wpmath0000016_3

Derived_algebraic_geometry.html

  1. n \mathcal{E}_{n}

Derived_scheme.html

  1. ( X , 𝒪 ) (X,\mathcal{O})
  2. 𝒪 \mathcal{O}
  3. ( X , π 0 𝒪 ) (X,\pi_{0}\mathcal{O})
  4. π k 𝒪 \pi_{k}\mathcal{O}
  5. π 0 𝒪 \pi_{0}\mathcal{O}
  6. E E_{\infty}

Descartes_number.html

  1. 22021 22021
  2. D D
  3. σ ( D ) = ( 3 2 + 3 + 1 ) ( 7 2 + 7 + 1 ) ( 11 2 + 11 + 1 ) ( 13 2 + 13 + 1 ) ( 22021 + 1 ) = ( 13 ) ( 3 19 ) ( 7 19 ) ( 3 61 ) ( 22 1001 ) = 3 2 7 13 19 2 61 ( 22 7 11 13 ) = 2 ( 3 2 7 2 11 2 13 2 ) ( 19 2 61 ) = 2 ( 3 2 7 2 11 2 13 2 ) 22021 = 2 D , \begin{aligned}\displaystyle\sigma(D)&\displaystyle=(3^{2}+3+1)\cdot(7^{2}+7+1% )\cdot(11^{2}+11+1)\cdot(13^{2}+13+1)\cdot(22021+1)=(13)\cdot(3\cdot 19)\cdot(% 7\cdot 19)\cdot(3\cdot 61)\cdot(22\cdot 1001)\\ &\displaystyle=3^{2}\cdot 7\cdot 13\cdot 19^{2}\cdot 61\cdot(22\cdot 7\cdot 11% \cdot 13)=2\cdot(3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2})\cdot(19^{2}\cdot 61% )=2\cdot(3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2})\cdot 22021=2D,\end{aligned}
  4. 19 < s u p > 2 61 = 22021 19<sup>2⋅61=22021

Descendant_tree_(group_theory).html

  1. p n p^{n}
  2. p p
  3. n 0 n\geq 0
  4. p n p^{n}
  5. c c
  6. r = n - c r=n-c
  7. r r
  8. π ( G ) \pi(G)
  9. G G
  10. π ( G ) = G / N \pi(G)=G/N
  11. G G
  12. N G N\triangleleft G
  13. N = ζ 1 ( G ) N=\zeta_{1}(G)
  14. G G
  15. π ( G ) = G / ζ 1 ( G ) \pi(G)=G/\zeta_{1}(G)
  16. G G
  17. N = γ c ( G ) N=\gamma_{c}(G)
  18. G G
  19. c c
  20. G G
  21. N = P c - 1 ( G ) N=P_{c-1}(G)
  22. G G
  23. c c
  24. G G
  25. N = G ( d - 1 ) N=G^{(d-1)}
  26. G G
  27. d d
  28. G G
  29. G G
  30. π ( G ) \pi(G)
  31. G π ( G ) G\to\pi(G)
  32. π : G π ( G ) \pi:G\to\pi(G)
  33. π ( G ) = G / N \pi(G)=G/N
  34. π ( G ) G \pi(G)\to G
  35. R R
  36. P P
  37. P P
  38. R R
  39. R R
  40. P P
  41. R = Q 0 Q 1 Q m - 1 Q m = P R=Q_{0}\to Q_{1}\to\cdots\to Q_{m-1}\to Q_{m}=P
  42. m 1 m\geq 1
  43. R R
  44. P P
  45. Q j = π j ( R ) Q_{j}=\pi^{j}(R)
  46. R R
  47. 0 j m 0\leq j\leq m
  48. R / γ c + 1 - j ( R ) R/\gamma_{c+1-j}(R)
  49. c - j c-j
  50. R R
  51. R R
  52. c m c\geq m
  53. 𝒯 ( G ) \mathcal{T}(G)
  54. G G
  55. G G
  56. G G
  57. 𝒯 ( 1 ) \mathcal{T}(1)
  58. 1 1
  59. 1 1
  60. p p
  61. γ j ( S ) \gamma_{j}(S)
  62. j 1 j\geq 1
  63. S S
  64. S / γ j ( S ) S/\gamma_{j}(S)
  65. S S
  66. cc ( S ) = r \mathrm{cc}(S)=r
  67. r = lim j cc ( S / γ j ( S ) ) r=\lim_{j\to\infty}\,\mathrm{cc}(S/\gamma_{j}(S))
  68. S S
  69. r r
  70. T T
  71. p \mathbb{Z}_{p}
  72. d d
  73. P = S / T P=S/T
  74. T T
  75. d = ( p - 1 ) p s d=(p-1)p^{s}
  76. 0 s < r 0\leq s<r
  77. r r
  78. r r
  79. p p
  80. r r
  81. S S
  82. r r
  83. i 1 i\geq 1
  84. j i j\geq i
  85. cc ( S / γ j ( S ) ) = r \mathrm{cc}(S/\gamma_{j}(S))=r
  86. S / γ j ( S ) S/\gamma_{j}(S)
  87. r r
  88. S S
  89. γ j / γ j + 1 ( S ) \gamma_{j}/\gamma_{j+1}(S)
  90. p p
  91. 𝒯 ( R ) \mathcal{T}(R)
  92. R = S / γ i ( S ) R=S/\gamma_{i}(S)
  93. i i
  94. 𝒯 ( S ) \mathcal{T}(S)
  95. S S
  96. R = S / γ i ( S ) S / γ i + 1 ( S ) R=S/\gamma_{i}(S)\leftarrow S/\gamma_{i+1}(S)\leftarrow\cdots
  97. 𝒯 ( R ) \mathcal{T}(R)
  98. R = R 0 R=R_{0}
  99. ( R n ) n 0 (R_{n})_{n\geq 0}
  100. n n
  101. ( n ) = 𝒯 ( R n ) 𝒯 ( R n + 1 ) \mathcal{B}(n)=\mathcal{T}(R_{n})\setminus\mathcal{T}(R_{n+1})
  102. ( R n ) \mathcal{B}(R_{n})
  103. R n R_{n}
  104. n 0 n\geq 0
  105. k 0 k\geq 0
  106. ( n ) \mathcal{B}(n)
  107. k ( n ) \mathcal{B}_{k}(n)
  108. 𝒯 k ( R ) \mathcal{T}_{k}(R)
  109. 𝒯 ( R ) \mathcal{T}(R)
  110. ( k ( n ) ) n 0 (\mathcal{B}_{k}(n))_{n\geq 0}
  111. ( ( n ) ) n 0 (\mathcal{B}(n))_{n\geq 0}
  112. R n R_{n}
  113. S S
  114. r 1 r\geq 1
  115. d d
  116. k 1 k\geq 1
  117. f ( k ) 1 f(k)\geq 1
  118. d d
  119. 𝒯 ( S ) \mathcal{T}(S)
  120. k ( n + d ) k ( n ) \mathcal{B}_{k}(n+d)\simeq\mathcal{B}_{k}(n)
  121. n f ( k ) n\geq f(k)
  122. P = R f ( k ) P=R_{f(k)}
  123. k k
  124. G G
  125. cc ( G ) = r \mathrm{cc}(G)=r
  126. 𝒯 ( G ) \mathcal{T}(G)
  127. r r
  128. 𝒯 r ( G ) \mathcal{T}^{r}(G)
  129. r r
  130. G G
  131. 𝒯 ( G ) = 𝒯 r ( G ) \mathcal{T}(G)=\mathcal{T}^{r}(G)
  132. ν ( G ) \nu(G)
  133. G G
  134. G G
  135. ν ( G ) = 0 \nu(G)=0
  136. ν ( G ) = 1 \nu(G)=1
  137. G G
  138. G G
  139. ν ( G ) = m 2 \nu(G)=m\geq 2
  140. G G
  141. G G
  142. r r
  143. ν ( G ) = m 2 \nu(G)=m\geq 2
  144. 𝒯 r ( G ) \mathcal{T}^{r}(G)
  145. m - 1 m-1
  146. 𝒯 r + j ( G ) \mathcal{T}^{r+j}(G)
  147. r + j r+j
  148. 1 j m - 1 1\leq j\leq m-1
  149. G G
  150. 𝒯 ( G ) = ˙ j = 0 m - 1 𝒯 r + j ( G ) \mathcal{T}(G)=\dot{\cup}_{j=0}^{m-1}\,\mathcal{T}^{r+j}(G)
  151. c = cl ( Q ) = cl ( P ) + 1 c=\mathrm{cl}(Q)=\mathrm{cl}(P)+1
  152. P = π ( Q ) P=\pi(Q)
  153. Q Q
  154. r = cc ( Q ) = cc ( P ) r=\mathrm{cc}(Q)=\mathrm{cc}(P)
  155. | γ c ( Q ) | = p |\gamma_{c}(Q)|=p
  156. Q Q
  157. P Q P\leftarrow Q
  158. m - 1 m-1
  159. | γ c ( Q ) | = p m |\gamma_{c}(Q)|=p^{m}
  160. m 2 m\geq 2
  161. Q Q
  162. m m
  163. 𝒯 ( 1 ) \mathcal{T}(1)
  164. 1 1
  165. ˙ r = 0 𝒢 ( p , r ) \dot{\cup}_{r=0}^{\infty}\,\mathcal{G}(p,r)
  166. 𝒢 ( p , r ) \mathcal{G}(p,r)
  167. 𝒢 ( p , r ) = ( ˙ i 𝒯 ( S i ) ) ˙ 𝒢 0 ( p , r ) \mathcal{G}(p,r)=\left(\dot{\cup}_{i}\,\mathcal{T}(S_{i})\right)\dot{\cup}% \mathcal{G}_{0}(p,r)
  168. 𝒯 ( S i ) \mathcal{T}(S_{i})
  169. S i S_{i}
  170. r r
  171. 𝒢 0 ( p , r ) \mathcal{G}_{0}(p,r)
  172. order , counting number \langle\,\text{order},\,\text{counting number}\rangle
  173. counting number \langle\,\text{counting number}\rangle
  174. p p
  175. 512 = 2 9 512=2^{9}
  176. p = 2 p=2
  177. 2187 = 3 7 2187=3^{7}
  178. p = 3 p=3
  179. 1 1
  180. P P
  181. P - # 1 ; counting number P-\#1;\,\text{counting number}
  182. d 2 d\geq 2
  183. P P
  184. P - # d ; counting number P-\#d;\,\text{counting number}
  185. 𝒢 ( p , 0 ) = 𝒢 0 ( p , 0 ) \mathcal{G}(p,0)=\mathcal{G}_{0}(p,0)
  186. 0
  187. 1 1
  188. C p C_{p}
  189. p p
  190. p = 2 p=2
  191. C p C_{p}
  192. 2 , 1 \langle 2,1\rangle
  193. p = 3 p=3
  194. 3 , 1 \langle 3,1\rangle
  195. 𝒢 ( p , 1 ) = 𝒯 1 ( R ) ˙ 𝒢 0 ( p , 1 ) \mathcal{G}(p,1)=\mathcal{T}^{1}(R)\dot{\cup}\mathcal{G}_{0}(p,1)
  196. 1 1
  197. R = C p × C p R=C_{p}\times C_{p}
  198. 2 2
  199. 1 1
  200. 2 2
  201. C p 2 C_{p^{2}}
  202. p 2 p^{2}
  203. 𝒢 0 ( p , 1 ) \mathcal{G}_{0}(p,1)
  204. 𝒯 1 ( R ) = 𝒯 1 ( S 1 ) \mathcal{T}^{1}(R)=\mathcal{T}^{1}(S_{1})
  205. S 1 S_{1}
  206. 1 1
  207. p = 2 p=2
  208. p = 3 p=3
  209. R R
  210. 4 , 2 \langle 4,2\rangle
  211. 9 , 2 \langle 9,2\rangle
  212. ( 2 ) \mathcal{B}(2)
  213. ( 7 ) \mathcal{B}(7)
  214. p 3 p^{3}
  215. 2 2
  216. p p
  217. 𝒢 ( 2 , 1 ) \mathcal{G}(2,1)
  218. 𝒢 ( 3 , 1 ) \mathcal{G}(3,1)
  219. 8 , 3 \langle 8,3\rangle
  220. 1 1
  221. ( 3 ) \mathcal{B}(3)
  222. 81 , 9 \langle 81,9\rangle
  223. 2 2
  224. ( 4 ) \mathcal{B}(4)
  225. 1 1
  226. 𝒢 ( p , 1 ) \mathcal{G}(p,1)
  227. p 5 p\geq 5
  228. 𝒢 ( p , 1 ) \mathcal{G}(p,1)
  229. p 7 p\geq 7
  230. 𝒢 ( 2 , 1 ) \mathcal{G}(2,1)
  231. 𝒢 ( 3 , 1 ) \mathcal{G}(3,1)
  232. G G
  233. x , y x,y
  234. s j s_{j}
  235. 2 j n - 1 = cl ( G ) 2\leq j\leq n-1=\mathrm{cl}(G)
  236. s 2 = [ y , x ] s_{2}=[y,x]
  237. s n = 1 s_{n}=1
  238. | G | = p n |G|=p^{n}
  239. p = 2 p=2
  240. 0 w , z 1 0\leq w,z\leq 1
  241. G n ( z , w ) = x , y , s 2 , , s n - 1 x 2 = s n - 1 w , y 2 = s 2 - 1 s n - 1 z , [ s 2 , y ] = 1 , s 2 = [ y , x ] , s j = [ s j - 1 , x ] for 3 j n - 1 \begin{aligned}\displaystyle G^{n}(z,w)=&\displaystyle\langle x,y,s_{2},\ldots% ,s_{n-1}\mid\\ &\displaystyle x^{2}=s_{n-1}^{w},\ y^{2}=s_{2}^{-1}s_{n-1}^{z},\ [s_{2},y]=1,% \\ &\displaystyle s_{2}=[y,x],\ s_{j}=[s_{j-1},x]\,\text{ for }3\leq j\leq n-1% \rangle\end{aligned}
  242. 1 1
  243. D ( 2 n ) = G n ( 0 , 0 ) D(2^{n})=G^{n}(0,0)
  244. n 3 n\geq 3
  245. Q ( 2 n ) = G n ( 0 , 1 ) Q(2^{n})=G^{n}(0,1)
  246. n 3 n\geq 3
  247. S ( 2 n ) = G n ( 1 , 0 ) S(2^{n})=G^{n}(1,0)
  248. n 4 n\geq 4
  249. p = 3 p=3
  250. 0 a 1 0\leq a\leq 1
  251. - 1 w , z 1 -1\leq w,z\leq 1
  252. G a n ( z , w ) = x , y , s 2 , , s n - 1 x 3 = s n - 1 w , y 3 = s 2 - 3 s 3 - 1 s n - 1 z , [ y , s 2 ] = s n - 1 a , s 2 = [ y , x ] , s j = [ s j - 1 , x ] for 3 j n - 1 \begin{aligned}\displaystyle G^{n}_{a}(z,w)=&\displaystyle\langle x,y,s_{2},% \ldots,s_{n-1}\mid\\ &\displaystyle x^{3}=s_{n-1}^{w},\ y^{3}=s_{2}^{-3}s_{3}^{-1}s_{n-1}^{z},\ [y,% s_{2}]=s_{n-1}^{a},\\ &\displaystyle s_{2}=[y,x],\ s_{j}=[s_{j-1},x]\,\text{ for }3\leq j\leq n-1% \rangle\end{aligned}
  253. a = 0 a=0
  254. a = 1 a=1
  255. G 0 3 ( 0 , 0 ) G^{3}_{0}(0,0)
  256. G 0 3 ( 0 , 1 ) G^{3}_{0}(0,1)
  257. r 2 r\geq 2
  258. 𝒢 ( p , 1 ) \mathcal{G}(p,1)
  259. G G
  260. G / G G/G^{\prime}
  261. ( p , p ) (p,p)
  262. p = 2 p=2
  263. 2 2
  264. ( 2 , 2 ) (2,2)
  265. 1 1
  266. 𝒢 ( p , r ) \mathcal{G}(p,r)
  267. r 2 r\geq 2
  268. r = 2 r=2
  269. G G
  270. G / G G/G^{\prime}
  271. ( p , p ) (p,p)
  272. ( p 2 , p ) (p^{2},p)
  273. ( p , p , p ) (p,p,p)
  274. p 3 p^{3}
  275. 2 2
  276. ( p 2 , p ) (p^{2},p)
  277. ( p , p , p ) (p,p,p)
  278. 2 2
  279. ( p , p ) (p,p)
  280. 1 1
  281. p = 2 p=2
  282. 8 , 3 \langle 8,3\rangle
  283. p 3 p\geq 3
  284. 2 2
  285. ( p , p ) (p,p)
  286. G 0 3 ( 0 , 0 ) G^{3}_{0}(0,0)
  287. 2 2
  288. 𝒯 ( G 0 3 ( 0 , 0 ) ) \mathcal{T}(G^{3}_{0}(0,0))
  289. 𝒯 1 ( G 0 3 ( 0 , 0 ) ) \mathcal{T}^{1}(G^{3}_{0}(0,0))
  290. 𝒯 1 ( C p × C p ) \mathcal{T}^{1}(C_{p}\times C_{p})
  291. 𝒢 ( p , 1 ) \mathcal{G}(p,1)
  292. 𝒯 2 ( G 0 3 ( 0 , 0 ) ) \mathcal{T}^{2}(G^{3}_{0}(0,0))
  293. 𝒢 = 𝒢 ( p , p ) ( p , 2 ) \mathcal{G}=\mathcal{G}_{(p,p)}(p,2)
  294. 𝒢 ( p , 2 ) \mathcal{G}(p,2)
  295. 2 2
  296. G 0 3 ( 0 , 0 ) G^{3}_{0}(0,0)
  297. p = 3 p=3
  298. 𝒢 \mathcal{G}
  299. 243 = 3 5 243=3^{5}
  300. 243 , 5 \langle 243,5\rangle
  301. 243 , 7 \langle 243,7\rangle
  302. 𝒢 0 ( 3 , 2 ) \mathcal{G}_{0}(3,2)
  303. 𝒢 ( 3 , 2 ) \mathcal{G}(3,2)
  304. G = 243 , 4 G=\langle 243,4\rangle
  305. G = 243 , 9 G=\langle 243,9\rangle
  306. 𝒯 2 ( G ) \mathcal{T}^{2}(G)
  307. 𝒢 0 ( 3 , 2 ) \mathcal{G}_{0}(3,2)
  308. 𝒯 ( G ) \mathcal{T}(G)
  309. 243 , 3 \langle 243,3\rangle
  310. 243 , 6 \langle 243,6\rangle
  311. 243 , 8 \langle 243,8\rangle
  312. 𝒯 2 ( 729 , 40 ) \mathcal{T}^{2}(\langle 729,40\rangle)
  313. 𝒯 2 ( 243 , 6 ) \mathcal{T}^{2}(\langle 243,6\rangle)
  314. 𝒯 2 ( 243 , 8 ) \mathcal{T}^{2}(\langle 243,8\rangle)
  315. 𝒢 ( 3 , 2 ) \mathcal{G}(3,2)
  316. G G
  317. r ( G ) = dim 𝔽 p ( H 2 ( G , 𝔽 p ) ) r(G)=\mathrm{dim}_{\mathbb{F}_{p}}(\mathrm{H}^{2}(G,\mathbb{F}_{p}))
  318. d ( G ) = dim 𝔽 p ( H 1 ( G , 𝔽 p ) ) d(G)=\mathrm{dim}_{\mathbb{F}_{p}}(\mathrm{H}^{1}(G,\mathbb{F}_{p}))
  319. G G
  320. σ Aut ( G ) \sigma\in\mathrm{Aut}(G)
  321. x x - 1 x\mapsto x^{-1}
  322. G / G G/G^{\prime}
  323. G G
  324. G / G G/G^{\prime}
  325. 243 , 3 \langle 243,3\rangle
  326. 729 , 40 \langle 729,40\rangle
  327. 729 , 35 \langle 729,35\rangle
  328. 729 , 34 \langle 729,34\rangle
  329. 3 3
  330. 5 5
  331. ( f , g , h ) (f,g,h)
  332. G / G G/G^{\prime}
  333. G / γ 3 ( G ) G/\gamma_{3}(G)
  334. G / γ 4 ( G ) G/\gamma_{4}(G)
  335. G / γ 5 ( G ) G/\gamma_{5}(G)
  336. ( 0 , 1 , 0 ) (0,1,0)
  337. ( 3 , 3 ) (3,3)
  338. 27 , 3 \langle 27,3\rangle
  339. 243 , 3 \langle 243,3\rangle
  340. 729 , 40 \langle 729,40\rangle
  341. ( 0 , 1 , 2 ) (0,1,2)
  342. ( 3 , 3 ) (3,3)
  343. 27 , 3 \langle 27,3\rangle
  344. 243 , 6 \langle 243,6\rangle
  345. 729 , 49 \langle 729,49\rangle
  346. ( 1 , 1 , 2 ) (1,1,2)
  347. ( 3 , 3 ) (3,3)
  348. 27 , 3 \langle 27,3\rangle
  349. 243 , 8 \langle 243,8\rangle
  350. 729 , 54 \langle 729,54\rangle
  351. ( 1 , 0 , 0 ) (1,0,0)
  352. ( 9 , 3 ) (9,3)
  353. 81 , 3 \langle 81,3\rangle
  354. 243 , 15 \langle 243,15\rangle
  355. 729 , 79 \langle 729,79\rangle
  356. ( 0 , 0 , 1 ) (0,0,1)
  357. ( 9 , 3 ) (9,3)
  358. 81 , 3 \langle 81,3\rangle
  359. 243 , 17 \langle 243,17\rangle
  360. 729 , 84 \langle 729,84\rangle
  361. ( 0 , 0 , 0 ) (0,0,0)
  362. ( 3 , 3 , 3 ) (3,3,3)
  363. 81 , 12 \langle 81,12\rangle
  364. 243 , 53 \langle 243,53\rangle
  365. 729 , 395 \langle 729,395\rangle
  366. 2 2
  367. 3 3
  368. ( f , g , h ) (f,g,h)
  369. ( 3 , 3 ) (3,3)
  370. 𝒢 ( 3 , 2 ) \mathcal{G}(3,2)
  371. G ( f , g , h ) = a , t , z a 3 = z f , [ t , t a ] = z g , t 1 + a + a 2 = z h , z 3 = 1 , [ z , a ] = 1 , [ z , t ] = 1 \begin{aligned}\displaystyle G(f,g,h)=&\displaystyle\langle a,t,z\mid\\ &\displaystyle a^{3}=z^{f},\ [t,t^{a}]=z^{g},\ t^{1+a+a^{2}}=z^{h},\\ &\displaystyle z^{3}=1,\ [z,a]=1,\ [z,t]=1\rangle\end{aligned}
  372. p = 3 p=3
  373. 𝒯 2 ( 27 , 2 ) \mathcal{T}^{2}(\langle 27,2\rangle)
  374. 𝒢 ( 3 , 2 ) \mathcal{G}(3,2)
  375. 81 , 3 \langle 81,3\rangle
  376. 243 , 20 \langle 243,20\rangle
  377. 243 , 19 \langle 243,19\rangle
  378. 243 , 16 \langle 243,16\rangle
  379. 9 9
  380. 243 , 18 \langle 243,18\rangle
  381. ( 3 , 3 ) (3,3)
  382. G = 243 , 14 G=\langle 243,14\rangle
  383. 𝒯 ( G ) = 𝒯 2 ( G ) \mathcal{T}(G)=\mathcal{T}^{2}(G)
  384. 243 , 13 \langle 243,13\rangle
  385. 243 , 15 \langle 243,15\rangle
  386. 243 , 17 \langle 243,17\rangle
  387. 𝒯 2 ( 2187 , 319 ) \mathcal{T}^{2}(\langle 2187,319\rangle)
  388. 𝒯 2 ( 243 , 15 ) \mathcal{T}^{2}(\langle 243,15\rangle)
  389. 𝒯 2 ( 243 , 17 ) \mathcal{T}^{2}(\langle 243,17\rangle)
  390. 3 3
  391. ( 3 , 3 ) (3,3)
  392. 243 , 13 \langle 243,13\rangle
  393. 2187 , 319 \langle 2187,319\rangle
  394. 3 3
  395. p = 2 p=2
  396. p = 3 p=3
  397. ( p , p , p ) (p,p,p)
  398. 𝒢 ( p , 2 ) \mathcal{G}(p,2)
  399. ( p , p , p ) (p,p,p)
  400. 8 , 5 \langle 8,5\rangle
  401. 27 , 5 \langle 27,5\rangle
  402. # 59 \#59
  403. ( f , g , h ) = ( 0 , 0 , 0 ) (f,g,h)=(0,0,0)
  404. p = 2 p=2
  405. 𝒢 ( p , 3 ) \mathcal{G}(p,3)
  406. G G
  407. G / G G/G^{\prime}
  408. ( p 3 , p ) (p^{3},p)
  409. ( p 2 , p 2 ) (p^{2},p^{2})
  410. ( p 2 , p , p ) (p^{2},p,p)
  411. ( p , p , p , p ) (p,p,p,p)
  412. ( p , p ) (p,p)
  413. ( p 2 , p ) (p^{2},p)
  414. ( p , p , p ) (p,p,p)
  415. p 4 p^{4}
  416. C p × C p × C p C_{p}\times C_{p}\times C_{p}
  417. 3 3
  418. 8 , 5 \langle 8,5\rangle
  419. 27 , 5 \langle 27,5\rangle
  420. p = 2 p=2
  421. p = 3 p=3
  422. 𝒯 2 ( C p × C p × C p ) \mathcal{T}^{2}(C_{p}\times C_{p}\times C_{p})
  423. 2 2
  424. 𝒯 3 ( C p × C p × C p ) \mathcal{T}^{3}(C_{p}\times C_{p}\times C_{p})
  425. 𝒢 = 𝒢 ( p , p , p ) ( p , 3 ) \mathcal{G}=\mathcal{G}_{(p,p,p)}(p,3)
  426. 𝒢 ( p , 3 ) \mathcal{G}(p,3)
  427. 2 2
  428. C p × C p × C p C_{p}\times C_{p}\times C_{p}
  429. p = 2 p=2
  430. 𝒢 \mathcal{G}
  431. 32 = 2 5 32=2^{5}
  432. 32 , 32 \langle 32,32\rangle
  433. 32 , 33 \langle 32,33\rangle
  434. 32 , 27..31 \langle 32,27..31\rangle
  435. 32 , 34..35 \langle 32,34..35\rangle
  436. 32 , 28 \langle 32,28\rangle
  437. 𝒯 3 ( 64 , 140 ) \mathcal{T}^{3}(\langle 64,140\rangle)
  438. # 73 \#73
  439. 𝒯 3 ( 64 , 147 ) \mathcal{T}^{3}(\langle 64,147\rangle)
  440. # 74 \#74
  441. 𝒯 3 ( 32 , 29 ) \mathcal{T}^{3}(\langle 32,29\rangle)
  442. # 75 \#75
  443. 𝒯 3 ( 32 , 30 ) \mathcal{T}^{3}(\langle 32,30\rangle)
  444. # 76 \#76
  445. 𝒯 3 ( 32 , 31 ) \mathcal{T}^{3}(\langle 32,31\rangle)
  446. # 77 \#77
  447. 32 , 34 \langle 32,34\rangle
  448. 𝒯 3 ( 64 , 174 ) \mathcal{T}^{3}(\langle 64,174\rangle)
  449. # 78 \#78
  450. 𝒯 3 ( 32 , 35 ) \mathcal{T}^{3}(\langle 32,35\rangle)
  451. # 79 \#79
  452. ( Q ) \mathcal{M}(Q)
  453. r 2 ( G ) r_{2}(G^{\prime})
  454. r 4 ( G ) r_{4}(G^{\prime})
  455. r 2 ( H i / H i ) r_{2}(H_{i}/H_{i}^{\prime})
  456. 32 , 32 \langle 32,32\rangle
  457. ( 2 ) (2)
  458. 2 2
  459. 0
  460. 2 2
  461. 32 , 33 \langle 32,33\rangle
  462. ( 2 ) (2)
  463. 2 2
  464. 0
  465. 2 2
  466. 32 , 29 \langle 32,29\rangle
  467. ( 2 , 2 ) (2,2)
  468. 2 2
  469. 1 1
  470. 3 3
  471. 32 , 30 \langle 32,30\rangle
  472. ( 2 , 2 ) (2,2)
  473. 2 2
  474. 1 1
  475. 3 3
  476. 32 , 35 \langle 32,35\rangle
  477. ( 2 , 2 ) (2,2)
  478. 2 2
  479. 1 1
  480. 3 3
  481. 32 , 28 \langle 32,28\rangle
  482. ( 2 , 2 , 2 ) (2,2,2)
  483. 2 2
  484. 2 2
  485. 3 3
  486. 32 , 27 \langle 32,27\rangle
  487. ( 2 , 2 , 2 , 2 ) (2,2,2,2)
  488. 3 3
  489. 2 2
  490. 3 3
  491. 4 4
  492. Q = G / γ 3 ( G ) Q=G/\gamma_{3}(G)
  493. G G
  494. ( 2 , 2 , 2 ) (2,2,2)
  495. 3 3
  496. 2 2
  497. G G
  498. H i H_{i}
  499. 1 i 7 1\leq i\leq 7

Descent_along_torsors.html

  1. Spec L Spec K \operatorname{Spec}L\to\operatorname{Spec}K

Desuspension.html

  1. X X
  2. Σ X \Sigma{X}
  3. Σ - 1 \Sigma^{-1}
  4. X X
  5. Σ - 1 X \Sigma^{-1}{X}

Determinant_method.html

  1. 𝔸 2 \mathbb{A}^{2}
  2. p p
  3. k k
  4. k k

Deterministic_scale-free_network.html

  1. P ( k ) = c k - λ P(k)=ck^{-\lambda}

Dialing_scales.html

  1. sin t sin t + cos t \sin t\over\sin t+\cos t
  2. sin φ 1 + sin 2 φ \sin\varphi\over\sqrt{1+\sin^{2}\varphi}

Diameter_of_a_finite_group.html

  1. ( G , ) \left(G,\circ\right)
  2. D S D_{S}
  3. Λ = ( G , S ) \Lambda=\left(G,S\right)
  4. ( G , ) \left(G,\circ\right)
  5. D S D_{S}
  6. diam ( G ) ( log | G | ) 𝒪 ( 1 ) . \operatorname{diam}(G)\leqslant\left(\log|G|\right)^{\mathcal{O}(1)}.

Dichlorochromopyrrolate_synthase.html

  1. \rightleftharpoons

Different_types_of_boundary_conditions_in_fluid_dynamics.html

  1. V n o r m a l = 0 V_{normal}=0
  2. V t a n g e n t i a l = V w a l l V_{tangential}=V_{wall}
  3. Q A d i a b a t i c W a l l s = 0 Q_{AdiabaticWalls}=0
  4. V r ( θ ) = C o n s t a n t \operatorname{V_{r}}(\theta)=Constant

Differentiable_stack.html

  1. Ω X p \Omega_{X}^{p}
  2. Ω X p ( x ) \Omega_{X}^{p}(x)
  3. Ω X 0 \Omega_{X}^{0}
  4. 𝒪 X \mathcal{O}_{X}
  5. Ω X * \Omega_{X}^{*}
  6. G X G\to X
  7. G G × X G G\to G\times_{X}G
  8. B S 1 × X X BS^{1}\times X\to X
  9. H 2 ( X , S 1 ) H^{2}(X,S^{1})
  10. B S 1 × X X BS^{1}\times X\to X

Differential_graded_Lie_algebra.html

  1. L = L i L=\bigoplus L_{i}
  2. [ , ] : L i L j L i + j [\cdot,\cdot]:L_{i}\otimes L_{j}\to L_{i+j}
  3. d : L i L i - 1 d:L_{i}\to L_{i-1}
  4. [ x , y ] = ( - 1 ) | x | | y | + 1 [ y , x ] , [x,y]=(-1)^{|x||y|+1}[y,x],
  5. ( - 1 ) | x | | z | [ x , [ y , z ] ] + ( - 1 ) | y | | x | [ y , [ z , x ] ] + ( - 1 ) | z | | y | [ z , [ x , y ] ] = 0 , (-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,
  6. d [ x , y ] = [ d x , y ] + ( - 1 ) | x | [ x , d y ] d[x,y]=[dx,y]+(-1)^{|x|}[x,dy]
  7. L i L^{i}
  8. L i = L - i L^{i}=L_{-i}
  9. L L_{\infty}
  10. f : L L f:L\to L^{\prime}
  11. f [ x , y ] L = [ f ( x ) , f ( y ) ] L f[x,y]_{L}=[f(x),f(y)]_{L^{\prime}}
  12. f ( d L x ) = d L f ( x ) f(d_{L}x)=d_{L^{\prime}}f(x)
  13. L × L L\times L^{\prime}
  14. L L L\oplus L^{\prime}
  15. [ ( x , x ) , ( y , y ) ] = ( [ x , y ] , [ x , x ] ) [(x,x^{\prime}),(y,y^{\prime})]=([x,y],[x,x^{\prime}])
  16. D ( x , x ) = ( d x , d x ) D(x,x^{\prime})=(dx,d^{\prime}x^{\prime})
  17. L * L L*L^{\prime}
  18. x L - 1 x\in L_{-1}
  19. d x + 1 2 [ x , x ] . dx+\frac{1}{2}[x,x].

Diffuse_series.html

  1. v = R [ 2 + p ] 2 - R [ m + d ] 2 w i t h m = 2 , 3 , 4 , 5 , 6 , v=\frac{R}{\left[2+p\right]^{2}}-\frac{R}{\left[m+d\right]^{2}}with\ m=2,3,4,5% ,6,...
  2. 2 P 3 2 2P_{\frac{3}{2}}
  3. 2 P 1 2 2P_{\frac{1}{2}}
  4. n d 2 D 3 2 nd^{2}D_{\frac{3}{2}}
  5. n d 2 D 5 2 nd^{2}D_{\frac{5}{2}}
  6. P 1 2 a n d P 3 2 P_{\frac{1}{2}}\ and\ P_{\frac{3}{2}}
  7. D 3 2 a n d D 5 2 D_{\frac{3}{2}}\ and\ D_{\frac{5}{2}}
  8. ν d = R ( Z 3 p 2 3 2 - Z n d 2 n 2 ) n = 3 , 4 , 5 , 6 , \nu_{d}=R\left(\frac{Z_{3p}^{2}}{3^{2}}-\frac{Z_{nd}^{2}}{n^{2}}\right)n=3,4,5% ,6,...
  9. ν s = R ( Z 3 p 2 3 2 - Z n s 2 n 2 ) n = 4 , 5 , 6 , \nu_{s}=R\left(\frac{Z_{3p}^{2}}{3^{2}}-\frac{Z_{ns}^{2}}{n^{2}}\right)n=4,5,6% ,...

Diffusive_gradients_in_thin_films.html

  1. C D G T C_{DGT}
  2. C D G T = M Δ g D t A C_{DGT}=\frac{M\Delta g}{DtA}
  3. M M
  4. Δ g \Delta g
  5. D D
  6. t t
  7. A A

Digital_signal_conditioning.html

  1. 1. 1.
  2. 2. 2.
  3. 3. 3.
  4. 4. 4.
  5. 101011 2 101011_{2}
  6. E x .1. Ex.1.
  7. 00100111 2 00100111_{2}
  8. 100111 100111
  9. n = 5 n=5
  10. N 1 0 = a 5 2 5 + a 4 2 4 + . + a 1 2 1 + a 0 2 0 N_{1}0=a_{5}2^{5}+a_{4}2^{4}+....+a_{1}2^{1}+a_{0}2^{0}
  11. N 1 0 = ( 1 ) 2 5 + ( 0 ) 2 4 + ( 0 ) 2 3 + ( 1 ) 2 2 + ( 1 ) 2 1 + ( 1 ) 2 0 N_{1}0=(1)2^{5}+(0)2^{4}+(0)2^{3}+(1)2^{2}+(1)2^{1}+(1)2^{0}
  12. N 1 0 = 32 + 4 + 2 + 1 N_{1}0=32+4+2+1
  13. N 1 0 = 39 N_{1}0=39

Digraph_realization_problem.html

  1. ( ( a 1 , b 1 ) , , ( a n , b n ) ) ((a_{1},b_{1}),\ldots,(a_{n},b_{n}))
  2. v i v_{i}
  3. a i a_{i}
  4. b i b_{i}
  5. n n
  6. ( a 1 , , a n ) (a_{1},\cdots,a_{n})
  7. ( b 1 , , b n ) (b_{1},\ldots,b_{n})

Dimensionless_numbers_in_fluid_mechanics.html

  1. Ar = g L 3 ρ ( ρ - ρ ) μ 2 \mathrm{Ar}=\frac{gL^{3}\rho_{\ell}(\rho-\rho_{\ell})}{\mu^{2}}
  2. A = ρ 1 - ρ 2 ρ 1 + ρ 2 \mathrm{A}=\frac{\rho_{1}-\rho_{2}}{\rho_{1}+\rho_{2}}
  3. Be = Δ P L 2 μ α \mathrm{Be}=\frac{\Delta PL^{2}}{\mu\alpha}
  4. Bm = τ y L μ V \mathrm{Bm}=\frac{\tau_{y}L}{\mu V}
  5. Bi = h L C k b \mathrm{Bi}=\frac{hL_{C}}{k_{b}}
  6. B = u ρ μ ( 1 - ϵ ) D \mathrm{B}=\frac{u\rho}{\mu(1-\epsilon)D}
  7. Bo = ρ a L 2 γ \mathrm{Bo}=\frac{\rho aL^{2}}{\gamma}
  8. Br = μ U 2 κ ( T w - T 0 ) \mathrm{Br}=\frac{\mu U^{2}}{\kappa(T_{w}-T_{0})}
  9. N BK = u μ k rw σ \mathrm{N}_{\mathrm{BK}}=\frac{u\mu}{k_{\mathrm{rw}}\sigma}
  10. Ca = μ V γ \mathrm{Ca}=\frac{\mu V}{\gamma}
  11. Da = k τ \mathrm{Da}=k\tau
  12. D = ρ V d μ ( d 2 R ) 1 / 2 \mathrm{D}=\frac{\rho Vd}{\mu}\left(\frac{d}{2R}\right)^{1/2}
  13. De = t c t p \mathrm{De}=\frac{t_{\mathrm{c}}}{t_{\mathrm{p}}}
  14. c d = 2 F d ρ v 2 A , c_{\mathrm{d}}=\dfrac{2F_{\mathrm{d}}}{\rho v^{2}A}\,,
  15. Ec = V 2 c p Δ T \mathrm{Ec}=\frac{V^{2}}{c_{p}\Delta T}
  16. Eo = Δ ρ g L 2 σ \mathrm{Eo}=\frac{\Delta\rho\,g\,L^{2}}{\sigma}
  17. Er = μ v L K \mathrm{Er}=\frac{\mu vL}{K}
  18. Eu = Δ p ρ V 2 \mathrm{Eu}=\frac{\Delta{}p}{\rho V^{2}}
  19. Θ r \Theta_{r}
  20. Θ r = c p ( T - T e ) U e 2 / 2 \Theta_{r}=\frac{c_{p}(T-T_{e})}{U_{e}^{2}/2}
  21. Fr = v g \mathrm{Fr}=\frac{v}{\sqrt{g\ell}}
  22. Ga = g L 3 ν 2 \mathrm{Ga}=\frac{g\,L^{3}}{\nu^{2}}
  23. G = U e θ ν ( θ R ) 1 / 2 \mathrm{G}=\frac{U_{e}\theta}{\nu}\left(\frac{\theta}{R}\right)^{1/2}
  24. Gz = D H L Re Pr \mathrm{Gz}={D_{H}\over L}\mathrm{Re}\,\mathrm{Pr}
  25. Gr L = g β ( T s - T ) L 3 ν 2 \mathrm{Gr}_{L}=\frac{g\beta(T_{s}-T_{\infty})L^{3}}{\nu^{2}}
  26. Hg = - 1 ρ d p d x L 3 ν 2 \mathrm{Hg}=-\frac{1}{\rho}\frac{\mathrm{d}p}{\mathrm{d}x}\frac{L^{3}}{\nu^{2}}
  27. Ir = tan α H / L 0 \mathrm{Ir}=\frac{\tan\alpha}{\sqrt{H/L_{0}}}
  28. Ka = k t c \mathrm{Ka}=kt_{c}
  29. K C = V T L \mathrm{K_{C}}=\frac{V\,T}{L}
  30. Kn = λ L \mathrm{Kn}=\frac{\lambda}{L}
  31. Ku = U h ρ g 1 / 2 ( σ g ( ρ l - ρ g ) ) 1 / 4 \mathrm{Ku}=\frac{U_{h}\rho_{g}^{1/2}}{\left({\sigma g(\rho_{l}-\rho_{g})}% \right)^{1/4}}
  32. La = σ ρ L μ 2 \mathrm{La}=\frac{\sigma\rho L}{\mu^{2}}
  33. Le = α D = Sc Pr \mathrm{Le}=\frac{\alpha}{D}=\frac{\mathrm{Sc}}{\mathrm{Pr}}
  34. C L = L q S C_{\mathrm{L}}=\frac{L}{q\,S}
  35. χ \chi
  36. χ = m m g ρ g ρ \chi=\frac{m_{\ell}}{m_{g}}\sqrt{\frac{\rho_{g}}{\rho_{\ell}}}
  37. M = < m t p l > v v sound \mathrm{M}=\frac{<}{m}tpl>{{v}}{{v_{\mathrm{sound}}}}
  38. Mg = - d σ d T L Δ T η α \mathrm{Mg}=-{\frac{\mathrm{d}\sigma}{\mathrm{d}T}}\frac{L\Delta T}{\eta\alpha}
  39. Mo = g μ c 4 Δ ρ ρ c 2 σ 3 \mathrm{Mo}=\frac{g\mu_{c}^{4}\,\Delta\rho}{\rho_{c}^{2}\sigma^{3}}
  40. Nu = h d k \mathrm{Nu}=\frac{hd}{k}
  41. Oh = μ ρ σ L = We Re \mathrm{Oh}=\frac{\mu}{\sqrt{\rho\sigma L}}=\frac{\sqrt{\mathrm{We}}}{\mathrm{% Re}}
  42. Pr = ν α = c p μ k \mathrm{Pr}=\frac{\nu}{\alpha}=\frac{c_{p}\mu}{k}
  43. C p = p - p 1 2 ρ V 2 C_{p}={p-p_{\infty}\over\frac{1}{2}\rho_{\infty}V_{\infty}^{2}}
  44. Ra x = g β ν α ( T s - T ) x 3 \mathrm{Ra}_{x}=\frac{g\beta}{\nu\alpha}(T_{s}-T_{\infty})x^{3}
  45. Re = v L ρ μ \mathrm{Re}=\frac{vL\rho}{\mu}
  46. Ri = g h u 2 = 1 Fr 2 \mathrm{Ri}=\frac{gh}{u^{2}}=\frac{1}{\mathrm{Fr}^{2}}
  47. Ro = f L 2 ν = St Re \mathrm{Ro}={fL^{2}\over\nu}=\mathrm{St}\,\mathrm{Re}
  48. Sc = ν D \mathrm{Sc}=\frac{\nu}{D}
  49. H = δ * θ H=\frac{\delta^{*}}{\theta}
  50. Sh = K L D \mathrm{Sh}=\frac{KL}{D}
  51. S = ( r c ) 2 μ N P \mathrm{S}=\left(\frac{r}{c}\right)^{2}\frac{\mu N}{P}
  52. St = h c p ρ V = Nu Re Pr \mathrm{St}=\frac{h}{c_{p}\rho V}=\frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}}
  53. Stk = τ U o d c \mathrm{Stk}=\frac{\tau U_{o}}{d_{c}}
  54. N = B 2 L c σ ρ U = Ha 2 Re \mathrm{N}=\frac{B^{2}L_{c}\sigma}{\rho U}=\frac{\mathrm{Ha}^{2}}{\mathrm{Re}}
  55. Ta = 4 Ω 2 R 4 ν 2 \mathrm{Ta}=\frac{4\Omega^{2}R^{4}}{\nu^{2}}
  56. U = H λ 2 h 3 \mathrm{U}=\frac{H\,\lambda^{2}}{h^{3}}
  57. j * = R ( ω ρ μ ) 1 2 j^{*}=R\left(\frac{\omega\rho}{\mu}\right)^{\frac{1}{2}}
  58. Wea = w w H 100 \mathrm{Wea}=\frac{w}{w_{\mathrm{H}}}100
  59. We = ρ v 2 l σ \mathrm{We}=\frac{\rho v^{2}l}{\sigma}
  60. Wi = γ ˙ λ \mathrm{Wi}=\dot{\gamma}\lambda
  61. α \alpha
  62. α = R ( ω ρ μ ) 1 2 \alpha=R\left(\frac{\omega\rho}{\mu}\right)^{\frac{1}{2}}

Direct_sum_of_topological_groups.html

  1. H 1 × H 1 G ( h 1 , h 2 ) h 1 h 2 \begin{aligned}\displaystyle H_{1}\times H_{1}&\displaystyle\longrightarrow G% \\ \displaystyle(h_{1},h_{2})&\displaystyle\longmapsto h_{1}h_{2}\end{aligned}
  2. H i , i = 1 , , n H_{i},i=1,\ldots,n
  3. i = 1 n H i G ( h i ) i I h 1 h 2 h n \begin{aligned}\displaystyle\prod^{n}_{i=1}H_{i}&\displaystyle\longrightarrow G% \\ \displaystyle(h_{i})_{i\in I}&\displaystyle\longmapsto h_{1}h_{2}\cdots h_{n}% \end{aligned}
  4. H i H_{i}
  5. H i H_{i}
  6. 0 H i G π G / H 0 0\to H\stackrel{i}{{}\to{}}G\stackrel{\pi}{{}\to{}}G/H\to 0
  7. i i
  8. π \pi
  9. G G
  10. 𝕋 \mathbb{T}
  11. 𝕋 \mathbb{T}
  12. \mathbb{R}

Discrete-stable_distribution.html

  1. Q ( s , ν , a ) = n = 0 P ( N , ν , a ) ( 1 - s ) N = exp ( - a s ν ) . Q(s,\nu,a)=\sum_{n=0}^{\infty}P(N,\nu,a)(1-s)^{N}=\exp(-as^{\nu}).
  2. a > 0 a>0
  3. 0 < ν 1 0<\nu\leq 1
  4. 0 < ν < 1 0<\nu<1
  5. lim N P ( N , ν , a ) 1 N ν + 1 . \lim_{N\to\infty}P(N,\nu,a)\sim\frac{1}{N^{\nu+1}}.
  6. ν = 1 \nu=1
  7. a a
  8. P ( N , ν , a ) = ( - 1 ) N N ! d N Q ( s , ν , a ) d s N | s = 1 . P(N,\nu,a)=\left.\frac{(-1)^{N}}{N!}\frac{d^{N}Q(s,\nu,a)}{ds^{N}}\right|_{s=1}.
  9. P ( N ; ν = 1 , a ) = a N e - a N ! . \!P(N;\nu=1,a)=\frac{a^{N}e^{-a}}{N!}.
  10. ν = 1 / 2 \nu=1/2
  11. ν = 1 / 3 \nu=1/3
  12. a a
  13. 0 < α < 1 0<\alpha<1
  14. c c
  15. ν = α \nu=\alpha
  16. a = c sec ( π α / 2 ) a=c\sec(\pi\alpha/2)
  17. P ( N , α , c sec ( π α / 2 ) ) = 0 P ( N , 1 , a ) p ( a , α , 1 , c , 0 ) d a P(N,\alpha,c\sec(\pi\alpha/2))=\int_{0}^{\infty}P(N,1,a)p(a,\alpha,1,c,0)\,da
  18. p ( a , α , 1 , c , 0 ) p(a,\alpha,1,c,0)
  19. β = 1 \beta=1
  20. μ = 0 \mu=0
  21. ν \nu
  22. α \alpha
  23. ν α \nu\cdot\alpha
  24. α \alpha
  25. P ( N , ν α , c sec ( π α / 2 ) = 0 P ( N , α , a ) p ( a , ν , 1 , c , 0 ) d a . P(N,\nu\cdot\alpha,c\sec(\pi\alpha/2)=\int_{0}^{\infty}P(N,\alpha,a)p(a,\nu,1,% c,0)\,da.
  26. ν 1 \nu\rightarrow 1
  27. a sec ( π ν / 2 ) a\sec(\pi\nu/2)
  28. N N
  29. N 1 N\gg 1
  30. P ( N ) 1 / N 1 + ν P(N)\sim 1/N^{1+\nu}
  31. ν 1 \nu\approx 1
  32. ν > 1 \nu>1
  33. P ( N , ν , a ) P(N,\nu,a)
  34. ν 1 \nu\leq 1

Discrete_spline_interpolation.html

  1. g ( x ) = { g 1 ( x ) x < x 1 g i ( x ) x i - 1 x < x i for i = 2 , 3 , , n - 1 g n ( x ) x x n - 1 g(x)=\begin{cases}g_{1}(x)&x<x_{1}\\ g_{i}(x)&x_{i-1}\leq x<x_{i}\,\text{ for }i=2,3,\ldots,n-1\\ g_{n}(x)&x\geq x_{n-1}\end{cases}
  2. ( g i + 1 - g i ) ( x i + j h ) = 0 for j = - 1 , 0 , 1 and i = 1 , 2 , , n - 1 (g_{i+1}-g_{i})(x_{i}+jh)=0\,\text{ for }j=-1,0,1\,\text{ and }i=1,2,\ldots,n-1
  3. g i + 1 ( x i - h ) = g i ( x i - h ) g_{i+1}(x_{i}-h)=g_{i}(x_{i}-h)
  4. g i + 1 ( x i ) = g i ( x i ) g_{i+1}(x_{i})=g_{i}(x_{i})
  5. g i + 1 ( x i + h ) = g i ( x i + h ) g_{i+1}(x_{i}+h)=g_{i}(x_{i}+h)
  6. D ( 0 ) f ( x ) = f ( x ) D^{(0)}f(x)=f(x)
  7. D ( 1 ) f ( x ) = f ( x + h ) - f ( x - h ) 2 h D^{(1)}f(x)=\frac{f(x+h)-f(x-h)}{2h}
  8. D ( 2 ) f ( x ) = f ( x + h ) - 2 f ( x ) + f ( x - h ) h 2 D^{(2)}f(x)=\frac{f(x+h)-2f(x)+f(x-h)}{h^{2}}
  9. D ( j ) g i + 1 ( x i ) = D ( j ) g i ( x i ) for j = 0 , 1 , 2 and i = 1 , 2 , , n - 1. D^{(j)}g_{i+1}(x_{i})=D^{(j)}g_{i}(x_{i})\,\text{ for }j=0,1,2\,\text{ and }i=% 1,2,\ldots,n-1.
  10. D ( j ) g ( x ) D^{(j)}g(x)
  11. g ( x ) = { x 3 x < 1 x 3 - 2 ( x - 1 ) ( ( x - 1 ) 2 - h 2 ) 1 x < 2 x 3 - 2 ( x - 1 ) ( ( x - 1 ) 2 - h 2 ) + ( x - 2 ) ( ( x - 2 ) 2 - h 2 ) x 2 g(x)=\begin{cases}x^{3}&x<1\\ x^{3}-2(x-1)((x-1)^{2}-h^{2})&1\leq x<2\\ x^{3}-2(x-1)((x-1)^{2}-h^{2})+(x-2)((x-2)^{2}-h^{2})&x\geq 2\end{cases}
  12. g ( x i ) = f ( x i ) for i = 0 , 1 , , n . g(x_{i})=f(x_{i})\,\text{ for }i=0,1,\ldots,n.
  13. D ( 1 ) g 1 ( x 0 ) = D ( 1 ) f ( x 0 ) . D^{(1)}g_{1}(x_{0})=D^{(1)}f(x_{0}).
  14. D ( 1 ) g n ( x n ) = D ( 1 ) f ( x n ) . D^{(1)}g_{n}(x_{n})=D^{(1)}f(x_{n}).

Discrete_time_and_continuous_time.html

  1. x t + 1 = r x t ( 1 - x t ) , x_{t+1}=rx_{t}(1-x_{t}),
  2. r = 4 r=4
  3. x 1 = 1 / 3 x_{1}=1/3
  4. x 2 = 4 ( 1 / 3 ) ( 2 / 3 ) = 8 / 9 x_{2}=4(1/3)(2/3)=8/9
  5. x 3 = 4 ( 8 / 9 ) ( 1 / 9 ) = 32 / 81 x_{3}=4(8/9)(1/9)=32/81
  6. P t + 1 = P t + δ f ( P t , ) P_{t+1}=P_{t}+\delta\cdot f(P_{t},...)
  7. δ \delta
  8. f f
  9. d P d t = λ f ( P , ) \frac{dP}{dt}=\lambda\cdot f(P,...)
  10. λ \lambda
  11. f f

Discretization_of_Navier–Stokes_equations.html

  1. u i t + u i u j x j = - P x i + ν 2 u i x j x j + f i \frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}=-% \frac{\partial P}{\partial x_{i}}+\nu\frac{\partial^{2}u_{i}}{\partial x_{j}% \partial x_{j}}+f_{i}
  2. V [ u i t + u i u j x j ] d V = V [ - P x i + ν 2 u i x j x j + f i ] d V \iiint_{V}\left[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{% \partial x_{j}}\right]dV=\iiint_{V}\left[-\frac{\partial P}{\partial x_{i}}+% \nu\frac{\partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}+f_{i}\right]dV
  3. u i t V + A u i u j n j d A = - A P n i d A + A ν u i x j n j d A + f i V \frac{\partial u_{i}}{\partial t}V+\iint_{A}u_{i}u_{j}n_{j}dA=-\iint_{A}Pn_{i}% dA+\iint_{A}\nu\frac{\partial u_{i}}{\partial x_{j}}n_{j}dA+f_{i}V
  4. u i t V + n b r ( u i u j n j A ) n b r = - n b r ( P n i A ) n b r + n b r ( ν u i x j n j A ) n b r + f i V \frac{\partial u_{i}}{\partial t}V+\sum_{nbr}\left(u_{i}u_{j}n_{j}A\right)_{% nbr}=-\sum_{nbr}\left(Pn_{i}A\right)_{nbr}+\sum_{nbr}\left(\nu\frac{\partial u% _{i}}{\partial x_{j}}n_{j}A\right)_{nbr}+f_{i}V
  5. u i t Δ x Δ y - ( u i u Δ y ) w + ( u i u Δ y ) e - ( u i v Δ x ) s + ( u i v Δ x ) n = - ( P n i Δ y ) w - ( P n i Δ y ) e - ( P n i Δ x ) s - ( P n i Δ x ) n - ( ν u i x Δ y ) w + ( ν u i x Δ y ) e - ( ν u i y Δ x ) s + ( ν u i y Δ x ) n + f i \begin{aligned}&\displaystyle\frac{\partial u_{i}}{\partial t}\Delta x\Delta y% -\left(u_{i}u\Delta y\right)_{w}+\left(u_{i}u\Delta y\right)_{e}-\left(u_{i}v% \Delta x\right)_{s}+\left(u_{i}v\Delta x\right)_{n}=\\ &\displaystyle-\left(Pn_{i}\Delta y\right)_{w}-\left(Pn_{i}\Delta y\right)_{e}% -\left(Pn_{i}\Delta x\right)_{s}-\left(Pn_{i}\Delta x\right)_{n}\\ &\displaystyle-\left(\nu\frac{\partial u_{i}}{\partial x}\Delta y\right)_{w}+% \left(\nu\frac{\partial u_{i}}{\partial x}\Delta y\right)_{e}-\left(\nu\frac{% \partial u_{i}}{\partial y}\Delta x\right)_{s}+\left(\nu\frac{\partial u_{i}}{% \partial y}\Delta x\right)_{n}+f_{i}\end{aligned}
  6. u t Δ x Δ y - ( u u Δ y ) w + ( u u Δ y ) e - ( u v Δ x ) s + ( u v Δ x ) n = + ( P Δ y ) w - ( P Δ y ) e - ( ν u x Δ y ) w + ( ν u x Δ y ) e - ( ν u y Δ x ) s + ( ν u y Δ x ) n + f x \begin{aligned}&\displaystyle\frac{\partial u}{\partial t}\Delta x\Delta y-% \left(uu\Delta y\right)_{w}+\left(uu\Delta y\right)_{e}-\left(uv\Delta x\right% )_{s}+\left(uv\Delta x\right)_{n}=\\ &\displaystyle+\left(P\Delta y\right)_{w}-\left(P\Delta y\right)_{e}-\left(\nu% \frac{\partial u}{\partial x}\Delta y\right)_{w}+\left(\nu\frac{\partial u}{% \partial x}\Delta y\right)_{e}-\left(\nu\frac{\partial u}{\partial y}\Delta x% \right)_{s}+\left(\nu\frac{\partial u}{\partial y}\Delta x\right)_{n}+f_{x}% \end{aligned}
  7. v t Δ x Δ y - ( v u Δ y ) w + ( v u Δ y ) e - ( v v Δ x ) s + ( v v Δ x ) n = + ( P Δ x ) s - ( P Δ x ) n - ( ν v x Δ y ) w + ( ν v x Δ y ) e - ( ν v y Δ x ) s + ( ν v y Δ x ) n + f y \begin{aligned}&\displaystyle\frac{\partial v}{\partial t}\Delta x\Delta y-% \left(vu\Delta y\right)_{w}+\left(vu\Delta y\right)_{e}-\left(vv\Delta x\right% )_{s}+\left(vv\Delta x\right)_{n}=\\ &\displaystyle+\left(P\Delta x\right)_{s}-\left(P\Delta x\right)_{n}-\left(\nu% \frac{\partial v}{\partial x}\Delta y\right)_{w}+\left(\nu\frac{\partial v}{% \partial x}\Delta y\right)_{e}-\left(\nu\frac{\partial v}{\partial y}\Delta x% \right)_{s}+\left(\nu\frac{\partial v}{\partial y}\Delta x\right)_{n}+f_{y}% \end{aligned}
  8. u i t = u i n - u i n - 1 Δ t \frac{\partial u_{i}}{\partial t}=\frac{u_{i}^{n}-u_{i}^{n-1}}{\Delta t}
  9. ( u x ) w = u I , J - u I - 1 , J Δ x \left(\frac{\partial u}{\partial x}\right)_{w}=\frac{u_{I,J}-u_{I-1,J}}{\Delta x}

Disparity_filter_algorithm_of_weighted_network.html

  1. ρ ( x ) d x = ( k - 1 ) ( 1 - x ) k - 2 d x \rho(x)\,dx=(k-1)(1-x)^{k-2}\,dx
  2. α i j = 1 - ( k - 1 ) 0 p i j ( 1 - x ) k - 2 d x \alpha_{ij}=1-(k-1)\int_{0}^{p_{ij}}(1-x)^{k-2}\,dx

Displaced_Poisson_distribution.html

  1. P ( X = n ) = { e - λ λ n + r ( n + r ) ! 1 I ( r , λ ) , n = 0 , 1 , 2 , if r 0 e - λ λ n + r ( n + r ) ! 1 I ( r + s , λ ) , n = s , s + 1 , s + 2 , otherwise P(X=n)=\begin{cases}e^{-\lambda}\dfrac{\lambda^{n+r}}{\left(n+r\right)!}\cdot% \dfrac{1}{I\left(r,\lambda\right)},\quad n=0,1,2,\ldots&\,\text{if }r\geq 0\\ e^{-\lambda}\dfrac{\lambda^{n+r}}{\left(n+r\right)!}\cdot\dfrac{1}{I\left(r+s,% \lambda\right)},\quad n=s,s+1,s+2,\ldots&\,\text{otherwise}\end{cases}
  2. λ > 0 \lambda>0
  3. I ( , ) I\left(\cdot,\cdot\right)
  4. P ( X = n ) / P ( X = n - 1 ) P(X=n)/P(X=n-1)
  5. λ / n \lambda/n
  6. n > 0 n>0
  7. λ / ( n + r ) \lambda/\left(n+r\right)

Distance_oracle.html

  1. O ( m + n log n ) O(m+n\log n)
  2. O ( 1 ) O(1)
  3. O ( n 2 ) O(n^{2})
  4. O ( n 3 ) O(n^{3})
  5. O ( n 2 ) O(n^{2})
  6. O ( m + n log n ) O(m+n\log n)
  7. O ( k n 1 + 1 / k ) O(kn^{1+1/k})
  8. O ( k ) O(k)
  9. 2 k - 1 2k-1
  10. 2 k - 1 2k-1
  11. O ( k m n 1 / k ) O(kmn^{1/k})
  12. k = 1 k=1
  13. k = 2 k=2
  14. O ( n 1.5 ) O(n^{1.5})
  15. k = log n k=\lfloor\log n\rfloor
  16. O ( n log n ) O(n\log n)
  17. O ( l o g n ) O(logn)
  18. O ( log n ) O(\log n)
  19. A 0 = V A_{0}=V
  20. i = 1 , , k - 1 i=1,\ldots,k-1
  21. A i A_{i}
  22. A i - 1 A_{i-1}
  23. n - 1 / k n^{-1/k}
  24. A i A_{i}
  25. n 1 - i / k n^{1-i/k}
  26. A i A_{i}
  27. A k = A_{k}=\emptyset
  28. i = 0 , , k - 1 i=0,\ldots,k-1
  29. d ( A i , v ) = min ( d ( w , v ) | w A i d(A_{i},v)=\min{(d(w,v)|w\in A_{i}}
  30. p i ( v ) = arg min ( d ( w , v ) | w A i p_{i}(v)=\arg\min{(d(w,v)|w\in A_{i}}
  31. p i ( v ) p_{i}(v)
  32. d ( A i , v ) d(A_{i},v)
  33. d ( A 0 , v ) = 0 a n d p 0 ( v ) = v d(A_{0},v)=0andp_{0}(v)=v
  34. d ( A k , v ) = d(A_{k},v)=\infty
  35. i = 0 , , k - 1 i=0,\ldots,k-1
  36. B i ( v ) = { w A i A i + 1 | d ( w , v ) < d ( A i + 1 , v ) } B_{i}(v)=\{w\in A_{i}\setminus A_{i+1}|d(w,v)<d(A_{i+1},v)\}
  37. B i ( v ) B_{i}(v)
  38. A i A_{i}
  39. A i + 1 A_{i+1}
  40. B i B_{i}
  41. B ( v ) = i = 0 k - 1 B i ( v ) . B(v)=\bigcup_{i=0}^{k-1}B_{i}(v).
  42. B ( v ) B(v)
  43. k n 1 / k kn^{1/k}
  44. B ( v ) B(v)
  45. w B ( V ) w\in B(V)
  46. d ( w , v ) d(w,v)
  47. O ( k n + Σ | B ( v ) | ) = O ( k n + n k n 1 / k ) = O ( k n 1 + 1 / k ) O(kn+\Sigma|B(v)|)=O(kn+nkn^{1/k})=O(kn^{1+1/k})
  48. w := u , i := 0 w:=u,i:=0
  49. w B ( v ) w\notin B(v)
  50. i := i + 1 i:=i+1
  51. ( u , v ) := ( v , u ) (u,v):=(v,u)
  52. w := p i ( u ) w:=p_{i}(u)
  53. d ( w , u ) + d ( w , v ) d(w,u)+d(w,v)
  54. d ( w , u ) d(w,u)
  55. d ( v , u ) d(v,u)
  56. A k - 1 B ( v ) A_{k-1}\subseteq B(v)
  57. d ( w , u ) ( k - 1 ) d ( v , u ) d(w,u)\leq(k-1)d(v,u)
  58. d ( w , v ) d ( w , u ) + d ( u , v ) k d ( v , u ) d(w,v)\leq d(w,u)+d(u,v)\leq kd(v,u)
  59. ( 2 k - 1 ) d ( u , v ) (2k-1)d(u,v)
  60. O ( n 4 / 3 m 1 / 3 ) O(n^{4/3}m^{1/3})
  61. O ( 1 ) O(1)
  62. O ( N + n ) O(N+n)
  63. Ω ( n 2 ) \Omega(n^{2})
  64. O ( 1 ) O(1)

Distributed_file_system_for_cloud.html

  1. a v a i l i = i = 0 | s i | j = i + 1 | s i | c o n f i . c o n f j . d i v e r s i t y ( s i , s j ) avail_{i}=\sum_{i=0}^{|s_{i}|}\sum_{j=i+1}^{|s_{i}|}conf_{i}.conf_{j}.% diversity(s_{i},s_{j})
  2. s i s_{i}
  3. c o n f i conf_{i}
  4. c o n f j conf_{j}
  5. i {}_{i}
  6. j {}_{j}
  7. s i s_{i}
  8. s j s_{j}

Distribution_learning_theory.html

  1. X \textstyle X
  2. X \textstyle X
  3. X = { 0 , 1 } n \textstyle X=\{0,1\}^{n}
  4. n \textstyle n
  5. y X \textstyle y\in X
  6. X \textstyle X
  7. D \textstyle D
  8. X \textstyle X
  9. E D \textstyle E_{D}
  10. D \textstyle D
  11. y X \textstyle y\in X
  12. E D [ y ] \textstyle E_{D}[y]
  13. y \textstyle y
  14. D \textstyle D
  15. E D [ y ] = Pr [ Y = y ] \textstyle E_{D}[y]=\Pr[Y=y]
  16. Y D \textstyle Y\sim D
  17. G D \textstyle G_{D}
  18. D \textstyle D
  19. y \textstyle y
  20. G D [ y ] X \textstyle G_{D}[y]\in X
  21. D \textstyle D
  22. D \textstyle D
  23. D \textstyle D
  24. C X \textstyle C_{X}
  25. C X \textstyle C_{X}
  26. D C X \textstyle D\in C_{X}
  27. X \textstyle X
  28. C X \textstyle C_{X}
  29. C \textstyle C
  30. D \textstyle D
  31. D \textstyle D
  32. D \textstyle D^{\prime}
  33. K L - d i s t a n c e ( D , D ) T V - d i s t a n c e ( D , D ) K o l m o g o r o v - d i s t a n c e ( D , D ) KL-distance(D,D^{\prime})\geq TV-distance(D,D^{\prime})\geq Kolmogorov-% distance(D,D^{\prime})
  34. D \textstyle D
  35. D \textstyle D^{\prime}
  36. d ( D , D ) \textstyle d(D,D^{\prime})
  37. D \textstyle D
  38. D \textstyle D^{\prime}
  39. G E N ( D ) \textstyle GEN(D)
  40. D \textstyle D
  41. D \textstyle D
  42. C \textstyle C
  43. S = { ( x 1 , y 1 ) , , ( x n , y n ) } \textstyle S=\{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  44. f : X Y \textstyle f:X\rightarrow Y
  45. f = arg min g V ( y , g ( x ) ) d ρ ( x , y ) f=\arg\min_{g}\int V(y,g(x))d\rho(x,y)
  46. V ( , ) V(\cdot,\cdot)
  47. V ( y , z ) = ( y - z ) 2 V(y,z)=(y-z)^{2}
  48. ρ ( x , y ) \rho(x,y)
  49. ρ x ( y ) \rho_{x}(y)
  50. f ( x ) = y y d ρ x ( y ) f(x)=\int_{y}yd\rho_{x}(y)
  51. S S
  52. ρ ( x , y ) \rho(x,y)
  53. ρ \rho
  54. S S
  55. f f
  56. C \textstyle C
  57. ϵ > 0 \textstyle\epsilon>0
  58. 0 < δ 1 \textstyle 0<\delta\leq 1
  59. G E N ( D ) \textstyle GEN(D)
  60. D C \textstyle D\in C
  61. A \textstyle A
  62. C \textstyle C
  63. D \textstyle D^{\prime}
  64. Pr [ d ( D , D ) ϵ ] 1 - δ \Pr[d(D,D^{\prime})\leq\epsilon]\geq 1-\delta
  65. D C \textstyle D^{\prime}\in C
  66. A \textstyle A
  67. C \textstyle C
  68. C \textstyle C
  69. N ( μ , σ 2 ) \textstyle N(\mu,\sigma^{2})
  70. A \textstyle A
  71. μ , σ \textstyle\mu,\sigma
  72. A \textstyle A
  73. A \textstyle A
  74. O R \textstyle OR
  75. # P P / poly \textstyle\#P\subseteq P/\,\text{poly}
  76. ϵ - \textstyle\epsilon-
  77. C \textstyle C
  78. ϵ - \textstyle\epsilon-
  79. C \textstyle C
  80. C ϵ \textstyle C_{\epsilon}
  81. ϵ \textstyle\epsilon
  82. C \textstyle C
  83. D C \textstyle D\in C
  84. D C ϵ \textstyle D^{\prime}\in C_{\epsilon}
  85. d ( D , D ) ϵ \textstyle d(D,D^{\prime})\leq\epsilon
  86. ϵ - \textstyle\epsilon-
  87. D \textstyle D
  88. ϵ > 0 \textstyle\epsilon>0
  89. ϵ - \textstyle\epsilon-
  90. C ϵ \textstyle C_{\epsilon}
  91. C ϵ \textstyle C_{\epsilon}
  92. D C ϵ \textstyle D^{\prime}\in C_{\epsilon}
  93. D C \textstyle D\in C
  94. D , D ′′ C ϵ \textstyle D^{\prime},D^{\prime\prime}\in C_{\epsilon}
  95. d ( D , D ) \textstyle d(D,D^{\prime})
  96. d ( D , D ′′ ) \textstyle d(D,D^{\prime\prime})
  97. D \textstyle D
  98. D \textstyle D
  99. D \textstyle D
  100. C ϵ \textstyle C_{\epsilon}
  101. D * \textstyle D^{*}
  102. ϵ - \textstyle\epsilon-
  103. D \textstyle D
  104. d ( D * , D ) ϵ \textstyle d(D^{*},D)\leq\epsilon
  105. 1 - δ \textstyle 1-\delta
  106. O ( log N / ϵ 2 ) \textstyle O(\log N/\epsilon^{2})
  107. D \textstyle D
  108. O ( N log N / ϵ 2 ) \textstyle O(N\log N/\epsilon^{2})
  109. N = | C ϵ | \textstyle N=|C_{\epsilon}|
  110. n \textstyle n
  111. X 1 , , X n \textstyle X_{1},\dots,X_{n}
  112. p 1 , , p n \textstyle p_{1},\dots,p_{n}
  113. n \textstyle n
  114. X = i X i \textstyle X=\sum_{i}X_{i}
  115. P B D = { D : D is a Poisson binomial distribution } \textstyle PBD=\{D:D~{}\,\text{ is a Poisson binomial distribution}\}
  116. P B D \textstyle PBD
  117. P B D \textstyle PBD
  118. D P B D \textstyle D\in PBD
  119. n \textstyle n
  120. ϵ > 0 \textstyle\epsilon>0
  121. 0 < δ 1 \textstyle 0<\delta\leq 1
  122. G E N ( D ) \textstyle GEN(D)
  123. D \textstyle D^{\prime}
  124. Pr [ d ( D , D ) ϵ ] 1 - δ \textstyle\Pr[d(D,D^{\prime})\leq\epsilon]\geq 1-\delta
  125. O ~ ( ( 1 / ϵ 3 ) log ( 1 / δ ) ) \textstyle\tilde{O}((1/\epsilon^{3})\log(1/\delta))
  126. O ~ ( ( 1 / ϵ 3 ) log n log 2 ( 1 / δ ) ) \textstyle\tilde{O}((1/\epsilon^{3})\log n\log^{2}(1/\delta))
  127. D P B D \textstyle D\in PBD
  128. n \textstyle n
  129. ϵ > 0 \textstyle\epsilon>0
  130. 0 < δ 1 \textstyle 0<\delta\leq 1
  131. G E N ( D ) \textstyle GEN(D)
  132. D P B D \textstyle D^{\prime}\in PBD
  133. Pr [ d ( D , D ) ϵ ] 1 - δ \textstyle\Pr[d(D,D^{\prime})\leq\epsilon]\geq 1-\delta
  134. O ~ ( ( 1 / ϵ 2 ) ) log ( 1 / δ ) \textstyle\tilde{O}((1/\epsilon^{2}))\log(1/\delta)
  135. ( 1 / ϵ ) O ( log 2 ( 1 / ϵ ) ) O ~ ( log n log ( 1 / δ ) ) \textstyle(1/\epsilon)^{O(\log^{2}(1/\epsilon))}\tilde{O}(\log n\log(1/\delta))
  136. n \textstyle n
  137. D \textstyle D
  138. n \textstyle n
  139. O ( 1 / ϵ 2 ) \textstyle O(1/\epsilon^{2})
  140. ϵ - \textstyle\epsilon-
  141. P B D \textstyle PBD
  142. n \textstyle n
  143. X 1 , , X n \textstyle X_{1},\dots,X_{n}
  144. { 0 , 1 , , k - 1 } \textstyle\{0,1,\dots,k-1\}
  145. k - \textstyle k-
  146. n \textstyle n
  147. X = i X i \textstyle X=\sum_{i}X_{i}
  148. k - S I I R V = { D : D is a k-sum of independent integer random variable } \textstyle k-SIIRV=\{D:D\,\text{is a k-sum of independent integer random % variable }\}
  149. D k - S I I R V \textstyle D\in k-SIIRV
  150. n \textstyle n
  151. ϵ > 0 \textstyle\epsilon>0
  152. G E N ( D ) \textstyle GEN(D)
  153. D \textstyle D^{\prime}
  154. Pr [ d ( D , D ) ϵ ] 1 - δ \textstyle\Pr[d(D,D^{\prime})\leq\epsilon]\geq 1-\delta
  155. poly ( k / ϵ ) \textstyle\,\text{poly}(k/\epsilon)
  156. poly ( k / ϵ ) \textstyle\,\text{poly}(k/\epsilon)
  157. n \textstyle n
  158. k = 2 \textstyle k=2
  159. X N ( μ 1 , Σ 1 ) \textstyle X\sim N(\mu_{1},\Sigma_{1})
  160. Y N ( μ 2 , Σ 2 ) \textstyle Y\sim N(\mu_{2},\Sigma_{2})
  161. Z \textstyle Z
  162. X \textstyle X
  163. w 1 \textstyle w_{1}
  164. Y \textstyle Y
  165. w 2 = 1 - w 1 \textstyle w_{2}=1-w_{1}
  166. F 1 \textstyle F_{1}
  167. X \textstyle X
  168. F 2 \textstyle F_{2}
  169. Y \textstyle Y
  170. Z \textstyle Z
  171. F = w 1 F 1 + w 2 F 2 \textstyle F=w_{1}F_{1}+w_{2}F_{2}
  172. Z \textstyle Z
  173. w 1 , w 2 , μ 1 , μ 2 , Σ 1 , Σ 2 \textstyle w_{1},w_{2},\mu_{1},\mu_{2},\Sigma_{1},\Sigma_{2}
  174. || μ 1 - μ 2 || \textstyle||\mu_{1}-\mu_{2}||
  175. μ i , Σ i \textstyle\mu_{i},\Sigma_{i}
  176. w i \textstyle w_{i}
  177. G M \textstyle GM
  178. D G M \textstyle D\in GM
  179. || μ 1 - μ 2 || c n max ( λ m a x ( Σ 1 ) , λ m a x ( Σ 2 ) \textstyle||\mu_{1}-\mu_{2}||\geq c\sqrt{n\max(\lambda_{max}(\Sigma_{1}),% \lambda_{max}(\Sigma_{2})}
  180. c > 1 / 2 \textstyle c>1/2
  181. λ m a x ( A ) \textstyle\lambda_{max}(A)
  182. A \textstyle A
  183. ϵ > 0 \textstyle\epsilon>0
  184. 0 < δ 1 \textstyle 0<\delta\leq 1
  185. G E N ( D ) \textstyle GEN(D)
  186. w i , μ i , Σ i \textstyle w^{\prime}_{i},\mu^{\prime}_{i},\Sigma^{\prime}_{i}
  187. Pr [ || w i - w i || ϵ ] 1 - δ \textstyle\Pr[||w_{i}-w^{\prime}_{i}||\leq\epsilon]\geq 1-\delta
  188. μ i \textstyle\mu_{i}
  189. Σ i \textstyle\Sigma_{i}
  190. M = 2 O ( log 2 ( 1 / ( ϵ δ ) ) ) \textstyle M=2^{O(\log^{2}(1/(\epsilon\delta)))}
  191. O ( M 2 d + M d n ) \textstyle O(M^{2}d+Mdn)
  192. k - \textstyle k-
  193. F G M \textstyle F\in GM
  194. ϵ > 0 \textstyle\epsilon>0
  195. 0 < δ 1 \textstyle 0<\delta\leq 1
  196. G E N ( D ) \textstyle GEN(D)
  197. w i , μ i , Σ i \textstyle w^{\prime}_{i},\mu^{\prime}_{i},\Sigma^{\prime}_{i}
  198. F = w 1 F 1 + w 2 F 2 \textstyle F^{\prime}=w^{\prime}_{1}F^{\prime}_{1}+w^{\prime}_{2}F^{\prime}_{2}
  199. F i = N ( μ i , Σ i ) \textstyle F^{\prime}_{i}=N(\mu^{\prime}_{i},\Sigma^{\prime}_{i})
  200. Pr [ d ( F , F ) ϵ ] 1 - δ \textstyle\Pr[d(F,F^{\prime})\leq\epsilon]\geq 1-\delta
  201. poly ( n , 1 / ϵ , 1 / δ , 1 / w 1 , 1 / w 2 , 1 / d ( F 1 , F 2 ) ) \textstyle\,\text{poly}(n,1/\epsilon,1/\delta,1/w_{1},1/w_{2},1/d(F_{1},F_{2}))
  202. F 1 \textstyle F_{1}
  203. F 2 \textstyle F_{2}

Divine_coincidence.html

  1. π t = β E t [ π t + 1 ] + κ ( y t - y t * ) \pi_{t}=\beta E_{t}[\pi_{t+1}]+\kappa(y_{t}-y_{t}^{*})\,
  2. π t \pi_{t}
  3. E t [ π t + 1 ] E_{t}[\pi_{t+1}]
  4. y t y_{t}
  5. y t * y_{t}^{*}
  6. ( y t - y t * ) (y_{t}-y_{t}^{*})

Dmitrii_Ivanovich_Zhuravskii.html

  1. τ = V Q I t , \tau={VQ\over It},

DNA-Functionalized_Quantum_Dots.html

  1. V t = V v d W + V e s \ V_{t}=V_{vdW}+V_{es}
  2. V e s = 2 π ϵ 0 ϵ R P ψ 2 l n [ 1 + e x p ( κ h ) ] \ V_{es}=2\pi\epsilon_{0}\epsilon R_{P}\psi^{2}ln[1+exp(\kappa h)]
  3. h h
  4. R P R_{P}
  5. ϵ ϵ 0 \epsilon\epsilon_{0}
  6. ψ \psi
  7. κ \kappa
  8. V v d W = A H 6 [ l n h ( 4 R P + h ) ( 2 R P + h ) 2 + ( 2 R P 2 ) h ( 4 R P + h ) + ( 2 R P 2 ) ( 2 R P + h ) 2 ] \ V_{vdW}={A_{H}\over 6}[ln{h(4R_{P}+h)\over(2R_{P}+h)^{2}}+{(2R_{P}^{2})\over h% (4R_{P}+h)}+{(2R_{P}^{2})\over(2R_{P}+h)^{2}}]
  9. A H A_{H}

Dodd-Bullough-Mikhailov_equation.html

  1. u x t + α * e u + γ * e - 2 * u = 0 u_{xt}+\alpha*e^{u}+\gamma*e^{-2*u}=0

Dodecahedral_number.html

  1. n ( 3 n - 1 ) ( 3 n - 2 ) 2 {n(3n-1)(3n-2)\over 2}

Domain_adaptation.html

  1. X X
  2. Y Y
  3. h : X Y h:X\to Y
  4. Y Y
  5. X X
  6. S = { ( x i , y i ) } i = 1 m ( X × Y ) m S=\{(x_{i},y_{i})\}_{i=1}^{m}\in(X\times Y)^{m}
  7. ( x i , y i ) S (x_{i},y_{i})\in S
  8. D S D_{S}
  9. X × Y X\times Y
  10. h h
  11. S S
  12. D S D_{S}
  13. D S D_{S}
  14. D T D_{T}
  15. X × Y X\times Y
  16. D S D_{S}
  17. D T D_{T}
  18. h h
  19. D T D_{T}
  20. h h
  21. h h

Dorfman–Steiner_theorem.html

  1. p A A p . q = p - c p . e A \frac{p_{A}A}{p.q}=\frac{p-c}{p}.e_{A}
  2. p A \ p_{A}
  3. A \ A
  4. p \ p
  5. q \ q
  6. c \ c
  7. e A \ e_{A}

Dose_from_radioactive_seeds.html

  1. S k S_{k}
  2. Λ \Lambda
  3. G ( r , θ ) G(r,\theta)
  4. g ( r ) g(r)
  5. F ( r , θ ) F(r,\theta)
  6. D ( r , θ ) = S k Λ G ( r , θ ) G ( r 0 , θ 0 ) g ( r ) F ( r , θ ) t D(r,\theta)=S_{k}\Lambda\frac{G(r,\theta)}{G(r_{0},\theta_{0})}g(r)F(r,\theta)t

Double-layer_capacitance.html

  1. E = U d = 2 V 0 , 4 nm = 5000 kV/mm E=\frac{U}{d}=\frac{2\ \,\text{V}}{0{,}4\ \,\text{nm}}=5000\ \,\text{kV/mm}
  2. C d = ϵ 4 π δ \ C_{d}=\frac{\epsilon}{4\pi\delta}
  3. C = ε A d C=\frac{\varepsilon A}{d}

Double-tuned_amplifier.html

  1. M = k L p L s M=k\sqrt{L_{\mathrm{p}}L_{\mathrm{s}}}
  2. 2 1 / n - 1 4 \sqrt[4]{2^{1/n}-1}
  3. ω 0 = ω 0 p = 1 L p C p = ω 0 s = 1 L s C s \omega_{0}=\omega_{0\mathrm{p}}={1\over\sqrt{L_{\mathrm{p}}C_{\mathrm{p}}}}=% \omega_{0\mathrm{s}}={1\over\sqrt{L_{\mathrm{s}}C_{\mathrm{s}}}}
  4. Q = Q p = 1 L p G o = Q s = 1 L s G i Q=Q_{\mathrm{p}}={1\over L_{\mathrm{p}}G_{\mathrm{o}}}=Q_{\mathrm{s}}={1\over L% _{\mathrm{s}}G_{\mathrm{i}}}
  5. A = A 0 2 k Q 4 Q δ - i ( 1 + k 2 Q 2 - 4 Q 2 δ 2 ) A=A_{0}\frac{2kQ}{4Q\delta-i(1+k^{2}Q^{2}-4Q^{2}\delta^{2})}
  6. i i
  7. A 0 = g m 2 G o G i A_{0}=\frac{g_{\mathrm{m}}}{2\sqrt{G_{\mathrm{o}}G_{\mathrm{i}}}}
  8. δ = ω - ω 0 ω 0 \delta=\frac{\omega-\omega_{0}}{\omega_{0}}
  9. δ H , δ L = ± 1 2 Q k 2 Q 2 - 1 \delta_{\mathrm{H}},\delta_{\mathrm{L}}=\pm{1\over 2Q}\sqrt{k^{2}Q^{2}-1}
  10. k 2 Q 2 - 1 = 0 k^{2}Q^{2}-1=0
  11. k = 1 Q k={1\over Q}

Double_exponential_moving_average.html

  1. 𝐷𝐸𝑀𝐴 = 2 * E M A - E M A ( E M A ) \,\textit{DEMA}={2*EMA-EMA(EMA)}

Double_vector_bundle.html

  1. T E TE
  2. E E
  3. T 2 M T^{2}M
  4. ( E , E H , E V , B ) (E,E^{H},E^{V},B)
  5. E H E^{H}
  6. E V E^{V}
  7. B B
  8. E E
  9. E H E^{H}
  10. E V E^{V}
  11. f E : E E f_{E}:E\mapsto E^{\prime}
  12. f H : E H E H f_{H}:E^{H}\mapsto E^{H}{}^{\prime}
  13. f V : E V E V f_{V}:E^{V}\mapsto E^{V}{}^{\prime}
  14. f B : B B f_{B}:B\mapsto B^{\prime}
  15. ( f E , f V ) (f_{E},f_{V})
  16. ( E , E V ) (E,E^{V})
  17. ( E , E V ) (E^{\prime},E^{V}{}^{\prime})
  18. ( f E , f H ) (f_{E},f_{H})
  19. ( E , E H ) (E,E^{H})
  20. ( E , E H ) (E^{\prime},E^{H}{}^{\prime})
  21. ( f V , f B ) (f_{V},f_{B})
  22. ( E V , B ) (E^{V},B)
  23. ( E V , B ) (E^{V}{}^{\prime},B^{\prime})
  24. ( f H , f B ) (f_{H},f_{B})
  25. ( E H , B ) (E^{H},B)
  26. ( E H , B ) (E^{H}{}^{\prime},B^{\prime})
  27. ( E , E H , E V , B ) (E,E^{H},E^{V},B)
  28. ( E , E V , E H , B ) (E,E^{V},E^{H},B)
  29. ( E , M ) (E,M)
  30. M M
  31. ( T E , E , T M , M ) (TE,E,TM,M)
  32. M M
  33. ( T T M , T M , T M , M ) (TTM,TM,TM,M)

Downside_beta.html

  1. r i r_{i}
  2. r m r_{m}
  3. i i
  4. m m
  5. u m u_{m}
  6. β - = c o v ( r i , r m | r m < u m ) v a r ( r m | r m < u m ) \beta^{-}=\frac{cov(r_{i},r_{m}|r_{m}<u_{m})}{var(r_{m}|r_{m}<u_{m})}
  7. β - \beta^{-}
  8. β + \beta^{+}
  9. i i

Draft:Aerodynamic_Whistles.html

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

Draft:Apprentissage_des_intérêts.html

  1. X = { x i } X=\{x_{i}\}\,\!
  2. Y = { y i | i = 1 , 2 , , k } Y=\{y_{i}|i=1,2,\cdots,k\}\,\!
  3. y i x y j y_{i}\succ_{x}y_{j}\,\!
  4. x x\,\!
  5. y i y_{i}\,\!
  6. y j y_{j}\,\!
  7. x x\,\!
  8. y i y_{i}\,\!
  9. j i , y i x y j \forall j\neq i,y_{i}\succ_{x}y_{j}\,\!
  10. x x\,\!
  11. L Y L\subseteq Y\,\!
  12. { y i x y j | y i L , y j Y \ L } \{y_{i}\succ_{x}y_{j}|y_{i}\in L,y_{j}\in Y\backslash L\}\,\!
  13. X X\,\!
  14. Y Y\,\!
  15. y 1 y 2 y k y_{1}\succ y_{2}\succ\cdots\succ y_{k}\,\!
  16. x l x_{l}\,\!
  17. y l y_{l}\,\!
  18. x i x j x_{i}\succ x_{j}\,\!
  19. A B A\succ B\,\!
  20. A A\,\!
  21. B B\,\!
  22. a a\,\!
  23. b b\,\!
  24. a > b a>b\,\!
  25. V ( A , B ) { 0 , 1 } V(A,B)\in\{0,1\}\,\!
  26. ( A , B ) (A,B)\,\!
  27. A B A\succ B\,\!
  28. B A B\succ A\,\!
  29. f : X × Y f:X\times Y\rightarrow\mathbb{R}\,\!
  30. y i x y j f ( x , y i ) > f ( x , y j ) y_{i}\succ_{x}y_{j}\Rightarrow f(x,y_{i})>f(x,y_{j})\,\!
  31. f : X f:X\rightarrow\mathbb{R}\,\!

Draft:Asymmetric_Simple_Inclusion_Process.html

  1. n n
  2. k = 1 , , n k=1,...,n
  3. k = 1 k=1
  4. λ \lambda
  5. k k
  6. μ k \mu_{k}
  7. n + 1 n+1
  8. 1 / λ 1/\lambda
  9. k k
  10. 1 / μ k 1/\mu_{k}
  11. k k
  12. k k
  13. k + 1 k+1
  14. k = n k=n
  15. n n

Draft:Basic_theorems_of_algebraic_K-theory.html

  1. C D C\subset D
  2. M N M\oplus N

Draft:Behrend's_formula.html

  1. C C
  2. # C = p 1 # Aut ( p ) , \#C=\sum_{p}{1\over\#\operatorname{Aut}(p)},
  3. 𝐅 q \mathbf{F}_{q}
  4. f : X X f:X\to X
  5. # X ( 𝐅 q ) = q dim X i 0 ( - 1 ) i tr ( f ; H i ( X ( 𝐅 q ) , l ) ) , \#X(\mathbf{F}_{q})=q^{\operatorname{dim}X}\sum_{i\geq 0}(-1)^{i}\operatorname% {tr}(f;H^{i}(X(\mathbf{F}_{q}),\mathbb{Q}_{l})),
  6. τ ( G ) \tau(G)
  7. # X ( 𝐅 q ) = τ ( G ) x 1 vol ( G ( 𝒪 x ) ) \#X(\mathbf{F}_{q})=\tau(G)\prod_{x}{1\over\operatorname{vol}(G(\mathcal{O}_{x% }))}

Draft:Bertheau's_law.html

  1. C = ( n ( n - 1 ) 2 ) 1. X C=(\frac{n(n-1)}{2})^{1.X}

Draft:Bertrand_curve.html

  1. 3 \mathbb{R}^{3}
  2. 3 \mathbb{R}^{3}
  3. r 1 ( t ) \vec{r}_{1}(t)
  4. r 2 ( t ) \vec{r}_{2}(t)
  5. 3 \mathbb{R}^{3}
  6. t t
  7. N 1 = N 2 \vec{N}_{1}=\vec{N}_{2}
  8. r 1 \vec{r}_{1}
  9. r 2 \vec{r}_{2}
  10. r 1 \vec{r}_{1}
  11. r 2 \vec{r}_{2}
  12. a κ + b τ = 1 a\kappa+b\tau=1
  13. a , b a,b
  14. a 0 a\neq 0

Draft:Blackbody_limit_for_solar_energy_conversion.html

  1. T s T_{s}
  2. T c T_{c}
  3. E s = W + Q c E_{s}=W+Q_{c}
  4. S s + S g = Q c / T c S_{s}+S_{g}=Q_{c}/T_{c}
  5. Q c Q_{c}
  6. W = E s - T c ( S s + S g ) W=E_{s}-T_{c}(S_{s}+S_{g})
  7. W = E s - T c ( E s / T s + S g ) W=E_{s}-T_{c}(E_{s}/T_{s}+S_{g})
  8. η = W / E s = 1 - E c / E s - Q c / E s \eta=W/E_{s}=1-E_{c}/E_{s}-Q_{c}/E_{s}
  9. η = W / E s = 1 - T c / T s - ( T c S g ) / E s \eta=W/E_{s}=1-T_{c}/T_{s}-(T_{c}S_{g})/E_{s}
  10. S s S_{s}
  11. d U = T d S - p d V dU=TdS-pdV
  12. S = d S = ( d U + p d V ) / T = ( u d V + p d V ) / T S=\int dS=\int(dU+pdV)/T=\int(udV+pdV)/T
  13. u u
  14. p p
  15. u = 8 π h ν 3 c 3 1 e h ν k T - 1 d ν = 4 π I c = a T 4 u=\int\frac{8\pi h\nu^{3}}{c^{3}}\frac{1}{e^{\frac{h\nu}{kT}-1}}d\nu=4\pi\frac% {I}{c}=aT^{4}
  16. h h
  17. k k
  18. c c
  19. ν \nu
  20. I I
  21. q = I = σ T 4 q=I=\sigma T^{4}
  22. σ \sigma
  23. h / λ h/\lambda
  24. F = 2 c o s θ / c I d A F=2cos\theta/cIdA
  25. θ \theta
  26. p = cos ( θ ) d f r a c Ω d A = 2 c I c o s ( θ ) 2 d Ω p=\int\cos(\theta)\frac{d}{frac{\Omega}{dA}}=\frac{2}{c}\int Icos(\theta)^{2}d\Omega
  27. p = 2 c 0 2 π d ϕ 0 π 2 I sin θ cos ( θ ) 2 d θ = 4 I π 3 c = u 3 p=\frac{2}{c}\int_{0}^{2}\pi d\phi\int_{0}^{\frac{\pi}{2}}I\sin\theta\cos(% \theta)^{2}d\theta=\frac{4I\pi}{3c}=\frac{u}{3}
  28. u u
  29. S = ( d U + p d V ) / T = ( u + u / 3 ) / T d V = ( a T 4 + a / 3 T 4 ) / T d V S=\int(dU+pdV)/T=\int(u+u/3)/TdV=\int(aT^{4}+a/3T^{4})/TdV
  30. s = S / V = ( 4 / 3 ) a T 3 s=S/V=(4/3)aT^{3}
  31. s = ( 4 / 3 ) σ T 3 s=(4/3)\sigma T^{3}
  32. T c T_{c}
  33. E s = W + Q c + E c E_{s}=W+Q_{c}+E_{c}
  34. S s + S g = Q c / T c + S c S_{s}+S_{g}=Q_{c}/T_{c}+S_{c}
  35. E c E_{c}
  36. S c S_{c}
  37. W = E s - T c ( S s + S g - S c ) - E c W=E_{s}-T_{c}(S_{s}+S_{g}-S_{c})-E_{c}
  38. E s E_{s}
  39. η = W / E s = 1 - T c / E s ( S s + S g - S c ) - E c / E s \eta=W/E_{s}=1-T_{c}/E_{s}(S_{s}+S_{g}-S_{c})-E_{c}/E_{s}
  40. S s S_{s}
  41. S c S_{c}
  42. η = 1 - T c / ( σ T s 4 ) ( 4 / 3 σ T s 3 + S g - 4 / 3 σ T c 3 ) - ( T c 4 ) / ( T s 4 ) \eta=1-T_{c}/(\sigma T_{s}^{4})(4/3\sigma T_{s}^{3}+S_{g}-4/3\sigma T_{c}^{3})% -(T_{c}^{4})/(T_{s}^{4})
  43. η = W / E s = 1 - T c / E s ( S s + S g - S c ) - E c / E s \eta=W/E_{s}=1-T_{c}/E_{s}(S_{s}+S_{g}-S_{c})-E_{c}/E_{s}
  44. η = ( 1 - 4 / 3 T c / T s ) + 1 / 3 ( T c 4 ) / ( T s 4 ) - ( T c S g ) / E s \eta=(1-4/3T_{c}/T_{s})+1/3(T_{c}^{4})/(T_{s}^{4})-(T_{c}S_{g})/E_{s}
  45. S g S_{g}
  46. T c T_{c}
  47. E s E_{s}
  48. S g e n , a b s = E s ( 1 / T c - 4 / 31 / T s ) S_{gen,abs}=E_{s}(1/T_{c}-4/31/T_{s})
  49. S g e n , e m s = E c ( 4 / 31 / T c - 1 / T c ) = 1 / 3 E c / T c S_{gen,ems}=E_{c}(4/31/T_{c}-1/T_{c})=1/3E_{c}/T_{c}
  50. 4 / 3 E s / T s + S g = Q c / T a + 4 / 3 E c / T c 4/3E_{s}/T_{s}+S_{g}=Q_{c}/T_{a}+4/3E_{c}/T_{c}
  51. E s / T s E_{s}/T_{s}
  52. 4 / 3 T a / T s + T a / E s S g = Q c / E s + 4 / 3 E c / E S T a / T c 4/3T_{a}/T_{s}+T_{a}/E_{s}S_{g}=Q_{c}/E_{s}+4/3E_{c}/E_{S}T_{a}/T_{c}
  53. T a T_{a}
  54. 4 / 3 T a / T s + T a / E s ( E s ( 1 / T c - 4 / 31 / T s ) + 1 / 3 E c / T s ) g = Q c / E s + 4 / 3 E c / E S T a / T c 4/3T_{a}/T_{s}+T_{a}/E_{s}(E_{s}(1/T_{c}-4/31/T_{s})+1/3E_{c}/T_{s})_{g}=Q_{c}% /E_{s}+4/3E_{c}/E_{S}T_{a}/T_{c}
  55. 4 / 3 T a / T s + T a / T c - 4 / 3 T a / T s + 1 / 3 E c / E s T a / T c = Q c / E s + 4 / 3 E c / E S T a / T c 4/3T_{a}/T_{s}+T_{a}/T_{c}-4/3T_{a}/T_{s}+1/3E_{c}/E_{s}T_{a}/T_{c}=Q_{c}/E_{s% }+4/3E_{c}/E_{S}T_{a}/T_{c}
  56. T a / T c + E c / E s T a / T c = Q c / E s = T a / T c ( 1 - ( T c 4 ) / ( T s 4 ) ) T_{a}/T_{c}+E_{c}/E_{s}T_{a}/T_{c}=Q_{c}/E_{s}=T_{a}/T_{c}(1-(T_{c}^{4})/(T_{s% }^{4}))
  57. η = W / E s = 1 - E c / E s - Q c / E s \eta=W/E_{s}=1-E_{c}/E_{s}-Q_{c}/E_{s}
  58. Q c / E s Q_{c}/E_{s}
  59. η = W / E s = 1 - ( T c 4 ) / ( T s 4 ) - T a / T c ( 1 - ( T c 4 ) / ( T s 4 ) ) \eta=W/E_{s}=1-(T_{c}^{4})/(T_{s}^{4})-T_{a}/T_{c}(1-(T_{c}^{4})/(T_{s}^{4}))
  60. η = W / E s = ( 1 - ( T c 4 ) / ( T s 4 ) ) ( 1 - T a / T c ) \eta=W/E_{s}=(1-(T_{c}^{4})/(T_{s}^{4}))(1-T_{a}/T_{c})

Draft:Bock's_Parametrization.html

  1. E E
  2. X 0 X_{0}
  3. λ \lambda
  4. x x
  5. Δ E ( x ) = E [ w G ( a , b x / X 0 ) + ( 1 - w ) G ( c , d x / λ ) ] \Delta E(x)=E\left[wG(a,bx/X_{0})+(1-w)G(c,dx/\lambda)\right]
  6. G ( q , p ) = 0 p t q - 1 e - t d t 0 t q - 1 e - t d t G(q,p)=\frac{\int_{0}^{p}t^{q-1}e^{-t}\,\,\text{d}t}{\int_{0}^{\infty}t^{q-1}e% ^{-t}\,\,\text{d}t}
  7. a = 0.6165 + 0.3183 ln E b = 0.2198 c = a d = 0.9099 - 0.0237 ln E w = 0.4634 \begin{aligned}\displaystyle a&\displaystyle=0.6165+0.3183\ln E\\ \displaystyle b&\displaystyle=0.2198\\ \displaystyle c&\displaystyle=a\\ \displaystyle d&\displaystyle=0.9099-0.0237\ln E\\ \displaystyle w&\displaystyle=0.4634\end{aligned}
  8. E E
  9. E - Δ E ( x ) E-\Delta E(x)
  10. G ( q , p ) = γ ( q , p ) / Γ ( q ) G(q,p)=\gamma(q,p)/\Gamma(q)
  11. γ ( q , p ) \gamma(q,p)
  12. Γ ( q ) \Gamma(q)

Draft:Bockstein_spectral_sequence.html

  1. 0 C 𝑝 C mod p C / p 0 0\to C\overset{p}{\to}C\overset{\,\text{mod }p}{\to}C\otimes\mathbb{Z}/p\to 0
  2. H * ( C ) i = p H * ( C ) 𝑗 H * ( C / p ) 𝑘 H_{*}(C)\overset{i=p}{\to}H_{*}(C)\overset{j}{\to}H_{*}(C\otimes\mathbb{Z}/p)% \overset{k}{\to}
  3. H * ( C ) s , t = H s - t ( C ) H_{*}(C)_{s,t}=H_{s-t}(C)
  4. H * ( C / p ) H_{*}(C\otimes\mathbb{Z}/p)
  5. deg i = ( 1 , - 1 ) \operatorname{deg}i=(1,-1)
  6. deg j = ( 0 , 0 ) \operatorname{deg}j=(0,0)
  7. deg k = ( - 1 , 0 ) \operatorname{deg}k=(-1,0)
  8. E s , t 1 = H s - t ( C / p ) E_{s,t}^{1}=H_{s-t}(C\otimes\mathbb{Z}/p)
  9. d 1 = j k {}^{1}d=j\circ k
  10. H * ( C ) p r - 1 p r - 1 H * ( C ) j r E r 𝑘 H_{*}(C)\overset{p^{r-1}}{\to}p^{r-1}H_{*}(C)\overset{{}^{r}j}{\to}E^{r}% \overset{k}{\to}
  11. j r {}^{r}j
  12. ( mod p ) p - r + 1 (\,\text{mod }p)\circ p^{-{r+1}}
  13. deg j r = ( - ( r - 1 ) , r - 1 ) \operatorname{deg}{}^{r}j=(-(r-1),r-1)

Draft:BVA-100.html

  1. ± % Desirable Weight = Actual Weight - Desirable Weight Desirable Weight × 100 \pm\%\,\text{ Desirable Weight}=\frac{\,\text{Actual Weight}-\,\text{Desirable% Weight}}{\,\text{Desirable Weight}}\times 100

Draft:Chern_class.html

  1. c i H 2 i ( B U ( n ) , ) c_{i}\in H^{2i}(BU(n),\mathbb{Z})
  2. c 0 = 1 c_{0}=1
  3. c i = 0 c_{i}=0
  4. c 1 c_{1}
  5. \mathbb{Z}
  6. i : B U ( n ) B U ( n + 1 ) i:BU(n)\to BU(n+1)
  7. i * c i ( n + 1 ) = c i ( n ) i^{*}c_{i}^{(n+1)}=c_{i}^{(n)}
  8. B U ( k ) × B U ( l ) B U ( k + l ) BU(k)\times BU(l)\hookrightarrow BU(k+l)
  9. i * c i = j = 0 i c i c j - i i^{*}c_{i}=\sum_{j=0}^{i}c_{i}\cup c_{j-i}
  10. H ( B U ( n ) , ) ( c 1 , , c n ) H^{\bullet}(BU(n),\mathbb{Z})\simeq\mathbb{Z}(c_{1},\ldots,c_{n})

Draft:Chirp:_Reducing_Spectral_Ripple.html

  1. Δ f p = 0.75 Δ F a n d δ = 1 / Δ F \Delta f_{p}=0.75\Delta F\qquad and\qquad\delta=1/\Delta F
  2. Δ f p = 0.73 Δ F a n d δ = 0.86 / Δ F \Delta f_{p}=0.73\Delta F\qquad and\qquad\delta=0.86/\Delta F
  3. | U ( ω ) | 2 = 0.42323 - 0.49755 c o s ( 2 π ω ω m a x ) + 0.07922 c o s ( 4 π ω ω m a x ) |U(\omega)|^{2}=0.42323-0.49755cos\left(\frac{2\pi\omega}{\omega_{max}}\right)% +0.07922cos\left(\frac{4\pi\omega}{\omega_{max}}\right)
  4. t T = 1 2 + ω ω m a x + 0.1871 s i n ( 2 π ω ω m a x ) + 0.014895 s i n ( 4 π ω ω m a x ) \frac{t}{T}=\frac{1}{2}+\frac{\omega}{\omega_{max}}+0.1871sin\left(\frac{2\pi% \omega}{\omega_{max}}\right)+0.014895sin\left(\frac{4\pi\omega}{\omega_{max}}\right)

Draft:Chow_group.html

  1. A k ( X ) = A d - k ( X ) A^{k}(X)=A_{d-k}(X)
  2. 0 d A k ( X ) \oplus_{0}^{d}A^{k}(X)
  3. Z k ( X ) Z_{k}(X)
  4. ( f ) = V W ord V ( f ) [ V ] (f)=\sum_{V\subset W}\operatorname{ord}_{V}(f)[V]
  5. ord V \operatorname{ord}_{V}
  6. Rat k ( X ) Z k ( X ) \operatorname{Rat}_{k}(X)\subset Z_{k}(X)
  7. Rat k ( X ) \operatorname{Rat}_{k}(X)
  8. x = ( f i ) x=\sum(f_{i})
  9. f i f_{i}
  10. A k ( X ) = Z k ( X ) / Rat k ( X ) ; A_{k}(X)=Z_{k}(X)/\operatorname{Rat}_{k}(X);
  11. A k ( X ) A_{k}(X)
  12. x y x\sim y
  13. [ X ] = V \sub X length 𝒪 X ( local ring at V ) [ V ] [X]=\sum_{V\sub X}\operatorname{length}_{\mathcal{O}_{X}}(\,\text{local ring % at }V)[V]
  14. f : X Y f:X\to Y
  15. f * : Z k ( X ) Z k ( Y ) f_{*}:Z_{k}(X)\to Z_{k}(Y)
  16. f * [ V ] = deg ( f | V ) [ f ( V ) ] f_{*}[V]=\operatorname{deg}(f|_{V})[f(V)]
  17. deg ( f | V ) \operatorname{deg}(f|_{V})
  18. f ( V ) f(V)
  19. x y f * ( x ) f * ( y ) x\sim y\Rightarrow f_{*}(x)\sim f_{*}(y)
  20. f * : A k ( X ) A k ( Y ) . f_{*}:A_{k}(X)\to A_{k}(Y).
  21. A * A_{*}
  22. F n G ( X ) = { [ F ] | dim Supp F n . } F^{n}G(X)=\{[F]|\dim\operatorname{Supp}F\leq n.\}
  23. gr 𝐆 ( X ) \operatorname{gr}\mathbf{G}(X)
  24. A * gr 𝐆 , [ V ] [ 𝒪 V ] A_{*}\to\operatorname{gr}\mathbf{G},\,[V]\mapsto[\mathcal{O}_{V}]
  25. 𝒪 V \mathcal{O}_{V}
  26. A k ( Z ) A k ( X ) A k ( X - Z ) 0 , A_{k}(Z)\to A_{k}(X)\to A_{k}(X-Z)\to 0,
  27. i : Z X i:Z\hookrightarrow X
  28. p : X X p:X^{\prime}\to X
  29. p : X - Z X - Z p:X^{\prime}-Z^{\prime}\to X-Z
  30. A k ( Z ) A k ( Z ) A k ( X ) A k ( X ) 0 A_{k}(Z^{\prime})\to A_{k}(Z)\oplus A_{k}(X^{\prime})\to A_{k}(X)\to 0
  31. p : E X p:E\to X
  32. p * : A k - r ( X ) A k ( E ) p^{*}:A_{k-r}(X)\to A_{k}(E)
  33. i t * : A k ( X × 1 ) A k - 1 ( X ) i^{*}_{t}:A_{k}(X\times\mathbb{P}^{1})\to A_{k-1}(X)
  34. D V D\cdot V
  35. A k - 1 ( Supp ( D ) V ) A_{k-1}(\operatorname{Supp}(D)\cap V)
  36. V Supp ( D ) V\not\subset\operatorname{Supp}(D)
  37. V Supp ( D ) V\subset\operatorname{Supp}(D)
  38. 𝒪 V ( C ) 𝒪 X ( D ) | V \mathcal{O}_{V}(C)\simeq\mathcal{O}_{X}(D)|_{V}
  39. Z k ( Y ) A k - 1 ( Supp ( D ) Y ) . Z_{k}(Y)\to A_{k-1}(\operatorname{Supp}(D)\cap Y).
  40. D x D\cdot x
  41. x 0 D x = 0 x\sim 0\Rightarrow D\cdot x=0
  42. A k ( Y ) A k - 1 ( Supp ( D ) Y ) A_{k}(Y)\to A_{k-1}(\operatorname{Supp}(D)\cap Y)
  43. Z k ( X ) A k - 1 ( D ) , x D x Z_{k}(X)\to A_{k-1}(D),\,x\mapsto D\cdot x
  44. D ( D x ) = ( D D ) X D\cdot(D^{\prime}\cdot x)=(D\cdot D^{\prime})\cdot X
  45. D 1 D k x D_{1}\cdot\dots\cdot D_{k}\cdot x
  46. ( D 1 D k x ) . (D_{1}\cdot\dots\cdot D_{k}\cdot x).
  47. cl : A * ( X ) H * ( X ) \operatorname{cl}:A_{*}(X)\to\operatorname{H}_{*}(X)
  48. i : X Y i:X\hookrightarrow Y
  49. C = Spec ( 0 I n / I n + 1 ) C=\operatorname{Spec}(\oplus_{0}^{\infty}I^{n}/I^{n+1})
  50. Pic ( X ) = H 1 ( X , 𝒪 X * ) = H 1 ( X , 𝒦 1 ) = A d - 1 ( X ) \operatorname{Pic}(X)=\operatorname{H}^{1}(X,\mathcal{O}_{X}^{*})=% \operatorname{H}^{1}(X,\mathcal{K}_{1})=A_{d-1}(X)
  51. 𝒦 p \mathcal{K}_{p}

Draft:Chow_variety.html

  1. V = Γ ( 𝐏 n , 𝒪 𝐏 n ( 1 ) ) V=\Gamma(\mathbf{P}^{n},\mathcal{O}_{\mathbf{P}^{n}}(1))
  2. 𝐏 n ˇ = 𝐏 ( V * ) . \check{\mathbf{P}^{n}}=\mathbf{P}(V^{*}).
  3. 𝐇𝐢𝐥𝐛 𝐂𝐲𝐜𝐥 , Z [ Z ] \mathbf{Hilb}\to\mathbf{Cycl},\,Z\mapsto[Z]
  4. M ¯ 0 , n \overline{M}_{0,n}
  5. Gr ( 2 , n ) \operatorname{Gr}(2,\mathbb{C}^{n})

Draft:Cobordism_ring.html

  1. Ω * S O = 0 Ω n S O \Omega^{SO}_{*}=\oplus_{0}^{\infty}\Omega^{SO}_{n}
  2. Ω n S O \Omega^{SO}_{n}
  3. Ω * O \Omega^{O}_{*}
  4. Ω n = π n M O \Omega_{n}=\pi_{n}MO

Draft:Component_size_distribution_(network_science).html

  1. k {\left\langle k\right\rangle}
  2. s {s}
  3. P s ( s k ) s - 1 s ! e - k s P_{s}\sim\frac{(s\left\langle k\right\rangle)^{s-1}}{s!}{e^{-\left\langle k% \right\rangle s}}
  4. k s - 1 {\left\langle k\right\rangle^{s-1}}
  5. e x p [ ( s - 1 ) l n k ] {exp\left[(s-1)ln\left\langle k\right\rangle\right]}
  6. s ! = 2 π s ( s e ) s {s!=\sqrt{2\pi s}\left(\frac{s}{e}\right)^{s}}
  7. s {s}
  8. P s s - 3 2 e - ( k - 1 ) s + ( s - 1 ) l n k ) {P_{s}\sim s^{-\frac{3}{2}}e^{-(\left\langle k\right\rangle-1)s+(s-1)ln\left% \langle k\right\rangle)}}
  9. s - 3 2 {s^{-\frac{3}{2}}}
  10. e - ( k - 1 ) s + ( s - 1 ) l n k ) {e^{-(\left\langle k\right\rangle-1)s+(s-1)ln\left\langle k\right\rangle)}}
  11. s {s}
  12. P s s - 3 2 e - ( k - 1 ) s + ( s - 1 ) l n k ) {P_{s}\sim s^{-\frac{3}{2}}e^{-(\left\langle k\right\rangle-1)s+(s-1)ln\left% \langle k\right\rangle)}}
  13. k = 1 {\left\langle k\right\rangle=1}
  14. P s {P_{s}}
  15. P s s - 3 2 {P_{s}\sim s^{-\frac{3}{2}}}
  16. ( f c ) {(f_{c})}
  17. f c {f_{c}}
  18. f t {f_{t}}
  19. t {t}
  20. C S t ( s ) = s n s / s s n s {CS_{t}(s)=sn_{s}/\sum_{s}sn_{s}}
  21. s {s}
  22. n s {n_{s}}
  23. s {s}
  24. t = t n {t=t_{n}}
  25. C S t ( s ) {CS_{t}(s)}
  26. t n {t_{n}}
  27. t n {t_{n}}
  28. f t n {f_{t_{n}}}

Draft:Continuous_Proportion.html

  1. R = m 3 - m 2 m 2 - m 1 generalized… R = m n + 1 - m n m n - m n - 1 R=\frac{m_{3}-m_{2}}{m_{2}-m_{1}}\qquad\,\text{generalized...}\qquad R=\frac{m% _{n+1}-m_{n}}{m_{n}-m_{n-1}}
  2. m 3 = 2 m 2 - m 1 generalized… m n + 1 = 2 m n - m n - 1 Example: 1 , 2 , 3 , , m n - 1 , m n , m n + 1 m_{3}=2m_{2}-m_{1}\qquad\,\text{generalized...}\qquad m_{n+1}=2m_{n}-m_{n-1}% \qquad\,\text{Example: }1,2,3,...,m_{n-1},m_{n},m_{n+1}
  3. R = m 3 - m 2 m 2 - m 1 = m 2 m 1 = m 3 m 2 generalized… R = m n + 1 - m n m n - m n - 1 = m n m n - 1 = m n + 1 m n R=\frac{m_{3}-m_{2}}{m_{2}-m_{1}}=\frac{m_{2}}{m_{1}}=\frac{m_{3}}{m_{2}}% \qquad\,\text{generalized...}\qquad R=\frac{m_{n+1}-m_{n}}{m_{n}-m_{n-1}}=% \frac{m_{n}}{m_{n-1}}=\frac{m_{n+1}}{m_{n}}
  4. m n m n - 1 = m n + 1 m n or m n 2 = m n + 1 m n - 1 \frac{m_{n}}{m_{n-1}}=\frac{m_{n+1}}{m_{n}}\qquad\,\text{or}\qquad m_{n}^{2}=m% _{n+1}m_{n-1}
  5. 2 1 = 4 2 = 8 4 = 16 8 = 32 16 = 64 32 = or 2 1 2 0 = 2 2 2 1 = 2 3 2 2 = 2 4 2 3 = 2 5 2 4 = 2 6 2 5 = 2 n + 1 2 n \frac{2}{1}=\frac{4}{2}=\frac{8}{4}=\frac{16}{8}=\frac{32}{16}=\frac{64}{32}=.% ..\qquad\,\text{or}\qquad\frac{2^{1}}{2^{0}}=\frac{2^{2}}{2^{1}}=\frac{2^{3}}{% 2^{2}}=\frac{2^{4}}{2^{3}}=\frac{2^{5}}{2^{4}}=\frac{2^{6}}{2^{5}}=\frac{2^{n+% 1}}{2^{n}}
  6. Let m n + 1 = m n + m n - 1 m n m n - 1 = m n + m n - 1 m n ( m n m n - 1 ) 2 = m n m n - 1 + 1 \,\text{Let }\quad m_{n+1}=m_{n}+m_{n-1}\quad\Rightarrow\quad\frac{m_{n}}{m_{n% -1}}=\frac{m_{n}+m_{n-1}}{m_{n}}\quad\Rightarrow\quad\left(\frac{m_{n}}{m_{n-1% }}\right)^{2}=\frac{m_{n}}{m_{n-1}}+1
  7. Let R = m n m n - 1 R 2 = R + 1 \,\text{Let }\quad R=\frac{m_{n}}{m_{n-1}}\quad\Rightarrow\quad R^{2}=R+1
  8. R = 1 ± 5 2 or φ + = 1 + 5 2 and φ - = 1 - 5 2 \therefore R=\frac{1\pm\sqrt{5}}{2}\qquad\,\text{or}\qquad\varphi_{+}=\frac{1+% \sqrt{5}}{2}\qquad\,\text{and}\qquad\varphi_{-}=\frac{1-\sqrt{5}}{2}
  9. Let m n = F n 0 , 1 , 1 , 2 , 3 , 5 , , F n - 1 , F n , F n - 1 + F n \text{Let }m_{n}=F_{n}\quad\Rightarrow\quad 0,1,1,2,3,5,...,F_{n-1},F_{n},F_{n% -1}+F_{n}
  10. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  11. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  12. F n = φ + n - φ - n φ + - φ - = φ + n - φ - n 5 and F n = F n + 1 - F n - 1 F_{n}=\frac{\varphi_{+}^{n}-\varphi_{-}^{n}}{\varphi_{+}-\varphi_{-}}=\frac{% \varphi_{+}^{n}-\varphi_{-}^{n}}{\sqrt{5}}\quad\,\text{and}\quad F_{n}=F_{n+1}% -F_{n-1}
  13. Let m n = L n 2 , 1 , 3 , 4 , 7 , 11 , , L n - 1 , L n , L n - 1 + L n \text{Let }m_{n}=L_{n}\quad\Rightarrow\quad 2,1,3,4,7,11,...,L_{n-1},L_{n},L_{% n-1}+L_{n}
  14. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  15. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  16. L n = φ + n + φ - n φ + + φ - = φ + n + φ - n 1 and L n = F n + 1 + F n - 1 L_{n}=\frac{\varphi_{+}^{n}+\varphi_{-}^{n}}{\varphi_{+}+\varphi_{-}}=\frac{% \varphi_{+}^{n}+\varphi_{-}^{n}}{1}\quad\,\text{and}\quad L_{n}=F_{n+1}+F_{n-1}
  17. m t + 1 s - 2 {}^{\ s-2}m_{t+1}\,\text{ }
  18. t - 2 m s + 1 \,\text{ }^{\ t-2}m_{s+1}
  19. m Δ t s s m t - 1 , s m t , s m t + 1 , {}^{s}m_{\Delta t}\ \Rightarrow\ ^{s}m_{t-1},\ ^{s}m_{t},\ ^{s}m_{t+1},\ ...
  20. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  21. where, s m t + 1 = s m t + s - 1 m t + 1 = 2 s m t - s m t - 1 + s - 2 m t + 1 \text{where, }\ ^{s}m_{t+1}=\ ^{s}m_{t}+\ ^{s-1}m_{t+1}=2\ ^{s}m_{t}-\ ^{s}m_{% t-1}+^{\ s-2}m_{t+1}
  22. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  23. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  24. Example: Let the o r i e n t a t i o n be s = 4 and the i n t e r v a l be t = 2 \text{Example: Let the}~{}orientation\ \,\text{be }\ s=4\ \text{ and the}~{}% interval\ \,\text{be }\ t=2
  25. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  26. 4 m 3 = 4 m 2 + 3 m 3 = 2 × 4 m 2 - 4 m 1 + 2 m 3 \Rightarrow\ ^{4}m_{3}=\ ^{4}m_{2}+\ ^{3}m_{3}=2\times\ ^{4}m_{2}-\ ^{4}m_{1}+% \ ^{2}m_{3}
  27. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  28. 4 m 3 = 4 + 6 = 2 × 4 - 1 + 3 = 10 \therefore\ \ ^{4}m_{3}=\ 4+6=2\times 4-1+3=10
  29. m Δ s t t m s - 1 , t m s , t m s + 1 , {}^{t}m_{\Delta s}\ \Rightarrow\ ^{t}m_{s-1},\ ^{t}m_{s},\ ^{t}m_{s+1},\ ...
  30. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  31. where, t m s + 1 = t m s + t - 1 m s + 1 = 2 t m s - t m s - 1 + t - 2 m s + 1 \text{where, }\ ^{t}m_{s+1}=\ ^{t}m_{s}+\ ^{t-1}m_{s+1}=2\ ^{t}m_{s}-\ ^{t}m_{% s-1}+^{\ t-2}m_{s+1}
  32. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  33. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  34. Example: Let the o r i e n t a t i o n be t = 3 and the i n t e r v a l be s = 1 \text{Example: Let the}~{}orientation\ \,\text{be }\ t=3\ \text{ and the}~{}% interval\ \,\text{be }\ s=1
  35. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  36. 3 m 2 = 3 m 1 + 2 m 2 = 2 × 3 m 1 - 3 m 0 + 1 m 2 \Rightarrow\ ^{3}m_{2}=\ ^{3}m_{1}+\ ^{2}m_{2}=2\times\ ^{3}m_{1}-\ ^{3}m_{0}+% \ ^{1}m_{2}
  37. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  38. 3 m 2 = 1 + 2 = 2 × 1 - 0 + 1 = 3 \therefore\ \ ^{3}m_{2}=\ 1+2=2\times 1-0+1=3
  39. R = m Σ m Σ - 1 = m Σ + 1 m Σ or 2 1 2 0 = 2 2 2 1 = 2 3 2 2 = 2 4 2 3 = 2 5 2 4 = 2 6 2 5 = 2 Σ 2 Σ - 1 = 2 Σ + 1 2 Σ \therefore\quad R=\frac{m_{\Sigma}}{m_{\Sigma-1}}=\frac{m_{\Sigma+1}}{m_{% \Sigma}}\qquad\,\text{or}\qquad\frac{2^{1}}{2^{0}}=\frac{2^{2}}{2^{1}}=\frac{2% ^{3}}{2^{2}}=\frac{2^{4}}{2^{3}}=\frac{2^{5}}{2^{4}}=\frac{2^{6}}{2^{5}}=\frac% {2^{\Sigma}}{2^{\Sigma-1}}=\frac{2^{\Sigma+1}}{2^{\Sigma}}\quad\circlearrowright
  40. R = m \Tau m \Tau - 1 = m \Tau + 1 m \Tau or 2 1 2 0 = 2 2 2 1 = 2 3 2 2 = 2 4 2 3 = 2 5 2 4 = 2 6 2 5 = 2 \Tau 2 \Tau - 1 = 2 \Tau + 1 2 \Tau \therefore\quad R=\frac{m_{\Tau}}{m_{\Tau-1}}=\frac{m_{\Tau+1}}{m_{\Tau}}% \qquad\,\text{or}\qquad\frac{2^{1}}{2^{0}}=\frac{2^{2}}{2^{1}}=\frac{2^{3}}{2^% {2}}=\frac{2^{4}}{2^{3}}=\frac{2^{5}}{2^{4}}=\frac{2^{6}}{2^{5}}=\frac{2^{\Tau% }}{2^{\Tau-1}}=\frac{2^{\Tau+1}}{2^{\Tau}}\quad\circlearrowleft
  41. Let interval n = Σ \Tau and Fibonacci number F n = E Σ \Tau E Σ \Tau = φ + Σ \Tau - φ - Σ \Tau φ + - φ - = E Σ \Tau + 1 - E Σ \Tau - 1 \,\text{Let interval }n=\tfrac{\Sigma}{\Tau}\ \,\text{ and Fibonacci number }F% _{n}=\overset{\rightharpoonup}{E}_{\frac{\Sigma}{\Tau}}\qquad\therefore\ % \overset{\rightharpoonup}{E}_{\frac{\Sigma}{\Tau}}=\frac{\varphi_{+}^{\frac{% \Sigma}{\Tau}}-\varphi_{-}^{\frac{\Sigma}{\Tau}}}{\varphi_{+}-\varphi_{-}}=% \overset{\rightharpoonup}{E}_{\frac{\Sigma}{\Tau}+1}-\overset{\rightharpoonup}% {E}_{\frac{\Sigma}{\Tau}-1}
  42. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  43. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  44. 0 , 1 , 1 , 2 , 3 , 5 , , E Σ \Tau - 1 , E Σ \Tau , E Σ \Tau + 1 , where E Σ \Tau + 1 = E Σ \Tau - 1 + E Σ \Tau \Rightarrow\quad 0,1,1,2,3,5,...,\overset{\rightharpoonup}{E}_{\frac{\Sigma}{% \Tau}-1},\overset{\rightharpoonup}{E}_{\frac{\Sigma}{\Tau}},\overset{% \rightharpoonup}{E}_{\frac{\Sigma}{\Tau}+1},\ \,\text{ where}\ \ \ \overset{% \rightharpoonup}{E}_{\frac{\Sigma}{\Tau}+1}=\overset{\rightharpoonup}{E}_{% \frac{\Sigma}{\Tau}-1}+\overset{\rightharpoonup}{E}_{\frac{\Sigma}{\Tau}}
  45. Let interval n = \Tau Σ and Fibonacci number F n = E \Tau Σ E \Tau Σ = φ + \Tau Σ - φ - \Tau Σ φ + - φ - = E \Tau Σ + 1 - E \Tau Σ - 1 \,\text{Let interval }n=\tfrac{\Tau}{\Sigma}\ \,\text{ and Fibonacci number }F% _{n}=\overset{\leftharpoondown}{E}_{\frac{\Tau}{\Sigma}}\qquad\therefore\ % \overset{\leftharpoondown}{E}_{\frac{\Tau}{\Sigma}}=\frac{\varphi_{+}^{\frac{% \Tau}{\Sigma}}-\varphi_{-}^{\frac{\Tau}{\Sigma}}}{\varphi_{+}-\varphi_{-}}=% \overset{\leftharpoondown}{E}_{\frac{\Tau}{\Sigma}+1}-\overset{% \leftharpoondown}{E}_{\frac{\Tau}{\Sigma}-1}
  46. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  47. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  48. 0 , 1 , 1 , 2 , 3 , 5 , , E \Tau Σ - 1 , E \Tau Σ , E \Tau Σ + 1 , where E \Tau Σ + 1 = E \Tau Σ - 1 + E \Tau Σ \Rightarrow\quad 0,1,1,2,3,5,...,\overset{\leftharpoondown}{E}_{\frac{\Tau}{% \Sigma}-1},\overset{\leftharpoondown}{E}_{\frac{\Tau}{\Sigma}},\overset{% \leftharpoondown}{E}_{\frac{\Tau}{\Sigma}+1},\ \,\text{ where}\ \ \ \overset{% \leftharpoondown}{E}_{\frac{\Tau}{\Sigma}+1}=\overset{\leftharpoondown}{E}_{% \frac{\Tau}{\Sigma}-1}+\overset{\leftharpoondown}{E}_{\frac{\Tau}{\Sigma}}
  49. Σ \Tau \tfrac{\Sigma}{\Tau}
  50. F I = φ + I + φ - I φ + + φ - = E I - 1 + E I + 1 \therefore\ \overset{\rightharpoonup}{F}\text{I}=\frac{\varphi_{+}\text{I}+% \varphi_{-}\text{I}}{\varphi_{+}+\varphi_{-}}=\overset{\rightharpoonup}{E}_{\,% \text{I}-1}+\overset{\rightharpoonup}{E}_{\,\text{I}+1}
  51. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  52. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  53. 2 , 1 , 3 , 4 , 7 , 11 , , F I - 1 , F I , F I + 1 , where F I + 1 = F I - 1 + F I \Rightarrow\quad 2,1,3,4,7,11,...,\overset{\rightharpoonup}{F}_{\,\text{I}-1},% \overset{\rightharpoonup}{F}\text{I},\overset{\rightharpoonup}{F}_{\,\text{I}+% 1},\ \,\text{where}\ \ \ \overset{\rightharpoonup}{F}_{\,\text{I}+1}=\overset{% \rightharpoonup}{F}_{\,\text{I}-1}+\overset{\rightharpoonup}{F}\text{I}
  54. F 2 I = F I 2 - 2 \overset{\rightharpoonup}{F}_{2\,\text{I}}=\overset{\rightharpoonup}{F}\text{I% }^{2}-2
  55. F 0 = 2 F 0 = F 0 2 - 2 = 2 2 - 2 = 2 \overset{\rightharpoonup}{F}_{0}=2\ \Rightarrow\ \overset{\rightharpoonup}{F}_% {0}=\overset{\rightharpoonup}{F}_{0}^{2}-2=2^{2}-2=2
  56. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  57. F 2 = 3 F 4 = F 2 2 - 2 = 3 2 - 2 = 7 \overset{\rightharpoonup}{F}_{2}=3\ \Rightarrow\ \overset{\rightharpoonup}{F}_% {4}=\overset{\rightharpoonup}{F}_{2}^{2}-2=3^{2}-2=7
  58. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  59. F 4 = 7 F 8 = F 4 2 - 2 = 7 2 - 2 = 47 \overset{\rightharpoonup}{F}_{4}=7\ \Rightarrow\ \overset{\rightharpoonup}{F}_% {8}=\overset{\rightharpoonup}{F}_{4}^{2}-2=7^{2}-2=47
  60. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  61. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  62. 2 ¯ , 1 , 3 , 4 , 7 ¯ , 11 , 18 , 29 , 47 ¯ , 76 , 123 , 199 , 322 ¯ , 521 , 843 , \Rightarrow\quad\underline{{2}},\ 1,\ 3,\ 4,\ \underline{{7}},\ 11,\ 18,\ 29,% \ \underline{{47}},\ 76,\ 123,\ 199,\ \underline{{322}},\ 521,\ 843,\ ...
  63. F 2 I = F I 2 + 2 \overset{\rightharpoonup}{F}_{2\,\text{I}}=\overset{\rightharpoonup}{F}\text{I% }^{2}+2
  64. F 1 = 1 F 2 = F 1 2 + 2 = 1 2 + 2 = 3 \overset{\rightharpoonup}{F}_{1}=1\ \ \Rightarrow\ \overset{\rightharpoonup}{F% }_{2}\ =\overset{\rightharpoonup}{F}_{1}^{2}+2=1^{2}+2\ =3
  65. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  66. F 3 = 4 F 6 = F 3 2 + 2 = 4 2 + 2 = 18 \overset{\rightharpoonup}{F}_{3}=4\ \ \Rightarrow\ \overset{\rightharpoonup}{F% }_{6}\ =\overset{\rightharpoonup}{F}_{3}^{2}+2=4^{2}+2\ =18
  67. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  68. F 5 = 11 F 10 = F 5 2 + 2 = 11 2 + 2 = 123 \overset{\rightharpoonup}{F}_{5}=11\ \Rightarrow\ \overset{\rightharpoonup}{F}% _{10}=\overset{\rightharpoonup}{F}_{5}^{2}+2=11^{2}+2=123
  69. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  70. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  71. 2 , 1 , 3 ¯ , 4 , 7 , 11 , 18 ¯ , 29 , 47 , 76 , 123 ¯ , 199 , 322 , 521 , 843 ¯ , \Rightarrow\quad 2,\ 1,\ \underline{{3}},\ 4,\ 7,\ 11,\ \underline{{18}},\ 29,% \ 47,\ 76,\ \underline{{123}},\ 199,\ 322,\ 521,\ \underline{{843}},\ ...
  72. E 2 I = E I F I \overset{\rightharpoonup}{E}_{2\,\text{I}}=\overset{\rightharpoonup}{E}\text{I% }\overset{\rightharpoonup}{F}\text{I}
  73. 0 ¯ , 1 , 1 ¯ , 2 , 3 ¯ , 5 , 8 ¯ , 13 , 21 ¯ , 34 , 55 ¯ , 89 , 144 ¯ , 233 , 377 ¯ , \underline{{0}},\ 1,\ \underline{{1}},\ 2,\ \underline{{3}},\ \ 5,\ \ % \underline{{8}},\ 13,\ \underline{{21}},\ 34,\ \ \underline{{55}},\ 89,\ % \underline{{144}},\ 233,\ \underline{{377}},\ ...
  74. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  75. 2 , 1 , 3 , 4 , 7 , 11 , 18 , 29 , 47 , 76 , 123 , 199 , 322 , 521 , 843 , 2,\ 1,\ 3,\ 4,\ 7,\ 11,\ 18,\ 29,\ 47,\ 76,\ 123,\ 199,\ 322,\ 521,\ 843,\ ...
  76. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  77. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  78. E 2 × 0 = E 0 = E 0 F 0 = 0 × 2 = 0 \Rightarrow\ \overset{\rightharpoonup}{E}_{2\times 0}=\overset{\rightharpoonup% }{E}_{0}=\overset{\rightharpoonup}{E}_{0}\overset{\rightharpoonup}{F}_{0}=0% \times 2=0
  79. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  80. E 2 × 1 = E 2 = E 1 F 1 = 1 × 1 = 1 \Rightarrow\ \overset{\rightharpoonup}{E}_{2\times 1}=\overset{\rightharpoonup% }{E}_{2}=\overset{\rightharpoonup}{E}_{1}\overset{\rightharpoonup}{F}_{1}=1% \times 1=1
  81. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  82. E 2 × 2 = E 4 = E 2 F 2 = 1 × 3 = 3 \Rightarrow\ \overset{\rightharpoonup}{E}_{2\times 2}=\overset{\rightharpoonup% }{E}_{4}=\overset{\rightharpoonup}{E}_{2}\overset{\rightharpoonup}{F}_{2}=1% \times 3=3
  83. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  84. E 2 × 3 = E 6 = E 3 F 3 = 2 × 4 = 8 \Rightarrow\ \overset{\rightharpoonup}{E}_{2\times 3}=\overset{\rightharpoonup% }{E}_{6}=\overset{\rightharpoonup}{E}_{3}\overset{\rightharpoonup}{F}_{3}=2% \times 4=8
  85. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  86. E 2 × 4 = E 8 = E 4 F 4 = 3 × 7 = 21 \Rightarrow\ \overset{\rightharpoonup}{E}_{2\times 4}=\overset{\rightharpoonup% }{E}_{8}=\overset{\rightharpoonup}{E}_{4}\overset{\rightharpoonup}{F}_{4}=3% \times 7=21
  87. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Draft:Contraction_theorem.html

  1. X X
  2. \mathbb{C}
  3. C C
  4. X X
  5. X X
  6. Y Y
  7. C C
  8. P P
  9. C C
  10. X C X\setminus C
  11. Y P Y\setminus P
  12. C C

Draft:Contributions_on_developping_measuring_tools_on_surfaces.html

  1. γ : ( a , b ) M \,\!\gamma:(a,b)\rightarrow M

Draft:Darrell_Kingee_Brown.html

  1. n = 0 x n n ! \sum_{n=0}^{\infty}\frac{x^{n}}{n!}

Draft:Deflection_Magnetometer.html

  1. τ D = m B c o s θ \tau_{D}=mBcos\theta
  2. τ R = m B H s i n θ \tau_{R}=mB_{H}sin\theta
  3. = > τ R = τ D =>\tau_{R}=\tau_{D}
  4. = > m B H s i n θ = m B c o s θ =>mB_{H}sin\theta=mBcos\theta
  5. B = B H t a n θ B=B_{H}tan\theta
  6. = > B = μ 0 4 π 2 M d ( d 2 - l 2 ) 2 =>B=\frac{\mu_{0}}{4\pi}\frac{2Md}{(d^{2}-l^{2})^{2}}
  7. B = B h t a n θ B=B_{h}tan\theta
  8. = > μ 0 4 π 2 M d ( d 2 - l 2 ) 2 = B h t a n θ =>\frac{\mu_{0}}{4\pi}\frac{2Md}{(d^{2}-l^{2})^{2}}=B_{h}tan\theta
  9. = > B = μ 0 4 π 2 M d ( d 2 + l 2 ) 3 2 =>B=\frac{\mu_{0}}{4\pi}\frac{2Md}{(d^{2}+l^{2})^{\frac{3}{2}}}
  10. B = B h t a n θ B=B_{h}tan\theta
  11. = > μ 0 4 π 2 M d ( d 2 + l 2 ) 3 2 = B h t a n θ =>\frac{\mu_{0}}{4\pi}\frac{2Md}{(d^{2}+l^{2})^{\frac{3}{2}}}=B_{h}tan\theta
  12. l 2 l^{2}
  13. = > μ 0 4 π M d 3 = B h t a n θ =>\frac{\mu_{0}}{4\pi}\frac{M}{d^{3}}=B_{h}tan\theta
  14. = > I θ + m B H s i n θ = 0 =>I\theta+mB_{H}sin\theta=0
  15. θ \theta
  16. = > T = 2 π l m B H =>T=2\pi\sqrt{\frac{l}{mB_{H}}}
  17. l = M L 2 + b 2 12 l=M\frac{L^{2}+b^{2}}{12}
  18. = > T 2 = 4 π 2 l m B H =>T^{2}=4\pi^{2}\frac{l}{mB_{H}}
  19. = > m B H = 4 π 2 l T 2 =>mB_{H}=4\pi^{2}\frac{l}{T^{2}}

Draft:Density_estimation_by_entropy_maximization_with_kernels.html

  1. p ( x ) p(x)
  2. x ( t ) , t = 1 , T x(t)\in\mathbb{R},t=1,\dots T
  3. max p ( x ) H ( p ( x ) ) = - - p ( x ) log p ( x ) d x s . t . - r i ( x ) p ( x ) d x = α i , for i = 0 , , M , \max_{p(x)}H(p(x))=-\int_{-\infty}^{\infty}p(x)\log p(x)\ dx\quad{\rm s.t.}% \int_{-\infty}^{\infty}r_{i}(x)p(x)\ dx=\alpha_{i},\,\text{ for }i=0,\dots,M,
  4. r i ( x ) r_{i}(x)
  5. α i = t = 1 T r i ( x t ) / T \alpha_{i}=\sum_{t=1}^{T}r_{i}(x_{t})/T
  6. i = 0 , , M i=0,\dots,M
  7. x x
  8. p p
  9. \mathbb{R}
  10. M + 1 M+1
  11. r i p = α i \int r_{i}p=\alpha_{i}
  12. p = 1 \int p=1
  13. r 0 = 1 r_{0}=1
  14. α 0 = 1 \alpha_{0}=1
  15. p p
  16. ( p ) = - p log p + i = 0 M λ i ( r i - α i ) p , \mathcal{L}(p)=-\int p\log p+\sum_{i=0}^{M}\lambda_{i}\int\left(r_{i}-\alpha_{% i}\right)p,
  17. λ i \lambda_{i}
  18. i = 0 , , M i=0,\dots,M
  19. p p
  20. L ( p ) / p = 0 {\partial L(p)}/{\partial p}=0
  21. p ( x ) = exp { - 1 + i = 0 M λ i r i ( x ) } , p(x)=\exp\left\{-1+\sum_{i=0}^{M}\lambda_{i}r_{i}(x)\right\},
  22. p p
  23. M + 1 M+1
  24. M + 1 M+1
  25. s y m b o l λ ( n + 1 ) = s y m b o l λ ( n ) - 𝐉 - 1 E p ( n ) { 𝐫 - s y m b o l α } , symbol{\lambda}^{(n+1)}=symbol{\lambda}^{(n)}-{\textbf{J}}^{-1}E_{p^{(n)}}% \left\{{\textbf{r}}-{symbol\alpha}\right\},
  26. p ( n ) p^{(n)}
  27. n n
  28. 𝐫 = [ r 0 , , r M ] {\textbf{r}}=[r_{0},\dots,r_{M}]^{\top}
  29. s y m b o l λ = [ λ 0 , , λ M ] M + 1 {symbol{\lambda}}=[\lambda_{0},\dots,\lambda_{M}]^{\top}\in\mathbb{R}^{M+1}
  30. s y m b o l α = [ α 0 , , α M ] {symbol{\alpha}}=[\alpha_{0},\dots,\alpha_{M}]^{\top}
  31. i i
  32. E p ( n ) { 𝐫 - s y m b o l α } E_{p^{(n)}}\left\{{\textbf{r}}-{symbol\alpha}\right\}
  33. E p ( n ) { r i - α i } = ( r i - α i ) p ( n ) , E_{p^{(n)}}\left\{r_{i}-\alpha_{i}\right\}=\int\left(r_{i}-\alpha_{i}\right)p^% {(n)},
  34. ( i , j ) (i,j)
  35. ${\textbf J}$
  36. 𝐉 i j = ( r i - α i ) p ( n ) λ j = r i r j p ( n ) . {\textbf{J}}_{ij}=\frac{\partial\int\left(r_{i}-\alpha_{i}\right)p^{(n)}}{% \partial\lambda_{j}}=\int r_{i}r_{j}p^{(n)}.
  37. R 1 R_{1}
  38. R 2 R_{2}
  39. R 3 R_{3}
  40. R 4 R_{4}
  41. x 4 x^{4}
  42. | x | 1 + | x | \frac{|x|}{1+|x|}
  43. x | x | 10 + | x | \frac{x|x|}{10+|x|}
  44. x 1 + x 2 \frac{x}{1+x^{2}}
  45. a b p = T a b / T \int_{a}^{b}p=T_{ab}/T
  46. 1 1
  47. x x
  48. x 2 x^{2}
  49. x / ( 1 + x 2 ) x/(1+x^{2})
  50. r ( x ) = exp ( - ( x - μ ) 2 2 σ 2 ) , r(x)=\exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right),
  51. p ( x ) = β 2 σ Γ ( 1 / β ) exp { - ( | x - μ | σ ) β } p(x)=\frac{\beta}{2\sigma\Gamma(1/\beta)}\exp\left\{-(\frac{|x-\mu|}{\sigma})^% {\beta}\right\}
  52. p ( x ) = 1 σ T t = 1 T 𝒩 ( x - x t σ ) p(x)=\frac{1}{\sigma T}\sum_{t=1}^{T}\mathcal{N}(\frac{x-x_{t}}{\sigma})
  53. p ( x ) = exp { - 1 + i = 0 M λ i r i ( x ) } p(x)=\exp\left\{-1+\sum_{i=0}^{M}\lambda_{i}r_{i}(x)\right\}
  54. T T
  55. M M
  56. T T
  57. T T
  58. M M
  59. μ 𝒰 ( - 2 , 2 ) \mu\sim\mathcal{U}(-2,2)
  60. σ 𝒰 ( 0.2 , 0.7 ) \sigma\sim\mathcal{U}(0.2,0.7)
  61. exp { 0.71 - 0.25 x - 1.07 x 2 - 0.5 x 1 + x 2 - 3.09 e - 2 ( x + 0.37 ) 2 } \exp\left\{0.71-0.25x-1.07x^{2}-0.5\frac{x}{1+x^{2}}-3.09e^{-2(x+0.37)^{2}}\right\}
  62. μ 𝒰 ( - 2 , 2 ) \mu\sim\mathcal{U}(-2,2)
  63. σ 𝒰 ( 0.2 , 0.7 ) \sigma\sim\mathcal{U}(0.2,0.7)
  64. - p * log p * / p \int_{-\infty}^{\infty}p_{*}\log p_{*}/p
  65. 512 × 512 512\times 512

Draft:Differential_graded_module.html

  1. d \mathbb{Z}^{d}
  2. R [ ϵ ] R[\epsilon]

Draft:Direct_limit_topology.html

  1. X n , Y n X_{n},Y_{n}
  2. lim X n × Y n = lim X n × lim Y n . \underrightarrow{\lim}X_{n}\times Y_{n}=\underrightarrow{\lim}X_{n}\times% \underrightarrow{\lim}Y_{n}.

Draft:Disparity_Filter_Algorithm_of_Weighted_Network.html

  1. ρ ( x ) d x = ( k - 1 ) ( 1 - x ) k - 2 d x \rho(x)dx=(k-1)(1-x)^{k-2}dx
  2. α i j = 1 - ( k - 1 ) 0 p i j ( 1 - x ) k - 2 d x \alpha_{ij}=1-(k-1)\int_{0}^{p_{ij}}(1-x)^{k-2}dx

Draft:Distributional_calculus.html

  1. δ ( ξ ) = e 2 π i x , ξ d x . \delta(\xi)=\int e^{2\pi i\langle x,\xi\rangle}\,dx.
  2. f t ( x ) = t 1 + t 2 x 2 f_{t}(x)={t\over 1+t^{2}x^{2}}
  3. f t , ϕ = - - 0 arctan ( t x ) ϕ ( x ) d x - 0 arctan ( t x ) ϕ ( x ) d x \langle f_{t},\phi\rangle=-\int_{-\infty}^{0}\arctan(tx)\phi^{\prime}(x)\,dx-% \int_{0}^{\infty}\arctan(tx)\phi^{\prime}(x)\,dx
  4. lim t f t , ϕ = π δ 0 , ϕ \displaystyle\lim_{t\to\infty}\langle f_{t},\phi\rangle=\langle\pi\delta_{0},\phi\rangle
  5. f t f_{t}
  6. t t\to\infty
  7. π δ 0 \pi\delta_{0}
  8. f ( x + i 0 ) f(x+i0)
  9. f ( x + i y ) f(x+iy)
  10. y 0 + y\to 0^{+}
  11. f ( x - i 0 ) f(x-i0)
  12. ( x - i 0 ) - 1 - ( x + i 0 ) - 1 = 2 π i δ 0 . (x-i0)^{-1}-(x+i0)^{-1}=2\pi i\delta_{0}.
  13. Γ N = [ - N - 1 / 2 , N + 1 / 2 ] 2 \Gamma_{N}=[-N-1/2,N+1/2]^{2}
  14. I N = def Γ N ϕ ^ ( z ) π cot ( π z ) d z = 2 π i - N N ϕ ^ ( n ) I_{N}\overset{\mathrm{def}}{=}\int_{\Gamma_{N}}\widehat{\phi}(z)\pi% \operatorname{cot}(\pi z)\,dz={2\pi i}\sum_{-N}^{N}\widehat{\phi}(n)
  15. - R R ϕ ^ ( ξ ) π cot ( π ξ ) d = - R R 0 ϕ ( x ) e - 2 π I x ξ d x d ξ + - R R - 0 ϕ ( x ) e - 2 π I x ξ d x d ξ = ϕ , cot ( - i 0 ) - cot ( - i 0 ) \begin{aligned}\displaystyle\int_{-R}^{R}\widehat{\phi}(\xi)\pi\operatorname{% cot}(\pi\xi)\,d&\displaystyle=\int_{-R}^{R}\int_{0}^{\infty}\phi(x)e^{-2\pi Ix% \xi}\,dxd\xi+\int_{-R}^{R}\int_{-\infty}^{0}\phi(x)e^{-2\pi Ix\xi}\,dxd\xi\\ &\displaystyle=\langle\phi,\cot(\cdot-i0)-\cot(\cdot-i0)\rangle\end{aligned}

Draft:Distributions_on_an_algebraic_group.html

  1. A k ( x ) A\to k(x)
  2. f ( I x n ) = 0 f(I_{x}^{n})=0
  3. k [ G ] Δ k [ G ] k [ G ] f g k k = k k[G]\overset{\Delta}{\to}k[G]\otimes k[G]\overset{f\otimes g}{\to}k\otimes k=k
  4. G × G G G\times G\to G
  5. 1 Δ Δ = Δ 1 Δ 1\otimes\Delta\circ\Delta=\Delta\otimes 1\circ\Delta
  6. Δ ( I 1 n ) r = 0 n I 1 r I 1 n - r \Delta(I_{1}^{n})\subset\sum_{r=0}^{n}I_{1}^{r}\otimes I^{n-r}_{1}
  7. k [ G ] k , ϕ ϕ ( 1 ) k[G]\to k,\phi\mapsto\phi(1)
  8. I 1 / I 1 2 I_{1}/I_{1}^{2}
  9. [ f , g ] = f * g - g * f [f,g]=f*g-g*f
  10. 𝔤 = Lie ( G ) \mathfrak{g}=\operatorname{Lie}(G)
  11. 𝔤 Dist ( G ) \mathfrak{g}\hookrightarrow\operatorname{Dist}(G)
  12. U ( 𝔤 ) Dist ( G ) U(\mathfrak{g})\to\operatorname{Dist}(G)
  13. G = 𝔾 a G=\mathbb{G}_{a}
  14. G = 𝔾 m G=\mathbb{G}_{m}
  15. H K Dist ( H ) Dist ( K ) H\subset K\Leftrightarrow\operatorname{Dist}(H)\subset\operatorname{Dist}(K)
  16. 𝔤 \mathfrak{g}

Draft:Divisor_(algebraic_geometry).html

  1. U i {U_{i}}
  2. f i f_{i}
  3. U i U_{i}
  4. f i f_{i}
  5. f i f_{i}
  6. O X ( D ) O_{X}(D)
  7. ( D ) \mathcal{L}(D)
  8. 1 𝒪 X * M X * M X * / 𝒪 X * 1 1\to\mathcal{O}^{*}_{X}\to M^{*}_{X}\to M^{*}_{X}/\mathcal{O}^{*}_{X}\to 1
  9. 1 Γ ( X , O X * ) Γ ( X , M X * ) Γ ( X , M X * / O X * ) 𝛿 H 1 ( X , O X * ) = Pic ( X ) . 1\to\Gamma(X,O^{*}_{X})\to\Gamma(X,M^{*}_{X})\to\Gamma(X,M^{*}_{X}/O^{*}_{X})% \overset{\delta}{\to}H^{1}(X,O^{*}_{X})=\operatorname{Pic}(X).
  10. CDiv ( X ) = Γ ( X , M X * / 𝒪 X * ) \operatorname{CDiv}(X)=\Gamma(X,M^{*}_{X}/\mathcal{O}^{*}_{X})
  11. CDiv ( U ) = Γ ( U , M U * / O U * ) = Γ ( U , M X * / O X * ) \operatorname{CDiv}(U)=\Gamma(U,M^{*}_{U}/O^{*}_{U})=\Gamma(U,M^{*}_{X}/O^{*}_% {X})
  12. CDiv \operatorname{CDiv}
  13. M X * / O X * M^{*}_{X}/O^{*}_{X}
  14. Γ ( X , M X * ) CDiv ( X ) \Gamma(X,M^{*}_{X})\to\operatorname{CDiv}(X)
  15. f ( f ) . f\mapsto(f).
  16. CCl ( X ) = CDiv ( X ) / { ( f ) | f Γ ( X , M X * ) } \operatorname{CCl}(X)=\operatorname{CDiv}(X)/\{(f)|f\in\Gamma(X,M^{*}_{X})\}
  17. Pic ( X ) \operatorname{Pic}(X)
  18. D O X ( D ) D\mapsto O_{X}(D)
  19. D ( D ) D\mapsto\mathcal{L}(D)
  20. ( f g ) = ( f ) + ( g ) . (fg)=(f)+(g).
  21. 𝒪 X X * \mathcal{O}_{X}\cap\mathcal{M}_{X}^{*}
  22. D D
  23. X X
  24. ( D ) \mathcal{L}(D)
  25. ( D ) \mathcal{L}(D)
  26. ( D ) \mathcal{L}(D)
  27. Γ ( X , ( D ) ) \mathbb{P}\Gamma(X,\mathcal{L}(D))
  28. D D
  29. X X
  30. k k
  31. Γ ( X , ( D ) ) \Gamma(X,\mathcal{L}(D))
  32. k k
  33. k k
  34. D D
  35. \mathbb{Q}
  36. \mathbb{Q}
  37. \mathbb{R}
  38. \mathbb{Q}
  39. \mathbb{Q}
  40. \mathbb{Q}
  41. \mathbb{Q}
  42. \mathbb{Q}
  43. D = a j Z j D=\sum a_{j}Z_{j}
  44. \mathbb{Q}
  45. [ D ] = [ a j ] Z j [D]=\sum[a_{j}]Z_{j}
  46. [ a j ] [a_{j}]
  47. a j a_{j}
  48. I ( D ) I(D)
  49. U i = Spec A i U_{i}=\operatorname{Spec}A_{i}
  50. f i A i f_{i}\in A_{i}
  51. D U i D\cap U_{i}
  52. f i = 0 f_{i}=0
  53. I ( D ) | U i = A / f i A I(D)|_{U_{i}}=A/f_{i}A
  54. D + D D+D^{\prime}
  55. f g = 0 fg=0
  56. R R R\to R^{\prime}
  57. D × R R D\times_{R}R^{\prime}
  58. X × R R X\times_{R}R^{\prime}
  59. f : X X f:X^{\prime}\to X
  60. D = D × X X D^{\prime}=D\times_{X}X^{\prime}
  61. I ( D ) = f * ( I ( D ) ) I(D^{\prime})=f^{*}(I(D))
  62. I ( D ) - 1 𝒪 X - I(D)^{-1}\otimes_{\mathcal{O}_{X}}-
  63. 0 I ( D ) 𝒪 X 𝒪 D 0 0\to I(D)\to\mathcal{O}_{X}\to\mathcal{O}_{D}\to 0
  64. 0 𝒪 X I ( D ) - 1 I ( D ) - 1 𝒪 D 0 0\to\mathcal{O}_{X}\to I(D)^{-1}\to I(D)^{-1}\otimes\mathcal{O}_{D}\to 0
  65. 𝒪 X \mathcal{O}_{X}
  66. I ( D ) - 1 I(D)^{-1}
  67. I ( D ) - 1 I(D)^{-1}
  68. L L
  69. 𝒪 X \mathcal{O}_{X}
  70. L / 𝒪 X L/\mathcal{O}_{X}
  71. s = 0 s=0
  72. Γ ( D , 𝒪 D ) \Gamma(D,\mathcal{O}_{D})
  73. deg D \operatorname{deg}D
  74. Spec R \operatorname{Spec}R
  75. D + D D+D^{\prime}
  76. deg ( D + D ) = deg ( D ) + deg ( D ) \operatorname{deg}(D+D^{\prime})=\operatorname{deg}(D)+\operatorname{deg}(D^{% \prime})
  77. f : X X f:X^{\prime}\to X
  78. deg ( f * D ) = deg ( f ) deg ( D ) \operatorname{deg}(f^{*}D)=\operatorname{deg}(f)\operatorname{deg}(D)
  79. deg ( D × R R ) = deg ( D ) \operatorname{deg}(D\times_{R}R^{\prime})=\operatorname{deg}(D)
  80. 0 E n E 0 F 0 0\to E_{n}\to\cdots\to E_{0}\to F\to 0
  81. 0 E n E 0 0 0\to E_{n}\to\cdots\to E_{0}\to 0

Draft:Divisor_class_group.html

  1. z n = x y z^{n}=xy

Draft:Energy_and_hydraulic_grade_line.html

  1. ρ g {\rho}g
  2. p + ρ v 2 2 + ρ g z = k = constant p+{\rho\ v^{2}\over 2}+{\rho}gz=\,\text{k}=\,\text{constant}
  3. g g\,
  4. v 2 2 g + p ρ g + z = H \,\frac{v^{2}}{2\,g}\,+\,\frac{p}{\rho g}\,+\,z\,=H\,
  5. v v\,
  6. g g\,
  7. z z\,
  8. p p\,
  9. ρ \rho\,
  10. v 2 2 g \frac{v^{2}}{2\,g}\,
  11. p ρ g \frac{p}{\rho g}\,
  12. z z\,
  13. z + p ρ g \ z\,+\,\frac{p}{\rho g}
  14. v 2 2 g \frac{v^{2}}{2\,g}\,
  15. v 2 2 g + p ρ g + z \,\frac{v^{2}}{2\,g}\,+\,\frac{p}{\rho g}\,+\,z\,
  16. v 2 2 g \frac{v^{2}}{2\,g}\,
  17. p ρ g \frac{p}{\rho g}\,

Draft:Exact_couple.html

  1. F p - 1 C F p C F_{p-1}C\subset F_{p}C
  2. gr C = - F p C / F p - 1 C , \operatorname{gr}C=\bigoplus_{-\infty}^{\infty}F_{p}C/F_{p-1}C,
  3. E p , q 0 = ( gr C ) p , q = ( F p C / F p - 1 C ) p + q . E^{0}_{p,q}=(\operatorname{gr}C)_{p,q}=(F_{p}C/F_{p-1}C)_{p+q}.
  4. 0 F p - 1 C F p C ( gr C ) p 0 0\to F_{p-1}C\to F_{p}C\to(\operatorname{gr}C)_{p}\to 0
  5. H n ( F p - 1 C ) 𝑖 H n ( F p C ) 𝑗 H n ( gr ( C ) p ) 𝑘 H n - 1 ( F p - 1 C ) \dots\to H_{n}(F_{p-1}C)\overset{i}{\to}H_{n}(F_{p}C)\overset{j}{\to}H_{n}(% \operatorname{gr}(C)_{p})\overset{k}{\to}H_{n-1}(F_{p-1}C)\to\dots
  6. D p , q = H p + q ( F p C ) , E p , q 1 = H p + q ( gr ( C ) p ) D_{p,q}=H_{p+q}(F_{p}C),\,E^{1}_{p,q}=H_{p+q}(\operatorname{gr}(C)_{p})
  7. D p - 1 , q + 1 𝑖 D p , q 𝑗 E p , q 1 𝑘 D p - 1 , q \dots\to D_{p-1,q+1}\overset{i}{\to}D_{p,q}\overset{j}{\to}E^{1}_{p,q}\overset% {k}{\to}D_{p-1,q}\to\dots
  8. E 1 E^{1}
  9. d = j k d=j\circ k
  10. E * , * r E^{r}_{*,*}
  11. E p , q r 𝑘 D p - 1 , q r j r E p - r , q + r - 1 r . E^{r}_{p,q}\overset{k}{\to}D^{r}_{p-1,q}\overset{{}^{r}j}{\to}E^{r}_{p-r,q+r-1}.
  12. d = j k d=j\circ k
  13. Z r = k - 1 ( im i r ) , B r = j ( ker i r ) Z^{r}=k^{-1}(\operatorname{im}i^{r}),\,B^{r}=j(\operatorname{ker}i^{r})
  14. E 1 E^{1}
  15. F p C F p C / F p - 1 C F_{p}C\to F_{p}C/F_{p-1}C
  16. [ x ¯ ] Z p , q r - 1 E p , q 1 [\overline{x}]\in Z^{r-1}_{p,q}\subset E^{1}_{p,q}
  17. k ( [ x ¯ ] ) = i r - 1 ( [ y ] ) k([\overline{x}])=i^{r-1}([y])
  18. [ y ] D p - r , q + r - 1 = H p + q - 1 ( F p C ) [y]\in D_{p-r,q+r-1}=H_{p+q-1}(F_{p}C)
  19. k ( [ x ¯ ] ) = [ d ( x ) ] k([\overline{x}])=[d(x)]
  20. ( F p C ) p + q (F_{p}C)_{p+q}
  21. d ( x ) - i r - 1 ( y ) = d ( x ) d(x)-i^{r-1}(y)=d(x^{\prime})
  22. x F p - 1 C x^{\prime}\in F_{p-1}C
  23. [ x ¯ ] Z p r x A p r [\overline{x}]\in Z^{r}_{p}\Leftrightarrow x\in A^{r}_{p}
  24. F p - 1 C F_{p-1}C
  25. Z p r ( A p r + F p - 1 C ) / F p - 1 C Z_{p}^{r}\simeq(A^{r}_{p}+F_{p-1}C)/F_{p-1}C
  26. ker ( i r - 1 : H p + q ( F p C ) H p + q ( F p + r - 1 C ) ) \operatorname{ker}(i^{r-1}:H_{p+q}(F_{p}C)\to H_{p+q}(F_{p+r-1}C))
  27. x d ( F p + r - 1 C ) x\in d(F_{p+r-1}C)
  28. ¯ \overline{\cdot}
  29. B p r - 1 = j ( ker i r - 1 ) ( d ( A p + r - 1 r - 1 ) + F p - 1 C ) / F p - 1 C B^{r-1}_{p}=j(\operatorname{ker}i^{r-1})\simeq(d(A^{r-1}_{p+r-1})+F_{p-1}C)/F_% {p-1}C
  30. A p r F p - 1 C = A p - 1 r - 1 A^{r}_{p}\cap F_{p-1}C=A^{r-1}_{p-1}
  31. E p , * r = Z p r - 1 B p r - 1 A p r + F p - 1 C d ( A p + r - 1 r - 1 ) + F p - 1 C A p r d ( A p + r - 1 r - 1 ) + A p - 1 r - 1 . E^{r}_{p,*}={Z^{r-1}_{p}\over B^{r-1}_{p}}\simeq{A^{r}_{p}+F_{p-1}C\over d(A^{% r-1}_{p+r-1})+F_{p-1}C}\simeq{A^{r}_{p}\over d(A^{r-1}_{p+r-1})+A^{r-1}_{p-1}}.
  32. \square
  33. \square
  34. K p , q K^{p,q}
  35. G p = i p K i , * G^{p}=\bigoplus_{i\geq p}K^{i,*}
  36. 0 G p + 1 G p K p , * 0 0\to G^{p+1}\to G^{p}\to K^{p,*}\to 0
  37. D p , q 𝑗 E 1 p , q 𝑘 \dots\to D^{p,q}\overset{j}{\to}E_{1}^{p,q}\overset{k}{\to}\dots
  38. F E B F\to E\to B

Draft:Faithfully_flat_descent.html

  1. A 2 A^{\otimes 2}
  2. α : N B A A B N \alpha:N\otimes_{B}A\simeq A\otimes_{B}N
  3. A B B A A B A A\simeq B\otimes_{B}A\to A\otimes_{B}A

Draft:Fiber_sequence.html

  1. F f Ff
  2. f : X Y f:X\to Y

Draft:Finite_field.html

  1. { n 1 | n 0 } \{n\cdot 1|n\neq 0\}
  2. ( x + y ) p = x p + y p (x+y)^{p}=x^{p}+y^{p}
  3. x x
  4. y y
  5. p p
  6. ( x + y ) p (x+y)^{p}
  7. p p
  8. 𝐅 p m \mathbf{F}_{p^{m}}
  9. X p m - X X^{p^{m}}-X
  10. 𝐅 p m \mathbf{F}_{p^{m}}
  11. p m p^{m}
  12. ( x + y ) p m = x p m + y p m = x + y (x+y)^{p^{m}}=x^{p^{m}}+y^{p^{m}}=x+y
  13. X p m - X X^{p^{m}}-X
  14. 𝐅 p \mathbf{F}_{p}
  15. F , n n 1 \mathbb{Z}\to F,\,n\mapsto n\cdot 1
  16. / p 𝐅 p \mathbb{Z}/p\mathbb{Z}\simeq\mathbf{F}_{p}
  17. 𝐅 p \mathbf{F}_{p}
  18. F = 𝐅 q F=\mathbf{F}_{q}
  19. 𝐅 p \mathbf{F}_{p}
  20. / p \mathbb{Z}/p\mathbb{Z}
  21. 𝐅 p n \mathbf{F}_{p^{n}}
  22. 𝐅 p ¯ \overline{\mathbf{F}_{p}}
  23. 𝐅 p ¯ \overline{\mathbf{F}_{p}}
  24. 𝐅 p n \mathbf{F}_{p^{n}}
  25. 𝐅 p \mathbf{F}_{p}
  26. Gal ( F / 𝔽 p ) \operatorname{Gal}(F/\mathbb{F}_{p})
  27. 𝔽 p \mathbb{F}_{p}
  28. σ : x x p \sigma:x\mapsto x^{p}
  29. Gal ( F / 𝔽 p ) / n , σ m m \operatorname{Gal}(F/\mathbb{F}_{p})\simeq\mathbb{Z}/n\mathbb{Z},\,\sigma^{m}\mapsto m
  30. K Gal ( F / K ) = { ϕ : F F | ϕ ( x ) = x , x K } K\to\operatorname{Gal}(F/K)=\{\phi:F\to F|\phi(x)=x,x\in K\}

Draft:Forces_between_polymer-coated_surfaces.html

  1. ϕ \phi
  2. V p l a t e = ( 64 c * Γ 0 2 k b T κ ) e - κ h V_{plate}=\left(\frac{64c^{*}\Gamma_{0}^{2}k_{b}T}{\kappa}\right)e^{-\kappa h}
  3. V p l a t e V_{plate}
  4. κ \kappa
  5. h h
  6. c * c^{*}
  7. k b k_{b}
  8. T T
  9. Γ 0 \Gamma_{0}
  10. Γ 0 = tanh ( z e ϕ 0 4 k b T ) \Gamma_{0}=\tanh\left(\frac{ze\phi_{0}}{4k_{b}T}\right)
  11. z z
  12. ϕ 0 \phi_{0}
  13. χ \chi
  14. Δ m i x S \Delta_{mix}S
  15. ϕ i \phi_{i}
  16. \Chi i \Chi_{i}
  17. Δ m i x H = 0 \Delta_{mix}H=0
  18. χ 0 \chi\neq 0
  19. Δ m i x G o \Delta_{mix}G^{o}
  20. Δ m i x G = R T [ n s ln ϕ s + n p ln ϕ p + n s ϕ p χ ] \Delta_{mix}G=RT[\,n_{s}\ln\phi_{s}+n_{p}\ln\phi_{p}+n_{s}\phi_{p}\chi\,]\,
  21. R R
  22. T T
  23. n i n_{i}
  24. ϕ i \phi_{i}
  25. χ \chi
  26. χ = ( δ s - δ p ) 2 V s e g R T \chi=(\delta_{s}-\delta_{p})^{2}\frac{V_{seg}}{RT}\,
  27. V s e g V_{seg}
  28. δ i \delta_{i}
  29. χ Δ m i x H \chi\propto\Delta_{mix}H
  30. Δ m i x H \Delta_{mix}H
  31. χ < 0 \chi<0
  32. χ = 0 \chi=0
  33. χ = 0.5 \chi=0.5
  34. χ > 0.5 \chi>0.5
  35. Θ \Theta
  36. Γ ( m o l m 2 ) \Gamma\left(\frac{mol}{m^{2}}\right)
  37. Θ = ω Γ \Theta=\omega\Gamma
  38. ω \omega
  39. ( m 2 m o l ) \left(\frac{m^{2}}{mol}\right)
  40. Γ \Gamma
  41. R g R_{g}
  42. Γ < 1 R g 2 \Gamma<\frac{1}{R_{g}^{2}}
  43. Γ 1 R g 2 \Gamma>>\frac{1}{R_{g}^{2}}
  44. x 3 2 R g ; x\leq 3\sqrt{2}R_{g};
  45. Π ( x ) = k B T Γ x ( 2 π 2 R g 2 x 2 - 1 ) \Pi(x)=\frac{k_{B}T\Gamma}{x}\left(\frac{2\pi^{2}R_{g}^{2}}{x^{2}}-1\right)
  46. x > 3 2 R g ; x>3\sqrt{2}R_{g};
  47. Π ( x ) = k B T Γ x R g 2 e - ( x 2 R g ) 2 \Pi(x)=\frac{k_{B}T\Gamma x}{R_{g}^{2}}e^{-\left(\frac{x}{2R_{g}}\right)^{2}}
  48. x < 2 L 0 ; x<2L_{0};
  49. Π ( x ) = k B T Γ 3 / 2 [ ( 2 L 0 x ) 9 / 4 - ( x 2 L 0 ) 3 / 4 ] \Pi(x)=k_{B}T\Gamma^{3/2}\left[\left(\frac{2L_{0}}{x}\right)^{9/4}-\left(\frac% {x}{2L_{0}}\right)^{3/4}\right]
  50. Π ( x ) \Pi(x)
  51. x x
  52. k B k_{B}
  53. T T
  54. Γ \Gamma
  55. x x
  56. R g R_{g}
  57. L 0 L_{0}

Draft:Formulae_(software).html

  1. π \pi
  2. NOT(A AND (B OR C)) \mbox{NOT(A AND (B OR C))}~{}
  3. ¬ ( A ( B C ) ) \lnot(\mbox{A}~{}\land(\mbox{B}~{}\lor\mbox{C}~{}))
  4. - ( A ( B + C ) ) -(\mbox{A}~{}(\mbox{B}~{}+\mbox{C}~{}))
  5. A ( B + C ) ¯ \overline{\mbox{A}~{}\cdot(\mbox{B}~{}+\mbox{C}~{})}
  6. 2 + 3 2+3
  7. 5 5

Draft:Foundations_of_algebraic_geometry.html

  1. G = Spec A G=\operatorname{Spec}A
  2. Lie ( G ) \operatorname{Lie}(G)
  3. α \alpha
  4. α ( D ) f = ( D f ) ( e ) \alpha(D)f=(Df)(e)
  5. π 1 ét X \pi_{1}^{\,\text{ét}}X
  6. X ( ) X(\mathbb{C})
  7. X = Spec F X=\operatorname{Spec}F
  8. π 1 ét X = Gal ( F ¯ / F ) \pi_{1}^{\,\text{ét}}X=\operatorname{Gal}(\overline{F}/F)
  9. c 1 ( L ) c_{1}(L)
  10. c 1 ( L ) F = F - L - 1 F c_{1}(L)F=F-L^{-1}\otimes F

Draft:Functorial_point.html

  1. R X ( R ) R\mapsto X(R)
  2. A = B [ t 1 , , t n ] / ( f 1 , , f m ) A=B[t_{1},\dots,t_{n}]/(f_{1},\dots,f_{m})
  3. B [ t 1 , , t n ] R , t i r i B[t_{1},\dots,t_{n}]\to R,\,t_{i}\mapsto r_{i}
  4. X ( R ) = { ( r 1 , , r n ) R n | f 1 ( r 1 , , r n ) = = f m ( r 1 , , r n ) = 0 } X(R)=\{(r_{1},\dots,r_{n})\in R^{n}|f_{1}(r_{1},\dots,r_{n})=\cdots=f_{m}(r_{1% },\dots,r_{n})=0\}
  5. 𝒪 X , x \mathcal{O}_{X,x}
  6. 𝔭 \mathfrak{p}
  7. Spec ( A / 𝔭 ) \operatorname{Spec}(A/\mathfrak{p})
  8. X = Spec ( A ) X=\operatorname{Spec}(A)
  9. A k ( x ) A\to k(x)
  10. A A 𝔭 k ( 𝔭 ) A\to A_{\mathfrak{p}}\to k(\mathfrak{p})
  11. x = 𝔭 x=\mathfrak{p}
  12. Spec ( R ) X R = def X × Spec ( B ) Spec ( R ) \operatorname{Spec}(R)\to X_{R}\overset{\mathrm{def}}{=}X\times_{\operatorname% {Spec}(B)}\operatorname{Spec}(R)
  13. X R Spec ( R ) X_{R}\to\operatorname{Spec}(R)
  14. | S | X R |S|\subset X_{R}
  15. R X ( R ) R\mapsto X(R)
  16. R Y ( R ) R\mapsto Y(R)

Draft:Generic_rank.html

  1. M R Q ( R ) M\otimes_{R}Q(R)
  2. Q ( R ) Q(R)

Draft:Geometry_of_an_algebraic_curve.html

  1. X i ~ \widetilde{X_{i}}
  2. X i X_{i}
  3. χ ( 𝒪 X ) = 1 ν χ ( 𝒪 X i ~ ) + δ = 1 ν ( 1 - g i ) + δ . \chi(\mathcal{O}_{X})=\sum_{1}^{\nu}\chi(\mathcal{O}_{\widetilde{X_{i}}})+% \delta=\sum_{1}^{\nu}(1-g_{i})+\delta.
  4. 0 * ( * ) r Γ ( X , ) Pic ( X ) Pic ( X ~ ) 0. 0\to\mathbb{C}^{*}\to(\mathbb{C}^{*})^{r}\to\Gamma(X,\mathcal{F})\to% \operatorname{Pic}(X)\to\operatorname{Pic}(\widetilde{X})\to 0.
  5. π : X ~ X \pi:\widetilde{X}\to X
  6. = π * 𝒪 X ~ / 𝒪 X \mathcal{F}=\pi_{*}\mathcal{O}_{\widetilde{X}}/\mathcal{O}_{X}
  7. Pic ( X ) = H 1 ( X , 𝒪 X * ) \operatorname{Pic}(X)=\operatorname{H}^{1}(X,\mathcal{O}_{X}^{*})
  8. Pic ( X ~ ) = H 1 ( X ~ , 𝒪 X ~ * ) = H 1 ( X , π * 𝒪 X ~ * ) . \operatorname{Pic}(\widetilde{X})=\operatorname{H}^{1}(\widetilde{X},\mathcal{% O}_{\widetilde{X}}^{*})=\operatorname{H}^{1}(X,\pi_{*}\mathcal{O}_{\widetilde{% X}}^{*}).
  9. deg : Pic ( X ) H 2 ( X ; ) r \operatorname{deg}:\operatorname{Pic}(X)\to\operatorname{H}^{2}(X;\mathbb{Z})% \simeq\mathbb{Z}^{r}
  10. 0 * ( * ) r Γ ( X ~ , ) J ( X ) J ( X ~ ) 0. 0\to\mathbb{C}^{*}\to(\mathbb{C}^{*})^{r}\to\Gamma(\widetilde{X},\mathcal{F})% \to J(X)\to J(\widetilde{X})\to 0.
  11. π : 𝒳 S \pi:\mathcal{X}\to S
  12. π \pi
  13. π - 1 ( t ) \pi^{-1}(t)
  14. ord p \operatorname{ord}_{p}
  15. a 0 ( V , p ) < a 1 ( V , p ) < a r ( V , p ) . a_{0}(V,p)<a_{1}(V,p)<\cdots a_{r}(V,p).
  16. a 0 ( V , p ) a_{0}(V,p)
  17. a i ( V , p ) a_{i}(V,p)
  18. ϕ V \phi_{V}
  19. b i ( V , p ) = a i ( V , p ) - i b_{i}(V,p)=a_{i}(V,p)-i
  20. d , g \mathcal{H}_{d,g}
  21. C , π : C 𝐏 1 C,\pi:C\to\mathbf{P}^{1}

Draft:GHz_Frequency_Ultrasonic_Interferometry.html

  1. K s K_{s}
  2. v p v_{p}
  3. v s v_{s}
  4. t t
  5. f f
  6. 1 2 \frac{1}{2}
  7. t t
  8. t t
  9. v v
  10. v v
  11. 2 l t \frac{2l}{t}
  12. l l
  13. K s K_{s}
  14. ρ \rho
  15. ( v p 2 - 4 3 v s 2 ) \left(v_{p}^{2}-\frac{4}{3}v_{s}^{2}\right)
  16. G G
  17. ρ v s 2 \rho v_{s}^{2}
  18. ρ \rho

Draft:Graph_Sparsification.html

  1. L G L_{G}
  2. G G
  3. ϵ \epsilon
  4. G G
  5. ( 1 - ϵ ) x t L H x x t L G x ( 1 - ϵ ) x t L H x (1-\epsilon)x^{t}L_{H}x\leq x^{t}L_{G}x\leq(1-\epsilon)x^{t}L_{H}x
  6. x x
  7. H H
  8. ϵ \epsilon
  9. G G

Draft:Hamiltonian_group_action.html

  1. ( M , ω ) (M,\omega)
  2. μ : M 𝔤 * \mu:M\to\mathfrak{g}^{*}
  3. X 𝔤 X\in\mathfrak{g}
  4. d μ X = ι X # ω , d\mu^{X}=\iota_{X^{\#}}\omega,
  5. μ X : M , p μ ( p ) , X \mu^{X}:M\to\mathbb{R},p\mapsto\langle\mu(p),X\rangle
  6. X # X^{\#}
  7. X X

Draft:Harder–Narasimhan_stratification.html

  1. 𝐅 q \mathbf{F}_{q}
  2. 𝐅 q \mathbf{F}_{q}
  3. d ( P ) : X ( G ) G d(P):X(G)\to G
  4. χ \chi
  5. P × G , χ 𝒪 X P\times_{G,\chi}\mathcal{O}_{X}
  6. P P
  7. Bun G \operatorname{Bun}_{G}
  8. Bun G d \operatorname{Bun}_{G}^{d}
  9. d : X ( G ) d:X(G)\to\mathbb{Z}

Draft:Hawkes_process.html

  1. d d
  2. N i ( t ) := j 1 1 t i , j t N_{i}\left(t\right):=\sum_{j\geq 1}1_{t_{i,j}\leq t}
  3. N t = [ N 1 ( t ) , , N d ( t ) ] T d N_{t}=\left[N_{1}\left(t\right),...,N_{d}\left(t\right)\right]^{T}\in\mathbb{N% }^{d}
  4. t 0 t\geq 0
  5. A = ( a i j ) A=\left(a_{ij}\right)
  6. d × d d\times d
  7. a i j 0 a_{ij}\geq 0
  8. j j
  9. i i
  10. A A
  11. i i
  12. j j
  13. j j
  14. i i
  15. μ i 0 \mu_{i}\geq 0
  16. i i
  17. h i , j : + + h_{i,j}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}
  18. j j
  19. i i
  20. t t
  21. h i j ( t ) = e - β i , j t h_{ij}\left(t\right)=e^{-\beta_{i,j}t}
  22. θ := ( μ , A ) \theta:=\left(\mu,A\right)
  23. λ i , θ ( t ) \lambda_{i,\theta}\left(t\right)
  24. N i N_{i}
  25. t t
  26. θ \theta
  27. j j
  28. θ \theta
  29. λ i , θ ( t ) := μ i + 0 t j = 1 d a i , j h i , j ( t - s ) d N j \lambda_{i,\theta}\left(t\right):=\mu_{i}+\int_{0}^{t}\sum_{j=1}^{d}a_{i,j}h_{% i,j}\left(t-s\right)dN_{j}
  30. λ i , θ ( t ) := lim s 0 P ( N j ( t + s ) - N ( j ) = 1 | t - ) s . \lambda_{i,\theta}\left(t\right):=\lim_{s\downarrow 0}\frac{P\left(N_{j}\left(% t+s\right)-N\left(j\right)=1|\mathcal{F}_{t^{-}}\right)}{s}.
  31. λ * = ( λ 1 * , , λ d * ) \lambda^{*}=\left(\lambda_{1}^{*},...,\lambda_{d}^{*}\right)
  32. λ * \lambda^{*}
  33. R T ( θ ) := λ θ T 2 - 2 T i = 1 d [ 0 , T ] λ i , θ ( t ) d N i ( t ) R_{T}\left(\theta\right):=\left\|\lambda_{\theta}\right\|_{T}^{2}-\frac{2}{T}% \sum_{i=1}^{d}\int_{[0,T]}\lambda_{i,\theta}\left(t\right)dN_{i}\left(t\right)
  34. λ θ T 2 := 1 T i = 1 d 0 T λ i , θ ( t ) 2 d t \left\|\lambda_{\theta}\right\|_{T}^{2}:=\frac{1}{T}\sum_{i=1}^{d}\int_{0}^{T}% \lambda_{i,\theta}\left(t\right)^{2}dt
  35. λ , λ T := 1 T i = 1 d 0 T λ i ( t ) λ i ( t ) d t \left\langle\lambda,\lambda^{{}^{\prime}}\right\rangle_{T}:=\frac{1}{T}\sum_{i% =1}^{d}\int_{0}^{T}\lambda_{i}\left(t\right)\lambda_{i}^{{}^{\prime}}\left(t% \right)dt
  36. N i N_{i}
  37. λ i , θ ( t ) \lambda_{i,\theta}\left(t\right)
  38. E [ R T ( θ ) ] = E ( λ θ T 2 ) - 2 T i = 1 d E ( [ 0 , T ] λ i , θ ( t ) d N i ( t ) ) E\left[R_{T}\left(\theta\right)\right]=E\left(\left\|\lambda_{\theta}\right\|_% {T}^{2}\right)-\frac{2}{T}\sum_{i=1}^{d}E\left(\int_{[0,T]}\lambda_{i,\theta}% \left(t\right)dN_{i}\left(t\right)\right)
  39. = E ( λ θ T 2 ) - 2 E ( 1 T i = 1 d [ 0 , T ] λ i , θ ( t ) λ i * ( t ) d t ) =E\left(\left\|\lambda_{\theta}\right\|_{T}^{2}\right)-2E\left(\frac{1}{T}\sum% _{i=1}^{d}\int_{[0,T]}\lambda_{i,\theta}\left(t\right)\lambda_{i}^{*}\left(t% \right)dt\right)
  40. = E ( λ θ - λ * T 2 ) + E ( λ θ * T 2 ) . =E\left(\left\|\lambda_{\theta}-\lambda^{*}\right\|_{T}^{2}\right)+E\left(% \left\|\lambda_{\theta}^{*}\right\|_{T}^{2}\right).
  41. M ( t ) := N i ( t ) - λ * ( t ) M\left(t\right):=N_{i}\left(t\right)-\lambda^{*}\left(t\right)
  42. [ 0 , T ] λ i , θ ( t ) λ i * ( t ) d M ( t ) \int_{[0,T]}\lambda_{i,\theta}\left(t\right)\lambda_{i}^{*}\left(t\right)dM% \left(t\right)
  43. 0
  44. λ θ = λ * \lambda_{\theta}=\lambda^{*}
  45. R T ( θ ) R_{T}\left(\theta\right)
  46. θ , ^ \hat{\theta,}
  47. A ^ \hat{A}
  48. θ ^ \hat{\theta}
  49. A ^ \hat{A}
  50. . * \left\|.\right\|_{*}
  51. A * := t r ( A * A ) \left\|A\right\|_{*}:=tr\left(\sqrt{A^{*}A}\right)
  52. A . A.
  53. θ ^ := argmin θ { R T ( θ ) + τ A * } \hat{\theta}:=\textrm{argmin}_{\theta}\left\{R_{T}\left(\theta\right)+\tau% \left\|A\right\|_{*}\right\}
  54. τ > 0 \tau>0
  55. λ θ ^ - λ θ * \lambda_{\hat{\theta}}-\lambda_{\theta^{*}}
  56. λ θ ^ \lambda_{\hat{\theta}}
  57. λ θ * \lambda_{\theta^{*}}
  58. . T \left\|.\right\|_{T}
  59. λ θ ^ - λ θ * T \left\|\lambda_{\hat{\theta}}-\lambda_{\theta}^{*}\right\|_{T}
  60. θ ^ \hat{\theta}

Draft:Hedge_behavior.html

  1. H 0.5 H\simeq 0.5

Draft:HedgeSPA.html

  1. A S R = i ( R i , A - R f ) w i Z π - σ π , A + 1 2 i w i ( Z i + σ i , A ) 2 Z π - σ π , A - 1 2 Z π - σ π , A ASR={\sum_{i}(R_{i,A}-R_{f})w_{i}\over Z_{\pi}^{-}\sigma_{\pi,A}}+{1\over 2}{% \sum_{i}w_{i}(Z_{i}^{+}\sigma_{i,A})^{2}\over Z_{\pi}^{-}\sigma_{\pi,A}}-{1% \over 2}Z_{\pi}^{-}\sigma_{\pi,A}
  2. Z π + = m a x ( Z c f π ( Z α + ) , 0 ) Z α + , Z π - = m i n ( Z c f π ( Z α - ) , 0 Z α - Z_{\pi}^{+}={max(Z_{cf\pi}(Z_{\alpha}^{+}),0)\over Z_{\alpha}^{+}},Z_{\pi}^{-}% ={min(Z_{cf\pi}(Z_{\alpha}^{-}),0\over Z_{\alpha}^{-}}
  3. Z c f π ( Z α ) = Z α + 1 6 ( Z α 2 - 1 ) S π + 1 24 ( Z α 3 - 3 Z α ) ( K π - 3 ) - 1 36 ( 2 Z α 3 - 5 Z α ) S π 2 Z_{cf\pi}(Z_{\alpha})=Z_{\alpha}+{1\over 6}(Z_{\alpha}^{2}-1)S_{\pi}+{1\over 2% 4}(Z_{\alpha}^{3}-3Z_{\alpha})(K_{\pi}-3)-{1\over 36}(2Z_{\alpha}^{3}-5Z_{% \alpha})S_{\pi}^{2}
  4. I m p l i e d R e t u r n = S R * Δ σ p f Δ W i + R f = R p f - R f σ p f * Δ σ p f Δ W i + R f ImpliedReturn=SR*{\Delta\sigma_{pf}\over\Delta W_{i}}+R_{f}={R_{pf}-R_{f}\over% {\sigma}_{pf}}*{\Delta\sigma_{pf}\over\Delta W_{i}}+R_{f}
  5. S R SR
  6. σ p f \sigma_{pf}
  7. W i W_{i}
  8. i t h i^{th}
  9. R f R_{f}
  10. V a R π , a := - F - 1 ( α ) , VaR_{\pi,a}:=-F^{-1}(\alpha),
  11. F ( y ) = - y 1 2 π σ π e - 1 2 σ π 2 ( x - μ π 2 ) d x F(y)=\int\limits_{-\infty}^{y}{1\over{\sqrt{2\pi}\sigma_{\pi}}}e^{-{1\over 2% \sigma_{\pi}^{2}}(x-\mu_{\pi}^{2})}\,dx
  12. c V a R cVaR
  13. 100 α % 100\alpha\%
  14. c V a R π , α = 𝔼 ( V a R π , γ | 0 < γ α ) cVaR_{\pi,\alpha}=\mathbb{E}(VaR_{\pi,\gamma}|0<\gamma\leqslant\alpha)
  15. S R = μ π , A - R f σ π , A SR={\mu_{\pi,A}-R_{f}\over\sigma_{\pi,A}}
  16. μ π , A \mu_{\pi,A}
  17. R f R_{f}
  18. σ π , A \sigma_{\pi,A}
  19. m i n h Σ 1 t T ( p h e r r ( t ) ) 2 min_{h}{\Sigma}_{1\leqslant t\leqslant T}(pherr(t))^{2}
  20. M a x D D π % = M a x ( 0 t T ) [ D r a w d o w n ] π % ( t ) MaxDD_{\pi}\%=Max_{(0\leqslant t\leqslant T)}{[Drawdown]_{\pi}\%(t)}
  21. ( 0 t T ) (0\leqslant t\leqslant T)
  22. D r a w d o w n π % ( t ) = D r a w d o w n π ( t ) N A V π , ( 0 τ t ) p e a k Drawdown_{\pi}\%(t)={Drawdown_{\pi}(t)\over NAV_{\pi,(0\leqslant\tau\leqslant t% )}^{peak}}
  23. D r a w d o w n π ( t ) = N A V π , ( 0 τ t ) p e a k - N A V π , t Drawdown_{\pi}(t)=NAV_{\pi,(0\leqslant\tau\leqslant t)}^{peak}-NAV_{\pi,t}
  24. N A V π , t = w T N A V t = [ w 1 w n ] [ N A V 1 , t N A V n , t ] NAV_{\pi,t}=w^{T}NAV_{t}=[w_{1}...w_{n}]\begin{bmatrix}NAV_{1,t}\\ \vdots\\ NAV_{n,t}\end{bmatrix}
  25. m a x U = w m k t T Π - ( λ 2 ) w m k t T Σ w m k t maxU=w_{mkt}^{T}{\Pi}-({\lambda\over 2})w_{mkt}^{T}\Sigma w_{mkt}
  26. U U
  27. w m k t w_{mkt}
  28. Π \Pi
  29. λ \lambda
  30. Σ \Sigma
  31. N N
  32. K K
  33. r f r_{f}
  34. M M
  35. i t h i^{th}
  36. r i r_{i}
  37. r i = B i , 0 + B i , 1 X 1 + + B i , N X N + ϵ i r_{i}=B_{i,0}+B_{i,1}X_{1}+...+B_{i,N}X_{N}+\epsilon_{i}
  38. r i r_{i}
  39. i t h i^{th}
  40. B i , j B_{i,j}
  41. i t h i^{th}
  42. j t h j^{th}
  43. X j X_{j}
  44. j t h j^{th}
  45. ϵ i \epsilon_{i}
  46. i t h i^{th}
  47. r p r_{p}
  48. r p r_{p}
  49. Σ i w i r i \Sigma_{i}w_{i}r_{i}
  50. w i w_{i}
  51. i t h i^{th}

Draft:Hidden_Variables_(Network_Science).html

  1. P ( k ) k - γ P\left(k\right)\sim k^{-\gamma}
  2. γ \gamma
  3. 2 < γ 3 2<\gamma\leq 3
  4. p i = i - α p_{i}=i^{-\alpha}
  5. α [ 0 , 1 ) \alpha\in[0,1)
  6. γ = 1 + α α \gamma=\frac{1+\alpha}{\alpha}
  7. P ( k , k ) = E k k k N P\left(k,k^{\prime}\right)=\frac{E_{kk^{\prime}}}{\langle k\rangle N}
  8. E k k E_{kk^{\prime}}
  9. k k
  10. k , k^{\prime},
  11. k \langle k\rangle
  12. N N
  13. P ( k ) = k k k P ( k , k ) P\left(k\right)=\frac{\langle k\rangle}{k}\sum\limits_{k^{\prime}}P\left(k,k^{% \prime}\right)
  14. P ( k | k ) P\left(k^{\prime}|k\right)
  15. k ¯ n n ( k ) = k k P ( k | k ) \bar{k}_{nn}\left(k\right)=\sum\limits_{k^{\prime}}k^{\prime}P\left(k^{\prime}% |k\right)
  16. P ( k | k ) = E k k k N k P\left(k^{\prime}|k\right)=\frac{E_{k^{\prime}k}}{kN_{k}}
  17. c ¯ ( k ) = k , k ′′ P ( k | k ) P ( k ′′ | k ) p k , k ′′ \bar{c}\left(k\right)=\sum\limits_{k^{\prime},k^{\prime\prime}}P\left(k^{% \prime}|k\right)P\left(k^{\prime\prime}|k\right)p_{k^{\prime},k^{\prime\prime}}
  18. p k , k ′′ p_{k^{\prime},k^{\prime\prime}}
  19. k k^{\prime}
  20. k ′′ k^{\prime\prime}
  21. h i h_{i}
  22. ρ ( h ) \rho(h)
  23. r ( h i , h j ) r(h_{i},h_{j})
  24. P ( k ) = h g ( k | h ) ρ ( h ) P(k)=\sum\limits_{h}g(k|h)\rho(h)
  25. g ( k | h ) g(k|h)
  26. h h
  27. k k
  28. k ¯ ( h ) = k k g ( k | h ) \bar{k}(h)=\sum\limits_{k}kg(k|h)
  29. k = h k ¯ ( h ) ρ ( h ) \langle k\rangle=\sum\limits_{h}\bar{k}(h)\rho(h)
  30. k ¯ ( h ) = N h ρ ( h ) r ( h , h ) \bar{k}(h)=N\sum\limits_{h^{\prime}}\rho(h^{\prime})r(h,h^{\prime})
  31. k = N h , h ρ ( h ) r ( h , h ) ρ ( h ) \langle k\rangle=N\sum\limits_{h,h^{\prime}}\rho(h)r(h,h^{\prime})\rho(h^{% \prime})
  32. k ¯ n n ( h ) = h = k ¯ ( h ) p ( h | h ) \bar{k}_{nn}(h)=\sum\limits_{h^{\prime}}=\bar{k}(h^{\prime})p(h^{\prime}|h)
  33. h h
  34. h h^{\prime}
  35. p ( h | h ) = ρ ( h ) r ( h , h ) h ′′ ρ ( h ′′ ) r ( h , h ′′ ) = N ρ ( h ) r ( h , h ) k ¯ ( h ) p(h^{\prime}|h)=\frac{\rho(h^{\prime})r(h,h^{\prime})}{\sum\limits_{h^{\prime% \prime}}\rho(h^{\prime\prime})r(h,h^{\prime\prime})}=\frac{N\rho(h^{\prime})r(% h,h^{\prime})}{\bar{k}(h)}
  36. c h = h , h ′′ p ( h | h ) r ( h , h ′′ ) p ( h ′′ | h ) c_{h}=\sum\limits_{h^{\prime},h^{\prime\prime}}p(h^{\prime}|h)r(h^{\prime},h^{% \prime\prime})p(h^{\prime\prime}|h)
  37. c ¯ ( k ) = 1 P ( k ) h ρ ( h ) g ( k | h ) c h \bar{c}(k)=\frac{1}{P(k)}\sum\limits_{h}\rho(h)g(k|h)c_{h}
  38. 2 \mathbb{H}^{2}

Draft:Hierarchical_testing_of_variables_in_high-dimensional_datasets.html

  1. H 0 , C H_{0,C}
  2. C C
  3. τ \tau
  4. N in ( b ) N\text{in}^{(b)}
  5. B B
  6. S ^ ( 1 ) , , S ^ ( B ) \hat{S}^{(1)},\ldots,\hat{S}^{(B)}
  7. N out ( b ) N\text{out}^{(b)}
  8. C C
  9. S ^ ( b ) \hat{S}^{(b)}
  10. p C , ( b ) = { p partial F-test C S ( b ) based on 𝐘 N out ( b ) , 𝐗 N out ( b ) , S ^ ( b ) , if C S ^ ( b ) 0 1 , if C S ^ ( b ) = 0 p^{C,(b)}=\begin{cases}\displaystyle p\text{partial F-test}^{C\ \cap\ S^{(b)}}% \,\text{ based on }\mathbf{Y}_{N\text{out}^{(b)}},\mathbf{X}_{N\text{out}^{(b)% },\hat{S}^{(b)}},&\,\text{if }C\cap\hat{S}^{(b)}\neq{}0\\ \displaystyle 1,&\,\text{if }C\cap\hat{S}^{(b)}=0\end{cases}
  11. p adj C , ( b ) = min ( 1 , p C , ( b ) | S ^ ( b ) | | C S ^ ( b ) | ) p_{\textrm{adj}}^{C,(b)}=\min\left(1,p^{C,{(b)}}\frac{|\hat{S}^{(b)}|}{|C\cap% \hat{S}^{(b)}|}\right)
  12. B B
  13. γ \gamma
  14. 1 , , B p adj C , ( b ) / γ 1,\ldots,B\ p_{\textrm{adj}}^{C,(b)}/\gamma
  15. γ \gamma
  16. Q c ( γ ) = min ( 1 , q γ ( p adj C , ( b ) / γ ; b = 1 , , B ) ) Q^{c}(\gamma)=\min\left(1,q_{\gamma}(p_{\textrm{adj}}^{C,(b)}/\gamma;b=1,% \ldots,B)\right)
  17. γ \gamma
  18. γ ( γ min , 1 ) \gamma\in(\gamma_{\textrm{min}},1)
  19. ( 1 - log ( γ min ) ) (1-\log(\gamma_{\textrm{min}}))
  20. P C = min ( 1 , ( 1 - log ( γ min ) ) inf γ ( γ min , 1 ) Q C ( γ ) ) P^{C}=\min\left(1,(1-\log(\gamma_{\min}))\inf_{\gamma\in(\gamma_{\min},1)}Q^{C% }(\gamma)\right)
  21. P h C = max D τ : C D P C P_{h}^{C}=\max_{D\in{\tau:C\subseteq{D}}}P^{C}

Draft:Hilbert–Kunz_multiplicity.html

  1. p > 0 p>0

Draft:Hypervelocity_Asteroid_Intercept_Vehicle.html

  1. M V t = G M m d 2 = T M\ \frac{V\vartriangle}{t\vartriangle}=\frac{GMm}{d^{2}}=T
  2. M V t = G m d 2 = T M = A M\ \frac{V\vartriangle}{t\vartriangle}=\frac{Gm}{d^{2}}=\frac{T}{M}=A
  3. A = 1.1579 * 10 - 9 m m / s 2 A=1.1579*10^{-9}mm/s^{2}
  4. V = A t \vartriangle V=A\vartriangle t
  5. X = 1 2 A ( t ) 2 \vartriangle X=\frac{1}{2}A(\vartriangle t)^{2}

Draft:Intelligent_scissors.html

  1. f Z f_{Z}
  2. f G f_{G}
  3. f D f_{D}
  4. l ( p , q ) = ω z f Z ( q ) + ω D f D ( p , q ) + ω G f G ( q ) l(p,q)=\omega_{z}\cdot f_{Z}(q)+\omega_{D}\cdot f_{D}(p,q)+\omega_{G}\cdot f_{% G}(q)

Draft:K-theory_of_a_category.html

  1. A r ( C ) Ar(C)
  2. [ n ] = { 0 < 1 < 2 < < n } [n]=\{0<1<2<\cdots<n\}
  3. S n C S_{n}C
  4. f : A r [ n ] C f:Ar[n]\to C
  5. i j k i\leq j\leq k
  6. f ( i = i ) = * f(i=i)=*
  7. f ( i j ) f ( i k ) f(i\leq j)\to f(i\leq k)
  8. f ( j k ) f(j\leq k)
  9. f ( i j ) f ( i k ) f(i\leq j)\to f(i\leq k)
  10. f ( i j ) f ( j = j ) = * f(i\leq j)\to f(j=j)=*
  11. S n C S_{n}C
  12. S n C S_{n}C
  13. S ( m ) C = S S C S^{(m)}C=S\cdots SC

Draft:Kane's_Method.html

  1. s s
  2. q i q_{i}
  3. f ( q 1 , q 2 , , q n , t ) = 0 f(q_{1},q_{2},...,q_{n},t)=0
  4. f ( q 1 , q 2 , , q n , q ˙ 1 , q ˙ 2 , , q ˙ n , t ) = 0 f(q_{1},q_{2},...,q_{n},\dot{q}_{1},\dot{q}_{2},...,\dot{q}_{n},t)=0
  5. s s
  6. s s
  7. n n
  8. { q ˙ 1 , q ˙ 2 , , q ˙ n } \{\dot{q}_{1},\dot{q}_{2},...,\dot{q}_{n}\}
  9. u r s = 1 n Y r s q ˙ s + Z r r = 1 , 2 , , n u_{r}\doteq\sum_{s=1}^{n}Y_{rs}\dot{q}_{s}+Z_{r}\qquad r=1,2,\cdots,n
  10. Y r s Y_{rs}
  11. Z r Z_{r}
  12. q ˙ s \dot{q}_{s}
  13. { q ˙ 1 , q ˙ 2 , , q ˙ n } \{\dot{q}_{1},\dot{q}_{2},...,\dot{q}_{n}\}
  14. Y r s = δ r s Y_{rs}=\delta_{rs}
  15. Z r = 0 Z_{r}=0
  16. δ r s \delta_{rs}
  17. s s
  18. u r = s = 1 p A r s u s + B r r = p + 1 , , n u_{r}=\sum_{s=1}^{p}A_{rs}u_{s}+B_{r}\qquad r=p+1,\cdots,n
  19. p n - m p\doteq n-m
  20. p p
  21. p i p_{i}
  22. s s
  23. n n
  24. p i p_{i}
  25. 𝐯 p i = r = 1 n 𝐯 r p i u r + 𝐯 t p i \mathbf{v}^{p_{i}}=\sum_{r=1}^{n}\mathbf{v}^{p_{i}}_{r}u_{r}+\mathbf{v}^{p_{i}% }_{t}
  26. u r u_{r}
  27. 𝐯 t p i \mathbf{v}^{p_{i}}_{t}
  28. 𝐯 r p i \mathbf{v}^{p_{i}}_{r}
  29. p i p_{i}
  30. s s
  31. s y m b o l ω B = r = 1 n s y m b o l ω r B u r + s y m b o l ω t B symbol{\omega}^{B}=\sum_{r=1}^{n}symbol{\omega}^{B}_{r}u_{r}+symbol{\omega}^{B% }_{t}
  32. s y m b o l ω t B symbol{\omega}^{B}_{t}
  33. s y m b o l ω r B symbol{\omega}^{B}_{r}
  34. s s
  35. p p
  36. p p
  37. p i p_{i}
  38. s y m b o l ω B = r = 1 p s y m b o l ω ~ r B u r + s y m b o l ω ~ t B symbol{\omega}^{B}=\sum_{r=1}^{p}\widetilde{symbol{\omega}}^{B}_{r}u_{r}+% \widetilde{symbol{\omega}}^{B}_{t}
  39. 𝐯 p i = r = 1 p 𝐯 ~ r p i u r + 𝐯 ~ t p i \mathbf{v}^{p_{i}}=\sum_{r=1}^{p}\widetilde{\mathbf{v}}^{p_{i}}_{r}u_{r}+% \widetilde{\mathbf{v}}^{p_{i}}_{t}
  40. 𝐯 ~ r p i , s y m b o l ω ~ r B \widetilde{\mathbf{v}}^{p_{i}}_{r},\widetilde{symbol{\omega}}^{B}_{r}
  41. s s
  42. s s
  43. s s
  44. F r i = 1 ν 𝐯 r p i 𝐑 i r = 1 , 2 , , n F_{r}\doteq\sum_{i=1}^{\nu}\mathbf{v}_{r}^{p_{i}}\cdot\mathbf{R}_{i}\qquad r=1% ,2,\cdots,n
  45. p i p_{i}
  46. s s
  47. 𝐯 r p i \mathbf{v}_{r}^{p_{i}}
  48. p i p_{i}
  49. 𝐑 i \mathbf{R}_{i}
  50. p i p_{i}
  51. s s
  52. F ~ r i = 1 ν 𝐯 ~ r p i 𝐑 i r = 1 , 2 , , p \widetilde{F}_{r}\doteq\sum_{i=1}^{\nu}\widetilde{\mathbf{v}}_{r}^{p_{i}}\cdot% \mathbf{R}_{i}\qquad r=1,2,\cdots,p
  53. s s
  54. R 1 , , R k R_{1},...,R_{k}
  55. p 1 , , p k p_{1},...,p_{k}
  56. ( F r ) B = i = 1 k 𝐯 r p i 𝐑 i (F_{r})_{B}=\sum_{i=1}^{k}\mathbf{v}_{r}^{p_{i}}\cdot\mathbf{R}_{i}
  57. Q Q
  58. 𝐯 p i A = 𝐯 Q A + s A y m b o l ω B × 𝐫 Q p i {}^{A}\mathbf{v}^{p_{i}}={}^{A}\mathbf{v}^{Q}+{}^{A}symbol{\omega}^{B}\times% \mathbf{r}^{Qp_{i}}
  59. 𝐯 Q A {}^{A}\mathbf{v}^{Q}
  60. Q Q
  61. s A y m b o l ω B {}^{A}symbol{\omega}^{B}
  62. 𝐫 Q p i \mathbf{r}^{Qp_{i}}
  63. Q Q
  64. p i p_{i}
  65. u r u_{r}
  66. r 𝐯 r p i A u r + 𝐯 t p i A = ( r 𝐯 r Q A u r + 𝐯 t Q A ) + ( r s A y m b o l ω r B u r + s A y m b o l ω t B ) × 𝐫 Q p i \sum_{r}{}^{A}\mathbf{v}_{r}^{p_{i}}u_{r}+{}^{A}\mathbf{v}_{t}^{p_{i}}=\left(% \sum_{r}{}^{A}\mathbf{v}_{r}^{Q}u_{r}+{}^{A}\mathbf{v}_{t}^{Q}\right)+\left(% \sum_{r}{}^{A}symbol{\omega}_{r}^{B}u_{r}+{}^{A}symbol{\omega}_{t}^{B}\right)% \times\mathbf{r}^{Qp_{i}}
  67. r ( 𝐯 r p i A - 𝐯 r Q A - s A y m b o l ω r B × 𝐫 Q p i ) u r = - 𝐯 t p i A + 𝐯 t Q A + s A y m b o l ω t B × 𝐫 Q p i \Rightarrow\sum_{r}\left({}^{A}\mathbf{v}_{r}^{p_{i}}-{}^{A}\mathbf{v}_{r}^{Q}% -{}^{A}symbol{\omega}_{r}^{B}\times\mathbf{r}^{Qp_{i}}\right)u_{r}=-{}^{A}% \mathbf{v}_{t}^{p_{i}}+{}^{A}\mathbf{v}_{t}^{Q}+{}^{A}symbol{\omega}_{t}^{B}% \times\mathbf{r}^{Qp_{i}}
  68. 𝐯 r p i A = 𝐯 r Q A + s A y m b o l ω r B × 𝐫 Q p i {}^{A}\mathbf{v}_{r}^{p_{i}}={}^{A}\mathbf{v}_{r}^{Q}+{}^{A}symbol{\omega}_{r}% ^{B}\times\mathbf{r}^{Qp_{i}}
  69. ( F r ) B = i = 1 k ( 𝐯 r Q A + s A y m b o l ω r B × 𝐫 Q p i ) 𝐑 i = 𝐯 r Q A i = 1 k 𝐑 i + s A y m b o l ω r B i = 1 k 𝐫 Q p i × 𝐑 i (F_{r})_{B}=\sum_{i=1}^{k}\left({}^{A}\mathbf{v}_{r}^{Q}+{}^{A}symbol{\omega}_% {r}^{B}\times\mathbf{r}^{Qp_{i}}\right)\cdot\mathbf{R}_{i}={}^{A}\mathbf{v}_{r% }^{Q}\cdot\sum_{i=1}^{k}\mathbf{R}_{i}+{}^{A}symbol{\omega}_{r}^{B}\cdot\sum_{% i=1}^{k}\mathbf{r}^{Qp_{i}}\times\mathbf{R}_{i}
  70. 𝐑 \mathbf{R}
  71. Q Q
  72. 𝐓 \mathbf{T}
  73. ( F r ) B = s A y m b o l ω r B 𝐓 + 𝐯 r Q A 𝐑 (F_{r})_{B}={}^{A}symbol{\omega}^{B}_{r}\cdot\mathbf{T}+{}^{A}\mathbf{v}^{Q}_{% r}\cdot\mathbf{R}
  74. p i p_{i}
  75. 𝐑 i * - m i 𝐚 i \mathbf{R}_{i}^{*}\doteq-m_{i}\mathbf{a}_{i}
  76. 𝐚 i \mathbf{a}_{i}
  77. p i p_{i}
  78. F r * i = 1 ν 𝐯 r p i 𝐑 i * F^{*}_{r}\doteq\sum_{i=1}^{\nu}\mathbf{v}_{r}^{p_{i}}\cdot\mathbf{R}_{i}^{*}
  79. F ~ r * i = 1 ν 𝐯 ~ r p i 𝐑 i * \widetilde{F}^{*}_{r}\doteq\sum_{i=1}^{\nu}\widetilde{\mathbf{v}}_{r}^{p_{i}}% \cdot\mathbf{R}_{i}^{*}
  80. s s
  81. ( F ~ r * ) B = s y m b o l ω ~ r B A 𝐓 * + 𝐯 ~ r * A 𝐑 * (\widetilde{F}_{r}^{*})_{B}={}^{A}\widetilde{symbol{\omega}}^{B}_{r}\cdot% \mathbf{T}^{*}+{}^{A}\widetilde{\mathbf{v}}^{*}_{r}\cdot\mathbf{R}^{*}
  82. s y m b o l ω ~ r B A {}^{A}\widetilde{symbol{\omega}}^{B}_{r}
  83. 𝐑 * - M 𝐚 * \mathbf{R}^{*}\doteq-M\mathbf{a}^{*}
  84. 𝐯 ~ r * \widetilde{\mathbf{v}}^{*}_{r}
  85. 𝐓 * = - s y m b o l α 𝐈 ^ - s y m b o l ω × 𝐈 ^ s y m b o l ω \mathbf{T}^{*}=-symbol{\alpha}\cdot\hat{\mathbf{I}}-symbol{\omega}\times\hat{% \mathbf{I}}\cdot symbol{\omega}
  86. s y m b o l α symbol{\alpha}
  87. s y m b o l ω symbol{\omega}
  88. 𝐈 ^ = I i j 𝐧 i 𝐧 j \hat{\mathbf{I}}=\sum I_{ij}\mathbf{n}_{i}\mathbf{n}_{j}
  89. I i j I_{ij}
  90. 𝐧 i , j \mathbf{n}_{i,j}
  91. s s
  92. F ~ r + F ~ r * = 0 r = 1 , 2 , , p \widetilde{F}_{r}+\widetilde{F}_{r}^{*}=0\qquad\quad r=1,2,\cdots,p
  93. F ~ r \widetilde{F}_{r}
  94. F ~ r * \widetilde{F}_{r}^{*}
  95. r r
  96. R R
  97. q 1 q_{1}
  98. u 1 = ( R - r ) q ˙ 1 u_{1}=(R-r)\dot{q}_{1}
  99. 𝐯 B A = ( R - r ) q ˙ 1 𝐛 1 {}^{A}\mathbf{v}^{B^{\prime}}=(R-r)\dot{q}_{1}\mathbf{b}_{1}
  100. s A y m b o l ω B = q ˙ 2 𝐛 3 {}^{A}symbol{\omega}^{B}=\dot{q}_{2}\mathbf{b}_{3}
  101. q 1 q_{1}
  102. q 2 q_{2}
  103. q ˙ 2 = R - r r q ˙ 1 \dot{q}_{2}=\dfrac{R-r}{r}\dot{q}_{1}
  104. 𝐯 1 B A = 𝐛 1 {}^{A}\mathbf{v}^{B^{\prime}}_{1}=\mathbf{b}_{1}
  105. s A y m b o l ω 1 B = 1 r 𝐛 3 {}^{A}symbol{\omega}^{B}_{1}=\dfrac{1}{r}\mathbf{b}_{3}
  106. 𝐑 = - m g 𝐚 2 \mathbf{R}=-mg\mathbf{a}_{2}
  107. F 1 = 𝐯 1 B A 𝐑 = ( 𝐛 1 ) ( - m g 𝐚 2 ) = - m g sin ( q 1 ) F_{1}={}^{A}\mathbf{v}^{B^{\prime}}_{1}\cdot\mathbf{R}=(\mathbf{b}_{1})\cdot(-% mg\mathbf{a}_{2})=-mg\sin(q_{1})
  108. 𝐚 B A = ( R - r ) ( q ¨ 1 𝐛 1 + q ˙ 1 ( s A y m b o l ω B × 𝐛 1 ) ) {}^{A}\mathbf{a}^{B^{\prime}}=(R-r)(\ddot{q}_{1}\mathbf{b}_{1}+\dot{q}_{1}({}^% {A}symbol{\omega}^{B}\times\mathbf{b}_{1}))
  109. s A y m b o l α B = q ¨ 2 𝐛 3 {}^{A}symbol{\alpha}^{B}=\ddot{q}_{2}\mathbf{b}_{3}
  110. 𝐑 * = - m 𝐚 B A = - m ( R - r ) ( q ¨ 1 𝐛 1 + q ˙ 1 ( s A y m b o l ω B × 𝐛 1 ) ) \mathbf{R}^{*}=-m{}^{A}\mathbf{a}^{B^{\prime}}=-m(R-r)(\ddot{q}_{1}\mathbf{b}_% {1}+\dot{q}_{1}({}^{A}symbol{\omega}^{B}\times\mathbf{b}_{1}))
  111. 𝐓 * = - s A y m b o l α B ( I 𝐛 3 𝐛 3 ) = - R - r r q ¨ 1 I 𝐛 3 \mathbf{T}^{*}=-{}^{A}symbol{\alpha}^{B}\cdot(I\mathbf{b}_{3}\mathbf{b}_{3})=-% \dfrac{R-r}{r}\ddot{q}_{1}I\mathbf{b}_{3}
  112. F 1 * = ( 1 r 𝐛 3 ) ( - R - r r q ¨ 1 I 𝐛 3 ) + ( - m ( R - r ) ( q ¨ 1 𝐛 1 + q ˙ 1 ( s A y m b o l ω B × 𝐛 1 ) ) ) ( 𝐛 1 ) F_{1}^{*}=\left(\dfrac{1}{r}\mathbf{b}_{3}\right)\cdot\left(-\dfrac{R-r}{r}% \ddot{q}_{1}I\mathbf{b}_{3}\right)+\left(-m(R-r)(\ddot{q}_{1}\mathbf{b}_{1}+% \dot{q}_{1}({}^{A}symbol{\omega}^{B}\times\mathbf{b}_{1}))\right)\cdot(\mathbf% {b}_{1})
  113. F 1 * = - ( I r 2 + m ) ( R - r ) q ¨ 1 \Rightarrow F_{1}^{*}=-\left(\dfrac{I}{r^{2}}+m\right)(R-r)\ddot{q}_{1}
  114. ( I r 2 + m ) ( R - r ) q ¨ 1 + m g sin ( q 1 ) = 0 \left(\dfrac{I}{r^{2}}+m\right)(R-r)\ddot{q}_{1}+mg\sin(q_{1})=0
  115. p p

Draft:Kernels_on_Graph.html

  1. A A
  2. 𝐆 \mathbf{G}
  3. A i j = 1 A_{ij}=1
  4. i i
  5. j j
  6. A i j = 0 A_{ij}=0
  7. A i i A_{ii}
  8. L = D - A L=D-A
  9. D D
  10. D i i D_{ii}
  11. i i
  12. x 1 x_{1}
  13. x 2 x_{2}
  14. k ( x 1 , x 2 ) = ϕ ( x 1 ) ϕ ( x 2 ) k(x_{1},x_{2})=\phi(x_{1})\cdot\phi(x_{2})
  15. x 1 x_{1}
  16. x 2 x_{2}
  17. ϕ ( x 1 ) , ϕ ( x 2 ) \phi(x_{1}),\phi(x_{2})
  18. ϕ \phi
  19. K V N = k = 0 γ k A k = ( 𝐈 - γ A ) - 1 K_{VN}=\sum_{k=0}^{\infty}\gamma^{k}A^{k}=(\mathbf{I}-\gamma A)^{-1}
  20. A i j k A^{k}_{ij}
  21. i i
  22. j j
  23. k k
  24. K V N K_{VN}
  25. γ \gamma
  26. i , j i,j
  27. i , j i,j
  28. K R L = k = 0 γ k ( - L ) k = ( I + γ L ) - 1 K_{RL}=\sum_{k=0}^{\infty}\gamma^{k}(-L)^{k}=(I+\gamma L)^{-1}
  29. γ \gamma
  30. x i ( t ) x_{i}(t)
  31. t t
  32. j j
  33. A i j A_{ij}
  34. Δ t \Delta t
  35. x i A i j Δ t x_{i}A_{ij}\Delta t
  36. i i
  37. j j
  38. Δ t \Delta t
  39. i i
  40. x i ( t + Δ t ) - x i ( t ) = j = 1 n x j A j i Δ t - j = 1 n x i A i j Δ t x_{i}(t+\Delta t)-x_{i}(t)=\sum_{j=1}^{n}x_{j}A_{ji}\Delta t-\sum_{j=1}^{n}x_{% i}A_{ij}\Delta t
  41. 𝐱 ( t ) \mathbf{x}(t)
  42. 𝐱 ( t ) = 𝐱 ( 0 ) exp ( - L t ) \mathbf{x}(t)=\mathbf{x}(0)\exp(-Lt)
  43. 𝐱 ( 0 ) \mathbf{x}(0)
  44. t = 0 t=0
  45. K D L = exp ( - γ L ) K_{DL}=\exp(-\gamma L)
  46. K D A = exp ( γ A ) = k = 0 γ k A k k ! K_{DA}=\exp(\gamma A)=\sum_{k=0}^{\infty}\frac{\gamma^{k}A^{k}}{k!}
  47. K D A K_{DA}
  48. γ \gamma
  49. γ k / k ! \gamma^{k}/k!
  50. P P
  51. P i j = A i j / j = 1 n A i j P_{ij}=A_{ij}/\sum_{j=1}^{n}A_{ij}
  52. i , j i,j
  53. i i
  54. j j
  55. i i
  56. n ( i , j ) = V G ( e i - e j ) T L + ( e i - e j ) n(i,j)=V_{G}(e_{i}-e_{j})^{T}L^{+}(e_{i}-e_{j})
  57. e i , e j e_{i},e_{j}
  58. i , j i,j
  59. V G V_{G}
  60. L + L^{+}
  61. L L
  62. n ( i , j ) n(i,j)
  63. i , j i,j
  64. L + L^{+}
  65. K C T = L + K_{CT}=L^{+}
  66. i i
  67. j j
  68. i i
  69. 1 - γ 1-\gamma
  70. K R W R = ( D - γ A ) - 1 D K_{RWR}=(D-\gamma A)^{-1}D
  71. x i k ( t ) x_{ik}(t)
  72. i i
  73. k k
  74. t t
  75. i , j i,j
  76. t t
  77. d i j ( t ) = k = 1 n ( x i k ( t ) - x j k ( t ) ) 2 = ( e i - e j ) T Z ( t ) Z T ( t ) ( e i - e j ) d_{ij}(t)=\sum_{k=1}^{n}(x_{ik}(t)-x_{jk}(t))^{2}=(e_{i}-e_{j})^{T}Z(t)Z^{T}(t% )(e_{i}-e_{j})
  78. K M D ( t ) = Z ( t ) Z T ( t ) K_{MD}(t)=Z(t)Z^{T}(t)
  79. Z ( t ) = 1 t τ = 1 t P τ = 1 t ( I - P ) - 1 ( I - P t ) P Z(t)=\frac{1}{t}\sum_{\tau=1}^{t}P^{\tau}=\frac{1}{t}(I-P)^{-1}(I-P^{t})P
  80. d i j d_{ij}

Draft:Kleiman's_intersection_theory.html

  1. c 1 ( L ) F = F - L - 1 F . c_{1}(L)F=F-L^{-1}\otimes F.
  2. c 1 ( L 1 ) c 1 ( L 2 ) = c 1 ( L 1 ) + c 1 ( L 2 ) - c 1 ( L 1 L 2 ) c_{1}(L_{1})c_{1}(L_{2})=c_{1}(L_{1})+c_{1}(L_{2})-c_{1}(L_{1}\otimes L_{2})
  3. c 1 ( L 1 ) c_{1}(L_{1})
  4. c 1 ( L 2 ) c_{1}(L_{2})
  5. c 1 ( L ) c 1 ( L - 1 ) = c 1 ( L ) + c 1 ( L - 1 ) . c_{1}(L)c_{1}(L^{-1})=c_{1}(L)+c_{1}(L^{-1}).
  6. dim supp c 1 ( L ) F dim supp F - 1 \dim\operatorname{supp}c_{1}(L)F\leq\dim\operatorname{supp}F-1
  7. L 1 L r L_{1}\cdot{\dots}\cdot L_{r}
  8. L 1 L r F = χ ( c 1 ( L 1 ) c 1 ( L r ) F ) L_{1}\cdot{\dots}\cdot L_{r}\cdot F=\chi(c_{1}(L_{1})\cdots c_{1}(L_{r})F)
  9. L 1 L r F = 0 r ( - 1 ) i χ ( i ( 0 r L j - 1 ) F ) . L_{1}\cdot{\dots}\cdot L_{r}\cdot F=\sum_{0}^{r}(-1)^{i}\chi(\wedge^{i}(\oplus% _{0}^{r}L_{j}^{-1})\otimes F).
  10. L 1 L r F L_{1}\cdot{\dots}\cdot L_{r}\cdot F
  11. D 1 D r D_{1}\cdot{\dots}\cdot D_{r}
  12. f : X Y f:X\to Y
  13. L i , 1 i m L_{i},1\leq i\leq m
  14. m dim supp F m\geq\dim\operatorname{supp}F
  15. f * L 1 f * L m F = L 1 L m f * F f^{*}L_{1}\cdots f^{*}L_{m}\cdot F=L_{1}\cdots L_{m}\cdot f_{*}F
  16. E G EG
  17. A G * ( X ) = A n ( E G ( n ) × G X ) A^{*}_{G}(X)=\bigoplus A^{n}(EG(n)\times_{G}X)
  18. E G ( n ) EG(n)
  19. A G * ( X ) A^{*}_{G}(X)
  20. A G * ( p t ) A^{*}_{G}(pt)
  21. X × G E G E G / G X\times_{G}EG\to EG/G
  22. cl : A G * ( X ) H G ( X ; ) \operatorname{cl}:A^{*}_{G}(X)\to H_{G}(X;\mathbb{Z})

Draft:Knockoff_filter.html

  1. 𝐲 = 𝐗 β + 𝐳 \mathbf{y}=\mathbf{X}\beta+\mathbf{z}
  2. 𝐲 n \mathbf{y}\in\mathbb{R}^{n}
  3. 𝐗 n × p \mathbf{X}\in\mathbb{R}^{n\times p}
  4. β p \mathbf{\beta}\in\mathbb{R}^{p}
  5. z 𝒩 ( 0 , σ 2 𝐈 ) z\sim\mathcal{N}(0,\sigma^{2}\mathbf{I})
  6. S ^ { 1 , , p } \hat{S}\subset\{1,\cdots,p\}
  7. FDR = 𝔼 [ # { j : β j = 0 but j S ^ } # { j : j S ^ 1 } ] \,\text{FDR}=\mathbb{E}\left[\dfrac{\#\{j:\beta_{j}=0\,\text{ but }j\in\hat{S}% \}}{\#\{j:j\in\hat{S}\vee 1\}}\right]
  8. x y := max { x , y } x\vee y:=\max\{x,y\}
  9. q q
  10. q \leq q
  11. β \beta
  12. X X
  13. 𝚺 = 𝐗 𝐓 𝐗 \mathbf{\Sigma=X^{T}X}
  14. 𝐗 ~ j \mathbf{\tilde{X}}_{j}
  15. X j X_{j}
  16. 𝐗 ~ T 𝐗 ~ = 𝚺 \mathbf{\tilde{X}}^{T}\mathbf{\tilde{X}}=\mathbf{\Sigma}
  17. 𝐗 T 𝐗 ~ = 𝚺 - diag { 𝐬 } \mathbf{X}^{T}\mathbf{\tilde{X}}=\mathbf{\Sigma}-\,\text{diag}\{\mathbf{s}\}
  18. s s
  19. 𝐗 ~ \mathbf{\tilde{X}}
  20. X X
  21. 𝐗 j T 𝐗 ~ k = 𝐗 j T 𝐗 k \mathbf{X}_{j}^{T}\mathbf{\tilde{X}}_{k}=\mathbf{X}_{j}^{T}\mathbf{X}_{k}
  22. k \neq k
  23. 𝐗 j T 𝐗 ~ j = Σ j j - s j = 1 - s j \mathbf{X}_{j}^{T}\mathbf{\tilde{X}}_{j}=\Sigma_{jj}-s_{j}=1-s_{j}
  24. 𝐀 := [ 𝐗 𝐗 ~ ] \mathbf{A}:=[\mathbf{X}\quad\mathbf{\tilde{X}}]
  25. β ^ ( λ ) = arg min β { 1 2 𝐲 - 𝐀 β 2 2 + λ β 1 } \hat{\beta}(\lambda)=\,\text{arg}\min_{\beta}\left\{\frac{1}{2}\|\mathbf{y}-% \mathbf{A}\mathbf{\beta}\|_{2}^{2}+\lambda\|\beta\|_{1}\right\}
  26. Z j = sup { λ : β ^ j ( λ ) 0 } Z_{j}=\sup\left\{\lambda:\hat{\beta}_{j}(\lambda)\neq 0\right\}
  27. 2 p 2p
  28. ( Z 1 , , Z p , Z ~ 1 , , Z ~ p ) (Z_{1},\cdots,Z_{p},\tilde{Z}_{1},\cdots,\tilde{Z}_{p})
  29. W j W_{j}
  30. W j = Z j Z ~ j { + 1 if Z j > Z ~ j - 1 if Z j < Z ~ j = { Z j if Z j > Z ~ j - Z ~ j if Z j < Z ~ j W_{j}=Z_{j}\vee\tilde{Z}_{j}\cdot\begin{cases}+1\quad\,\text{if }Z_{j}>\tilde{% Z}_{j}\\ -1\quad\,\text{if }Z_{j}<\tilde{Z}_{j}\end{cases}=\begin{cases}Z_{j}\qquad\,% \text{if }Z_{j}>\tilde{Z}_{j}\\ -\tilde{Z}_{j}\quad\,\text{if }Z_{j}<\tilde{Z}_{j}\end{cases}
  31. T = min { t 𝒲 : # { j : W j - t } # { j : W j t } 1 q } T=\min\left\{t\in\mathcal{W}:\frac{\#\{j:W_{j}\leq-t\}}{\#\{j:W_{j}\geq t\}% \vee 1}\leq q\right\}
  32. S ^ \hat{S}
  33. S ^ = { j : W j > T } \hat{S}=\{j:W_{j}>T\}
  34. q ( 0 , 1 ] q\in(0,1]
  35. 𝔼 [ # { j : β j = 0 but j S ^ } # { j : j S ^ 1 } ] q \mathbb{E}\left[\dfrac{\#\{j:\beta_{j}=0\,\text{ but }j\in\hat{S}\}}{\#\{j:j% \in\hat{S}\vee 1\}}\right]\leq q

Draft:Krein_space.html

  1. F F
  2. 𝐑 \mathbf{R}
  3. 𝐂 \mathbf{C}
  4. V V
  5. F F
  6. V V
  7. , : V × V F \langle\cdot,\cdot\rangle:V\times V\to F
  8. x , y , z V x,y,z\in V
  9. a F a\in F
  10. x , y = y , x ¯ . \langle x,y\rangle=\overline{\langle y,x\rangle}.
  11. F = 𝐑 F=\mathbf{R}
  12. x , y = y , x \langle x,y\rangle=\langle y,x\rangle
  13. F = 𝐑 F=\mathbf{R}
  14. F = 𝐂 F=\mathbf{C}
  15. x , y \langle x,y\rangle
  16. a + b i a+bi
  17. a - b i a-bi
  18. z z
  19. z ¯ \overline{z}
  20. a x , y = a x , y . \langle ax,y\rangle=a\langle x,y\rangle.
  21. x + y , z = x , z + y , z . \langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle.
  22. x V , x , x = 0 x = 0 \forall x^{\prime}\in V,\langle x,x^{\prime}\rangle=0\Rightarrow x=0
  23. x V , x , x 0 \forall x\in V,\langle x,x\rangle\geq 0
  24. V V
  25. , \langle\cdot,\cdot\rangle
  26. V + V_{+}
  27. V - V_{-}
  28. , , + \langle\cdot,\cdot,\rangle_{+}
  29. , , - \langle\cdot,\cdot,\rangle_{-}
  30. V V
  31. v V v\in V
  32. v = v + - v - v=v_{+}-v_{-}
  33. v + V + v_{+}\in V_{+}
  34. v - V - v_{-}\in V_{-}
  35. v , v V v,v^{\prime}\in V
  36. v , v = v , v + - v , v - \langle v,v^{\prime}\rangle=\langle v,v^{\prime}\rangle_{+}-\langle v,v^{% \prime}\rangle_{-}
  37. V - = { 0 } V_{-}=\{0\}

Draft:Leaving_Certificate_Mathematics_Ordinary_Level.html

  1. m x n = ( 2 ) x ( 6 ) = 12 mxn=(2)x(6)=12
  2. 6 ! 6!
  3. 6 x 5 x 4 x 3 x 2 x 1 6x5x4x3x2x1
  4. 10 ! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3 , 628 , 800 10!=10x9x8x7x6x5x4x3x2x1=3,628,800
  5. 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7 ! = 5 , 040 7x6x5x4x3x2x1=7!=5,040
  6. 1 x 6 x 5 x 4 x 3 x 2 x 1 = 1 x 6 ! = 720 1x6x5x4x3x2x1=1x6!=720
  7. 5 , 040 - 720 = 4320 5,040-720=4320
  8. 3 x ( 5 x 4 x 3 x 2 x 1 ) x 4 = 3 x 5 ! x 4 = 12 x 5 ! = 1440 3x(5x4x3x2x1)x4=3x5!x4=12x5!=1440
  9. 3 x 2 x 1 = 3 ! = 6 3x2x1=3!=6
  10. P ( 6 ) = 1 6 P(6)=\frac{1}{6}
  11. P ( o d d ) = 3 6 P(odd)=\frac{3}{6}
  12. = 1 2 =\frac{1}{2}
  13. a 1 , a 2 , a 3 , a n a_{1},a_{2},a_{3},...a_{n}
  14. a , a + d , a + 2 d , a + 3 d , a + ( n - 1 ) d a,a+d,a+2d,a+3d,...a+(n-1)d
  15. T n = a + ( n - 1 ) d T_{n}=a+(n-1)d
  16. T n T_{n}
  17. a a
  18. d d
  19. 1 4 , 1 2 , 3 4 , 1 , 5 4 \frac{1}{4},\frac{1}{2},\frac{3}{4},1,\frac{5}{4}
  20. 1 4 \frac{1}{4}
  21. a = 1 4 a=\frac{1}{4}
  22. 1 2 - 1 4 = 1 4 \frac{1}{2}-\frac{1}{4}=\frac{1}{4}
  23. d = 1 4 d=\frac{1}{4}
  24. T n = a + ( n - 1 ) d T_{n}=a+(n-1)d
  25. T n = 1 4 + ( n - 1 ) 1 4 T_{n}=\frac{1}{4}+(n-1)\frac{1}{4}
  26. T n = 1 4 + 1 4 n - 1 4 T_{n}=\frac{1}{4}+\frac{1}{4}n-\frac{1}{4}
  27. T n = 1 4 n T_{n}=\frac{1}{4}n
  28. T 3 = 1 4 ( 3 ) T_{3}=\frac{1}{4}(3)
  29. T 3 = 3 4 T_{3}=\frac{3}{4}
  30. T 100 = 1 4 ( 100 ) T_{100}=\frac{1}{4}(100)
  31. x 2 + 2 x + 5 = 0 x^{2}+2x+5=0
  32. x = - b ± b 2 - 4 a c 2 a x={-b\pm\sqrt{b^{2}-4ac}\over 2a}
  33. x = - 2 ± 2 2 - 4 ( 1 x 5 ) 2 x 1 x={-2\pm\sqrt{2^{2}-4(1x5)}\over 2x1}
  34. x = - 2 ± 4 - 4 ( 5 ) 2 x={-2\pm\sqrt{4-4(5)}\over 2}
  35. x = - 2 ± 4 - 20 2 x={-2\pm\sqrt{4-20}\over 2}
  36. x = - 2 ± - 16 2 x={-2\pm\sqrt{-16}\over 2}
  37. x = - 2 ± 16 . - 1 2 x={-2\pm\sqrt{16}.\sqrt{-1}\over 2}
  38. x = - 2 ± 4 - 1 2 x={-2\pm 4\sqrt{-1}\over 2}
  39. x = - 1 ± 2 - 1 x={-1\pm 2\sqrt{-1}}
  40. - 1 \sqrt{-1}
  41. i i
  42. i = - 1 i=\sqrt{-1}
  43. x = - 1 ± 2 i x=-1\pm{2i}
  44. i i
  45. - 1 + 2 i -1+2i
  46. - 1 - 2 i -1-2i
  47. a + b i a+bi
  48. a a
  49. b i bi