wpmath0000016_4

Draft:Lie's_formula.html

  1. p p
  2. M M
  3. X X
  4. φ φ
  5. φ ( exp ( t X ) p ) = k = 0 t k k ! X k φ ( p ) \varphi(\mathrm{exp}(tX)p)=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}X^{k}\varphi(p)
  6. φ φ
  7. φ φ
  8. X X
  9. e x p exp
  10. X X
  11. τ τ
  12. φ φ
  13. exp ( τ X ) \mathrm{exp}(\tau X)
  14. τ 𝐑 τ∈\mathbf{R}
  15. exp ( τ X ) p \mathrm{exp}(\tau X)p
  16. τ τ
  17. p p
  18. X X
  19. p p
  20. exp ( τ X ) \mathrm{exp}(\tau X)
  21. τ τ
  22. 𝐑 \mathbf{R}
  23. X X
  24. X X
  25. X X
  26. X X

Draft:List_of_shape_topics_in_various_fields.html

  1. λ = 3.570 \scriptstyle{\lambda_{\infty}=3.570}
  2. log ( 2 ) log ( 3 ) \textstyle{\frac{\log(2)}{\log(3)}}
  3. log ( φ ) log ( 2 ) = log ( 1 + 5 ) log ( 2 ) - 1 \textstyle{\frac{\log(\scriptstyle\varphi)}{\log(2)}=\frac{\log(1+\sqrt{5})}{% \log(2)}-1}
  4. log ( 2 ) log ( 8 3 ) \textstyle{\frac{\log(2)}{\log(\tfrac{8}{3})}}
  5. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  6. log ( 5 ) log ( 10 ) = 1 - log ( 2 ) log ( 10 ) \textstyle{\frac{\log(5)}{\log(10)}=1-\frac{\log(2)}{\log(10)}}
  7. log ( 1 + 2 ) \log{(1+\sqrt{2})}
  8. - log ( 2 ) log ( 1 - γ 2 ) \textstyle{-\frac{\log(2)}{\log(\frac{1-\gamma}{2})}}
  9. γ l m - 1 \gamma\,l_{m-1}
  10. l m - 1 = ( 1 - γ ) m - 1 / 2 m - 1 \scriptstyle l_{m-1}=(1-\gamma)^{m-1}/2^{m-1}
  11. γ = 1 / 3 \scriptstyle\gamma=1/3
  12. γ \scriptstyle\gamma
  13. 0 < D < 1 \scriptstyle 0\,<\,D\,<\,1
  14. 1 \textstyle{1}
  15. 1 / 2 2 n 1/2^{2n}
  16. 2 + log ( 1 / 2 ) log ( 2 ) = 1 \textstyle{2+\frac{\log(1/2)}{\log(2)}=1}
  17. f ( x ) = n = 0 s ( 2 n x ) 2 n \textstyle{f(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}}}
  18. s ( x ) s(x)
  19. f ( x ) = n = 0 w n s ( 2 n x ) \textstyle{f(x)=\sum_{n=0}^{\infty}{w^{n}s(2^{n}x)}}
  20. w = 1 / 2 \scriptstyle{w=1/2}
  21. 2 + l o g ( w ) / l o g ( 2 ) 2+log(w)/log(2)
  22. w w
  23. [ 1 / 2 , 1 ] \scriptstyle{\left[1/2,1\right]}
  24. 2 | α | 3 s + | α | 4 s = 1 2|\alpha|^{3s}+|\alpha|^{4s}=1
  25. 1 12 \scriptstyle{1\mapsto 12}
  26. 2 13 \scriptstyle{2\mapsto 13}
  27. 3 1 \scriptstyle{3}\mapsto 1
  28. α \alpha
  29. z 3 - z 2 - z - 1 = 0 z^{3}-z^{2}-z-1=0
  30. 2 log ( 3 ) log ( 7 ) \textstyle{2\frac{\log(3)}{\log(7)}}
  31. 3 log ( φ ) log ( 3 + 13 2 ) \textstyle{3\frac{\log(\varphi)}{\log\left(\frac{3+\sqrt{13}}{2}\right)}}
  32. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  33. 2 log ( 27 - 3 78 3 + 27 + 3 78 3 3 ) log ( 2 ) , or root of 2 x - 1 = 2 ( 2 - x ) / 2 \begin{aligned}&\displaystyle\textstyle{\frac{2\log\left(\frac{\sqrt[3]{27-3% \sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right)}{\log(2)}},\\ &{}^{\,\text{or root of}}\\ &\displaystyle 2^{x}-1=2^{(2-x)/2}\\ \end{aligned}
  34. log ( 4 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}}
  35. log ( 4 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}}
  36. log ( 4 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}}
  37. log ( 4 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}}
  38. log ( 4 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}}
  39. log ( 5 ) log ( 3 ) \textstyle{\frac{\log(5)}{\log(3)}}
  40. log ( 5 ) log ( 3 ) \textstyle{\frac{\log(5)}{\log(3)}}
  41. log ( 10 3 ) log ( 5 ) \textstyle{\frac{\log(\frac{10}{3})}{\log(\sqrt{5})}}
  42. 2 - log ( 2 ) log ( 2 ) = 3 2 \textstyle{2-\frac{\log(\sqrt{2})}{\log(2)}=\frac{3}{2}}
  43. f ( x ) = k = 1 sin ( 2 k x ) 2 k \textstyle{f(x)=\sum_{k=1}^{\infty}\frac{\sin(2^{k}x)}{\sqrt{2}^{k}}}
  44. f : [ 0 , 1 ] \scriptstyle{f:[0,1]\to\mathbb{R}}
  45. f ( x ) = k = 1 sin ( b k x ) a k \textstyle{f(x)=\sum_{k=1}^{\infty}\frac{\sin(b^{k}x)}{a^{k}}}
  46. 1 < a < 2 1<a<2
  47. b > 1 b>1
  48. 2 - log ( a ) / log ( b ) \scriptstyle{2-\log(a)/\log(b)}
  49. log ( 8 ) log ( 4 ) = 3 2 \textstyle{\frac{\log(8)}{\log(4)}=\frac{3}{2}}
  50. log ( 1 + 73 - 6 87 3 + 73 + 6 87 3 3 ) log ( 2 ) \textstyle{\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{% 87}}}{3}\right)}{\log(2)}}
  51. log ( 1 + 73 - 6 87 3 + 73 + 6 87 3 3 ) log ( 2 ) \textstyle{\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{% 87}}}{3}\right)}{\log(2)}}
  52. log ( 3 ) log ( 2 ) \textstyle{\frac{\log(3)}{\log(2)}}
  53. log ( 3 ) log ( 2 ) \textstyle{\frac{\log(3)}{\log(2)}}
  54. log ( 3 ) log ( 2 ) \textstyle{\frac{\log(3)}{\log(2)}}
  55. log ( 3 ) log ( 2 ) \textstyle{\frac{\log(3)}{\log(2)}}
  56. log ( 4 ) log ( 2 ) \textstyle{\frac{\log(4)}{\log(2)}}
  57. log φ log φ φ = φ \textstyle{\frac{\log{\varphi}}{\log{\sqrt[\varphi]{\varphi}}}=\varphi}
  58. r r
  59. r 2 r^{2}
  60. r = 1 / φ 1 / φ \scriptstyle{r=1/\varphi^{1/\varphi}}
  61. φ \scriptstyle{\varphi}
  62. ( r 2 ) φ + r φ = 1 \scriptstyle{({r^{2}})^{\varphi}+r^{\varphi}=1}
  63. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  64. 1 + log 2 log 3 \textstyle{1+\frac{\log 2}{\log 3}}
  65. 1 + log k ( k + 1 2 ) \scriptstyle{1+\log_{k}\left(\frac{k+1}{2}\right)}
  66. log ( 6 ) log ( 3 ) \textstyle{\frac{\log(6)}{\log(3)}}
  67. 3 log ( φ ) log ( 1 + 2 ) \textstyle{3\frac{\log(\varphi)}{\log(1+\sqrt{2})}}
  68. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  69. ( 1 / 3 ) s + ( 1 / 2 ) s + ( 2 / 3 ) s = 1 \scriptstyle{(1/3)^{s}+(1/2)^{s}+(2/3)^{s}=1}
  70. n n
  71. c n c_{n}
  72. s s
  73. k = 1 n c k s = 1 \scriptstyle{\sum_{k=1}^{n}c_{k}^{s}=1}
  74. 1 + log 3 log 5 \textstyle{1+\frac{\log 3}{\log 5}}
  75. 1 + log k ( k + 1 2 ) \scriptstyle{1+\log_{k}\left(\frac{k+1}{2}\right)}
  76. z n + 1 = a + b z n e x p [ i [ k - p / ( 1 + z n 2 ) ] ] \scriptstyle{z_{n+1}=a+bz_{n}exp[i[k-p/(1+\lfloor z_{n}\rfloor^{2})]]}
  77. log ( 50 ) log ( 10 ) \textstyle{\frac{\log(50)}{\log(10)}}
  78. 4 log ( 2 ) log ( 5 ) \textstyle{\frac{4\log(2)}{\log(5)}}
  79. log ( 7 ) log ( 3 ) \textstyle{\frac{\log(7)}{\log(3)}}
  80. log ( 4 ) log ( 2 ( 1 + cos ( 85 ) ) ) \textstyle{\frac{\log(4)}{\log(2(1+\cos(85^{\circ})))}}
  81. log ( 4 ) log ( 2 ( 1 + cos ( a ) ) ) [ 1 , 2 ] \scriptstyle{\frac{\log(4)}{\log(2(1+\cos(a)))}}\in[1,2]
  82. log ( 3 0.63 + 2 0.63 ) log 2 \textstyle{\frac{\log{(3^{0.63}+2^{0.63})}}{\log{2}}}
  83. p × q \scriptstyle{p\times q}
  84. p q \scriptstyle{p\leq q}
  85. log ( k = 1 p n k a ) log p \scriptstyle{\frac{\log{\left(\sum_{k=1}^{p}n_{k}^{a}\right)}}{\log{p}}}
  86. a = log p l o g q \scriptstyle{a=\frac{\log{p}}{log{q}}}
  87. n k n_{k}
  88. k t h k^{th}
  89. log ( 6 ) log ( 1 + φ ) \textstyle{\frac{\log(6)}{\log(1+\varphi)}}
  90. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  91. 6 ( 1 / 3 ) s + 5 ( 1 / 3 3 ) s = 1 \scriptstyle{6(1/3)^{s}+5{(1/3\sqrt{3})}^{s}=1}
  92. 1 / 3 1/3
  93. 1 / 3 3 \scriptstyle{1/{3\sqrt{3}}}
  94. log ( 8 ) log ( 3 ) \textstyle{\frac{\log(8)}{\log(3)}}
  95. log ( 8 ) log ( 3 ) \textstyle{\frac{\log(8)}{\log(3)}}
  96. log ( 4 ) log ( 3 ) + log ( 2 ) log ( 3 ) = log ( 8 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}+\frac{\log(2)}{\log(3)}=\frac{\log(8)}{\log% (3)}}
  97. D i m H ( F × G ) = D i m H ( F ) + D i m H ( G ) Dim_{H}(F\times G)=Dim_{H}(F)+Dim_{H}(G)
  98. 2 \textstyle{2}
  99. 2 \textstyle{2}
  100. 2 \textstyle{2}
  101. 2 \textstyle{2}
  102. 2 \textstyle{2}
  103. 2 \textstyle{2}
  104. log ( 2 ) log ( 2 ) = 2 \textstyle{\frac{\log(2)}{\log(\sqrt{2})}=2}
  105. log ( 4 ) log ( 2 ) = 2 \textstyle{\frac{\log(4)}{\log(2)}=2}
  106. 7 ( 1 / 3 ) s + 6 ( 1 / 3 3 ) s = 1 \scriptstyle{7({1/3})^{s}+6({1/3\sqrt{3}})^{s}=1}
  107. 1 / 3 3 \scriptstyle{1/3\sqrt{3}}
  108. log ( 4 ) log ( 2 ) = 2 \textstyle{\frac{\log(4)}{\log(2)}=2}
  109. log ( 4 ) log ( 2 ) = 2 \textstyle{\frac{\log(4)}{\log(2)}=2}
  110. log ( 2 ) log ( 2 / 2 ) = 2 \textstyle{\frac{\log(2)}{\log(2/\sqrt{2})}=2}
  111. log ( 4 ) log ( 2 ) = 2 \textstyle{\frac{\log(4)}{\log(2)}=2}
  112. σ \sigma
  113. log ( 5 ) log ( 2 ) \textstyle{\frac{\log(5)}{\log(2)}}
  114. log ( 20 ) log ( 2 + φ ) \textstyle{\frac{\log(20)}{\log(2+\varphi)}}
  115. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  116. log ( 13 ) log ( 3 ) \textstyle{\frac{\log(13)}{\log(3)}}
  117. log ( 32 ) log ( 4 ) = 5 2 \textstyle{\frac{\log(32)}{\log(4)}=\frac{5}{2}}
  118. log ( 16 ) log ( 3 ) \textstyle{\frac{\log(16)}{\log(3)}}
  119. n log ( 2 ) log ( 3 ) \scriptstyle{n\frac{\log(2)}{\log(3)}}
  120. log ( 7 6 - 1 3 ) log ( 2 - 1 ) \textstyle{\frac{\log(\frac{\sqrt{7}}{6}-\frac{1}{3})}{\log(\sqrt{2}-1)}}
  121. 2 - 1 \scriptstyle\sqrt{2}-1
  122. log ( 12 ) log ( 1 + φ ) \textstyle{\frac{\log(12)}{\log(1+\varphi)}}
  123. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  124. log ( 6 ) log ( 2 ) \textstyle{\frac{\log(6)}{\log(2)}}
  125. log ( 6 ) log ( 2 ) \textstyle{\frac{\log(6)}{\log(2)}}
  126. log ( 6 ) log ( 2 ) \textstyle{\frac{\log(6)}{\log(2)}}
  127. log ( 20 ) log ( 3 ) \textstyle{\frac{\log(20)}{\log(3)}}
  128. log ( 20 ) log ( 3 ) = 2.7268 \scriptstyle{\frac{\log(20)}{\log(3)}=2.7268}
  129. log ( 8 ) log ( 2 ) = 3 \textstyle{\frac{\log(8)}{\log(2)}=3}
  130. log ( 8 ) log ( 2 ) = 3 \textstyle{\frac{\log(8)}{\log(2)}=3}
  131. log ( 8 ) log ( 2 ) = 3 \textstyle{\frac{\log(8)}{\log(2)}=3}
  132. log ( 8 ) log ( 2 ) = 3 \textstyle{\frac{\log(8)}{\log(2)}=3}
  133. 3 \textstyle{3}
  134. E ( C 1 s + C 2 s ) = 1 \scriptstyle{E(C_{1}^{s}+C_{2}^{s})=1}
  135. E ( C 1 ) = 0.5 \scriptstyle{E(C_{1})=0.5}
  136. E ( C 2 ) = 0.3 \scriptstyle{E(C_{2})=0.3}
  137. C 1 C_{1}
  138. C 2 C_{2}
  139. s s
  140. E ( C 1 s + C 2 s ) = 1 \scriptstyle{E(C_{1}^{s}+C_{2}^{s})=1}
  141. E ( X ) E(X)
  142. X X
  143. s + 1 = 12 * 2 - ( s + 1 ) - 6 * 3 - ( s + 1 ) s+1=12*2^{-(s+1)}-6*3^{-(s+1)}
  144. log ( 4 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}}
  145. 4 3 \textstyle{\frac{4}{3}}
  146. 4 3 \textstyle{\frac{4}{3}}
  147. 4 3 \textstyle{\frac{4}{3}}
  148. 2 - 1 2 \textstyle{2-\frac{1}{2}}
  149. f ( x + h ) - f ( x ) f(x+h)-f(x)
  150. α \alpha
  151. = h 2 α =h^{2\alpha}
  152. 2 - α 2-\alpha
  153. 5 3 \textstyle{\frac{5}{3}}
  154. log ( 9 * 0.75 ) log ( 3 ) \textstyle{\frac{\log(9*0.75)}{\log(3)}}
  155. log ( 9 p ) log ( 3 ) \textstyle{\frac{\log(9p)}{\log(3)}}
  156. 91 48 \textstyle{\frac{91}{48}}
  157. log ( 2 ) log ( 2 ) = 2 \textstyle{\frac{\log(2)}{\log(\sqrt{2})}=2}
  158. log ( 13 ) log ( 3 ) \textstyle{\frac{\log(13)}{\log(3)}}
  159. 3 - 1 2 \textstyle{3-\frac{1}{2}}
  160. f : 2 - > \scriptstyle{f:\mathbb{R}^{2}->\mathbb{R}}
  161. ( x , y ) (x,y)
  162. h h
  163. k k
  164. f ( x + h , y + k ) - f ( x , y ) \scriptstyle{f(x+h,y+k)-f(x,y)}
  165. h 2 + k 2 \scriptstyle{\sqrt{h^{2}+k^{2}}}
  166. α \alpha
  167. ( h 2 + k 2 ) α (h^{2}+k^{2})^{\alpha}
  168. 3 - α 3-\alpha
  169. ( 0 , 2 ) \textstyle{\in(0,2)}
  170. { p , q } \{p,q\}
  171. { p } \{p\}
  172. { q } \{q\}
  173. { p , q } \{p,q\}
  174. 1 / p + 1 / q > 1 / 2 1/p+1/q>1/2
  175. 1 / p + 1 / q = 1 / 2 1/p+1/q=1/2
  176. 1 / p + 1 / q < 1 / 2 1/p+1/q<1/2
  177. { p } \{p\}
  178. { q } \{q\}
  179. { 3 } , { 4 } , { 5 } , { 5 2 } \{3\},\{4\},\{5\},\{\frac{5}{2}\}
  180. { 6 } \{6\}
  181. { p , q , r } \{p,q,r\}
  182. { p , q } \{p,q\}
  183. { p } \{p\}
  184. { r } \{r\}
  185. { q , r } \{q,r\}
  186. { p , q , r } \{p,q,r\}
  187. { p , q } , { q , r } \{p,q\},\{q,r\}
  188. sin ( π p ) sin ( π r ) - cos ( π q ) \sin\left(\frac{\pi}{p}\right)\sin\left(\frac{\pi}{r}\right)-\cos\left(\frac{% \pi}{q}\right)
  189. > 0 >0
  190. = 0 =0
  191. < 0 <0
  192. χ \chi
  193. χ = V + F - E - C \chi=V+F-E-C
  194. { p , q , r , s } \{p,q,r,s\}
  195. { p , q , r } \{p,q,r\}
  196. { p , q } \{p,q\}
  197. { p } \{p\}
  198. { s } \{s\}
  199. { r , s } \{r,s\}
  200. { q , r , s } \{q,r,s\}
  201. { p , q , r , s } \{p,q,r,s\}
  202. { p , q , r } \{p,q,r\}
  203. { q , r , s } \{q,r,s\}
  204. cos 2 ( π q ) sin 2 ( π p ) + cos 2 ( π r ) sin 2 ( π s ) \frac{\cos^{2}\left(\frac{\pi}{q}\right)}{\sin^{2}\left(\frac{\pi}{p}\right)}+% \frac{\cos^{2}\left(\frac{\pi}{r}\right)}{\sin^{2}\left(\frac{\pi}{s}\right)}
  205. < 1 <1
  206. = 1 =1
  207. > 1 >1
  208. ( n + 1 k + 1 ) {{n+1}\choose{k+1}}
  209. 2 n - k ( n k ) 2^{n-k}{n\choose k}
  210. 2 k + 1 ( n k + 1 ) 2^{k+1}{n\choose{k+1}}
  211. \infty

Draft:Locally_free_sheaf_and_an_algebraic_vector_bundle.html

  1. E ~ = Γ ( - , E ) \tilde{E}=\Gamma(-,E)
  2. 𝒪 X \mathcal{O}_{X}
  3. E ~ U i 𝒪 U i n . \tilde{E}_{U_{i}}\simeq\mathcal{O}_{U_{i}}^{\oplus n}.
  4. F | U i 𝒪 U i n . F|_{U_{i}}\simeq\mathcal{O}_{U_{i}}^{\oplus n}.
  5. g i j Aut ( Γ ( U i U j , 𝒪 ) ) , g_{ij}\in\operatorname{Aut}(\Gamma(U_{i}\cap U_{j},\mathcal{O})),
  6. g i k g k j = g i j . g_{ik}\circ g_{kj}=g_{ij}.
  7. V Spec Sym ( V * ) V\mapsto\operatorname{Spec}\operatorname{Sym}(V^{*})

Draft:Main_theorem_of_elimination_theory.html

  1. p : 𝐏 R Spec R p:\mathbf{P}_{R}\to\operatorname{Spec}R
  2. X 𝐏 R X\subset\mathbf{P}_{R}
  3. R [ x 0 , , x n ] R[x_{0},\dots,x_{n}]
  4. Z d = { y Spec R | I y ( x 0 , , x n ) d } Z_{d}=\{y\in\operatorname{Spec}R|I_{y}\not\supset(x_{0},\dots,x_{n})^{d}\}
  5. I y I_{y}
  6. p ( X ) = d Z d \textstyle p(X)=\cap_{d}Z_{d}
  7. Z d Z_{d}
  8. x i x_{i}
  9. x 0 i 0 x n i n f x_{0}^{i_{0}}\cdots x_{n}^{i_{n}}f
  10. i 0 + + i n + deg f = d i_{0}+\dots+i_{n}+\operatorname{deg}f=d
  11. x i x_{i}
  12. y Z d M ( y ) y\in Z_{d}\Leftrightarrow M(y)
  13. < q <q\Leftrightarrow
  14. q × q q\times q

Draft:Matsusaka's_big_theorem.html

  1. L m L^{m}

Draft:Moduli_stack_of_principal_bundles.html

  1. 𝐅 q \mathbf{F}_{q}
  2. Bun G \operatorname{Bun}_{G}
  3. 𝐅 q \mathbf{F}_{q}
  4. R R\mapsto
  5. X × 𝐅 q Spec R X\times_{\mathbf{F}_{q}}\operatorname{Spec}R
  6. 𝐅 q \mathbf{F}_{q}
  7. Bun G \operatorname{Bun}_{G}
  8. Bun G ( 𝐅 q ) \operatorname{Bun}_{G}(\mathbf{F}_{q})
  9. Bun G \operatorname{Bun}_{G}

Draft:Multidimensional_system_simulation.html

  1. y ( 𝐱 , t ) y(\mathbf{x},t)
  2. y ˙ ( 𝐱 , t ) + L { y ( 𝐱 , t ) } = v ( 𝐱 , t ) , 𝐱 𝐕 y ( 𝐱 , 0 ) = y i ( 𝐱 ) , 𝐱 𝐕 f b { y ( 𝐱 , t ) } = ϕ ( 𝐱 , t ) , 𝐱 𝐒 \begin{aligned}\displaystyle\dot{y}(\,\textbf{x},t)+L\left\{y(\,\textbf{x},t)% \right\}&\displaystyle=v(\,\textbf{x},t),\,\textbf{x}\in{\,\textbf{V}}\\ \displaystyle y(\,\textbf{x},0)&\displaystyle={y}_{i}(\,\textbf{x}),\,\textbf{% x}\in{\,\textbf{V}}\\ \displaystyle{f}_{b}\left\{y(\,\textbf{x},t)\right\}&\displaystyle=\phi(\,% \textbf{x},t),\,\textbf{x}\in{\,\textbf{S}}\end{aligned}
  3. 𝐱 \mathbf{x}
  4. V V
  5. L L
  6. v ( 𝐱 , t ) v(\mathbf{x},t)
  7. Φ ( 𝐱 , t ) Φ(\mathbf{x},t)
  8. c c
  9. λ λ
  10. y ˙ ( x , t ) + 1 c ( λ y ( x , t ) ) = 0 , x 0 x x 1 y ( x 0 , t ) = y b ( t ) y ( x 1 , t ) = 0 \begin{aligned}\displaystyle\dot{y}(x,t)+\frac{1}{c}\big(\lambda{y^{\prime}}(x% ,t)\big)^{\prime}&\displaystyle=0,x_{0}\leq x\leq x_{1}\\ \displaystyle y({x}_{0},t)&\displaystyle={y}_{b}(t)\\ \displaystyle y({x}_{1},t)&\displaystyle=0\end{aligned}
  11. c c
  12. λ λ
  13. y y
  14. [ u F r a c t i o n , u 1 , u 100 ] [u^{\prime}Fraction^{\prime},u^{\prime}1^{\prime},u^{\prime}100^{\prime}]
  15. a = λ / c a=λ/c
  16. y ˙ ( x , t ) - a y ′′ ( x , t ) = 0 , x 0 x x 1 \dot{y}(x,t)-ay^{\prime\prime}(x,t)=0,x_{0}\leq x\leq x_{1}
  17. T { y ( x , t ) } \displaystyle T\left\{y(x,t)\right\}
  18. T { y ′′ ( x , t ) } = β μ a y b ( t ) - β μ 2 a y ¯ ( β μ , t ) T\left\{y^{\prime\prime}(x,t)\right\}=\frac{\beta_{\mu}}{\sqrt{a}}y_{b}(t)-% \frac{\beta_{\mu}^{2}}{a}\bar{y}(\beta_{\mu},t)
  19. y ¯ ˙ ( β μ , t ) = - β μ 2 y ¯ ( β μ , t ) + a β μ y b ( t ) = f ( y ¯ , y b ) \dot{\bar{y}}(\beta_{\mu},t)=-\beta_{\mu}^{2}\bar{y}(\beta_{\mu},t)+\sqrt{a}% \beta_{\mu}y_{b}(t)=f(\bar{y},y_{b})
  20. y ( x , t ) y(x,t)
  21. T - 1 { y ¯ ( β μ , t ) } = y ( x , t ) = μ = 1 1 N μ y ¯ ( β μ , t ) K ( x , β μ ) T^{-1}\left\{\bar{y}(\beta_{\mu},t)\right\}=y(x,t)=\sum_{\mu=1}^{\infty}\frac{% 1}{N_{\mu}}\bar{y}(\beta_{\mu},t)K(x,\beta_{\mu})
  22. K ( x , β < s u b > μ ) K(x,β<sub>μ)

Draft:Multiplicity_theory.html

  1. 𝐞 I ( M ) \mathbf{e}_{I}(M)
  2. F ( t ) = 1 d a d - i ( 1 - t ) d + r ( t ) . F(t)=\sum_{1}^{d}{a_{d-i}\over(1-t)^{d}}+r(t).
  3. a d - i a_{d-i}
  4. 𝐞 ( M ) = a 0 \mathbf{e}(M)=a_{0}

Draft:Multirate_Filter_Bank_and_Multidimensional_Directional_Filter_Banks.html

  1. M M
  2. M t h M^{th}
  3. M M
  4. x ( n ) M = x ( M . n ) {x(n)}_{\downarrow{}M}=x(M.n)
  5. X ( z ) M = 1 M m = 0 M - 1 X ( W M m . Z 1 M ) X(z)_{\downarrow M}=\frac{1}{M}\sum_{m=0}^{M-1}X(W_{M}^{m}.Z^{\frac{1}{M}})
  6. W M = e - j 2 π M W^{M}=e^{-j\frac{2\pi{}}{M}}
  7. x ( n ) M = { x ( n M ) 0 n M N o t h e r w i s e x(n)_{\uparrow M}=\begin{cases}\begin{array}[]{c}x(\frac{n}{M})\\ 0\end{array}&\begin{array}[]{c}\frac{n}{M}\in N\\ otherwise\end{array}\end{cases}
  8. X ( z ) M = X ( z M ) {X(z)}_{\uparrow{}M}=X(z^{M})
  9. ω 0 \omega_{0}
  10. x ( n ) x\left(n\right)
  11. x 1 ( n ) , x 2 ( n ) , x 3 ( n ) , x_{1}(n),x_{2}(n),x_{3}(n),...
  12. x ( n ) x\left(n\right)
  13. x 1 ( n ) , x 2 ( n ) , x 3 ( n ) , x_{1}(n),x_{2}(n),x_{3}(n),...
  14. B W 1 , B W 2 , B W 3 , BW_{1},BW_{2},BW_{3},...
  15. f c 1 , f c 2 , f c 3 , f_{c1},f_{c2},f_{c3},...
  16. H k ( z ) H_{k}(z)
  17. F k ( z ) F_{k}(z)

Draft:Murray_polygon.html

  1. [ r n , r n - 1 , , r 2 , r 1 ] [r_{n},r_{n-1},\ldots,r_{2},r_{1}]
  2. [ d n , d n - 1 , , d 2 , d 1 ] [d_{n},d_{n-1},\ldots,d_{2},d_{1}]
  3. 0 d i < r i 0\leq d_{i}<r_{i}
  4. i = 1 , 2 , , n i=1,2,\ldots,n
  5. d d
  6. N N
  7. d 1 d_{1}
  8. d i := N mod r i d_{i}:=N\mod r_{i}
  9. N := N / r i N:=\lfloor N/r_{i}\rfloor
  10. d n d_{n}
  11. d n - 1 d_{n-1}
  12. d 1 d_{1}
  13. N := N r i + d i N:=N\cdot r_{i}+d_{i}
  14. a i + b i > r i a_{i}+b_{i}>r_{i}
  15. d i = ( a i + b i ) mod r i d_{i}=(a_{i}+b_{i})\mod r_{i}
  16. d = [ d n , d n - 1 , , d 2 , d 1 ] d=[d_{n},d_{n-1},\ldots,d_{2},d_{1}]
  17. e = [ e n , e n - 1 , , e 2 , e 1 ] e=[e_{n},e_{n-1},\ldots,e_{2},e_{1}]
  18. e i = r i - 1 - d i e_{i}=r_{i}-1-d_{i}
  19. d i := r i - 1 - d i d_{i}:=r_{i}-1-d_{i}
  20. r = [ 5 , 3 , 5 , 3 ] r=[5,3,5,3]
  21. [ 1 ] , [ 2 ] , [ 10 ] , [ 11 ] , [ 12 ] , [ 20 ] , [ 21 ] , [ 22 ] , [ 30 ] , [ 31 ] , [ 32 ] , [ 40 ] , [ 41 ] , [ 42 ] , [ 100 ] [1],[2],[10],[11],[12],[20],[21],[22],[30],[31],[32],[40],[41],[42],[100]
  22. d = [ 1230 ] d=[1230]
  23. N = ( ( ( 1 ) 3 + 2 ) 5 + 3 ) 3 + 0 = 84 N=(((1)\cdot 3+2)\cdot 5+3)\cdot 3+0=84
  24. e = [ 3012 ] e=[3012]
  25. m , n m,n
  26. j j
  27. m = k = 1 j m k m=\prod_{k=1}^{j}m_{k}
  28. n = k = 1 j n k n=\prod_{k=1}^{j}n_{k}
  29. 0 d m n - 1 0\leq d\leq mn-1
  30. ( x , y ) (x,y)
  31. m - 1 , n - 1 m-1,n-1
  32. p = [ p n , p n - 1 , , p 2 , p 1 ] p=[p_{n},p_{n-1},\ldots,p_{2},p_{1}]
  33. n = 2 j n=2j
  34. r i ( 1 < i < 2 j ) r_{i}(1<i<2j)
  35. p i p_{i}
  36. m k m_{k}
  37. i = 2 k - 1 i=2k-1
  38. n k n_{k}
  39. i = 2 k i=2k
  40. e = G ( p ) = [ e n , e n - 1 , , e 2 , e 1 ] g e=G(p)=[e_{n},e_{n-1},\ldots,e_{2},e_{1}]_{g}
  41. f = [ e n - 1 , e n - 3 , , e 3 , e 1 ] g f=[e_{n-1},e_{n-3},\ldots,e_{3},e_{1}]_{g}
  42. g = [ e n , e n - 2 , , e 4 , e 2 ] g g=[e_{n},e_{n-2},\ldots,e_{4},e_{2}]_{g}
  43. x = G - 1 ( f ) = [ x n , x n - 1 , , x 2 , x 1 ] x=G^{-1}(f)=[x_{n},x_{n-1},\ldots,x_{2},x_{1}]
  44. y = G - 1 ( g ) = [ y n , y n - 1 , , y 2 , y 1 ] y=G^{-1}(g)=[y_{n},y_{n-1},\ldots,y_{2},y_{1}]

Draft:Mutual_energy_theorem.html

  1. u 12 = 1 2 ( 𝐄 𝟏 𝐃 𝟐 + 𝐇 𝟏 𝐁 𝟐 ) u_{12}=\frac{1}{2}\left(\mathbf{E_{1}}\cdot\mathbf{D_{2}}+\mathbf{H_{1}}\cdot% \mathbf{B_{2}}\right)
  2. u 21 = 1 2 ( 𝐄 𝟐 𝐃 𝟏 + 𝐇 𝟐 𝐁 𝟏 ) u_{21}=\frac{1}{2}\left(\mathbf{E_{2}}\cdot\mathbf{D_{1}}+\mathbf{H_{2}}\cdot% \mathbf{B_{1}}\right)
  3. 𝐒 𝟏𝟐 = 𝐄 𝟏 × 𝐇 𝟐 , \mathbf{S_{12}}=\mathbf{E_{1}}\times\mathbf{H_{2}},
  4. 𝐒 𝟐𝟏 = 𝐄 𝟐 × 𝐇 𝟏 , \mathbf{S_{21}}=\mathbf{E_{2}}\times\mathbf{H_{1}},
  5. V \partial V\!
  6. - ( E × H ) = J E + K H + E t D + H t B -\nabla\cdot(E\times H)=J\cdot E+K\cdot H+E\cdot\partial_{t}\,D+H\cdot\partial% _{t}\,B
  7. S ( E 1 ( ω ) × H 2 ( ω ) - E 2 ( ω ) × H 1 ( ω ) ) d t n ^ d S \int_{S}(E_{1}(\omega)\times H_{2}(\omega)-E_{2}(\omega)\times H_{1}(\omega))% \,dt\,\hat{n}dS
  8. = V ( J 1 ( ω ) E 2 ( ω ) - J 2 ( ω ) E 1 ( ω ) - K 1 ( ω ) H 2 ( ω ) + K 2 ( ω ) H 1 ( ω ) d V = 0 =\int_{V}\,(J_{1}(\omega)\cdot E_{2}(\omega)-J_{2}(\omega)\cdot E_{1}(\omega)-% K_{1}(\omega)\cdot H_{2}(\omega)+K_{2}(\omega)\cdot H_{1}(\omega)\,\,dV=0
  9. - S t = - ( E 1 ( t ) × H 2 ( t ) + E 2 ( t ) × H 1 ( t ) ) d t n ^ d S -\int_{S}\int_{t=-\infty}^{\infty}(E_{1}(t)\times H_{2}(t)+E_{2}(t)\times H_{1% }(t))\,dt\,\hat{n}dS
  10. = V t = - ( J 1 ( t ) E 2 ( t ) + K 1 ( t ) H 2 ( t ) + J 2 ( t ) E 1 ( t ) + K 2 ( t ) H 1 ( t ) ) d t d V =\int_{V}\int_{t=-\infty}^{\infty}(J_{1}(t)\cdot E_{2}(t)+K_{1}(t)\cdot H_{2}(% t)+J_{2}(t)\cdot E_{1}(t)+K_{2}(t)\cdot H_{1}(t))\,dt\,dV
  11. S t = - ( E 1 ( t ) × H 2 ( t ) - E 2 ( t ) × H 1 ( t ) ) d t n ^ d S \int_{S}\int_{t=-\infty}^{\infty}(E_{1}(t)\times H_{2}(t)-E_{2}(t)\times H_{1}% (t))\,dt\,\hat{n}dS
  12. = V t = - ( J 1 ( t ) E 2 ( t ) - K 1 ( t ) H 2 ( t ) - J 2 ( t ) E 1 ( t ) + K 2 ( t ) H 1 ( t ) ) d t d V =\int_{V}\int_{t=-\infty}^{\infty}(J_{1}(t)\cdot E_{2}(t)-K_{1}(t)\cdot H_{2}(% t)-J_{2}(t)\cdot E_{1}(t)+K_{2}(t)\cdot H_{1}(t))\,dt\,dV
  13. V ( J 1 ( ω ) E 2 * ( ω ) + J 2 * ( ω ) E 1 ( ω ) + K 1 ( ω ) H 2 * ( ω ) + K 2 * ( ω ) H 1 ( ω ) ) d V = 0 \int_{V}(J_{1}(\omega)\cdot E_{2}^{*}(\omega)+J_{2}^{*}(\omega)\cdot E_{1}(% \omega)+K_{1}(\omega)\cdot H_{2}^{*}(\omega)+K_{2}^{*}(\omega)\cdot H_{1}(% \omega))\,dV=0
  14. ( E l × H ^ l + E l ^ × H l ) + v ^ e l p e l + v e l p ^ e l + v ^ i l p i l + v i l p ^ i l + + v ^ n l p n l + v i n ] m p ^ i l = 0 \nabla\cdot(E_{l}\times\hat{H}_{l^{\prime}}+\hat{E_{l^{\prime}}}\times H_{l})+% \hat{v}_{el^{\prime}}\cdot p_{el}+v_{el}\hat{p}_{el^{\prime}}+\hat{v}_{il^{% \prime}}\cdot p_{il}+v_{il}\hat{p}_{il^{\prime}}++\hat{v}_{nl^{\prime}}\cdot p% _{nl}+v_{in]m}\hat{p}_{il^{\prime}}=0
  15. V - ( J 1 ( τ - t ) E 2 ( t ) - J 2 ( t ) E 1 ( τ - t ) - K 1 ( τ - t ) H 2 ( t ) + K 2 ( t ) H 1 ( τ - t ) d t d V = 0 \int_{V}\,\int_{-\infty}^{\infty}(J_{1}(\tau-t)\cdot E_{2}(t)-J_{2}(t)\cdot E_% {1}(\tau-t)-K_{1}(\tau-t)\cdot H_{2}(t)+K_{2}(t)\cdot H_{1}(\tau-t)\,dt\,dV=0
  16. ϵ \textstyle\epsilon
  17. μ \textstyle\mu
  18. - S ( E 1 ( ω ) × H 2 * ( ω ) + E 2 * ( ω ) × H 1 ( ω ) ) n ^ d S -\int_{S}(E_{1}(\omega)\times H_{2}^{*}(\omega)+E_{2}^{*}(\omega)\times H_{1}(% \omega))\cdot\hat{n}dS
  19. = V ( J 1 ( ω ) E 2 * ( ω ) + J 2 * ( ω ) E 1 ( ω ) + K 1 ( ω ) H 2 * ( ω ) + K 2 * ( ω ) H 1 ( ω ) ) d V =\int_{V}(J_{1}(\omega)\cdot E_{2}^{*}(\omega)+J_{2}^{*}(\omega)\cdot E_{1}(% \omega)+K_{1}(\omega)\cdot H_{2}^{*}(\omega)+K_{2}^{*}(\omega)\cdot H_{1}(% \omega))\,dV
  20. - S t = - ( E 1 ( t + τ ) × H 2 ( t ) + E 2 ( t ) × H 1 ( t + τ ) ) d t n ^ d S -\int_{S}\int_{t=-\infty}^{\infty}(E_{1}(t+\tau)\times H_{2}(t)+E_{2}(t)\times H% _{1}(t+\tau))\,dt\,\hat{n}dS
  21. = V t = - ( J 1 ( t + τ ) E 2 ( t ) + K 1 ( t + τ ) H 2 ( t ) + J 2 ( t ) E 1 ( t + τ ) + K 2 ( t ) H 1 ( t + τ ) ) d t d V =\int_{V}\int_{t=-\infty}^{\infty}(J_{1}(t+\tau)\cdot E_{2}(t)+K_{1}(t+\tau)% \cdot H_{2}(t)+J_{2}(t)\cdot E_{1}(t+\tau)+K_{2}(t)\cdot H_{1}(t+\tau))\,dt\,dV
  22. τ = 0 \textstyle\tau=0
  23. ϵ \textstyle\epsilon
  24. μ \textstyle\mu
  25. ( E , H ) \textstyle(E,H)
  26. ϵ \textstyle\epsilon
  27. μ \textstyle\mu
  28. [ E r ( t ) , H r ( t ) , J r ( t ) , K r ( t ) , ϵ r ( t ) , μ r ( t ) ] r [ E ( t ) , H ( t ) , J ( t ) , K ( t ) , ϵ ( t ) , μ ( t ) ] [E_{r}(t),H_{r}(t),J_{r}(t),K_{r}(t),\epsilon_{r}(t),\mu_{r}(t)]\equiv r[E(t),% H(t),J(t),K(t),\epsilon(t),\mu(t)]
  29. = [ E ( - t ) , H ( - t ) , J ( - t ) , K ( - t ) , - ϵ ( - t ) , - μ ( - t ) ] =[E(-t),H(-t),J(-t),K(-t),-\epsilon(-t),-\mu(-t)]
  30. - ( E 1 × H 2 + E 2 × H 1 ) -\nabla\cdot(E_{1}\times H_{2}+E_{2}\times H_{1})
  31. = J 1 E 2 + J 2 E 1 + K 1 H 2 + K 2 H 1 + E 1 D 2 + E 2 D 1 + H 1 B 2 + H 2 B 1 =J_{1}\cdot E_{2}+J_{2}\cdot E_{1}+K_{1}\cdot H_{2}+K_{2}\cdot H_{1}+E_{1}% \cdot\partial D_{2}+E_{2}\cdot\partial D_{1}+H_{1}\cdot\partial B_{2}+H_{2}% \cdot\partial B_{1}
  32. - S ( E 1 ( ω ) × H 2 ( ω ) + E 2 ( ω ) × H 1 ( ω ) ) n ^ d S -\int_{S}(E_{1}(\omega)\times H_{2}(\omega)+E_{2}(\omega)\times H_{1}(\omega))% \,\hat{n}dS
  33. = V ( J 1 ( ω ) E 2 ( ω ) + J 2 ( ω ) E 1 ( ω ) + K 1 ( ω ) H 2 ( ω ) + K 2 ( ω ) H 1 ( ω ) ) d V =\int_{V}(J_{1}(\omega)\cdot E_{2}(\omega)+J_{2}(\omega)\cdot E_{1}(\omega)+K_% {1}(\omega)\cdot H_{2}(\omega)+K_{2}(\omega)\cdot H_{1}(\omega))\,dV
  34. - S t = - ( E 1 ( τ - t ) × H 2 ( t ) + E 2 ( t ) × H 1 ( τ - t ) ) n ^ d S -\int_{S}\int_{t=-\infty}^{\infty}(E_{1}(\tau-t)\times H_{2}(t)+E_{2}(t)\times H% _{1}(\tau-t))\,\hat{n}dS
  35. = V t = - ( J 1 ( τ - t ) E 2 ( t ) + J 2 ( t ) E 1 ( τ - t ) + K 1 ( τ - t ) H 2 ( t ) + K 2 ( t ) H 1 ( τ - t ) ) d V =\int_{V}\int_{t=-\infty}^{\infty}(J_{1}(\tau-t)\cdot E_{2}(t)+J_{2}(t)\cdot E% _{1}(\tau-t)+K_{1}(\tau-t)\cdot H_{2}(t)+K_{2}(t)\cdot H_{1}(\tau-t))\,dV
  36. - S ( E 1 ( ω ) × H 2 * ( ω ) + E 2 * ( ω ) × H 1 ( ω ) ) n ^ d S -\int_{S}(E_{1}(\omega)\times H_{2}^{*}(\omega)+E_{2}^{*}(\omega)\times H_{1}(% \omega))\cdot\hat{n}dS
  37. = V ( J 1 ( ω ) E 2 * ( ω ) + J 2 * ( ω ) E 1 ( ω ) + K 1 ( ω ) H 2 * ( ω ) + K 2 * ( ω ) H 1 ( ω ) ) d V =\int_{V}(J_{1}(\omega)\cdot E_{2}^{*}(\omega)+J_{2}^{*}(\omega)\cdot E_{1}(% \omega)+K_{1}(\omega)\cdot H_{2}^{*}(\omega)+K_{2}^{*}(\omega)\cdot H_{1}(% \omega))\,dV
  38. - S t = - ( E 1 ( t + τ ) × H 2 ( t ) + E 2 ( t ) × H 1 ( t + τ ) ) d t n ^ d S -\int_{S}\int_{t=-\infty}^{\infty}(E_{1}(t+\tau)\times H_{2}(t)+E_{2}(t)\times H% _{1}(t+\tau))\,dt\,\hat{n}dS
  39. = V t = - ( J 1 ( t + τ ) E 2 ( t ) + K 1 ( t + τ ) H 2 ( t ) + J 2 ( t ) E 1 ( t + τ ) + K 2 ( t ) H 1 ( t + τ ) ) d t d V =\int_{V}\int_{t=-\infty}^{\infty}(J_{1}(t+\tau)\cdot E_{2}(t)+K_{1}(t+\tau)% \cdot H_{2}(t)+J_{2}(t)\cdot E_{1}(t+\tau)+K_{2}(t)\cdot H_{1}(t+\tau))\,dt\,dV
  40. S ( E 1 ( ω ) × H 2 * ( ω ) - E 2 * ( ω ) × H 1 ( ω ) ) d t n ^ d S \int_{S}(E_{1}(\omega)\times H_{2}^{*}(\omega)-E_{2}^{*}(\omega)\times H_{1}(% \omega))\,dt\,\hat{n}dS
  41. = V ( J 1 ( ω ) E 2 * ( ω ) - J 2 * ( ω ) E 1 ( ω ) - K 1 ( ω ) H 2 * ( ω ) + K 2 * ( ω ) H 1 ( ω ) d V = 0 =\int_{V}\,(J_{1}(\omega)\cdot E_{2}^{*}(\omega)-J_{2}^{*}(\omega)\cdot E_{1}(% \omega)-K_{1}(\omega)\cdot H_{2}^{*}(\omega)+K_{2}^{*}(\omega)\cdot H_{1}(% \omega)\,\,dV=0
  42. S t = - ( E 1 ( t + τ ) × H 2 ( t ) - E 2 ( t ) × H 1 ( t + τ ) ) d t n ^ d S \int_{S}\int_{t=-\infty}^{\infty}(E_{1}(t+\tau)\times H_{2}(t)-E_{2}(t)\times H% _{1}(t+\tau))\,dt\,\hat{n}dS
  43. = V t = - ( J 1 ( t + τ ) E 2 ( t ) - K 1 ( t + τ ) H 2 ( t ) - J 2 ( t ) E 1 ( t + τ ) + K 2 ( t ) H 1 ( t + τ ) ) d t d V =\int_{V}\int_{t=-\infty}^{\infty}(J_{1}(t+\tau)\cdot E_{2}(t)-K_{1}(t+\tau)% \cdot H_{2}(t)-J_{2}(t)\cdot E_{1}(t+\tau)+K_{2}(t)\cdot H_{1}(t+\tau))\,dt\,dV
  44. - S ( E 1 × H 2 * + E 2 × H 1 * ) n ^ d S -\int_{S}(E_{1}\times H_{2}^{*}+E_{2}\times H_{1}^{*})\cdot\hat{n}dS
  45. = V ( E 1 J 2 * + E 2 J 1 * + H 1 * K 2 + H 2 * K 1 ) d V =\int_{V}(E_{1}\cdot J_{2}^{*}+E_{2}\cdot J_{1}^{*}+H_{1}^{*}\cdot K_{2}+H_{2}% ^{*}\cdot K_{1})dV
  46. + j ω V ( H 1 * μ 2 H 2 + H 2 * μ 1 H 1 - E 1 ϵ 2 * E 2 * - E 2 ϵ 1 * E 1 * ) d V +j\omega\int_{V}(H_{1}^{*}\cdot\mu_{2}H_{2}+H_{2}^{*}\cdot\mu_{1}H_{1}-E_{1}% \cdot\epsilon_{2}^{*}E_{2}^{*}-E_{2}\cdot\epsilon_{1}^{*}E_{1}^{*})dV
  47. - S t = - ( E 1 ( t + τ ) × H 2 * ( t ) + E 2 ( t + τ ) × H 1 * ( τ ) ) d t n ^ d S -\int_{S}\int_{t=-\infty}^{\infty}(E_{1}(t+\tau)\times H_{2}^{*}(t)+E_{2}(t+% \tau)\times H_{1}^{*}(\tau))dt\cdot\hat{n}dS
  48. = V t = - ( E 1 ( t + τ ) J 2 * ( t ) + E 2 ( t + τ ) J 1 * ( t ) + H 1 * ( t ) K 2 ( t + τ ) + H 2 * ( t ) K 1 ( t + τ ) ) d t d V =\int_{V}\int_{t=-\infty}^{\infty}(E_{1}(t+\tau)\cdot J_{2}^{*}(t)+E_{2}(t+% \tau)\cdot J_{1}^{*}(t)+H_{1}^{*}(t)\cdot K_{2}(t+\tau)+H_{2}^{*}(t)\cdot K_{1% }(t+\tau))\,dt\,dV
  49. + τ V t = - ( H 1 * ( t ) ( μ 2 * H 2 ) ( t + τ ) + H 2 * ( t ) ( μ 1 * H 1 ) ( t + τ ) +\partial_{\tau}\int_{V}\int_{t=-\infty}^{\infty}(H_{1}^{*}(t)\cdot(\mu_{2}*H_% {2})(t+\tau)+H_{2}^{*}(t)\cdot(\mu_{1}*H_{1})(t+\tau)
  50. - E 1 ( t + τ ) ( ϵ 2 * * E 2 * ) ( t ) - E 2 ( t + τ ) ( ϵ 1 * * E 1 * ) ( t ) ) d t d V -E_{1}(t+\tau)\cdot(\epsilon_{2}^{*}*E_{2}^{*})(t)-E_{2}(t+\tau)\cdot(\epsilon% _{1}^{*}*E_{1}^{*})(t))\,dt\,dV
  51. ( ζ 1 , ζ 2 ) τ = S t = - ( E 1 ( t + τ ) × H 2 ( t ) + E 2 ( t ) × H 1 ( t + τ ) ) d t n ^ d S (\zeta_{1},\zeta_{2})_{\tau}=\int_{S}\int_{t=-\infty}^{\infty}(E_{1}(t+\tau)% \times H_{2}(t)+E_{2}(t)\times H_{1}(t+\tau))\,dt\,\hat{n}dS
  52. ( ζ 1 , ζ 2 ) τ \textstyle(\zeta_{1},\zeta_{2})_{\tau}
  53. ( ζ 1 , ζ 2 ) τ = 0 \textstyle(\zeta_{1},\zeta_{2})_{\tau=0}
  54. ζ s = s ζ = [ Z H , 1 Z E , - 1 Z K , - Z J , - 1 Z 2 μ , - Z 2 ϵ ] \zeta_{s}=s\zeta=[ZH,\frac{1}{Z}E,-\frac{1}{Z}K,-ZJ,-\frac{1}{Z^{2}}\mu,-Z^{2}\epsilon]

Draft:Net_Reclassification_Improvement.html

  1. N R I = nbr events going up - nbr events going down nbr events - nbr nonevents going up - nbr nonevents going down nbr nonevents NRI=\frac{\mathrm{nbr\;events\;going\;up}-\mathrm{nbr\;events\;going\;down}}{% \mathrm{nbr\;events}}-\frac{\mathrm{nbr\;nonevents\;going\;up}-\mathrm{nbr\;% nonevents\;going\;down}}{\mathrm{nbr\;nonevents}}

Draft:Noncommutative_ring.html

  1. A I AI
  2. A I = A AI=A
  3. a i A a_{i}\in A
  4. 1 a 1 I + + a k I 1\in a_{1}I+\cdots+a_{k}I
  5. I k A , ( y 1 , , y k ) a 1 y 1 + + a k y k . I^{\oplus k}\to A,\,(y_{1},\dots,y_{k})\mapsto a_{1}y_{1}+\cdots+a_{k}y_{k}.
  6. a 1 y 1 = a 2 y 2 + + a k y k a_{1}y_{1}=a_{2}y_{2}+\cdots+a_{k}y_{k}
  7. y 1 y_{1}
  8. y 1 A = I y_{1}A=I
  9. a 1 I = a 1 y 1 A a 2 I + + a k I a_{1}I=a_{1}y_{1}A\subset a_{2}I+\cdots+a_{k}I
  10. I k A I^{\oplus k}\simeq A
  11. A End A ( A ) M k ( End A ( I ) ) A\simeq\operatorname{End}_{A}(A)\simeq M_{k}(\operatorname{End}_{A}(I))
  12. R = n = 0 F n A R=\bigoplus_{n=0}^{\infty}F^{n}A

Draft:Nonlinear_wave_groups_on_deep_water.html

  1. z = cos ( k x - ω t ) z=\cos(kx-\omega t)
  2. ( z , x ) (z,x)
  3. z z
  4. x x
  5. ( z , x ) (z,x)
  6. ( z , x ) (z,x)
  7. z = f < 0 , x = 0 z=f<0,\,x=0
  8. θ \theta
  9. z z
  10. t t
  11. z = c W ( θ , t ) , x = ( W - f ) tan θ , - π / 2 < θ < π / 2. ( 1 ) z=cW(\theta,t),\,\,\,\,x=(W-f)\,\tan\,\theta,\,\,\,\,-\pi/2<\theta<\pi/2.(1)
  12. W W
  13. f f
  14. t t
  15. f f
  16. c c
  17. ( z , x ) (z,x)
  18. ( σ , θ ) (\sigma,\theta)
  19. z = σ + W ( θ , t ) , x = ( σ + W - f ) tan θ z=\sigma+W(\theta,t),\ \ x=(\sigma+W-f)\,\tan\,\theta
  20. σ = 0 \sigma=0
  21. σ < 0 \sigma<0
  22. Φ \Phi
  23. ( σ , θ ) (\sigma,\theta)
  24. W ( θ , t ) W(\theta,t)
  25. ν ( θ , t ) \nu(\theta,t)
  26. ε = c / f \,\varepsilon=c/f
  27. ε \,\varepsilon
  28. W ( θ , t ) = W 0 ( θ , t ) + ε W 1 ( θ , t ) + W(\theta,t)=W_{0}(\theta,t)+\varepsilon W_{1}(\theta,t)+...
  29. z = P 2 n ( τ , θ ) , x = ( z - f ) tan θ , z=P_{2n}(\tau,\theta),\,\,\,\,x=(z-f)\tan\theta,
  30. z = Q 2 n ( τ , θ ) , x = ( z - f ) tan θ , z=Q_{2n}(\tau,\theta),\,\,\,\,x=(z-f)\tan\theta,
  31. z = I 2 n + 1 ( τ , θ ) , x = ( z - f ) tan θ , z=I_{2n+1}(\tau,\theta),\,\,\,\,x=(z-f)\tan\theta,
  32. z = J 2 n + 1 ( τ , θ ) , x = ( z - f ) tan θ , z=J_{2n+1}(\tau,\theta),\,\,\,\,x=(z-f)\tan\theta,
  33. P 2 n ( τ , θ ) = 0 + x 2 n e - x 2 / 2 cos ( 1 2 x 2 tan θ ) sin ( τ x ) d x ; ( 2 ) P_{2n}(\tau,\theta)=\int\limits_{0}^{+\infty}x^{2n}e^{-x^{2}/2}\cos(\frac{1}{2% }x^{2}\tan\theta)\sin(\tau x)\,dx;(2)
  34. Q 2 n ( τ , θ ) = 0 + x 2 n e - x 2 / 2 sin ( 1 2 x 2 tan θ ) sin ( τ x ) d x . ( 3 ) Q_{2n}(\tau,\theta)=\int\limits_{0}^{+\infty}x^{2n}e^{-x^{2}/2}\sin(\frac{1}{2% }x^{2}\tan\theta)\sin(\tau x)\,dx.(3)
  35. I 2 n + 1 ( τ , θ ) = P 2 n ( τ , θ ) τ , ( 4 ) I_{2n+1}(\tau,\theta)=\frac{\partial P_{2n}(\tau,\theta)}{\partial\tau},(4)
  36. J 2 n + 1 ( τ , θ ) = Q 2 n ( τ , θ ) τ , ( 5 ) J_{2n+1}(\tau,\theta)=\frac{\partial Q_{2n}(\tau,\theta)}{\partial\tau},(5)
  37. z = c W 0 ( θ , t ) , x = ( z - f ) tan θ , - π / 2 < θ < π / 2 ( 6 ) z=cW_{0}(\theta,t),\,\,\,x=(z-f)\tan\theta,\,\,\,-\pi/2<\theta<\pi/2(6)
  38. W 0 ( θ , t ) = n = 1 + ( - 1 ) n - 1 1 2 n [ a n I 2 n + 1 ( τ , θ ) + b n J 2 n + 1 ( τ , θ ) ] + W_{0}(\theta,t)=\sum_{n=1}^{+\infty}(-1)^{n-1}\frac{1}{2n}[a_{n}I_{2n+1}(\tau,% \theta)+b_{n}J_{2n+1}(\tau,\theta)]+
  39. 1 2 | f | n = 1 + ( - 1 ) n [ ρ n P 2 n ( τ , θ ) + e n Q 2 n ( τ , θ ) ] , t = τ 2 | f | \frac{1}{\sqrt{2|f|}}\sum_{n=1}^{+\infty}(-1)^{n}[\rho_{n}P_{2n}(\tau,\theta)+% e_{n}Q_{2n}(\tau,\theta)],\,\,\,\,\,\,t=\tau\,{\sqrt{2|f|}}
  40. a n a_{n}
  41. b n b_{n}
  42. ρ n \rho_{n}
  43. e n e_{n}
  44. f f
  45. W 0 ( θ , t ) W_{0}(\theta,t)
  46. z = c W 0 ( arc tangent ( x / ( z - f ) ) , t ) . z=cW_{0}(\hbox{arc tangent}(x/(z-f)),t).
  47. ε 0 \,\varepsilon\rightarrow 0

Draft:Normal_homomorphism.html

  1. R S R\to S
  2. κ ( 𝔭 ) \kappa(\mathfrak{p})
  3. 𝔭 \mathfrak{p}
  4. L R S L\otimes_{R}S

Draft:Performance_Comparisons_of_Spatial_Rotations.html

  1. R R
  2. 𝐪 = ( w , [ u v e c , u r ] ) \mathbf{q}=(w,[u^{\prime}vec^{\prime},u^{\prime}r^{\prime}])
  3. w w
  4. v new = v + 2 r × ( r × v + w v ) \vec{v}\text{new}=\vec{v}+2\vec{r}\times(\vec{r}\times\vec{v}+w\vec{v})
  5. v new = q v q - 1 \vec{v}\text{new}=q\vec{v}q^{-1}
  6. R R
  7. R R
  8. R R
  9. R R
  10. R R

Draft:Portfolio_performance_contributions.html

  1. Π t \Pi_{t}
  2. t t
  3. \Epsilon t \Epsilon_{t}
  4. t t
  5. \Epsilon 0 = 0 \Epsilon_{0}=0
  6. m k , t m_{k,t}
  7. k k
  8. t t
  9. m k , 0 m_{k,0}
  10. t = 0 t=0
  11. p k , t p_{k,t}
  12. k k
  13. t t
  14. d k , t d_{k,t}
  15. k k
  16. t t
  17. d k , 0 = 0 d_{k,0}=0
  18. f k , t f_{k,t}
  19. k k
  20. t t
  21. t t
  22. ϕ k , t \phi_{k,t}
  23. k k
  24. t t
  25. ϕ k , t = 1 α k , t 1 f k , t + ( 1 - α k , t ) i = 0 t - 1 m k , i ϕ k , i i = 0 t - 1 m k , i \phi_{k,t}=\cfrac{1}{\alpha_{k,t}\cdot\cfrac{1}{f_{k,t}}+(1-\alpha_{k,t})\cdot% \cfrac{\sum\limits_{i=0}^{t-1}\cfrac{m_{k,i}}{\phi_{k,i}}}{\sum\limits_{i=0}^{% t-1}m_{k,i}}}
  26. ϕ k , 0 = f k , 0 \phi_{k,0}=f_{k,0}
  27. ϕ k , t = f k , t if m k , t = 0 or i = 0 t - 1 m k , i = 0 \phi_{k,t}=f_{k,t}\,\text{ if }m_{k,t}=0\,\text{ or }\sum\limits_{i=0}^{t-1}m_% {k,i}=0
  28. α k , t \alpha_{k,t}
  29. k k
  30. t t
  31. k k
  32. α k , t = max { 0 , min { 1 , i = 0 t m k , i m k , t } } \alpha_{k,t}=\max\left\{0,\min\left\{1,\cfrac{\sum\limits_{i=0}^{t}m_{k,i}}{m_% {k,t}}\right\}\right\}
  33. Λ t \Lambda_{t}
  34. t t
  35. Λ t = i = 1 t ( 1 - \Epsilon i Π i ) with Λ 0 = 1 \Lambda_{t}=\prod_{i=1}^{t}\left(1-\cfrac{\Epsilon_{i}}{\Pi_{i}}\right)\,\text% { with }\Lambda_{0}=1
  36. λ k , t \lambda_{k,t}
  37. k k
  38. k k
  39. t t
  40. λ k , t = i = 1 t ( 1 - \Epsilon i Π i + j = 1 i m k , j ( p k , i - p k , j ) ( 1 ϕ k , j - 1 f k , i ) price adjustement factor + j = 1 i l = 0 j - 1 m k , l d k , j ( 1 ϕ k , l - 1 f k , i ) dividend adjustement factor ) with λ k , 0 = 1 \lambda_{k,t}=\prod_{i=1}^{t}\left(1-\cfrac{\Epsilon_{i}}{\Pi_{i}+\underbrace{% \sum\limits_{j=1}^{i}m_{k,j}\left(p_{k,i}-p_{k,j}\right)\left(\cfrac{1}{\phi_{% k,j}}-\cfrac{1}{f_{k,i}}\right)}_{\,\text{price adjustement factor}}+% \underbrace{\sum\limits_{j=1}^{i}\sum\limits_{l=0}^{j-1}m_{k,l}d_{k,j}\left(% \cfrac{1}{\phi_{k,l}}-\cfrac{1}{f_{k,i}}\right)}_{\,\text{dividend adjustement% factor}}}\right)\,\text{ with }\lambda_{k,0}=1
  41. r t r_{t}
  42. r k , t r_{k,t}
  43. k k
  44. k [ 1 , n ] k\in[1,n]
  45. r k , t base r_{k,t}^{\,\text{base}}
  46. k k
  47. r k , t crcy r_{k,t}^{\,\text{crcy}}
  48. k k
  49. r t = k = 1 n ( r k , t base + r k , t crcy ) = r k , t r_{t}=\sum\limits_{k=1}^{n}\underbrace{\left(r_{k,t}^{\,\text{base}}+r_{k,t}^{% \,\text{crcy}}\right)}_{=r_{k,t}}
  50. k [ 1 , n ] k\in[1,n]
  51. k k
  52. r k , t = 1 Π 0 i = 1 t Λ i - 1 ( j = 0 i - 1 m k , j ( p k , i - p k , j f k , i - p k , i - 1 - p k , j f k , i - 1 ) price + j = 1 i - 1 l = 0 j - 1 m k , l d k , j ( 1 f k , i - 1 f k , i - 1 ) paid + j = 0 i - 1 m k , j d k , i f k , i to be paid dividend ) r_{k,t}=\cfrac{1}{\Pi_{0}}\sum\limits_{i=1}^{t}\Lambda_{i-1}\left(\underbrace{% \sum\limits_{j=0}^{i-1}m_{k,j}\left(\cfrac{p_{k,i}-p_{k,j}}{f_{k,i}}-\cfrac{p_% {k,i-1}-p_{k,j}}{f_{k,i-1}}\right)}_{\,\text{price}}+\underbrace{\underbrace{% \sum\limits_{j=1}^{i-1}\sum\limits_{l=0}^{j-1}m_{k,l}d_{k,j}\left(\cfrac{1}{f_% {k,i}}-\cfrac{1}{f_{k,i-1}}\right)}_{\,\text{paid}}+\underbrace{\sum\limits_{j% =0}^{i-1}m_{k,j}\cfrac{d_{k,i}}{f_{k,i}}}_{\,\text{to be paid}}}_{\,\text{% dividend}}\right)
  53. r k , t base = 1 Π 0 i = 1 t λ k , i - 1 ( j = 0 i - 1 m k , j ( p k , i - p k , i - 1 ) ϕ k , j price + j = 0 i - 1 m k , j d k , i ϕ k , j dividend ) r_{k,t}^{\,\text{base}}=\cfrac{1}{\Pi_{0}}\sum\limits_{i=1}^{t}\lambda_{k,i-1}% \left(\underbrace{\sum\limits_{j=0}^{i-1}m_{k,j}\cfrac{\left(p_{k,i}-p_{k,i-1}% \right)}{\phi_{k,j}}}_{\,\text{price}}+\underbrace{\sum\limits_{j=0}^{i-1}m_{k% ,j}\cfrac{d_{k,i}}{\phi_{k,j}}}_{\,\text{dividend}}\right)
  54. r k , t crcy = r k , t - r k , t base r_{k,t}^{\,\text{crcy}}=r_{k,t}-r_{k,t}^{\,\text{base}}

Draft:ProgArchives's_Top_Progressive_Rock_Albums_List.html

  1. N a × R a + N 1 × R 1 N a + N 1 {N_{a}\times R_{a}+N_{1}\times R_{1}\over N_{a}+N_{1}}
  2. N a N_{a}
  3. R a R_{a}
  4. N 1 N_{1}
  5. R 1 R_{1}

Draft:Ramification_theory_of_local_rings.html

  1. L / K L/K
  2. Frac ( A ) = K \operatorname{Frac}(A)=K
  3. Frac ( B ) = L \operatorname{Frac}(B)=L
  4. G = Gal ( L / K ) G=\operatorname{Gal}(L/K)
  5. Gal S ( B / A ) = { σ G | σ ( B ) B } . \operatorname{Gal}^{S}(B/A)=\{\sigma\in G|\sigma(B)\subset B\}.
  6. Gal S ( B / A ) Gal ( k ( B ) / k ( A ) ) \operatorname{Gal}^{S}(B/A)\to\operatorname{Gal}(k(B)/k(A))
  7. L / K L/K
  8. K S K i K^{S}\subset K^{i}
  9. A S A i A^{S}\subset A^{i}

Draft:Relativistic_Global_Non-Inertial_Reference_Frames.html

  1. r r
  2. ω r = c \omega\,r=c
  3. ω \omega
  4. c c
  5. ω r = c \omega\,r=c
  6. x 0 = c t = c o n s t . x^{0}=c\,t=const.
  7. x i o x_{i}^{o}
  8. x f o x_{f}^{o}
  9. x P o = x i o + 1 2 ( x f o - x i o ) = 1 2 ( x i o + x f o ) x_{P}^{o}=x_{i}^{o}+{\frac{1}{2}}\,(x_{f}^{o}-x_{i}^{o})={\frac{1}{2}}\,(x_{i}% ^{o}+x_{f}^{o})
  10. c c
  11. c c
  12. x μ ( τ ) x^{\mu}(\tau)
  13. τ \tau
  14. Σ τ \Sigma_{\tau}
  15. Σ τ \Sigma_{\tau}
  16. τ \tau
  17. Σ τ \Sigma_{\tau}
  18. σ r \sigma^{r}
  19. σ A = ( τ ; σ r ) \sigma^{A}=(\tau;\sigma^{r})
  20. z μ ( τ , σ r ) z^{\mu}(\tau,\sigma^{r})
  21. Σ τ \Sigma_{\tau}
  22. x μ σ A ( x ) x^{\mu}\mapsto\sigma^{A}(x)
  23. x μ x^{\mu}
  24. σ A x μ = z μ ( τ , σ r ) \sigma^{A}\mapsto x^{\mu}=z^{\mu}(\tau,\sigma^{r})
  25. Σ τ \Sigma_{\tau}
  26. z μ ( τ , σ r ) z^{\mu}(\tau,\sigma^{r})
  27. g A B 4 ( τ , σ r ) = [ z A μ η μ ν z B ν ] ( τ , σ r ) {}^{4}g_{AB}(\tau,\sigma^{r})=[z^{\mu}_{A}\,\eta_{\mu\nu}\,z^{\nu}_{B}](\tau,% \sigma^{r})
  28. z A μ = z μ / σ A z^{\mu}_{A}=\partial\,z^{\mu}/\partial\,\sigma^{A}
  29. η μ ν 4 = ϵ ( + - - - ) {}^{4}\eta_{\mu\nu}=\epsilon\,(+---)
  30. ϵ = 1 \epsilon=1
  31. ϵ = - 1 \epsilon=-1
  32. z r μ ( τ , σ u ) z^{\mu}_{r}(\tau,\sigma^{u})
  33. Σ τ \Sigma_{\tau}
  34. l μ ( τ , σ u ) l^{\mu}(\tau,\sigma^{u})
  35. ϵ μ [ z 1 α z 2 β z 3 γ ] α β γ ( τ , σ u ) \epsilon^{\mu}{}_{\alpha\beta\gamma}\,[z^{\alpha}_{1}\,z^{\beta}_{2}\,z^{% \gamma}_{3}](\tau,\sigma^{u})
  36. z τ μ ( τ , σ r ) = [ N l μ + N r z r μ ] ( τ , σ r ) z^{\mu}_{\tau}(\tau,\sigma^{r})=[N\,l^{\mu}+N^{r}\,z^{\mu}_{r}](\tau,\sigma^{r})
  37. N ( τ , σ r ) = ϵ [ z τ μ l μ ] ( τ , σ r ) N(\tau,\sigma^{r})=\epsilon\,[z^{\mu}_{\tau}\,l_{\mu}](\tau,\sigma^{r})
  38. N r ( τ , σ r ) = - ϵ g τ r 4 ( τ , σ r ) N_{r}(\tau,\sigma^{r})=-\epsilon\,{}^{4}g_{\tau r}(\tau,\sigma^{r})
  39. N ( τ , σ r ) d τ N(\tau,\sigma^{r})\,d\tau
  40. Σ τ \Sigma_{\tau}
  41. Σ τ + d τ \Sigma_{\tau+d\tau}
  42. Σ τ + d τ \Sigma_{\tau+d\tau}
  43. Σ τ \Sigma_{\tau}
  44. l μ ( τ , σ u ) l^{\mu}(\tau,\sigma^{u})
  45. z r μ ( τ , σ u ) z^{\mu}_{r}(\tau,\sigma^{u})
  46. Σ τ \Sigma_{\tau}
  47. N ( τ , σ r ) > 0 N(\tau,\sigma^{r})>0
  48. ϵ g τ τ 4 ( τ , σ r ) = ( N 2 - N u N u ) ( τ , σ r ) > 0 \epsilon\,{}^{4}g_{\tau\tau}(\tau,\sigma^{r})=(N^{2}-N_{u}\,N^{u})(\tau,\sigma% ^{r})>0
  49. h r s ( τ , σ u ) = - ϵ g r s 4 ( τ , σ u ) h_{rs}(\tau,\sigma^{u})=-\epsilon\,{}^{4}g_{rs}(\tau,\sigma^{u})
  50. Σ τ \Sigma_{\tau}
  51. ϵ A μ \epsilon^{\mu}_{A}
  52. ϵ τ μ \epsilon^{\mu}_{\tau}
  53. ϵ r μ \epsilon^{\mu}_{r}
  54. ϵ A μ \epsilon^{\mu}_{A}
  55. z μ ( τ , σ r ) = x μ ( τ ) + ϵ A μ F A ( τ , σ r ) , F A ( τ , 0 ) = 0 , z^{\mu}(\tau,\sigma^{r})=x^{\mu}(\tau)+\epsilon^{\mu}_{A}\,F^{A}(\tau,\sigma^{% r}),\qquad F^{A}(\tau,0)=0,
  56. x μ ( τ ) = x o μ + ϵ A μ f A ( τ ) , x^{\mu}(\tau)=x^{\mu}_{o}+\epsilon^{\mu}_{A}\,f^{A}(\tau),
  57. x μ ( τ ) x^{\mu}(\tau)
  58. f A ( τ ) f^{A}(\tau)
  59. u μ ( τ ) = x ˙ μ ( τ ) / ϵ x ˙ 2 ( τ ) u^{\mu}(\tau)={\dot{x}}^{\mu}(\tau)/\sqrt{\epsilon\,{\dot{x}}^{2}(\tau)}
  60. x ˙ μ ( τ ) = d x μ ( τ ) < m t p l > d τ {\dot{x}}^{\mu}(\tau)={\frac{{dx^{\mu}(\tau)}}{<}mtpl>{{d\tau}}}
  61. a μ ( τ ) = d u μ ( τ ) < m t p l > d τ a^{\mu}(\tau)={\frac{{du^{\mu}(\tau)}}{<}mtpl>{{d\tau}}}
  62. x μ ( τ ) = x o μ + ϵ τ μ τ x^{\mu}(\tau)=x_{o}^{\mu}+\epsilon^{\mu}_{\tau}\,\tau
  63. z μ ( τ , σ r ) = x μ ( τ ) + ϵ r μ σ r z^{\mu}(\tau,\sigma^{r})=x^{\mu}(\tau)+\epsilon^{\mu}_{r}\,\sigma^{r}
  64. ϵ A μ \epsilon^{\mu}_{A}
  65. f A ( τ ) f^{A}(\tau)
  66. F A ( τ , σ r ) F^{A}(\tau,\sigma^{r})
  67. x μ ( τ ) = x o μ + ϵ τ μ f ( τ ) x^{\mu}(\tau)=x^{\mu}_{o}+\epsilon^{\mu}_{\tau}\,f(\tau)
  68. z μ ( τ , σ u ) = x o μ + ϵ τ μ f ( τ ) + ϵ r μ σ r z^{\mu}(\tau,\sigma^{u})=x_{o}^{\mu}+\epsilon_{\tau}^{\mu}\,f(\tau)+\epsilon_{% r}^{\mu}\,\sigma^{r}
  69. f ( τ ) = τ f(\tau)=\tau
  70. x μ ( τ ) x^{\mu}(\tau)
  71. σ = | σ | \sigma=|\vec{\sigma}|
  72. z μ ( τ , σ u ) = x μ ( τ ) + ϵ r μ R r ( α i ( τ , σ ) ) s σ s σ x μ ( τ ) + ϵ r μ σ r z^{\mu}(\tau,\sigma^{u})=x^{\mu}(\tau)+\epsilon_{r}^{\mu}\,R^{r}{}_{s}(\alpha_% {i}(\tau,\sigma))\,\sigma^{s}\,\rightarrow_{\sigma\rightarrow\infty}\,x^{\mu}(% \tau)+\epsilon_{r}^{\mu}\,\sigma^{r}
  73. R r ( α i ( τ , σ ) ) s R^{r}{}_{s}(\alpha_{i}(\tau,\sigma))
  74. σ \sigma\rightarrow\infty
  75. α i ( τ , σ ) = f ( σ ) α ~ i ( τ ) \alpha_{i}(\tau,\sigma)=f(\sigma)\,{\tilde{\alpha}}_{i}(\tau)
  76. i = 1 , 2 , 3 i=1,2,3
  77. 0 a n d < m a t h > d f ( σ ) d σ 0 0and<math>{{d\,f(\sigma)}\over{d\,\sigma}}\not=0
  78. R r ( α i ( τ ) ) s R^{r}{}_{s}(\alpha_{i}(\tau))
  79. σ = r ( σ r ) 2 \sigma=\sqrt{\sum_{r}\,(\sigma^{r})^{2}}
  80. z μ ( τ , σ r ) = x μ ( τ ) + ϵ A μ Λ A ( τ , σ ) r σ r σ x μ ( τ ) + ϵ r μ σ r z^{\mu}(\tau,\sigma^{r})=x^{\mu}(\tau)+\epsilon^{\mu}_{A}\,\Lambda^{A}{}_{r}(% \tau,\sigma)\,\sigma^{r}\,{\rightarrow}_{\sigma\rightarrow\infty}\,x^{\mu}(% \tau)+\epsilon^{\mu}_{r}\,\sigma^{r}
  81. Λ A ( τ , σ ) B \Lambda^{A}{}_{B}(\tau,\sigma)
  82. α i ( τ , σ ) = f ( σ ) α ~ i ( τ ) \alpha_{i}(\tau,\sigma)=f(\sigma)\,{\tilde{\alpha}}_{i}(\tau)
  83. i = 1 , 2 , 3 i=1,2,3
  84. φ i ( τ , σ ) = g ( σ ) φ ~ i ( τ ) \varphi_{i}(\tau,\sigma)=g(\sigma)\,{\tilde{\varphi}}_{i}(\tau)
  85. i = 1 , 2 , 3 i=1,2,3
  86. f ( σ ) f(\sigma)
  87. g ( σ ) g(\sigma)
  88. g A B 4 ( τ , σ u ) {}^{4}g_{AB}(\tau,\sigma^{u})
  89. K r s 3 ( τ , σ u ) {}^{3}K_{rs}(\tau,\sigma^{u})
  90. Σ τ \Sigma_{\tau}
  91. z μ ( τ , σ u ) z^{\mu}(\tau,\sigma^{u})
  92. ϕ ~ ( x μ ) \tilde{\phi}(x^{\mu})
  93. ϕ ( τ , σ u ) = ϕ ~ ( z μ ( τ , σ u ) ) \phi(\tau,\sigma^{u})=\tilde{\phi}(z^{\mu}(\tau,\sigma^{u}))
  94. x μ ( τ ) x^{\mu}(\tau)
  95. i = 1 , . . , N i=1,..,N
  96. η i r ( τ ) \eta^{r}_{i}(\tau)
  97. i = 1 , . . , N i=1,..,N
  98. x i μ ( τ ) = z μ ( τ , η i r ( τ ) ) x^{\mu}_{i}(\tau)=z^{\mu}(\tau,\eta^{r}_{i}(\tau))
  99. g A B 4 ( τ , σ r ) {}^{4}g_{AB}(\tau,\sigma^{r})
  100. z μ ( τ , σ r ) z^{\mu}(\tau,\sigma^{r})
  101. ρ μ ( τ , σ r ) \rho_{\mu}(\tau,\sigma^{r})
  102. P μ = d 3 σ ρ μ ( τ , σ u ) P^{\mu}=\int d^{3}\sigma\rho^{\mu}(\tau,\sigma^{u})
  103. J μ ν = d 3 σ ( z μ ρ ν - z ν ρ μ ) ( τ , σ u ) J^{\mu\nu}=\int d^{3}\sigma(z^{\mu}\rho^{\nu}-z^{\nu}\rho^{\mu})(\tau,\sigma^{% u})
  104. M c = ϵ P 2 Mc=\sqrt{\epsilon\,P^{2}}
  105. P μ P^{\mu}
  106. Σ τ \Sigma_{\tau}
  107. M M
  108. S \vec{S}
  109. M M
  110. S \vec{S}
  111. M c 2 M\,c^{2}
  112. M c Mc
  113. S \vec{S}
  114. Σ τ \Sigma_{\tau}

Draft:Ring_of_invariants.html

  1. g x = x g\cdot x=x
  2. F G ( t ) F_{G}(t)
  3. S S

Draft:Ringed_topos.html

  1. ( T , 𝒪 T ) ( T , 𝒪 T ) (T,\mathcal{O}_{T})\to(T^{\prime},\mathcal{O}_{T^{\prime}})
  2. f : T T f:T\to T^{\prime}
  3. 𝒪 T f * 𝒪 T \mathcal{O}_{T^{\prime}}\to f_{*}\mathcal{O}_{T}
  4. Spec R = Map O - alg ( R , - ) \operatorname{Spec}R=\operatorname{Map}_{O-\,\text{alg}}(R,-)
  5. n \mathcal{E}_{n}

Draft:Rollett_Stability_Factor.html

  1. ( S 11 S 12 S 21 S 22 ) \begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}
  2. S 12 S_{12}\,
  3. K = 1 - | S 11 | 2 - | S 22 | 2 + | Δ | 2 2 | S 12 S 21 | K=\frac{1-\left|S_{11}\right|^{2}-\left|S_{22}\right|^{2}+\left|\Delta\right|^% {2}}{2\left|S_{12}S_{21}\right|}\,
  4. Δ \Delta\,
  5. S 11 S_{11}\,
  6. S 22 S_{22}\,
  7. S 12 S_{12}\,
  8. S 21 S_{21}\,
  9. S 12 S_{12}\,

Draft:Self-healing_Materials.html

  1. η ¯ = n i = 1 n ( 1 η i ( P ) ) \bar{\eta}=\frac{n}{\sum_{i=1}^{n}\left(\frac{1}{\eta_{i}(P)}\right)}

Draft:Sheaf_(infinity_category).html

  1. U i U_{i}
  2. F ( U ) lim F ( U i ) F(U)\to\underleftarrow{\lim}F(U_{i})
  3. F ( U V ) F(U\cap V)
  4. F ( U V ) F ( U ) F(U\cup V)\to F(U)
  5. F ( U V ) F ( V ) F(U\cup V)\to F(V)
  6. c M c_{M}
  7. c M ( X ) c_{M}(X)
  8. c M c_{M}
  9. C * ( - ; M ) C^{*}(-;M)
  10. C * ( X ; M ) C^{*}(X;M)
  11. C * ( X ; M ) C^{*}(X;M)

Draft:Shields_diagram.html

  1. τ * = τ ( ρ s - ρ ) ( g ) ( D ) \tau*=\frac{\tau}{(\rho_{s}-\rho)(g)(D)}
  2. R e * = u * D v Re^{*}=\frac{u^{*}D}{v}
  3. τ * c = f ( R e * c ) \tau_{*}c=f(Re_{*}c)
  4. τ * c = 0.11 R e * c + 0.054 ( 1 - e - 4 R e * c 0.52 25 ) \tau_{*}c=\frac{0.11}{Re_{*}c}+0.054(1-e^{-\frac{4Re_{*}c^{0.52}}{25}})
  5. τ * c = S h i e l d s N u m b e r = τ c ( s - 1 ) ( ρ ) ( g ) ( D ) \tau_{*}c=ShieldsNumber=\frac{\tau_{c}}{(s-1)(\rho)(g)(D)}
  6. R e * c = ( u * c ) ( D ) v Re_{*}c=\frac{(u_{*}c)(D)}{v}
  7. R e = d v ( ( 0.1 ) ( ρ s ρ - 1 ) ( g ) ( D ) ) 0.5 Re=\frac{d}{v}({(0.1)(\frac{\rho_{s}}{\rho}-1)(g)(D)})^{0.5}
  8. R e = d v ( ( 0.1 ) ( ρ s ρ - 1 ) ( g ) ( D ) ) 0.5 = 114 Re=\frac{d}{v}({(0.1)(\frac{\rho_{s}}{\rho}-1)(g)(D)})^{0.5}=114
  9. τ * = 0.045 = τ ( ρ s - ρ ) ( g ) ( D ) \tau*=0.045=\frac{\tau}{(\rho_{s}-\rho)(g)(D)}
  10. τ = 1.5 N m 2 \tau=1.5\frac{N}{m^{2}}
  11. τ b = ρ g h S o \tau_{b}=\rho ghS_{o}
  12. τ b = ρ g h S o = 1.5 N m 2 = ( 998.2 k g m 3 ) * ( 9.81 m s 2 ) * h * ( 0.004 m m ) \tau_{b}=\rho ghS_{o}=1.5\frac{N}{m^{2}}=(998.2\frac{kg}{m^{3}})*(9.81\frac{m}% {s^{2}})*h*(0.004\frac{m}{m})
  13. h = 0.15 m = 15 c m h=0.15m=15cm

Draft:Signed_particle_formulation_of_quantum_mechanics.html

  1. V = V ( < m t p l > x ) V=V(<mtpl>{{x}})
  2. d t dt
  3. γ ( < m t p l > x ( t ) ) d t \gamma(<mtpl>{{x}}(t))dt
  4. Γ ( x ) = - + D < m t p l > p V W + ( x ; p ) \Gamma(x)=\int_{-\infty}^{+\infty}{D}<mtpl>{{p}}^{\prime}V_{W}^{+}({{x}};{{p}}% ^{\prime})
  5. lim Δ p 0 + M = - + V W + ( < m t p l > x ; M Δ p ) \lim_{\Delta{p}^{\prime}\to 0+}\sum_{M=-\infty}^{+\infty}V_{W}^{+}(<mtpl>{{x}}% ;{{M\Delta p^{\prime}}})
  6. V W + ( < m t p l > x ; p ) V_{W}^{+}(<mtpl>{{x}};{{p}})
  7. V W ( < m t p l > x ; p ) = i π d d + 1 - + d x e - 2 i x p [ V ( x + x ) - V ( x - x ) ] V_{W}(<mtpl>{{x}};{{p}})=\frac{i}{\pi^{d}{\hbar^{d+1}}}\int_{-\infty}^{+\infty% }{d}{{x}}^{\prime}{e}^{-{2i\over\hbar}}{x}^{\prime}\cdot{{p}}\left[V(x+x^{% \prime})-V(x-x^{\prime})\right]
  8. p + p p+p^{\prime}
  9. p - p p-p^{\prime}
  10. V W + ( x ; p ) Γ ( x ) \frac{{V_{W}^{+}}(x;p)}{\Gamma(x)}
  11. Δ x Δ p \Delta x\Delta p
  12. 2 \frac{\hbar}{2}
  13. 2 \frac{\hbar}{2}
  14. \hbar
  15. - + D < m t p l > p V W + ( x ; p ) \int_{-\infty}^{+\infty}{D}<mtpl>{{p}}^{\prime}V_{W}^{+}({{x}};{{p}}^{\prime})
  16. lim Δ p 0 + M = - + V W + ( < m t p l > x ; M Δ p ) \lim_{\Delta{p}^{\prime}\to 0+}\sum_{M=-\infty}^{+\infty}V_{W}^{+}(<mtpl>{{x}}% ;{{M\Delta p^{\prime}}})

Draft:Six_Energies_-_Symmetry_and_Spaces.html

  1. C 3 C^{3}
  2. ψ \psi
  3. ψ \psi
  4. S 2 S^{2}
  5. S 2 S^{2}
  6. C 3 C^{3}
  7. e i φ e{i\varphi}
  8. C / t = v N \partial C/\partial t=vN
  9. [ 1 - 1 1 0 ] \begin{bmatrix}1&-1\\ 1&0\end{bmatrix}
  10. D 3 D_{3}
  11. S 2 S^{2}
  12. [ r U , R S ] σ 1 [ R S , - r b ] t r = R S 2 - r U r b = 0 [r_{U},R_{S}]\sigma_{1}[R_{S},-r_{b}]^{tr}=R_{S}^{2}-r_{U}r_{b}=0
  13. r b r_{b}
  14. S x S_{x}
  15. σ 2 \sigma_{2}
  16. S y S_{y}
  17. z = x + i y z=x+iy
  18. C C
  19. S 2 S^{2}
  20. S 2 S^{2}
  21. S 2 S^{2}
  22. σ j \sigma_{j}
  23. S 2 S^{2}
  24. S 2 S^{2}
  25. S z , σ 3 S_{z},\sigma_{3}
  26. z = x + i y z=x+iy
  27. S 2 S^{2}
  28. E r o t E_{rot}
  29. S 2 S^{2}
  30. ω \omega
  31. E r o t E_{rot}
  32. S 2 S^{2}
  33. S x , S y S_{x},S_{y}
  34. E M p o t EM_{pot}
  35. S x S_{x}
  36. S y S_{y}
  37. E h e a t E_{heat}
  38. S y S_{y}
  39. r U r_{U}
  40. r b r_{b}
  41. S 2 S^{2}
  42. E m a g n E_{magn}
  43. μ \mu
  44. s s
  45. μ = ± γ s \mu=\pm\gamma s
  46. p = m v p=mv
  47. S 2 S^{2}
  48. S 2 S^{2}
  49. μ \mu
  50. s s
  51. p p
  52. f f
  53. h f = m 2 c 2 hf=m^{2}c^{2}
  54. E k i n E_{kin}
  55. f f
  56. S 2 S^{2}
  57. E p o t E_{pot}
  58. S 2 S^{2}
  59. S 2 S^{2}
  60. C 3 C^{3}
  61. D 3 D_{3}
  62. E h e a t , m a g n E_{heat,magn}
  63. E k i n , p E_{kin},p
  64. D 3 D_{3}
  65. M = [ - 1 1 0 1 ] M=\begin{bmatrix}-1&1\\ 0&1\end{bmatrix}
  66. D 3 D_{3}
  67. i d , C id,C
  68. m P , m Q m_{P},m_{Q}
  69. m P , m Q m_{P},m_{Q}

Draft:Strategarchery.html

  1. i = 1 | A | x i - i = 1 | O | y i \sum_{i=1}^{|A|}x_{i}-\sum_{i=1}^{|O|}y_{i}
  2. A A
  3. O O

Draft:Sukeima.html

  1. n × n n\times n
  2. n × n n\times n
  3. U t + 1 ( N i , j ) = U t ( N i , j ) + 2 - N G ( N i , j ) V t ( N ) U_{t+1}(N_{i,j})=U_{t}(N_{i,j})+2-\sum_{N\in G(N_{i,j})}V_{t}(N)
  4. V t + 1 ( N i , j ) = { 1 if U t + 1 ( N i , j ) > 3 0 if U t + 1 ( N i , j ) < 0 V t ( N i , j ) otherwise , V_{t+1}(N_{i,j})=\left\{\begin{array}[]{ll}1&\mbox{if}~{}\,\,U_{t+1}(N_{i,j})>% 3\\ 0&\mbox{if}~{}\,\,U_{t+1}(N_{i,j})<0\\ V_{t}(N_{i,j})&\mbox{otherwise}~{},\end{array}\right.
  5. t t
  6. U ( N i , j ) U(N_{i,j})
  7. i i
  8. j j
  9. V ( N i , j ) V(N_{i,j})
  10. i i
  11. j j
  12. G ( N i , j ) G(N_{i,j})
  13. t t
  14. t + 1 t+1

Draft:Suslin_homology.html

  1. H i ( X , A ) = Tor i ( C , A ) H_{i}(X,A)=\operatorname{Tor}_{i}^{\mathbb{Z}}(C,A)
  2. i × X \triangle^{i}\times X
  3. i \triangle^{i}
  4. i \triangle^{i}

Draft:Tautological_bundle.html

  1. 𝒪 ( i ) \mathcal{O}(i)
  2. 𝒪 ( 1 ) \mathcal{O}(1)
  3. Pic 𝐏 𝐤 n = \mathrm{Pic}\ \mathbf{P}^{n}_{\mathbf{k}}=\mathbb{Z}

Draft:TEMA_indicator.html

  1. 𝑇𝐸𝑀𝐴 = 3 * E M A - 3 * E M A ( E M A ) + E M A ( E M A ( E M A ) ) \,\textit{TEMA}={3*EMA-3*EMA(EMA)+EMA(EMA(EMA))}

Draft:Template:calculate::dz=d0÷(1+z)::doc.html

  1. d z = d 0 1 + z d_{z}=\frac{d_{0}}{1+z}

Draft:Template:calculate::v=c*(((1+z)^2-1)÷((1+z)^2+1))::doc.html

  1. v = c * ( 1 + z ) 2 - 1 ( 1 + z ) 2 + 1 v=c*{(1+z)^{2}-1\over(1+z)^{2}+1}

Draft:Template:calculate::z=√((c+v)÷(c-v))-1::doc.html

  1. z = c + v c - v - 1 z=\sqrt{\frac{c+v}{c-v}\ }-1

Draft:Template:convert::kps::Mly::doc.html

  1. D = v H 0 D=\frac{v}{H_{0}}

Draft:Template:convert::kps::Mpc::doc.html

  1. D = v H 0 D=\frac{v}{H_{0}}

Draft:Template:convert::kps::z::doc.html

  1. z = c + v c - v - 1 z=\sqrt{\frac{c+v}{c-v}\ }-1

Draft:Template:convert::Mly::kps::doc.html

  1. v = D * H 0 v=D*H_{0}

Draft:Template:convert::Mpc::kps::doc.html

  1. v = H 0 * D v=H_{0}*D

Draft:Template:convert::Mpc::yr::doc.html

  1. t = D * 3261633.44 t=D*3261633.44

Draft:Template:convert::z::kps::doc.html

  1. v = c * ( 1 + z ) 2 - 1 ( 1 + z ) 2 + 1 v=c*{(1+z)^{2}-1\over(1+z)^{2}+1}

Draft:Tensor_product_of_representations.html

  1. V 1 , V n V_{1},\cdots V_{n}
  2. g ( v 1 v n ) = g v 1 g v n . g\cdot(v_{1}\otimes\dots\otimes v_{n})=gv_{1}\otimes\dots\otimes gv_{n}.
  3. End ( V ) V * V \operatorname{End}(V)\simeq V^{*}\otimes V
  4. ( g T ) ( v ) = g T ( g - 1 v ) (g\cdot T)(v)=gT(g^{-1}v)
  5. E m E^{\otimes m}
  6. GL ( n , ) \operatorname{GL}(n,\mathbb{C})

Draft:The_Cannon_Effect.html

  1. f t f^{t}
  2. f t f^{t}
  3. 0 < d ( x , y ) < δ 0<d(x,y)<\delta
  4. d ( f τ ( x ) , f τ ( y ) ) > e a τ d ( x , y ) . d(f^{\tau}(x),f^{\tau}(y))>\mathrm{e}^{a\tau}\,d(x,y).

Draft:The_study_of_the_momentum_transfer_due_to_fluid_friction_and_heat_transfer_between_a_stream_and_a_solid_object_in_internal_flow,.html

  1. ( X / D R e ) = 0.026 \left(\frac{X/D}{Re}\right)=0.026
  2. ( X / D R e ) = 0.01 \left(\frac{X/D}{Re}\right)=0.01
  3. \sim
  4. \sim
  5. v D U L v\sim\frac{DU}{L}
  6. d P d y = 0 \frac{dP}{dy}=0
  7. d P d x = μ d 2 u d y 2 = constant \frac{dP}{dx}=\mu\frac{d^{2}u}{dy^{2}}=\,\text{constant}
  8. ± \pm
  9. u = 1.5 U [ 1 - ( y 0.5 D ) 2 ] u=1.5U\left[1-\left(\frac{y}{0.5D}\right)^{2}\right]
  10. U = D 2 12 μ ( - d P d x ) U=\frac{D^{2}}{12\mu}\left(\frac{-dP}{dx}\right)
  11. d P d x = μ ( d 2 u d r 2 + d u r d r ) \frac{dP}{dx}=\mu(\frac{d^{2}u}{dr^{2}}+\frac{du}{rdr})
  12. u = U [ 1 - ( r 2 R 2 ) 2 ] u=U\left[1-\left(\frac{r^{2}}{R^{2}}\right)^{2}\right]
  13. U = R 2 4 μ ( - d P d x ) U=\frac{R^{2}}{4\mu}\left(\frac{-dP}{dx}\right)

Draft:Thom_spectrum.html

  1. Ω n U = lim r π n + r M U ( r ) \Omega_{n}^{U}=\underrightarrow{\lim}_{r}\pi_{n+r}MU(r)

Draft:Trace_morphism.html

  1. f * 𝒪 X 𝒪 S f_{*}\mathcal{O}_{X}\to\mathcal{O}_{S}

Draft:Ultra-Small_World.html

  1. L log log N L\propto\log\log N
  2. L log N L\propto\log N
  3. L log log N L\propto\log\log N
  4. L log log N L\propto\log\log N

Draft:Uniform_Stability_and_Generalization_in_learning_theory.html

  1. L L
  2. L L
  3. S Z m , i { 1 , , m } , sup z Z | V ( f S , z ) - V ( f S i , z ) | β \forall S\in Z^{m},\forall i\in\{1,...,m\},\sup_{z\in Z}|V(f_{S},z)-V(f_{S^{i}% },z)|\leq\beta
  4. S Z m , i { 1 , , m } , S { sup z Z | V ( f S , z ) - V ( f S i , z ) | β } 1 - δ \forall S\in Z^{m},\forall i\in\{1,...,m\},\mathbb{P}_{S}\{\sup_{z\in Z}|V(f_{% S},z)-V(f_{S^{i}},z)|\leq\beta\}\geq 1-\delta
  5. L L
  6. 1 - δ 1-\delta
  7. I [ f S ] - I S [ f S ] β n + ( 2 n β n + M ) l n ( 2 δ ) 2 n I[f_{S}]-I_{S}[f_{S}]\leq\beta_{n}+(2n\beta_{n}+M)\sqrt{\frac{ln(\frac{2}{% \delta})}{2n}}
  8. β n = O ( 1 n ) \beta_{n}=O\left(\frac{1}{n}\right)
  9. I [ f S ] - I S [ f S ] O ( 1 n ) I[f_{S}]-I_{S}[f_{S}]\leq O\left(\frac{1}{\sqrt{n}}\right)
  10. z z
  11. f S λ = a r g min f ( 1 n i = 1 n V ( f ( x i ) , y i ) + λ || f || k 2 ) f^{\lambda}_{S}=arg\min_{f\in\mathcal{H}}\left(\frac{1}{n}\sum^{n}_{i=1}V(f(x_% {i}),y_{i})+\lambda||f||^{2}_{\mathbb{R}^{k}}\right)
  12. | V ( f i ( x ) , y ) - V ( f 2 ( x ) , y ) | L || f 1 - f 2 || |V(f_{i}(x),y^{\prime})-V(f_{2}(x),y^{\prime})|\leq L||f_{1}-f_{2}||_{\infty}
  13. || f - f || κ || f - f || K ||f-f^{\prime}||_{\infty}\leq\kappa||f-f^{\prime}||_{\mathbb{R}^{K}}
  14. f , f f,f^{\prime}\in\mathcal{H}
  15. || f S λ - f S i , z λ || K 2 L || f S λ - f S i , z λ || λ n ||f^{\lambda}_{S}-f^{\lambda}_{S^{i,z}}||^{2}_{\mathbb{R}^{K}}\leq\frac{L||f^{% \lambda}_{S}-f^{\lambda}_{S^{i,z}}||_{\infty}}{\lambda n}
  16. | I [ f S λ ] - I S [ f S λ ] | L 2 κ 2 λ n + ( 2 L 2 κ 2 λ n + M ) 2 l n ( 2 δ ) n |I[f^{\lambda}_{S}]-I_{S}[f^{\lambda}_{S}]|\leq\frac{L^{2}\kappa^{2}}{\lambda n% }+(\frac{2L^{2}\kappa^{2}}{\lambda n}+M)\sqrt{\frac{2ln(\frac{2}{\delta})}{n}}
  17. 1 - δ 1-\delta
  18. λ \lambda
  19. O ( 1 n ) O\left(\frac{1}{\sqrt{n}}\right)
  20. λ \lambda
  21. λ \lambda
  22. λ \lambda
  23. L L
  24. lim n I S [ f S ] = I [ f S ] \lim_{n\to\infty}I_{S}[f_{S}]=I[f_{S}]
  25. L L
  26. P r [ | I S [ f S ] - I [ f S ] | ϵ ( n ) ] 1 - δ Pr[|I_{S}[f_{S}]-I[f_{S}]|\leq\epsilon(n)]\geq 1-\delta
  27. ϵ ( n ) \epsilon(n)
  28. L L
  29. ( S , z ) Z n + 1 , i { 1 , , n } , sup z Z | I [ f S ] - I [ f S i , z ] | β P ( | I [ f S ] - 𝔼 S [ I [ f S ] ] | ϵ ) 2 e ( - 2 ϵ 2 n β 2 ) \forall(S,z)\in Z^{n+1},\forall i\in\{1,...,n\},\sup_{z\in Z}|I[f_{S}]-I[f_{S^% {i,z}}]|\leq\beta\implies P(|I[f_{S}]-\mathbb{E}_{S}[I[f_{S}]]|\geq\epsilon)% \leq 2e^{\left(\frac{-2\epsilon^{2}}{n\beta^{2}}\right)}
  30. 1 - δ 1-\delta
  31. I [ f S ] I[f_{S}]
  32. 𝔼 S [ I [ f S ] ] \mathbb{E}_{S}[I[f_{S}]]
  33. δ \delta
  34. ϵ \epsilon
  35. ϵ = n β l n ( 2 δ ) 2 n \epsilon=n\beta\sqrt{\frac{ln(\frac{2}{\delta})}{2n}}
  36. 1 - δ 1-\delta
  37. | I [ f S ] - 𝔼 S [ I [ f S ] ] | n β l n ( 2 δ ) 2 n I [ f S ] 𝔼 S [ I [ f S ] ] + n β l n ( 2 δ ) 2 n |I[f_{S}]-\mathbb{E}_{S}[I[f_{S}]]|\leq n\beta\sqrt{\frac{ln(\frac{2}{\delta})% }{2n}}\implies I[f_{S}]\leq\mathbb{E}_{S}[I[f_{S}]]+n\beta\sqrt{\frac{ln(\frac% {2}{\delta})}{2n}}
  38. I S [ f S ] I_{S}[f_{S}]
  39. I [ f S ] - I S [ f S ] 𝔼 S [ I [ f S ] ] - I S [ f S ] + n β l n ( 2 δ ) 2 n I[f_{S}]-I_{S}[f_{S}]\leq\mathbb{E}_{S}[I[f_{S}]]-I_{S}[f_{S}]+n\beta\sqrt{% \frac{ln(\frac{2}{\delta})}{2n}}
  40. 𝔼 S [ I [ f S ] ] - I S [ f S ] \mathbb{E}_{S}[I[f_{S}]]-I_{S}[f_{S}]
  41. 𝔼 S [ I [ f S ] ] - I S [ f S ] \mathbb{E}_{S}[I[f_{S}]]-I_{S}[f_{S}]
  42. 𝔼 S [ I [ f S ] ] - I S [ f S ] β + ( n β + M ) l n ( 2 δ ) 2 n \mathbb{E}_{S}[I[f_{S}]]-I_{S}[f_{S}]\leq\beta+(n\beta+M)\sqrt{\frac{ln(\frac{% 2}{\delta})}{2n}}
  43. V ( f , z ) V(f,z)
  44. I [ f S ] - I S [ f S ] β + ( 2 n β + M ) l n ( 2 δ ) 2 n I[f_{S}]-I_{S}[f_{S}]\leq\beta+(2n\beta+M)\sqrt{\frac{ln(\frac{2}{\delta})}{2n}}
  45. V ( f , z ) V(f,z)
  46. 1 - δ 1-\delta
  47. 𝔼 S [ I [ f S ] ] - I S [ f S ] β + ( n β + M ) l n ( 2 δ ) 2 n \mathbb{E}_{S}[I[f_{S}]]-I_{S}[f_{S}]\leq\beta+(n\beta+M)\sqrt{\frac{ln(\frac{% 2}{\delta})}{2n}}
  48. I S [ f S ] I_{S}[f_{S}]
  49. I S [ f S ] I_{S}[f_{S}]
  50. I S [ f S ] I_{S}[f_{S}]
  51. i , sup S , z | I S [ f S ] - I S i , z [ f S i , z ] | c i \forall i,\sup_{S,z}|I_{S}[f_{S}]-I_{S^{i,z}}[f_{S^{i,z}}]|\leq c_{i}
  52. 𝔼 S [ I S [ f S ] ] \mathbb{E}_{S}[I_{S}[f_{S}]]
  53. c i c_{i}
  54. i , sup S , z | I S [ f S ] - I S i , z [ f S i , z ] | = | 1 n i = 1 n V ( f s , z i ) - ( 1 n j i V ( f S i , z , z j ) + 1 n V ( f S i , z , z ) ) | \forall i,\sup_{S,z}|I_{S}[f_{S}]-I_{S^{i,z}}[f_{S}^{i,z}]|=|\frac{1}{n}\sum^{% n}_{i=1}V(f_{s},z_{i})-(\frac{1}{n}\sum_{j\neq i}V(f_{S^{i,z}},z_{j})+\frac{1}% {n}V(f_{S^{i,z}},z))|
  55. z i z j z_{i}\neq z_{j}
  56. 1 n j i | V ( f S , z j ) - V ( f S i , z , z j ) | + 1 n | V ( f S , z i ) - V ( f S i , z , z ) | \leq\frac{1}{n}\sum_{j\neq i}|V(f_{S},z_{j})-V(f_{S^{i,z}},z_{j})|+\frac{1}{n}% |V(f_{S},z_{i})-V(f_{S^{i,z}},z)|
  57. V ( f S , z j ) - V ( f S i , z , z j ) V(f_{S},z_{j})-V(f_{S^{i,z}},z_{j})
  58. β \beta
  59. V ( f S , z j ) - V ( f S i , z , z j ) β V(f_{S},z_{j})-V(f_{S^{i,z}},z_{j})\leq\beta
  60. V ( f S , z i ) - V ( f S i , z , z ) M V(f_{S},z_{i})-V(f_{S^{i,z}},z)\leq M
  61. n - 1 n β + M n β + M n \leq\frac{n-1}{n}\beta+\frac{M}{n}\leq\beta+\frac{M}{n}
  62. c i c_{i}
  63. β + M n , i \beta+\frac{M}{n},\forall i
  64. i , sup S , z | I S [ f S ] - I S < m t p l > i , z [ f S i , z ] | β + M n P r [ | I S [ f S ] - 𝔼 S [ I S [ f S ] ] | ϵ ] 2 e - 2 n ϵ 2 ( n β + M ) 2 \forall i,\sup_{S,z}|I_{S}[f_{S}]-I_{S^{<}mtpl>{{i,z}}}[f_{S}^{i,z}]|\leq\beta% +\frac{M}{n}\implies Pr[|I_{S}[f_{S}]-\mathbb{E}_{S}[I_{S}[f_{S}]]|\geq% \epsilon]\leq 2e^{\frac{-2n\epsilon^{2}}{(n\beta+M)^{2}}}
  65. 1 - δ 1-\delta
  66. ϵ \epsilon
  67. ϵ = ( n β + M ) l n ( 2 δ ) 2 n \epsilon=(n\beta+M)\sqrt{\frac{ln(\frac{2}{\delta})}{2n}}
  68. | I S [ f S ] - 𝔼 S [ I S [ f S ] ] | ( n β + M ) l n ( 2 δ ) 2 n I S [ f S ] 𝔼 S [ I S [ f S ] ] + ( n β + M ) l n ( 2 δ ) 2 n |I_{S}[f_{S}]-\mathbb{E}_{S}[I_{S}[f_{S}]]|\leq(n\beta+M)\sqrt{\frac{ln(\frac{% 2}{\delta})}{2n}}\implies I_{S}[f_{S}]\leq\mathbb{E}_{S}[I_{S}[f_{S}]]+(n\beta% +M)\sqrt{\frac{ln(\frac{2}{\delta})}{2n}}
  69. 𝔼 S [ I [ f S ] ] \mathbb{E}_{S}[I[f_{S}]]
  70. I S [ f S ] - 𝔼 S [ I [ f S ] ] 𝔼 S [ I S [ f S ] ] - 𝔼 S [ I [ f S ] ] + ( n β + M ) l n ( 2 δ ) 2 n I_{S}[f_{S}]-\mathbb{E}_{S}[I[f_{S}]]\leq\mathbb{E}_{S}[I_{S}[f_{S}]]-\mathbb{% E}_{S}[I[f_{S}]]+(n\beta+M)\sqrt{\frac{ln(\frac{2}{\delta})}{2n}}
  71. 𝔼 S [ I S [ f S ] ] - 𝔼 S [ I [ f S ] ] \mathbb{E}_{S}[I_{S}[f_{S}]]-\mathbb{E}_{S}[I[f_{S}]]
  72. β \beta
  73. 𝔼 S [ I S [ f S ] ] - 𝔼 S [ I [ f S ] ] = 𝔼 z [ 𝔼 S [ I S [ f S ] ] ] - 𝔼 S [ I [ f S ] ] = 𝔼 z [ 𝔼 S [ 1 n i = 1 n V ( f , z i ) ] ] - 𝔼 S [ 𝔼 z [ V ( f S , z ) ] ] \mathbb{E}_{S}[I_{S}[f_{S}]]-\mathbb{E}_{S}[I[f_{S}]]=\mathbb{E}_{z}[\mathbb{E% }_{S}[I_{S}[f_{S}]]]-\mathbb{E}_{S}[I[f_{S}]]=\mathbb{E}_{z}[\mathbb{E}_{S}[% \frac{1}{n}\sum^{n}_{i=1}V(f,z_{i})]]-\mathbb{E}_{S}[\mathbb{E}_{z}[V(f_{S},z)]]
  74. 𝔼 z [ 𝔼 S [ 1 n i = 1 n V ( f , z i ) - V ( f S , z ) ] ] = 𝔼 z [ 𝔼 S [ 1 n i = 1 n ( V ( f S i , z , z ) - V ( f S , z ) ) ] ] \mathbb{E}_{z}[\mathbb{E}_{S}[\frac{1}{n}\sum^{n}_{i=1}V(f,z_{i})-V(f_{S},z)]]% =\mathbb{E}_{z}[\mathbb{E}_{S}[\frac{1}{n}\sum^{n}_{i=1}\left(V(f_{S^{i,z}},z)% -V(f_{S},z))\right]]
  75. V ( f S i , z , z ) - V ( f S , z ) β V(f_{S^{i,z}},z)-V(f_{S},z)\leq\beta
  76. 𝔼 S [ I S [ f S ] ] - 𝔼 S [ I [ f S ] ] β \mathbb{E}_{S}[I_{S}[f_{S}]]-\mathbb{E}_{S}[I[f_{S}]]\leq\beta
  77. 1 - δ 1-\delta
  78. 𝔼 S [ I [ f S ] ] - I S [ f S ] β + ( n β + M ) l n ( 2 δ ) 2 n \mathbb{E}_{S}[I[f_{S}]]-I_{S}[f_{S}]\leq\beta+(n\beta+M)\sqrt{\frac{ln(\frac{% 2}{\delta})}{2n}}

Draft:Wang_sequence.html

  1. p : E B p:E\to B
  2. H t ( F ; G ) \dots\to H_{t}(F;G)\to

Draft:XTX.html

  1. X T X X^{T}X
  2. f ( β ) = 1 2 ( y - X β ) 2 f(\beta)=\frac{1}{2}(y-X\beta)^{2}

Draft:Алгоритъм_на_Белман-Форд.html

  1. | V | - 1 |V|-1
  2. | V | |V|
  3. O ( | V | | E | ) O(|V|\cdot|E|)
  4. | V | |V|
  5. | E | |E|
  6. i i
  7. | V | - 1 |V|-1
  8. | V | - 1 |V|-1
  9. | V | |V|

Draft_tube.html

  1. ( V 2 2 - V 3 2 ) - 2 g h d V 2 2 \frac{(V_{2}^{2}-V_{3}^{2})-2gh_{d}}{V_{2}^{2}}
  2. z 2 + p 2 ρ g + V 2 2 2 g = z 3 + p 3 ρ g + V 3 2 2 g , z_{2}\,+\,\frac{p_{2}}{\rho g}\,+\,\frac{V_{2}^{2}}{2\,g}\,=\,z_{3}\,+\,\frac{% p_{3}}{\rho g}\,+\,\frac{V_{3}^{2}}{2\,g},
  3. p 2 ρ g = - [ z + V 2 2 - V 3 2 2 g ] \frac{p_{2}}{\rho g}\,=\,-\,\left[z\,+\,\frac{V_{2}^{2}-V_{3}^{2}}{2\,g}\right]\,

Drinfeld-Sokolov-Wilson_equation.html

  1. u t + 3 v v x = 0 v t = 2 3 v x 3 + u x v + 2 u v x \begin{aligned}&\displaystyle\frac{\partial u}{\partial t}+3v\frac{\partial v}% {\partial x}=0\\ &\displaystyle\frac{\partial v}{\partial t}=2\frac{\partial^{3}v}{\partial x^{% 3}}+\frac{\partial u}{\partial x}v+2u\frac{\partial v}{\partial x}\end{aligned}

Drop_impact.html

  1. σ / R 2 \sigma/R^{2}
  2. ρ R / τ 2 σ / R 2 \rho R/\tau^{2}\propto\sigma/R^{2}
  3. ρ \rho
  4. τ \tau
  5. σ \sigma
  6. τ ρ / σ R 3 / 2 \tau\propto\sqrt{\rho/\sigma}R^{3/2}

Droplet-shaped_wave.html

  1. t = 0 t=0
  2. v = β c v=βc
  3. c c
  4. β > 1 β>1
  5. τ = c t , ρ , φ , z τ=ct,ρ,φ,z
  6. s ( τ , ρ , z ) = δ ( ρ ) 2 π ρ J ( τ , z ) H ( β τ - z ) H ( z ) , s(\tau,\rho,z)=\frac{\delta(\rho)}{2\pi\rho}J(\tau,z)H(\beta\tau-z)H(z),
  7. δ ( ) δ(•)
  8. H ( ) H(•)
  9. J ( τ , z ) J(τ,z)
  10. H ( β τ - z ) H ( z ) = 0 H(βτ-z)H(z)=0
  11. s ( τ , ρ , z ) = 0 s(τ,ρ,z)=0
  12. τ = 0 τ=0
  13. ψ ( τ , ρ , z ) ψ(τ,ρ,z)
  14. ψ ψ
  15. [ τ 2 - ρ - 1 ρ ( ρ ρ ) - z 2 ] ψ ( τ , ρ , z ) = s ( τ , ρ , z ) ψ ( τ , ρ , z ) = 0 for τ < 0 \begin{aligned}&\displaystyle\left[{\partial_{\tau}^{2}-{\rho^{-1}}{\partial_{% \rho}}\left({\rho{\partial_{\rho}}}\right)-\partial_{z}^{2}}\right]\psi\left(% \tau,\rho,z\right)=s\left(\tau,\rho,z\right)\\ &\displaystyle\psi\left(\tau,\rho,z\right)=0\quad\mathrm{for}\quad\tau<0\end{aligned}

Droz-Farny_line_theorem.html

  1. T T
  2. A A
  3. B B
  4. C C
  5. H H
  6. L 1 L_{1}
  7. L 2 L_{2}
  8. H H
  9. A 1 A_{1}
  10. B 1 B_{1}
  11. C 1 C_{1}
  12. L 1 L_{1}
  13. B C BC
  14. C A CA
  15. A B AB
  16. A 2 A_{2}
  17. B 2 B_{2}
  18. C 2 C_{2}
  19. L 2 L_{2}
  20. A 1 A 2 A_{1}A_{2}
  21. B 1 B 2 B_{1}B_{2}
  22. C 1 C 2 C_{1}C_{2}
  23. T T
  24. A A
  25. B B
  26. C C
  27. P P
  28. A A
  29. B B
  30. C C
  31. L L
  32. P P
  33. A 1 A_{1}
  34. B 1 B_{1}
  35. C 1 C_{1}
  36. B C BC
  37. C A CA
  38. A B AB
  39. P A 1 PA_{1}
  40. P B 1 PB_{1}
  41. P C 1 PC_{1}
  42. P A PA
  43. P B PB
  44. P C PC
  45. L L
  46. A 1 A_{1}
  47. B 1 B_{1}
  48. C 1 C_{1}
  49. P P
  50. T T

Drucker_stability.html

  1. d s y m b o l σ : d s y m b o l ε 0 . \,\text{d}symbol{\sigma}:\,\text{d}symbol{\varepsilon}\geq 0\,.
  2. d s y m b o l σ : d s y m b o l ε p 0 \,\text{d}symbol{\sigma}:\,\text{d}symbol{\varepsilon}_{p}\geq 0

Dryden_Wind_Turbulence_Model.html

  1. Φ u g ( Ω ) = σ u 2 2 L u π 1 1 + ( L u Ω ) 2 Φ v g ( Ω ) = σ v 2 2 L v π 1 + 12 ( L v Ω ) 2 ( 1 + 4 ( L v Ω ) 2 ) 2 Φ w g ( Ω ) = σ w 2 2 L w π 1 + 12 ( L w Ω ) 2 ( 1 + 4 ( L w Ω ) 2 ) 2 \begin{aligned}\displaystyle\Phi_{u_{g}}(\Omega)&\displaystyle=\sigma_{u}^{2}% \frac{2L_{u}}{\pi}\frac{1}{1+(L_{u}\Omega)^{2}}\\ \displaystyle\Phi_{v_{g}}(\Omega)&\displaystyle=\sigma_{v}^{2}\frac{2L_{v}}{% \pi}\frac{1+12(L_{v}\Omega)^{2}}{\left(1+4(L_{v}\Omega)^{2}\right)^{2}}\\ \displaystyle\Phi_{w_{g}}(\Omega)&\displaystyle=\sigma_{w}^{2}\frac{2L_{w}}{% \pi}\frac{1+12(L_{w}\Omega)^{2}}{\left(1+4(L_{w}\Omega)^{2}\right)^{2}}\end{aligned}
  2. Ω = ω V Φ i ( Ω ) = V Φ i ( ω V ) \begin{aligned}\displaystyle\Omega&\displaystyle=\frac{\omega}{V}\\ \displaystyle\Phi_{i}(\Omega)&\displaystyle=V\Phi_{i}\left(\frac{\omega}{V}% \right)\end{aligned}
  3. p g = w g y q g = w g x r g = - v g x \begin{aligned}\displaystyle p_{g}&\displaystyle=\frac{\partial w_{g}}{% \partial y}\\ \displaystyle q_{g}&\displaystyle=\frac{\partial w_{g}}{\partial x}\\ \displaystyle r_{g}&\displaystyle=-\frac{\partial v_{g}}{\partial x}\end{aligned}
  4. Φ p g ( ω ) = σ w 2 2 V L w 0.8 ( 2 π L w 4 b ) 1 3 1 + ( 4 b ω π V ) 2 Φ q g ( ω ) = ± ( ω V ) 2 1 + ( 4 b ω π V ) 2 Φ w g ( ω ) Φ r g ( ω ) = ( ω V ) 2 1 + ( 3 b ω π V ) 2 Φ v g ( ω ) \begin{aligned}\displaystyle\Phi_{p_{g}}(\omega)&\displaystyle=\frac{\sigma_{w% }^{2}}{2VL_{w}}\frac{0.8\left(\frac{2\pi L_{w}}{4b}\right)^{\frac{1}{3}}}{1+% \left(\frac{4b\omega}{\pi V}\right)^{2}}\\ \displaystyle\Phi_{q_{g}}(\omega)&\displaystyle=\frac{\pm\left(\frac{\omega}{V% }\right)^{2}}{1+\left(\frac{4b\omega}{\pi V}\right)^{2}}\Phi_{w_{g}}(\omega)\\ \displaystyle\Phi_{r_{g}}(\omega)&\displaystyle=\frac{\mp\left(\frac{\omega}{V% }\right)^{2}}{1+\left(\frac{3b\omega}{\pi V}\right)^{2}}\Phi_{v_{g}}(\omega)% \end{aligned}
  5. Φ y ( ω ) = | G ( i ω ) | 2 \Phi_{y}(\omega)=|G(i\omega)|^{2}
  6. G u g ( s ) = σ u 2 L u π V 1 1 + L u V s G v g ( s ) = σ v 2 L v π V 1 + 2 3 L v V s ( 1 + 2 L v V s ) 2 G w g ( s ) = σ w 2 L w π V 1 + 2 3 L w V s ( 1 + 2 L w V s ) 2 G p g ( s ) = σ w 0.8 V ( π 4 b ) 1 6 ( 2 L w ) 1 3 ( 1 + 4 b π V s ) G q g ( s ) = ± s V 1 + 4 b π V s G w g ( s ) G r g ( s ) = s V 1 + 3 b π V s G v g ( s ) \begin{aligned}\displaystyle G_{u_{g}}(s)&\displaystyle=\sigma_{u}\sqrt{\frac{% 2L_{u}}{\pi V}}\frac{1}{1+\frac{L_{u}}{V}s}\\ \displaystyle G_{v_{g}}(s)&\displaystyle=\sigma_{v}\sqrt{\frac{2L_{v}}{\pi V}}% \frac{1+\frac{2\sqrt{3}L_{v}}{V}s}{\left(1+\frac{2L_{v}}{V}s\right)^{2}}\\ \displaystyle G_{w_{g}}(s)&\displaystyle=\sigma_{w}\sqrt{\frac{2L_{w}}{\pi V}}% \frac{1+\frac{2\sqrt{3}L_{w}}{V}s}{\left(1+\frac{2L_{w}}{V}s\right)^{2}}\\ \displaystyle G_{p_{g}}(s)&\displaystyle=\sigma_{w}\sqrt{\frac{0.8}{V}}\frac{% \left(\frac{\pi}{4b}\right)^{\frac{1}{6}}}{(2L_{w})^{\frac{1}{3}}\left(1+\frac% {4b}{\pi V}s\right)}\\ \displaystyle G_{q_{g}}(s)&\displaystyle=\frac{\pm\frac{s}{V}}{1+\frac{4b}{\pi V% }s}G_{w_{g}}(s)\\ \displaystyle G_{r_{g}}(s)&\displaystyle=\frac{\mp\frac{s}{V}}{1+\frac{3b}{\pi V% }s}G_{v_{g}}(s)\end{aligned}

DTDP-L-rhamnose_4-epimerase.html

  1. \rightleftharpoons

Dual-beta.html

  1. ( r j - r f ) t = a j + D + β j + ( r m + - r f ) t D + a j - ( 1 - D ) + β j - ( r m - - r f ) t ( 1 - D ) + ϵ t (r_{j}-r_{f})_{t}=a_{j}^{+}D+\beta_{j}^{+}(r_{m}^{+}-r_{f})_{t}D+a_{j}^{-}(1-D% )+\beta_{j}^{-}(r_{m}^{-}-r_{f})_{t}(1-D)+\epsilon_{t}
  2. ( r j - r f ) t (r_{j}-r_{f})_{t}
  3. a j + a_{j}^{+}
  4. a j - a_{j}^{-}
  5. β j + ( r m + - r f ) t \beta_{j}^{+}(r_{m}^{+}-r_{f})_{t}
  6. β j - ( r m - - r f ) t \beta_{j}^{-}(r_{m}^{-}-r_{f})_{t}
  7. a j + a_{j}^{+}
  8. β j + \beta_{j}^{+}
  9. a j - a_{j}^{-}
  10. β j - \beta_{j}^{-}
  11. r m + = r m r_{m}^{+}=r_{m}
  12. r m - = r m r_{m}^{-}=r_{m}
  13. D D
  14. ϵ t \epsilon_{t}

Dual-phase_evolution.html

  1. G = N , E \textstyle G=\langle N,E\rangle
  2. N \textstyle N
  3. E { ( x , y ) x , y N } \textstyle E\subset\{(x,y)\mid x,y\in N\}
  4. ( x , y ) \textstyle(x,y)
  5. x \textstyle x
  6. y \textstyle y

Duct_modes.html

  1. p = α i ϕ i p=\sum\alpha_{i}\phi_{i}
  2. α i \alpha_{i}
  3. ϕ i \phi_{i}

Dwell_time_(radar).html

  1. T D = θ A Z 6 n T_{D}=\frac{\theta_{AZ}}{6\cdot n}

Dynamic_connectivity.html

  1. O ( α ( n ) ) O(\alpha(n))
  2. O ( log ( n ) ) O(\log(n))
  3. O ( log ( n ) ) O(\log(n))
  4. O ( n ) O(n)
  5. O ( l g 2 n ) O(lg^{2}n)
  6. O ( l g 2 n ) O(lg^{2}n)
  7. O ( l g 2 n ) O(lg^{2}n)
  8. O ( l g n ) O(lgn)
  9. O ( l g n ) O(lgn)
  10. O ( l g 2 n ) O(lg^{2}n)
  11. O ( l g n / l g l g n ) O(lgn/lglgn)
  12. O ( n ) O(n)
  13. ( 1 - 1 / 9 ) C lg n = 2 - 0.17 C lg n = n - 0.17 C (1-1/9)^{C\lg{n}}=2^{-0.17C\lg{n}}=n^{-0.17C}

Dynamical_pictures_(quantum_mechanics).html

  1. ψ | p ^ | ψ \langle\psi|\hat{p}|\psi\rangle
  2. p ^ \hat{p}
  3. t H = 0 \partial_{t}H=0
  4. | ψ ( t ) = U ( t , t 0 ) | ψ ( t 0 ) . |\psi(t)\rangle=U(t,t_{0})|\psi(t_{0})\rangle.
  5. ψ ( t ) | = ψ ( t 0 ) | U ( t , t 0 ) . \langle\psi(t)|=\langle\psi(t_{0})|U^{\dagger}(t,t_{0}).
  6. ψ ( t ) | ψ ( t ) = ψ ( t 0 ) | U ( t , t 0 ) U ( t , t 0 ) | ψ ( t 0 ) = ψ ( t 0 ) | ψ ( t 0 ) . \langle\psi(t)|\psi(t)\rangle=\langle\psi(t_{0})|U^{\dagger}(t,t_{0})U(t,t_{0}% )|\psi(t_{0})\rangle=\langle\psi(t_{0})|\psi(t_{0})\rangle.
  7. U ( t , t 0 ) U ( t , t 0 ) = I . U^{\dagger}(t,t_{0})U(t,t_{0})=I.
  8. | ψ ( t 0 ) = U ( t 0 , t 0 ) | ψ ( t 0 ) . |\psi(t_{0})\rangle=U(t_{0},t_{0})|\psi(t_{0})\rangle.
  9. U ( t , t 0 ) = U ( t , t 1 ) U ( t 1 , t 0 ) . U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}).
  10. i d d t | ψ ( t ) = H | ψ ( t ) , i\hbar\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle,
  11. | ψ ( t ) = U ( t ) | ψ ( 0 ) |\psi(t)\rangle=U(t)|\psi(0)\rangle
  12. i d d t U ( t ) | ψ ( 0 ) = H U ( t ) | ψ ( 0 ) . i\hbar{d\over dt}U(t)|\psi(0)\rangle=HU(t)|\psi(0)\rangle.
  13. | ψ ( 0 ) |\psi(0)\rangle
  14. i d d t U ( t ) = H U ( t ) . i\hbar{d\over dt}U(t)=HU(t).
  15. U ( t ) = e - i H t / . U(t)=e^{-iHt/\hbar}.
  16. e - i H t / = 1 - i H t - 1 2 ( H t ) 2 + . e^{-iHt/\hbar}=1-\frac{iHt}{\hbar}-\frac{1}{2}\left(\frac{Ht}{\hbar}\right)^{2% }+\cdots.
  17. | ψ ( t ) = e - i H t / | ψ ( 0 ) . |\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle.
  18. | ψ ( 0 ) |\psi(0)\rangle
  19. | ψ ( t ) = e - i E t / | ψ ( 0 ) . |\psi(t)\rangle=e^{-iEt/\hbar}|\psi(0)\rangle.
  20. U ( t ) = exp ( - i 0 t H ( t ) d t ) , U(t)=\exp\left({-\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime})\,dt^{\prime}}\right),
  21. U ( t ) = T exp ( - i 0 t H ( t ) d t ) , U(t)=\mathrm{T}\exp\left({-\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime})\,dt^{% \prime}}\right),
  22. | ψ |\psi\rangle
  23. | ψ ( t ) |\psi(t)\rangle
  24. A t = ψ ( t ) | A | ψ ( t ) . \langle A\rangle_{t}=\langle\psi(t)|A|\psi(t)\rangle.
  25. | ψ |\psi\rangle
  26. | ψ |\psi\rangle
  27. U ( t ) U(t)
  28. | ψ ( t ) = U ( t ) | ψ ( 0 ) . |\psi(t)\rangle=U(t)|\psi(0)\rangle.
  29. U ( t ) = e - i H t / , U(t)=e^{-iHt/\hbar},
  30. A t = ψ ( 0 ) | e i H t / A e - i H t / | ψ ( 0 ) . \langle A\rangle_{t}=\langle\psi(0)|e^{iHt/\hbar}Ae^{-iHt/\hbar}|\psi(0)\rangle.
  31. A ( t ) := e i H t / A e - i H t / . A(t):=e^{iHt/\hbar}Ae^{-iHt/\hbar}.
  32. d d t A ( t ) = i H e i H t / A e - i H t / + e i H t / ( A t ) e - i H t / + i e i H t / A ( - H ) e - i H t / {d\over dt}A(t)={i\over\hbar}He^{iHt/\hbar}Ae^{-iHt/\hbar}+e^{iHt/\hbar}\left(% \frac{\partial A}{\partial t}\right)e^{-iHt/\hbar}+{i\over\hbar}e^{iHt/\hbar}A% \cdot(-H)e^{-iHt/\hbar}
  33. = i e i H t / ( H A - A H ) e - i H t / + e i H t / ( A t ) e - i H t / ={i\over\hbar}e^{iHt/\hbar}\left(HA-AH\right)e^{-iHt/\hbar}+e^{iHt/\hbar}\left% (\frac{\partial A}{\partial t}\right)e^{-iHt/\hbar}
  34. = i ( H A ( t ) - A ( t ) H ) + e i H t / ( A t ) e - i H t / . ={i\over\hbar}\left(HA(t)-A(t)H\right)+e^{iHt/\hbar}\left(\frac{\partial A}{% \partial t}\right)e^{-iHt/\hbar}.
  35. d d t A ( t ) = i [ H , A ( t ) ] + e i H t / ( A t ) e - i H t / , {d\over dt}A(t)={i\over\hbar}[H,A(t)]+e^{iHt/\hbar}\left(\frac{\partial A}{% \partial t}\right)e^{-iHt/\hbar},
  36. e B A e - B = A + [ B , A ] + 1 2 ! [ B , [ B , A ] ] + 1 3 ! [ B , [ B , [ B , A ] ] ] + . {e^{B}Ae^{-B}}=A+[B,A]+\frac{1}{2!}[B,[B,A]]+\frac{1}{3!}[B,[B,[B,A]]]+\cdots.
  37. A ( t ) = A + i t [ H , A ] - t 2 2 ! 2 [ H , [ H , A ] ] - i t 3 3 ! 3 [ H , [ H , [ H , A ] ] ] + A(t)=A+\frac{it}{\hbar}[H,A]-\frac{t^{2}}{2!\hbar^{2}}[H,[H,A]]-\frac{it^{3}}{% 3!\hbar^{3}}[H,[H,[H,A]]]+\dots
  38. [ A , H ] i { A , H } [A,H]\leftrightarrow i\hbar\{A,H\}
  39. { A , H } = d d t A , \{A,H\}={d\over dt}A~{},
  40. H = p 2 2 m + m ω 2 x 2 2 H=\frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}
  41. d d t x ( t ) = i [ H , x ( t ) ] = p m {d\over dt}x(t)={i\over\hbar}[H,x(t)]=\frac{p}{m}
  42. d d t p ( t ) = i [ H , p ( t ) ] = - m ω 2 x {d\over dt}p(t)={i\over\hbar}[H,p(t)]=-m\omega^{2}x
  43. p ˙ ( 0 ) = - m ω 2 x 0 , \dot{p}(0)=-m\omega^{2}x_{0},
  44. x ˙ ( 0 ) = p 0 m , \dot{x}(0)=\frac{p_{0}}{m},
  45. x ( t ) = x 0 cos ( ω t ) + p 0 ω m sin ( ω t ) x(t)=x_{0}\cos(\omega t)+\frac{p_{0}}{\omega m}\sin(\omega t)
  46. p ( t ) = p 0 cos ( ω t ) - m ω x 0 sin ( ω t ) p(t)=p_{0}\cos(\omega t)-m\omega\!x_{0}\sin(\omega t)
  47. [ x ( t 1 ) , x ( t 2 ) ] = i m ω sin ( ω t 2 - ω t 1 ) [x(t_{1}),x(t_{2})]=\frac{i\hbar}{m\omega}\sin(\omega t_{2}-\omega t_{1})
  48. [ p ( t 1 ) , p ( t 2 ) ] = i m ω sin ( ω t 2 - ω t 1 ) [p(t_{1}),p(t_{2})]=i\hbar m\omega\sin(\omega t_{2}-\omega t_{1})
  49. [ x ( t 1 ) , p ( t 2 ) ] = i cos ( ω t 2 - ω t 1 ) [x(t_{1}),p(t_{2})]=i\hbar\cos(\omega t_{2}-\omega t_{1})
  50. t 1 = t 2 t_{1}=t_{2}
  51. H 0 , S H_{0,S}
  52. H 1 , S H_{1,S}
  53. H 1 , S H_{1,S}
  54. H 0 , S H_{0,S}
  55. H 0 , S H_{0,S}
  56. e ± i H 0 , S t / e^{\pm iH_{0,S}t/\hbar}
  57. | ψ S ( t ) |\psi_{S}(t)\rangle
  58. A S ( t ) A_{S}(t)
  59. A S A_{S}
  60. H 0 H_{0}
  61. H 0 , I ( t ) = e i H 0 , S t / H 0 , S e - i H 0 , S t / = H 0 , S . H_{0,I}(t)=e^{iH_{0,S}t/\hbar}H_{0,S}e^{-iH_{0,S}t/\hbar}=H_{0,S}.
  62. H 1 , I ( t ) = e i H 0 , S t / H 1 , S e - i H 0 , S t / , H_{1,I}(t)=e^{iH_{0,S}t/\hbar}H_{1,S}e^{-iH_{0,S}t/\hbar},
  63. ρ I \rho_{I}
  64. ρ S \rho_{S}
  65. p n p_{n}
  66. | ψ n |\psi_{n}\rangle
  67. ρ I ( t ) = n p n ( t ) | ψ n , I ( t ) ψ n , I ( t ) | = n p n ( t ) e i H 0 , S t / | ψ n , S ( t ) ψ n , S ( t ) | e - i H 0 , S t / = e i H 0 , S t / ρ S ( t ) e - i H 0 , S t / . \rho_{I}(t)=\sum_{n}p_{n}(t)|\psi_{n,I}(t)\rangle\langle\psi_{n,I}(t)|=\sum_{n% }p_{n}(t)e^{iH_{0,S}t/\hbar}|\psi_{n,S}(t)\rangle\langle\psi_{n,S}(t)|e^{-iH_{% 0,S}t/\hbar}=e^{iH_{0,S}t/\hbar}\rho_{S}(t)e^{-iH_{0,S}t/\hbar}.
  68. i d d t | ψ I ( t ) = H 1 , I ( t ) | ψ I ( t ) . i\hbar\frac{d}{dt}|\psi_{I}(t)\rangle=H_{1,I}(t)|\psi_{I}(t)\rangle.
  69. A S A_{S}
  70. A I ( t ) A_{I}(t)
  71. i d d t A I ( t ) = [ A I ( t ) , H 0 ] . i\hbar\frac{d}{dt}A_{I}(t)=\left[A_{I}(t),H_{0}\right].\;
  72. H = H 0 H^{\prime}=H_{0}
  73. i d d t ρ I ( t ) = [ H 1 , I ( t ) , ρ I ( t ) ] . i\hbar\frac{d}{dt}\rho_{I}(t)=\left[H_{1,I}(t),\rho_{I}(t)\right].
  74. V V
  75. H int ( t ) e ( i / ) t H 0 V e ( - i / ) t H 0 H_{\rm int}(t)\equiv e^{{(i/\hbar})tH_{0}}\,V\,e^{{(-i/\hbar})tH_{0}}
  76. | ψ I ( t ) = e i H 0 , S t / | ψ S ( t ) |\psi_{I}(t)\rangle=e^{iH_{0,S}~{}t/\hbar}|\psi_{S}(t)\rangle
  77. | ψ S ( t ) = e - i H S t / | ψ S ( 0 ) |\psi_{S}(t)\rangle=e^{-iH_{S}~{}t/\hbar}|\psi_{S}(0)\rangle
  78. A H ( t ) = e i H S t / A S e - i H S t / A_{H}(t)=e^{iH_{S}~{}t/\hbar}A_{S}e^{-iH_{S}~{}t/\hbar}
  79. A I ( t ) = e i H 0 , S t / A S e - i H 0 , S t / A_{I}(t)=e^{iH_{0,S}~{}t/\hbar}A_{S}e^{-iH_{0,S}~{}t/\hbar}
  80. ρ I ( t ) = e i H 0 , S t / ρ S ( t ) e - i H 0 , S t / \rho_{I}(t)=e^{iH_{0,S}~{}t/\hbar}\rho_{S}(t)e^{-iH_{0,S}~{}t/\hbar}
  81. ρ S ( t ) = e - i H S t / ρ S ( 0 ) e i H S t / \rho_{S}(t)=e^{-iH_{S}~{}t/\hbar}\rho_{S}(0)e^{iH_{S}~{}t/\hbar}
  82. ψ A ( t ) ψ = ψ ( t ) A ψ ( t ) = ψ I ( t ) A I ( t ) ψ I ( t ) . \langle\psi\mid A(t)\mid\psi\rangle=\langle\psi(t)\mid A\mid\psi(t)\rangle=% \langle\psi_{I}(t)\mid A_{I}(t)\mid\psi_{I}(t)\rangle~{}.

Earnings_surprise.html

  1. S U E t = Q t - E ( Q t ) σ ( Q t - E ( Q t ) SUE_{t}=\frac{Q_{t}-E(Q_{t})}{\sigma(Q_{t}-E(Q_{t})}
  2. E ( Q t ) = δ + Q t - 4 E(Q_{t})=\delta+Q_{t-4}
  3. S U E = E P S - F o r e c a s t σ ( E P S - F o r e c a s t ) SUE=\frac{EPS-Forecast}{\sigma(EPS-Forecast)}

Eckhaus_equation.html

  1. i ψ t + ψ x x + 2 ( | ψ | 2 ) x ψ + | ψ | 4 ψ = 0. i\psi_{t}+\psi_{xx}+2\left(|\psi|^{2}\right)_{x}\,\psi+|\psi|^{4}\,\psi=0.
  2. i φ t + φ x x = 0 , i\varphi_{t}+\varphi_{xx}=0,
  3. φ ( x , t ) = ψ ( x , t ) exp ( - x | ψ ( x , t ) | 2 d x ) . \varphi(x,t)=\psi(x,t)\,\exp\left(\int_{-\infty}^{x}|\psi(x^{\prime},t)|^{2}\;% \,\text{d}x^{\prime}\right).
  4. ψ ( x , t ) = φ ( x , t ) ( 1 + 2 - x | φ ( x , t ) | 2 d x ) 1 / 2 . \psi(x,t)=\frac{\varphi(x,t)}{\displaystyle\left(1+2\,\int_{-\infty}^{x}|% \varphi(x^{\prime},t)|^{2}\;\,\text{d}x^{\prime}\right)^{1/2}}.

EdDSA.html

  1. - x 2 + y 2 = 1 - 121665 121666 x 2 y 2 -x^{2}+y^{2}=1-\frac{121665}{121666}x^{2}y^{2}
  2. x = - 486664 u / v x=\sqrt{-486664}u/v
  3. y = ( u - 1 ) / ( u + 1 ) y=(u-1)/(u+1)

Edmonds–Pruhs_protocol.html

  1. Σ k ( 2 d n ) ! ( 2 d n - k ) ! ( a n ) k = O ( 1 a 2 ) \Sigma_{k}\frac{(2dn)!}{(2dn-k)!(an)^{k}}=O(\frac{1}{a^{2}})
  2. 32 d 5 a 2 ( a - 4 d 2 ) \frac{32d^{5}}{a^{2}\cdot(a-4d^{2})}
  3. 16 d 3 a 3 + 8 d 2 a 2 \frac{16d^{3}}{a^{3}}+\frac{8d^{2}}{a^{2}}
  4. O ( 1 a 2 ) O(\frac{1}{a^{2}})
  5. O ( 1 n ) O(\frac{1}{n})
  6. O ( 1 n 2 ) O(\frac{1}{n^{2}})
  7. O ( n n 2 ) = O ( 1 n ) O(\frac{n}{n^{2}})=O(\frac{1}{n})

Edmund_Schuster.html

  1. Θ ν \Theta_{\nu}
  2. Θ ν = 2 ( 1 ν + ν ) sin ( 2 3 arctan ν ) \Theta_{\nu}=2\left(\frac{1}{\sqrt{\nu}}+\sqrt{\nu}\right)\sin\left(\frac{2}{3% }\arctan\sqrt{\nu}\right)
  3. ν \nu
  4. z = 1 + i ν z=1+i\sqrt{\nu}
  5. Θ ν = 2 ( z z ¯ ) 2 / 3 z 2 / 3 - z ¯ 2 / 3 z - z ¯ \Theta_{\nu}=2(z\bar{z})^{2/3}\frac{z^{2/3}-\bar{z}^{2/3}}{z-\bar{z}}
  6. z ¯ = 1 - i ν \bar{z}=1-i\sqrt{\nu}
  7. z {z}

Effaceable_functor.html

  1. u : A M u:A\to M
  2. F ( u ) = 0 F(u)=0

Effect_Model_law.html

  1. ( R c ) \scriptstyle{(Rc)}
  2. ( R t ) \scriptstyle{(Rt)}
  3. A B \scriptstyle{AB}
  4. R R \scriptstyle{RR}
  5. A B \scriptstyle{AB}
  6. a \scriptstyle{a}
  7. R c \scriptstyle{Rc}
  8. b \scriptstyle{b}
  9. R c \scriptstyle{Rc}
  10. R t = a × R c + b \scriptstyle{Rt=a\times Rc+b}
  11. ( R c ) \scriptstyle{(Rc)}
  12. ( R t ) \scriptstyle{(Rt)}
  13. ( R c , R t ) \scriptstyle{(Rc,Rt)}
  14. R t \scriptstyle{Rt}
  15. R t = f ( R c , X ) \scriptstyle{Rt=f(Rc,X)}
  16. R t = g ( Y , X ) \scriptstyle{Rt=g(Y,X)}
  17. R c \scriptstyle{Rc}
  18. A B \scriptstyle{AB}
  19. A B = R c - R t = h ( Y , X ) \scriptstyle{AB=Rc-Rt=h(Y,X)}
  20. R t \scriptstyle{Rt}
  21. R c \scriptstyle{Rc}
  22. ( R c , R t ) \scriptstyle{(Rc,Rt)}
  23. A B \scriptstyle{AB}
  24. X \scriptstyle{X}
  25. Y \scriptstyle{Y}
  26. R c \scriptstyle{Rc}
  27. × \scriptstyle{\times}
  28. A B i \scriptstyle{AB_{i}}
  29. A B i = R t i - R c i \scriptstyle{AB_{i}=Rt_{i}-Rc_{i}}
  30. R c < 0.1 \scriptstyle{Rc<0.1}
  31. R c > 0.9 \scriptstyle{Rc>0.9}
  32. R t \scriptstyle{Rt}
  33. R c \scriptstyle{Rc}
  34. A B i \scriptstyle{AB_{i}}
  35. X \scriptstyle{X}
  36. Y \scriptstyle{Y}
  37. ( R c , R t ) \scriptstyle{(Rc,Rt)}
  38. A B i \scriptstyle{AB_{i}}

Effect_of_gait_parameters_on_energetic_cost.html

  1. s α v 0.42 \textstyle s\;\alpha\;v^{0.42}
  2. s \textstyle s
  3. v \textstyle v
  4. 0.12 L \textstyle 0.12L
  5. L \textstyle L
  6. W ˙ α l 4 \textstyle\dot{W}\;\alpha\;l^{4}
  7. W ˙ \textstyle\dot{W}
  8. l \textstyle l
  9. W ˙ α f 3 \textstyle\dot{W}\;\alpha\;f^{3}
  10. W ˙ \textstyle\dot{W}
  11. f \textstyle f
  12. E ˙ α f 4 \textstyle\dot{E}\;\alpha\;f^{4}
  13. E ˙ \textstyle\dot{E}
  14. f \textstyle f
  15. E ˙ α W ˙ α w 2 \textstyle\dot{E}\;\alpha\;\dot{W}\;\alpha\;w^{2}
  16. E ˙ \textstyle\dot{E}
  17. W ˙ \textstyle\dot{W}
  18. w \textstyle w

Effective_field_goal_percentage.html

  1. e F G % = F G + 0.5 * 3 P F G A eFG\%=\frac{FG+0.5*3P}{FGA}

Efficiency_(Network_Science).html

  1. i i
  2. G G
  3. E ( G ) = 2 n ( n - 1 ) i < j G n 1 d ( i , j ) E(G)=\frac{2}{n(n-1)}\sum_{i<j\in G}^{n}\frac{1}{d(i,j)}
  4. n n
  5. d ( i , j ) d(i,j)
  6. i i
  7. j j
  8. L L
  9. E g l o b ( G ) = E ( G ) E ( G i d e a l ) E_{glob}(G)=\frac{E(G)}{E(G^{ideal})}
  10. 1 / L 1/L
  11. 1 / L 1/L
  12. E g l o b ( G ) E_{glob}(G)
  13. E l o c ( G ) = 1 N i G n E ( G i ) E_{loc}(G)=\frac{1}{N}\sum_{i\in G}^{n}E(G_{i})
  14. G i G_{i}
  15. i i
  16. i i

Efficient_cake-cutting.html

  1. C = P 1 P n C=P_{1}\sqcup...\sqcup P_{n}
  2. i : V i ( Q i ) V i ( P i ) \forall{i}:\ V_{i}(Q_{i})\geq V_{i}(P_{i})
  3. i : V i ( Q i ) > V i ( P i ) \exists{i}:V_{i}(Q_{i})>V_{i}(P_{i})
  4. i = 1 n a i V i ( P i ) \sum_{i=1}^{n}{a_{i}\cdot V_{i}(P_{i})}
  5. i = 1 n V i ( P i ) x i \sum_{i=1}^{n}{\frac{V_{i}(P_{i})}{x_{i}}}
  6. V i ( P i ) x i \frac{V_{i}(P_{i})}{x_{i}}

Eigenoperator.html

  1. H A = λ A HA=\lambda A\,
  2. λ \lambda

Eigenstate_thermalization_hypothesis.html

  1. H ^ \hat{H}
  2. H ^ | E α = E α | E α , \hat{H}|E_{\alpha}\rangle=E_{\alpha}|E_{\alpha}\rangle,
  3. | E α |E_{\alpha}\rangle
  4. E α E_{\alpha}
  5. A ^ \hat{A}
  6. A α β E α | A ^ | E β . A_{\alpha\beta}\equiv\langle E_{\alpha}|\hat{A}|E_{\beta}\rangle.
  7. A ^ \hat{A}
  8. A ^ \hat{A}
  9. A α α A_{\alpha\alpha}
  10. A α + 1 , α + 1 - A α , α A_{\alpha+1,\alpha+1}-A_{\alpha,\alpha}
  11. A α β A_{\alpha\beta}
  12. α β \alpha\neq\beta
  13. i t | Ψ ( t ) = H ^ | Ψ ( t ) , i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle,
  14. H ^ \hat{H}
  15. | Ψ ( t ) |\Psi(t)\rangle
  16. t t
  17. | Ψ ( 0 ) = α = 1 D c α | E α , |\Psi(0)\rangle=\sum_{\alpha=1}^{D}c_{\alpha}|E_{\alpha}\rangle,
  18. | Ψ ( t ) = α = 1 D c α e - i E α t / | E α . |\Psi(t)\rangle=\sum_{\alpha=1}^{D}c_{\alpha}e^{-iE_{\alpha}t/\hbar}|E_{\alpha% }\rangle.
  19. A ^ \hat{A}
  20. Ψ ( t ) | A ^ | Ψ ( t ) = α , β = 1 D c α * c β A α β e - i ( E β - E α ) t / . \langle\Psi(t)|\hat{A}|\Psi(t)\rangle=\sum_{\alpha,\beta=1}^{D}c_{\alpha}^{*}c% _{\beta}A_{\alpha\beta}e^{-i\left(E_{\beta}-E_{\alpha}\right)t/\hbar}.
  21. A ^ \hat{A}
  22. A ¯ lim τ 1 τ 0 τ Ψ ( t ) | A ^ | Ψ ( t ) d t . \overline{A}\equiv\lim_{\tau\to\infty}\frac{1}{\tau}\int_{0}^{\tau}\langle\Psi% (t)|\hat{A}|\Psi(t)\rangle~{}dt.
  23. A ¯ = lim τ 1 τ 0 τ [ α , β = 1 D c α * c β A α β e - i ( E β - E α ) t / ] d t . \overline{A}=\lim_{\tau\to\infty}\frac{1}{\tau}\int_{0}^{\tau}\left[\sum_{% \alpha,\beta=1}^{D}c_{\alpha}^{*}c_{\beta}A_{\alpha\beta}e^{-i\left(E_{\beta}-% E_{\alpha}\right)t/\hbar}\right]~{}dt.
  24. A ¯ = α = 1 D | c α | 2 A α α + i lim τ [ α β D c α * c β A α β E β - E α ( e - i ( E β - E α ) τ / - 1 τ ) ] . \overline{A}=\sum_{\alpha=1}^{D}|c_{\alpha}|^{2}A_{\alpha\alpha}+i\hbar\lim_{% \tau\to\infty}\left[\sum_{\alpha\neq\beta}^{D}\frac{c_{\alpha}^{*}c_{\beta}A_{% \alpha\beta}}{E_{\beta}-E_{\alpha}}\left(\frac{e^{-i\left(E_{\beta}-E_{\alpha}% \right)\tau/\hbar}-1}{\tau}\right)\right].
  25. A ¯ = α = 1 D | c α | 2 A α α . \overline{A}=\sum_{\alpha=1}^{D}|c_{\alpha}|^{2}A_{\alpha\alpha}.
  26. A ^ \hat{A}
  27. A mc = 1 𝒩 α = 1 𝒩 A α α , \langle A\rangle_{\,\text{mc}}=\frac{1}{\mathcal{N}}\sum_{\alpha^{\prime}=1}^{% \mathcal{N}}A_{\alpha^{\prime}\alpha^{\prime}},
  28. 𝒩 \mathcal{N}
  29. A α α A_{\alpha\alpha}
  30. A ¯ = α = 1 D | c α | 2 A α α A α = 1 D | c α | 2 = A , \overline{A}=\sum_{\alpha=1}^{D}|c_{\alpha}|^{2}A_{\alpha\alpha}\approx A\sum_% {\alpha=1}^{D}|c_{\alpha}|^{2}=A,
  31. A mc = 1 𝒩 α = 1 𝒩 A α α 1 𝒩 α = 1 𝒩 A = A . \langle A\rangle_{\,\text{mc}}=\frac{1}{\mathcal{N}}\sum_{\alpha^{\prime}=1}^{% \mathcal{N}}A_{\alpha^{\prime}\alpha^{\prime}}\approx\frac{1}{\mathcal{N}}\sum% _{\alpha^{\prime}=1}^{\mathcal{N}}A=A.
  32. A α α A_{\alpha\alpha}
  33. A ^ \hat{A}
  34. f M B ( 𝐩 , T α ) = ( 2 π m k T ) - 3 / 2 e - 𝐩 2 / 2 m k T α , f_{MB}\left(\mathbf{p},T_{\alpha}\right)=\left(2\pi mkT\right)^{-3/2}e^{-% \mathbf{p}^{2}/2mkT_{\alpha}},
  35. 𝐩 \mathbf{p}
  36. T α T_{\alpha}
  37. E α = 3 2 N k T α , E_{\alpha}=\frac{3}{2}NkT_{\alpha},
  38. c α c_{\alpha}
  39. A α α A_{\alpha\alpha}
  40. A α α A_{\alpha\alpha}
  41. A α α A_{\alpha\alpha}
  42. A ^ \hat{A}
  43. A ^ \hat{A}
  44. A α α A_{\alpha\alpha}
  45. c α c_{\alpha}
  46. A ^ \hat{A}
  47. ( A t - A ¯ ) 2 ¯ lim τ 1 τ 0 τ ( A t - A ¯ ) 2 d t , \overline{\left(A_{t}-\overline{A}\right)^{2}}\equiv\lim_{\tau\to\infty}\frac{% 1}{\tau}\int_{0}^{\tau}\left(A_{t}-\overline{A}\right)^{2}dt,
  48. A t A_{t}
  49. A ^ \hat{A}
  50. ( A t - A ¯ ) 2 ¯ = α β | c α | 2 | c β | 2 | A α β | 2 . \overline{\left(A_{t}-\overline{A}\right)^{2}}=\sum_{\alpha\neq\beta}|c_{% \alpha}|^{2}|c_{\beta}|^{2}|A_{\alpha\beta}|^{2}.
  51. A ^ \hat{A}

Eisenstein_triple.html

  1. c 2 = a 2 - a b + b 2 . c^{2}=a^{2}-ab+b^{2}.
  2. c 2 = a 2 + a b + b 2 . c^{2}=a^{2}+ab+b^{2}.

Elasticity_(cloud_computing).html

  1. t 0 t_{0}
  2. t 1 t_{1}
  3. t 2 t_{2}

Elasto-capillarity.html

  1. 1 / 2 1/2
  2. 1 / 2 1/2
  3. Δ p = - γ n ^ = γ ( 1 R 1 + 1 R 2 ) \Delta p=-\gamma\nabla\cdot\hat{n}=\gamma(\cfrac{1}{R_{1}}+\cfrac{1}{R_{2}})
  4. Δ p \Delta p
  5. γ \gamma
  6. n ^ \hat{n}
  7. R 1 , R 2 R_{1},R_{2}
  8. Δ p = - γ ( cos θ A + cos θ B ) / h f s \Delta p=-\gamma(\cos\theta_{A}+\cos\theta_{B})/h_{fs}
  9. θ A , θ B \theta_{A},\theta_{B}
  10. h f s h_{fs}
  11. P l i q P_{liq}
  12. A w e t A_{wet}

Electric_dipole_spin_resonance.html

  1. e e
  2. s y m b o l μ symbol{\mu}
  3. μ B \mu_{B}
  4. a c ac
  5. s y m b o l μ symbol{\mu}
  6. s y m b o l B symbol{B}
  7. H = - ( s y m b o l μ \cdotsymbol B ) H=-(symbol{\mu}\cdotsymbol{B})
  8. s y m b o l μ symbol{\mu}
  9. s y m b o l S symbol{S}
  10. s y m b o l μ = g μ B s y m b o l S symbol{\mu}=g{\mu_{B}}symbol{S}
  11. g g
  12. g 2 g\approx 2
  13. B B
  14. E ± = ± 1 2 g μ B B E_{\pm}=\pm\frac{1}{2}g\mu_{B}B
  15. s y m b o l B ~ ( t ) \tilde{symbol{B}}(t)
  16. ω S = g μ B B / \omega_{S}=g\mu_{B}B/\hbar
  17. E ~ ( t ) {\tilde{E}}(t)
  18. B ~ ( t ) ( v / c ) E ~ ( t ) {\tilde{B}}(t)\approx(v/c){\tilde{E}}(t)
  19. v / c 1 v/c\ll 1
  20. e 2 / c 1 / 137 e^{2}/\hbar c\approx 1/137
  21. Z Z
  22. Z e 2 / c Ze^{2}/\hbar c
  23. s y m b o l k = s y m b o l p / symbol{k}=symbol{p}/\hbar
  24. s y m b o l σ symbol{\sigma}
  25. s y m b o l k \rightarrowsymbol k - ( e / c ) s y m b o l A symbol{k}\rightarrowsymbol{k}-(e/\hbar c)symbol{A}
  26. s y m b o l A symbol{A}
  27. s y m b o l E = - 1 c \partialsymbol A / t symbol{E}=-\frac{1}{c}\partialsymbol{A}/\partial t
  28. \tildesymbol E {\tildesymbol{E}}
  29. λ C = / m c 4 × 10 - 11 \lambda_{C}=\hbar/mc\approx 4\times 10^{-11}
  30. μ B = e λ C / 2 \mu_{B}=e\lambda_{C}/2
  31. \tildesymbol B {\tildesymbol{B}}
  32. s y m b o l σ symbol{\sigma}
  33. s y m b o l k symbol{k}
  34. \tildesymbol A {\tildesymbol{A}}
  35. ω S \omega_{S}
  36. ω C \omega_{C}
  37. \tildesymbol E {\tildesymbol{E}}
  38. s y m b o l r S O symbol{r}_{SO}
  39. ω S / Δ E \hbar\omega_{S}/\Delta E
  40. Δ E \Delta E
  41. Δ s o \Delta_{so}
  42. E G E_{G}
  43. m * m^{*}
  44. m * 2 E G P 2 , | g | m 0 P 2 2 E G m^{*}\approx\frac{\hbar^{2}E_{G}}{P^{2}},\,\,\,|g|\approx\frac{m_{0}P^{2}}{% \hbar^{2}E_{G}}
  45. P 10 e V Å P\approx 10eV\AA
  46. m 0 m_{0}
  47. s y m b o l r s o {symbol{r}_{so}}
  48. Δ s o E G \Delta_{so}\approx E_{G}
  49. r s o 2 | g | k m 0 E G r_{so}\approx\frac{\hbar^{2}|g|k}{m_{0}E_{G}}
  50. k k
  51. E ~ {\tilde{E}}
  52. U = e r s o E ~ e E ~ P 2 E G 2 k e E ~ 2 k m * E G . U=er_{so}{\tilde{E}}\approx e{\tilde{E}}\frac{P^{2}}{E_{G}^{2}}k\approx e{% \tilde{E}}\frac{\hbar^{2}k}{m^{*}E_{G}}.
  53. k / m 0 \hbar k/m_{0}
  54. E ~ {\tilde{E}}
  55. B ~ = v / c E ~ {\tilde{B}}={v/c}{\tilde{E}}
  56. U v = μ B B ~ = e E ~ 2 k m 0 2 c 2 , U_{v}=\mu_{B}{\tilde{B}}=e{\tilde{E}}\frac{\hbar^{2}k}{m_{0}^{2}c^{2}},
  57. μ B \mu_{B}
  58. c c
  59. U U v m 0 m * m 0 c 2 E G \frac{U}{U_{v}}\approx\frac{m_{0}}{m^{*}}\frac{m_{0}c^{2}}{E_{G}}
  60. m 0 c 2 m_{0}c^{2}\approx
  61. E G E_{G}
  62. E G E_{G}\approx
  63. Z e 2 / c Ze^{2}/\hbar c
  64. Z Z
  65. e 2 / c 1 / 137 e^{2}/\hbar c\approx 1/137
  66. ω S \omega_{S}
  67. ( s y m b o l r S O s y m b o l E ~ ) (symbol{r}_{SO}\cdot{\tilde{symbol{E}}})
  68. ω C + ω S \omega_{C}+\omega_{S}
  69. ω C \omega_{C}
  70. k 3 k^{3}
  71. ω S \omega_{S}
  72. s y m b o l E ( s y m b o l r ) {symbolE}({symbolr})
  73. 1 - x {}_{1-x}
  74. x {}_{x}
  75. s y m b o l B ( t ) symbol{B}(t)
  76. s y m b o l E ( t ) symbol{E}(t)
  77. 3 {}_{3}
  78. 5 {}_{5}
  79. H R = α ( σ x k y - σ y k x ) H_{R}=\alpha(\sigma_{x}k_{y}-\sigma_{y}k_{x})
  80. α \alpha
  81. α \alpha
  82. Å \AA
  83. § \S
  84. 1 - x {}_{1-x}
  85. x {}_{x}

Electrodynamic_droplet_deformation.html

  1. D = d 1 - d 2 d 1 + d 2 D=\frac{d_{1}-d_{2}}{d_{1}+d_{2}}
  2. V = | V | e i ω t V=|V|e^{i\omega t}
  3. σ i j = ϵ 0 ( E i E j - 1 2 δ i j E 2 ) + 1 μ 0 ( B i B j - 1 2 δ i j B 2 ) \sigma_{ij}=\epsilon_{0}\left(E_{i}E_{j}-\frac{1}{2}\delta_{ij}E^{2}\right)+% \frac{1}{\mu_{0}}\left(B_{i}B_{j}-\frac{1}{2}\delta_{ij}B^{2}\right)
  4. 0 = [ 2 r 2 + sin θ r 2 θ ( 1 sin ( θ ) 2 θ ) ] 2 ψ . 0=\left[\frac{\partial^{2}}{\partial r^{2}}+\frac{\sin\theta}{r^{2}}\frac{% \partial}{\partial\theta}\left(\frac{1}{\sin(\theta)^{2}}\frac{\partial}{% \partial\theta}\right)\right]^{2}\psi.
  5. ψ = [ C 1 b 4 r 2 + C 2 b 2 + C 3 b r 3 + C 4 r 5 b 3 ) ] sin ( θ ) 2 cos ( θ ) . \psi=\left[\frac{C_{1}b^{4}}{r^{2}}+C_{2}b^{2}+\frac{C_{3}}{br^{3}}+\frac{C_{4% }r^{5}}{b^{3}})\right]\sin(\theta)^{2}\cos(\theta).
  6. τ i j = μ ( v j x i + v i x j ) . \tau_{ij}=\mu\left(\frac{\partial v_{j}}{\partial x_{i}}+\frac{\partial v_{i}}% {\partial x_{j}}\right).
  7. Δ τ r r + Δ σ r r = γ ( 1 R 1 + 1 R 2 ) \Delta\tau_{rr}+\Delta\sigma_{rr}=\gamma\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)
  8. D = 9 ϵ 0 K 2 16 γ ϕ ( E 0 2 b ) + 9 ϵ 0 K 2 32 γ I cos ( 2 ω t + α ) 1 + k 2 λ 2 ( E 0 2 b ) D=\frac{9\epsilon_{0}K_{2}}{16\gamma}\phi(E_{0}^{2}b)+\frac{9\epsilon_{0}K_{2}% }{32\gamma}\frac{I\cos(2\omega t+\alpha)}{\sqrt{1+k^{2}\lambda^{2}}}(E_{0}^{2}b)
  9. ϕ \phi

Electrokinematics_theorem.html

  1. A 0 A_{0}
  2. E = - A 0 E=-\nabla A_{0}
  3. ε \varepsilon
  4. J q J_{q}
  5. J d = ε E / t J_{d}=\varepsilon\partial E/\partial t
  6. J = J q + J d J=J_{q}+J_{d}
  7. F = - Φ F=-\nabla\Phi
  8. Ω \Omega
  9. ( ε F ) = 0 \nabla(\varepsilon F)=0
  10. Ω \Omega
  11. F F
  12. a γ = ( γ a ) - γ a a\cdot\nabla\gamma=\nabla\cdot(\gamma a)-\gamma\nabla\cdot a
  13. J = 0 \nabla\cdot J=0
  14. - S Φ J d S = Ω J q F d 3 r - S ε A 0 t F d S -\int_{S}\Phi J\cdot dS=\int_{\Omega}J_{q}\cdot Fd^{3}r-\int_{S}\varepsilon% \frac{\partial A_{0}}{\partial t}F\cdot dS
  15. J q = j = 1 N ( t ) q j δ ( r - r j ) v j J_{q}=\sum_{j=1}^{N(t)}q_{j}\delta(r-r_{j})v_{j}
  16. δ ( r - r j ) \delta(r-r_{j})
  17. Ω \Omega
  18. - S Φ J d S = j = 1 N ( t ) q j v j F ( r j ) - S ε A 0 t F d S -\int_{S}\Phi J\cdot dS=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F(r_{j})-\int_{S}% \varepsilon\frac{\partial A_{0}}{\partial t}F\cdot dS
  19. A V k [ r , V k ( t ) ] = V k ( t ) ψ k ( r ) A_{Vk}[r,V_{k}(t)]=V_{k}(t)\psi_{k}(r)
  20. A 0 = A V k + A q j A_{0}=A_{Vk}+A_{qj}
  21. V k ( t ) V_{k}(t)
  22. ψ k ( r ) = 1 \psi_{k}(r)=1
  23. ψ k ( r ) = ψ k ( ) = 0 \psi_{k}(r)=\psi_{k}(\infty)=0
  24. r r\to\infty
  25. A q j [ r , r j ( t ) ] A_{qj}[r,r_{j}(t)]
  26. A q j [ r , r j ( t ) ] = 0 A_{qj}[r,r_{j}(t)]=0
  27. r r
  28. r j ( t ) r_{j}(t)
  29. r r\to\infty
  30. Ω \Omega
  31. S E = k = 1 n S k S_{E}=\sum_{k=1}^{n}S_{k}
  32. S R S_{R}
  33. - S E Φ J q d S = j = 1 N ( t ) q j v j F ( r j ) + j = 1 M ( t ) S R ε ( Φ E j t - A q j t F ) d S -\int_{S_{E}}\Phi J_{q}\cdot dS=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F(r_{j})+\sum% _{j=1}^{M(t)}\int_{S_{R}}\varepsilon(\Phi\frac{\partial E_{j}}{\partial t}-% \frac{\partial A_{qj}}{\partial t}F)\cdot dS
  34. - S E Φ J V d S = k = 1 n S R ε Φ E k t d S - k = 1 n S ε A V k t F d S -\int_{S_{E}}\Phi J_{V}\cdot dS=\sum_{k=1}^{n}\int_{S_{R}}\varepsilon\Phi\frac% {\partial E_{k}}{\partial t}\cdot dS-\sum_{k=1}^{n}\int_{S}\varepsilon\frac{% \partial A_{Vk}}{\partial t}F\cdot dS
  35. M ( t ) M(t)
  36. Ω \Omega
  37. E j = - A q j E_{j}=-\nabla A_{qj}
  38. E k = - A V k E_{k}=-\nabla A_{Vk}
  39. S R S_{R}
  40. i h - S h J d S = i q h + i V h i_{h}\equiv-\int_{S_{h}}J\cdot dS=i_{qh}+i_{Vh}
  41. S h S_{h}
  42. i q h i_{qh}
  43. i V h i_{Vh}
  44. S R 0 S_{R}\neq 0
  45. Φ = Φ h = 1 \Phi=\Phi_{h}=1
  46. Φ h = 0 \Phi_{h}=0
  47. i q h = j = 1 N ( t ) q j v j F h ( r j ) + j = 1 M ( t ) S R ε ( Φ h E j t - A q j t F h ) d S = i_{qh}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_{h}(r_{j})+\sum_{j=1}^{M(t)}\int_{S_% {R}}\varepsilon(\Phi_{h}\frac{\partial E_{j}}{\partial t}-\frac{\partial A_{qj% }}{\partial t}F_{h})\cdot dS=
  48. i d h = k = 1 n C h k d V k d t i_{dh}=\sum_{k=1}^{n}C_{hk}\frac{dV_{k}}{dt}
  49. F = F h ( r j ) F=F_{h}(r_{j})
  50. C h k C_{hk}
  51. C h k = - ( S k ε F h d S + S R ε ( Φ h ψ k + ψ k F h ) d S ) C_{hk}=-(\int_{S_{k}}\varepsilon F_{h}\cdot dS+\int_{S_{R}}\varepsilon(\Phi_{h% }\nabla\psi_{k}+\psi_{k}F_{h})\cdot dS)
  52. V h V_{h}
  53. Φ h \Phi_{h}
  54. S R S_{R}
  55. Φ h = 0 \Phi_{h}=0
  56. S R S_{R}
  57. i q h = j = 1 N ( t ) q j v j F h ( r j ) - j = 1 M ( t ) S R ε A 0 j t F h d S i_{qh}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_{h}(r_{j})-\sum_{j=1}^{M(t)}\int_{S_% {R}}\varepsilon\frac{\partial A_{0j}}{\partial t}F_{h}\cdot dS
  58. C h k = - ( S k ε F h d S + S R ε Ψ k F h d S ) C_{hk}=-(\int_{S_{k}}\varepsilon F_{h}\cdot dS+\int_{S_{R}}\varepsilon\Psi_{k}% F_{h}\cdot dS)
  59. S R S_{R}
  60. F h F_{h}
  61. S R S_{R}
  62. i q h = j = 1 N ( t ) q j v j F h ( r j ) - j = 1 M ( t ) S R ε Φ h E j t d S i_{qh}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_{h}(r_{j})-\sum_{j=1}^{M(t)}\int_{S_% {R}}\varepsilon\Phi_{h}\frac{\partial E_{j}}{\partial t}\cdot dS
  63. C h k = - ( S k ε F h d S + S R ε Ψ h d S ) C_{hk}=-(\int_{S_{k}}\varepsilon F_{h}\cdot dS+\int_{S_{R}}\varepsilon\nabla% \Psi_{h}\cdot dS)
  64. S R S_{R}
  65. S E S_{E}
  66. Φ = 1 \Phi=1
  67. Ω \Omega
  68. F = 0 F=0
  69. h = 1 n i h - S R ε ( j = 1 M ( t ) E j t + k = 1 n E k t ) d S = 0 \sum_{h=1}^{n}i_{h}-\int_{S_{R}}\varepsilon(\sum_{j=1}^{M(t)}\frac{\partial E_% {j}}{\partial t}+\sum_{k=1}^{n}\frac{\partial E_{k}}{\partial t})\cdot dS=0
  70. S R S_{R}
  71. Ω \Omega
  72. S R = 0 S_{R}=0
  73. Φ h = 1 \Phi_{h}=1
  74. S h S_{h}
  75. Φ h = 0 \Phi_{h}=0
  76. i h = j = 1 N ( t ) q j v j F h ( r j ) + k = 1 n C h k d V h d t i_{h}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_{h}(r_{j})+\sum_{k=1}^{n}C_{hk}\frac{% dV_{h}}{dt}
  77. C h k = - S k ε F h d S C_{hk}=-\int_{S_{k}}\varepsilon F_{h}\cdot dS
  78. F ( t ) F(t)
  79. F = F V = - k = 1 n V k ( t ) ψ k ( r ) F=F_{V}=-\sum_{k=1}^{n}V_{k}(t)\nabla\psi_{k}(r)
  80. Ω \Omega
  81. - S Φ J d S = Ω J F d 3 r -\int_{S}\Phi J\cdot dS=\int_{\Omega}J\cdot Fd^{3}r
  82. h = 1 n V h i h = Ω J F V d 3 r W \sum_{h=1}^{n}V_{h}i_{h}=\int_{\Omega}J\cdot F_{V}d^{3}r\equiv W
  83. W W
  84. Ω \Omega
  85. Ω ( E J q + E ε E t ) d 3 r = Ω E J d 3 r d Ξ d t \int_{\Omega}(E\cdot J_{q}+E\cdot\frac{\varepsilon\partial E}{\partial t})d^{3% }r=\int_{\Omega}E\cdot Jd^{3}r\equiv\frac{d\Xi}{dt}
  86. Ξ \Xi
  87. Ω \Omega
  88. E = F V + E q E=F_{V}+E_{q}
  89. F V F_{V}
  90. E q = - ψ q ( r , t ) E_{q}=-\nabla\psi_{q}(r,t)
  91. Ω \Omega
  92. ψ q ( r , t ) = 0 \psi_{q}(r,t)=0
  93. Ω E q J d 3 r = 0 \int_{\Omega}E_{q}\cdot Jd^{3}r=0
  94. W = d Ξ / d t W=d\Xi/dt
  95. S R S_{R}
  96. i h = i q h i_{h}=i_{qh}
  97. f = ω / ( 2 π ) 1 / ( 2 π t j ) f=\omega/(2\pi)\ll 1/(2\pi t_{j})
  98. t j t_{j}
  99. 0 t j e x p ( - j ω t ) ( Q / t ) d t Q ( t j ) - Q ( 0 ) \int_{0}^{t_{j}}exp(-j\omega t)(\partial Q/\partial t)dt\approx Q(t_{j})-Q(0)
  100. Q ( t j ) = Q ( 0 ) = 0 Q(t_{j})=Q(0)=0
  101. i q h = j = 1 N ( t ) q j v j F h ( r j ) = - j = 1 N ( t ) q j d Φ h [ r j ( t ) ] d t = Ω J q F d 3 r i_{qh}=\sum_{j=1}^{N(t)}q_{j}v_{j}\cdot F_{h}(r_{j})=-\sum_{j=1}^{N(t)}q_{j}% \frac{d\Phi_{h}[r_{j}(t)]}{dt}=\int_{\Omega}J_{q}\cdot Fd^{3}r
  102. f 1 / ( 2 π t j ) f\gg 1/(2\pi t_{j})
  103. τ j \tau_{j}
  104. Ω \Omega
  105. Ω \Omega
  106. S R S_{R}
  107. S E 1 S_{E1}
  108. S E 2 S_{E2}
  109. S E 1 u S E 2 S_{E1}\rightarrow u\rightarrow S_{E2}
  110. F h = F = - u / L F_{h}=F=-u/L
  111. i = i 1 = i q 1 = - i 2 i=i_{1}=i_{q1}=-i_{2}
  112. i = 1 L j = 1 N ( t ) q j v j u = 1 L Ω J q u d 3 r i=\frac{1}{L}\sum_{j=1}^{N(t)}q_{j}v_{ju}=\frac{1}{L}\int_{\Omega}J_{qu}d^{3}r
  113. v j u v_{ju}
  114. J q u J_{qu}
  115. v v
  116. J q J_{q}
  117. u u
  118. S S S_{S}
  119. i = i q h i=i_{qh}
  120. V h V_{h}
  121. D ( ω l ) 1 T - T / 2 T / 2 Δ i ( t ) e x p ( - j ω l t ) d t D(\omega_{l})\equiv\frac{1}{T^{^{\prime}}}\int_{-T^{^{\prime}}/2}^{T^{^{\prime% }}/2}\Delta i(t)exp(-j\omega_{l}t)dt
  122. S S ( ω l ) lim Δ f 0 D ( ω l ) D * ( ω l ) Δ f = lim T ( 2 T D ( ω l ) D * ( ω l ) ) S_{S}(\omega_{l})\equiv\lim_{\Delta f\to 0}\frac{\left\langle D(\omega_{l})D^{% *}(\omega_{l})\right\rangle}{\Delta f}=\lim_{T^{^{\prime}}\to\infty}(2T^{^{% \prime}}\left\langle D(\omega_{l})D^{*}(\omega_{l})\right\rangle)
  123. ω l = l ( 2 π / T ) \omega_{l}=l(2\pi/T^{^{\prime}})
  124. l = , - 2 , - 1 , 1 , 2 , l=...,-2,-1,1,2,...
  125. l = 1 , 2 , l=1,2,...
  126. t b j t_{bj}
  127. ( t b j + t j ) (t_{bj}+t_{j})
  128. Ω \Omega
  129. Φ h [ r j ( t b j ) ] = 1 \Phi_{h}[r_{j}(t_{bj})]=1
  130. Φ h [ r j ( t b j + t j ) ] = 0 \Phi_{h}[r_{j}(t_{bj}+t_{j})]=0
  131. Φ h [ r j ( t b j ) ] = Φ h [ r j ( t b j + t j ) ] \Phi_{h}[r_{j}(t_{bj})]=\Phi_{h}[r_{j}(t_{bj}+t_{j})]
  132. D ( ω l ) q T ( Δ N + - Δ N - ) D(\omega_{l})\equiv\frac{q}{T^{^{\prime}}}(\Delta N^{+}-\Delta N^{-})
  133. N + ( N - ) N^{+}(N^{-})
  134. - T / 2 , T / 2 -T^{^{\prime}}/2,T^{^{\prime}}/2
  135. τ c t j m i n \tau_{c}\ll t_{jmin}
  136. τ c \tau_{c}
  137. Δ N + Δ N - = 0 \left\langle\Delta N^{+}\Delta N^{-}\right\rangle=0
  138. Δ N + Δ N + = N + \left\langle\Delta N^{+}\Delta N^{+}\right\rangle=\left\langle N^{+}\right\rangle
  139. Δ N - Δ N - = N - \left\langle\Delta N^{-}\Delta N^{-}\right\rangle=\left\langle N^{-}\right\rangle
  140. S S = 2 q ( I + + I - ) S_{S}=2q(I^{+}+I^{-})
  141. I + ( I - ) I^{+}(I^{-})
  142. I + = I 0 e x p ( q v / k B T ) I^{+}=I_{0}exp(qv/k_{B}T)
  143. I - = I 0 I^{-}=I_{0}
  144. k B k_{B}
  145. I = I + - I - I=I^{+}-I^{-}
  146. v = 0 v=0
  147. g = ( d I / d v ) = q I 0 / ( k B T ) g=(dI/dv)=qI_{0}/(k_{B}T)
  148. S S = 4 k B T g S_{S}=4k_{B}Tg
  149. I + I - I^{+}\gg I^{-}
  150. S S = F a ( 2 q I ) S_{S}=F_{a}(2qI)
  151. F a F_{a}
  152. i ( t ) i ( t + θ ) \left\langle i(t)i(t+\theta)\right\rangle
  153. i ( t ) i(t)
  154. V 1 = V 2 = 0 V_{1}=V_{2}=0
  155. N ( t ) = N ¯ N(t)=\overline{N}
  156. i ( t ) i ( t + θ ) = q 2 L 2 j = 1 N ¯ v j u 2 ( t ) t e x p ( - | θ | / τ c ) = q 2 N ¯ k B T L 2 m e x p ( - | θ | / τ c ) \left\langle i(t)i(t+\theta)\right\rangle=\frac{q^{2}}{L^{2}}\sum_{j=1}^{% \overline{N}}\left\langle v_{ju}^{2}(t)\right\rangle_{t}exp(-\left|\theta% \right|/\tau_{c})=\frac{q^{2}\overline{N}k_{B}T}{L^{2}m}exp(-\left|\theta% \right|/\tau_{c})
  157. τ c τ j m i n \tau_{c}\ll\tau_{jmin}
  158. μ = q τ c / [ m ( 1 + j ω ) ] \mu=q\tau_{c}/[m(1+j\omega)]
  159. G = q μ N ¯ / L 2 G=q\mu\overline{N}/L^{2}
  160. S T = 4 k B T R e { G ( j ω ) } S_{T}=4k_{B}TRe\{G(j\omega)\}
  161. J q u = q μ n q E , ( E E u ) J_{qu}=q\mu n_{q}E,(E\equiv E_{u})
  162. i = 1 L Ω q μ n q E d 3 r i=\frac{1}{L}\int_{\Omega}q\mu n_{q}Ed^{3}r
  163. n q n_{q}
  164. i ¯ I = q μ n q E A \overline{i}\equiv I=q\mu n_{q}EA
  165. A A
  166. Δ i I = 1 Ω ( 1 n q Ω Δ n q d 3 r + 1 E Ω Δ E d 3 r + 1 μ Ω Δ μ d 3 r ) \frac{\Delta i}{I}=\frac{1}{\Omega}(\frac{1}{n_{q}}\int_{\Omega}\Delta n_{q}d^% {3}r+\frac{1}{E}\int_{\Omega}\Delta Ed^{3}r+\frac{1}{\mu}\int_{\Omega}\Delta% \mu d^{3}r)
  167. Δ i \Delta i
  168. χ = 0 , 1 \chi=0,1
  169. ε t \varepsilon_{t}
  170. q Δ χ q\Delta\chi
  171. n q n_{q}
  172. E E
  173. Δ E \Delta E
  174. Δ i \Delta i
  175. Δ i I = 1 Ω n q Ω Δ n q d 3 r \frac{\Delta i}{I}=\frac{1}{\Omega n_{q}}\int_{\Omega}\Delta n_{q}d^{3}r
  176. Δ i I = 1 Ω n q δ Ω Δ n q a d 3 r = - 1 Ω n q Δ χ \frac{\Delta i}{I}=\frac{1}{\Omega n_{q}}\int_{\delta\Omega}\Delta n_{q}ad^{3}% r=-\frac{1}{\Omega n_{q}}\Delta\chi
  177. Ω \Omega
  178. δ Ω \delta\Omega
  179. Δ n q \Delta n_{q}
  180. Δ E \Delta E
  181. δ Ω Δ n q d 3 r = - Δ χ \int_{\delta\Omega}\Delta n_{q}d^{3}r=-\Delta\chi
  182. Δ χ \Delta\chi
  183. χ ¯ \overline{\chi}
  184. χ ¯ ϕ = { [ 1 + e x p [ ( ε t - ε f ) / k B T ] } - 1 \overline{\chi}\equiv\phi=\{[1+exp[(\varepsilon_{t}-\varepsilon_{f})/k_{B}T]\}% ^{-1}
  185. ε f \varepsilon_{f}
  186. S t S_{t}
  187. Δ i \Delta i
  188. S t / I 2 = [ 1 / ( Ω n q ) ] 2 S χ S_{t}/I^{2}=[1/(\Omega n_{q})]^{2}S_{\chi}
  189. S χ = 4 ϕ ( 1 - ϕ ) τ / [ 1 + ( ω τ ) 2 ] S_{\chi}=4\phi(1-\phi)\tau/[1+(\omega\tau)^{2}]
  190. χ \chi
  191. τ \tau
  192. n t n_{t}
  193. S g r S_{gr}
  194. S g r = 4 I 2 n t ϕ ( 1 - ϕ ) τ Ω n q 2 [ 1 + ( ω τ ) 2 ] S_{gr}=\frac{4I^{2}n_{t}\phi(1-\phi)\tau}{\Omega n^{2}_{q}[1+(\omega\tau)^{2}]}
  195. τ \tau
  196. τ \tau
  197. S f S_{f}
  198. S t S_{t}
  199. n t Ω n_{t}\Omega
  200. S f = n t B Ω n q 2 I 2 f γ S_{f}=\frac{n_{t}B}{\Omega n_{q}^{2}}\frac{I^{2}}{f^{\gamma}}
  201. 0.85 < γ < 1.15 0.85<\gamma<1.15
  202. 1 / 2 π τ M 1/2\pi\tau_{M}
  203. τ M \tau_{M}
  204. τ \tau
  205. B ( ε f / k B T ) B(\varepsilon_{f}/k_{B}T)
  206. n q e x p ( ε f / k B T ) n_{q}\propto exp(\varepsilon_{f}/k_{B}T)
  207. ε t > ε f \varepsilon_{t}>\varepsilon_{f}
  208. S χ ϕ = e x p ( ε f / k B T ) S_{\chi}\propto\phi=exp(\varepsilon_{f}/k_{B}T)
  209. B ( ε f / k B T ) e x p ( ε f / k B T ) B(\varepsilon_{f}/k_{B}T)\propto exp(\varepsilon_{f}/k_{B}T)
  210. S f = α I 2 N q f γ S_{f}=\frac{\alpha I^{2}}{N_{q}f^{\gamma}}
  211. N q N_{q}
  212. α \alpha
  213. ω \omega
  214. S R S_{R}
  215. F ( r , t ) = - Φ F(r,t)=-\nabla\Phi
  216. 1 / f γ 1/f^{\gamma}
  217. 1 / f γ 1/f^{\gamma}

Electronic_entropy.html

  1. i i
  2. S = - k B i p i ln p i S=-k_{B}\sum_{i}p_{i}\ln p_{i}
  3. S = - k B 0 n ( E ) [ p ( E ) ln p ( E ) + ( 1 - p ( E ) ) ln ( 1 - p ( E ) ) ] d E S=-k_{B}\int_{0}^{\infty}n(E)\left[p(E)\ln p(E)+(1-p(E))\ln\left(1-p(E)\right)% \right]dE
  4. n ( E ) n(E)
  5. f f
  6. p ( E ) = f = 1 e ( E - E F ) / k T + 1 p(E)=f=\frac{1}{e^{(E-E_{F})/kT}+1}
  7. T T
  8. S = - k B 0 n ( E ) [ f ln f + ( 1 - f ) ln ( 1 - f ) ] d E S=-k_{B}\int_{0}^{\infty}n(E)\left[f\ln f+(1-f)\ln\left(1-f\right)\right]dE
  9. S = π 3 k B 2 T n ( E F ) S=\frac{\pi}{3}k_{B}^{2}Tn(E_{F})
  10. S n sites [ x ln x + ( 1 - x ) ln ( 1 - x ) ] S\approx n\text{sites}\left[x\ln x+(1-x)\ln(1-x)\right]
  11. n < s u b > s i t e s n<sub>sites

Elimination_(pharmacology).html

  1. C L o = Q ( C A - C V ) C A CL_{o}=Q\cdot\frac{(C_{A}-C_{V})}{C_{A}}
  2. C L o CL_{o}
  3. C A C_{A}
  4. C V C_{V}
  5. Q Q

Elisabethatriene_synthase.html

  1. \rightleftharpoons

Elliptic_pseudoprime.html

  1. ( - d ) \mathbb{Q}\big(\sqrt{-d}\big)
  2. X / exp ( ( 1 / 3 ) log X log log log X / log log X ) . X/\exp((1/3)\log X\log\log\log X/\log\log X)\ .

En-ring.html

  1. n \mathcal{E}_{n}
  2. A ( U ) A(U)
  3. μ : A ( U 1 ) A ( U m ) A ( V ) \mu:A(U_{1})\otimes\cdots\otimes A(U_{m})\to A(V)
  4. U j U_{j}
  5. μ \mu
  6. m = 1 m=1
  7. n \mathcal{E}_{n}
  8. n \mathcal{E}_{n}
  9. X C * ( Ω n X ; Λ ) X\mapsto C_{*}(\Omega^{n}X;\Lambda)
  10. n \mathcal{E}_{n}
  11. Λ \Lambda

Engelbart's_Law.html

  1. 𝒜 , , 𝒞 \mathcal{A},\mathcal{B},\mathcal{C}
  2. C o D I A K t t + 1 = 1 + CoDIAK_{t\to t+1}=1+\mathcal{B}
  3. C o D I A K t t + 1 = e CoDIAK_{t\to t+1}=e^{\mathcal{B}}

Ent-cassa-12,15-diene_synthase.html

  1. \rightleftharpoons

Ent-isokaurene_synthase.html

  1. \rightleftharpoons

Ent-pimara-8(14),15-diene_synthase.html

  1. \rightleftharpoons

Ent-pimara-9(11),15-diene_synthase.html

  1. \rightleftharpoons

Ent-sandaracopimaradiene_synthase.html

  1. \rightleftharpoons

Entrance_length.html

  1. u ( r , x ) x = 0 u = u ( r ) \frac{\partial u(r,x)}{\partial x}=0\quad\Rightarrow u=u(r)
  2. x x
  3. ( τ w ) (\tau_{w})
  4. L h , l a m i n a r = 0.05 R e D L_{h,laminar}=0.05R_{e}D
  5. R e R_{e}
  6. D D
  7. L h , t u r b u l e n t = 1.359 D ( R e ) 1 / 4 L_{h,turbulent}=1.359D(R_{e})^{1/4}
  8. L h , t u r b u l e n t 10 D L_{h,turbulent}\approx 10D
  9. D h = 4 A P D_{h}=\frac{4A}{P}
  10. A A
  11. P P
  12. u ( r ) = - R 2 4 μ d P d x ( 1 - r 2 R 2 ) u(r)=-\frac{R^{2}}{4\mu}\frac{dP}{dx}(1-\frac{r^{2}}{R^{2}})
  13. d P d x = c o n s t a n t \frac{dP}{dx}=constant
  14. V a v g = u d A A c V_{avg}=\frac{\int u\mathrm{d}A}{A_{c}}
  15. A c A_{c}
  16. V a v g = 2 R 2 0 R u ( r ) r d r V_{avg}=\frac{2}{R^{2}}\int_{0}^{R}u(r)r\mathrm{d}r
  17. = - 2 R 2 0 R R 2 4 μ d P d x ( 1 - r 2 R 2 ) r d r =-\frac{2}{R^{2}}\int_{0}^{R}\frac{R^{2}}{4\mu}\frac{dP}{dx}(1-\frac{r^{2}}{R^% {2}})r\mathrm{d}r
  18. = - R 2 8 μ d P d x =-\frac{R^{2}}{8\mu}\frac{dP}{dx}
  19. U m a x = 2 V a v g U_{max}=2V_{avg}

Environmentally_extended_input-output_analysis.html

  1. 𝐙 \mathbf{Z}
  2. 𝐱 \mathbf{x}
  3. 𝐙 \mathbf{Z}
  4. 𝐀 \mathbf{A}
  5. A = Z × x ^ - 1 A=Z\times\hat{x}^{-1}
  6. x ^ \hat{x}
  7. 𝐱 \mathbf{x}
  8. x ^ = I x \hat{x}=I\vec{x}
  9. 𝐀 \mathbf{A}
  10. 𝐱 \mathbf{x}
  11. 𝐲 \mathbf{y}
  12. 𝐲 \mathbf{y}
  13. 𝐱 \mathbf{x}
  14. 𝐈 \mathbf{I}
  15. x = ( I - A ) - 1 × y \vec{x}=\left(I-A\right)^{-1}\times\vec{y}
  16. 𝐙 \mathbf{Z}
  17. 𝐌 \mathbf{M}
  18. m m
  19. n n
  20. 𝐌 \mathbf{M}
  21. 𝐅 \mathbf{F}
  22. 𝐀 \mathbf{A}
  23. F = M × x ^ - 1 F=M\times\hat{x}^{-1}
  24. E = F ( I - A ) - 1 × y E=F(I-A)^{-1}\times\vec{y}
  25. 𝐲 \mathbf{y}
  26. 𝐅 \mathbf{F}
  27. C O 2 CO_{2}

Epi-cedrol_synthase.html

  1. \rightleftharpoons

Epi-isozizaene_synthase.html

  1. \rightleftharpoons

Equation-free_modeling.html

  1. U ( t k ) U(t_{k})
  2. t k t_{k}
  3. u ( t k ) u(t_{k})
  4. U ( t k ) U(t_{k})
  5. u ( t ) u(t)
  6. t k t t k + δ t t_{k}\leq t\leq t_{k}+\delta t
  7. U ( t k + δ t ) U(t_{k}+\delta t)
  8. u ( t ) u(t)
  9. U U
  10. t k t_{k}
  11. t k + 1 = t k + Δ t t_{k+1}=t_{k}+\Delta t
  12. ( U ( t ) , u ( t ) ) (U(t),u(t))
  13. d U d t = - U - u + 2 , d u d t = 100 ( U 3 - u ) . \frac{dU}{dt}=-U-u+2\,,\quad\frac{du}{dt}=100(U^{3}-u).
  14. U U
  15. u u
  16. d U / d t = G ( U ) dU/dt=G(U)
  17. U U
  18. ( U , u ) = ( U , 0.5 ) (U,u)=(U,0.5)
  19. u U 3 u\approx U^{3}
  20. ( U , u ) (U,u)
  21. t = n δ t t=n\delta t
  22. u ( n δ t ) = 0.5 u(n\delta t)=0.5
  23. d U / d t dU/dt
  24. U n + 1 = S ( U n , λ ; δ t ) , \vec{U}^{n+1}=\vec{S}(\vec{U}^{n},\lambda;\delta t),
  25. λ \lambda
  26. λ \lambda
  27. U - S ( U , λ ; δ t ) = 0 . \vec{U}-\vec{S}(\vec{U},\lambda;\delta t)=\vec{0}.
  28. Δ t δ t \Delta t\gg\delta t
  29. U n U ( n Δ t ) U^{n}\approx U(n\Delta t)
  30. U n , k U ( n Δ t + k δ t ) U^{n,k}\approx U(n\Delta t+k\delta t)
  31. U ( n Δ t ) = U n = U n , 0 U(n\Delta t)=U^{n}=U^{n,0}
  32. U ( ( n + 1 ) Δ t ) U((n+1)\Delta t)
  33. U n + 1 = U n , k + ( Δ t - k δ t ) F ( U n ) U^{n+1}=U^{n,k}+(\Delta t-k\delta t)F(\vec{U}^{n})
  34. F ( U n ) = ( U n , k - U n , k - 1 ) / δ t F(\vec{U}^{n})=(U^{n,k}-U^{n,k-1})/\delta t
  35. k k

Equilibrant_Force.html

  1. 𝐅 1 + 𝐅 2 = 𝐅 R = 𝐅 E \mathbf{F}_{1}+\mathbf{F}_{2}=\mathbf{F}_{R}=\mathbf{F}_{E}
  2. c 2 = a 2 + b 2 c^{2}=a^{2}+b^{2}
  3. c 2 c^{2}
  4. o p p / a d j opp/adj
  5. 10 / 8 10/8

Equitable_division.html

  1. i i
  2. j j
  3. V i ( P i ) = V j ( P j ) V_{i}(P_{i})=V_{j}(P_{j})
  4. P i P_{i}
  5. i i
  6. V i V_{i}
  7. i i
  8. i i
  9. V i ( ) = 0 V_{i}(\emptyset)=0
  10. V i ( E n t i r e C a k e ) = 1 V_{i}(EntireCake)=1
  11. i i
  12. x [ 0 , 1 ] x\in[0,1]
  13. u 1 ( [ 0 , x ] ) u_{1}([0,x])
  14. u 2 ( [ x , 1 ] ) u_{2}([x,1])
  15. a a
  16. b b
  17. 0 < a < b < 1 0<a<b<1
  18. [ 0 , a ] [0,a]
  19. [ b , 1 ] [b,1]
  20. [ a , b ] [a,b]
  21. [ a , b ] [a,b]
  22. 1 / n 1/n
  23. 1 / n 1/n
  24. n n
  25. n ! n!
  26. n - 1 n-1
  27. n - 1 n-1
  28. n - 1 n-1
  29. x A B x_{AB}
  30. x B C x_{BC}
  31. V A ( 0 , x A B ) = V B ( x A B , x B C ) V_{A}(0,x_{AB})=V_{B}(x_{AB},x_{BC})
  32. V B ( x A B , x B C ) = V C ( x B C , 1 ) V_{B}(x_{AB},x_{BC})=V_{C}(x_{BC},1)
  33. n ! n!
  34. 1 / n 1/n
  35. 1 / n 1/n

Equivalent_definitions_of_mathematical_structures.html

  1. ( X , 𝒪 ) (X,\mathcal{O})

Equivariant_bundle.html

  1. π \pi
  2. π g = g π \pi\circ g=g\circ\pi

Equivariant_differential_form.html

  1. α : 𝔤 Ω * ( M ) \alpha:\mathfrak{g}\to\Omega^{*}(M)
  2. 𝔤 = Lie ( G ) \mathfrak{g}=\operatorname{Lie}(G)
  3. α ( Ad ( g ) X ) = g α ( X ) . \alpha(\operatorname{Ad}(g)X)=g\alpha(X).
  4. [ 𝔤 ] Ω * ( M ) = Sym ( 𝔤 * ) Ω * ( M ) . \mathbb{C}[\mathfrak{g}]\otimes\Omega^{*}(M)=\operatorname{Sym}(\mathfrak{g}^{% *})\otimes\Omega^{*}(M).
  5. α \alpha
  6. d 𝔤 α d_{\mathfrak{g}}\alpha
  7. α \alpha
  8. ( d 𝔤 α ) ( X ) = d ( α ( X ) ) - i X # ( α ( X ) ) (d_{\mathfrak{g}}\alpha)(X)=d(\alpha(X))-i_{X^{\#}}(\alpha(X))
  9. i X # i_{X^{\#}}
  10. d 𝔤 d 𝔤 = 0 d_{\mathfrak{g}}\circ d_{\mathfrak{g}}=0
  11. α ( X ) \alpha(X)
  12. X # X^{\#}
  13. H G * ( X ) = ker d 𝔤 / im d 𝔤 H^{*}_{G}(X)=\operatorname{ker}d_{\mathfrak{g}}/\operatorname{im}d_{\mathfrak{% g}}
  14. d 𝔤 d_{\mathfrak{g}}
  15. d 𝔤 d_{\mathfrak{g}}
  16. V = Ω * ( M ) V=\Omega^{*}(M)
  17. Mor G ( 𝔤 , V ) = Mor ( 𝔤 , V ) G = ( Mor ( 𝔤 , ) V ) G . \operatorname{Mor}_{G}(\mathfrak{g},V)=\operatorname{Mor}(\mathfrak{g},V)^{G}=% (\operatorname{Mor}(\mathfrak{g},\mathbb{C})\otimes V)^{G}.
  18. [ 𝔤 ] \mathbb{C}[\mathfrak{g}]
  19. 𝔤 \mathfrak{g}

Equivariant_sheaf.html

  1. σ : G × S X X \sigma:G\times_{S}X\to X
  2. 𝒪 X \mathcal{O}_{X}
  3. 𝒪 G × S X \mathcal{O}_{G\times_{S}X}
  4. ϕ : σ * F p 2 * F \phi:\sigma^{*}F\simeq p_{2}^{*}F
  5. p 23 * ϕ ( 1 G × σ ) * ϕ = ( m × 1 X ) * ϕ p_{23}^{*}\phi\circ(1_{G}\times\sigma)^{*}\phi=(m\times 1_{X})^{*}\phi
  6. F g h x F x F_{gh\cdot x}\simeq F_{x}
  7. F g h x F h x F x F_{g\cdot h\cdot x}\simeq F_{h\cdot x}\simeq F_{x}
  8. ( e × e × 1 ) * , e : S G (e\times e\times 1)^{*},e:S\to G
  9. ( e × 1 ) * ( e × 1 ) * ϕ = ( e × 1 ) * ϕ (e\times 1)^{*}\circ(e\times 1)^{*}\phi=(e\times 1)^{*}\phi
  10. ( e × 1 ) * ϕ (e\times 1)^{*}\phi
  11. ϕ \phi
  12. ϕ \phi
  13. 𝒪 X \mathcal{O}_{X}
  14. S S
  15. G ( R ) Aut ( X × S Spec R , F S R ) G(R)\to\operatorname{Aut}(X\times_{S}\operatorname{Spec}R,F\otimes_{S}R)
  16. g : E x E g x g:E_{x}\to E_{gx}
  17. G × X X G\times X\to X
  18. G × E E G\times E\to E
  19. E X E\to X
  20. ϕ \phi
  21. V x V g x V_{x}\overset{\simeq}{\to}V_{gx}
  22. g - 1 g^{-1}

Erdős–Hajnal_conjecture.html

  1. H H
  2. H \mathcal{F}_{H}
  3. H H
  4. δ H > 0 \delta_{H}>0
  5. n n
  6. H \mathcal{F}_{H}
  7. Ω ( n δ H ) \Omega(n^{\delta_{H}})
  8. n n
  9. H H

Erdős–Nicolas_number.html

  1. m m
  2. d n , d m d = n . \sum_{d\mid n,\ d\leq m}d=n.

Erodability.html

  1. A A
  2. A = R K L S C P A=RKLSCP
  3. d z d t = - K τ ( τ - τ c ) a {dz\over dt}=-K_{\tau}(\tau\ -\tau\ _{c})^{a}
  4. τ \tau
  5. τ \tau
  6. τ = ρ g D S \tau\ =\rho\ gDS
  7. K τ K_{\tau}
  8. K τ K_{\tau}
  9. d z d t = - K ω ω {dz\over dt}=-K_{\omega}\ \omega
  10. K ω K_{\omega}
  11. ω \omega
  12. ω = ρ g Q S / W \omega\ =\rho\ gQS/W

Error_analysis_(mathematics).html

  1. z = f ( x , y ) \scriptstyle z\,=\,f(x,y)
  2. x \scriptstyle x
  3. y \scriptstyle y
  4. x ¯ \scriptstyle\bar{x}
  5. y ¯ \scriptstyle\bar{y}
  6. z \scriptstyle z
  7. z ¯ \scriptstyle\bar{z}
  8. z = f ( a 0 , a 1 , , a n ) \scriptstyle z^{\prime}=f^{\prime}(a_{0},\,a_{1},\,\dots,\,a_{n})
  9. z = f ( a 0 , a 1 , , a n ) \scriptstyle z\,=\,f(a_{0},a_{1},\dots,a_{n})
  10. ϵ \scriptstyle\epsilon
  11. 0 | z - z | ϵ \scriptstyle 0\,\leq\,|z-z^{\prime}|\,\leq\,\epsilon
  12. z = f ( a 0 , a 1 , , a n ) \scriptstyle z^{\prime}\,=\,f^{\prime}(a_{0},\,a_{1},\,\dots,\,a_{n})
  13. a i = a i ¯ ± ϵ i \scriptstyle a_{i}\,=\,\bar{a_{i}}\,\pm\,\epsilon_{i}
  14. z = z \scriptstyle z^{\prime}\,=\,z
  15. A = 1 M μ = 1 M A μ . \langle A\rangle=\frac{1}{M}\sum_{\mu=1}^{M}A_{\mu}.
  16. σ 2 ( A ) = 1 M σ 2 ( A ) , \sigma^{2}(\langle A\rangle)=\frac{1}{M}\sigma^{2}(A),
  17. σ 2 ( A ) = 1 M σ 2 A [ 1 + 2 μ ( 1 - μ M ) ϕ μ ] , \sigma^{2}(\langle A\rangle)=\frac{1}{M}\sigma^{2}A\left[1+2\sum_{\mu}\left(1-% \frac{\mu}{M}\right)\phi_{\mu}\right],
  18. ϕ μ \phi_{\mu}
  19. ϕ μ = A μ A 0 - A 2 A 2 - A 2 . \phi_{\mu}=\frac{\langle A_{\mu}A_{0}\rangle-\langle A\rangle^{2}}{\langle A^{% 2}\rangle-\langle A\rangle^{2}}.

Estevez-Mansfield-Clarkson_equation.html

  1. U t y y y + β * U y * U y t + β * U y y * U t + U t t = 0 U_{tyyy}+\beta*U_{y}*U_{yt}+\beta*U_{yy}*U_{t}+U_{tt}=0
  2. U = u ( x , y , t ) U=u(x,y,t)

Eternal_dominating_set.html

  1. γ ( G H ) γ ( G ) γ ( H ) . \gamma_{\infty}(G\,\Box\,H)\geq\gamma_{\infty}(G)\gamma_{\infty}(H).

Etherington's_reciprocity_theorem.html

  1. d L = ( 1 + z ) 2 d A d_{L}=(1+z)^{2}d_{A}
  2. d L d_{L}
  3. d A d_{A}

Evaluation_of_binary_classifiers.html

  1. 𝑇𝑃𝑅 = 𝑇𝑃 / P = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑁 ) \mathit{TPR}=\mathit{TP}/P=\mathit{TP}/(\mathit{TP}+\mathit{FN})
  2. 𝑆𝑃𝐶 = 𝑇𝑁 / N = 𝑇𝑁 / ( 𝐹𝑃 + 𝑇𝑁 ) \mathit{SPC}=\mathit{TN}/N=\mathit{TN}/(\mathit{FP}+\mathit{TN})
  3. 𝑃𝑃𝑉 = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑃 ) \mathit{PPV}=\mathit{TP}/(\mathit{TP}+\mathit{FP})
  4. 𝑁𝑃𝑉 = 𝑇𝑁 / ( 𝑇𝑁 + 𝐹𝑁 ) \mathit{NPV}=\mathit{TN}/(\mathit{TN}+\mathit{FN})
  5. 𝐹𝑃𝑅 = 𝐹𝑃 / N = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑁 ) \mathit{FPR}=\mathit{FP}/N=\mathit{FP}/(\mathit{FP}+\mathit{TN})
  6. 𝐹𝐷𝑅 = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑃 ) = 1 - 𝑃𝑃𝑉 \mathit{FDR}=\mathit{FP}/(\mathit{FP}+\mathit{TP})=1-\mathit{PPV}
  7. 𝐹𝑁𝑅 = 𝐹𝑁 / ( 𝐹𝑁 + 𝑇𝑃 ) \mathit{FNR}=\mathit{FN}/(\mathit{FN}+\mathit{TP})
  8. 𝐴𝐶𝐶 = ( 𝑇𝑃 + 𝑇𝑁 ) / ( P + N ) \mathit{ACC}=(\mathit{TP}+\mathit{TN})/(P+N)
  9. F1 = 2 𝑇𝑃 / ( 2 𝑇𝑃 + 𝐹𝑃 + 𝐹𝑁 ) \mathit{F1}=2\mathit{TP}/(2\mathit{TP}+\mathit{FP}+\mathit{FN})
  10. T P × T N - F P × F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) \frac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}
  11. L = ( P + N ) × log ( P + N ) L T P = T P × log T P ( T P + F P ) ( T P + F N ) L F P = F P × log F P ( F P + T P ) ( F P + T N ) L F N = F N × log F N ( F N + T P ) ( F N + T N ) L T N = T N × log T N ( T N + F P ) ( T N + F N ) L P = P × log P P + N L N = N × log N P + N U C = L + L T P + L F P + L F N + L T N L + L P + L N \begin{aligned}\displaystyle L&\displaystyle=(P+N)\times\log(P+N)\\ \displaystyle LTP&\displaystyle=TP\times\log\frac{TP}{(TP+FP)(TP+FN)}\\ \displaystyle LFP&\displaystyle=FP\times\log\frac{FP}{(FP+TP)(FP+TN)}\\ \displaystyle LFN&\displaystyle=FN\times\log\frac{FN}{(FN+TP)(FN+TN)}\\ \displaystyle LTN&\displaystyle=TN\times\log\frac{TN}{(TN+FP)(TN+FN)}\\ \displaystyle LP&\displaystyle=P\times\log\frac{P}{P+N}\\ \displaystyle LN&\displaystyle=N\times\log\frac{N}{P+N}\\ \displaystyle UC&\displaystyle=\frac{L+LTP+LFP+LFN+LTN}{L+LP+LN}\end{aligned}
  12. PPV = ( sensitivity ) ( prevalence ) ( sensitivity ) ( prevalence ) + ( 1 - specificity ) ( 1 - prevalence ) \,\text{PPV}=\frac{(\,\text{sensitivity})(\,\text{prevalence})}{(\,\text{% sensitivity})(\,\text{prevalence})+(1-\,\text{specificity})(1-\,\text{% prevalence})}
  13. NPV = ( specificity ) ( 1 - prevalence ) ( specificity ) ( 1 - prevalence ) + ( 1 - sensitivity ) ( prevalence ) . \,\text{NPV}=\frac{(\,\text{specificity})(1-\,\text{prevalence})}{(\,\text{% specificity})(1-\,\text{prevalence})+(1-\,\text{sensitivity})(\,\text{% prevalence})}.
  14. F 1 = 2 precision recall precision + recall F_{1}=2\cdot\frac{\mathrm{precision}\cdot\mathrm{recall}}{\mathrm{precision}+% \mathrm{recall}}

Evapoporometry.html

  1. l n p A p A = - 4 y V A R T d ln{p^{\prime}A\over p^{\circ}A}=-{4yV\text{A}\over RTd}
  2. p A p^{\prime}A
  3. p A p^{\circ}A
  4. y y
  5. V A V\text{A}
  6. R R
  7. T T
  8. d d
  9. X A0 X\text{A0}
  10. X A0 X^{\circ}\text{A0}
  11. W A W\text{A}
  12. W A W^{\circ}\text{A}
  13. d c d\text{c}
  14. k x k\text{x}

Evidence-based_subjective_logic.html

  1. ( b , d , u ) = ( p , n , c ) p + n + c ( p , n ) = c ( b , d ) u . ( 1 ) (b,d,u)=\frac{(p,n,c)}{p+n+c}\quad\quad(p,n)=c\frac{(b,d)}{u}.\quad\quad(1)
  2. p p\to\infty
  3. u 0 u\to 0
  4. ( p z , n z ) = ( p x , n x ) + ( p y , n y ) . (p_{z},n_{z})=(p_{x},n_{x})+(p_{y},n_{y}).
  5. x y = ( p x + p y , n x + n y , c ) p x + p y + n x + n y + c x\oplus y=\frac{(p_{x}+p_{y},n_{x}+n_{y},c)}{p_{x}+p_{y}+n_{x}+n_{y}+c}
  6. x y = ( x u y b + y u x b , x u y d + y u x d , x u y u ) x u + y u - x u y u . ( 2 ) x\oplus y=\frac{(x_{\rm u}y_{\rm b}+y_{\rm u}x_{\rm b},x_{\rm u}y_{\rm d}+y_{% \rm u}x_{\rm d},x_{\rm u}y_{\rm u})}{x_{\rm u}+y_{\rm u}-x_{\rm u}y_{\rm u}}.% \quad\quad(2)
  7. x y = def ( x b y b , x b y d , 1 - x b y b - x b y d ) . x\otimes y\stackrel{\rm def}{=}(x_{\rm b}y_{\rm b},x_{\rm b}y_{\rm d},1-x_{\rm b% }y_{\rm b}-x_{\rm b}y_{\rm d}).
  8. ( x y 1 ) ( x y 2 ) (x\otimes y_{1})\oplus(x\otimes y_{2})
  9. y 1 y 2 y_{1}\oplus y_{2}
  10. y 1 y 2 y_{1}\oplus y_{2}
  11. x ( y 1 y 2 ) x\otimes(y_{1}\oplus y_{2})
  12. x ( y 1 y 2 ) ( x y 1 ) ( x y 2 ) . x\otimes(y_{1}\oplus y_{2})\neq(x\otimes y_{1})\oplus(x\otimes y_{2}).
  13. λ x = ( λ x b , λ x d , x u ) λ ( x b + x d ) + x u ( 3 ) \lambda\cdot x=\frac{(\lambda x_{\rm b},\lambda x_{\rm d},x_{\rm u})}{\lambda(% x_{\rm b}+x_{\rm d})+x_{\rm u}}\quad\quad(3)
  14. x y = def g ( x ) y , ( 4 ) \quad x\boxtimes y\stackrel{\rm def}{=}g(x)\cdot y,\quad\quad(4)
  15. x ( y 1 y 2 ) = ( x y 1 ) ( x y 2 ) x\boxtimes(y_{1}\oplus y_{2})=(x\boxtimes y_{1})\oplus(x\boxtimes y_{2})
  16. x 1 ( x 2 y ) = x 2 ( x 1 y ) x_{1}\boxtimes(x_{2}\boxtimes y)=x_{2}\boxtimes(x_{1}\boxtimes y)
  17. x 1 ( x 2 y ) ( x 1 x 2 ) y x_{1}\boxtimes(x_{2}\boxtimes y)\neq(x_{1}\boxtimes x_{2})\boxtimes y
  18. ( x 1 x 2 ) y ( x 1 y ) ( x 2 y ) . (x_{1}\oplus x_{2})\boxtimes y\neq(x_{1}\boxtimes y)\oplus(x_{2}\boxtimes y).
  19. For i j : R i j = A i j k : k i R i k A k j ( 5 ) \mbox{For }~{}i\neq j:\quad R_{ij}=A_{ij}\oplus\sum_{k:\,k\neq i}R_{ik}% \boxtimes A_{kj}\quad\quad(5)
  20. Diagonal: R i i = B . \mbox{Diagonal: }~{}R_{ii}=B.
  21. R = B 𝟏 ( R A ) . ( 6 ) R=B{\mathbf{1}}\oplus(R\boxtimes A).\quad\quad(6)
  22. 𝟏 {\mathbf{1}}
  23. f ( X ) = B 𝟏 ( X A ) f(X)=B{\mathbf{1}}\oplus(X\boxtimes A)
  24. X 0 X_{0}
  25. R = f ( R ) R=f(R)
  26. X 0 X_{0}
  27. R = 𝟏 + R A R={\mathbf{1}}+RA
  28. R = ( 𝟏 - A ) - 1 = 𝟏 + k 1 A k R=({\mathbf{1}}-A)^{-1}={\mathbf{1}}+\sum_{k\geq 1}A^{k}

Evolution_of_a_random_network.html

  1. N G {N_{G}}
  2. k {\left\langle k\right\rangle}
  3. k = 1 {\left\langle k\right\rangle=1}
  4. k {\left\langle k\right\rangle}
  5. p {p}
  6. P c = 1 N - 1 1 N {P_{c}=\frac{1}{N-1}\approx\frac{1}{N}}
  7. N {N}
  8. P c {P_{c}}
  9. p {p}
  10. 0 < k < 1 {0<\left\langle k\right\rangle<1}
  11. ( p < 1 n ) {\left(p<\frac{1}{n}\right)}
  12. k = 0 {\left\langle k\right\rangle=0}
  13. N {N}
  14. k {\left\langle k\right\rangle}
  15. N k = p N ( N - 1 ) / 2 {N\left\langle k\right\rangle=pN(N-1)/2}
  16. k < 1 {\left\langle k\right\rangle<1}
  17. N G N {\frac{N_{G}}{N}}
  18. k < 1 {\left\langle k\right\rangle<1}
  19. N G l n N {N_{G}\sim lnN}
  20. N G / N = l n N / N 0 {N_{G}/N=lnN/N\rightarrow 0}
  21. N {N\rightarrow\infty}
  22. 0 < k = 1 {0<\left\langle k\right\rangle=1}
  23. ( p = 1 n ) {\left(p=\frac{1}{n}\right)}
  24. 0 < k < 1 {0<\left\langle k\right\rangle<1}
  25. k > 1 {\left\langle k\right\rangle>1}
  26. N G N 2 3 {N_{G}\sim N^{\tfrac{2}{3}}}
  27. N G {N_{G}}
  28. N G N N - 1 3 {\frac{N_{G}}{N}\sim N^{-\tfrac{1}{3}}}
  29. N {N\rightarrow\infty}
  30. k = 1 {\left\langle k\right\rangle=1}
  31. N = 7 109 {N=7\ast 109}
  32. k = 1 {\left\langle k\right\rangle=1}
  33. N G l n N = l n ( 7 109 ) 22.7 {N_{G}\simeq lnN=ln(7\ast 109)\simeq 22.7}
  34. k = 1 {\left\langle k\right\rangle=1}
  35. N G N 2 3 = ( 7 109 ) 2 3 3 106 {N_{G}\simeq N^{\tfrac{2}{3}}=(7\ast 109){\tfrac{2}{3}}\sim 3\ast 106}
  36. 0 < k > 1 {0<\left\langle k\right\rangle>1}
  37. ( p > 1 n ) {\left(p>\frac{1}{n}\right)}
  38. N G / N k - 1 {N_{G}/N\sim\left\langle k\right\rangle-1}
  39. N G ( p - p c ) N {N_{G}\sim(p-p_{c})N}
  40. 0 < k = 1 {0<\left\langle k\right\rangle=1}
  41. 0 < k {0<\left\langle k\right\rangle}
  42. N G {N_{G}}
  43. 0 < k {0<\left\langle k\right\rangle}
  44. 0 < k > l n N {0<\left\langle k\right\rangle>lnN}
  45. ( p > l n N N ) {\left(p>\frac{lnN}{N}\right)}
  46. N G N {N_{G}\simeq N}
  47. N {N}
  48. ( l n N N ) 0 {\left(\frac{lnN}{N}\right)\rightarrow 0}
  49. ( l n N N ) 0 {\left(\frac{lnN}{N}\right)\rightarrow 0}
  50. k = N - 1 {\left\langle k\right\rangle=N-1}

Excess_demand_function.html

  1. d P d t = λ f ( P , ) \frac{dP}{dt}=\lambda\cdot f(P,...)
  2. λ \lambda
  3. P t + 1 = P t + δ f ( P t , ) P_{t+1}=P_{t}+\delta\cdot f(P_{t},...)
  4. P t + 1 - P t P_{t+1}-P_{t}
  5. d P d t \frac{dP}{dt}
  6. δ \delta
  7. δ = 1 \delta=1

Excimer_lamp.html

  1. R g 2 * 𝜏 R g + R g + h ν ( U V p h o t o n ) , {Rg_{2}^{*}\ \xrightarrow[]{\tau}\ Rg+Rg+h\nu(UVphoton)},
  2. R g X * 𝜏 R g + X + h ν ( U V p h o t o n ) , {RgX^{*}\ \xrightarrow[]{\tau}\ Rg+X+h\nu(UVphoton)},

Excisive_triad.html

  1. ( X ; A , B ) (X;A,B)

Exo-(1-4)-α-D-glucan_lyase.html

  1. \rightleftharpoons

Exo-alpha-bergamotene_synthase.html

  1. \rightleftharpoons

Exotic_affine_space.html

  1. n \mathbb{C}^{n}

Exp_algebra.html

  1. exp ( g t ) = 1 + g 1 t + g 2 t 2 + g 3 t 3 + . \exp(gt)=1+g_{1}t+g_{2}t^{2}+g_{3}t^{3}+\cdots.
  2. exp ( ( g + h ) t ) = exp ( g t ) exp ( h t ) \exp((g+h)t)=\exp(gt)\exp(ht)

Expanded_cuboctahedron.html

  1. { 4 3 } \begin{Bmatrix}4\\ 3\end{Bmatrix}

Expanded_icosidodecahedron.html

  1. { 5 3 } \begin{Bmatrix}5\\ 3\end{Bmatrix}

Explicit_algebraic_stress_model.html

  1. b i j = λ = 1 10 G ( λ ) T i j ( λ ) b_{ij}=\sum_{\lambda=1}^{10}G^{(\lambda)}T_{ij}^{(\lambda)}
  2. T i j ( 1 ) = s i j T_{ij}^{(1)}=s_{ij}
  3. T i j ( 2 ) = s i k w k j - w i k s k j T_{ij}^{(2)}=s_{ik}w_{kj}-w_{ik}s{kj}
  4. T i j ( 3 ) = s i k s k j - s m k s k m 1 3 δ i j T_{ij}^{(3)}=s_{ik}s_{kj}-s_{mk}s_{km}\frac{1}{3}\delta_{ij}
  5. T i j ( 5 ) = w i k s k l s l j - s i k s k l w l j T_{ij}^{(5)}=w_{ik}s_{kl}s_{lj}-s_{ik}s_{kl}w_{lj}
  6. C μ = - A 1 g g 2 - 2 3 A 3 2 η 1 - 2 A 2 η 2 C\mu=\frac{-A_{1}g}{g^{2}-\frac{2}{3}A_{3}^{2}\eta_{1}-2A^{2}\eta_{2}}
  7. g = C 1 - 2 b i j g=C_{1}-2b_{ij}\,

Exponential_integrator.html

  1. y ( t ) = L y ( t ) + 𝒩 ( y ( t ) ) , y ( t 0 ) = y 0 , ( 1 ) y^{\prime}(t)=Ly(t)+\mathcal{N}(y(t)),\qquad y(t_{0})=y_{0},\qquad\qquad(1)
  2. L L
  3. 𝒩 \mathcal{N}
  4. y ( t ) = f ( y ( t ) ) , y ( t 0 ) = y 0 , y^{\prime}(t)=f(y(t)),\qquad y(t_{0})=y_{0},
  5. y * y^{*}
  6. L = f y ( y * ) ; 𝒩 = f ( y ) - L y . L=\frac{\partial f}{\partial y}(y^{*});\qquad\mathcal{N}=f(y)-Ly.
  7. f y \frac{\partial f}{\partial y}
  8. f f
  9. y y
  10. t t
  11. y ( t ) = e L t y 0 + 0 t e L ( t - τ ) 𝒩 ( y ( τ ) ) d τ . ( 2 ) y(t)=e^{Lt}y_{0}+\int_{0}^{t}e^{L(t-\tau)}\mathcal{N}\left(y\left(\tau\right)% \right)\,d\tau.\qquad(2)
  12. 𝒩 0 \mathcal{N}\equiv 0
  13. 𝒩 ( y ( τ ) ) 𝒩 ( y ( 0 ) ) \mathcal{N}(y(\tau))\approx\mathcal{N}(y(0))
  14. e L ( t - τ ) e^{L(t-\tau)}
  15. y ( t ) = e L t y 0 + L - 1 ( e L t - 1 ) 𝒩 ( y ( t 0 ) ) . y(t)=e^{Lt}y_{0}+L^{-1}(e^{Lt}-1)\mathcal{N}(y(t_{0})).
  16. φ 0 ( z ) = e z ; φ 1 ( z ) = e z - 1 z , φ 2 ( z ) = e z - 1 - z z 2 . \varphi_{0}(z)=e^{z};\qquad\varphi_{1}(z)=\frac{e^{z}-1}{z},\qquad\varphi_{2}(% z)=\frac{e^{z}-1-z}{z^{2}}.
  17. u u
  18. u n u_{n}
  19. t n t_{n}
  20. 𝒩 = 𝒩 ( u , t ) \mathcal{N}=\mathcal{N}(u,t)
  21. a n = e L h / 2 u n + L - 1 ( e L h / 2 - I ) 𝒩 ( u n , t n ) a_{n}=e^{Lh/2}u_{n}+L^{-1}\left(e^{Lh/2}-I\right)\mathcal{N}(u_{n},t_{n})
  22. b n = e L h / 2 u n + L - 1 ( e L h / 2 - I ) 𝒩 ( a n , t n + h / 2 ) b_{n}=e^{Lh/2}u_{n}+L^{-1}\left(e^{Lh/2}-I\right)\mathcal{N}(a_{n},t_{n}+h/2)
  23. c n = e L h / 2 a n + L - 1 ( e L h / 2 - I ) ( 2 𝒩 ( b n , t n + h / 2 ) - 𝒩 ( u n , t n ) ) c_{n}=e^{Lh/2}a_{n}+L^{-1}\left(e^{Lh/2}-I\right)\left(2\mathcal{N}(b_{n},t_{n% }+h/2)-\mathcal{N}(u_{n},t_{n})\right)
  24. u n + 1 = e L h u n + h - 2 L - 3 { [ - 4 - L h + e L h ( 4 - 3 L h + ( L h ) 2 ) ] 𝒩 ( u n , t n ) + 2 [ 2 + L h + e L h ( - 2 + L h ) ] ( 𝒩 ( a n , t n + h / 2 ) + 𝒩 ( b n , t n + h / 2 ) ) + [ - 4 - 3 L h - ( L h ) 2 + e L h ( 4 - L h ) ] 𝒩 ( c n , t n + h ) } . u_{n+1}=e^{Lh}u_{n}+h^{-2}L^{-3}\left\{\left[-4-Lh+e^{Lh}\left(4-3Lh+(Lh)^{2}% \right)\right]\mathcal{N}(u_{n},t_{n})+2\left[2+Lh+e^{Lh}\left(-2+Lh\right)% \right]\left(\mathcal{N}(a_{n},t_{n}+h/2)+\mathcal{N}(b_{n},t_{n}+h/2)\right)+% \left[-4-3Lh-(Lh)^{2}+e^{Lh}\left(4-Lh\right)\right]\mathcal{N}(c_{n},t_{n}+h)% \right\}.
  25. φ 1 ( z ) = 1 - e z z , \varphi_{1}(z)=\frac{1-e^{z}}{z},
  26. z z
  27. φ 1 \varphi_{1}

Exponential_map.html

  1. X γ X ( 1 ) X\mapsto\gamma_{X}(1)
  2. γ X \gamma_{X}

Exponential_map_(Lie_theory).html

  1. G G
  2. 𝔤 \mathfrak{g}
  3. G G
  4. exp : 𝔤 G \exp\colon\mathfrak{g}\to G
  5. exp ( X ) = γ ( 1 ) \exp(X)=\gamma(1)
  6. γ : G \gamma\colon\mathbb{R}\to G
  7. G G
  8. X X
  9. exp ( t X ) = γ ( t ) \exp(tX)=\gamma(t)
  10. γ \gamma
  11. X X
  12. exp ( X ) = γ ( 1 ) \exp(X)=\gamma(1)
  13. γ \gamma
  14. G G
  15. exp ( X ) = k = 0 X k k ! = I + X + 1 2 X 2 + 1 6 X 3 + \exp(X)=\sum_{k=0}^{\infty}\frac{X^{k}}{k!}=I+X+\frac{1}{2}X^{2}+\frac{1}{6}X^% {3}+\cdots
  16. I I
  17. 𝔤 \mathfrak{g}
  18. t exp ( t X ) t\mapsto\exp(tX)
  19. t t X . t\mapsto tX.
  20. Lie ( ) = \operatorname{Lie}(\mathbb{R})=\mathbb{R}
  21. { i t : t } . \{it:t\in\mathbb{R}\}.
  22. i t exp ( i t ) = e i t = cos ( t ) + i sin ( t ) , it\mapsto\exp(it)=e^{it}=\cos(t)+i\sin(t),\,
  23. z = x + y ȷ , ȷ 2 = + 1 , z=x+y\jmath,\quad\jmath^{2}=+1,
  24. { ȷ t : t } \{\jmath t:t\in\mathbb{R}\}
  25. { cosh t + ȷ sinh t : t } \{\cosh t+\jmath\ \sinh t:t\in\mathbb{R}\}
  26. ȷ t exp ( ȷ t ) = cosh t + ȷ sinh t . \jmath t\mapsto\exp(\jmath t)=\cosh t+\jmath\ \sinh t.
  27. S S
  28. S U ( 2 ) SU(2)
  29. { i t + j u + k v : t , u , v } . \{it+ju+kv:t,u,v\in\mathbb{R}\}.
  30. w = ( i t + j u + k v ) exp ( i t + j u + k v ) = cos ( | w | ) + sin ( | w | ) w | w | . {w}=(it+ju+kv)\mapsto\exp(it+ju+kv)=\cos(|{w}|)+\sin(|{w}|)\frac{{w}}{|{w}|}.\,
  31. R R
  32. { s S 3 H : Re ( s ) = cos ( R ) } \{s\in S^{3}\subset{H}:\operatorname{Re}(s)=\cos(R)\}
  33. sin ( R ) \sin(R)
  34. R 0 ( mod 2 π ) R\not\equiv 0\;\;(\mathop{{\rm mod}}2\pi)
  35. Lie ( V ) = V \operatorname{Lie}(V)=V
  36. exp : Lie ( V ) = V V \operatorname{exp}:\operatorname{Lie}(V)=V\to V
  37. X 𝔤 X\in\mathfrak{g}
  38. γ ( t ) = exp ( t X ) \gamma(t)=\exp(tX)
  39. G G
  40. X X
  41. exp ( t + s ) X = ( exp t X ) ( exp s X ) \exp(t+s)X=(\exp tX)(\exp sX)\,
  42. exp ( - X ) = ( exp X ) - 1 . \exp(-X)=(\exp X)^{-1}.\,
  43. exp : 𝔤 G \exp\colon\mathfrak{g}\to G
  44. exp * : 𝔤 𝔤 \exp_{*}\colon\mathfrak{g}\to\mathfrak{g}
  45. 𝔤 \mathfrak{g}
  46. G G
  47. G G
  48. G G
  49. G = G L n ( ) G=GL_{n}(\mathbb{C})
  50. γ ( t ) = exp ( t X ) \gamma(t)=\exp(tX)
  51. X X
  52. g G g\in G
  53. X L X^{L}
  54. X X
  55. g exp ( t X ) g\exp(tX)
  56. g g
  57. X R X^{R}
  58. exp ( t X ) g \exp(tX)g
  59. ξ L , R \xi^{L,R}
  60. X L , R X^{L,R}
  61. G G
  62. ϕ : G H \phi\colon G\to H
  63. ϕ * \phi_{*}
  64. G G
  65. g ( exp X ) g - 1 = exp ( Ad g X ) g(\exp X)g^{-1}=\exp(\mathrm{Ad}_{g}X)\,
  66. Ad exp X = exp ( ad X ) . \mathrm{Ad}_{\exp X}=\exp(\mathrm{ad}_{X}).\,

Exponential_search.html

  1. j {}^{j}
  2. j {}^{j}
  3. j - 1 {}^{j-1}

Exposed_point.html

  1. C C
  2. x C x\in C
  3. C C
  4. x x
  5. x x
  6. C C
  7. exp ( C ) \exp(C)
  8. C C
  9. x C x\in C
  10. f f
  11. x x
  12. C C
  13. x x
  14. ( x n ) C (x_{n})\subset C
  15. f ( x n ) max f ( C ) x n - x 0 f(x_{n})\to\max f(C)\Longrightarrow\|x_{n}-x\|\to 0
  16. C C
  17. str exp ( C ) \operatorname{str}\exp(C)
  18. C C

Extended_theories_of_gravity.html

  1. f ( χ ) = χ 3 2 f(\chi)=\chi^{\frac{3}{2}}

Extension_of_a_topological_group.html

  1. 0 H ı X π G 0 0\to H\stackrel{\imath}{\to}X\stackrel{\pi}{\to}G\to 0
  2. H , X H,X
  3. G G
  4. i i
  5. π \pi
  6. 0 H i X π G 0 0\rightarrow H\stackrel{i}{\rightarrow}X\stackrel{\pi}{\rightarrow}G\rightarrow 0
  7. 0 H i X π G 0 0\to H\stackrel{i^{\prime}}{\rightarrow}X^{\prime}\stackrel{\pi^{\prime}}{% \rightarrow}G\rightarrow 0
  8. T : X X T:X\to X^{\prime}
  9. 0 H i X π G 0 0\rightarrow H\stackrel{i}{\rightarrow}X\stackrel{\pi}{\rightarrow}G\rightarrow 0
  10. 0 H i H H × G π G G 0 0\rightarrow H\stackrel{i_{H}}{\rightarrow}H\times G\stackrel{\pi_{G}}{% \rightarrow}G\rightarrow 0
  11. i H : H H × G i_{H}:H\to H\times G
  12. π G : H × G G \pi_{G}:H\times G\to G
  13. 0 H i X π G 0 0\rightarrow H\stackrel{i}{\rightarrow}X\stackrel{\pi}{\rightarrow}G\rightarrow 0
  14. R : X H R:X\rightarrow H
  15. R i R\circ i
  16. H H
  17. 0 H i X π G 0 0\rightarrow H\stackrel{i}{\rightarrow}X\stackrel{\pi}{\rightarrow}G\rightarrow 0
  18. i ( H ) i(H)
  19. X X
  20. \mathbb{R}
  21. \mathbb{Z}
  22. ı \imath
  23. π \pi
  24. 0 ı π / 0 0\to\mathbb{Z}\stackrel{\imath}{\to}\mathbb{R}\stackrel{\pi}{\to}\mathbb{R}/% \mathbb{Z}\to 0
  25. 0 H ı X π G 0 0\to H\stackrel{\imath}{\to}X\stackrel{\pi}{\to}G\to 0
  26. H , X H,X
  27. G G
  28. i i
  29. π \pi
  30. 0 H ı X π G 0. 0\to H\stackrel{\imath}{\to}X\stackrel{\pi}{\to}G\to 0.
  31. H , X H^{\wedge},X^{\wedge}
  32. G G^{\wedge}
  33. H , X H,X
  34. G G
  35. i i^{\wedge}
  36. π \pi^{\wedge}
  37. i i
  38. π \pi
  39. 0 G π X ı H 0 0\to G^{\wedge}\stackrel{\pi^{\wedge}}{\to}X^{\wedge}\stackrel{\imath^{\wedge}% }{\to}H^{\wedge}\to 0

Extreme_learning_machine.html

  1. 𝐘 ^ = 𝐖 2 σ ( 𝐖 1 x ) \mathbf{\hat{Y}}=\mathbf{W}_{2}\sigma(\mathbf{W}_{1}x)
  2. σ σ
  3. 𝐖 < s u b > 1 \mathbf{W}<sub>1
  4. 𝐘 \mathbf{Y}
  5. 𝐗 \mathbf{X}
  6. 𝐖 2 = σ ( 𝐖 1 𝐗 ) + 𝐘 \mathbf{W}_{2}=\sigma(\mathbf{W}_{1}\mathbf{X})^{+}\mathbf{Y}

Étale_homotopy_type.html

  1. U X U\rightarrow X
  2. U n := U × X U × X × X U U_{n}:=U\times_{X}U\times_{X}\dots\times_{X}U
  3. n 0 n\geq 0
  4. ( U n ) n 0 (U_{n})_{n\geq 0}

Étale_topos.html

  1. { φ i : U i X } i I \{\varphi_{i}:U_{i}\to X\}_{i\in I}
  2. φ i \varphi_{i}
  3. X = i I φ i ( U i ) X=\bigcup_{i\in I}\varphi_{i}(U_{i})
  4. X ét X\text{ét}
  5. \mathcal{F}
  6. { φ i : U i U } \{\varphi_{i}:U_{i}\to U\}
  7. 0 ( U ) i I ( U i ) i , j I ( U i j ) 0\to\mathcal{F}(U)\to\prod_{i\in I}\mathcal{F}(U_{i}){{{}\atop\longrightarrow}% \atop{\longrightarrow\atop{}}}\prod_{i,j\in I}\mathcal{F}(U_{ij})
  8. U i j = U i × U U j U_{ij}=U_{i}\times_{U}U_{j}

Fabry_gap_theorem.html

  1. f ( z ) = j 𝐍 α j z p j f(z)=\sum_{j\in\mathbf{N}}\alpha_{j}z^{p_{j}}

Factor_analysis_of_mixed_data.html

  1. K K
  2. k = 1 , K {k=1,K}
  3. Q Q
  4. q = 1 , Q {q=1,Q}
  5. z z
  6. r ( z , k ) r(z,k)
  7. k k
  8. z z
  9. η 2 ( z , q ) \eta^{2}(z,q)
  10. z z
  11. q q
  12. K K
  13. I I
  14. I I
  15. K K
  16. k r 2 ( z , k ) \sum_{k}r^{2}(z,k)
  17. I I
  18. Q Q
  19. q η 2 ( z , q ) \sum_{q}\eta^{2}(z,q)
  20. { K , Q } \{K,Q\}
  21. I I
  22. K + Q K+Q
  23. k r 2 ( z , k ) + q η 2 ( z , q ) \sum_{k}r^{2}(z,k)+\sum_{q}\eta^{2}(z,q)
  24. I I
  25. j j
  26. s s
  27. j j
  28. s s
  29. η 2 ( j , s ) \eta^{2}(j,s)
  30. k k
  31. s s
  32. k k
  33. s s
  34. r 2 ( k , s ) r^{2}(k,s)
  35. l l
  36. c c
  37. l l
  38. c c
  39. l l
  40. c c
  41. l l
  42. c c
  43. l l
  44. c c
  45. l l
  46. c c
  47. ϕ 2 \phi^{2}
  48. l l
  49. c c
  50. k 1 k_{1}
  51. k 2 k_{2}
  52. k 3 k_{3}
  53. q 1 q_{1}
  54. q 2 q_{2}
  55. q 3 q_{3}
  56. i 1 i_{1}
  57. q 1 q_{1}
  58. q 2 q_{2}
  59. q 3 q_{3}
  60. i 2 i_{2}
  61. q 1 q_{1}
  62. q 2 q_{2}
  63. q 3 q_{3}
  64. i 3 i_{3}
  65. q 1 q_{1}
  66. q 2 q_{2}
  67. q 3 q_{3}
  68. i 4 i_{4}
  69. q 1 q_{1}
  70. q 2 q_{2}
  71. q 3 q_{3}
  72. i 5 i_{5}
  73. q 1 q_{1}
  74. q 2 q_{2}
  75. q 3 q_{3}
  76. i 6 i_{6}
  77. q 1 q_{1}
  78. q 2 q_{2}
  79. q 3 q_{3}
  80. k 1 k_{1}
  81. k 2 k_{2}
  82. k 3 k_{3}
  83. q 1 q_{1}
  84. q 2 q_{2}
  85. q 3 q_{3}
  86. k 1 k_{1}
  87. k 2 k_{2}
  88. k 3 k_{3}
  89. q 1 q_{1}
  90. q 2 q_{2}
  91. q 3 q_{3}
  92. R 2 R^{2}
  93. ϕ 2 \phi^{2}
  94. η 2 \eta^{2}
  95. F 1 F1
  96. k 1 k_{1}
  97. k 2 k_{2}
  98. Q 3 Q_{3}
  99. F 1 F1
  100. k 2 k_{2}
  101. k 3 k_{3}
  102. F 1 F1
  103. Q 3 Q_{3}
  104. k 2 k_{2}
  105. k 3 k_{3}
  106. c c
  107. Q 3 Q_{3}

Factorial_moment_measure.html

  1. x \textstyle x
  2. x N , \textstyle x\in{N},
  3. N ( B ) , \textstyle{N}(B),
  4. n = 1 , 2 , \textstyle n=1,2,\dots
  5. n \textstyle n
  6. N \textstyle{N}
  7. 𝐑 d \textstyle\,\textbf{R}^{d}
  8. N ( n ) ( B 1 × , , × B n ) = ( x 1 , , x n ) N i = 1 n 𝟏 B i ( x i ) {N}^{(n)}(B_{1}\times,\dots,\times B_{n})=\sum_{(x_{1}\neq,\dots,\neq x_{n})% \in{N}}\prod_{i=1}^{n}\mathbf{1}_{B_{i}}(x_{i})
  9. B 1 , , B n \textstyle B_{1},...,B_{n}
  10. 𝐑 d \textstyle\,\textbf{R}^{d}
  11. n \textstyle n
  12. B 1 × , , × B n . B_{1}\times,\dots,\times B_{n}.
  13. 𝟏 \textstyle\mathbf{1}
  14. 𝟏 B 1 \textstyle\mathbf{1}_{B_{1}}
  15. B n \textstyle B_{n}
  16. n \textstyle n
  17. Π \textstyle\Pi
  18. M ( n ) ( B 1 × , , × B n ) = E [ N ( n ) ( B 1 × , , × B n ) ] , M^{(n)}(B_{1}\times,\dots,\times B_{n})=E[{N}^{(n)}(B_{1}\times,\dots,\times B% _{n})],
  19. 𝐑 n d f ( x 1 , , x n ) M ( n ) ( d x 1 , , d x n ) = E [ ( x 1 , , x n ) N f ( x 1 , , x n ) ] , \int_{\,\textbf{R}^{nd}}f(x_{1},\dots,x_{n})M^{(n)}(dx_{1},\dots,dx_{n})=E% \left[\sum_{(x_{1}\neq,\dots,\neq x_{n})\in{N}}f(x_{1},\dots,x_{n})\right],
  20. M 1 \textstyle M^{1}
  21. M ( 1 ) ( B ) = M 1 ( B ) = E [ N ( B ) ] , M^{(1)}(B)=M^{1}(B)=E[{N}(B)],
  22. M 1 \textstyle M^{1}
  23. N \textstyle{N}
  24. B \textstyle B
  25. A \textstyle A
  26. B \textstyle B
  27. M ( 2 ) ( A × B ) = M 1 ( A × B ) - M 1 ( A B ) . M^{(2)}(A\times B)=M^{1}(A\times B)-M^{1}(A\cap B).
  28. B \textstyle B
  29. n \textstyle n\,
  30. M ( n ) ( B × , , × B ) = E [ N ( B ) ( N ( B ) - 1 ) ( N ( B ) - n + 1 ) ] , M^{(n)}(B\times,\dots,\times B)=E[{N}(B)({N}(B)-1)\dots({N}(B)-n+1)],
  31. n \textstyle n\,
  32. N ( B ) \textstyle{N}(B)
  33. n \textstyle n
  34. μ ( n ) ( x 1 , , x n ) \textstyle\mu^{(n)}(x_{1},\dots,x_{n})
  35. M ( n ) ( B 1 × , , × B n ) = B 1 B n μ ( n ) ( x 1 , , x n ) d x 1 d x n . M^{(n)}(B_{1}\times,\dots,\times B_{n})=\int_{B_{1}}\dots\int_{B_{n}}\mu^{(n)}% (x_{1},\dots,x_{n})dx_{1}\dots dx_{n}.
  36. E [ ( x 1 , , x n ) N f ( x 1 , , x n ) ] = 𝐑 n d f ( x 1 , , x n ) μ ( n ) ( x 1 , , x n ) d x 1 d x n , E\left[\sum_{(x_{1}\neq,\dots,\neq x_{n})\in{N}}f(x_{1},\dots,x_{n})\right]=% \int_{\,\textbf{R}^{nd}}f(x_{1},\dots,x_{n})\mu^{(n)}(x_{1},\dots,x_{n})dx_{1}% \dots dx_{n},
  37. f \textstyle f
  38. 𝐑 n \textstyle\,\textbf{R}^{n}
  39. N {N}
  40. ρ ( x 1 , x 2 ) = μ ( 2 ) ( x 1 , x 2 ) μ ( 1 ) ( x 1 ) μ ( 1 ) ( x 2 ) , \rho(x_{1},x_{2})=\frac{\mu^{(2)}(x_{1},x_{2})}{\mu^{(1)}(x_{1})\mu^{(1)}(x_{2% })},
  41. x 1 , x 2 R d x_{1},x_{2}\in R^{d}
  42. ρ ( x 1 , x 2 ) 0 \rho(x_{1},x_{2})\geq 0
  43. ρ ( x 1 , x 2 ) = 1 \rho(x_{1},x_{2})=1
  44. Λ \textstyle\Lambda
  45. n \textstyle n
  46. M ( n ) ( B 1 × , , × B n ) = i = 1 n [ Λ ( B i ) ] , M^{(n)}(B_{1}\times,\dots,\times B_{n})=\prod_{i=1}^{n}[\Lambda(B_{i})],
  47. Λ \textstyle\Lambda
  48. N \textstyle{N}
  49. B \textstyle B
  50. Λ ( B ) = M 1 ( B ) = E [ N ( B ) ] . \Lambda(B)=M^{1}(B)=E[{N}(B)].
  51. n \textstyle n
  52. M ( n ) ( B 1 × , , × B n ) = λ n i = 1 n | B i | , M^{(n)}(B_{1}\times,\dots,\times B_{n})=\lambda^{n}\prod_{i=1}^{n}|B_{i}|,
  53. | B i | \textstyle|B_{i}|
  54. B i \textstyle B_{i}
  55. n \textstyle n
  56. μ ( n ) ( x 1 , , x n ) = λ n . \mu^{(n)}(x_{1},\dots,x_{n})=\lambda^{n}.
  57. ρ ( x 1 , x 2 ) = 1 , \rho(x_{1},x_{2})=1,

Fair_cake-cutting.html

  1. C = P 1 P n C=P_{1}\sqcup\cdots\sqcup P_{n}
  2. i : V i ( P i ) 1 / n \forall{i}:\ V_{i}(P_{i})\geq 1/n
  3. i : V i ( P i ) w i \forall i:\ V_{i}(P_{i})\geq w_{i}
  4. i , j : V i ( P i ) V i ( P j ) \forall i,j:\ V_{i}(P_{i})\geq V_{i}(P_{j})
  5. i , j : V i ( P i ) = V j ( P j ) \forall i,j:\ V_{i}(P_{i})=V_{j}(P_{j})
  6. i , j : V i ( P j ) = w j \forall{i,j}:\ V_{i}(P_{j})=w_{j}
  7. i , j : V i ( P j ) = 1 / n \forall i,j:\ V_{i}(P_{j})=1/n
  8. n = 2 n=2
  9. ϵ > 0 \epsilon>0
  10. ϵ \epsilon
  11. O ( n 2 / ϵ ) O(n^{2}/\epsilon)

Fair_item_assignment.html

  1. 1 / n 1/n

Farley–Buneman_instability.html

  1. ν e n \nu_{en}
  2. ν i n \nu_{in}
  3. f f
  4. t t
  5. x x
  6. k k
  7. f exp ( i ω t + i k x ) f\sim\exp(i\omega t+ikx)
  8. ω \omega
  9. ω \omega
  10. ω ( 1 + i ψ 0 ω - i ν i n ν i n ) = k v E + i ψ 0 k 2 c i 2 ν i n \omega\left(1+i\psi_{0}\frac{\omega-i\nu_{in}}{\nu_{in}}\right)=kv_{E}+i\psi_{% 0}\frac{k^{2}c_{i}^{2}}{\nu_{in}}
  11. v E v_{E}
  12. E × B E\times B
  13. c i c_{i}
  14. ψ 0 \psi_{0}
  15. Ω i \Omega_{i}
  16. Ω e \Omega_{e}
  17. ψ 0 = ν i n ν e n Ω i Ω e \psi_{0}=\frac{\nu_{in}\nu_{en}}{\Omega_{i}\Omega_{e}}
  18. ω = ω r + i γ \omega=\omega_{r}+i\gamma
  19. γ \gamma
  20. ω r = k v E 1 + ψ 0 \omega_{r}=\frac{kv_{E}}{1+\psi_{0}}
  21. γ = ψ 0 ν i n ω r 2 - k 2 c i 2 1 + ψ 0 \gamma=\frac{\psi_{0}}{\nu_{in}}\frac{\omega_{r}^{2}-k^{2}c_{i}^{2}}{1+\psi_{0}}

Fat_object.html

  1. side of smallest cube enclosing o side of largest cube enclosed in o \frac{\,\text{side of smallest cube enclosing}\ o}{\,\text{side of largest % cube enclosed in}\ o}
  2. ( volume of smallest ball enclosing o volume of o ) 1 / d \left(\frac{\,\text{volume of smallest ball enclosing}\ o}{\,\text{volume of}% \ o}\right)^{1/d}
  3. ( volume of o volume of largest ball enclosed in o ) 1 / d \left(\frac{\,\text{volume of}\ o}{\,\text{volume of largest ball enclosed in}% \ o}\right)^{1/d}
  4. V d r d V_{d}\cdot r^{d}
  5. V d = π d / 2 Γ ( d 2 + 1 ) V_{d}=\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}
  6. V d 1 / d d / 2 {V_{d}}^{1/d}\cdot{\sqrt{d}}/2
  7. 0.5 π k / ( k ! ) 1 / 2 k \sqrt{0.5\pi k}/{{(k!)}^{1/2k}}
  8. V d 1 / d d / 2 {V_{d}}^{1/d}\cdot{\sqrt{d}}/2
  9. 2 / V d 1 / d 2/{V_{d}}^{1/d}
  10. 2 / ( k ! ) 1 / 2 k / π 2/{(k!)}^{1/2k}/{\sqrt{\pi}}
  11. 2 / V d 1 / d 2/{V_{d}}^{1/d}
  12. 1 2 sup b B ( volume of B volume of B o ) 1 / d \frac{1}{2}\cdot\sup_{b\in B}\left(\frac{\,\text{volume of}\ B}{\,\text{volume% of}\ B\cap o}\right)^{1/d}
  13. volume ( b o ) volume ( b ) volume ( B o ) volume ( B ) \frac{\,\text{volume}\ (b\cap o)}{\,\text{volume}\ (b)}\geq\frac{\,\text{% volume}\ (B\cap o)}{\,\text{volume}\ (B)}
  14. f ( θ ) = min ( r ( θ ) radius ( b ) , 1 ) f(\theta)=\min{(\frac{r(\theta)}{\,\text{radius}\ (b)},1)}
  15. F ( θ ) = min ( r ( θ ) radius ( B ) , 1 ) F(\theta)=\min{(\frac{r(\theta)}{\,\text{radius}\ (B)},1)}
  16. f ( θ ) F ( θ ) f(\theta)\geq F(\theta)
  17. volume ( b ) C d diameter ( o ) d \,\text{volume}\ (b)\leq C_{d}\cdot\,\text{diameter}\ (o)^{d}
  18. C d C_{d}
  19. C d ( diameter(smallest ball enclosing o ) / 2 ) d C_{d}\cdot(\,\text{diameter(smallest ball enclosing}\ o)/2)^{d}
  20. Δ = r 2 ( cot A 2 + cot B 2 + cot C 2 ) \Delta=r^{2}\cdot(\cot\frac{\angle A}{2}+\cot\frac{\angle B}{2}+\cot\frac{% \angle C}{2})
  21. cot A 2 + cot B 2 + cot C 2 π \sqrt{\frac{\cot\frac{\angle A}{2}+\cot\frac{\angle B}{2}+\cot\frac{\angle C}{% 2}}{\pi}}
  22. Δ = R 2 2 sin A sin B sin C \Delta=R^{2}\cdot 2\sin A\sin B\sin C
  23. π 2 sin A sin B sin C \sqrt{\frac{\pi}{2\sin A\sin B\sin C}}
  24. Δ = c 2 2 ( cot A + cot B ) = c 2 ( sin A ) ( sin B ) 2 sin ( A + B ) \Delta=\frac{c^{2}}{2(\cot\angle{A}+\cot\angle{B})}=\frac{c^{2}(\sin\angle{A})% (\sin\angle{B})}{2\sin(\angle{A}+\angle{B})}
  25. π ( cot A + cot B ) 2 = π sin ( A + B ) 2 ( sin A ) ( sin B ) \sqrt{\frac{\pi\cdot(\cot\angle{A}+\cot\angle{B})}{2}}=\sqrt{\frac{\pi\cdot% \sin(\angle{A}+\angle{B})}{2(\sin\angle{A})(\sin\angle{B})}}
  26. sin C = sin A + B = 1 \sin{\angle{C}}=\sin{\angle{A}+\angle{B}}=1
  27. R r = 1 4 sin ( A 2 ) sin ( B 2 ) sin ( C 2 ) = 1 cos A + cos B + cos C - 1 \frac{R}{r}=\frac{1}{4\sin(\frac{\angle{A}}{2})\sin(\frac{\angle{B}}{2})\sin(% \frac{\angle{C}}{2})}=\frac{1}{\cos\angle{A}+\cos\angle{B}+\cos\angle{C}-1}
  28. c 2 = R sin C \frac{c}{2}=R\sin{\angle{C}}
  29. c / 2 r = sin C 4 sin ( A 2 ) sin ( B 2 ) sin ( C 2 ) = sin C cos A + cos B + cos C - 1 \frac{c/2}{r}=\frac{\sin{\angle{C}}}{4\sin(\frac{\angle{A}}{2})\sin(\frac{% \angle{B}}{2})\sin(\frac{\angle{C}}{2})}=\frac{\sin{\angle{C}}}{\cos\angle{A}+% \cos\angle{B}+\cos\angle{C}-1}
  30. sin C = 1 \sin{\angle{C}}=1
  31. sin max ( A , B , C , π / 2 ) 4 sin ( A 2 ) sin ( B 2 ) sin ( π - A - B 2 ) = sin max ( A , B , C , π / 2 ) cos A + cos B - cos ( A + B ) - 1 \frac{\sin{\max(\angle{A},\angle{B},\angle{C},\pi/2)}}{4\sin(\frac{\angle{A}}{% 2})\sin(\frac{\angle{B}}{2})\sin(\frac{\pi-\angle{A}-\angle{B}}{2})}=\frac{% \sin{\max(\angle{A},\angle{B},\angle{C},\pi/2)}}{\cos\angle{A}+\cos\angle{B}-% \cos(\angle{A}+\angle{B})-1}
  32. sin max ( θ , π / 2 ) 4 sin 2 ( π - θ 4 ) sin ( θ 2 ) 1 4 1 / 2 2 θ / 2 = 1 θ \frac{\sin{\max(\theta,\pi/2)}}{4\sin^{2}(\frac{\pi-\theta}{4})\sin(\frac{% \theta}{2})}\approx\frac{1}{4{\sqrt{1/2}}^{2}\theta/2}=\frac{1}{\theta}
  33. length of chord height of segment = 2 R sin θ 2 R ( 1 - cos θ 2 ) = 2 sin θ 2 ( 1 - cos θ 2 ) θ θ 2 / 8 = 8 θ \frac{\,\text{length of chord}}{\,\text{height of segment}}=\frac{2R\sin\frac{% \theta}{2}}{R\left(1-\cos\frac{\theta}{2}\right)}=\frac{2\sin\frac{\theta}{2}}% {\left(1-\cos\frac{\theta}{2}\right)}\approx\frac{\theta}{\theta^{2}/8}=\frac{% 8}{\theta}
  34. radius of circle length of chord = R 2 R sin θ 2 = 1 2 sin θ 2 1 2 θ / 2 = 1 θ \frac{\,\text{radius of circle}}{\,\text{length of chord}}=\frac{R}{2R\sin% \frac{\theta}{2}}=\frac{1}{2\sin\frac{\theta}{2}}\approx\frac{1}{2\theta/2}=% \frac{1}{\theta}
  35. 2 a sin θ 2 2a\sin\frac{\theta}{2}
  36. 2 b sin θ 2 2b\sin\frac{\theta}{2}
  37. b ( 1 - cos θ 2 ) b\left(1-\cos\frac{\theta}{2}\right)
  38. a ( 1 - cos θ 2 ) a\left(1-\cos\frac{\theta}{2}\right)
  39. 2 a sin θ 2 b ( 1 - cos θ 2 ) 8 a b θ \frac{2a\sin\frac{\theta}{2}}{b\left(1-\cos\frac{\theta}{2}\right)}\approx% \frac{8a}{b\theta}
  40. 2 b sin θ 2 a ( 1 - cos θ 2 ) 8 b a θ \frac{2b\sin\frac{\theta}{2}}{a\left(1-\cos\frac{\theta}{2}\right)}\approx% \frac{8b}{a\theta}
  41. 2 sin θ 2 ( 1 - cos θ 2 ) \frac{2\sin\frac{\theta}{2}}{\left(1-\cos\frac{\theta}{2}\right)}
  42. ( b a cos 2 ( Θ + θ 2 ) + a b sin 2 ( Θ + θ 2 ) ) \left(\frac{b}{a}\cos^{2}(\Theta+\frac{\theta}{2})+\frac{a}{b}\sin^{2}(\Theta+% \frac{\theta}{2})\right)

Ferrers_function.html

  1. P v μ ( x ) = ( 1 + x 1 - x ) μ / 2 F ( v + 1 , - v ; 1 - μ ; 1 / 2 - x / 2 ) Γ ( 1 - μ ) P_{v}^{\mu}(x)=\left(\frac{1+x}{1-x}\right)^{\mu/2}\cdot\frac{F(v+1,-v;1-\mu;1% /2-x/2)}{\Gamma(1-\mu)}
  2. Q v μ ( x ) = cos ( μ π ) ( 1 + x 1 - x ) μ / 2 F ( v + 1 , - v ; 1 - μ ; 1 / 2 - 2 / x ) Γ ( 1 - μ ) Q_{v}^{\mu}(x)=\cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;% 1-\mu;1/2-2/x)}{\Gamma(1-\mu)}

Fifth-order_Korteweg–de_Vries_equation.html

  1. u t + α u x x x + β u x x x x x = x f ( u , u x , u x x ) u_{t}+\alpha u_{xxx}+\beta u_{xxxxx}=\frac{\partial}{\partial x}f(u,u_{x},u_{% xx})
  2. f f
  3. α \alpha
  4. β \beta
  5. β 0 \beta\neq 0

File:CMB_Wiener_filter_example,_wiener_filtered.png.html

  1. s ^ = ( S - 1 + N - 1 ) - 1 N - 1 d \hat{s}=(S^{-1}+N^{-1})^{-1}N^{-1}d

File:CMB_Wiener_filter_example.png.html

  1. d d
  2. S S
  3. N N
  4. s ^ = ( S - 1 + N - 1 ) - 1 N - 1 d \hat{s}=(S^{-1}+N^{-1})^{-1}N^{-1}d

File:Half-Width_plot_for_Life_time.png.html

  1. \B 1 \B_{1}

File:Lorentz_curve.png.html

  1. B 0 B_{0}

Final_functor.html

  1. F : C D F:C\to D
  2. G : D S e t G:D\to Set
  3. G F G\circ F
  4. { * } 𝑑 D \{*\}\xrightarrow{d}D

Finger_search.html

  1. O ( i = 1 m log d ( f , a i ) ) O(\sum_{i=1}^{m}\log d(f,a_{i}))

Finite_Legendre_transform.html

  1. L x ( k ) = 2 k + 1 N t = - 1 t = 1 x ( t ) P k ( t ) , L_{x}(k)=\frac{2k+1}{N}\sum_{t=-1}^{t=1}x(t)P_{k}(t),
  2. x ( t ) = k L x ( k ) P k ( t ) . x(t)=\sum_{k}L_{x}(k)P_{k}(t).

Finite_volume_method_for_one-dimensional_steady_state_diffusion.html

  1. ρ ϕ t + div ( ρ ϕ υ ) = div ( Γ grad ϕ ) + S ϕ \frac{\partial\rho\phi}{\partial t}+\operatorname{div}(\rho\phi\upsilon)=% \operatorname{div}(\Gamma\operatorname{grad}\phi)+S_{\phi}
  2. ρ \rho
  3. ϕ \phi
  4. Γ \Gamma
  5. S S
  6. div ( ρ ϕ υ ) \operatorname{div}(\rho\phi\upsilon)
  7. ϕ \phi
  8. div ( Γ grad ϕ ) \operatorname{div}(\Gamma\operatorname{grad}\phi)
  9. ϕ \phi
  10. S ϕ S_{\phi}
  11. ϕ \phi
  12. ρ ϕ t \frac{\partial\rho\phi}{\partial t}
  13. ϕ \phi
  14. div ( Γ grad ϕ ) + S ϕ = 0 \operatorname{div}(\Gamma\operatorname{grad}\phi)+S_{\phi}=0
  15. d d x ( Γ grad ϕ ) + S ϕ = 0 \frac{d}{dx}(\Gamma\operatorname{grad}\phi)+S_{\phi}=0
  16. δ x W P \delta x_{WP}
  17. δ x w P \delta x_{wP}
  18. δ x P e \delta x_{Pe}
  19. δ x P E \delta x_{PE}
  20. Δ V d d x ( Γ d ϕ d x ) d V + Δ V S d V = ( Γ A d ϕ d x ) e - ( Γ A d ϕ d x ) w + S Δ V = 0 \int_{\Delta V}\frac{d}{dx}\left(\Gamma\frac{d\phi}{dx}\right)dV+\int_{\Delta V% }SdV=\left(\Gamma A\frac{d\phi}{dx}\right)_{e}-\left(\Gamma A\frac{d\phi}{dx}% \right)_{w}+\overrightarrow{S}\Delta V=0
  21. A A
  22. Δ V \Delta V
  23. S \overrightarrow{S}
  24. ϕ \phi
  25. ϕ \phi
  26. d ϕ d x \frac{d\phi}{dx}
  27. ϕ \phi
  28. Γ w = Γ W + Γ P 2 \Gamma_{w}=\frac{\Gamma_{W}+\Gamma_{P}}{2}
  29. Γ w = Γ P + Γ E 2 \Gamma_{w}=\frac{\Gamma_{P}+\Gamma_{E}}{2}
  30. d ϕ d x \frac{d\phi}{dx}
  31. ( d ϕ d x ) e = ϕ E - ϕ P δ x P E \left(\frac{d\phi}{dx}\right)_{e}=\frac{\phi_{E}-\phi_{P}}{\delta x_{PE}}
  32. ( d ϕ d x ) w = ϕ P - ϕ W δ x W P \left(\frac{d\phi}{dx}\right)_{w}=\frac{\phi_{P}-\phi_{W}}{\delta x_{WP}}
  33. S Δ V = S u + S p ϕ p \overrightarrow{S}\Delta V=S_{u}+S_{p}\phi_{p}
  34. Γ e A e ( ϕ E - ϕ P δ x P E ) - Γ w A w ( ϕ P - ϕ W δ x P E ) + ( S u + S p ϕ p ) \Gamma_{e}A_{e}\left(\frac{\phi_{E}-\phi_{P}}{\delta x_{PE}}\right)-\Gamma_{w}% A_{w}\left(\frac{\phi_{P}-\phi_{W}}{\delta x_{PE}}\right)+(S_{u}+S_{p}\phi_{p})
  35. ( Γ e δ x P E A e + Γ w δ x W P A w - S p ) ϕ P = ( Γ w δ x W P A w ) ϕ W + ( Γ e δ x W P A e ) ϕ E + S u \left(\frac{\Gamma_{e}}{\delta x_{PE}}A_{e}+\frac{\Gamma_{w}}{\delta x_{WP}}A_% {w}-S_{p}\right)\phi_{P}=\left(\frac{\Gamma_{w}}{\delta x_{WP}}A_{w}\right)% \phi_{W}+\left(\frac{\Gamma_{e}}{\delta x_{WP}}A_{e}\right)\phi_{E}+S_{u}
  36. a P ϕ P = a W ϕ W + a E ϕ E + S u a_{P}\phi_{P}=a_{W}\phi_{W}+a_{E}\phi_{E}+S_{u}
  37. a W a_{W}
  38. a E a_{E}
  39. a P a_{P}
  40. Γ w δ x W P A w \frac{\Gamma_{w}}{\delta x_{WP}}A_{w}
  41. Γ e δ x P E A e \frac{\Gamma_{e}}{\delta x_{PE}}A_{e}
  42. a W + a E - S P a_{W}+a_{E}-S_{P}
  43. ϕ \phi

Finite_volume_method_for_three-dimensional_diffusion_problem.html

  1. ϕ \phi
  2. ρ ϕ t + div ( ρ ϕ υ ) = div ( Γ grad ϕ ) + S ϕ \frac{\partial\rho\phi}{\partial t}+\operatorname{div}(\rho\phi\upsilon)=% \operatorname{div}(\Gamma\operatorname{grad}\phi)+S_{\phi}
  3. ϕ \phi
  4. ρ \rho
  5. div ( ρ ϕ υ ) \operatorname{div}(\rho\phi\upsilon)
  6. ϕ \phi
  7. ρ ϕ t \frac{\partial\rho\phi}{\partial t}
  8. div ( Γ grad ϕ ) \operatorname{div}(\Gamma\operatorname{grad}\phi)
  9. ϕ \phi
  10. S ϕ S_{\phi}
  11. ϕ \phi
  12. div ( Γ grad ϕ ) + S ϕ = 0 \operatorname{div}(\Gamma\operatorname{grad}\phi)+S_{\phi}=0
  13. x ( Γ ϕ x ) + y ( Γ ϕ y ) + z ( Γ ϕ z ) + S ϕ = 0 \frac{\partial}{\partial x}\left(\Gamma\frac{\partial\phi}{\partial x}\right)+% \frac{\partial}{\partial y}\left(\Gamma\frac{\partial\phi}{\partial y}\right)+% \frac{\partial}{\partial z}\left(\Gamma\frac{\partial\phi}{\partial z}\right)+% S_{\phi}=0
  14. ( ρ 𝐮 ) = 0 \nabla\cdot(\rho\mathbf{u})=0
  15. δ x W P , δ x P E , δ x P N , δ x S P , δ x P T , δ x B P {\delta x_{WP}},{\delta x_{PE}},{\delta x_{PN}},{\delta x_{SP}},{\delta x_{PT}% },{\delta x_{BP}}
  16. ( Γ A d ϕ d x ) e - ( Γ A d ϕ d x ) w ] + [ ( Γ A d ϕ d x ) n - ( Γ A d ϕ d x ) s ] + [ ( Γ A d ϕ d x ) t - ( Γ A d ϕ d x ) b ] + S Δ V = 0 \left(\Gamma A\frac{d\phi}{dx}\right)_{e}-\left(\Gamma A\frac{d\phi}{dx}\right% )_{w}]+[\left(\Gamma A\frac{d\phi}{dx}\right)_{n}-\left(\Gamma A\frac{d\phi}{% dx}\right)_{s}]+[\left(\Gamma A\frac{d\phi}{dx}\right)_{t}-\left(\Gamma A\frac% {d\phi}{dx}\right)_{b}]+\overrightarrow{S}\Delta V=0
  17. Γ e A e ( ϕ E - ϕ P δ x P E ) - Γ w A w ( ϕ P - ϕ W δ x P E ) ] + [ Γ n A n ( ϕ E - ϕ P δ x P E ) - Γ s A s ( ϕ P - ϕ W δ x P E ) ] + [ Γ t A t ( ϕ E - ϕ P δ x P E ) - Γ b A b ( ϕ P - ϕ W δ x P E ) ] + ( S u + S p ϕ p ) = 0 \Gamma_{e}A_{e}\left(\frac{\phi_{E}-\phi_{P}}{\delta x_{PE}}\right)-\Gamma_{w}% A_{w}\left(\frac{\phi_{P}-\phi_{W}}{\delta x_{PE}}\right)]+[\Gamma_{n}A_{n}% \left(\frac{\phi_{E}-\phi_{P}}{\delta x_{PE}}\right)-\Gamma_{s}A_{s}\left(% \frac{\phi_{P}-\phi_{W}}{\delta x_{PE}}\right)]+[\Gamma_{t}A_{t}\left(\frac{% \phi_{E}-\phi_{P}}{\delta x_{PE}}\right)-\Gamma_{b}A_{b}\left(\frac{\phi_{P}-% \phi_{W}}{\delta x_{PE}}\right)]+(S_{u}+S_{p}\phi_{p})=0
  18. a P ϕ P = a W ϕ W + a E ϕ E + a S ϕ S + a N ϕ N + a B ϕ B + a T ϕ T + S u a_{P}\phi_{P}=a_{W}\phi_{W}+a_{E}\phi_{E}+a_{S}\phi_{S}+a_{N}\phi_{N}+a_{B}% \phi_{B}+a_{T}\phi_{T}+S_{u}
  19. a W a_{W}
  20. a E a_{E}
  21. a S a_{S}
  22. a N a_{N}
  23. a B a_{B}
  24. a T a_{T}
  25. a P a_{P}
  26. Γ w δ x W P A w \frac{\Gamma_{w}}{\delta x_{WP}}A_{w}
  27. Γ e δ x P E A e \frac{\Gamma_{e}}{\delta x_{PE}}A_{e}
  28. Γ s δ x S P A s \frac{\Gamma_{s}}{\delta x_{SP}}A_{s}
  29. Γ n δ x P N A n \frac{\Gamma_{n}}{\delta x_{PN}}A_{n}
  30. Γ b δ x B P A w \frac{\Gamma_{b}}{\delta x_{BP}}A_{w}
  31. Γ t δ x P T A t \frac{\Gamma_{t}}{\delta x_{PT}}A_{t}
  32. a W + a E + a S + a N + a B + a T - S P a_{W}+a_{E}+a_{S}+a_{N}+a_{B}+a_{T}-S_{P}
  33. ϕ \phi