wpmath0000013_2

Color_killer.html

  1. E ( t ) = E y ( t ) + a 1 ( E b ( t ) - E y ( t ) ) sin ( ω t ) + a 2 ( E r ( t ) - E y ( t ) ) cos ( ω t ) E(t)=E_{y}(t)+a_{1}\cdot(E_{b}(t)-E_{y}(t))\cdot\sin(\omega t)+a_{2}\cdot(E_{r% }(t)-E_{y}(t))\cdot\cos(\omega t)
  2. a 1 a_{1}
  3. a 2 a_{2}
  4. E y E_{y}
  5. ( E b - E y ) (E_{b}-E_{y})
  6. ( E r - E y ) (E_{r}-E_{y})
  7. ω \omega
  8. ω \omega

Community_matrix.html

  1. d x d t = x ( α - β y ) d y d t = - y ( γ - δ x ) , \begin{array}[]{rcl}\frac{dx}{dt}&=x(\alpha-\beta y)\\ \frac{dy}{dt}&=-y(\gamma-\delta x),\end{array}
  2. [ d u d t d v d t ] = A [ u v ] , \begin{bmatrix}\frac{du}{dt}\\ \frac{dv}{dt}\end{bmatrix}=A\begin{bmatrix}u\\ v\end{bmatrix},

Commutant-associative_algebra.html

  1. ( [ A 1 , A 2 ] , [ A 3 , A 4 ] , [ A 5 , A 6 ] ) = 0 ([A_{1},A_{2}],[A_{3},A_{4}],[A_{5},A_{6}])=0

Commuting_matrices.html

  1. A A
  2. B B
  3. A B = B A AB=BA
  4. [ A , B ] = A B - B A [A,B]=AB-BA
  5. A 1 , , A k A_{1},\ldots,A_{k}
  6. α i β i \alpha_{i}\leftrightarrow\beta_{i}
  7. P ( A , B ) P(A,B)
  8. P ( α i , β i ) P(\alpha_{i},\beta_{i})
  9. A A
  10. B B
  11. A A
  12. B B
  13. A = U Λ 1 U A=U\Lambda_{1}U^{\dagger}
  14. B = U Λ 2 U B=U\Lambda_{2}U^{\dagger}
  15. A B = U Λ 1 U U Λ 2 U = U Λ 1 Λ 2 U = U Λ 2 Λ 1 U = U Λ 2 U U Λ 1 U = B A . AB=U\Lambda_{1}U^{\dagger}U\Lambda_{2}U^{\dagger}=U\Lambda_{1}\Lambda_{2}U^{% \dagger}=U\Lambda_{2}\Lambda_{1}U^{\dagger}=U\Lambda_{2}U^{\dagger}U\Lambda_{1% }U^{\dagger}=BA.
  16. A A
  17. B B
  18. C C
  19. B B
  20. C C

Comonotonicity.html

  1. F ( x 1 , , x n ) := μ ( { ( y 1 , , y n ) n y 1 x 1 , , y n x n } ) , ( x 1 , , x n ) n . F(x_{1},\ldots,x_{n}):=\mu\bigl(\{(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}\mid y% _{1}\leq x_{1},\ldots,y_{n}\leq x_{n}\}\bigr),\qquad(x_{1},\ldots,x_{n})\in{% \mathbb{R}}^{n}.
  2. F i ( x ) := μ ( { ( y 1 , , y n ) n y i x } ) , x F_{i}(x):=\mu\bigl(\{(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}\mid y_{i}\leq x\}% \bigr),\qquad x\in{\mathbb{R}}
  3. F ( x 1 , , x n ) = min i { 1 , , n } F i ( x i ) , ( x 1 , , x n ) n . F(x_{1},\ldots,x_{n})=\min_{i\in\{1,\ldots,n\}}F_{i}(x_{i}),\qquad(x_{1},% \ldots,x_{n})\in{\mathbb{R}}^{n}.
  4. ( X 1 x 1 , , X n x n ) = min i { 1 , , n } ( X i x i ) , ( x 1 , , x n ) n . {\mathbb{P}}(X_{1}\leq x_{1},\ldots,X_{n}\leq x_{n})=\min_{i\in\{1,\ldots,n\}}% {\mathbb{P}}(X_{i}\leq x_{i}),\qquad(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}.
  5. ( X 1 , , X n ) = d ( F X 1 - 1 ( U ) , , F X n - 1 ( U ) ) , (X_{1},\ldots,X_{n})=\text{d}(F_{X_{1}}^{-1}(U),\ldots,F_{X_{n}}^{-1}(U)),\,
  6. ( X 1 x 1 , , X n x n ) ( X i x i ) , ( x 1 , , x n ) n , {\mathbb{P}}(X_{1}\leq x_{1},\ldots,X_{n}\leq x_{n})\leq{\mathbb{P}}(X_{i}\leq x% _{i}),\qquad(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n},
  7. ( X 1 x 1 , , X n x n ) min i { 1 , , n } ( X i x i ) , ( x 1 , , x n ) n , {\mathbb{P}}(X_{1}\leq x_{1},\ldots,X_{n}\leq x_{n})\leq\min_{i\in\{1,\ldots,n% \}}{\mathbb{P}}(X_{i}\leq x_{i}),\qquad(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n},
  8. Cov ( X , Y ) Cov ( X * , Y * ) \,\text{Cov}(X,Y)\leq\,\text{Cov}(X^{*},Y^{*})
  9. 𝔼 [ X Y ] 𝔼 [ X * Y * ] {\mathbb{E}}[XY]\leq{\mathbb{E}}[X^{*}Y^{*}]
  10. ( < v a r > X , < v a r > Y < / v a r > ) (<var>X,<var>Y</var>)
  11. < v a r > n = 2 <var>n=2
  12. < v a r > n = 2 <var>n=2
  13. ( < v a r > X * , < v a r > Y < / v a r > * ) (<var>X^{*},<var>Y</var>^{*})

Compact_Lie_algebra.html

  1. 𝔤 \mathfrak{g}
  2. SO ( 𝔤 ) \operatorname{SO}(\mathfrak{g})
  3. ad 𝔤 \operatorname{ad}\ \mathfrak{g}
  4. 𝔰 𝔬 ( 𝔤 ) \mathfrak{so}(\mathfrak{g})
  5. 𝔤 𝔩 , \mathfrak{gl},
  6. 𝔰 𝔬 . \mathfrak{so}.
  7. A n : A_{n}:
  8. 𝔰 𝔲 n + 1 , \mathfrak{su}_{n+1},
  9. B n : B_{n}:
  10. 𝔰 𝔬 2 n + 1 , \mathfrak{so}_{2n+1},
  11. 𝔬 2 n + 1 , \mathfrak{o}_{2n+1},
  12. C n : C_{n}:
  13. 𝔰 𝔭 n , \mathfrak{sp}_{n},
  14. 𝔲 𝔰 𝔭 n , \mathfrak{usp}_{n},
  15. D n : D_{n}:
  16. 𝔰 𝔬 2 n , \mathfrak{so}_{2n},
  17. 𝔬 2 n , \mathfrak{o}_{2n},
  18. E 6 , E 7 , E 8 , F 4 , G 2 . E_{6},E_{7},E_{8},F_{4},G_{2}.
  19. n 1 n\geq 1
  20. A n , A_{n},
  21. n 2 n\geq 2
  22. B n , B_{n},
  23. n 3 n\geq 3
  24. C n , C_{n},
  25. n 4 n\geq 4
  26. D n . D_{n}.
  27. n 0 n\geq 0
  28. n 1 n\geq 1
  29. n = 0 , n=0,
  30. A 0 B 0 C 0 D 0 A_{0}\cong B_{0}\cong C_{0}\cong D_{0}
  31. SU ( 1 ) SO ( 1 ) Sp ( 0 ) SO ( 0 ) . \operatorname{SU}(1)\cong\operatorname{SO}(1)\cong\operatorname{Sp}(0)\cong% \operatorname{SO}(0).
  32. n = 1 , n=1,
  33. 𝔰 𝔲 2 𝔰 𝔬 3 𝔰 𝔭 1 \mathfrak{su}_{2}\cong\mathfrak{so}_{3}\cong\mathfrak{sp}_{1}
  34. A 1 B 1 C 1 A_{1}\cong B_{1}\cong C_{1}
  35. SU ( 2 ) Spin ( 3 ) Sp ( 1 ) \operatorname{SU}(2)\cong\operatorname{Spin}(3)\cong\operatorname{Sp}(1)
  36. n = 2 , n=2,
  37. 𝔰 𝔬 5 𝔰 𝔭 2 \mathfrak{so}_{5}\cong\mathfrak{sp}_{2}
  38. B 2 C 2 , B_{2}\cong C_{2},
  39. Sp ( 2 ) Spin ( 5 ) . \operatorname{Sp}(2)\cong\operatorname{Spin}(5).
  40. n = 3 , n=3,
  41. 𝔰 𝔲 4 𝔰 𝔬 6 \mathfrak{su}_{4}\cong\mathfrak{so}_{6}
  42. A 3 D 3 , A_{3}\cong D_{3},
  43. SU ( 4 ) Spin ( 6 ) . \operatorname{SU}(4)\cong\operatorname{Spin}(6).
  44. E 4 E_{4}
  45. E 5 E_{5}
  46. A 4 A_{4}
  47. D 5 , D_{5},

Compact_tension_specimen.html

  1. K I = P B π W [ 16.7 ( a W ) 1 / 2 - 104.7 ( a W ) 3 / 2 + 369.9 ( a W ) 5 / 2 - 573.8 ( a W ) 7 / 2 + 360.5 ( a W ) 9 / 2 ] \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\frac{P}{B}\sqrt{\frac{% \pi}{W}}\left[16.7\left(\frac{a}{W}\right)^{1/2}-104.7\left(\frac{a}{W}\right)% ^{3/2}+369.9\left(\frac{a}{W}\right)^{5/2}\right.\\ &\displaystyle\qquad\left.-573.8\left(\frac{a}{W}\right)^{7/2}+360.5\left(% \frac{a}{W}\right)^{9/2}\right]\end{aligned}
  2. P P
  3. B B
  4. a a
  5. W W

Comparison_of_programming_paradigms.html

  1. π r 2 . \pi r^{2}.\,

Comparison_of_vector_algebra_and_geometric_algebra.html

  1. 3 {\mathbb{R}}^{3}
  2. 𝐯 × 𝐮 = - ( 𝐮 × 𝐯 ) \mathbf{v}\times\mathbf{u}=-(\mathbf{u}\times\mathbf{v})
  3. 𝐯 𝐮 = - ( 𝐮 𝐯 ) \mathbf{v}\wedge\mathbf{u}=-(\mathbf{u}\wedge\mathbf{v})
  4. ( 𝐮 + 𝐯 ) × 𝐰 = 𝐮 × 𝐰 + 𝐯 × 𝐰 (\mathbf{u}+\mathbf{v})\times\mathbf{w}=\mathbf{u}\times\mathbf{w}+\mathbf{v}% \times\mathbf{w}
  5. ( 𝐮 + 𝐯 ) 𝐰 = 𝐮 𝐰 + 𝐯 𝐰 (\mathbf{u}+\mathbf{v})\wedge\mathbf{w}=\mathbf{u}\wedge\mathbf{w}+\mathbf{v}% \wedge\mathbf{w}
  6. 𝐮 × ( 𝐯 + 𝐰 ) = 𝐮 × 𝐯 + 𝐮 × 𝐰 \mathbf{u}\times(\mathbf{v}+\mathbf{w})=\mathbf{u}\times\mathbf{v}+\mathbf{u}% \times\mathbf{w}
  7. 𝐮 ( 𝐯 + 𝐰 ) = 𝐮 𝐯 + 𝐮 𝐰 \mathbf{u}\wedge(\mathbf{v}+\mathbf{w})=\mathbf{u}\wedge\mathbf{v}+\mathbf{u}% \wedge\mathbf{w}
  8. ( 𝐮 × 𝐯 ) × 𝐰 𝐮 × ( 𝐯 × 𝐰 ) (\mathbf{u}\times\mathbf{v})\times\mathbf{w}\neq\mathbf{u}\times(\mathbf{v}% \times\mathbf{w})
  9. ( 𝐮 𝐯 ) 𝐰 = 𝐮 ( 𝐯 𝐰 ) (\mathbf{u}\wedge\mathbf{v})\wedge\mathbf{w}=\mathbf{u}\wedge(\mathbf{v}\wedge% \mathbf{w})
  10. 𝐮 × 𝐮 = 0 \mathbf{u}\times\mathbf{u}=0
  11. 𝐮 𝐮 = 0 \mathbf{u}\wedge\mathbf{u}=0
  12. 𝐮 × 𝐯 \mathbf{u}\times\mathbf{v}
  13. 𝐮 \mathbf{u}
  14. 𝐯 \mathbf{v}
  15. 𝐮 𝐯 \mathbf{u}\wedge\mathbf{v}
  16. 3 \mathbb{R}^{3}
  17. 𝒢 3 \mathcal{G}_{3}
  18. a × b = - i ( a b ) {a}\times{b}=-i({a}\wedge{b})
  19. i i
  20. i 2 = ( e 1 e 2 e 3 ) 2 = e 1 e 2 e 3 e 1 e 2 e 3 = - e 1 e 2 e 1 e 3 e 2 e 3 = e 1 e 1 e 2 e 3 e 2 e 3 = - e 3 e 2 e 2 e 3 = - 1 i^{2}=(e_{1}e_{2}e_{3})^{2}=e_{1}e_{2}e_{3}e_{1}e_{2}e_{3}=-e_{1}e_{2}e_{1}e_{% 3}e_{2}e_{3}=e_{1}e_{1}e_{2}e_{3}e_{2}e_{3}=-e_{3}e_{2}e_{2}e_{3}=-1
  21. 3 \mathbb{R}^{3}
  22. - i = - e 1 e 2 e 3 -i=-{e_{1}}{e_{2}}{e_{3}}
  23. u v = 1 i < j 3 ( u i v j - v i u j ) e i e j = 1 i < j 3 ( u i v j - v i u j ) e i e j u\wedge v=\sum_{1\leq i<j\leq 3}(u_{i}v_{j}-v_{i}u_{j}){e_{i}}\wedge{e_{j}}=% \sum_{1\leq i<j\leq 3}(u_{i}v_{j}-v_{i}u_{j}){e_{i}}{e_{j}}
  24. 𝐮 2 = 𝐮 𝐮 {\|\mathbf{u}\|}^{2}=\mathbf{u}\cdot\mathbf{u}
  25. 𝐮 2 = 𝐮 2 {\|\mathbf{u}\|}^{2}={\mathbf{u}}^{2}
  26. 𝐮 𝐮 = 𝐮 𝐮 + 𝐮 𝐮 = 𝐮 𝐮 \mathbf{u}\,\mathbf{u}=\mathbf{u}\cdot\mathbf{u}+\mathbf{u}\wedge\mathbf{u}=% \mathbf{u}\cdot\mathbf{u}
  27. 𝐮 2 𝐯 2 = ( 𝐮 𝐯 ) 2 + 𝐮 × 𝐯 2 {\|\mathbf{u}\|}^{2}{\|\mathbf{v}\|}^{2}=({\mathbf{u}\cdot\mathbf{v}})^{2}+{\|% \mathbf{u}\times\mathbf{v}\|}^{2}
  28. 𝐮 2 𝐯 2 = ( 𝐮 𝐯 ) 2 - ( 𝐮 𝐯 ) 2 {\|\mathbf{u}\|}^{2}{\|\mathbf{v}\|}^{2}=({\mathbf{u}\cdot\mathbf{v}})^{2}-(% \mathbf{u}\wedge\mathbf{v})^{2}
  29. ( 𝐮𝐯 ) ( 𝐯𝐮 ) = ( 𝐮 𝐯 + 𝐮 𝐯 ) ( 𝐮 𝐯 - 𝐮 𝐯 ) (\mathbf{u}\mathbf{v})(\mathbf{v}\mathbf{u})=({\mathbf{u}\cdot\mathbf{v}}+{% \mathbf{u}\wedge\mathbf{v}})({\mathbf{u}\cdot\mathbf{v}}-{\mathbf{u}\wedge% \mathbf{v}})
  30. 𝐮 × 𝐯 = i < j | u i u j v i v j | 𝐞 i × 𝐞 j \mathbf{u}\times\mathbf{v}=\sum_{i<j}{\begin{vmatrix}u_{i}&u_{j}\\ v_{i}&v_{j}\end{vmatrix}{\mathbf{e}}_{i}\times{\mathbf{e}}_{j}}
  31. 𝐮 𝐯 = i < j | u i u j v i v j | 𝐞 i 𝐞 j \mathbf{u}\wedge\mathbf{v}=\sum_{i<j}{\begin{vmatrix}u_{i}&u_{j}\\ v_{i}&v_{j}\end{vmatrix}{\mathbf{e}}_{i}\wedge{\mathbf{e}}_{j}}
  32. 𝐞 i 𝐞 j {\mathbf{e}}_{i}\wedge{\mathbf{e}}_{j}
  33. 𝐞 1 \mathbf{e}_{1}
  34. 1 \mathbb{R}^{1}
  35. 𝐞 1 𝐞 2 \mathbf{e}_{1}\wedge\mathbf{e}_{2}
  36. 2 \mathbb{R}^{2}
  37. 𝐞 1 𝐞 2 𝐞 3 \mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3}
  38. 3 \mathbb{R}^{3}
  39. A x = b Ax=b
  40. x = 1 | A | adj ( A ) b x=\frac{1}{|A|}\operatorname{adj}(A)b
  41. N {\mathbb{R}}^{N}
  42. [ a b ] [ x y ] = = a x + b y = = c \begin{bmatrix}a&b\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}==ax+by==c
  43. a a
  44. b b
  45. ( a x + b y ) b = ( a b ) x = c b (ax+by)\wedge b=(a\wedge b)x=c\wedge b
  46. a ( a x + b y ) = ( a b ) y = a c a\wedge(ax+by)=(a\wedge b)y=a\wedge c
  47. a b 0 a\wedge b\neq 0
  48. [ x y ] = 1 a b [ c b a c ] \begin{bmatrix}x\\ y\end{bmatrix}=\frac{1}{a\wedge b}\begin{bmatrix}c\wedge b\\ a\wedge c\end{bmatrix}
  49. a , b 2 a,b\in{\mathbb{R}}^{2}
  50. e 1 e 2 {e}_{1}\wedge{e}_{2}
  51. u v = | u 1 u 2 v 1 v 2 | e 1 e 2 u\wedge v=\begin{vmatrix}u_{1}&u_{2}\\ v_{1}&v_{2}\end{vmatrix}{e}_{1}\wedge{e}_{2}
  52. [ a b c ] [ x y z ] = d \begin{bmatrix}a&b&c\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=d
  53. [ x y z ] = 1 a b c [ d b c a d c a b d ] \begin{bmatrix}x\\ y\\ z\end{bmatrix}=\frac{1}{a\wedge b\wedge c}\begin{bmatrix}d\wedge b\wedge c\\ a\wedge d\wedge c\\ a\wedge b\wedge d\end{bmatrix}
  54. e 1 e 2 e 3 {e}_{1}\wedge{e}_{2}\wedge{e}_{3}
  55. [ 1 1 0 ] x + [ 1 1 1 ] y = [ 1 1 2 ] \begin{bmatrix}1\\ 1\\ 0\end{bmatrix}x+\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}y=\begin{bmatrix}1\\ 1\\ 2\end{bmatrix}
  56. ( 1 , 1 , 1 ) (1,1,1)
  57. x x
  58. [ 1 1 0 ] [ 1 1 1 ] x = [ 1 1 2 ] [ 1 1 1 ] \begin{bmatrix}1\\ 1\\ 0\end{bmatrix}\wedge\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}x=\begin{bmatrix}1\\ 1\\ 2\end{bmatrix}\wedge\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}
  59. ( 1 , 1 , 0 ) (1,1,0)
  60. y y
  61. [ 1 1 0 ] [ 1 1 1 ] y = [ 1 1 0 ] [ 1 1 2 ] . \begin{bmatrix}1\\ 1\\ 0\end{bmatrix}\wedge\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}y=\begin{bmatrix}1\\ 1\\ 0\end{bmatrix}\wedge\begin{bmatrix}1\\ 1\\ 2\end{bmatrix}.
  62. x x
  63. y y
  64. [ x y ] = 1 ( 1 , 1 , 0 ) ( 1 , 1 , 1 ) [ ( 1 , 1 , 2 ) ( 1 , 1 , 1 ) ( 1 , 1 , 0 ) ( 1 , 1 , 2 ) ] . \begin{bmatrix}x\\ y\end{bmatrix}=\frac{1}{(1,1,0)\wedge(1,1,1)}\begin{bmatrix}(1,1,2)\wedge(1,1,% 1)\\ (1,1,0)\wedge(1,1,2)\end{bmatrix}.
  65. e i e j = e i j {e}_{i}\wedge{e}_{j}={e}_{ij}
  66. [ x y ] = 1 e 13 + e 23 [ - e 13 - e 23 2 e 13 + 2 e 23 ] = [ - 1 2 ] . \begin{bmatrix}x\\ y\end{bmatrix}=\frac{1}{{e}_{13}+{e}_{23}}\begin{bmatrix}{-{e}_{13}-{e}_{23}}% \\ {2{e}_{13}+2{e}_{23}}\\ \end{bmatrix}=\begin{bmatrix}-1\\ 2\end{bmatrix}.
  67. 𝐫 {\mathbf{r}}
  68. 𝐫 0 {\mathbf{r}}_{0}
  69. 𝐫 1 {\mathbf{r}}_{1}
  70. 𝐫 2 {\mathbf{r}}_{2}
  71. ( ( 𝐫 2 - 𝐫 0 ) × ( 𝐫 1 - 𝐫 0 ) ) ( 𝐫 - 𝐫 0 ) = 0. (({\mathbf{r}}_{2}-{\mathbf{r}}_{0})\times({\mathbf{r}}_{1}-{\mathbf{r}}_{0}))% \cdot({\mathbf{r}}-{\mathbf{r}}_{0})=0.
  72. ( 𝐫 2 - 𝐫 0 ) ( 𝐫 1 - 𝐫 0 ) ( 𝐫 - 𝐫 0 ) = 0. ({\mathbf{r}}_{2}-{\mathbf{r}}_{0})\wedge({\mathbf{r}}_{1}-{\mathbf{r}}_{0})% \wedge({\mathbf{r}}-{\mathbf{r}}_{0})=0.
  73. u ^ = u / u \hat{u}=u/{\|u\|}
  74. v v
  75. u ^ \hat{u}
  76. Proj u ^ v = u ^ ( u ^ v ) \mathrm{Proj}_{{\hat{u}}}\,{v}=\hat{u}(\hat{u}\cdot v)
  77. v - u ^ ( u ^ v ) = 1 u 2 ( u 2 v - u ( u v ) ) v-\hat{u}(\hat{u}\cdot v)=\frac{1}{{\|u\|}^{2}}({\|u\|}^{2}v-u(u\cdot v))
  78. u u 2 ( u v - u v ) = 1 u ( u v ) = u ^ ( u ^ v ) = ( v u ^ ) u ^ \frac{u}{{u}^{2}}(uv-u\cdot v)=\frac{1}{u}(u\wedge v)=\hat{u}(\hat{u}\wedge v)% =(v\wedge\hat{u})\hat{u}
  79. v = u ^ ( u ^ v ) + u ^ ( u ^ v ) v=\hat{u}(\hat{u}\cdot v)+\hat{u}(\hat{u}\wedge v)
  80. v = 1 u ( u v ) + 1 u ( u v ) = ( v u ) 1 u + ( v u ) 1 u v=\frac{1}{u}({u}\cdot v)+\frac{1}{u}({u}\wedge v)=({v}\cdot u)\frac{1}{u}+(v% \wedge u)\frac{1}{u}
  81. v = u ^ u ^ v = u ^ ( u ^ v + u ^ v ) v=\hat{u}\hat{u}v=\hat{u}(\hat{u}\cdot v+\hat{u}\wedge v)
  82. v = a u + x v=au+x
  83. u x = 0 u\cdot x=0
  84. v v
  85. u u
  86. u v = u ( a u + x ) = u x u\wedge v=u\wedge(au+x)=u\wedge x
  87. u v = u x = u x - u x = u x u\wedge v=u\wedge x=ux-u\cdot x=ux
  88. x = 1 u ( u v ) x=\frac{1}{u}(u\wedge v)
  89. 𝐯 = ( 𝐯 𝐮 ^ ) 𝐮 ^ + 𝐮 ^ × ( 𝐯 × 𝐮 ^ ) \mathbf{v}=(\mathbf{v}\cdot\hat{\mathbf{u}})\hat{\mathbf{u}}+\hat{\mathbf{u}}% \times(\mathbf{v}\times\hat{\mathbf{u}})
  90. 𝐯 = ( 𝐯 𝐮 ^ ) 𝐮 ^ + ( 𝐯 𝐮 ^ ) 𝐮 ^ \mathbf{v}=(\mathbf{v}\cdot\hat{\mathbf{u}})\hat{\mathbf{u}}+(\mathbf{v}\wedge% \hat{\mathbf{u}})\hat{\mathbf{u}}
  91. 𝐯 = ( 𝐯 𝐮 ) 1 𝐮 + ( 𝐯 𝐮 ) 1 𝐮 \mathbf{v}=(\mathbf{v}\cdot\mathbf{u})\frac{1}{\mathbf{u}}+(\mathbf{v}\wedge% \mathbf{u})\frac{1}{\mathbf{u}}
  92. 𝐯 = 1 𝐮 ( 𝐮 𝐯 ) + 1 𝐮 ( 𝐮 𝐯 ) \mathbf{v}=\frac{1}{\mathbf{u}}(\mathbf{u}\cdot\mathbf{v})+\frac{1}{\mathbf{u}% }(\mathbf{u}\wedge\mathbf{v})
  93. w = a u + b v + x w=au+bv+x
  94. u x = v x = 0 u\cdot x=v\cdot x=0
  95. w w
  96. u u
  97. v v
  98. w u v = ( a u + b v + x ) u v = x u v . w\wedge u\wedge v=(au+bv+x)\wedge u\wedge v=x\wedge u\wedge v.
  99. x ( u v ) x(u\wedge v)
  100. x = ( w u v ) 1 u v = 1 u v ( u v w ) . x=(w\wedge u\wedge v)\frac{1}{u\wedge v}=\frac{1}{u\wedge v}(u\wedge v\wedge w).
  101. x = 1 ( A u , v ) 2 i < j < k | w i w j w k u i u j u k v i v j v k | | u i u j u k v i v j v k e i e j e k | x=\frac{1}{(A_{u,v})^{2}}\sum_{i<j<k}\begin{vmatrix}w_{i}&w_{j}&w_{k}\\ u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ \end{vmatrix}\begin{vmatrix}u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ {e}_{i}&{e}_{j}&{e}_{k}\\ \end{vmatrix}
  102. ( A u , v ) 2 = i < j | u i u j v i v j | = - ( u v ) 2 (A_{u,v})^{2}=\sum_{i<j}\begin{vmatrix}u_{i}&u_{j}\\ v_{i}&v_{j}\end{vmatrix}=-(u\wedge v)^{2}
  103. u u
  104. v v
  105. x x
  106. x 2 = x w = 1 ( A u , v ) 2 i < j < k | w i w j w k u i u j u k v i v j v k | 2 {\|x\|}^{2}=x\cdot w=\frac{1}{(A_{u,v})^{2}}\sum_{i<j<k}{\begin{vmatrix}w_{i}&% w_{j}&w_{k}\\ u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ \end{vmatrix}}^{2}
  107. i < j < k | w i w j w k u i u j u k v i v j v k | 2 \sum_{i<j<k}{\begin{vmatrix}w_{i}&w_{j}&w_{k}\\ u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ \end{vmatrix}}^{2}
  108. i < j < k | w i w j w k u i u j u k v i v j v k | e i e j e k \sum_{i<j<k}{\begin{vmatrix}w_{i}&w_{j}&w_{k}\\ u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ \end{vmatrix}}{e}_{i}\wedge{e}_{j}\wedge{e}_{k}
  109. e i e j e k {e}_{i}\wedge{e}_{j}\wedge{e}_{k}
  110. v = 1 u ( u v + u v ) v=\frac{1}{u}(u\cdot v+u\wedge v)
  111. r = 1 u ( u v ) = u u 2 ( u v ) = 1 u 2 u ( u v ) r=\frac{1}{u}(u\wedge v)=\frac{u}{u^{2}}(u\wedge v)=\frac{1}{{\|u\|}^{2}}u(u% \wedge v)
  112. r = 1 u 2 i < j | u i u j v i v j | | u i u j e i e j | r=\frac{1}{{\|{u}\|}^{2}}\sum_{i<j}\begin{vmatrix}u_{i}&u_{j}\\ v_{i}&v_{j}\end{vmatrix}\begin{vmatrix}u_{i}&u_{j}\\ e_{i}&e_{j}\end{vmatrix}
  113. r = v - u ^ ( u ^ v ) r=v-\hat{u}(\hat{u}\cdot v)
  114. u u
  115. | u i u j u i u j | = 0 \begin{vmatrix}u_{i}&u_{j}\\ u_{i}&u_{j}\end{vmatrix}=0
  116. r u = 0 \ r\cdot u=0
  117. r r
  118. r 2 = r v = 1 u 2 i < j | u i u j v i v j | 2 {\|r\|}^{2}=r\cdot v=\frac{1}{{\|{u}\|}^{2}}\sum_{i<j}\begin{vmatrix}u_{i}&u_{% j}\\ v_{i}&v_{j}\end{vmatrix}^{2}
  119. r 2 u 2 = i < j | u i u j v i v j | 2 {\|r\|}^{2}{\|{u}\|}^{2}=\sum_{i<j}\begin{vmatrix}u_{i}&u_{j}\\ v_{i}&v_{j}\end{vmatrix}^{2}
  120. u u
  121. v v
  122. u v = i < j | u i u j v i v j | e i e j u\wedge v=\sum_{i<j}{\begin{vmatrix}u_{i}&u_{j}\\ v_{i}&v_{j}\end{vmatrix}e_{i}\wedge e_{j}}
  123. e i e j e_{i}\wedge e_{j}
  124. 1 u ( u v ) \frac{1}{u}(u\wedge v)
  125. A 2 = 𝐮 × 𝐯 2 = i < j | u i u j v i v j | 2 , A^{2}={\|\mathbf{u}\times\mathbf{v}\|}^{2}=\sum_{i<j}{\begin{vmatrix}u_{i}&u_{% j}\\ v_{i}&v_{j}\end{vmatrix}}^{2},
  126. A 2 = - ( 𝐮 𝐯 ) 2 = i < j | u i u j v i v j | 2 . A^{2}=-(\mathbf{u}\wedge\mathbf{v})^{2}=\sum_{i<j}{\begin{vmatrix}u_{i}&u_{j}% \\ v_{i}&v_{j}\end{vmatrix}}^{2}.
  127. ( sin θ ) 2 = 𝐮 × 𝐯 2 𝐮 2 𝐯 2 ({\sin\theta})^{2}=\frac{{\|\mathbf{u}\times\mathbf{v}\|}^{2}}{{\|\mathbf{u}\|% }^{2}{\|\mathbf{v}\|}^{2}}
  128. ( sin θ ) 2 = - ( 𝐮 𝐯 ) 2 𝐮 2 𝐯 2 ({\sin\theta})^{2}=-\frac{(\mathbf{u}\wedge\mathbf{v})^{2}}{{\mathbf{u}}^{2}{% \mathbf{v}}^{2}}
  129. V 2 = ( 𝐮 × 𝐯 ) 𝐰 2 = | u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 | 2 V^{2}={\|(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}\|}^{2}={\begin{vmatrix}u_% {1}&u_{2}&u_{3}\\ v_{1}&v_{2}&v_{3}\\ w_{1}&w_{2}&w_{3}\\ \end{vmatrix}}^{2}
  130. V 2 = - ( 𝐮 𝐯 𝐰 ) 2 = - ( i < j < k | u i u j u k v i v j v k w i w j w k | 𝐞 ^ i 𝐞 ^ j 𝐞 ^ k ) 2 = i < j < k | u i u j u k v i v j v k w i w j w k | 2 V^{2}=-(\mathbf{u}\wedge\mathbf{v}\wedge\mathbf{w})^{2}=-\left(\sum_{i<j<k}% \begin{vmatrix}u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ w_{i}&w_{j}&w_{k}\\ \end{vmatrix}\hat{\mathbf{e}}_{i}\wedge\hat{\mathbf{e}}_{j}\wedge\hat{\mathbf{% e}}_{k}\right)^{2}=\sum_{i<j<k}{\begin{vmatrix}u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ w_{i}&w_{j}&w_{k}\\ \end{vmatrix}}^{2}
  131. w ( u v ) = i , j < k w i e i | u j u k v j v k | e j e k w(u\wedge v)=\sum_{i,j<k}w_{i}{e}_{i}{\begin{vmatrix}u_{j}&u_{k}\\ v_{j}&v_{k}\\ \end{vmatrix}}{e}_{j}\wedge{e}_{k}
  132. i = j i=j
  133. i = k i=k
  134. w ( u v ) = i < j ( w i e j - w j e i ) | u i u j v i v j | + i < j < k | w i w j w k u i u j u k v i v j v k | e i e j e k w(u\wedge v)=\sum_{i<j}(w_{i}{e}_{j}-w_{j}{e}_{i}){\begin{vmatrix}u_{i}&u_{j}% \\ v_{i}&v_{j}\\ \end{vmatrix}}+\sum_{i<j<k}{\begin{vmatrix}w_{i}&w_{j}&w_{k}\\ u_{i}&u_{j}&u_{k}\\ v_{i}&v_{j}&v_{k}\\ \end{vmatrix}}{e}_{i}\wedge{e}_{j}\wedge{e}_{k}
  135. w u v w\wedge u\wedge v
  136. ( u v ) w (u\wedge v)w
  137. w ( u v ) = w ( u v ) + w u v w(u\wedge v)=w\cdot(u\wedge v)+w\wedge u\wedge v
  138. ( u v ) w = - w ( u v ) + w u v (u\wedge v)w=-w\cdot(u\wedge v)+w\wedge u\wedge v
  139. w u v = 1 2 ( w ( u v ) + ( u v ) w ) w\wedge u\wedge v=\frac{1}{2}(w(u\wedge v)+(u\wedge v)w)
  140. w ( u v ) = 1 2 ( w ( u v ) - ( u v ) w ) w\cdot(u\wedge v)=\frac{1}{2}(w(u\wedge v)-(u\wedge v)w)
  141. w = x + y w=x+y
  142. x = a u + b v x=au+bv
  143. y u = y v = 0 y\cdot u=y\cdot v=0
  144. w w
  145. u v u\wedge v
  146. w ( u v ) = x ( u v ) + y ( u v ) = x ( u v ) + y ( u v ) + y u v w(u\wedge v)=x(u\wedge v)+y(u\wedge v)=x\cdot(u\wedge v)+y\cdot(u\wedge v)+y% \wedge u\wedge v
  147. y ( u v ) = 0 y\cdot(u\wedge v)=0
  148. d d t ( 𝐫 𝐫 ) = 1 𝐫 3 ( 𝐫 × d 𝐫 d t ) × 𝐫 = ( 𝐫 ^ × 1 < m t p l > 𝐫 d 𝐫 d t ) × 𝐫 ^ \frac{d}{dt}\left(\frac{\mathbf{r}}{\|\mathbf{r}\|}\right)=\frac{1}{{\|\mathbf% {r}\|}^{3}}\left(\mathbf{r}\times\frac{d\mathbf{r}}{dt}\right)\times\mathbf{r}% =\left(\hat{\mathbf{r}}\times\frac{1}{<}mtpl>{{\|\mathbf{r}\|}}\frac{d\mathbf{% r}}{dt}\right)\times\hat{\mathbf{r}}
  149. d d t ( 𝐫 𝐫 ) = 1 𝐫 3 𝐫 ( 𝐫 d 𝐫 d t ) = 1 < m t p l > 𝐫 ( 𝐫 ^ d 𝐫 d t ) \frac{d}{dt}\left(\frac{\mathbf{r}}{\|\mathbf{r}\|}\right)=\frac{1}{{\|\mathbf% {r}\|}^{3}}\mathbf{r}\left(\mathbf{r}\wedge\frac{d\mathbf{r}}{dt}\right)=\frac% {1}{<}mtpl>{{\mathbf{r}}}\left(\hat{\mathbf{r}}\wedge\frac{d\mathbf{r}}{dt}\right)
  150. 1 < m t p l > 𝐫 d 𝐫 d t \frac{1}{<}mtpl>{{\|\mathbf{r}\|}}\frac{d\mathbf{r}}{dt}
  151. 𝐫 \mathbf{r}
  152. 1 < m t p l > 𝐫 d 𝐫 d t \frac{1}{<}mtpl>{{\|\mathbf{r}\|}}\frac{d\mathbf{r}}{dt}
  153. 𝐫 ^ \hat{\mathbf{r}}
  154. 𝐫 ^ \hat{\mathbf{r}}
  155. d 𝐫 d t \frac{d\mathbf{r}}{dt}
  156. < m t p l > 𝐫 d 𝐫 ^ d t = 𝐫 ^ d 𝐫 d t <mtpl>{{\mathbf{r}}}\frac{d\hat{\mathbf{r}}}{dt}=\hat{\mathbf{r}}\wedge\frac{d% \mathbf{r}}{dt}

Compatibility_(mechanics).html

  1. s y m b o l R ( s y m b o l ε ) symbol{R}(symbol{\varepsilon})
  2. s y m b o l ε symbol{\varepsilon}
  3. s y m b o l R := s y m b o l × ( s y m b o l \timessymbol ε ) . symbol{R}:=symbol{\nabla}\times(symbol{\nabla}\timessymbol{\varepsilon})~{}.
  4. s y m b o l R := s y m b o l \timessymbol F = s y m b o l 0 symbol{R}:=symbol{\nabla}\timessymbol{F}=symbol{0}
  5. s y m b o l F symbol{F}
  6. ε 11 = u 1 x 1 ; ε 12 = 1 2 [ u 1 x 2 + u 2 x 1 ] ; ε 22 = u 2 x 2 \varepsilon_{11}=\cfrac{\partial u_{1}}{\partial x_{1}}~{};~{}~{}\varepsilon_{% 12}=\cfrac{1}{2}\left[\cfrac{\partial u_{1}}{\partial x_{2}}+\cfrac{\partial u% _{2}}{\partial x_{1}}\right]~{};~{}~{}\varepsilon_{22}=\cfrac{\partial u_{2}}{% \partial x_{2}}
  7. 2 ε 11 x 2 2 - 2 2 ε 12 x 1 x 2 + 2 ε 22 x 1 2 = 0 \cfrac{\partial^{2}\varepsilon_{11}}{\partial x_{2}^{2}}-2\cfrac{\partial^{2}% \varepsilon_{12}}{\partial x_{1}\partial x_{2}}+\cfrac{\partial^{2}\varepsilon% _{22}}{\partial x_{1}^{2}}=0
  8. 𝐮 = 𝐮 ( x 1 , x 2 ) \mathbf{u}=\mathbf{u}(x_{1},x_{2})
  9. 2 ε 33 x 1 x 2 = x 3 [ ε 23 x 1 + ε 31 x 2 - ε 12 x 3 ] \cfrac{\partial^{2}\varepsilon_{33}}{\partial x_{1}\partial x_{2}}=\cfrac{% \partial}{\partial x_{3}}\left[\cfrac{\partial\varepsilon_{23}}{\partial x_{1}% }+\cfrac{\partial\varepsilon_{31}}{\partial x_{2}}-\cfrac{\partial\varepsilon_% {12}}{\partial x_{3}}\right]
  10. e i k r e j l s ε i j , k l = 0 e_{ikr}~{}e_{jls}~{}\varepsilon_{ij,kl}=0
  11. e i j k e_{ijk}
  12. s y m b o l × ( s y m b o l \timessymbol ε ) = s y m b o l 0 symbol{\nabla}\times(symbol{\nabla}\timessymbol{\varepsilon})=symbol{0}
  13. s y m b o l \timessymbol ε = e i j k ε r j , i 𝐞 k 𝐞 r symbol{\nabla}\timessymbol{\varepsilon}=e_{ijk}\varepsilon_{rj,i}\mathbf{e}_{k% }\otimes\mathbf{e}_{r}
  14. s y m b o l R := s y m b o l × ( s y m b o l \timessymbol ε ) ; R r s := e i k r e j l s ε i j , k l symbol{R}:=symbol{\nabla}\times(symbol{\nabla}\timessymbol{\varepsilon})~{};~{% }~{}R_{rs}:=e_{ikr}~{}e_{jls}~{}\varepsilon_{ij,kl}
  15. s y m b o l \timessymbol F = s y m b o l 0 symbol{\nabla}\timessymbol{F}=symbol{0}
  16. s y m b o l F symbol{F}
  17. e A B C F i B X A = 0 e_{ABC}~{}\cfrac{\partial F_{iB}}{\partial X_{A}}=0
  18. 𝐱 = s y m b o l χ ( 𝐗 , t ) \mathbf{x}=symbol{\chi}(\mathbf{X},t)
  19. R α β ρ γ := X ρ [ Γ α β γ ] - X β [ Γ α ρ γ ] + Γ μ ρ γ Γ α β μ - Γ μ β γ Γ α ρ μ = 0 R^{\gamma}_{\alpha\beta\rho}:=\frac{\partial}{\partial X^{\rho}}[\Gamma^{% \gamma}_{\alpha\beta}]-\frac{\partial}{\partial X^{\beta}}[\Gamma^{\gamma}_{% \alpha\rho}]+\Gamma^{\gamma}_{\mu\rho}~{}\Gamma^{\mu}_{\alpha\beta}-\Gamma^{% \gamma}_{\mu\beta}~{}\Gamma^{\mu}_{\alpha\rho}=0
  20. Γ i j k \Gamma^{k}_{ij}
  21. R i j k m R^{m}_{ijk}
  22. { ( 𝐄 1 , 𝐄 2 , 𝐄 3 ) , O } \{(\mathbf{E}_{1},\mathbf{E}_{2},\mathbf{E}_{3}),O\}
  23. 𝐮 = 𝐱 - 𝐗 ; u i = x i - X i \mathbf{u}=\mathbf{x}-\mathbf{X}~{};~{}~{}u_{i}=x_{i}-X_{i}
  24. s y m b o l 𝐮 = 𝐮 𝐗 ; s y m b o l 𝐱 = 𝐱 𝐗 symbol{\nabla}\mathbf{u}=\frac{\partial\mathbf{u}}{\partial\mathbf{X}}~{};~{}~% {}symbol{\nabla}\mathbf{x}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}}
  25. s y m b o l A ( 𝐗 ) symbol{A}(\mathbf{X})
  26. 𝐯 ( 𝐗 ) \mathbf{v}(\mathbf{X})
  27. s y m b o l 𝐯 = s y m b o l A v i , j = A i j symbol{\nabla}\mathbf{v}=symbol{A}\quad\equiv\quad v_{i,j}=A_{ij}
  28. 𝐯 \mathbf{v}
  29. v i , j = A i j v_{i,j}=A_{ij}
  30. v i , j k = A i j , k ; v i , k j = A i k , j v_{i,jk}=A_{ij,k}~{};~{}~{}v_{i,kj}=A_{ik,j}
  31. v i , j k = v i , k j v_{i,jk}=v_{i,kj}
  32. A i j , k = A i k , j A_{ij,k}=A_{ik,j}
  33. s y m b o l × s y m b o l A = s y m b o l 0 symbol{\nabla}\times symbol{A}=symbol{0}
  34. s y m b o l A symbol{A}
  35. s y m b o l × s y m b o l A = s y m b o l 0 symbol{\nabla}\times symbol{A}=symbol{0}
  36. 𝐯 \mathbf{v}
  37. A A
  38. B B
  39. 𝐯 ( 𝐗 B ) - 𝐯 ( 𝐗 A ) = 𝐗 A 𝐗 B s y m b o l 𝐯 d 𝐗 = 𝐗 A 𝐗 B s y m b o l A ( 𝐗 ) d 𝐗 \mathbf{v}(\mathbf{X}_{B})-\mathbf{v}(\mathbf{X}_{A})=\int_{\mathbf{X}_{A}}^{% \mathbf{X}_{B}}symbol{\nabla}\mathbf{v}\cdot~{}d\mathbf{X}=\int_{\mathbf{X}_{A% }}^{\mathbf{X}_{B}}symbol{A}(\mathbf{X})\cdot d\mathbf{X}
  40. 𝐯 \mathbf{v}
  41. A A
  42. B B
  43. Ω s y m b o l A d s = Ω 𝐧 ( s y m b o l × s y m b o l A ) d a \oint_{\partial\Omega}symbol{A}~{}ds=\int_{\Omega}\mathbf{n}\cdot(symbol{% \nabla}\times symbol{A})~{}da
  44. s y m b o l A symbol{A}
  45. Ω s y m b o l A d s = 0 A B s y m b o l A d 𝐗 + B A s y m b o l A d 𝐗 = 0 \oint_{\partial\Omega}symbol{A}~{}ds=0\quad\implies\quad\int_{AB}symbol{A}% \cdot d\mathbf{X}+\int_{BA}symbol{A}\cdot d\mathbf{X}=0
  46. 𝐯 \mathbf{v}
  47. s y m b o l F = 𝐱 𝐗 = s y m b o l 𝐱 symbol{F}=\cfrac{\partial\mathbf{x}}{\partial\mathbf{X}}=symbol{\nabla}\mathbf% {x}
  48. s y m b o l F symbol{F}
  49. s y m b o l \timessymbol F = s y m b o l 0 symbol{\nabla}\timessymbol{F}=symbol{0}
  50. s y m b o l ϵ symbol{\epsilon}
  51. 𝐮 \mathbf{u}
  52. s y m b o l ϵ = 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] symbol{\epsilon}=\frac{1}{2}[symbol{\nabla}\mathbf{u}+(symbol{\nabla}\mathbf{u% })^{T}]
  53. 𝐮 \mathbf{u}
  54. s y m b o l ϵ symbol{\epsilon}
  55. s y m b o l 𝐮 = s y m b o l ϵ + s y m b o l ω symbol{\nabla}\mathbf{u}=symbol{\epsilon}+symbol{\omega}
  56. s y m b o l ω := 1 2 [ s y m b o l 𝐮 - ( s y m b o l 𝐮 ) T ] symbol{\omega}:=\frac{1}{2}[symbol{\nabla}\mathbf{u}-(symbol{\nabla}\mathbf{u}% )^{T}]
  57. s y m b o l s y m b o l ω ω i j , k = 1 2 ( u i , j k - u j , i k ) = 1 2 ( u i , j k + u k , j i - u j , i k - u k , j i ) = ε i k , j - ε j k , i symbol{\nabla}symbol{\omega}\equiv\omega_{ij,k}=\frac{1}{2}(u_{i,jk}-u_{j,ik})% =\frac{1}{2}(u_{i,jk}+u_{k,ji}-u_{j,ik}-u_{k,ji})=\varepsilon_{ik,j}-% \varepsilon_{jk,i}
  58. s y m b o l ω symbol{\omega}
  59. ω i j , k l = ω i j , l k \omega_{ij,kl}=\omega_{ij,lk}
  60. ε i k , j l - ε j k , i l - ε i l , j k + ε j l , i k = 0 \varepsilon_{ik,jl}-\varepsilon_{jk,il}-\varepsilon_{il,jk}+\varepsilon_{jl,ik% }=0
  61. s y m b o l × ( s y m b o l \timessymbol ϵ ) = s y m b o l 0 symbol{\nabla}\times(symbol{\nabla}\timessymbol{\epsilon})=symbol{0}
  62. 𝐰 \mathbf{w}
  63. s y m b o l × s y m b o l ϵ = s y m b o l 𝐰 + s y m b o l 𝐰 T symbol{\nabla}\times symbol{\epsilon}=symbol{\nabla}\mathbf{w}+symbol{\nabla}% \mathbf{w}^{T}
  64. s y m b o l × ( s y m b o l 𝐰 + s y m b o l 𝐰 T ) = s y m b o l 0 symbol{\nabla}\times(symbol{\nabla}\mathbf{w}+symbol{\nabla}\mathbf{w}^{T})=% symbol{0}
  65. s y m b o l × ( s y m b o l \timessymbol ϵ ) = s y m b o l 0 symbol{\nabla}\times(symbol{\nabla}\timessymbol{\epsilon})=symbol{0}
  66. 𝐮 \mathbf{u}
  67. s y m b o l ω symbol{\omega}
  68. s y m b o l 𝐰 symbol{\nabla}\mathbf{w}
  69. 𝐗 A \mathbf{X}_{A}
  70. 𝐗 B \mathbf{X}_{B}
  71. 𝐰 ( 𝐗 B ) - 𝐰 ( 𝐗 A ) = 𝐗 A 𝐗 B s y m b o l 𝐰 d 𝐗 = 𝐗 A 𝐗 B ( s y m b o l × s y m b o l ϵ ) d 𝐗 \mathbf{w}(\mathbf{X}_{B})-\mathbf{w}(\mathbf{X}_{A})=\int_{\mathbf{X}_{A}}^{% \mathbf{X}_{B}}symbol{\nabla}\mathbf{w}\cdot d\mathbf{X}=\int_{\mathbf{X}_{A}}% ^{\mathbf{X}_{B}}(symbol{\nabla}\times symbol{\epsilon})\cdot d\mathbf{X}
  72. 𝐰 ( 𝐗 A ) \mathbf{w}(\mathbf{X}_{A})
  73. 𝐰 ( 𝐗 ) \mathbf{w}(\mathbf{X})
  74. 𝐗 A \mathbf{X}_{A}
  75. 𝐗 b \mathbf{X}_{b}
  76. 𝐗 A 𝐗 B ( s y m b o l × s y m b o l ϵ ) d 𝐗 = s y m b o l 0 \oint_{\mathbf{X}_{A}}^{\mathbf{X}_{B}}(symbol{\nabla}\times symbol{\epsilon})% \cdot d\mathbf{X}=symbol{0}
  77. 𝐗 A 𝐗 B ( s y m b o l × s y m b o l ϵ ) d 𝐗 = Ω A B 𝐧 ( s y m b o l × s y m b o l \timessymbol ϵ ) d a = s y m b o l 0 \oint_{\mathbf{X}_{A}}^{\mathbf{X}_{B}}(symbol{\nabla}\times symbol{\epsilon})% \cdot d\mathbf{X}=\int_{\Omega_{AB}}\mathbf{n}\cdot(symbol{\nabla}\times symbol% {\nabla}\timessymbol{\epsilon})~{}da=symbol{0}
  78. 𝐰 \mathbf{w}
  79. s y m b o l ω symbol{\omega}
  80. 𝐮 \mathbf{u}
  81. 𝐮 ( 𝐗 B ) - 𝐮 ( 𝐗 A ) = 𝐗 A 𝐗 B s y m b o l 𝐮 d 𝐗 = 𝐗 A 𝐗 B ( s y m b o l ϵ + s y m b o l ω ) d 𝐗 \mathbf{u}(\mathbf{X}_{B})-\mathbf{u}(\mathbf{X}_{A})=\int_{\mathbf{X}_{A}}^{% \mathbf{X}_{B}}symbol{\nabla}\mathbf{u}\cdot d\mathbf{X}=\int_{\mathbf{X}_{A}}% ^{\mathbf{X}_{B}}(symbol{\epsilon}+symbol{\omega})\cdot d\mathbf{X}
  82. s y m b o l × s y m b o l ϵ = s y m b o l 𝐰 = - s y m b o l × Ω symbol{\nabla}\times symbol{\epsilon}=symbol{\nabla}\mathbf{w}=-symbol{\nabla}\times\Omega
  83. 𝐗 A 𝐗 B ( s y m b o l ϵ + s y m b o l ω ) d 𝐗 = Ω A B 𝐧 ( s y m b o l × s y m b o l ϵ + s y m b o l × s y m b o l ω ) d a = s y m b o l 0 \oint_{\mathbf{X}_{A}}^{\mathbf{X}_{B}}(symbol{\epsilon}+symbol{\omega})\cdot d% \mathbf{X}=\int_{\Omega_{AB}}\mathbf{n}\cdot(symbol{\nabla}\times symbol{% \epsilon}+symbol{\nabla}\times symbol{\omega})~{}da=symbol{0}
  84. 𝐮 \mathbf{u}
  85. 𝐮 \mathbf{u}
  86. s y m b o l C ( 𝐗 ) symbol{C}(\mathbf{X})
  87. s y m b o l C symbol{C}
  88. 𝐱 ( 𝐗 ) \mathbf{x}(\mathbf{X})
  89. ( 1 ) ( 𝐱 𝐗 ) T ( 𝐱 𝐗 ) = s y m b o l C (1)\quad\left(\frac{\partial\mathbf{x}}{\partial\mathbf{X}}\right)^{T}\left(% \frac{\partial\mathbf{x}}{\partial\mathbf{X}}\right)=symbol{C}
  90. 𝐱 ( 𝐗 ) \mathbf{x}(\mathbf{X})
  91. x i X α x i X β = C α β \frac{\partial x^{i}}{\partial X^{\alpha}}\frac{\partial x^{i}}{\partial X^{% \beta}}=C_{\alpha\beta}
  92. C α β = g α β C_{\alpha\beta}=g_{\alpha\beta}
  93. δ i j x i X α x j X β = g α β \delta_{ij}~{}\frac{\partial x^{i}}{\partial X^{\alpha}}~{}\frac{\partial x^{j% }}{\partial X^{\beta}}=g_{\alpha\beta}
  94. G i j = X α x i X β x j g α β G_{ij}=\frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{% \partial x^{j}}~{}g_{\alpha\beta}
  95. G i j G_{ij}
  96. g α β g_{\alpha\beta}
  97. δ i j = G i j \delta_{ij}=G_{ij}
  98. Γ i j k ( x ) = 0 {}_{(x)}\Gamma_{ij}^{k}=0
  99. 2 x m X α X β = x m X μ ( X ) Γ α β μ - x i X α x j X β ( x ) Γ i j m \frac{\partial^{2}x^{m}}{\partial X^{\alpha}\partial X^{\beta}}=\frac{\partial x% ^{m}}{\partial X^{\mu}}\,_{(X)}\Gamma^{\mu}_{\alpha\beta}-\frac{\partial x^{i}% }{\partial X^{\alpha}}~{}\frac{\partial x^{j}}{\partial X^{\beta}}\,_{(x)}% \Gamma^{m}_{ij}
  100. F α m X β = F μ m Γ α β μ ( X ) ; F α i := x i X α \frac{\partial F^{m}_{~{}\alpha}}{\partial X^{\beta}}=F^{m}_{~{}\mu}\,{}_{(X)}% \Gamma^{\mu}_{\alpha\beta}\qquad;~{}~{}F^{i}_{~{}\alpha}:=\frac{\partial x^{i}% }{\partial X^{\alpha}}
  101. Γ α β γ ( X ) = 1 2 ( g α γ X β + g β γ X α - g α β X γ ) ; Γ α β ν ( X ) = g ( X ) ν γ Γ α β γ ; g α β = C α β ; g α β = C α β {}_{(X)}\Gamma_{\alpha\beta\gamma}=\frac{1}{2}\left(\frac{\partial g_{\alpha% \gamma}}{\partial X^{\beta}}+\frac{\partial g_{\beta\gamma}}{\partial X^{% \alpha}}-\frac{\partial g_{\alpha\beta}}{\partial X^{\gamma}}\right)~{};~{}~{}% _{(X)}\Gamma^{\nu}_{\alpha\beta}=g^{\nu\gamma}\,_{(X)}\Gamma_{\alpha\beta% \gamma}~{};~{}~{}g_{\alpha\beta}=C_{\alpha\beta}~{};~{}~{}g^{\alpha\beta}=C^{% \alpha\beta}
  102. Γ α β μ ( X ) = C μ γ 2 ( C α γ X β + C β γ X α - C α β X γ ) \,{}_{(X)}\Gamma^{\mu}_{\alpha\beta}=\cfrac{C^{\mu\gamma}}{2}\left(\frac{% \partial C_{\alpha\gamma}}{\partial X^{\beta}}+\frac{\partial C_{\beta\gamma}}% {\partial X^{\alpha}}-\frac{\partial C_{\alpha\beta}}{\partial X^{\gamma}}\right)
  103. F α m X β = F μ m C μ γ 2 ( C α γ X β + C β γ X α - C α β X γ ) \frac{\partial F^{m}_{~{}\alpha}}{\partial X^{\beta}}=F^{m}_{~{}\mu}~{}\cfrac{% C^{\mu\gamma}}{2}\left(\frac{\partial C_{\alpha\gamma}}{\partial X^{\beta}}+% \frac{\partial C_{\beta\gamma}}{\partial X^{\alpha}}-\frac{\partial C_{\alpha% \beta}}{\partial X^{\gamma}}\right)
  104. 2 F α m X β X ρ = 2 F α m X ρ X β F μ m X ρ ( X ) Γ α β μ + F μ m X ρ [ ( X ) Γ α β μ ] = F μ m X β ( X ) Γ α ρ μ + F μ m X β [ ( X ) Γ α ρ μ ] \frac{\partial^{2}F^{m}_{~{}\alpha}}{\partial X^{\beta}\partial X^{\rho}}=% \frac{\partial^{2}F^{m}_{~{}\alpha}}{\partial X^{\rho}\partial X^{\beta}}% \implies\frac{\partial F^{m}_{~{}\mu}}{\partial X^{\rho}}\,_{(X)}\Gamma^{\mu}_% {\alpha\beta}+F^{m}_{~{}\mu}~{}\frac{\partial}{\partial X^{\rho}}[\,_{(X)}% \Gamma^{\mu}_{\alpha\beta}]=\frac{\partial F^{m}_{~{}\mu}}{\partial X^{\beta}}% \,_{(X)}\Gamma^{\mu}_{\alpha\rho}+F^{m}_{~{}\mu}~{}\frac{\partial}{\partial X^% {\beta}}[\,_{(X)}\Gamma^{\mu}_{\alpha\rho}]
  105. F γ m Γ μ ρ γ ( X ) Γ α β μ ( X ) + F μ m X ρ [ ( X ) Γ α β μ ] = F γ m Γ μ β γ ( X ) Γ α ρ μ ( X ) + F μ m X β [ ( X ) Γ α ρ μ ] F^{m}_{~{}\gamma}\,{}_{(X)}\Gamma^{\gamma}_{\mu\rho}\,{}_{(X)}\Gamma^{\mu}_{% \alpha\beta}+F^{m}_{~{}\mu}~{}\frac{\partial}{\partial X^{\rho}}[\,_{(X)}% \Gamma^{\mu}_{\alpha\beta}]=F^{m}_{~{}\gamma}\,{}_{(X)}\Gamma^{\gamma}_{\mu% \beta}\,{}_{(X)}\Gamma^{\mu}_{\alpha\rho}+F^{m}_{~{}\mu}~{}\frac{\partial}{% \partial X^{\beta}}[\,_{(X)}\Gamma^{\mu}_{\alpha\rho}]
  106. F γ m ( Γ μ ρ γ ( X ) Γ α β μ ( X ) + X ρ [ ( X ) Γ α β γ ] - ( X ) Γ μ β γ Γ α ρ μ ( X ) - X β [ ( X ) Γ α ρ γ ] ) = 0 F^{m}_{~{}\gamma}\left(\,{}_{(X)}\Gamma^{\gamma}_{\mu\rho}\,{}_{(X)}\Gamma^{% \mu}_{\alpha\beta}+\frac{\partial}{\partial X^{\rho}}[\,_{(X)}\Gamma^{\gamma}_% {\alpha\beta}]-\,_{(X)}\Gamma^{\gamma}_{\mu\beta}\,{}_{(X)}\Gamma^{\mu}_{% \alpha\rho}-\frac{\partial}{\partial X^{\beta}}[\,_{(X)}\Gamma^{\gamma}_{% \alpha\rho}]\right)=0
  107. F γ m F^{m}_{\gamma}
  108. R α β ρ γ := X ρ [ ( X ) Γ α β γ ] - X β [ ( X ) Γ α ρ γ ] + ( X ) Γ μ ρ γ Γ α β μ ( X ) - ( X ) Γ μ β γ Γ α ρ μ ( X ) = 0 R^{\gamma}_{\alpha\beta\rho}:=\frac{\partial}{\partial X^{\rho}}[\,_{(X)}% \Gamma^{\gamma}_{\alpha\beta}]-\frac{\partial}{\partial X^{\beta}}[\,_{(X)}% \Gamma^{\gamma}_{\alpha\rho}]+\,_{(X)}\Gamma^{\gamma}_{\mu\rho}\,{}_{(X)}% \Gamma^{\mu}_{\alpha\beta}-\,_{(X)}\Gamma^{\gamma}_{\mu\beta}\,{}_{(X)}\Gamma^% {\mu}_{\alpha\rho}=0
  109. s y m b o l C symbol{C}
  110. R α β ρ γ = 0 ; g α β = C α β R^{\gamma}_{\alpha\beta\rho}=0~{};~{}~{}g_{\alpha\beta}=C_{\alpha\beta}
  111. 𝐱 \mathbf{x}
  112. 𝐗 \mathbf{X}
  113. x i X α x i X β = C α β \frac{\partial x^{i}}{\partial X^{\alpha}}\frac{\partial x^{i}}{\partial X^{% \beta}}=C_{\alpha\beta}
  114. F α i X β = F γ i Γ α β γ ( X ) \frac{\partial F^{i}_{~{}\alpha}}{\partial X^{\beta}}=F^{i}_{~{}\gamma}~\,{}_{% (X)}\Gamma^{\gamma}_{\alpha\beta}
  115. F α i F^{i}_{~{}\alpha}
  116. Γ α β γ ( X ) = ( X ) Γ β α γ ; R α β ρ γ = 0 {}_{(X)}\Gamma^{\gamma}_{\alpha\beta}=_{(X)}\Gamma^{\gamma}_{\beta\alpha}~{};~% {}~{}R^{\gamma}_{\alpha\beta\rho}=0
  117. Γ j k i \Gamma^{i}_{jk}
  118. F α i F^{i}_{~{}\alpha}
  119. C 2 C^{2}
  120. x i X α = F α i \frac{\partial x^{i}}{\partial X^{\alpha}}=F^{i}_{~{}\alpha}
  121. F α i F^{i}_{~{}\alpha}
  122. C 2 C^{2}
  123. x i ( X α ) x^{i}(X^{\alpha})
  124. x i x^{i}
  125. det | x i X α | 0 \det\left|\frac{\partial x^{i}}{\partial X^{\alpha}}\right|\neq 0
  126. x i X α g α β x j X β = δ i j \frac{\partial x^{i}}{\partial X^{\alpha}}~{}g^{\alpha\beta}~{}\frac{\partial x% ^{j}}{\partial X^{\beta}}=\delta^{ij}
  127. g α β = C α β = x k X α x k X β g_{\alpha\beta}=C_{\alpha\beta}=\frac{\partial x^{k}}{\partial X^{\alpha}}~{}% \frac{\partial x^{k}}{\partial X^{\beta}}
  128. 𝐱 𝐗 \frac{\partial\mathbf{x}}{\partial\mathbf{X}}
  129. s y m b o l C symbol{C}

Competitive_learning.html

  1. 𝐰 i = ( w i 1 , . . , w i d ) T , i = 1 , . . , M {\mathbf{w}}_{i}=\left({w_{i1},..,w_{id}}\right)^{T},i=1,..,M
  2. 𝐱 n = ( x n 1 , . . , x n d ) T d {\mathbf{x}}^{n}=\left({x_{n1},..,x_{nd}}\right)^{T}\in\mathbb{R}^{d}
  3. 𝐰 i {\mathbf{w}}_{i}
  4. o i = 1 o_{i}=1
  5. o i = 0 , i = 1 , . . , M , i m o_{i}=0,i=1,..,M,i\neq m
  6. 𝐱 - 𝐰 i \left\|{{\mathbf{x}}-{\mathbf{w}}_{i}}\right\|
  7. 𝐱 n {\mathbf{x}}^{n}
  8. 𝐰 i {\mathbf{w}}_{i}

Complete_spatial_randomness.html

  1. ρ \rho
  2. k k
  3. V V
  4. ρ \rho
  5. P ( k , ρ , V ) = ( V ρ ) k e - ( V ρ ) k ! . P(k,\rho,V)=\frac{(V\rho)^{k}e^{-(V\rho)}}{k!}.\,\!
  6. ρ V \rho V
  7. N th N^{\mathrm{th}}
  8. r r
  9. P N ( r ) = D ( N - 1 ) ! λ N r D N - 1 e - λ r D , P_{N}(r)=\frac{D}{(N-1)!}{\lambda}^{N}r^{DN-1}e^{-\lambda r^{D}},
  10. D D
  11. λ \lambda
  12. λ = ρ π D 2 Γ ( D 2 + 1 ) \lambda=\frac{\rho\pi^{\frac{D}{2}}}{\Gamma(\frac{D}{2}+1)}
  13. Γ \Gamma
  14. P N ( r ) P_{N}(r)
  15. D D

Complex_lamellar_vector_field.html

  1. 𝐅 ( × 𝐅 ) = 0. \mathbf{F}\cdot(\nabla\times\mathbf{F})=0.
  2. × 𝐅 = 0. \nabla\times\mathbf{F}=\mathbf{0}.

Complex_normal_distribution.html

  1. μ = 0 and C = 0 \mu=0\ \,\text{and}\ C=0
  2. Z = X + i Y Z=X+iY\,
  3. μ = E [ Z ] , Γ = E [ ( Z - μ ) ( Z ¯ - μ ¯ ) ] , C = E [ ( Z - μ ) ( Z - μ ) ] , \mu=\operatorname{E}[Z],\quad\Gamma=\operatorname{E}[(Z-\mu)(\overline{Z}-% \overline{\mu})^{\prime}],\quad C=\operatorname{E}[(Z-\mu)(Z-\mu)^{\prime}],
  4. P = Γ ¯ - C ¯ Γ - 1 C P=\overline{\Gamma}-\overline{C}^{\prime}\Gamma^{-1}C
  5. V x x E [ ( X - μ x ) ( X - μ x ) ] = 1 2 Re [ Γ + C ] , V x y E [ ( X - μ x ) ( Y - μ y ) ] = 1 2 Im [ - Γ + C ] , \displaystyle V_{xx}\equiv\operatorname{E}[(X-\mu_{x})(X-\mu_{x})^{\prime}]=% \tfrac{1}{2}\operatorname{Re}[\Gamma+C],\quad V_{xy}\equiv\operatorname{E}[(X-% \mu_{x})(Y-\mu_{y})^{\prime}]=\tfrac{1}{2}\operatorname{Im}[-\Gamma+C],
  6. Γ = V x x + V y y + i ( V y x - V x y ) , \displaystyle\Gamma=V_{xx}+V_{yy}+i(V_{yx}-V_{xy}),
  7. f ( z ) \displaystyle f(z)
  8. φ ( w ) = exp { i Re ( w ¯ μ ) - 1 4 ( w ¯ Γ w + Re ( w ¯ C w ¯ ) ) } , \varphi(w)=\exp\!\big\{i\operatorname{Re}(\overline{w}^{\prime}\mu)-\tfrac{1}{% 4}\big(\overline{w}^{\prime}\Gamma w+\operatorname{Re}(\overline{w}^{\prime}C% \overline{w})\big)\big\},
  9. w w
  10. Z 𝒞 𝒩 ( μ , Γ , C ) A Z + b 𝒞 𝒩 ( A μ + b , A Γ A ¯ , A C A ) Z\ \sim\ \mathcal{CN}(\mu,\,\Gamma,\,C)\quad\Rightarrow\quad AZ+b\ \sim\ % \mathcal{CN}(A\mu+b,\,A\Gamma\overline{A}^{\prime},\,ACA^{\prime})
  11. 2 [ ( Z ¯ - μ ¯ ) P - 1 ¯ ( Z - μ ) - Re ( ( Z - μ ) R P - 1 ¯ ( Z - μ ) ) ] χ 2 ( 2 k ) 2\Big[(\overline{Z}-\overline{\mu})^{\prime}\overline{P^{-1}}(Z-\mu)-% \operatorname{Re}\big((Z-\mu)^{\prime}R^{\prime}\overline{P^{-1}}(Z-\mu)\big)% \Big]\ \sim\ \chi^{2}(2k)
  12. T ( 1 T t = 1 T z t - E [ z t ] ) 𝑑 𝒞 𝒩 ( 0 , Γ , C ) , \sqrt{T}\Big(\tfrac{1}{T}\textstyle\sum_{t=1}^{T}z_{t}-\operatorname{E}[z_{t}]% \Big)\ \xrightarrow{d}\ \mathcal{CN}(0,\,\Gamma,\,C),
  13. ( X Y ) 𝒩 ( [ Re μ Im μ ] , 1 2 [ Re Γ - Im Γ Im Γ Re Γ ] ) \begin{pmatrix}X\\ Y\end{pmatrix}\ \sim\ \mathcal{N}\Big(\begin{bmatrix}\operatorname{Re}\,\mu\\ \operatorname{Im}\,\mu\end{bmatrix},\ \tfrac{1}{2}\begin{bmatrix}\operatorname% {Re}\,\Gamma&-\operatorname{Im}\,\Gamma\\ \operatorname{Im}\,\Gamma&\operatorname{Re}\,\Gamma\end{bmatrix}\Big)
  14. Z 𝒞 𝒩 ( 0 , Γ ) Z\sim\mathcal{CN}(0,\,\Gamma)
  15. f ( z ) = 1 π k det ( Γ ) e - z ¯ Γ - 1 z . f(z)=\tfrac{1}{\pi^{k}\det(\Gamma)}\,e^{-\overline{z}^{\prime}\;\Gamma^{-1}\;z}.
  16. μ \mu
  17. Γ \Gamma
  18. z z
  19. ln ( L ( μ , Γ ) ) = - ln ( det ( Γ ) ) - ( z - μ ) ¯ Γ - 1 ( z - μ ) - k ln ( π ) . \ln(L(\mu,\Gamma))=-\ln(\det(\Gamma))-\overline{(z-\mu)}^{\prime}\Gamma^{-1}(z% -\mu)-k\ln(\pi).
  20. f ( z ) = 1 π e - z ¯ z = 1 π e - | z | 2 . f(z)=\tfrac{1}{\pi}e^{-\overline{z}z}=\tfrac{1}{\pi}e^{-|z|^{2}}.
  21. Q = j = 1 n z j ¯ z j = j = 1 n z j 2 Q=\sum_{j=1}^{n}\overline{z_{j}^{\prime}}z_{j}=\sum_{j=1}^{n}\|z_{j}\|^{2}
  22. W = j = 1 n z j z j ¯ W=\sum_{j=1}^{n}z_{j}\overline{z_{j}^{\prime}}
  23. f ( w ) = det ( Γ - 1 ) n det ( w ) n - k π k ( k - 1 ) / 2 j = 1 p ( n - j ) ! e - tr ( Γ - 1 w ) f(w)=\frac{\det(\Gamma^{-1})^{n}\det(w)^{n-k}}{\pi^{k(k-1)/2}\prod_{j=1}^{p}(n% -j)!}\ e^{-\operatorname{tr}(\Gamma^{-1}w)}

Complex_squaring_map.html

  1. z n + 1 = z n 2 \qquad z_{n+1}=z_{n}^{2}
  2. z n z_{n}
  3. z 0 z_{0}
  4. z n = z 0 2 n \qquad z_{n}=z_{0}^{2^{n}}
  5. z 0 = exp ( i θ ) z_{0}=\exp(i\theta)
  6. z n = exp ( i 2 n θ ) z_{n}=\exp(i2^{n}\theta)
  7. z n = cos ( 2 n θ ) + i sin ( 2 n θ ) z_{n}=\cos(2^{n}\theta)+i\sin(2^{n}\theta)
  8. z n + 1 = z n p z_{n+1}=z_{n}^{p}
  9. z n = z 0 p n z_{n}=z_{0}^{p^{n}}

Composite_fermion.html

  1. B * = B - 2 p ρ ϕ 0 , B^{*}=B-2p\rho\phi_{0},
  2. B B
  3. 2 p 2p
  4. ρ \rho
  5. ϕ 0 = h c / e \phi_{0}=hc/e
  6. 2 p 2p
  7. B * . B^{*}.
  8. B * . B^{*}.
  9. B * . B^{*}.
  10. ν = ρ ϕ 0 / B . \nu=\rho\phi_{0}/B.
  11. B * , B^{*},
  12. Λ \Lambda
  13. ν = ρ ϕ 0 / | B * | . \nu=\rho\phi_{0}/|B^{*}|.
  14. ν = ν * 2 p ν * ± 1 . \nu=\frac{\nu^{*}}{2p\nu^{*}\pm 1}.
  15. B B
  16. ν \nu
  17. B * B^{*}
  18. ν * \nu^{*}
  19. B = 2 p ρ ϕ 0 B=2p\rho\phi_{0}
  20. ν = 1 / 2 p \nu=1/2p
  21. B * = 0 B^{*}=0
  22. B * = 0 B^{*}=0
  23. B B
  24. B * B^{*}
  25. B B
  26. B * = 0 B^{*}=0
  27. 1 / B * . 1/B^{*}.
  28. | B * | |B^{*}|
  29. ν * = n \nu^{*}=n
  30. ν = n 2 p n ± 1 . \nu=\frac{n}{2pn\pm 1}.
  31. ν = 1 - n 2 p n ± 1 , \nu=1-\frac{n}{2pn\pm 1},
  32. n 2 n + 1 = 1 3 , 2 5 , 3 7 , 4 9 , 5 11 , {n\over 2n+1}={1\over 3},\,{2\over 5},\,{3\over 7},\,{4\over 9},\,{5\over 11},\cdots
  33. n 2 n - 1 = 2 3 , 3 5 , 4 7 , 5 9 , 6 11 , {n\over 2n-1}={2\over 3},\,{3\over 5},\,{4\over 7},\,{5\over 9},\,{6\over 11},\cdots
  34. n 4 n + 1 = 1 5 , 2 9 , 3 13 , 4 17 , {n\over 4n+1}={1\over 5},\,{2\over 9},\,{3\over 13},\,{4\over 17},\cdots
  35. R H = h ν e 2 , R_{H}={h\over\nu e^{2}},
  36. ν \nu
  37. ν * = 4 / 3 \nu^{*}=4/3
  38. ν = 5 / 2 , \nu=5/2,
  39. Ψ ν FQHE = P Ψ ν * IQHE 1 j < k N ( z j - z k ) 2 p \Psi^{\rm FQHE}_{\nu}=P\;\;\Psi^{\rm IQHE}_{\nu^{*}}\prod_{1\leq j<k\leq N}(z_% {j}-z_{k})^{2p}
  40. Ψ ν FQHE \Psi^{\rm FQHE}_{\nu}
  41. ν \nu
  42. Ψ ν * IQHE \Psi^{\rm IQHE}_{\nu^{*}}
  43. ν * \nu^{*}
  44. N N
  45. z j = x j + i y j z_{j}=x_{j}+iy_{j}
  46. j j
  47. P P
  48. j < k = 1 N ( z j - z k ) 2 p \prod_{j<k=1}^{N}(z_{j}-z_{k})^{2p}
  49. 2 p 2p
  50. ν * \nu^{*}
  51. ν * = n \nu^{*}=n
  52. ν = n / ( 2 p n ± 1 ) \nu=n/(2pn\pm 1)
  53. ν * = 1 \nu^{*}=1
  54. ν = 1 / ( 2 p + 1 ) \nu=1/(2p+1)

Composite_Index_of_National_Capability.html

  1. C o u n t r y W o r l d \frac{Country}{World}
  2. T P R + U P R + I S P R + E C R + M E R + M P R 6 \frac{TPR+UPR+ISPR+ECR+MER+MPR}{6}

Compressed_suffix_array.html

  1. O ( n H k ( T ) ) + o ( n ) O(nH_{k}(T))+o(n)
  2. H k ( T ) H_{k}(T)
  3. O ( n log | Σ | ) O(n\,{\log|\Sigma|})
  4. Ω ( n log n ) \Omega(n\,{\log n})

Computable_analysis.html

  1. f : f\colon\mathbb{R}\to\mathbb{R}
  2. { x i } i = 1 \{x_{i}\}_{i=1}^{\infty}
  3. { f ( x i ) } i = 1 \{f(x_{i})\}_{i=1}^{\infty}

Computed_tomography_dose_index.html

  1. C T D I = 1 n T - 7 T 7 T D ( z ) d z CTDI=\frac{1}{nT}\int_{-7T}^{7T}{D(z)dz}
  2. n n
  3. T T
  4. D ( z ) D(z)
  5. z z
  6. C T D I 100 = 1 n T - 50 m m 50 m m D a ( z ) d z CTDI_{100}=\frac{1}{nT}\int_{-50mm}^{50mm}{D_{a}(z)dz}
  7. D a ( z ) D_{a}(z)
  8. C T D I w = 1 3 C T D I 100 c e n t r a l + 2 3 C T D I 100 p e r i p h e r a l . CTDI_{w}=\frac{1}{3}CTDI_{100}^{central}+\frac{2}{3}CTDI_{100}^{peripheral}.

Computer-automated_design.html

  1. J [ 0 , ) J\in[0,\infty)
  2. f ( 0 , 1 ] f\in(0,1]
  3. f = J 1 + J f=\tfrac{J}{1+J}

Computer_stereo_vision.html

  1. Therefore displacement d = E F + G H = B F ( E F B F + G H B F ) = B F ( E F B F + G H D G ) = B F ( B C + C D A C ) = B F B D A C = k z , where \begin{aligned}\displaystyle\,\text{Therefore displacement }d&\displaystyle=EF% +GH\\ &\displaystyle=BF(\frac{EF}{BF}+\frac{GH}{BF})\\ &\displaystyle=BF(\frac{EF}{BF}+\frac{GH}{DG})\\ &\displaystyle=BF(\frac{BC+CD}{AC})\\ &\displaystyle=BF\frac{BD}{AC}\\ &\displaystyle=\frac{k}{z}\,\text{, where}\\ \end{aligned}
  2. d = k z d=\frac{k}{z}
  3. z 1 z_{1}
  4. z 2 z_{2}
  5. z 2 ( x , y ) = min { v : v = z 1 ( x , y - k z 1 ( x , y ) ) } z_{2}(x,y)=\min\left\{v:v=z_{1}(x,y-\frac{k}{z_{1}(x,y)})\right\}
  6. z 1 ( x , y ) = min { v : v = z 2 ( x , y + k z 2 ( x , y ) ) } z_{1}(x,y)=\min\left\{v:v=z_{2}(x,y+\frac{k}{z_{2}(x,y)})\right\}
  7. P ( x , μ , σ ) = 1 σ 2 π e - ( x - μ ) 2 2 σ 2 P(x,\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}
  8. P ( x ) = 2 - L ( x ) P(x)=2^{-L(x)}
  9. L ( x ) = - log 2 P ( x ) L(x)=-\log_{2}{P(x)}
  10. L ( x , μ , σ ) = log 2 ( σ 2 π ) + ( x - μ ) 2 2 σ 2 log 2 e L(x,\mu,\sigma)=\log_{2}(\sigma\sqrt{2\pi})+\frac{(x-\mu)^{2}}{2\sigma^{2}}% \log_{2}e
  11. I ( x , μ , σ ) = ( x - μ ) 2 σ 2 I(x,\mu,\sigma)=\frac{(x-\mu)^{2}}{\sigma^{2}}
  12. L ( x , μ , σ ) = log 2 ( σ 2 π ) + I ( x , μ , σ ) log 2 e 2 L(x,\mu,\sigma)=\log_{2}(\sigma\sqrt{2\pi})+I(x,\mu,\sigma)\frac{\log_{2}e}{2}
  13. z ( x , y ) z(x,y)
  14. I m I_{m}
  15. I m ( z 1 , z 2 ) = 1 σ m 2 x , y cd ( color 1 ( x , y + k z 1 ( x , y ) ) , color 2 ( x , y ) ) 2 I_{m}(z_{1},z_{2})=\frac{1}{\sigma_{m}^{2}}\sum_{x,y}\operatorname{cd}(% \operatorname{color}_{1}(x,y+\frac{k}{z_{1}(x,y)}),\operatorname{color}_{2}(x,% y))^{2}
  16. I s ( z 1 , z 2 ) = 1 2 σ h 2 i : { 1 , 2 } x 1 , y 1 x 2 , y 2 cd ( color i ( x 1 , y 1 ) , color i ( x 2 , y 2 ) ) 2 ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 + ( z i ( x 1 , y 1 ) - z i ( x 2 , y 2 ) ) 2 I_{s}(z_{1},z_{2})=\frac{1}{2\sigma_{h}^{2}}\sum_{i:\{1,2\}}\sum_{x_{1},y_{1}}% \sum_{x_{2},y_{2}}\frac{\operatorname{cd}(\operatorname{color}_{i}(x_{1},y_{1}% ),\operatorname{color}_{i}(x_{2},y_{2}))^{2}}{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^% {2}+(z_{i}(x_{1},y_{1})-z_{i}(x_{2},y_{2}))^{2}}
  17. I t ( z 1 , z 2 ) = I m ( z 1 , z 2 ) + I s ( z 1 , z 2 ) I_{t}(z_{1},z_{2})=I_{m}(z_{1},z_{2})+I_{s}(z_{1},z_{2})
  18. I min = min { i : i = I t ( z 1 , z 2 ) } } I_{\operatorname{min}}=\min{\{i:i=I_{t}(z_{1},z_{2})\}}\}
  19. ( z 1 , z 2 ) { ( z 1 , z 2 ) : I t ( z 1 , z 2 ) = I min } (z_{1},z_{2})\in\{(z_{1},z_{2}):I_{t}(z_{1},z_{2})=I_{\operatorname{min}}\}

Concentration_dimension.html

  1. σ ( X ) = sup { E [ , X 2 ] | B , 1 } . \sigma(X)=\sup\left\{\left.\sqrt{\operatorname{E}[\langle\ell,X\rangle^{2}]}\,% \right|\,\ell\in B^{\ast},\|\ell\|\leq 1\right\}.
  2. d ( X ) = E [ X 2 ] σ ( X ) 2 . d(X)=\frac{\operatorname{E}[\|X\|^{2}]}{\sigma(X)^{2}}.

Concentric_tube_heat_exchanger.html

  1. D eo = 4 A r e a W e t t e d P e r i m e t e r D_{\mathrm{eo}}=\frac{4\cdot Area}{WettedPerimeter}
  2. 1 U o = R f o + R f i D o D i + D o 2 k w ln D o D i + 1 h o + 1 h i D o D i {1\over U_{o}}={R_{fo}}+{R_{fi}}\cdot\frac{D_{o}}{D_{i}}+\frac{D_{o}}{2k_{w}}% \cdot\ln{\frac{D_{o}}{D_{i}}}+{1\over h_{o}}+{1\over h_{i}}\cdot\frac{D_{o}}{D% _{i}}
  3. A = Q U Δ T A=\frac{Q}{U\Delta T}
  4. q m a x C m i n ( T h , i - T c , i ) q_{max}\equiv C_{min}(T_{h,i}-T_{c,i})
  5. E q q m a x E\equiv\frac{q}{q_{max}}

Condition_of_average.html

  1. P a y o u t = C l a i m × S u m I n s u r e d C u r r e n t V a l u e Payout=Claim\times\frac{Sum\ Insured}{Current\ Value}\!
  2. P a y o u t = C l a i m × S u m I n s u r e d C u r r e n t V a l u e = $ 3 M × $ 5 M $ 10 M = $ 1.5 M Payout=Claim\times\frac{Sum\ Insured}{Current\ Value}=\$3\mbox{M}~{}\times% \frac{\$5\mbox{M}~{}}{\$10\mbox{M}~{}}=\$1.5\mbox{M}~{}\!

Conditional_probability.html

  1. B {}_{B}
  2. 1 {}_{1}
  3. 2 {}_{2}
  4. 3 {}_{3}
  5. P ( A | B ) = P ( A B ) P ( B ) P(A|B)=\frac{P(A\cap B)}{P(B)}
  6. P ( A B ) / P ( B ) P(A\cap B)/P(B)
  7. P ( A | B ) P(A|B)
  8. P ( A B ) = P ( A | B ) P ( B ) P(A\cap B)=P(A|B)P(B)
  9. P ( A B ) = P ( A ) + P ( B ) - \cancelto 0 P ( A B ) P(A\cup B)=P(A)+P(B)-\cancelto 0{P(A\cap B)}
  10. X , Y {}_{X,Y}
  11. P ( X A Y B ) = y B x A f X , Y ( x , y ) d x d y y B x Ω f X , Y ( x , y ) d x d y . P(X\in A\mid Y\in B)=\frac{\int_{y\in B}\int_{x\in A}f_{X,Y}(x,y)\,dx\,dy}{% \int_{y\in B}\int_{x\in\Omega}f_{X,Y}(x,y)\,dx\,dy}.
  12. 0 {}_{0}
  13. P ( X A Y = y 0 ) = x A f X , Y ( x , y 0 ) d x x Ω f X , Y ( x , y 0 ) d x . P(X\in A\mid Y=y_{0})=\frac{\int_{x\in A}f_{X,Y}(x,y_{0})\,dx}{\int_{x\in% \Omega}f_{X,Y}(x,y_{0})\,dx}.
  14. i {}_{i}
  15. P ( X A Y i [ y i , y i + δ y i ] ) i x A f X , Y ( x , y i ) d x δ y i i x Ω f X , Y ( x , y i ) d x δ y i , P(X\in A\mid Y\in\cup_{i}[y_{i},y_{i}+\delta y_{i}])\approxeq\frac{\sum_{i}% \int_{x\in A}f_{X,Y}(x,y_{i})\,dx\,\delta y_{i}}{\sum_{i}\int_{x\in\Omega}f_{X% ,Y}(x,y_{i})\,dx\,\delta y_{i}},
  16. P ( A X = x ) P(A\mid X=x)
  17. X = x . X=x.
  18. P ( A | X ) ( ω ) = P ( A X = X ( ω ) ) . P(A|X)(\omega)=P(A\mid X=X(\omega)).
  19. g ( x ) = P ( A X = x ) g(x)=P(A\mid X=x)
  20. P ( A | X ) = g X P(A|X)=g\circ X
  21. 6 / 36 {6}/{36}
  22. 1 / 6 {1}/{6}
  23. 10 / 36 {10}/{36}
  24. 3 / 10 {3}/{10}
  25. A B A\cap B
  26. A B A\cap B
  27. \cap
  28. P ( A B ) \displaystyle P(A\cap B)
  29. P ( B | A ) = P ( A | B ) P ( B ) P ( A ) . P(B|A)=\frac{P(A|B)P(B)}{P(A)}.
  30. P ( A | B ) P ( B ) = P ( A B ) = P ( B A ) = P ( B | A ) P ( A ) P(A|B)P(B)=P(A\cap B)=P(B\cap A)=P(B|A)P(A)
  31. P ( A ) = n P ( A B n ) = n P ( A | B n ) P ( B n ) P(A)=\sum_{n}P(A\cap B_{n})=\sum_{n}P(A|B_{n})P(B_{n})
  32. ( B n ) (B_{n})
  33. A A
  34. C {}_{C}
  35. C {}_{C}
  36. C {}_{C}
  37. C {}_{C}
  38. 1. ω B : P ( ω | B ) = α P ( ω ) \displaystyle\,\text{1. }\omega\in B:P(\omega|B)=\alpha P(\omega)
  39. ω Ω P ( ω | B ) \displaystyle\sum_{\omega\in\Omega}{P(\omega|B)}
  40. 1. ω B : P ( ω | B ) = P ( ω ) P ( B ) 2. ω B : P ( ω | B ) = 0 \begin{aligned}\displaystyle\,\text{1. }\omega\in B&\displaystyle:P(\omega|B)=% \frac{P(\omega)}{P(B)}\\ \displaystyle\,\text{2. }\omega\notin B&\displaystyle:P(\omega|B)=0\end{aligned}
  41. P ( A | B ) = ω A B P ( ω | B ) + \cancelto 0 ω A B c P ( ω | B ) = ω A B P ( ω ) P ( B ) = P ( A B ) P ( B ) \begin{aligned}\displaystyle P(A|B)&\displaystyle=\sum_{\omega\in A\cap B}{P(% \omega|B)}+\cancelto{0}{\sum_{\omega\in A\cap B^{c}}P(\omega|B)}\\ &\displaystyle=\sum_{\omega\in A\cap B}{\frac{P(\omega)}{P(B)}}\\ &\displaystyle=\frac{P(A\cap B)}{P(B)}\end{aligned}

Conductivity_(electrolytic).html

  1. R = l A ρ . R=\frac{l}{A}\rho.
  2. R * = C × ρ * R^{*}=C\times\rho^{*}
  3. κ = 1 ρ = C R \kappa=\frac{1}{\rho}=\frac{C}{R}
  4. κ = G * × G \kappa=G^{*}\times G
  5. Λ m = κ c \Lambda_{m}=\frac{\kappa}{c}
  6. Λ m = Λ m 0 - K c \Lambda_{m}=\Lambda_{m}^{0}-K\sqrt{c}
  7. Λ m 0 \Lambda_{m}^{0}
  8. Λ m 0 > K c \Lambda_{m}^{0}>K\sqrt{c}
  9. Λ m 0 = ν + λ + 0 + ν - λ - 0 \Lambda_{m}^{0}=\nu_{+}\lambda_{+}^{0}+\nu_{-}\lambda_{-}^{0}
  10. ν + \nu_{+}
  11. ν - \nu_{-}
  12. λ + 0 \lambda_{+}^{0}
  13. λ - 0 \lambda_{-}^{0}
  14. Λ m = Λ m 0 - ( A + B Λ m 0 ) c \Lambda_{m}=\Lambda_{m}^{0}-(A+B\Lambda_{m}^{0})\sqrt{c}
  15. 1 Λ m = 1 Λ m 0 + Λ m c K a ( Λ m 0 ) 2 \frac{1}{\Lambda_{m}}=\frac{1}{\Lambda_{m}^{0}}+\frac{\Lambda_{m}c}{K_{a}(% \Lambda_{m}^{0})^{2}}

Conductivity_of_transparency.html

  1. σ g t = ϵ g r a p h e n e - l g ( I I 0 ) ρ s a m p l e \sigma_{gt}=\frac{\epsilon_{graphene}}{-lg(\frac{I}{I_{0}})\rho_{sample}}
  2. σ g t \sigma_{gt}
  3. ϵ g r a p h e n e \epsilon_{graphene}
  4. ρ s a m p l e \rho_{sample}
  5. I I
  6. I 0 I_{0}
  7. d g r a p h i t e = 0.335 n m d_{graphite}=0.335nm
  8. 301655 c m - 1 301655cm^{-1}
  9. - l g ( I I 0 ) = ϵ * c d ; ϵ * c = ϵ g r a p h e n e -lg(\frac{I}{I_{0}})=\epsilon^{*}cd;\epsilon^{*}c=\epsilon_{graphene}
  10. ϵ g r a p h e n e = - l g ( I I 0 ) d g r a p h i t e = - l g ( 97.7 % 100 % ) 3.35 × 10 - 8 c m = 301655 c m - 1 \epsilon_{graphene}=\frac{-lg(\frac{I}{I_{0}})}{d_{graphite}}=\frac{-lg(\frac{% 97.7\%}{100\%})}{3.35\times 10^{-8}cm}=301655cm^{-1}
  11. σ g t = 301655 c m - 1 - l g ( I 100 % ) ρ s a m p l e \sigma_{gt}=\frac{301655cm^{-1}}{-lg(\frac{I}{100\%})\rho_{sample}}
  12. ρ \rho
  13. σ g t \sigma_{gt}
  14. 3.0 10 11 3.0\cdot 10^{11}
  15. 5.7 10 - 5 5.7\cdot 10^{-5}
  16. 1 10 5 1\cdot 10^{5}
  17. 1.8 10 4 1.8\cdot 10^{4}
  18. 6.5 10 4 6.5\cdot 10^{4}
  19. 2.7 10 4 2.7\cdot 10^{4}

Conductor_(class_field_theory).html

  1. 𝔣 ( L / K ) \mathfrak{f}(L/K)
  2. U ( n ) = 1 + 𝔪 n = { u 𝒪 × : u 1 ( mod 𝔪 K n ) } U^{(n)}=1+\mathfrak{m}^{n}=\left\{u\in\mathcal{O}^{\times}:u\equiv 1\,(\mathrm% {mod}\,\mathfrak{m}_{K}^{n})\right\}
  3. 𝔪 K \mathfrak{m}_{K}
  4. U K ( n ) U_{K}^{(n)}
  5. 𝔪 K n \mathfrak{m}_{K}^{n}
  6. 𝔣 ( L / K ) = η L / K ( s ) + 1 \mathfrak{f}(L/K)=\eta_{L/K}(s)+1
  7. 𝔪 K 𝔣 ( L / K ) = lcm 𝜒 𝔪 K 𝔣 χ \mathfrak{m}_{K}^{\mathfrak{f}(L/K)}=\underset{\chi}{\mathrm{lcm}}\,\mathfrak{% m}_{K}^{\mathfrak{f}_{\chi}}
  8. 𝔣 χ \mathfrak{f}_{\chi}
  9. N L / K ( L × ) = N L ab / K ( ( L ab ) × ) . N_{L/K}(L^{\times})=N_{L^{\,\text{ab}}/K}\left((L^{\,\text{ab}})^{\times}% \right).
  10. 𝔣 ( L / K ) \mathfrak{f}(L/K)
  11. 𝔣 ( L / K ) \mathfrak{f}(L/K)
  12. 𝐐 ( ζ n ) \mathbf{Q}(\zeta_{n})
  13. 𝐐 ( d ) / 𝐐 \mathbf{Q}(\sqrt{d})/\mathbf{Q}
  14. 𝔣 ( 𝐐 ( d ) / 𝐐 ) = { | Δ 𝐐 ( d ) | for d > 0 | Δ 𝐐 ( d ) | for d < 0 \mathfrak{f}\left(\mathbf{Q}(\sqrt{d})/\mathbf{Q}\right)=\begin{cases}\left|% \Delta_{\mathbf{Q}(\sqrt{d})}\right|&\,\text{for }d>0\\ \infty\left|\Delta_{\mathbf{Q}(\sqrt{d})}\right|&\,\text{for }d<0\end{cases}
  15. Δ 𝐐 ( d ) \Delta_{\mathbf{Q}(\sqrt{d})}
  16. 𝐐 ( d ) / 𝐐 \mathbf{Q}(\sqrt{d})/\mathbf{Q}
  17. 𝔣 ( L / K ) = 𝔭 𝔭 𝔣 ( L 𝔭 / K 𝔭 ) . \displaystyle\mathfrak{f}(L/K)=\prod_{\mathfrak{p}}\mathfrak{p}^{\mathfrak{f}(% L_{\mathfrak{p}}/K_{\mathfrak{p}})}.
  18. 𝔣 ( L / K ) \mathfrak{f}(L/K)

Confederate_war_finance.html

  1. M V = P Y MV=PY

Configural_frequency_analysis.html

  1. e ( c ) = n i = 1 m p i ( c i ) . e(c)=n\prod_{i=1}^{m}p_{i}(c_{i}).
  2. α \alpha

Confirmed_line_item_performance.html

  1. C L I P w e e k l y , p CLIP_{weekly,p}
  2. m i n [ v i t u a l d e l i v e r y p , D W v i r t u a l l y c o m m i t t e d o r d e r p , D W , 1 ] min[\frac{vitualdelivery_{p,DW}}{virtuallycommittedorder_{p,DW}},1]
  3. C L I P w e e k l y , p CLIP_{weekly,p}
  4. [ p = 1 n u m b e r o f o r d e r e d p r o d u c t s ( C L I P w e e k l y , p ) n u m b e r o f o r d e r e d p r o d u c t s ] [\frac{\sum_{p=1}^{numberoforderedproducts}(CLIP_{weekly,p})}{% numberoforderedproducts}]

Conformal_pictures.html

  1. ( x , y ) 2 (x,y)\in\mathbb{R}^{2}
  2. x + i y x+i\,y\in\mathbb{C}
  3. e i θ e^{i\theta}
  4. z = r e i θ z=r\,e^{i\theta}
  5. f ( z ) = a ( z - z 0 ) + b + o ( z - z 0 ) f(z)=a\,(z-z_{0})+b+o(z-z_{0})
  6. a = f ( z 0 ) a=f^{\prime}(z_{0})
  7. b = f ( z 0 ) b=f(z_{0})
  8. z z k z\mapsto z^{k}
  9. z k z k - 1 z\mapsto k\,z^{k-1}
  10. z k z^{k}
  11. z z 2 z\mapsto z^{2}
  12. z z 3 z\mapsto z^{3}
  13. z 1 / z z\mapsto 1/z
  14. z 1 / z z\mapsto 1/z
  15. z a z + b c z + d z\mapsto\frac{az+b}{cz+d}
  16. a d - b c 0 ad-bc\neq 0
  17. z 1 / z 2 z\mapsto 1/z^{2}
  18. z e z z\mapsto e^{z}
  19. z log ( z ) z\mapsto\log(z)
  20. log ( exp ( z ) ) = z \log(\exp(z))=z
  21. log ( r e i θ ) = log ( r ) + i θ \log(r\,e^{i\theta})=\log(r)+i\theta
  22. exp ( x + i y ) = exp ( x ) e i y \exp(x+i\,y)=\exp(x)\,e^{i\,y}
  23. z e 1 / z z\mapsto e^{1/z}
  24. z e 1 / z z\mapsto e^{1/z}
  25. z = 0 - z=0^{-}
  26. z = 0 + z=0^{+}

Conformal_radius.html

  1. rad ( z , D ) := 1 f ( z ) . \mathrm{rad}(z,D):=\frac{1}{f^{\prime}(z)}\,.
  2. rad ( φ ( z ) , D ) = | φ ( z ) | rad ( z , D ) \mathrm{rad}(\varphi(z),D^{\prime})=|\varphi^{\prime}(z)|\,\mathrm{rad}(z,D)
  3. rad ( z , D ) = 2 Im ( g ( z ) ) | g ( z ) | . \mathrm{rad}(z,D)=\frac{2\,\mathrm{Im}(g(z))}{|g^{\prime}(z)|}\,.
  4. f ( z ) = i z - i z + i , f(z)=i\frac{z-i}{z+i},
  5. rad ( z , D ) 4 dist ( z , D ) rad ( z , D ) , \frac{\mathrm{rad}(z,D)}{4}\leq\mathrm{dist}(z,\partial D)\leq\mathrm{rad}(z,D),
  6. φ ( - r ) = i r \varphi(-r)=i\sqrt{r}
  7. | φ ( - r ) | = 1 2 r |\varphi^{\prime}(-r)|=\frac{1}{2\sqrt{r}}
  8. rad ( i r , ) = 2 r \mathrm{rad}(i\sqrt{r},\mathbb{H})=2\sqrt{r}
  9. rad ( , D ) := 1 rad ( , E ) := lim z f ( z ) z , \mathrm{rad}(\infty,D):=\frac{1}{\mathrm{rad}(\infty,E)}:=\lim_{z\to\infty}% \frac{f(z)}{z},
  10. f ( z ) = c 1 z + c 0 + c - 1 z - 1 + , c 1 𝐑 + . f(z)=c_{1}z+c_{0}+c_{-1}z^{-1}+\dots,\qquad c_{1}\in\mathbf{R}_{+}.
  11. D { z : | z - c 0 | 2 c 1 } , D\subseteq\{z:|z-c_{0}|\leq 2c_{1}\}\,,
  12. d ( z 1 , , z k ) := 1 i < j k | z i - z j | d(z_{1},\ldots,z_{k}):=\prod_{1\leq i<j\leq k}|z_{i}-z_{j}|
  13. z 1 , , z k z_{1},\ldots,z_{k}
  14. d n ( D ) := sup z 1 , , z n D d ( z 1 , , z n ) 1 ( n 2 ) d_{n}(D):=\sup_{z_{1},\ldots,z_{n}\in D}d(z_{1},\ldots,z_{n})^{\frac{1}{{\left% ({{n}\atop{2}}\right)}}}
  15. d n ( D ) d_{n}(D)
  16. d ( D ) := lim n d n ( D ) d(D):=\lim_{n\to\infty}d_{n}(D)
  17. 𝒫 n \mathcal{P}_{n}
  18. 𝒬 n \mathcal{Q}_{n}
  19. 𝒫 n \mathcal{P}_{n}
  20. μ n ( D ) := inf p 𝒫 sup z D | p ( z ) | \mu_{n}(D):=\inf_{p\in\mathcal{P}}\sup_{z\in D}|p(z)|
  21. μ ~ n ( D ) := inf p 𝒬 sup z D | p ( z ) | \tilde{\mu}_{n}(D):=\inf_{p\in\mathcal{Q}}\sup_{z\in D}|p(z)|
  22. μ ( D ) := lim n μ n ( D ) 1 n \mu(D):=\lim_{n\to\infty}\mu_{n}(D)^{\frac{1}{n}}
  23. μ ( D ) := lim n μ ~ n ( D ) 1 n \mu(D):=\lim_{n\to\infty}\tilde{\mu}_{n}(D)^{\frac{1}{n}}

Conformal_vector_field.html

  1. ( M , g ) (M,g)
  2. X X
  3. X g = φ g \mathcal{L}_{X}g=\varphi g
  4. φ \varphi
  5. M M
  6. X g \mathcal{L}_{X}g
  7. g g
  8. X X
  9. φ \varphi
  10. X X

Congestion_game.html

  1. E E
  2. n n
  3. S i S_{i}
  4. P S i P\in S_{i}
  5. E E
  6. e e
  7. ( P 1 , P 2 , , P n ) (P_{1},P_{2},\ldots,P_{n})
  8. x e = # { i : e P i } x_{e}=\#\{i:e\in P_{i}\}
  9. e e
  10. d e : d_{e}:\mathbb{N}\longrightarrow\mathbb{R}
  11. P i P_{i}
  12. i i
  13. e P i d e ( x e ) \textstyle\sum_{e\in P_{i}}d_{e}(x_{e})
  14. d e d_{e}
  15. Φ = e E k = 1 x e d e ( k ) \textstyle\Phi=\sum_{e\in E}\sum_{k=1}^{x_{e}}d_{e}(k)
  16. e E x e d e ( x e ) \textstyle\sum_{e\in E}x_{e}d_{e}(x_{e})
  17. i i
  18. P i P_{i}
  19. Q i Q_{i}
  20. e P i - Q i e\in P_{i}-Q_{i}
  21. d e ( x e ) d_{e}(x_{e})
  22. e Q i - P i e\in Q_{i}-P_{i}
  23. d e ( x e + 1 ) d_{e}(x_{e}+1)
  24. i i
  25. Φ \Phi
  26. Φ \Phi
  27. Φ \Phi
  28. d e d_{e}
  29. n n\rightarrow\infty
  30. E E
  31. n n
  32. n n
  33. i i
  34. r i r_{i}
  35. S i S_{i}
  36. d e d_{e}
  37. P S i P\in S_{i}
  38. f P f_{P}
  39. i i
  40. P P
  41. P S i f P = r i \textstyle\sum_{P\in S_{i}}f_{P}=r_{i}
  42. f P f_{P}
  43. S i S_{i}
  44. n n
  45. [ 0 , r i ] n [0,r_{i}]^{n}
  46. Φ = e E 0 x e d e ( z ) d z \textstyle\Phi=\sum_{e\in E}\int_{0}^{x_{e}}d_{e}(z)\,dz
  47. Φ \Phi
  48. d e d_{e}
  49. x e x_{e}
  50. Φ \Phi
  51. Φ \Phi
  52. f P f_{P}
  53. Φ \Phi
  54. i i
  55. Q Q
  56. P P
  57. e P d e ( x e ) > e Q d e ( x e ) \textstyle\sum_{e\in P}d_{e}(x_{e})>\sum_{e\in Q}d_{e}(x_{e})
  58. δ < f P \delta<f_{P}
  59. P P
  60. Q Q
  61. x e Q x_{e}\in Q
  62. δ \delta
  63. Φ \Phi
  64. 0 x e + δ d e ( z ) d z \textstyle\int_{0}^{x_{e}+\delta}d_{e}(z)dz
  65. δ d e ( x e ) \delta\cdot d_{e}(x_{e})
  66. δ 2 \delta^{2}
  67. P P
  68. δ ( e Q d e ( x e ) - e P d e ( x e ) ) \textstyle\delta(\sum_{e\in Q}d_{e}(x_{e})-\sum_{e\in P}d_{e}(x_{e}))
  69. Φ \Phi
  70. Φ \Phi
  71. ( λ , μ ) (\lambda,\mu)
  72. x , y > 0 x,y>0
  73. y d ( x ) λ y d ( y ) + μ x d ( x ) yd(x)\leq\lambda yd(y)+\mu xd(x)
  74. ( λ , μ ) (\lambda,\mu)
  75. f f
  76. f * f^{*}
  77. e x e d e ( x e ) λ 1 - μ e x e * d e ( x e * ) \textstyle\sum_{e}x_{e}d_{e}(x_{e})\leq\frac{\lambda}{1-\mu}\sum_{e}x_{e}^{*}d% _{e}(x_{e}^{*})
  78. λ 1 - μ \textstyle\frac{\lambda}{1-\mu}

Conic_bundle.html

  1. X 2 + a X Y + b Y 2 = P ( T ) . X^{2}+aXY+bY^{2}=P(T).\,
  2. ( a , P ) (a,P)
  3. k k
  4. X 2 - a Y 2 = P ( T ) . X^{2}-aY^{2}=P(T).\,
  5. X 2 - a Y 2 = P ( T ) Z 2 . X^{2}-aY^{2}=P(T)Z^{2}.\,
  6. T T = 1 T T\mapsto T^{\prime}=\frac{1}{T}
  7. T = 0 T=0
  8. T = 0 T^{\prime}=0
  9. X 2 - a Y 2 = P * ( T ) Z 2 X^{\prime 2}-aY^{\prime 2}=P^{*}(T^{\prime})Z^{\prime 2}
  10. P * ( T ) P^{*}(T^{\prime})
  11. P P
  12. [ x : y : z ] [x^{\prime}:y^{\prime}:z^{\prime}]
  13. k k
  14. m m
  15. k k
  16. P * ( T ) = T 2 m P ( 1 T ) P^{*}(T^{\prime})=T^{2m}P(\frac{1}{T})
  17. F a , P F_{a,P}
  18. U U
  19. U U^{\prime}
  20. X 2 - a Y 2 = P ( T ) Z 2 X^{2}-aY^{2}=P(T)Z^{2}
  21. X 2 - Y 2 = P ( T ) Z 2 X^{\prime 2}-Y^{\prime 2}=P(T^{\prime})Z^{\prime 2}
  22. x = x , , y = y , x^{\prime}=x,,y^{\prime}=y,
  23. z = z t m z^{\prime}=zt^{m}
  24. p : U P 1 , k p:U\to P_{1,k}
  25. ( [ x : y : z ] , t ) t ([x:y:z],t)\mapsto t
  26. U U^{\prime}

Connection_(algebraic_framework).html

  1. E X E\to X
  2. C ( X ) C^{\infty}(X)
  3. E X E\to X
  4. A A
  5. P P
  6. A A
  7. P P
  8. D ( A ) D(A)
  9. A A
  10. A A
  11. P P
  12. A A
  13. : D ( A ) u u Diff 1 ( P , P ) \nabla:D(A)\ni u\to\nabla_{u}\in\mathrm{Diff}_{1}(P,P)
  14. u \nabla_{u}
  15. P P
  16. u ( a p ) = u ( a ) p + a u ( p ) , a A , p P . \nabla_{u}(ap)=u(a)p+a\nabla_{u}(p),\quad a\in A,\quad p\in P.
  17. \nabla
  18. R ( u , u ) = [ u , u ] - [ u , u ] R(u,u^{\prime})=[\nabla_{u},\nabla_{u^{\prime}}]-\nabla_{[u,u^{\prime}]}\,
  19. P P
  20. u , u D ( A ) u,u^{\prime}\in D(A)
  21. E X E\to X
  22. Γ \Gamma
  23. E X E\to X
  24. \nabla
  25. C ( X ) C^{\infty}(X)
  26. E X E\to X
  27. \nabla
  28. E X E\to X
  29. A A
  30. A A
  31. R - S R-S
  32. R R
  33. S S
  34. R - S R-S
  35. P P
  36. : D ( A ) u u Diff 1 ( P , P ) \nabla:D(A)\ni u\to\nabla_{u}\in\mathrm{Diff}_{1}(P,P)
  37. u ( a p b ) = u ( a ) p b + a u ( p ) b + a p u ( b ) , a R , b S , p P . \nabla_{u}(apb)=u(a)pb+a\nabla_{u}(p)b+apu(b),\quad a\in R,\quad b\in S,\quad p% \in P.

Connes_embedding_problem.html

  1. ω \omega
  2. τ \tau
  3. R ω R^{\omega}
  4. l ( R ) = { ( x n ) n R : s u p n || x n || < } l^{\infty}(R)=\{(x_{n})_{n}\subseteq R:sup_{n}||x_{n}||<\infty\}
  5. I ω = { ( x n ) l ( R ) : l i m n ω τ ( x n * x n ) 1 2 = 0 } I_{\omega}=\{(x_{n})\in l^{\infty}(R):lim_{n\rightarrow\omega}\tau(x_{n}^{*}x_% {n})^{\frac{1}{2}}=0\}
  6. l ( R ) / I ω l^{\infty}(R)/I_{\omega}
  7. τ R ω ( x ) = l i m n ω τ ( x n + I ω ) \tau_{R^{\omega}}(x)=lim_{n\rightarrow\omega}\tau(x_{n}+I_{\omega})
  8. ( x n ) n (x_{n})_{n}
  9. x x
  10. R ω R^{\omega}
  11. R ω R^{\omega}

CONQUEST.html

  1. ρ ( 𝐫 , 𝐫 ) \rho(\mathbf{r},\mathbf{r}^{\prime})
  2. ρ ( 𝐫 , 𝐫 ) = i α j β ϕ i α ( 𝐫 ) K i α j β ϕ j β ( 𝐫 ) \rho(\mathbf{r},\mathbf{r}^{\prime})=\sum_{i\alpha j\beta}\phi_{i\alpha}(% \mathbf{r})K_{i\alpha j\beta}\phi_{j\beta}(\mathbf{r}^{\prime})
  3. ϕ i α ( 𝐫 ) \phi_{i\alpha}(\mathbf{r})
  4. α \alpha
  5. K i α j β K_{i\alpha j\beta}
  6. K i j = 0 , | 𝐑 i - 𝐑 j | > R c K_{ij}=0,|\mathbf{R}_{i}-\mathbf{R}_{j}|>R_{c}

Constant_(mathematics).html

  1. a x 2 + b x + c , ax^{2}+bx+c\,,
  2. x a x 2 + b x + c , x\mapsto ax^{2}+bx+c\,,
  3. f ( x ) = 5 f(x)=5
  4. d d x 2 x = lim h 0 2 x + h - 2 x h = lim h 0 2 x 2 h - 1 h = 2 x lim h 0 2 h - 1 h since x is constant (i.e. does not depend on h ) = 2 x 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 , where 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 means not depending on x . \begin{array}[]{lll}\frac{d}{dx}2^{x}&=\lim_{h\to 0}\frac{2^{x+h}-2^{x}}{h}&=% \lim_{h\to 0}2^{x}\frac{2^{h}-1}{h}\\ &=2^{x}\lim_{h\to 0}\frac{2^{h}-1}{h}&\,\text{since }x\,\text{ is constant (i.% e. does not depend on }h\,\text{)}\\ &=2^{x}\cdot\mathbf{constant,}&\,\text{ where }\mathbf{constant}\,\text{ means% not depending on }x.\end{array}
  5. 2 \sqrt{2}
  6. 1 + 5 2 1+\sqrt{5}\over 2
  7. f ( x ) = 72 f ( x ) = 0 f(x)=72\Rightarrow f^{\prime}(x)=0
  8. f ( x ) = 72 72 d x = 72 x + c f(x)=72\Rightarrow\int 72\,dx=72x+c
  9. f ( x ) = 72 lim x 72 = 72 f(x)=72\Rightarrow\lim_{x\to\infty}72=72

Constraint_graph_(layout).html

  1. O ( n 2 ) O(n^{2})

Constructible_set_(topology).html

  1. f : X Y f:X\to Y
  2. 𝐀 2 𝐀 2 \mathbf{A}^{2}\rightarrow\mathbf{A}^{2}
  3. ( x , y ) (x,y)
  4. ( x , x y ) (x,xy)
  5. { x 0 } { x = y = 0 } \{x\neq 0\}\cup\{x=y=0\}

Contiguity_(probability_theory).html

  1. ( Ω n , n ) (\Omega_{n},\mathcal{F}_{n})
  2. Q ( A ) = A f d P , Q(A)=\int_{A}f\,\mathrm{d}P,\,

Continuity_set.html

  1. μ ( B ) = 0 . \mu(\partial B)=0\,.
  2. B \partial B
  3. | μ | ( B ) = 0 . |\mu|(\partial B)=0\,.
  4. Pr [ X B ] = 0 , \Pr[X\in\partial B]=0,

Continuous-repayment_mortgage.html

  1. P 0 ( 1 + i ) n = k = 1 n x ( 1 + i ) n - k = x [ ( 1 + i ) n - 1 ] i P_{0}(1+i)^{n}=\sum_{k=1}^{n}x(1+i)^{n-k}=\frac{x[(1+i)^{n}-1]}{i}
  2. P 0 e r T = 0 T M a e r ( T - t ) d t = M a ( e r T - 1 ) r . P_{0}e^{rT}=\int\limits_{0}^{T}M_{a}e^{r(T-t)}\,dt=\frac{M_{a}(e^{rT}-1)}{r}.
  3. P v ( n ) = x ( 1 - ( 1 + i ) - n ) i P_{v}(n)=\frac{x(1-(1+i)^{-n})}{i}
  4. x = P 0 i 1 - ( 1 + i ) - n x=\frac{P_{0}\cdot i}{1-(1+i)^{-n}}
  5. x ( N ) = P 0 r N ( 1 - ( 1 + r N ) - N T ) x(N)=\frac{P_{0}\cdot r}{N(1-(1+\frac{r}{N})^{-NT})}
  6. N x ( N ) = P 0 r 1 - ( 1 + r N ) - N T N\cdot x(N)=\frac{P_{0}\cdot r}{1-(1+\frac{r}{N})^{-NT}}
  7. lim N ( 1 + r N ) N t = e r t \lim_{N\to\infty}\left(1+\frac{r}{N}\right)^{Nt}=e^{rt}
  8. M a = lim N N x ( N ) = lim N P 0 r 1 - ( 1 + r N ) - N T = P 0 r 1 - e - r T . M_{a}=\lim_{N\to\infty}N\cdot x(N)=\lim_{N\to\infty}\frac{P_{0}\cdot r}{1-(1+% \frac{r}{N})^{-NT}}=\frac{P_{0}\cdot r}{1-e^{-rT}}.
  9. P v ( N , t ) = N x ( N ) ( 1 - ( 1 + r N ) - N t ) r P_{v}(N,t)=\frac{N\cdot x(N)(1-(1+\frac{r}{N})^{-Nt})}{r}
  10. P v ( t ) = lim N P v ( N , t ) = M a r ( 1 - e - r t ) P_{v}(t)=\lim_{N\to\infty}P_{v}(N,t)=\frac{M_{a}}{r}(1-e^{-rt})
  11. P ( t ) = M a r ( 1 - e - r ( T - t ) ) = P 0 ( 1 - e - r ( T - t ) ) 1 - e - r T P(t)=\frac{M_{a}}{r}(1-e^{-r(T-t)})=\frac{P_{0}(1-e^{-r(T-t)})}{1-e^{-rT}}
  12. P ( z ) = M a r ( 1 - e - r z ) . P(z)=\frac{M_{a}}{r}(1-e^{-rz}).
  13. M a r = 1 r d P ( z ) d z + P ( z ) . \frac{M_{a}}{r}=\frac{1}{r}\frac{dP(z)}{dz}+P(z).
  14. V 0 = R C d V ( t ) d t + V ( t ) . V_{0}=RC\frac{dV(t)}{dt}+V(t).
  15. P t + Δ t \displaystyle P_{t+\Delta t}
  16. d P ( t ) d t = r P ( t ) - M a {\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_{a}
  17. P 0 M a r P_{0}\leqslant\frac{M_{a}}{r}
  18. P t + Δ t = P t ( 1 + r Δ t ) - M N Δ t P_{t+\Delta t}=P_{t}(1+r\Delta t)-M_{N}\Delta t\;
  19. P 1 = P 0 ( 1 + r Δ t ) - M N Δ t P_{1}=P_{0}(1+r\Delta t)-M_{N}\Delta t\;
  20. P 2 = [ P 0 ( 1 + r Δ t ) - M N Δ t ] ( 1 + r Δ t ) - M N Δ t = P 0 ( 1 + r Δ t ) 2 - M N Δ t ( 1 + r Δ t ) - M N Δ t \begin{aligned}\displaystyle P_{2}&\displaystyle=[P_{0}(1+r\Delta t)-M_{N}% \Delta t](1+r\Delta t)-M_{N}\Delta t\\ &\displaystyle=P_{0}(1+r\Delta t)^{2}-M_{N}\Delta t(1+r\Delta t)-M_{N}\Delta t% \end{aligned}
  21. P n \displaystyle P_{n}
  22. M N Δ t = x M_{N}\Delta t=x
  23. P n = P 0 ( 1 + i ) n - x [ ( 1 + i ) n - 1 ] i P_{n}=P_{0}(1+i)^{n}-\dfrac{x[(1+i)^{n}-1]}{i}
  24. P 0 = x [ 1 - ( 1 + i ) - m ] i P_{0}=\dfrac{x[1-(1+i)^{-m}]}{i}
  25. d P ( t ) d t = r P ( t ) - M a {\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_{a}
  26. P ( s ) = M a s ( r - s ) = M a r × ( - r ) s ( s - r ) . P(s)=\frac{M_{a}}{s(r-s)}=\frac{M_{a}}{r}\times\frac{(-r)}{s(s-r)}.
  27. P ( t ) = M a r ( 1 - e r t ) . P(t)=\frac{M_{a}}{r}(1-e^{rt}).
  28. P ( t ) = M a r ( 1 - e r ( t - T ) ) \displaystyle P(t)=\frac{M_{a}}{r}(1-e^{r(t-T)})
  29. P ( t ) = P 0 e r t - M a r ( e r t - 1 ) P(t)=P_{0}e^{rt}-\frac{M_{a}}{r}(e^{rt}-1)
  30. M a = r P ( t ) - d P ( t ) d t M_{a}=rP(t)-\frac{dP(t)}{dt}
  31. M a . t = 0 t r P ( t ) d t - 0 t d P ( t ) d t d t M_{a}.t=\int_{0}^{t}rP(t)\,dt\,-\int_{0}^{t}\frac{dP(t)}{dt}\,dt\,
  32. I ( t ) = M a t - M a ( e r t - 1 ) r e r T I(t)=M_{a}t-\frac{M_{a}(e^{rt}-1)}{re^{rT}}
  33. I ( t ) = M a t - P 0 + P ( t ) I(t)=M_{a}t-P_{0}+P(t)\,
  34. C = M a T = P 0 r T 1 - e - r T C=M_{a}T=\frac{P_{0}rT}{1-e^{-rT}}
  35. C ( s ) = r T 1 - e - r T = s 1 - e - s C(s)=\frac{rT}{1-e^{-rT}}=\frac{s}{1-e^{-s}}
  36. C = 1000000 × 2 1 - e - 2 2.313 × 10 6 C=1000000\times\frac{2}{1-e^{-2}}\approx 2.313\times 10^{6}
  37. s 1 - e - s = 1 + s e \frac{s}{1-e^{-s}}=1+s_{e}
  38. M a = P 0 r 1 - e - r T \displaystyle M_{a}=\frac{P_{0}r}{1-e^{-rT}}
  39. k = M min M a = P 0 r M a k=\frac{M_{\min}}{M_{a}}=\frac{P_{0}r}{M_{a}}
  40. T = - 1 r ln ( 1 - P 0 r M a ) T=-\frac{1}{r}\ln\left(1-\frac{P_{0}r}{M_{a}}\right)
  41. r T = s ( k ) = - ln ( 1 - k ) rT=s(k)=-\ln(1-k)\;
  42. P ( t ) P 0 = 1 2 = 1 - e - r ( T - t ) 1 - e - r T \frac{P(t)}{P_{0}}=\frac{1}{2}=\frac{1-e^{-r(T-t)}}{1-e^{-rT}}
  43. t 1 2 = 1 r ln ( 1 + e r T 2 ) t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right)
  44. P 0 = M a r ( 1 - e - r T ) P_{0}=\frac{M_{a}}{r}(1-e^{-rT})
  45. r P 0 = M a ( 1 - e - r T ) rP_{0}=M_{a}(1-e^{-rT})\,
  46. r P 0 - M a + M a e - r T = 0 \Rightarrow rP_{0}-M_{a}+M_{a}e^{-rT}=0
  47. f ( r ) = 0 f^{\prime}(r)=0\,
  48. P 0 - M a T e - r T = 0 \Rightarrow P_{0}-M_{a}Te^{-rT}=0
  49. r = 1 T ln M a T P 0 \Rightarrow r=\frac{1}{T}\ln{\frac{M_{a}T}{P_{0}}}
  50. r 2 T ln M a T P 0 r\approx\frac{2}{T}\ln{\frac{M_{a}T}{P_{0}}}
  51. r 1 = r 0 - f ( r 0 ) f ( r 0 ) . r_{1}=r_{0}-\frac{f(r_{0})}{f^{\prime}(r_{0})}.\,\!
  52. r = 1 T ( W ( - s e - s ) + s ) r=\frac{1}{T}\left(W(-se^{-s})+s\right)
  53. P v ( t ) = M a r ( 1 - e - r t ) . P_{v}(t)=\frac{M_{a}}{r}(1-e^{-rt}).
  54. F v ( t ) = M a r ( e r t - 1 ) . F_{v}(t)=\frac{M_{a}}{r}(e^{rt}-1).
  55. M a = lim N N x ( N ) = lim N P T r ( 1 + r N ) N T - 1 = P T r e r T - 1 . M_{a}=\lim_{N\to\infty}N\cdot x(N)=\lim_{N\to\infty}\frac{P_{T}\cdot r}{(1+% \frac{r}{N})^{NT}-1}=\frac{P_{T}\cdot r}{e^{rT}-1}.
  56. F v ( t ) = P v ( t ) × e r t . F_{v}(t)=P_{v}(t)\times e^{rt}.\,
  57. P ( t ) = P 0 e r t - M a r ( e r t - 1 ) . P(t)=P_{0}e^{rt}-\frac{M_{a}}{r}(e^{rt}-1).
  58. x ( 12 ) = P M T ( 1 % , 120 , 500000 ) = 2173.55 x(12)=PMT(1\%,120,500000)=2173.55
  59. M a = 500000 × 12 % e 0.12 10 - 1 = 25860.77 M_{a}=\frac{500000\times 12\%}{e^{0.12\cdot 10}-1}=25860.77
  60. M a = lim N N x ( N ) M_{a}=\lim_{N\to\infty}N\cdot x(N)\,
  61. M a = lim N N x ( N ) = P 0 r 1 - e - r T M_{a}=\lim_{N\to\infty}N\cdot x(N)=\frac{P_{0}\cdot r}{1-e^{-rT}}
  62. M a = lim N N x ( N ) = P T r e r T - 1 M_{a}=\lim_{N\to\infty}N\cdot x(N)=\frac{P_{T}\cdot r}{e^{rT}-1}
  63. F v ( t ) = M a r ( e r t - 1 ) F_{v}(t)=\frac{M_{a}}{r}(e^{rt}-1)
  64. P v ( t ) = M a r ( 1 - e - r t ) P_{v}(t)=\frac{M_{a}}{r}(1-e^{-rt})
  65. P ( t ) = M a r ( 1 - e - r ( T - t ) ) P(t)=\frac{M_{a}}{r}(1-e^{-r(T-t)})
  66. T = - 1 r ln ( 1 - P 0 r M a ) T=-\frac{1}{r}\ln\left(1-\frac{P_{0}r}{M_{a}}\right)
  67. t 1 2 = 1 r ln ( 1 + e r T 2 ) t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right)
  68. r 2 T ln M a T P 0 r\approx\frac{2}{T}\ln{\frac{M_{a}T}{P_{0}}}
  69. r = 1 T ( W ( - s e - s ) + s ) with s = M a t P 0 r=\frac{1}{T}\left(W(-se^{-s})+s\right)\,\text{ with }s=\frac{M_{a}t}{P_{0}}

Continuous_graph.html

  1. n n\to\infty
  2. x [ 0 , 1 / 2 ) , y [ 1 / 2 , 1 ] x\in[0,1/2),y\in[1/2,1]
  3. y [ 0 , 1 / 2 ) , x [ 1 / 2 , 1 ] y\in[0,1/2),x\in[1/2,1]

Continuum_(set_theory).html

  1. 𝔠 \mathfrak{c}
  2. 𝔠 \mathfrak{c}
  3. 0 \aleph_{0}
  4. 𝔠 \mathfrak{c}
  5. 2 0 2^{\aleph_{0}}
  6. 0 \aleph_{0}

Contour_advection.html

  1. 1 r = ( d 2 x d s 2 ) 2 + ( d 2 y d s 2 ) 2 \frac{1}{r}=\sqrt{\left(\frac{\mathrm{d}^{2}x}{\mathrm{d}s^{2}}\right)^{2}+% \left(\frac{\mathrm{d}^{2}y}{\mathrm{d}s^{2}}\right)^{2}}
  2. r r
  3. s s
  4. r Δ s r\Delta s
  5. Δ s \Delta s

Contrast-to-noise_ratio.html

  1. C = | S A - S B | σ o C=\frac{|S_{A}-S_{B}|}{\sigma_{o}}

Contrast_transfer_function.html

  1. τ ( r , z ) = τ o e x p [ - i π λ d z U ( r , z ) ] \tau(r,z)=\tau_{o}exp[-i\pi\lambda\int dz^{\prime}U(r,z^{\prime})]
  2. τ o = τ ( r , 0 ) \tau_{o}=\tau(r,0)
  3. U ( r , z ) = 2 m V ( r , z ) / h 2 U(r,z)=2mV(r,z)/h^{2}
  4. r r
  5. z z
  6. τ o \tau_{o}
  7. λ \lambda
  8. U U
  9. V V
  10. ϕ ( r ) = π λ d z U ( r , z ) \phi(r)=\pi\lambda\int dz^{\prime}U(r,z^{\prime})
  11. τ ( r , z ) = τ o [ 1 + i ϕ ( r ) ] \tau(r,z)=\tau_{o}[1+i\phi(r)]
  12. I ( θ ) = δ ( θ ) + Φ K ( θ ) I(\theta)=\delta(\theta)+\Phi K(\theta)
  13. θ \theta
  14. δ \delta
  15. Φ \Phi
  16. K ( θ ) K(\theta)
  17. K ( θ ) = s i n [ ( 2 π / λ ) W ( θ ) ] K(\theta)=sin[(2\pi/\lambda)W(\theta)]
  18. W ( θ ) = - z θ 2 / 2 + C s θ 4 / 4 W(\theta)=-z\theta^{2}/2+C_{s}\theta^{4}/4
  19. λ \lambda
  20. C s C_{s}
  21. θ = λ k \theta=\lambda k
  22. K ( k ) = s i n [ 2 π λ ) W ( k ) ] K(k)=sin[2\pi\lambda)W(k)]
  23. W ( k ) = - z λ k 2 / 2 + C s λ 3 k 4 W(k)=-z\lambda k^{2}/2+C_{s}\lambda^{3}k^{4}
  24. z z
  25. λ \lambda
  26. C s C_{s}
  27. k k
  28. r s = C s θ 3 M r_{s}=C_{s}\cdot\theta^{3}\cdot M
  29. C s C_{s}
  30. M M
  31. α s = arctan ( b R ) - arctan ( b R + r s ) \alpha_{s}=\arctan\left(\frac{b}{R}\right)-\arctan\left(\frac{b}{R+r_{s}}\right)
  32. b b
  33. R R
  34. α s = arctan ( b r s R 2 + R r s + b 2 ) \alpha_{s}=\arctan\left(\frac{br_{s}}{R^{2}+Rr_{s}+b^{2}}\right)
  35. r s r_{s}
  36. R R
  37. b b
  38. arctan ( ) \arctan()
  39. α s arctan ( b r s b 2 ) b r s b 2 = r s b = C s θ 3 M b \alpha_{s}\approx\arctan\left(\frac{br_{s}}{b^{2}}\right)\approx\frac{br_{s}}{% b^{2}}=\frac{r_{s}}{b}=\frac{C_{s}\cdot\theta^{3}\cdot M}{b}
  40. θ \theta
  41. R f tan ( θ ) θ and M b f \frac{R}{f}\approx\tan\left(\theta\right)\approx\theta~{}~{}\,\text{and}~{}~{}% M\approx\frac{b}{f}
  42. α s C s R 3 f 4 \alpha_{s}\approx\frac{C_{s}\cdot R^{3}}{f^{4}}
  43. Δ b \Delta b
  44. α f \alpha_{f}
  45. R 2 + b 2 sin ( α f ) = Δ b sin ( θ - α f ) \sqrt{R^{2}+b^{2}}\cdot\sin(\alpha_{f})=\Delta b\cdot\sin(\theta^{\prime}-% \alpha_{f})
  46. R R
  47. b b
  48. α f θ \alpha_{f}<<\theta^{\prime}
  49. | b sin ( α f ) | | R | |b\cdot\sin(\alpha_{f})|<<|R|
  50. sin ( α f ) Δ b sin ( θ ) R 2 + b 2 = Δ b R R 2 + b 2 \sin(\alpha_{f})\approx\frac{\Delta b\sin(\theta^{\prime})}{\sqrt{R^{2}+b^{2}}% }=\frac{\Delta b\cdot R}{R^{2}+b^{2}}
  51. α f \alpha_{f}
  52. θ \theta
  53. R b R<<b
  54. α f \alpha_{f}
  55. α f Δ b R b 2 \alpha_{f}\approx\frac{\Delta b\cdot R}{b^{2}}
  56. Δ b / b 2 Δ f / f 2 \Delta b/b^{2}\approx\Delta f/f^{2}
  57. α f Δ f R f 2 \alpha_{f}\approx\frac{\Delta f\cdot R}{f^{2}}
  58. z z
  59. z s = ( C s λ ) 1 / 2 z_{s}=(C_{s}\lambda)^{1/2}
  60. z s z_{s}
  61. C s C_{s}
  62. K e f f ( k ) = E t E s ( s i n [ ( 2 π / λ ) W ( k ) ] K_{eff}(k)=E_{t}E_{s}(sin[(2\pi/\lambda)W(k)]

Controlled_grammar.html

  1. G = ( N , T , S , P ) G=(N,T,S,P)
  2. X α X\to\alpha
  3. α \alpha
  4. ( N T ) * (N\cup T)^{*}
  5. { p 1 p 2 p n : n 0 } \{p_{1}p_{2}...p_{n}:n\geq 0\}
  6. L ( G ) = { w T * : S p 1 p n w } L(G)=\{w\in T^{*}:S\Rightarrow_{p_{1}}...\Rightarrow_{p_{n}}w\}
  7. G = ( N , T , S , P , R ) G=(N,T,S,P,R)
  8. p 1 p 2 p n p_{1}p_{2}...p_{n}
  9. G = ( N , T , S , P , R , F ) G=(N,T,S,P,R,F)
  10. G = ( N , T , S , P , R , F ) G=(N,T,S,P,R,F)
  11. p i \Rightarrow_{p_{i}}
  12. ( N T ) * (N\cup T)^{*}
  13. p = A w P p=A\to w\in P
  14. x p a c y x\Rightarrow^{ac}_{p}y
  15. x = x 1 A x 2 x=x_{1}Ax_{2}
  16. y = y 1 w y 2 y=y_{1}wy_{2}
  17. x = y x=y
  18. p F p\in F
  19. L ( G ) = { w T * : S p 1 a c w 1 p 2 a c w 2 p 3 a c p n a c w , f o r s o m e p 1 p 2 p n R } L(G)=\{w\in T^{*}:S\Rightarrow^{ac}_{p_{1}}w_{1}\Rightarrow^{ac}_{p_{2}}w_{2}% \Rightarrow^{ac}_{p_{3}}...\Rightarrow^{ac}_{p_{n}}w,\ for\ some\ p_{1}p_{2}..% .p_{n}\in R\}
  20. { a n : n 1 } \{a^{n}:n\geq 1\}
  21. G = ( { S , A , X } , { a } , S , { f , g , h , k , l } ) G=(\{S,A,X\},\{a\},S,\{f,g,h,k,l\})
  22. f : S A A f:S\to AA
  23. g : S X g:S\to X
  24. h : A S h:A\to S
  25. k : A X k:A\to X
  26. l : S a l:S\to a
  27. G = ( { S , A , X } , { a } , S , { f , g , h , k , l } , ( f | g | h | k | l ) * , { f , g , h , k , l } ) G^{\prime}=(\{S,A,X\},\{a\},S,\{f,g,h,k,l\},(f|g|h|k|l)^{*},\{f,g,h,k,l\})
  28. ( f | g | h | k | l ) * (f|g|h|k|l)^{*}
  29. ( f * g h * k ) * l * (f^{*}gh^{*}k)^{*}l^{*}
  30. { g , k } \{g,k\}
  31. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  32. S n S^{n}
  33. S 1 S^{1}
  34. a 2 0 a^{2^{0}}
  35. f u g h v k f^{u}gh^{v}k...
  36. n = u n=u
  37. n < u n<u
  38. n > u n>u
  39. n = u n=u
  40. n < u n<u
  41. n < u n<u
  42. n > u n>u
  43. n = u n=u
  44. S n S^{n}
  45. S n f f A 2 n g A 2 n h h S 2 n k S 2 n S^{n}\Rightarrow_{f}...\Rightarrow_{f}A^{2n}\Rightarrow{g}A^{2n}\Rightarrow{h}% ...\Rightarrow{h}S^{2n}\Rightarrow{k}S^{2n}
  46. f 8 g h * k f^{8}gh^{*}k
  47. s 1 = f f g h k l l s_{1}=ffghkll
  48. S f a c A A f a c failure: f cannot apply, no S to rewrite S\Rightarrow^{ac}_{f}AA\Rightarrow^{ac}_{f}\,\text{failure: f cannot apply, no% S to rewrite}
  49. s 2 = f g h h h k l l s_{2}=fghhhkll
  50. S f a c A A g a c A A h a c S A h a c S S h a c failure: h cannot apply, no A to rewrite S\Rightarrow^{ac}_{f}AA\Rightarrow^{ac}_{g}AA\Rightarrow^{ac}_{h}SA\Rightarrow% ^{ac}_{h}SS\Rightarrow^{ac}_{h}\,\text{failure: h cannot apply, no A to rewrite}
  51. s 3 = f g h h k l l s_{3}=fghhkll
  52. S f a c A A g a c A A h a c S A h a c S S k a c S S l a c a S l a c a a S\Rightarrow^{ac}_{f}AA\Rightarrow^{ac}_{g}AA\Rightarrow^{ac}_{h}SA\Rightarrow% ^{ac}_{h}SS\Rightarrow^{ac}_{k}SS\Rightarrow^{ac}_{l}aS\Rightarrow^{ac}_{l}aa
  53. f * g h * k f^{*}gh^{*}k
  54. S S f a c A A S f a c A A A A g a c A A A A ...\Rightarrow SS\Rightarrow^{ac}_{f}AAS\Rightarrow^{ac}_{f}AAAA\Rightarrow^{% ac}_{g}AAAA
  55. h a c S A A A h a c S S A A h a c S S S A h a c S S S S k a c S S S S \Rightarrow^{ac}_{h}SAAA\Rightarrow^{ac}_{h}SSAA\Rightarrow^{ac}_{h}SSSA% \Rightarrow^{ac}_{h}SSSS\Rightarrow^{ac}_{k}SSSS
  56. l a c a S S S l a c a a S S l a c a a a S l a c a a a a \Rightarrow^{ac}_{l}aSSS\Rightarrow^{ac}_{l}aaSS\Rightarrow^{ac}_{l}aaaS% \Rightarrow^{ac}_{l}aaaa
  57. ( m 1 | m 2 | | m n ) * (m_{1}|m_{2}|...|m_{n})^{*}
  58. m i m_{i}
  59. G = ( N , T , M , S , F ) G=(N,T,M,S,F)
  60. m i = p i , 1 p i , 2 p i , n i m_{i}=p_{i,1}p_{i,2}...p_{i,n_{i}}
  61. p i , j p_{i,j}
  62. ( N T ) * (N\cup T)^{*}
  63. m = p 1 p 2 p n M m=p_{1}p_{2}...p_{n}\in M
  64. x m a c y x\Rightarrow^{ac}_{m}y
  65. x = x 1 A x 2 x=x_{1}Ax_{2}
  66. y = y 1 w y 2 y=y_{1}wy_{2}
  67. A p 1 a c w 1 p 2 a c w 2 p 3 a c p n a c w A\Rightarrow^{ac}_{p_{1}}w_{1}\Rightarrow^{ac}_{p_{2}}w_{2}\Rightarrow^{ac}_{p% _{3}}...\Rightarrow^{ac}_{p_{n}}w
  68. x = y x=y
  69. m F m\in F
  70. m M m\in M
  71. p ( m ) p(m)
  72. G = ( N , T , M , S , F ) G=(N,T,M,S,F)
  73. m i = { p 1 , p 2 , , p n } m_{i}=\{p_{1},p_{2},...,p_{n}\}
  74. ( N T ) * (N\cup T)^{*}
  75. m = { p 1 , p 2 , , p n } M m=\{p_{1},p_{2},...,p_{n}\}\in M
  76. x m a c y x\Rightarrow^{ac}_{m}y
  77. x = x 1 A x 2 x=x_{1}Ax_{2}
  78. y = y 1 w y 2 y=y_{1}wy_{2}
  79. A p i 1 a c w 1 p i 2 a c w 2 p i 3 a c p i n a c w A\Rightarrow^{ac}_{p_{i_{1}}}w_{1}\Rightarrow^{ac}_{p_{i_{2}}}w_{2}\Rightarrow% ^{ac}_{p_{i_{3}}}...\Rightarrow^{ac}_{p_{i_{n}}}w
  80. m = { p i 1 , p i 2 , , p i n } m=\{p_{i_{1}},p_{i_{2}},...,p_{i_{n}}\}
  81. x = y x=y
  82. m F m\in F
  83. G = ( N , T , S , P ) G=(N,T,S,P)
  84. ( p , σ , ϕ ) (p,\sigma,\phi)
  85. σ \sigma
  86. ϕ \phi
  87. x , y ( N T ) * x,y\in(N\cup T)^{*}
  88. p = ( A w , σ , ϕ ) P p=(A\to w,\sigma,\phi)\in P
  89. x p y x\Rightarrow_{p}y
  90. x = x A x ′′ , y = x w x ′′ x=x^{\prime}Ax^{\prime\prime},y=x^{\prime}wx^{\prime\prime}
  91. x = y x=y
  92. L ( G ) = { w ( N T ) * : S p 1 w 1 p 2 p n w } L(G)=\{w\in(N\cup T)^{*}:S\Rightarrow_{p_{1}}w_{1}\Rightarrow_{p_{2}}...% \Rightarrow_{p_{n}}w\}
  93. p i = ( A i v i , σ i , ϕ i ) p_{i}=(A_{i}\to v_{i},\sigma_{i},\phi_{i})
  94. w i - 1 = x i - 1 A x i - 1 , w i = x i - 1 v i x i - 1 , a n d p i + 1 σ i w_{i-1}=x_{i-1}Ax^{\prime}_{i-1},w_{i}=x_{i-1}v_{i}x^{\prime}_{i-1},\ and\ p_{% i+1}\in\sigma_{i}
  95. w i - 1 = w i , p i + 1 ϕ i w_{i-1}=w_{i},p_{i+1}\in\phi_{i}
  96. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  97. G = ( { S , A } , { a } , S , { r 1 , r 2 , r 3 } ) G=(\{S,A\},\{a\},S,\{r_{1},r_{2},r_{3}\})
  98. r 1 = ( S A A , { r 1 } , { r 2 } ) r_{1}=(S\to AA,\{r_{1}\},\{r_{2}\})
  99. r 2 = ( A S , { r 2 } , { r 1 , r 3 } ) r_{2}=(A\to S,\{r_{2}\},\{r_{1},r_{3}\})
  100. r 3 = ( S a , { r 3 } , ) r_{3}=(S\to a,\{r_{3}\},\emptyset)
  101. S r 1 A A r 1 A A r 2 S A r 2 S S r 2 S S S\Rightarrow_{r_{1}}AA\Rightarrow_{r_{1}}AA\Rightarrow_{r_{2}}SA\Rightarrow_{r% _{2}}SS\Rightarrow_{r_{2}}SS
  102. r 1 A A S r 1 A A A A r 1 A A A A \Rightarrow_{r_{1}}AAS\Rightarrow_{r_{1}}AAAA\Rightarrow_{r_{1}}AAAA
  103. r 2 S A A A r 2 S S A A r 2 S S S A r 2 S S S S r 2 S S S S \Rightarrow_{r_{2}}SAAA\Rightarrow_{r_{2}}SSAA\Rightarrow_{r_{2}}SSSA% \Rightarrow_{r_{2}}SSSS\Rightarrow_{r_{2}}SSSS
  104. r 3 a S S S r 3 a a S S r 3 a a a S r 3 a a a a r 3 a a a a \Rightarrow_{r_{3}}aSSS\Rightarrow_{r_{3}}aaSS\Rightarrow_{r_{3}}aaaS% \Rightarrow_{r_{3}}aaaa\Rightarrow_{r_{3}}aaaa
  105. r 1 r_{1}
  106. r 2 r_{2}
  107. r 1 r_{1}
  108. r 2 r_{2}
  109. r 2 r_{2}
  110. r 1 r_{1}
  111. r 3 r_{3}
  112. r 1 r_{1}
  113. r 2 r_{2}
  114. r 1 r_{1}
  115. r 2 r_{2}
  116. r 3 r_{3}
  117. G = ( N , T , S , P ) G=(N,T,S,P)
  118. ( p , R ) (p,R)
  119. N T N\cup T
  120. x , y ( N T ) * x,y\in(N\cup T)^{*}
  121. p = ( A w , R ) P p=(A\to w,R)\in P
  122. x p y x\Rightarrow_{p}y
  123. x = x A x ′′ x=x^{\prime}Ax^{\prime\prime}
  124. y = x w x ′′ y=x^{\prime}wx^{\prime\prime}
  125. x R x\in R
  126. ( S x , a * S b * ) (S\to x,a^{*}Sb^{*})
  127. { a m A b n : m , n 0 } \{a^{m}Ab^{n}:m,n\geq 0\}
  128. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  129. G = ( { S , S } , { a } , { f , g , h } , S ) G=(\{S,S^{\prime}\},\{a\},\{f,g,h\},S)
  130. f = ( S A A , A * S + ) f=(S\to AA,A^{*}S^{+})
  131. g = ( A B , B * A + ) g=(A\to B,B^{*}A^{+})
  132. h = ( B S , S * B + ) h=(B\to S,S^{*}B^{+})
  133. k = ( S a , a * S + ) k=(S\to a,a^{*}S^{+})
  134. S f A A g B A g B B S\Rightarrow_{f}AA\Rightarrow_{g}BA\Rightarrow_{g}BB
  135. h S B h S S f A A S f A A A A \Rightarrow_{h}SB\Rightarrow_{h}SS\Rightarrow_{f}AAS\Rightarrow_{f}AAAA
  136. g B A A A g B B A A g B B B A g B B B B \Rightarrow_{g}BAAA\Rightarrow_{g}BBAA\Rightarrow_{g}BBBA\Rightarrow_{g}BBBB
  137. h S B B B h S S B B h S S S B h S S S S \Rightarrow_{h}SBBB\Rightarrow_{h}SSBB\Rightarrow_{h}SSSB\Rightarrow_{h}SSSS
  138. k a S S S k a a S S k a a a S k a a a a \Rightarrow_{k}aSSS\Rightarrow_{k}aaSS\Rightarrow_{k}aaaS\Rightarrow_{k}aaaa
  139. G = ( N , T , S , P ) G=(N,T,S,P)
  140. ( p , R , Q ) (p,R,Q)
  141. x A x , x w x ( N T ) * xAx^{\prime},xwx^{\prime}\in(N\cup T)^{*}
  142. p = ( A w , R , Q ) P p=(A\to w,R,Q)\in P
  143. x A x p x w x xAx^{\prime}\Rightarrow_{p}xwx^{\prime}
  144. x A x xAx^{\prime}
  145. x A x xAx^{\prime}
  146. L ( G ) = { w T * : S * w } L(G)=\{w\in T^{*}:S\Rightarrow^{*}w\}
  147. ( N T ) * (N\cup T)^{*}
  148. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  149. G = ( { S , X , Y , A } , { a } , S , { r 1 , r 2 , r 3 , r 4 , r 5 } ) G=(\{S,X,Y,A\},\{a\},S,\{r_{1},r_{2},r_{3},r_{4},r_{5}\})
  150. r 1 = ( S X X , , { Y , A } ) r_{1}=(S\to XX,\emptyset,\{Y,A\})
  151. r 2 = ( X Y , , { S } ) r_{2}=(X\to Y,\emptyset,\{S\})
  152. r 3 = ( Y S , , { X } ) r_{3}=(Y\to S,\emptyset,\{X\})
  153. r 4 = ( S A , , { X } ) r_{4}=(S\to A,\emptyset,\{X\})
  154. r 5 = ( A a , , { S } ) r_{5}=(A\to a,\emptyset,\{S\})
  155. S r 1 X X r 2 Y X r 2 Y Y r 3 S Y r 3 S S S\Rightarrow_{r_{1}}XX\Rightarrow_{r_{2}}YX\Rightarrow_{r_{2}}YY\Rightarrow_{r% _{3}}SY\Rightarrow_{r_{3}}SS
  156. r 1 X X S r 1 X X X X r 2 Y X X X r 2 Y Y X X r 2 Y Y Y X r 2 Y Y Y Y \Rightarrow_{r_{1}}XXS\Rightarrow_{r_{1}}XXXX\Rightarrow_{r_{2}}YXXX% \Rightarrow_{r_{2}}YYXX\Rightarrow_{r_{2}}YYYX\Rightarrow_{r_{2}}YYYY
  157. r 3 S Y Y Y r 3 S S Y Y r 3 S S S Y r 3 S S S S \Rightarrow_{r_{3}}SYYY\Rightarrow_{r_{3}}SSYY\Rightarrow_{r_{3}}SSSY% \Rightarrow_{r_{3}}SSSS
  158. r 4 A S S S r 4 A A S S r 4 A A A S r 4 A A A A \Rightarrow_{r_{4}}ASSS\Rightarrow_{r_{4}}AASS\Rightarrow_{r_{4}}AAAS% \Rightarrow_{r_{4}}AAAA
  159. r 5 a A A A r 5 a a A A r 5 a a a A r 5 a a a a \Rightarrow_{r_{5}}aAAA\Rightarrow_{r_{5}}aaAA\Rightarrow_{r_{5}}aaaA% \Rightarrow_{r_{5}}aaaa
  160. r 1 r_{1}
  161. r 2 r_{2}
  162. r 3 r_{3}
  163. G = ( N , T , S , P ) G=(N,T,S,P)
  164. x A x , x w x ( N T ) * xAx^{\prime},xwx^{\prime}\in(N\cup T)^{*}
  165. p = A w P p=A\to w\in P
  166. x A x p x w x xAx^{\prime}\Rightarrow_{p}xwx^{\prime}
  167. p = A w P p^{\prime}=A\to w^{\prime}\in P
  168. p < p p<p^{\prime}
  169. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  170. G = ( { S , X , Y , Z , A } , { a } , S , P ) G=(\{S,X,Y,Z,A\},\{a\},S,P)
  171. S S X X X X X Y Y X X Y Y Y Y S S Y Y S Y Y S\Rightarrow_{S\to XX}\ XX\ \Rightarrow_{X\to Y}\ YX\ \Rightarrow_{X\to Y}\ YY% \ \Rightarrow_{Y\to S}\ SY\ \Rightarrow_{Y\to S}\ YY
  172. S X X X X S S X X X X X X \Rightarrow_{S\to XX}\ XXS\ \Rightarrow_{S\to XX}\ XXXX
  173. X Y Y X X X X Y Y Y X X X Y Y Y Y X X Y Y Y Y Y \Rightarrow_{X\to Y}\ YXXX\ \Rightarrow_{X\to Y}\ YYXX\ \Rightarrow_{X\to Y}\ % YYYX\ \Rightarrow_{X\to Y}\ YYYY
  174. Y S S Y Y Y Y S S S Y Y Y S S S S Y Y S S S S S \Rightarrow_{Y\to S}\ SYYY\ \Rightarrow_{Y\to S}\ SSYY\ \Rightarrow_{Y\to S}\ % SSSY\ \Rightarrow_{Y\to S}\ SSSS
  175. S A A S S S S A A A S S S A A A A S S A A A A A \Rightarrow_{S\to A}\ ASSS\ \Rightarrow_{S\to A}\ AASS\ \Rightarrow_{S\to A}\ % AAAS\ \Rightarrow_{S\to A}\ AAAA
  176. A a a A A A A a a a A A A a a a a A A a a a a a \Rightarrow_{A\to a}\ aAAA\ \Rightarrow_{A\to a}\ aaAA\ \Rightarrow_{A\to a}\ % aaaA\ \Rightarrow_{A\to a}\ aaaa
  177. Y Z , S Z , Y S , S A Y\to Z,S\to Z,Y\to S,S\to A
  178. Y S Y\to S
  179. S A S\to A
  180. Y Z Y\to Z
  181. A Z A\to Z
  182. S n S^{n}
  183. A n A^{n}
  184. S 2 n S^{2n}
  185. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  186. X w X\to w
  187. { w w : w { a , b } * } \{ww:w\in\{a,b\}^{*}\}
  188. G = ( { S , A } , { a , b } , S , { f , g , h , k } ) G=(\{S,A\},\{a,b\},S,\{f,g,h,k\})
  189. f = S A A f=S\to AA
  190. g = A a A g=A\to aA
  191. h = A b A h=A\to bA
  192. k = A ϵ k=A\to\epsilon
  193. S f A A g a A a A g a a A a a A h a a b A a a b A k a a b a a b S\Rightarrow_{f}AA\Rightarrow_{g}aAaA\Rightarrow_{g}aaAaaA\Rightarrow_{h}% aabAaabA\Rightarrow_{k}aabaab
  194. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  195. G = ( { S } , { a } , S , P ) G=(\{S\},\{a\},S,P)
  196. S S S S\to SS
  197. S a S\to a
  198. S S 2 S 4 S 8 S\Rightarrow S^{2}\Rightarrow S^{4}\Rightarrow S^{8}\Rightarrow...
  199. { a 2 n : n 0 } \{a^{2^{n}}:n\geq 0\}
  200. { a n b n c n : n 0 } \{a^{n}b^{n}c^{n}:n\geq 0\}
  201. G = ( { S , A , B , C } , { a , b , c } , S , P ) G=(\{S,A,B,C\},\{a,b,c\},S,P)
  202. S A B C S\to ABC
  203. A a A A\to aA
  204. A a A\to a
  205. B b B B\to bB
  206. B b B\to b
  207. C c C C\to cC
  208. C c C\to c
  209. S A B C a A b B c C a a A b b B c c C a a a b b b c c c S\Rightarrow ABC\Rightarrow aAbBcC\Rightarrow aaAbbBccC\Rightarrow aaabbbccc
  210. A a A , B b B , C c C A\to aA,B\to bB,C\to cC
  211. A a , B b B , C c C A\to a,B\to bB,C\to cC
  212. G = ( N , T , S , P ) G=(N,T,S,P)
  213. ( A w , k ) (A\to w,k)
  214. A w A\to w
  215. p = ( A w , k ) p=(A\to w,k)
  216. G = ( N , T , S , P ) G=(N,T,S,P)
  217. p = ( A 1 w 1 , , A n w n ) p=(A_{1}\to w_{1},...,A_{n}\to w_{n})
  218. n > 0 n>0
  219. x p y x\Rightarrow_{p}y
  220. p = ( A 1 w 1 , , A n w n ) P p=(A_{1}\to w_{1},...,A_{n}\to w_{n})\in P
  221. x = x 1 A 1 x 2 x n A n x n + 1 , y = x 1 w 1 x 2 x n w n x n + 1 x=x_{1}A_{1}x_{2}...x_{n}A_{n}x_{n+1},y=x_{1}w_{1}x_{2}...x_{n}w_{n}x_{n+1}
  222. x i ( N T ) * x_{i}\in(N\cup T)^{*}
  223. { a n b n c n : n 0 } \{a^{n}b^{n}c^{n}:n\geq 0\}
  224. G = ( { S } , { a , b , c } , S , { r 1 , r 2 , r 3 } ) G=(\{S\},\{a,b,c\},S,\{r_{1},r_{2},r_{3}\})
  225. r 1 = ( S S S S ) r_{1}=(S\to SSS)
  226. r 2 = ( S a S , S b S , S c S ) r_{2}=(S\to aS,S\to bS,S\to cS)
  227. r 3 = ( S ϵ , S ϵ , S ϵ ) r_{3}=(S\to\epsilon,S\to\epsilon,S\to\epsilon)
  228. S r 1 S S S r 2 a S b S c S r 2 a a S b b S c c S r 2 a a a S b b b S c c c S r 3 a a a b b b c c c S\Rightarrow_{r_{1}}SSS\Rightarrow_{r_{2}}aSbScS\Rightarrow_{r_{2}}aaSbbSccS% \Rightarrow_{r_{2}}aaaSbbbScccS\Rightarrow_{r_{3}}aaabbbccc

Convexoid_operator.html

  1. T - λ T-\lambda
  2. λ \lambda

Convolution_for_optical_broad-beam_responses_in_scattering_media.html

  1. C ( x , y , z ) = - - G ( x - x , y - y , z ) S ( x , y ) d x d y . ( 1 ) C(x,y,z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\ G(x-x^{\prime},y-y^{% \prime},z)S(x^{\prime},y^{\prime})\,dx^{\prime}\,dy^{\prime}.\qquad(1)
  2. x ′′ = x - x x^{\prime\prime}=x-x^{\prime}\,
  3. y ′′ = y - y y^{\prime\prime}=y-y^{\prime}\,
  4. C ( x , y , z ) = - - G ( x ′′ , y ′′ , z ) S ( x - x ′′ , y - y ′′ ) d x ′′ d y ′′ . ( 2 ) C(x,y,z)=\int_{-\infty}^{\infty}\!\int_{-\infty}^{\infty}\ G(x^{\prime\prime},% y^{\prime\prime},z)S(x-x^{\prime\prime},y-y^{\prime\prime})\,dx^{\prime\prime}% \,dy^{\prime\prime}.\qquad(2)
  5. C ( x , y , z ) C(x,y,z)\,
  6. C ( r , z ) = 0 S ( r ) r [ 0 2 π G ( r 2 + r - 2 2 r r c o s ϕ , z ) d ϕ ] d r ( 3 ) C(r,z)=\int_{0}^{\infty}\ S(r^{\prime})r^{\prime}\left[\int_{0}^{2\pi}\ G\left% (\sqrt{r^{2}+r^{\prime}\,{}^{2}-2rr^{\prime}cos\phi^{\prime}},z\right)\,d\phi^% {\prime}\right]dr^{\prime}\qquad(3)
  7. C ( r , z ) = 0 G ( r ′′ , z ) r ′′ [ 0 2 π S ( r 2 + r ′′ - 2 2 r r ′′ c o s ϕ ′′ ) d ϕ ′′ ] d r ′′ ( 4 ) C(r,z)=\int_{0}^{\infty}G(r^{\prime\prime},z)r^{\prime\prime}\left[\int_{0}^{2% \pi}S\left(\sqrt{r^{2}+r^{\prime\prime}\,{}^{2}-2rr^{\prime\prime}cos\phi^{% \prime\prime}}\right)\,d\phi^{\prime\prime}\right]dr^{\prime\prime}\qquad(4)
  8. r = x 2 + y 2 r^{\prime}=\sqrt{x^{\prime 2}+y^{\prime 2}}
  9. S ( r ) = S 0 exp [ - 2 ( r R ) 2 ] . ( 5 ) S(r^{\prime})=S_{0}\exp\left[-2\left(\frac{r^{\prime}}{R}\right)^{2}\right].% \qquad(5)
  10. 1 e 2 \tfrac{1}{e^{2}}\,
  11. S 0 = 2 P 0 π R 2 . ( 6 ) S_{0}=\frac{2P_{0}}{\pi R^{2}}.\qquad(6)
  12. C ( r , z ) = 2 π S ( r ) 0 G ( r ′′ , z ) exp [ - 2 ( r ′′ R ) 2 ] I 0 ( 4 r r ′′ R 2 ) r ′′ d r ′′ , ( 7 ) C(r,z)=2\pi S(r)\int_{0}^{\infty}G(r^{\prime\prime},z)\exp\left[-2\left(\frac{% r^{\prime\prime}}{R}\right)^{2}\right]I_{0}\left(\frac{4rr^{\prime\prime}}{R^{% 2}}\right)r^{\prime\prime}\,dr^{\prime\prime},\qquad(7)
  13. S ( r ) = { S 0 , if r R 0 , if r > R ( 8 ) S(r^{\prime})=\begin{cases}S_{0},&\,\text{if }r^{\prime}\leq R\\ \,0,&\,\text{if }r^{\prime}>R\end{cases}\qquad(8)
  14. S 0 = P 0 π R 2 . ( 9 ) S_{0}=\frac{P_{0}}{\pi R^{2}}.\qquad(9)
  15. C ( r , z ) = 2 π S 0 0 G ( r ′′ , z ) I ϕ ( r , r ′′ ) r ′′ d r ′′ , ( 10 ) C(r,z)=2\pi S_{0}\int_{0}^{\infty}G(r^{\prime\prime},z)I_{\phi}(r,r^{\prime% \prime})r^{\prime\prime}\,dr^{\prime\prime},\qquad(10)
  16. I ϕ ( r , r ′′ ) = { 1 , if R r + r ′′ , 1 π cos - 1 ( r 2 + r ′′ 2 - R 2 2 r r ′′ ) , if | r - r ′′ | R < r + r ′′ , 0 , if R < | r + r ′′ | . ( 11 ) I_{\phi}(r,r^{\prime\prime})=\begin{cases}1,&\mbox{if }~{}R\geq r+r^{\prime% \prime},\\ \tfrac{1}{\pi}\cos^{-1}\left(\tfrac{r^{2}+r^{\prime\prime 2}-R^{2}}{2rr^{% \prime\prime}}\right),&\mbox{if }~{}\left|r-r^{\prime\prime}\right|\leq R<r+r^% {\prime\prime},\\ 0,&\mbox{if }~{}R<\left|r+r^{\prime\prime}\right|.\end{cases}\qquad(11)
  17. C ( r , z ) = G 1 ( 0 , z ) δ ( r ) 2 π r + G 2 ( r , z ) , ( 12 ) C(r,z)=G_{1}(0,z)\frac{\delta(r)}{2\pi r}+G_{2}(r,z),\qquad(12)
  18. C ( r , z ) = G 1 ( 0 , z ) S ( r ) + 2 π S 0 0 G 2 ( r ′′ , z ) e x p [ - 2 ( r ′′ - r R ) 2 ] I 0 e ( 4 r r ′′ R 2 ) r ′′ d r ′′ . ( 13 ) C(r,z)=G_{1}(0,z)S(r)+2\pi S_{0}\int_{0}^{\infty}G_{2}(r^{\prime\prime},z)\,% exp\left[-2\left(\frac{r^{\prime\prime}-r}{R}\right)^{2}\right]I_{0e}\left(% \frac{4rr^{\prime\prime}}{R^{2}}\right)r^{\prime\prime}\,dr^{\prime\prime}.% \qquad(13)
  19. C ( r , z ) = G 1 ( 0 , z ) S ( r ) + 2 π S 0 0 G 2 ( r ′′ , z ) I ϕ ( r , r ′′ ) r ′′ d r ′′ . ( 14 ) C(r,z)=G_{1}(0,z)S(r)+2\pi S_{0}\int_{0}^{\infty}G_{2}(r^{\prime\prime},z)I_{% \phi}(r,r^{\prime\prime})r^{\prime\prime}\,dr^{\prime\prime}.\qquad(14)

Coppersmith_method.html

  1. 𝐑 n \mathbf{R}^{n}
  2. L = 𝐙 b 1 𝐙 b k L=\mathbf{Z}b_{1}\oplus\ldots\oplus\mathbf{Z}b_{k}
  3. B = ( b 1 , b 2 , , b k ) B=(b_{1},b_{2},\ldots,b_{k})
  4. ( b 1 * , b 2 * , , b k * ) (b_{1}^{*},b_{2}^{*},\dots,b_{k}^{*})
  5. det ( L ) || b i * || \mathrm{det}(L)\leq\prod||b_{i}^{*}||
  6. ( b 1 * , b 2 * , , b k * ) (b_{1}^{*},b_{2}^{*},\dots,b_{k}^{*})
  7. || b k * || ( det ( L ) ) 1 / k 2 ( 1 - k ) / 4 ||b_{k}^{*}||\geq(\mathrm{det}(L))^{1/k}\cdot 2^{(1-k)/4}
  8. F ( x ) = x n + a n - 1 x n - 1 + + a 1 x + a 0 F(x)=x^{n}+a_{n-1}x^{n-1}+\ldots+a_{1}x+a_{0}
  9. F ( x 0 ) 0 mod M F(x_{0})\equiv 0\mod M
  10. | x 0 | < M 1 / n |x_{0}|<M^{1/n}
  11. x 0 x_{0}
  12. F 2 F_{2}
  13. x 0 x_{0}
  14. x 0 x_{0}
  15. F 2 ( x 0 ) < M F_{2}(x_{0})<M
  16. x 0 x_{0}
  17. F 2 F_{2}
  18. p 1 ( x ) , p 2 ( x ) , , p n ( x ) p_{1}(x),p_{2}(x),\dots,p_{n}(x)
  19. x 0 x_{0}
  20. M a M^{a}
  21. x 0 x_{0}
  22. x 0 x_{0}
  23. M a M^{a}
  24. F 2 ( x ) = c i p i ( x ) F_{2}(x)=\sum c_{i}p_{i}(x)
  25. p i p_{i}
  26. | F 2 ( x ) | < M a |F_{2}(x)|<M^{a}
  27. F 2 ( x ) F_{2}(x)

Coreflexive_relation.html

  1. a , b X , a R b a = b . \forall a,b\in X,\ aRb\Rightarrow\;a=b.

Cornacchia's_algorithm.html

  1. x 2 + d y 2 = m x^{2}+dy^{2}=m
  2. 1 d < m 1\leq d<m
  3. r 0 2 - d ( mod m ) r_{0}^{2}\equiv-d\;\;(\mathop{{\rm mod}}m)
  4. r 0 r_{0}
  5. - d -d
  6. r 1 m ( mod r 0 ) r_{1}\equiv m\;\;(\mathop{{\rm mod}}r_{0})
  7. r 2 r 0 ( mod r 1 ) r_{2}\equiv r_{0}\;\;(\mathop{{\rm mod}}r_{1})
  8. r k < m r_{k}<\sqrt{m}
  9. s = m - r k 2 d s=\sqrt{\tfrac{m-r_{k}^{2}}{d}}
  10. x = r k , y = s x=r_{k},y=s
  11. ( x , y ) (x,y)
  12. g c d ( x , y ) = g 1 gcd(x,y)=g≠1
  13. m m
  14. m m
  15. ( u , v ) (u,v)
  16. ( g u , g v ) (gu,gv)
  17. x 2 + 6 y 2 = 103 x^{2}+6y^{2}=103
  18. 7 2 < 103 7^{2}<103
  19. 103 - 7 2 6 = 3 \sqrt{\tfrac{103-7^{2}}{6}}=3

Corner-point_grid.html

  1. ( i , j , k ) (i,j,k)
  2. k k
  3. i i
  4. j j
  5. i i
  6. k k

Cost_of_electricity_by_source.html

  1. LCOE = sum of costs over lifetime sum of electricity produced over lifetime = t = 1 n I t + M t + F t ( 1 + r ) t t = 1 n E t ( 1 + r ) t \mathrm{LCOE}=\frac{\,\text{sum of costs over lifetime}}{\,\text{sum of % electricity produced over lifetime}}=\frac{\sum_{t=1}^{n}\frac{I_{t}+M_{t}+F_{% t}}{\left({1+r}\right)^{t}}}{\sum_{t=1}^{n}\frac{E_{t}}{\left({1+r}\right)^{t}}}

Coulomb's_law.html

  1. | 𝐅 | = k e | q 1 q 2 | r 2 |\mathbf{F}|=k_{e}{|q_{1}q_{2}|\over r^{2}}\qquad
  2. 𝐅 1 = k e q 1 q 2 | 𝐫 21 | 2 𝐫 ^ 21 , \qquad\mathbf{F}_{1}=k_{e}\frac{q_{1}q_{2}}{{|\mathbf{r}_{21}|}^{2}}\mathbf{% \hat{r}}_{21},\qquad
  3. k e k_{e}
  4. k e = 8.987 551 787 368 176 4 × 10 9 N m 2 C - 2 k_{e}=8.987\,551\,787\,368\,176\,4\times 10^{9}\ \mathrm{N\cdot m^{2}\cdot C}^% {-2}
  5. q 1 q_{1}
  6. q 2 q_{2}
  7. r r
  8. s y m b o l r 21 = s y m b o l r 1 - r 2 symbol{r_{21}}=symbol{r_{1}-r_{2}}
  9. s y m b o l r ^ 21 = s y m b o l r 21 / | s y m b o l r 21 | symbol{\hat{r}_{21}}={symbol{r_{21}}/|symbol{r_{21}}|}
  10. q 2 q_{2}
  11. q 1 q_{1}
  12. 𝐅 1 \mathbf{F}_{1}
  13. q 1 q_{1}
  14. q 2 q_{2}
  15. 𝐫 12 \mathbf{r}_{12}
  16. q 2 q_{2}
  17. 𝐅 2 = - 𝐅 1 \mathbf{F}_{2}=-\mathbf{F}_{1}
  18. k e = 1 / ( 4 π ε 0 ) k_{e}=1/(4\pi\varepsilon_{0})
  19. ε 0 \varepsilon_{0}
  20. ε \varepsilon
  21. k k
  22. s y m b o l F symbol{F}
  23. q t q_{t}
  24. s y m b o l E symbol{E}
  25. s y m b o l F = q t s y m b o l E symbol{F}=q_{t}symbol{E}
  26. q q
  27. q t q_{t}
  28. s y m b o l E symbol{E}
  29. s y m b o l E symbol{E}
  30. q q
  31. r r
  32. | s y m b o l E | = 1 4 π ε 0 | q | r 2 |symbol{E}|={1\over 4\pi\varepsilon_{0}}{|q|\over r^{2}}
  33. k e k_{e}
  34. e e
  35. k e \displaystyle k_{e}
  36. s y m b o l F symbol{F}
  37. q q
  38. Q Q
  39. s y m b o l F symbol{F}
  40. q 1 q_{1}
  41. q 2 q_{2}
  42. | s y m b o l F | = k e | q 1 q 2 | r 2 |symbol{F}|=k_{e}{|q_{1}q_{2}|\over r^{2}}
  43. r r
  44. k e k_{e}
  45. q 1 q 2 q_{1}q_{2}
  46. s y m b o l F 1 symbol{F}_{1}
  47. q 1 q_{1}
  48. s y m b o l F 2 symbol{F}_{2}
  49. q 2 q_{2}
  50. q 1 q 2 > 0 q_{1}q_{2}>0
  51. q 1 q 2 < 0 q_{1}q_{2}<0
  52. s y m b o l F 1 symbol{F_{1}}
  53. q 1 q_{1}
  54. s y m b o l r 1 symbol{r_{1}}
  55. q 2 q_{2}
  56. s y m b o l r 2 symbol{r_{2}}
  57. s y m b o l F 1 = q 1 q 2 4 π ε 0 ( s y m b o l r 1 - r 2 ) | s y m b o l r 1 - r 2 | 3 = q 1 q 2 4 π ε 0 s y m b o l r ^ 21 | s y m b o l r 21 | 2 , symbol{F_{1}}={q_{1}q_{2}\over 4\pi\varepsilon_{0}}{(symbol{r_{1}-r_{2}})\over% |symbol{r_{1}-r_{2}}|^{3}}={q_{1}q_{2}\over 4\pi\varepsilon_{0}}{symbol{\hat{r% }_{21}}\over|symbol{r_{21}}|^{2}},
  58. s y m b o l r 21 = s y m b o l r 1 - r 2 symbol{r_{21}}=symbol{r_{1}-r_{2}}
  59. s y m b o l r ^ 21 = s y m b o l r 21 / | s y m b o l r 21 | symbol{\hat{r}_{21}}={symbol{r_{21}}/|symbol{r_{21}}|}
  60. ε 0 \varepsilon_{0}
  61. s y m b o l r ^ 21 symbol{\hat{r}_{21}}
  62. q 2 q_{2}
  63. q 1 q_{1}
  64. q 1 q 2 q_{1}q_{2}
  65. q 1 q_{1}
  66. s y m b o l r ^ 21 symbol{\hat{r}_{21}}
  67. q 1 q 2 q_{1}q_{2}
  68. q 1 q_{1}
  69. - s y m b o l r ^ 21 -symbol{\hat{r}_{21}}
  70. s y m b o l F 2 symbol{F_{2}}
  71. q 2 q_{2}
  72. s y m b o l F 2 = - s y m b o l F 1 symbol{F_{2}}=-symbol{F_{1}}
  73. s y m b o l F symbol{F}
  74. q q
  75. s y m b o l r symbol{r}
  76. N N
  77. s y m b o l F ( r ) = q 4 π ε 0 i = 1 N q i s y m b o l r - r i | s y m b o l r - r i | 3 = q 4 π ε 0 i = 1 N q i s y m b o l R i ^ | s y m b o l R i | 2 , symbol{F(r)}={q\over 4\pi\varepsilon_{0}}\sum_{i=1}^{N}q_{i}{symbol{r-r_{i}}% \over|symbol{r-r_{i}}|^{3}}={q\over 4\pi\varepsilon_{0}}\sum_{i=1}^{N}q_{i}{% symbol{\widehat{R_{i}}}\over|symbol{R_{i}}|^{2}},
  78. q i q_{i}
  79. s y m b o l r i symbol{r_{i}}
  80. i t h i^{th}
  81. s y m b o l R i ^ symbol{\widehat{R_{i}}}
  82. s y m b o l R i = s y m b o l r - s y m b o l r i symbol{R}_{i}=symbol{r}-symbol{r}_{i}
  83. q i q_{i}
  84. q q
  85. d q dq
  86. λ ( s y m b o l r ) \lambda(symbol{r^{\prime}})
  87. s y m b o l r symbol{r^{\prime}}
  88. d l dl^{\prime}
  89. d q = λ ( s y m b o l r ) d l dq=\lambda(symbol{r^{\prime}})dl^{\prime}
  90. σ ( s y m b o l r ) \sigma(symbol{r^{\prime}})
  91. s y m b o l r symbol{r^{\prime}}
  92. d A dA^{\prime}
  93. d q = σ ( s y m b o l r ) d A . dq=\sigma(symbol{r^{\prime}})\,dA^{\prime}.
  94. ρ ( s y m b o l r ) \rho(symbol{r^{\prime}})
  95. s y m b o l r symbol{r^{\prime}}
  96. d V dV^{\prime}
  97. d q = ρ ( s y m b o l r ) d V . dq=\rho(symbol{r^{\prime}})\,dV^{\prime}.
  98. q q^{\prime}
  99. s y m b o l r symbol{r}
  100. s y m b o l F = q 4 π ε 0 d q s y m b o l r - s y m b o l r | s y m b o l r - s y m b o l r | 3 . symbol{F}={q^{\prime}\over 4\pi\varepsilon_{0}}\int dq{symbol{r}-symbol{r^{% \prime}}\over|symbol{r}-symbol{r^{\prime}}|^{3}}.
  101. m m
  102. q q
  103. l l
  104. m g mg
  105. T T
  106. s y m b o l F symbol{F}
  107. L 1 L_{1}\,\!
  108. F 1 F_{1}\,\!
  109. L 2 < L 1 L_{2}<L_{1}\,\!
  110. F 2 = m g . tan θ 2 F_{2}=mg.\tan\theta_{2}\,\!
  111. q 2 4 4 π ϵ 0 L 2 2 = m g . tan θ 2 \frac{\frac{q^{2}}{4}}{4\pi\epsilon_{0}L_{2}^{2}}=mg.\tan\theta_{2}
  112. θ 1 \theta_{1}\,\!
  113. θ 2 \theta_{2}\,\!
  114. L 1 L_{1}\,\!
  115. L 2 L_{2}\,\!
  116. L 1 L 2 4 ( L 2 L 1 ) 2 L 1 L 2 4 3 \frac{L_{1}}{L_{2}}\approx 4{\left(\frac{L_{2}}{L_{1}}\right)}^{2}% \Longrightarrow\frac{L_{1}}{L_{2}}\approx\sqrt[3]{4}\,\!

Coupled_map_lattice.html

  1. x n + 1 = r x n ( 1 - x n ) \qquad x_{n+1}=rx_{n}(1-x_{n})
  2. s s
  3. s - 1 s-1
  4. ϵ = 0.5 \epsilon=0.5
  5. x n + 1 = ( ϵ ) [ r x n ( 1 - x n ) ] s + ( 1 - ϵ ) [ r x n ( 1 - x n ) ] s - 1 \qquad x_{n+1}=(\epsilon)[rx_{n}(1-x_{n})]_{s}+(1-\epsilon)[rx_{n}(1-x_{n})]_{% s-1}
  6. u s t + 1 = ( 1 - ε ) f ( u s t ) + ε 2 ( f ( u s + 1 t ) + f ( u s - 1 t ) ) t , ε [ 0 , 1 ] u_{s}^{t+1}=(1-\varepsilon)f(u_{s}^{t})+\frac{\varepsilon}{2}\left(f(u_{s+1}^{% t})+f(u_{s-1}^{t})\right)\ \ \ t\in\mathbb{N},\ \varepsilon\in[0,1]
  7. u s t , u_{s}^{t}\in{\mathbb{R}}\ ,
  8. f f
  9. f ( x n ) = 1 - a x 2 {f(x_{n})}=1-ax^{2}

Cousin's_theorem.html

  1. n \mathbb{R}^{n}
  2. 𝒞 \mathcal{C}
  3. 𝒞 \mathcal{C}
  4. 𝒞 \mathcal{C}

Covariance_intersection.html

  1. a ^ \hat{a}
  2. A A
  3. b ^ \hat{b}
  4. B B
  5. C - 1 = ω A - 1 + ( 1 - ω ) B - 1 , C^{-1}=\omega A^{-1}+(1-\omega)B^{-1}\,,
  6. c ^ = C ( ω A - 1 a ^ + ( 1 - ω ) B - 1 b ^ ) . \hat{c}=C(\omega A^{-1}\hat{a}+(1-\omega)B^{-1}\hat{b})\,.

Cover's_theorem.html

  1. n n
  2. n - 1 n-1

Covering_groups_of_the_alternating_and_symmetric_groups.html

  1. 2 S n - 2\cdot S_{n}^{-}
  2. 2 S n - 2\cdot S_{n}^{-}
  3. 2 S n + 2\cdot S_{n}^{+}
  4. 2 S n + 2\cdot S_{n}^{+}
  5. 2 S n - 2\cdot S_{n}^{-}
  6. 2 S n + 2\cdot S_{n}^{+}
  7. A n SO ( n - 1 ) A_{n}\hookrightarrow\operatorname{SO}(n-1)
  8. S n O ( n - 1 ) S_{n}\hookrightarrow\operatorname{O}(n-1)
  9. Spin ( n ) SO ( n ) , \operatorname{Spin}(n)\to\operatorname{SO}(n),
  10. A n A_{n}
  11. 2 A n A n . 2\cdot A_{n}\to A_{n}.
  12. Pin ± ( n ) O ( n ) . \operatorname{Pin}_{\pm}(n)\to\operatorname{O}(n).
  13. S ~ n \tilde{S}_{n}
  14. S ^ n \hat{S}_{n}
  15. A 6 , A_{6},
  16. 3 A 6 , 3\cdot A_{6},
  17. S 6 , S_{6},
  18. 3 S 6 , 3\cdot S_{6},
  19. U 4 ( 3 ) U_{4}(3)
  20. L 3 ( 4 ) , L_{3}(4),
  21. G 2 ( 4 ) . G_{2}(4).
  22. S n Aut ( A n ) , S_{n}\cong\operatorname{Aut}(A_{n}),

Credit_valuation_adjustment.html

  1. CVA = E Q [ L * ] = ( 1 - R ) 0 T E Q [ B 0 B t E ( t ) | τ = t ] d PD ( 0 , t ) \mathrm{CVA}=E^{Q}[L^{*}]=(1-R)\int_{0}^{T}E^{Q}\left[\frac{B_{0}}{B_{t}}E(t)|% \tau=t\right]d\mathrm{PD}(0,t)
  2. T T
  3. B t B_{t}
  4. t t
  5. R R
  6. τ \tau
  7. E ( t ) E(t)
  8. t t
  9. PD ( s , t ) \mathrm{PD}(s,t)
  10. s s
  11. t t
  12. CVA = ( 1 - R ) 0 T EE * ( t ) d PD ( 0 , t ) \mathrm{CVA}=(1-R)\int_{0}^{T}\mathrm{EE}^{*}(t)~{}d\mathrm{PD}(0,t)
  13. EE * \mathrm{EE}^{*}

Critical_taper.html

  1. τ b \tau_{b}
  2. τ b \!\tau_{b}
  3. ρ \rho
  4. β \beta
  5. ρ g H s i n β \!\rho gHsin\beta
  6. ρ w * g * D * s i n ( α + β ) \rho_{w}*g*D*sin(\alpha+\beta)
  7. α + β \alpha+\beta
  8. ρ w \rho_{w}
  9. ρ w g D s i n ( α + β ) \!\rho_{w}gDsin(\alpha+\beta)
  10. τ b \tau_{b}
  11. τ b = S 0 + μ ( σ n - P f ) \!\tau_{b}=S_{0}+\mu(\sigma_{n}-P_{f})
  12. μ \mu
  13. σ n \sigma_{n}
  14. ρ g H s i n β + ρ w g D s i n ( α + β ) + τ b = d d x 0 H σ x d z \rho gHsin\beta+\rho_{w}gDsin(\alpha+\beta)+\tau_{b}=\frac{d}{dx}\int_{0}^{H}% \sigma_{x}dz

Crossing_sequence_(Turing_machines).html

  1. 𝒞 i ( x ) \mathcal{C}_{i}(x)
  2. c s ( x , i ) cs(x,i)
  3. q i 1 , q i 2 , , q i k , q_{i_{1}},q_{i_{2}},...,q_{i_{k}},

CryoEDM.html

  1. ν \nu
  2. h ν = 2 d E ± 2 μ B h\nu=2dE\pm 2\mu B
  3. μ \mu
  4. ± \pm
  5. π / 2 \pi/2
  6. π / 2 \pi/2

Crystal_oscillator_frequencies.html

  1. 135 11 \tfrac{135}{11}
  2. 59 4 \tfrac{59}{4}

CSMP_III.html

  1. X = 6 Y / W + ( Z - 2 ) 2 X=6Y/W+(Z-2)^{2}

Cubic_field.html

  1. 𝐐 [ x ] / ( f ( x ) ) \mathbf{Q}[x]/(f(x))
  2. n 3 \sqrt[3]{n}
  3. 𝐐 ( 2 3 ) \mathbf{Q}(\sqrt[3]{2})
  4. f ( X ) = X 3 - a X + b f(X)=X^{3}-aX+b
  5. N ± ( X ) A ± 12 ζ ( 3 ) X + 4 ζ ( 1 3 ) B ± 5 Γ ( 2 3 ) 3 ζ ( 5 3 ) X 5 6 N^{\pm}(X)\sim\frac{A_{\pm}}{12\zeta(3)}X+\frac{4\zeta(\frac{1}{3})B_{\pm}}{5% \Gamma(\frac{2}{3})^{3}\zeta(\frac{5}{3})}X^{\frac{5}{6}}
  6. 3 \sqrt{3}

Cuckoo_search.html

  1. p a ( 0 , 1 ) p_{a}\in(0,1)
  2. f ( 𝐱 ) , 𝐱 = ( x 1 , x 2 , , x d ) ; f(\mathbf{x}),\quad\mathbf{x}=(x_{1},x_{2},\dots,x_{d});\,
  3. n n
  4. F i F_{i}
  5. F i f ( 𝐱 i ) F_{i}\propto f(\mathbf{x}_{i})
  6. F i > F j F_{i}>F_{j}
  7. p a p_{a}
  8. p a p_{a}
  9. n n
  10. 𝐱 t + 1 = 𝐱 t + s E t , \mathbf{x}_{t+1}=\mathbf{x}_{t}+sE_{t},
  11. E t E_{t}
  12. s s
  13. r r
  14. r 2 = 2 d D t , r^{2}=2dDt,
  15. D = s 2 / 2 τ D=s^{2}/2\tau
  16. s s
  17. τ \tau
  18. s 2 = τ r 2 t d . s^{2}=\frac{\tau\;r^{2}}{t\;d}.
  19. r = L / 10 r=L/10
  20. τ = 1 \tau=1
  21. s 0.01 L s\approx 0.01L
  22. s 0.001 L s\approx 0.001L

Cunningham_function.html

  1. ω m , n ( x ) = e - x + π i ( m / 2 - n ) Γ ( 1 + n - m / 2 ) U ( m / 2 - n , 1 + m , x ) . \displaystyle\omega_{m,n}(x)=\frac{e^{-x+\pi i(m/2-n)}}{\Gamma(1+n-m/2)}U(m/2-% n,1+m,x).
  2. x X ′′ + ( x + 1 + m ) X + ( n + 1 2 m + 1 ) X . xX^{\prime\prime}+(x+1+m)X^{\prime}+(n+\tfrac{1}{2}m+1)X.
  3. ω 2 n ( x ) = ω 0 , n ( x ) . \omega_{2n}(x)=\omega_{0,n}(x).

Cup_(disambiguation).html

  1. \smile

CURE_data_clustering_algorithm.html

  1. E = i = 1 k p C i ( p - m i ) 2 , E=\sum_{i=1}^{k}\sum_{p\in C_{i}}(p-m_{i})^{2},
  2. d m i n , d m e a n d_{min},d_{mean}

Currencies_of_the_European_Union.html

  1. S \mathrm{S}\!\!\!\|

Cybernetical_physics.html

  1. 10 - 15 10^{-15}
  2. T h o t T_{hot}
  3. T c o l d T_{cold}
  4. η C a r n o t = 1 - T c o l d T h o t . \eta_{Carnot}=1-\frac{T_{cold}}{T_{hot}}.
  5. η N C A = 1 - ( T c o l d T h o t ) \eta_{NCA}=1-\sqrt{(}\frac{T_{cold}}{T_{hot}})
  6. x ( t ) x(t)
  7. y ( t ) y(t)
  8. x * x*
  9. y * y*
  10. x ( t ) x(t)
  11. x * ( t ) x*(t)
  12. y ( t ) y(t)
  13. y * ( t ) y*(t)
  14. x * x*
  15. x * ( t ) x*(t)
  16. x * ( t ) x*(t)
  17. x * ( t ) x*(t)
  18. x * ( t ) x*(t)
  19. x * ( t ) x*(t)

Cycle_basis.html

  1. m - n + c m-n+c
  2. m m
  3. n n
  4. c c
  5. T T
  6. G G
  7. e e
  8. T T
  9. C e C_{e}
  10. e e
  11. e e
  12. T T
  13. e e
  14. m - n + c m-n+c
  15. T T
  16. e e
  17. m - n + 1 m-n+1
  18. H 2 ( S , \Z 2 ) H_{2}(S,\Z_{2})
  19. S S
  20. H 1 ( G , \Z 2 ) H_{1}(G,\Z_{2})
  21. H 1 ( G , R ) H_{1}(G,R)
  22. R R
  23. H 1 ( G , \Z ) H_{1}(G,\Z)
  24. H 1 ( G , \Z ) H_{1}(G,\Z)
  25. O ( m n ) O(mn)
  26. m m
  27. n n
  28. O ( m 2 n / log n ) O(m^{2}n/\log n)
  29. O ( n log 4 n ) O(n\log^{4}n)

Cycle_decomposition_(graph_theory).html

  1. K n K_{n}
  2. K n - I K_{n}-I
  3. m m
  4. n n
  5. 4 m n 4\leq m\leq n
  6. K n - I K_{n}-I
  7. I I
  8. m m
  9. K n - I K_{n}-I
  10. m m
  11. m m
  12. n n
  13. K n K_{n}
  14. m m
  15. K n K_{n}
  16. m m

Cycle_rank.html

  1. r ( G ) = 1 + min v V r ( G - v ) , r(G)=1+\min_{v\in V}r(G-v),\,
  2. P n P_{n}
  3. log n \lfloor\log n\rfloor
  4. ( m × n ) (m\times n)
  5. T m , n T_{m,n}
  6. r ( T n , n ) = n r(T_{n,n})=n
  7. r ( T m , n ) = min { m , n } + 1 r(T_{m,n})=\min\{m,n\}+1
  8. O ( 1.9129 n ) O(1.9129^{n})
  9. O * ( 2 n ) O^{*}(2^{n})
  10. O ( ( log n ) 3 2 ) O((\log n)^{\frac{3}{2}})
  11. ( n × n ) (n\times n)
  12. n n

Cyclic_polytope.html

  1. d \mathbb{R}^{d}
  2. 𝐱 : d , 𝐱 ( t ) := [ t , t 2 , , t d ] T \mathbf{x}:\mathbb{R}\rightarrow\mathbb{R}^{d},\mathbf{x}(t):=\begin{bmatrix}t% ,t^{2},\ldots,t^{d}\end{bmatrix}^{T}
  3. d d
  4. n n
  5. C ( n , d ) := 𝐜𝐨𝐧𝐯 { 𝐱 ( t 1 ) , 𝐱 ( t 2 ) , , 𝐱 ( t n ) } C(n,d):=\mathbf{conv}\{\mathbf{x}(t_{1}),\mathbf{x}(t_{2}),\ldots,\mathbf{x}(t% _{n})\}
  6. n > d 2 n>d\geq 2
  7. 𝐱 ( t i ) \mathbf{x}(t_{i})
  8. t 1 < t 2 < < t n t_{1}<t_{2}<\ldots<t_{n}
  9. T := { t 1 , t 2 , , t n } T:=\{t_{1},t_{2},\ldots,t_{n}\}
  10. d d
  11. T d T T_{d}\subseteq T
  12. C ( n , d ) C(n,d)
  13. T T d T\setminus T_{d}
  14. T d T_{d}
  15. ( t 1 , t 2 , , t n ) (t_{1},t_{2},\ldots,t_{n})
  16. f i ( Δ ( n , d ) ) = ( n i + 1 ) for 0 i < [ d 2 ] f_{i}(\Delta(n,d))={\left({{n}\atop{i+1}}\right)}\quad\textrm{for}\quad 0\leq i% <\left[\frac{d}{2}\right]
  17. ( f 0 , , f [ d 2 ] - 1 ) (f_{0},\ldots,f_{[\frac{d}{2}]-1})
  18. ( f [ d 2 ] , , f d - 1 ) (f_{[\frac{d}{2}]},\ldots,f_{d-1})
  19. f i ( Δ ) f i ( Δ ( n , d ) ) for i = 0 , 1 , , d - 1. f_{i}(\Delta)\leq f_{i}(\Delta(n,d))\quad\textrm{for}\quad i=0,1,\ldots,d-1.

Cylindrical_equal-area_projection.html

  1. x = ( λ - λ 0 ) cos ϕ 0 x=(\lambda-\lambda_{0})\cos\phi_{0}\,
  2. y = sin ϕ / cos ϕ 0 y=\sin\phi/\cos\phi_{0}\,
  3. λ \lambda\,
  4. λ 0 \lambda_{0}\,
  5. ϕ \phi\,
  6. ϕ 0 \phi_{0}\,
  7. x = ( λ - λ 0 ) ( cos ϕ 0 ) π / 180 x=(\lambda-\lambda_{0})(\cos\phi_{0})\pi/180^{\circ}
  8. y = sin ϕ / cos ϕ 0 y=\sin\phi/\cos\phi_{0}\,
  9. x = ( λ - λ 0 ) S x=(\lambda-\lambda_{0})S\,
  10. y = sin ϕ y=\sin\phi\,
  11. ϕ 0 \phi_{0}\,
  12. π ( cos ϕ 0 ) 2 \pi(\cos\phi_{0})^{2}\,
  13. π 3.14 \pi\approx 3.14
  14. 3 π / 4 2.36 3\pi/4\approx 2.36\approx
  15. δ S \delta_{S}
  16. 2 2
  17. 37 04 17 ′′ arccos ( 2 / π ) 37^{\circ}04^{\prime}17^{\prime\prime}\approx\arccos(\sqrt{2/\pi})
  18. 2 \approx 2
  19. 2 \approx 2
  20. π / 2 1.57 \pi/2\approx 1.57
  21. 1.3 \approx 1.3
  22. 1 1
  23. 55 39 14 ′′ arccos ( 1 / π ) 55^{\circ}39^{\prime}14^{\prime\prime}\approx\arccos(\sqrt{1/\pi})

Danny_Calegari.html

  1. \mathbb{R}

Data_stream_clustering.html

  1. \ell
  2. \ell
  3. \ell
  4. \ell
  5. \ell
  6. \ell

Data_validation_and_reconciliation.html

  1. F ( y ) = 0 F(y)=0\,
  2. y = ( y 1 , , y n ) y=(y_{1},\ldots,y_{n})
  3. y y\,
  4. F ( y ) = 0 F(y)=0\,\!
  5. y y\,\!
  6. y * y^{*}\,\!
  7. y * y^{*}\,\!
  8. y y\,\!
  9. y ¯ \bar{y}\,\!
  10. y * y^{*}\,
  11. n n
  12. y i y_{i}
  13. min x , y * i = 1 n ( y i * - y i σ i ) 2 subject to F ( x , y * ) = 0 y min y * y max x min x x max , \begin{aligned}\displaystyle\min_{x,y^{*}}&\displaystyle\sum_{i=1}^{n}\left(% \frac{y_{i}^{*}-y_{i}}{\sigma_{i}}\right)^{2}\\ \displaystyle\,\text{subject to }&\displaystyle F(x,y^{*})=0\\ &\displaystyle y_{\min}\leq y^{*}\leq y_{\max}\\ &\displaystyle x_{\min}\leq x\leq x_{\max},\end{aligned}\,\!
  14. y i * y_{i}^{*}\,\!
  15. i i
  16. i = 1 , , n i=1,\ldots,n\,\!
  17. y i y_{i}\,\!
  18. i i
  19. i = 1 , , n i=1,\ldots,n\,\!
  20. x j x_{j}\,\!
  21. j j
  22. j = 1 , , m j=1,\ldots,m\,\!
  23. σ i \sigma_{i}\,\!
  24. i i
  25. i = 1 , , n i=1,\ldots,n\,\!
  26. F ( x , y * ) = 0 F(x,y^{*})=0\,\!
  27. p p\,\!
  28. x min , x max , y min , y max x_{\min},x_{\max},y_{\min},y_{\max}\,\!
  29. ( y i * - y i σ i ) 2 \left(\frac{y_{i}^{*}-y_{i}}{\sigma_{i}}\right)^{2}\,\!
  30. f ( y * ) = i = 1 n ( y i * - y i σ i ) 2 f(y^{*})=\sum_{i=1}^{n}\left(\frac{y_{i}^{*}-y_{i}}{\sigma_{i}}\right)^{2}
  31. a = b + c a=b+c\,\!
  32. c c\,\!
  33. a a\,\!
  34. b b\,\!
  35. d o f dof\,\!
  36. a = b + c a=b+c\,
  37. x x\,\!
  38. y y\,\!
  39. F ( x , y ) = 0 F(x,y)=0\,\!
  40. y y\,\!
  41. x x\,\!
  42. F ( x , y ) = 0 F(x,y)=0\,\!
  43. n n\,
  44. r e d = n - d o f red=n-dof\,\!
  45. d o f dof\,
  46. r e d = n - d o f = n - ( n + m - p ) = p - m , \begin{aligned}\displaystyle red=n-dof=n-(n+m-p)=p-m,\end{aligned}
  47. p p\,
  48. m m\,
  49. d d\,\!
  50. c c\,\!
  51. a a\,\!
  52. b b\,\!
  53. c c\,\!
  54. a a\,\!
  55. b b\,\!
  56. a + b = c a+b=c\,\!
  57. c = d c=d\,\!
  58. F ( x , y ) = 0 F(x,y)=0\,\!
  59. p - m 0 p-m\geq 0\,\!
  60. c c\,\!
  61. d d\,\!
  62. a a\,\!
  63. b b\,\!
  64. c c\,\!
  65. a a\,\!
  66. b b\,\!
  67. a a\,\!
  68. c c\,\!
  69. b b\,\!
  70. d d\,\!
  71. y y\,\!
  72. x x\,\!
  73. f ( y * ) f(y^{*})\,\!
  74. P α P_{\alpha}\,
  75. f ( y * ) P 95 f(y^{*})\leq P_{95}
  76. i i
  77. p - m p-m
  78. p p

David_Mount.html

  1. ϵ \epsilon
  2. ( 1 + ϵ ) (1+\epsilon)
  3. n n
  4. O ( n l o g n ) O(nlogn)
  5. O ( n ) O(n)
  6. H + O ( H + 1 ) H+O(\sqrt{H}+1)
  7. H H
  8. O ( c d , ϵ l o g ( n ) ) O(c_{d,\epsilon}log(n))
  9. c d , ϵ c_{d,\epsilon}
  10. d d
  11. ϵ \epsilon
  12. ( 1 + ϵ ) (1+\epsilon)
  13. O ( n 2 l o g ( n ) ) O(n^{2}log(n))
  14. O ( l o g n ) O(logn)
  15. n n

David_Shmoys.html

  1. n n
  2. J J
  3. m m
  4. M M
  5. j j
  6. p i , j p_{i,j}
  7. i i
  8. c i , j , i = 1 , 2 , . . , m ; j = 1 , 2 , . . , n ; n m c_{i,j},i=1,2,..,m;j=1,2,..,n;n\geq m
  9. C C
  10. T i , i = 1 , 2 , . . , m T_{i},i=1,2,..,m
  11. C C
  12. i i
  13. T i , i = 1 , 2 , . . , m T_{i},i=1,2,..,m
  14. T 1 = T 2 = . . = T m = T T_{1}=T_{2}=..=T_{m}=T
  15. T T
  16. T T
  17. L P ( T ) : : i = 1 m j = 1 n c i j x i j C LP(T)::\sum_{i=1}^{m}\sum_{j=1}^{n}c_{ij}x_{ij}\leq C
  18. i = 1 m x i j = 1 j = 1 , , n \sum_{i=1}^{m}x_{ij}=1\qquad j=1,\ldots,n
  19. i = 1 m p i j x i j T i = 1 , , m \sum_{i=1}^{m}p_{ij}x_{ij}\leq T\qquad i=1,\ldots,m
  20. x i j 0 i = 1 , , m , j = 1 , , n x_{ij}\geq 0\qquad i=1,\ldots,m,\quad j=1,\ldots,n
  21. x i j = 0 if p i j T , i = 1 , , m , j = 1 , , n x_{ij}=0\qquad\,\text{if}\qquad p_{ij}\geq T,\qquad i=1,\ldots,m,\quad j=1,% \ldots,n
  22. G = ( W V , E ) G=(W\cup V,E)
  23. W = { w j | j J } W=\{w_{j}|j\in J\}
  24. V = { v i , s | i = 1 , 2 , . . , m ; s = 1 , 2 , . . , k i } V=\{v_{i,s}|i=1,2,..,m;s=1,2,..,k_{i}\}
  25. k i = j x i j k_{i}=\lceil\sum_{j}x_{ij}\rceil
  26. i i
  27. p i j p_{ij}
  28. p i 1 p i 2 p i n p_{i1}\geq p_{i2}\geq\ldots\geq p_{in}
  29. j 1 j_{1}
  30. i j 1 x i j 1 \sum_{i}^{j_{1}}x_{ij}\geq 1
  31. E E
  32. ( w j , v i 1 , j = 1 , 2 , . . , j 1 - 1 ) (w_{j},v_{i1},j=1,2,..,j_{1}-1)
  33. x i j x_{ij}
  34. x v i 1 j = x i j x^{\prime}_{v_{i1}j}=x_{ij}
  35. ( w j 1 , v i 1 ) (w_{j_{1}},v_{i1})
  36. x v i 1 j 1 = 1 - i = 1 j 1 - 1 x v i 1 j x^{\prime}_{v_{i1}j_{1}}=1-\sum_{i=1}^{j_{1}-1}x^{\prime}_{v_{i1}j}
  37. v i 1 v_{i1}
  38. x v i 1 j 1 < x i j 1 x^{\prime}_{v_{i1}j_{1}}<x_{ij_{1}}
  39. ( w j 1 , v i 2 ) (w_{j_{1}},v_{i2})
  40. x v i 2 j 1 = x i j - x v i 1 j 1 x^{\prime}_{v_{i2}j_{1}}=x_{ij}-x^{\prime}_{v_{i1}j_{1}}
  41. v i 2 v_{i2}
  42. W W
  43. V V
  44. x x^{\prime}
  45. G G
  46. G G
  47. L P ( T ) LP(T)
  48. T + m a x i , j p i , j T+max_{i,j}p_{i,j}
  49. C C
  50. m a x i , j p i , j T max_{i,j}p_{i,j}\leq T
  51. 6 2 3 6\frac{2}{3}
  52. O ( log k log log k ) O(\log{k}\ \log{\log{k}})
  53. 3 3
  54. 5.69 5.69
  55. ( 1 + 2 / e ) 1.736 (1+2/e)\approx 1.736

Davies_equation.html

  1. f ± f_{\pm}
  2. - log f ± = 0.5 z 1 z 2 ( I 1 + I - 0.15 I ) -\log f_{\pm}=0.5z_{1}z_{2}\left(\frac{\sqrt{I}}{1+\sqrt{I}}-0.15I\right)

Davies–Bouldin_index.html

  1. S i = 1 T i j = 1 T i || X j - A i || p S_{i}=\frac{1}{T_{i}}\sum_{j=1}^{T_{i}}{\left|\left|X_{j}-A_{i}\right|\right|_% {p}}
  2. A i A_{i}
  3. M i , j = || A i - A j || p = ( k = 1 n | a k , i - a k , j | p ) 1 p M_{i,j}=\left|\left|A_{i}-A_{j}\right|\right|_{p}=\Bigl(\displaystyle\sum_{k=1% }^{n}\left|a_{k,i}-a_{k,j}\right|^{p}\Bigr)^{\frac{1}{p}}
  4. M i , j M_{i,j}
  5. C i C_{i}
  6. C j C_{j}
  7. a k , i a_{k,i}
  8. A i A_{i}
  9. R i , j 0 R_{i,j}\geqslant 0
  10. R i , j = R j , i R_{i,j}=R_{j,i}
  11. S j S k S_{j}\geqslant S_{k}
  12. M i , j = M i , k M_{i,j}=M_{i,k}
  13. R i , j > R i , k R_{i,j}>R_{i,k}
  14. S j = S k S_{j}=S_{k}
  15. M i , j M i , k M_{i,j}\leqslant M_{i,k}
  16. R i , j > R i , k R_{i,j}>R_{i,k}
  17. R i , j = S i + S j M i , j R_{i,j}=\frac{S_{i}+S_{j}}{M_{i,j}}
  18. D i max j i R i , j D_{i}\equiv\max_{j\neq i}R_{i,j}
  19. 𝐷𝐵 1 N i = 1 N D i \mathit{DB}\equiv\frac{1}{N}\displaystyle\sum_{i=1}^{N}D_{i}

Days_in_inventory.html

  1. D I I = a v e r a g e i n v e n t o r y C O G S / D a y s DII=\dfrac{average~{}inventory}{COGS/Days}
  2. Average days to sell the inventory = 365 days Inventory Turnover Ratio \mbox{Average days to sell the inventory}~{}=\frac{\mbox{365 days}~{}}{\mbox{% Inventory Turnover Ratio}~{}}

Days_payable_outstanding.html

  1. D P O = e n d i n g A / P P u r c h a s e / d a y DPO=\dfrac{ending~{}A/P}{Purchase/day}

De_Bruijn's_theorem.html

  1. 1 × 2 × 4 \scriptstyle 1\times 2\times 4
  2. 6 × 6 × 6 \scriptstyle 6\times 6\times 6
  3. 27 \scriptstyle 27
  4. 26 \scriptstyle 26
  5. 27 \scriptstyle 27
  6. 1 × 2 × 4 \scriptstyle 1\times 2\times 4
  7. d \scriptstyle d
  8. A 1 × A 2 × × A d \scriptstyle A_{1}\times A_{2}\times\dots\times A_{d}
  9. a 1 × a 2 × × a d \scriptstyle a_{1}\times a_{2}\times\dots\times a_{d}
  10. b i \scriptstyle b_{i}
  11. a 1 b 1 , a 2 b 2 , a d b d \scriptstyle a_{1}b_{1},a_{2}b_{2},\dots a_{d}b_{d}
  12. A 1 , A 2 , , A d \scriptstyle A_{1},A_{2},\dots,A_{d}
  13. 5 × 6 \scriptstyle 5\times 6
  14. 2 × 3 \scriptstyle 2\times 3
  15. a i , a_{i},
  16. A i A_{i}
  17. 6 6
  18. 2 2
  19. 3 3
  20. 2 1 4 × 4 × 8 \scriptstyle 2\frac{1}{4}\times 4\times 8
  21. 2 × 4 × 12 \scriptstyle 2\times 4\times 12
  22. 2 × 3 \scriptstyle 2\times 3
  23. 5 × 6 \scriptstyle 5\times 6
  24. A 1 = a 1 \scriptstyle A_{1}=a_{1}
  25. A 2 = a 1 a 2 \scriptstyle A_{2}=a_{1}a_{2}
  26. a 1 \scriptstyle a_{1}
  27. a 1 , a 2 \scriptstyle a_{1},a_{2}
  28. A 1 = a 1 a 2 \scriptstyle A_{1}=a_{1}a_{2}
  29. A 2 = a 2 \scriptstyle A_{2}=a_{2}
  30. A 1 = a 1 + a 2 \scriptstyle A_{1}=a_{1}+a_{2}
  31. A 2 = a 1 a 2 \scriptstyle A_{2}=a_{1}a_{2}

De_Rham_invariant.html

  1. 𝐙 / 2 \mathbf{Z}/2
  2. L 4 k + 1 , L^{4k+1},
  3. L 4 k L 4 k L^{4k}\cong L_{4k}
  4. L 4 k + 2 . L_{4k+2}.
  5. H 2 k ( M ) , H_{2k}(M),
  6. w 2 w 4 k - 1 w_{2}w_{4k-1}
  7. v 2 k S q 1 v 2 k , v_{2k}Sq^{1}v_{2k},
  8. v 2 k H 2 k ( M ; Z 2 ) v_{2k}\in H^{2k}(M;Z_{2})
  9. M M
  10. S q 1 Sq^{1}
  11. ( v 2 k S q 1 v 2 k , [ M ] ) (v_{2k}Sq^{1}v_{2k},[M])

De_Sitter–Schwarzschild_metric.html

  1. d s 2 = - f ( r ) d t 2 + d r 2 f ( r ) + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}=-f(r)dt^{2}+{dr^{2}\over f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^% {2})\,
  2. f ( r ) = 1 - 2 a / r f(r)=1-2a/r\,
  3. f ( r ) = 1 - b r 2 f(r)=1-br^{2}\,
  4. f ( r ) = 1 - 2 a r - b r 2 f(r)=1-{2a\over r}-br^{2}\,
  5. d s 2 = - f ( r ) d t 2 + d r 2 f ( r ) + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}=-f(r)\,dt^{2}+{dr^{2}\over f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d% \phi^{2})\,
  6. f ( r ) = 1 - 2 a r - b r 2 f(r)=1-{2a\over r}-br^{2}\,
  7. f ( r ) = u 2 - ϵ 2 R 2 f(r)={u^{2}-\epsilon^{2}\over R^{2}}\,
  8. d s 2 = - ( R 2 - z 2 ) d t 2 + d z 2 ( R 2 - z 2 ) + R 2 d Ω 2 ds^{2}=-(R^{2}-z^{2})\,dt^{2}+{dz^{2}\over(R^{2}-z^{2})}+R^{2}\,d\Omega^{2}\,
  9. d S 2 × S 2 dS_{2}\times S_{2}
  10. d S 2 = - d t 2 + cosh 2 t d x 2 + R 2 d Ω 2 dS^{2}=-dt^{2}+\cosh^{2}t\,dx^{2}+R^{2}\,d\Omega^{2}\,

Decimal128_floating-point_format.html

  1. ( - 1 ) signbit × 10 exponentbits 2 - 6176 10 × truesignificand 10 (-1)\text{signbit}\times 10^{\,\text{exponentbits}_{2}-6176_{10}}\times\,\text% {truesignificand}_{10}

Decimal32_floating-point_format.html

  1. ( - 1 ) signbit × 10 exponentbits 2 - 101 10 × truesignificand 10 (-1)\text{signbit}\times 10^{\,\text{exponentbits}_{2}-101_{10}}\times\,\text{% truesignificand}_{10}

Decimal64_floating-point_format.html

  1. ( - 1 ) signbit × 10 exponentbits 2 - 398 10 × truesignificand 10 (-1)\text{signbit}\times 10^{\,\text{exponentbits}_{2}-398_{10}}\times\,\text{% truesignificand}_{10}

Decision-theoretic_rough_sets.html

  1. α \textstyle\alpha
  2. β \textstyle\beta
  3. A = { a 1 , , a m } \textstyle A=\{a_{1},\ldots,a_{m}\}
  4. m \textstyle m
  5. Ω = { w 1 , , w s } \textstyle\Omega=\{w_{1},\ldots,w_{s}\}
  6. s s
  7. P ( w j [ x ] ) \textstyle P(w_{j}\mid[x])
  8. x \textstyle x
  9. w j \textstyle w_{j}
  10. [ x ] \textstyle[x]
  11. λ ( a i w j ) \textstyle\lambda(a_{i}\mid w_{j})
  12. a i \textstyle a_{i}
  13. w j \textstyle w_{j}
  14. a i \textstyle a_{i}
  15. R ( a i [ x ] ) = j = 1 s λ ( a i w j ) P ( w j [ x ] ) . R(a_{i}\mid[x])=\sum_{j=1}^{s}\lambda(a_{i}\mid w_{j})P(w_{j}\mid[x]).
  16. A = { a P , a N , a B } \textstyle A=\{a_{P},a_{N},a_{B}\}
  17. a P \textstyle a_{P}
  18. a N \textstyle a_{N}
  19. a B \textstyle a_{B}
  20. A \textstyle A
  21. A \textstyle A
  22. A \textstyle A
  23. A \textstyle A
  24. A \textstyle A
  25. Ω = { A , A c } \textstyle\Omega=\{A,A^{c}\}
  26. λ ( a A ) \textstyle\lambda(a_{\diamond}\mid A)
  27. a \textstyle a_{\diamond}
  28. A \textstyle A
  29. λ ( a A c ) \textstyle\lambda(a_{\diamond}\mid A^{c})
  30. A c \textstyle A^{c}
  31. λ P P \textstyle\lambda_{PP}
  32. A \textstyle A
  33. λ B P \textstyle\lambda_{BP}
  34. A \textstyle A
  35. λ N P \textstyle\lambda_{NP}
  36. A \textstyle A
  37. λ N \textstyle\lambda_{\diamond N}
  38. A \textstyle A
  39. \textstyle\diamond
  40. R ( a [ x ] ) \textstyle R(a_{\diamond}\mid[x])
  41. R ( a P [ x ] ) = λ P P P ( A [ x ] ) + λ P N P ( A c [ x ] ) , \textstyle R(a_{P}\mid[x])=\lambda_{PP}P(A\mid[x])+\lambda_{PN}P(A^{c}\mid[x]),
  42. R ( a N [ x ] ) = λ N P P ( A [ x ] ) + λ N N P ( A c [ x ] ) , \textstyle R(a_{N}\mid[x])=\lambda_{NP}P(A\mid[x])+\lambda_{NN}P(A^{c}\mid[x]),
  43. R ( a B [ x ] ) = λ B P P ( A [ x ] ) + λ B N P ( A c [ x ] ) , \textstyle R(a_{B}\mid[x])=\lambda_{BP}P(A\mid[x])+\lambda_{BN}P(A^{c}\mid[x]),
  44. λ P = λ ( a A ) \textstyle\lambda_{\diamond P}=\lambda(a_{\diamond}\mid A)
  45. λ N = λ ( a A c ) \textstyle\lambda_{\diamond N}=\lambda(a_{\diamond}\mid A^{c})
  46. = P \textstyle\diamond=P
  47. N \textstyle N
  48. B \textstyle B
  49. λ P P λ B P < λ N P \textstyle\lambda_{PP}\leq\lambda_{BP}<\lambda_{NP}
  50. λ N N λ B N < λ P N \textstyle\lambda_{NN}\leq\lambda_{BN}<\lambda_{PN}
  51. P ( A [ x ] ) γ \textstyle P(A\mid[x])\geq\gamma
  52. P ( A [ x ] ) α \textstyle P(A\mid[x])\geq\alpha
  53. A \textstyle A
  54. P ( A [ x ] ) β \textstyle P(A\mid[x])\leq\beta
  55. P ( A [ x ] ) γ \textstyle P(A\mid[x])\leq\gamma
  56. A \textstyle A
  57. β P ( A [ x ] ) α \textstyle\beta\leq P(A\mid[x])\leq\alpha
  58. A \textstyle A
  59. α = λ P N - λ B N ( λ B P - λ B N ) - ( λ P P - λ P N ) , \alpha=\frac{\lambda_{PN}-\lambda_{BN}}{(\lambda_{BP}-\lambda_{BN})-(\lambda_{% PP}-\lambda_{PN})},
  60. γ = λ P N - λ N N ( λ N P - λ N N ) - ( λ P P - λ P N ) , \gamma=\frac{\lambda_{PN}-\lambda_{NN}}{(\lambda_{NP}-\lambda_{NN})-(\lambda_{% PP}-\lambda_{PN})},
  61. β = λ B N - λ N N ( λ N P - λ N N ) - ( λ B P - λ B N ) . \beta=\frac{\lambda_{BN}-\lambda_{NN}}{(\lambda_{NP}-\lambda_{NN})-(\lambda_{% BP}-\lambda_{BN})}.
  62. α \textstyle\alpha
  63. β \textstyle\beta
  64. γ \textstyle\gamma
  65. α > β \textstyle\alpha>\beta
  66. α > γ > β \textstyle\alpha>\gamma>\beta
  67. P ( A [ x ] ) α \textstyle P(A\mid[x])\geq\alpha
  68. A \textstyle A
  69. P ( A [ x ] ) β \textstyle P(A\mid[x])\leq\beta
  70. A \textstyle A
  71. β < P ( A [ x ] ) < α \textstyle\beta<P(A\mid[x])<\alpha
  72. A \textstyle A
  73. α = β = γ \textstyle\alpha=\beta=\gamma
  74. α \textstyle\alpha
  75. P ( A [ x ] ) > α \textstyle P(A\mid[x])>\alpha
  76. A \textstyle A
  77. P ( A [ x ] ) < α \textstyle P(A\mid[x])<\alpha
  78. A \textstyle A
  79. P ( A [ x ] ) = α \textstyle P(A\mid[x])=\alpha
  80. A \textstyle A

Decision_rule.html

  1. ( 𝒳 , Σ , P θ ) \scriptstyle(\mathcal{X},\Sigma,P_{\theta})
  2. 𝒳 \scriptstyle\mathcal{X}
  3. θ \theta
  4. θ \theta
  5. \mathcal{R}
  6. θ \theta
  7. θ \theta
  8. θ ^ \hat{\theta}
  9. θ ^ \hat{\theta}
  10. θ ^ \hat{\theta}

Decision_tree_model.html

  1. f : { 0 , 1 } n { 0 , 1 } f:\{0,1\}^{n}\rightarrow\{0,1\}
  2. log n \log n
  3. Ω ( log n ) \Omega(\log n)
  4. Ω ( n log n ) \Omega(n\log n)
  5. f ( x 1 , , x i ) f(x_{1},\dots,x_{i})
  6. f ( x ) f(x)
  7. f ( x ) f(x)
  8. x { 0 , 1 } n x\in\{0,1\}^{n}
  9. f f
  10. D ( f ) D(f)
  11. f f
  12. f f
  13. p i p_{i}
  14. R 2 ( f ) R_{2}(f)
  15. f ( x ) f(x)
  16. 2 / 3 2/3
  17. x { 0 , 1 } n x\in\{0,1\}^{n}
  18. R 2 ( f ) R_{2}(f)
  19. R 0 ( f ) R_{0}(f)
  20. R 1 ( f ) R_{1}(f)
  21. Q 2 ( f ) Q_{2}(f)
  22. f ( x ) f(x)
  23. 2 / 3 2/3
  24. x { 0 , 1 } n x\in\{0,1\}^{n}
  25. Q E ( f ) Q_{E}(f)
  26. f ( x ) f(x)
  27. f f
  28. Q 2 ( f ) Q_{2}(f)
  29. Q E ( f ) Q_{E}(f)
  30. Q 0 ( f ) Q_{0}(f)
  31. Q 1 ( f ) Q_{1}(f)
  32. n n
  33. f f
  34. Q 2 ( f ) R 2 ( f ) R 1 ( f ) R 0 ( f ) D ( f ) n Q_{2}(f)\leq R_{2}(f)\leq R_{1}(f)\leq R_{0}(f)\leq D(f)\leq n
  35. Q 2 ( f ) Q E ( f ) D ( f ) n Q_{2}(f)\leq Q_{E}(f)\leq D(f)\leq n
  36. D ( f ) R 0 ( f ) 2 D(f)\leq R_{0}(f)^{2}
  37. D ( f ) = O ( R 2 ( f ) 3 ) D(f)=O(R_{2}(f)^{3})
  38. D ( f ) = O ( R 1 ( f ) 2 ) D(f)=O(R_{1}(f)^{2})
  39. R 0 ( f ) = O ( R 2 ( f ) 3 ) R_{0}(f)=O(R_{2}(f)^{3})
  40. R 0 ( f ) R_{0}(f)
  41. R 2 ( f ) R_{2}(f)
  42. Q 2 ( f ) Q_{2}(f)
  43. D ( f ) D(f)
  44. D ( f ) = O ( Q E ( f ) 3 ) D(f)=O(Q_{E}(f)^{3})
  45. D ( f ) = O ( Q 2 ( f ) 6 ) D(f)=O(Q_{2}(f)^{6})
  46. D ( O R n ) = n D(OR_{n})=n
  47. Q 2 ( O R n ) = Θ ( n ) Q_{2}(OR_{n})=\Theta(\sqrt{n})
  48. { 0 , 1 } n \{0,1\}^{n}
  49. Q E ( f ) Q_{E}(f)
  50. D ( f ) D(f)
  51. R 2 ( f ) R_{2}(f)
  52. D ( f ) D(f)
  53. D ( f ) R 0 ( f ) 2 D(f)\leq R_{0}(f)^{2}
  54. D ( f ) R 1 ( f ) R 2 ( f ) D(f)\leq R_{1}(f)R_{2}(f)
  55. D ( f ) R 2 ( f ) 3 D(f)\leq R_{2}(f)^{3}
  56. R 0 ( f ) R 2 ( f ) 2 log N R_{0}(f)\leq R_{2}(f)^{2}\log N
  57. D ( f ) Q 2 ( f ) 6 D(f)\leq Q_{2}(f)^{6}
  58. D ( f ) Q E ( f ) 2 Q 2 ( f ) 2 D(f)\leq Q_{E}(f)^{2}Q_{2}(f)^{2}
  59. D ( f ) Q 1 ( f ) 2 Q 2 ( f ) 2 D(f)\leq Q_{1}(f)^{2}Q_{2}(f)^{2}
  60. R 0 ( f ) Q 1 ( f ) Q 2 ( f ) 2 log N R_{0}(f)\leq Q_{1}(f)Q_{2}(f)^{2}\log N
  61. D ( f ) Q 1 ( f ) Q 2 ( f ) 2 D(f)\leq Q_{1}(f)Q_{2}(f)^{2}

Decisional_composite_residuosity_assumption.html

  1. z y n ( mod n 2 ) . z\equiv y^{n}\;\;(\mathop{{\rm mod}}n^{2}).\,

Dedekind_number.html

  1. M ( n ) = k = 1 2 2 n j = 1 2 n - 1 i = 0 j - 1 ( 1 - b i k b j k m = 0 log 2 i ( 1 - b m i + b m i b m j ) ) , M(n)=\sum_{k=1}^{2^{2^{n}}}\prod_{j=1}^{2^{n}-1}\prod_{i=0}^{j-1}\left(1-b_{i}% ^{k}b_{j}^{k}\prod_{m=0}^{\log_{2}i}(1-b_{m}^{i}+b_{m}^{i}b_{m}^{j})\right),
  2. b i k b_{i}^{k}
  3. i i
  4. k k
  5. b i k = k 2 i - 2 k 2 i + 1 . b_{i}^{k}=\left\lfloor\frac{k}{2^{i}}\right\rfloor-2\left\lfloor\frac{k}{2^{i+% 1}}\right\rfloor.
  6. ( n n / 2 ) log 2 M ( n ) ( n n / 2 ) ( 1 + O ( log n n ) ) . {n\choose\lfloor n/2\rfloor}\leq\log_{2}M(n)\leq{n\choose\lfloor n/2\rfloor}% \left(1+O\left(\frac{\log n}{n}\right)\right).
  7. n / 2 \lfloor n/2\rfloor
  8. M ( n ) = ( 1 + o ( 1 ) ) 2 ( n n / 2 ) exp a ( n ) M(n)=(1+o(1))2^{n\choose\lfloor n/2\rfloor}\exp a(n)
  9. M ( n ) = ( 1 + o ( 1 ) ) 2 ( n n / 2 + 1 ) exp ( b ( n ) + c ( n ) ) M(n)=(1+o(1))2^{n\choose\lfloor n/2\rfloor+1}\exp(b(n)+c(n))
  10. a ( n ) = ( n n / 2 - 1 ) ( 2 - n / 2 + n 2 2 - n - 5 - n 2 - n - 4 ) , a(n)={n\choose n/2-1}(2^{-n/2}+n^{2}2^{-n-5}-n2^{-n-4}),
  11. b ( n ) = ( n ( n - 3 ) / 2 ) ( 2 - ( n + 3 ) / 2 + n 2 2 - n - 6 - n 2 - n - 3 ) , b(n)={n\choose(n-3)/2}(2^{-(n+3)/2}+n^{2}2^{-n-6}-n2^{-n-3}),
  12. c ( n ) = ( n ( n - 1 ) / 2 ) ( 2 - ( n + 1 ) / 2 + n 2 2 - n - 4 ) . c(n)={n\choose(n-1)/2}(2^{-(n+1)/2}+n^{2}2^{-n-4}).

Degrees_of_freedom_(physics_and_chemistry).html

  1. x x
  2. y y
  3. z z
  4. 3 N = 6 = 3 + 2 + 1. 3N=6=3+2+1.
  5. N > 2 N>2
  6. 3 N = 3 + 3 + ( 3 N - 6 ) 3N=3+3+(3N-6)
  7. N N
  8. 3 N 6 3N−6
  9. N > 2 N>2
  10. d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}
  11. x x
  12. y y
  13. z z
  14. x x
  15. y y
  16. z z
  17. 3 N 5 3N−5
  18. 3 N 6 3N−6
  19. E = i = 1 N E i ( X i ) , E=\sum_{i=1}^{N}E_{i}(X_{i}),
  20. E E
  21. E = X 1 4 + X 2 4 E=X_{1}^{4}+X_{2}^{4}
  22. E = X 1 4 + X 1 X 2 + X 2 4 E=X_{1}^{4}+X_{1}X_{2}+X_{2}^{4}
  23. i i
  24. N N
  25. i i
  26. p i ( X i ) = e - E i k B T d X i e - E i k B T p_{i}(X_{i})=\frac{e^{-\frac{E_{i}}{k_{B}T}}}{\int dX_{i}\,e^{-\frac{E_{i}}{k_% {B}T}}}
  27. \langle\rangle
  28. E = i = 1 N E i . \langle E\rangle=\sum_{i=1}^{N}\langle E_{i}\rangle.
  29. E = α i X i 2 + β i X i Y E=\alpha_{i}\,\,X_{i}^{2}+\beta_{i}\,\,X_{i}Y
  30. Y Y
  31. E E
  32. E = X 1 4 + X 1 3 X 2 + X 2 4 E=X_{1}^{4}+X_{1}^{3}X_{2}+X_{2}^{4}
  33. E = X 1 4 + X 2 4 E=X_{1}^{4}+X_{2}^{4}
  34. E = X 1 2 + X 1 X 2 + 2 X 2 2 E=X_{1}^{2}+X_{1}X_{2}+2X_{2}^{2}
  35. E = X 1 2 + 2 X 2 2 E=X_{1}^{2}+2X_{2}^{2}
  36. E = i = 1 N α i X i 2 E=\sum_{i=1}^{N}\alpha_{i}X_{i}^{2}
  37. N N
  38. U = E = N k B T 2 U=\langle E\rangle=N\,\frac{k_{B}T}{2}
  39. E i = d X i α i X i 2 p i ( X i ) = d X i α i X i 2 e - α i X i 2 k B T d X i e - α i X i 2 k B T \langle E_{i}\rangle=\int dX_{i}\,\,\alpha_{i}X_{i}^{2}\,\,p_{i}(X_{i})=\frac{% \int dX_{i}\,\,\alpha_{i}X_{i}^{2}\,\,e^{-\frac{\alpha_{i}X_{i}^{2}}{k_{B}T}}}% {\int dX_{i}\,\,e^{-\frac{\alpha_{i}X_{i}^{2}}{k_{B}T}}}
  40. E i = k B T 2 d x x 2 e - x 2 2 d x e - x 2 2 = k B T 2 \langle E_{i}\rangle=\frac{k_{B}T}{2}\frac{\int dx\,\,x^{2}\,\,e^{-\frac{x^{2}% }{2}}}{\int dx\,\,e^{-\frac{x^{2}}{2}}}=\frac{k_{B}T}{2}

Delivery_Performance.html

  1. D P T V DP_{T}^{V}
  2. D P D S DP_{D}^{S}
  3. D P T S DP_{T}^{S}
  4. ( D e m a n d p , c + B a c k l o g p - 1 , c ) > 0 (Demand_{p,c}+Backlog_{p-1,c})>0
  5. D P T V DP_{T}^{V}
  6. D e l i v e r e d p , c + P r e d e l i v e r y p - 1 , c D e m a n d p , c + B a c k l o g p - 1 , c \frac{Delivered_{p,c}+Predelivery_{p-1,c}}{Demand_{p,c}+Backlog_{p-1,c}}
  7. D P p , c = p , c ( D P T V ) c o u n t p , c ( D P T V < > N U L L ) DP_{p,c}=\frac{\sum_{p,c}(DP_{T}^{V})}{count_{p,c}(DP_{T}^{V}<>NULL)}
  8. D P * S DP_{*}^{S}
  9. D P D S DP_{D}^{S}
  10. D P p , c = p , c ( D P ) c o u n t p , c ( s i n g u l a r c a s e s ) DP_{p,c}=\frac{\sum_{p,c}(DP)}{count_{p,c}(singularcases)}
  11. D P T S DP_{T}^{S}
  12. D P p , c = p , c ( D P ) c o u n t p , c ( s i n g u l a r c a s e s ) DP_{p,c}=\frac{\sum_{p,c}(DP)}{count_{p,c}(singularcases)}
  13. D P T / D S / V DP_{T/D}^{S/V}

Delivery_Reliability.html

  1. D R T V DR_{T}^{V}
  2. D R D S DR_{D}^{S}
  3. D R T S DR_{T}^{S}
  4. D e m a n d p , c + B a c k l o g p - 1 , c > 0 Demand_{p,c}+Backlog_{p-1,c}>0
  5. D R T V DR_{T}^{V}
  6. m i n ( D e l i v e r e d p , c + P r e d e l i v e r y p - 1 , c D e m a n d p , c + B a c k l o g p - 1 , c , 1 ) min(\frac{Delivered_{p,c}+Predelivery_{p-1,c}}{Demand_{p,c}+Backlog_{p-1,c}},1)
  7. D R p , c = p , c ( D R T V ) c o u n t p , c ( D R T V < > N U L L ) DR_{p,c}=\frac{\sum_{p,c}(DR_{T}^{V})}{count_{p,c}(DR_{T}^{V}<>NULL)}
  8. D R * S DR_{*}^{S}
  9. D R D S DR_{D}^{S}
  10. D R p , c = p , c ( D R ) c o u n t p , c ( s i n g u l a r c a s e s ) DR_{p,c}=\frac{\sum_{p,c}(DR)}{count_{p,c}(singularcases)}
  11. D R T S DR_{T}^{S}
  12. D R p , c = p , c ( D R ) c o u n t p , c ( s i n g u l a r c a s e s ) DR_{p,c}=\frac{\sum_{p,c}(DR)}{count_{p,c}(singularcases)}
  13. D R T / D S / V DR_{T/D}^{S/V}

Demonic_composition.html

  1. { ( x , z ) | x ( S R ) z y Y ( x R y y S z ) } . \{(x,z)\ |\ x\,(S\circ R)\,z\wedge\forall y\in Y\ (x\,R\,y\Rightarrow y\,S\,z)\}.
  2. { ( x , z ) | x ( S R ) z y Y ( y S z x R y ) } . \{(x,z)\ |\ x\,(S\circ R)\,z\wedge\forall y\in Y\ (y\,S\,z\Rightarrow x\,R\,y)\}.

Dense_set.html

  1. A ¯ \displaystyle\overline{A}
  2. A ¯ = A { lim n a n : n 0 , a n A } \overline{A}=A\cup\{\lim_{n}a_{n}:\forall n\geq 0,\ a_{n}\in A\}
  3. A ¯ = X \overline{A}=X
  4. A { lim n a n : n 0 , a n A } \displaystyle A\;\subseteq\;\{\lim_{n}a_{n}:\,\forall n\;\geq\;0,\,\ a_{n}\in A\}
  5. { U n } \displaystyle\{U_{n}\}
  6. n = 1 U n \displaystyle\bigcap^{\infty}_{n=1}U_{n}
  7. α α
  8. C ( 0 , 11 < s u p > α , 𝐑 ) C(0,11<sup>α,\mathbf{R})

Depolarizing_pre-pulse.html

  1. g N a + = g ¯ N a + m 3 h , {g}_{Na^{+}}=\bar{g}_{Na^{+}}m^{3}h,
  2. g ¯ N a + \bar{g}_{Na^{+}}

Derivation_of_the_conjugate_gradient_method.html

  1. s y m b o l A x = s y m b o l b symbol{Ax}=symbol{b}
  2. s y m b o l A symbol{A}
  3. f ( s y m b o l x ) = s y m b o l x T s y m b o l A s y m b o l x - 2 s y m b o l b T s y m b o l x . f(symbol{x})=symbol{x}^{\mathrm{T}}symbol{A}symbol{x}-2symbol{b}^{\mathrm{T}}% symbol{x}\,\text{.}
  4. f ( s y m b o l x ) = s y m b o l x T s y m b o l A s y m b o l x - 2 s y m b o l b T s y m b o l x . f(symbol{x})=symbol{x}^{\mathrm{T}}symbol{A}symbol{x}-2symbol{b}^{\mathrm{T}}% symbol{x}\,\text{.}
  5. s y m b o l x 0 symbol{x}_{0}
  6. s y m b o l r 0 = s y m b o l b - s y m b o l A x 0 symbol{r}_{0}=symbol{b}-symbol{Ax}_{0}
  7. α i = s y m b o l p i T s y m b o l r i s y m b o l p i T s y m b o l A p i , s y m b o l x i + 1 = s y m b o l x i + α i s y m b o l p i , s y m b o l r i + 1 = s y m b o l r i - α i s y m b o l A p i \begin{aligned}\displaystyle\alpha_{i}&\displaystyle=\frac{symbol{p}_{i}^{% \mathrm{T}}symbol{r}_{i}}{symbol{p}_{i}^{\mathrm{T}}symbol{Ap}_{i}}\,\text{,}% \\ \displaystyle symbol{x}_{i+1}&\displaystyle=symbol{x}_{i}+\alpha_{i}symbol{p}_% {i}\,\text{,}\\ \displaystyle symbol{r}_{i+1}&\displaystyle=symbol{r}_{i}-\alpha_{i}symbol{Ap}% _{i}\end{aligned}
  8. s y m b o l p 0 , s y m b o l p 1 , s y m b o l p 2 , symbol{p}_{0},symbol{p}_{1},symbol{p}_{2},\ldots
  9. s y m b o l p i T s y m b o l A p j = 0 symbol{p}_{i}^{\mathrm{T}}symbol{Ap}_{j}=0
  10. i j i\neq j
  11. s y m b o l p 0 , s y m b o l p 1 , s y m b o l p 2 , symbol{p}_{0},symbol{p}_{1},symbol{p}_{2},\ldots
  12. s y m b o l r 0 symbol{r}_{0}
  13. { s y m b o l v 1 , s y m b o l v 2 , s y m b o l v 3 , } \{symbol{v}_{1},symbol{v}_{2},symbol{v}_{3},\ldots\}
  14. 𝒦 ( s y m b o l A , s y m b o l r 0 ) = { s y m b o l r 0 , s y m b o l A r 0 , s y m b o l A 2 s y m b o l r 0 , } \mathcal{K}(symbol{A},symbol{r}_{0})=\{symbol{r}_{0},symbol{Ar}_{0},symbol{A}^% {2}symbol{r}_{0},\ldots\}
  15. s y m b o l v i = s y m b o l w i / \lVertsymbol w i 2 symbol{v}_{i}=symbol{w}_{i}/\lVertsymbol{w}_{i}\rVert_{2}
  16. s y m b o l w i = { s y m b o l r 0 if i = 1 , s y m b o l A v i - 1 - j = 1 i - 1 ( s y m b o l v j T s y m b o l A v i - 1 ) s y m b o l v j if i > 1 . symbol{w}_{i}=\begin{cases}symbol{r}_{0}&\,\text{if }i=1\,\text{,}\\ symbol{Av}_{i-1}-\sum_{j=1}^{i-1}(symbol{v}_{j}^{\mathrm{T}}symbol{Av}_{i-1})% symbol{v}_{j}&\,\text{if }i>1\,\text{.}\end{cases}
  17. i > 1 i>1
  18. s y m b o l v i symbol{v}_{i}
  19. s y m b o l A v i - 1 symbol{Av}_{i-1}
  20. { s y m b o l v 1 , s y m b o l v 2 , , s y m b o l v i - 1 } \{symbol{v}_{1},symbol{v}_{2},\ldots,symbol{v}_{i-1}\}
  21. s y m b o l A V i = s y m b o l V i + 1 s y m b o l H ~ i symbol{AV}_{i}=symbol{V}_{i+1}symbol{\tilde{H}}_{i}
  22. s y m b o l V i = [ s y m b o l v 1 s y m b o l v 2 s y m b o l v i ] , s y m b o l H ~ i = [ h 11 h 12 h 13 h 1 , i h 21 h 22 h 23 h 2 , i h 32 h 33 h 3 , i h i , i - 1 h i , i h i + 1 , i ] = [ s y m b o l H i h i + 1 , i s y m b o l e i T ] \begin{aligned}\displaystyle symbol{V}_{i}&\displaystyle=\begin{bmatrix}symbol% {v}_{1}&symbol{v}_{2}&\cdots&symbol{v}_{i}\end{bmatrix}\,\text{,}\\ \displaystyle symbol{\tilde{H}}_{i}&\displaystyle=\begin{bmatrix}h_{11}&h_{12}% &h_{13}&\cdots&h_{1,i}\\ h_{21}&h_{22}&h_{23}&\cdots&h_{2,i}\\ &h_{32}&h_{33}&\cdots&h_{3,i}\\ &&\ddots&\ddots&\vdots\\ &&&h_{i,i-1}&h_{i,i}\\ &&&&h_{i+1,i}\end{bmatrix}=\begin{bmatrix}symbol{H}_{i}\\ h_{i+1,i}symbol{e}_{i}^{\mathrm{T}}\end{bmatrix}\end{aligned}
  23. h j i = { s y m b o l v j T s y m b o l A v i if j i , \lVertsymbol w i + 1 2 if j = i + 1 , 0 if j > i + 1 . h_{ji}=\begin{cases}symbol{v}_{j}^{\mathrm{T}}symbol{Av}_{i}&\,\text{if }j\leq i% \,\text{,}\\ \lVertsymbol{w}_{i+1}\rVert_{2}&\,\text{if }j=i+1\,\text{,}\\ 0&\,\text{if }j>i+1\,\text{.}\end{cases}
  24. s y m b o l r 0 = s y m b o l b - s y m b o l A x 0 symbol{r}_{0}=symbol{b}-symbol{Ax}_{0}
  25. s y m b o l x 0 symbol{x}_{0}
  26. s y m b o l y i = s y m b o l H i - 1 ( \lVertsymbol r 0 2 s y m b o l e 1 ) symbol{y}_{i}=symbol{H}_{i}^{-1}(\lVertsymbol{r}_{0}\rVert_{2}symbol{e}_{1})
  27. s y m b o l x i = s y m b o l x 0 + s y m b o l V i s y m b o l y i symbol{x}_{i}=symbol{x}_{0}+symbol{V}_{i}symbol{y}_{i}
  28. s y m b o l A symbol{A}
  29. s y m b o l A symbol{A}
  30. s y m b o l H i = s y m b o l V i T s y m b o l A V i symbol{H}_{i}=symbol{V}_{i}^{\mathrm{T}}symbol{AV}_{i}
  31. s y m b o l H i = [ a 1 b 2 b 2 a 2 b 3 b i - 1 a i - 1 b i b i a i ] . symbol{H}_{i}=\begin{bmatrix}a_{1}&b_{2}\\ b_{2}&a_{2}&b_{3}\\ &\ddots&\ddots&\ddots\\ &&b_{i-1}&a_{i-1}&b_{i}\\ &&&b_{i}&a_{i}\end{bmatrix}\,\text{.}
  32. s y m b o l v i symbol{v}_{i}
  33. s y m b o l A symbol{A}
  34. s y m b o l H i symbol{H}_{i}
  35. s y m b o l H i symbol{H}_{i}
  36. s y m b o l H i = s y m b o l L i s y m b o l U i = [ 1 c 2 1 c i - 1 1 c i 1 ] [ d 1 b 2 d 2 b 3 d i - 1 b i d i ] symbol{H}_{i}=symbol{L}_{i}symbol{U}_{i}=\begin{bmatrix}1\\ c_{2}&1\\ &\ddots&\ddots\\ &&c_{i-1}&1\\ &&&c_{i}&1\end{bmatrix}\begin{bmatrix}d_{1}&b_{2}\\ &d_{2}&b_{3}\\ &&\ddots&\ddots\\ &&&d_{i-1}&b_{i}\\ &&&&d_{i}\end{bmatrix}
  37. c i c_{i}
  38. d i d_{i}
  39. c i \displaystyle c_{i}
  40. s y m b o l x i = s y m b o l x 0 + s y m b o l V i s y m b o l y i symbol{x}_{i}=symbol{x}_{0}+symbol{V}_{i}symbol{y}_{i}
  41. s y m b o l x i = s y m b o l x 0 + s y m b o l V i s y m b o l H i - 1 ( \lVertsymbol r 0 2 s y m b o l e 1 ) = s y m b o l x 0 + s y m b o l V i s y m b o l U i - 1 s y m b o l L i - 1 ( \lVertsymbol r 0 2 s y m b o l e 1 ) = s y m b o l x 0 + s y m b o l P i s y m b o l z i \begin{aligned}\displaystyle symbol{x}_{i}&\displaystyle=symbol{x}_{0}+symbol{% V}_{i}symbol{H}_{i}^{-1}(\lVertsymbol{r}_{0}\rVert_{2}symbol{e}_{1})\\ &\displaystyle=symbol{x}_{0}+symbol{V}_{i}symbol{U}_{i}^{-1}symbol{L}_{i}^{-1}% (\lVertsymbol{r}_{0}\rVert_{2}symbol{e}_{1})\\ &\displaystyle=symbol{x}_{0}+symbol{P}_{i}symbol{z}_{i}\end{aligned}
  42. s y m b o l P i = s y m b o l V i s y m b o l U i - 1 , s y m b o l z i = s y m b o l L i - 1 ( \lVertsymbol r 0 2 s y m b o l e 1 ) . \begin{aligned}\displaystyle symbol{P}_{i}&\displaystyle=symbol{V}_{i}symbol{U% }_{i}^{-1}\,\text{,}\\ \displaystyle symbol{z}_{i}&\displaystyle=symbol{L}_{i}^{-1}(\lVertsymbol{r}_{% 0}\rVert_{2}symbol{e}_{1})\,\text{.}\end{aligned}
  43. s y m b o l P i = [ s y m b o l P i - 1 s y m b o l p i ] , s y m b o l z i = [ s y m b o l z i - 1 ζ i ] . \begin{aligned}\displaystyle symbol{P}_{i}&\displaystyle=\begin{bmatrix}symbol% {P}_{i-1}&symbol{p}_{i}\end{bmatrix}\,\text{,}\\ \displaystyle symbol{z}_{i}&\displaystyle=\begin{bmatrix}symbol{z}_{i-1}\\ \zeta_{i}\end{bmatrix}\,\text{.}\end{aligned}
  44. s y m b o l p i symbol{p}_{i}
  45. ζ i \zeta_{i}
  46. s y m b o l p i \displaystyle symbol{p}_{i}
  47. s y m b o l x i symbol{x}_{i}
  48. s y m b o l x i \displaystyle symbol{x}_{i}
  49. s y m b o l p i symbol{p}_{i}
  50. s y m b o l x i \displaystyle symbol{x}_{i}
  51. s y m b o l r i symbol{r}_{i}
  52. s y m b o l p i symbol{p}_{i}
  53. i j i\neq j
  54. s y m b o l r i T s y m b o l r j = 0 , s y m b o l p i T s y m b o l A p j = 0 . \begin{aligned}\displaystyle symbol{r}_{i}^{\mathrm{T}}symbol{r}_{j}&% \displaystyle=0\,\text{,}\\ \displaystyle symbol{p}_{i}^{\mathrm{T}}symbol{Ap}_{j}&\displaystyle=0\,\text{% .}\end{aligned}
  55. s y m b o l r i symbol{r}_{i}
  56. s y m b o l v i + 1 symbol{v}_{i+1}
  57. i = 0 i=0
  58. s y m b o l r 0 = \lVertsymbol r 0 2 s y m b o l v 1 symbol{r}_{0}=\lVertsymbol{r}_{0}\rVert_{2}symbol{v}_{1}
  59. i > 0 i>0
  60. s y m b o l r i = s y m b o l b - s y m b o l A x i = s y m b o l b - s y m b o l A ( s y m b o l x 0 + s y m b o l V i s y m b o l y i ) = s y m b o l r 0 - s y m b o l A V i s y m b o l y i = s y m b o l r 0 - s y m b o l V i + 1 s y m b o l H ~ i s y m b o l y i = s y m b o l r 0 - s y m b o l V i s y m b o l H i s y m b o l y i - h i + 1 , i ( s y m b o l e i T s y m b o l y i ) s y m b o l v i + 1 = \lVertsymbol r 0 2 s y m b o l v 1 - s y m b o l V i ( \lVertsymbol r 0 2 s y m b o l e 1 ) - h i + 1 , i ( s y m b o l e i T s y m b o l y i ) s y m b o l v i + 1 = - h i + 1 , i ( s y m b o l e i T s y m b o l y i ) s y m b o l v i + 1 . \begin{aligned}\displaystyle symbol{r}_{i}&\displaystyle=symbol{b}-symbol{Ax}_% {i}\\ &\displaystyle=symbol{b}-symbol{A}(symbol{x}_{0}+symbol{V}_{i}symbol{y}_{i})\\ &\displaystyle=symbol{r}_{0}-symbol{AV}_{i}symbol{y}_{i}\\ &\displaystyle=symbol{r}_{0}-symbol{V}_{i+1}symbol{\tilde{H}}_{i}symbol{y}_{i}% \\ &\displaystyle=symbol{r}_{0}-symbol{V}_{i}symbol{H}_{i}symbol{y}_{i}-h_{i+1,i}% (symbol{e}_{i}^{\mathrm{T}}symbol{y}_{i})symbol{v}_{i+1}\\ &\displaystyle=\lVertsymbol{r}_{0}\rVert_{2}symbol{v}_{1}-symbol{V}_{i}(% \lVertsymbol{r}_{0}\rVert_{2}symbol{e}_{1})-h_{i+1,i}(symbol{e}_{i}^{\mathrm{T% }}symbol{y}_{i})symbol{v}_{i+1}\\ &\displaystyle=-h_{i+1,i}(symbol{e}_{i}^{\mathrm{T}}symbol{y}_{i})symbol{v}_{i% +1}\,\text{.}\end{aligned}
  61. s y m b o l p i symbol{p}_{i}
  62. s y m b o l P i T s y m b o l A P i symbol{P}_{i}^{\mathrm{T}}symbol{AP}_{i}
  63. s y m b o l P i T s y m b o l A P i \displaystyle symbol{P}_{i}^{\mathrm{T}}symbol{AP}_{i}
  64. α i \alpha_{i}
  65. β i \beta_{i}
  66. s y m b o l p i symbol{p}_{i}
  67. s y m b o l r i symbol{r}_{i}
  68. s y m b o l p i symbol{p}_{i}
  69. s y m b o l r i symbol{r}_{i}
  70. s y m b o l r i + 1 T s y m b o l r i = ( s y m b o l r i - α i s y m b o l A p i ) T s y m b o l r i = 0 symbol{r}_{i+1}^{\mathrm{T}}symbol{r}_{i}=(symbol{r}_{i}-\alpha_{i}symbol{Ap}_% {i})^{\mathrm{T}}symbol{r}_{i}=0
  71. α i = s y m b o l r i T s y m b o l r i s y m b o l r i T s y m b o l A p i = s y m b o l r i T s y m b o l r i ( s y m b o l p i - β i - 1 s y m b o l p i - 1 ) T s y m b o l A p i = s y m b o l r i T s y m b o l r i s y m b o l p i T s y m b o l A p i . \begin{aligned}\displaystyle\alpha_{i}&\displaystyle=\frac{symbol{r}_{i}^{% \mathrm{T}}symbol{r}_{i}}{symbol{r}_{i}^{\mathrm{T}}symbol{Ap}_{i}}\\ &\displaystyle=\frac{symbol{r}_{i}^{\mathrm{T}}symbol{r}_{i}}{(symbol{p}_{i}-% \beta_{i-1}symbol{p}_{i-1})^{\mathrm{T}}symbol{Ap}_{i}}\\ &\displaystyle=\frac{symbol{r}_{i}^{\mathrm{T}}symbol{r}_{i}}{symbol{p}_{i}^{% \mathrm{T}}symbol{Ap}_{i}}\,\text{.}\end{aligned}
  72. s y m b o l p i symbol{p}_{i}
  73. s y m b o l p i + 1 T s y m b o l A p i = ( s y m b o l r i + 1 + β i s y m b o l p i ) T s y m b o l A p i = 0 symbol{p}_{i+1}^{\mathrm{T}}symbol{Ap}_{i}=(symbol{r}_{i+1}+\beta_{i}symbol{p}% _{i})^{\mathrm{T}}symbol{Ap}_{i}=0
  74. β i = - s y m b o l r i + 1 T s y m b o l A p i s y m b o l p i T s y m b o l A p i = - s y m b o l r i + 1 T ( s y m b o l r i - s y m b o l r i + 1 ) α i s y m b o l p i T s y m b o l A p i = s y m b o l r i + 1 T s y m b o l r i + 1 s y m b o l r i T s y m b o l r i . \begin{aligned}\displaystyle\beta_{i}&\displaystyle=-\frac{symbol{r}_{i+1}^{% \mathrm{T}}symbol{Ap}_{i}}{symbol{p}_{i}^{\mathrm{T}}symbol{Ap}_{i}}\\ &\displaystyle=-\frac{symbol{r}_{i+1}^{\mathrm{T}}(symbol{r}_{i}-symbol{r}_{i+% 1})}{\alpha_{i}symbol{p}_{i}^{\mathrm{T}}symbol{Ap}_{i}}\\ &\displaystyle=\frac{symbol{r}_{i+1}^{\mathrm{T}}symbol{r}_{i+1}}{symbol{r}_{i% }^{\mathrm{T}}symbol{r}_{i}}\,\text{.}\end{aligned}

Design_effect.html

  1. ρ \rho
  2. D eff = 1 + ( m - 1 ) ρ . D\text{eff}=1+(m-1)\rho.

Desmic_system.html

  1. ( w 2 - x 2 ) ( y 2 - z 2 ) = 0 \displaystyle(w^{2}-x^{2})(y^{2}-z^{2})=0
  2. ( w 2 - y 2 ) ( x 2 - z 2 ) = 0 \displaystyle(w^{2}-y^{2})(x^{2}-z^{2})=0
  3. ( w 2 - z 2 ) ( y 2 - x 2 ) = 0 \displaystyle(w^{2}-z^{2})(y^{2}-x^{2})=0
  4. a ( w 2 x 2 + y 2 z 2 ) + b ( w 2 y 2 + x 2 z 2 ) + c ( w 2 z 2 + x 2 y 2 ) = 0 \displaystyle a(w^{2}x^{2}+y^{2}z^{2})+b(w^{2}y^{2}+x^{2}z^{2})+c(w^{2}z^{2}+x% ^{2}y^{2})=0

Detective_quantum_efficiency.html

  1. DQE = NEQ q \mathrm{DQE}=\frac{\mathrm{NEQ}}{q}
  2. DQE ( u ) = NEQ ( u ) q = q G 2 T 2 ( u ) W ( u ) \mathrm{DQE}(u)=\frac{\mathrm{NEQ}(u)}{q}=\frac{qG^{2}\mathrm{T^{2}}(u)}{% \mathrm{W}(u)}
  3. SNR i n 2 ( u ) = q \mathrm{SNR}_{in}^{2}(u)=q
  4. SNR o u t 2 ( u ) = q 2 G 2 T 2 ( u ) W ( u ) \mathrm{SNR}_{out}^{2}(u)=\frac{q^{2}G^{2}\mathrm{T}^{2}(u)}{\mathrm{W}(u)}
  5. DQE ( u ) = SNR o u t 2 ( u ) SNR i n 2 ( u ) . \mathrm{DQE}(u)=\frac{\mathrm{SNR}_{out}^{2}(u)}{\mathrm{SNR}_{in}^{2}(u)}.

Determinantal_variety.html

  1. 𝐀 m n × 𝐆𝐫 ( r , m ) \mathbf{A}^{mn}\times\mathbf{Gr}(r,m)
  2. 𝐀 m n \mathbf{A}^{mn}
  3. 𝐆𝐫 ( r , m ) \mathbf{Gr}(r,m)
  4. Z r = { ( A , W ) A ( k n ) W } Z_{r}=\{(A,W)\mid A(k^{n})\subseteq W\}
  5. Y r Y_{r}
  6. Z r Z_{r}
  7. 𝐆𝐫 ( r , m ) \mathbf{Gr}(r,m)
  8. Hom ( k n , ) \mathrm{Hom}(k^{n},\mathcal{R})
  9. \mathcal{R}
  10. dim Y r = dim Z r \dim Y_{r}=\dim Z_{r}
  11. dim Z r = dim 𝐆𝐫 ( r , m ) + n r = r ( m - r ) + n r \dim Z_{r}=\dim\mathbf{Gr}(r,m)+nr=r(m-r)+nr
  12. Hom ( k n , ) \mathrm{Hom}(k^{n},\mathcal{R})
  13. G = 𝐆𝐋 ( m ) × 𝐆𝐋 ( n ) G=\mathbf{GL}(m)\times\mathbf{GL}(n)
  14. Y r Y_{r}

Determining_the_number_of_clusters_in_a_data_set.html

  1. k n / 2 k\approx\sqrt{n/2}
  2. Γ \Gamma
  3. c 1 c K c_{1}...c_{K}
  4. c X c_{X}
  5. d K = 1 p min c 1 c K E [ ( X - c X ) T Γ - 1 ( X - c X ) ] d_{K}=\frac{1}{p}\min_{c_{1}...c_{K}}{E[(X-c_{X})^{T}\Gamma^{-1}(X-c_{X})]}
  6. Y = ( p / 2 ) Y=(p/2)
  7. α p \alpha^{p}
  8. α \alpha
  9. α - 2 \alpha^{-2}
  10. ( - p / 2 ) (-p/2)
  11. α p \alpha^{p}
  12. α p \alpha^{p}
  13. α - 2 \alpha^{-2}
  14. d K - p / 2 d_{K}^{-p/2}
  15. ( m × n ) / t (m\times n)/t

Deutsche_Wertungszahl.html

  1. Z n = Z 0 + 800 E + n ( W - W e ) Z_{n}=Z_{0}+\frac{800}{E+n}(W-W_{e})
  2. W e = 1 1 + 10 ( Z G - Z A ) / 400 W_{e}=\frac{1}{1+10^{(Z_{G}-Z_{A})/400}}
  3. E = a E 0 + B E=a\cdot E_{0}+B
  4. E 0 = ( Z A 1000 ) 4 + J E_{0}=\left(\frac{Z_{A}}{1000}\right)^{4}+J
  5. a = Z A 2000 a=\frac{Z_{A}}{2000}
  6. B = e 1300 - Z A 150 - 1 B=e^{\frac{1300-Z_{A}}{150}}-1
  7. 5 E { min ( 30 ; 5 i ) , if B = 0 150 , if B > 0 5\leq E\leq\begin{cases}\min(30;5i),\;\mathrm{if}\;B=0\\ 150,\;\mathrm{if}\;B>0\end{cases}

Dieudonné_module.html

  1. X X
  2. / p \mathbb{Z}/p\mathbb{Z}
  3. k k
  4. 𝐃 ( X ) \mathbf{D}(X)
  5. k k
  6. F = Frob k F=\mathrm{Frob}_{k}
  7. V = 0 V=0
  8. X = μ p X=\mu_{p}
  9. 𝐃 ( X ) = k \mathbf{D}(X)=k
  10. F = 0 F=0
  11. V = Frob k - 1 V=\mathrm{Frob}_{k}^{-1}
  12. X = α p X=\alpha_{p}
  13. 𝔾 a 𝔾 a \mathbb{G}_{a}\to\mathbb{G}_{a}
  14. 𝐃 ( X ) = k \mathbf{D}(X)=k
  15. F = V = 0 F=V=0
  16. X = E [ p ] X=E[p]

Difference_algebra.html

  1. R R
  2. σ : R R \sigma\colon R\to R
  3. σ \sigma
  4. R R
  5. K = ( x ) K=\mathbb{C}(x)
  6. σ \sigma
  7. σ ( f ( x ) ) = f ( x + 1 ) \sigma(f(x))=f(x+1)
  8. σ \sigma
  9. K K
  10. R R
  11. K K
  12. K R K\to R
  13. σ : R R \sigma\colon R\to R
  14. σ : K K \sigma\colon K\to K
  15. K { y } = K { y 1 , , y n } K\{y\}=K\{y_{1},\ldots,y_{n}\}
  16. K K
  17. y 1 , , y n y_{1},\ldots,y_{n}
  18. K K
  19. σ i ( y j ) , ( i , 1 j n ) \sigma^{i}(y_{j}),\ (i\in\mathbb{N},1\leq j\leq n)
  20. K K
  21. σ \sigma
  22. K K
  23. K { y } K\{y\}
  24. K K
  25. F F
  26. K { y } K\{y\}
  27. R R
  28. K K
  29. F F
  30. R R
  31. 𝕍 R ( F ) = { a R n | f ( a ) = 0 for all f F } . \mathbb{V}_{R}(F)=\{a\in R^{n}|\ f(a)=0\,\text{ for all }f\in F\}.
  32. K K
  33. K = ( x ) K=\mathbb{C}(x)
  34. R R
  35. \mathbb{C}
  36. σ \sigma
  37. σ ( f ( x ) ) = f ( x + 1 ) \sigma(f(x))=f(x+1)
  38. Γ \Gamma
  39. Γ ( x + 1 ) = x Γ ( x ) \Gamma(x+1)=x\Gamma(x)
  40. Γ 𝕍 R ( σ ( y 1 ) - x y 1 ) \Gamma\in\mathbb{V}_{R}(\sigma(y_{1})-xy_{1})
  41. K K
  42. K K
  43. K K
  44. K K
  45. R 𝕍 R ( F ) R\rightsquigarrow\mathbb{V}_{R}(F)
  46. F K { y } . F\subset K\{y\}.
  47. y 1 , , y n y_{1},\ldots,y_{n}
  48. K { y } K\{y\}
  49. K { y } K\{y\}
  50. K { y } K\{y\}

Differential_gain.html

  1. CCVS = CVS + color \mbox{CCVS}~{}=\mbox{CVS}~{}+\mbox{color}
  2. A ( C V S + c o l o r ) = A ( C V S ) + A ( c o l o r ) A\cdot(CVS+color)=A\cdot(CVS)+A\cdot(color)
  3. DG = a - b a \mbox{DG}~{}=\frac{a-b}{a}
  4. α \alpha
  5. f(t) = 1 2 ( sin ( ω t + α ) + sin ( ω t - α ) ) = sin ( ω t ) cos ( α ) \mbox{f(t)}~{}=\frac{1}{2}(\sin(\omega t+\alpha)+\sin(\omega t-\alpha))=\sin(% \omega t)\cdot\cos(\alpha)

Differential_group_delay.html

  1. E T = ( E i , x t x ) 2 + ( E i , y t y ) 2 E_{T}=(E_{i,x}\cdot t_{x})^{2}+(E_{i,y}\cdot t_{y})^{2}\,

Differential_invariant.html

  1. I ( x , y , d y d x , , d k y d x k ) I\left(x,y,\frac{dy}{dx},\dots,\frac{d^{k}y}{dx^{k}}\right)
  2. ( x ¯ , y ¯ ) = g ( x , y ) , (\overline{x},\overline{y})=g\cdot(x,y),
  3. g ( x , y , d y d x ) = def ( x ¯ , y ¯ , d y ¯ d x ¯ ) . g\cdot\left(x,y,\frac{dy}{dx}\right)\stackrel{\,\text{def}}{=}\left(\overline{% x},\overline{y},\frac{d\overline{y}}{d\overline{x}}\right).

Differential_of_a_function.html

  1. d y = f ( x ) d x , dy=f^{\prime}(x)\,dx,
  2. f ( x ) f^{\prime}(x)
  3. d y = d y d x d x dy=\frac{dy}{dx}\,dx
  4. d f ( x ) = f ( x ) d x . df(x)=f^{\prime}(x)\,dx.
  5. d y d x \frac{dy}{dx}
  6. d y = f ( x ) d x dy=f^{\prime}(x)\,dx
  7. d f ( x , Δ x ) = def f ( x ) Δ x . df(x,\Delta x)\stackrel{\rm{def}}{=}f^{\prime}(x)\,\Delta x.
  8. d f ( x ) = f ( x ) d x df(x)=f^{\prime}(x)\,dx
  9. Δ y = def f ( x + Δ x ) - f ( x ) \Delta y\stackrel{\rm{def}}{=}f(x+\Delta x)-f(x)
  10. Δ y = f ( x ) Δ x + ε = d f ( x ) + ε \Delta y=f^{\prime}(x)\,\Delta x+\varepsilon=df(x)+\varepsilon\,
  11. Δ y d y \Delta y\approx dy
  12. Δ y - d y Δ x 0 \frac{\Delta y-dy}{\Delta x}\to 0
  13. y = f ( x 1 , , x n ) , y=f(x_{1},\dots,x_{n}),\,
  14. y x 1 d x 1 \frac{\partial y}{\partial x_{1}}dx_{1}
  15. d y = y x 1 d x 1 + + y x n d x n , dy=\frac{\partial y}{\partial x_{1}}dx_{1}+\cdots+\frac{\partial y}{\partial x% _{n}}dx_{n},
  16. Δ y = def f ( x 1 + Δ x 1 , , x n + Δ x n ) - f ( x 1 , , x n ) = y x 1 Δ x 1 + + y x n Δ x n + ε 1 Δ x 1 + + ε n Δ x n \begin{aligned}\displaystyle\Delta y&\displaystyle{}\stackrel{\mathrm{def}}{=}% f(x_{1}+\Delta x_{1},\dots,x_{n}+\Delta x_{n})-f(x_{1},\dots,x_{n})\\ &\displaystyle{}=\frac{\partial y}{\partial x_{1}}\Delta x_{1}+\cdots+\frac{% \partial y}{\partial x_{n}}\Delta x_{n}+\varepsilon_{1}\Delta x_{1}+\cdots+% \varepsilon_{n}\Delta x_{n}\end{aligned}
  17. d y = y x 1 Δ x 1 + + y x n Δ x n . dy=\frac{\partial y}{\partial x_{1}}\Delta x_{1}+\cdots+\frac{\partial y}{% \partial x_{n}}\Delta x_{n}.
  18. d x i ( Δ x 1 , , Δ x n ) = Δ x i , dx_{i}(\Delta x_{1},\dots,\Delta x_{n})=\Delta x_{i},
  19. d y = y x 1 d x 1 + + y x n d x n . dy=\frac{\partial y}{\partial x_{1}}\,dx_{1}+\cdots+\frac{\partial y}{\partial x% _{n}}\,dx_{n}.
  20. d y Δ y dy\approx\Delta y
  21. Δ x 1 2 + + Δ x n 2 \sqrt{\Delta x_{1}^{2}+\cdots+\Delta x_{n}^{2}}
  22. Δ f = f x Δ x + f y Δ y + \Delta f=f_{x}\Delta x+f_{y}\Delta y+\cdots
  23. d 2 y = d ( d y ) = d ( f ( x ) d x ) = f ′′ ( x ) ( d x ) 2 , d^{2}y=d(dy)=d(f^{\prime}(x)dx)=f^{\prime\prime}(x)\,(dx)^{2},
  24. d n y = f ( n ) ( x ) ( d x ) n . d^{n}y=f^{(n)}(x)\,(dx)^{n}.
  25. f ( n ) ( x ) = d n f d x n . f^{(n)}(x)=\frac{d^{n}f}{dx^{n}}.
  26. d 2 y = f ′′ ( x ) ( d x ) 2 + f ( x ) d 2 x d 3 y = f ′′′ ( x ) ( d x ) 3 + 3 f ′′ ( x ) d x d 2 x + f ( x ) d 3 x \begin{aligned}\displaystyle d^{2}y&\displaystyle=f^{\prime\prime}(x)\,(dx)^{2% }+f^{\prime}(x)d^{2}x\\ \displaystyle d^{3}y&\displaystyle=f^{\prime\prime\prime}(x)\,(dx)^{3}+3f^{% \prime\prime}(x)dx\,d^{2}x+f^{\prime}(x)d^{3}x\end{aligned}
  27. d n f = k = 0 n ( n k ) n f x k y n - k ( d x ) k ( d y ) n - k , d^{n}f=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\frac{\partial^{n}f}{\partial x% ^{k}\partial y^{n-k}}(dx)^{k}(dy)^{n-k},
  28. ( n k ) \scriptstyle{{\left({{n}\atop{k}}\right)}}
  29. d 2 f = ( 2 f x 2 ( d x ) 2 + 2 2 f x y d x d y + 2 f y 2 ( d y ) 2 ) + f x d 2 x + f y d 2 y . d^{2}f=\left(\frac{\partial^{2}f}{\partial x^{2}}(dx)^{2}+2\frac{\partial^{2}f% }{\partial x\partial y}dx\,dy+\frac{\partial^{2}f}{\partial y^{2}}(dy)^{2}% \right)+\frac{\partial f}{\partial x}d^{2}x+\frac{\partial f}{\partial y}d^{2}y.
  30. d 2 z = r d x 2 + 2 s d x d y + t d y 2 ? d^{2}z=r\,dx^{2}+2s\,dx\,dy+t\,dy^{2}\,?
  31. d n f ( x , Δ x ) = d n d t n f ( x + t Δ x ) | t = 0 d^{n}f(x,\Delta x)=\left.\frac{d^{n}}{dt^{n}}f(x+t\Delta x)\right|_{t=0}
  32. lim t 0 Δ t Δ x n f t n \lim_{t\to 0}\frac{\Delta^{n}_{t\Delta x}f}{t^{n}}
  33. Δ t Δ x n f \Delta^{n}_{t\Delta x}f
  34. f ( x + Δ x ) f ( x ) + d f ( x , Δ x ) + 1 2 d 2 f ( x , Δ x ) + + 1 n ! d n f ( x , Δ x ) + f(x+\Delta x)\sim f(x)+df(x,\Delta x)+\frac{1}{2}d^{2}f(x,\Delta x)+\cdots+% \frac{1}{n!}d^{n}f(x,\Delta x)+\cdots
  35. d ( a f + b g ) = a d f + b d g . d(af+bg)=a\,df+b\,dg.
  36. d ( f g ) = f d g + g d f . d(fg)=f\,dg+g\,df.
  37. d ( f n ) = n f n - 1 d f d(f^{n})=nf^{n-1}df
  38. d y = f ( u ) d u = f ( g ( x ) ) g ( x ) d x . dy=f^{\prime}(u)\,du=f^{\prime}(g(x))g^{\prime}(x)\,dx.
  39. d y \displaystyle dy
  40. Δ f = f ( 𝐱 + Δ 𝐱 ) - f ( 𝐱 ) . \Delta f=f(\mathbf{x}+\Delta\mathbf{x})-f(\mathbf{x}).
  41. Δ f = A Δ 𝐱 + Δ 𝐱 s y m b o l ε \Delta f=A\Delta\mathbf{x}+\|\Delta\mathbf{x}\|symbol{\varepsilon}
  42. d f ( 𝐱 , 𝐡 ) = lim t 0 f ( 𝐱 + t 𝐡 ) - f ( 𝐱 ) t = d d t f ( 𝐱 + t 𝐡 ) | t = 0 , df(\mathbf{x},\mathbf{h})=\lim_{t\to 0}\frac{f(\mathbf{x}+t\mathbf{h})-f(% \mathbf{x})}{t}=\left.\frac{d}{dt}f(\mathbf{x}+t\mathbf{h})\right|_{t=0},
  43. Δ y = f ( x ) Δ x + ( Δ x ) 2 2 f ′′ ( ξ ) \Delta y=f^{\prime}(x)\Delta x+\frac{(\Delta x)^{2}}{2}f^{\prime\prime}(\xi)
  44. d y = g ( x ) d x , dy=g(x)\,dx,

Differential_phase.html

  1. θ \theta
  2. β \beta
  3. f(t) = s i n ( ω t + θ ) \mbox{f(t)}~{}=sin(\omega t+\theta)
  4. f(t) = s i n ( ω t + θ + β ) \mbox{f(t)}~{}=sin(\omega t+\theta+\beta)
  5. f(t) = s i n ( ω t + θ - β ) \mbox{f(t)}~{}=sin(\omega t+\theta-\beta)
  6. f(t) = 1 2 ( sin ( ω t + θ + β ) + sin ( ω t + θ - β ) ) = sin ( ω t + θ ) cos ( β ) \mbox{f(t)}~{}=\frac{1}{2}(\sin(\omega t+\theta+\beta)+\sin(\omega t+\theta-% \beta))=\sin(\omega t+\theta)\cdot\cos(\beta)
  7. β \beta
  8. c o s ( β ) cos(\beta)

Differential_privacy.html

  1. 𝒜 \mathcal{A}\,\!
  2. 𝒜 \mathcal{A}\,\!
  3. ϵ \epsilon\,\!
  4. D 1 D_{1}\,\!
  5. D 2 D_{2}\,\!
  6. S Range ( 𝒜 ) S\subseteq\mathrm{Range}(\mathcal{A})\,\!
  7. Pr [ 𝒜 ( D 1 ) S ] e ϵ × Pr [ 𝒜 ( D 2 ) S ] \Pr[\mathcal{A}(D_{1})\in S]\leq e^{\epsilon}\times\Pr[\mathcal{A}(D_{2})\in S% ]\,\!
  8. Range ( 𝒜 ) \mathrm{Range}(\mathcal{A})\,\!
  9. 𝒜 \mathcal{A}\,\!
  10. 𝒜 \mathcal{A}\,\!
  11. D 1 D_{1}\,\!
  12. X { 0 , 1 } X\in\{0,1\}\,\!
  13. Q ( i ) Q(i)\,\!
  14. i i\,\!
  15. X X\,\!
  16. Q ( 5 ) - Q ( 4 ) Q(5)-Q(4)\,\!
  17. D 2 D_{2}\,\!
  18. Q ( i ) Q(i)\,\!
  19. 𝒜 \mathcal{A}\,\!
  20. 𝒜 \mathcal{A}\,\!
  21. ϵ \epsilon\,\!
  22. S S\,\!
  23. { 3.5 } , { 4 } \{3.5\},\{4\}\,\!
  24. 𝒜 \mathcal{A}\,\!
  25. S S\,\!
  26. 3.5 𝒜 ( D 1 ) 3.7 3.5\leq\mathcal{A}(D_{1})\leq 3.7\,\!
  27. 𝒜 \mathcal{A}\,\!
  28. Δ f \Delta f\,\!
  29. f : 𝒟 d f:\mathcal{D}\rightarrow\mathbb{R}^{d}\,\!
  30. Δ f = max D 1 , D 2 f ( D 1 ) - f ( D 2 ) 1 \Delta f=\max_{D_{1},D_{2}}\lVert f(D_{1})-f(D_{2})\rVert_{1}\,\!
  31. D 1 D_{1}\,\!
  32. D 2 D_{2}\,\!
  33. D 1 , D 2 𝒟 D_{1},D_{2}\in\mathcal{D}\,\!
  34. Q ( i ) Q(i)\,\!
  35. f f\,\!
  36. i i\,\!
  37. Q ( i ) Q(i)\,\!
  38. noise ( y ) exp ( - | y | / λ ) \,\text{noise}(y)\propto\exp(-|y|/\lambda)\,\!
  39. λ \lambda\,\!
  40. 𝒜 \mathcal{A}\,\!
  41. 𝒜 \mathcal{A}\,\!
  42. 𝒯 𝒜 ( x ) = f ( x ) + Y \mathcal{T}_{\mathcal{A}}(x)=f(x)+Y\,\!
  43. Y Lap ( λ ) Y\sim\,\text{Lap}(\lambda)\,\!\,\!
  44. f f\,\!
  45. 𝒯 𝒜 ( x ) \mathcal{T}_{\mathcal{A}}(x)\,\!
  46. pdf ( 𝒯 𝒜 , D 1 ( x ) = t ) pdf ( 𝒯 𝒜 , D 2 ( x ) = t ) = noise ( t - f ( D 1 ) ) noise ( t - f ( D 2 ) ) \frac{\mathrm{pdf}(\mathcal{T}_{\mathcal{A},D_{1}}(x)=t)}{\mathrm{pdf}(% \mathcal{T}_{\mathcal{A},D_{2}}(x)=t)}=\frac{\,\text{noise}(t-f(D_{1}))}{\,% \text{noise}(t-f(D_{2}))}\,\!
  47. e | f ( D 1 ) - f ( D 2 ) | λ e Δ ( f ) λ e^{\frac{|f(D_{1})-f(D_{2})|}{\lambda}}\leq e^{\frac{\Delta(f)}{\lambda}}\,\!
  48. Δ ( f ) λ \frac{\Delta(f)}{\lambda}\,\!
  49. ϵ \epsilon\,\!
  50. 𝒯 \mathcal{T}\,\!
  51. 𝒜 \mathcal{A}\,\!
  52. ϵ \epsilon\,\!
  53. λ = 1 / ϵ \lambda=1/\epsilon\,\!
  54. t t
  55. ϵ t \epsilon t
  56. n n
  57. 1 , , n \mathcal{M}_{1},\dots,\mathcal{M}_{n}
  58. ϵ 1 , , ϵ n \epsilon_{1},\dots,\epsilon_{n}
  59. g g
  60. g ( 1 , , n ) g(\mathcal{M}_{1},\dots,\mathcal{M}_{n})
  61. ( i = 1 n ϵ i ) (\sum\limits_{i=1}^{n}\epsilon_{i})
  62. g g
  63. ( max i ϵ i ) (\max_{i}\epsilon_{i})
  64. c c
  65. c c
  66. c c
  67. exp ( ϵ c ) \exp(\epsilon c)
  68. exp ( ϵ ) \exp(\epsilon)
  69. c c
  70. Pr [ 𝒜 ( D 1 ) S ] exp ( ϵ c ) × Pr [ 𝒜 ( D 2 ) S ] \Pr[\mathcal{A}(D_{1})\in S]\leq\exp(\epsilon c)\times\Pr[\mathcal{A}(D_{2})% \in S]\,\!
  71. ϵ / c \epsilon/c
  72. c c
  73. c c
  74. ( ϵ / c ) (\epsilon/c)
  75. 𝒜 \mathcal{A}
  76. Pr [ 𝒜 ( D 1 ) S ] exp ( ϵ ) × Pr [ 𝒜 ( D 2 ) S ] \Pr[\mathcal{A}(D_{1})\in S]\leq\exp(\epsilon)\times\Pr[\mathcal{A}(D_{2})\in S% ]\,\!
  77. Pr [ 𝒜 ( D 2 ) S ] exp ( ϵ ) × Pr [ 𝒜 ( D 3 ) S ] \Pr[\mathcal{A}(D_{2})\in S]\leq\exp(\epsilon)\times\Pr[\mathcal{A}(D_{3})\in S% ]\,\!
  78. Pr [ 𝒜 ( D 1 ) S ] exp ( ϵ ) × ( exp ( ϵ ) × Pr [ 𝒜 ( D 3 ) S ] ) = exp ( 2 ϵ ) × Pr [ 𝒜 ( D 3 ) S ] \Pr[\mathcal{A}(D_{1})\in S]\leq\exp(\epsilon)\times(\exp(\epsilon)\times\Pr[% \mathcal{A}(D_{3})\in S])=\exp(2\epsilon)\times\Pr[\mathcal{A}(D_{3})\in S]\,\!
  79. c c
  80. T T
  81. c c
  82. T ( A ) T(A)
  83. T ( B ) T(B)
  84. c c
  85. A A
  86. B B
  87. A , B A,B
  88. M M
  89. ϵ \epsilon
  90. M T M\circ T
  91. ( ϵ × c ) (\epsilon\times c)
  92. h h
  93. A A
  94. B B
  95. A A
  96. B B
  97. M T M\circ T
  98. ( ϵ × c × h ) (\epsilon\times c\times h)