wpmath0000013_9

Nauru_graph.html

  1. G ( 4 , 1 ) G(4,1)
  2. G ( 5 , 2 ) G(5,2)
  3. G ( 8 , 3 ) G(8,3)
  4. G ( 10 , 2 ) G(10,2)
  5. G ( 10 , 3 ) G(10,3)
  6. ( x - 3 ) ( x - 2 ) 6 ( x - 1 ) 3 x 4 ( x + 1 ) 3 ( x + 2 ) 6 ( x + 3 ) , (x-3)(x-2)^{6}(x-1)^{3}x^{4}(x+1)^{3}(x+2)^{6}(x+3),

Near-infrared_window_in_biological_tissue.html

  1. μ a \mu_{a}
  2. μ a \mu_{a}
  3. μ a \mu_{a}
  4. ε \varepsilon\,
  5. ε \varepsilon\,
  6. μ a \mu_{a}\,
  7. H b O 2 HbO_{2}
  8. H b Hb
  9. H b O 2 HbO2
  10. H b O 2 HbO_{2}
  11. H b Hb
  12. H b O 2 HbO_{2}
  13. C H b O 2 C_{HbO2}
  14. C H b C_{Hb}
  15. μ a ( λ 1 ) = ln ( 10 ) ε H b O 2 ( λ 1 ) C H b O 2 + ln ( 10 ) ε H b ( λ 1 ) C H b \mu_{a}(\lambda_{1})=\ln(10)\varepsilon_{HbO2}(\lambda_{1})C_{HbO2}+\ln(10)% \varepsilon_{Hb}(\lambda_{1})C_{Hb}\,
  16. μ a ( λ 2 ) = ln ( 10 ) ε H b O 2 ( λ 2 ) C H b O 2 + ln ( 10 ) ε H b ( λ 2 ) C H b \mu_{a}(\lambda_{2})=\ln(10)\varepsilon_{HbO2}(\lambda_{2})C_{HbO2}+\ln(10)% \varepsilon_{Hb}(\lambda_{2})C_{Hb}\,
  17. λ 1 \lambda_{1}
  18. λ 2 \lambda_{2}
  19. ε H b O 2 \varepsilon_{HbO2}
  20. ε H b \varepsilon_{Hb}
  21. H b O 2 HbO_{2}
  22. H b Hb
  23. C H b O 2 C_{HbO2}
  24. C H b C_{Hb}
  25. H b O 2 HbO_{2}
  26. H b Hb
  27. S O 2 SO_{2}
  28. S O 2 = C H b O 2 C H b O 2 + C H b SO_{2}=\frac{C_{HbO2}}{C_{HbO2}+C_{Hb}}
  29. μ s \mu_{s}
  30. μ e f f \mu_{eff}
  31. μ e f f = 3 μ a ( μ a + μ s ) \mu_{eff}=\sqrt{3\mu_{a}(\mu_{a}+\mu^{\prime}_{s})}
  32. μ s \mu^{\prime}_{s}
  33. μ s = μ s ( 1 - g ) \mu^{\prime}_{s}=\mu_{s}(1-g)\,
  34. g g
  35. λ - 0.7 \lambda\,^{-0.7}
  36. d d
  37. μ e f f \mu_{eff}
  38. S a O 2 SaO_{2}\,
  39. S v O 2 SvO_{2}\,
  40. S t O 2 StO_{2}\,
  41. T S I TSI\,
  42. S t O 2 StO_{2}\,
  43. S a O 2 SaO_{2}\,
  44. S v O 2 SvO_{2}\,

Near_sets.html

  1. δ \delta
  2. s y m b o l ε A N e a r symbol{\varepsilon{ANear}}
  3. s y m b o l ε A M e r symbol{\varepsilon{AMer}}
  4. s y m b o l s T o p symbol{sTop}
  5. s y m b o l M e t symbol{Met^{\infty}}
  6. s y m b o l A \hookrightarrowsymbol B symbol{A}\hookrightarrowsymbol{B}
  7. s y m b o l A symbol{A}
  8. s y m b o l B symbol{B}
  9. s y m b o l ε A M e r symbol{\varepsilon AMer}
  10. s y m b o l ε A N e a r symbol{\varepsilon ANear}
  11. s y m b o l ε A N e a r symbol{\varepsilon{ANear}}
  12. ε \varepsilon
  13. s y m b o l ε A M e r symbol{\varepsilon AMer}
  14. ε \varepsilon
  15. s y m b o l s T o p symbol{sTop}
  16. s y m b o l T o p symbol{Top}
  17. s y m b o l M e t symbol{Met^{\infty}}
  18. s y m b o l ε A P symbol{\varepsilon AP}
  19. ε \varepsilon
  20. ρ X , ρ Y \rho_{X},\rho_{Y}
  21. X , Y X,Y
  22. f : ( X , ρ X ) ( Y , ρ Y ) f:(X,\rho_{X})\longrightarrow(Y,\rho_{Y})
  23. f : ( X , ν D ρ X ) ( Y , ν D ρ Y ) f:(X,\nu_{D_{\rho_{X}}})\longrightarrow(Y,\nu_{D_{\rho_{Y}}})
  24. A , B 2 X A,B\in 2^{X}
  25. D ρ : 2 X × 2 X [ 0 , ] D_{\rho}:2^{X}\times 2^{X}\longrightarrow[0,\infty]
  26. D ρ ( A , B ) = { inf { ρ ( a , b ) : a A , b B } , if A and B are not empty , , if A or B is empty . D_{\rho}(A,B)=\begin{cases}\inf{\{\rho(a,b):a\in A,b\in B\}},&\,\text{if }A\,% \text{ and }B\,\text{ are not empty},\\ \infty,&\,\text{if }A\,\text{ or }B\,\text{ is empty}.\end{cases}
  27. s y m b o l ε symbol{\varepsilon}
  28. s y m b o l ε A N e a r symbol{\varepsilon{ANear}}
  29. F : s y m b o l ε A P s y m b o l ε A N e a r F:symbol{\varepsilon{AP}}\longrightarrow symbol{\varepsilon{ANear}}
  30. F ( ( X , ρ ) ) = ( X , ν D ρ ) F((X,\rho))=(X,\nu_{D_{\rho}})
  31. F ( f ) = f F(f)=f
  32. f : ( X , ρ X ) ( Y , ρ Y ) f:(X,\rho_{X})\longrightarrow(Y,\rho_{Y})
  33. f : ( X , ν D ρ X ) ( Y , ν D ρ Y ) f:(X,\nu_{D_{\rho_{X}}})\longrightarrow(Y,\nu_{D_{\rho_{Y}}})
  34. s y m b o l ε A P symbol{\varepsilon{AP}}
  35. s y m b o l ε A N e a r symbol{\varepsilon{ANear}}
  36. F : s y m b o l ε A P s y m b o l ε A N e a r F:symbol{\varepsilon{AP}}\longrightarrow symbol{\varepsilon{ANear}}
  37. F ( ( X , ρ ) ) = ( X , ν D ρ ) F((X,\rho))=(X,\nu_{D_{\rho}})
  38. F ( f ) = f . F(f)=f.
  39. s y m b o l M e t symbol{Met^{\infty}}
  40. s y m b o l ε A P symbol{\varepsilon{AP}}
  41. s y m b o l ε A N e a r symbol{\varepsilon{ANear}}
  42. s y m b o l M e t symbol{Met^{\infty}}
  43. s y m b o l ε A N e a r symbol{\varepsilon{ANear}}
  44. f : f:\mathbb{R}\rightarrow\mathbb{R}
  45. c c
  46. { x } \{x\}
  47. c c
  48. { f ( x ) } \{f(x)\}
  49. f ( c ) f(c)
  50. X X
  51. 2 X 2^{X}
  52. X X
  53. 2 X 2^{X}
  54. X X
  55. A , B X A,B\subset X
  56. A A
  57. B B
  58. A δ B A\ \delta\ B
  59. A 2 X A\in 2^{X}
  60. cl ( A ) \mbox{cl}~{}(A)
  61. cl ( A ) = { x X : D ( x , A ) = 0 } , where D ( x , A ) = i n f { d ( x , a ) : a A } . \begin{aligned}\displaystyle\mbox{cl}~{}(A)&\displaystyle=\left\{x\in X:D(x,A)% =0\right\},\ \mbox{where}\\ \displaystyle D(x,A)&\displaystyle=inf\left\{d(x,a):a\in A\right\}.\end{aligned}
  62. cl ( A ) \mbox{cl}~{}(A)
  63. x x
  64. X X
  65. A A
  66. D ( x , A ) D(x,A)
  67. x x
  68. A A
  69. d ( x , a ) = | x - a | d(x,a)=\left|x-a\right|
  70. δ = { ( A , B ) 2 X × 2 X : cl ( A ) cl ( B ) } . \delta=\left\{(A,B)\in 2^{X}\times 2^{X}:\mbox{cl}~{}(A)\ \cap\ \mbox{cl}~{}(B% )\neq\emptyset\right\}.
  71. A A
  72. B B
  73. A δ ¯ B A\ \underline{\delta}\ B
  74. A , B , E 2 X A,B,E\in 2^{X}
  75. A A
  76. B B
  77. B B
  78. A A
  79. A B A\cup B
  80. E E
  81. A A
  82. B B
  83. E E
  84. \emptyset
  85. A A
  86. B B
  87. C , D 2 X C,D\in 2^{X}
  88. C D = X C\cup D=X
  89. A A
  90. C C
  91. B B
  92. D D
  93. ( X , δ ) (X,\delta)
  94. X X
  95. X X
  96. δ \delta
  97. X X
  98. A A
  99. X X
  100. A A
  101. A A
  102. X X
  103. x X x\in X
  104. A A
  105. A , B A,B
  106. C c C^{c}
  107. X X
  108. A , B A,B
  109. X X
  110. C c = X \ C C^{c}=X\backslash C
  111. C C
  112. A \displaystyle A
  113. X X
  114. Φ = { ϕ 1 , , ϕ n } \Phi=\left\{\phi_{1},\dots,\phi_{n}\right\}
  115. x X x\in X
  116. X X
  117. ϕ : X \phi:X\rightarrow\mathbb{R}
  118. X X
  119. Φ : X n \Phi:X\longrightarrow\mathbb{R}^{n}
  120. Φ ( x ) = ( ϕ 1 ( x ) , , ϕ n ( x ) ) \Phi(x)=(\phi_{1}(x),\dots,\phi_{n}(x))
  121. n \mathbb{R}^{n}
  122. Φ ( x ) \Phi(x)
  123. x x
  124. x X x\in X
  125. δ Φ \delta_{\Phi}
  126. 𝒬 : 2 X 2 R n \mathcal{Q}:2^{X}\longrightarrow 2^{R^{n}}
  127. 2 X 2^{X}
  128. 2 R n 2^{R^{n}}
  129. A , B 2 X A,B\in 2^{X}
  130. 𝒬 ( A ) , 𝒬 ( B ) \mathcal{Q}(A),\mathcal{Q}(B)
  131. A , B A,B
  132. 𝒬 ( A ) = { Φ ( a ) : a A } , 𝒬 ( B ) = { Φ ( b ) : b B } . \begin{aligned}\displaystyle\mathcal{Q}(A)&\displaystyle=\left\{\Phi(a):a\in A% \right\},\\ \displaystyle\mathcal{Q}(B)&\displaystyle=\left\{\Phi(b):b\in B\right\}.\end{aligned}
  133. A δ Φ B A\ \delta_{\Phi}\ B
  134. A A
  135. B B
  136. A δ ¯ Φ B A\ \underline{\delta}_{\Phi}\ B
  137. A A
  138. B B
  139. A A
  140. B B
  141. A δ Φ B 𝒬 ( cl ( A ) ) δ 𝒬 ( cl ( B ) ) . A\ \delta_{\Phi}\ B\Leftrightarrow\mathcal{Q}(\mbox{cl}~{}(A))\;\delta\;% \mathcal{Q}(\mbox{cl}~{}(B))\neq\emptyset.
  142. Φ \mathop{\cap}_{\Phi}
  143. A A
  144. B B
  145. A Φ B = { x A B : 𝒬 ( A ) δ 𝒬 ( B ) } . A\ \mathop{\cap}_{\Phi}\ B=\left\{x\in A\cup B:\mathcal{Q}(A)\;\delta\;% \mathcal{Q}(B)\right\}.
  146. x A B x\in A\cup B
  147. A Φ B A\ \mathop{\cap}_{\Phi}\ B
  148. Φ ( x ) = Φ ( a ) = Φ ( b ) \Phi(x)=\Phi(a)=\Phi(b)
  149. a A , b B a\in A,b\in B
  150. A A
  151. B B
  152. A Φ B A\ \mathop{\cap}_{\Phi}\ B
  153. δ Φ \delta_{\Phi}
  154. δ Φ = { ( A , B ) 2 X × 2 X : cl ( A ) Φ cl ( B ) } . \delta_{\Phi}=\left\{(A,B)\in 2^{X}\times 2^{X}:\mbox{cl}~{}(A)\ \mathop{\cap}% _{\Phi}\ \mbox{cl}~{}(B)\neq\emptyset\right\}.
  155. A A
  156. B B
  157. A δ ¯ Φ B A\ \underline{\delta}_{\Phi}\ B
  158. δ Φ \delta_{\Phi}
  159. A , B , E X A,B,E\subset X
  160. A A
  161. B B
  162. B B
  163. A A
  164. A B A\cup B
  165. E E
  166. A A
  167. B B
  168. E E
  169. x , y X x,y\in X
  170. x x
  171. y y
  172. \emptyset
  173. A A
  174. B B
  175. C , D 2 X C,D\in 2^{X}
  176. C D = X C\cup D=X
  177. A A
  178. C C
  179. B B
  180. D D
  181. ( X , δ Φ ) (X,\delta_{\Phi})
  182. \mathcal{R}
  183. X X
  184. ( X , ) (X,\mathcal{R})
  185. X ( ) X(\mathcal{R})
  186. δ \mathcal{R}_{\delta}
  187. X X
  188. ( X , δ ) (X,\mathcal{R}_{\delta})
  189. δ \delta
  190. δ Φ \delta_{\Phi}
  191. δ Φ \mathcal{R}_{\delta_{\Phi}}
  192. ( X , δ Φ ) (X,\mathcal{R}_{\delta_{\Phi}})
  193. X X
  194. ( X , δ Φ ) (X,\mathcal{R}_{\delta_{\Phi}})
  195. X X
  196. A A
  197. cl ( A ) Φ \mbox{cl}~{}_{\Phi}(A)
  198. cl ( A ) Φ = { x X : Φ ( x ) δ 𝒬 ( cl ( A ) ) } . \mbox{cl}~{}_{\Phi}(A)=\left\{x\in X:{\Phi(x)}\delta\mathcal{Q}(\mbox{cl}~{}(A% ))\right\}.
  199. x X x\in X
  200. A A
  201. Φ ( x ) \Phi(x)
  202. 𝒬 ( cl ( A ) ) \mathcal{Q}(\mbox{cl}~{}(A))
  203. A A
  204. ( X , δ Φ ) (X,\mathcal{R}_{\delta_{\Phi}})
  205. x X x\in X
  206. A A
  207. A A
  208. A A
  209. ( X , δ Φ ) (X,\mathcal{R}_{\delta_{\Phi}})
  210. A X A\subset X
  211. cl ( A ) cl ( A ) Φ \mbox{cl}~{}(A)\subseteq\mbox{cl}~{}_{\Phi}(A)
  212. Φ ( x ) 𝒬 ( X cl ( A ) ) \Phi(x)\in\mathcal{Q}(X\setminus\mbox{cl}~{}(A))
  213. Φ ( x ) = Φ ( a ) \Phi(x)=\Phi(a)
  214. a cl A a\in\mbox{cl}~{}A
  215. Φ ( x ) 𝒬 ( cl ( A ) Φ ) \Phi(x)\in\mathcal{Q}(\mbox{cl}~{}_{\Phi}(A))
  216. cl ( A ) cl ( A ) Φ \mbox{cl}~{}(A)\subseteq\mbox{cl}~{}_{\Phi}(A)
  217. δ \delta
  218. δ Φ \delta_{\Phi}
  219. ( X , δ Φ ) (X,\mathcal{R}_{\delta_{\Phi}})
  220. A , B , C X A,B,C\subset X
  221. {}^{\circ}
  222. A δ B implies A δ Φ B A\ \delta\ B\ \mbox{implies}~{}\ A\ \delta_{\Phi}\ B
  223. {}^{\circ}
  224. ( A B ) δ C implies ( A B ) δ Φ C (A\cup B)\ \delta\ C\ \mbox{implies}~{}\ (A\cup B)\ \delta_{\Phi}\ C
  225. {}^{\circ}
  226. cl A δ cl B implies cl A δ Φ cl B \mbox{cl}~{}A\ \delta\ \mbox{cl}~{}B\ \mbox{implies}~{}\ \mbox{cl}~{}A\ \delta% _{\Phi}\ \mbox{cl}~{}B
  227. {}^{\circ}
  228. A δ B A B A\ \delta\ B\Leftrightarrow A\cap B\neq\emptyset
  229. x A B , Φ ( x ) 𝒬 ( A ) x\in A\cap B,\Phi(x)\in\mathcal{Q}(A)
  230. Φ ( x ) 𝒬 ( B ) \Phi(x)\in\mathcal{Q}(B)
  231. A δ Φ B A\ \delta_{\Phi}\ B
  232. 1 2 1^{\circ}\Rightarrow\ 2^{\circ}
  233. {}^{\circ}
  234. cl A δ cl B \mbox{cl}~{}A\ \delta\ \mbox{cl}~{}B
  235. cl A \mbox{cl}~{}A
  236. cl A \mbox{cl}~{}A
  237. o 3 o {}^{o}\Rightarrow\ 3^{o}
  238. \qquad\blacksquare
  239. δ \delta
  240. δ \delta
  241. X X
  242. x X x\in X
  243. N x , ε N_{x,\varepsilon}
  244. ε > 0 \varepsilon>0
  245. N x , ε = { y X : d ( x , y ) < ε } . N_{x,\varepsilon}=\left\{y\in X:d(x,y)<\varepsilon\right\}.
  246. A A
  247. int ( A ) \mbox{int}~{}(A)
  248. A A
  249. bdy ( A ) \mbox{bdy}~{}(A)
  250. X X
  251. int ( A ) = { x X : N x , ε A } . \mbox{int}~{}(A)=\left\{x\in X:N_{x,\varepsilon}\subseteq A\right\}.
  252. bdy ( A ) = cl ( A ) int ( A ) . \mbox{bdy}~{}(A)=\mbox{cl}~{}(A)\setminus\mbox{int}~{}(A).
  253. A A
  254. B B
  255. δ \delta
  256. A δ B A\ \ll_{\delta}\ B
  257. A int B A\subset\ \mbox{int}~{}B
  258. A δ ¯ X int B A\ \underline{\delta}\ X\setminus\mbox{int}~{}B
  259. A A
  260. int B \mbox{int}~{}B
  261. A A
  262. B B
  263. δ Φ \delta_{\Phi}
  264. A Φ B A\ \mathop{\ll}_{\Phi}\ B
  265. 𝒬 ( A ) 𝒬 ( int B ) \mathcal{Q}(A)\subset\ \mathcal{Q}(\mbox{int}~{}B)
  266. A δ ¯ Φ X int B A\ \underline{\delta}_{\Phi}\ X\setminus\mbox{int}~{}B
  267. 𝒬 ( A ) \mathcal{Q}(A)
  268. int B \mbox{int}~{}B
  269. Φ \mathop{\ll}_{\Phi}
  270. δ \delta
  271. Φ = { ( A , B ) 2 X × 2 X : 𝒬 ( A ) 𝒬 ( int B ) } . \mathop{\ll}_{\Phi}=\left\{(A,B)\in 2^{X}\times 2^{X}:\mathcal{Q}(A)\subset% \mathcal{Q}(\mbox{int}~{}B)\right\}.
  272. A Φ B A\ \mathop{\ll}_{\Phi}\ B
  273. a A a\in A
  274. b int B b\in\mbox{int}~{}B
  275. A , B A,B
  276. X X
  277. A δ ¯ Φ B A\ \underline{\delta}_{\Phi}\ B
  278. δ Φ \delta_{\Phi}
  279. A δ ¯ Φ B A Φ E 1 , B Φ E 2 , for some E 1 , E 2 X (See Fig. 6). A\ \underline{\delta}_{\Phi}\ B\Leftrightarrow A\ \mathop{\ll}_{\Phi}\ E1,B\ % \mathop{\ll}_{\Phi}\ E2,\ \mbox{for some}~{}\ E1,E2\subset X\ \mbox{(See Fig. % 6).}~{}
  280. δ Φ \delta_{\Phi}
  281. X X
  282. ε \varepsilon
  283. δ Φ \mathcal{R}_{\delta_{\Phi}}
  284. δ Φ , ε \delta_{\Phi,\varepsilon}
  285. D Φ ( A , B ) = i n f { d ( Φ ( a ) , Φ ( a ) ) : Φ ( a ) 𝒬 ( A ) , Φ ( a ) 𝒬 ( B ) } , d ( Φ ( a ) , Φ ( a ) ) = i = 1 n | ϕ i ( a ) - ϕ i ( b ) | , δ Φ , ε = { ( A , B ) 2 X × 2 X : | D ( cl ( A ) , cl ( B ) ) | < ε } . \begin{aligned}\displaystyle D_{\Phi}(A,B)&\displaystyle=inf\left\{d(\Phi(a),% \Phi(a)):\Phi(a)\in\mathcal{Q}(A),\Phi(a)\in\mathcal{Q}(B)\right\},\\ \displaystyle d(\Phi(a),\Phi(a))&\displaystyle=\mathop{\sum}_{i=1}^{n}|\phi_{i% }(a)-\phi_{i}(b)|,\\ \displaystyle\delta_{\Phi,\varepsilon}&\displaystyle=\left\{(A,B)\in 2^{X}% \times 2^{X}:|D(\mbox{cl}~{}(A),\mbox{cl}~{}(B))|<\varepsilon\right\}.\end{aligned}
  286. δ Φ , ε = δ Φ { δ Φ , ε } \mathcal{R}_{\delta_{\Phi,\varepsilon}}=\mathcal{R}_{\delta_{\Phi}}\cup\left\{% \delta_{\Phi,\varepsilon}\right\}
  287. δ Φ , ε \mathcal{R}_{\delta_{\Phi,\varepsilon}}
  288. δ Φ \mathcal{R}_{\delta_{\Phi}}
  289. X X
  290. A , B A,B
  291. ( X , δ Φ , ε ) (X,\mathcal{R}_{\delta_{\Phi,\varepsilon}})
  292. A δ Φ , ε B A\ \delta_{\Phi,\varepsilon}\ B
  293. D Φ ( A , B ) < ε . D_{\Phi}(A,B)<\varepsilon.
  294. τ \tau
  295. O O
  296. τ O × O \tau\subseteq O\times O
  297. O O
  298. τ \tau
  299. ( O , τ ) (O,\tau)
  300. A O A\subseteq O
  301. τ \tau
  302. τ \tau
  303. x , y A x,y\in A
  304. ( x , y ) τ (x,y)\in\tau
  305. τ \tau
  306. τ \tau
  307. ( O , τ ) (O,\tau)
  308. H τ ( O ) H_{\tau}(O)
  309. H τ ( O ) H_{\tau}(O)
  310. O O
  311. ε ( 0 , ] \varepsilon\in(0,\infty]
  312. | Φ | = 1 |\Phi|=1
  313. x i x_{i}
  314. ϕ ( x ) \phi(x)
  315. x i x_{i}
  316. ϕ ( x ) \phi(x)
  317. x i x_{i}
  318. ϕ ( x ) \phi(x)
  319. x i x_{i}
  320. ϕ ( x ) \phi(x)
  321. x 1 x_{1}
  322. x 6 x_{6}
  323. x 11 x_{11}
  324. x 16 x_{16}
  325. x 2 x_{2}
  326. x 7 x_{7}
  327. x 12 x_{12}
  328. x 17 x_{17}
  329. x 3 x_{3}
  330. x 8 x_{8}
  331. x 13 x_{13}
  332. x 18 x_{18}
  333. x 4 x_{4}
  334. x 9 x_{9}
  335. x 14 x_{14}
  336. x 19 x_{19}
  337. x 5 x_{5}
  338. x 10 x_{10}
  339. x 15 x_{15}
  340. x 20 x_{20}
  341. ε = { ( x , y ) O × O : Φ ( x ) - Φ ( y ) 2 ε } \cong_{\varepsilon}=\{(x,y)\in O\times O:\;\parallel\Phi(x)-\Phi(y)\parallel_{% {}_{2}}\leq\varepsilon\}
  342. ε = 0.1 \varepsilon=0.1
  343. H ε ( O ) = \displaystyle H_{\cong_{\varepsilon}}(O)=
  344. Φ ( x ) - Φ ( y ) 2 ε \parallel\Phi(x)-\Phi(y)\parallel_{2}\leq\varepsilon
  345. X X
  346. Y Y
  347. O = { X Y } O=\{X\cup Y\}
  348. ε \varepsilon
  349. ( U , δ Φ , ε ) (U,\mathcal{R}_{\delta_{\Phi,\varepsilon}})
  350. δ Φ , ε \delta_{\Phi,\varepsilon}
  351. X , Y 2 U X,Y\in 2^{U}
  352. Φ , ε \cong_{\Phi,\varepsilon}
  353. Φ \Phi
  354. ε ( 0 , ] \varepsilon\in(0,\infty]
  355. Φ , ε = { ( x , y ) U × U | Φ ( x ) - Φ ( y ) | ε } . \simeq_{\Phi,\varepsilon}=\{(x,y)\in U\times U\mid\ |\Phi(x)-\Phi(y)|\leq% \varepsilon\}.
  356. Z = X Y Z=X\cup Y
  357. H τ Φ , ε ( Z ) H_{\tau_{\Phi,\varepsilon}}(Z)
  358. ( Z , Φ , ε ) (Z,\simeq_{\Phi,\varepsilon})
  359. A X , B Y A\subseteq X,B\subseteq Y
  360. D t N M : 2 U × 2 U : [ 0 , ] D_{{}_{tNM}}:2^{U}\times 2^{U}:\longrightarrow[0,\infty]
  361. D t N M ( X , Y ) = { 1 - t N M ( A , B ) , if X and Y are not empty , , if X or Y is empty , D_{{}_{tNM}}(X,Y)=\begin{cases}1-tNM(A,B),&\mbox{if }~{}X\mbox{ and }~{}Y\mbox% { are not empty}~{},\\ \infty,&\mbox{if }~{}X\mbox{ or }~{}Y\mbox{ is empty}~{},\end{cases}
  362. t N M ( A , B ) = ( C H τ Φ , ε ( Z ) | C | ) - 1 C H τ Φ , ε ( Z ) | C | min ( | C A | , | [ C B | ) max ( | C A | , | C B | ) . tNM(A,B)=\Biggl(\sum_{C\in H_{\tau_{\Phi,\varepsilon}}(Z)}|C|\Biggr)^{-1}\cdot% \sum_{C\in H_{\tau_{\Phi,\varepsilon}}(Z)}|C|\frac{\min(|C\cap A|,|[C\cap B|)}% {\max(|C\cap A|,|C\cap B|)}.
  363. t N M tNM
  364. t N M tNM
  365. Z = X Y Z=X\cup Y
  366. t N M tNM
  367. X X
  368. Y Y
  369. t N M tNM
  370. X X
  371. Y Y
  372. t N M tNM

Negative_index_metamaterials.html

  1. n = ± ϵ μ \scriptstyle n=\pm\sqrt{\epsilon\mu}

Negative_multinomial_distribution.html

  1. k 0 p 0 p \tfrac{k_{0}}{p_{0}}\,p
  2. k 0 p 0 2 p p + k 0 p 0 diag ( p ) \tfrac{k_{0}}{p_{0}^{2}}\,pp^{\prime}+\tfrac{k_{0}}{p_{0}}\,\operatorname{diag% }(p)
  3. ( p 0 1 - p e i t ) k 0 \bigg(\frac{p_{0}}{1-p^{\prime}e^{it}}\bigg)^{\!k_{0}}
  4. x i , j x_{i,j}
  5. 0 i 2 0\leq i\leq 2
  6. 0 j 3 0\leq j\leq 3
  7. i 0 i_{0}
  8. X = { X 1 , X 2 , X 3 } N M ( k 0 , { p 1 , p 2 , p 3 } ) X=\{X_{1},X_{2},X_{3}\}\sim NM(k_{0},\{p_{1},p_{2},p_{3}\})
  9. μ ^ i , j = x i , . × x . , j x . , . \hat{\mu}_{i,j}=\frac{x_{i,.}\times x_{.,j}}{x_{.,.}}
  10. x i , . = j = 0 3 x i , j x_{i,.}=\sum_{j=0}^{3}{x_{i,j}}
  11. x . , j = i = 0 2 x i , j x_{.,j}=\sum_{i=0}^{2}{x_{i,j}}
  12. x . , . = i = 0 2 j = 0 3 x i , j x_{.,.}=\sum_{i=0}^{2}\sum_{j=0}^{3}{{x_{i,j}}}
  13. μ ^ 1 , 1 = x 1 , . × x . , 1 x . , . = 34 × 68 400 = 5.78 \hat{\mu}_{1,1}=\frac{x_{1,.}\times x_{.,1}}{x_{.,.}}=\frac{34\times 68}{400}=% 5.78
  14. X = { X 1 = 5 , X 2 = 1 , X 3 = 5 } X=\left\{X_{1}=5,X_{2}=1,X_{3}=5\right\}
  15. X N M ( k 0 = 10 , { p 1 = 0.2 , p 2 = 0.1 , p 3 = 0.2 } ) X\sim NM(k_{0}=10,\{p_{1}=0.2,p_{2}=0.1,p_{3}=0.2\})
  16. p 0 = 1 - i = 1 3 p i = 0.5 p_{0}=1-\sum_{i=1}^{3}{p_{i}}=0.5
  17. N M ( X | k 0 , { p 1 , p 2 , p 3 } ) = 0.00465585119998784 NM(X|k_{0},\{p_{1},p_{2},p_{3}\})=0.00465585119998784
  18. c o v [ X 1 , X 3 ] = 10 × 0.2 × 0.2 0.5 2 = 1.6 cov[X_{1},X_{3}]=\frac{10\times 0.2\times 0.2}{0.5^{2}}=1.6
  19. μ 2 = k 0 p 2 p 0 = 10 × 0.1 0.5 = 2.0 \mu_{2}=\frac{k_{0}p_{2}}{p_{0}}=\frac{10\times 0.1}{0.5}=2.0
  20. μ 3 = k 0 p 3 p 0 = 10 × 0.2 0.5 = 4.0 \mu_{3}=\frac{k_{0}p_{3}}{p_{0}}=\frac{10\times 0.2}{0.5}=4.0
  21. c o r r [ X 2 , X 3 ] = ( μ 2 × μ 3 ( k 0 + μ 2 ) ( k 0 + μ 3 ) ) 1 2 corr[X_{2},X_{3}]=\left(\frac{\mu_{2}\times\mu_{3}}{(k_{0}+\mu_{2})(k_{0}+\mu_% {3})}\right)^{\frac{1}{2}}
  22. c o r r [ X 2 , X 3 ] = ( 2 × 4 ( 10 + 2 ) ( 10 + 4 ) ) 1 2 = 0.21821789023599242. corr[X_{2},X_{3}]=\left(\frac{2\times 4}{(10+2)(10+4)}\right)^{\frac{1}{2}}=0.% 21821789023599242.
  23. k 0 k_{0}
  24. k 0 k_{0}
  25. X i X_{i}
  26. ( μ i = k 0 p i p 0 ) \left(\mu_{i}=k_{0}\frac{p_{i}}{p_{0}}\right)
  27. X i X_{i}
  28. X i X_{i}
  29. X = { X 1 , , X m } X=\{X_{1},\cdots,X_{m}\}
  30. X N M ( k 0 , { p 1 , , p m } ) X\sim NM(k_{0},\{p_{1},\cdots,p_{m}\})
  31. μ j \mu_{j}
  32. X j X_{j}
  33. { x 1 , , x m } \{x_{1},\cdots,x_{m}\}
  34. μ ^ i = x i . \hat{\mu}_{i}=x_{i}.
  35. μ ^ j = x j , . I \hat{\mu}_{j}=\frac{x_{j,.}}{I}
  36. 0 j J 0\leq j\leq J
  37. X 0 X_{0}
  38. μ ^ 0 = 34 / 3 = 11.33 \hat{\mu}_{0}=34/3=11.33
  39. X 1 X_{1}
  40. μ ^ 1 = 185 / 3 = 61.67 \hat{\mu}_{1}=185/3=61.67
  41. X 2 X_{2}
  42. μ ^ 2 = 125 / 3 = 41.67 \hat{\mu}_{2}=125/3=41.67
  43. X 3 X_{3}
  44. μ ^ 3 = 56 / 3 = 18.67 \hat{\mu}_{3}=56/3=18.67
  45. k 0 k_{0}
  46. k 0 k_{0}
  47. \Chi 2 = i ( x i - μ i ) 2 μ i \Chi^{2}=\sum_{i}{\frac{(x_{i}-\mu_{i})^{2}}{\mu_{i}}}
  48. μ i \mu_{i}
  49. μ i ^ \hat{\mu_{i}}
  50. \Chi 2 ( k 0 ) = i ( x i - μ i ^ ) 2 μ i ^ ( 1 + μ i ^ k 0 ) \Chi^{2}(k_{0})=\sum_{i}{\frac{(x_{i}-\hat{\mu_{i}})^{2}}{\hat{\mu_{i}}\left(1% +\frac{\hat{\mu_{i}}}{k_{0}}\right)}}
  51. k 0 k_{0}
  52. k 0 k_{0}
  53. \Chi 2 ( k 0 ) \Chi^{2}(k_{0})
  54. μ j \mu_{j}
  55. μ ^ 1 = 185 / 3 = 61.67 \hat{\mu}_{1}=185/3=61.67
  56. μ ^ 2 = 125 / 3 = 41.67 \hat{\mu}_{2}=125/3=41.67
  57. μ ^ 3 = 56 / 3 = 18.67 \hat{\mu}_{3}=56/3=18.67
  58. \Chi 2 ( k 0 ) = 5.261948 \Chi^{2}(k_{0})=5.261948
  59. k 0 k_{0}
  60. x = { x 1 = 5 , x 2 = 1 , x 3 = 5 } x=\{x_{1}=5,x_{2}=1,x_{3}=5\}
  61. k 0 k_{0}
  62. \Chi 2 ( k 0 ) = i = 1 3 ( x i - μ i ^ ) 2 μ i ^ ( 1 + μ i ^ k 0 ) \Chi^{2}(k_{0})=\sum_{i=1}^{3}{\frac{(x_{i}-\hat{\mu_{i}})^{2}}{\hat{\mu_{i}}% \left(1+\frac{\hat{\mu_{i}}}{k_{0}}\right)}}
  63. \Chi 2 ( k 0 ) = ( 5 - 61.67 ) 2 61.67 ( 1 + 61.67 / k 0 ) + ( 1 - 41.67 ) 2 41.67 ( 1 + 41.67 / k 0 ) + ( 5 - 18.67 ) 2 18.67 ( 1 + 18.67 / k 0 ) = 5.261948. \Chi^{2}(k_{0})=\frac{(5-61.67)^{2}}{61.67(1+61.67/k_{0})}+\frac{(1-41.67)^{2}% }{41.67(1+41.67/k_{0})}+\frac{(5-18.67)^{2}}{18.67(1+18.67/k_{0})}=5.261948.
  64. k 0 k_{0}
  65. k 0 k_{0}
  66. k 0 > 0 k_{0}>0
  67. k 0 = 2 k_{0}=2
  68. μ i k 0 p 0 = p i \frac{\mu_{i}}{k_{0}}p_{0}=p_{i}
  69. 61.67 k 0 p 0 = 31 p 0 = p 1 \frac{61.67}{k_{0}}p_{0}=31p_{0}=p_{1}
  70. 20 p 0 = p 2 20p_{0}=p_{2}
  71. 9 p 0 = p 3 9p_{0}=p_{3}
  72. 1 - p 0 = p 1 + p 2 + p 3 = 60 p 0 1-p_{0}=p_{1}+p_{2}+p_{3}=60p_{0}
  73. p 0 = 1 61 p_{0}=\frac{1}{61}
  74. p 1 = 31 61 p_{1}=\frac{31}{61}
  75. p 2 = 20 61 p_{2}=\frac{20}{61}
  76. p 3 = 9 61 p_{3}=\frac{9}{61}
  77. x = { x 1 = 5 , x 2 = 1 , x 3 = 5 } x=\{x_{1}=5,x_{2}=1,x_{3}=5\}
  78. X N M ( 2 , { 31 61 , 20 61 , 9 61 } ) . X\sim NM\left(2,\left\{\frac{31}{61},\frac{20}{61},\frac{9}{61}\right\}\right).

Nehari_manifold.html

  1. - u = | u | p - 1 u , with u | Ω = 0. -\triangle u=|u|^{p-1}u,\,\text{ with }u\mid_{\partial\Omega}=0.
  2. J ( v ) = 1 2 Ω | v | 2 d μ - 1 p + 1 Ω | v | p + 1 d μ J(v)=\frac{1}{2}\int_{\Omega}{|\nabla v|^{2}\,d\mu}-\frac{1}{p+1}\int_{\Omega}% {|v|^{p+1}\,d\mu}
  3. v L 2 ( Ω ) 2 = v L p + 1 ( Ω ) p + 1 > 0. \|\nabla v\|^{2}_{L^{2}(\Omega)}=\|v\|^{p+1}_{L^{p+1}(\Omega)}>0.
  4. J ( u ) , u = 0. \langle J^{\prime}(u),u\rangle=0.\,

Neighborly_polytope.html

  1. k = d / 2 k=\lfloor d/2\rfloor
  2. k 1 + d / 2 k\geq 1+\lfloor d/2\rfloor
  3. f k - 1 i = 0 d / 2 ( ( d - i k - i ) + ( i k - d + i ) ) * ( n - d - 1 + i i ) , f_{k-1}\leq\sum_{i=0}^{d/2}{}^{*}\left({\left({{d-i}\atop{k-i}}\right)}+{\left% ({{i}\atop{k-d+i}}\right)}\right){\left({{n-d-1+i}\atop{i}}\right)},
  4. i = d / 2 i=\lfloor d/2\rfloor

Neighbourhood_components_analysis.html

  1. A A
  2. A A
  3. k k
  4. A A
  5. A A
  6. A * A^{*}
  7. A * = argmax f A ( A ) A^{*}=\mbox{argmax}~{}_{A}f(A)
  8. k k
  9. C i C_{i}
  10. A A
  11. f ( ) f(\cdot)
  12. k k
  13. p i j = { e - || A x i - A x j || 2 k e - || A x i - A x k || 2 , if j i 0 , if j = i p_{ij}=\begin{cases}\frac{e^{-||Ax_{i}-Ax_{j}||^{2}}}{\sum_{k}e^{-||Ax_{i}-Ax_% {k}||^{2}}},&\mbox{if}~{}j\neq i\\ 0,&\mbox{if}~{}j=i\end{cases}
  14. i i
  15. C i C_{i}
  16. p i = j C i p i j p_{i}=\sum_{j\in C_{i}}p_{ij}\quad
  17. p i j p_{ij}
  18. j j
  19. i i
  20. f ( A ) = i j C i p i j = i p i f(A)=\sum_{i}\sum_{j\in C_{i}}p_{ij}=\sum_{i}p_{i}
  21. i i
  22. j C i j\in C_{i}
  23. P ( C l a s s ( X i ) = C l a s s ( X j ) ) = p i j P(Class(X_{i})=Class(X_{j}))=p_{ij}
  24. j C j j\in C_{j}
  25. C j C_{j}
  26. A A
  27. f A = - 2 A i j C i p i j ( x i j x i j T - k p i k x i k x i k T ) \frac{\partial f}{\partial A}=-2A\sum_{i}\sum_{j\in C_{i}}p_{ij}\left(x_{ij}x_% {ij}^{T}-\sum_{k}p_{ik}x_{ik}x_{ik}^{T}\right)
  28. = 2 A i ( p i k p i k x i k x i k T - j C i p i j x i j x i j T ) =2A\sum_{i}\left(p_{i}\sum_{k}p_{ik}x_{ik}x_{ik}^{T}-\sum_{j\in C_{i}}p_{ij}x_% {ij}x_{ij}^{T}\right)
  29. A A
  30. f ( ) f(\cdot)
  31. L 1 L_{1}
  32. p i p_{i}
  33. A A
  34. g ( A ) = i log ( j C i p i j ) = i log ( p i ) g(A)=\sum_{i}\log\left(\sum_{j\in C_{i}}p_{ij}\right)=\sum_{i}\log(p_{i})
  35. g A = 2 A i ( k p i k x i k x i k T - j C i p i j x i j x i j T j C i p i j ) \frac{\partial g}{\partial A}=2A\sum_{i}\left(\sum_{k}p_{ik}x_{ik}x_{ik}^{T}-% \frac{\sum_{j\in C_{i}}p_{ij}x_{ij}x_{ij}^{T}}{\sum_{j\in C_{i}}p_{ij}}\right)
  36. A A

Nekhoroshev_estimates.html

  1. H ( I ) + ϵ h ( I , θ ) H(I)+\epsilon h(I,\theta)
  2. n n
  3. ( I , θ ) (I,\theta)
  4. | I ( t ) - I ( 0 ) | < ε 1 / ( 2 n ) |I(t)-I(0)|<\varepsilon^{1/(2n)}
  5. | t | < exp ( c ( 1 ε ) 1 / ( 2 n ) ) |t|<\exp\left({c\left({1\over\varepsilon}\right)^{1/(2n)}}\right)
  6. c c

Nelson–Aalen_estimator.html

  1. H ~ ( t ) = t i t d i n i , \tilde{H}(t)=\sum_{t_{i}\leq t}\frac{d_{i}}{n_{i}},
  2. d i d_{i}
  3. t i t_{i}
  4. n i n_{i}
  5. t i t_{i}

Nested_word.html

  1. \ell
  2. [ ] [\ell]
  3. { 1 , 2 , , - 1 , } \{1,2,\ldots,\ell-1,\ell\}
  4. [ 0 ] = [0]=\emptyset
  5. 0 \ell\geq 0
  6. { - , 1 , 2 , , - 1 , } × { 1 , 2 , , - 1 , , } \{-\infty,1,2,\ldots,\ell-1,\ell\}\times\{1,2,\ldots,\ell-1,\ell,\infty\}
  7. \ell
  8. \ell
  9. \ell
  10. Σ = { a 1 , a 2 , , a n } \Sigma=\{a_{1},a_{2},\ldots,a_{n}\}
  11. Σ ^ \hat{\Sigma}
  12. Σ ^ \hat{\Sigma}
  13. w 1 w 2 w w_{1}w_{2}\cdots w_{\ell}
  14. x 1 x 2 x x_{1}x_{2}...x_{\ell}
  15. x i x_{i}
  16. w i = a w_{i}=a
  17. w = w 1 w w=w_{1}\cdots w_{\ell}
  18. w j w_{j}
  19. w j w_{j}
  20. j j
  21. w j w_{j}
  22. w j w_{j}
  23. M = ( Q , Σ ^ , Γ , δ , q 0 , F ) M=(Q,\hat{\Sigma},\Gamma,\delta,q_{0},F)
  24. Q Q
  25. Σ ^ \hat{\Sigma}
  26. Σ c \Sigma\text{c}
  27. Σ r \Sigma\text{r}
  28. Σ int \Sigma\text{int}
  29. Σ c \Sigma\text{c}
  30. Σ r \Sigma\text{r}
  31. Σ int \Sigma\text{int}
  32. Γ \Gamma
  33. Γ \bot\in\Gamma
  34. δ = δ c δ r δ int \delta=\delta\text{c}\cup\delta\text{r}\cup\delta\text{int}
  35. δ c : Q × Σ c Q × Γ \delta\text{c}\colon Q\times\Sigma\text{c}\to Q\times\Gamma
  36. δ r : Q × Σ r × Γ Q \delta\text{r}\colon Q\times\Sigma\text{r}\times\Gamma\to Q
  37. δ int : Q × Σ int Q \delta\text{int}:Q\times\Sigma\text{int}\to Q
  38. q 0 Q q_{0}\in\,Q
  39. F Q F\subseteq Q
  40. a c Σ c a\text{c}\in\Sigma\text{c}
  41. a r Σ r a\text{r}\in\Sigma\text{r}
  42. a i Σ int a\text{i}\in\Sigma\text{int}
  43. L = { a n b a n n N } L=\{a^{n}ba^{n}\mid n\in\mathrm{N}\}
  44. Σ \Sigma
  45. L L
  46. Σ ^ \hat{\Sigma}
  47. L L
  48. s s
  49. 2 s 2 2^{s^{2}}
  50. | A | |A|
  51. A A
  52. O ( | A | 3 ) O(|A|^{3}\ell)
  53. O ( | A | 3 ) O(|A|^{3})
  54. A A
  55. O ( ) O(\ell)
  56. O ( d ) O(d)
  57. d d
  58. O ( log ( ) ) O(\log(\ell))
  59. O ( 2 log ( ) ) O(\ell^{2}\log(\ell))
  60. O ( log ) O(\log\ell)
  61. Σ ^ \,\hat{\Sigma}
  62. Σ ^ \,\hat{\Sigma}
  63. M 1 M_{1}
  64. M 2 M_{2}
  65. i = 1 , 2 i=1,2
  66. M i M_{i}
  67. ( Q i , Σ ^ , Γ i , δ i , s i , Z i , F i ) (Q_{i},\ \hat{\Sigma},\ \Gamma_{i},\ \delta_{i},\ s_{i},\ Z_{i},\ F_{i})
  68. Q 1 × Q 2 \,Q_{1}\times Q_{2}
  69. ( s 1 , s 2 ) \left(s_{1},s_{2}\right)
  70. F 1 × F 2 F_{1}\times F_{2}
  71. Γ 1 × Γ 2 \,\Gamma_{1}\times\Gamma_{2}
  72. ( Z 1 , Z 2 ) (Z_{1},Z_{2})
  73. M M
  74. ( p 1 , p 2 ) (p_{1},p_{2})
  75. a \left\langle a\right.
  76. M M
  77. ( γ 1 , γ 2 ) (\gamma_{1},\gamma_{2})
  78. ( q 1 , q 2 ) (q_{1},q_{2})
  79. γ i \gamma_{i}
  80. M i M_{i}
  81. p i p_{i}
  82. q i q_{i}
  83. a \left\langle a\right.
  84. M M
  85. ( p 1 , p 2 ) (p_{1},p_{2})
  86. a a
  87. M M
  88. ( q 1 , q 2 ) (q_{1},q_{2})
  89. M i M_{i}
  90. p i p_{i}
  91. q i q_{i}
  92. M M
  93. ( p 1 , p 2 ) (p_{1},p_{2})
  94. a \left.a\right\rangle
  95. M M
  96. ( γ 1 , γ 2 ) (\gamma_{1},\gamma_{2})
  97. ( q 1 , q 2 ) (q_{1},q_{2})
  98. γ i \gamma_{i}
  99. M i M_{i}
  100. p i p_{i}
  101. q i q_{i}
  102. a \left.a\right\rangle
  103. M 1 M_{1}
  104. M 2 M_{2}
  105. G = ( V = V 0 V 1 , Σ , R , S ) G=(V=V^{0}\cup V^{1}\,,\Sigma\,,R\,,S\,)
  106. V 0 V^{0}\,
  107. V 1 V^{1}\,
  108. v V v\in V
  109. G G\,
  110. V 0 V^{0}\,
  111. Σ \Sigma\,
  112. V V\,
  113. G G\,
  114. S S\,
  115. V V\,
  116. R R\,
  117. V V\,
  118. ( V Σ ) * (V\cup\Sigma)^{*}
  119. \exist w ( V Σ ) * : ( S , w ) R \exist\,w\in(V\cup\Sigma)^{*}:(S,w)\in R
  120. R R\,
  121. X , Y V , Z V 0 X,Y\in V,Z\in V^{0}
  122. a Σ ^ a\in\hat{\Sigma}
  123. b Σ ^ b\in\hat{\Sigma}
  124. X ϵ X\to\epsilon
  125. X a Y X\to aY
  126. X V 0 X\in V^{0}
  127. Y V 0 Y\in V^{0}
  128. a Σ a\in\Sigma
  129. X a Z b Y X\to\langle aZb\rangle Y
  130. X V 0 X\in V^{0}
  131. Y V 0 Y\in V^{0}
  132. ϵ \epsilon
  133. \ell
  134. \Omicron ( log ) \Omicron(\log\ell)

Net_foreign_assets.html

  1. Change in NFA = Current Account \begin{aligned}\displaystyle\mbox{Change in NFA}&\displaystyle=\mbox{Current % Account}\\ \end{aligned}
  2. = Current Account + Valuation Effects \displaystyle=\mbox{Current Account}~{}+\mbox{Valuation Effects}

Net_income_attributable.html

  1. N I A = C o n t r i b u t i o n × A d j u s t e d C l o s i n g B a l a n c e - A d j u s t e d O p e n i n g B a l a n c e A d j u s t e d O p e n i n g B a l a n c e NIA=Contribution\times\frac{Adjusted\ Closing\ Balance-Adjusted\ Opening\ % Balance}{Adjusted\ Opening\ Balance}
  2. 2000 × 6000 - 9000 9000 = - 666. 66 ¯ 2000\times\frac{6000-9000}{9000}=-666.\overline{66}

Net_operating_assets.html

  1. NOA = operating assets - operating liabilities \mathrm{NOA}={\mbox{operating assets}~{}}-{\mbox{operating liabilities}~{}}
  2. Operating assets = total assets - cash \mbox{Operating assets}~{}=\mbox{total assets}~{}-\mbox{cash}~{}
  3. Operating liabilities = total liabilities - short-term notes - long-term notes \mbox{Operating liabilities}~{}=\mbox{total liabilities}~{}-\mbox{short-term % notes}~{}-\mbox{long-term notes}~{}
  4. DAOE = NOPAT ( t ) - WACC × NOA ( t - 1 ) WACC + NOA - BVD \mathrm{DAOE}=\frac{\mbox{NOPAT}~{}(t)-\mbox{WACC}~{}\times\mbox{NOA}~{}(t-1)}% {\mbox{WACC}~{}}+{\mbox{NOA}~{}}-{\mbox{BVD}~{}}
  5. FCF = NOPAT - Change in NOA \mathrm{FCF}={\mbox{NOPAT}~{}}-{\mbox{Change in NOA}~{}}
  6. DCF = FCF WACC - BVD \mathrm{DCF}=\frac{\mbox{FCF}~{}}{\mbox{WACC}~{}}-{\mbox{BVD}~{}}

Network_Effectiveness_Ratio.html

  1. N E R = 100 A n s w e r e d C a l l s + U s e r B u s y + R i n g N o A n s w e r + T e r m i n a l R e j e c t S e i z u r e s NER=100\ \frac{Answered\ Calls\ +\ User\ Busy\ +\ Ring\ No\ Answer\ +\ % Terminal\ Reject}{Seizures}

Neumann_polynomial.html

  1. α = 0 \alpha=0
  2. O 0 ( α ) ( t ) = 1 t , O_{0}^{(\alpha)}(t)=\frac{1}{t},
  3. O 1 ( α ) ( t ) = 2 α + 1 t 2 , O_{1}^{(\alpha)}(t)=2\frac{\alpha+1}{t^{2}},
  4. O 2 ( α ) ( t ) = 2 + α t + 4 ( 2 + α ) ( 1 + α ) t 3 , O_{2}^{(\alpha)}(t)=\frac{2+\alpha}{t}+4\frac{(2+\alpha)(1+\alpha)}{t^{3}},
  5. O 3 ( α ) ( t ) = 2 ( 1 + α ) ( 3 + α ) t 2 + 8 ( 1 + α ) ( 2 + α ) ( 3 + α ) t 4 , O_{3}^{(\alpha)}(t)=2\frac{(1+\alpha)(3+\alpha)}{t^{2}}+8\frac{(1+\alpha)(2+% \alpha)(3+\alpha)}{t^{4}},
  6. O 4 ( α ) ( t ) = ( 1 + α ) ( 4 + α ) 2 t + 4 ( 1 + α ) ( 2 + α ) ( 4 + α ) t 3 + 16 ( 1 + α ) ( 2 + α ) ( 3 + α ) ( 4 + α ) t 5 . O_{4}^{(\alpha)}(t)=\frac{(1+\alpha)(4+\alpha)}{2t}+4\frac{(1+\alpha)(2+\alpha% )(4+\alpha)}{t^{3}}+16\frac{(1+\alpha)(2+\alpha)(3+\alpha)(4+\alpha)}{t^{5}}.
  7. O n ( α ) ( t ) = α + n 2 α k = 0 n / 2 ( - 1 ) n - k ( n - k ) ! k ! ( - α n - k ) ( 2 t ) n + 1 - 2 k , O_{n}^{(\alpha)}(t)=\frac{\alpha+n}{2\alpha}\sum_{k=0}^{\lfloor n/2\rfloor}(-1% )^{n-k}\frac{(n-k)!}{k!}{-\alpha\choose n-k}\left(\frac{2}{t}\right)^{n+1-2k},
  8. ( z 2 ) α Γ ( α + 1 ) 1 t - z = n = 0 O n ( α ) ( t ) J α + n ( z ) , \frac{\left(\frac{z}{2}\right)^{\alpha}}{\Gamma(\alpha+1)}\frac{1}{t-z}=\sum_{% n=0}O_{n}^{(\alpha)}(t)J_{\alpha+n}(z),
  9. f ( z ) = n = 0 a n J α + n ( z ) f(z)=\sum_{n=0}a_{n}J_{\alpha+n}(z)\,
  10. | z | < c |z|<c
  11. a n = 1 2 π i | z | = c Γ ( α + 1 ) ( z 2 ) α f ( z ) O n ( α ) ( z ) d z , a_{n}=\frac{1}{2\pi i}\oint_{|z|=c^{\prime}}\frac{\Gamma(\alpha+1)}{\left(% \frac{z}{2}\right)^{\alpha}}f(z)O_{n}^{(\alpha)}(z)\mathrm{d}z,
  12. c < c c^{\prime}<c
  13. z - α f ( z ) z^{-\alpha}f(z)
  14. z = 0 z=0
  15. ( 1 2 z ) s = Γ ( s ) k = 0 ( - 1 ) k J s + 2 k ( z ) ( s + 2 k ) ( - s k ) \left(\tfrac{1}{2}z\right)^{s}=\Gamma(s)\cdot\sum_{k=0}(-1)^{k}J_{s+2k}(z)(s+2% k){-s\choose k}
  16. e i γ z = Γ ( s ) k = 0 i k C k ( s ) ( γ ) ( s + k ) J s + k ( z ) ( z 2 ) s . e^{i\gamma z}=\Gamma(s)\cdot\sum_{k=0}i^{k}C_{k}^{(s)}(\gamma)(s+k)\frac{J_{s+% k}(z)}{\left(\frac{z}{2}\right)^{s}}.
  17. C k ( s ) C_{k}^{(s)}
  18. ( z 2 ) 2 k ( 2 k - 1 ) ! J s ( z ) = i = k ( - 1 ) i - k ( i + k - 1 2 k - 1 ) ( i + k + s - 1 2 k - 1 ) ( s + 2 i ) J s + 2 i ( z ) , \frac{\left(\frac{z}{2}\right)^{2k}}{(2k-1)!}J_{s}(z)=\sum_{i=k}(-1)^{i-k}{i+k% -1\choose 2k-1}{i+k+s-1\choose 2k-1}(s+2i)J_{s+2i}(z),
  19. n = 0 t n J s + n ( z ) = e t z 2 t s j = 0 ( - z 2 t ) j j ! γ ( j + s , t z 2 ) Γ ( j + s ) = 0 e - z x 2 2 t z x t J s ( z 1 - x 2 ) 1 - x 2 s d x , \sum_{n=0}t^{n}J_{s+n}(z)=\frac{e^{\frac{tz}{2}}}{t^{s}}\sum_{j=0}\frac{\left(% -\frac{z}{2t}\right)^{j}}{j!}\frac{\gamma\left(j+s,\frac{tz}{2}\right)}{\,% \Gamma(j+s)}=\int_{0}^{\infty}e^{-\frac{zx^{2}}{2t}}\frac{zx}{t}\frac{J_{s}(z% \sqrt{1-x^{2}})}{\sqrt{1-x^{2}}^{s}}\,dx,
  20. M ( a , s , z ) = Γ ( s ) k = 0 ( - 1 t ) k L k ( - a - k ) ( t ) J s + k - 1 ( 2 t z ) ( t z ) s - k - 1 M(a,s,z)=\Gamma(s)\sum_{k=0}^{\infty}\left(-\frac{1}{t}\right)^{k}L_{k}^{(-a-k% )}(t)\frac{J_{s+k-1}\left(2\sqrt{tz}\right)}{(\sqrt{tz})^{s-k-1}}
  21. J s ( 2 z ) z s = 4 s Γ ( s + 1 2 ) π e 2 i z k = 0 L k ( - s - 1 / 2 - k ) ( i t 4 ) ( 4 i z ) k J 2 s + k ( 2 t z ) t z 2 s + k , \frac{J_{s}(2z)}{z^{s}}=\frac{4^{s}\Gamma\left(s+\frac{1}{2}\right)}{\sqrt{\pi% }}e^{2iz}\sum_{k=0}L_{k}^{(-s-1/2-k)}\left(\frac{it}{4}\right)(4iz)^{k}\frac{J% _{2s+k}\left(2\sqrt{tz}\right)}{\sqrt{tz}^{2s+k}},
  22. Γ ( ν - μ ) J ν ( z ) = Γ ( μ + 1 ) n = 0 Γ ( ν - μ + n ) n ! Γ ( ν + n + 1 ) ( z 2 ) ν - μ + n J μ + n ( z ) , \Gamma(\nu-\mu)J_{\nu}(z)=\Gamma(\mu+1)\sum_{n=0}\frac{\Gamma(\nu-\mu+n)}{n!% \Gamma(\nu+n+1)}\left(\frac{z}{2}\right)^{\nu-\mu+n}J_{\mu+n}(z),
  23. J s ( z 2 - 2 u z ) ( z 2 - 2 u z ) ± s = k = 0 ( ± u ) k k ! J s ± k ( z ) z ± s \frac{J_{s}\left(\sqrt{z^{2}-2uz}\right)}{\left(\sqrt{z^{2}-2uz}\right)^{\pm s% }}=\sum_{k=0}\frac{(\pm u)^{k}}{k!}\frac{J_{s\pm k}(z)}{z^{\pm s}}
  24. J s ( z ) d z = 2 k = 0 J s + 2 k + 1 ( z ) \int J_{s}(z)dz=2\sum_{k=0}J_{s+2k+1}(z)

Neuman–Stubblebine_protocol.html

  1. K A S K_{AS}
  2. K B S K_{BS}
  3. N A N_{A}
  4. N B N_{B}
  5. T A T_{A}
  6. T B T_{B}
  7. K A B K_{AB}
  8. A B : A , N A A\rightarrow B:A,N_{A}
  9. B S : B , N B , { A , N A , T B } K B S B\rightarrow S:B,N_{B},\{A,N_{A},T_{B}\}_{K_{BS}}
  10. S A : { B , N A , K A B , T B } K A S , { A , K A B , T B } K B S , N B S\rightarrow A:\{B,N_{A},K_{AB},T_{B}\}_{K_{AS}},\{A,K_{AB},T_{B}\}_{K_{BS}},N% _{B}
  11. A B : { A , K A B , T B } K B S , { N B } K A B A\rightarrow B:\{A,K_{AB},T_{B}\}_{K_{BS}},\{N_{B}\}_{K_{AB}}
  12. N A N_{A}
  13. T B T_{B}
  14. N B N_{B}
  15. A B : { A , K A B , T B } K B S , N A A\rightarrow B:\{A,K_{AB},T_{B}\}_{K_{BS}},N^{\prime}_{A}
  16. B A : N B , { N A } K A B B\rightarrow A:N^{\prime}_{B},\{N^{\prime}_{A}\}_{K_{AB}}
  17. A B : { N B } K A B A\rightarrow B:\{N^{\prime}_{B}\}_{K_{AB}}

Neuronal_encoding_of_sound.html

  1. C sin ( 2 π f t ) C\sin(2\pi ft)

Neutrino_decoupling.html

  1. e - + e + ν e + ν ¯ e e^{-}+e^{+}\longleftrightarrow\nu_{e}+\bar{\nu}_{e}
  2. n n
  3. T T
  4. n T 3 n\propto T^{3}
  5. σ v G F 2 T 2 \langle\sigma v\rangle\sim G_{F}^{2}T^{2}
  6. G F G_{F}
  7. c c
  8. Γ \Gamma
  9. Γ = n σ v G F 2 T 5 \Gamma=n\langle\sigma v\rangle\sim G_{F}^{2}T^{5}
  10. H H
  11. H = 8 π 3 G ρ H=\sqrt{\frac{8\pi}{3}G\rho}
  12. G G
  13. ρ \rho
  14. ρ T 4 \rho\propto T^{4}
  15. G F 2 T 5 G T 4 G_{F}^{2}T^{5}\sim\sqrt{GT^{4}}
  16. T ( G G F 2 ) 1 / 3 1 MeV T\sim\left(\frac{\sqrt{G}}{G_{F}^{2}}\right)^{1/3}\sim 1~{}\textrm{MeV}
  17. n p + e - + ν ¯ e n\leftrightarrow p+e^{-}+\bar{\nu}_{e}
  18. p + e - ν e + n p+e^{-}\leftrightarrow\nu_{e}+n
  19. [ n n + p ] = 0.21 \left[\frac{n}{n+p}\right]=0.21
  20. n n ( T ) n p ( T ) = exp ( - Δ m T ) \frac{n_{n}(T)}{n_{p}(T)}=\exp\left(\frac{-\Delta m}{T}\right)
  21. Δ m \Delta m
  22. T T

New_algebra.html

  1. A plano B subducere Z quadratum G residua erit A planum in G - Z quadrato in B B in G \frac{A\,\text{ plano}}{B}\,\text{ subducere }\frac{Z\,\text{ quadratum}}{G}\,% \text{ residua erit }\frac{A\,\text{ planum in }G-Z\,\text{ quadrato in }B}{B% \,\text{ in }G}
  2. A B - Z G = A G - Z B B G \frac{A}{B}-\frac{Z}{G}=\frac{AG-ZB}{BG}
  3. A 6 , A 5 B , A 4 B 2 , A 3 B 3 , A 2 B 4 , A B 5 , B 6 A^{6},A^{5}B,A^{4}B^{2},A^{3}B^{3},A^{2}B^{4},AB^{5},B^{6}\,
  4. ( A - B ) 6 = A 6 - 6 A 5 B + 15 A 4 B 2 - 20 A 3 B 3 + 15 A 2 B 4 - 6 A B 5 + B 6 . (A-B)^{6}=A^{6}-6A^{5}B+15A^{4}B^{2}-20A^{3}B^{3}+15A^{2}B^{4}-6AB^{5}+B^{6}.\,
  5. S in A planum + Rbis in A planum R aequabitur B plano \frac{S\,\text{ in }A\,\text{ planum }+\,\text{ Rbis in }A\,\text{ planum}}{R}% \,\text{ aequabitur }B\,\text{ plano}
  6. S A + 2 R A R = B . \frac{SA+2RA}{R}=B.

Newey–West_estimator.html

  1. Q * Q*
  2. σ ( i j ) \sigma_{(ij)}
  3. X X
  4. b b
  5. β \beta
  6. e i e_{i}
  7. E i E_{i}
  8. X X
  9. e e
  10. Q * Q*
  11. w = 1 - L + 1 w_{\ell}=1-\frac{\ell}{L+1}

Newman–Keuls_method.html

  1. X ¯ 1 \bar{X}_{1}
  2. X ¯ 2 \bar{X}_{2}
  3. X ¯ 3 \bar{X}_{3}
  4. X ¯ 4 \bar{X}_{4}
  5. q = X ¯ A - X ¯ B M S E n , q=\frac{\bar{X}_{A}-\bar{X}_{B}}{\sqrt{}}{\frac{MSE}{n}},
  6. q q
  7. X ¯ A \bar{X}_{A}
  8. X ¯ B \bar{X}_{B}
  9. M S E MSE
  10. n n
  11. n A n B {n_{A}}\neq{n_{B}}
  12. q = X ¯ A - X ¯ B M S E 2 ( 1 n A + 1 n B ) , q=\frac{\bar{X}_{A}-\bar{X}_{B}}{\sqrt{}}{\frac{MSE}{2}(\frac{1}{n_{A}}+\frac{% 1}{n_{B}})},
  13. n A n_{A}
  14. n B n_{B}
  15. q α p ν q_{\alpha}\,{}_{\nu}\,{}_{p}
  16. α \alpha
  17. ν \nu
  18. p p

Niche_apportionment_models.html

  1. P i = k ( 1 - k ) i - 1 Pi=k(1-k)^{i}-1
  2. R ( x i ) = μ i ± r σ i n R(x_{i})=\mu_{i}\pm\frac{r\sigma_{i}}{\sqrt{n}}

Nilradical_of_a_Lie_algebra.html

  1. 𝔫 𝔦 𝔩 ( 𝔤 ) \mathfrak{nil}(\mathfrak{g})
  2. 𝔤 \mathfrak{g}
  3. 𝔯 𝔞 𝔡 ( 𝔤 ) \mathfrak{rad}(\mathfrak{g})
  4. 𝔤 \mathfrak{g}
  5. 𝔤 red \mathfrak{g}^{\mathrm{red}}
  6. 0 𝔫 𝔦 𝔩 ( 𝔤 ) 𝔤 𝔤 red 0 0\to\mathfrak{nil}(\mathfrak{g})\to\mathfrak{g}\to\mathfrak{g}^{\mathrm{red}}\to 0
  7. 𝔫 𝔦 𝔩 ( 𝔤 ) \mathfrak{nil}(\mathfrak{g})
  8. 𝔤 \mathfrak{g}
  9. 0 𝔯 𝔞 𝔡 ( 𝔤 ) 𝔤 𝔤 ss 0 0\to\mathfrak{rad}(\mathfrak{g})\to\mathfrak{g}\to\mathfrak{g}^{\mathrm{ss}}\to 0
  10. 𝔤 ss \mathfrak{g}^{\mathrm{ss}}

Nodal_precession.html

  1. ω p = - 3 2 R E 2 ( a ( 1 - e 2 ) ) 2 J 2 ω cos i \omega_{p}=-\frac{3}{2}\frac{R_{E}^{2}}{(a(1-e^{2}))^{2}}J_{2}\omega\cos i
  2. ω p \omega_{p}\,
  3. R E R_{E}\,
  4. a a\,
  5. e e\,
  6. ω \omega\,
  7. 2 π 2\pi
  8. i i\,
  9. J 2 J_{2}\,
  10. - 5 C 20 = 1.08262668 × 10 - 3 \scriptstyle-\sqrt{5}C_{20}=1.08262668\times 10^{-3}
  11. J 2 = 2 ε E 3 - R E 3 ω E 2 3 G M E J_{2}=\frac{2\varepsilon_{E}}{3}-\frac{R_{E}^{3}\omega_{E}^{2}}{3GM_{E}}
  12. ε E \varepsilon_{E}\,
  13. R E R_{E}\,
  14. ω E \omega_{E}\,
  15. G M E GM_{E}\,
  16. R E = 6.378137 × 10 6 m R_{E}=6.378137\times 10^{6}m
  17. J 2 = 1.08262668 × 10 - 3 J_{2}=1.08262668\times 10^{-3}
  18. ω p = - 3 2 6378137 2 ( 7178137 ( 1 - 0 2 ) ) 2 1.08262668 × 10 - 3 0.001038 cos 56 \omega_{p}=-\frac{3}{2}\frac{6378137^{2}}{(7178137(1-0^{2}))^{2}}\ 1.08262668% \times 10^{-3}\ 0.001038\ \cos 56^{\circ}
  19. ω p = - 7.44 × 10 - 7 r a d / s \omega_{p}=-7.44\times 10^{-7}\ rad/s
  20. ω \omega
  21. ω \omega

Noether_inequality.html

  1. p g 1 2 c 1 ( X ) 2 + 2. p_{g}\leq\frac{1}{2}c_{1}(X)^{2}+2.
  2. b + 2 e + 3 σ + 5 b_{+}\leq 2e+3\sigma+5\,
  3. b - + 4 b 1 4 b + + 9. b_{-}+4b_{1}\leq 4b_{+}+9.\,
  4. 5 c 1 ( X ) 2 - c 2 ( X ) + 36 12 q 5c_{1}(X)^{2}-c_{2}(X)+36\geq 12q
  5. 5 c 1 ( X ) 2 - c 2 ( X ) + 36 0 ( c 1 2 ( X ) even ) 5c_{1}(X)^{2}-c_{2}(X)+36\geq 0\quad(c_{1}^{2}(X)\,\text{ even})
  6. 5 c 1 ( X ) 2 - c 2 ( X ) + 30 0 ( c 1 2 ( X ) odd ) . 5c_{1}(X)^{2}-c_{2}(X)+30\geq 0\quad(c_{1}^{2}(X)\,\text{ odd}).
  7. 0 H 0 ( 𝒪 X ) H 0 ( K ) H 0 ( K | D ) H 1 ( 𝒪 X ) 0\to H^{0}(\mathcal{O}_{X})\to H^{0}(K)\to H^{0}(K|_{D})\to H^{1}(\mathcal{O}_% {X})\to
  8. p g - 1 h 0 ( K | D ) . p_{g}-1\leq h^{0}(K|_{D}).
  9. 𝒪 D ( 2 K ) \mathcal{O}_{D}(2K)
  10. K | D K|_{D}
  11. h 0 ( K | D ) - 1 1 2 deg D ( K ) = 1 2 K 2 . h^{0}(K|_{D})-1\leq\frac{1}{2}\mathrm{deg}_{D}(K)=\frac{1}{2}K^{2}.\,

Nominal_impedance.html

  1. Z nom = L C Z_{\mathrm{nom}}=\sqrt{\frac{L}{C}}
  2. α R 2 Z 0 \alpha\approx\frac{R}{2Z_{0}}
  3. 50 30 × 77 Ω 50\approx\sqrt{30\times 77}\mathrm{\ }\Omega

Non-autonomous_mechanics.html

  1. Q Q\to\mathbb{R}
  2. \mathbb{R}
  3. ( t , q i ) (t,q^{i})
  4. Q = × M Q=\mathbb{R}\times M
  5. Γ \Gamma
  6. Q Q\to\mathbb{R}
  7. Γ i = 0 \Gamma^{i}=0
  8. ( q t i - Γ i ) i (q^{i}_{t}-\Gamma^{i})\partial_{i}
  9. Γ \Gamma
  10. X = X=\mathbb{R}
  11. J 1 Q J^{1}Q
  12. Q Q\to\mathbb{R}
  13. ( t , q i , q t i ) (t,q^{i},q^{i}_{t})
  14. V Q VQ
  15. Q Q\to\mathbb{R}
  16. ( t , q i , p i ) (t,q^{i},p_{i})
  17. p i d q i - H ( t , q i , p i ) d t p_{i}dq^{i}-H(t,q^{i},p_{i})dt
  18. T Q TQ
  19. Q Q
  20. ( t , q i , p , p i ) (t,q^{i},p,p_{i})
  21. p - H p-H

Non-minimally_coupled_inflation.html

  1. ξ \xi
  2. S = d 4 x - g [ m P 2 2 R - 1 2 μ ϕ μ ϕ - V ( ϕ ) - ξ 2 R ϕ 2 ] S=\int d^{4}x\sqrt{-g}\left[{m_{P}^{2}\over 2}R-{1\over 2}\partial^{\mu}\phi% \partial_{\mu}\phi-V(\phi)-{\xi\over 2}R\phi^{2}\right]
  3. ξ \xi
  4. R R
  5. ϕ \phi

Non-squeezing_theorem.html

  1. 2 n = { z = ( x 1 , , x n , y 1 , , y n ) } , \mathbb{R}^{2n}=\{z=(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})\},
  2. B ( R ) = { z 2 n | z < R } , B(R)=\{z\in\mathbb{R}^{2n}|\|z\|<R\},
  3. Z ( r ) = { z 2 n | x 1 2 + y 1 2 < r 2 } , Z(r)=\{z\in\mathbb{R}^{2n}|x_{1}^{2}+y_{1}^{2}<r^{2}\},
  4. ω = d x 1 d y 1 + + d x n d y n . \omega=dx_{1}\wedge dy_{1}+\cdots+dx_{n}\wedge dy_{n}.\,
  5. p j p_{j}
  6. v a r ( Q ) v a r ( P ) c o v 2 ( Q , P ) + ( 2 ) 2 var(Q)var(P)\geq cov^{2}(Q,P)+\left(\frac{\hbar}{2}\right)^{2}

Non-topological_soliton.html

  1. = | μ Φ | 2 - U ( | Φ | ) \mathcal{L}=|\partial_{\mu}\Phi|^{2}-U(|\Phi|)\,
  2. Φ = ( ϕ 0 / 2 ) e i ω t \Phi=(\phi_{0}/\sqrt{2})e^{i\omega t}
  3. ϕ 0 \phi_{0}
  4. U ( | Φ | ) U(|\Phi|)
  5. ϕ 0 \phi_{0}
  6. E = ( ϕ 0 2 ω 2 2 + U ( ϕ 0 2 ) ) 4 3 π R 3 . ( 1 ) E=\Big(\frac{\phi_{0}^{2}\omega^{2}}{2}+U(\frac{\phi_{0}}{\sqrt{2}})\Big)\frac% {4}{3}\pi R^{3}.\,\,\,\,\,\,\,\,\,\,\,\,\,(1)
  7. j μ = - i ( Φ * μ Φ - μ Φ * Φ ) , j_{\mu}=-i(\Phi^{*}\partial_{\mu}\Phi-\partial_{\mu}\Phi^{*}\Phi),\,
  8. Q = j 0 d 3 x = ω φ 0 2 4 3 π R 3 . Q=\int j_{0}d^{3}x=\omega\varphi_{0}^{2}\frac{4}{3}\pi R^{3}.
  9. E = Q 2 U ( φ 0 / 2 ) ϕ 0 2 . ( 2 ) E=Q\sqrt{\frac{2U(\varphi_{0}/\sqrt{2})}{\phi_{0}^{2}}}.\,\,\,\,\,\,\,\,\,\,\,% \,(2)
  10. 2 U ( φ 0 / 2 ) φ 0 2 U ( | Φ | ) | Φ | 2 | min < m 2 . \frac{2U(\varphi_{0}/\sqrt{2})}{\varphi_{0}^{2}}\equiv\frac{U(|\Phi|)}{|\Phi|^% {2}}\Bigg|_{\min}<m^{2}.
  11. E surface E\text{surface}
  12. E surface U ( φ 0 / 2 ) R 2 Δ R U ( φ 0 / 2 ) R 3 E\text{surface}\sim U(\varphi_{0}/\sqrt{2})R^{2}\Delta R\ll U(\varphi_{0}/% \sqrt{2})R^{3}
  13. Q 1 Q\gg 1
  14. = | μ Φ | 2 + 1 2 ( μ σ ) 2 - g | Φ | 2 σ 2 - λ 4 ( σ 2 - σ 0 2 ) 2 \mathcal{L}=|\partial_{\mu}\Phi|^{2}+\frac{1}{2}(\partial_{\mu}\sigma)^{2}-g|% \Phi|^{2}\sigma^{2}-\frac{\lambda}{4}(\sigma^{2}-\sigma_{0}^{2})^{2}
  15. Φ . \Phi.
  16. Φ ( r , t ) = ( ϕ ( r ) / 2 ) e i ω t \Phi(\vec{r},t)=(\phi(r)/\sqrt{2})e^{i\omega t}
  17. Q = 4 π ω 0 ϕ 2 ( r ) r 2 d r Q=4\pi\omega\int\limits_{0}^{\infty}\phi^{2}(r)r^{2}dr
  18. ϕ ( r ) = φ 0 ( sin ω r / r ) \phi(r)=\varphi_{0}(\sin\omega r/r)
  19. σ ( r ) = 0 \sigma(r)=0
  20. E π Q R + π 3 λ σ 0 4 R 3 E\approx\frac{\pi Q}{R}+\frac{\pi}{3}\lambda\sigma_{0}^{4}R^{3}
  21. E u p = 4 3 π λ 1 / 4 Q 3 / 4 σ 0 E_{up}=\frac{4}{3}\pi\lambda^{1/4}Q^{3/4}\sigma_{0}
  22. ω 2 ϕ + 2 ϕ - g σ 2 ϕ = 0 \omega^{2}\phi+\vec{\partial^{2}}\phi-g\sigma^{2}\phi=0
  23. 2 σ - g ϕ 2 σ - λ σ ( σ 2 - σ 0 2 ) = 0 \vec{\partial^{2}}\sigma-g\phi^{2}\sigma-\lambda\sigma(\sigma^{2}-\sigma_{0}^{% 2})=0
  24. Q g σ 0 Qg\sigma_{0}
  25. Q min = ( 4 π 3 g ) 4 λ . Q_{\min}={\Big(\frac{4\pi}{3g}\Big)}^{4}\lambda.
  26. = k = 1 N ( i 2 Ψ ¯ k γ μ μ Ψ k - ( m + g σ ) Ψ ¯ k Ψ k ) - U ( σ ) + 1 2 ( μ σ ) 2 . ( 3 ) \mathcal{L}=\sum_{k=1}^{N}\left(\frac{i}{2}\overleftrightarrow{\overline{\Psi}% _{k}\gamma^{\mu}\partial_{\mu}\Psi_{k}}-(m+g\sigma)\overline{\Psi}_{k}\Psi_{k}% \right)-U(\sigma)+\frac{1}{2}(\partial_{\mu}\sigma)^{2}.\,\,\,\,\,(3)
  27. E up = 4 3 π 2 U 1 / 4 ( - m g ) N 3 / 4 , E\text{up}=\frac{4}{3}\pi\sqrt{2}U^{1/4}\left(-\frac{m}{g}\right)N^{3/4},
  28. E ( Q ) < Q m E(Q)<Qm
  29. E ( Q ) E(Q)
  30. E ( Q ) E(Q)
  31. d 2 E ( Q ) d Q 2 < 0. ( 4 ) \frac{d^{2}E(Q)}{dQ^{2}}<0.\,\,\,\,\,\,\,\,\,\,\,\,\,(4)
  32. E ( Q 1 ) + E ( Q 2 ) > E ( Q 1 + Q 2 ) E(Q_{1})+E(Q_{2})>E(Q_{1}+Q_{2})
  33. E surface R 2 ( Q ) Q 2 / 3 E\text{surface}\propto R^{2}(Q)\propto Q^{2/3}
  34. d 2 ( E surface + Q 2 U ( ϕ 0 / 2 ) / ϕ 0 2 ) d Q 2 < 0. \frac{d^{2}(E\text{surface}+Q\sqrt{2U(\phi_{0}/\sqrt{2})/\phi_{0}^{2}})}{dQ^{2% }}<0.
  35. E surface E\text{surface}
  36. Q min Q_{\min}
  37. E surface E\text{surface}
  38. Q ( m - U ( | Φ | ) / | Φ | 2 | min ) Q(m-\sqrt{U(|\Phi|)/|\Phi|^{2}}\Big|_{\min})
  39. m - E ( Q ) / Q m-E(Q)/Q
  40. S eff = - i ln det ( i γ μ μ - g σ ( x ) i γ μ μ - g σ 0 ) . S_{\,\text{eff}}=-i\ln\det\Big(\frac{i\gamma^{\mu}{\partial}_{\mu}-g\sigma(x)}% {i\gamma^{\mu}{\partial}_{\mu}-g\sigma_{0}}\Big).
  41. Φ = 1 / 2 ( ϕ c l ( r - R ( t ) ) + n q n ( t ) β ( r ) ) e i ζ ( t ) , σ = σ c l ( r - R ( t ) ) + n q n ( t ) α ( r ) . \Phi=1/\sqrt{2}(\phi_{cl}(\vec{r}-\vec{R(t)})+\sum_{n}q_{n}(t)\beta(\vec{r}))e% ^{i\zeta(t)},\sigma=\sigma_{cl}(\vec{r}-\vec{R(t)})+\sum_{n}q_{n}(t)\alpha(% \vec{r}).
  42. ϕ c l \phi_{cl}\,
  43. σ c l \sigma_{cl}
  44. R ( t ) \vec{R(t)}
  45. ζ ( t ) \zeta(t)
  46. q n , n = 5 , 6 , , q_{n},n=5,6,\dots,\infty
  47. A μ = k a k μ ( t ) e i k r + c . c . A^{\mu}=\sum_{k}a_{k}^{\mu}(t)e^{i\vec{k}\vec{r}}+c.c.
  48. Q min Q_{\min}
  49. Q max : E ( Q > Q max ) > Q m Q_{\max}:E(Q>Q_{\max})>Qm
  50. E ( Q ) ( C 1 Q 4 / 3 + C 2 Q 2 ) 2 / 3 E(Q)\propto(C_{1}Q^{4/3}+C_{2}Q^{2})^{2/3}
  51. E ( Q ) / M P l 2 R ( Q ) < 1 E(Q)/M_{Pl}^{2}R(Q)<1
  52. Ψ \Psi
  53. Ψ + Ψ Φ = Ψ + ( i 0 + i σ ) Ψ - i g Φ Ψ + σ 2 Ψ * + i g Φ * Ψ T σ 2 Ψ \mathcal{L}_{\Psi}+\mathcal{L}_{\Psi\Phi}=\Psi^{+}(i\partial_{0}+i\vec{% \partial}\vec{\sigma})\Psi-ig\Phi\Psi^{+}\sigma_{2}\Psi^{*}+ig\Phi^{*}\Psi^{T}% \sigma_{2}\Psi
  54. Ψ \Psi
  55. d N / d t d S < ( U ( | Φ | ) / | Φ | 2 | min ) 3 / 2 / 192 π 2 dN/dtdS<\Big(U(|\Phi|)/|\Phi|^{2}\Big|_{\min}\Big)^{3/2}/192\pi^{2}
  56. S U ( 2 ) L S U ( 2 ) R SU(2)_{L}SU(2)_{R}
  57. m ( Ψ ¯ R Ψ L + Ψ ¯ L Ψ R ) m(\overline{\Psi}_{R}\Psi_{L}+\overline{\Psi}_{L}\Psi_{R})
  58. M P l 2 R ( Q ) M^{2}_{Pl}R(Q)
  59. d 2 E ( Q ) / d Q 2 < 0 d^{2}E(Q)/dQ^{2}<0
  60. = i 2 Ψ ¯ γ μ μ Ψ - m ( σ ) Ψ ¯ Ψ - U ( σ ) + 1 2 ( μ σ ) 2 . \mathcal{L}=\frac{i}{2}\overleftrightarrow{\overline{\Psi}\gamma^{\mu}\partial% _{\mu}\Psi}-m(\sigma)\overline{\Psi}\Psi-U(\sigma)+\frac{1}{2}(\partial_{\mu}% \sigma)^{2}.
  61. σ \sigma
  62. k F ( r ) k_{F}(r)
  63. Ψ ( r ) \Psi(r)
  64. Ψ ( r ) \Psi(r)
  65. σ ( r ) \sigma(r)
  66. 2 σ - ( d m ( σ ) / d σ ) Ψ ¯ Ψ - d U ( σ ) / d σ = 0 {\vec{\partial}}^{2}\sigma-(dm(\sigma)/d\sigma)\langle\overline{\Psi}\Psi% \rangle-dU(\sigma)/d\sigma=0
  67. Ψ ¯ Ψ \langle\overline{\Psi}\Psi\rangle
  68. Ψ ¯ Ψ = m ( k 2 + m 2 ) 1 / 2 2 d 3 k ( 2 π ) 3 , 0 < k < k F . \langle\overline{\Psi}\Psi\rangle=\int\frac{m}{(k^{2}+m^{2})^{1/2}}\frac{2d^{3% }k}{(2\pi)^{3}},\quad 0<k<k_{F}.
  69. ε F = ( k F 2 + m 2 ) 1 / 2 \varepsilon_{F}=(k^{2}_{F}+m^{2})^{1/2}
  70. σ ( r ) \sigma(r)
  71. P f - U = 0 P_{f}-U=0
  72. k F k_{F}
  73. σ \sigma
  74. P f = k 2 3 ( k 2 + m 2 ) 1 / 2 2 d 3 k ( 2 π ) 3 , 0 < k < k F . P_{f}=\int\frac{k^{2}}{3(k^{2}+m^{2})^{1/2}}\frac{2d^{3}k}{(2\pi)^{3}},\quad 0% <k<k_{F}.
  75. m ( σ ) = g σ m(\sigma)=g\sigma
  76. U ( σ ) = λ / 4 ( σ 2 - σ 0 2 ) 2 U(\sigma)=\lambda/4(\sigma^{2}-\sigma^{2}_{0})^{2}
  77. σ = 0 \sigma=0
  78. k F = ε F = ( 3 π 2 λ ) 1 / 4 σ 0 k_{F}=\varepsilon_{F}=(3\pi^{2}\lambda)^{1/4}\sigma_{0}
  79. E f = 3 P f E_{f}=3P_{f}
  80. V V
  81. Q = V ( k F 3 / 3 π 2 ) Q=V(k^{3}_{F}/3\pi^{2})
  82. E ( Q ) = ( U + E f ) V = ε F Q E(Q)=(U+E_{f})V=\varepsilon_{F}Q
  83. U ( σ ) U(\sigma)
  84. σ \sigma
  85. ν \nu
  86. U ( σ ) = μ 2 σ 2 ( 1 - σ / σ 0 ) 2 / 2 U(\sigma)=\mu^{2}\sigma^{2}(1-\sigma/\sigma_{0})^{2}/2
  87. g σ Ψ ¯ Ψ g\sigma\overline{\Psi}\Psi
  88. σ 0 \sigma\simeq 0
  89. U ( 0 ) = λ σ 0 4 / 4 U(0)=\lambda\sigma^{4}_{0}/4
  90. m H = σ 0 2 λ m_{H}=\sigma_{0}\sqrt{2\lambda}
  91. g g
  92. g > 2.3 g>2.3
  93. m H m_{H}
  94. m H m_{H}
  95. λ = 1 \lambda=1
  96. σ 0 = \sigma_{0}=
  97. g g
  98. E = ( 4 π ) 1 / 3 ( 3 / 4 ) 5 / 3 Q 4 / 3 / R + β e 2 Q 2 / R + ( 4 π R 3 / 3 ) λ σ 0 4 / 4. E=(4\pi)^{1/3}(3/4)^{5/3}Q^{4/3}/R+\beta e^{2}Q^{2}/R+(4\pi R^{3}/3)\lambda% \sigma^{4}_{0}/4.
  99. R R
  100. Q Q
  101. β \beta
  102. R R
  103. E ( Q ) = 4 σ 0 ( π λ ) 1 / 4 / 3 ( ( 4 π ) 1 / 3 ( 3 / 4 ) 5 / 3 Q 4 / 3 + β e 2 Q 2 ) 3 / 4 . E(Q)=4\sigma_{0}(\pi\lambda)^{1/4}/3\left((4\pi)^{1/3}(3/4)^{5/3}Q^{4/3}+\beta e% ^{2}Q^{2}\right)^{3/4}.
  104. E ( Q ) E(Q)
  105. Q Q
  106. Q Q
  107. E ( Q ) E(Q)
  108. Q Q
  109. σ \sigma
  110. σ \sigma
  111. η = ( | n Φ - n Φ ¯ | ) / ( n Φ + n Φ ¯ ) \eta=(|n_{\Phi}-n_{\overline{\Phi}}|)/(n_{\Phi}+n_{\overline{\Phi}})
  112. T c T_{c}
  113. ξ T c - 1 \xi\propto T^{-1}_{c}
  114. T G T_{G}
  115. Λ > 0.8 U M \Lambda>0.8U_{M}
  116. Q Q
  117. Q Q
  118. n ( r ) = b r - 3 / 2 exp ( - c r ) / V ξ n(r)=br^{-3/2}\exp{(-cr)}/V_{\xi}
  119. r ( L / 2 ξ ) 3 r\simeq(L/2\xi)^{3}
  120. V ξ V_{\xi}
  121. L L
  122. Q ( r ) r V ξ η n Q Q(r)\simeq rV_{\xi}\eta n_{Q}
  123. n Q n_{Q}
  124. Q min Q_{\min}
  125. Q min Q_{\min}
  126. = | μ Φ | 2 + ( μ σ ) 2 / 2 - λ 1 ( σ 2 - σ 0 2 ) 2 / 8 - λ 2 ( σ - σ 0 ) 3 σ 0 / 3 - h | Φ | 2 ( σ - σ 0 ) 2 - g | Φ | 4 - Λ \mathcal{L}=|\partial_{\mu}\Phi|^{2}+(\partial_{\mu}\sigma)^{2}/2-\lambda_{1}(% \sigma^{2}-\sigma_{0}^{2})^{2}/8-\lambda_{2}(\sigma-\sigma_{0})^{3}\sigma_{0}/% 3-h|\Phi|^{2}(\sigma-\sigma_{0})^{2}-g|\Phi|^{4}-\Lambda
  127. λ 1 / λ 2 = 0.15 , \lambda_{1}/\lambda_{2}=0.15,\,
  128. Λ = 0.6 λ 1 σ 0 4 \Lambda=0.6\lambda_{1}\sigma_{0}^{4}
  129. ξ = 1 / λ 1 T G , \xi=1/\lambda_{1}T_{G},
  130. Q min = 18 λ 1 / h 2 Q_{\min}=18\lambda_{1}/h^{2}
  131. n ( Q min ) ( η / λ 1 Q min ) 3 / 2 exp ( - c Q min λ 1 3 / η ) . n(Q_{\min})\propto(\eta/\lambda_{1}Q_{\min})^{3/2}\exp{(-cQ_{\min}\lambda_{1}^% {3}/\eta)}.
  132. Φ \langle\Phi\rangle
  133. Φ = f ( t ) exp ( i θ ( t ) ) / 2 \Phi=f(t)\exp{(i\theta(t))}/\sqrt{2}
  134. | Φ | 2 = f 2 ( t ) / 2 |\Phi|^{2}=f^{2}(t)/2
  135. j 0 = f 2 0 θ ( t ) j_{0}=f^{2}\partial_{0}\theta(t)
  136. θ ( t ) \theta(t)
  137. j 0 j_{0}
  138. a ( t ) a(t)
  139. d s 2 = d t 2 - a 2 ( t ) ( d x 2 + d y 2 + d z 2 ) ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2})
  140. Q = j 0 a 3 d 3 x Q=\int j_{0}a^{3}d^{3}x
  141. Φ \Phi
  142. m ( T ) = m 2 - λ 2 / 2 m(T)=m^{2}-\lambda^{2}/2
  143. U ( | Φ | ) = m 2 | Φ | 2 - λ | Φ | 4 / 2 + γ | Φ | 6 U(|\Phi|)=m^{2}|\Phi|^{2}-\lambda|\Phi|^{4}/2+\gamma|\Phi|^{6}
  144. h χ 2 | Φ | 2 h\chi^{2}|\Phi|^{2}
  145. χ \chi
  146. U ( | Φ | ) / | Φ | 2 | min < m 2 U(|\Phi|)/|\Phi|^{2}\Big|_{\min}<m^{2}
  147. χ \chi
  148. T G T_{G}

Non-uniform_discrete_Fourier_transform.html

  1. X ( z k ) = X ( z ) | z = z k = n = 0 N - 1 x [ n ] z k - n , k = 0 , 1 , , N - 1 , X(z_{k})=X(z)|_{z=z_{k}}=\sum_{n=0}^{N-1}x[n]z_{k}^{-n},\quad k=0,1,...,N-1,
  2. X ( z ) X(z)
  3. x [ n ] x[n]
  4. { z i } i = 0 , 1 , , N - 1 \{z_{i}\}_{i=0,1,...,N-1}
  5. 𝐗 = 𝐃𝐱 \mathbf{X}=\mathbf{D}\mathbf{x}
  6. 𝐗 = [ X ( z 0 ) X ( z 1 ) X ( z N - 1 ) ] , 𝐱 = [ x [ 0 ] x [ 1 ] x [ N - 1 ] ] , and 𝐃 = [ 1 z 0 - 1 z 0 - 2 z 0 - ( N - 1 ) 1 z 1 - 1 z 1 - 2 z 1 - ( N - 1 ) 1 z N - 1 - 1 z N - 1 - 2 z N - 1 - ( N - 1 ) ] . \mathbf{X}=\begin{bmatrix}X(z_{0})\\ X(z_{1})\\ \vdots\\ X(z_{N-1})\end{bmatrix},\quad\mathbf{x}=\begin{bmatrix}x[0]\\ x[1]\\ \vdots\\ x[N-1]\end{bmatrix},\,\text{ and}\quad\mathbf{D}=\begin{bmatrix}1&z_{0}^{-1}&z% _{0}^{-2}&\cdots&z_{0}^{-(N-1)}\\ 1&z_{1}^{-1}&z_{1}^{-2}&\cdots&z_{1}^{-(N-1)}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&z_{N-1}^{-1}&z_{N-1}^{-2}&\cdots&z_{N-1}^{-(N-1)}\end{bmatrix}.
  7. 𝐃 \mathbf{D}
  8. z k {z_{k}}
  9. d e t ( 𝐃 ) det(\mathbf{D})
  10. 𝐃 \mathbf{D}
  11. z k {z_{k}}
  12. 𝐃 \mathbf{D}
  13. 𝐱 = 𝐃 - 𝟏 𝐗 \mathbf{x}=\mathbf{D^{-1}}\mathbf{X}
  14. 𝐗 and 𝐃 \mathbf{X}\,\text{ and }\mathbf{D}
  15. 𝐱 \mathbf{x}
  16. O ( N 3 ) O(N^{3})
  17. X ( z ) X(z)
  18. X ^ [ k ] = X ( z k ) , k = 0 , 1 , , N - 1 , \hat{X}[k]=X(z_{k}),\quad k=0,1,...,N-1,
  19. X ( z ) X(z)
  20. X ( z ) = k = 0 N - 1 L k ( z ) L k ( z k ) X ^ [ k ] , X(z)=\sum_{k=0}^{N-1}\frac{L_{k}(z)}{L_{k}(z_{k})}\hat{X}[k],
  21. { L i ( z ) } i = 0 , 1 , , N - 1 \{L_{i}(z)\}_{i=0,1,...,N-1}
  22. L k ( z ) = i k ( 1 - z i z - 1 ) , k = 0 , 1 , , N - 1 L_{k}(z)=\prod_{i\neq k}(1-z_{i}z^{-1}),\quad k=0,1,...,N-1
  23. X ( z ) X(z)
  24. X ( z ) = c 0 + c 1 ( 1 - z 0 z - 1 ) + c 2 ( 1 - z 0 z - 1 ) ( 1 - z 1 z - 1 ) + + C N - 1 k = 0 N - 2 ( 1 - z k z - 1 ) , X(z)=c_{0}+c_{1}(1-z_{0}z^{-1})+c_{2}(1-z_{0}z^{-1})(1-z_{1}z^{-1})+...+C_{N-1% }\prod_{k=0}^{N-2}(1-z_{k}z^{-1}),
  25. c j c_{j}
  26. X ^ [ 0 ] , X ^ [ 1 ] , , X ^ [ j ] \hat{X}[0],\hat{X}[1],...,\hat{X}[j]
  27. z 0 , z 1 , , z j z_{0},z_{1},...,z_{j}
  28. c 0 = X ^ [ 0 ] , c_{0}=\hat{X}[0],
  29. c 1 = X ^ [ 1 ] - c 0 1 - z 0 z 1 - 1 , c_{1}=\frac{\hat{X}[1]-c_{0}}{1-z_{0}z_{1}^{-1}},
  30. c 2 = X ^ [ 2 ] - c 0 - c 1 ( 1 - z 0 z - 1 ) ( 1 - z 0 z 2 - 1 ) ( 1 - z 1 z 2 - 1 ) , c_{2}=\frac{\hat{X}[2]-c_{0}-c_{1}(1-z_{0}z^{-1})}{(1-z_{0}z_{2}^{-1})(1-z_{1}% z_{2}^{-1})},
  31. \vdots
  32. { c j } \{c_{j}\}
  33. 𝐋𝐜 = 𝐗 \mathbf{L}\mathbf{c}=\mathbf{X}
  34. 𝐗 = [ X ^ [ 0 ] X ^ [ 1 ] X ^ [ N - 1 ] ] , 𝐜 = [ c 0 c 1 c N - 1 ] , and 𝐋 = [ 1 0 0 0 0 1 ( 1 - z 0 z 1 - 1 ) 0 0 0 1 ( 1 - z 0 z 2 - 1 ) ( 1 - z 0 z 2 - 1 ) ( 1 - z 1 z 2 - 1 ) 0 0 1 ( 1 - z 0 z N - 1 - 1 ) ( 1 - z 0 z N - 1 - 1 ) ( 1 - z 1 z N - 1 - 1 ) k = 0 N - 2 ( 1 - z k z N - 1 - 1 ) ] . \mathbf{X}=\begin{bmatrix}\hat{X}[0]\\ \hat{X}[1]\\ \vdots\\ \hat{X}[N-1]\end{bmatrix},\quad\mathbf{c}=\begin{bmatrix}c_{0}\\ c_{1}\\ \vdots\\ c_{N-1}\end{bmatrix},\,\text{ and}\quad\mathbf{L}=\begin{bmatrix}1&0&0&0&% \cdots&0\\ 1&(1-z_{0}z_{1}^{-1})&0&0&\cdots&0\\ 1&(1-z_{0}z_{2}^{-1})&(1-z_{0}z_{2}^{-1})(1-z_{1}z_{2}^{-1})&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&(1-z_{0}z_{N-1}^{-1})&(1-z_{0}z_{N-1}^{-1})(1-z_{1}z_{N-1}^{-1})&\cdots&% \prod_{k=0}^{N-2}(1-z_{k}z_{N-1}^{-1})\end{bmatrix}.
  35. { c j } \{c_{j}\}
  36. O ( N 3 ) O(N^{3})
  37. L k + 1 ( z ) = ( 1 - z k + 1 z - 1 ) ( 1 - z k z - 1 ) L k ( z ) , k = 0 , 1 , , N - 1 L_{k+1}(z)=\frac{(1-z_{k+1}z^{-1})}{(1-z_{k}z^{-1})}L_{k}(z),\quad k=0,1,...,N-1
  38. x [ n 1 , n 2 ] x[n_{1},n_{2}]
  39. N 1 × N 2 N_{1}\times N_{2}
  40. X ^ ( z 1 k , z 2 k ) = n 1 = 0 N 1 - 1 n 2 = 0 N 2 - 1 x [ n 1 , n 2 ] z 1 k - n 1 z 2 k - n 2 , k = 0 , 1 , , N 1 N 2 - 1 , \hat{X}(z_{1k},z_{2k})=\sum_{n_{1}=0}^{N_{1}-1}\sum_{n_{2}=0}^{N_{2}-1}x[n_{1}% ,n_{2}]z_{1k}^{-n_{1}}z_{2k}^{-n_{2}},\quad k=0,1,...,N_{1}N_{2}-1,
  41. X ^ ( z 1 k , z 2 k ) \hat{X}(z_{1k},z_{2k})
  42. x [ n 1 , n 2 ] x[n_{1},n_{2}]
  43. ( z 1 k , z 2 k ) (z_{1k},z_{2k})
  44. N 1 N 2 N_{1}N_{2}
  45. ( z 1 , z 2 ) (z_{1},z_{2})
  46. 𝐗 ^ = 𝐃𝐗 , \mathbf{\hat{X}}=\mathbf{D}\mathbf{X},
  47. 𝐃 \mathbf{D}
  48. 𝐃 \mathbf{D}
  49. d e t ( 𝐃 ) det(\mathbf{D})
  50. O ( N 1 3 N 2 3 ) O(N_{1}^{3}N_{2}^{3})

Nonlinear_complementarity_problem.html

  1. x 0 , f ( x ) 0 and x T f ( x ) = 0 x\geq 0,\ f(x)\geq 0\,\text{ and }x^{T}f(x)=0\,

Nonlinear_resonance.html

  1. ω n = ω 1 + ω 2 + + ω n - 1 , \omega_{n}=\omega_{1}+\omega_{2}+\cdots+\omega_{n-1},
  2. ω i = ω ( 𝐤 i ) , \omega_{i}=\omega(\mathbf{k}_{i}),
  3. 𝐤 i \mathbf{k}_{i}
  4. i i
  5. ω = ω ( 𝐤 ) , \omega=\omega(\mathbf{k}),
  6. ω \omega
  7. ω 0 \omega_{0}
  8. ω = ω 0 + κ A 2 , \omega=\omega_{0}+\kappa A^{2},
  9. A A
  10. κ \kappa
  11. F F
  12. F crit F_{\mathrm{crit}}
  13. F crit = 4 m 2 ω 0 2 γ 3 3 3 κ , F_{\mathrm{crit}}=\frac{4m^{2}\omega_{0}^{2}\gamma^{3}}{3\sqrt{3}\kappa},
  14. m m
  15. γ \gamma
  16. ω 0 \omega_{0}
  17. ω 0 . \omega_{0}.

Norm_residue_isomorphism_theorem.html

  1. \ell
  2. n n
  3. = 2 \ell=2
  4. n n
  5. : k * H 1 ( k , μ ) \partial:k^{*}\rightarrow H^{1}(k,\mu_{\ell})
  6. μ \mu_{\ell}
  7. k × / ( k × ) H 1 ( k , μ ) k^{\times}/(k^{\times})^{\ell}\cong H^{1}(k,\mu_{\ell})
  8. k × k^{\times}
  9. \partial
  10. n : k × k × H e ´ t n ( k , μ n ) . \partial^{n}:k^{\times}\otimes\cdots\otimes k^{\times}\rightarrow H^{n}_{\rm% \acute{e}t}(k,\mu_{\ell}^{\otimes n}).
  11. k { 0 , 1 } k\setminus\{0,1\}
  12. n ( , a , , 1 - a , ) \partial^{n}(\ldots,a,\ldots,1-a,\ldots)
  13. K * M ( k ) = T ( k × ) / ( { a ( 1 - a ) : a k { 0 , 1 } } ) K^{M}_{*}(k)=T(k^{\times})/(\{a\otimes(1-a)\colon a\in k\setminus\{0,1\}\})
  14. T ( k × ) T(k^{\times})
  15. a ( 1 - a ) a\otimes(1-a)
  16. n \partial^{n}
  17. n : K n M ( k ) H e ´ t n ( k , μ n ) . \partial^{n}\colon K^{M}_{n}(k)\to H^{n}_{\rm\acute{e}t}(k,\mu_{\ell}^{\otimes n% }).
  18. K n M ( k ) / K^{M}_{n}(k)/\ell
  19. n : K n M ( k ) / H e ´ t n ( k , μ n ) \partial^{n}:K_{n}^{M}(k)/\ell\to H^{n}_{\rm\acute{e}t}(k,\mu_{\ell}^{\otimes n})
  20. ( a 1 , a 2 ) (a_{1},a_{2})
  21. 1 / 1/\ell
  22. H p , q ( X , 𝐙 / ) H^{p,q}(X,\mathbf{Z}/\ell)
  23. H e ´ t p ( k , μ q ) H^{p}_{\rm\acute{e}t}(k,\mu^{\otimes q}_{\ell})

Normal-exponential-gamma_distribution.html

  1. μ \mu
  2. θ \theta
  3. k k
  4. f ( x ; μ , k , θ ) exp ( ( x - μ ) 2 4 θ 2 ) D - 2 k - 1 ( | x - μ | θ ) f(x;\mu,k,\theta)\propto\exp{\left(\frac{(x-\mu)^{2}}{4\theta^{2}}\right)}D_{-% 2k-1}\left(\frac{|x-\mu|}{\theta}\right)\,\!
  5. f ( x ; μ , k , θ ) = 0 0 N ( x | μ , σ 2 ) Exp ( σ 2 | ψ ) Gamma ( ψ | k , 1 / θ 2 ) d σ 2 d ψ , f(x;\mu,k,\theta)=\int_{0}^{\infty}\int_{0}^{\infty}\ \mathrm{N}(x|\mu,\sigma^% {2})\mathrm{Exp}(\sigma^{2}|\psi)\mathrm{Gamma}(\psi|k,1/\theta^{2})\,d\sigma^% {2}\,d\psi,
  6. μ \mu

Normal_crossings.html

  1. Z = i Z i Z=\cup_{i}Z_{i}
  2. Z i Z_{i}
  3. Z i Z_{i}
  4. Z k Z_{k}
  5. ( Z - Z k ) | Z k (Z-Z_{k})|_{Z_{k}}
  6. x y = 0 xy=0

Normally_hyperbolic_invariant_manifold.html

  1. Λ \Lambda
  2. Λ \Lambda
  3. Λ \Lambda
  4. T Λ T\Lambda
  5. 0 < μ - 1 < λ < 1 0<\mu^{-1}<\lambda<1
  6. T Λ M = T Λ E s E u T_{\Lambda}M=T\Lambda\oplus E^{s}\oplus E^{u}
  7. ( D f ) x E x s = E f ( x ) s and ( D f ) x E x u = E f ( x ) u for all x Λ , (Df)_{x}E^{s}_{x}=E^{s}_{f(x)}\,\text{ and }(Df)_{x}E^{u}_{x}=E^{u}_{f(x)}\,% \text{ for all }x\in\Lambda,
  8. D f n v c λ n v for all v E s and n > 0 , \|Df^{n}v\|\leq c\lambda^{n}\|v\|\,\text{ for all }v\in E^{s}\,\text{ and }n>0,
  9. D f - n v c λ n v for all v E u and n > 0 , \|Df^{-n}v\|\leq c\lambda^{n}\|v\|\,\text{ for all }v\in E^{u}\,\text{ and }n>0,
  10. D f n v c μ | n | v for all v T Λ and n . \|Df^{n}v\|\leq c\mu^{|n|}\|v\|\,\text{ for all }v\in T\Lambda\,\text{ and }n% \in\mathbb{Z}.

Noro–Frenkel_law_of_corresponding_states.html

  1. B 2 = 2 π 0 r 2 [ 1 - e - V ( r ) / k B T ] d r B_{2}=2\pi\int_{0}^{\infty}r^{2}\left[1-e^{-V(r)/k\text{B}T}\right]dr

Nosé–Hoover_thermostat.html

  1. ( P , R , p s , s ) = i 𝐩 i 2 2 m s 2 + 1 2 i j , i j U ( 𝐫 𝐢 - 𝐫 𝐣 ) + p s 2 2 Q + g k T ln ( s ) , \mathcal{H}(P,R,p_{s},s)=\sum_{i}\frac{\mathbf{p}_{i}^{2}}{2ms^{2}}+\frac{1}{2% }\sum_{ij,i\not=j}U\left(\mathbf{r_{i}}-\mathbf{r_{j}}\right)+\frac{p_{s}^{2}}% {2Q}+gkT\ln\left(s\right),
  2. 𝐫 𝐢 \mathbf{r_{i}}
  3. 𝐩 𝐢 \mathbf{p_{i}}
  4. R = R , P = P s and t = t d τ s R^{\prime}=R,~{}P^{\prime}=\frac{P}{s}~{}\,\text{and}~{}t^{\prime}=\int^{t}% \frac{\mathrm{d}\tau}{s}
  5. g = 3 N g=3N

Novikov's_compact_leaf_theorem.html

  1. π 1 ( M 3 ) \pi_{1}(M^{3})
  2. π 2 ( M 3 ) 0 \pi_{2}(M^{3})\neq 0
  3. L F L\in F
  4. π 1 ( L ) π 1 ( M 3 ) \pi_{1}(L)\to\pi_{1}(M^{3})

NP-complete.html

  1. C \scriptstyle C
  2. C \scriptstyle C
  3. C \scriptstyle C
  4. C \scriptstyle C
  5. C \scriptstyle C
  6. C \scriptstyle C
  7. O ( n log n ) \scriptstyle O(n\log n)
  8. X \scriptstyle X
  9. Y \scriptstyle Y
  10. Y \scriptstyle Y
  11. X \scriptstyle X

Nuclear_C*-algebra.html

  1. 𝒪 2 \mathcal{O}_{2}
  2. 𝒪 2 \mathcal{O}_{2}

Nuclear_drip_line.html

  1. E F n = m n c 2 E_{F}^{n}=m_{n}c^{2}\,
  2. E F n = ( p F n ) 2 c 2 + m n 2 c 4 E_{F}^{n}=\sqrt{(p_{F}^{n})^{2}c^{2}+m_{n}^{2}c^{4}}\,

Nuclear_magnetic_resonance.html

  1. 1 / 2 {1}/{2}
  2. μ z = γ S z = γ m . \mu_{\mathrm{z}}=\gamma S_{\mathrm{z}}=\gamma m\hbar.
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. E = - s y m b o l μ 𝐁 0 = - μ x B 0 x - μ y B 0 y - μ z B 0 z . E=-symbol{\mu}\cdot\mathbf{B}_{0}=-\mu_{\mathrm{x}}B_{0x}-\mu_{\mathrm{y}}B_{0% y}-\mu_{\mathrm{z}}B_{0z}.
  6. E = - μ z B 0 , E=-\mu_{\mathrm{z}}B_{0}\ ,
  7. E = - γ m B 0 . E=-\gamma m\hbar B_{0}\ .
  8. 1 / 2 {1}/{2}
  9. Δ E = γ B 0 , \Delta{E}=\gamma\hbar B_{0}\ ,
  10. T 1 T_{1}
  11. T 2 T_{2}
  12. T 2 * T^{*}_{2}
  13. T 2 * T^{*}_{2}
  14. π \pi
  15. T 2 T_{2}
  16. 3 / 2 {3}/{2}
  17. 1 / 2 {1}/{2}
  18. P u r i t y = W t ( S t d ) × n [ H ] ( S t d ) × M W ( S p l ) W t ( S p l ) × M W ( S t d ) × n [ H ] ( S p l ) × P Purity=\frac{Wt(Std)\times n[H](Std)\times MW(Spl)}{Wt(Spl)\times MW(Std)% \times n[H](Spl)}\times P
  19. W t ( S t d ) Wt(Std)
  20. W t ( S p l ) Wt(Spl)
  21. n [ H ] ( S t d ) n[H](Std)
  22. n [ H ] ( S p l ) n[H](Spl)
  23. M W ( S t d ) MW(Std)
  24. M W ( S p l ) MW(Spl)
  25. P P
  26. 1 / 2 {1}/{2}

Nuclear_magnetic_resonance_in_porous_media.html

  1. S S
  2. V V
  3. V V
  4. S S
  5. δ \delta
  6. T i T_{i}
  7. V V
  8. S S
  9. d d
  10. 1 T i = ( 1 - δ S V ) 1 T i b + δ S V 1 T i s + D ( γ G t E ) 2 12 \frac{1}{T_{i}}=\left(1-\frac{\delta S}{V}\right)\frac{1}{T_{ib}}+\frac{\delta S% }{V}\frac{1}{T_{is}}+D\frac{\left({\gamma Gt_{E}}\right)^{2}}{12}
  11. i = 1 , 2 i=1,2
  12. δ \delta
  13. S S
  14. V V
  15. T i b T_{ib}
  16. T i s T_{is}
  17. γ \gamma
  18. G G
  19. t E t_{E}
  20. D D
  21. T 2 T_{2}
  22. T 2 T_{2}
  23. ρ r / D \rho r/D
  24. ρ \rho
  25. r r
  26. D D
  27. M ( t ) = M 0 e - t / T 2 M(t)=M_{0}\mathrm{e}^{-t/T_{2}}
  28. M 0 M_{0}
  29. T 2 {T_{2}}
  30. 1 T 2 = 1 T 2 b + ρ S V \frac{1}{T_{2}}=\frac{1}{T_{2b}}+\rho\frac{S}{V}
  31. S / V S/V
  32. T 2 b T_{2b}
  33. ρ \rho
  34. ρ \rho
  35. 1 T 2 = ρ S V \frac{1}{T_{2}}=\frac{\rho S}{V}
  36. M ( t ) = M 0 i = 1 n a i e - t / T 2 M(t)=M_{0}\sum_{i=1}^{n}{a_{i}}\mathrm{e}^{-t/T_{2}}
  37. a i a_{i}
  38. i i
  39. T 2 i {T_{2i}}
  40. n n
  41. S / V S/V
  42. T 2 l m T_{2lm}
  43. T 2 l m = exp ( a i ln T 2 i a i ) = T 2 i a i a i T_{2lm}=\exp\left(\frac{\sum{a_{i}}\cdot\ln{T_{2i}}}{\sum{a_{i}}}\right)=\sqrt% [\sum{a_{i}}]{\prod T_{2i}^{a_{i}}}
  44. T 2 l m {T_{2lm}}
  45. S / V S/V
  46. k a Φ b ( T 2 l m ) c k\approx a\Phi^{b}(T_{2lm})^{c}
  47. Φ \Phi
  48. b b
  49. c c
  50. k Φ τ ( V S ) 2 k\approx\frac{\Phi}{\tau}\left(\frac{V}{S}\right)^{2}
  51. τ \tau
  52. Φ 1 - b \Phi^{1-b}
  53. F = τ / Φ F=\tau/\Phi
  54. k a F b ( T 2 l m ) c k\approx aF^{b}(T_{2lm})^{c}
  55. b = - 1 b=-1
  56. c = 2 c=2
  57. F F
  58. ρ \rho
  59. ρ \rho
  60. ρ \rho
  61. k a F b ( ρ T 2 l m ) c k\approx aF^{b}(\rho T_{2lm})^{c}
  62. T 2 T_{2}
  63. δ M δ t = D 0 2 M - M T 2 b \frac{\delta M}{\delta t}=D_{0}\nabla^{2}M-\frac{M}{T_{2b}}
  64. D 0 M + ρ M = 0 D_{0}\nabla M+\rho M=0
  65. t = 0 t=0
  66. M = M 0 M=M_{0}
  67. D 0 D_{0}
  68. Δ t \Delta t
  69. x ( t ) x(t)
  70. x ( t + Δ t ) x(t+\Delta t)
  71. ε \varepsilon
  72. δ t = ε 2 6 D 0 \delta t=\frac{\varepsilon^{2}}{6D_{0}}
  73. x ( t + Δ t ) = x ( t ) ε sin θ cos Φ x(t+\Delta t)=x(t)\varepsilon\sin\theta\cos\Phi
  74. y ( t + Δ t ) = y ( t ) ε sin θ cos Φ y(t+\Delta t)=y(t)\varepsilon\sin\theta\cos\Phi
  75. z ( t + Δ t ) = z ( t ) ε cos θ z(t+\Delta t)=z(t)\varepsilon\cos\theta
  76. θ ( 0 θ π ) \theta(0\leqslant\theta\leqslant\pi)
  77. Φ ( 0 Φ 2 π ) \Phi(0\leqslant\Phi\leqslant 2\pi)
  78. θ \theta
  79. π \pi
  80. δ \delta
  81. δ \delta
  82. δ = 2 ε ρ 3 D 0 \delta=\frac{2\varepsilon\rho}{3D_{0}}
  83. p ( t ) p(t)
  84. 2 Δ t 2\Delta t
  85. G G
  86. g ( t ) = e - γ 2 G 2 D 0 ( Δ τ ) 2 t g(t)=\mathrm{e}^{-\gamma^{2}G^{2}D_{0}(\Delta\tau)^{2}t}
  87. γ \gamma
  88. M ( t ) = M 0 ( ( p ( t ) g ( t ) e - t / T 2 b ) M(t)=M_{0}\left((p(t)g(t)\mathrm{e}^{-t/T_{2b}}\right)
  89. T 2 T_{2}

Nucleic_acid_structure.html

  1. L k = T w + W r Lk=Tw+Wr

Null_sign.html

  1. \emptyset

Number_bond.html

  1. 5 + 2 = 7 5+2=7\;

Numerical_response.html

  1. d P / d t = a c V P - m P dP/dt=acVP-mP

NumXL.html

  1. 𝐗 s y m b o l β = μ \mathbf{X}symbol{\beta}=\mu\,\!
  2. 𝐗 s y m b o l β = ln ( μ 1 - μ ) \mathbf{X}symbol{\beta}=\ln{\left(\frac{\mu}{1-\mu}\right)}\,\!
  3. 𝐗 s y m b o l β = ln ( μ 1 - μ ) \mathbf{X}symbol{\beta}=\ln{\left(\frac{\mu}{1-\mu}\right)}\,\!
  4. 𝐗 s y m b o l β = Φ - 1 ( μ ) \mathbf{X}symbol{\beta}=\Phi^{-1}{\left(\mu\right)}\,\!
  5. 𝐗 s y m b o l β = ln ( - ln ( 1 - μ ) ) \mathbf{X}symbol{\beta}=\ln{\left(-\ln{\left(1-\mu\right)}\right)}\,\!

Octave_(electronics).html

  1. octaves = log 2 ( f 2 f 1 ) \mathrm{octaves}=\log_{2}\left(\frac{f_{2}}{f_{1}}\right)
  2. 10 log 10 ( 4 ) 6.0206 10\log_{10}(4)\approx 6.0206
  3. number of octaves = log 2 ( 13 4 ) = 1.7 \,\text{number of octaves}=\log_{2}\left(\frac{13}{4}\right)=1.7
  4. Mag 13 kHz = 52 dB + ( 1.7 octaves × - 2 dB/octave ) = 48.6 dB . \,\text{Mag}_{13\,\text{ kHz}}=52\,\text{ dB}+(1.7\,\text{ octaves}\times-2\,% \text{ dB/octave})=48.6\,\text{ dB}.\,

Odd_graph.html

  1. ( 2 n - 1 n - 1 ) {\textstyle\left({{2n-1}\atop{n-1}}\right)}
  2. n ( 2 n - 1 n - 1 ) / 2 n{\textstyle\left({{2n-1}\atop{n-1}}\right)}/2
  3. ( 2 n - 2 n - 2 ) {\textstyle\left({{2n-2}\atop{n-2}}\right)}
  4. ( 2 n - 2 n - 2 ) {\textstyle\left({{2n-2}\atop{n-2}}\right)}
  5. ( 2 n - 2 n - 2 ) . {\textstyle\left({{2n-2}\atop{n-2}}\right)}.

Office_Open_XML_file_formats.html

  1. π 2 \frac{\pi}{2}

Olation.html

  1. \overrightarrow{\leftarrow}
  2. \overrightarrow{\leftarrow}
  3. \overrightarrow{\leftarrow}
  4. \overrightarrow{\leftarrow}
  5. \overrightarrow{\leftarrow}

Omega-categorical_theory.html

  1. 0 \aleph_{0}

Omega_and_agemo_subgroup.html

  1. Ω i ( G ) = { g : g p i = 1 } . \Omega_{i}(G)=\langle\{g:g^{p^{i}}=1\}\rangle.
  2. i ( G ) = { g p i : g G } . \mho^{i}(G)=\langle\{g^{p^{i}}:g\in G\}\rangle.
  3. K = a , b : a 4 = b 4 = 1 , b a = a b 3 , K=\langle a,b:a^{4}=b^{4}=1,ba=ab^{3}\rangle,

Omega_ratio.html

  1. Ω ( r ) = r ( 1 - F ( x ) ) d x - r F ( x ) d x \Omega(r)=\frac{\int_{r}^{\infty}(1-F(x))\,dx}{\int_{-\infty}^{r}F(x)dx}

One-dimensional_space.html

  1. L = 2 r L=2r
  2. r r

One-third_hypothesis.html

  1. ( n r ) {\textstyle\left({{n}\atop{r}}\right)}
  2. ( 1 + x ) n = r = 0 ( n r ) x r . (1+x)^{n}=\sum_{r=0}^{\infty}{n\choose r}x^{r}.\qquad
  3. ( n r ) p r q n - r {\textstyle\left({{n}\atop{r}}\right)}p^{r}q^{n-r}\!
  4. q r q^{r}\!
  5. ( n r ) p r q n {\textstyle\left({{n}\atop{r}}\right)}p^{r}q^{n}\!
  6. ( n r ) / 2 n + r {\textstyle\left({{n}\atop{r}}\right)}/2^{n+r}\!

One-way_speed_of_light.html

  1. 1 / 299 , 792 , 458 {1}/{299,792,458}
  2. t 1 t_{1}
  3. t 2 t_{2}
  4. t 3 t_{3}
  5. t 2 = t 1 + 1 2 ( t 3 - t 1 ) t_{2}=t_{1}+\tfrac{1}{2}\left(t_{3}-t_{1}\right)
  6. t 2 = t 1 + ε ( t 3 - t 1 ) t_{2}=t_{1}+\varepsilon\left(t_{3}-t_{1}\right)
  7. Δ c / c \Delta c/c
  8. < 3 × 10 - 8 <3\times 10^{-8}
  9. 10 - 8 \sim 10^{-8}\!
  10. < 3 × 10 - 9 <3\times 10^{-9}
  11. < 1.5 × 10 - 6 <1.5\times 10^{-6}
  12. < 5 × 10 - 9 <5\times 10^{-9}
  13. κ ~ e - \tilde{\kappa}_{e-}
  14. κ ~ o + \tilde{\kappa}_{o+}
  15. κ ~ t r \tilde{\kappa}_{tr}
  16. κ ~ e - = ( - 0.31 ± 0.73 ) × 10 - 17 \tilde{\kappa}_{e-}=\scriptstyle(-0.31\pm 0.73)\times 10^{-17}
  17. κ ~ o + = 0.7 ± 1 × 10 - 14 \tilde{\kappa}_{o+}=\scriptstyle 0.7\pm 1\times 10^{-14}
  18. κ ~ t r = - 0.4 ± 0.9 × 10 - 10 \tilde{\kappa}_{tr}=\scriptstyle-0.4\pm 0.9\times 10^{-10}
  19. c ± = c 1 ± κ c_{\pm}=\frac{c}{1\pm\kappa}
  20. d t ~ = γ ~ [ 1 + κ 𝐯 ~ / c - κ 𝐯 ~ / c ] d t ~ - ( κ + γ ~ 𝐯 ~ ) d 𝐱 ~ / c - [ γ ~ ( 1 + κ 𝐯 ~ / c ) - 1 ] κ 𝐯 ~ 𝐯 ~ 2 c 𝐯 ~ d 𝐱 ~ + γ ~ κ 𝐯 ~ ( κ d 𝐱 ~ ) / c , d 𝐱 ~ = - γ ~ 𝐯 ~ d t ~ + d 𝐱 ~ + [ γ ~ ( 1 + κ 𝐯 ~ / c ) - 1 ] 𝐯 ~ d 𝐱 𝐯 ~ 2 𝐯 ~ - γ ~ 𝐯 ~ ( κ d 𝐱 ~ ) / c , γ ~ = γ ( 1 - κ 𝐯 / c ) , 𝐯 ~ = 𝐯 1 - κ 𝐯 / c , \begin{aligned}\displaystyle d\tilde{t}^{\prime}=&\displaystyle\tilde{\gamma}% \left[1+\kappa\cdot\tilde{\mathbf{v}}/c-\kappa^{\prime}\cdot\tilde{\mathbf{v}}% ^{\prime}/c\right]d\tilde{t}-\left(\kappa^{\prime}+\tilde{\gamma}\tilde{% \mathbf{v}}^{\prime}\right)\cdot d\tilde{\mathbf{x}}/c\\ &\displaystyle-\left[\tilde{\gamma}\left(1+\kappa\cdot\tilde{\mathbf{v}}/c% \right)-1\right]\frac{\kappa^{\prime}\cdot\tilde{\mathbf{v}}}{\tilde{\mathbf{v% }}^{2}c}\tilde{\mathbf{v}}\cdot d\tilde{\mathbf{x}}+\tilde{\gamma}\kappa\cdot% \tilde{\mathbf{v}}\left(\kappa\cdot d\tilde{\mathbf{x}}\right)/c,\\ \displaystyle d\tilde{\mathbf{x}}^{\prime}=&\displaystyle-\tilde{\gamma}\tilde% {\mathbf{v}}d\tilde{t}+d\tilde{\mathbf{x}}+\left[\tilde{\gamma}\left(1+\kappa% \cdot\tilde{\mathbf{v}}/c\right)-1\right]\frac{\tilde{\mathbf{v}}\cdot d% \mathbf{x}}{\tilde{\mathbf{v}}^{2}}\tilde{\mathbf{v}}-\tilde{\gamma}\tilde{% \mathbf{v}}\left(\kappa\cdot d\tilde{\mathbf{x}}\right)/c,\\ \displaystyle\tilde{\gamma}=&\displaystyle\gamma\left(1-\kappa\cdot\mathbf{v}/% c\right),\\ \displaystyle\tilde{\mathbf{v}}=&\displaystyle\frac{\mathbf{v}}{1-\kappa\cdot% \mathbf{v}/c},\end{aligned}

Open_statistical_ensemble.html

  1. p m v = z m m ! Υ v t = 0 z t t ! [ i = 1 m j = m + 1 m + t ψ i v χ j v ] 1... m + t ( m , t ) d s y m b o l r 1 d s y m b o l r m + t , p^{v}_{m}=\frac{z^{m}}{m!\Upsilon_{v}}\sum_{t=0}^{\infty}\frac{z^{t}}{t!}\int% \left[\prod_{i=1}^{m}\prod_{j=m+1}^{m+t}\psi^{v}_{i}\chi^{v}_{j}\right]{% \mathcal{B}}^{(m,t)}_{1...m+t}dsymbol{r}_{1}...dsymbol{r}_{m+t},
  2. p m v p^{v}_{m}
  3. m ~{}m
  4. v ~{}v
  5. m 1 ~{}m\geq 1
  6. z ~{}z
  7. Υ v ~{}\Upsilon_{v}
  8. ψ i v ~{}\psi^{v}_{i}
  9. χ j v = 1 - ψ j v ~{}\chi^{v}_{j}=1-\psi^{v}_{j}
  10. 1... m + t ( m , t ) ~{}{\mathcal{B}}^{(m,t)}_{1...m+t}
  11. p m v = 1 m ! Υ v [ i = 1 m ψ i v ] ϱ G , 1... m ( m ) ( χ v ) d s y m b o l r 1 d s y m b o l r m , p^{v}_{m}=\frac{1}{m!\Upsilon_{v}}\int\left[\prod_{i=1}^{m}\psi^{v}_{i}\right]% \varrho^{(m)}_{G,1...m}(\chi^{v})dsymbol{r}_{1}...dsymbol{r}_{m},
  12. ϱ G , 1... m ( m ) ~{}\varrho^{(m)}_{G,1...m}
  13. z ~{}z
  14. χ v ~{}\chi^{v}
  15. 1... m + t ( m , t ) ~{}{\mathcal{B}}^{(m,t)}_{1...m+t}
  16. ϱ G , 1... m ( m ) z m e - β U 1... m m \varrho^{(m)}_{G,1...m}\approx z^{m}e^{-\beta U^{m}_{1...m}}
  17. U 1... m m ~{}U^{m}_{1...m}
  18. m ~{}m
  19. 1 / β = k B T ~{}1/\beta=k_{B}T
  20. k B ~{}k_{B}
  21. T ~{}T
  22. Υ v = exp t = 1 z t t ! [ 1 - i = 1 t χ i v ] 𝒰 1... t ( t ) d s y m b o l r 1 d s y m b o l r t , \Upsilon_{v}=\exp{\sum_{t=1}^{\infty}\frac{z^{t}}{t!}\int\left[1-\prod_{i=1}^{% t}\chi^{v}_{i}\right]{\mathcal{U}}^{(t)}_{1...t}dsymbol{r}_{1}...dsymbol{r}_{t% }},
  23. Ξ v = exp t = 1 z t t ! [ i = 1 t ψ i v ] 𝒰 1... t ( t ) d s y m b o l r 1 d s y m b o l r t , \Xi_{v}=\exp{\sum_{t=1}^{\infty}\frac{z^{t}}{t!}\int\left[\prod_{i=1}^{t}\psi^% {v}_{i}\right]{\mathcal{U}}^{(t)}_{1...t}dsymbol{r}_{1}...dsymbol{r}_{t}},
  24. 𝒰 1... t ( t ) ~{}{\mathcal{U}}^{(t)}_{1...t}
  25. Υ v ~{}\Upsilon_{v}
  26. Υ v = exp β [ v P ( z , T ) + a σ ( z , T ) ] , ~{}\Upsilon_{v}=\exp{\beta[vP(z,T)+a\sigma(z,T)]},
  27. P ( z , T ) ~{}P(z,T)
  28. σ ( z , T ) ~{}\sigma(z,T)
  29. a ~{}a
  30. v ~{}v
  31. p 0 v = e - β ( v P ( z , T ) + a σ ( z , T ) ) ~{}p^{v}_{0}=e^{-\beta\left(vP{\left(z,T\right)}+a\sigma{\left(z,T\right)}% \right)}
  32. p 0 v e - β R m i n , ~{}p^{v}_{0}\propto e^{-\beta R_{min}},
  33. R m i n ~{}R_{min}
  34. p = ϱ v ~{}p=\varrho v

OpenSSH.html

  1. 2 - 18 2^{-18}

Operating_temperature.html

  1. T J = T a + P D × R j a T_{J}=T_{a}+P_{D}\times R_{ja}

Optical_Multi-Tree_with_Shuffle_Exchange.html

  1. 3 n 3 / 2 3n^{3}/2
  2. n 2 n^{2}
  3. ( n 3 ) / 4 (n^{3})/4

Optical_sectioning.html

  1. D z = λ n ( NA ) 2 D_{z}=\frac{\lambda n}{(\mathrm{NA})^{2}}
  2. D x = D y = 0.61 λ NA D_{x}=D_{y}=\frac{0.61\lambda}{\mathrm{NA}}

OPTICS_algorithm.html

  1. ε \varepsilon
  2. M i n P t s MinPts
  3. p p
  4. M i n P t s MinPts
  5. ε \varepsilon
  6. N ε ( p ) N_{\varepsilon}(p)
  7. M i n P t s MinPts
  8. core-dist ε , M i n P t s ( p ) = { UNDEFINED if | N ε ( p ) | < M i n P t s M i n P t s -th smallest distance to N ε ( p ) otherwise \,\text{core-dist}_{\varepsilon,MinPts}(p)=\begin{cases}\,\text{UNDEFINED}&\,% \text{if }|N_{\varepsilon}(p)|<MinPts\\ MinPts\,\text{-th smallest distance to }N_{\varepsilon}(p)&\,\text{otherwise}% \end{cases}
  9. o o
  10. p p
  11. o o
  12. p p
  13. p p
  14. reachability-dist ε , M i n P t s ( o , p ) = { UNDEFINED if | N ε ( p ) | < M i n P t s max ( core-dist ε , M i n P t s ( p ) , dist ( p , o ) ) otherwise \,\text{reachability-dist}_{\varepsilon,MinPts}(o,p)=\begin{cases}\,\text{% UNDEFINED}&\,\text{if }|N_{\varepsilon}(p)|<MinPts\\ \max(\,\text{core-dist}_{\varepsilon,MinPts}(p),\,\text{dist}(p,o))&\,\text{% otherwise}\end{cases}
  15. p p
  16. o o
  17. ε < ε \varepsilon^{\prime}<\varepsilon
  18. p p
  19. o o
  20. ε \varepsilon
  21. ε \varepsilon
  22. ε \varepsilon
  23. O ( n 2 ) O(n^{2})
  24. ε \varepsilon
  25. ε \varepsilon
  26. ε \varepsilon
  27. p p
  28. q q
  29. ε \varepsilon
  30. O ( log n ) O(\log n)
  31. O ( n log n ) O(n\cdot\log n)
  32. ε \varepsilon
  33. ε > max x , y d ( x , y ) \varepsilon>\max_{x,y}d(x,y)
  34. ε \varepsilon
  35. ε \varepsilon

Optimum_HDTV_viewing_distance.html

  1. VD = DS ( NHR NVR ) 2 + 1 CVR tan 1 60 \textrm{VD}=\frac{\textrm{DS}}{\sqrt{\left(\frac{\textrm{NHR}}{\textrm{NVR}}% \right)^{2}+1}\cdot\textrm{CVR}\cdot\tan{\frac{1}{60}}}
  2. VD = 32 ( 1920 1080 ) 2 + 1 480 tan 1 60 = 112.36 \textrm{VD}=\frac{\textrm{32}}{\sqrt{\left(\frac{\textrm{1920}}{\textrm{1080}}% \right)^{2}+1}\cdot\textrm{480}\cdot\tan{\frac{1}{60}}}=112.36
  3. VD = 32 ( 1920 1080 ) 2 + 1 1080 tan 1 60 = 49.94 \textrm{VD}=\frac{\textrm{32}}{\sqrt{\left(\frac{\textrm{1920}}{\textrm{1080}}% \right)^{2}+1}\cdot\textrm{1080}\cdot\tan{\frac{1}{60}}}=49.94

Option_type.html

  1. A ? = A + 1 A^{?}=A+1
  2. return : A A ? = a Just a \,\text{return}\colon A\to A^{?}=a\mapsto\,\text{Just}\,a
  3. bind : A ? ( A B ? ) B ? = a f { Nothing if a = Nothing f a if a = Just a \,\text{bind}\colon A^{?}\to(A\to B^{?})\to B^{?}=a\mapsto f\mapsto\begin{% cases}\,\text{Nothing}&\,\text{if}\ a=\,\text{Nothing}\\ f\,a^{\prime}&\,\text{if}\ a=\,\text{Just}\,a^{\prime}\end{cases}
  4. fmap : ( A B ) A ? B ? = f a { Nothing if a = Nothing Just f a if a = Just a \,\text{fmap}\colon(A\to B)\to A^{?}\to B^{?}=f\mapsto a\mapsto\begin{cases}\,% \text{Nothing}&\,\text{if}\ a=\,\text{Nothing}\\ \,\text{Just}\,f\,a^{\prime}&\,\text{if}\ a=\,\text{Just}\,a^{\prime}\end{cases}
  5. join : A ? ? A ? = a { Nothing if a = Nothing Nothing if a = Just Nothing Just a if a = Just Just a \,\text{join}\colon{A^{?}}^{?}\to A^{?}=a\mapsto\begin{cases}\,\text{Nothing}&% \,\text{if}\ a=\,\text{Nothing}\\ \,\text{Nothing}&\,\text{if}\ a=\,\text{Just}\,\,\text{Nothing}\\ \,\text{Just}\,a^{\prime}&\,\text{if}\ a=\,\text{Just}\,\,\text{Just}\,a^{% \prime}\end{cases}
  6. mplus : A ? A ? A ? = a 1 a 2 { Nothing if a 1 = Nothing and a 2 = Nothing Just a 2 if a 1 = Nothing and a 2 = Just a 2 Just a 1 if a 1 = Just a 1 \,\text{mplus}\colon A^{?}\to A^{?}\to A^{?}=a_{1}\mapsto a_{2}\mapsto\begin{% cases}\,\text{Nothing}&\,\text{if}\ a_{1}=\,\text{Nothing}\and a_{2}=\,\text{% Nothing}\\ \,\text{Just}\,a^{\prime}_{2}&\,\text{if}\ a_{1}=\,\text{Nothing}\and a_{2}=\,% \text{Just}\,a^{\prime}_{2}\\ \,\text{Just}\,a^{\prime}_{1}&\,\text{if}\ a_{1}=\,\text{Just}\,a^{\prime}_{1}% \end{cases}

Orbital_integral.html

  1. M r f ( x ) = K f ( g k y ) d k , M^{r}f(x)=\int_{K}f(gk\cdot y)\,dk,
  2. K f ( g k y ) d k \int_{K}f(gk\cdot y)\,dk
  3. f ( x ) = lim r 0 + M r f ( x ) . f(x)=\lim_{r\to 0^{+}}M^{r}f(x).\,

Order_isomorphism.html

  1. ( S , S ) (S,\leq_{S})
  2. ( T , T ) (T,\leq_{T})
  3. ( S , S ) (S,\leq_{S})
  4. ( T , T ) (T,\leq_{T})
  5. f f
  6. S S
  7. T T
  8. x x
  9. y y
  10. S S
  11. x S y x\leq_{S}y
  12. f ( x ) T f ( y ) f(x)\leq_{T}f(y)
  13. f f
  14. T T
  15. f f
  16. f ( x ) = f ( y ) f(x)=f(y)
  17. f f
  18. x y x\leq y
  19. y x y\leq x
  20. x = y x=y
  21. ( , ) (\mathbb{R},\leq)
  22. ( , ) (\mathbb{R},\geq)
  23. \mathbb{R}
  24. \leq
  25. ( 0 , 1 ) (0,1)
  26. [ 0 , 1 ] [0,1]
  27. f f
  28. f f
  29. ( S , S ) (S,\leq_{S})
  30. ( T , T ) (T,\leq_{T})
  31. g g
  32. ( T , T ) (T,\leq_{T})
  33. ( U , U ) (U,\leq_{U})
  34. f f
  35. g g
  36. ( S , S ) (S,\leq_{S})
  37. ( U , U ) (U,\leq_{U})

Ordinal_number.html

  1. 0 \aleph_{0}
  2. 1 \aleph_{1}
  3. 0 \aleph_{0}
  4. 0 , 1 , 2 , , 41 {0,1,2,…,41}
  5. ${ }$
  6. 0 {0}
  7. ${ { } }$
  8. 0 , 1 {0,1}
  9. , {{},{{}}}
  10. 0 , 1 , 2 {0,1,2}
  11. , , , {{},{{}},{{},{{}}}}
  12. 0 , 1 , 2 , 3 {0,1,2,3}
  13. , , , , , , , {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}
  14. 0 , 1 , 2 , 3 {0,1,2,3}
  15. 0 , 1 {0,1}
  16. 0 , 1 , 2 , 3 {0,1,2,3}
  17. α { α } \alpha\cup\{\alpha\}
  18. α ι | ι < γ \langle\alpha_{\iota}|\iota<\gamma\rangle
  19. α ι < α ρ \alpha_{\iota}<\alpha_{\rho}\!
  20. ι < ρ , \iota<\rho,\!
  21. { α ι | ι < γ } , \{\alpha_{\iota}|\iota<\gamma\},\!
  22. α \alpha
  23. α \alpha
  24. α \alpha
  25. γ \gamma
  26. γ \gamma
  27. β < γ \beta<\gamma
  28. β \beta
  29. β < γ \beta<\gamma
  30. γ \gamma
  31. ω γ \omega\cdot\gamma
  32. γ \gamma
  33. ω γ \omega^{\gamma}
  34. γ \gamma
  35. α \alpha
  36. ω α = α \omega^{\alpha}=\alpha
  37. ε γ \varepsilon_{\gamma}
  38. C C
  39. α \alpha
  40. β \beta
  41. C C
  42. α < β \alpha<\beta
  43. F F
  44. δ \delta
  45. F ( δ ) F(\delta)
  46. δ \delta
  47. F ( γ ) F(\gamma)
  48. γ < δ \gamma<\delta
  49. ε \varepsilon_{\cdot}
  50. α \alpha
  51. α \alpha
  52. α \alpha
  53. α \alpha
  54. α \alpha
  55. α \alpha
  56. α \alpha
  57. α \alpha
  58. α \alpha
  59. α \alpha
  60. 0 \aleph_{0}
  61. n - 2 \aleph_{n-2}
  62. ω α \omega_{\alpha}
  63. α \aleph_{\alpha}
  64. 0 \aleph_{0}
  65. 0 \aleph_{0}
  66. 0 \aleph_{0}
  67. 0 2 \aleph_{0}^{2}
  68. 0 \aleph_{0}
  69. ω 2 > ω \omega^{2}>\omega
  70. ω 1 \omega_{1}
  71. ω 1 \omega_{1}
  72. ω 2 \omega_{2}
  73. 1 \aleph_{1}
  74. ω ω \omega_{\omega}
  75. ω n \omega_{n}
  76. ω n \omega_{n}
  77. α \alpha
  78. δ \delta
  79. α \alpha
  80. δ \delta
  81. α \alpha
  82. ω ω \omega_{\omega}
  83. ω \omega
  84. ω α + 1 \omega_{\alpha+1}
  85. ω \omega
  86. ω 1 \omega_{1}
  87. ω 2 \omega_{2}
  88. ω ω \omega_{\omega}
  89. ω α = α \omega^{\alpha}=\alpha
  90. ω \omega
  91. ω ω \omega^{\omega}
  92. ω ω ω \omega^{\omega^{\omega}}
  93. ι \iota
  94. ω α = α \omega^{\alpha}=\alpha
  95. ε ι \varepsilon_{\iota}
  96. ι \iota
  97. ε α = α \varepsilon_{\alpha}=\alpha
  98. ω 1 CK \omega_{1}^{\mathrm{CK}}
  99. ω 1 \omega_{1}
  100. ω 1 CK \omega_{1}^{\mathrm{CK}}
  101. ε 0 \varepsilon_{0}
  102. 0 , 1 , 2 {0,1,2}

Oregonator.html

  1. \longrightarrow
  2. \longrightarrow
  3. d [ X ] d t = k I [ A ] [ Y ] - k I I [ X ] [ Y ] + k I I I [ A ] [ X ] - 2 k I V [ X ] 2 \frac{d[X]}{dt}=k_{I}[A][Y]-k_{II}[X][Y]+k_{III}[A][X]-2k_{IV}[X]^{2}
  4. d [ Y ] d t = - k I [ A ] [ Y ] - k I I [ X ] [ Y ] + 1 2 f k V [ B ] [ Z ] \frac{d[Y]}{dt}=-k_{I}[A][Y]-k_{II}[X][Y]+\frac{1}{2}fk_{V}[B][Z]
  5. d [ Z ] d t = 2 k I I I [ A ] [ X ] - k V [ B ] [ Z ] \frac{d[Z]}{dt}=2k_{III}[A][X]-k_{V}[B][Z]

Organorhodium_chemistry.html

  1. RHC = CH 2 + CO + H 2 Rh 4 ( CO ) 10 / PPh 3 RCH 2 CH 2 CHO \mathrm{RHC=CH_{2}\ +\ CO\ +\ H_{2}\ \xrightarrow{Rh_{4}(CO)_{10}/PPh_{3}}\ \ % RCH_{2}CH_{2}CHO}
  2. PhNO 2 + C 6 H 6 + 3 CO Rh 6 ( CO ) 16 PhNHCOPh + 2 CO 2 \mathrm{PhNO_{2}\ +\ C_{6}H_{6}\ +\ 3\ CO\ \xrightarrow{Rh_{6}(CO)_{16}}\ \ % PhNHCOPh\ +\ 2\ CO_{2}}

Organoruthenium_chemistry.html

  1. \overrightarrow{\leftarrow}

Oriented_matroid.html

  1. X X
  2. X ¯ \underline{X}
  3. X + X^{+}
  4. X - X^{-}
  5. X + X^{+}
  6. X - X^{-}
  7. X ¯ = X + X - \underline{X}=X^{+}\cup X^{-}
  8. X X
  9. $\empty$
  10. Y Y
  11. X X
  12. Y = - X Y=-X
  13. Y + = X - Y^{+}=X^{-}
  14. Y - = X + Y^{-}=X^{+}
  15. x x
  16. x x
  17. - x -x
  18. E E
  19. E E
  20. 𝒞 \mathcal{C}
  21. E E
  22. 𝒞 \mathcal{C}
  23. 𝒞 \mathcal{C}
  24. E E
  25. 𝒞 \notin\mathcal{C}
  26. 𝒞 = - 𝒞 \mathcal{C}=-\mathcal{C}
  27. for all X , Y 𝒞 if X ¯ Y ¯ , then X = Y or X = - Y \mbox{ for all }~{}X,Y\in\mathcal{C}\mbox{ if }~{}\underline{X}\subseteq% \underline{Y},\mbox{ then }~{}X=Y\mbox{ or }~{}X=-Y
  28. for all X , Y 𝒞 , X - Y , and e X + Y - there is a Z 𝒞 such that \mbox{ for all }~{}X,Y\in\mathcal{C},X\neq-Y,\mbox{ and }~{}e\in X^{+}\cap Y^{% -}\mbox{ there is a }~{}Z\in\mathcal{C}\mbox{ such that }~{}
  29. Z + ( X + Y + ) { e } and Z^{+}\subseteq(X^{+}\cup Y^{+})\setminus\{e\}\mbox{ and }~{}
  30. Z - ( X - Y - ) { e } . Z^{-}\subseteq(X^{-}\cup Y^{-})\setminus\{e\}.
  31. E E
  32. r r
  33. χ : E r { - 1 , 0 , 1 } \chi\colon E^{r}\to\{-1,0,1\}
  34. χ \chi
  35. σ \sigma
  36. x 1 , , x r E x_{1},\dots,x_{r}\in E
  37. χ ( x σ ( 1 ) , , x σ ( r ) ) = sgn ( σ ) χ ( x 1 , , x r ) \chi\left(x_{\sigma(1)},\dots,x_{\sigma(r)}\right)=\,\text{sgn}(\sigma)\chi% \left(x_{1},\dots,x_{r}\right)
  38. sgn ( σ ) \,\text{sgn}(\sigma)
  39. x 1 , , x r , y 1 , , y r E x_{1},\dots,x_{r},y_{1},\dots,y_{r}\in E
  40. χ ( y i , x 2 , , x r ) χ ( y 1 , , y i - 1 , x 1 , y i + 1 , , y r ) 0 \chi(y_{i},x_{2},\dots,x_{r})\chi(y_{1},\dots,y_{i-1},x_{1},y_{i+1},\dots,y_{r% })\geq 0
  41. i i
  42. χ ( x 1 , , x r ) χ ( y 1 , , y r ) 0 \chi\left(x_{1},\dots,x_{r}\right)\chi\left(y_{1},\dots,y_{r}\right)\geq 0
  43. r r
  44. E E
  45. r r
  46. χ \chi
  47. C C
  48. C { x 1 , , x r , x r + 1 } C\subset\{x_{1},\dots,x_{r},x_{r+1}\}
  49. { x 1 , , x r } \{x_{1},\dots,x_{r}\}
  50. C C
  51. C + = { x i : ( - 1 ) i χ ( x 1 , , x i - 1 , , x i + 1 , , x r + 1 ) = 1 } C^{+}=\{x_{i}:(-1)^{i}\chi(x_{1},\dots,x_{i-1},\dots,x_{i+1},\dots,x_{r+1})=1\}
  52. X ¯ \textstyle\underline{X}
  53. X + \textstyle X^{+}
  54. X - \textstyle X^{-}
  55. 𝒞 \textstyle\mathcal{C}
  56. X \textstyle X
  57. 𝒞 \textstyle\mathcal{C}
  58. { ( 1 , 2 ) , ( 1 , 3 ) , ( 3 , 2 ) } \textstyle\{(1,2),(1,3),(3,2)\}
  59. { ( 3 , 4 ) , ( 4 , 3 ) } \textstyle\{(3,4),(4,3)\}
  60. { ( 1 , 2 ) , - ( 1 , 3 ) , - ( 3 , 2 ) } \textstyle\{(1,2),-(1,3),-(3,2)\}
  61. { - ( 1 , 2 ) , ( 1 , 3 ) , ( 3 , 2 ) } \textstyle\{-(1,2),(1,3),(3,2)\}
  62. { ( 3 , 4 ) , ( 4 , 3 ) } \textstyle\{(3,4),(4,3)\}
  63. { - ( 3 , 4 ) , - ( 4 , 3 ) } \textstyle\{-(3,4),-(4,3)\}
  64. { ( 1 , 2 ) , ( 1 , 3 ) , ( 3 , 2 ) , ( 3 , 4 ) , ( 4 , 3 ) } \textstyle\{(1,2),(1,3),(3,2),(3,4),(4,3)\}
  65. E \textstyle E
  66. n \textstyle\mathbb{R}^{n}
  67. E \textstyle E
  68. { v 1 , , v m } \textstyle\{v_{1},\dots,v_{m}\}
  69. i = 1 m λ i v i = 0 \sum_{i=1}^{m}\lambda_{i}v_{i}=0
  70. λ i \textstyle\lambda_{i}\in\mathbb{R}
  71. X = { X + , X - } \textstyle X=\{X^{+},X^{-}\}
  72. X + = { v i : λ i > 0 } \textstyle X^{+}=\{v_{i}:\lambda_{i}>0\}
  73. X - = { v i : λ i < 0 } \textstyle X^{-}=\{v_{i}:\lambda_{i}<0\}
  74. X \textstyle X
  75. E \textstyle E
  76. E E
  77. χ : E r { - 1 , 0 , 1 } \chi:E^{r}\rightarrow\{-1,0,1\}
  78. x 1 , , x r E x_{1},\dots,x_{r}\in E
  79. χ ( x 1 , , x r ) = sgn ( det ( x 1 , , x r ) ) \chi(x_{1},\dots,x_{r})=\,\text{sgn}(\det(x_{1},\dots,x_{r}))
  80. χ \chi
  81. E E
  82. χ : E r { - 1 , 0 , 1 } \chi:E^{r}\rightarrow\{-1,0,1\}
  83. x 1 , , x k E x_{1},\dots,x_{k}\in E
  84. sgn ( x 1 , , x k ) \,\text{sgn}(x_{1},\dots,x_{k})
  85. χ * : E | E | - r { - 1 , 0 , 1 } \chi^{*}:E^{|E|-r}\rightarrow\{-1,0,1\}
  86. χ * ( x 1 , , x r ) χ ( x r + 1 , , x | E | ) sgn ( x 1 , , x r , x r + 1 , , x | E | ) , \chi^{*}(x_{1},\dots,x_{r})\mapsto\chi(x_{r+1},\dots,x_{|E|})\,\text{sgn}(x_{1% },\dots,x_{r},x_{r+1},\dots,x_{|E|}),
  87. χ * \chi^{*}
  88. d d
  89. e : S d S d + 1 e:S^{d}\hookrightarrow S^{d+1}
  90. h : S d + 1 S d + 1 h:S^{d+1}\rightarrow S^{d+1}
  91. h e h\circ e
  92. S d S^{d}
  93. S d + 1 S^{d+1}
  94. S d S^{d}
  95. d + 1 d+1
  96. S d S^{d}
  97. R n R^{n}

Original_sin_(economics).html

  1. O S I N 1 i = 1 - securities issued by country i in currency i Securities issued by country i OSIN1_{i}=1-\frac{\,\text{securities issued by country i in currency i}}{\,% \text{Securities issued by country i}}
  2. O S I N 2 i = m a x ( I N D E X A i , O S I N 3 i ) OSIN2_{i}=max(INDEXA_{i},OSIN3_{i})
  3. I N D E X A i = Securities + loans issued by country i in major currencies Securities + loans issued by country i INDEXA_{i}={\,\text{Securities}+\,\text{loans issued by country i in major % currencies}\over\,\text{Securities}+\,\text{loans issued by country i}}
  4. O S I N 3 i = m a x ( I N D E X B i , 0 ) OSIN3_{i}=max(INDEXB_{i},0)
  5. I N D E X B i = 1 - Securities in currency i (regardless of the nationality of the issuer) Securities issued by country i INDEXB_{i}=1-\frac{\,\text{Securities in currency i (regardless of the % nationality of the issuer)}}{\,\text{Securities issued by country i}}

Orthogonal_diagonalization.html

  1. Δ ( t ) . \Delta(t).
  2. Δ ( t ) \Delta(t)
  3. λ \lambda
  4. P T A P P^{T}AP
  5. λ 1 , , λ n \lambda_{1},\dots,\lambda_{n}

Oscillator_linewidth.html

  1. S v ( f ) = A 2 2 c f 0 2 c 2 π 2 f 0 4 + ( f - f 0 ) 2 , S_{v}(f)=\frac{A^{2}}{2}\frac{cf_{0}^{2}}{c^{2}\pi^{2}f_{0}^{4}+(f-f_{0})^{2}},
  2. S v ( f 0 + Δ f ) = A 2 2 π f Δ f Δ 2 + Δ f 2 . S_{v}(f_{0}+\Delta f)=\frac{A^{2}}{2\pi}\frac{f_{\Delta}}{f_{\Delta}^{2}+% \Delta f^{2}}.
  3. L ( Δ f ) = 1 π f Δ f Δ 2 + Δ f 2 . L(\Delta f)=\frac{1}{\pi}\frac{f_{\Delta}}{f_{\Delta}^{2}+\Delta f^{2}}.
  4. - L ( Δ f ) d Δ f = f Δ π - d Δ f f Δ 2 + Δ f 2 = 1 π tan - 1 ( Δ f f Δ ) | - = 1 \int_{-\infty}^{\infty}L(\Delta f)d\Delta f=\frac{f_{\Delta}}{\pi}\int_{-% \infty}^{\infty}\frac{d\Delta f}{f_{\Delta}^{2}+\Delta f^{2}}=\left.\frac{1}{% \pi}\tan^{-1}(\frac{\Delta f}{f_{\Delta}})\right|_{-\infty}^{\infty}=1

Oscillatory_integral_operator.html

  1. T λ u ( x ) = 𝐑 n e i λ S ( x , y ) a ( x , y ) u ( y ) d y , x 𝐑 m , y 𝐑 n , T_{\lambda}u(x)=\int_{\mathbf{R}^{n}}e^{i\lambda S(x,y)}a(x,y)u(y)\,dy,\qquad x% \in\mathbf{R}^{m},\quad y\in\mathbf{R}^{n},
  2. det j , k 2 S x j y k ( x , y ) 0 \mathop{\rm det}_{j,k}\frac{\partial^{2}S}{\partial x_{j}\partial y_{k}}(x,y)\neq 0
  3. C λ - n / 2 C\lambda^{-n/2}\,
  4. || T λ || L 2 ( 𝐑 n ) L 2 ( 𝐑 n ) C λ - n / 2 . ||T_{\lambda}||_{L^{2}(\mathbf{R}^{n})\to L^{2}(\mathbf{R}^{n})}\leq C\lambda^% {-n/2}.

Output_power_of_an_analog_TV_transmitter.html

  1. P = 1 T 0 T i ( t ) e ( t ) d t . P=\frac{1}{T}\int_{0}^{T}i(t)\cdot e(t)dt\,\!.
  2. P = 1 T R 0 T e ( t ) 2 d t . P=\frac{1}{T\cdot R}\int_{0}^{T}e(t)^{2}dt\,\!.
  3. R R
  4. e ( t ) e(t)
  5. P n = E p 2 2 R P_{n}=\frac{E_{p}^{2}}{2\cdot R}\,\!
  6. P n = E 2 R P_{n}=\frac{E^{2}}{R}\,\!
  7. E E
  8. P t = E 2 64 R ( 4.7 ( 100 % ) 2 + 59.3 ( 73 % ) 2 ) 57 % E 2 R P_{t}=\frac{E^{2}}{64\cdot R}\cdot(4.7\cdot(100\%)^{2}+59.3\cdot(73\%)^{2})% \approx 57\%\cdot\frac{E^{2}}{R}\,\!

Outside_air_temperature.html

  1. c = 38.945 K {c}=38.945\sqrt{K}
  2. c c
  3. K K

Overcompleteness.html

  1. { ϕ i } i J \{\phi_{i}\}_{i\in J}
  2. X X
  3. X X
  4. { ϕ i } i J \{\phi_{i}\}_{i\in J}
  5. ϕ j \phi_{j}
  6. { ϕ i } i J \ { j } \{\phi_{i}\}_{i\in J\backslash\{j\}}
  7. { ϕ i } i J \{\phi_{i}\}_{i\in J}
  8. f f\in\mathcal{H}
  9. A f 2 i J | f , ϕ i | 2 B f 2 A\|f\|^{2}\leq\sum_{i\in J}|\langle f,\phi_{i}\rangle|^{2}\leq B\|f\|^{2}
  10. , \langle\cdot,\cdot\rangle
  11. A A
  12. B B
  13. A A
  14. B B
  15. A = B A=B
  16. = span { ϕ i } \mathcal{H}=\operatorname{span}\{\phi_{i}\}
  17. { α i } i = 1 \{\alpha_{i}\}_{i=1}^{\infty}
  18. { β i } i = 1 \{\beta_{i}\}_{i=1}^{\infty}
  19. \mathcal{H}
  20. { ϕ i } i = 1 = { α i } i = 1 { β i } i = 1 \{\phi_{i}\}_{i=1}^{\infty}=\{\alpha_{i}\}_{i=1}^{\infty}\cup\{\beta_{i}\}_{i=% 1}^{\infty}
  21. \mathcal{H}
  22. A = B = 2 A=B=2
  23. S S
  24. S f = i J f , ϕ i ϕ i Sf=\sum_{i\in J}\langle f,\phi_{i}\rangle\phi_{i}
  25. f f\in\mathcal{H}
  26. ϵ \epsilon
  27. F = { ϕ i } i J F=\{\phi_{i}\}_{i\in J}
  28. L 2 ( ) L^{2}(\mathbb{R})
  29. f L 2 ( ) f\in L^{2}(\mathbb{R})
  30. f - f ^ < ϵ \|f-\hat{f}\|<\epsilon
  31. N ( f , ϵ ) = { f ^ : f ^ = i = 1 k β i ϕ i , f - f ^ < ϵ } N(f,\epsilon)=\{\hat{f}:\hat{f}=\sum_{i=1}^{k}\beta_{i}\phi_{i},\|f-\hat{f}\|<\epsilon\}
  32. k F ( f , ϵ ) = inf { k : f ^ N ( f , ϵ ) } k_{F}(f,\epsilon)=\inf\{k:\hat{f}\in N(f,\epsilon)\}
  33. k ( f , ϵ ) k(f,\epsilon)
  34. F F
  35. f f
  36. f f
  37. k k
  38. L 2 ( ) L^{2}(\mathbb{R})
  39. k F ( ϵ ) = sup f L 2 ( ) { k F ( f , ϵ ) } k_{F}(\epsilon)=\sup_{f\in L^{2}(\mathbb{R})}\{k_{F}(f,\epsilon)\}
  40. G G
  41. k F ( ϵ ) < k G ( ϵ ) k_{F}(\epsilon)<k_{G}(\epsilon)
  42. F F
  43. G G
  44. ϵ \epsilon
  45. γ \gamma
  46. ϵ < γ \epsilon<\gamma
  47. k F ( ϵ ) < k G ( ϵ ) k_{F}(\epsilon)<k_{G}(\epsilon)
  48. F F
  49. G G
  50. f = A x f=Ax\,
  51. f f
  52. A A
  53. x x
  54. f f
  55. A A
  56. x x
  57. L 2 ( ) L^{2}(\mathbb{R})
  58. x x
  59. L 1 ( ) L^{1}(\mathbb{R})
  60. L 2 ( ) L^{2}(\mathbb{R})
  61. g g
  62. g g
  63. T a : L 2 ( R ) L 2 ( R ) , ( T a f ) ( x ) = f ( x - a ) T_{a}:L^{2}(R)\rightarrow L^{2}(R),(T_{a}f)(x)=f(x-a)
  64. E b : L 2 ( R ) L 2 ( R ) , ( E b f ) ( x ) = e 2 π i b x f ( x ) E_{b}:L^{2}(R)\rightarrow L^{2}(R),(E_{b}f)(x)=e^{2\pi ibx}f(x)
  65. D c : L 2 ( R ) L 2 ( R ) , ( D c f ) ( x ) = 1 c 1 2 f ( x c ) D_{c}:L^{2}(R)\rightarrow L^{2}(R),(D_{c}f)(x)=\frac{1}{c^{\frac{1}{2}}}f(% \frac{x}{c})
  66. L 2 ( R ) L^{2}(R)
  67. { E m b T n a g } m , n Z \{E_{mb}T_{na}g\}_{m,n\in Z}
  68. a , b > 0 a,b>0
  69. g L 2 ( R ) g\in L^{2}(R)
  70. a a
  71. b b
  72. { E m b T n a g } m , n Z \{E_{mb}T_{na}g\}_{m,n\in Z}
  73. L 2 ( R ) L^{2}(R)
  74. a b > 1 ab>1
  75. L 2 ( R ) L^{2}(R)
  76. a b = 1 ab=1
  77. { E m b T n a g } m , n Z \{E_{mb}T_{na}g\}_{m,n\in Z}
  78. { E m b T n a g } m , n Z \{E_{mb}T_{na}g\}_{m,n\in Z}
  79. a b < 1 ab<1
  80. { E m b / c T n a c g c } m , n Z \{E_{mb/c}T_{nac}g_{c}\}_{m,n\in Z}
  81. { E m b T n a g } m , n Z . \{E_{mb}T_{na}g\}_{m,n\in Z}.\,
  82. g g
  83. g ( x ) = e - x 2 g(x)=e^{-x^{2}}
  84. { E m b T n a g } m , n Z \{E_{mb}T_{na}g\}_{m,n\in Z}
  85. a b < 0.994 ab<0.994
  86. g ( x ) = 1 c o s h ( π x ) g(x)=\frac{1}{cosh(\pi x)}
  87. { E m b T n a g } m , n Z \{E_{mb}T_{na}g\}_{m,n\in Z}
  88. a b < 1 ab<1
  89. g ( x ) = I [ 0 , c ) ( x ) g(x)=I_{[0,c)}(x)
  90. I ( x ) I(x)
  91. { E m b T n a g } m , n Z \{E_{mb}T_{na}g\}_{m,n\in Z}
  92. a > c a>c
  93. a > 1 a>1
  94. c > 1 c>1
  95. a = 1 a=1
  96. a c 1 a\leq c\leq 1
  97. a < 1 a<1
  98. c ( 1 , 2 ) c\in(1,2)
  99. a = p q < 1 a=\frac{p}{q}<1
  100. p p
  101. q q
  102. 2 - 1 q < c < 2 2-\frac{1}{q}<c<2
  103. 3 4 < a < 1 \frac{3}{4}<a<1
  104. c = L - 1 + L ( 1 - a ) c=L-1+L(1-a)
  105. L 3 L\geq 3
  106. a < 1 a<1
  107. c > 1 c>1
  108. | c - [ c ] - 1 2 | < 1 2 - a |c-[c]-\frac{1}{2}|<\frac{1}{2}-a
  109. [ c ] [c]
  110. c c
  111. ψ \psi
  112. { 2 j 2 ψ ( 2 j x - k ) } j , k Z \{2^{\frac{j}{2}}\psi(2^{j}x-k)\}_{j,k\in Z}
  113. L 2 ( R ) L^{2}(R)
  114. j , k j,k
  115. R R
  116. L 2 ( R ) L^{2}(R)
  117. { a j 2 ψ ( a j x - k b ) } j , k Z \{a^{\frac{j}{2}}\psi(a^{j}x-kb)\}_{j,k\in Z}
  118. a > 1 a>1
  119. b > 0 b>0
  120. ψ L 2 ( R ) \psi\in L^{2}(R)
  121. ψ ^ ( γ ) \hat{\psi}(\gamma)
  122. ψ L 1 ( R ) \psi\in L^{1}(R)
  123. ψ ^ ( γ ) = R ψ ( x ) e - 2 π i x γ d x \hat{\psi}(\gamma)=\int_{R}\psi(x)e^{-2\pi ix\gamma}dx
  124. a , b a,b
  125. G 0 ( γ ) = j Z | ψ ^ ( a j γ ) | 2 G_{0}(\gamma)=\sum_{j\in Z}|\hat{\psi}(a^{j}\gamma)|^{2}
  126. G 1 ( γ ) = k 0 j Z | ψ ^ ( a j γ ) ψ ^ ( a j γ + k b ) | G_{1}(\gamma)=\sum_{k\neq 0}\sum_{j\in Z}|\hat{\psi}(a^{j}\gamma)\hat{\psi}(a^% {j}\gamma+\frac{k}{b})|
  127. B = 1 b sup | γ | [ 1 , a ] ( G 0 ( γ ) + G 1 ( γ ) ) < B=\frac{1}{b}\sup_{|\gamma|\in[1,a]}(G_{0}(\gamma)+G_{1}(\gamma))<\infty
  128. A = 1 b inf | γ | [ 1 , a ] ( G 0 ( γ ) - G 1 ( γ ) ) > 0 A=\frac{1}{b}\inf_{|\gamma|\in[1,a]}(G_{0}(\gamma)-G_{1}(\gamma))>0
  129. j Z | ψ ^ ( 2 j γ ) | 2 = A \sum_{j\in Z}|\hat{\psi}(2^{j}\gamma)|^{2}=A
  130. j = 0 ψ ^ ( 2 j γ ) ψ ^ ( 2 j ( γ + q ) ) ¯ = 0 \sum_{j=0}^{\infty}\hat{\psi}(2^{j}\gamma)\overline{\hat{\psi}(2^{j}(\gamma+q)% )}=0
  131. q q
  132. { ψ j , k } j , k Z \{\psi_{j,k}\}_{j,k\in Z}

Overfull_graph.html

  1. | E | > Δ ( G ) | V | / 2 |E|>\Delta(G)\lfloor|V|/2\rfloor
  2. | E | |E|
  3. Δ ( G ) \displaystyle\Delta(G)
  4. | V | |V|
  5. Δ ( G ) = Δ ( S ) \displaystyle\Delta(G)=\Delta(S)
  6. Δ + 1 Δ+1
  7. Δ ( G ) = Δ ( S ) \displaystyle\Delta(G)=\Delta(S)
  8. Δ ( G ) n / 3 \Delta(G)\geq n/3
  9. Δ ( G ) = Δ ( S ) \displaystyle\Delta(G)=\Delta(S)
  10. Δ n 3 \Delta\geq\frac{n}{3}
  11. Δ n 2 \Delta\geq\frac{n}{2}

Overlap_coefficient.html

  1. overlap ( X , Y ) = | X Y | min ( | X | , | Y | ) \mathrm{overlap}(X,Y)=\frac{|X\cap Y|}{\min(|X|,|Y|)}

Overring.html

  1. A B K A\subseteq B\subseteq K

Overshoot_(signal).html

  1. P O = 100 e ( - ζ π 1 - ζ 2 ) PO=100\cdot e^{\left({\frac{-\zeta\pi}{\sqrt{1-\zeta^{2}}}}\right)}
  2. ζ = ( ln P O 100 ) 2 π 2 + ( ln P O 100 ) 2 \zeta=\sqrt{\frac{(\ln\frac{PO}{100})^{2}}{\pi^{2}+(\ln\frac{PO}{100})^{2}}}

Owen's_T_function.html

  1. T ( h , a ) = 1 2 π 0 a e - 1 2 h 2 ( 1 + x 2 ) 1 + x 2 d x ( - < h , a < + ) . T(h,a)=\frac{1}{2\pi}\int_{0}^{a}\frac{e^{-\frac{1}{2}h^{2}(1+x^{2})}}{1+x^{2}% }dx\quad\left(-\infty<h,a<+\infty\right).
  2. T ( h , 0 ) = 0 T(h,0)=0
  3. T ( 0 , a ) = 1 2 π arctan ( a ) T(0,a)=\frac{1}{2\pi}\arctan(a)
  4. T ( - h , a ) = T ( h , a ) T(-h,a)=T(h,a)
  5. T ( h , - a ) = - T ( h , a ) T(h,-a)=-T(h,a)
  6. T ( h , a ) + T ( a h , 1 a ) = 1 2 ( Φ ( h ) + Φ ( a h ) ) - Φ ( h ) Φ ( a h ) if a 0 T(h,a)+T(ah,\frac{1}{a})=\frac{1}{2}\left(\Phi(h)+\Phi(ah)\right)-\Phi(h)\Phi(% ah)\quad\mbox{if}~{}\quad a\geq 0
  7. T ( h , a ) + T ( a h , 1 a ) = 1 2 ( Φ ( h ) + Φ ( a h ) ) - Φ ( h ) Φ ( a h ) - 1 2 if a < 0 T(h,a)+T(ah,\frac{1}{a})=\frac{1}{2}\left(\Phi(h)+\Phi(ah)\right)-\Phi(h)\Phi(% ah)-\frac{1}{2}\quad\mbox{if}~{}\quad a<0
  8. T ( h , 1 ) = 1 2 Φ ( h ) ( 1 - Φ ( h ) ) T(h,1)=\frac{1}{2}\Phi(h)\left(1-\Phi(h)\right)
  9. T ( 0 , x ) d x = x T ( 0 , x ) - 1 4 π ln ( 1 + x 2 ) + C \int T(0,x)\mathrm{d}x=xT(0,x)-\frac{1}{4\pi}\ln(1+x^{2})+C
  10. Φ ( x ) = 1 2 π - x exp ( - t 2 / 2 ) d y \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\exp\left(-t^{2}/2\right)% \mathrm{d}y

Oxygen_saturation_(medicine).html

  1. S p O 2 = H b O 2 H b O 2 + H b S_{\mathrm{p}}O_{\mathrm{2}}=\frac{HbO_{\mathrm{2}}}{HbO_{\mathrm{2}}+Hb}

Oxygenation_(environmental).html

  1. l n ( D O ) = A 1 + A 2 * 100 / T + A 3 * l n ( T / 100 ) + A 4 * T / 100 + S * [ B 1 + B 2 * T / 100 + B 3 * ( T / 100 ) 2 ] ln(DO)=A1+A2*100/T+A3*ln(T/100)+A4*T/100+S*[B1+B2*T/100+B3*(T/100)^{2}]

P-adic_Hodge_theory.html

  1. Rep 𝐐 p ( K ) \mathrm{Rep}_{\mathbf{Q}_{p}}(K)
  2. Rep c r i s ( K ) Rep s t ( K ) Rep d R ( K ) Rep H T ( K ) Rep 𝐐 p ( K ) \mathrm{Rep}_{cris}(K)\subsetneq\mathrm{Rep}_{st}(K)\subsetneq\mathrm{Rep}_{dR% }(K)\subsetneq\mathrm{Rep}_{HT}(K)\subsetneq\mathrm{Rep}_{\mathbf{Q}_{p}}(K)
  3. D B ( V ) = ( B 𝐐 p V ) G K D_{B}(V)=(B\otimes_{\mathbf{Q}_{p}}V)^{G_{K}}
  4. E := B G K E:=B^{G_{K}}
  5. dim E D B ( V ) = dim 𝐐 p V \dim_{E}D_{B_{\ast}}(V)=\dim_{\mathbf{Q}_{p}}V
  6. α V : B E D B ( V ) B 𝐐 p V \alpha_{V}:B_{\ast}\otimes_{E}D_{B_{\ast}}(V)\longrightarrow B_{\ast}\otimes_{% \mathbf{Q}_{p}}V
  7. H dR ( X / 𝐂 ) H ( X ( 𝐂 ) , 𝐐 ) 𝐐 𝐂 . H^{\ast}_{\mathrm{dR}}(X/\mathbf{C})\cong H^{\ast}(X(\mathbf{C}),\mathbf{Q})% \otimes_{\mathbf{Q}}\mathbf{C}.
  8. B HT := i 𝐙 𝐂 K ( i ) B_{\mathrm{HT}}:=\oplus_{i\in\mathbf{Z}}\mathbf{C}_{K}(i)
  9. B HT K gr H dR ( X / K ) B HT 𝐐 p H e ´ t ( X × K K ¯ , 𝐐 p ) B_{\mathrm{HT}}\otimes_{K}\mathrm{gr}H^{\ast}_{\mathrm{dR}}(X/K)\cong B_{% \mathrm{HT}}\otimes_{\mathbf{Q}_{p}}H^{\ast}_{\mathrm{\acute{e}t}}(X\times_{K}% \overline{K},\mathbf{Q}_{p})
  10. gr H dR \mathrm{gr}H^{\ast}_{\mathrm{dR}}
  11. B dR K H dR ( X / K ) B dR 𝐐 p H e ´ t ( X × K K ¯ , 𝐐 p ) B_{\mathrm{dR}}\otimes_{K}H^{\ast}_{\mathrm{dR}}(X/K)\cong B_{\mathrm{dR}}% \otimes_{\mathbf{Q}_{p}}H^{\ast}_{\mathrm{\acute{e}t}}(X\times_{K}\overline{K}% ,\mathbf{Q}_{p})
  12. B cris K 0 H dR ( X / K ) B cris 𝐐 p H e ´ t ( X × K K ¯ , 𝐐 p ) B_{\mathrm{cris}}\otimes_{K_{0}}H^{\ast}_{\mathrm{dR}}(X/K)\cong B_{\mathrm{% cris}}\otimes_{\mathbf{Q}_{p}}H^{\ast}_{\mathrm{\acute{e}t}}(X\times_{K}% \overline{K},\mathbf{Q}_{p})
  13. H dR ( X / K ) H^{\ast}_{\mathrm{dR}}(X/K)
  14. D B ( V ) = H dR i ( X / K ) . D_{B_{\ast}}(V)=H^{i}_{\mathrm{dR}}(X/K).
  15. B st K 0 H dR ( X / K ) B st 𝐐 p H e ´ t ( X × K K ¯ , 𝐐 p ) B_{\mathrm{st}}\otimes_{K_{0}}H^{\ast}_{\mathrm{dR}}(X/K)\cong B_{\mathrm{st}}% \otimes_{\mathbf{Q}_{p}}H^{\ast}_{\mathrm{\acute{e}t}}(X\times_{K}\overline{K}% ,\mathbf{Q}_{p})
  16. B G K B^{G_{K}}

P-adic_L-function.html

  1. L ( s , χ ) = n χ ( n ) n s = p prime 1 1 - χ ( p ) p - s L(s,\chi)=\sum_{n}\frac{\chi(n)}{n^{s}}=\prod_{p\,\text{ prime}}\frac{1}{1-% \chi(p)p^{-s}}
  2. L ( 1 - n , χ ) = - B n , χ n L(1-n,\chi)=-\frac{B_{n,\chi}}{n}
  3. n = 0 B n , χ t n n ! = a = 1 f χ ( a ) t e a t e f t - 1 \displaystyle\sum_{n=0}^{\infty}B_{n,\chi}\frac{t^{n}}{n!}=\sum_{a=1}^{f}\frac% {\chi(a)te^{at}}{e^{ft}-1}
  4. L p ( 1 - n , χ ) = ( 1 - χ ( p ) p n - 1 ) L ( 1 - n , χ ) \displaystyle L_{p}(1-n,\chi)=(1-\chi(p)p^{n-1})L(1-n,\chi)
  5. L p ( 1 - n , χ ) = ( 1 - χ ω - n ( p ) p n - 1 ) L ( 1 - n , χ ω - n ) \displaystyle L_{p}(1-n,\chi)=(1-\chi\omega^{-n}(p)p^{n-1})L(1-n,\chi\omega^{-% n})

P-derivation.html

  1. R R
  2. δ : R R \delta:R\to R
  3. δ p ( a b ) = δ p ( a ) b p + a p δ p ( b ) + p δ p ( a ) δ p ( b ) \delta_{p}(ab)=\delta_{p}(a)b^{p}+a^{p}\delta_{p}(b)+p\delta_{p}(a)\delta_{p}(b)
  4. δ p ( a + b ) = δ p ( a ) + δ p ( b ) + a p + b p - ( a + b ) p p \delta_{p}(a+b)=\delta_{p}(a)+\delta_{p}(b)+\frac{a^{p}+b^{p}-(a+b)^{p}}{p}
  5. δ p ( 1 ) = 0 \delta_{p}(1)=0
  6. R R
  7. σ : R R \sigma:R\to R
  8. σ ( x ) = x p mod p R \sigma(x)=x^{p}\mod pR
  9. ( R , δ ) (R,\delta)
  10. σ ( x ) := x p + p δ ( x ) \sigma(x):=x^{p}+p\delta(x)
  11. R = R=\mathbb{Z}
  12. δ ( x ) = x - x p p . \delta(x)=\frac{x-x^{p}}{p}.
  13. σ : R R \sigma:R\to R
  14. δ ( x ) = σ ( x ) - x p p \delta(x)=\frac{\sigma(x)-x^{p}}{p}

PA_degree.html

  1. ϕ e \phi_{e}
  2. ϕ e B \phi^{B}_{e}
  3. χ A = ϕ e B \chi_{A}=\phi^{B}_{e}
  4. f ( n ) ϕ n ( n ) f(n)\not=\phi_{n}(n)
  5. ϕ n ( n ) \phi_{n}(n)

Pachner_moves.html

  1. Δ n + 1 \Delta_{n+1}
  2. ( n + 1 ) (n+1)
  3. Δ n + 1 \partial\Delta_{n+1}
  4. N N
  5. C N C\subset N
  6. ϕ : C C Δ n + 1 \phi:C\to C^{\prime}\subset\partial\Delta_{n+1}
  7. ( N C ) ϕ ( Δ n + 1 C ) (N\setminus C)\cup_{\phi}(\partial\Delta_{n+1}\setminus C^{\prime})
  8. N N

Packing_in_a_hypergraph.html

  1. P P
  2. H H
  3. D 0 D_{0}
  4. ϵ \epsilon
  5. D 0 D_{0}
  6. x , y V x,y\in V
  7. D D 0 D\geq D_{0}
  8. D ( 1 - ϵ ) deg ( x ) D ( 1 + ϵ ) D(1-\epsilon)\leq\,\text{deg}(x)\leq D(1+\epsilon)
  9. codeg ( x , y ) ϵ D \,\text{codeg}(x,y)\leq\epsilon D
  10. n K + 1 ( 1 - o ( 1 ) ) \frac{n}{K+1}(1-o(1))
  11. ( K + 1 ) (K+1)
  12. D ( 1 + o ( 1 ) ) D(1+o(1))
  13. D D
  14. o ( D ) o(D)
  15. E H E\in H
  16. t E [ 0 , D ] t_{E}\in[0,D]
  17. E E
  18. P P
  19. P P
  20. | P | = n K + 1 |P|=\frac{n}{K+1}
  21. c c
  22. γ > 0 \gamma>0
  23. c , D 0 , ϵ c,D_{0},\epsilon
  24. ( D 0 , ϵ ) (D_{0},\epsilon)
  25. f x , H ( c ) < γ 2 f_{x,H}(c)<\gamma^{2}
  26. f x , H ( c ) f_{x,H}(c)
  27. x x
  28. P P
  29. c c
  30. x x
  31. c c
  32. γ 2 n \gamma^{2}n
  33. x x
  34. γ n \gamma n
  35. 1 - γ 1-\gamma
  36. P c P_{c}
  37. ( 1 - γ ) n (1-\gamma)n
  38. | P | ( 1 - γ ) n K + 1 |P|\geq(1-\gamma)\frac{n}{K+1}
  39. lim c lim x , H f x , H ( c ) = 0 \lim_{c\rightarrow\infty}\lim_{x,H}f_{x,H}(c)=0
  40. x x
  41. c > 0 c>0
  42. c c
  43. [ 0 , c ) [0,c)
  44. k k
  45. e - c c k k ! \frac{e^{-c}c^{k}}{k!}
  46. k k
  47. x 1 , , x k x_{1},...,x_{k}
  48. [ 0 , c ) [0,c)
  49. Q Q
  50. a a
  51. ϵ > 0 \epsilon>0
  52. K K
  53. ( 1 - ϵ ) (1-\epsilon)
  54. K K
  55. T T
  56. f ( c ) f(c)
  57. T T
  58. lim c f ( c ) = 0 \lim_{c\rightarrow\infty}f(c)=0
  59. c c
  60. lim * f x , H ( c ) = f ( c ) \lim^{*}f_{x,H}(c)=f(c)
  61. f ( c ) = 0 f(c)=0
  62. c 0 , Δ c > 0 c\geq 0,\Delta c>0
  63. Δ c \Delta c
  64. f ( c + Δ c ) - f ( c ) - ( Δ c ) f ( c ) Q + 1 f(c+\Delta c)-f(c)\approx-(\Delta c)f(c)^{Q+1}
  65. c + Δ c c+\Delta c
  66. [ c , c + Δ c ) [c,c+\Delta c)
  67. [ 0 , c ) [0,c)
  68. Δ c 0 \Delta c\rightarrow 0
  69. f ( c ) = - f ( c ) Q + 1 f^{\prime}(c)=-f(c)^{Q+1}
  70. f ( 0 ) = 1 f(0)=1
  71. f ( c ) = ( 1 + Q c ) - 1 / Q f(c)=(1+Qc)^{-1/Q}
  72. lim c f ( c ) = 0 \lim_{c\rightarrow\infty}f(c)=0
  73. lim * f x , H ( c ) = f ( c ) \lim^{*}f_{x,H}(c)=f(c)
  74. T T
  75. T = { x } T=\{x\}
  76. T T
  77. x x
  78. y T y\in T
  79. x x
  80. y T y\in T
  81. t y t_{y}
  82. t x = c t_{x}=c
  83. y T y\in T
  84. t E t_{E}
  85. y E y\in E
  86. x E x\in E
  87. E E
  88. t E < t y t_{E}<t_{y}
  89. y , z E y,z\in E
  90. z T z\in T
  91. E , E E,E^{\prime}
  92. t E , t E < t y t_{E},t_{E^{\prime}}<t_{y}
  93. y E , E y\in E,E^{\prime}
  94. | E E | > 1 |E\cup E^{\prime}|>1
  95. E E
  96. t E < t y t_{E}<t_{y}
  97. z E - { y } z\in E-\{y\}
  98. T T
  99. y y
  100. t E t_{E}
  101. y y
  102. y T y\in T
  103. x x
  104. T T
  105. x x
  106. c c
  107. f ( T , c ) f(T,c)
  108. T T
  109. f ( T , c ) f(T,c)
  110. f ( T , c ) = 1 \sum f(T,c)=1
  111. T T
  112. l i m * lim^{*}
  113. lim * f x , H ( c ) = f ( c ) \lim^{*}f_{x,H}(c)=f(c)
  114. 2 l < k < n 2\leq l<k<n
  115. M ( n , k , l ) M(n,k,l)
  116. κ \kappa
  117. { 1 , , n } \{1,...,n\}
  118. A κ A\in\kappa
  119. lim n M ( n , k , l ) ( n l ) / ( k l ) = 1 \lim_{n\rightarrow\infty}\frac{M(n,k,l)}{{n\choose l}/{k\choose l}}=1
  120. 2 l < k 2\leq l<k
  121. m ( n , k , l ) m(n,k,l)
  122. κ \kappa
  123. { 1 , , n } \{1,...,n\}
  124. A κ A\in\kappa
  125. γ \gamma
  126. v , v V v,v^{\prime}\in V
  127. O ( n D - 1 / ( k - 1 ) ) O(nD^{-1/(k-1)})
  128. O ( n D - 1 / 2 ln 3 / 2 D ) O(nD^{-1/2}\ln^{3/2}D)
  129. O ( n 1 / 2 ln 3 / 2 n ) O(n^{1/2}\ln^{3/2}n)

PageRank.html

  1. P R ( E ) . PR(E).
  2. P R ( A ) = P R ( B ) + P R ( C ) + P R ( D ) . PR(A)=PR(B)+PR(C)+PR(D).\,
  3. P R ( A ) = P R ( B ) 2 + P R ( C ) 1 + P R ( D ) 3 . PR(A)=\frac{PR(B)}{2}+\frac{PR(C)}{1}+\frac{PR(D)}{3}.\,
  4. P R ( A ) = P R ( B ) L ( B ) + P R ( C ) L ( C ) + P R ( D ) L ( D ) . PR(A)=\frac{PR(B)}{L(B)}+\frac{PR(C)}{L(C)}+\frac{PR(D)}{L(D)}.\,
  5. P R ( u ) = v B u P R ( v ) L ( v ) PR(u)=\sum_{v\in B_{u}}\frac{PR(v)}{L(v)}
  6. P R ( A ) = 1 - d N + d ( P R ( B ) L ( B ) + P R ( C ) L ( C ) + P R ( D ) L ( D ) + ) . PR(A)={1-d\over N}+d\left(\frac{PR(B)}{L(B)}+\frac{PR(C)}{L(C)}+\frac{PR(D)}{L% (D)}+\,\cdots\right).
  7. P R ( A ) = 1 - d + d ( P R ( B ) L ( B ) + P R ( C ) L ( C ) + P R ( D ) L ( D ) + ) . PR(A)=1-d+d\left(\frac{PR(B)}{L(B)}+\frac{PR(C)}{L(C)}+\frac{PR(D)}{L(D)}+\,% \cdots\right).
  8. P R ( p i ) = 1 - d N + d p j M ( p i ) P R ( p j ) L ( p j ) PR(p_{i})=\frac{1-d}{N}+d\sum_{p_{j}\in M(p_{i})}\frac{PR(p_{j})}{L(p_{j})}
  9. p 1 , p 2 , , p N p_{1},p_{2},...,p_{N}
  10. M ( p i ) M(p_{i})
  11. p i p_{i}
  12. L ( p j ) L(p_{j})
  13. p j p_{j}
  14. 𝐑 = [ P R ( p 1 ) P R ( p 2 ) P R ( p N ) ] \mathbf{R}=\begin{bmatrix}PR(p_{1})\\ PR(p_{2})\\ \vdots\\ PR(p_{N})\end{bmatrix}
  15. 𝐑 = [ ( 1 - d ) / N ( 1 - d ) / N ( 1 - d ) / N ] + d [ ( p 1 , p 1 ) ( p 1 , p 2 ) ( p 1 , p N ) ( p 2 , p 1 ) ( p i , p j ) ( p N , p 1 ) ( p N , p N ) ] 𝐑 \mathbf{R}=\begin{bmatrix}{(1-d)/N}\\ {(1-d)/N}\\ \vdots\\ {(1-d)/N}\end{bmatrix}+d\begin{bmatrix}\ell(p_{1},p_{1})&\ell(p_{1},p_{2})&% \cdots&\ell(p_{1},p_{N})\\ \ell(p_{2},p_{1})&\ddots&&\vdots\\ \vdots&&\ell(p_{i},p_{j})&\\ \ell(p_{N},p_{1})&\cdots&&\ell(p_{N},p_{N})\end{bmatrix}\mathbf{R}
  16. ( p i , p j ) \ell(p_{i},p_{j})
  17. p j p_{j}
  18. p i p_{i}
  19. i = 1 N ( p i , p j ) = 1 \sum_{i=1}^{N}\ell(p_{i},p_{j})=1
  20. t - 1 t^{-1}
  21. t t
  22. t = 0 t=0
  23. P R ( p i ; 0 ) = 1 N PR(p_{i};0)=\frac{1}{N}
  24. P R ( p i ; t + 1 ) = 1 - d N + d p j M ( p i ) P R ( p j ; t ) L ( p j ) PR(p_{i};t+1)=\frac{1-d}{N}+d\sum_{p_{j}\in M(p_{i})}\frac{PR(p_{j};t)}{L(p_{j% })}
  25. 𝐑 ( t + 1 ) = d 𝐑 ( t ) + 1 - d N 𝟏 \mathbf{R}(t+1)=d\mathcal{M}\mathbf{R}(t)+\frac{1-d}{N}\mathbf{1}
  26. 𝐑 i ( t ) = P R ( p i ; t ) \mathbf{R}_{i}(t)=PR(p_{i};t)
  27. 𝟏 \mathbf{1}
  28. N N
  29. \mathcal{M}
  30. i j = { 1 / L ( p j ) , if j links to i 0 , otherwise \mathcal{M}_{ij}=\begin{cases}1/L(p_{j}),&\mbox{if }~{}j\mbox{ links to }~{}i% \\ 0,&\mbox{otherwise}\end{cases}
  31. := ( K - 1 A ) T \mathcal{M}:=(K^{-1}A)^{T}
  32. A A
  33. K K
  34. ϵ \epsilon
  35. | 𝐑 ( t + 1 ) - 𝐑 ( t ) | < ϵ |\mathbf{R}(t+1)-\mathbf{R}(t)|<\epsilon
  36. t t\to\infty
  37. 𝐑 = d 𝐑 + 1 - d N 𝟏 \mathbf{R}=d\mathcal{M}\mathbf{R}+\frac{1-d}{N}\mathbf{1}
  38. 𝐑 = ( 𝐈 - d ) - 1 1 - d N 𝟏 \mathbf{R}=(\mathbf{I}-d\mathcal{M})^{-1}\frac{1-d}{N}\mathbf{1}
  39. 𝐈 \mathbf{I}
  40. 0 < d < 1 0<d<1
  41. \mathcal{M}
  42. \mathcal{M}
  43. 𝐑 \mathbf{R}
  44. | 𝐑 | = 1 |\mathbf{R}|=1
  45. 𝐄𝐑 = 𝟏 \mathbf{E}\mathbf{R}=\mathbf{1}
  46. 𝐄 \mathbf{E}
  47. 𝐑 = ( d + 1 - d N 𝐄 ) 𝐑 = : ^ 𝐑 \mathbf{R}=\left(d\mathcal{M}+\frac{1-d}{N}\mathbf{E}\right)\mathbf{R}=:% \widehat{\mathcal{M}}\mathbf{R}
  48. 𝐑 \mathbf{R}
  49. ^ \widehat{\mathcal{M}}
  50. x ( 0 ) x(0)
  51. ^ \widehat{\mathcal{M}}
  52. x ( t + 1 ) = ^ x ( t ) x(t+1)=\widehat{\mathcal{M}}x(t)
  53. | x ( t + 1 ) - x ( t ) | < ϵ |x(t+1)-x(t)|<\epsilon
  54. 1 - d N 𝐄 = ( 1 - d ) 𝐏𝟏 t \frac{1-d}{N}\mathbf{E}=(1-d)\mathbf{P}\mathbf{1}^{t}
  55. 𝐏 \mathbf{P}
  56. 𝐏 := 1 N 𝟏 \mathbf{P}:=\frac{1}{N}\mathbf{1}
  57. \mathcal{M}
  58. 𝐏 \mathbf{P}
  59. := + 𝒟 \mathcal{M}^{\prime}:=\mathcal{M}+\mathcal{D}
  60. 𝒟 \mathcal{D}
  61. 𝒟 := 𝐏𝐃 t \mathcal{D}:=\mathbf{P}\mathbf{D}^{t}
  62. 𝐃 i = { 1 , if L ( p i ) = 0 0 , otherwise \mathbf{D}_{i}=\begin{cases}1,&\mbox{if }~{}L(p_{i})=0\\ 0,&\mbox{otherwise}\end{cases}
  63. \mathcal{M}
  64. 𝐑 power = 𝐑 iterative | 𝐑 iterative | = 𝐑 algebraic | 𝐑 algebraic | \mathbf{R}_{\textrm{power}}=\frac{\mathbf{R}_{\textrm{iterative}}}{|\mathbf{R}% _{\textrm{iterative}}|}=\frac{\mathbf{R}_{\textrm{algebraic}}}{|\mathbf{R}_{% \textrm{algebraic}}|}
  65. D = 1 2 | E | [ d e g ( p 1 ) d e g ( p 2 ) d e g ( p N ) ] D={1\over 2|E|}\begin{bmatrix}deg(p_{1})\\ deg(p_{2})\\ \vdots\\ deg(p_{N})\end{bmatrix}
  66. d e g ( p i ) deg(p_{i})
  67. p i p_{i}
  68. Y = 1 N 𝟏 Y={1\over N}\mathbf{1}
  69. 1 - d 1 + d Y - D 1 R - D 1 Y - D 1 , {1-d\over 1+d}\|Y-D\|_{1}\leq\|R-D\|_{1}\leq\|Y-D\|_{1},
  70. O ( log n / ϵ ) O(\log n/\epsilon)
  71. ϵ \epsilon
  72. 1 - ϵ 1-\epsilon
  73. O ( log n / ϵ ) O(\sqrt{\log n}/\epsilon)
  74. O ( log n / ϵ ) O(\sqrt{\log n/\epsilon})

Paired_difference_test.html

  1. D ¯ = 1 n i ( Y i 2 - Y i 1 ) = 1 n i Y i 2 - 1 n i Y i 1 = Y ¯ 2 - Y ¯ 1 , \bar{D}=\frac{1}{n}\sum_{i}(Y_{i2}-Y_{i1})=\frac{1}{n}\sum_{i}Y_{i2}-\frac{1}{% n}\sum_{i}Y_{i1}=\bar{Y}_{2}-\bar{Y}_{1},
  2. var ( D ¯ ) = var ( Y ¯ 2 - Y ¯ 1 ) = var ( Y ¯ 2 ) + var ( Y ¯ 1 ) - 2 c o v ( Y ¯ 1 , Y ¯ 2 ) = σ 1 2 / n + σ 2 2 / n - 2 σ 1 σ 2 corr ( Y i 1 , Y i 2 ) / n , \begin{array}[]{ccl}{\rm var}(\bar{D})&=&{\rm var}(\bar{Y}_{2}-\bar{Y}_{1})\\ &=&{\rm var}(\bar{Y}_{2})+{\rm var}(\bar{Y}_{1})-2{\rm cov}(\bar{Y}_{1},\bar{Y% }_{2})\\ &=&\sigma_{1}^{2}/n+\sigma_{2}^{2}/n-2\sigma_{1}\sigma_{2}{\rm corr}(Y_{i1},Y_% {i2})/n,\end{array}
  3. Y ¯ 2 - Y ¯ 1 σ 1 2 / n + σ 2 2 / n , \frac{\bar{Y}_{2}-\bar{Y}_{1}}{\sqrt{\sigma_{1}^{2}/n+\sigma_{2}^{2}/n}},
  4. P ( Y ¯ 2 - Y ¯ 1 σ 1 2 / n + σ 2 2 / n > 1.64 ) = P ( Y ¯ 2 - Y ¯ 1 S > 1.64 σ 1 2 / n + σ 2 2 / n / S ) = P ( Y ¯ 2 - Y ¯ 1 - δ + δ S > 1.64 σ 1 2 / n + σ 2 2 / n / S ) = P ( Y ¯ 2 - Y ¯ 1 - δ S > 1.64 σ 1 2 / n + σ 2 2 / n / S - δ / S ) = 1 - Φ ( 1.64 σ 1 2 / n + σ 2 2 / n / S - δ / S ) , \begin{array}[]{lcl}P\left(\frac{\bar{Y}_{2}-\bar{Y}_{1}}{\sqrt{\sigma_{1}^{2}% /n+\sigma_{2}^{2}/n}}>1.64\right)&=&P\left(\frac{\bar{Y}_{2}-\bar{Y}_{1}}{S}>1% .64\sqrt{\sigma_{1}^{2}/n+\sigma_{2}^{2}/n}/S\right)\\ &=&P\left(\frac{\bar{Y}_{2}-\bar{Y}_{1}-\delta+\delta}{S}>1.64\sqrt{\sigma_{1}% ^{2}/n+\sigma_{2}^{2}/n}/S\right)\\ &=&P\left(\frac{\bar{Y}_{2}-\bar{Y}_{1}-\delta}{S}>1.64\sqrt{\sigma_{1}^{2}/n+% \sigma_{2}^{2}/n}/S-\delta/S\right)\\ &=&1-\Phi(1.64\sqrt{\sigma_{1}^{2}/n+\sigma_{2}^{2}/n}/S-\delta/S),\end{array}
  5. 1 - Φ ( 1.64 - δ / S ) . 1-\Phi(1.64-\delta/S).
  6. σ 1 2 / n + σ 2 2 / n / S = σ 1 2 + σ 2 2 σ 1 2 + σ 2 2 - 2 σ 1 σ 2 ρ > 1 where ρ := corr ( Y i 1 , Y i 2 ) . \sqrt{\sigma_{1}^{2}/n+\sigma_{2}^{2}/n}/S=\sqrt{\frac{\sigma_{1}^{2}+\sigma_{% 2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{1}\sigma_{2}\rho}}>1~{}~{}\,% \text{where}~{}~{}\rho:={\rm corr}(Y_{i1},Y_{i2}).
  7. ρ \rho
  8. Y i j = μ j + α i + ϵ i j Y_{ij}=\mu_{j}+\alpha_{i}+\epsilon_{ij}
  9. cov ( Y i 1 , Y i 2 ) = var ( α i ) . {\rm cov}(Y_{i1},Y_{i2})={\rm var}(\alpha_{i}).
  10. μ H A \mu_{HA}
  11. μ H B \mu_{HB}
  12. μ L A \mu_{LA}
  13. μ L B \mu_{LB}
  14. p H A p_{HA}
  15. p H B p_{HB}
  16. p L A p_{LA}
  17. p L B p_{LB}
  18. E X ¯ A = μ H A p H A p H A + p L A + μ L A p L A p H A + p L A , E\bar{X}_{A}=\mu_{HA}\frac{p_{HA}}{p_{HA}+p_{LA}}+\mu_{LA}\frac{p_{LA}}{p_{HA}% +p_{LA}},
  19. E X ¯ B = μ H B p H B p H B + p L B + μ L B p L B p H B + p L B . E\bar{X}_{B}=\mu_{HB}\frac{p_{HB}}{p_{HB}+p_{LB}}+\mu_{LB}\frac{p_{LB}}{p_{HB}% +p_{LB}}.
  20. μ H A p H A p H A + p L A - μ H B p H B p H B + p L B + μ L A p L A p H A + p L A - μ L B p L B p H B + p L B . \mu_{HA}\frac{p_{HA}}{p_{HA}+p_{LA}}-\mu_{HB}\frac{p_{HB}}{p_{HB}+p_{LB}}+\mu_% {LA}\frac{p_{LA}}{p_{HA}+p_{LA}}-\mu_{LB}\frac{p_{LB}}{p_{HB}+p_{LB}}.
  21. p H A = ( p H A + p L A ) ( p H A + p H B ) p_{HA}=(p_{HA}+p_{LA})(p_{HA}+p_{HB})
  22. p H B = ( p H B + p L B ) ( p H A + p H B ) . p_{HB}=(p_{HB}+p_{LB})(p_{HA}+p_{HB}).

Pairwise_Stone_space.html

  1. ( X , τ 1 , τ 2 ) \scriptstyle(X,\tau_{1},\tau_{2})
  2. ( X , τ ) \scriptstyle(X,\tau)
  3. ( X , τ , τ * ) \scriptstyle(X,\tau,\tau^{*})
  4. τ * \scriptstyle\tau^{*}
  5. τ \scriptstyle\tau
  6. ( X , τ 1 , τ 2 ) \scriptstyle(X,\tau_{1},\tau_{2})
  7. ( X , τ 1 ) \scriptstyle(X,\tau_{1})
  8. ( X , τ 2 ) \scriptstyle(X,\tau_{2})

Pairwise_summation.html

  1. O ( ε log n ) O(\varepsilon\sqrt{\log n})
  2. S n = i = 1 n x i S_{n}=\sum_{i=1}^{n}x_{i}
  3. S n + E n S_{n}+E_{n}
  4. E n E_{n}
  5. | E n | ε log 2 n 1 - ε log 2 n i = 1 n | x i | |E_{n}|\leq\frac{\varepsilon\log_{2}n}{1-\varepsilon\log_{2}n}\sum_{i=1}^{n}|x% _{i}|
  6. | E n | / | S n | |E_{n}|/|S_{n}|
  7. | E n | | S n | ε log 2 n 1 - ε log 2 n ( i = 1 n | x i | | i = 1 n x i | ) . \frac{|E_{n}|}{|S_{n}|}\leq\frac{\varepsilon\log_{2}n}{1-\varepsilon\log_{2}n}% \left(\frac{\sum_{i=1}^{n}|x_{i}|}{\left|\sum_{i=1}^{n}x_{i}\right|}\right).
  8. n \sqrt{n}
  9. n n\to\infty
  10. 1 - ε log 2 n 1-\varepsilon\log_{2}n
  11. ε log 2 n \varepsilon\log_{2}n
  12. O ( ε n ) O(\varepsilon n)
  13. O ( ε n ) O(\varepsilon\sqrt{n})
  14. O ( ε log n ) O(\varepsilon\sqrt{\log n})

Paleoneurology.html

  1. E Q = Brainweight 0.12 Bodyweight 0.66 EQ=\operatorname{Brainweight}\over\operatorname{0.12}\cdot\operatorname{% Bodyweight}^{0.66}

Paleosalinity.html

  1. Δ ρ ρ = α Δ T - β Δ S \frac{\Delta\rho}{\rho}=\alpha\Delta T-\beta\Delta S
  2. α \alpha
  3. β \beta
  4. β α \frac{\beta}{\alpha}
  5. β α \frac{\beta}{\alpha}
  6. β α \frac{\beta}{\alpha}

Paneitz_operator.html

  1. P = Δ 2 + δ { ( n - 2 ) J - 4 V } d + ( n - 4 ) Q P=\Delta^{2}+\delta\left\{(n-2)J-4V\cdot\right\}d+(n-4)Q
  2. ( - 4 | V | 2 + n J 2 + 2 Δ J ) / 4 (-4|V|^{2}+nJ^{2}+2\Delta J)/4\,
  3. Ω n / 2 + 2 P ( g ) ϕ = P ( Ω 2 g ) Ω n / 2 - 2 ϕ . \Omega^{n/2+2}P(g)\phi=P(\Omega^{2}g)\Omega^{n/2-2}\phi.\,

Papal_conclave,_1513.html

  1. m ( m + r - 14 ) ! r ! ( r - 13 ) ! ( m + r - 1 ) ! \frac{m(m+r-14)!r!}{(r-13)!(m+r-1)!}
  2. m ( m + r - 18 ) ! r ! ( r - 17 ) ! ( m + r - 1 ) ! . \frac{m(m+r-18)!r!}{(r-17)!(m+r-1)!}.

Papyrus_125.html

  1. 𝔓 \mathfrak{P}

Papyrus_126.html

  1. 𝔓 \mathfrak{P}

Papyrus_127.html

  1. 𝔓 \mathfrak{P}

Par_yield.html

  1. 100 c 1 + R ( 0 , 1 ) + 100 c ( 1 + R ( 0 , 2 ) ) 2 + + 100 + 100 c ( 1 + R ( 0 , n ) ) n = 100. \frac{100c}{1+R(0,1)}+\frac{100c}{(1+R(0,2))^{2}}+\cdots+\frac{100+100c}{(1+R(% 0,n))^{n}}=100.
  2. c = 1 - P n k = 1 n P k c=\frac{1-P_{n}}{\sum_{k=1}^{n}P_{k}}
  3. R ( 0 , n ) R(0,n)
  4. n n
  5. P n P_{n}
  6. n n

Paradox_of_the_pesticides.html

  1. H \displaystyle H
  2. d H d t \displaystyle\frac{dH}{dt}
  3. P = r c and H = m a c P=\frac{r}{c}\quad\,\text{and}\quad H=\frac{m}{ac}
  4. d H d t \displaystyle\frac{dH}{dt}
  5. P = r - q c and H = m + q a c P=\frac{r-q}{c}\quad\,\text{and}\quad H=\frac{m+q}{ac}

Paranematic_susceptibility.html

  1. P 2 = η 𝐇 2 \langle P_{2}\rangle=\eta\mathbf{H}^{2}
  2. η \eta
  3. ( T - T C * ) - 1 (T-T^{*}_{C})^{-1}
  4. T C * T^{*}_{C}

Paranormal_operator.html

  1. T 2 x T x 2 \|T^{2}x\|\geq\|Tx\|^{2}\,

Parikh's_theorem.html

  1. Σ = { a 1 , a 2 , , a k } \Sigma=\{a_{1},a_{2},\ldots,a_{k}\}
  2. p : Σ * k p:\Sigma^{*}\to\mathbb{N}^{k}
  3. p ( w ) = ( | w | a 1 , | w | a 2 , , | w | a k ) p(w)=(|w|_{a_{1}},|w|_{a_{2}},\ldots,|w|_{a_{k}})
  4. | w | a i |w|_{a_{i}}
  5. a i a_{i}
  6. w w
  7. k \mathbb{N}^{k}
  8. u 0 + u 1 , , u m = { u 0 + t 1 u 1 + + t m u m t 1 , , t m } u_{0}+\langle u_{1},\ldots,u_{m}\rangle=\{u_{0}+t_{1}u_{1}+\ldots+t_{m}u_{m}% \mid t_{1},\ldots,t_{m}\in\mathbb{N}\}
  9. u 0 , , u m u_{0},\ldots,u_{m}
  10. k \mathbb{N}^{k}
  11. L L
  12. P ( L ) P(L)
  13. L L
  14. P ( L ) = { p ( w ) w L } P(L)=\{p(w)\mid w\in L\}
  15. P ( L ) P(L)
  16. S S
  17. S S

Parity_function.html

  1. n n
  2. f : { 0 , 1 } n { 0 , 1 } f:\{0,1\}^{n}\to\{0,1\}
  3. f ( x ) = 1 f(x)=1
  4. x { 0 , 1 } n x\in\{0,1\}^{n}
  5. f f
  6. f ( x ) = x 1 x 2 x n f(x)=x_{1}\oplus x_{2}\oplus\dots\oplus x_{n}
  7. k k
  8. exp ( Ω ( n 1 k - 1 ) ) \exp\left(\Omega\left(n^{\frac{1}{k-1}}\right)\right)
  9. k k
  10. exp ( O ( n 1 k - 1 ) ) \exp\left(O\left(n^{\frac{1}{k-1}}\right)\right)
  11. f : { 0 , 1 } ω { 0 , 1 } f\colon\{0,1\}^{\omega}\to\{0,1\}
  12. w w
  13. v v
  14. f ( w ) = f ( v ) f(w)=f(v)
  15. w w
  16. v v
  17. 2 𝔠 2^{\mathfrak{c}}
  18. { 0 , 1 } ω \{0,1\}^{\omega}
  19. { 0 , 1 } \{0,1\}
  20. \approx
  21. w v w\approx v
  22. w w
  23. v v
  24. f f
  25. f - 1 [ 0 ] f^{-1}[0]
  26. Σ 1 1 \Sigma^{1}_{1}

Partial_application.html

  1. f : ( X × Y × Z ) N \scriptstyle f\colon(X\times Y\times Z)\to N
  2. partial ( f ) : ( Y × Z ) N \scriptstyle\,\text{partial}(f)\colon(Y\times Z)\to N
  3. f p a r t i a l ( 2 , 3 ) f_{partial}(2,3)

Partially_ordered_ring.html

  1. \leq
  2. x y x\leq y
  3. x + z y + z x+z\leq y+z
  4. 0 x 0\leq x
  5. 0 y 0\leq y
  6. 0 x y 0\leq x\cdot y
  7. x , y , z A x,y,z\in A
  8. ( A , ) (A,\leq)
  9. A A
  10. ( A , ) (A,\leq)
  11. \leq
  12. ( A , ) (A,\leq)
  13. \leq
  14. 0 x 0\leq x
  15. P + P P P+P\subseteq P
  16. P P P P\cdot P\subseteq P
  17. P ( - P ) = { 0 } P\cap(-P)=\{0\}
  18. 0 S 0\in S
  19. S ( - S ) = { 0 } S\cap(-S)=\{0\}
  20. S + S S S+S\subseteq S
  21. S S S S\cdot S\subseteq S
  22. \leq
  23. x y x\leq y
  24. y - x S y-x\in S
  25. ( A , ) (A,\leq)
  26. | x | |x|
  27. x ( - x ) x\vee(-x)
  28. x y x\vee y
  29. | x y | | x | | y | |x\cdot y|\leq|x|\cdot|y|
  30. ( A , ) (A,\leq)
  31. x y = 0 x\wedge y=0
  32. 0 z 0\leq z
  33. z x y = x z y = 0 zx\wedge y=xz\wedge y=0
  34. x , y , z A x,y,z\in A
  35. 𝒞 ( X ) \mathcal{C}(X)
  36. 𝒞 ( X ) \mathcal{C}(X)
  37. [ f + g ] ( x ) = f ( x ) + g ( x ) [f+g](x)=f(x)+g(x)
  38. [ f g ] ( x ) = f ( x ) g ( x ) [fg](x)=f(x)\cdot g(x)
  39. [ f g ] ( x ) = f ( x ) g ( x ) . [f\wedge g](x)=f(x)\wedge g(x).
  40. 𝒞 ( X ) \mathcal{C}(X)
  41. 𝒞 ( X ) \mathcal{C}(X)
  42. | x y | = | x | | y | |xy|=|x||y|
  43. ( A , ) (A,\leq)
  44. x , y , z A x,y,z\in A
  45. x y x\leq y
  46. x - y 0 x-y\leq 0
  47. x y x\leq y
  48. 0 z 0\leq z
  49. x z y z xz\leq yz
  50. z x z y zx\leq zy
  51. 0 1 0\leq 1
  52. | x y | = | x | | y | |xy|=|x||y|
  53. | x + y | | x | + | y | |x+y|\leq|x|+|y|
  54. ( A , ) (A,\leq)
  55. 0 1 0\neq 1
  56. \wedge

Particle_physics_experiments.html

  1. f ( x ) = x 1 - ( x c ) 2 exp { - χ ( 1 - ( x c ) 2 ) } for x > 0. f(x)=x\cdot\sqrt{1-\left(\frac{x}{c}\right)^{2}}\exp\left\{-\chi\cdot\left(1-% \left(\frac{x}{c}\right)^{2}\right)\right\}\,\text{ for }x>0.
  2. f ( x ) = x [ 1 - ( x c ) 2 ] p exp { - χ ( 1 - ( x c ) 2 ) } f(x)=x\cdot\left[1-\left(\frac{x}{c}\right)^{2}\right]^{p}\exp\left\{-\chi% \cdot\left(1-\left(\frac{x}{c}\right)^{2}\right)\right\}

Pascal's_law.html

  1. Δ P = ρ g ( Δ h ) \Delta P=\rho g(\Delta h)\,
  2. < m a t h > Δ P <math>\Delta P

Passenger_load_factor.html

  1. ( 5 f l i g h t s ) ( 200 k m / f l i g h t ) ( 60 p a s s e n g e r s ) ( 5 f l i g h t s ) ( 200 k m / f l i g h t ) ( 100 s e a t s ) = 60 , 000 p a s s e n g e r k m 100 , 000 s e a t k m = 0.6 = 60 % \frac{(5\ flights)(200\ km/flight)(60\ passengers)}{(5\ flights)(200\ km/% flight)(100\ seats)}=\frac{60,000\ passenger\cdot km}{100,000\ seat\cdot km}=0% .6=60\%

Pasting_lemma.html

  1. A B Int A A\setminus B\subseteq\operatorname{Int}A
  2. B A Int B B\setminus A\subseteq\operatorname{Int}B
  3. X , Y X,Y
  4. A = X Y A=X\cup Y
  5. f : A B f:A\to B
  6. f - 1 ( U ) X f^{-1}(U)\cap X
  7. f - 1 ( U ) Y f^{-1}(U)\cap Y
  8. f - 1 ( U ) f^{-1}(U)
  9. \Box
  10. A = X 1 X 2 X 3 A=X_{1}\cup X_{2}\cup X_{3}\cup\cdots
  11. X 1 , X 2 , X 3 X_{1},X_{2},X_{3}\ldots
  12. ι : Z R \iota:Z\rightarrow R
  13. X 1 , X 2 , X 3 X_{1},X_{2},X_{3}\ldots

Pauli–Lubanski_pseudovector.html

  1. W W
  2. S S
  3. ε μ ν ρ σ \varepsilon_{\mu\nu\rho\sigma}
  4. J ν ρ J^{\nu\rho}
  5. P σ P^{\sigma}
  6. 𝐖 = ( 𝐉 𝐩 ) . \mathbf{W}=\star(\mathbf{J}\wedge\mathbf{p}).
  7. P μ W μ = 0 , P^{\mu}W_{\mu}=0,
  8. [ P μ , W ν ] = 0 , \left[P^{\mu},W^{\nu}\right]=0,
  9. [ J μ ν , W ρ ] = i ( g ρ ν W μ - g ρ μ W ν ) , \left[J^{\mu\nu},W^{\rho}\right]=i\left(g^{\rho\nu}W^{\mu}-g^{\rho\mu}W^{\nu}% \right),
  10. [ W μ , W ν ] = - i ϵ μ ν ρ σ W ρ P σ . \left[W_{\mu},W_{\nu}\right]=-i\epsilon_{\mu\nu\rho\sigma}W^{\rho}P^{\sigma}.
  11. W 2 = W μ W μ = - m 2 s ( s + 1 ) , W^{2}=W_{\mu}W^{\mu}=-m^{2}s(s+1),
  12. s s
  13. m m
  14. [ u v e c , u W ] = m [ u v e c , u J ] [u^{\prime}vec^{\prime},u^{\prime}W^{\prime}]=m[u^{\prime}vec^{\prime},u^{% \prime}J^{\prime}]
  15. W μ W μ = - m 2 J J . W_{\mu}W^{\mu}=-m^{2}\vec{J}\cdot\vec{J}.
  16. [ u v e c , u P ] [u^{\prime}vec^{\prime},u^{\prime}P^{\prime}]
  17. [ u v e c , u P ] [u^{\prime}vec^{\prime},u^{\prime}P^{\prime}]
  18. [ u v e c , u S ] [u^{\prime}vec^{\prime},u^{\prime}S^{\prime}]
  19. W 0 = P S 3 , W 1 = m S 1 , W 2 = m S 2 , W 3 = E c 2 S 3 , W_{0}=PS_{3},\qquad W_{1}=mS_{1},\qquad W_{2}=mS_{2},\qquad W_{3}=\frac{E}{c^{% 2}}S_{3},
  20. E 2 = P 2 c 2 + m 2 c 4 E^{2}=P^{2}c^{2}+m^{2}c^{4}
  21. [ W 1 , W 2 ] = i h 2 π ( ( E / c 2 ) 2 - ( P / c ) 2 ) S 3 , [ W 2 , S 3 ] = i h 2 π W 1 , [ S 3 , W 1 ] = i h 2 π W 2 . [W_{1},W_{2}]=\tfrac{ih}{2\pi}((E/c^{2})^{2}-(P/c)^{2})S_{3},\qquad[W_{2},S_{3% }]=\tfrac{ih}{2\pi}W_{1},\qquad[S_{3},W_{1}]=\tfrac{ih}{2\pi}W_{2}.
  22. [ W 1 , W 2 ] = i h 2 π m 2 S 3 . [W_{1},W_{2}]=\tfrac{ih}{2\pi}m^{2}S_{3}.
  23. W 2 = W μ W μ = - E 2 ( ( K 2 - J 1 ) 2 + ( K 1 + J 2 ) 2 ) = def - E 2 ( A 2 + B 2 ) . W^{2}=W_{\mu}W^{\mu}=-E^{2}((K_{2}-J_{1})^{2}+(K_{1}+J_{2})^{2})\stackrel{% \mathrm{def}}{=}-E^{2}(A^{2}+B^{2}).
  24. P P
  25. W W
  26. [ u v e c , u W ] [u^{\prime}vec^{\prime},u^{\prime}W^{\prime}]
  27. [ u v e c , u P ] [u^{\prime}vec^{\prime},u^{\prime}P^{\prime}]
  28. [ u v e c , u W ] [u^{\prime}vec^{\prime},u^{\prime}W^{\prime}]
  29. I S O ( 2 ) ISO(2)
  30. [ u v e c , u W ] [u^{\prime}vec^{\prime},u^{\prime}W^{\prime}]
  31. S O ( 3 ) SO(3)
  32. [ u v e c , u W ] [u^{\prime}vec^{\prime},u^{\prime}W^{\prime}]
  33. [ u v e c , u P ] [u^{\prime}vec^{\prime},u^{\prime}P^{\prime}]
  34. [ u v e c , u W ] [u^{\prime}vec^{\prime},u^{\prime}W^{\prime}]
  35. [ u v e c , u W ] [u^{\prime}vec^{\prime},u^{\prime}W^{\prime}]
  36. [ u v e c , u J ] · [ u v e c , u P ] [u^{\prime}vec^{\prime},u^{\prime}J^{\prime}]·[u^{\prime}vec^{\prime},u^{% \prime}P^{\prime}]
  37. W 0 / P , W^{0}/P,
  38. W 0 = - J P . W^{0}=-\vec{J}\cdot\vec{P}.
  39. W W
  40. Z Z

Pea_galaxy.html

  1. L L_{\odot}

Pebble_motion_problems.html

  1. G = ( V , E ) G=(V,E)
  2. n n
  3. P = { 1 , , k } P=\{1,\ldots,k\}
  4. k < n k<n
  5. S : P V S:P\rightarrow V
  6. S ( i ) S ( j ) S(i)\neq S(j)
  7. i j i\neq j
  8. m = ( p , u , v ) m=(p,u,v)
  9. p p
  10. u u
  11. v v
  12. S 0 S_{0}
  13. S + S_{+}
  14. S 0 S_{0}
  15. S + S_{+}

Penrose_tiling.html

  1. ( B L B S ) = ( 1 1 1 0 ) ( A L A S ) . \begin{pmatrix}B_{L}\\ B_{S}\end{pmatrix}=\begin{pmatrix}1&1\\ 1&0\end{pmatrix}\begin{pmatrix}A_{L}\\ A_{S}\end{pmatrix}\,.
  2. ( φ A L φ A S ) = ( 1 1 1 0 ) ( B L B S ) = ( 2 1 1 1 ) ( A L A S ) , \begin{pmatrix}\varphi A_{L}\\ \varphi A_{S}\end{pmatrix}=\begin{pmatrix}1&1\\ 1&0\end{pmatrix}\begin{pmatrix}B_{L}\\ B_{S}\end{pmatrix}=\begin{pmatrix}2&1\\ 1&1\end{pmatrix}\begin{pmatrix}A_{L}\\ A_{S}\end{pmatrix}\,,
  3. φ n ( A L A S ) = ( 2 1 1 1 ) n ( A L A S ) . \varphi^{n}\begin{pmatrix}A_{L}\\ A_{S}\end{pmatrix}=\begin{pmatrix}2&1\\ 1&1\end{pmatrix}^{n}\begin{pmatrix}A_{L}\\ A_{S}\end{pmatrix}\,.
  4. ( 2 1 1 1 ) n = ( F 2 n + 1 F 2 n F 2 n F 2 n - 1 ) , \begin{pmatrix}2&1\\ 1&1\end{pmatrix}^{n}=\begin{pmatrix}F_{2n+1}&F_{2n}\\ F_{2n}&F_{2n-1}\end{pmatrix}\,,

Pentagram_map.html

  1. P 1 , P 3 , P 5 , P_{1},P_{3},P_{5},\ldots
  2. Q 2 , Q 4 , Q 6 , Q_{2},Q_{4},Q_{6},\ldots
  3. Q 4 Q_{4}
  4. ( P 1 P 5 ) (P_{1}P_{5})
  5. ( P 3 P 7 ) (P_{3}P_{7})
  6. P Q P\to Q
  7. Q k Q_{k}
  8. P k P_{k}
  9. P k P_{k}
  10. P N + k P_{N+k}
  11. T 2 T^{2}
  12. H H
  13. T 2 ( H ) T^{2}(H)
  14. H H^{\prime}
  15. T ( H ) T(H)
  16. H H^{\prime}
  17. H H
  18. K a n Ka^{n}
  19. K > 0 K>0
  20. 0 < a < 1 0<a<1
  21. x , y , z x,y,z
  22. x + y + z = 180. x+y+z=180.
  23. ( x , y , z ) (x,y,z)
  24. ( x , y , z ) (x,y,z)
  25. 7 7
  26. 6 6
  27. 3 3
  28. t 1 , t 2 , t 3 , t 4 t_{1},t_{2},t_{3},t_{4}
  29. X = ( t 1 - t 2 ) ( t 3 - t 4 ) ( t 1 - t 3 ) ( t 2 - t 4 ) . X=\frac{(t_{1}-t_{2})(t_{3}-t_{4})}{(t_{1}-t_{3})(t_{2}-t_{4})}.
  30. F 1 , , F 2 N F_{1},\ldots,F_{2N}
  31. t 1 , t 2 , t 3 , t 4 t_{1},t_{2},t_{3},t_{4}
  32. x 1 , , x 2 n x_{1},\ldots,x_{2n}
  33. x 1 , y 1 , x 2 , y 2 , x_{1},y_{1},x_{2},y_{2},\ldots
  34. x 1 , x 2 , x 3 , x 4 , . x_{1},x_{2},x_{3},x_{4},\ldots\,.
  35. F 2 N F^{2N}
  36. F 2 N F^{2N}
  37. V 1 , V 2 , V 3 , \ldots V_{1},V_{2},V_{3},\ldots
  38. R 3 R^{3}
  39. V i + 3 = a i V i + 2 + b i V i + 1 + V i V_{i+3}=a_{i}V_{i+2}+b_{i}V_{i+1}+V_{i}
  40. a 1 , b 1 , a 2 , b 2 , a_{1},b_{1},a_{2},b_{2},\ldots
  41. B A B\circ A
  42. A A
  43. B B
  44. A ( x 1 , , x 2 N ) = ( a 1 , , a 2 N ) A(x_{1},\ldots,x_{2N})=(a_{1},\ldots,a_{2N})
  45. B ( x 1 , , x 2 N ) = ( b 1 , , b 2 N ) B(x_{1},\ldots,x_{2N})=(b_{1},\ldots,b_{2N})
  46. a 2 k - 1 = ( 1 - x 2 k + 1 x 2 k + 2 ) ( 1 - x 2 k - 3 x 2 k - 2 ) x 2 k + 0 a_{2k-1}=\frac{(1-x_{2k+1}x_{2k+2})}{(1-x_{2k-3}x_{2k-2})}x_{2k+0}
  47. a 2 k + 0 = ( 1 - x 2 k - 3 x 2 k - 2 ) ( 1 - x 2 k + 1 x 2 k + 2 ) x 2 k - 1 a_{2k+0}=\frac{(1-x_{2k-3}x_{2k-2})}{(1-x_{2k+1}x_{2k+2})}x_{2k-1}
  48. b 2 k + 1 = ( 1 - x 2 k - 2 x 2 k - 1 ) ( 1 - x 2 k + 2 x 2 k + 3 ) x 2 k + 0 b_{2k+1}=\frac{(1-x_{2k-2}x_{2k-1})}{(1-x_{2k+2}x_{2k+3})}x_{2k+0}
  49. b 2 k + 0 = ( 1 - x 2 k + 2 x 2 k + 3 ) ( 1 - x 2 k - 2 x 2 k - 1 ) x 2 k - 1 b_{2k+0}=\frac{(1-x_{2k+2}x_{2k+3})}{(1-x_{2k-2}x_{2k-1})}x_{2k-1}
  50. F 2 N F^{2N}
  51. A ( P ) A(P)
  52. B ( A ( P ) ) B(A(P))
  53. A ( B ( A ( P ) ) ) A(B(A(P)))
  54. O N = x 1 x 3 x 2 N - 1 O_{N}=x_{1}x_{3}\cdots x_{2N-1}
  55. E N = x 2 x 4 x 2 N E_{N}=x_{2}x_{4}\cdots x_{2N}
  56. O k = x 1 x 5 x 9 x 2 N - 3 + x 3 x 7 x 11 x 2 N - 1 O_{k}=x_{1}x_{5}x_{9}\cdots x_{2N-3}+x_{3}x_{7}x_{11}\cdots x_{2N-1}
  57. E k = x 2 x 6 x 10 x 2 N - 2 + x 4 x 8 x 12 x 2 N E_{k}=x_{2}x_{6}x_{10}\cdots x_{2N-2}+x_{4}x_{8}x_{12}\cdots x_{2N}
  58. O k O_{k}
  59. E k E_{k}
  60. f = O N E N f=O_{N}E_{N}
  61. f = O N E N f=O_{N}E_{N}
  62. - x 1 x 5 x 6 x 7 -x_{1}x_{5}x_{6}x_{7}
  63. + x 1 x 2 x 3 x 7 x 8 x 9 +x_{1}x_{2}x_{3}x_{7}x_{8}x_{9}
  64. O k O_{k}
  65. E k E_{k}
  66. O N O_{N}
  67. E N E_{N}
  68. O k ( P ) = E k ( P ) O_{k}(P)=E_{k}(P)
  69. { , } \{\cdot,\cdot\}
  70. { O i , O j } = { O i , E j } = { E i , E j } = 0 \{O_{i},O_{j}\}=\{O_{i},E_{j}\}=\{E_{i},E_{j}\}=0
  71. x 1 , y 1 , x 2 , y 2 , . x_{1},y_{1},x_{2},y_{2},\ldots\,.
  72. { x i , x i + 1 } = - x i x i + 1 \{x_{i},x_{i+1}\}=-x_{i}\,x_{i+1}
  73. { x i , x i - 1 } = x i x i - 1 \{x_{i},x_{i-1}\}=x_{i}\,x_{i-1}
  74. { y i , y i + 1 } = y i y i + 1 \{y_{i},y_{i+1}\}=y_{i}\,y_{i+1}
  75. { y i , y i - 1 } = - y i y i - 1 \{y_{i},y_{i-1}\}=-y_{i}\,y_{i-1}
  76. { x i , x j } = { y i , y j } = { x i , y j } = 0 \{x_{i},x_{j}\}=\{y_{i},y_{j}\}=\{x_{i},y_{j}\}=0
  77. i , j . i,j.
  78. f f
  79. H ( f ) = ( x i + 1 f / x i + 1 - x i - 1 f / x i - 1 ) x i / x i + ( y i - 1 f / y i - 1 - y i + 1 f / y i + 1 ) y i / y i H(f)=(x_{i+1}\partial f/\partial x_{i+1}-x_{i-1}\partial f/\partial x_{i-1})x_% {i}\partial/\partial x_{i}+(y_{i-1}\partial f/\partial y_{i-1}-y_{i+1}\partial f% /\partial y_{i+1})y_{i}\partial/\partial y_{i}
  80. H ( f ) g = { f , g } H(f)g=\{f,g\}
  81. g g
  82. H ( f ) H(f)
  83. O n = x 1 x n O_{n}=x_{1}\cdots x_{n}
  84. E n = y 1 y n E_{n}=y_{1}\cdots y_{n}
  85. { O n , f } = { E n , f } = 0 \{O_{n},f\}=\{E_{n},f\}=0
  86. O n O_{n}
  87. E n E_{n}
  88. a 1 b 1 + a 2 b 2 = a 3 b 3 a_{1}b_{1}+a_{2}b_{2}=a_{3}b_{3}
  89. a i a_{i}
  90. b i b_{i}
  91. a 2 , b 2 , a 3 , b 3 a_{2},b_{2},a_{3},b_{3}
  92. T T
  93. π : T R 2 \pi:T\to R^{2}
  94. T T
  95. T T
  96. G = π ( T ) G=\pi(T)
  97. G G
  98. G G
  99. G G
  100. v = A D / B C v=AD/BC
  101. C : R - > R 2 C:R->R^{2}
  102. C ( x - t ) C(x-t)
  103. C ( x + t ) C(x+t)
  104. C t ( x ) C_{t}(x)
  105. C t ( x ) C_{t}(x)
  106. C 0 ( x ) C_{0}(x)

Percolation_critical_exponents.html

  1. p p\,\!
  2. p c p_{c}\,\!
  3. p c p_{c}\,\!
  4. | p - p c | |p-p_{c}|\,\!
  5. { σ , τ } \{\sigma,\,\tau\}\,\!
  6. { d f , ν } \{d\text{f},\,\nu\}\,\!
  7. p c p_{c}\,\!
  8. d f d\text{f}\,\!
  9. D D\,\!
  10. M ( L ) L d f M(L)\sim L^{d\text{f}}\,\!
  11. p = p c p=p_{c}\,\!
  12. L L\to\infty\,\!
  13. τ \tau\,\!
  14. n s n_{s}\,\!
  15. s s\,\!
  16. n s s - τ n_{s}\sim s^{-\tau}\,\!
  17. s s\to\infty\,\!
  18. r \vec{r}\,\!
  19. g ( r ) | r | - 2 ( d - d f ) g(\vec{r})\sim|\vec{r}|^{-2(d-d\text{f})}\,\!
  20. g ( r ) | r | - d + ( 2 - η ) g(\vec{r})\sim|\vec{r}|^{-d+(2-\eta)}\,\!
  21. η \eta\,\!
  22. Ω \Omega\,\!
  23. n s s - τ ( 1 + const × s - Ω ) n_{s}\sim s^{-\tau}(1+\,\text{const}\times s^{-\Omega})\,\!
  24. s s\to\infty\,\!
  25. d min d_{\mathrm{min}}
  26. \langle\ell\rangle
  27. r r
  28. r d min \langle\ell\rangle\sim r^{d_{\mathrm{min}}}
  29. d d_{\ell}
  30. \ell
  31. M d M\sim\ell^{d_{\ell}}
  32. d f d_{f}
  33. d = d f / d min d_{\ell}=d_{f}/d_{\mathrm{min}}
  34. p c p_{c}\,\!
  35. ν \nu\,\!
  36. ξ \xi\,\!
  37. ξ | p - p c | - ν \xi\sim|p-p_{c}|^{-\nu}\,\!
  38. s max s_{\max}\,\!
  39. n s s - τ f ( s / s max ) n_{s}\sim s^{-\tau}f(s/s_{\max})\,\!
  40. σ \sigma
  41. s max | p - p c | - 1 / σ s_{\max}\sim|p-p_{c}|^{-1/\sigma}\,\!
  42. s max ξ d f s_{\max}\sim\xi^{d\text{f}}\,\!
  43. σ = 1 / ν d f \sigma=1/\nu d\text{f}\,\!
  44. n c n_{c}
  45. α \alpha
  46. n c A + B ( p - p c ) + C ( p - p c ) 2 + D ± | p - p c | 2 - α + n_{c}\sim A+B(p-p_{c})+C(p-p_{c})^{2}+D_{\pm}|p-p_{c}|^{2-\alpha}+\cdots
  47. D ± D_{\pm}
  48. P P\,\!
  49. P P_{\infty}
  50. P | p - p c | β P\sim|p-p_{c}|^{\beta}\,\!
  51. β \beta\,\!
  52. P \ P
  53. S = s s 2 n s / p c | p - p c | - γ S=\sum_{s}s^{2}n_{s}/p_{c}\sim|p-p_{c}|^{-\gamma}\,\!
  54. γ \gamma\,\!
  55. τ = d d f + 1 \tau=\frac{d}{d\text{f}}+1\,\!
  56. d f = d - β ν d\text{f}=d-\frac{\beta}{\nu}\,\!
  57. 2 d f - d = 2 - η 2d\text{f}-d=2-\eta\,\!
  58. { σ , τ } \{\sigma,\tau\}
  59. α = 2 - τ - 1 σ \alpha=2-\frac{\tau-1}{\sigma}\,\!
  60. β = τ - 2 σ \beta=\frac{\tau-2}{\sigma}\,\!
  61. γ = 3 - τ σ \gamma=\frac{3-\tau}{\sigma}\,\!
  62. ν = τ - 1 σ d \nu=\frac{\tau-1}{\sigma d}\,\!
  63. δ = 1 τ - 2 \delta=\frac{1}{\tau-2}\,\!
  64. { d f , ν } \{d\text{f},\nu\}
  65. α = 2 - ν d \alpha=2-\nu d\,\!
  66. β = ν ( d - d f ) \beta=\nu(d-d\text{f})\,\!
  67. γ = ν ( 2 d f - d ) \gamma=\nu(2d\text{f}-d)\,\!
  68. σ = 1 ν d f \sigma=\frac{1}{\nu d\text{f}}\,\!
  69. 2 2
  70. 3 3
  71. 4 4
  72. 5 5
  73. 6 < v a r > ε < / v a r > < r e f n a m e = " E s s a m 80 " > [ u c i t e j o u r n a l , u t i t l e = P e r c o l a t i o n t h e o r y , u l a s t = E s s a m , u f i r s t = J . W . , u y e a r = 1980 , u p a g e = 833 , u j o u r n a l = R e p . P r o g . P h y s . , u v o l u m e = 43 , u i s s u e = 7 , u u r l = h t t p : / / s t a c k s . i o p . o r g / 0034 - 4885 / 43 / i = 7 / a = 001 , u b i b c o d e = 1980 R P P h 43..833 E , u d o i = 10.1088 / 0034 - 4885 / 43 / 7 / 001 ] < / r e f > 6–<var>ε</var><refname="Essam80">[u^{\prime}citejournal^{\prime},u^{\prime}% title=Percolationtheory^{\prime},u^{\prime}last=Essam^{\prime},u^{\prime}first% =J.W.^{\prime},u^{\prime}year=1980^{\prime},u^{\prime}page=833^{\prime},u^{% \prime}journal=Rep.Prog.Phys.^{\prime},u^{\prime}volume=43^{\prime},u^{\prime}% issue=7^{\prime},u^{\prime}url=http://stacks.iop.org/0034-4885/43/i=7/a=001^{% \prime},u^{\prime}bibcode=1980RPPh...43..833E^{\prime},u^{\prime}doi=10.1088/0% 034-4885/43/7/001^{\prime}]</ref>
  74. 6 + 6+
  75. d f d\text{f}
  76. t t
  77. d b d\text{b}
  78. d min d\text{min}