wpmath0000005_12

Scattering_parameters.html

  1. N 2 N^{2}\,
  2. S n n S_{nn}\,
  3. S n n S_{nn}\,
  4. a n a_{n}\,
  5. b n b_{n}\,
  6. a = 1 2 k ( V + Z p I ) a=\frac{1}{2}\,k(V+Z_{p}I)\,
  7. b = 1 2 k ( V - Z p * I ) b=\frac{1}{2}\,k(V-Z_{p}^{*}I)\,
  8. Z p Z_{p}\,
  9. Z p * Z_{p}^{*}\,
  10. Z p Z_{p}\,
  11. V V\,
  12. I I\,
  13. k = ( | \real { Z p } | ) - 1 k=\left(\sqrt{\left|\real\{Z_{p}\}\right|}\right)^{-1}\,
  14. a = 1 2 ( V + Z 0 I ) | \real { Z 0 } | a=\frac{1}{2}\,\frac{(V+Z_{0}I)}{\sqrt{\left|\real\{Z_{0}\}\right|}}\,
  15. b = 1 2 ( V - Z 0 * I ) | \real { Z 0 } | b=\frac{1}{2}\,\frac{(V-Z_{0}^{*}I)}{\sqrt{\left|\real\{Z_{0}\}\right|}}\,
  16. b = S a b=Sa\,
  17. S m n = S n m S_{mn}=S_{nm}\,
  18. Σ | a n | 2 = Σ | b n | 2 \Sigma\left|a_{n}\right|^{2}=\Sigma\left|b_{n}\right|^{2}\,
  19. ( S ) H ( S ) = ( I ) (S)^{H}(S)=(I)\,
  20. ( S ) H (S)^{H}\,
  21. ( S ) (S)\,
  22. ( I ) (I)\,
  23. Σ | a n | 2 Σ | b n | 2 \Sigma\left|a_{n}\right|^{2}\neq\Sigma\left|b_{n}\right|^{2}\,
  24. Σ | a n | 2 > Σ | b n | 2 \Sigma\left|a_{n}\right|^{2}>\Sigma\left|b_{n}\right|^{2}\,
  25. ( I ) - ( S ) H ( S ) (I)-(S)^{H}(S)\,
  26. ( b 1 b 2 ) = ( S 11 S 12 S 21 S 22 ) ( a 1 a 2 ) \begin{pmatrix}b_{1}\\ b_{2}\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}a_{1}\\ a_{2}\end{pmatrix}\,
  27. b 1 = S 11 a 1 + S 12 a 2 b_{1}=S_{11}a_{1}+S_{12}a_{2}\,
  28. b 2 = S 21 a 1 + S 22 a 2 b_{2}=S_{21}a_{1}+S_{22}a_{2}\,
  29. S 11 S_{11}\,
  30. S 12 S_{12}\,
  31. S 21 S_{21}\,
  32. S 22 S_{22}\,
  33. a 1 a_{1}\,
  34. b 1 b_{1}\,
  35. b 2 b_{2}\,
  36. Z 0 Z_{0}\,
  37. b 2 b_{2}\,
  38. a 2 a_{2}\,
  39. a 1 = V 1 + a_{1}=V_{1}^{+}
  40. a 2 = V 2 + a_{2}=V_{2}^{+}
  41. b 1 = V 1 - b_{1}=V_{1}^{-}
  42. b 2 = V 2 - b_{2}=V_{2}^{-}
  43. S 11 = b 1 a 1 = V 1 - V 1 + S_{11}=\frac{b_{1}}{a_{1}}=\frac{V_{1}^{-}}{V_{1}^{+}}
  44. S 21 = b 2 a 1 = V 2 - V 1 + S_{21}=\frac{b_{2}}{a_{1}}=\frac{V_{2}^{-}}{V_{1}^{+}}\,
  45. a 1 a_{1}\,
  46. S 12 = b 1 a 2 = V 1 - V 2 + S_{12}=\frac{b_{1}}{a_{2}}=\frac{V_{1}^{-}}{V_{2}^{+}}\,
  47. S 22 = b 2 a 2 = V 2 - V 2 + S_{22}=\frac{b_{2}}{a_{2}}=\frac{V_{2}^{-}}{V_{2}^{+}}\,
  48. S 11 S_{11}\,
  49. S 12 S_{12}\,
  50. S 21 S_{21}\,
  51. S 22 S_{22}\,
  52. V + V^{+}
  53. V - V^{-}
  54. b 2 = V 2 + b_{2}=V_{2}^{+}
  55. a 2 = V 2 - a_{2}=V_{2}^{-}
  56. S 21 = V 2 + / V 1 + S_{21}=V_{2}^{+}/V_{1}^{+}
  57. G = S 21 G=S_{21}\,
  58. | G | = | S 21 | \left|G\right|=\left|S_{21}\right|\,
  59. g = 20 log 10 | S 21 | g=20\log_{10}\left|S_{21}\right|\,
  60. I L IL
  61. I L = - 20 log 10 | S 21 | IL=-20\log_{10}\left|S_{21}\right|\,
  62. R L in = 10 log 10 | 1 S 11 2 | = - 20 log 10 | S 11 | RL_{\mathrm{in}}=10\log_{10}\left|\frac{1}{S_{11}^{2}}\right|=-20\log_{10}% \left|S_{11}\right|\,
  63. R L out = - 20 log 10 | S 22 | RL_{\mathrm{out}}=-20\log_{10}\left|S_{22}\right|\,
  64. g rev g_{\mathrm{rev}}\,
  65. g rev = 20 log 10 | S 12 | g_{\mathrm{rev}}=20\log_{10}\left|S_{12}\right|\,
  66. I rev I_{\mathrm{rev}}\,
  67. g rev g_{\mathrm{rev}}\,
  68. I rev = | g rev | = | 20 log 10 | S 12 | | I_{\mathrm{rev}}=\left|g_{\mathrm{rev}}\right|=\left|20\log_{10}\left|S_{12}% \right|\right|\,
  69. ρ in \rho_{\mathrm{in}}\,
  70. ρ out \rho_{\mathrm{out}}\,
  71. S 11 S_{11}\,
  72. S 22 S_{22}\,
  73. ρ in = S 11 \rho_{\mathrm{in}}=S_{11}\,
  74. ρ out = S 22 \rho_{\mathrm{out}}=S_{22}\,
  75. S 11 S_{11}\,
  76. S 22 S_{22}\,
  77. ρ in \rho_{\mathrm{in}}\,
  78. ρ out \rho_{\mathrm{out}}\,
  79. S 11 S_{11}\,
  80. S 22 S_{22}\,
  81. s in s_{\mathrm{in}}\,
  82. s in = 1 + | S 11 | 1 - | S 11 | s_{\mathrm{in}}=\frac{1+\left|S_{11}\right|}{1-\left|S_{11}\right|}\,
  83. s out s_{\mathrm{out}}\,
  84. s out = 1 + | S 22 | 1 - | S 22 | s_{\mathrm{out}}=\frac{1+\left|S_{22}\right|}{1-\left|S_{22}\right|}\,
  85. s k = 1 + | S k k | | 1 - | S k k | | s_{k}=\frac{1+\left|S_{kk}\right|}{|1-\left|S_{kk}\right||}\,
  86. ( S 11 S 12 S 13 S 14 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44 ) \begin{pmatrix}S_{11}&S_{12}&S_{13}&S_{14}\\ S_{21}&S_{22}&S_{23}&S_{24}\\ S_{31}&S_{32}&S_{33}&S_{34}\\ S_{41}&S_{42}&S_{43}&S_{44}\end{pmatrix}
  87. S 12 S_{12}\,
  88. S 21 S_{21}\,
  89. S 12 S_{12}\,
  90. | S 12 | \left|S_{12}\right|\,
  91. ρ L \rho_{L}\,
  92. ρ in \rho_{\mathrm{in}}\,
  93. ρ in = S 11 + S 12 S 21 ρ L 1 - S 22 ρ L \rho_{\mathrm{in}}=S_{11}+\frac{S_{12}S_{21}\rho_{L}}{1-S_{22}\rho_{L}}\,
  94. S 12 = 0 S_{12}=0\,
  95. ρ in = S 11 \rho_{\mathrm{in}}=S_{11}\,
  96. ρ out \rho_{\mathrm{out}}\,
  97. ρ s \rho_{s}\,
  98. ρ o u t = S 22 + S 12 S 21 ρ s 1 - S 11 ρ s \rho_{out}=S_{22}+\frac{S_{12}S_{21}\rho_{s}}{1-S_{11}\rho_{s}}\,
  99. | ρ S | < 1 \left|\rho_{S}\right|<1\,
  100. | ρ L | < 1 \left|\rho_{L}\right|<1\,
  101. | ρ in | < 1 \left|\rho_{\mathrm{in}}\right|<1\,
  102. | ρ out | < 1 \left|\rho_{\mathrm{out}}\right|<1\,
  103. ρ L \rho_{L}\,
  104. | ρ i n | = 1 \left|\rho_{in}\right|=1\,
  105. r L = | S 12 S 21 | S 22 | 2 - | Δ | 2 | r_{L}=\left|\frac{S_{12}S_{21}}{\left|S_{22}\right|^{2}-\left|\Delta\right|^{2% }}\right|\,
  106. c L = ( S 22 - Δ S 11 * ) * | S 22 | 2 - | Δ | 2 c_{L}=\frac{(S_{22}-\Delta S_{11}^{*})^{*}}{\left|S_{22}\right|^{2}-\left|% \Delta\right|^{2}}\,
  107. ρ S \rho_{S}\,
  108. | ρ o u t | = 1 \left|\rho_{out}\right|=1\,
  109. r s = | S 12 S 21 | S 11 | 2 - | Δ | 2 | r_{s}=\left|\frac{S_{12}S_{21}}{\left|S_{11}\right|^{2}-\left|\Delta\right|^{2% }}\right|\,
  110. c s = ( S 11 - Δ S 22 * ) * | S 11 | 2 - | Δ | 2 c_{s}=\frac{(S_{11}-\Delta S_{22}^{*})^{*}}{\left|S_{11}\right|^{2}-\left|% \Delta\right|^{2}}\,
  111. Δ = S 11 S 22 - S 12 S 21 \Delta=S_{11}S_{22}-S_{12}S_{21}\,
  112. K K\,
  113. K = 1 - | S 11 | 2 - | S 22 | 2 + | Δ | 2 2 | S 12 S 21 | K=\frac{1-\left|S_{11}\right|^{2}-\left|S_{22}\right|^{2}+\left|\Delta\right|^% {2}}{2\left|S_{12}S_{21}\right|}\,
  114. K > 1 K>1\,
  115. | Δ | < 1 \left|\Delta\right|<1\,
  116. ( b 1 a 1 ) = ( T 11 T 12 T 21 T 22 ) ( a 2 b 2 ) \begin{pmatrix}b_{1}\\ a_{1}\end{pmatrix}=\begin{pmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{pmatrix}\begin{pmatrix}a_{2}\\ b_{2}\end{pmatrix}\,
  117. ( a 1 b 1 ) = ( T 11 T 12 T 21 T 22 ) ( b 2 a 2 ) \begin{pmatrix}a_{1}\\ b_{1}\end{pmatrix}=\begin{pmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{pmatrix}\begin{pmatrix}b_{2}\\ a_{2}\end{pmatrix}\,
  118. ( T 1 ) \begin{pmatrix}T_{1}\end{pmatrix}\,
  119. ( T 2 ) \begin{pmatrix}T_{2}\end{pmatrix}\,
  120. ( T 3 ) \begin{pmatrix}T_{3}\end{pmatrix}\,
  121. ( T T ) \begin{pmatrix}T_{T}\end{pmatrix}\,
  122. ( T T ) = ( T 1 ) ( T 2 ) ( T 3 ) \begin{pmatrix}T_{T}\end{pmatrix}=\begin{pmatrix}T_{1}\end{pmatrix}\begin{% pmatrix}T_{2}\end{pmatrix}\begin{pmatrix}T_{3}\end{pmatrix}\,
  123. T 11 = - det ( S ) S 21 T_{11}=\frac{-\det\begin{pmatrix}S\end{pmatrix}}{S_{21}}\,
  124. T 12 = S 11 S 21 T_{12}=\frac{S_{11}}{S_{21}}\,
  125. T 21 = - S 22 S 21 T_{21}=\frac{-S_{22}}{S_{21}}\,
  126. T 22 = 1 S 21 T_{22}=\frac{1}{S_{21}}\,
  127. S 11 = T 12 T 22 S_{11}=\frac{T_{12}}{T_{22}}\,
  128. S 12 = det ( T ) T 22 S_{12}=\frac{\det\begin{pmatrix}T\end{pmatrix}}{T_{22}}\,
  129. S 21 = 1 T 22 S_{21}=\frac{1}{T_{22}}\,
  130. S 22 = - T 21 T 22 S_{22}=\frac{-T_{21}}{T_{22}}\,
  131. det ( S ) \det\begin{pmatrix}S\end{pmatrix}\,
  132. ( S ) \begin{pmatrix}S\end{pmatrix}\,
  133. ( s n n ) (s_{nn})\,
  134. S m n S_{mn}\,
  135. m n m\neq\;n\,
  136. S m m S_{mm}\,
  137. S m n = b m a n S_{mn}=\frac{b_{m}}{a_{n}}\,
  138. S m m = b m a m S_{mm}=\frac{b_{m}}{a_{m}}\,
  139. S 11 = b 1 a 1 = V 1 - V 1 + S_{11}=\frac{b_{1}}{a_{1}}=\frac{V_{1}^{-}}{V_{1}^{+}}\,
  140. S 33 = b 3 a 3 = V 3 - V 3 + S_{33}=\frac{b_{3}}{a_{3}}=\frac{V_{3}^{-}}{V_{3}^{+}}\,
  141. S 32 = b 3 a 2 = V 3 - V 2 + S_{32}=\frac{b_{3}}{a_{2}}=\frac{V_{3}^{-}}{V_{2}^{+}}\,
  142. S 23 = b 2 a 3 = V 2 - V 3 + S_{23}=\frac{b_{2}}{a_{3}}=\frac{V_{2}^{-}}{V_{3}^{+}}\,
  143. S 11 S_{11}\,
  144. S 21 S_{21}\,
  145. S 22 S_{22}\,
  146. S 12 S_{12}\,
  147. S 11 S_{11}\,
  148. S 21 S_{21}\,
  149. S 12 S_{12}\,
  150. S 22 S_{22}\,
  151. S 11 S_{11}\,
  152. S 21 S_{21}\,
  153. S 12 S_{12}\,
  154. S 22 S_{22}\,
  155. S 11 S_{11}\,
  156. S 22 S_{22}\,
  157. S n n S_{nn}\,
  158. S 11 S_{11}\,
  159. S 12 S_{12}\,
  160. S 21 S_{21}\,
  161. S 22 S_{22}\,
  162. S 33 S_{33}\,
  163. S 35 S_{35}\,
  164. S 53 S_{53}\,
  165. S 55 S_{55}\,

Schiffler_point.html

  1. [ 1 cos B + cos C , 1 cos C + cos A , 1 cos A + cos B ] \left[\frac{1}{\cos B+\cos C},\frac{1}{\cos C+\cos A},\frac{1}{\cos A+\cos B}\right]
  2. [ b + c - a b + c , c + a - b c + a , a + b - c a + b ] \left[\frac{b+c-a}{b+c},\frac{c+a-b}{c+a},\frac{a+b-c}{a+b}\right]

Schönhage–Strassen_algorithm.html

  1. ( d k - 1 , , d 1 , d 0 ) (d_{k-1},\dots,d_{1},d_{0})
  2. i = 0 k - 1 d i ( 2 n ) i \sum_{i=0}^{k-1}d_{i}\cdot(2^{n})^{i}
  3. d i d_{i}
  4. 0 d i < 2 n 0\leq d_{i}<2^{n}
  5. N \sqrt{N}

Schrödinger–Newton_equation.html

  1. i Ψ t = - 2 2 m 2 Ψ + V Ψ + m Φ Ψ , \mathrm{i}\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla^{2}% \Psi+V\Psi+m\Phi\Psi\,,
  2. Φ \Phi
  3. 2 Φ = 4 π G m | Ψ | 2 . \nabla^{2}\Phi=4\pi Gm|\Psi|^{2}\,.
  4. i Ψ t = [ - 2 2 m 2 + V - G m 2 | Ψ ( t , 𝐲 ) | 2 | 𝐱 - 𝐲 | d 3 𝐲 ] Ψ . \mathrm{i}\hbar\frac{\partial\Psi}{\partial t}=\left[-\frac{\hbar^{2}}{2m}% \nabla^{2}+V-Gm^{2}\int\frac{|\Psi(t,\mathbf{y})|^{2}}{|\mathbf{x}-\mathbf{y}|% }\,\mathrm{d}^{3}\mathbf{y}\right]\Psi\,.
  5. m μ m , t μ - 5 t , 𝐱 μ - 3 𝐱 , ψ ( t , 𝐱 ) μ 9 / 2 ψ ( μ 5 t , μ 3 𝐱 ) m\to\mu m,\;t\to\mu^{-5}t,\;\;\mathbf{x}\to\mu^{-3}\mathbf{x},\;\;\psi(t,% \mathbf{x})\to\mu^{9/2}\psi(\mu^{5}t,\mu^{3}\mathbf{x})
  6. i Ψ ( t , 𝐱 𝟏 , , 𝐱 𝐍 ) t = ( \displaystyle\mathrm{i}\hbar\frac{\partial\Psi(t,\mathbf{x_{1}},\dots,\mathbf{% x_{N}})}{\partial t}=\Bigg(
  7. V i j V_{ij}
  8. V i j V_{ij}
  9. V i j V_{ij}
  10. i ψ c ( t , 𝐑 ) t = ( 2 2 M 2 - G d 3 𝐑 d 3 𝐲 d 3 𝐳 | ψ c ( t , 𝐑 ) | 2 ρ c ( 𝐲 ) ρ c ( 𝐳 ) | 𝐑 - 𝐑 - 𝐲 + 𝐳 | ) ψ c ( t , 𝐑 ) \mathrm{i}\hbar\frac{\partial\psi_{c}(t,\mathbf{R})}{\partial t}=\Bigg(\frac{% \hbar^{2}}{2M}\nabla^{2}-G\int\mathrm{d}^{3}\mathbf{R^{\prime}}\,\int\mathrm{d% }^{3}\mathbf{y}\,\int\mathrm{d}^{3}\mathbf{z}\,\frac{|\psi_{c}(t,\mathbf{R^{% \prime}})|^{2}\,\rho_{c}(\mathbf{y})\rho_{c}(\mathbf{z})}{|\mathbf{R}-\mathbf{% R^{\prime}}-\mathbf{y}+\mathbf{z}|}\Bigg)\psi_{c}(t,\mathbf{R})\,
  11. ψ c \psi_{c}
  12. ρ c \rho_{c}
  13. Ψ ( t = 0 , r ) = ( π σ 2 ) - 3 / 4 exp ( - r 2 2 σ 2 ) , \Psi(t=0,r)=(\pi\sigma^{2})^{-3/4}\exp\left(-\frac{r^{2}}{2\sigma^{2}}\right)\,,
  14. Ψ ( t , r ) = ( π σ 2 ) - 3 / 4 ( 1 + i t m σ 2 ) - 3 / 2 exp ( - r 2 2 σ 2 ( 1 + i t m σ 2 ) ) . \Psi(t,r)=(\pi\sigma^{2})^{-3/4}\left(1+\frac{\mathrm{i}\hbar t}{m\sigma^{2}}% \right)^{-3/2}\exp\left(-\frac{r^{2}}{2\sigma^{2}\left(1+\frac{\mathrm{i}\hbar t% }{m\sigma^{2}}\right)}\right)\,.
  15. 4 π r 2 | Ψ | 2 4\pi r^{2}|\Psi|^{2}
  16. r p = σ 1 + 2 t 2 m 2 σ 4 . r_{p}=\sigma\sqrt{1+\frac{\hbar^{2}t^{2}}{m^{2}\sigma^{4}}}\,.
  17. r ¨ p = 2 m 2 r p 3 \ddot{r}_{p}=\frac{\hbar^{2}}{m^{2}r_{p}^{3}}
  18. r ¨ = - G m r 2 , \ddot{r}=-\frac{Gm}{r^{2}}\,,
  19. r p = σ r_{p}=\sigma
  20. t = 0 t=0
  21. m 3 σ = 2 G 1.7 × 10 - 58 m kg 3 , m^{3}\sigma=\frac{\hbar^{2}}{G}\approx 1.7\times 10^{-58}\mathrm{m\,kg^{3}}\,,
  22. a 0 2 G m 3 . a_{0}\approx\frac{\hbar^{2}}{Gm^{3}}\,.
  23. a 0 ( R ) a 0 1 / 4 R 3 / 4 . a_{0}^{(R)}\approx a_{0}^{1/4}R^{3/4}\,.
  24. a 0 ( R ) R a_{0}^{(R)}\approx R

Schur's_theorem.html

  1. x + y = z . x+y=z.
  2. { 1 , , S ( c ) } \{1,\ldots,S(c)\}
  3. x + y = z . x+y=z.
  4. { a 1 , , a n } \{a_{1},\ldots,a_{n}\}
  5. gcd ( a 1 , , a n ) = 1 \gcd(a_{1},\ldots,a_{n})=1
  6. ( c 1 , , c n ) (c_{1},\ldots,c_{n})
  7. x = c 1 a 1 + + c n a n x=c_{1}a_{1}+\cdots+c_{n}a_{n}
  8. x x
  9. x n - 1 ( n - 1 ) ! a 1 a n ( 1 + o ( 1 ) ) . \frac{x^{n-1}}{(n-1)!a_{1}\cdots a_{n}}(1+o(1)).
  10. { a 1 , , a n } \{a_{1},\ldots,a_{n}\}
  11. x x
  12. { a 1 , , a n } \{a_{1},\ldots,a_{n}\}
  13. C * C^{*}
  14. C C
  15. C ( s ) C(s)
  16. κ ( s ) \kappa(s)
  17. C * ( s ) C^{*}(s)
  18. κ * ( s ) \kappa^{*}(s)
  19. d d
  20. C C
  21. d * d^{*}
  22. C * C^{*}
  23. κ * ( s ) κ ( s ) \kappa^{*}(s)\leq\kappa(s)
  24. d * d d^{*}\geq d
  25. C 2 C^{2}
  26. { p ( n ) 0 : n } \begin{Bmatrix}p(n)\neq 0:n\in\mathbb{N}\end{Bmatrix}

Schwarz_lemma.html

  1. D = { z : | z | < 1 } D=\{z:\ |z|<1\}
  2. C C
  3. f : D D f:D\mapsto D
  4. f ( 0 ) = 0 f(0)=0
  5. | f ( z ) | | z | |f(z)|\leq|z|
  6. z D z\in D
  7. | f ( 0 ) | 1 |f^{\prime}(0)|\leq 1
  8. | f ( z ) | = | z | |f(z)|=|z|
  9. z z
  10. | f ( 0 ) | = 1 |f^{\prime}(0)|=1
  11. f ( z ) = a z f(z)=az
  12. a C a\in C
  13. | a | = 1 |a|=1
  14. g ( z ) = { f ( z ) z if z 0 f ( 0 ) if z = 0 , g(z)=\begin{cases}\frac{f(z)}{z}&\mbox{if }~{}z\neq 0\\ f^{\prime}(0)&\mbox{if }~{}z=0,\end{cases}
  15. | g ( z ) | | g ( z r ) | = | f ( z r ) | | z r | 1 r . |g(z)|\leq|g(z_{r})|=\frac{|f(z_{r})|}{|z_{r}|}\leq\frac{1}{r}.
  16. r 1 r\rightarrow 1
  17. | g ( z ) | 1 |g(z)|\leq 1
  18. | f ( z 1 ) - f ( z 2 ) 1 - f ( z 1 ) ¯ f ( z 2 ) | | z 1 - z 2 1 - z 1 ¯ z 2 | \left|\frac{f(z_{1})-f(z_{2})}{1-\overline{f(z_{1})}f(z_{2})}\right|\leq\left|% \frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\right|
  19. | f ( z ) | 1 - | f ( z ) | 2 1 1 - | z | 2 . \frac{\left|f^{\prime}(z)\right|}{1-\left|f(z)\right|^{2}}\leq\frac{1}{1-\left% |z\right|^{2}}.
  20. d ( z 1 , z 2 ) = tanh - 1 | z 1 - z 2 1 - z 1 ¯ z 2 | d(z_{1},z_{2})=\tanh^{-1}\left|\frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\right|
  21. | f ( z 1 ) - f ( z 2 ) f ( z 1 ) ¯ - f ( z 2 ) | | z 1 - z 2 | | z 1 ¯ - z 2 | . \left|\frac{f(z_{1})-f(z_{2})}{\overline{f(z_{1})}-f(z_{2})}\right|\leq\frac{% \left|z_{1}-z_{2}\right|}{\left|\overline{z_{1}}-z_{2}\right|}.
  22. | f ( z ) | Im ( f ( z ) ) 1 Im ( z ) . \frac{\left|f^{\prime}(z)\right|}{\,\text{Im}(f(z))}\leq\frac{1}{\,\text{Im}(z% )}.
  23. f ( z ) = a z + b c z + d f(z)=\frac{az+b}{cz+d}
  24. z - z 0 z 0 ¯ z - 1 , | z 0 | < 1 , \frac{z-z_{0}}{\overline{z_{0}}z-1},\qquad|z_{0}|<1,
  25. M ( z ) = z 1 - z 1 - z 1 ¯ z , φ ( z ) = f ( z 1 ) - z 1 - f ( z 1 ) ¯ z . M(z)=\frac{z_{1}-z}{1-\overline{z_{1}}z},\qquad\varphi(z)=\frac{f(z_{1})-z}{1-% \overline{f(z_{1})}z}.
  26. | φ ( f ( M - 1 ( z ) ) ) | = | f ( z 1 ) - f ( M - 1 ( z ) ) 1 - f ( z 1 ) ¯ f ( M - 1 ( z ) ) | | z | . \left|\varphi\left(f(M^{-1}(z))\right)\right|=\left|\frac{f(z_{1})-f(M^{-1}(z)% )}{1-\overline{f(z_{1})}f(M^{-1}(z))}\right|\leq|z|.
  27. | f ( z 1 ) - f ( z 2 ) 1 - f ( z 1 ) ¯ f ( z 2 ) | | z 1 - z 2 1 - z 1 ¯ z 2 | . \left|\frac{f(z_{1})-f(z_{2})}{1-\overline{f(z_{1})}f(z_{2})}\right|\leq\left|% \frac{z_{1}-z_{2}}{1-\overline{z_{1}}z_{2}}\right|.

Schwarzschild_geodesics.html

  1. c 2 d τ 2 = ( 1 - r s r ) c 2 d t 2 - d r 2 1 - r s r - r 2 d θ 2 - r 2 sin 2 θ d φ 2 c^{2}{d\tau}^{2}=\left(1-\frac{r_{s}}{r}\right)c^{2}dt^{2}-\frac{dr^{2}}{1-% \frac{r_{s}}{r}}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta\,d\varphi^{2}
  2. r s = 2 G M c 2 r_{s}=\frac{2GM}{c^{2}}
  3. 3 / 8 {3}/{8}
  4. c 2 d τ 2 = ( 1 - r s r ) c 2 d t 2 - d r 2 1 - r s r - r 2 d φ 2 . c^{2}d\tau^{2}=\left(1-\frac{r_{s}}{r}\right)c^{2}dt^{2}-\frac{dr^{2}}{1-\frac% {r_{s}}{r}}-r^{2}d\varphi^{2}.
  5. ( 1 - r s r ) d t d τ = E m c 2 . \left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}=\frac{E}{mc^{2}}.
  6. h = L μ = r 2 d φ d τ , h=\frac{L}{\mu}=r^{2}\frac{d\varphi}{d\tau},
  7. μ \mu
  8. μ \mu
  9. c 2 = ( 1 - r s r ) c 2 ( d t d τ ) 2 - 1 1 - r s r ( d r d τ ) 2 - r 2 ( d φ d τ ) 2 , c^{2}=\left(1-\frac{r_{s}}{r}\right)c^{2}\left(\frac{dt}{d\tau}\right)^{2}-% \frac{1}{1-\frac{r_{s}}{r}}\left(\frac{dr}{d\tau}\right)^{2}-r^{2}\left(\frac{% d\varphi}{d\tau}\right)^{2},
  10. ( d r d τ ) 2 = E 2 m 2 c 2 - ( 1 - r s r ) ( c 2 + h 2 r 2 ) . \left(\frac{dr}{d\tau}\right)^{2}=\frac{E^{2}}{m^{2}c^{2}}-\left(1-\frac{r_{s}% }{r}\right)\left(c^{2}+\frac{h^{2}}{r^{2}}\right).
  11. τ = d r ± E 2 m 2 c 2 - ( 1 - r s r ) ( c 2 + h 2 r 2 ) . \tau=\int\frac{dr}{\pm\sqrt{\frac{E^{2}}{m^{2}c^{2}}-\left(1-\frac{r_{s}}{r}% \right)\left(c^{2}+\frac{h^{2}}{r^{2}}\right)}}.
  12. d t / d τ dt/d\tau
  13. E E
  14. t = d r ± c ( 1 - r s r ) 1 - ( 1 - r s r ) ( c 2 + h 2 r 2 ) m 2 c 2 / E 2 . t=\int\frac{dr}{\pm c\left(1-\frac{r_{s}}{r}\right)\sqrt{1-\left(1-\frac{r_{s}% }{r}\right)\left(c^{2}+\frac{h^{2}}{r^{2}}\right)m^{2}c^{2}/E^{2}}}.
  15. r - r s r-r_{s}
  16. t = constant ± r s c ( 2 3 ( r r s ) 3 / 2 + 2 r r s + ln | r / r s - 1 | r / r s + 1 ) t=\,\text{constant}\pm\frac{r_{s}}{c}\left(\frac{2}{3}\left(\frac{r}{r_{s}}% \right)^{3/2}+2\sqrt{\frac{r}{r_{s}}}+\ln\frac{|\sqrt{r/r_{s}}-1|}{\sqrt{r/r_{% s}}+1}\right)
  17. τ = constant ± 2 3 r s c ( r r s ) 3 / 2 \tau=\,\text{constant}\pm\frac{2}{3}\frac{r_{s}}{c}\left(\frac{r}{r_{s}}\right% )^{3/2}
  18. t = constant ± 1 c ( r + r s ln | r / r s - 1 | ) t=\,\text{constant}\pm\frac{1}{c}\left(r+r_{s}\ln|r/r_{s}-1|\right)
  19. τ = constant . \tau=\,\text{constant}.
  20. τ = constant ± r s c ( arcsin r r s - r r s ( 1 - r r s ) ) . \tau=\,\text{constant}\pm\frac{r_{s}}{c}\left(\arcsin\sqrt{\frac{r}{r_{s}}}-% \sqrt{\frac{r}{r_{s}}\left(1-\frac{r}{r_{s}}\right)}\right).
  21. τ = constant ± i r s c ( ln ( r r s + r r s - 1 ) + r r s ( r r s - 1 ) ) . \tau=\,\text{constant}\pm i\frac{r_{s}}{c}\left(\ln\left(\sqrt{\frac{r}{r_{s}}% }+\sqrt{\frac{r}{r_{s}}-1}\right)+\sqrt{\frac{r}{r_{s}}\left(\frac{r}{r_{s}}-1% \right)}\right).
  22. ( d r d φ ) 2 = ( d r d τ ) 2 ( d τ d φ ) 2 = ( d r d τ ) 2 ( r 2 h ) 2 , \left(\frac{dr}{d\varphi}\right)^{2}=\left(\frac{dr}{d\tau}\right)^{2}\left(% \frac{d\tau}{d\varphi}\right)^{2}=\left(\frac{dr}{d\tau}\right)^{2}\left(\frac% {r^{2}}{h}\right)^{2},
  23. ( d r d φ ) 2 = r 4 b 2 - ( 1 - r s r ) ( r 4 a 2 + r 2 ) \left(\frac{dr}{d\varphi}\right)^{2}=\frac{r^{4}}{b^{2}}-\left(1-\frac{r_{s}}{% r}\right)\left(\frac{r^{4}}{a^{2}}+r^{2}\right)
  24. a = h c , a=\frac{h}{c},
  25. b = c L E = h m c E . b=\frac{cL}{E}=\frac{hmc}{E}.
  26. φ = d r ± r 2 1 b 2 - ( 1 - r s r ) ( 1 a 2 + 1 r 2 ) . \varphi=\int\frac{dr}{\pm r^{2}\sqrt{\frac{1}{b^{2}}-\left(1-\frac{r_{s}}{r}% \right)\left(\frac{1}{a^{2}}+\frac{1}{r^{2}}\right)}}.
  27. ( d u d φ ) 2 = 1 b 2 - ( 1 - u r s ) ( 1 a 2 + u 2 ) \left(\frac{du}{d\varphi}\right)^{2}=\frac{1}{b^{2}}-\left(1-ur_{s}\right)% \left(\frac{1}{a^{2}}+u^{2}\right)
  28. ( d u d φ ) 2 = r s ( u - u 1 ) ( u - u 2 ) ( u - u 3 ) \left(\frac{du}{d\varphi}\right)^{2}=r_{s}\left(u-u_{1}\right)\left(u-u_{2}% \right)\left(u-u_{3}\right)
  29. u 1 + u 2 + u 3 = 1 r s u_{1}+u_{2}+u_{3}=\frac{1}{r_{s}}
  30. u = u 1 + ( u 2 - u 1 ) sn 2 ( 1 2 φ r s ( u 3 - u 1 ) + δ ) u=u_{1}+\left(u_{2}-u_{1}\right)\,\mathrm{sn}^{2}\left(\frac{1}{2}\varphi\sqrt% {r_{s}\left(u_{3}-u_{1}\right)}+\delta\right)
  31. k = u 2 - u 1 u 3 - u 1 k=\sqrt{\frac{u_{2}-u_{1}}{u_{3}-u_{1}}}
  32. u = u 1 + ( u 2 - u 1 ) sin 2 ( 1 2 φ + δ ) u=u_{1}+\left(u_{2}-u_{1}\right)\,\sin^{2}\left(\frac{1}{2}\varphi+\delta\right)
  33. e = u 2 - u 1 u 2 + u 1 e=\frac{u_{2}-u_{1}}{u_{2}+u_{1}}
  34. d 2 u d φ 2 = r s 2 [ ( u - u 2 ) ( u - u 3 ) + ( u - u 1 ) ( u - u 3 ) + ( u - u 1 ) ( u - u 2 ) ] \frac{d^{2}u}{d\varphi^{2}}=\frac{r_{s}}{2}\left[\left(u-u_{2}\right)\left(u-u% _{3}\right)+\left(u-u_{1}\right)\left(u-u_{3}\right)+\left(u-u_{1}\right)\left% (u-u_{2}\right)\right]
  35. r = 1 / u 2 = 1 / u 3 r=1/u_{2}=1/u_{3}
  36. K = 0 1 d y ( 1 - y 2 ) ( 1 - k 2 y 2 ) K=\int_{0}^{1}\frac{dy}{\sqrt{\left(1-y^{2}\right)\left(1-k^{2}y^{2}\right)}}
  37. Δ φ = 4 K r s ( u 3 - u 1 ) \Delta\varphi=\frac{4K}{\sqrt{r_{s}\left(u_{3}-u_{1}\right)}}
  38. k 2 = u 2 - u 1 u 3 - u 1 r s ( u 2 - u 1 ) 1 k^{2}=\frac{u_{2}-u_{1}}{u_{3}-u_{1}}\approx r_{s}\left(u_{2}-u_{1}\right)\ll 1
  39. 1 r s ( u 3 - u 1 ) = 1 1 - r s ( 2 u 1 + u 2 ) 1 + 1 2 r s ( 2 u 1 + u 2 ) \frac{1}{\sqrt{r_{s}\left(u_{3}-u_{1}\right)}}=\frac{1}{\sqrt{1-r_{s}\left(2u_% {1}+u_{2}\right)}}\approx 1+\frac{1}{2}r_{s}\left(2u_{1}+u_{2}\right)
  40. K 0 1 d y 1 - y 2 ( 1 + 1 2 k 2 y 2 ) = π 2 ( 1 + k 2 4 ) K\approx\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\left(1+\frac{1}{2}k^{2}y^{2}% \right)=\frac{\pi}{2}\left(1+\frac{k^{2}}{4}\right)
  41. δ φ = Δ φ - 2 π 3 2 π r s ( u 1 + u 2 ) \delta\varphi=\Delta\varphi-2\pi\approx\frac{3}{2}\pi r_{s}\left(u_{1}+u_{2}\right)
  42. r max = 1 u 1 = A ( 1 + e ) r_{\mathrm{max}}=\frac{1}{u_{1}}=A(1+e)
  43. r min = 1 u 2 = A ( 1 - e ) r_{\mathrm{min}}=\frac{1}{u_{2}}=A(1-e)
  44. u 1 + u 2 = 2 A ( 1 - e 2 ) u_{1}+u_{2}=\frac{2}{A\left(1-e^{2}\right)}
  45. δ φ 6 π G M c 2 A ( 1 - e 2 ) \delta\varphi\approx\frac{6\pi GM}{c^{2}A\left(1-e^{2}\right)}
  46. φ = d r r 2 1 b 2 - ( 1 - r s r ) 1 r 2 \varphi=\int\frac{dr}{r^{2}\sqrt{\frac{1}{b^{2}}-\left(1-\frac{r_{s}}{r}\right% )\frac{1}{r^{2}}}}
  47. δ φ 2 r s b = 4 G M c 2 b . \delta\varphi\approx\frac{2r_{s}}{b}=\frac{4GM}{c^{2}b}.
  48. b = r 3 r 3 / ( r 3 - r s ) b=r_{3}\sqrt{r_{3}/(r_{3}-r_{s})}
  49. ( d r d τ ) 2 = E 2 m 2 c 2 - c 2 + r s c 2 r - L 2 m μ r 2 + r s L 2 m μ r 3 \left(\frac{dr}{d\tau}\right)^{2}=\frac{E^{2}}{m^{2}c^{2}}-c^{2}+\frac{r_{s}c^% {2}}{r}-\frac{L^{2}}{m\mu r^{2}}+\frac{r_{s}L^{2}}{m\mu r^{3}}
  50. 1 2 m ( d r d τ ) 2 = [ E 2 2 m c 2 - 1 2 m c 2 ] + G M m r - L 2 2 μ r 2 + G ( M + m ) L 2 c 2 μ r 3 , \frac{1}{2}m\left(\frac{dr}{d\tau}\right)^{2}=\left[\frac{E^{2}}{2mc^{2}}-% \frac{1}{2}mc^{2}\right]+\frac{GMm}{r}-\frac{L^{2}}{2\mu r^{2}}+\frac{G(M+m)L^% {2}}{c^{2}\mu r^{3}},
  51. V ( r ) = - G M m r + L 2 2 μ r 2 - G ( M + m ) L 2 c 2 μ r 3 V(r)=-\frac{GMm}{r}+\frac{L^{2}}{2\mu r^{2}}-\frac{G(M+m)L^{2}}{c^{2}\mu r^{3}}
  52. δ φ 6 π G ( M + m ) c 2 A ( 1 - e 2 ) \delta\varphi\approx\frac{6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}
  53. V ( r ) = μ c 2 2 [ - r s r + a 2 r 2 - r s a 2 r 3 ] V(r)=\frac{\mu c^{2}}{2}\left[-\frac{r_{s}}{r}+\frac{a^{2}}{r^{2}}-\frac{r_{s}% a^{2}}{r^{3}}\right]
  54. F = - d V d r = - μ c 2 2 r 4 [ r s r 2 - 2 a 2 r + 3 r s a 2 ] = 0 F=-\frac{dV}{dr}=-\frac{\mu c^{2}}{2r^{4}}\left[r_{s}r^{2}-2a^{2}r+3r_{s}a^{2}% \right]=0
  55. r outer = a 2 r s ( 1 + 1 - 3 r s 2 a 2 ) r_{\mathrm{outer}}=\frac{a^{2}}{r_{s}}\left(1+\sqrt{1-\frac{3r_{s}^{2}}{a^{2}}% }\right)
  56. r inner = a 2 r s ( 1 - 1 - 3 r s 2 a 2 ) = 3 a 2 r outer r_{\mathrm{inner}}=\frac{a^{2}}{r_{s}}\left(1-\sqrt{1-\frac{3r_{s}^{2}}{a^{2}}% }\right)=\frac{3a^{2}}{r_{\mathrm{outer}}}
  57. r outer 2 a 2 r s r_{\mathrm{outer}}\approx\frac{2a^{2}}{r_{s}}
  58. r inner 3 2 r s r_{\mathrm{inner}}\approx\frac{3}{2}r_{s}
  59. r outer 3 = G ( M + m ) ω φ 2 r_{\mathrm{outer}}^{3}=\frac{G(M+m)}{\omega_{\varphi}^{2}}
  60. G M m r 2 = μ ω φ 2 r \frac{GMm}{r^{2}}=\mu\omega_{\varphi}^{2}r
  61. μ \mu
  62. ω φ 2 G M r outer 3 = ( r s c 2 2 r outer 3 ) = ( r s c 2 2 ) ( r s 3 8 a 6 ) = c 2 r s 4 16 a 6 \omega_{\varphi}^{2}\approx\frac{GM}{r_{\mathrm{outer}}^{3}}=\left(\frac{r_{s}% c^{2}}{2r_{\mathrm{outer}}^{3}}\right)=\left(\frac{r_{s}c^{2}}{2}\right)\left(% \frac{r_{s}^{3}}{8a^{6}}\right)=\frac{c^{2}r_{s}^{4}}{16a^{6}}
  63. r outer r inner 3 r s r_{\mathrm{outer}}\approx r_{\mathrm{inner}}\approx 3r_{s}
  64. 3 / 2 {3}/{2}
  65. 3 / 2 {3}/{2}
  66. 3 / 2 {3}/{2}
  67. ω r 2 = 1 m [ d 2 V d r 2 ] r = r outer \omega_{r}^{2}=\frac{1}{m}\left[\frac{d^{2}V}{dr^{2}}\right]_{r=r_{\mathrm{% outer}}}
  68. ω r 2 = ( c 2 r s 2 r outer 4 ) ( r outer - r inner ) = ω φ 2 1 - 3 r s 2 a 2 \omega_{r}^{2}=\left(\frac{c^{2}r_{s}}{2r_{\mathrm{outer}}^{4}}\right)\left(r_% {\mathrm{outer}}-r_{\mathrm{inner}}\right)=\omega_{\varphi}^{2}\sqrt{1-\frac{3% r_{s}^{2}}{a^{2}}}
  69. ω r = ω φ ( 1 - 3 r s 2 4 a 2 + ) \omega_{r}=\omega_{\varphi}\left(1-\frac{3r_{s}^{2}}{4a^{2}}+\cdots\right)
  70. δ φ = T ( ω φ - ω r ) 2 π ( 3 r s 2 4 a 2 ) = 3 π m 2 c 2 2 L 2 r s 2 \delta\varphi=T\left(\omega_{\varphi}-\omega_{r}\right)\approx 2\pi\left(\frac% {3r_{s}^{2}}{4a^{2}}\right)=\frac{3\pi m^{2}c^{2}}{2L^{2}}r_{s}^{2}
  71. δ φ 3 π m 2 c 2 2 L 2 ( 4 G 2 M 2 c 4 ) = 6 π G 2 M 2 m 2 c 2 L 2 \delta\varphi\approx\frac{3\pi m^{2}c^{2}}{2L^{2}}\left(\frac{4G^{2}M^{2}}{c^{% 4}}\right)=\frac{6\pi G^{2}M^{2}m^{2}}{c^{2}L^{2}}
  72. h 2 G ( M + m ) = A ( 1 - e 2 ) \frac{h^{2}}{G(M+m)}=A\left(1-e^{2}\right)
  73. δ φ 6 π G ( M + m ) c 2 A ( 1 - e 2 ) \delta\varphi\approx\frac{6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}
  74. d 2 x λ d q 2 + Γ μ ν λ d x μ d q d x ν d q = 0 \frac{d^{2}x^{\lambda}}{dq^{2}}+\Gamma^{\lambda}_{\mu\nu}\frac{dx^{\mu}}{dq}% \frac{dx^{\nu}}{dq}=0
  75. c 2 d τ 2 = w ( r ) c 2 d t 2 - v ( r ) d r 2 - r 2 d θ 2 - r 2 sin 2 θ d ϕ 2 c^{2}d\tau^{2}=w(r)c^{2}dt^{2}-v(r)dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta d% \phi^{2}\,
  76. 0 = d 2 θ d q 2 + 2 r d θ d q d r d q - sin θ cos θ ( d ϕ d q ) 2 0=\frac{d^{2}\theta}{dq^{2}}+\frac{2}{r}\frac{d\theta}{dq}\frac{dr}{dq}-\sin% \theta\cos\theta\left(\frac{d\phi}{dq}\right)^{2}
  77. 0 = d 2 ϕ d q 2 + 2 r d ϕ d q d r d q + 2 cot θ d ϕ d q d θ d q 0=\frac{d^{2}\phi}{dq^{2}}+\frac{2}{r}\frac{d\phi}{dq}\frac{dr}{dq}+2\cot% \theta\frac{d\phi}{dq}\frac{d\theta}{dq}
  78. 0 = d 2 t d q 2 + 1 w d w d r d t d q d r d q 0=\frac{d^{2}t}{dq^{2}}+\frac{1}{w}\frac{dw}{dr}\frac{dt}{dq}\frac{dr}{dq}
  79. 0 = d 2 r d q 2 + 1 2 v d v d r ( d r d q ) 2 - r v ( d θ d q ) 2 - r sin 2 θ v ( d ϕ d q ) 2 + c 2 2 v d w d r ( d t d q ) 2 0=\frac{d^{2}r}{dq^{2}}+\frac{1}{2v}\frac{dv}{dr}\left(\frac{dr}{dq}\right)^{2% }-\frac{r}{v}\left(\frac{d\theta}{dq}\right)^{2}-\frac{r\sin^{2}\theta}{v}% \left(\frac{d\phi}{dq}\right)^{2}+\frac{c^{2}}{2v}\frac{dw}{dr}\left(\frac{dt}% {dq}\right)^{2}
  80. 0 = d d q [ ln d ϕ d q + ln r 2 ] 0=\frac{d}{dq}\left[\ln\frac{d\phi}{dq}+\ln r^{2}\right]
  81. 0 = d d q [ ln d t d q + ln w ] 0=\frac{d}{dq}\left[\ln\frac{dt}{dq}+\ln w\right]
  82. 0 = δ s = δ d s = δ g μ ν d x μ d τ d x ν d τ d τ = δ 2 T d τ 0=\delta s=\delta\int ds=\delta\int\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac% {dx^{\nu}}{d\tau}}d\tau=\delta\int\sqrt{2T}d\tau
  83. 2 T = c 2 = ( d s d τ ) 2 = g μ ν d x μ d τ d x ν d τ = ( 1 - r s r ) c 2 ( d t d τ ) 2 - 1 1 - r s r ( d r d τ ) 2 - r 2 ( d φ d τ ) 2 2T=c^{2}=\left(\frac{ds}{d\tau}\right)^{2}=g_{\mu\nu}\frac{dx^{\mu}}{d\tau}% \frac{dx^{\nu}}{d\tau}=\left(1-\frac{r_{s}}{r}\right)c^{2}\left(\frac{dt}{d% \tau}\right)^{2}-\frac{1}{1-\frac{r_{s}}{r}}\left(\frac{dr}{d\tau}\right)^{2}-% r^{2}\left(\frac{d\varphi}{d\tau}\right)^{2}
  84. x ˙ μ = d x μ d τ \dot{x}^{\mu}=\frac{dx^{\mu}}{d\tau}
  85. 2 T = c 2 = ( 1 - r s r ) c 2 ( t ˙ ) 2 - 1 1 - r s r ( r ˙ ) 2 - r 2 ( φ ˙ ) 2 2T=c^{2}=\left(1-\frac{r_{s}}{r}\right)c^{2}\left(\dot{t}\right)^{2}-\frac{1}{% 1-\frac{r_{s}}{r}}\left(\dot{r}\right)^{2}-r^{2}\left(\dot{\varphi}\right)^{2}
  86. 0 = δ 2 T d τ = δ T 2 T d τ = 1 c δ T d τ . 0=\delta\int\sqrt{2T}d\tau=\int\frac{\delta T}{\sqrt{2T}}d\tau=\frac{1}{c}% \delta\int Td\tau.
  87. d d τ ( T x ˙ σ ) = T x σ . \frac{d}{d\tau}\left(\frac{\partial T}{\partial\dot{x}^{\sigma}}\right)=\frac{% \partial T}{\partial x^{\sigma}}.
  88. d d τ [ r 2 d φ d τ ] = 0 , \frac{d}{d\tau}\left[r^{2}\frac{d\varphi}{d\tau}\right]=0,
  89. d d τ [ ( 1 - r s r ) d t d τ ] = 0 , \frac{d}{d\tau}\left[\left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}\right]=0,
  90. r 2 d φ d τ = a c , r^{2}\frac{d\varphi}{d\tau}=ac,
  91. ( 1 - r s r ) d t d τ = a b . \left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}=\frac{a}{b}.
  92. 2 H = c 2 = p t 2 c 2 ( 1 - r s r ) - ( 1 - r s r ) p r 2 - p θ 2 r 2 - p φ 2 r 2 sin 2 θ 2H=c^{2}=\frac{p_{t}^{2}}{c^{2}\left(1-\frac{r_{s}}{r}\right)}-\left(1-\frac{r% _{s}}{r}\right)p_{r}^{2}-\frac{p_{\theta}^{2}}{r^{2}}-\frac{p_{\varphi}^{2}}{r% ^{2}\sin^{2}\theta}
  93. p φ = - a c p_{\varphi}=-ac
  94. p θ = 0 p_{\theta}=0
  95. p t = a c 2 b p_{t}=\frac{ac^{2}}{b}
  96. d r d τ = H p r = - ( 1 - r s r ) p r \frac{dr}{d\tau}=\frac{\partial H}{\partial p_{r}}=-\left(1-\frac{r_{s}}{r}% \right)p_{r}
  97. d φ d τ = H p φ = - p φ r 2 = a c r 2 \frac{d\varphi}{d\tau}=\frac{\partial H}{\partial p_{\varphi}}=\frac{-p_{% \varphi}}{r^{2}}=\frac{ac}{r^{2}}
  98. d t d τ = H p t = p t c 2 ( 1 - r s r ) = a b ( 1 - r s r ) \frac{dt}{d\tau}=\frac{\partial H}{\partial p_{t}}=\frac{p_{t}}{c^{2}\left(1-% \frac{r_{s}}{r}\right)}=\frac{a}{b\left(1-\frac{r_{s}}{r}\right)}
  99. d r d φ = - r 2 a c ( 1 - r s r ) p r \frac{dr}{d\varphi}=-\frac{r^{2}}{ac}\left(1-\frac{r_{s}}{r}\right)p_{r}
  100. ( d r d φ ) 2 = r 4 b 2 - ( 1 - r s r ) ( r 4 a 2 + r 2 ) \left(\frac{dr}{d\varphi}\right)^{2}=\frac{r^{4}}{b^{2}}-\left(1-\frac{r_{s}}{% r}\right)\left(\frac{r^{4}}{a^{2}}+r^{2}\right)
  101. g μ ν S x μ S x ν = c 2 . g^{\mu\nu}\frac{\partial S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\nu% }}=c^{2}.
  102. 1 c 2 ( 1 - r s r ) ( S t ) 2 - ( 1 - r s r ) ( S r ) 2 - 1 r 2 ( S φ ) 2 = c 2 \frac{1}{c^{2}\left(1-\frac{r_{s}}{r}\right)}\left(\frac{\partial S}{\partial t% }\right)^{2}-\left(1-\frac{r_{s}}{r}\right)\left(\frac{\partial S}{\partial r}% \right)^{2}-\frac{1}{r^{2}}\left(\frac{\partial S}{\partial\varphi}\right)^{2}% =c^{2}
  103. S = - p t t + p φ φ + S r ( r ) S=-p_{t}t+p_{\varphi}\varphi+S_{r}(r)\,
  104. S r ( r ) = r d r 1 - r s r p t 2 c 2 - ( 1 - r s r ) ( c 2 + p φ 2 r 2 ) . S_{r}(r)=\int^{r}\frac{dr}{1-\frac{r_{s}}{r}}\sqrt{\frac{p_{t}^{2}}{c^{2}}-% \left(1-\frac{r_{s}}{r}\right)\left(c^{2}+\frac{p_{\varphi}^{2}}{r^{2}}\right)}.
  105. S p φ = φ + S r p φ = constant \frac{\partial S}{\partial p_{\varphi}}=\varphi+\frac{\partial S_{r}}{\partial p% _{\varphi}}=\mathrm{constant}
  106. φ - r p φ d r r 2 p t 2 c 2 - ( 1 - r s r ) ( c 2 + p φ 2 r 2 ) = constant \varphi-\int^{r}\frac{p_{\varphi}dr}{r^{2}\sqrt{\frac{p_{t}^{2}}{c^{2}}-\left(% 1-\frac{r_{s}}{r}\right)\left(c^{2}+\frac{p_{\varphi}^{2}}{r^{2}}\right)}}=% \mathrm{constant}
  107. ( d r d φ ) 2 = r 4 b 2 - ( 1 - r s r ) ( r 4 a 2 + r 2 ) . \left(\frac{dr}{d\varphi}\right)^{2}=\frac{r^{4}}{b^{2}}-\left(1-\frac{r_{s}}{% r}\right)\left(\frac{r^{4}}{a^{2}}+r^{2}\right).
  108. S φ = p φ = - a c \frac{\partial S}{\partial\varphi}=p_{\varphi}=-ac
  109. S t = p t = a c 2 b \frac{\partial S}{\partial t}=p_{t}=\frac{ac^{2}}{b}
  110. S = - m c 2 d τ = - m c c d τ d q d q = - m c - g μ ν d x μ d q d x ν d q d q S=\int{-mc^{2}d\tau}=-mc\int{c\frac{d\tau}{dq}dq}=-mc\int{\sqrt{-g_{\mu\nu}% \frac{dx^{\mu}}{dq}\frac{dx^{\nu}}{dq}}dq}
  111. ( c d τ d q ) 2 = - g μ ν d x μ d q d x ν d q = ( 1 - r s r ) c 2 ( d t d q ) 2 - 1 1 - r s r ( d r d q ) 2 - r 2 ( d φ d q ) 2 . \left(c\frac{d\tau}{dq}\right)^{2}=-g_{\mu\nu}\frac{dx^{\mu}}{dq}\frac{dx^{\nu% }}{dq}=\left(1-\frac{r_{s}}{r}\right)c^{2}\left(\frac{dt}{dq}\right)^{2}-\frac% {1}{1-\frac{r_{s}}{r}}\left(\frac{dr}{dq}\right)^{2}-r^{2}\left(\frac{d\varphi% }{dq}\right)^{2}\,.
  112. δ ( c d τ d q ) 2 = 2 c 2 d τ d q δ d τ d q = δ [ ( 1 - r s r ) c 2 ( d t d q ) 2 - 1 1 - r s r ( d r d q ) 2 - r 2 ( d φ d q ) 2 ] . \delta\left(c\frac{d\tau}{dq}\right)^{2}=2c^{2}\frac{d\tau}{dq}\delta\frac{d% \tau}{dq}=\delta\left[\left(1-\frac{r_{s}}{r}\right)c^{2}\left(\frac{dt}{dq}% \right)^{2}-\frac{1}{1-\frac{r_{s}}{r}}\left(\frac{dr}{dq}\right)^{2}-r^{2}% \left(\frac{d\varphi}{dq}\right)^{2}\right]\,.
  113. 2 c 2 d τ d q δ d τ d q = - 2 r 2 d φ d q δ d φ d q . 2c^{2}\frac{d\tau}{dq}\delta\frac{d\tau}{dq}=-2r^{2}\frac{d\varphi}{dq}\delta% \frac{d\varphi}{dq}\,.
  114. 2 c d τ d q 2c\frac{d\tau}{dq}
  115. c δ d τ d q = - r 2 c d φ d τ δ d φ d q = - r 2 c d φ d τ d δ φ d q . c\,\delta\frac{d\tau}{dq}=-\frac{r^{2}}{c}\frac{d\varphi}{d\tau}\delta\frac{d% \varphi}{dq}=-\frac{r^{2}}{c}\frac{d\varphi}{d\tau}\frac{d\delta\varphi}{dq}\,.
  116. 0 = δ c d τ d q d q = c δ d τ d q d q = - r 2 c d φ d τ d δ φ d q d q . 0=\delta\int{c\frac{d\tau}{dq}dq}=\int{c\delta\frac{d\tau}{dq}dq}=\int{-\frac{% r^{2}}{c}\frac{d\varphi}{d\tau}\frac{d\delta\varphi}{dq}dq}\,.
  117. 0 = - r 2 c d φ d τ δ φ - d d q [ - r 2 c d φ d τ ] δ φ d q . 0=-\frac{r^{2}}{c}\frac{d\varphi}{d\tau}\delta\varphi-\int{\frac{d}{dq}\left[-% \frac{r^{2}}{c}\frac{d\varphi}{d\tau}\right]\delta\varphi dq}\,.
  118. d d q [ - r 2 c d φ d τ ] = 0 . \frac{d}{dq}\left[-\frac{r^{2}}{c}\frac{d\varphi}{d\tau}\right]=0\,.
  119. 2 c 2 d τ d q δ d τ d q = 2 ( 1 - r s r ) c 2 d t d q δ d t d q . 2c^{2}\frac{d\tau}{dq}\delta\frac{d\tau}{dq}=2\left(1-\frac{r_{s}}{r}\right)c^% {2}\frac{dt}{dq}\delta\frac{dt}{dq}\,.
  120. 2 c d τ d q 2c\frac{d\tau}{dq}
  121. c δ d τ d q = c ( 1 - r s r ) d t d τ δ d t d q = c ( 1 - r s r ) d t d τ d δ t d q . c\delta\frac{d\tau}{dq}=c\left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}\delta% \frac{dt}{dq}=c\left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}\frac{d\delta t}{% dq}\,.
  122. 0 = δ c d τ d q d q = c ( 1 - r s r ) d t d τ d δ t d q d q . 0=\delta\int{c\frac{d\tau}{dq}dq}=\int{c\left(1-\frac{r_{s}}{r}\right)\frac{dt% }{d\tau}\frac{d\delta t}{dq}dq}\,.
  123. 0 = c ( 1 - r s r ) d t d τ δ t - d d q [ c ( 1 - r s r ) d t d τ ] δ t d q . 0=c\left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}\delta t-\int{\frac{d}{dq}% \left[c\left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}\right]\delta tdq}\,.
  124. d d q [ c ( 1 - r s r ) d t d τ ] = 0 . \frac{d}{dq}\left[c\left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}\right]=0\,.
  125. L = p ϕ = m r 2 d φ d τ , L=p_{\phi}=m\,r^{2}\frac{d\varphi}{d\tau}\,,
  126. E = - p t = m c 2 ( 1 - r s r ) d t d τ . E=-p_{t}=m\,c^{2}\left(1-\frac{r_{s}}{r}\right)\frac{dt}{d\tau}\,.
  127. d φ d t = ( 1 - r s r ) L c 2 E r 2 . \frac{d\varphi}{dt}=\left(1-\frac{r_{s}}{r}\right)\frac{L\,c^{2}}{E\,r^{2}}\,.
  128. d φ d τ = L m r 2 \frac{d\varphi}{d\tau}=\frac{L}{m\,r^{2}}\,
  129. d t d τ = E ( 1 - r s r ) m c 2 \frac{dt}{d\tau}=\frac{E}{\left(1-\frac{r_{s}}{r}\right)m\,c^{2}}\,
  130. c 2 = 1 1 - r s r E 2 m 2 c 2 - 1 1 - r s r ( d r d τ ) 2 - 1 r 2 L 2 m 2 , c^{2}=\frac{1}{1-\frac{r_{s}}{r}}\,\frac{E^{2}}{m^{2}c^{2}}-\frac{1}{1-\frac{r% _{s}}{r}}\left(\frac{dr}{d\tau}\right)^{2}-\frac{1}{r^{2}}\,\frac{L^{2}}{m^{2}% }\,,
  131. ( d r d τ ) 2 = E 2 m 2 c 2 - ( 1 - r s r ) ( c 2 + L 2 m 2 r 2 ) , {\left(\frac{dr}{d\tau}\right)}^{2}=\frac{E^{2}}{m^{2}c^{2}}-\left(1-\frac{r_{% s}}{r}\right)\left(c^{2}+\frac{L^{2}}{m^{2}r^{2}}\right)\,,
  132. ( d φ d τ ) 2 = L 2 m 2 r 4 {\left(\frac{d\varphi}{d\tau}\right)}^{2}=\frac{L^{2}}{m^{2}r^{4}}
  133. ( d r d φ ) 2 = E 2 r 4 L 2 c 2 - ( 1 - r s r ) ( m 2 c 2 r 4 L 2 + r 2 ) {\left(\frac{dr}{d\varphi}\right)}^{2}=\frac{E^{2}r^{4}}{L^{2}c^{2}}-\left(1-% \frac{r_{s}}{r}\right)\left(\frac{m^{2}c^{2}r^{4}}{L^{2}}+r^{2}\right)\,
  134. a = L m c a=\frac{L}{m\,c}
  135. b = L c E , b=\frac{L\,c}{E}\,,
  136. ( d r d φ ) 2 = r 4 b 2 - ( 1 - r s r ) ( r 4 a 2 + r 2 ) . {\left(\frac{dr}{d\varphi}\right)}^{2}=\frac{r^{4}}{b^{2}}-\left(1-\frac{r_{s}% }{r}\right)\left(\frac{r^{4}}{a^{2}}+r^{2}\right)\,.

Schwarz–Ahlfors–Pick_theorem.html

  1. ρ \rho
  2. σ \sigma
  3. f : U S f:U\rightarrow S
  4. σ ( f ( z 1 ) , f ( z 2 ) ) ρ ( z 1 , z 2 ) \sigma(f(z_{1}),f(z_{2}))\leq\rho(z_{1},z_{2})
  5. z 1 , z 2 U . z_{1},z_{2}\in U.

Schwarz–Christoffel_mapping.html

  1. { ζ : Im ζ > 0 } \{\zeta\in\mathbb{C}:\operatorname{Im}\,\zeta>0\}
  2. α , β , γ , \alpha,\beta,\gamma,\ldots
  3. f ( ζ ) = ζ K ( w - a ) 1 - ( α / π ) ( w - b ) 1 - ( β / π ) ( w - c ) 1 - ( γ / π ) d w f(\zeta)=\int^{\zeta}\frac{K}{(w-a)^{1-(\alpha/\pi)}(w-b)^{1-(\beta/\pi)}(w-c)% ^{1-(\gamma/\pi)}\cdots}\,\mbox{d}~{}w
  4. K K
  5. a < b < c < a<b<c<...
  6. ζ \zeta
  7. z z
  8. ζ \zeta
  9. z z
  10. α \alpha
  11. K K
  12. α = 0 α=0
  13. β = γ = π / 2 β=γ={π}/{2}
  14. f ( ζ ) = ζ K ( w - 1 ) 1 / 2 ( w + 1 ) 1 / 2 d w . f(\zeta)=\int^{\zeta}\frac{K}{(w-1)^{1/2}(w+1)^{1/2}}\,\mbox{d}~{}w.\,
  15. z = f ( ζ ) = C + K a r c c o s h ζ z=f(ζ)=C+Karccoshζ
  16. z = a r c c o s h ζ z=arccoshζ
  17. π a , π b \pi a,\,\pi b
  18. π ( 1 - a - b ) \pi(1-a-b)
  19. z = f ( ζ ) = ζ d w ( w - 1 ) 1 - a ( w + 1 ) 1 - b . z=f(\zeta)=\int^{\zeta}\frac{dw}{(w-1)^{1-a}(w+1)^{1-b}}.
  20. z = f ( ζ ) = ζ d w w ( w 2 - 1 ) = 2 F ( ζ + 1 ; 2 / 2 ) , z=f(\zeta)=\int^{\zeta}\frac{\mbox{d}~{}w}{\sqrt{w(w^{2}-1)}}=\sqrt{2}\,F\left% (\sqrt{\zeta+1};\sqrt{2}/2\right),

Schwinger_function.html

  1. x ¯ \scriptstyle\bar{x}
  2. m , n d d x 1 d d x m d d y 1 d d y n S m + n ( x 1 , , x m , y 1 , , y n ) f m ( x ¯ 1 , , x ¯ m ) * f n ( y 1 , , y n ) 0 \sum_{m,n}\int d^{d}x_{1}\cdots d^{d}x_{m}\,d^{d}y_{1}\cdots d^{d}y_{n}S_{m+n}% (x_{1},\dots,x_{m},y_{1},\dots,y_{n})f_{m}(\bar{x}_{1},\dots,\bar{x}_{m})^{*}f% _{n}(y_{1},\dots,y_{n})\geq 0
  3. 𝒟 ϕ F [ ϕ ( x ) ] F [ ϕ ( x ¯ ) ] * e - S [ ϕ ] = 𝒟 ϕ 0 ϕ + ( τ = 0 ) = ϕ 0 𝒟 ϕ + F [ ϕ + ] e - S + [ ϕ + ] ϕ - ( τ = 0 ) = ϕ 0 𝒟 ϕ - F [ ϕ ¯ - ] * e - S - [ ϕ - ] . \int\mathcal{D}\phi F[\phi(x)]F[\phi(\bar{x})]^{*}e^{-S[\phi]}=\int\mathcal{D}% \phi_{0}\int_{\phi_{+}(\tau=0)=\phi_{0}}\mathcal{D}\phi_{+}F[\phi_{+}]e^{-S_{+% }[\phi_{+}]}\int_{\phi_{-}(\tau=0)=\phi_{0}}\mathcal{D}\phi_{-}F[\bar{\phi}_{-% }]^{*}e^{-S_{-}[\phi_{-}]}.

Score_test.html

  1. θ \theta
  2. θ 0 \theta_{0}
  3. θ \theta
  4. θ 0 \theta_{0}
  5. L L
  6. θ \theta
  7. x x
  8. U ( θ ) U(\theta)
  9. U ( θ ) = log L ( θ | x ) θ . U(\theta)=\frac{\partial\log L(\theta|x)}{\partial\theta}.
  10. ( θ ) = - E [ 2 θ 2 log L ( X ; θ ) | θ ] . \mathcal{I}(\theta)=-\operatorname{E}\left[\left.\frac{\partial^{2}}{\partial% \theta^{2}}\log L(X;\theta)\right|\theta\right]\,.
  11. 0 : θ = θ 0 \mathcal{H}_{0}:\theta=\theta_{0}
  12. S ( θ 0 ) = U ( θ 0 ) 2 I ( θ 0 ) S(\theta_{0})=\frac{U(\theta_{0})^{2}}{I(\theta_{0})}
  13. χ 1 2 \chi^{2}_{1}
  14. 0 \mathcal{H}_{0}
  15. S * ( θ ) = S ( θ ) S^{*}(\theta)=\sqrt{S(\theta)}
  16. ( log L ( θ | x ) θ ) θ = θ 0 C \left(\frac{\partial\log L(\theta|x)}{\partial\theta}\right)_{\theta=\theta_{0% }}\geq C
  17. L L
  18. θ 0 \theta_{0}
  19. C C
  20. H 0 H_{0}
  21. H 0 H_{0}
  22. H 0 H_{0}
  23. θ = θ 0 \theta=\theta_{0}
  24. θ = θ 0 + h \theta=\theta_{0}+h
  25. L ( θ 0 + h | x ) L ( θ 0 | x ) K ; \frac{L(\theta_{0}+h|x)}{L(\theta_{0}|x)}\geq K;
  26. log L ( θ 0 + h | x ) - log L ( θ 0 | x ) log K . \log L(\theta_{0}+h|x)-\log L(\theta_{0}|x)\geq\log K.
  27. log L ( θ 0 + h | x ) log L ( θ 0 | x ) + h × ( log L ( θ | x ) θ ) θ = θ 0 \log L(\theta_{0}+h|x)\approx\log L(\theta_{0}|x)+h\times\left(\frac{\partial% \log L(\theta|x)}{\partial\theta}\right)_{\theta=\theta_{0}}
  28. C C
  29. log ( K ) \log(K)
  30. θ ^ 0 \hat{\theta}_{0}
  31. θ \theta
  32. H 0 H_{0}
  33. U T ( θ ^ 0 ) I - 1 ( θ ^ 0 ) U ( θ ^ 0 ) χ k 2 U^{T}(\hat{\theta}_{0})I^{-1}(\hat{\theta}_{0})U(\hat{\theta}_{0})\sim\chi^{2}% _{k}
  34. H 0 H_{0}
  35. k k
  36. U ( θ ^ 0 ) = log L ( θ ^ 0 | x ) θ U(\hat{\theta}_{0})=\frac{\partial\log L(\hat{\theta}_{0}|x)}{\partial\theta}
  37. I ( θ ^ 0 ) = - E ( 2 log L ( θ ^ 0 | x ) θ θ ) . I(\hat{\theta}_{0})=-E\left(\frac{\partial^{2}\log L(\hat{\theta}_{0}|x)}{% \partial\theta\partial\theta^{\prime}}\right).
  38. H 0 H_{0}

Screw_theory.html

  1. 𝖲 = ( 𝐒 , 𝐕 ) , \mathsf{S}=(\mathbf{S},\mathbf{V}),
  2. a ^ c ^ = ( a , b ) ( c , d ) = ( a c , a d + b c ) . \hat{a}\hat{c}=(a,b)(c,d)=(ac,ad+bc).\!
  3. a ^ 𝖲 = ( a , b ) ( 𝐒 , 𝐕 ) = ( a 𝐒 , a 𝐕 + b 𝐒 ) . \hat{a}\mathsf{S}=(a,b)(\mathbf{S},\mathbf{V})=(a\mathbf{S},a\mathbf{V}+b% \mathbf{S}).\!
  4. 𝖲 𝖳 = ( 𝐒 , 𝐕 ) ( 𝐓 , 𝐖 ) = ( 𝐒 𝐓 , 𝐒 𝐖 + 𝐕 𝐓 ) , \mathsf{S}\cdot\mathsf{T}=(\mathbf{S},\mathbf{V})\cdot(\mathbf{T},\mathbf{W})=% (\mathbf{S}\cdot\mathbf{T},\,\,\mathbf{S}\cdot\mathbf{W}+\mathbf{V}\cdot% \mathbf{T}),
  5. 𝖲 × 𝖳 = ( 𝐒 , 𝐕 ) × ( 𝐓 , 𝐖 ) = ( 𝐒 × 𝐓 , 𝐒 × 𝐖 + 𝐕 × 𝐓 ) . \mathsf{S}\times\mathsf{T}=(\mathbf{S},\mathbf{V})\times(\mathbf{T},\mathbf{W}% )=(\mathbf{S}\times\mathbf{T},\,\,\mathbf{S}\times\mathbf{W}+\mathbf{V}\times% \mathbf{T}).
  6. sin z ^ = sin ϕ + d cos ϕ , cos z ^ = cos ϕ - d sin ϕ . \sin\hat{z}=\sin\phi+d\cos\phi,\,\,\,\cos\hat{z}=\cos\phi-d\sin\phi.\!
  7. | 𝖲 | = 𝖲 𝖲 = 1 ; |\mathsf{S}|=\sqrt{\mathsf{S}\cdot\mathsf{S}}=1;
  8. 𝖲 𝖳 = | 𝖲 | | 𝖳 | cos z ^ ; \mathsf{S}\cdot\mathsf{T}=|\mathsf{S}||\mathsf{T}|\cos\hat{z};
  9. 𝖲 × 𝖳 = | 𝖲 | | 𝖳 | sin z ^ 𝖭 . \mathsf{S}\times\mathsf{T}=|\mathsf{S}||\mathsf{T}|\sin\hat{z}\mathsf{N}.
  10. 𝖱 = i = 1 n 𝖶 i = i = 1 n ( 𝐅 i , 𝐏 i × 𝐅 i ) . \mathsf{R}=\sum_{i=1}^{n}\mathsf{W}_{i}=\sum_{i=1}^{n}(\mathbf{F}_{i},\mathbf{% P}_{i}\times\mathbf{F}_{i}).
  11. 𝖱 = ( 𝐅 - 𝐅 , 𝐀 × 𝐅 - 𝐁 × 𝐅 ) = ( 0 , ( 𝐀 - 𝐁 ) × 𝐅 ) . \mathsf{R}=(\mathbf{F}-\mathbf{F},\mathbf{A}\times\mathbf{F}-\mathbf{B}\times% \mathbf{F})=(0,(\mathbf{A}-\mathbf{B})\times\mathbf{F}).
  12. 𝖬 = ( 0 , 𝐌 ) , \mathsf{M}=(0,\mathbf{M}),
  13. 𝐏 ( t ) = [ A ( t ) ] 𝐩 + 𝐝 ( t ) . \mathbf{P}(t)=[A(t)]\mathbf{p}+\mathbf{d}(t).
  14. 𝐕 P ( t ) = [ d A ( t ) d t ] 𝐩 + 𝐯 ( t ) , \mathbf{V}_{P}(t)=\left[\frac{dA(t)}{dt}\right]\mathbf{p}+\mathbf{v}(t),
  15. 𝐕 P ( t ) = [ Ω ] 𝐏 + 𝐯 - [ Ω ] 𝐝 or 𝐕 P ( t ) = ω × 𝐏 + 𝐯 + 𝐝 × ω , \mathbf{V}_{P}(t)=[\Omega]\mathbf{P}+\mathbf{v}-[\Omega]\mathbf{d}\quad\mbox{% or}~{}\quad\mathbf{V}_{P}(t)=\mathbf{\omega}\times\mathbf{P}+\mathbf{v}+% \mathbf{d}\times\mathbf{\omega},
  16. 𝖳 = ( ω , 𝐯 + 𝐝 × ω ) , \mathsf{T}=(\vec{\omega},\mathbf{v}+\mathbf{d}\times\vec{\omega}),\!
  17. 𝖫 = ( ω , 𝐝 × ω ) , \mathsf{L}=(\omega,\mathbf{d}\times\omega),
  18. 𝖳 = ( 0 , 𝐯 ) . \mathsf{T}=(0,\mathbf{v}).
  19. ξ = { ω q × ω } . \xi=\begin{Bmatrix}\omega\\ q\times\omega\end{Bmatrix}.
  20. ξ = { 0 v } . \xi=\begin{Bmatrix}0\\ v\end{Bmatrix}.
  21. 𝗊 = ( 𝐪 - 𝐩 , 𝐩 × 𝐪 ) , \mathsf{q}=(\mathbf{q}-\mathbf{p},\mathbf{p}\times\mathbf{q}),
  22. 𝖰 = ( 𝐐 - 𝐏 , 𝐏 × 𝐐 ) = ( [ A ] ( 𝐪 - 𝐩 ) , [ A ] ( 𝐩 × 𝐪 ) + 𝐝 × [ A ] ( 𝐪 - 𝐩 ) ) \mathsf{Q}=(\mathbf{Q}-\mathbf{P},\mathbf{P}\times\mathbf{Q})=([A](\mathbf{q}-% \mathbf{p}),[A](\mathbf{p}\times\mathbf{q})+\mathbf{d}\times[A](\mathbf{q}-% \mathbf{p}))
  23. { 𝐐 - 𝐏 𝐏 × 𝐐 } = [ A 0 D A A ] { 𝐪 - 𝐩 𝐩 × 𝐪 } . \begin{Bmatrix}\mathbf{Q}-\mathbf{P}\\ \mathbf{P}\times\mathbf{Q}\end{Bmatrix}=\begin{bmatrix}A&0\\ DA&A\end{bmatrix}\begin{Bmatrix}\mathbf{q}-\mathbf{p}\\ \mathbf{p}\times\mathbf{q}\end{Bmatrix}.
  24. [ A ^ ] = ( [ A ] , [ D A ] ) , [\hat{A}]=([A],[DA]),
  25. 𝖲 = [ A ^ ] 𝗌 , ( 𝐒 , 𝐕 ) = ( [ A ] , [ D A ] ) ( 𝐬 , 𝐯 ) = ( [ A ] 𝐬 , [ A ] 𝐯 + [ D A ] 𝐬 ) . \mathsf{S}=[\hat{A}]\mathsf{s},\quad(\mathbf{S},\mathbf{V})=([A],[DA])(\mathbf% {s},\mathbf{v})=([A]\mathbf{s},[A]\mathbf{v}+[DA]\mathbf{s}).
  26. 𝐏 ( t ) = [ T ( t ) ] 𝐩 = { 𝐏 1 } = [ A ( t ) 𝐝 ( t ) 0 1 ] { 𝐩 1 } . \,\textbf{P}(t)=[T(t)]\,\textbf{p}=\begin{Bmatrix}\,\textbf{P}\\ 1\end{Bmatrix}=\begin{bmatrix}A(t)&\,\textbf{d}(t)\\ 0&1\end{bmatrix}\begin{Bmatrix}\,\textbf{p}\\ 1\end{Bmatrix}.
  27. 𝐕 P = [ T ˙ ( t ) ] 𝐩 = { 𝐕 P 0 } = [ A ˙ ( t ) 𝐝 ˙ ( t ) 0 0 ] { 𝐩 1 } . \,\textbf{V}_{P}=[\dot{T}(t)]\,\textbf{p}=\begin{Bmatrix}\,\textbf{V}_{P}\\ 0\end{Bmatrix}=\begin{bmatrix}\dot{A}(t)&\dot{\,\textbf{d}}(t)\\ 0&0\end{bmatrix}\begin{Bmatrix}\,\textbf{p}\\ 1\end{Bmatrix}.
  28. 𝐕 P = [ T ˙ ( t ) ] [ T ( t ) ] - 1 𝐏 ( t ) = [ S ] 𝐏 , \,\textbf{V}_{P}=[\dot{T}(t)][T(t)]^{-1}\,\textbf{P}(t)=[S]\,\textbf{P},
  29. [ S ] = [ Ω - Ω 𝐝 + 𝐝 ˙ 0 0 ] = [ Ω 𝐝 × ω + 𝐯 0 0 ] . [S]=\begin{bmatrix}\Omega&-\Omega\,\textbf{d}+\dot{\,\textbf{d}}\\ 0&0\end{bmatrix}=\begin{bmatrix}\Omega&\mathbf{d}\times\omega+\mathbf{v}\\ 0&0\end{bmatrix}.
  30. [ T ˙ ( t ) ] = [ S ] [ T ( t ) ] , [\dot{T}(t)]=[S][T(t)],
  31. [ T ( t ) ] = e [ S ] t . [T(t)]=e^{[S]t}.
  32. g ( θ ) = exp ( ξ θ ) g ( 0 ) , g\left(\theta\right)=\exp(\xi\theta)g\left(0\right),
  33. U ( q , 1 ) ( z 0 0 z * ) = U ( q z , z * ) U ( ( z * ) - 1 q z , 1 ) . U(q,1)\begin{pmatrix}z&0\\ 0&z^{*}\end{pmatrix}=U(qz,z^{*})\thicksim U((z^{*})^{-1}qz,1).
  34. 1 / ( exp ( a r - b ϵ r ) ) = 1/(\exp(ar-b\epsilon r))=
  35. ( e a r e - b r ϵ ) - 1 = (e^{ar}e^{-br\epsilon})^{-1}=
  36. e b r ϵ e - a r , e^{br\epsilon}e^{-ar},
  37. ( e b ϵ e - a r ) q ( e a r e b ϵ r ) = e b ϵ r ( e - a r q e a r ) e b ϵ r = e 2 b ϵ r ( e - a r q e a r ) . (e^{b\epsilon}e^{-ar})q(e^{ar}e^{b\epsilon r})=e^{b\epsilon r}(e^{-ar}qe^{ar})% e^{b\epsilon r}=e^{2b\epsilon r}(e^{-ar}qe^{ar}).
  38. 𝐗 i ( t ) = [ A ( t ) ] 𝐱 i + 𝐝 ( t ) i = 1 , , n , \mathbf{X}_{i}(t)=[A(t)]\mathbf{x}_{i}+\mathbf{d}(t)\quad i=1,\ldots,n,
  39. 𝐕 i = ω × ( 𝐗 i - 𝐝 ) + 𝐯 , \mathbf{V}_{i}=\vec{\omega}\times(\mathbf{X}_{i}-\mathbf{d})+\mathbf{v},
  40. δ W = 𝐅 1 𝐕 1 δ t + 𝐅 2 𝐕 2 δ t + + 𝐅 n 𝐕 n δ t . \delta W=\mathbf{F}_{1}\cdot\mathbf{V}_{1}\delta t+\mathbf{F}_{2}\cdot\mathbf{% V}_{2}\delta t+\ldots+\mathbf{F}_{n}\cdot\mathbf{V}_{n}\delta t.
  41. δ W = i = 1 n 𝐅 i ( ω × ( 𝐗 i - 𝐝 ) + 𝐯 ) δ t . \delta W=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot(\vec{\omega}\times(\mathbf{X}_{i}-% \mathbf{d})+\mathbf{v})\delta t.
  42. δ W = ( i = 1 n 𝐅 i ) 𝐝 × ω δ t + ( i = 1 n 𝐅 i ) 𝐯 δ t + ( i = 1 n 𝐗 i × 𝐅 i ) ω δ t = ( i = 1 n 𝐅 i ) ( 𝐯 + 𝐝 × ω ) δ t + ( i = 1 n 𝐗 i × 𝐅 i ) ω δ t . \delta W=(\sum_{i=1}^{n}\mathbf{F}_{i})\cdot\mathbf{d}\times\vec{\omega}\delta t% +(\sum_{i=1}^{n}\mathbf{F}_{i})\cdot\mathbf{v}\delta t+(\sum_{i=1}^{n}\mathbf{% X}_{i}\times\mathbf{F}_{i})\cdot\vec{\omega}\delta t=(\sum_{i=1}^{n}\mathbf{F}% _{i})\cdot(\mathbf{v}+\mathbf{d}\times\vec{\omega})\delta t+(\sum_{i=1}^{n}% \mathbf{X}_{i}\times\mathbf{F}_{i})\cdot\vec{\omega}\delta t.
  43. 𝖳 = ( ω , 𝐝 × ω + 𝐯 ) = ( 𝐓 , 𝐓 ) , 𝖶 = ( i = 1 n 𝐅 i , i = 1 n 𝐗 i × 𝐅 i ) = ( 𝐖 , 𝐖 ) , \mathsf{T}=(\vec{\omega},\mathbf{d}\times\vec{\omega}+\mathbf{v})=(\mathbf{T},% \mathbf{T}^{\circ}),\quad\mathsf{W}=(\sum_{i=1}^{n}\mathbf{F}_{i},\sum_{i=1}^{% n}\mathbf{X}_{i}\times\mathbf{F}_{i})=(\mathbf{W},\mathbf{W}^{\circ}),
  44. δ W = ( 𝐖 𝐓 + 𝐖 𝐓 ) δ t . \delta W=(\mathbf{W}\cdot\mathbf{T}^{\circ}+\mathbf{W}^{\circ}\cdot\mathbf{T})% \delta t.
  45. δ W = ( 𝐖 𝐓 + 𝐖 𝐓 ) δ t = 𝖶 [ Π ] 𝖳 δ t , \delta W=(\mathbf{W}\cdot\mathbf{T}^{\circ}+\mathbf{W}^{\circ}\cdot\mathbf{T})% \delta t=\mathsf{W}[\Pi]\mathsf{T}\delta t,
  46. [ Π ] = [ 0 I I 0 ] , [\Pi]=\begin{bmatrix}0&I\\ I&0\end{bmatrix},
  47. δ W = 𝖶 [ Π ] 𝖳 δ t = 0 , \delta W=\mathsf{W}[\Pi]\mathsf{T}\delta t=0,
  48. 𝖳 ˇ = ( 𝐝 × ω + 𝐯 , ω ) , \check{\mathsf{T}}=(\mathbf{d}\times\vec{\omega}+\mathbf{v},\vec{\omega}),
  49. δ W = 𝖶 𝖳 ˇ δ t . \delta W=\mathsf{W}\cdot\check{\mathsf{T}}\delta t.
  50. δ W = 𝖶 𝖳 ˇ δ t = 0 , \delta W=\mathsf{W}\cdot\check{\mathsf{T}}\delta t=0,

Sea_state.html

  1. S ( ω , Θ ) S(\omega,\Theta)
  2. S ( ω ) S(\omega)
  3. f ( Θ ) f(\Theta)
  4. { S ( ω ) } = { length 2 time } \{S(\omega)\}=\{{\,\text{length}}^{2}\cdot\,\text{time}\}
  5. S ( ω j ) S(\omega_{j})
  6. A j A_{j}
  7. j j
  8. 1 2 A j 2 = S ( ω j ) Δ ω \frac{1}{2}A_{j}^{2}=S(\omega_{j})\,\Delta\omega
  9. S ( ω ) H 1 / 3 2 T 1 = 0.11 2 π ( ω T 1 2 π ) - 5 exp [ - 0.44 ( ω T 1 2 π ) - 4 ] \frac{S(\omega)}{H_{1/3}^{2}T_{1}}=\frac{0.11}{2\pi}\left(\frac{\omega T_{1}}{% 2\pi}\right)^{-5}\mathrm{exp}\left[-0.44\left(\frac{\omega T_{1}}{2\pi}\right)% ^{-4}\right]
  10. S ( ω ) = 155 H 1 / 3 2 T 1 4 ω 5 exp ( - 944 T 1 4 ω 4 ) ( 3.3 ) Y , S(\omega)=155\frac{H_{1/3}^{2}}{T_{1}^{4}\omega^{5}}\mathrm{exp}\left(\frac{-9% 44}{T_{1}^{4}\omega^{4}}\right)(3.3)^{Y},
  11. Y = exp [ - ( 0.191 ω T 1 - 1 2 1 / 2 σ ) 2 ] Y=\exp\left[-\left(\frac{0.191\omega T_{1}-1}{2^{1/2}\sigma}\right)^{2}\right]
  12. σ = { 0.07 if ω 5.24 / T 1 , 0.09 if ω > 5.24 / T 1 . \sigma=\begin{cases}0.07&\,\text{if }\omega\leq 5.24/T_{1},\\ 0.09&\,\text{if }\omega>5.24/T_{1}.\end{cases}
  13. f ( Θ ) f(\Theta)
  14. f ( Θ ) = 2 π cos 2 Θ , - π / 2 Θ π / 2 f(\Theta)=\frac{2}{\pi}\cos^{2}\Theta,\qquad-\pi/2\leq\Theta\leq\pi/2
  15. ζ \zeta
  16. ϵ j \epsilon_{j}
  17. 2 π 2\pi
  18. Θ j \Theta_{j}
  19. f ( Θ ) : \sqrt{f(\Theta)}:
  20. ζ = j = 1 N 2 S ( ω j ) Δ ω j sin ( ω j t - k j x cos Θ j - k j y sin Θ j + ϵ j ) . \zeta=\sum_{j=1}^{N}\sqrt{2S(\omega_{j})\Delta\omega_{j}}\;\sin(\omega_{j}t-k_% {j}x\cos\Theta_{j}-k_{j}y\sin\Theta_{j}+\epsilon_{j}).

Search_problem.html

  1. L ( R ) = { x y R ( x , y ) } . L(R)=\{x\mid\exists yR(x,y)\}.\,

SeaWiFS.html

  1. C h l = a n t i l o g ( 0.366 - 3.067 𝖱 + 1.93 𝖱 2 + 0.64 𝖱 3 - 1.53 𝖱 4 ) Chl=antilog(0.366-3.067\mathsf{R}+1.93\mathsf{R}^{2}+0.64\mathsf{R}^{3}-1.53% \mathsf{R}^{4})
  2. L T ( λ ) = L r ( λ ) + L a ( λ ) + L r a ( λ ) + T L g ( λ ) + t ( L f ( λ ) + L W ( λ ) ) L_{T}(\lambda)=L_{r}(\lambda)+L_{a}(\lambda)+L_{ra}(\lambda)+TL_{g}(\lambda)+t% (L_{f}(\lambda)+L_{W}(\lambda))

Second_moment_of_area.html

  1. I I
  2. J J
  3. I = A x 2 d A I=\int_{A}x^{2}\,\mathrm{d}A
  4. I = A r 2 d A I=\int_{A}r^{2}\,\mathrm{d}A
  5. I = m r 2 d m I=\int_{m}r^{2}\mathrm{d}m
  6. B B BB
  7. J B B = A ρ 2 d A J_{BB}=\int_{A}{\rho}^{2}\,\mathrm{d}A
  8. d A \mathrm{d}A
  9. ρ \rho
  10. I x x I_{xx}
  11. I x I_{x}
  12. I x = A y 2 d x d y I_{x}=\iint_{A}y^{2}\,\mathrm{d}x\,\mathrm{d}y
  13. I x y = A x y d x d y I_{xy}=\iint_{A}xy\,\mathrm{d}x\,\mathrm{d}y
  14. x x^{\prime}
  15. x x
  16. I x = I x + A d y 2 I_{x}=I_{x^{\prime}}+Ad_{y}^{2}
  17. A A
  18. d y d_{y}
  19. x x^{\prime}
  20. x x
  21. y y
  22. y y^{\prime}
  23. B B^{\prime}
  24. B B
  25. J z J_{z}
  26. I x I_{x}
  27. I y I_{y}
  28. J z = A ρ 2 d A = A ( x 2 + y 2 ) d A = A x 2 d A + A y 2 d A = I x + I y J_{z}=\int_{A}\rho^{2}\,\mathrm{d}A=\int_{A}(x^{2}+y^{2})\,\mathrm{d}A=\int_{A% }x^{2}\,\mathrm{d}A+\int_{A}y^{2}\,\mathrm{d}A=I_{x}+I_{y}
  29. x x
  30. y y
  31. ρ \rho
  32. b b
  33. h h
  34. I x I_{x}
  35. I y I_{y}
  36. J z J_{z}
  37. I x = A y 2 d A = - b / 2 b / 2 - h / 2 h / 2 y 2 d y d x = - b / 2 b / 2 1 3 h 3 4 d x = b h 3 12 I_{x}=\int_{A}y^{2}\,\mathrm{d}A=\int^{b/2}_{-b/2}\int^{h/2}_{-h/2}y^{2}\,% \mathrm{d}y\,\mathrm{d}x=\int^{b/2}_{-b/2}\frac{1}{3}\frac{h^{3}}{4}\,\mathrm{% d}x=\frac{bh^{3}}{12}
  38. I y = A x 2 d A = - b / 2 b / 2 - h / 2 h / 2 x 2 d y d x = - b / 2 b / 2 h x 2 d x = b 3 h 12 I_{y}=\int_{A}x^{2}\,\mathrm{d}A=\int^{b/2}_{-b/2}\int^{h/2}_{-h/2}x^{2}\,% \mathrm{d}y\,\mathrm{d}x=\int^{b/2}_{-b/2}hx^{2}\,\mathrm{d}x=\frac{b^{3}h}{12}
  39. J z = I x + I y = b h 3 12 + h b 3 12 = b h 12 ( b 2 + h 2 ) J_{z}=I_{x}+I_{y}=\frac{bh^{3}}{12}+\frac{hb^{3}}{12}=\frac{bh}{12}(b^{2}+h^{2})
  40. r o r_{o}
  41. r i r_{i}
  42. J z J_{z}
  43. z z
  44. r o r_{o}
  45. r i r_{i}
  46. r r
  47. J z J_{z}
  48. r 2 r^{2}
  49. x x
  50. y y
  51. I x I_{x}
  52. J z J_{z}
  53. I x , c i r c l e = y 2 d A = ( r sin θ ) 2 d A = 0 2 π 0 r ( r sin θ ) 2 ( r d r d θ ) = 0 2 π 0 r r 3 sin 2 θ d r d θ = 0 2 π r 4 sin 2 θ 4 d θ = π 4 r 4 I_{x,circle}=\iint y^{2}\,dA=\iint(r\sin{\theta)}^{2}\,dA=\int_{0}^{2\pi}\int_% {0}^{r}(r\sin{\theta})^{2}\left(r\,dr\,d\theta\right)=\int_{0}^{2\pi}\int_{0}^% {r}r^{3}\sin^{2}{\theta}\,dr\,d\theta=\int_{0}^{2\pi}\frac{r^{4}\sin^{2}{% \theta}}{4}\,d\theta=\frac{\pi}{4}r^{4}
  54. J z , c i r c l e = r 2 d A = 0 2 π 0 r r 2 ( r d r d θ ) = 0 2 π 0 r r 3 d r d θ = 0 2 π r 4 4 d θ = π 2 r 4 J_{z,circle}=\iint r^{2}\,dA=\int_{0}^{2\pi}\int_{0}^{r}r^{2}\left(r\,dr\,d% \theta\right)=\int_{0}^{2\pi}\int_{0}^{r}r^{3}\,dr\,d\theta=\int_{0}^{2\pi}% \frac{r^{4}}{4}\,d\theta=\frac{\pi}{2}r^{4}
  55. z z
  56. r o r_{o}
  57. r i r_{i}
  58. J z = J z , r o - J z , r i = π 2 r o 4 - π 2 r i 4 = π 2 ( r o 4 - r i 4 ) J_{z}=J_{z,r_{o}}-J_{z,r_{i}}=\frac{\pi}{2}r_{o}^{4}-\frac{\pi}{2}r_{i}^{4}=% \frac{\pi}{2}({r_{o}}^{4}-{r_{i}}^{4})
  59. d r dr
  60. J z = r 2 d A = 0 2 π r i r o r 2 ( r d r d θ ) = 0 2 π r i r o r 3 d r d θ = 0 2 π [ r o 4 4 - r i 4 4 ] d θ = π 2 ( r o 4 - r i 4 ) J_{z}=\iint r^{2}\,dA=\int_{0}^{2\pi}\int_{r_{i}}^{r_{o}}r^{2}\left(r\,dr\,d% \theta\right)=\int_{0}^{2\pi}\int_{r_{i}}^{r_{o}}r^{3}\,dr\,d\theta=\int_{0}^{% 2\pi}\left[\frac{r_{o}^{4}}{4}-\frac{r_{i}^{4}}{4}\right]\,d\theta=\frac{\pi}{% 2}\left(r_{o}^{4}-r_{i}^{4}\right)
  61. I x = 1 12 i = 1 i = N ( y i 2 + y i y i + 1 + y i + 1 2 ) ( x i y i + 1 - x i + 1 y i ) I_{x}=\frac{1}{12}\sum_{i=1}^{i=N}(y_{i}^{2}+y_{i}y_{i+1}+y_{i+1}^{2})(x_{i}y_% {i+1}-x_{i+1}y_{i})
  62. I y = 1 12 i = 1 i = N ( x i 2 + x i x i + 1 + x i + 1 2 ) ( x i y i + 1 - x i + 1 y i ) I_{y}=\frac{1}{12}\sum_{i=1}^{i=N}(x_{i}^{2}+x_{i}x_{i+1}+x_{i+1}^{2})(x_{i}y_% {i+1}-x_{i+1}y_{i})
  63. I x y = 1 24 i = 1 i = N ( x i y i + 1 + 2 x i y i + 2 x i + 1 y i + 1 + x i + 1 y i ) ( x i y i + 1 - x i + 1 y i ) I_{xy}=\frac{1}{24}\sum_{i=1}^{i=N}(x_{i}y_{i+1}+2x_{i}y_{i}+2x_{i+1}y_{i+1}+x% _{i+1}y_{i})(x_{i}y_{i+1}-x_{i+1}y_{i})
  64. x i , y i x_{i},y_{i}
  65. x n + 1 = x 1 , y n + 1 = y 1 x_{n+1}=x_{1},y_{n+1}=y_{1}

Secretary_problem.html

  1. n n
  2. n / e n/e
  3. 1 / e 1/e
  4. 1 / e 1/e
  5. n n
  6. n n
  7. 1 / e 1/e
  8. P ( r ) = i = 1 n P ( applicant i is selected applicant i is the best ) = i = 1 n P ( applicant i is selected | applicant i is the best ) × P ( applicant i is the best ) = [ i = 1 r - 1 0 + i = r n P ( the second best of the first i applicants is in the first r - 1 applicants | applicant i is the best ) ] × 1 n = i = r n r - 1 i - 1 × 1 n = r - 1 n i = r n 1 i - 1 . \begin{aligned}\displaystyle P(r)&\displaystyle=\sum_{i=1}^{n}P\left(\,\text{% applicant }i\,\text{ is selected}\cap\,\text{applicant }i\,\text{ is the best}% \right)\\ &\displaystyle=\sum_{i=1}^{n}P\left(\,\text{applicant }i\,\text{ is selected}|% \,\text{applicant }i\,\text{ is the best}\right)\times P\left(\,\text{% applicant }i\,\text{ is the best}\right)\\ &\displaystyle=\left[\sum_{i=1}^{r-1}0+\sum_{i=r}^{n}P\left(\left.\begin{array% }[]{l}\,\text{the second best of the first }i\,\text{ applicants}\\ \,\text{is in the first }r-1\,\text{ applicants}\end{array}\right|\,\text{% applicant }i\,\text{ is the best}\right)\right]\times\frac{1}{n}\\ &\displaystyle=\sum_{i=r}^{n}\frac{r-1}{i-1}\times\frac{1}{n}\quad=\quad\frac{% r-1}{n}\sum_{i=r}^{n}\frac{1}{i-1}.\end{aligned}
  9. x x
  10. P ( x ) = x x 1 1 t d t = - x log ( x ) . P(x)=x\int_{x}^{1}\frac{1}{t}\,dt=-x\log(x).
  11. x x
  12. n n
  13. r r
  14. P P
  15. 1 / e 0.368 1/e\approx 0.368
  16. n n
  17. N N
  18. P ( N = k ) k = 1 , 2 , P(N=k)_{k=1,2,\cdots}
  19. N N
  20. [ 0 , T ] [0,T]
  21. N N
  22. f f
  23. [ 0 , T ] [0,T]
  24. F F
  25. F ( t ) = 0 t f ( s ) d s F(t)=\int_{0}^{t}f(s)ds
  26. 0 t T \,0\leq t\leq T
  27. τ \tau
  28. F ( τ ) = 1 / e . F(\tau)=1/e.
  29. τ \tau
  30. τ \tau
  31. N N
  32. N N
  33. n n
  34. n = 2 n=2
  35. n > 2 n>2
  36. n n
  37. x t = max { x 1 , x 2 , , x t } x_{t}=\max\left\{x_{1},x_{2},\ldots,x_{t}\right\}
  38. E t = E ( X t | I t = 1 ) = t t + 1 . E_{t}=E\left(X_{t}|I_{t}=1\right)=\frac{t}{t+1}.
  39. c c
  40. n \lfloor\sqrt{n}\rfloor
  41. n \lceil\sqrt{n}\rceil
  42. n \sqrt{n}
  43. n n
  44. 1 c n 1\leq c\leq n
  45. V n ( c ) = t = c n - 1 [ s = c t - 1 ( s - 1 s ) ] ( 1 t + 1 ) + [ s = c n - 1 ( s - 1 s ) ] 1 2 = 2 c n - c 2 + c - n 2 c n . V_{n}(c)=\sum_{t=c}^{n-1}\left[\prod_{s=c}^{t-1}\left(\frac{s-1}{s}\right)% \right]\left(\frac{1}{t+1}\right)+\left[\prod_{s=c}^{n-1}\left(\frac{s-1}{s}% \right)\right]\frac{1}{2}={\frac{2cn-{c}^{2}+c-n}{2cn}}.
  46. V n ( c ) V_{n}(c)
  47. V c = - c 2 + n 2 c 2 n . \frac{\partial V}{\partial c}=\frac{-{c}^{\,2}+n}{2{c}^{\,2}n}.
  48. 2 V / c 2 < 0 \partial^{\,2}V/\partial c^{\,2}<0
  49. c c
  50. V V
  51. c = n c=\sqrt{n}
  52. c c
  53. n \lfloor\sqrt{n}\rfloor
  54. n \lceil\sqrt{n}\rceil
  55. n n
  56. n n
  57. 0.25 n 2 n ( n - 1 ) \frac{0.25n^{2}}{n(n-1)}
  58. 1 n / 2 + 1 \frac{1}{n/2+1}
  59. n n
  60. n n
  61. e e

Secure_Remote_Password_protocol.html

  1. N \scriptstyle\mathbb{Z}_{N}

Seesaw_mechanism.html

  1. A = ( 0 M M B ) , A=\begin{pmatrix}0&M\\ M&B\end{pmatrix}\,\text{,}
  2. λ ± = B ± B 2 + 4 M 2 2 . \lambda_{\pm}=\frac{B\pm\sqrt{B^{2}+4M^{2}}}{2}\,\text{.}
  3. B B
  4. λ - - M 2 B . \lambda_{-}\approx-\frac{M^{2}}{B}.
  5. L = ( χ χ ) , L=\begin{pmatrix}\chi\\ \chi^{\prime}\end{pmatrix}~{},
  6. 1 2 B χ α χ α , 1 2 B η α η α , or M η α χ α , \frac{1}{2}\,B^{\prime}\,\chi^{\alpha}\chi_{\alpha}\,\,\text{,}\quad\frac{1}{2% }\,B\,\eta^{\alpha}\eta_{\alpha}\,\,\text{,}\quad\,\text{or}\quad M\,\eta^{% \alpha}\chi_{\alpha}\,\,\text{,}
  7. 1 2 ( χ η ) ( B M M B ) ( χ η ) . \frac{1}{2}\,\begin{pmatrix}\chi&\eta\end{pmatrix}\begin{pmatrix}B^{\prime}&M% \\ M&B\end{pmatrix}\begin{pmatrix}\chi\\ \eta\end{pmatrix}.
  8. y u k = y η L ϵ H * + \mathcal{L}_{yuk}=y\,\eta L\epsilon H^{*}+...
  9. VEV v 246 GeV , | H | = v / 2 \,\text{VEV }v\approx 246\,\text{ GeV},\qquad\qquad|\langle H\rangle|=v/\sqrt{2}
  10. M t = O ( v / 2 ) 174 GeV , M_{t}=O(v/\sqrt{2})\approx 174\,\text{ GeV}~{},

Segre_embedding.html

  1. σ : P n × P m P ( n + 1 ) ( m + 1 ) - 1 \sigma:P^{n}\times P^{m}\to P^{(n+1)(m+1)-1}
  2. ( [ X ] , [ Y ] ) P n × P m ([X],[Y])\in P^{n}\times P^{m}
  3. σ : ( [ X 0 : X 1 : : X n ] , [ Y 0 : Y 1 : : Y m ] ) [ X 0 Y 0 : X 0 Y 1 : : X i Y j : : X n Y m ] \sigma:([X_{0}:X_{1}:\cdots:X_{n}],[Y_{0}:Y_{1}:\cdots:Y_{m}])\mapsto[X_{0}Y_{% 0}:X_{0}Y_{1}:\cdots:X_{i}Y_{j}:\cdots:X_{n}Y_{m}]
  4. P n P^{n}
  5. P m P^{m}
  6. [ X 0 : X 1 : : X n ] [X_{0}:X_{1}:\cdots:X_{n}]
  7. Σ n , m \Sigma_{n,m}
  8. φ : U × V U V . \varphi:U\times V\to U\otimes V.
  9. u u
  10. U U
  11. v v
  12. V V
  13. c c
  14. K K
  15. φ ( u , v ) = u v = c u c - 1 v = φ ( c u , c - 1 v ) . \varphi(u,v)=u\otimes v=cu\otimes c^{-1}v=\varphi(cu,c^{-1}v).
  16. σ : P ( U ) × P ( V ) P ( U V ) . \sigma:P(U)\times P(V)\to P(U\otimes V).
  17. ( m + 1 ) ( n + 1 ) - 1 = m n + m + n . (m+1)(n+1)-1=mn+m+n.
  18. ( Z i , j ) (Z_{i,j})
  19. Z i , j Z k , l - Z i , l Z k , j . Z_{i,j}Z_{k,l}-Z_{i,l}Z_{k,j}.
  20. Z i , j Z_{i,j}
  21. Σ n , m \Sigma_{n,m}
  22. P n P^{n}
  23. P m P^{m}
  24. π X : Σ n , m P n \pi_{X}:\Sigma_{n,m}\to P^{n}
  25. j 0 j_{0}
  26. [ Z i , j ] [Z_{i,j}]
  27. [ Z i , j 0 ] [Z_{i,j_{0}}]
  28. Z i , j Z k , l = Z i , l Z k , j Z_{i,j}Z_{k,l}=Z_{i,l}Z_{k,j}
  29. Z i 0 , j 0 0 Z_{i_{0},j_{0}}\neq 0
  30. [ Z i , j 1 ] = [ Z i 0 , j 0 Z i , j 1 ] = [ Z i 0 , j 1 Z i , j 0 ] = [ Z i , j 0 ] [Z_{i,j_{1}}]=[Z_{i_{0},j_{0}}Z_{i,j_{1}}]=[Z_{i_{0},j_{1}}Z_{i,j_{0}}]=[Z_{i,% j_{0}}]
  31. π X : Σ n , m P n \pi_{X}:\Sigma_{n,m}\to P^{n}
  32. π Y \pi_{Y}
  33. σ ( π X ( ) , π Y ( p ) ) : Σ n , m P ( n + 1 ) ( m + 1 ) - 1 \sigma(\pi_{X}(\cdot),\pi_{Y}(p)):\Sigma_{n,m}\to P^{(n+1)(m+1)-1}
  34. [ Z 0 : Z 1 : Z 2 : Z 3 ] [Z_{0}:Z_{1}:Z_{2}:Z_{3}]
  35. det ( Z 0 Z 1 Z 2 Z 3 ) = Z 0 Z 3 - Z 1 Z 2 . \det\left(\begin{matrix}Z_{0}&Z_{1}\\ Z_{2}&Z_{3}\end{matrix}\right)=Z_{0}Z_{3}-Z_{1}Z_{2}.
  36. σ : P 2 × P 1 P 5 \sigma:P^{2}\times P^{1}\to P^{5}
  37. P 3 P^{3}
  38. Δ P n × P n \Delta\subset P^{n}\times P^{n}
  39. ν 2 : P n P n 2 + 2 n . \nu_{2}:P^{n}\to P^{n^{2}+2n}.

Seifert_conjecture.html

  1. C 1 C^{1}
  2. C 2 + δ C^{2+\delta}
  3. δ > 0 \delta>0
  4. C C^{\infty}
  5. C 2 C^{2}
  6. R 4 R^{4}
  7. C 2 C^{2}

Seifert_fiber_space.html

  1. S 1 S^{1}
  2. { b , ( ε , g ) ; ( a 1 , b 1 ) , , ( a r , b r ) } \{b,(\varepsilon,g);(a_{1},b_{1}),\dots,(a_{r},b_{r})\}\,
  3. ε \varepsilon
  4. o 1 , o 2 , n 1 , n 2 , n 3 , n 4 o_{1},o_{2},n_{1},n_{2},n_{3},n_{4}\,
  5. { b , ( ϵ , g ) ; ( a 1 , b 1 ) , , ( a r , b r ) } \{b,(\epsilon,g);(a_{1},b_{1}),...,(a_{r},b_{r})\}
  6. { 0 , ( ϵ , g ) ; } \{0,(\epsilon,g);\}
  7. { - b - r , ( ϵ , g ) ; ( a 1 , a 1 - b 1 ) , , ( a r , a r - b r ) } \{-b-r,(\epsilon,g);(a_{1},a_{1}-b_{1}),...,(a_{r},a_{r}-b_{r})\}
  8. π 1 ( S 1 ) π 1 ( M ) π 1 ( B ) 1 \pi_{1}(S^{1})\rightarrow\pi_{1}(M)\rightarrow\pi_{1}(B)\rightarrow 1
  9. u 1 , v 1 , u g , v g , q 1 , q r , h | u i h = h ϵ u i , v i h = h ϵ v i , q i h = h q i , q j a j h b j = 1 , q 1 q r [ u 1 , v 1 ] [ u g , v g ] = h b \langle u_{1},v_{1},...u_{g},v_{g},q_{1},...q_{r},h|u_{i}h=h^{\epsilon}u_{i},v% _{i}h=h^{\epsilon}v_{i},q_{i}h=hq_{i},q_{j}^{a_{j}}h^{b_{j}}=1,q_{1}...q_{r}[u% _{1},v_{1}]...[u_{g},v_{g}]=h^{b}\rangle
  10. v 1 , , v g , q 1 , q r , h | v i h = h ϵ i v i , q i h = h q i , q j a j h b j = 1 , q 1 q r v 1 2 v g 2 = h b \langle v_{1},...,v_{g},q_{1},...q_{r},h|v_{i}h=h^{\epsilon_{i}}v_{i},q_{i}h=% hq_{i},q_{j}^{a_{j}}h^{b_{j}}=1,q_{1}...q_{r}v_{1}^{2}...v_{g}^{2}=h^{b}\rangle

Seifert_surface.html

  1. V = ( 1 - 1 0 1 ) . V=\begin{pmatrix}1&-1\\ 0&1\end{pmatrix}.
  2. S S
  3. S S
  4. H 1 ( S ) H_{1}(S)
  5. S S
  6. H 1 ( S ) H_{1}(S)
  7. ( 0 - 1 1 0 ) \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  8. × \times
  9. v ( i , j ) v(i,j)
  10. V - V V-V
  11. = Q =Q
  12. × \times
  13. V V
  14. V - V V-V
  15. = Q =Q
  16. A ( t ) = d e t ( V - t V A(t)=det(V-tV
  17. t t
  18. 2 g \leq 2g
  19. S S
  20. V + V V+V^{\top}
  21. ( 0 1 1 0 ) \oplus\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  22. g ( K 1 # K 2 ) = g ( K 1 ) + g ( K 2 ) g(K_{1}\#K_{2})=g(K_{1})+g(K_{2})

Seifert–Weber_space.html

  1. 5 3 \mathbb{Z}_{5}^{3}

Selection_rule.html

  1. ψ 1 * μ ψ 2 d τ \int\psi_{1}^{*}\mu\psi_{2}d\tau
  2. ψ 1 \psi_{1}
  3. ψ 2 \psi_{2}
  4. ψ 1 * μ ψ 2 \psi_{1}^{*}\mu\psi_{2}
  5. ψ 1 * μ ψ 2 \psi_{1}^{*}\mu\psi_{2}
  6. 𝐉 i = 𝐉 f + s y m b o l λ \mathbf{J}_{\mathrm{i}}=\mathbf{J}_{\mathrm{f}}+symbol{\lambda}
  7. s y m b o l λ = λ ( λ + 1 ) \|symbol{\lambda}\|=\sqrt{\lambda(\lambda+1)}\,\hbar
  8. λ z = μ \lambda_{z}=\mu\,\hbar
  9. 𝐉 i \mathbf{J}_{\mathrm{i}}
  10. 𝐉 f \mathbf{J}_{\mathrm{f}}
  11. | J i - J f | λ J i + J f |J_{\mathrm{i}}-J_{\mathrm{f}}|\leq\lambda\leq J_{\mathrm{i}}+J_{\mathrm{f}}
  12. μ = M i - M f . \mu=M_{\mbox{i}~{}}-M_{\mbox{f}~{}}\,.
  13. π ( E λ ) = π i π f = ( - 1 ) λ \pi(\mathrm{E}\lambda)=\pi_{\mathrm{i}}\pi_{\mathrm{f}}=(-1)^{\lambda}\,
  14. π ( M λ ) = π i π f = ( - 1 ) λ + 1 . \pi(\mathrm{M}\lambda)=\pi_{\mathrm{i}}\pi_{\mathrm{f}}=(-1)^{\lambda+1}\,.
  15. J = L + S J=L+S
  16. L L
  17. S S
  18. M J M_{J}
  19. ↮ \not\leftrightarrow
  20. Δ J = 0 , ± 1 ( J = 0 ↮ 0 ) \begin{matrix}\Delta J=0,\pm 1\\ (J=0\not\leftrightarrow 0)\end{matrix}
  21. Δ J = 0 , ± 1 , ± 2 ( J = 0 ↮ 0 , 1 ; 1 2 ↮ 1 2 ) \begin{matrix}\Delta J=0,\pm 1,\pm 2\\ (J=0\not\leftrightarrow 0,1;\ \begin{matrix}{1\over 2}\end{matrix}\not% \leftrightarrow\begin{matrix}{1\over 2}\end{matrix})\end{matrix}
  22. Δ J = 0 , ± 1 , ± 2 , ± 3 ( 0 ↮ 0 , 1 , 2 ; 1 2 ↮ 1 2 , 3 2 ; 1 ↮ 1 ) \begin{matrix}\Delta J=0,\pm 1,\pm 2,\pm 3\\ (0\not\leftrightarrow 0,1,2;\ \begin{matrix}{1\over 2}\end{matrix}\not% \leftrightarrow\begin{matrix}{1\over 2}\end{matrix},\begin{matrix}{3\over 2}% \end{matrix};\ 1\not\leftrightarrow 1)\end{matrix}
  23. Δ M J = 0 , ± 1 \Delta M_{J}=0,\pm 1
  24. Δ M J = 0 , ± 1 , ± 2 \Delta M_{J}=0,\pm 1,\pm 2
  25. Δ M J = 0 , ± 1 , ± 2 , ± 3 \Delta M_{J}=0,\pm 1,\pm 2,\pm 3
  26. π f = - π i \pi_{\mathrm{f}}=-\pi_{\mathrm{i}}\,
  27. π f = π i \pi_{\mathrm{f}}=\pi_{\mathrm{i}}\,
  28. π f = - π i \pi_{\mathrm{f}}=-\pi_{\mathrm{i}}\,
  29. π f = π i \pi_{\mathrm{f}}=\pi_{\mathrm{i}}\,
  30. Δ L = 0 , ± 1 ( L = 0 ↮ 0 ) \begin{matrix}\Delta L=0,\pm 1\\ (L=0\not\leftrightarrow 0)\end{matrix}
  31. Δ L = 0 \Delta L=0\,
  32. Δ L = 0 , ± 1 , ± 2 ( L = 0 ↮ 0 , 1 ) \begin{matrix}\Delta L=0,\pm 1,\pm 2\\ (L=0\not\leftrightarrow 0,1)\end{matrix}
  33. Δ L = 0 , ± 1 , ± 2 , ± 3 ( L = 0 ↮ 0 , 1 , 2 ; 1 ↮ 1 ) \begin{matrix}\Delta L=0,\pm 1,\pm 2,\pm 3\\ (L=0\not\leftrightarrow 0,1,2;\ 1\not\leftrightarrow 1)\end{matrix}
  34. Δ L = 0 , ± 1 , ± 2 \Delta L=0,\pm 1,\pm 2\,
  35. Δ L = 0 , ± 1 , ± 2 , ± 3 ( L = 0 ↮ 0 ) \begin{matrix}\Delta L=0,\pm 1,\\ \pm 2,\pm 3\\ (L=0\not\leftrightarrow 0)\end{matrix}
  36. Δ L = 0 , ± 1 ( L = 0 ↮ 0 ) \begin{matrix}\Delta L=0,\pm 1\\ (L=0\not\leftrightarrow 0)\end{matrix}
  37. Δ L = 0 , ± 1 , ± 2 , ± 3 , ± 4 ( L = 0 ↮ 0 , 1 ) \begin{matrix}\Delta L=0,\pm 1,\\ \pm 2,\pm 3,\pm 4\\ (L=0\not\leftrightarrow 0,1)\end{matrix}
  38. Δ L = 0 , ± 1 , ± 2 ( L = 0 ↮ 0 ) \begin{matrix}\Delta L=0,\pm 1,\\ \pm 2\\ (L=0\not\leftrightarrow 0)\end{matrix}

Self-assembled_monolayer.html

  1. 𝐤 ( 𝟏 - θ ) = d θ d t . \mathbf{k(1-\theta)}=\frac{d\theta}{dt}.

Self-descriptive_number.html

  1. ( b - 4 ) b b - 1 + 2 b b - 2 + b b - 3 + b 3 (b-4)b^{b-1}+2b^{b-2}+b^{b-3}+b^{3}

Self-energy.html

  1. Σ \Sigma
  2. \hbar
  3. Σ \Sigma
  4. M M
  5. G 0 G_{0}
  6. G G
  7. G = G 0 + G 0 Σ G . G=G_{0}+G_{0}\Sigma G.
  8. G 0 - 1 G_{0}^{-1}
  9. G 0 G_{0}
  10. G - 1 G^{-1}
  11. Σ = G 0 - 1 - G - 1 . \Sigma=G_{0}^{-1}-G^{-1}.

Self-Indication_Assumption_Doomsday_argument_rebuttal.html

  1. P ( n N ) = P ( b N ) N P(n\mid N)=\frac{P(b\mid N)}{N}
  2. P ( n N ) = 1 c P(n\mid N)=\frac{1}{c}
  3. Ω \Omega
  4. Ω \Omega
  5. Ω \Omega
  6. P ( N ) = { 1 N ln ( Ω ) , N Ω 0 , N > Ω P(N)=\left\{\begin{matrix}\frac{1}{N\ln(\Omega)},\;\;N\leq\Omega\\ 0,\;\;N>\Omega\end{matrix}\right.
  7. ln ( Ω ) \ln(\Omega)
  8. Ω \Omega
  9. P ( n ) = N = n Ω P ( n N ) P ( N ) d N = n Ω [ 1 c ] 1 N ln ( Ω ) d N P(n)=\int_{N=n}^{\Omega}P(n\mid N)P(N)\,dN=\int_{n}^{\Omega}\left[\frac{1}{c}% \right]\frac{1}{N\ln(\Omega)}\,dN
  10. P ( n ) = ln ( Ω n ) 1 ln ( Ω ) c P(n)=\ln\left(\frac{\Omega}{n}\right)\frac{1}{\ln(\Omega)c}
  11. Ω \Omega
  12. P ( N n ) = P ( n N ) P ( N ) P ( n ) = 1 c 1 N ln ( Ω ) ln ( Ω ) c ln ( Ω n ) = 1 N ln ( Ω n ) P(N\mid n)=\frac{P(n\mid N)P(N)}{P(n)}=\frac{1}{c}\frac{1}{N\ln(\Omega)}\frac{% \ln(\Omega)c}{\ln(\frac{\Omega}{n})}=\frac{1}{N\ln(\frac{\Omega}{n})}
  13. P ( ( N > x n ) n ) = x n Ω 1 N ln ( Ω n ) d N = ln ( Ω ) - ln ( x n ) ln ( Ω ) - ln ( n ) P((N>xn)\mid n)=\int_{xn}^{\Omega}\frac{1}{N\ln(\frac{\Omega}{n})}\,dN=\frac{% \ln(\Omega)-\ln(xn)}{\ln(\Omega)-\ln(n)}
  14. lim Ω P ( N x n ) = 0 \lim_{\Omega\to\infty}P(N<=xn)=0
  15. Ω \Omega
  16. Ω \Omega
  17. Ω \Omega
  18. Ω \Omega
  19. 10 12 10^{12}
  20. Ω \Omega
  21. 10 24 10^{24}
  22. Ω \Omega
  23. Ω \Omega

Selmer_group.html

  1. Sel ( f ) ( A / K ) = v ker ( H 1 ( G K , ker ( f ) ) H 1 ( G K v , A v [ f ] ) / im ( κ v ) ) \mathrm{Sel}^{(f)}(A/K)=\bigcap_{v}\mathrm{ker}(H^{1}(G_{K},\mathrm{ker}(f))% \rightarrow H^{1}(G_{K_{v}},A_{v}[f])/\mathrm{im}(\kappa_{v}))
  2. κ v \kappa_{v}
  3. B v ( K v ) / f ( A v ( K v ) ) H 1 ( G K v , A v [ f ] ) B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])
  4. H 1 ( G K v , A v [ f ] ) / im ( κ v ) H^{1}(G_{K_{v}},A_{v}[f])/\mathrm{im}(\kappa_{v})
  5. H 1 ( G K v , A v ) [ f ] H^{1}(G_{K_{v}},A_{v})[f]

Semantics_encoding.html

  1. [ ] : A B [\cdot]:A\longrightarrow B
  2. o p A op_{A}
  3. o p B op_{B}
  4. T A 1 , T A 2 , , T A n , [ o p A ( T A 1 , T A 2 , , T A n ) ] = o p B ( [ T A 1 ] , [ T A 2 ] , , [ T A n ] ) \forall T_{A}^{1},T_{A}^{2},\dots,T_{A}^{n},[op_{A}(T_{A}^{1},T_{A}^{2},\cdots% ,T_{A}^{n})]=op_{B}([T_{A}^{1}],[T_{A}^{2}],\cdots,[T_{A}^{n}])
  5. o p A op_{A}
  6. o p B op_{B}
  7. T B 1 , T B 2 , , T B n , T A 1 , , T A n , o p B ( T B 1 , , T B N ) = [ o p A ( T A 1 , T A 2 , , T A n ) ] \forall T_{B}^{1},T_{B}^{2},\dots,T_{B}^{n},\exists T_{A}^{1},\dots,T_{A}^{n},% op_{B}(T_{B}^{1},\cdots,T_{B}^{N})=[op_{A}(T_{A}^{1},T_{A}^{2},\cdots,T_{A}^{n% })]
  8. \longrightarrow
  9. * \longrightarrow^{*}
  10. T A 1 , T A 2 T_{A}^{1},T_{A}^{2}
  11. T A 1 * T A 2 T_{A}^{1}\longrightarrow^{*}T_{A}^{2}
  12. [ T A 1 ] * [ T A 2 ] [T_{A}^{1}]\longrightarrow^{*}[T_{A}^{2}]
  13. T A 1 T_{A}^{1}
  14. T B 2 T_{B}^{2}
  15. [ T A 1 ] * T B 2 [T_{A}^{1}]\longrightarrow^{*}T_{B}^{2}
  16. T A 2 T_{A}^{2}
  17. T B 2 = [ T A 2 ] T_{B}^{2}=[T_{A}^{2}]
  18. T A T_{A}
  19. T A T_{A}
  20. [ T A ] [T_{A}]
  21. [ T A ] [T_{A}]
  22. [ T A ] [T_{A}]
  23. T A T_{A}
  24. o b s A obs_{A}
  25. o b s B obs_{B}
  26. T A T_{A}
  27. o b s A obs_{A}
  28. [ T A ] [T_{A}]
  29. o b s B obs_{B}
  30. o b s A obs_{A}
  31. o b s B obs_{B}
  32. [ T A ] [T_{A}]
  33. o b s B obs_{B}
  34. T A T_{A}
  35. o b s A obs_{A}
  36. T A 1 , T A 2 T_{A}^{1},T_{A}^{2}
  37. T A 2 T_{A}^{2}
  38. T A 1 T_{A}^{1}
  39. [ T A 2 ] [T_{A}^{2}]
  40. [ T A 1 ] [T_{A}^{1}]
  41. T A 1 , T A 2 T_{A}^{1},T_{A}^{2}
  42. [ T A 2 ] [T_{A}^{2}]
  43. [ T A 1 ] [T_{A}^{1}]
  44. T A 2 T_{A}^{2}
  45. T A 1 T_{A}^{1}
  46. T A 1 T_{A}^{1}
  47. T A 2 T_{A}^{2}
  48. [ T A 1 ] [T_{A}^{1}]
  49. [ T A 2 ] [T_{A}^{2}]
  50. [ T A 1 ] [T_{A}^{1}]
  51. [ T A 2 ] [T_{A}^{2}]
  52. T A 1 T_{A}^{1}
  53. T A 2 T_{A}^{2}
  54. T A T_{A}
  55. T A 1 | T A 2 T_{A}^{1}~{}|~{}T_{A}^{2}
  56. [ T A ] [T_{A}]
  57. [ T A 1 ] | [ T A 2 ] [T_{A}^{1}]~{}|~{}[T_{A}^{2}]
  58. [ T A ] [T_{A}]
  59. T B 1 | T B 2 T_{B}^{1}~{}|~{}T_{B}^{2}
  60. T B T_{B}
  61. T A 1 | T A 2 T_{A}^{1}~{}|~{}T_{A}^{2}
  62. [ T A 1 ] = T B 1 [T_{A}^{1}]=T_{B}^{1}
  63. [ T A 2 ] = T B 2 [T_{A}^{2}]=T_{B}^{2}

Semisimple_Lie_algebra.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 \mathfrak{g}
  3. 𝔤 \mathfrak{g}
  4. 𝔤 \mathfrak{g}
  5. 𝔤 \mathfrak{g}
  6. 𝔤 \mathfrak{g}
  7. 𝔤 \mathfrak{g}
  8. A n : A_{n}:
  9. 𝔰 𝔩 n + 1 \mathfrak{sl}_{n+1}
  10. B n : B_{n}:
  11. 𝔰 𝔬 2 n + 1 \mathfrak{so}_{2n+1}
  12. C n : C_{n}:
  13. 𝔰 𝔭 2 n \mathfrak{sp}_{2n}
  14. D n : D_{n}:
  15. 𝔰 𝔬 2 n \mathfrak{so}_{2n}
  16. n 1 n\geq 1
  17. n 2 n\geq 2
  18. n 3 n\geq 3
  19. n 4 n\geq 4
  20. 𝔤 \mathfrak{g}
  21. 𝔤 \mathfrak{g}
  22. 𝔤 𝔩 n \mathfrak{gl}_{n}
  23. ad \operatorname{ad}
  24. Der ( 𝔤 ) \operatorname{Der}(\mathfrak{g})
  25. 𝔤 \mathfrak{g}
  26. 𝔤 \mathfrak{g}
  27. 𝔤 \mathfrak{g}
  28. Der ( 𝔤 ) \operatorname{Der}(\mathfrak{g})
  29. x = s + n x=s+n
  30. x 𝔤 x\in\mathfrak{g}
  31. ad ( x ) = ad ( s ) + ad ( n ) . \operatorname{ad}(x)=\operatorname{ad}(s)+\operatorname{ad}(n).
  32. 𝔤 \mathfrak{g}
  33. 𝔤 \mathfrak{g}
  34. ρ ( x ) = ρ ( s ) + ρ ( n ) \rho(x)=\rho(s)+\rho(n)\,
  35. 𝔤 \mathfrak{g}
  36. 𝔤 = [ 𝔤 , 𝔤 ] \mathfrak{g}=[\mathfrak{g},\mathfrak{g}]
  37. 𝔰 𝔩 \mathfrak{sl}
  38. 𝔰 𝔩 \mathfrak{sl}
  39. 𝔰 𝔩 n \mathfrak{sl}_{n}
  40. 𝔤 𝔩 n \mathfrak{gl}_{n}

Sensor_fusion.html

  1. 𝐱 1 {\,\textbf{x}}_{1}
  2. 𝐱 2 {\,\textbf{x}}_{2}
  3. σ 1 2 \scriptstyle\sigma_{1}^{2}
  4. σ 2 2 \scriptstyle\sigma_{2}^{2}
  5. 𝐱 3 {\,\textbf{x}}_{3}
  6. 𝐱 3 = σ 3 2 ( σ 1 - 2 𝐱 1 + σ 2 - 2 𝐱 2 ) {\,\textbf{x}}_{3}=\scriptstyle\sigma_{3}^{2}(\scriptstyle\sigma_{1}^{-2}{\,% \textbf{x}}_{1}+\scriptstyle\sigma_{2}^{-2}{\,\textbf{x}}_{2})
  7. σ 3 2 = ( σ 1 - 2 + σ 2 - 2 ) - 1 \scriptstyle\sigma_{3}^{2}=(\scriptstyle\sigma_{1}^{-2}+\scriptstyle\sigma_{2}% ^{-2})^{-1}
  8. 𝐏 k {\,\textbf{P}}_{k}
  9. 𝐋 k = [ σ 2 2 𝐏 k σ 2 2 𝐏 k + σ 1 2 𝐏 k + σ 1 2 σ 2 2 σ 1 2 𝐏 k σ 2 2 𝐏 k + σ 1 2 𝐏 k + σ 1 2 σ 2 2 ] . {\,\textbf{L}}_{k}=\begin{bmatrix}\tfrac{\scriptstyle\sigma_{2}^{2}{\,\textbf{% P}}_{k}}{\scriptstyle\sigma_{2}^{2}{\,\textbf{P}}_{k}+\scriptstyle\sigma_{1}^{% 2}{\,\textbf{P}}_{k}+\scriptstyle\sigma_{1}^{2}\scriptstyle\sigma_{2}^{2}}&% \tfrac{\scriptstyle\sigma_{1}^{2}{\,\textbf{P}}_{k}}{\scriptstyle\sigma_{2}^{2% }{\,\textbf{P}}_{k}+\scriptstyle\sigma_{1}^{2}{\,\textbf{P}}_{k}+\scriptstyle% \sigma_{1}^{2}\scriptstyle\sigma_{2}^{2}}\end{bmatrix}.

Sequence_space.html

  1. ( x n ) n 𝐍 , x n 𝐊 . (x_{n})_{n\in\mathbf{N}},\quad x_{n}\in\mathbf{K}.
  2. ( x n ) n 𝐍 + ( y n ) n 𝐍 = def ( x n + y n ) n 𝐍 (x_{n})_{n\in\mathbf{N}}+(y_{n})_{n\in\mathbf{N}}\stackrel{\rm{def}}{=}(x_{n}+% y_{n})_{n\in\mathbf{N}}
  3. α ( x n ) n 𝐍 := ( α x n ) n 𝐍 . \alpha(x_{n})_{n\in\mathbf{N}}:=(\alpha x_{n})_{n\in\mathbf{N}}.
  4. n | x n | p < . \sum_{n}|x_{n}|^{p}<\infty.
  5. p \|\cdot\|_{p}
  6. x p = ( n | x n | p ) 1 / p \|x\|_{p}=\left(\sum_{n}|x_{n}|^{p}\right)^{1/p}
  7. d ( x , y ) = n | x n - y n | p . d(x,y)=\sum_{n}|x_{n}-y_{n}|^{p}.\,
  8. x = sup n | x n | , \|x\|_{\infty}=\sup_{n}|x_{n}|,
  9. sup n | i = 0 n x i | < . \sup_{n}\left|\sum_{i=0}^{n}x_{i}\right|<\infty.
  10. x b s = sup n | i = 0 n x i | , \|x\|_{bs}=\sup_{n}\left|\sum_{i=0}^{n}x_{i}\right|,
  11. ( x n ) n 𝐍 ( i = 0 n x i ) n 𝐍 . (x_{n})_{n\in\mathbf{N}}\mapsto\left(\sum_{i=0}^{n}x_{i}\right)_{n\in\mathbf{N% }}.
  12. c 00 c_{00}
  13. x + y p 2 + x - y p 2 = 2 x p 2 + 2 y p 2 \|x+y\|_{p}^{2}+\|x-y\|_{p}^{2}=2\|x\|_{p}^{2}+2\|y\|_{p}^{2}
  14. L x ( y ) = n x n y n L_{x}(y)=\sum_{n}x_{n}y_{n}
  15. | L x ( y ) | x q y p |L_{x}(y)|\leq\|x\|_{q}\,\|y\|_{p}
  16. L x ( p ) * = def sup y p , y 0 | L x ( y ) | y p x q . \|L_{x}\|_{(\ell^{p})^{*}}\stackrel{\rm{def}}{=}\sup_{y\in\ell^{p},y\not=0}% \frac{|L_{x}(y)|}{\|y\|_{p}}\leq\|x\|_{q}.
  17. y n = { 0 if x n = 0 x n - 1 | x n | q if x n 0 y_{n}=\begin{cases}0&\rm{if}\ x_{n}=0\\ x_{n}^{-1}|x_{n}|^{q}&\rm{if}\ x_{n}\not=0\end{cases}
  18. L x ( p ) * = x q . \|L_{x}\|_{(\ell^{p})^{*}}=\|x\|_{q}.
  19. x L x x\mapsto L_{x}
  20. κ q : q ( p ) * . \kappa_{q}:\ell^{q}\to(\ell^{p})^{*}.
  21. q κ q ( p ) * ( κ q * ) - 1 \ell^{q}\xrightarrow{\kappa_{q}}(\ell^{p})^{*}\xrightarrow{(\kappa_{q}^{*})^{-% 1}}
  22. Q : 1 X Q:\ell^{1}\to X
  23. 1 / ker Q \ell^{1}/\ker Q
  24. 1 = Y ker Q \ell^{1}=Y\oplus\ker Q
  25. X = p X=\ell^{p}
  26. p [ 1 , + ] p\in[1,+\infty]
  27. p \ell^{p}
  28. p p
  29. 1 p < q + 1\leq p<q\leq+\infty
  30. f q f p \|f\|_{q}\leq\|f\|_{p}
  31. F := f f p F:=\frac{f}{\|f\|_{p}}
  32. f p f\in\ell^{p}
  33. | F ( m ) | 1 |F(m)|\leq 1
  34. m m\in\mathbb{N}
  35. F q q 1 \|F\|_{q}^{q}\leq 1

Set_packing.html

  1. 𝒰 \mathcal{U}
  2. 𝒮 \mathcal{S}
  3. 𝒰 \mathcal{U}
  4. 𝒞 𝒮 \mathcal{C}\subseteq\mathcal{S}
  5. 𝒞 \mathcal{C}
  6. | 𝒞 | |\mathcal{C}|
  7. ( 𝒰 , 𝒮 ) (\mathcal{U},\mathcal{S})
  8. k k
  9. k k
  10. ( 𝒰 , 𝒮 ) (\mathcal{U},\mathcal{S})
  11. S 𝒮 x S \sum_{S\in\mathcal{S}}x_{S}
  12. S : e S x S 1 \sum_{S\colon e\in S}x_{S}\leqslant 1
  13. e 𝒰 e\in\mathcal{U}
  14. x S { 0 , 1 } x_{S}\in\{0,1\}
  15. S 𝒮 S\in\mathcal{S}
  16. 𝒰 \mathcal{U}
  17. 𝒮 \mathcal{S}
  18. 𝒞 \mathcal{C}
  19. 𝒰 , 𝒮 \mathcal{U},\mathcal{S}
  20. S 𝒮 w ( S ) x S \sum_{S\in\mathcal{S}}w(S)\cdot x_{S}
  21. O ( | U | ) O(\sqrt{|U|})
  22. 𝒮 \mathcal{S}
  23. S 𝒮 S\in\mathcal{S}
  24. v S v_{S}
  25. v S v_{S}
  26. v T v_{T}
  27. S T ϕ S\cap T\neq\phi
  28. 𝒮 \mathcal{S}
  29. G ( V , E ) G(V,E)
  30. v v
  31. S v S_{v}
  32. v v
  33. G ( V , E ) G(V,E)
  34. O ( n 1 - ϵ ) O(n^{1-\epsilon})

Severi–Brauer_variety.html

  1. z 2 = a x 2 + b y 2 z^{2}=ax^{2}+by^{2}
  2. 0 Pic ( X ) δ Br ( K ) Br ( K ) ( X ) 0 . 0\rightarrow\mathrm{Pic}(X)\rightarrow\mathbb{Z}\stackrel{\delta}{\rightarrow}% \mathrm{Br}(K)\rightarrow\mathrm{Br}(K)(X)\rightarrow 0\ .

SHA-2.html

  1. Ch ( E , F , G ) = ( E and F ) ( ¬ E and G ) \operatorname{Ch}(E,F,G)=(E\and F)\oplus(\neg E\and G)
  2. Ma ( A , B , C ) = ( A and B ) ( A and C ) ( B and C ) \operatorname{Ma}(A,B,C)=(A\and B)\oplus(A\and C)\oplus(B\and C)
  3. Σ 0 ( A ) = ( A 2 ) ( A 13 ) ( A 22 ) \Sigma_{0}(A)=(A\!\ggg\!2)\oplus(A\!\ggg\!13)\oplus(A\!\ggg\!22)
  4. Σ 1 ( E ) = ( E 6 ) ( E 11 ) ( E 25 ) \Sigma_{1}(E)=(E\!\ggg\!6)\oplus(E\!\ggg\!11)\oplus(E\!\ggg\!25)
  5. \color r e d \color{red}\boxplus

Shadow_price.html

  1. p 1 , p 2 \,\!p_{1},p_{2}
  2. m \,\!m
  3. max { u ( x 1 , x 2 ) : p 1 x 1 + p 2 x 2 = m } \max\{\,\!u(x_{1},x_{2})\mbox{ }~{}:\mbox{ }~{}p_{1}x_{1}+p_{2}x_{2}=m\}
  4. L ( x 1 , x 2 , λ ) := u ( x 1 , x 2 ) + λ ( m - p 1 x 1 - p 2 x 2 ) \,\!L(x_{1},x_{2},\lambda):=u(x_{1},x_{2})+\lambda(m-p_{1}x_{1}-p_{2}x_{2})
  5. x 1 * , x 2 * , λ * \,\!x^{*}_{1}\mbox{, }~{}x^{*}_{2}\mbox{, }~{}\lambda^{*}
  6. λ * = u ( x 1 * , x 2 * ) x 1 p 1 = u ( x 1 * , x 2 * ) x 2 p 2 \lambda^{*}=\frac{\frac{\partial u(x^{*}_{1},x^{*}_{2})}{\partial x_{1}}}{p_{1% }}=\frac{\frac{\partial u(x^{*}_{1},x^{*}_{2})}{\partial x_{2}}}{p_{2}}
  7. λ * \,\!\lambda^{*}
  8. λ * \,\!\lambda^{*}
  9. U ( p 1 , p 2 , m ) = max { u ( x 1 , x 2 ) : p 1 x 1 + p 2 x 2 = m } U(p_{1},p_{2},m)=\max\{\,\!u(x_{1},x_{2})\mbox{ }~{}:\mbox{ }~{}p_{1}x_{1}+p_{% 2}x_{2}=m\}
  10. U ( p 1 , p 2 , m ) = u ( x 1 * ( p 1 , p 2 , m ) , x 2 * ( p 1 , p 2 , m ) ) \,\!U(p_{1},p_{2},m)=u(x_{1}^{*}(p_{1},p_{2},m),x_{2}^{*}(p_{1},p_{2},m))
  11. x 1 * ( , , ) , x 2 * ( , , ) \,\!x_{1}^{*}(\cdot,\cdot,\cdot),x_{2}^{*}(\cdot,\cdot,\cdot)
  12. x i * ( p 1 , p 2 , m ) = arg max { u ( x 1 , x 2 ) : p 1 x 1 + p 2 x 2 = m } for i = 1 , 2 x_{i}^{*}(p_{1},p_{2},m)=\arg\max\{\,\!u(x_{1},x_{2})\mbox{ }~{}:\mbox{ }~{}p_% {1}x_{1}+p_{2}x_{2}=m\}\mbox{ for }~{}i=1,2
  13. E ( p 1 , p 2 , m ) = p 1 x 1 * ( p 1 , p 2 , m ) + p 2 x 2 * ( p 1 , p 2 , m ) \,\!E(p_{1},p_{2},m)=p_{1}x_{1}^{*}(p_{1},p_{2},m)+p_{2}x_{2}^{*}(p_{1},p_{2},m)
  14. λ * \,\!\lambda^{*}
  15. p 1 , p 2 , m \,\!p_{1},p_{2},m
  16. U m = u x 1 x 1 * m + u x 2 x 2 * m = λ * p 1 x 1 * m + λ * p 2 x 2 * m = λ * ( p 1 x 1 * m + p 2 x 2 * m ) = λ * E m \,\!\frac{\partial U}{\partial m}=\frac{\partial u}{\partial x_{1}}\frac{% \partial x_{1}^{*}}{\partial m}+\frac{\partial u}{\partial x_{2}}\frac{% \partial x_{2}^{*}}{\partial m}=\lambda^{*}p_{1}\frac{\partial x_{1}^{*}}{% \partial m}+\lambda^{*}p_{2}\frac{\partial x_{2}^{*}}{\partial m}=\lambda^{*}% \left(p_{1}\frac{\partial x_{1}^{*}}{\partial m}+p_{2}\frac{\partial x_{2}^{*}% }{\partial m}\right)=\lambda^{*}\frac{\partial E}{\partial m}
  17. λ * = U / m E / m Δ Optimal Utility Δ Optimal Expenditure \,\!\lambda^{*}=\frac{\partial U/\partial m}{\partial E/\partial m}\approx% \frac{\Delta\mbox{Optimal Utility }~{}}{\Delta\mbox{Optimal Expenditure}~{}}
  18. λ * \,\!\lambda^{*}

Shannon's_source_coding_theorem.html

  1. N N
  2. H ( X ) H(X)
  3. N H ( X ) N H(X)
  4. N N→∞
  5. N H ( X ) N H(X)
  6. Σ [ u s u , u b = 1 , u p = 217 ] Σ[u^{\prime}su^{\prime},u^{\prime}b=1^{\prime},u^{\prime}p=\u{2}217^{\prime}]
  7. Σ [ u s u , u b = 2 , u p = 217 ] Σ[u^{\prime}su^{\prime},u^{\prime}b=2^{\prime},u^{\prime}p=\u{2}217^{\prime}]
  8. X X
  9. f f
  10. Σ [ u s u , u b = 1 , u p = 217 ] Σ[u^{\prime}su^{\prime},u^{\prime}b=1^{\prime},u^{\prime}p=\u{2}217^{\prime}]
  11. Σ [ u s u , u b = 2 , u p = 217 ] Σ[u^{\prime}su^{\prime},u^{\prime}b=2^{\prime},u^{\prime}p=\u{2}217^{\prime}]
  12. S S
  13. f ( X ) f(X)
  14. f f
  15. X X
  16. H ( X ) log 2 a 𝔼 S < H ( X ) log 2 a + 1 \frac{H(X)}{\log_{2}a}\leq\mathbb{E}S<\frac{H(X)}{\log_{2}a}+1
  17. X X
  18. H ( X ) H(X)
  19. ε > 0 ε>0
  20. n n
  21. n n
  22. n ( H ( X ) + ε ) n(H(X)+ε)
  23. 1 ε 1−ε
  24. ε > 0 ε>0
  25. p ( x 1 , , x n ) = Pr [ X 1 = x 1 , , X n = x n ] . p(x_{1},\ldots,x_{n})=\Pr\left[X_{1}=x_{1},\cdots,X_{n}=x_{n}\right].
  26. A [ u s u , u b = , u n , u p = , u 3 b 5 ] A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =^{\prime},u^{\prime}\u{0}3b5^{\prime}]
  27. A n ε = { ( x 1 , , x n ) : | - 1 n log p ( x 1 , , x n ) - H n ( X ) | < ε } . A_{n}^{\varepsilon}=\left\{(x_{1},\cdots,x_{n})\ :\ \left|-\frac{1}{n}\log p(x% _{1},\cdots,x_{n})-H_{n}(X)\right|<\varepsilon\right\}.
  28. n n
  29. A [ u s u , u b = , u n , u p = , u 3 b 5 ] A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =^{\prime},u^{\prime}\u{0}3b5^{\prime}]
  30. n n
  31. P ( A n ε ) > 1 - ε P(A_{n}^{\varepsilon})>1-\varepsilon
  32. 2 - n ( H ( X ) + ε ) p ( x 1 , , x n ) 2 - n ( H ( X ) - ε ) 2^{-n(H(X)+\varepsilon)}\leq p\left(x_{1},\cdots,x_{n}\right)\leq 2^{-n(H(X)-% \varepsilon)}
  33. X X
  34. A [ u s u , u b = , u n , u p = , u 3 b 5 ] A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =^{\prime},u^{\prime}\u{0}3b5^{\prime}]
  35. 1 ε 1−ε
  36. | A n ε | 2 n ( H ( X ) + ε ) \left|A_{n}^{\varepsilon}\right|\leq 2^{n(H(X)+\varepsilon)}
  37. A [ u s u , u b = , u n , u p = , u 3 b 5 ] A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =^{\prime},u^{\prime}\u{0}3b5^{\prime}]
  38. | A n ε | ( 1 - ε ) 2 n ( H ( X ) - ε ) \left|A_{n}^{\varepsilon}\right|\geq(1-\varepsilon)2^{n(H(X)-\varepsilon)}
  39. A [ u s u , u b = , u n , u p = , u 3 b 5 ] A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =^{\prime},u^{\prime}\u{0}3b5^{\prime}]
  40. | A n ε | 2 n ( H ( X ) + ε ) , n . ( H ( X ) + ε ) \left|A_{n}^{\varepsilon}\right|\leq 2^{n(H(X)+\varepsilon)},n.(H(X)+\varepsilon)
  41. n ( H ( X ) + ε ) n(H(X)+ε)
  42. 1 ε 1−ε
  43. ε ε
  44. A [ u s u , u b = , u n , u p = , u 3 b 5 ] A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =^{\prime},u^{\prime}\u{0}3b5^{\prime}]
  45. 1 1
  46. 1 i n 1≤i≤n
  47. q i = a - s i / C q_{i}=a^{-s_{i}}/C
  48. C C
  49. H ( X ) = - i = 1 n p i log 2 p i - i = 1 n p i log 2 q i = - i = 1 n p i log 2 a - s i + i = 1 n p i log 2 C = - i = 1 n p i log 2 a - s i + log 2 C - i = 1 n - s i p i log 2 a 𝔼 S log 2 a \begin{aligned}\displaystyle H(X)&\displaystyle=-\sum_{i=1}^{n}p_{i}\log_{2}p_% {i}\\ &\displaystyle\leq-\sum_{i=1}^{n}p_{i}\log_{2}q_{i}\\ &\displaystyle=-\sum_{i=1}^{n}p_{i}\log_{2}a^{-s_{i}}+\sum_{i=1}^{n}p_{i}\log_% {2}C\\ &\displaystyle=-\sum_{i=1}^{n}p_{i}\log_{2}a^{-s_{i}}+\log_{2}C\\ &\displaystyle\leq-\sum_{i=1}^{n}-s_{i}p_{i}\log_{2}a\\ &\displaystyle\leq\mathbb{E}S\log_{2}a\\ \end{aligned}
  50. C = i = 1 n a - s i 1 C=\sum_{i=1}^{n}a^{-s_{i}}\leq 1
  51. l o g C 0 logC≤0
  52. s i = - log a p i s_{i}=\lceil-\log_{a}p_{i}\rceil
  53. - log a p i s i < - log a p i + 1 -\log_{a}p_{i}\leq s_{i}<-\log_{a}p_{i}+1
  54. a - s i p i a^{-s_{i}}\leq p_{i}
  55. a - s i p i = 1 \sum a^{-s_{i}}\leq\sum p_{i}=1
  56. S S
  57. 𝔼 S = p i s i < p i ( - log a p i + 1 ) = - p i log 2 p i log 2 a + 1 = H ( X ) log 2 a + 1 \begin{aligned}\displaystyle\mathbb{E}S&\displaystyle=\sum p_{i}s_{i}\\ &\displaystyle<\sum p_{i}\left(-\log_{a}p_{i}+1\right)\\ &\displaystyle=\sum-p_{i}\frac{\log_{2}p_{i}}{\log_{2}a}+1\\ &\displaystyle=\frac{H(X)}{\log_{2}a}+1\\ \end{aligned}
  58. A [ u s u , u b = , u n , u p = , u 3 b 5 ] A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =^{\prime},u^{\prime}\u{0}3b5^{\prime}]
  59. A n ε = { x 1 n : | - 1 n log p ( X 1 , , X n ) - H n ¯ ( X ) | < ε } . A_{n}^{\varepsilon}=\left\{x_{1}^{n}\ :\ \left|-\frac{1}{n}\log p\left(X_{1},% \cdots,X_{n}\right)-\overline{H_{n}}(X)\right|<\varepsilon\right\}.
  60. δ > 0 δ>0
  61. n n
  62. P r ( A [ u s u , u b = , u n , u p = , u 3 b 5 ] ) > 1 δ Pr(A[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{% \prime}p=^{\prime},u^{\prime}\u{0}3b5^{\prime}])>1−δ
  63. 2 n ( H n ¯ ( X ) + ε ) 2^{n(\overline{H_{n}}(X)+\varepsilon)}
  64. H < s u b > n ¯ ( X ) + ε \overline{H<sub>n}(X)+ε

Shape_optimization.html

  1. Ω \Omega
  2. ( Ω ) \mathcal{F}(\Omega)
  3. 𝒢 ( Ω ) = 0. \mathcal{G}(\Omega)=0.
  4. Ω \Omega
  5. Ω 0 \Omega_{0}
  6. Ω t \Omega_{t}
  7. f t : Ω 0 Ω t , for 0 t t 0 . f_{t}:\Omega_{0}\to\Omega_{t},\mbox{ for }~{}0\leq t\leq t_{0}.
  8. V V
  9. T s T_{s}
  10. Ω 0 \Omega_{0}
  11. V V
  12. x ( 0 ) = x 0 Ω 0 , x ( s ) = V ( x ( s ) ) , T s ( x 0 ) = x ( s ) , s 0 x(0)=x_{0}\in\Omega_{0},\quad x^{\prime}(s)=V(x(s)),\quad T_{s}(x_{0})=x(s),% \quad s\geq 0
  13. Ω 0 T s ( Ω 0 ) = Ω s . \Omega_{0}\mapsto T_{s}(\Omega_{0})=\Omega_{s}.
  14. ( Ω ) \mathcal{F}(\Omega)
  15. Ω 0 \Omega_{0}
  16. d ( Ω 0 ; V ) = lim s 0 ( Ω s ) - ( Ω 0 ) s d\mathcal{F}(\Omega_{0};V)=\lim_{s\to 0}\frac{\mathcal{F}(\Omega_{s})-\mathcal% {F}(\Omega_{0})}{s}
  17. V V
  18. L 2 ( Ω 0 ) \nabla\mathcal{F}\in L^{2}(\partial\Omega_{0})
  19. d ( Ω 0 ; V ) = , V Ω 0 d\mathcal{F}(\Omega_{0};V)=\langle\nabla\mathcal{F},V\rangle_{\partial\Omega_{% 0}}
  20. \nabla\mathcal{F}
  21. Ω \partial\Omega
  22. \mathcal{F}
  23. 𝒢 \mathcal{G}

Shattered_set.html

  1. U A = T . U\cap A=T.\,
  2. S C ( n ) = max x 1 , x 2 , , x n Ω card { { x 1 , x 2 , , x n } s , s C } S_{C}(n)=\max_{x_{1},x_{2},\dots,x_{n}\in\Omega}\operatorname{card}\{\,\{\,x_{% 1},x_{2},\dots,x_{n}\}\cap s,s\in C\}
  3. card \operatorname{card}
  4. S C ( n ) S_{C}(n)
  5. S C ( n ) S_{C}(n)
  6. S C ( n ) 2 n S_{C}(n)\leq 2^{n}
  7. { s A | s C } P ( A ) \{s\cap A|s\in C\}\subseteq P(A)
  8. A Ω A\subseteq\Omega
  9. S C ( n ) = 2 n S_{C}(n)=2^{n}
  10. S C ( N ) < 2 N S_{C}(N)<2^{N}
  11. N > 1 N>1
  12. S C ( n ) < 2 n S_{C}(n)<2^{n}
  13. n N n\geq N
  14. V C ( C ) = min 𝑛 { n : S C ( n ) < 2 n } VC(C)=\underset{n}{\min}\{n:S_{C}(n)<2^{n}\}\,
  15. V C 0 ( C ) = max 𝑛 { n : S C ( n ) = 2 n } . VC_{0}(C)=\underset{n}{\max}\{n:S_{C}(n)=2^{n}\}.\,
  16. V C ( C ) = V C 0 ( C ) + 1. VC(C)=VC_{0}(C)+1.
  17. S C ( n ) = 2 n S_{C}(n)=2^{n}

Shear_modulus.html

  1. G = def τ x y γ x y = F / A Δ x / l = F l A Δ x G\ \stackrel{\mathrm{def}}{=}\ \frac{\tau_{xy}}{\gamma_{xy}}=\frac{F/A}{\Delta x% /l}=\frac{Fl}{A\Delta x}
  2. τ x y = F / A \tau_{xy}=F/A\,
  3. F F
  4. A A
  5. γ x y = Δ x / l = tan θ \gamma_{xy}=\Delta x/l=\tan\theta\,
  6. γ x y = θ \gamma_{xy}=\theta
  7. Δ x \Delta x
  8. l l
  9. ( v s ) (v_{s})
  10. v s = G ρ v_{s}=\sqrt{\frac{G}{\rho}}
  11. ρ \rho
  12. μ ( T ) = μ 0 - D exp ( T 0 / T ) - 1 \mu(T)=\mu_{0}-\frac{D}{\exp(T_{0}/T)-1}
  13. μ ( p , T ) = μ 0 + μ p p η 1 / 3 + μ T ( T - 300 ) ; η := ρ / ρ 0 \mu(p,T)=\mu_{0}+\frac{\partial\mu}{\partial p}\frac{p}{\eta^{1/3}}+\frac{% \partial\mu}{\partial T}(T-300);\quad\eta:=\rho/\rho_{0}
  14. μ ( p , T ) = 1 𝒥 ( T ^ ) [ ( μ 0 + μ p p η 1 / 3 ) ( 1 - T ^ ) + ρ C m k b T ] ; C := ( 6 π 2 ) 2 / 3 3 f 2 \mu(p,T)=\frac{1}{\mathcal{J}(\hat{T})}\left[\left(\mu_{0}+\frac{\partial\mu}{% \partial p}\cfrac{p}{\eta^{1/3}}\right)(1-\hat{T})+\frac{\rho}{Cm}~{}k_{b}~{}T% \right];\quad C:=\cfrac{(6\pi^{2})^{2/3}}{3}f^{2}
  15. 𝒥 ( T ^ ) := 1 + exp [ - 1 + 1 / ζ 1 + ζ / ( 1 - T ^ ) ] for T ^ := T T m [ 0 , 1 + ζ ] , \mathcal{J}(\hat{T}):=1+\exp\left[-\cfrac{1+1/\zeta}{1+\zeta/(1-\hat{T})}% \right]\quad\,\text{for}\quad\hat{T}:=\frac{T}{T_{m}}\in[0,1+\zeta],

Sheet_metal.html

  1. F M a x = k T L t 2 W F_{Max}=k\frac{TLt^{2}}{W}

Sheet_resistance.html

  1. R = ρ L A = ρ L W t R=\rho\frac{L}{A}=\rho\frac{L}{Wt}
  2. ρ \rho
  3. A A
  4. L L
  5. W W
  6. t t
  7. R = ρ t L W = R s L W R=\frac{\rho}{t}\frac{L}{W}=R_{s}\frac{L}{W}
  8. R s R_{s}
  9. ρ \rho
  10. ρ = R s t \rho=R_{s}\cdot t
  11. Ω / \Omega/\Box
  12. L = W L=W
  13. R S = R R_{S}=R
  14. ρ ¯ = 1 σ ¯ \overline{\rho}=\frac{1}{\overline{\sigma}}
  15. R s = ρ ¯ / x j = ( σ ¯ x j ) - 1 = 1 0 x j σ ( x ) d x R_{s}=\overline{\rho}/x_{j}=(\overline{\sigma}x_{j})^{-1}=\frac{1}{\int_{0}^{x% _{j}}\sigma(x)dx}
  16. R s = 1 0 x j μ q N ( x ) d x R_{s}=\frac{1}{\int_{0}^{x_{j}}\mu qN(x)dx}
  17. x j x_{j}
  18. μ \mu
  19. q q
  20. N ( x ) N(x)
  21. N B N_{B}
  22. R s x j R_{s}x_{j}

Shekel_function.html

  1. n n
  2. m m
  3. f ( x ) = i = 1 m ( c i + j = 1 n ( x j - a j i ) 2 ) - 1 f(\vec{x})=\sum_{i=1}^{m}\;\left(c_{i}+\sum\limits_{j=1}^{n}(x_{j}-a_{ji})^{2}% \right)^{-1}
  4. f ( x 1 , x 2 , , x n - 1 , x n ) = i = 1 m ( c i + j = 1 n ( x j - a i j ) 2 ) - 1 f(x_{1},x_{2},...,x_{n-1},x_{n})=\sum_{i=1}^{m}\;\left(c_{i}+\sum\limits_{j=1}% ^{n}(x_{j}-a_{ij})^{2}\right)^{-1}

Shell_theorem.html

  1. r r
  2. m m
  3. r 3 r^{3}
  4. m / r 2 m/r^{2}
  5. r 3 / r 2 = r r^{3}/r^{2}=r
  6. r r
  7. d F = G m d M s 2 . dF=\frac{Gm\;dM}{s^{2}}.
  8. d F r = G m d M s 2 cos ϕ . dF_{r}=\frac{Gm\;dM}{s^{2}}\cos\phi.
  9. F r = d F r F_{r}=\int dF_{r}
  10. F r = G m d M cos ϕ s 2 . F_{r}=Gm\int\frac{dM\cos\phi}{s^{2}}.
  11. 4 π R 2 4\pi R^{2}\,
  12. 2 π R 2 sin θ d θ 2\pi R^{2}\sin\theta\,d\theta
  13. d M = 2 π R 2 sin θ 4 π R 2 M d θ = 1 2 M sin θ d θ dM=\frac{2\pi R^{2}\sin\theta}{4\pi R^{2}}M\,d\theta=\textstyle\frac{1}{2}M% \sin\theta\,d\theta
  14. F r = G M m 2 sin θ cos ϕ s 2 d θ F_{r}=\frac{GMm}{2}\int\frac{\sin\theta\cos\phi}{s^{2}}\,d\theta
  15. cos ϕ = r 2 + s 2 - R 2 2 r s \cos\phi=\frac{r^{2}+s^{2}-R^{2}}{2rs}
  16. cos θ = r 2 + R 2 - s 2 2 r R . \cos\theta=\frac{r^{2}+R^{2}-s^{2}}{2rR}.
  17. sin θ d θ = s r R d s . \sin\theta\;d\theta=\frac{s}{rR}ds.
  18. F r = G M m 2 r R cos ϕ s d s F_{r}=\frac{GMm}{2rR}\int\frac{\cos\phi}{s}\,ds
  19. F r = G M m 4 r 2 R ( 1 + r 2 - R 2 s 2 ) d s . F_{r}=\frac{GMm}{4r^{2}R}\int\left(1+\frac{r^{2}-R^{2}}{s^{2}}\right)\,ds.
  20. s - r 2 - R 2 s s-\frac{r^{2}-R^{2}}{s}
  21. F r = G M m r 2 , F_{r}=\frac{GMm}{r^{2}},
  22. F t o t a l = d F r = G m r 2 d M . F_{total}=\int dF_{r}=\frac{Gm}{r^{2}}\int dM.
  23. d M = 4 π x 2 d x 4 3 π R 3 M = 3 M x 2 d x R 3 dM=\frac{4\pi x^{2}dx}{\frac{4}{3}\pi R^{3}}M=\frac{3Mx^{2}dx}{R^{3}}
  24. F t o t a l = 3 G M m r 2 R 3 0 R x 2 d x = G M m r 2 F_{total}=\frac{3GMm}{r^{2}R^{3}}\int_{0}^{R}x^{2}dx=\frac{GMm}{r^{2}}
  25. s - r 2 - R 2 s s-\frac{r^{2}-R^{2}}{s}
  26. F r = 0 F_{r}=0\,
  27. f = k / r p f=k/r^{p}
  28. F ( r ) = G M m 4 r 2 R R - r R + r ( 1 s p - 2 + r 2 - R 2 s p ) d s F(r)=\frac{GMm}{4r^{2}R}\int_{R-r}^{R+r}\left(\frac{1}{s^{p-2}}+\frac{r^{2}-R^% {2}}{s^{p}}\right)\,ds
  29. F ( r ) F(r)
  30. p = 2 p=2
  31. S 𝐠 d 𝐒 = - 4 π G M \int_{S}{\mathbf{g}}\cdot\,d{\mathbf{S}}=-4\pi GM
  32. S 𝐠 d 𝐒 = S 𝐠 𝐧 ^ d S \int_{S}{\mathbf{g}}\cdot\,d{\mathbf{S}}=\int_{S}{\mathbf{g}}\cdot{\mathbf{% \hat{n}}}\,dS
  33. 𝐧 ^ \mathbf{\hat{n}}\,
  34. 𝐧 ^ \mathbf{\hat{n}}
  35. 𝐠 = g ( r ) 𝐧 ^ \mathbf{g}=g(r)\mathbf{\hat{n}}
  36. 𝐧 ^ \mathbf{\hat{n}}
  37. 𝐠 = g ( r ) 𝐧 ^ \mathbf{g}=g(r)\mathbf{\hat{n}}
  38. S 𝐠 d 𝐒 = g ( r ) S d S = g ( r ) 4 π r 2 \int_{S}\mathbf{g}\cdot\,d{\mathbf{S}}=g(r)\int_{S}\,d{S}=g(r)4\pi r^{2}
  39. g ( r ) 4 π r 2 = - 4 π G M g(r)4\pi r^{2}=-4\pi GM\,
  40. g ( r ) = - G M r 2 . g(r)=-\frac{GM}{r^{2}}.
  41. F F
  42. F = M m f ( r ) F=Mmf(r)
  43. f f
  44. F = - G M m r 2 - Λ M m r 3 F=-\frac{GMm}{r^{2}}-\frac{\Lambda Mmr}{3}
  45. Λ \Lambda
  46. d M = R 2 / 2 d θ sin 2 θ π R 2 M = sin 2 θ 2 π M d θ dM=\frac{R^{2}/2d\theta\sin^{2}{\theta}}{\pi R^{2}}M=\frac{\sin^{2}{\theta}}{2% \pi}Md\theta
  47. F r = G M m 2 π sin 2 θ cos ϕ s 2 d θ . F_{r}=\frac{GMm}{2\pi}\int\frac{\sin^{2}{\theta}\cos\phi}{s^{2}}d\theta.
  48. M = π R 2 ρ M=\pi R^{2}\rho
  49. F ( r ) = G m ρ 8 r 3 R - r R + r ( r 2 + s 2 - R 2 ) 2 ( r 2 R 2 + r 2 s 2 + R 2 s 2 ) - s 4 - r 4 - R 4 s 2 d s F(r)=\frac{Gm\rho}{8r^{3}}\int_{R-r}^{R+r}{\frac{(r^{2}+s^{2}-R^{2})\sqrt{2(r^% {2}R^{2}+r^{2}s^{2}+R^{2}s^{2})-s^{4}-r^{4}-R^{4}}}{s^{2}}}\,ds
  50. ρ \rho
  51. k g m 2 \frac{kg}{m^{2}}

Short-rate_model.html

  1. r t r_{t}\,
  2. r t r_{t}\,
  3. t t
  4. r t r_{t}\,
  5. Q Q
  6. t t
  7. T T
  8. P ( t , T ) = 𝔼 Q [ exp ( - t T r s d s ) | t ] P(t,T)=\mathbb{E}^{Q}\left[\left.\exp{\left(-\int_{t}^{T}r_{s}\,ds\right)}% \right|\mathcal{F}_{t}\right]
  9. \mathcal{F}
  10. f ( t , T ) = - T ln ( P ( t , T ) ) . f(t,T)=-\frac{\partial}{\partial T}\ln(P(t,T)).
  11. W t W_{t}\,
  12. d W t dW_{t}\,
  13. X t X_{t}\,
  14. r t r_{t}\,
  15. r t = exp X t r_{t}=\exp{X_{t}}\,
  16. r t = r 0 + a t + σ W t * r_{t}=r_{0}+at+\sigma W^{*}_{t}
  17. W t * W^{*}_{t}
  18. d r t = ( θ - α r t ) d t + σ d W t dr_{t}=(\theta-\alpha r_{t})\,dt+\sigma\,dW_{t}
  19. d r t = a ( b - r t ) d t + σ d W t dr_{t}=a(b-r_{t})\,dt+\sigma\,dW_{t}
  20. d r t = θ r t d t + σ r t d W t dr_{t}=\theta r_{t}\,dt+\sigma r_{t}\,dW_{t}
  21. d r t = ( θ - α r t ) d t + r t σ d W t dr_{t}=(\theta-\alpha r_{t})\,dt+\sqrt{r_{t}}\,\sigma\,dW_{t}
  22. d r t = a ( b - r t ) d t + r t σ d W t dr_{t}=a(b-r_{t})\,dt+\sqrt{r_{t}}\,\sigma\,dW_{t}
  23. σ r t \sigma\sqrt{r_{t}}
  24. d r t = θ t d t + σ d W t dr_{t}=\theta_{t}\,dt+\sigma\,dW_{t}
  25. d r t = ( θ t - α r t ) d t + σ t d W t dr_{t}=(\theta_{t}-\alpha r_{t})\,dt+\sigma_{t}\,dW_{t}
  26. θ , α \theta,\alpha
  27. σ \sigma
  28. d ln ( r ) = [ θ t + σ t σ t ln ( r ) ] d t + σ t d W t d\ln(r)=[\theta_{t}+\frac{\sigma^{\prime}_{t}}{\sigma_{t}}\ln(r)]dt+\sigma_{t}% \,dW_{t}
  29. d ln ( r ) = θ t d t + σ d W t d\ln(r)=\theta_{t}\,dt+\sigma\,dW_{t}
  30. d ln ( r ) = [ θ t - ϕ t ln ( r ) ] d t + σ t d W t d\ln(r)=[\theta_{t}-\phi_{t}\ln(r)]\,dt+\sigma_{t}\,dW_{t}
  31. d ln ( r t ) = θ t d t + σ d W t d\ln(r_{t})=\theta_{t}\,dt+\sigma\,dW_{t}
  32. d X t = ( a t - b X t ) d t + X t c t d W 1 t dX_{t}=(a_{t}-bX_{t})\,dt+\sqrt{X_{t}}\,c_{t}\,dW_{1t}
  33. d Y t = ( d t - e Y t ) d t + Y t f t d W 2 t dY_{t}=(d_{t}-eY_{t})\,dt+\sqrt{Y_{t}}\,f_{t}\,dW_{2t}
  34. d r t = ( μ X + θ Y ) d t + σ t Y d W 3 t dr_{t}=(\mu X+\theta Y)dt+\sigma_{t}\sqrt{Y}dW_{3t}
  35. d r t = ( θ t - α t ) d t + r t σ t d W t dr_{t}=(\theta_{t}-\alpha_{t})\,dt+\sqrt{r_{t}}\,\sigma_{t}\,dW_{t}
  36. d α t = ( ζ t - α t ) d t + α t σ t d W t d\alpha_{t}=(\zeta_{t}-\alpha_{t})\,dt+\sqrt{\alpha_{t}}\,\sigma_{t}\,dW_{t}
  37. d σ t = ( β t - σ t ) d t + σ t η t d W t d\sigma_{t}=(\beta_{t}-\sigma_{t})\,dt+\sqrt{\sigma_{t}}\,\eta_{t}\,dW_{t}

Shulba_Sutras.html

  1. ( 3 , 4 , 5 ) (3,4,5)
  2. ( 5 , 12 , 13 ) (5,12,13)
  3. ( 8 , 15 , 17 ) (8,15,17)
  4. ( 12 , 35 , 37 ) (12,35,37)
  5. 2 1 + 1 3 + 1 3 4 - 1 3 4 34 = 577 408 = 1.4142... \sqrt{2}\approx 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}=% \frac{577}{408}=1.4142...
  6. a 2 + r a + r 2 a - ( r / 2 a ) 2 2 ( a + r 2 a ) , \sqrt{a^{2}+r}\approx a+\frac{r}{2a}-\frac{(r/2a)^{2}}{2(a+\frac{r}{2a})},
  7. a = 4 / 3 a=4/3
  8. r = 2 / 9 r=2/9
  9. 2 = 1 + 24 60 + 51 60 2 + 10 60 3 = 1.41421297. \sqrt{2}=1+\frac{24}{60}+\frac{51}{60^{2}}+\frac{10}{60^{3}}=1.41421297.
  10. 2 \sqrt{2}
  11. x x - 1 + 1 2 x - 1 \sqrt{x}\approx\sqrt{x-1}+\frac{1}{2\cdot\sqrt{x-1}}
  12. a 2 + r a + r 2 a \sqrt{a^{2}+r}\approx a+\frac{r}{2\cdot a}

Side_lobe.html

  1. Radiation Pattern (in units of dB) 20 log 10 ( | sin ( X ) X | ) \displaystyle\mbox{Radiation Pattern (in units of dB)}~{}\propto 20\log_{10}% \left(\left|\frac{\sin(X)}{X}\right|\right)
  2. X \displaystyle X
  3. X \displaystyle X
  4. 0 \displaystyle 0
  5. 0 dB \displaystyle 0~{}\mbox{dB}~{}
  6. 3.14 = π \displaystyle 3.14=\pi
  7. - dB -\infty~{}\mbox{dB}~{}
  8. 4.49 3 π 2 4.49\approx\frac{3\pi}{2}
  9. - 13.26 dB \displaystyle-13.26~{}\mbox{dB}~{}
  10. 6.28 = 2 π \displaystyle 6.28=2\pi
  11. - dB -\infty~{}\mbox{dB}~{}
  12. 7.72 5 π 2 7.72\approx\frac{5\pi}{2}
  13. - 17.83 dB \displaystyle-17.83~{}\mbox{dB}~{}
  14. Radiation Pattern (in units of dB) 10 log 10 ( | 2 J 1 ( X ) X | 2 ) \displaystyle\mbox{Radiation Pattern (in units of dB)}~{}\propto 10\log_{10}% \left(\left|2\frac{J_{1}(X)}{X}\right|^{2}\right)
  15. J 1 ( x ) \displaystyle J_{1}(x)
  16. X \displaystyle X
  17. X \displaystyle X
  18. 0 \displaystyle 0
  19. 0 dB \displaystyle 0~{}\mbox{dB}~{}
  20. 3.83 \displaystyle 3.83
  21. - dB -\infty~{}\mbox{dB}~{}
  22. 5.14 \displaystyle 5.14
  23. - 17.57 dB \displaystyle-17.57~{}\mbox{dB}~{}
  24. 7.02 \displaystyle 7.02
  25. - dB -\infty~{}\mbox{dB}~{}
  26. 8.42 \displaystyle 8.42
  27. - 23.81 dB \displaystyle-23.81~{}\mbox{dB}~{}

Sierpiński's_constant.html

  1. K = lim n [ k = 1 n r 2 ( k ) k - π ln n ] K=\lim_{n\to\infty}\left[\sum_{k=1}^{n}{r_{2}(k)\over k}-\pi\ln n\right]
  2. K \displaystyle K
  3. G G
  4. γ \gamma

Sigma-ideal.html

  1. { A n } n N n A n N . \left\{A_{n}\right\}_{n\in\mathbb{N}}\subseteq N\Rightarrow\bigcup_{n\in% \mathbb{N}}A_{n}\in N.

Sigma_approximation.html

  1. s ( θ ) = 1 2 a 0 + k = 1 m - 1 sinc ( k m ) [ a k cos ( 2 π k T θ ) + b k sin ( 2 π k T θ ) ] , s(\theta)=\frac{1}{2}a_{0}+\sum_{k=1}^{m-1}\mathrm{sinc}\Bigl(\frac{k}{m}\Bigr% )\cdot\left[a_{k}\cos\Bigl(\frac{2\pi k}{T}\theta\Bigr)+b_{k}\sin\Bigl(\frac{2% \pi k}{T}\theta\Bigr)\right],
  2. sinc x = sin π x π x . \mathrm{sinc}\,x=\frac{\sin\pi x}{\pi x}.
  3. sinc ( k m ) \mathrm{sinc}\Bigl(\frac{k}{m}\Bigr)

Signaling_game.html

  1. t j t_{j}
  2. m * ( t j ) m^{*}(t_{j})
  3. m ( t j ) m(t_{j})
  4. t j t_{j}
  5. a * ( m ) a^{*}(m)
  6. μ ( t i | m ) \mu(t_{i}|m)
  7. t i t_{i}
  8. m m
  9. t i t_{i}
  10. m m
  11. μ ( t | m ) \mu(t|m)
  12. t i μ ( t i | m ) U R ( t i , m , a ) \sum_{t_{i}}\mu(t_{i}|m)U_{R}(t_{i},m,a)
  13. a a
  14. a * ( m ) a^{*}(m)
  15. t t
  16. m * m^{*}
  17. U S ( t , m , a * ( m ) ) U_{S}(t,m,a^{*}(m))
  18. a * a^{*}
  19. m m
  20. t t
  21. m * ( t ) m^{*}(t)
  22. m m
  23. m m
  24. μ ( t | m ) \mu(t|m)
  25. μ ( t | m ) = p ( t ) / t i p ( t i ) \mu(t|m)=p(t)/\sum_{t_{i}}p(t_{i})

Signalling_(economics).html

  1. q 1 q_{1}
  2. 1 - q 1 1-q_{1}
  3. W ( y ) = { 1 , y < y * 2 , y > y * W(y)=\begin{cases}1,&y<y*\\ 2,&y>y*\end{cases}

Signature_of_a_knot.html

  1. ϕ : H 1 ( S ) × H 1 ( S ) \phi:H_{1}(S)\times H_{1}(S)\to\mathbb{Z}
  2. l k ( a + , b - ) lk(a^{+},b^{-})
  3. a , b H 1 ( S ) a,b\in H_{1}(S)
  4. a + , b - a^{+},b^{-}
  5. b 1 , , b 2 g b_{1},...,b_{2g}
  6. H 1 ( S ) H_{1}(S)
  7. V i j = ϕ ( b i , b j ) V_{ij}=\phi(b_{i},b_{j})
  8. V + V V+V^{\perp}
  9. X X
  10. H 1 ( X ; ) H_{1}(X;\mathbb{Q})
  11. [ ] \mathbb{Q}[\mathbb{Z}]
  12. V V
  13. V ¯ \overline{V}
  14. [ ] \mathbb{Q}[\mathbb{Z}]
  15. \mathbb{Q}
  16. V V
  17. \mathbb{Z}
  18. X X
  19. H 1 ( X ; ) H 2 ( X ; ) ¯ H_{1}(X;\mathbb{Q})\simeq\overline{H^{2}(X;\mathbb{Q})}
  20. H 2 ( X ; ) H^{2}(X;\mathbb{Q})
  21. X X
  22. \mathbb{Q}
  23. H 2 ( X ; ) H^{2}(X;\mathbb{Q})
  24. E x t [ ] ( H 1 ( X ; ) , [ ] ) Ext_{\mathbb{Q}[\mathbb{Z}]}(H_{1}(X;\mathbb{Q}),\mathbb{Q}[\mathbb{Z}])
  25. [ ] \mathbb{Q}[\mathbb{Z}]
  26. [ ] \mathbb{Q}[\mathbb{Z}]
  27. E x t [ ] ( H 1 ( X ; ) , [ ] ) H o m [ ] ( H 1 ( X ; ) , [ [ ] ] / [ ] ) Ext_{\mathbb{Q}[\mathbb{Z}]}(H_{1}(X;\mathbb{Q}),\mathbb{Q}[\mathbb{Z}])\simeq Hom% _{\mathbb{Q}[\mathbb{Z}]}(H_{1}(X;\mathbb{Q}),[\mathbb{Q}[\mathbb{Z}]]/\mathbb% {Q}[\mathbb{Z}])
  28. [ [ ] ] [\mathbb{Q}[\mathbb{Z}]]
  29. [ ] \mathbb{Q}[\mathbb{Z}]
  30. H 1 ( X ; ) × H 1 ( X ; ) [ [ ] ] / [ ] H_{1}(X;\mathbb{Q})\times H_{1}(X;\mathbb{Q})\to[\mathbb{Q}[\mathbb{Z}]]/% \mathbb{Q}[\mathbb{Z}]
  31. [ [ ] ] [\mathbb{Q}[\mathbb{Z}]]
  32. [ ] \mathbb{Q}[\mathbb{Z}]
  33. [ ] \mathbb{Q}[\mathbb{Z}]
  34. [ ] / Δ K \mathbb{Q}[\mathbb{Z}]/\Delta K
  35. t r : [ ] / Δ K tr:\mathbb{Q}[\mathbb{Z}]/\Delta K\to\mathbb{Q}
  36. t t - 1 t\longmapsto t^{-1}
  37. H 1 ( X ; ) H_{1}(X;\mathbb{Q})
  38. H 1 ( X ; ) H_{1}(X;\mathbb{Q})
  39. H 1 ( X ; ) H_{1}(X;\mathbb{R})

Significance_arithmetic.html

  1. | x f / f | ; \left|xf^{\prime}/f\right|;
  2. e x e^{x}
  3. | x | |x|
  4. ln ( x ) \ln(x)
  5. 1 | ln ( x ) | \frac{1}{|\ln(x)|}
  6. sin ( x ) \sin(x)
  7. | x cot ( x ) | |x\cot(x)|
  8. cos ( x ) \cos(x)
  9. | x tan ( x ) | |x\tan(x)|
  10. tan ( x ) \tan(x)
  11. | x ( tan ( x ) + cot ( x ) ) | |x(\tan(x)+\cot(x))|
  12. arcsin ( x ) \arcsin(x)
  13. | x 1 - x 2 arcsin ( x ) | \left|\frac{x}{\sqrt{1-x^{2}}\arcsin(x)}\right|
  14. arccos ( x ) \arccos(x)
  15. | x 1 - x 2 arccos ( x ) | \left|\frac{x}{\sqrt{1-x^{2}}\arccos(x)}\right|
  16. arctan ( x ) \arctan(x)
  17. | x ( 1 + x 2 ) arctan ( x ) | \left|\frac{x}{(1+x^{2})\arctan(x)}\right|

Silhouette_edge.html

  1. a x + b y + c z + d = { > 0 front facing = 0 parallel < 0 back facing ax+by+cz+d=\begin{cases}>0&\,\text{front facing}\\ =0&\,\text{parallel}\\ <0&\,\text{back facing}\end{cases}
  2. 𝐧𝐨𝐫𝐦𝐚𝐥 ( light position ) + plane D = a , b , c , d L x , L y , L z , L w + plane D \,\textbf{normal}\cdot(\,\textbf{light position})+\,\text{plane}_{D}=\langle a% ,b,c,d\rangle\cdot\langle L_{x},L_{y},L_{z},L_{w}\rangle+\,\text{plane}_{D}
  3. a L x + b L y + c L z + d L w + plane D = indicator aL_{x}+bL_{y}+cL_{z}+dL_{w}+\,\text{plane}_{D}=\,\text{indicator}
  4. plane D = PointOnPlane ( 𝐧𝐨𝐫𝐦𝐚𝐥 ) \,\text{plane}_{D}=\,\text{PointOnPlane}\cdot(\,\textbf{normal})
  5. indicator = { > 0 front-facing = 0 parallel < 0 back-facing \,\text{indicator}=\begin{cases}>0&\,\text{front-facing}\\ =0&\,\text{parallel}\\ <0&\,\text{back-facing}\end{cases}

Silver_chloride_electrode.html

  1. A g + + e - A g ( s ) Ag^{+}+e^{-}\rightleftharpoons Ag(s)
  2. A g C l ( s ) A g + + C l - AgCl(s)\rightleftharpoons Ag^{+}+Cl^{-}
  3. A g C l ( s ) + e - A g ( s ) + C l - AgCl(s)+e^{-}\rightleftharpoons Ag(s)+Cl^{-}
  4. E = E 0 - R T F ln a C l - E=E^{0}-\frac{RT}{F}\ln a_{Cl^{-}}

Silverman–Toeplitz_theorem.html

  1. ( a i , j ) i , j (a_{i,j})_{i,j\in\mathbb{N}}
  2. lim i a i , j = 0 j \lim_{i\to\infty}a_{i,j}=0\quad j\in\mathbb{N}
  3. lim i j = 0 a i , j = 1 \lim_{i\to\infty}\sum_{j=0}^{\infty}a_{i,j}=1
  4. sup i j = 0 | a i , j | < \sup_{i}\sum_{j=0}^{\infty}|a_{i,j}|<\infty

SIMCOS.html

  1. e - s T e^{-sT}
  2. e - s T = ˙ ( s T ) 2 - 6 s T + 12 ( s T ) 2 + 6 s T + 12 e^{-sT}\dot{=}\frac{(sT)^{2}-6sT+12}{(sT)^{2}+6sT+12}
  3. e - s T = ˙ ( s T ) 4 - 20 ( s T ) 3 + 180 ( s T ) 2 - 840 s T + 1680 ( s T ) 4 + 20 ( s T ) 3 + 180 ( s T ) 2 + 840 s T + 1680 . e^{-sT}\dot{=}\frac{(sT)^{4}-20(sT)^{3}+180(sT)^{2}-840sT+1680}{(sT)^{4}+20(sT% )^{3}+180(sT)^{2}+840sT+1680}.

Similarity_invariance.html

  1. f f
  2. f ( A ) = f ( B - 1 A B ) f(A)=f(B^{-1}AB)
  3. B - 1 A B B^{-1}AB
  4. B - 1 A B B^{-1}AB
  5. B B

Simple_function.html

  1. f : X f:X\to\mathbb{C}
  2. f ( x ) = k = 1 n a k 𝟏 A k ( x ) , f(x)=\sum_{k=1}^{n}a_{k}{\mathbf{1}}_{A_{k}}(x),
  3. 𝟏 A {\mathbf{1}}_{A}
  4. \mathbb{C}
  5. k = 1 n a k μ ( A k ) , \sum_{k=1}^{n}a_{k}\mu(A_{k}),
  6. f : X + f\colon X\to\mathbb{R}^{+}
  7. f f
  8. ( X , Σ , μ ) (X,\Sigma,\mu)
  9. n n\in\mathbb{N}
  10. f f
  11. 2 2 n + 1 2^{2n}+1
  12. 2 2 n 2^{2n}
  13. 2 - n 2^{-n}
  14. n n
  15. I n , k = [ k - 1 2 n , k 2 n ) I_{n,k}=\left[\frac{k-1}{2^{n}},\frac{k}{2^{n}}\right)
  16. k = 1 , 2 , , 2 2 n k=1,2,\ldots,2^{2n}
  17. I n , 2 2 n + 1 = [ 2 n , ) I_{n,2^{2n}+1}=[2^{n},\infty)
  18. n n
  19. I n , k I_{n,k}
  20. A n , k = f - 1 ( I n , k ) A_{n,k}=f^{-1}(I_{n,k})\,
  21. k = 1 , 2 , , 2 2 n + 1 k=1,2,\ldots,2^{2n}+1
  22. f n = k = 1 2 2 n + 1 k - 1 2 n 𝟏 A n , k f_{n}=\sum_{k=1}^{2^{2n}+1}\frac{k-1}{2^{n}}{\mathbf{1}}_{A_{n,k}}
  23. f f
  24. n n\to\infty
  25. f f
  26. f f
  27. f f

Simple_precedence_grammar.html

  1. \in
  2. S a S S b | c S\to aSSb|c
  3. = ˙ \dot{=}
  4. \lessdot
  5. = ˙ \dot{=}
  6. \lessdot
  7. = ˙ \dot{=}
  8. \lessdot
  9. \lessdot
  10. \gtrdot
  11. \gtrdot
  12. \gtrdot
  13. \gtrdot
  14. \gtrdot
  15. \gtrdot
  16. \gtrdot
  17. \lessdot
  18. \lessdot

Simple_precedence_parser.html

  1. \lessdot
  2. = ˙ \dot{=}
  3. \gtrdot
  4. = ˙ \dot{=}
  5. \lessdot
  6. \gtrdot
  7. \lessdot
  8. = ˙ \dot{=}
  9. \gtrdot
  10. \gtrdot
  11. = ˙ \dot{=}
  12. \gtrdot
  13. = ˙ \dot{=}
  14. \gtrdot
  15. \gtrdot
  16. \gtrdot
  17. \gtrdot
  18. \gtrdot
  19. \gtrdot
  20. \gtrdot
  21. \gtrdot
  22. \gtrdot
  23. \lessdot
  24. = ˙ \dot{=}
  25. \lessdot
  26. \lessdot
  27. \lessdot
  28. = ˙ \dot{=}
  29. \lessdot
  30. \lessdot
  31. \lessdot
  32. = ˙ \dot{=}
  33. \lessdot
  34. \lessdot
  35. \lessdot
  36. \lessdot
  37. \lessdot
  38. \gtrdot
  39. \gtrdot
  40. \gtrdot
  41. \gtrdot
  42. \gtrdot
  43. \gtrdot
  44. \gtrdot
  45. \gtrdot
  46. \lessdot
  47. \lessdot
  48. \lessdot
  49. \lessdot
  50. \lessdot
  51. \lessdot

Simply_typed_lambda_calculus.html

  1. λ \lambda^{\to}
  2. \to
  3. \to
  4. σ \sigma
  5. τ \tau
  6. σ τ \sigma\to\tau
  7. σ \sigma
  8. τ \tau
  9. \to
  10. σ τ ρ \sigma\to\tau\to\rho
  11. σ ( τ ρ ) \sigma\to(\tau\to\rho)
  12. B B
  13. τ : := τ τ T where T B \tau::=\tau\to\tau\mid T\quad\mathrm{where}\quad T\in B
  14. B = { a , b } B=\{a,b\}
  15. a a
  16. b b
  17. a a a\to a
  18. a b a\to b
  19. a b a a\to b\to a
  20. o o
  21. ι \iota
  22. o o
  23. ι \iota
  24. o o
  25. : x , τ x\mathbin{:}\tau
  26. x x
  27. τ \tau
  28. e : := x λ x : τ . e e e c e::=x\mid\lambda x\mathbin{:}\tau.e\mid e\,e\mid c
  29. c c
  30. x x
  31. x x
  32. e : := x λ x . e e e e::=x\mid\lambda x.e\mid e\,e
  33. Γ , Δ , \Gamma,\Delta,\dots
  34. : x , σ x\mathbin{:}\sigma
  35. x x
  36. σ \sigma
  37. Γ : e , σ \Gamma\vdash e\mathbin{:}\sigma
  38. e e
  39. σ \sigma
  40. Γ \Gamma
  41. e e
  42. σ \sigma
  43. : x , σ Γ Γ : x , σ {\frac{x\mathbin{:}\sigma\in\Gamma}{\Gamma\vdash x\mathbin{:}\sigma}}
  44. c is a constant of type T Γ : c , T {\frac{c\,\text{ is a constant of type }T}{\Gamma\vdash c\mathbin{:}T}}
  45. Γ , : x , σ : e , τ Γ ( λ x : σ . e ) : ( σ τ ) {\frac{\Gamma,x\mathbin{:}\sigma\vdash e\mathbin{:}\tau}{\Gamma\vdash(\lambda x% \mathbin{:}\sigma.~{}e)\mathbin{:}(\sigma\to\tau)}}
  46. Γ : e 1 , σ τ Γ : e 2 , σ Γ : e 1 e 2 , τ {\frac{\Gamma\vdash e_{1}\mathbin{:}\sigma\to\tau\quad\Gamma\vdash e_{2}% \mathbin{:}\sigma}{\Gamma\vdash e_{1}~{}e_{2}\mathbin{:}\tau}}
  47. x x
  48. σ \sigma
  49. x x
  50. σ \sigma
  51. x x
  52. σ \sigma
  53. e e
  54. τ \tau
  55. x x
  56. : λ x , σ . e \lambda x\mathbin{:}\sigma.~{}e
  57. σ τ \sigma\to\tau
  58. e 1 e_{1}
  59. σ τ \sigma\to\tau
  60. e 2 e_{2}
  61. σ \sigma
  62. e 1 e 2 e_{1}~{}e_{2}
  63. τ \tau
  64. τ \tau
  65. : λ x , τ . : x , τ τ \lambda x\mathbin{:}\tau.x\mathbin{:}\tau\to\tau
  66. σ , τ \sigma,\tau
  67. : λ x , σ . : λ y , τ . : x , σ τ σ \lambda x\mathbin{:}\sigma.\lambda y\mathbin{:}\tau.x\mathbin{:}\sigma\to\tau\to\sigma
  68. τ , τ , τ ′′ \tau,\tau^{\prime},\tau^{\prime\prime}
  69. λ x : τ τ τ ′′ . λ y : τ τ . λ z : τ . x z ( y z ) : ( τ τ τ ′′ ) ( τ τ ) τ τ ′′ \lambda x\mathbin{:}\tau\to\tau^{\prime}\to\tau^{\prime\prime}.\lambda y% \mathbin{:}\tau\to\tau^{\prime}.\lambda z\mathbin{:}\tau.xz(yz):(\tau\to\tau^{% \prime}\to\tau^{\prime\prime})\to(\tau\to\tau^{\prime})\to\tau\to\tau^{\prime\prime}
  70. τ \tau
  71. o ( τ ) o(\tau)
  72. o ( T ) = 0 o(T)=0
  73. o ( σ τ ) = max ( o ( σ ) + 1 , o ( τ ) ) o(\sigma\to\tau)=\mbox{max}~{}(o(\sigma)+1,o(\tau))
  74. o ( ι ι ι ) = 1 o(\iota\to\iota\to\iota)=1
  75. o ( ( ι ι ) ι ) = 2 o((\iota\to\iota)\to\iota)=2
  76. : λ x , 𝚒𝚗𝚝 . x \lambda x\mathbin{:}\mathtt{int}.~{}x
  77. : λ x , 𝚋𝚘𝚘𝚕 . x \lambda x\mathbin{:}\mathtt{bool}.~{}x
  78. : λ x , 𝚒𝚗𝚝 . x \lambda x\mathbin{:}\mathtt{int}.~{}x
  79. : λ x , 𝚋𝚘𝚘𝚕 . x \lambda x\mathbin{:}\mathtt{bool}.~{}x
  80. λ x . x \lambda x.~{}x
  81. ( λ x : σ . t ) u = β t [ x := u ] (\lambda x\mathbin{:}\sigma.~{}t)\,u=_{\beta}t[x:=u]
  82. Γ \Gamma
  83. Γ , : x , σ : t , τ \Gamma,x\mathbin{:}\sigma\vdash t\mathbin{:}\tau
  84. Γ : u , σ \Gamma\vdash u\mathbin{:}\sigma
  85. : λ x , σ . t x = η t \lambda x\mathbin{:}\sigma.~{}t\,x=_{\eta}t
  86. Γ t : σ τ \Gamma\vdash t\!:\sigma\to\tau
  87. x x
  88. t t
  89. β η \beta\eta
  90. : s , σ s\mathbin{:}\sigma
  91. : t , τ t\mathbin{:}\tau
  92. ( s , t ) (s,t)
  93. σ × τ \sigma\times\tau
  94. : u , τ 1 × τ 2 u\mathbin{:}\tau_{1}\times\tau_{2}
  95. : π 1 ( u ) , τ 1 \pi_{1}(u)\mathbin{:}\tau_{1}
  96. : π 2 ( u ) , τ 2 \pi_{2}(u)\mathbin{:}\tau_{2}
  97. π i \pi_{i}
  98. ( ) ()
  99. π 1 ( : s , σ , : t , τ ) = : s , σ \pi_{1}(s\mathbin{:}\sigma,t\mathbin{:}\tau)=s\mathbin{:}\sigma
  100. π 2 ( : s , σ , : t , τ ) = : t , τ \pi_{2}(s\mathbin{:}\sigma,t\mathbin{:}\tau)=t\mathbin{:}\tau
  101. ( π 1 ( : u , σ × τ ) , π 2 ( : u , σ × τ ) ) = : u , σ × τ (\pi_{1}(u\mathbin{:}\sigma\times\tau),\pi_{2}(u\mathbin{:}\sigma\times\tau))=% u\mathbin{:}\sigma\times\tau
  102. : t , 1 = ( ) t\mathbin{:}1=()
  103. σ τ \sigma\to\tau
  104. ( : x , σ , : t , τ ) (x\mathbin{:}\sigma,t\mathbin{:}\tau)
  105. σ \sigma
  106. τ \tau
  107. λ x . x \lambda x.~{}x
  108. 𝚒𝚗𝚝 𝚒𝚗𝚝 \mathtt{int}\to\mathtt{int}
  109. 𝚋𝚘𝚘𝚕 𝚋𝚘𝚘𝚕 \mathtt{bool}\to\mathtt{bool}
  110. α α \alpha\to\alpha
  111. Γ e τ \Gamma\vdash e\Leftarrow\tau
  112. Γ e τ \Gamma\vdash e\Rightarrow\tau
  113. Γ \Gamma
  114. e e
  115. τ \tau
  116. Γ e τ \Gamma\vdash e\Leftarrow\tau
  117. Γ e τ \Gamma\vdash e\Rightarrow\tau
  118. Γ \Gamma
  119. e e
  120. τ \tau
  121. : x , σ Γ Γ x σ {\frac{x\mathbin{:}\sigma\in\Gamma}{\Gamma\vdash x\Rightarrow\sigma}}
  122. c is a constant of type T Γ c T {\frac{c\,\text{ is a constant of type }T}{\Gamma\vdash c\Rightarrow T}}
  123. Γ , : x , σ e τ Γ λ x . e σ τ {\frac{\Gamma,x\mathbin{:}\sigma\vdash e\Leftarrow\tau}{\Gamma\vdash\lambda x.% ~{}e\Leftarrow\sigma\to\tau}}
  124. Γ e 1 σ τ Γ e 2 σ Γ e 1 e 2 τ {\frac{\Gamma\vdash e_{1}\Rightarrow\sigma\to\tau\quad\Gamma\vdash e_{2}% \Leftarrow\sigma}{\Gamma\vdash e_{1}~{}e_{2}\Rightarrow\tau}}
  125. Γ e τ Γ e τ {\frac{\Gamma\vdash e\Rightarrow\tau}{\Gamma\vdash e\Leftarrow\tau}}
  126. Γ e τ Γ ( : e , τ ) τ {\frac{\Gamma\vdash e\Leftarrow\tau}{\Gamma\vdash(e\mathbin{:}\tau)\Rightarrow% \tau}}
  127. : x , σ x\mathbin{:}\sigma
  128. σ \sigma
  129. x x
  130. λ x . e \lambda x.~{}e
  131. σ τ \sigma\to\tau
  132. : x , σ x\mathbin{:}\sigma
  133. e e
  134. τ \tau
  135. e 1 e_{1}
  136. σ τ \sigma\to\tau
  137. e 2 e_{2}
  138. σ \sigma
  139. e 1 e 2 e_{1}~{}e_{2}
  140. τ \tau
  141. e e
  142. τ \tau
  143. τ \tau
  144. e e
  145. τ \tau
  146. ( : e , τ ) (e\mathbin{:}\tau)
  147. τ \tau
  148. λ \lambda
  149. Ω = ( λ x . x x ) ( λ x . x x ) \Omega=(\lambda x.~{}x~{}x)(\lambda x.~{}x~{}x)
  150. 𝚏𝚒𝚡 α \mathtt{fix}_{\alpha}
  151. ( α α ) α (\alpha\to\alpha)\to\alpha
  152. β \beta
  153. β η \beta\eta
  154. β η \beta\eta
  155. ( o o ) ( o o ) (o\to o)\to(o\to o)
  156. λ \lambda^{\to}
  157. λ \lambda^{\to}
  158. β η \beta\eta
  159. β η \beta\eta
  160. β η \beta\eta
  161. λ \lambda
  162. β η \beta\eta

Sine_bar.html

  1. sin ( a n g l e ) = p e r p e n d i c u l a r h y p o t e n u s e \sin\left(angle\right)={perpendicular\over hypotenuse}

Single-molecule_magnet.html

  1. ^ H B = i < j J i , j 𝐒 i 𝐒 j \mathcal{\hat{H}}_{HB}=\sum_{i<j}J_{i,j}\mathbf{S}_{i}\cdot\mathbf{S}_{j}
  2. J i , j J_{i,j}
  3. 𝐒 i \mathbf{S}_{i}
  4. 𝐒 j \mathbf{S}_{j}
  5. U = S 2 | D | \ U=S^{2}|D|\,

Single-stock_futures.html

  1. F = [ S - P V ( D i v ) ] ( 1 + r ) ( T - t ) F=[S-PV(Div)]\cdot(1+r)^{(T-t)}
  2. F = [ S - P V ( D i v ) ] e r ( T - t ) F=[S-PV(Div)]\cdot e^{r\cdot(T-t)}

Single_displacement_reaction.html

  1. \color G r e e n F e 0 + \color O r a n g e C u + 2 \color B l u e S O 4 - 2 \color O r a n g e C u 0 + \color G r e e n F e + 2 \color B l u e S O 4 - 2 \overset{0}{{\color{Green}Fe}}+\overset{+2}{{\color{Orange}Cu}}\overset{-2}{{% \color{Blue}SO_{4}}}\rightarrow\overset{0}{{\color{Orange}Cu}}+\overset{+2}{{% \color{Green}Fe}}\overset{-2}{{\color{Blue}SO_{4}}}

Singleton_bound.html

  1. C C
  2. n n
  3. M M
  4. d d
  5. C C
  6. n n
  7. d = min { x , y C : x y } d ( x , y ) d=\min_{\{x,y\in C:x\neq y\}}d(x,y)
  8. d ( x , y ) d(x,y)
  9. x x
  10. y y
  11. A q ( n , d ) A_{q}(n,d)
  12. n n
  13. d d
  14. A q ( n , d ) q n - d + 1 . A_{q}(n,d)\leq q^{n-d+1}.
  15. q q
  16. n n
  17. q n q^{n}
  18. q q
  19. C C
  20. d d
  21. c C c\in C
  22. d - 1 d-1
  23. C C
  24. d d
  25. n - ( d - 1 ) = n - d + 1 n-(d-1)=n-d+1
  26. q n - d + 1 q^{n-d+1}
  27. C C
  28. | C | A q ( n , d ) q n - d + 1 . |C|\leq A_{q}(n,d)\leq q^{n-d+1}.
  29. C C
  30. n n
  31. k k
  32. d d
  33. q q
  34. q k q^{k}
  35. q k q n - d + 1 q^{k}\leq q^{n-d+1}
  36. k n - d + 1 k\leq n-d+1
  37. d n - k + 1 d\leq n-k+1
  38. n - k n-k
  39. ( 𝔽 q ) n (\mathbb{F}_{q})^{n}
  40. n n
  41. k k
  42. C C
  43. n , k , d n,k,d
  44. 𝔽 q \mathbb{F}_{q}
  45. C C
  46. k k
  47. C C
  48. n - k n-k
  49. C C
  50. C C^{\perp}
  51. G = ( I | A ) G=(I|A)
  52. C C
  53. A A
  54. d d
  55. C C
  56. n , k , d n,k,d
  57. 𝔽 q \mathbb{F}_{q}
  58. A w A_{w}
  59. C C
  60. w w
  61. A w = ( n w ) j = 0 w - d ( - 1 ) j ( w j ) ( q w - d + 1 - j - 1 ) = ( n w ) ( q - 1 ) j = 0 w - d ( - 1 ) j ( w - 1 j ) q w - d - j . A_{w}={\left({{n}\atop{w}}\right)}\sum_{j=0}^{w-d}(-1)^{j}{\left({{w}\atop{j}}% \right)}(q^{w-d+1-j}-1)={\left({{n}\atop{w}}\right)}(q-1)\sum_{j=0}^{w-d}(-1)^% {j}{\left({{w-1}\atop{j}}\right)}q^{w-d-j}.
  62. P G ( N , q ) PG(N,q)
  63. N N
  64. 𝔽 q \mathbb{F}_{q}
  65. K = { P 1 , P 2 , , P m } K=\{P_{1},P_{2},\dots,P_{m}\}
  66. ( N + 1 ) × m (N+1)\times m
  67. G G
  68. K K
  69. G G
  70. [ m , N + 1 , m - N ] [m,N+1,m-N]
  71. 𝔽 q \mathbb{F}_{q}

Sinusoidal_projection.html

  1. x = ( λ - λ 0 ) cos φ x=\left(\lambda-\lambda_{0}\right)\cos\varphi
  2. y = φ y=\varphi\,

SITOR.html

  1. ( 7 3 ) = ( 7 4 ) = 35 \textstyle{\left({{7}\atop{3}}\right)}={\left({{7}\atop{4}}\right)}=35

Skeleton_(category_theory).html

  1. h o m D ( d 1 , d 2 ) = h o m C ( d 1 , d 2 ) hom_{D}(d_{1},d_{2})=hom_{C}(d_{1},d_{2})
  2. K K
  3. K n K^{n}
  4. K m K n K^{m}\to K^{n}
  5. A , B A,B
  6. H o m ( A , B ) Hom(A,B)

Skew_lines.html

  1. 𝐚 \mathbf{a}
  2. 𝐛 \mathbf{b}
  3. 𝐜 \mathbf{c}
  4. 𝐝 \mathbf{d}
  5. 𝐚 \mathbf{a}
  6. 𝐛 \mathbf{b}
  7. 𝐜 \mathbf{c}
  8. 𝐝 \mathbf{d}
  9. V = 1 6 | det [ 𝐚 - 𝐛 𝐛 - 𝐜 𝐜 - 𝐝 ] | . V=\frac{1}{6}\left|\det\left[\begin{matrix}\mathbf{a}-\mathbf{b}\\ \mathbf{b}-\mathbf{c}\\ \mathbf{c}-\mathbf{d}\end{matrix}\right]\right|.
  10. 𝐱 = 𝐚 + λ 𝐛 ; \mathbf{x}=\mathbf{a}+\lambda\mathbf{b};
  11. 𝐲 = 𝐜 + μ 𝐝 . \mathbf{y}=\mathbf{c}+\mu\mathbf{d}.
  12. 𝐱 \mathbf{x}
  13. 𝐚 \mathbf{a}
  14. 𝐛 \mathbf{b}
  15. λ \lambda
  16. 𝐲 \mathbf{y}
  17. 𝐜 \mathbf{c}
  18. 𝐝 \mathbf{d}
  19. 𝐧 = 𝐛 × 𝐝 | 𝐛 × 𝐝 | \mathbf{n}=\frac{\mathbf{b}\times\mathbf{d}}{|\mathbf{b}\times\mathbf{d}|}
  20. d = | 𝐧 ( 𝐜 - 𝐚 ) | . d=|\mathbf{n}\cdot(\mathbf{c}-\mathbf{a})|.
  21. i + j Align l t ; d i+j&lt;d
  22. I I
  23. J J
  24. i + j d i+j≥d
  25. I I
  26. J J
  27. I I
  28. J J
  29. k 0 k≥0
  30. I J I∪J

SKI_combinator_calculus.html

  1. \ldots
  2. \to
  3. \to
  4. \to
  5. \to
  6. \to
  7. \to
  8. \to
  9. \to
  10. \to
  11. \to

Skyrmion.html

  1. ( S U ( N ) L × S U ( N ) R S U ( N ) diag ) \left(\frac{SU(N)_{L}\times SU(N)_{R}}{SU(N)\text{diag}}\right)
  2. π 3 ( S U ( N ) L × S U ( N ) R S U ( N ) diag S U ( N ) ) \pi_{3}\left(\frac{SU(N)_{L}\times SU(N)_{R}}{SU(N)\text{diag}}\cong SU(N)\right)

Slide_attack.html

  1. K 1 K m K_{1}\cdots K_{m}
  2. K 1 K_{1}
  3. K 2 K_{2}
  4. K i K_{i}
  5. 2 n / 2 2^{n/2}
  6. 2 n / 2 2^{n/2}
  7. ( P , C ) (P,C)
  8. ( P 0 , C 0 ) ( P 1 , C 1 ) (P_{0},C_{0})(P_{1},C_{1})
  9. P 0 = F ( P 1 ) P_{0}=F(P_{1})
  10. C 0 = F ( C 1 ) C_{0}=F(C_{1})
  11. ( P 0 , C 0 ) ( P 1 , C 1 ) (P_{0},C_{0})(P_{1},C_{1})
  12. P 0 = F ( P 1 ) P_{0}=F(P_{1})
  13. C 0 = F ( C 1 ) C_{0}=F(C_{1})
  14. 2 n / 2 2^{n/2}
  15. P = ( L 0 , R 0 ) P=(L_{0},R_{0})
  16. P 0 = ( R 0 , L 0 F ( R 0 , K ) ) P_{0}=(R_{0},L_{0}\bigoplus F(R_{0},K))
  17. 2 n 2^{n}
  18. 2 n / 2 2^{n/2}
  19. 2 n / 4 2^{n/4}

Slope_field.html

  1. y = f ( x , y ) y^{\prime}=f(x,y)
  2. f ( x , y ) f(x,y)
  3. x , y x,y
  4. [ 1 , f ( x , y ) ] [1,f(x,y)]
  5. x , y x,y
  6. x , y x,y
  7. [ 1 , f ( x , y ) ] [1,f(x,y)]
  8. x , y x,y
  9. y = f ( x , y ) y^{\prime}=f(x,y)
  10. x , y x,y
  11. f ( x , y ) f(x,y)
  12. d x 1 d t = f 1 ( t , x 1 , x 2 , , x n ) \frac{dx_{1}}{dt}=f_{1}(t,x_{1},x_{2},\ldots,x_{n})
  13. d x 2 d t = f 2 ( t , x 1 , x 2 , , x n ) \frac{dx_{2}}{dt}=f_{2}(t,x_{1},x_{2},\ldots,x_{n})
  14. \vdots
  15. d x n d t = f n ( t , x 1 , x 2 , , x n ) \frac{dx_{n}}{dt}=f_{n}(t,x_{1},x_{2},\ldots,x_{n})
  16. ( t , x 1 , x 2 , , x n ) (t,x_{1},x_{2},\ldots,x_{n})
  17. ( 1 f 1 ( t , x 1 , x 2 , , x n ) f 2 ( t , x 1 , x 2 , , x n ) f n ( t , x 1 , x 2 , , x n ) ) \begin{pmatrix}1\\ f_{1}(t,x_{1},x_{2},\ldots,x_{n})\\ f_{2}(t,x_{1},x_{2},\ldots,x_{n})\\ \vdots\\ f_{n}(t,x_{1},x_{2},\ldots,x_{n})\end{pmatrix}
  18. ( t , x 1 , x 2 , , x n ) (t,x_{1},x_{2},\ldots,x_{n})
  19. n = 1 n=1
  20. n > 2 n>2

Slutsky_equation.html

  1. x i ( 𝐩 , w ) p j = h i ( 𝐩 , u ) p j - x i ( 𝐩 , w ) w x j ( 𝐩 , w ) , {\partial x_{i}(\mathbf{p},w)\over\partial p_{j}}={\partial h_{i}(\mathbf{p},u% )\over\partial p_{j}}-{\partial x_{i}(\mathbf{p},w)\over\partial w}x_{j}(% \mathbf{p},w),\,
  2. h ( 𝐩 , u ) h(\mathbf{p},u)
  3. x ( 𝐩 , w ) x(\mathbf{p},w)
  4. 𝐩 \mathbf{p}
  5. w w
  6. u u
  7. v ( 𝐩 , w ) v(\mathbf{p},w)
  8. ϵ p , i j = ϵ p , i j h - ϵ w , i b j \epsilon_{p,ij}=\epsilon_{p,ij}^{h}-\epsilon_{w,i}b_{j}
  9. 𝐃 𝐩 𝐱 ( 𝐩 , w ) = 𝐃 𝐩 𝐡 ( 𝐩 , u ) - 𝐃 𝐰 𝐱 ( 𝐩 , w ) 𝐱 ( 𝐩 , w ) , \mathbf{D_{p}x}(\mathbf{p},w)=\mathbf{D_{p}h}(\mathbf{p},u)-\mathbf{D_{w}x}(% \mathbf{p},w)\mathbf{x}(\mathbf{p},w)^{\top},\,
  10. 𝐃 𝐩 𝐡 ( 𝐩 , u ) \mathbf{D_{p}h}(\mathbf{p},u)
  11. h i ( 𝐩 , u ) = x i ( 𝐩 , e ( 𝐩 , u ) ) h_{i}(\mathbf{p},u)=x_{i}(\mathbf{p},e(\mathbf{p},u))
  12. e ( 𝐩 , u ) e(\mathbf{p},u)
  13. h i ( 𝐩 , u ) p j = x i ( 𝐩 , e ( 𝐩 , u ) ) p j + x i ( 𝐩 , e ( 𝐩 , u ) ) e ( 𝐩 , u ) e ( 𝐩 , u ) p j \frac{\partial h_{i}(\mathbf{p},u)}{\partial p_{j}}=\frac{\partial x_{i}(% \mathbf{p},e(\mathbf{p},u))}{\partial p_{j}}+\frac{\partial x_{i}(\mathbf{p},e% (\mathbf{p},u))}{\partial e(\mathbf{p},u)}\cdot\frac{\partial e(\mathbf{p},u)}% {\partial p_{j}}
  14. e ( 𝐩 , u ) p j = h j ( 𝐩 , u ) \frac{\partial e(\mathbf{p},u)}{\partial p_{j}}=h_{j}(\mathbf{p},u)
  15. h j ( 𝐩 , u ) = h j ( 𝐩 , v ( 𝐩 , w ) ) = x j ( 𝐩 , w ) , h_{j}(\mathbf{p},u)=h_{j}(\mathbf{p},v(\mathbf{p},w))=x_{j}(\mathbf{p},w),
  16. v ( 𝐩 , w ) v(\mathbf{p},w)

Small-angle_approximation.html

  1. sin θ θ \sin\theta\approx\theta
  2. cos θ 1 - θ 2 2 \cos\theta\approx 1-\frac{\theta^{2}}{2}
  3. tan θ θ \tan\theta\approx\theta
  4. θ 2 / 2 \theta^{2}/2
  5. cos θ 1 - θ 2 2 \cos{\theta}\approx 1-\frac{\theta^{2}}{2}
  6. O s O\approx s
  7. H A H\approx A
  8. sin θ = O H O A = tan θ = O A s A = A * θ A = θ \sin\theta={O\over H}\approx{O\over A}=\tan\theta={O\over A}\approx{s\over A}=% {{A*\theta}\over A}=\theta
  9. sin θ tan θ θ \sin\theta\approx\tan\theta\approx\theta
  10. sin θ = n = 0 ( - 1 ) n ( 2 n + 1 ) ! θ 2 n + 1 = θ - θ 3 3 ! + θ 5 5 ! - θ 7 7 ! + \sin\theta=\sum^{\infty}_{n=0}\frac{(-1)^{n}}{(2n+1)!}\theta^{2n+1}=\theta-% \frac{\theta^{3}}{3!}+\frac{\theta^{5}}{5!}-\frac{\theta^{7}}{7!}+\cdots
  11. sin θ = θ - θ 3 6 + θ 5 120 - θ 7 5040 + \sin\theta=\theta-\frac{\theta^{3}}{6}+\frac{\theta^{5}}{120}-\frac{\theta^{7}% }{5040}+\cdots
  12. sin θ θ \sin\theta\approx\theta
  13. tan θ sin θ θ \tan\theta\approx\sin\theta\approx\theta

Small-signal_model.html

  1. V IN V_{\mathrm{IN}}
  2. v in v_{\mathrm{in}}
  3. v IN ( t ) = V IN + v in ( t ) v_{\mathrm{IN}}(t)=V_{\mathrm{IN}}+v_{\mathrm{in}}(t)

Small-world_network.html

  1. L log N L\propto\log N
  2. ω \omega
  3. ω = L r L - C C l \omega=\tfrac{L_{r}}{L}-\tfrac{C}{C_{l}}
  4. L log log N L\propto\log\log N

Smith–Waterman_algorithm.html

  1. H H
  2. H ( i , 0 ) = 0 , 0 i m H(i,0)=0,\;0\leq i\leq m
  3. H ( 0 , j ) = 0 , 0 j n H(0,j)=0,\;0\leq j\leq n
  4. H ( i , j ) = max { 0 H ( i - 1 , j - 1 ) + s ( a i , b j ) Match/Mismatch max k 1 { H ( i - k , j ) + W k } Deletion max l 1 { H ( i , j - l ) + W l } Insertion } , 1 i m , 1 j n H(i,j)=\max\begin{Bmatrix}0\\ H(i-1,j-1)+\ s(a_{i},b_{j})&\,\text{Match/Mismatch}\\ \max_{k\geq 1}\{H(i-k,j)+\ W_{k}\}&\,\text{Deletion}\\ \max_{l\geq 1}\{H(i,j-l)+\ W_{l}\}&\,\text{Insertion}\end{Bmatrix},\;1\leq i% \leq m,1\leq j\leq n
  5. a , b a,b
  6. Σ \Sigma
  7. m = length ( a ) m=\,\text{length}(a)
  8. n = length ( b ) n=\,\text{length}(b)
  9. s ( a , b ) s(a,b)
  10. H ( i , j ) H(i,j)
  11. W i W_{i}
  12. s ( a , b ) = + 2 if a = b (match), - 1 if a b (mismatch) s(a,b)=+2\,\text{ if }a=b\,\text{ (match), }-1\,\text{ if }a\neq b\text{ (% mismatch)}
  13. W i = - i W_{i}=-i
  14. H = ( - A C A C A C T A - \color b l u e 0 0 0 0 0 0 0 0 0 A 0 \color b l u e 2 1 2 1 2 1 0 2 G 0 \color b l u e 1 1 1 1 1 1 0 1 C 0 0 \color b l u e 3 2 3 2 3 2 1 A 0 2 2 \color b l u e 5 4 5 4 3 4 C 0 1 4 4 \color b l u e 7 6 7 6 5 A 0 2 3 6 6 \color b l u e 9 8 7 8 C 0 1 4 5 8 8 \color b l u e 11 \color b l u e 10 9 A 0 2 3 6 7 10 10 10 \color b l u e 12 ) H=\begin{pmatrix}&-&A&C&A&C&A&C&T&A\\ -&\color{blue}0&0&0&0&0&0&0&0&0\\ A&0&\color{blue}2&1&2&1&2&1&0&2\\ G&0&\color{blue}1&1&1&1&1&1&0&1\\ C&0&0&\color{blue}3&2&3&2&3&2&1\\ A&0&2&2&\color{blue}5&4&5&4&3&4\\ C&0&1&4&4&\color{blue}7&6&7&6&5\\ A&0&2&3&6&6&\color{blue}9&8&7&8\\ C&0&1&4&5&8&8&\color{blue}11&\color{blue}10&9\\ A&0&2&3&6&7&10&10&10&\color{blue}12\end{pmatrix}
  15. T = ( - A C A C A C T A - \color b l u e 0 0 0 0 0 0 0 0 0 A 0 \color b l u e G 0 \color b l u e C 0 \color b l u e A 0 \color b l u e C 0 \color b l u e A 0 \color b l u e C 0 \color b l u e \color b l u e A 0 \color b l u e ) T=\begin{pmatrix}&-&A&C&A&C&A&C&T&A\\ -&\color{blue}0&0&0&0&0&0&0&0&0\\ A&0&\color{blue}\nwarrow&\leftarrow&\nwarrow&\leftarrow&\nwarrow&\leftarrow&% \leftarrow&\nwarrow\\ G&0&\color{blue}\uparrow&\nwarrow&\uparrow&\nwarrow&\uparrow&\nwarrow&\nwarrow% &\uparrow\\ C&0&\uparrow&\color{blue}\nwarrow&\leftarrow&\nwarrow&\leftarrow&\nwarrow&% \leftarrow&\leftarrow\\ A&0&\nwarrow&\uparrow&\color{blue}\nwarrow&\leftarrow&\nwarrow&\leftarrow&% \leftarrow&\nwarrow\\ C&0&\uparrow&\nwarrow&\uparrow&\color{blue}\nwarrow&\leftarrow&\nwarrow&% \leftarrow&\leftarrow\\ A&0&\nwarrow&\uparrow&\nwarrow&\uparrow&\color{blue}\nwarrow&\leftarrow&% \leftarrow&\nwarrow\\ C&0&\uparrow&\nwarrow&\uparrow&\nwarrow&\uparrow&\color{blue}\nwarrow&\color{% blue}\leftarrow&\leftarrow\\ A&0&\nwarrow&\uparrow&\nwarrow&\uparrow&\nwarrow&\uparrow&\nwarrow&\color{blue% }\nwarrow\end{pmatrix}

Smooth_structure.html

  1. μ \mu
  2. ν \nu
  3. μ \mu
  4. ν \nu
  5. f : M M f:M\rightarrow M
  6. μ f = ν \mu\circ f=\nu
  7. C k C^{k}

Smoothed-particle_hydrodynamics.html

  1. h h
  2. 𝐫 \mathbf{r}
  3. 2 h 2h
  4. 𝐫 \mathbf{r}
  5. W W
  6. A A
  7. 𝐫 \mathbf{r}
  8. A ( 𝐫 ) = j m j A j ρ j W ( | 𝐫 - 𝐫 j | , h ) , A(\mathbf{r})=\sum_{j}m_{j}\frac{A_{j}}{\rho_{j}}W(|\mathbf{r}-\mathbf{r}_{j}|% ,h),
  9. m j m_{j}
  10. j j
  11. A j A_{j}
  12. A A
  13. j j
  14. ρ j \rho_{j}
  15. j j
  16. 𝐫 \mathbf{r}
  17. W W
  18. i i
  19. ρ i \rho_{i}
  20. ρ i = ρ ( 𝐫 i ) = j m j ρ j ρ j W ( | 𝐫 i - 𝐫 j | , h ) = j m j W ( | 𝐫 i - 𝐫 j | , h ) , \rho_{i}=\rho(\mathbf{r}_{i})=\sum_{j}m_{j}\frac{\rho_{j}}{\rho_{j}}W(|\mathbf% {r}_{i}-\mathbf{r}_{j}|,h)=\sum_{j}m_{j}W(|\mathbf{r}_{i}-\mathbf{r}_{j}|,h),
  21. j j
  22. \nabla
  23. A ( 𝐫 ) = j m j A j ρ j W ( | 𝐫 - 𝐫 j | , h ) . \nabla A(\mathbf{r})=\sum_{j}m_{j}\frac{A_{j}}{\rho_{j}}\nabla W(|\mathbf{r}-% \mathbf{r}_{j}|,h).

Smoothness.html

  1. f ( x ) = { x if x 0 , 0 if x < 0 f(x)=\begin{cases}x&\mbox{if }~{}x\geq 0,\\ 0&\mbox{if }~{}x<0\end{cases}
  2. f ( x ) = { x 2 sin ( 1 x ) if x 0 , 0 if x = 0 f(x)=\begin{cases}x^{2}\sin{(\tfrac{1}{x})}&\mbox{if }~{}x\neq 0,\\ 0&\mbox{if }~{}x=0\end{cases}
  3. f ( x ) = { - cos ( 1 x ) + 2 x sin ( 1 x ) if x 0 , 0 if x = 0. f^{\prime}(x)=\begin{cases}-\mathord{\cos(\tfrac{1}{x})}+2x\sin(\tfrac{1}{x})&% \mbox{if }~{}x\neq 0,\\ 0&\mbox{if }~{}x=0.\end{cases}
  4. f ( x ) = | x | k + 1 f(x)=|x|^{k+1}
  5. f ( x ) = { e - 1 1 - x 2 if | x | < 1 , 0 otherwise f(x)=\begin{cases}e^{-\frac{1}{1-x^{2}}}&\mbox{ if }~{}|x|<1,\\ 0&\mbox{ otherwise }\end{cases}
  6. f i x i 1 1 x i 2 2 x i k k \frac{\partial^{\ell}f_{i}}{\partial x_{i_{1}}^{\ell_{1}}\,\partial x_{i_{2}}^% {\ell_{2}}\cdots\partial x_{i_{k}}^{\ell_{k}}}
  7. k k
  8. i i
  9. i 1 , i 2 , , i k i_{1},i_{2},\ldots,i_{k}
  10. , 1 , 2 , , k \ell,\ell_{1},\ell_{2},\ldots,\ell_{k}
  11. 1 + 2 + + k = \ell_{1}+\ell_{2}+\cdots+\ell_{k}=\ell
  12. p K , m = sup x K | f ( m ) ( x ) | p_{K,m}=\sup_{x\in K}\left|f^{(m)}(x)\right|
  13. d n s d t n \displaystyle\frac{d^{n}s}{dt^{n}}
  14. f ( x ) > 0 for a < x < b . f(x)>0\quad\,\text{ for }\quad a<x<b.\,
  15. ψ F φ - 1 \scriptstyle\psi\circ F\circ\varphi^{-1}

SO(10)_(physics).html

  1. 45 24 0 10 - 4 10 ¯ 4 1 0 45\rightarrow 24_{0}\oplus 10_{-4}\oplus\overline{10}_{4}\oplus 1_{0}
  2. 16 10 1 5 ¯ - 3 1 5 . 16\rightarrow 10_{1}\oplus\bar{5}_{-3}\oplus 1_{5}.
  3. 10 5 - 2 5 ¯ 2 . 10\rightarrow 5_{-2}\oplus\bar{5}_{2}.
  4. 16 ( 4 , 2 , 1 ) ( 4 ¯ , 1 , 2 ) . 16\rightarrow(4,2,1)\oplus(\bar{4},1,2).
  5. 16 ¯ H \overline{16}_{H}
  6. 126 ¯ H \overline{126}_{H}
  7. 16 ¯ H \overline{16}_{H}
  8. 126 ¯ H \overline{126}_{H}
  9. 16 ¯ \overline{16}
  10. 126 ¯ \overline{126}
  11. 16 ¯ \overline{16}
  12. 126 ¯ \overline{126}
  13. < 16 ¯ H > 16 f ϕ <\overline{16}_{H}>16_{f}\phi
  14. < 126 ¯ H > 16 f 16 f <\overline{126}_{H}>16_{f}16_{f}
  15. < 16 ¯ H > < 16 ¯ H > 16 f 16 f <\overline{16}_{H}><\overline{16}_{H}>16_{f}16_{f}
  16. ( 3 , 2 ) - 5 6 (3,2)_{-\frac{5}{6}}
  17. ( 3 , 2 ) 1 6 (3,2)_{\frac{1}{6}}

Sobol_sequence.html

  1. lim n 1 n i = 1 n f ( x i ) = I s f \lim_{n\to\infty}\;\frac{1}{n}\sum_{i=1}^{n}f(x_{i})\;=\;\int_{I^{s}}f
  2. j = 1 s [ a j b d j , a j + 1 b d j ] \prod_{j=1}^{s}\left[\frac{a_{j}}{b^{d_{j}}},\frac{a_{j}+1}{b^{d_{j}}}\right]
  3. a j < b d j a_{j}<b^{d_{j}}
  4. 0 t m 0\leq t\leq m
  5. Card P { x 1 , , x b m } = b t \,\text{Card}\,P\cap\{x_{1},...,x_{b^{m}}\}=b^{t}
  6. k 0 , m t k\geq 0,\;m\geq t
  7. { x k b m , , x ( k + 1 ) b m - 1 } \{x_{kb^{m}},...,x_{(k+1)b^{m}-1}\}
  8. G ( n ) = n n / 2 G(n)=n\oplus\lfloor n/2\rfloor
  9. x n , i = x n - 1 , i v k , i . x_{n,i}=x_{n-1,i}\oplus v_{k,i}.\,
  10. det ( V d ) 1 mod 2 , \det({V}_{d})\equiv 1(\mod 2),
  11. V d := ( v 1 , 1 , 1 v 2 , 1 , 1 v d , 1 , 1 v 1 , 2 , 1 v 2 , 2 , 1 v d , 2 , 1 v 1 , d , 1 v 2 , d , 1 v d , d , 1 ) {V}_{d}:=\begin{pmatrix}{v_{1,1,1}}&{v_{2,1,1}}&{\dots}&{v_{d,1,1}}\\ {v_{1,2,1}}&{v_{2,2,1}}&{\dots}&{v_{d,2,1}}\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {v_{1,d,1}}&{v_{2,d,1}}&{\dots}&{v_{d,d,1}}\end{pmatrix}
  12. det ( U d ) 1 mod 2 , \det({U}_{d})\equiv 1\mod 2,
  13. U d := ( v 1 , 1 , 1 v 1 , 1 , 2 v 2 , 1 , 1 v 2 , 1 , 2 v d , 1 , 1 v d , 1 , 2 v 1 , 2 , 1 v 1 , 2 , 2 v 2 , 2 , 1 v 2 , 2 , 2 v d , 2 , 1 v d , 2 , 2 v 1 , 2 d , 1 v 1 , 2 d , 2 v 2 , 2 d , 1 v 2 , 2 d , 2 v d , 2 d , 1 v d , 2 d , 2 ) {U}_{d}:=\begin{pmatrix}{v_{1,1,1}}&{v_{1,1,2}}&{v_{2,1,1}}&{v_{2,1,2}}&{\dots% }&{v_{d,1,1}}&{v_{d,1,2}}\\ {v_{1,2,1}}&{v_{1,2,2}}&{v_{2,2,1}}&{v_{2,2,2}}&{\dots}&{v_{d,2,1}}&{v_{d,2,2}% }\\ {\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\\ {v_{1,2d,1}}&{v_{1,2d,2}}&{v_{2,2d,1}}&{v_{2,2d,2}}&{\dots}&{v_{d,2d,1}}&{v_{d% ,2d,2}}\end{pmatrix}

Soil_mechanics.html

  1. D 50 D_{50}
  2. D 10 D_{10}
  3. V a V_{a}
  4. V w V_{w}
  5. V s V_{s}
  6. W a W_{a}
  7. W w W_{w}
  8. W s W_{s}
  9. M a M_{a}
  10. M w M_{w}
  11. M s Ms
  12. ρ a \rho_{a}
  13. ρ w \rho_{w}
  14. ρ s \rho_{s}
  15. W s = M s g W_{s}=M_{s}g
  16. ρ w = 1 g / c m 3 \rho_{w}=1g/cm^{3}
  17. G s = ρ s ρ w G_{s}=\frac{\rho_{s}}{\rho_{w}}
  18. γ \gamma
  19. ρ \rho
  20. g g
  21. ρ \rho
  22. ρ = M s + M w V s + V w + V a = M t V t \rho=\frac{M_{s}+M_{w}}{V_{s}+V_{w}+V_{a}}=\frac{M_{t}}{V_{t}}
  23. ρ d \rho_{d}
  24. ρ d = M s V s + V w + V a = M s V t \rho_{d}=\frac{M_{s}}{V_{s}+V_{w}+V_{a}}=\frac{M_{s}}{V_{t}}
  25. ρ \rho^{\prime}
  26. ρ = ρ - ρ w \rho^{\prime}=\rho\ -\rho_{w}
  27. ρ w \rho_{w}
  28. w w
  29. w = M w M s = W w W s w=\frac{M_{w}}{M_{s}}=\frac{W_{w}}{W_{s}}
  30. e e
  31. e = V V V S = V V V T - V V = n 1 - n e=\frac{V_{V}}{V_{S}}=\frac{V_{V}}{V_{T}-V_{V}}=\frac{n}{1-n}
  32. n n
  33. n = V v V t = V v V s + V v = e 1 + e n=\frac{V_{v}}{V_{t}}=\frac{V_{v}}{V_{s}+V_{v}}=\frac{e}{1+e}
  34. S S
  35. S = V w V v S=\frac{V_{w}}{V_{v}}
  36. ρ = ( G s + S e ) ρ w 1 + e \rho=\frac{(G_{s}+Se)\rho_{w}}{1+e}
  37. ρ = ( 1 + w ) G s ρ w 1 + e \rho=\frac{(1+w)G_{s}\rho_{w}}{1+e}
  38. w = S e G s w=\frac{Se}{G_{s}}
  39. σ = σ - u \sigma^{\prime}=\sigma-u\,
  40. σ v \sigma_{v}
  41. ρ \rho
  42. H H
  43. σ v = ρ g H = γ H \sigma_{v}=\rho gH=\gamma H
  44. g g
  45. γ \gamma
  46. u = ρ w g z w u=\rho_{w}gz_{w}
  47. ρ w \rho_{w}
  48. z w z_{w}
  49. u = ρ w g z w u=\rho_{w}gz_{w}
  50. z w z_{w}
  51. σ = σ - u , \sigma^{\prime}=\sigma-u,
  52. w l w_{l}
  53. w p w_{p}
  54. L I = w - P L L L - P L LI=\frac{w-PL}{LL-PL}
  55. D r D_{r}
  56. D r = e m a x - e e m a x - e m i n 100 % D_{r}=\frac{e_{max}-e}{e_{max}-e_{min}}100\%
  57. e m a x e_{max}
  58. e m i n e_{min}
  59. e e
  60. D r = 100 % D_{r}=100\%
  61. D r = 0 % D_{r}=0\%
  62. u = ρ w g z w u=\rho_{w}gz_{w}
  63. z w z_{w}
  64. Q = - K A μ ( u b - u a ) L Q=\frac{-KA}{\mu}\frac{(u_{b}-u_{a})}{L}
  65. Q Q
  66. K K
  67. A A
  68. u b - u a L \frac{u_{b}-u_{a}}{L}
  69. μ \mu
  70. x x
  71. x x
  72. u e u_{e}
  73. u e = u - ρ w g z u_{e}=u-\rho_{w}gz
  74. z z
  75. u u
  76. u e u_{e}
  77. Q = - K A μ ( u e , b - u e , a ) L Q=\frac{-KA}{\mu}\frac{(u_{e,b}-u_{e,a})}{L}
  78. A A
  79. v x = - K μ d u e d x v_{x}=\frac{-K}{\mu}\frac{du_{e}}{dx}
  80. v x = Q / A v_{x}=Q/A
  81. v p x v_{px}
  82. n n
  83. v p x = v x n v_{px}=\frac{v_{x}}{n}
  84. v = k i v=ki
  85. k k
  86. k = K ρ w g μ w k=\frac{K\rho_{w}g}{\mu_{w}}
  87. i i
  88. h h
  89. u e = ρ w g h + C o n s t a n t u_{e}=\rho_{w}gh+Constant
  90. C o n s t a n t Constant
  91. u e u_{e}
  92. k k
  93. 10 - 12 m s 10^{-12}\frac{m}{s}
  94. 10 - 1 m s 10^{-1}\frac{m}{s}
  95. τ c r i t \tau_{crit}
  96. σ n \sigma_{n}^{\prime}
  97. ϕ c r i t \phi_{crit}^{\prime}
  98. τ c r i t = σ n tan ϕ c r i t \tau_{crit}=\sigma_{n}^{\prime}\tan\phi_{crit}^{\prime}
  99. τ p e a k = σ n tan ϕ p e a k \tau_{peak}=\sigma_{n}^{\prime}\tan\phi_{peak}^{\prime}
  100. ϕ p e a k > ϕ c r i t \phi_{peak}^{\prime}>\phi_{crit}^{\prime}
  101. τ f = c + σ f tan ϕ \tau_{f}=c^{\prime}+\sigma_{f}^{\prime}\tan\phi^{\prime}\,
  102. c c^{\prime}
  103. ϕ \phi^{\prime}
  104. c c^{\prime}
  105. ϕ \phi^{\prime}
  106. c c^{\prime}
  107. c c^{\prime}
  108. c c^{\prime}
  109. ϕ \phi^{\prime}
  110. σ = σ - u \sigma^{\prime}=\sigma-u\,

Solar_irradiance.html

  1. cos ( c ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) cos ( C ) \cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)\,
  2. C = h C=h\,
  3. c = Θ c=\Theta\,
  4. a = 1 2 π - ϕ a=\tfrac{1}{2}\pi-\phi\,
  5. b = 1 2 π - δ b=\tfrac{1}{2}\pi-\delta\,
  6. cos ( Θ ) = sin ( ϕ ) sin ( δ ) + cos ( ϕ ) cos ( δ ) cos ( h ) \cos(\Theta)=\sin(\phi)\sin(\delta)+\cos(\phi)\cos(\delta)\cos(h)\,
  7. Q = S o R o 2 R E 2 cos ( Θ ) when cos ( Θ ) > 0 Q=S_{o}\frac{R_{o}^{2}}{R_{E}^{2}}\cos(\Theta)\,\text{ when }\cos(\Theta)>0
  8. Q = 0 when cos ( Θ ) 0 Q=0\,\text{ when }\cos(\Theta)\leq 0\,
  9. Q ¯ day = - 1 2 π π - π Q d h \overline{Q}^{\,\text{day}}=-\frac{1}{2\pi}{\int_{\pi}^{-\pi}Q\,dh}
  10. Θ = 1 2 π \Theta=\tfrac{1}{2}\pi
  11. sin ( ϕ ) sin ( δ ) + cos ( ϕ ) cos ( δ ) cos ( h o ) = 0 \sin(\phi)\sin(\delta)+\cos(\phi)\cos(\delta)\cos(h_{o})=0\,
  12. cos ( h o ) = - tan ( ϕ ) tan ( δ ) \cos(h_{o})=-\tan(\phi)\tan(\delta)
  13. R o 2 R E 2 \frac{R_{o}^{2}}{R_{E}^{2}}
  14. π - π Q d h = h o - h o Q d h = S o R o 2 R E 2 h o - h o cos ( Θ ) d h \int_{\pi}^{-\pi}Q\,dh=\int_{h_{o}}^{-h_{o}}Q\,dh=S_{o}\frac{R_{o}^{2}}{R_{E}^% {2}}\int_{h_{o}}^{-h_{o}}\cos(\Theta)\,dh
  15. π - π Q d h = S o R o 2 R E 2 [ h sin ( ϕ ) sin ( δ ) + cos ( ϕ ) cos ( δ ) sin ( h ) ] h = h o h = - h o \int_{\pi}^{-\pi}Q\,dh=S_{o}\frac{R_{o}^{2}}{R_{E}^{2}}\left[h\sin(\phi)\sin(% \delta)+\cos(\phi)\cos(\delta)\sin(h)\right]_{h=h_{o}}^{h=-h_{o}}
  16. π - π Q d h = - 2 S o R o 2 R E 2 [ h o sin ( ϕ ) sin ( δ ) + cos ( ϕ ) cos ( δ ) sin ( h o ) ] \int_{\pi}^{-\pi}Q\,dh=-2S_{o}\frac{R_{o}^{2}}{R_{E}^{2}}\left[h_{o}\sin(\phi)% \sin(\delta)+\cos(\phi)\cos(\delta)\sin(h_{o})\right]
  17. Q ¯ day = S o π R o 2 R E 2 [ h o sin ( ϕ ) sin ( δ ) + cos ( ϕ ) cos ( δ ) sin ( h o ) ] \overline{Q}^{\,\text{day}}=\frac{S_{o}}{\pi}\frac{R_{o}^{2}}{R_{E}^{2}}\left[% h_{o}\sin(\phi)\sin(\delta)+\cos(\phi)\cos(\delta)\sin(h_{o})\right]
  18. sin δ = sin ε sin ( θ - ϖ ) \sin\delta=\sin\varepsilon~{}\sin(\theta-\varpi)\,
  19. R E = R o 1 + e cos ( θ - ϖ ) R_{E}=\frac{R_{o}}{1+e\cos(\theta-\varpi)}
  20. R o R E = 1 + e cos ( θ - ϖ ) \frac{R_{o}}{R_{E}}={1+e\cos(\theta-\varpi)}
  21. Q ¯ day \overline{Q}^{\mathrm{day}}
  22. Q ¯ day \overline{Q}^{\mathrm{day}}
  23. R o R E = 1 + e cos ( θ - ϖ ) = 1 + e cos ( π 2 - ϖ ) = 1 + e sin ( ϖ ) \frac{R_{o}}{R_{E}}=1+e\cos(\theta-\varpi)=1+e\cos(\tfrac{\pi}{2}-\varpi)=1+e% \sin(\varpi)
  24. e sin ( ϖ ) e\sin(\varpi)

Solar_neutrino.html

  1. p + p d + e + + ν e p+p\to\,\text{d}+e^{+}+\nu_{e}\!
  2. \to
  3. d + p H 3 e + γ d+p\to{{}^{3}}He+\gamma
  4. H 3 e + H 3 e H 4 e + 2 p {{}^{3}}He+{{}^{3}}He\to{{}^{4}}He+2p
  5. H 3 e + H 4 e B 7 e + γ {{}^{3}}He+{{}^{4}}He\to{{}^{7}}Be+\gamma
  6. B 7 e + e - L 7 i + ν e {{}^{7}}Be+e^{-}\to{{}^{7}}Li+\nu_{e}
  7. B 7 e + p B 8 + γ {{}^{7}}Be+p\to{{}^{8}}B+\gamma
  8. B 8 B 8 e + e + + ν e {{}^{8}}B\to{{}^{8}}Be+e^{+}+\nu_{e}

Solar_rotation.html

  1. ω = A + B sin 2 ( φ ) + C sin 4 ( φ ) \omega=A+B\,\sin^{2}(\varphi)+C\,\sin^{4}(\varphi)

Solid-state_nuclear_magnetic_resonance.html

  1. d = μ 0 4 π γ 1 γ 2 r 3 d=\frac{\hbar\mu_{0}}{4\pi}\frac{\gamma_{1}\gamma_{2}}{r^{3}}
  2. D 3 cos 2 θ - 1 D\propto 3\cos^{2}\theta-1
  3. ( 3 cos 2 θ - 1 ) (3\cos^{2}\theta-1)
  4. θ = arctan 2 \theta=\arctan\sqrt{2}

Solid_torus.html

  1. S 1 × D 2 S^{1}\times D^{2}
  2. S 1 × S 1 S^{1}\times S^{1}
  3. D 2 D^{2}
  4. S 1 S^{1}
  5. π 1 ( S 1 × D 2 ) π 1 ( S 1 ) , \pi_{1}(S^{1}\times D^{2})\cong\pi_{1}(S^{1})\cong\mathbb{Z},
  6. H k ( S 1 × D 2 ) H k ( S 1 ) { if k = 0 , 1 0 otherwise . H_{k}(S^{1}\times D^{2})\cong H_{k}(S^{1})\cong\begin{cases}\mathbb{Z}&\mbox{ % if }~{}k=0,1\\ 0&\mbox{ otherwise }\end{cases}.

Solow_residual.html

  1. Y ( t ) = [ K ( t ) ] α [ A ( t ) L ( t ) ] 1 - α Y(t)=[K(t)]^{\alpha}[A(t)L(t)]^{1-\alpha}\,
  2. Y t = Y K K t + Y L L t + Y A A t \frac{\partial Y}{\partial t}=\frac{\partial Y}{\partial K}\frac{\partial K}{% \partial t}+\frac{\partial Y}{\partial L}\frac{\partial L}{\partial t}+\frac{% \partial Y}{\partial A}\frac{\partial A}{\partial t}
  3. Y K = α [ K ( t ) ] α - 1 [ A ( t ) L ( t ) ] 1 - α = α Y [ K ( t ) ] \frac{\partial Y}{\partial K}={\alpha}[K(t)]^{\alpha-1}\cdot[A(t)L(t)]^{1-% \alpha}=\frac{{\alpha}Y}{[K(t)]}
  4. Y L = ( 1 - α ) Y [ L ( t ) ] and Y A = ( 1 - α ) Y [ A ( t ) ] \frac{\partial Y}{\partial L}=\frac{(1-{\alpha})Y}{[L(t)]}\,\text{ and }\frac{% \partial Y}{\partial A}=\frac{(1-{\alpha})Y}{[A(t)]}
  5. Y t = α Y [ K ( t ) ] K t + ( 1 - α ) Y [ L ( t ) ] L t + ( 1 - α ) Y [ A ( t ) ] A t \frac{\partial Y}{\partial t}=\frac{{\alpha}Y}{[K(t)]}\frac{\partial K}{% \partial t}+\frac{(1-{\alpha})Y}{[L(t)]}\frac{\partial L}{\partial t}+\frac{(1% -{\alpha})Y}{[A(t)]}\frac{\partial A}{\partial t}
  6. Y t Y = α K t K ( t ) + ( 1 - α ) L t L ( t ) + ( 1 - α ) A t A ( t ) \frac{\frac{\partial Y}{\partial t}}{Y}=\alpha\frac{\frac{\partial K}{\partial t% }}{K(t)}+(1-{\alpha})\frac{\frac{\partial L}{\partial t}}{L(t)}+(1-{\alpha})% \frac{\frac{\partial A}{\partial t}}{A(t)}
  7. S R ( t ) = Y t Y - ( α K t K ( t ) + ( 1 - α ) L t L ( t ) ) SR(t)=\frac{\frac{\partial Y}{\partial t}}{Y}-\left(\alpha\frac{\frac{\partial K% }{\partial t}}{K(t)}+(1-{\alpha})\frac{\frac{\partial L}{\partial t}}{L(t)}\right)
  8. ε \varepsilon
  9. ln ( Y ( t ) ) = α ln ( K ( t ) ) + ( 1 - α ) [ ln ( L ( t ) ) ] + ( 1 - α ) [ ln ( A ( t ) ) ] + ε . \ln(Y(t))=\alpha\ln(K(t))+(1-\alpha)[\ln(L(t))]+(1-\alpha)[\ln(A(t))]+% \varepsilon.\,
  10. y = C + β k + γ + ε . y=C+\beta k+\gamma\ell+\varepsilon.\,
  11. Y ( t ) = [ K ( t ) ] α [ H ( t ) ] β [ A ( t ) L ( t ) ] 1 - α - β Y(t)=[K(t)]^{\alpha}[H(t)]^{\beta}[A(t)L(t)]^{1-\alpha-\beta}\,
  12. ln ( Y ( t ) ) = α ln ( K ( t ) ) + β ln ( H ( t ) ) + ( 1 - α - β ) [ ln ( L ( t ) ) ] + ( 1 - α - β ) [ ln ( A ( t ) ) ] + ε . \ln(Y(t))=\alpha\ln(K(t))+\beta\ln(H(t))+(1-\alpha-\beta)[\ln(L(t))]+(1-\alpha% -\beta)[\ln(A(t))]+\varepsilon.\,
  13. A ( t ) A(t)
  14. A ( t ) 1 - α - β A(t)^{1-\alpha-\beta}

Solow–Swan_model.html

  1. L L
  2. K K
  3. Y ( t ) = K ( t ) α ( A ( t ) L ( t ) ) 1 - α Y(t)=K(t)^{\alpha}(A(t)L(t))^{1-\alpha}\,
  4. t t
  5. 0 < α < 1 0<\alpha<1
  6. Y ( t ) Y(t)
  7. A A
  8. A L AL
  9. A ( 0 ) A(0)
  10. K ( 0 ) K(0)
  11. L ( 0 ) L(0)
  12. n n
  13. g g
  14. L ( t ) = L ( 0 ) e n t L(t)=L(0)e^{nt}
  15. A ( t ) = A ( 0 ) e g t A(t)=A(0)e^{gt}
  16. A ( t ) L ( t ) A(t)L(t)
  17. ( n + g ) (n+g)
  18. δ \delta
  19. c Y ( t ) cY(t)
  20. 0 < c < 1 0<c<1
  21. s = 1 - c s=1-c
  22. K ˙ ( t ) = s Y ( t ) - δ K ( t ) \dot{K}(t)=s\cdot Y(t)-{\delta}\cdot K(t)\,
  23. K ˙ \dot{K}
  24. d K ( t ) d t \frac{dK(t)}{dt}
  25. Y ( K , A L ) Y(K,AL)
  26. y ( t ) = Y ( t ) A ( t ) L ( t ) = k ( t ) α y(t)=\frac{Y(t)}{A(t)L(t)}=k(t)^{\alpha}
  27. k k
  28. k ˙ ( t ) = s k ( t ) α - ( n + g + δ ) k ( t ) \dot{k}(t)=sk(t)^{\alpha}-(n+g+\delta)k(t)
  29. s k ( t ) α = s y ( t ) sk(t)^{\alpha}=sy(t)
  30. s s
  31. y ( t ) y(t)
  32. ( n + g + δ ) k ( t ) (n+g+\delta)k(t)
  33. k k
  34. k ( t ) k(t)
  35. k * k^{*}
  36. s k ( t ) α = ( n + g + δ ) k ( t ) sk(t)^{\alpha}=(n+g+\delta)k(t)
  37. k * = ( s n + g + δ ) 1 1 - α k^{*}=\left(\frac{s}{n+g+\delta}\right)^{\frac{1}{1-\alpha}}\,
  38. K K
  39. A L AL
  40. ( n + g ) (n+g)
  41. Y Y
  42. K ( t ) Y ( t ) = k ( t ) 1 - α \frac{K(t)}{Y(t)}=k(t)^{1-\alpha}
  43. k * k^{*}
  44. K ( t ) Y ( t ) = s n + g + δ \frac{K(t)}{Y(t)}=\frac{s}{n+g+\delta}
  45. α < 1 {\alpha}<1
  46. t t
  47. K ( t ) K(t)
  48. M P K = Y K = α A 1 - α / ( K / L ) 1 - α MPK=\frac{\partial Y}{\partial K}={\alpha}A^{1-\alpha}/(K/L)^{1-\alpha}
  49. A A
  50. K / L K/L
  51. K / L K/L
  52. Y / L Y/L
  53. Y K \frac{\partial Y}{\partial K}
  54. r r
  55. α = K Y K Y = r K Y \alpha=\frac{K\frac{\partial Y}{\partial K}}{Y}=\frac{rK}{Y}\,
  56. α \alpha
  57. Y ( t ) = K ( t ) α H ( t ) β ( A ( t ) L ( t ) ) 1 - α - β Y(t)=K(t)^{\alpha}H(t)^{\beta}(A(t)L(t))^{1-\alpha-\beta}
  58. H ( t ) H(t)
  59. δ \delta
  60. s Y ( t ) sY(t)
  61. s = s K + s H s=s_{K}+s_{H}
  62. k ˙ = s K k α h β - ( n + g + δ ) k \dot{k}=s_{K}k^{\alpha}h^{\beta}-(n+g+\delta)k
  63. h ˙ = s H k α h β - ( n + g + δ ) h \dot{h}=s_{H}k^{\alpha}h^{\beta}-(n+g+\delta)h
  64. k ˙ = h ˙ = 0 \dot{k}=\dot{h}=0
  65. s K k α h β - ( n + g + δ ) k = 0 s_{K}k^{\alpha}h^{\beta}-(n+g+\delta)k=0
  66. s H k α h β - ( n + g + δ ) h = 0 s_{H}k^{\alpha}h^{\beta}-(n+g+\delta)h=0
  67. k k
  68. h h
  69. k * = ( s K 1 - β s H β n + g + δ ) 1 1 - α - β k^{*}=\left(\frac{s_{K}^{1-\beta}s_{H}^{\beta}}{n+g+\delta}\right)^{\frac{1}{1% -\alpha-\beta}}
  70. h * = ( s K α s H 1 - α n + g + δ ) 1 1 - α - β h^{*}=\left(\frac{s_{K}^{\alpha}s_{H}^{1-\alpha}}{n+g+\delta}\right)^{\frac{1}% {1-\alpha-\beta}}
  71. y * = ( k * ) α ( h * ) β y^{*}=(k^{*})^{\alpha}(h^{*})^{\beta}
  72. β {\beta}
  73. β {\beta}
  74. M P K = Y K = α A 1 - α ( H / L ) β / ( K / L ) 1 - α MPK=\frac{\partial Y}{\partial K}={\alpha}A^{1-\alpha}(H/L)^{\beta}/(K/L)^{1-\alpha}
  75. y ( t ) = Y ( t ) A ( t ) L ( t ) = K ( t ) α ( A ( t ) L ( t ) ) 1 - α A ( t ) L ( t ) = K ( t ) α ( A ( t ) L ( t ) ) α = k ( t ) α y(t)=\frac{Y(t)}{A(t)L(t)}=\frac{K(t)^{\alpha}(A(t)L(t))^{1-\alpha}}{A(t)L(t)}% =\frac{K(t)^{\alpha}}{(A(t)L(t))^{\alpha}}=k(t)^{\alpha}
  76. k ˙ ( t ) = K ˙ ( t ) A ( t ) L ( t ) - K ( t ) [ A ( t ) L ( t ) ] 2 [ A ( t ) L ˙ ( t ) + L ( t ) A ˙ ( t ) ] = K ˙ ( t ) A ( t ) L ( t ) - K ( t ) A ( t ) L ( t ) L ˙ ( t ) L ( t ) - K ( t ) A ( t ) L ( t ) A ˙ ( t ) A ( t ) \dot{k}(t)=\frac{\dot{K}(t)}{A(t)L(t)}-\frac{K(t)}{[A(t)L(t)]^{2}}[A(t)\dot{L}% (t)+L(t)\dot{A}(t)]=\frac{\dot{K}(t)}{A(t)L(t)}-\frac{K(t)}{A(t)L(t)}\frac{% \dot{L}(t)}{L(t)}-\frac{K(t)}{A(t)L(t)}\frac{\dot{A}(t)}{A(t)}
  77. K ˙ ( t ) = s Y ( t ) - δ K ( t ) \dot{K}(t)=sY(t)-{\delta}K(t)\,
  78. L ˙ ( t ) L ( t ) \frac{\dot{L}(t)}{L(t)}
  79. A ˙ ( t ) A ( t ) \frac{\dot{A}(t)}{A(t)}
  80. n n
  81. g g
  82. k ˙ ( t ) = s Y ( t ) A ( t ) L ( t ) - δ K ( t ) A ( t ) L ( t ) - n K ( t ) A ( t ) L ( t ) - g K ( t ) A ( t ) L ( t ) = s y ( t ) - δ k ( t ) - n k ( t ) - g k ( t ) \dot{k}(t)=s\frac{Y(t)}{A(t)L(t)}-\delta\frac{K(t)}{A(t)L(t)}-n\frac{K(t)}{A(t% )L(t)}-g\frac{K(t)}{A(t)L(t)}=sy(t)-{\delta}k(t)-nk(t)-gk(t)
  83. y ( t ) = k ( t ) α y(t)=k(t)^{\alpha}

Solubility_pump.html

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

Solution_concept.html

  1. \subset
  2. Γ \Gamma
  3. G Γ G\in\Gamma
  4. S G S_{G}
  5. G G
  6. Π G Γ 2 S G ; \Pi_{G\in\Gamma}2^{S_{G}};
  7. F : Γ G Γ 2 S G F:\Gamma\rightarrow\bigcup\nolimits_{G\in\Gamma}2^{S_{G}}
  8. F ( G ) S G F(G)\subseteq S_{G}
  9. G Γ . G\in\Gamma.

Solutions_of_the_Einstein_field_equations.html

  1. G a b = κ T a b G_{ab}\,=\kappa T_{ab}
  2. G a b + Λ g a b = κ T a b G_{ab}+\Lambda g_{ab}\,=\kappa T_{ab}
  3. κ \kappa
  4. T a b = ; b 0 . T^{ab}{}_{;b}\,=0\,.
  5. T a b = 0 T_{ab}\,=0
  6. T a b = ( ρ + p ) u a u b + p g a b T_{ab}\,=(\rho+p)u_{a}u_{b}+pg_{ab}
  7. u a u a = - 1 u^{a}u_{a}=-1\!
  8. ρ \rho
  9. u a u_{a}
  10. p p
  11. T a b = ρ u a u b T_{ab}\,=\rho u_{a}u_{b}
  12. ρ \rho
  13. p p
  14. T a b = ; b 0 T^{ab}{}_{;b}=0
  15. g μ ν Γ μ ν σ = 0 . g^{\mu\nu}\Gamma^{\sigma}_{\mu\nu}=0\,.
  16. d s 2 = ( - N + N i N j γ i j ) d t 2 + 2 N i γ i j d t d x j + γ i j d x i d x j ds^{2}\,=(-N+N^{i}N^{j}\gamma_{ij})dt^{2}+2N^{i}\gamma_{ij}dtdx^{j}+\gamma_{ij% }dx^{i}dx^{j}
  17. i , j = 1 3 . i,j=1\dots 3\,.
  18. N N
  19. N i N^{i}
  20. γ i j \gamma_{ij}
  21. t = c o n s t t=const

Somer–Lucas_pseudoprime.html

  1. U ( P , Q ) U(P,Q)
  2. D = P 2 - 4 Q , D=P^{2}-4Q,
  3. gcd ( N , D ) = 1 \gcd(N,D)=1
  4. 1 d ( N - ( D N ) ) , \frac{1}{d}\left(N-\left(\frac{D}{N}\right)\right),
  5. ( D N ) \left(\frac{D}{N}\right)

Sonometer.html

  1. f = 1 2 l T m f=\frac{1}{2l}\sqrt{\frac{T}{m}}

Sort-merge_join.html

  1. R R
  2. S S
  3. | R | < | S | |R|<|S|
  4. R R
  5. P r P_{r}
  6. S S
  7. P s P_{s}
  8. O ( P r + P s ) O(P_{r}+P_{s})
  9. R R
  10. S S
  11. O ( P r + P s + P r log ( P r ) + P s log ( P s ) ) O(P_{r}+P_{s}+P_{r}\log(P_{r})+P_{s}\log(P_{s}))
  12. O ( P r log ( P r ) + P s log ( P s ) ) O(P_{r}\log(P_{r})+P_{s}\log(P_{s}))

Sound_energy_density_level.html

  1. L ( E ) = 10 log 10 ( E 1 E 0 ) dB L(E)=10\,\log_{10}\left(\frac{E_{1}}{E_{0}}\right){\rm dB}
  2. E 0 = 10 - 12 J m 3 E_{0}=10^{-12}\mathrm{\frac{J}{m^{3}}}

Space_diagonal.html

  1. a 3 . a\sqrt{3}.
  2. a 2 + b 2 + c 2 . \sqrt{a^{2}+b^{2}+c^{2}}.
  3. a 2 a\sqrt{2}
  4. a 1 + φ 2 a\sqrt{1+\varphi^{2}}
  5. ( 1 + 5 ) / 2 (1+\sqrt{5})/2
  6. a r a\sqrt{r}

Space_gun.html

  1. l l
  2. v e v_{e}
  3. a a
  4. a = v e 2 2 l a=\frac{v_{e}^{2}}{2l}
  5. l l
  6. v e v_{e}
  7. a a

Spacetime_symmetries.html

  1. X T = 0 \mathcal{L}_{X}T=0
  2. [ X , Y ] T = X ( Y T ) - Y ( X T ) \mathcal{L}_{[X,Y]}T=\mathcal{L}_{X}(\mathcal{L}_{Y}T)-\mathcal{L}_{Y}(% \mathcal{L}_{X}T)
  3. [ X , Y ] T [\mathcal{L}_{X},\mathcal{L}_{Y}]T
  4. X g a b = 0 \mathcal{L}_{X}g_{ab}=0
  5. X a ; b + X b ; a = 0 X_{a;b}+X_{b;a}\,=0
  6. X g a b = 2 c g a b \mathcal{L}_{X}g_{ab}=2cg_{ab}
  7. ( X g a b ) ; c = 0 (\mathcal{L}_{X}g_{ab})_{;c}=0
  8. X g a b = ϕ g a b \mathcal{L}_{X}g_{ab}=\phi g_{ab}
  9. M M
  10. X R a = b c d 0 \mathcal{L}_{X}R^{a}{}_{bcd}=0
  11. X T a b = 0 \mathcal{L}_{X}T_{ab}=0

Span_(architecture).html

  1. M m a x M_{max}
  2. δ m a x \delta_{max}
  3. M m a x = q L 2 8 M_{max}=\frac{qL^{2}}{8}
  4. δ m a x = 5 M m a x L 2 48 E I = 5 q L 4 384 E I \delta_{max}=\frac{5M_{max}L^{2}}{48EI}=\frac{5qL^{4}}{384EI}
  5. q q
  6. L L
  7. E E
  8. I I

Sparsely_totient_number.html

  1. φ ( m ) > φ ( n ) \varphi(m)>\varphi(n)
  2. φ \varphi
  3. lim inf P ( n ) / log n = 1 \liminf P(n)/\log n=1
  4. P ( n ) log δ n P(n)\ll\log^{\delta}n
  5. δ = 37 / 20 \delta=37/20
  6. lim sup P ( n ) / log n = 2 \limsup P(n)/\log n=2

Specific_rotation.html

  1. [ α ] λ T = α l × ρ [\alpha]_{\lambda}^{T}=\frac{\alpha}{l\times\rho}
  2. [ α ] D 20 + 6.2 [\alpha]_{D}^{20}+6.2
  3. [ α ] λ T = α l × c [\alpha]_{\lambda}^{T}=\frac{\alpha}{l\times c}
  4. [ α ] λ T = 100 × α l × c [\alpha]_{\lambda}^{T}=\frac{100\times\alpha}{l\times c}
  5. [ α ] D 20 + 6.2 [\alpha]_{D}^{20}+6.2
  6. e e ( % ) = α obs × 100 [ α ] λ ee(\%)=\frac{\alpha\text{obs}\times 100}{[\alpha]_{\lambda}}

Specific_storage.html

  1. S = d V w d h 1 A = S s b + S y S=\frac{dV_{w}}{dh}\frac{1}{A}=S_{s}b+S_{y}\,
  2. V w V_{w}
  3. h h
  4. S s S_{s}
  5. S y S_{y}
  6. b b
  7. S = S s b S=S_{s}b\,
  8. S y S_{y}
  9. S s S_{s}
  10. S s b S y S_{s}b\ll\!\ S_{y}
  11. S = S y S=S_{y}\,
  12. ( S s ) m = 1 m a d m w d h (S_{s})_{m}=\frac{1}{m_{a}}\frac{dm_{w}}{dh}
  13. ( S s ) m (S_{s})_{m}
  14. m a m_{a}
  15. d m w dm_{w}
  16. d h dh
  17. S s = 1 V a d V w d h = 1 V a d V w d p d p d h = 1 V a d V w d p γ w S_{s}=\frac{1}{V_{a}}\frac{dV_{w}}{dh}=\frac{1}{V_{a}}\frac{dV_{w}}{dp}\frac{% dp}{dh}=\frac{1}{V_{a}}\frac{dV_{w}}{dp}\gamma_{w}
  18. S s S_{s}
  19. V a V_{a}
  20. d V w dV_{w}
  21. d p dp
  22. d h dh
  23. γ w \gamma_{w}
  24. S s = γ w ( β p + n β w ) S_{s}=\gamma_{w}(\beta_{p}+n\cdot\beta_{w})
  25. γ w \gamma_{w}
  26. n n
  27. β p \beta_{p}
  28. β w \beta_{w}
  29. β p = - d V t d σ e 1 V t \beta_{p}=-\frac{dV_{t}}{d\sigma_{e}}\frac{1}{V_{t}}
  30. β w = - d V w d p 1 V w \beta_{w}=-\frac{dV_{w}}{dp}\frac{1}{V_{w}}
  31. σ e \sigma_{e}
  32. V t V_{t}
  33. V w V_{w}
  34. σ e \sigma_{e}
  35. p p
  36. S y = V w d V T S_{y}=\frac{V_{wd}}{V_{T}}
  37. V w d V_{wd}
  38. V T V_{T}
  39. S s S_{s}

Specified_complexity.html

  1. σ = - log 2 [ R × φ ( T ) × P ( T ) ] , \sigma=-\log_{2}[R\times\varphi(T)\times\operatorname{P}(T)],
  2. 10 120 × φ ( T ) × P ( T ) < 1 2 . 10^{120}\times\varphi(T)\times\operatorname{P}(T)<\frac{1}{2}.
  3. P ( T ) < 1 2 × 10 - 140 . \operatorname{P}(T)<\frac{1}{2}\times 10^{-140}.
  4. 10 20 = 10 5 × 10 5 × 10 5 × 10 5 10^{20}=10^{5}\times 10^{5}\times 10^{5}\times 10^{5}

Spectral_power_distribution.html

  1. M ( λ ) = 2 Φ A λ Φ A Δ λ M(\lambda)=\frac{\partial^{2}\Phi}{\partial A\partial\lambda}\approx\frac{\Phi% }{A\Delta\lambda}
  2. M ( λ ) M(\lambda)
  3. Φ \Phi
  4. A A
  5. λ \lambda
  6. M r e l ( λ ) = M ( λ ) M ( λ o ) M_{rel}(\lambda)=\frac{M(\lambda)}{M({\lambda_{o}})}
  7. R ( λ ) = S ( λ ) M ( λ ) R(\lambda)=\frac{S(\lambda)}{M(\lambda)}

Spectral_resolution.html

  1. Δ λ \Delta\lambda
  2. R = λ Δ λ R={\lambda\over\Delta\lambda}
  3. Δ λ \Delta\lambda
  4. λ \lambda
  5. Δ v \Delta v
  6. Δ v \Delta v
  7. R = c Δ v R={c\over\Delta v}
  8. c c

Spectroradiometer.html

  1. E ( λ ) = Δ Φ Δ A Δ λ E(\lambda)=\frac{\Delta\Phi}{\Delta A\Delta\lambda}
  2. E E
  3. Φ \Phi
  4. Δ λ \Delta\lambda
  5. A A
  6. λ \lambda

Specular_highlight.html

  1. ( A ^ B ^ ) (\hat{A}\cdot\hat{B})
  2. max ( 0 , ( A ^ B ^ ) ) \max(0,(\hat{A}\cdot\hat{B}))
  3. k spec = R V cos n β = ( R ^ V ^ ) n k_{\mathrm{spec}}=\|R\|\|V\|\cos^{n}\beta=(\hat{R}\cdot\hat{V})^{n}
  4. k spec = N H cos n β = ( N ^ H ^ ) n k_{\mathrm{spec}}=\|N\|\|H\|\cos^{n}\beta=(\hat{N}\cdot\hat{H})^{n}
  5. n n
  6. k = ( L R ) n = [ L ( E - 2 N ( N E ) ) ] n , k=(\vec{L}\cdot\vec{R})^{n}=[\vec{L}\cdot(\vec{E}-2\vec{N}(\vec{N}\cdot\vec{E}% ))]^{n},
  7. E = N = 1 \|\vec{E}\|=\|\vec{N}\|=1
  8. N = { 0 ; 1 ; 0 } ; E = { 3 2 ; 1 2 ; 0 } ; L = { - 0.6 ; 0.8 ; 0 } ; n = 3 \vec{N}=\{0;\;1;\;0\};\;\;\vec{E}=\{\frac{\sqrt{3}}{2};\;\frac{1}{2};\;0\};\;% \;\vec{L}=\{-0.6;\;0.8;\;0\};\;\;n=3
  9. k = [ L ( E - 2 N ( N E ) ) ] n = [ L ( E - 2 N ( 0 3 2 + 1 0.5 + 0 0 ) ) ] 3 = k=[\vec{L}\cdot(\vec{E}-2\vec{N}(\vec{N}\cdot\vec{E}))]^{n}=[\vec{L}\cdot(\vec% {E}-2\vec{N}(0\cdot\frac{\sqrt{3}}{2}+1\cdot 0.5+0\cdot 0))]^{3}=
  10. = [ L ( E - N ) ] 3 = [ L ( { 3 2 - 0 ; 1 2 - 1 ; 0 - 0 } ) ] 3 = [ - 0.6 3 2 + 0.8 ( - 0.5 ) + 0 0 ] 3 = ( - 0.5196 - 0.4 ) 3 = 0.9196 3 = 0.7777. =[\vec{L}\cdot(\vec{E}-\vec{N})]^{3}=[\vec{L}\cdot(\{\frac{\sqrt{3}}{2}-0;\;% \frac{1}{2}-1;\;0-0\})]^{3}=[-0.6\cdot\frac{\sqrt{3}}{2}+0.8\cdot(-0.5)+0\cdot 0% ]^{3}=(-0.5196-0.4)^{3}=0.9196^{3}=0.7777.
  11. k = ( N H ) n = ( N ( ( L + E ) / 2 ) ) n = ( N ( ( { - 0.6 + 3 2 ; 0.8 + 0.5 ; 0 + 0 } ) / 2 ) ) 3 = ( N ( ( { 0.266 ; 1.3 ; 0 } ) / 2 ) ) 3 = k=(\vec{N}\cdot\vec{H})^{n}=(\vec{N}\cdot((\vec{L}+\vec{E})/2))^{n}=(\vec{N}% \cdot((\{-0.6+\frac{\sqrt{3}}{2};\;0.8+0.5;\;0+0\})/2))^{3}=(\vec{N}\cdot((\{0% .266;\;1.3;\;0\})/2))^{3}=
  12. = ( N ( { 0.133 ; 0.65 ; 0 } ) ) 3 = ( 0 0.133 + 1 0.65 + 0 ) 3 = 0.65 3 = 0.274625. =(\vec{N}\cdot(\{0.133;\;0.65;\;0\}))^{3}=(0\cdot 0.133+1\cdot 0.65+0)^{3}=0.6% 5^{3}=0.274625.
  13. H { 0.133 ; 0.65 ; 0 } H = H { 0.133 ; 0.65 ; 0 } 0.133 2 + 0.65 2 = H { 0.133 ; 0.65 ; 0 } 0.668 = { 0.20048 ; 0.979701 ; 0 } , \frac{\vec{H}\{0.133;\;0.65;\;0\}}{\|\vec{H}\|}=\frac{\vec{H}\{0.133;\;0.65;\;% 0\}}{\sqrt{0.133^{2}+0.65^{2}}}=\frac{\vec{H}\{0.133;\;0.65;\;0\}}{0.668}=\{0.% 20048;0.979701;0\},
  14. k = ( N H ) n = ( 0 0.2 + 1 0.9797 + 0 0 ) 3 = 0.979701 3 = 0.940332. k=(\vec{N}\cdot\vec{H})^{n}=(0\cdot 0.2+1\cdot 0.9797+0\cdot 0)^{3}=0.979701^{% 3}=0.940332.
  15. k spec = e - ( ( N , H ) m ) 2 k_{\mathrm{spec}}=e^{-\left(\frac{\angle(N,H)}{m}\right)^{2}}
  16. k spec = exp ( - tan 2 ( α ) / m 2 ) π m 2 cos 4 ( α ) , α = arccos ( N H ) k_{\mathrm{spec}}=\frac{\exp{\left(-\tan^{2}(\alpha)/m^{2}\right)}}{\pi m^{2}% \cos^{4}(\alpha)},~{}\alpha=\arccos(N\cdot H)
  17. tan 2 ( α ) / m 2 = 1 - cos 2 ( α ) cos 2 ( α ) m 2 \tan^{2}(\alpha)/m^{2}=\frac{1-\cos^{2}(\alpha)}{\cos^{2}(\alpha)m^{2}}
  18. cos ( α ) \cos(\alpha)
  19. 0 < N V 0<N\cdot V
  20. 0 < P V 0<P\cdot V
  21. 0 < R V 0<R\cdot V
  22. T = D + ( - D N ) * N D + ( - D T ) * N T=\frac{D+(-D\cdot N)*N}{\|D+(-D\cdot T)*N\|}
  23. P = L + ( - L T ) * T L + ( - L T ) * T P=\frac{L+(-L\cdot T)*T}{\|L+(-L\cdot T)*T\|}
  24. R = - L + 2 * ( L T ) * T - L + 2 * ( L T ) * T R=\frac{-L+2*(L\cdot T)*T}{\|-L+2*(L\cdot T)*T\|}
  25. k diff = L P k_{\mathrm{diff}}=L\cdot P
  26. k spec = ( V R ) s k_{\mathrm{spec}}=(V\cdot R)^{s}
  27. k diff = L P = L L + ( - L T ) * T L + ( - L T ) * T = = 1 - ( L T ) 2 k_{\mathrm{diff}}=L\cdot P=L\cdot\frac{L+(-L\cdot T)*T}{\|L+(-L\cdot T)*T\|}=.% ..=\sqrt{1-(L\cdot T)^{2}}
  28. k spec = ( V R ) s = ( 1 - ( L T ) 2 * 1 - ( V T ) 2 - ( L T ) * ( V T ) ) s = [ sin ( ( L , T ) ) sin ( ( V , T ) ) - cos ( ( L , T ) ) cos ( ( V , T ) ) ] s = ( - cos ( ( L , T ) + ( V , T ) ) ) s = [ cos ( ( L , T ) ) cos ( ( V , T ) ) - sin ( ( L , T ) ) sin ( ( V , T ) ) ] s = cos s ( ( L , T ) + ( V , T ) ) \begin{aligned}\displaystyle k_{\mathrm{spec}}&\displaystyle{}=(V\cdot R)^{s}% \\ &\displaystyle{}=(\sqrt{1-(L\cdot T)^{2}}*\sqrt{1-(V\cdot T)^{2}}-(L\cdot T)*(% V\cdot T))^{s}\\ &\displaystyle{}=\left[\sin(\angle(L,T))\sin(\angle(V,T))-\cos(\angle(L,T))% \cos(\angle(V,T))\right]^{s}=(-\cos(\angle(L,T)+\angle(V,T)))^{s}\\ &\displaystyle{}=\left[\cos(\angle(L,T))\cos(\angle(V,T))-\sin(\angle(L,T))% \sin(\angle(V,T))\right]^{s}\\ &\displaystyle{}=\cos^{s}(\angle(L,T)+\angle(V,T))\end{aligned}
  29. k spec k_{\mathrm{spec}}
  30. k spec = 1 ( N L ) ( N V ) N L 4 π α x α y exp [ - 2 ( H X α x ) 2 + ( H Y α y ) 2 1 + ( H N ) ] k_{\mathrm{spec}}=\frac{1}{\sqrt{(N\cdot L)(N\cdot V)}}\frac{N\cdot L}{4\pi% \alpha_{x}\alpha_{y}}\exp\left[-2\frac{\left(\frac{H\cdot X}{\alpha_{x}}\right% )^{2}+\left(\frac{H\cdot Y}{\alpha_{y}}\right)^{2}}{1+(H\cdot N)}\right]
  31. k spec = D F G 4 ( V N ) ( N L ) k_{\mathrm{spec}}=\frac{DFG}{4(V\cdot N)(N\cdot L)}
  32. F = ( 1 + V N ) λ . F=(1+V\cdot N)^{\lambda}.
  33. G = min ( 1 , 2 ( H N ) ( V N ) V H , 2 ( H N ) ( L N ) V H ) G=\min{\left(1,\frac{2(H\cdot N)(V\cdot N)}{V\cdot H},\frac{2(H\cdot N)(L\cdot N% )}{V\cdot H}\right)}

Speeds_and_feeds.html

  1. V T n = C VT^{n}=C
  2. π {\pi}
  3. R P M = C u t t i n g S p e e d × 12 π × D i a m e t e r RPM={CuttingSpeed\times 12\over\pi\times Diameter}
  4. R P M = C u t t i n g S p e e d × 12 π × D i a m e t e r = 12 × 100 f t / m i n 3 × 10 i n c h e s = 40 r e v s / m i n RPM={CuttingSpeed\times 12\over\pi\times Diameter}={12\times 100ft/min\over 3% \times 10inches}={40revs/min}
  5. R P M = S p e e d π × D i a m e t e r = 1000 × 30 m / m i n 3 × 10 m m = 1000 r e v s / m i n RPM={Speed\over\pi\times Diameter}={1000\times 30m/min\over 3\times 10mm}={100% 0revs/min}
  6. R P M = S p e e d C i r c u m f e r e n c e = S p e e d π × D i a m e t e r RPM={Speed\over Circumference}={Speed\over\pi\times Diameter}
  7. R P M = 100 f t / m i n π × 10 i n c h e s ( 1 f t 12 i n c h e s ) = 100 2.62 = 38.2 r e v s / m i n RPM={100ft/min\over\pi\times 10\,inches\left(\frac{1ft}{12\,inches}\right)}={1% 00\over 2.62}=38.2revs/min
  8. R P M = 30 m / m i n π × 10 m m ( 1 m 1000 m m ) = 1000 * 30 π * 10 = 955 r e v s / m i n RPM={30m/min\over\pi\times 10\,mm\left(\frac{1m}{1000\,mm}\right)}={1000*30% \over\pi*10}=955revs/min
  9. F R = R P M × T × C L FR={RPM\times T\times CL}

Speedup.html

  1. S = T o l d T n e w S=\frac{T_{old}}{T_{new}}
  2. S S
  3. T o l d T_{old}
  4. T n e w T_{new}
  5. S = P n e w P o l d S=\frac{P_{new}}{P_{old}}
  6. S S
  7. P o l d P_{old}
  8. P n e w P_{new}
  9. S = T o l d T n e w = 2.25 s 1.50 s = 1.5 S=\frac{T_{old}}{T_{new}}=\frac{2.25\ \mathrm{s}}{1.50\ \mathrm{s}}=1.5
  10. S = C P I o l d C P I n e w = 3 CPI 2 CPI = 1.5 S=\frac{CPI_{old}}{CPI_{new}}=\frac{3\ \mathrm{CPI}}{2\ \mathrm{CPI}}=1.5
  11. S = I P C n e w I P C o l d = 0.5 IPC 0.333 IPC = 1.5 S=\frac{IPC_{new}}{IPC_{old}}=\frac{0.5\ \mathrm{IPC}}{0.333\ \mathrm{IPC}}=1.5
  12. S p S_{p}
  13. p p
  14. S p = p \,S_{p}=p
  15. E p = S p p = T 1 p T p E_{p}=\frac{S_{p}}{p}=\frac{T_{1}}{pT_{p}}
  16. 1 ln p \frac{1}{\ln p}

Sphaleron.html

  1. τ τ
  2. θ W \theta_{W}
  3. A = ν f ( ξ ) ξ r ^ × σ , ϕ = ν 2 h ( ξ ) r ^ σ ϕ 0 {A}=\nu\frac{f(\xi)}{\xi}\hat{{r}}\times{\sigma}\,,\,\phi=\frac{\nu}{\sqrt{2}}% h(\xi)\hat{{r}}\cdot{\sigma}\phi_{0}
  4. ξ = r g ν \xi=rg\nu
  5. ϕ 0 = [ 1 0 ] \phi_{0}=\begin{bmatrix}1\\ 0\end{bmatrix}
  6. \infty
  7. ξ \xi\rightarrow\infty
  8. r σ r \frac{{r}\cdot{\sigma}}{r}
  9. r r\rightarrow\infty

Spherical_3-manifold.html

  1. M = S 3 / Γ M=S^{3}/\Gamma
  2. Γ \Gamma
  3. S 3 S^{3}
  4. S 3 / Γ S^{3}/\Gamma
  5. S 3 2 S^{3}\subset\mathbb{C}^{2}
  6. ( ω 0 0 ω q ) . \begin{pmatrix}\omega&0\\ 0&\omega^{q}\end{pmatrix}.
  7. ω = e 2 π i / p \omega=e^{2\pi i/p}
  8. L ( p ; q ) L(p;q)
  9. / p \mathbb{Z}/p\mathbb{Z}
  10. q q
  11. p p
  12. L ( p ; q 1 ) L(p;q_{1})
  13. L ( p ; q 2 ) L(p;q_{2})
  14. q 1 q 2 ± n 2 ( mod p ) q_{1}q_{2}\equiv\pm n^{2}\;\;(\mathop{{\rm mod}}p)
  15. n ; n\in\mathbb{N};
  16. q 1 ± q 2 ± 1 ( mod p ) . q_{1}\equiv\pm q_{2}^{\pm 1}\;\;(\mathop{{\rm mod}}p).
  17. x , y x y x - 1 = y - 1 , x 2 k = y n \langle x,y\mid xyx^{-1}=y^{-1},x^{2^{k}}=y^{n}\rangle
  18. x , y x y x - 1 = y - 1 , x 2 m = y n \langle x,y\mid xyx^{-1}=y^{-1},x^{2m}=y^{n}\rangle
  19. x , y , z ( x y ) 2 = x 2 = y 2 , z x z - 1 = y , z y z - 1 = x y , z 3 k = 1 \langle x,y,z\mid(xy)^{2}=x^{2}=y^{2},zxz^{-1}=y,zyz^{-1}=xy,z^{3^{k}}=1\rangle
  20. x , y , z ( x y ) 2 = x 2 = y 2 , z x z - 1 = y , z y z - 1 = x y , z 3 m = 1 \langle x,y,z\mid(xy)^{2}=x^{2}=y^{2},zxz^{-1}=y,zyz^{-1}=xy,z^{3m}=1\rangle
  21. x , y ( x y ) 2 = x 3 = y 4 . \langle x,y\mid(xy)^{2}=x^{3}=y^{4}\rangle.
  22. x , y ( x y ) 2 = x 3 = y 5 . \langle x,y\mid(xy)^{2}=x^{3}=y^{5}\rangle.

Spheromak.html

  1. H H
  2. H H
  3. × B = α B v = ± β B \begin{array}[]{rcl}\vec{\nabla}\times\vec{B}&=&\alpha\vec{B}\\ \vec{v}&=&\pm\beta\vec{B}\\ \end{array}
  4. j × B \vec{j}\times\vec{B}
  5. ρ × v \rho\vec{\nabla}\times\vec{v}
  6. ρ \rho

Spigot_algorithm.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π \pi
  5. ln ( 2 ) = k = 1 1 k 2 k . \ln(2)=\sum_{k=1}^{\infty}\frac{1}{k2^{k}}\,.
  6. 2 7 ln ( 2 ) = 2 7 k = 1 1 k 2 k . 2^{7}\ln(2)=2^{7}\sum_{k=1}^{\infty}\frac{1}{k2^{k}}\,.
  7. 2 7 ln ( 2 ) = k = 1 7 2 7 - k k + k = 8 1 k 2 k - 7 . 2^{7}\ln(2)=\sum_{k=1}^{7}\frac{2^{7-k}}{k}+\sum_{k=8}^{\infty}\frac{1}{k2^{k-% 7}}\,.
  8. 2 7 - k mod k k . \frac{2^{7-k}\mod k}{k}\,.
  9. 2 7 ln ( 2 ) mod 1 64 105 + 37 360 = 0.10011100 2 + 0.00011010 2 = 0.1011 2 , 2^{7}\ln(2)\mod{1}\approx\frac{64}{105}+\frac{37}{360}=0.10011100\cdots_{2}+0.% 00011010\cdots_{2}=0.1011\cdots_{2}\,,
  10. π \pi

Spin_label.html

  1. \neq

Spin–orbit_interaction.html

  1. Δ H = - s y m b o l μ \cdotsymbol B , \Delta H=-symbol{\mu}\cdotsymbol{B},
  2. s y m b o l B = - s y m b o l v × s y m b o l E c 2 , symbol{B}=-{symbol{v}\times symbol{E}\over c^{2}},
  3. s y m b o l E = | E / r | s y m b o l r symbol{E}=\left|E/r\right|symbol{r}
  4. s y m b o l p = m e s y m b o l v symbol{p}=m\text{e}symbol{v}
  5. s y m b o l B = s y m b o l r \timessymbol p m e c 2 | E r | . symbol{B}={symbol{r}\timessymbol{p}\over m\text{e}c^{2}}\left|{E\over r}\right|.
  6. s y m b o l E = - s y m b o l V symbol{E}=-symbol{\nabla}V
  7. | E | = V r = 1 e U ( r ) r , \left|E\right|={\partial V\over\partial r}={1\over e}{\partial U(r)\over% \partial r},
  8. U = e V U=eV
  9. s y m b o l L = s y m b o l r \timessymbol p symbol{L}=symbol{r}\timessymbol{p}
  10. s y m b o l B = 1 m e e c 2 1 r U ( r ) r s y m b o l L . symbol{B}={1\over m\text{e}ec^{2}}{1\over r}{\partial U(r)\over\partial r}% symbol{L}.
  11. s y m b o l μ S = - g S μ B 𝐒 . symbol{\mu}_{S}=-g_{S}\mu_{B}\frac{\mathbf{S}}{\hbar}.
  12. s y m b o l S symbol{S}
  13. μ B \mu\text{B}
  14. g s 2 g\text{s}\approx 2
  15. s y m b o l μ symbol{\mu}
  16. Δ H L = - s y m b o l μ \cdotsymbol B . \Delta H\text{L}=-symbol{\mu}\cdotsymbol{B}.
  17. Δ H L = 2 μ B m e e c 2 1 r U ( r ) r s y m b o l L \cdotsymbol S . \Delta H\text{L}={2\mu\text{B}\over\hbar m\text{e}ec^{2}}{1\over r}{\partial U% (r)\over\partial r}symbol{L}\cdotsymbol{S}.
  18. s y m b o l Ω T symbol{\Omega}\text{T}
  19. s y m b o l ω symbol{\omega}
  20. s y m b o l Ω T = s y m b o l ω ( γ - 1 ) , symbol{\Omega}\text{T}=symbol{\omega}(\gamma-1),
  21. γ \gamma
  22. s y m b o l Ω T symbol{\Omega}\text{T}
  23. Δ H T = s y m b o l Ω T s y m b o l S . \Delta H\text{T}=symbol{\Omega}\text{T}\cdot symbol{S}.
  24. ( v / c ) 2 (v/c)^{2}
  25. Δ H T = - μ B m e e c 2 1 r U ( r ) r s y m b o l L \cdotsymbol S . \Delta H\text{T}=-{\mu\text{B}\over\hbar m\text{e}ec^{2}}{1\over r}{\partial U% (r)\over\partial r}symbol{L}\cdotsymbol{S}.
  26. Δ H Δ H L + Δ H T = μ B m e e c 2 1 r U ( r ) r s y m b o l L \cdotsymbol S . \Delta H\equiv\Delta H\text{L}+\Delta H\text{T}={\mu\text{B}\over\hbar m\text{% e}ec^{2}}{1\over r}{\partial U(r)\over\partial r}symbol{L}\cdotsymbol{S}.
  27. s y m b o l J = s y m b o l L + s y m b o l S . symbol{J}=symbol{L}+symbol{S}.
  28. s y m b o l J 2 = s y m b o l L 2 + s y m b o l S 2 + 2 s y m b o l L s y m b o l S symbol{J}^{2}=symbol{L}^{2}+symbol{S}^{2}+2symbol{L}\cdot symbol{S}
  29. s y m b o l L s y m b o l S = 1 2 ( s y m b o l J 2 - s y m b o l L 2 - s y m b o l S 2 ) = 2 2 ( L z L - L + - L z ) symbol{L}\cdot symbol{S}=\frac{1}{2}(symbol{J}^{2}-symbol{L}^{2}-symbol{S}^{2}% )=\frac{\hbar^{2}}{2}\begin{pmatrix}L_{\mathrm{z}}&L_{-}\\ L_{+}&-L_{\mathrm{z}}\end{pmatrix}
  30. 1 r 3 = 2 a 3 n 3 l ( l + 1 ) ( 2 l + 1 ) \left\langle{1\over r^{3}}\right\rangle=\frac{2}{a^{3}n^{3}l(l+1)(2l+1)}
  31. a = / Z α m e c a=\hbar/Z\alpha m\text{e}c
  32. s y m b o l L \cdotsymbol S = 1 2 ( \langlesymbol J 2 - \langlesymbol L 2 - \langlesymbol S 2 ) \left\langle symbol{L}\cdotsymbol{S}\right\rangle={1\over 2}(\langlesymbol{J}^% {2}\rangle-\langlesymbol{L}^{2}\rangle-\langlesymbol{S}^{2}\rangle)
  33. = 2 2 ( j ( j + 1 ) - l ( l + 1 ) - s ( s + 1 ) ) ={\hbar^{2}\over 2}(j(j+1)-l(l+1)-s(s+1))
  34. Δ E = β 2 ( j ( j + 1 ) - l ( l + 1 ) - s ( s + 1 ) ) \Delta E={\beta\over 2}(j(j+1)-l(l+1)-s(s+1))
  35. β = β ( n , l ) = Z 4 μ 0 4 π g s μ B 2 1 n 3 a 0 3 l ( l + 1 / 2 ) ( l + 1 ) \beta=\beta(n,l)=Z^{4}{\mu_{0}\over 4{\pi}}g\text{s}\mu\text{B}^{2}{1\over n^{% 3}a_{0}^{3}l(l+1/2)(l+1)}
  36. E F E\text{F}
  37. s y m b o l L s y m b o l S symbol{L}\cdot symbol{S}
  38. Δ 0 \Delta_{0}
  39. Γ 8 \Gamma_{8}
  40. Γ \Gamma
  41. Γ 7 \Gamma_{7}
  42. Γ 6 \Gamma_{6}
  43. Γ \Gamma
  44. E F Δ 0 E\text{F}\ll\Delta_{0}
  45. H KL ( k x , k y , k z ) = - 2 2 m [ ( γ 1 + 5 2 γ 2 ) k 2 - 2 γ 2 ( J x 2 k x 2 + J y 2 k y 2 + J z 2 k z 2 ) - 2 γ 3 m n J m J n k m k n ] H\text{KL}(k\text{x},k\text{y},k\text{z})=-\frac{\hbar^{2}}{2m}\left[(\gamma_{% 1}+{\textstyle\frac{5}{2}\gamma_{2}})k^{2}-2\gamma_{2}(J\text{x}^{2}k\text{x}^% {2}+J\text{y}^{2}k\text{y}^{2}+J\text{z}^{2}k\text{z}^{2})-2\gamma_{3}\sum_{m% \neq n}J_{m}J_{n}k_{m}k_{n}\right]
  46. γ 1 , 2 , 3 \gamma_{1,2,3}
  47. J x , y , z J_{\,\text{x},\,\text{y},\,\text{z}}
  48. m m
  49. H D 3 = b 41 8 v 8 v [ ( k x k y 2 - k x k z 2 ) J x + ( k y k z 2 - k y k x 2 ) J y + ( k z k x 2 - k z k y 2 ) J z ] H_{\,\text{D}3}=b_{41}^{8\,\text{v}8\,\text{v}}[(k\text{x}k\text{y}^{2}-k\text% {x}k\text{z}^{2})J\text{x}+(k\text{y}k\text{z}^{2}-k\text{y}k\text{x}^{2})J% \text{y}+(k\text{z}k\text{x}^{2}-k\text{z}k\text{y}^{2})J\text{z}]
  50. b 41 8 v 8 v = - 81.93 meV nm 3 b_{41}^{8\,\text{v}8\,\text{v}}=-81.93\,\,\text{meV}\cdot\,\text{nm}^{3}
  51. H KL + H D 3 H\text{KL}+H_{\,\text{D}3}
  52. H 0 + H R = 2 k 2 2 m * σ 0 + α ( k y σ x - k x σ y ) H_{0}+H\text{R}=\frac{\hbar^{2}k^{2}}{2m^{*}}\sigma_{0}+\alpha(k\text{y}\sigma% \text{x}-k\text{x}\sigma\text{y})
  53. σ 0 \sigma_{0}
  54. σ x , y \sigma_{\,\text{x},\,\text{y}}
  55. m * m^{*}
  56. H R H\text{R}
  57. α \alpha
  58. s y m b o l J symbol{J}
  59. s y m b o l σ symbol{\sigma}
  60. s y m b o l k symbol{k}
  61. s y m b o l A symbol{A}
  62. s y m b o l k = - i - ( e c ) s y m b o l A {symbol{k}}=-i\nabla-(\frac{e}{\hbar c}){symbol{A}}
  63. s y m b o l k s y m b o l p symbol{k}\cdot{symbol{p}}
  64. k k
  65. s y m b o l r symbol{r}
  66. ( s y m b o l σ × s y m b o l k ) (symbol{\sigma}\times{symbol{k}})
  67. s y m b o l r SO {symbol{r}}_{\,\text{SO}}
  68. E G E_{G}
  69. s y m b o l r SO = 2 g 4 m 0 ( 1 E G + 1 E G + Δ 0 ) ( s y m b o l σ × s y m b o l k ) {symbol{r}}_{\,\text{SO}}=\frac{\hbar^{2}g}{4m_{0}}\left(\frac{1}{E_{G}}+\frac% {1}{E_{G}+\Delta_{0}}\right)(symbol{\sigma}\times{symbol{k}})
  70. m 0 m_{0}
  71. g g
  72. g g
  73. s y m b o l S = 1 2 s y m b o l σ {symbol{S}}=\frac{1}{2}{symbol{\sigma}}
  74. s y m b o l E symbol{E}
  75. - e ( s y m b o l r SO s y m b o l E ) -e({symbol{r}}_{\,\text{SO}}\cdot{symbol{E}})
  76. s y m b o l L symbol{L}
  77. s y m b o l S symbol{S}
  78. s y m b o l k symbol{k}
  79. s y m b o l σ symbol{\sigma}
  80. - s y m b o l μ \cdotsymbol B -symbol{\mu}\cdotsymbol{B}
  81. s y m b o l B = s y m b o l B ( s y m b o l r ) symbol{B}=symbol{B}(symbol{r})
  82. - s y m b o l μ x s y m b o l B ( s y m b o l r ) -symbol{\mu}\cdot\partial_{x}symbol{B}(symbol{r})
  83. - s y m b o l μ \cdotsymbol B -symbol{\mu}\cdotsymbol{B}
  84. 1 2 μ B ( s y m b o l σ g ^ s y m b o l B ( s y m b o l r ) ) \frac{1}{2}\mu_{B}(symbol{\sigma}{\hat{g}}symbol{B(symbol{r})})
  85. μ B \mu_{B}
  86. g ^ {\hat{g}}
  87. g g
  88. g ^ {\hat{g}}
  89. s y m b o l r symbol{r}
  90. g g
  91. s y m b o l 18 symbol{18}
  92. s y m b o l 96 symbol{96}
  93. s y m b o l 299 symbol{299}
  94. s y m b o l 4 symbol{4}