wpmath0000012_3

Diffusion_damping.html

  1. 𝒟 \mathcal{D}
  2. 𝒟 ( k ) = 0 η 0 τ ˙ e - [ k / k D ( η ) ] 2 d η . \mathcal{D}(\mathit{k})=\int_{0}^{\eta_{0}}\dot{\tau}e^{-[\mathit{k}/{\mathit{% k}_{\mathit{D}}(\eta)}]^{2}}\;d\eta.
  3. η \eta
  4. τ ˙ \dot{\tau}
  5. k \mathit{k}
  6. ( τ ˙ e - [ k / k D ( η ) ] 2 ) (\dot{\tau}e^{-[\mathit{k}/{\mathit{k}_{\mathit{D}}(\eta)}]^{2}})
  7. k D ( η ) = 2 π / λ D {\mathit{k}_{\mathit{D}}}(\eta)={2\pi}/\lambda_{\mathit{D}}
  8. 𝒟 \mathcal{D}
  9. [ Θ 0 + Ψ ] ( η ) = [ Θ ^ 0 + Ψ ] ( η ) 𝒟 ( k ) . [\Theta_{0}+\Psi](\eta_{\ast})=[\hat{\Theta}_{0}+\Psi](\eta_{\ast})\mathcal{D}% (\mathit{k}).
  10. η \mathit{\eta}_{\ast}
  11. Θ 0 \Theta_{0}
  12. Ψ \Psi
  13. [ Θ 0 + Ψ ] ( η ) [\Theta_{0}+\Psi](\eta)
  14. N \mathit{N}
  15. λ D = N λ C \lambda_{D}=\sqrt{\mathit{N}}\lambda_{C}
  16. k D \mathit{k}_{\mathit{D}}
  17. λ D \lambda_{\mathit{D}}
  18. λ C \lambda_{\mathit{C}}
  19. λ D \lambda_{\mathit{D}}
  20. N λ C \sqrt{\mathit{N}}\lambda_{\mathit{C}}
  21. N \mathit{N}
  22. η \mathit{\eta}_{\ast}
  23. x e \mathit{x}_{\mathit{e}}
  24. λ C ( x e n b ) - 1 \lambda_{\mathit{C}}\varpropto{(\mathit{x}_{\mathit{e}}\mathit{n}_{\mathit{b}}% )}^{-1}
  25. n b \mathit{n}_{\mathit{b}}
  26. l 800 \mathit{l}\gtrsim 800
  27. 10 13 10^{13}
  28. M s m p t 𝑟𝑒𝑐 3 / 2 n 𝑟𝑒𝑐 σ 3 . \mathit{M}_{\mathit{s}}\approx\frac{\mathit{m}_{\mathit{p}}{\mathit{t}_{% \mathit{rec}}}^{3/2}}{\sqrt{\mathit{n}_{\mathit{rec}}\sigma^{3}}}.
  29. 10 l 100 10\lesssim\mathit{l}\lesssim 100
  30. l 10 \mathit{l}\lessapprox 10

Dihydrogen_cation.html

  1. H 2 + H_{2}^{+}
  2. ( - 2 2 m 2 + V ) ψ = E ψ , \left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V\right)\psi=E\psi~{},
  3. V V
  4. V = - e 2 4 π ε 0 ( 1 r a + 1 r b ) V=-\frac{e^{2}}{4\pi\varepsilon_{0}}\left(\frac{1}{r_{a}}+\frac{1}{r_{b}}\right)
  5. ψ = ψ ( 𝐫 ) \psi=\psi(\mathbf{r})
  6. 1 / R 1/R
  7. R R
  8. V V
  9. r a r_{a}
  10. r b r_{b}
  11. ( = m = e = 4 π ε 0 = 1 ) (\hbar=m=e=4\pi\varepsilon_{0}=1)
  12. ( - 1 2 2 + V ) ψ = E ψ with V = - 1 r a - 1 r b . \left({}-\frac{1}{2}\nabla^{2}+V\right)\psi=E\psi\qquad\mbox{with}~{}\qquad V=% {}-\frac{1}{r_{a}}-\frac{1}{r_{b}}\;.
  13. \to
  14. ψ + ( 𝐫 ) \psi_{+}(\mathbf{r})
  15. ψ - ( 𝐫 ) \psi_{-}(\mathbf{r})
  16. ψ ± ( - 𝐫 ) = ± ψ ± ( 𝐫 ) . \psi_{\pm}(-{\mathbf{r}})={}\pm\psi_{\pm}({\mathbf{r}})\;.
  17. H 2 + H_{2}^{+}
  18. X 2 Σ g + {\rm X}^{2}\Sigma_{\rm g}^{+}
  19. 1 s σ g 1s\sigma_{\rm g}
  20. A 2 Σ u + {\rm A}^{2}\Sigma_{\rm u}^{+}
  21. 2 p σ u {\rm 2p}\sigma_{\rm u}
  22. E ± E_{\pm}
  23. E ± = - 1 2 - 9 4 R 4 + O ( R - 6 ) + E_{\pm}={}-\frac{1}{2}-\frac{9}{4R^{4}}+O(R^{-6})+\cdots
  24. Δ E = E - - E + = 4 e R e - R [ 1 + 1 2 R + O ( R - 2 ) ] \Delta E=E_{-}-E_{+}=\frac{4}{e}\,R\,e^{-R}\left[\,1+\frac{1}{2R}+O(R^{-2})\,\right]
  25. 4 e R e - R {\textstyle\frac{4}{e}}Re^{-R}
  26. ( 3 ) (3)
  27. Σ g + 2 {}^{2}\Sigma_{\rm g}^{+}
  28. Σ u + 2 {}^{2}\Sigma_{\rm u}^{+}
  29. Π u 2 {}^{2}\Pi_{\rm u}
  30. Π g 2 {}^{2}\Pi_{\rm g}

Dimension_theory_(algebra).html

  1. dim \operatorname{dim}
  2. ht \operatorname{ht}
  3. dim R [ x ] = dim R + 1. \operatorname{dim}R[x]=\operatorname{dim}R+1.
  4. 𝔭 \mathfrak{p}
  5. ht ( 𝔭 R [ x ] ) = ht ( 𝔭 ) \operatorname{ht}(\mathfrak{p}R[x])=\operatorname{ht}(\mathfrak{p})
  6. ht ( 𝔮 ) = ht ( 𝔭 ) + 1 \operatorname{ht}(\mathfrak{q})=\operatorname{ht}(\mathfrak{p})+1
  7. 𝔮 𝔭 R [ x ] \mathfrak{q}\supsetneq\mathfrak{p}R[x]
  8. R [ x ] R[x]
  9. 𝔭 \mathfrak{p}
  10. Spec R [ x ] Spec R \operatorname{Spec}R[x]\to\operatorname{Spec}R
  11. 2 \geq 2
  12. dim R [ x 1 , , x n ] = n . \operatorname{dim}R[x_{1},\dots,x_{n}]=n.
  13. ( R , 𝔪 ) (R,\mathfrak{m})
  14. 𝔪 \mathfrak{m}
  15. 𝔪 \mathfrak{m}
  16. 𝔪 \mathfrak{m}
  17. F ( t ) F(t)
  18. gr I R = 0 I n / I n + 1 \operatorname{gr}_{I}R=\oplus_{0}^{\infty}I^{n}/I^{n+1}
  19. F ( t ) = 0 ( I n / I n + 1 ) t n F(t)=\sum_{0}^{\infty}\ell(I^{n}/I^{n+1})t^{n}
  20. \ell
  21. ( gr I R ) 0 = R / I (\operatorname{gr}_{I}R)_{0}=R/I
  22. x 1 , , x s x_{1},\dots,x_{s}
  23. I / I 2 I/I^{2}
  24. gr I R \operatorname{gr}_{I}R
  25. R / I R/I
  26. t = 1 t=1
  27. d s d\leq s
  28. ( 1 - t ) - d = 0 ( d - 1 + j d - 1 ) t j (1-t)^{-d}=\sum_{0}^{\infty}{\left({{d-1+j}\atop{d-1}}\right)}t^{j}
  29. t n t^{n}
  30. F ( t ) = ( 1 - t ) d F ( t ) ( 1 - t ) - d F(t)=(1-t)^{d}F(t)(1-t)^{-d}
  31. 0 N a k ( d - 1 + n - k d - 1 ) = ( 1 - t ) d F ( t ) | t = 1 n d - 1 d - 1 ! + O ( n d - 2 ) . \sum_{0}^{N}a_{k}{\left({{d-1+n-k}\atop{d-1}}\right)}=(1-t)^{d}F(t)|_{t=1}{n^{% d-1}\over{d-1}!}+O(n^{d-2}).
  32. ( I n / I n + 1 ) \ell(I^{n}/I^{n+1})
  33. P P
  34. d - 1 d-1
  35. gr I R \operatorname{gr}_{I}R
  36. d ( R ) = d d(R)=d
  37. δ ( R ) \delta(R)
  38. 𝔪 \mathfrak{m}
  39. δ ( R ) = d ( R ) = dim R \delta(R)=d(R)=\dim R
  40. δ ( R ) \delta(R)
  41. δ ( R ) d ( R ) \delta(R)\geq d(R)
  42. d ( R ) dim R d(R)\geq\operatorname{dim}R
  43. d ( R ) d(R)
  44. 𝔭 0 𝔭 m \mathfrak{p}_{0}\subsetneq\cdots\subsetneq\mathfrak{p}_{m}
  45. D = R / 𝔭 0 D=R/\mathfrak{p}_{0}
  46. 0 D 𝑥 D D / x D 0 0\to D\overset{x}{\to}D\to D/xD\to 0
  47. d ( D ) > d ( D / x D ) d ( R / 𝔭 1 ) d(D)>d(D/xD)\geq d(R/\mathfrak{p}_{1})
  48. R / 𝔭 1 R/\mathfrak{p}_{1}
  49. 𝔭 i \mathfrak{p}_{i}
  50. m - 1 m-1
  51. m - 1 dim ( R / 𝔭 1 ) d ( R / 𝔭 1 ) d ( D ) - 1 d ( R ) - 1 m-1\leq\operatorname{dim}(R/\mathfrak{p}_{1})\leq d(R/\mathfrak{p}_{1})\leq d(% D)-1\leq d(R)-1
  52. dim R δ ( R ) . \operatorname{dim}R\geq\delta(R).
  53. x 1 , , x s x_{1},\dots,x_{s}
  54. ( x 1 , , x i ) (x_{1},\dots,x_{i})
  55. i \geq i
  56. ( x 1 , , x s ) (x_{1},\dots,x_{s})
  57. 𝔪 \mathfrak{m}
  58. ( R , 𝔪 ) (R,\mathfrak{m})
  59. k = R / 𝔪 k=R/\mathfrak{m}
  60. dim R dim k 𝔪 / 𝔪 2 \operatorname{dim}R\leq\operatorname{dim}_{k}\mathfrak{m}/\mathfrak{m}^{2}
  61. 𝔪 / 𝔪 2 \mathfrak{m}/\mathfrak{m}^{2}
  62. 𝔪 \mathfrak{m}
  63. dim R ^ = dim R \operatorname{dim}\widehat{R}=\operatorname{dim}R
  64. gr R = gr R ^ \operatorname{gr}R=\operatorname{gr}\widehat{R}
  65. x 1 , , x s x_{1},\dots,x_{s}
  66. 𝔭 \mathfrak{p}
  67. s dim R 𝔭 = ht 𝔭 s\geq\operatorname{dim}R_{\mathfrak{p}}=\operatorname{ht}\mathfrak{p}
  68. x 1 , , x n x_{1},\dots,x_{n}
  69. 𝔪 A \mathfrak{m}_{A}
  70. y 1 , , y m y_{1},\dots,y_{m}
  71. 𝔪 B / 𝔪 A B \mathfrak{m}_{B}/\mathfrak{m}_{A}B
  72. 𝔪 B s ( y 1 , , y m ) + 𝔪 A B {\mathfrak{m}_{B}}^{s}\subset(y_{1},\dots,y_{m})+\mathfrak{m}_{A}B
  73. 𝔪 B \mathfrak{m}_{B}
  74. ( y 1 , , y m , x 1 , , x n ) (y_{1},\dots,y_{m},x_{1},\dots,x_{n})
  75. 𝔪 B \mathfrak{m}_{B}
  76. m + n dim B m+n\geq\dim B
  77. \square
  78. 𝔭 0 𝔭 1 𝔭 n \mathfrak{p}_{0}\subsetneq\mathfrak{p}_{1}\subsetneq\cdots\subsetneq\mathfrak{% p}_{n}
  79. 𝔭 i R [ x ] \mathfrak{p}_{i}R[x]
  80. R [ x ] R[x]
  81. 𝔭 n R [ x ] \mathfrak{p}_{n}R[x]
  82. dim R + 1 dim R [ x ] \dim R+1\leq\dim R[x]
  83. 𝔪 \mathfrak{m}
  84. R [ x ] R[x]
  85. 𝔭 = R 𝔪 \mathfrak{p}=R\cap\mathfrak{m}
  86. R [ x ] 𝔪 = R 𝔭 [ x ] 𝔪 R[x]_{\mathfrak{m}}=R_{\mathfrak{p}}[x]_{\mathfrak{m}}
  87. R [ x ] 𝔪 / 𝔭 R 𝔭 R [ x ] 𝔪 = ( R 𝔭 / 𝔭 R 𝔭 ) [ x ] 𝔪 R[x]_{\mathfrak{m}}/\mathfrak{p}R_{\mathfrak{p}}R[x]_{\mathfrak{m}}=(R_{% \mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}})[x]_{\mathfrak{m}}
  88. 1 + dim R 1 + dim R 𝔭 dim R [ x ] 𝔪 1+\operatorname{dim}R\geq 1+\operatorname{dim}R_{\mathfrak{p}}\geq% \operatorname{dim}R[x]_{\mathfrak{m}}
  89. 𝔪 \mathfrak{m}
  90. 1 + dim R dim R [ x ] 1+\operatorname{dim}R\geq\operatorname{dim}R[x]
  91. \square
  92. R R^{\prime}
  93. R = R [ x ] R^{\prime}=R[x]
  94. dim R 𝔭 = dim R 𝔭 + dim κ ( 𝔭 ) R R 𝔭 \dim R^{\prime}_{\mathfrak{p^{\prime}}}=\dim R_{\mathfrak{p}}+\dim\kappa(% \mathfrak{p})\otimes_{R}{R^{\prime}}_{\mathfrak{p}^{\prime}}
  95. dim κ ( 𝔭 ) R R - dim κ ( 𝔭 ) R R / 𝔭 = 1 - tr . deg κ ( 𝔭 ) κ ( 𝔭 ) = tr . deg R R - tr . deg κ ( 𝔭 ) . \dim\kappa(\mathfrak{p})\otimes_{R}R^{\prime}-\dim\kappa(\mathfrak{p})\otimes_% {R}R^{\prime}/\mathfrak{p}^{\prime}=1-\operatorname{tr.deg}_{\kappa(\mathfrak{% p})}\kappa(\mathfrak{p}^{\prime})=\operatorname{tr.deg}_{R}R^{\prime}-% \operatorname{tr.deg}\kappa(\mathfrak{p}^{\prime}).
  96. R R^{\prime}
  97. R = R [ x ] / I R^{\prime}=R[x]/I
  98. R R^{\prime}
  99. tr . deg R R = 0 \operatorname{tr.deg}_{R}R^{\prime}=0
  100. I R = 0 I\cap R=0
  101. ht I = dim R [ x ] I = dim Q ( R ) [ x ] I = 1 - tr . deg Q ( R ) κ ( I ) = 1 \operatorname{ht}I=\dim R[x]_{I}=\dim Q(R)[x]_{I}=1-\operatorname{tr.deg}_{Q(R% )}\kappa(I)=1
  102. κ ( I ) = Q ( R ) \kappa(I)=Q(R^{\prime})
  103. Q ( R ) Q(R)
  104. 𝔭 c \mathfrak{p}^{\prime c}
  105. R [ x ] R[x]
  106. 𝔭 \mathfrak{p}^{\prime}
  107. κ ( 𝔭 c ) = κ ( 𝔭 ) \kappa(\mathfrak{p}^{\prime c})=\kappa(\mathfrak{p})
  108. ht 𝔭 = ht 𝔭 c / I ht 𝔭 c - ht I = dim R 𝔭 - tr . deg κ ( 𝔭 ) κ ( 𝔭 ) . \operatorname{ht}{\mathfrak{p}^{\prime}}=\operatorname{ht}{\mathfrak{p}^{% \prime c}/I}\leq\operatorname{ht}{\mathfrak{p}^{\prime c}}-\operatorname{ht}{I% }=\dim R_{\mathfrak{p}}-\operatorname{tr.deg}_{\kappa(\mathfrak{p})}\kappa(% \mathfrak{p}^{\prime}).
  109. \square
  110. pd R M \operatorname{pd}_{R}M
  111. gl . dim R = sup { pd R M | M is a finite module } \operatorname{gl.dim}R=\sup\{\operatorname{pd}_{R}M|\,\text{M is a finite % module}\}
  112. pd R M n Tor n + 1 R ( M , k ) = 0 \operatorname{pd}_{R}M\leq n\Leftrightarrow\operatorname{Tor}^{R}_{n+1}(M,k)=0
  113. n = 0 n=0
  114. Tor 1 R ( M , k ) = 0 M flat M free pd R ( M ) 0. \operatorname{Tor}^{R}_{1}(M,k)=0\Rightarrow M\,\text{ flat }\Rightarrow M\,% \text{ free }\Rightarrow\operatorname{pd}_{R}(M)\leq 0.
  115. gl . dim R n pd R k n Tor n + 1 R ( - , k ) = 0 pd R - n gl . dim R n , \operatorname{gl.dim}R\leq n\Rightarrow\operatorname{pd}_{R}k\leq n\Rightarrow% \operatorname{Tor}^{R}_{n+1}(-,k)=0\Rightarrow\operatorname{pd}_{R}-\leq n% \Rightarrow\operatorname{gl.dim}R\leq n,
  116. \square
  117. pd R K = pd R M - 1 \operatorname{pd}_{R}K=\operatorname{pd}_{R}M-1
  118. K K
  119. pd R M = 0 \operatorname{pd}_{R}M=0
  120. M R 1 M\otimes R_{1}
  121. R 1 R_{1}
  122. pd R M > 0 \operatorname{pd}_{R}M>0
  123. pd R K = pd R M - 1 \operatorname{pd}_{R}K=\operatorname{pd}_{R}M-1
  124. pd R M = 1 \operatorname{pd}_{R}M=1
  125. 0 P 1 P 0 M 0 0\to P_{1}\to P_{0}\to M\to 0
  126. Tor 1 R ( M , R 1 ) P 1 R 1 P 0 R 1 M R 1 0 \operatorname{Tor}^{R}_{1}(M,R_{1})\to P_{1}\otimes R_{1}\to P_{0}\otimes R_{1% }\to M\otimes R_{1}\to 0
  127. Tor 1 R ( M , R 1 ) = M f = { m M | f m = 0 } = 0. \operatorname{Tor}^{R}_{1}(M,R_{1})={}_{f}M=\{m\in M|fm=0\}=0.
  128. pd R ( M R 1 ) \operatorname{pd}_{R}(M\otimes R_{1})
  129. \square
  130. k = R / ( f 1 , , f n ) k=R/(f_{1},\dots,f_{n})
  131. f i f_{i}
  132. 0 M 𝑓 M M 1 0 0\to M\overset{f}{\to}M\to M_{1}\to 0
  133. pd R M < \operatorname{pd}_{R}M<\infty
  134. 0 = Tor i + 1 R ( M , k ) Tor i + 1 R ( M 1 , k ) Tor i R ( M , k ) 𝑓 Tor i R ( M , k ) , i pd R M . 0=\operatorname{Tor}^{R}_{i+1}(M,k)\to\operatorname{Tor}^{R}_{i+1}(M_{1},k)\to% \operatorname{Tor}^{R}_{i}(M,k)\overset{f}{\to}\operatorname{Tor}^{R}_{i}(M,k)% ,\quad i\geq\operatorname{pd}_{R}M.
  135. Tor i + 1 R ( M 1 , k ) Tor i R ( M , k ) \operatorname{Tor}^{R}_{i+1}(M_{1},k)\simeq\operatorname{Tor}^{R}_{i}(M,k)
  136. pd R M 1 = 1 + pd R M \operatorname{pd}_{R}M_{1}=1+\operatorname{pd}_{R}M
  137. pd R k = 1 + pd R ( R / ( f 1 , , f n - 1 ) ) = = n . \operatorname{pd}_{R}k=1+\operatorname{pd}_{R}(R/(f_{1},\dots,f_{n-1}))=\cdots% =n.
  138. dim R \operatorname{dim}R
  139. R 1 = R / f 1 R R_{1}=R/f_{1}R
  140. f 1 f_{1}
  141. R 1 R_{1}
  142. dim R 1 < dim R \dim R_{1}<\dim R
  143. M = 𝔪 M=\mathfrak{m}
  144. gl . dim R < gl . dim R 1 = pd R 1 k pd R 1 𝔪 / f 1 𝔪 < R 1 regular . \operatorname{gl.dim}R<\infty\Rightarrow\operatorname{gl.dim}R_{1}=% \operatorname{pd}_{R_{1}}k\leq\operatorname{pd}_{R_{1}}\mathfrak{m}/f_{1}% \mathfrak{m}<\infty\Rightarrow R_{1}\,\text{ regular}.
  145. dim R = 0 \operatorname{dim}R=0
  146. gl . dim R = 0 \operatorname{gl.dim}R=0
  147. M M
  148. 0 < pd R M < 0<\operatorname{pd}_{R}M<\infty
  149. pd R M = 1 \operatorname{pd}_{R}M=1
  150. F M F\to M
  151. 𝔪 F \mathfrak{m}F
  152. dim R = 0 \operatorname{dim}R=0
  153. 𝔪 \mathfrak{m}
  154. 𝔪 = ann ( s ) \mathfrak{m}=\operatorname{ann}(s)
  155. K 𝔪 F K\subset\mathfrak{m}F
  156. s K = 0 sK=0
  157. s = 0 s=0
  158. \square
  159. gr R k [ x 1 , , x d ] \operatorname{gr}R\simeq k[x_{1},\dots,x_{d}]
  160. \square
  161. x 1 , , x n x_{1},\dots,x_{n}
  162. R R
  163. x 1 x_{1}
  164. M M
  165. x i x_{i}
  166. M / ( x 1 , , x i - 1 ) M M/(x_{1},\dots,x_{i-1})M
  167. i = 2 , , n i=2,\dots,n
  168. 𝔪 \mathfrak{m}
  169. k = R / 𝔪 k=R/\mathfrak{m}
  170. 𝔪 \mathfrak{m}
  171. depth M = 0 𝔪 \operatorname{depth}M=0\Leftrightarrow\mathfrak{m}
  172. 𝔪 \Leftrightarrow\mathfrak{m}
  173. depth M dim R / 𝔭 \operatorname{depth}M\leq\dim R/{\mathfrak{p}}
  174. 𝔭 \mathfrak{p}
  175. depth M dim M \operatorname{depth}M\leq\operatorname{dim}M
  176. k [ x 1 , , x d ] k[x_{1},\dots,x_{d}]
  177. x 1 , , x n x_{1},\dots,x_{n}
  178. 𝔪 \mathfrak{m}
  179. Ext R n ( N , M ) Hom R ( N , M / ( x 1 , , x n ) M ) . \operatorname{Ext}_{R}^{n}(N,M)\simeq\operatorname{Hom}_{R}(N,M/(x_{1},\dots,x% _{n})M).
  180. Ext R n - 1 ( N , M ) Hom R ( N , M / ( x 1 , , x n - 1 ) M ) \operatorname{Ext}_{R}^{n-1}(N,M)\simeq\operatorname{Hom}_{R}(N,M/(x_{1},\dots% ,x_{n-1})M)
  181. x n x_{n}
  182. 0 M x 1 M M 1 0 0\to M\overset{x_{1}}{\to}M\to M_{1}\to 0
  183. x 1 x_{1}
  184. Ext R n ( N , M ) Ext R n - 1 ( N , M / x 1 M ) Hom R ( N , M / ( x 1 , , x n ) M ) \operatorname{Ext}^{n}_{R}(N,M)\simeq\operatorname{Ext}^{n-1}_{R}(N,M/x_{1}M)% \simeq\operatorname{Hom}_{R}(N,M/(x_{1},\dots,x_{n})M)
  185. n < depth M n<\operatorname{depth}M
  186. Ext R n ( N , M ) = 0 \operatorname{Ext}_{R}^{n}(N,M)=0
  187. Ext R n ( N , M ) 0 \operatorname{Ext}_{R}^{n}(N,M)\neq 0
  188. n = depth M n=\operatorname{depth}M
  189. 𝔪 \mathfrak{m}
  190. 𝔪 Supp ( N ) . \mathfrak{m}\in\operatorname{Supp}(N).
  191. 𝔪 \mathfrak{m}
  192. \square
  193. pd R M \operatorname{pd}_{R}M
  194. 0 K 𝑓 F M 0 0\to K\overset{f}{\to}F\to M\to 0
  195. 𝔪 F \mathfrak{m}F
  196. pd R K = pd R M - 1 , \operatorname{pd}_{R}K=\operatorname{pd}_{R}M-1,
  197. depth K = depth M + 1 \operatorname{depth}K=\operatorname{depth}M+1
  198. Ext R i ( k , F ) Ext R i ( k , M ) Ext R i + 1 ( k , K ) 0. \operatorname{Ext}_{R}^{i}(k,F)\to\operatorname{Ext}_{R}^{i}(k,M)\to% \operatorname{Ext}_{R}^{i+1}(k,K)\to 0.
  199. i < depth R i<\operatorname{depth}R
  200. i < depth K - 1 i<\operatorname{depth}K-1
  201. depth K depth R \operatorname{depth}K\leq\operatorname{depth}R
  202. Ext R i ( k , M ) = 0. \operatorname{Ext}_{R}^{i}(k,M)=0.
  203. i = depth K - 1 i=\operatorname{depth}K-1
  204. Ext R i + 1 ( k , K ) 0 \operatorname{Ext}_{R}^{i+1}(k,K)\neq 0
  205. Ext R i ( k , M ) 0. \operatorname{Ext}_{R}^{i}(k,M)\neq 0.
  206. \square
  207. Γ 𝔪 ( M ) = { s M | supp ( s ) { 𝔪 } } = { s M | 𝔪 j s = 0 for some j } . \Gamma_{\mathfrak{m}}(M)=\{s\in M|\operatorname{supp}(s)\subset\{\mathfrak{m}% \}\}=\{s\in M|\mathfrak{m}^{j}s=0\,\text{ for some }j\}.
  208. Γ 𝔪 \Gamma_{\mathfrak{m}}
  209. H 𝔪 j = R j Γ 𝔪 H^{j}_{\mathfrak{m}}=R^{j}\Gamma_{\mathfrak{m}}
  210. Γ 𝔪 ( M ) = lim Hom R ( R / 𝔪 j , M ) \Gamma_{\mathfrak{m}}(M)=\underrightarrow{\lim}\operatorname{Hom}_{R}(R/% \mathfrak{m}^{j},M)
  211. H 𝔪 i ( M ) = lim Ext R i ( R / 𝔪 j , M ) H^{i}_{\mathfrak{m}}(M)=\underrightarrow{\lim}\operatorname{Ext}^{i}_{R}(R/{% \mathfrak{m}}^{j},M)
  212. H 𝔪 i ( M ) = 0 , i > dim M H_{\mathfrak{m}}^{i}(M)=0,i>\dim M
  213. 0 \neq 0
  214. i = dim M . i=\dim M.
  215. Hom R ( H 𝔪 d ( - ) , E ) \operatorname{Hom}_{R}(H^{d}_{\mathfrak{m}}(-),E)
  216. \square
  217. K ( x ) i = R K(x)_{i}=R
  218. K ( x ) i = 0 K(x)_{i}=0
  219. d : K 1 ( R ) K 0 ( R ) , r x r . d:K_{1}(R)\to K_{0}(R),\,r\mapsto xr.
  220. K ( x , M ) = K ( x ) R M K(x,M)=K(x)\otimes_{R}M
  221. d 1 d\otimes 1
  222. H * ( x , M ) = H * ( K ( x , M ) ) \operatorname{H}_{*}(x,M)=\operatorname{H}_{*}(K(x,M))
  223. H 0 ( x , M ) = M / x M \operatorname{H}_{0}(x,M)=M/xM
  224. H 1 ( x , M ) = M x = { m M | x m = 0 } \operatorname{H}_{1}(x,M)={}_{x}M=\{m\in M|xm=0\}
  225. x 1 , , x n x_{1},\dots,x_{n}
  226. K ( x 1 , , x n ) = K ( x 1 ) K ( x n ) K(x_{1},\dots,x_{n})=K(x_{1})\otimes\dots\otimes K(x_{n})
  227. H * ( x 1 , , x n , M ) = H * ( K ( x 1 , , x n , M ) ) \operatorname{H}_{*}(x_{1},\dots,x_{n},M)=\operatorname{H}_{*}(K(x_{1},\dots,x% _{n},M))
  228. H 0 ( x ¯ , M ) = M / ( x 1 , , x n ) M \operatorname{H}_{0}(\underline{x},M)=M/(x_{1},\dots,x_{n})M
  229. H n ( x ¯ , M ) = Ann M ( ( x 1 , , x n ) ) \operatorname{H}_{n}(\underline{x},M)=\operatorname{Ann}_{M}((x_{1},\dots,x_{n% }))
  230. H 𝔪 i ( M ) lim H i ( K ( x 1 j , , x n j ; M ) ) \operatorname{H}^{i}_{\mathfrak{m}}(M)\simeq\underrightarrow{\lim}% \operatorname{H}^{i}(K(x_{1}^{j},\dots,x_{n}^{j};M))
  231. Tor s R ( k , k ) 0 \operatorname{Tor}^{R}_{s}(k,k)\neq 0
  232. gl . dim R s \operatorname{gl.dim}R\geq s
  233. gl . dim R = pd R k \operatorname{gl.dim}R=\operatorname{pd}_{R}k
  234. gl . dim R = dim R \operatorname{gl.dim}R=\dim R
  235. dim R s gl . dim R = dim R \dim R\leq s\leq\operatorname{gl.dim}R=\dim R
  236. ϵ 1 ( R ) = dim k H 1 ( x ¯ ) \epsilon_{1}(R)=\dim_{k}\operatorname{H}_{1}(\underline{x})
  237. x ¯ = ( x 1 , , x d ) \underline{x}=(x_{1},\dots,x_{d})
  238. dim R + ϵ 1 ( R ) \dim R+\epsilon_{1}(R)
  239. id R M \operatorname{id}_{R}M
  240. Mod R \operatorname{Mod}_{R}
  241. gl . dim R n \operatorname{gl.dim}R\leq n
  242. 0 M I 0 ϕ 0 I 1 I n - 1 ϕ n - 1 N 0 0\to M\to I_{0}\overset{\phi_{0}}{\to}I_{1}\to\dots\to I_{n-1}\overset{\phi_{n% -1}}{\to}N\to 0
  243. I i I_{i}
  244. Ext R 1 ( R / I , N ) Ext R 2 ( R / I , ker ( ϕ n - 1 ) ) Ext R n + 1 ( R / I , M ) , \operatorname{Ext}^{1}_{R}(R/I,N)\simeq\operatorname{Ext}^{2}_{R}(R/I,% \operatorname{ker}(\phi_{n-1}))\simeq\dots\simeq\operatorname{Ext}_{R}^{n+1}(R% /I,M),
  245. Ext R n + 1 ( R / I , - ) \operatorname{Ext}_{R}^{n+1}(R/I,-)
  246. R / I R/I
  247. sup { id R M | M } n \sup\{\operatorname{id}_{R}M|M\}\leq n
  248. \square
  249. w . gl . dim = inf { n | Tor i R ( M , N ) = 0 , i > n , M , N Mod R } \operatorname{w.gl.dim}=\inf\{n|\operatorname{Tor}^{R}_{i}(M,N)=0,\,i>n,M,N\in% \operatorname{Mod}_{R}\}
  250. w . gl . dim R gl . dim R \operatorname{w.gl.dim}R\leq\operatorname{gl.dim}R
  251. f ( n ) = dim k V n f(n)=\dim_{k}V^{n}
  252. gk ( A ) = lim sup n log f ( n ) log n \operatorname{gk}(A)=\limsup_{n\to\infty}{\log f(n)\over\log n}
  253. gk ( A ) \operatorname{gk}(A)

Dimensional_operator.html

  1. d : P ( E ) P ( E ) d:P(E)\rightarrow P(E)\,

Dining_cryptographers_problem.html

  1. 1 \scriptstyle 1
  2. 0 \scriptstyle 0
  3. 1 \scriptstyle 1
  4. 1 0 = 1 \scriptstyle 1\,\oplus\,0\;=\;1
  5. 1 1 = 0 \scriptstyle 1\,\oplus\,1\;=\;0
  6. 0 1 = 1 \scriptstyle 0\,\oplus\,1\;=\;1
  7. ¬ ( 1 0 ) = 0 \scriptstyle\lnot{(1\,\oplus\,0)}\;=\;0
  8. 0 \scriptstyle 0
  9. n \scriptstyle n
  10. n > 3 \scriptstyle n>3
  11. n \scriptstyle n
  12. m \scriptstyle m
  13. n \scriptstyle n
  14. m \scriptstyle m
  15. n × m \scriptstyle n\times m
  16. m \scriptstyle m

Diphenylphosphine.html

  1. \overrightarrow{\leftarrow}

Direct_multiple_shooting_method.html

  1. y ′′ ( t ) = f ( t , y ( t ) ) , y ( t a ) = y a , y ( t b ) = y b , y^{\prime\prime}(t)=f(t,y(t)),\quad y(t_{a})=y_{a},\quad y(t_{b})=y_{b},
  2. y ( t ) , t ( t a , t b ) . y(t),\quad t\in(t_{a},t_{b}).
  3. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , y ( t 0 ) = p y^{\prime}(t)=f(t,y(t)),\quad y(t_{0})=y_{0},y^{\prime}(t_{0})=p
  4. t a = t 0 < t 1 < < t N = t b t_{a}=t_{0}<t_{1}<\cdots<t_{N}=t_{b}
  5. y ( t ) = f ( t , y ( t ) ) , y ( t k ) = y k . y^{\prime}(t)=f(t,y(t)),\quad y(t_{k})=y_{k}.
  6. y ( t 1 ; t 0 , y 0 ) = y 1 \displaystyle y(t_{1};t_{0},y_{0})=y_{1}

Directed_graph.html

  1. G = ( V , A ) G=(V,A)
  2. G = ( V , E ) G=(V,E)
  3. e = ( x , y ) e=(x,y)
  4. x x
  5. y y
  6. y y
  7. x x
  8. y y
  9. x x
  10. x x
  11. y y
  12. x x
  13. y y
  14. y y
  15. x x
  16. x x
  17. y y
  18. ( y , x ) (y,x)
  19. ( x , y ) (x,y)
  20. a i j a_{ij}
  21. a i i a_{ii}
  22. G = ( V , E ) G=(V,E)
  23. v V v∈V
  24. deg - ( v ) \deg^{-}(v)
  25. deg + ( v ) . \deg^{+}(v).
  26. deg - ( v ) = 0 \deg^{-}(v)=0
  27. deg + ( v ) = 0 \deg^{+}(v)=0
  28. v V deg + ( v ) = v V deg - ( v ) = | E | . \sum_{v\in V}\deg^{+}(v)=\sum_{v\in V}\deg^{-}(v)=|E|\,.
  29. v V v∈V
  30. deg + ( v ) = deg - ( v ) \deg^{+}(v)=\deg^{-}(v)

Director_string.html

  1. ( λ x . E ) y E [ x := y ] (\lambda x.E)y\equiv E[x:=y]\,
  2. ( λ x . E ) y E [ x / y ] (\lambda x.E)y\equiv E[x/y]
  3. t : := f ( t 1 , t 2 , , t n ) t::=f(t_{1},t_{2},\dots,t_{n})
  4. t i t_{i}
  5. σ t : V P ( { 1 , 2 , , n } ) \sigma_{t}:V\to P(\{1,2,\dots,n\})
  6. P ( X ) P(X)
  7. X = { 1 , 2 , , n } X=\{1,2,\dots,n\}
  8. σ t \sigma_{t}
  9. t i t_{i}
  10. x V x\in V
  11. t 3 t_{3}
  12. t 5 t_{5}
  13. σ t ( x ) = { 3 , 5 } \sigma_{t}(x)=\{3,5\}
  14. t T t\in T
  15. σ t \sigma_{t}
  16. ( t , σ t ) (t,\sigma_{t})

Discounted_cumulative_gain.html

  1. p p
  2. CG p = i = 1 p r e l i \mathrm{CG_{p}}=\sum_{i=1}^{p}rel_{i}
  3. r e l i rel_{i}
  4. i i
  5. d i d_{i}
  6. d j d_{j}
  7. p p
  8. DCG p = r e l 1 + i = 2 p r e l i log 2 ( i ) \mathrm{DCG_{p}}=rel_{1}+\sum_{i=2}^{p}\frac{rel_{i}}{\log_{2}(i)}
  9. DCG p = i = 1 p 2 r e l i - 1 log 2 ( i + 1 ) \mathrm{DCG_{p}}=\sum_{i=1}^{p}\frac{2^{rel_{i}}-1}{\log_{2}(i+1)}
  10. r e l i { 0 , 1 } rel_{i}\in\{0,1\}
  11. p p
  12. p p
  13. nDCG p = D C G p I D C G p \mathrm{nDCG_{p}}=\frac{DCG_{p}}{IDCG_{p}}
  14. D C G p DCG_{p}
  15. I D C G p IDCG_{p}
  16. D 1 , D 2 , D 3 , D 4 , D 5 , D 6 D_{1},D_{2},D_{3},D_{4},D_{5},D_{6}
  17. 3 , 2 , 3 , 0 , 1 , 2 3,2,3,0,1,2
  18. CG 6 = i = 1 6 r e l i = 3 + 2 + 3 + 0 + 1 + 2 = 11 \mathrm{CG_{6}}=\sum_{i=1}^{6}rel_{i}=3+2+3+0+1+2=11
  19. D 3 D_{3}
  20. D 4 D_{4}
  21. i i
  22. r e l i rel_{i}
  23. log 2 i \log_{2}i
  24. r e l i log 2 i \frac{rel_{i}}{\log_{2}i}
  25. D C G 6 DCG_{6}
  26. DCG 6 = r e l 1 + i = 2 6 r e l i log 2 i = 3 + ( 2 + 1.892 + 0 + 0.431 + 0.774 ) = 8.10 \mathrm{DCG_{6}}=rel_{1}+\sum_{i=2}^{6}\frac{rel_{i}}{\log_{2}i}=3+(2+1.892+0+% 0.431+0.774)=8.10
  27. D 3 D_{3}
  28. D 4 D_{4}
  29. 3 , 3 , 2 , 2 , 1 , 0 3,3,2,2,1,0
  30. IDCG 6 = 8.69 \mathrm{IDCG_{6}}=8.69
  31. nDCG 6 = D C G 6 I D C G 6 = 8.10 8.69 = 0.932 \mathrm{nDCG_{6}}=\frac{DCG_{6}}{IDCG_{6}}=\frac{8.10}{8.69}=0.932
  32. 1 , 1 , 1 1,1,1
  33. 1 , 1 , 1 , 0 1,1,1,0
  34. 1 - 2 r e l i 1-2^{rel_{i}}
  35. 2 r e l i - 1 2^{rel_{i}}-1
  36. E x c e l l e n t , F a i r , B a d Excellent,Fair,Bad
  37. 1 , 0 , - 1 1,0,-1
  38. 2 , 1 , 0 2,1,0
  39. 1 , 1 , 1 1,1,1
  40. 1 , 1 , 1 , 1 , 1 1,1,1,1,1
  41. 1 , 1 , 1 , 0 , 0 1,1,1,0,0
  42. 1 , 1 , 1 , 1 , 1 1,1,1,1,1

Discrete_Fourier_transform_(general).html

  1. R R
  2. n 1 n\geq 1
  3. α R \alpha\in R
  4. α n = 1 \alpha^{n}=1
  5. j = 0 n - 1 α j k = 0 \sum_{j=0}^{n-1}\alpha^{jk}=0
  6. 1 k < n ( 1 ) 1\leq k<n\qquad(1)
  7. ( v 0 , , v n - 1 ) (v_{0},\ldots,v_{n-1})
  8. R R
  9. ( f 0 , , f n - 1 ) (f_{0},\ldots,f_{n-1})
  10. R R
  11. f k = j = 0 n - 1 v j α j k . ( 2 ) f_{k}=\sum_{j=0}^{n-1}v_{j}\alpha^{jk}.\qquad(2)
  12. ( v 0 , , v n - 1 ) (v_{0},\ldots,v_{n-1})
  13. j j
  14. ( f 0 , , f n - 1 ) (f_{0},\ldots,f_{n-1})
  15. k k
  16. ( f 0 , , f n - 1 ) (f_{0},\ldots,f_{n-1})
  17. ( v 0 , , v n - 1 ) (v_{0},\ldots,v_{n-1})
  18. R R
  19. α \alpha
  20. α k 1 \alpha^{k}\neq 1
  21. 1 k < n 1\leq k<n
  22. β = α k \beta=\alpha^{k}
  23. 1 k < n 1\leq k<n
  24. α n = 1 \alpha^{n}=1
  25. β n = ( α n ) k = 1 \beta^{n}=(\alpha^{n})^{k}=1
  26. β n - 1 = ( β - 1 ) ( j = 0 n - 1 β j ) = 0 \beta^{n}-1=(\beta-1)\left(\sum_{j=0}^{n-1}\beta^{j}\right)=0
  27. α \alpha
  28. β - 1 0 \beta-1\neq 0
  29. R R
  30. α n / 2 = - 1 \alpha^{n/2}=-1
  31. v j = 1 n k = 0 n - 1 f k α - j k . ( 3 ) v_{j}=\frac{1}{n}\sum_{k=0}^{n-1}f_{k}\alpha^{-jk}.\qquad(3)
  32. 1 / n 1/n
  33. n n
  34. R R
  35. 1 n k = 0 n - 1 f k α - j k = 1 n k = 0 n - 1 j = 0 n - 1 v j α j k α - j k = 1 n j = 0 n - 1 v j k = 0 n - 1 α ( j - j ) k . \begin{aligned}&\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}f_{k}\alpha^{-jk}\\ &\displaystyle{}=\frac{1}{n}\sum_{k=0}^{n-1}\sum_{j^{\prime}=0}^{n-1}v_{j^{% \prime}}\alpha^{j^{\prime}k}\alpha^{-jk}\\ &\displaystyle{}=\frac{1}{n}\sum_{j^{\prime}=0}^{n-1}v_{j^{\prime}}\sum_{k=0}^% {n-1}\alpha^{(j^{\prime}-j)k}.\end{aligned}
  36. v j v_{j}
  37. k = 0 n - 1 α ( j - j ) k = 0 \sum_{k=0}^{n-1}\alpha^{(j^{\prime}-j)k}=0
  38. j j j^{\prime}\neq j
  39. k = j - j k=j^{\prime}-j
  40. k = 0 n - 1 α ( j - j ) k = n \sum_{k=0}^{n-1}\alpha^{(j^{\prime}-j)k}=n
  41. j = j j^{\prime}=j
  42. [ f 0 f 1 f n - 1 ] = [ 1 1 1 1 1 α α 2 α n - 1 1 α 2 α 4 α 2 ( n - 1 ) 1 α n - 1 α 2 ( n - 1 ) α ( n - 1 ) ( n - 1 ) ] [ v 0 v 1 v n - 1 ] . \begin{bmatrix}f_{0}\\ f_{1}\\ \vdots\\ f_{n-1}\end{bmatrix}=\begin{bmatrix}1&1&1&\cdots&1\\ 1&\alpha&\alpha^{2}&\cdots&\alpha^{n-1}\\ 1&\alpha^{2}&\alpha^{4}&\cdots&\alpha^{2(n-1)}\\ \vdots&\vdots&\vdots&&\vdots\\ 1&\alpha^{n-1}&\alpha^{2(n-1)}&\cdots&\alpha^{(n-1)(n-1)}\\ \end{bmatrix}\begin{bmatrix}v_{0}\\ v_{1}\\ \vdots\\ v_{n-1}\end{bmatrix}.
  43. [ v 0 v 1 v n - 1 ] = 1 n [ 1 1 1 1 1 α - 1 α - 2 α - ( n - 1 ) 1 α - 2 α - 4 α - 2 ( n - 1 ) 1 α - ( n - 1 ) α - 2 ( n - 1 ) α - ( n - 1 ) ( n - 1 ) ] [ f 0 f 1 f n - 1 ] . \begin{bmatrix}v_{0}\\ v_{1}\\ \vdots\\ v_{n-1}\end{bmatrix}=\frac{1}{n}\begin{bmatrix}1&1&1&\cdots&1\\ 1&\alpha^{-1}&\alpha^{-2}&\cdots&\alpha^{-(n-1)}\\ 1&\alpha^{-2}&\alpha^{-4}&\cdots&\alpha^{-2(n-1)}\\ \vdots&\vdots&\vdots&&\vdots\\ 1&\alpha^{-(n-1)}&\alpha^{-2(n-1)}&\cdots&\alpha^{-(n-1)(n-1)}\end{bmatrix}% \begin{bmatrix}f_{0}\\ f_{1}\\ \vdots\\ f_{n-1}\end{bmatrix}.
  44. n n
  45. ( v 0 , , v n - 1 ) (v_{0},\ldots,v_{n-1})
  46. p v ( x ) = v 0 + v 1 x + v 2 x 2 + + v n - 1 x n - 1 . p_{v}(x)=v_{0}+v_{1}x+v_{2}x^{2}+\cdots+v_{n-1}x^{n-1}.\,
  47. f k = v 0 + v 1 α k + v 2 α 2 k + + v n - 1 α ( n - 1 ) k . f_{k}=v_{0}+v_{1}\alpha^{k}+v_{2}\alpha^{2k}+\cdots+v_{n-1}\alpha^{(n-1)k}.\,
  48. f k f_{k}
  49. p v ( x ) p_{v}(x)
  50. x = α k x=\alpha^{k}
  51. f k = p v ( α k ) . f_{k}=p_{v}(\alpha^{k}).\,
  52. n n
  53. α \alpha
  54. v j = 1 n ( f 0 + f 1 α - j + f 2 α - 2 j + + f n - 1 α - ( n - 1 ) j ) . ( 5 ) v_{j}=\frac{1}{n}(f_{0}+f_{1}\alpha^{-j}+f_{2}\alpha^{-2j}+\cdots+f_{n-1}% \alpha^{-(n-1)j}).\qquad(5)
  55. p f ( x ) = f 0 + f 1 x + f 2 x 2 + + f n - 1 x n - 1 , p_{f}(x)=f_{0}+f_{1}x+f_{2}x^{2}+\cdots+f_{n-1}x^{n-1},
  56. v j = 1 n p f ( α - j ) . v_{j}=\frac{1}{n}p_{f}(\alpha^{-j}).
  57. p ( x ) p(x)
  58. q ( x ) q(x)
  59. q ( x ) q(x)
  60. p ( x ) p(x)
  61. F = F={\mathbb{C}}
  62. n n
  63. α = e - 2 π i n , \alpha=e^{\frac{-2\pi i}{n}},
  64. f k = j = 0 n - 1 v j e - 2 π i n j k . f_{k}=\sum_{j=0}^{n-1}v_{j}e^{\frac{-2\pi i}{n}jk}.
  65. 1 n \frac{1}{\sqrt{n}}
  66. 1 1
  67. 1 n \frac{1}{n}
  68. n \sqrt{n}
  69. F = G F ( q ) F=GF(q)
  70. q q
  71. n n
  72. n n
  73. q - 1 q-1
  74. F F
  75. q - 1 q-1
  76. n = 1 + 1 + + 1 n times n=\underbrace{1+1+\cdots+1}_{n\ \rm times}
  77. 1 n \frac{1}{n}
  78. G F ( q ) GF(q)
  79. F = / p F={\mathbb{Z}}/p
  80. p p
  81. n n
  82. n n
  83. p - 1 p-1
  84. p = ξ n + 1 p=\xi n+1
  85. ξ \xi
  86. ω \omega
  87. ( p - 1 ) (p-1)
  88. n n
  89. α \alpha
  90. α = ω ξ \alpha=\omega^{\xi}
  91. p = 5 p=5
  92. α = 2 \alpha=2
  93. 2 1 = 2 ( m o d 5 ) 2^{1}=2(mod5)
  94. 2 2 = 4 ( m o d 5 ) 2^{2}=4(mod5)
  95. 2 3 = 3 ( m o d 5 ) 2^{3}=3(mod5)
  96. 2 4 = 1 ( m o d 5 ) 2^{4}=1(mod5)
  97. N = 4 N=4
  98. [ F ( 0 ) F ( 1 ) F ( 2 ) F ( 3 ) ] = [ 1 1 1 1 1 2 4 3 1 4 1 4 1 3 4 2 ] [ f ( 0 ) f ( 1 ) f ( 2 ) f ( 3 ) ] \begin{bmatrix}F(0)\\ F(1)\\ F(2)\\ F(3)\end{bmatrix}=\begin{bmatrix}1&1&1&1\\ 1&2&4&3\\ 1&4&1&4\\ 1&3&4&2\end{bmatrix}\begin{bmatrix}f(0)\\ f(1)\\ f(2)\\ f(3)\end{bmatrix}
  99. / m \mathbb{Z}/m
  100. m m
  101. 𝐅 1 n . \mathbf{F}_{1^{n}}.
  102. O ( n log n ) O(n\log n)

Discrete_Morse_theory.html

  1. 𝒳 \mathcal{X}
  2. κ : 𝒳 × 𝒳 \kappa:\mathcal{X}\times\mathcal{X}\to\mathbb{Z}
  3. σ \sigma
  4. τ \tau
  5. 𝒳 \mathcal{X}
  6. κ ( σ , τ ) \kappa(\sigma,~{}\tau)
  7. σ \sigma
  8. τ \tau
  9. \partial
  10. 𝒳 \mathcal{X}
  11. ( σ ) = τ 𝒳 κ ( σ , τ ) τ \partial(\sigma)=\sum_{\tau\in\mathcal{X}}\kappa(\sigma,\tau)\tau
  12. 0 \partial\circ\partial\equiv 0
  13. σ , τ 𝒳 \forall\sigma,\tau^{\prime}\in\mathcal{X}
  14. τ 𝒳 κ ( σ , τ ) κ ( τ , τ ) = 0 \sum_{\tau\in\mathcal{X}}\kappa(\sigma,\tau)\kappa(\tau,\tau^{\prime})=0
  15. 0 \partial\circ\partial\equiv 0
  16. μ : 𝒳 \mu:\mathcal{X}\to\mathbb{R}
  17. σ 𝒳 \sigma\in\mathcal{X}
  18. τ 𝒳 \tau\in\mathcal{X}
  19. σ \sigma
  20. μ ( σ ) μ ( τ ) \mu(\sigma)\leq\mu(\tau)
  21. σ 𝒳 \sigma\in\mathcal{X}
  22. τ 𝒳 \tau\in\mathcal{X}
  23. σ \sigma
  24. μ ( σ ) μ ( τ ) \mu(\sigma)\geq\mu(\tau)
  25. σ \sigma
  26. 𝒳 \mathcal{X}
  27. σ 𝒳 \sigma\in\mathcal{X}
  28. τ 𝒳 \tau\in\mathcal{X}
  29. μ \mu
  30. μ \mu
  31. 𝒳 = 𝒜 𝒦 𝒬 \mathcal{X}=\mathcal{A}\sqcup\mathcal{K}\sqcup\mathcal{Q}
  32. 𝒜 \mathcal{A}
  33. 𝒦 \mathcal{K}
  34. 𝒬 \mathcal{Q}
  35. k k
  36. 𝒦 \mathcal{K}
  37. ( k - 1 ) (k-1)
  38. 𝒬 \mathcal{Q}
  39. p k : 𝒦 k 𝒬 k - 1 p^{k}:\mathcal{K}^{k}\to\mathcal{Q}^{k-1}
  40. k k
  41. K 𝒦 k K\in\mathcal{K}^{k}
  42. K K
  43. p k ( K ) 𝒬 p^{k}(K)\in\mathcal{Q}
  44. 𝒳 \mathcal{X}
  45. \mathbb{Z}
  46. ± 1 \pm 1
  47. 𝒳 \mathcal{X}
  48. \mathbb{Z}
  49. 𝒳 \mathcal{X}
  50. 𝒜 \mathcal{A}
  51. 𝒦 \mathcal{K}
  52. 𝒬 \mathcal{Q}
  53. 𝒜 \mathcal{A}
  54. ρ = ( Q 1 , K 1 , Q 2 , K 2 , , Q M , K M ) \rho=(Q_{1},K_{1},Q_{2},K_{2},\ldots,Q_{M},K_{M})
  55. Q m = p ( K m ) Q_{m}=p(K_{m})
  56. κ ( K m , Q m + 1 ) 0 \kappa(K_{m},~{}Q_{m+1})\neq 0
  57. ν ( ρ ) = m = 1 M - 1 - κ ( K m , Q m + 1 ) m = 1 M κ ( K m , Q m ) \nu(\rho)=\frac{\sum_{m=1}^{M-1}-\kappa(K_{m},Q_{m+1})}{\sum_{m=1}^{M}\kappa(K% _{m},Q_{m})}
  58. ± 1 \pm 1
  59. μ \mu
  60. ρ \rho
  61. ρ \rho
  62. A , A 𝒜 A,A^{\prime}\in\mathcal{A}
  63. κ ( A , Q 1 ) 0 κ ( K M , A ) \kappa(A,Q_{1})\neq 0\neq\kappa(K_{M},A^{\prime})
  64. A ρ A A\stackrel{\rho}{\to}A^{\prime}
  65. m ( ρ ) = κ ( A , Q 1 ) ν ( ρ ) κ ( K M , A ) m(\rho)=\kappa(A,Q_{1})\cdot\nu(\rho)\cdot\kappa(K_{M},A)
  66. 𝒜 \mathcal{A}
  67. Δ ( A ) = κ ( A , A ) + A ρ A m ( ρ ) A \Delta(A)=\kappa(A,A^{\prime})+\sum_{A\stackrel{\rho}{\to}A^{\prime}}m(\rho)A^% {\prime}
  68. A A
  69. A A^{\prime}
  70. 𝒜 \mathcal{A}
  71. 𝒳 \mathcal{X}
  72. m q = | 𝒜 q | m_{q}=|\mathcal{A}_{q}|
  73. q q
  74. 𝒜 \mathcal{A}
  75. q t h q^{th}
  76. β q \beta_{q}
  77. q t h q^{th}
  78. 𝒳 \mathcal{X}
  79. N > 0 N>0
  80. m N β N m_{N}\geq\beta_{N}
  81. m N - m N - 1 + ± m 0 β N - β N - 1 + ± β 0 m_{N}-m_{N-1}+\ldots\pm m_{0}\geq\beta_{N}-\beta_{N-1}+\ldots\pm\beta_{0}
  82. χ ( 𝒳 ) \chi(\mathcal{X})
  83. 𝒳 \mathcal{X}
  84. χ ( 𝒳 ) = m 0 - m 1 + ± m dim 𝒳 \chi(\mathcal{X})=m_{0}-m_{1}+\ldots\pm m_{\dim\mathcal{X}}
  85. 𝒳 \mathcal{X}
  86. \partial
  87. μ : 𝒳 \mu:\mathcal{X}\to\mathbb{R}
  88. 𝒜 \mathcal{A}
  89. Δ \Delta
  90. H * ( 𝒳 , ) H * ( 𝒜 , Δ ) H_{*}(\mathcal{X},\partial)\simeq H_{*}(\mathcal{A},\Delta)

Discursive_dilemma.html

  1. C P and Q C\equiv P\and Q

Dispersionless_equation.html

  1. ( u t + u u x ) x + u y y = 0 , ( 1 ) (u_{t}+uu_{x})_{x}+u_{yy}=0,\qquad(1)
  2. [ L 1 , L 2 ] = 0. ( 2 ) [L_{1},L_{2}]=0.\qquad(2)
  3. L 1 = y + λ x - u x λ , ( 3 a ) L_{1}=\partial_{y}+\lambda\partial_{x}-u_{x}\partial_{\lambda},\qquad(3a)
  4. L 2 = t + ( λ 2 + u ) x + ( - λ u x + u y ) λ , ( 3 b ) L_{2}=\partial_{t}+(\lambda^{2}+u)\partial_{x}+(-\lambda u_{x}+u_{y})\partial_% {\lambda},\qquad(3b)
  5. λ \lambda
  6. x x
  7. A t 2 n + A x n + 1 + n A n - 1 A x 0 = 0. A^{n}_{t_{2}}+A^{n+1}_{x}+nA^{n-1}A^{0}_{x}=0.
  8. λ = p + n = 0 A n / p n + 1 , \lambda=p+\sum_{n=0}^{\infty}A^{n}/p^{n+1},
  9. p t 2 + p p x + A x 0 = 0 , p_{t_{2}}+pp_{x}+A^{0}_{x}=0,
  10. p t 3 + p 2 p x + ( p A 0 + A 1 ) x = 0 , p_{t_{3}}+p^{2}p_{x}+(pA^{0}+A^{1})_{x}=0,
  11. u = A 0 , u=A^{0},
  12. y = t 2 y=t_{2}
  13. t = t 3 t=t_{3}
  14. A n = h v n A^{n}=hv^{n}
  15. A n A^{n}
  16. h y + ( h v ) x = 0 , h_{y}+(hv)_{x}=0,
  17. v y + v v x + h x = 0. v_{y}+vv_{x}+h_{x}=0.
  18. u t 3 = u u x . ( 4 ) u_{t_{3}}=uu_{x}.\qquad(4)
  19. t 2 t_{2}
  20. t 3 t_{3}
  21. λ 2 = p 2 + 2 A 0 . \lambda^{2}=p^{2}+2A^{0}.
  22. v = v ( x 1 , x 2 , t ) v=v(x_{1},x_{2},t)
  23. t v = z ( v w ) + z ¯ ( v w ¯ ) , \displaystyle\partial_{t}v=\partial_{z}(vw)+\partial_{\bar{z}}(v\bar{w}),
  24. z = 1 2 ( x 1 - i x 2 ) \partial_{z}=\frac{1}{2}(\partial_{x_{1}}-i\partial_{x_{2}})
  25. z ¯ = 1 2 ( x 1 + i x 2 ) \partial_{\bar{z}}=\frac{1}{2}(\partial_{x_{1}}+i\partial_{x_{2}})
  26. w w
  27. v v

Divergence_(computer_science).html

  1. C l o c k = t i c k C l o c k Clock=tick\rightarrow Clock
  2. traces ( C l o c k ) = { , t i c k , t i c k , t i c k , } = { t i c k } * \operatorname{traces}(Clock)=\{\langle\rangle,\langle tick\rangle,\langle tick% ,tick\rangle,\cdots\}=\{tick\}^{*}
  3. P = C l o c k \ t i c k P=Clock\backslash tick

Dividend_discount_model.html

  1. P P
  2. g g
  3. r r
  4. D 1 D_{1}
  5. P = D 1 r - g P=\frac{D_{1}}{r-g}
  6. D 0 ( 1 + g ) t D_{0}(1+g)^{t}
  7. t t
  8. D 0 ( 1 + g ) t ( 1 + r ) t \frac{D_{0}(1+g)^{t}}{{(1+r)}^{t}}
  9. P P
  10. P = t = 1 D 0 ( 1 + g ) t ( 1 + r ) t P=\sum_{t=1}^{\infty}{D_{0}}\frac{(1+g)^{t}}{(1+r)^{t}}
  11. P = D 0 r ( 1 + r + r 2 + r 3 + . ) P={D_{0}}r^{\prime}(1+r^{\prime}+{r^{\prime}}^{2}+{r^{\prime}}^{3}+....)
  12. r = ( 1 + g ) ( 1 + r ) . r^{\prime}=\frac{(1+g)}{(1+r)}.
  13. r r^{\prime}
  14. 1 1 - r \frac{1}{1-r^{\prime}}
  15. r < 1 r^{\prime}<1
  16. P = D 0 r 1 - r P=\frac{D_{0}r^{\prime}}{1-r^{\prime}}
  17. r r^{\prime}
  18. P = D 0 ( 1 + g ) ( 1 + r ) 1 - 1 + g 1 + r P=\frac{\frac{{D_{0}}(1+g)}{(1+r)}}{1-\frac{1+g}{1+r}}
  19. 1 + r 1 + r \frac{1+r}{1+r}
  20. P = D 0 ( 1 + g ) r - g P=\frac{D_{0}(1+g)}{r-g}
  21. Income + Capital Gain = Total Return \,\text{Income}+\,\text{Capital Gain}=\,\text{Total Return}
  22. Dividend Yield + Growth = Cost Of Equity \,\text{Dividend Yield}+\,\text{Growth}=\,\text{Cost Of Equity}
  23. D P + g = r \frac{D}{P}+g=r
  24. D P = r - g \frac{D}{P}=r-g
  25. D r - g = P \frac{D}{r-g}=P
  26. r - g r-g
  27. P = t = 1 N D 0 ( 1 + g ) t ( 1 + r ) t + P N ( 1 + r ) N P=\sum_{t=1}^{N}\frac{D_{0}\left(1+g\right)^{t}}{\left(1+r\right)^{t}}+\frac{P% _{N}}{\left(1+r\right)^{N}}
  28. P = D 0 ( 1 + g ) r - g [ 1 - ( 1 + g ) N ( 1 + r ) N ] + D 0 ( 1 + g ) N ( 1 + g ) ( 1 + r ) N ( r - g ) , P=\frac{D_{0}\left(1+g\right)}{r-g}\left[1-\frac{\left(1+g\right)^{N}}{\left(1% +r\right)^{N}}\right]+\frac{D_{0}\left(1+g\right)^{N}\left(1+g_{\infty}\right)% }{\left(1+r\right)^{N}\left(r-g_{\infty}\right)},
  29. g g
  30. g g_{\infty}
  31. N N
  32. P 0 = D 1 r P_{0}=\frac{D_{1}}{r}
  33. r r
  34. r = D 1 P 0 + g . r=\frac{D_{1}}{P_{0}}+g.
  35. g g

Division_polynomials.html

  1. [ x , y , A , B ] \mathbb{Z}[x,y,A,B]
  2. x , y , A , B x,y,A,B
  3. ψ 0 = 0 \psi_{0}=0
  4. ψ 1 = 1 \psi_{1}=1
  5. ψ 2 = 2 y \psi_{2}=2y
  6. ψ 3 = 3 x 4 + 6 A x 2 + 12 B x - A 2 \psi_{3}=3x^{4}+6Ax^{2}+12Bx-A^{2}
  7. ψ 4 = 4 y ( x 6 + 5 A x 4 + 20 B x 3 - 5 A 2 x 2 - 4 A B x - 8 B 2 - A 3 ) \psi_{4}=4y(x^{6}+5Ax^{4}+20Bx^{3}-5A^{2}x^{2}-4ABx-8B^{2}-A^{3})
  8. \vdots
  9. ψ 2 m + 1 = ψ m + 2 ψ m 3 - ψ m - 1 ψ m + 1 3 for m 2 \psi_{2m+1}=\psi_{m+2}\psi_{m}^{3}-\psi_{m-1}\psi^{3}_{m+1}\,\text{ for }m\geq 2
  10. ψ 2 m = ( ψ m 2 y ) ( ψ m + 2 ψ m - 1 2 - ψ m - 2 ψ m + 1 2 ) for m 3 \psi_{2m}=\left(\frac{\psi_{m}}{2y}\right)\cdot(\psi_{m+2}\psi^{2}_{m-1}-\psi_% {m-2}\psi^{2}_{m+1})\,\text{ for }m\geq 3
  11. ψ n \psi_{n}
  12. y 2 = x 3 + A x + B y^{2}=x^{3}+Ax+B
  13. ψ 2 m + 1 [ x , A , B ] \psi_{2m+1}\in\mathbb{Z}[x,A,B]
  14. ψ 2 m 2 y [ x , A , B ] \psi_{2m}\in 2y\mathbb{Z}[x,A,B]
  15. [ x , y , A , B ] / ( y 2 - x 3 - A x - B ) \mathbb{Q}[x,y,A,B]/(y^{2}-x^{3}-Ax-B)
  16. E E
  17. y 2 = x 3 + A x + B y^{2}=x^{3}+Ax+B
  18. K K
  19. A , B K A,B\in K
  20. A , B A,B
  21. E E
  22. ψ 2 n + 1 \psi_{2n+1}
  23. x x
  24. E [ 2 n + 1 ] { O } E[2n+1]\setminus\{O\}
  25. E [ 2 n + 1 ] E[2n+1]
  26. ( 2 n + 1 ) th (2n+1)^{\,\text{th}}
  27. E E
  28. ψ 2 n / y \psi_{2n}/y
  29. x x
  30. E [ 2 n ] E [ 2 ] E[2n]\setminus E[2]
  31. P = ( x P , y P ) P=(x_{P},y_{P})
  32. E : y 2 = x 3 + A x + B E:y^{2}=x^{3}+Ax+B
  33. K K
  34. P P
  35. n P = ( ϕ n ( x ) ψ n 2 ( x ) , ω n ( x , y ) ψ n 3 ( x , y ) ) = ( x - ψ n - 1 ψ n + 1 ψ n 2 ( x ) , ψ 2 n ( x , y ) 2 ψ n 4 ( x ) ) nP=\left(\frac{\phi_{n}(x)}{\psi_{n}^{2}(x)},\frac{\omega_{n}(x,y)}{\psi^{3}_{% n}(x,y)}\right)=\left(x-\frac{\psi_{n-1}\psi_{n+1}}{\psi^{2}_{n}(x)},\frac{% \psi_{2n}(x,y)}{2\psi^{4}_{n}(x)}\right)
  36. ϕ n \phi_{n}
  37. ω n \omega_{n}
  38. ϕ n = x ψ n 2 - ψ n + 1 ψ n - 1 , \phi_{n}=x\psi_{n}^{2}-\psi_{n+1}\psi_{n-1},
  39. ω n = ψ n + 2 ψ n - 1 2 - ψ n - 2 ψ n + 1 2 4 y . \omega_{n}=\frac{\psi_{n+2}\psi_{n-1}^{2}-\psi_{n-2}\psi_{n+1}^{2}}{4y}.
  40. ψ 2 m \psi_{2m}
  41. ψ 2 m + 1 \psi_{2m+1}
  42. ψ n 2 \psi_{n}^{2}
  43. ψ 2 n y , ψ 2 n + 1 \frac{\psi_{2n}}{y},\psi_{2n+1}
  44. ϕ n \phi_{n}
  45. K [ x ] K[x]
  46. p > 3 p>3
  47. E : y 2 = x 3 + A x + B E:y^{2}=x^{3}+Ax+B
  48. 𝔽 p \mathbb{F}_{p}
  49. A , B 𝔽 p A,B\in\mathbb{F}_{p}
  50. \ell
  51. E E
  52. 𝔽 ¯ p \bar{\mathbb{F}}_{p}
  53. / × / \mathbb{Z}/\ell\times\mathbb{Z}/\ell
  54. p \ell\neq p
  55. / \mathbb{Z}/\ell
  56. { 0 } \{0\}
  57. = p \ell=p
  58. ψ \psi_{\ell}
  59. 1 2 ( l 2 - 1 ) \frac{1}{2}(l^{2}-1)
  60. 1 2 ( l - 1 ) \frac{1}{2}(l-1)
  61. \ell
  62. \ell
  63. E ( 𝔽 q ) E(\mathbb{F}_{q})

Dixmier_trace.html

  1. T 1 , = sup N i = 1 N μ i ( T ) log ( N ) \|T\|_{1,\infty}=\sup_{N}\frac{\sum_{i=1}^{N}\mu_{i}(T)}{\log(N)}
  2. a N = i = 1 N μ i ( T ) log ( N ) a_{N}=\frac{\sum_{i=1}^{N}\mu_{i}(T)}{\log(N)}
  3. Tr ω ( T ) = lim ω a N \operatorname{Tr}_{\omega}(T)=\lim_{\omega}a_{N}
  4. L 1 , ( H ) L^{1,\infty}(H)
  5. ζ T ( s ) = Tr ( T s ) = μ i s \zeta_{T}(s)=\operatorname{Tr}(T^{s})=\sum{\mu_{i}^{s}}

Dixon's_identity.html

  1. k = - a a ( - 1 ) k ( 2 a k + a ) 3 = ( 3 a ) ! ( a ! ) 3 . \sum_{k=-a}^{a}(-1)^{k}{2a\choose k+a}^{3}=\frac{(3a)!}{(a!)^{3}}.
  2. k = - a a ( - 1 ) k ( a + b a + k ) ( b + c b + k ) ( c + a c + k ) = ( a + b + c ) ! a ! b ! c ! \sum_{k=-a}^{a}(-1)^{k}{a+b\choose a+k}{b+c\choose b+k}{c+a\choose c+k}=\frac{% (a+b+c)!}{a!b!c!}
  3. ( b + c b - a ) ( c + a c - a ) F 2 3 ( - 2 a , - a - b , - a - c ; 1 + b - a , 1 + c - a ; 1 ) {b+c\choose b-a}{c+a\choose c-a}{}_{3}F_{2}(-2a,-a-b,-a-c;1+b-a,1+c-a;1)
  4. F 2 3 ( a , b , c ; 1 + a - b , 1 + a - c ; 1 ) = Γ ( 1 + a / 2 ) Γ ( 1 + a / 2 - b - c ) Γ ( 1 + a - b ) Γ ( 1 + a - c ) Γ ( 1 + a ) Γ ( 1 + a - b - c ) Γ ( 1 + a / 2 - b ) Γ ( 1 + a / 2 - c ) . \;{}_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)=\frac{\Gamma(1+a/2)\Gamma(1+a/2-b-c)\Gamma(% 1+a-b)\Gamma(1+a-c)}{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a/2-b)\Gamma(1+a/2-c)}.
  5. 1 / 2 {1}/{2}
  6. ϕ 3 4 [ a - q a 1 / 2 b c - a 1 / 2 a q / b a q / c ; q , q a 1 / 2 / b c ] = ( a q , a q / b c , q a 1 / 2 / b , q a 1 / 2 / c ; q ) ( a q / b , a q / c , q a 1 / 2 , q a 1 / 2 / b c ; q ) \;{}_{4}\phi_{3}\left[\begin{matrix}a&-qa^{1/2}&b&c\\ &-a^{1/2}&aq/b&aq/c\end{matrix};q,qa^{1/2}/bc\right]=\frac{(aq,aq/bc,qa^{1/2}/% b,qa^{1/2}/c;q)_{\infty}}{(aq/b,aq/c,qa^{1/2},qa^{1/2}/bc;q)_{\infty}}

Dm3.html

  1. d m 3 dm^{3}

DNA_sequencing_theory.html

  1. L L
  2. G G
  3. L / G L/G
  4. L G L\ll G
  5. 1 - L / G 1-L/G
  6. [ 1 - L / G ] N \left[1-L/G\right]^{N}
  7. N N
  8. P = 1 - [ 1 - L G ] N . P=1-\left[1-\frac{L}{G}\right]^{N}.
  9. N 1 N\gg 1
  10. [ 1 - L G ] N exp ( - N L / G ) , \left[1-\frac{L}{G}\right]^{N}\sim\exp(-NL/G),
  11. R = N L / G R=NL/G
  12. C C
  13. E C = 1 - e - R , E\langle C\rangle=1-e^{-R},
  14. E c o n t i g s = N e - R . E\langle contigs\rangle=Ne^{-R}.
  15. 1 / G 1/G
  16. L / G L/G
  17. R < 1 R<1

Doléans-Dade_exponential.html

  1. d log ( Y ) = 1 Y d Y - 1 2 Y 2 d [ Y ] = d X - 1 2 d [ X ] . \begin{aligned}\displaystyle d\log(Y)&\displaystyle=\frac{1}{Y}\,dY-\frac{1}{2% Y^{2}}\,d[Y]\\ &\displaystyle=dX-\frac{1}{2}\,d[X].\end{aligned}
  2. Y t = exp ( X t - X 0 - 1 2 [ X ] t ) , t 0. Y_{t}=\exp\Bigl(X_{t}-X_{0}-\frac{1}{2}[X]_{t}\Bigr),\qquad t\geq 0.
  3. Y t = exp ( X t - X 0 - 1 2 [ X ] t ) s t ( 1 + Δ X s ) exp ( - Δ X s + 1 2 Δ X s 2 ) , t 0 , Y_{t}=\exp\Bigl(X_{t}-X_{0}-\frac{1}{2}[X]_{t}\Bigr)\prod_{s\leq t}(1+\Delta X% _{s})\exp\Bigl(-\Delta X_{s}+\frac{1}{2}\Delta X_{s}^{2}\Bigr),\qquad t\geq 0,

Domain_decomposition_methods.html

  1. u ′′ ( x ) - u ( x ) = 0 u^{\prime\prime}(x)-u(x)=0
  2. u ( 0 ) = 0 , u ( 1 ) = 1 u(0)=0,u(1)=1
  3. u ( x ) = e x - e - x e 1 - e - 1 u(x)=\frac{e^{x}-e^{-x}}{e^{1}-e^{-1}}
  4. [ 0 , 1 2 ] [0,\frac{1}{2}]
  5. [ 1 2 , 1 ] [\frac{1}{2},1]
  6. v 1 ( x ) v_{1}(x)
  7. v 2 ( x ) v_{2}(x)
  8. v 1 ( 1 2 ) = v 2 ( 1 2 ) v_{1}\left(\frac{1}{2}\right)=v_{2}\left(\frac{1}{2}\right)
  9. v 1 ( 1 2 ) = v 2 ( 1 2 ) v_{1}^{\prime}\left(\frac{1}{2}\right)=v_{2}^{\prime}\left(\frac{1}{2}\right)
  10. v 1 ( x ) = n = 0 N u n T n ( y 1 ( x ) ) v_{1}(x)=\sum_{n=0}^{N}u_{n}T_{n}(y_{1}(x))
  11. v 2 ( x ) = n = 0 N u n + N T n ( y 2 ( x ) ) v_{2}(x)=\sum_{n=0}^{N}u_{n+N}T_{n}(y_{2}(x))
  12. y 1 ( x ) = 4 x - 1 y_{1}(x)=4x-1
  13. y 2 ( x ) = 4 x - 3 y_{2}(x)=4x-3
  14. T n ( y ) T_{n}(y)
  15. u 1 = 0.06236 u_{1}=0.06236
  16. u 2 = 0.21495 u_{2}=0.21495
  17. u 3 = 0.37428 u_{3}=0.37428
  18. u 4 = 0.44341 u_{4}=0.44341
  19. u 5 = 0.51492 u_{5}=0.51492
  20. u 6 = 0.69972 u_{6}=0.69972
  21. u 7 = 0.90645 u_{7}=0.90645

Donkey_sentence.html

  1. x ( FARMER ( x ) and y ( DONKEY ( y ) and OWNS ( x , y ) ) BEAT ( x , y ) ) \forall x\,(\,\text{FARMER}(x)\and\exists y\,(\,\text{DONKEY}(y)\and\,\text{% OWNS}(x,y))\rightarrow\,\text{BEAT}(x,y))
  2. x y ( FARMER ( x ) and DONKEY ( y ) and OWNS ( x , y ) BEAT ( x , y ) ) \forall x\,\exists y\,(\,\text{FARMER}(x)\and\,\text{DONKEY}(y)\and\,\text{% OWNS}(x,y)\rightarrow\,\text{BEAT}(x,y))
  3. x y ( FARMER ( x ) and DONKEY ( y ) and OWNS ( x , y ) BEAT ( x , y ) ) \forall x\,\forall y\,(\,\text{FARMER}(x)\and\,\text{DONKEY}(y)\and\,\text{% OWNS}(x,y)\rightarrow\,\text{BEAT}(x,y))

Doppler_imaging.html

  1. V sin i = 10 - 100 k m s - 1 V\sin i=10-100kms^{-1}
  2. V sin i < 10 k m s - 1 V\sin i<10kms^{-1}
  3. V sin i > 100 k m s - 1 V\sin i>100kms^{-1}
  4. Δ λ = v sin i cos l sin L k m s - 1 \Delta\,\lambda\,=v\sin i\cos l\sin Lkms^{-1}

Douady_rabbit.html

  1. \mathcal{M}
  2. z n + 1 = z n = γ z n ( 1 - z n ) , z_{n+1}=\mathcal{M}z_{n}=\gamma z_{n}\left(1-z_{n}\right),
  3. z z
  4. γ \gamma
  5. w n + 1 = w n = w n 2 - μ . w_{n+1}=\mathcal{M}w_{n}=w_{n}^{2}-\mu.
  6. w w
  7. μ \mu
  8. z z
  9. w w
  10. z = - w γ + 1 2 , z=-\frac{w}{\gamma}+\frac{1}{2},
  11. γ \gamma
  12. μ \mu
  13. μ = ( γ - 1 2 ) 2 - 1 4 , γ = 1 ± 1 + 4 μ . \mu=\left(\frac{\gamma-1}{2}\right)^{2}-\frac{1}{4}\quad,\quad\gamma=1\pm\sqrt% {1+4\mu}.
  14. μ \mu
  15. γ 2 - γ \gamma\to 2-\gamma
  16. \mathcal{M}
  17. z z
  18. w w
  19. \mathcal{M}
  20. γ \gamma
  21. μ \mu
  22. \mathcal{M}
  23. γ \gamma
  24. μ \mu
  25. \mathcal{M}
  26. γ \gamma
  27. μ \mu
  28. z z
  29. w w
  30. z z
  31. w w
  32. γ \gamma
  33. γ \gamma
  34. \mathcal{M}
  35. z = 0 z=0
  36. z = z=\infty
  37. 3 {\mathcal{M}}^{3}
  38. z 1 z_{1}
  39. z 2 z_{2}
  40. z 3 z_{3}
  41. z z
  42. γ = γ D = 2.55268 - 0.959456 i \gamma=\gamma_{D}=2.55268-0.959456i
  43. γ \gamma
  44. \mathcal{M}
  45. z = 0 z=0
  46. z = .656747 - .129015 i z=.656747-.129015i
  47. 3 {\mathcal{M}}^{3}
  48. z 1 = 0.499997032420304 - ( 1.221880225696050 × 10 - 6 ) i ( red ) , z^{1}=0.499997032420304-(1.221880225696050\times 10^{-6})i{\;}{\;}{\mathrm{(% red)}},
  49. z 2 = 0.638169999974373 - ( 0.239864000011495 ) i ( green ) , z^{2}=0.638169999974373-(0.239864000011495)i{\;}{\;}{\mathrm{(green)}},
  50. z 3 = 0.799901291393262 - ( 0.107547238170383 ) i ( yellow ) . z^{3}=0.799901291393262-(0.107547238170383)i{\;}{\;}{\mathrm{(yellow)}}.
  51. B ( z 1 ) B(z^{1})
  52. B ( z 2 ) B(z^{2})
  53. B ( z 3 ) B(z^{3})
  54. 3 {\mathcal{M}}^{3}
  55. B ( ) B(\infty)
  56. \mathcal{M}
  57. \mathcal{M}
  58. z 1 = z 2 , {\mathcal{M}}z^{1}=z^{2},
  59. z 2 = z 3 , {\mathcal{M}}z^{2}=z^{3},
  60. z 3 = z 1 . {\mathcal{M}}z^{3}=z^{1}.
  61. B ( z 1 ) = B ( z 2 ) or red green , {\mathcal{M}}B(z^{1})=B(z^{2}){\;}{\mathrm{or}}{\;}{\mathcal{M}}{\;}{\mathrm{% red}}\subseteq{\mathrm{green}},
  62. B ( z 2 ) = B ( z 3 ) or green yellow , {\mathcal{M}}B(z^{2})=B(z^{3}){\;}{\mathrm{or}}{\;}{\mathcal{M}}{\;}{\mathrm{% green}}\subseteq{\mathrm{yellow}},
  63. B ( z 3 ) = B ( z 1 ) or yellow red . {\mathcal{M}}B(z^{3})=B(z^{1}){\;}{\mathrm{or}}{\;}{\mathcal{M}}{\;}{\mathrm{% yellow}}\subseteq{\mathrm{red}}.
  64. γ = 2.55268 - 0.959456 i \gamma=2.55268-0.959456i
  65. μ = 0.122565 - 0.744864 i \mu=0.122565-0.744864i
  66. γ = 2 - γ D = - .55268 + .959456 i \gamma=2-\gamma_{D}=-.55268+.959456i
  67. μ \mu
  68. z 1 = 0.500003730675024 + ( 6.968273875812428 × 10 - 6 ) i ( red ) , z^{1}=0.500003730675024+(6.968273875812428\times 10^{-6})i{\;}{\;}({\mathrm{% red}}),
  69. z 2 = - 0.138169999969259 + ( 0.239864000061970 ) i ( green ) , z^{2}=-0.138169999969259+(0.239864000061970)i{\;}{\;}({\mathrm{green}}),
  70. z 3 = - 0.238618870661709 - ( 0.264884797354373 ) i ( yellow ) , z^{3}=-0.238618870661709-(0.264884797354373)i{\;}{\;}({\mathrm{yellow}}),
  71. \mathcal{M}
  72. z = 0 z=0
  73. z = 1.450795 + 0.7825835 i z=1.450795+0.7825835i
  74. z 1 z^{1}
  75. z 2 z^{2}
  76. z 3 z^{3}
  77. z = 0 z=0
  78. z = 1 z=1
  79. \mathcal{M}
  80. arg ( γ ) \arg(\gamma)
  81. 120 120^{\circ}
  82. | γ | = 1.1072538 |\gamma|=1.1072538
  83. γ = - 0.55268 + 0.959456 i \gamma=-0.55268+0.959456i
  84. μ = 0.122565 - 0.744864 i \mu=0.122565-0.744864i

Double_(manifold).html

  1. M M
  2. M M
  3. M × { 0 , 1 } / M\times\{0,1\}/\sim
  4. ( x , 0 ) ( x , 1 ) (x,0)\sim(x,1)
  5. x M x\in\partial M
  6. M \partial M
  7. M M
  8. M M
  9. M M
  10. M × [ 0 , 1 ] M\times[0,1]
  11. M M
  12. M × D k M\times D^{k}
  13. M × S k M\times S^{k}
  14. M M
  15. M M^{\prime}
  16. M M
  17. M # - M M\mathrel{\#}-M
  18. M M^{\prime}

Double_tangent_bundle.html

  1. ξ = ξ k x k | x T x M , X = X k x k | x T x M \xi=\xi^{k}\frac{\partial}{\partial x^{k}}\Big|_{x}\in T_{x}M,\qquad X=X^{k}% \frac{\partial}{\partial x^{k}}\Big|_{x}\in T_{x}M
  2. ξ ( x 1 , , x n , ξ 1 , , ξ n ) \xi\mapsto(x^{1},\ldots,x^{n},\xi^{1},\ldots,\xi^{n})
  3. ( π T M ) * - 1 ( X ) = { X k x k | ξ + Y k ξ k | ξ | ξ T x M , Y 1 , , Y n \R } . (\pi_{TM})^{-1}_{*}(X)=\Big\{\ X^{k}\frac{\partial}{\partial x^{k}}\Big|_{\xi}% +Y^{k}\frac{\partial}{\partial\xi^{k}}\Big|_{\xi}\ \Big|\ \xi\in T_{x}M\ ,\ Y^% {1},\ldots,Y^{n}\in\R\ \Big\}.
  4. j ( X k x k | ξ + Y k ξ k | ξ ) = ξ k x k | X + Y k ξ k | X . j\Big(X^{k}\frac{\partial}{\partial x^{k}}\Big|_{\xi}+Y^{k}\frac{\partial}{% \partial\xi^{k}}\Big|_{\xi}\Big)=\xi^{k}\frac{\partial}{\partial x^{k}}\Big|_{% X}+Y^{k}\frac{\partial}{\partial\xi^{k}}\Big|_{X}.
  5. f t s = j f s t \frac{\partial f}{{\partial t}{\partial s}}=j\circ\frac{\partial f}{{\partial s% }{\partial t}}
  6. p ( [ f ] ) = f t s ( 0 , 0 ) p([f])=\frac{\partial f}{{\partial t}{\partial s}}(0,0)
  7. J : J 0 2 ( 2 , M ) J 0 2 ( 2 , M ) / J ( [ f ] ) = [ f α ] J:J^{2}_{0}(\mathbb{R}^{2},M)\to J^{2}_{0}(\mathbb{R}^{2},M)\quad/\quad J([f])% =[f\circ\alpha]
  8. ( vl ξ X ) [ f ] := d d t | t = 0 f ( x , ξ + t X ) , f C ( T M ) . (\operatorname{vl}_{\xi}X)[f]:=\frac{d}{dt}\Big|_{t=0}f(x,\xi+tX),\qquad f\in C% ^{\infty}(TM).
  9. V T M := Ker ( π T M ) * T T M . VTM:=\operatorname{Ker}(\pi_{TM})_{*}\subset TTM.
  10. V : T M T T M ; V ξ := vl ξ ξ , V:TM\to TTM;\qquad V_{\xi}:=\operatorname{vl}_{\xi}\xi,
  11. × ( T M 0 ) T M 0 ; ( t , ξ ) e t ξ . \mathbb{R}\times(TM\setminus 0)\to TM\setminus 0;\qquad(t,\xi)\mapsto e^{t}\xi.
  12. J : T T M T T M ; J ξ X := vl ξ ( π T M ) * X , X T ξ T M J:TTM\to TTM;\qquad J_{\xi}X:=\operatorname{vl}_{\xi}(\pi_{TM})_{*}X,\qquad X% \in T_{\xi}TM
  13. Ran ( J ) = Ker ( J ) = V T M , V J = - J , J [ X , Y ] = J [ J X , Y ] + J [ X , J Y ] , \operatorname{Ran}(J)=\operatorname{Ker}(J)=VTM,\qquad\mathcal{L}_{V}J=-J,% \qquad J[X,Y]=J[JX,Y]+J[X,JY],
  14. Ran ( J ) = Ker ( J ) , J [ X , Y ] = J [ J X , Y ] + J [ X , J Y ] , \operatorname{Ran}(J)=\operatorname{Ker}(J),\qquad J[X,Y]=J[JX,Y]+J[X,JY],
  15. V = ξ k ξ k , J = d x k ξ k . V=\xi^{k}\frac{\partial}{\partial\xi^{k}},\qquad J=dx^{k}\otimes\frac{\partial% }{\partial\xi^{k}}.
  16. T ( T M 0 ) = H ( T M 0 ) V ( T M 0 ) T(TM\setminus 0)=H(TM\setminus 0)\oplus V(TM\setminus 0)
  17. D : ( T M 0 ) × Γ ( T M ) T M ; D X Y := ( κ j ) ( Y * X ) , D:(TM\setminus 0)\times\Gamma(TM)\to TM;\quad D_{X}Y:=(\kappa\circ j)(Y_{*}X),
  18. D X ( α Y + β Z ) = α D X Y + β D X Z , α , β D_{X}(\alpha Y+\beta Z)=\alpha D_{X}Y+\beta D_{X}Z,\qquad\alpha,\beta\in% \mathbb{R}
  19. D X ( f Y ) = X [ f ] Y + f D X Y , f C ( M ) D_{X}(fY)=X[f]Y+fD_{X}Y,\qquad\qquad\qquad f\in C^{\infty}(M)

Dqo_transformation.html

  1. x d q o = K x a b c = 2 3 [ cos ( θ ) cos ( θ - 2 π 3 ) cos ( θ + 2 π 3 ) - sin ( θ ) - sin ( θ - 2 π 3 ) - sin ( θ + 2 π 3 ) 2 2 2 2 2 2 ] [ x a x b x c ] x_{dqo}=Kx_{abc}=\sqrt{\frac{2}{3}}\begin{bmatrix}\cos(\theta)&\cos(\theta-% \frac{2\pi}{3})&\cos(\theta+\frac{2\pi}{3})\\ -\sin(\theta)&-\sin(\theta-\frac{2\pi}{3})&-\sin(\theta+\frac{2\pi}{3})\\ \frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{bmatrix}\cdot% \begin{bmatrix}x_{a}\\ x_{b}\\ x_{c}\end{bmatrix}
  2. x a b c = K - 1 x d q o = 2 3 [ cos ( θ ) - sin ( θ ) 2 2 cos ( θ - 2 π 3 ) - sin ( θ - 2 π 3 ) 2 2 cos ( θ + 2 π 3 ) - sin ( θ + 2 π 3 ) 2 2 ] [ x d x q x o ] x_{abc}=K^{-1}x_{dqo}=\sqrt{\frac{2}{3}}\begin{bmatrix}\cos(\theta)&-\sin(% \theta)&\frac{\sqrt{2}}{2}\\ \cos(\theta-\frac{2\pi}{3})&-\sin(\theta-\frac{2\pi}{3})&\frac{\sqrt{2}}{2}\\ \cos(\theta+\frac{2\pi}{3})&-\sin(\theta+\frac{2\pi}{3})&\frac{\sqrt{2}}{2}% \end{bmatrix}\cdot\begin{bmatrix}x_{d}\\ x_{q}\\ x_{o}\end{bmatrix}
  3. [ 1 0 0 0 cos ( - π / 4 ) sin ( - π / 4 ) 0 - sin ( - π / 4 ) cos ( - π / 4 ) ] \begin{bmatrix}1&0&0\\ 0&\cos(-\pi/4)&\sin(-\pi/4)\\ 0&-\sin(-\pi/4)&\cos(-\pi/4)\end{bmatrix}
  4. [ 1 0 0 0 1 2 - 1 2 0 1 2 1 2 ] \begin{bmatrix}1&0&0\\ 0&\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}
  5. ϕ = cos - 1 2 3 \phi=\cos^{-1}{\sqrt{\frac{2}{3}}}
  6. [ cos ( ϕ ) 0 - sin ( ϕ ) 0 1 0 sin ( ϕ ) 0 cos ( ϕ ) ] \begin{bmatrix}\cos(\phi)&0&-\sin(\phi)\\ 0&1&0\\ \sin(\phi)&0&\cos(\phi)\end{bmatrix}
  7. [ 2 3 0 - 1 3 0 1 0 1 3 0 2 3 ] \begin{bmatrix}\frac{\sqrt{2}}{\sqrt{3}}&0&\frac{-1}{\sqrt{3}}\\ 0&1&0\\ \frac{1}{\sqrt{3}}&0&\frac{\sqrt{2}}{\sqrt{3}}\end{bmatrix}
  8. 2 3 [ 1 - 1 2 - 1 2 0 3 2 - 3 2 1 2 1 2 1 2 ] = [ 2 3 0 - 1 3 0 1 0 1 3 0 2 3 ] [ 1 0 0 0 1 2 - 1 2 0 1 2 1 2 ] \sqrt{\frac{2}{3}}\begin{bmatrix}1&\frac{-1}{2}&\frac{-1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}=\begin{% bmatrix}\frac{\sqrt{2}}{\sqrt{3}}&0&\frac{-1}{\sqrt{3}}\\ 0&1&0\\ \frac{1}{\sqrt{3}}&0&\frac{\sqrt{2}}{\sqrt{3}}\end{bmatrix}\cdot\begin{bmatrix% }1&0&0\\ 0&\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}
  9. [ cos ( θ ) sin ( θ ) 0 - sin ( θ ) cos ( θ ) 0 0 0 1 ] \begin{bmatrix}\cos(\theta)&\sin(\theta)&0\\ -\sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}
  10. 2 3 [ cos ( θ ) cos ( θ - 2 π 3 ) cos ( θ + 2 π 3 ) - sin ( θ ) - sin ( θ - 2 π 3 ) - sin ( θ + 2 π 3 ) 2 2 2 2 2 2 ] = [ cos ( θ ) sin ( θ ) 0 - sin ( θ ) cos ( θ ) 0 0 0 1 ] 2 3 [ 1 - 1 2 - 1 2 0 3 2 - 3 2 1 2 1 2 1 2 ] \sqrt{\frac{2}{3}}\begin{bmatrix}\cos(\theta)&\cos(\theta-\frac{2\pi}{3})&\cos% (\theta+\frac{2\pi}{3})\\ -\sin(\theta)&-\sin(\theta-\frac{2\pi}{3})&-\sin(\theta+\frac{2\pi}{3})\\ \frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{bmatrix}=\begin{% bmatrix}\cos(\theta)&\sin(\theta)&0\\ -\sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}\cdot\sqrt{\frac{2}{3}}\begin{bmatrix}1&\frac{-1}{2}&\frac{-% 1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}
  11. δ \delta
  12. ω \omega
  13. θ = ω t \theta=\omega t
  14. I d I_{d}
  15. I q I_{q}
  16. P = 2 3 [ cos ( θ ) cos ( θ - 2 π 3 ) cos ( θ + 2 π 3 ) - sin ( θ ) - sin ( θ - 2 π 3 ) - sin ( θ + 2 π 3 ) 1 2 1 2 1 2 ] P=\frac{2}{3}\begin{bmatrix}\cos(\theta)&\cos(\theta-\frac{2\pi}{3})&\cos(% \theta+\frac{2\pi}{3})\\ -\sin(\theta)&-\sin(\theta-\frac{2\pi}{3})&-\sin(\theta+\frac{2\pi}{3})\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{bmatrix}
  17. P - 1 = [ cos ( θ ) - sin ( θ ) 1 cos ( θ - 2 π 3 ) - sin ( θ - 2 π 3 ) 1 cos ( θ + 2 π 3 ) - sin ( θ + 2 π 3 ) 1 ] P^{-1}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)&1\\ \cos(\theta-\frac{2\pi}{3})&-\sin(\theta-\frac{2\pi}{3})&1\\ \cos(\theta+\frac{2\pi}{3})&-\sin(\theta+\frac{2\pi}{3})&1\end{bmatrix}
  18. 3 \sqrt{3}
  19. α β γ \alpha\beta\gamma

Drift_current.html

  1. J n = q n μ n E ( A / c m 2 ) J_{n}=qn\mu_{n}E\quad(A/cm^{2})
  2. J p = q p μ p E ( A / c m 2 ) J_{p}=qp\mu_{p}E\quad(A/cm^{2})
  3. μ n \mu_{n}
  4. c m 2 / V s cm^{2}/Vs
  5. μ p \mu_{p}
  6. c m 2 / V s cm^{2}/Vs

Droop_speed_control.html

  1. F = P * N 120 F=\frac{P*N}{120}
  2. Droop % = No load speed - Full load speed No load speed \mathrm{Droop\%}=\frac{\mathrm{No\ load\ speed-Full\ load\ speed}}{\mathrm{No% \ load\ speed}}
  3. S = Δ f N f N S=\frac{\Delta f_{N}}{f_{N}}

Drucker–Prager_yield_criterion.html

  1. c = 2 , ϕ = - 20 c=2,\phi=-20^{\circ}
  2. J 2 = A + B I 1 \sqrt{J_{2}}=A+B~{}I_{1}
  3. I 1 I_{1}
  4. J 2 J_{2}
  5. A , B A,B
  6. σ e = a + b σ m \sigma_{e}=a+b~{}\sigma_{m}
  7. σ e \sigma_{e}
  8. σ m \sigma_{m}
  9. a , b a,b
  10. 1 2 ρ - 3 B ξ = A \tfrac{1}{\sqrt{2}}\rho-\sqrt{3}~{}B\xi=A
  11. 1 6 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] = A + B ( σ 1 + σ 2 + σ 3 ) . \sqrt{\cfrac{1}{6}\left[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2% }+(\sigma_{3}-\sigma_{1})^{2}\right]}=A+B~{}(\sigma_{1}+\sigma_{2}+\sigma_{3})% ~{}.
  12. σ t \sigma_{t}
  13. 1 3 σ t = A + B σ t . \cfrac{1}{\sqrt{3}}~{}\sigma_{t}=A+B~{}\sigma_{t}~{}.
  14. σ c \sigma_{c}
  15. 1 3 σ c = A - B σ c . \cfrac{1}{\sqrt{3}}~{}\sigma_{c}=A-B~{}\sigma_{c}~{}.
  16. A = 2 3 ( σ c σ t σ c + σ t ) ; B = 1 3 ( σ t - σ c σ c + σ t ) . A=\cfrac{2}{\sqrt{3}}~{}\left(\cfrac{\sigma_{c}~{}\sigma_{t}}{\sigma_{c}+% \sigma_{t}}\right)~{};~{}~{}B=\cfrac{1}{\sqrt{3}}~{}\left(\cfrac{\sigma_{t}-% \sigma_{c}}{\sigma_{c}+\sigma_{t}}\right)~{}.
  17. β = σ c σ t = 1 - 3 B 1 + 3 B . \beta=\cfrac{\sigma_{\mathrm{c}}}{\sigma_{\mathrm{t}}}=\cfrac{1-\sqrt{3}~{}B}{% 1+\sqrt{3}~{}B}~{}.
  18. c c
  19. ϕ \phi
  20. A A
  21. B B
  22. A = 6 c cos ϕ 3 ( 3 - sin ϕ ) ; B = 2 sin ϕ 3 ( 3 - sin ϕ ) A=\cfrac{6~{}c~{}\cos\phi}{\sqrt{3}(3-\sin\phi)}~{};~{}~{}B=\cfrac{2~{}\sin% \phi}{\sqrt{3}(3-\sin\phi)}
  23. A = 6 c cos ϕ 3 ( 3 + sin ϕ ) ; B = 2 sin ϕ 3 ( 3 + sin ϕ ) A=\cfrac{6~{}c~{}\cos\phi}{\sqrt{3}(3+\sin\phi)}~{};~{}~{}B=\cfrac{2~{}\sin% \phi}{\sqrt{3}(3+\sin\phi)}
  24. A = 3 c cos ϕ 9 + 3 sin 2 ϕ ; B = sin ϕ 9 + 3 sin 2 ϕ A=\cfrac{3~{}c~{}\cos\phi}{\sqrt{9+3~{}\sin^{2}\phi}}~{};~{}~{}B=\cfrac{\sin% \phi}{\sqrt{9+3~{}\sin^{2}\phi}}
  25. A , B A,B
  26. c , ϕ c,\phi
  27. [ 3 sin ( θ + π 3 ) - sin ϕ cos ( θ + π 3 ) ] ρ - 2 sin ( ϕ ) ξ = 6 c cos ϕ \left[\sqrt{3}~{}\sin\left(\theta+\tfrac{\pi}{3}\right)-\sin\phi\cos\left(% \theta+\tfrac{\pi}{3}\right)\right]\rho-\sqrt{2}\sin(\phi)\xi=\sqrt{6}c\cos\phi
  28. θ = π 3 \theta=\tfrac{\pi}{3}
  29. [ 3 sin 2 π 3 - sin ϕ cos 2 π 3 ] ρ - 2 sin ( ϕ ) ξ = 6 c cos ϕ \left[\sqrt{3}~{}\sin\tfrac{2\pi}{3}-\sin\phi\cos\tfrac{2\pi}{3}\right]\rho-% \sqrt{2}\sin(\phi)\xi=\sqrt{6}c\cos\phi
  30. 1 2 ρ - 2 sin ϕ 3 + sin ϕ ξ = 12 c cos ϕ 3 + sin ϕ ( 1.1 ) \tfrac{1}{\sqrt{2}}\rho-\cfrac{2\sin\phi}{3+\sin\phi}\xi=\cfrac{\sqrt{12}c\cos% \phi}{3+\sin\phi}\qquad\qquad(1.1)
  31. 1 2 ρ - 3 B ξ = A ( 1.2 ) \tfrac{1}{\sqrt{2}}\rho-\sqrt{3}~{}B\xi=A\qquad\qquad(1.2)
  32. A = 12 c cos ϕ 3 + sin ϕ = 6 c cos ϕ 3 ( 3 + sin ϕ ) ; B = 2 sin ϕ 3 ( 3 + sin ϕ ) A=\cfrac{\sqrt{12}c\cos\phi}{3+\sin\phi}=\cfrac{6c\cos\phi}{\sqrt{3}(3+\sin% \phi)}~{};~{}~{}B=\cfrac{2\sin\phi}{\sqrt{3}(3+\sin\phi)}
  33. A , B A,B
  34. c , ϕ c,\phi
  35. θ = 0 \theta=0
  36. A = 6 c cos ϕ 3 ( 3 - sin ϕ ) ; B = 2 sin ϕ 3 ( 3 - sin ϕ ) A=\cfrac{6c\cos\phi}{\sqrt{3}(3-\sin\phi)}~{};~{}~{}B=\cfrac{2\sin\phi}{\sqrt{% 3}(3-\sin\phi)}
  37. π \pi
  38. c = 2 , ϕ = 20 c=2,\phi=20^{\circ}
  39. σ 1 - σ 2 \sigma_{1}-\sigma_{2}
  40. c = 2 , ϕ = 20 c=2,\phi=20^{\circ}
  41. A = ± σ y 3 ; B = 1 3 ( ρ 5 ρ s ) A=\pm\cfrac{\sigma_{y}}{\sqrt{3}}~{};~{}~{}B=\mp\cfrac{1}{\sqrt{3}}~{}\left(% \cfrac{\rho}{5~{}\rho_{s}}\right)
  42. σ y \sigma_{y}
  43. ρ \rho
  44. ρ s \rho_{s}
  45. J 2 = ( A + B I 1 ) 2 = a + b I 1 + c I 1 2 . J_{2}=(A+B~{}I_{1})^{2}=a+b~{}I_{1}+c~{}I_{1}^{2}~{}.
  46. a , b , c a,b,c
  47. a = ( 1 + β 2 ) σ y 2 , b = 0 , c = - β 2 3 a=(1+\beta^{2})~{}\sigma_{y}^{2}~{},~{}~{}b=0~{},~{}~{}c=-\cfrac{\beta^{2}}{3}
  48. β \beta
  49. σ y \sigma_{y}
  50. f := F ( σ 22 - σ 33 ) 2 + G ( σ 33 - σ 11 ) 2 + H ( σ 11 - σ 22 ) 2 + 2 L σ 23 2 + 2 M σ 31 2 + 2 N σ 12 2 + I σ 11 + J σ 22 + K σ 33 - 1 0 \begin{aligned}\displaystyle f:=&\displaystyle\sqrt{F(\sigma_{22}-\sigma_{33})% ^{2}+G(\sigma_{33}-\sigma_{11})^{2}+H(\sigma_{11}-\sigma_{22})^{2}+2L\sigma_{2% 3}^{2}+2M\sigma_{31}^{2}+2N\sigma_{12}^{2}}\\ &\displaystyle+I\sigma_{11}+J\sigma_{22}+K\sigma_{33}-1\leq 0\end{aligned}
  51. F , G , H , L , M , N , I , J , K F,G,H,L,M,N,I,J,K
  52. F = 1 2 [ Σ 2 2 + Σ 3 2 - Σ 1 2 ] ; G = 1 2 [ Σ 3 2 + Σ 1 2 - Σ 2 2 ] ; H = 1 2 [ Σ 1 2 + Σ 2 2 - Σ 3 2 ] L = 1 2 ( σ 23 y ) 2 ; M = 1 2 ( σ 31 y ) 2 ; N = 1 2 ( σ 12 y ) 2 I = σ 1 c - σ 1 t 2 σ 1 c σ 1 t ; J = σ 2 c - σ 2 t 2 σ 2 c σ 2 t ; K = σ 3 c - σ 3 t 2 σ 3 c σ 3 t \begin{aligned}\displaystyle F=&\displaystyle\cfrac{1}{2}\left[\Sigma_{2}^{2}+% \Sigma_{3}^{2}-\Sigma_{1}^{2}\right]~{};~{}~{}G=\cfrac{1}{2}\left[\Sigma_{3}^{% 2}+\Sigma_{1}^{2}-\Sigma_{2}^{2}\right]~{};~{}~{}H=\cfrac{1}{2}\left[\Sigma_{1% }^{2}+\Sigma_{2}^{2}-\Sigma_{3}^{2}\right]\\ \displaystyle L=&\displaystyle\cfrac{1}{2(\sigma_{23}^{y})^{2}}~{};~{}~{}M=% \cfrac{1}{2(\sigma_{31}^{y})^{2}}~{};~{}~{}N=\cfrac{1}{2(\sigma_{12}^{y})^{2}}% \\ \displaystyle I=&\displaystyle\cfrac{\sigma_{1c}-\sigma_{1t}}{2\sigma_{1c}% \sigma_{1t}}~{};~{}~{}J=\cfrac{\sigma_{2c}-\sigma_{2t}}{2\sigma_{2c}\sigma_{2t% }}~{};~{}~{}K=\cfrac{\sigma_{3c}-\sigma_{3t}}{2\sigma_{3c}\sigma_{3t}}\end{aligned}
  53. Σ 1 := σ 1 c + σ 1 t 2 σ 1 c σ 1 t ; Σ 2 := σ 2 c + σ 2 t 2 σ 2 c σ 2 t ; Σ 3 := σ 3 c + σ 3 t 2 σ 3 c σ 3 t \Sigma_{1}:=\cfrac{\sigma_{1c}+\sigma_{1t}}{2\sigma_{1c}\sigma_{1t}}~{};~{}~{}% \Sigma_{2}:=\cfrac{\sigma_{2c}+\sigma_{2t}}{2\sigma_{2c}\sigma_{2t}}~{};~{}~{}% \Sigma_{3}:=\cfrac{\sigma_{3c}+\sigma_{3t}}{2\sigma_{3c}\sigma_{3t}}
  54. σ i c , i = 1 , 2 , 3 \sigma_{ic},i=1,2,3
  55. σ i t , i = 1 , 2 , 3 \sigma_{it},i=1,2,3
  56. σ 23 y , σ 31 y , σ 12 y \sigma_{23}^{y},\sigma_{31}^{y},\sigma_{12}^{y}
  57. σ 1 c , σ 2 c , σ 3 c \sigma_{1c},\sigma_{2c},\sigma_{3c}
  58. σ 1 t , σ 2 t , σ 3 t \sigma_{1t},\sigma_{2t},\sigma_{3t}
  59. I 1 I_{1}
  60. f := J 2 3 - α J 3 2 - k 2 0 f:=J_{2}^{3}-\alpha~{}J_{3}^{2}-k^{2}\leq 0
  61. J 2 J_{2}
  62. J 3 J_{3}
  63. α \alpha
  64. k k
  65. α \alpha
  66. α = 0 \alpha=0
  67. k 2 = σ y 6 27 k^{2}=\cfrac{\sigma_{y}^{6}}{27}
  68. σ y \sigma_{y}
  69. f := ( J 2 0 ) 3 - α ( J 3 0 ) 2 - k 2 0 f:=(J_{2}^{0})^{3}-\alpha~{}(J_{3}^{0})^{2}-k^{2}\leq 0
  70. J 2 0 , J 3 0 J_{2}^{0},J_{3}^{0}
  71. J 2 0 := 1 6 [ a 1 ( σ 22 - σ 33 ) 2 + a 2 ( σ 33 - σ 11 ) 2 + a 3 ( σ 11 - σ 22 ) 2 ] + a 4 σ 23 2 + a 5 σ 31 2 + a 6 σ 12 2 J 3 0 := 1 27 [ ( b 1 + b 2 ) σ 11 3 + ( b 3 + b 4 ) σ 22 3 + { 2 ( b 1 + b 4 ) - ( b 2 + b 3 ) } σ 33 3 ] - 1 9 [ ( b 1 σ 22 + b 2 σ 33 ) σ 11 2 + ( b 3 σ 33 + b 4 σ 11 ) σ 22 2 + { ( b 1 - b 2 + b 4 ) σ 11 + ( b 1 - b 3 + b 4 ) σ 22 } σ 33 2 ] + 2 9 ( b 1 + b 4 ) σ 11 σ 22 σ 33 + 2 b 11 σ 12 σ 23 σ 31 - 1 3 [ { 2 b 9 σ 22 - b 8 σ 33 - ( 2 b 9 - b 8 ) σ 11 } σ 31 2 + { 2 b 10 σ 33 - b 5 σ 22 - ( 2 b 10 - b 5 ) σ 11 } σ 12 2 { ( b 6 + b 7 ) σ 11 - b 6 σ 22 - b 7 σ 33 } σ 23 2 ] \begin{aligned}\displaystyle J_{2}^{0}:=&\displaystyle\cfrac{1}{6}\left[a_{1}(% \sigma_{22}-\sigma_{33})^{2}+a_{2}(\sigma_{33}-\sigma_{11})^{2}+a_{3}(\sigma_{% 11}-\sigma_{22})^{2}\right]+a_{4}\sigma_{23}^{2}+a_{5}\sigma_{31}^{2}+a_{6}% \sigma_{12}^{2}\\ \displaystyle J_{3}^{0}:=&\displaystyle\cfrac{1}{27}\left[(b_{1}+b_{2})\sigma_% {11}^{3}+(b_{3}+b_{4})\sigma_{22}^{3}+\{2(b_{1}+b_{4})-(b_{2}+b_{3})\}\sigma_{% 33}^{3}\right]\\ &\displaystyle-\cfrac{1}{9}\left[(b_{1}\sigma_{22}+b_{2}\sigma_{33})\sigma_{11% }^{2}+(b_{3}\sigma_{33}+b_{4}\sigma_{11})\sigma_{22}^{2}+\{(b_{1}-b_{2}+b_{4})% \sigma_{11}+(b_{1}-b_{3}+b_{4})\sigma_{22}\}\sigma_{33}^{2}\right]\\ &\displaystyle+\cfrac{2}{9}(b_{1}+b_{4})\sigma_{11}\sigma_{22}\sigma_{33}+2b_{% 11}\sigma_{12}\sigma_{23}\sigma_{31}\\ &\displaystyle-\cfrac{1}{3}\left[\{2b_{9}\sigma_{22}-b_{8}\sigma_{33}-(2b_{9}-% b_{8})\sigma_{11}\}\sigma_{31}^{2}+\{2b_{10}\sigma_{33}-b_{5}\sigma_{22}-(2b_{% 10}-b_{5})\sigma_{11}\}\sigma_{12}^{2}\right.\\ &\displaystyle\qquad\qquad\left.\{(b_{6}+b_{7})\sigma_{11}-b_{6}\sigma_{22}-b_% {7}\sigma_{33}\}\sigma_{23}^{2}\right]\end{aligned}
  72. J 2 0 = 1 6 [ ( a 2 + a 3 ) σ 11 2 + ( a 1 + a 3 ) σ 22 2 - 2 a 3 σ 1 σ 2 ] + a 6 σ 12 2 J 3 0 = 1 27 [ ( b 1 + b 2 ) σ 11 3 + ( b 3 + b 4 ) σ 22 3 ] - 1 9 [ b 1 σ 11 + b 4 σ 22 ] σ 11 σ 22 + 1 3 [ b 5 σ 22 + ( 2 b 10 - b 5 ) σ 11 ] σ 12 2 \begin{aligned}\displaystyle J_{2}^{0}=&\displaystyle\cfrac{1}{6}\left[(a_{2}+% a_{3})\sigma_{11}^{2}+(a_{1}+a_{3})\sigma_{22}^{2}-2a_{3}\sigma_{1}\sigma_{2}% \right]+a_{6}\sigma_{12}^{2}\\ \displaystyle J_{3}^{0}=&\displaystyle\cfrac{1}{27}\left[(b_{1}+b_{2})\sigma_{% 11}^{3}+(b_{3}+b_{4})\sigma_{22}^{3}\right]-\cfrac{1}{9}\left[b_{1}\sigma_{11}% +b_{4}\sigma_{22}\right]\sigma_{11}\sigma_{22}+\cfrac{1}{3}\left[b_{5}\sigma_{% 22}+(2b_{10}-b_{5})\sigma_{11}\right]\sigma_{12}^{2}\end{aligned}
  73. a 1 a_{1}
  74. a 2 a_{2}
  75. a 3 a_{3}
  76. a 6 a_{6}
  77. b 1 b_{1}
  78. b 2 b_{2}
  79. b 3 b_{3}
  80. b 4 b_{4}
  81. b 5 b_{5}
  82. b 10 b_{10}
  83. α \alpha
  84. β = α / 3 \beta=\alpha/3
  85. α \alpha

Dual-band_blade_antenna.html

  1. D m o n o = 4 π Ω A , m o n o = 2 D d i p o l e D_{mono}=\frac{4\pi}{\Omega_{A,mono}}=2D_{dipole}
  2. G = ϵ D G=\epsilon D
  3. ϵ \epsilon
  4. Z A , m o n o = 1 2 Z A , D i p o l e Z_{A,mono}=\frac{1}{2}Z_{A,Dipole}
  5. R r , m o n o = 1 2 R r , m o n o R_{r,mono}=\frac{1}{2}R_{r,mono}
  6. E θ s = H θ c E ϕ s = H ϕ c H θ s = - E θ c η o 2 H ϕ s = - E ϕ c η o 2 E_{\theta s}=H_{\theta c}\qquad E_{\phi s}=H_{\phi c}\qquad H_{\theta s}=\,-% \frac{E_{\theta c}}{\eta_{o}^{2}}\qquad H_{\phi s}=\,-\frac{E_{\phi}c}{\eta_{o% }^{2}}
  7. η = μ ϵ \eta=\sqrt{\tfrac{\mu}{\epsilon}}
  8. μ \mu
  9. ϵ \epsilon
  10. < λ 10 <\tfrac{\lambda}{10}
  11. λ 10 \tfrac{\lambda}{10}
  12. λ 10 \tfrac{\lambda}{10}

Dual_impedance.html

  1. Z Z
  2. Z = 1 Z Z^{\prime}=\frac{1}{Z}
  3. Z Z
  4. Y = Z Y=Z
  5. Z = V I Z=\frac{V}{I}
  6. Z = I V Z^{\prime}=\frac{I}{V}
  7. Z Z 0 = Z 0 Z \frac{Z^{\prime}}{Z_{0}}=\frac{Z_{0}}{Z}
  8. R R\,\!
  9. 1 R \frac{1}{R}
  10. 1 G \frac{1}{G}
  11. G G\,\!
  12. i ω L i\omega L\,\!
  13. 1 i ω L \frac{1}{i\omega L}
  14. 1 i ω C \frac{1}{i\omega C}
  15. i ω C i\omega C\,\!
  16. Z 1 + Z 2 Z_{1}+Z_{2}\,\!
  17. 1 Z 1 + Z 2 \frac{1}{Z_{1}+Z_{2}}
  18. Z = Z 1 Z 2 Z 1 + Z 2 Z=\frac{Z_{1}Z_{2}}{Z_{1}+Z_{2}}
  19. 1 Z 1 + 1 Z 2 \frac{1}{Z_{1}}+\frac{1}{Z_{2}}

Dual_speed_focuser.html

  1. ω = ω r 2 ( R + r ) \omega^{\prime}=\frac{\omega r}{2(R+r)}
  2. ω \omega
  3. ω \omega^{\prime}

Duality_(mechanical_engineering).html

  1. σ = E ε ε = 1 E σ \sigma=E\varepsilon\iff\varepsilon=\frac{1}{E}\sigma\,

Dunford–Pettis_property.html

  1. e n , e n = 1. \langle e_{n},e_{n}\rangle=1.
  2. f n , x n = - π π 1 d x = 2 π . \langle f_{n},x_{n}\rangle=\int\limits_{-\pi}^{\pi}1\,{\rm d}x=2\pi.

Dunford–Schwartz_theorem.html

  1. Let T be a linear operator from L 1 to L 1 with T 1 1 and T 1 . Then \,\text{Let }T\,\text{ be a linear operator from }L^{1}\,\text{ to }L^{1}\,% \text{ with }\|T\|_{1}\leq 1\,\text{ and }\|T\|_{\infty}\leq 1\,\text{. Then}
  2. lim n 1 n k = 0 n - 1 T k f \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}T^{k}f
  3. exists almost everywhere for all f L 1 . \,\text{exists almost everywhere for all }f\in L^{1}\,\text{.}
  4. T 1 + ε \|T\|_{\infty}\leq 1+\varepsilon

Durbin_test.html

  1. \cdots
  2. \cdots
  3. \cdots
  4. \cdots
  5. \vdots
  6. \vdots
  7. \vdots
  8. \ddots
  9. \vdots
  10. \cdots
  11. R j = i = 1 b R ( X i j ) R_{j}=\sum_{i=1}^{b}R(X_{ij})
  12. T 2 = T 1 / ( t - 1 ) ( b k - b - T 1 ) / ( b k - b - t + 1 ) T_{2}=\frac{T_{1}/\left(t-1\right)}{\left(bk-b-T_{1}\right)/\left(bk-b-t+1% \right)}
  13. T 1 = t - 1 A - C j = 1 t ( R j 2 - r C ) T_{1}=\frac{t-1}{A-C}\sum_{j=1}^{t}\left(R_{j}^{2}-rC\right)
  14. A = i = 1 b j = 1 j R ( X i j ) 2 A=\sum_{i=1}^{b}\sum_{j=1}^{j}R(X_{ij})^{2}
  15. C = 1 4 b k ( k + 1 ) 2 C=\frac{1}{4}bk\left(k+1\right)^{2}
  16. T 2 > F α , k - 1 , b k - b - t + 1 T_{2}>F_{\alpha,k-1,bk-b-t+1}
  17. | R j - R i | > t 1 - α / 2 , b k - b - t + 1 2 ( A - C ) r b k - k - t + 1 ( 1 - T 1 b ( k - 1 ) ) |R_{j}-R_{i}|>t_{1-\alpha/2,bk-b-t+1}\sqrt{\frac{2\left(A-C\right)r}{bk-k-t+1}% \left(1-\frac{T_{1}}{b\left(k-1\right)}\right)}
  18. χ t - 1 2 \chi_{t-1}^{2}
  19. t - 1 t-1

Dyadics.html

  1. 𝐚 = a 1 𝐢 + a 2 𝐣 + a 3 𝐤 \mathbf{a}=a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}
  2. 𝐛 = b 1 𝐢 + b 2 𝐣 + b 3 𝐤 \mathbf{b}=b_{1}\mathbf{i}+b_{2}\mathbf{j}+b_{3}\mathbf{k}
  3. 𝐚𝐛 = a 1 b 1 𝐢𝐢 + a 1 b 2 𝐢𝐣 + a 1 b 3 𝐢𝐤 + a 2 b 1 𝐣𝐢 + a 2 b 2 𝐣𝐣 + a 2 b 3 𝐣𝐤 + a 3 b 1 𝐤𝐢 + a 3 b 2 𝐤𝐣 + a 3 b 3 𝐤𝐤 \begin{array}[]{llll}\mathbf{ab}=&a_{1}b_{1}\mathbf{ii}&+a_{1}b_{2}\mathbf{ij}% &+a_{1}b_{3}\mathbf{ik}\\ &+a_{2}b_{1}\mathbf{ji}&+a_{2}b_{2}\mathbf{jj}&+a_{2}b_{3}\mathbf{jk}\\ &+a_{3}b_{1}\mathbf{ki}&+a_{3}b_{2}\mathbf{kj}&+a_{3}b_{3}\mathbf{kk}\end{array}
  4. 𝐚𝐛 𝐚 𝐛 𝐚𝐛 T = ( a 1 a 2 a 3 ) ( b 1 b 2 b 3 ) = ( a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 b 3 ) . \mathbf{ab}\equiv\mathbf{a}\otimes\mathbf{b}\equiv\mathbf{ab}^{\mathrm{T}}=% \begin{pmatrix}a_{1}\\ a_{2}\\ a_{3}\end{pmatrix}\begin{pmatrix}b_{1}&b_{2}&b_{3}\end{pmatrix}=\begin{pmatrix% }a_{1}b_{1}&a_{1}b_{2}&a_{1}b_{3}\\ a_{2}b_{1}&a_{2}b_{2}&a_{2}b_{3}\\ a_{3}b_{1}&a_{3}b_{2}&a_{3}b_{3}\end{pmatrix}.
  5. 𝐢 = ( 1 0 0 ) , 𝐣 = ( 0 1 0 ) , 𝐤 = ( 0 0 1 ) \mathbf{i}=\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\mathbf{j}=\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\mathbf{k}=\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}
  6. 𝐢𝐢 = ( 1 0 0 0 0 0 0 0 0 ) , 𝐣𝐢 = ( 0 0 0 1 0 0 0 0 0 ) , 𝐣𝐤 = ( 0 0 0 0 0 1 0 0 0 ) \mathbf{ii}=\begin{pmatrix}1&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix},\cdots\mathbf{ji}=\begin{pmatrix}0&0&0\\ 1&0&0\\ 0&0&0\end{pmatrix},\cdots\mathbf{jk}=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&0&0\end{pmatrix}\cdots
  7. 𝐀 \displaystyle\mathbf{A}
  8. 𝐚 = i = 1 N a i 𝐞 i = a 1 𝐞 1 + a 2 𝐞 2 + a N 𝐞 N \mathbf{a}=\sum_{i=1}^{N}a_{i}\mathbf{e}_{i}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{% e}_{2}+\cdots a_{N}\mathbf{e}_{N}
  9. 𝐛 = j = 1 N b j 𝐞 j = b 1 𝐞 1 + b 2 𝐞 2 + b N 𝐞 N \mathbf{b}=\sum_{j=1}^{N}b_{j}\mathbf{e}_{j}=b_{1}\mathbf{e}_{1}+b_{2}\mathbf{% e}_{2}+\cdots b_{N}\mathbf{e}_{N}
  10. 𝐀 = j = 1 N i = 1 N a i b j 𝐞 i 𝐞 j . \mathbf{A}=\sum_{j=1}^{N}\sum_{i=1}^{N}a_{i}b_{j}{\mathbf{e}}_{i}\mathbf{e}_{j}.
  11. 𝐚𝐛 = 𝐚𝐛 T = ( a 1 a 2 a N ) ( b 1 b 2 b N ) = ( a 1 b 1 a 1 b 2 a 1 b N a 2 b 1 a 2 b 2 a 2 b N a N b 1 a N b 2 a N b N ) . \mathbf{ab}=\mathbf{ab}^{\mathrm{T}}=\begin{pmatrix}a_{1}\\ a_{2}\\ \vdots\\ a_{N}\end{pmatrix}\begin{pmatrix}b_{1}&b_{2}&\cdots&b_{N}\end{pmatrix}=\begin{% pmatrix}a_{1}b_{1}&a_{1}b_{2}&\cdots&a_{1}b_{N}\\ a_{2}b_{1}&a_{2}b_{2}&\cdots&a_{2}b_{N}\\ \vdots&\vdots&\ddots&\vdots\\ a_{N}b_{1}&a_{N}b_{2}&\cdots&a_{N}b_{N}\end{pmatrix}.
  12. 𝐀 = i 𝐚 i 𝐛 i = 𝐚 1 𝐛 1 + 𝐚 2 𝐛 2 + 𝐚 3 𝐛 3 + \mathbf{A}=\sum_{i}\mathbf{a}_{i}\mathbf{b}_{i}=\mathbf{a}_{1}\mathbf{b}_{1}+% \mathbf{a}_{2}\mathbf{b}_{2}+\mathbf{a}_{3}\mathbf{b}_{3}+\cdots
  13. 𝐜 ( 𝐚𝐛 ) = ( 𝐜 𝐚 ) 𝐛 \mathbf{c}\cdot\left(\mathbf{a}\mathbf{b}\right)=\left(\mathbf{c}\cdot\mathbf{% a}\right)\mathbf{b}
  14. ( 𝐚𝐛 ) 𝐜 = 𝐚 ( 𝐛 𝐜 ) \left(\mathbf{a}\mathbf{b}\right)\cdot\mathbf{c}=\mathbf{a}\left(\mathbf{b}% \cdot\mathbf{c}\right)
  15. 𝐜 × ( 𝐚𝐛 ) = ( 𝐜 × 𝐚 ) 𝐛 \mathbf{c}\times\left(\mathbf{ab}\right)=\left(\mathbf{c}\times\mathbf{a}% \right)\mathbf{b}
  16. ( 𝐚𝐛 ) × 𝐜 = 𝐚 ( 𝐛 × 𝐜 ) \left(\mathbf{ab}\right)\times\mathbf{c}=\mathbf{a}\left(\mathbf{b}\times% \mathbf{c}\right)
  17. ( 𝐚𝐛 ) ( 𝐜𝐝 ) = 𝐚 ( 𝐛 𝐜 ) 𝐝 = ( 𝐛 𝐜 ) 𝐚𝐝 \left(\mathbf{a}\mathbf{b}\right)\cdot\left(\mathbf{c}\mathbf{d}\right)=% \mathbf{a}\left(\mathbf{b}\cdot\mathbf{c}\right)\mathbf{d}=\left(\mathbf{b}% \cdot\mathbf{c}\right)\mathbf{a}\mathbf{d}
  18. 𝐚𝐛 : 𝐜𝐝 = ( 𝐚 𝐝 ) ( 𝐛 𝐜 ) \mathbf{ab}\colon\mathbf{cd}=\left(\mathbf{a}\cdot\mathbf{d}\right)\left(% \mathbf{b}\cdot\mathbf{c}\right)
  19. ( 𝐚𝐛 ) : ( 𝐜𝐝 ) = 𝐜 ( 𝐚𝐛 ) 𝐝 = ( 𝐚 𝐜 ) ( 𝐛 𝐝 ) \left(\mathbf{ab}\right):\left(\mathbf{cd}\right)=\mathbf{c}\cdot\left(\mathbf% {ab}\right)\cdot\mathbf{d}=\left(\mathbf{a}\cdot\mathbf{c}\right)\left(\mathbf% {b}\cdot\mathbf{d}\right)
  20. ( 𝐚𝐛 ) × ( 𝐜𝐝 ) = ( 𝐚 𝐜 ) ( 𝐛 × 𝐝 ) \left(\mathbf{ab}\right)\!\!\!\begin{array}[]{c}{}_{\cdot}\\ {}^{\times}\end{array}\!\!\!\left(\mathbf{c}\mathbf{d}\right)=\left(\mathbf{a}% \cdot\mathbf{c}\right)\left(\mathbf{b}\times\mathbf{d}\right)
  21. ( 𝐚𝐛 ) × ( 𝐜𝐝 ) = ( 𝐚 × 𝐜 ) ( 𝐛 𝐝 ) \left(\mathbf{ab}\right)\!\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\cdot}\end{array}\!\!\!\left(\mathbf{cd}\right)=\left(\mathbf{a}\times% \mathbf{c}\right)\left(\mathbf{b}\cdot\mathbf{d}\right)
  22. ( 𝐚𝐛 ) × × ( 𝐜𝐝 ) = ( 𝐚 × 𝐜 ) ( 𝐛 × 𝐝 ) \left(\mathbf{ab}\right)\!\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\times}\end{array}\!\!\!\left(\mathbf{cd}\right)=\left(\mathbf{a}\times% \mathbf{c}\right)\left(\mathbf{b}\times\mathbf{d}\right)
  23. 𝐀 = i 𝐚 i 𝐛 i 𝐁 = i 𝐜 i 𝐝 i \mathbf{A}=\sum_{i}\mathbf{a}_{i}\mathbf{b}_{i}\quad\mathbf{B}=\sum_{i}\mathbf% {c}_{i}\mathbf{d}_{i}
  24. 𝐀 𝐁 = j i ( 𝐛 i 𝐜 j ) 𝐚 i 𝐝 j \mathbf{A}\cdot\mathbf{B}=\sum_{j}\sum_{i}\left(\mathbf{b}_{i}\cdot\mathbf{c}_% {j}\right)\mathbf{a}_{i}\mathbf{d}_{j}
  25. 𝐀 : 𝐁 = j i ( 𝐚 i 𝐝 j ) ( 𝐛 i 𝐜 j ) \mathbf{A}\colon\mathbf{B}=\sum_{j}\sum_{i}\left(\mathbf{a}_{i}\cdot\mathbf{d}% _{j}\right)\left(\mathbf{b}_{i}\cdot\mathbf{c}_{j}\right)
  26. 𝐀 : 𝐁 = j i ( 𝐚 i 𝐜 j ) ( 𝐛 i 𝐝 j ) \mathbf{A}\colon\mathbf{B}=\sum_{j}\sum_{i}\left(\mathbf{a}_{i}\cdot\mathbf{c}% _{j}\right)\left(\mathbf{b}_{i}\cdot\mathbf{d}_{j}\right)
  27. 𝐀 × 𝐁 = j i ( 𝐚 i 𝐜 j ) ( 𝐛 i × 𝐝 j ) \mathbf{A}\!\!\!\begin{array}[]{c}{}_{\cdot}\\ {}^{\times}\end{array}\!\!\!\mathbf{B}=\sum_{j}\sum_{i}\left(\mathbf{a}_{i}% \cdot\mathbf{c}_{j}\right)\left(\mathbf{b}_{i}\times\mathbf{d}_{j}\right)
  28. 𝐀 × 𝐁 = j i ( 𝐚 i × 𝐜 j ) ( 𝐛 i 𝐝 j ) \mathbf{A}\!\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\cdot}\end{array}\!\!\!\mathbf{B}=\sum_{j}\sum_{i}\left(\mathbf{a}_{i}% \times\mathbf{c}_{j}\right)\left(\mathbf{b}_{i}\cdot\mathbf{d}_{j}\right)
  29. 𝐀 × × 𝐁 = i , j ( 𝐚 i × 𝐜 j ) ( 𝐛 i × 𝐝 j ) \mathbf{A}\!\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\times}\end{array}\!\!\!\mathbf{B}=\sum_{i,j}\left(\mathbf{a}_{i}\times% \mathbf{c}_{j}\right)\left(\mathbf{b}_{i}\times\mathbf{d}_{j}\right)
  30. 𝐀 : 𝐁 = 𝐁 : 𝐀 \mathbf{A}\colon\!\mathbf{B}=\mathbf{B}\colon\!\mathbf{A}
  31. 𝐀 : 𝐁 T = 𝐀 T : 𝐁 \mathbf{A}\colon\!\mathbf{B}^{\mathrm{T}}=\mathbf{A}^{\mathrm{T}}\colon\!% \mathbf{B}
  32. 𝐀 : 𝐁 = ( 𝐀 𝐁 T ) : 𝐈 = ( 𝐁 𝐀 T ) : 𝐈 \mathbf{A}\colon\mathbf{B}=\left(\mathbf{A}\cdot\mathbf{B}^{\mathrm{T}}\right)% \colon\mathbf{I}=\left(\mathbf{B}\cdot\mathbf{A}^{\mathrm{T}}\right)\colon% \mathbf{I}
  33. ( 𝐚𝐛 ) × × ( 𝐚𝐛 ) = ( 𝐚 × 𝐚 ) ( 𝐛 × 𝐛 ) = 0 \left(\mathbf{ab}\right)\!\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\times}\end{array}\!\!\!\left(\mathbf{ab}\right)=\left(\mathbf{a}\times% \mathbf{a}\right)\left(\mathbf{b}\times\mathbf{b}\right)=0
  34. 𝐀 = i = 1 3 𝐚 i 𝐛 i \mathbf{A}=\sum_{i=1}^{3}\mathbf{a}_{i}\mathbf{b}_{i}
  35. 𝐀 × × 𝐀 = 2 [ ( 𝐚 1 × 𝐚 2 ) ( 𝐛 1 × 𝐛 2 ) + ( 𝐚 2 × 𝐚 3 ) ( 𝐛 2 × 𝐛 3 ) + ( 𝐚 3 × 𝐚 1 ) ( 𝐛 3 × 𝐛 1 ) ] \mathbf{A}\!\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\times}\end{array}\!\!\!\mathbf{A}=2\left[\left(\mathbf{a}_{1}\times% \mathbf{a}_{2}\right)\left(\mathbf{b}_{1}\times\mathbf{b}_{2}\right)+\left(% \mathbf{a}_{2}\times\mathbf{a}_{3}\right)\left(\mathbf{b}_{2}\times\mathbf{b}_% {3}\right)+\left(\mathbf{a}_{3}\times\mathbf{a}_{1}\right)\left(\mathbf{b}_{3}% \times\mathbf{b}_{1}\right)\right]
  36. | 𝐀 | = A 11 𝐢 𝐢 + A 12 𝐢 𝐣 + A 31 𝐢 𝐤 + A 21 𝐣 𝐢 + A 22 𝐣 𝐣 + A 23 𝐣 𝐤 + A 31 𝐤 𝐢 + A 32 𝐤 𝐣 + A 33 𝐤 𝐤 = A 11 + A 22 + A 33 \begin{array}[]{llll}|\mathbf{A}|&=A_{11}\mathbf{i}\cdot\mathbf{i}+A_{12}% \mathbf{i}\cdot\mathbf{j}+A_{31}\mathbf{i}\cdot\mathbf{k}\\ &+A_{21}\mathbf{j}\cdot\mathbf{i}+A_{22}\mathbf{j}\cdot\mathbf{j}+A_{23}% \mathbf{j}\cdot\mathbf{k}\\ &+A_{31}\mathbf{k}\cdot\mathbf{i}+A_{32}\mathbf{k}\cdot\mathbf{j}+A_{33}% \mathbf{k}\cdot\mathbf{k}\\ \\ &=A_{11}+A_{22}+A_{33}\\ \end{array}
  37. | 𝐀 | = i A i i |\mathbf{A}|=\sum_{i}A_{i}{}^{i}
  38. 𝐀 = A 11 𝐢 × 𝐢 + A 12 𝐢 × 𝐣 + A 31 𝐢 × 𝐤 + A 21 𝐣 × 𝐢 + A 22 𝐣 × 𝐣 + A 23 𝐣 × 𝐤 + A 31 𝐤 × 𝐢 + A 32 𝐤 × 𝐣 + A 33 𝐤 × 𝐤 = A 12 𝐤 - A 31 𝐣 - A 21 𝐤 + A 23 𝐢 + A 31 𝐣 - A 32 𝐢 = ( A 23 - A 32 ) 𝐢 + ( A 31 - A 13 ) 𝐣 + ( A 12 - A 21 ) 𝐤 \begin{array}[]{llll}\langle\mathbf{A}\rangle&=A_{11}\mathbf{i}\times\mathbf{i% }+A_{12}\mathbf{i}\times\mathbf{j}+A_{31}\mathbf{i}\times\mathbf{k}\\ &+A_{21}\mathbf{j}\times\mathbf{i}+A_{22}\mathbf{j}\times\mathbf{j}+A_{23}% \mathbf{j}\times\mathbf{k}\\ &+A_{31}\mathbf{k}\times\mathbf{i}+A_{32}\mathbf{k}\times\mathbf{j}+A_{33}% \mathbf{k}\times\mathbf{k}\\ \\ &=A_{12}\mathbf{k}-A_{31}\mathbf{j}-A_{21}\mathbf{k}\\ &+A_{23}\mathbf{i}+A_{31}\mathbf{j}-A_{32}\mathbf{i}\\ \\ &=(A_{23}-A_{32})\mathbf{i}+(A_{31}-A_{13})\mathbf{j}+(A_{12}-A_{21})\mathbf{k% }\\ \end{array}
  39. 𝐀 = j k ϵ i j k A j k . \langle\mathbf{A}\rangle=\sum_{jk}{\epsilon_{i}}^{jk}A_{jk}.
  40. 𝐈 𝐚 = 𝐚 𝐈 = 𝐚 \mathbf{I}\cdot\mathbf{a}=\mathbf{a}\cdot\mathbf{I}=\mathbf{a}
  41. 𝐚 ^ , 𝐛 ^ , 𝐜 ^ \hat{{\mathbf{a}}},\hat{\mathbf{b}},\hat{\mathbf{c}}
  42. 𝐈 = 𝐚 𝐚 ^ + 𝐛 𝐛 ^ + 𝐜 𝐜 ^ \mathbf{I}=\mathbf{a}\hat{\mathbf{a}}+\mathbf{b}\hat{\mathbf{b}}+\mathbf{c}% \hat{\mathbf{c}}
  43. 𝐈 = 𝐢𝐢 + 𝐣𝐣 + 𝐤𝐤 \mathbf{I}=\mathbf{ii}+\mathbf{jj}+\mathbf{kk}
  44. 𝐈 = ( 1 0 0 0 1 0 0 0 1 ) \mathbf{I}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}
  45. ( 𝐚 × 𝐈 ) ( 𝐛 × 𝐈 ) = 𝐚𝐛 - ( 𝐚 𝐛 ) 𝐈 \left(\mathbf{a}\times\mathbf{I}\right)\cdot\left(\mathbf{b}\times\mathbf{I}% \right)=\mathbf{ab}-\left(\mathbf{a}\cdot\mathbf{b}\right)\mathbf{I}
  46. 𝐈 × ( 𝐚𝐛 ) = 𝐛 × 𝐚 \mathbf{I}\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\cdot}\end{array}\!\!\!\left(\mathbf{ab}\right)=\mathbf{b}\times\mathbf{a}
  47. 𝐈 × × 𝐀 = ( 𝐀 × × 𝐈 ) 𝐈 - 𝐀 T \mathbf{I}\!\!\begin{array}[]{c}{}_{\times}\\ {}^{\times}\end{array}\!\!\mathbf{A}=(\mathbf{A}\!\!\begin{array}[]{c}{}_{% \times}\\ {}^{\times}\end{array}\!\!\mathbf{I})\mathbf{I}-\mathbf{A}^{\mathrm{T}}
  48. 𝐈 : ( 𝐚𝐛 ) = ( 𝐈 𝐚 ) 𝐛 = 𝐚 𝐛 = tr ( 𝐚𝐛 ) \mathbf{I}\;\colon\left(\mathbf{ab}\right)=\left(\mathbf{I}\cdot\mathbf{a}% \right)\cdot\mathbf{b}=\mathbf{a}\cdot\mathbf{b}=\mathrm{tr}\left(\mathbf{ab}\right)
  49. 𝐚 × 𝐈 \mathbf{a}\times\mathbf{I}
  50. ( 0 - 1 1 0 ) \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  51. ( 𝐣𝐢 - 𝐢𝐣 ) ( x 𝐢 + y 𝐣 ) = x 𝐣𝐢 𝐢 - x 𝐢𝐣 𝐢 + y 𝐣𝐢 𝐣 - y 𝐢𝐣 𝐣 = - y 𝐢 + x 𝐣 , (\mathbf{ji}-\mathbf{ij})\cdot(x\mathbf{i}+y\mathbf{j})=x\mathbf{ji}\cdot% \mathbf{i}-x\mathbf{ij}\cdot\mathbf{i}+y\mathbf{ji}\cdot\mathbf{j}-y\mathbf{ij% }\cdot\mathbf{j}=-y\mathbf{i}+x\mathbf{j},
  52. ( 0 - 1 1 0 ) ( x y ) = ( - y x ) . \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}-y\\ x\end{pmatrix}.
  53. 𝐈 cos θ + 𝐉 sin θ = ( cos θ - sin θ sin θ cos θ ) \mathbf{I}\cos\theta+\mathbf{J}\sin\theta=\begin{pmatrix}\cos\theta&-\sin% \theta\\ \sin\theta&\;\cos\theta\end{pmatrix}

Dynamic_mode_decomposition.html

  1. e Δ t A e^{\Delta tA}
  2. q ( t + Δ t ) e Δ t A q ( t ) {q(t+\Delta t)}\approx e^{\Delta tA}q(t)
  3. A ~ := e Δ t A \tilde{A}:=e^{\Delta tA}
  4. V 0 n = { q 0 , q 1 , q 2 , , q n } V_{0\dots n}=\{q_{0},q_{1},q_{2},\dots,q_{n}\}
  5. V 1 n + 1 = A ~ V 0 n V_{1\dots n+1}=\tilde{A}V_{0\dots n}
  6. { q 0 , q 1 , q 2 , , q n } \{q_{0},q_{1},q_{2},\dots,q_{n}\}
  7. A ~ \tilde{A}
  8. A ~ \tilde{A}
  9. { q 0 , q 1 , q 2 , , q n } \{q_{0},q_{1},q_{2},\dots,q_{n}\}
  10. n n
  11. n n
  12. q n + 1 q_{n+1}
  13. q n + 1 = c 0 q 0 + + c n q n = { q 0 , q 1 , q 2 , , q n } c q_{n+1}=c_{0}q_{0}+\cdots+c_{n}q_{n}=\{q_{0},q_{1},q_{2},\dots,q_{n}\}c
  14. V 1 n + 1 = V 0 n S V_{1\dots n+1}=V_{0\dots n}S
  15. S = ( 0 0 0 c 0 1 0 0 c 1 0 1 0 c 2 0 0 1 c n ) . S=\begin{pmatrix}0&0&\dots&0&c_{0}\\ 1&0&\dots&0&c_{1}\\ 0&1&\dots&0&c_{2}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\dots&1&c_{n}\end{pmatrix}.
  16. A ~ \tilde{A}
  17. A ~ \tilde{A}
  18. λ r \lambda_{r}
  19. q ( x 1 , x 2 , x 3 , ) = e c x 1 q ^ ( x 2 , x 3 , ) q(x_{1},x_{2},x_{3},\ldots)=e^{cx_{1}}\hat{q}(x_{2},x_{3},\ldots)
  20. x 1 x_{1}
  21. q ( x , y , t ) = e - i ω t q ^ ( x , t ) e - ( y / b ) 2 { e i ( k x - ω t ) } + random noise q(x,y,t)=e^{-i\omega t}\hat{q}(x,t)e^{-(y/b)^{2}}\Re\left\{e^{i(kx-\omega t)}% \right\}+\,\text{random noise}
  22. ω = 2 π / 0.1 \omega=2\pi/0.1
  23. b = 0.02 b=0.02
  24. k = 2 π / b k=2\pi/b
  25. Δ t = 1 / 90 s \Delta t=1/90\,\text{ s}
  26. f = 45 Hz f=45\,\text{ Hz}
  27. ω 1 = - 0.201 , ω 2 / 3 = - 0.223 ± i 62.768 \omega_{1}=-0.201,\omega_{2/3}=-0.223\pm i62.768
  28. ω 2 / 3 \omega_{2/3}
  29. f = 10 Hz f=10\,\text{ Hz}

Dynamic_rectangle.html

  1. φ \sqrt{φ}
  2. 2 \sqrt{2}
  3. 3 \sqrt{3}
  4. 5 \sqrt{5}
  5. 1 , 2 , 3 , 4 , 5 \scriptstyle\sqrt{1},\sqrt{2},\sqrt{3},\sqrt{4},\sqrt{5}
  6. 2 \sqrt{2}
  7. 3 \sqrt{3}
  8. 1 \sqrt{1}
  9. 2 \sqrt{2}
  10. 3 \sqrt{3}
  11. 4 \sqrt{4}
  12. 5 \sqrt{5}
  13. φ \sqrt{φ}
  14. 3 \sqrt{3}
  15. φ \sqrt{φ}
  16. 5 \sqrt{5}
  17. 5 \sqrt{5}

Dynamic_relaxation.html

  1. x x
  2. i i
  3. t t
  4. R i x ( t ) = M i A i x ( t ) R_{ix}(t)=M_{i}A_{ix}(t)\frac{}{}
  5. R R
  6. M M
  7. A A
  8. V V
  9. X X
  10. V i x ( t + Δ t 2 ) = V i x ( t - Δ t 2 ) + Δ t M i R i x ( t ) V_{ix}\left(t+\frac{\Delta t}{2}\right)=V_{ix}\left(t-\frac{\Delta t}{2}\right% )+\frac{\Delta t}{M_{i}}R_{ix}(t)
  11. X i ( t + Δ t ) = X i ( t ) + Δ t × V i x ( t + Δ t 2 ) X_{i}(t+\Delta t)=X_{i}(t)+\Delta t\times V_{ix}\left(t+\frac{\Delta t}{2}\right)
  12. Δ t \Delta t
  13. R i x ( t + Δ t ) = P i x ( t + Δ t ) + T m ( t + Δ t ) l m ( t + Δ t ) × ( X j ( t + Δ t ) - X i ( t + Δ t ) ) R_{ix}(t+\Delta t)=P_{ix}(t+\Delta t)+\sum\frac{T_{m}(t+\Delta t)}{l_{m}(t+% \Delta t)}\times(X_{j}(t+\Delta t)-X_{i}(t+\Delta t))
  14. P P
  15. T T
  16. m m
  17. i i
  18. j j
  19. l l
  20. E k ( t = 0 ) = 0 E_{k}(t=0)=0\frac{}{}
  21. V i ( t = 0 ) = 0 V_{i}(t=0)=0\frac{}{}
  22. X i ( t = 0 ) X_{i}(t=0)\frac{}{}
  23. P i ( t = 0 ) P_{i}(t=0)\frac{}{}
  24. T m ( t ) T_{m}(t)\frac{}{}
  25. R i ( t ) R_{i}(t)\frac{}{}
  26. V i ( t + Δ t 2 ) V_{i}(t+\frac{\Delta t}{2})\frac{}{}
  27. X i ( t + Δ t ) X_{i}(t+\Delta t)\frac{}{}

Dynamic_response_index.html

  1. d 2 X d t 2 + 2 ζ ω d X d t + ω 2 X = d 2 z d t 2 {d^{2}X\over dt^{2}}+2\cdot\zeta\cdot\omega\cdot{dX\over dt}+\omega^{2}\cdot X% ={d^{2}z\over dt^{2}}

Dyson_conjecture.html

  1. 1 i j n ( 1 - t i / t j ) a i \prod_{1\leq i\neq j\leq n}(1-t_{i}/t_{j})^{a_{i}}
  2. ( a 1 + a 2 + + a n ) ! a 1 ! a 2 ! a n ! . \frac{(a_{1}+a_{2}+\cdots+a_{n})!}{a_{1}!a_{2}!\cdots a_{n}!}.
  3. F ( a 1 , , a n ) = i = 1 n F ( a 1 , , a i - 1 , , a n ) . F(a_{1},\dots,a_{n})=\sum_{i=1}^{n}F(a_{1},\dots,a_{i}-1,\dots,a_{n}).
  4. 1 ( 2 π ) n 0 2 π 0 2 π 1 j < k n | e i θ j - e i θ k | β d θ 1 d θ n . \frac{1}{(2\pi)^{n}}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\prod_{1\leq j<k\leq n% }|e^{i\theta_{j}}-e^{i\theta_{k}}|^{\beta}\,d\theta_{1}\cdots d\theta_{n}.
  5. Γ ( 1 + β n / 2 ) Γ ( 1 + β / 2 ) n \frac{\Gamma(1+\beta n/2)}{\Gamma(1+\beta/2)^{n}}
  6. 1 i < j n ( x i x j ; q ) a i ( q x j x i ; q ) a j \prod_{1\leq i<j\leq n}\left(\frac{x_{i}}{x_{j}};q\right)_{a_{i}}\left(\frac{% qx_{j}}{x_{i}};q\right)_{a_{j}}
  7. ( q ; q ) a 1 + + a n ( q ; q ) a 1 ( q ; q ) a n . \frac{(q;q)_{a_{1}+\cdots+a_{n}}}{(q;q)_{a_{1}}\cdots(q;q)_{a_{n}}}.

Earth-centered_inertial.html

  1. ϵ \epsilon
  2. ϵ 23.4 \epsilon\approx 23.4^{\circ}

Earth_ellipsoid.html

  1. a a
  2. b b
  3. 1 / f 1/f
  4. f = ( a - b ) / a f=(a-b)/a
  5. a a
  6. G M GM
  7. J 2 J_{2}
  8. ω \omega
  9. 1 / f 1/f
  10. 1 / f 1/f

Effective_exchange_rate.html

  1. N N
  2. T r a d e i Trade_{i}
  3. E i E_{i}
  4. i i
  5. E e f f e c t i v e = E 1 T r a d e 1 T r a d e + + E N T r a d e N T r a d e E_{effective}=E_{1}\frac{Trade_{1}}{Trade}+...+E_{N}\frac{Trade_{N}}{Trade}

Effective_number_of_parties.html

  1. N = 1 i = 1 n p i 2 N=\frac{1}{\sum_{i=1}^{n}p_{i}^{2}}
  2. p i 2 p_{i}^{2}
  3. N = i = 1 n p i p i + p 1 2 - p i 2 N=\sum_{i=1}^{n}\frac{p_{i}}{p_{i}+p_{1}^{2}-p_{i}^{2}}
  4. N = i = 1 n 1 1 + ( p 1 2 / p i ) - p i N=\sum_{i=1}^{n}\frac{1}{1+(p_{1}^{2}/p_{i})-p_{i}}
  5. p i 2 p_{i}^{2}
  6. p 1 2 p_{1}^{2}

Efficiency_of_conversion.html

  1. E C I = A D × E C D ECI=AD\times ECD\!

Ehrenfest_model.html

  1. q i , i - 1 = i λ q_{i,i-1}=i\,\lambda
  2. q i , i + 1 = ( N - i ) λ q_{i,i+1}=(N-i\,)\lambda
  3. π i = 2 - N ( N i ) \pi_{i}=2^{-N}{\textstyle\left({{N}\atop{i}}\right)}
  4. H ( t ) = - i P ( X ( t ) = i ) log ( P ( X ( t ) = i ) π i ) , H(t)=-\sum_{i}P(X(t)=i)\log\left(\frac{P(X(t)=i)}{\pi_{i}}\right),

Eigenvector_slew.html

  1. x ^ , y ^ , z ^ \hat{x}\ ,\ \hat{y}\ ,\ \hat{z}
  2. x ^ = a ^ \hat{x}=\hat{a}
  3. y ^ = b ^ \hat{y}=\hat{b}
  4. z ^ = c ^ . \hat{z}=\hat{c}.
  5. r ^ = r x x ^ + r y y ^ + r z z ^ \hat{r}=r_{x}\cdot\hat{x}+r_{y}\cdot\hat{y}+r_{z}\cdot\hat{z}
  6. α \alpha
  7. α \alpha
  8. x ^ = d ^ \hat{x}=\hat{d}
  9. y ^ = e ^ \hat{y}=\hat{e}
  10. z ^ = f ^ \hat{z}=\hat{f}
  11. d ^ , e ^ , f ^ \hat{d}\ ,\ \hat{e}\ ,\ \hat{f}
  12. d ^ = r a r ^ + cos α ( a ^ - r a r ^ ) + sin α r ^ × a ^ \hat{d}=r_{a}\cdot\hat{r}+\cos\alpha\cdot(\hat{a}-r_{a}\cdot\hat{r})+\sin% \alpha\cdot\hat{r}\times\hat{a}
  13. e ^ = r b r ^ + cos α ( b ^ - r b r ^ ) + sin α r ^ × b ^ \hat{e}=r_{b}\cdot\hat{r}+\cos\alpha\cdot(\hat{b}-r_{b}\cdot\hat{r})+\sin% \alpha\cdot\hat{r}\times\hat{b}
  14. f ^ = r c r ^ + cos α ( c ^ - r c r ^ ) + sin α r ^ × c ^ . \hat{f}=r_{c}\cdot\hat{r}+\cos\alpha\cdot(\hat{c}-r_{c}\cdot\hat{r})+\sin% \alpha\cdot\hat{r}\times\hat{c}.
  15. a ^ d ^ \hat{a}\longrightarrow\hat{d}
  16. b ^ e ^ \hat{b}\longrightarrow\hat{e}
  17. c ^ f ^ \hat{c}\longrightarrow\hat{f}
  18. a ^ , b ^ , c ^ \hat{a}\ ,\ \hat{b}\ ,\ \hat{c}
  19. [ d ^ | a ^ e ^ | a ^ f ^ | a ^ d ^ | b ^ e ^ | b ^ f ^ | b ^ d ^ | c ^ e ^ | c ^ f ^ | c ^ ] \begin{bmatrix}\langle\hat{d}|\hat{a}\rangle&\langle\hat{e}|\hat{a}\rangle&% \langle\hat{f}|\hat{a}\rangle\\ \langle\hat{d}|\hat{b}\rangle&\langle\hat{e}|\hat{b}\rangle&\langle\hat{f}|% \hat{b}\rangle\\ \langle\hat{d}|\hat{c}\rangle&\langle\hat{e}|\hat{c}\rangle&\langle\hat{f}|% \hat{c}\rangle\end{bmatrix}
  20. a ^ , b ^ , c ^ \hat{a}\ ,\ \hat{b}\ ,\ \hat{c}
  21. cos α = d ^ | a ^ + e ^ | b ^ + f ^ | c ^ - 1 2 \cos\alpha=\frac{\langle\hat{d}|\hat{a}\rangle+\langle\hat{e}|\hat{b}\rangle+% \langle\hat{f}|\hat{c}\rangle-1}{2}
  22. r a = f ^ | b ^ - e ^ | c ^ r_{a}=\langle\hat{f}|\hat{b}\rangle-\langle\hat{e}|\hat{c}\rangle
  23. r b = d ^ | c ^ - f ^ | a ^ r_{b}=\langle\hat{d}|\hat{c}\rangle-\langle\hat{f}|\hat{a}\rangle
  24. r c = e ^ | a ^ - d ^ | b ^ r_{c}=\langle\hat{e}|\hat{a}\rangle-\langle\hat{d}|\hat{b}\rangle
  25. | r ¯ | = r a 2 + r b 2 + r c 2 |\bar{r}|=\sqrt{{r_{a}}^{2}+{r_{b}}^{2}+{r_{c}}^{2}}
  26. sin α = | r ¯ | 2 \sin\alpha=\frac{|\bar{r}|}{2}
  27. α \alpha
  28. α = arg ( cos α , sin α ) \alpha=\operatorname{arg}(\cos\alpha,\sin\alpha)
  29. arg ( x , y ) \operatorname{arg}(x\ ,\ y)
  30. ( x , y ) (\ x\ ,\ y\ )
  31. α \alpha
  32. 0 α π 0\leq\alpha\leq\pi
  33. 0 < α < π 0<\alpha<\pi
  34. | r ¯ | > 0 |\bar{r}|>0
  35. r ^ = r ¯ | r ¯ | \hat{r}=\frac{\bar{r}}{|\bar{r}|}
  36. d ^ | a ^ + e ^ | b ^ + f ^ | c ^ \langle\hat{d}|\hat{a}\rangle+\langle\hat{e}|\hat{b}\rangle+\langle\hat{f}|% \hat{c}\rangle
  37. r ^ = r x x ^ ( t ) + r y y ^ ( t ) + r z z ^ ( t ) = r x a ^ + r y b ^ + r z c ^ = r x d ^ + r y e ^ + r z f ^ \hat{r}=r_{x}\cdot\hat{x}(t)+r_{y}\cdot\hat{y}(t)+r_{z}\cdot\hat{z}(t)=r_{x}% \cdot\hat{a}+r_{y}\cdot\hat{b}+r_{z}\cdot\hat{c}=r_{x}\cdot\hat{d}+r_{y}\cdot% \hat{e}+r_{z}\cdot\hat{f}
  38. x ^ , y ^ , z ^ \hat{x}\ ,\ \hat{y}\ ,\ \hat{z}
  39. t t

Eight-vertex_model.html

  1. N × N N\times N
  2. N 2 N^{2}
  3. 2 N 2 2N^{2}
  4. j j
  5. ϵ j \epsilon_{j}
  6. w j = e - ϵ j k T w_{j}=e^{-\frac{\epsilon_{j}}{kT}}
  7. Z = exp ( - j n j ϵ j k T ) Z=\sum\exp\left(-\frac{\sum_{j}n_{j}\epsilon_{j}}{kT}\right)
  8. w 1 = w 2 = a w 3 = w 4 = b w 5 = w 6 = c w 7 = w 8 = d . \begin{aligned}\displaystyle w_{1}=w_{2}&\displaystyle=a\\ \displaystyle w_{3}=w_{4}&\displaystyle=b\\ \displaystyle w_{5}=w_{6}&\displaystyle=c\\ \displaystyle w_{7}=w_{8}&\displaystyle=d.\end{aligned}
  9. Δ = Δ \Delta^{\prime}=\Delta
  10. Γ = Γ \Gamma^{\prime}=\Gamma
  11. Δ = a 2 + b 2 - c 2 - d 2 2 ( a b + c d ) Γ = a b - c d a b + c d \begin{aligned}\displaystyle\Delta&\displaystyle=\frac{a^{2}+b^{2}-c^{2}-d^{2}% }{2(ab+cd)}\\ \displaystyle\Gamma&\displaystyle=\frac{ab-cd}{ab+cd}\end{aligned}
  12. T T
  13. T T^{\prime}
  14. a a
  15. b b
  16. c c
  17. d d
  18. a a^{\prime}
  19. b b^{\prime}
  20. c c^{\prime}
  21. d d^{\prime}
  22. a : b : c : d = snh ( η - u ) : snh ( η + u ) : snh ( 2 η ) : k snh ( 2 η ) snh ( η - u ) snh ( η + u ) a:b:c:d=\operatorname{snh}(\eta-u):\operatorname{snh}(\eta+u):\operatorname{% snh}(2\eta):k\operatorname{snh}(2\eta)\operatorname{snh}(\eta-u)\operatorname{% snh}(\eta+u)
  23. k k
  24. η \eta
  25. u u
  26. snh ( u ) = - i snh ( i u ) where snh ( u ) = H ( u ) k 1 / 2 Θ ( u ) \begin{aligned}\displaystyle\operatorname{snh}(u)&\displaystyle=-i% \operatorname{snh}(iu)\\ \displaystyle\,\text{where }\operatorname{snh}(u)&\displaystyle=\frac{H(u)}{k^% {1/2}\Theta(u)}\end{aligned}
  27. H ( u ) H(u)
  28. Θ ( u ) \Theta(u)
  29. k k
  30. T T
  31. u u
  32. u u
  33. v v
  34. T ( u ) T ( v ) = T ( v ) T ( u ) . T(u)T(v)=T(v)T(u).
  35. Q ( u ) Q(u)
  36. Q Q
  37. u u
  38. Q ( u ) , Q ( u ) Q(u),Q(u^{\prime})
  39. ζ ( u ) = [ c - 1 H ( 2 η ) Θ ( u - η ) Θ ( u + η ) ] N ϕ ( u ) = [ Θ ( 0 ) H ( u ) Θ ( u ) ] N . \begin{aligned}\displaystyle\zeta(u)&\displaystyle=[c^{-1}H(2\eta)\Theta(u-% \eta)\Theta(u+\eta)]^{N}\\ \displaystyle\phi(u)&\displaystyle=[\Theta(0)H(u)\Theta(u)]^{N}.\end{aligned}
  40. f = ϵ 5 - 2 k T n = 1 sinh 2 ( ( τ - λ ) n ) ( cosh ( n λ ) - cosh ( n α ) ) n sinh ( 2 n τ ) cosh ( n λ ) \begin{aligned}\displaystyle f=\epsilon_{5}-2kT\sum_{n=1}^{\infty}\frac{\sinh^% {2}((\tau-\lambda)n)(\cosh(n\lambda)-\cosh(n\alpha))}{n\sinh(2n\tau)\cosh(n% \lambda)}\end{aligned}
  41. τ = π K 2 K λ = π η i K α = π u i K \begin{aligned}\displaystyle\tau&\displaystyle=\frac{\pi K^{\prime}}{2K}\\ \displaystyle\lambda&\displaystyle=\frac{\pi\eta}{iK}\\ \displaystyle\alpha&\displaystyle=\frac{\pi u}{iK}\end{aligned}
  42. K K
  43. K K^{\prime}
  44. k k
  45. k k^{\prime}
  46. σ = ± 1 \sigma=\pm 1
  47. α i j \displaystyle\alpha_{ij}
  48. ϵ = - i j ( J h μ i j + J v α i j + J α i j μ i j + J α i + 1 , j μ i j + J ′′ α i j α i + 1 , j ) \begin{aligned}\displaystyle\epsilon&\displaystyle=-\sum_{ij}(J_{h}\mu_{ij}+J_% {v}\alpha_{ij}+J\alpha_{ij}\mu_{ij}+J^{\prime}\alpha_{i+1,j}\mu_{ij}+J^{\prime% \prime}\alpha_{ij}\alpha_{i+1,j})\end{aligned}
  49. J h J_{h}
  50. J v J_{v}
  51. J J
  52. J J^{\prime}
  53. J ′′ J^{\prime\prime}
  54. μ \mu
  55. α \alpha
  56. σ \sigma
  57. μ \mu
  58. α \alpha
  59. μ \mu
  60. α \alpha
  61. σ \sigma
  62. j j
  63. ϵ j \epsilon_{j}
  64. J h J_{h}
  65. J v J_{v}
  66. J J
  67. J J^{\prime}
  68. J ′′ J^{\prime\prime}
  69. ϵ 1 \displaystyle\epsilon_{1}
  70. Z I = 2 Z 8 V Z_{I}=2Z_{8V}

Einasto_profile.html

  1. ρ \rho
  2. r r
  3. ρ ( r ) exp ( - A r α ) . \rho(r)\propto\exp{(-Ar^{\alpha})}.
  4. α \alpha
  5. d ( log ρ ) / d ( log r ) - r α . d\ (\log\rho)/d\ (\log r)\propto-r^{\alpha}.
  6. α \alpha
  7. ρ r - N \rho\propto r^{-N}

Einstein_protocol.html

  1. a a
  2. b b
  3. b b
  4. a a
  5. D = c t 3 - t 1 2 D=c{{t_{3}-t_{1}}\over 2}
  6. a a
  7. b b

EIOLCA.html

  1. X i j X_{ij}
  2. j j
  3. i i
  4. y i y_{i}
  5. i i
  6. x i x_{i}
  7. i i
  8. x i = y i + j X i j x_{i}=y_{i}+\sum_{j}X_{ij}
  9. A i j A_{ij}
  10. A i j = X i j / x j A_{ij}=X_{ij}/x_{j}
  11. x i = y i + j A i j x j x_{i}=y_{i}+\sum_{j}A_{ij}x_{j}
  12. 𝐱 = 𝐲 + 𝐀𝐱 \mathbf{x}=\mathbf{y}+\mathbf{Ax}
  13. 𝐲 = ( 𝐈 - 𝐀 ) 𝐱 \mathbf{y}=(\mathbf{I-A})\mathbf{x}
  14. 𝐱 = ( 𝐈 - 𝐀 ) - 1 𝐲 \mathbf{x}=(\mathbf{I-A})^{-1}\mathbf{y}
  15. 𝐲 \mathbf{y}
  16. 𝐀 \mathbf{A}
  17. 𝐱 \mathbf{x}
  18. 𝐑 \mathbf{R}
  19. Δ 𝐛 \Delta\mathbf{b}
  20. Δ 𝐲 \Delta\mathbf{y}
  21. Δ 𝐛 = 𝐑 T Δ 𝐱 = 𝐑 T ( 𝐈 - 𝐀 ) - 1 Δ 𝐲 \Delta\mathbf{b}=\mathbf{R}^{T}\Delta\mathbf{x}=\mathbf{R}^{T}(\mathbf{I-A})^{% -1}\Delta\mathbf{y}
  22. Δ 𝐲 \Delta\mathbf{y}
  23. 𝐛 \mathbf{b}
  24. 𝐛 \mathbf{b}
  25. 𝐦 \mathbf{m}
  26. m m
  27. Δ m = 𝐦 T Δ 𝐛 = 𝐦 T 𝐑 T Δ 𝐱 = 𝐦 T 𝐑 T ( 𝐈 - 𝐀 ) - 1 Δ 𝐲 \Delta m=\mathbf{m}^{T}\Delta\mathbf{b}=\mathbf{m}^{T}\mathbf{R}^{T}\Delta% \mathbf{x}=\mathbf{m}^{T}\mathbf{R}^{T}(\mathbf{I-A})^{-1}\Delta\mathbf{y}
  28. 𝐦 \mathbf{m}
  29. 𝐛 \mathbf{b}
  30. 𝐀 \mathbf{A}
  31. 𝐑 \mathbf{R}
  32. 𝐲 \mathbf{y}

Ekeland's_variational_principle.html

  1. F ( u ) ε + inf x X F ( x ) . F(u)\leq\varepsilon+\inf_{x\in X}F(x).
  2. F ( v ) F ( u ) , F(v)\leq F(u),
  3. d ( u , v ) 1 , d(u,v)\leq 1,
  4. F ( w ) > F ( v ) - ε d ( v , w ) . F(w)>F(v)-\varepsilon d(v,w).

Elasticity_coefficient.html

  1. ε S v = v S S v = ln v ln S \varepsilon^{v}_{S}=\frac{\partial v}{\partial S}\frac{S}{v}=\frac{\partial\ln v% }{\partial\ln S}
  2. v v
  3. S S
  4. v = k S 1 n 1 S 2 n 2 v=k\ S_{1}^{n_{1}}S_{2}^{n_{2}}
  5. v v
  6. k k
  7. S i S_{i}
  8. n i n_{i}
  9. ε S 1 v \varepsilon^{v}_{S_{1}}
  10. S 1 S_{1}
  11. ε S 1 v = v S 1 S 1 v = n 1 k S 1 n 1 - 1 S 2 n 2 S 1 k S 1 n 1 S 2 n 2 = n 1 \varepsilon^{v}_{S_{1}}=\frac{\partial v}{\partial S_{1}}\frac{S_{1}}{v}=n_{1}% \ k\ S_{1}^{n_{1}-1}S_{2}^{n_{2}}\frac{S_{1}}{k\ S_{1}^{n_{1}}S_{2}^{n_{2}}}=n% _{1}
  12. v = V max S K m + S v=\frac{V_{\max}S}{K_{m}+S}
  13. ε S v = K m K m + S \varepsilon^{v}_{S}=\frac{K_{m}}{K_{m}+S}
  14. v = V max / K m 1 ( S - P / K e q ) 1 + S / K m 1 + P / K m 2 v=\frac{V_{\max}/K_{m1}(S-P/K_{eq})}{1+S/K_{m1}+P/K_{m2}}
  15. V max V_{\max}
  16. V m a x V_{max}
  17. K m 1 K_{m1}
  18. K m K_{m}
  19. K e q K_{eq}
  20. K m 2 K_{m2}
  21. K m K_{m}
  22. ε S v = 1 1 - Γ / K e q - S / K m 1 1 + S / K m 1 + P / K m 2 \varepsilon^{v}_{S}=\frac{1}{1-\Gamma/K_{eq}}-\frac{S/K_{m1}}{1+S/K_{m1}+P/K_{% m2}}
  23. ε P v = - Γ / K e q 1 - Γ / K e q - P / K m 2 1 + S / K m 1 + P / K m 2 \varepsilon^{v}_{P}=\frac{-\Gamma/K_{eq}}{1-\Gamma/K_{eq}}-\frac{P/K_{m2}}{1+S% /K_{m1}+P/K_{m2}}
  24. Γ \Gamma
  25. Γ = P / S \Gamma=P/S
  26. v = V max ( S / K s ) n 1 + ( S / K s ) n v=\frac{V_{\max}(S/K_{s})^{n}}{1+(S/K_{s})^{n}}
  27. K s K_{s}
  28. ε S v = n 1 + ( S / K s ) n \varepsilon^{v}_{S}=\frac{n}{1+(S/K_{s})^{n}}
  29. ε S v = ln v ln S \varepsilon^{v}_{S}=\frac{\partial\ln v}{\partial\ln S}
  30. ε = [ v 1 S 1 v 1 S m v n S 1 v n S m ] . \mathbf{\varepsilon}=\begin{bmatrix}\dfrac{\partial v_{1}}{\partial S_{1}}&% \cdots&\dfrac{\partial v_{1}}{\partial S_{m}}\\ \vdots&\ddots&\vdots\\ \dfrac{\partial v_{n}}{\partial S_{1}}&\cdots&\dfrac{\partial v_{n}}{\partial S% _{m}}\end{bmatrix}.

Elasticity_of_complementarity.html

  1. f ( x 1 , x 2 ) f(x_{1},x_{2})
  2. c = d ln ( d f d x 1 / d f d x 2 ) d ln ( x 2 / x 1 ) = d ( d f d x 1 / d f d x 2 ) d f d x 1 / d f d x 2 d ( x 2 / x 1 ) x 2 / x 1 . c=\frac{d\ln\left(\displaystyle\frac{df}{dx_{1}}/\displaystyle\frac{df}{dx_{2}% }\right)}{d\ln(x_{2}/x_{1})}=\frac{\displaystyle\frac{d(\frac{df}{dx_{1}}/% \frac{df}{dx_{2}})}{\frac{df}{dx_{1}}/\frac{df}{dx_{2}}}}{\displaystyle\frac{d% (x_{2}/x_{1})}{x_{2}/x_{1}}}.

Elasticity_of_intertemporal_substitution.html

  1. U = t = 0 T β t u ( c t ) U=\sum_{t=0}^{T}\beta^{t}u(c_{t})
  2. Q u ( c t ) = Q β R u ( c t + 1 ) Qu^{\prime}(c_{t})=Q\beta Ru^{\prime}(c_{t+1})
  3. Q Q
  4. Q u ( c t ) Qu^{\prime}(c_{t})
  5. R R
  6. R = u ( c t ) β u ( c t + 1 ) R=\frac{u^{\prime}(c_{t})}{\beta u^{\prime}(c_{t+1})}
  7. r = - ln [ u ( c t + 1 ) u ( c t ) ] - ln β r=-\ln{\left[\frac{u^{\prime}(c_{t+1})}{u^{\prime}(c_{t})}\right]}-\ln{\beta}
  8. r r
  9. R R
  10. ln ( c t + 1 / c t ) r \frac{\partial\ln(c_{t+1}/c_{t})}{\partial r}
  11. - ln ( c t + 1 / c t ) ln ( u ( c t + 1 ) / u ( c t ) ) -\frac{\partial\ln(c_{t+1}/c_{t})}{\partial\ln(u^{\prime}(c_{t+1})/u^{\prime}(% c_{t}))}
  12. t t
  13. u ( c t ) = c t 1 - σ 1 - σ . u(c_{t})=\frac{c_{t}^{1-\sigma}}{1-\sigma}.
  14. u ( c t ) = c t - σ . u^{\prime}(c_{t})=c_{t}^{-\sigma}.
  15. ln [ u ( c t + 1 ) u ( c t ) ] = - σ ln [ c t + 1 c t ] . \ln\left[\frac{u^{\prime}(c_{t+1})}{u^{\prime}(c_{t})}\right]=-\sigma\ln\left[% \frac{c_{t+1}}{c_{t}}\right].
  16. ln [ c t + 1 c t ] = - 1 σ ln [ u ( c t + 1 ) u ( c t ) ] \ln\left[\frac{c_{t+1}}{c_{t}}\right]=-\frac{1}{\sigma}\ln\left[\frac{u^{% \prime}(c_{t+1})}{u^{\prime}(c_{t})}\right]
  17. - ln ( c t + 1 / c t ) ln ( u ( c t + 1 ) / u ( c t ) ) = - [ - 1 σ ] = 1 σ . -\frac{\partial\ln(c_{t+1}/c_{t})}{\partial\ln(u^{\prime}(c_{t+1})/u^{\prime}(% c_{t}))}=-\left[-\frac{1}{\sigma}\right]=\frac{1}{\sigma}.
  18. U = 0 T e - ρ t u ( c t ) d t U=\int_{0}^{T}e^{-\rho t}u(c_{t})dt
  19. c t c_{t}
  20. c ( t ) c(t)
  21. u ( c ( t ) ) u(c(t))
  22. ρ \rho
  23. R R A = - d ( u ( c t ) ) d ( c t ) c t u ( c t ) = - u ′′ ( c t ) c t u ( c t ) RRA=-\frac{d(u^{\prime}(c_{t}))}{d(c_{t})}\frac{c_{t}}{u^{\prime}(c_{t})}=-u^{% \prime\prime}(c_{t})\frac{c_{t}}{u^{\prime}(c_{t})}
  24. E I S = - ( c ˙ t / c t ) ( u ˙ ( c t ) / u ( c t ) ) = - ( c ˙ t / c t ) ( u ′′ ( c t ) c ˙ t / u ( c t ) ) = ( c ˙ t / c t ) ( R R A ( c ˙ t / c t ) ) = 1 R R A = - u ( c t ) u ′′ ( c t ) c t EIS=-\frac{\partial(\dot{c}_{t}/c_{t})}{\partial(\dot{u}^{\prime}(c_{t})/u^{% \prime}(c_{t}))}=-\frac{\partial(\dot{c}_{t}/c_{t})}{\partial(u^{\prime\prime}% (c_{t})\dot{c}_{t}/u^{\prime}(c_{t}))}=\frac{\partial(\dot{c}_{t}/c_{t})}{% \partial(RRA\cdot(\dot{c}_{t}/c_{t}))}=\frac{1}{RRA}=-\frac{u^{\prime}(c_{t})}% {u^{\prime\prime}(c_{t})\cdot c_{t}}
  25. u ( c ) u(c)
  26. u ( c ) = c 1 - θ - 1 1 - θ u(c)=\frac{c^{1-\theta}-1}{1-\theta}
  27. θ = 1 \theta=1
  28. u ( c ) = l n ( c ) u(c)=ln(c)
  29. 1 θ \frac{1}{\theta}

Electromagnetically_induced_grating.html

  1. θ \theta
  2. sin β = n ( ω 1 ω 2 ) sin ( θ / 2 ) \sin\beta=n\Bigl(\frac{\omega_{1}}{\omega_{2}}\Bigr)\sin(\theta/2)
  3. ω 1 \omega_{1}
  4. ω 2 \omega_{2}

Electron-capture_mass_spectrometry.html

  1. A B + e - A B - AB+e^{-}\to AB^{-\bullet}
  2. A B + e - A - + B AB+e^{-}\to A^{-}+B^{\bullet}

Electron_capture_ionization.html

  1. A + e - 𝑀 A - A+e^{-}\overset{M}{\to}A^{-}

Electron_localization_function.html

  1. D σ ( 𝐫 ) = τ σ ( 𝐫 ) - 1 4 ( ρ σ ( 𝐫 ) ) 2 ρ σ ( 𝐫 ) , D_{\sigma}(\mathbf{r})=\tau_{\sigma}(\mathbf{r})-\tfrac{1}{4}\frac{(\nabla\rho% _{\sigma}(\mathbf{r}))^{2}}{\rho_{\sigma}(\mathbf{r})},
  2. D σ 0 ( 𝐫 ) = 3 5 ( 6 π 2 ) 2 / 3 ρ σ 5 / 3 ( 𝐫 ) . D^{0}_{\sigma}(\mathbf{r})=\tfrac{3}{5}(6\pi^{2})^{2/3}\rho^{5/3}_{\sigma}(% \mathbf{r}).
  3. χ σ ( 𝐫 ) = D σ ( 𝐫 ) D σ 0 ( 𝐫 ) , \chi_{\sigma}(\mathbf{r})=\frac{D_{\sigma}(\mathbf{r})}{D^{0}_{\sigma}(\mathbf% {r})},
  4. ELF ( 𝐫 ) = 1 1 + χ σ 2 ( 𝐫 ) . \mathrm{ELF}(\mathbf{r})=\frac{1}{1+\chi^{2}_{\sigma}(\mathbf{r})}.

Electron_rest_mass.html

  1. R = m e c α 2 2 h m e = 2 R h c α 2 R_{\infty}=\frac{m_{\rm e}c\alpha^{2}}{2h}\Rightarrow m_{\rm e}=\frac{2R_{% \infty}h}{c\alpha^{2}}
  2. × 10 8 \times 10^{−}8
  3. × 10 10 \times 10^{−}10
  4. N A = M u A r ( e ) m e = M u A r ( e ) c α 2 2 R h N_{\rm A}=\frac{M_{\rm u}A_{\rm r}({\rm e})}{m_{\rm e}}=\frac{M_{\rm u}A_{\rm r% }({\rm e})c\alpha^{2}}{2R_{\infty}h}
  5. m u = N A M u = A r ( e ) m e = A r ( e ) c α 2 2 R h m_{\rm u}=\frac{N_{\rm A}}{M_{\rm u}}=\frac{A_{\rm r}({\rm e})}{m_{\rm e}}=% \frac{A_{\rm r}({\rm e})c\alpha^{2}}{2R_{\infty}h}
  6. A r ( X ) = A r ( X Z + ) + Z A r ( e ) - E b / m u c 2 A_{\rm r}({\rm X})=A_{\rm r}({\rm X}^{Z+})+ZA_{\rm r}({\rm e})-E_{\rm b}/m_{% \rm u}c^{2}\,
  7. ν c ( C 6 + 12 ) ν c ( e ) = 6 A r ( e ) A r ( C 6 + 12 ) = 0.000 274 365 185 89 ( 58 ) \frac{\nu_{c}({}^{12}{\rm C}^{6+})}{\nu_{c}({\rm e})}=\frac{6A_{\rm r}({\rm e}% )}{A_{\rm r}({}^{12}{\rm C}^{6+})}=0.000\,274\,365\,185\,89(58)
  8. × 10 9 \times 10^{−}9

Electrothermal_instability.html

  1. β = Ω e ν = e B m e ν \beta\,=\,\frac{\Omega_{e}}{\nu}\,=\,\frac{e\ B}{m_{e}\ \nu}
  2. β = tan θ \ \beta\,=\,\tan\theta
  3. 𝐉 = σ 𝐄 \mathbf{J}=\sigma\mathbf{E}
  4. σ = σ s 1 1 + β 2 - β 1 + β 2 β 1 + β 2 1 1 + β 2 \sigma=\sigma_{s}\begin{Vmatrix}\dfrac{1}{1+\beta^{2}}&\dfrac{-\beta}{1+\beta^% {2}}\\ \dfrac{\beta}{1+\beta^{2}}&\dfrac{1}{1+\beta^{2}}\end{Vmatrix}
  5. σ s = n e e 2 m e ν \sigma_{s}=\frac{n_{e}\ e^{2}}{m_{e}\ \nu}
  6. J = n e e 2 m e ν E 1 + β 2 and J = - n e e 2 m e ν β E 1 + β 2 J_{\parallel}=\frac{n_{e}\ e^{2}}{m_{e}\ \nu}\ \frac{E}{1+\beta^{2}}\qquad\,% \text{and}\qquad J_{\perp}=\frac{-n_{e}\ e^{2}}{m_{e}\ \nu}\ \frac{\beta\ E}{1% +\beta^{2}}
  7. J = J β J_{\perp}=J_{\parallel}\ \beta
  8. 1 1 + β 2 1 \frac{1}{1+\beta^{2}}\ll 1
  9. σ σ s β 2 and σ σ s β \sigma_{\parallel}\approx\frac{\sigma_{s}}{\beta^{2}}\qquad\,\text{and}\qquad% \sigma_{\perp}\approx\frac{\sigma_{s}}{\beta}
  10. f = ( δ μ μ ) ( δ n e n e ) f=\frac{\left(\frac{\delta\mu}{\mu}\right)}{\left(\frac{\delta n_{e}}{n_{e}}% \right)}
  11. s = 2 k T e 2 E i ( T e - T g ) × 1 1 + 3 2 k T e E i s=\frac{2\ k\ T_{e}^{2}}{E_{i}\;(T_{e}-T_{g})}\times\frac{1}{1+\dfrac{3}{2}\ % \dfrac{k\;T_{e}}{E_{i}}}
  12. g = σ E 2 n e ( E i + 3 2 k T e ) ( 1 + β 2 ) ( β - β c r ) g=\frac{\sigma\ E^{2}}{n_{e}\;\left(E_{i}+\frac{3}{2}k\;T_{e}\right)\;\left(1+% \beta^{2}\right)}\;(\beta-\beta_{cr})
  13. β c r = 1.935 f + 0.065 + s \beta_{cr}=1.935f+0.065+s~{}
  14. α = n i n n \alpha=\frac{n_{i}}{n_{n}}
  15. β c r ( s 2 + 2 s ) 1 2 \beta_{cr}\approx(s^{2}+2s)^{\frac{1}{2}}
  16. β c r ( 2 + s ) \beta_{cr}\approx(2+s)

Elementary_Calculus:_An_Infinitesimal_Approach.html

  1. ( x - y ) 2 = x 2 - 2 x y + y 2 (x-y)^{2}=x^{2}-2xy+y^{2}
  2. sin 2 x + cos 2 x = 1 \sin^{2}x+\cos^{2}x=1
  3. log 10 ( x y ) = log 10 x + log 10 y \log_{10}(xy)=\log_{10}x+\log_{10}y
  4. ( d x ) 2 + ( d y ) 2 \int\sqrt{(dx)^{2}+(dy)^{2}}

Elimination_rate_constant.html

  1. C t + d t = C t - C t K d t C_{t+dt}=C_{t}-C_{t}\cdot K\cdot dt
  2. C t C_{t}
  3. t t
  4. d t dt
  5. C t + d t C_{t+dt}
  6. C t = C 0 e - K t C_{t}=C_{0}\cdot e^{-Kt}\,

Elliptic_divisibility_sequence.html

  1. W 2 n + 1 W 1 3 = W n + 2 W n 3 - W n + 1 3 W n - 1 , n 2 , W 2 n W 2 W 1 2 = W n + 2 W n W n - 1 2 - W n W n - 2 W n + 1 2 , n 3 , \begin{aligned}\displaystyle W_{2n+1}W_{1}^{3}&\displaystyle=W_{n+2}W_{n}^{3}-% W_{n+1}^{3}W_{n-1},\qquad n\geq 2,\\ \displaystyle W_{2n}W_{2}W_{1}^{2}&\displaystyle=W_{n+2}W_{n}W_{n-1}^{2}-W_{n}% W_{n-2}W_{n+1}^{2},\qquad n\geq 3,\\ \end{aligned}
  2. m n W m W n . m\mid n\Longrightarrow W_{m}\mid W_{n}.
  3. W 1 = 1 , W 2 = b , W 3 = c , W 4 = d . W_{1}=1,\quad W_{2}=b,\quad W_{3}=c,\quad W_{4}=d.
  4. W n + m W n - m W r 2 = W n + r W n - r W m 2 - W m + r W m - r W n 2 for all n > m > r . W_{n+m}W_{n-m}W_{r}^{2}=W_{n+r}W_{n-r}W_{m}^{2}-W_{m+r}W_{m-r}W_{n}^{2}\quad\,% \text{for all}\quad n>m>r.
  5. Δ = W 4 W 2 15 - W 3 3 W 2 12 + 3 W 4 2 W 2 10 - 20 W 4 W 3 3 W 2 7 + 3 W 4 3 W 2 5 + 16 W 3 6 W 2 4 + 8 W 4 2 W 3 3 W 2 2 + W 4 4 . \Delta=W_{4}W_{2}^{15}-W_{3}^{3}W_{2}^{12}+3W_{4}^{2}W_{2}^{10}-20W_{4}W_{3}^{% 3}W_{2}^{7}+3W_{4}^{3}W_{2}^{5}+16W_{3}^{6}W_{2}^{4}+8W_{4}^{2}W_{3}^{3}W_{2}^% {2}+W_{4}^{4}.
  6. 1 , 1 , - 1 , 1 , 2 , - 1 , - 3 , - 5 , 7 , - 4 , - 23 , 29 , 59 , 129 , - 314 , - 65 , 1529 , - 3689 , - 8209 , - 16264 , . \begin{aligned}&\displaystyle 1,\,1,\,-1,\,1,\,2,\,-1,\,-3,\,-5,\,7,\,-4,\,-23% ,\,29,\,59,\,129,\\ &\displaystyle-314,\,-65,\,1529,\,-3689,\,-8209,\,-16264,\dots.\\ \end{aligned}
  7. W n = ψ n ( P ) for all n 1. W_{n}=\psi_{n}(P)\qquad\,\text{for all}~{}n\geq 1.
  8. x ( n P ) = A n D n 2 with gcd ( A n , D n ) = 1 and D n 1. x(nP)=\frac{A_{n}}{D_{n}^{2}}\quad\,\text{with}~{}\gcd(A_{n},D_{n})=1~{}\,% \text{and}~{}D_{n}\geq 1.
  9. < v a r > n <var>n
  10. lim n log | W n | n 2 = h > 0. \lim_{n\to\infty}\frac{\log|W_{n}|}{n^{2}}=h>0.
  11. < v a r > q <var>q
  12. < v a r > q <var>q
  13. r ( q + 1 ) 2 and t q - 1. r\leq\left(\sqrt{q}+1\right)^{2}\quad\,\text{and}\quad t\mid q-1.
  14. < v a r > q <var>q
  15. W r i + j = W j A i j B j 2 for all i 0 and all j 1. W_{ri+j}=W_{j}\cdot A^{ij}\cdot B^{j^{2}}\quad\,\text{for all}~{}i\geq 0~{}\,% \text{and all}~{}j\geq 1.
  16. < v a r > A <var>A

Elliptic_hypergeometric_series.html

  1. ( a ; q ) n = k = 0 n - 1 ( 1 - a q k ) = ( 1 - a ) ( 1 - a q ) ( 1 - a q 2 ) ( 1 - a q n - 1 ) . \displaystyle(a;q)_{n}=\prod_{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots% (1-aq^{n-1}).
  2. ( a 1 , a 2 , , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n ( a m ; q ) n . \displaystyle(a_{1},a_{2},\ldots,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots% (a_{m};q)_{n}.
  3. θ ( x ; p ) = ( x , p / x ; p ) \displaystyle\theta(x;p)=(x,p/x;p)_{\infty}
  4. θ ( x 1 , , x m ; p ) = θ ( x 1 ; p ) θ ( x m ; p ) \displaystyle\theta(x_{1},...,x_{m};p)=\theta(x_{1};p)...\theta(x_{m};p)
  5. ( a ; q , p ) n = θ ( a ; p ) θ ( a q ; p ) θ ( a q n - 1 ; p ) \displaystyle(a;q,p)_{n}=\theta(a;p)\theta(aq;p)...\theta(aq^{n-1};p)
  6. ( a 1 , , a m ; q , p ) n = ( a 1 ; q , p ) n ( a m ; q , p ) n \displaystyle(a_{1},...,a_{m};q,p)_{n}=(a_{1};q,p)_{n}\cdots(a_{m};q,p)_{n}
  7. E r r + 1 ( a 1 , a r + 1 ; b 1 , , b r ; q , p ; z ) = n = 0 ( a 1 , , a r + 1 ; q ; p ) n ( q , b 1 , , b r ; q , p ) n z n \displaystyle{}_{r+1}E_{r}(a_{1},...a_{r+1};b_{1},...,b_{r};q,p;z)=\sum_{n=0}^% {\infty}\frac{(a_{1},...,a_{r+1};q;p)_{n}}{(q,b_{1},...,b_{r};q,p)_{n}}z^{n}
  8. V r r + 1 ( a 1 ; a 6 , a 7 , a r + 1 ; q , p ; z ) = n = 0 θ ( a 1 q 2 n ; p ) θ ( a 1 ; p ) ( a 1 , a 6 , a 7 , , a r + 1 ; q ; p ) n ( q , a 1 q / a 6 , a 1 q / a 7 , , a 1 q / a r + 1 ; q , p ) n ( q z ) n \displaystyle{}_{r+1}V_{r}(a_{1};a_{6},a_{7},...a_{r+1};q,p;z)=\sum_{n=0}^{% \infty}\frac{\theta(a_{1}q^{2n};p)}{\theta(a_{1};p)}\frac{(a_{1},a_{6},a_{7},.% ..,a_{r+1};q;p)_{n}}{(q,a_{1}q/a_{6},a_{1}q/a_{7},...,a_{1}q/a_{r+1};q,p)_{n}}% (qz)^{n}
  9. G r r ( a 1 , a r ; b 1 , , b r ; q , p ; z ) = n = - ( a 1 , , a r ; q ; p ) n ( b 1 , , b r ; q , p ) n z n \displaystyle{}_{r}G_{r}(a_{1},...a_{r};b_{1},...,b_{r};q,p;z)=\sum_{n=-\infty% }^{\infty}\frac{(a_{1},...,a_{r};q;p)_{n}}{(b_{1},...,b_{r};q,p)_{n}}z^{n}
  10. [ a ; σ , τ ] = θ 1 ( π σ a , e π i τ ) θ 1 ( π σ , e π i τ ) [a;\sigma,\tau]=\frac{\theta_{1}(\pi\sigma a,e^{\pi i\tau})}{\theta_{1}(\pi% \sigma,e^{\pi i\tau})}
  11. θ 1 ( x , q ) = n = - ( - 1 ) n q ( n + 1 / 2 ) 2 e ( 2 n + 1 ) i x \theta_{1}(x,q)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(n+1/2)^{2}}e^{(2n+1)ix}
  12. [ a ; σ , τ ] n = [ a ; σ , τ ] [ a + 1 ; σ , τ ] [ a + n - 1 ; σ , τ ] [a;\sigma,\tau]_{n}=[a;\sigma,\tau][a+1;\sigma,\tau]...[a+n-1;\sigma,\tau]
  13. [ a 1 , , a m ; σ , τ ] = [ a 1 ; σ , τ ] [ a m ; σ , τ ] [a_{1},...,a_{m};\sigma,\tau]=[a_{1};\sigma,\tau]...[a_{m};\sigma,\tau]
  14. e r r + 1 ( a 1 , a r + 1 ; b 1 , , b r ; σ , τ ; z ) = n = 0 [ a 1 , , a r + 1 ; σ ; τ ] n [ 1 , b 1 , , b r ; σ , τ ] n z n \displaystyle{}_{r+1}e_{r}(a_{1},...a_{r+1};b_{1},...,b_{r};\sigma,\tau;z)=% \sum_{n=0}^{\infty}\frac{[a_{1},...,a_{r+1};\sigma;\tau]_{n}}{[1,b_{1},...,b_{% r};\sigma,\tau]_{n}}z^{n}
  15. v r r + 1 ( a 1 ; a 6 , a r + 1 ; σ , τ ; z ) = n = 0 [ a 1 + 2 n ; σ , τ ] [ a 1 ; σ , τ ] [ a 1 , a 6 , , a r + 1 ; σ , τ ] n [ 1 , 1 + a 1 - a 6 , , 1 + a 1 - a r + 1 ; σ , τ ] n z n \displaystyle{}_{r+1}v_{r}(a_{1};a_{6},...a_{r+1};\sigma,\tau;z)=\sum_{n=0}^{% \infty}\frac{[a_{1}+2n;\sigma,\tau]}{[a_{1};\sigma,\tau]}\frac{[a_{1},a_{6},..% .,a_{r+1};\sigma,\tau]_{n}}{[1,1+a_{1}-a_{6},...,1+a_{1}-a_{r+1};\sigma,\tau]_% {n}}z^{n}

Ellrod_index.html

  1. D S H = d v d x + d u d y DSH=\frac{dv}{dx}+\frac{du}{dy}
  2. D S T = d u d x - d v d y DST=\frac{du}{dx}-\frac{dv}{dy}
  3. D E F = D S H 2 + D S T 2 DEF=\sqrt{DSH^{2}+DST^{2}}
  4. C V G = - ( d u d x + d v d y ) CVG=-(\frac{du}{dx}+\frac{dv}{dy})
  5. V W S = Δ V Δ Z VWS=\frac{\Delta V}{\Delta Z}
  6. E I = V W S × ( D E F + C V G ) EI=VWS\times(DEF+CVG)

End_extension.html

  1. 𝔅 = B , F \mathfrak{B}=\langle B,F\rangle
  2. T T\,
  3. 𝔄 = A , E \mathfrak{A}=\langle A,E\rangle
  4. 𝔄 end 𝔅 \mathfrak{A}\subseteq\text{end}\mathfrak{B}
  5. 𝔄 \mathfrak{A}
  6. 𝔅 \mathfrak{B}
  7. b A b\in A
  8. a A a\in A
  9. b F a bFa\,
  10. 𝔅 \mathfrak{B}
  11. 𝔄 \mathfrak{A}
  12. 𝔄 \mathfrak{A}
  13. 𝔅 \mathfrak{B}
  14. { b A : b E a } = { b B : b F a } \{b\in A:bEa\}=\{b\in B:bFa\}
  15. a A a\in A
  16. B , \langle B,\in\rangle
  17. A , \langle A,\in\rangle
  18. A A\,
  19. B B\,
  20. A B A\subseteq B

Energy_cannibalism.html

  1. C T C_{T}
  2. E T = t C T = t n = 1 N C n E_{T}=t\cdot C_{T}=t\cdot\sum_{n=1}^{N}C_{n}
  3. t t
  4. C n C_{n}
  5. N N
  6. r r
  7. E P E_{P}
  8. E a n n E_{ann}
  9. E P / E a n n E_{P}/E_{ann}
  10. E C a n E_{Can}
  11. E C a n = E P E a n n r C T t E_{Can}=\frac{E_{P}}{E_{ann}}\cdot r\cdot C_{T}\cdot t
  12. E P E a n n r C T t = C T t \frac{E_{P}}{E_{ann}}\cdot r\cdot C_{T}\cdot t=C_{T}\cdot t
  13. E P E a n n = 1 r \frac{E_{P}}{E_{ann}}=\frac{1}{r}

Entanglement_distillation.html

  1. ρ \rho
  2. | 00 , | 01 , | 10 , | 11 |00\rangle,|01\rangle,|10\rangle,|11\rangle
  3. α \alpha\,\!
  4. | ψ = α 00 | 00 + α 01 | 01 + α 10 | 10 + α 11 | 11 |\psi\rangle=\alpha_{00}|00\rangle+\alpha_{01}|01\rangle+\alpha_{10}|10\rangle% +\alpha_{11}|11\rangle
  5. | x |x\rangle
  6. | α x | 2 |\alpha_{x}|^{2}\,\!
  7. x ϵ 0 , 1 | α x | 2 = 1 \sum_{x\epsilon{0,1}}|\alpha_{x}|^{2}=1
  8. | 00 + | 11 2 \frac{|00\rangle+|11\rangle}{\sqrt{2}}
  9. | ψ |\psi\rangle
  10. n / m n/m
  11. | ϕ |\phi\rangle
  12. S ( p ) = - Tr ( p ln p ) S(p)=-\mathrm{Tr}(p\ln p)
  13. p A p_{A}
  14. p B p_{B}
  15. E = - Tr ( p A ln p A ) = - Tr ( p B ln p B ) , E=-\mathrm{Tr}(p_{A}\ln p_{A})=-\mathrm{Tr}(p_{B}\ln p_{B}),
  16. ln 2 \ln 2
  17. ln \ln
  18. log 2 \log_{2}
  19. p 1 p 2 p_{1}\otimes p_{2}\otimes
  20. p p\,\!
  21. ϵ \epsilon
  22. ψ \psi
  23. p p
  24. F = ψ | p | ψ F=\langle\psi|p|\psi\rangle
  25. ρ \rho
  26. S ( ρ ) N S(\rho)N
  27. ϕ \phi
  28. m n \frac{m}{n}
  29. 1 E ( ϕ ) \frac{1}{E(\phi)}
  30. n n\rightarrow\infty
  31. | ψ |\psi\rangle
  32. | ψ = x p ( x ) | x A | x B |\psi\rangle=\sum_{x}\sqrt{p(x)}|x_{A}\rangle|x_{B}\rangle
  33. | ψ m = x 1 , x 2 , , x m p ( x 1 ) p ( x 2 ) p ( x m ) | x 1 A x 2 A x m A | x 1 B x 2 B x m B |\psi\rangle^{\otimes m}=\sum_{x_{1},x_{2},...,x_{m}}\sqrt{p(x_{1})p(x_{2})...% p(x_{m})}|x_{1A}x_{2A}...x_{mA}\rangle|x_{1B}x_{2B}...x_{mB}\rangle
  34. x 1 , , x m x_{1},...,x_{m}\,\!
  35. A ϵ ( n ) A_{\epsilon}^{(n)}
  36. | ϕ m = x ϵ A ϵ ( n ) p ( x 1 ) p ( x 2 ) p ( x m ) | x 1 A x 2 A x m A | x 1 B x 2 B x m B |\phi_{m}\rangle=\sum_{x\epsilon A_{\epsilon}^{(n)}}\sqrt{p(x_{1})p(x_{2})...p% (x_{m})}|x_{1A}x_{2A}...x_{mA}\rangle|x_{1B}x_{2B}...x_{mB}\rangle
  37. | ϕ m = | ϕ m ϕ m | ϕ m |\phi_{m}^{{}^{\prime}}\rangle=\frac{|\phi_{m}\rangle}{\sqrt{\langle\phi_{m}|% \phi_{m}\rangle}}
  38. F ( | ψ m , | ϕ m ) 1 F(|\psi\rangle^{\otimes m},|\phi_{m}^{{}^{\prime}}\rangle)\rightarrow 1
  39. m m\rightarrow\infty
  40. | ψ |\psi\rangle
  41. A ϵ ( n ) A_{\epsilon}^{(n)}
  42. p ψ p_{\psi}\,\!
  43. | ψ m | ϕ m |\psi\rangle^{\otimes m}\rightarrow|\phi_{m}\rangle
  44. 1 - δ 1-\delta\,\!
  45. | ϕ m |\phi_{m}^{{}^{\prime}}\rangle
  46. 1 1 - δ \frac{1}{\sqrt{1-\delta}}
  47. | ϕ m |\phi_{m}^{{}^{\prime}}\rangle
  48. D ( p ) D(p)
  49. p p
  50. p p
  51. m m
  52. p p
  53. m D ( p ) m\cdot D(p)
  54. p p
  55. m D ( p ) m\cdot D(p)
  56. m D ( p ) m\cdot D(p)
  57. M M
  58. ψ - = ( - ) / 2 \psi^{-}=(\uparrow\downarrow-\downarrow\uparrow)/\sqrt{2}
  59. F = ψ - | M | ψ - F=\langle\psi^{-}|M|\psi^{-}\rangle
  60. M = | ϕ ϕ | M=|\phi\rangle\langle\phi|
  61. ϕ \phi
  62. ϕ \phi
  63. E ( ϕ ) = S ( p A ) = S ( p B ) E(\phi)=S(p_{A})=S(p_{B})
  64. p A = T r B ( | ϕ ϕ | ) p_{A}=Tr_{B}(|\phi\rangle\langle\phi|)
  65. p B p_{B}
  66. ϕ - \phi^{-}
  67. ϕ + \phi^{+}
  68. ψ + - \psi^{\frac{+}{-}}
  69. W F = F | ψ - ψ - | + 1 - F 3 | ϕ + ϕ + | + 1 - F 3 | ψ + ψ + | 1 - F 3 | ϕ - ϕ - | W_{F}=F\cdot|\psi^{-}\rangle\langle\psi^{-}|+\frac{1-F}{3}|\phi^{+}\rangle% \langle\phi^{+}|+\frac{1-F}{3}|\psi^{+}\rangle\langle\psi^{+}|\frac{1-F}{3}|% \phi^{-}\rangle\langle\phi^{-}|
  70. W F W_{F}
  71. σ y \sigma_{y}
  72. ϕ - \phi^{-}
  73. ψ + \psi^{+}
  74. F > 1 2 F>\frac{1}{2}
  75. ψ + \psi^{+}
  76. ψ + \psi^{+}
  77. ψ + \psi^{+}
  78. ϕ - \phi^{-}
  79. σ y \sigma_{y}
  80. F o u t < 1 F_{out}<1
  81. F i n > 1 2 F_{in}>\frac{1}{2}
  82. F o u t 1 F_{out}\rightarrow 1
  83. k ( F ) 1 1 - F k(F)\approx\frac{1}{\sqrt{1-F}}
  84. F o u t 1 F_{out}\rightarrow 1
  85. θ \theta
  86. c o s θ | A | B - s i n θ | A | B cos\theta|\uparrow_{A}\rangle\otimes|\downarrow_{B}\rangle-sin\theta|% \downarrow_{A}\rangle\otimes|\uparrow_{B}\rangle
  87. θ \theta
  88. π / 4 \pi/4
  89. t a n 2 θ tan^{2}\theta
  90. θ π / 4 \theta\neq\pi/4
  91. [ n , k ] \left[n,k\right]
  92. k k
  93. n n
  94. 0 k n 0\leq k\leq n
  95. k / n k/n
  96. n n
  97. | Φ + n \left|\Phi^{+}\right\rangle^{\otimes n}
  98. | Φ n + \left|\Phi_{n}^{+}\right\rangle
  99. | Φ + n \left|\Phi^{+}\right\rangle^{\otimes n}
  100. Π n \mathcal{E}\subset\Pi^{n}
  101. n n
  102. n n
  103. ( 𝐈 𝐀 ) | Φ n + \left(\mathbf{I}\otimes\mathbf{A}\right)\left|\Phi_{n}^{+}\right\rangle
  104. 𝐈 \mathbf{I}
  105. 𝐀 \mathbf{A}
  106. \mathcal{E}
  107. 𝒮 \mathcal{S}
  108. [ n , k ] \left[n,k\right]
  109. g 1 , , g n - k g_{1},\ldots,g_{n-k}
  110. n - k n-k
  111. 𝒮 \mathcal{S}
  112. { 𝐏 i } \left\{\mathbf{P}_{i}\right\}
  113. 2 n - k 2^{n-k}
  114. 2 n - k 2^{n-k}
  115. 𝒮 \mathcal{S}
  116. | Φ n + \left|\Phi_{n}^{+}\right\rangle
  117. i i
  118. 𝐏 i \mathbf{P}_{i}
  119. 𝐀 \mathbf{A}
  120. ( 𝐏 i 𝐈 ) ( 𝐈 𝐀 ) | Φ n + = ( 𝐈 𝐀 ) ( 𝐏 i 𝐈 ) | Φ n + . \left(\mathbf{P}_{i}\otimes\mathbf{I}\right)\left(\mathbf{I}\otimes\mathbf{A}% \right)\left|\Phi_{n}^{+}\right\rangle=\left(\mathbf{I}\otimes\mathbf{A}\right% )\left(\mathbf{P}_{i}\otimes\mathbf{I}\right)\left|\Phi_{n}^{+}\right\rangle.
  121. 𝐌 \mathbf{M}
  122. ( 𝐌 𝐈 ) | Φ n + = ( 𝐈 𝐌 T ) | Φ n + . \left(\mathbf{M}\otimes\mathbf{I}\right)\left|\Phi_{n}^{+}\right\rangle=\left(% \mathbf{I}\otimes\mathbf{M}^{T}\right)\left|\Phi_{n}^{+}\right\rangle.
  123. ( 𝐈 𝐀 ) ( 𝐏 i 𝐈 ) | Φ n + = ( 𝐈 𝐀 ) ( 𝐏 i 2 𝐈 ) | Φ n + = ( 𝐈 𝐀 ) ( 𝐏 i 𝐏 i T ) | Φ n + . \left(\mathbf{I}\otimes\mathbf{A}\right)\left(\mathbf{P}_{i}\otimes\mathbf{I}% \right)\left|\Phi_{n}^{+}\right\rangle=\left(\mathbf{I}\otimes\mathbf{A}\right% )\left(\mathbf{P}_{i}^{2}\otimes\mathbf{I}\right)\left|\Phi_{n}^{+}\right% \rangle=\left(\mathbf{I}\otimes\mathbf{A}\right)\left(\mathbf{P}_{i}\otimes% \mathbf{P}_{i}^{T}\right)\left|\Phi_{n}^{+}\right\rangle.
  124. 𝐏 i \mathbf{P}_{i}
  125. 𝐏 i T \mathbf{P}_{i}^{T}
  126. 𝐏 i \mathbf{P}_{i}
  127. 𝒮 \mathcal{S}
  128. 𝒮 \mathcal{S}
  129. 𝒮 \mathcal{S}
  130. 𝒮 \mathcal{S}
  131. k k
  132. k k
  133. c c
  134. n n
  135. ( 𝐈 A ( 𝐀 𝐈 ) B ) | Φ n + c + (\mathbf{I}^{A}\otimes\left(\mathbf{A\otimes I}\right)^{B})\left|\Phi_{n+c}^{+% }\right\rangle
  136. 𝐈 A \mathbf{I}^{A}
  137. 2 n + c × 2 n + c 2^{n+c}\times 2^{n+c}
  138. ( 𝐀 𝐈 ) B \left(\mathbf{A\otimes I}\right)^{B}
  139. n n
  140. c c
  141. n n
  142. n + c n+c
  143. c c
  144. n - k n-k
  145. n + c n+c
  146. k + c k+c
  147. c c
  148. k / n k/n
  149. ϵ \epsilon

Entropic_vector.html

  1. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  2. n N n\in N
  3. R 2 n - 1 R^{2^{n}-1}
  4. n n
  5. X = X 1 , X 2 , , X n \overrightarrow{X}=X_{1},X_{2},\ldots,X_{n}
  6. h X h_{\overrightarrow{X}}
  7. h X ( I ) = H ( X I ) = H ( X i 1 , X i 2 , , X i k ) h_{\overrightarrow{X}}(I)=H(X_{I})=H(X_{i_{1}},X_{i_{2}},\dots,X_{i_{k}})
  8. I = { i 1 , i 2 , , i k } I=\{i_{1},i_{2},\dots,i_{k}\}
  9. h = h x h=h_{\overrightarrow{x}}
  10. n n
  11. Γ n * \Gamma_{n}^{*}
  12. H : P n R + H:P_{n}\rightarrow R^{+}
  13. x x
  14. H ( x ) = 0 H(x)=0
  15. α R + \alpha\in R^{+}
  16. x x
  17. H ( x ) = α H(x)=\alpha
  18. P P
  19. [ n ] [n]
  20. H ( P ) log 2 n H(P)\leq\log_{2}n
  21. { 0 , 1 } \{0,1\}
  22. H ( X ) = H ( Y ) = 1 , I ( X ; Y ) = 0 H\left(X\right)=H(Y)=1,I\left(X;Y\right)=0
  23. H ( X , Y ) = H ( X ) + H ( Y ) - I ( X ; Y ) = 2 H(X,Y)=H(X)+H(Y)-I\left(X;Y\right)=2
  24. v = ( 1 , 1 , 2 ) T Γ 2 * v=\left(1,1,2\right)^{T}\in\Gamma_{2}^{*}
  25. 1 ) H ( ) = 0 1)\quad H()=0
  26. 2 ) α β : H ( α ) H ( β ) 2)\quad\alpha\subseteq\beta:H(\alpha)\leq H(\beta)
  27. 3 ) H ( X α ) + H ( X β ) H ( X α β ) + H ( X α β ) 3)\quad H(X_{\alpha})+H(X_{\beta})\leq H(X_{\alpha\cup\beta})+H(X_{\alpha\cap% \beta})
  28. Γ n \Gamma_{n}
  29. Γ n * \Gamma_{n}^{*}
  30. L n = Γ n = Γ n * = Γ n ¯ * L_{n}=\Gamma_{n}=\Gamma_{n}^{*}=\overline{\Gamma_{n}}^{*}
  31. L n o = Γ n o = Γ n ¯ * o = Shannon n + L_{n}^{o}=\Gamma_{n}^{o}=\overline{\Gamma_{n}}^{*o}=\langle\mathrm{Shannon}_{n% }\rangle^{+}
  32. n 4 n\geq 4
  33. Γ n * \Gamma_{n}^{*}
  34. Γ n * \overrightarrow{\Gamma_{n}^{*}}
  35. Γ n * \overrightarrow{\Gamma_{n}^{*}}
  36. Γ n * = Γ n \overrightarrow{\Gamma_{n}^{*}}=\Gamma_{n}
  37. Γ n \Gamma_{n}
  38. n 3 n\leq 3
  39. L n Γ n ¯ * Γ n L_{n}\subseteq\overline{\Gamma_{n}}^{*}\subseteq\Gamma_{n}
  40. Γ n o Γ n ¯ * o L n o \Gamma_{n}^{o}\subseteq\overline{\Gamma_{n}}^{*o}\subseteq L_{n}^{o}
  41. Γ n o = Shannon n + \Gamma_{n}^{o}=\langle\mathrm{Shannon}_{n}\rangle^{+}
  42. I ( X 3 , X 4 ) = I ( X 4 , X 2 ) + I ( X 1 : X 3 , X 4 ) + 3 I ( X 3 : X 4 | X 1 ) + I ( X 3 : X 4 | X 2 ) I(X_{3},X_{4})=I(X4,X_{2})+I(X_{1}:X_{3},X_{4})+3I(X_{3}:X_{4}|X_{1})+I(X_{3}:% X_{4}|X_{2})
  43. Γ 4 * ¯ \overline{\Gamma_{4}^{*}}
  44. Γ n * \Gamma_{n}^{*}
  45. 2 n R 2^{n}\rightarrow R
  46. G G
  47. G 1 , G 2 , , G n G_{1},G_{2},\dots,G_{n}
  48. α n \alpha\subset n
  49. H ( α ) = | G i | | G | H(\alpha)=\frac{|Gi|}{|G|}
  50. G α G_{\alpha}
  51. \quad\quad
  52. G = i α G i G=\cap_{i\in\alpha}G_{i}
  53. γ n \gamma^{n}
  54. n n
  55. Γ n \Gamma^{n}
  56. γ n Γ n \gamma^{n}\subset\Gamma^{n}
  57. v R 2 n - 1 v\in R^{2^{n}-1}
  58. n n
  59. v v
  60. n = 2 , 3 n=2,3
  61. n 4 n\geq 4
  62. v R 2 n - 1 v\in R^{2^{n}-1}
  63. n n
  64. Γ n * {\Gamma}^{*}_{n}

Entropy_(astrophysics).html

  1. d Q = d U - d W . dQ=dU-dW.\,
  2. d Q = C V d T + P d V . dQ=C_{V}dT+P\,dV.
  3. d Q = C P d T - V d P . dQ=C_{P}dT-V\,dP.
  4. d Q = 0 dQ=0\,
  5. γ = C P C V \gamma=\frac{C_{P}}{C_{V}}\,
  6. V d P = C P d T P d V = - C V d T \frac{V\,dP=C_{P}dT}{P\,dV=-C_{V}dT}
  7. d P P = - d V V γ . \frac{dP}{P}=-\frac{dV}{V}\gamma.
  8. P V γ = constant = K PV^{\gamma}=\,\text{constant}=K\,
  9. P = ρ k B T μ m H , P=\frac{\rho k_{B}T}{\mu m_{H}},
  10. k B k_{B}\,
  11. V = [ g r a m s ] / ρ V=[grams]/\rho\,
  12. γ = 5 / 3 \gamma=5/3\,
  13. K = k B T μ m H ρ 2 / 3 , K=\frac{k_{B}T}{\mu m_{H}\rho^{2/3}},
  14. μ \mu\,
  15. m H m_{H}\,
  16. m p m_{p}\,
  17. S = k B ln Ω + S 0 S=k_{B}\ln\Omega+S_{0}\,
  18. Ω \Omega\,

Entropy_estimation.html

  1. H ( X ) = - 𝕏 f ( x ) log f ( x ) d x H(X)=-\int_{\mathbb{X}}f(x)\log f(x)\,dx
  2. H ( X ) = - i = 1 n f ( x i ) log ( f ( x i ) w ( x i ) ) H(X)=-\sum_{i=1}^{n}f(x_{i})\log\left(\frac{f(x_{i})}{w(x_{i})}\right)

Equal_incircles_theorem.html

  1. γ n \gamma\,_{n}
  2. γ n \gamma\,_{n}
  3. tan γ n = sinh θ n \tan\gamma\,_{n}=\sinh\theta\,_{n}
  4. θ n = a + n b \theta\,_{n}=a+nb
  5. a \!a
  6. b \!b
  7. a \!a
  8. b \!b
  9. γ n \gamma\,_{n}
  10. γ n + 1 \gamma\,_{n+1}
  11. \triangle
  12. h - r \!h-r
  13. r \!r
  14. \triangle
  15. \triangle
  16. \triangle
  17. \triangle
  18. ( h - r ) ( tan γ n + 1 - tan γ n ) = r ( sec γ n + sec γ n + 1 ) . (h-r)(\tan\gamma\,_{n+1}-\tan\gamma\,_{n})=r(\sec\gamma\,_{n}+\sec\gamma\,_{n+% 1}).
  19. { γ m } \{\gamma\,_{m}\}
  20. tan γ n = sinh ( a + n b ) \tan\gamma\,_{n}=\sinh(a+nb)
  21. sec γ n = cosh ( a + n b ) \sec\gamma\,_{n}=\cosh\,(a+nb)
  22. a + ( n + 1 ) b = ( a + n b ) + b \!a+(n+1)b=(a+nb)+b
  23. sinh \!\sinh
  24. cosh \!\cosh
  25. r h - r = tanh b 2 . \frac{r}{h-r}=\tanh\frac{b}{2}.
  26. b \!b
  27. h \!h
  28. r \!r
  29. b \!b
  30. r N \!r_{N}
  31. r N h - r N = tanh N b 2 . \frac{r_{N}}{h-r_{N}}=\tanh\frac{Nb}{2}.

Equatorial_waves.html

  1. × \times
  2. f y = β \frac{\partial f}{\partial y}=\beta
  3. ϕ t + c 2 ( v y + u x ) = 0 \frac{\partial\phi}{\partial t}+c^{2}\left(\frac{\partial v}{\partial y}+\frac% {\partial u}{\partial x}\right)=0
  4. u t - v β y = - ϕ x \frac{\partial u}{\partial t}-v\beta y=-\frac{\partial\phi}{\partial x}
  5. v t + u β y = - ϕ y \frac{\partial v}{\partial t}+u\beta y=-\frac{\partial\phi}{\partial y}
  6. { u , v , ϕ } = { u ^ ( y ) , v ^ ( y ) , ϕ ^ ( y ) } e i ( k x - ω t ) \begin{Bmatrix}u,v,\phi\end{Bmatrix}=\begin{Bmatrix}\hat{u}(y),\hat{v}(y),\hat% {\phi}(y)\end{Bmatrix}e^{i(kx-\omega t)}
  7. ϕ t + c 2 u x = 0 \frac{\partial\phi}{\partial t}+c^{2}\frac{\partial u}{\partial x}=0
  8. u t = - ϕ x \frac{\partial u}{\partial t}=-\frac{\partial\phi}{\partial x}
  9. u β y = - ϕ y . u\beta y=-\frac{\partial\phi}{\partial y}.
  10. u t - v β y = τ x ρ h , \frac{\partial u}{\partial t}-v\beta y=\frac{\tau_{x}}{\rho h},
  11. v t - u β y = τ y ρ h , \frac{\partial v}{\partial t}-u\beta y=\frac{\tau_{y}}{\rho h},
  12. h t + h ( u x + v y ) - K E T = 0 , \frac{\partial h}{\partial t}+h\left(\frac{\partial u}{\partial x}+\frac{% \partial v}{\partial y}\right)-K_{E}T=0,
  13. T t + u T x - K T h = 0. \frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}-K_{T}h=0.

Equilateral_pentagon.html

  1. a = 2 sin ( β 2 ) a=2\sin\left(\frac{\beta}{2}\right)
  2. b 2 \displaystyle b^{2}
  3. cos ( δ ) = 1 2 + 1 2 - b 2 2 ( 1 ) ( 1 ) . \cos(\delta)=\frac{1^{2}+1^{2}-b^{2}}{2(1)(1)}\ .
  4. δ = arccos [ cos ( α ) + cos ( β ) - cos ( α + β ) - 1 2 ] \delta=\arccos\left[\cos(\alpha)+\cos(\beta)-\cos(\alpha+\beta)-\frac{1}{2}\right]

Equisatisfiability.html

  1. a b a\vee b
  2. ( ( a n ) ( ¬ n b ) ) ((a\vee n)\wedge(\neg n\vee b))
  3. n n
  4. b b
  5. a a
  6. n n
  7. n n

Equivalent_impedance_transforms.html

  1. p 1 = 1 + m 1 , p_{1}=1+m_{1}\ ,
  2. p 2 = m 1 ( 1 + m 1 ) , p_{2}=m_{1}(1+m_{1})\ ,
  3. p 3 = ( 1 + m 1 ) 2 . p_{3}=(1+m_{1})^{2}\ .
  4. p 1 = m 1 2 1 + m 1 , p_{1}=\frac{{m_{1}}^{2}}{1+m_{1}}\ ,
  5. p 2 = m 1 1 + m 1 , p_{2}=\frac{m_{1}}{1+m_{1}}\ ,
  6. p 3 = ( m 1 1 + m 1 ) 2 . p_{3}=\left(\frac{m_{1}}{1+m_{1}}\right)^{2}\ .
  7. m 1 = 0.5 , m_{1}=0.5\ ,
  8. p 1 = 1 6 , p_{1}=\textstyle\frac{1}{6}\ ,
  9. p 2 = 1 3 , p_{2}=\textstyle\frac{1}{3}\ ,
  10. p 3 = 1 9 . p_{3}=\textstyle\frac{1}{9}\ .
  11. q 1 := 1 + m 1 + m 2 q_{1}:=1+m_{1}+m_{2}\,\!
  12. q 2 := q 1 2 - 4 m 1 m 2 q_{2}:=\sqrt{{q_{1}}^{2}-4m_{1}m_{2}}\,\!
  13. q 3 := ( 1 + m 1 ) ( 1 + m 2 ) ( m 1 - m 2 ) 2 q_{3}:=\frac{(1+m_{1})(1+m_{2})}{(m_{1}-m_{2})^{2}}
  14. q 4 := q 2 - q 1 + 2 m 2 2 q 2 q_{4}:=\frac{q_{2}-q_{1}+2m_{2}}{2q_{2}}
  15. q 5 := q 2 + q 1 - 2 m 2 2 q 2 q_{5}:=\frac{q_{2}+q_{1}-2m_{2}}{2q_{2}}
  16. p 1 = q 1 + q 2 2 q 5 , p_{1}=\frac{q_{1}+q_{2}}{2q_{5}}\ ,
  17. p 2 = q 1 - q 2 2 q 4 , p_{2}=\frac{q_{1}-q_{2}}{2q_{4}}\ ,
  18. p 3 = m 2 q 5 , p_{3}=\frac{m_{2}}{q_{5}}\ ,
  19. p 4 = m 2 q 4 . p_{4}=\frac{m_{2}}{q_{4}}\ .
  20. p 1 = 1 q 3 ( 1 + m 2 ) , p_{1}=\frac{1}{q_{3}(1+m_{2})}\ ,
  21. p 2 = m 1 1 + m 1 , p_{2}=\frac{m_{1}}{1+m_{1}}\ ,
  22. p 3 = 1 q 3 ( 1 + m 1 ) , p_{3}=\frac{1}{q_{3}(1+m_{1})}\ ,
  23. p 4 = m 2 1 + m 2 . p_{4}=\frac{m_{2}}{1+m_{2}}\ .
  24. p 1 = q 4 ( q 1 + q 2 ) 2 m 2 , p_{1}=\frac{q_{4}(q_{1}+q_{2})}{2m_{2}}\ ,
  25. p 2 = q 5 ( q 1 - q 2 ) 2 m 2 , p_{2}=\frac{q_{5}(q_{1}-q_{2})}{2m_{2}}\ ,
  26. p 3 = q 4 , p_{3}=q_{4}\ ,
  27. p 4 = q 5 . p_{4}=q_{5}\ .
  28. p 1 = 1 + m 1 , p_{1}=1+m_{1}\ ,
  29. p 2 = m 1 q 3 ( 1 + m 1 ) , p_{2}=m_{1}q_{3}(1+m_{1})\ ,
  30. p 3 = 1 + m 2 , p_{3}=1+m_{2}\ ,
  31. p 4 = m 1 q 3 ( 1 + m 2 ) . p_{4}=m_{1}q_{3}(1+m_{2})\ .
  32. m 1 = 3 , m_{1}=3\ ,
  33. m 2 = 1 , m_{2}=1\ ,
  34. q 3 = 2 , q_{3}=2\ ,
  35. p 1 = 1 4 , p_{1}=\textstyle\frac{1}{4}\ ,
  36. p 2 = 3 4 , p_{2}=\textstyle\frac{3}{4}\ ,
  37. p 3 = 1 8 , p_{3}=\textstyle\frac{1}{8}\ ,
  38. p 4 = 1 2 . p_{4}=\textstyle\frac{1}{2}\ .
  39. [ 𝐙 ] = [ Z 11 Z 12 Z 13 Z 21 Z 22 Z 23 Z 31 Z 32 Z 33 ] \mathbf{[Z]}=\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\ Z_{21}&Z_{22}&Z_{23}\\ Z_{31}&Z_{32}&Z_{33}\end{bmatrix}
  40. [ 𝐕 ] = [ 𝐙 ] [ 𝐈 ] \mathbf{[V]}=\mathbf{[Z][I]}
  41. [ 𝐙 ] = [ R 1 + R 2 - R 2 - R 2 R 2 + R 3 ] \mathbf{[Z]}=\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\ -R_{2}&R_{2}+R_{3}\end{bmatrix}
  42. [ V 1 0 ] = [ R 1 + R 2 - R 2 - R 2 R 2 + R 3 ] [ I 1 I 2 ] \begin{bmatrix}V_{1}\\ 0\end{bmatrix}=\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\ -R_{2}&R_{2}+R_{3}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}
  43. Z p = s L p + R p + 1 s C p Z_{\mathrm{p}}=sL_{\mathrm{p}}+R_{\mathrm{p}}+{1\over sC_{\mathrm{p}}}
  44. s = σ + i ω \scriptstyle s=\sigma+i\omega
  45. s Z p = s 2 L p + s R p + D p sZ_{\mathrm{p}}=s^{2}L_{\mathrm{p}}+sR_{\mathrm{p}}+D_{\mathrm{p}}\,\!
  46. s [ 𝐙 ] = s 2 [ 𝐋 ] + s [ 𝐑 ] + [ 𝐃 ] s\mathbf{[Z]}=s^{2}\mathbf{[L]}+s\mathbf{[R]}+\mathbf{[D]}
  47. Z ( s ) = | 𝐙 | z 11 Z(s)=\frac{|\mathbf{Z}|}{z_{11}}
  48. | 𝐙 | = ( R 1 + R 2 ) ( R 2 + R 3 ) - R 2 2 = R 1 R 2 + R 1 R 3 + R 2 R 3 , |\mathbf{Z}|=(R_{1}+R_{2})(R_{2}+R_{3})-{R_{2}}^{2}=R_{1}R_{2}+R_{1}R_{3}+R_{2% }R_{3}\ ,
  49. z 11 = Z 22 = R 2 + R 3 , z_{11}=Z_{22}=R_{2}+R_{3}\ ,
  50. Z ( s ) = R 1 + R 2 R 3 R 2 + R 3 . Z(s)=R_{1}+\frac{R_{2}R_{3}}{R_{2}+R_{3}}\ .
  51. [ 𝐙 ] = [ 𝐓 ] T [ 𝐙 ] [ 𝐓 ] \mathbf{[Z^{\prime}]}=\mathbf{[T]}^{T}\mathbf{[Z]}\mathbf{[T]}
  52. [ 𝐓 ] = [ 1 0 0 T 21 T 22 T 2 n T n 1 T n 2 T n n ] \mathbf{[T]}=\begin{bmatrix}1&0\cdots 0\\ T_{21}&T_{22}\cdots T_{2n}\\ \cdot&\cdots\\ T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}
  53. Z A = Z 1 , Z_{\mathrm{A}}=Z_{1}\ ,\!
  54. Z B = Z 1 + 2 Z 2 . Z_{\mathrm{B}}=Z_{1}+2Z_{2}\ .\!
  55. Z A = Z 1 Z 2 Z 1 + 2 Z 2 , Z_{\mathrm{A}}=\frac{Z_{1}Z_{2}}{Z_{1}+2Z_{2}}\ ,\!
  56. Z B = Z 2 . Z_{\mathrm{B}}=Z_{2}\ .
  57. Z A = Z 1 Z 0 Z 1 + 2 Z 0 , Z_{\mathrm{A}}=\frac{Z_{1}Z_{0}}{Z_{1}+2Z_{0}}\ ,\!
  58. Z B = Z 0 + 2 Z 2 . Z_{\mathrm{B}}=Z_{0}+2Z_{2}\ .\!

Equivariant_K-theory.html

  1. Coh G ( X ) \operatorname{Coh}^{G}(X)
  2. K i G ( X ) = π i ( B + Coh G ( X ) ) . K_{i}^{G}(X)=\pi_{i}(B^{+}\operatorname{Coh}^{G}(X)).
  3. K 0 G ( C ) K_{0}^{G}(C)
  4. Coh G ( X ) \operatorname{Coh}^{G}(X)
  5. K i G ( X ) K_{i}^{G}(X)
  6. K i K_{i}
  7. [ X / G ] [X/G]

Erdős–Fuchs_theorem.html

  1. R ( n ) = r ( 1 ) + r ( 2 ) + + r ( n ) n . R(n)=\frac{r(1)+r(2)+\cdots+r(n)}{n}.
  2. R ( n ) = C + O ( n - 3 / 4 - ε ) R(n)=C+O\left(n^{-3/4-\varepsilon}\right)

Ergodic_process.html

  1. X ( t ) X(t)
  2. μ X = E [ X ( t ) ] \mu_{X}=E[X(t)]
  3. r X ( τ ) = E [ ( X ( t ) - μ X ) ( X ( t + τ ) - μ X ) ] r_{X}(\tau)=E[(X(t)-\mu_{X})(X(t+\tau)-\mu_{X})]
  4. τ \tau
  5. t t
  6. μ X \mu_{X}
  7. r X ( τ ) r_{X}(\tau)
  8. X ( t ) X(t)
  9. μ ^ X = 1 T 0 T X ( t ) d t \hat{\mu}_{X}=\frac{1}{T}\int_{0}^{T}X(t)\,dt
  10. μ X \mu_{X}
  11. T T\rightarrow\infty
  12. r ^ X ( τ ) = 1 T 0 T [ X ( t + τ ) - μ X ] [ X ( t ) - μ x ] d t \hat{r}_{X}(\tau)=\frac{1}{T}\int_{0}^{T}[X(t+\tau)-\mu_{X}][X(t)-\mu_{x}]\,dt
  13. r X ( τ ) r_{X}(\tau)
  14. T T\rightarrow\infty
  15. X [ n ] X[n]
  16. n n
  17. X [ n ] X[n]
  18. μ ^ X = 1 N n = 1 N X [ n ] \hat{\mu}_{X}=\frac{1}{N}\sum_{n=1}^{N}X[n]
  19. E [ X ] E[X]
  20. N N\rightarrow\infty

Ernst_equation.html

  1. ( u ) ( u r r + u r / r + u z z ) = ( u r ) 2 + ( u z ) 2 . \displaystyle\Re(u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.

Esscher_principle.html

  1. π [ X , h ] = E [ X e h X ] / E [ e h X ] \pi[X,h]=E[Xe^{hX}]/E[e^{hX}]
  2. h h
  3. Y = X e h X / m X ( h ) Y=Xe^{hX}/m_{X}(h)
  4. m X ( h ) m_{X}(h)
  5. h > 0 h>0

Essential_dimension.html

  1. q ( x i e i ) = a i j x i x j q(\sum x_{i}e_{i})=\sum a_{ij}x_{i}x_{j}
  2. q K : V L K K q_{K}:V\otimes_{L}K\to K
  3. \mathcal{F}
  4. A f f / k Aff/k
  5. p : A f f / k . p:\mathcal{F}\to Aff/k.
  6. \mathcal{F}
  7. g \mathcal{M}_{g}
  8. 𝒢 \mathcal{BG}
  9. A A f f / k A\in Aff/k
  10. p - 1 ( S p e c ( K ) ) p^{-1}(Spec(K))
  11. \mathcal{F}
  12. = 𝒢 \mathcal{F}=\mathcal{BG}

Essential_manifold.html

  1. H n ( M ) H n ( K ( π , 1 ) ) H_{n}(M)\to H_{n}(K(\pi,1))
  2. n \mathbb{RP}^{n}\to\mathbb{RP}^{\infty}
  3. = K ( 2 , 1 ) \mathbb{RP}^{\infty}=K(\mathbb{Z}_{2},1)

Essential_range.html

  1. ( X , 𝔄 , μ ) (X,\mathfrak{A},\mu)
  2. ess . im ( f ) = { z for all ε > 0 : μ ( { x : | f ( x ) - z | < ε } ) > 0 } \operatorname{ess.im}(f)=\left\{z\in\mathbb{C}\mid\,\text{for all}\ % \varepsilon>0:\mu(\{x:|f(x)-z|<\varepsilon\})>0\right\}
  3. im ( f ) ¯ \overline{\operatorname{im}(f)}
  4. f = g f=g
  5. μ \mu
  6. ess . im ( f ) = ess . im ( g ) \operatorname{ess.im}(f)=\operatorname{ess.im}(g)
  7. im ( g ) \operatorname{im}(g)
  8. ess . im ( f ) = f = g a.e. im ( g ) ¯ \operatorname{ess.im}(f)=\bigcap_{f=g\,\,\text{a.e.}}\overline{\operatorname{% im}(g)}
  9. A X : f ( A ) ess . im ( f ) = μ ( A ) = 0 \forall A\subseteq X:f(A)\cap\operatorname{ess.im}(f)=\emptyset\implies\mu(A)=0
  10. \mathbb{C}
  11. σ ( f ) \sigma(f)
  12. L ( μ ) L^{\infty}(\mu)
  13. μ \mu
  14. X n X\subseteq\mathbb{R}^{n}
  15. f : X f:X\to\mathbb{C}
  16. μ \mu
  17. ess . im ( f ) = im ( f ) ¯ \operatorname{ess.im}(f)=\overline{\operatorname{im}(f)}

Essential_subgroup.html

  1. S S
  2. G G

Estimating_equations.html

  1. f ( x ; λ ) = { λ e - λ x , x 0 , 0 , x < 0. f(x;\lambda)=\left\{\begin{matrix}\lambda e^{-\lambda x},&\;x\geq 0,\\ 0,&\;x<0.\end{matrix}\right.
  2. x ¯ \bar{x}
  3. x ¯ = λ - 1 , \bar{x}=\lambda^{-1},
  4. m = λ - 1 ln 2. m=\lambda^{-1}\ln 2.

Euler's_factorization_method.html

  1. 1000009 1000009
  2. 1000 2 + 3 2 1000^{2}+3^{2}
  3. 972 2 + 235 2 972^{2}+235^{2}
  4. 1000009 = 293 3413 1000009=293\cdot 3413
  5. 1000009 1000009
  6. n = a 2 + b 2 = c 2 + d 2 n=a^{2}+b^{2}=c^{2}+d^{2}
  7. n n
  8. a 2 - c 2 = d 2 - b 2 a^{2}-c^{2}=d^{2}-b^{2}
  9. ( a - c ) ( a + c ) = ( d - b ) ( d + b ) (a-c)(a+c)=(d-b)(d+b)
  10. k = gcd ( a - c , d - b ) k=\operatorname{gcd}(a-c,d-b)
  11. h = gcd ( a + c , d + b ) h=\operatorname{gcd}(a+c,d+b)
  12. l , m , l , m l,m,l^{\prime},m^{\prime}
  13. ( a - c ) = k l (a-c)=kl
  14. ( d - b ) = k m (d-b)=km
  15. gcd ( l , m ) = 1 \operatorname{gcd}(l,m)=1
  16. ( a + c ) = h m (a+c)=hm^{\prime}
  17. ( d + b ) = h l (d+b)=hl^{\prime}
  18. gcd ( l , m ) = 1 \operatorname{gcd}(l^{\prime},m^{\prime})=1
  19. k l h m = k m h l klhm^{\prime}=kmhl^{\prime}
  20. l m = l m lm^{\prime}=l^{\prime}m
  21. ( l , m ) (l,m)
  22. ( l , m ) (l^{\prime},m^{\prime})
  23. l = l l=l^{\prime}
  24. m = m m=m^{\prime}
  25. ( a - c ) = k l (a-c)=kl
  26. ( d - b ) = k m (d-b)=km
  27. ( a + c ) = h m (a+c)=hm
  28. ( d + b ) = h l (d+b)=hl
  29. m = gcd ( a + c , d - b ) m=\operatorname{gcd}(a+c,d-b)
  30. l = gcd ( a - c , d + b ) l=\operatorname{gcd}(a-c,d+b)
  31. ( k 2 + h 2 ) ( l 2 + m 2 ) = ( k l - h m ) 2 + ( k m + h l ) 2 = ( ( a - c ) - ( a + c ) ) 2 + ( ( d - b ) + ( d + b ) ) 2 = ( 2 c ) 2 + ( 2 d ) 2 = 4 n , (k^{2}+h^{2})(l^{2}+m^{2})=(kl-hm)^{2}+(km+hl)^{2}=((a-c)-(a+c))^{2}+((d-b)+(d% +b))^{2}=(2c)^{2}+(2d)^{2}=4n,
  32. ( k 2 + h 2 ) ( l 2 + m 2 ) = ( k l + h m ) 2 + ( k m - h l ) 2 = ( ( a - c ) + ( a + c ) ) 2 + ( ( d - b ) - ( d + b ) ) 2 = ( 2 a ) 2 + ( 2 b ) 2 = 4 n . (k^{2}+h^{2})(l^{2}+m^{2})=(kl+hm)^{2}+(km-hl)^{2}=((a-c)+(a+c))^{2}+((d-b)-(d% +b))^{2}=(2a)^{2}+(2b)^{2}=4n.
  33. ( k , h ) (k,h)
  34. ( l , m ) (l,m)
  35. ( k , h ) (k,h)
  36. n = ( ( k 2 ) 2 + ( h 2 ) 2 ) ( l 2 + m 2 ) . n=((\tfrac{k}{2})^{2}+(\tfrac{h}{2})^{2})(l^{2}+m^{2}).\,
  37. 1000009 = 1000 2 + 3 2 = 972 2 + 235 2 \ 1000009=1000^{2}+3^{2}=972^{2}+235^{2}
  38. 1000009 = [ ( 4 2 ) 2 + ( 34 2 ) 2 ] ( 7 2 + 58 2 ) 1000009=\left[\left(\frac{4}{2}\right)^{2}+\left(\frac{34}{2}\right)^{2}\right% ]\cdot\left(7^{2}+58^{2}\right)\,
  39. = ( 2 2 + 17 2 ) ( 7 2 + 58 2 ) =\left(2^{2}+17^{2}\right)\cdot\left(7^{2}+58^{2}\right)\,
  40. = ( 4 + 289 ) ( 49 + 3364 ) =(4+289)\cdot(49+3364)\,
  41. = 293 3413 =293\cdot 3413\,

Euler_sequence.html

  1. 0 Ω A n / A 1 𝒪 A n ( - 1 ) n + 1 𝒪 A n 0. 0\to\Omega^{1}_{\mathbb{P}^{n}_{A}/A}\to\mathcal{O}_{\mathbb{P}^{n}_{A}}(-1)^{% \oplus n+1}\to\mathcal{O}_{\mathbb{P}^{n}_{A}}\to 0.
  2. S ( - 1 ) n + 1 S , e i x i S(-1)^{\oplus n+1}\to S,e_{i}\mapsto x_{i}
  3. S = A [ x 0 , , x n ] S=A[x_{0},\ldots,x_{n}]
  4. e i = 1 e_{i}=1
  5. 1 \geq 1
  6. 0 𝒪 n 𝒪 ( 1 ) ( n + 1 ) 𝒯 n 0 0\to\mathcal{O}_{\mathbb{P}^{n}}\to\mathcal{O}(1)^{\oplus(n+1)}\to\mathcal{T}_% {\mathbb{P}^{n}}\to 0
  7. 0 𝒪 ( V ) 𝒪 ( V ) ( 1 ) V 𝒯 ( V ) 0 0\to\mathcal{O}_{\mathbb{P}(V)}\to\mathcal{O}_{\mathbb{P}(V)}(1)\otimes V\to% \mathcal{T}_{\mathbb{P}(V)}\to 0
  8. ( V ) \mathbb{P}(V)
  9. ( V ) \mathbb{P}(V)
  10. ( V ) \mathbb{P}(V)
  11. ω A n / A = 𝒪 A n ( - ( n + 1 ) ) \omega_{\mathbb{P}^{n}_{A}/A}=\mathcal{O}_{\mathbb{P}^{n}_{A}}(-(n+1))

Euler_spiral.html

  1. V ² / R V²/R
  2. 1 / R 1/R
  3. R R\,
  4. R c R_{c}\,
  5. θ \theta\,
  6. R R
  7. θ s \theta_{s}\,
  8. L , s L,s\,
  9. L s , s o L_{s},s_{o}\,
  10. 1 R = d θ d s s \frac{1}{R}=\frac{d\theta}{ds}\propto s
  11. R s = constant = R c s o Rs=\,\text{constant}=R_{c}s_{o}\,
  12. d θ d s = s R c s o \frac{d\theta}{ds}=\frac{s}{R_{c}s_{o}}
  13. d θ d s = 2 a 2 s \frac{d\theta}{ds}=2a^{2}s
  14. 2 a 2 = 1 R c s o 2a^{2}=\frac{1}{R_{c}s_{o}}
  15. a = 1 2 R c s o a=\frac{1}{\sqrt{2R_{c}s_{o}}}
  16. θ = ( a s ) 2 \theta=(as)^{2}\,
  17. x = 0 L cos θ d s = 0 L cos [ ( a s ) 2 ] d s \begin{aligned}\displaystyle x&\displaystyle=\int_{0}^{L}\cos\theta\,ds\\ &\displaystyle=\int_{0}^{L}\cos\left[(as)^{2}\right]ds\end{aligned}
  18. s = a s s^{\prime}=as\,
  19. d s = d s a ds=\frac{ds^{\prime}}{a}\,
  20. x = 1 a 0 L cos s 2 d s x=\frac{1}{a}\int_{0}^{L^{\prime}}\cos{s}^{2}ds
  21. y = 0 L sin θ d s = 0 L sin [ ( a s ) 2 ] d s = 1 a 0 L sin s 2 d s \begin{aligned}\displaystyle y&\displaystyle=\int_{0}^{L}\sin\theta\,ds\\ &\displaystyle=\int_{0}^{L}\sin\left[(as)^{2}\right]ds\\ &\displaystyle=\frac{1}{a}\int_{0}^{L^{\prime}}\sin{s}^{2}\,ds\end{aligned}
  22. C ( L ) \displaystyle C(L)
  23. cos θ = 1 - θ 2 2 ! + θ 4 4 ! - θ 6 6 ! + \cos\theta=1-\frac{\theta^{2}}{2!}+\frac{\theta^{4}}{4!}-\frac{\theta^{6}}{6!}+\cdots
  24. C ( L ) = 0 L cos s 2 d s = 0 L ( 1 - s 4 2 ! + s 8 4 ! - s 12 6 ! + ) d s = L - L 5 5 × 2 ! + L 9 9 × 4 ! - L 13 13 × 6 ! + \begin{aligned}\displaystyle C(L)&\displaystyle=\int_{0}^{L}\cos s^{2}\,ds\\ &\displaystyle=\int_{0}^{L}\left(1-\frac{s^{4}}{2!}+\frac{s^{8}}{4!}-\frac{s^{% 12}}{6!}+\cdots\right)\,ds\\ &\displaystyle=L-\frac{L^{5}}{5\times 2!}+\frac{L^{9}}{9\times 4!}-\frac{L^{13% }}{13\times 6!}+\cdots\end{aligned}
  25. sin θ = θ - θ 3 3 ! + θ 5 5 ! - θ 7 7 ! + \sin\theta=\theta-\frac{\theta^{3}}{3!}+\frac{\theta^{5}}{5!}-\frac{\theta^{7}% }{7!}+\cdots
  26. S ( L ) = 0 L sin s 2 d s = 0 L ( s 2 - s 6 3 ! + s 10 5 ! - s 14 7 ! + ) d s = L 3 3 - L 7 7 × 3 ! + L 11 11 × 5 ! - L 15 15 × 7 ! + \begin{aligned}\displaystyle S(L)&\displaystyle=\int_{0}^{L}\sin s^{2}\,ds\\ &\displaystyle=\int_{0}^{L}\left(s^{2}-\frac{s^{6}}{3!}+\frac{s^{10}}{5!}-% \frac{s^{14}}{7!}+\cdots\right)\,ds\\ &\displaystyle=\frac{L^{3}}{3}-\frac{L^{7}}{7\times 3!}+\frac{L^{11}}{11\times 5% !}-\frac{L^{15}}{15\times 7!}+\cdots\end{aligned}
  27. 2 R L = 2 R c L s = 1 a 2 2RL=2R_{c}L_{s}=\frac{1}{a^{2}}\,
  28. 1 R = L R c L s = 2 a 2 L \frac{1}{R}=\frac{L}{R_{c}L_{s}}=2a^{2}L\,
  29. x = 1 a 0 L cos s 2 d s x=\frac{1}{a}\int_{0}^{L^{\prime}}\cos s^{2}\,ds
  30. y = 1 a 0 L sin s 2 d s y=\frac{1}{a}\int_{0}^{L^{\prime}}\sin s^{2}\,ds\,
  31. L = a L L^{\prime}=aL\,
  32. a = 1 2 R c L s a=\frac{1}{\sqrt{2R_{c}L_{s}}}
  33. ( x , y ) (x,y)
  34. ( x , y ) (x′,y′)
  35. ( x , y ) (x′,y′)
  36. ( x , y ) (x,y)
  37. 1 / a 1/a
  38. 1 / a > 1 1/a>1
  39. R c = R c 2 R c L s = R c 2 L s \begin{aligned}\displaystyle R^{\prime}_{c}&\displaystyle=\frac{R_{c}}{\sqrt{2% R_{c}L_{s}}}\\ &\displaystyle=\sqrt{\frac{R_{c}}{2L_{s}}}\\ \end{aligned}
  40. L s = L s 2 R c L s = L s 2 R c \begin{aligned}\displaystyle L^{\prime}_{s}&\displaystyle=\frac{L_{s}}{\sqrt{2% R_{c}L_{s}}}\\ &\displaystyle=\sqrt{\frac{L_{s}}{2R_{c}}}\end{aligned}
  41. 2 R c L s = 2 R c 2 L s L s 2 R c = 2 2 = 1 \begin{aligned}\displaystyle 2R^{\prime}_{c}L^{\prime}_{s}&\displaystyle=2% \sqrt{\frac{R_{c}}{2L_{s}}}\sqrt{\frac{L_{s}}{2R_{c}}}\\ &\displaystyle=\tfrac{2}{2}\\ &\displaystyle=1\end{aligned}
  42. R c \displaystyle R_{c}
  43. θ s = L s 2 R c = 100 2 × 300 = 0.1667 radian \begin{aligned}\displaystyle\theta_{s}&\displaystyle=\frac{L_{s}}{2R_{c}}\\ &\displaystyle=\frac{100}{2\times 300}\\ &\displaystyle=0.1667\ \mbox{radian}\\ \end{aligned}
  44. 2 R c L s = 60 , 000 2R_{c}L_{s}=60,000\,
  45. R c = 3 6 m \displaystyle R^{\prime}_{c}=\tfrac{3}{\sqrt{6}}\mbox{m}
  46. 2 R c L s = 2 × 3 6 × 1 6 = 1 \begin{aligned}\displaystyle 2R^{\prime}_{c}L^{\prime}_{s}&\displaystyle=2% \times\tfrac{3}{\sqrt{6}}\times\tfrac{1}{\sqrt{6}}\\ &\displaystyle=1\end{aligned}
  47. θ s = L s 2 R c = 1 6 2 × 3 6 = 0.1667 radian \begin{aligned}\displaystyle\theta_{s}&\displaystyle=\frac{L^{\prime}_{s}}{2R^% {\prime}_{c}}\\ &\displaystyle=\frac{\tfrac{1}{\sqrt{6}}}{2\times\tfrac{3}{\sqrt{6}}}\\ &\displaystyle=0.1667\ \mbox{radian}\\ \end{aligned}
  48. θ s \theta_{s}\,
  49. x = 0 L cos s 2 d s x=\int_{0}^{L}\cos s^{2}ds
  50. y = 0 L sin s 2 d s y=\int_{0}^{L}\sin s^{2}ds
  51. 2 R c L s = 1 2R_{c}L_{s}=1\,\!
  52. θ s = L s 2 R c = L s 2 \theta_{s}=\frac{L_{s}}{2R_{c}}=L_{s}^{2}
  53. θ = θ s L 2 L s 2 = L 2 \theta=\theta_{s}\cdot\frac{L^{2}}{L_{s}^{2}}=L^{2}
  54. 1 R = d θ d L = 2 L \frac{1}{R}=\frac{d\theta}{dL}=2L
  55. 2 R c L s = 1 2R_{c}L_{s}=1
  56. 1 / R c = 2 L s 1/R_{c}=2L_{s}

Euler–Poisson–Darboux_equation.html

  1. u x , y + N ( u x + u y ) x + y = 0. u_{x,y}+\frac{N(u_{x}+u_{y})}{x+y}=0.

Even-hole-free_graph.html

  1. 𝒪 ( n 40 ) {\mathcal{O}}(n^{40})
  2. 𝒪 ( n 19 ) {\mathcal{O}}(n^{19})
  3. 𝒪 ( n 11 ) {\mathcal{O}}(n^{11})
  4. 𝒪 ( n 40 ) {\mathcal{O}}(n^{40})

Event_partitioning.html

  1. { x } m n {}_{m}^{n}\{x\}
  2. { x } 0 1 {}_{0}^{1}\{x\}

Event_segment.html

  1. Z Z
  2. 𝕋 \mathbb{T}
  3. 𝕋 = [ 0 , ) \mathbb{T}=[0,\infty)
  4. Z Z
  5. ϵ Z \epsilon\not\in Z
  6. ( t , z ) (t,z)
  7. t 𝕋 t\in\mathbb{T}
  8. z Z z\in Z
  9. z Z z\in Z
  10. t 𝕋 t\in\mathbb{T}
  11. [ t l , t u ] 𝕋 [t_{l},t_{u}]\subset\mathbb{T}
  12. ϵ [ t l , t u ] \epsilon_{[t_{l},t_{u}]}
  13. Z Z
  14. [ t l , t u ] [t_{l},t_{u}]
  15. Z Z
  16. ω \omega
  17. [ t 1 , t 2 ] [t_{1},t_{2}]
  18. ω \omega^{\prime}
  19. [ t 3 , t 4 ] [t_{3},t_{4}]
  20. ω ω \omega\omega^{\prime}
  21. [ t 1 , t 4 ] [t_{1},t_{4}]
  22. t 2 = t 3 t_{2}=t_{3}
  23. ( t 1 , z 1 ) ( t 2 , z 2 ) ( t n , z n ) (t_{1},z_{1})(t_{2},z_{2})\cdots(t_{n},z_{n})
  24. Z Z
  25. [ t l , t u ] 𝕋 [t_{l},t_{u}]\subset\mathbb{T}
  26. ϵ [ t l , t 1 ] , ( t 1 , z 1 ) , ϵ [ t 1 , t 2 ] , ( t 2 , z 2 ) , , ( t n , z n ) , \epsilon_{[t_{l},t_{1}]},(t_{1},z_{1}),\epsilon_{[t_{1},t_{2}]},(t_{2},z_{2}),% \ldots,(t_{n},z_{n}),
  27. ϵ [ t n , t u ] \epsilon_{[t_{n},t_{u}]}
  28. t l t 1 t 2 t n - 1 t n t u t_{l}\leq t_{1}\leq t_{2}\leq\cdots\leq t_{n-1}\leq t_{n}\leq t_{u}
  29. ω \omega
  30. [ t l , t u ] 𝕋 [t_{l},t_{u}]\subseteq\mathbb{T}
  31. Z Z
  32. ω : [ t l , t u ] Z * . \omega:[t_{l},t_{u}]\rightarrow Z^{*}.
  33. Ω Z , [ t l , t u ] \Omega_{Z,[t_{l},t_{u}]}
  34. Z Z
  35. [ t l , t u ] 𝕋 [t_{l},t_{u}]\subset\mathbb{T}
  36. Z Z
  37. [ t l , t u ] [t_{l},t_{u}]
  38. L L
  39. Z Z
  40. [ t l , t u ] [t_{l},t_{u}]
  41. Z Z
  42. [ t l , t u ] [t_{l},t_{u}]
  43. L Ω Z , [ t l , t u ] L\subseteq\Omega_{Z,[t_{l},t_{u}]}

Exact_division.html

  1. w 1 , w 2 , , w k w_{1},w_{2},...,w_{k}
  2. w 1 , w 2 , , w k w_{1},w_{2},...,w_{k}
  3. C = P 1 P k C=P_{1}\sqcup...\sqcup P_{k}
  4. V i ( P j ) = w j V_{i}(P_{j})=w_{j}
  5. w j w_{j}
  6. ϵ > 0 \epsilon>0
  7. ϵ \epsilon
  8. w 1 , w 2 , , w k w_{1},w_{2},...,w_{k}
  9. | V i ( P j ) - w j | < ϵ |V_{i}(P_{j})-w_{j}|<\epsilon
  10. w j w_{j}
  11. ϵ \epsilon
  12. w j w_{j}
  13. k R kR
  14. n × k n\times k
  15. ( i , j ) (i,j)
  16. M j M_{j}
  17. M j M_{j}
  18. w j w_{j}
  19. R R\to\infty
  20. n ( k - 1 ) n\cdot(k-1)
  21. 1 / n 1/n
  22. p [ 0 , 1 ] p\in[0,1]
  23. 2 ( k - 1 ) 2(k-1)
  24. k k
  25. k = 2 k=2
  26. n ( k - 1 ) = n n(k-1)=n
  27. n n
  28. n + 1 n+1
  29. S n S^{n}
  30. V : S n n V:S^{n}\to\mathbb{R}^{n}
  31. x x
  32. V ( x ) V(x)
  33. i i
  34. i i
  35. V V
  36. x x
  37. V ( x ) + V ( - x ) = 0 V(x)+V(-x)=0
  38. x x
  39. V ( x ) = 0 V(x)=0
  40. k = 2 k=2
  41. C C
  42. w [ 0 , 1 ] w\in[0,1]
  43. C w C_{w}
  44. n - 1 n-1
  45. w w
  46. i = 1 , , n : u i ( C w ) = w \forall i=1,\cdots,n:\,\,\,\,\,u_{i}(C_{w})=w
  47. W [ 0 , 1 ] W\subseteq[0,1]
  48. 1 W 1\in W
  49. C 1 := C C_{1}:=C
  50. w W w\in W
  51. 1 - w W 1-w\in W
  52. C 1 - w := C C w C_{1-w}:=C\setminus C_{w}
  53. C w C_{w}
  54. n - 1 n-1
  55. C 1 - w C_{1-w}
  56. n - 1 n-1
  57. W W
  58. n - 1 n-1
  59. w W w\in W
  60. w / 2 W w/2\in W
  61. W = [ 0 , 1 ] W=[0,1]
  62. C w C_{w}
  63. n - 1 n-1
  64. n n
  65. w w
  66. f : C n f:C\to\mathbb{R}^{n}
  67. f ( t ) = ( t , t 2 , , t n ) t [ 0 , 1 ] f(t)=(t,t^{2},...,t^{n})\,\,\,\,\,\,t\in[0,1]
  68. n \mathbb{R}^{n}
  69. v i ( U ) = u i ( f - 1 ( U ) C w ) U n v_{i}(U)=u_{i}(f^{-1}(U)\cap C_{w})\,\,\,\,\,\,\,\,\,U\subseteq\mathbb{R}^{n}
  70. f - 1 ( n ) = C f^{-1}(\mathbb{R}^{n})=C
  71. i i
  72. v i ( n ) = w v_{i}(\mathbb{R}^{n})=w
  73. n \mathbb{R}^{n}
  74. H , H H,H^{\prime}
  75. i = 1 , , n : v i ( H ) = v i ( H ) = w / 2 \forall i=1,\cdots,n:\,\,\,\,\,v_{i}(H)=v_{i}(H^{\prime})=w/2
  76. M = f - 1 ( H ) C w M=f^{-1}(H)\cap C_{w}
  77. M = f - 1 ( H ) C w M^{\prime}=f^{-1}(H^{\prime})\cap C_{w}
  78. v i v_{i}
  79. i = 1 , , n : u i ( M ) = u i ( M ) = w / 2 \forall i=1,\cdots,n:\,\,\,\,\,u_{i}(M)=u_{i}(M^{\prime})=w/2
  80. C w C_{w}
  81. n - 1 n-1
  82. f ( C w ) f(C_{w})
  83. n - 1 n-1
  84. n \mathbb{R}^{n}
  85. H H
  86. H H^{\prime}
  87. f ( C w ) f(C_{w})
  88. n n
  89. H f ( C w ) H\cap f(C_{w})
  90. H f ( C w ) H^{\prime}\cap f(C_{w})
  91. 2 n - 1 2n-1
  92. n - 1 n-1
  93. H H
  94. n - 1 n-1
  95. M M
  96. n - 1 n-1
  97. C w / 2 = M C_{w/2}=M
  98. w W w\in W
  99. n n
  100. n n
  101. ( n 1 ) (n−1)
  102. n \mathbb{R}^{n}
  103. n - 1 n-1
  104. ϵ > 0 \epsilon>0
  105. ϵ \epsilon
  106. | V i ( P j ) - w j | < ϵ |V_{i}(P_{j})-w_{j}|<\epsilon
  107. || v || n / k ||v||\leq\sqrt{n}/k
  108. 1 / n 1/n
  109. 1 / n 1/n
  110. n n
  111. k k
  112. 2 ( k - 1 ) 2(k-1)
  113. n n
  114. n ( k - 1 ) n(k-1)
  115. 2 n 2n
  116. n n
  117. n n
  118. 1 / n 1/n

Exact_statistics.html

  1. μ \mu
  2. σ 2 \sigma^{2}
  3. X ¯ \overline{X}
  4. S 2 S^{2}
  5. Z = n ( X ¯ - μ ) / σ N ( 0 , 1 ) Z=\sqrt{n}(\overline{X}-\mu)/\sigma\sim N(0,1)
  6. U = n S 2 / σ 2 χ n - 1 2 U=nS^{2}/\sigma^{2}\sim\chi^{2}_{n-1}
  7. ρ = μ / σ \rho=\mu/\sigma
  8. ρ \rho
  9. R = x ¯ S s σ - X ¯ - μ σ = x ¯ s U n - Z n R=\frac{\overline{x}S}{s\sigma}-\frac{\overline{X}-\mu}{\sigma}=\frac{% \overline{x}}{s}\frac{\sqrt{U}}{\sqrt{n}}~{}-~{}\frac{Z}{\sqrt{n}}
  10. x ¯ \overline{x}
  11. X ¯ \overline{X}
  12. S S
  13. s s
  14. ρ \rho
  15. R R

Example_of_a_game_without_a_value.html

  1. x x
  2. y y
  3. x , y [ 0 , 1 ] x,y\in[0,1]
  4. K ( x , y ) = { - 1 if x < y < x + 1 / 2 0 if x = y or y = x + 1 / 2 1 otherwise K(x,y)=\begin{cases}-1&\,\text{if }x<y<x+1/2\\ 0&\,\text{if }x=y\,\text{ or }y=x+1/2\\ 1&\,\text{otherwise}\end{cases}
  5. K ( x , y ) K(x,y)
  6. ( x , y ) (x,y)
  7. f f
  8. g g
  9. sup f inf g K d f d g = 1 3 \sup_{f}\inf_{g}\iint K\,df\,dg=\frac{1}{3}
  10. inf g sup f K d f d g = 3 7 . \inf_{g}\sup_{f}\iint K\,df\,dg=\frac{3}{7}.
  11. sup \sup
  12. inf \inf
  13. ϵ \epsilon
  14. ϵ < 1 2 ( 3 7 - 1 3 ) 0.0476 \epsilon<\frac{1}{2}\left(\frac{3}{7}-\frac{1}{3}\right)\simeq 0.0476
  15. { 0 , 1 / 2 , 1 } \left\{0,1/2,1\right\}
  16. { 1 / 4 , 1 / 2 , 1 } \left\{1/4,1/2,1\right\}
  17. { P | K ( P ) < c } \{P|K(P)<c\}
  18. { P | K ( P ) > c } \{P|K(P)>c\}

Exchange-rate_pass_through.html

  1. 25 % 50 % = .5. \frac{25\%}{50\%}=.5.
  2. Δ ln p t = α + i = 0 N γ i Δ ln e t - i + δ Δ ln c t + ψ Δ ln d t + ϵ t , \Delta\ln p_{t}=\alpha+\sum_{i=0}^{N}\gamma_{i}\Delta\ln e_{t-i}+\delta\Delta% \ln c_{t}+\psi\Delta\ln d_{t}+\epsilon_{t},
  3. p p
  4. e e
  5. c c
  6. d d
  7. Δ \Delta
  8. N N
  9. i = 0 N γ i . \sum_{i=0}^{N}\gamma_{i}.

Expected_transmission_count.html

  1. e p t e_{pt}
  2. ETX = 1 1 - e p t \mathrm{ETX}=\frac{1}{1-e_{pt}}

Experiment_(probability_theory).html

  1. \scriptstyle\mathcal{F}
  2. \scriptstyle\mathcal{F}
  3. \scriptstyle\mathcal{F}

Exploration_problem.html

  1. H p ( x ) = - p ( x ) log p ( x ) d x . H_{p}(x)=-\int p(x)\log p(x)\,dx.
  2. H p ( x ) H_{p}(x)
  3. I b ( u ) = H p ( x ) - E z [ H b ( x | z , u ) ] . I_{b}(u)=H_{p}(x)-E_{z}\left[H_{b}(x^{\prime}|z,u)\right].

Exponential_field.html

  1. E ( a + b ) = E ( a ) E ( b ) , E ( 0 F ) = 1 F \begin{aligned}&\displaystyle E(a+b)=E(a)\cdot E(b),\\ &\displaystyle E(0_{F})=1_{F}\end{aligned}
  2. 1 = E ( 0 ) = E ( x + x + + x p of these ) = E ( x ) E ( x ) E ( x ) = E ( x ) p . 1=E(0)=E(\underbrace{x+x+\ldots+x}_{p\,\text{ of these}})=E(x)E(x)\cdots E(x)=% E(x)^{p}.
  3. ( E ( x ) - 1 ) p = E ( x ) p - 1 p = E ( x ) p - 1 = 0. (E(x)-1)^{p}=E(x)^{p}-1^{p}=E(x)^{p}-1=0.\,
  4. n ( - 1 ) n . n\mapsto(-1)^{n}.

Exponentially_equivalent_measures.html

  1. { ω Ω | d ( Y ε ( ω ) , Z ε ( ω ) ) > δ } Σ ε , \big\{\omega\in\Omega\big|d(Y_{\varepsilon}(\omega),Z_{\varepsilon}(\omega))>% \delta\big\}\in\Sigma_{\varepsilon},
  2. lim sup ε 0 ε log 𝐏 ε [ d ( Y ε , Z ε ) > δ ] = - . \limsup_{\varepsilon\downarrow 0}\varepsilon\log\mathbf{P}_{\varepsilon}\big[d% (Y_{\varepsilon},Z_{\varepsilon})>\delta\big]=-\infty.

Extended_Kalman_filter.html

  1. s y m b o l x k = f ( s y m b o l x k - 1 , s y m b o l u k - 1 ) + s y m b o l w k - 1 symbol{x}_{k}=f(symbol{x}_{k-1},symbol{u}_{k-1})+symbol{w}_{k-1}
  2. s y m b o l z k = h ( s y m b o l x k ) + s y m b o l v k symbol{z}_{k}=h(symbol{x}_{k})+symbol{v}_{k}
  3. s y m b o l x ^ k | k - 1 = f ( s y m b o l x ^ k - 1 | k - 1 , s y m b o l u k - 1 ) \hat{symbol{x}}_{k|k-1}=f(\hat{symbol{x}}_{k-1|k-1},symbol{u}_{k-1})
  4. s y m b o l P k | k - 1 = s y m b o l F k - 1 s y m b o l P k - 1 | k - 1 s y m b o l F k - 1 + s y m b o l Q k - 1 symbol{P}_{k|k-1}={{symbol{F}_{k-1}}}symbol{P}_{k-1|k-1}{{symbol{F}_{k-1}^{% \top}}}+symbol{Q}_{k-1}
  5. s y m b o l y ~ k = s y m b o l z k - h ( s y m b o l x ^ k | k - 1 ) \tilde{symbol{y}}_{k}=symbol{z}_{k}-h(\hat{symbol{x}}_{k|k-1})
  6. s y m b o l S k = s y m b o l H k s y m b o l P k | k - 1 s y m b o l H k + s y m b o l R k symbol{S}_{k}={{symbol{H}_{k}}}symbol{P}_{k|k-1}{{symbol{H}_{k}^{\top}}}+% symbol{R}_{k}
  7. s y m b o l K k = s y m b o l P k | k - 1 s y m b o l H k s y m b o l S k - 1 symbol{K}_{k}=symbol{P}_{k|k-1}{{symbol{H}_{k}^{\top}}}symbol{S}_{k}^{-1}
  8. s y m b o l P k | k = ( s y m b o l I - s y m b o l K k s y m b o l H k ) s y m b o l P k | k - 1 symbol{P}_{k|k}=(symbol{I}-symbol{K}_{k}{{symbol{H}_{k}}})symbol{P}_{k|k-1}
  9. s y m b o l x ^ k | k = s y m b o l x ^ k | k - 1 + s y m b o l K k s y m b o l y ~ k \hat{symbol{x}}_{k|k}=\hat{symbol{x}}_{k|k-1}+symbol{K}_{k}\tilde{symbol{y}}_{k}
  10. s y m b o l F k - 1 = f s y m b o l x | s y m b o l x ^ k - 1 | k - 1 , s y m b o l u k - 1 {{symbol{F}_{k-1}}}=\left.\frac{\partial f}{\partial symbol{x}}\right|_{\hat{% symbol{x}}_{k-1|k-1},symbol{u}_{k-1}}
  11. s y m b o l H k = h s y m b o l x | s y m b o l x ^ k | k - 1 {{symbol{H}_{k}}}=\left.\frac{\partial h}{\partial symbol{x}}\right|_{\hat{% symbol{x}}_{k|k-1}}
  12. s y m b o l x k = f ( s y m b o l x k - 1 , s y m b o l u k - 1 , s y m b o l w k - 1 ) symbol{x}_{k}=f(symbol{x}_{k-1},symbol{u}_{k-1},symbol{w}_{k-1})
  13. s y m b o l z k = h ( s y m b o l x k , s y m b o l v k ) symbol{z}_{k}=h(symbol{x}_{k},symbol{v}_{k})
  14. s y m b o l P k | k - 1 = s y m b o l F k - 1 s y m b o l P k - 1 | k - 1 s y m b o l F k - 1 + s y m b o l L k - 1 s y m b o l Q k - 1 s y m b o l L k - 1 T symbol{P}_{k|k-1}={{{symbol{F}_{k-1}}}}{symbol{P}_{k-1|k-1}}{{{symbol{F}_{k-1}% ^{\top}}}}{+}{symbol{L}_{k-1}}{symbol{Q}_{k-1}}{symbol{L}^{T}_{k-1}}
  15. s y m b o l S k = s y m b o l H k s y m b o l P k | k - 1 s y m b o l H k + s y m b o l M k s y m b o l R k s y m b o l M k T symbol{S}_{k}={{symbol{H}_{k}}}{symbol{P}_{k|k-1}}{{symbol{H}_{k}^{\top}}}{+}{% symbol{M}_{k}}{symbol{R}_{k}}{symbol{M}_{k}^{T}}
  16. s y m b o l L k - 1 symbol{L}_{k-1}
  17. s y m b o l M k symbol{M}_{k}
  18. s y m b o l L k - 1 = f s y m b o l w | s y m b o l x ^ k - 1 | k - 1 , s y m b o l u k - 1 {{symbol{L}_{k-1}}}=\left.\frac{\partial f}{\partial symbol{w}}\right|_{\hat{% symbol{x}}_{k-1|k-1},symbol{u}_{k-1}}
  19. s y m b o l M k = h s y m b o l v | s y m b o l x ^ k | k - 1 {{symbol{M}_{k}}}=\left.\frac{\partial h}{\partial symbol{v}}\right|_{\hat{% symbol{x}}_{k|k-1}}
  20. 𝐱 ˙ ( t ) = f ( 𝐱 ( t ) , 𝐮 ( t ) ) + 𝐰 ( t ) , 𝐰 ( t ) N ( 𝟎 , 𝐐 ( t ) ) 𝐳 ( t ) = h ( 𝐱 ( t ) ) + 𝐯 ( t ) , 𝐯 ( t ) N ( 𝟎 , 𝐑 ( t ) ) \begin{aligned}\displaystyle\dot{\mathbf{x}}(t)&\displaystyle=f\bigl(\mathbf{x% }(t),\mathbf{u}(t)\bigr)+\mathbf{w}(t),&\displaystyle\mathbf{w}(t)&% \displaystyle\sim N\bigl(\mathbf{0},\mathbf{Q}(t)\bigr)\\ \displaystyle\mathbf{z}(t)&\displaystyle=h\bigl(\mathbf{x}(t)\bigr)+\mathbf{v}% (t),&\displaystyle\mathbf{v}(t)&\displaystyle\sim N\bigl(\mathbf{0},\mathbf{R}% (t)\bigr)\end{aligned}
  21. 𝐱 ^ ( t 0 ) = E [ 𝐱 ( t 0 ) ] , 𝐏 ( t 0 ) = V a r [ 𝐱 ( t 0 ) ] \hat{\mathbf{x}}(t_{0})=E\bigl[\mathbf{x}(t_{0})\bigr]\,\text{, }\mathbf{P}(t_% {0})=Var\bigl[\mathbf{x}(t_{0})\bigr]
  22. 𝐱 ^ ˙ ( t ) = f ( 𝐱 ^ ( t ) , 𝐮 ( t ) ) + 𝐊 ( t ) ( 𝐳 ( t ) - h ( 𝐱 ^ ( t ) ) ) 𝐏 ˙ ( t ) = 𝐅 ( t ) 𝐏 ( t ) + 𝐏 ( t ) 𝐅 ( t ) - 𝐊 ( t ) 𝐇 ( t ) 𝐏 ( t ) + 𝐐 ( t ) 𝐊 ( t ) = 𝐏 ( t ) 𝐇 ( t ) 𝐑 ( t ) - 1 𝐅 ( t ) = f 𝐱 | 𝐱 ^ ( t ) , 𝐮 ( t ) 𝐇 ( t ) = h 𝐱 | 𝐱 ^ ( t ) \begin{aligned}\displaystyle\dot{\hat{\mathbf{x}}}(t)&\displaystyle=f\bigl(% \hat{\mathbf{x}}(t),\mathbf{u}(t)\bigr)+\mathbf{K}(t)\Bigl(\mathbf{z}(t)-h% \bigl(\hat{\mathbf{x}}(t)\bigr)\Bigr)\\ \displaystyle\dot{\mathbf{P}}(t)&\displaystyle=\mathbf{F}(t)\mathbf{P}(t)+% \mathbf{P}(t)\mathbf{F}(t)^{\top}-\mathbf{K}(t)\mathbf{H}(t)\mathbf{P}(t)+% \mathbf{Q}(t)\\ \displaystyle\mathbf{K}(t)&\displaystyle=\mathbf{P}(t)\mathbf{H}(t)^{\top}% \mathbf{R}(t)^{-1}\\ \displaystyle\mathbf{F}(t)&\displaystyle=\left.\frac{\partial f}{\partial% \mathbf{x}}\right|_{\hat{\mathbf{x}}(t),\mathbf{u}(t)}\\ \displaystyle\mathbf{H}(t)&\displaystyle=\left.\frac{\partial h}{\partial% \mathbf{x}}\right|_{\hat{\mathbf{x}}(t)}\end{aligned}
  23. 𝐱 ˙ ( t ) = f ( 𝐱 ( t ) , 𝐮 ( t ) ) + 𝐰 ( t ) , 𝐰 ( t ) N ( 𝟎 , 𝐐 ( t ) ) 𝐳 k = h ( 𝐱 k ) + 𝐯 k , 𝐯 k N ( 𝟎 , 𝐑 k ) \begin{aligned}\displaystyle\dot{\mathbf{x}}(t)&\displaystyle=f\bigl(\mathbf{x% }(t),\mathbf{u}(t)\bigr)+\mathbf{w}(t),&\displaystyle\mathbf{w}(t)&% \displaystyle\sim N\bigl(\mathbf{0},\mathbf{Q}(t)\bigr)\\ \displaystyle\mathbf{z}_{k}&\displaystyle=h(\mathbf{x}_{k})+\mathbf{v}_{k},&% \displaystyle\mathbf{v}_{k}&\displaystyle\sim N(\mathbf{0},\mathbf{R}_{k})\end% {aligned}
  24. 𝐱 k = 𝐱 ( t k ) \mathbf{x}_{k}=\mathbf{x}(t_{k})
  25. 𝐱 ^ 0 | 0 = E [ 𝐱 ( t 0 ) ] , 𝐏 0 | 0 = V a r [ 𝐱 ( t 0 ) ] \hat{\mathbf{x}}_{0|0}=E\bigl[\mathbf{x}(t_{0})\bigr],\mathbf{P}_{0|0}=Var% \bigl[\mathbf{x}(t_{0})\bigr]
  26. solve { 𝐱 ^ ˙ ( t ) = f ( 𝐱 ^ ( t ) , 𝐮 ( t ) ) , 𝐏 ˙ ( t ) = 𝐅 ( t ) 𝐏 ( t ) + 𝐏 ( t ) 𝐅 ( t ) + 𝐐 ( t ) , with { 𝐱 ^ ( t k - 1 ) = 𝐱 ^ k - 1 | k - 1 , 𝐏 ( t k - 1 ) = 𝐏 k - 1 | k - 1 , { 𝐱 ^ k | k - 1 = 𝐱 ^ ( t k ) 𝐏 k | k - 1 = 𝐏 ( t k ) \begin{aligned}\displaystyle\,\text{solve }&\displaystyle\begin{cases}\dot{% \hat{\mathbf{x}}}(t)=f\bigl(\hat{\mathbf{x}}(t),\mathbf{u}(t)\bigr),\\ \dot{\mathbf{P}}(t)=\mathbf{F}(t)\mathbf{P}(t)+\mathbf{P}(t)\mathbf{F}(t)^{% \top}+\mathbf{Q}(t),\end{cases}\qquad\,\text{with }\begin{cases}\hat{\mathbf{x% }}(t_{k-1})=\hat{\mathbf{x}}_{k-1|k-1},\\ \mathbf{P}(t_{k-1})=\mathbf{P}_{k-1|k-1},\end{cases}\\ \displaystyle\Rightarrow&\displaystyle\begin{cases}\hat{\mathbf{x}}_{k|k-1}=% \hat{\mathbf{x}}(t_{k})\\ \mathbf{P}_{k|k-1}=\mathbf{P}(t_{k})\end{cases}\end{aligned}
  27. 𝐅 ( t ) = f 𝐱 | 𝐱 ^ ( t ) , 𝐮 ( t ) \mathbf{F}(t)=\left.\frac{\partial f}{\partial\mathbf{x}}\right|_{\hat{\mathbf% {x}}(t),\mathbf{u}(t)}
  28. 𝐊 k = 𝐏 k | k - 1 𝐇 k ( 𝐇 k 𝐏 k | k - 1 𝐇 k + 𝐑 k ) - 1 \mathbf{K}_{k}=\mathbf{P}_{k|k-1}\mathbf{H}_{k}^{\top}\bigl(\mathbf{H}_{k}% \mathbf{P}_{k|k-1}\mathbf{H}_{k}^{\top}+\mathbf{R}_{k}\bigr)^{-1}
  29. 𝐱 ^ k | k = 𝐱 ^ k | k - 1 + 𝐊 k ( 𝐳 k - h ( 𝐱 ^ k | k - 1 ) ) \hat{\mathbf{x}}_{k|k}=\hat{\mathbf{x}}_{k|k-1}+\mathbf{K}_{k}\bigl(\mathbf{z}% _{k}-h(\hat{\mathbf{x}}_{k|k-1})\bigr)
  30. 𝐏 k | k = ( 𝐈 - 𝐊 k 𝐇 k ) 𝐏 k | k - 1 \mathbf{P}_{k|k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k|k-1}
  31. 𝐇 k = h 𝐱 | 𝐱 ^ k | k - 1 \,\textbf{H}_{k}=\left.\frac{\partial h}{\partial\,\textbf{x}}\right|_{\hat{\,% \textbf{x}}_{k|k-1}}

Extended_negative_binomial_distribution.html

  1. m = 1 m=1
  2. m 1 m≥1
  3. p p
  4. r r
  5. r r
  6. p p
  7. f ( k ; m , r , p ) = 0 for k { 0 , 1 , , m - 1 } f(k;m,r,p)=0\qquad\,\text{ for }k\in\{0,1,\ldots,m-1\}
  8. f ( k ; m , r , p ) = ( k + r - 1 k ) p k ( 1 - p ) - r - j = 0 m - 1 ( j + r - 1 j ) p j for k with k m , f(k;m,r,p)=\frac{{k+r-1\choose k}p^{k}}{(1-p)^{-r}-\sum_{j=0}^{m-1}{j+r-1% \choose j}p^{j}}\quad\,\text{for }k\in{\mathbb{N}}\,\text{ with }k\geq m,
  9. ( k + r - 1 k ) = Γ ( k + r ) k ! Γ ( r ) = ( - 1 ) k ( - r k ) ( 1 ) {k+r-1\choose k}=\frac{\Gamma(k+r)}{k!\,\Gamma(r)}=(-1)^{k}\,{-r\choose k}% \qquad\qquad(1)
  10. Γ Γ
  11. f ( . ; m , r , p s ) f( . ;m,r,ps)
  12. s s∈
  13. ( 0 , 1 ] (0, 1]
  14. φ ( s ) = k = m f ( k ; m , r , p ) s k = ( 1 - p s ) - r - j = 0 m - 1 ( j + r - 1 j ) ( p s ) j ( 1 - p ) - r - j = 0 m - 1 ( j + r - 1 j ) p j for | s | 1 p . \begin{aligned}\displaystyle\varphi(s)&\displaystyle=\sum_{k=m}^{\infty}f(k;m,% r,p)s^{k}\\ &\displaystyle=\frac{(1-ps)^{-r}-\sum_{j=0}^{m-1}{\left({{j+r-1}\atop{j}}% \right)}(ps)^{j}}{(1-p)^{-r}-\sum_{j=0}^{m-1}{\left({{j+r-1}\atop{j}}\right)}p% ^{j}}\qquad\,\text{for }|s|\leq\frac{1}{p}.\end{aligned}
  15. m = 1 m=1
  16. r r∈
  17. ( 1 , 0 ) (–1, 0)
  18. φ ( s ) = 1 - ( 1 - p s ) - r 1 - ( 1 - p ) - r for | s | 1 p . \varphi(s)=\frac{1-(1-ps)^{-r}}{1-(1-p)^{-r}}\qquad\,\text{for }|s|\leq\frac{1% }{p}.

Exterior_angle_theorem.html

  1. b + d = 180 b+d=180^{\circ}
  2. b + d = b + a + c b+d=b+a+c
  3. d = a + c . \therefore d=a+c.
  4. d = a + c d=a+c
  5. b + d = 180 b+d=180^{\circ}
  6. b + a + c = 180 . \therefore b+a+c=180^{\circ}.

External_ventricular_drain.html

  1. C P P = M A P - I C P CPP=MAP-ICP

Extremal_length.html

  1. Γ \Gamma
  2. Γ \Gamma
  3. D D
  4. Γ \Gamma
  5. D D
  6. f : D D f:D\to D^{\prime}
  7. Γ \Gamma
  8. Γ \Gamma
  9. f f
  10. Γ \Gamma
  11. Γ \Gamma
  12. D D
  13. Γ \Gamma
  14. D D
  15. ρ : D [ 0 , ] \rho:D\to[0,\infty]
  16. γ \gamma
  17. L ρ ( γ ) := γ ρ | d z | L_{\rho}(\gamma):=\int_{\gamma}\rho\,|dz|
  18. ρ \rho
  19. γ \gamma
  20. | d z | |dz|
  21. L ρ ( γ ) = L_{\rho}(\gamma)=\infty
  22. γ : I D \gamma:I\to D
  23. I I
  24. γ ρ | d z | \int_{\gamma}\rho\,|dz|
  25. ρ ( γ ( t ) ) \rho(\gamma(t))
  26. I I
  27. J I J\subset I
  28. γ \gamma
  29. J J
  30. I ρ ( γ ( t ) ) d length γ ( t ) \int_{I}\rho(\gamma(t))\,d{\mathrm{length}}_{\gamma}(t)
  31. length γ ( t ) {\mathrm{length}}_{\gamma}(t)
  32. γ \gamma
  33. { s I : s t } \{s\in I:s\leq t\}
  34. L ρ ( Γ ) := inf γ Γ L ρ ( γ ) . L_{\rho}(\Gamma):=\inf_{\gamma\in\Gamma}L_{\rho}(\gamma).
  35. ρ \rho
  36. A ( ρ ) := D ρ 2 d x d y , A(\rho):=\int_{D}\rho^{2}\,dx\,dy,
  37. Γ \Gamma
  38. E L ( Γ ) := sup ρ L ρ ( Γ ) 2 A ( ρ ) , EL(\Gamma):=\sup_{\rho}\frac{L_{\rho}(\Gamma)^{2}}{A(\rho)}\,,
  39. ρ : D [ 0 , ] \rho:D\to[0,\infty]
  40. 0 < A ( ρ ) < 0<A(\rho)<\infty
  41. Γ \Gamma
  42. Γ 0 \Gamma_{0}
  43. Γ \Gamma
  44. E L ( Γ ) EL(\Gamma)
  45. E L ( Γ 0 ) EL(\Gamma_{0})
  46. Γ \Gamma
  47. 1 / E L ( Γ ) 1/EL(\Gamma)
  48. D D
  49. D ¯ \overline{D}
  50. D D
  51. w , h > 0 w,h>0
  52. R R
  53. R = ( 0 , w ) × ( 0 , h ) R=(0,w)\times(0,h)
  54. Γ \Gamma
  55. γ : ( 0 , 1 ) R \gamma:(0,1)\to R
  56. lim t 0 γ ( t ) \lim_{t\to 0}\gamma(t)
  57. { 0 } × [ 0 , h ] \{0\}\times[0,h]
  58. lim t 1 γ ( t ) \lim_{t\to 1}\gamma(t)
  59. { 1 } × [ 0 , h ] \{1\}\times[0,h]
  60. γ \gamma
  61. E L ( Γ ) = w / h EL(\Gamma)=w/h
  62. ρ = 1 \rho=1
  63. R R
  64. ρ \rho
  65. A ( ρ ) = w h A(\rho)=w\,h
  66. L ρ ( Γ ) = w L_{\rho}(\Gamma)=w
  67. E L ( Γ ) EL(\Gamma)
  68. E L ( Γ ) w / h EL(\Gamma)\geq w/h
  69. ρ : R [ 0 , ] \rho:R\to[0,\infty]
  70. := L ρ ( Γ ) > 0 \ell:=L_{\rho}(\Gamma)>0
  71. y ( 0 , h ) y\in(0,h)
  72. γ y ( t ) = i y + w t \gamma_{y}(t)=i\,y+w\,t
  73. \R 2 \R^{2}
  74. γ y Γ \gamma_{y}\in\Gamma
  75. L ρ ( γ y ) \ell\leq L_{\rho}(\gamma_{y})
  76. 0 1 ρ ( i y + w t ) w d t . \ell\leq\int_{0}^{1}\rho(i\,y+w\,t)\,w\,dt.
  77. y ( 0 , h ) y\in(0,h)
  78. h 0 h 0 1 ρ ( i y + w t ) w d t d y h\,\ell\leq\int_{0}^{h}\int_{0}^{1}\rho(i\,y+w\,t)\,w\,dt\,dy
  79. x = w t x=w\,t
  80. h 0 h 0 w ρ ( x + i y ) d x d y ( R ρ 2 d x d y R d x d y ) 1 / 2 = ( w h A ( ρ ) ) 1 / 2 h\,\ell\leq\int_{0}^{h}\int_{0}^{w}\rho(x+i\,y)\,dx\,dy\leq\Bigl(\int_{R}\rho^% {2}\,dx\,dy\int_{R}\,dx\,dy\Bigr)^{1/2}=\bigl(w\,h\,A(\rho)\bigr)^{1/2}
  81. 2 / A ( ρ ) w / h \ell^{2}/A(\rho)\leq w/h
  82. E L ( Γ ) w / h EL(\Gamma)\leq w/h
  83. Γ \Gamma
  84. { γ y : y ( 0 , h ) } \{\gamma_{y}:y\in(0,h)\}
  85. Γ \Gamma\,^{\prime}
  86. R R
  87. R R
  88. E L ( Γ ) = h / w EL(\Gamma\,^{\prime})=h/w
  89. E L ( Γ ) E L ( Γ ) = 1 EL(\Gamma)\,EL(\Gamma\,^{\prime})=1
  90. E L ( Γ ) EL(\Gamma)
  91. ρ \rho
  92. L ρ ( Γ ) 2 / A ( ρ ) L_{\rho}(\Gamma)^{2}/A(\rho)
  93. ρ \rho
  94. E L ( Γ ) E L ( Γ ) = 1 EL(\Gamma)\,EL(\Gamma\,^{\prime})=1
  95. E L ( Γ ) EL(\Gamma\,^{\prime})
  96. E L ( Γ ) EL(\Gamma)
  97. r 1 r_{1}
  98. r 2 r_{2}
  99. 0 < r 1 < r 2 < 0<r_{1}<r_{2}<\infty
  100. A A
  101. A := { z : r 1 < | z | < r 2 } A:=\{z\in\mathbb{C}:r_{1}<|z|<r_{2}\}
  102. C 1 C_{1}
  103. C 2 C_{2}
  104. A A
  105. C 1 := { z : | z | = r 1 } C_{1}:=\{z:|z|=r_{1}\}
  106. C 2 := { z : | z | = r 2 } C_{2}:=\{z:|z|=r_{2}\}
  107. A A
  108. C 1 C_{1}
  109. C 2 C_{2}
  110. Γ \Gamma
  111. γ A \gamma\subset A
  112. C 1 C_{1}
  113. C 2 C_{2}
  114. E L ( Γ ) EL(\Gamma)
  115. ρ ( z ) = 1 / | z | \rho(z)=1/|z|
  116. γ Γ \gamma\in\Gamma
  117. C 1 C_{1}
  118. C 2 C_{2}
  119. γ | z | - 1 d s γ | z | - 1 d | z | = γ d log | z | = log ( r 2 / r 1 ) . \int_{\gamma}|z|^{-1}\,ds\geq\int_{\gamma}|z|^{-1}\,d|z|=\int_{\gamma}d\log|z|% =\log(r_{2}/r_{1}).
  120. A ( ρ ) = A | z | - 2 d x d y = 0 2 π r 1 r 2 r - 2 r d r d θ = 2 π log ( r 2 / r 1 ) . A(\rho)=\int_{A}|z|^{-2}\,dx\,dy=\int_{0}^{2\pi}\int_{r_{1}}^{r_{2}}r^{-2}\,r% \,dr\,d\theta=2\,\pi\,\log(r_{2}/r_{1}).
  121. E L ( Γ ) log ( r 2 / r 1 ) 2 π . EL(\Gamma)\geq\frac{\log(r_{2}/r_{1})}{2\pi}.
  122. ρ \rho
  123. := L ρ ( Γ ) > 0 \ell:=L_{\rho}(\Gamma)>0
  124. θ [ 0 , 2 π ) \theta\in[0,2\,\pi)
  125. γ θ : ( r 1 , r 2 ) A \gamma_{\theta}:(r_{1},r_{2})\to A
  126. γ θ ( r ) = e i θ r \gamma_{\theta}(r)=e^{i\theta}r
  127. γ θ ρ d s = r 1 r 2 ρ ( e i θ r ) d r . \ell\leq\int_{\gamma_{\theta}}\rho\,ds=\int_{r_{1}}^{r_{2}}\rho(e^{i\theta}r)% \,dr.
  128. θ \theta
  129. 2 π A ρ d r d θ ( A ρ 2 r d r d θ ) 1 / 2 ( 0 2 π r 1 r 2 1 r d r d θ ) 1 / 2 . 2\,\pi\,\ell\leq\int_{A}\rho\,dr\,d\theta\leq\Bigl(\int_{A}\rho^{2}\,r\,dr\,d% \theta\Bigr)^{1/2}\Bigl(\int_{0}^{2\pi}\int_{r_{1}}^{r_{2}}\frac{1}{r}\,dr\,d% \theta\Bigr)^{1/2}.
  130. 4 π 2 2 A ( ρ ) 2 π log ( r 2 / r 1 ) . 4\,\pi^{2}\,\ell^{2}\leq A(\rho)\cdot\,2\,\pi\,\log(r_{2}/r_{1}).
  131. E L ( Γ ) ( 2 π ) - 1 log ( r 2 / r 1 ) EL(\Gamma)\leq(2\,\pi)^{-1}\,\log(r_{2}/r_{1})
  132. E L ( Γ ) = log ( r 2 / r 1 ) 2 π . EL(\Gamma)=\frac{\log(r_{2}/r_{1})}{2\pi}.
  133. r 1 , r 2 , C 1 , C 2 , Γ r_{1},r_{2},C_{1},C_{2},\Gamma
  134. A A
  135. Γ * \Gamma^{*}
  136. C 1 C_{1}
  137. C 2 C_{2}
  138. E L ( Γ * ) = 2 π log ( r 2 / r 1 ) = E L ( Γ ) - 1 . EL(\Gamma^{*})=\frac{2\pi}{\log(r_{2}/r_{1})}=EL(\Gamma)^{-1}.
  139. ρ \rho
  140. L ρ ( Γ ) 2 / A ( ρ ) L_{\rho}(\Gamma)^{2}/A(\rho)
  141. ρ \rho
  142. \R 3 \R^{3}
  143. x - x x\mapsto-x
  144. Γ \Gamma
  145. Γ \Gamma
  146. π 2 / ( 2 π ) = π / 2 \pi^{2}/(2\,\pi)=\pi/2
  147. Γ \Gamma
  148. z 0 z_{0}
  149. E L ( Γ ) = EL(\Gamma)=\infty
  150. ρ ( z ) := { ( - | z - z 0 | log | z - z 0 | ) - 1 | z - z 0 | < 1 / 2 , 0 | z - z 0 | 1 / 2 , \rho(z):=\begin{cases}(-|z-z_{0}|\,\log|z-z_{0}|)^{-1}&|z-z_{0}|<1/2,\\ 0&|z-z_{0}|\geq 1/2,\end{cases}
  151. A ( ρ ) < A(\rho)<\infty
  152. L ρ ( γ ) = L_{\rho}(\gamma)=\infty
  153. γ Γ \gamma\in\Gamma
  154. Γ 1 Γ 2 \Gamma_{1}\subset\Gamma_{2}
  155. E L ( Γ 1 ) E L ( Γ 2 ) EL(\Gamma_{1})\geq EL(\Gamma_{2})
  156. γ 1 Γ 1 \gamma_{1}\in\Gamma_{1}
  157. γ 2 Γ 2 \gamma_{2}\in\Gamma_{2}
  158. γ 2 \gamma_{2}
  159. γ 1 \gamma_{1}
  160. E L ( Γ 1 Γ 2 ) ( E L ( Γ 1 ) - 1 + E L ( Γ 2 ) - 1 ) - 1 . EL(\Gamma_{1}\cup\Gamma_{2})\geq\bigl(EL(\Gamma_{1})^{-1}+EL(\Gamma_{2})^{-1}% \bigr)^{-1}.
  161. E L ( Γ 1 ) = 0 EL(\Gamma_{1})=0
  162. E L ( Γ 2 ) = 0 EL(\Gamma_{2})=0
  163. 0
  164. Γ 1 Γ 2 \Gamma_{1}\cup\Gamma_{2}
  165. ρ 1 , ρ 2 \rho_{1},\rho_{2}
  166. L ρ j ( Γ j ) 1 L_{\rho_{j}}(\Gamma_{j})\geq 1
  167. j = 1 , 2 j=1,2
  168. ρ = max { ρ 1 , ρ 2 } \rho=\max\{\rho_{1},\rho_{2}\}
  169. L ρ ( Γ 1 Γ 2 ) 1 L_{\rho}(\Gamma_{1}\cup\Gamma_{2})\geq 1
  170. A ( ρ ) = ρ 2 d x d y ( ρ 1 2 + ρ 2 2 ) d x d y = A ( ρ 1 ) + A ( ρ 2 ) A(\rho)=\int\rho^{2}\,dx\,dy\leq\int(\rho_{1}^{2}+\rho_{2}^{2})\,dx\,dy=A(\rho% _{1})+A(\rho_{2})
  171. f : D D * f:D\to D^{*}
  172. Γ \Gamma
  173. D D
  174. Γ * := { f γ : γ Γ } \Gamma^{*}:=\{f\circ\gamma:\gamma\in\Gamma\}
  175. f f
  176. E L ( Γ ) = E L ( Γ * ) EL(\Gamma)=EL(\Gamma^{*})
  177. Γ 0 \Gamma_{0}
  178. γ Γ \gamma\in\Gamma
  179. f γ f\circ\gamma
  180. Γ 0 * = { f γ : γ Γ 0 } \Gamma_{0}^{*}=\{f\circ\gamma:\gamma\in\Gamma_{0}\}
  181. Γ * \Gamma^{*}
  182. ρ * : D * [ 0 , ] \rho^{*}:D^{*}\to[0,\infty]
  183. ρ ( z ) = | f ( z ) | ρ * ( f ( z ) ) . \rho(z)=|f\,^{\prime}(z)|\,\rho^{*}\bigl(f(z)\bigr).
  184. w = f ( z ) w=f(z)
  185. A ( ρ ) = D ρ ( z ) 2 d z d z ¯ = D ρ * ( f ( z ) ) 2 | f ( z ) | 2 d z d z ¯ = D * ρ * ( w ) 2 d w d w ¯ = A ( ρ * ) . A(\rho)=\int_{D}\rho(z)^{2}\,dz\,d\bar{z}=\int_{D}\rho^{*}(f(z))^{2}\,|f\,^{% \prime}(z)|^{2}\,dz\,d\bar{z}=\int_{D^{*}}\rho^{*}(w)^{2}\,dw\,d\bar{w}=A(\rho% ^{*}).
  186. γ Γ 0 \gamma\in\Gamma_{0}
  187. γ * := f γ \gamma^{*}:=f\circ\gamma
  188. L ρ ( γ ) = γ ρ * ( f ( z ) ) | f ( z ) | | d z | = γ * ρ ( w ) | d w | = L ρ * ( γ * ) . L_{\rho}(\gamma)=\int_{\gamma}\rho^{*}\bigl(f(z)\bigr)\,|f\,^{\prime}(z)|\,|dz% |=\int_{\gamma^{*}}\rho(w)\,|dw|=L_{\rho^{*}}(\gamma^{*}).
  189. γ \gamma
  190. I I
  191. ( t ) \ell(t)
  192. γ \gamma
  193. I ( - , t ] I\cap(-\infty,t]
  194. * ( t ) \ell^{*}(t)
  195. γ * \gamma^{*}
  196. γ \gamma
  197. d * ( t ) = | f ( γ ( t ) ) | d ( t ) d\ell^{*}(t)=|f\,^{\prime}(\gamma(t))|\,d\ell(t)
  198. L ρ ( γ ) = L ρ * ( γ * ) L_{\rho}(\gamma)=L_{\rho^{*}}(\gamma^{*})
  199. E L ( Γ 0 ) E L ( Γ 0 * ) = E L ( Γ * ) . EL(\Gamma_{0})\geq EL(\Gamma_{0}^{*})=EL(\Gamma^{*}).
  200. Γ \Gamma
  201. Γ * \Gamma^{*}
  202. E L ( Γ ) = E L ( Γ * ) EL(\Gamma)=EL(\Gamma^{*})
  203. f f
  204. Γ \Gamma
  205. Γ * \Gamma^{*}
  206. Γ ^ \hat{\Gamma}
  207. γ Γ \gamma\in\Gamma
  208. f γ f\circ\gamma
  209. E L ( Γ ^ ) = EL(\hat{\Gamma})=\infty
  210. ρ ( z ) = | f ( z ) | h ( | f ( z ) | ) \rho(z)=|f\,^{\prime}(z)|\,h(|f(z)|)
  211. h ( r ) = ( r log ( r + 2 ) ) - 1 h(r)=\bigl(r\,\log(r+2)\bigr)^{-1}
  212. A ( ρ ) = D * h ( | w | ) 2 d w d w ¯ 0 2 π 0 ( r log ( r + 2 ) ) - 2 r d r d θ < . A(\rho)=\int_{D^{*}}h(|w|)^{2}\,dw\,d\bar{w}\leq\int_{0}^{2\pi}\int_{0}^{% \infty}(r\,\log(r+2))^{-2}\,r\,dr\,d\theta<\infty.
  213. γ Γ ^ \gamma\in\hat{\Gamma}
  214. r ( 0 , ) r\in(0,\infty)
  215. f γ f\circ\gamma
  216. { z : | z | < r } \{z:|z|<r\}
  217. L ρ ( γ ) inf { h ( s ) : s [ 0 , r ] } length ( f γ ) = L_{\rho}(\gamma)\geq\inf\{h(s):s\in[0,r]\}\,\mathrm{length}(f\circ\gamma)=\infty
  218. γ Γ ^ \gamma\in\hat{\Gamma}
  219. f γ f\circ\gamma
  220. H ( t ) := 0 t h ( s ) d s H(t):=\int_{0}^{t}h(s)\,ds
  221. L ρ ( γ ) L_{\rho}(\gamma)
  222. t H ( | f γ ( t ) | ) t\mapsto H(|f\circ\gamma(t)|)
  223. \R \R
  224. \R \R
  225. lim t H ( t ) = \lim_{t\to\infty}H(t)=\infty
  226. L ρ ( γ ) = L_{\rho}(\gamma)=\infty
  227. E L ( Γ ^ ) = EL(\hat{\Gamma})=\infty
  228. E L ( Γ ) = E L ( Γ 0 Γ ^ ) E L ( Γ 0 ) EL(\Gamma)=EL(\Gamma_{0}\cup\hat{\Gamma})\geq EL(\Gamma_{0})
  229. E L ( Γ 0 ) E L ( Γ * ) EL(\Gamma_{0})\geq EL(\Gamma^{*})
  230. E L ( Γ ) E L ( Γ * ) EL(\Gamma)\geq EL(\Gamma^{*})
  231. { z : r < | z | < R } \{z:r<|z|<R\}
  232. 0 r < R 0\leq r<R\leq\infty
  233. { w : r * < | w | < R * } \{w:r^{*}<|w|<R^{*}\}
  234. R r R * r * \frac{R}{r}\neq\frac{R^{*}}{r^{*}}
  235. G = ( V , E ) G=(V,E)
  236. Γ \Gamma
  237. G G
  238. ρ : E [ 0 , ) \rho:E\to[0,\infty)
  239. ρ \rho
  240. ρ ( e ) \rho(e)
  241. A ( ρ ) A(\rho)
  242. e E ρ ( e ) 2 \sum_{e\in E}\rho(e)^{2}
  243. Γ \Gamma
  244. G G
  245. ρ : V [ 0 , ) \rho:V\to[0,\infty)
  246. A ( ρ ) := v V ρ ( v ) 2 A(\rho):=\sum_{v\in V}\rho(v)^{2}
  247. ρ ( v ) \rho(v)