wpmath0000013_8

Matrix_consimilarity.html

  1. A = S B S ¯ - 1 A=SB\bar{S}^{-1}\,
  2. n × n n\times n
  3. S S
  4. S ¯ \bar{S}
  5. S S
  6. n × n n\times n

Matrix_multiplication_algorithm.html

  1. n × n n×n
  2. C = A B C=AB
  3. n × m n×m
  4. A A
  5. m × p m×p
  6. B B
  7. C C
  8. n × p n×p
  9. c i j = k = 1 m a i k b k j c_{ij}=\sum_{k=1}^{m}a_{ik}b_{kj}
  10. i i
  11. n n
  12. j j
  13. p p
  14. A A
  15. B B
  16. C C
  17. i i
  18. n n
  19. j j
  20. p p
  21. s u m = 0 sum=0
  22. k k
  23. m m
  24. s u m s u m + A < s u b > i k × B k j sum←sum+A<sub>ik×B_{kj}
  25. Θ ( n m p ) Θ(nmp)
  26. n × n n×n
  27. M M
  28. b b
  29. A A
  30. B B
  31. n > M b n>M∕b
  32. A A
  33. B B
  34. B B
  35. C = ( C 11 C 12 C 21 C 22 ) , A = ( A 11 A 12 A 21 A 22 ) , B = ( B 11 B 12 B 21 B 22 ) C=\begin{pmatrix}C_{11}&C_{12}\\ C_{21}&C_{22}\\ \end{pmatrix},\,A=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{pmatrix},\,B=\begin{pmatrix}B_{11}&B_{12}\\ B_{21}&B_{22}\\ \end{pmatrix}
  36. n n
  37. ( C 11 C 12 C 21 C 22 ) = ( A 11 A 12 A 21 A 22 ) ( B 11 B 12 B 21 B 22 ) = ( A 11 B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22 ) \begin{pmatrix}C_{11}&C_{12}\\ C_{21}&C_{22}\\ \end{pmatrix}=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\\ \end{pmatrix}\begin{pmatrix}B_{11}&B_{12}\\ B_{21}&B_{22}\\ \end{pmatrix}=\begin{pmatrix}A_{11}B_{11}+A_{12}B_{21}&A_{11}B_{12}+A_{12}B_{2% 2}\\ A_{21}B_{11}+A_{22}B_{21}&A_{21}B_{12}+A_{22}B_{22}\\ \end{pmatrix}
  38. n n
  39. T ( 1 ) = Θ ( 1 ) T(1)=\Theta(1)
  40. T ( n ) = 8 T ( n / 2 ) + Θ ( n 2 ) T(n)=8T(n/2)+\Theta(n^{2})
  41. n / 2 n/2
  42. n × m n×m
  43. A A
  44. m × p m×p
  45. B B
  46. m a x ( n , m , p ) max(n,m,p)
  47. m a x ( n , m , p ) = n max(n,m,p)=n
  48. A A
  49. C = ( A 1 A 2 ) B = ( A 1 B A 2 B ) C=\begin{pmatrix}A_{1}\\ A_{2}\end{pmatrix}{B}=\begin{pmatrix}A_{1}B\\ A_{2}B\end{pmatrix}
  50. m a x ( n , m , p ) = p max(n,m,p)=p
  51. B B
  52. C = A ( B 1 B 2 ) = ( A B 1 A B 2 ) C=A\begin{pmatrix}B_{1}&B_{2}\end{pmatrix}=\begin{pmatrix}AB_{1}&AB_{2}\end{pmatrix}
  53. m a x ( n , m , p ) = m max(n,m,p)=m
  54. A A
  55. B B
  56. C = ( A 1 A 2 ) ( B 1 B 2 ) = A 1 B 1 + A 2 B 2 C=\begin{pmatrix}A_{1}&A_{2}\end{pmatrix}\begin{pmatrix}B_{1}\\ B_{2}\end{pmatrix}=A_{1}B_{1}+A_{2}B_{2}
  57. ω ω
  58. 2 × 2 2×2
  59. O ( n log 2 7 ) O ( n 2.807 ) O(n^{\log_{2}7})\approx O(n^{2.807})
  60. n > 100 n>100
  61. k k
  62. k × k k×k
  63. n × n n×n
  64. A A
  65. B B
  66. C C
  67. A B = C AB=C
  68. ( A 11 B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22 ) \begin{pmatrix}A_{11}B_{11}+A_{12}B_{21}&A_{11}B_{12}+A_{12}B_{22}\\ A_{21}B_{11}+A_{22}B_{21}&A_{21}B_{12}+A_{22}B_{22}\\ \end{pmatrix}
  69. m u l t i p l y ( C , A , B ) multiply(C,A,B)
  70. n = 1 n=1
  71. c < s u b > 11 a 11 × b 11 c<sub>11←a_{11}×b_{11}
  72. p a r t i t i o n partition
  73. Θ ( l o g < s u p > 2 n ) Θ(log<sup>2n)
  74. O ( n < s u p > 2.376 ) O(n<sup>2.376)

Matsumoto_zeta_function.html

  1. ϕ ( s ) = p 1 A p ( p - s ) \phi(s)=\prod_{p}\frac{1}{A_{p}(p^{-s})}

Matter_collineation.html

  1. X T a b = 0 \mathcal{L}_{X}T_{ab}=0
  2. T a b T_{ab}
  3. X X
  4. X X

Max-dominated_strategy.html

  1. s i S i s_{i}\in S_{i}
  2. i i
  3. s - i S - i s_{-i}\in S_{-i}
  4. s i S i s^{\prime}_{i}\in S_{i}
  5. u i ( s i , s - i ) > u i ( s i , s - i ) u_{i}(s^{\prime}_{i},s_{-i})>u_{i}(s_{i},s_{-i})
  6. s i s_{i}
  7. s - i s_{-i}
  8. s i s^{\prime}_{i}
  9. s i s_{i}
  10. i i
  11. s i S i s_{i}\in S_{i}
  12. s i S i s^{\prime}_{i}\in S_{i}
  13. s - i S - i s_{-i}\in S_{-i}
  14. s i s^{\prime}_{i}
  15. u i ( s i , s - i ) > u i ( s i , s - i ) u_{i}(s^{\prime}_{i},s_{-i})>u_{i}(s_{i},s_{-i})
  16. s i s_{i}
  17. s i S i s_{i}\in S_{i}
  18. i i
  19. s - i S - i s_{-i}\in S_{-i}
  20. s i S i s^{\prime}_{i}\in S_{i}
  21. u i ( s i , s - i ) u i ( s i , s - i ) u_{i}(s^{\prime}_{i},s_{-i})\geq u_{i}(s_{i},s_{-i})
  22. s i s_{i}
  23. s - i s_{-i}
  24. s i s^{\prime}_{i}
  25. s i s_{i}
  26. i i
  27. s i S i s_{i}\in S_{i}
  28. s i S i s^{\prime}_{i}\in S_{i}
  29. s - i S - i s_{-i}\in S_{-i}
  30. s i s^{\prime}_{i}
  31. u i ( s i , s - i ) u i ( s i , s - i ) u_{i}(s^{\prime}_{i},s_{-i})\geq u_{i}(s_{i},s_{-i})
  32. s i s_{i}
  33. G G
  34. G G
  35. G 0 , , G r G_{0},...,G_{r}
  36. G 0 = G G_{0}=G
  37. G k + 1 G_{k+1}
  38. G k G_{k}
  39. G r G_{r}
  40. G r G_{r}
  41. s 1 , s 2 , s 3 , , s k s_{1},s_{2},s_{3},...,s_{k}
  42. s 1 s_{1}
  43. s 2 s_{2}
  44. s 1 s_{1}
  45. s 2 s_{2}
  46. s 1 s_{1}
  47. s - i s_{-i}
  48. s 1 s_{1}
  49. s 1 s_{1}

MAXEkSAT.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. ( 1 2 ) k = 1 2 k \textstyle(\frac{1}{2})^{k}=\frac{1}{2^{k}}
  4. 1 - 1 2 k \textstyle 1-\frac{1}{2^{k}}
  5. 1 c \textstyle 1_{c}
  6. 1 - 1 2 k \textstyle 1-\frac{1}{2^{k}}
  7. | C | \textstyle|C|
  8. ( 1 - 1 2 k ) | C | (\textstyle 1-\frac{1}{2^{k}})|C|
  9. ( 1 - 1 2 k ) \textstyle(1-\frac{1}{2^{k}})
  10. | C | \textstyle|C|
  11. A L G = ( 1 - 1 2 k ) | C | > ( 1 - 1 2 k ) O P T \textstyle ALG=(1-\frac{1}{2^{k}})\cdot|C|>(1-\frac{1}{2^{k}})\cdot OPT
  12. ( 1 - 1 2 k ) \textstyle\geq(1-\frac{1}{2^{k}})
  13. ( 1 - 1 2 k ) \textstyle(1-\frac{1}{2^{k}})
  14. ( 1 - 1 2 k ) \textstyle(1-\frac{1}{2^{k}})
  15. ( 1 - 2 2 k ) \textstyle(1-\frac{2}{2^{k}})
  16. ( 1 - 2 2 k ) \textstyle(1-\frac{2}{2^{k}})
  17. ( 1 1 + 2 k ϵ ) \textstyle(\frac{1}{1+2^{k}\epsilon})
  18. ( 1 - 1 2 k - ϵ ) \textstyle(1-\frac{1}{2^{k}}-\epsilon)
  19. ϵ \textstyle\epsilon
  20. ( 1 - 1 2 k - ϵ ) \textstyle(1-\frac{1}{2^{k}}-\epsilon)
  21. ( 1 - 1 2 k ) \textstyle(1-\frac{1}{2^{k}})
  22. ϵ \textstyle\epsilon
  23. ( 1 - 1 2 k ) \textstyle(1-\frac{1}{2^{k}})
  24. S { 0 , 1 } n S\subseteq\{0,1\}^{n}
  25. [ n = 2 m , n - 1 - d - 2 / 2 m , d ] 2 [n=2^{m},n-1-\lceil{d-2}/2\rceil m,d]_{2}
  26. O ( n / 2 ) O(n^{\lfloor\ell/2\rfloor})

Maximal_common_divisor.html

  1. c d c\simeq d

Maximum_coverage_problem.html

  1. k k
  2. k k
  3. k k
  4. S = S 1 , S 2 , , S m S=S_{1},S_{2},\ldots,S_{m}
  5. S S S^{{}^{\prime}}\subseteq S
  6. | S | k \left|S^{{}^{\prime}}\right|\leq k
  7. | S i S S i | \left|\bigcup_{S_{i}\in S^{{}^{\prime}}}{S_{i}}\right|
  8. 1 - 1 e + o ( 1 ) 0.632 1-\frac{1}{e}+o(1)\approx 0.632
  9. e j E y j \sum_{e_{j}\in E}y_{j}
  10. x i k \sum{x_{i}}\leq k
  11. k k
  12. e j S i x i y j \sum_{e_{j}\in S_{i}}x_{i}\geq y_{j}
  13. y j > 0 y_{j}>0
  14. e j S i e_{j}\in S_{i}
  15. 0 y j 1 0\leq y_{j}\leq 1
  16. y j = 1 y_{j}=1
  17. e j e_{j}
  18. x i { 0 , 1 } x_{i}\in\{0,1\}
  19. x i = 1 x_{i}=1
  20. S i S_{i}
  21. 1 - 1 e 1-\frac{1}{e}
  22. e j e_{j}
  23. w ( e j ) w(e_{j})
  24. 1 1
  25. e E w ( e j ) y j \sum_{e\in E}w(e_{j})\cdot y_{j}
  26. x i k \sum{x_{i}}\leq k
  27. k k
  28. e j S i x i y j \sum_{e_{j}\in S_{i}}x_{i}\geq y_{j}
  29. y j 0 y_{j}\geq 0
  30. e j S i e_{j}\in S_{i}
  31. 0 y j 1 0\leq y_{j}\leq 1
  32. y j = 1 y_{j}=1
  33. e j e_{j}
  34. x i { 0 , 1 } x_{i}\in\{0,1\}
  35. x i = 1 x_{i}=1
  36. S i S_{i}
  37. 1 - 1 e 1-\frac{1}{e}
  38. e j e_{j}
  39. w ( e j ) w(e_{j})
  40. S i S_{i}
  41. c ( S i ) c(S_{i})
  42. k k
  43. B B
  44. B B
  45. e E w ( e j ) y j \sum_{e\in E}w(e_{j})\cdot y_{j}
  46. c ( S i ) x i B \sum{c(S_{i})\cdot x_{i}}\leq B
  47. B B
  48. e j S i x i y j \sum_{e_{j}\in S_{i}}x_{i}\geq y_{j}
  49. y j 0 y_{j}\geq 0
  50. e j S i e_{j}\in S_{i}
  51. 0 y j 1 0\leq y_{j}\leq 1
  52. y j = 1 y_{j}=1
  53. e j e_{j}
  54. x i { 0 , 1 } x_{i}\in\{0,1\}
  55. x i = 1 x_{i}=1
  56. S i S_{i}
  57. 1 - 1 / e 1-1/e
  58. N P D T I M E ( n O ( log log n ) ) NP\subseteq DTIME(n^{O(\log\log n)})
  59. S i S_{i}
  60. c ( S i ) c(S_{i})
  61. e j e_{j}
  62. e j e_{j}
  63. S i S_{i}
  64. e j e_{j}
  65. w i ( e j ) w_{i}(e_{j})
  66. c i ( e j ) c_{i}(e_{j})
  67. B B
  68. e E , S i w i ( e j ) y i j \sum_{e\in E,S_{i}}w_{i}(e_{j})\cdot y_{ij}
  69. c i ( e j ) y i j + c ( S i ) x i B \sum{c_{i}(e_{j})\cdot y_{ij}}+\sum{c(S_{i})\cdot x_{i}}\leq B
  70. B B
  71. i y i j 1 \sum_{i}y_{ij}\leq 1
  72. e j = 1 e_{j}=1
  73. S i x i y i j \sum_{S_{i}}x_{i}\geq y_{ij}
  74. y j 0 y_{j}\geq 0
  75. e j S i e_{j}\in S_{i}
  76. y i j { 0 , 1 } y_{ij}\in\{0,1\}
  77. y i j = 1 y_{ij}=1
  78. e j e_{j}
  79. S i S_{i}
  80. x i { 0 , 1 } x_{i}\in\{0,1\}
  81. x i = 1 x_{i}=1
  82. S i S_{i}
  83. 1 - 1 / e - o ( 1 ) 1-1/e-o(1)

Maximum_theorem.html

  1. X X
  2. Θ \Theta
  3. f : X × Θ f:X\times\Theta\to\mathbb{R}
  4. C : Θ X C:\Theta\twoheadrightarrow X
  5. x x
  6. X X
  7. θ \theta
  8. Θ \Theta
  9. f * ( θ ) = max { f ( x , θ ) | x C ( θ ) } f^{*}(\theta)=\max\{f(x,\theta)|x\in C(\theta)\}
  10. C * ( θ ) = arg max { f ( x , θ ) | x C ( θ ) } = { x C ( θ ) | f ( x , θ ) = f * ( θ ) } C^{*}(\theta)=\mathrm{arg}\max\{f(x,\theta)|x\in C(\theta)\}=\{x\in C(\theta)% \,|\,f(x,\theta)=f^{*}(\theta)\}
  11. C C
  12. θ \theta
  13. f * f^{*}
  14. θ \theta
  15. C * C^{*}
  16. θ \theta
  17. Θ \Theta
  18. f ( x , θ ) f(x,\theta)
  19. C ( θ ) C(\theta)
  20. f f
  21. f * ( θ ) f^{*}(\theta)
  22. C * C^{*}
  23. f f
  24. C C
  25. f f
  26. f f
  27. C * ( θ ) C^{*}(\theta)
  28. θ \theta
  29. Θ \Theta
  30. θ n \theta_{n}
  31. θ \theta
  32. x n C * ( θ n ) x_{n}\in C^{*}(\theta_{n})
  33. X X
  34. C C
  35. x n k x C ( θ ) x_{n_{k}}\to x\in C(\theta)
  36. x C * ( θ ) x\in C^{*}(\theta)
  37. lim k f * ( θ n k ) = lim k f ( x n k , θ n k ) = f ( x , θ ) = f * ( θ ) \lim_{k\to\infty}f^{*}(\theta_{n_{k}})=\lim_{k\to\infty}f(x_{n_{k}},\theta_{n_% {k}})=f(x,\theta)=f^{*}(\theta)
  38. f * f^{*}
  39. C * C^{*}
  40. x C * ( θ ) x\not\in C^{*}(\theta)
  41. x ^ C ( θ ) \hat{x}\in C(\theta)
  42. f ( x ^ , θ ) > f ( x , θ ) f(\hat{x},\theta)>f(x,\theta)
  43. C C
  44. n k n_{k}
  45. x ^ n j C ( θ n j ) \hat{x}_{n_{j}}\in C(\theta_{n_{j}})
  46. x ^ n j x ^ \hat{x}_{n_{j}}\to\hat{x}
  47. f f
  48. lim j f ( x ^ n j , θ n j ) = f ( x ^ , θ ) > f ( x , θ ) = lim j f ( x n j , θ n j ) \lim_{j\to\infty}f(\hat{x}_{n_{j}},\theta_{n_{j}})=f(\hat{x},\theta)>f(x,% \theta)=\lim_{j\to\infty}f(x_{n_{j}},\theta_{n_{j}})
  49. j j
  50. f ( x ^ n j , θ n j ) > f ( x n j , θ n j ) f(\hat{x}_{n_{j}},\theta_{n_{j}})>f(x_{n_{j}},\theta_{n_{j}})
  51. x n j x_{n_{j}}
  52. x n C * ( θ n ) x_{n}\in C^{*}(\theta_{n})
  53. f * f^{*}
  54. C * C^{*}
  55. C * ( θ ) C ( θ ) C^{*}(\theta)\subset C(\theta)
  56. C ( θ ) C(\theta)
  57. C * C^{*}
  58. f f
  59. x x
  60. θ \theta
  61. C C
  62. C * C^{*}
  63. f f
  64. x x
  65. θ \theta
  66. C C
  67. C * C^{*}
  68. f f
  69. C C
  70. f * f^{*}
  71. C * C^{*}
  72. f f
  73. C * C^{*}
  74. X = + l X=\mathbb{R}_{+}^{l}
  75. l l
  76. Θ = + + l × + + \Theta=\mathbb{R}_{++}^{l}\times\mathbb{R}_{++}
  77. p p
  78. w w
  79. f ( x , θ ) = u ( x ) f(x,\theta)=u(x)
  80. C ( θ ) = B ( p , w ) = { x | p x w } C(\theta)=B(p,w)=\{x\,|\,px\leq w\}
  81. f * ( θ ) = v ( p , w ) f^{*}(\theta)=v(p,w)
  82. C * ( θ ) = x ( p , w ) C^{*}(\theta)=x(p,w)

Maxwell–Stefan_diffusion.html

  1. μ i R T = ln a i = = j = = 1 j i n χ i χ j 𝔇 i j ( v j - v i ) = = j = = 1 j i n c i c j c 2 𝔇 i j ( J j c j - J i c i ) \frac{\nabla\mu_{i}}{R\,T}=\nabla\ln a_{i}==\sum_{j==1\atop j\neq i}^{n}{\frac% {\chi_{i}\chi_{j}}{\mathfrak{D}_{ij}}(\vec{v}_{j}-\vec{v}_{i})}==\sum_{j==1% \atop j\neq i}^{n}{\frac{c_{i}c_{j}}{c^{2}\mathfrak{D}_{ij}}\left(\frac{\vec{J% }_{j}}{c_{j}}-\frac{\vec{J}_{i}}{c_{i}}\right)}
  2. 𝔇 i j \mathfrak{D}_{ij}
  3. v i \vec{v}_{i}
  4. c i c_{i}
  5. J i \vec{J}_{i}

Maxwell–Wagner–Sillars_polarization.html

  1. ϵ 1 , ϵ 2 \epsilon^{\prime}_{1},\epsilon^{\prime}_{2}
  2. σ 1 , σ 2 \sigma_{1},\sigma_{2}
  3. τ M W = ϵ 0 ϵ 1 + ϵ 2 σ 1 + σ 2 \tau_{MW}=\epsilon_{0}\frac{\epsilon_{1}+\epsilon_{2}}{\sigma_{1}+\sigma_{2}}
  4. ϵ 2 \epsilon^{\prime}_{2}
  5. R R
  6. ϵ 1 \epsilon_{1}
  7. ϵ 1 \epsilon_{1}

McCallum_rule.html

  1. m t + 1 = m t - Δ v ¯ t - 16 + 1.5 ( + Δ p D + Δ q ¯ ) - 0.5 Δ x t - 1 , m_{t+1}=m_{t}-\overline{\Delta v}_{t-16}+1.5\left(+{\Delta p}_{D}+\overline{% \Delta q}\right)-0.5{\Delta x}_{t-1}\,,
  2. m t m_{t}\,
  3. Δ v ¯ t - 16 \overline{\Delta v}_{t-16}\,
  4. Δ p D {\Delta p}_{D}\,
  5. Δ q ¯ \overline{\Delta q}\,
  6. Δ x t - 1 {\Delta x}_{t-1}\,
  7. V = X M , V=\frac{X}{M}\,,
  8. P = X Q , P=\frac{X}{Q}\,,
  9. M V = X = P Q . MV=X=PQ\,.
  10. m + v = x = p + q . m+v=x=p+q\,.
  11. Δ , \Delta\,,
  12. Δ m t = m t + 1 - m t . {\Delta m}_{t}=m_{t+1}-m_{t}\,.
  13. Δ m t + Δ v t = Δ x t = Δ p t + Δ q t , {\Delta m}_{t}+{\Delta v}_{t}={\Delta x}_{t}={\Delta p}_{t}+{\Delta q}_{t}\,,
  14. m t + 1 = m t - Δ v t + Δ x t . m_{t+1}=m_{t}-{\Delta v}_{t}+{\Delta x}_{t}\,.
  15. Δ v t Δ v ¯ t - 16 = v t - v t - 16 16 . {\Delta v}_{t}\approx\overline{\Delta v}_{t-16}=\frac{v_{t}-v_{t-16}}{16}\,.
  16. Δ p D , {\Delta p}_{D}\,,
  17. Δ q ¯ . \overline{\Delta q}\,.
  18. Δ x t = Δ p t + Δ q t Δ p D + Δ q ¯ . {\Delta x}_{t}={\Delta p}_{t}+{\Delta q}_{t}\approx{\Delta p}_{D}+\overline{% \Delta q}\,.
  19. Δ q ¯ = 0.0075 \overline{\Delta q}=0.0075\,
  20. Δ p D = 0.0050 {\Delta p}_{D}=0.0050\,
  21. m t + 1 = m t - Δ v t + Δ x t = m t - Δ v ¯ t - 16 + Δ p D + Δ q ¯ + ε t , m_{t+1}=m_{t}-{\Delta v}_{t}+{\Delta x}_{t}=m_{t}-\overline{\Delta v}_{t-16}+{% \Delta p}_{D}+\overline{\Delta q}+\varepsilon_{t}\,,
  22. ε t \varepsilon_{t}\,
  23. ε t = 0.5 ( ( + Δ p D + Δ q ¯ ) - Δ x t - 1 ) , \varepsilon_{t}=0.5\left((+{\Delta p}_{D}+\overline{\Delta q})-{\Delta x}_{t-1% }\right)\,,

McCutcheon_index.html

  1. McCutcheon index = Displacement Distance . \,\text{McCutcheon index}={\,\text{Displacement}\over\,\text{Distance}}.

McGee_graph.html

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

Mean-preserving_spread.html

  1. 1 / 100 1/100
  2. x A i x_{Ai}
  3. x A i = 198 x_{Ai}=198
  4. i = 1 , , 50 i=1,\dots,50
  5. x A i = 202 x_{Ai}=202
  6. i = 51 , , 100 i=51,\dots,100
  7. 1 / 100 1/100
  8. x B i x_{Bi}
  9. x B 1 = 100 x_{B1}=100
  10. x B i = 200 x_{Bi}=200
  11. i = 2 , , 99 i=2,\dots,99
  12. x B 100 = 300 x_{B100}=300
  13. x A x_{A}
  14. x B x_{B}
  15. x B = 𝑑 ( x A + z ) x_{B}\overset{d}{=}(x_{A}+z)
  16. z z
  17. E ( z x A ) = 0 E(z\mid x_{A})=0
  18. x A x_{A}
  19. = 𝑑 \overset{d}{=}
  20. F A F_{A}
  21. F B F_{B}
  22. F A F_{A}
  23. x x
  24. F B F_{B}
  25. x x
  26. x x
  27. x x

Mean_dependence.html

  1. f X 1 f X 2 f_{X_{1}}\perp{}f_{X_{2}}
  2. Cov ( X 1 , X 2 ) = 0 \,\text{Cov}(X_{1},X_{2})=0
  3. X 1 X_{1}
  4. X 2 X_{2}

Mean_percentage_error.html

  1. MPE = 100 % n t = 1 n a t - f t a t \,\text{MPE}=\frac{100\%}{n}\sum_{t=1}^{n}\frac{a_{t}-f_{t}}{a_{t}}

Mean_signed_deviation.html

  1. θ ^ \hat{\theta}
  2. θ \theta
  3. ( θ ^ i , θ i ) (\hat{\theta}_{i},\theta_{i})
  4. θ ^ i \hat{\theta}_{i}
  5. θ \theta
  6. θ = θ i \theta=\theta_{i}
  7. θ i \theta_{i}
  8. θ ^ i \hat{\theta}_{i}
  9. θ i \theta_{i}
  10. MSD ( θ ^ ) = i = 1 n θ i ^ - θ i n . \operatorname{MSD}(\hat{\theta})=\sum^{n}_{i=1}\frac{\hat{\theta_{i}}-\theta_{% i}}{n}.

Mean_systemic_pressure.html

  1. V R M S P - R A P VR\simeq MSP-RAP
  2. S V R = M S P - R A P C O SVR=\frac{MSP-RAP}{CO}

Mean_value_analysis.html

  1. W k ( m ) = L k ( m - 1 ) + 1 μ k . W_{k}(m)=\frac{L_{k}\left(m-1\right)+1}{\mu_{k}}.
  2. λ m = m k = 1 K W k ( m ) v k . \lambda_{m}=\frac{m}{\sum_{k=1}^{K}W_{k}(m)v_{k}}.
  3. L k ( m ) = v k λ m W k ( m ) . L_{k}(m)=v_{k}\lambda_{m}W_{k}(m).
  4. L k ( m - 1 ) m - 1 m L k ( m ) L_{k}(m-1)\approx\frac{m-1}{m}L_{k}(m)

Mechanical_filter.html

  1. S = F x S={F\over x}
  2. Z = S j ω Z={S\over j\omega}
  3. M = F d v / d t = F a M=\frac{F}{dv/dt}={F\over a}
  4. Z = j ω M Z=j\omega M\,
  5. D = F v D={F\over v}
  6. Z = D Z=D\,
  7. [ V F ] = [ z 11 z 12 z 21 z 22 ] [ I v ] \begin{bmatrix}V\\ F\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I\\ v\end{bmatrix}
  8. z 22 z_{22}\,
  9. z 11 z_{11}\,
  10. z 21 z_{21}\,
  11. z 12 z_{12}\,

Mechanical_similarity.html

  1. U ( r ) r k U(r)\propto r^{k}
  2. r r
  3. t t
  4. l l
  5. t l 1 - k / 2 . t\propto l^{1-k/2}.
  6. k = 2 k=2
  7. k = 1 k=1
  8. k = - 1 k=-1

Membrane.html

  1. Q p = F w A Q_{p}=F_{w}\cdot A
  2. k = F w P T M P k={F_{w}\over P_{TMP}}
  3. P T M P = ( P f + P c ) 2 - P p P_{TMP}={(P_{f}+P_{c})\over 2}-P_{p}
  4. r = ( C f - C p ) C f 100 r={(C_{f}-C_{p})\over C_{f}}\cdot 100
  5. Q f = Q p + Q c Q_{f}=Q_{p}+Q_{c}
  6. Q f C f = Q p C p + Q c C c Q_{f}\cdot C_{f}=Q_{p}\cdot C_{p}+Q_{c}\cdot C_{c}
  7. S = Q p e r m e a t e Q f e e d = 1 - Q c o n c e n t r a t e Q f e e d S={Q_{permeate}\over Q_{feed}}=1-{Q_{concentrate}\over Q_{feed}}

Membrane_curvature.html

  1. c 1 = 1 / R 1 c1=1/R1
  2. c 2 = 1 / R 2 c2=1/R2

Meredith_graph.html

  1. ( x - 4 ) ( x - 1 ) 10 x 21 ( x + 1 ) 11 ( x + 3 ) ( x 2 - 13 ) ( x 6 - 26 x 4 + 3 x 3 + 169 x 2 - 39 x - 45 ) 4 (x-4)(x-1)^{10}x^{21}(x+1)^{11}(x+3)(x^{2}-13)(x^{6}-26x^{4}+3x^{3}+169x^{2}-3% 9x-45)^{4}

Metal-mesh_optical_filter.html

  1. t a t<<a
  2. λ > g \lambda>g
  3. Γ ( ω ) \Gamma(\omega)
  4. Γ \Gamma
  5. ω = g / λ \omega=g/\lambda
  6. τ ( ω ) \tau(\omega)
  7. τ ( ω ) = [ 1 + Γ ( ω ) ] \tau(\omega)=\left[1+\Gamma(\omega)\right]
  8. | Γ ( w ) | 2 + | τ ( ω ) | 2 = 1 \left|\Gamma(w)\right|^{2}+\left|\tau(\omega)\right|^{2}=1
  9. ϕ Γ ( ω ) \phi_{\Gamma}(\omega)
  10. ϕ τ ( ω ) \phi_{\tau}(\omega)
  11. | τ ( ω ) | 2 \left|\tau(\omega)\right|^{2}
  12. sin 2 ϕ Γ = 1 - | τ ( ω ) | 2 \sin^{2}\phi_{\Gamma}=1-\left|\tau(\omega)\right|^{2}
  13. sin 2 ϕ τ = | τ ( ω ) | 2 \sin^{2}\phi_{\tau}=\left|\tau(\omega)\right|^{2}
  14. | Γ ( ω ) | 2 = sin 2 ϕ Γ ( ω ) = 1 - sin 2 ϕ τ ( ω ) \left|\Gamma(\omega)\right|^{2}=\sin^{2}\phi_{\Gamma}(\omega)=1-\sin^{2}\phi_{% \tau}(\omega)
  15. Γ ( ω ) \Gamma(\omega)
  16. ω \omega
  17. [ R e ( - 1 / 2 ) , I m ( 0 ) ] \left[Re(-1/2),Im(0)\right]
  18. ( I m ( Γ ( ω ) ) > 0 ) \left(Im(\Gamma(\omega))>0\right)
  19. ( I m ( Γ ( ω ) ) < 0 ) \left(Im(\Gamma(\omega))<0\right)
  20. ω \omega
  21. ( ϕ τ ( ω ) ϕ Γ ( ω ) ) \left(\phi_{\tau}(\omega)\neq\phi_{\Gamma}(\omega)\right)
  22. τ ( ω ) \tau(\omega)
  23. Γ ( ω ) \Gamma(\omega)
  24. [ τ i n d + τ c a p ] = 1 \left[\tau_{ind}+\tau_{cap}\right]=1
  25. τ i n d ( ω ) = - Γ c a p ( ω ) \tau_{ind}\left(\omega\right)=-\Gamma_{cap}(\omega)
  26. τ c a p ( ω ) = - Γ i n d ( ω ) \tau_{cap}\left(\omega\right)=-\Gamma_{ind}(\omega)
  27. | τ c a p ( w ) | 2 + | τ i n d ( ω ) | 2 = 1 \left|\tau_{cap}(w)\right|^{2}+\left|\tau_{ind}(\omega)\right|^{2}=1
  28. τ c a p ( ω ) \tau_{cap}\left(\omega\right)
  29. τ i n d ( ω ) \tau_{ind}\left(\omega\right)
  30. ω 0 \omega\rightarrow 0
  31. τ i n d ( ω 0 ) = 0 \tau_{ind}\left(\omega\rightarrow 0\right)=0
  32. τ c a p ( ω 0 ) = 1 \tau_{cap}\left(\omega\rightarrow 0\right)=1
  33. ρ = 1 / δ σ \rho=1/\delta\sigma
  34. σ \sigma
  35. δ \delta
  36. Γ ( ω ) \Gamma(\omega)
  37. 2 Γ ( ω ) 2\Gamma(\omega)
  38. J ¯ = Γ ( ω ) * c / 4 π \bar{J}=\Gamma(\omega)*c/4\pi
  39. P D = 2 ρ J ¯ 2 P_{D}=2\rho\bar{J}^{2}
  40. η \eta
  41. η = g / 2 a \eta=g/2a
  42. η = 1 / ( 1 - 2 a / g ) \eta=1/(1-2a/g)
  43. P D = 2 ρ η J ¯ 2 P_{D}=2\rho\eta\bar{J}^{2}
  44. A = P d / P o A=P_{d}/P_{o}
  45. P o P_{o}
  46. A = | Γ | 2 2 ρ η = | Γ | 2 η ( c λ σ ) 1 / 2 A=\left|\Gamma\right|^{2}2\rho\eta=\left|\Gamma\right|^{2}\eta\left(\frac{c}{% \lambda\sigma}\right)^{1/2}
  47. 10 - 4 10^{-4}
  48. 10 - 2 10^{-2}
  49. | τ c a p ( ω ) | 2 \left|\tau_{cap}(\omega)\right|^{2}
  50. | τ i n d ( ω ) | 2 \left|\tau_{ind}(\omega)\right|^{2}
  51. ϕ c a p ( ω ) \phi_{cap}\left(\omega\right)
  52. ϕ i n d ( ω ) \phi_{ind}\left(\omega\right)
  53. Γ ( ω ) ϕ ( ω ) \Gamma(\omega)\phi(\omega)
  54. 2 Y ( ω ) 2Y(\omega)
  55. ω < 1 \omega<1
  56. 2 Y ( ω ) 2Y(\omega)
  57. Γ ( ω ) \Gamma(\omega)
  58. τ ( ω ) \tau(\omega)
  59. Γ ( ω ) = - Y ( ω ) 1 + Y ( ω ) \Gamma(\omega)=\frac{-Y(\omega)}{1+Y(\omega)}
  60. τ ( ω ) = 1 1 + Y ( ω ) \tau(\omega)=\frac{1}{1+Y(\omega)}
  61. τ ( ω ) = [ 1 + Γ ( ω ) ] \tau(\omega)=\left[1+\Gamma(\omega)\right]
  62. Y ( ω ) = i B ( ω ) Y\left(\omega\right)=iB(\omega)
  63. B ( ω ) B\left(\omega\right)
  64. ω \omega
  65. B i n d ( ω ) B c a p ( ω ) = - 1 B_{ind}\left(\omega\right)B_{cap}(\omega)=-1
  66. B ( ω ) B\left(\omega\right)
  67. B ( ω ) B\left(\omega\right)
  68. L L
  69. C C
  70. R R
  71. 2 C 2C
  72. L / 2 L/2
  73. L i n d = C c a p L_{ind}=C_{cap}
  74. ω 1 \omega\rightarrow 1
  75. τ c a p ( ω 1 ) = 0 \tau_{cap}\left(\omega\rightarrow 1\right)=0
  76. τ i n d ( ω 1 ) = 1 \tau_{ind}\left(\omega\rightarrow 1\right)=1
  77. ω 0 \omega\rightarrow 0
  78. ω 1 \omega\rightarrow 1
  79. R R
  80. ( ω = ω o ) \left(\omega=\omega_{o}\right)
  81. Z o = i ω L = 1 / i ω C Z_{o}=i\omega L=1/i\omega C
  82. Z o Z_{o}
  83. ω o \omega_{o}
  84. a / g a/g
  85. R R
  86. R = η / 2 ( c λ σ ) 1 / 2 R=\eta/2\left(\frac{c}{\lambda\sigma}\right)^{1/2}
  87. C C
  88. L L
  89. R R
  90. Z o ( ω o ) Z_{o}\left(\omega_{o}\right)
  91. Z o = i ω L = 1 / ( i ω C ) Z_{o}=i\omega L=1/\left(i\omega C\right)
  92. Ω ( ω ) \Omega\left(\omega\right)
  93. Ω ( ω ) = ( ω / ω o ) - ( ω o / ω ) = ( λ o / λ ) - ( λ / λ o ) \Omega\left(\omega\right)=\left(\omega/\omega_{o}\right)-\left(\omega_{o}/% \omega\right)=\left(\lambda_{o}/\lambda\right)-\left(\lambda/\lambda_{o}\right)
  94. Y ( ω ) Y\left(\omega\right)
  95. 1 1 + i Z o Ω \frac{1}{1+iZ_{o}\Omega}
  96. 1 1 - i Z o / Ω \frac{1}{1-iZ_{o}/\Omega}
  97. | Γ ( ω ) | 2 \left|\Gamma\left(\omega\right)\right|^{2}
  98. 1 ( 1 + R ) 2 + Z o 2 Ω 2 \frac{1}{(1+R)^{2}+Z_{o}^{2}\Omega^{2}}
  99. 1 ( 1 + R ) 2 + Z o 2 / Ω 2 \frac{1}{(1+R)^{2}+Z_{o}^{2}/\Omega^{2}}
  100. | τ ( ω ) | 2 \left|\tau\left(\omega\right)\right|^{2}
  101. R 2 + Z o 2 Ω 2 ( 1 + R ) 2 + Z o 2 Ω 2 \frac{R^{2}+Z_{o}^{2}\Omega^{2}}{(1+R)^{2}+Z_{o}^{2}\Omega^{2}}
  102. R 2 + Z o 2 / Ω 2 ( 1 + R ) 2 + Z o 2 / Ω 2 \frac{R^{2}+Z_{o}^{2}/\Omega^{2}}{(1+R)^{2}+Z_{o}^{2}/\Omega^{2}}
  103. ϕ Γ ( ω ) \phi_{\Gamma}\left(\omega\right)
  104. π - arctan ( Z o Ω ( 1 + R ) ) \pi-\arctan{\left(\frac{Z_{o}\Omega}{(1+R)}\right)}
  105. π + arctan ( Z o ( 1 + R ) Ω ) \pi+\arctan{\left(\frac{Z_{o}}{(1+R)\Omega}\right)}
  106. ϕ τ ( ω ) \phi_{\tau}\left(\omega\right)
  107. arctan ( Z o Ω R ( 1 + R ) + Z o 2 Ω 2 ) \arctan{\left(\frac{Z_{o}\Omega}{R(1+R)+Z_{o}^{2}\Omega^{2}}\right)}
  108. - arctan ( Z o / Ω R ( 1 + R ) + Z o 2 / Ω 2 ) -\arctan{\left(\frac{Z_{o}/\Omega}{R(1+R)+Z_{o}^{2}/\Omega^{2}}\right)}
  109. A ( ω ) A\left(\omega\right)
  110. 2 R | Γ | 2 2R\left|\Gamma\right|^{2}
  111. λ < g \lambda<g
  112. a a
  113. g g
  114. t t
  115. .4 μ m \approx.4\mu m
  116. .9 μ m .9\mu m
  117. 1.5 μ m 1.5\mu m

Metamaterial_antenna.html

  1. R R
  2. L L
  3. G G
  4. C C
  5. j j
  6. ω \omega

METATOY.html

  1. f f
  2. 2 f 2f
  3. f 1 f_{1}
  4. f 2 f_{2}
  5. f 1 + f 2 f_{1}+f_{2}
  6. f 1 f_{1}
  7. f 2 f_{2}
  8. f 1 + f 2 f_{1}+f_{2}
  9. ( x , y , z ) (x,y,z)
  10. u ( x , y , z ) u(x,y,z)
  11. 𝐫 ( x , y , z ) = ϕ ( x , y , z ) , \mathbf{r}(x,y,z)=\nabla\phi(x,y,z),
  12. ϕ ( x , y , z ) \phi(x,y,z)
  13. u ( x , y , z ) = A ( x , y , z ) exp ( i ϕ ( x , y , z ) ) u(x,y,z)=A(x,y,z)\exp(i\phi(x,y,z))
  14. × ϕ ( x , y , z ) = 0 , \nabla\times\nabla\phi(x,y,z)=0,
  15. × 𝐫 ( x , y , z ) = 0. \nabla\times\mathbf{r}(x,y,z)=0.

Method_of_conditional_probabilities.html

  1. u V 1 d ( u ) + 1 | V | D + 1 . \sum_{u\in V}\frac{1}{d(u)+1}~{}\geq~{}\frac{|V|}{D+1}.
  2. | S ( t ) | + w R ( t ) 1 d ( w ) + 1 . |S^{(t)}|~{}+~{}\sum_{w\in R^{(t)}}\frac{1}{d(w)+1}.
  3. w N ( t ) ( u ) { u } 1 d ( w ) + 1 \sum_{w\in N^{(t)}(u)\cup\{u\}}\frac{1}{d(w)+1}
  4. 1 - w N ( t ) ( u ) { u } 1 d ( w ) + 1 , 1-\sum_{w\in N^{(t)}(u)\cup\{u\}}\frac{1}{d(w)+1},
  5. w N ( t ) ( u ) { u } 1 d ( w ) + 1 ( d ( u ) + 1 ) 1 d ( u ) + 1 = 1 \sum_{w\in N^{(t)}(u)\cup\{u\}}\frac{1}{d(w)+1}\leq(d(u)+1)\frac{1}{d(u)+1}=1
  6. w N ( t ) ( u ) { u } 1 d ( w ) + 1 ( d ( u ) + 1 ) 1 d ( u ) + 1 = 1 \sum_{w\in N^{(t)}(u)\cup\{u\}}\frac{1}{d(w)+1}\leq(d^{\prime}(u)+1)\frac{1}{d% ^{\prime}(u)+1}=1

Method_of_continuity.html

  1. ( L t ) t [ 0 , 1 ] (L_{t})_{t\in[0,1]}
  2. t [ 0 , 1 ] t\in[0,1]
  3. x B x\in B
  4. || x || B C || L t ( x ) || V . ||x||_{B}\leq C||L_{t}(x)||_{V}.
  5. L 0 L_{0}
  6. L 1 L_{1}
  7. L 0 L_{0}
  8. L 1 L_{1}
  9. || L 0 - L 1 || 1 / ( 3 C ) ||L_{0}-L_{1}||\leq 1/(3C)
  10. L 0 L_{0}
  11. L 1 ( B ) V L_{1}(B)\subseteq V
  12. L 1 ( B ) V L_{1}(B)\subseteq V
  13. y V y\in V
  14. || y || V 1 ||y||_{V}\leq 1
  15. dist ( y , L 1 ( B ) ) > 2 / 3 \mathrm{dist}(y,L_{1}(B))>2/3
  16. y = L 0 ( x ) y=L_{0}(x)
  17. x B x\in B
  18. || x || B C || y || V ||x||_{B}\leq C||y||_{V}
  19. || y - L 1 ( x ) || V = || ( L 0 - L 1 ) ( x ) || V || L 0 - L 1 || || x || B 1 / 3 , ||y-L_{1}(x)||_{V}=||(L_{0}-L_{1})(x)||_{V}\leq||L_{0}-L_{1}||||x||_{B}\leq 1/3,
  20. L 1 ( x ) L 1 ( B ) L_{1}(x)\in L_{1}(B)

Method_of_quantum_characteristics.html

  1. n n
  2. 2 n 2n
  3. ξ i = ( x 1 , , x n , p 1 , , p n ) 2 n , \xi^{i}=(x^{1},...,x^{n},p_{1},...,p_{n})\in\mathbb{R}^{2n},
  4. { ξ k , ξ l } = - I k l . \{\xi^{k},\xi^{l}\}=-I^{kl}.
  5. I k l I^{kl}
  6. I = 0 - E n E n 0 , \left\|I\right\|=\left\|\begin{array}[]{ll}0&-E_{n}\\ E_{n}&0\end{array}\right\|,
  7. E n E_{n}
  8. n × n n\times n
  9. ξ \xi
  10. ξ ^ i = ( x ^ 1 , , x ^ n , p ^ 1 , , p ^ n ) Op ( L 2 ( n ) ) . \hat{\xi}^{i}=(\hat{x}^{1},...,\hat{x}^{n},\hat{p}_{1},...,\hat{p}_{n})\in% \operatorname{Op}(L^{2}(\mathbb{R}^{n})).
  11. [ ξ ^ k , ξ ^ l ] = - i I k l . [\hat{\xi}^{k},\hat{\xi}^{l}]=-i\hbar I^{kl}.
  12. ξ i ξ ^ i \xi^{i}\rightarrow\hat{\xi}^{i}
  13. f ( ξ ) f ^ f(\xi)\to\hat{f}
  14. f ^ = f ( ξ ^ ) s = 0 1 s ! s f ( 0 ) ξ i 1 ξ i s ξ ^ i 1 ξ ^ i s . \hat{f}=f(\hat{\xi})\equiv\sum_{s=0}^{\infty}\frac{1}{s!}\frac{\partial^{s}f(0% )}{\partial\xi^{i_{1}}...\partial\xi^{i_{s}}}\hat{\xi}^{i_{1}}...\hat{\xi}^{i_% {s}}.
  15. ξ ^ \hat{\xi}
  16. f ( ξ ) f(\xi)
  17. f ^ \hat{f}
  18. f ( ξ ) f ^ f(\xi)\leftarrow\hat{f}
  19. c c
  20. 𝕍 \mathbb{V}
  21. f ( ξ ) f ^ g ( ξ ) g ^ c × f ( ξ ) c × f ^ f ( ξ ) + g ( ξ ) f ^ + g ^ } vector space 𝕍 f ( ξ ) g ( ξ ) f ^ g ^ } algebra \left.\begin{array}[]{c}\begin{array}[]{c}\left.\begin{array}[]{ccc}f(\xi)&% \longleftrightarrow&\hat{f}\\ g(\xi)&\longleftrightarrow&\hat{g}\\ c\times f(\xi)&\longleftrightarrow&c\times\hat{f}\\ f(\xi)+g(\xi)&\longleftrightarrow&\hat{f}+\hat{g}\end{array}\right\}\;\,\text{% vector space}\;\;\mathbb{V}\end{array}\\ \begin{array}[]{ccc}{f(\xi)\star g(\xi)}&{\longleftrightarrow}&\;\;{\hat{f}% \hat{g}}\end{array}\end{array}\right\}{\,\text{algebra}}
  22. f ( ξ ) f(\xi)
  23. g ( ξ ) g(\xi)
  24. f ^ \hat{f}
  25. g ^ \hat{g}
  26. 𝕍 \mathbb{V}
  27. ξ i \xi_{i}
  28. B ^ ( ξ ) = d 2 n η ( 2 π ) n exp ( - i η k ( ξ - ξ ^ ) k ) 𝕍 . \hat{B}(\xi)=\int\frac{d^{2n}\eta}{(2\pi\hbar)^{n}}\exp(-\frac{i}{\hbar}\eta_{% k}(\xi-\hat{\xi})^{k})\in\mathbb{V}.
  29. f ( ξ ) f(\xi)
  30. f ^ \hat{f}
  31. f ( ξ ) = Tr [ B ^ ( ξ ) f ^ ] , f(\xi)=\operatorname{Tr}[\hat{B}(\xi)\hat{f}],
  32. f ^ = d 2 n ξ ( 2 π ) n f ( ξ ) B ^ ( ξ ) . \hat{f}=\int\frac{d^{2n}\xi}{(2\pi\hbar)^{n}}f(\xi)\hat{B}(\xi).
  33. f ( ξ ) f(\xi)
  34. f ^ \hat{f}
  35. B ^ ( ξ ) \hat{B}(\xi)
  36. d 2 n ξ ( 2 π ) n B ^ ( ξ ) Tr [ B ^ ( ξ ) f ^ ] = f ^ , \int\frac{d^{2n}\xi}{(2\pi\hbar)^{n}}\hat{B}(\xi)\operatorname{Tr}[\hat{B}(\xi% )\hat{f}]=\hat{f},
  37. Tr [ B ^ ( ξ ) B ^ ( ξ ) ] = ( 2 π ) n δ 2 n ( ξ - ξ ) . \operatorname{Tr}[\hat{B}(\xi)\hat{B}(\xi^{\prime})]=(2\pi\hbar)^{n}\delta^{2n% }(\xi-\xi^{\prime}).
  38. O p ( L 2 ( n ) ) Op(L^{2}(\mathbb{R}^{n}))
  39. 𝕍 \mathbb{V}
  40. f ( ξ ) = T r [ B ^ ( ξ ) f ^ ] and g ( ξ ) = T r [ B ^ ( ξ ) g ^ ] , f(\xi)=Tr[\hat{B}(\xi)\hat{f}]~{}~{}\mathrm{and}~{}~{}g(\xi)=Tr[\hat{B}(\xi)% \hat{g}],
  41. f ( ξ ) g ( ξ ) = T r [ B ^ ( ξ ) f ^ g ^ ] f(\xi)\star g(\xi)=Tr[\hat{B}(\xi)\hat{f}\hat{g}]
  42. \star
  43. f ( ξ ) g ( ξ ) = f ( ξ ) exp ( i 2 𝒫 ) g ( ξ ) . f(\xi)\star g(\xi)=f(\xi)\exp(\frac{i\hbar}{2}\mathcal{P})g(\xi).
  44. 𝒫 = - I k l ξ k ξ l \mathcal{P}=-{I}^{kl}\overleftarrow{\frac{\partial}{\partial\xi^{k}}}% \overrightarrow{\frac{\partial}{\partial\xi^{l}}}
  45. \star
  46. f g = f g + i 2 f g . f\star g=f\circ g+\frac{i\hbar}{2}f\wedge g.
  47. \circ
  48. \circ
  49. f g f\wedge g
  50. ξ ξ ^ \xi\leftrightarrow\hat{\xi}
  51. 𝐔 ^ \mathbf{\hat{U}}
  52. U ^ = exp ( - i H ^ τ ) , \hat{U}=\exp\Bigl(-\frac{i}{\hbar}\hat{H}\tau\Bigr),
  53. H ^ \hat{H}
  54. ξ q ξ ´ \xi\stackrel{q}{\longrightarrow}\acute{\xi}
  55. \updownarrow\;\;\;\;\;\;\updownarrow
  56. ξ ^ U ^ ξ ^ ´ , \hat{\xi}\stackrel{\hat{U}}{\longrightarrow}\acute{\hat{\xi}},
  57. ξ ^ i ξ ^ i ´ = U ^ + ξ ^ i U ^ . \hat{\xi}^{i}\rightarrow\acute{\hat{\xi}^{i}}=\hat{U}^{+}\hat{\xi}^{i}\hat{U}.
  58. ξ ´ i \acute{\xi}^{i}
  59. ξ ^ i ´ \acute{\hat{\xi}^{i}}
  60. B ^ ( ξ ) \hat{B}(\xi)
  61. ξ i ξ ´ i = q i ( ξ , τ ) = T r [ B ^ ( ξ ) U ^ + ξ ^ i U ^ ] , \xi^{i}\rightarrow\acute{\xi}^{i}=q^{i}(\xi,\tau)=Tr[\hat{B}(\xi)\hat{U}^{+}% \hat{\xi}^{i}\hat{U}],
  62. q i ( ξ , 0 ) = ξ i . q^{i}(\xi,0)=\xi^{i}.
  63. q i ( ξ , τ ) q^{i}(\xi,\tau)
  64. τ \tau
  65. ξ ^ \hat{\xi}
  66. f ^ f ^ ´ = U ^ + f ^ U ^ \hat{f}\rightarrow\acute{\hat{f}}=\hat{U}^{+}\hat{f}\hat{U}
  67. f ( ξ ) q f ´ ( ξ ) = T r [ B ^ ( ξ ) U ^ + f ^ U ^ ] f(\xi)\stackrel{q}{\longrightarrow}\acute{f}(\xi)=Tr[\hat{B}(\xi)\hat{U}^{+}% \hat{f}\hat{U}]
  68. \updownarrow\;\;\;\;\;\;\;\;\;\;\,\updownarrow
  69. f ^ U ^ f ^ ´ = U ^ + f ^ U ^ \hat{f}\;\;\;\;\stackrel{\hat{U}}{\longrightarrow}\,\acute{\hat{f}}\;\;\;\;\;=% \hat{U}^{+}\hat{f}\hat{U}
  70. f ( ξ ) f(\xi)
  71. f ( ξ ) f ´ ( ξ ) T r [ B ^ ( ξ ) U + ^ f ( ξ ^ ) U ^ ] = s = 0 1 s ! s f ( 0 ) ξ i 1 ξ i s q i 1 ( ξ , τ ) q i s ( ξ , τ ) f ( q ( ξ , τ ) ) . f(\xi)\rightarrow\acute{f}(\xi)\equiv Tr[\hat{B}(\xi)\hat{U^{+}}f(\hat{\xi})% \hat{U}]=\sum_{s=0}^{\infty}\frac{1}{s!}\frac{\partial^{s}f(0)}{\partial\xi^{i% _{1}}...\partial\xi^{i_{s}}}q^{i_{1}}(\xi,\tau)\star...\star q^{i_{s}}(\xi,% \tau)\equiv f(\star q(\xi,\tau)).
  72. \star
  73. f ( q ( ξ , τ ) ) f(\star q(\xi,\tau))
  74. f ( q ( ξ , τ ) ) f(q(\xi,\tau))
  75. \hbar
  76. q i ( ξ , τ ) q^{i}(\xi,\tau)
  77. τ f ^ = - i [ f ^ , H ^ ] , \frac{\partial}{\partial\tau}\hat{f}=-\frac{i}{\hbar}[\hat{f},\hat{H}],
  78. τ f ( ξ , τ ) = f ( ξ , τ ) H ( ξ ) . \frac{\partial}{\partial\tau}f(\xi,\tau)=f(\xi,\tau)\wedge H(\xi).
  79. \star
  80. f ( ξ , τ ) = f ( q ( ξ , τ ) , 0 ) . f(\xi,\tau)=f(\star q(\xi,\tau),0).
  81. W ( ξ , τ ) = W ( q ( ξ , - τ ) , 0 ) . W(\xi,\tau)=W(\star q(\xi,-\tau),0).
  82. τ q i ( ξ , τ ) = { ζ i , H ( ζ ) } | ζ = q ( ξ , τ ) . \frac{\partial}{\partial\tau}q^{i}(\xi,\tau)=\{\zeta^{i},H(\zeta)\}|_{\zeta=% \star q(\xi,\tau)}.
  83. \star
  84. \star
  85. τ \tau
  86. q i ( ξ , τ ) q j ( ξ , τ ) = ξ i ξ j = - I i j . q^{i}(\xi,\tau)\wedge q^{j}(\xi,\tau)=\xi^{i}\wedge\xi^{j}=-{I}^{ij}.
  87. ξ ξ ´ = q ( ξ , τ ) , \xi\rightarrow\acute{\xi}=q(\xi,\tau),
  88. q ( ξ , τ 1 + τ 2 ) = q ( q ( ξ , τ 1 ) , τ 2 ) , q(\xi,\tau_{1}+\tau_{2})=q(\star q(\xi,\tau_{1}),\tau_{2}),
  89. H ( ξ ) = H ( q ( ξ , τ ) ) H(\xi)=H(\star q(\xi,\tau))
  90. H ( ξ ) = T r [ B ^ ( ξ ) H ^ ] H(\xi)=Tr[\hat{B}(\xi)\hat{H}]
  91. H ( ξ ) H(\xi)
  92. ξ ^ i ξ ^ i ( τ ) = U ^ + ξ ^ i U ^ . \hat{\xi}^{i}\rightarrow\hat{\xi}^{i}(\tau)=\hat{U}^{+}\hat{\xi}^{i}\hat{U}.
  93. f ^ \hat{f}
  94. f ^ \hat{f}
  95. f ( ξ ^ ) f(\hat{\xi})
  96. f ^ \hat{f}
  97. f ^ ( τ ) = U + f ^ U = U + f ( ξ ^ ) U = f ( U + ξ ^ U ) = f ( ξ ^ ( τ ) ) . \hat{f}(\tau)=U^{+}\hat{f}U=U^{+}f(\hat{\xi})U=f(U^{+}\hat{\xi}U)=f(\hat{\xi}(% \tau)).
  98. ξ ^ ( τ ) \hat{\xi}(\tau)
  99. CLASSICAL DYNAMICS \mathrm{CLASSICAL\;DYNAMICS}
  100. QUANTUM DYNAMICS \mathrm{QUANTUM\;DYNAMICS}
  101. τ ρ ( ξ , τ ) = - { ρ ( ξ , τ ) , ( ξ ) } \frac{\partial}{\partial\tau}\rho(\xi,\tau)=-\{\rho(\xi,\tau),\mathcal{H}(\xi)\}
  102. τ W ( ξ , τ ) = - W ( ξ , τ ) H ( ξ ) \frac{\partial}{\partial\tau}W(\xi,\tau)=-W(\xi,\tau)\wedge H(\xi)
  103. τ c i ( ξ , τ ) = { ζ i , ( ζ ) } | ζ = c ( ξ , τ ) \frac{\partial}{\partial\tau}c^{i}(\xi,\tau)=\{\zeta^{i},\mathcal{H}(\zeta)\}|% _{\zeta=c(\xi,\tau)}
  104. τ q i ( ξ , τ ) = { ζ i , H ( ζ ) } | ζ = q ( ξ , τ ) \frac{\partial}{\partial\tau}q^{i}(\xi,\tau)=\{\zeta^{i},H(\zeta)\}|_{\zeta=% \star q(\xi,\tau)}
  105. c i ( ξ , 0 ) = ξ i c^{i}(\xi,0)=\xi^{i}
  106. q i ( ξ , 0 ) = ξ i q^{i}(\xi,0)=\xi^{i}
  107. \star
  108. c ( ξ , τ 1 + τ 2 ) = c ( c ( ξ , τ 1 ) , τ 2 ) c(\xi,\tau_{1}+\tau_{2})=c(c(\xi,\tau_{1}),\tau_{2})
  109. q ( ξ , τ 1 + τ 2 ) = q ( q ( ξ , τ 1 ) , τ 2 ) q(\xi,\tau_{1}+\tau_{2})=q(\star q(\xi,\tau_{1}),\tau_{2})
  110. { c i ( ξ , τ ) , c j ( ξ , τ ) } = { ξ i , ξ j } \{c^{i}(\xi,\tau),c^{j}(\xi,\tau)\}=\{\xi^{i},\xi^{j}\}
  111. q i ( ξ , τ ) q j ( ξ , τ ) = ξ i ξ j q^{i}(\xi,\tau)\wedge q^{j}(\xi,\tau)=\xi^{i}\wedge\xi^{j}
  112. H ( ξ ) = H ( c ( ξ , τ ) ) H(\xi)=H(c(\xi,\tau))
  113. H ( ξ ) = H ( q ( ξ , τ ) ) H(\xi)=H(\star q(\xi,\tau))
  114. ρ ( ξ , τ ) = ρ ( c ( ξ , - τ ) , 0 ) \rho(\xi,\tau)=\rho(c(\xi,-\tau),0)
  115. W ( ξ , τ ) = W ( q ( ξ , - τ ) , 0 ) W(\xi,\tau)=W(\star q(\xi,-\tau),0)
  116. c i ( ξ , τ ) c^{i}(\xi,\tau)
  117. q i ( ξ , τ ) q^{i}(\xi,\tau)
  118. q i ( ξ , τ ) q^{i}(\xi,\tau)
  119. f ( ξ , τ ) f(\xi,\tau)
  120. \star
  121. \hbar
  122. \hbar

Method_of_steepest_descent.html

  1. C f ( z ) e λ g ( z ) d z \int_{C}f(z)e^{\lambda g(z)}dz
  2. M = sup x C ( S ( x ) ) < , M=\sup_{x\in C}\Re(S(x))<\infty,
  3. ( ) \Re(\cdot)
  4. C | f ( x ) e λ 0 S ( x ) | d x < , \int_{C}\left|f(x)e^{\lambda_{0}S(x)}\right|dx<\infty,
  5. | C f ( x ) e λ S ( x ) d x | const e λ M , λ , λ λ 0 . \left|\int_{C}f(x)e^{\lambda S(x)}dx\right|\leqslant\,\text{const}\cdot e^{% \lambda M},\qquad\forall\lambda\in\mathbb{R},\quad\lambda\geqslant\lambda_{0}.
  6. | C f ( x ) e λ S ( x ) d x | C | f ( x ) | | e λ S ( x ) | d x C | f ( x ) | e λ M | e λ 0 ( S ( x ) - M ) e ( λ - λ 0 ) ( S ( x ) - M ) | d x C | f ( x ) | e λ M | e λ 0 ( S ( x ) - M ) | d x | e ( λ - λ 0 ) ( S ( x ) - M ) | 1 = e - λ 0 M C | f ( x ) e λ 0 S ( x ) | d x const e λ M . \begin{aligned}\displaystyle\left|\int_{C}f(x)e^{\lambda S(x)}dx\right|&% \displaystyle\leqslant\int_{C}|f(x)|\left|e^{\lambda S(x)}\right|dx\\ &\displaystyle\equiv\int_{C}|f(x)|e^{\lambda M}\left|e^{\lambda_{0}(S(x)-M)}e^% {(\lambda-\lambda_{0})(S(x)-M)}\right|dx\\ &\displaystyle\leqslant\int_{C}|f(x)|e^{\lambda M}\left|e^{\lambda_{0}(S(x)-M)% }\right|dx&&\displaystyle\left|e^{(\lambda-\lambda_{0})(S(x)-M)}\right|% \leqslant 1\\ &\displaystyle=\underbrace{e^{-\lambda_{0}M}\int_{C}\left|f(x)e^{\lambda_{0}S(% x)}\right|dx}_{\,\text{const}}\cdot e^{\lambda M}.\end{aligned}
  7. x x
  8. n n
  9. S x x ′′ ( x ) ( 2 S ( x ) x i x j ) , 1 i , j n , S^{\prime\prime}_{xx}(x)\equiv\left(\frac{\partial^{2}S(x)}{\partial x_{i}% \partial x_{j}}\right),\qquad 1\leqslant i,\,j\leqslant n,
  10. S ( x ) S(x)
  11. s y m b o l φ ( x ) = ( φ 1 ( x ) , φ 2 ( x ) , , φ k ( x ) ) symbol{\varphi}(x)=(\varphi_{1}(x),\varphi_{2}(x),\ldots,\varphi_{k}(x))
  12. s y m b o l φ x ( x ) ( φ i ( x ) x j ) , 1 i k , 1 j n . symbol{\varphi}_{x}^{\prime}(x)\equiv\left(\frac{\partial\varphi_{i}(x)}{% \partial x_{j}}\right),\qquad 1\leqslant i\leqslant k,\quad 1\leqslant j% \leqslant n.
  13. S ( z ) S(z)
  14. det S z z ′′ ( z 0 ) 0 \det S^{\prime\prime}_{zz}(z^{0})\neq 0
  15. S ( z ) S(z)
  16. S S
  17. W W
  18. S S
  19. det S z z ′′ ( z 0 ) 0 \det S^{\prime\prime}_{zz}(z^{0})\neq 0
  20. U W U⊂W
  21. w = 0 w=0
  22. φ : V U \mathbf{φ}:V→U
  23. w V : S ( s y m b o l φ ( w ) ) = S ( z 0 ) + 1 2 j = 1 n μ j w j 2 , \detsymbol φ w ( 0 ) = 1 , \forall w\in V:\qquad S(symbol{\varphi}(w))=S(z^{0})+\frac{1}{2}\sum_{j=1}^{n}% \mu_{j}w_{j}^{2},\quad\detsymbol{\varphi}_{w}^{\prime}(0)=1,
  24. S z z ′′ ( z 0 ) S_{zz}^{\prime\prime}(z^{0})
  25. f ( 0 ) = 0 f(0)=0
  26. f ( z ) = i = 1 n z i g i ( z ) , f(z)=\sum_{i=1}^{n}z_{i}g_{i}(z),
  27. g i ( 0 ) = f ( z ) z i | z = 0 . g_{i}(0)=\left.\tfrac{\partial f(z)}{\partial z_{i}}\right|_{z=0}.
  28. f ( z ) = 0 1 d d t f ( t z 1 , , t z n ) d t = i = 1 n z i 0 1 f ( z ) z i | z = ( t z 1 , , t z n ) d t , f(z)=\int_{0}^{1}\frac{d}{dt}f\left(tz_{1},\cdots,tz_{n}\right)dt=\sum_{i=1}^{% n}z_{i}\int_{0}^{1}\left.\frac{\partial f(z)}{\partial z_{i}}\right|_{z=(tz_{1% },\ldots,tz_{n})}dt,
  29. g i ( z ) = 0 1 f ( z ) z i | z = ( t z 1 , , t z n ) d t g_{i}(z)=\int_{0}^{1}\left.\frac{\partial f(z)}{\partial z_{i}}\right|_{z=(tz_% {1},\ldots,tz_{n})}dt
  30. g i ( 0 ) = f ( z ) z i | z = 0 . g_{i}(0)=\left.\frac{\partial f(z)}{\partial z_{i}}\right|_{z=0}.
  31. S ( 0 ) = 0 S(0)=0
  32. S ( z ) = i = 1 n z i g i ( z ) . S(z)=\sum_{i=1}^{n}z_{i}g_{i}(z).
  33. S ( z ) z i | z = 0 = g i ( 0 ) = 0 , \left.\frac{\partial S(z)}{\partial z_{i}}\right|_{z=0}=g_{i}(0)=0,
  34. S ( z ) = i , j = 1 n z i z j h i j ( z ) . S(z)=\sum_{i,j=1}^{n}z_{i}z_{j}h_{ij}(z).
  35. A A
  36. A i j = A i j ( s ) + A i j ( a ) , A i j ( s ) = 1 2 ( A i j + A j i ) , A i j ( a ) = 1 2 ( A i j - A j i ) . A_{ij}=A_{ij}^{(s)}+A_{ij}^{(a)},\qquad A_{ij}^{(s)}=\tfrac{1}{2}\left(A_{ij}+% A_{ji}\right),\qquad A_{ij}^{(a)}=\tfrac{1}{2}\left(A_{ij}-A_{ji}\right).
  37. A A
  38. i , j B i j A i j = i , j B i j A i j ( s ) , \sum_{i,j}B_{ij}A_{ij}=\sum_{i,j}B_{ij}A_{ij}^{(s)},
  39. A A
  40. i , j B i j C i j = i , j B j i C j i = - i , j B i j C i j = 0. \sum_{i,j}B_{ij}C_{ij}=\sum_{i,j}B_{ji}C_{ji}=-\sum_{i,j}B_{ij}C_{ij}=0.
  41. i i
  42. j j
  43. 2 S ( z ) z i z j | z = 0 = 2 h i j ( 0 ) ; \left.\frac{\partial^{2}S(z)}{\partial z_{i}\partial z_{j}}\right|_{z=0}=2h_{% ij}(0);
  44. S ( s y m b o l ψ ( u ) ) = i = 1 n u i 2 . S(symbol{\psi}(u))=\sum_{i=1}^{n}u_{i}^{2}.
  45. S ( s y m b o l ϕ ( y ) ) = y 1 2 + + y r - 1 2 + i , j = r n y i y j H i j ( y ) , S(symbol{\phi}(y))=y_{1}^{2}+\cdots+y_{r-1}^{2}+\sum_{i,j=r}^{n}y_{i}y_{j}H_{% ij}(y),
  46. 2 S ( s y m b o l ϕ ( y ) ) y i y j = l , k = 1 n 2 S ( z ) z k z l | z = s y m b o l ϕ ( y ) ϕ k y i ϕ l y j + k = 1 n S ( z ) z k | z = s y m b o l ϕ ( y ) 2 ϕ k y i y j \frac{\partial^{2}S(symbol{\phi}(y))}{\partial y_{i}\partial y_{j}}=\sum_{l,k=% 1}^{n}\left.\frac{\partial^{2}S(z)}{\partial z_{k}\partial z_{l}}\right|_{z=% symbol{\phi}(y)}\frac{\partial\phi_{k}}{\partial y_{i}}\frac{\partial\phi_{l}}% {\partial y_{j}}+\sum_{k=1}^{n}\left.\frac{\partial S(z)}{\partial z_{k}}% \right|_{z=symbol{\phi}(y)}\frac{\partial^{2}\phi_{k}}{\partial y_{i}\partial y% _{j}}
  47. S y y ′′ ( s y m b o l ϕ ( 0 ) ) = s y m b o l ϕ y ( 0 ) T S z z ′′ ( 0 ) s y m b o l ϕ y ( 0 ) , det s y m b o l ϕ y ( 0 ) 0 ; S^{\prime\prime}_{yy}(symbol{\phi}(0))=symbol{\phi}^{\prime}_{y}(0)^{T}S^{% \prime\prime}_{zz}(0)symbol{\phi}^{\prime}_{y}(0),\qquad\det symbol{\phi}^{% \prime}_{y}(0)\neq 0;
  48. 0 det S y y ′′ ( s y m b o l ϕ ( 0 ) ) = 2 r - 1 det ( 2 H i j ( 0 ) ) . 0\neq\det S^{\prime\prime}_{yy}(symbol{\phi}(0))=2^{r-1}\det\left(2H_{ij}(0)% \right).
  49. L L
  50. J J
  51. H ~ i j ( y ) = H i j ( y ) / H r r ( y ) \tilde{H}_{ij}(y)=H_{ij}(y)/H_{rr}(y)
  52. S ( s y m b o l φ ( y ) ) = y 1 2 + + y r - 1 2 + H r r ( y ) i , j = r n y i y j H ~ i j ( y ) = y 1 2 + + y r - 1 2 + H r r ( y ) [ y r 2 + 2 y r j = r + 1 n y j H ~ r j ( y ) + i , j = r + 1 n y i y j H ~ i j ( y ) ] = y 1 2 + + y r - 1 2 + H r r ( y ) [ ( y r + j = r + 1 n y j H ~ r j ( y ) ) 2 - ( j = r + 1 n y j H ~ r j ( y ) ) 2 ] + H r r ( y ) i , j = r + 1 n y i y j H ~ i j ( y ) \begin{aligned}\displaystyle S(symbol{\varphi}(y))=&\displaystyle y_{1}^{2}+% \cdots+y_{r-1}^{2}+H_{rr}(y)\sum_{i,j=r}^{n}y_{i}y_{j}\tilde{H}_{ij}(y)\\ \displaystyle=&\displaystyle y_{1}^{2}+\cdots+y_{r-1}^{2}+H_{rr}(y)\left[y_{r}% ^{2}+2y_{r}\sum_{j=r+1}^{n}y_{j}\tilde{H}_{rj}(y)+\sum_{i,j=r+1}^{n}y_{i}y_{j}% \tilde{H}_{ij}(y)\right]\\ \displaystyle=&\displaystyle y_{1}^{2}+\cdots+y_{r-1}^{2}+H_{rr}(y)\left[\left% (y_{r}+\sum_{j=r+1}^{n}y_{j}\tilde{H}_{rj}(y)\right)^{2}-\left(\sum_{j=r+1}^{n% }y_{j}\tilde{H}_{rj}(y)\right)^{2}\right]+H_{rr}(y)\sum_{i,j=r+1}^{n}y_{i}y_{j% }\tilde{H}_{ij}(y)\end{aligned}
  53. z = η ( x ) , 0 = η ( 0 ) , z=\mathbf{η}(x),0=\mathbf{η}(0),
  54. x r = H r r ( y ) ( y r + j = r + 1 n y j H ~ r j ( y ) ) , x j = y j , j r . x_{r}=\sqrt{H_{rr}(y)}\left(y_{r}+\sum_{j=r+1}^{n}y_{j}\tilde{H}_{rj}(y)\right% ),\qquad x_{j}=y_{j},\quad\forall j\neq r.
  55. y x y↔x
  56. x r y k | y = 0 = H r r ( 0 ) [ δ r , k + j = r + 1 n δ j , k H ~ j r ( 0 ) ] . \left.\frac{\partial x_{r}}{\partial y_{k}}\right|_{y=0}=\sqrt{H_{rr}(0)}\left% [\delta_{r,\,k}+\sum_{j=r+1}^{n}\delta_{j,\,k}\tilde{H}_{jr}(0)\right].
  57. S ( s y m b o l η ( x ) ) = x 1 2 + + x r 2 + i , j = r + 1 n x i x j W i j ( x ) . S(symbol{\eta}(x))={x}_{1}^{2}+\cdots+{x}_{r}^{2}+\sum_{i,j=r+1}^{n}{x}_{i}{x}% _{j}W_{ij}(x).
  58. S z z ′′ ( 0 ) S^{\prime\prime}_{zz}(0)
  59. S ( s y m b o l φ ( w ) ) = 1 2 j = 1 n μ j w j 2 . S(symbol{\varphi}(w))=\frac{1}{2}\sum_{j=1}^{n}\mu_{j}w_{j}^{2}.
  60. S w w ′′ ( s y m b o l φ ( 0 ) ) = s y m b o l φ w ( 0 ) T S z z ′′ ( 0 ) s y m b o l φ w ( 0 ) , S^{\prime\prime}_{ww}(symbol{\varphi}(0))=symbol{\varphi}^{\prime}_{w}(0)^{T}S% ^{\prime\prime}_{zz}(0)symbol{\varphi}^{\prime}_{w}(0),
  61. det S w w ′′ ( s y m b o l φ ( 0 ) ) = μ 1 μ n \det S^{\prime\prime}_{ww}(symbol{\varphi}(0))=\mu_{1}\cdots\mu_{n}
  62. S z z ′′ ( 0 ) S^{\prime\prime}_{zz}(0)
  63. S z z ′′ ( 0 ) = P J z P - 1 S^{\prime\prime}_{zz}(0)=PJ_{z}P^{-1}
  64. d e t P 0 detP≠0
  65. det S z z ′′ ( 0 ) = μ 1 μ n \det S^{\prime\prime}_{zz}(0)=\mu_{1}\cdots\mu_{n}
  66. det S w w ′′ ( s y m b o l φ ( 0 ) ) = [ det s y m b o l φ w ( 0 ) ] 2 det S z z ′′ ( 0 ) det s y m b o l φ w ( 0 ) = ± 1. \det S^{\prime\prime}_{ww}(symbol{\varphi}(0))=\left[\det symbol{\varphi}^{% \prime}_{w}(0)\right]^{2}\det S^{\prime\prime}_{zz}(0)\Longrightarrow\det symbol% {\varphi}^{\prime}_{w}(0)=\pm 1.
  67. det s y m b o l φ w ( 0 ) = - 1 \det symbol{\varphi}^{\prime}_{w}(0)=-1
  68. det s y m b o l φ w ( 0 ) = + 1 \det symbol{\varphi}^{\prime}_{w}(0)=+1
  69. f ( z ) f(z)
  70. S ( z ) S(z)
  71. ( S ( z ) ) \Re(S(z))
  72. max z I x ( S ( z ) ) = ( S ( x 0 ) ) \max_{z\in I_{x}}\Re(S(z))=\Re(S(x^{0}))
  73. det S x x ′′ ( x 0 ) 0 \det S^{\prime\prime}_{xx}(x^{0})\neq 0
  74. I ( λ ) I x f ( x ) e λ S ( x ) d x = ( 2 π λ ) n 2 e λ S ( x 0 ) ( f ( x 0 ) + O ( λ - 1 ) ) j = 1 n ( - μ j ) - 1 2 , λ , I(\lambda)\equiv\int_{I_{x}}f(x)e^{\lambda S(x)}dx=\left(\frac{2\pi}{\lambda}% \right)^{\frac{n}{2}}e^{\lambda S(x^{0})}\left(f(x^{0})+O\left(\lambda^{-1}% \right)\right)\prod_{j=1}^{n}(-\mu_{j})^{-\frac{1}{2}},\qquad\lambda\to\infty,
  75. S x x ′′ ( x 0 ) S^{\prime\prime}_{xx}(x^{0})
  76. ( - μ j ) - 1 2 (-\mu_{j})^{-\frac{1}{2}}
  77. | arg - μ j | < π 4 . \left|\arg\sqrt{-\mu_{j}}\right|<\tfrac{\pi}{4}.
  78. I x Ω x I^{\prime}_{x}\subset\Omega_{x}
  79. I ( λ ) I(λ)
  80. φ ( w ) \mathbf{φ}(w)
  81. I ( λ ) I(λ)
  82. U I x U\cap I^{\prime}_{x}
  83. I x ( U I x ) I^{\prime}_{x}\setminus(U\cap I^{\prime}_{x})
  84. λ λ→∞
  85. U I x = s y m b o l φ ( I w ) U\cap I^{\prime}_{x}=symbol{\varphi}(I_{w})
  86. I 0 ( λ ) = e λ S ( x 0 ) I w f [ s y m b o l φ ( w ) ] exp ( λ j = 1 n μ j 2 w j 2 ) | \detsymbol φ w ( w ) | d w . I_{0}(\lambda)=e^{\lambda S(x^{0})}\int_{I_{w}}f[symbol{\varphi}(w)]\exp\left(% \lambda\sum_{j=1}^{n}\tfrac{\mu_{j}}{2}w_{j}^{2}\right)\left|\detsymbol{% \varphi}_{w}^{\prime}(w)\right|dw.
  87. det s y m b o l φ w ( 0 ) = 1 \det symbol{\varphi}_{w}^{\prime}(0)=1
  88. I 0 ( λ ) f ( x 0 ) e λ S ( x 0 ) 𝐑 n exp ( λ j = 1 n μ j 2 w j 2 ) d w = f ( x 0 ) e λ S ( x 0 ) j = 1 n - e 1 2 λ μ j y 2 d y . I_{0}(\lambda)\approx f(x^{0})e^{\lambda S(x^{0})}\int_{\mathbf{R}^{n}}\exp% \left(\lambda\sum_{j=1}^{n}\tfrac{\mu_{j}}{2}w_{j}^{2}\right)dw=f(x^{0})e^{% \lambda S(x^{0})}\prod_{j=1}^{n}\int_{-\infty}^{\infty}e^{\frac{1}{2}\lambda% \mu_{j}y^{2}}dy.
  89. j = - e 1 2 λ μ j y 2 d y = 2 0 e - 1 2 λ ( - μ j y ) 2 d y = 2 0 e - 1 2 λ | - μ j | 2 y 2 exp ( 2 i arg - μ j ) d y . \mathcal{I}_{j}=\int_{-\infty}^{\infty}e^{\frac{1}{2}\lambda\mu_{j}y^{2}}dy=2% \int_{0}^{\infty}e^{-\frac{1}{2}\lambda\left(\sqrt{-\mu_{j}}y\right)^{2}}dy=2% \int_{0}^{\infty}e^{-\frac{1}{2}\lambda\left|\sqrt{-\mu_{j}}\right|^{2}y^{2}% \exp\left(2i\arg\sqrt{-\mu_{j}}\right)}dy.
  90. ( S x x ′′ ( x 0 ) ) \Re\left(S_{xx}^{\prime\prime}(x^{0})\right)
  91. ( μ j ) < 0 \Re(\mu_{j})<0
  92. j \mathcal{I}_{j}
  93. j = 2 - μ j λ 0 e - ξ 2 2 d ξ = 2 π λ ( - μ j ) - 1 2 . \mathcal{I}_{j}=\frac{2}{\sqrt{-\mu_{j}}\sqrt{\lambda}}\int_{0}^{\infty}e^{-% \frac{\xi^{2}}{2}}d\xi=\sqrt{\frac{2\pi}{\lambda}}(-\mu_{j})^{-\frac{1}{2}}.
  94. I ( λ ) = ( 2 π λ ) n 2 e λ S ( x 0 ) ( det ( - S x x ′′ ( x 0 ) ) ) - 1 2 ( f ( x 0 ) + O ( λ - 1 ) ) , I(\lambda)=\left(\frac{2\pi}{\lambda}\right)^{\frac{n}{2}}e^{\lambda S(x^{0})}% \left(\det(-S_{xx}^{\prime\prime}(x^{0}))\right)^{-\frac{1}{2}}\left(f(x^{0})+% O\left(\lambda^{-1}\right)\right),
  95. det ( - S x x ′′ ( x 0 ) ) \sqrt{\det\left(-S_{xx}^{\prime\prime}(x^{0})\right)}
  96. ( det ( - S x x ′′ ( x 0 ) ) ) - 1 2 = exp ( - i Ind ( - S x x ′′ ( x 0 ) ) ) j = 1 n | μ j | - 1 2 , Ind ( - S x x ′′ ( x 0 ) ) = 1 2 j = 1 n arg ( - μ j ) , | arg ( - μ j ) | < π 2 . \begin{aligned}\displaystyle\left(\det\left(-S_{xx}^{\prime\prime}(x^{0})% \right)\right)^{-\frac{1}{2}}&\displaystyle=\exp\left(-i\,\text{ Ind}\left(-S_% {xx}^{\prime\prime}(x^{0})\right)\right)\prod_{j=1}^{n}\left|\mu_{j}\right|^{-% \frac{1}{2}},\\ \displaystyle\,\text{Ind}\left(-S_{xx}^{\prime\prime}(x^{0})\right)&% \displaystyle=\tfrac{1}{2}\sum_{j=1}^{n}\arg(-\mu_{j}),&&\displaystyle|\arg(-% \mu_{j})|<\tfrac{\pi}{2}.\end{aligned}
  97. S ( x ) S(x)
  98. x x
  99. Ind ( - S x x ′′ ( x 0 ) ) = 0. \,\text{Ind}\left(-S_{xx}^{\prime\prime}(x^{0})\right)=0.
  100. S ( x ) S(x)
  101. x x
  102. ( S ( x ) ) = 0 \Re(S(x))=0
  103. x x
  104. Ind ( - S x x ′′ ( x 0 ) ) = π 4 sign S x x ′′ ( x 0 ) , \,\text{Ind}\left(-S_{xx}^{\prime\prime}(x^{0})\right)=\frac{\pi}{4}\,\text{% sign }S_{xx}^{\prime\prime}(x_{0}),
  105. sign S x x ′′ ( x 0 ) \,\text{sign }S_{xx}^{\prime\prime}(x_{0})
  106. S x x ′′ ( x 0 ) S_{xx}^{\prime\prime}(x_{0})
  107. I n d Ind
  108. S ( x ) S(x)
  109. S ( x ( k ) ) = 0 , det S x x ′′ ( x ( k ) ) 0 , x ( k ) Ω x ( k ) , \nabla S\left(x^{(k)}\right)=0,\quad\det S^{\prime\prime}_{xx}\left(x^{(k)}% \right)\neq 0,\quad x^{(k)}\in\Omega_{x}^{(k)},
  110. { Ω x ( k ) } k = 1 K \left\{\Omega_{x}^{(k)}\right\}_{k=1}^{K}
  111. k = 1 K ρ k ( x ) = 1 , x Ω x , ρ k ( x ) = 0 x Ω x Ω x ( k ) . \begin{aligned}\displaystyle\sum_{k=1}^{K}\rho_{k}(x)&\displaystyle=1,&&% \displaystyle\forall x\in\Omega_{x},\\ \displaystyle\rho_{k}(x)&\displaystyle=0&&\displaystyle\forall x\in\Omega_{x}% \setminus\Omega_{x}^{(k)}.\end{aligned}
  112. I x Ω x f ( x ) e λ S ( x ) d x k = 1 K I x Ω x ρ k ( x ) f ( x ) e λ S ( x ) d x . \int_{I_{x}\subset\Omega_{x}}f(x)e^{\lambda S(x)}dx\equiv\sum_{k=1}^{K}\int_{I% _{x}\subset\Omega_{x}}\rho_{k}(x)f(x)e^{\lambda S(x)}dx.
  113. λ λ→∞
  114. k = 1 K a neighborhood of x ( k ) f ( x ) e λ S ( x ) d x = ( 2 π λ ) n 2 k = 1 K e λ S ( x ( k ) ) ( det ( - S x x ′′ ( x ( k ) ) ) ) - 1 2 f ( x ( k ) ) , \sum_{k=1}^{K}\int_{\,\text{a neighborhood of }x^{(k)}}f(x)e^{\lambda S(x)}dx=% \left(\frac{2\pi}{\lambda}\right)^{\frac{n}{2}}\sum_{k=1}^{K}e^{\lambda S\left% (x^{(k)}\right)}\left(\det\left(-S_{xx}^{\prime\prime}\left(x^{(k)}\right)% \right)\right)^{-\frac{1}{2}}f\left(x^{(k)}\right),
  115. f ( x ) f(x)
  116. det S z z ′′ ( z 0 ) = 0 \det S^{\prime\prime}_{zz}(z^{0})=0
  117. S ( z ) S(z)
  118. f ( x ) e λ S ( x ) d x , \int f(x)e^{\lambda S(x)}dx,
  119. λ , f ( x ) λ→∞,f(x)
  120. S ( z ) S(z)
  121. S ( z ) S(z)
  122. f ( x ) f(x)
  123. S ( x ) S(x)
  124. S ( x ) S(x)
  125. I < s u b > 0 ( λ ) I<sub>0(λ)
  126. 𝐑 < s u p > n \mathbf{R}<sup>n
  127. | arg - μ j | π 4 . \left|\arg\sqrt{-\mu_{j}}\right|\leqslant\tfrac{\pi}{4}.

Metric_k-center.html

  1. max v V min s S d ( v , s ) \max_{v\in V}\min_{s\in S}d(v,s)

Meyer_set.html

  1. X ϵ = { y | x y - 1 | ε for all x X } . X^{\epsilon}=\{y\mid|x\cdot y-1|\leq\varepsilon\,\text{ for all }x\in X\}.

Michael_Saks_(mathematician).html

  1. n + 1 n+1
  2. P 0 , P 1 , , P n P_{0},P_{1},\ldots,P_{n}
  3. x i x_{i}
  4. P 0 P_{0}
  5. f ( x 1 , x 2 , , x n ) f(x_{1},x_{2},\ldots,x_{n})
  6. f f
  7. Ω ( n log ( n ) ) \Omega(n\log(n))

Michael_Stifel.html

  1. q m q n = q m + n q^{m}q^{n}=q^{m+n}
  2. q m q n = q m - n \tfrac{q^{m}}{q^{n}}=q^{m-n}

Microrheology.html

  1. G ( ω ) = G ( ω ) + i G ′′ ( ω ) G(\omega)=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)\,
  2. G ~ ( s ) = k B T π a s Δ r ~ 2 ( s ) \tilde{G}(s)=\frac{k_{\mathrm{B}}T}{\pi as\langle\Delta\tilde{r}^{2}(s)\rangle}
  3. G ~ ( s ) \tilde{G}(s)
  4. Δ r ~ 2 ( s ) \langle\Delta\tilde{r}^{2}(s)\rangle
  5. x ( t ) x(t)
  6. < x ω 2 > <x_{\omega}^{2}>
  7. α ( ω ) \alpha(\omega)
  8. G ( ω ) G(\omega)
  9. G ( ω ) = 1 6 π a α ( ω ) G(\omega)=\frac{1}{6\pi a\alpha(\omega)}

Microwave_cavity.html

  1. Q \scriptstyle Q
  2. Q c \scriptstyle Q_{c}
  3. Q c = ( k a d ) 3 b η 2 π 2 R s 1 l 2 a 3 ( 2 b + d ) + ( 2 b + a ) d 3 Q_{c}=\frac{(kad)^{3}b\eta}{2\pi^{2}R_{s}}\cdot\frac{1}{l^{2}a^{3}\left(2b+d% \right)+\left(2b+a\right)d^{3}}\,
  4. Q d \scriptstyle Q_{d}
  5. Q d = 1 tan δ Q_{d}=\frac{1}{\tan\delta}\,
  6. Q e x t \scriptstyle Q_{ext}
  7. Q = ( 1 Q c + 1 Q d ) - 1 Q=\left(\frac{1}{Q_{c}}+\frac{1}{Q_{d}}\right)^{-1}\,
  8. η \scriptstyle\eta
  9. R s \scriptstyle R_{s}
  10. μ r \scriptstyle\mu_{r}
  11. ϵ r \scriptstyle\epsilon_{r}
  12. tan δ \scriptstyle\tan\delta
  13. T E m n l \scriptstyle TE_{mnl}
  14. T M m n l \scriptstyle TM_{mnl}
  15. f m n l = c 2 π μ r ϵ r k m n l = c 2 π μ r ϵ r ( m π a ) 2 + ( n π b ) 2 + ( l π d ) 2 = c 2 μ r ϵ r ( m a ) 2 + ( n b ) 2 + ( l d ) 2 \begin{aligned}\displaystyle f_{mnl}&\displaystyle=\frac{c}{2\pi\sqrt{\mu_{r}% \epsilon_{r}}}\cdot k_{mnl}\\ &\displaystyle=\frac{c}{2\pi\sqrt{\mu_{r}\epsilon_{r}}}\sqrt{\left(\frac{m\pi}% {a}\right)^{2}+\left(\frac{n\pi}{b}\right)^{2}+\left(\frac{l\pi}{d}\right)^{2}% }\\ &\displaystyle=\frac{c}{2\sqrt{\mu_{r}\epsilon_{r}}}\sqrt{\left(\frac{m}{a}% \right)^{2}+\left(\frac{n}{b}\right)^{2}+\left(\frac{l}{d}\right)^{2}}\end{aligned}
  16. k m n l \scriptstyle k_{mnl}
  17. m \scriptstyle m
  18. n \scriptstyle n
  19. l \scriptstyle l
  20. a \scriptstyle a
  21. b \scriptstyle b
  22. d \scriptstyle d
  23. μ r \scriptstyle\mu_{r}
  24. ϵ r \scriptstyle\epsilon_{r}
  25. L \scriptstyle L
  26. R \scriptstyle R
  27. f m n p = c 2 π μ r ϵ r ( X m n R ) 2 + ( p π L ) 2 f_{mnp}=\frac{c}{2\pi\sqrt{\mu_{r}\epsilon_{r}}}\sqrt{\left(\frac{X_{mn}}{R}% \right)^{2}+\left(\frac{p\pi}{L}\right)^{2}}
  28. f m n p = c 2 π μ r ϵ r ( X m n R ) 2 + ( p π L ) 2 f_{mnp}=\frac{c}{2\pi\sqrt{\mu_{r}\epsilon_{r}}}\sqrt{\left(\frac{X^{\prime}_{% mn}}{R}\right)^{2}+\left(\frac{p\pi}{L}\right)^{2}}
  29. X m n \scriptstyle X_{mn}
  30. n \scriptstyle n
  31. m \scriptstyle m
  32. X m n \scriptstyle X^{\prime}_{mn}
  33. n \scriptstyle n
  34. m \scriptstyle m
  35. m n l \scriptstyle mnl
  36. L m n l = μ k m n l 2 V L_{mnl}=\mu k_{mnl}^{2}V\,
  37. C m n l = ϵ k m n l 4 V C_{mnl}=\frac{\epsilon}{k_{mnl}^{4}V}\,
  38. f m n l = 1 2 π L m n l C m n l = 1 2 π 1 k m n l 2 μ ϵ \begin{aligned}\displaystyle f_{mnl}&\displaystyle=\frac{1}{2\pi\sqrt{L_{mnl}C% _{mnl}}}\\ &\displaystyle=\frac{1}{2\pi\sqrt{\frac{1}{k_{mnl}^{2}}\mu\epsilon}}\end{aligned}
  39. k m n l \scriptstyle k_{mnl}
  40. ϵ \scriptstyle\epsilon
  41. μ \scriptstyle\mu

Mie–Gruneisen_equation_of_state.html

  1. Γ = V ( d p d e ) V \Gamma=V\left(\frac{dp}{de}\right)_{V}
  2. p - p 0 = Γ V ( e - e 0 ) p-p_{0}=\frac{\Gamma}{V}(e-e_{0})
  3. p = ρ 0 C 0 2 χ [ 1 - Γ 0 2 χ ] ( 1 - s χ ) 2 + Γ 0 E ; χ := 1 - ρ 0 ρ p=\frac{\rho_{0}C_{0}^{2}\chi\left[1-\frac{\Gamma_{0}}{2}\,\chi\right]}{\left(% 1-s\chi\right)^{2}}+\Gamma_{0}E;\quad\chi:=1-\cfrac{\rho_{0}}{\rho}
  4. C 0 C_{0}
  5. ρ 0 \rho_{0}
  6. ρ \rho
  7. Γ 0 \Gamma_{0}
  8. s = d U s / d U p s=dU_{s}/dU_{p}
  9. U s U_{s}
  10. U p U_{p}
  11. E E
  12. p = ρ 0 C 0 2 ( η - 1 ) [ η - Γ 0 2 ( η - 1 ) ] [ η - s ( η - 1 ) ] 2 + Γ 0 E ; η := ρ ρ 0 . p=\frac{\rho_{0}C_{0}^{2}(\eta-1)\left[\eta-\frac{\Gamma_{0}}{2}(\eta-1)\right% ]}{\left[\eta-s(\eta-1)\right]^{2}}+\Gamma_{0}E;\quad\eta:=\cfrac{\rho}{\rho_{% 0}}\,.
  13. E = 1 V 0 C v d T C v ( T - T 0 ) V 0 = ρ 0 c v ( T - T 0 ) E=\frac{1}{V_{0}}\int C_{v}dT\approx\frac{C_{v}(T-T_{0})}{V_{0}}=\rho_{0}c_{v}% (T-T_{0})
  14. V 0 V_{0}
  15. T = T 0 T=T_{0}
  16. C v C_{v}
  17. c v c_{v}
  18. C p C_{p}
  19. C v C_{v}
  20. T < T 1 T<T_{1}
  21. T T 1 T>=T_{1}
  22. d d
  23. Γ 0 = - 1 2 d Π ′′′ ( a ) a 2 + ( d - 1 ) [ Π ′′ ( a ) a - Π ( a ) ] Π ′′ ( a ) a + ( d - 1 ) Π ( a ) , \Gamma_{0}=-\frac{1}{2d}\frac{\Pi^{\prime\prime\prime}(a)a^{2}+(d-1)\left[\Pi^% {\prime\prime}(a)a-\Pi^{\prime}(a)\right]}{\Pi^{\prime\prime}(a)a+(d-1)\Pi^{% \prime}(a)},
  24. Π \Pi
  25. a a
  26. d d
  27. d = 1 d=1
  28. 10 1 2 10\frac{1}{2}
  29. m + n + 3 2 \frac{m+n+3}{2}
  30. 3 α a 2 \frac{3\alpha a}{2}
  31. d = 2 d=2
  32. 5 5
  33. m + n + 2 4 \frac{m+n+2}{4}
  34. 3 α a - 1 4 \frac{3\alpha a-1}{4}
  35. d = 3 d=3
  36. 19 6 \frac{19}{6}
  37. n + m + 1 6 \frac{n+m+1}{6}
  38. 3 α a - 2 6 \frac{3\alpha a-2}{6}
  39. d = d=\infty
  40. - 1 2 -\frac{1}{2}
  41. - 1 2 -\frac{1}{2}
  42. - 1 2 -\frac{1}{2}
  43. d d
  44. 11 d - 1 2 \frac{11}{d}-\frac{1}{2}
  45. m + n + 4 2 d - 1 2 \frac{m+n+4}{2d}-\frac{1}{2}
  46. 3 α a + 1 2 d - 1 2 \frac{3\alpha a+1}{2d}-\frac{1}{2}
  47. Π ′′′ ( a ) a > - ( d - 1 ) Π ′′ ( a ) , \Pi^{\prime\prime\prime}(a)a>-(d-1)\Pi^{\prime\prime}(a),
  48. ( 1 ) p - p 0 = Γ V ( e - e 0 ) (1)\qquad p-p_{0}=\frac{\Gamma}{V}(e-e_{0})
  49. ρ 0 U s = ρ ( U s - U p ) , p H - p H 0 = ρ 0 U s U p and p H U p = ρ 0 U s ( U p 2 2 + E H - E H 0 ) \rho_{0}U_{s}=\rho(U_{s}-U_{p})~{}~{},\quad p_{H}-p_{H0}=\rho_{0}U_{s}U_{p}% \quad\,\text{and}\quad p_{H}U_{p}=\rho_{0}U_{s}\left(\frac{U_{p}^{2}}{2}+E_{H}% -E_{H0}\right)
  50. U p U s = 1 - ρ 0 ρ = 1 - V V 0 = : χ . \frac{U_{p}}{U_{s}}=1-\frac{\rho_{0}}{\rho}=1-\frac{V}{V_{0}}=:\chi\,.
  51. V = 1 / ρ V=1/\rho
  52. U s = C 0 + s χ U s or U s = C 0 1 - s χ . U_{s}=C_{0}+s\chi U_{s}\quad\,\text{or}\quad U_{s}=\frac{C_{0}}{1-s\chi}\,.
  53. p H = ρ 0 χ U s 2 = ρ C 0 2 χ ( 1 - s χ ) 2 . p_{H}=\rho_{0}\chi U_{s}^{2}=\frac{\rho C_{0}^{2}\chi}{(1-s\chi)^{2}}\,.
  54. p H χ U s = 1 2 ρ χ 2 U s 3 + ρ 0 U s E H = 1 2 p H χ U s + ρ 0 U s E H . p_{H}\chi U_{s}=\tfrac{1}{2}\rho\chi^{2}U_{s}^{3}+\rho_{0}U_{s}E_{H}=\tfrac{1}% {2}p_{H}\chi U_{s}+\rho_{0}U_{s}E_{H}\,.
  55. E H = 1 2 p H χ ρ 0 = 1 2 p H ( V 0 - V ) E_{H}=\tfrac{1}{2}\frac{p_{H}\chi}{\rho_{0}}=\tfrac{1}{2}p_{H}(V_{0}-V)
  56. p H - p 0 = Γ V ( p H χ V 0 2 - e 0 ) or ρ C 0 2 χ ( 1 - s χ ) 2 ( 1 - χ 2 Γ V V 0 ) - p 0 = - Γ V e 0 . p_{H}-p_{0}=\frac{\Gamma}{V}\left(\frac{p_{H}\chi V_{0}}{2}-e_{0}\right)\quad% \,\text{or}\quad\frac{\rho C_{0}^{2}\chi}{(1-s\chi)^{2}}\left(1-\frac{\chi}{2}% \,\frac{\Gamma}{V}\,V_{0}\right)-p_{0}=-\frac{\Gamma}{V}e_{0}\,.
  57. p 0 = - d e 0 / d V p_{0}=-de_{0}/dV
  58. ( 2 ) ρ C 0 2 χ ( 1 - s χ ) 2 ( 1 - Γ 0 χ 2 ) + d e 0 d V + Γ 0 V 0 e 0 = 0 . (2)\qquad\frac{\rho C_{0}^{2}\chi}{(1-s\chi)^{2}}\left(1-\frac{\Gamma_{0}\chi}% {2}\right)+\frac{de_{0}}{dV}+\frac{\Gamma_{0}}{V_{0}}e_{0}=0\,.
  59. e 0 = ρ C 0 2 V 0 2 s 4 [ exp ( Γ 0 χ ) ( Γ 0 s - 3 ) s 2 - [ Γ 0 s - ( 3 - s χ ) ] s 2 1 - s χ + exp [ - Γ 0 s ( 1 - s χ ) ] ( Γ 0 2 - 4 Γ 0 s + 2 s 2 ) ( Ei [ Γ 0 s ( 1 - s χ ) ] - Ei [ Γ 0 s ] ) ] \begin{aligned}\displaystyle e_{0}=\frac{\rho C_{0}^{2}V_{0}}{2s^{4}}\Biggl[&% \displaystyle\exp(\Gamma_{0}\chi)(\tfrac{\Gamma_{0}}{s}-3)s^{2}-\frac{[\tfrac{% \Gamma_{0}}{s}-(3-s\chi)]s^{2}}{1-s\chi}+\\ &\displaystyle\exp\left[-\tfrac{\Gamma_{0}}{s}(1-s\chi)\right](\Gamma_{0}^{2}-% 4\Gamma_{0}s+2s^{2})(\,\text{Ei}[\tfrac{\Gamma_{0}}{s}(1-s\chi)]-\,\text{Ei}[% \tfrac{\Gamma_{0}}{s}])\Biggr]\end{aligned}
  60. p 0 = - d e 0 d V = ρ C 0 2 2 s 4 ( 1 - χ ) [ s ( 1 - s χ ) 2 ( - Γ 0 2 ( 1 - χ ) ( 1 - s χ ) + Γ 0 [ s { 4 ( χ - 1 ) χ s - 2 χ + 3 } - 1 ] - exp ( Γ 0 χ ) [ Γ 0 ( χ - 1 ) - 1 ] ( 1 - s χ ) 2 ( Γ 0 - 3 s ) + s [ 3 - χ s { ( χ - 2 ) s + 4 } ] ) - exp [ - Γ 0 s ( 1 - s χ ) ] [ Γ 0 ( χ - 1 ) - 1 ] ( Γ 0 2 - 4 Γ 0 s + 2 s 2 ) ( Ei [ Γ 0 s ( 1 - s χ ) ] - Ei [ Γ 0 s ] ) ] . \begin{aligned}\displaystyle p_{0}=-\frac{de_{0}}{dV}=\frac{\rho C_{0}^{2}}{2s% ^{4}(1-\chi)}\Biggl[&\displaystyle\frac{s}{(1-s\chi)^{2}}\Bigl(-\Gamma_{0}^{2}% (1-\chi)(1-s\chi)+\Gamma_{0}[s\{4(\chi-1)\chi s-2\chi+3\}-1]\\ &\displaystyle-\exp(\Gamma_{0}\chi)[\Gamma_{0}(\chi-1)-1](1-s\chi)^{2}(\Gamma_% {0}-3s)+s[3-\chi s\{(\chi-2)s+4\}]\Bigr)\\ &\displaystyle-\exp\left[-\tfrac{\Gamma_{0}}{s}(1-s\chi)\right][\Gamma_{0}(% \chi-1)-1](\Gamma_{0}^{2}-4\Gamma_{0}s+2s^{2})(\,\text{Ei}[\tfrac{\Gamma_{0}}{% s}(1-s\chi)]-\,\text{Ei}[\tfrac{\Gamma_{0}}{s}])\Biggr]\,.\end{aligned}
  61. e 0 ( V ) = A + B χ ( V ) + C χ 2 ( V ) + D χ 3 ( V ) + e_{0}(V)=A+B\chi(V)+C\chi^{2}(V)+D\chi^{3}(V)+\dots
  62. p 0 ( V ) = - d e 0 d V = - d e 0 d χ d χ d V = 1 V 0 ( B + 2 C χ + 3 D χ 2 + ) . p_{0}(V)=-\frac{de_{0}}{dV}=-\frac{de_{0}}{d\chi}\,\frac{d\chi}{dV}=\frac{1}{V% _{0}}\,(B+2C\chi+3D\chi^{2}+\dots)\,.
  63. p = 1 V 0 ( B + 2 C χ + 3 D χ 2 + ) + Γ 0 V 0 [ e - ( A + B χ + C χ 2 + D χ 3 + ) ] . p=\frac{1}{V_{0}}\,(B+2C\chi+3D\chi^{2}+\dots)+\frac{\Gamma_{0}}{V_{0}}\left[e% -(A+B\chi+C\chi^{2}+D\chi^{3}+\dots)\right]\,.
  64. p = 1 V 0 [ 2 C χ ( 1 - Γ 0 2 χ ) + 3 D χ 2 ( 1 - Γ 0 3 χ ) + ] + Γ 0 E p=\frac{1}{V_{0}}\left[2C\chi\left(1-\tfrac{\Gamma_{0}}{2}\chi\right)+3D\chi^{% 2}\left(1-\tfrac{\Gamma_{0}}{3}\chi\right)+\dots\right]+\Gamma_{0}E
  65. C = ρ C 0 2 V 0 2 ( 1 - s χ ) 2 . C=\frac{\rho C_{0}^{2}V_{0}}{2(1-s\chi)^{2}}\,.
  66. p = ρ C 0 2 χ ( 1 - s χ ) 2 ( 1 - Γ 0 2 χ ) + Γ 0 E . p=\frac{\rho C_{0}^{2}\chi}{(1-s\chi)^{2}}\left(1-\tfrac{\Gamma_{0}}{2}\chi% \right)+\Gamma_{0}E\,.

Mills_ratio.html

  1. X X
  2. m ( x ) := F ¯ ( x ) f ( x ) , m(x):=\frac{\bar{F}(x)}{f(x)},
  3. f ( x ) f(x)
  4. F ¯ ( x ) := Pr [ X > x ] = x + f ( u ) d u \bar{F}(x):=\Pr[X>x]=\int_{x}^{+\infty}f(u)\,du
  5. h ( x ) := lim δ 0 1 δ Pr [ x < X x + δ | X > x ] h(x):=\lim_{\delta\to 0}\frac{1}{\delta}\Pr[x<X\leq x+\delta|X>x]
  6. m ( x ) = 1 h ( x ) . m(x)=\frac{1}{h(x)}.
  7. X X
  8. m ( x ) 1 / x , m(x)\sim 1/x,\,
  9. \sim
  10. x + x\to+\infty

Milner–Rado_paradox.html

  1. α \alpha
  2. β < α \beta<\alpha
  3. { X n β } n \{X^{\beta}_{n}\}_{n}
  4. β \beta
  5. { β γ } γ < cf ( α ) \{\beta_{\gamma}\}_{\gamma<\mathrm{cf}\,(\alpha)}
  6. α \alpha
  7. β 0 = 0 \beta_{0}=0
  8. cf ( α ) κ \mathrm{cf}\,(\alpha)\leq\kappa
  9. X 0 α = { 0 } ; X n + 1 α = γ X n β γ + 1 β γ X^{\alpha}_{0}=\{0\};\ \ X^{\alpha}_{n+1}=\bigcup_{\gamma}X^{\beta_{\gamma+1}}% _{n}\setminus\beta_{\gamma}
  10. n > 0 X n α = n γ X n β γ + 1 β γ = γ n X n β γ + 1 β γ = γ β γ + 1 β γ = α β 0 \bigcup_{n>0}X^{\alpha}_{n}=\bigcup_{n}\bigcup_{\gamma}X^{\beta_{\gamma+1}}_{n% }\setminus\beta_{\gamma}=\bigcup_{\gamma}\bigcup_{n}X^{\beta_{\gamma+1}}_{n}% \setminus\beta_{\gamma}=\bigcup_{\gamma}\beta_{\gamma+1}\setminus\beta_{\gamma% }=\alpha\setminus\beta_{0}
  11. n X n α = α \bigcup_{n}X^{\alpha}_{n}=\alpha
  12. ot ( A ) \mathrm{ot}\,(A)
  13. A A
  14. ot ( X 0 α ) = 1 = κ 0 \mathrm{ot}(X^{\alpha}_{0})=1=\kappa^{0}
  15. β γ + 1 β γ \beta_{\gamma+1}\setminus\beta_{\gamma}
  16. X n β γ + 1 β γ X^{\beta_{\gamma+1}}_{n}\setminus\beta_{\gamma}
  17. X n β γ + 1 X^{\beta_{\gamma+1}}_{n}
  18. ot ( X n + 1 α ) = γ ot ( X n β γ + 1 β γ ) γ κ n = κ n cf ( α ) κ n κ = κ n + 1 \mathrm{ot}(X^{\alpha}_{n+1})=\sum_{\gamma}\mathrm{ot}(X^{\beta_{\gamma+1}}_{n% }\setminus\beta_{\gamma})\leq\sum_{\gamma}\kappa^{n}=\kappa^{n}\cdot\mathrm{cf% }(\alpha)\leq\kappa^{n}\cdot\kappa=\kappa^{n+1}

Milnor_ring.html

  1. K * M ( F ) K_{*}^{M}(F)
  2. ( a ) \ell(a)
  3. ( a b ) = ( a ) + ( b ) , ( a ) ( 1 - a ) = 0. \ell(ab)=\ell(a)+\ell(b),\quad\ell(a)\ell(1-a)=0.\,
  4. K 0 M ( F ) = K_{0}^{M}(F)={\mathbb{Z}}
  5. K 1 M ( F ) = F - { 0 } K_{1}^{M}(F)=F-\{0\}

Minimal_logic.html

  1. ¬ A , A B \neg A,A\vdash B
  2. ¬ A , A ¬ B \neg A,A\vdash\neg B
  3. ¬ A ( A B ) \neg A\to(A\to B)
  4. ¬ ¬ A A \neg\neg A\to A

Minimalist_grammar.html

  1. G = ( C , F , L ) G=(C,F,L)
  2. w : t w:t
  3. X Y X\circ Y
  4. w : X \vdash w:X
  5. w : X L w:X\in L
  6. w : X w : X w:X\vdash w:X
  7. w : X L w:X\notin L
  8. Γ a : X / Y Γ b : Y Γ ; Γ a b : X [ / E ] \frac{\Gamma\vdash a:X/Y\qquad\Gamma^{\prime}\vdash b:Y}{\Gamma;\Gamma^{\prime% }\vdash ab:X}[/E]
  9. Γ b : Y Γ a : X \ Y Γ ; Γ b a : X [ \ E ] \frac{\Gamma^{\prime}\vdash b:Y\qquad\Gamma\vdash a:X\backslash Y}{\Gamma^{% \prime};\Gamma\vdash ba:X}[\backslash E]
  10. Γ ; Γ α Γ , Γ α e n t r o p y \frac{\Gamma;\Gamma^{\prime}\vdash\alpha}{\Gamma,\Gamma^{\prime}\vdash\alpha}entropy
  11. Γ a : X Y Δ , b : X , c : Y , Δ d : Z Δ , Γ , Δ d [ b := a , c := a ] : Z [ E ] \frac{\Gamma\vdash a:X\circ Y\qquad\Delta,b:X,c:Y,\Delta^{\prime}\vdash d:Z}{% \Delta,\Gamma,\Delta^{\prime}\vdash d[b:=a,c:=a]:Z}[\circ E]
  12. Γ a : X Y * Δ , b : X , c : Y * , Δ d : Z Δ , Γ , Δ d [ b := ϵ , c := a ] : Z [ E s t r o n g ] \frac{\Gamma\vdash a:X\circ Y^{*}\qquad\Delta,b:X,c:Y^{*},\Delta^{\prime}% \vdash d:Z}{\Delta,\Gamma,\Delta^{\prime}\vdash d[b:=\epsilon,c:=a]:Z}[\circ E% _{strong}]
  13. X C , Y * F X\in C,Y^{*}\in F
  14. Γ a : X Y Δ , b : X , c : Y , Δ d : Z Δ , Γ , Δ d [ b := a , c := ϵ ] : Z [ E w e a k ] \frac{\Gamma\vdash a:X\circ Y\qquad\Delta,b:X,c:Y,\Delta^{\prime}\vdash d:Z}{% \Delta,\Gamma,\Delta^{\prime}\vdash d[b:=a,c:=\epsilon]:Z}[\circ E_{weak}]
  15. X C , Y F X\in C,Y\in F
  16. E \circ E
  17. G = ( { N , S } , { W } , L ) G=(\{N,S\},\{W\},L)
  18. John : N \,\text{John}:N
  19. see : ( S \ N ) / N \,\text{see}:(S\backslash N)/N
  20. did : ( S \ W ) / S \,\text{did}:(S\backslash W)/S
  21. who : N W \,\text{who}:N\circ W
  22. who : N W x : W x : W did : ( S \ W ) / S John : N y : N y : N see : ( S \ N ) / N y : N see y : S \ N [ / E ] y : N John see y : S [ \ E ] y : N did John see y : S \ W [ / E ] x : W , y : N x did John see y : S [ \ E ] who did John see : S [ E ] \dfrac{\vdash\,\text{who}:N\circ W\quad\dfrac{\,\text{x}:W\vdash\,\text{x}:W% \quad\dfrac{\vdash\,\text{did}:(S\backslash W)/S\quad\dfrac{\vdash\,\text{John% }:N\quad\dfrac{\,\text{y}:N\vdash\,\text{y}:N\quad\vdash\,\text{see}:(S% \backslash N)/N}{\,\text{y}:N\vdash\,\text{see y}:S\backslash N}[/E]}{\,\text{% y}:N\vdash\,\text{John see y}:S}[\backslash E]}{\,\text{y}:N\vdash\,\text{did % John see y}:S\backslash W}[/E]}{\,\text{x}:W,\,\text{y}:N\vdash\,\text{x did % John see y}:S}[\backslash E]}{\vdash\,\text{who did John see}:S}[\circ E]

Minimum-weight_triangulation.html

  1. O ( n ) O(\sqrt{n})
  2. 2 O ( n log n ) 2^{O(\sqrt{n}\log n)}
  3. Θ ( n ) \Theta(n)
  4. Θ ( n ) \Theta(\sqrt{n})
  5. Ω ( n ) \Omega(\sqrt{n})

Minimum_k-cut.html

  1. i = 1 k - 1 j = i + 1 k v 1 C i v 2 C j w ( { v 1 , v 2 } ) \sum_{i=1}^{k-1}\sum_{j=i+1}^{k}\sum_{\begin{smallmatrix}v_{1}\in C_{i}\\ v_{2}\in C_{j}\end{smallmatrix}}w(\left\{v_{1},v_{2}\right\})
  2. k k
  3. k k

Minuscule_397.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Minuscule_436.html

  1. 𝔓 \mathfrak{P}

Minuscule_451.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Minuscule_623.html

  1. 𝔓 \mathfrak{P}

Minuscule_629.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Minuscule_630.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Miraclebet.html

  1. s 1 s_{1}
  2. s 2 s_{2}
  3. o 1 o_{1}
  4. o 2 o_{2}
  5. r 1 r_{1}
  6. r 2 r_{2}
  7. 1.25 - 1 + 3.9 - 1 = 1.056 1.25^{-1}+3.9^{-1}=1.056
  8. 1.43 - 1 + 2.85 - 1 = 1.051 1.43^{-1}+2.85^{-1}=1.051
  9. 1.43 - 1 + 3.9 - 1 = 0.956 1.43^{-1}+3.9^{-1}=0.956
  10. $ 100 * 1.43 / 3.9 = 36.67 \$100*1.43/3.9=36.67
  11. r 1 = $ 100 * 1.43 = $ 143 r_{1}=\$100*1.43=\$143
  12. r 2 = $ 36.67 * 3.9 = $ 143 r_{2}=\$36.67*3.9=\$143
  13. o 1 o_{1}
  14. o 2 o_{2}
  15. o 1 - 1 + o 2 - 1 < 1 o_{1}^{-1}+o_{2}^{-1}<1
  16. s 1 s_{1}
  17. s 2 = s 1 * o 1 / o 2 s_{2}=s_{1}*o_{1}/o_{2}

MIS_capacitor.html

  1. C MIS ( max ) = ε 0 ε r A d C_{\mathrm{MIS(max)}}=\varepsilon_{0}\varepsilon_{r}\cdot{{A}\over{d}}

Misorientation.html

  1. g B = Δ g A B g A g_{B}=\Delta g_{AB}g_{A}
  2. Δ g A B = g B g A - 1 \Delta g_{AB}=g_{B}g_{A}^{-1}
  3. Δ g A B = Δ g B A \Delta g_{AB}=\Delta g_{BA}
  4. 1 24 * 24 * 2 = 1 1152 \frac{1}{24*24*2}=\frac{1}{1152}
  5. Δ g A B = O B c r y s g B ( O A c r y s g A ) - 1 \Delta g_{AB}=O_{B}^{crys}g_{B}(O_{A}^{crys}g_{A})^{-1}
  6. d Δ g d\Delta g
  7. Δ g \Delta g
  8. [ c ϕ 1 c ϕ 2 - s ϕ 1 s ϕ 2 c Φ s ϕ 1 c ϕ 2 + c ϕ 1 s ϕ 2 c Φ s ϕ 2 s Φ - c ϕ 1 s ϕ 2 - s ϕ 1 c ϕ 2 c Φ - s ϕ 1 s ϕ 2 + c ϕ 1 c ϕ 2 c Φ c ϕ 2 s Φ s ϕ 1 s Φ - c ϕ 1 s Φ c Φ ] \begin{bmatrix}c\phi_{1}c\phi_{2}-s\phi_{1}s\phi_{2}c\Phi&s\phi_{1}c\phi_{2}+c% \phi_{1}s\phi_{2}c\Phi&s\phi_{2}s\Phi\\ -c\phi_{1}s\phi_{2}-s\phi_{1}c\phi_{2}c\Phi&-s\phi_{1}s\phi_{2}+c\phi_{1}c\phi% _{2}c\Phi&c\phi_{2}s\Phi\\ s\phi_{1}s\Phi&-c\phi_{1}s\Phi&c\Phi\end{bmatrix}
  9. g c o p p e r = [ - 0.579 0.707 0.406 - 0.579 - 0.707 0.406 0.574 0 0.819 ] g_{copper}=\begin{bmatrix}-0.579&0.707&0.406\\ -0.579&-0.707&0.406\\ 0.574&0&0.819\\ \end{bmatrix}
  10. g S 3 = [ - 0.376 0.756 0.536 - 0.770 - 0.577 0.273 0.516 - 0.310 0.799 ] g_{S3}=\begin{bmatrix}-0.376&0.756&0.536\\ -0.770&-0.577&0.273\\ 0.516&-0.310&0.799\\ \end{bmatrix}
  11. Δ g A B = g c o p p e r g S 3 - 1 = [ 0.970 0.149 - 0.194 - 0.099 0.965 0.244 0.224 - 0.218 0.950 ] \Delta g_{AB}=g_{copper}g_{S3}^{-1}=\begin{bmatrix}0.970&0.149&-0.194\\ -0.099&0.965&0.244\\ 0.224&-0.218&0.950\\ \end{bmatrix}
  12. c o s Θ = ( g 11 + g 22 + g 33 - 1 ) / 2 cos\Theta=(g_{11}+g_{22}+g_{33}-1)/2
  13. r 1 = ( g 23 - g 32 ) / ( 2 s i n Θ ) r_{1}=(g_{23}-g_{32})/(2sin\Theta)
  14. r 2 = ( g 31 - g 13 ) / ( 2 s i n Θ ) r_{2}=(g_{31}-g_{13})/(2sin\Theta)
  15. r 3 = ( g 12 - g 21 ) / ( 2 s i n Θ ) r_{3}=(g_{12}-g_{21})/(2sin\Theta)

Mnev's_universality_theorem.html

  1. S n S\subset{\mathbb{R}}^{n}
  2. n {\mathbb{R}}^{n}
  3. S n S\subset{\mathbb{R}}^{n}
  4. U = U 1 U k U=U_{1}\coprod\cdots\coprod U_{k}
  5. V = V 1 V k V=V_{1}\coprod\cdots\coprod V_{k}
  6. U i ϕ i V i U_{i}\stackrel{\phi_{i}}{\mapsto}V_{i}
  7. U n + d , V n U\subset{\mathbb{R}}^{n+d},V\subset{\mathbb{R}}^{n}
  8. U = U 1 U k U=U_{1}\coprod\cdots\coprod U_{k}
  9. V = V 1 V k V=V_{1}\coprod\cdots\coprod V_{k}
  10. U i U_{i}
  11. V i V_{i}
  12. π \pi
  13. π : U V \pi:\;U\mapsto V
  14. ϕ 1 , , ϕ l , ψ 1 , , ψ m : n ( d ) * \phi_{1},\dots,\phi_{l},\psi_{1},\dots,\psi_{m}:\;{\mathbb{R}}^{n}\mapsto({% \mathbb{R}}^{d})^{*}
  15. U i = { ( v , v ) n + d | v V i U_{i}=\{(v,v^{\prime})\in{\mathbb{R}}^{n+d}\ \ |\ \ v\in V_{i}
  16. ϕ a ( v ) , v > 0 , ψ b ( v ) , v = 0 \langle\phi_{a}(v),v^{\prime}\rangle>0,\langle\psi_{b}(v),v^{\prime}\rangle=0
  17. a = 1 , , l , b = 1 , , m } . a=1,\dots,l,b=1,\dots,m\}.
  18. n {\mathbb{R}}^{n}

Mobile_Membranes.html

  1. R R
  2. {\mathcal{M}}
  3. M , N , M,N,\dots
  4. V * V^{*}
  5. u , v , u,v,\dots
  6. V V
  7. a , b , a,b,\dots
  8. M : := u [ M ] u M M \qquad\qquad M::=u\;\mid\;[\;M\;]_{u}\;\mid\;M\|M
  9. M M
  10. N N
  11. M M
  12. N N
  13. M N M\rightarrow N
  14. R R
  15. M M
  16. N N
  17. R R
  18. ( Comp1 ) M M M N M N ; ( Comp2 ) M M N N M N M N \qquad{\it(Comp1)}\quad\frac{\displaystyle M\ \rightarrow\ M^{\prime}}{% \displaystyle M\|N\ \rightarrow\ M^{\prime}\|N};\qquad\qquad{\it(Comp2)}\quad% \frac{\displaystyle M\ \rightarrow\ M^{\prime}\qquad\displaystyle N\ % \rightarrow\ N^{\prime}}{\displaystyle M\|N\ \rightarrow\ M^{\prime}\|N^{% \prime}}
  19. ( 𝑀𝑒𝑚 ) M M [ M ] u [ M ] u ; ( 𝑆𝑡𝑟𝑢𝑐 ) M m e m M M N N m e m N M N \qquad{\it(Mem)}\ \frac{\displaystyle M\ \rightarrow\ M^{\prime}}{% \displaystyle[\;M\;]_{u}\rightarrow[\;M^{\prime}\;]_{u}};\qquad\qquad{\it(% Struc)}\ \frac{\displaystyle M\equiv_{mem}M^{\prime}\quad M^{\prime}% \rightarrow N^{\prime}\quad\ N^{\prime}\equiv_{mem}N}{\displaystyle M% \rightarrow N}
  20. M 0 M_{0}
  21. R R
  22. 𝑜𝑏𝑗𝑒𝑐𝑡 𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛 {\it object~{}evolution}
  23. 𝑒𝑛𝑑𝑜𝑐𝑦𝑡𝑜𝑠𝑖𝑠 {\it endocytosis}
  24. 𝑒𝑥𝑜𝑐𝑦𝑡𝑜𝑠𝑖𝑠 {\it exocytosis}
  25. 𝑝𝑖𝑛𝑜𝑐𝑦𝑡𝑜𝑠𝑖𝑠 {\it pinocytosis}
  26. 𝑝ℎ𝑎𝑔𝑜𝑐𝑦𝑡𝑜𝑠𝑖𝑠 {\it phagocytosis}
  27. \mathcal{M}
  28. R R
  29. [ [ a M ] m N ] k [ [ v M ] m N ] k [[a\;\|\;M]_{m}\;\|\;N]_{k}\rightarrow[[v\;\|\;M]_{m}\;\|\;N]_{k}
  30. k , m 𝒩 k,m\in\mathcal{N}
  31. a V a\in V
  32. v V * v\in V^{*}
  33. [ a M ] m [ v M ] m [a\;\|\;M]_{m}\rightarrow[v\;\|\;M]_{m}
  34. m 𝒩 m\in\mathcal{N}
  35. a V a\in V
  36. v V * v\in V^{*}
  37. [ a M 1 ] h [ M ] m [ [ b M 1 ] h M ] m [a\;\|\;M_{1}]_{h}\;\|\;[M]_{m}\rightarrow[[b\;\|\;M_{1}]_{h}\;\|\;M]_{m}
  38. h , m 𝒩 h,m\in\mathcal{N}
  39. a , b V a,b\in V
  40. [ [ a M 1 ] h M ] m [ b M 1 ] h [ M ] m [[a\;\|\;M_{1}]_{h}\;\|\;M]_{m}\rightarrow[b\;\|\;M_{1}]_{h}\;\|\;[M]_{m}
  41. h , m 𝒩 h,m\in\mathcal{N}
  42. a , b V a,b\in V
  43. M 1 M_{1}
  44. M M
  45. N N
  46. S M ( l e v o l , e n d o , e x o ) SM(levol,endo,exo)
  47. l e v o l levol
  48. g e v o l gevol
  49. l e v o l levol
  50. g e v o l gevol
  51. S M n ( g e v o l , e n d o , e x o ) SM_{n}(gevol,endo,exo)
  52. R E RE
  53. S M 4 ( g e v o l , e n d o , e x o ) = R E SM_{4}(gevol,endo,exo)=RE
  54. S M 3 ( l e v o l , e n d o , e x o ) = R E SM_{3}(levol,endo,exo)=RE
  55. R R
  56. [ u v M ] h [ v N ] m [ w [ w ] h ] m [u\;\|\;v\;\|\;M]_{h}\;\|\;[v^{\prime}\;\|\;N]_{m}\!\rightarrow\![w^{\prime}\;% \|\;[w]_{h}]_{m}
  57. h , m 𝒩 ; u V + , v , v , w , w V * h,m\!\in\!\mathcal{N};u\!\in\!V^{+},v,v^{\prime},w,w^{\prime}\!\in\!V^{*}
  58. [ v N [ u v M ] h ] m [ w M ] h [ w N ] m [v^{\prime}\;\|\;N\;\|\;[u\;\|\;v\;\|\;M]_{h}]_{m}\!\rightarrow\![w\;\|\;M]_{h% }\;\|\;[w^{\prime}\;\|\;N]_{m}
  59. h , m 𝒩 ; u V + , v , v , w , w V * h,m\!\in\!\!\mathcal{N};u\in V^{+},v,v^{\prime},w,w^{\prime}\in V^{*}
  60. [ v M ] h [ u v N ] m [ [ w M ] h w N ] m [v\;\|\;M]_{h}\;\|\;[u\;\|\;v^{\prime}\;\|\;N]_{m}\!\!\rightarrow\!\![[w\;\|\;% M]_{h}\;\|\;w^{\prime}\;\|\;N]_{m}
  61. h , m 𝒩 h,m\!\!\in\!\!\mathcal{N}
  62. u V + , v , v , w , w V * u\!\in\!V^{+},v,v^{\prime}\!,w,w^{\prime}\!\in\!V^{*}
  63. [ u v [ v M ] h N ] m [ w M ] h [ w N ] m [u\;\|\;v^{\prime}\;\|\;[v\;\|\;M]_{h}\;\|\;N]_{m}\!\!\rightarrow\!\![w\;\|\;M% ]_{h}\;\|\;[w^{\prime}\;\|\;N]_{m}
  64. h , m 𝒩 , u V + , v , v , w , w V * h,m\!\in\!\mathcal{N},u\in V^{+},v,v^{\prime},w,w^{\prime}\in V^{*}
  65. M M
  66. N N
  67. n n
  68. α { e x o , e n d o , f e n d o , f e x o } \alpha\subseteq\{exo,endo,fendo,fexo\}
  69. E M n ( α ) EM_{n}(\alpha)
  70. E M 3 ( e n d o , e x o ) = E M 3 ( f e n d o , f e x o ) EM_{3}(endo,exo)=EM_{3}(fendo,fexo)
  71. E M 12 ( e n d o , e x o , f e n d o , f e x o ) = R E EM_{12}(endo,exo,fendo,fexo)=RE
  72. E M 9 ( e n d o , e x o , f e n d o , f e x o ) = R E EM_{9}(endo,exo,fendo,fexo)=RE
  73. a a
  74. a ¯ \overline{a}
  75. R R
  76. [ u v M ] h [ u ¯ v N ] m [ [ w M ] h w N ] m [u\;\|\;v\;\|\;M]_{h}\;\|\;[\overline{u}\;\|\;v^{\prime}\;\|\;N]_{m}% \rightarrow[\;[w\;\|\;M]_{h}\;\|\;w^{\prime}\;\|\;N]_{m}
  77. h , m 𝒩 , u , u ¯ V + , v , v , w , w V * h,m\in\mathcal{N},u,\overline{u}\in V^{+},v,v^{\prime},w,w^{\prime}\!\!\in V^{*}
  78. [ u ¯ v N [ u v M ] h ] m [ w M ] h [ w N ] m [\overline{u}\;\|\;v^{\prime}\;\|\;N\;\|\;[u\;\|\;v\;\|\;M]_{h}]_{m}% \rightarrow[w\;\|\;M]_{h}\;\|\;[w^{\prime}\;\|\;N]_{m}
  79. h , m 𝒩 , u , u ¯ V + , v , v , w , w V * h,m\in\mathcal{N},u,\overline{u}\in V^{+},v,v^{\prime},w,w^{\prime}\!\!\in V^{*}
  80. M M
  81. N N
  82. n n
  83. M M n ( m e n d o , m e x o ) MM_{n}(mendo,mexo)
  84. M M 3 ( m e n d o , m e x o ) = R E MM_{3}(mendo,mexo)=RE
  85. R R
  86. [ M ] v a u [ [ ] u x M ] v y [M]_{v\;\|\;a\;\|\;u}\rightarrow[[~{}]_{u\;\|\;x}\;\|\;M]_{v\;\|\;y}
  87. a V , u , v , x , y V * , u x , v y V + a\in V,u,v,x,y\in V^{*},ux,vy\in V^{+}
  88. [ [ M ] a u N ] a ¯ v M [ N ] u v x [[M]_{a\;\|\;u}\;\|\;N]_{\overline{a}\;\|\;v}\rightarrow M\;\|\;[N]_{u\;\|\;v% \;\|\;x}
  89. , a ¯ V , u , v , x V * , u v x V + ,\overline{a}\in V,u,v,x\in V^{*},uvx\in V^{+}
  90. [ M 1 ] a u [ N ] a ¯ b v [ [ [ M 1 ] u x ] b N ] v y [M_{1}]_{a\;\|\;u}\;\|\;[N]_{\overline{a}\;\|\;b\;\|\;v}\rightarrow[[[M_{1}]_{% u\;\|\;x}]_{b}\;\|\;N]_{v\;\|\;y}
  91. a , a ¯ , b V , u , v , x , y V * , u x , v y V + a,\overline{a},b\!\in\!V,u,v,x,y\in V^{*},ux,vy\in V^{+}
  92. M 1 M_{1}
  93. M M
  94. N N
  95. \qquad
  96. \qquad
  97. \qquad
  98. \qquad
  99. n n
  100. r 1 , r 2 { p i n o , e x o , p h a g o } r_{1},r_{2}\in\{pino,exo,phago\}
  101. r , s r,s
  102. M M O S n ( r 1 ( r ) , r 2 ( s ) MMOS_{n}(r_{1}(r),r_{2}(s)
  103. * *
  104. M M O S m ( p i n o ( r ) , e x o ( s ) ) = R E MMOS_{m}(pino(r),exo(s))=RE
  105. m 3 , r 5 , s 4 m\geq 3,r\geq 5,s\geq 4
  106. M M O S m ( p h a g o ( r ) , e x o ( s ) ) = R E MMOS_{m}(phago(r),exo(s))=RE
  107. m 4 , r 6 , s 3 m\geq 4,r\geq 6,s\geq 3
  108. 𝒩 {\mathcal{N}}
  109. m , n , m,n,\dots
  110. 𝒜 {\mathcal{A}}
  111. A , A , B , B , A,A^{\prime},B,B^{\prime},\dots
  112. C , C , C,C^{\prime},\dots
  113. C : := i n n i n ¯ n o u t n o u t ¯ n o p e n n o p e n ¯ n \qquad C::=in\ n\;\mid\;\overline{in}\ n\;\mid\;out\ n\;\mid\;\overline{out}\ % n\;\mid\;open\ n\;\mid\;\overline{open}\ n
  114. A : := 0 A B C . A n [ A ] \qquad A::=0\quad\mid\quad A\;\mid\;B\quad\mid\quad C.A\quad\mid\quad n[\;A\;]
  115. 0
  116. C . A C.\,A
  117. C C
  118. A A
  119. n [ A ] n[\;A\;]
  120. n n
  121. A A
  122. A B A\,\mid\,B
  123. A A
  124. B B
  125. ( ν n ) A (\nu n)A
  126. n n
  127. A A
  128. a m b \equiv_{amb}
  129. ( 𝒜 , , 0 ) (\mathcal{A},\mid,0)
  130. a m b \Rightarrow_{amb}
  131. ( I n ) n [ i n m . A A ] m [ i n ¯ m . B B ] a m b m [ n [ A A ] B B ] (In)\quad n[\;in\ m.A\mid A^{\prime}\;]\mid m[\;\overline{in}\ m.B\mid B^{% \prime}\;]\Rightarrow_{amb}m[\;n[\;A\mid A^{\prime}\;]\mid B\mid B^{\prime}\;]
  132. ( O u t ) m [ n [ o u t m . A A ] o u t ¯ m . B B ] a m b n [ A A ] m [ B B ] (Out)\quad m[\;n[\;out\ m.A\mid A^{\prime}\;]\mid\overline{out}\ m.B\mid B^{% \prime}\;]\Rightarrow_{amb}n[\;A\mid A^{\prime}\;]\mid m[\;B\mid B^{\prime}\;]
  133. ( O p e n ) o p e n n . A n [ o p e n ¯ n . B B ] a m b A B B (Open)\quad open\ n.A\;\mid\;n[\;\overline{open}\ n.B\mid B^{\prime}\;]% \Rightarrow_{amb}A\;\mid\;B\;\mid\;B^{\prime}
  134. ( C o m p 1 ) A a m b A A B a m b A B ; ( C o m p 2 ) A a m b A B a m b B A B a m b A B (Comp1)\quad\frac{\displaystyle A\Rightarrow_{amb}A^{\prime}}{\displaystyle A% \;\mid\;B\Rightarrow_{amb}A^{\prime}\;\mid\;B};\qquad\qquad\qquad\qquad(Comp2)% \quad\frac{\displaystyle A\Rightarrow_{amb}A^{\prime}\quad\displaystyle B% \Rightarrow_{amb}B^{\prime}}{\displaystyle A\;\mid\;B\Rightarrow_{amb}A^{% \prime}\;\mid\;B^{\prime}}
  135. ( A m b ) A a m b A n [ A ] a m b n [ A ] ; ( S t r u c ) A A , A a m b B , B B A a m b B (Amb)\quad\frac{\displaystyle A\Rightarrow_{amb}A^{\prime}}{\displaystyle n[\;% A\;]\Rightarrow_{amb}n[\;A^{\prime}\;]};\qquad\qquad\qquad\qquad(Struc)\quad% \frac{\displaystyle A\equiv A^{\prime},\ A^{\prime}\Rightarrow_{amb}B^{\prime}% ,\ B^{\prime}\equiv B}{\displaystyle A\Rightarrow_{amb}B}
  136. a m b * \Rightarrow^{*}_{amb}
  137. a m b \Rightarrow_{amb}
  138. 𝒜 \mathcal{A}
  139. \mathcal{M}
  140. 𝒯 : 𝒜 \mathcal{T}:\mathcal{A}\rightarrow\mathcal{M}
  141. 𝒯 ( A ) = d l o c k 𝒯 1 ( A ) \mathcal{T}(A)=dlock~{}\mathcal{T}_{1}(A)
  142. 𝒯 1 : 𝒜 \mathcal{T}_{1}:\mathcal{A}\rightarrow\mathcal{M}
  143. 𝒯 1 ( A ) = \qquad\mathcal{T}_{1}(A)=
  144. { c a p n [ ] c a p n if A = c a p n c a p n [ 𝒯 1 ( A 1 ) ] c a p n if A = c a p n . A 1 [ 𝒯 1 ( A 1 ) ] n if A = n [ A 1 ] [ ] n if A = n [ ] 𝒯 1 ( A 1 ) 𝒯 1 ( A 2 ) if A = A 1 A 2 λ if A = 0 \begin{cases}cap~{}n\;\|\;[~{}]_{cap~{}n}&\mbox{if }~{}A=cap~{}n\\ cap~{}n\;\|\;[\;\mathcal{T}_{1}(A_{1})\;]_{cap~{}n}&\mbox{if }~{}A=cap~{}n.\,A% _{1}\\ \;[\;\mathcal{T}_{1}(A_{1})\;]_{n}&\mbox{if }~{}A=n[\;A_{1}\;]\\ \;[~{}]_{n}&\mbox{if }~{}A=n[~{}]\\ \mathcal{T}_{1}(A_{1})\;\|\;\mathcal{T}_{1}(A_{2})&\mbox{if }~{}A=A_{1}\,\mid% \,A_{2}\\ \lambda&\mbox{if }~{}A=0\end{cases}
  145. d l o c k dlock
  146. A a m b B \qquad\qquad A\equiv_{amb}B
  147. 𝒯 ( A ) m e m 𝒯 ( B ) \mathcal{T}(A)\equiv_{mem}\mathcal{T}(B)
  148. M M
  149. N N
  150. d l o c k dlock
  151. M m e m N M\Rightarrow_{mem}N
  152. r 1 , , r i r_{1},\ldots,r_{i}
  153. M M
  154. N N
  155. A A
  156. B B
  157. M M
  158. A a m b B A\Rightarrow_{amb}B
  159. M = 𝒯 ( A ) M=\mathcal{T}(A)
  160. M M
  161. M m e m N M\Rightarrow_{mem}N
  162. N = 𝒯 ( B ) N=\mathcal{T}(B)
  163. M M
  164. N N
  165. d l o c k dlock
  166. A A
  167. M = 𝒯 ( A ) M=\mathcal{T}(A)
  168. M M
  169. M m e m N M\Rightarrow_{mem}N
  170. B B
  171. A a m b * B A\Rightarrow_{amb}^{*}B
  172. N = 𝒯 ( B ) N=\mathcal{T}(B)
  173. A a m b B A\Rightarrow_{amb}B
  174. 𝒯 ( A ) m e m 𝒯 ( B ) \mathcal{T}(A)\Rightarrow_{mem}\mathcal{T}(B)
  175. 𝒯 ( A ) m e m M \mathcal{T}(A)\Rightarrow_{mem}M
  176. B B
  177. A a m b B A\Rightarrow_{amb}B
  178. M = 𝒯 ( B ) M=\mathcal{T}(B)
  179. P , Q P,Q
  180. : : = ::\,=
  181. P Q σ ( ) σ ( P ) P\circ Q\;\mid\;\sigma(~{})\;\mid\;\sigma(P)
  182. σ , τ \sigma,\tau
  183. : : = ::\,=
  184. O σ τ a . σ O\;\mid\;\sigma\mid\tau\;\mid\;a.\sigma
  185. a , b a,b
  186. : : = ::\,=
  187. n n ¯ ( σ ) n n ¯ p i n o ( σ ) n^{\searrow}\;\mid\;\overline{n}^{\searrow}(\sigma)\;\mid\;n^{\nwarrow}\;\mid% \;\overline{n}^{\nwarrow}\;\mid\;pino(\sigma)
  188. \searrow
  189. \nwarrow
  190. s s
  191. s = s 1 s 2 s=s_{1}\;\mid\;s_{2}
  192. s s
  193. a a
  194. s 1 s = a . s 1 s_{1}s=a.s_{1}
  195. n n
  196. 𝒫 \mathcal{P}
  197. a .0 a.0
  198. a a
  199. 0 ( P ) 0(P)
  200. ( P ) (P)
  201. 0 ( ) 0(~{})
  202. ( ) (~{})
  203. P Q b Q P P\circ Q\equiv_{b}Q\circ P
  204. \qquad\qquad
  205. σ τ b τ σ \sigma\;\mid\;\tau\equiv_{b}\tau\;\mid\;\sigma
  206. P ( Q R ) b ( P Q ) R P\circ(Q\circ R)\equiv_{b}(P\circ Q)\circ R
  207. \qquad\qquad
  208. σ ( τ ρ ) b ( σ τ ) ρ \sigma\;\mid\;(\tau\;\mid\;\rho)\equiv_{b}(\sigma\;\mid\;\tau)\;\mid\;\rho
  209. \qquad
  210. \qquad\qquad
  211. σ 0 b σ \sigma\;\mid\;0\equiv_{b}\sigma
  212. \qquad
  213. \qquad\qquad
  214. \qquad
  215. P b Q implies P R b Q R P\equiv_{b}Q\mbox{implies }~{}P\circ R\equiv_{b}Q\circ R
  216. \qquad\qquad
  217. σ b τ implies σ ρ b τ ρ \sigma\equiv_{b}\tau\mbox{implies }~{}\sigma\;\mid\;\rho\equiv_{b}\tau\;\mid\;\rho
  218. P b Q and σ b τ implies σ ( P ) b τ ( Q ) P\equiv_{b}Q\mbox{and }~{}\sigma\equiv_{b}\tau\mbox{implies }~{}\sigma(P)% \equiv_{b}\tau(Q)
  219. \qquad\qquad
  220. σ b τ implies a . σ b a . τ \sigma\equiv_{b}\tau\mbox{implies }~{}a.\sigma\equiv_{b}a.\tau
  221. p i n o ( ρ ) . σ σ 0 ( P ) b σ σ 0 ( ρ ( ) P ) pino(\rho).\sigma\mid\sigma_{0}(P)\rightarrow_{b}\sigma\mid\sigma_{0}(\rho(~{}% )\circ P)
  222. \qquad\qquad
  223. n ¯ . τ τ 0 ( n . σ σ 0 ( P ) Q ) b P σ σ 0 τ τ 0 ( Q ) \overline{n}^{\nwarrow}.\tau\mid\tau_{0}(n^{\nwarrow}.\sigma\mid\sigma_{0}(P)% \circ Q)\rightarrow_{b}P\circ\sigma\mid\sigma_{0}\mid\tau\mid\tau_{0}(Q)
  224. \qquad\qquad
  225. n . σ σ 0 ( P ) n ¯ ( ρ ) . τ τ 0 ( Q ) b τ τ 0 ( ρ ( σ σ 0 ( P ) ) Q ) n^{\searrow}.\sigma\mid\sigma_{0}(P)\circ\overline{n}^{\searrow}(\rho).\tau% \mid\tau_{0}(Q)\rightarrow_{b}\tau\mid\tau_{0}(\rho(\sigma\mid\sigma_{0}(P))% \circ Q)
  226. \qquad\qquad
  227. P b Q implies P R b Q R P\rightarrow_{b}Q\mbox{implies }~{}P\circ R\rightarrow_{b}Q\circ R
  228. \qquad\qquad
  229. P b Q implies σ ( P ) b σ ( Q ) P\rightarrow_{b}Q\mbox{implies }~{}\sigma(P)\rightarrow_{b}\sigma(Q)
  230. \qquad\qquad
  231. P b P and P b Q and Q b Q implies P b Q P\equiv_{b}P^{\prime}\mbox{and }~{}P^{\prime}\rightarrow_{b}Q^{\prime}\mbox{% and }~{}Q^{\prime}\equiv_{b}Q\mbox{implies }~{}P\rightarrow_{b}Q
  232. \qquad\qquad
  233. p i n o ( ρ ) pino(\rho)
  234. p i n o pino
  235. σ \sigma
  236. p i n o pino
  237. n n^{\nwarrow}
  238. n ¯ \overline{n}^{\nwarrow}
  239. P P
  240. n n^{\searrow}
  241. n ¯ ( ρ ) \overline{n}^{\searrow}(\rho)
  242. Q Q
  243. P P
  244. Q Q
  245. P P
  246. ρ \rho
  247. 𝒫 \mathcal{P}
  248. \mathcal{M}
  249. 𝒯 : 𝒫 \mathcal{T}:\mathcal{P}\rightarrow\mathcal{M}
  250. 𝒯 ( P ) = \mathcal{T}(P)=
  251. { [ ] 𝒮 ( σ ) if P = σ ( ) [ 𝒯 ( R ) ] 𝒮 ( σ ) if P = σ ( R ) 𝒯 ( Q ) 𝒯 ( R ) if P = Q R \begin{cases}\;[~{}]_{\mathcal{S}(\sigma)}&\mbox{if }~{}P=\sigma(~{})\\ \;[\mathcal{T}(R)]_{\mathcal{S}(\sigma)}&\mbox{if }~{}P=\sigma(R)\\ \mathcal{T}(Q)\;\|\;\mathcal{T}(R)&\mbox{if }~{}P=Q\,\mid\,R\\ \end{cases}
  252. 𝒮 : 𝒫 V * \mathcal{S}:\mathcal{P}\rightarrow V^{*}
  253. 𝒮 ( σ ) = \mathcal{S}(\sigma)=
  254. { σ if σ = n o r σ = n o r σ = n ¯ n ¯ S ( ρ ) if σ = n ¯ ( ρ ) p i n o S ( ρ ) if σ = p i n o ( ρ ) 𝒮 ( a ) 𝒮 ( ρ ) if σ = a . ρ 𝒮 ( τ ) 𝒮 ( ρ ) if σ = τ ρ λ if σ = 0 \begin{cases}\sigma&\mbox{if }~{}\sigma=n^{\searrow}or\sigma=n^{\nwarrow}or% \sigma=\overline{n}^{\nwarrow}\\ \overline{n}^{\searrow}\;\|\;S(\rho)&\mbox{if }~{}\sigma=\overline{n}^{% \searrow}(\rho)\\ pino\;\|\;S(\rho)&\mbox{if }~{}\sigma=pino(\rho)\\ \mathcal{S}(a)\;\|\;\mathcal{S}(\rho)&\mbox{if }~{}\sigma=a.\rho\\ \mathcal{S}(\tau)\;\|\;\mathcal{S}(\rho)&\mbox{if }~{}\sigma=\tau\,\mid\,\rho% \\ \lambda&\mbox{if }~{}\sigma=0\end{cases}
  255. \qquad\qquad
  256. [ M ] S ( p i n o ( ρ ) . σ σ 0 ) [ [ ] S ( ρ ) M ] S ( σ σ 0 ) [M]_{S(pino(\rho).\sigma\mid\sigma_{0})}\rightarrow[[~{}]_{S(\rho)}\;\|\;M]_{S% (\sigma\mid\sigma_{0})}
  257. \qquad\qquad
  258. [ [ M ] S ( n . σ σ 0 ) N ] S ( n ¯ . τ τ 0 ) M [ N ] S ( σ σ 0 τ τ 0 ) [[M]_{S(n^{\nwarrow}.\sigma\mid\sigma_{0})}\;\|\;N]_{S(\overline{n}^{\nwarrow}% .\tau\mid\tau_{0})}\rightarrow M\;\|\;[N]_{S(\sigma\mid\sigma_{0}\mid\tau\mid% \tau_{0})}
  259. \qquad\qquad
  260. [ M 1 ] S ( n . σ σ 0 ) [ N ] S ( n ¯ ( ρ ) . τ τ 0 ) [ [ [ M 1 ] S ( σ σ 0 ) ] S ( ρ ) N ] S ( τ τ 0 ) [M_{1}]_{S(n^{\searrow}.\sigma\mid\sigma_{0})}\;\|\;[N]_{S(\overline{n}^{% \searrow}(\rho).\tau\mid\tau_{0})}\rightarrow[[[M_{1}]_{S(\sigma\mid\sigma_{0}% )}]_{S(\rho)}\;\|\;N]_{S(\tau\mid\tau_{0})}
  261. M 1 M_{1}
  262. M M
  263. N N
  264. P P
  265. M = 𝒯 ( P ) M=\mathcal{T}(P)
  266. N N
  267. M m N M\equiv_{m}N
  268. N = 𝒯 ( Q ) N=\mathcal{T}(Q)
  269. P b Q P\equiv_{b}Q
  270. P P
  271. M = 𝒯 ( P ) M=\mathcal{T}(P)
  272. Q Q
  273. N = 𝒯 ( Q ) N=\mathcal{T}(Q)
  274. M m N M\equiv_{m}N
  275. P b Q P\not\equiv_{b}Q
  276. P = n . n ( ) P=n^{\searrow}.n^{\nwarrow}(~{})
  277. M = 𝒯 = [ ] n n m [ ] n n = N M=\mathcal{T}=[~{}]_{n^{\searrow}\;\|\;n^{\nwarrow}}\equiv_{m}[~{}]_{n^{% \nwarrow}\;\|\;n^{\searrow}}=N
  278. Q = n . n ( ) Q=n^{\nwarrow}.n^{\searrow}(~{})
  279. Q = n n ( ) Q=n^{\nwarrow}\mid n^{\searrow}(~{})
  280. N = 𝒯 ( Q ) N=\mathcal{T}(Q)
  281. P b Q P\not\equiv_{b}Q
  282. P P
  283. M = 𝒯 ( P ) M=\mathcal{T}(P)
  284. N N
  285. M N M\rightarrow N
  286. N = 𝒯 ( Q ) N=\mathcal{T}(Q)
  287. P b Q P\rightarrow_{b}Q
  288. P P
  289. M = 𝒯 ( P ) M=\mathcal{T}(P)
  290. Q Q
  291. N = 𝒯 ( Q ) N=\mathcal{T}(Q)
  292. M N M\rightarrow N
  293. P ↛ b Q P\not\rightarrow_{b}Q
  294. P = n ¯ . n ¯ ( n . n ( ) ) P=\overline{n}^{\nwarrow}.\overline{n}^{\nwarrow}(n^{\nwarrow}.n^{\searrow}(~{% }))
  295. M = ( ( ) n n ) n ¯ n ¯ M=((~{})_{n^{\nwarrow}\;\|\;n^{\searrow}})_{\overline{n}^{\nwarrow}\;\|\;% \overline{n}^{\nwarrow}}
  296. M [ ] n n ¯ M\rightarrow[~{}]_{n^{\searrow}\;\|\;\overline{n}^{\nwarrow}}
  297. Q = n . n ¯ ( ) Q=n^{\searrow}.\overline{n}^{\nwarrow}(~{})
  298. N = 𝒯 ( Q ) N=\mathcal{T}(Q)
  299. P ↛ b Q P\not\rightarrow_{b}Q

Modal_depth.html

  1. \Box
  2. \Diamond
  3. M D ( ϕ ) MD(\phi)
  4. ϕ \phi
  5. M D ( p ) = 0 MD(p)=0
  6. p p
  7. M D ( ) = 0 MD(\top)=0
  8. M D ( ) = 0 MD(\bot)=0
  9. M D ( ¬ φ ) = M D ( φ ) MD(\neg\varphi)=MD(\varphi)
  10. M D ( φ ψ ) = m a x ( M D ( φ ) , M D ( ψ ) ) MD(\varphi\wedge\psi)=max(MD(\varphi),MD(\psi))
  11. M D ( φ ψ ) = m a x ( M D ( φ ) , M D ( ψ ) ) MD(\varphi\vee\psi)=max(MD(\varphi),MD(\psi))
  12. M D ( φ ψ ) = m a x ( M D ( φ ) , M D ( ψ ) ) MD(\varphi\rightarrow\psi)=max(MD(\varphi),MD(\psi))
  13. M D ( φ ) = 1 + M D ( φ ) MD(\Box\varphi)=1+MD(\varphi)
  14. M D ( φ ) = 1 + M D ( φ ) MD(\Diamond\varphi)=1+MD(\varphi)
  15. ( p p ) \Box(\Box p\rightarrow p)
  16. M D ( ( p p ) ) = MD(\Box(\Box p\rightarrow p))=
  17. 1 + M D ( p p ) = 1+MD(\Box p\rightarrow p)=
  18. 1 + m a x ( M D ( p ) , M D ( p ) ) = 1+max(MD(\Box p),MD(p))=
  19. 1 + m a x ( 1 + M D ( p ) , 0 ) = 1+max(1+MD(p),0)=
  20. 1 + m a x ( 1 + 0 , 0 ) = 1+max(1+0,0)=
  21. 1 + 1 = 1+1=
  22. M , w φ M,w\models\Diamond\Diamond\varphi
  23. v v
  24. M , v φ M,v\models\Diamond\varphi
  25. u u
  26. M , u φ M,u\models\varphi
  27. u u
  28. v v
  29. w w
  30. w w
  31. v v
  32. v v
  33. u u
  34. φ \Box\varphi
  35. φ \varphi
  36. w w
  37. v W ( w , v ) R \forall v\in W\ (w,v)\not\in R
  38. W W
  39. R R
  40. M , w φ M,w\models\Box\Box\varphi
  41. w w

Models_of_neural_computation.html

  1. 10 6 10^{6}
  2. r p r_{p}
  3. r j r_{j}
  4. r p = | [ e p - j = 1 , j p n k p j | r j - r p j o | ] | r_{p}=\left|\left[e_{p}-\sum_{j=1,j\neq p}^{n}k_{pj}\left|r_{j}-r_{pj}^{o}% \right|\right]\right|
  5. e p e_{p}
  6. r p j o r_{pj}^{o}
  7. k p j k_{pj}
  8. y R ( t ) - y L ( t ) y_{R}\left(t\right)-y_{L}\left(t\right)
  9. y L ( t ) = 0 τ u L ( σ ) w ( t - σ ) d σ y_{L}\left(t\right)=\int_{0}^{\tau}u_{L}\left(\sigma\right)w\left(t-\sigma% \right)d\sigma
  10. y R ( t ) = 0 τ u R ( σ ) w ( t - σ ) d σ y_{R}\left(t\right)=\int_{0}^{\tau}u_{R}\left(\sigma\right)w\left(t-\sigma% \right)d\sigma
  11. w ( t - σ ) w\left(t-\sigma\right)
  12. R = A 1 ( t - τ ) B 2 ( t ) - A 2 ( t - τ ) B 1 ( t ) R=A_{1}(t-\tau)B_{2}(t)-A_{2}(t-\tau)B_{1}(t)
  13. w w^{\prime}
  14. g g
  15. f j = i g ( w j i x i + b j ) f_{j}=\sum_{i}g\left(w_{ji}^{\prime}x_{i}+b_{j}\right)

Modified_Allan_variance.html

  1. mod σ y 2 ( n τ 0 ) = 1 2 τ 2 [ 1 n i = 0 n - 1 x i + 2 n - 2 x i + n + x i ] 2 \operatorname{mod}\sigma_{y}^{2}(n\tau_{0})=\frac{1}{2\tau^{2}}\left\langle% \left[\frac{1}{n}\sum_{i=0}^{n-1}x_{i+2n}-2x_{i+n}+x_{i}\right]^{2}\right\rangle
  2. mod σ y 2 ( n τ 0 ) = 1 2 [ 1 n i = 0 n - 1 y ¯ i + n - y ¯ i ] 2 \operatorname{mod}\sigma_{y}^{2}(n\tau_{0})=\frac{1}{2}\left\langle\left[\frac% {1}{n}\sum_{i=0}^{n-1}\bar{y}_{i+n}-\bar{y}_{i}\right]^{2}\right\rangle
  3. mod σ y 2 ( n τ 0 ) = 1 2 n 4 τ 0 2 ( N - 3 n + 1 ) j = 0 N - 3 n { i = j j + n - 1 x i + 2 n - 2 x i + n + x i } 2 \operatorname{mod}\sigma_{y}^{2}(n\tau_{0})=\frac{1}{2n^{4}\tau_{0}^{2}(N-3n+1% )}\sum_{j=0}^{N-3n}\left\{\sum_{i=j}^{j+n-1}x_{i+2n}-2x_{i+n}+x_{i}\right\}^{2}
  4. mod σ y 2 ( n τ 0 ) = 1 2 n 4 ( M - 3 n + 2 ) j = 0 M - 3 n + 1 { i = j j + n - 1 ( k = i i + n - 1 y ¯ k + n - y ¯ k ) } 2 \operatorname{mod}\sigma_{y}^{2}(n\tau_{0})=\frac{1}{2n^{4}(M-3n+2)}\sum_{j=0}% ^{M-3n+1}\left\{\sum_{i=j}^{j+n-1}\left(\sum_{k=i}^{i+n-1}\bar{y}_{k+n}-\bar{y% }_{k}\right)\right\}^{2}

Modified_models_of_gravity.html

  1. a 0 1 × 10 - 10 m s - 2 a_{0}\approx 1\times 10^{-10}{\rm m\ s}^{-2}

Modified_Newtonian_dynamics.html

  1. μ ( a a 0 ) = ( 1 + a 0 a ) - 1 \mu{\left(\frac{a}{a_{0}}\right)}=\left(1+\frac{a_{0}}{a}\right)^{-1}
  2. μ ( a a 0 ) = ( 1 + ( a 0 a ) 2 ) - 1 / 2 \mu{\left(\frac{a}{a_{0}}\right)}=\left(1+\left(\frac{a_{0}}{a}\right)^{2}% \right)^{-1/2}
  3. F N = m a 2 / a 0 F_{N}=ma^{2}/a_{0}
  4. G M m r 2 = m ( v 2 r ) 2 a 0 v 4 = G M a 0 \frac{GMm}{r^{2}}=m\frac{\left(\frac{v^{2}}{r}\right)^{2}}{a_{0}}\Rightarrow v% ^{4}=GMa_{0}
  5. N e w t o n = - ϕ 2 8 π G \mathcal{L}_{Newton}=-\frac{\|\nabla\phi\|^{2}}{8\pi G}
  6. A Q U A L = - a 0 2 F ( ϕ 2 / a 0 2 ) 8 π G \mathcal{L}_{AQUAL}=-\frac{a_{0}^{2}F(\|\nabla\phi\|^{2}/a_{0}^{2})}{8\pi G}
  7. [ μ ( ϕ a 0 ) ϕ ] = 4 π G ρ \nabla\cdot\left[\mu\left(\frac{\left\|\nabla\phi\right\|}{a_{0}}\right)\nabla% \phi\right]=4\pi G\rho
  8. a i n > a 0 a_{in}>a_{0}
  9. a e x < a i n < a 0 a_{ex}<a_{in}<a_{0}
  10. a i n < a 0 < a e x a_{in}<a_{0}<a_{ex}
  11. a i n < a e x < a 0 a_{in}<a_{ex}<a_{0}

Modulational_instability.html

  1. A z + i β 2 2 A t 2 = i γ | A | 2 A \frac{\partial A}{\partial z}+i\beta_{2}\frac{\partial^{2}A}{\partial t^{2}}=i% \gamma|A|^{2}A
  2. A A
  3. t t
  4. z z
  5. β 2 \beta_{2}
  6. γ \gamma
  7. P P
  8. A = P e i γ P z A=\sqrt{P}e^{i\gamma Pz}
  9. e i γ P z e^{i\gamma Pz}
  10. A = ( P + ϵ ( t , z ) ) e i γ P z A=\left(\sqrt{P}+\epsilon\left(t,z\right)\right)e^{i\gamma Pz}
  11. ϵ ( t , z ) \epsilon\left(t,z\right)
  12. A A
  13. ϵ z + i β 2 2 ϵ t 2 = i γ P ( ϵ + ϵ * ) \frac{\partial\epsilon}{\partial z}+i\beta_{2}\frac{\partial^{2}\epsilon}{% \partial t^{2}}=i\gamma P\left(\epsilon+\epsilon^{*}\right)
  14. ϵ 2 0 \epsilon^{2}\approx 0
  15. ϵ = c 1 e i k m z - i ω m t + c 2 e - i k m z + i ω m t \epsilon=c_{1}e^{ik_{m}z-i\omega_{m}t}+c_{2}e^{-ik_{m}z+i\omega_{m}t}
  16. ω m \omega_{m}
  17. k m k_{m}
  18. c 1 c_{1}
  19. c 2 c_{2}
  20. ω m \omega_{m}
  21. k m k_{m}
  22. k m = ± β 2 2 ω m 4 + 2 γ P β 2 ω m 2 k_{m}=\pm\sqrt{\beta_{2}^{2}\omega_{m}^{4}+2\gamma P\beta_{2}\omega_{m}^{2}}
  23. β 2 2 ω m 2 + 2 γ P β 2 < 0 \beta_{2}^{2}\omega_{m}^{2}+2\gamma P\beta_{2}<0
  24. β 2 \beta_{2}
  25. g g
  26. \equiv
  27. [ 2 | k m | ] \Im\left[2|k_{m}|\right]
  28. P P
  29. \propto
  30. e g z e^{gz}
  31. g = { 2 - β 2 2 ω m 4 - 2 γ P β 2 ω m 2 ; - β 2 2 ω m 2 - 2 γ P β 2 > 0 0 ; - β 2 2 ω m 2 - 2 γ P β 2 0 g=\begin{cases}2\sqrt{-\beta_{2}^{2}\omega_{m}^{4}-2\gamma P\beta_{2}\omega_{m% }^{2}}&;\,-\beta_{2}^{2}\omega_{m}^{2}-2\gamma P\beta_{2}>0\\ 0&;\,-\beta_{2}^{2}\omega_{m}^{2}-2\gamma P\beta_{2}\leq 0\end{cases}
  32. ω m \omega_{m}

Moffat_distribution.html

  1. R = X 2 + Y 2 . R=\sqrt{X^{2}+Y^{2}}.
  2. f ( x , y ; α , β ) = ( β - 1 ) ( π α 2 ) - 1 [ 1 + ( x 2 + y 2 α 2 ) ] - β , f(x,y;\alpha,\beta)=\left(\beta-1\right)\left(\pi\alpha^{2}\right)^{-1}\left[1% +\left(\frac{x^{2}+y^{2}}{\alpha^{2}}\right)\right]^{-\beta},\,
  3. α \alpha
  4. β \beta
  5. f ( r ; α , β ) = 2 r β - 1 α 2 [ 1 + ( r 2 α 2 ) ] - β . f(r;\alpha,\beta)=2r\frac{\beta-1}{\alpha^{2}}\left[1+\left(\frac{r^{2}}{% \alpha^{2}}\right)\right]^{-\beta}.\,
  6. { ( r 3 + α 2 r ) f ( r ) + f ( r ) ( - α 2 + 2 β r 2 - r 2 ) = 0 , f ( 1 ) = 2 ( β - 1 ) ( 1 α 2 + 1 ) - β α 2 } \left\{\begin{array}[]{l}\left(r^{3}+\alpha^{2}r\right)f^{\prime}(r)+f(r)\left% (-\alpha^{2}+2\beta r^{2}-r^{2}\right)=0,\\ f(1)=\frac{2(\beta-1)\left(\frac{1}{\alpha^{2}}+1\right)^{-\beta}}{\alpha^{2}}% \end{array}\right\}

Mollweide's_formula.html

  1. a + b c = cos ( α - β 2 ) sin ( γ 2 ) \frac{a+b}{c}=\frac{\cos\left(\frac{\alpha-\beta}{2}\right)}{\sin\left(\frac{% \gamma}{2}\right)}
  2. a - b c = sin ( α - β 2 ) cos ( γ 2 ) . \frac{a-b}{c}=\frac{\sin\left(\frac{\alpha-\beta}{2}\right)}{\cos\left(\frac{% \gamma}{2}\right)}.

Molybdopterin_synthase.html

  1. \rightleftharpoons

Monk's_formula.html

  1. 𝔖 s r 𝔖 w = i r < j ( w t i j ) = ( w ) + 1 𝔖 w t i j , \mathfrak{S}_{s_{r}}\mathfrak{S}_{w}=\sum_{{i\leq r<j}\atop{\ell(wt_{ij})=\ell% (w)+1}}\mathfrak{S}_{wt_{ij}},
  2. ( w ) \ell(w)

Monomial_conjecture.html

  1. x 1 t x d t ( x 1 t + 1 , , x d t + 1 ) . x_{1}^{t}\cdots x_{d}^{t}\not\in(x_{1}^{t+1},\dots,x_{d}^{t+1}).\,

Montgomery's_pair_correlation_conjecture.html

  1. 1 - ( sin ( π u ) π u ) 2 + δ ( u ) , 1-\left(\frac{\sin(\pi u)}{\pi u}\right)^{2}+\delta(u),
  2. δ n = ( γ n + 1 - γ n ) log γ n 2 π 2 π . \delta_{n}=(\gamma_{n+1}-\gamma_{n})\frac{\log{\frac{\gamma_{n}}{2\pi}}}{2\pi}.
  3. 1 M { ( n , k ) | N n N + M , \frac{1}{M}\{(n,k)|N\leq n\leq N+M,\,
  4. k 0 , δ n + δ n + 1 + + δ n + k [ α , β ] } k\geq 0,\,\delta_{n}+\delta_{n+1}+\cdots+\delta_{n+k}\in[\alpha,\beta]\}
  5. α β ( 1 - ( sin π u π u ) 2 ) d u \sim\int_{\alpha}^{\beta}\left(1-\biggl(\frac{\sin{\pi u}}{\pi u}\biggr)^{2}% \right)du

Montgomery_curve.html

  1. K K
  2. M A , B M_{A,B}
  3. B y 2 = x 3 + A x 2 + x By^{2}=x^{3}+Ax^{2}+x
  4. A , B K A,B∈K
  5. q q
  6. A A
  7. B B
  8. P , Q P,Q
  9. R R
  10. R = P + Q R=P+Q
  11. [ 2 ] P = P + P [2]P=P+P
  12. P = ( x , y ) P=(x,y)
  13. B y 2 = x 3 + A x 2 + x By^{2}=x^{3}+Ax^{2}+x
  14. P = ( X : Z ) P=(X:Z)
  15. P = ( X : Z ) P=(X:Z)
  16. x = X / Z x=X/Z
  17. Z 0 Z\neq 0
  18. ( x , y ) (x,y)
  19. ( x , - y ) (x,-y)
  20. ( X : Z ) (X:Z)
  21. P = ( X : Z ) P=(X:Z)
  22. [ n ] P = ( X n : Z n ) [n]P=(X_{n}:Z_{n})
  23. P n = [ n ] P = ( X n : Z n ) P_{n}=[n]P=(X_{n}:Z_{n})
  24. P m = [ m ] P = ( X m : Z m ) P_{m}=[m]P=(X_{m}:Z_{m})
  25. P m + n = P m + P n = ( X m + n : Z m + n ) P_{m+n}=P_{m}+P_{n}=(X_{m+n}:Z_{m+n})
  26. X m + n = Z m - n ( ( X m - Z m ) ( X n + Z n ) + ( X m + Z m ) ( X n - Z n ) ) 2 X_{m+n}=Z_{m-n}((X_{m}-Z_{m})(X_{n}+Z_{n})+(X_{m}+Z_{m})(X_{n}-Z_{n}))^{2}
  27. Z m + n = X m - n ( ( X m - Z m ) ( X n + Z n ) - ( X m + Z m ) ( X n - Z n ) ) 2 Z_{m+n}=X_{m-n}((X_{m}-Z_{m})(X_{n}+Z_{n})-(X_{m}+Z_{m})(X_{n}-Z_{n}))^{2}
  28. m = n m=n
  29. [ 2 ] P n = P n + P n = P 2 n = ( X 2 n : Z 2 n ) [2]P_{n}=P_{n}+P_{n}=P_{2n}=(X_{2n}:Z_{2n})
  30. 4 X n Z n = ( X n + Z n ) 2 - ( X n - Z n ) 2 4X_{n}Z_{n}=(X_{n}+Z_{n})^{2}-(X_{n}-Z_{n})^{2}
  31. X 2 n = ( X n + Z n ) 2 ( X n - Z n ) 2 X_{2n}=(X_{n}+Z_{n})^{2}(X_{n}-Z_{n})^{2}
  32. Z 2 n = ( 4 X n Z n ) ( ( X n - Z n ) 2 + ( ( A + 2 ) / 4 ) ( 4 X n Z n ) ) Z_{2n}=(4X_{n}Z_{n})((X_{n}-Z_{n})^{2}+((A+2)/4)(4X_{n}Z_{n}))
  33. ( A + 2 ) / 4 (A+2)/4
  34. A A
  35. P 1 = ( X 1 : Z 1 ) P_{1}=(X_{1}:Z_{1})
  36. Z 1 = 1 Z_{1}=1
  37. X X 1 = X 1 2 XX_{1}=X_{1}^{2}\,
  38. X 3 = ( X X 1 - 1 ) 2 X_{3}=(XX_{1}-1)^{2}\,
  39. Z 3 = 4 X 1 ( X X 1 + a X 1 + 1 ) Z_{3}=4X_{1}(XX_{1}+aX_{1}+1)\,
  40. P 1 = ( 2 , 3 ) P_{1}=(2,\sqrt{3})
  41. 2 y 2 = x 3 - x 2 + x 2y^{2}=x^{3}-x^{2}+x
  42. ( X 1 : Z 1 ) (X_{1}:Z_{1})
  43. x 1 = X 1 / Z 1 x_{1}=X_{1}/Z_{1}
  44. P 1 = ( 2 : 1 ) P_{1}=(2:1)
  45. X X 1 = X 1 2 = 4 XX_{1}=X_{1}^{2}=4\,
  46. X 3 = ( X X 1 - 1 ) 2 = 9 X_{3}=(XX_{1}-1)^{2}=9\,
  47. Z 3 = 4 X 1 ( X X 1 + A X 1 + 1 ) = 24 Z_{3}=4X_{1}(XX_{1}+AX_{1}+1)=24\,
  48. P 3 = ( X 3 : Z 3 ) = ( 9 : 24 ) P_{3}=(X_{3}:Z_{3})=(9:24)
  49. P 3 = 2 P 1 P_{3}=2P_{1}
  50. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  51. P 2 = ( x 2 , y 2 ) P_{2}=(x_{2},y_{2})
  52. M A , B M_{A,B}
  53. P 3 = P 1 + P 2 P_{3}=P_{1}+P_{2}
  54. M A , B M_{A,B}
  55. P 1 P_{1}
  56. P 2 P_{2}
  57. ( x 3 , y 3 ) (x_{3},y_{3})
  58. P 3 P_{3}
  59. y = l x + m ~{}y=lx+m
  60. P 1 P_{1}
  61. P 2 P_{2}
  62. l = y 2 - y 1 x 2 - x 1 l=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}
  63. m = y 1 - ( y 2 - y 1 x 2 - x 1 ) x 1 m=y_{1}-\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)x_{1}
  64. M A , B M_{A,B}
  65. y ~{}y
  66. y = l x + m ~{}y=lx+m
  67. x 3 + ( A - B l 2 ) x 2 + ( 1 - 2 B l m ) x - B m 2 = 0 x^{3}+(A-Bl^{2})x^{2}+(1-2Blm)x-Bm^{2}=0
  68. x ~{}x
  69. P 1 P_{1}
  70. P 2 P_{2}
  71. P 3 P_{3}
  72. ( x - x 1 ) ( x - x 2 ) ( x - x 3 ) = 0 (x-x_{1})(x-x_{2})(x-x_{3})=0
  73. - x 1 - x 2 - x 3 = A - B l 2 -x_{1}-x_{2}-x_{3}=A-Bl^{2}
  74. x 3 x_{3}
  75. x 1 x_{1}
  76. y 1 y_{1}
  77. x 2 x_{2}
  78. y 2 y_{2}
  79. x 3 = B ( y 2 - y 1 x 2 - x 1 ) 2 - A - x 1 - x 2 x_{3}=B\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)^{2}-A-x_{1}-x_{2}
  80. y ~{}y
  81. P 3 P_{3}
  82. x 3 x_{3}
  83. y = l x + m ~{}y=lx+m
  84. P 3 P_{3}
  85. R ~{}R
  86. R + P 1 + P 2 = P R+P_{1}+P_{2}=P_{\infty}
  87. P 1 P_{1}
  88. P 2 P_{2}
  89. R + P 1 + P 2 = P R+P_{1}+P_{2}=P_{\infty}
  90. - R = P 1 + P 2 -R=P_{1}+P_{2}
  91. R ~{}R
  92. - R ~{}-R
  93. y ~{}y
  94. R ~{}R
  95. y ~{}y
  96. x 3 x_{3}
  97. P 3 = ( x 3 , y 3 ) P_{3}=(x_{3},y_{3})
  98. P 3 = P 1 + P 2 P_{3}=P_{1}+P_{2}
  99. x 3 = B ( y 2 - y 1 ) 2 ( x 2 - x 1 ) 2 - A - x 1 - x 2 = B ( x 2 y 1 - x 1 y 2 ) 2 x 1 x 2 ( x 2 - x 1 ) 2 x_{3}=\frac{B(y_{2}-y_{1})^{2}}{(x_{2}-x_{1})^{2}}-A-x_{1}-x_{2}=\frac{B(x_{2}% y_{1}-x_{1}y_{2})^{2}}{x_{1}x_{2}(x_{2}-x_{1})^{2}}
  100. y 3 = ( 2 x 1 + x 2 + A ) ( y 2 - y 1 ) x 2 - x 1 - B ( y 2 - y 1 ) 3 ( x 2 - x 1 ) 3 - y 1 y_{3}=\frac{(2x_{1}+x_{2}+A)(y_{2}-y_{1})}{x_{2}-x_{1}}-\frac{B(y_{2}-y_{1})^{% 3}}{(x_{2}-x_{1})^{3}}-y_{1}
  101. P 1 P_{1}
  102. M A , B M_{A,B}
  103. [ 2 ] P 1 [2]P_{1}
  104. P 1 P_{1}
  105. P 3 = 2 P 1 P_{3}=2P_{1}
  106. P 1 P_{1}
  107. M A , B : f ( x , y ) = 0 M_{A,B}:f(x,y)=0
  108. f ( x , y ) = x 3 + A x 2 + x - B y 2 f(x,y)=x^{3}+Ax^{2}+x-By^{2}
  109. l = - f x f y l=-\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}
  110. l = 3 x 1 2 + 2 A x 1 + 1 2 B y 1 l=\frac{3x_{1}^{2}+2Ax_{1}+1}{2By_{1}}
  111. P 3 P_{3}
  112. P 3 = 2 P 1 P_{3}=2P_{1}
  113. x 3 = B l 2 - A - x 1 - x 1 = B ( 3 x 1 2 + 2 A x 1 + 1 ) 2 ( 2 B y 1 ) 2 - A - x 1 - x 1 = ( x 1 2 - 1 ) 2 4 B y 1 2 = ( x 1 2 - 1 ) 2 4 x 1 ( x 1 2 + A x 1 + 1 ) x_{3}=Bl^{2}-A-x_{1}-x_{1}=\frac{B(3x_{1}^{2}+2Ax_{1}+1)^{2}}{(2By_{1})^{2}}-A% -x_{1}-x_{1}=\frac{(x_{1}^{2}-1)^{2}}{4By_{1}^{2}}=\frac{(x_{1}^{2}-1)^{2}}{4x% _{1}(x_{1}^{2}+Ax_{1}+1)}
  114. y 3 = ( 2 x 1 + x 1 + A ) l - B l 3 - y 1 = ( 2 x 1 + x 1 + A ) ( 3 x 1 2 + 2 A x 1 + 1 ) 2 B y 1 - B ( 3 x 1 2 + 2 A x 1 + 1 ) 3 ( 2 B y 1 ) 3 - y 1 y_{3}=(2x_{1}+x_{1}+A)l-Bl^{3}-y_{1}=\frac{(2x_{1}+x_{1}+A)(3{x_{1}}^{2}+2Ax_{% 1}+1)}{2By_{1}}-\frac{B(3{x_{1}}^{2}+2Ax_{1}+1)^{3}}{(2By_{1})^{3}}-y_{1}
  115. K K
  116. M A , B M_{A,B}
  117. M A , B M_{A,B}
  118. B v 2 = u 3 + A u 2 + u Bv^{2}=u^{3}+Au^{2}+u
  119. A K \ { - 2 , 2 } A\in K\backslash\{-2,2\}
  120. B K \ { 0 } B\in K\backslash\{0\}
  121. E a , d E_{a,d}
  122. E a , d : a x 2 + y 2 = 1 + d x 2 y 2 , E_{a,d}\ :\ ax^{2}+y^{2}=1+dx^{2}y^{2},\,
  123. a , d K \ { 0 } a,d\in K\backslash\{0\}
  124. a d a\neq d
  125. K K
  126. E a , d E_{a,d}
  127. M A , B M_{A,B}
  128. A = 2 ( a + d ) a - d A=\frac{2(a+d)}{a-d}
  129. B = 4 a - d B=\frac{4}{a-d}
  130. ψ : E a , d M A , B \psi\,:\,E_{a,d}\rightarrow M_{A,B}
  131. ( x , y ) ( u , v ) = ( 1 + y 1 - y , 1 + y ( 1 - y ) x ) (x,y)\mapsto(u,v)=\left(\frac{1+y}{1-y},\frac{1+y}{(1-y)x}\right)
  132. E a , d E_{a,d}
  133. M A , B M_{A,B}
  134. ψ - 1 \psi^{-1}
  135. M A , B E a , d M_{A,B}\rightarrow E_{a,d}
  136. ( u , v ) ( x , y ) = ( u v , u - 1 u + 1 ) , a = A + 2 B , d = A - 2 B (u,v)\mapsto(x,y)=\left(\frac{u}{v},\frac{u-1}{u+1}\right),a=\frac{A+2}{B},d=% \frac{A-2}{B}
  137. ψ \psi
  138. v = 0 v=0
  139. u + 1 = 0 u+1=0
  140. M A , B M_{A,B}
  141. M A , B M_{A,B}
  142. B y 2 = x 3 + A x 2 + x By^{2}=x^{3}+Ax^{2}+x
  143. M A , B M_{A,B}
  144. B 3 B^{3}
  145. u = x B u=\frac{x}{B}
  146. v = y B v=\frac{y}{B}
  147. v 2 = u 3 + A B u 2 + 1 B 2 u v^{2}=u^{3}+\frac{A}{B}u^{2}+\frac{1}{B^{2}}u
  148. t - A 3 B t-\frac{A}{3B}
  149. v 2 = ( t - A 3 B ) 3 + A B ( t - A 3 B ) 2 + 1 B 2 ( t - A 3 B ) v^{2}=\left(t-\frac{A}{3B}\right)^{3}+\frac{A}{B}\left(t-\frac{A}{3B}\right)^{% 2}+\frac{1}{B^{2}}\left(t-\frac{A}{3B}\right)
  150. v 2 = t 3 + ( 3 - A 2 3 B 2 ) t + ( 2 A 3 - 9 A 27 B 3 ) v^{2}=t^{3}+\left(\frac{3-A^{2}}{3B^{2}}\right)t+\left(\frac{2A^{3}-9A}{27B^{3% }}\right)
  151. ψ \psi
  152. M A , B E M_{A,B}\rightarrow E
  153. ( x , y ) ( u , v ) = ( x B + A 3 B , y B ) , a = 3 - A 2 3 B 2 , b = 2 A 3 - 9 A 27 B 3 (x,y)\mapsto(u,v)=\left(\frac{x}{B}+\frac{A}{3B},\frac{y}{B}\right),a=\frac{3-% A^{2}}{3B^{2}},b=\frac{2A^{3}-9A}{27B^{3}}

Moran_process.html

  1. i = 0 , , N i=0,...,N
  2. i 1 , i i−1,i
  3. i + 1 i+1
  4. P 0 , 0 = 1 P i , i - 1 = N - i N i N P i , i = 1 - P i , i - 1 - P i , i + 1 P i , i + 1 = i N N - i N P N , N = 1. \begin{aligned}\displaystyle P_{0,0}&\displaystyle=1\\ \displaystyle P_{i,i-1}&\displaystyle=\frac{N-i}{N}\frac{i}{N}\\ \displaystyle P_{i,i}&\displaystyle=1-P_{i,i-1}-P_{i,i+1}\\ \displaystyle P_{i,i+1}&\displaystyle=\frac{i}{N}\frac{N-i}{N}\\ \displaystyle P_{N,N}&\displaystyle=1.\end{aligned}
  5. P i , j P_{i,j}
  6. P 0 , 0 = 1 P_{0,0}=1
  7. P N , N = 1 P_{N,N}=1
  8. 1 , , N 1 1,...,N−1
  9. X ( t ) X(t)
  10. X ( 0 ) = i X(0)=i
  11. E [ X ( t ) | X ( 0 ) = i ] \displaystyle E[X(t)|X(0)=i]
  12. V t = V 1 1 - ( 1 - 2 N 2 ) t 2 N 2 V_{t}=V_{1}\frac{1-\left(1-\frac{2}{N^{2}}\right)^{t}}{\frac{2}{N^{2}}}
  13. 1 N \frac{1}{N}
  14. i N . \frac{i}{N}.
  15. k i = N [ j = 1 i N - i N - j + j = i + 1 N - 1 i j ] k_{i}=N\left[\sum_{j=1}^{i}\frac{N-i}{N-j}+\sum_{j=i+1}^{N-1}\frac{i}{j}\right]
  16. y i j = k i j - k i - 1 j y_{i}^{j}=k_{i}^{j}-k_{i-1}^{j}
  17. y i + 1 j \displaystyle y_{i+1}^{j}
  18. k N j = 0 k_{N}^{j}=0
  19. q j = P j , j + 1 = j N N - j N q_{j}=P_{j,j+1}=\frac{j}{N}\frac{N-j}{N}
  20. k 1 j k_{1}^{j}
  21. k N j = i = 1 m y i j = N k 1 j \displaystyle k_{N}^{j}=\sum_{i=1}^{m}y_{i}^{j}=N\cdot k_{1}^{j}
  22. k i j = { i j k j j j i N - i N - j k j j j i k_{i}^{j}=\begin{cases}\frac{i}{j}\cdot k_{j}^{j}&j\geq i\\ \frac{N-i}{N-j}\cdot k_{j}^{j}&j\leq i\end{cases}
  23. k j j = N k_{j}^{j}=N
  24. k i = j = 1 N - 1 k i j \displaystyle k_{i}=\sum_{j=1}^{N-1}k_{i}^{j}
  25. lim N k i - N 2 [ ( 1 - x i ) ln ( 1 - x i ) + x i ln ( x i ) ] \lim_{N\to\infty}k_{i}\approx-N^{2}\left[(1-x_{i})\ln(1-x_{i})+x_{i}\ln(x_{i})\right]
  26. f i f_{i}
  27. P 0 , 0 \displaystyle P_{0,0}
  28. P i , j P_{i,j}
  29. f i i f i i + g i ( N - i ) . \frac{f_{i}\cdot i}{f_{i}\cdot i+g_{i}(N-i)}.
  30. x i = { 0 i = 0 β i x i - 1 + ( 1 - α i - β i ) x i + α i x i + 1 1 i N - 1 1 i = N x_{i}=\begin{cases}0&i=0\\ \beta_{i}x_{i-1}+(1-\alpha_{i}-\beta_{i})x_{i}+\alpha_{i}x_{i+1}&1\leq i\leq N% -1\\ 1&i=N\end{cases}
  31. x i = 1 + j = 1 i - 1 k = 1 j γ k 1 + j = 1 N - 1 k = 1 j γ k (1) x_{i}=\frac{{\displaystyle 1+\sum_{j=1}^{i-1}\prod_{k=1}^{j}\gamma_{k}}}{{% \displaystyle 1+\sum_{j=1}^{N-1}\prod_{k=1}^{j}\gamma_{k}}}\qquad\,\text{(1)}
  32. γ i = P i , i - 1 / P i , i + 1 \gamma_{i}=P_{i,i-1}/P_{i,i+1}
  33. g i / f i g_{i}/f_{i}
  34. x 1 ( 1 + j = 1 N - 1 k = 1 j γ k ) = x N = 1. x_{1}\left(1+\sum_{j=1}^{N-1}\prod_{k=1}^{j}\gamma_{k}\right)=x_{N}=1.
  35. x 1 = 1 1 + j = 1 N - 1 k = 1 j γ k x_{1}=\frac{1}{1+\sum_{j=1}^{N-1}\prod_{k=1}^{j}\gamma_{k}}
  36. x i = 1 + j = 1 i - 1 k = 1 j γ k 1 + j = 1 N - 1 k = 1 j γ k x_{i}=\frac{{\displaystyle 1+\sum_{j=1}^{i-1}\prod_{k=1}^{j}\gamma_{k}}}{{% \displaystyle 1+\sum_{j=1}^{N-1}\prod_{k=1}^{j}\gamma_{k}}}
  37. P 0 , 0 \displaystyle P_{0,0}
  38. γ i = 1 / r \gamma_{i}=1/r
  39. x i = 1 - r - i 1 - r - N x 1 = ρ = 1 - r - 1 1 - r - N (2) x_{i}=\frac{1-r^{-i}}{1-r^{-N}}\quad\Rightarrow\quad x_{1}=\rho=\frac{1-r^{-1}% }{1-r^{-N}}\qquad\,\text{(2)}
  40. ρ ρ
  41. E [ X ( t ) | X ( t - 1 ) = i ] \displaystyle E[X(t)|X(t-1)=i]
  42. p = i N , p=\frac{i}{N},
  43. r = 1 + s r=1+s
  44. ρ = 1 - r - 1 1 - r - N . (2) \rho=\frac{1-r^{-1}}{1-r^{-N}}.\qquad\,\text{(2)}
  45. N × u N×u
  46. R = N u ρ = u if ρ = 1 N . R=N\cdot u\cdot\rho=u\quad\,\text{if}\quad\rho=\frac{1}{N}.

Morisita's_overlap_index.html

  1. C D = 2 i = 1 S x i y i ( D x + D y ) X Y C_{D}=\frac{2\sum_{i=1}^{S}x_{i}y_{i}}{(D_{x}+D_{y})XY}
  2. C H = 2 i = 1 S x i y i ( i = 1 S x i 2 X 2 + i = 1 S y i 2 Y 2 ) X Y . C_{H}=\frac{2\sum_{i=1}^{S}x_{i}y_{i}}{\left({\sum_{i=1}^{S}x_{i}^{2}\over X^{% 2}}+{\sum_{i=1}^{S}y_{i}^{2}\over Y^{2}}\right)XY}\,.

Morison_equation.html

  1. u ( t ) u(t)
  2. F = ρ C m V u ˙ F I + 1 2 ρ C d A u | u | F D , F\,=\,\underbrace{\rho\,C_{m}\,V\,\dot{u}}_{F_{I}}+\underbrace{\frac{1}{2}\,% \rho\,C_{d}\,A\,u\,|u|}_{F_{D}},
  3. F ( t ) F(t)
  4. u ˙ d u / d t \dot{u}\equiv\,\text{d}u/\,\text{d}t
  5. u ( t ) , u(t),
  6. F I = ρ C m V u ˙ F_{I}\,=\,\rho\,C_{m}\,V\,\dot{u}
  7. ρ V u ˙ \rho\,V\,\dot{u}
  8. ρ C a V u ˙ , \rho\,C_{a}\,V\,\dot{u},
  9. F D = 1 2 ρ C d A u | u | F_{D}\,=\,{\scriptstyle\frac{1}{2}}\,\rho\,C_{d}\,A\,u\,|u|
  10. C m = 1 + C a C_{m}=1+C_{a}
  11. C a C_{a}
  12. A = D A=D
  13. V = 1 4 π D 2 V={\scriptstyle\frac{1}{4}}\pi{D^{2}}
  14. F ( t ) F(t)
  15. F = C m ρ π 4 D 2 u ˙ + C d 1 2 ρ D u | u | . F\,=\,C_{m}\,\rho\,\frac{\pi}{4}D^{2}\,\dot{u}\,+\,C_{d}\,\frac{1}{2}\,\rho\,D% \,u\,|u|.
  16. v ( t ) v(t)
  17. F = ρ V u ˙ a + ρ C a V ( u ˙ - v ˙ ) b + 1 2 ρ C d A ( u - v ) | u - v | c . F=\underbrace{\rho\,V\dot{u}}_{a}+\underbrace{\rho\,C_{a}V\left(\dot{u}-\dot{v% }\right)}_{b}+\underbrace{\frac{1}{2}\rho\,C_{d}A\left(u-v\right)\left|u-v% \right|}_{c}.
  18. C a C_{a}
  19. C m C_{m}
  20. C m = 1 + C a C_{m}=1+C_{a}

Morphism_of_varieties.html

  1. f = ( f 1 , , f m ) f=(f_{1},\dots,f_{m})
  2. f i f_{i}
  3. k [ x 1 , , x n ] / I k[x_{1},\dots,x_{n}]/I
  4. f ( X ) f(X)
  5. t t p t\mapsto t^{p}
  6. | |
  7. χ ( f * F ) = deg ( f ) χ ( F ) . \chi(f^{*}F)=\deg(f)\chi(F).
  8. H p ( Y , R q f * f * F ) H p + q ( X , f * F ) \operatorname{H}^{p}(Y,R^{q}f_{*}f^{*}F)\Rightarrow\operatorname{H}^{p+q}(X,f^% {*}F)
  9. χ ( f * F ) = q = 0 ( - 1 ) q χ ( R q f * f * F ) \chi(f^{*}F)=\sum_{q=0}^{\infty}(-1)^{q}\chi(R^{q}f_{*}f^{*}F)
  10. L n L^{\otimes n}
  11. R q f * ( f * F ) = R q f * 𝒪 X L n R^{q}f_{*}(f^{*}F)=R^{q}f_{*}\mathcal{O}_{X}\otimes L^{\otimes n}
  12. R q f * 𝒪 X R^{q}f_{*}\mathcal{O}_{X}
  13. deg ( f * L ) = deg ( f ) deg ( L ) \operatorname{deg}(f^{*}L)=\operatorname{deg}(f)\operatorname{deg}(L)
  14. f * 𝒪 X f_{*}\mathcal{O}_{X}

Motivic_zeta_function.html

  1. X X
  2. Z ( X , t ) = n = 0 [ X ( n ) ] t n Z(X,t)=\sum_{n=0}^{\infty}[X^{(n)}]t^{n}
  3. X ( n ) X^{(n)}
  4. n n
  5. X X
  6. X n X^{n}
  7. S n S_{n}
  8. [ X ( n ) ] [X^{(n)}]
  9. X ( n ) X^{(n)}
  10. Z ( X , t ) Z(X,t)
  11. X X
  12. Z ( X , t ) Z(X,t)
  13. 1 / ( 1 - t ) χ ( X ) 1/(1-t)^{\chi(X)}
  14. μ \mu
  15. k k
  16. A A
  17. μ ( X ) \mu(X)\,
  18. X X
  19. μ ( X ) = μ ( Z ) + μ ( X Z ) \mu(X)=\mu(Z)+\mu(X\setminus Z)
  20. Z Z
  21. X X
  22. μ ( X 1 × X 2 ) = μ ( X 1 ) μ ( X 2 ) \mu(X_{1}\times X_{2})=\mu(X_{1})\mu(X_{2})
  23. k k
  24. A = A={\mathbb{Z}}
  25. μ ( X ) = # ( X ( k ) ) \mu(X)=\#(X(k))
  26. μ \mu
  27. A [ [ t ] ] A[[t]]
  28. Z μ ( X , t ) = n = 0 μ ( X ( n ) ) t n Z_{\mu}(X,t)=\sum_{n=0}^{\infty}\mu(X^{(n)})t^{n}
  29. A = K ( V ) A=K(V)
  30. [ X ] [X]
  31. X X
  32. [ X ] = [ X ] [X^{\prime}]=[X]\,
  33. X X^{\prime}
  34. X X
  35. [ X ] = [ Z ] + [ X Z ] [X]=[Z]+[X\setminus Z]
  36. Z Z
  37. X X
  38. [ X 1 × X 2 ] = [ X 1 ] [ X 2 ] [X_{1}\times X_{2}]=[X_{1}]\cdot[X_{2}]
  39. 𝕃 = [ 𝔸 1 ] \mathbb{L}=[{\mathbb{A}}^{1}]
  40. Z ( 𝔸 n , t ) = 1 1 - 𝕃 n t Z({\mathbb{A}}^{n},t)=\frac{1}{1-{\mathbb{L}}^{n}t}
  41. Z ( n , t ) = i = 0 n 1 1 - 𝕃 i t Z({\mathbb{P}}^{n},t)=\prod_{i=0}^{n}\frac{1}{1-{\mathbb{L}}^{i}t}
  42. X X
  43. g g
  44. 𝕃 {\mathbb{L}}
  45. Z ( X , t ) = P ( t ) ( 1 - t ) ( 1 - 𝕃 t ) , Z(X,t)=\frac{P(t)}{(1-t)(1-{\mathbb{L}}t)}\,,
  46. P ( t ) P(t)
  47. 2 g 2g
  48. S S
  49. 0
  50. S S
  51. n = 0 [ S [ n ] ] t n = m = 1 Z ( S , 𝕃 m - 1 t m ) \sum_{n=0}^{\infty}[S^{[n]}]t^{n}=\prod_{m=1}^{\infty}Z(S,{\mathbb{L}}^{m-1}t^% {m})
  52. S [ n ] S^{[n]}
  53. n n
  54. S S
  55. n = 0 [ ( 𝔸 2 ) [ n ] ] t n = m = 1 1 1 - 𝕃 m + 1 t m \sum_{n=0}^{\infty}[({\mathbb{A}}^{2})^{[n]}]t^{n}=\prod_{m=1}^{\infty}\frac{1% }{1-{\mathbb{L}}^{m+1}t^{m}}

Movable_cellular_automaton.html

  1. d 2 h i j d t 2 = ( 1 m i + 1 m j ) p i j + k j C ( i j , i k ) ψ ( α i j , i k ) 1 m i p i k + l i C ( i j , j l ) ψ ( α i j , j l ) 1 m j p j l {d^{2}h^{ij}\over dt^{2}}=\left({1\over m^{i}}+{1\over m^{j}}\right)p^{ij}+% \sum_{k\neq j}C(ij,ik)\psi(\alpha_{ij,ik}){1\over m^{i}}p^{ik}+\sum_{l\neq i}C% (ij,jl)\psi(\alpha_{ij,jl}){1\over m^{j}}p^{jl}
  2. d 2 θ i j d t 2 = ( q i j J i + q j i J j ) τ i j + k j S ( i j , i k ) q i k J i τ i k + l j S ( i j , j l ) q j l J j τ j l {d^{2}\theta^{ij}\over dt^{2}}=\left({q^{ij}\over J^{i}}+{q^{ji}\over J^{j}}% \right)\tau^{ij}+\sum_{k\neq j}S(ij,ik){q^{ik}\over J^{i}}\tau^{ik}+\sum_{l% \neq j}S(ij,jl){q^{jl}\over J^{j}}\tau^{jl}
  3. ε i j = h i j r 0 i j = ( q i j + q j i ) - ( d i + d j ) / 2 ( d i + d j ) / 2 \varepsilon^{ij}={h^{ij}\over r_{0}^{ij}}={\left(q^{ij}+q^{ji}\right)-\left(d^% {i}+d^{j}\right)\big/2\over\left(d^{i}+d^{j}\right)\big/2}
  4. ( Δ ε i ( j ) + Δ ε j ( i ) ) ( d i + d j ) 2 = V n i j Δ t \left(\Delta{\varepsilon^{i(j)}}+\Delta{\varepsilon^{j(i)}}\right){\left(d^{i}% +d^{j}\right)\over 2}=V_{n}^{ij}\Delta{t}

MRB_constant.html

  1. k = 1 ( - 1 ) k k 1 / k . \sum_{k=1}^{\infty}(-1)^{k}k^{1/k}.
  2. s n = k = 1 n ( - 1 ) k k 1 / k s_{n}=\sum_{k=1}^{n}(-1)^{k}k^{1/k}
  3. 0.187859 = k = 1 ( - 1 ) k ( k 1 / k - 1 ) = k = 1 ( ( 2 k ) 1 / ( 2 k ) - ( 2 k - 1 ) 1 / ( 2 k - 1 ) ) . 0.187859\ldots=\sum_{k=1}^{\infty}(-1)^{k}(k^{1/k}-1)=\sum_{k=1}^{\infty}\left% ((2k)^{1/(2k)}-(2k-1)^{1/(2k-1)}\right).

Mudrock_line.html

  1. V p = 1.16 V s + 1.36 V_{p}=1.16V_{s}+1.36
  2. V p V_{p}
  3. V s V_{s}

Multi-particle_collision_dynamics.html

  1. N N
  2. m m
  3. r i \vec{r}_{i}
  4. v i \vec{v}_{i}
  5. r i ( t + δ t MPC ) = r i ( t ) + v i ( t ) δ t MPC \vec{r}_{i}(t+\delta t_{\mathrm{MPC}})=\vec{r}_{i}(t)+\vec{v}_{i}(t)\delta t_{% \mathrm{MPC}}
  6. δ t MPC \delta t_{\mathrm{MPC}}
  7. a a
  8. v i v CMS + 𝐑 ^ ( v i - v CMS ) \vec{v}_{i}\rightarrow\vec{v}_{\mathrm{CMS}}+\hat{\mathbf{R}}(\vec{v}_{i}-\vec% {v}_{\mathrm{CMS}})
  9. v CMS \vec{v}_{\mathrm{CMS}}
  10. 𝐑 ^ \hat{\mathbf{R}}
  11. 𝐑 ^ \hat{\mathbf{R}}
  12. + α +\alpha
  13. - α -\alpha
  14. 1 / 2 1/2
  15. α \alpha
  16. m m
  17. n s n_{s}
  18. a a
  19. α \alpha
  20. δ t MPC \delta t_{\mathrm{MPC}}
  21. λ = δ t MPC k T / m \lambda=\delta t_{\mathrm{MPC}}\sqrt{kT/m}
  22. D = k T δ t M P C 2 m [ d n s ( 1 - cos ( α ) ) ( n s - 1 + exp - n s ) - 1 ] D=\frac{kT\delta t_{\mathrm{M}PC}}{2m}\Bigg[\frac{dn_{s}}{(1-\cos(\alpha))(n_{% s}-1+\exp^{-n_{s}})}-1\Bigg]
  23. ν \nu
  24. D T D_{T}
  25. d d
  26. a = 1 , k T = 1 , m = 1 a=1,\;kT=1,\;m=1
  27. α = 130 o , n s = 10 , δ t MPC [ 0.01 ; 0.1 ] \alpha=130^{o},\;n_{s}=10,\;\delta t_{\mathrm{MPC}}\in[0.01;0.1]

Multi-track_Turing_machine.html

  1. M = Q , Σ , Γ , δ , q 0 , F M=\langle Q,\Sigma,\Gamma,\delta,q_{0},F\rangle
  2. Q Q
  3. Σ \Sigma
  4. Γ Q \Gamma\in Q
  5. q 0 Q q_{0}\in Q
  6. F Q F\subseteq Q
  7. δ ( Q \ F × Σ ) × ( Q × Σ × d ) \delta\subseteq\left(Q\backslash F\times\Sigma\right)\times\left(Q\times\Sigma% \times d\right)
  8. δ ( Q i , [ x 1 , x 2 x n ] ) = ( Q j , [ y 1 , y 2 y n ] , d ) \delta\left(Q_{i},[x_{1},x_{2}...x_{n}]\right)=(Q_{j},[y_{1},y_{2}...y_{n}],d)
  9. d { L , R } d\in\{L,R\}
  10. Q , Σ , Γ , δ , q 0 , F \langle Q,\Sigma,\Gamma,\delta,q_{0},F\rangle
  11. \subseteq
  12. \subseteq
  13. M M M\subseteq M^{\prime}
  14. M M M^{\prime}\subseteq M
  15. Q , Σ × B , Γ × Γ , δ , q 0 , F \langle Q,\Sigma\times{B},\Gamma\times\Gamma,\delta^{\prime},q_{0},F\rangle
  16. δ ( q i , [ x 1 , x 2 ] ) = δ ( q i , [ x 1 , x 2 ] ) \delta\left(q_{i},[x_{1},x_{2}]\right)=\delta^{\prime}\left(q_{i},[x_{1},x_{2}% ]\right)

Multiangle_light_scattering.html

  1. I ( θ ) = < m t p l > I 0 N Δ V ( k r ) 2 i ( θ ) I(\theta)=\frac{<}{m}tpl>{{I_{0}N\Delta V}}{{(kr)^{2}}}i(\theta)
  2. R ( θ ) = < m t p l > I ( θ ) r 2 I 0 Δ V = N i ( θ ) / k 2 R(\theta)=\frac{<}{m}tpl>{{I(\theta)r^{2}}}{{I_{0}\Delta V}}=Ni(\theta)/k^{2}
  3. i ( θ ) = k 2 V 2 | m - 1 | 2 < m t p l > 4 π 2 G 2 ( 2 k a sin θ 2 ) i(\theta)=\frac{{k^{2}V^{2}\left|{m-1}\right|^{2}}}{<}mtpl>{{4\pi^{2}}}G^{2}% \left({2ka\sin\frac{\theta}{2}}\right)
  4. G ( ξ ) = 3 < m t p l > ξ 2 ( sin ξ - ξ cos ξ ) G(\xi)=\frac{3}{<}mtpl>{{\xi^{2}}}(\sin\xi-\xi\cos\xi)
  5. k = < m t p l > 2 π n 0 λ 0 k=\frac{<}{m}tpl>{{2\pi n_{0}}}{{\lambda_{0}}}
  6. V = 4 3 π a 3 V=\frac{4}{3}\pi a^{3}

Multiclass_classification.html

  1. L L
  2. L L
  3. X X
  4. y y
  5. y y
  6. K K
  7. X X
  8. f f
  9. k k
  10. K K
  11. k k
  12. K K
  13. y i = 1 y_{i}=1
  14. y i = k y_{i}=k
  15. L L
  16. X X
  17. y y
  18. f f
  19. x x
  20. k k
  21. y ^ = arg max k 1 K f k ( x ) \hat{y}=\arg\max_{k\in 1\ldots K}f_{k}(x)
  22. K ( K 1 ) / 2 K(K−1)/2
  23. K K
  24. K ( K 1 ) / 2 K(K−1)/2

Multidimensional_system.html

  1. ( i , j ) (i,j)
  2. u ( i , j ) u(i,j)
  3. y ( i , j ) y(i,j)
  4. R ( i , j ) R(i,j)
  5. S ( i , j ) S(i,j)
  6. R ( i + 1 , j ) \displaystyle R(i+1,j)
  7. A 1 , A 2 , A 3 , A 4 , B 1 , B 2 , C 1 , C 2 A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},C_{1},C_{2}
  8. D D
  9. [ R ( i + 1 , j ) S ( i , j + 1 ) y ( i , j ) ] = [ A 1 A 2 B 1 A 3 A 4 B 2 C 1 C 2 D ] [ R ( i , j ) S ( i , j ) u ( i , j ) ] \begin{bmatrix}R(i+1,j)\\ S(i,j+1)\\ y(i,j)\end{bmatrix}=\begin{bmatrix}A_{1}&A_{2}&B_{1}\\ A_{3}&A_{4}&B_{2}\\ C_{1}&C_{2}&D\end{bmatrix}\begin{bmatrix}R(i,j)\\ S(i,j)\\ u(i,j)\end{bmatrix}
  10. u ( i , j ) u(i,j)
  11. p , q = 0 , 0 m , n a p , q y ( i - p , j - q ) = p , q = 0 , 0 m , n b p , q x ( i - p , j - q ) \sum_{p,q=0,0}^{m,n}a_{p,q}y(i-p,j-q)=\sum_{p,q=0,0}^{m,n}b_{p,q}x(i-p,j-q)
  12. x ( i , j ) x(i,j)
  13. y ( i , j ) y(i,j)
  14. ( i , j ) (i,j)
  15. a p , q a_{p,q}
  16. b p , q b_{p,q}
  17. p , q = 0 , 0 m , n a p , q z 1 - p z 2 - q Y ( z 1 , z 2 ) = p , q = 0 , 0 m , n b p , q z 1 - p z 2 - q X ( z 1 , z 2 ) \sum_{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}Y(z_{1},z_{2})=\sum_{p,q=0,0}^{% m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}X(z_{1},z_{2})
  18. T ( z 1 , z 2 ) T(z_{1},z_{2})
  19. T ( z 1 , z 2 ) = Y ( z 1 , z 2 ) X ( z 1 , z 2 ) = p , q = 0 , 0 m , n b p , q z 1 - p z 2 - q p , q = 0 , 0 m , n a p , q z 1 - p z 2 - q T(z_{1},z_{2})={Y(z_{1},z_{2})\over X(z_{1},z_{2})}={\sum_{p,q=0,0}^{m,n}b_{p,% q}z_{1}^{-p}z_{2}^{-q}\over\sum_{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}}
  20. T ( z 1 , z 2 ) T(z_{1},z_{2})
  21. Y ( z 1 , z 2 ) = p , q = 0 , 0 m , n b p , q z 1 - p z 2 - q i , j = 0 , 0 m , n a p , q z 1 - p z 2 - q X ( z 1 , z 2 ) Y(z_{1},z_{2})={\sum_{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}\over\sum_{i,j=% 0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}}X(z_{1},z_{2})
  22. k k
  23. Y ( z 1 , z 2 ) = p , q = 0 , 0 m , n b p , q z 1 - p z 2 - q X ( z 1 , z 2 ) Y(z_{1},z_{2})=\sum_{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}X(z_{1},z_{2})
  24. R ( 1 × m ) , S ( 1 × n ) , x ( 1 × 1 ) R(1\times m),\quad S(1\times n),\quad x(1\times 1)
  25. y ( 1 × 1 ) y(1\times 1)
  26. z 1 z_{1}
  27. z 2 z_{2}
  28. x ( i , j ) x(i,j)
  29. 1 1
  30. A 1 A_{1}
  31. A 4 A_{4}
  32. b i , j b_{i,j}
  33. A 2 A_{2}
  34. b 0 , 0 b_{0,0}
  35. B 1 B_{1}
  36. x ( i , j ) x(i,j)
  37. R i , j R_{i,j}
  38. b 0 , 0 b_{0,0}
  39. D D
  40. x ( i , j ) x(i,j)
  41. y y
  42. A 1 = [ 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 ] A_{1}=\begin{bmatrix}0&0&0&\cdots&0&0\\ 1&0&0&\cdots&0&0\\ 0&1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&0&0\\ 0&0&0&\cdots&1&0\end{bmatrix}
  43. A 2 = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] A_{2}=\begin{bmatrix}0&0&0&\cdots&0&0\\ 0&0&0&\cdots&0&0\\ 0&0&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&0&0\\ 0&0&0&\cdots&0&0\end{bmatrix}
  44. A 3 = [ b 1 , n b 2 , n b 3 , n b m - 1 , n b m , n b 1 , n - 1 b 2 , n - 1 b 3 , n - 1 b m - 1 , n - 1 b m , n - 1 b 1 , n - 2 b 2 , n - 2 b 3 , n - 2 b m - 1 , n - 2 b m , n - 2 b 1 , 2 b 2 , 2 b 3 , 2 b m - 1 , 2 b m , 2 b 1 , 1 b 2 , 1 b 3 , 1 b m - 1 , 1 b m , 1 ] A_{3}=\begin{bmatrix}b_{1,n}&b_{2,n}&b_{3,n}&\cdots&b_{m-1,n}&b_{m,n}\\ b_{1,n-1}&b_{2,n-1}&b_{3,n-1}&\cdots&b_{m-1,n-1}&b_{m,n-1}\\ b_{1,n-2}&b_{2,n-2}&b_{3,n-2}&\cdots&b_{m-1,n-2}&b_{m,n-2}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ b_{1,2}&b_{2,2}&b_{3,2}&\cdots&b_{m-1,2}&b_{m,2}\\ b_{1,1}&b_{2,1}&b_{3,1}&\cdots&b_{m-1,1}&b_{m,1}\end{bmatrix}
  45. A 4 = [ 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 ] A_{4}=\begin{bmatrix}0&0&0&\cdots&0&0\\ 1&0&0&\cdots&0&0\\ 0&1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&0&0\\ 0&0&0&\cdots&1&0\end{bmatrix}
  46. B 1 = [ 1 0 0 0 0 0 ] B_{1}=\begin{bmatrix}1\\ 0\\ 0\\ 0\\ \vdots\\ 0\\ 0\end{bmatrix}
  47. B 2 = [ b 0 , n b 0 , n - 1 b 0 , n - 2 b 0 , 2 b 0 , 1 ] B_{2}=\begin{bmatrix}b_{0,n}\\ b_{0,n-1}\\ b_{0,n-2}\\ \vdots\\ b_{0,2}\\ b_{0,1}\end{bmatrix}
  48. C 1 = [ b 1 , 0 b 2 , 0 b 3 , 0 b m - 1 , 0 b m , 0 ] C_{1}=\begin{bmatrix}b_{1,0}&b_{2,0}&b_{3,0}&\cdots&b_{m-1,0}&b_{m,0}\\ \end{bmatrix}
  49. C 2 = [ 0 0 0 0 1 ] C_{2}=\begin{bmatrix}0&0&0&\cdots&0&1\\ \end{bmatrix}
  50. D = [ b 0 , 0 ] D=\begin{bmatrix}b_{0,0}\end{bmatrix}

Multiphoton_lithography.html

  1. χ ( 3 ) \chi^{(3)}

Multiple-prism_dispersion_theory.html

  1. ( ϕ 2 , m / λ ) = H 2 , m ( n m / λ ) + ( k 1 , m k 2 , m ) - 1 ( H 1 , m ( n m / λ ) ± ( ϕ 2 , ( m - 1 ) / λ ) ) (\partial\phi_{2,m}/\partial\lambda)=H_{2,m}(\partial n_{m}/\partial\lambda)+(% k_{1,m}k_{2,m})^{-1}\bigg(H_{1,m}(\partial n_{m}/\partial\lambda)\pm\ (% \partial\phi_{2,(m-1)}/\partial\lambda)\bigg)
  2. λ ϕ 2 , m = H 2 , m λ n m + ( k 1 , m k 2 , m ) - 1 ( H 1 , m λ n m ± λ ϕ 2 , ( m - 1 ) ) \nabla_{\lambda}\phi_{2,m}=H_{2,m}\nabla_{\lambda}n_{m}+(k_{1,m}k_{2,m})^{-1}% \bigg(H_{1,m}\nabla_{\lambda}n_{m}\pm\nabla_{\lambda}\phi_{2,(m-1)}\bigg)
  3. λ = / λ \nabla_{\lambda}=\partial/\partial\lambda
  4. k 1 , m = c o s ψ 1 , m / c o s ϕ 1 , m \,k_{1,m}=cos\psi_{1,m}/cos\phi_{1,m}
  5. k 2 , m = c o s ϕ 2 , m / c o s ψ 2 , m \,k_{2,m}=cos\phi_{2,m}/cos\psi_{2,m}
  6. H 1 , m = ( t a n ϕ 1 , m ) / n m \,H_{1,m}=(tan\phi_{1,m})/n_{m}
  7. H 2 , m = ( t a n ϕ 2 , m ) / n m \,H_{2,m}=(tan\phi_{2,m})/n_{m}
  8. ϕ 1 , m \phi_{1,m}
  9. ψ 1 , m \psi_{1,m}
  10. ϕ 2 , m \phi_{2,m}
  11. ψ 2 , m \psi_{2,m}
  12. ( ϕ 2 , 1 / λ ) = ( s i n ψ 2 , 1 / c o s ϕ 2 , 1 ) ( n 1 / λ ) + ( c o s ψ 2 , 1 / c o s ϕ 2 , 1 ) t a n ψ 1 , 1 ( n 1 / λ ) (\partial\phi_{2,1}/\partial\lambda)=(sin\psi_{2,1}/cos\phi_{2,1})(\partial n_% {1}/\partial\lambda)+(cos\psi_{2,1}/cos\phi_{2,1})tan\psi_{1,1}(\partial n_{1}% /\partial\lambda)
  13. ϕ 2 , m \phi_{2,m}
  14. ( ϕ 2 , 1 / λ ) = t a n ψ 1 , 1 ( n 1 / λ ) (\partial\phi_{2,1}/\partial\lambda)=tan\psi_{1,1}(\partial n_{1}/\partial\lambda)
  15. Δ λ Δ θ ( Θ λ ) - 1 \Delta\lambda\approx\Delta\theta\left({\partial\Theta\over\partial\lambda}% \right)^{-1}
  16. Δ θ \Delta\theta
  17. Δ λ Δ θ ( M θ λ ) - 1 \Delta\lambda\approx\Delta\theta\left(M{\partial\theta\over\partial\lambda}% \right)^{-1}
  18. Δ λ Δ θ ( M θ λ + ϕ 2 , m λ ) - 1 \Delta\lambda\approx\Delta\theta\left(M{\partial\theta\over\partial\lambda}+{% \partial\phi_{2,m}\over\partial\lambda}\right)^{-1}

Multiple-scale_analysis.html

  1. d 2 y d t 2 + y + ε y 3 = 0 , \frac{d^{2}y}{dt^{2}}+y+\varepsilon y^{3}=0,
  2. y ( 0 ) = 1 , d y d t ( 0 ) = 0 , y(0)=1,\qquad\frac{dy}{dt}(0)=0,
  3. | y ( t ) | 1 + 1 2 ε and | d y d t | 1 + 1 2 ε for all t . \left|y(t)\right|\leq\sqrt{1+\tfrac{1}{2}\varepsilon}\quad\,\text{ and }\quad% \left|\frac{dy}{dt}\right|\leq\sqrt{1+\tfrac{1}{2}\varepsilon}\qquad\,\text{ % for all }t.
  4. y ( t ) = cos ( t ) + ε [ 1 32 cos ( 3 t ) - 1 32 cos ( t ) - 3 8 t sin ( t ) secular ] + 𝒪 ( ε 2 ) . y(t)=\cos(t)+\varepsilon\left[\tfrac{1}{32}\cos(3t)-\tfrac{1}{32}\cos(t)-% \underbrace{\tfrac{3}{8}\,t\,\sin(t)}\text{secular}\right]+\mathcal{O}(% \varepsilon^{2}).
  5. t 1 = ε t t_{1}=\varepsilon t\,
  6. y ( t ) = Y 0 ( t , t 1 ) + ε Y 1 ( t , t 1 ) + . y(t)=Y_{0}(t,t_{1})+\varepsilon Y_{1}(t,t_{1})+\cdots.
  7. d y d t \displaystyle\frac{dy}{dt}
  8. d 2 y d t 2 = 2 Y 0 t 2 + ε ( 2 2 Y 0 t t 1 + 2 Y 1 t 2 ) + 𝒪 ( ε 2 ) . \frac{d^{2}y}{dt^{2}}=\frac{\partial^{2}Y_{0}}{\partial t^{2}}+\varepsilon% \left(2\frac{\partial^{2}Y_{0}}{\partial t\,\partial t_{1}}+\frac{\partial^{2}% Y_{1}}{\partial t^{2}}\right)+\mathcal{O}(\varepsilon^{2}).
  9. 2 Y 0 t 2 + Y 0 \displaystyle\frac{\partial^{2}Y_{0}}{\partial t^{2}}+Y_{0}
  10. Y 0 ( t , t 1 ) = A ( t 1 ) e + i t + A ( t 1 ) e - i t , Y_{0}(t,t_{1})=A(t_{1})\,e^{+it}+A^{\ast}(t_{1})\,e^{-it},
  11. [ - 3 A 2 A - 2 i d A d t 1 ] e + i t - A 3 e + 3 i t + c . c . \left[-3\,A^{2}\,A^{\ast}-2\,i\,\frac{dA}{dt_{1}}\right]\,e^{+it}-A^{3}\,e^{+3% it}+c.c.
  12. - 3 A 2 A - 2 i d A d t 1 = 0. -3\,A^{2}\,A^{\ast}-2\,i\,\frac{dA}{dt_{1}}=0.
  13. A = 1 2 exp ( 3 8 i t 1 ) . A=\tfrac{1}{2}\,\exp\left(\tfrac{3}{8}\,i\,t_{1}\right).
  14. y ( t ) = cos [ ( 1 + 3 8 ε ) t ] + 𝒪 ( ε ) , y(t)=\cos\left[\left(1+\tfrac{3}{8}\,\varepsilon\right)t\right]+\mathcal{O}(% \varepsilon),
  15. y r cos θ y\approx r\cos\theta
  16. ( r , θ ) (r,\theta)
  17. r ( t ) r(t)
  18. θ ( t ) \theta(t)
  19. d θ / d t 1 d\theta/dt\approx 1
  20. y = r cos θ + 1 32 ε r 3 cos 3 θ + 1 1024 ε 2 r 5 ( - 21 cos 3 θ + cos 5 θ ) + 𝒪 ( ε 3 ) y=r\cos\theta+\frac{1}{32}\varepsilon r^{3}\cos 3\theta+\frac{1}{1024}% \varepsilon^{2}r^{5}(-21\cos 3\theta+\cos 5\theta)+\mathcal{O}(\varepsilon^{3})
  21. d r / d t = 0 dr/dt=0
  22. d θ d t = 1 + 3 8 ε r 2 - 15 256 ε 2 r 4 + 𝒪 ( ε 3 ) . \frac{d\theta}{dt}=1+\frac{3}{8}\varepsilon r^{2}-\frac{15}{256}\varepsilon^{2% }r^{4}+\mathcal{O}(\varepsilon^{3}).

Multiple_correspondence_analysis.html

  1. y i k y_{ik}
  2. y i k y_{ik}
  3. i i
  4. k k
  5. p k p_{k}
  6. k k
  7. x i k = y i k / p k - 1 x_{ik}=y_{ik}/p_{k}-1
  8. k k
  9. p k p_{k}

Multiple_gamma_function.html

  1. Γ N ( w | a 1 , , a N ) = exp ( s ζ N ( s , w | a 1 , , a N ) | s = 0 ) \Gamma_{N}(w|a_{1},...,a_{N})=\exp\left(\frac{\partial}{\partial s}\zeta_{N}(s% ,w|a_{1},...,a_{N})|_{s=0}\right)
  2. Γ N ( w | a 1 , , a N ) = Γ N - 1 ( w | a 1 , , a N - 1 ) Γ N ( w + a N | a 1 , , a N ) \Gamma_{N}(w|a_{1},...,a_{N})=\Gamma_{N-1}(w|a_{1},...,a_{N-1})\Gamma_{N}(w+a_% {N}|a_{1},...,a_{N})

Multiplexed_binary_offset_carrier.html

  1. Φ ( f ) \Phi(f)
  2. Φ ( f ) = 10 11 B O C ( 1 , 1 ) + 1 11 B O C ( 6 , 1 ) \Phi(f)=\frac{10}{11}BOC(1,1)+\frac{1}{11}BOC(6,1)

Multiplicatively_closed_set.html

  1. 1 S 1\in S
  2. { 1 , x , x 2 , x 3 , } \{1,x,x^{2},x^{3},\dots\}

Multiplicity-one_theorem.html

  1. L 0 2 ( G ( K ) \ G ( 𝐀 ) , ω ) = ^ ( π , V π ) m π V π L^{2}_{0}(G(K)\backslash G(\mathbf{A}),\omega)=\hat{\bigoplus}_{(\pi,V_{\pi})}% m_{\pi}V_{\pi}

Multislice.html

  1. - 2 2 m 2 Ψ ( x , t ) x 2 + V ( x , t ) Ψ ( x , t ) \displaystyle-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi(x,t)}{\partial x^{2}}% +V(x,t)\Psi(x,t)
  2. Ψ ( 𝐫 ) \displaystyle\Psi({\mathbf{r}})
  3. G ( 𝐫 , 𝐫 ) G(\mathbf{r,r^{\prime}})
  4. 𝐫 \mathbf{r}
  5. 𝐫 \mathbf{r^{\prime}}
  6. Ψ ( r ) = exp ( i 𝐤 𝐫 ) \Psi(r)=\exp(i\mathbf{k\cdot r})
  7. Ψ ( 𝐫 ) = exp ( i 𝐤 𝐫 ) - m 2 π 2 exp ( i k | 𝐫 - 𝐫 | ) | 𝐫 - 𝐫 | V ( 𝐫 ) Ψ ( 𝐫 ) d r \displaystyle\Psi({\mathbf{r}})=\exp(i{\mathbf{k\cdot r}})-\frac{m}{2\pi\hbar^% {2}}\int\frac{\exp(ik\cdot{\mathbf{|r-r^{\prime}|}})}{{\mathbf{|r-r^{\prime}|}% }}V({\mathbf{r^{\prime}}})\Psi({\mathbf{r^{\prime}}})dr^{\prime}
  8. z ^ \hat{z}
  9. ϕ ( 𝐫 ) \phi({\mathbf{r}})
  10. ϕ ( 𝐫 ) \displaystyle\phi({\mathbf{r}})
  11. 𝐤 ( 𝐫 - 𝐫 ) = k ( z - z ) & | 𝐫 - 𝐫 | ( z - z ) + ( 𝐗 - 𝐗 ) 2 / 2 ( z - z ) \begin{aligned}\displaystyle{\mathbf{k}}\cdot({\mathbf{r-r^{\prime}}})&% \displaystyle=k(z-z^{\prime})\quad\&\quad|{\mathbf{r-r^{\prime}}}|\approx(z-z^% {\prime})+({\mathbf{X-X^{\prime}}})^{2}/{2(z-z^{\prime})}\end{aligned}
  12. ϕ ( 𝐫 ) = 1 - i π E λ z = - z = z V ( 𝐗 , z ) ϕ ( 𝐗 , z ) 1 i λ ( z - z ) exp ( i k | 𝐗 - 𝐗 | 2 2 ( z - z ) ) d 𝐗 d z \begin{aligned}\displaystyle\phi({\mathbf{r}})=1-i\frac{\pi}{E\lambda}\int\int% \limits_{z^{\prime}=-\infty}^{z^{\prime}=z}V({\mathbf{X^{\prime}}},z^{\prime})% \phi({\mathbf{X^{\prime}}},z^{\prime})\frac{1}{i\lambda(z-z^{\prime})}\exp% \left(ik\frac{|{\mathbf{X-X^{\prime}}}|^{2}}{2(z-z^{\prime})}\right)d{\mathbf{% X^{\prime}}}dz^{\prime}\end{aligned}
  13. λ = 2 π / k \lambda=2\pi/k
  14. ( E ) = 2 k 2 / 2 m (E)=\hbar^{2}k^{2}/{2m}
  15. σ = π / E λ \begin{aligned}\displaystyle\sigma=\pi/E\lambda\end{aligned}
  16. p ( 𝐗 , z ) = 1 i z λ exp ( i k 𝐗 2 2 z ) \begin{aligned}\displaystyle p({\mathbf{X}},z)=\frac{1}{iz\lambda}\exp\left(ik% \frac{{\mathbf{X}}^{2}}{2z}\right)\end{aligned}
  17. V ( 𝐗 , z ) V({\mathbf{X^{\prime}}},z)
  18. ϕ ( 𝐗 , z n + 1 ) = p ( 𝐗 - 𝐗 , z n + 1 - z n ) ϕ ( 𝐗 , z n ) exp ( - i σ z n z n + 1 V ( 𝐗 , z ) d z ) d X \displaystyle\phi({\mathbf{X}},z_{n+1})=\int p({\mathbf{X}}-{\mathbf{X^{\prime% }}},z_{n+1}-z_{n})\phi({\mathbf{X}},z_{n})\exp\left(-i\sigma\int\limits_{z_{n}% }^{z_{n+1}}V({\mathbf{X^{\prime}}},z^{\prime})dz^{\prime}\right)dX^{\prime}
  19. ϕ n + 1 = ϕ ( 𝐗 , z n + 1 ) = [ q n ϕ n ] * p n \begin{aligned}\displaystyle\phi_{n+1}=\phi({\mathbf{X}},z_{n+1})=[q_{n}\phi_{% n}]*p_{n}\end{aligned}
  20. p n = p ( 𝐗 , z n + 1 - z n ) p_{n}=p({\mathbf{X}},z_{n+1}-z_{n})
  21. q n ( 𝐗 ) q_{n}({\mathbf{X}})
  22. q n ( 𝐗 ) = exp { - i σ z n z n + 1 V ( 𝐗 , z ) d z } \begin{aligned}\displaystyle q_{n}({\mathbf{X}})=\exp\{-i\sigma\int\limits_{z_% {n}}^{z_{n+1}}V({\mathbf{X}},z^{\prime})dz^{\prime}\}\end{aligned}
  23. V ( 𝐗 , z ) V(\mathbf{X},z)
  24. 𝐥𝐨𝐠 𝟐 \mathbf{log_{2}}
  25. N log N N\log N
  26. N 2 N^{2}
  27. N N
  28. ϕ ~ ( 𝐮 , z ) = ϕ ~ ( 𝐮 , z = 0 ) exp ( π i λ 𝐮 2 z ) \tilde{\phi}(\mathbf{u},z)=\tilde{\phi}(\mathbf{u},z=0)\exp(\pi i\lambda% \mathbf{u}^{2}z)
  29. 𝐮 \mathbf{u}
  30. k = 2 π / λ k=2\pi/\lambda
  31. θ \theta\sim
  32. d - S Δ z / cos θ - Δ z θ d-S\approx\Delta z/\cos\theta-\Delta z\theta
  33. cos ( 0.01 ) = 0.99995 \cos(0.01)=0.99995
  34. Δ z / cos θ Δ z θ \Delta z/\cos\theta\approx\Delta z\theta
  35. Δ z \Delta z
  36. C s C_{s}

Multitree.html

  1. 2 lim n D ( n ) / ( n n / 2 ) 2 3 11 2\leq\lim_{n\to\infty}D(n)\Big/{\left({{n}\atop{\lfloor n/2\rfloor}}\right)}% \leq 2\frac{3}{11}

Multivariate_mutual_information.html

  1. I ( X 1 ; ; X n + 1 ) = I ( X 1 ; ; X n ) - I ( X 1 ; ; X n | X n + 1 ) , I(X_{1};\ldots;X_{n+1})=I(X_{1};\ldots;X_{n})-I(X_{1};\ldots;X_{n}|X_{n+1}),
  2. I ( X 1 ; ; X n | X n + 1 ) = 𝔼 X n + 1 ( I ( X 1 ; ; X n ) | X n + 1 ) . I(X_{1};\ldots;X_{n}|X_{n+1})=\mathbb{E}_{X_{n+1}}\big(I(X_{1};\ldots;X_{n})|X% _{n+1}\big).
  3. μ ( X ~ i ) \mu(\tilde{X}_{i})
  4. I ( X 1 ; X 2 ; ; X n + 1 ) = μ ( i = 1 n + 1 X ~ i ) I(X_{1};X_{2};...;X_{n+1})=\mu\left(\bigcap_{i=1}^{n+1}\tilde{X}_{i}\right)
  5. Y ~ = i = 1 n X ~ i \tilde{Y}=\bigcap_{i=1}^{n}\tilde{X}_{i}
  6. A ~ = ( A ~ B ~ ) ( A ~ \ B ~ ) \tilde{A}=(\tilde{A}\cap\tilde{B})\cup(\tilde{A}\backslash\tilde{B})
  7. μ ( A ~ ) = μ ( A ~ B ~ ) + μ ( A ~ \ B ~ ) \mu(\tilde{A})=\mu(\tilde{A}\cap\tilde{B})+\mu(\tilde{A}\backslash\tilde{B})
  8. I ( X 1 ; X 2 ; ; X n + 1 ) = μ ( Y ~ X ~ n + 1 ) = μ ( Y ~ ) - μ ( Y ~ \ X ~ n + 1 ) I(X_{1};X_{2};...;X_{n+1})=\mu(\tilde{Y}\cap\tilde{X}_{n+1})=\mu(\tilde{Y})-% \mu(\tilde{Y}\backslash\tilde{X}_{n+1})
  9. 𝒱 = { X 1 , X 2 , , X n } \mathcal{V}=\{X_{1},X_{2},\ldots,X_{n}\}
  10. I ( 𝒱 ) - 𝒯 𝒱 ( - 1 ) | 𝒱 | - | 𝒯 | H ( 𝒯 ) I(\mathcal{V})\equiv-\sum_{\mathcal{T}\subseteq\mathcal{V}}(-1)^{\left|% \mathcal{V}\right|-\left|\mathcal{T}\right|}H(\mathcal{T})
  11. 𝒯 𝒱 \mathcal{T}\subseteq\mathcal{V}
  12. | 𝒱 | = n \left|\mathcal{V}\right|=n
  13. I ( X 1 ; ; X n | Y ) = - T { 1 , , n } ( - 1 ) | T | H ( T | Y ) I(X_{1};\ldots;X_{n}|Y)=-\sum_{T\subseteq\{1,\ldots,n\}}(-1)^{|T|}H(T|Y)
  14. I ( r a i n ; d a r k | c l o u d ) I ( r a i n ; d a r k ) I(rain;dark|cloud)\leq I(rain;dark)
  15. I ( r a i n ; d a r k ; c l o u d ) I(rain;dark;cloud)
  16. I ( X ; Y ; Z ) I(X;Y;Z)
  17. X X
  18. Y Y
  19. Z Z
  20. I ( Y ; Z ) I(Y;Z)
  21. I ( Y ; Z | X ) I(Y;Z|X)
  22. X X
  23. Y Y
  24. Z Z
  25. I ( Y ; Z | X ) > I ( Y ; Z ) I(Y;Z|X)>I(Y;Z)
  26. I ( X ; Y ; Z ) I(X;Y;Z)
  27. X , Y , Z X,Y,Z
  28. I ( X ; Y ; Z ) I(X;Y;Z)
  29. Y Y
  30. X X
  31. Z Z
  32. X X
  33. Y Y
  34. Z Z
  35. ( X ) (X)
  36. ( Y ) (Y)
  37. ( Z ) (Z)
  38. X X
  39. I ( X ; Y ; Z ) I(X;Y;Z)
  40. I ( X ; Y ; Z ) I(X;Y;Z)
  41. \rightarrow
  42. - m i n { I ( X ; Y | Z ) , I ( Y ; Z | X ) , I ( X ; Z | Y ) } I ( X ; Y ; Z ) m i n { I ( X ; Y ) , I ( Y ; Z ) , I ( X ; Z ) } -min\ \{I(X;Y|Z),I(Y;Z|X),I(X;Z|Y)\}\leq I(X;Y;Z)\leq min\ \{I(X;Y),I(Y;Z),I(X% ;Z)\}
  43. 2 n - 1 2^{n}-1

Mumford_vanishing_theorem.html

  1. H i ( X , L - 1 ) = 0 for i = 0 , 1. H^{i}(X,L^{-1})=0\,\text{ for }i=0,1.

Mumford–Tate_group.html

  1. [ a b - b a ] . \begin{bmatrix}a&b\\ -b&a\end{bmatrix}.

MurmurHash.html

  1. 2 32 2^{32}
  2. \leftarrow
  3. \leftarrow
  4. \leftarrow
  5. \leftarrow
  6. \leftarrow
  7. \leftarrow
  8. \leftarrow
  9. \leftarrow
  10. \leftarrow
  11. × \times
  12. \leftarrow
  13. \leftarrow
  14. × \times
  15. \leftarrow
  16. \leftarrow
  17. \leftarrow
  18. × \times
  19. \leftarrow
  20. \leftarrow
  21. × \times
  22. \leftarrow
  23. \leftarrow
  24. × \times
  25. \leftarrow
  26. \leftarrow
  27. \leftarrow
  28. \leftarrow
  29. × \times
  30. \leftarrow
  31. \leftarrow
  32. × \times
  33. \leftarrow

Murray's_law.html

  1. n n
  2. r p 3 = r d 1 3 + r d 2 3 + r d 3 3 + + r d n 3 r_{p}^{3}=r_{d_{1}}^{3}+r_{d_{2}}^{3}+r_{d_{3}}^{3}+...+r_{d_{n}}^{3}
  3. r p r_{p}
  4. r d 1 r_{d_{1}}
  5. r d 2 r_{d_{2}}
  6. r d 3 r_{d_{3}}
  7. r d n r_{d_{n}}
  8. d i d_{i}
  9. d i 6 = 1024 Q 2 μ π 2 k [ ρ tube ( c 2 + 2 c ) + ρ fluid ] d_{i}^{6}=\frac{1024Q^{2}\mu}{\pi^{2}k[\rho_{\,\text{tube}}(c^{2}+2c)+\rho_{\,% \text{fluid}}]}
  10. Q Q
  11. μ \mu
  12. k k
  13. ρ tube \rho_{\,\text{tube}}
  14. c c
  15. ρ fluid \rho_{\,\text{fluid}}
  16. d i 7 = 80 Q 3 ρ fluid f π 3 k [ ρ tube ( c 2 + 2 c ) + ρ fluid ] d_{i}^{7}=\frac{80Q^{3}\rho_{\,\text{fluid}}f}{\pi^{3}k[\rho_{\,\text{tube}}(c% ^{2}+2c)+\rho_{\,\text{fluid}}]}
  17. r p ( 7 / 3 ) = r d 1 ( 7 / 3 ) + r d 2 ( 7 / 3 ) + r d 3 ( 7 / 3 ) + + r d n ( 7 / 3 ) r_{p}^{(7/3)}=r_{d_{1}}^{(7/3)}+r_{d_{2}}^{(7/3)}+r_{d_{3}}^{(7/3)}+...+r_{d_{% n}}^{(7/3)}

N-ary_group.html

  1. a a a = a , a a b = b , a a c = c , a b a = c , a b b = a , a b c = b , a c a = b , a c b = c , a c c = a , aaa=a,aab=b,aac=c,aba=c,abb=a,abc=b,aca=b,acb=c,acc=a,
  2. b a a = b , b a b = c , b a c = a , b b a = a , b b b = b , b b c = c , b c a = c , b c b = a , b c c = b , baa=b,bab=c,bac=a,bba=a,bbb=b,bbc=c,bca=c,bcb=a,bcc=b,
  3. c a a = c , c a b = a , c a c = b , c b a = b , c b b = c , c b c = a , c c a = a , c c b = b , c c c = c . caa=c,cab=a,cac=b,cba=b,cbb=c,cbc=a,cca=a,ccb=b,ccc=c.

N-body_problem.html

  1. N N
  2. m i , i = 1 , 2 , , N m_{i},i=1,2,\ldots,N
  3. 3 \mathbb{R}^{3}
  4. m i m_{i}
  5. 𝐪 i \mathbf{q}_{i}
  6. m i d 2 𝐪 i / d t 2 m_{i}d^{2}\mathbf{q}_{i}/dt^{2}
  7. m i m_{i}
  8. m j m_{j}
  9. 𝐅 i j = G m i m j ( 𝐪 j - 𝐪 i ) 𝐪 j - 𝐪 i 3 , \mathbf{F}_{ij}=\frac{Gm_{i}m_{j}(\mathbf{q}_{j}-\mathbf{q}_{i})}{\left\|% \mathbf{q}_{j}-\mathbf{q}_{i}\right\|^{3}},
  10. G G
  11. 𝐪 j - 𝐪 i \left\|\mathbf{q}_{j}-\mathbf{q}_{i}\right\|
  12. 𝐪 i \mathbf{q}_{i}
  13. 𝐪 j \mathbf{q}_{j}
  14. m i d 2 𝐪 i d t 2 = j = 1 , j i N G m i m j ( 𝐪 j - 𝐪 i ) 𝐪 j - 𝐪 i 3 = U 𝐪 i m_{i}\frac{d^{2}\mathbf{q}_{i}}{dt^{2}}=\sum_{j=1,j\neq i}^{N}\frac{Gm_{i}m_{j% }(\mathbf{q}_{j}-\mathbf{q}_{i})}{\left\|\mathbf{q}_{j}-\mathbf{q}_{i}\right\|% ^{3}}=\frac{\partial U}{\partial\mathbf{q}_{i}}
  15. U U
  16. U = 1 i < j N G m i m j 𝐪 j - 𝐪 i . U=\sum_{1\leq i<j\leq N}\frac{Gm_{i}m_{j}}{\left\|\mathbf{q}_{j}-\mathbf{q}_{i% }\right\|}.
  17. 𝐩 i = m i d 𝐪 i / d t \mathbf{p}_{i}=m_{i}d\mathbf{q}_{i}/dt
  18. d 𝐪 i d t = H 𝐩 i d 𝐩 i d t = - H 𝐪 i , \frac{d\mathbf{q}_{i}}{dt}=\frac{\partial H}{\partial\mathbf{p}_{i}}\qquad% \frac{d\mathbf{p}_{i}}{dt}=-\frac{\partial H}{\partial\mathbf{q}_{i}},
  19. H = T + U H=T+U
  20. T = i = 1 N 𝐩 i 2 2 m i . T=\sum_{i=1}^{N}\frac{\left\|\mathbf{p}_{i}\right\|^{2}}{2m_{i}}.
  21. 6 N 6N
  22. 3 N 3N
  23. 𝐂 = i = 1 N m i 𝐪 i i = 1 N m i \mathbf{C}=\frac{\sum_{i=1}^{N}m_{i}\mathbf{q}_{i}}{\sum_{i=1}^{N}m_{i}}
  24. 𝐂 = 𝐋 0 t + 𝐂 0 \mathbf{C}=\mathbf{L}_{0}t+\mathbf{C}_{0}
  25. 𝐋 0 \mathbf{L}_{0}
  26. 𝐂 0 \mathbf{C}_{0}
  27. 𝐀 = i = 1 N 𝐪 i × 𝐩 i , \mathbf{A}=\sum_{i=1}^{N}\mathbf{q}_{i}\times\mathbf{p}_{i},
  28. × \times
  29. 𝐀 \mathbf{A}
  30. H H
  31. T T
  32. U U
  33. 𝐪 i ( t ) \mathbf{q}_{i}(t)
  34. λ - 2 / 3 𝐪 i ( λ t ) \lambda^{-2/3}\mathbf{q}_{i}(\lambda t)
  35. λ > 0 \lambda>0
  36. I = i = 1 N m i 𝐪 i 𝐪 i = i = 1 N m i 𝐪 i 2 I=\sum_{i=1}^{N}m_{i}\mathbf{q}_{i}\cdot\mathbf{q}_{i}=\sum_{i=1}^{N}m_{i}\|% \mathbf{q}_{i}\|^{2}
  37. Q = ( 1 / 2 ) d I / d t Q=(1/2)dI/dt
  38. d 2 I d t 2 = 2 T - U . \frac{d^{2}I}{dt^{2}}=2T-U.
  39. d 2 I / d t 2 \langle d^{2}I/dt^{2}\rangle
  40. T = U / 2 \langle T\rangle=\langle U\rangle/2
  41. t cr = G M / R 3 t_{\rm cr}=\sqrt{GM/R^{3}}
  42. ( N = 2 ) (N=2)
  43. m 1 𝐚 1 = G m 1 m 2 r 12 3 ( 𝐫 2 - 𝐫 1 ) Sun-Earth m_{1}\mathbf{a}_{1}=\frac{Gm_{1}m_{2}}{r_{12}^{3}}(\mathbf{r}_{2}-\mathbf{r}_{% 1})\quad\,\text{Sun-Earth}
  44. m 2 𝐚 2 = G m 1 m 2 r 21 3 ( 𝐫 1 - 𝐫 2 ) Earth-Sun m_{2}\mathbf{a}_{2}=\frac{Gm_{1}m_{2}}{r_{21}^{3}}(\mathbf{r}_{1}-\mathbf{r}_{% 2})\quad\,\text{Earth-Sun}
  45. m 2 m_{2}
  46. m 1 m_{1}
  47. α + ( η / r 3 ) 𝐫 = 0 \alpha+(\eta/r^{3})\mathbf{r}=0
  48. 𝐫 = 𝐫 2 - 𝐫 1 \mathbf{r}=\mathbf{r}_{2}-\mathbf{r}_{1}
  49. m 2 m_{2}
  50. m 1 m_{1}
  51. α \alpha
  52. d 2 𝐫 / d t 2 d^{2}\mathbf{r}/dt^{2}
  53. η = G ( m 1 + m 2 ) \eta=G(m_{1}+m_{2})
  54. α + ( η / r 3 ) 𝐫 = 0 \alpha+(\eta/r^{3})\mathbf{r}=0
  55. m 1 m_{1}
  56. q ¨ = k q \ddot{q}=kq
  57. N - 1 N-1
  58. 𝐪 1 ( 0 ) , , 𝐪 N ( 0 ) \mathbf{q}_{1}(0),\ldots,\mathbf{q}_{N}(0)
  59. 𝐂 \mathbf{C}
  60. e e
  61. e = 1 e=1
  62. e = 0 e=0
  63. N = 3 N=3
  64. N = 3 N=3
  65. N 3 N\geq 3
  66. n n
  67. n n
  68. n n
  69. d 2 𝐱 i ( t ) d t 2 = G k = 1 , k i n m k ( 𝐱 k ( t ) - 𝐱 i ( t ) ) | 𝐱 k ( t ) - 𝐱 i ( t ) | 3 \frac{d^{2}\mathbf{x}_{i}(t)}{dt^{2}}=G\sum_{k=1,k\neq i}^{n}\frac{m_{k}\left(% \mathbf{x}_{k}(t)-\mathbf{x}_{i}(t)\right)}{\left|\mathbf{x}_{k}(t)-\mathbf{x}% _{i}(t)\right|^{3}}
  70. [ 0 , ) [0,\infty)
  71. N > 2 N>2
  72. U ϵ = 1 i < j N G m i m j ( 𝐪 j - 𝐪 i 2 + ϵ ) 1 / 2 . U_{\epsilon}=\sum_{1\leq i<j\leq N}\frac{Gm_{i}m_{j}}{\left(\left\|\mathbf{q}_% {j}-\mathbf{q}_{i}\right\|^{2}+\epsilon\right)^{1/2}}.
  73. N 2 / 2 N^{2}/2
  74. O ( N 2 ) O(N^{2})
  75. O ( N 2 ) O(N^{2})
  76. O ( N log ( N ) ) O(N\log(N))
  77. O ( N ) O(N)
  78. O ( N log ( N ) ) O(N\log(N))
  79. P 3 M P^{3}M
  80. O ( N 2 ) O(N^{2})
  81. O ( N ) O(N)

N-body_units.html

  1. 1 R = 1 M 2 i , j i N m i m j | r j - r i | \frac{1}{R}=\frac{1}{M^{2}}\sum_{i,j\neq i}^{N}\frac{m_{i}m_{j}}{\left|\vec{r_% {j}}-\vec{r_{i}}\right|}
  2. M = i = 1 N m i M=\sum_{i=1}^{N}m_{i}
  3. 1 2 2 \scriptstyle\frac{1}{2}\sqrt{2}
  4. 2 2 \scriptstyle 2\sqrt{2}

N-curve.html

  1. γ n \gamma_{n}
  2. γ - 1 \gamma^{-1}\,
  3. γ ( 0 ) γ ( 1 ) 0. \gamma(0)\gamma(1)\neq 0.\,
  4. γ * = ( γ ( 0 ) + γ ( 1 ) ) e - γ \gamma^{*}=(\gamma(0)+\gamma(1))e-\gamma
  5. e ( t ) = 1 , t [ 0 , 1 ] e(t)=1,\forall t\in[0,1]
  6. γ - 1 = γ * γ ( 0 ) γ ( 1 ) . \gamma^{-1}=\frac{\gamma^{*}}{\gamma(0)\gamma(1)}.
  7. γ H \gamma\in H
  8. α γ - 1 α γ \alpha\to\gamma^{-1}\cdot\alpha\cdot\gamma
  9. x - [ x ] [ 0 , 1 ] . x-[x]\in[0,1].
  10. γ H \gamma\in H
  11. γ n \gamma_{n}
  12. γ n ( t ) = γ ( n t - [ n t ] ) . \gamma_{n}(t)=\gamma(nt-[nt]).\,
  13. γ n \gamma_{n}
  14. α , β H . \alpha,\beta\in H.
  15. α ( 0 ) = β ( 1 ) = 1 , the f-product α β = β + α - e \alpha(0)=\beta(1)=1,\mbox{ the f-product }~{}\alpha\cdot\beta=\beta+\alpha-e
  16. u n ( t ) = cos ( 2 π n t ) + i sin ( 2 π n t ) u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,
  17. α ( t ) = cos 3 ( 2 π t ) + i sin 3 ( 2 π t ) , 0 t 1 \alpha(t)=\cos^{3}(2\pi t)+i\sin^{3}(2\pi t),0\leq t\leq 1
  18. α u n \alpha\cdot u_{n}
  19. x = cos 3 ( 2 π t ) + cos ( 2 π n t ) - 1 , x=\cos^{3}(2\pi t)+\cos(2\pi nt)-1,
  20. y = sin 3 ( 2 π t ) + sin ( 2 π n t ) y=\sin^{3}(2\pi t)+\sin(2\pi nt)
  21. α and u n \alpha\mbox{ and }~{}u_{n}
  22. N = 53 N=53
  23. u ( t ) = cos ( 2 π t ) + i sin ( 2 π t ) u(t)=\cos(2\pi t)+i\sin(2\pi t)\,
  24. u n ( t ) = cos ( 2 π n t ) + i sin ( 2 π n t ) u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,
  25. u u n u\cdot u_{n}
  26. x = cos ( 2 π n t ) + cos ( 2 π t ) - 1 , x=\cos(2\pi nt)+\cos(2\pi t)-1,
  27. y = sin ( 2 π n t ) + sin ( 2 π t ) y=\sin(2\pi nt)+\sin(2\pi t)
  28. r = cos ( 3 θ ) r=\cos(3\theta)
  29. ρ \rho
  30. ρ ( t ) = cos ( 6 π t ) [ cos ( 2 π t ) + i sin ( 2 π t ) ] , 0 t 1 \rho(t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1
  31. ρ n - ρ \rho_{n}-\rho
  32. x = cos ( 6 π n t ) cos ( 2 π n t ) - cos ( 6 π t ) cos ( 2 π t ) , x=\cos(6\pi nt)\cos(2\pi nt)-\cos(6\pi t)\cos(2\pi t),
  33. y = cos ( 6 π n t ) sin ( 2 π n t ) - cos ( 6 π t ) sin ( 2 π t ) , 0 t 1 y=\cos(6\pi nt)\sin(2\pi nt)-\cos(6\pi t)\sin(2\pi t),0\leq t\leq 1
  34. γ H \gamma\in H
  35. γ n also H \gamma_{n}\mbox{ also }~{}\in H
  36. α γ n - 1 α γ n \alpha\to\gamma_{n}^{-1}\cdot\alpha\cdot\gamma_{n}
  37. ϕ γ n , e \phi_{\gamma_{n},e}
  38. ϕ γ n , e ( α ) = α + [ α ( 1 ) - α ( 0 ) ] ( γ n - 1 ) e . \phi_{\gamma_{n},e}(\alpha)=\alpha+[\alpha(1)-\alpha(0)](\gamma_{n}-1)e.
  39. r = cos ( 2 θ ) r=\cos(2\theta)
  40. x = cos ( 4 π t ) cos ( 2 π t ) , x=\cos(4\pi t)\cos(2\pi t),
  41. y = cos ( 4 π t ) sin ( 2 π t ) , 0 t 1 y=\cos(4\pi t)\sin(2\pi t),0\leq t\leq 1
  42. c ( t ) = 2 π t + i cos ( 2 π t ) , 0 t 1. c(t)=2\pi t+i\cos(2\pi t),\quad 0\leq t\leq 1.\,
  43. ϕ ρ n , e ( c ) \phi_{\rho_{n},e}(c)
  44. x = 2 π [ t - 1 + cos ( 4 π n t ) cos ( 2 π n t ) ] , y = cos ( 2 π t ) + 2 π cos ( 4 π n t ) sin ( 2 π n t ) x=2\pi[t-1+\cos(4\pi nt)\cos(2\pi nt)],\quad y=\cos(2\pi t)+2\pi\cos(4\pi nt)% \sin(2\pi nt)
  45. χ ( t ) = 2 π t + i cos ( 2 π t ) , 0 t 1 \chi(t)=2\pi t+i\cos(2\pi t),0\leq t\leq 1
  46. ρ = cos ( 3 θ ) \rho=\cos(3\theta)
  47. ρ ( t ) = cos ( 6 π t ) [ cos ( 2 π t ) + i sin ( 2 π t ) ] , 0 t 1 \rho(t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1
  48. ϕ ρ n , e ( χ ) \phi_{\rho_{n},e}(\chi)
  49. x = 2 π t + 2 π [ cos ( 6 π n t ) cos ( 2 π n t ) - 1 ] , x=2\pi t+2\pi[\cos(6\pi nt)\cos(2\pi nt)-1],
  50. y = cos ( 2 π t ) + 2 π cos ( 6 π n t ) sin ( 2 π n t ) , 0 t 1 y=\cos(2\pi t)+2\pi\cos(6\pi nt)\sin(2\pi nt),0\leq t\leq 1
  51. n = 15 n=15
  52. β \beta
  53. L 1 ( β ) = L 2 ( β ) = 1 L_{1}(\beta)=L_{2}(\beta)=1
  54. γ * = ( γ ( 0 ) + γ ( 1 ) ) β - γ \gamma^{*}=(\gamma(0)+\gamma(1))\beta-\gamma
  55. γ - 1 = γ * γ ( 0 ) γ ( 1 ) . \gamma^{-1}=\frac{\gamma^{*}}{\gamma(0)\gamma(1)}.
  56. α H \alpha\in H
  57. ϕ α n , β \phi_{\alpha_{n},\beta}
  58. ϕ α n , β ( γ ) = α n - 1 γ α n \phi_{\alpha_{n},\beta}(\gamma)=\alpha_{n}^{-1}\cdot\gamma\cdot\alpha_{n}
  59. ϕ α n , β ( γ ) = γ + [ γ ( 1 ) - γ ( 0 ) ] ( α n - β ) . \phi_{\alpha_{n},\beta}(\gamma)=\gamma+[\gamma(1)-\gamma(0)](\alpha_{n}-\beta).
  60. α \alpha
  61. β \beta
  62. γ \gamma
  63. α \alpha
  64. β \beta
  65. γ \gamma
  66. γ ( t ) = 4 π t + i cos ( 4 π t ) 0 t 1 \gamma(t)=4\pi t+i\cos(4\pi t)0\leq t\leq 1
  67. γ ( 1 ) - γ ( 0 ) = 4 π \gamma(1)-\gamma(0)=4\pi
  68. n = 40 n=40
  69. ϕ u n , u ( γ ) \phi_{u_{n},u}(\gamma)
  70. η \eta
  71. r = cos 3 θ + sin 3 θ r=\cos^{3}\theta+\sin^{3}\theta
  72. x = cos ( 2 π t ) ( cos 3 2 π t + sin 3 2 π t ) , x=\cos(2\pi t)(\cos^{3}2\pi t+\sin^{3}2\pi t),
  73. y = sin ( 2 π t ) ( cos 3 2 π t + sin 3 2 π t ) y=\sin(2\pi t)(\cos^{3}2\pi t+\sin^{3}2\pi t)
  74. α = η \alpha=\eta
  75. β = u , \beta=u,
  76. u u
  77. x = 2 π t cos ( 2 π t ) + 2 π [ ( cos 3 2 π n t + sin 3 2 π n t ) cos ( 2 π n t ) - cos ( 2 π t ) ] , x=2\pi t\cos(2\pi t)+2\pi[(\cos^{3}2\pi nt+\sin^{3}2\pi nt)\cos(2\pi nt)-\cos(% 2\pi t)],
  78. y = 2 π t sin ( 2 π t ) + 2 π [ ( cos 3 2 π n t ) + sin 3 2 π n t ) sin ( 2 π n t ) - sin ( 2 π t ) ] y=2\pi t\sin(2\pi t)+2\pi[(\cos^{3}2\pi nt)+\sin^{3}2\pi nt)\sin(2\pi nt)-\sin% (2\pi t)]
  79. n = 20 n=20

N-slit_interferometric_equation.html

  1. x | s = j = 1 𝒩 x | j j | s \langle x|s\rangle=\sum_{j=1}^{\mathcal{N}}\,\langle x|j\rangle\langle j|s\rangle
  2. j | s = Ψ ( r j , s ) e - i θ j \langle j|s\rangle=\Psi(r_{j,s})e^{-i\theta_{j}}
  3. x | j = Ψ ( r x , j ) e - i ϕ j \langle x|j\rangle=\Psi(r_{x,j})e^{-i\phi_{j}}
  4. θ j \theta_{j}
  5. ϕ j \phi_{j}
  6. x | s = j = 1 𝒩 Ψ ( r j ) e - i Ω j \langle x|s\rangle=\sum_{j=1}^{\mathcal{N}}\,\Psi(r_{j})e^{-i\Omega_{j}}
  7. Ψ ( r j ) = Ψ ( r x , j ) Ψ ( r j , s ) \Psi(r_{j})=\Psi(r_{x,j})\Psi(r_{j,s})
  8. Ω j = θ j + ϕ j \Omega_{j}=\theta_{j}+\phi_{j}
  9. | x | s | 2 = j = 1 𝒩 Ψ ( r j ) 2 + 2 j = 1 𝒩 Ψ ( r j ) ( m = j + 1 𝒩 Ψ ( r m ) cos ( Ω m - Ω j ) ) |\langle x|s\rangle|^{2}\ =\sum_{j=1}^{\mathcal{N}}\,\Psi(r_{j})^{2}\ +2\sum_{% j=1}^{\mathcal{N}}\,\Psi(r_{j})\bigg(\sum_{m=j+1}^{\mathcal{N}}\,\Psi(r_{m})% \cos(\Omega_{m}-\Omega_{j})\bigg)
  10. cos ( Ω m - Ω j ) = cos k | L m - L m - 1 | \cos(\Omega_{m}-\Omega_{j})=\cos k|L_{m}-L_{m-1}|
  11. k k
  12. L m L_{m}
  13. L m - 1 L_{m-1}
  14. d m ( sin θ m + sin ϕ m ) = M λ d_{m}\left(\sin{\theta_{m}}+\sin{\phi_{m}}\right)=M\lambda
  15. θ m \theta_{m}
  16. ϕ m \phi_{m}
  17. λ \lambda
  18. Δ λ Δ θ ( Θ λ ) - 1 \Delta\lambda\approx\Delta\theta\left({\partial\Theta\over\partial\lambda}% \right)^{-1}
  19. Δ θ \Delta\theta

Nanofluidic_circuitry.html

  1. 2 ϕ = - 1 ε 0 ε a z a e n a \nabla^{2}\phi=-\frac{1}{\varepsilon_{0}\varepsilon}\displaystyle\sum_{a}z_{a}% en_{a}
  2. a a
  3. s y m b o l J a = - D a ( n a + z a e n a k T ϕ ) symbol{J}_{a}=-D_{a}(\nabla n_{a}+\frac{z_{a}en_{a}}{kT}\nabla\phi)
  4. ϕ \phi
  5. e e
  6. ε 0 \varepsilon_{0}
  7. ε \varepsilon
  8. D a D_{a}
  9. n a n_{a}
  10. z a z_{a}
  11. a a
  12. ( n a s y m b o l u + s y m b o l J a ) = 0 \nabla\cdot(n_{a}symbol{u}+symbol{J}_{a})=0
  13. s y m b o l u = 0 \nabla\cdot symbol{u}=0
  14. s y m b o l u s y m b o l u = 1 ρ [ - p + μ 2 s y m b o l u - ( a z a e n a ) ϕ ] symbol{u}\cdot\nabla symbol{u}=\frac{1}{\rho}[-\nabla p+\mu\nabla^{2}symbol{u}% -(\displaystyle\sum_{a}z_{a}en_{a})\nabla\phi]
  15. p p
  16. s y m b o l u symbol{u}
  17. μ \mu
  18. ρ \rho
  19. I I
  20. S = I + - I - I + + I - S=\frac{I^{+}-I^{-}}{I^{+}+I^{-}}

Napkin_ring_problem.html

  1. V = π h 3 6 . V=\frac{\pi h^{3}}{6}.
  2. R R
  3. h h
  4. R 2 - ( h 2 ) 2 , ( 1 ) \sqrt{R^{2}-\left(\frac{h}{2}\right)^{2}},\qquad\qquad(1)
  5. R 2 - y 2 . ( 2 ) \sqrt{R^{2}-y^{2}}.\qquad\qquad(2)\,
  6. π ( larger radius ) 2 - π ( smaller radius ) 2 \displaystyle{}\quad\pi(\,\text{larger radius})^{2}-\pi(\,\text{smaller radius% })^{2}
  7. - h / 2 h / 2 ( area of cross-section at height y ) d y , \int_{-h/2}^{h/2}(\,\text{area of cross-section at height }y)\,dy,
  8. 4 3 π ( h 2 ) 3 = π h 3 6 . \frac{4}{3}\pi\left(\frac{h}{2}\right)^{3}=\frac{\pi h^{3}}{6}.

Narrow_escape_problem.html

  1. d X t = 2 D d B t + 1 γ F ( x ) d t dX_{t}=\sqrt{2D}\,dB_{t}+\frac{1}{\gamma}F(x)dt
  2. D D
  3. γ \gamma
  4. F ( x ) F(x)
  5. B t B_{t}
  6. Ω \Omega
  7. Ω a \partial\Omega_{a}
  8. Ω \partial\Omega
  9. ε = | Ω a | | Ω | 1 \varepsilon=\frac{|\partial\Omega_{a}|}{|\partial\Omega|}\ll 1
  10. p ε ( x , t ) p_{\varepsilon}(x,t)
  11. x x
  12. t t
  13. t p ε ( x , t ) = D Δ p ε ( x , t ) - 1 γ ( p ε ( x , t ) F ( x ) ) \frac{\partial}{\partial t}p_{\varepsilon}(x,t)=D\Delta p_{\varepsilon}(x,t)-% \frac{1}{\gamma}\nabla(p_{\varepsilon}(x,t)F(x))
  14. p ε ( x , 0 ) = ρ 0 ( x ) p_{\varepsilon}(x,0)=\rho_{0}(x)\,
  15. t > 0 t>0
  16. p ε ( x , t ) = 0 for x Ω a p_{\varepsilon}(x,t)=0\,\text{ for }x\in\partial\Omega_{a}
  17. D n p ε ( x , t ) - p ε ( x , t ) γ F ( x ) n ( x ) = 0 for x Ω - Ω a D\frac{\partial}{\partial n}p_{\varepsilon}(x,t)-\frac{p_{\varepsilon}(x,t)}{% \gamma}F(x)\cdot n(x)=0\,\text{ for }x\in\partial\Omega-\partial\Omega_{a}
  18. u ε ( y ) = Ω 0 p ε ( x , t y ) d t d x u_{\varepsilon}(y)=\int_{\Omega}\int_{0}^{\infty}p_{\varepsilon}(x,ty)\,dt\,dx
  19. y y
  20. D Δ u ε ( y ) + 1 γ F ( y ) u ε ( y ) = - 1 D\Delta u_{\varepsilon}(y)+\frac{1}{\gamma}F(y)\cdot\nabla u_{\varepsilon}(y)=-1
  21. u ε ( y ) = 0 for y Ω a u_{\varepsilon}(y)=0\,\text{ for }y\in\partial\Omega_{a}
  22. u ε ( y ) n = 0 for y Ω r \frac{\partial u_{\varepsilon}(y)}{\partial n}=0\,\text{ for }y\in\partial% \Omega_{r}
  23. u ε ( y ) = A π D ln 1 ε + O ( 1 ) , u_{\varepsilon}(y)=\frac{A}{\pi D}\ln\frac{1}{\varepsilon}+O(1),
  24. A A
  25. u ϵ ( y ) u_{\epsilon}(y)
  26. y y
  27. R R
  28. E ( τ | x ( 0 ) = 0 ) = R 2 D ( log ( 1 ε ) + log 2 + 1 4 + O ( ε ) ) . E(\tau|x(0)=0)=\frac{R^{2}}{D}\left(\log\left(\frac{1}{\varepsilon}\right)+% \log 2+\frac{1}{4}+O(\varepsilon)\right).
  29. E ( τ ) = R 2 D ( log ( 1 ε ) + log 2 + 1 8 + O ( ε ) ) . E(\tau)=\frac{R^{2}}{D}\left(\log\left(\frac{1}{\varepsilon}\right)+\log 2+% \frac{1}{8}+O(\varepsilon)\right).
  30. α \alpha
  31. E τ = | Ω | α D [ log 1 ε + O ( 1 ) ] . E\tau=\frac{|\Omega|}{\alpha D}\left[\log\frac{1}{\varepsilon}+O(1)\right].
  32. E τ E\tau
  33. E τ = | Ω | ( d - 1 ) D ( 1 ε + O ( 1 ) ) , E\tau=\frac{|\Omega|}{(d-1)D}\left(\frac{1}{\varepsilon}+O(1)\right),
  34. β = R 1 R 2 < 1 , \beta={\frac{R_{1}}{R_{2}}}<1,
  35. E τ = ( R 2 2 - R 1 2 ) D [ log 1 ε + log 2 + 2 β 2 ] + 1 2 R 2 2 1 - β 2 log 1 β - 1 4 R 2 2 + O ( ε , β 4 ) R 2 2 . E\tau=\frac{(R_{2}^{2}-R_{1}^{2})}{D}\left[\log\frac{1}{\varepsilon}+\log 2+2% \beta^{2}\right]+\frac{1}{2}\frac{R_{2}^{2}}{1-\beta^{2}}\log\frac{1}{\beta}-% \frac{1}{4}R_{2}^{2}+O(\varepsilon,\beta^{4})R_{2}^{2}.
  36. E τ E\tau
  37. 2 ϵ 2\epsilon
  38. β \beta
  39. F ( x ) 0 F(x)\neq 0\,
  40. Ω \Omega
  41. Ω \partial\Omega
  42. Γ \Gamma
  43. Ω \partial\Omega
  44. x Ω x\in\Omega
  45. τ x \tau_{x}
  46. Γ \Gamma
  47. x x
  48. Ω \Omega
  49. Ω \partial\Omega
  50. T ( x ) := 𝔼 [ τ x ] T(x):=\mathbb{E}[\tau_{x}]
  51. v ( x ) := 𝔼 [ ( τ x - T ( x ) ) 2 ] v(x):=\mathbb{E}[(\tau_{x}-T(x))^{2}]
  52. - Δ T = 2 in Ω , T = 0 on Γ , n T = 0 on Ω Γ -\Delta T=2\,\text{ in }\Omega,\,\text{ }T=0\,\text{ on }\Gamma,\,\text{ }% \partial_{n}T=0\,\text{ on }\partial\Omega\setminus\Gamma
  53. - Δ v = 2 | T | 2 in Ω , v = 0 on Γ , n v = 0 on Ω Γ -\Delta v=2|\nabla T|^{2}\,\text{ in }\Omega,\,\text{ }v=0\text{ on }\Gamma,\,% \text{ }\partial_{n}v=0\,\text{ on }\partial\Omega\setminus\Gamma
  54. n := n \partial_{n}:=n\cdot\nabla
  55. n n
  56. Ω . \partial\Omega.
  57. v ¯ := 1 | Ω | Ω v ( x ) d x = 1 | Ω | Ω T 2 ( x ) d x = : T 2 \bar{v}:=\frac{1}{|\Omega|}\int_{\Omega}v(x)dx=\frac{1}{|\Omega|}\int_{\Omega}% T^{2}(x)dx=:T^{2}
  58. Ω := { r e i θ | 0 r < 1 , - ε θ 2 π - ε } , Γ := { e i θ | | θ | ε } \Omega:=\left\{re^{i\theta}|0\leq r<1,\,\text{ }-\varepsilon\leq\theta\leq 2% \pi-\varepsilon\right\},\,\text{ }\Gamma:=\left\{e^{i\theta}||\theta|\leq% \varepsilon\right\}
  59. T ( z ) T(z)
  60. z Ω ¯ z\in\bar{\Omega}
  61. T ( z ) = 1 - | z | 2 2 + 2 log | 1 - z + ( 1 - z e - i ε ) ( 1 - z e i ε ) 2 sin ε 2 | T(z)=\frac{1-|z|^{2}}{2}+2\log{\left|\frac{1-z+\sqrt{(1-ze^{-i\varepsilon})(1-% ze^{i\varepsilon})}}{2\sin{\frac{\varepsilon}{2}}}\right|}
  62. j ¯ ( e i θ ) := - 1 2 π r T ( e i θ ) = { 0 , if ε < θ < 2 π - ε 1 2 π cos θ 2 sin 2 ε 2 - sin 2 θ 2 , if | θ | < ε \bar{j}(e^{i\theta}):=-\frac{1}{2\pi}\frac{\partial}{\partial r}T(e^{i\theta})% =\begin{cases}0,&\,\text{if }\varepsilon<\theta<2\pi-\varepsilon\\ \frac{1}{2\pi}\frac{\cos{\frac{\theta}{2}}}{\sqrt{\sin^{2}{\frac{\varepsilon}{% 2}}-\sin^{2}{\frac{\theta}{2}}}},&\,\text{if }|\theta|<\varepsilon\end{cases}
  63. γ Ω \gamma\subset\partial\Omega
  64. Ω \Omega
  65. Ω \Omega
  66. Ω Γ \partial\Omega\setminus\Gamma
  67. Γ \Gamma
  68. γ \gamma
  69. P ( γ ) = γ j ¯ ( y ) d S y P(\gamma)=\int_{\gamma}\bar{j}(y)dS_{y}
  70. d S y dS_{y}
  71. Ω \partial\Omega
  72. y Ω y\in\partial\Omega

Nash_blowing-up.html

  1. Sing ( X ) \,\text{Sing}(X)
  2. X reg := X Sing ( X ) X\text{reg}:=X\setminus\,\text{Sing}(X)
  3. τ : X reg X × G r n \tau:X\text{reg}\rightarrow X\times G_{r}^{n}
  4. G r n G_{r}^{n}
  5. τ ( a ) := ( a , T X , a ) \tau(a):=(a,T_{X,a})
  6. T X , a T_{X,a}
  7. f 1 , f 2 , , f n - r f_{1},f_{2},\ldots,f_{n-r}
  8. f i / x j \partial f_{i}/\partial x_{j}
  9. y 2 - x q = 0 y^{2}-x^{q}=0
  10. ( x q ) (x^{q})
  11. ( y 2 ) (y^{2})
  12. q > 2 q>2

Natural_computing.html

  1. f A f_{A}
  2. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  3. w 1 , w 2 , , w n w_{1},w_{2},\ldots,w_{n}
  4. f A ( w 1 x 1 + w 2 x 2 + + w n x n ) f_{A}(w_{1}x_{1}+w_{2}x_{2}+\ldots+w_{n}x_{n})

Natural_logarithm_of_2.html

  1. ln 2 0.69314718056 \ln 2\approx 0.69314718056
  2. log b 2 = ln 2 ln b . \log_{b}2=\frac{\ln 2}{\ln b}.
  3. log 10 2 0.301029995663981195. \log_{10}2\approx 0.301029995663981195.
  4. log 2 10 = 1 / log 10 2 3.321928095 \log_{2}10=1/\log_{10}2\approx 3.321928095
  5. n = 1 ( - 1 ) n + 1 n = n = 0 1 ( 2 n + 1 ) ( 2 n + 2 ) = ln 2. \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}=\sum_{n=0}^{\infty}\frac{1}{(2n+1)(2n+% 2)}=\ln 2.
  6. n = 0 ( - 1 ) n ( n + 1 ) ( n + 2 ) = 2 ln 2 - 1. \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)(n+2)}=2\ln 2-1.
  7. n = 1 1 n ( 4 n 2 - 1 ) = 2 ln 2 - 1. \sum_{n=1}^{\infty}\frac{1}{n(4n^{2}-1)}=2\ln 2-1.
  8. n = 1 ( - 1 ) n n ( 4 n 2 - 1 ) = ln 2 - 1. \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n(4n^{2}-1)}=\ln 2-1.
  9. n = 1 ( - 1 ) n n ( 9 n 2 - 1 ) = 2 ln 2 - 3 2 . \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n(9n^{2}-1)}=2\ln 2-\frac{3}{2}.
  10. n = 2 1 2 n [ ζ ( n ) - 1 ] = ln 2 - 1 2 . \sum_{n=2}^{\infty}\frac{1}{2^{n}}[\zeta(n)-1]=\ln 2-\frac{1}{2}.
  11. n = 1 1 2 n + 1 [ ζ ( n ) - 1 ] = 1 - γ - 1 2 ln 2. \sum_{n=1}^{\infty}\frac{1}{2n+1}[\zeta(n)-1]=1-\gamma-\frac{1}{2}\ln 2.
  12. n = 1 1 2 2 n ( 2 n + 1 ) ζ ( 2 n ) = 1 2 ( 1 - ln 2 ) . \sum_{n=1}^{\infty}\frac{1}{2^{2n}(2n+1)}\zeta(2n)=\frac{1}{2}(1-\ln 2).
  13. ln 2 = k 1 1 k 2 k . \ln 2=\sum_{k\geq 1}\frac{1}{k2^{k}}.
  14. ln 2 = k 1 ( 1 3 k + 1 4 k ) 1 k . \ln 2=\sum_{k\geq 1}\left(\frac{1}{3^{k}}+\frac{1}{4^{k}}\right)\frac{1}{k}.
  15. ln 2 = 2 3 + 1 2 k 1 ( 1 2 k + 1 4 k + 1 + 1 8 k + 4 + 1 16 k + 12 ) 1 16 k . \ln 2=\frac{2}{3}+\frac{1}{2}\sum_{k\geq 1}\left(\frac{1}{2k}+\frac{1}{4k+1}+% \frac{1}{8k+4}+\frac{1}{16k+12}\right)\frac{1}{16^{k}}.
  16. ln 2 = 2 3 k 0 1 ( 2 k + 1 ) 9 k . \ln 2=\frac{2}{3}\sum_{k\geq 0}\frac{1}{(2k+1)9^{k}}.
  17. ln 2 = k 0 ( 14 ( 2 k + 1 ) 31 2 k + 1 + 6 ( 2 k + 1 ) 161 2 k + 1 + 10 ( 2 k + 1 ) 49 2 k + 1 ) . \ln 2=\sum_{k\geq 0}\left(\frac{14}{(2k+1)31^{2k+1}}+\frac{6}{(2k+1)161^{2k+1}% }+\frac{10}{(2k+1)49^{2k+1}}\right).
  18. γ \gamma
  19. ζ \zeta
  20. 0 1 d x 1 + x = ln 2 , or, equivalently, 1 2 d x x = ln 2. \int_{0}^{1}\frac{dx}{1+x}=\ln 2,\,\text{ or, equivalently, }\int_{1}^{2}\frac% {dx}{x}=\ln 2.
  21. 1 d x ( 1 + x 2 ) ( 1 + x ) 2 = 1 4 ( 1 - ln 2 ) . \int_{1}^{\infty}\frac{dx}{(1+x^{2})(1+x)^{2}}=\frac{1}{4}(1-\ln 2).
  22. 0 d x 1 + e n x = 1 n ln 2 ; 0 d x 3 + e n x = 2 3 n ln 2. \int_{0}^{\infty}\frac{dx}{1+e^{nx}}=\frac{1}{n}\ln 2;\int_{0}^{\infty}\frac{% dx}{3+e^{nx}}=\frac{2}{3n}\ln 2.
  23. 0 1 e x - 1 - 2 e 2 x - 1 d x = ln 2. \int_{0}^{\infty}\frac{1}{e^{x}-1}-\frac{2}{e^{2x}-1}\,dx=\ln 2.
  24. 0 e - x 1 - e - x x d x = ln 2. \int_{0}^{\infty}e^{-x}\frac{1-e^{-x}}{x}\,dx=\ln 2.
  25. 0 1 ln x 2 - 1 x ln x d x = - 1 + ln 2 + γ . \int_{0}^{1}\ln\frac{x^{2}-1}{x\ln x}dx=-1+\ln 2+\gamma.
  26. 0 π / 3 tan x d x = 2 0 π / 4 tan x d x = ln 2. \int_{0}^{\pi/3}\tan x\,dx=2\int_{0}^{\pi/4}\tan x\,dx=\ln 2.
  27. - π / 4 π / 4 ln ( sin x + cos x ) d x = - π 4 ln 2. \int_{-\pi/4}^{\pi/4}\ln(\sin x+\cos x)\,dx=-\frac{\pi}{4}\ln 2.
  28. 0 1 x 2 ln ( 1 + x ) d x = 2 3 ln 2 - 5 18 . \int_{0}^{1}x^{2}\ln(1+x)\,dx=\frac{2}{3}\ln 2-\frac{5}{18}.
  29. 0 1 x ln ( 1 + x ) ln ( 1 - x ) d x = 1 4 - ln 2. \int_{0}^{1}x\ln(1+x)\ln(1-x)\,dx=\frac{1}{4}-\ln 2.
  30. 0 1 x 3 ln ( 1 + x ) ln ( 1 - x ) d x = 13 96 - 2 3 ln 2. \int_{0}^{1}x^{3}\ln(1+x)\ln(1-x)\,dx=\frac{13}{96}-\frac{2}{3}\ln 2.
  31. 0 1 ln x ( 1 + x ) 2 d x = - ln 2. \int_{0}^{1}\frac{\ln x}{(1+x)^{2}}\,dx=-\ln 2.
  32. 0 1 ln ( 1 + x ) - x x 2 d x = 1 - 2 ln 2. \int_{0}^{1}\frac{\ln(1+x)-x}{x^{2}}\,dx=1-2\ln 2.
  33. 0 1 d x x ( 1 - ln x ) ( 1 - 2 ln x ) = ln 2. \int_{0}^{1}\frac{dx}{x(1-\ln x)(1-2\ln x)}=\ln 2.
  34. 1 ln ln x x 3 d x = - 1 2 ( γ + ln 2 ) . \int_{1}^{\infty}\frac{\ln\ln x}{x^{3}}\,dx=-\frac{1}{2}(\gamma+\ln 2).
  35. γ \gamma
  36. ln 2 = 1 1 - 1 1 3 + 1 1 3 12 - . \ln 2=\frac{1}{1}-\frac{1}{1\cdot 3}+\frac{1}{1\cdot 3\cdot 12}-\cdots.
  37. ln 2 = 1 2 + 1 2 3 + 1 2 3 7 + 1 2 3 7 9 + . \ln 2=\frac{1}{2}+\frac{1}{2\cdot 3}+\frac{1}{2\cdot 3\cdot 7}+\frac{1}{2\cdot 3% \cdot 7\cdot 9}+\cdots.
  38. ln 2 = cot ( arccot 0 - arccot 1 + arccot 5 - arccot 55 + arccot 14187 - ) . \ln 2=\cot(\operatorname{arccot}0-\operatorname{arccot}1+\operatorname{arccot}% 5-\operatorname{arccot}55+\operatorname{arccot}14187-\cdots).
  39. ln 2 = 1 1 - 1 2 + 1 3 - 1 4 + 1 5 - . \ln 2=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots.
  40. ln 2 = [ 0 ; 1 , 2 , 3 , 1 , 5 , 2 3 , 7 , 1 2 , 9 , 2 5 , , 2 k - 1 , 2 k , ] \ln 2=\left[0;1,2,3,1,5,\frac{2}{3},7,\frac{1}{2},9,\frac{2}{5},...,2k-1,\frac% {2}{k},...\right]
  41. ln 2 = 1 1 + 1 2 + 1 3 + 2 2 + 2 5 + 3 2 + 3 7 + 4 2 + = 2 3 - 1 2 9 - 2 2 15 - 3 2 21 - \ln 2=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+\cfrac{3}{2+% \cfrac{3}{7+\cfrac{4}{2+\ddots}}}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2% }}{15-\cfrac{3^{2}}{21-\ddots}}}}
  42. ln 2 \ln 2
  43. c c
  44. c = 2 i 3 j 5 k 7 ln c = i ln 2 + j ln 3 + k ln 5 + ln 7 + c=2^{i}3^{j}5^{k}7^{\ell}\cdots\rightarrow\ln c=i\ln 2+j\ln 3+k\ln 5+\ell\ln 7+\cdots
  45. r = a / b r=a/b
  46. ln r = ln a - ln b \ln r=\ln a-\ln b
  47. ln c n = 1 n ln c \ln\sqrt[n]{c}=\frac{1}{n}\ln c
  48. 2 i 2^{i}
  49. b j b^{j}
  50. b b
  51. ln b \ln b
  52. 2 2
  53. b b
  54. p s = q t + d p^{s}=q^{t}+d
  55. d d
  56. p s / q t = 1 + d / q t p^{s}/q^{t}=1+d/q^{t}
  57. s ln p - t ln q = ln ( 1 + d q t ) = m = 1 ( - 1 ) m + 1 ( d / q t ) m m . s\ln p-t\ln q=\ln\left(1+\frac{d}{q^{t}}\right)=\sum_{m=1}^{\infty}(-1)^{m+1}% \frac{(d/q^{t})^{m}}{m}.
  58. q = 2 q=2
  59. ln p \ln p
  60. ln 2 \ln 2
  61. d / q t d/q^{t}
  62. 3 2 = 2 3 + 1 3^{2}=2^{3}+1
  63. 2 ln 3 = 3 ln 2 - k 1 ( - 1 ) k k 8 k . 2\ln 3=3\ln 2-\sum_{k\geq 1}\frac{(-1)^{k}}{k8^{k}}.
  64. q = 10 q=10