wpmath0000015_2

Deviation_of_a_local_ring.html

  1. P ( x ) = n 0 x n Tor n R ( k , k ) = n 0 ( 1 + t 2 n + 1 ) ε 2 n ( 1 - t 2 n + 2 ) ε 2 n + 1 . P(x)=\sum_{n\geq 0}x^{n}\operatorname{Tor}^{R}_{n}(k,k)=\prod_{n\geq 0}\frac{(% 1+t^{2n+1})^{\varepsilon_{2n}}}{(1-t^{2n+2})^{\varepsilon_{2n+1}}}.

Dewar_reactivity_number.html

  1. Δ E = 2 β ( a r + a s ) = β N i \Delta E=2\beta(a_{r}+a_{s})=\beta N_{i}
  2. N i = 2 ( a r + a s ) N_{i}=2(a_{r}+a_{s})

Diacetyl_reductase_((R)-acetoin_forming).html

  1. \rightleftharpoons

Diacetyl_reductase_((S)-acetoin_forming).html

  1. \rightleftharpoons

Diacylglycerol_diphosphate_phosphatase.html

  1. \rightleftharpoons

Diacylglycerol_kinase_(CTP_dependent).html

  1. \rightleftharpoons

Diadenosine_hexaphosphate_hydrolase_(AMP-forming).html

  1. \rightleftharpoons
  2. \rightleftharpoons

Diadenosine_hexaphosphate_hydrolase_(ATP-forming).html

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

Diapolycopene_oxygenase.html

  1. \rightleftharpoons

Dichloroarcyriaflavin_A_synthase.html

  1. \rightleftharpoons

Diffraction_efficiency.html

  1. P P
  2. P 0 P_{0}
  3. η \eta
  4. η = P P 0 . \eta=\frac{P}{P_{0}}\ .

Dihydrocarveol_dehydrogenase.html

  1. \rightleftharpoons

Dihydroorotate_dehydrogenase_(fumarate).html

  1. \rightleftharpoons

Dihydroorotate_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Dimension_(graph_theory).html

  1. n n
  2. n n
  3. G G
  4. d i m G dim\,G
  5. E 2 E^{2}
  6. E 1 E^{1}
  7. K n K_{n}
  8. n - 1 n-1
  9. K 3 K_{3}
  10. K 4 K_{4}
  11. K 4 , 2 K_{4,2}
  12. K m , 1 K_{m,1}
  13. m 3 m\geq 3
  14. m m
  15. K m , 2 K_{m,2}
  16. m 3 m\geq 3
  17. m m
  18. K 2 , 2 K_{2,2}
  19. w , x , y , z w,x,y,z
  20. w 2 + x 2 = a , y = 0 , z = 0 w^{2}+x^{2}=a,y=0,z=0
  21. 0 < a < 1 0<a<1
  22. y 2 + z 2 = 1 - a , w = 0 , x = 0 y^{2}+z^{2}=1-a,w=0,x=0
  23. w 2 + x 2 + y 2 + z 2 = a + 1 - a = 1 {\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}}={\sqrt{a+1-a}}=1
  24. K 3 , 3 K_{3,3}
  25. A 1 A_{1}
  26. A 2 A_{2}
  27. A 3 A_{3}
  28. B 1 B_{1}
  29. B 2 B_{2}
  30. B 3 B_{3}
  31. A 1 A_{1}
  32. A 2 A_{2}
  33. A 3 A_{3}
  34. B 1 B_{1}
  35. B 2 B_{2}
  36. B 3 B_{3}
  37. A 1 A_{1}
  38. A 2 A_{2}
  39. A 3 A_{3}
  40. d i m K m , n = 1 , 2 , 3 or 4 dim\,K_{m,n}=1,2,3\,\text{ or }4
  41. m m
  42. n n
  43. n n
  44. G G
  45. ( 1.. n ) (1..n)
  46. n n
  47. 2 n 2n
  48. x 1 , x 2 , . . x 2 n x_{1},x_{2},..x_{2n}
  49. n n
  50. x 2 n - 2 2 + x 2 n - 1 2 = 1 / 2 , x i | ( i 2 n - 2 , i 2 n - 1 ) = 0 x_{2n-2}^{2}+x_{2n-1}^{2}=1/2,\quad x_{i}|(i\neq 2{n-2},i\neq 2{n-1})=0
  51. p p
  52. q q
  53. x 2 p - 2 2 + x 2 p - 1 2 + x 2 q - 2 2 + x 2 q - 1 2 = 1 / 2 + 1 / 2 = 1 {\sqrt{x_{2p-2}^{2}+x_{2p-1}^{2}+x_{2q-2}^{2}+x_{2q-1}^{2}}}={\sqrt{1/2+1/2}}=1
  54. n n
  55. G G
  56. n n
  57. E d i m G Edim\,G
  58. d i m G E d i m G dim\,G\leq Edim\,G

Dimethyl-sulfide_monooxygenase.html

  1. \rightleftharpoons

Dimethyl_sulfide:cytochrome_c2_reductase.html

  1. \rightleftharpoons

Dimethylamine-corrinoid_protein_Co-methyltransferase.html

  1. \rightleftharpoons

Dimethylglycine_N-methyltransferase.html

  1. \rightleftharpoons

Dimethylsulfone_reductase.html

  1. \rightleftharpoons

Dinoflagellate_luciferase.html

  1. \rightleftharpoons

Diophantine_quintuple.html

  1. { a 1 , a 2 , a 3 , a 4 , , a m } \{a_{1},a_{2},a_{3},a_{4},\ldots,a_{m}\}
  2. a i a j + 1 a_{i}a_{j}+1
  3. 1 i < j m 1\leq i<j\leq m
  4. { 1 16 , 33 16 , 17 4 , 105 16 } \left\{\frac{1}{16},\frac{33}{16},\frac{17}{4},\frac{105}{16}\right\}
  5. a i a j + 1 a_{i}a_{j}+1
  6. { 1 , 3 , 8 , 120 } \{1,3,8,120\}
  7. 777480 8288641 \frac{777480}{8288641}

Dirac_equation_in_curved_spacetime.html

  1. i γ a e a μ D μ Ψ - m Ψ = 0. i\gamma^{a}e_{a}^{\mu}D_{\mu}\Psi-m\Psi=0.
  2. D μ = μ - i 4 ω μ a b σ a b , D_{\mu}=\partial_{\mu}-\frac{i}{4}\omega_{\mu}^{ab}\sigma_{ab},
  3. σ a b = i 2 [ γ a , γ b ] , \sigma_{ab}=\frac{i}{2}\left[\gamma_{a},\gamma_{b}\right],
  4. ω < s u b > μ a b ω<sub>μ^{ab}

Discrete_Universal_Denoiser.html

  1. x n = ( x 1 x n ) x^{n}=\left(x_{1}\ldots x_{n}\right)
  2. z n = ( z 1 z n ) z^{n}=\left(z_{1}\ldots z_{n}\right)
  3. z n z^{n}
  4. x n x^{n}
  5. k k
  6. 2 k + 1 2k+1
  7. z n z^{n}
  8. x ^ i \hat{x}_{i}
  9. k k
  10. ( z i - k , , z i - 1 , z i + 1 , , z i + k ) \left(z_{i-k},\ldots,z_{i-1},z_{i+1},\ldots,z_{i+k}\right)
  11. z i z_{i}
  12. z n z^{n}
  13. x n x^{n}
  14. X n X^{n}
  15. X i | Z i - k , , Z i - 1 , Z i + 1 , , Z i + k X_{i}|Z_{i-k},\ldots,Z_{i-1},Z_{i+1},\ldots,Z_{i+k}
  16. X i X_{i}
  17. ( Z i - k , , Z i - 1 , Z i + 1 , , Z i + k ) \left(Z_{i-k},\ldots,Z_{i-1},Z_{i+1},\ldots,Z_{i+k}\right)
  18. X ^ i \hat{X}_{i}
  19. X i | Z i - k , , Z i - 1 , Z i + 1 , , Z i + k X_{i}|Z_{i-k},\ldots,Z_{i-1},Z_{i+1},\ldots,Z_{i+k}
  20. Z i | Z i - k , , Z i - 1 , Z i + 1 , , Z i + k Z_{i}|Z_{i-k},\ldots,Z_{i-1},Z_{i+1},\ldots,Z_{i+k}
  21. Z i Z_{i}
  22. Z n Z^{n}
  23. n n
  24. k k
  25. n n
  26. 𝒵 \mathcal{Z}
  27. O ( n ) O(n)
  28. O ( min ( n , | 𝒵 | 2 k ) ) O\left(\min(n,|\mathcal{Z}|^{2k})\right)
  29. n n
  30. k = k n k=k_{n}
  31. 𝐱 \mathbf{x}
  32. 𝐗 \mathbf{X}
  33. 𝐗 \mathbf{X}
  34. 𝐱 \mathbf{x}
  35. x ^ i \hat{x}_{i}
  36. ( z i - k , , z i + k ) \left(z_{i-k},\ldots,z_{i+k}\right)
  37. 𝐱 \mathbf{x}
  38. 𝒳 \mathcal{X}
  39. x n = ( x 1 , , x n ) 𝒳 n x^{n}=\left(x_{1},\ldots,x_{n}\right)\in\mathcal{X}^{n}
  40. x i x_{i}
  41. Z i Z_{i}
  42. 𝒵 \mathcal{Z}
  43. 𝒳 \mathcal{X}
  44. 𝒵 \mathcal{Z}
  45. Π \Pi
  46. π ( x , z ) = ( Z = z | X = x ) \pi(x,z)=\mathbb{P}\left(Z=z\,|\,X=x\right)
  47. π z \pi_{z}
  48. z z
  49. Π \Pi
  50. Z n = ( z 1 , , z n ) 𝒵 n Z^{n}=\left(z_{1},\ldots,z_{n}\right)\in\mathcal{Z}^{n}
  51. z n z^{n}
  52. X ^ n : 𝒵 n 𝒳 n \hat{X}^{n}:\mathcal{Z}^{n}\to\mathcal{X}^{n}
  53. x n x^{n}
  54. z n z^{n}
  55. x ^ n = X ^ n ( z n ) = ( X ^ 1 ( z n ) , , X ^ n ( z n ) ) \hat{x}^{n}=\hat{X}^{n}\left(z^{n}\right)=\left(\hat{X}_{1}(z^{n}),\ldots,\hat% {X}_{n}(z^{n})\right)
  56. X ^ n \hat{X}^{n}
  57. Λ : 𝒳 × 𝒳 [ 0 , ) \Lambda:\mathcal{X}\times\mathcal{X}\to[0,\infty)
  58. X ^ n \hat{X}^{n}
  59. ( x n , z n ) (x^{n},z^{n})
  60. L X ^ n ( x n , z n ) = 1 n i = 1 n Λ ( x i , X ^ i ( z n ) ) . \begin{aligned}\displaystyle L_{\hat{X}^{n}}\left(x^{n},z^{n}\right)=\frac{1}{% n}\sum_{i=1}^{n}\Lambda\left(x_{i}\,,\,\hat{X}_{i}(z^{n})\right)\,.\end{aligned}
  61. 𝒳 \mathcal{X}
  62. 𝒳 = ( a 1 , , a | 𝒳 | ) \mathcal{X}=\left(a_{1},\ldots,a_{|\mathcal{X}|}\right)
  63. | 𝒳 | |\mathcal{X}|
  64. | 𝒳 | |\mathcal{X}|
  65. λ x ^ = ( Λ ( a 1 , x ^ ) Λ ( a | 𝒳 | , x ^ ) ) . \begin{aligned}\displaystyle\lambda_{\hat{x}}=\left(\begin{array}[]{c}\Lambda(% a_{1},\hat{x})\\ \vdots\\ \Lambda(a_{|\mathcal{X}|},\hat{x})\end{array}\right)\,.\end{aligned}
  66. k k
  67. k + 1 i n - k k+1\leq i\leq n-k
  68. i i
  69. z n z^{n}
  70. l k ( z n , i ) = ( z i - k , , z i - 1 ) l^{k}(z^{n},i)=\left(z_{i-k},\ldots,z_{i-1}\right)
  71. r k ( z n , i ) = ( z i + 1 , , z i + k ) r^{k}(z^{n},i)=\left(z_{i+1},\ldots,z_{i+k}\right)
  72. ( l k , r k ) (l^{k},r^{k})
  73. z n z^{n}
  74. ( l k , r k ) 𝒵 k × 𝒵 k (l^{k},r^{k})\in\mathcal{Z}^{k}\times\mathcal{Z}^{k}
  75. z n z^{n}
  76. 𝒵 \mathcal{Z}
  77. z z
  78. μ ( z n , l k , r k ) [ z ] = | { k + 1 i n - k | ( z i - k , , z i + k ) = l k z r k } | | { k + 1 i n - k | l k ( z n , i ) = l k and r k ( z n , i ) = r k } | . \begin{aligned}\displaystyle\mu\left(z^{n},l^{k},r^{k}\right)[z]=\frac{\Big|% \left\{k+1\leq i\leq n-k\,\,|\,\,(z_{i-k},\ldots,z_{i+k})=l^{k}zr^{k}\right\}% \Big|}{\Big|\left\{k+1\leq i\leq n-k\,\,|\,\,l^{k}(z^{n},i)=l^{k}\,\text{ and % }r^{k}(z^{n},i)=r^{k}\right\}\Big|}\,.\end{aligned}
  79. k k
  80. z n z^{n}
  81. | 𝒵 | |\mathcal{Z}|
  82. μ ( z n , l k , r k ) \mu\left(z^{n},l^{k},r^{k}\right)
  83. z n z^{n}
  84. N n , k = min ( n , | 𝒵 | 2 k ) N_{n,k}=\min\left(n,|\mathcal{Z}|^{2k}\right)
  85. z n z^{n}
  86. O ( n ) O(n)
  87. O ( N n , k ) O(N_{n,k})
  88. Λ \Lambda
  89. x ^ 𝒳 \hat{x}\in\mathcal{X}
  90. λ x ^ \lambda_{\hat{x}}
  91. 𝐯 \mathbf{v}
  92. | 𝒳 | |\mathcal{X}|
  93. X ^ B a y e s ( 𝐯 ) = argmin x ^ 𝒳 λ x ^ 𝐯 . \begin{aligned}\displaystyle\hat{X}_{Bayes}(\mathbf{v})=\,\text{argmin}_{\hat{% x}\in\mathcal{X}}\lambda_{\hat{x}}^{\top}\mathbf{v}\,.\end{aligned}
  94. ( l k , r k ) (l^{k},r^{k})
  95. z n z^{n}
  96. z 𝒵 z\in\mathcal{Z}
  97. z z
  98. l r z r k l^{r}zr^{k}
  99. z n z^{n}
  100. Π - μ ( z n , l k , r k ) π z \Pi^{-\top}\mu\left(z^{n}\,,\,l^{k}\,,\,r^{k}\right)\odot\pi_{z}
  101. g ( l k , z , r k ) := X ^ B a y e s ( Π - μ ( z n , l k , r k ) π z ) . \begin{aligned}\displaystyle g(l^{k},z,r^{k}):=\hat{X}_{Bayes}\left(\Pi^{-\top% }\mu\left(z^{n}\,,\,l^{k}\,,\,r^{k}\right)\odot\pi_{z}\right)\,.\end{aligned}
  102. z n z^{n}
  103. k k
  104. π z \pi_{z}
  105. z z
  106. Π \Pi
  107. 𝐚 \mathbf{a}
  108. 𝐛 \mathbf{b}
  109. 𝐚 𝐛 \mathbf{a}\odot\mathbf{b}
  110. ( 𝐚 𝐛 ) i = a i b i \left(\mathbf{a}\odot\mathbf{b}\right)_{i}=a_{i}b_{i}
  111. Π - μ π z \Pi^{-\top}\mu\odot\pi_{z}
  112. ( Π - μ ) π z (\Pi^{-\top}\mu)\odot\pi_{z}
  113. Π \Pi
  114. | 𝒳 | = | 𝒵 | |\mathcal{X}|=|\mathcal{Z}|
  115. | 𝒳 | | 𝒵 | |\mathcal{X}|\leq|\mathcal{Z}|
  116. Π \Pi
  117. ( Π ) - 1 (\Pi^{\top})^{-1}
  118. ( Π Π ) - 1 Π \left(\Pi\Pi^{\top}\right)^{-1}\Pi
  119. g ( l k , z , r k ) := X ^ B a y e s ( ( Π Π ) - 1 Π μ ( z n , l k , r k ) π z ) . \begin{aligned}\displaystyle g(l^{k},z,r^{k}):=\hat{X}_{Bayes}\left((\Pi\Pi^{% \top})^{-1}\Pi\mu\left(z^{n},l^{k},r^{k}\right)\odot\pi_{z}\right)\,.\end{aligned}
  120. Π - \Pi^{-\top}
  121. λ x ^ π z \lambda_{\hat{x}}\odot\pi_{z}
  122. ( x ^ , z ) 𝒳 × 𝒵 (\hat{x},z)\in\mathcal{X}\times\mathcal{Z}
  123. O ( N k , n ) O(N_{k,n})
  124. O ( N k , n ) O(N_{k,n})
  125. z n z^{n}
  126. X ^ n ( z n ) = ( X ^ 1 ( z n ) , , X ^ n ( z n ) ) \hat{X}^{n}(z^{n})=\left(\hat{X}_{1}(z^{n}),\ldots,\hat{X}_{n}(z^{n})\right)
  127. z i z_{i}
  128. X ^ i ( z n ) := g ( l k ( z n , i ) , z i , r k ( z n , i ) ) . \begin{aligned}\displaystyle\hat{X}_{i}(z^{n}):=g\left(l^{k}(z^{n},i)\,,\,z_{i% }\,,\,r^{k}(z^{n},i)\right)\,.\end{aligned}
  129. O ( n ) O(n)
  130. O ( n ) O(n)
  131. O ( N k , n ) O(N_{k,n})
  132. x n x^{n}
  133. X ^ D U D E n : 𝒵 n 𝒳 n \hat{X}^{n}_{DUDE}:\mathcal{Z}^{n}\to\mathcal{X}^{n}
  134. X ^ D U D E n \hat{X}^{n}_{DUDE}
  135. k n k_{n}
  136. lim n k n = \lim_{n\to\infty}k_{n}=\infty
  137. k n | 𝒵 | 2 K n = o ( n log n ) k_{n}|\mathcal{Z}|^{2K_{n}}=o\left(\frac{n}{\log n}\right)
  138. 𝒟 n \mathcal{D}_{n}
  139. n n
  140. X ^ n : 𝒵 n 𝒳 n \hat{X}^{n}:\mathcal{Z}^{n}\to\mathcal{X}^{n}
  141. 𝐗 \mathbf{X}
  142. 𝐙 \mathbf{Z}
  143. lim n 𝐄 [ L X ^ D U D E n ( X n , Z n ) ] = lim n min X ^ n 𝒟 n 𝐄 [ L X ^ n ( X n , Z n ) ] , \begin{aligned}\displaystyle\lim_{n\to\infty}\mathbf{E}\left[L_{\hat{X}^{n}_{% DUDE}}\left(X^{n},Z^{n}\right)\right]=\lim_{n\to\infty}\min_{\hat{X}^{n}\in% \mathcal{D}_{n}}\mathbf{E}\left[L_{\hat{X}^{n}}\left(X^{n},Z^{n}\right)\right]% \,,\end{aligned}
  144. 𝐗 \mathbf{X}
  145. lim sup n L X ^ D U D E n ( X n , Z n ) = lim n min X ^ n 𝒟 n 𝐄 [ L X ^ n ( X n , Z n ) ] , almost surely . \begin{aligned}\displaystyle\limsup_{n\to\infty}L_{\hat{X}^{n}_{DUDE}}\left(X^% {n},Z^{n}\right)=\lim_{n\to\infty}\min_{\hat{X}^{n}\in\mathcal{D}_{n}}\mathbf{% E}\left[L_{\hat{X}^{n}}\left(X^{n},Z^{n}\right)\right]\,,\,\,\text{ almost % surely}\,.\end{aligned}
  146. 𝒟 n , k \mathcal{D}_{n,k}
  147. n n
  148. k k
  149. X ^ n : 𝒵 𝒳 \hat{X}^{n}:\mathcal{Z}\to\mathcal{X}
  150. X ^ i ( z n ) = f ( z i - k , , z i + k ) \hat{X}_{i}(z^{n})=f\left(z_{i-k},\ldots,z_{i+k}\right)
  151. f : 𝒵 2 k + 1 𝒳 f:\mathcal{Z}^{2k+1}\to\mathcal{X}
  152. 𝐱 𝒳 \mathbf{x}\in\mathcal{X}^{\infty}
  153. 𝐙 \mathbf{Z}
  154. lim n [ L X ^ D U D E n ( x n , Z n ) - min X ^ n 𝒟 n , k L X ^ n ( x n , Z n ) ] = 0 , almost surely . \begin{aligned}\displaystyle\lim_{n\to\infty}\left[L_{\hat{X}^{n}_{DUDE}}\left% (x^{n},Z^{n}\right)-\min_{\hat{X}^{n}\in\mathcal{D}_{n,k}}L_{\hat{X}^{n}}\left% (x^{n},Z^{n}\right)\right]=0\,,\,\,\text{ almost surely}\,.\end{aligned}
  155. X ^ k n \hat{X}^{n}_{k}
  156. k k
  157. n n
  158. A , C > 0 A,C>0
  159. B > 1 B>1
  160. ( Π , Λ ) \left(\Pi,\Lambda\right)
  161. n , k n,k
  162. x n 𝒳 n x^{n}\in\mathcal{X}^{n}
  163. A n B k 𝐄 [ L X ^ k n ( x n , Z n ) - min X ^ n 𝒟 n , k L X ^ n ( x n , Z n ) ] k C n | 𝒵 | k , \begin{aligned}\displaystyle\frac{A}{\sqrt{n}}B^{k}\,\leq\mathbf{E}\left[L_{% \hat{X}^{n}_{k}}\left(x^{n},Z^{n}\right)-\min_{\hat{X}^{n}\in\mathcal{D}_{n,k}% }L_{\hat{X}^{n}}\left(x^{n},Z^{n}\right)\right]\leq\sqrt{k}\frac{C}{\sqrt{n}}|% \mathcal{Z}|^{k}\,,\end{aligned}
  164. Z n Z^{n}
  165. x n x^{n}
  166. A , B A,B
  167. n n
  168. X ^ n 𝒟 n \hat{X}^{n}\in\mathcal{D}^{n}
  169. Π \Pi
  170. ( Π , Λ ) \left(\Pi,\Lambda\right)
  171. x n x^{n}
  172. X n X^{n}
  173. n = 1 n=1
  174. ( X , Z ) (X,Z)
  175. z z
  176. X 𝒳 X\in\mathcal{X}
  177. ( X = x | Z = z ) \mathbb{P}(X=x|Z=z)
  178. 𝒳 \mathcal{X}
  179. 𝒳 \mathcal{X}
  180. 𝐏 X | z \mathbf{P}_{X|z}
  181. 𝒳 \mathcal{X}
  182. x x
  183. ( X = x | Z = z ) \mathbb{P}\left(X=x|Z=z\right)
  184. x ^ \hat{x}
  185. λ x ^ 𝐏 X | z \lambda_{\hat{x}}^{\top}\mathbf{P}_{X|z}
  186. 𝐯 \mathbf{v}
  187. 𝒳 \mathcal{X}
  188. U ( 𝐯 ) = min x ^ 𝒳 𝐯 λ x ^ U(\mathbf{v})=\min_{\hat{x}\in\mathcal{X}}\mathbf{v}^{\top}\lambda_{\hat{x}}
  189. 𝐯 \mathbf{v}
  190. X ^ B a y e s ( 𝐯 ) = argmin x ^ 𝒳 𝐯 λ x ^ \hat{X}_{Bayes}(\mathbf{v})=\,\text{argmin}_{\hat{x}\in\mathcal{X}}\mathbf{v}^% {\top}\lambda_{\hat{x}}
  191. X ^ B a y e s ( 𝐯 ) = X ^ B a y e s ( α 𝐯 ) \hat{X}_{Bayes}(\mathbf{v})=\hat{X}_{Bayes}(\alpha\mathbf{v})
  192. α > 0 \alpha>0
  193. n = 1 n=1
  194. X ^ ( z ) = X ^ B a y e s ( 𝐏 X | z ) \hat{X}(z)=\hat{X}_{Bayes}\left(\mathbf{P}_{X|z}\right)
  195. Z Z
  196. Π \Pi
  197. 𝐏 X | z Π - P Z π z \mathbf{P}_{X|z}\propto\Pi^{-\top}P_{Z}\odot\pi_{z}
  198. π z \pi_{z}
  199. z z
  200. Π \Pi
  201. X ^ ( z ) = X ^ B a y e s ( Π - 𝐏 Z π z ) \hat{X}(z)=\hat{X}_{Bayes}\left(\Pi^{-\top}\mathbf{P}_{Z}\odot\pi_{z}\right)
  202. | 𝒳 | | 𝒵 | |\mathcal{X}|\leq|\mathcal{Z}|
  203. Π \Pi
  204. Π - 1 \Pi^{-1}
  205. X ^ ( z ) = X ^ B a y e s ( ( Π Π ) - 1 Π 𝐏 Z π z ) . \hat{X}(z)=\hat{X}_{Bayes}\left((\Pi\Pi^{\top})^{-1}\Pi\mathbf{P}_{Z}\odot\pi_% {z}\right)\,.
  206. n n
  207. X ^ o p t ( z n ) \hat{X}^{opt}(z^{n})
  208. 𝐏 X i | z n \mathbf{P}_{X_{i}|z^{n}}
  209. X ^ i o p t ( z n ) = X ^ B a y e s 𝐏 X i | z n = argmin x ^ 𝒳 λ x ^ 𝐏 X i | z n , \begin{aligned}\displaystyle\hat{X}^{opt}_{i}(z^{n})=\hat{X}_{Bayes}\mathbf{P}% _{X_{i}|z^{n}}=\,\text{argmin}_{\hat{x}\in\mathcal{X}}\lambda_{\hat{x}}^{\top}% \mathbf{P}_{X_{i}|z^{n}}\,,\end{aligned}
  210. 𝐏 X i | z n \mathbf{P}_{X_{i}|z^{n}}
  211. 𝒳 \mathcal{X}
  212. x x
  213. ( X i = x | Z n = z n ) \mathbb{P}\left(X_{i}=x|Z^{n}=z^{n}\right)
  214. 𝐏 X i | z n \mathbf{P}_{X_{i}|z^{n}}
  215. n = 1 n=1
  216. X ^ i o p t ( z n ) = X ^ B a y e s ( Π - 𝐏 Z i , z n \ i π z i ) \hat{X}^{opt}_{i}(z^{n})=\hat{X}_{Bayes}\left(\Pi^{-\top}\mathbf{P}_{Z_{i},z^{% n\backslash i}}\odot\pi_{z_{i}}\right)
  217. z n \ i = ( z 1 , , z i - 1 , z i + 1 , , z n ) 𝒵 n - 1 z^{n\backslash i}=\left(z_{1},\ldots,z_{i-1},z_{i+1},\ldots,z_{n}\right)\in% \mathcal{Z}^{n-1}
  218. 𝐏 Z i , z n \ i \mathbf{P}_{Z_{i},z^{n\backslash i}}
  219. 𝒵 \mathcal{Z}
  220. z z
  221. ( ( Z 1 , , Z n ) = ( z 1 , , z i - 1 , z , z i + 1 , , z n ) ) . \mathbb{P}\left((Z_{1},\ldots,Z_{n})=(z_{1},\ldots,z_{i-1},z,z_{i+1},\ldots,z_% {n})\right)\,.
  222. Π - \Pi^{-\top}
  223. Π \Pi
  224. X X
  225. Z Z
  226. 𝐏 Z i , z n \ i \mathbf{P}_{Z_{i},z^{n\backslash i}}
  227. z n z^{n}
  228. μ ( Z i , l k ( Z n , i ) , r k ( Z n , i ) ) \mu\left(Z_{i},l^{k}(Z^{n},i),r^{k}(Z^{n},i)\right)
  229. k k
  230. k k

Dispersive_body_waves.html

  1. d U ( r , w ) d r - i k U ( r , w ) = 0 ( 1.1 ) \frac{dU(r,w)}{dr}-ikU(r,w)=0\quad(1.1)
  2. U ( r + r , w ) = U ( r , w ) exp ( i k r ) ( 1.2 ) U(r+\bigtriangleup r,w)=U(r,w)\exp(ik\bigtriangleup r)\quad(1.2)
  3. K ( i w ) = k ( w ) + i a ( w ) ( 1.3 ) K(iw)=k(w)+ia(w)\quad(1.3)
  4. K ( i w ) = H ( a ( w ) ) + i a ( w ) ( 1.4 ) K(iw)=H(a(w))+ia(w)\quad(1.4)
  5. U ( r + r , w ) = U ( r , w ) exp ( [ - b w + i H ( b w ) ] r ) ( 1.5 ) U(r+\bigtriangleup r,w)=U(r,w)\exp([-bw+iH(bw)]\bigtriangleup r)\quad(1.5)
  6. U ( r , w ) = U 0 e x p ( i w t ) ( 1.6 ) U(r,w)=U_{0}exp(iwt)\quad(1.6)
  7. U ( t + t , w ) = U e x p ( i w t ) exp ( [ - b w + i H ( b w ) ] t ) ( 1.7 ) U(t+\bigtriangleup t,w)=U^{\prime}exp(iw\bigtriangleup t)\exp([-bw+iH(bw)]% \bigtriangleup t)\quad(1.7)
  8. U ( t + t ) = 0 U ( t + t , w ) d w . ( 1.8 ) U(t+\bigtriangleup t)=\int_{0}^{\infty}U(t+\bigtriangleup t,w)dw.\quad(1.8)
  9. U ( t + t , w ) = U ( t , w ) exp ( | w w h | - γ | w | t 2 Q ( w ) ) exp ( i | w w h | - γ w t ) ( 1.8. a ) U(t+\bigtriangleup t,w)=U(t,w)\exp\bigg(|\frac{w}{w_{h}}|^{-\gamma}\frac{|w|% \bigtriangleup t}{2Q(w)}\bigg)\exp\bigg(i|\frac{w}{w_{h}}|^{-\gamma}w% \bigtriangleup t\bigg)\quad(1.8.a)
  10. γ = ( π Q r ) - 1 \gamma=(\pi Q_{r})^{-1}
  11. b = ( 1 2 Q ) b=(\frac{1}{2Q})

Distance_between_two_straight_lines.html

  1. y = m x + b 1 y=mx+b_{1}\,
  2. y = m x + b 2 , y=mx+b_{2}\,,
  3. y = - x / m , y=-x/m\,,
  4. { y = m x + b 1 y = - x / m , \begin{cases}y=mx+b_{1}\\ y=-x/m\,,\end{cases}
  5. { y = m x + b 2 y = - x / m , \begin{cases}y=mx+b_{2}\\ y=-x/m\,,\end{cases}
  6. ( x 1 , y 1 ) = ( - b 1 m m 2 + 1 , b 1 m 2 + 1 ) , \left(x_{1},y_{1}\right)\ =\left(\frac{-b_{1}m}{m^{2}+1},\frac{b_{1}}{m^{2}+1}% \right)\,,
  7. ( x 2 , y 2 ) = ( - b 2 m m 2 + 1 , b 2 m 2 + 1 ) . \left(x_{2},y_{2}\right)\ =\left(\frac{-b_{2}m}{m^{2}+1},\frac{b_{2}}{m^{2}+1}% \right)\,.
  8. d = ( b 1 m - b 2 m m 2 + 1 ) 2 + ( b 2 - b 1 m 2 + 1 ) 2 , d=\sqrt{\left(\frac{b_{1}m-b_{2}m}{m^{2}+1}\right)^{2}+\left(\frac{b_{2}-b_{1}% }{m^{2}+1}\right)^{2}}\,,
  9. d = | b 2 - b 1 | m 2 + 1 . d=\frac{|b_{2}-b_{1}|}{\sqrt{m^{2}+1}}\,.
  10. a x + b y + c 1 = 0 ax+by+c_{1}=0\,
  11. a x + b y + c 2 = 0 , ax+by+c_{2}=0,\,
  12. d = | c 2 - c 1 | a 2 + b 2 . d=\frac{|c_{2}-c_{1}|}{\sqrt{a^{2}+b^{2}}}.

Distorted_Schwarzschild_metric.html

  1. ( 1 ) d s 2 = - e 2 ψ ( ρ , z ) d t 2 + e 2 γ ( ρ , z ) - 2 ψ ( ρ , z ) ( d ρ 2 + d z 2 ) + e - 2 ψ ( ρ , z ) ρ 2 d ϕ 2 , (1)\quad ds^{2}=-e^{2\psi(\rho,z)}dt^{2}+e^{2\gamma(\rho,z)-2\psi(\rho,z)}(d% \rho^{2}+dz^{2})+e^{-2\psi(\rho,z)}\rho^{2}d\phi^{2}\,,
  2. ( 2 ) ψ S S = 1 2 ln L - M L + M , γ S S = 1 2 ln L 2 - M 2 l + l - , (2)\quad\psi_{SS}=\frac{1}{2}\ln\frac{L-M}{L+M}\,,\quad\gamma_{SS}=\frac{1}{2}% \ln\frac{L^{2}-M^{2}}{l_{+}l_{-}}\,,
  3. ( 3 ) L = 1 2 ( l + + l - ) , l + = ρ 2 + ( z + M ) 2 , l - = ρ 2 + ( z - M ) 2 , (3)\quad L=\frac{1}{2}\big(l_{+}+l_{-}\big)\,,\quad l_{+}=\sqrt{\rho^{2}+(z+M)% ^{2}}\,,\quad l_{-}=\sqrt{\rho^{2}+(z-M)^{2}}\,,
  4. ( 4 ) d s 2 = - L - M L + M d t 2 + ( L + M ) 2 l + l - ( d ρ 2 + d z 2 ) + L + M L - M ρ 2 d ϕ 2 . (4)\quad ds^{2}=-\frac{L-M}{L+M}dt^{2}+\frac{(L+M)^{2}}{l_{+}l_{-}}(d\rho^{2}+% dz^{2})+\frac{L+M}{L-M}\,\rho^{2}d\phi^{2}\,.
  5. ( 5. a ) 2 ψ = 0 , (5.a)\quad\nabla^{2}\psi=0\,,
  6. ( 5. b ) γ , ρ = ρ ( ψ , ρ 2 - ψ , z 2 ) , (5.b)\quad\gamma_{,\,\rho}=\rho\,\Big(\psi^{2}_{,\,\rho}-\psi^{2}_{,\,z}\Big)\,,
  7. ( 5. c ) γ , z = 2 ρ ψ , ρ ψ , z , (5.c)\quad\gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z}\,,
  8. ( 5. d ) γ , ρ ρ + γ , z z = - ( ψ , ρ 2 + ψ , z 2 ) , (5.d)\quad\gamma_{,\,\rho\rho}+\gamma_{,\,zz}=-\big(\psi^{2}_{,\,\rho}+\psi^{2% }_{,\,z}\big)\,,
  9. 2 := ρ ρ + 1 ρ ρ + z z \nabla^{2}:=\partial_{\rho\rho}+\frac{1}{\rho}\partial_{\rho}+\partial_{zz}
  10. R a b = 0 R_{ab}=0
  11. R = 0 R=0
  12. { ψ 1 , ψ 2 } \{\psi^{\langle 1\rangle},\psi^{\langle 2\rangle}\}
  13. ( 6 ) ψ ~ = ψ 1 + ψ 2 , (6)\quad\tilde{\psi}\,=\,\psi^{\langle 1\rangle}+\psi^{\langle 2\rangle}\,,
  14. ( 7 ) γ ~ = γ 1 + γ 2 + 2 ρ { ( ψ , ρ 1 ψ , ρ 2 - ψ , z 1 ψ , z 2 ) d ρ + ( ψ , ρ 1 ψ , z 2 + ψ , z 1 ψ , ρ 2 ) d z } . (7)\quad\tilde{\gamma}\,=\,\gamma^{\langle 1\rangle}+\gamma^{\langle 2\rangle}% +2\int\rho\,\Big\{\,\Big(\psi^{\langle 1\rangle}_{,\,\rho}\psi^{\langle 2% \rangle}_{,\,\rho}-\psi^{\langle 1\rangle}_{,\,z}\psi^{\langle 2\rangle}_{,\,z% }\Big)\,d\rho+\Big(\psi^{\langle 1\rangle}_{,\,\rho}\psi^{\langle 2\rangle}_{,% \,z}+\psi^{\langle 1\rangle}_{,\,z}\psi^{\langle 2\rangle}_{,\,\rho}\Big)\,dz% \,\Big\}\,.
  15. ψ 1 = ψ S S \psi^{\langle 1\rangle}=\psi_{SS}
  16. γ 1 = γ S S \gamma^{\langle 1\rangle}=\gamma_{SS}
  17. ψ 2 = ψ \psi^{\langle 2\rangle}=\psi
  18. γ 2 = γ \gamma^{\langle 2\rangle}=\gamma
  19. { ψ ~ , γ ~ } \{\tilde{\psi},\tilde{\gamma}\}
  20. ( 8 ) d s 2 = - e 2 ψ ( ρ , z ) L - M L + M d t 2 + e 2 γ ( ρ , z ) - 2 ψ ( ρ , z ) ( L + M ) 2 l + l - ( d ρ 2 + d z 2 ) + e - 2 ψ ( ρ , z ) L + M L - M ρ 2 d ϕ 2 . (8)\quad ds^{2}=-e^{2\psi(\rho,z)}\frac{L-M}{L+M}dt^{2}+e^{2\gamma(\rho,z)-2% \psi(\rho,z)}\frac{(L+M)^{2}}{l_{+}l_{-}}(d\rho^{2}+dz^{2})+e^{-2\psi(\rho,z)}% \frac{L+M}{L-M}\,\rho^{2}d\phi^{2}\,.
  21. ( 9 ) L + M = r , l + + l - = 2 M cos θ , z = ( r - M ) cos θ , (9)\quad L+M=r\,,\quad l_{+}+l_{-}=2M\cos\theta\,,\quad z=(r-M)\cos\theta\,,
  22. ρ = r 2 - 2 M r sin θ , l + l - = ( r - M ) 2 - M 2 cos 2 θ , \;\;\quad\rho=\sqrt{r^{2}-2Mr}\,\sin\theta\,,\quad l_{+}l_{-}=(r-M)^{2}-M^{2}% \cos^{2}\theta\,,
  23. { t , r , θ , ϕ } \{t,r,\theta,\phi\}
  24. ( 10 ) d s 2 = - e 2 ψ ( r , θ ) ( 1 - 2 M r ) d t 2 + e 2 γ ( r , θ ) - 2 ψ ( r , θ ) { ( 1 - 2 M r ) - 1 d r 2 + r 2 d θ 2 } + e - 2 ψ ( r , θ ) r 2 sin 2 θ d ϕ 2 . (10)\quad ds^{2}=-e^{2\psi(r,\theta)}\,\Big(1-\frac{2M}{r}\Big)\,dt^{2}+e^{2% \gamma(r,\theta)-2\psi(r,\theta)}\Big\{\,\Big(1-\frac{2M}{r}\Big)^{-1}dr^{2}+r% ^{2}d\theta^{2}\,\Big\}+e^{-2\psi(r,\theta)}r^{2}\sin^{2}\theta\,d\phi^{2}\,.
  25. { ψ ( ρ , z ) = 0 , γ ( ρ , z ) = 0 } \{\psi(\rho,z)=0,\gamma(\rho,z)=0\}
  26. ( 11 ) d s 2 = - ( 1 - 2 M r ) d t 2 + ( 1 - 2 M r ) - 1 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 . (11)\quad ds^{2}=-\Big(1-\frac{2M}{r}\Big)\,dt^{2}+\Big(1-\frac{2M}{r}\Big)^{-% 1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}\,.
  27. ψ ( r , θ ) \psi(r,\theta)
  28. ( 12 ) ψ ( r , θ ) = - i = 1 a i ( R n ( cos θ ) M ) P i (12)\quad\psi(r,\theta)\,=-\sum_{i=1}^{\infty}a_{i}\Big(\frac{R_{n}(\cos\theta% )}{M}\Big)P_{i}
  29. R := [ ( 1 - 2 M r ) r 2 + M 2 cos 2 θ ] 1 / 2 R:=\Big[\Big(1-\frac{2M}{r}\Big)r^{2}+M^{2}\cos^{2}\theta\Big]^{1/2}
  30. ( 13 ) P i := p i ( ( r - m ) cos θ R ) (13)\quad P_{i}:=p_{i}\Big(\frac{(r-m)\cos\theta}{R}\Big)
  31. a i a_{i}
  32. γ ( r , θ ) \gamma(r,\theta)
  33. ( 14 ) γ ( r , θ ) = i = 1 j = 0 a i a j (14)\quad\gamma(r,\theta)\,=\sum_{i=1}^{\infty}\sum_{j=0}^{\infty}a_{i}a_{j}
  34. ( i j i + j ) \Big(\frac{ij}{i+j}\Big)
  35. ( R M ) i + j \Big(\frac{R}{M}\Big)^{i+j}
  36. ( P i P j - P i - 1 P j - 1 ) (P_{i}P_{j}-P_{i-1}P_{j-1})
  37. - 1 M i = 1 α i j = 0 i - 1 -\frac{1}{M}\sum_{i=1}^{\infty}\alpha_{i}\sum_{j=0}^{i-1}
  38. [ ( - 1 ) i + j ( r - M ( 1 - cos θ ) ) + r - M ( 1 + cos θ ) ] \Big[(-1)^{i+j}(r-M(1-\cos\theta))+r-M(1+\cos\theta)\Big]
  39. ( R M ) j P j . \Big(\frac{R}{M}\Big)^{j}P_{j}\,.

Distribution_(number_theory).html

  1. r = 0 N - 1 ϕ ( x + r N ) = ϕ ( N x ) . \sum_{r=0}^{N-1}\phi\left(x+\frac{r}{N}\right)=\phi(Nx)\ .
  2. w ( m , n ) y x ϕ ( y ) = ϕ ( x ) w(m,n)\sum_{y\mapsto x}\phi(y)=\phi(x)
  3. f d ϕ = x X n f ( x ) ϕ n ( x ) . \int f\,d\phi=\sum_{x\in X_{n}}f(x)\phi_{n}(x)\ .
  4. p = 0 q - 1 ζ ( s , a + p / q ) = q s ζ ( s , q a ) . \sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^{s}\,\zeta(s,qa)\ .
  5. t ζ ( s , { t } ) t\mapsto\zeta(s,\{t\})
  6. B n ( x ) = k = 0 n ( n n - k ) b k x n - k , B_{n}(x)=\sum_{k=0}^{n}{n\choose n-k}b_{k}x^{n-k}\ ,
  7. t e x t e t - 1 = n = 0 B n ( x ) t n n ! . \frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!}\ .
  8. B k ( x ) = n k - 1 a = 0 n - 1 b k ( x + a n ) . B_{k}(x)=n^{k-1}\sum_{a=0}^{n-1}b_{k}\left({\frac{x+a}{n}}\right)\ .
  9. ϕ n : 1 n / \phi_{n}:\frac{1}{n}\mathbb{Z}/\mathbb{Z}\rightarrow\mathbb{Q}
  10. ϕ n : x n k - 1 B k ( x ) \phi_{n}:x\mapsto n^{k-1}B_{k}(\langle x\rangle)
  11. p b = a g b = g a . \prod_{pb=a}g_{b}=g_{a}\ .
  12. g N ( r ) = 1 | G ( N ) | a G ( N ) h ( r a N ) σ a - 1 . g_{N}(r)=\frac{1}{|G(N)|}\sum_{a\in G(N)}h\left({\left\langle{\frac{ra}{N}}% \right\rangle}\right)\sigma_{a}^{-1}\ .
  13. ( T l f ) ( a b ) = f ( l a b ) + k = 0 l - 1 f ( a + k b l b ) - k = 0 l - 1 f ( k l ) . (T_{l}f)\left(\frac{a}{b}\right)=f\left(\frac{la}{b}\right)+\sum_{k=0}^{l-1}f% \left({\frac{a+kb}{lb}}\right)-\sum_{k=0}^{l-1}f\left(\frac{k}{l}\right)\ .
  14. a k + 2 = λ p a k + 1 - p a k , a_{k+2}=\lambda_{p}a_{k+1}-pa_{k}\ ,
  15. a k = π 1 k + π 2 k . a_{k}=\pi_{1}^{k}+\pi_{2}^{k}\ .

Distribution_algebra.html

  1. D ( G , K ) D(G,K)

Ditrans,polycis-polyprenyl_diphosphate_synthase_((2E,6E)-farnesyl_diphosphate_specific).html

  1. \rightleftharpoons
  2. \rightleftharpoons

Divided_domain.html

  1. 𝔭 \mathfrak{p}
  2. 𝔭 = 𝔭 R 𝔭 \mathfrak{p}=\mathfrak{p}R_{\mathfrak{p}}

DNA_oxidative_demethylase.html

  1. \rightleftharpoons

Dold–Kan_correspondence.html

  1. K ( A , n ) K(A,n)

Dolichyl-P-Man:Man5GlcNAc2-PP-dolichol_alpha-1,3-mannosyltransferase.html

  1. \rightleftharpoons

Dolichyl-P-Man:Man6GlcNAc2-PP-dolichol_alpha-1,2-mannosyltransferase.html

  1. \rightleftharpoons

Dolichyl-P-Man:Man7GlcNAc2-PP-dolichol_alpha-1,6-mannosyltransferase.html

  1. \rightleftharpoons

Dolichyl-P-Man:Man8GlcNAc2-PP-dolichol_alpha-1,2-mannosyltransferase.html

  1. \rightleftharpoons

Domain_wall_(magnetism).html

  1. J ex J_{\mathrm{ex}}

Douady–Earle_extension.html

  1. μ ( z ) = z ¯ F f z F f , \displaystyle{\mu(z)={\partial_{\overline{z}}F_{f}\over\partial_{z}F_{f}},}
  2. sup | z | < 1 | μ ( z ) | < 1. \displaystyle{\sup_{|z|<1}|\mu(z)|<1.}
  3. E ( f ) = H f - 1 \displaystyle{E(f)=H_{f^{-1}}}
  4. F f g = F f g . \displaystyle{F_{f\circ g}=F_{f}\circ g.}
  5. H f ( a ) = f - 1 ( a ) . \displaystyle{H_{f}(a)=f^{-1}(a).}
  6. H g f h = h - 1 H f g - 1 \displaystyle{H_{g\circ f\circ h}=h^{-1}\circ H_{f}\circ g^{-1}}
  7. Φ ( a , b ) = F g a f ( b ) = 1 2 π 0 2 π g a f g - b ( e i θ ) d θ = 1 2 π 0 2 π ( f ( e i θ ) - b 1 - b ¯ f ( e i θ ) ) 1 - | a | 2 | a - e i θ | 2 d θ \displaystyle{\Phi(a,b)=F_{g_{a}\circ f}(b)={1\over 2\pi}\int_{0}^{2\pi}g_{a}% \circ f\circ g_{-b}(e^{i\theta})\,d\theta={1\over 2\pi}\int_{0}^{2\pi}\left({f% (e^{i\theta})-b\over 1-\overline{b}f(e^{i\theta})}\right){1-|a|^{2}\over|a-e^{% i\theta}|^{2}}\,d\theta}
  8. f n = g z n f g - w n , \displaystyle{f_{n}=g_{z_{n}}\circ f\circ g_{-w_{n}},}
  9. f ( e i θ ) = m a m e i m θ , \displaystyle{f(e^{i\theta})=\sum_{m}a_{m}e^{im\theta},}
  10. z F f ( 0 ) = a 1 , z ¯ F f ( 0 ) = a - 1 . \displaystyle{\partial_{z}F_{f}(0)=a_{1},\,\,\,\partial_{\overline{z}}F_{f}(0)% =a_{-1}.}
  11. | z F f ( 0 ) | 2 - | z ¯ F f ( 0 ) | 2 = | a 1 | 2 - | a - 1 | 2 . \displaystyle{|\partial_{z}F_{f}(0)|^{2}-|\partial_{\overline{z}}F_{f}(0)|^{2}% =|a_{1}|^{2}-|a_{-1}|^{2}.}
  12. | a 1 | 2 - | a - 1 | 2 > 0. \displaystyle{|a_{1}|^{2}-|a_{-1}|^{2}>0.}
  13. Φ z ( 0 , 0 ) = a - 1 , Φ z ¯ ( 0 , 0 ) = a 1 , Φ w ( 0 , 0 ) = - 1 , Φ w ¯ ( 0 , 0 ) = 1 2 π 0 2 π f ( e i θ ) 2 d θ = b . \displaystyle{\Phi_{z}(0,0)=a_{-1},\,\,\Phi_{\overline{z}}(0,0)=a_{1},\,\,\Phi% _{w}(0,0)=-1,\,\,\Phi_{\overline{w}}(0,0)={1\over 2\pi}\int_{0}^{2\pi}f(e^{i% \theta})^{2}\,d\theta=b.}
  14. | Φ w ( 0 , 0 ) | 2 - | Φ w ¯ ( 0 , 0 ) | 2 = 1 - | 1 2 π 0 2 π f ( e i θ ) 2 d θ | 2 0. \displaystyle{|\Phi_{w}(0,0)|^{2}-|\Phi_{\overline{w}}(0,0)|^{2}=1-\left|{1% \over 2\pi}\int_{0}^{2\pi}f(e^{i\theta})^{2}\,d\theta\right|^{2}\geq 0.}
  15. f ( e i θ ) = ζ f ( e i θ ) ¯ \displaystyle{f(e^{i\theta})=\zeta\overline{f(e^{i\theta})}}
  16. | z H f ( 0 ) | 2 - | z ¯ H f ( 0 ) | 2 = | Φ z ( 0 , 0 ) | 2 - | Φ z ¯ ( 0 , 0 ) | 2 | Φ w ( 0 , 0 ) | 2 - | Φ w ¯ ( 0 , 0 ) | 2 > 0. \displaystyle{|\partial_{z}H_{f}(0)|^{2}-|\partial_{\overline{z}}H_{f}(0)|^{2}% ={|\Phi_{z}(0,0)|^{2}-|\Phi_{\overline{z}}(0,0)|^{2}\over|\Phi_{w}(0,0)|^{2}-|% \Phi_{\overline{w}}(0,0)|^{2}}>0.}
  17. z ¯ H f ( 0 ) z H f ( 0 ) = g b ( - a - 1 a 1 ¯ ) . \displaystyle{{\partial_{\overline{z}}H_{f}(0)\over\partial_{z}H_{f}(0)}=g_{b}% \left(-{a_{-1}\over\overline{a_{1}}}\right).}
  18. | f ( z 1 ) - f ( z 2 ) | | f ( z 1 ) - f ( z 3 ) | a | z 1 - z 2 | b | z 1 - z 3 | b . \displaystyle{{|f(z_{1})-f(z_{2})|\over|f(z_{1})-f(z_{3})|}\leq a{|z_{1}-z_{2}% |^{b}\over|z_{1}-z_{3}|^{b}}.}
  19. | ( f ( z 1 ) , f ( z 2 ) ; f ( z 3 ) , f ( z 4 ) ) | c | ( z 1 , z 2 ; z 3 , z 4 ) | d , \displaystyle{|(f(z_{1}),f(z_{2});f(z_{3}),f(z_{4}))|\leq c|(z_{1},z_{2};z_{3}% ,z_{4})|^{d},}
  20. ( z 1 , z 2 ; z 3 , z 4 ) = ( z 1 - z 3 ) ( z 2 - z 4 ) ( z 2 - z 3 ) ( z 1 - z 4 ) \displaystyle{(z_{1},z_{2};z_{3},z_{4})={(z_{1}-z_{3})(z_{2}-z_{4})\over(z_{2}% -z_{3})(z_{1}-z_{4})}}
  21. μ F ( z ) = z ¯ F ( z ) z F ( z ) . \displaystyle{\mu_{F}(z)={\partial_{\overline{z}}F(z)\over\partial_{z}F(z)}.}
  22. μ G F - 1 F = F z F z ¯ μ G - μ F 1 - μ F ¯ μ G . \displaystyle{\mu_{G\circ F^{-1}}\circ F={F_{z}\over\overline{F_{z}}}{\mu_{G}-% \mu_{F}\over 1-\overline{\mu_{F}}\mu_{G}}.}
  23. μ F G - 1 G = G z G z ¯ μ F , μ G - 1 F = μ F . \displaystyle{\mu_{F\circ G^{-1}}\circ G={G_{z}\over\overline{G_{z}}}\mu_{F},% \,\,\,\mu_{G^{-1}\circ F}=\mu_{F}.}
  24. μ F ( 0 ) = g b ( - a - 1 a 1 ¯ ) , \displaystyle{\mu_{F}(0)=g_{b}\left(-{a_{-1}\over\overline{a_{1}}}\right),}
  25. a ± 1 = 1 2 π 0 2 π f ( e i θ ) e i θ d θ , b = 1 2 π 0 2 π f ( e i θ ) 2 d θ . \displaystyle{a_{\pm 1}={1\over 2\pi}\int_{0}^{2\pi}f(e^{i\theta})e^{\mp i% \theta}\,d\theta,\,\,\,b={1\over 2\pi}\int_{0}^{2\pi}f(e^{i\theta})^{2}\,d% \theta.}
  26. μ F < 1. \displaystyle{\|\mu_{F}\|_{\infty}<1.}
  27. Λ ( f ) = | g b ( - a - 1 a 1 ¯ ) | \displaystyle{\Lambda(f)=\left|g_{b}\left(-{a_{-1}\over\overline{a_{1}}}\right% )\right|}
  28. | f ( z ) - f ( w ) | C | z - w | d , \displaystyle{|f(z)-f(w)|\leq C|z-w|^{d},}
  29. | ( w , i ; z , - i ) | 16 | z - w | , | ( f ( w ) , i ; f ( z ) , - i ) | | f ( z ) - f ( w ) | / 8 , \displaystyle{|(w,i;z,-i)|\leq 16|z-w|,\,\,\,|(f(w),i;f(z),-i)|\geq|f(z)-f(w)|% /8,}
  30. | ( z , ζ ; w , 1 ) | 8 | z - w | , | ( f ( z ) , ζ ; f ( w ) , 1 ) | | f ( z ) - f ( w ) | / 8 , \displaystyle{|(z,\zeta;w,1)|\leq 8|z-w|,\,\,\,|(f(z),\zeta;f(w),1)|\geq|f(z)-% f(w)|/8,}

Double-Blind_FROG.html

  1. I ( ω , τ ) = | e - i ω t E 1 ( t ) | E 2 ( t - τ ) | 2 d t | 2 I(\omega,\tau)=|\int e^{-i\omega t}E_{1}(t)|E_{2}(t-\tau)|^{2}dt|^{2}
  2. | e - i ω t E 1 ( t ) | E 2 ( t - τ ) | 2 d t | 2 |\int e^{-i\omega t}E_{1}(t)|E_{2}(t-\tau)|^{2}dt|^{2}
  3. E 1 E_{1}
  4. E 2 E_{2}
  5. | e - i ω t E 2 ( t ) | E 1 ( t - τ ) | 2 d t | 2 |\int e^{-i\omega t}E_{2}(t)|E_{1}(t-\tau)|^{2}dt|^{2}
  6. E 1 E_{1}
  7. E 2 E_{2}
  8. E 2 E_{2}
  9. E 1 E_{1}
  10. E 1 E_{1}
  11. E 1 E_{1}
  12. E 1 E_{1}
  13. E 1 E_{1}
  14. E 2 E_{2}
  15. E 1 E_{1}
  16. E 2 E_{2}
  17. E 2 E_{2}
  18. E 1 E_{1}
  19. E 2 E_{2}
  20. E 1 E_{1}
  21. E 2 E_{2}

Double_integrator.html

  1. 𝐮 \,\textbf{u}
  2. 𝐱 ˙ ( t ) = [ 0 1 0 0 ] 𝐱 ( t ) + [ 0 1 ] 𝐮 ( t ) \dot{\,\textbf{x}}(t)=\begin{bmatrix}0&1\\ 0&0\\ \end{bmatrix}\,\textbf{x}(t)+\begin{bmatrix}0\\ 1\end{bmatrix}\,\textbf{u}(t)
  3. 𝐲 ( t ) = [ 1 0 ] 𝐱 ( t ) . \,\textbf{y}(t)=\begin{bmatrix}1&0\end{bmatrix}\,\textbf{x}(t).
  4. 𝐮 \,\textbf{u}
  5. 𝐲 \,\textbf{y}
  6. Y ( s ) U ( s ) = 1 s 2 . \frac{Y(s)}{U(s)}=\frac{1}{s^{2}}.

Double_ionization.html

  1. 3.2 U p \sim 3.2U_{p}
  2. U p = F 2 4 ω 2 U_{p}=\frac{F^{2}}{4\omega^{2}}
  3. F F
  4. ω \omega
  5. 3.2 U p 3.2U_{p}
  6. I p I_{p}
  7. U p U_{p}
  8. 3.2 U p > I p 3.2U_{p}>I_{p}
  9. U p U_{p}
  10. 3.2 U p < I p 3.2U_{p}<I_{p}
  11. U p U_{p}

Double_layer_forces.html

  1. d 2 ψ d z 2 = - ρ ϵ 0 ϵ \frac{d^{2}\psi}{dz^{2}}=-\frac{\rho}{\epsilon_{0}\epsilon}
  2. ρ = q ( c + - c - ) \rho=q(c_{+}-c_{-})
  3. μ ± = μ + ( 0 ) + k T ln c ± ± q ψ \mu_{\pm}=\mu_{+}^{(0)}+kT\ln c_{\pm}\pm q\psi
  4. μ ± ( 0 ) \mu_{\pm}^{(0)}
  5. c ± = c B e β q ψ c_{\pm}=c_{\rm B}e^{\mp\beta q\psi}
  6. d 2 ψ d z 2 = q c B ϵ 0 ϵ [ e + β q ψ - e - β q ψ ] \frac{d^{2}\psi}{dz^{2}}=\frac{qc_{\rm B}}{\epsilon_{0}\epsilon}[e^{+\beta q% \psi}-e^{-\beta q\psi}]
  7. - V d Π + N + d μ + + N - d μ - = 0 -Vd\Pi+N_{+}d\mu_{+}+N_{-}d\mu_{-}=0
  8. d Π = k T ( d c + + d c - ) + q ( c + - c - ) d ψ d\Pi=kT(dc_{+}+dc_{-})+q(c_{+}-c_{-})d\psi
  9. 2 ( d 2 ψ / d z 2 ) d ψ = d ( d ψ / d z ) 2 2(d^{2}\psi/dz^{2})d\psi=d(d\psi/dz)^{2}
  10. Π = k T c B ( e + β q ψ + e - β q ψ - 2 ) - ϵ 0 ϵ 2 ( d ψ d z ) 2 \Pi=kTc_{\rm B}(e^{+\beta q\psi}+e^{-\beta q\psi}-2)-\frac{\epsilon_{0}% \epsilon}{2}\left(\frac{d\psi}{dz}\right)^{2}
  11. d 2 ψ d z 2 = κ 2 ψ \frac{d^{2}\psi}{dz^{2}}=\kappa^{2}\psi
  12. κ 2 = 2 β q 2 c B ϵ 0 ϵ \;\;\;\kappa^{2}=\frac{2\beta q^{2}c_{\rm B}}{\epsilon_{0}\epsilon}
  13. β q ψ 1 or ψ k T q 26 mV \beta q\psi\ll 1\;\;\;{\rm or}\;\;\;\psi\ll\frac{kT}{q}\simeq 26\;{\rm mV}
  14. Π = ϵ 0 ϵ 2 [ κ 2 ψ 2 - ( d ψ d z ) 2 ] \Pi=\frac{\epsilon_{0}\epsilon}{2}\left[\kappa^{2}\psi^{2}-\left(\frac{d\psi}{% dz}\right)^{2}\right]
  15. ψ ( z ) = ψ D e - κ z \psi(z)=\psi_{\rm D}e^{-\kappa z}
  16. Π = 2 ϵ ϵ 0 κ 2 ψ D 2 e - κ h \Pi=2\epsilon\epsilon_{0}\kappa^{2}\psi_{\rm D}^{2}e^{-\kappa h}
  17. ψ eff = 4 β q tanh ( β q ψ D ) \psi_{\rm eff}=\frac{4}{\beta q}\tanh(\beta q\psi_{\rm D})
  18. Π ( h ) = 2 ϵ ϵ 0 κ 2 ψ eff ( 1 ) ψ eff ( 2 ) e - κ h \Pi(h)=2\epsilon\epsilon_{0}\kappa^{2}\psi^{(1)}_{\rm eff}\psi^{(2)}_{\rm eff}% e^{-\kappa h}
  19. ψ ¯ D \bar{\psi}_{\rm D}
  20. σ ¯ \bar{\sigma}
  21. p = C D C I + C D p=\frac{C_{\rm D}}{C_{\rm I}+C_{\rm D}}
  22. ψ ¯ D = 1 1 - p + p tanh ( κ h / 2 ) ψ D \bar{\psi}_{\rm D}=\frac{1}{1-p+p\tanh(\kappa h/2)}\psi_{\rm D}
  23. σ ¯ = tanh ( κ h / 2 ) 1 - p + p tanh ( κ h / 2 ) σ \bar{\sigma}=\frac{\tanh(\kappa h/2)}{1-p+p\tanh(\kappa h/2)}\sigma
  24. Π = 2 ϵ ϵ 0 κ 2 ψ D 2 e - κ h [ 1 + ( 1 - 2 p ) e - κ h ] 2 \Pi=2\epsilon\epsilon_{0}\kappa^{2}\psi_{\rm D}^{2}\frac{e^{-\kappa h}}{[1+(1-% 2p)e^{-\kappa h}]^{2}}
  25. U = Q 2 4 π ϵ ϵ 0 ( e κ a 1 + κ a ) 2 e - κ r r U=\frac{Q^{2}}{4\pi\epsilon\epsilon_{0}}\left(\frac{e^{\kappa a}}{1+\kappa a}% \right)^{2}\frac{e^{-\kappa r}}{r}
  26. q 2 = κ 2 + ( 2 π b ) 2 q^{2}=\kappa^{2}+\left(\frac{2\pi}{b}\right)^{2}
  27. F 123 = F 12 + F 12 + F 12 + Δ F 123 F_{123}=F_{12}+F_{12}+F_{12}+\Delta F_{123}

Double_pushout_graph_rewriting.html

  1. G G
  2. \mathcal{M}

Draft:Adaptive_filtering_by_optimal_projection.html

  1. 𝐗 \mathbf{X}
  2. ( n , t ) (n,t)
  3. n n
  4. t t
  5. 𝐅 \mathbf{F}
  6. ( n , n ) (n,n)
  7. 𝐗 ^ = 𝐅𝐗 \hat{\mathbf{X}}=\mathbf{FX}
  8. 𝐅𝐅 = 𝐅 \mathbf{FF}=\mathbf{F}
  9. E 1 E_{1}
  10. n 1 n_{1}
  11. 𝐅 \mathbf{F}
  12. E 0 E_{0}
  13. n 0 = n - n 1 n_{0}=n-n_{1}
  14. 𝐅 \mathbf{F}
  15. F 1 = E 0 F_{1}=E_{0}^{\bot}
  16. n 1 n_{1}
  17. F 0 = E 1 F_{0}=E_{1}^{\bot}
  18. n 0 = n - n 1 n_{0}=n-n_{1}
  19. E 1 E_{1}
  20. E 0 E_{0}
  21. F 1 F_{1}
  22. F 0 F_{0}
  23. F 1 F_{1}
  24. E 0 E_{0}
  25. 𝐗 \mathbf{X}
  26. 𝐗 1 \mathbf{X}_{1}
  27. 𝐗 2 \mathbf{X}_{2}
  28. ( n , t 1 ) (n,t_{1})
  29. ( n , t 2 ) (n,t_{2})
  30. 𝐗 1 \mathbf{X}_{1}
  31. 𝐗 2 \mathbf{X}_{2}
  32. F 1 = arg min { F n with dim ( F ) = n 1 } ( max p F 𝐩 𝐓 𝐗 2 2 𝐩 𝐓 𝐗 1 2 ) F_{1}={\arg\min}_{\{F\in\Re^{n}\,\text{ with }\dim(F)=n_{1}\}}\begin{pmatrix}% \max_{p\in F}\frac{\|\mathbf{p^{T}X}_{2}\|^{2}}{\|\mathbf{p^{T}X}_{1}\|^{2}}% \end{pmatrix}
  33. 𝐑 2 - 1 𝐑 1 \mathbf{R}_{2}^{-1}\mathbf{R}_{1}
  34. 𝐑 1 = 𝐗 1 𝐗 1 T t 1 \mathbf{R}_{1}=\frac{\mathbf{X}_{1}\mathbf{X}_{1}^{T}}{t_{1}}
  35. 𝐑 2 = 𝐗 2 𝐗 2 T t 2 \mathbf{R}_{2}=\frac{\mathbf{X}_{2}\mathbf{X}_{2}^{T}}{t_{2}}
  36. 𝐑 2 - 1 𝐑 1 = 𝐏𝐃𝐏 - 1 \mathbf{R}_{2}^{-1}\mathbf{R}_{1}=\mathbf{PDP}^{-1}
  37. 𝐏 = [ 𝐩 1 𝐩 n ] \mathbf{P}=\begin{bmatrix}\mathbf{p}_{1}&\cdots&\mathbf{p}_{n}\end{bmatrix}
  38. 𝐃 \mathbf{D}
  39. { λ 1 , , λ n } \{\lambda_{1},\cdots,\lambda_{n}\}
  40. F 1 F_{1}
  41. n 1 n_{1}
  42. 𝐏 \mathbf{P}
  43. 𝐗 \mathbf{X}
  44. E 1 E_{1}
  45. E 1 = E 0 C - 1 = ( F 1 ) 𝐑 - 1 E_{1}=E_{0}^{\bot_{C^{-1}}}=(F_{1}^{\bot})^{\bot_{\mathbf{R}^{-1}}}
  46. E 1 E_{1}
  47. E 0 E_{0}
  48. 𝐑 , \langle\mathbf{R}\cdot,\cdot\rangle
  49. 𝐑 = 𝐗𝐗 T t \mathbf{R}=\frac{\mathbf{X}\mathbf{X}^{T}}{t}
  50. 𝐗 \mathbf{X}
  51. 𝐗 \mathbf{X}
  52. 𝐗 2 \mathbf{X}_{2}
  53. E 0 E_{0}

Draft:Tau_(proposed_mathematical_constant).html

  1. π \pi
  2. τ \tau
  3. π \pi
  4. π \pi
  5. π \pi
  6. τ \tau
  7. π \pi
  8. τ \tau
  9. π \pi
  10. π \pi
  11. π π \pi\!\;\!\!\!\pi
  12. π \pi
  13. τ \tau
  14. τ \tau
  15. π \pi
  16. τ \tau
  17. π \pi
  18. π \pi
  19. τ \tau
  20. π \pi
  21. τ \tau
  22. 0
  23. 1 12 \tfrac{1}{12}
  24. 1 8 \tfrac{1}{8}
  25. 1 6 \tfrac{1}{6}
  26. 1 4 \tfrac{1}{4}
  27. 1 2 \tfrac{1}{2}
  28. 3 4 \tfrac{3}{4}
  29. 1 1
  30. τ \tau
  31. 0
  32. 1 12 τ \tfrac{1}{12}\tau
  33. 1 8 τ \tfrac{1}{8}\tau
  34. 1 6 τ \tfrac{1}{6}\tau
  35. 1 4 τ \tfrac{1}{4}\tau
  36. 1 2 τ \tfrac{1}{2}\tau
  37. 3 4 τ \tfrac{3}{4}\tau
  38. 1 τ 1\tau
  39. π \pi
  40. 0
  41. 1 6 π \tfrac{1}{6}\pi
  42. 1 4 π \tfrac{1}{4}\pi
  43. 1 3 π \tfrac{1}{3}\pi
  44. 1 2 π \tfrac{1}{2}\pi
  45. π \pi
  46. 3 2 π \tfrac{3}{2}\pi
  47. 2 π 2\pi
  48. τ \tau
  49. τ \tau
  50. i π {}^{i\pi}
  51. π \pi
  52. τ \tau
  53. i x {}^{ix}
  54. π \pi
  55. τ \tau
  56. i τ {}^{i\tau}
  57. π \pi
  58. π \pi
  59. τ \tau
  60. f ( x ) = 1 σ 2 π e - 1 2 ( x - μ σ ) 2 f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}% \right)^{2}}
  61. f ( x ) = 1 σ τ e - 1 2 ( x - μ σ ) 2 f(x)=\frac{1}{\sigma\sqrt{\tau}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}% \right)^{2}}
  62. f ^ ( ξ ) = - f ( x ) e - 2 π i x ξ d x \hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)\ e^{-2\pi ix\xi}\,dx
  63. f ^ ( ξ ) = - f ( x ) e - i τ x ξ d x \hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)\ e^{-i\tau x\xi}\,dx
  64. e 2 π i k n e^{2\pi i\frac{k}{n}}\;
  65. e τ i k n e^{\tau i\frac{k}{n}}\;
  66. f ( a ) = 1 2 π i γ f ( z ) z - a d z f(a)=\frac{1}{2\pi i}\oint_{\gamma}\frac{f(z)}{z-a}\,dz
  67. f ( a ) = 1 τ i γ f ( z ) z - a d z f(a)=\frac{1}{\tau i}\oint_{\gamma}\frac{f(z)}{z-a}\,dz
  68. ω = 2 π f \scriptstyle\omega\;=\;2\pi f
  69. ω = τ f \scriptstyle\omega\;=\;\tau f
  70. k = 2 π λ \scriptstyle k\;=\;\frac{2\pi}{\lambda}
  71. k = τ λ \scriptstyle k\;=\;\frac{\tau}{\lambda}
  72. π \pi
  73. π \pi
  74. π \pi
  75. τ \tau
  76. π \pi
  77. π \pi
  78. τ \tau
  79. π \pi
  80. τ \tau
  81. π \pi
  82. π \pi
  83. π \pi
  84. τ \tau
  85. π \pi
  86. π \pi
  87. π \pi
  88. π \pi
  89. π \pi
  90. π \pi
  91. τ \tau
  92. erf ( x ) = 2 π 0 x e - t 2 d t . \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\,\mathrm{d}t.
  93. erf ( x ) = 2 2 τ 0 x e - t 2 d t . \operatorname{erf}(x)=\frac{2\sqrt{2}}{\sqrt{\tau}}\int_{0}^{x}e^{-t^{2}}\,% \mathrm{d}t.
  94. π ( n - 2 ) \pi(n-2)
  95. 1 2 τ ( n - 2 ) \frac{1}{2}\tau(n-2)
  96. π a b \pi ab
  97. 1 2 τ a b \frac{1}{2}\tau ab
  98. - e - x 2 d x = π . \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}.
  99. - e - x 2 d x = 2 τ 2 . \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\frac{\sqrt{2\tau}}{2}.
  100. π r 2 \pi{r^{2}}
  101. 1 2 τ r 2 \frac{1}{2}\tau r^{2}
  102. n = 1 1 n 2 = π 2 6 \sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}
  103. n = 1 1 n 2 = τ 2 24 \sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\tau^{2}}{24}
  104. π \pi

Drift_plus_penalty.html

  1. Δ ( t ) = L ( t + 1 ) - L ( t ) \Delta(t)=L(t+1)-L(t)
  2. Δ ( t ) + V p ( t ) , \Delta(t)+Vp(t),
  3. p ( t ) = penalty function whose time average must be minimized p(t)=\,\text{penalty function whose time average must be minimized}
  4. y 1 ( t ) , y 2 ( t ) , , y K ( t ) = other functions whose time averages must be non-positive y_{1}(t),y_{2}(t),...,y_{K}(t)=\,\text{other functions whose time averages % must be non-positive}
  5. ω ( t ) = random event on slot t (assumed i.i.d. over slots) \omega(t)=\,\text{random event on slot t (assumed i.i.d. over slots)}
  6. α ( t ) = control action on slot t (chosen after observing ω ( t ) ) \alpha(t)=\,\text{control action on slot t (chosen after observing }\omega(t)% \,\text{)}
  7. p ( t ) = P ( α ( t ) , ω ( t ) ) (a deterministic function of α ( t ) , ω ( t ) ) p(t)=P(\alpha(t),\omega(t))\,\text{ (a deterministic function of }\alpha(t),% \omega(t)\,\text{)}
  8. y i ( t ) = Y i ( α ( t ) , ω ( t ) ) i { 1 , , K } (deterministic functions of α ( t ) , ω ( t ) ) y_{i}(t)=Y_{i}(\alpha(t),\omega(t))\,\text{ }\forall i\in\{1,...,K\}\,\text{ (% deterministic functions of }\alpha(t),\omega(t)\,\text{)}
  9. ω ( t ) \omega(t)
  10. Ω \Omega
  11. α ( t ) \alpha(t)
  12. A A
  13. Ω \Omega
  14. A A
  15. A A
  16. ω ( t ) \omega(t)
  17. α ( t ) \alpha(t)
  18. ω ( t ) \omega(t)
  19. Minimize: lim t 1 t τ = 0 t - 1 E [ p ( τ ) ] \,\text{Minimize: }\lim_{t\rightarrow\infty}\frac{1}{t}\sum_{\tau=0}^{t-1}E[p(% \tau)]
  20. Subject to: lim t 1 t τ = 0 t - 1 E [ y i ( τ ) ] 0 i { 1 , , K } \,\text{Subject to: }\lim_{t\rightarrow\infty}\frac{1}{t}\sum_{\tau=0}^{t-1}E[% y_{i}(\tau)]\leq 0\,\text{ }\forall i\in\{1,...,K\}
  21. ( E q .1 ) Q i ( t + 1 ) = max [ Q i ( t ) + y i ( t ) , 0 ] (Eq.1)\,\text{ }Q_{i}(t+1)=\max[Q_{i}(t)+y_{i}(t),0]
  22. Q i ( t + 1 ) Q i ( t ) + y i ( t ) Q_{i}(t+1)\geq Q_{i}(t)+y_{i}(t)
  23. y i ( t ) Q i ( t + 1 ) - Q i ( t ) y_{i}(t)\leq Q_{i}(t+1)-Q_{i}(t)
  24. τ = 0 t - 1 y i ( τ ) Q i ( t ) - Q i ( 0 ) = Q i ( t ) \sum_{\tau=0}^{t-1}y_{i}(\tau)\leq Q_{i}(t)-Q_{i}(0)=Q_{i}(t)
  25. 1 t τ = 0 t - 1 E [ y i ( τ ) ] E [ Q i ( t ) ] t \frac{1}{t}\sum_{\tau=0}^{t-1}E[y_{i}(\tau)]\leq\frac{E[Q_{i}(t)]}{t}
  26. lim t E [ Q i ( t ) ] t = 0 \lim_{t\rightarrow\infty}\frac{E[Q_{i}(t)]}{t}=0
  27. L ( t ) = 1 2 i = 1 K Q i ( t ) 2 L(t)=\frac{1}{2}\sum_{i=1}^{K}Q_{i}(t)^{2}
  28. Q i ( t + 1 ) 2 ( Q i ( t ) + y i ( t ) ) 2 = Q i ( t ) 2 + y i ( t ) 2 + 2 Q i ( t ) y i ( t ) Q_{i}(t+1)^{2}\leq(Q_{i}(t)+y_{i}(t))^{2}=Q_{i}(t)^{2}+y_{i}(t)^{2}+2Q_{i}(t)y% _{i}(t)
  29. 1 2 i = 1 K Q i ( t + 1 ) 2 1 2 i = 1 K Q i ( t ) 2 + 1 2 i = 1 K y i ( t ) 2 + i = 1 K Q i ( t ) y i ( t ) \frac{1}{2}\sum_{i=1}^{K}Q_{i}(t+1)^{2}\leq\frac{1}{2}\sum_{i=1}^{K}Q_{i}(t)^{% 2}+\frac{1}{2}\sum_{i=1}^{K}y_{i}(t)^{2}+\sum_{i=1}^{K}Q_{i}(t)y_{i}(t)
  30. Δ ( t ) = L ( t + 1 ) - L ( t ) 1 2 i = 1 k y i ( t ) 2 + i = 1 K Q i ( t ) y i ( t ) \Delta(t)=L(t+1)-L(t)\leq\frac{1}{2}\sum_{i=1}^{k}y_{i}(t)^{2}+\sum_{i=1}^{K}Q% _{i}(t)y_{i}(t)
  31. Δ ( t ) B + i = 1 K Q i ( t ) y i ( t ) \Delta(t)\leq B+\sum_{i=1}^{K}Q_{i}(t)y_{i}(t)
  32. ( E q .2 ) Δ ( t ) + V p ( t ) B + V p ( t ) + i = 1 K Q i ( t ) y i ( t ) (Eq.2)\,\text{ }\Delta(t)+Vp(t)\leq B+Vp(t)+\sum_{i=1}^{K}Q_{i}(t)y_{i}(t)
  33. A A
  34. Observe: ω ( t ) , Q 1 ( t ) , , Q K ( t ) \,\text{Observe: }\omega(t),Q_{1}(t),...,Q_{K}(t)
  35. α ( t ) A \alpha(t)\in A
  36. V P ( α ( t ) , ω ( t ) ) + i = 1 K Q i ( t ) Y i ( α ( t ) , ω ( t ) ) VP(\alpha(t),\omega(t))+\sum_{i=1}^{K}Q_{i}(t)Y_{i}(\alpha(t),\omega(t))
  37. α ( t ) \alpha(t)
  38. V P ( α ( t ) , ω ( t ) ) + i = 1 K Q i ( t ) Y i ( α ( t ) , ω ( t ) ) C + inf α A [ V P ( α , ω ( t ) ) + i = 1 K Q i ( t ) Y i ( α , ω ( t ) ) ] VP(\alpha(t),\omega(t))+\sum_{i=1}^{K}Q_{i}(t)Y_{i}(\alpha(t),\omega(t))\leq C% +\inf_{\alpha\in A}[VP(\alpha,\omega(t))+\sum_{i=1}^{K}Q_{i}(t)Y_{i}(\alpha,% \omega(t))]
  39. ω \omega
  40. α ( t ) \alpha(t)
  41. ω ( t ) \omega(t)
  42. ω \omega
  43. ω Ω \omega\in\Omega
  44. α ( t ) A \alpha(t)\in A
  45. ω ( t ) = ω \omega(t)=\omega
  46. ω \omega
  47. α * ( t ) \alpha^{*}(t)
  48. ( E q .3 ) E [ P ( α * ( t ) , ω ( t ) ) ] = p * = optimal time average penalty for the problem (Eq.3)\,\text{ }E[P(\alpha^{*}(t),\omega(t))]=p^{*}=\,\text{ optimal time % average penalty for the problem}
  49. ( E q .4 ) E [ Y i ( α * ( t ) , ω ( t ) ) ] 0 i { 1 , , K } (Eq.4)\text{ }E[Y_{i}(\alpha^{*}(t),\omega(t))]\leq 0\,\text{ }\forall i\in\{1% ,...,K\}
  50. ω ( t ) \omega(t)
  51. α ( t ) \alpha(t)
  52. ω ( t ) \omega(t)
  53. α * ( t ) \alpha^{*}(t)
  54. ω ( t ) \omega(t)
  55. α ( t ) \alpha(t)
  56. α ( t ) \alpha(t)
  57. α * ( t ) \alpha^{*}(t)
  58. ω \omega
  59. α ( t ) = drift-plus-penalty action for slot t \alpha(t)=\,\text{drift-plus-penalty action for slot t}
  60. α * ( t ) = ω -only action that satisfies (Eq.3)-(Eq.4) \alpha^{*}(t)=\omega\,\text{-only action that satisfies (Eq.3)-(Eq.4)}
  61. α ( t ) \alpha(t)
  62. α ( t ) \alpha(t)
  63. Δ ( t ) + V p ( t ) \Delta(t)+Vp(t)
  64. B + V p ( t ) + i = 1 K Q i ( t ) y i ( t ) \leq B+Vp(t)+\sum_{i=1}^{K}Q_{i}(t)y_{i}(t)
  65. = B + V P ( α ( t ) , ω ( t ) ) + i = 1 K Q i ( t ) Y i ( α ( t ) , ω ( t ) ) =B+VP(\alpha(t),\omega(t))+\sum_{i=1}^{K}Q_{i}(t)Y_{i}(\alpha(t),\omega(t))
  66. B + C + V P ( α * ( t ) , ω ( t ) ) + i = 1 K Q i ( t ) Y i ( α * ( t ) , ω ( t ) ) \leq B+C+VP(\alpha^{*}(t),\omega(t))+\sum_{i=1}^{K}Q_{i}(t)Y_{i}(\alpha^{*}(t)% ,\omega(t))
  67. α ( t ) \alpha(t)
  68. α * ( t ) \alpha^{*}(t)
  69. E [ Δ ( t ) + V p ( t ) ] E[\Delta(t)+Vp(t)]
  70. B + C + V E [ P ( α * ( t ) , ω ( t ) ) ] + i = 1 K E [ Q i ( t ) Y i ( α * ( t ) , ω ( t ) ) ] \leq B+C+VE[P(\alpha^{*}(t),\omega(t))]+\sum_{i=1}^{K}E[Q_{i}(t)Y_{i}(\alpha^{% *}(t),\omega(t))]
  71. = B + C + V E [ P ( α * ( t ) , ω ( t ) ) ] + i = 1 K E [ Q i ( t ) ] E [ Y i ( α * ( t ) , ω ( t ) ) ] =B+C+VE[P(\alpha^{*}(t),\omega(t))]+\sum_{i=1}^{K}E[Q_{i}(t)]E[Y_{i}(\alpha^{*% }(t),\omega(t))]
  72. B + C + V p * \leq B+C+Vp^{*}
  73. α * ( t ) , ω ( t ) \alpha^{*}(t),\omega(t)
  74. Q i ( t ) Q_{i}(t)
  75. α * ( t ) \alpha^{*}(t)
  76. τ = 0 t - 1 E [ Δ ( τ ) + V p ( τ ) ] ( B + C + V p * ) t \sum_{\tau=0}^{t-1}E[\Delta(\tau)+Vp(\tau)]\leq(B+C+Vp^{*})t
  77. Δ ( τ ) = L ( τ + 1 ) - L ( τ ) \Delta(\tau)=L(\tau+1)-L(\tau)
  78. E [ L ( t ) ] - E [ L ( 0 ) ] + V τ = 0 t - 1 E [ p ( τ ) ] ( B + C + V p * ) t E[L(t)]-E[L(0)]+V\sum_{\tau=0}^{t-1}E[p(\tau)]\leq(B+C+Vp^{*})t
  79. V τ = 0 t - 1 E [ p ( τ ) ] ( B + C + V p * ) t V\sum_{\tau=0}^{t-1}E[p(\tau)]\leq(B+C+Vp^{*})t
  80. 1 t τ = 0 t - 1 E [ p ( τ ) ] p * + ( B + C ) / V \frac{1}{t}\sum_{\tau=0}^{t-1}E[p(\tau)]\leq p^{*}+(B+C)/V
  81. ω \omega
  82. α * ( t ) \alpha^{*}(t)
  83. ϵ > 0 \epsilon>0
  84. ( E q .5 ) E [ Y i ( α * ( t ) , ω ( t ) ) ] - ϵ i { 1 , , K } (Eq.5)\text{ }E[Y_{i}(\alpha^{*}(t),\omega(t))]\leq-\epsilon\,\text{ }\forall i% \in\{1,...,K\}
  85. Δ ( t ) + V p ( t ) B + C + V P ( α * ( t ) , ω ( t ) ) + i = 1 K Q i ( t ) Y i ( α * ( t ) , ω ( t ) ) \Delta(t)+Vp(t)\leq B+C+VP(\alpha^{*}(t),\omega(t))+\sum_{i=1}^{K}Q_{i}(t)Y_{i% }(\alpha^{*}(t),\omega(t))
  86. p m i n P ( ) p m a x p_{min}\leq P(\cdot)\leq p_{max}
  87. Δ ( t ) + V p m i n B + C + V p m a x + i = 1 K Q i ( t ) Y i ( α * ( t ) , ω ( t ) ) \Delta(t)+Vp_{min}\leq B+C+Vp_{max}+\sum_{i=1}^{K}Q_{i}(t)Y_{i}(\alpha^{*}(t),% \omega(t))
  88. E [ Δ ( t ) ] + V p m i n B + C + V p m a x + i = 1 K E [ Q i ( t ) ] ( - ϵ ) E[\Delta(t)]+Vp_{min}\leq B+C+Vp_{max}+\sum_{i=1}^{K}E[Q_{i}(t)](-\epsilon)
  89. 1 t τ = 0 t - 1 i = 1 K E [ Q i ( τ ) ] B + C + V ( p m a x - p m i n ) ϵ \frac{1}{t}\sum_{\tau=0}^{t-1}\sum_{i=1}^{K}E[Q_{i}(\tau)]\leq\frac{B+C+V(p_{% max}-p_{min})}{\epsilon}
  90. Minimize: f ( y ¯ 1 , y ¯ 2 , , y ¯ K ) \,\text{Minimize: }f(\overline{y}_{1},\overline{y}_{2},...,\overline{y}_{K})
  91. Subject to: g i ( y ¯ 1 , y ¯ 2 , , y ¯ K ) 0 i { 1 , , N } \,\text{Subject to: }g_{i}(\overline{y}_{1},\overline{y}_{2},...,\overline{y}_% {K})\leq 0\,\text{ }\forall i\in\{1,...,N\}
  92. y ¯ i = lim t 1 t τ = 0 t - 1 E [ y i ( τ ) ] \overline{y}_{i}=\lim_{t\rightarrow\infty}\frac{1}{t}\sum_{\tau=0}^{t-1}E[y_{i% }(\tau)]
  93. ω ( t ) \omega(t)
  94. ω ( t ) \omega(t)
  95. ω ( t ) \omega(t)
  96. Δ [ r ] \Delta[r]
  97. p [ r ] p[r]
  98. E [ Δ [ r ] + V p [ r ] | Q [ r ] ] E [ T [ r ] | Q [ r ] ] \frac{E[\Delta[r]+Vp[r]|Q[r]]}{E[T[r]|Q[r]]}
  99. A = { ( x 1 , x 2 , , x N ) | x m i n , i x i x m a x , i i { 1 , , N } } A=\{(x_{1},x_{2},...,x_{N})\,\text{ }|\,\text{ }x_{min,i}\leq x_{i}\leq x_{max% ,i}\,\text{ }\forall i\in\{1,...,N\}\}
  100. x m i n , i < x m a x , i x_{min,i}<x_{max,i}
  101. Y i ( x ) Y_{i}(x)
  102. ( E q .6 ) Minimize: P ( x ) (Eq.6)\,\text{ }\,\text{Minimize: }P(x)
  103. ( E q .7 ) Subject to: Y i ( x ) 0 i { 1 , , K } , x = ( x 1 , , x N ) A (Eq.7)\,\text{ }\,\text{Subject to: }Y_{i}(x)\leq 0\,\text{ }\forall i\in\{1,.% ..,K\}\,\text{ },\,\text{ }x=(x_{1},...,x_{N})\in A
  104. ω ( t ) \omega(t)
  105. α ( t ) \alpha(t)
  106. α ( t ) = x ( t ) = ( x 1 ( t ) , x 2 ( t ) , , x N ( t ) ) \alpha(t)=x(t)=(x_{1}(t),x_{2}(t),...,x_{N}(t))
  107. p ( t ) = P ( x 1 ( t ) , , x N ( t ) ) p(t)=P(x_{1}(t),...,x_{N}(t))
  108. y i ( t ) = Y i ( x 1 ( t ) , , x N ( t ) ) i { 1 , , K } y_{i}(t)=Y_{i}(x_{1}(t),...,x_{N}(t))\,\text{ }\forall i\in\{1,\ldots,K\}
  109. x ¯ ( t ) = 1 t τ = 0 t - 1 ( x 1 ( τ ) , , x N ( τ ) ) \overline{x}(t)=\frac{1}{t}\sum_{\tau=0}^{t-1}(x_{1}(\tau),...,x_{N}(\tau))
  110. P ¯ ( t ) = 1 t τ = 0 t - 1 P ( x 1 ( τ ) , , x N ( τ ) ) \overline{P}(t)=\frac{1}{t}\sum_{\tau=0}^{t-1}P(x_{1}(\tau),...,x_{N}(\tau))
  111. Y ¯ i ( t ) = 1 t τ = 0 t - 1 Y i ( x 1 ( τ ) , , x N ( τ ) ) \overline{Y}_{i}(t)=\frac{1}{t}\sum_{\tau=0}^{t-1}Y_{i}(x_{1}(\tau),...,x_{N}(% \tau))
  112. ( E q .8 ) Minimize: lim t P ¯ ( t ) (Eq.8)\,\text{ }\,\text{Minimize: }\lim_{t\rightarrow\infty}\overline{P}(t)
  113. ( E q .9 ) Subject to: lim t Y ¯ i ( t ) 0 i { 1 , , K } (Eq.9)\,\text{ }\,\text{Subject to: }\lim_{t\rightarrow\infty}\overline{Y}_{i}% (t)\leq 0\,\text{ }\forall i\in\{1,...,K\}
  114. P ( x ¯ ( t ) ) P ¯ ( t ) , Y i ( x ¯ ( t ) ) Y ¯ i ( t ) i { 1 , , K } P(\overline{x}(t))\leq\overline{P}(t)\,\text{ },\,\text{ }Y_{i}(\overline{x}(t% ))\leq\overline{Y}_{i}(t)\,\text{ }\forall i\in\{1,...,K\}
  115. lim t x ¯ ( t ) \lim_{t\rightarrow\infty}\overline{x}(t)
  116. x ( t ) = ( x 1 ( t ) , , x N ( t ) ) A x(t)=(x_{1}(t),...,x_{N}(t))\in A
  117. V P ( x ( t ) ) + i = 1 K Q i ( t ) Y i ( x ( t ) ) VP(x(t))+\sum_{i=1}^{K}Q_{i}(t)Y_{i}(x(t))
  118. Q i ( t + 1 ) = max [ Q i ( t ) + Y i ( x ( t ) ) , 0 ] i { 1 , , K } Q_{i}(t+1)=\max[Q_{i}(t)+Y_{i}(x(t)),0]\,\text{ }\forall i\in\{1,...,K\}
  119. x ¯ ( t ) \overline{x}(t)
  120. P ( x ( t ) ) = n = 1 N c n x n ( t ) P(x(t))=\sum_{n=1}^{N}c_{n}x_{n}(t)
  121. Y i ( x ( t ) ) = n = 1 N a i n x n ( t ) - b i i { 1 , , K } Y_{i}(x(t))=\sum_{n=1}^{N}a_{in}x_{n}(t)-b_{i}\,\text{ }\forall i\in\{1,...,K\}
  122. [ V c n + i = 1 K Q i ( t ) a i n ] x n ( t ) [Vc_{n}+\sum_{i=1}^{K}Q_{i}(t)a_{in}]x_{n}(t)
  123. Choose x i ( t ) = x m i n , i if V c n + i = 1 K Q i ( t ) a i n 0 \,\text{Choose }x_{i}(t)=x_{min,i}\,\text{ if }Vc_{n}+\sum_{i=1}^{K}Q_{i}(t)a_% {in}\geq 0
  124. Choose x i ( t ) = x m a x , i if V c n + i = 1 K Q i ( t ) a i n < 0 \,\text{Choose }x_{i}(t)=x_{max,i}\,\text{ if }Vc_{n}+\sum_{i=1}^{K}Q_{i}(t)a_% {in}<0

Drimenol_cyclase.html

  1. \rightleftharpoons

DTDP-3-amino-3,4,6-trideoxy-alpha-D-glucopyranose_N,N-dimethyltransferase.html

  1. \rightleftharpoons

DTDP-3-amino-3,6-dideoxy-alpha-D-galactopyranose_3-N-acetyltransferase.html

  1. \rightleftharpoons

DTDP-3-amino-3,6-dideoxy-alpha-D-galactopyranose_N,N-dimethyltransferase.html

  1. \rightleftharpoons

DTDP-3-amino-3,6-dideoxy-alpha-D-galactopyranose_transaminase.html

  1. \rightleftharpoons

DTDP-3-amino-3,6-dideoxy-alpha-D-glucopyranose_N,N-dimethyltransferase.html

  1. \rightleftharpoons

DTDP-3-amino-3,6-dideoxy-alpha-D-glucopyranose_transaminase.html

  1. \rightleftharpoons

DTDP-4-amino-4,6-dideoxy-D-galactose_acyltransferase.html

  1. \rightleftharpoons

DTDP-4-amino-4,6-dideoxy-D-glucose_acyltransferase.html

  1. \rightleftharpoons

DTDP-6-deoxy-L-talose_4-dehydrogenase_(NAD+).html

  1. \rightleftharpoons

DTDP-dihydrostreptose—streptidine-6-phosphate_dihydrostreptosyltransferase.html

  1. \rightleftharpoons

Dubins_path.html

  1. x ˙ \displaystyle\dot{x}
  2. ( x , y ) (x,y)
  3. θ \theta
  4. V V
  5. u u

Dudley_triangle.html

  1. 2 2 2 2 1 2 2 0 0 2 2 6 5 6 2 \begin{matrix}&&&&&2&&&&\\ &&&&2&&2&&&\\ &&&2&&1&&2&&\\ &&2&&0&&0&&2&\\ &2&&6&&5&&6&&2\\ &&&&&\vdots&&&&\\ \end{matrix}
  2. a ( n , m ) = m 2 + m n + n 2 - 1 mod n + m + 1 a(n,m)=m^{2}+mn+n^{2}-1\mod n+m+1\,

Dunham_expansion.html

  1. E ( v , J ) = k , l Y k , l ( v + 1 / 2 ) k [ J ( J + 1 ) ] l , E(v,J)=\sum_{k,l}Y_{k,l}(v+1/2)^{k}[J(J+1)]^{l},
  2. Y k , l Y_{k,l}
  3. Y 0 , 0 Y_{0,0}
  4. Y 0 , 1 = B e Y_{0,1}=B_{e}
  5. Y 0 , 2 = - D e Y_{0,2}=-D_{e}
  6. Y 0 , 3 = H e Y_{0,3}=H_{e}
  7. Y 0 , 4 = L e Y_{0,4}=L_{e}
  8. Y 1 , 0 = ω e Y_{1,0}=\omega_{e}
  9. Y 1 , 1 = - α e Y_{1,1}=-\alpha_{e}
  10. Y 1 , 2 = - β e Y_{1,2}=-\beta_{e}
  11. Y 2 , 0 = - ω e x e Y_{2,0}=-\omega_{e}x_{e}
  12. Y 2 , 1 = γ e Y_{2,1}=\gamma_{e}
  13. Y 3 , 0 = ω e y e Y_{3,0}=\omega_{e}y_{e}
  14. Y 4 , 0 = ω e z e Y_{4,0}=\omega_{e}z_{e}

Dunnett's_test.html

  1. α \alpha
  2. k ( k - 1 ) 2 \frac{k(k-1)}{2}
  3. ( k - 1 ) (k-1)
  4. ( p + 1 ) (p+1)
  5. ( X 0 ¯ , , X p ¯ ) (\bar{X_{0}},...,\bar{X_{p}})
  6. ( X 1 ¯ , , X p ¯ ) (\bar{X_{1}},...,\bar{X_{p}})
  7. X 0 ¯ \bar{X_{0}}
  8. s s
  9. p + 1 p+1
  10. X i ¯ \bar{X_{i}}
  11. p + 1 p+1
  12. σ 2 \sigma^{2}
  13. μ i \mu_{i}
  14. s 2 s^{2}
  15. σ 2 \sigma^{2}
  16. p p
  17. X i ¯ - X 0 ¯ \bar{X_{i}}-\bar{X_{0}}
  18. p p
  19. X i ¯ - X 0 ¯ \bar{X_{i}}-\bar{X_{0}}
  20. P P
  21. P P
  22. P P
  23. X i j X_{ij}
  24. i = 1... p i=1...p
  25. j = 1... N i j=1...N_{i}
  26. s 2 = i = 0 p j = 1 N i ( X i j - X i ¯ ) n s^{2}=\frac{\sum_{i=0}^{p}\sum_{j=1}^{N_{i}}(X_{ij}-\bar{X_{i}})}{n}
  27. X i ¯ \bar{X_{i}}
  28. i i
  29. N i N_{i}
  30. i i
  31. n = i = 0 p N i - ( p + 1 ) n=\sum_{i=0}^{p}N_{i}-(p+1)
  32. m i - m 0 , ( i = 1... p ) m_{i}-m_{0},(i=1...p)
  33. p p
  34. m i - m 0 m_{i}-m_{0}
  35. P P
  36. p p
  37. z i = X i ¯ - X 0 ¯ - ( m i - m 0 ) 1 N i + 1 N 0 z_{i}=\cfrac{\bar{X_{i}}-\bar{X_{0}}-(m_{i}-m_{0})}{\sqrt{\cfrac{1}{N_{i}}+% \cfrac{1}{N_{0}}}}
  38. D i = X i ¯ - X 0 ¯ - ( m i - m 0 ) s 1 N i + 1 N 0 D_{i}=\cfrac{\bar{X_{i}}-\bar{X_{0}}-(m_{i}-m_{0})}{s\sqrt{\cfrac{1}{N_{i}}+% \cfrac{1}{N_{0}}}}
  39. D i = z i s D_{i}=\frac{z_{i}}{s}
  40. P P
  41. p p
  42. m i - m 0 , ( i = 1... p ) m_{i}-m_{0},(i=1...p)
  43. X i ¯ - X 0 ¯ - d i s 1 N i + 1 N 0 , i = 1... p \bar{X_{i}}-\bar{X_{0}}-d_{i}^{\prime}s\sqrt{\frac{1}{N_{i}}+\frac{1}{N_{0}}},% i=1...p
  44. p p
  45. d i d_{i}^{\prime}
  46. P r o b ( t 1 < d 1 , , t p < d p ) Prob(t_{1}<d_{1}^{\prime},...,t_{p}<d_{p}^{\prime})
  47. X i ¯ - X 0 ¯ + d i s 1 N i + 1 N 0 , i = 1... p \bar{X_{i}}-\bar{X_{0}}+d_{i}^{\prime}s\sqrt{\frac{1}{N_{i}}+\frac{1}{N_{0}}},% i=1...p
  48. m i - m 0 m_{i}-m_{0}
  49. X i ¯ - X 0 ¯ ± d i s 1 N i + 1 N 0 , i = 1... p \bar{X_{i}}-\bar{X_{0}}\pm d_{i}^{\prime}s\sqrt{\frac{1}{N_{i}}+\frac{1}{N_{0}% }},i=1...p
  50. d i ′′ d_{i}^{\prime\prime}
  51. P r o b ( | t 1 | < d 1 , , | t p | < d p ) Prob(|t_{1}|<d_{1}^{\prime},...,|t_{p}|<d_{p}^{\prime})
  52. d i ′′ d_{i}^{\prime\prime}
  53. d i d_{i}^{\prime}
  54. s 2 = 19 s^{2}=19
  55. 55 2 + 47 2 + 48 2 + 55 2 + + 41 2 - 3 ( 50 2 + 61 2 + 52 2 + 45 2 ) 8 = 152 8 = 19 \frac{55^{2}+47^{2}+48^{2}+55^{2}+...+41^{2}-3(50^{2}+61^{2}+52^{2}+45^{2})}{8% }=\frac{152}{8}=19
  56. s = 19 = 4.36 s=\sqrt{19}=4.36
  57. s 2 N = 4.36 2 N = 3.56 s\sqrt{\frac{2}{N}}=4.36\sqrt{\frac{2}{N}}=3.56
  58. A = t s 2 N A=t_{s}\sqrt{\frac{2}{N}}
  59. 61 - 50 - 9 = 2 l b s . 61-50-9=2lbs.
  60. 52 - 50 - 9 = - 7 l b s 52-50-9=-7lbs
  61. 45 - 50 - 9 = - 14 l b s 45-50-9=-14lbs
  62. 61 - 50 - 11 = 0 l b s . 61-50-11=0lbs.
  63. 61 - 50 + 11 = 22 l b s . 61-50+11=22lbs.
  64. 52 - 50 - 11 = - 9 l b s 52-50-11=-9lbs
  65. 52 - 50 + 11 = 13 l b s 52-50+11=13lbs
  66. 45 - 50 - 11 = - 16 l b s 45-50-11=-16lbs
  67. 45 - 50 + 11 = 6 l b s 45-50+11=6lbs

Dye_decolorizing_peroxidase.html

  1. \rightleftharpoons

Earle–Hamilton_fixed-point_theorem.html

  1. α ( z , v ) = sup g | g ( z ) v | , \displaystyle{\alpha(z,v)=\sup_{g}|g^{\prime}(z)v|,}
  2. ( γ ) = 0 1 α ( γ ( t ) , γ ˙ ( t ) ) d t . \displaystyle{\ell(\gamma)=\int_{0}^{1}\alpha(\gamma(t),\dot{\gamma}(t))\,dt.}
  3. d ( x , y ) = inf γ ( 0 ) = x , γ ( 1 ) = y ( γ ) \displaystyle{d(x,y)=\inf_{\gamma(0)=x,\gamma(1)=y}\ell(\gamma)}
  4. g ( z ) = a ( z ) + b \displaystyle{g(z)=a(z)+b}
  5. α ( z , v ) v / R , \displaystyle{\alpha(z,v)\geq\|v\|/R,}
  6. d ( x , y ) x - y / R . \displaystyle{d(x,y)\geq\|x-y\|/R.}
  7. ( g f ) ( z ) = g ( f ( z ) ) f ( z ) \displaystyle{(g\circ f)^{\prime}(z)=g^{\prime}(f(z))f^{\prime}(z)}
  8. α ( f ( z ) , f ( z ) v ) α ( z , v ) \displaystyle{\alpha(f(z),f^{\prime}(z)v)\leq\alpha(z,v)}
  9. d ( f ( x ) , f ( y ) ) d ( x , y ) . \displaystyle{d(f(x),f(y))\leq d(x,y).}
  10. f δ , y ( z ) = f ( z ) + δ ( f ( z ) - f ( y ) ) \displaystyle{f_{\delta,y}(z)=f(z)+\delta(f(z)-f(y))}
  11. d ( f ( x ) , f ( y ) ) ( 1 + δ ) - 1 d ( x , y ) . \displaystyle{d(f(x),f(y))\leq(1+\delta)^{-1}d(x,y).}

Eaton's_inequality.html

  1. i = 1 n a i 2 = 1. \sum_{i=1}^{n}a_{i}^{2}=1.
  2. P ( | i = 1 n a i X i | k ) 2 inf 0 c k c ( z - c k - c ) 3 ϕ ( z ) d z = 2 B E ( k ) , P\left(\left|\sum_{i=1}^{n}a_{i}X_{i}\right|\geq k\right)\leq 2\inf_{0\leq c% \leq k}\int_{c}^{\infty}\left(\frac{z-c}{k-c}\right)^{3}\phi(z)\,dz=2B_{E}(k),
  3. P ( | i = 1 n a i X i | k ) 2 ( 1 - Φ [ k - 1.5 k ] ) = 2 B E d ( k ) , P\left(\left|\sum_{i=1}^{n}a_{i}X_{i}\right|\geq k\right)\leq 2\left(1-\Phi% \left[k-\frac{1.5}{k}\right]\right)=2B_{Ed}(k),
  4. B E P = min { 1 , k - 2 , 2 B E } B_{EP}=\min\{1,k^{-2},2B_{E}\}
  5. i = 1 n b i 2 = 1. \sum_{i=1}^{n}b_{i}^{2}=1.
  6. a i b i + + a n b n a_{i}b_{i}+\cdots+a_{n}b_{n}
  7. i = 1 n b i 2 \sum_{i=1}^{n}b_{i}^{2}
  8. E f ( a i b i + + a n b n ) E f ( Z ) Ef(a_{i}b_{i}+\cdots+a_{n}b_{n})\leq Ef(Z)
  9. E f ( a i b i + + a n b n ) inf [ E ( e λ Z ) e λ x ] = e - x 2 / 2 Ef(a_{i}b_{i}+\cdots+a_{n}b_{n})\leq\inf\left[\frac{E(e^{\lambda Z})}{e^{% \lambda x}}\right]=e^{-x^{2}/2}
  10. S n = a i b i + + a n b n S_{n}=a_{i}b_{i}+\cdots+a_{n}b_{n}
  11. P ( S n x ) 2 e 3 9 P ( Z x ) P(S_{n}\geq x)\leq\frac{2e^{3}}{9}P(Z\geq x)
  12. P ( S n x ) e - x 2 / 2 P(S_{n}\geq x)\leq e^{-x^{2}/2}
  13. P ( | μ - σ | ) 0.5 P(|\mu-\sigma|)\leq 0.5\,

Eddy_break-up_model_for_combustion.html

  1. ρ = P R T for all j m j M j {\rho=\frac{P}{RT\sum_{\,\text{for all j}}{\frac{m_{j}}{M_{j}}}}}
  2. R f u = - C R ρ m f u ε k {R_{fu}=-C_{R}\rho m_{fu}\frac{\varepsilon}{k}}
  3. R o x = - C R ρ m o x s ε k {R_{ox}=-C_{R}\rho\frac{m_{ox}}{s}\frac{\varepsilon}{k}}
  4. R p r = - C R ρ m p r ( 1 + s ) ε k {R_{pr}=-C_{R}^{{}^{\prime}}\rho\frac{m_{pr}}{(1+s)}\frac{\varepsilon}{k}}
  5. f = [ s m f u - m o x ] - [ s m f u - m o x ] 0 [ s m f u - m o x ] 1 - [ s m f u - m o x ] 0 {f=\frac{[{sm}_{fu}-m_{ox}]-{[{sm}_{fu}-m_{ox}]}_{0}}{{[{sm}_{fu}-m_{ox}]}_{1}% -{[{sm}_{fu}-m_{ox}]}_{0}}}
  6. f s t = m o x , 0 s m f u , 1 + m o x , 0 {f_{st}=\frac{m_{ox,0}}{{sm}_{fu,1}+m_{ox,0}}}
  7. I f f < f s t , f = - m o x + m o x , 0 s m f u , 1 + m o x , 0 {{If}f<f_{st},f=\frac{-m_{ox}+m_{ox,0}}{{sm}_{fu,1}+m_{ox,0}}}
  8. I f f > f s t , f = - s m f u , 1 + m o x , 0 s m f u , 1 + m o x , 0 {{If}f>f_{st},f=\frac{-{sm}_{fu,1}+m_{ox,0}}{{sm}_{fu,1}+m_{ox,0}}}

EdgeRank.html

  1. edges e u e w e d e \sum_{\mathrm{edges\,}e}u_{e}w_{e}d_{e}
  2. u e u_{e}
  3. w e w_{e}
  4. d e d_{e}

Edwards_equation.html

  1. log k B = β n p K b + C \log k_{B}=\beta_{n}pK_{b}+C
  2. log 10 ( k k 0 ) = s n \log_{10}\left(\frac{k}{k_{0}}\right)=sn
  3. log k k 0 = α E n + β H \log\frac{k}{k_{0}}=\alpha E_{n}+\beta H\,
  4. H = p K a + 1.74 \ H=pK_{a}+1.74
  5. 2 X - X 2 + 2 e - \mathrm{2X^{-}\rightleftharpoons X_{2}+2e^{-}}
  6. E n = E 0 + 2.60 \ E_{n}=E_{0}+2.60
  7. E n = a P + b H \ E_{n}=aP+bH
  8. P log R N R H 2 0 \ P\equiv\log\frac{R_{N}}{R_{H_{2}0}}
  9. log k k 0 = A P + B H \log\frac{k}{k_{0}}=AP+BH\,
  10. log k k 0 = α P + β H \log\frac{k}{k_{0}}=\alpha P+\beta H\,

Effects_of_global_warming_on_human_health.html

  1. \leftrightarrow
  2. \leftrightarrow
  3. \leftrightarrow

Eilenberg–Ganea_theorem.html

  1. δ n + 1 C n ( E ) δ n C n - 1 ( E ) C 1 ( E ) δ 1 C 0 ( E ) 𝜀 Z 0 , \cdots\xrightarrow{\delta_{n}+1}C_{n}(E)\xrightarrow{\delta_{n}}C_{n-1}(E)% \rightarrow\cdots\rightarrow C_{1}(E)\xrightarrow{\delta_{1}}C_{0}(E)% \xrightarrow{\varepsilon}Z\rightarrow 0,
  2. n = sup { k : There exists a Z [ G ] module M with H k ( G , M ) 0 } . n=\sup\{k:\,\text{There exists a }Z[G]\,\text{ module }M\,\text{ with }H^{k}(G% ,M)\neq 0\}.

Eisenstein_reciprocity.html

  1. m > 1 m>1
  2. 𝒪 m \mathcal{O}_{m}
  3. ( ζ m ) , \mathbb{Q}(\zeta_{m}),
  4. ζ m = e 2 π i 1 m \zeta_{m}=e^{2\pi i\frac{1}{m}}
  5. ζ m , ζ m 2 , ζ m m = 1 \zeta_{m},\zeta_{m}^{2},\dots\zeta_{m}^{m}=1
  6. 𝒪 m . \mathcal{O}_{m}.
  7. α 𝒪 m \alpha\in\mathcal{O}_{m}
  8. m m
  9. \mathbb{Z}
  10. ( mod ( 1 - ζ m ) 2 ) . \;\;(\mathop{{\rm mod}}(1-\zeta_{m})^{2}).
  11. 𝒪 m \mathcal{O}_{m}
  12. . \mathbb{Z}.
  13. α , β 𝒪 m \alpha,\beta\in\mathcal{O}_{m}
  14. α \alpha
  15. β \beta
  16. m . m.
  17. c c
  18. ζ m c α \zeta_{m}^{c}\alpha
  19. ( mod m ) . \;\;(\mathop{{\rm mod}}m).
  20. α \alpha
  21. β \beta
  22. α ± β \alpha\pm\beta
  23. α ± β \alpha\pm\beta
  24. m m
  25. α \alpha
  26. β \beta
  27. α β \alpha\beta
  28. α m \alpha^{m}
  29. 1 - ζ m 1-\zeta_{m}
  30. m = l m=l
  31. l = ( 1 - ζ l ) ( 1 - ζ l 2 ) ( 1 - ζ l l - 1 ) . l=(1-\zeta_{l})(1-\zeta_{l}^{2})\dots(1-\zeta_{l}^{l-1}).
  32. ( l ) (l)
  33. \mathbb{Z}
  34. ( ζ l ) \mathbb{Q}(\zeta_{l})
  35. ( l ) = ( 1 - ζ l ) l - 1 , (l)=(1-\zeta_{l})^{l-1},
  36. ( 1 - ζ l ) (1-\zeta_{l})
  37. α , β 𝒪 m , \alpha,\beta\in\mathcal{O}_{m},
  38. 𝒪 m \mathcal{O}_{m}
  39. ( α β ) m = { ζ where ζ m = 1 if α and β are relatively prime 0 otherwise . \left(\frac{\alpha}{\beta}\right)_{m}=\begin{cases}\zeta\mbox{ where }~{}\zeta% ^{m}=1&\mbox{ if }~{}\alpha\mbox{ and }~{}\beta\mbox{ are relatively prime}\\ 0&\mbox{ otherwise}~{}.\\ \end{cases}
  40. α \alpha
  41. β \beta
  42. η 𝒪 m \eta\in\mathcal{O}_{m}
  43. α η m ( mod β ) \alpha\equiv\eta^{m}\;\;(\mathop{{\rm mod}}\beta)
  44. ( α β ) m = 1. \left(\frac{\alpha}{\beta}\right)_{m}=1.
  45. ( α β ) m 1 \left(\frac{\alpha}{\beta}\right)_{m}\neq 1
  46. α \alpha
  47. ( mod β ) . \;\;(\mathop{{\rm mod}}\beta).
  48. ( α β ) m = 1 \left(\frac{\alpha}{\beta}\right)_{m}=1
  49. α \alpha
  50. ( mod β ) . \;\;(\mathop{{\rm mod}}\beta).
  51. m m\in\mathbb{Z}
  52. a a\in\mathbb{Z}
  53. m . m.
  54. ( ζ m a ) m = ζ m a m - 1 - 1 m . \left(\frac{\zeta_{m}}{a}\right)_{m}=\zeta_{m}^{\frac{a^{m-1}-1}{m}}.
  55. ( 1 - ζ m a ) m = ( ζ m a ) m m - 1 2 . \left(\frac{1-\zeta_{m}}{a}\right)_{m}=\left(\frac{\zeta_{m}}{a}\right)_{m}^{% \frac{m-1}{2}}.
  56. α 𝒪 m \alpha\in\mathcal{O}_{m}
  57. m m
  58. α \alpha
  59. a a
  60. ( α a ) m = ( a α ) m . \left(\frac{\alpha}{a}\right)_{m}=\left(\frac{a}{\alpha}\right)_{m}.
  61. K ( ζ l ) K\supset\mathbb{Q}(\zeta_{l})
  62. l l
  63. l l
  64. l l
  65. K . K.
  66. p p
  67. x p + y p + z p = 0 x^{p}+y^{p}+z^{p}=0\;\;
  68. \mathbb{Z}
  69. x , y , z x,y,z
  70. p x y z . p\nmid xyz.\;\;
  71. p x y z . p\mid xyz.\;
  72. 2 p - 1 1 ( mod p 2 ) . 2^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{2}).\;\;
  73. 3 p - 1 1 ( mod p 2 ) . 3^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{2}).
  74. r x , r p - 1 1 ( mod p 2 ) . r\mid x,\;\;\;r^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{2}).
  75. r ( x - y ) , r p - 1 1 ( mod p 2 ) . r\mid(x-y),\;\;\;r^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{2}).
  76. p > 3 , p>3,
  77. x p x , y p y x^{p}\equiv x,\;y^{p}\equiv y
  78. z p z ( mod p 3 ) . z^{p}\equiv z\;\;(\mathop{{\rm mod}}p^{3}).
  79. a a\in\mathbb{Z}
  80. l a l\nmid a
  81. l l
  82. x l a ( mod p ) x^{l}\equiv a\;\;(\mathop{{\rm mod}}p)
  83. p p
  84. a = b l . a=b^{l}.
  85. x n a ( mod p ) x^{n}\equiv a\;\;(\mathop{{\rm mod}}p)
  86. p . p.
  87. 8 n 8\nmid n
  88. a = b n a=b^{n}
  89. 8 | n 8|n
  90. a = b n a=b^{n}
  91. a = 2 n 2 b n . a=2^{\frac{n}{2}}b^{n}.

Eisenstein_sum.html

  1. E ( χ , α ) = T r F / K t = α χ ( t ) E(\chi,\alpha)=\sum_{Tr_{F/K}t=\alpha}\chi(t)

Electrically_small_antenna.html

  1. 2 π r λ 2\pi r\over\lambda
  2. 1 \scriptstyle\ll 1
  3. Q 1 r 3 Q\propto\frac{1}{r^{3}}
  4. r r

Electropermanent_magnet.html

  1. H c i H_{ci}
  2. A = π \sdot r 2 A=\pi\sdot r^{2}
  3. s = r \sdot π s=r\sdot\sqrt{\pi}
  4. \mathcal{R}
  5. M M F A l N i C o = H C - A l N i C o \sdot L A l N i C o MMF_{AlNiCo}=H_{C-AlNiCo}\sdot L_{AlNiCo}
  6. A l N i C o = L A l N i C o B r - A l N i C o H C - A l N i C o \sdot A \mathcal{R}_{AlNiCo}=\frac{L_{AlNiCo}}{\frac{B_{r-AlNiCo}}{H_{C-AlNiCo}}\sdot A}
  7. Φ A l N i C o = M M F A l N i C o A l N i C o \Phi_{AlNiCo}=\frac{MMF_{AlNiCo}}{\mathcal{R}_{AlNiCo}}
  8. \mathcal{R}
  9. M M F N d F e B = H C - N d F e B \sdot L N d F e B MMF_{NdFeB}=H_{C-NdFeB}\sdot L_{NdFeB}
  10. N d F e B = L N d F e B B r - N d F e B H C - N d F e B \sdot A \mathcal{R}_{NdFeB}=\frac{L_{NdFeB}}{\frac{B_{r-NdFeB}}{H_{C-NdFeB}}\sdot A}
  11. Φ N d F e B = M M F N d F e B N d F e B \Phi_{NdFeB}=\frac{MMF_{NdFeB}}{\mathcal{R}_{NdFeB}}
  12. \mathcal{R}
  13. g a p = L g a p μ 0 \sdot A \mathcal{R}_{gap}=\frac{L_{gap}}{\mu_{0}\sdot A}
  14. h i p e r c o = L h i p e r c o μ r - h i p e r c o \sdot A \mathcal{R}_{hiperco}=\frac{L_{hiperco}}{\mu_{r-hiperco}\sdot A}
  15. e q u i v a l e n t \mathcal{R}_{equivalent}
  16. e q u i v a l e n t = ( 1 N d F e B + 1 A l N i C o ) - 1 \mathcal{R}_{equivalent}=\left(\frac{1}{\mathcal{R}_{NdFeB}}+\frac{1}{\mathcal% {R}_{AlNiCo}}\right)^{-1}
  17. \color g r e e n M M F e q u i v a l e n t = e q u i v a l e n t \sdot ( Φ N d F e B + Φ A l N i C o ) \color{green}MMF_{equivalent}=\mathcal{R}_{equivalent}\sdot\left(\Phi_{NdFeB}+% \Phi_{AlNiCo}\right)
  18. \color r e d M M F e q u i v a l e n t = e q u i v a l e n t \sdot ( Φ N d F e B - Φ A l N i C o ) \color{red}MMF_{equivalent}=\mathcal{R}_{equivalent}\sdot\left(\Phi_{NdFeB}-% \Phi_{AlNiCo}\right)
  19. Φ g a p = M M F e q u i v a l e n t 2 \sdot g a p + h i p e r c o + e q u i v a l e n t \Phi_{gap}=\frac{MMF_{equivalent}}{2\sdot\mathcal{R}_{gap}+\mathcal{R}_{% hiperco}+\mathcal{R}_{equivalent}}
  20. 𝐁 g a p = Φ g a p A \mathbf{B}_{gap}=\frac{\Phi_{gap}}{A}
  21. 𝐅 ( g a p - d i s t a n c e ) = ( 𝐁 g a p ) 2 \sdot A μ 0 \mathbf{F}_{(gap-distance)}=\frac{(\mathbf{B}_{gap})^{2}\sdot A}{\mu_{0}}
  22. B a p p l y = μ 0 H c i = 62.8 m T B_{apply}=\mu_{0}H_{ci}=62.8mT
  23. B z = μ 0 H z B_{z}=\mu_{0}H_{z}
  24. H z ( 0 ) = N \sdot i D 2 - D 1 \sdot l n ( D 2 + D 2 2 + L 2 D 1 + D 1 2 + L 2 ) H_{z}(0)=\frac{N\sdot i}{D_{2}-D_{1}}\sdot ln\left(\frac{D_{2}+\sqrt{{D_{2}}^{% 2}+L^{2}}}{D_{1}+\sqrt{{D_{1}}^{2}+L^{2}}}\right)
  25. N = L d w i r e \sdot ( D 2 - D 1 ) 2 \sdot d w i r e N=\frac{L}{d_{wire}}\sdot\frac{(D_{2}-D_{1})}{2\sdot d_{wire}}
  26. L w i r e = 0.515 \sdot N \sdot π \sdot ( D 2 + D 1 ) L_{wire}=0.515\sdot N\sdot\pi\sdot(D_{2}+D_{1})
  27. P = I 2 \sdot R P=I^{2}\sdot R
  28. V = P / I V=P/I
  29. H z ( Z ) = N \sdot i 2 L \sdot ( D 2 - D 1 ) \sdot [ ( L + 2 z ) \sdot l n ( D 2 + D 2 2 + ( L + 2 z ) 2 D 1 + D 1 2 + ( L + 2 z ) 2 ) + ( L - 2 z ) \sdot l n ( D 2 + D 2 2 + ( L - 2 z ) 2 D 1 + D 1 2 + ( L - 2 z ) 2 ) ] H_{z}(Z)=\frac{N\sdot i}{2L\sdot(D_{2}-D_{1})}\sdot\left[(L+2z)\sdot ln\left(% \frac{D_{2}+\sqrt{{D_{2}}^{2}+(L+2z)^{2}}}{D_{1}+\sqrt{{D_{1}}^{2}+(L+2z)^{2}}% }\right)+(L-2z)\sdot ln\left(\frac{D_{2}+\sqrt{{D_{2}}^{2}+(L-2z)^{2}}}{D_{1}+% \sqrt{{D_{1}}^{2}+(L-2z)^{2}}}\right)\right]
  30. 300 μ m 300\mu m
  31. 400 μ m 400\mu m

Elliott_formula.html

  1. ϕ λ ( 𝐤 ) \phi_{\lambda}({\mathbf{k}})
  2. λ \lambda
  3. E λ E_{\lambda}
  4. 𝐤 \hbar{\mathbf{k}}
  5. α ( E ) \alpha(E)
  6. γ = 0.13 meV \hbar\gamma=0.13\,\mathrm{meV}
  7. α ( E ) \alpha(E)
  8. E E
  9. E gap = 1.490 meV E_{\mathrm{gap}}=1.490\,\mathrm{meV}
  10. ω \hbar\omega
  11. F λ F_{\lambda}
  12. λ \lambda
  13. γ λ \gamma_{\lambda}
  14. λ \lambda
  15. γ λ \gamma_{\lambda}
  16. γ λ γ \gamma_{\lambda}\rightarrow\gamma
  17. γ λ ( ω ) \gamma_{\lambda}(\omega)
  18. λ \lambda
  19. γ λ ( ω ) \gamma_{\lambda}(\omega)
  20. E λ E_{\lambda}
  21. ω \omega
  22. E λ E_{\lambda}
  23. | 𝐤 ϕ λ ( 𝐤 ) | 2 |\sum_{\mathbf{k}}\phi_{\lambda}({\mathbf{k}})|^{2}
  24. F λ F_{\lambda}
  25. s s
  26. s s
  27. γ λ \gamma_{\lambda}
  28. E λ E_{\lambda}
  29. E λ E_{\lambda}
  30. F λ F_{\lambda}
  31. γ λ ( ω ) \gamma_{\lambda}(\omega)
  32. γ 1 / , meV \hbar\gamma\approx 1/,\mathrm{meV}
  33. E gap E_{\mathrm{gap}}
  34. S λ = 𝐤 | ϕ λ ( 𝐤 ) | 2 f 𝐤 e f 𝐤 h + Δ N λ S_{\lambda}=\sum_{\mathbf{k}}|\phi_{\lambda}({\mathbf{k}})|^{2}f_{\mathbf{k}}^% {e}f_{\mathbf{k}}^{h}+\Delta N_{\lambda}\;
  35. f 𝐤 e f_{\mathbf{k}}^{e}
  36. f 𝐤 h f_{\mathbf{k}}^{h}
  37. 𝐤 \hbar{\mathbf{k}}
  38. S λ S_{\lambda}
  39. Δ N λ \Delta N_{\lambda}
  40. f 𝐤 e f 𝐤 h f^{e}_{\mathbf{k}}f^{h}_{\mathbf{k}}
  41. 𝐤 \mathbf{k}
  42. 𝐤 \mathbf{k}
  43. f 𝐤 e f 𝐤 h f^{e}_{\mathbf{k}}f^{h}_{\mathbf{k}}
  44. Δ N λ \Delta N_{\lambda}
  45. s s
  46. S λ S_{\lambda}
  47. λ = 1 s \lambda=1s
  48. γ = 1.7 meV \hbar\gamma=1.7\,\mathrm{meV}
  49. Δ N λ , λ \Delta N_{\lambda,\lambda}
  50. λ \lambda
  51. Δ N ν , λ ν \Delta N_{\nu,\lambda\neq\nu}
  52. ν \nu
  53. λ ν \lambda\neq\nu
  54. Δ N ν , λ ν \Delta N_{\nu,\lambda\neq\nu}
  55. S ν , λ ( ω ) = β ( E β - E ν ) J ν β J β λ E β - E ν - ω - i γ λ , ν ( ω ) S^{\nu,\lambda}(\omega)=\sum_{\beta}\frac{(E_{\beta}-E_{\nu})J_{\nu\beta}J_{% \beta\lambda}}{E_{\beta}-E_{\nu}-\hbar\omega-\mathrm{i}\gamma_{\lambda,\nu}(% \omega)}
  56. J ν β 𝐤 ϕ ν ( 𝐤 ) 𝐤 𝐤 THz ϕ β ( 𝐤 ) J_{\nu\beta}\propto\sum_{\mathbf{k}}\phi^{\star}_{\nu}({\mathbf{k}}){\mathbf{k% }}\cdot{\mathbf{k}}_{\rm THz}\phi_{\beta}({\mathbf{k}})
  57. 𝐤 THz {\mathbf{k}}_{\rm THz}
  58. S ν , λ ( ω ) S^{\nu,\lambda}(\omega)
  59. γ λ , ν ( ω ) \gamma_{\lambda,\nu}(\omega)
  60. ω \omega
  61. γ ( ω ) \gamma(\omega)
  62. γ ν , λ \gamma_{\nu,\lambda}

Elliptic_Gauss_sum.html

  1. - t χ ( t ) ϕ ( t π ) ( p - 1 ) / m -\sum_{t}\chi(t)\phi\left(\frac{t}{\pi}\right)^{(p-1)/m}

Empirical_likelihood.html

  1. max π i , θ i = 1 n ln π i \max_{\pi_{i},\theta}\sum_{i=1}^{n}\ln\pi_{i}
  2. s . t . i = 1 n π i = 1 , i = 1 n π i h ( y i ; θ ) = 0 s.t.\sum_{i=1}^{n}\pi_{i}=1,\sum_{i=1}^{n}\pi_{i}h(y_{i};\theta)=0
  3. = i = 1 n ln π i + μ ( 1 - i = 1 n π i ) - n τ i = 1 n π i h ( y i ; θ ) \mathcal{L}=\sum_{i=1}^{n}\ln\pi_{i}+\mu(1-\sum_{i=1}^{n}\pi_{i})-n\tau^{% \prime}\sum_{i=1}^{n}\pi_{i}h(y_{i};\theta)

Emulsion_stabilization_using_polyelectrolytes.html

  1. h h
  2. V = 64 π R C k B T Γ e - K h K 2 , V=\frac{64\pi RCk_{B}T\Gamma e^{-Kh}}{K^{2}},
  3. R R
  4. C C
  5. k B k_{B}
  6. Γ \Gamma
  7. h h
  8. T T
  9. K K
  10. k B T k_{B}T
  11. λ B = e 2 4 π ε r ε 0 k B T , \lambda_{B}=\frac{e^{2}}{4\pi\varepsilon_{r}\varepsilon_{0}\ k_{B}T},
  12. e e
  13. ε 0 \varepsilon_{0}
  14. ε r \varepsilon_{r}
  15. σ = ε r ε 0 ϕ 0 K , \sigma={\varepsilon_{r}\varepsilon_{0}\phi_{0}}{K},
  16. ϕ 0 \phi_{0}
  17. ( C m 2 ) (\frac{C}{m^{2}})

Encoder_receiver_transmitter.html

  1. p ( x ) = x 16 + x 14 + x 13 + x 11 + x 10 + x 9 + x 8 + x 6 + x 5 + x + 1 p(x)=x^{16}+x^{14}+x^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+x^{5}+x+1

Engel_identity.html

  1. L L
  2. [ x , y ] [x,y]
  3. x , y x,y
  4. L L
  5. L L
  6. x , y x,y
  7. L L
  8. [ x , [ x , , [ x , [ x , y ] ] , ] ] = 0 [x,[x,\ldots,[x,[x,y]],\ldots]]=0
  9. x x
  10. G G
  11. 0
  12. 1 1
  13. 1 1
  14. G G

Ent-cassa-12,15-diene_11-hydroxylase.html

  1. \rightleftharpoons

Ent-isokaurene_C2-hydroxylase.html

  1. \rightleftharpoons

Ent-kaurene_oxidase.html

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

Ent-kaurenoic_acid_oxidase.html

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

Entropic_value_at_risk.html

  1. ( Ω , , P ) (\Omega,\mathcal{F},P)
  2. Ω \Omega
  3. \mathcal{F}
  4. σ \sigma
  5. Ω \Omega
  6. P P
  7. \mathcal{F}
  8. X X
  9. 𝐋 M + \mathbf{L}_{M^{+}}
  10. X : Ω \R X:\Omega\rightarrow\R
  11. M X ( z ) M_{X}(z)
  12. z 0 z\geq 0
  13. X 𝐋 M + X\in\mathbf{L}_{M^{+}}
  14. 1 - α 1-\alpha
  15. EVaR 1 - α ( X ) := inf z > 0 { z - 1 ln ( M X ( z ) / α ) } . \,\text{EVaR}_{1-\alpha}(X):=\inf_{z>0}\{z^{-1}\ln(M_{X}(z)/\alpha)\}.\,
  16. X 𝐋 M + X\in\mathbf{L}_{M^{+}}
  17. Pr ( X a ) e - z a M X ( z ) , z > 0. \,\text{Pr}(X\geq a)\leq e^{-za}M_{X}(z),\quad\forall z>0.\,
  18. e - z a M X ( z ) = α e^{-za}M_{X}(z)=\alpha
  19. a a
  20. a X ( α , z ) := z - 1 ln ( M X ( z ) / α ) a_{X}(\alpha,z):=z^{-1}\ln(M_{X}(z)/\alpha)
  21. EVaR 1 - α ( X ) := inf z > 0 { a X ( α , z ) } \,\text{EVaR}_{1-\alpha}(X):=\inf_{z>0}\{a_{X}(\alpha,z)\}
  22. a X ( 1 , z ) a_{X}(1,z)
  23. 𝐋 M \mathbf{L}_{M}
  24. X : Ω \R X:\Omega\rightarrow\R
  25. M X ( z ) M_{X}(z)
  26. z z
  27. EVaR 1 - α ( X ) = sup Q ( E Q ( X ) ) \,\text{EVaR}_{1-\alpha}(X)=\sup_{Q\in\Im}(E_{Q}(X))\,
  28. X 𝐋 M X\in\mathbf{L}_{M}
  29. \Im
  30. ( Ω , ) (\Omega,\mathcal{F})
  31. = { Q P : D K L ( Q | | P ) - ln α } \Im=\{Q\ll P:D_{KL}(Q||P)\leq-\ln\alpha\}
  32. D K L ( Q | | P ) := d Q d P ( ln d Q d P ) d P D_{KL}(Q||P):=\int\frac{dQ}{dP}(\ln\frac{dQ}{dP})dP
  33. Q Q
  34. P P
  35. M X ( z ) M_{X}(z)
  36. X 𝐋 M + X\in\mathbf{L}_{M^{+}}
  37. z > 0 z>0
  38. M X ( z ) = sup 0 < α 1 { α exp ( z EVaR 1 - α ( X ) ) } . M_{X}(z)=\sup_{0<\alpha\leq 1}\{\alpha\exp(z\,\text{EVaR}_{1-\alpha}(X))\}.\,
  39. X , Y 𝐋 M X,Y\in\mathbf{L}_{M}
  40. EVaR 1 - α ( X ) = EVaR 1 - α ( Y ) \,\text{EVaR}_{1-\alpha}(X)=\,\text{EVaR}_{1-\alpha}(Y)
  41. α ] 0 , 1 ] \alpha\in]0,1]
  42. F X ( b ) = F Y ( b ) F_{X}(b)=F_{Y}(b)
  43. b \R b\in\R
  44. θ \theta
  45. X 𝐋 M + X\in\mathbf{L}_{M^{+}}
  46. θ > 0 \theta>0
  47. θ - 1 ln M X ( θ ) = a X ( 1 , θ ) = sup 0 < α 1 { EVaR 1 - α ( X ) + θ - 1 ln α } . \theta^{-1}\ln M_{X}(\theta)=a_{X}(1,\theta)=\sup_{0<\alpha\leq 1}\{\,\text{% EVaR}_{1-\alpha}(X)+\theta^{-1}\ln\alpha\}.\,
  48. 1 - α 1-\alpha
  49. 1 - α 1-\alpha
  50. VaR ( X ) CVaR ( X ) EVaR ( X ) . \,\text{VaR}(X)\leq\,\text{CVaR}(X)\leq\,\text{EVaR}(X).\,
  51. E ( X ) EVaR 1 - α ( X ) esssup ( X ) \,\text{E}(X)\leq\,\text{EVaR}_{1-\alpha}(X)\leq\,\text{esssup}(X)\,
  52. E ( X ) \,\text{E}(X)
  53. X X
  54. esssup ( X ) \,\text{esssup}(X)
  55. X X
  56. inf t \R { t : Pr ( X t ) = 1 } \inf_{t\in\R}\{t:\,\text{Pr}(X\leq t)=1\}
  57. EVaR 0 ( X ) = E ( X ) \,\text{EVaR}_{0}(X)=\,\text{E}(X)
  58. lim α 0 EVaR 1 - α ( X ) = esssup ( X ) \lim_{\alpha\rightarrow 0}\,\text{EVaR}_{1-\alpha}(X)=\,\text{esssup}(X)
  59. X N ( μ , σ ) X\sim N(\mu,\sigma)
  60. EVaR 1 - α ( X ) = μ + - 2 ln α σ . \,\text{EVaR}_{1-\alpha}(X)=\mu+\sqrt{-2\ln\alpha}\sigma.\,
  61. X U ( a , b ) X\sim U(a,b)
  62. EVaR 1 - α ( X ) = inf t > 0 { t ln ( t e t - 1 b - e t - 1 a b - a ) - t ln α } . \,\text{EVaR}_{1-\alpha}(X)=\inf_{t>0}\left\{t\ln\left(t\frac{e^{t^{-1}b}-e^{t% ^{-1}a}}{b-a}\right)-t\ln\alpha\right\}.\,
  63. N ( 0 , 1 ) N(0,1)
  64. U ( 0 , 1 ) U(0,1)
  65. ρ \rho
  66. min s y m b o l w s y m b o l W ρ ( G ( s y m b o l w , s y m b o l ψ ) ) \min_{symbol{w}\in symbol{W}}\rho(G(symbol{w},symbol{\psi}))\,
  67. s y m b o l w \insymbol W \R n symbol{w}\insymbol{W}\subseteq\R^{n}
  68. n n
  69. s y m b o l ψ symbol{\psi}
  70. m m
  71. G ( s y m b o l w , . ) : \R m \R G(symbol{w},.):\R^{m}\rightarrow\R
  72. s y m b o l w \insymbol W symbol{w}\insymbol{W}
  73. ρ \rho
  74. EVaR \,\text{EVaR}
  75. min s y m b o l w \insymbol W , t > 0 { t ln M G ( s y m b o l w , s y m b o l ψ ) ( t - 1 ) - t ln α } . \min_{symbol{w}\insymbol{W},t>0}\{t\ln M_{G(symbol{w},symbol{\psi})}(t^{-1})-t% \ln\alpha\}.\,
  76. s y m b o l S s y m b o l ψ symbol{S}_{symbol{\psi}}
  77. s y m b o l ψ symbol{\psi}
  78. G ( . , s y m b o l s ) G(.,symbol{s})
  79. s y m b o l s \insymbol S s y m b o l ψ symbol{s}\insymbol{S}_{symbol{\psi}}
  80. G ( s y m b o l w , s y m b o l ψ ) G(symbol{w},symbol{\psi})
  81. G ( s y m b o l w , s y m b o l ψ ) = g 0 ( s y m b o l w ) + i = 1 m g i ( s y m b o l w ) ψ i , g i : \R n \R , i = 0 , 1 , , m , G(symbol{w},symbol{\psi})=g_{0}(symbol{w})+\sum_{i=1}^{m}g_{i}(symbol{w})\psi_% {i},\quad g_{i}:\R^{n}\rightarrow\R,i=0,1,\dots,m,\,
  82. ψ 1 , , ψ m \psi_{1},\dots,\psi_{m}
  83. 𝐋 M \mathbf{L}_{M}
  84. min s y m b o l w \insymbol W , t > 0 { g 0 ( s y m b o l ( w ) ) + t i = 1 m ln M g i ( s y m b o l ( w ) ) ψ i ( t - 1 ) - t ln α } . \min_{symbol{w}\insymbol{W},t>0}\left\{g_{0}(symbol(w))+t\sum_{i=1}^{m}\ln M_{% g_{i}(symbol(w))\psi_{i}}(t^{-1})-t\ln\alpha\right\}.\,
  85. min s y m b o l w \insymbol W , t \R { t + 1 α E [ g 0 ( s y m b o l w ) + i = 1 m g i ( s y m b o l w ) ψ i - t ] + } . \min_{symbol{w}\insymbol{W},t\in\R}\left\{t+\frac{1}{\alpha}\,\text{E}\left[g_% {0}(symbol{w})+\sum_{i=1}^{m}g_{i}(symbol{w})\psi_{i}-t\right]_{+}\right\}.\,
  86. ψ \psi
  87. ψ 1 , , ψ m \psi_{1},\dots,\psi_{m}
  88. k k
  89. s y m b o l w symbol{w}
  90. t t
  91. m k mk
  92. k m k^{m}
  93. k = 2 k=2
  94. m = 100 m=100
  95. 10 - 12 10^{-12}
  96. 4 × 10 10 4\times 10^{10}
  97. 10 - 10 10^{-10}
  98. g g
  99. g ( 1 ) = 0 g(1)=0
  100. β \beta
  101. g g
  102. β \beta
  103. ER g , β ( X ) := sup Q E Q ( X ) \,\text{ER}_{g,\beta}(X):=\sup_{Q\in\Im}\,\text{E}_{Q}(X)\,
  104. = { Q P : H g ( P , Q ) β } \Im=\{Q\ll P:H_{g}(P,Q)\leq\beta\}
  105. H g ( P , Q ) H_{g}(P,Q)
  106. Q Q
  107. P P
  108. g g
  109. ER g , β ( X ) = inf t > 0 , μ \R { t [ μ + E P ( g * ( X t - μ + β ) ) ] } \,\text{ER}_{g,\beta}(X)=\inf_{t>0,\mu\in\R}\left\{t\left[\mu+\,\text{E}_{P}% \left(g^{*}\left(\frac{X}{t}-\mu+\beta\right)\right)\right]\right\}\,
  110. g * g^{*}
  111. g g
  112. g ( x ) = { x ln x x > 0 0 x = 0 + x < 0 , g(x)=\begin{cases}x\ln x&x>0\\ 0&x=0\\ +\infty&x<0,\end{cases}\,
  113. g * ( x ) = e x - 1 g^{*}(x)=e^{x-1}
  114. β = - ln α \beta=-\ln\alpha
  115. g g
  116. g ( x ) = { 0 0 x 1 α + otherwise , g(x)=\left\{\begin{array}[]{lr}0&0\leq x\leq\frac{1}{\alpha}\\ +\infty&\,\text{otherwise},\end{array}\right.\,
  117. g * ( x ) = 1 α max { 0 , x } g^{*}(x)=\frac{1}{\alpha}\max\{0,x\}
  118. β = 0 \beta=0
  119. g g

Entropy_of_entanglement.html

  1. A A
  2. B B
  3. S S
  4. ρ A B = | Ψ Ψ | A B \rho_{AB}=|\Psi\rangle\langle\Psi|_{AB}
  5. 𝒮 ( ρ A ) = - Tr [ ρ A log ρ A ] = - Tr [ ρ B log ρ B ] = 𝒮 ( ρ B ) \mathcal{S}(\rho_{A})=-\,\text{Tr}[\rho_{A}\,\text{log}\rho_{A}]=-\,\text{Tr}[% \rho_{B}\,\text{log}\rho_{B}]=\mathcal{S}(\rho_{B})
  6. ρ A = Tr B ( ρ A B ) \rho_{A}=\mathrm{Tr}_{B}(\rho_{AB})
  7. ρ B = Tr A ( ρ A B ) \rho_{B}=\mathrm{Tr}_{A}(\rho_{AB})
  8. 𝒮 α \mathcal{S}_{\alpha}
  9. α 0 \alpha\geq 0
  10. 𝒮 α ( ρ A ) = 1 1 - α log Tr ( ρ α ) = 𝒮 α ( ρ B ) \mathcal{S}_{\alpha}(\rho_{A})=\frac{1}{1-\alpha}\,\text{log}\,\text{Tr}(\rho^% {\alpha})=\mathcal{S}_{\alpha}(\rho_{B})
  11. α 1 \alpha\rightarrow 1

Entropy_production.html

  1. N = S - S 0 - d Q T . N=S-S_{0}-\int\frac{dQ}{T}.
  2. S ˙ i \dot{S}_{i}
  3. d S d t = Σ k Q ˙ k T k + Σ k S ˙ k + Σ k S ˙ i k with S ˙ i k 0. \frac{\mathrm{d}S}{\mathrm{d}t}=\Sigma_{k}\frac{\dot{Q}_{k}}{T_{k}}+\Sigma_{k}% \dot{S}_{k}+\Sigma_{k}\dot{S}_{ik}\,\text{ with }\dot{S}_{ik}\geq 0.
  4. Q ˙ k \dot{Q}_{k}
  5. S ˙ k = n ˙ k S m k = m ˙ k s k \dot{S}_{k}=\dot{n}_{k}S_{mk}=\dot{m}_{k}s_{k}
  6. n ˙ k , m ˙ k \dot{n}_{k},\dot{m}_{k}
  7. S ˙ i k \dot{S}_{ik}
  8. S ˙ i k \dot{S}_{ik}
  9. d U d t = Σ k Q ˙ k + Σ k H ˙ k - Σ k p k d V k d t + P , \frac{\mathrm{d}U}{\mathrm{d}t}=\Sigma_{k}\dot{Q}_{k}+\Sigma_{k}\dot{H}_{k}-% \Sigma_{k}p_{k}\frac{\mathrm{d}V_{k}}{\mathrm{d}t}+P,
  10. H ˙ k = n ˙ k H m k = m ˙ k h k \dot{H}_{k}=\dot{n}_{k}H_{mk}=\dot{m}_{k}h_{k}
  11. Q ˙ H \dot{Q}_{H}
  12. Q ˙ a \dot{Q}_{a}
  13. S ˙ i \dot{S}_{i}
  14. Q ˙ L \dot{Q}_{L}
  15. Q ˙ a \dot{Q}_{a}
  16. S ˙ i \dot{S}_{i}
  17. n ˙ = 0 \dot{n}=0
  18. 0 = Σ k Q ˙ k + P 0=\Sigma_{k}\dot{Q}_{k}+P
  19. 0 = Σ k Q ˙ k T k + S ˙ i . 0=\Sigma_{k}\frac{\dot{Q}_{k}}{T_{k}}+\dot{S}_{i}.
  20. 0 = Q ˙ H - Q ˙ a - P 0=\dot{Q}_{H}-\dot{Q}_{a}-P
  21. 0 = Q ˙ H T H - Q ˙ a T a + S ˙ i . 0=\frac{\dot{Q}_{H}}{T_{H}}-\frac{\dot{Q}_{a}}{T_{a}}+\dot{S}_{i}.
  22. Q ˙ H \dot{Q}_{H}
  23. Q ˙ a \dot{Q}_{a}
  24. Q ˙ a \dot{Q}_{a}
  25. P = T H - T a T H Q ˙ H - T a S ˙ i . P=\frac{T_{H}-T_{a}}{T_{H}}\dot{Q}_{H}-T_{a}\dot{S}_{i}.
  26. η = P Q ˙ H . \eta=\frac{P}{\dot{Q}_{H}}.
  27. S ˙ i = 0 \dot{S}_{i}=0
  28. η C = T H - T a T H . \eta_{C}=\frac{T_{H}-T_{a}}{T_{H}}.
  29. 0 = Q ˙ L - Q ˙ a + P 0=\dot{Q}_{L}-\dot{Q}_{a}+P
  30. 0 = Q ˙ L T L - Q ˙ a T a + S ˙ i . 0=\frac{\dot{Q}_{L}}{T_{L}}-\frac{\dot{Q}_{a}}{T_{a}}+\dot{S}_{i}.
  31. Q ˙ L \dot{Q}_{L}
  32. Q ˙ a \dot{Q}_{a}
  33. Q ˙ L = T L T a - T L ( P - T a S ˙ i ) . \dot{Q}_{L}=\frac{T_{L}}{T_{a}-T_{L}}(P-T_{a}\dot{S}_{i}).
  34. ξ = Q ˙ L P . \xi=\frac{\dot{Q}_{L}}{P}.
  35. S ˙ i = 0 \dot{S}_{i}=0
  36. ξ C = T L T a - T L . \xi_{C}=\frac{T_{L}}{T_{a}-T_{L}}.
  37. T a S ˙ i T_{a}\dot{S}_{i}
  38. P d i s s = T a S ˙ i P_{diss}=T_{a}\dot{S}_{i}
  39. Q ˙ a = 0 \dot{Q}_{a}=0
  40. Q ˙ H \dot{Q}_{H}
  41. 0 = Q ˙ H T H + S ˙ i . 0=\frac{\dot{Q}_{H}}{T_{H}}+\dot{S}_{i}.
  42. Q ˙ H 0 \dot{Q}_{H}\geq 0
  43. T H > 0 T_{H}>0
  44. S ˙ i 0 \dot{S}_{i}\leq 0
  45. Q ˙ L = Q ˙ a \dot{Q}_{L}=\dot{Q}_{a}
  46. 0 = Q ˙ L T L - Q ˙ L T a + S ˙ i 0=\frac{\dot{Q}_{L}}{T_{L}}-\frac{\dot{Q}_{L}}{T_{a}}+\dot{S}_{i}
  47. S ˙ i = Q ˙ L ( 1 T a - 1 T L ) . \dot{S}_{i}=\dot{Q}_{L}\left(\frac{1}{T_{a}}-\frac{1}{T_{L}}\right).
  48. Q ˙ L 0 \dot{Q}_{L}\geq 0
  49. T a > T L T_{a}>T_{L}
  50. S ˙ i 0 \dot{S}_{i}\leq 0
  51. Q ˙ \dot{Q}
  52. S ˙ i = Q ˙ ( 1 T 2 - 1 T 1 ) . \dot{S}_{i}=\dot{Q}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right).
  53. Q ˙ = κ A L ( T 1 - T 2 ) \dot{Q}=\kappa\frac{A}{L}(T_{1}-T_{2})
  54. S ˙ i = κ A L ( T 1 - T 2 ) 2 T 1 T 2 . \dot{S}_{i}=\kappa\frac{A}{L}\frac{(T_{1}-T_{2})^{2}}{T_{1}T_{2}}.
  55. V ˙ \dot{V}
  56. S ˙ i = - 1 2 V ˙ T d p . \dot{S}_{i}=-\int_{1}^{2}\frac{\dot{V}}{T}\mathrm{d}p.
  57. V ˙ = C ( p 1 - p 2 ) \dot{V}=C(p_{1}-p_{2})
  58. S ˙ i = C ( p 1 - p 2 ) 2 T . \dot{S}_{i}=C\frac{(p_{1}-p_{2})^{2}}{T}.
  59. S ˙ i \dot{S}_{i}
  60. S t 1 = S a 1 + S b 1 . S_{t1}=S_{a1}+S_{b1}.
  61. S = n C V ln T T 0 + n R ln V V 0 S=nC_{V}\ln\frac{T}{T_{0}}+nR\ln\frac{V}{V_{0}}
  62. S i = S t 2 - S t 1 . S_{i}=S_{t2}-S_{t1}.
  63. S i = n a R ln V t V a + n b R ln V t V b . S_{i}=n_{a}R\ln\frac{V_{t}}{V_{a}}+n_{b}R\ln\frac{V_{t}}{V_{b}}.
  64. S i = - n t R [ x ln x + ( 1 - x ) ln ( 1 - x ) ] . S_{i}=-n_{t}R[x\ln x+(1-x)\ln(1-x)].
  65. S m = C V ln T T 0 + R ln V m V 0 . S_{m}=C_{V}\ln\frac{T}{T_{0}}+R\ln\frac{V_{m}}{V_{0}}.
  66. S m i = R ln 2. S_{mi}=R\ln 2.
  67. S m = k ln Ω , S_{m}=k\ln\Omega,
  68. S m i = k ln ( 2 N A ) = k N A ln 2 = R ln 2. S_{mi}=k\ln(2^{N_{A}})=kN_{A}\ln 2=R\ln 2.
  69. d U d t = Q ˙ - p d V d t + P . \frac{\mathrm{d}U}{\mathrm{d}t}=\dot{Q}-p\frac{\mathrm{d}V}{\mathrm{d}t}+P.
  70. d S d t - Q ˙ T 0. \frac{\mathrm{d}S}{\mathrm{d}t}-\frac{\dot{Q}}{T}\geq 0.
  71. Q ˙ = 0 \dot{Q}=0
  72. Q ˙ = d U / d t - P \dot{Q}=\mathrm{d}U/\mathrm{d}t-P
  73. d ( T S ) d t - d U d t + P 0. \frac{\mathrm{d}(TS)}{\mathrm{d}t}-\frac{\mathrm{d}U}{\mathrm{d}t}+P\geq 0.
  74. F = U - T S , F=U-TS,
  75. d F d t - P 0. \frac{\mathrm{d}F}{\mathrm{d}t}-P\leq 0.
  76. W S F i - F f W_{S}\leq F_{i}-F_{f}
  77. d U d t = Q ˙ - d ( p V ) d t . \frac{\mathrm{d}U}{\mathrm{d}t}=\dot{Q}-\frac{\mathrm{d}(pV)}{\mathrm{d}t}.
  78. d ( T S ) d t - d U d t - d ( p V ) d t 0. \frac{\mathrm{d}(TS)}{\mathrm{d}t}-\frac{\mathrm{d}U}{\mathrm{d}t}-\frac{% \mathrm{d}(pV)}{\mathrm{d}t}\geq 0.
  79. G = U + p V - T S , G=U+pV-TS,
  80. d G d t 0. \frac{\mathrm{d}G}{\mathrm{d}t}\leq 0.
  81. S ˙ i = 0 \dot{S}_{i}=0
  82. T d S d t = Q ˙ + n ˙ T S m . T\frac{\mathrm{d}S}{\mathrm{d}t}=\dot{Q}+\dot{n}TS_{m}.
  83. d U d t = Q ˙ + n ˙ H m - p d V d t . \frac{\mathrm{d}U}{\mathrm{d}t}=\dot{Q}+\dot{n}H_{m}-p\frac{\mathrm{d}V}{% \mathrm{d}t}.
  84. Q ˙ \dot{Q}
  85. d U = T d S - p d V + ( H m - T S m ) d n . \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V+(H_{m}-TS_{m})\mathrm{d}n.
  86. H m - T S m = G m = μ H_{m}-TS_{m}=G_{m}=\mu
  87. d U = T d S - p d V + μ d n . \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V+\mu\mathrm{d}n.

Envirome.html

  1. ϵ 4 \epsilon 4

Epi-isozizaene_5-monooxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons
  5. \rightleftharpoons

Epigroup.html

  1. K e = { x S | n > 0 : x n G e } K_{e}=\{x\in S\;|\;\exists n>0:\;x^{n}\in G_{e}\}

Equated_Monthly_Installment.html

  1. P = A 1 - ( 1 + r ) - n r P\,=\,A\cdot\frac{1-\left({1+r}\right)^{-n}}{r}
  2. A = P r ( 1 + r ) n ( 1 + r ) n - 1 A\,=\,P\cdot\frac{r(1+r)^{n}}{(1+r)^{n}-1}

Equidissection.html

  1. m \langle m\rangle
  2. 1 \langle 1\rangle
  3. 2 \langle 2\rangle
  4. n \langle n\rangle
  5. n ! \langle n!\rangle
  6. S ( T ( r / s ) ) = r + s S(T(r/s))=\langle r+s\rangle
  7. n ! \langle n!\rangle
  8. n \langle n\rangle

Equivalence_principle_(geometric).html

  1. X \scriptstyle{X}\,
  2. T X \scriptstyle{TX}\,
  3. X \scriptstyle{X}\,
  4. F X \scriptstyle{FX}\,
  5. T X \scriptstyle{TX}\,
  6. SO ( 1 , 3 ) \scriptstyle{\mathrm{SO}(1,3)}\,
  7. F X / SO ( 1 , 3 ) X \scriptstyle{FX/\mathrm{SO}(1,3)}\to\scriptstyle{X}\,
  8. X \scriptstyle{X}\,

Ermakov–Lewis_invariant.html

  1. H ^ = 1 2 [ p ^ 2 + Ω 2 ( t ) q ^ 2 ] . \hat{H}=\frac{1}{2}\left[\hat{p}^{2}+\Omega^{2}(t)\hat{q}^{2}\right].
  2. I ^ = 1 2 [ ( q ^ ρ ) 2 + ( ρ p ^ - ρ ˙ q ^ ) 2 ] , \hat{I}=\frac{1}{2}\left[\left(\frac{\hat{q}}{\rho}\right)^{2}+(\rho\hat{p}-% \dot{\rho}\hat{q})^{2}\right],
  3. ρ \rho
  4. ρ ¨ + Ω 2 ρ = ρ - 3 . \ddot{\rho}+\Omega^{2}\rho=\rho^{-3}.
  5. I ^ \hat{I}
  6. T ^ = e i ln ρ 2 ( q ^ p ^ + p ^ q ^ ) e - i ρ ˙ 2 ρ q ^ 2 = e i ln ρ 2 d q ^ 2 d t e - i q ^ 2 2 d ln ρ d t , \hat{T}=e^{i\frac{\ln\rho}{2}(\hat{q}\hat{p}+\hat{p}\hat{q})}e^{-i\frac{\dot{% \rho}}{2\rho}\hat{q}^{2}}=e^{i\frac{\ln\rho}{2}\frac{d\hat{q}^{2}}{dt}}e^{-i% \frac{\hat{q}^{2}}{2}\frac{d\ln\rho}{dt}},
  7. 1 2 [ p ^ 2 + q ^ 2 ] = T ^ I ^ T ^ . \frac{1}{2}\left[\hat{p}^{2}+\hat{q}^{2}\right]=\hat{T}\hat{I}\hat{T}^{\dagger}.

Erythromycin_12_hydroxylase.html

  1. \rightleftharpoons

Ethylmalonyl-CoA_decarboxylase.html

  1. \rightleftharpoons

Eugenol_synthase.html

  1. \rightleftharpoons

Euler_substitution.html

  1. R ( x , a x 2 + b x + c ) d x \int\!R(x,\sqrt{ax^{2}+bx+c})\,\mathrm{d}x
  2. R R
  3. x x
  4. a x 2 + b x + c \sqrt{ax^{2}+bx+c}
  5. a > 0 a>0
  6. a x 2 + b x + c = ± x a + t \sqrt{ax^{2}+bx+c}=\pm x\sqrt{a}+t
  7. x x
  8. x = c - t 2 ± 2 t a - b x=\frac{c-t^{2}}{\pm 2t\sqrt{a}-b}
  9. d x \mathrm{d}x
  10. t t
  11. c > 0 c>0
  12. a x 2 + b x + c = x t ± c . \begin{aligned}\displaystyle\sqrt{ax^{2}+bx+c}=xt\pm\sqrt{c}.\end{aligned}
  13. x x
  14. x = ± 2 t c - b a - t 2 . x=\frac{\pm 2t\sqrt{c}-b}{a-t^{2}}.
  15. a x 2 + b x + c ax^{2}+bx+c
  16. α \alpha
  17. β \beta
  18. a x 2 + b x + c = a ( x - α ) ( x - β ) = ( x - α ) t \sqrt{ax^{2}+bx+c}=\sqrt{a(x-\alpha)(x-\beta)}=(x-\alpha)t
  19. x = a β - α t 2 a - t 2 x=\frac{a\beta-\alpha t^{2}}{a-t^{2}}
  20. t t
  21. d x x 2 + c \int\!\frac{\mathrm{d}x}{\sqrt{x^{2}+c}}
  22. x 2 + c = - x + t \sqrt{x^{2}+c}=-x+t
  23. x = t 2 - c 2 t d x = t 2 + c 2 t 2 d t x=\frac{t^{2}-c}{2t}\quad\quad\mathrm{d}x=\frac{t^{2}+c}{2t^{2}}\,\mathrm{d}t
  24. x 2 + c = - t 2 - c 2 t + t = t 2 + c 2 t \sqrt{x^{2}+c}=-\frac{t^{2}-c}{2t}+t=\frac{t^{2}+c}{2t}
  25. d x x 2 + c = t 2 + c 2 t 2 t 2 + c 2 t d t \int\frac{\mathrm{d}x}{\sqrt{x^{2}+c}}=\int\frac{\frac{t^{2}+c}{2t^{2}}}{\frac% {t^{2}+c}{2t}}\,\mathrm{d}t
  26. = d t t = ln | t | + C = ln | x + x 2 + c | + C =\int\!\frac{\mathrm{d}t}{t}=\ln|t|+C=\ln|x+\sqrt{x^{2}+c}|+C
  27. c = ± 1 c=\pm 1
  28. d x x 2 + 1 = arsinh ( x ) + C \int\frac{\mathrm{d}x}{\sqrt{x^{2}+1}}=\mbox{arsinh}~{}(x)+C
  29. d x x 2 - 1 = arcosh ( x ) + C ( x > 1 ) \int\frac{\mathrm{d}x}{\sqrt{x^{2}-1}}=\mbox{arcosh}~{}(x)+C\quad(x>1)
  30. d x - x 2 + c \textstyle\int\!\frac{\mathrm{d}x}{\sqrt{-x^{2}+c}}
  31. x 2 + c = ± i x + t \sqrt{x^{2}+c}=\pm ix+t
  32. R 1 ( x , a x 2 + b x + c ) log ( R 2 ( x , a x 2 + b x + c ) ) d x \int\!R_{1}(x,\sqrt{ax^{2}+bx+c})\,\log(R_{2}(x,\sqrt{ax^{2}+bx+c}))\,\mathrm{% d}x
  33. R 1 R_{1}
  34. R 2 R_{2}
  35. x x
  36. a x 2 + b x + c \sqrt{ax^{2}+bx+c}
  37. a x 2 + b x + c = a + x t \sqrt{ax^{2}+bx+c}=\sqrt{a}+xt
  38. R 1 ( t ) log ( R 2 ( t ) ) d t \int\!\overset{\sim}{R_{1}}(t)\,\log(\overset{\sim}{R_{2}}(t))\,\mathrm{d}t
  39. R 1 ( t ) \overset{\sim}{R_{1}}(t)
  40. R 2 ( t ) \overset{\sim}{R_{2}}(t)
  41. t t

Eulerian_matroid.html

  1. U n r U{}^{r}_{n}
  2. r + 1 r+1
  3. r + 1 r+1
  4. n n
  5. n n
  6. U n 2 U{}^{2}_{n}
  7. n n
  8. G G
  9. H H
  10. C C
  11. H H
  12. H H
  13. C C
  14. C C
  15. U 6 2 U{}^{2}_{6}
  16. U 6 4 U{}^{4}_{6}
  17. U 6 3 U{}^{3}_{6}
  18. M M
  19. M ¯ \bar{M}
  20. M M
  21. e e
  22. M ¯ \bar{M}
  23. M M
  24. M M
  25. M ¯ \bar{M}
  26. M M
  27. e e

Exact_C*-algebra.html

  1. 0 A 𝑓 B 𝑔 C 0 0\;\xrightarrow{}\;A\;\xrightarrow{f}\;B\;\xrightarrow{g}\;C\;\xrightarrow{}\;0
  2. 0 A min E f id B min E g id C min E 0 , 0\;\xrightarrow{}\;A\otimes_{\min}E\;\xrightarrow{f\otimes\operatorname{id}}\;% B\otimes_{\min}E\;\xrightarrow{g\otimes\operatorname{id}}\;C\otimes_{\min}E\;% \xrightarrow{}\;0,
  3. 𝒪 2 \mathcal{O}_{2}

Exalcomm.html

  1. 0 Der B ( C , L ) Der A ( C , L ) Der A ( B , L ) Exalcomm B ( C , L ) Exalcomm A ( C , L ) Exalcomm A ( B , L ) , 0\rightarrow\operatorname{Der}_{B}(C,L)\rightarrow\operatorname{Der}_{A}(C,L)% \rightarrow\operatorname{Der}_{A}(B,L)\rightarrow\operatorname{Exalcomm}_{B}(C% ,L)\rightarrow\operatorname{Exalcomm}_{A}(C,L)\rightarrow\operatorname{% Exalcomm}_{A}(B,L),

Exchange_spring_magnet.html

  1. E x = A ( d θ d x ) 2 E_{x}=A(\frac{d\theta}{dx})^{2}
  2. A = 1 6 n J S 2 Δ r j 2 A=\frac{1}{6}nJS^{2}\sum\Delta r_{j}^{2}
  3. Δ r j = r j - r i \Delta r_{j}=r_{j}-r_{i}
  4. j j
  5. i i
  6. A A
  7. 10 - 11 10^{-11}
  8. E a = K 1 s i n 2 ( θ ) E_{a}=K_{1}sin^{2}(\theta)
  9. E e x t = - M B s i n ( θ ) E_{ext}=-MBsin(\theta)

Excursion_probability.html

  1. { sup t T f ( t ) u } . \mathbb{P}\left\{\sup_{t\in T}f(t)\geq u\right\}.

Exp4j.html

  1. 3 * sin ( π ) - 2 e 3*\frac{\sin{(\pi)}-2}{e}

Expected_linear_time_MST_algorithm.html

  1. E [ Y ] E [ X ] / 2 E[Y]\leq E[X]/2
  2. d = 0 k 2 d = 2 k \sum_{d=0}^{\infty}\frac{k}{2^{d}}=2k
  3. d = 1 2 d - 1 n 4 d = n / 2 \sum_{d=1}^{\infty}\frac{2^{d-1}n}{4^{d}}=n/2

Explicit_reciprocity_law.html

  1. ( a , b ) = ω ( ( - 1 ) ord ( a ) ord ( b ) b ord ( a ) / a ord ( b ) ) ( q - 1 ) / n (a,b)=\omega((-1)^{\operatorname{ord}(a)\operatorname{ord}(b)}b^{\operatorname% {ord}(a)}/a^{\operatorname{ord}(b)})^{(q-1)/n}
  2. a = p α u a=p^{\alpha}u
  3. b = p β v b=p^{\beta}v
  4. ( a , b ) p = ( - 1 ) α β ε ( p ) ( u p ) β ( v p ) α (a,b)_{p}=(-1)^{\alpha\beta\varepsilon(p)}\left(\frac{u}{p}\right)^{\beta}% \left(\frac{v}{p}\right)^{\alpha}
  5. ε ( p ) = ( p - 1 ) / 2 \varepsilon(p)=(p-1)/2
  6. a = 2 α u a=2^{\alpha}u
  7. b = 2 β v b=2^{\beta}v
  8. ( a , b ) 2 = ( - 1 ) ϵ ( u ) ϵ ( v ) + α ω ( v ) + β ω ( u ) (a,b)_{2}=(-1)^{\epsilon(u)\epsilon(v)+\alpha\omega(v)+\beta\omega(u)}
  9. ω ( x ) = ( x 2 - 1 ) / 8. \omega(x)=(x^{2}-1)/8.

Factor_system.html

  1. c ( h , k ) g c ( h k , g ) = c ( h , k g ) c ( k , g ) . c(h,k)^{g}c(hk,g)=c(h,kg)c(k,g).
  2. c ( g , h ) = c ( g , h ) ( a g h a h a g h - 1 ) . c^{\prime}(g,h)=c(g,h)(a_{g}^{h}a_{h}a_{gh}^{-1}).
  3. c ( g , h ) = a g h a h a g h - 1 c(g,h)=a_{g}^{h}a_{h}a_{gh}^{-1}
  4. λ u g = u g λ g , \lambda u_{g}=u_{g}\lambda^{g},
  5. u g u h = u g h c ( g , h ) . u_{g}u_{h}=u_{gh}c(g,h).
  6. A = ( L , G , c ) . A=(L,G,c).
  7. u t i = u i u_{t^{i}}=u^{i}\,
  8. c ( t i , t j ) = { 1 if i + j < n , a if i + j n . c(t^{i},t^{j})=\begin{cases}1&\,\text{if }i+j<n,\\ a&\,\text{if }i+j\geq n.\end{cases}
  9. Br ( L / K ) K * / N L / K L * H 2 ( G , L * ) . \operatorname{Br}(L/K)\equiv K^{*}/N_{L/K}L^{*}\equiv H^{2}(G,L^{*}).

Factorization_of_the_mean.html

  1. 10 6 10^{6}
  2. 10 13 10^{13}
  3. m 3 {m}^{3}

FAD-dependent_urate_hydroxylase.html

  1. \rightleftharpoons

FAD_reductase_(NAD(P)H).html

  1. \rightleftharpoons

FAD_reductase_(NADH).html

  1. \rightleftharpoons

Falconer's_conjecture.html

  1. ( - ϵ , ϵ ) (-\epsilon,\epsilon)
  2. ϵ > 0 \epsilon>0
  3. d 2 + 1 3 \tfrac{d}{2}+\tfrac{1}{3}

False_coverage_rate.html

  1. FCR > q \,\text{FCR}>q
  2. q = V R = α m 0 R q=\frac{V}{R}=\frac{\alpha m_{0}}{R}
  3. m 0 m_{0}
  4. R R
  5. 1 - q 1-q
  6. q q
  7. Pr [ θ CI , CI constructed ] Pr [ θ CI ] α \Pr[\theta\in\mathrm{CI},\ \,\text{CI constructed}]\leq\Pr[\theta\in\mathrm{CI% }]\leq\alpha

False_diffusion.html

  1. τ f c = ρ u Δ x Δ y sin 2 θ 4 ( Δ y sin 3 θ + Δ x cos 3 θ ) {\tau_{fc}^{\star}=\frac{\rho u\,\Delta x\,\Delta y\sin 2\theta}{4(\Delta y% \sin^{3}\theta+\Delta x\cos^{3}\theta)}}
  2. ϕ W \phi_{W}
  3. ϕ P \phi_{P}
  4. ϕ W k = ϕ w k - ( δ x i 2 ) ( ϕ x ) w k + 1 2 ! ( δ x i 2 ) 2 ( 2 ϕ x 2 ) w k + . \phi_{Wk}=\phi_{wk}-\left(\frac{\delta x_{i}}{2}\right)\left(\frac{\partial% \phi}{\partial x}\right)_{wk}+\frac{1}{2!}\left(\frac{\delta x_{i}}{2}\right)^% {2}\left(\frac{\partial^{2}\phi}{\partial x^{2}}\right)_{wk}+\cdots.
  5. ϕ P k = ϕ w k - ( δ x i 2 ) ( ϕ x ) w k + 1 2 ! ( δ x i 2 ) 2 ( 2 ϕ x 2 ) w k + . \phi_{Pk}=\phi_{wk}-\left(\frac{\delta x_{i}}{2}\right)\left(\frac{\partial% \phi}{\partial x}\right)_{wk}+\frac{1}{2!}\left(\frac{\delta x_{i}}{2}\right)^% {2}\left(\frac{\partial^{2}\phi}{\partial x^{2}}\right)_{wk}+\cdots.
  6. ϕ w k = ϕ W k {\phi_{wk}=\phi_{Wk}}
  7. - ρ w u w δ y i ( δ x i 2 ) ( ϕ x ) w k {-\rho_{w}u_{w}\delta y_{i}\left(\frac{\delta x_{i}}{2}\right)\left(\frac{% \partial\phi}{\partial x}\right)_{wk}}
  8. ϕ {\phi}
  9. τ f c , U A C = ρ w u w ( Δ x i 2 ) \tau_{fc,UAC}^{\star}={\rho_{w}u_{w}\left(\frac{\Delta x_{i}}{2}\right)}
  10. t + k Δ t t+k\,\Delta t
  11. ϕ {\phi}
  12. C P ϕ P = ( m ˙ w - ( m ˙ s ) 2 m ˙ w ) ϕ W + ( m ˙ s + ( m ˙ s ) 2 m ˙ w ) ϕ S W + 0. ϕ S for 0 < θ 45 C_{P}\phi_{P}=\left(\dot{m}_{w}-\frac{(\dot{m}_{s})^{2}}{\dot{m}_{w}}\right)% \phi_{W}+\left(\dot{m}_{s}+\frac{(\dot{m}_{s})^{2}}{\dot{m}_{w}}\right)\phi_{% SW}+0.\phi_{S}\,\text{ for }0<\theta\leq 45
  13. C P ϕ P = C w ϕ W + C s w ϕ S W C_{P}\phi_{P}=C_{w}\phi_{W}+C_{s}w\phi_{SW}
  14. C P ϕ P = ( m ˙ s - ( m ˙ w ) 2 m ˙ s ) ϕ W + ( m ˙ w + ( m ˙ w ) 2 m ˙ s ) ϕ S W + 0. ϕ W for 45 < θ < 90 C_{P}\phi_{P}=\left(\dot{m}_{s}-\frac{(\dot{m}_{w})^{2}}{\dot{m}_{s}}\right)% \phi_{W}+{\left(\dot{m}_{w}+\frac{(\dot{m}_{w})^{2}}{\dot{m}_{s}}\right)\phi_{% SW}}+0.\phi_{W}\,\text{ for }45<\theta<90
  15. C P ϕ P = C s ϕ S + C s w ϕ S W C_{P}\phi_{P}=C_{s}\phi_{S}+C_{s}w\phi_{SW}

Farnesyl_diphosphatase.html

  1. \rightleftharpoons

Farnesylcysteine_lyase.html

  1. \rightleftharpoons

Farrell–Markushevich_theorem.html

  1. g n Ω n 2 = g Ω 2 . \displaystyle{\|g_{n}\|^{2}_{\Omega_{n}}=\|g\|_{\Omega}^{2}.}

Fatty-acid_peroxygenase.html

  1. \rightleftharpoons

Fault_reporting.html

  1. A v a i l a b i l i t y R e d u c t i o n = T o t a l T i m e T o t a l T i m e + M a i n t e n a n c e D o w n T i m e Availability\ Reduction=\frac{Total\ Time}{Total\ Time+Maintenance\ Down\ Time}
  2. D i a g n o s t i c C o s t = M a i n t e n a n c e D o w n T i m e × L a b o r R a t e × T e a m S i z e Diagnostic\ Cost=Maintenance\ Down\ Time\times Labor\ Rate\times\ Team\ Size

Fbsp_wavelet.html

  1. f b s p ( m - fb - fc ) ( t ) := f b . sinc m ( t f b m ) . e j 2 π f c t fbsp^{(\operatorname{m-fb-fc})}(t):={\sqrt{fb}}.\operatorname{sinc}^{m}\left(% \frac{t}{fb^{m}}\right).e^{j2\pi fct}

Feature_hashing.html

  1. i , j i,j
  2. j j
  3. i i
  4. ( John likes to watch movies Mary too also football 1 1 1 1 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 ) \begin{pmatrix}\textrm{John}&\textrm{likes}&\textrm{to}&\textrm{watch}&\textrm% {movies}&\textrm{Mary}&\textrm{too}&\textrm{also}&\textrm{football}\\ 1&1&1&1&1&0&0&0&0\\ 0&1&0&0&1&1&1&0&0\\ 1&1&0&0&0&0&0&1&1\end{pmatrix}
  5. h h
  6. ξ ξ
  7. h h
  8. ξ ξ
  9. 𝔼 [ φ ( x ) , φ ( x ) ] = x , x \mathbb{E}[\langle\varphi(x),\varphi(x^{\prime})\rangle]=\langle x,x^{\prime}\rangle
  10. φ ( x ) , φ ( x ) \langle\varphi(x),\varphi(x^{\prime})\rangle

Feature_learning.html

  1. p p

Ferredoxin:protochlorophyllide_reductase_(ATP-dependent).html

  1. \rightleftharpoons

Ferredoxin:thioredoxin_reductase.html

  1. \rightleftharpoons

Ferrero–Washington_theorem.html

  1. K ^ \hat{K}
  2. K ^ \hat{K}
  3. T p ( K ) = Gal ( A ( p ) / K ^ ) . T_{p}(K)=\mathrm{Gal}(A^{(p)}/\hat{K})\ .
  4. λ m + μ p m + κ . \lambda m+\mu p^{m}+\kappa\ .

Ferric-chelate_reductase_(NADPH).html

  1. \rightleftharpoons

Festuclavine_dehydrogenase.html

  1. \rightleftharpoons

Finger_rafting.html

  1. h r f h_{rf}
  2. h r f = 14.2 ( 1 - ν 2 ) ρ w g σ t 2 Y h_{rf}=\frac{14.2(1-\nu^{2})}{\rho_{w}g}\frac{\sigma_{t}^{2}}{Y}
  3. ν \nu
  4. σ t \sigma_{t}
  5. ρ w \rho_{w}

Finite_subdivision_rule.html

  1. R R
  2. S R S_{R}
  3. S R S_{R}
  4. s ~ \tilde{s}
  5. S R S_{R}
  6. s s
  7. s s
  8. s s
  9. s \partial s
  10. ψ s : s S R \psi_{s}:s\rightarrow S_{R}
  11. s ~ \tilde{s}
  12. R ( S R ) R(S_{R})
  13. S R S_{R}
  14. ϕ R : R ( S R ) S R \phi_{R}:R(S_{R})\rightarrow S_{R}
  15. s s
  16. ψ s \psi_{s}
  17. R R
  18. R R
  19. X X
  20. f : X S R f:X\rightarrow S_{R}
  21. X X
  22. R ( X ) R(X)
  23. f : R ( X ) R ( S R ) f:R(X)\rightarrow R(S_{R})
  24. R ( X ) R(X)
  25. R R
  26. ϕ R f : R ( X ) S R \phi_{R}\circ f:R(X)\rightarrow S_{R}
  27. R R
  28. R n ( X ) R^{n}(X)
  29. ϕ R n f : R n ( X ) S R \phi_{R}^{n}\circ f:R^{n}(X)\rightarrow S_{R}
  30. S R S_{R}
  31. ϕ \phi
  32. f : 2 R ( S R ) f:\mathbb{R}^{2}\rightarrow R(S_{R})
  33. R R
  34. X X
  35. R n ( X ) R^{n}(X)
  36. R n ( X ) R^{n}(X)
  37. R n + 1 ( X ) R^{n+1}(X)
  38. T T
  39. R R
  40. M s u p ( R , T ) M_{sup}(R,T)
  41. m i n f ( R , T ) m_{inf}(R,T)
  42. ρ \rho
  43. T T
  44. R R
  45. H ( ρ ) H(\rho)
  46. R R
  47. ρ \rho
  48. R R
  49. C ( ρ ) C(\rho)
  50. R R
  51. ρ \rho
  52. A ( ρ ) A(\rho)
  53. R R
  54. ρ \rho
  55. R R
  56. M s u p ( R , T ) = sup H ( ρ ) 2 A ( ρ ) M_{sup}(R,T)=\sup\frac{H(\rho)^{2}}{A(\rho)}
  57. m i n f ( R , T ) = inf A ( ρ ) C ( ρ ) 2 m_{inf}(R,T)=\inf\frac{A(\rho)}{C(\rho)^{2}}
  58. T 1 , T 2 , T_{1},T_{2},...
  59. K K
  60. R R
  61. M s u p ( R , T i ) M_{sup}(R,T_{i})
  62. m i n f ( R , T i ) m_{inf}(R,T_{i})
  63. i i
  64. [ r , K r ] [r,Kr]
  65. x x
  66. N N
  67. x x
  68. I I
  69. R R
  70. N { x } N\setminus\{x\}
  71. N N
  72. i i
  73. R R
  74. I I
  75. T 1 , T 2 , T_{1},T_{2},...
  76. K K
  77. K K^{\prime}
  78. K K
  79. T i T_{i}
  80. i i
  81. K K^{\prime}
  82. [ r , K r ] [r,K^{\prime}r]
  83. G G
  84. 3 \mathbb{H}^{3}

Firoozbakht's_conjecture.html

  1. p n 1 / n p_{n}^{1/n}\,
  2. p n p_{n}\,
  3. p n + 1 1 / ( n + 1 ) < p n 1 / n for all n 1. p_{n+1}^{1/(n+1)}<p_{n}^{1/n}\,\text{ for all }n\geq 1.
  4. p n + 1 < p n 1 + 1 n for all n 1 , p_{n+1}<p_{n}^{1+\frac{1}{n}}\,\text{ for all }n\geq 1,
  5. × 10 1 2 \times 10^{1}2
  6. × 10 1 8 \times 10^{1}8
  7. g n = p n + 1 - p n g_{n}=p_{n+1}-p_{n}
  8. g n < ( log p n ) 2 - log p n for all n > 4. g_{n}<(\log p_{n})^{2}-\log p_{n}\,\text{ for all }n>4.
  9. g n > 2 - ε e γ ( log p n ) 2 g_{n}>\frac{2-\varepsilon}{e^{\gamma}}(\log p_{n})^{2}
  10. ε > 0 , \varepsilon>0,
  11. γ \gamma
  12. ( log ( p n + 1 ) log ( p n ) ) n < e \left(\frac{\log(p_{n+1})}{\log(p_{n})}\right)^{n}<e
  13. ( p n + 1 p n ) n < n * log ( n ) \left(\frac{p_{n+1}}{p_{n}}\right)^{n}<n*\log(n)
  14. n > 5 n>5

Fischer_group_Fi22.html

  1. × 10 1 3 \times 10^{1}3
  2. T 6 A ( τ ) T_{6A}(\tau)
  3. j 6 A ( τ ) = T 6 A ( τ ) + 10 = ( ( η ( τ ) η ( 3 τ ) η ( 2 τ ) η ( 6 τ ) ) 3 + 2 3 ( η ( 2 τ ) η ( 6 τ ) η ( τ ) η ( 3 τ ) ) 3 ) 2 = ( ( η ( τ ) η ( 2 τ ) η ( 3 τ ) η ( 6 τ ) ) 2 + 3 2 ( η ( 3 τ ) η ( 6 τ ) η ( τ ) η ( 2 τ ) ) 2 ) 2 - 4 = 1 q + 10 + 79 q + 352 q 2 + 1431 q 3 + 4160 q 4 + 13015 q 5 + \begin{aligned}\displaystyle j_{6A}(\tau)&\displaystyle=T_{6A}(\tau)+10\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)\,\eta(3\tau)}{\eta(2\tau)\,\eta(6% \tau)}\big)^{3}+2^{3}\big(\tfrac{\eta(2\tau)\,\eta(6\tau)}{\eta(\tau)\,\eta(3% \tau)}\big)^{3}\Big)^{2}\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)\,\eta(2\tau)}{\eta(3\tau)\,\eta(6% \tau)}\big)^{2}+3^{2}\big(\tfrac{\eta(3\tau)\,\eta(6\tau)}{\eta(\tau)\,\eta(2% \tau)}\big)^{2}\Big)^{2}-4\\ &\displaystyle=\frac{1}{q}+10+79q+352q^{2}+1431q^{3}+4160q^{4}+13015q^{5}+% \dots\end{aligned}

Fischer_group_Fi23.html

  1. × 10 1 8 \times 10^{1}8
  2. T 3 A ( τ ) T_{3A}(\tau)
  3. j 3 A ( τ ) = T 3 A ( τ ) + 42 = ( ( η ( τ ) η ( 3 τ ) ) 6 + 3 3 ( η ( 2 τ ) η ( τ ) ) 6 ) 2 = 1 q + 42 + 783 q + 8672 q 2 + 65367 q 3 + 371520 q 4 + \begin{aligned}\displaystyle j_{3A}(\tau)&\displaystyle=T_{3A}(\tau)+42\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)}{\eta(3\tau)}\big)^{6}+3^{3}\big(% \tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{6}\Big)^{2}\\ &\displaystyle=\frac{1}{q}+42+783q+8672q^{2}+65367q^{3}+371520q^{4}+\dots\end{aligned}

Fischer_group_Fi24.html

  1. × 10 2 4 \times 10^{2}4
  2. T 3 A ( τ ) T_{3A}(\tau)
  3. j 3 A ( τ ) = T 3 A ( τ ) + 42 = ( ( η ( τ ) η ( 3 τ ) ) 6 + 3 3 ( η ( 2 τ ) η ( τ ) ) 6 ) 2 = 1 q + 42 + 783 q + 8672 q 2 + 65367 q 3 + 371520 q 4 + 1741655 q 5 + \begin{aligned}\displaystyle j_{3A}(\tau)&\displaystyle=T_{3A}(\tau)+42\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)}{\eta(3\tau)}\big)^{6}+3^{3}\big(% \tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{6}\Big)^{2}\\ &\displaystyle=\frac{1}{q}+42+783q+8672q^{2}+65367q^{3}+371520q^{4}+1741655q^{% 5}+\dots\end{aligned}

Flat_convergence.html

  1. T \partial{T}
  2. T = A + B T=A+\partial B
  3. T i T_{i}
  4. M ( T i ) + M ( T i ) M(T_{i})+M(\partial T_{i})

Flat_pseudospectral_method.html

  1. D D
  2. y y
  3. D D
  4. y ˙ \displaystyle\dot{y}
  5. Y Y
  6. β \beta
  7. a a
  8. b b
  9. x \displaystyle x
  10. x = a ( Y , D Y , , D β Y ) x=a(Y,DY,\ldots,D^{\beta}Y)
  11. u = b ( Y , D Y , , D β + 1 Y ) u=b(Y,DY,\ldots,D^{\beta+1}Y)

Flavin_reductase_(NADH).html

  1. \rightleftharpoons

Flavonoid_3',5'-hydroxylase.html

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

Flavonoid_4'-O-methyltransferase.html

  1. \rightleftharpoons

Flexible_algebra.html

  1. \circ
  2. a ( b a ) = ( a b ) a . a\circ\left(b\circ a\right)=\left(a\circ b\right)\circ a.
  3. a a
  4. b b

Flory–Schulz_distribution.html

  1. - 1 log ( 1 - a ) -\frac{1}{\log(1-a)}
  2. 2 - 2 a a 2 \frac{2-2a}{a^{2}}
  3. 2 - a 2 - 2 a \frac{2-a}{\sqrt{2-2a}}
  4. ( a - 6 ) a + 6 2 - 2 a \frac{(a-6)a+6}{2-2a}
  5. a 2 e t ( ( a - 1 ) e t + 1 ) 2 \frac{a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}
  6. a 2 e i t ( 1 + ( a - 1 ) e i t ) 2 \frac{a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}
  7. a 2 z ( ( a - 1 ) z + 1 ) 2 \frac{a^{2}z}{((a-1)z+1)^{2}}
  8. f a ( k ) = a 2 k ( 1 - a ) k - 1 f_{a}(k)=a^{2}k(1-a)^{k-1}\,
  9. { ( a - 1 ) ( k + 1 ) f ( k ) + k f ( k + 1 ) = 0 , f ( 0 ) = 0 , f ( 1 ) = a 2 } \left\{(a-1)(k+1)f(k)+kf(k+1)=0,f(0)=0,f(1)=a^{2}\right\}

Flow_distribution_in_manifolds.html

  1. Δ P + ρ f 2 D W 2 Δ X + ρ 2 Δ W 2 = 0 \Delta\,P+\tfrac{\rho f}{2\,D}\,W^{2}\Delta\,X+\tfrac{\rho}{2}\Delta\,W^{2}\,=\,0
  2. W W\,
  3. P P\,
  4. ρ \rho
  5. D D\,
  6. f f\,
  7. X X\,
  8. R = 128 μ L π d 4 \,R\,=\tfrac{\,128\mu\,L}{\pi\,d^{4}}
  9. 1 ρ d P d X + f 2 D W 2 + ( 2 - β 2 ) d W 2 d X = 0 \frac{1}{\rho}\frac{\,d\,P}{\,d\,X}+\tfrac{\,f}{2\,D}\,W^{2}+\left(\frac{2-% \beta}{2}\right)\frac{\,d\,W^{2}}{\,d\,X}\,=\,0
  10. Δ P + ρ f 2 D W 2 Δ X + ρ ( 2 - β 2 ) Δ W 2 = 0 \Delta\,P+\tfrac{\rho f}{2\,D}\,W^{2}\Delta\,X+\rho\left(\frac{2-\beta}{2}% \right)\Delta\,W^{2}=0
  11. f = 0.3164 / R e 0.25 = f 0 W - 0.25 \,f\,=\,0.3164\,/\,Re^{0.25}\,=\,f_{0}\,W^{-0.25}
  12. 1 ρ d P d X - f 2 D W 2 + ( 2 - β 2 ) d W 2 d X = 0 \frac{1}{\rho}\frac{\,d\,P}{\,d\,X}-\tfrac{\,f}{2\,D}\,W^{2}+\left(\frac{2-% \beta}{2}\right)\frac{\,d\,W^{2}}{\,d\,X}\,=\,0
  13. Δ P - ρ f 2 D W 2 Δ X + ρ ( 2 - β 2 ) Δ W 2 = 0 \Delta\,P-\tfrac{\rho f}{2\,D}\,W^{2}\Delta\,X+\rho\left(\frac{2-\beta}{2}% \right)\Delta\,W^{2}=0
  14. 1 ρ d ( P - P e ) d X + 1 2 [ f D + f e D e ( F F e ) 2 ] W 2 + [ ( 2 - β ) - ( 2 - β e ) ( F F e ) 2 ] W d W d X = 0 \frac{1}{\rho}\frac{\,d\left(\,P-P_{e}\right)}{\,d\,X}+\tfrac{1}{2}\,\left[% \frac{\,f}{\,D}+\frac{f_{e}}{D_{e}}\left(\frac{F}{F_{e}}\right)^{2}\right]\,W^% {2}+\left[\left(\,2-\beta\right)\,-\left(\,2-\beta_{e}\right)\left(\frac{\,F}{% F_{e}}\right)^{2}\right]\,W\tfrac{\,dW}{\,dX}\,=\,0
  15. Δ ( P - P e ) + ρ 2 [ f D + f e D e ( F F e ) 2 ] W 2 Δ X + ρ 2 [ ( 2 - β ) - ( 2 - β e ) ( F F e ) 2 ] Δ W 2 = 0 \Delta\left(\,P-P_{e}\right)+\frac{\rho}{2}\,\left[\frac{\,f}{\,D}+\frac{f_{e}% }{D_{e}}\left(\frac{F}{F_{e}}\right)^{2}\right]\,W^{2}\Delta\,X+\frac{\rho}{2}% \left[\left(\,2-\beta\right)\,-\left(\,2-\beta_{e}\right)\left(\frac{\,F}{F_{e% }}\right)^{2}\right]\Delta\,W^{2}\,=\,0
  16. 1 ρ d ( P - P e ) d X + 1 2 [ f D - ( 1 - W 0 W ) f e D e ( F F e ) 2 ] W 2 + [ ( 2 - β ) - ( 2 - β e ) ( 1 - W 0 W ) ( F F e ) 2 ] W d W d X = f e 2 D e W 0 2 ( F F e ) 2 \frac{1}{\rho}\frac{\,d\left(\,P-P_{e}\right)}{\,d\,X}+\tfrac{1}{2}\,\left[% \frac{\,f}{\,D}-\left(\,1-\frac{W_{0}}{W}\right)\frac{f_{e}}{D_{e}}\left(\frac% {F}{F_{e}}\right)^{2}\right]\,W^{2}+\left[\left(\,2-\beta\right)\,-\left(\,2-% \beta_{e}\right)\left(\,1-\frac{W_{0}}{W}\right)\left(\frac{\,F}{F_{e}}\right)% ^{2}\right]\,W\tfrac{\,dW}{\,dX}\,=\frac{f_{e}}{\,2D_{e}}\,W_{0}^{2}\left(% \frac{F}{F_{e}}\right)^{2}
  17. Δ ( P - P e ) + ρ 2 [ f D - ( 1 - W 0 W ) f e D e ( F F e ) 2 ] W 2 Δ X + ρ 2 [ ( 2 - β ) - ( 2 - β e ) ( 1 - W 0 W ) ( F F e ) 2 ] Δ W 2 = ρ f e 2 D e W 0 2 ( F F e ) 2 Δ X \Delta\left(\,P-P_{e}\right)+\frac{\rho}{2}\,\left[\frac{\,f}{\,D}-\left(\,1-% \frac{W_{0}}{W}\right)\frac{f_{e}}{D_{e}}\left(\frac{F}{F_{e}}\right)^{2}% \right]\,W^{2}\Delta\,X+\frac{\rho}{2}\left[\left(\,2-\beta\right)\,-\left(\,2% -\beta_{e}\right)\left(\,1-\frac{W_{0}}{W}\right)\left(\frac{\,F}{F_{e}}\right% )^{2}\right]\Delta\,W^{2}\,=\frac{\rho\,f_{e}}{\,2D_{e}}\,W_{0}^{2}\left(\frac% {F}{F_{e}}\right)^{2}\Delta\,X

Flow_through_cascades.html

  1. c 2 c_{2}
  2. d m ˙ = ρ c x d y d\dot{m}=\rho c_{x}dy
  3. m ˙ = ρ c x d y \dot{m}=\int\rho c_{x}dy
  4. M y = c y d m ˙ M_{y}=\int c_{y}d\dot{m}
  5. M x = ρ c x 2 d y M_{x}=\int\rho c_{x2}dy
  6. tan α = M y M x \tan\alpha=\frac{M_{y}}{M_{x}}
  7. Δ p 0 = p 01 - p 02 d m ˙ \Delta p_{0}=\frac{p_{01}-p_{02}}{d\dot{m}}
  8. C x m = 0.5 ( c x 1 + c x 2 ) C_{xm}=0.5(c_{x1}+c_{x2})
  9. C y m = 0.5 ( c y 2 - c y 1 ) C_{ym}=0.5(c_{y2}-c_{y1})
  10. tan α = C y m C x m \tan\alpha=\frac{C_{ym}}{C_{xm}}
  11. tan α m = 0.5 ( tan α 2 - tan α 1 ) \tan\alpha_{m}=0.5(\tan\alpha_{2}-\tan\alpha_{1})
  12. c m c_{m}
  13. F y = 0.5 ( ρ l ( c x m ) 2 ) 2 ( s l ) ( tan α 1 + tan α 2 ) F_{y}=0.5(\rho l(c_{xm})^{2})2(\frac{s}{l})(\tan\alpha_{1}+\tan\alpha_{2})
  14. F x = ( 0.5 ρ l ( c 2 ) 2 ) ( s l ) cos 2 α 2 ( tan 2 α 2 - tan 2 α 1 ) + s Δ p 0 F_{x}=(0.5\rho l(c_{2})^{2})(\frac{s}{l})\cos^{2}\alpha_{2}(\tan^{2}\alpha_{2}% -\tan^{2}\alpha_{1})+s\Delta p_{0}
  15. L = ( 0.5 ρ l ( c m ) 2 ) 2 ( s l ) cos α m ( tan α 2 + tan α 1 ) + s Δ p 0 sin α m L=(0.5\rho l(c_{m})^{2})2(\frac{s}{l})\cos\alpha_{m}(\tan\alpha_{2}+\tan\alpha% _{1})+s\Delta p_{0}\sin\alpha_{m}
  16. D = s Δ p 0 cos α m D=s\Delta p_{0}\cos\alpha_{m}

Fluorescence_intensity_decay_shape_microscopy.html

  1. F ( t ) = A e - t / τ F(t)=Ae^{-t/\tau}
  2. A A
  3. τ \tau
  4. F ( t ) = ( A 1 e - t / τ 1 ) F(t)=\sum(A_{1}e^{-t/\tau_{1}})
  5. χ 2 \chi^{2}

Fluoroacetyl-CoA_thioesterase.html

  1. \rightleftharpoons

FMN_reductase_(NAD(P)H).html

  1. \rightleftharpoons

FMN_reductase_(NADH).html

  1. \rightleftharpoons

FMN_reductase_(NADPH).html

  1. \rightleftharpoons

Folded_Reed–Solomon_code.html

  1. m m
  2. 1 - R 1-\sqrt{R}
  3. 1 - R 1-\sqrt{R}
  4. R R
  5. 1 - R 1-\sqrt{R}
  6. R < 1 / 16 R<1/16
  7. R 0 R\to 0
  8. 1 - O ( R log ( 1 / R ) ) 1-O(R\log(1/R))
  9. ( 1 - R - ϵ ) (1-R-\epsilon)
  10. ϵ > 0 \epsilon>0
  11. f ( X ) [ f ( 1 ) f ( γ ) f ( γ m - 1 ) ] , [ f ( γ m ) f ( γ m + 1 ) f ( γ 2 m - 1 ) ] , , [ f ( γ n - m ) f ( γ n - m + 1 ) f ( γ n - 1 ) ] f(X)\mapsto\begin{bmatrix}f(1)\\ f(\gamma)\\ \vdots\\ f(\gamma^{m-1})\end{bmatrix},\begin{bmatrix}f(\gamma^{m})\\ f(\gamma^{m+1})\\ \vdots\\ f(\gamma^{2m-1})\end{bmatrix},\ldots,\begin{bmatrix}f(\gamma^{n-m})\\ f(\gamma^{n-m+1})\\ \vdots\\ f(\gamma^{n-1})\end{bmatrix}
  12. [ n = q - 1 , k ] q [n=q-1,k]_{q}
  13. n n
  14. k k
  15. m 1 m\geq 1
  16. m m
  17. n n
  18. f f ( γ 0 ) , f ( γ 1 ) , f ( γ 2 ) , , f ( γ n - 1 ) f\mapsto\langle f\left(\gamma^{0}\right),f(\gamma^{1}),f(\gamma^{2}),\ldots,f(% \gamma^{n-1})\rangle
  19. γ 𝔽 q \gamma\in\mathbb{F}_{q}
  20. 𝔽 q = { 0 , 1 , γ , γ 2 , , γ n - 1 } \mathbb{F}_{q}=\left\{0,1,\gamma,\gamma^{2},\ldots,\gamma^{n-1}\right\}
  21. m m
  22. C C
  23. F R S 𝔽 , γ , m , k FRS_{\mathbb{F},\gamma,m,k}
  24. N = n / m N=n/m
  25. 𝔽 m \mathbb{F}^{m}
  26. F R S 𝔽 , γ , m , k FRS_{\mathbb{F},\gamma,m,k}
  27. [ q - 1 , k ] [q-1,k]
  28. m m
  29. m = 3 m=3
  30. m m
  31. f ( X ) f(X)
  32. f f
  33. x 0 , x 1 , x 2 , , x n - 1 x_{0},x_{1},x_{2},\ldots,x_{n-1}
  34. x i = γ i x_{i}=\gamma^{i}
  35. n / 3 n/3
  36. 𝔽 q 3 \mathbb{F}_{q}^{3}
  37. R R
  38. [ n , k , d ] q [n,k,d]_{q}
  39. n n
  40. k k
  41. d d
  42. m m
  43. [ n m , k m , d m ] q m \left[\dfrac{n}{m},\dfrac{k}{m},\dfrac{d}{m}\right]_{q^{m}}
  44. R = n k R=\dfrac{n}{k}
  45. δ \delta
  46. R R\leq
  47. 1 - δ + o ( 1 ) 1-\delta+o(1)
  48. R R
  49. R R
  50. δ 1 - R \delta\geq 1-R
  51. m = 3 m=3
  52. ρ \rho
  53. ρ \rho
  54. 1 / 3 1/3
  55. 𝔽 q 3 \mathbb{F}_{q}^{3}
  56. ρ \rho
  57. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  58. f f
  59. k k
  60. s s
  61. \geq
  62. f 0 = f , f 1 , , f s - 1 f_{0}=f,f_{1},\ldots,f_{s-1}
  63. f i ( X ) = f i - 1 ( X ) d mod E ( X ) f_{i}(X)=f_{i-1}(X)^{d}\mod E(X)
  64. E ( X ) E(X)
  65. E ( X ) = X q - γ E(X)=X^{q}-\gamma
  66. d d
  67. f f
  68. k k
  69. f ( γ X ) = f ( X ) d mod E ( X ) f(\gamma X)=f(X)^{d}\mod E(X)
  70. f ( γ X ) f(\gamma X)
  71. f ( X ) f(X)
  72. γ \gamma
  73. F q F_{q}
  74. s = m s=m
  75. { 1 , γ , γ 2 m , , γ ( n / m - 1 ) m } \{1,\gamma,\gamma^{2m},\ldots,\gamma^{(n/m-1)m}\}
  76. { 1 , γ , , γ m - 2 , γ m , γ m + 1 , , γ 2 m - 2 , , γ n - m , γ n - m + 1 , , γ n - 2 } \{1,\gamma,\ldots,\gamma^{m-2},\gamma^{m},\gamma^{m+1},\ldots,\gamma^{2m-2},% \ldots,\gamma^{n-m},\gamma^{n-m+1},\ldots,\gamma^{n-2}\}
  77. f f
  78. 0 i n / m - 1 0\leq i\leq n/m-1
  79. 0 < j < m - 1 , f ( γ m i + j ) 0<j<m-1,f(\gamma^{mi+j})
  80. f ( γ m i + j ) f(\gamma^{mi+j})
  81. f 1 ( γ - 1 γ m i + j ) f_{1}(\gamma^{-1}\gamma^{mi+j})
  82. 2 ( m - 1 ) / m 2(m-1)/m
  83. R R
  84. 1 - R s / [ s + 1 ] 1-R^{s/[s+1]}
  85. s 1 s\geq 1
  86. 1 - R - ϵ 1-R-\epsilon
  87. Q ( X , Y 1 , Y 2 , , Y s ) = A 0 ( X ) + A 1 ( X ) Y 1 + A 2 ( X ) Y 2 + + A s ( X ) Y s Q(X,Y_{1},Y_{2},\ldots,Y_{s})=A_{0}(X)+A_{1}(X)Y_{1}+A_{2}(X)Y_{2}+\cdots+A_{s% }(X)Y_{s}
  88. f 𝔽 q [ X ] f\in{\mathbb{F}_{q}[X]}
  89. k - 1 k-1
  90. q s q^{s}
  91. n Ω ( 1 / ε 2 ) n^{\Omega(1/\varepsilon^{2})}
  92. 1 - R - ε 1-R-\varepsilon
  93. n O ( 1 / ε 2 ) {n^{O(1/\varepsilon^{2})}}
  94. f ( x ) f(x)
  95. y y
  96. m m
  97. ( [ y 0 y 1 y 2 y m - 1 ] , [ y m y m + 1 y m + 2 y 2 m - 1 ] , , [ y n - m y n - m + 1 y n - m + 2 y n - 1 ] ) \left(\begin{bmatrix}y_{0}\\ y_{1}\\ y_{2}\\ \cdots\\ y_{m-1}\end{bmatrix},\begin{bmatrix}y_{m}\\ y_{m+1}\\ y_{m+2}\\ \cdots\\ y_{2m-1}\end{bmatrix},\cdots,\begin{bmatrix}y_{n-m}\\ y_{n-m+1}\\ y_{n-m+2}\\ \cdots\\ y_{n-1}\end{bmatrix}\right)
  98. Q ( X , Y 1 , Y 2 , , Y s ) = A 0 ( X ) + A 1 ( X ) Y 1 + A 2 ( X ) Y 2 + + A s ( X ) Y s Q(X,Y_{1},Y_{2},\ldots,Y_{s})=A_{0}(X)+A_{1}(X)Y_{1}+A_{2}(X)Y_{2}+\cdots+A_{s% }(X)Y_{s}
  99. deg ( A i ) D for i = 1 , 2 , , s and deg ( A 0 ) D + k - 1 \deg(A_{i})\leq D\,\text{ for }i=1,2,\ldots,s\,\text{ and }\deg(A_{0})\leq D+k-1
  100. D D
  101. D = N ( m - s + 1 ) - k + 1 s + 1 D=\lfloor{{N(m-s+1)-k+1}\over{s+1}}\rfloor
  102. Q ( γ i m + j , y i m + j , y i m + j + 1 , , y i m + j + s - 1 ) = 0 Q(\gamma^{im+j},y_{im+j},y_{im+j+1},\cdots,y_{im+j+s-1})=0
  103. i = 0 , 1 , , n / m - 1 and j = 0 , 1 , , m - s i=0,1,\ldots,n/m-1\,\text{ and }j=0,1,\ldots,m-s
  104. Q ( X , Y 1 , Y 2 , , Y s ) Q(X,Y_{1},Y_{2},\ldots,Y_{s})
  105. ( D + 1 ) s + D + k = ( D + 1 ) ( s + 1 ) + k - 1 > N ( m - s + 1 ) (D+1)s+D+k=(D+1)(s+1)+k-1>N(m-s+1)
  106. Q ( X , Y 1 , Y 2 , , Y s ) Q(X,Y_{1},Y_{2},\ldots,Y_{s})
  107. Q 𝔽 q [ X , Y 1 , , Y s ] Q\in\mathbb{F}_{q}[X,Y_{1},\cdots,Y_{s}]
  108. Q ( X , Y 1 , Y 2 , , Y s ) Q(X,Y_{1},Y_{2},\ldots,Y_{s})
  109. 𝔽 q \mathbb{F}_{q}
  110. N m Nm
  111. O ( N m l o g 2 ( N m ) l o g l o g ( N m ) ) O(Nmlog^{2}(Nm)loglog(Nm))
  112. 𝔽 q \mathbb{F}_{q}
  113. Q ( X , Y 1 , Y 2 , , Y s ) Q(X,Y_{1},Y_{2},\ldots,Y_{s})
  114. f ( X ) f(X)
  115. f ( X ) 𝔽 [ X ] f(X)\in\mathbb{F}[X]
  116. k - 1 k-1
  117. y y
  118. t t
  119. t > D + k - 1 m - s + 1 t>{{D+k-1}\over{m-s+1}}
  120. Q ( X , f ( X ) , f ( γ X ) , , f ( γ s - 1 X ) ) = 0 Q(X,f(X),f(\gamma X),\cdots,f(\gamma_{s-1}X))=0
  121. m m
  122. y y
  123. Q ( X , Y 1 , Y 2 , , Y s ) Q(X,Y_{1},Y_{2},\ldots,Y_{s})
  124. f ( x ) f(x)
  125. D D
  126. t ( m - s + 1 ) > N ( m - s + 1 ) + s ( k - 1 ) s + 1 t(m-s+1)>\dfrac{N(m-s+1)+s(k-1)}{s+1}
  127. t N s + 1 + s s + 1 k m - s + 1 = N ( 1 s + 1 + s s + 1 m R m - s + 1 ) t\geq\frac{N}{s+1}+\frac{s}{s+1}\cdot\frac{k}{m-s+1}=N\left(\frac{1}{s+1}+% \frac{s}{s+1}\cdot\frac{mR}{m-s+1}\right)
  128. 1 s + 1 + s s + 1 m R m - s + 1 \dfrac{1}{s+1}+\dfrac{s}{s+1}\cdot\dfrac{mR}{m-s+1}
  129. f 𝔽 q [ X ] f\in{\mathbb{F}_{q}[X]}
  130. k - 1 k-1
  131. A 0 ( X ) + A 1 ( X ) f ( X ) + A 2 ( X ) f ( γ X ) + + A s ( X ) f ( γ s - 1 X ) = 0 A_{0}(X)+A_{1}(X)f(X)+A_{2}(X)f(\gamma X)+\cdots+A_{s}(X)f(\gamma^{s-1}X)=0
  132. 𝔽 q \mathbb{F}_{q}
  133. f 0 , f 1 , , f k - 1 f_{0},f_{1},\cdots,f_{k-1}
  134. f ( X ) = f 0 + f 1 X + + f k - 1 X k - 1 f(X)=f_{0}+f_{1}X+\cdots+f_{k-1}X^{k-1}
  135. 𝔽 q k \mathbb{F}^{k}_{q}
  136. γ \gamma
  137. k k
  138. γ \gamma
  139. s - 1 s-1
  140. F R S q ( m ) [ n , k ] FRS^{(m)}_{q}[n,k]
  141. N = n / m N=n/m
  142. R = k / n R=k/n
  143. s s
  144. 1 s m 1\leq s\leq m
  145. y ( 𝔽 q m ) N y\in(\mathbb{F}_{q}^{m})^{N}
  146. O ( ( N m l o g q ) 2 ) O((Nmlogq)^{2})
  147. s - 1 s-1
  148. f 𝔽 q [ X ] f\in\mathbb{F}_{q}[X]
  149. k k
  150. y y
  151. s s + 1 ( 1 - m R m - s + 1 ) \frac{s}{s+1}\left(1-\frac{mR}{m-s+1}\right)
  152. N N
  153. s = m = 1 s=m=1
  154. ( 1 - R ) / 2 (1-R)/2
  155. n O ( 1 / ε ) n^{O(1/\varepsilon)}
  156. 1 - R - ε 1-R-\varepsilon
  157. n Ω ( 1 / ε ) n^{\Omega(1/\varepsilon)}
  158. q s q^{s}
  159. k - 1 k-1
  160. k - 1 k-1
  161. ( f 0 , f 1 , , f k - 1 ) (f_{0},f_{1},\ldots,f_{k-1})
  162. ν 𝔽 q k \nu\subseteq\mathbb{F}_{q}^{k}
  163. ν \nu
  164. | ν | q ( 1 - ε ) k |\nu|\geq q^{(1-\varepsilon)k}
  165. ( 1 - ε ) (1-\varepsilon)
  166. ν \nu
  167. s s
  168. S 𝔽 q k {S\subset\mathbb{F}_{q}^{k}}
  169. | S ν | L {|S\cap\nu|\leq L}
  170. n Ω ( 1 / ε ) n^{\Omega(1/\varepsilon)}
  171. O ( 1 / ε 2 ) O(1/\varepsilon^{2})
  172. q s q^{s}
  173. s s
  174. n Ω ( 1 / ε 2 ) n^{\Omega(1/\varepsilon^{2})}
  175. n O ( 1 / ε 2 ) n^{O(1/\varepsilon^{2})}

Folding_(DSP_implementation).html

  1. D F \scriptstyle D_{F}
  2. U , V \scriptstyle U,V
  3. w ( e ) \scriptstyle w(e)
  4. U , V \scriptstyle U,V
  5. u \scriptstyle u
  6. U \scriptstyle U
  7. v \scriptstyle v
  8. V \scriptstyle V
  9. P U \scriptstyle P_{U}
  10. U \scriptstyle U
  11. { S i | j } \scriptstyle\{S_{i}|j\}
  12. S i \scriptstyle S_{i}
  13. j \scriptstyle j
  14. S 1 , S 2 \scriptstyle S_{1},S_{2}
  15. D F ( 1 2 ) = 4 ( 1 ) - 1 + 1 - 3 = 1 \scriptstyle D_{F}(1\rightarrow 2)=4(1)-1+1-3=1
  16. D F ( 1 5 ) = 4 ( 1 ) - 1 + 0 - 3 = 0 \scriptstyle D_{F}(1\rightarrow 5)=4(1)-1+0-3=0
  17. D F ( 1 6 ) = 4 ( 1 ) - 1 + 2 - 3 = 2 \scriptstyle D_{F}(1\rightarrow 6)=4(1)-1+2-3=2
  18. D F ( 1 7 ) = 4 ( 1 ) - 1 + 3 - 3 = 3 \scriptstyle D_{F}(1\rightarrow 7)=4(1)-1+3-3=3
  19. D F ( 1 8 ) = 4 ( 2 ) - 1 + 1 - 3 = 5 \scriptstyle D_{F}(1\rightarrow 8)=4(2)-1+1-3=5
  20. D F ( 3 1 ) = 4 ( 0 ) - 1 + 3 - 2 = 0 \scriptstyle D_{F}(3\rightarrow 1)=4(0)-1+3-2=0
  21. D F ( 4 2 ) = 4 ( 0 ) - 1 + 1 - 0 = 0 \scriptstyle D_{F}(4\rightarrow 2)=4(0)-1+1-0=0
  22. D F ( 5 3 ) = 4 ( 0 ) - 2 + 2 - 0 = 0 \scriptstyle D_{F}(5\rightarrow 3)=4(0)-2+2-0=0
  23. D F ( 6 4 ) = 4 ( 1 ) - 2 + 0 - 2 = 0 \scriptstyle D_{F}(6\rightarrow 4)=4(1)-2+0-2=0
  24. D F ( 7 3 ) = 4 ( 1 ) - 2 + 2 - 3 = 1 \scriptstyle D_{F}(7\rightarrow 3)=4(1)-2+2-3=1
  25. D F ( 8 4 ) = 4 ( 1 ) - 2 + 0 - 1 = 1 \scriptstyle D_{F}(8\rightarrow 4)=4(1)-2+0-1=1
  26. { i , j } \scriptstyle\{i,j\}
  27. T i n p u t \scriptstyle T_{input}
  28. T o u t p u t \scriptstyle T_{output}
  29. T i n p u t = u + P U \scriptstyle T_{input}=u+P_{U}
  30. U \scriptstyle U
  31. P U \scriptstyle P_{U}
  32. u \scriptstyle u
  33. T o u t p u t \scriptstyle T_{output}
  34. U \scriptstyle U
  35. u + P U + m a x V { D F ( U V ) } \scriptstyle u+P_{U}+max_{V}\{D_{F}(U\rightarrow V)\}
  36. T i n p u t \scriptstyle T_{input}
  37. T o u t p u t \scriptstyle T_{output}

FOMP.html

  1. E A = K 1 sin 2 θ + K 2 sin 4 θ + K 3 sin 6 θ \displaystyle E_{A}=K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta+K_{3}\sin^{6}\theta
  2. E H = - H M s cos ( θ - γ ) \displaystyle E_{H}=-HM_{s}\cos(\theta-\gamma)
  3. K 1 , K 2 , K 3 \,K_{1},K_{2},K_{3}
  4. H \,H
  5. M s \,M_{s}
  6. θ \,\theta
  7. γ \,\gamma
  8. E T = E A + E H \displaystyle E_{T}=E_{A}+E_{H}
  9. E A θ = 0 \frac{\partial E_{A}}{\partial\theta}=0
  10. 2 E A θ 2 > 0. \frac{\partial^{2}E_{A}}{\partial\theta^{2}}>0.
  11. sin θ c = [ - K 2 ± ( K 2 2 - 3 K 1 K 3 ) 1 / 2 ] / 3 K 3 1 / 2 \displaystyle\sin\theta_{c}={[-K_{2}\pm(K_{2}^{2}-3K1K_{3})^{1/2}]/3K_{3}}^{1/2}
  12. K R K\rightleftharpoons R
  13. K R \scriptstyle K\rightleftharpoons R
  14. \Leftarrow
  15. \Rightarrow
  16. K 1 K_{1}
  17. R 1 = - K 1 - 2 K 2 - 3 K 3 R_{1}=-K_{1}-2K_{2}-3K_{3}
  18. \rightleftharpoons
  19. R 1 R_{1}
  20. K 1 = - R 1 - 2 R 2 - 3 R 3 K_{1}=-R_{1}-2R_{2}-3R_{3}
  21. K 2 K_{2}
  22. R 2 = K 2 + 3 K 3 R_{2}=K_{2}+3K_{3}
  23. \rightleftharpoons
  24. R 2 R_{2}
  25. K 2 = R 2 + 3 R 3 K_{2}=R_{2}+3R_{3}
  26. K 3 K_{3}
  27. R 3 = - K 3 R_{3}=-K_{3}
  28. \rightleftharpoons
  29. R 3 R_{3}
  30. K 3 = - R 3 K_{3}=-R_{3}
  31. γ γ
  32. γ γ
  33. γ γ
  34. E t \displaystyle E_{t}
  35. J \,J
  36. K 1 A , K 2 A \,K_{1A},\ K_{2A}
  37. K 1 B , K 2 B \,K_{1B},\ K_{2B}
  38. H \,H
  39. M A , M B \,M_{A},\ M_{B}
  40. θ A , θ B \,\theta_{A},\ \theta_{B}
  41. < v a r > A <var>A
  42. E A = K 1 * sin 2 θ + K 2 * sin 4 θ + K 3 * sin 6 θ \displaystyle E_{A}=K_{1}^{*}\sin^{2}\theta+K_{2}^{*}\sin^{4}\theta+K_{3}^{*}% \sin^{6}\theta
  43. K 1 * , K 2 * , K 3 * \,K_{1}^{*},K_{2}^{*},K_{3}^{*}
  44. θ \,\theta
  45. E A = K 1 sin 2 θ + K 2 sin 4 θ + K t sin 3 θ cos θ sin 3 ϕ \displaystyle E_{A}=K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta+K_{t}\sin^{3}% \theta\cos\theta\sin 3\phi
  46. < v a r > θ <var>θ

Food_systems_on_space_exploration_missions.html

  1. - d [ A ] d T = k [ A n ] -{d[A]\over dT}=k[A^{n}]
  2. Q 10 = Shelf Life at temperature T°C Shelf Life at temperature ( T°C + 10°C ) Q_{10}=\frac{\,\text{Shelf Life at temperature T°C}}{\,\text{Shelf Life at % temperature ( T°C + 10°C )}}
  3. t s = t 0 e - a T t_{s}=t_{0}e^{-aT}
  4. ln Q 10 10 \ln{Q_{10}\over 10}

Forecast_verification.html

  1. R p i ln p i q i R\approx\sum p_{i}\ln\frac{p_{i}}{q_{i}}

Formate_dehydrogenase-N.html

  1. \rightleftharpoons

Formate_dehydrogenase_(acceptor).html

  1. \rightleftharpoons

Fourier_coordinates.html

  1. ω = e 2 π i / N \omega=e^{2\pi i/N}
  2. ω N = 1 , ω 1 \omega^{N}=1,\quad\omega\neq 1
  3. 𝐅 N = [ 1 1 1 1 1 ω ω 2 ω ( N - 1 ) 1 ω 2 ω 2 ( N - 1 ) 1 ω ( N - 1 ) ω 2 ( N - 1 ) ω ( N - 1 ) 2 ] \mathbf{F}_{N}=\begin{bmatrix}1&1&1&\ldots&1\\ 1&\omega&\omega^{2}&\ldots&\omega^{(N-1)}\\ 1&\omega^{2}&\vdots&\ldots&\omega^{2(N-1)}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega^{(N-1)}&\omega^{2(N-1)}&\ldots&\omega^{(N-1)^{2}}\\ \end{bmatrix}
  4. 𝐅 5 = [ 1 1 1 1 1 1 ω ω 2 ω 3 ω 4 1 ω 2 ω 4 ω 6 ω 8 1 ω 3 ω 6 ω 9 ω 12 1 ω 4 ω 8 ω 12 ω 16 ] = [ 1 1 1 1 1 1 ω ω 2 ω 3 ω 4 1 ω 2 ω 4 ω ω 3 1 ω 3 ω ω 4 ω 2 1 ω 4 ω 3 ω 2 ω ] . \mathbf{F}_{5}=\begin{bmatrix}1&1&1&1&1\\ 1&\omega&\omega^{2}&\omega^{3}&\omega^{4}\\ 1&\omega^{2}&\omega^{4}&\omega^{6}&\omega^{8}\\ 1&\omega^{3}&\omega^{6}&\omega^{9}&\omega^{12}\\ 1&\omega^{4}&\omega^{8}&\omega^{12}&\omega^{16}\\ \end{bmatrix}=\begin{bmatrix}1&1&1&1&1\\ 1&\omega&\omega^{2}&\omega^{3}&\omega^{4}\\ 1&\omega^{2}&\omega^{4}&\omega&\omega^{3}\\ 1&\omega^{3}&\omega&\omega^{4}&\omega^{2}\\ 1&\omega^{4}&\omega^{3}&\omega^{2}&\omega\\ \end{bmatrix}.
  5. 𝐅 2 = 𝐏 . \mathbf{F}^{2}=\mathbf{P}.
  6. 𝐅 4 = 𝐈 . \mathbf{F}^{4}=\mathbf{I}.
  7. 1 k N - 1 1\leq k\leq N-1
  8. 𝐅 ( ω 𝐤 ) = 𝐏𝐅 ( ω ) . \mathbf{F(\omega^{k})}=\mathbf{P}\mathbf{F(\omega)}.

Fpqc_morphism.html

  1. f : X Y f:X\to Y
  2. V i V_{i}
  3. V i V_{i}
  4. x X x\in X
  5. U U
  6. f ( U ) f(U)
  7. f : U f ( U ) f:U\to f(U)
  8. x X x\in X
  9. f ( U ) f(U)
  10. f : X Y f:X\to Y
  11. V i V_{i}
  12. f : f - 1 ( V i ) V i f:f^{-1}(V_{i})\to V_{i}
  13. f : X Y f:X\to Y

Fractal_derivative.html

  1. v = d x d t = d x β d t α , α , β > 0 v^{\prime}=\frac{dx^{\prime}}{dt^{\prime}}=\frac{dx^{\beta}}{dt^{\alpha}}\,,% \quad\alpha,\beta>0
  2. f ( t ) t α = lim t 1 t f ( t 1 ) - f ( t ) t 1 α - t α , α > 0 \frac{\partial f(t)}{\partial t^{\alpha}}=\lim_{t_{1}\rightarrow t}\frac{f(t_{% 1})-f(t)}{t_{1}^{\alpha}-t^{\alpha}}\,,\quad\alpha>0
  3. β f ( t ) t α = lim t 1 t f β ( t 1 ) - f β ( t ) t 1 α - t α , α > 0 , β > 0 \frac{\partial^{\beta}f(t)}{\partial t^{\alpha}}=\lim_{t_{1}\rightarrow t}% \frac{f^{\beta}(t_{1})-f^{\beta}(t)}{t_{1}^{\alpha}-t^{\alpha}}\,,\quad\alpha>% 0,\beta>0
  4. d u ( x , t ) d t α = D x β ( u ( x , t ) x β ) , - < x < + , ( 1 ) \frac{du(x,t)}{dt^{\alpha}}=D\frac{\partial}{\partial x^{\beta}}\left(\frac{% \partial u(x,t)}{\partial x^{\beta}}\right),-\infty<x<+\infty\,,\quad(1)
  5. u ( x , 0 ) = δ ( x ) . u(x,0)=\delta(x).
  6. u ( x , t ) = 1 2 π t α e - x 2 β 4 t α u(x,t)=\frac{1}{2\sqrt{\pi t^{\alpha}}}e^{-\frac{x^{2\beta}}{4t^{\alpha}}}
  7. x 2 ( t ) t ( 3 α - α β ) / 2 β . \left\langle x^{2}(t)\right\rangle\propto t^{(3\alpha-\alpha\beta)/2\beta}.

Fractional_Poisson_process.html

  1. ν \nu
  2. [ ν ] = sec - μ [\nu]=\sec^{-\mu}
  3. 0 < μ 1 0<\mu\leq 1
  4. P μ ( n , t ) = ( ν t μ ) n n ! k = 0 ( k + n ) ! k ! ( - ν t μ ) k Γ ( μ ( k + n ) + 1 ) , 0 < μ 1 , P_{\mu}(n,t)=\frac{(\nu t^{\mu})^{n}}{n!}\sum\limits_{k=0}^{\infty}\frac{(k+n)% !}{k!}\frac{(-\nu t^{\mu})^{k}}{\Gamma(\mu(k+n)+1)},\qquad 0<\mu\leq 1,
  5. ν \nu
  6. [ ν ] = sec - μ [\nu]=\sec^{-\mu}
  7. Γ ( μ ( k + n ) + 1 ) {\Gamma(\mu(k+n)+1)}
  8. P μ ( n , t ) P_{\mu}(n,t)
  9. [ 0 , t ] [0,t]
  10. P μ ( n , t ) P_{\mu}(n,t)
  11. E μ ( z ) E_{\mu}(z)
  12. P μ ( n , t ) = ( ( - z ) n n ! d n d z n E μ ( z ) ) | z = - ν t μ , P_{\mu}(n,t)=(\frac{(-z)^{n}}{n!}\frac{d^{n}}{dz^{n}}E_{\mu}(z))|_{z=-\nu t^{% \mu}},
  13. P μ ( n = 0 , t ) = E μ ( - ν t μ ) . P_{\mu}(n=0,t)=E_{\mu}(-\nu t^{\mu}).
  14. μ = 1 \mu=1
  15. P μ ( n , t ) P_{\mu}(n,t)
  16. P ( n , t ) = P 1 ( n , t ) P(n,t)=P_{1}(n,t)
  17. P ( n , t ) = ( ν ¯ t ) n n ! exp ( - ν ¯ t ) , P(n,t)=\frac{(\overline{\nu}t)^{n}}{n!}\exp(-\overline{\nu}t),
  18. P ( n = 0 , t ) = exp ( - ν ¯ t ) , P(n=0,t)=\exp(-\overline{\nu}t),
  19. ν ¯ \overline{\nu}
  20. [ ν ¯ ] = sec - 1 [\overline{\nu}]=\sec^{-1}
  21. P μ ( n , t ) P_{\mu}(n,t)
  22. μ \mu
  23. n ¯ μ \overline{n}_{\mu}
  24. n ¯ μ = n = 0 n P μ ( n , t ) = ν t μ Γ ( μ + 1 ) . \overline{n}_{\mu}=\sum\limits_{n=0}^{\infty}nP_{\mu}(n,t)=\frac{\nu t^{\mu}}{% \Gamma(\mu+1)}.
  25. n 2 ¯ μ \overline{n^{2}}_{\mu}
  26. n μ 2 ¯ = n = 0 n 2 P μ ( n , t ) = n ¯ μ + n ¯ μ 2 π Γ ( μ + 1 ) 2 2 μ - 1 Γ ( μ + 1 2 ) . \overline{n_{\mu}^{2}}=\sum\limits_{n=0}^{\infty}n^{2}P_{\mu}(n,t)=\overline{n% }_{\mu}+\overline{n}_{\mu}^{2}\frac{\sqrt{\pi}\Gamma(\mu+1)}{2^{2\mu-1}\Gamma(% \mu+\frac{1}{2})}.
  27. σ μ = n μ 2 ¯ - n ¯ μ 2 = n ¯ μ + n ¯ μ 2 { μ B ( μ , 1 2 ) 2 2 μ - 1 - 1 } , \sigma_{\mu}=\overline{n_{\mu}^{2}}-\overline{n}_{\mu}^{2}=\overline{n}_{\mu}+% \overline{n}_{\mu}^{2}\left\{\frac{\mu B(\mu,\frac{1}{2})}{2^{2\mu-1}}-1\right\},
  28. B ( μ , 1 2 ) B(\mu,\frac{1}{2})
  29. C μ ( s , t ) = n = 0 e i s n P μ ( n , t ) = E μ ( ν t μ ( e i s - 1 ) ) . C_{\mu}(s,t)=\sum\limits_{n=0}^{\infty}e^{isn}P_{\mu}(n,t)=E_{\mu}(\nu t^{\mu}% (e^{is}-1)).
  30. C μ ( s , t ) = m = 0 1 Γ ( m μ + 1 ) ( ν t μ ( e i s - 1 ) ) m , C_{\mu}(s,t)=\sum\limits_{m=0}^{\infty}\frac{1}{\Gamma(m\mu+1)}\left(\nu t^{% \mu}(e^{is}-1)\right)^{m},
  31. k < m t p l > th k^{<}mtpl>{{\rm th}}
  32. n μ k ¯ = ( 1 / i k ) k C μ ( s , t ) s k | s = 0 . \overline{n_{\mu}^{k}}=(1/i^{k})\frac{\partial^{k}C_{\mu}(s,t)}{\partial s^{k}% }|_{s=0}.
  33. G μ ( s , t ) G_{\mu}(s,t)
  34. G μ ( s , t ) = n = 0 s n P μ ( n , t ) . G_{\mu}(s,t)=\sum\limits_{n=0}^{\infty}s^{n}P_{\mu}(n,t).
  35. G μ ( s , t ) = E μ ( ν t μ ( s - 1 ) ) , G_{\mu}(s,t)=E_{\mu}(\nu t^{\mu}(s-1)),
  36. E μ ( z ) E_{\mu}(z)
  37. E μ ( z ) = m = 0 z m Γ ( μ m + 1 ) . E_{\mu}(z)=\sum\limits_{m=0}^{\infty}\frac{z^{m}}{\Gamma(\mu m+1)}.
  38. H μ ( s , t ) H_{\mu}(s,t)
  39. H μ ( s , t ) = n = 0 e - s n P μ ( n , t ) . H_{\mu}(s,t)=\sum\limits_{n=0}^{\infty}e^{-sn}P_{\mu}(n,t).
  40. k < m t p l > th k^{<}mtpl>{{\rm th}}
  41. n μ k ¯ = ( - 1 ) k k H μ ( s , t ) s k | s = 0 . \overline{n_{\mu}^{k}}=(-1)^{k}\frac{\partial^{k}H_{\mu}(s,t)}{\partial s^{k}}% |_{s=0}.
  42. H μ ( s , t ) H_{\mu}(s,t)
  43. H μ ( s , t ) = E μ ( ν t μ ( e - s - 1 ) ) , H_{\mu}(s,t)=E_{\mu}(\nu t^{\mu}(e^{-s}-1)),
  44. H μ ( s , t ) = m = 0 1 Γ ( m μ + 1 ) ( ν t μ ( e - s - 1 ) ) m , H_{\mu}(s,t)=\sum\limits_{m=0}^{\infty}\frac{1}{\Gamma(m\mu+1)}\left(\nu t^{% \mu}(e^{-s}-1)\right)^{m},
  45. ψ μ ( τ ) \psi_{\mu}(\tau)
  46. ψ μ ( τ ) = - d d τ P μ ( τ ) , \psi_{\mu}(\tau)=-\frac{d}{d\tau}P_{\mu}(\tau),
  47. P μ ( τ ) P_{\mu}(\tau)
  48. τ \tau
  49. P μ ( τ ) = 1 - n = 1 P μ ( n , τ ) = E μ ( - ν τ μ ) , P_{\mu}(\tau)=1-\sum\limits_{n=1}^{\infty}P_{\mu}(n,\tau)=E_{\mu}(-\nu\tau^{% \mu}),
  50. P μ ( n , τ ) P_{\mu}(n,\tau)
  51. ψ μ ( τ ) \psi_{\mu}(\tau)
  52. ψ μ ( τ ) = ν τ μ - 1 E μ , μ ( - ν τ μ ) , t 0 , 0 < μ 1 , \psi_{\mu}(\tau)=\nu\tau^{\mu-1}E_{\mu,\mu}(-\nu\tau^{\mu}),\qquad t\geq 0,% \qquad 0<\mu\leq 1,
  53. E α , β ( z ) E_{\alpha,\beta}(z)
  54. E α , β ( z ) = m = 0 z m Γ ( α m + β ) , E α , 1 ( z ) = E α ( z ) . E_{\alpha,\beta}(z)=\sum\limits_{m=0}^{\infty}\frac{z^{m}}{\Gamma(\alpha m+% \beta)},\qquad E_{\alpha,1}(z)=E_{\alpha}(z).
  55. ψ μ ( τ ) \psi_{\mu}(\tau)
  56. ψ μ ( τ ) 1 / ν τ μ + 1 , τ , \psi_{\mu}(\tau)\simeq 1/\nu\tau^{\mu+1},\qquad\tau\rightarrow\infty,
  57. ψ μ ( τ ) ν τ μ - 1 , τ 0. \psi_{\mu}(\tau)\simeq\nu\tau^{\mu-1},\qquad\tau\rightarrow 0.
  58. { X ( t ) \{X(t)
  59. t 0 } t\geq 0\}
  60. X ( t ) = i = 1 N ( t ) Y i , X(t)=\sum\limits_{i=1}^{N(t)}Y_{i},
  61. { N ( t ) \{N(t)
  62. t 0 } t\geq 0\}
  63. { Y i \{Y_{i}
  64. i = 1 , 2 , } i=1,2,\ldots\}
  65. p ( Y ) p(Y)
  66. Y i Y_{i}
  67. { N ( t ) \{N(t)
  68. t 0 } t\geq 0\}
  69. { Y i \{Y_{i}
  70. i = 1 , 2 , } i=1,2,\ldots\}
  71. | ς Align g t ; |\varsigma&gt;
  72. | ς = n = 0 ( μ ς μ ) n n ! ( E μ ( n ) ( - μ | ς | 2 μ ) ) 1 / 2 | n , |\varsigma\rangle=\sum\limits_{n=0}^{\infty}\frac{(\sqrt{\mu}\varsigma^{\mu})^% {n}}{\sqrt{n!}}(E_{\mu}^{(n)}(-\mu|\varsigma|^{2\mu}))^{1/2}|n\rangle,
  73. | n |n\rangle
  74. ς \varsigma
  75. E μ ( n ) ( - μ | ς | 2 μ ) = d n d z n E μ ( z ) | z = - μ | ς | 2 μ E_{\mu}^{(n)}(-\mu|\varsigma|^{2\mu})=\frac{d^{n}}{dz^{n}}E_{\mu}(z)|_{z=-\mu|% \varsigma|^{2\mu}}
  76. E μ ( x ) E_{\mu}(x)
  77. P μ ( n ) P_{\mu}(n)
  78. P μ ( n ) = | n | ς | 2 = ( μ | ς | 2 μ ) n n ! ( E μ ( n ) ( - μ | ς | 2 μ ) ) , P_{\mu}(n)=|\langle n|\varsigma\rangle|^{2}=\frac{(\mu|\varsigma|^{2\mu})^{n}}% {n!}\left(E_{\mu}^{(n)}(-\mu|\varsigma|^{2\mu})\right),
  79. a + a^{+}
  80. a a
  81. [ a , a + ] = a a + - a + a = 1 [a,a^{+}]=aa^{+}-a^{+}a=1
  82. n ¯ \bar{n}
  83. | ς |\varsigma\rangle
  84. n ¯ = ς | a + a | ς = n = 0 n P μ ( n ) = ( μ | ς | 2 μ ) / Γ ( μ + 1 ) \bar{n}=\langle\varsigma|a^{+}a|\varsigma\rangle=\sum\limits_{n=0}^{\infty}nP_% {\mu}(n)=(\mu|\varsigma|^{2\mu})/\Gamma(\mu+1)

Fractional_wavelet_transform.html

  1. W α ( a , b ) = 1 a 𝒦 α ( u , t ) f ( t ) ψ ( u - b a ) d t d u = 1 a ( f ( t ) 𝒦 α ( u , t ) d t ) ψ ( u - b a ) d u = 1 a F α ( u ) ψ ( u - b a ) d u \begin{aligned}\displaystyle W^{\alpha}(a,b)&\displaystyle=\frac{1}{\sqrt{a}}% \int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}{\mathcal{K}}_{\alpha}(u,t)f(t% )\psi^{\ast}\left(\frac{u-b}{a}\right)dtdu\\ &\displaystyle=\frac{1}{\sqrt{a}}\int\limits_{\mathbb{R}}\left(\int\limits_{% \mathbb{R}}f(t){\mathcal{K}}_{\alpha}(u,t)dt\right)\psi^{\ast}\left(\frac{u-b}% {a}\right)du\\ &\displaystyle=\frac{1}{\sqrt{a}}\int\limits_{\mathbb{R}}F_{\alpha}(u)\psi^{% \ast}\left(\frac{u-b}{a}\right)du\\ \end{aligned}
  2. 𝒦 α ( u , t ) \mathcal{K}_{\alpha}(u,t)
  3. 𝒦 α ( u , t ) = { A α e j u 2 + t 2 2 cot α - j u t csc α , α k π δ ( t - u ) , α = 2 k π δ ( t + u ) , α = ( 2 k - 1 ) π \mathcal{K}_{\alpha}(u,t)=\begin{cases}A_{\alpha}e^{j\frac{u^{2}+t^{2}}{2}\cot% \alpha-jut\csc\alpha},&\alpha\neq k\pi\\ \delta(t-u),&\alpha=2k\pi\\ \delta(t+u),&\alpha=(2k-1)\pi\\ \end{cases}
  4. A α = ( 1 - j cot α ) / 2 π A_{\alpha}=\sqrt{{(1-j\cot\alpha)}/{2\pi}}
  5. F α ( u ) F_{\alpha}(u)
  6. f ( t ) f(t)
  7. ( W ψ α f ) ( b , a ) = 1 a f ( t ) ψ ( t - b a ) d t = csc α 4 π 2 F ( u sin α ) Ψ ( a u sin α ) e j u 2 4 ( 1 - a 2 ) sin 2 α - j b u d u \begin{aligned}\displaystyle\left(W_{\psi}^{\alpha}f\right)(b,a)&\displaystyle% =\frac{1}{\sqrt{a}}\int\limits_{\mathbb{R}}f(t)\psi^{\ast}\left(\frac{t-b}{a}% \right)dt\\ &\displaystyle=\frac{\csc\alpha}{4\pi^{2}}\int\limits_{\mathbb{R}}F(u\sin% \alpha)\Psi^{\ast}(au\sin\alpha)e^{j\frac{u^{2}}{4}(1-a^{2})\sin 2\alpha-jbu}% du\\ \end{aligned}
  8. F ( u sin α ) F(u\sin\alpha)
  9. Ψ ( u sin α ) \Psi(u\sin\alpha)
  10. sin α \sin\alpha
  11. f ( t ) f(t)
  12. ψ ( t ) \psi(t)
  13. f ( t ) L 2 ( ) f(t)\in L^{2}(\mathbb{R})
  14. W f α ( a , b ) = 𝒲 α [ f ( t ) ] ( a , b ) = f ( t ) ψ α , a , b ( t ) d t W_{f}^{\alpha}(a,b)=\mathcal{W}^{\alpha}[f(t)](a,b)=\int\limits_{\mathbb{R}}f(% t)\psi_{\alpha,a,b}^{\ast}(t)\,dt
  15. ψ α , a , b ( t ) \psi_{\alpha,a,b}(t)
  16. ψ ( t ) \psi(t)
  17. ψ α , a , b ( t ) = 1 a ψ ( t - b a ) e - j t 2 - b 2 2 cot α \psi_{\alpha,a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right)e^{-j% \frac{t^{2}-b^{2}}{2}\cot\alpha}
  18. a + a\in\mathbb{R^{+}}
  19. b b\in\mathbb{R}
  20. f ( t ) = 1 2 π C ψ + W f α ( a , b ) ψ α , a , b ( t ) d a a 2 d b f(t)=\frac{1}{2\pi C_{\psi}}\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{+% }}W_{f}^{\alpha}(a,b)\psi_{\alpha,a,b}(t)\frac{da}{a^{2}}db
  21. C ψ C_{\psi}
  22. C ψ = | Ψ ( Ω ) | 2 | Ω | d Ω < C_{\psi}=\int\limits_{\mathbb{R}}{\frac{|\Psi(\Omega)|^{2}}{|\Omega|}}\,d% \Omega<\infty
  23. Ψ ( Ω ) \Psi(\Omega)
  24. ψ ( t ) \psi(t)
  25. Ψ ( 0 ) = 0 \Psi(0)=0
  26. ψ ( t ) d t = 0 \int_{\mathbb{R}}\psi(t)dt=0
  27. f ( t ) f(t)
  28. W f α ( a , b ) = 2 π a F α ( u ) Ψ ( a u csc α ) 𝒦 α ( u , b ) d u W_{f}^{\alpha}(a,b)=\int\limits_{\mathbb{R}}{\sqrt{2\pi a}F_{\alpha}(u)\Psi^{% \ast}(au\csc\alpha)}\mathcal{K}^{\ast}_{\alpha}(u,b)du
  29. F α ( u ) F_{\alpha}(u)
  30. f ( t ) f(t)
  31. Ψ ( u csc α ) \Psi(u\csc\alpha)
  32. csc α \csc\alpha
  33. ψ ( t ) \psi(t)
  34. α = π / 2 \alpha={\pi}/{2}

Free_category.html

  1. V 0 E 0 V 1 E 1 E n - 1 V n V_{0}\xrightarrow{\;\;E_{0}\;\;}V_{1}\xrightarrow{\;\;E_{1}\;\;}\cdots% \xrightarrow{E_{n-1}}V_{n}
  2. V k V_{k}
  3. E k E_{k}

Free_energy_principle.html

  1. ( Ω , Ψ , S , A , R , q , p ) (\Omega,\Psi,S,A,R,q,p)
  2. Ω \Omega
  3. ω Ω \omega\in\Omega
  4. Ψ : Ψ × A × Ω \Psi:\Psi\times A\times\Omega\to\mathbb{R}
  5. S : Ψ × A × Ω S:\Psi\times A\times\Omega\to\mathbb{R}
  6. A : S × R A:S\times R\to\mathbb{R}
  7. R : R × S R:R\times S\to\mathbb{R}
  8. p ( s , ψ | m ) p(s,\psi|m)
  9. m m
  10. q ( ψ | μ ) q(\psi|\mu)
  11. ψ Ψ \psi\in\Psi
  12. μ R \mu\in R
  13. p ( s | m ) p(s|m)
  14. - log p ( s | m ) -\log p(s|m)
  15. a ( t ) = arg min 𝑎 { F ( s ( t ) , μ ( t ) ) } a(t)=\underset{a}{\operatorname{arg\,min}}\{F(s(t),\mu(t))\}
  16. μ ( t ) = arg min 𝜇 { F ( s ( t ) , μ ) ) } \mu(t)=\underset{\mu}{\operatorname{arg\,min}}\{F(s(t),\mu))\}
  17. F ( s , μ ) free - energy = E q [ - log p ( s , ψ m ) ] e n e r g y - H [ q ( ψ | μ ) ] entropy = - log p ( s | m ) s u r p r i s e + D KL [ q ( ψ | μ ) p ( ψ s , m ) ] d i v e r g e n c e - log p ( s | m ) s u r p r i s e \underset{\mathrm{free-energy}}{\underbrace{F(s,\mu)}}=\underset{energy}{% \underbrace{E_{q}[-\log p(s,\psi\mid m)]}}-\underset{\mathrm{entropy}}{% \underbrace{H[q(\psi|\mu)]}}=\underset{surprise}{\underbrace{-\log p(s|m)}}+% \underset{divergence}{\underbrace{D_{\mathrm{KL}}[q(\psi|\mu)\|p(\psi\mid s,m)% ]}}\geq\underset{surprise}{\underbrace{-\log p(s|m)}}
  18. lim T 1 T 0 T F ( s ( t ) , μ ( t ) ) d t f r e e - a c t i o n lim T 1 T 0 T - log p ( s ( t ) | m ) s u r p r i s e d t = H [ p ( s | m ) ] \lim_{T\to\infty}\frac{1}{T}\underset{free-action}{\underbrace{\int_{0}^{T}F(s% (t),\mu(t))dt}}\geq\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\underset{surprise}% {\underbrace{-\log p(s(t)|m)}}dt=H[p(s|m)]
  19. F ( s , μ ) f r e e - e n e r g y = D KL [ q ( ψ | μ ) p ( ψ | m ) ] c o m p l e x i t y - E q [ log p ( s | ψ , m ) ] a c c u r a c y \underset{free-energy}{\underbrace{F(s,\mu)}}=\underset{complexity}{% \underbrace{D_{\mathrm{KL}}[q(\psi|\mu)\|p(\psi|m)]}}-\underset{accuracy}{% \underbrace{E_{q}[\log p(s|\psi,m)]}}
  20. Ψ = X × Θ × Π \Psi=X\times\Theta\times\Pi
  21. D D
  22. μ ~ ˙ = D μ ~ - μ ~ F ( s , μ ~ ) \dot{\tilde{\mu}}=D\tilde{\mu}-\partial_{\tilde{\mu}}F(s,\tilde{\mu})
  23. a ˙ = - a F ( s , μ ~ ) \dot{a}=-\partial_{a}F(s,\tilde{\mu})
  24. f = Γ V + × W f=\Gamma\cdot\nabla V+\nabla\times W
  25. V ( x ) V(x)
  26. W ( x ) W(x)
  27. Γ \Gamma
  28. c ( x ) = f V + Γ V c(x)=f\cdot\nabla V+\nabla\cdot\Gamma\cdot V
  29. p ( x ~ | m ) p(\tilde{x}|m)
  30. p ( x | m ) = exp ( V ( x ) ) p(x|m)=\exp(V(x))
  31. W = 0 W=0

Free_motion_equation.html

  1. Q Q\to\mathbb{R}
  2. Q Q\to\mathbb{R}
  3. q ¯ t t i = 0 \overline{q}^{i}_{tt}=0
  4. ( t , q ¯ i ) (t,\overline{q}^{i})
  5. Q Q\to\mathbb{R}
  6. ( t , q i ) (t,q^{i})
  7. Q Q\to\mathbb{R}
  8. q t t i = d t Γ i + j Γ i ( q t j - Γ j ) - q i q ¯ m q ¯ m q j q k ( q t j - Γ j ) ( q t k - Γ k ) , q^{i}_{tt}=d_{t}\Gamma^{i}+\partial_{j}\Gamma^{i}(q^{j}_{t}-\Gamma^{j})-\frac{% \partial q^{i}}{\partial\overline{q}^{m}}\frac{\partial\overline{q}^{m}}{% \partial q^{j}\partial q^{k}}(q^{j}_{t}-\Gamma^{j})(q^{k}_{t}-\Gamma^{k}),
  9. Γ i = t q i ( t , q ¯ j ) \Gamma^{i}=\partial_{t}q^{i}(t,\overline{q}^{j})
  10. Q Q\to\mathbb{R}
  11. ( t , q ¯ i ) (t,\overline{q}^{i})
  12. Q Q\to\mathbb{R}
  13. T m × k T^{m}\times\mathbb{R}^{k}

Freeze-casting.html

  1. Δ σ = σ p s - ( σ p l + σ s l ) \Delta\sigma=\sigma_{ps}-(\sigma_{pl}+\sigma_{sl})
  2. v c 1 R v_{c}\propto\tfrac{1}{R}
  3. v c = Δ σ d 3 η R ( a 0 d ) z v_{c}=\frac{\Delta\sigma d}{3\eta R}\left(\frac{a_{0}}{d}\right)^{z}
  4. λ = A ν - n \lambda=A\nu^{-n}

Frits_Beukers.html

  1. ζ ( 2 ) \zeta(2)
  2. ζ ( 3 ) \zeta(3)

Froissart_Stora_equation.html

  1. P y = P y 0 ( 2 e - π | ϵ | 2 2 α 0 - 1 ) P_{y}=P_{y0}(2e^{-\frac{\pi|\epsilon|^{2}}{2\alpha_{0}}}-1)
  2. ϵ \epsilon
  3. α 0 \alpha_{0}
  4. P y 0 P_{y0}

Fujisaki_Model.html

  1. F 0 ( t ) = l n ( F b ) + i = 1 I A p i G p ( t - T 0 i ) + j = 1 J A a j { G a ( t - T 1 j ) - G a ( t - T 2 j ) } F_{0}(t)=ln(F_{b})+\sum_{i=1}^{I}A_{pi}G_{p}(t-T_{0i})+\sum_{j=1}^{J}A_{aj}\{G% _{a}(t-T_{1j})-G_{a}(t-T_{2j})\}
  2. G p ( t ) = α 2 t e x p ( - α t ) f o r t 0 G_{p}(t)=\alpha^{2}texp(-\alpha t)\quad for\ t\geq 0
  3. G a ( t ) = m i n [ 1 - ( 1 + β t ) e x p ( - β t ) , γ ] f o r t 0 G_{a}(t)=min[1-(1+\beta t)exp(-\beta t),\gamma]\quad for\ t\geq 0

Fumarate_reductase_(menaquinone).html

  1. \rightleftharpoons

Fumigaclavine_A_dimethylallyltransferase.html

  1. \rightleftharpoons

Fumigaclavine_B_O-acetyltransferase.html

  1. \rightleftharpoons

Fumonisin_B1_esterase.html

  1. \rightleftharpoons

Functional_encryption.html

  1. F F
  2. ( p k , m s k ) S e t u p ( 1 λ ) (pk,msk)\leftarrow Setup(1^{\lambda})
  3. p k pk
  4. m s k msk
  5. s k K e y g e n ( m s k , k ) sk\leftarrow Keygen(msk,k)
  6. s k sk
  7. k k
  8. c E n c ( p k , x ) c\leftarrow Enc(pk,x)
  9. x x
  10. F ( k , x ) D e c ( s k , c ) F(k,x)\leftarrow Dec(sk,c)
  11. c c
  12. F ( k , x ) F(k,x)
  13. x x
  14. k k
  15. \perp
  16. F ( k , x ) = x F(k,x)=x
  17. k k
  18. \perp

Fundamental_increment_lemma.html

  1. f ( a ) = lim h 0 f ( a + h ) - f ( a ) h . f^{\prime}(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.
  2. φ \varphi
  3. lim h 0 φ ( h ) = 0 and f ( a + h ) = f ( a ) + f ( a ) h + φ ( h ) h \lim_{h\to 0}\varphi(h)=0\qquad\,\text{and}\qquad f(a+h)=f(a)+f^{\prime}(a)h+% \varphi(h)h
  4. φ ( h ) = f ( a + h ) - f ( a ) h - f ( a ) \varphi(h)=\frac{f(a+h)-f(a)}{h}-f^{\prime}(a)
  5. φ \varphi
  6. φ \varphi
  7. f ( a ) f^{\prime}(a)
  8. n \mathbb{R}^{n}
  9. \mathbb{R}
  10. M : n M:\mathbb{R}^{n}\to\mathbb{R}
  11. Φ : D , D n { 0 } , \Phi:D\to\mathbb{R},\qquad D\subseteq\mathbb{R}^{n}\smallsetminus\{{0}\},
  12. lim h 0 Φ ( h ) = 0 and f ( a + h ) = f ( a ) + M ( h ) + Φ ( h ) h \lim_{{h}\to 0}\Phi({h})=0\qquad\,\text{and}\qquad f({a}+{h})=f({a})+M({h})+% \Phi({h})\cdot\|{h}\|

Fusion_frame.html

  1. \mathcal{H}
  2. { W i } i \{W_{i}\}_{i\in\mathcal{I}}
  3. \mathcal{H}
  4. \mathcal{I}
  5. { v i } i \{v_{i}\}_{i\in\mathcal{I}}
  6. { W i , v i } i \{W_{i},v_{i}\}_{i\in\mathcal{I}}
  7. \mathcal{H}
  8. 0 < A B 0<A\leq B
  9. f f\in\mathcal{H}
  10. A f 2 i v i 2 P W i f 2 B f 2 A\|f\|^{2}\leq\sum_{i\in\mathcal{I}}v_{i}^{2}\big\|P_{W_{i}}f\big\|^{2}\leq B% \|f\|^{2}
  11. P W i P_{W_{i}}
  12. W i W_{i}
  13. W W\subset\mathcal{H}
  14. { x n } \{x_{n}\}
  15. W W
  16. f f\in\mathcal{H}
  17. f f
  18. W W
  19. P W f = f , x n x n P_{W}f=\sum\langle f,x_{n}\rangle x_{n}
  20. dim = : N < \dim\mathcal{H}=:N<\infty
  21. | | < |\mathcal{I}|<\infty
  22. { W i , v i } i \{W_{i},v_{i}\}_{i\in\mathcal{I}}
  23. N \mathcal{H}_{N}
  24. { f i j } j 𝒥 i \{f_{ij}\}_{j\in\mathcal{J}_{i}}
  25. W i W_{i}
  26. J i J_{i}
  27. i i\in\mathcal{I}
  28. F i = [ f i 1 f i 2 f i | J i | ] N × | J i | F_{i}=\begin{bmatrix}\vdots&\vdots&&\vdots\\ f_{i1}&f_{i2}&\cdots&f_{i|J_{i}|}\\ \vdots&\vdots&&\vdots\\ \end{bmatrix}_{N\times|J_{i}|}
  29. F ~ i = [ f ~ i 1 f ~ i 2 f ~ i | J i | ] N × | J i | , \tilde{F}_{i}=\begin{bmatrix}\vdots&\vdots&&\vdots\\ \tilde{f}_{i1}&\tilde{f}_{i2}&\cdots&\tilde{f}_{i|J_{i}|}\\ \vdots&\vdots&&\vdots\\ \end{bmatrix}_{N\times|J_{i}|},
  30. f ~ i j \tilde{f}_{ij}
  31. f i j f_{ij}
  32. S : S:\mathcal{H}\to\mathcal{H}
  33. S = i v i 2 F i F ~ i T S=\sum_{i\in\mathcal{I}}v_{i}^{2}F_{i}\tilde{F}_{i}^{T}
  34. S S
  35. N × N N\times N

Futalosine_hydrolase.html

  1. \rightleftharpoons

Fuzzy_extractor.html

  1. R R
  2. w w
  3. w w^{\prime}
  4. w w
  5. R R
  6. R R
  7. P P
  8. R R
  9. P P
  10. R R
  11. P P
  12. A A
  13. max a P [ A = a ] \max_{\mathrm{a}}P[A=a]
  14. A A
  15. B B
  16. b b
  17. B B
  18. A A
  19. max a P [ A = a | B = b ] \max_{\mathrm{a}}P[A=a|B=b]
  20. A A
  21. E b B [ max a P [ A = a | B = b ] ] E_{b\leftarrow B}[\max_{\mathrm{a}}P[A=a|B=b]]
  22. B B
  23. b b
  24. A A
  25. A A
  26. H ( A ) = - log ( max a P [ A = a ] ) H_{\infty}(A)=-\log(\max_{\mathrm{a}}P[A=a])
  27. m m
  28. m m
  29. A A
  30. B B
  31. S D [ A , B ] SD[A,B]
  32. 1 2 v | P [ A = v ] - P [ B = v ] | \frac{1}{2}\sum_{\mathrm{v}}|P[A=v]-P[B=v]|
  33. A A
  34. B B
  35. 1 - S D [ A , B ] 1-SD[A,B]
  36. M M
  37. M { 0 , 1 } l M\rightarrow\{0,1\}^{l}
  38. r r
  39. ( m , l , ϵ ) (m,l,\epsilon)
  40. m m
  41. m m
  42. m m
  43. W W
  44. M ( E x t ( W ; I ) , I ) ϵ ( U l , U r ) , M(Ext(W;I),I)\approx_{\epsilon}(U_{l},U_{r}),
  45. I = U r I=U_{r}
  46. W W
  47. w W w\leftarrow W
  48. i I i\leftarrow I
  49. 1 - ϵ 1-\epsilon
  50. l = m - 2 l o g 1 ϵ + O ( 1 ) l=m-2log\frac{1}{\epsilon}+O(1)
  51. m m
  52. w w
  53. s s
  54. s s
  55. w w^{\prime}
  56. w w
  57. w w
  58. s s
  59. w w
  60. 𝕄 \mathbb{M}
  61. w 𝕄 w\in\mathbb{M}
  62. w 𝕄 w^{\prime}\in\mathbb{M}
  63. w w
  64. ( m , m ~ , t ) (m,\tilde{m},t)
  65. w 𝕄 w\in\mathbb{M}
  66. s { 0 , 1 } * s\in{\{0,1\}^{*}}
  67. w 𝕄 w^{\prime}\in\mathbb{M}
  68. s { 0 , 1 } * s\in{\{0,1\}^{*}}
  69. d i s ( w , w ) t dis(w,w^{\prime})\leq t
  70. R e c ( w , S S ( w ) ) = w Rec(w^{\prime},SS(w))=w
  71. m m
  72. M M
  73. W W
  74. s s
  75. ( W , E ) (W,E)
  76. H ~ ( W | E ) m \tilde{H}_{\mathrm{\infty}}(W|E)\geq m
  77. H ~ ( W | S S ( W ) , E ) m ~ \tilde{H}_{\mathrm{\infty}}(W|SS(W),E)\geq\tilde{m}
  78. R R
  79. w w
  80. P P
  81. w w^{\prime}
  82. w w
  83. t t
  84. P = I P=I
  85. ( m , l , t , ϵ ) (m,l,t,\epsilon)
  86. w 𝕄 w\in\mathbb{M}
  87. R { 0 , 1 } l R\in{\mathbb{\{}0,1\}^{l}}
  88. P { 0 , 1 } * P\in{\mathbb{\{}0,1\}^{*}}
  89. d i s ( w , w ) t dis(w,w^{\prime})\leq t
  90. ( R , P ) G e n ( w ) (R,P)\leftarrow Gen(w)
  91. R e p ( w , P ) = R Rep(w^{\prime},P)=R
  92. W W
  93. M M
  94. R R
  95. P P
  96. H ~ ( W | E ) m \tilde{H}_{\mathrm{\infty}}(W|E)\geq m
  97. ( R , P , E ) ( U l , P , E ) (R,P,E)\approx(U_{\mathrm{l}},P,E)
  98. ϵ \epsilon
  99. w w
  100. s s
  101. x x
  102. w w
  103. R R
  104. ( s , x ) (s,x)
  105. P P
  106. R R
  107. w w^{\prime}
  108. P = ( s , x ) P=(s,x)
  109. R e c ( w , s ) Rec(w^{\prime},s)
  110. w w
  111. E x t ( w , x ) Ext(w,x)
  112. R R
  113. ( M , m , m ~ , t ) (M,m,\tilde{m},t)
  114. ( n , m ~ , l , ϵ ) (n,\tilde{m},l,\epsilon)
  115. ( M , m , l , t , ϵ ) (M,m,l,t,\epsilon)
  116. ( w , r , x ) : s e t P = ( S S ( w ; r ) , x ) , R = E x t ( w ; x ) , (w,r,x):setP=(SS(w;r),x),R=Ext(w;x),
  117. ( R , P ) (R,P)
  118. ( w , ( s , x ) ) (w^{\prime},(s,x))
  119. w = R e c ( w , s ) w=Rec(w^{\prime},s)
  120. R = E x t ( w ; x ) R=Ext(w;x)
  121. H ( W | S S ( W ) ) m ~ H_{\infty}(W|SS(W))\geq\tilde{m}
  122. ( n , m , l , ϵ ) (n,m,l,\epsilon)
  123. S D ( ( E x t ( W ; X ) , S S ( W ) , X ) , ( U l , S S ( W ) , X ) ) = S D ( ( R , P ) , ( U l , P ) ) ϵ . SD((Ext(W;X),SS(W),X),(U_{l},SS(W),X))=SD((R,P),(U_{l},P))\leq\epsilon.
  124. ( M , m , m ~ , t ) (M,m,\tilde{m},t)
  125. ( n , m ~ - l o g ( 1 δ ) , l , ϵ ) (n,\tilde{m}-log(\frac{1}{\delta}),l,\epsilon)
  126. ( M , m , l , t , ϵ + δ ) (M,m,l,t,\epsilon+\delta)
  127. ( n , k , d ) (n,k,d)_{\mathcal{F}}
  128. [ n , k , d ] [n,k,d]_{\mathcal{F}}
  129. n n
  130. k k
  131. d d
  132. \mathcal{F}
  133. n \mathcal{F}^{n}
  134. C n C\in\mathcal{F}^{n}
  135. c C c\in C
  136. w n w\in\mathcal{F}^{n}
  137. d i s H a m ( c , w ) ( d - 1 ) / 2 dis_{Ham}(c,w)\leq(d-1)/2
  138. 2 t + 1 2t+1
  139. t {t}
  140. ( n , k , 2 t + 1 ) (n,k,2t+1)_{\mathcal{F}}
  141. c C c\in C
  142. w w
  143. S S ( w ) = s = w - c SS(w)=s=w-c
  144. c c
  145. w w
  146. w w^{\prime}
  147. s s
  148. w w^{\prime}
  149. c c
  150. s s
  151. c c
  152. w w
  153. R e c ( w , s ) = s + d e c ( w - s ) = w Rec(w^{\prime},s)=s+dec(w^{\prime}-s)=w
  154. n \mathcal{F}\geq n
  155. 2 t log ( ) 2t\log(\mathcal{F})
  156. [ n , k , 2 t + 1 ] [n,k,2t+1]_{\mathcal{F}}
  157. S S ( w ) = s SS(w)=s
  158. w w
  159. w w^{\prime}
  160. e e
  161. s y n ( e ) = s y n ( w ) - s syn(e)=syn(w^{\prime})-s
  162. w = w - e w=w^{\prime}-e
  163. 𝒰 \mathcal{U}
  164. w w
  165. w w^{\prime}
  166. w w
  167. x w x_{w}
  168. n n
  169. a 𝒰 a\in\mathcal{U}
  170. a w a\in w
  171. a w a\notin w
  172. n n
  173. k k
  174. n - k n-k
  175. [ n , n - t α , 2 t + 1 ] 2 [n,n-t\alpha,2t+1]_{2}
  176. n = 2 α - 1 n=2^{\alpha}-1
  177. t n t\ll n
  178. k n - l o g ( n t ) k\leq n-log{n\choose{t}}
  179. S S ( w ) = s = s y n ( x w ) SS(w)=s=syn(x_{w})
  180. w w^{\prime}
  181. S S ( w ) = s = s y n ( x w ) SS(w^{\prime})=s^{\prime}=syn(x_{w}^{\prime})
  182. s y n ( x v ) = s - s syn(x_{v})=s^{\prime}-s
  183. R e c ( w , s ) = w v = w Rec(w^{\prime},s)=w^{\prime}\triangle v=w
  184. n / 4 n/4
  185. n / 2 n/2
  186. p {}_{p}
  187. w w^{\prime}
  188. p p
  189. w w^{\prime}
  190. n H ( p ) - o ( n ) nH(p)-o(n)
  191. H H
  192. m n ( H ( 1 2 - γ ) ) + ε m\geq n(H(\frac{1}{2}-\gamma))+\varepsilon
  193. n ( 1 2 - γ ) n(\frac{1}{2}-\gamma)
  194. γ > 0 \gamma>0
  195. d i s err t dis\text{err}\leq t
  196. w w
  197. t t
  198. w w
  199. f ( W ) f(W)
  200. P P
  201. w w
  202. w w^{\prime}
  203. w w
  204. w w
  205. w w^{\prime}
  206. w w
  207. P P
  208. w w
  209. w w^{\prime}
  210. P P
  211. w w
  212. P P
  213. w w
  214. f ( W ) f(W)
  215. Y ( ) Y()
  216. ϵ \epsilon
  217. | P r [ A 1 ( Y ( W ) ) = f ( W ) ] - P r [ A 2 ( ) = f ( W ) ] | ϵ |Pr[A_{1}(Y(W))=f(W)]-Pr[A_{2}()=f(W)]|\leq\epsilon
  218. G e n ( W ) Gen(W)
  219. P P
  220. R R
  221. R R
  222. ϵ \epsilon
  223. f ( W ) f(W)
  224. Y = G e n ( W ) = R , P Y=Gen(W)=R,P
  225. | Pr [ A 1 ( R , P ) = f ( W ) ] - Pr [ A 2 ( ) = f ( W ) ] | ϵ |\Pr[A_{1}(R,P)=f(W)]-\Pr[A_{2}()=f(W)]|\leq\epsilon
  226. A 1 A_{1}
  227. ( R , P ) (R,P)
  228. A 2 A_{2}
  229. f ( W ) f(W)
  230. ϵ \epsilon
  231. ( R , P ) (R,P)
  232. G e n ( W ) Gen(W)
  233. ( R , P ) ϵ ( U , U | P | ) (R,P)\approx_{\epsilon}(U_{\ell},U_{|P|})
  234. S S ( w ) SS(w)
  235. E x t ( w ; i ) Ext(w;i)
  236. w w
  237. i i
  238. ( m , t , ϵ ) (m,t,\epsilon)
  239. w w
  240. P , R P,R
  241. G e n ( w ) Gen(w)
  242. w w^{\prime}
  243. t t
  244. w w
  245. R e p ( w , P ) = R Rep(w^{\prime},P)=R
  246. ( m , t , ϵ ) (m,t,\epsilon)
  247. t t
  248. t t
  249. P P
  250. P P
  251. R e p ( W , P ) Rep(W,P)
  252. R e p ( W , P ) Rep(W,P)
  253. R ~ \tilde{R}
  254. H 1 H_{1}
  255. H 2 H_{2}
  256. G e n ( W ) Gen(W)
  257. P P
  258. s = S S ( w ) s=SS(w)
  259. w w
  260. s s
  261. R R
  262. w w
  263. s s
  264. G e n ( w ) : s = S S ( w ) , r e t u r n : P = ( s , H 1 ( w , s ) ) , R = H 2 ( w , s ) Gen(w):s=SS(w),return:P=(s,H_{1}(w,s)),R=H_{2}(w,s)
  265. R e p ( W , P ) Rep(W,P)
  266. H 1 H_{1}
  267. H 2 H_{2}
  268. R e c ( W , S ) Rec(W,S)
  269. P P
  270. w w
  271. s s
  272. R R
  273. w w
  274. s s
  275. R e p ( w , P ~ ) : Rep(w^{\prime},\tilde{P}):
  276. s ~ \tilde{s}
  277. h ~ \tilde{h}
  278. P ~ ; w ~ = R e c ( w , s ~ ) . \tilde{P};\tilde{w}=Rec(w^{\prime},\tilde{s}).
  279. Δ ( w ~ , w ) t \Delta(\tilde{w},w^{\prime})\leq t
  280. h ~ = H 1 ( w ~ , s ~ ) \tilde{h}=H_{1}(\tilde{w},\tilde{s})
  281. r e t u r n : H 2 ( w ~ , s ~ ) return:H_{2}(\tilde{w},\tilde{s})
  282. r e t u r n : f a i l return:fail
  283. P P
  284. R e p Rep
  285. P P
  286. w ~ \tilde{w}
  287. H 1 ( w , s ) = H 1 ( w ~ , s ~ ) H_{1}(w,s)=H_{1}(\tilde{w},\tilde{s})
  288. w ~ \tilde{w}
  289. P P
  290. w w
  291. P P
  292. R R