wpmath0000016_11

Point_process_operation.html

  1. x \textstyle x
  2. N \textstyle{N}
  3. x N , \textstyle x\in{N},
  4. N \textstyle{N}
  5. B \textstyle B
  6. N ( B ) , \textstyle{N}(B),
  7. 𝐑 d \textstyle\,\textbf{R}^{d}
  8. N \textstyle{N}
  9. N p \textstyle{N}_{p}
  10. p \textstyle p
  11. N \textstyle{N}
  12. p \textstyle p
  13. 1 - p \textstyle 1-p
  14. p ( x ) 1 \textstyle p(x)\leq 1
  15. p ( x ) \textstyle p(x)
  16. p ( x ) \textstyle p(x)
  17. N \textstyle{N}
  18. p \textstyle p
  19. N 1 , N 2 \textstyle{N}_{1},{N}_{2}\dots
  20. Λ 1 , Λ 2 , \textstyle\Lambda_{1},\Lambda_{2},\dots
  21. N = i = 1 N i , {N}=\bigcup_{i=1}^{\infty}{N}_{i},
  22. N i \textstyle{N}_{i}
  23. N \textstyle{N}
  24. Λ = i = 1 Λ i . \Lambda=\sum\limits_{i=1}^{\infty}\Lambda_{i}.
  25. x \textstyle x
  26. N \textstyle{N}
  27. N x \textstyle N^{x}
  28. N c = x N N x . {N}_{c}=\bigcup_{x\in{N}}N^{x}.
  29. N x \textstyle N^{x}
  30. N \textstyle{N}
  31. λ \textstyle\lambda
  32. N c \textstyle{N}_{c}
  33. λ c = c λ , \lambda_{c}=c\lambda,
  34. c \textstyle c
  35. N x \textstyle N^{x}

Point_set_registration.html

  1. { , 𝒮 } \{\mathcal{M},\mathcal{S}\}
  2. d \mathbb{R}^{d}
  3. M M
  4. N N
  5. \mathcal{M}
  6. \mathcal{M}
  7. 𝒮 \mathcal{S}
  8. d \mathbb{R}^{d}
  9. d \mathbb{R}^{d}
  10. T T
  11. T ( ) T(\mathcal{M})
  12. θ \theta
  13. T ( , θ ) T(\mathcal{M},\theta)
  14. θ \theta
  15. θ \theta
  16. T T
  17. \mathcal{M}
  18. 𝒮 \mathcal{S}
  19. T ( ) T(\mathcal{M})
  20. 𝒮 \mathcal{S}
  21. dist \operatorname{dist}
  22. dist ( T ( ) , 𝒮 ) = m T ( ) s 𝒮 ( m - s ) 2 \operatorname{dist}(T(\mathcal{M}),\mathcal{S})=\sum_{m\in T(\mathcal{M})}\sum% _{s\in\mathcal{S}}(m-s)^{2}
  23. g g
  24. dist robust ( T ( ) , 𝒮 ) = m T ( ) s 𝒮 g ( ( m - s ) 2 ) \operatorname{dist}_{\operatorname{robust}}(T(\mathcal{M}),\mathcal{S})=\sum_{% m\in T(\mathcal{M})}\sum_{s\in\mathcal{S}}g((m-s)^{2})
  25. g g
  26. \mathcal{M}
  27. 𝒮 \mathcal{S}
  28. \mathcal{M}
  29. 𝒮 \mathcal{S}
  30. ( , 𝒮 ) (\mathcal{M},\mathcal{S})
  31. θ \theta
  32. θ 0 \theta_{0}
  33. X X
  34. \emptyset
  35. m i T ( , θ ) m_{i}\in T(\mathcal{M},\theta)
  36. s ^ j \hat{s}_{j}
  37. 𝒮 \mathcal{S}
  38. m i m_{i}
  39. X X
  40. X X
  41. m i , s ^ j \langle m_{i},\hat{s}_{j}\rangle
  42. θ \theta
  43. ( X ) (X)
  44. θ \theta
  45. m i , s ^ j \langle m_{i},\hat{s}_{j}\rangle
  46. 𝒮 \mathcal{S}
  47. \mathcal{M}
  48. m i m_{i}
  49. i i
  50. \mathcal{M}
  51. s j s_{j}
  52. j j
  53. 𝒮 \mathcal{S}
  54. μ \mathbf{\mu}
  55. μ i j = { 1 if point m i corresponds to point s j 0 otherwise \mu_{ij}=\left\{\begin{matrix}1&\,\text{if point }m_{i}\,\text{ corresponds to% point }s_{j}\\ 0&\,\text{otherwise}\end{matrix}\right.
  56. \mathcal{M}
  57. 𝒮 \mathcal{S}
  58. T T
  59. μ \mathbf{\mu}
  60. T ( m ) = 𝐀 m + 𝐭 T(m)=\mathbf{A}m+\mathbf{t}
  61. 𝐀 \mathbf{A}
  62. { a , θ , b , c } \{a,\theta,b,c\}
  63. cost = j = 1 N i = 1 M μ i j s j - 𝐭 - 𝐀 m i 2 + g ( 𝐀 ) - α j = 1 N i = 1 M μ i j \operatorname{cost}=\sum_{j=1}^{N}\sum_{i=1}^{M}\mu_{ij}\lVert s_{j}-\mathbf{t% }-\mathbf{A}m_{i}\rVert^{2}+g(\mathbf{A})-\alpha\sum_{j=1}^{N}\sum_{i=1}^{M}% \mu_{ij}
  64. j i = 1 M μ i j 1 \forall j~{}\sum_{i=1}^{M}\mu_{ij}\leq 1
  65. i j = 1 N μ i j 1 \forall i~{}\sum_{j=1}^{N}\mu_{ij}\leq 1
  66. i j μ i j { 0 , 1 } \forall ij~{}\mu_{ij}\in\{0,1\}
  67. α \alpha
  68. g ( 𝐀 ) g(\mathbf{A})
  69. g ( 𝐀 ( a , θ , b , c ) ) = γ ( a 2 + b 2 + c 2 ) g(\mathbf{A}(a,\theta,b,c))=\gamma(a^{2}+b^{2}+c^{2})
  70. γ \gamma
  71. { Q j } \{Q_{j}\}
  72. Q j 1 Q_{j}\in\mathbb{R}^{1}
  73. μ j \mu_{j}
  74. Q j Q_{j}
  75. j = 1 J μ j = 1 \sum_{j=1}^{J}\mu_{j}=1
  76. μ \mathbf{\mu}
  77. j = 1 J μ j Q j \sum_{j=1}^{J}\mu_{j}Q_{j}
  78. β > 0 \beta>0
  79. β \beta
  80. μ \mathbf{\mu}
  81. μ j ^ = exp ( β Q j ^ ) j = 1 J exp ( β Q j ) \mu_{\hat{j}}=\frac{\exp{(\beta Q_{\hat{j}})}}{\sum_{j=1}^{J}\exp{(\beta Q_{j}% )}}
  82. β \beta
  83. j = 1 J μ j Q j \sum_{j=1}^{J}\mu_{j}Q_{j}
  84. E ( μ ) = j = 1 N i = 0 M μ i j Q i j E(\mu)=\sum_{j=1}^{N}\sum_{i=0}^{M}\mu_{ij}Q_{ij}
  85. Q i j = - ( s j - 𝐭 - 𝐀 m i 2 - α ) = - cost μ i j Q_{ij}=-(\lVert s_{j}-\mathbf{t}-\mathbf{A}m_{i}\rVert^{2}-\alpha)=-\frac{% \partial\operatorname{cost}}{\partial\mu_{ij}}
  86. μ \mu
  87. j i = 1 M μ i j = 1 \forall j~{}\sum_{i=1}^{M}\mu_{ij}=1
  88. i j = 1 N μ i j = 1 \forall i~{}\sum_{j=1}^{N}\mu_{ij}=1
  89. ( , 𝒮 ) (\mathcal{M},\mathcal{S})
  90. 𝐭 \mathbf{t}
  91. a , θ , b , c a,\theta,b,c
  92. β \beta
  93. β 0 \beta_{0}
  94. μ ^ i j \hat{\mu}_{ij}
  95. 1 + ϵ 1+\epsilon
  96. β < β f \beta<\beta_{f}
  97. μ \mu
  98. Q i j Q_{ij}
  99. - cost μ i j -\frac{\partial\operatorname{cost}}{\partial\mu_{ij}}
  100. μ i j 0 \mu^{0}_{ij}
  101. exp ( β Q i j ) \exp(\beta Q_{ij})
  102. μ ^ \hat{\mu}
  103. μ ^ \hat{\mu}
  104. μ ^ i j 1 \hat{\mu}^{1}_{ij}
  105. μ ^ i j 0 i = 1 M + 1 μ ^ i j 0 \frac{\hat{\mu}^{0}_{ij}}{\sum_{i=1}^{M+1}\hat{\mu}^{0}_{ij}}
  106. μ ^ \hat{\mu}
  107. μ ^ i j 0 \hat{\mu}^{0}_{ij}
  108. μ ^ i j 1 j = 1 N + 1 μ ^ i j 1 \frac{\hat{\mu}^{1}_{ij}}{\sum_{j=1}^{N+1}\hat{\mu}^{1}_{ij}}
  109. θ \theta
  110. 𝐭 \mathbf{t}
  111. a , b , c a,b,c
  112. β \beta
  113. β r β \beta_{r}\beta
  114. γ \gamma
  115. γ β r \frac{\gamma}{\beta_{r}}
  116. a , b , c , θ a,b,c,\theta
  117. 𝐭 \mathbf{t}
  118. β \beta
  119. β 0 \beta_{0}
  120. β r \beta_{r}
  121. β f \beta_{f}
  122. M + 1 M+1
  123. N + 1 N+1
  124. M M
  125. N N
  126. μ \mu
  127. M + 1 M+1
  128. N + 1 N+1
  129. μ \mathbf{\mu}
  130. K K
  131. K C KC
  132. x i , x j x_{i},x_{j}
  133. K K
  134. χ \mathcal{\chi}
  135. K C ( 𝒳 ) = i j K C ( x i , x j ) = 2 i < j K C ( x i , x j ) KC(\mathcal{X})=\sum_{i\neq j}KC(x_{i},x_{j})=2\sum_{i<j}KC(x_{i},x_{j})
  136. θ \theta
  137. cost ( 𝒮 , , θ ) = - m s 𝒮 K C ( s , T ( m , θ ) ) \operatorname{cost}(\mathcal{S},\mathcal{M},\theta)=-\sum_{m\in\mathcal{M}}% \sum_{s\in\mathcal{S}}KC(s,T(m,\theta))
  138. K C ( 𝒮 T ( , θ ) ) = K C ( 𝒮 ) + K C ( T ( , θ ) ) - 2 cost ( 𝒮 , , θ ) KC(\mathcal{S}\cup T(\mathcal{M},\theta))=KC(\mathcal{S})+KC(T(\mathcal{M},% \theta))-2\operatorname{cost}(\mathcal{S},\mathcal{M},\theta)
  139. K C ( 𝒮 ) KC(\mathcal{S})
  140. θ \theta
  141. K C ( T ( , θ ) ) KC(T(\mathcal{M},\theta))
  142. θ \theta
  143. K C ( 𝒮 T ( , θ ) ) = C - 2 cost ( 𝒮 , , θ ) KC(\mathcal{S}\cup T(\mathcal{M},\theta))=C-2\operatorname{cost}(\mathcal{S},% \mathcal{M},\theta)
  144. P ( x , θ ) = 1 N m K ( x , T ( m , θ ) ) P_{\mathcal{M}}(x,\theta)=\frac{1}{N}\sum_{m\in\mathcal{M}}K(x,T(m,\theta))
  145. P 𝒮 ( x ) = 1 N s 𝒮 K ( x , s ) P_{\mathcal{S}}(x)=\frac{1}{N}\sum_{s\in\mathcal{S}}K(x,s)
  146. cost ( 𝒮 , , θ ) = - N 2 x P P 𝒮 d x \operatorname{cost}(\mathcal{S},\mathcal{M},\theta)=-N^{2}\int_{x}P_{\mathcal{% M}}\cdot P_{\mathcal{S}}~{}dx
  147. P , P 𝒮 P_{\mathcal{M}},P_{\mathcal{S}}
  148. \mathcal{M}
  149. M M
  150. \mathcal{M}
  151. N N
  152. 𝒮 \mathcal{S}
  153. s s
  154. p ( s ) = i = 1 M + 1 P ( i ) p ( s | i ) p(s)=\sum_{i=1}^{M+1}P(i)p(s|i)
  155. D D
  156. p ( s | i ) p(s|i)
  157. m i m_{i}\in\mathcal{M}
  158. p ( s | i ) = 1 ( 2 π σ 2 ) D / 2 ) exp ( - s - m i 2 2 σ 2 ) p(s|i)=\frac{1}{(2\pi\sigma^{2})^{D/2)}}\exp{\left(-\frac{\lVert s-m_{i}\rVert% ^{2}}{2\sigma^{2}}\right)}
  159. P ( i ) = 1 M P(i)=\frac{1}{M}
  160. w [ 0 , 1 ] w\in[0,1]
  161. p ( s ) = w 1 N + ( 1 - w ) i = 1 M 1 M p ( s | i ) p(s)=w\frac{1}{N}+(1-w)\sum_{i=1}^{M}\frac{1}{M}p(s|i)
  162. θ \theta
  163. E ( θ , σ 2 ) = - j = 1 N log i = 1 M + 1 P ( i ) p ( s | i ) E(\theta,\sigma^{2})=-\sum_{j=1}^{N}\log\sum_{i=1}^{M+1}P(i)p(s|i)
  164. m i m_{i}
  165. s j s_{j}
  166. P ( i | s j ) = P ( i ) p ( s j | i ) p ( s j ) P(i|s_{j})=\frac{P(i)p(s_{j}|i)}{p(s_{j})}
  167. θ \theta
  168. σ 2 \sigma^{2}
  169. P old ( i , s j ) P^{\,\text{old}}(i,s_{j})
  170. cost = - j = 1 N i = 1 M + 1 P old ( i | s j ) log ( P new ( i ) p new ( s j | i ) ) \operatorname{cost}=-\sum_{j=1}^{N}\sum_{i=1}^{M+1}P^{\,\text{old}}(i|s_{j})% \log(P^{\,\text{new}}(i)p^{\,\text{new}}(s_{j}|i))
  171. θ \theta
  172. σ \sigma
  173. cost ( θ , σ 2 ) = 1 2 σ 2 j = 1 N i = 1 M + 1 P old ( i | s j ) s j - T ( m i , θ ) 2 + N 𝐏 D 2 log σ 2 \operatorname{cost}(\theta,\sigma^{2})=\frac{1}{2\sigma^{2}}\sum_{j=1}^{N}\sum% _{i=1}^{M+1}P^{\,\text{old}}(i|s_{j})\lVert s_{j}-T(m_{i},\theta)\rVert^{2}+% \frac{N_{\mathbf{P}}D}{2}\log{\sigma^{2}}
  174. N 𝐏 = j = 0 N i = 0 M P old ( i | s j ) N N_{\mathbf{P}}=\sum_{j=0}^{N}\sum_{i=0}^{M}P^{\,\text{old}}(i|s_{j})\leq N
  175. N = N 𝐏 N=N_{\mathbf{P}}
  176. w = 0 w=0
  177. P old P^{\,\text{old}}
  178. P old ( i | s j ) = exp ( - 1 2 σ old 2 s j - T ( m i , θ old ) 2 ) k = 1 M exp ( - 1 2 σ old 2 s j - T ( m k , θ old ) 2 ) + ( 2 π σ 2 ) D 2 w 1 - w M N P^{\,\text{old}}(i|s_{j})=\frac{\exp\left(-\frac{1}{2\sigma^{\,\text{old}2}}% \lVert s_{j}-T(m_{i},\theta^{\,\text{old}})\rVert^{2}\right)}{\sum_{k=1}^{M}% \exp\left(-\frac{1}{2\sigma^{\,\text{old}2}}\lVert s_{j}-T(m_{k},\theta^{\,% \text{old}})\rVert^{2}\right)+(2\pi\sigma^{2})^{\frac{D}{2}}\frac{w}{1-w}\frac% {M}{N}}
  179. E E
  180. \mathcal{M}
  181. 𝒮 \mathcal{S}
  182. M × D M\times D
  183. N × D N\times D
  184. 𝐌 \mathbf{M}
  185. 𝐒 \mathbf{S}
  186. ( , 𝒮 ) (\mathcal{M},\mathcal{S})
  187. θ \theta
  188. θ 0 \theta_{0}
  189. 0 w 1 0\leq w\leq 1
  190. σ 2 \sigma^{2}
  191. 1 D N M j = 1 N i = 1 M s j - m i 2 \frac{1}{DNM}\sum_{j=1}^{N}\sum_{i=1}^{M}\lVert s_{j}-m_{i}\rVert^{2}
  192. 𝐏 \mathbf{P}
  193. i [ 1 , M ] i\in[1,M]
  194. j [ 1 , N ] j\in[1,N]
  195. p i j p_{ij}
  196. exp ( - 1 2 σ 2 s j - T ( m i , θ ) 2 ) k = 1 M exp ( - 1 2 σ 2 s j - T ( m k , θ ) 2 ) + ( 2 π σ 2 ) D 2 w 1 - w M N \frac{\exp\left(-\frac{1}{2\sigma^{2}}\lVert s_{j}-T(m_{i},\theta)\rVert^{2}% \right)}{\sum_{k=1}^{M}\exp\left(-\frac{1}{2\sigma^{2}}\lVert s_{j}-T(m_{k},% \theta)\rVert^{2}\right)+(2\pi\sigma^{2})^{\frac{D}{2}}\frac{w}{1-w}\frac{M}{N}}
  197. { θ , σ 2 } \{\theta,\sigma^{2}\}
  198. ( 𝐒 , 𝐌 , 𝐏 ) (\mathbf{S},\mathbf{M},\mathbf{P})
  199. θ \theta
  200. 𝟏 \mathbf{1}
  201. a a
  202. 𝐑 \mathbf{R}
  203. 𝐭 \mathbf{t}
  204. θ \theta
  205. θ = { a , 𝐑 , 𝐭 } \theta=\{a,\mathbf{R},\mathbf{t}\}
  206. θ 0 = { 1 , 𝐈 , 𝟎 } \theta_{0}=\{1,\mathbf{I},\mathbf{0}\}
  207. T ( 𝐌 ) = a 𝐌𝐑 T + 𝟏 𝐭 T T(\mathbf{M})=a\mathbf{M}\mathbf{R}^{T}+\mathbf{1}\mathbf{t}^{T}
  208. ( 𝐒 , 𝐌 , 𝐏 ) (\mathbf{S},\mathbf{M},\mathbf{P})
  209. N 𝐏 N_{\mathbf{P}}
  210. 𝟏 T 𝐏𝟏 \mathbf{1}^{T}\mathbf{P}\mathbf{1}
  211. μ s \mu_{s}
  212. 1 N 𝐏 𝐒 T 𝐏 T 𝟏 \frac{1}{N_{\mathbf{P}}}\mathbf{S}^{T}\mathbf{P}^{T}\mathbf{1}
  213. μ m \mu_{m}
  214. 1 N 𝐏 𝐌 T 𝐏𝟏 \frac{1}{N_{\mathbf{P}}}\mathbf{M}^{T}\mathbf{P}\mathbf{1}
  215. 𝐒 ^ \hat{\mathbf{S}}
  216. 𝐒 - 𝟏 μ s T \mathbf{S}-\mathbf{1}\mu_{s}^{T}
  217. 𝐌 ^ \hat{\mathbf{M}}
  218. 𝐌 - 𝟏 μ m T \mathbf{M}-\mathbf{1}\mu_{m}^{T}
  219. 𝐀 \mathbf{A}
  220. 𝐒 T ^ 𝐏 T 𝐌 ^ \hat{\mathbf{S}^{T}}\mathbf{P}^{T}\hat{\mathbf{M}}
  221. 𝐔 , 𝐕 \mathbf{U},\mathbf{V}
  222. ( 𝐀 ) (\mathbf{A})
  223. 𝐀 = 𝐔 Σ 𝐕 T \mathbf{A}=\mathbf{U}\Sigma\mathbf{V}^{T}
  224. 𝐂 \mathbf{C}
  225. diag ( 1 , , 1 , det ( 𝐔𝐕 T ) ) \operatorname{diag}(1,...,1,\det(\mathbf{UV}^{T}))
  226. diag ( ξ ) \operatorname{diag}(\xi)
  227. ξ \xi
  228. 𝐑 \mathbf{R}
  229. 𝐔𝐂𝐕 T \mathbf{UCV}^{T}
  230. a a
  231. tr ( 𝐀 T 𝐑 ) tr ( 𝐌 ^ 𝐓 diag ( 𝐏𝟏 ) 𝐌 ^ ) \frac{\operatorname{tr}(\mathbf{A}^{T}\mathbf{R})}{\operatorname{tr}(\mathbf{% \hat{\mathbf{M}}^{T}\operatorname{diag}(\mathbf{P}\mathbf{1})\hat{\mathbf{M}}})}
  232. tr \operatorname{tr}
  233. 𝐭 \mathbf{t}
  234. μ s - a 𝐑 μ m \mu_{s}-a\mathbf{R}\mu_{m}
  235. σ 2 \sigma^{2}
  236. 1 N 𝐏 D ( tr ( 𝐒 ^ 𝐓 diag ( 𝐏 𝐓 𝟏 ) 𝐒 ^ ) ) - a tr ( 𝐀 T 𝐑 ) \frac{1}{N_{\mathbf{P}}D}(\operatorname{tr}(\mathbf{\hat{\mathbf{S}}^{T}% \operatorname{diag}(\mathbf{P}^{T}\mathbf{1})\hat{\mathbf{S}}}))-a% \operatorname{tr}(\mathbf{A}^{T}\mathbf{R})
  237. { a , 𝐑 , 𝐭 } , σ 2 \{a,\mathbf{R},\mathbf{t}\},\sigma^{2}
  238. 𝐁 \mathbf{B}
  239. 𝐭 \mathbf{t}
  240. T ( 𝐌 ) = 𝐌𝐁 T + 𝟏 𝐭 T T(\mathbf{M})=\mathbf{M}\mathbf{B}^{T}+\mathbf{1}\mathbf{t}^{T}
  241. ( 𝐒 , 𝐌 , 𝐏 ) (\mathbf{S},\mathbf{M},\mathbf{P})
  242. N 𝐏 N_{\mathbf{P}}
  243. 𝟏 T 𝐏𝟏 \mathbf{1}^{T}\mathbf{P}\mathbf{1}
  244. μ s \mu_{s}
  245. 1 N 𝐏 𝐒 T 𝐏 T 𝟏 \frac{1}{N_{\mathbf{P}}}\mathbf{S}^{T}\mathbf{P}^{T}\mathbf{1}
  246. μ m \mu_{m}
  247. 1 N 𝐏 𝐌 T 𝐏𝟏 \frac{1}{N_{\mathbf{P}}}\mathbf{M}^{T}\mathbf{P}\mathbf{1}
  248. 𝐒 ^ \hat{\mathbf{S}}
  249. 𝐒 - 𝟏 μ s T \mathbf{S}-\mathbf{1}\mu_{s}^{T}
  250. 𝐌 ^ \hat{\mathbf{M}}
  251. 𝐌 - 𝟏 μ s T \mathbf{M}-\mathbf{1}\mu_{s}^{T}
  252. 𝐁 \mathbf{B}
  253. ( 𝐒 T ^ 𝐏 T 𝐌 ^ ) ( 𝐌 T ^ diag ( 𝐏𝟏 ) 𝐌 ^ ) - 1 (\hat{\mathbf{S}^{T}}\mathbf{P}^{T}\hat{\mathbf{M}})(\hat{\mathbf{M}^{T}}% \operatorname{diag}(\mathbf{P}\mathbf{1})\hat{\mathbf{M}})^{-1}
  254. 𝐭 \mathbf{t}
  255. μ s - 𝐁 μ m \mu_{s}-\mathbf{B}\mu_{m}
  256. σ 2 \sigma^{2}
  257. 1 N 𝐏 D ( tr ( 𝐒 ^ diag ( 𝐏 𝐓 𝟏 ) 𝐒 ^ ) ) - tr ( 𝐒 T ^ 𝐏 T 𝐌 ^ 𝐁 T ) \frac{1}{N_{\mathbf{P}}D}(\operatorname{tr}(\mathbf{\hat{\mathbf{S}}% \operatorname{diag}(\mathbf{P}^{T}\mathbf{1})\hat{\mathbf{S}}}))-\operatorname% {tr}(\hat{\mathbf{S}^{T}}\mathbf{P}^{T}\hat{\mathbf{M}}\mathbf{B}^{T})
  258. 𝐁 , 𝐭 } , σ 2 \mathbf{B},\mathbf{t}\},\sigma^{2}
  259. O ( M + N ) O(M+N)
  260. O ( M N ) O(MN)

Poisson_point_process.html

  1. Pr ( X = x ) = ( μ ( A ) ) x e - μ ( A ) x ! \Pr(X=x)=\frac{(\mu(A))^{x}e^{-\mu(A)}}{x!}

Poisson_wavelet.html

  1. ψ n ( t ) \psi_{n}(t)
  2. ψ n ( t ) = { ( t - n n ! ) t n - 1 e - t for t 0 0 for t < 0. \psi_{n}(t)=\begin{cases}\left(\frac{t-n}{n!}\right)t^{n-1}e^{-t}&\,\text{ for% }t\geq 0\\ 0&\,\text{ for }t<0.\end{cases}
  3. p n ( t ) = t n n ! e - t . p_{n}(t)=\frac{t^{n}}{n!}e^{-t}.
  4. ψ n ( t ) \psi_{n}(t)
  5. ψ n ( t ) = - d d t p n ( t ) . \psi_{n}(t)=-\frac{d}{dt}p_{n}(t).
  6. ψ n ( t ) \psi_{n}(t)
  7. ψ n ( t ) = p n ( t ) - p n - 1 ( t ) . \psi_{n}(t)=p_{n}(t)-p_{n-1}(t).
  8. - ψ n ( t ) d t = 0. \int_{-\infty}^{\infty}\psi_{n}(t)\,dt=0.
  9. ψ n ( t ) \psi_{n}(t)
  10. Ψ ( ω ) = - i ω ( 1 + i ω ) n + 1 . \Psi(\omega)=\frac{-i\omega}{(1+i\omega)^{n+1}}.
  11. ψ n ( t ) \psi_{n}(t)
  12. C ψ n = - | Ψ n ( ω ) | 2 | ω | d ω = 1 n . C_{\psi_{n}}=\int_{-\infty}^{\infty}\frac{\left|\Psi_{n}(\omega)\right|^{2}}{|% \omega|}\,d\omega=\frac{1}{n}.
  13. ( W n f ) ( a , b ) = 1 | a | - f ( t ) ψ n ( t - b a ) d t (W_{n}f)(a,b)=\frac{1}{\sqrt{|a|}}\int_{-\infty}^{\infty}f(t)\psi_{n}\left(% \frac{t-b}{a}\right)\,dt
  14. ( W n f ) ( a , b ) (W_{n}f)(a,b)
  15. f ( t ) = 1 C ψ n - [ - { ( W n f ) ( a , b ) 1 | a | ψ n ( t - b a ) } d b ] d a a 2 f(t)=\frac{1}{C_{\psi_{n}}}\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty% }\,\left\{(W_{n}f)(a,b)\frac{1}{\sqrt{|a|}}\psi_{n}\left(\frac{t-b}{a}\right)% \,\right\}db\right]\frac{da}{a^{2}}
  16. ψ ( t ) = 1 π 1 - t 2 ( 1 + t 2 ) 2 \psi(t)=\frac{1}{\pi}\frac{1-t^{2}}{(1+t^{2})^{2}}
  17. ψ ( t ) = P ( t ) + t d d t P ( t ) \psi(t)=P(t)+t\frac{d}{dt}P(t)
  18. P ( t ) = 1 π 1 1 + t 2 P(t)=\frac{1}{\pi}\frac{1}{1+t^{2}}
  19. P ( t ) P(t)
  20. s ( x ) s(x)
  21. L p ( ) L^{p}(\mathbb{R})
  22. ϕ ( x , y ) \phi(x,y)
  23. - | ϕ ( x , y ) | p d x c < \int_{-\infty}^{\infty}|\phi(x,y)|^{p}\,dx\leq c<\infty
  24. ϕ ( x , y ) s ( x ) \phi(x,y)\rightarrow s(x)
  25. y 0 y\rightarrow 0
  26. L p ( ) L^{p}(\mathbb{R})
  27. ϕ ( x , y ) \phi(x,y)
  28. ϕ ( t , y ) = P y ( t ) s ( t ) \phi(t,y)=P_{y}(t)\star s(t)
  29. P y ( t ) = 1 y P ( t y ) = 1 π y t 2 + y 2 P_{y}(t)=\frac{1}{y}P\left(\frac{t}{y}\right)=\frac{1}{\pi}\frac{y}{t^{2}+y^{2}}
  30. \star
  31. P y ( t ) P_{y}(t)
  32. ϕ ( x , y ) \phi(x,y)
  33. ϕ ( x , y ) \phi(x,y)
  34. s ( x ) s(x)
  35. - ψ ( t ) d t = 0 \int_{-\infty}^{\infty}\psi(t)\,dt=0
  36. ψ ( t ) \psi(t)
  37. Ψ ( ω ) = | ω | e - | ω | \Psi(\omega)=|\omega|e^{-|\omega|}
  38. C ψ = - | Ψ ( ω ) | 2 | ω | d ω = 2. C_{\psi}=\int_{-\infty}^{\infty}\frac{\left|\Psi(\omega)\right|^{2}}{|\omega|}% \,d\omega=2.
  39. ψ n ( t ) \psi_{n}(t)
  40. n = 1 , 2 , 3 , 4 n=1,2,3,4
  41. ψ n ( t ) \psi_{n}(t)
  42. n = 1 , 2 , 3 , 4 n=1,2,3,4
  43. ψ n ( t ) = 1 2 π ( 1 - i t ) - ( n + 1 ) \psi_{n}(t)=\frac{1}{2\pi}(1-it)^{-(n+1)}
  44. n = 1 , 2 , 3 , n=1,2,3,\ldots
  45. ψ n ( t ) \psi_{n}(t)
  46. ψ n ( t ) = 1 2 π 1 n ! i n d n d t n ( ( 1 - i t ) - 1 ) \psi_{n}(t)=\frac{1}{2\pi}\frac{1}{n!\,i^{n}}\frac{d^{n}}{dt^{n}}\left((1-it)^% {-1}\right)
  47. ( 1 - i t ) - 1 (1-it)^{-1}
  48. P ( t ) = 1 1 + t 2 P(t)=\frac{1}{1+t^{2}}
  49. ( 1 - i t ) - 1 = P ( t ) + i t P ( t ) (1-it)^{-1}=P(t)+itP(t)
  50. ψ n ( t ) = 1 2 π 1 n ! i n d n d t n P ( t ) + i ( 1 2 π 1 n ! i n d n d t n ( t P ( t ) ) ) \psi_{n}(t)=\frac{1}{2\pi}\frac{1}{n!\,i^{n}}\frac{d^{n}}{dt^{n}}P(t)+i\left(% \frac{1}{2\pi}\frac{1}{n!\,i^{n}}\frac{d^{n}}{dt^{n}}\left(tP(t)\right)\right)
  51. ψ n ( t ) \psi_{n}(t)
  52. ψ n ( t ) \psi_{n}(t)
  53. Ψ n ( ω ) = 1 Γ ( n + 1 ) ω n e - ω u ( ω ) \Psi_{n}(\omega)=\frac{1}{\Gamma(n+1)}\omega^{n}e^{-\omega}u(\omega)
  54. u ( ω ) u(\omega)

Polar_circle_(geometry).html

  1. r 2 = H A × H D = H B × H E = H C × H F = - 4 R 2 cos A cos B cos C = 1 2 ( a 2 + b 2 + c 2 ) - 4 R 2 , \begin{aligned}\displaystyle r^{2}&\displaystyle=HA\times HD=HB\times HE=HC% \times HF\\ &\displaystyle=-4R^{2}\cos A\cos B\cos C=\frac{1}{2}(a^{2}+b^{2}+c^{2})-4R^{2}% ,\end{aligned}

Polling_system.html

  1. 𝔼 ( W i ) = 1 + ρ i 2 𝔼 ( C ) + ( 1 + ρ i ) Var ( C i ) 2 𝔼 ( C ) \mathbb{E}(W_{i})=\frac{1+\rho_{i}}{2}\mathbb{E}(C)+\frac{(1+\rho_{i})\,\text{% Var}(C_{i})}{2\mathbb{E}(C)}
  2. 𝔼 ( W i ) = 1 - ρ i 2 𝔼 ( C ) + ( 1 - ρ i ) Var ( C i + 1 ) 2 𝔼 ( C ) \mathbb{E}(W_{i})=\frac{1-\rho_{i}}{2}\mathbb{E}(C)+\frac{(1-\rho_{i})\,\text{% Var}(C_{i+1})}{2\mathbb{E}(C)}
  3. 𝔼 ( C ) = i = 1 n 𝔼 ( d i ) 1 - ρ \mathbb{E}(C)=\sum_{i=1}^{n}\frac{\mathbb{E}(d_{i})}{1-\rho}

Polyadic_space.html

  1. ( X , τ X ) (X,\tau_{X})
  2. { ω } X \left\{\omega\right\}\cup X
  3. ω X \omega X
  4. ω X \omega\notin X
  5. τ ω X \tau_{\omega X}
  6. τ X τ ω X \tau_{X}\subseteq\tau_{\omega X}
  7. X C { ω } τ ω X X\setminus C\cup\left\{\omega\right\}\in\tau_{\omega X}
  8. C X C\subseteq X
  9. X X
  10. ω X \omega X
  11. X X
  12. P P
  13. λ \lambda
  14. f : ω X λ P f:\omega X^{\lambda}\rightarrow P
  15. ω X λ \omega X^{\lambda}
  16. ω X \omega X
  17. λ \lambda
  18. + \mathbb{Z}+
  19. ω + \omega\mathbb{Z}+
  20. λ = 1 \lambda=1
  21. h : ω + [ 0 , 1 ] h:\omega\mathbb{Z}+\rightarrow\left[0,1\right]
  22. h ( x ) = { 1 / x , if x + 0 , if x = ω h(x)=\begin{cases}1/x,&\,\text{if }x\in\mathbb{Z}+\\ 0,&\,\text{if }x=\omega\end{cases}
  23. { 0 } n { 1 / n } \left\{0\right\}\cup\bigcup_{n\in\mathbb{N}}\left\{1/n\right\}
  24. c ( X ) c(X)
  25. X X
  26. sup { | B | : B is a disjoint collection of open sets of X } \sup\left\{|B|:B\,\text{ is a disjoint collection of open sets of }X\right\}
  27. t ( X ) t(X)
  28. X X
  29. A X A\subset X
  30. p A ¯ p\in\bar{A}
  31. a ( p , A ) := min { | B | : p B ¯ , B A } a(p,A):=\min\left\{|B|:p\in\bar{B},B\subset A\right\}
  32. t ( p , X ) := sup { a ( p , A ) : A X , p A ¯ } t(p,X):=\sup\left\{a(p,A):A\subset X,p\in\bar{A}\right\}
  33. t ( X ) := sup { t ( p , X ) : p X } . t(X):=\sup\left\{t(p,X):p\in X\right\}.
  34. w ( X ) w(X)
  35. X X
  36. w ( X ) = c ( X ) t ( X ) w(X)=c(X)\cdot t(X)
  37. X X
  38. A X A\subset X
  39. P X P\subset X
  40. A P A\subset P
  41. c ( P ) c ( A ) c(P)\leq c(A)
  42. X X
  43. 2 ω \leq 2^{\omega}
  44. + \mathbb{Z}+
  45. d i s ( X ) dis(X)
  46. X X
  47. Δ ( X ) \Delta(X)
  48. X X
  49. X X
  50. d i s ( X ) Δ ( X ) dis(X)\geq\Delta(X)
  51. C O ( X ) CO(X)
  52. X X
  53. C O ( X ) CO(X)
  54. G C O ( X ) G\in CO(X)^{\prime}
  55. G = C O ( X ) \langle\langle G\rangle\rangle=CO(X)
  56. C O ( X ) CO(X)
  57. G G
  58. ( τ , κ ) (\tau,\kappa)
  59. G G
  60. τ \tau
  61. G α G_{\alpha}
  62. α \alpha
  63. G α G_{\alpha}
  64. κ \kappa
  65. X X
  66. G G
  67. C O ( X ) CO(X)
  68. ( τ , κ ) (\tau,\kappa)
  69. X X
  70. α κ τ \alpha\kappa^{\tau}
  71. X X
  72. cmpn X \operatorname{cmpn}\,X
  73. n n
  74. X X
  75. 𝒮 \mathcal{S}
  76. S S
  77. { : is a finite subset of 𝒮 } \{\bigcap\mathcal{F}:\mathcal{F}\,\text{ is a finite subset of }\mathcal{S}\}
  78. 𝒮 ^ \mathcal{S}^{\widehat{\mathcal{F}}}
  79. S S
  80. n n
  81. [ S ] n [S]^{n}
  82. n n
  83. [ S ] < = n [S]^{<=n}
  84. 2 n < ω 2\leq n<\omega
  85. \bigcap\mathcal{F}\neq
  86. [ 𝒮 ] n \mathcal{F}\in[\mathcal{S}]^{n}
  87. 𝒮 \mathcal{S}
  88. 𝒮 \mathcal{S}
  89. 𝒮 \mathcal{S}
  90. 𝒮 \mathcal{S}
  91. 𝒮 ^ \mathcal{S}^{\widehat{\mathcal{F}}}
  92. X X
  93. cmpn X n \operatorname{cmpn}\,X\leq n
  94. 𝒮 \mathcal{S}
  95. 𝒮 = 𝒮 ^ \mathcal{S}=\mathcal{S}^{\widehat{\mathcal{F}}}
  96. 𝒮 = 𝒮 ^ \mathcal{S}=\mathcal{S}^{\widehat{\mathcal{F}}}
  97. X X
  98. K K
  99. U U
  100. \mathcal{F}
  101. 𝒮 \mathcal{F}\subset\mathcal{S}
  102. K U K\subset\bigcup\mathcal{F}\subset U
  103. S S
  104. n n
  105. 1 n < ω 1\leq n<\omega
  106. [ S ] n [S]^{\leq n}
  107. s S s\in S
  108. s - = { F [ S ] n : s F } s^{-}=\{F\in[S]^{\leq n}:s\in F\}
  109. s + = { F [ S ] n : s F } s^{+}=\{F\in[S]^{\leq n}:s\notin F\}
  110. 𝒮 \mathcal{S}
  111. 𝒮 = s S { s + , s - } \mathcal{S}=\bigcup_{s\in S}\{s^{+},s^{-}\}
  112. 𝒮 \mathcal{S}
  113. [ S ] n [S]^{\leq n}
  114. k k
  115. n n
  116. 0 k n 0\leq k\leq n
  117. [ S ] k [S]^{k}
  118. [ S ] n [S]^{\leq n}
  119. [ S ] n [S]^{\leq n}
  120. n + 1 n+1
  121. cmpn [ S ] n \operatorname{cmpn}\,[S]^{\leq n}
  122. < <
  123. S S
  124. n + 1 n+1
  125. \mathcal{R}
  126. [ S ] 2 n [S]^{\leq 2n}
  127. s S s\in S
  128. L s = { F s + : | { t F : t < s } | n - 1 } L_{s}=\{F\in s^{+}:|\{t\in F:t<s\}|\leq n-1\}
  129. R s = { F s + : | { t F : t > s } | n - 1 } R_{s}=\{F\in s^{+}:|\{t\in F:t>s\}|\leq n-1\}
  130. = s S { L s , R s , s + } \mathcal{R}=\bigcup_{s\in S}\{L_{s},R_{s},s^{+}\}
  131. A A
  132. B B
  133. C C
  134. A B C A\cup B\cup C\neq
  135. = { L s : s A } { R s : s B } { s - : s C } \mathcal{F}=\{L_{s}:s\in A\}\cup\{R_{s}:s\in B\}\cup\{s^{-}:s\in C\}
  136. \mathcal{F}
  137. n + 1 n+1
  138. \mathcal{R}
  139. A B A\cup B\in\bigcap\mathcal{F}
  140. \blacksquare
  141. X X
  142. A X A\in X
  143. r : X A r:X\rightarrow A
  144. r | A r|_{A}
  145. A A
  146. A A
  147. X X
  148. U U
  149. A U X A\subset U\subset X
  150. A A
  151. U U
  152. A A
  153. X X
  154. cmpn [ S ] n \operatorname{cmpn}\,[S]^{\leq n}
  155. n n
  156. 2 n < ω 2\leq n<\omega
  157. [ ω 1 ] 2 n - 1 [\omega_{1}]^{\leq 2n-1}
  158. K K
  159. cmpn K n \operatorname{cmpn}\,K\leq n
  160. n n
  161. 1 n < ω 1\leq n<\omega
  162. cmpn [ ω 1 ] 2 n - 1 = n + 1 = cmpn [ ω 1 ] 2 n \operatorname{cmpn}\,[\omega_{1}]^{\leq 2n-1}=n+1=\operatorname{cmpn}\,[\omega% _{1}]^{\leq 2n}
  163. A A
  164. S S
  165. A = S { } A=S\cup\{\infty\}
  166. g : A n [ S ] n g:A^{n}\rightarrow[S]^{\leq n}
  167. g ( ( x 1 , , x n ) ) = { x 1 , , x n } S g((x_{1},...,x_{n}))=\{x_{1},\ldots,x_{n}\}\cap S
  168. [ S ] n [S]^{\leq n}
  169. [ ω 1 ] 2 n - 1 [\omega_{1}]^{\leq 2n-1}
  170. cmpn [ ω 1 ] 2 n - 1 = n + 1 \operatorname{cmpn}\,[\omega_{1}]^{\leq 2n-1}=n+1
  171. 𝒮 \mathcal{S}
  172. 𝒮 \mathcal{S}
  173. \bigcap\mathcal{F}\neq
  174. 𝒮 \mathcal{F}\subseteq\mathcal{S}
  175. C e n ( 𝒮 ) = { χ T : T is a centred subcollection of 𝒮 } Cen(\mathcal{S})=\{\chi_{T}:T\,\text{ is a centred subcollection of }\mathcal{% S}\}
  176. 2 𝒮 2^{\mathcal{S}}
  177. X X
  178. 𝒮 \mathcal{S}
  179. X X
  180. C e n ( 𝒮 ) Cen(\mathcal{S})
  181. X X
  182. 𝒜 𝒫 ( X ) \mathcal{A}\subseteq\mathcal{P}(X)
  183. 𝒜 \mathcal{A}
  184. A 𝒜 and B 𝒜 B 𝒜 A\in\mathcal{A}\and B\subseteq\mathcal{A}\Rightarrow B\in\mathcal{A}
  185. A X A\subseteq X
  186. A A
  187. 𝒜 \mathcal{A}
  188. A 𝒜 A\in\mathcal{A}
  189. 𝒜 \mathcal{A}
  190. D X D^{X}
  191. K ( 𝒜 ) K(\mathcal{A})
  192. K K
  193. X X
  194. 𝒜 𝒫 ( X ) \mathcal{A}\subseteq\mathcal{P}(X)
  195. K K
  196. K ( 𝒜 ) K(\mathcal{A})
  197. K K
  198. κ \kappa
  199. τ \tau
  200. X X
  201. ( κ + 1 ) τ (\kappa+1)^{\tau}
  202. X X
  203. X X
  204. 𝔫 \mathfrak{n}
  205. X X
  206. X X
  207. 𝐁 ( 𝔫 ) \mathbf{B}(\mathfrak{n})
  208. { G α : α A } \{G_{\alpha}:\alpha\in A\}
  209. X X
  210. | A | = 𝔫 |A|=\mathfrak{n}
  211. B A B\subset A
  212. p X p\in X
  213. | B | = 𝔫 |B|=\mathfrak{n}
  214. N N
  215. p p
  216. | { β B : N G β = } | < 𝔫 |\{\beta\in B:N\cap G_{\beta}=\}|<\mathfrak{n}
  217. X X
  218. X X
  219. 𝐁 ( 𝔫 ) \mathbf{B}(\mathfrak{n})
  220. 𝔫 \mathfrak{n}
  221. X X
  222. H ( X ) H(X)
  223. X X
  224. J 2 ( X ) J_{2}(X)
  225. H ( X ) H(X)
  226. { F H ( X ) : | F | 2 } \{F\in H(X):|F|\leq 2\}
  227. H ( X ) H(X)
  228. U 0 , , U n = { F H ( X ) : F U 0 U n , F U i for 0 i n } \langle U_{0},\dots,U_{n}\rangle=\{F\in H(X):F\subseteq U_{0}\cup\dots\cup U_{% n},F\cap U_{i}\neq\,\text{ for }0\leq i\leq n\}
  229. n n
  230. U i U_{i}
  231. X X
  232. X X
  233. Y Y
  234. H ( X ) H(X)
  235. Y Y

Polykay.html

  1. k r , s k_{r,s}

Polynomial_decomposition.html

  1. g h g\circ h
  2. f = x 6 - 3 x 3 + 1 f=x^{6}-3x^{3}+1
  3. g = x 2 - 3 x + 1 g=x^{2}-3x+1
  4. h = x 3 h=x^{3}
  5. f ( x ) = ( g h ) ( x ) = g ( h ( x ) ) = g ( x 3 ) = ( x 3 ) 2 - 3 ( x 3 ) + 1. f(x)=(g\circ h)(x)=g(h(x))=g(x^{3})=(x^{3})^{2}-3(x^{3})+1.
  6. x 6 - 6 x 5 + 21 x 4 - 44 x 3 + 68 x 2 - 64 x + 41 \displaystyle x^{6}-6x^{5}+21x^{4}-44x^{3}+68x^{2}-64x+41
  7. f = g 1 g 2 g m = h 1 h 2 h n f=g_{1}\circ g_{2}\circ\cdots\circ g_{m}=h_{1}\circ h_{2}\circ\cdots\circ h_{n}
  8. g i h i g_{i}\neq h_{i}
  9. i i
  10. m = n m=n
  11. x 2 x 3 = x 3 x 2 x^{2}\circ x^{3}=x^{3}\circ x^{2}
  12. x 6 - 6 x 5 + 15 x 4 - 20 x 3 + 15 x 2 - 6 x - 1 = ( x 3 - 2 ) ( x 2 - 2 x + 1 ) \begin{aligned}&\displaystyle x^{6}-6x^{5}+15x^{4}-20x^{3}+15x^{2}-6x-1\\ \displaystyle=&\displaystyle(x^{3}-2)\circ(x^{2}-2x+1)\end{aligned}
  13. 1 ± 2 1 / 6 , 1 ± - 1 ± 3 i 2 1 / 3 1\pm 2^{1/6},1\pm\frac{\sqrt{-1\pm\sqrt{3}i}}{2^{1/3}}
  14. x 4 - 8 x 3 + 18 x 2 - 8 x + 2 = ( x 2 + 1 ) ( x 2 - 4 x + 1 ) \begin{aligned}&\displaystyle x^{4}-8x^{3}+18x^{2}-8x+2\\ \displaystyle=&\displaystyle(x^{2}+1)\circ(x^{2}-4x+1)\end{aligned}
  15. 2 ± 3 ± i 2\pm\sqrt{3\pm i}
  16. - 9 ( 8 10 i 3 3 / 2 + 72 ) 2 / 3 + 36 ( 8 10 i 3 3 / 2 + 72 ) 1 / 3 + 156 ( 8 10 i 3 3 / 2 + 72 ) 1 / 3 6 - - ( 8 10 i 3 3 / 2 + 72 ) 1 / 3 - 52 3 ( 8 10 i 3 3 / 2 + 72 ) 1 / 3 + 8 2 + 2 -{{\sqrt{{{9\left({{8\sqrt{10}i}\over{3^{{{3}/{2}}}}}+72\right)^{{{2}/{3}}}+36% \left({{8\sqrt{10}i}\over{3^{{{3}/{2}}}}}+72\right)^{{{1}/{3}}}+156}\over{% \left({{8\sqrt{10}i}\over{3^{{{3}/{2}}}}}+72\right)^{{{1}/{3}}}}}}}\over{6}}-{% {\sqrt{-\left({{8\sqrt{10}i}\over{3^{{{3}/{2}}}}}+72\right)^{{{1}/{3}}}-{{52}% \over{3\left({{8\sqrt{10}i}\over{3^{{{3}/{2}}}}}+72\right)^{{{1}/{3}}}}}+8}}% \over{2}}+2
  17. ± \pm

Polynomial_least_squares.html

  1. y y
  2. α \alpha
  3. β \beta
  4. y = α + β t y=\alpha+\beta t
  5. ε \varepsilon
  6. z = y + ε = α + β t + ε z=y+\varepsilon=\alpha+\beta t+\varepsilon
  7. z n z_{n}
  8. n n
  9. y ( t ) y(t)
  10. f ( a x + b y ) = a f ( x ) + b f ( y ) f(ax+by)=af(x)+bf(y)
  11. a a
  12. b b
  13. x x
  14. y y
  15. E [ f ( x ) ] = f ( E [ x ] ) E[f(x)]=f(E[x])
  16. E E
  17. ε \varepsilon
  18. z ( t ) z(t)
  19. z ^ ( t ) = j = 1 J a j t j - 1 \hat{z}(t)=\sum_{j=1}^{J}a_{j}t^{j-1}
  20. e e
  21. e = n = 1 N ( z n - z ^ n ) 2 e=\sum_{n=1}^{N}(z_{n}-\hat{z}_{n})^{2}
  22. e e
  23. z - z ^ z-\hat{z}
  24. z ^ \hat{z}
  25. n = 1 N ( z n - z ^ n ) z ^ n = 0 \sum_{n=1}^{N}(z_{n}-\hat{z}_{n})\hat{z}_{n}=0
  26. z n z_{n}
  27. z ^ ( t ) \hat{z}(t)
  28. e e
  29. e min = n = 1 N ( z n - z ^ n ) z n e_{\min}=\sum_{n=1}^{N}(z_{n}-\hat{z}_{n})z_{n}
  30. e min e_{\min}
  31. w n ( τ ) w_{n}(\tau)
  32. y ( t ) y(t)
  33. y ^ ( τ ) = 1 N n = 1 N z n w n ( τ ) = 1 N n = 1 N ( α + β t n + ε n ) w n ( τ ) \hat{y}(\tau)=\frac{1}{N}\sum_{n=1}^{N}z_{n}w_{n}(\tau)=\frac{1}{N}\sum_{n=1}^% {N}(\alpha+\beta t_{n}+\varepsilon_{n})w_{n}(\tau)
  34. z n z_{n}
  35. z z
  36. w n ( τ ) [ t 2 ¯ - t ¯ t n + ( t n - t ¯ ) τ ] ( t 2 ¯ - t ¯ 2 ) w_{n}(\tau)\equiv\frac{[\bar{t^{2}}-\bar{t}t_{n}+(t_{n}-\bar{t})\tau]}{(\bar{t% ^{2}}-\bar{t}^{2})}
  37. τ \tau
  38. t t
  39. y ( t ) y(t)
  40. τ \tau
  41. t t
  42. u n u_{n}
  43. u ¯ = def 1 N n = 1 N u n \bar{u}\overset{\underset{\mathrm{def}}{}}{=}\frac{1}{N}\sum_{n=1}^{N}u_{n}
  44. w n ( τ ) w_{n}(\tau)
  45. y ^ ( τ ) = α ^ + β ^ τ \hat{y}(\tau)=\hat{\alpha}+\hat{\beta}\tau
  46. α ^ = ( z ¯ t 2 ¯ - z t ¯ t ¯ ) ( t 2 ¯ - t ¯ 2 ) = α + ( ε ¯ t 2 ¯ - ε t ¯ t ¯ ) ( t 2 ¯ - t ¯ 2 ) \hat{\alpha}=\frac{(\bar{z}\bar{t^{2}}-\bar{zt}\bar{t})}{(\bar{t^{2}}-\bar{t}^% {2})}=\alpha+\frac{(\bar{\varepsilon}\bar{t^{2}}-\bar{{\varepsilon}t}\bar{t})}% {(\bar{t^{2}}-\bar{t}^{2})}
  47. β ^ = ( z t ¯ - z ¯ t ¯ ) ( t 2 ¯ - t ¯ 2 ) = β + ( ε t ¯ - ε ¯ t ¯ ) ( t 2 ¯ - t ¯ 2 ) \hat{\beta}=\frac{(\bar{zt}-\bar{z}\bar{t})}{(\bar{t^{2}}-\bar{t}^{2})}=\beta+% \frac{(\bar{\varepsilon t}-\bar{\varepsilon}\bar{t})}{(\bar{t^{2}}-\bar{t}^{2})}
  48. ε n \varepsilon_{n}
  49. α ^ \hat{\alpha}
  50. β ^ \hat{\beta}
  51. ε n \varepsilon_{n}
  52. α ^ = α \hat{\alpha}=\alpha
  53. β ^ = β \hat{\beta}=\beta
  54. ε n \varepsilon_{n}
  55. y ^ \hat{y}
  56. E [ y ^ ( τ ) ] = α + β τ + 1 N n = 1 N E [ ε n ] w n ( τ ) = α + β τ = α + β t E[\hat{y}(\tau)]=\alpha+\beta\tau+\frac{1}{N}\sum_{n=1}^{N}E[\varepsilon_{n}]w% _{n}(\tau)=\alpha+\beta\tau=\alpha+\beta t
  57. y ^ \hat{y}
  58. σ y ^ 2 = E [ ( y ^ ) - E [ y ^ ] ) 2 ] = 1 N 1 N n = 1 N i = 1 N w n ( τ ) E [ ε n ε i ] w i ( τ ) \sigma_{\hat{y}}^{2}=E[(\hat{y})-E[\hat{y}])^{2}]=\frac{1}{N}\frac{1}{N}\sum_{% n=1}^{N}\sum_{i=1}^{N}w_{n}(\tau)E[\varepsilon_{n}\varepsilon_{i}]w_{i}(\tau)
  59. = σ ε 2 1 N 1 N n = 1 N i = 1 N w n 2 ( τ ) =\sigma_{\varepsilon}^{2}\frac{1}{N}\frac{1}{N}\sum_{n=1}^{N}\sum_{i=1}^{N}w_{% n}^{2}(\tau)
  60. i = 1 N E [ ε n ε i ] w i ( τ ) = σ ε 2 w n ( τ ) \sum_{i=1}^{N}E[\varepsilon_{n}\varepsilon_{i}]w_{i}(\tau)=\sigma_{\varepsilon% }^{2}w_{n}(\tau)
  61. σ ε 2 \sigma_{\varepsilon}^{2}
  62. ε n \varepsilon_{n}
  63. E [ ε n ε i ] = σ ε 2 E[\varepsilon_{n}\varepsilon_{i}]=\sigma_{\varepsilon}^{2}
  64. ε n \varepsilon_{n}
  65. σ ε 2 = 0 \sigma_{\varepsilon}^{2}=0
  66. i n i\neq n
  67. σ y ^ 2 \sigma_{\hat{y}}^{2}
  68. σ y ^ 2 = σ ε 2 ( t 2 ¯ - 2 t ¯ τ + τ 2 ) N ( t 2 ¯ - t ¯ 2 ) \sigma_{\hat{y}}^{2}=\sigma_{\varepsilon}^{2}\frac{(\bar{t^{2}}-2\bar{t}\tau+% \tau^{2})}{N(\bar{t^{2}}-\bar{t}^{2})}
  69. σ ε 2 \sigma_{\varepsilon}^{2}
  70. σ ε 2 \sigma_{\varepsilon}^{2}
  71. e min e_{\min}
  72. σ ε 2 σ ε 2 ^ = ( z 2 ¯ - α ^ z ¯ - β ^ z t ¯ ) \sigma_{\varepsilon}^{2}\approx\hat{\sigma_{\varepsilon}^{2}}=(\bar{z^{2}}-% \hat{\alpha}\bar{z}-\hat{\beta}\bar{zt})
  73. α ^ \hat{\alpha}
  74. β ^ \hat{\beta}
  75. z z
  76. t t
  77. σ y ^ 2 [ ( z 2 ¯ - z ¯ 2 ) ( t 2 ¯ - t ¯ 2 ) - ( ( z t ¯ - z ¯ t ¯ ) ( t 2 ¯ - t ¯ ) ) 2 ] ( t 2 ¯ - 2 t ¯ τ + τ 2 ) N \sigma_{\hat{y}}^{2}\approx\bigg[\frac{(\bar{z^{2}}-\bar{z}^{2})}{(\bar{t^{2}}% -\bar{t}^{2})}-\Biggl(\frac{(\bar{zt}-\bar{z}\bar{t})}{(\bar{t^{2}}-\bar{t})}% \Biggl)^{2}\bigg]{\frac{(\bar{t^{2}}-2\bar{t}\tau+\tau^{2})}{N}}
  78. ε n \varepsilon_{n}
  79. z n z_{n}
  80. y ^ ( τ ) \hat{y}(\tau)
  81. E [ y ^ ] E[\hat{y}]
  82. σ y ^ 2 \sigma_{\hat{y}}^{2}
  83. σ y ^ 2 \sigma_{\hat{y}}^{2}
  84. σ ε 2 \sigma_{\varepsilon}^{2}
  85. τ \tau
  86. σ y ^ 2 \sigma_{\hat{y}}^{2}
  87. τ \tau
  88. τ = t ¯ \tau=\bar{t}
  89. t n t_{n}
  90. σ ε 2 N \frac{\sigma_{\varepsilon}^{2}}{N}
  91. α \alpha
  92. σ y ^ 2 \sigma_{\hat{y}}^{2}
  93. τ 2 \tau^{2}
  94. τ \tau
  95. t ¯ \bar{t}
  96. σ y ^ 2 \sigma_{\hat{y}}^{2}
  97. ε n \varepsilon_{n}
  98. τ \tau
  99. t t
  100. σ y ^ 2 \sigma_{\hat{y}}^{2}
  101. ε n \varepsilon_{n}
  102. ε n \varepsilon_{n}
  103. α \alpha
  104. β \beta
  105. σ ε 2 σ ε 2 ^ = ( z 2 ¯ - α ^ z ¯ - β ^ z t ¯ ) \sigma_{\varepsilon}^{2}\approx\hat{\sigma_{\varepsilon}^{2}}=(\bar{z^{2}}-% \hat{\alpha}\bar{z}-\hat{\beta}\bar{zt})
  106. σ y ^ 2 \sigma_{\hat{y}}^{2}
  107. σ ε 2 \sigma_{\varepsilon}^{2}
  108. w n ( τ ) w_{n}(\tau)

Polynomial_Wigner–Ville_distribution.html

  1. z ( t ) = e j 2 π ϕ ( t ) z(t)=e^{j2\pi\phi(t)}
  2. ϕ ( t ) \phi(t)
  3. j = - 1 j=\sqrt{-1}
  4. W z g ( t , f ) W^{g}_{z}(t,f)
  5. W z g ( t , f ) = τ f [ K z g ( t , τ ) ] W^{g}_{z}(t,f)=\mathcal{F}_{\tau\to f}\left[K^{g}_{z}(t,\tau)\right]
  6. τ f \mathcal{F}_{\tau\to f}
  7. τ \tau
  8. K z g ( t , τ ) K^{g}_{z}(t,\tau)
  9. K z g ( t , τ ) = k = - q 2 q 2 [ z ( t + c k τ ) ] b k K^{g}_{z}(t,\tau)=\prod_{k=-\frac{q}{2}}^{\frac{q}{2}}\left[z\left(t+c_{k}\tau% \right)\right]^{b_{k}}
  10. z ( t ) z(t)
  11. p p
  12. K z g ( t , τ ) = k = 0 q 2 [ z ( t + c k τ ) ] b k [ z * ( t + c - k τ ) ] - b - k K^{g}_{z}(t,\tau)=\prod_{k=0}^{\frac{q}{2}}\left[z\left(t+c_{k}\tau\right)% \right]^{b_{k}}\left[z^{*}\left(t+c_{-k}\tau\right)\right]^{-b_{-k}}
  13. K z g ( n , m ) = k = 0 q 2 [ z ( n + c k m ) ] b k [ z * ( n + c - k m ) ] - b - k K^{g}_{z}(n,m)=\prod_{k=0}^{\frac{q}{2}}\left[z\left(n+c_{k}m\right)\right]^{b% _{k}}\left[z^{*}\left(n+c_{-k}m\right)\right]^{-b_{-k}}
  14. n = t f s , m = τ f s , n=t{f}_{s},m={\tau}{f}_{s},
  15. f s f_{s}
  16. q = 2 , b - 1 = - 1 , b 1 = 1 , b 0 = 0 , c - 1 = - 1 2 , c 0 = 0 , c 1 = 1 2 q=2,b_{-1}=-1,b_{1}=1,b_{0}=0,c_{-1}=-\frac{1}{2},c_{0}=0,c_{1}=\frac{1}{2}
  17. q = 4 q=4
  18. b k b_{k}
  19. c k c_{k}
  20. b 1 = - b - 1 = 2 , b 2 = b - 2 = 1 , b 0 = 0 b_{1}=-b_{-1}=2,b_{2}=b_{-2}=1,b_{0}=0
  21. c 1 = - c - 1 = 0.675 , c 2 = - c - 2 = - 0.85 c_{1}=-c_{-1}=0.675,c_{2}=-c_{-2}=-0.85
  22. K z g ( n , m ) = [ z ( n + 0.675 m ) z * ( n - 0.675 m ) ] 2 z * ( n + 0.85 m ) z ( n - 0.85 m ) K^{g}_{z}(n,m)=\left[z\left(n+0.675m\right)z^{*}\left(n-0.675m\right)\right]^{% 2}z^{*}\left(n+0.85m\right)z\left(n-0.85m\right)

Pop_(motion).html

  1. p = d c d t = d 2 s d t 2 = d 3 j d t 3 = d 4 a d t 4 = d 5 v d t 5 = d 6 r d t 6 \vec{p}=\frac{d\vec{c}}{dt}=\frac{d^{2}\vec{s}}{dt^{2}}=\frac{d^{3}\vec{j}}{dt% ^{3}}=\frac{d^{4}\vec{a}}{dt^{4}}=\frac{d^{5}\vec{v}}{dt^{5}}=\frac{d^{6}\vec{% r}}{dt^{6}}
  2. c = c 0 + p t \vec{c}=\vec{c}_{0}+\vec{p}\,t
  3. s = s 0 + c 0 t + 1 2 p t 2 \vec{s}=\vec{s}_{0}+\vec{c}_{0}\,t+\frac{1}{2}\vec{p}\,t^{2}
  4. j = j 0 + s 0 t + 1 2 c 0 t 2 + 1 6 p t 3 \vec{j}=\vec{j}_{0}+\vec{s}_{0}\,t+\frac{1}{2}\vec{c}_{0}\,t^{2}+\frac{1}{6}% \vec{p}\,t^{3}
  5. a = a 0 + j 0 t + 1 2 s 0 t 2 + 1 6 c 0 t 3 + 1 24 p t 4 \vec{a}=\vec{a}_{0}+\vec{j}_{0}\,t+\frac{1}{2}\vec{s}_{0}\,t^{2}+\frac{1}{6}% \vec{c}_{0}\,t^{3}+\frac{1}{24}\vec{p}\,t^{4}
  6. v = v 0 + a 0 t + 1 2 j 0 t 2 + 1 6 s 0 t 3 + 1 24 c 0 t 4 + 1 120 p t 5 \vec{v}=\vec{v}_{0}+\vec{a}_{0}\,t+\frac{1}{2}\vec{j}_{0}\,t^{2}+\frac{1}{6}% \vec{s}_{0}\,t^{3}+\frac{1}{24}\vec{c}_{0}\,t^{4}+\frac{1}{120}\vec{p}\,t^{5}
  7. r = r 0 + v 0 t + 1 2 a 0 t 2 + 1 6 j 0 t 3 + 1 24 s 0 t 4 + 1 120 c 0 t 5 + 1 720 p t 6 \vec{r}=\vec{r}_{0}+\vec{v}_{0}\,t+\frac{1}{2}\vec{a}_{0}\,t^{2}+\frac{1}{6}% \vec{j}_{0}\,t^{3}+\frac{1}{24}\vec{s}_{0}\,t^{4}+\frac{1}{120}\vec{c}_{0}\,t^% {5}+\frac{1}{720}\vec{p}\,t^{6}
  8. p \vec{p}
  9. c 0 \vec{c}_{0}
  10. c \vec{c}
  11. s 0 \vec{s}_{0}
  12. s \vec{s}
  13. j 0 \vec{j}_{0}
  14. j \vec{j}
  15. a 0 \vec{a}_{0}
  16. a \vec{a}
  17. v 0 \vec{v}_{0}
  18. v \vec{v}
  19. r 0 \vec{r}_{0}
  20. r \vec{r}
  21. t t

Poretsky's_law_of_forms.html

  1. f ( X ) = 0 f(X)=0
  2. g ( X ) = h ( X ) g(X)=h(X)
  3. g = f h g=f\oplus h
  4. \oplus

Port_(circuit_theory).html

  1. [ V 1 V 2 ] = [ z 11 z 12 z 21 z 22 ] [ I 1 I 2 ] \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}

Portal:Mathematics::Selected_picture::6.html

  1. 1 \aleph_{1}

Positive_real_numbers.html

  1. > 0 = { x x > 0 } \mathbb{R}_{>0}=\left\{x\in\mathbb{R}\mid x>0\right\}
  2. > 0 \mathbb{R}_{>0}
  3. + \mathbb{R}_{+}
  4. + \mathbb{R}^{+}
  5. 0 = { x x 0 } \mathbb{R}_{\geq 0}=\left\{x\in\mathbb{R}\mid x\geq 0\right\}
  6. > 0 \mathbb{R}_{>0}
  7. > 0 = ( 0 , 1 ) { 1 } ( 1 , ) \mathbb{R}_{>0}=(0,1)\cup\{1\}\cup(1,\infty)
  8. floor : [ 1 , ) , x x \operatorname{floor}:[1,\infty)\to\mathbb{N},\,x\mapsto\lfloor x\rfloor
  9. excess : [ 1 , ) ( 0 , 1 ) , x x - x \operatorname{excess}:[1,\infty)\to(0,1),\,x\mapsto x-\lfloor x\rfloor
  10. x > 0 x\in\mathbb{R}_{>0}
  11. [ n 1 ; n 2 , n 3 , ] [n_{1};n_{2},n_{3},\ldots]
  12. x x\in\mathbb{Q}
  13. x x
  14. x x
  15. > 0 \mathbb{R}_{>0}
  16. z = | z | e i φ z=|z|\mathrm{e}^{\mathrm{i}\varphi}
  17. φ = 0 \varphi=0
  18. n n\in\mathbb{N}
  19. > 0 \mathbb{R}_{>0}
  20. [ a , b ] R > 0 [a,b]\subseteq\mathrm{R}_{>0}
  21. μ ( [ a , b ] ) = log ( b / a ) \mu([a,b])=\log(b/a)
  22. > 0 \mathbb{R}_{>0}
  23. [ a , b ] [ a z , b z ] [a,b]\to[az,bz]
  24. z > 0 z\in\mathbb{R}_{>0}

Posner's_theorem.html

  1. A Z Z ( 0 ) A\otimes_{Z}Z_{(0)}
  2. Z ( 0 ) Z_{(0)}

Potential_predictability.html

  1. σ \sigma
  2. σ 2 \sigma^{2}
  3. σ ν 2 \sigma_{\nu}^{2}
  4. σ ε 2 \sigma_{\varepsilon}^{2}
  5. σ 2 = σ ν 2 + σ ε 2 \sigma^{2}=\sigma_{\nu}^{2}+\sigma_{\varepsilon}^{2}
  6. p = σ ν 2 σ 2 = σ ν 2 σ ν 2 + σ ε 2 p=\frac{\sigma_{\nu}^{2}}{\sigma^{2}}=\frac{\sigma_{\nu}^{2}}{\sigma_{\nu}^{2}% +\sigma_{\varepsilon}^{2}}
  7. γ = σ ν 2 σ ε 2 \gamma=\frac{\sigma_{\nu}^{2}}{\sigma_{\varepsilon}^{2}}
  8. p p
  9. γ \gamma
  10. p = γ 1 + γ , γ = p 1 - p p=\frac{\gamma}{1+\gamma}\qquad,\qquad\gamma=\frac{p}{1-p}
  11. 0 < p < 1 0<p<1
  12. p p
  13. p p

Potential_theory_of_Polanyi.html

  1. d μ = - S m d T + V m d P + d U m d\mu={-S_{m}dT}+{V_{m}dP}+{dU_{m}}
  2. μ \mu
  3. S m S_{m}
  4. V m V_{m}
  5. U m U_{m}
  6. x x
  7. μ ( x , P x ) {\mu(x,P_{x})}
  8. μ ( , P ) {\mu(\infty,P)}
  9. μ ( , P ) μ ( x , P x ) d μ = μ ( x , P x ) - μ ( , P ) = 0 \int_{\mu(\infty,P)}^{\mu(x,P_{x})}d\mu={\mu(x,P_{x})}-{\mu(\infty,P)}=0
  10. P x P_{x}
  11. P P
  12. P P
  13. P x P_{x}
  14. P P x V m d P + U m ( x ) - U m ( ) = 0 \int_{P}^{P_{x}}V_{m}dP+U_{m}(x)-U_{m}(\infty)=0
  15. U m ( ) U_{m}(\infty)
  16. - U m ( x ) = P P x V m d P -U_{m}(x)=\int_{P}^{P_{x}}V_{m}dP
  17. P V m = R T PV_{m}=RT
  18. - U m ( x ) = P P x R T P d P -U_{m}(x)=\int_{P}^{P_{x}}{RT\over P}dP
  19. - U m ( x ) = R T ( ln P x - ln P ) -U_{m}(x)=RT\cdot(\ln{P_{x}}-\ln{P})
  20. - U m ( x ) = R T ln P x P -U_{m}(x)=RT\cdot\ln{P_{x}\over P}
  21. P 0 P_{0}
  22. x f x_{f}
  23. P 0 P_{0}
  24. U m ( x f ) = - R T ln P 0 P U_{m}(x_{f})=-RT\cdot\ln{P_{0}\over P}
  25. ϵ s \epsilon_{s}
  26. ϵ s = - R T ln C s C \epsilon_{s}=-RT\ln{C_{s}\over C}
  27. C s C_{s}
  28. C C
  29. θ \theta
  30. θ = a / a 0 = e ( A / E ) b \theta=a/a_{0}=e^{{({A/E})}^{b}}
  31. a a
  32. a 0 a_{0}
  33. E E
  34. A A
  35. - Δ G = R T l o g p 0 p -\Delta G=RTlog{p_{0}\over p}
  36. b b
  37. b b
  38. b b
  39. log a = log a 0 + 0.434 * ( A E ) b \log a=\log a_{0}+0.434*({A\over E})^{b}
  40. log q e = log Q 0 + ( ϵ s w E ) b \log q_{e}=\log Q^{0}+({\epsilon_{sw}\over E})^{b}
  41. q e q_{e}
  42. Q 0 Q^{0}
  43. ϵ s w \epsilon_{sw}
  44. - R T ln C e C s -RT\ln{C_{e}\over C_{s}}
  45. C e C_{e}
  46. C 0 C_{0}
  47. E 0 E_{0}
  48. β \beta
  49. E = β E 0 E=\beta E_{0}
  50. β = α α 0 \beta={\alpha\over\alpha_{0}}
  51. α \alpha
  52. α 0 \alpha_{0}
  53. b b
  54. β \beta
  55. b b
  56. β E \beta E
  57. β \beta

Prandtl_condition.html

  1. ρ x . U x = ρ y . U y \rho_{x}.U_{x}=\rho_{y}.U_{y}
  2. P x - P y = ρ x . U x 2 - ρ y . U y 2 P_{x}-P_{y}=\rho_{x}.U_{x}^{2}-\rho_{y}.U{y}^{2}
  3. C p . T x + U x 2 2 = C p . T y + U y 2 2 C_{p}.T_{x}+\frac{U_{x}^{2}}{2}=C_{p}.T_{y}+\frac{U_{y}^{2}}{2}
  4. c * = k R T * c^{*}=\sqrt{kRT^{*}}
  5. M * = U c * = c M c * M^{*}=\frac{U}{c^{*}}=\frac{cM}{c^{*}}
  6. c 2 k - 1 + U 2 2 = c * 2 k - 1 + c * 2 2 = ( k + 1 ) c * 2 2 ( k - 1 ) \frac{c^{2}}{k-1}+\frac{U^{2}}{2}=\frac{c^{*^{2}}}{k-1}+\frac{c^{*^{2}}}{2}=% \frac{(k+1)c^{*^{2}}}{2(k-1)}
  7. c 1 2 k U 1 + U 1 = c 2 2 k U 2 + U 2 \frac{c_{1}^{2}}{kU_{1}}+U_{1}=\frac{c_{2}^{2}}{kU_{2}}+U_{2}
  8. 1 k U 1 [ ( k + 1 ) c * 2 2 - ( k - 1 ) U 1 2 ] + U 1 = 1 k U 2 [ ( k + 1 ) c * 2 2 - ( k - 1 ) U 2 2 ] + U 2 \frac{1}{kU_{1}}[\frac{(k+1)c^{*^{2}}}{2}-\frac{(k-1)U_{1}}{2}]+U_{1}=\frac{1}% {kU_{2}}[\frac{(k+1)c^{*^{2}}}{2}-\frac{(k-1)U_{2}}{2}]+U_{2}
  9. U 1 . U 2 = 1 U_{1}.U_{2}=1

Precession_Electron_Diffraction.html

  1. d 4 C s ϕ 2 α d\propto 4C_{s}\phi^{2}\alpha
  2. 𝐠 \mathbf{g}
  3. I 𝐠 k i n e m a t i c a l I_{\mathbf{g}}^{kinematical}
  4. F 𝐠 F_{\mathbf{g}}
  5. I 𝐠 k i n e m a t i c a l = | F 𝐠 | 2 \begin{aligned}\displaystyle I_{\mathbf{g}}^{kinematical}=|F_{\mathbf{g}}|^{2}% \end{aligned}
  6. I 𝐠 k i n e m a t i c a l I 𝐠 e x p e r i m e n t a l g 1 - g 2 R o I_{\mathbf{g}}^{kinematical}\propto I_{\mathbf{g}}^{experimental}\cdot g\sqrt{% 1-\frac{g}{2R_{o}}}
  7. I 𝐠 k i n e m a t i c a l I 𝐠 e x p e r i m e n t a l g 1 - g 2 R o 0 A 𝐠 J 0 ( 2 x ) d x I_{\mathbf{g}}^{kinematical}\propto I_{\mathbf{g}}^{experimental}\cdot g\sqrt{% 1-\frac{g}{2R_{o}}}\cdot\int\limits_{0}^{A_{\mathbf{g}}}J_{0}(2x)\,dx
  8. A 𝐠 = 2 π t F 𝐠 k A_{\mathbf{g}}=\frac{2\pi tF_{\mathbf{g}}}{k}
  9. t t
  10. k k
  11. J 0 J_{0}

Predictive_genomics.html

  1. K K
  2. h L 2 h^{2}_{L}
  3. A U C AUC
  4. K K
  5. h L 2 h^{2}_{L}
  6. A U C AUC

Presheaf_of_spaces.html

  1. C P S h v ( C ) C\to PShv(C)

Presilphiperfolanol_synthase.html

  1. \rightleftharpoons

Pressuron.html

  1. Φ \Phi
  2. S = 1 c d 4 x - g [ Φ m ( g μ ν , Ψ ) + 1 2 κ ( Φ R - ω ( Φ ) Φ ( σ Φ ) 2 - V ( Φ ) ) ] , S=\frac{1}{c}\int d^{4}x\sqrt{-g}\left[\sqrt{\Phi}\mathcal{L}_{m}(g_{\mu\nu},% \Psi)+\frac{1}{2\kappa}\left(\Phi R-\frac{\omega(\Phi)}{\Phi}(\partial_{\sigma% }\Phi)^{2}-V(\Phi)\right)\right],
  3. R R
  4. g μ ν g_{\mu\nu}
  5. g g
  6. κ = 8 π G c 4 \kappa=\frac{8\pi G}{c^{4}}
  7. G G
  8. c c
  9. V ( Φ ) V(\Phi)
  10. m \mathcal{L}_{m}
  11. Ψ \Psi
  12. R μ ν - 1 2 g μ ν R = κ 1 Φ T μ ν + 1 Φ [ μ ν - g μ ν ] Φ + ω ( Φ ) Φ 2 [ μ Φ ν Φ - 1 2 g μ ν ( α Φ ) 2 ] - g μ ν V ( Φ ) 2 Φ , R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa~{}\frac{1}{\sqrt{\Phi}}T_{\mu\nu}+% \frac{1}{\Phi}[\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box]\Phi+\frac{\omega(\Phi)% }{\Phi^{2}}\left[\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}(% \partial_{\alpha}\Phi)^{2}\right]-g_{\mu\nu}\frac{V(\Phi)}{2\Phi},
  13. 2 ω ( Φ ) + 3 Φ Φ = κ 1 Φ ( T - m ) - ω ( Φ ) Φ ( σ Φ ) 2 + V ( Φ ) - 2 V ( Φ ) Φ \frac{2\omega(\Phi)+3}{\Phi}\Box\Phi=\kappa\frac{1}{\sqrt{\Phi}}\left(T-% \mathcal{L}_{m}\right)-\frac{\omega^{\prime}(\Phi)}{\Phi}(\partial_{\sigma}% \Phi)^{2}+V^{\prime}(\Phi)-2\frac{V(\Phi)}{\Phi}
  14. m = - c 2 i μ i δ ( x i α ) \mathcal{L}_{m}=-c^{2}\sum_{i}\mu_{i}\delta(x^{\alpha}_{i})
  15. μ i \mu_{i}
  16. x i α x^{\alpha}_{i}
  17. δ ( x i α ) \delta(x^{\alpha}_{i})
  18. T = - c 2 i μ i δ ( x i α ) T=-c^{2}\sum_{i}\mu_{i}\delta(x^{\alpha}_{i})
  19. ( T - m ) \left(T-\mathcal{L}_{m}\right)
  20. 1 ω 0 P c 2 ρ 10 - 6 ω 0 \frac{1}{\omega_{0}}\frac{P}{c^{2}\rho}\sim\frac{10^{-6}}{\omega_{0}}
  21. ω 0 \omega_{0}
  22. ω ( Φ ) \omega(\Phi)
  23. P P
  24. ρ \rho
  25. 10 - 4 10^{-4}
  26. V ( Φ ) V(\Phi)

Price's_model.html

  1. p k p_{k}
  2. k p k = 1 \sum_{k}{p_{k}}=1
  3. k k p k = m \sum_{k}{kp_{k}}=m
  4. k + k 0 k+k_{0}
  5. k 0 k_{0}
  6. k 0 k_{0}
  7. k 0 = 1 k_{0}=1
  8. ( k + 1 ) p k k ( k + 1 ) p k = ( k + 1 ) p k m + 1 \frac{(k+1)p_{k}}{\sum_{k}(k+1)p_{k}}=\frac{(k+1)p_{k}}{m+1}
  9. p k , n p_{k,n}
  10. ( n + 1 ) p k , n + 1 - n p k , n = [ k p k - 1 , n - ( k + 1 ) p k , n ] m m + 1 for k 1 , (n+1)p_{k,n+1}-np_{k,n}=[kp_{k-1,n}-(k+1)p_{k,n}]\frac{m}{m+1}\,\text{ for }k% \geq 1,
  11. ( n + 1 ) p 0 , n + 1 - n p 0 , n = 1 - p 0 , n m m + 1 for k = 0. (n+1)p_{0,n+1}-np_{0,n}=1-p_{0,n}\frac{m}{m+1}\,\text{ for }k=0.
  12. p k , n + 1 = p k , n = p k p_{k,n+1}=p_{k,n}=p_{k}
  13. p k p_{k}
  14. p k = { [ k p k - 1 - ( k + 1 ) p k ] m m + 1 for k 1 1 - p 0 m m + 1 for k = 0 p_{k}=\begin{cases}[kp_{k-1}-(k+1)p_{k}]\frac{m}{m+1}&\,\text{for }k\geq 1\\ 1-p_{0}\frac{m}{m+1}&\,\text{for }k=0\end{cases}
  15. p 0 = m + 1 2 m + 1 p_{0}=\frac{m+1}{2m+1}
  16. p k = k ! ( k + 2 + 1 / m ) ( 3 + 1 / m ) p 0 = ( 1 + 1 / m ) 𝐁 ( k + 1 , 2 + 1 / m ) , p_{k}=\frac{k!}{(k+2+1/m)\cdots(3+1/m)}p_{0}=(1+1/m)\mathbf{B}(k+1,2+1/m),
  17. 𝐁 ( a , b ) \mathbf{B}(a,b)
  18. p k k - ( 2 + 1 / m ) p_{k}\sim k^{-(2+1/m)}
  19. p k p_{k}
  20. α = 2 + 1 / m \alpha=2+1/m
  21. k 0 1 k_{0}\neq 1
  22. p k = m + k 0 m ( k 0 + 1 ) + k 0 𝐁 ( k + k 0 , 2 + k 0 / m ) 𝐁 ( k 0 , 2 + k 0 / m ) , p_{k}=\frac{m+k_{0}}{m(k_{0}+1)+k_{0}}\frac{\mathbf{B}(k+k_{0},2+k_{0}/m)}{% \mathbf{B}(k_{0},2+k_{0}/m)},
  23. p k p_{k}
  24. α = 2 + k 0 / m \alpha=2+k_{0}/m
  25. k 0 k_{0}

Price_of_fairness.html

  1. P O F = max D D i v i s i o n s ( w e l f a r e ( D ) ) max D F a i r D i v i s i o n s ( w e l f a r e ( D ) ) POF=\frac{\max_{D\in Divisions}{(welfare(D))}}{\max_{D\in FairDivisions}{(% welfare(D))}}

Prime_avoidance_lemma.html

  1. I 1 , I 2 , , I n , n 1 I_{1},I_{2},\dots,I_{n},n\geq 1
  2. I i I_{i}
  3. i 3 i\geq 3
  4. I i I_{i}
  5. I i \cup I_{i}
  6. I i I_{i}
  7. z i E - j i I j z_{i}\in E-\cup_{j\neq i}I_{j}
  8. z i I i z_{i}\in I_{i}
  9. z i z_{i}
  10. I i I_{i}
  11. z = z 1 z n - 1 + z n z=z_{1}\dots z_{n-1}+z_{n}
  12. I i I_{i}
  13. I i I_{i}
  14. i n - 1 i\leq n-1
  15. z n z_{n}
  16. I i I_{i}
  17. I n I_{n}
  18. z 1 z n - 1 z_{1}\dots z_{n-1}
  19. I n I_{n}
  20. I n I_{n}
  21. z i , i < n z_{i},i<n
  22. I n I_{n}

Primitive_element_(co-algebra).html

  1. μ ( x ) = x g + g x \mu(x)=x\otimes g+g\otimes x
  2. μ \mu
  3. [ x , y ] = x y - y x [x,y]=xy-yx

Principal_factor.html

  1. 𝒥 \mathcal{J}
  2. J { 0 } J\cup\{0\}

Principalization_(algebra).html

  1. K K
  2. L | K L|K
  3. ι L | K : K L , 𝔞 𝔞 𝒪 L \iota_{L|K}:\ \mathcal{I}_{K}\to\mathcal{I}_{L},\ \mathfrak{a}\mapsto\mathfrak% {a}\mathcal{O}_{L}
  4. 𝒪 L \mathcal{O}_{L}
  5. L L
  6. j L | K : K / 𝒫 K L / 𝒫 L , 𝔞 𝒫 K ( 𝔞 𝒪 L ) 𝒫 L j_{L|K}:\ \mathcal{I}_{K}/\mathcal{P}_{K}\to\mathcal{I}_{L}/\mathcal{P}_{L},\ % \mathfrak{a}\mathcal{P}_{K}\mapsto(\mathfrak{a}\mathcal{O}_{L})\mathcal{P}_{L}
  7. 𝒫 K \mathcal{P}_{K}
  8. 𝒫 L \mathcal{P}_{L}
  9. 𝔞 K \mathfrak{a}\in\mathcal{I}_{K}
  10. 𝔞 𝒫 K 𝒫 K \mathfrak{a}\mathcal{P}_{K}\neq\mathcal{P}_{K}
  11. L L
  12. 𝔞 𝒪 L = A 𝒪 L \mathfrak{a}\mathcal{O}_{L}=A\mathcal{O}_{L}
  13. A L A\in L
  14. ( 𝔞 𝒪 L ) 𝒫 L = 𝒫 L (\mathfrak{a}\mathcal{O}_{L})\mathcal{P}_{L}=\mathcal{P}_{L}
  15. L | K L|K
  16. 𝔞 \mathfrak{a}
  17. 𝔞 𝒫 K \mathfrak{a}\mathcal{P}_{K}
  18. L L
  19. ker ( j L | K ) \ker(j_{L|K})
  20. 𝔪 = 𝔪 0 𝔪 \mathfrak{m}=\mathfrak{m}_{0}\mathfrak{m}_{\infty}
  21. K K
  22. 𝔪 0 \mathfrak{m}_{0}
  23. 𝒪 K \mathcal{O}_{K}
  24. 𝔪 \mathfrak{m}_{\infty}
  25. K K
  26. 𝒮 K , 𝔪 = α 𝒪 K α 1 ( mod 𝔪 ) K ( 𝔪 ) \mathcal{S}_{K,\mathfrak{m}}=\langle\alpha\mathcal{O}_{K}\mid\alpha\equiv 1\;% \;(\mathop{{\rm mod}}\mathfrak{m})\rangle\leq\mathcal{I}_{K}(\mathfrak{m})
  27. 𝔪 \mathfrak{m}
  28. K ( 𝔪 ) = K ( 𝔪 0 ) \mathcal{I}_{K}(\mathfrak{m})=\mathcal{I}_{K}(\mathfrak{m}_{0})
  29. K K
  30. 𝔪 0 \mathfrak{m}_{0}
  31. α 1 ( mod 𝔪 ) \alpha\equiv 1\;\;(\mathop{{\rm mod}}\mathfrak{m})
  32. α 1 ( mod 𝔪 0 ) \alpha\equiv 1\;\;(\mathop{{\rm mod}}\mathfrak{m}_{0})
  33. ( α ) > 0 \infty(\alpha)>0
  34. \infty
  35. 𝔪 \mathfrak{m}_{\infty}
  36. 𝒮 K , 𝔪 K ( 𝔪 ) \mathcal{S}_{K,\mathfrak{m}}\leq\mathcal{H}\leq\mathcal{I}_{K}(\mathfrak{m})
  37. K ( 𝔪 ) / \mathcal{I}_{K}(\mathfrak{m})/\mathcal{H}
  38. 𝔪 \mathfrak{m}
  39. K ( 𝔪 K ) / K \mathcal{I}_{K}(\mathfrak{m}_{K})/\mathcal{H}_{K}
  40. L ( 𝔪 L ) / L \mathcal{I}_{L}(\mathfrak{m}_{L})/\mathcal{H}_{L}
  41. 𝔞 𝒪 L L ( 𝔪 L ) \mathfrak{a}\mathcal{O}_{L}\in\mathcal{I}_{L}(\mathfrak{m}_{L})
  42. 𝔞 K ( 𝔪 K ) \mathfrak{a}\in\mathcal{I}_{K}(\mathfrak{m}_{K})
  43. 𝔞 𝒪 L L \mathfrak{a}\mathcal{O}_{L}\in\mathcal{H}_{L}
  44. 𝔞 K \mathfrak{a}\in\mathcal{H}_{K}
  45. ι L | K \iota_{L|K}
  46. j L | K : K ( 𝔪 K ) / K L ( 𝔪 L ) / L , 𝔞 K ( 𝔞 𝒪 L ) L j_{L|K}:\ \mathcal{I}_{K}(\mathfrak{m}_{K})/\mathcal{H}_{K}\to\mathcal{I}_{L}(% \mathfrak{m}_{L})/\mathcal{H}_{L},\ \mathfrak{a}\mathcal{H}_{K}\mapsto(% \mathfrak{a}\mathcal{O}_{L})\mathcal{H}_{L}
  47. F | K F|K
  48. G = Gal ( F | K ) G=\mathrm{Gal}(F|K)
  49. 𝔭 K \mathfrak{p}\in\mathbb{P}_{K}
  50. K K
  51. 𝔡 = 𝔡 ( F | K ) \mathfrak{d}=\mathfrak{d}(F|K)
  52. F F
  53. 𝔓 F \mathfrak{P}\in\mathbb{P}_{F}
  54. F F
  55. 𝔭 \mathfrak{p}
  56. σ G \sigma\in G
  57. A Norm K | ( 𝔭 ) σ ( A ) ( mod 𝔓 ) A^{\mathrm{Norm}_{K|\mathbb{Q}}(\mathfrak{p})}\equiv\sigma(A)\;\;(\mathop{{\rm mod% }}\mathfrak{P})
  58. A 𝒪 F A\in\mathcal{O}_{F}
  59. [ F | K 𝔓 ] := σ \left[\frac{F|K}{\mathfrak{P}}\right]:=\sigma
  60. 𝔓 \mathfrak{P}
  61. D 𝔓 = { σ G σ ( 𝔓 ) = 𝔓 } D_{\mathfrak{P}}=\{\sigma\in G\mid\sigma(\mathfrak{P})=\mathfrak{P}\}
  62. 𝔓 \mathfrak{P}
  63. f := f ( 𝔓 | 𝔭 ) = [ 𝒪 F / 𝔓 : 𝒪 K / 𝔭 ] f:=f(\mathfrak{P}|\mathfrak{p})=[\mathcal{O}_{F}/\mathfrak{P}:\mathcal{O}_{K}/% \mathfrak{p}]
  64. 𝔓 \mathfrak{P}
  65. 𝔭 \mathfrak{p}
  66. 𝔭 \mathfrak{p}
  67. [ F | K 𝔓 ] \left[\frac{F|K}{\mathfrak{P}}\right]
  68. D 𝔓 D_{\mathfrak{P}}
  69. I 𝔓 = { σ G σ ( A ) A ( mod 𝔓 ) for all A 𝒪 F } = ker ( D 𝔓 Gal ( 𝒪 F / 𝔓 | 𝒪 K / 𝔭 ) ) I_{\mathfrak{P}}=\{\sigma\in G\mid\sigma(A)\equiv A\;\;(\mathop{{\rm mod}}% \mathfrak{P})\,\text{ for all }A\in\mathcal{O}_{F}\}=\ker(D_{\mathfrak{P}}\to% \mathrm{Gal}(\mathcal{O}_{F}/\mathfrak{P}|\mathcal{O}_{K}/\mathfrak{p}))
  70. e ( 𝔓 | 𝔭 ) e(\mathfrak{P}|\mathfrak{p})
  71. 𝔓 \mathfrak{P}
  72. 𝔭 \mathfrak{p}
  73. F F
  74. 𝔭 \mathfrak{p}
  75. τ ( 𝔓 ) \tau(\mathfrak{P})
  76. τ G \tau\in G
  77. [ F | K τ ( 𝔓 ) ] = τ [ F | K 𝔓 ] τ - 1 \left[\frac{F|K}{\tau(\mathfrak{P})}\right]=\tau\left[\frac{F|K}{\mathfrak{P}}% \right]\tau^{-1}
  78. τ ( A ) Norm K | ( 𝔭 ) ( τ σ τ - 1 ) ( τ ( A ) ) ( mod τ ( 𝔓 ) ) \tau(A)^{\mathrm{Norm}_{K|\mathbb{Q}}(\mathfrak{p})}\equiv(\tau\sigma\tau^{-1}% )(\tau(A))\;\;(\mathop{{\rm mod}}\tau(\mathfrak{P}))
  79. A 𝒪 F A\in\mathcal{O}_{F}
  80. D τ ( 𝔓 ) = τ D 𝔓 τ - 1 D_{\tau(\mathfrak{P})}=\tau D_{\mathfrak{P}}\tau^{-1}
  81. D 𝔓 D_{\mathfrak{P}}
  82. 𝔭 ( F | K 𝔭 ) := { τ [ F | K 𝔓 ] τ - 1 τ G } \mathfrak{p}\mapsto\left(\frac{F|K}{\mathfrak{p}}\right):=\left\{\tau\left[% \frac{F|K}{\mathfrak{P}}\right]\tau^{-1}\mid\tau\in G\right\}
  83. 𝔭 ∤ 𝔡 \mathfrak{p}\not\mid\mathfrak{d}
  84. ( F | K 𝔭 ) = 1 \left(\frac{F|K}{\mathfrak{p}}\right)=1
  85. 𝔭 \mathfrak{p}
  86. F F
  87. K L F K\subseteq L\subseteq F
  88. H = Gal ( F | L ) G H=\mathrm{Gal}(F|L)\leq G
  89. ι L | K \iota_{L|K}
  90. j L | K j_{L|K}
  91. 𝔭 \mathfrak{p}
  92. 𝔭 \mathfrak{p}
  93. F F
  94. 𝒪 L \mathcal{O}_{L}
  95. 𝒪 F \mathcal{O}_{F}
  96. 𝒪 F \mathcal{O}_{F}
  97. 𝔭 \mathfrak{p}
  98. G G
  99. G G
  100. G / D 𝔓 G/D_{\mathfrak{P}}
  101. τ ( 𝔓 ) \tau(\mathfrak{P})
  102. τ D 𝔓 \tau D_{\mathfrak{P}}
  103. 𝔮 \mathfrak{q}
  104. 𝒪 L \mathcal{O}_{L}
  105. 𝔭 \mathfrak{p}
  106. H H
  107. 𝒪 F \mathcal{O}_{F}
  108. 𝔮 \mathfrak{q}
  109. 𝔮 \mathfrak{q}
  110. H H
  111. G / D 𝔓 G/D_{\mathfrak{P}}
  112. H \ G / D 𝔓 H\backslash G/D_{\mathfrak{P}}
  113. 𝔭 \mathfrak{p}
  114. 𝒪 L \mathcal{O}_{L}
  115. 𝔭 𝒪 L = i = 1 g 𝔮 i \mathfrak{p}\mathcal{O}_{L}=\prod_{i=1}^{g}\,\mathfrak{q}_{i}
  116. 𝔮 i L \mathfrak{q}_{i}\in\mathbb{P}_{L}
  117. 1 i g 1\leq i\leq g
  118. 𝔭 \mathfrak{p}
  119. L L
  120. 𝔮 i 𝒪 F = ϱ H τ i D ϱ ( 𝔓 ) \mathfrak{q}_{i}\mathcal{O}_{F}=\prod_{\varrho\in H\tau_{i}D}\,\varrho(% \mathfrak{P})
  121. G = ˙ i = 1 g H τ i D 𝔓 G=\dot{\cup}_{i=1}^{g}\,H\tau_{i}D_{\mathfrak{P}}
  122. G G
  123. H H
  124. D 𝔓 D_{\mathfrak{P}}
  125. ( τ 1 , , τ g ) (\tau_{1},\ldots,\tau_{g})
  126. H τ i D 𝔓 H\cdot\tau_{i}D_{\mathfrak{P}}
  127. τ i D 𝔓 \tau_{i}D_{\mathfrak{P}}
  128. H H
  129. G / D 𝔓 G/D_{\mathfrak{P}}
  130. H τ i D 𝔓 H\tau_{i}\cdot D_{\mathfrak{P}}
  131. H τ i H\tau_{i}
  132. D 𝔓 D_{\mathfrak{P}}
  133. H \ G H\backslash G
  134. # ( H τ i D 𝔓 ) # D 𝔓 = # H τ i D 𝔓 = # ( H τ i D 𝔓 ) # H \#(H\cdot\tau_{i}D_{\mathfrak{P}})\cdot\#D_{\mathfrak{P}}=\#H\tau_{i}D_{% \mathfrak{P}}=\#(H\tau_{i}\cdot D_{\mathfrak{P}})\cdot\#H
  135. D i D_{i}
  136. τ i ( 𝔓 ) \tau_{i}(\mathfrak{P})
  137. L L
  138. # D i = f ( τ i ( 𝔓 ) | 𝔮 i ) \#D_{i}=f(\tau_{i}(\mathfrak{P})|\mathfrak{q}_{i})
  139. τ i D 𝔓 \tau_{i}D_{\mathfrak{P}}
  140. H τ i D 𝔓 = { σ H σ τ i D 𝔓 = τ i D 𝔓 } = { σ H τ i - 1 σ τ i D 𝔓 } = H τ i D 𝔓 τ i - 1 = H D τ i ( 𝔓 ) = D i H_{\tau_{i}D_{\mathfrak{P}}}=\{\sigma\in H\mid\sigma\tau_{i}D_{\mathfrak{P}}=% \tau_{i}D_{\mathfrak{P}}\}=\{\sigma\in H\mid\tau_{i}^{-1}\sigma\tau_{i}\in D_{% \mathfrak{P}}\}=H\cap\tau_{i}D_{\mathfrak{P}}\tau_{i}^{-1}=H\cap D_{\tau_{i}(% \mathfrak{P})}=D_{i}
  141. # D i = # H τ i D 𝔓 = # H / # ( H τ i D 𝔓 ) \#D_{i}=\#H_{\tau_{i}D_{\mathfrak{P}}}=\#H/\#(H\cdot\tau_{i}D_{\mathfrak{P}})
  142. f ( 𝔮 i | 𝔭 ) = f ( τ i ( 𝔓 ) | 𝔭 ) f ( τ i ( 𝔓 ) | 𝔮 i ) = f ( 𝔓 | 𝔭 ) # D i = # D 𝔓 # H / # ( H τ i D 𝔓 ) = # ( H τ i D 𝔓 ) # D 𝔓 # H = # H τ i D 𝔓 # H = # ( H τ i D 𝔓 ) . f(\mathfrak{q}_{i}|\mathfrak{p})=\frac{f(\tau_{i}(\mathfrak{P})|\mathfrak{p})}% {f(\tau_{i}(\mathfrak{P})|\mathfrak{q}_{i})}=\frac{f(\mathfrak{P}|\mathfrak{p}% )}{\#D_{i}}=\frac{\#D_{\mathfrak{P}}}{\#H/\#(H\cdot\tau_{i}D_{\mathfrak{P}})}=% \frac{\#(H\cdot\tau_{i}D_{\mathfrak{P}})\cdot\#D_{\mathfrak{P}}}{\#H}=\frac{\#% H\tau_{i}D_{\mathfrak{P}}}{\#H}=\#(H\tau_{i}\cdot D_{\mathfrak{P}}).
  143. f i := f ( 𝔮 i | 𝔭 ) f_{i}:=f(\mathfrak{q}_{i}|\mathfrak{p})
  144. H τ i H\tau_{i}
  145. [ F | K 𝔓 ] \left[\frac{F|K}{\mathfrak{P}}\right]
  146. H \ G H\backslash G
  147. D 𝔓 τ i - 1 H D_{\mathfrak{P}}\cdot\tau_{i}^{-1}H
  148. τ i - 1 H \tau_{i}^{-1}H
  149. [ F | K 𝔓 ] \left[\frac{F|K}{\mathfrak{P}}\right]
  150. G / H G/H
  151. 𝒪 L \mathcal{O}_{L}
  152. 𝔭 \mathfrak{p}
  153. ι L | K ( 𝔭 ) = 𝔭 𝒪 L = i = 1 g 𝔮 i \iota_{L|K}(\mathfrak{p})=\mathfrak{p}\mathcal{O}_{L}=\prod_{i=1}^{g}\,% \mathfrak{q}_{i}
  154. j L | K ( 𝔭 K ) = ( 𝔭 𝒪 L ) L = i = 1 g 𝔮 i L j_{L|K}(\mathfrak{p}\mathcal{H}_{K})=(\mathfrak{p}\mathcal{O}_{L})\mathcal{H}_% {L}=\prod_{i=1}^{g}\,\mathfrak{q}_{i}\mathcal{H}_{L}
  155. F | K F|K
  156. G G
  157. F F
  158. 𝔭 \mathfrak{p}
  159. D τ ( 𝔓 ) = : D 𝔭 D_{\tau(\mathfrak{P})}=:D_{\mathfrak{p}}
  160. τ G \tau\in G
  161. ( F | K 𝔭 ) = [ F | K 𝔓 ] \left(\frac{F|K}{\mathfrak{p}}\right)=\left[\frac{F|K}{\mathfrak{P}}\right]
  162. 𝔓 𝔭 \mathfrak{P}\mid\mathfrak{p}
  163. A Norm K | ( 𝔭 ) ( F | K 𝔭 ) ( A ) ( mod 𝔭 𝒪 F ) A^{\mathrm{Norm}_{K|\mathbb{Q}}(\mathfrak{p})}\equiv\left(\frac{F|K}{\mathfrak% {p}}\right)(A)\;\;(\mathop{{\rm mod}}\mathfrak{p}\mathcal{O}_{F})
  164. A 𝒪 F A\in\mathcal{O}_{F}
  165. F | K F|K
  166. 𝒮 K , 𝔣 K ( 𝔣 ) \mathcal{S}_{K,\mathfrak{f}}\leq\mathcal{H}\leq\mathcal{I}_{K}(\mathfrak{f})
  167. 𝔣 \mathfrak{f}
  168. K K
  169. 𝒮 K , 𝔣 = α 𝒪 K α 1 ( mod 𝔣 ) \mathcal{S}_{K,\mathfrak{f}}=\langle\alpha\mathcal{O}_{K}\mid\alpha\equiv 1\;% \;(\mathop{{\rm mod}}\mathfrak{f})\rangle
  170. 𝔣 \mathfrak{f}
  171. K K
  172. 𝔣 = 𝔣 ( F | K ) \mathfrak{f}=\mathfrak{f}(F|K)
  173. 𝔭 𝔣 \mathfrak{p}\mid\mathfrak{f}
  174. 𝔭 𝔡 \mathfrak{p}\mid\mathfrak{d}
  175. 𝔣 \mathfrak{f}
  176. K ( 𝔣 ) G , 𝔭 ( F | K 𝔭 ) \mathbb{P}_{K}(\mathfrak{f})\to G,\ \mathfrak{p}\mapsto\left(\frac{F|K}{% \mathfrak{p}}\right)
  177. 𝔭 \mathfrak{p}
  178. 𝔭 \mathfrak{p}
  179. K K
  180. F F
  181. K ( 𝔣 ) G , 𝔞 = 𝔭 v 𝔭 ( 𝔞 ) ( F | K 𝔞 ) := ( F | K 𝔭 ) v 𝔭 ( 𝔞 ) \mathcal{I}_{K}(\mathfrak{f})\to G,\ \mathfrak{a}=\prod\,\mathfrak{p}^{v_{% \mathfrak{p}}(\mathfrak{a})}\mapsto\left(\frac{F|K}{\mathfrak{a}}\right):=% \prod\,\left(\frac{F|K}{\mathfrak{p}}\right)^{v_{\mathfrak{p}}(\mathfrak{a})}
  182. = 𝒮 K , 𝔣 Norm F | K ( F ( 𝔣 ) ) \mathcal{H}=\mathcal{S}_{K,\mathfrak{f}}\cdot\mathrm{Norm}_{F|K}(\mathcal{I}_{% F}(\mathfrak{f}))
  183. K ( 𝔣 ) / G = Gal ( F | K ) , 𝔞 ( F | K 𝔞 ) \mathcal{I}_{K}(\mathfrak{f})/\mathcal{H}\to G=\mathrm{Gal}(F|K),\ \mathfrak{a% }\mathcal{H}\mapsto\left(\frac{F|K}{\mathfrak{a}}\right)
  184. K ( 𝔣 ) / \mathcal{I}_{K}(\mathfrak{f})/\mathcal{H}
  185. G G
  186. 𝔞 \mathfrak{a}\mathcal{H}
  187. 𝔞 \mathfrak{a}
  188. ( F | K 𝔞 ) \left(\frac{F|K}{\mathfrak{a}}\right)
  189. 𝔞 \mathfrak{a}
  190. K L F K\subseteq L\subseteq F
  191. F | K F|K
  192. G = Gal ( F | K ) G=\mathrm{Gal}(F|K)
  193. K L F K\leq L\leq F
  194. H = Gal ( F | L ) G H=\mathrm{Gal}(F|L)\leq G
  195. K | K K^{\prime}|K
  196. L | L L^{\prime}|L
  197. K K
  198. L L
  199. F F
  200. G = Gal ( F | K ) G G^{\prime}=\mathrm{Gal}(F|K^{\prime})\leq G
  201. H = Gal ( F | L ) H H^{\prime}=\mathrm{Gal}(F|L^{\prime})\leq H
  202. 𝒮 K , 𝔪 K K K ( 𝔡 ) \mathcal{S}_{K,\mathfrak{m}_{K}}\leq\mathcal{H}_{K}\leq\mathcal{I}_{K}(% \mathfrak{d})
  203. 𝒮 L , 𝔪 L L L ( 𝔡 ) \mathcal{S}_{L,\mathfrak{m}_{L}}\leq\mathcal{H}_{L}\leq\mathcal{I}_{L}(% \mathfrak{d})
  204. ( K | K ) : K ( 𝔡 ) / K Gal ( K | K ) G / G \left(\frac{K^{\prime}|K}{\cdot}\right):\,\mathcal{I}_{K}(\mathfrak{d})/% \mathcal{H}_{K}\to\mathrm{Gal}(K^{\prime}|K)\simeq G/G^{\prime}
  205. ( L | L ) : L ( 𝔡 ) / L Gal ( L | L ) H / H \left(\frac{L^{\prime}|L}{\cdot}\right):\,\mathcal{I}_{L}(\mathfrak{d})/% \mathcal{H}_{L}\to\mathrm{Gal}(L^{\prime}|L)\simeq H/H^{\prime}
  206. 𝔡 = 𝔡 ( F | K ) \mathfrak{d}=\mathfrak{d}(F|K)
  207. L ( 𝔡 ) \mathcal{I}_{L}(\mathfrak{d})
  208. L ( 𝔡 𝒪 L ) \mathcal{I}_{L}(\mathfrak{d}\mathcal{O}_{L})
  209. 𝔪 K , 𝔪 L \mathfrak{m}_{K},\mathfrak{m}_{L}
  210. 𝔣 ( K | K ) , 𝔣 ( L | L ) \mathfrak{f}(K^{\prime}|K),\mathfrak{f}(L^{\prime}|L)
  211. 𝔡 , 𝔡 𝒪 L \mathfrak{d},\mathfrak{d}\mathcal{O}_{L}
  212. 𝔞 𝒪 L L \mathfrak{a}\mathcal{O}_{L}\in\mathcal{H}_{L}
  213. 𝔞 K \mathfrak{a}\in\mathcal{H}_{K}
  214. 𝔭 K \mathfrak{p}\in\mathbb{P}_{K}
  215. K K
  216. 𝔡 \mathfrak{d}
  217. 𝔓 F \mathfrak{P}\in\mathbb{P}_{F}
  218. F F
  219. 𝔭 \mathfrak{p}
  220. j L | K : K ( 𝔡 ) / K L ( 𝔡 ) / L j_{L|K}:\,\mathcal{I}_{K}(\mathfrak{d})/\mathcal{H}_{K}\to\mathcal{I}_{L}(% \mathfrak{d})/\mathcal{H}_{L}
  221. T ~ G , H \tilde{T}_{G,H}
  222. T ~ G , H ( K | K ) = ( L | L ) j L | K \tilde{T}_{G,H}\circ\left(\frac{K^{\prime}|K}{\cdot}\right)=\left(\frac{L^{% \prime}|L}{\cdot}\right)\circ j_{L|K}
  223. j L | K j_{L|K}
  224. 𝔭 K \mathfrak{p}\mathcal{H}_{K}
  225. K K
  226. j L | K ( 𝔭 K ) = ( 𝔭 𝒪 L ) L = i = 1 g 𝔮 i L j_{L|K}(\mathfrak{p}\mathcal{H}_{K})=(\mathfrak{p}\mathcal{O}_{L})\mathcal{H}_% {L}=\prod_{i=1}^{g}\,\mathfrak{q}_{i}\mathcal{H}_{L}
  227. L L
  228. ( L | L ) \left(\frac{L^{\prime}|L}{\cdot}\right)
  229. L L
  230. i = 1 g ( L | L 𝔮 i ) = i = 1 g [ F | L τ i ( 𝔓 ) ] H = i = 1 g τ i [ F | L 𝔓 ] τ i - 1 H = i = 1 g τ i [ F | K 𝔓 ] f i τ i - 1 H , \prod_{i=1}^{g}\,\left(\frac{L^{\prime}|L}{\mathfrak{q}_{i}}\right)=\prod_{i=1% }^{g}\,\left[\frac{F|L}{\tau_{i}(\mathfrak{P})}\right]\cdot H^{\prime}=\prod_{% i=1}^{g}\,\tau_{i}\left[\frac{F|L}{\mathfrak{P}}\right]\tau_{i}^{-1}\cdot H^{% \prime}=\prod_{i=1}^{g}\,\tau_{i}\left[\frac{F|K}{\mathfrak{P}}\right]^{f_{i}}% \tau_{i}^{-1}\cdot H^{\prime},
  231. ( K | K ) \left(\frac{K^{\prime}|K}{\cdot}\right)
  232. K K
  233. 𝔭 K \mathfrak{p}\mathcal{H}_{K}
  234. ( K | K 𝔭 ) = [ F | K 𝔓 ] G \left(\frac{K^{\prime}|K}{\mathfrak{p}}\right)=\left[\frac{F|K}{\mathfrak{P}}% \right]\cdot G^{\prime}
  235. g g
  236. ( τ 1 - 1 , , τ g - 1 ) (\tau_{1}^{-1},\ldots,\tau_{g}^{-1})
  237. D 𝔓 \ G / H D_{\mathfrak{P}}\backslash G/H
  238. [ F | K 𝔓 ] \left[\frac{F|K}{\mathfrak{P}}\right]
  239. G / H G/H
  240. f i = # ( H τ i D 𝔓 ) = # ( D 𝔓 τ i - 1 H ) f_{i}=\#(H\tau_{i}\cdot D_{\mathfrak{P}})=\#(D_{\mathfrak{P}}\cdot\tau_{i}^{-1% }H)
  241. τ i - 1 H \tau_{i}^{-1}H
  242. [ F | K 𝔓 ] G \left[\frac{F|K}{\mathfrak{P}}\right]\cdot G^{\prime}
  243. T ~ G , H ( [ F | K 𝔓 ] G ) = T G , H ( [ F | K 𝔓 ] ) = i = 1 g ( τ i - 1 ) - 1 [ F | K 𝔓 ] f i τ i - 1 H = i = 1 g τ i [ F | K 𝔓 ] f i τ i - 1 H \tilde{T}_{G,H}\left(\left[\frac{F|K}{\mathfrak{P}}\right]\cdot G^{\prime}% \right)=T_{G,H}\left(\left[\frac{F|K}{\mathfrak{P}}\right]\right)=\prod_{i=1}^% {g}\,(\tau_{i}^{-1})^{-1}\left[\frac{F|K}{\mathfrak{P}}\right]^{f_{i}}\tau_{i}% ^{-1}\cdot H^{\prime}=\prod_{i=1}^{g}\,\tau_{i}\left[\frac{F|K}{\mathfrak{P}}% \right]^{f_{i}}\tau_{i}^{-1}\cdot H^{\prime}
  244. j L | K j_{L|K}
  245. T G , H T_{G,H}
  246. L = F 1 ( K ) L=F^{1}(K)
  247. K K
  248. K K
  249. F = F 2 ( K ) F=F^{2}(K)
  250. K K
  251. K K
  252. F 1 ( K ) F^{1}(K)
  253. K = L K^{\prime}=L
  254. L = F L^{\prime}=F
  255. 𝔡 = 𝒪 K \mathfrak{d}=\mathcal{O}_{K}
  256. K = 𝒫 K \mathcal{H}_{K}=\mathcal{P}_{K}
  257. L = 𝒫 L \mathcal{H}_{L}=\mathcal{P}_{L}
  258. H = G H=G^{\prime}
  259. G G
  260. T G , G T_{G,G^{\prime}}
  261. G G
  262. G G^{\prime}
  263. G G
  264. G G^{\prime}
  265. G G
  266. G G
  267. G / G ′′ G/G^{\prime\prime}
  268. G G
  269. [ G : G ] < [G:G^{\prime}]<\infty
  270. p p
  271. F = F p 2 ( K ) F=F^{2}_{p}(K)
  272. K K
  273. K K
  274. p p
  275. L L
  276. K K
  277. F p 1 ( K ) F^{1}_{p}(K)
  278. H = Gal ( F p 2 ( K ) | L ) G = Gal ( F p 2 ( K ) | K ) H=\mathrm{Gal}(F^{2}_{p}(K)|L)\leq G=\mathrm{Gal}(F^{2}_{p}(K)|K)
  279. G G
  280. G G^{\prime}
  281. ker ( j L | K ) \ker(j_{L|K})
  282. Cl p ( L ) \mathrm{Cl}_{p}(L)
  283. ker ( T G , H ) \ker(T_{G,H})
  284. H / H H/H^{\prime}
  285. T G , H T_{G,H}
  286. G = Gal ( F p 2 ( K ) | K ) G=\mathrm{Gal}(F^{2}_{p}(K)|K)
  287. K K
  288. K K
  289. Gal ( F p ( K ) | K ) \mathrm{Gal}(F^{\infty}_{p}(K)|K)
  290. F p ( K ) F^{\infty}_{p}(K)
  291. K K
  292. L | K L|K
  293. G = Gal ( L | K ) = σ G=\mathrm{Gal}(L|K)=\langle\sigma\rangle
  294. σ \sigma
  295. σ = 1 \sigma^{\ell}=1
  296. = [ L : K ] \ell=[L:K]
  297. U = U L U=U_{L}
  298. G G
  299. G G
  300. Δ : U U , E E σ - 1 := σ ( E ) / E \Delta:\,U\to U,\ E\mapsto E^{\sigma-1}:=\sigma(E)/E
  301. σ - 1 [ G ] \sigma-1\in\mathbb{Z}[G]
  302. N : U U , E E T G := i = 0 - 1 σ i ( E ) N:\,\,U\to U,\ E\mapsto E^{T_{G}}:=\prod_{i=0}^{\ell-1}\,\sigma^{i}(E)
  303. T G = i = 0 - 1 σ i [ G ] T_{G}=\sum_{i=0}^{\ell-1}\,\sigma^{i}\in\mathbb{Z}[G]
  304. U K U_{K}
  305. N ( E ) = N L | K ( E ) N(E)=\mathrm{N}_{L|K}(E)
  306. Δ N = 1 \Delta\circ N=1
  307. N Δ = 1 N\circ\Delta=1
  308. G G
  309. U L U_{L}
  310. H 0 ( G , U L ) := ker ( Δ ) / im ( N ) = U K / N L | K ( U L ) H^{0}(G,U_{L}):=\ker(\Delta)/\mathrm{im}(N)=U_{K}/\mathrm{N}_{L|K}(U_{L})
  311. U K U_{K}
  312. G G
  313. U L U_{L}
  314. H - 1 ( G , U L ) := ker ( N ) / im ( Δ ) = E L | K / U L σ - 1 H^{-1}(G,U_{L}):=\ker(N)/\mathrm{im}(\Delta)=E_{L|K}/U_{L}^{\sigma-1}
  315. E L | K = { E U L N ( E ) = 1 } E_{L|K}=\{E\in U_{L}\mid N(E)=1\}
  316. L | K L|K
  317. σ - 1 \sigma-1
  318. H E L | K H\in E_{L|K}
  319. H = σ ( E ) / E H=\sigma(E)/E
  320. E U L E\in U_{L}
  321. H - 1 ( G , U L ) = E L | K / U L σ - 1 H^{-1}(G,U_{L})=E_{L|K}/U_{L}^{\sigma-1}
  322. \ell
  323. H - 1 ( G , L × ) = { A L × N ( A ) = 1 } / ( L × ) σ - 1 H^{-1}(G,L^{\times})=\{A\in L^{\times}\mid N(A)=1\}/(L^{\times})^{\sigma-1}
  324. G G
  325. L × = L { 0 } L^{\times}=L\setminus\{0\}
  326. L L
  327. H - 1 ( G , L × ) = 1 H^{-1}(G,L^{\times})=1
  328. L | K L|K
  329. \ell
  330. 𝔡 L | K = 𝒪 K \mathfrak{d}_{L|K}=\mathcal{O}_{K}
  331. 𝔧 K 𝒫 K \mathfrak{j}\in\mathcal{I}_{K}\setminus\mathcal{P}_{K}
  332. K K
  333. L L
  334. 𝔧 𝒪 L = A 𝒪 L 𝒫 L \mathfrak{j}\mathcal{O}_{L}=A\mathcal{O}_{L}\in\mathcal{P}_{L}
  335. A 𝒪 L A\in\mathcal{O}_{L}
  336. \ell
  337. K K
  338. 𝔧 = N L | K ( A ) 𝒪 K 𝒫 K \mathfrak{j}^{\ell}=\mathrm{N}_{L|K}(A)\mathcal{O}_{K}\in\mathcal{P}_{K}
  339. \ell
  340. L L
  341. K K
  342. H E L | K U L σ - 1 H\in E_{L|K}\setminus U_{L}^{\sigma-1}
  343. A L × A\in L^{\times}
  344. H = A σ - 1 H=A^{\sigma-1}
  345. A σ = A H A^{\sigma}=A\cdot H
  346. A A
  347. A A
  348. A A
  349. L | K L|K
  350. ( A 𝒪 L ) σ = A σ 𝒪 L = A H 𝒪 L = A 𝒪 L (A\mathcal{O}_{L})^{\sigma}=A^{\sigma}\mathcal{O}_{L}=A\cdot H\mathcal{O}_{L}=% A\mathcal{O}_{L}
  351. 𝔧 := ( A 𝒪 L ) 𝒪 K \mathfrak{j}:=(A\mathcal{O}_{L})\cap\mathcal{O}_{K}
  352. K K
  353. 𝔧 = β 𝒪 K \mathfrak{j}=\beta\mathcal{O}_{K}
  354. β 𝒪 K \beta\in\mathcal{O}_{K}
  355. L | K L|K
  356. 𝔞 \mathfrak{a}
  357. 𝒪 L \mathcal{O}_{L}
  358. 𝒪 K \mathcal{O}_{K}
  359. 𝔞 = ( 𝔞 𝒪 K ) 𝒪 L \mathfrak{a}=(\mathfrak{a}\cap\mathcal{O}_{K})\mathcal{O}_{L}
  360. β 𝒪 L = 𝔧 𝒪 L = A 𝒪 L \beta\mathcal{O}_{L}=\mathfrak{j}\mathcal{O}_{L}=A\mathcal{O}_{L}
  361. A = β E A=\beta E
  362. E U L E\in U_{L}
  363. H = A σ - 1 = ( β E ) σ - 1 = E σ - 1 H=A^{\sigma-1}=(\beta E)^{\sigma-1}=E^{\sigma-1}
  364. β σ - 1 = 1 \beta^{\sigma-1}=1
  365. 𝔧 𝒪 L = ( 𝔧 𝒪 L ) = N L | K ( 𝔧 𝒪 L ) 𝒪 L = N L | K ( A 𝒪 L ) 𝒪 L = N L | K ( A ) 𝒪 L \mathfrak{j}^{\ell}\mathcal{O}_{L}=(\mathfrak{j}\mathcal{O}_{L})^{\ell}=% \mathrm{N}_{L|K}(\mathfrak{j}\mathcal{O}_{L})\mathcal{O}_{L}=\mathrm{N}_{L|K}(% A\mathcal{O}_{L})\mathcal{O}_{L}=\mathrm{N}_{L|K}(A)\mathcal{O}_{L}
  366. 𝔧 = N L | K ( A ) 𝒪 K \mathfrak{j}^{\ell}=\mathrm{N}_{L|K}(A)\mathcal{O}_{K}
  367. K K
  368. = 2 \ell=2
  369. K K
  370. L L
  371. L L
  372. [ L : K ] [L:K]
  373. = 2 \ell=2
  374. L L
  375. K K
  376. L | K L|K
  377. \ell
  378. L | K L|K
  379. # ker ( j L | K ) = [ L : K ] \#\ker(j_{L|K})\geq\ell=[L:K]
  380. L | K L|K
  381. h ( G , U L ) h(G,U_{L})
  382. G G
  383. U L U_{L}
  384. h ( G , U L ) := # H - 1 ( G , U L ) / # H 0 ( G , U L ) = ( ker ( N ) : im ( Δ ) ) / ( ker ( Δ ) : im ( N ) ) = ( E L | K : U L σ - 1 ) / ( U K : N L | K ( U L ) ) . h(G,U_{L}):=\#H^{-1}(G,U_{L})/\#H^{0}(G,U_{L})=(\ker(N):\mathrm{im}(\Delta))/(% \ker(\Delta):\mathrm{im}(N))=(E_{L|K}:U_{L}^{\sigma-1})/(U_{K}:\mathrm{N}_{L|K% }(U_{L})).
  385. h ( G , U L ) = [ L : K ] h(G,U_{L})=[L:K]
  386. L | K L|K
  387. L G = K 𝒪 L \mathcal{I}^{G}_{L}=\mathcal{I}_{K}\mathcal{O}_{L}
  388. 𝒫 L G = 𝒫 L K 𝒪 L \mathcal{P}^{G}_{L}=\mathcal{P}_{L}\cap\mathcal{I}_{K}\mathcal{O}_{L}
  389. H 1 ( G , U L ) 𝒫 L G / 𝒫 K 𝒪 L H^{1}(G,U_{L})\cong\mathcal{P}^{G}_{L}/\mathcal{P}_{K}\mathcal{O}_{L}
  390. 2 2
  391. # ker ( j L | K ) = # ( 𝒫 L K 𝒪 L / 𝒫 K 𝒪 L ) = # ( 𝒫 L G / 𝒫 K 𝒪 L ) = # H 1 ( G , U L ) = # H - 1 ( G , U L ) \#\ker(j_{L|K})=\#(\mathcal{P}_{L}\cap\mathcal{I}_{K}\mathcal{O}_{L}/\mathcal{% P}_{K}\mathcal{O}_{L})=\#(\mathcal{P}^{G}_{L}/\mathcal{P}_{K}\mathcal{O}_{L})=% \#H^{1}(G,U_{L})=\#H^{-1}(G,U_{L})
  392. = h ( G , U L ) # H 0 ( G , U L ) = [ L : K ] # H 0 ( G , U L ) = [ L : K ] ( U K : N L | K ( U L ) ) . =h(G,U_{L})\cdot\#H^{0}(G,U_{L})=[L:K]\cdot\#H^{0}(G,U_{L})=[L:K]\cdot(U_{K}:% \mathrm{N}_{L|K}(U_{L})).
  393. ( U K : N L | K ( U L ) ) (U_{K}:\mathrm{N}_{L|K}(U_{L}))
  394. 3 3
  395. K = ( d ) K=\mathbb{Q}(\sqrt{d})
  396. 3 3
  397. d { - 3299 , - 4027 , - 9748 } d\in\{-3299,-4027,-9748\}
  398. - 2 10 4 < d < 10 5 -2\cdot 10^{4}<d<10^{5}
  399. 27 27
  400. 66 66
  401. 4596 4596
  402. - 10 6 < d < 10 7 -10^{6}<d<10^{7}
  403. 3 3
  404. ( 3 , 3 ) (3,3)
  405. 2 2
  406. 2 2
  407. ( 2 , 2 ) (2,2)
  408. 2 2
  409. 2 2
  410. ( 2 , 2 ) (2,2)
  411. 3 3
  412. K ( ζ f ) K\subset\mathbb{Q}(\zeta_{f})
  413. ζ f f = 1 \zeta_{f}^{f}=1
  414. 3 3
  415. ( 3 , 3 ) (3,3)
  416. f f
  417. 3 3
  418. K = ( D 3 ) K=\mathbb{Q}(\sqrt[3]{D})
  419. N = K ( ζ 3 ) N=K(\zeta_{3})
  420. ζ 3 3 = 1 \zeta_{3}^{3}=1
  421. 3 3
  422. ( 3 , 3 ) (3,3)
  423. 2 2
  424. K = ( d , - 1 ) K=\mathbb{Q}(\sqrt{d},\sqrt{-1})
  425. 2 2
  426. ( 2 , 2 ) (2,2)
  427. 2 2
  428. ( 2 , 2 , 2 ) (2,2,2)
  429. 2 2
  430. 2 2

Principles_of_grid_generation.html

  1. y = x 2 y=x^{2}
  2. ξ = x \xi=x\,
  3. η = y y max \eta=\frac{y}{y_{\max}}\,
  4. y max y_{\max}
  5. ξ \xi
  6. η \eta
  7. x x
  8. y y
  9. ξ x x + ξ y y = P ( ξ , η ) \xi_{xx}+\xi_{yy}=P(\xi,\eta)
  10. η x x + η y y = Q ( ξ , η ) \eta_{xx}+\eta_{yy}=Q(\xi,\eta)
  11. α x ξ ξ - 2 β x ξ η + γ x η η = - I 2 ( P x ξ + Q x η ) \alpha x_{\xi\xi}-2\beta x_{\xi\eta}+\gamma x_{\eta\eta}=-I^{2}(Px_{\xi}+Qx_{% \eta})
  12. α y ξ ξ - 2 β y ξ η + γ y η η = - I 2 ( P y ξ + Q y η ) \alpha y_{\xi\xi}-2\beta y_{\xi\eta}+\gamma y_{\eta\eta}=-I^{2}(Py_{\xi}+Qy_{% \eta})
  13. α \displaystyle\alpha
  14. Δ ξ = Δ η = 1 \Delta\xi=\Delta\eta=1
  15. x ξ y η - x η y ξ = I x_{\xi}y_{\eta}-x_{\eta}y_{\xi}=I
  16. I I
  17. d ξ = 0 = ξ x d x + ξ y d y . d\xi=0=\xi_{x}\,dx+\xi_{y}\,dy.
  18. ξ \xi
  19. η \eta
  20. x ξ x η + y ξ y η = 0. x_{\xi}x_{\eta}+y_{\xi}y_{\eta}=0.
  21. I I
  22. I I
  23. η = 0 \eta=0
  24. ξ \xi

Probabilistic_classification.html

  1. x x
  2. ŷ ŷ
  3. y ^ = f ( x ) \hat{y}=f(x)
  4. X X
  5. Y Y
  6. Pr ( Y | X ) \Pr(Y|X)
  7. x X x\in X
  8. y Y y\in Y
  9. y ^ = arg max y Pr ( Y = y | X ) \hat{y}=\operatorname{\arg\max}_{y}\Pr(Y=y|X)
  10. Pr ( Y | X ) \Pr(Y|X)
  11. Pr ( X | Y ) \Pr(X|Y)
  12. Pr ( Y ) \Pr(Y)
  13. Pr ( Y | X ) \Pr(Y|X)
  14. P r ( y | 𝐱 ) Pr(y|\mathbf{x})
  15. y y
  16. 𝐱 \mathbf{x}

Process_chemistry.html

  1. A E = MW(product) MW(raw materials) × 100 % AE=\frac{\,\text{MW(product)}}{\sum\,\text{MW(raw materials)}}\times 100\%
  2. V T O = nominal volume of all reactors [ m 3 ] * time per batch [ h ] output per step [ k g ] VTO=\frac{\,\text{nominal volume of all reactors}[m^{3}]*\,\text{time per % batch}[h]}{\,\text{output per step}[kg]}
  3. E = mass of waste mass of isolated product = mass of materials - mass of isolated product mass of isolated product E=\frac{\sum\,\text{mass of waste}}{\,\text{mass of isolated product}}=\frac{% \sum\,\text{mass of materials}-\,\text{mass of isolated product}}{\,\text{mass% of isolated product}}
  4. P M I = mass of materials mass of isolated product = E + 1 PMI=\frac{\sum\,\text{mass of materials}}{\,\text{mass of isolated product}}=E+1
  5. PEI yield = average yield * 100 % aspiration level yield = average yield * 100 % median yield + best yield 2 \,\text{PEI yield}=\frac{\,\text{average yield}*100\%}{\,\text{aspiration % level yield}}=\frac{\,\text{average yield}*100\%}{\frac{\,\text{median yield}+% \,\text{best yield}}{2}}
  6. PEI cycle time = aspiration level cycle time * 100 % average cycle time = median cycle time + best cycle time 2 average cycle time \,\text{PEI cycle time}=\frac{\,\text{aspiration level cycle time}*100\%}{\,% \text{average cycle time}}=\frac{\frac{\,\text{median cycle time}+\,\text{best% cycle time}}{2}}{\,\text{average cycle time}}
  7. VTO = 6 m 3 × 48 h 35 kg = 8.2 m 3 h / kg \,\text{VTO}=\frac{6\,\text{ m}^{3}\times 48\,\text{ h}}{35\,\text{ kg}}=8.2\,% \text{ m}^{3}\cdot\,\text{ h / kg}
  8. VTO = 6 m 3 × 13 h 799 kg = 0.1 m 3 h / kg \,\text{VTO}=\frac{6\,\text{ m}^{3}\times 13\,\text{ h}}{799\,\text{ kg}}=0.1% \,\text{ m}^{3}\cdot\,\text{ h / kg}

Production_flow_analysis.html

  1. b i p b_{ip}
  2. p = 1 m b i p * 2 m - p \sum_{p=1}^{m}b_{ip}*2^{m-p}
  3. i = 1 n b i p * 2 n - i \sum_{i=1}^{n}b_{ip}*2^{n-i}
  4. s i j = n i j / ( n i j + u ) s_{ij}=n_{ij}/(n_{ij}+u)
  5. n i j n_{ij}
  6. s r k = m a x ( s r i * , s r j * ) s_{rk}=max(s_{ri*},s_{rj*})

Productive_matrix.html

  1. A A
  2. n n
  3. n × 1 n\times 1
  4. P P
  5. P - A P P-AP
  6. A M n ( \R ) A\in\mathrm{M}_{n}(\R)
  7. A 0 A\geqslant 0
  8. P M n , 1 ( \R ) , P > 0 \exists P\in\mathrm{M}_{n,1}(\R),P>0
  9. P - A P > 0 P-AP>0
  10. A = ( 0 1 0 0 1 / 2 1 / 2 1 / 4 1 / 2 0 ) A=\begin{pmatrix}0&1&0\\ 0&1/2&1/2\\ 1/4&1/2&0\\ \end{pmatrix}
  11. a \R + \forall a\in\R_{+}
  12. A = ( 0 a 0 0 ) A=\begin{pmatrix}0&a\\ 0&0\\ \end{pmatrix}
  13. P = ( a + 1 1 ) P=\begin{pmatrix}a+1\\ 1\\ \end{pmatrix}
  14. A M n ( \R ) A\in\mathrm{M}_{n}(\R)
  15. I n - A I_{n}-A
  16. U M n , 1 ( \R ) , P > 0 U\in\mathrm{M}_{n,1}(\R),P>0
  17. P = ( I n - A ) - 1 U P=(I_{n}-A)^{-1}U
  18. P - A P = ( I n - A ) P = ( I n - A ) ( I n - A ) - 1 U = U P-AP=(I_{n}-A)P=(I_{n}-A)(I_{n}-A)^{-1}U=U
  19. P - A P > 0 P-AP>0
  20. A A
  21. P > 0 \exists P>0
  22. V = P - A P > 0 V=P-AP>0
  23. I n - A I_{n}-A
  24. I n - A I_{n}-A
  25. Z M n , 1 ( \R ) \exists Z\in\mathrm{M}_{n,1}(\R)
  26. ( I n - A ) Z = 0 (I_{n}-A)Z=0
  27. - Z -Z
  28. Z Z
  29. Z Z
  30. c = sup i [ | 1 , n | ] z i p i c=\sup_{i\in[|1,n|]}\frac{z_{i}}{p_{i}}
  31. k [ | 1 , n | ] k\in[|1,n|]
  32. V V
  33. Z Z
  34. c v k = c ( p k - i = 1 n a k i p i ) = c p k - i = 1 n a k i c p i cv_{k}=c(p_{k}-\sum_{i=1}^{n}a_{ki}p_{i})=cp_{k}-\sum_{i=1}^{n}a_{ki}cp_{i}
  35. c p k = z k = i = 1 n a k i z i cp_{k}=z_{k}=\sum_{i=1}^{n}a_{ki}z_{i}
  36. c v k = i = 1 n a k i ( z j - c p j ) 0 cv_{k}=\sum_{i=1}^{n}a_{ki}(z_{j}-cp_{j})\leq\ 0
  37. c > 0 c>0
  38. v k > 0 v_{k}>0
  39. I n - A I_{n}-A
  40. I n - A I_{n}-A
  41. X M n , 1 ( \R ) , X 0 \exists X\in\mathrm{M}_{n,1}(\R),X\geqslant 0
  42. Y = ( I n - A ) - 1 X Y=(I_{n}-A)^{-1}X
  43. c = sup i [ | 1 , n | ] - y i p i c=\sup_{i\in[|1,n|]}-\frac{y_{i}}{p_{i}}
  44. k [ | 1 , n | ] k\in[|1,n|]
  45. V V
  46. X X
  47. c v k = c ( p k - i = 1 n a k i p i ) = - y k - i = 1 n a k i c p i cv_{k}=c(p_{k}-\sum_{i=1}^{n}a_{ki}p_{i})=-y_{k}-\sum_{i=1}^{n}a_{ki}cp_{i}
  48. x k = y k - i = 1 n a k i y i x_{k}=y_{k}-\sum_{i=1}^{n}a_{ki}y_{i}
  49. c v k + x k = - i = 1 n a k i ( c p i + y i ) cv_{k}+x_{k}=-\sum_{i=1}^{n}a_{ki}(cp_{i}+y_{i})
  50. x k - c v k < 0 x_{k}\leq\ -cv_{k}<0
  51. i [ | 1 , n | ] , a k i 0 , c p i + y i 0 \forall i\in[|1,n|],a_{k}i\geqslant 0,cp_{i}+y_{i}\geqslant 0
  52. X 0 X\geqslant 0
  53. ( I n - A ) - 1 (I_{n}-A)^{-1}
  54. A M n ( \R ) A\in\mathrm{M}_{n}(\R)
  55. ( I n - A ) - 1 (I_{n}-A)^{-1}
  56. ( ( I n - t A ) ) - 1 = ( t ( I n - A ) ) - 1 = t ( ( I n - A ) - 1 ) ((I_{n}-^{\operatorname{t}}\!A))^{-1}=(^{\operatorname{t}}\!(I_{n}-A))^{-1}=^{% \operatorname{t}}\!((I_{n}-A)^{-1})
  57. ( I n - t A ) (I_{n}-^{\operatorname{t}}\!A)
  58. A t {}^{\operatorname{t}}\!A

Profinite_integer.html

  1. ^ = p p \widehat{\mathbb{Z}}=\prod_{p}\mathbb{Z}_{p}
  2. p \mathbb{Z}_{p}
  3. ^ = lim / n \widehat{\mathbb{Z}}=\underleftarrow{\lim}\mathbb{Z}/n
  4. 𝐅 ¯ q \overline{\mathbf{F}}_{q}
  5. 𝐅 q \mathbf{F}_{q}
  6. Gal ( 𝐅 ¯ q / 𝐅 q ) = ^ \operatorname{Gal}(\overline{\mathbf{F}}_{q}/\mathbf{F}_{q})=\widehat{\mathbb{% Z}}
  7. ^ , n ( n , n , ) . \mathbb{Z}\hookrightarrow\widehat{\mathbb{Z}},\,n\mapsto(n,n,\dots).
  8. ^ \widehat{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Q}
  9. 𝐀 , f = p p \mathbf{A}_{\mathbb{Q},f}=\prod_{p}{}^{{}^{\prime}}\mathbb{Q}_{p}
  10. \mathbb{Q}
  11. / × ^ U ( 1 ) , ( q , a ) χ ( q a ) \mathbb{Q}/\mathbb{Z}\times\widehat{\mathbb{Z}}\to U(1),\,(q,a)\mapsto\chi(qa)
  12. χ \chi
  13. 𝐀 , f \mathbf{A}_{\mathbb{Q},f}
  14. / U ( 1 ) , α e 2 π i α \mathbb{Q}/\mathbb{Z}\to U(1),\,\alpha\mapsto e^{2\pi i\alpha}
  15. ^ \widehat{\mathbb{Z}}
  16. / \mathbb{Q}/\mathbb{Z}

Projective_Set_(game).html

  1. 2 6 - 1 = 63 2^{6}-1=63
  2. 𝔽 6 \mathbb{F}^{6}
  3. n n

Prolycopene_isomerase.html

  1. \rightleftharpoons

Proportional_approval_voting.html

  1. 1 + 1 2 + 1 3 + + 1 n 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}
  2. c ! s ! ( c - s ) ! \frac{c!}{s!(c-s)!}

Prosolanapyrone-III_cycloisomerase.html

  1. \rightleftharpoons

Proton_magnetic_moment.html

  1. μ N = e 2 m p \mu_{\mathrm{N}}={{e\hbar}\over{2m_{\mathrm{p}}}}
  2. s y m b o l μ = g μ N s y m b o l I symbol{\mu}=\frac{g\mu_{\mathrm{N}}}{\hbar}symbol{I}
  3. s y m b o l μ = \gammasymbol I symbol{\mu}=\gammasymbol{I}
  4. γ = g μ N = g e 2 m p \gamma=\frac{g\mu_{\mathrm{N}}}{\hbar}=g\frac{e}{2m_{p}}
  5. μ N \mu_{\mathrm{N}}
  6. μ N \mu_{\mathrm{N}}

Protostadienol_synthase.html

  1. \rightleftharpoons

Provenance_(geology).html

  1. ϵ N d = 10 4 [ 43 1 N d / 1 44 N d s a m p l e ( T ) 43 1 N d / 1 44 N d C H U R ( T ) - 1 ] \epsilon Nd=10^{4}\left[\frac{{}^{1}43Nd/^{1}44Nd_{sample}(T)}{{}^{1}43Nd/^{1}% 44Nd_{CHUR}(T)}-1\right]

Proximal_gradient_method.html

  1. minimize x N f 1 ( x ) + f 2 ( x ) + + f n - 1 ( x ) + f n ( x ) \operatorname{minimize}_{x\in\mathbb{R}^{N}}\qquad f_{1}(x)+f_{2}(x)+\cdots+f_% {n-1}(x)+f_{n}(x)
  2. f 1 , f 2 , , f n f_{1},f_{2},...,f_{n}
  3. f : N f:\mathbb{R}^{N}\rightarrow\mathbb{R}
  4. f 1 , , f n f_{1},...,f_{n}
  5. f 1 , , f n f_{1},...,f_{n}
  6. N \mathbb{R}^{N}
  7. N N
  8. f : N [ - , + ] f:\mathbb{R}^{N}\rightarrow[-\infty,+\infty]
  9. C C
  10. N \mathbb{R}^{N}
  11. C C
  12. i C : x { 0 if x C + if x C i_{C}:x\mapsto\begin{cases}0&\,\text{if }x\in C\\ +\infty&\,\text{if }x\notin C\end{cases}
  13. p p
  14. p \|\cdot\|_{p}
  15. x p = ( | x 1 | p + | x 2 | p + + | x N | p ) 1 / p \|x\|_{p}=(|x_{1}|^{p}+|x_{2}|^{p}+\cdots+|x_{N}|^{p})^{1/p}
  16. x N x\in\mathbb{R}^{N}
  17. C C
  18. D C ( x ) = min y C x - y D_{C}(x)=\min_{y\in C}\|x-y\|
  19. C C
  20. x N x\in\mathbb{R}^{N}
  21. C C
  22. P C x C P_{C}x\in C
  23. D C ( x ) = x - P C x 2 D_{C}(x)=\|x-P_{C}x\|_{2}
  24. f f
  25. f : N 2 N : x { u N y N , ( y - x ) T u + f ( x ) f ( y ) ) . } \partial f:\mathbb{R}^{N}\rightarrow 2^{\mathbb{R}^{N}}:x\mapsto\{u\in\mathbb{% R}^{N}\mid\forall y\in\mathbb{R}^{N},(y-x)^{\mathrm{T}}u+f(x)\leq f(y)).\}
  26. f i f_{i}
  27. C i C_{i}
  28. C i C_{i}
  29. C i C_{i}
  30. P C i P_{C_{i}}
  31. x x
  32. x k + 1 = P C 1 P C 2 P C n x k x_{k+1}=P_{C_{1}}P_{C_{2}}\cdots P_{C_{n}}x_{k}
  33. f f
  34. x x
  35. x N x\in\mathbb{R}^{N}
  36. minimize y C f ( y ) + 1 2 x - y 2 2 \,\text{minimize}_{y\in C}\qquad f(y)+\frac{1}{2}\left\|x-y\right\|_{2}^{2}
  37. prox f ( x ) \operatorname{prox}_{f}(x)
  38. prox f ( x ) : N N \operatorname{prox}_{f}(x):\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}
  39. f f
  40. p = prox f ( x ) x - p f ( p ) ( ( x , p ) N × N ) p=\operatorname{prox}_{f}(x)\Leftrightarrow x-p\in\partial f(p)\qquad(\forall(% x,p)\in\mathbb{R}^{N}\times\mathbb{R}^{N})
  41. f f
  42. p = prox f ( x ) x - p f ( p ) ( ( x , p ) N × N ) p=\operatorname{prox}_{f}(x)\Leftrightarrow x-p\in\nabla f(p)\quad(\forall(x,p% )\in\mathbb{R}^{N}\times\mathbb{R}^{N})

Proximal_gradient_methods_for_learning.html

  1. 1 \ell_{1}
  2. min w d 1 n i = 1 n ( y i - w , x i ) 2 + λ w 1 , where x i d and y i . \min_{w\in\mathbb{R}^{d}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\langle w,x_{i}% \rangle)^{2}+\lambda\|w\|_{1},\quad\,\text{ where }x_{i}\in\mathbb{R}^{d}\,% \text{ and }y_{i}\in\mathbb{R}.
  3. min x F ( x ) + R ( x ) , \min_{x\in\mathcal{H}}F(x)+R(x),
  4. F F
  5. R R
  6. \mathcal{H}
  7. x x
  8. F ( x ) + R ( x ) F(x)+R(x)
  9. ( F + R ) ( x ) = 0 \nabla(F+R)(x)=0
  10. 0 ( F + R ) ( x ) , 0\in\partial(F+R)(x),
  11. φ \partial\varphi
  12. φ \varphi
  13. φ : \varphi:\mathcal{H}\to\mathbb{R}
  14. prox φ : \operatorname{prox}_{\varphi}:\mathcal{H}\to\mathcal{H}
  15. prox φ ( u ) = arg min x φ ( x ) + 1 2 u - x 2 2 , \operatorname{prox}_{\varphi}(u)=\operatorname{arg}\min_{x\in\mathcal{H}}% \varphi(x)+\frac{1}{2}\|u-x\|_{2}^{2},
  16. 2 \ell_{2}
  17. x * x^{*}
  18. min x F ( x ) + R ( x ) \min_{x\in\mathcal{H}}F(x)+R(x)
  19. x * = prox γ R ( x * - γ F ( x * ) ) , x^{*}=\operatorname{prox}_{\gamma R}\left(x^{*}-\gamma\nabla F(x^{*})\right),
  20. γ > 0 \gamma>0
  21. φ : 𝒳 \varphi:\mathcal{X}\to\mathbb{R}
  22. 𝒳 \mathcal{X}
  23. φ * : 𝒳 \varphi^{*}:\mathcal{X}\to\mathbb{R}
  24. φ * ( u ) := sup x 𝒳 x , u - φ ( x ) . \varphi^{*}(u):=\sup_{x\in\mathcal{X}}\langle x,u\rangle-\varphi(x).
  25. x 𝒳 x\in\mathcal{X}
  26. γ > 0 \gamma>0
  27. x = prox γ φ ( x ) + γ prox φ * / γ ( x / γ ) , x=\operatorname{prox}_{\gamma\varphi}(x)+\gamma\operatorname{prox}_{\varphi^{*% }/\gamma}(x/\gamma),
  28. γ = 1 \gamma=1
  29. x = prox φ ( x ) + prox φ * ( x ) x=\operatorname{prox}_{\varphi}(x)+\operatorname{prox}_{\varphi^{*}}(x)
  30. φ * \varphi^{*}
  31. φ \varphi
  32. 1 \ell_{1}
  33. min w d 1 n i = 1 n ( y i - w , x i ) 2 + λ w 1 , \min_{w\in\mathbb{R}^{d}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\langle w,x_{i}% \rangle)^{2}+\lambda\|w\|_{1},
  34. x i d and y i . x_{i}\in\mathbb{R}^{d}\,\text{ and }y_{i}\in\mathbb{R}.
  35. 1 \ell_{1}
  36. 1 \ell_{1}
  37. w w
  38. min w d 1 n i = 1 n ( y i - w , x i ) 2 + λ w 0 , \min_{w\in\mathbb{R}^{d}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\langle w,x_{i}% \rangle)^{2}+\lambda\|w\|_{0},
  39. w 0 \|w\|_{0}
  40. 0 \ell_{0}
  41. w w
  42. 1 \ell_{1}
  43. λ = 1 \lambda=1
  44. min w d 1 n i = 1 n ( y i - w , x i ) 2 + w 1 , \min_{w\in\mathbb{R}^{d}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\langle w,x_{i}% \rangle)^{2}+\|w\|_{1},
  45. F ( w ) = 1 n i = 1 n ( y i - w , x i ) 2 F(w)=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle)^{2}
  46. R ( w ) = w 1 R(w)=\|w\|_{1}
  47. R R
  48. R ( w ) R(w)
  49. prox R ( x ) \operatorname{prox}_{R}(x)
  50. u = prox R ( x ) \displaystyle u=\operatorname{prox}_{R}(x)\iff
  51. R ( w ) = w 1 R(w)=\|w\|_{1}
  52. R ( w ) \partial R(w)
  53. i i
  54. R ( w ) \partial R(w)
  55. | w i | = { 1 , w i > 0 - 1 , w i < 0 [ - 1 , 1 ] , w i = 0. \partial|w_{i}|=\begin{cases}1,&w_{i}>0\\ -1,&w_{i}<0\\ \left[-1,1\right],&w_{i}=0.\end{cases}
  56. R ( w ) = w 1 R(w)=\|w\|_{1}
  57. γ > 0 \gamma>0
  58. prox γ R ( x ) \operatorname{prox}_{\gamma R}(x)
  59. ( prox γ R ( x ) ) i = { x i - γ , x i > γ 0 , | x i | γ x i + γ , x i < - γ , \left(\operatorname{prox}_{\gamma R}(x)\right)_{i}=\begin{cases}x_{i}-\gamma,&% x_{i}>\gamma\\ 0,&|x_{i}|\leq\gamma\\ x_{i}+\gamma,&x_{i}<-\gamma,\end{cases}
  60. S γ ( x ) = prox γ 1 ( x ) S_{\gamma}(x)=\operatorname{prox}_{\gamma\|\cdot\|_{1}}(x)
  61. x * = prox γ R ( x * - γ F ( x * ) ) . x^{*}=\operatorname{prox}_{\gamma R}\left(x^{*}-\gamma\nabla F(x^{*})\right).
  62. w 0 d w^{0}\in\mathbb{R}^{d}
  63. k = 1 , 2 , k=1,2,\ldots
  64. w k + 1 = S γ ( w k - γ F ( w k ) ) . w^{k+1}=S_{\gamma}\left(w^{k}-\gamma\nabla F\left(w^{k}\right)\right).
  65. F ( w ) F(w)
  66. R ( w ) R(w)
  67. w k - γ F ( w k ) w^{k}-\gamma\nabla F\left(w^{k}\right)
  68. S γ S_{\gamma}
  69. γ \gamma
  70. F F
  71. R R
  72. w k + 1 = prox γ R ( w k - γ F ( w k ) ) , w^{k+1}=\operatorname{prox}_{\gamma R}\left(w^{k}-\gamma\nabla F\left(w^{k}% \right)\right),
  73. γ k \gamma_{k}
  74. γ \gamma
  75. 1 \ell_{1}
  76. 1 \ell_{1}
  77. R ( w ) = w 1 R(w)=\|w\|_{1}
  78. min w F ( w ) + R ( w ) , \min_{w}F(w)+R(w),
  79. F F
  80. 2 \ell_{2}
  81. min w d 1 n i = 1 n ( y i - w , x i ) 2 + λ ( ( 1 - μ ) w 1 + μ w 2 2 ) , \min_{w\in\mathbb{R}^{d}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\langle w,x_{i}% \rangle)^{2}+\lambda\left((1-\mu)\|w\|_{1}+\mu\|w\|_{2}^{2}\right),
  82. x i d and y i . x_{i}\in\mathbb{R}^{d}\,\text{ and }y_{i}\in\mathbb{R}.
  83. 0 < μ 1 0<\mu\leq 1
  84. λ ( ( 1 - μ ) w 1 + μ w 2 2 ) \lambda\left((1-\mu)\|w\|_{1}+\mu\|w\|_{2}^{2}\right)
  85. μ > 0 \mu>0
  86. μ w 2 2 \mu\|w\|_{2}^{2}
  87. { w 1 , , w G } \{w_{1},\ldots,w_{G}\}
  88. R ( w ) = g = 1 G w g 2 , R(w)=\sum_{g=1}^{G}\|w_{g}\|_{2},
  89. 2 \ell_{2}
  90. w g w_{g}
  91. λ γ ( g = 1 G w g 2 ) \lambda\gamma\left(\sum_{g=1}^{G}\|w_{g}\|_{2}\right)
  92. S ~ λ γ ( w g ) = { w g - λ γ w g w g 2 , w g 2 > λ γ 0 , w g 2 λ γ \widetilde{S}_{\lambda\gamma}(w_{g})=\begin{cases}w_{g}-\lambda\gamma\frac{w_{% g}}{\|w_{g}\|_{2}},&\|w_{g}\|_{2}>\lambda\gamma\\ 0,&\|w_{g}\|_{2}\leq\lambda\gamma\end{cases}
  93. w g w_{g}
  94. g g

Pseudocapacitance.html

  1. RuO 2 + xH + + xe - RuO 2 - x ( OH ) x \mathrm{RuO_{2}+xH^{+}+xe^{-}\leftrightarrow RuO_{2-x}(OH)_{x}}
  2. 0 x 2 0\leq x\leq 2

Pseudofunctor.html

  1. f ( x y ) = f ( x ) f ( y ) f(x\circ y)=f(x)\circ f(y)
  2. f ( 1 ) = 1 f(1)=1

Public_Market_Equivalent.html

  1. N A V P M E = s T C s × I T I s NAV_{PME}=\sum_{s}^{T}C_{s}\times\cfrac{I_{T}}{I_{s}}
  2. C s C_{s}
  3. I s I_{s}
  4. P M E = I R R ( C s , N A V P M E ) PME=IRR(C_{s},NAV_{PME})
  5. N A V P M E + , T = s = 0 t ( c o n t r i b u t i o n s - λ T . d i s t r i b u t i o n s ) . I t I s NAV_{PME+,T}=\sum_{s=0}^{t}(contribution_{s}-\lambda_{T}.distribution_{s}).% \cfrac{I_{t}}{I_{s}}
  6. λ T = ( S c - N A V P E , T ) S d \lambda_{T}=\cfrac{(S_{c}-NAV_{PE,T})}{S_{d}}
  7. S c = s = 0 T ( c o n t r i b u t i o n s . I T I s ) S_{c}=\sum_{s=0}^{T}(contribution_{s}.\cfrac{I_{T}}{I_{s}})
  8. S d = s = 0 T ( d i s t r i b u t i o n s . I T I s ) S_{d}=\sum_{s=0}^{T}(distribution_{s}.\cfrac{I_{T}}{I_{s}})
  9. N A V P M E + , T = N A V P E , T NAV_{PME+,T}=NAV_{PE,T}
  10. P M E + T = I R R ( C o n t r i b u t i o n s , λ T . D i s t r i b u t i o n s , N A V P E , T ) PME+_{T}=IRR(Contributions,\lambda_{T}.Distributions,NAV_{PE,T})
  11. D w e i g h t , t = D t D t + N A V t D_{weight,t}=\cfrac{D_{t}}{D_{t}+NAV_{t}}
  12. N A V m P M E , t = ( 1 - D w e i g h t , t ) × ( N A V m P M E , t - 1 * I t I t - 1 + C a l l i ) NAV_{mPME,t}=(1-D_{weight,t})\times(NAV_{mPME,t-1}*\cfrac{I_{t}}{I_{t-1}}+Call% _{i})
  13. D i s t m P M E , t = ( D w e i g h t , t ) × ( N A V m P M E , t - 1 * I t I t - 1 + C a l l i ) Dist_{mPME,t}=(D_{weight,t})\times(NAV_{mPME,t-1}*\cfrac{I_{t}}{I_{t-1}}+Call_% {i})
  14. I R R m P M E = I R R ( C a l l , D i s t m P M E , N A V m P M E , T ) IRR_{mPME}=IRR(Call,Dist_{mPME},NAV_{mPME,T})
  15. K S - P M E = F V ( D i s t ) F V ( C a l l ) KS-PME=\cfrac{FV(Dist)}{FV(Call)}
  16. F V ( D i s t ) = t ( d i s t ( t ) × I T I t ) FV(Dist)=\sum_{t}(dist(t)\times\cfrac{I_{T}}{I_{t}})
  17. F V ( C a l l ) = t ( c a l l ( t ) × I T I t ) FV(Call)=\sum_{t}(call(t)\times\cfrac{I_{T}}{I_{t}})
  18. I T I_{T}
  19. K S - P M E = t d i s t ( t ) I t t c a l l ( t ) I t KS-PME=\frac{\sum_{t}\frac{dist(t)}{I_{t}}}{\sum_{t}\frac{call(t)}{I_{t}}}
  20. K S - P M E = F V ( D i s t ) F V ( C a l l ) KS-PME=\cfrac{FV(Dist)}{FV(Call)}
  21. N A V P M E = t T C t I T I t NAV_{PME}=\sum_{t}^{T}C_{t}\cfrac{I_{T}}{I_{t}}
  22. N A V P M E = t T c a l l ( t ) I T I t - t T d i s t ( t ) I T I t NAV_{PME}=\sum_{t}^{T}call(t)\cfrac{I_{T}}{I_{t}}-\sum_{t}^{T}dist(t)\cfrac{I_% {T}}{I_{t}}
  23. N A V P M E = F V ( C a l l ) - F V ( D i s t ) NAV_{PME}=FV(Call)-FV(Dist)
  24. K S - P M E = 1 - N A V P M E F V ( C a l l ) KS-PME=1-\cfrac{NAV_{PME}}{FV(Call)}
  25. a = I R R ( F V ( C ) , F V ( D ) , N A V P E ) a=IRR(FV(C),FV(D),NAV_{PE})
  26. α = l n ( 1 + a ) Δ \alpha=\cfrac{ln(1+a)}{\Delta}
  27. Δ \Delta
  28. C 0.. n C_{0..n}
  29. N A V NAV
  30. r r
  31. N A V P E = i = 0 n c i . ( 1 + r ) n - i NAV_{PE}=\sum_{i=0}^{n}{c_{i}.(1+r)^{n-i}}
  32. α \alpha
  33. r r
  34. r ( t ) = b ( t ) + α r(t)=b(t)+\alpha
  35. c i c_{i}
  36. t n t_{n}
  37. v i ( t n ) = c i . e t i t n ( b ( t ) + α ) d t v_{i}(t_{n})=c_{i}.e^{\int_{t_{i}}^{t_{n}}{(b(t)+\alpha)dt}}
  38. I n I i = e t i t n b ( t ) d t \frac{I_{n}}{I_{i}}=e^{\int_{t_{i}}^{t_{n}}{b(t)dt}}
  39. v i ( t n ) = c i . I n I i . e t i t n α d t v_{i}(t_{n})=c_{i}.\frac{I_{n}}{I_{i}}.e^{\int_{t_{i}}^{t_{n}}{\alpha dt}}
  40. t i = i Δ t_{i}=i\Delta
  41. v i ( t n ) = c i . I n I i . e α . ( n - i ) Δ v_{i}(t_{n})=c_{i}.\frac{I_{n}}{I_{i}}.e^{\alpha.(n-i)\Delta}
  42. N A V P E = i = 0 n c i . I n I i . e α . ( n - i ) Δ NAV_{PE}=\sum_{i=0}^{n}{c_{i}.\frac{I_{n}}{I_{i}}.e^{\alpha.(n-i)\Delta}}
  43. 1 + a = e α . Δ 1+a=e^{\alpha.\Delta}
  44. N A V P E = i = 0 n c i . I n I i . ( 1 + a ) n - i NAV_{PE}=\sum_{i=0}^{n}{c_{i}.\frac{I_{n}}{I_{i}}.(1+a)^{n-i}}
  45. α \alpha
  46. α = l n ( 1 + a ) Δ \alpha=\cfrac{ln(1+a)}{\Delta}
  47. α \alpha
  48. N A V P E = i = 0 n c i . ( ( 1 + b i ) 1 t n - t i + α ) t n - t i NAV_{PE}=\sum_{i=0}^{n}{c_{i}.((1+b_{i})^{\frac{1}{t_{n}-t_{i}}}+\alpha)^{t_{n% }-t_{i}}}
  49. b i = I n I i - 1 b_{i}=\frac{I_{n}}{I_{i}}-1
  50. 1 = i = 1 n d i ( 1 + b T i , T N + r p p ) T N - T i i = 1 n c j ( 1 + b T j , T N + r p p ) T N - T j , 1=\frac{\sum_{i=1}^{n}d_{i}(1+b_{T_{i},T_{N}}+r_{pp})^{T_{N}-T_{i}}}{\sum_{i=1% }^{n}c_{j}(1+b_{T_{j},T_{N}}+r_{pp})^{T_{N}-T_{j}}},
  51. c i c_{i}
  52. d j d_{j}
  53. T i T_{i}
  54. T j T_{j}
  55. b T i , T N b_{T_{i},T_{N}}
  56. T i T_{i}
  57. T N T_{N}
  58. r p p r_{pp}
  59. r r
  60. N A V P E = i = 0 n c i . ( 1 + r ) t n - t i NAV_{PE}=\sum_{i=0}^{n}{c_{i}.(1+r)^{t_{n}-t_{i}}}
  61. r ( t i ) = β i , n + α r(t_{i})=\beta_{i,n}+\alpha
  62. β i , n \beta_{i,n}
  63. t i t_{i}
  64. t n t_{n}
  65. β i , n = ( I n I i ) 1 t n - t i - 1 \beta_{i,n}=(\frac{I_{n}}{I_{i}})^{\frac{1}{t_{n}-t_{i}}}-1
  66. β i , n = ( 1 + b i ) 1 t n - t i - 1 \beta_{i,n}=(1+b_{i})^{\frac{1}{t_{n}-t_{i}}}-1
  67. N A V P E = i = 0 n c i . ( ( 1 + b i ) 1 t n - t i + α ) t n - t i NAV_{PE}=\sum_{i=0}^{n}{c_{i}.((1+b_{i})^{\frac{1}{t_{n}-t_{i}}}+\alpha)^{t_{n% }-t_{i}}}
  68. V a l u e I n d e x E n d i n g = F V R e t u r n e d - F V I n v e s t e d Value_{Index_{Ending}}=FV_{Returned}-FV_{Invested}
  69. I 0 I_{0}
  70. A d j . P M E = E [ e - r T ( L P 1 ( A t , T ) + L P 2 ( A T , T ) + L P 3 ( A T , T ) ) ] I 0 + E [ 0 T e - r s m X 0 d s ] Adj.PME=\cfrac{E[e^{-rT}(LP_{1}(A_{t},T)+LP_{2}(A_{T},T)+LP_{3}(A_{T},T))]}{I_% {0}+E[\int_{0}^{T}e^{-rs}mX_{0}ds]}

Pulsatile_flow_generator.html

  1. k = F l k=\frac{F}{\triangle l}\,

Pulsational_pair-instability_supernova.html

  1. E = m c 2 E=mc^{2}

Pulse_vaccination_strategy.html

  1. d S d t = μ N - μ S - β I N S , S ( n T + ) = ( 1 - p ) S ( n T - ) n = 0 , 1 , 2 , \frac{dS}{dt}=\mu N-\mu S-\beta\frac{I}{N}S,S(nT^{+})=(1-p)S(nT^{-})n=0,1,2,\dots
  2. d V d t = - μ V , V ( n T + ) = V ( n T - ) + p S ( n T - ) n = 0 , 1 , 2 , \frac{dV}{dt}=-\mu V,V(nT^{+})=V(nT^{-})+pS(nT^{-})n=0,1,2,\dots
  3. I = 0 I=0
  4. S * ( t ) = 1 - p 1 - ( 1 - p ) E - μ T E - μ M O D ( t , T ) S^{*}(t)=1-\frac{p}{1-(1-p)E^{-\mu T}}E^{-\mu MOD(t,T)}
  5. R 0 0 T S * ( t ) d t < 1 R_{0}\int_{0}^{T}{S^{*}(t)dt}<1

Pupil_function.html

  1. P ( u , v ) \mathrm{P}(u,v)
  2. P ( u , v ) = A ( u , v ) exp ( i Θ ( u , v ) ) \mathrm{P}(u,v)=\mathrm{A}(u,v)\cdot\mathrm{exp}(i\,\mathrm{\Theta}(u,v))
  3. Θ ( u , v ) \mathrm{\Theta}(u,v)
  4. ( u , v ) (u,v)
  5. A ( u , v ) \mathrm{A}(u,v)
  6. P ( u , v ) = 1 , u , v : u 2 + v 2 R \mathrm{P}(u,v)=1,\forall u,v:\sqrt{u^{2}+v^{2}}\leq R
  7. P ( u , v ) = 0 , u , v : u 2 + v 2 > R , \mathrm{P}(u,v)=0,\forall u,v:\sqrt{u^{2}+v^{2}}>R,
  8. R R
  9. k ( u 2 + v 2 ) k(u^{2}+v^{2})
  10. P ( u , v ) = exp ( i k ( u 2 + v 2 ) ) , u , v : u 2 + v 2 R \mathrm{P}(u,v)=\mathrm{exp}(i\,k(u^{2}+v^{2})),\forall u,v:\sqrt{u^{2}+v^{2}}\leq R
  11. P ( u , v ) = 0 , \mathrm{P}(u,v)=0,
  12. k u 2 ku^{2}
  13. P ( u , v ) = exp ( i k u 2 ) , u , v : u 2 + v 2 R \mathrm{P}(u,v)=\mathrm{exp}(i\,ku^{2}),\forall u,v:\sqrt{u^{2}+v^{2}}\leq R
  14. P ( u , v ) = 0 , \mathrm{P}(u,v)=0,

Pyridoxal_5'-phosphate_synthase_(glutamine_hydrolyzing).html

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

Pyrrolysine—tRNAPyl_ligase.html

  1. \rightleftharpoons

Q-construction.html

  1. B + C B^{+}C
  2. π 0 ( B + C ) \pi_{0}(B^{+}C)
  3. i = 0 , 1 , 2 i=0,1,2
  4. π i ( B + C ) \pi_{i}(B^{+}C)
  5. K i ( C ) = π i ( B + C ) K_{i}(C)=\pi_{i}(B^{+}C)
  6. K i ( C ; G ) = π i ( B + C ; G ) K_{i}(C;G)=\pi_{i}(B^{+}C;G)
  7. π * \pi_{*}
  8. B + B^{+}
  9. R S R\to S
  10. B + P ( R ) B + P ( S ) B^{+}P(R)\to B^{+}P(S)
  11. K i ( P ( R ) ) = K i ( R ) K i ( S ) K_{i}(P(R))=K_{i}(R)\to K_{i}(S)
  12. P ( R ) P(R)
  13. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  14. X Z Y X\leftarrow Z\to Y
  15. B + C B^{+}C
  16. B + C = Ω B Q C B^{+}C=\Omega BQC
  17. Ω \Omega
  18. B Q C BQC
  19. π i ( B + C ) \pi_{i}(B^{+}C)
  20. i = 0 , 1 , 2 i=0,1,2
  21. S - 1 S S^{-1}S
  22. S = iso C S=\operatorname{iso}C
  23. Ω B Q C B ( S - 1 S ) \Omega BQC\simeq B(S^{-1}S)
  24. f : E Q C f:E\to QC
  25. f - 1 ( X ) f^{-1}(X)
  26. S - 1 f S^{-1}f
  27. S - 1 E Q C S^{-1}E\to QC
  28. Ω B Q C \Omega BQC
  29. F ( B S - 1 f ) F(BS^{-1}f)
  30. B ( S - 1 S ) B(S^{-1}S)
  31. B S - 1 f BS^{-1}f
  32. * B Q C *\to BQC
  33. F ( B S - 1 f ) F(BS^{-1}f)
  34. π i B ( S - 1 S ) \pi_{i}B(S^{-1}S)
  35. K i K_{i}
  36. i = 0 , 1 , 2 i=0,1,2
  37. π 0 B ( S - 1 S ) = K 0 ( R ) \pi_{0}B(S^{-1}S)=K_{0}(R)
  38. G L n ( R ) = Aut ( R n ) S - 1 S GL_{n}(R)=\operatorname{Aut}(R^{n})\to S^{-1}S
  39. B G L ( R ) = lim B G L n ( R ) B ( S - 1 S ) BGL(R)=\underrightarrow{\lim}BGL_{n}(R)\to B(S^{-1}S)
  40. B G L ( R ) BGL(R)
  41. G L ( R ) GL(R)
  42. K ( G L ( R ) , 1 ) K(GL(R),1)
  43. B ( S - 1 S ) B(S^{-1}S)
  44. f : B G L ( R ) B ( S - 1 S ) 0 . f:BGL(R)\to B(S^{-1}S)^{0}.
  45. S n S_{n}
  46. R n R^{n}
  47. B S n BS_{n}
  48. R n R^{n}
  49. e π 0 ( B S ) e\in\pi_{0}(BS)
  50. H p ( B ( S - 1 S ) 0 ) H p ( B ( S - 1 S ) ) = H p ( B S ) [ π 0 ( B S ) - 1 ] = H p ( B S ) [ e - 1 ] . H_{p}(B(S^{-1}S)^{0})\subset H_{p}(B(S^{-1}S))=H_{p}(BS)[\pi_{0}(BS)^{-1}]=H_{% p}(BS)[e^{-1}].
  51. x e - n xe^{-n}
  52. x x e m x\mapsto xe^{m}
  53. R m S R^{m}\in S
  54. H p ( B ( S - 1 S ) 0 ) = lim H p ( B S n ) = lim H p ( B G L n ( R ) ) = H p ( B G L ( R ) ) , p 0 H_{p}(B(S^{-1}S)^{0})=\underrightarrow{\lim}H_{p}(BS_{n})=\underrightarrow{% \lim}H_{p}(BGL_{n}(R))=H_{p}(BGL(R)),\quad p\geq 0
  55. B ( S - 1 S ) 0 B(S^{-1}S)^{0}
  56. π 1 ( B ( S - 1 S ) 0 ) = π 1 ( B ( S - 1 S ) 0 ) ab = H 1 ( B ( S - 1 S ) 0 ) = H 1 ( B G L ( R ) ) = H 1 ( G L ( R ) ) = G L ( R ) ab = K 1 ( R ) . \pi_{1}(B(S^{-1}S)^{0})=\pi_{1}(B(S^{-1}S)^{0})\text{ab}=H_{1}(B(S^{-1}S)^{0})% =H_{1}(BGL(R))=H_{1}(GL(R))=GL(R)^{\,\text{ab}}=K_{1}(R).
  57. π 2 \pi_{2}
  58. K 2 K_{2}
  59. F f Ff
  60. π 2 ( B G L ( R ) ) = 0 π 2 ( B ( S - 1 S ) 0 ) π 1 ( F f ) π 1 ( B G L ( R ) ) = G L ( R ) K 1 ( R ) \pi_{2}(BGL(R))=0\to\pi_{2}(B(S^{-1}S)^{0})\to\pi_{1}(Ff)\to\pi_{1}(BGL(R))=GL% (R)\to K_{1}(R)
  61. π 1 ( F f ) E ( R ) \pi_{1}(Ff)\to E(R)
  62. π 1 ( F f ) \pi_{1}(Ff)
  63. π 1 ( F f ) \pi_{1}(Ff)
  64. K 2 ( R ) K_{2}(R)
  65. F f Ff
  66. f ~ \widetilde{f}
  67. Y \to Y
  68. H p ( X , ) H p ( Y , ) , p 0 H_{p}(X,\mathbb{Z})\simeq H_{p}(Y,\mathbb{Z}),p\geq 0
  69. F f X Y Ff\to X\to Y
  70. * Y Y *\to Y\to Y
  71. E p q 2 = H p ( Y , H q ( F f , ) ) H p + q ( X , ) , {}^{2}E_{pq}=H_{p}(Y,H_{q}(Ff,\mathbb{Z}))\Rightarrow H_{p+q}(X,\mathbb{Z}),
  72. E p q 2 = H p ( Y , H q ( * , ) ) H p + q ( Y , ) . {}^{2}E^{\prime}_{pq}=H_{p}(Y,H_{q}(*,\mathbb{Z}))\Rightarrow H_{p+q}(Y,% \mathbb{Z}).
  73. E 0 q 2 = E 0 q 2 {}^{2}E_{0q}={}^{2}E^{\prime}_{0q}
  74. F f Ff
  75. H p ( X , ) H p ( Y , ) H_{p}(X,\mathbb{Z})\simeq H_{p}(Y,\mathbb{Z})
  76. F f ~ F f \widetilde{Ff}\to Ff
  77. G = π 1 ( F f ) G=\pi_{1}(Ff)
  78. E p q 2 = H p ( G , H q ( F f ~ , ) ) H p + q ( F f , ) = H p + q ( * , ) {}^{2}E_{pq}=H_{p}(G,H_{q}(\widetilde{Ff},\mathbb{Z}))\Rightarrow H_{p+q}(Ff,% \mathbb{Z})=H_{p+q}(*,\mathbb{Z})

Q-Weibull_distribution.html

  1. { ( 2 - q ) κ λ ( x λ ) κ - 1 e q - ( x / λ ) κ x 0 0 x < 0 \begin{cases}(2-q)\frac{\kappa}{\lambda}\left(\frac{x}{\lambda}\right)^{\kappa% -1}e_{q}^{-(x/\lambda)^{\kappa}}&x\geq 0\\ 0&x<0\end{cases}
  2. { 1 - e q - ( x / λ ) κ x 0 0 x < 0 \begin{cases}1-e_{q^{\prime}}^{-(x/\lambda^{\prime})^{\kappa}}&x\geq 0\\ 0&x<0\end{cases}
  3. f ( x ; q , λ , κ ) = { ( 2 - q ) κ λ ( x λ ) κ - 1 e q ( - ( x / λ ) κ ) x 0 , 0 x < 0 , f(x;q,\lambda,\kappa)=\begin{cases}(2-q)\frac{\kappa}{\lambda}\left(\frac{x}{% \lambda}\right)^{\kappa-1}e_{q}(-(x/\lambda)^{\kappa})&x\geq 0,\\ 0&x<0,\end{cases}
  4. e q ( x ) = { exp ( x ) if q = 1 , [ 1 + ( 1 - q ) x ] 1 / ( 1 - q ) if q 1 and 1 + ( 1 - q ) x > 0 , 0 1 / ( 1 - q ) if q 1 and 1 + ( 1 - q ) x 0 , e_{q}(x)=\begin{cases}\exp(x)&\,\text{if }q=1,\\ [1+(1-q)x]^{1/(1-q)}&\,\text{if }q\neq 1\,\text{ and }1+(1-q)x>0,\\ 0^{1/(1-q)}&\,\text{if }q\neq 1\,\text{ and }1+(1-q)x\leq 0,\\ \end{cases}
  5. { 1 - e q - ( x / λ ) κ x 0 0 x < 0 \begin{cases}1-e_{q^{\prime}}^{-(x/\lambda^{\prime})^{\kappa}}&x\geq 0\\ 0&x<0\end{cases}
  6. λ = λ ( 2 - q ) 1 κ \lambda^{\prime}={\lambda\over(2-q)^{1\over\kappa}}
  7. q = 1 ( 2 - q ) q^{\prime}={1\over(2-q)}
  8. μ ( q , κ , λ ) = { λ ( 2 + 1 1 - q + 1 κ ) ( 1 - q ) - 1 κ B [ 1 + 1 κ , 2 + 1 1 - q ] q < 1 λ Γ ( 1 + 1 κ ) q = 1 λ ( 2 - q ) ( q - 1 ) - 1 + κ κ B [ 1 + 1 κ , - ( 1 + 1 q - 1 + 1 κ ) ] 1 < q < 1 + 1 + 2 κ 1 + κ 1 + κ κ + 1 q < 2 \mu(q,\kappa,\lambda)=\begin{cases}\lambda\,\left(2+\frac{1}{1-q}+\frac{1}{% \kappa}\right)(1-q)^{-\frac{1}{\kappa}}\,B\left[1+\frac{1}{\kappa},2+\frac{1}{% 1-q}\right]&q<1\\ \lambda\,\Gamma(1+\frac{1}{\kappa})&q=1\\ \lambda\,(2-q)(q-1)^{-\frac{1+\kappa}{\kappa}}\,B\left[1+\frac{1}{\kappa},-% \left(1+\frac{1}{q-1}+\frac{1}{\kappa}\right)\right]&1<q<1+\frac{1+2\kappa}{1+% \kappa}\\ \infty&1+\frac{\kappa}{\kappa+1}\leq q<2\end{cases}
  9. B ( ) B()
  10. Γ ( ) \Gamma()
  11. κ = 1 \kappa=1
  12. κ \kappa
  13. α = 2 - q q - 1 , λ Lomax = 1 λ ( q - 1 ) \alpha={{2-q}\over{q-1}}~{},~{}\lambda\text{Lomax}={1\over{\lambda(q-1)}}
  14. κ = 1 \kappa=1
  15. If X qWeibull ( q , λ , κ = 1 ) and Y [ Pareto ( x m = 1 λ ( q - 1 ) , α = 2 - q q - 1 ) - x m ] , then X Y \,\text{If }X\sim\mathrm{qWeibull}(q,\lambda,\kappa=1)\,\text{ and }Y\sim\left% [\,\text{Pareto}\left(x_{m}={1\over{\lambda(q-1)}},\alpha={{2-q}\over{q-1}}% \right)-x_{m}\right],\,\text{ then }X\sim Y\,

QCO.html

  1. S V V = ( S V m a x - S V m i n ) / S V m e a n SVV=(SV_{max}-SV_{min})/SV_{mean}

Quadratic_integrate_and_fire.html

  1. d x d t = x 2 + I \frac{dx}{dt}=x^{2}+I
  2. I I
  3. V t V_{t}
  4. V r V_{r}
  5. x ( t ) V t x(t)\geq V_{t}
  6. V r V_{r}

Quadratrix_of_Hippias.html

  1. γ : ( 0 , π 2 ] 2 \gamma:(0,\tfrac{\pi}{2}]\rightarrow\mathbb{R}^{2}
  2. γ ( t ) = ( x ( t ) y ( t ) ) = ( 2 a π t cot ( t ) 2 a π t ) \gamma(t)=\begin{pmatrix}x(t)\\ y(t)\end{pmatrix}=\begin{pmatrix}\frac{2a}{\pi}t\cot(t)\\ \frac{2a}{\pi}t\end{pmatrix}
  3. ( 0 , π 2 ] (0,\tfrac{\pi}{2}]
  4. cot ( t ) \cot(t)
  5. t = 0 t=0
  6. lim t 0 t cot ( t ) = 1 \lim_{t\to 0}t\cot(t)=1
  7. ( - π , π ) (-\pi,\pi)
  8. f ( x ) = x cot ( π 2 a x ) f(x)=x\cdot\cot\left(\frac{\pi}{2a}\cdot x\right)
  9. 2 π \tfrac{2}{\pi}
  10. | A J ¯ | = 2 π r \left|\overline{AJ}\right|=\tfrac{2}{\pi}r
  11. | B L ¯ | = π 2 r \left|\overline{BL}\right|=\tfrac{\pi}{2}r
  12. | B O ¯ | = r 2 \left|\overline{BO}\right|=\tfrac{r}{2}
  13. | O Q ¯ | = | B O ¯ | = r 2 \left|\overline{OQ}\right|=\left|\overline{BO}\right|=\tfrac{r}{2}

Quadrature_based_moment_methods.html

  1. ( K n ) (Kn)
  2. ( S t ) (St)
  3. ( ξ ) (\xi)
  4. ( T ) (T)
  5. ( v ) ({v})
  6. ( K n ) (Kn)
  7. ( S t ) (St)
  8. f ( v ) f({v})
  9. f ( ξ ) f(\xi)
  10. K n O ( 1 ) Kn\approx O(1)
  11. S t > 1 St>1
  12. f ( t , x , v ) f(t,{x},{v})
  13. f t + v f x + v ( v ˙ f ) = C \frac{\partial f}{\partial t}+{v}\frac{\partial f}{\partial{x}}+\frac{\partial% }{\partial{v}}\cdot({\dot{v}}f)=C
  14. f ( t , x , v ) f(t,{x},{v})
  15. M i , j , k ( γ ) ( t , x ) = v 1 i v 2 j v 3 k f ( t , x , v ) d v M^{(\gamma)}_{i,j,k}(t,{x})=\int v_{1}^{i}v_{2}^{j}v_{3}^{k}f(t,{x},{v})d{v}
  16. v d v_{d}
  17. i , j , k 0 i,j,k\geq 0
  18. γ i + j + k \gamma\equiv i+j+k
  19. M M
  20. t M γ + x M γ + 1 = i v ˙ M γ - 1 + 𝒞 \frac{\partial}{\partial t}M^{\gamma}+\nabla_{{x}}\cdot M^{\gamma+1}={i}\dot{{% v}}\cdot M^{\gamma-1}+\mathcal{C}
  21. i { i , j , k } {i}\equiv\{i,j,k\}
  22. i v ˙ { i v ˙ 1 , j v ˙ 2 , k v ˙ 3 } {i}\dot{{v}}\equiv\{i\dot{v}_{1},j\dot{v}_{2},k\dot{v}_{3}\}
  23. γ + 1 \gamma+1
  24. M i , j , k γ ( t , x ) α = 1 β U α i V α j W α k ϕ α ( x , t ) γ = 0 , 1 , 2 , M^{\gamma}_{i,j,k}(t,{x})\approx\sum_{\alpha=1}^{\beta}U_{\alpha}^{i}V_{\alpha% }^{j}W_{\alpha}^{k}\phi_{\alpha}({x},t)\;\;\;\;\;\;\;\;\;\;\gamma=0,1,2,...
  25. U , V , W U,V,W
  26. ϕ \phi
  27. α \alpha
  28. β \beta
  29. K n 1 Kn<<1
  30. S t O ( 1 ) St\approx O(1)

Quadric_geometric_algebra.html

  1. 𝒢 6 , 3 \mathcal{G}_{6,3}
  2. 𝒞 6 , 3 \mathcal{C}\ell_{6,3}
  3. 𝒢 4 , 1 \mathcal{G}_{4,1}
  4. 𝒢 1 , 3 \mathcal{G}_{1,3}
  5. A x 2 + B y 2 + C z 2 + D x y + E y z + F z x + G x + H y + I z + J = 0 Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fzx+Gx+Hy+Iz+J=0
  6. A x 2 + B y 2 + C z 2 + G x + H y + I z + J = 0. Ax^{2}+By^{2}+Cz^{2}+Gx+Hy+Iz+J=0.
  7. π / 2 \pi/2
  8. y = ± x y=\pm x
  9. z = ± x z=\pm x
  10. z = ± y z=\pm y

Quadrisecant.html

  1. ( d - 2 ) ( d - 3 ) 2 ( d - 4 ) 12 - g ( d 2 - 7 d + 13 - g ) 2 . \frac{(d-2)(d-3)^{2}(d-4)}{12}-\frac{g(d^{2}-7d+13-g)}{2}.

Quadrupole_formula.html

  1. h ¯ i j ( t , r ) = 2 G c 4 r I ¨ i j ( t - r ) , \bar{h}_{ij}(t,r)=\frac{2G}{c^{4}r}\ddot{I}_{ij}(t-r),
  2. h ¯ i j \bar{h}_{ij}
  3. I i j I_{ij}

Quantal_translative_momentum_transfer.html

  1. λ λ
  2. θ θ
  3. d d
  4. n d nd
  5. n n
  6. θ θ
  7. 2 d sin θ = n λ , 2d\sin\theta=n\lambda\,,
  8. p p
  9. θ θ
  10. n n
  11. n P nP
  12. P P
  13. p s i n θ psinθ
  14. 2 p s i n θ 2psinθ
  15. 2 p sin θ = n P . 2p\sin\theta=nP\,.
  16. θ θ
  17. p / d = P / λ p/d=P/\lambda
  18. p λ = P d . p\lambda=Pd\,.
  19. p p
  20. λ λ
  21. p λ = h . p\lambda=h\,.
  22. P P
  23. P = h / d . P=h/d\,.
  24. p = k \vec{p}=\hbar\vec{k}
  25. p i 1 , p i 2 \vec{p}_{i1},\vec{p}_{i2}
  26. p f 1 , p f 2 \vec{p}_{f1},\vec{p}_{f2}
  27. q = p i 1 - p f 1 = p f 2 - p i 2 \vec{q}=\vec{p}_{i1}-\vec{p}_{f1}=\vec{p}_{f2}-\vec{p}_{i2}
  28. k = q / \vec{k}=\vec{q}/\hbar
  29. k = 2 π / λ k=2\pi/\lambda
  30. Q = k f - k i Q=k_{f}-k_{i}
  31. Δ x = / | q | \Delta x=\hbar/|q|
  32. k f k_{f}
  33. k i k_{i}
  34. G = Q = k f - k i \vec{G}=\vec{Q}=\vec{k}_{f}-\vec{k}_{i}
  35. G = 2 π / d G=2\pi/d
  36. Q Q
  37. 2 θ 2\theta
  38. α \alpha
  39. Q Q
  40. Q = 4 π sin θ λ Q=\frac{4\pi\sin\theta}{\lambda}
  41. 2 θ 2\theta
  42. Q Q

Quantifier_(logic).html

  1. x X , P ( x ) Q ( x ) \forall{x}{\in}X,P(x)Q(x)
  2. x P x P \exists{x}\,P\qquad\forall{x}\,P
  3. ( x ) P ( x . P ) x P ( x : P ) x ( P ) x P x , P x X P x : X P (\exists{x})P\qquad(\exists x\ .\ P)\qquad\exists x\ \cdot\ P\qquad(\exists x:% P)\qquad\exists{x}(P)\qquad\exists_{x}\,P\qquad\exists{x}{,}\,P\qquad\exists{x% }{\in}X\,P\qquad\exists\,x{:}X\,P
  4. ( x ) P x P (x)\,P\qquad\bigwedge_{x}P
  5. x D P ( x ) \forall x\!\in\!D\;P(x)
  6. x ( x D P ( x ) ) \forall x\;(x\!\in\!D\to P(x))
  7. x D P ( x ) \exists x\!\in\!D\;P(x)
  8. x ( x D and P ( x ) ) \exists x\;(x\!\in\!\!D\and P(x))
  9. ¬ ( x D P ( x ) ) x D ¬ P ( x ) , \neg(\forall x\!\in\!D\;P(x))\equiv\exists x\!\in\!D\;\neg P(x),
  10. ¬ ( x D P ( x ) ) x D ¬ P ( x ) , \neg(\exists x\!\in\!D\;P(x))\equiv\forall x\!\in\!D\;\neg P(x),
  11. x ( y B ( x , y ) ) C ( y , x ) \forall x(\exists yB(x,y))\vee C(y,x)
  12. x n A ( x 1 , , x n ) \forall x_{n}A(x_{1},\ldots,x_{n})
  13. x n A ( x 1 , , x n ) \exists x_{n}A(x_{1},\ldots,x_{n})
  14. ! x n A ( x 1 , , x n ) \exists!x_{n}A(x_{1},\ldots,x_{n})
  15. x n A ( x 1 , , x n ) \exists x_{n}A(x_{1},\ldots,x_{n})
  16. y , z { A ( x 1 , , x n - 1 , y ) A ( x 1 , , x n - 1 , z ) y = z } . \forall y,z\left\{A(x_{1},\ldots,x_{n-1},y)\wedge A(x_{1},\ldots,x_{n-1},z)% \implies y=z\right\}.
  17. many x n A ( x 1 , , x n - 1 , x n ) \exists^{\mathrm{many}}x_{n}A(x_{1},\ldots,x_{n-1},x_{n})
  18. P { w : F ( v 1 , , v n - 1 , w ) = 𝐓 } b \operatorname{P}\{w:F(v_{1},\ldots,v_{n-1},w)=\mathbf{T}\}\geq b
  19. few x n A ( x 1 , , x n - 1 , x n ) \exists^{\mathrm{few}}x_{n}A(x_{1},\ldots,x_{n-1},x_{n})
  20. 0 < P { w : F ( v 1 , , v n - 1 , w ) = 𝐓 } a 0<\operatorname{P}\{w:F(v_{1},\ldots,v_{n-1},w)=\mathbf{T}\}\leq a
  21. [ § n n 2 4 ] = { 0 , 1 , 2 } \left[\S n\in\mathbb{N}\quad n^{2}\leq 4\right]=\left\{0,1,2\right\}
  22. { n 𝐍 : n 2 4 } = { 0 , 1 , 2 } \{n\in\mathbf{N}:n^{2}\leq 4\}=\left\{0,1,2\right\}\,\!

Quantum_algorithm_for_linear_systems_of_equations.html

  1. κ \kappa
  2. O ( log ( N ) κ 2 ) O(\log(N)\kappa^{2})
  3. N N
  4. O ( N κ ) O(N\kappa)
  5. O ( N κ ) O(N\sqrt{\kappa})
  6. N × N N\times N
  7. A A
  8. b \overrightarrow{b}
  9. x \overrightarrow{x}
  10. A x = b A\overrightarrow{x}=\overrightarrow{b}
  11. x \overrightarrow{x}
  12. M M
  13. x | M | x \langle x|M|x\rangle
  14. b \overrightarrow{b}
  15. | b = i = 1 N b i | i . |b\rangle=\sum_{i\mathop{=}1}^{N}b_{i}|i\rangle.
  16. e i A t e^{iAt}
  17. | b |b\rangle
  18. t t
  19. | b |b\rangle
  20. A A
  21. λ j \lambda_{j}
  22. j = 1 N β j | u j | λ j , \sum_{j\mathop{=}1}^{N}\beta_{j}|u_{j}\rangle|\lambda_{j}\rangle,
  23. u j u_{j}
  24. A A
  25. | b = j = 1 N β j | u j |b\rangle=\sum_{j\mathop{=}1}^{N}\beta_{j}|u_{j}\rangle
  26. | λ j |\lambda_{j}\rangle
  27. C λ j - 1 | λ j C\lambda^{-1}_{j}|\lambda_{j}\rangle
  28. C C
  29. | λ j |\lambda_{j}\rangle
  30. i = 1 N β i λ j - 1 | u j = A - 1 | b = | x , \sum_{i\mathop{=}1}^{N}\beta_{i}\lambda^{-1}_{j}|u_{j}\rangle=A^{-1}|b\rangle=% |x\rangle,
  31. | x |x\rangle
  32. x x
  33. x | M | x \langle x|M|x\rangle
  34. A A
  35. A A
  36. 𝐂 = [ 0 A A t 0 ] . \mathbf{C}=\begin{bmatrix}0&A\\ A^{t}&0\end{bmatrix}.
  37. C C
  38. C y = [ b 0 ] . Cy=\begin{bmatrix}b\\ 0\end{bmatrix}.
  39. y = [ 0 x ] y=\begin{bmatrix}0\\ x\end{bmatrix}
  40. | b |b\rangle
  41. B B
  42. | initial |\mathrm{initial}\rangle
  43. | b |b\rangle
  44. | b |b\rangle
  45. | b |b\rangle
  46. | ψ 0 |\psi_{0}\rangle
  47. | ψ 0 := 2 / T τ = 0 T - 1 sin π ( τ + 1 2 T ) | τ |\psi_{0}\rangle:=\sqrt{2/T}\sum_{\tau\mathop{=}0}^{T-1}\sin\pi\left(\tfrac{% \tau+\tfrac{1}{2}}{T}\right)|\tau\rangle
  48. T T
  49. | ψ 0 |\psi_{0}\rangle
  50. U invert U_{\mathrm{invert}}
  51. A A
  52. e i A t e^{i}At
  53. O ( log ( N ) s 2 t ) O(\log(N)s^{2}t)
  54. U invert U_{\mathrm{invert}}
  55. | ψ 0 C |\psi_{0}\rangle^{C}
  56. | k |k\rangle
  57. λ k := 2 π k / t 0 \lambda_{k}:=2\pi k/t_{0}
  58. | h ( λ k ) S := 1 - f ( λ k ) 2 - g ( λ k ) 2 | nothing S + f ( λ k ) | well S + g ( λ k ) | ill S , |h(\lambda_{k})\rangle^{S}:=\sqrt{1-f(\lambda_{k})^{2}-g(\lambda_{k})^{2}}|% \mathrm{nothing}\rangle^{S}+f(\lambda_{k})|\mathrm{well}\rangle^{S}+g(\lambda_% {k})|\mathrm{ill}\rangle^{S},
  59. | b |b\rangle
  60. U invert B | initial U_{\mathrm{invert}}B|\mathrm{initial}\rangle
  61. U invert B R init B U invert R succ U_{\mathrm{invert}}BR_{\mathrm{init}}B^{\dagger}U^{\dagger}_{\mathrm{invert}}R% _{\mathrm{succ}}
  62. R succ = I 2 - 2 | well well | 2 , R_{\mathrm{succ}}=I^{2}-2|\mathrm{well}\rangle\langle\mathrm{well}|^{2},
  63. R init = I 2 - 2 | initial initial | . R_{\mathrm{init}}=I^{2}-2|\mathrm{initial}\rangle\langle\mathrm{initial}|.
  64. S S
  65. S S
  66. p p
  67. 1 p \frac{1}{p}
  68. O ( 1 p ) O\left(\frac{1}{\sqrt{p}}\right)
  69. S S
  70. i = 1 N β i λ j - 1 | u j = A - 1 | b = | x . \sum_{i\mathop{=}1}^{N}\beta_{i}\lambda^{-1}_{j}|u_{j}\rangle=A^{-1}|b\rangle=% |x\rangle.
  71. x | M | x \langle x|M|x\rangle
  72. x \overrightarrow{x}
  73. O ( N 3 ) O(N^{3})
  74. x \overrightarrow{x}
  75. O ( N s κ ) O(Ns\kappa)
  76. | A x - b | 2 |A\overrightarrow{x}-\overrightarrow{b}|^{2}
  77. x \overrightarrow{x}
  78. x M x \overrightarrow{x}^{\dagger}M\overrightarrow{x}
  79. O ( N κ ) O(N\sqrt{\kappa})
  80. O ( κ 2 log N ) O(\kappa^{2}\log N)
  81. O ( κ log 3 κ log N ) O(\kappa\log^{3}\kappa\log N)
  82. A A
  83. κ \kappa
  84. A A
  85. A A
  86. A A
  87. 1 κ \frac{1}{\kappa}
  88. κ 2 \kappa^{2}
  89. κ \kappa
  90. poly ( log ( N ) ) \mathrm{poly}(\log(N))
  91. κ \kappa
  92. e i A t e^{iAt}
  93. A A
  94. ε \varepsilon
  95. | x |x\rangle
  96. O ( 1 t 0 ) O\left(\frac{1}{t_{0}}\right)
  97. λ \lambda
  98. O ( 1 λ t 0 ) O\left(\frac{1}{\lambda t_{0}}\right)
  99. λ - 1 \lambda^{-1}
  100. λ 1 / κ \lambda\geq 1/\kappa
  101. t 0 = O ( κ ε ) t_{0}=O(\kappa\varepsilon)
  102. ε \varepsilon
  103. O ( 1 ε ) O\left(\frac{1}{\varepsilon}\right)

Quantum_complex_network.html

  1. i i
  2. j j
  3. i i
  4. j j
  5. | 0 |0\rangle
  6. | 1 |1\rangle
  7. | ψ i j |\psi_{ij}\rangle
  8. | ψ i j = | ϕ i | ϕ j , |\psi_{ij}\rangle=|\phi\rangle_{i}\otimes|\phi\rangle_{j},
  9. | ϕ i |\phi\rangle_{i}
  10. | ϕ j |\phi\rangle_{j}
  11. | Φ i j + = 1 2 ( | 0 i | 0 j + | 1 i | 1 j ) , |\Phi_{ij}^{+}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{i}\otimes|0\rangle_{j}+|1% \rangle_{i}\otimes|1\rangle_{j}),
  12. | Φ i j - = 1 2 ( | 0 i | 0 j - | 1 i | 1 j ) , |\Phi_{ij}^{-}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{i}\otimes|0\rangle_{j}-|1% \rangle_{i}\otimes|1\rangle_{j}),
  13. | Ψ i j + = 1 2 ( | 0 i | 1 j + | 1 i | 0 j ) , |\Psi_{ij}^{+}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{i}\otimes|1\rangle_{j}+|1% \rangle_{i}\otimes|0\rangle_{j}),
  14. | Ψ i j - = 1 2 ( | 0 i | 1 j - | 1 i | 0 j ) . |\Psi_{ij}^{-}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{i}\otimes|1\rangle_{j}-|1% \rangle_{i}\otimes|0\rangle_{j}).
  15. N - 1 N-1
  16. p p
  17. p p
  18. p p
  19. p p
  20. i i
  21. j j
  22. | ψ i j = 1 - p / 2 | 0 i | 0 j + p / 2 | 1 i | 1 j , |\psi_{ij}\rangle=\sqrt{1-p/2}|0\rangle_{i}\otimes|0\rangle_{j}+\sqrt{p/2}|1% \rangle_{i}\otimes|1\rangle_{j},
  23. p = 0 p=0
  24. | ψ i j = | 0 i | 0 j , |\psi_{ij}\rangle=|0\rangle_{i}\otimes|0\rangle_{j},
  25. p = 1 p=1
  26. | ψ i j = 1 / 2 ( | 0 i | 0 j + | 1 i | 1 j ) |\psi_{ij}\rangle=\sqrt{1/2}(|0\rangle_{i}\otimes|0\rangle_{j}+|1\rangle_{i}% \otimes|1\rangle_{j})
  27. p p
  28. 0 < p < 1 0<p<1
  29. p p
  30. p p
  31. N N
  32. p N Z , p\sim N^{Z},
  33. Z - 2 Z\geq-2
  34. z z
  35. p p
  36. p p
  37. p p
  38. p c p_{c}
  39. p > p c p>p_{c}
  40. p < p c p<p_{c}
  41. p c p_{c}
  42. p < p c p<p_{c}
  43. p > p c p>p_{c}
  44. p c p_{c}
  45. p c p_{c}
  46. p c p_{c}

Quantum_dynamical_semigroup.html

  1. ϕ t \phi_{t}
  2. t 0 t\geq 0
  3. ϕ s ( ϕ t ( ρ ) ) = ϕ t + s ( ρ ) , t , s 0. \phi_{s}(\phi_{t}(\rho))=\phi_{t+s}(\rho),\qquad t,s\geq 0.
  4. ( ρ ) = lim Δ t 0 ϕ Δ t ( ρ ) - ϕ 0 ( ρ ) Δ t \mathcal{L}(\rho)=\mathrm{lim}_{\Delta t\to 0}\frac{\phi_{\Delta t}(\rho)-\phi% _{0}(\rho)}{\Delta t}
  5. ϕ t \phi_{t}
  6. ϕ t + s ( ρ ) = e s ϕ t ( ρ ) . \phi_{t+s}(\rho)=e^{\mathcal{L}s}\phi_{t}(\rho).

Quantum_excitation_(accelerator_physics).html

  1. S ( ξ ) = 9 3 8 π ξ 0 K 5 / 3 ( ξ ¯ ) d ξ ¯ S(\xi)=\frac{9\sqrt{3}}{8\pi}\xi\int_{0}^{\infty}K_{5/3}(\bar{\xi})d\bar{\xi}
  2. d = N ˙ u 2 d=\dot{N}\langle u^{2}\rangle
  3. d = 55 24 3 α m c γ 5 | ρ | 3 d=\frac{55}{24\sqrt{3}}\alpha\frac{\hbar}{mc}\frac{\gamma^{5}}{|\rho|^{3}}

Quantum_heat_engines_and_refrigerators.html

  1. T h T_{h}
  2. ω h = E 3 - E 1 \hbar\omega_{h}=E_{3}-E_{1}
  3. N h N g = e - ω h k b T h \frac{N_{h}}{N_{g}}=e^{-\frac{\hbar\omega_{h}}{k_{b}T_{h}}}
  4. = h 2 π \hbar=\frac{h}{2\pi}
  5. k b k_{b}
  6. T c T_{c}
  7. E 2 - E 1 = ω c E_{2}-E_{1}=\hbar\omega_{c}
  8. N c N g = e - ω c k b T c \frac{N_{c}}{N_{g}}=e^{-\frac{\hbar\omega_{c}}{k_{b}T_{c}}}
  9. ν \nu
  10. ν = ω h - ω c \nu=\omega_{h}-\omega_{c}
  11. η = ν ω h = 1 - ω c ω h \eta=\frac{\hbar\nu}{\hbar\omega_{h}}=1-\frac{\omega_{c}}{\omega_{h}}
  12. G = N h - N c 0 G=N_{h}-N_{c}\geq 0
  13. ω c k b T c ω c k b T c \frac{\hbar\omega_{c}}{k_{b}T_{c}}\geq\frac{\hbar\omega_{c}}{k_{b}T_{c}}
  14. η = 1 - ω c ω h 1 - T c T h = η c \eta=1-\frac{\omega_{c}}{\omega_{h}}\leq 1-\frac{T_{c}}{T_{h}}=\eta_{c}
  15. η c \eta_{c}
  16. G = 0 G=0
  17. ϵ = ω c ν T c T h - T c \epsilon=\frac{\omega_{c}}{\nu}\leq\frac{T_{c}}{T_{h}-T_{c}}
  18. A B A\rightarrow B
  19. U c {U}_{c}
  20. B C B\rightarrow C
  21. U c h {U}_{ch}
  22. C D C\rightarrow D
  23. U h U_{h}
  24. D A D\rightarrow A
  25. U h c U_{hc}
  26. U g l o b a l U_{global}
  27. U g l o b a l = U h c U h U c h U c {U}_{global}~{}~{}=~{}~{}{U}_{hc}{U}_{h}{U}_{ch}{U}_{c}
  28. [ U i , U j ] 0 [{\ U}_{i},{U}_{j}]\neq 0
  29. τ c y c \tau_{cyc}
  30. 2 π / ω 2\pi/\omega
  31. τ c y c 2 π / ω \tau_{cyc}\gg 2\pi/\omega
  32. ω \hbar\omega
  33. k b T k_{b}T
  34. η = 1 - ω c ω h \eta=1-\frac{\omega_{c}}{\omega_{h}}
  35. η c \eta_{c}
  36. η = 1 - T c T h \eta=1-\sqrt{\frac{T_{c}}{T_{h}}}
  37. τ c y c 2 π / ω \tau_{cyc}\ll 2\pi/\omega
  38. ω h \omega_{h}
  39. ω c \omega_{c}
  40. H = H s + H c + H h + H s c + H s h H=H_{s}+H_{c}+H_{h}+H_{sc}+H_{sh}
  41. H s ( t ) H_{s}(t)
  42. d d t ρ = - i [ H s , ρ ] + L h ( ρ ) + L c ( ρ ) \frac{d}{dt}\rho=-\frac{i}{\hbar}[H_{s},\rho]+L_{h}(\rho)+L_{c}(\rho)
  43. ρ \rho
  44. L h / c L_{h/c}
  45. d d t E = H s t + L h ( H s ) + L c ( H s ) \frac{d}{dt}E=\langle\frac{\partial H_{s}}{\partial t}\rangle+\langle L_{h}(H_% {s})\rangle+\langle L_{c}(H_{s})\rangle
  46. P = H t P=\langle\frac{\partial H}{\partial t}\rangle
  47. J h = L h ( H s ) J_{h}=\langle L_{h}(H_{s})\rangle
  48. J c = L c ( H s ) J_{c}=\langle L_{c}(H_{s})\rangle
  49. d S d t = - J h T h - J c T c 0 \frac{dS}{dt}=-\frac{J_{h}}{T_{h}}-\frac{J_{c}}{T_{c}}\geq 0
  50. ρ = ρ s ρ h ρ c . \rho=\rho_{s}\otimes\rho_{h}\otimes\rho_{c}~{}.
  51. d d t ρ s = L ρ s , \frac{d}{dt}\rho_{s}={L}\rho_{s}~{},
  52. L {L}
  53. L L
  54. T h T_{h}
  55. T c T_{c}
  56. T d T_{d}
  57. H 0 = ω h a a + ω c b b + ω d c c , H_{0}=\hbar\omega_{h}a^{\dagger}a+\hbar\omega_{c}b^{\dagger}b+\hbar\omega_{d}c% ^{\dagger}c~{}~{},
  58. ω h \omega_{h}
  59. ω c \omega_{c}
  60. ω d \omega_{d}
  61. ω d = ω h - ω c \omega_{d}=\omega_{h}-\omega_{c}
  62. a b c a^{\dagger}bc
  63. H I = ϵ ( a b c + a b c ) , H_{I}=\hbar\epsilon\left(ab^{\dagger}c^{\dagger}+a^{\dagger}bc\right)~{}~{},
  64. ϵ \epsilon
  65. d E s d t = J h + J c + J d . \frac{dE_{s}}{dt}={J}_{h}+{J}_{c}+{J}_{d}~{}~{}.
  66. d E s d t = 0 \frac{dE_{s}}{dt}=0
  67. d d t Δ S u = - J h T h - J c T c - J d T d 0 . \frac{d}{dt}\Delta{S}_{u}~{}=~{}-\frac{{J}_{h}}{T_{h}}-\frac{{J}_{c}}{T_{c}}-% \frac{{J}_{d}}{T_{d}}~{}\geq~{}0~{}~{}.
  68. T d T_{d}\rightarrow\infty
  69. P = J d {P}={J}_{d}
  70. P {P}
  71. T c 0 T_{c}\rightarrow 0
  72. S ˙ c - T c α , α 0 . \dot{S}_{c}\propto-T_{c}^{\alpha}~{}~{}~{},~{}~{}~{}~{}\alpha\geq 0~{}~{}.
  73. α = 0 \alpha=0
  74. α 0 \alpha\geq 0
  75. α > 0 \alpha>0
  76. S ˙ c = 0 \dot{S}_{c}=0
  77. J c T c α + 1 {J}_{c}\propto T_{c}^{\alpha+1}
  78. J c ( T c ( t ) ) = - c V ( T c ( t ) ) d T c ( t ) d t . {J}_{c}(T_{c}(t))=-c_{V}(T_{c}(t))\frac{dT_{c}(t)}{dt}~{}~{}.
  79. c V ( T c ) c_{V}(T_{c})
  80. J c T c α + 1 {J}_{c}\propto T_{c}^{\alpha+1}
  81. c V T c η c_{V}\sim T_{c}^{\eta}
  82. η 0 {\eta}\geq 0
  83. ζ \zeta
  84. d T c ( t ) d t - T c ζ , T c 0 , ζ = α - η + 1 \frac{dT_{c}(t)}{dt}\propto-T_{c}^{\zeta},~{}~{}~{}~{}~{}T_{c}\rightarrow 0,~{% }~{}~{}~{}~{}{\zeta=\alpha-\eta+1}
  85. ζ \zeta
  86. α \alpha
  87. ζ < 0 \zeta<0

Quantum_machine_learning.html

  1. ψ | φ \langle\psi|\varphi\rangle
  2. O ( M l o g ( M N ) ) O(Mlog(MN))
  3. O ( k l o g ( M N ) ) O(k\;log(MN))

Quantum_mechanical_scattering_of_photon_and_nucleus.html

  1. d 4 σ = Z 2 α fine 3 c 2 ( 2 π ) 2 | 𝐩 + | | 𝐩 - | d E + ω 3 d Ω + d Ω - d Φ | 𝐪 | 4 × × [ - 𝐩 - 2 sin 2 Θ - ( E - - c | 𝐩 - | cos Θ - ) 2 ( 4 E + 2 - c 2 𝐪 2 ) - 𝐩 + 2 sin 2 Θ + ( E + - c | 𝐩 + | cos Θ + ) 2 ( 4 E - 2 - c 2 𝐪 2 ) + 2 2 ω 2 𝐩 + 2 sin 2 Θ + + 𝐩 - 2 sin 2 Θ - ( E + - c | 𝐩 + | cos Θ + ) ( E - - c | 𝐩 - | cos Θ - ) + 2 | 𝐩 + | | 𝐩 - | sin Θ + sin Θ - cos Φ ( E + - c | 𝐩 + | cos Θ + ) ( E - - c | 𝐩 - | cos Θ - ) ( 2 E + 2 + 2 E - 2 - c 2 𝐪 2 ) ] . \begin{aligned}\displaystyle d^{4}\sigma&\displaystyle=\frac{Z^{2}\alpha_{% \textrm{fine}}^{3}c^{2}}{(2\pi)^{2}\hbar}|\mathbf{p}_{+}||\mathbf{p}_{-}|\frac% {dE_{+}}{\omega^{3}}\frac{d\Omega_{+}d\Omega_{-}d\Phi}{|\mathbf{q}|^{4}}\times% \\ &\displaystyle\times\left[-\frac{\mathbf{p}_{-}^{2}\sin^{2}\Theta_{-}}{(E_{-}-% c|\mathbf{p}_{-}|\cos\Theta_{-})^{2}}\left(4E_{+}^{2}-c^{2}\mathbf{q}^{2}% \right)\right.\\ &\displaystyle-\frac{\mathbf{p}_{+}^{2}\sin^{2}\Theta_{+}}{(E_{+}-c|\mathbf{p}% _{+}|\cos\Theta_{+})^{2}}\left(4E_{-}^{2}-c^{2}\mathbf{q}^{2}\right)\\ &\displaystyle+2\hbar^{2}\omega^{2}\frac{\mathbf{p}_{+}^{2}\sin^{2}\Theta_{+}+% \mathbf{p}_{-}^{2}\sin^{2}\Theta_{-}}{(E_{+}-c|\mathbf{p}_{+}|\cos\Theta_{+})(% E_{-}-c|\mathbf{p}_{-}|\cos\Theta_{-})}\\ &\displaystyle+2\left.\frac{|\mathbf{p}_{+}||\mathbf{p}_{-}|\sin\Theta_{+}\sin% \Theta_{-}\cos\Phi}{(E_{+}-c|\mathbf{p}_{+}|\cos\Theta_{+})(E_{-}-c|\mathbf{p}% _{-}|\cos\Theta_{-})}\left(2E_{+}^{2}+2E_{-}^{2}-c^{2}\mathbf{q}^{2}\right)% \right].\\ \end{aligned}
  2. d Ω + = sin Θ + d Θ + , d Ω - = sin Θ - d Θ - . \begin{aligned}\displaystyle d\Omega_{+}&\displaystyle=\sin\Theta_{+}\ d\Theta% _{+},\\ \displaystyle d\Omega_{-}&\displaystyle=\sin\Theta_{-}\ d\Theta_{-}.\end{aligned}
  3. Z Z
  4. α f i n e 1 / 137 \alpha_{fine}\approx 1/137
  5. \hbar
  6. c c
  7. E k i n , + / - E_{kin,+/-}
  8. E + , - E_{+,-}
  9. 𝐩 + , - \mathbf{p}_{+,-}
  10. E + , - = E k i n , + / - + m e c 2 = m e 2 c 4 + 𝐩 + , - 2 c 2 . E_{+,-}=E_{kin,+/-}+m_{e}c^{2}=\sqrt{m_{e}^{2}c^{4}+\mathbf{p}_{+,-}^{2}c^{2}}.
  11. ω = E + + E - . \hbar\omega=E_{+}+E_{-}.
  12. 𝐪 \mathbf{q}
  13. - 𝐪 2 \displaystyle-\mathbf{q}^{2}
  14. Θ + \displaystyle\Theta_{+}
  15. 𝐤 \mathbf{k}
  16. E + E_{+}
  17. Θ + \Theta_{+}
  18. Θ - \Theta_{-}
  19. Φ \Phi
  20. d 2 σ ( E + , ω , Θ + ) d E + d Ω + = j = 1 6 I j \displaystyle\frac{d^{2}\sigma(E_{+},\omega,\Theta_{+})}{dE_{+}d\Omega_{+}}=% \sum\limits_{j=1}^{6}I_{j}
  21. I 1 = 2 π A ( Δ 2 ( p ) ) 2 + 4 p + 2 p - 2 sin 2 Θ + × ln ( ( Δ 2 ( p ) ) 2 + 4 p + 2 p - 2 sin 2 Θ + - ( Δ 2 ( p ) ) 2 + 4 p + 2 p - 2 sin 2 Θ + ( Δ 1 ( p ) + Δ 2 ( p ) ) + Δ 1 ( p ) Δ 2 ( p ) - ( Δ 2 ( p ) ) 2 - 4 p + 2 p - 2 sin 2 Θ + - ( Δ 2 ( p ) ) 2 + 4 p + 2 p - 2 sin 2 Θ + ( Δ 1 ( p ) - Δ 2 ( p ) ) + Δ 1 ( p ) Δ 2 ( p ) ) × [ - 1 - c Δ 2 ( p ) p - ( E + - c p + cos Θ + ) + p + 2 c 2 sin 2 Θ + ( E + - c p + cos Θ + ) 2 - 2 2 ω 2 p - Δ 2 ( p ) c ( E + - c p + cos Θ + ) ( ( Δ 2 ( p ) ) 2 + 4 p + 2 p - 2 sin 2 Θ + ) ] , I 2 = 2 π A c p - ( E + - c p + cos Θ + ) ln ( E - + p - c E - - p - c ) , I 3 = 2 π A ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + × ln ( ( ( E - + p - c ) ( 4 p + 2 p - 2 sin 2 Θ + ( E - - p - c ) + ( Δ 1 ( p ) + Δ 2 ( p ) ) ( ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) - ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ) ) ) ( ( E - - p - c ) ( 4 p + 2 p - 2 sin 2 Θ + ( - E - - p - c ) + ( Δ 1 ( p ) - Δ 2 ( p ) ) ( ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) - ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ) ) ) - 1 ) × [ c ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) p - ( E + - c p + cos Θ + ) + [ ( ( Δ 2 ( p ) ) 2 + 4 p + 2 p - 2 sin 2 Θ + ) ( E - 3 + E - p - c ) + p - c ( 2 ( ( Δ 1 ( p ) ) 2 - 4 p + 2 p - 2 sin 2 Θ + ) E - p - c + Δ 1 ( p ) Δ 2 ( p ) ( 3 E - 2 + p - 2 c 2 ) ) ] [ ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ] - 1 + [ - 8 p + 2 p - 2 m 2 c 4 sin 2 Θ + ( E + 2 + E - 2 ) - 2 2 ω 2 p + 2 sin 2 Θ + p - c ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) + 2 2 ω 2 p - m 2 c 3 ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) ] [ ( E + - c p + cos Θ + ) ( ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ) ] - 1 + 4 E + 2 p - 2 ( 2 ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 - 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ) ( Δ 1 ( p ) E - + Δ 2 ( p ) p - c ) ( ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ) 2 ] , I 4 = 4 π A p - c ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + + 16 π E + 2 p - 2 A ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 ( ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ) 2 , I 5 = 4 π A ( - ( Δ 2 ( p ) ) 2 + ( Δ 1 ( p ) ) 2 - 4 p + 2 p - 2 sin 2 Θ + ) ( ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ) × [ 2 ω 2 p - 2 E + c p + cos Θ + [ E - [ 2 ( Δ 2 ( p ) ) 2 ( ( Δ 2 ( p ) ) 2 - ( Δ 1 ( p ) ) 2 ) + 8 p + 2 p - 2 sin 2 Θ + ( ( Δ 2 ( p ) ) 2 + ( Δ 1 ( p ) ) 2 ) ] + p - c [ 2 Δ 1 ( p ) Δ 2 ( p ) ( ( Δ 2 ( p ) ) 2 - ( Δ 1 ( p ) ) 2 ) + 16 Δ 1 ( p ) Δ 2 ( p ) p + 2 p - 2 sin 2 Θ + ] ] [ ( Δ 2 ( p ) ) 2 + 4 p + 2 p - 2 sin 2 Θ + ] - 1 + 2 2 ω 2 p + 2 sin 2 Θ + ( 2 Δ 1 ( p ) Δ 2 ( p ) p - c + 2 ( Δ 2 ( p ) ) 2 E - + 8 p + 2 p - 2 sin 2 Θ + E - ) E + - c p + cos Θ + - [ 2 E + 2 p - 2 { 2 ( ( Δ 2 ( p ) ) 2 - ( Δ 1 ( p ) ) 2 ) ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 8 p + 2 p - 2 sin 2 Θ + [ ( ( Δ 1 ( p ) ) 2 + ( Δ 2 ( p ) ) 2 ) ( E - 2 + p - 2 c 2 ) + 4 Δ 1 ( p ) Δ 2 ( p ) E - p - c ] } ] [ ( Δ 2 ( p ) E - + Δ 1 ( p ) p - c ) 2 + 4 m 2 c 4 p + 2 p - 2 sin 2 Θ + ] - 1 - 8 p + 2 p - 2 sin 2 Θ + ( E + 2 + E - 2 ) ( Δ 2 ( p ) p - c + Δ 1 ( p ) E - ) E + - c p + cos Θ + ] , I 6 = - 16 π E - 2 p + 2 sin 2 Θ + A ( E + - c p + cos Θ + ) 2 ( - ( Δ 2 ( p ) ) 2 + ( Δ 1 ( p ) ) 2 - 4 p + 2 p - 2 sin 2 Θ + ) \begin{aligned}\displaystyle I_{1}&\displaystyle=\frac{2\pi A}{\sqrt{(\Delta^{% (p)}_{2})^{2}+4p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}}}\\ &\displaystyle\times\ln\left(\frac{(\Delta^{(p)}_{2})^{2}+4p_{+}^{2}p_{-}^{2}% \sin^{2}\Theta_{+}-\sqrt{(\Delta^{(p)}_{2})^{2}+4p_{+}^{2}p_{-}^{2}\sin^{2}% \Theta_{+}}(\Delta^{(p)}_{1}+\Delta^{(p)}_{2})+\Delta^{(p)}_{1}\Delta^{(p)}_{2% }}{-(\Delta^{(p)}_{2})^{2}-4p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}-\sqrt{(\Delta% ^{(p)}_{2})^{2}+4p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}}(\Delta^{(p)}_{1}-\Delta% ^{(p)}_{2})+\Delta^{(p)}_{1}\Delta^{(p)}_{2}}\right)\\ &\displaystyle\times\left[-1-\frac{c\Delta^{(p)}_{2}}{p_{-}(E_{+}-cp_{+}\cos% \Theta_{+})}+\frac{p_{+}^{2}c^{2}\sin^{2}\Theta_{+}}{(E_{+}-cp_{+}\cos\Theta_{% +})^{2}}-\frac{2\hbar^{2}\omega^{2}p_{-}\Delta^{(p)}_{2}}{c(E_{+}-cp_{+}\cos% \Theta_{+})((\Delta^{(p)}_{2})^{2}+4p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+})}% \right],\\ \displaystyle I_{2}&\displaystyle=\frac{2\pi Ac}{p_{-}(E_{+}-cp_{+}\cos\Theta_% {+})}\ln\left(\frac{E_{-}+p_{-}c}{E_{-}-p_{-}c}\right),\\ \displaystyle I_{3}&\displaystyle=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_{2}E_{-}+% \Delta^{(p)}_{1}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}}}% \\ &\displaystyle\times\ln\Bigg(\Big((E_{-}+p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin^{2}% \Theta_{+}(E_{-}-p_{-}c)+(\Delta^{(p)}_{1}+\Delta^{(p)}_{2})((\Delta^{(p)}_{2}% E_{-}+\Delta^{(p)}_{1}p_{-}c)\\ &\displaystyle-\sqrt{(\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}+4m^{2}% c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}}))\Big)\Big((E_{-}-p_{-}c)(4p_{+}^{2% }p_{-}^{2}\sin^{2}\Theta_{+}(-E_{-}-p_{-}c)\\ &\displaystyle+(\Delta^{(p)}_{1}-\Delta^{(p)}_{2})((\Delta^{(p)}_{2}E_{-}+% \Delta^{(p)}_{1}p_{-}c)-\sqrt{(\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)^{% 2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}}))\Big)^{-1}\Bigg)\\ &\displaystyle\times\left[\frac{c(\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c% )}{p_{-}(E_{+}-cp_{+}\cos\Theta_{+})}\right.\\ &\displaystyle+\Big[((\Delta^{(p)}_{2})^{2}+4p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_% {+})(E_{-}^{3}+E_{-}p_{-}c)+p_{-}c(2((\Delta^{(p)}_{1})^{2}-4p_{+}^{2}p_{-}^{2% }\sin^{2}\Theta_{+})E_{-}p_{-}c\\ &\displaystyle+\Delta^{(p)}_{1}\Delta^{(p)}_{2}(3E_{-}^{2}+p_{-}^{2}c^{2}))% \Big]\Big[(\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^% {2}p_{-}^{2}\sin^{2}\Theta_{+}\Big]^{-1}\\ &\displaystyle+\Big[-8p_{+}^{2}p_{-}^{2}m^{2}c^{4}\sin^{2}\Theta_{+}(E_{+}^{2}% +E_{-}^{2})-2\hbar^{2}\omega^{2}p_{+}^{2}\sin^{2}\Theta_{+}p_{-}c(\Delta^{(p)}% _{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)\\ &\displaystyle+2\hbar^{2}\omega^{2}p_{-}m^{2}c^{3}(\Delta^{(p)}_{2}E_{-}+% \Delta^{(p)}_{1}p_{-}c)\Big]\Big[(E_{+}-cp_{+}\cos\Theta_{+})((\Delta^{(p)}_{2% }E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta% _{+})\Big]^{-1}\\ &\displaystyle+\left.\frac{4E_{+}^{2}p_{-}^{2}(2(\Delta^{(p)}_{2}E_{-}+\Delta^% {(p)}_{1}p_{-}c)^{2}-4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+})(\Delta^{% (p)}_{1}E_{-}+\Delta^{(p)}_{2}p_{-}c)}{((\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1% }p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+})^{2}}\right],\\ \displaystyle I_{4}&\displaystyle=\frac{4\pi Ap_{-}c(\Delta^{(p)}_{2}E_{-}+% \Delta^{(p)}_{1}p_{-}c)}{(\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}+4m% ^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}}+\frac{16\pi E_{+}^{2}p_{-}^{2}A% (\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}}{((\Delta^{(p)}_{2}E_{-}+% \Delta^{(p)}_{1}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+})^{% 2}},\\ \displaystyle I_{5}&\displaystyle=\frac{4\pi A}{(-(\Delta^{(p)}_{2})^{2}+(% \Delta^{(p)}_{1})^{2}-4p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+})((\Delta^{(p)}_{2}% E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_% {+})}\\ &\displaystyle\times\left[\frac{\hbar^{2}\omega^{2}p_{-}^{2}}{E_{+}cp_{+}\cos% \Theta_{+}}\Big[E_{-}[2(\Delta^{(p)}_{2})^{2}((\Delta^{(p)}_{2})^{2}-(\Delta^{% (p)}_{1})^{2})+8p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}((\Delta^{(p)}_{2})^{2}+(% \Delta^{(p)}_{1})^{2})]\right.\\ &\displaystyle+p_{-}c[2\Delta^{(p)}_{1}\Delta^{(p)}_{2}((\Delta^{(p)}_{2})^{2}% -(\Delta^{(p)}_{1})^{2})+16\Delta^{(p)}_{1}\Delta^{(p)}_{2}p_{+}^{2}p_{-}^{2}% \sin^{2}\Theta_{+}]\Big]\Big[(\Delta^{(p)}_{2})^{2}+4p_{+}^{2}p_{-}^{2}\sin^{2% }\Theta_{+}\Big]^{-1}\\ &\displaystyle+\frac{2\hbar^{2}\omega^{2}p_{+}^{2}\sin^{2}\Theta_{+}(2\Delta^{% (p)}_{1}\Delta^{(p)}_{2}p_{-}c+2(\Delta^{(p)}_{2})^{2}E_{-}+8p_{+}^{2}p_{-}^{2% }\sin^{2}\Theta_{+}E_{-})}{E_{+}-cp_{+}\cos\Theta_{+}}\\ &\displaystyle-\Big[2E_{+}^{2}p_{-}^{2}\{2((\Delta^{(p)}_{2})^{2}-(\Delta^{(p)% }_{1})^{2})(\Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}+8p_{+}^{2}p_{-}^% {2}\sin^{2}\Theta_{+}[((\Delta^{(p)}_{1})^{2}+(\Delta^{(p)}_{2})^{2})(E_{-}^{2% }+p_{-}^{2}c^{2})\\ &\displaystyle+4\Delta^{(p)}_{1}\Delta^{(p)}_{2}E_{-}p_{-}c]\}\Big]\Big[(% \Delta^{(p)}_{2}E_{-}+\Delta^{(p)}_{1}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2% }\sin^{2}\Theta_{+}\Big]^{-1}\\ &\displaystyle-\left.\frac{8p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+}(E_{+}^{2}+E_{% -}^{2})(\Delta^{(p)}_{2}p_{-}c+\Delta^{(p)}_{1}E_{-})}{E_{+}-cp_{+}\cos\Theta_% {+}}\right],\\ \displaystyle I_{6}&\displaystyle=-\frac{16\pi E_{-}^{2}p_{+}^{2}\sin^{2}% \Theta_{+}A}{(E_{+}-cp_{+}\cos\Theta_{+})^{2}(-(\Delta^{(p)}_{2})^{2}+(\Delta^% {(p)}_{1})^{2}-4p_{+}^{2}p_{-}^{2}\sin^{2}\Theta_{+})}\end{aligned}
  22. A = Z 2 α f i n e 3 c 2 ( 2 π ) 2 | 𝐩 + | | 𝐩 - | ω 3 , Δ 1 ( p ) : = - | 𝐩 + | 2 - | 𝐩 - | 2 - ( c ω ) + 2 c ω | 𝐩 + | cos Θ + , Δ 2 ( p ) : = 2 c ω | 𝐩 i | - 2 | 𝐩 + | | 𝐩 - | cos Θ + + 2. \begin{aligned}\displaystyle A&\displaystyle=\frac{Z^{2}\alpha_{fine}^{3}c^{2}% }{(2\pi)^{2}\hbar}\frac{|\mathbf{p}_{+}||\mathbf{p}_{-}|}{\omega^{3}},\\ \displaystyle\Delta^{(p)}_{1}&\displaystyle:=-|\mathbf{p}_{+}|^{2}-|\mathbf{p}% _{-}|^{2}-\left(\frac{\hbar}{c}\omega\right)+2\frac{\hbar}{c}\omega|\mathbf{p}% _{+}|\cos\Theta_{+},\\ \displaystyle\Delta^{(p)}_{2}&\displaystyle:=2\frac{\hbar}{c}\omega|\mathbf{p}% _{i}|-2|\mathbf{p}_{+}||\mathbf{p}_{-}|\cos\Theta_{+}+2.\end{aligned}

Quantum_refereed_game.html

  1. n n
  2. n n
  3. n n
  4. { R a : a Σ } \{R_{a}:a\in\Sigma\}
  5. 𝒳 1 , , 𝒳 n \mathcal{X}_{1},\cdots,\mathcal{X}_{n}
  6. 𝒴 1 , , 𝒴 n \mathcal{Y}_{1},\cdots,\mathcal{Y}_{n}
  7. 𝒳 k = 𝒜 k k \mathcal{X}_{k}=\mathcal{A}_{k}\otimes\mathcal{B}_{k}
  8. 𝒴 k = 𝒞 k 𝒟 k \mathcal{Y}_{k}=\mathcal{C}_{k}\otimes\mathcal{D}_{k}
  9. 𝒜 k , k , 𝒞 k \mathcal{A}_{k},\mathcal{B}_{k},\mathcal{C}_{k}
  10. 𝒟 k , 1 k n \mathcal{D}_{k},\ 1\leq k\leq n
  11. 𝒜 k , k \mathcal{A}_{k},\mathcal{B}_{k}
  12. k k
  13. 𝒞 k , 𝒟 k \mathcal{C}_{k},\mathcal{D}_{k}
  14. n n
  15. a Σ a\in\Sigma
  16. n n
  17. V A , V B : Σ V_{A},V_{B}:\Sigma\mapsto\mathbb{R}
  18. a a
  19. V A ( a ) + V B ( a ) = 0 V_{A}(a)+V_{B}(a)=0
  20. A 𝒮 n ( 𝒜 1 n , 𝒞 1 n ) A\in\mathcal{S}_{n}(\mathcal{A}_{1\cdots n},\mathcal{C}_{1\cdots n})
  21. B 𝒮 n ( 1 n , 𝒟 1 n ) B\in\mathcal{S}_{n}(\mathcal{B}_{1\cdots n},\mathcal{D}_{1\cdots n})
  22. 𝒮 n ( 𝒳 1 n , 𝒴 1 n ) \mathcal{S}_{n}(\mathcal{X}_{1\cdots n},\mathcal{Y}_{1\cdots n})
  23. 𝒳 1 , , 𝒳 n \mathcal{X}_{1},\cdots,\mathcal{X}_{n}
  24. 𝒴 1 , , 𝒴 n \mathcal{Y}_{1},\cdots,\mathcal{Y}_{n}
  25. A B A\otimes B
  26. V ( a ) = V A ( a ) = - V B ( a ) V(a)=V_{A}(a)=-V_{B}(a)
  27. R = a Σ V ( a ) R a R=\sum_{a\in\Sigma}V(a)R_{a}
  28. a Σ V ( a ) A B , R a = A B , R \sum_{a\in\Sigma}V(a)\langle A\otimes B,R_{a}\rangle=\langle A\otimes B,R\rangle
  29. max A min B A B , R = min B max A A B , R \max_{A}\min_{B}\langle A\otimes B,R\rangle=\min_{B}\max_{A}\langle A\otimes B% ,R\rangle
  30. A , B A,B
  31. 𝒮 n ( 𝒜 1 n , 𝒞 1 n ) \mathcal{S}_{n}(\mathcal{A}_{1\cdots n},\mathcal{C}_{1\cdots n})
  32. 𝒮 n ( 1 n , 𝒟 1 n ) \mathcal{S}_{n}(\mathcal{B}_{1\cdots n},\mathcal{D}_{1\cdots n})
  33. L = ( L yes , L no ) L=(L_{\,\text{yes}},L_{\,\text{no}})
  34. x L yes x\in L_{\,\text{yes}}
  35. x L no x\in L_{\,\text{no}}
  36. n n
  37. x x
  38. L = ( L yes , L no ) L=(L_{\,\text{yes}},L_{\,\text{no}})
  39. x L yes x\in L_{\,\text{yes}}
  40. x L no x\in L_{\,\text{no}}
  41. { a , b } \{a,b\}
  42. ( a ) (a)
  43. ( b ) (b)
  44. V ( a ) = 1 , V ( b ) = 0 V(a)=1,V(b)=0
  45. { R a , R b } \{R_{a},R_{b}\}
  46. A 𝒮 n ( 𝒜 1 n , 𝒞 1 n ) A\in\mathcal{S}_{n}(\mathcal{A}_{1\cdots n},\mathcal{C}_{1\cdots n})
  47. B 𝒮 n ( 1 n , 𝒟 1 n ) B\in\mathcal{S}_{n}(\mathcal{B}_{1\cdots n},\mathcal{D}_{1\cdots n})
  48. Ω a ( A ) = Tr 𝒞 1 n 𝒜 1 n ( ( A I 𝒟 1 n B 1 n ) R a ) \Omega_{a}(A)=\operatorname{Tr}_{\mathcal{C}_{1\cdots n}\otimes\mathcal{A}_{1% \cdots n}}((A\otimes I_{\mathcal{D}_{1\cdots n}\otimes B_{1\cdots n}})R_{a})
  49. Ω b ( A ) = Tr 𝒞 1 n 𝒜 1 n ( ( A I 𝒟 1 n B 1 n ) R b ) \Omega_{b}(A)=\operatorname{Tr}_{\mathcal{C}_{1\cdots n}\otimes\mathcal{A}_{1% \cdots n}}((A\otimes I_{\mathcal{D}_{1\cdots n}\otimes B_{1\cdots n}})R_{b})
  50. Tr 𝒳 ( Z ) \operatorname{Tr}_{\mathcal{X}}(Z)
  51. a a
  52. A B , R a = B , Ω a ( A ) \langle A\otimes B,R_{a}\rangle=\langle B,\Omega_{a}(A)\rangle
  53. b b
  54. A B , R b = B , Ω b ( A ) \langle A\otimes B,R_{b}\rangle=\langle B,\Omega_{b}(A)\rangle
  55. { Ω a ( A ) , Ω b ( A ) } \{\Omega_{a}(A),\Omega_{b}(A)\}
  56. A A
  57. max B B , Ω b ( A ) \max_{B}\langle B,\Omega_{b}(A)\rangle
  58. min { p 0 : Ω b ( A ) p Q , Q co- 𝒮 n ( 1 n , 𝒟 1 n ) } \min\{p\geq 0:\Omega_{b}(A)\leq pQ,\ Q\in\,\text{co-}\mathcal{S}_{n}(\mathcal{% B}_{1\cdots n},\mathcal{D}_{1\cdots n})\}
  59. p p
  60. min p subject to Ω b ( A ) p Q , A 𝒮 n ( 𝒜 1 n , 𝒞 1 n ) , Q co- 𝒮 n ( 1 n , 𝒟 1 n ) \begin{array}[]{rl}\min&p\\ \,\text{subject to}&\Omega_{b}(A)\leq pQ,\\ &A\in\mathcal{S}_{n}(\mathcal{A}_{1\cdots n},\mathcal{C}_{1\cdots n}),\\ &Q\in\,\text{co-}\mathcal{S}_{n}(\mathcal{B}_{1\cdots n},\mathcal{D}_{1\cdots n% })\end{array}
  61. min Tr ( P 1 ) subject to Ω b ( A n ) Q , Tr 𝒞 k ( A k ) = A k - 1 I 𝒜 k ( 2 k n ) , Tr 𝒞 1 ( A 1 ) = I 𝒜 1 , Q k = P k I 𝒟 k ( 1 k n ) , Tr k ( P k ) = Q k - 1 ( 2 k n ) , A k Pos ( 𝒞 1 k A 1 k ) ( 1 k n ) , Q k Pos ( 𝒟 1 k B 1 k ) ( 1 k n ) , P k Pos ( 𝒟 1 k B 1 k ) ( 1 k n ) , \begin{array}[]{rll}\min&\operatorname{Tr}(P_{1})\\ \,\text{subject to}&\Omega_{b}(A_{n})\leq Q,\\ &\operatorname{Tr}_{\mathcal{C}_{k}}(A_{k})=A_{k-1}\otimes I_{\mathcal{A}_{k}}% &(2\leq k\leq n),\\ &\operatorname{Tr}_{\mathcal{C}_{1}}(A_{1})=I_{\mathcal{A}_{1}},\\ &Q_{k}=P_{k}\otimes I_{\mathcal{D}_{k}}&(1\leq k\leq n),\\ &\operatorname{Tr}_{\mathcal{B}_{k}}(P_{k})=Q_{k-1}&(2\leq k\leq n),\\ &A_{k}\in\operatorname{Pos}(\mathcal{C}_{1\cdots k}\otimes A_{1\cdots k})&(1% \leq k\leq n),\\ &Q_{k}\in\operatorname{Pos}(\mathcal{D}_{1\cdots k}\otimes B_{1\cdots k})&(1% \leq k\leq n),\\ &P_{k}\in\operatorname{Pos}(\mathcal{D}_{1\cdots k}\otimes B_{1\cdots k})&(1% \leq k\leq n),\\ \end{array}
  62. n n

Quantum_stochastic_calculus.html

  1. H = H sys ( 𝐙 ) + 1 2 n ( ( p n - κ n X ) 2 + ω n 2 q n 2 ) , H=H_{\mathrm{sys}}(\mathbf{Z})+\frac{1}{2}\sum_{n}\left((p_{n}-\kappa_{n}X)^{2% }+\omega_{n}^{2}q_{n}^{2}\right)\,,
  2. H sys H_{\mathrm{sys}}
  3. 𝐙 \mathbf{Z}
  4. n n
  5. ω n \omega_{n}
  6. p n p_{n}
  7. q n q_{n}
  8. X X
  9. κ n \kappa_{n}
  10. Y Y
  11. [ , ] [\cdot,\cdot]
  12. { , } \{\cdot,\cdot\}
  13. f f
  14. f ( t ) n κ n 2 cos ( ω n t ) , f(t)\equiv\sum_{n}\kappa_{n}^{2}\cos(\omega_{n}t)\,,
  15. ξ \xi
  16. ξ ( t ) i n κ n ω n 2 ( - a n ( t 0 ) e - i ω n ( t - t 0 ) + a n ( t 0 ) e i ω n ( t - t 0 ) ) , \xi(t)\equiv i\sum_{n}\kappa_{n}\sqrt{\frac{\hbar\omega_{n}}{2}}\left(-a_{n}(t% _{0})e^{-i\omega_{n}(t-t_{0})}+a^{\dagger}_{n}(t_{0})e^{i\omega_{n}(t-t_{0})}% \right)\,,
  17. a n a_{n}
  18. a n ω n q n + i p n 2 ω n . a_{n}\equiv\frac{\omega_{n}q_{n}+ip_{n}}{\sqrt{2\hbar\omega_{n}}}\,.
  19. H = H sys + H B + H int H=H_{\mathrm{sys}}+H_{B}+H_{\mathrm{int}}
  20. H B = - d ω ω b ( ω ) b ( ω ) , H_{B}=\hbar\int_{-\infty}^{\infty}\mathrm{d}\omega\,\omega b^{\dagger}(\omega)% b(\omega)\,,
  21. H int = i - d ω κ ( ω ) ( b ( ω ) c - c b ( ω ) ) , H_{\mathrm{int}}=i\hbar\int_{-\infty}^{\infty}\mathrm{d}\omega\,\kappa(\omega)% \left(b^{\dagger}(\omega)c-c^{\dagger}b(\omega)\right)\,,
  22. b ( ω ) b(\omega)
  23. [ b ( ω ) , b ( ω ) ] = δ ( ω - ω ) [b(\omega),b^{\dagger}(\omega^{\prime})]=\delta(\omega-\omega^{\prime})
  24. c c
  25. κ ( ω ) \kappa(\omega)
  26. H sys H_{\mathrm{sys}}
  27. ω \omega
  28. - -\infty
  29. κ ( ω ) = γ 2 π . \kappa(\omega)=\sqrt{\frac{\gamma}{2\pi}}\,.
  30. t t
  31. b in ( t ) = 1 2 π - d ω e - i ω ( t - t 0 ) b 0 ( ω ) , b_{\mathrm{in}}(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{d}% \omega\,e^{-i\omega(t-t_{0})}b_{0}(\omega)\,,
  32. b 0 ( ω ) = b ( ω ) | t = t 0 b_{0}(\omega)=\left.b(\omega)\right|_{t=t_{0}}
  33. [ b in ( t ) , b in ( t ) ] = δ ( t - t ) [b_{\mathrm{in}}(t),b_{\mathrm{in}}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})
  34. a a
  35. b in ( t ) b in ( t ) ρ in = N δ ( t - t ) . \langle b^{\dagger}_{\mathrm{in}}(t)b_{\mathrm{in}}(t^{\prime})\rangle_{\rho_{% \mathrm{in}}}=N\delta(t-t^{\prime})\,.
  36. B ( t , t 0 ) = t 0 t b in ( t ) d t . B(t,t_{0})=\int_{t_{0}}^{t}b_{\mathrm{in}}(t^{\prime})\mathrm{d}t^{\prime}\,.
  37. [ B ( t , t 0 ) , B ( t , t 0 ) ] = t - t 0 [B(t,t_{0}),B^{\dagger}(t,t_{0})]=t-t_{0}
  38. B ( t , t 0 ) B ( t , t 0 ) ρ ( t , t 0 ) = N ( t - t 0 ) . \langle B^{\dagger}(t,t_{0})B(t,t_{0})\rangle_{\rho(t,t_{0})}=N(t-t_{0})\,.
  39. ρ ( t , t 0 ) = ( 1 - e - κ ) exp [ - κ B ( t , t 0 ) B ( t , t 0 ) t - t 0 ] , \rho(t,t_{0})=(1-e^{-\kappa})\exp\left[-\frac{\kappa B^{\dagger}(t,t_{0})B(t,t% _{0})}{t-t_{0}}\right]\,,
  40. N = 1 / ( e κ - 1 ) N=1/(e^{\kappa}-1)
  41. g ( t ) g(t)
  42. ( 𝐈 ) t 0 t g ( t ) d B ( t ) = lim n i = 1 n g ( t i ) ( B ( t i + 1 , t 0 ) - B ( t i , t 0 ) ) , (\mathbf{I})\int_{t_{0}}^{t}g(t^{\prime})\mathrm{d}B(t^{\prime})=\lim_{n\to% \infty}\sum_{i=1}^{n}g(t_{i})\left(B(t_{i+1},t_{0})-B(t_{i},t_{0})\right)\,,
  43. d B \mathrm{d}B
  44. d B \mathrm{d}B^{\dagger}
  45. ( 𝐈 ) d a = - i [ a , H sys ] d t + γ ( ( N + 1 ) 𝒟 [ c ] a + N 𝒟 [ c ] a ) d t - γ ( [ a , c ] d B ( t ) - d B ( t ) [ a , c ] ) , (\mathbf{I})\,\mathrm{d}a=-\frac{i}{\hbar}[a,H_{\mathrm{sys}}]\mathrm{d}t+% \gamma\left((N+1)\mathcal{D}[c^{\dagger}]a+N\mathcal{D}[c]a\right)\mathrm{d}t-% \sqrt{\gamma}\left([a,c^{\dagger}]\mathrm{d}B(t)-\mathrm{d}B^{\dagger}(t)[a,c]% \right)\,,
  46. 𝒟 [ A ] a A a A - 1 2 ( A A a + a A A ) . \mathcal{D}[A]a\equiv AaA^{\dagger}-\frac{1}{2}\left(A^{\dagger}Aa+aA^{\dagger% }A\right)\,.
  47. a a
  48. g ( t ) g(t)
  49. ( 𝐒 ) t 0 t g ( t ) d B ( t ) = lim n i = 1 n g ( t i ) + g ( t i + 1 ) 2 ( B ( t i + 1 , t 0 ) - B ( t i , t 0 ) ) , (\mathbf{S})\int_{t_{0}}^{t}g(t^{\prime})\mathrm{d}B(t^{\prime})=\lim_{n\to% \infty}\sum_{i=1}^{n}\frac{g(t_{i})+g(t_{i+1})}{2}\left(B(t_{i+1},t_{0})-B(t_{% i},t_{0})\right)\,,
  50. ( 𝐒 ) t 0 t g ( t ) d B ( t ) - ( 𝐒 ) t 0 t d B ( t ) g ( t ) = γ 2 t 0 t d t [ g ( t ) , c ( t ) ] . (\mathbf{S})\int_{t_{0}}^{t}g(t^{\prime})\mathrm{d}B(t^{\prime})-(\mathbf{S})% \int_{t_{0}}^{t}\mathrm{d}B(t^{\prime})g(t^{\prime})=\frac{\sqrt{\gamma}}{2}% \int_{t_{0}}^{t}\mathrm{d}t^{\prime}\,[g(t^{\prime}),c(t^{\prime})]\,.
  51. ( 𝐒 ) d a = - i [ a , H sys ] d t - γ 2 ( [ a , c ] c - c [ a , c ] ) d t - γ ( [ a , c ] d B ( t ) - d B ( t ) [ a , c ] ) . (\mathbf{S})\,\mathrm{d}a=-\frac{i}{\hbar}[a,H_{\mathrm{sys}}]\mathrm{d}t-% \frac{\gamma}{2}\left([a,c^{\dagger}]c-c^{\dagger}[a,c]\right)\mathrm{d}t-% \sqrt{\gamma}\left([a,c^{\dagger}]\mathrm{d}B(t)-\mathrm{d}B^{\dagger}(t)[a,c]% \right)\,.
  52. a a
  53. N N
  54. ( 𝐒 ) t 0 t g ( t ) d B ( t ) = ( 𝐈 ) t 0 t g ( t ) d B ( t ) + 1 2 γ N t 0 t d t [ g ( t ) , c ( t ) ] . (\mathbf{S})\int_{t_{0}}^{t}g(t^{\prime})\mathrm{d}B(t^{\prime})=(\mathbf{I})% \int_{t_{0}}^{t}g(t^{\prime})\mathrm{d}B(t^{\prime})+\frac{1}{2}\sqrt{\gamma}N% \int_{t_{0}}^{t}\mathrm{d}t^{\prime}\,[g(t^{\prime}),c(t^{\prime})]\,.
  55. ( 𝐈 ) d ( a b ) = a d b + b d a + d a d b , (\mathbf{I})\,\mathrm{d}(ab)=a\,\mathrm{d}b+b\,\mathrm{d}a+\mathrm{d}a\,% \mathrm{d}b\,,
  56. ( 𝐒 ) d ( a b ) = a d b + d a b . (\mathbf{S})\,\mathrm{d}(ab)=a\,\mathrm{d}b+\mathrm{d}a\,b\,.
  57. ρ ˙ = 𝒟 [ c ] ρ - i [ H sys , ρ ] . \dot{\rho}=\mathcal{D}[c]\rho-i[H_{\mathrm{sys}},\rho]\,.
  58. d ρ I ( t ) = ( d N ( t ) 𝒢 [ c ] - d t [ i H sys + 1 2 c c ] ) ρ I ( t ) , \mathrm{d}\rho_{I}(t)=\left(\mathrm{d}N(t)\mathcal{G}[c]-\mathrm{d}t\mathcal{H% }[iH_{\mathrm{sys}}+\frac{1}{2}c^{\dagger}c]\right)\rho_{I}(t)\,,
  59. 𝒢 [ r ] ρ r ρ r Tr [ r ρ r ] - ρ [ r ] ρ r ρ + ρ r - Tr [ r ρ + ρ r ] ρ \begin{array}[]{rcl}\mathcal{G}[r]\rho&\equiv&\frac{r\rho r^{\dagger}}{% \operatorname{Tr}[r\rho r^{\dagger}]}-\rho\\ \mathcal{H}[r]\rho&\equiv&r\rho+\rho r^{\dagger}-\operatorname{Tr}[r\rho+\rho r% ^{\dagger}]\rho\end{array}
  60. N ( t ) N(t)
  61. t t
  62. E [ d N ( t ) ] = d t Tr [ c c ρ I ( t ) ] , \operatorname{E}[\mathrm{d}N(t)]=\mathrm{d}t\operatorname{Tr}[c^{\dagger}c\rho% _{I}(t)]\,,
  63. E [ ] \operatorname{E}[\cdot]
  64. d ρ J ( t ) = - i [ H sys , ρ J ( t ) ] d t + d t 𝒟 [ c ] ρ J ( t ) + d W ( t ) [ c ] ρ J ( t ) , \mathrm{d}\rho_{J}(t)=-i[H_{\mathrm{sys}},\rho_{J}(t)]\mathrm{d}t+\mathrm{d}t% \mathcal{D}[c]\rho_{J}(t)+\mathrm{d}W(t)\mathcal{H}[c]\rho_{J}(t)\,,
  65. d W ( t ) \mathrm{d}W(t)
  66. d W ( t ) 2 = d t E [ d W ( t ) ] = 0 . \begin{array}[]{rcl}\mathrm{d}W(t)^{2}&=&\mathrm{d}t\\ \operatorname{E}[\mathrm{d}W(t)]&=&0\,.\end{array}
  67. E [ d ρ I ( t ) ] = E [ d ρ J ( t ) ] = ρ ˙ d t . \operatorname{E}[\mathrm{d}\rho_{I}(t)]=\operatorname{E}[\mathrm{d}\rho_{J}(t)% ]=\dot{\rho}\mathrm{d}t\,.

Quantum_thermodynamics.html

  1. E = h ν E=h\nu
  2. H = H S + H B + H S B H=H_{S}+H_{B}+H_{SB}
  3. H S H_{S}
  4. H B H_{B}
  5. H S B H_{SB}
  6. ρ S ( t ) = T r B ( ρ S B ( t ) ) \rho_{S}(t)=Tr_{B}(\rho_{SB}(t))
  7. ρ ˙ S = - i [ H S , ρ S ] + L D ( ρ S ) \dot{\rho}_{S}=-{i\over\hbar}[H_{S},\rho_{S}]+L_{D}(\rho_{S})
  8. H S H_{S}
  9. L D L_{D}
  10. L D ( ρ S ) = n ( V n ρ S V n - 1 2 ( ρ S V n V n + V n V n ρ S ) ) L_{D}(\rho_{S})=\sum_{n}\left(V_{n}\rho_{S}V_{n}^{\dagger}-\frac{1}{2}\left(% \rho_{S}V_{n}^{\dagger}V_{n}+V_{n}^{\dagger}V_{n}\rho_{S}\right)\right)
  11. V n V_{n}
  12. ρ S B = ρ s ρ B \rho_{SB}=\rho_{s}\otimes\rho_{B}
  13. ρ S \rho_{S}
  14. ρ ˙ S ( t ) = 0 \dot{\rho}_{S}(t\rightarrow\infty)=0
  15. O O
  16. d O d t = i [ H S , O ] + L D * ( O ) + O t \frac{dO}{dt}=\frac{i}{\hbar}[H_{S},O]+L_{D}^{*}(O)+\frac{\partial O}{\partial t}
  17. O O
  18. O = H S O=H_{S}
  19. d E d t = H S t + L D * ( H S ) \frac{dE}{dt}=\langle\frac{\partial H_{S}}{\partial t}\rangle+\langle L_{D}^{*% }(H_{S})\rangle
  20. P = H S t P=\langle\frac{\partial H_{S}}{\partial t}\rangle
  21. J = L D * ( H S ) J=\langle L_{D}^{*}(H_{S})\rangle
  22. L D L_{D}
  23. ρ S ( ) \rho_{S}(\infty)
  24. L D L_{D}
  25. H S H_{S}
  26. L D L_{D}
  27. H S H_{S}
  28. L D L_{D}
  29. A \langle A\rangle
  30. A A
  31. A = j α i P j A=\sum_{j}\alpha_{i}P_{j}
  32. P j P_{j}
  33. α j \alpha_{j}
  34. p j = T r ( ρ P j ) p_{j}=Tr(\rho P_{j})
  35. A \langle A\rangle
  36. S A = - j p j ln p j S_{A}=-\sum_{j}p_{j}\ln p_{j}
  37. H H
  38. S E S_{E}
  39. S v n = - T r ( ρ ln ρ ) S_{vn}=-Tr(\rho\ln\rho)
  40. S A S v n S_{A}\geq S_{vn}
  41. S E = S v n S_{E}=S_{vn}
  42. S v n S_{vn}
  43. S v n S_{vn}
  44. ρ = Π j ρ j \rho=\Pi_{j}\otimes\rho_{j}
  45. n J n T n 0 \sum_{n}\frac{J_{n}}{T_{n}}\geq 0
  46. T r ( L D ρ [ ln ρ - ln ρ ( ) ] ) 0 Tr\left(L_{D}\rho[\ln\rho-\ln\rho(\infty)]\right)\geq 0
  47. ρ ( ) \rho(\infty)
  48. H ( t ) H(t)
  49. S v n S_{vn}
  50. S E S_{E}
  51. [ H ( t ) , H ( t ) ] = 0 [H(t),H(t^{\prime})]=0
  52. T c 0 T_{c}\rightarrow 0
  53. S ˙ c - T c α , α 0 . \dot{S}_{c}\propto-T_{c}^{\alpha}~{}~{}~{},~{}~{}~{}~{}\alpha\geq 0~{}~{}.
  54. α = 0 \alpha=0
  55. α 0 \alpha\geq 0
  56. α > 0 \alpha>0
  57. S ˙ c = 0 \dot{S}_{c}=0
  58. J c T c α + 1 {J}_{c}\propto T_{c}^{\alpha+1}
  59. J c ( T c ( t ) ) = - c V ( T c ( t ) ) d T c ( t ) d t . {J}_{c}(T_{c}(t))=-c_{V}(T_{c}(t))\frac{dT_{c}(t)}{dt}~{}~{}.
  60. c V ( T c ) c_{V}(T_{c})
  61. J c T c α + 1 {J}_{c}\propto T_{c}^{\alpha+1}
  62. c V T c η c_{V}\sim T_{c}^{\eta}
  63. η 0 {\eta}\geq 0
  64. ζ \zeta
  65. d T c ( t ) d t - T c ζ , T c 0 , ζ = α - η + 1 \frac{dT_{c}(t)}{dt}\propto-T_{c}^{\zeta},~{}~{}~{}~{}~{}T_{c}\rightarrow 0,~{% }~{}~{}~{}~{}{\zeta=\alpha-\eta+1}
  66. ζ \zeta
  67. α \alpha
  68. ζ < 0 \zeta<0
  69. | ψ t |\psi_{t}

Quarter_5-cubic_honeycomb.html

  1. D ~ 5 {\tilde{D}}_{5}

Quarter_6-cubic_honeycomb.html

  1. D ~ 6 {\tilde{D}}_{6}

Quarter_7-cubic_honeycomb.html

  1. D ~ 7 {\tilde{D}}_{7}

Quarter_hypercubic_honeycomb.html

  1. D ~ n - 1 {\tilde{D}}_{n-1}
  2. D ~ 4 {\tilde{D}}_{4}
  3. A ~ 4 {\tilde{A}}_{4}
  4. D ~ 5 {\tilde{D}}_{5}
  5. B ~ 5 {\tilde{B}}_{5}

Quasi-commutative_property.html

  1. p q = q p pq=qp
  2. x y - y x = z xy-yx=z
  3. x z = z x xz=zx
  4. y z = z y yz=zy
  5. f : X × Y X f:X\times Y\rightarrow X
  6. x X x\in X
  7. y 1 , y 2 Y y_{1},y_{2}\in Y
  8. f ( f ( x , y 1 ) , y 2 ) = f ( f ( x , y 2 ) , y 1 ) f(f(x,y_{1}),y_{2})=f(f(x,y_{2}),y_{1})

Quasi-fibration.html

  1. f : X Y f:X\to Y
  2. f - 1 ( y ) f^{-1}(y)

Quickprop.html

  1. Δ ( k ) w i j = Δ ( k - 1 ) w i j ( i j E ( k ) i j E ( k - 1 ) - i j E ( k ) ) \Delta^{(k)}\,w_{ij}=\Delta^{(k-1)}\,w_{ij}\left(\frac{\nabla_{ij}\,E^{(k)}}{% \nabla_{ij}\,E^{(k-1)}-\nabla_{ij}\,E^{(k)}}\right)
  2. w i j w_{ij}

Quillen's_theorems_A_and_B.html

  1. B f : B C B D Bf:BC\to BD
  2. F f Ff
  3. F B f = B F f FBf=BFf

Quot_scheme.html

  1. Quot F ( X ) \operatorname{Quot}_{F}(X)
  2. Quot F ( X ) ( T ) = Mor S ( T , Quot F ( X ) ) \operatorname{Quot}_{F}(X)(T)=\operatorname{Mor}_{S}(T,\operatorname{Quot}_{F}% (X))
  3. F × S T F\times_{S}T
  4. 𝒪 X \mathcal{O}_{X}

Quotient_by_an_equivalence_relation.html

  1. f : R X × X f:R\to X\times X
  2. R 𝑓 X × X pr i X , i = 1 , 2 , R\overset{f}{\to}X\times X\overset{\operatorname{pr}_{i}}{\to}X,\,i=1,2,
  3. f : R ( T ) = Mor ( T , R ) X ( T ) × X ( T ) f:R(T)=\operatorname{Mor}(T,R)\to X(T)\times X(T)
  4. ( x , y ) (x,y)
  5. ( y , x ) (y,x)
  6. q : X Q q:X\to Q
  7. q : Z Q q:Z\to Q

Quotient_stack.html

  1. [ X / G ] [X/G]
  2. X / G X/G
  3. [ X / G ] X / G [X/G]\to X/G
  4. X / G X/G
  5. [ X / G ] [X/G]
  6. X = S X=S
  7. [ S / G ] [S/G]
  8. L = π * MU L=\pi_{*}\operatorname{MU}
  9. [ Spec L / G ] [\operatorname{Spec}L/G]
  10. G G
  11. G ( R ) = { g R [ [ t ] ] | g ( t ) = b 0 t + b 1 t 2 , b 0 R × } G(R)=\{g\in R[\![t]\!]|g(t)=b_{0}t+b_{1}t^{2}\dots,b_{0}\in R^{\times}\}
  12. FG \mathcal{M}\text{FG}
  13. T P X X / G T\leftarrow P\to X\to X/G