. α = i 1 < i 2 < < i p a i 1 i p w i 1 w i p + j B j w j \alpha=\sum_{i_{1}<i_{2}<\cdots<i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i% _{p}}+\sum_{j}B_{j}w^{\prime}_{j}
  50. α ω = i 1 < i 2 < < i p a i 1 i p w i 1 w i p ω . \alpha\omega=\sum_{i_{1}<i_{2}<\cdots<i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}% \cdots w_{i_{p}}\omega.
  51. a = a i 1 i m a x w i 1 w i m a x a=a_{i_{1}\dots i_{max}}w_{i_{1}}\dots w_{i_{max}}
  52. w i m a x w i 1 α ω = a i 1 i m a x ω w^{\prime}_{i_{max}}\cdots w^{\prime}_{i_{1}}\alpha\omega=a_{i_{1}\dots i_{max% }}\omega
  53. Δ = C ( W ) ω = ( Λ * W ) ω \Delta=\mathrm{C}\ell(W)\omega=(\Lambda^{*}W)\omega
  54. c ( v ) w 1 w n = ( ϵ ( w ) + i ( w ) ) ( w 1 w n ) c(v)w_{1}\wedge\cdots\wedge w_{n}=(\epsilon(w)+i(w^{\prime}))\left(w_{1}\wedge% \cdots\wedge w_{n}\right)
  55. Δ + = Λ e v e n W , Δ - = Λ o d d W \Delta_{+}=\Lambda^{even}W,\,\Delta_{-}=\Lambda^{odd}W
  56. c ( u ) α = { α if α Λ e v e n W - α if α Λ o d d W c(u)\alpha=\left\{\begin{matrix}\alpha&\hbox{if }\alpha\in\Lambda^{even}W\\ -\alpha&\hbox{if }\alpha\in\Lambda^{odd}W\end{matrix}\right.
  57. V ¯ \overline{V}
  58. Δ Δ * p = 0 n Γ p p = 0 k - 1 ( Γ p σ Γ p ) Γ k \Delta\otimes\Delta^{*}\cong\bigoplus_{p=0}^{n}\Gamma_{p}\cong\bigoplus_{p=0}^% {k-1}\left(\Gamma_{p}\oplus\sigma\Gamma_{p}\right)\,\oplus\Gamma_{k}
  59. ( α ω ) * = ω ( α * ) . (\alpha\omega)^{*}=\omega(\alpha^{*}).
  60. Δ + Δ + * Δ - Δ - * p = 0 k Γ 2 p Δ + Δ - * Δ - Δ + * p = 0 k - 1 Γ 2 p + 1 \begin{matrix}\Delta_{+}\otimes\Delta^{*}_{+}\cong\Delta_{-}\otimes\Delta^{*}_% {-}&\cong&\bigoplus_{p=0}^{k}\Gamma_{2p}\\ \Delta_{+}\otimes\Delta^{*}_{-}\cong\Delta_{-}\otimes\Delta^{*}_{+}&\cong&% \bigoplus_{p=0}^{k-1}\Gamma_{2p+1}\end{matrix}
  61. Δ ¯ \overline{Δ}
  62. Δ ¯ σ - Δ * \bar{\Delta}\cong\sigma_{-}\Delta^{*}
  63. Δ Δ ¯ p = 0 k ( σ - Γ p σ + Γ p ) . \Delta\otimes\bar{\Delta}\cong\bigoplus_{p=0}^{k}\left(\sigma_{-}\Gamma_{p}% \oplus\sigma_{+}\Gamma_{p}\right).
  64. Δ ¯ + σ - Δ + * \bar{\Delta}_{+}\cong\sigma_{-}\otimes\Delta_{+}^{*}
  65. Δ ¯ - σ - Δ - * . \bar{\Delta}_{-}\cong\sigma_{-}\otimes\Delta_{-}^{*}.
  66. Δ ¯ + σ - Δ - * \bar{\Delta}_{+}\cong\sigma_{-}\otimes\Delta_{-}^{*}
  67. Δ ¯ - σ - Δ + * . \bar{\Delta}_{-}\cong\sigma_{-}\otimes\Delta_{+}^{*}.
  68. Δ Δ * p = 0 k Γ 2 p . \Delta\otimes\Delta^{*}\cong\bigoplus_{p=0}^{k}\Gamma_{2p}.
  69. Δ ¯ σ - Δ * . \bar{\Delta}\cong\sigma_{-}\Delta^{*}.
  70. Δ Δ ¯ σ - Γ 0 σ + Γ 1 σ ± Γ k \Delta\otimes\bar{\Delta}\cong\sigma_{-}\Gamma_{0}\oplus\sigma_{+}\Gamma_{1}% \oplus\dots\oplus\sigma_{\pm}\Gamma_{k}
  71. ϕ ¯ \overline{ϕ}
  72. ϕ ¯ \overline{ϕ}
  73. ψ ¯ \overline{ψ}
  74. n n
  75. 2 ν 2^{\nu}
  76. n = 2 ν + 1 n=2\nu+1
  77. 2 ν 2\nu
  78. { ± 1 } \{\pm 1\}

Spintronics.html

  1. 1 2 \frac{1}{2}\hbar
  2. μ = 3 2 q m e \mu=\frac{\sqrt{3}}{2}\frac{q}{m_{e}}\hbar
  3. P X = X - X X + X P_{X}=\frac{X_{\uparrow}-X_{\downarrow}}{X_{\uparrow}+X_{\downarrow}}
  4. τ \tau
  5. λ \lambda
  6. M C = I c , p - I c , a p I c , a p MC=\frac{I_{c,p}-I_{c,ap}}{I_{c,ap}}
  7. T R = I C I E TR=\frac{I_{C}}{I_{E}}

Spiral.html

  1. r = a + b θ r=a+b\cdot\theta
  2. r = θ 1 / 2 r=\theta^{1/2}
  3. r = a / θ r=a/\theta
  4. r = θ - 1 / 2 r=\theta^{-1/2}
  5. r = a e b θ r=a\cdot e^{b\theta}
  6. θ = n × 137.5 , r = c n \theta=n\times 137.5^{\circ},\ r=c\sqrt{n}

Splay_tree.html

  1. T amortized ( m ) = O ( m log n ) T_{\mathrm{amortized}}(m)=O(m\log n)
  2. Φ i - Φ f = x rank i ( x ) - rank f ( x ) = O ( n log n ) \Phi_{i}-\Phi_{f}=\sum_{x}{\mathrm{rank}_{i}(x)-\mathrm{rank}_{f}(x)}=O(n\log n)
  3. T actual ( m ) = O ( m log n + n log n ) T_{\mathrm{actual}}(m)=O(m\log n+n\log n)
  4. rank ( r o o t ) - rank ( x ) = O ( log W - log w ( x ) ) = O ( log W w ( x ) ) \mathrm{rank}(root)-\mathrm{rank}(x)=O(\log{W}-\log{w(x)})=O(\log{\frac{W}{w(x% )}})
  5. Φ i - Φ f x t r e e log W w ( x ) \Phi_{i}-\Phi_{f}\leq\sum_{x\in tree}{\log{\frac{W}{w(x)}}}
  6. O ( x s e q u e n c e ( log W w ( x ) ) + x t r e e ( log W w ( x ) ) ) O(\sum_{x\in sequence}{(\log{\frac{W}{w(x)}})}+\sum_{x\in tree}{(\log{\frac{W}% {w(x)}})})
  7. O [ m log n + n log n ] O\left[m\log n+n\log n\right]
  8. q x q_{x}
  9. O [ m + x t r e e q x log m q x ] O\left[m+\sum_{x\in tree}q_{x}\log\frac{m}{q_{x}}\right]
  10. w ( x ) = q x w(x)=q_{x}
  11. W = m W=m
  12. O [ m + n log n + x s e q u e n c e log ( | x - f | + 1 ) ] O\left[m+n\log n+\sum_{x\in sequence}\log(|x-f|+1)\right]
  13. w ( x ) = 1 / ( | x - f | + 1 ) 2 w(x)=1/(|x-f|+1)^{2}
  14. O [ m + n + x , y s e q u e n c e m log ( | y - x | + 1 ) ] O\left[m+n+\sum_{x,y\in sequence}^{m}\log(|y-x|+1)\right]
  15. t ( x ) t(x)
  16. O [ m + n log n + x s e q u e n c e log ( t ( x ) + 1 ) ] O\left[m+n\log n+\sum_{x\in sequence}\log(t(x)+1)\right]
  17. w ( x ) = 1 / ( t ( x ) + 1 ) 2 w(x)=1/(t(x)+1)^{2}
  18. 4.5 n 4.5n
  19. A A
  20. x x
  21. x x
  22. d ( x ) + 1 d(x)+1
  23. A ( S ) A(S)
  24. A A
  25. S S
  26. O [ n + A ( S ) ] O[n+A(S)]
  27. T 1 T_{1}
  28. T 2 T_{2}
  29. S S
  30. T 2 T_{2}
  31. S S
  32. T 1 T_{1}
  33. O ( n ) O(n)
  34. S S
  35. m m
  36. S S
  37. O ( m + n ) O(m+n)
  38. S S
  39. S S
  40. O ( n ) O(n)

Spontaneous_emission.html

  1. E 2 E_{2}
  2. E 1 E_{1}
  3. ω \omega
  4. ω \hbar\omega
  5. h ν h\nu
  6. h h
  7. ν \nu
  8. E 2 - E 1 = ω , E_{2}-E_{1}=\hbar\omega,
  9. \hbar
  10. t t
  11. N ( t ) N(t)
  12. N N
  13. N ( t ) t = - A 21 N ( t ) , \frac{\partial N(t)}{\partial t}=-A_{21}N(t),
  14. A 21 A_{21}
  15. A 21 A_{21}
  16. s - 1 s^{-1}
  17. N ( t ) = N ( 0 ) e - A 21 t = N ( 0 ) e - Γ r a d t , N(t)=N(0)e^{-A_{21}t}=N(0)e^{-\Gamma_{rad}t},
  18. N ( 0 ) N(0)
  19. t t
  20. Γ r a d \Gamma_{rad}
  21. N N
  22. 1 e \frac{1}{e}
  23. Γ r a d \Gamma_{rad}
  24. τ 21 \tau_{21}
  25. A 21 = Γ 21 = 1 τ 21 . A_{21}=\Gamma_{21}=\frac{1}{\tau_{21}}.
  26. | ψ ( t ) = a ( t ) e - i ω 0 t | e ; 0 + k , s b k s ( t ) e - i ω k t | g ; 1 k s |\psi(t)\rangle=a(t)e^{-i\omega_{0}t}|e;0\rangle+\sum_{k,s}b_{ks}(t)e^{-i% \omega_{k}t}|g;1_{ks}\rangle
  27. | e ; 0 |e;0\rangle
  28. a ( t ) a(t)
  29. | g ; 1 k s |g;1_{ks}\rangle
  30. b k s ( t ) b_{ks}(t)
  31. k s ks
  32. ω 0 \omega_{0}
  33. ω k = c | k | \omega_{k}=c|k|
  34. k k
  35. s s
  36. | b ( t ) | 2 |b(t)|^{2}
  37. Γ r a d ( ω ) = ω 3 n | μ 12 | 2 3 π ε 0 c 3 = 4 α ω 3 n | 1 | 𝐫 | 2 | 2 3 c 2 \Gamma_{rad}(\omega)=\frac{\omega^{3}n|\mu_{12}|^{2}}{3\pi\varepsilon_{0}\hbar c% ^{3}}=\frac{4\alpha\omega^{3}n|\langle 1|\mathbf{r}|2\rangle|^{2}}{3c^{2}}
  38. | μ 12 | 2 π ε 0 c = 4 α | 1 | 𝐫 | 2 | 2 \frac{|\mu_{12}|^{2}}{\pi\varepsilon_{0}\hbar c}=4\alpha|\langle 1|\mathbf{r}|% 2\rangle|^{2}
  39. ω \omega
  40. n n
  41. μ 12 \mu_{12}
  42. ε 0 \varepsilon_{0}
  43. \hbar
  44. c c
  45. α \alpha
  46. ω 3 \omega^{3}
  47. ω 3 \omega^{3}
  48. N N
  49. Γ t o t \Gamma_{tot}
  50. Γ t o t = Γ r a d + Γ n r a d \Gamma_{tot}=\Gamma_{rad}+\Gamma_{nrad}
  51. Γ t o t \Gamma_{tot}
  52. Γ r a d \Gamma_{rad}
  53. Γ n r a d \Gamma_{nrad}
  54. Q E = Γ r a d Γ n r a d + Γ r a d . QE=\frac{\Gamma_{rad}}{\Gamma_{nrad}+\Gamma_{rad}}.

Sprague–Grundy_theorem.html

  1. { } \{\}
  2. { { } } \{\{\}\}
  3. * n *n
  4. n n
  5. * 0 *0
  6. { } \{\}
  7. * ( n + 1 ) = * n { * n } *(n+1)=*n\cup\{*n\}
  8. * 1 = { * 0 } *1=\{*0\}
  9. * 2 = { * 0 , * 1 } *2=\{*0,*1\}
  10. * n *n
  11. n n
  12. G G
  13. H H
  14. G + H G+H
  15. G G
  16. H H
  17. G + H G+H
  18. G + H = { G + h h H } { g + H g G } G+H=\{G+h\mid h\in H\}\cup\{g+H\mid g\in G\}
  19. G G
  20. G G^{\prime}
  21. H H
  22. G + H G+H
  23. G + H G^{\prime}+H
  24. G G G\approx G^{\prime}
  25. G G
  26. A A
  27. G A + G G\approx A+G
  28. G + H G+H
  29. A + G + H A+G+H
  30. H H
  31. G + H G+H
  32. A + G + H A+G+H
  33. A A
  34. A A
  35. A A
  36. G + H G+H
  37. G + H G+H
  38. A + G + H A+G+H
  39. G + H G+H
  40. G + H G+H
  41. A + G + H A+G+H
  42. G + H G+H
  43. G G G\approx G^{\prime}
  44. G + G G+G^{\prime}
  45. G G G\approx G^{\prime}
  46. H = G H=G
  47. G + G G^{\prime}+G
  48. G + G G+G^{\prime}
  49. G + G G+G
  50. G + G G+G
  51. G G
  52. A = G + G A=G+G^{\prime}
  53. G G
  54. G G + ( G + G ) G\approx G+(G+G^{\prime})
  55. A = G + G A=G+G
  56. G G^{\prime}
  57. G G + ( G + G ) G^{\prime}\approx G^{\prime}+(G+G)
  58. \approx
  59. \approx
  60. G G G\approx G^{\prime}
  61. G = { G 1 , G 2 , , G k } G=\{G_{1},G_{2},\ldots,G_{k}\}
  62. G i * n i G_{i}\approx*n_{i}
  63. G = { * n 1 , * n 2 , , * n k } G^{\prime}=\{*n_{1},*n_{2},\ldots,*n_{k}\}
  64. G * m G\approx*m
  65. m m
  66. n 1 , n 2 , , n k n_{1},n_{2},\ldots,n_{k}
  67. n i n_{i}
  68. G G G\approx G^{\prime}
  69. k k
  70. G + G G+G^{\prime}
  71. G i G_{i}
  72. G G
  73. * n i *n_{i}
  74. G G^{\prime}
  75. G G^{\prime}
  76. G + G G+G^{\prime}
  77. G G G\approx G^{\prime}
  78. G + * m G^{\prime}+*m
  79. G * m G^{\prime}\approx*m
  80. G G^{\prime}
  81. * m *m
  82. G + * m G^{\prime}+*m
  83. * m *m
  84. * m *m^{\prime}
  85. m < m m^{\prime}<m
  86. m m
  87. G G^{\prime}
  88. * m *m^{\prime}
  89. G G^{\prime}
  90. * n i *n_{i}
  91. n i < m n_{i}<m
  92. * m *m
  93. * n i *n_{i}
  94. n i > m n_{i}>m
  95. * n i *n_{i}
  96. * m *m
  97. n i = m n_{i}=m
  98. m m
  99. n i n_{i}
  100. G G G\approx G^{\prime}
  101. G * m G^{\prime}\approx*m
  102. G * m G\approx*m
  103. G G
  104. m m
  105. G * m G\approx*m
  106. m m
  107. * m *m
  108. m m
  109. G G
  110. m m
  111. G + H G+H
  112. * m + H *m+H
  113. H H

Sprouts_(game).html

  1. n + m = 3 n - m + 2 ( 3 n - m ) + p n+m=3n-m+2(3n-m)+p
  2. m = 2 n + p / 4 m=2n+p/4

Square-free_integer.html

  1. ζ ( s ) ζ ( 2 s ) = n = 1 | μ ( n ) | n s \frac{\zeta(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{s}}
  2. ζ ( s ) ζ ( 2 s ) = p ( 1 - p - 2 s ) ( 1 - p - s ) = p ( 1 + p - s ) . \frac{\zeta(s)}{\zeta(2s)}=\prod_{p}\frac{(1-p^{-2s})}{(1-p^{-s})}=\prod_{p}(1% +p^{-s}).
  3. Q ( x ) x p prime ( 1 - 1 p 2 ) = x p prime 1 ( 1 - 1 p 2 ) - 1 Q(x)\approx x\prod_{p\ \,\text{prime}}\left(1-\frac{1}{p^{2}}\right)=x\prod_{p% \ \,\text{prime}}\frac{1}{(1-\frac{1}{p^{2}})^{-1}}
  4. Q ( x ) x p prime 1 1 + 1 p 2 + 1 p 4 + = x k = 1 1 k 2 = x ζ ( 2 ) Q(x)\approx x\prod_{p\ \,\text{prime}}\frac{1}{1+\frac{1}{p^{2}}+\frac{1}{p^{4% }}+\cdots}=\frac{x}{\sum_{k=1}^{\infty}\frac{1}{k^{2}}}=\frac{x}{\zeta(2)}
  5. Q ( x ) = x ζ ( 2 ) + O ( x ) = 6 x π 2 + O ( x ) Q(x)=\frac{x}{\zeta(2)}+O\left(\sqrt{x}\right)=\frac{6x}{\pi^{2}}+O\left(\sqrt% {x}\right)
  6. Q ( x ) = 6 x π 2 + O ( x 1 / 2 exp ( - c ( log x ) 3 / 5 ( log log x ) 1 / 5 ) ) . Q(x)=\frac{6x}{\pi^{2}}+O\left(x^{1/2}\exp\left(-c\frac{(\log x)^{3/5}}{(\log% \log x)^{1/5}}\right)\right).
  7. Q ( x ) = x ζ ( 2 ) + O ( x 17 / 54 + ε ) = 6 x π 2 + O ( x 17 / 54 + ε ) . Q(x)=\frac{x}{\zeta(2)}+O\left(x^{17/54+\varepsilon}\right)=\frac{6x}{\pi^{2}}% +O\left(x^{17/54+\varepsilon}\right).
  8. lim x Q ( x ) x = 6 π 2 = 1 ζ ( 2 ) \lim_{x\to\infty}\frac{Q(x)}{x}=\frac{6}{\pi^{2}}=\frac{1}{\zeta(2)}
  9. Q ( x , n ) = x k = 1 1 k n + O ( x n ) = x ζ ( n ) + O ( x n ) . Q(x,n)=\frac{x}{\sum_{k=1}^{\infty}\frac{1}{k^{n}}}+O\left(\sqrt[n]{x}\right)=% \frac{x}{\zeta(n)}+O\left(\sqrt[n]{x}\right).
  10. n = 0 p n + 1 a n , a n { 0 , 1 } , and p n is the n th prime . \prod_{n=0}^{\infty}{p_{n+1}}^{a_{n}},a_{n}\in\{0,1\},\,\text{ and }p_{n}\,% \text{ is the }n\,\text{th prime}.
  11. a n a_{n}
  12. n = 0 a n 2 n \sum_{n=0}^{\infty}{a_{n}}\cdot 2^{n}
  13. ( 2 n n ) {2n\choose n}
  14. core t ( n ) \mathrm{core}_{t}(n)
  15. core t ( p e ) = p e mod t . \mathrm{core}_{t}(p^{e})=p^{e\mod t}.
  16. core 2 \mathrm{core}_{2}
  17. n 1 core t ( n ) n s = ζ ( t s ) ζ ( s - 1 ) ζ ( t s - t ) \sum_{n\geq 1}\frac{\mathrm{core}_{t}(n)}{n^{s}}=\frac{\zeta(ts)\zeta(s-1)}{% \zeta(ts-t)}

Square_(disambiguation).html

  1. \Box

Square_root.html

  1. a \sqrt{a}
  2. 9 \sqrt{9}
  3. a \sqrt{a}
  4. a \sqrt{a}
  5. a \sqrt{a}
  6. 2 \sqrt{2}
  7. 2 \sqrt{2}
  8. 2 \sqrt{2}
  9. x \sqrt{x}
  10. x 2 = | x | = { x , if x 0 - x , if x < 0. \sqrt{x^{2}}=\left|x\right|=\begin{cases}x,&\mbox{if }~{}x\geq 0\\ -x,&\mbox{if }~{}x<0.\end{cases}
  11. x y = x y \sqrt{xy}=\sqrt{x}\sqrt{y}
  12. x = x 1 / 2 . \sqrt{x}=x^{1/2}.
  13. f ( x ) = 1 2 x . f^{\prime}(x)=\frac{1}{2\sqrt{x}}.
  14. 1 + x \sqrt{1+}{x}
  15. | x | |x|
  16. 1 + x = n = 0 ( - 1 ) n ( 2 n ) ! ( 1 - 2 n ) ( n ! ) 2 ( 4 n ) x n = 1 + 1 2 x - 1 8 x 2 + 1 16 x 3 - 5 128 x 4 + , \sqrt{1+x}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}x^{n}% =1+\textstyle\frac{1}{2}x-\frac{1}{8}x^{2}+\frac{1}{16}x^{3}-\frac{5}{128}x^{4% }+\dots,\!
  17. a = e ( ln a ) / 2 \sqrt{a}=e^{(\ln a)/2}
  18. a = 10 ( log 10 a ) / 2 . \sqrt{a}=10^{(\log_{10}a)/2}.
  19. ln \ln
  20. log 10 \log_{10}
  21. a \sqrt{a}
  22. ( x + c ) 2 = x 2 + 2 x c + c 2 (x+c)^{2}=x^{2}+2xc+c^{2}
  23. ( x + c ) 2 x 2 + 2 x c (x+c)^{2}\approx x^{2}+2xc
  24. x 2 + 2 x c + c 2 x^{2}+2xc+c^{2}
  25. y = 2 x c + x 2 y=2xc+x^{2}
  26. 2 x c 2xc
  27. a a
  28. c = a 2 x c=\frac{a}{2x}
  29. d y / d x = f ( x ) = 2 x , dy/dx=f^{\prime}(x)=2x,
  30. a \sqrt{a}
  31. x 0 x_{0}
  32. a \sqrt{a}
  33. a = 2 - n 4 n a , \sqrt{a}=2^{-n}\sqrt{4^{n}a},
  34. [ 1 , 4 ) [1,4)
  35. - x = i x . \sqrt{-x}=i\sqrt{x}.
  36. ( i x ) 2 = i 2 ( x ) 2 = ( - 1 ) x = - x . (i\sqrt{x})^{2}=i^{2}(\sqrt{x})^{2}=(-1)x=-x.
  37. i = 1 2 2 + i 1 2 2 = 2 2 ( 1 + i ) . \sqrt{i}=\frac{1}{2}\sqrt{2}+i\frac{1}{2}\sqrt{2}=\frac{\sqrt{2}}{2}(1+i).
  38. i = ( a + b i ) 2 i=(a+bi)^{2}\!
  39. i = a 2 + 2 a b i - b 2 . i=a^{2}+2abi-b^{2}.\!
  40. { 2 a b = 1 a 2 - b 2 = 0 \begin{cases}2ab=1\\ a^{2}-b^{2}=0\end{cases}
  41. a = b = ± 1 2 . a=b=\pm\frac{1}{\sqrt{2}}.
  42. a = b = 1 2 . a=b=\frac{1}{\sqrt{2}}.
  43. i = cos ( π 2 ) + i sin ( π 2 ) i=\cos\left(\frac{\pi}{2}\right)+i\sin\left(\frac{\pi}{2}\right)
  44. i = ( cos ( π 2 ) + i sin ( π 2 ) ) 1 2 = cos ( π 4 ) + i sin ( π 4 ) = 1 2 + i ( 1 2 ) = 1 2 ( 1 + i ) . \begin{aligned}\displaystyle\sqrt{i}&\displaystyle=\left(\cos\left(\frac{\pi}{% 2}\right)+i\sin\left(\frac{\pi}{2}\right)\right)^{\frac{1}{2}}\\ &\displaystyle=\cos\left(\frac{\pi}{4}\right)+i\sin\left(\frac{\pi}{4}\right)% \\ &\displaystyle=\frac{1}{\sqrt{2}}+i\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{% \sqrt{2}}(1+i).\\ \end{aligned}
  45. z = r e i φ with - π < φ π , z=re^{i\varphi}\,\text{ with }-\pi<\varphi\leq\pi,
  46. z = r e i φ / 2 . \sqrt{z}=\sqrt{r}\,e^{i\varphi/2}.
  47. 1 + x \sqrt{1+}{x}
  48. r ( cos φ + i sin φ ) = r [ cos φ 2 + i sin φ 2 ] . \sqrt{r\left(\cos\varphi+i\,\sin\varphi\right)}=\sqrt{r}\left[\cos\frac{% \varphi}{2}+i\sin\frac{\varphi}{2}\right].
  49. z = | z | + Re ( z ) 2 + i sgn ( Im ( z ) ) | z | - Re ( z ) 2 , \sqrt{z}=\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}+i\ \operatorname{sgn}(% \operatorname{Im}(z))\ \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}},
  50. ± ( | z | + Re ( z ) 2 + i sgn ( Im ( z ) ) | z | - Re ( z ) 2 ) \pm\left(\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}+i\ \operatorname{sgn}(% \operatorname{Im}(z))\ \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right)
  51. z \sqrt{z}
  52. w \sqrt{w}
  53. - 1 \displaystyle-1
  54. 1 \sqrt{−1}
  55. 1 \sqrt{−1}
  56. - 1 - 1 = i i = - 1 \sqrt{-1}\cdot\sqrt{-1}=i\cdot i=-1
  57. - 1 - 1 = ( - i ) ( - i ) = - 1 \sqrt{-1}\cdot\sqrt{-1}=(-i)\cdot(-i)=-1
  58. - 1 - 1 = 1 = - 1 , \sqrt{-1\cdot-1}=\sqrt{1}=-1,
  59. u u
  60. v v
  61. u = v u=v
  62. u + v = 0 u+v=0
  63. p p
  64. e e
  65. q q
  66. ( q 1 ) / 2 (q−1)/2
  67. ( q 1 ) / 2 (q−1)/2
  68. / 8 \mathbb{Z}/8\mathbb{Z}
  69. \mathbb{H}
  70. { a i + b j + c k a 2 + b 2 + c 2 = 1 } . \{ai+bj+ck\mid a^{2}+b^{2}+c^{2}=1\}.
  71. n n
  72. 0 \sqrt{0}
  73. 1 \sqrt{1}
  74. 2 \sqrt{2}
  75. 3 \sqrt{3}
  76. 4 \sqrt{4}
  77. 5 \sqrt{5}
  78. 6 \sqrt{6}
  79. 7 \sqrt{7}
  80. 8 \sqrt{8}
  81. 9 \sqrt{9}
  82. 10 \sqrt{10}
  83. 11 \sqrt{11}
  84. 12 \sqrt{12}
  85. 13 \sqrt{13}
  86. 14 \sqrt{14}
  87. 15 \sqrt{15}
  88. 16 \sqrt{16}
  89. 17 \sqrt{17}
  90. 18 \sqrt{18}
  91. 19 \sqrt{19}
  92. 20 \sqrt{20}
  93. 21 \sqrt{21}
  94. 8 = 4 2 = 2 2 \sqrt{8}\ =\ \sqrt{4}\sqrt{2}\ =\ 2\sqrt{2}
  95. 12 = 4 3 = 2 3 \sqrt{12}\ =\ \sqrt{4}\sqrt{3}\ =\ 2\sqrt{3}
  96. 18 = 9 2 = 3 2 \sqrt{18}\ =\ \sqrt{9}\sqrt{2}\ =\ 3\sqrt{2}
  97. 20 = 4 5 = 2 5 \sqrt{20}\ =\ \sqrt{4}\sqrt{5}\ =\ 2\sqrt{5}
  98. 2 \sqrt{2}
  99. 3 \sqrt{3}
  100. 4 \sqrt{4}
  101. 5 \sqrt{5}
  102. 6 \sqrt{6}
  103. 7 \sqrt{7}
  104. 8 \sqrt{8}
  105. 9 \sqrt{9}
  106. 10 \sqrt{10}
  107. 11 \sqrt{11}
  108. 12 \sqrt{12}
  109. 13 \sqrt{13}
  110. 14 \sqrt{14}
  111. 15 \sqrt{15}
  112. 16 \sqrt{16}
  113. 17 \sqrt{17}
  114. 18 \sqrt{18}
  115. 19 \sqrt{19}
  116. 20 \sqrt{20}
  117. 11 = 3 + 1 3 + 1 6 + 1 3 + 1 6 + 1 3 + \sqrt{11}=3+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\ddots% }}}}}
  118. 11 = 3 + 2 6 + 2 6 + 2 6 + 2 6 + 2 6 + = 3 + 6 1 20 - 1 - 1 20 - 1 20 - 1 20 - . \sqrt{11}=3+\cfrac{2}{6+\cfrac{2}{6+\cfrac{2}{6+\cfrac{2}{6+\cfrac{2}{6+\ddots% }}}}}=3+\cfrac{6\cdot 1}{20-1-\cfrac{1}{20-\cfrac{1}{20-\cfrac{1}{20-\ddots}}}}.
  119. a \sqrt{a}
  120. a b \sqrt{ab}
  121. a \sqrt{a}
  122. a / h = h / b , \ a/h=h/b,
  123. h 2 = a b , \ h^{2}=ab,
  124. h = a b h=\sqrt{ab}
  125. ( a + b ) / 2 (a+b)/2
  126. ( a + b ) / 2 a b (a+b)/2\geq\sqrt{ab}
  127. 1 \sqrt{1}
  128. x \sqrt{x}
  129. x \sqrt{x}
  130. x + 1 \sqrt{x}{+1}

SQUID.html

  1. I I
  2. I a I_{a}
  3. I b I_{b}
  4. Φ \Phi
  5. I I
  6. I s I_{s}
  7. I I
  8. I I
  9. I / 2 + I s I/2+I_{s}
  10. I / 2 - I s I/2-I_{s}
  11. I c I_{c}
  12. Φ 0 / 2 \Phi_{0}/2
  13. Φ 0 \Phi_{0}
  14. Φ 0 \Phi_{0}
  15. I c I_{c}
  16. Φ 0 \Phi_{0}
  17. R R
  18. Δ Φ \Delta\Phi
  19. Δ V \Delta V
  20. λ = i c L Φ 0 \lambda=\frac{i_{c}L}{\Phi_{0}}

Srinivasa_Ramanujan.html

  1. 1 + 2 1 + 3 1 + . \sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}.
  2. x + n + a = a x + ( n + a ) 2 + x a ( x + n ) + ( n + a ) 2 + ( x + n ) x+n+a=\sqrt{ax+(n+a)^{2}+x\sqrt{a(x+n)+(n+a)^{2}+(x+n)\sqrt{\cdots}}}
  3. B n n {B_{n}\over n}
  4. 2 n ( 2 n - 1 ) b n n 2^{n}(2^{n}-1){b_{n}\over n}
  5. 2 ( 2 n - 1 ) B n 2(2^{n}-1)B_{n}\,
  6. 1 - 5 ( 1 2 ) 3 + 9 ( 1 × 3 2 × 4 ) 3 - 13 ( 1 × 3 × 5 2 × 4 × 6 ) 3 + = 2 π 1-5\left(\frac{1}{2}\right)^{3}+9\left(\frac{1\times 3}{2\times 4}\right)^{3}-% 13\left(\frac{1\times 3\times 5}{2\times 4\times 6}\right)^{3}+\cdots=\frac{2}% {\pi}
  7. 1 + 9 ( 1 4 ) 4 + 17 ( 1 × 5 4 × 8 ) 4 + 25 ( 1 × 5 × 9 4 × 8 × 12 ) 4 + = 2 3 2 π 1 2 Γ 2 ( 3 4 ) . 1+9\left(\frac{1}{4}\right)^{4}+17\left(\frac{1\times 5}{4\times 8}\right)^{4}% +25\left(\frac{1\times 5\times 9}{4\times 8\times 12}\right)^{4}+\cdots=\frac{% 2^{\frac{3}{2}}}{\pi^{\frac{1}{2}}\Gamma^{2}\left(\frac{3}{4}\right)}.
  8. 1 π = 2 2 9801 k = 0 ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k . \frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum^{\infty}_{k=0}\frac{(4k)!(1103+26390k% )}{(k!)^{4}396^{4k}}.
  9. e π 58 = 396 4 - 104.000000177 . e^{\pi\sqrt{58}}=396^{4}-104.000000177\dots.
  10. 9801 2 / 4412 9801\sqrt{2}/4412
  11. [ 1 + 2 n = 1 cos ( n θ ) cosh ( n π ) ] - 2 + [ 1 + 2 n = 1 cosh ( n θ ) cosh ( n π ) ] - 2 = 2 Γ 4 ( 3 4 ) π = 8 π 3 Γ 4 ( 1 4 ) \left[1+2\sum_{n=1}^{\infty}\frac{\cos(n\theta)}{\cosh(n\pi)}\right]^{-2}+% \left[1+2\sum_{n=1}^{\infty}\frac{\cosh(n\theta)}{\cosh(n\pi)}\right]^{-2}=% \frac{2\Gamma^{4}\left(\frac{3}{4}\right)}{\pi}=\frac{8\pi^{3}}{\Gamma^{4}% \left(\frac{1}{4}\right)}
  12. θ \theta
  13. Γ ( z ) \Gamma(z)
  14. θ 0 \theta^{0}
  15. θ 4 \theta^{4}
  16. θ 8 \theta^{8}

Stainless_steel.html

  1. σ \sigma

Standard_deviation.html

  1. 2 , 4 , 4 , 4 , 5 , 5 , 7 , 9. 2,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9.
  2. 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 8 = 5. \frac{2+4+4+4+5+5+7+9}{8}=5.
  3. ( 2 - 5 ) 2 = ( - 3 ) 2 = 9 ( 5 - 5 ) 2 = 0 2 = 0 ( 4 - 5 ) 2 = ( - 1 ) 2 = 1 ( 5 - 5 ) 2 = 0 2 = 0 ( 4 - 5 ) 2 = ( - 1 ) 2 = 1 ( 7 - 5 ) 2 = 2 2 = 4 ( 4 - 5 ) 2 = ( - 1 ) 2 = 1 ( 9 - 5 ) 2 = 4 2 = 16. \begin{array}[]{lll}(2-5)^{2}=(-3)^{2}=9&&(5-5)^{2}=0^{2}=0\\ (4-5)^{2}=(-1)^{2}=1&&(5-5)^{2}=0^{2}=0\\ (4-5)^{2}=(-1)^{2}=1&&(7-5)^{2}=2^{2}=4\\ (4-5)^{2}=(-1)^{2}=1&&(9-5)^{2}=4^{2}=16.\\ \end{array}
  4. 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 8 = 4. \frac{9+1+1+1+0+0+4+16}{8}=4.
  5. 4 = 2. \sqrt{4}=2.
  6. E [ X ] = μ . \operatorname{E}[X]=\mu.\,\!
  7. σ = E [ ( X - μ ) 2 ] = E [ X 2 ] + E [ ( - 2 μ X ) ] + E [ μ 2 ] = E [ X 2 ] - 2 μ E [ X ] + μ 2 = E [ X 2 ] - 2 μ 2 + μ 2 = E [ X 2 ] - μ 2 = E [ X 2 ] - ( E [ X ] ) 2 \begin{aligned}\displaystyle\sigma&\displaystyle=\sqrt{\operatorname{E}[(X-\mu% )^{2}]}\\ &\displaystyle=\sqrt{\operatorname{E}[X^{2}]+\operatorname{E}[(-2\mu X)]+% \operatorname{E}[\mu^{2}]}=\sqrt{\operatorname{E}[X^{2}]-2\mu\operatorname{E}[% X]+\mu^{2}}\\ &\displaystyle=\sqrt{\operatorname{E}[X^{2}]-2\mu^{2}+\mu^{2}}=\sqrt{% \operatorname{E}[X^{2}]-\mu^{2}}\\ &\displaystyle=\sqrt{\operatorname{E}[X^{2}]-(\operatorname{E}[X])^{2}}\end{aligned}
  8. σ = 1 N [ ( x 1 - μ ) 2 + ( x 2 - μ ) 2 + + ( x N - μ ) 2 ] , where μ = 1 N ( x 1 + + x N ) , \sigma=\sqrt{\frac{1}{N}\left[(x_{1}-\mu)^{2}+(x_{2}-\mu)^{2}+\cdots+(x_{N}-% \mu)^{2}\right]},{\rm\ \ where\ \ }\mu=\frac{1}{N}(x_{1}+\cdots+x_{N}),
  9. σ = 1 N i = 1 N ( x i - μ ) 2 , where μ = 1 N i = 1 N x i . \sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\mu)^{2}},{\rm\ \ where\ \ }\mu=% \frac{1}{N}\sum_{i=1}^{N}x_{i}.
  10. σ = i = 1 N p i ( x i - μ ) 2 , where μ = i = 1 N p i x i . \sigma=\sqrt{\sum_{i=1}^{N}p_{i}(x_{i}-\mu)^{2}},{\rm\ \ where\ \ }\mu=\sum_{i% =1}^{N}p_{i}x_{i}.
  11. σ = 𝐗 ( x - μ ) 2 p ( x ) d x , where μ = 𝐗 x p ( x ) d x , \sigma=\sqrt{\int_{\mathbf{X}}(x-\mu)^{2}\,p(x)\,dx},{\rm\ \ where\ \ }\mu=% \int_{\mathbf{X}}x\,p(x)\,dx,
  12. s N = 1 N i = 1 N ( x i - x ¯ ) 2 , s_{N}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}},
  13. { x 1 , x 2 , , x N } \scriptstyle\{x_{1},\,x_{2},\,\ldots,\,x_{N}\}
  14. x ¯ \scriptstyle\overline{x}
  15. n > 75 n>75
  16. s N = 1 N i = 1 N ( x i - x ¯ ) 2 . s_{N}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}}.
  17. s 2 = 1 N - 1 i = 1 N ( x i - x ¯ ) 2 . s^{2}=\frac{1}{N-1}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}.
  18. ( x 1 - x ¯ , , x n - x ¯ ) . \scriptstyle(x_{1}-\overline{x},\;\dots,\;x_{n}-\overline{x}).
  19. s = 1 N - 1 i = 1 N ( x i - x ¯ ) 2 . s=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}}.
  20. c 4 ( N ) = 2 N - 1 Γ ( N 2 ) Γ ( N - 1 2 ) . c_{4}(N)\,=\,\sqrt{\frac{2}{N-1}}\,\,\,\frac{\Gamma\left(\frac{N}{2}\right)}{% \Gamma\left(\frac{N-1}{2}\right)}.
  21. σ ^ = 1 N - 1.5 i = 1 n ( x i - x ¯ ) 2 , \hat{\sigma}=\sqrt{\frac{1}{N-1.5}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}},
  22. σ ^ = 1 n - 1.5 - 1 4 γ 2 i = 1 n ( x i - x ¯ ) 2 , \hat{\sigma}=\sqrt{\frac{1}{n-1.5-\tfrac{1}{4}\gamma_{2}}\sum_{i=1}^{n}(x_{i}-% \bar{x})^{2}},
  23. σ ( c ) = 0 \sigma(c)=0\,
  24. σ ( X + c ) = σ ( X ) , \sigma(X+c)=\sigma(X),\,
  25. σ ( c X ) = | c | σ ( X ) . \sigma(cX)=|c|\sigma(X).\,
  26. σ ( X + Y ) = var ( X ) + var ( Y ) + 2 cov ( X , Y ) . \sigma(X+Y)=\sqrt{\operatorname{var}(X)+\operatorname{var}(Y)+2\,\operatorname% {cov}(X,Y)}.\,
  27. var = σ 2 \scriptstyle\operatorname{var}\,=\,\sigma^{2}
  28. cov \scriptstyle\operatorname{cov}
  29. σ ( X ) = E [ ( X - E ( X ) ) 2 ] = E [ X 2 ] - ( E [ X ] ) 2 . \sigma(X)=\sqrt{E[(X-E(X))^{2}]}=\sqrt{E[X^{2}]-(E[X])^{2}}.
  30. σ ( X ) = N N - 1 E [ ( X - E ( X ) ) 2 ] . \sigma(X)=\sqrt{\frac{N}{N-1}}\sqrt{E[(X-E(X))^{2}]}.
  31. 1 N i = 1 N ( x i - x ¯ ) 2 = 1 N ( i = 1 N x i 2 ) - x ¯ 2 = ( 1 N i = 1 N x i 2 ) - ( 1 N i = 1 N x i ) 2 . \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}}=\sqrt{\frac{1}{N}% \left(\sum_{i=1}^{N}x_{i}^{2}\right)-\overline{x}^{2}}=\sqrt{\left(\frac{1}{N}% \sum_{i=1}^{N}x_{i}^{2}\right)-\left(\frac{1}{N}\sum_{i=1}^{N}x_{i}\right)^{2}}.
  32. M = ( x ¯ , x ¯ , x ¯ ) M=(\overline{x},\overline{x},\overline{x})
  33. 1 - 1 k 2 \scriptstyle 1-\frac{1}{k^{2}}
  34. k \scriptstyle k
  35. l \scriptstyle l
  36. 1 1 - l \scriptstyle\frac{1}{\sqrt{1-l}}
  37. f ( x ; μ , σ 2 ) = 1 σ 2 π e - 1 2 ( x - μ σ ) 2 f(x;\mu,\sigma^{2})=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-% \mu}{\sigma}\right)^{2}}
  38. erf ( z 2 ) \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)
  39. erf \scriptstyle\operatorname{erf}
  40. x = 1 2 [ 1 + erf ( x - μ σ 2 ) ] = 1 2 [ 1 + erf ( z 2 ) ] x=\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}% \right)\right]=\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{z}{\sqrt{2}}% \right)\right]
  41. σ ( r ) = 1 N - 1 i = 1 N ( x i - r ) 2 . \sigma(r)=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_{i}-r)^{2}}.
  42. r = x ¯ . r=\overline{x}.\,
  43. σ mean = 1 N σ \sigma_{\,\text{mean}}=\frac{1}{\sqrt{N}}\sigma
  44. var ( X ) \displaystyle\operatorname{var}(X)
  45. var ( mean ) = var ( 1 N i = 1 N X i ) = 1 N 2 var ( i = 1 N X i ) = 1 N 2 i = 1 N var ( X i ) = N N 2 var ( X ) = 1 N var ( X ) . \begin{aligned}\displaystyle\operatorname{var}(\,\text{mean})&\displaystyle=% \operatorname{var}\left(\frac{1}{N}\sum_{i=1}^{N}X_{i}\right)=\frac{1}{N^{2}}% \operatorname{var}\left(\sum_{i=1}^{N}X_{i}\right)\\ &\displaystyle=\frac{1}{N^{2}}\sum_{i=1}^{N}\operatorname{var}(X_{i})=\frac{N}% {N^{2}}\operatorname{var}(X)=\frac{1}{N}\operatorname{var}(X).\end{aligned}
  46. σ mean = σ N . \sigma\text{mean}=\frac{\sigma}{\sqrt{N}}.
  47. σ mean \sigma\text{mean}
  48. σ \sigma
  49. s j = k = 1 N x k j . \ s_{j}=\sum_{k=1}^{N}{x_{k}^{j}}.
  50. σ = N s 2 - s 1 2 N \sigma=\frac{\sqrt{Ns_{2}-s_{1}^{2}}}{N}
  51. s = N s 2 - s 1 2 N ( N - 1 ) . s=\sqrt{\frac{Ns_{2}-s_{1}^{2}}{N(N-1)}}.
  52. A 0 \displaystyle A_{0}
  53. Q 0 = 0 Q k = Q k - 1 + k - 1 k ( x k - A k - 1 ) 2 = Q k - 1 + ( x k - A k - 1 ) ( x k - A k ) \begin{aligned}\displaystyle Q_{0}&\displaystyle=0\\ \displaystyle Q_{k}&\displaystyle=Q_{k-1}+\frac{k-1}{k}(x_{k}-A_{k-1})^{2}=Q_{% k-1}+(x_{k}-A_{k-1})(x_{k}-A_{k})\\ \end{aligned}
  54. Q 1 = 0 Q_{1}=0
  55. k - 1 = 0 k-1=0
  56. x 1 = A 1 x_{1}=A_{1}
  57. s n 2 = Q n n - 1 s^{2}_{n}=\frac{Q_{n}}{n-1}
  58. σ n 2 = Q n n \sigma^{2}_{n}=\frac{Q_{n}}{n}
  59. s j = k = 1 N w k x k j . \ s_{j}=\sum_{k=1}^{N}{w_{k}x_{k}^{j}}.\,
  60. W 0 \displaystyle W_{0}
  61. A 0 \displaystyle A_{0}
  62. σ n 2 = Q n W n \sigma^{2}_{n}=\frac{Q_{n}}{W_{n}}\,
  63. s n 2 = n n - 1 σ n 2 s^{2}_{n}=\frac{n^{\prime}}{n^{\prime}-1}\sigma^{2}_{n}\,
  64. N X Y \displaystyle N_{X\cup Y}
  65. μ X Y \displaystyle\mu_{X\cup Y}
  66. μ \displaystyle\mu
  67. X = i X i \scriptstyle X\,=\,\bigcup_{i}X_{i}
  68. μ X \displaystyle\mu_{X}
  69. X i X j = , i < j . X_{i}\cap X_{j}=\varnothing,\quad\forall\ i<j.
  70. μ X Y \displaystyle\mu_{X\cup Y}
  71. σ X = i σ X i 2 + i , j cov ( X i , X j ) \sigma_{X}=\sqrt{\sum_{i}{\sigma_{X_{i}}^{2}}+\sum_{i,j}\operatorname{cov}(X_{% i},X_{j})}
  72. cov ( X i , X j ) = 0 , i < j \displaystyle\operatorname{cov}(X_{i},X_{j})=0,\quad\forall i<j
  73. μ X Y \displaystyle\mu_{X\cup Y}
  74. X = i X i \scriptstyle X\,=\,\bigcup_{i}X_{i}
  75. μ X \displaystyle\mu_{X}
  76. X i X j = , i < j . X_{i}\cap X_{j}=\varnothing,\quad\forall i<j.
  77. μ X Y \displaystyle\mu_{X\cup Y}