wpmath0000009_6

Kantorovich_theorem.html

  1. x x
  2. f ( x ) = 0 f(x)=0
  3. F ( x ) = 0 F(x)=0
  4. X \R n X\subset\R^{n}
  5. F : \R n X \R n F:\R^{n}\supset X\to\R^{n}
  6. F ( x ) F^{\prime}(x)
  7. U X U\subset X
  8. L > 0 L>0
  9. 𝐱 , 𝐲 U \mathbf{x},\mathbf{y}\in U
  10. F ( 𝐱 ) - F ( 𝐲 ) L 𝐱 - 𝐲 \|F^{\prime}(\mathbf{x})-F^{\prime}(\mathbf{y})\|\leq L\;\|\mathbf{x}-\mathbf{% y}\|
  11. v \R n v\in\R^{n}
  12. F ( 𝐱 ) ( v ) - F ( 𝐲 ) ( v ) L 𝐱 - 𝐲 v \|F^{\prime}(\mathbf{x})(v)-F^{\prime}(\mathbf{y})(v)\|\leq L\;\|\mathbf{x}-% \mathbf{y}\|\,\|v\|
  13. 𝐱 0 X \mathbf{x}_{0}\in X
  14. F ( 𝐱 0 ) F^{\prime}(\mathbf{x}_{0})
  15. 𝐡 0 = - F ( 𝐱 0 ) - 1 F ( 𝐱 0 ) . \mathbf{h}_{0}=-F^{\prime}(\mathbf{x}_{0})^{-1}F(\mathbf{x}_{0}).
  16. 𝐱 1 = 𝐱 0 + 𝐡 0 \mathbf{x}_{1}=\mathbf{x}_{0}+\mathbf{h}_{0}
  17. B ( 𝐱 1 , 𝐡 0 ) B(\mathbf{x}_{1},\|\mathbf{h}_{0}\|)
  18. M L M\leq L
  19. ( 𝐱 k ) k (\mathbf{x}_{k})_{k}
  20. ( 𝐡 k ) k (\mathbf{h}_{k})_{k}
  21. ( α k ) k (\alpha_{k})_{k}
  22. 𝐡 k = - F ( 𝐱 k ) - 1 F ( 𝐱 k ) α k = M F ( 𝐱 k ) - 1 h k 𝐱 k + 1 = 𝐱 k + 𝐡 k . \begin{aligned}\displaystyle\mathbf{h}_{k}&\displaystyle=-F^{\prime}(\mathbf{x% }_{k})^{-1}F(\mathbf{x}_{k})\\ \displaystyle\alpha_{k}&\displaystyle=M\,\|F^{\prime}(\mathbf{x}_{k})^{-1}\|\,% \|h_{k}\|\\ \displaystyle\mathbf{x}_{k+1}&\displaystyle=\mathbf{x}_{k}+\mathbf{h}_{k}.\end% {aligned}
  23. α 0 1 2 \alpha_{0}\leq\tfrac{1}{2}
  24. 𝐱 * \mathbf{x}^{*}
  25. F ( 𝐱 * ) = 0 F(\mathbf{x}^{*})=0
  26. B ¯ ( 𝐱 1 , 𝐡 0 ) \bar{B}(\mathbf{x}_{1},\|\mathbf{h}_{0}\|)
  27. 𝐱 0 \mathbf{x}_{0}
  28. 𝐱 * \mathbf{x}^{*}
  29. t t * * t^{\ast}\leq t^{**}
  30. p ( t ) = ( 1 2 L F ( 𝐱 0 ) - 1 - 1 ) t 2 - t + 𝐡 0 p(t)=\left(\tfrac{1}{2}L\|F^{\prime}(\mathbf{x}_{0})^{-1}\|^{-1}\right)t^{2}-t% +\|\mathbf{h}_{0}\|
  31. t / * * = 2 𝐡 0 1 ± 1 - 2 α t^{\ast/**}=\frac{2\|\mathbf{h}_{0}\|}{1\pm\sqrt{1-2\alpha}}
  32. θ = t * t * * = 1 - 1 - 2 α 1 + 1 - 2 α . \theta=\frac{t^{*}}{t^{**}}=\frac{1-\sqrt{1-2\alpha}}{1+\sqrt{1-2\alpha}}.
  33. 𝐱 * \mathbf{x}^{*}
  34. B ¯ ( 𝐱 1 , θ 𝐡 0 ) B ¯ ( 𝐱 0 , t * ) \bar{B}(\mathbf{x}_{1},\theta\|\mathbf{h}_{0}\|)\subset\bar{B}(\mathbf{x}_{0},% t^{*})
  35. B ( 𝐱 0 , t * ) B(\mathbf{x}_{0},t^{*\ast})
  36. F F
  37. p ( t ) p(t)
  38. t t^{\ast}
  39. t 0 = 0 , t k + 1 = t k - p ( t k ) p ( t k ) t_{0}=0,\,t_{k+1}=t_{k}-\tfrac{p(t_{k})}{p^{\prime}(t_{k})}
  40. 𝐱 k + p - 𝐱 k t k + p - t k . \|\mathbf{x}_{k+p}-\mathbf{x}_{k}\|\leq t_{k+p}-t_{k}.
  41. 𝐱 n + 1 - 𝐱 * θ 2 n 𝐱 n + 1 - 𝐱 n θ 2 n 2 n 𝐡 0 . \|\mathbf{x}_{n+1}-\mathbf{x}^{*}\|\leq\theta^{2^{n}}\|\mathbf{x}_{n+1}-% \mathbf{x}_{n}\|\leq\frac{\theta^{2^{n}}}{2^{n}}\|\mathbf{h}_{0}\|.

Kaplansky_density_theorem.html

  1. A A
  2. B ( H ) B(H)
  3. a a
  4. A A
  5. A A
  6. ( A ) 1 - = ( A - ) 1 (A)_{1}^{-}=(A^{-})_{1}
  7. h h
  8. ( A - ) 1 (A^{-})_{1}
  9. h h
  10. ( A ) 1 (A)_{1}
  11. lim f ( a α ) = f ( lim a α ) \lim f(a_{\alpha})=f(\lim a_{\alpha})

Kapustinskii_equation.html

  1. U L = K ν | z + | | z - | r + + r - ( 1 - d r + + r - ) U_{L}={K}\cdot\frac{\nu\cdot|z^{+}|\cdot|z^{-}|}{r^{+}+r^{-}}\cdot\biggl(1-% \frac{d}{r^{+}+r^{-}}\biggr)
  2. × 10 4 \times 10^{−}4
  3. × 10 11 \times 10^{−}11

Karamata's_inequality.html

  1. I I
  2. f f
  3. I I
  4. I I
  5. y 1 y 2 y n , y_{1}\geq y_{2}\geq\cdots\geq y_{n},
  6. f f
  7. f f
  8. f f
  9. f −f
  10. a := x 1 + x 2 + + x n n a:=\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}
  11. n n
  12. ( a , a , , a ) (a,a,...,a)
  13. i i
  14. a a
  15. n n
  16. f f
  17. f ( x 1 ) + f ( x 2 ) + + f ( x n ) f ( a ) + f ( a ) + + f ( a ) = n f ( a ) . f(x_{1})+f(x_{2})+\cdots+f(x_{n})\geq f(a)+f(a)+\cdots+f(a)=nf(a).
  18. n n
  19. f f
  20. i i
  21. x y x≠y
  22. I I
  23. f ( x ) - f ( y ) x - y \frac{f(x)-f(y)}{x-y}
  24. ( x , f ( x ) ) (x,f(x))
  25. ( y , f ( y ) ) (y,f(y))
  26. f f
  27. x x
  28. y y
  29. c i + 1 := f ( x i + 1 ) - f ( y i + 1 ) x i + 1 - y i + 1 f ( x i ) - f ( y i ) x i - y i = : c i c_{i+1}:=\frac{f(x_{i+1})-f(y_{i+1})}{x_{i+1}-y_{i+1}}\leq\frac{f(x_{i})-f(y_{% i})}{x_{i}-y_{i}}=:c_{i}
  30. A i = x 1 + + x i , B i = y 1 + + y i A_{i}=x_{1}+\cdots+x_{i},\qquad B_{i}=y_{1}+\cdots+y_{i}
  31. i = 1 n ( f ( x i ) - f ( y i ) ) = i = 1 n c i ( x i - y i ) = i = 1 n c i ( A i - A i - 1 = x i - ( B i - B i - 1 = y i ) ) = i = 1 n c i ( A i - B i ) - i = 1 n c i ( A i - 1 - B i - 1 ) = c n ( A n - B n = 0 ) + i = 1 n - 1 ( c i - c i + 1 0 ) ( A i - B i 0 ) - c 1 ( A 0 - B 0 = 0 ) 0 , \begin{aligned}\displaystyle\sum_{i=1}^{n}\bigl(f(x_{i})-f(y_{i})\bigr)&% \displaystyle=\sum_{i=1}^{n}c_{i}(x_{i}-y_{i})\\ &\displaystyle=\sum_{i=1}^{n}c_{i}\bigl(\underbrace{A_{i}-A_{i-1}}_{=\,x_{i}}{% }-(\underbrace{B_{i}-B_{i-1}}_{=\,y_{i}})\bigr)\\ &\displaystyle=\sum_{i=1}^{n}c_{i}(A_{i}-B_{i})-\sum_{i=1}^{n}c_{i}(A_{i-1}-B_% {i-1})\\ &\displaystyle=c_{n}(\underbrace{A_{n}-B_{n}}_{=\,0})+\sum_{i=1}^{n-1}(% \underbrace{c_{i}-c_{i+1}}_{\geq\,0})(\underbrace{A_{i}-B_{i}}_{\geq\,0})-c_{1% }(\underbrace{A_{0}-B_{0}}_{=\,0})\\ &\displaystyle\geq 0,\end{aligned}
  32. x < s u b > 1 > y 1 x<sub>1>y_{1}

Karger's_algorithm.html

  1. ( u , v ) (u,v)
  2. G = ( V , E ) G=(V,E)
  3. u u
  4. v v
  5. u u
  6. v v
  7. ( S , T ) (S,T)
  8. G = ( V , E ) G=(V,E)
  9. V V
  10. S T = V S\cup T=V
  11. { u v E : u S , v T } \{\,uv\in E\colon u\in S,v\in T\,\}
  12. w ( S , T ) = | { u v E : u S , v T } | . w(S,T)=|\{\,uv\in E\colon u\in S,v\in T\,\}|\,.
  13. 2 | V | 2^{|V|}
  14. S S
  15. T T
  16. S S
  17. T T
  18. S S
  19. T T
  20. 2 | V | - 1 - 1 2^{|V|-1}-1
  21. w : E 𝐑 + w\colon E\rightarrow\mathbf{R}^{+}
  22. w ( S , T ) = u v E : u S , v T w ( u v ) , w(S,T)=\sum_{uv\in E\colon u\in S,v\in T}w(uv)\,,
  23. w = 1 w=1
  24. s s
  25. t t
  26. s S s\in S
  27. t T t\in T
  28. s s
  29. t t
  30. s , t V s,t\in V
  31. s , t V s,t\in V
  32. s s
  33. t t
  34. O ( m n + n 2 log n ) O(mn+n^{2}\log n)
  35. e = { u , v } e=\{u,v\}
  36. u v uv
  37. { w , u } \{w,u\}
  38. { w , v } \{w,v\}
  39. w { u , v } w\notin\{u,v\}
  40. { w , u v } \{w,uv\}
  41. u u
  42. v v
  43. e e
  44. G / e G/e
  45. G = ( V , E ) G=(V,E)
  46. | V | > 2 |V|>2
  47. e E e\in E
  48. G G / e G\leftarrow G/e
  49. G G
  50. O ( | V | 2 ) O(|V|^{2})
  51. w ( e i ) = π ( i ) w(e_{i})=\pi(i)
  52. π \pi
  53. O ( | E | log | E | ) O(|E|\log|E|)
  54. O ( | E | ) O(|E|)
  55. O ( | E | log | E | ) O(|E|\log|E|)
  56. O ( | V | ) O(|V|)
  57. G = ( V , E ) G=(V,E)
  58. n = | V | n=|V|
  59. ( n 2 ) - 1 {\left({{n}\atop{2}}\right)}^{-1}
  60. 2 n - 1 - 1 2^{n-1}-1
  61. ( n 2 ) {\textstyle\left({{n}\atop{2}}\right)}
  62. ( n 2 ) / ( 2 n - 1 - 1 ) {\textstyle\left({{n}\atop{2}}\right)}/(2^{n-1}-1)
  63. n n
  64. ( n 2 ) {\left({{n}\atop{2}}\right)}
  65. C C
  66. k k
  67. C C
  68. C C
  69. C C
  70. 1 - k / | E | 1-k/|E|
  71. G G
  72. k k
  73. | E | n k / 2 |E|\geq nk/2
  74. C C
  75. k | E | k n k / 2 = 2 n . \frac{k}{|E|}\leq\frac{k}{nk/2}=\frac{2}{n}.
  76. p n p_{n}
  77. n n
  78. C C
  79. p n ( 1 - 2 n ) p n - 1 p_{n}\geq\bigl(1-\frac{2}{n}\bigr)p_{n-1}
  80. p 2 = 1 p_{2}=1
  81. p n i = 0 n - 3 ( 1 - 2 n - i ) = i = 0 n - 3 n - i - 2 n - i = n - 2 n n - 3 n - 1 n - 4 n - 2 3 5 2 4 1 3 = ( n 2 ) - 1 . p_{n}\geq\prod_{i=0}^{n-3}\Bigl(1-\frac{2}{n-i}\Bigr)=\prod_{i=0}^{n-3}{\frac{% n-i-2}{n-i}}=\frac{n-2}{n}\cdot\frac{n-3}{n-1}\cdot\frac{n-4}{n-2}\cdots\frac{% 3}{5}\cdot\frac{2}{4}\cdot\frac{1}{3}={\left({{n}\atop{2}}\right)}^{-1}\,.
  82. T = ( n 2 ) ln n T={\left({{n}\atop{2}}\right)}\ln n
  83. [ 1 - ( n 2 ) - 1 ] T 1 e ln n = 1 n . \Bigl[1-{\left({{n}\atop{2}}\right)}^{-1}\Bigr]^{T}\leq\frac{1}{e^{\ln n}}=% \frac{1}{n}\,.
  84. T T
  85. n n
  86. m m
  87. O ( T m ) = O ( n 2 m log n ) O(Tm)=O(n^{2}m\log n)
  88. t t
  89. G = ( V , E ) G=(V,E)
  90. t t
  91. | V | > t |V|>t
  92. e E e\in E
  93. G G / e G\leftarrow G/e
  94. G G
  95. p n , t p_{n,t}
  96. C C
  97. n n
  98. p n , t i = 0 n - t - 1 ( 1 - 2 n - i ) = ( t 2 ) / ( n 2 ) . p_{n,t}\geq\prod_{i=0}^{n-t-1}\Bigl(1-\frac{2}{n-i}\Bigr)={\left({{t}\atop{2}}% \right)}\Bigg/{\left({{n}\atop{2}}\right)}\,.
  99. Ω ( t 2 / n 2 ) \Omega(t^{2}/n^{2})
  100. 1 2 \frac{1}{2}
  101. t = 1 + n / 2 t=\lceil 1+n/\sqrt{2}\rceil
  102. C C
  103. G = ( V , E ) G=(V,E)
  104. | V | < 6 |V|<6
  105. V V
  106. t 1 + | V | / 2 t\leftarrow\lceil 1+|V|/\sqrt{2}\rceil
  107. G 1 G_{1}\leftarrow
  108. G G
  109. t t
  110. G 2 G_{2}\leftarrow
  111. G G
  112. t t
  113. G 1 G_{1}
  114. G 2 G_{2}
  115. P ( n ) P(n)
  116. C C
  117. P ( n ) = 1 - ( 1 - 1 2 P ( 1 + n 2 ) ) 2 P(n)=1-\left(1-\frac{1}{2}P\left(\Bigl\lceil 1+\frac{n}{\sqrt{2}}\Bigr\rceil% \right)\right)^{2}
  118. P ( n ) = O ( 1 log n ) P(n)=O\left(\frac{1}{\log n}\right)
  119. T ( n ) = 2 T ( 1 + n 2 ) + O ( n 2 ) T(n)=2T\left(\Bigl\lceil 1+\frac{n}{\sqrt{2}}\Bigr\rceil\right)+O(n^{2})
  120. T ( n ) = O ( n 2 log n ) T(n)=O(n^{2}\log n)
  121. O ( 1 / n ) O(1/n)
  122. O ( log n / P ( n ) ) O(\log n/P(n))
  123. T ( n ) log n P ( n ) = O ( n 2 log 3 n ) T(n)\cdot\frac{\log n}{P(n)}=O(n^{2}\log^{3}n)
  124. O ( n 2 ln 3 n ) O(n^{2}\ln^{3}n)
  125. P ( n ) = O ( 1 ln n ) P(n)=O\left(\frac{1}{\ln n}\right)
  126. O ( ln 2 n ) O(\ln^{2}n)
  127. Pr [ miss a specific min-cut ] = ( 1 - P ( n ) ) O ( ln 2 n ) ( 1 - c ln n ) 3 ln 2 n c e - 3 ln n = 1 n 3 \Pr[\,\text{miss a specific min-cut}]=(1-P(n))^{O(\ln^{2}n)}\leq\left(1-\frac{% c}{\ln n}\right)^{\frac{3\ln^{2}n}{c}}\leq e^{-3\ln n}=\frac{1}{n^{3}}
  128. ( n 2 ) {\left({{n}\atop{2}}\right)}
  129. Pr [ miss any min-cut ] ( n 2 ) 1 n 3 = O ( 1 n ) . \Pr[\,\text{miss any min-cut}]\leq{\left({{n}\atop{2}}\right)}\cdot\frac{1}{n^% {3}}=O\left(\frac{1}{n}\right).
  130. O ( n 2 ) O(n^{2})
  131. O ( n 2 ln O ( 1 ) n ) O(n^{2}\ln^{O(1)}n)

Karnaugh_map.html

  1. A A
  2. B B
  3. C C
  4. D D
  5. f ( A , B , C , D ) = m i , i { 6 , 8 , 9 , 10 , 11 , 12 , 13 , 14 } f(A,B,C,D)=\sum m_{i},i\in\{6,8,9,10,11,12,13,14\}
  6. m i m_{i}
  7. f ( A , B , C , D ) = M i , i { 0 , 1 , 2 , 3 , 4 , 5 , 7 , 15 } f(A,B,C,D)=\prod M_{i},i\in\{0,1,2,3,4,5,7,15\}
  8. M i M_{i}
  9. A D AD
  10. A A
  11. D D
  12. A D ¯ A\overline{D}
  13. A A
  14. D D
  15. D ¯ \overline{D}
  16. A D ¯ A\overline{D}
  17. B ¯ D ¯ \overline{B}\,\overline{D}
  18. C ¯ \overline{C}
  19. A C ¯ A\overline{C}
  20. A B ¯ A\overline{B}
  21. B C D ¯ BC\overline{D}
  22. A C ¯ + A B ¯ + B C D ¯ A\overline{C}+A\overline{B}+BC\overline{D}
  23. f ( A , B , C , D ) = A ¯ B C D ¯ + A B ¯ C ¯ D ¯ + A B ¯ C ¯ D + A B ¯ C D ¯ + A B ¯ C D + A B C ¯ D ¯ + A B C ¯ D + A B C D ¯ = A C ¯ + A B ¯ + B C D ¯ \begin{aligned}\displaystyle f(A,B,C,D)=&\displaystyle\overline{A}BC\overline{% D}+A\overline{B}\,\overline{C}\,\overline{D}+A\overline{B}\,\overline{C}D+A% \overline{B}C\overline{D}+\\ &\displaystyle A\overline{B}CD+AB\overline{C}\,\overline{D}+AB\overline{C}D+% ABC\overline{D}\\ \displaystyle=&\displaystyle A\overline{C}+A\overline{B}+BC\overline{D}\end{aligned}
  24. A ¯ B ¯ \overline{A}\,\overline{B}
  25. A ¯ C ¯ \overline{A}\,\overline{C}
  26. B C D BCD
  27. f ( A , B , C , D ) ¯ = A ¯ B ¯ + A ¯ C ¯ + B C D \overline{f(A,B,C,D)}=\overline{A}\,\overline{B}+\overline{A}\,\overline{C}+BCD
  28. f ( A , B , C , D ) ¯ ¯ \displaystyle\overline{\overline{f(A,B,C,D)}}
  29. f ( A , B , C , D ) = A + B C D ¯ f(A,B,C,D)=A+BC\overline{D}
  30. A A
  31. A C ¯ A\overline{C}
  32. f ( A , B , C , D ) ¯ = A ¯ B ¯ + A ¯ C ¯ + A ¯ D \overline{f(A,B,C,D)}=\overline{A}\,\overline{B}+\overline{A}\,\overline{C}+% \overline{A}D
  33. A D ¯ A\overline{D}
  34. A ¯ D \overline{A}D
  35. ( A + D ¯ ) \left(A+\overline{D}\right)
  36. m ( ) \sum m()
  37. \sum
  38. \sum
  39. \sum
  40. \sum
  41. \sum
  42. \sum
  43. \sum
  44. \sum
  45. \sum
  46. \sum
  47. \sum
  48. \sum
  49. \sum
  50. \sum
  51. \sum
  52. \sum

Karp–Flatt_metric.html

  1. ψ \psi
  2. p p
  3. p p
  4. e e
  5. e = 1 ψ - 1 p 1 - 1 p e=\frac{\frac{1}{\psi}-\frac{1}{p}}{1-\frac{1}{p}}
  6. e e
  7. T ( p ) = T s + T p p T(p)=T_{s}+\frac{T_{p}}{p}
  8. T ( p ) T(p)
  9. p p
  10. T s T_{s}
  11. T p T_{p}
  12. p p
  13. p p
  14. T ( 1 ) = T s + T p T(1)=T_{s}+T_{p}
  15. e e
  16. T s T ( 1 ) \frac{T_{s}}{T(1)}
  17. T ( p ) = T ( 1 ) e + T ( 1 ) ( 1 - e ) p T(p)=T(1)e+\frac{T(1)(1-e)}{p}
  18. ψ \psi
  19. T ( 1 ) T ( p ) \frac{T(1)}{T(p)}
  20. 1 ψ = e + 1 - e p \frac{1}{\psi}=e+\frac{1-e}{p}

Kauffman_polynomial.html

  1. F ( K ) ( a , z ) = a - w ( K ) L ( K ) F(K)(a,z)=a^{-w(K)}L(K)\,
  2. w ( K ) w(K)
  3. L ( K ) L(K)
  4. L ( O ) = 1 L(O)=1
  5. L ( s r ) = a L ( s ) , L ( s ) = a - 1 L ( s ) . L(s_{r})=aL(s),\qquad L(s_{\ell})=a^{-1}L(s).
  6. s s
  7. s r s_{r}
  8. s s_{\ell}

Kaufmann–Bucherer–Neumann_experiments.html

  1. m T m_{T}
  2. m T m = p m v = E m c 2 = 1 1 - v 2 c 2 \frac{m_{T}}{m}=\frac{p}{mv}=\frac{E}{mc^{2}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^% {2}}}}
  3. ϵ / m 0 1.76 × 10 7 \scriptstyle\epsilon/m_{0}\sim 1.76\times 10^{7}
  4. ϕ ( β ) = 3 4 β 2 [ 1 β lg 1 - β 1 + β + 2 1 - β 2 ] , β = v c \phi(\beta)=\frac{3}{4\beta^{2}}\left[\frac{1}{\beta}\lg\frac{1-\beta}{1+\beta% }+\frac{2}{1-\beta^{2}}\right],\;\beta=\frac{v}{c}
  5. ϕ ( β ) = 3 4 β 2 ( 1 + β 2 2 β lg 1 + β 1 - β - 1 ) , \phi(\beta)=\frac{3}{4\beta^{2}}\left(\frac{1+\beta^{2}}{2\beta}\lg\frac{1+% \beta}{1-\beta}-1\right),
  6. Abraham \displaystyle\,\text{Abraham}

Kautz_filter.html

  1. { - α 1 , - α 2 , , - α n } \{-\alpha_{1},-\alpha_{2},\ldots,-\alpha_{n}\}
  2. Φ 1 ( s ) = 2 α 1 ( s + α 1 ) \Phi_{1}(s)=\frac{\sqrt{2\alpha_{1}}}{(s+\alpha_{1})}
  3. Φ 2 ( s ) = 2 α 2 ( s + α 2 ) ( s - α 1 ) ( s + α 1 ) \Phi_{2}(s)=\frac{\sqrt{2\alpha_{2}}}{(s+\alpha_{2})}\cdot\frac{(s-\alpha_{1})% }{(s+\alpha_{1})}
  4. Φ n ( s ) = 2 α n ( s + α n ) ( s - α 1 ) ( s - α 2 ) ( s - α n - 1 ) ( s + α 1 ) ( s + α 2 ) ( s + α n - 1 ) \Phi_{n}(s)=\frac{\sqrt{2\alpha_{n}}}{(s+\alpha_{n})}\cdot\frac{(s-\alpha_{1})% (s-\alpha_{2})\cdots(s-\alpha_{n-1})}{(s+\alpha_{1})(s+\alpha_{2})\cdots(s+% \alpha_{n-1})}
  5. ϕ n ( t ) = a n 1 e - α 1 t + a n 2 e - α 2 t + + a n n e - α n t \phi_{n}(t)=a_{n1}e^{-\alpha_{1}t}+a_{n2}e^{-\alpha_{2}t}+\cdots+a_{nn}e^{-% \alpha_{n}t}
  6. Φ n ( s ) = i = 1 n a n i s + α i \Phi_{n}(s)=\sum_{i=1}^{n}\frac{a_{ni}}{s+\alpha_{i}}
  7. ϕ k ( t ) = 2 a ( - 1 ) k - 1 e - a t L k - 1 ( 2 a t ) \phi_{k}(t)=\sqrt{2a}(-1)^{k-1}e^{-at}L_{k-1}(2at)

Kavrayskiy_VII_projection.html

  1. x = 3 λ 2 1 3 - ( ϕ π ) 2 x=\frac{3\lambda}{2}\sqrt{\frac{1}{3}-\left(\frac{\phi}{\pi}\right)^{2}}
  2. y = ϕ y=\phi\,
  3. λ \lambda
  4. ϕ \phi

Kaya_identity.html

  1. F = P × G P × E G × F E F=P\times\frac{G}{P}\times\frac{E}{G}\times\frac{F}{E}

Kazhdan–Lusztig_polynomial.html

  1. w W w∈W
  2. T y T w = T y w , if ( y w ) = ( y ) + ( w ) ( T s + 1 ) ( T s - q ) = 0 , if s S . \begin{aligned}\displaystyle T_{y}T_{w}&\displaystyle=T_{yw},&&\displaystyle% \mbox{if }~{}\ell(yw)=\ell(y)+\ell(w)\\ \displaystyle(T_{s}+1)(T_{s}-q)&\displaystyle=0,&&\displaystyle\mbox{if }~{}s% \in S.\end{aligned}
  3. D ( T w ) = T w - 1 - 1 D(T_{w})=T_{w^{-1}}^{-1}
  4. C w = q - ( w ) 2 y w P y , w T y C^{\prime}_{w}=q^{-\frac{\ell(w)}{2}}\sum_{y\leq w}P_{y,w}T_{y}
  5. C w C^{\prime}_{w}
  6. T y - 1 - 1 = x D ( R x , y ) q - ( x ) T x . T_{y^{-1}}^{-1}=\sum_{x}D(R_{x,y})q^{-\ell(x)}T_{x}.
  7. R x , y = { 0 , if x y 1 , if x = y R s x , s y , if s x < x and s y < y R x s , y s , if x s < x and y s < y ( q - 1 ) R s x , y + q R s x , s y , if s x > x and s y < y R_{x,y}=\begin{cases}0,&\mbox{if }~{}x\not\leq y\\ 1,&\mbox{if }~{}x=y\\ R_{sx,sy},&\mbox{if }~{}sx<x\mbox{ and }~{}sy<y\\ R_{xs,ys},&\mbox{if }~{}xs<x\mbox{ and }~{}ys<y\\ (q-1)R_{sx,y}+qR_{sx,sy},&\mbox{if }~{}sx>x\mbox{ and }~{}sy<y\end{cases}
  8. q 1 2 ( ( w ) - ( x ) ) D ( P x , w ) - q 1 2 ( ( x ) - ( w ) ) P x , w = x < y w ( - 1 ) ( x ) + ( y ) q 1 2 ( - ( x ) + 2 ( y ) - ( w ) ) D ( R x , y ) P y , w q^{\frac{1}{2}(\ell(w)-\ell(x))}D(P_{x,w})-q^{\frac{1}{2}(\ell(x)-\ell(w))}P_{% x,w}=\sum_{x<y\leq w}(-1)^{\ell(x)+\ell(y)}q^{\frac{1}{2}(-\ell(x)+2\ell(y)-% \ell(w))}D(R_{x,y})P_{y,w}
  9. 152 q 22 \displaystyle 152q^{22}
  10. w ( ρ ) ρ −w(ρ)−ρ
  11. w ( ρ ) ρ −w(ρ)−ρ
  12. ch ( L w ) = y w ( - 1 ) ( w ) - ( y ) P y , w ( 1 ) ch ( M y ) \operatorname{ch}(L_{w})=\sum_{y\leq w}(-1)^{\ell(w)-\ell(y)}P_{y,w}(1)% \operatorname{ch}(M_{y})
  13. ch ( M w ) = y w P w 0 w , w 0 y ( 1 ) ch ( L y ) \operatorname{ch}(M_{w})=\sum_{y\leq w}P_{w_{0}w,w_{0}y}(1)\operatorname{ch}(L% _{y})
  14. w ( ρ ) ρ w(ρ)−ρ
  15. w ( λ + ρ ) ρ w(λ+ρ)−ρ
  16. λ λ
  17. P y , w ( q ) = i q i dim ( Ext ( w ) - ( y ) - 2 i ( M y , L w ) ) P_{y,w}(q)=\sum_{i}q^{i}\dim(\operatorname{Ext}^{\ell(w)-\ell(y)-2i}(M_{y},L_{% w}))
  18. j + ( w ) + ( y ) j+ℓ(w)+ℓ(y)
  19. P y , w 0 P_{y,w_{0}}
  20. P y , w ( q ) = i q i dim I H X y 2 i ( X w ¯ ) P_{y,w}(q)=\sum_{i}q^{i}\dim IH^{2i}_{X_{y}}(\overline{X_{w}})
  21. X < s u b > w X<sub>w

Kelvin_bridge.html

  1. R x R s = R 2 R 1 + R p a r R s ( R 1 R 1 + R 2 + R p a r ) ( R 2 R 1 - R 2 R 1 ) \frac{Rx}{Rs}=\frac{R2}{R1}+\frac{Rpar}{Rs}\cdot\left(\frac{R^{\prime}1}{R^{% \prime}1+R^{\prime}2+Rpar}\right)\cdot\left(\frac{R2}{R1}-\frac{R^{\prime}2}{R% ^{\prime}1}\right)
  2. R x R s = R 2 R 1 \frac{Rx}{Rs}=\frac{R2}{R1}
  3. R x = R 2 R s R 1 Rx=R2\cdot\frac{Rs}{R1}

Kelvin_functions.html

  1. J ν ( x e 3 π i 4 ) , J_{\nu}\left(xe^{\frac{3\pi i}{4}}\right),\,
  2. K ν ( x e π i 4 ) , K_{\nu}\left(xe^{\frac{\pi i}{4}}\right),\,
  3. ber n ( x ) = ( x 2 ) n k 0 cos [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) k \mathrm{ber}_{n}(x)=\left(\frac{x}{2}\right)^{n}\sum_{k\geq 0}\frac{\cos\left[% \left(\frac{3n}{4}+\frac{k}{2}\right)\pi\right]}{k!\Gamma(n+k+1)}\left(\frac{x% ^{2}}{4}\right)^{k}
  4. Γ ( z ) Γ(z)
  5. ber ( x ) = 1 + k 1 ( - 1 ) k [ ( 2 k ) ! ] 2 ( x 2 ) 4 k \mathrm{ber}(x)=1+\sum_{k\geq 1}\frac{(-1)^{k}}{[(2k)!]^{2}}\left(\frac{x}{2}% \right)^{4k}
  6. ber ( x ) e x 2 2 π x ( f 1 ( x ) cos α + g 1 ( x ) sin α ) - kei ( x ) π \mathrm{ber}(x)\sim\frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2\pi x}}\left(f_{1}(x)% \cos\alpha+g_{1}(x)\sin\alpha\right)-\frac{\mathrm{kei}(x)}{\pi}
  7. α = x 2 - π 8 , \alpha=\frac{x}{\sqrt{2}}-\frac{\pi}{8},
  8. f 1 ( x ) = 1 + k 1 cos ( k π / 4 ) k ! ( 8 x ) k l = 1 k ( 2 l - 1 ) 2 f_{1}(x)=1+\sum_{k\geq 1}\frac{\cos(k\pi/4)}{k!(8x)^{k}}\prod_{l=1}^{k}(2l-1)^% {2}
  9. g 1 ( x ) = k 1 sin ( k π / 4 ) k ! ( 8 x ) k l = 1 k ( 2 l - 1 ) 2 g_{1}(x)=\sum_{k\geq 1}\frac{\sin(k\pi/4)}{k!(8x)^{k}}\prod_{l=1}^{k}(2l-1)^{2}
  10. bei n ( x ) = ( x 2 ) n k 0 sin [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) k \mathrm{bei}_{n}(x)=\left(\frac{x}{2}\right)^{n}\sum_{k\geq 0}\frac{\sin\left[% \left(\frac{3n}{4}+\frac{k}{2}\right)\pi\right]}{k!\Gamma(n+k+1)}\left(\frac{x% ^{2}}{4}\right)^{k}
  11. bei ( x ) = k 0 ( - 1 ) k [ ( 2 k + 1 ) ! ] 2 ( x 2 ) 4 k + 2 \mathrm{bei}(x)=\sum_{k\geq 0}\frac{(-1)^{k}}{[(2k+1)!]^{2}}\left(\frac{x}{2}% \right)^{4k+2}
  12. bei ( x ) e x 2 2 π x [ f 1 ( x ) sin α - g 1 ( x ) cos α ] - ker ( x ) π , \mathrm{bei}(x)\sim\frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2\pi x}}[f_{1}(x)\sin% \alpha-g_{1}(x)\cos\alpha]-\frac{\mathrm{ker}(x)}{\pi},
  13. f 1 ( x ) f_{1}(x)
  14. g 1 ( x ) g_{1}(x)
  15. ker n ( x ) = - ln ( x 2 ) ber n ( x ) + π 4 bei n ( x ) + 1 2 ( x 2 ) - n k = 0 n - 1 cos [ ( 3 n 4 + k 2 ) π ] ( n - k - 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n k 0 cos [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k \begin{aligned}\displaystyle\mathrm{ker}_{n}(x)&\displaystyle=-\ln\left(\frac{% x}{2}\right)\mathrm{ber}_{n}(x)+\frac{\pi}{4}\mathrm{bei}_{n}(x)\\ &\displaystyle\qquad+\frac{1}{2}\left(\frac{x}{2}\right)^{-n}\sum_{k=0}^{n-1}% \cos\left[\left(\frac{3n}{4}+\frac{k}{2}\right)\pi\right]\frac{(n-k-1)!}{k!}% \left(\frac{x^{2}}{4}\right)^{k}\\ &\displaystyle\qquad\qquad+\frac{1}{2}\left(\frac{x}{2}\right)^{n}\sum_{k\geq 0% }\cos\left[\left(\frac{3n}{4}+\frac{k}{2}\right)\pi\right]\frac{\psi(k+1)+\psi% (n+k+1)}{k!(n+k)!}\left(\frac{x^{2}}{4}\right)^{k}\end{aligned}
  16. ker ( x ) = - ln ( x 2 ) ber ( x ) + π 4 bei ( x ) + k 0 ( - 1 ) k ψ ( 2 k + 1 ) [ ( 2 k ) ! ] 2 ( x 2 4 ) 2 k \mathrm{ker}(x)=-\ln\left(\frac{x}{2}\right)\mathrm{ber}(x)+\frac{\pi}{4}% \mathrm{bei}(x)+\sum_{k\geq 0}(-1)^{k}\frac{\psi(2k+1)}{[(2k)!]^{2}}\left(% \frac{x^{2}}{4}\right)^{2k}
  17. ker ( x ) π 2 x e - x 2 [ f 2 ( x ) cos β + g 2 ( x ) sin β ] , \mathrm{ker}(x)\sim\sqrt{\frac{\pi}{2x}}e^{-\frac{x}{\sqrt{2}}}[f_{2}(x)\cos% \beta+g_{2}(x)\sin\beta],
  18. β = x 2 + π 8 , \beta=\frac{x}{\sqrt{2}}+\frac{\pi}{8},
  19. f 2 ( x ) = 1 + k 1 ( - 1 ) k cos ( k π / 4 ) k ! ( 8 x ) k l = 1 k ( 2 l - 1 ) 2 f_{2}(x)=1+\sum_{k\geq 1}(-1)^{k}\frac{\cos(k\pi/4)}{k!(8x)^{k}}\prod_{l=1}^{k% }(2l-1)^{2}
  20. g 2 ( x ) = k 1 ( - 1 ) k sin ( k π / 4 ) k ! ( 8 x ) k l = 1 k ( 2 l - 1 ) 2 . g_{2}(x)=\sum_{k\geq 1}(-1)^{k}\frac{\sin(k\pi/4)}{k!(8x)^{k}}\prod_{l=1}^{k}(% 2l-1)^{2}.
  21. kei n ( x ) = - ln ( x 2 ) bei n ( x ) - π 4 ber n ( x ) - 1 2 ( x 2 ) - n k = 0 n - 1 sin [ ( 3 n 4 + k 2 ) π ] ( n - k - 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n k 0 sin [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k \begin{aligned}\displaystyle\mathrm{kei}_{n}(x)&\displaystyle=-\ln\left(\frac{% x}{2}\right)\mathrm{bei}_{n}(x)-\frac{\pi}{4}\mathrm{ber}_{n}(x)\\ &\displaystyle\qquad-\frac{1}{2}\left(\frac{x}{2}\right)^{-n}\sum_{k=0}^{n-1}% \sin\left[\left(\frac{3n}{4}+\frac{k}{2}\right)\pi\right]\frac{(n-k-1)!}{k!}% \left(\frac{x^{2}}{4}\right)^{k}\\ &\displaystyle\qquad\qquad+\frac{1}{2}\left(\frac{x}{2}\right)^{n}\sum_{k\geq 0% }\sin\left[\left(\frac{3n}{4}+\frac{k}{2}\right)\pi\right]\frac{\psi(k+1)+\psi% (n+k+1)}{k!(n+k)!}\left(\frac{x^{2}}{4}\right)^{k}\end{aligned}
  22. kei ( x ) = - ln ( x 2 ) bei ( x ) - π 4 ber ( x ) + k 0 ( - 1 ) k ψ ( 2 k + 2 ) [ ( 2 k + 1 ) ! ] 2 ( x 2 4 ) 2 k + 1 \mathrm{kei}(x)=-\ln\left(\frac{x}{2}\right)\mathrm{bei}(x)-\frac{\pi}{4}% \mathrm{ber}(x)+\sum_{k\geq 0}(-1)^{k}\frac{\psi(2k+2)}{[(2k+1)!]^{2}}\left(% \frac{x^{2}}{4}\right)^{2k+1}
  23. kei ( x ) - π 2 x e - x 2 [ f 2 ( x ) sin β + g 2 ( x ) cos β ] , \mathrm{kei}(x)\sim-\sqrt{\frac{\pi}{2x}}e^{-\frac{x}{\sqrt{2}}}[f_{2}(x)\sin% \beta+g_{2}(x)\cos\beta],
  24. f 2 ( x ) f_{2}(x)
  25. g 2 ( x ) g_{2}(x)

Kempner_function.html

  1. n = 2 1 / [ S ( n ) ] ! = 1.09317 \sum_{n=2}^{\infty}1/[S(n)]!=1.09317\ldots
  2. n = 2 S ( n ) / n ! 1.71400629359162 \sum_{n=2}^{\infty}S(n)/n!\approx 1.71400629359162\ldots
  3. n = 2 1 / i = 2 n S ( i ) 0.719960700043 \sum_{n=2}^{\infty}1/\prod_{i=2}^{n}S(i)\approx 0.719960700043\ldots
  4. n S ( n ) - α S ( n ) ! - 1 / 2 < ( α > 1 ) . \sum_{n}S(n)^{-\alpha}{S(n)!}^{-1/2}<\infty\,(\alpha>1).

Kenneth_Stewart_Cole.html

  1. ϵ * - ϵ = ϵ 0 - ϵ 1 + ( i ω τ 0 ) 1 - α \epsilon^{*}-\epsilon_{\infty}=\dfrac{\epsilon_{0}-\epsilon_{\infty}}{1+(i% \omega\tau_{0})^{1-\alpha}}
  2. ϵ * \epsilon^{*}
  3. ϵ 0 {\epsilon_{0}}
  4. ϵ \epsilon_{\infty}
  5. ω = 2 π \omega=2\pi
  6. τ 0 \tau_{0}
  7. α \alpha
  8. Z Z
  9. Z = R R 0 - R 1 + ( j f f c ) 1 - α Z=R_{\infty}\frac{R_{0}-R_{\infty}}{1+(\tfrac{jf}{f_{c}})^{1-\alpha}}
  10. R 0 R_{0}
  11. R R_{\infty}
  12. f c f_{c}
  13. S S
  14. C C
  15. R R
  16. R 0 = R R_{0}=R
  17. R = R S R + S R_{\infty}\ =\tfrac{RS}{R+S}
  18. f c f_{c}
  19. f c = 1 2 π C ( R + S ) f_{c}=\tfrac{1}{2\pi C(R+S)}

Kernel_(statistics).html

  1. p ( x | μ , σ 2 ) = 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 p(x|\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2% \sigma^{2}}}
  2. p ( x | μ , σ 2 ) e - ( x - μ ) 2 2 σ 2 p(x|\mu,\sigma^{2})\propto e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}
  3. σ 2 \sigma^{2}
  4. x x
  5. - + K ( u ) d u = 1 ; \int_{-\infty}^{+\infty}K(u)\,du=1\,;
  6. K ( - u ) = K ( u ) for all values of u . K(-u)=K(u)\mbox{ for all values of }~{}u\,.
  7. u 2 K ( u ) d u \textstyle\int u^{2}K(u)du
  8. K ( u ) 2 d u \textstyle\int K(u)^{2}du
  9. K ( u ) = 1 2 1 { | u | 1 } K(u)=\frac{1}{2}\,\mathbf{1}_{\{|u|\leq 1\}}
  10. 1 3 \frac{1}{3}
  11. 1 2 \frac{1}{2}
  12. K ( u ) = ( 1 - | u | ) 1 { | u | 1 } K(u)=(1-|u|)\,\mathbf{1}_{\{|u|\leq 1\}}
  13. 1 6 \frac{1}{6}
  14. 2 3 \frac{2}{3}
  15. K ( u ) = 3 4 ( 1 - u 2 ) 1 { | u | 1 } K(u)=\frac{3}{4}(1-u^{2})\,\mathbf{1}_{\{|u|\leq 1\}}
  16. 1 5 \frac{1}{5}
  17. 3 5 \frac{3}{5}
  18. K ( u ) = 15 16 ( 1 - u 2 ) 2 1 { | u | 1 } K(u)=\frac{15}{16}(1-u^{2})^{2}\,\mathbf{1}_{\{|u|\leq 1\}}
  19. 1 7 \frac{1}{7}
  20. 5 7 \frac{5}{7}
  21. K ( u ) = 35 32 ( 1 - u 2 ) 3 1 { | u | 1 } K(u)=\frac{35}{32}(1-u^{2})^{3}\,\mathbf{1}_{\{|u|\leq 1\}}
  22. 1 9 \frac{1}{9}
  23. 350 429 \frac{350}{429}
  24. K ( u ) = 70 81 ( 1 - | u | 3 ) 3 1 { | u | 1 } K(u)=\frac{70}{81}(1-{\left|u\right|}^{3})^{3}\,\mathbf{1}_{\{|u|\leq 1\}}
  25. 35 243 \frac{35}{243}
  26. 175 247 \frac{175}{247}
  27. K ( u ) = 1 2 π e - 1 2 u 2 K(u)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^{2}}
  28. 1 1\,
  29. 1 2 π \frac{1}{2\sqrt{\pi}}
  30. K ( u ) = π 4 cos ( π 2 u ) 𝟏 { | u | 1 } K(u)=\frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right)\mathbf{1}_{\{|u|\leq 1\}}
  31. 1 - 8 π 2 1-\frac{8}{\pi^{2}}
  32. π 2 16 \frac{\pi^{2}}{16}
  33. K ( u ) = 1 e u + 2 + e - u K(u)=\frac{1}{e^{u}+2+e^{-u}}
  34. π 2 3 \frac{\pi^{2}}{3}
  35. 1 6 \frac{1}{6}
  36. K ( u ) = 1 2 e - | u | 2 sin ( | u | 2 + π 4 ) K(u)=\frac{1}{2}e^{-\frac{|u|}{\sqrt{2}}}\cdot\sin\left(\frac{|u|}{\sqrt{2}}+% \frac{\pi}{4}\right)
  37. 0
  38. 3 2 16 \frac{3\sqrt{2}}{16}
  39. ( u 2 K ( u ) d u ) 1 2 K ( u ) 2 d u \left(\int u^{2}K(u)du\right)^{\frac{1}{2}}\cdot\int K(u)^{2}du

Kernel_regression.html

  1. Y Y
  2. X X
  3. E ( Y | X ) = m ( X ) \operatorname{E}(Y|X)=m(X)
  4. m m
  5. m m
  6. m ^ h ( x ) = i = 1 n K h ( x - x i ) y i i = 1 n K h ( x - x i ) \widehat{m}_{h}(x)=\frac{\sum_{i=1}^{n}K_{h}(x-x_{i})y_{i}}{\sum_{i=1}^{n}K_{h% }(x-x_{i})}
  7. K K
  8. h h
  9. E ( Y | X = x ) = y f ( y | x ) d y = y f ( x , y ) f ( x ) d y \operatorname{E}(Y|X=x)=\int yf(y|x)dy=\int y\frac{f(x,y)}{f(x)}dy
  10. f ^ ( x , y ) = n - 1 h - 2 i = 1 n K ( x - x i h ) K ( y - y i h ) \hat{f}(x,y)=n^{-1}h^{-2}\sum_{i=1}^{n}K\left(\frac{x-x_{i}}{h}\right)K\left(% \frac{y-y_{i}}{h}\right)
  11. f ^ ( x ) = n - 1 h - 1 i = 1 n K ( x - x i h ) \hat{f}(x)=n^{-1}h^{-1}\sum_{i=1}^{n}K\left(\frac{x-x_{i}}{h}\right)
  12. m ^ P C ( x ) = h - 1 i = 1 n ( x i - x i - 1 ) K ( x - x i h ) y i \widehat{m}_{PC}(x)=h^{-1}\sum_{i=1}^{n}(x_{i}-x_{i-1})K\left(\frac{x-x_{i}}{h% }\right)y_{i}
  13. m ^ G M ( x ) = h - 1 i = 1 n [ s i - 1 s i K ( x - u h ) d u ] y i \widehat{m}_{GM}(x)=h^{-1}\sum_{i=1}^{n}\left[\int_{s_{i-1}}^{s_{i}}K\left(% \frac{x-u}{h}\right)du\right]y_{i}
  14. s i = x i - 1 + x i 2 s_{i}=\frac{x_{i-1}+x_{i}}{2}

KeY.html

  1. ϕ [ α ] ψ \phi\rightarrow[\alpha]\psi
  2. ψ \psi
  3. α \alpha
  4. ϕ \phi
  5. { ϕ } α { ψ } \{\phi\}\alpha\{\psi\}
  6. ϕ \phi
  7. ψ \psi
  8. [ α ] [\alpha]
  9. α \langle\alpha\rangle
  10. α \alpha
  11. [ α ] [\alpha]
  12. α \langle\alpha\rangle
  13. Γ Δ \Gamma\vdash\Delta
  14. Γ \Gamma
  15. Δ \Delta
  16. γ Γ γ δ Δ δ \bigwedge_{\gamma\in\Gamma}\gamma\rightarrow\bigvee_{\delta\in\Delta}\delta
  17. e = ˙ e e\ \dot{=}\ e
  18. x = ˙ 0 [ x + + ; ] x = ˙ 1 x\ \dot{=}\ 0\rightarrow[x++;]x\ \dot{=}\ 1
  19. x = ˙ 0 x = ˙ 0 x\ \dot{=}\ 0\rightarrow x\ \dot{=}\ 0
  20. [ α ] ψ [\alpha]\psi
  21. w p ( α , ψ ) wp(\alpha,\psi)
  22. w p wp
  23. [ α ] ψ [\alpha]\psi
  24. [ x = 3 ; x = x + 1 ; ] x = ˙ 4 [x=3;x=x+1;]x\ \dot{=}\ 4
  25. { x := 3 } [ x = x + 1 ; ] x = ˙ 4 \{x:=3\}[x=x+1;]x\ \dot{=}\ 4
  26. { x := 4 } [ ] x = ˙ 4 \{x:=4\}[]x\ \dot{=}\ 4
  27. x x
  28. x x
  29. y y
  30. x 0 y 0 x\geq 0\land y\geq 0
  31. z = ˙ x y z\ \dot{=}\ x\cdot y

Key_encapsulation.html

  1. ( n , e ) (n,e)
  2. M e < n M^{e}<n
  3. 1 < m < n 1<m<n
  4. c c
  5. c m e ( mod n ) . c\equiv m^{e}\;\;(\mathop{{\rm mod}}n).
  6. m m
  7. c c
  8. d d
  9. m c d ( mod n ) . m\equiv c^{d}\;\;(\mathop{{\rm mod}}n).
  10. m m
  11. 1 < m < n 1<m<n
  12. M = K D F ( m ) M=KDF(m)
  13. c c
  14. c m e ( mod n ) . c\equiv m^{e}\;\;(\mathop{{\rm mod}}n).
  15. m m
  16. c c
  17. d d
  18. m c d ( mod n ) . m\equiv c^{d}\;\;(\mathop{{\rm mod}}n).
  19. m m
  20. M = K D F ( m ) M=KDF(m)

Key_selection_vector.html

  1. A 1 A_{1}
  2. A 40 A_{40}
  3. A i A_{i}
  4. A 1 A_{1}
  5. A n A_{n}
  6. A i A_{i}
  7. A i A_{i}
  8. A i A_{i}
  9. A i A_{i}

Keystone_effect.html

  1. cos ( ε - ( α / 2 ) ) cos ( ε + ( α / 2 ) ) \frac{\cos(\varepsilon-(\alpha/2))}{\cos(\varepsilon+(\alpha/2))}
  2. ϵ \epsilon
  3. α \alpha
  4. ϵ \epsilon

Kharitonov_region.html

  1. D D
  2. P P
  3. D D
  4. V T n ( V S n ) V_{T}^{n}(V_{S}^{n})
  5. P . P.
  6. V T n V_{T}^{n}
  7. ( T n ) (T^{n})
  8. V S n V_{S}^{n}
  9. ( S n ) . (S^{n}).

Khintchine_inequality.html

  1. N N
  2. x 1 , , x N x_{1},\dots,x_{N}\in\mathbb{C}
  3. ± 1 \pm 1
  4. | x 1 | 2 + + | x N | 2 \sqrt{|x_{1}|^{2}+\cdots+|x_{N}|^{2}}
  5. { ϵ n } n = 1 N \{\epsilon_{n}\}_{n=1}^{N}
  6. P ( ϵ n = ± 1 ) = 1 2 P(\epsilon_{n}=\pm 1)=\frac{1}{2}
  7. n = 1 N n=1\ldots N
  8. 0 < p < 0<p<\infty
  9. x 1 , , x N x_{1},...,x_{N}\in\mathbb{C}
  10. A p ( n = 1 N | x n | 2 ) 1 2 ( 𝔼 | n = 1 N ϵ n x n | p ) 1 / p B p ( n = 1 N | x n | 2 ) 1 2 A_{p}\left(\sum_{n=1}^{N}|x_{n}|^{2}\right)^{\frac{1}{2}}\leq\left(\mathbb{E}% \Big|\sum_{n=1}^{N}\epsilon_{n}x_{n}\Big|^{p}\right)^{1/p}\leq B_{p}\left(\sum% _{n=1}^{N}|x_{n}|^{2}\right)^{\frac{1}{2}}
  11. A p , B p > 0 A_{p},B_{p}>0
  12. p p
  13. A p , B p A_{p},B_{p}
  14. A p = 1 A_{p}=1
  15. p 2 p\geq 2
  16. B p = 1 B_{p}=1
  17. 0 < p 2 0<p\leq 2
  18. T T
  19. L p ( X , μ ) L^{p}(X,\mu)
  20. L p ( Y , ν ) L^{p}(Y,\nu)
  21. 1 p < 1\leq p<\infty
  22. T < \|T\|<\infty
  23. ( n = 1 N | T f n | 2 ) 1 2 L p ( Y , ν ) C p ( n = 1 N | f n | 2 ) 1 2 L p ( X , μ ) \left\|\left(\sum_{n=1}^{N}|Tf_{n}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(Y,% \nu)}\leq C_{p}\left\|\left(\sum_{n=1}^{N}|f_{n}|^{2}\right)^{\frac{1}{2}}% \right\|_{L^{p}(X,\mu)}
  24. C p > 0 C_{p}>0
  25. p p
  26. T \|T\|

Killed_process.html

  1. Y t = X t for t < ζ , Y_{t}=X_{t}\mbox{ for }~{}t<\zeta,

Kilpatrick_limit.html

  1. E E
  2. f f
  3. f = 1.64 MHz ( E E 0 ) 2 exp ( - 8.5 E 0 E ) , f=1.64\,\mathrm{MHz}\cdot\left(\frac{E}{E_{0}}\right)^{2}\cdot\exp\left(-8.5% \frac{E_{0}}{E}\right),\quad
  4. E 0 = 1 MV m E_{0}=1\mathrm{\frac{MV}{m}}

Kinematic_chain.html

  1. M = 6 n - i = 1 j ( 6 - f i ) = 6 ( N - 1 - j ) + i = 1 j f i M=6n-\sum_{i=1}^{j}\ (6-f_{i})=6(N-1-j)+\sum_{i=1}^{j}\ f_{i}
  2. [ T ] = [ Z 1 ] [ X 1 ] [ Z 2 ] [ X 2 ] [ X n - 1 ] [ Z n ] , [T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\ldots[X_{n-1}][Z_{n}],\!

Kirchhoff_equations.html

  1. d d t T ω \displaystyle{d\over{dt}}{{\partial T}\over{\partial\vec{\omega}}}
  2. ω \vec{\omega}
  3. v \vec{v}
  4. x \vec{x}
  5. I ~ \tilde{I}
  6. m m
  7. n ^ \hat{n}
  8. x \vec{x}
  9. p p
  10. Q h \vec{Q}_{h}
  11. F h \vec{F}_{h}
  12. Q \vec{Q}
  13. F \vec{F}
  14. Q h \vec{Q}_{h}
  15. F h \vec{F}_{h}
  16. d d t L ω = L ω × ω + L v × v , d d t L v = L v × ω , {d\over{dt}}{{\partial L}\over{\partial\vec{\omega}}}={{\partial L}\over{% \partial\vec{\omega}}}\times\vec{\omega}+{{\partial L}\over{\partial\vec{v}}}% \times\vec{v},\quad{d\over{dt}}{{\partial L}\over{\partial\vec{v}}}={{\partial L% }\over{\partial\vec{v}}}\times\vec{\omega},
  17. L ( ω , v ) = 1 2 ( A ω , ω ) + ( B ω , v ) + 1 2 ( C v , v ) + ( k , ω ) + ( l , v ) . L(\vec{\omega},\vec{v})={1\over 2}(A\vec{\omega},\vec{\omega})+(B\vec{\omega},% \vec{v})+{1\over 2}(C\vec{v},\vec{v})+(\vec{k},\vec{\omega})+(\vec{l},\vec{v}).
  18. J 0 = ( L ω , ω ) + ( L v , v ) - L , J 1 = ( L ω , L v ) , J 2 = ( L v , L v ) J_{0}=\left({{\partial L}\over{\partial\vec{\omega}}},\vec{\omega}\right)+% \left({{\partial L}\over{\partial\vec{v}}},\vec{v}\right)-L,\quad J_{1}=\left(% {{\partial L}\over{\partial\vec{\omega}}},{{\partial L}\over{\partial\vec{v}}}% \right),\quad J_{2}=\left({{\partial L}\over{\partial\vec{v}}},{{\partial L}% \over{\partial\vec{v}}}\right)

Kirkman's_schoolgirl_problem.html

  1. n n
  2. n n
  3. n 3 ( mod 6 ) n\equiv 3\;\;(\mathop{{\rm mod}}6)
  4. 1 2 ( n - 1 ) \frac{1}{2}(n-1)
  5. n = 15 n=15

Kleene–Brouwer_order.html

  1. ( X , < ) (X,<)
  2. t t
  3. s s
  4. X X
  5. t < K B s t<_{KB}s\,
  6. n n
  7. t n = s n t\upharpoonright n=s\upharpoonright n
  8. t ( n ) t(n)
  9. s ( n ) s(n)
  10. t t
  11. s s
  12. s ( n ) s(n)
  13. t ( n ) t(n)
  14. t ( n ) < s ( n ) t(n)<s(n)
  15. t n = s n t\upharpoonright n=s\upharpoonright n
  16. t n t\upharpoonright n
  17. t t
  18. t ( n ) t(n)
  19. t < K B s t<_{KB}s\,
  20. s s
  21. t t
  22. s s
  23. t t
  24. t t
  25. s s
  26. X X
  27. X X
  28. x x
  29. || x || ||x||
  30. 1 + || y || 1+||y||
  31. y y
  32. x x

Kneser's_theorem_(differential_equations).html

  1. y ′′ + q ( x ) y = 0 y^{\prime\prime}+q(x)y=0\,
  2. q : [ 0 , + ) q:[0,+\infty)\to\mathbb{R}
  3. lim sup x + x 2 q ( x ) < 1 4 \limsup_{x\to+\infty}x^{2}q(x)<\tfrac{1}{4}
  4. lim inf x + x 2 q ( x ) > 1 4 . \liminf_{x\to+\infty}x^{2}q(x)>\tfrac{1}{4}.
  5. q ( x ) = ( 1 4 - a ) x - 2 for x > 0 q(x)=\left(\frac{1}{4}-a\right)x^{-2}\quad\,\text{for}\quad x>0
  6. a a
  7. a a
  8. lim sup x + x 2 q ( x ) = lim inf x + x 2 q ( x ) = 1 4 - a \limsup_{x\to+\infty}x^{2}q(x)=\liminf_{x\to+\infty}x^{2}q(x)=\frac{1}{4}-a
  9. q ( x ) q(x)
  10. y ( x ) = x n y(x)=x^{n}\,
  11. n ( n - 1 ) + 1 4 - a = ( n - 1 2 ) 2 - a = 0 n(n-1)+\frac{1}{4}-a=\left(n-\frac{1}{2}\right)^{2}-a=0
  12. a a
  13. y ( x ) = A x 1 2 + a + B x 1 2 - a y(x)=Ax^{\frac{1}{2}+\sqrt{a}}+Bx^{\frac{1}{2}-\sqrt{a}}
  14. A A
  15. B B
  16. a a
  17. a = - ω 2 a=-\omega^{2}
  18. x 1 2 ± i ω = x e ± ( i ω ) ln x = x ( cos ( ω ln x ) ± i sin ( ω ln x ) ) x^{\frac{1}{2}\pm i\omega}=\sqrt{x}\ e^{\pm(i\omega)\ln{x}}=\sqrt{x}\ (\cos{(% \omega\ln x)}\pm i\sin{(\omega\ln x)})

Knot_thickness.html

  1. τ ( x ) = inf r ( x , y , z ) , \tau(x)=\inf r(x,y,z),\,
  2. τ ( L ) = inf τ ( x ) . \tau(L)=\inf\tau(x).

Knowledge_space.html

  1. ( Q , K ) (Q,K)
  2. Q Q
  3. K K
  4. Q Q
  5. K K
  6. Q Q
  7. S , T Q S,T\in Q
  8. S T Q S\cup T\in Q
  9. S , T Q S,T\in Q
  10. S T Q S\cap T\in Q
  11. S S
  12. T T
  13. S T S\cup T
  14. S S
  15. S { x } S\setminus\{x\}

Knödel_number.html

  1. n n
  2. m - φ ( n ) m-\varphi(n)

Kobon_triangle_problem.html

  1. k n + 1 = 2 k n - 1 , k_{n+1}=2\cdot k_{n}-1,\!\,

Kohn_anomaly.html

  1. 2 k F 2k_{F}
  2. Re ( ϵ ( 𝐪 , ω ) ) \operatorname{Re}(\epsilon(\mathbf{q},\omega))
  3. 𝐪 = 2 𝐤 F \mathbf{q}=2\mathbf{k}_{F}
  4. 𝐤 F \mathbf{k}_{F}
  5. Re ( ϵ ( 𝐫 , ω ) ) \operatorname{Re}(\epsilon(\mathbf{r},\omega))
  6. ω 2 ( 𝐪 ) \omega^{2}(\mathbf{q})
  7. 𝐪 = 2 𝐤 F \mathbf{q}=2\mathbf{k}_{F}

Kravchuk_polynomials.html

  1. 𝒦 0 ( x ; n ) = 1 \mathcal{K}_{0}(x;n)=1
  2. 𝒦 1 ( x ; n ) = - 2 x + n \mathcal{K}_{1}(x;n)=-2x+n
  3. 𝒦 2 ( x ; n ) = 2 x 2 - 2 n x + ( n 2 ) \mathcal{K}_{2}(x;n)=2x^{2}-2nx+{n\choose 2}
  4. 𝒦 3 ( x ; n ) = - 4 3 x 3 + 2 n x 2 - ( n 2 - n + 2 3 ) x + ( n 3 ) . \mathcal{K}_{3}(x;n)=-\frac{4}{3}x^{3}+2nx^{2}-(n^{2}-n+\frac{2}{3})x+{n% \choose 3}.
  5. 𝒦 k ( x ; n ) = 𝒦 k ( x ) = j = 0 k ( - 1 ) j ( q - 1 ) k - j ( x j ) ( n - x k - j ) , k = 0 , 1 , , n . \mathcal{K}_{k}(x;n)=\mathcal{K}_{k}(x)=\sum_{j=0}^{k}(-1)^{j}(q-1)^{k-j}{% \left({{x}\atop{j}}\right)}{\left({{n-x}\atop{k-j}}\right)},\quad k=0,1,\ldots% ,n.
  6. 𝒦 k ( x ; n ) = j = 0 k ( - q ) j ( q - 1 ) k - j ( n - j k - j ) ( x j ) . \mathcal{K}_{k}(x;n)=\sum_{j=0}^{k}(-q)^{j}(q-1)^{k-j}{\left({{n-j}\atop{k-j}}% \right)}{\left({{x}\atop{j}}\right)}.
  7. 𝒦 k ( x ; n ) = j = 0 k ( - 1 ) j q k - j ( n - k + j j ) ( n - x k - j ) . \mathcal{K}_{k}(x;n)=\sum_{j=0}^{k}(-1)^{j}q^{k-j}{\left({{n-k+j}\atop{j}}% \right)}{\left({{n-x}\atop{k-j}}\right)}.
  8. i = 0 n ( n i ) ( q - 1 ) i 𝒦 r ( i ; n ) 𝒦 s ( i ; n ) = q n ( q - 1 ) r ( n r ) δ r , s . \sum_{i=0}^{n}{\left({{n}\atop{i}}\right)}(q-1)^{i}\mathcal{K}_{r}(i;n)% \mathcal{K}_{s}(i;n)=q^{n}(q-1)^{r}{\left({{n}\atop{r}}\right)}\delta_{r,s}.

Krogmann's_salt.html

  1. d z 2 d_{z^{2}}

Kuratowski's_closure-complement_problem.html

  1. ( 0 , 1 ) ( 1 , 2 ) { 3 } ( [ 4 , 5 ] \Q ) , (0,1)\cup(1,2)\cup\{3\}\cup\bigl([4,5]\cap\Q\bigr),
  2. ( 1 , 2 ) (1,2)
  3. [ 4 , 5 ] [4,5]

Kuratowski's_free_set_theorem.html

  1. [ X ] < ω [X]^{<\omega}
  2. X X
  3. n n
  4. [ X ] n [X]^{n}
  5. n n
  6. X X
  7. Φ : [ X ] n [ X ] < ω \Phi\colon[X]^{n}\to[X]^{<\omega}
  8. U U
  9. X X
  10. Φ \Phi
  11. n n
  12. V V
  13. U U
  14. u U V u\in U\setminus V
  15. u Φ ( V ) u\notin\Phi(V)
  16. n \aleph_{n}
  17. n n
  18. X X
  19. X X
  20. n \aleph_{n}
  21. Φ \Phi
  22. [ X ] n [X]^{n}
  23. [ X ] < ω [X]^{<\omega}
  24. ( n + 1 ) (n+1)
  25. X X
  26. Φ \Phi
  27. n = 1 n=1

Kutta–Joukowski_theorem.html

  1. Γ \Gamma\,
  2. L L\,
  3. L L^{\prime}\,
  4. ρ \rho_{\infty}\,
  5. V V_{\infty}\,
  6. Γ \Gamma\,
  7. Γ = C V d 𝐬 = C V cos θ d s \Gamma=\oint_{C}V\cdot d\mathbf{s}=\oint_{C}V\cos\theta\;ds\,
  8. C C
  9. V cos θ V\cos\theta\,
  10. C C\,
  11. d s ds\,
  12. C C\,
  13. - ρ V Γ -\rho_{\infty}V_{\infty}\Gamma
  14. V . V_{\infty}.
  15. Γ \Gamma
  16. α > 0 \alpha>0\,
  17. R e = ρ V c A μ Re=\frac{\rho V_{\infty}c_{A}}{\mu}\,
  18. c c
  19. ρ \rho
  20. V V
  21. V + v V+v
  22. Γ = V c - ( V + v ) c = - v c . \Gamma=Vc-(V+v)c=-vc.\,
  23. Δ P \Delta P
  24. ρ 2 ( V ) 2 + ( P + Δ P ) = ρ 2 ( V + v ) 2 + P , \frac{\rho}{2}(V)^{2}+(P+\Delta P)=\frac{\rho}{2}(V+v)^{2}+P,\,
  25. ρ 2 ( V ) 2 + Δ P = ρ 2 ( V 2 + 2 V v + v 2 ) , \frac{\rho}{2}(V)^{2}+\Delta P=\frac{\rho}{2}(V^{2}+2Vv+v^{2}),\,
  26. Δ P = ρ V v (ignoring ρ 2 v 2 ) , \Delta P=\rho Vv\qquad\,\text{(ignoring }\frac{\rho}{2}v^{2}),\,
  27. L = c Δ P = ρ V v c = - ρ V Γ . L=c\Delta P=\rho Vvc=-\rho V\Gamma.\,
  28. 𝐅 \mathbf{F}\,
  29. 𝐅 = - C p 𝐧 d s , \mathbf{F}=-\oint_{C}p\mathbf{n}\,ds,
  30. p p
  31. 𝐧 \mathbf{n}\,
  32. ϕ \phi
  33. F x = - C p sin ϕ d s , F y = C p cos ϕ d s . F_{x}=-\oint_{C}p\sin\phi\,ds\quad,\qquad F_{y}=\oint_{C}p\cos\phi\,ds.
  34. F = F x + i F y = - C p ( sin ϕ - i cos ϕ ) d s . F=F_{x}+iF_{y}=-\oint_{C}p(\sin\phi-i\cos\phi)\,ds.
  35. F F
  36. F ¯ = - C p ( sin ϕ + i cos ϕ ) d s = - i C p ( cos ϕ - i sin ϕ ) d s = - i C p e - i ϕ d s . \bar{F}=-\oint_{C}p(\sin\phi+i\cos\phi)\,ds=-i\oint_{C}p(\cos\phi-i\sin\phi)\,% ds=-i\oint_{C}pe^{-i\phi}\,ds.
  37. d z = d x + i d y = d s ( cos ϕ + i sin ϕ ) = d s e i ϕ d z ¯ = e - i ϕ d s . dz=dx+idy=ds(\cos\phi+i\sin\phi)=ds\,e^{i\phi}\qquad\Rightarrow\qquad d\bar{z}% =e^{-i\phi}ds.
  38. F ¯ = - i C p d z ¯ . \bar{F}=-i\oint_{C}p\,d\bar{z}.
  39. ρ . \rho.
  40. p p
  41. v = v x + i v y v=v_{x}+iv_{y}
  42. p = p 0 - ρ | v | 2 2 . p=p_{0}-\frac{\rho|v|^{2}}{2}.
  43. F F
  44. F ¯ = - i p 0 C d z ¯ + i ρ 2 C | v | 2 d z ¯ = i ρ 2 C | v | 2 d z ¯ . \bar{F}=-ip_{0}\oint_{C}d\bar{z}+i\frac{\rho}{2}\oint_{C}|v|^{2}\,d\bar{z}=% \frac{i\rho}{2}\oint_{C}|v|^{2}\,d\bar{z}.
  45. w = f ( z ) , w=f(z),
  46. w = v x - i v y = v ¯ , w^{\prime}=v_{x}-iv_{y}=\bar{v},
  47. v = ± | v | e i ϕ . v=\pm|v|e^{i\phi}.
  48. v 2 d z ¯ = | v | 2 d z , v^{2}d\bar{z}=|v|^{2}dz,\,
  49. F ¯ = i ρ 2 C w 2 d z , \bar{F}=\frac{i\rho}{2}\oint_{C}w^{\prime 2}\,dz,
  50. w w
  51. w ( z ) = a 0 + a 1 z + a 2 z 2 + . w^{\prime}(z)=a_{0}+\frac{a_{1}}{z}+\frac{a_{2}}{z^{2}}+\dots.
  52. a 0 a_{0}\,
  53. a 0 = v x - i v y a_{0}=v_{x\infty}-iv_{y\infty}\,
  54. a 1 a_{1}\,
  55. a 1 = 1 2 π i C w d z . a_{1}=\frac{1}{2\pi i}\oint_{C}w^{\prime}\,dz.
  56. C w ( z ) d z = C ( v x - i v y ) ( d x + i d y ) = C ( v x d x + v y d y ) + i C ( v x d y - v y d x ) = C 𝐯 d s + i C ( v x d y - v y d x ) . \oint_{C}w^{\prime}(z)\,dz=\oint_{C}(v_{x}-iv_{y})(dx+idy)=\oint_{C}(v_{x}\,dx% +v_{y}\,dy)+i\oint_{C}(v_{x}\,dy-v_{y}\,dx)=\oint_{C}\mathbf{v}\,{ds}+i\oint_{% C}(v_{x}\,dy-v_{y}\,dx).
  57. Γ . \Gamma.
  58. C ( v x d y - v y d x ) = C ( ψ y d y + ψ x d x ) = C d ψ = 0. \oint_{C}(v_{x}\,dy-v_{y}\,dx)=\oint_{C}\left(\frac{\partial\psi}{\partial y}% dy+\frac{\partial\psi}{\partial x}dx\right)=\oint_{C}d\psi=0.
  59. ψ \psi\,
  60. d ψ = 0 d\psi=0\,
  61. a 1 = Γ 2 π i . a_{1}=\frac{\Gamma}{2\pi i}.
  62. w 2 ( z ) = a 0 2 + a 0 Γ π i z + . w^{\prime 2}(z)=a_{0}^{2}+\frac{a_{0}\Gamma}{\pi iz}+\dots.
  63. F ¯ = i ρ 2 [ 2 π i a 0 Γ π i ] = i ρ a 0 Γ = i ρ Γ ( v x - i v y ) = ρ Γ v y + i ρ Γ v x = F x - i F y . \bar{F}=\frac{i\rho}{2}\left[2\pi i\frac{a_{0}\Gamma}{\pi i}\right]=i\rho a_{0% }\Gamma=i\rho\Gamma(v_{x\infty}-iv_{y\infty})=\rho\Gamma v_{y\infty}+i\rho% \Gamma v_{x\infty}=F_{x}-iF_{y}.
  64. F x = ρ Γ v y , F y = - ρ Γ v x . F_{x}=\rho\Gamma v_{y\infty}\quad,\qquad F_{y}=-\rho\Gamma v_{x\infty}.

Kylee.html

  1. \cdot
  2. \cdot
  3. \cdot
  4. \cdot
  5. \cdot
  6. \cdot
  7. \cdot
  8. \cdot
  9. \cdot
  10. \cdot
  11. \cdot
  12. \cdot
  13. \cdot
  14. \cdot
  15. \cdot
  16. \cdot
  17. \cdot
  18. \cdot
  19. \cdot
  20. \cdot
  21. \cdot
  22. \cdot
  23. \cdot
  24. \cdot
  25. \cdot
  26. \cdot
  27. \cdot
  28. \cdot
  29. \cdot
  30. \cdot
  31. \cdot
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  33. \cdot
  34. \cdot
  35. \cdot
  36. \cdot
  37. \cdot
  38. \cdot
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  40. \cdot
  41. \cdot
  42. \cdot
  43. \cdot
  44. \cdot
  45. \cdot
  46. \cdot
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  48. \cdot
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  50. \cdot
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  58. \cdot
  59. \cdot
  60. \cdot
  61. \cdot
  62. \cdot

L-xylulose_reductase.html

  1. \rightleftharpoons

Lacunary_function.html

  1. f ( z ) = n = 0 z 2 n = z + z 2 + z 4 + z 8 + f(z)=\sum_{n=0}^{\infty}z^{2^{n}}=z+z^{2}+z^{4}+z^{8}+\cdots\,
  2. f ( z 2 ) = f ( z ) - z f ( z 4 ) = f ( z 2 ) - z 2 f ( z 8 ) = f ( z 4 ) - z 4 f(z^{2})=f(z)-z\qquad f(z^{4})=f(z^{2})-z^{2}\qquad f(z^{8})=f(z^{4})-z^{4}\cdots\,
  3. f ( z ) = n = 0 z 3 n = z + z 3 + z 9 + z 27 + g ( z ) = n = 0 z 4 n = z + z 4 + z 16 + z 64 + f(z)=\sum_{n=0}^{\infty}z^{3^{n}}=z+z^{3}+z^{9}+z^{27}+\cdots\qquad g(z)=\sum_% {n=0}^{\infty}z^{4^{n}}=z+z^{4}+z^{16}+z^{64}+\cdots\,
  4. f ( z ) = k = 1 a k z λ k = n = 1 b n z n f(z)=\sum_{k=1}^{\infty}a_{k}z^{\lambda_{k}}=\sum_{n=1}^{\infty}b_{n}z^{n}\,
  5. lim k λ k λ k - 1 > 1 + δ \lim_{k\to\infty}\frac{\lambda_{k}}{\lambda_{k-1}}>1+\delta\,
  6. S ( λ k , θ ) = k = 1 a k cos ( λ k θ ) S ( λ k , θ , ω ) = k = 1 a k cos ( λ k θ + ω ) S(\lambda_{k},\theta)=\sum_{k=1}^{\infty}a_{k}\cos(\lambda_{k}\theta)\qquad S(% \lambda_{k},\theta,\omega)=\sum_{k=1}^{\infty}a_{k}\cos(\lambda_{k}\theta+% \omega)\,
  7. k = 1 a k 2 \sum_{k=1}^{\infty}a_{k}^{2}\,
  8. g ( z ) = n = 1 z n g(z)=\sum_{n=1}^{\infty}z^{n}\,
  9. g ( z ) = n = 1 e i n θ = n = 1 ( cos n θ + i sin n θ ) g(z)=\sum_{n=1}^{\infty}e^{in\theta}=\sum_{n=1}^{\infty}\left(\cos n\theta+i% \sin n\theta\right)\,

Lagrange_multipliers_on_Banach_spaces.html

  1. g - 1 ( 0 ) = { x U g ( x ) = 0 Y } U . g^{-1}(0)=\{x\in U\mid g(x)=0\in Y\}\subseteq U.
  2. D f ( u 0 ) = λ D g ( u 0 ) . (L) \mathrm{D}f(u_{0})=\lambda\circ\mathrm{D}g(u_{0}).\quad\mbox{(L)}~{}
  3. D f ( u 0 ) = ( D g ( u 0 ) ) * ( λ ) , \mathrm{D}f(u_{0})=\left(\mathrm{D}g(u_{0})\right)^{*}(\lambda),
  4. ( D g ( u 0 ) ) * ( λ ) = λ D g ( u 0 ) . \left(\mathrm{D}g(u_{0})\right)^{*}(\lambda)=\lambda\circ\mathrm{D}g(u_{0}).
  5. f ( u ) = - 1 + 1 u ( x ) 2 d x . f(u)=\int_{-1}^{+1}u^{\prime}(x)^{2}\,\mathrm{d}x.
  6. g ( u ) = 1 2 - 1 + 1 u ( x ) d x - 1. g(u)=\frac{1}{2}\int_{-1}^{+1}u(x)\,\mathrm{d}x-1.
  7. λ \lambda

Lambek–Moser_theorem.html

  1. f ( n ) = n - 1 \scriptstyle f^{\ast}(n)=\lfloor\sqrt{n-1}\rfloor
  2. G ( n ) = n - 1 + n . \scriptstyle G(n)=\lfloor\sqrt{n-1}\rfloor+n.
  3. f ( n ) = r n - n \scriptstyle f(n)=\lfloor rn\rfloor-n
  4. f ( n ) = s n - n , \scriptstyle f^{\ast}(n)=\lfloor sn\rfloor-n,
  5. 1 r + 1 s = 1 \scriptstyle\tfrac{1}{r}+\tfrac{1}{s}=1
  6. F ( n ) = r n \scriptstyle F(n)=\lfloor rn\rfloor
  7. G ( n ) = s n . \scriptstyle G(n)=\lfloor sn\rfloor.

Lamé_parameters.html

  1. s y m b o l σ = 2 μ s y m b o l ε + λ tr ( s y m b o l ε ) I symbol{\sigma}=2\mu symbol{\varepsilon}+\lambda\;\mathrm{tr}(symbol{% \varepsilon})I
  2. I \scriptstyle I
  3. tr ( ) \scriptstyle\mathrm{tr}(\cdot)
  4. K = λ + ( 2 / 3 ) μ K=\lambda+(2/3)\mu

Landau–Kolmogorov_inequality.html

  1. f ( k ) L ( T ) C ( n , k , T ) f L ( T ) 1 - k / n f ( n ) L ( T ) k / n for 1 k < n . \|f^{(k)}\|_{L_{\infty}(T)}\leq C(n,k,T){\|f\|_{L_{\infty}(T)}}^{1-k/n}{\|f^{(% n)}\|_{L_{\infty}(T)}}^{k/n}\,\text{ for }1\leq k<n.
  2. C ( n , k , ) = a n - k a n - 1 + k / n , C(n,k,\mathbb{R})=a_{n-k}a_{n}^{-1+k/n}~{},
  3. f ( k ) L q ( T ) K f L p ( T ) α f ( n ) L r ( T ) 1 - α for 1 k < n . \|f^{(k)}\|_{L_{q}(T)}\leq K\cdot{\|f\|^{\alpha}_{L_{p}(T)}}\cdot{\|f^{(n)}\|^% {1-\alpha}_{L_{r}(T)}}\,\text{ for }1\leq k<n.

Langlands_decomposition.html

  1. P = M A N P=MAN
  2. G G
  3. P = M A N P=MAN
  4. M A MA
  5. P P
  6. N N
  7. P P
  8. G G

Laplace_expansion_(potential).html

  1. 1 | 𝐫 - 𝐫 | = = 0 4 π 2 + 1 m = - ( - 1 ) m r < m t p l > < r > + 1 Y - m ( θ , φ ) Y m ( θ , φ ) . \frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|}=\sum_{\ell=0}^{\infty}\frac{4\pi}{2% \ell+1}\sum_{m=-\ell}^{\ell}(-1)^{m}\frac{r_{<}mtpl>{{\scriptscriptstyle<}}^{% \ell}}{r_{{\scriptscriptstyle>}}^{\ell+1}}Y^{-m}_{\ell}(\theta,\varphi)Y^{m}_{% \ell}(\theta^{\prime},\varphi^{\prime}).
  2. Y m Y^{m}_{\ell}
  3. 1 | 𝐫 - 𝐫 | = = 0 m = - ( - 1 ) m I - m ( 𝐫 ) R m ( 𝐫 ) with | 𝐫 | > | 𝐫 | . \frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|}=\sum_{\ell=0}^{\infty}\sum_{m=-\ell% }^{\ell}(-1)^{m}I^{-m}_{\ell}(\mathbf{r})R^{m}_{\ell}(\mathbf{r}^{\prime})% \quad\hbox{with}\quad|\mathbf{r}|>|\mathbf{r}^{\prime}|.
  4. 1 | 𝐫 - 𝐫 | = 1 r 2 + ( r ) 2 - 2 r r cos γ = 1 r 1 + h 2 - 2 h cos γ with h r r . \frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|}=\frac{1}{\sqrt{r^{2}+(r^{\prime})^{% 2}-2rr^{\prime}\cos\gamma}}=\frac{1}{r\sqrt{1+h^{2}-2h\cos\gamma}}\quad\hbox{% with}\quad h\equiv\frac{r^{\prime}}{r}.
  5. P ( cos γ ) P_{\ell}(\cos\gamma)
  6. 1 1 + h 2 - 2 h cos γ = = 0 h P ( cos γ ) . \frac{1}{\sqrt{1+h^{2}-2h\cos\gamma}}=\sum_{\ell=0}^{\infty}h^{\ell}P_{\ell}(% \cos\gamma).
  7. P ( cos γ ) = 4 π 2 + 1 m = - ( - 1 ) m Y - m ( θ , φ ) Y m ( θ , φ ) P_{\ell}(\cos\gamma)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}(-1)^{m}Y^{-m}_{% \ell}(\theta,\varphi)Y^{m}_{\ell}(\theta^{\prime},\varphi^{\prime})

Laplace_formula.html

  1. r t r_{t}
  2. r b r_{b}
  3. Δ P = α ( 1 r t + 1 r b ) \Delta P=\alpha\left(\frac{1}{r_{t}}+\frac{1}{r_{b}}\right)
  4. α \alpha

Laplacian_smoothing.html

  1. x ¯ i = 1 N j = 1 N x ¯ j \bar{x}_{i}=\frac{1}{N}\sum_{j=1}^{N}\bar{x}_{j}
  2. N N
  3. i i
  4. x ¯ j \bar{x}_{j}
  5. j j
  6. x ¯ i \bar{x}_{i}
  7. i i

Large_deviations_of_Gaussian_random_functions.html

  1. M M
  2. X X
  3. X X
  4. 0
  5. X X
  6. 1 1
  7. a > 0 a>0
  8. P ( M > a ) P(M>a)
  9. C a exp ( - a 2 / 2 ) + 2 P ( ξ > a ) Ca\exp(-a^{2}/2)+2P(\xi>a)
  10. ξ \xi
  11. N ( 0 , 1 ) N(0,1)
  12. C C
  13. a a
  14. X X
  15. a a
  16. C C
  17. X X
  18. ( π / 2 ) 1 / 4 C 1 / 2 (\pi/2)^{1/4}C^{1/2}
  19. 1 1
  20. 2 2
  21. P ( ξ > a ) P(\xi>a)
  22. X X
  23. χ a \chi_{a}
  24. { X > a } \{X>a\}
  25. t t
  26. X ( t ) > a X(t)>a
  27. E ( χ a ) E(\chi_{a})
  28. E ( χ a ) = C a exp ( - a 2 / 2 ) + 2 P ( ξ > a ) E(\chi_{a})=Ca\exp(-a^{2}/2)+2P(\xi>a)
  29. { X > a } \{X>a\}
  30. M < a M<a
  31. χ a = 0 \chi_{a}=0
  32. M > a M>a
  33. { X > a } \{X>a\}
  34. a a
  35. M > a M>a
  36. { X > a } \{X>a\}
  37. χ a \chi_{a}
  38. 1 1
  39. M > a M>a
  40. E ( χ a ) E(\chi_{a})
  41. P ( M > a ) P(M>a)

Lateral_earth_pressure.html

  1. K 0 ( N C ) = 1 - sin ϕ K_{0(NC)}=1-\sin\phi^{\prime}
  2. K 0 ( O C ) = K 0 ( N C ) * O C R ( sin ϕ ) K_{0(OC)}=K_{0(NC)}*OCR^{(\sin\phi^{\prime})}
  3. ϕ \phi^{\prime}
  4. K a = cos β cos β - ( cos 2 β - cos 2 ϕ ) 1 / 2 cos β + ( cos 2 β - cos 2 ϕ ) 1 / 2 K_{a}=\cos\beta\frac{\cos\beta-\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}{% \cos\beta+\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}
  5. K p = cos β cos β + ( cos 2 β - cos 2 ϕ ) 1 / 2 cos β - ( cos 2 β - cos 2 ϕ ) 1 / 2 K_{p}=\cos\beta\frac{\cos\beta+\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}{% \cos\beta-\left(\cos^{2}\beta-\cos^{2}\phi\right)^{1/2}}
  6. K a = tan 2 ( 45 - ϕ 2 ) = 1 - sin ( ϕ ) 1 + sin ( ϕ ) K_{a}=\tan^{2}\left(45-\frac{\phi}{2}\right)=\frac{1-\sin(\phi)}{1+\sin(\phi)}
  7. K p = tan 2 ( 45 + ϕ 2 ) = 1 + sin ( ϕ ) 1 - sin ( ϕ ) K_{p}=\tan^{2}\left(45+\frac{\phi}{2}\right)=\frac{1+\sin(\phi)}{1-\sin(\phi)}
  8. K a = cos 2 ( ϕ - θ ) cos 2 θ cos ( δ + θ ) ( 1 + sin ( δ + ϕ ) sin ( ϕ - β ) cos ( δ + θ ) cos ( β - θ ) ) 2 K_{a}=\frac{\cos^{2}\left(\phi-\theta\right)}{\cos^{2}\theta\cos\left(\delta+% \theta\right)\left(1+\sqrt{\frac{\sin\left(\delta+\phi\right)\sin\left(\phi-% \beta\right)}{\cos\left(\delta+\theta\right)\cos\left(\beta-\theta\right)}}\ % \right)^{2}}
  9. K p = cos 2 ( ϕ + θ ) cos 2 θ cos ( δ - θ ) ( 1 - sin ( δ + ϕ ) sin ( ϕ + β ) cos ( δ - θ ) cos ( β - θ ) ) 2 K_{p}=\frac{\cos^{2}\left(\phi+\theta\right)}{\cos^{2}\theta\cos\left(\delta-% \theta\right)\left(1-\sqrt{\frac{\sin\left(\delta+\phi\right)\sin\left(\phi+% \beta\right)}{\cos\left(\delta-\theta\right)\cos\left(\beta-\theta\right)}}\ % \right)^{2}}
  10. σ h = K a σ v - 2 c K a \sigma_{h}=K_{a}\sigma_{v}-2c\sqrt{K_{a}}
  11. σ h = K p σ v + 2 c K p \sigma_{h}=K_{p}\sigma_{v}+2c\sqrt{K_{p}}

Lax–Wendroff_method.html

  1. u ( x , t ) t + f ( u ( x , t ) ) x = 0 \frac{\partial u(x,t)}{\partial t}+\frac{\partial f(u(x,t))}{\partial x}=0\,
  2. u i n + 1 = u i n - Δ t 2 Δ x A [ u i + 1 n - u i - 1 n ] + Δ t 2 2 Δ x 2 A 2 [ u i + 1 n - 2 u i n + u i - 1 n ] . u_{i}^{n+1}=u_{i}^{n}-\frac{\Delta t}{2\Delta x}A\left[u_{i+1}^{n}-u_{i-1}^{n}% \right]+\frac{\Delta t^{2}}{2\Delta x^{2}}A^{2}\left[u_{i+1}^{n}-2u_{i}^{n}+u_% {i-1}^{n}\right].
  3. A ( u ) = f ( u ) = f u A(u)=f^{\prime}(u)=\frac{\partial f}{\partial u}
  4. u i n + 1 = u i n - Δ t 2 Δ x [ f ( u i + 1 n ) - f ( u i - 1 n ) ] + Δ t 2 2 Δ x 2 [ A i + 1 / 2 ( f ( u i + 1 n ) - f ( u i n ) ) - A i - 1 / 2 ( f ( u i n ) - f ( u i - 1 n ) ) ] . u_{i}^{n+1}=u_{i}^{n}-\frac{\Delta t}{2\Delta x}\left[f(u_{i+1}^{n})-f(u_{i-1}% ^{n})\right]+\frac{\Delta t^{2}}{2\Delta x^{2}}\left[A_{i+1/2}\left(f(u_{i+1}^% {n})-f(u_{i}^{n})\right)-A_{i-1/2}\left(f(u_{i}^{n})-f(u_{i-1}^{n})\right)% \right].
  5. A i ± 1 / 2 A_{i\pm 1/2}
  6. 1 2 ( U i n + U i ± 1 / 2 n ) \frac{1}{2}(U^{n}_{i}+U^{n}_{i\pm 1/2})
  7. u i + 1 / 2 n + 1 / 2 = 1 2 ( u i + 1 n + u i n ) - Δ t 2 Δ x ( f ( u i + 1 n ) - f ( u i n ) ) , u_{i+1/2}^{n+1/2}=\frac{1}{2}(u_{i+1}^{n}+u_{i}^{n})-\frac{\Delta t}{2\,\Delta x% }(f(u_{i+1}^{n})-f(u_{i}^{n})),
  8. u i - 1 / 2 n + 1 / 2 = 1 2 ( u i n + u i - 1 n ) - Δ t 2 Δ x ( f ( u i n ) - f ( u i - 1 n ) ) . u_{i-1/2}^{n+1/2}=\frac{1}{2}(u_{i}^{n}+u_{i-1}^{n})-\frac{\Delta t}{2\,\Delta x% }(f(u_{i}^{n})-f(u_{i-1}^{n})).
  9. u i n + 1 = u i n - Δ t Δ x [ f ( u i + 1 / 2 n + 1 / 2 ) - f ( u i - 1 / 2 n + 1 / 2 ) ] . u_{i}^{n+1}=u_{i}^{n}-\frac{\Delta t}{\Delta x}\left[f(u_{i+1/2}^{n+1/2})-f(u_% {i-1/2}^{n+1/2})\right].
  10. u i * = u i n - Δ t Δ x ( f ( u i + 1 n ) - f ( u i n ) ) . u_{i}^{*}=u_{i}^{n}-\frac{\Delta t}{\Delta x}(f(u_{i+1}^{n})-f(u_{i}^{n})).
  11. u i n + 1 = 1 2 ( u i n + u i * ) - Δ t 2 Δ x [ f ( u i * ) - f ( u i - 1 * ) ] . u_{i}^{n+1}=\frac{1}{2}(u_{i}^{n}+u_{i}^{*})-\frac{\Delta t}{2\Delta x}\left[f% (u_{i}^{*})-f(u_{i-1}^{*})\right].
  12. u i * = u i n - Δ t Δ x ( f ( u i n ) - f ( u i - 1 n ) ) . u_{i}^{*}=u_{i}^{n}-\frac{\Delta t}{\Delta x}(f(u_{i}^{n})-f(u_{i-1}^{n})).
  13. u i n + 1 = 1 2 ( u i n + u i * ) - Δ t 2 Δ x [ f ( u i + 1 * ) - f ( u i * ) ] . u_{i}^{n+1}=\frac{1}{2}(u_{i}^{n}+u_{i}^{*})-\frac{\Delta t}{2\Delta x}\left[f% (u_{i+1}^{*})-f(u_{i}^{*})\right].

Layer_cake_representation.html

  1. f ( x ) = 0 + 1 L ( f , t ) ( x ) d t for all x n , f(x)=\int_{0}^{+\infty}1_{L(f,t)}(x)\,\mathrm{d}t\,\text{ for all }x\in\mathbb% {R}^{n},
  2. L ( f , t ) = { y n | f ( y ) t } . L(f,t)=\{y\in\mathbb{R}^{n}|f(y)\geq t\}.
  3. 1 L ( f , t ) ( x ) = 1 [ 0 , f ( x ) ] ( t ) 1_{L(f,t)}(x)=1_{[0,f(x)]}(t)
  4. f ( x ) = 0 f ( x ) d t . f(x)=\int_{0}^{f(x)}\mathrm{d}t.

Layered_hidden_Markov_model.html

  1. N N
  2. N + 1 N+1
  3. N N
  4. i i
  5. K i K_{i}
  6. L L
  7. T L T_{L}
  8. 𝐨 L = { o 1 , o 2 , , o T L } \mathbf{o}_{L}=\{o_{1},o_{2},\dots,o_{T_{L}}\}
  9. K L K_{L}
  10. K L K_{L}
  11. L L
  12. L - 1 L-1
  13. N N
  14. 𝐨 p = { o 1 , o 2 , , o T p } \mathbf{o}_{p}=\{o_{1},o_{2},\dots,o_{T_{p}}\}
  15. L + 1 L+1
  16. L L
  17. L ( i ) L(i)
  18. i i
  19. L L
  20. L + 1 L+1
  21. n + 1 n+1
  22. L L

LBOZ.html

  1. 𝐗 \mathbf{X}
  2. ξ k = 1 𝐗 k - r o w 𝐗 k - c o l + \xi_{k}=\frac{1}{\|\mathbf{X}_{k-row}\|\|\mathbf{X}^{+}_{k-col}\|}
  3. 𝐗 + \mathbf{X}^{+}
  4. \|\cdots\|
  5. 1 / ξ 1/\xi

Least_distance_of_distinct_vision.html

  1. M = 250 f . {M}=\frac{250}{f}.

Least_squares_inference_in_phylogeny.html

  1. D i j D_{ij}
  2. T i j T_{ij}
  3. i i
  4. j j
  5. S = i j w i j ( D i j - T i j ) 2 S=\sum_{ij}w_{ij}(D_{ij}-T_{ij})^{2}
  6. w i j w_{ij}
  7. ( D i j - T i j ) 2 (D_{ij}-T_{ij})^{2}
  8. w i j w_{ij}
  9. i j , k l w i j , k l ( D i j - T i j ) ( D k l - T k l ) \sum_{ij,kl}w_{ij,kl}(D_{ij}-T_{ij})(D_{kl}-T_{kl})
  10. w i j , k l w_{ij,kl}
  11. O ( n 2 ) O(n^{2})
  12. O ( n 3 ) O(n^{3})
  13. O ( n 4 ) O(n^{4})

Ledinegg_instability.html

  1. J 2 / ρ J^{2}/\rho

Legendre_wavelet.html

  1. P n ( z ) P_{n}(z)
  2. 2 n d 2^{nd}
  3. ( 1 - z 2 ) d 2 y d z 2 - 2 z d y d z + n ( n + 1 ) y = 0. (1-z^{2})\frac{d^{2}y}{dz^{2}}-2z\frac{dy}{dz}+n(n+1)y=0.
  4. P n ( cos θ ) P_{n}(\cos{\theta})
  5. H ( ω ) H(\omega)
  6. | H ( 0 ) | = 1 |H(0)|=1
  7. | H ( π ) | = 0 |H(\pi)|=0
  8. | H ( ω ) | |H(\omega)|
  9. ν = 2 n + 1 \nu=2n+1
  10. | H ν ( ω ) | = | P ν ( cos ω 2 ) P ν cos ( 0 ) | |H_{\nu}(\omega)|=|\frac{P_{\nu}(\cos{\frac{\omega}{2})}}{P_{\nu}\cos(0)}|
  11. ν \nu
  12. - π < ω < π -\pi<\omega<\pi
  13. ν \nu
  14. | H ν ( ω ) | |H_{\nu}(\omega)|
  15. ν \nu
  16. ν \nu
  17. ν \nu
  18. H ν ( ω ) = - e - j ν ω - π 2 P ν ( cos ( ω 2 ) ) H_{\nu}(\omega)=-e^{-j\nu\frac{\omega-\pi}{2}}P_{\nu}(\cos(\frac{\omega}{2}))
  19. G ν ( ω ) G_{\nu}(\omega)
  20. H ν ( ω ) = - e - j ( ν - 2 ) ω 2 P ν ( sin ( ω 2 ) ) H_{\nu}(\omega)=-e^{-j{(\nu-2)}\frac{\omega}{2}}P_{\nu}(\sin(\frac{\omega}{2}))
  21. | G ν ( 0 ) | = 0 |G_{\nu}(0)|=0
  22. | G ν ( π ) | = 1 |G_{\nu}(\pi)|=1
  23. H ν ( ω ) H_{\nu}(\omega)
  24. H ν ( ω ) = 1 2 k Z h k ν e - j ω k H_{\nu}(\omega)=\frac{1}{\sqrt{2}}\sum_{k\in Z}h_{k}^{\nu}e^{-j\omega k}
  25. { h k } \{h_{k}\}
  26. k Z k\in Z
  27. h k ν 2 = - 1 2 2 ν . ( 2 k k ) . ( 2 ν - 2 k ν - k ) \frac{h_{k}^{\nu}}{\sqrt{2}}=-\frac{1}{2^{2\nu}}.{\left({{2k}\atop{k}}\right)}% .{\left({{2\nu-2k}\atop{\nu-k}}\right)}
  28. h k ν = h ν - k ν {h_{k}^{\nu}}={h_{\nu-k}^{\nu}}
  29. ν + 1 \nu+1
  30. H n ( ω ) H_{n}(\omega)
  31. ν \nu
  32. ν \nu
  33. ν = 1 \nu=1
  34. ν = 3 \nu=3
  35. ν = 5 \nu=5
  36. h 0 h_{0}
  37. - 2 / 2 -\sqrt{2}/2
  38. - 5 2 / 16 -5\sqrt{2}/16
  39. - 63 2 / 256 -63\sqrt{2}/256
  40. h 1 h_{1}
  41. - 2 / 2 -\sqrt{2}/2
  42. - 3 2 / 16 -3\sqrt{2}/16
  43. - 35 2 / 256 -35\sqrt{2}/256
  44. h 2 h_{2}
  45. - 3 2 / 16 -3\sqrt{2}/16
  46. - 30 2 / 256 -30\sqrt{2}/256
  47. h 3 h_{3}
  48. - 5 2 / 16 -5\sqrt{2}/16
  49. - 30 2 / 256 -30\sqrt{2}/256
  50. h 4 h_{4}
  51. - 35 2 / 256 -35\sqrt{2}/256
  52. h 5 h_{5}
  53. - 63 2 / 256 -63\sqrt{2}/256
  54. ν \nu

Lehmer's_GCD_algorithm.html

  1. [ A B x C D y ] \textstyle\begin{bmatrix}A&B&x\\ C&D&y\end{bmatrix}
  2. $\textstyle$
  3. [ A B x C D y ] \textstyle\begin{bmatrix}A&B&x\\ C&D&y\end{bmatrix}
  4. [ 0 1 1 - w ] [ A B x C D y ] = [ C D y A - w C B - w D x - w y ] \textstyle\begin{bmatrix}0&1\\ 1&-w\end{bmatrix}\cdot\begin{bmatrix}A&B&x\\ C&D&y\end{bmatrix}=\begin{bmatrix}C&D&y\\ A-wC&B-wD&x-wy\end{bmatrix}

Lehmer_sieve.html

  1. 2 93 + 1 = 3 × 3 × 529510939 × 715827883 × 2903110321 2^{93}+1=3\times 3\times 529510939\times 715827883\times 2903110321

Leibniz_harmonic_triangle.html

  1. L ( r , 1 ) = 1 / r L(r,1)=1/r
  2. r r
  3. c c
  4. L ( r , c ) = L ( r - 1 , c - 1 ) L ( r , c - 1 ) . L(r,c)=L(r-1,c-1)−L(r,c-1).
  5. 1 1 2 1 2 1 3 1 6 1 3 1 4 1 12 1 12 1 4 1 5 1 20 1 30 1 20 1 5 1 6 1 30 1 60 1 60 1 30 1 6 1 7 1 42 1 105 1 140 1 105 1 42 1 7 1 8 1 56 1 168 1 280 1 280 1 168 1 56 1 8 \begin{array}[]{cccccccccccccccccc}&&&&&&&&&1&&&&&&&&\\ &&&&&&&&\frac{1}{2}&&\frac{1}{2}&&&&&&&\\ &&&&&&&\frac{1}{3}&&\frac{1}{6}&&\frac{1}{3}&&&&&&\\ &&&&&&\frac{1}{4}&&\frac{1}{12}&&\frac{1}{12}&&\frac{1}{4}&&&&&\\ &&&&&\frac{1}{5}&&\frac{1}{20}&&\frac{1}{30}&&\frac{1}{20}&&\frac{1}{5}&&&&\\ &&&&\frac{1}{6}&&\frac{1}{30}&&\frac{1}{60}&&\frac{1}{60}&&\frac{1}{30}&&\frac% {1}{6}&&&\\ &&&\frac{1}{7}&&\frac{1}{42}&&\frac{1}{105}&&\frac{1}{140}&&\frac{1}{105}&&% \frac{1}{42}&&\frac{1}{7}&&\\ &&\frac{1}{8}&&\frac{1}{56}&&\frac{1}{168}&&\frac{1}{280}&&\frac{1}{280}&&% \frac{1}{168}&&\frac{1}{56}&&\frac{1}{8}&\\ &&&&&\vdots&&&&\vdots&&&&\vdots&&&&\\ \end{array}
  6. L ( r , c ) = 1 r ( r - 1 c - 1 ) L(r,c)=\frac{1}{r{r-1\choose c-1}}
  7. n 2 n - 1 n2^{n-1}
  8. L ( r , c ) = 0 1 x c - 1 ( 1 - x ) r - c d x . L(r,c)=\int_{0}^{1}\!x^{c-1}(1-x)^{r-c}\,dx\,.

Lemaître–Tolman_metric.html

  1. d s 2 = d t 2 - ( R ) 2 1 + 2 E d r 2 - R 2 d Ω 2 \mathrm{d}s^{2}=\mathrm{d}t^{2}-\frac{(R^{\prime})^{2}}{1+2E}\mathrm{d}r^{2}-R% ^{2}\,\mathrm{d}\Omega^{2}
  2. d Ω 2 = d θ 2 + sin 2 θ d ϕ 2 \mathrm{d}\Omega^{2}=\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi^{2}
  3. R = R ( t , r ) , R = R / r , E = E ( r ) > 1 2 R=R(t,r)~{},~{}~{}~{}~{}~{}~{}~{}~{}R^{\prime}=\partial R/\partial r~{},~{}~{}% ~{}~{}~{}~{}~{}~{}E=E(r)>\frac{1}{2}
  4. u a = δ 0 a = ( 1 , 0 , 0 , 0 ) u^{a}=\delta^{a}_{0}=(1,0,0,0)
  5. ( r , θ , ϕ ) (r,\theta,\phi)
  6. 8 π ρ = 2 M R 2 R 8\pi\rho=\frac{2M^{\prime}}{R^{2}\,R^{\prime}}
  7. R ˙ 2 = 2 M R + 2 E \dot{R}^{2}=\frac{2M}{R}+2E
  8. R ˙ = R / t \dot{R}=\partial R/\partial t
  9. E E
  10. E > 0 : R = M 2 E ( cosh η - 1 ) , ( sinh η - η ) = ( 2 E ) 3 / 2 ( t - t B ) M ; E>0:~{}~{}~{}~{}~{}~{}~{}~{}R=\frac{M}{2E}(\cosh\eta-1)~{},~{}~{}~{}~{}~{}~{}~% {}~{}(\sinh\eta-\eta)=\frac{(2E)^{3/2}(t-t_{B})}{M}~{};
  11. E = 0 : R = ( 9 M ( t - t B ) 2 2 ) 1 / 3 ; E=0:~{}~{}~{}~{}~{}~{}~{}~{}R=\left(\frac{9M(t-t_{B})^{2}}{2}\right)^{1/3}~{};
  12. E < 0 : R = M 2 E ( 1 - cos η ) , ( η - sin η ) = ( - 2 E ) 3 / 2 ( t - t B ) M ; E<0:~{}~{}~{}~{}~{}~{}~{}~{}R=\frac{M}{2E}(1-\cos\eta)~{},~{}~{}~{}~{}~{}~{}~{% }~{}(\eta-\sin\eta)=\frac{(-2E)^{3/2}(t-t_{B})}{M}~{};
  13. r r
  14. E ( r ) E(r)
  15. r r
  16. M ( r ) M(r)
  17. r r
  18. t B ( r ) t_{B}(r)
  19. r r
  20. M = M=
  21. E = 0 , t B = E=0~{},~{}~{}t_{B}=

Lens_clock.html

  1. ϕ \phi
  2. ϕ = 2 ( n - 1 ) s ( D / 2 ) 2 , \phi={2(n-1)s\over(D/2)^{2}},
  3. n n
  4. s s
  5. D D
  6. ϕ \phi
  7. s s
  8. D D
  9. R R
  10. R = ( n - 1 ) ϕ , R={(n-1)\over\phi},
  11. n n
  12. n 2 n_{2}
  13. ϕ = ( n 2 - 1 ) R . \phi={(n_{2}-1)\over R}.
  14. R 1 = ( 1.523 - 1 ) - 3.0 dpt = - 0.174 m R_{1}={(1.523-1)\over-3.0\ \mathrm{dpt}}=-0.174\ \mathrm{m}
  15. R 2 = ( 1.523 - 1 ) - 7.0 dpt = - 0.0747 m R_{2}={(1.523-1)\over-7.0\ \mathrm{dpt}}=-0.0747\ \mathrm{m}
  16. ϕ 1 = ( 1.7 - 1 ) - 0.174 m = - 4.02 dpt \phi_{1}={(1.7-1)\over-0.174\ \mathrm{m}}=-4.02\ \mathrm{dpt}
  17. ϕ 2 = ( 1.7 - 1 ) - 0.0747 m = - 9.37 dpt \phi_{2}={(1.7-1)\over-0.0747\ \mathrm{m}}=-9.37\ \mathrm{dpt}
  18. s s

Lense–Thirring_precession.html

  1. S S
  2. s y m b o l B = 3 5 R 2 q ( s y m b o l ω \cdotsymbol r s y m b o l r r 5 - 1 3 s y m b o l ω r 3 ) . symbol{B}=\frac{3}{5}R^{2}q\Big(symbol{\omega}\cdotsymbol{r}\frac{symbol{r}}{r% ^{5}}-\frac{1}{3}\frac{symbol{\omega}}{r^{3}}\Big).
  3. s y m b o l ω = - 4 \rhosymbol u d V r . symbol{\omega}=-4\int\frac{\rhosymbol{u}\,dV}{r}.
  4. s y m b o l B = 12 5 R 2 q ( s y m b o l ω \cdotsymbol r s y m b o l r r 5 - 1 3 s y m b o l ω r 3 ) . symbol{B}=\frac{12}{5}R^{2}q\Big(symbol{\omega}\cdotsymbol{r}\frac{symbol{r}}{% r^{5}}-\frac{1}{3}\frac{symbol{\omega}}{r^{3}}\Big).
  5. r r
  6. R R
  7. θ \theta
  8. s y m b o l B = - ( 1 3 s y m b o l ω r 3 cos θ ) . symbol{B}=-\left(\frac{1}{3}\frac{symbol{\omega}}{r^{3}}\cos\theta\right).
  9. s y m b o l B = - 4 5 s y m b o l ω m R 2 r 3 cos θ . symbol{B}=-\frac{4}{5}\frac{symbol{\omega}mR^{2}}{r^{3}}\cos\theta.
  10. s y m b o l Ω LIF symbol{\Omega}_{\,\text{LIF}}
  11. Ω LT = - 2 5 G m ω c 2 R cos θ . \Omega\text{LT}=-\frac{2}{5}\frac{Gm\omega}{c^{2}R}\cos\theta.
  12. Ω LT = - 2.2 10 - 4 arcseconds / day . \Omega\text{LT}=-2.2\cdot 10^{-4}\,\text{ arcseconds}/\,\text{day}.
  13. Ω rel = 3 π G m c 2 r . \Omega\text{rel}=\frac{3\pi Gm}{c^{2}r}.
  14. d Ω d t = 2 G S c 2 a 3 ( 1 - e 2 ) 3 / 2 = 2 G 2 M 2 χ c 3 a 3 ( 1 - e 2 ) 3 / 2 \frac{d\Omega}{dt}=\frac{2GS}{c^{2}a^{3}(1-e^{2})^{3/2}}=\frac{2G^{2}M^{2}\chi% }{c^{3}a^{3}(1-e^{2})^{3/2}}
  15. d s y m b o l S d t = 2 G c 2 j s y m b o l L j \timessymbol S a j 3 ( 1 - e j 2 ) 3 / 2 \frac{dsymbol{S}}{dt}=\frac{2G}{c^{2}}\sum_{j}\frac{symbol{L}_{j}\timessymbol{% S}}{a_{j}^{3}(1-e_{j}^{2})^{3/2}}

Leray–Hirsch_theorem.html

  1. π : E B \pi:E\longrightarrow B
  2. p p
  3. H p ( F ) = H p ( F ; ) H^{p}(F)=H^{p}(F;\mathbb{Q})
  4. ι : F E \iota:F\longrightarrow E
  5. ι * : H * ( E ) H * ( F ) \iota^{*}:H^{*}(E)\longrightarrow H^{*}(F)
  6. s : H * ( F ) H * ( E ) s:H^{*}(F)\longrightarrow H^{*}(E)
  7. ι * s = Id \iota^{*}\circ s=\mathrm{Id}
  8. H * ( F ) H * ( B ) H * ( E ) α β s ( α ) π * ( β ) \begin{array}[]{ccc}H^{*}(F)\otimes H^{*}(B)&\longrightarrow&H^{*}(E)\\ \alpha\otimes\beta&\longmapsto&s(\alpha)\cup\pi^{*}(\beta)\end{array}
  9. p p
  10. c 1 , p , , c m p , p H p ( E ) c_{1,p},\ldots,c_{m_{p},p}\in H^{p}(E)
  11. p p
  12. H * ( B ) H^{*}(B)
  13. H * ( F ) H * ( B ) H * ( E ) i , j , k a i , j , k ι * ( c i , j ) b k i , j , k a i , j , k c i , j π * ( b k ) \begin{array}[]{ccc}H^{*}(F)\otimes H^{*}(B)&\longrightarrow&H^{*}(E)\\ \sum_{i,j,k}a_{i,j,k}\iota^{*}(c_{i,j})\otimes b_{k}&\longmapsto&\sum_{i,j,k}a% _{i,j,k}c_{i,j}\wedge\pi^{*}(b_{k})\end{array}
  14. { b k } \{b_{k}\}
  15. H * ( B ) H^{*}(B)
  16. { ι * ( c i , j ) b k } \{\iota^{*}(c_{i,j})\otimes b_{k}\}
  17. H * ( F ) H * ( B ) . H^{*}(F)\otimes H^{*}(B).

Lever_rule.html

  1. X α = c - b a - b X_{\alpha}=\frac{c-b}{a-b}
  2. W α W_{\alpha}
  3. W β = W T O T - W α W_{\beta}=W_{TOT}-W_{\alpha}
  4. W T O T W_{TOT}
  5. W α , B = a W α W_{\alpha,B}=aW_{\alpha}
  6. W β , B = b ( W T O T - W α ) W_{\beta,B}=b\left(W_{TOT}-W_{\alpha}\right)
  7. W B = c W T O T W_{B}=cW_{TOT}
  8. c W T O T = W B = W α , B + W β , B = a W α + b ( W T O T - W α ) cW_{TOT}=W_{B}=W_{\alpha,B}+W_{\beta,B}=aW_{\alpha}+b\left(W_{TOT}-W_{\alpha}\right)
  9. W α W T O T = c - b a - b \frac{W_{\alpha}}{W_{TOT}}=\frac{c-b}{a-b}
  10. Percent weight of the solid phase = X s = w o - w l w s - w l \,\text{Percent weight of the solid phase}=X_{s}=\frac{w_{o}-w_{l}}{w_{s}-w_{l}}
  11. Percent weight of the liquid phase = X l = w s - w o w s - w l \,\text{Percent weight of the liquid phase}=X_{l}=\frac{w_{s}-w_{o}}{w_{s}-w_{% l}}

Leverett_J-function.html

  1. J ( S w ) = p c ( S w ) k / ϕ γ cos θ J(S_{w})=\frac{p_{c}(S_{w})\sqrt{k/\phi}}{\gamma\cos\theta}
  2. S w S_{w}
  3. p c p_{c}
  4. k k
  5. ϕ \phi
  6. γ \gamma
  7. θ \theta
  8. k / ϕ \sqrt{k/\phi}

Levi's_lemma.html

  1. f - 1 ( 0 ) = { 1 M } f^{-1}(0)=\{1_{M}\}

Levy–Mises_equations.html

  1. 𝐝 ε 1 σ 1 = 𝐝 ε 2 σ 2 = 𝐝 ε 3 σ 3 = 𝐝 λ \frac{\mathbf{d}\varepsilon_{1}}{\sigma^{\prime}_{1}}=\frac{\mathbf{d}% \varepsilon_{2}}{\sigma^{\prime}_{2}}=\frac{\mathbf{d}\varepsilon_{3}}{\sigma^% {\prime}_{3}}=\mathbf{d}\lambda

Lewin's_equation.html

  1. B = f ( P , E ) B=f(P,E)
  2. B B
  3. P P
  4. E E

Lexis_ratio.html

  1. L 2 = Q 2 = s 2 σ 0 2 . L^{2}=Q^{2}=\frac{s^{2}}{\sigma_{0}^{2}}.
  2. s 2 s^{2}\,
  3. σ 0 2 \sigma_{0}^{2}

Lie_algebra_bundle.html

  1. ξ = ( ξ , p , X , θ ) \xi=(\xi,p,X,\theta)\,
  2. ξ \xi\,
  3. θ : ξ ξ ξ \theta:\xi\otimes\xi\rightarrow\xi
  4. ξ x \xi_{x}\,
  5. ξ = ( ξ , p , X ) \xi=(\xi,p,X)\,
  6. U U
  7. ϕ : U × L p - 1 ( U ) \phi:U\times L\to p^{-1}(U)\,
  8. ϕ x : x × L p - 1 ( x ) \phi_{x}:x\times L\rightarrow p^{-1}(x)\,
  9. 𝔰 𝔬 ( 3 ) × \mathfrak{so}(3)\times\mathbb{R}
  10. \mathbb{R}
  11. 𝔰 𝔬 ( 3 ) \mathfrak{so}(3)
  12. [ X , Y ] x = x [ X , Y ] [X,Y]_{x}=x\cdot[X,Y]
  13. X , Y 𝔰 𝔬 ( 3 ) X,Y\in\mathfrak{so}(3)
  14. x x\in\mathbb{R}

Lie_bracket_of_vector_fields.html

  1. X Y \mathcal{L}_{X}Y
  2. δ 1 δ 2 - δ 2 δ 1 \delta_{1}\circ\delta_{2}-\delta_{2}\circ\delta_{1}
  3. δ 1 \delta_{1}
  4. δ 2 \delta_{2}
  5. [ X , Y ] ( f ) = X ( Y ( f ) ) - Y ( X ( f ) ) for all f C ( M ) . [X,Y](f)=X(Y(f))-Y(X(f))\;\;\,\text{ for all }f\in C^{\infty}(M).
  6. Φ t X \Phi^{X}_{t}
  7. [ X , Y ] x := lim t 0 ( d Φ - t X ) Y Φ t X ( x ) - Y x t = d d t | t = 0 ( d Φ - t X ) Y Φ t X ( x ) [X,Y]_{x}:=\lim_{t\to 0}\frac{(\mathrm{d}\Phi^{X}_{-t})Y_{\Phi^{X}_{t}(x)}-Y_{% x}}{t}=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}(\mathrm{d}\Phi^{X}_{-% t})Y_{\Phi^{X}_{t}(x)}
  8. [ X , Y ] = X Y [X,Y]=\mathcal{L}_{X}Y
  9. [ X , Y ] x := 1 2 d 2 dt 2 | t = 0 ( Φ - t Y Φ - t X Φ t Y Φ t X ) ( x ) = d d t | t = 0 ( Φ - t Y Φ - t X Φ t Y Φ t X ) ( x ) [X,Y]_{x}:=\left.\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{dt}^{2}}\right|_{t=0% }(\Phi^{Y}_{-t}\circ\Phi^{X}_{-t}\circ\Phi^{Y}_{t}\circ\Phi^{X}_{t})(x)=\left.% \frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}(\Phi^{Y}_{-\sqrt{t}}\circ\Phi^{X}_% {-\sqrt{t}}\circ\Phi^{Y}_{\sqrt{t}}\circ\Phi^{X}_{\sqrt{t}})(x)
  10. { x i } \{x^{i}\}
  11. i = x i \partial_{i}=\frac{\partial}{\partial x^{i}}
  12. X = i = 1 n X i i X=\sum_{i=1}^{n}X^{i}\partial_{i}
  13. Y = i = 1 n Y i i Y=\sum_{i=1}^{n}Y^{i}\partial_{i}
  14. X i : M X^{i}:M\to\mathbb{R}
  15. Y i : M Y^{i}:M\to\mathbb{R}
  16. [ X , Y ] := i = 1 n ( X ( Y i ) - Y ( X i ) ) i = i = 1 n j = 1 n ( X j j Y i - Y j j X i ) i [X,Y]:=\sum_{i=1}^{n}\left(X(Y^{i})-Y(X^{i})\right)\partial_{i}=\sum_{i=1}^{n}% \sum_{j=1}^{n}\left(X^{j}\partial_{j}Y^{i}-Y^{j}\partial_{j}X^{i}\right)% \partial_{i}
  17. X : M n X:M\to\mathbb{R}^{n}
  18. Y : M n Y:M\to\mathbb{R}^{n}
  19. [ X , Y ] : M n [X,Y]:M\to\mathbb{R}^{n}
  20. [ X , Y ] := J Y X - J X Y [X,Y]:=J_{Y}X-J_{X}Y
  21. J Y J_{Y}
  22. J X J_{X}
  23. Y Y
  24. X X
  25. V = Γ ( T M ) V=\Gamma(TM)
  26. T M TM
  27. M M
  28. V × V V\times V
  29. V V
  30. [ , ] [\cdot,\cdot]
  31. [ X , Y ] = - [ Y , X ] [X,Y]=-[Y,X]\,
  32. [ X , [ Y , Z ] ] + [ Z , [ X , Y ] ] + [ Y , [ Z , X ] ] = 0. [X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0.\,
  33. [ X , X ] = 0 [X,X]=0
  34. X X
  35. [ X , f Y ] = X ( f ) Y + f [ X , Y ] [X,fY]=X(f)Y+f[X,Y]
  36. [ X , Y ] = 0 [X,Y]=0\,
  37. ( Φ t Y Φ s X ) ( x ) = ( Φ s X Φ t Y ) ( x ) (\Phi^{Y}_{t}\Phi^{X}_{s})(x)=(\Phi^{X}_{s}\,\Phi^{Y}_{t})(x)
  38. [ X , Y ] = X Y - Y X [X,Y]=XY-YX

Liénard–Wiechert_potential.html

  1. 𝐫 \mathbf{r}
  2. t t
  3. 𝐫 \mathbf{r}
  4. t t
  5. t r = t - R ( t r ) c t_{r}=t-\frac{R(t_{r})}{c}
  6. R ( t r ) R(t_{r})
  7. v M < c v_{M}<c
  8. t r t_{r}
  9. f ( t ) = | 𝐫 - 𝐫 s ( t ) | - c ( t - t ) f(t^{\prime})=|\mathbf{r}-\mathbf{r}_{s}(t^{\prime})|-c(t-t^{\prime})
  10. t = t t^{\prime}=t
  11. f ( t ) = | 𝐫 - 𝐫 s ( t ) | - c ( t - t ) = | 𝐫 - 𝐫 s ( t ) | 0 f(t^{\prime})=|\mathbf{r}-\mathbf{r}_{s}(t^{\prime})|-c(t-t^{\prime})=|\mathbf% {r}-\mathbf{r}_{s}(t^{\prime})|\geq 0
  12. f ( t ) f^{\prime}(t^{\prime})
  13. f ( t ) = 𝐫 - 𝐫 s ( t r ) | 𝐫 - 𝐫 s ( t r ) | ( - 𝐯 s ( t ) ) + c c - | 𝐫 - 𝐫 s ( t r ) | 𝐫 - 𝐫 s ( t r ) | | | 𝐯 s ( t ) | = c - | 𝐯 s ( t ) | c - v M > 0 f^{\prime}(t^{\prime})=\frac{\mathbf{r}-\mathbf{r}_{s}(t_{r})}{|\mathbf{r}-% \mathbf{r}_{s}(t_{r})|}\cdot(-\mathbf{v}_{s}(t^{\prime}))+c\geq c-\left|\frac{% \mathbf{r}-\mathbf{r}_{s}(t_{r})}{|\mathbf{r}-\mathbf{r}_{s}(t_{r})|}\right|\,% |\mathbf{v}_{s}(t^{\prime})|=c-|\mathbf{v}_{s}(t^{\prime})|\geq c-v_{M}>0
  14. f ( t - Δ t ) f ( t ) - f ( t ) Δ t f ( t ) - ( c - v M ) Δ t f(t-\Delta t)\leq f(t)-f^{\prime}(t)\Delta t\leq f(t)-(c-v_{M})\Delta t
  15. Δ t \Delta t
  16. f ( t ) < 0 f(t^{\prime})<0
  17. t r t_{r}
  18. f ( t r ) = 0 f(t_{r})=0
  19. 𝐫 \mathbf{r}
  20. c c
  21. v < c v<c
  22. 𝐫 \mathbf{r}
  23. ( 𝐫 , t ) (\mathbf{r},t)
  24. 𝐫 s ( t ) \mathbf{r}_{s}(t^{\prime})
  25. t r t_{r}
  26. t r t_{r}
  27. | 𝐫 - 𝐫 s ( t r ) | = c ( t - t r ) |\mathbf{r}-\mathbf{r}_{s}(t_{r})|=c(t-t_{r})
  28. t 1 t_{1}
  29. t 2 t_{2}
  30. t 1 t 2 t_{1}\leq t_{2}
  31. | 𝐫 - 𝐫 s ( t 1 ) | = c ( t - t 1 ) |\mathbf{r}-\mathbf{r}_{s}(t_{1})|=c(t-t_{1})
  32. | 𝐫 - 𝐫 s ( t 2 ) | = c ( t - t 2 ) |\mathbf{r}-\mathbf{r}_{s}(t_{2})|=c(t-t_{2})
  33. c ( t 2 - t 1 ) = | 𝐫 - 𝐫 s ( t 1 ) | - | 𝐫 - 𝐫 s ( t 2 ) | | 𝐫 s ( t 2 ) - 𝐫 s ( t 1 ) | c(t_{2}-t_{1})=|\mathbf{r}-\mathbf{r}_{s}(t_{1})|-|\mathbf{r}-\mathbf{r}_{s}(t% _{2})|\leq|\mathbf{r}_{s}(t_{2})-\mathbf{r}_{s}(t_{1})|
  34. t 2 = t 1 t_{2}=t_{1}
  35. t 1 t_{1}
  36. t 2 t_{2}
  37. | 𝐫 s ( t 2 ) - 𝐫 s ( t 1 ) | / ( t 2 - t 1 ) c |\mathbf{r}_{s}(t_{2})-\mathbf{r}_{s}(t_{1})|/(t_{2}-t_{1})\geq c
  38. 𝐫 \mathbf{r}
  39. φ \varphi
  40. 𝐀 \mathbf{A}
  41. q q
  42. 𝐫 s \mathbf{r}_{s}
  43. 𝐯 s \mathbf{v}_{s}
  44. φ ( 𝐫 , t ) = 1 4 π ϵ 0 ( q ( 1 - 𝐧 s y m b o l β s ) | 𝐫 - 𝐫 s | ) t r \varphi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\left(\frac{q}{(1-\mathbf{n}% \cdot symbol{\beta}_{s})|\mathbf{r}-\mathbf{r}_{s}|}\right)_{t_{r}}
  45. 𝐀 ( 𝐫 , t ) = μ 0 c 4 π ( q s y m b o l β s ( 1 - 𝐧 s y m b o l β s ) | 𝐫 - 𝐫 s | ) t r = s y m b o l β s ( t r ) c φ ( 𝐫 , t ) \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}c}{4\pi}\left(\frac{qsymbol{\beta}_{s}}{% (1-\mathbf{n}\cdot symbol{\beta}_{s})|\mathbf{r}-\mathbf{r}_{s}|}\right)_{t_{r% }}=\frac{symbol{\beta}_{s}(t_{r})}{c}\varphi(\mathbf{r},t)
  46. s y m b o l β s ( t ) = 𝐯 s ( t ) c symbol{\beta}_{s}(t)=\frac{\mathbf{v}_{s}(t)}{c}
  47. 𝐧 = 𝐫 - 𝐫 s | 𝐫 - 𝐫 s | \mathbf{n}=\frac{\mathbf{r}-\mathbf{r}_{s}}{|\mathbf{r}-\mathbf{r}_{s}|}
  48. 𝐄 = - φ - 𝐀 t \mathbf{E}=-\nabla\varphi-\dfrac{\partial\mathbf{A}}{\partial t}
  49. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  50. 𝐄 ( 𝐫 , t ) = 1 4 π ϵ 0 ( q ( 𝐧 - s y m b o l β ) γ 2 ( 1 - 𝐧 s y m b o l β ) 3 | 𝐫 - 𝐫 s | 2 + q 𝐧 × ( ( 𝐧 - s y m b o l β ) × s y m b o l β ˙ ) c ( 1 - 𝐧 s y m b o l β ) 3 | 𝐫 - 𝐫 s | ) t r \mathbf{E}(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\left(\frac{q(\mathbf{n}-% symbol{\beta})}{\gamma^{2}(1-\mathbf{n}\cdot symbol{\beta})^{3}|\mathbf{r}-% \mathbf{r}_{s}|^{2}}+\frac{q\mathbf{n}\times\big((\mathbf{n}-symbol{\beta})% \times\dot{symbol{\beta}}\big)}{c(1-\mathbf{n}\cdot symbol{\beta})^{3}|\mathbf% {r}-\mathbf{r}_{s}|}\right)_{t_{r}}
  51. 𝐁 ( 𝐫 , t ) = μ 0 4 π ( q c ( s y m b o l β × 𝐧 ) γ 2 ( 1 - 𝐧 s y m b o l β ) 3 | 𝐫 - 𝐫 s | 2 + q 𝐧 × ( 𝐧 × ( ( 𝐧 - s y m b o l β ) × s y m b o l β ˙ ) ) ( 1 - 𝐧 s y m b o l β ) 3 | 𝐫 - 𝐫 s | ) t r = 𝐧 ( t r ) c × 𝐄 ( 𝐫 , t ) \mathbf{B}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\left(\frac{qc(symbol{\beta}% \times\mathbf{n})}{\gamma^{2}(1-\mathbf{n}\cdot symbol{\beta})^{3}|\mathbf{r}-% \mathbf{r}_{s}|^{2}}+\frac{q\mathbf{n}\times\Big(\mathbf{n}\times\big((\mathbf% {n}-symbol{\beta})\times\dot{symbol{\beta}}\big)\Big)}{(1-\mathbf{n}\cdot symbol% {\beta})^{3}|\mathbf{r}-\mathbf{r}_{s}|}\right)_{t_{r}}=\frac{\mathbf{n}(t_{r}% )}{c}\times\mathbf{E}(\mathbf{r},t)
  52. s y m b o l β ( t ) = 𝐯 s ( t ) c symbol{\beta}(t)=\frac{\mathbf{v}_{s}(t)}{c}
  53. 𝐧 ( t ) = 𝐫 - 𝐫 s ( t ) | 𝐫 - 𝐫 s ( t ) | \mathbf{n}(t)=\frac{\mathbf{r}-\mathbf{r}_{s}(t)}{|\mathbf{r}-\mathbf{r}_{s}(t% )|}
  54. γ ( t ) = 1 1 - | s y m b o l β ( t ) | 2 \gamma(t)=\frac{1}{\sqrt{1-|symbol{\beta}(t)|^{2}}}
  55. 𝐧 - s y m b o l β \mathbf{n}-symbol{\beta}
  56. c s y m b o l β csymbol{\beta}
  57. s y m b o l β ˙ \dot{symbol{\beta}}
  58. q q
  59. 𝐄 ( 𝐫 , t ) \mathbf{E}(\mathbf{r},t)
  60. ρ ( 𝐫 , t ) \rho(\mathbf{r},t)
  61. 𝐉 ( 𝐫 , t ) \mathbf{J}(\mathbf{r},t)
  62. φ ( 𝐫 , t ) = 1 4 π ϵ 0 ρ ( 𝐫 , t r ) | 𝐫 - 𝐫 | d 3 𝐫 \varphi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(\mathbf{r}^{% \prime},t_{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}d^{3}\mathbf{r}^{\prime}
  63. 𝐀 ( 𝐫 , t ) = μ 0 4 π 𝐉 ( 𝐫 , t r ) | 𝐫 - 𝐫 | d 3 𝐫 \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}^{% \prime},t_{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}d^{3}\mathbf{r}^{\prime}
  64. t r = t - 1 c | 𝐫 - 𝐫 | t_{r}^{\prime}=t-\frac{1}{c}|\mathbf{r}-\mathbf{r}^{\prime}|
  65. 𝐫 s ( t ) \mathbf{r}_{s}(t^{\prime})
  66. ρ ( 𝐫 , t ) = q δ 3 ( 𝐫 - 𝐫 s ( t ) ) \rho(\mathbf{r}^{\prime},t^{\prime})=q\delta^{3}(\mathbf{r^{\prime}}-\mathbf{r% }_{s}(t^{\prime}))
  67. 𝐉 ( 𝐫 , t ) = q 𝐯 s ( t ) δ 3 ( 𝐫 - 𝐫 s ( t ) ) \mathbf{J}(\mathbf{r}^{\prime},t^{\prime})=q\mathbf{v}_{s}(t^{\prime})\delta^{% 3}(\mathbf{r^{\prime}}-\mathbf{r}_{s}(t^{\prime}))
  68. δ 3 \delta^{3}
  69. 𝐯 s ( t ) \mathbf{v}_{s}(t^{\prime})
  70. φ ( 𝐫 , t ) = 1 4 π ϵ 0 q δ 3 ( 𝐫 - 𝐫 s ( t r ) ) | 𝐫 - 𝐫 | d 3 𝐫 \varphi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\int\frac{q\delta^{3}(\mathbf{% r^{\prime}}-\mathbf{r}_{s}(t_{r}^{\prime}))}{|\mathbf{r}-\mathbf{r}^{\prime}|}% d^{3}\mathbf{r}^{\prime}
  71. 𝐀 ( 𝐫 , t ) = μ 0 4 π q 𝐯 s ( t r ) δ 3 ( 𝐫 - 𝐫 s ( t r ) ) | 𝐫 - 𝐫 | d 3 𝐫 \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int\frac{q\mathbf{v}_{s}(t_{r}^{% \prime})\delta^{3}(\mathbf{r^{\prime}}-\mathbf{r}_{s}(t_{r}^{\prime}))}{|% \mathbf{r}-\mathbf{r}^{\prime}|}d^{3}\mathbf{r}^{\prime}
  72. t r t_{r}^{\prime}
  73. t t^{\prime}
  74. δ ( t - t r ) \delta(t^{\prime}-t_{r}^{\prime})
  75. φ ( 𝐫 , t ) = 1 4 π ϵ 0 q δ 3 ( 𝐫 - 𝐫 s ( t ) ) | 𝐫 - 𝐫 | δ ( t - t r ) d t d 3 𝐫 \varphi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\iint\frac{q\delta^{3}(\mathbf% {r^{\prime}}-\mathbf{r}_{s}(t^{\prime}))}{|\mathbf{r}-\mathbf{r}^{\prime}|}% \delta(t^{\prime}-t_{r}^{\prime})\,dt^{\prime}\,d^{3}\mathbf{r}^{\prime}
  76. 𝐀 ( 𝐫 , t ) = μ 0 4 π q 𝐯 s ( t ) δ 3 ( 𝐫 - 𝐫 s ( t ) ) | 𝐫 - 𝐫 | δ ( t - t r ) d t d 3 𝐫 \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\iint\frac{q\mathbf{v}_{s}(t^{% \prime})\delta^{3}(\mathbf{r^{\prime}}-\mathbf{r}_{s}(t^{\prime}))}{|\mathbf{r% }-\mathbf{r}^{\prime}|}\delta(t^{\prime}-t_{r}^{\prime})\,dt^{\prime}\,d^{3}% \mathbf{r}^{\prime}
  77. φ ( 𝐫 , t ) = 1 4 π ϵ 0 δ ( t - t r ) | 𝐫 - 𝐫 | q δ 3 ( 𝐫 - 𝐫 s ( t ) ) d 3 𝐫 d t \varphi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\iint\frac{\delta(t^{\prime}-t% _{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}q\delta^{3}(\mathbf{r^{\prime% }}-\mathbf{r}_{s}(t^{\prime}))\,d^{3}\mathbf{r}^{\prime}dt^{\prime}
  78. 𝐀 ( 𝐫 , t ) = μ 0 4 π δ ( t - t r ) | 𝐫 - 𝐫 | q 𝐯 s ( t ) δ 3 ( 𝐫 - 𝐫 s ( t ) ) d 3 𝐫 d t \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\iint\frac{\delta(t^{\prime}-t_{r% }^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}q\mathbf{v}_{s}(t^{\prime})% \delta^{3}(\mathbf{r^{\prime}}-\mathbf{r}_{s}(t^{\prime}))\,d^{3}\mathbf{r}^{% \prime}dt^{\prime}
  79. 𝐫 = 𝐫 s ( t ) \mathbf{r}^{\prime}=\mathbf{r}_{s}(t^{\prime})
  80. t r t_{r}^{\prime}
  81. 𝐫 \mathbf{r}^{\prime}
  82. t r = t r ( 𝐫 s ( t ) , t ) t_{r}=t_{r}(\mathbf{r}_{s}(t^{\prime}),t^{\prime})
  83. φ ( 𝐫 , t ) = 1 4 π ϵ 0 q δ ( t - t r ) | 𝐫 - 𝐫 s ( t ) | d t \varphi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\int q\frac{\delta(t^{\prime}-% t_{r}^{\prime})}{|\mathbf{r}-\mathbf{r}_{s}(t^{\prime})|}dt^{\prime}
  84. 𝐀 ( 𝐫 , t ) = μ 0 4 π q 𝐯 s ( t ) δ ( t - t r ) | 𝐫 - 𝐫 s ( t ) | d t \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int q\mathbf{v}_{s}(t^{\prime})% \frac{\delta(t^{\prime}-t_{r}^{\prime})}{|\mathbf{r}-\mathbf{r}_{s}(t^{\prime}% )|}\,dt^{\prime}
  85. t r t_{r}^{\prime}
  86. ( 𝐫 , t ) (\mathbf{r},t)
  87. 𝐫 s ( t ) \mathbf{r}_{s}(t^{\prime})
  88. t t^{\prime}
  89. δ ( f ( t ) ) = i δ ( t - t i ) | f ( t i ) | \delta(f(t^{\prime}))=\sum_{i}\frac{\delta(t^{\prime}-t_{i})}{|f^{\prime}(t_{i% })|}
  90. t i t_{i}
  91. f f
  92. t r t_{r}
  93. ( 𝐫 , t ) (\mathbf{r},t)
  94. 𝐫 s ( t ) \mathbf{r}_{s}(t^{\prime})
  95. δ ( t - t r ) = δ ( t - t r ) t ( t - t r ) | t = t r = \displaystyle\delta(t^{\prime}-t_{r}^{\prime})=\frac{\delta(t^{\prime}-t_{r})}% {\frac{\partial}{\partial t^{\prime}}(t^{\prime}-t_{r}^{\prime})|_{t^{\prime}=% t_{r}}}=
  96. s y m b o l β s = 𝐯 s / c symbol{\beta}_{s}=\mathbf{v}_{s}/c
  97. 𝐫 s \mathbf{r}_{s}
  98. | 𝐱 | = 𝐱 ^ 𝐯 |\mathbf{x}|^{\prime}=\hat{\mathbf{x}}\cdot\mathbf{v}
  99. t = t r t^{\prime}=t_{r}
  100. φ ( 𝐫 , t ) = 1 4 π ϵ 0 ( q | 𝐫 - 𝐫 s | ( 1 - s y m b o l β s ( 𝐫 - 𝐫 s ) / | 𝐫 - 𝐫 s | ) ) t r = 1 4 π ϵ 0 ( q ( 1 - 𝐧 s y m b o l β s ) | 𝐫 - 𝐫 s | ) t r \varphi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\left(\frac{q}{|\mathbf{r}-% \mathbf{r}_{s}|(1-symbol{\beta}_{s}\cdot(\mathbf{r}-\mathbf{r}_{s})/|\mathbf{r% }-\mathbf{r}_{s}|)}\right)_{t_{r}}=\frac{1}{4\pi\epsilon_{0}}\left(\frac{q}{(1% -\mathbf{n}\cdot symbol{\beta}_{s})|\mathbf{r}-\mathbf{r}_{s}|}\right)_{t_{r}}
  101. 𝐀 ( 𝐫 , t ) = μ 0 4 π ( q 𝐯 | 𝐫 - 𝐫 s | ( 1 - s y m b o l β s ( 𝐫 - 𝐫 s ) / | 𝐫 - 𝐫 s | ) ) t r = μ 0 c 4 π ( q s y m b o l β s ( 1 - 𝐧 s y m b o l β s ) | 𝐫 - 𝐫 s | ) t r \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\left(\frac{q\mathbf{v}}{|\mathbf% {r}-\mathbf{r}_{s}|(1-symbol{\beta}_{s}\cdot(\mathbf{r}-\mathbf{r}_{s})/|% \mathbf{r}-\mathbf{r}_{s}|)}\right)_{t_{r}}=\frac{\mu_{0}c}{4\pi}\left(\frac{% qsymbol{\beta}_{s}}{(1-\mathbf{n}\cdot symbol{\beta}_{s})|\mathbf{r}-\mathbf{r% }_{s}|}\right)_{t_{r}}
  102. φ \varphi
  103. 𝐀 \mathbf{A}
  104. 𝐫 𝐬 = 𝐫 𝐬 ( t r ) \mathbf{r_{s}}=\mathbf{r_{s}}(t_{r})
  105. t r + 1 c | 𝐫 - 𝐫 𝐬 | = t t_{r}+\frac{1}{c}|\mathbf{r}-\mathbf{r_{s}}|=t
  106. d t r d t + 1 c d t r d t d | 𝐫 - 𝐫 𝐬 | d t r = 1 \frac{dt_{r}}{dt}+\frac{1}{c}\frac{dt_{r}}{dt}\frac{d|\mathbf{r}-\mathbf{r_{s}% }|}{dt_{r}}=1
  107. d t r d t ( 1 - 𝐧 s y m b o l β s ) = 1 \frac{dt_{r}}{dt}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)=1
  108. d t r d t = 1 ( 1 - 𝐧 s y m b o l β s ) \frac{dt_{r}}{dt}=\frac{1}{\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)}
  109. 𝐫 \mathbf{r}
  110. s y m b o l t r + 1 c s y m b o l | 𝐫 - 𝐫 𝐬 | = 0 {symbol\nabla}t_{r}+\frac{1}{c}{symbol\nabla}|\mathbf{r}-\mathbf{r_{s}}|=0
  111. s y m b o l t r + 1 c ( s y m b o l t r d | 𝐫 - 𝐫 𝐬 | d t r + 𝐧 ) = 0 {symbol\nabla}t_{r}+\frac{1}{c}\left({symbol\nabla}t_{r}\frac{d|\mathbf{r}-% \mathbf{r_{s}}|}{dt_{r}}+\mathbf{n}\right)=0
  112. s y m b o l t r ( 1 - 𝐧 s y m b o l β s ) = - 𝐧 / c {symbol\nabla}t_{r}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)=-\mathbf{n}/c
  113. s y m b o l t r = - 𝐧 / c ( 1 - 𝐧 s y m b o l β s ) {symbol\nabla}t_{r}=-\frac{\mathbf{n}/c}{\left(1-\mathbf{n}\cdot{symbol\beta}_% {s}\right)}
  114. d | 𝐫 - 𝐫 𝐬 | d t = d t r d t d | 𝐫 - 𝐫 𝐬 | d t r = - 𝐧 s y m b o l β s c ( 1 - 𝐧 s y m b o l β s ) \frac{d|\mathbf{r}-\mathbf{r_{s}}|}{dt}=\frac{dt_{r}}{dt}\frac{d|\mathbf{r}-% \mathbf{r_{s}}|}{dt_{r}}=\frac{-\mathbf{n}\cdot{symbol\beta}_{s}c}{\left(1-% \mathbf{n}\cdot{symbol\beta}_{s}\right)}
  115. s y m b o l | 𝐫 - 𝐫 𝐬 | = s y m b o l t r d | 𝐫 - 𝐫 𝐬 | d t r + 𝐧 = 𝐧 ( 1 - 𝐧 s y m b o l β s ) {symbol\nabla}|\mathbf{r}-\mathbf{r_{s}}|={symbol\nabla}t_{r}\frac{d|\mathbf{r% }-\mathbf{r_{s}}|}{dt_{r}}+\mathbf{n}=\frac{\mathbf{n}}{\left(1-\mathbf{n}% \cdot{symbol\beta}_{s}\right)}
  116. d φ d t = - q 4 π ϵ 0 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 2 d d t [ ( | 𝐫 - 𝐫 𝐬 | ( 1 - 𝐧 s y m b o l β s ) ] = - q 4 π ϵ 0 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 2 d d t [ | 𝐫 - 𝐫 𝐬 | - ( 𝐫 - 𝐫 𝐬 ) s y m b o l β s ] = - q c 4 π ϵ 0 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 3 [ - 𝐧 s y m b o l β s + β s 2 - ( 𝐫 - 𝐫 𝐬 ) s y m b o l β ˙ s / c ] \begin{aligned}\displaystyle\frac{d\varphi}{dt}=&\displaystyle-\frac{q}{4\pi% \epsilon_{0}}\frac{1}{|\mathbf{r}-\mathbf{r_{s}}|^{2}\left(1-\mathbf{n}\cdot{% symbol\beta}_{s}\right)^{2}}\frac{d}{dt}\left[(|\mathbf{r}-\mathbf{r_{s}}|(1-% \mathbf{n}\cdot{symbol\beta}_{s})\right]\\ \displaystyle=&\displaystyle-\frac{q}{4\pi\epsilon_{0}}\frac{1}{|\mathbf{r}-% \mathbf{r_{s}}|^{2}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)^{2}}\frac{d% }{dt}\left[|\mathbf{r}-\mathbf{r_{s}}|-(\mathbf{r}-\mathbf{r_{s}})\cdot{symbol% \beta}_{s}\right]\\ \displaystyle=&\displaystyle-\frac{qc}{4\pi\epsilon_{0}}\frac{1}{|\mathbf{r}-% \mathbf{r_{s}}|^{2}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)^{3}}\left[-% \mathbf{n}\cdot{symbol\beta}_{s}+{\beta_{s}}^{2}-(\mathbf{r}-\mathbf{r_{s}})% \cdot\dot{symbol\beta}_{s}/c\right]\end{aligned}
  117. s y m b o l 𝐀 = - q 4 π ϵ 0 c 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 2 ( s y m b o l [ ( | 𝐫 - 𝐫 𝐬 | - ( 𝐫 - 𝐫 𝐬 ) s y m b o l β s ) ] s y m b o l β s - [ ( | 𝐫 - 𝐫 𝐬 | - ( 𝐫 - 𝐫 𝐬 ) s y m b o l β s ) ] s y m b o l s y m b o l β s ) = - q 4 π ϵ 0 c 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 3 [ ( 𝐧 s y m b o l β s ) - β s 2 ( 1 - 𝐧 s y m b o l β s ) - β s 2 𝐧 s y m b o l β s + ( ( 𝐫 - 𝐫 𝐬 ) s y m b o l β ˙ s / c ) ( 𝐧 s y m b o l β s ) + ( | 𝐫 - 𝐫 𝐬 | - ( 𝐫 - 𝐫 𝐬 ) s y m b o l β s ) ( 𝐧 s y m b o l β ˙ s / c ) ] = q 4 π ϵ 0 c 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 3 [ β s 2 - 𝐧 s y m b o l β s - ( 𝐫 - 𝐫 𝐬 ) s y m b o l β ˙ s / c ] \begin{aligned}\displaystyle{symbol\nabla}\cdot\mathbf{A}=&\displaystyle-\frac% {q}{4\pi\epsilon_{0}c}\frac{1}{|\mathbf{r}-\mathbf{r_{s}}|^{2}\left(1-\mathbf{% n}\cdot{symbol\beta}_{s}\right)^{2}}\big({symbol\nabla}\left[\left(|\mathbf{r}% -\mathbf{r_{s}}|-(\mathbf{r}-\mathbf{r_{s}})\cdot{symbol\beta}_{s}\right)% \right]\cdot{symbol\beta}_{s}-\left[\left(|\mathbf{r}-\mathbf{r_{s}}|-(\mathbf% {r}-\mathbf{r_{s}})\cdot{symbol\beta}_{s}\right)\right]{symbol\nabla}\cdot{% symbol\beta}_{s}\big)\\ \displaystyle=&\displaystyle-\frac{q}{4\pi\epsilon_{0}c}\frac{1}{|\mathbf{r}-% \mathbf{r_{s}}|^{2}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)^{3}}\cdot\\ &\displaystyle\left[(\mathbf{n}\cdot{symbol\beta}_{s})-{\beta}_{s}^{2}(1-% \mathbf{n}\cdot{symbol\beta}_{s})-{\beta}_{s}^{2}\mathbf{n}\cdot{symbol\beta}_% {s}+\left((\mathbf{r}-\mathbf{r_{s}})\cdot\dot{symbol\beta}_{s}/c\right)(% \mathbf{n}\cdot{symbol\beta}_{s})+\big(|\mathbf{r}-\mathbf{r_{s}}|-(\mathbf{r}% -\mathbf{r_{s}})\cdot{symbol\beta}_{s}\big)(\mathbf{n}\cdot\dot{symbol\beta}_{% s}/c)\right]\\ \displaystyle=&\displaystyle\frac{q}{4\pi\epsilon_{0}c}\frac{1}{|\mathbf{r}-% \mathbf{r_{s}}|^{2}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)^{3}}\left[% \beta_{s}^{2}-\mathbf{n}\cdot{symbol\beta}_{s}-(\mathbf{r}-\mathbf{r_{s}})% \cdot\dot{symbol\beta}_{s}/c\right]\end{aligned}
  118. d φ d t + c 2 s y m b o l 𝐀 = 0 \frac{d\varphi}{dt}+c^{2}{symbol\nabla}\cdot\mathbf{A}=0
  119. s y m b o l φ = - q 4 π ϵ 0 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 3 [ 𝐧 ( 1 - β s 2 + ( 𝐫 - 𝐫 𝐬 ) s y m b o l β ˙ s / c ) - s y m b o l β s ( 1 - 𝐧 s y m b o l β s ) ] {symbol\nabla}\varphi=-\frac{q}{4\pi\epsilon_{0}}\frac{1}{|\mathbf{r}-\mathbf{% r_{s}}|^{2}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)^{3}}\left[\mathbf{n% }\left(1-{\beta_{s}}^{2}+(\mathbf{r}-\mathbf{r_{s}})\cdot\dot{symbol\beta}_{s}% /c\right)-{symbol\beta}_{s}(1-\mathbf{n}\cdot{symbol\beta}_{s})\right]
  120. d 𝐀 d t = q 4 π ϵ 0 1 | 𝐫 - 𝐫 𝐬 | 2 ( 1 - 𝐧 s y m b o l β s ) 3 [ s y m b o l β s ( 𝐧 s y m b o l β s - β s 2 + ( 𝐫 - 𝐫 𝐬 ) s y m b o l β ˙ s / c ) + | 𝐫 - 𝐫 𝐬 | s y m b o l β ˙ s ( 1 - 𝐧 s y m b o l β s ) / c ] \frac{d\mathbf{A}}{dt}=\frac{q}{4\pi\epsilon_{0}}\frac{1}{|\mathbf{r}-\mathbf{% r_{s}}|^{2}\left(1-\mathbf{n}\cdot{symbol\beta}_{s}\right)^{3}}\left[{symbol% \beta}_{s}\left(\mathbf{n}\cdot{symbol\beta}_{s}-{\beta_{s}}^{2}+(\mathbf{r}-% \mathbf{r_{s}})\cdot\dot{symbol\beta}_{s}/c\right)+|\mathbf{r}-\mathbf{r_{s}}|% \dot{symbol\beta}_{s}(1-\mathbf{n}\cdot{symbol\beta}_{s})/c\right]
  121. 𝐮 \mathbf{u}
  122. 𝐯 \mathbf{v}
  123. 𝐰 \mathbf{w}
  124. 𝐮 × ( 𝐯 × 𝐰 ) = ( 𝐮 𝐰 ) 𝐯 - ( 𝐮 𝐯 ) 𝐰 \mathbf{u}\times(\mathbf{v}\times\mathbf{w})=(\mathbf{u}\cdot\mathbf{w})% \mathbf{v}-(\mathbf{u}\cdot\mathbf{v})\mathbf{w}
  125. 𝐄 ( 𝐫 , t ) = q 4 π ϵ 0 1 | 𝐫 - 𝐫 s | 2 ( 1 - 𝐧 s y m b o l β s ) 3 [ ( 𝐧 - s y m b o l β s ) ( 1 - β s 2 ) + | 𝐫 - 𝐫 s | ( 𝐧 s y m b o l β ˙ s / c ) ( 𝐧 - s y m b o l β s ) - | 𝐫 - 𝐫 s | ( 𝐧 ( 𝐧 - s y m b o l β s ) ) s y m b o l β ˙ s / c ] \begin{aligned}\displaystyle\mathbf{E}(\mathbf{r},t)=&\displaystyle\frac{q}{4% \pi\epsilon_{0}}\frac{1}{|\mathbf{r}-\mathbf{r}_{s}|^{2}(1-\mathbf{n}\cdot{% symbol\beta}_{s})^{3}}\cdot\\ &\displaystyle\left[\left(\mathbf{n}-{symbol\beta}_{s}\right)(1-{\beta_{s}}^{2% })+|\mathbf{r}-\mathbf{r}_{s}|(\mathbf{n}\cdot\dot{symbol\beta}_{s}/c)(\mathbf% {n}-{symbol\beta}_{s})-|\mathbf{r}-\mathbf{r}_{s}|\big(\mathbf{n}\cdot(\mathbf% {n}-{symbol\beta}_{s})\big)\dot{symbol\beta}_{s}/c\right]\end{aligned}
  126. - s y m b o l φ - d 𝐀 d t -{symbol\nabla}\varphi-\frac{d\mathbf{A}}{dt}
  127. s y m b o l × 𝐀 {symbol\nabla}\times\mathbf{A}
  128. 𝐁 = \displaystyle{\mathbf{B}}=

Lifson–Roig_model.html

  1. n c \sqrt{nc}
  2. Z = V ( i = 0 N + 1 M ( i ) ) V ~ Z=V\left(\prod_{i=0}^{N+1}M(i)\right)\tilde{V}
  3. V = [ 0001 ] V=[0001]

Lift_(data_mining).html

  1. s u p p ( A 0 ) = P ( A and 0 ) = P ( A ) P ( 0 | A ) = P ( 0 ) P ( A | 0 ) supp(A\Rightarrow 0)=P(A\and 0)=P(A)P(0|A)=P(0)P(A|0)
  2. s u p p ( B 1 ) = P ( B and 1 ) = P ( B ) P ( 1 | B ) = P ( 1 ) P ( B | 1 ) supp(B\Rightarrow 1)=P(B\and 1)=P(B)P(1|B)=P(1)P(B|1)
  3. c o n f ( A 0 ) = P ( 0 | A ) conf(A\Rightarrow 0)=P(0|A)
  4. c o n f ( B 1 ) = P ( 1 | B ) conf(B\Rightarrow 1)=P(1|B)
  5. l i f t ( A 0 ) = P ( 0 | A ) P ( 0 ) = P ( A and 0 ) P ( A ) P ( 0 ) lift(A\Rightarrow 0)=\frac{P(0|A)}{P(0)}=\frac{P(A\and 0)}{P(A)P(0)}
  6. l i f t ( B 1 ) = P ( 1 | B ) P ( 1 ) = P ( B and 1 ) P ( B ) P ( 1 ) lift(B\Rightarrow 1)=\frac{P(1|B)}{P(1)}=\frac{P(B\and 1)}{P(B)P(1)}

Lift_(mathematics).html

  1. \circ
  2. f : [ 0 , 1 ] 2 , (projective plane path) g : S 2 2 , (covering map) h : [ 0 , 1 ] S 2 . (sphere path) \begin{aligned}\displaystyle f\colon&\displaystyle[0,1]\to\mathbb{RP}^{2},&&% \displaystyle\,\text{(projective plane path)}\\ \displaystyle g\colon&\displaystyle S^{2}\to\mathbb{RP}^{2},&&\displaystyle\,% \text{(covering map)}\\ \displaystyle h\colon&\displaystyle[0,1]\to S^{2}.&&\displaystyle\,\text{(% sphere path)}\end{aligned}

Lifting-line_theory.html

  1. l l
  2. L ~ \tilde{L}
  3. L ~ ( y ) \tilde{L}_{(y)}
  4. ρ \rho
  5. V V_{\infty}
  6. α \alpha_{\infty}
  7. L ~ ( y ) = ρ V Γ ( y ) \tilde{L}_{(y)}=\rho V\Gamma_{(y)}
  8. Γ ( y ) \Gamma_{(y)}
  9. d Γ d y {\operatorname{d}\Gamma\over\operatorname{d}y}
  10. ω i \omega_{i}
  11. α i \alpha_{i}
  12. L t o t a l = ρ V t i p t i p Γ ( y ) d y L_{total}=\rho V_{\infty}\int_{tip}^{tip}\Gamma_{(y)}\operatorname{d}y
  13. Γ ( y ) \Gamma_{(y)}
  14. d Γ ( y ) d y {\operatorname{d}\Gamma_{(y)}\over\operatorname{d}y}
  15. Γ ( y ) \Gamma_{(y)}
  16. Γ \ \Gamma
  17. C L \ C_{L}
  18. A R \ AR
  19. α \ \alpha_{\infty}
  20. V \ V_{\infty}
  21. C D i \ C_{D_{i}}
  22. e \ e
  23. y y
  24. C l \ C_{l}
  25. γ \ \gamma
  26. c \ c
  27. α g e o \ \alpha_{geo}
  28. α 0 \ \alpha_{0}
  29. C l α \ C_{l_{\alpha}}
  30. α i \ \alpha_{i}
  31. w i \ w_{i}
  32. y y
  33. y = s c o s θ y=scos{\theta}
  34. Γ ( y ) = Γ ( θ ) = γ = 4 s V n A n sin ( n θ ) ( 1 ) \Gamma(y)=\Gamma(\theta)=\gamma=4sV_{\infty}\sum_{n}{A_{n}\sin(n\theta})\qquad% (1)
  35. C l C_{l}
  36. C l = 2 γ V c ( 2 ) C_{l}=\frac{2\gamma}{V_{\infty}c}\qquad(2)
  37. C l = C l α ( α + α g e o - α 0 - α i ) ( 3 ) C_{l}=C_{l_{\alpha}}(\alpha_{\infty}+\alpha_{geo}-\alpha_{0}-\alpha_{i})\qquad% (3)
  38. γ = 1 2 V c C l α ( α + α g e o - α 0 - α i ) ( 4 ) \gamma=\frac{1}{2}V_{\infty}cC_{l_{\alpha}}(\alpha_{\infty}+\alpha_{geo}-% \alpha_{0}-\alpha_{i})\qquad(4)
  39. γ \gamma
  40. α i \alpha_{i}
  41. α i \alpha_{i}
  42. Γ ( y ) \Gamma(y)
  43. d Γ = 4 s V n = 1 n A n cos ( n θ ) ( 5 ) d\Gamma=4sV_{\infty}\sum_{n=1}^{\infty}nA_{n}\cos(n\theta)\qquad(5)
  44. d w i = d Γ 4 π r ( 6 ) dw_{i}=\frac{d\Gamma}{4\pi r}\qquad(6)
  45. w i = - s s 1 y - y 0 d Γ ( 7 ) w_{i}=\int_{-s}^{s}\frac{1}{y-y_{0}}d\Gamma\qquad(7)
  46. w i = V n = 1 n A n sin ( n θ ) sin ( θ ) ( 8 ) w_{i}=V_{\infty}\sum_{n=1}^{\infty}\frac{nA_{n}\sin(n\theta)}{\sin(\theta)}% \qquad(8)
  47. α i = w i V ( 9 ) \alpha_{i}=\frac{w_{i}}{V_{\infty}}\qquad(9)
  48. 4 s V n = 1 A n sin ( n θ ) = 1 2 V c C l α [ α + α g e o - α 0 - n = 1 n A n sin ( n θ ) sin ( θ ) ] ( 10 ) 4sV_{\infty}\sum_{n=1}^{\infty}A_{n}\sin(n\theta)=\frac{1}{2}V_{\infty}cC_{l_{% \alpha}}\left[\alpha_{\infty}+\alpha_{geo}-\alpha_{0}-\sum_{n=1}^{\infty}\frac% {nA_{n}\sin(n\theta)}{\sin(\theta)}\right]\qquad(10)
  49. n = 1 A n sin ( n θ ) ( sin ( θ ) + n C l α c 8 s ) = C l α c 8 s sin ( θ ) ( α + α g e o - α 0 ) ( 11 ) \sum_{n=1}^{\infty}A_{n}\sin(n\theta)\bigg(\sin(\theta)+\frac{nC_{l\alpha}c}{8% s}\bigg)=\frac{C_{l\alpha}c}{8s}\sin(\theta)(\alpha_{\infty}+\alpha_{geo}-% \alpha_{0})\qquad(11)
  50. θ \theta
  51. ( - π , π ) (-\pi,\pi)
  52. π \pi
  53. Lift = ρ V - s s Γ d y \,\text{Lift}=\rho V_{\infty}\int_{-s}^{s}\Gamma dy
  54. C L = π A R A 1 C_{L}=\pi A\!RA_{1}
  55. A 1 A_{1}
  56. Draginduced = ρ V - s s Γ sin α i d y \,\text{Drag}\text{induced}=\rho V_{\infty}\int_{-s}^{s}\Gamma\sin{\alpha_{i}}dy
  57. C D , induced = π A R n = 1 n A n 2 C_{D,\,\text{induced}}=\pi A\!R\sum_{n=1}^{\infty}nA_{n}^{2}
  58. C l = C l α ( α + α g e o - α 0 - α i + p y s ) ( 3 ) C_{l}=C_{l_{\alpha}}\left(\alpha_{\infty}+\alpha_{geo}-\alpha_{0}-\alpha_{i}+% \frac{py}{s}\right)\qquad(3)
  59. p \ p
  60. α 0 \alpha_{0}
  61. α 0 \alpha_{0}
  62. c ( θ ) = c r o o t cos ( θ ) c(\theta)=c_{root}\cos(\theta)
  63. C L 3 D = C l α ( AR AR + 2 ) α \ C_{L3D}=C_{l_{\alpha}}\left(\frac{\,\text{AR}}{\,\text{AR}+2}\right)\alpha
  64. C L3D \ C_{\,\text{L3D}}
  65. C l α \ C_{l_{\alpha}}
  66. AR \ \,\text{AR}
  67. α \ \alpha
  68. C l α \ C_{l_{\alpha}}
  69. π \pi
  70. C D i = C L 2 π AR e \ C_{D_{i}}=\frac{{C_{L}}^{2}}{\pi\,\text{AR}e}
  71. C D i \ C_{D_{i}}
  72. C L \ C_{L}
  73. AR \ \,\text{AR}
  74. e \ e

Ligand_cone_angle.html

  1. θ = 2 3 Σ θ i 2 \theta=\frac{2}{3}\Sigma\frac{\theta_{i}}{2}

Light_scattering_by_particles.html

  1. x = 2 π r λ . x=\frac{2\pi r}{\lambda}.
  2. x 1 x\ll 1

Lightness.html

  1. V = 10 Y V=10\sqrt{Y}
  2. V 2 = 1.4742 Y - 0.004743 Y 2 V^{2}=1.4742Y-0.004743Y^{2}
  3. Y = 1.2219 V - 0.23111 V 2 + 0.23951 V 3 - 0.021009 V 4 + 0.0008404 V 5 Y=1.2219V-0.23111V^{2}+0.23951V^{3}-0.021009V^{4}+0.0008404V^{5}
  4. V = 5 ( Y / 19.77 ) 0.426 = 1.4 Y 0.426 V=5(Y/19.77)^{0.426}=1.4Y^{0.426}
  5. V = 2.357 Y 0.343 - 1.52 V=2.357Y^{0.343}-1.52
  6. V = 2.217 Y 0.352 - 1.324 V=2.217Y^{0.352}-1.324
  7. V = 2.468 Y 1 / 3 - 1.636 V=2.468Y^{1/3}-1.636
  8. L * = 25.29 Y 1 / 3 - 18.38 L^{*}=25.29Y^{1/3}-18.38
  9. W * = 25 Y 1 / 3 - 17 W^{*}=25Y^{1/3}-17
  10. 1 % < Y < 98 % 1\%<Y<98\%
  11. L * = 116 ( Y / Y n ) 1 / 3 - 16 L^{*}=116(Y/Y_{n})^{1/3}-16
  12. Y n Y_{n}
  13. Y / Y n > 0.01 Y/Y_{n}>0.01
  14. Y / Y n = ( 6 / 29 ) 3 0.008856 Y/Y_{n}=(6/29)^{3}\approx 0.008856
  15. ( 29 / 3 ) 3 903.3 (29/3)^{3}\approx 903.3
  16. f ( Y / Y n ) = { 841 108 Y / Y n + 4 29 , Y / Y n ( 6 / 29 ) 3 ( Y / Y n ) 1 / 3 , Y / Y n > ( 6 / 29 ) 3 f(Y/Y_{n})=\left\{\begin{array}[]{ll}\frac{841}{108}Y/Y_{n}+\frac{4}{29},&Y/Y_% {n}\leq(6/29)^{3}\\ \\ (Y/Y_{n})^{1/3},&Y/Y_{n}>(6/29)^{3}\end{array}\right.
  17. L * = 116 f ( Y / Y n ) - 16 L^{*}=116f(Y/Y_{n})-16
  18. ( 33 / 58 ) 3 \left(33/58\right)^{3}

Limit_comparison_test.html

  1. Σ n a n \Sigma_{n}a_{n}
  2. Σ n b n \Sigma_{n}b_{n}
  3. a n , b n 0 a_{n},b_{n}\geq 0
  4. n n
  5. lim n a n b n = c \lim_{n\to\infty}\frac{a_{n}}{b_{n}}=c
  6. 0 < c < 0<c<\infty
  7. lim a n b n = c \lim\frac{a_{n}}{b_{n}}=c
  8. ε \varepsilon
  9. n 0 n_{0}
  10. n n 0 n\geq n_{0}
  11. | a n b n - c | < ε \left|\frac{a_{n}}{b_{n}}-c\right|<\varepsilon
  12. - ε < a n b n - c < ε -\varepsilon<\frac{a_{n}}{b_{n}}-c<\varepsilon
  13. c - ε < a n b n < c + ε c-\varepsilon<\frac{a_{n}}{b_{n}}<c+\varepsilon
  14. ( c - ε ) b n < a n < ( c + ε ) b n (c-\varepsilon)b_{n}<a_{n}<(c+\varepsilon)b_{n}
  15. c > 0 c>0
  16. ε \varepsilon
  17. c - ε c-\varepsilon
  18. b n < 1 c - ε a n b_{n}<\frac{1}{c-\varepsilon}a_{n}
  19. a n a_{n}
  20. b n b_{n}
  21. a n < ( c + ε ) b n a_{n}<(c+\varepsilon)b_{n}
  22. b n b_{n}
  23. a n a_{n}
  24. n = 1 1 n 2 + 2 n \sum_{n=1}^{\infty}\frac{1}{n^{2}+2n}
  25. n = 1 1 n 2 = π 2 6 \sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}
  26. lim n 1 n 2 + 2 n n 2 1 = 1 > 0 \lim_{n\to\infty}\frac{1}{n^{2}+2n}\frac{n^{2}}{1}=1>0
  27. a n , b n 0 a_{n},b_{n}\geq 0
  28. n n
  29. lim sup n a n b n = c \limsup_{n\to\infty}\frac{a_{n}}{b_{n}}=c
  30. 0 c < 0\leq c<\infty
  31. Σ n b n \Sigma_{n}b_{n}
  32. Σ n a n \Sigma_{n}a_{n}
  33. a n = ( 1 - ( - 1 ) n n 2 a_{n}=\frac{(1-(-1)^{n}}{n^{2}}
  34. b n = 1 n 2 b_{n}=\frac{1}{n^{2}}
  35. n n
  36. lim n a n b n = lim n ( 1 - ( - 1 ) n ) \lim_{n\to\infty}\frac{a_{n}}{b_{n}}=\lim_{n\to\infty}(1-(-1)^{n})
  37. lim sup n a n b n = lim sup n ( 1 - ( - 1 ) n ) = 2 [ 0 , ) \limsup_{n\to\infty}\frac{a_{n}}{b_{n}}=\limsup_{n\to\infty}(1-(-1)^{n})=2\in[% 0,\infty)
  38. n = 1 1 n 2 \sum_{n=1}^{\infty}\frac{1}{n^{2}}
  39. n = 1 1 - ( - 1 ) n n 2 \sum_{n=1}^{\infty}\frac{1-(-1)^{n}}{n^{2}}
  40. a n , b n 0 a_{n},b_{n}\geq 0
  41. n n
  42. Σ n a n \Sigma_{n}a_{n}
  43. Σ n b n \Sigma_{n}b_{n}
  44. lim sup n a n b n = \limsup_{n\to\infty}\frac{a_{n}}{b_{n}}=\infty
  45. lim inf n b n a n = 0 \liminf_{n\to\infty}\frac{b_{n}}{a_{n}}=0
  46. a n a_{n}
  47. b n b_{n}
  48. f ( z ) = n = 0 a n z n f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}
  49. D = { z : | z | < 1 } D=\{z\in\mathbb{C}:|z|<1\}
  50. f f
  51. n = 1 n | a n | 2 \sum_{n=1}^{\infty}n|a_{n}|^{2}
  52. n = 1 1 / n \sum_{n=1}^{\infty}1/n
  53. lim inf n n | a n | 2 1 / n = lim inf n ( n | a n | ) 2 = 0 \liminf_{n\to\infty}\frac{n|a_{n}|^{2}}{1/n}=\liminf_{n\to\infty}(n|a_{n}|)^{2% }=0
  54. lim inf n n | a n | = 0 \liminf_{n\to\infty}n|a_{n}|=0

Limit_point_compact.html

  1. X × X\times\mathbb{Z}
  2. \mathbb{Z}
  3. f = π f=\pi_{\mathbb{Z}}
  4. X × X\times\mathbb{Z}
  5. X × X\times\mathbb{Z}
  6. f ( X × ) = f(X\times\mathbb{Z})=\mathbb{Z}

Limits_of_integration.html

  1. a b f ( x ) d x \int_{a}^{b}f(x)\,dx
  2. lim z a + z b f ( x ) d x \lim_{z\rightarrow a^{+}}\int_{z}^{b}f(x)\,dx
  3. lim z b - a z f ( x ) d x \lim_{z\rightarrow b^{-}}\int_{a}^{z}f(x)\,dx
  4. a f ( x ) d x \int_{a}^{\infty}f(x)\,dx
  5. - b f ( x ) d x \int_{-\infty}^{b}f(x)\,dx

Line_moiré.html

  1. p m = p b p r p b - p r . p_{m}=\frac{p_{b}\cdot p_{r}}{p_{b}-p_{r}}.
  2. v m v r = p m p r . \frac{v_{m}}{v_{r}}=\frac{p_{m}}{p_{r}}.
  3. v m v r = p b p b - p r . \frac{v_{m}}{v_{r}}=\frac{p_{b}}{p_{b}-p_{r}}.
  4. { tan α m = p b + l tan α b l tan α r = p b - p r + l tan α b l \begin{cases}\tan\alpha_{m}=\frac{p_{b}+l\cdot\tan\alpha_{b}}{l}\\ \tan\alpha_{r}=\frac{p_{b}-p_{r}+l\cdot\tan\alpha_{b}}{l}\end{cases}
  5. tan α m = p b tan α r - p r tan α b p b - p r \tan\alpha_{m}=\frac{p_{b}\cdot\tan\alpha_{r}-p_{r}\cdot\tan\alpha_{b}}{p_{b}-% p_{r}}
  6. T b = p b cos α b , T r = p r cos α r , T m = p m cos α m T_{b}=p_{b}\cdot\cos\alpha_{b},\ T_{r}=p_{r}\cdot\cos\alpha_{r},\ T_{m}=p_{m}% \cdot\cos\alpha_{m}
  7. α m = arctan ( T b sin α r - T r sin α b T b cos α r - T r cos α b ) \alpha_{m}=\arctan\left(\frac{T_{b}\cdot\sin\alpha_{r}-T_{r}\cdot\sin\alpha_{b% }}{T_{b}\cdot\cos\alpha_{r}-T_{r}\cdot\cos\alpha_{b}}\right)
  8. T m = T b T r T b 2 + T r 2 - 2 T b T r cos ( α r - α b ) T_{m}=\frac{T_{b}\cdot T_{r}}{\sqrt{T_{b}^{2}+T_{r}^{2}-2\cdot T_{b}\cdot T_{r% }\cdot\cos(\alpha_{r}-\alpha_{b})}}
  9. T m = T 2 sin ( α r - α b 2 ) T_{m}=\frac{T}{2\cdot\sin\left(\frac{\alpha_{r}-\alpha_{b}}{2}\right)}
  10. α m = 90 + α r + α b 2 \alpha_{m}=90^{\circ}+\frac{\alpha_{r}+\alpha_{b}}{2}
  11. tan α r = p r p b tan α b + ( 1 - p r p b ) tan α m . \tan\alpha_{r}=\frac{p_{r}}{p_{b}}\cdot\tan\alpha_{b}+\left(1-\frac{p_{r}}{p_{% b}}\right)\cdot\tan\alpha_{m}.

Linear_response_function.html

  1. h ( t ) h(t)
  2. x ( t ) x(t)
  3. x ( t ) x(t)
  4. h ( t ) h(t)
  5. x ( t ) x(t)
  6. h ( t ) h(t^{\prime})
  7. χ ( t - t ) \chi(t-t^{\prime})
  8. x ( t ) = - t d t χ ( t - t ) h ( t ) + . x(t)=\int_{-\infty}^{t}dt^{\prime}\,\chi(t-t^{\prime})h(t^{\prime})+\dots\,.
  9. χ ~ ( ω ) \tilde{\chi}(\omega)
  10. h ( t ) = h 0 sin ( ω t ) h(t)=h_{0}\cdot\sin(\omega t)
  11. ω \omega
  12. x ( t ) = | χ ~ ( ω ) | h 0 sin ( ω t + arg χ ~ ( ω ) ) , x(t)=|\tilde{\chi}(\omega)|\cdot h_{0}\cdot\sin(\omega t+\arg\tilde{\chi}(% \omega))\,,
  13. | χ ~ ( ω ) | |\tilde{\chi}(\omega)|
  14. arg χ ~ ( ω ) \arg\tilde{\chi}(\omega)
  15. h ( t ) h(t)
  16. x ¨ ( t ) + γ x ˙ ( t ) + ω 0 2 x ( t ) = h ( t ) . \ddot{x}(t)+\gamma\dot{x}(t)+\omega_{0}^{2}x(t)=h(t).\,
  17. χ ~ ( ω ) = x ~ ( ω ) h ~ ( ω ) = 1 ω 0 2 - ω 2 + i γ ω . \tilde{\chi}(\omega)=\frac{\tilde{x}(\omega)}{\tilde{h}(\omega)}=\frac{1}{% \omega_{0}^{2}-\omega^{2}+i\gamma\omega}.\,
  18. χ ~ ( ω ) , \tilde{\chi}(\omega),
  19. γ \gamma
  20. χ ~ ( ω ) \tilde{\chi}(\omega)
  21. ω ω 0 \omega\approx\omega_{0}
  22. , Δ ω , ,\Delta\omega,
  23. ω 0 , \omega_{0},
  24. S := ω 0 / Δ ω S:=\omega_{0}/\Delta\omega
  25. H ^ 0 H ^ 0 - h ( t ) B ^ ( t ) \hat{H}_{0}\to\hat{H}_{0}-h(t^{\prime})\hat{B}(t^{\prime})\,
  26. B ^ \hat{B}
  27. A ^ ( t ) . \hat{A}(t).
  28. χ ( t - t ) \chi(t-t^{\prime})
  29. χ ~ ( ω ) \tilde{\chi}(\omega)
  30. χ ~ ( ω ) \tilde{\chi}(\omega)

Link::cut_tree.html

  1. 1 / 2 {1}/{2}
  2. \leq
  3. O ( log 2 n ) O(\log^{2}n)
  4. Φ = v log s ( v ) \Phi=\sum_{v}\log{s(v)}
  5. c o s t ( s p l a y ( v ) ) 3 ( log s ( r o o t ( v ) ) - log s ( v ) ) + 1 cost(splay(v))\leq 3\left(\log{s(root(v))}-\log{s(v)}\right)+1
  6. s ( v ) s ( w ) s(v)\leq s(w)
  7. 3 ( log s ( R ) - log s ( v ) ) 3\left(\log{s(R)}-\log{s(v)}\right)

Linner_hue_index.html

  1. H L H_{L}
  2. H L H_{L}
  3. H L = 10 × log 10 ( A 510 / A 610 ) H_{L}=10\times\log_{10}(A_{510}/A_{610})

Liouville's_theorem_(conformal_mappings).html

  1. D f T D f = | det D f | 2 / n I Df^{T}Df=\left|\det Df\right|^{2/n}I
  2. f ( x ) = b + α A ( x - a ) | x - a | ϵ f(x)=b+\frac{\alpha A(x-a)}{|x-a|^{\epsilon}}

Liouville's_theorem_(differential_algebra).html

  1. e - x 2 , e^{-x^{2}},
  2. sin ( x ) x \frac{\sin(x)}{x}
  3. x x x^{x}
  4. a = c 1 D u 1 u 1 + + c n D u n u n + D v . a=c_{1}\frac{Du_{1}}{u_{1}}+\cdots+c_{n}\frac{Du_{n}}{u_{n}}+Dv.
  5. 1 x \frac{1}{x}
  6. 1 x 2 + 1 \frac{1}{x^{2}+1}
  7. e i θ = cos θ + i sin θ e - i θ = cos θ - i sin θ e 2 i θ = e i θ e - i θ = cos θ + i sin θ cos θ - i sin θ = 1 + i tan θ 1 - i tan θ 2 i θ = ln 1 + i tan θ 1 - i tan θ 2 i tan - 1 x = ln 1 + i x 1 - i x tan - 1 x = 1 2 i ln 1 + i x 1 - i x \begin{aligned}\displaystyle e^{i\theta}&\displaystyle=\cos\theta+i\sin\theta% \\ \displaystyle e^{-i\theta}&\displaystyle=\cos\theta-i\sin\theta\\ \displaystyle e^{2i\theta}&\displaystyle=\frac{e^{i\theta}}{e^{-i\theta}}=% \frac{\cos\theta+i\sin\theta}{\cos\theta-i\sin\theta}\\ &\displaystyle=\frac{1+i\tan\theta}{1-i\tan\theta}\\ \displaystyle 2i\theta&\displaystyle=\ln\frac{1+i\tan\theta}{1-i\tan\theta}\\ \displaystyle 2i\tan^{-1}x&\displaystyle=\ln\frac{1+ix}{1-ix}\\ \displaystyle\tan^{-1}x&\displaystyle=\frac{1}{2i}\ln\frac{1+ix}{1-ix}\end{aligned}

Liouville_dynamical_system.html

  1. T = 1 2 { u 1 ( q 1 ) + u 2 ( q 2 ) + + u s ( q s ) } { v 1 ( q 1 ) q ˙ 1 2 + v 2 ( q 2 ) q ˙ 2 2 + + v s ( q s ) q ˙ s 2 } T=\frac{1}{2}\left\{u_{1}(q_{1})+u_{2}(q_{2})+\cdots+u_{s}(q_{s})\right\}\left% \{v_{1}(q_{1})\dot{q}_{1}^{2}+v_{2}(q_{2})\dot{q}_{2}^{2}+\cdots+v_{s}(q_{s})% \dot{q}_{s}^{2}\right\}
  2. V = w 1 ( q 1 ) + w 2 ( q 2 ) + + w s ( q s ) u 1 ( q 1 ) + u 2 ( q 2 ) + + u s ( q s ) V=\frac{w_{1}(q_{1})+w_{2}(q_{2})+\cdots+w_{s}(q_{s})}{u_{1}(q_{1})+u_{2}(q_{2% })+\cdots+u_{s}(q_{s})}
  3. 2 Y d t = d φ 1 E χ 1 - ω 1 + γ 1 = d φ 2 E χ 2 - ω 2 + γ 2 = = d φ s E χ s - ω s + γ s \frac{\sqrt{2}}{Y}\,dt=\frac{d\varphi_{1}}{\sqrt{E\chi_{1}-\omega_{1}+\gamma_{% 1}}}=\frac{d\varphi_{2}}{\sqrt{E\chi_{2}-\omega_{2}+\gamma_{2}}}=\cdots=\frac{% d\varphi_{s}}{\sqrt{E\chi_{s}-\omega_{s}+\gamma_{s}}}
  4. γ s \gamma_{s}
  5. V ( x , y ) = - μ 1 ( x - a ) 2 + y 2 - μ 2 ( x + a ) 2 + y 2 . V(x,y)=\frac{-\mu_{1}}{\sqrt{\left(x-a\right)^{2}+y^{2}}}-\frac{\mu_{2}}{\sqrt% {\left(x+a\right)^{2}+y^{2}}}.
  6. x = a cosh ξ cos η , x=a\cosh\xi\cos\eta,
  7. y = a sinh ξ sin η , y=a\sinh\xi\sin\eta,
  8. V ( ξ , η ) = - μ 1 a ( cosh ξ - cos η ) - μ 2 a ( cosh ξ + cos η ) = - μ 1 ( cosh ξ + cos η ) - μ 2 ( cosh ξ - cos η ) a ( cosh 2 ξ - cos 2 η ) , V(\xi,\eta)=\frac{-\mu_{1}}{a\left(\cosh\xi-\cos\eta\right)}-\frac{\mu_{2}}{a% \left(\cosh\xi+\cos\eta\right)}=\frac{-\mu_{1}\left(\cosh\xi+\cos\eta\right)-% \mu_{2}\left(\cosh\xi-\cos\eta\right)}{a\left(\cosh^{2}\xi-\cos^{2}\eta\right)},
  9. T = m a 2 2 ( cosh 2 ξ - cos 2 η ) ( ξ ˙ 2 + η ˙ 2 ) . T=\frac{ma^{2}}{2}\left(\cosh^{2}\xi-\cos^{2}\eta\right)\left(\dot{\xi}^{2}+% \dot{\eta}^{2}\right).
  10. Y = cosh 2 ξ - cos 2 η Y=\cosh^{2}\xi-\cos^{2}\eta
  11. W = - μ 1 ( cosh ξ + cos η ) - μ 2 ( cosh ξ - cos η ) W=-\mu_{1}\left(\cosh\xi+\cos\eta\right)-\mu_{2}\left(\cosh\xi-\cos\eta\right)
  12. m a 2 2 ( cosh 2 ξ - cos 2 η ) 2 ξ ˙ 2 = E cosh 2 ξ + ( μ 1 + μ 2 a ) cosh ξ - γ \frac{ma^{2}}{2}\left(\cosh^{2}\xi-\cos^{2}\eta\right)^{2}\dot{\xi}^{2}=E\cosh% ^{2}\xi+\left(\frac{\mu_{1}+\mu_{2}}{a}\right)\cosh\xi-\gamma
  13. m a 2 2 ( cosh 2 ξ - cos 2 η ) 2 η ˙ 2 = - E cos 2 η + ( μ 1 - μ 2 a ) cos η + γ \frac{ma^{2}}{2}\left(\cosh^{2}\xi-\cos^{2}\eta\right)^{2}\dot{\eta}^{2}=-E% \cos^{2}\eta+\left(\frac{\mu_{1}-\mu_{2}}{a}\right)\cos\eta+\gamma
  14. d u = d ξ E cosh 2 ξ + ( μ 1 + μ 2 a ) cosh ξ - γ = d η - E cos 2 η + ( μ 1 - μ 2 a ) cos η + γ , du=\frac{d\xi}{\sqrt{E\cosh^{2}\xi+\left(\frac{\mu_{1}+\mu_{2}}{a}\right)\cosh% \xi-\gamma}}=\frac{d\eta}{\sqrt{-E\cos^{2}\eta+\left(\frac{\mu_{1}-\mu_{2}}{a}% \right)\cos\eta+\gamma}},
  15. u = d ξ E cosh 2 ξ + ( μ 1 + μ 2 a ) cosh ξ - γ = d η - E cos 2 η + ( μ 1 - μ 2 a ) cos η + γ . u=\int\frac{d\xi}{\sqrt{E\cosh^{2}\xi+\left(\frac{\mu_{1}+\mu_{2}}{a}\right)% \cosh\xi-\gamma}}=\int\frac{d\eta}{\sqrt{-E\cos^{2}\eta+\left(\frac{\mu_{1}-% \mu_{2}}{a}\right)\cos\eta+\gamma}}.
  16. r 1 2 r 2 2 ( d θ 1 d t ) ( d θ 2 d t ) - 2 c [ μ 1 cos θ 1 + μ 2 cos θ 2 ] , r_{1}^{2}r_{2}^{2}\left(\frac{d\theta_{1}}{dt}\right)\left(\frac{d\theta_{2}}{% dt}\right)-2c\left[\mu_{1}\cos\theta_{1}+\mu_{2}\cos\theta_{2}\right],
  17. φ r = d q r v r ( q r ) , \varphi_{r}=\int dq_{r}\sqrt{v_{r}(q_{r})},
  18. v 1 ( q 1 ) q ˙ 1 2 + v 2 ( q 2 ) q ˙ 2 2 + + v s ( q s ) q ˙ s 2 = φ ˙ 1 2 + φ ˙ 2 2 + + φ ˙ s 2 = F , v_{1}(q_{1})\dot{q}_{1}^{2}+v_{2}(q_{2})\dot{q}_{2}^{2}+\cdots+v_{s}(q_{s})% \dot{q}_{s}^{2}=\dot{\varphi}_{1}^{2}+\dot{\varphi}_{2}^{2}+\cdots+\dot{% \varphi}_{s}^{2}=F,
  19. Y = χ 1 ( φ 1 ) + χ 2 ( φ 2 ) + + χ s ( φ s ) , Y=\chi_{1}(\varphi_{1})+\chi_{2}(\varphi_{2})+\cdots+\chi_{s}(\varphi_{s}),
  20. T = 1 2 Y F . T=\frac{1}{2}YF.
  21. W = ω 1 ( φ 1 ) + ω 2 ( φ 2 ) + + ω s ( φ s ) , W=\omega_{1}(\varphi_{1})+\omega_{2}(\varphi_{2})+\cdots+\omega_{s}(\varphi_{s% }),
  22. V = W Y . V=\frac{W}{Y}.
  23. φ r \varphi_{r}
  24. d d t ( T φ ˙ r ) = d d t ( Y φ ˙ r ) = 1 2 F Y φ r - V φ r . \frac{d}{dt}\left(\frac{\partial T}{\partial\dot{\varphi}_{r}}\right)=\frac{d}% {dt}\left(Y\dot{\varphi}_{r}\right)=\frac{1}{2}F\frac{\partial Y}{\partial% \varphi_{r}}-\frac{\partial V}{\partial\varphi_{r}}.
  25. 2 Y φ ˙ r 2Y\dot{\varphi}_{r}
  26. 2 Y φ ˙ r d d t ( Y φ ˙ r ) = 2 T φ ˙ r Y φ r - 2 Y φ ˙ r V φ r = 2 φ ˙ r φ r [ ( E - V ) Y ] , 2Y\dot{\varphi}_{r}\frac{d}{dt}\left(Y\dot{\varphi}_{r}\right)=2T\dot{\varphi}% _{r}\frac{\partial Y}{\partial\varphi_{r}}-2Y\dot{\varphi}_{r}\frac{\partial V% }{\partial\varphi_{r}}=2\dot{\varphi}_{r}\frac{\partial}{\partial\varphi_{r}}% \left[(E-V)Y\right],
  27. d d t ( Y 2 φ ˙ r 2 ) = 2 E φ ˙ r Y φ r - 2 φ ˙ r W φ r = 2 E φ ˙ r d χ r d φ r - 2 φ ˙ r d ω r d φ r , \frac{d}{dt}\left(Y^{2}\dot{\varphi}_{r}^{2}\right)=2E\dot{\varphi}_{r}\frac{% \partial Y}{\partial\varphi_{r}}-2\dot{\varphi}_{r}\frac{\partial W}{\partial% \varphi_{r}}=2E\dot{\varphi}_{r}\frac{d\chi_{r}}{d\varphi_{r}}-2\dot{\varphi}_% {r}\frac{d\omega_{r}}{d\varphi_{r}},
  28. d d t ( Y 2 φ ˙ r 2 ) = 2 d d t ( E χ r - ω r ) , \frac{d}{dt}\left(Y^{2}\dot{\varphi}_{r}^{2}\right)=2\frac{d}{dt}\left(E\chi_{% r}-\omega_{r}\right),
  29. 1 2 Y 2 φ ˙ r 2 = E χ r - ω r + γ r , \frac{1}{2}Y^{2}\dot{\varphi}_{r}^{2}=E\chi_{r}-\omega_{r}+\gamma_{r},
  30. γ r \gamma_{r}
  31. r = 1 s γ r = 0. \sum_{r=1}^{s}\gamma_{r}=0.
  32. 2 Y d t = d φ 1 E χ 1 - ω 1 + γ 1 = d φ 2 E χ 2 - ω 2 + γ 2 = = d φ s E χ s - ω s + γ s . \frac{\sqrt{2}}{Y}dt=\frac{d\varphi_{1}}{\sqrt{E\chi_{1}-\omega_{1}+\gamma_{1}% }}=\frac{d\varphi_{2}}{\sqrt{E\chi_{2}-\omega_{2}+\gamma_{2}}}=\cdots=\frac{d% \varphi_{s}}{\sqrt{E\chi_{s}-\omega_{s}+\gamma_{s}}}.

Lissajous_knot.html

  1. x = cos ( n x t + ϕ x ) , y = cos ( n y t + ϕ y ) , z = cos ( n z t + ϕ z ) , x=\cos(n_{x}t+\phi_{x}),\qquad y=\cos(n_{y}t+\phi_{y}),\qquad z=\cos(n_{z}t+% \phi_{z}),
  2. n x n_{x}
  3. n y n_{y}
  4. n z n_{z}
  5. ϕ x \phi_{x}
  6. ϕ y \phi_{y}
  7. ϕ z \phi_{z}
  8. n x , n y , n z n_{x},n_{y},n_{z}
  9. n x ϕ y - n y ϕ x , n y ϕ z - n z ϕ y , n z ϕ x - n x ϕ z n_{x}\phi_{y}-n_{y}\phi_{x},\quad n_{y}\phi_{z}-n_{z}\phi_{y},\quad n_{z}\phi_% {x}-n_{x}\phi_{z}
  10. t = t + c t^{\prime}=t+c
  11. ϕ x \phi_{x}
  12. ϕ y \phi_{y}
  13. ϕ z \phi_{z}
  14. ϕ z = 0 \phi_{z}=0
  15. ( n x , n y , n z ) = ( 3 , 2 , 7 ) (n_{x},n_{y},n_{z})=(3,2,7)
  16. ( ϕ x , ϕ y ) = ( 0.7 , 0.2 ) (\phi_{x},\phi_{y})=(0.7,0.2)
  17. ( n x , n y , n z ) = ( 3 , 2 , 5 ) (n_{x},n_{y},n_{z})=(3,2,5)
  18. ( ϕ x , ϕ y ) = ( 1.5 , 0.2 ) (\phi_{x},\phi_{y})=(1.5,0.2)
  19. ( n x , n y , n z ) = ( 3 , 5 , 7 ) (n_{x},n_{y},n_{z})=(3,5,7)
  20. ( ϕ x , ϕ y ) = ( 0.7 , 1.0 ) (\phi_{x},\phi_{y})=(0.7,1.0)
  21. ( n x , n y , n z ) = ( 3 , 4 , 7 ) (n_{x},n_{y},n_{z})=(3,4,7)
  22. ( ϕ x , ϕ y ) = ( 0.1 , 0.7 ) (\phi_{x},\phi_{y})=(0.1,0.7)
  23. n x n_{x}
  24. n y n_{y}
  25. n z n_{z}
  26. n x n_{x}
  27. n y n_{y}
  28. n z n_{z}
  29. ( x , y , z ) ( - x , - y , - z ) (x,y,z)\mapsto(-x,-y,-z)
  30. n x n_{x}
  31. ( x , y , z ) ( x , - y , - z ) (x,y,z)\mapsto(x,-y,-z)

List_decoding.html

  1. 𝒞 \mathcal{C}
  2. ( n , k , d ) q (n,k,d)_{q}
  3. 𝒞 \mathcal{C}
  4. n n
  5. k k
  6. d d
  7. Σ \Sigma
  8. q q
  9. x Σ n x\in\Sigma^{n}
  10. e e
  11. x 1 , x 2 , , x m 𝒞 x_{1},x_{2},\ldots,x_{m}\in\mathcal{C}
  12. x x
  13. e e
  14. y y
  15. c c
  16. d d
  17. 𝒞 \mathcal{C}
  18. c 1 c_{1}
  19. c 2 c_{2}
  20. d d
  21. y y
  22. c 1 c_{1}
  23. c 2 c_{2}
  24. c 1 c_{1}
  25. c 2 c_{2}
  26. y y
  27. e e
  28. c c
  29. e e
  30. p n pn
  31. r r
  32. p p
  33. n n
  34. n n
  35. R R
  36. 1 - R 1-R
  37. 1 - R 1-R
  38. R R
  39. 0 p 1 0\leq p\leq 1
  40. L 1 L\geq 1
  41. 𝒞 Σ n \mathcal{C}\subseteq\Sigma^{n}
  42. p p
  43. L L
  44. y Σ n y\in\Sigma^{n}
  45. c C c\in C
  46. p n pn
  47. y y
  48. L L
  49. 𝒞 \mathcal{C}
  50. p p
  51. L L
  52. p p
  53. L L
  54. d / 2 d/2
  55. d / 2 d/2
  56. H q ( p ) = p log q ( q - 1 ) - p log q p - ( 1 - p ) log q ( 1 - p ) {H_{q}}(p)=p{\log_{q}}(q-1)-p{\log_{q}}p-(1-p){\log_{q}}\left({1-p}\right)
  57. 0 < p < 1 0<p<1
  58. 0 p 1 0\leq p\leq 1
  59. q 2 q\geq 2
  60. 0 p 1 - 1 / q 0\leq p\leq 1-1/q
  61. ϵ 0 \epsilon\geq 0
  62. n n
  63. R 1 - H q ( p ) - ϵ R\leq 1-H_{q}(p)-\epsilon
  64. p p
  65. O ( 1 / ϵ ) O(1/\epsilon)
  66. R 1 - H q ( p ) + ϵ R\geq 1-H_{q}(p)+\epsilon
  67. p p
  68. L L
  69. L = q Ω ( n ) L=q^{\Omega(n)}
  70. q q
  71. q S C p qSC_{p}
  72. B S C p BSC_{p}
  73. p p
  74. L L
  75. R 1 - H q ( p ) - 1 / L R\leq 1-H_{q}(p)-1/L
  76. L L
  77. y [ q ] n y\in[q]^{n}
  78. L + 1 L+1
  79. m 0 , , m L [ q ] k m_{0},\dots,m_{L}\in[q]^{k}
  80. 𝒞 ( m i ) B ( y , p n ) \mathcal{C}(m_{i})\in B(y,pn)
  81. 0 i L 0\leq i\leq L
  82. p p
  83. B ( y , p n ) B(y,pn)
  84. p n pn
  85. y y
  86. 𝒞 ( m i ) \mathcal{C}(m_{i})
  87. m i [ q ] k m_{i}\in[q]^{k}
  88. B ( y , p n ) B(y,pn)
  89. Pr [ C ( m i ) B ( y , p n ) ] = Vol q ( y , p n ) / q n q - n ( 1 - H q ( p ) ) , \Pr[C(m_{i})\in B(y,pn)]=\mathrm{Vol}_{q}(y,pn)/q^{n}\leq q^{-n(1-H_{q}(p))},
  90. V o l q ( y , p n ) Vol_{q}(y,pn)
  91. p n pn
  92. y y
  93. q H q ( p ) q^{H_{q}(p)}
  94. p p
  95. [ q ] n [q]^{n}
  96. ( y , m 0 , , m L ) (y,m_{0},\dots,m_{L})
  97. q - n ( L + 1 ) ( 1 - H q ( p ) ) q^{-n(L+1)(1-H_{q}(p))}
  98. 1 1
  99. y [ q ] n y\in[q]^{n}
  100. L L
  101. [ q ] k [q]^{k}
  102. y [ q ] n y\in[q]^{n}
  103. | 𝒞 B ( y , p n ) | |\mathcal{C}\cap B(y,pn)|
  104. R 1 - H q ( p ) + ϵ R\geq 1-H_{q}(p)+\epsilon
  105. c 𝒞 c\in\mathcal{C}
  106. y [ q ] n y\in[q]^{n}
  107. Pr [ c B ( y , p n ) ] = Pr [ y B ( c , p n ) ] \Pr[c\in B(y,pn)]=\Pr[y\in B(c,pn)]\,
  108. y y
  109. [ q ] n [q]^{n}
  110. Pr [ c B ( y , p n ) ] = Pr [ y B ( c , p n ) ] = Vol ( y , p n ) / q n q - n ( 1 - H q ( p ) ) - o ( n ) \Pr[c\in B(y,pn)]=\Pr[y\in B(c,pn)]=\mathrm{Vol}(y,pn)/q^{n}\geq q^{-n(1-H_{q}% (p))-o(n)}\,
  111. X c X_{c}
  112. X c = 1 if c B ( y , p n ) , X_{c}=1\,\text{ if }c\in B(y,pn),
  113. E [ | B ( y , p n ) | ] \displaystyle E[|B(y,pn)|]
  114. 1 - 1 - δ 1-\sqrt{1-\delta}
  115. δ \delta
  116. δ = 1 - R \delta=1-R
  117. 1 - R 1-\sqrt{R}
  118. 1 - 2 R 1-\sqrt{2R}
  119. 1 - R 1-\sqrt{R}
  120. 1 - R 1-\sqrt{R}
  121. R R
  122. m 1 m\geq 1
  123. 1 1
  124. 1 - R - ϵ 1-R-\epsilon
  125. ϵ > 0 \epsilon>0
  126. [ n , k + 1 ] q [n,k+1]_{q}
  127. ( α i , y i ) (\alpha_{i},y_{i})
  128. 1 i n 1\leq i\leq n
  129. y i y_{i}
  130. i i
  131. α i \alpha_{i}
  132. F q F_{q}
  133. e = n - t e=n-t
  134. P ( X ) F q [ X ] P(X)\in F_{q}[X]
  135. k k
  136. p ( α i ) = y i p(\alpha_{i})=y_{i}
  137. t t
  138. i i
  139. t t
  140. Q ( X , Y ) Q(X,Y)
  141. Q ( α i , y i ) = 0 Q(\alpha_{i},y_{i})=0
  142. 1 i n 1\leq i\leq n
  143. k k
  144. p ( X ) p(X)
  145. Y - p ( X ) Y-p(X)
  146. Q ( X , Y ) Q(X,Y)
  147. Q ( X , p ( X ) ) = 0 Q(X,p(X))=0
  148. p ( α i ) = y i p(\alpha_{i})=y_{i}
  149. t t
  150. i [ n ] i\in[n]
  151. p ( X ) p(X)

List_of_area_moments_of_inertia.html

  1. I x = π 4 r 4 I_{x}=\frac{\pi}{4}r^{4}
  2. I y = π 4 r 4 I_{y}=\frac{\pi}{4}r^{4}
  3. I z = π 2 r 4 I_{z}=\frac{\pi}{2}r^{4}
  4. I x = π 4 ( r 2 4 - r 1 4 ) I_{x}=\frac{\pi}{4}\left({r_{2}}^{4}-{r_{1}}^{4}\right)
  5. I y = π 4 ( r 2 4 - r 1 4 ) I_{y}=\frac{\pi}{4}\left({r_{2}}^{4}-{r_{1}}^{4}\right)
  6. I z = π 2 ( r 2 4 - r 1 4 ) I_{z}=\frac{\pi}{2}\left({r_{2}}^{4}-{r_{1}}^{4}\right)
  7. r r 1 r 2 r\equiv r_{1}\approx r_{2}
  8. r 2 r 1 + t r_{2}\equiv r_{1}+t
  9. ( r 2 4 - r 1 4 ) = ( ( r 1 + t ) 4 - r 1 4 ) = ( 4 r 1 3 t + 6 r 1 2 t 2 + 4 r 1 t 3 + t 4 ) \left(r_{2}^{4}-r_{1}^{4}\right)=\left(\left(r_{1}+t\right)^{4}-r_{1}^{4}% \right)=\left(4r_{1}^{3}t+6r_{1}^{2}t^{2}+4r_{1}t^{3}+t^{4}\right)
  10. r 1 t r_{1}>>t
  11. ( 4 r 1 3 t + 6 r 1 2 t 2 + 4 r 1 t 3 + t 4 ) 4 r 1 3 t \left(4r_{1}^{3}t+6r_{1}^{2}t^{2}+4r_{1}t^{3}+t^{4}\right)\approx 4r_{1}^{3}t
  12. I x = I y = π r 3 t I_{x}=I_{y}=\pi{r}^{3}{t}
  13. I 0 = ( θ - sin θ ) r 4 8 I_{0}=\left(\theta-\sin\theta\right)\frac{r^{4}}{8}
  14. θ \theta
  15. π \pi
  16. I 0 = ( π 8 - 8 9 π ) r 4 0.1098 r 4 I_{0}=\left(\frac{\pi}{8}-\frac{8}{9\pi}\right)r^{4}\approx 0.1098r^{4}
  17. I = π r 4 8 I=\frac{\pi r^{4}}{8}
  18. 4 r 3 π \frac{4r}{3\pi}
  19. I 0 = π r 4 8 I_{0}=\frac{\pi r^{4}}{8}
  20. I = π r 4 16 I=\frac{\pi r^{4}}{16}
  21. I 0 = ( π 16 - 4 9 π ) r 4 0.0549 r 4 I_{0}=\left(\frac{\pi}{16}-\frac{4}{9\pi}\right)r^{4}\approx 0.0549r^{4}
  22. 4 r 3 π \frac{4r}{3\pi}
  23. I x = π 4 a b 3 I_{x}=\frac{\pi}{4}ab^{3}
  24. I y = π 4 a 3 b I_{y}=\frac{\pi}{4}a^{3}b
  25. I x = b h 3 12 I_{x}=\frac{bh^{3}}{12}
  26. I y = b 3 h 12 I_{y}=\frac{b^{3}h}{12}
  27. I = b h 3 3 I=\frac{bh^{3}}{3}
  28. I = b h 3 12 + b h r 2 I=\frac{bh^{3}}{12}+bhr^{2}
  29. I 0 = b h 3 36 I_{0}=\frac{bh^{3}}{36}
  30. I = b h 3 12 I=\frac{bh^{3}}{12}
  31. I 0 = 5 3 16 a 4 I_{0}=\frac{5\sqrt{3}}{16}a^{4}
  32. I x = I y = t ( 5 L 2 - 5 L t + t 2 ) ( L 2 - L t + t 2 ) 12 ( 2 L - t ) I_{x}=I_{y}=\frac{t(5L^{2}-5Lt+t^{2})(L^{2}-Lt+t^{2})}{12(2L-t)}
  33. I ( x y ) = L 2 t ( L - t ) 2 4 ( t - 2 L ) I_{(}xy)=\frac{L^{2}t(L-t)^{2}}{4(t-2L)}
  34. I a = t ( 2 L - t ) ( 2 L 2 - 2 L t + t 2 ) 12 I_{a}=\frac{t(2L-t)(2L^{2}-2Lt+t^{2})}{12}
  35. I b = t ( 2 L 4 - 4 L 3 t + 8 L 2 t 2 - 6 L t 3 + t 4 ) 12 ( 2 L - t ) I_{b}=\frac{t(2L^{4}-4L^{3}t+8L^{2}t^{2}-6Lt^{3}+t^{4})}{12(2L-t)}
  36. I ( x y ) I_{(}xy)
  37. I z = I x + A r 2 I_{z}=I_{x}+Ar^{2}

List_of_Battlestar_Galactica_objects.html

  1. 5 × 10 14 J/kg 5\times 10^{14}\,\text{J/kg}
  2. 4.7 × 10 7 J/kg 4.7\times 10^{7}\,\text{J/kg}

List_of_deadly_earthquakes_since_1900.html

  1. M w M_{\mathrm{w}}
  2. M s M_{s}
  3. M w M_{\mathrm{w}}
  4. M s M_{s}
  5. M s M_{s}
  6. M w M_{\mathrm{w}}
  7. M s M_{s}
  8. M w M_{\mathrm{w}}
  9. M L M\text{L}
  10. M s M_{s}
  11. M s M_{s}
  12. M w M_{\mathrm{w}}
  13. m b m_{b}
  14. M s M_{s}
  15. M s M_{s}
  16. M s M_{s}
  17. M w M_{\mathrm{w}}
  18. M w M_{\mathrm{w}}
  19. M w M_{\mathrm{w}}
  20. M w M_{\mathrm{w}}
  21. M w M_{\mathrm{w}}
  22. M s M_{s}
  23. M s M_{s}
  24. m b m_{b}
  25. M w M_{\mathrm{w}}
  26. M s M_{s}
  27. M s M_{s}
  28. M s M_{s}
  29. m b m_{b}
  30. m b m_{b}
  31. m b m_{b}
  32. M w M_{\mathrm{w}}
  33. M w M_{\mathrm{w}}
  34. M w M_{\mathrm{w}}
  35. m b m_{b}
  36. M s M_{s}
  37. M w M_{\mathrm{w}}
  38. M w M_{\mathrm{w}}
  39. M s M_{s}
  40. M w M_{\mathrm{w}}
  41. M w M_{\mathrm{w}}
  42. M w M_{\mathrm{w}}
  43. M s M_{s}
  44. M s M_{s}
  45. M w M_{\mathrm{w}}
  46. M w M_{\mathrm{w}}
  47. M w M_{\mathrm{w}}
  48. M w M_{\mathrm{w}}
  49. M s M_{s}
  50. m b m_{b}
  51. m b m_{b}
  52. M s M_{s}
  53. M s M_{s}
  54. M s M_{s}
  55. M w M_{\mathrm{w}}
  56. M s M_{s}
  57. M w M_{\mathrm{w}}
  58. M w M_{\mathrm{w}}
  59. m b m_{b}
  60. M w M_{\mathrm{w}}
  61. m b m_{b}
  62. M w M_{\mathrm{w}}
  63. M s M_{s}
  64. M s M_{s}
  65. M w M_{\mathrm{w}}
  66. M w M_{\mathrm{w}}
  67. M w M_{\mathrm{w}}
  68. M w M_{\mathrm{w}}
  69. M w M_{\mathrm{w}}
  70. M w M_{\mathrm{w}}
  71. M s M_{s}
  72. M w M_{\mathrm{w}}
  73. M s M_{s}
  74. m b m_{b}
  75. m b m_{b}
  76. M s M_{s}
  77. m b m_{b}
  78. M s M_{s}
  79. M w M_{\mathrm{w}}
  80. M s M_{s}
  81. M w M_{\mathrm{w}}
  82. M w M_{\mathrm{w}}
  83. M w M_{\mathrm{w}}
  84. M w M_{\mathrm{w}}
  85. M w M_{\mathrm{w}}
  86. m b m_{b}
  87. M w M_{\mathrm{w}}
  88. M w M_{\mathrm{w}}
  89. M w M_{\mathrm{w}}
  90. M w M_{\mathrm{w}}
  91. M w M_{\mathrm{w}}
  92. M w M_{\mathrm{w}}
  93. M w M_{\mathrm{w}}
  94. M w M_{\mathrm{w}}
  95. M w M_{\mathrm{w}}
  96. M w M_{\mathrm{w}}
  97. M w M_{\mathrm{w}}
  98. M w M_{\mathrm{w}}
  99. M w M_{\mathrm{w}}
  100. M w M_{\mathrm{w}}
  101. M w M_{\mathrm{w}}
  102. M w M_{\mathrm{w}}
  103. M w M_{\mathrm{w}}
  104. M w M_{\mathrm{w}}
  105. M s M_{s}
  106. M w M_{\mathrm{w}}
  107. M s M_{s}
  108. m b m_{b}
  109. M w M_{\mathrm{w}}
  110. M w M_{\mathrm{w}}
  111. M w M_{\mathrm{w}}
  112. M w M_{\mathrm{w}}
  113. M w M_{\mathrm{w}}
  114. M w M_{\mathrm{w}}
  115. M s M_{s}
  116. M s M_{s}
  117. M w M_{\mathrm{w}}
  118. M w M_{\mathrm{w}}
  119. m b m_{b}
  120. M w M_{\mathrm{w}}
  121. M w M_{\mathrm{w}}
  122. M w M_{\mathrm{w}}
  123. M w M_{\mathrm{w}}
  124. M s M_{s}
  125. M w M_{\mathrm{w}}
  126. M w M_{\mathrm{w}}
  127. M w M_{\mathrm{w}}
  128. M w M_{\mathrm{w}}
  129. M w M_{\mathrm{w}}
  130. M w M_{\mathrm{w}}
  131. m b m_{b}
  132. M w M_{\mathrm{w}}
  133. M w M_{\mathrm{w}}
  134. M s M_{s}
  135. M w M_{\mathrm{w}}
  136. M w M_{\mathrm{w}}
  137. M s M_{s}
  138. M s M_{s}
  139. M s M_{s}
  140. m b m_{b}
  141. M w M_{\mathrm{w}}
  142. M s M_{s}
  143. M w M_{\mathrm{w}}
  144. M s M_{s}
  145. M w M_{\mathrm{w}}
  146. M w M_{\mathrm{w}}
  147. M w M_{\mathrm{w}}
  148. M w M_{\mathrm{w}}
  149. M s M_{s}
  150. m b m_{b}
  151. M s M_{s}
  152. M w M_{\mathrm{w}}
  153. M w M_{\mathrm{w}}
  154. m b m_{b}
  155. M w M_{\mathrm{w}}
  156. M w M_{\mathrm{w}}
  157. M s M_{s}
  158. M w M_{\mathrm{w}}
  159. M w M_{\mathrm{w}}
  160. M s M_{s}
  161. m b m_{b}
  162. M w M_{\mathrm{w}}
  163. M w M_{\mathrm{w}}
  164. M w M_{\mathrm{w}}
  165. M w M_{\mathrm{w}}
  166. m b m_{b}
  167. M w M_{\mathrm{w}}
  168. M w M_{\mathrm{w}}
  169. M w M_{\mathrm{w}}
  170. M s M_{s}
  171. M w M_{\mathrm{w}}
  172. M w M_{\mathrm{w}}
  173. m b m_{b}
  174. M w M_{\mathrm{w}}
  175. M w M_{\mathrm{w}}
  176. M w M_{\mathrm{w}}
  177. M s M_{s}
  178. M s M_{s}
  179. M s M_{s}
  180. M w M_{\mathrm{w}}
  181. M w M_{\mathrm{w}}
  182. M w M_{\mathrm{w}}
  183. M s M_{s}
  184. M s M_{s}
  185. M s M_{s}
  186. M s M_{s}
  187. M w M_{\mathrm{w}}
  188. M w M_{\mathrm{w}}
  189. M w M_{\mathrm{w}}
  190. M s M_{s}
  191. M s M_{s}
  192. M w M_{\mathrm{w}}
  193. M s M_{s}
  194. m b m_{b}
  195. M w M_{\mathrm{w}}
  196. M w M_{\mathrm{w}}
  197. M w M_{\mathrm{w}}
  198. M s M_{s}
  199. m b m_{b}
  200. M w M_{\mathrm{w}}
  201. M s M_{s}
  202. m b m_{b}
  203. M s M_{s}
  204. M s M_{s}
  205. M s M_{s}
  206. m b m_{b}
  207. M w M_{\mathrm{w}}
  208. m b m_{b}
  209. m b m_{b}
  210. M w M_{\mathrm{w}}
  211. M s M_{s}
  212. m b m_{b}
  213. m b m_{b}
  214. M w M_{\mathrm{w}}
  215. M w M_{\mathrm{w}}
  216. M w M_{\mathrm{w}}
  217. M w M_{\mathrm{w}}
  218. M s M_{s}
  219. M s M_{s}
  220. M s M_{s}
  221. M s M_{s}
  222. M w M_{\mathrm{w}}
  223. M s M_{s}
  224. M s M_{s}
  225. M s M_{s}
  226. M s M_{s}
  227. M w M_{\mathrm{w}}
  228. M s M_{s}
  229. m b m_{b}
  230. M w M_{\mathrm{w}}
  231. m b m_{b}
  232. m b m_{b}
  233. m b m_{b}
  234. m b m_{b}
  235. m b m_{b}
  236. m b m_{b}
  237. m b m_{b}
  238. M w M_{\mathrm{w}}
  239. M s M_{s}
  240. m b m_{b}
  241. m b m_{b}
  242. m b m_{b}
  243. m b m_{b}
  244. m b m_{b}
  245. m b m_{b}
  246. M w M_{\mathrm{w}}
  247. m b m_{b}
  248. M w M_{\mathrm{w}}
  249. M w M_{\mathrm{w}}
  250. m b m_{b}
  251. M L M\text{L}
  252. m b m_{b}
  253. m b m_{b}
  254. m b m_{b}
  255. m b m_{b}
  256. M w M_{\mathrm{w}}
  257. m b m_{b}
  258. M w M_{\mathrm{w}}
  259. M w M_{\mathrm{w}}
  260. m b m_{b}
  261. m b m_{b}
  262. m b m_{b}
  263. M w M_{\mathrm{w}}
  264. m b m_{b}
  265. m b m_{b}
  266. m b m_{b}
  267. m b m_{b}
  268. m b m_{b}
  269. M w M_{\mathrm{w}}
  270. M w M_{\mathrm{w}}
  271. M w M_{\mathrm{w}}
  272. M w M_{\mathrm{w}}
  273. M w M_{\mathrm{w}}
  274. M s M_{s}
  275. m b m_{b}
  276. M w M_{\mathrm{w}}
  277. M w M_{\mathrm{w}}
  278. M w M_{\mathrm{w}}
  279. M w M_{\mathrm{w}}
  280. M s M_{s}
  281. m b m_{b}
  282. m b m_{b}
  283. m b m_{b}
  284. m b m_{b}
  285. M w M_{\mathrm{w}}
  286. M s M_{s}
  287. M s M_{s}
  288. M s M_{s}
  289. M w M_{\mathrm{w}}
  290. M s M_{s}
  291. M s M_{s}
  292. M s M_{s}
  293. M s M_{s}
  294. M s M_{s}
  295. M s M_{s}
  296. M w M_{\mathrm{w}}
  297. M s M_{s}
  298. M s M_{s}
  299. M w M_{\mathrm{w}}
  300. m b m_{b}
  301. M s M_{s}
  302. M s M_{s}
  303. M s M_{s}
  304. M s M_{s}
  305. M w M_{\mathrm{w}}
  306. m b m_{b}
  307. M s M_{s}
  308. M s M_{s}
  309. M s M_{s}
  310. M s M_{s}
  311. m b m_{b}
  312. M w M_{\mathrm{w}}
  313. M w M_{\mathrm{w}}
  314. m b m_{b}
  315. m b m_{b}
  316. m b m_{b}
  317. M s M_{s}
  318. m b m_{b}
  319. m b m_{b}
  320. M w M_{\mathrm{w}}
  321. M s M_{s}
  322. M s M_{s}
  323. M s M_{s}
  324. M s M_{s}
  325. M s M_{s}
  326. M s M_{s}
  327. M s M_{s}
  328. m b m_{b}
  329. m b m_{b}
  330. M L M\text{L}
  331. M s M_{s}
  332. M s M_{s}
  333. M s M_{s}
  334. m b m_{b}
  335. m b m_{b}
  336. M s M_{s}
  337. M s M_{s}
  338. M s M_{s}
  339. M w M_{\mathrm{w}}
  340. M s M_{s}
  341. M s M_{s}
  342. M s M_{s}
  343. m b m_{b}
  344. m b m_{b}
  345. m b m_{b}
  346. M s M_{s}
  347. m b m_{b}
  348. M s M_{s}
  349. m b m_{b}
  350. M s M_{s}
  351. M w M_{\mathrm{w}}
  352. M w M_{\mathrm{w}}
  353. m b m_{b}
  354. M s M_{s}
  355. m b m_{b}
  356. m b m_{b}
  357. m b m_{b}
  358. m b m_{b}
  359. M s M_{s}
  360. M w M_{\mathrm{w}}
  361. M s M_{s}
  362. M w M_{\mathrm{w}}
  363. M w M_{\mathrm{w}}
  364. M w M_{\mathrm{w}}
  365. M w M_{\mathrm{w}}
  366. M w M_{\mathrm{w}}
  367. M w M_{\mathrm{w}}
  368. m b m_{b}
  369. M w M_{\mathrm{w}}
  370. M w M_{\mathrm{w}}
  371. M w M_{\mathrm{w}}
  372. M w M_{\mathrm{w}}
  373. M w M_{\mathrm{w}}
  374. M w M_{\mathrm{w}}
  375. M w M_{\mathrm{w}}
  376. m b m_{b}
  377. M w M_{\mathrm{w}}
  378. M s M_{s}
  379. M w M_{\mathrm{w}}
  380. M s M_{s}
  381. m b m_{b}
  382. M w M_{\mathrm{w}}
  383. M w M_{\mathrm{w}}
  384. M w M_{\mathrm{w}}
  385. M w M_{\mathrm{w}}
  386. M w M_{\mathrm{w}}
  387. m b m_{b}
  388. M s M_{s}
  389. M w M_{\mathrm{w}}
  390. M w M_{\mathrm{w}}
  391. M w M_{\mathrm{w}}
  392. M w M_{\mathrm{w}}
  393. M w M_{\mathrm{w}}
  394. M w M_{\mathrm{w}}
  395. M w M_{\mathrm{w}}
  396. M w M_{\mathrm{w}}
  397. M w M_{\mathrm{w}}
  398. M w M_{\mathrm{w}}
  399. M w M_{\mathrm{w}}
  400. M w M_{\mathrm{w}}
  401. M w M_{\mathrm{w}}
  402. M w M_{\mathrm{w}}
  403. M w M_{\mathrm{w}}
  404. M w M_{\mathrm{w}}
  405. M w M_{\mathrm{w}}
  406. M w M_{\mathrm{w}}
  407. M w M_{\mathrm{w}}
  408. M w M_{\mathrm{w}}
  409. M w M_{\mathrm{w}}
  410. m b m_{b}
  411. M w M_{\mathrm{w}}
  412. M w M_{\mathrm{w}}
  413. M w M_{\mathrm{w}}
  414. M L M\text{L}
  415. m b m_{b}
  416. M w M_{\mathrm{w}}
  417. M w M_{\mathrm{w}}
  418. M w M_{\mathrm{w}}
  419. M w M_{\mathrm{w}}
  420. M w M_{\mathrm{w}}
  421. M w M_{\mathrm{w}}
  422. M w M_{\mathrm{w}}
  423. M w M_{\mathrm{w}}
  424. M w M_{\mathrm{w}}
  425. M w M_{\mathrm{w}}
  426. m b m_{b}
  427. M w M_{\mathrm{w}}
  428. M w M_{\mathrm{w}}
  429. M w M_{\mathrm{w}}
  430. M w M_{\mathrm{w}}
  431. M w M_{\mathrm{w}}
  432. M w M_{\mathrm{w}}
  433. m b m_{b}
  434. M w M_{\mathrm{w}}
  435. M w M_{\mathrm{w}}
  436. M w M_{\mathrm{w}}
  437. M w M_{\mathrm{w}}
  438. M w M_{\mathrm{w}}
  439. M w M_{\mathrm{w}}
  440. M w M_{\mathrm{w}}
  441. M w M_{\mathrm{w}}
  442. M L M\text{L}
  443. M w M_{\mathrm{w}}
  444. M w M_{\mathrm{w}}
  445. M w M_{\mathrm{w}}
  446. M w M_{\mathrm{w}}
  447. M w M_{\mathrm{w}}
  448. M w M_{\mathrm{w}}
  449. M w M_{\mathrm{w}}
  450. M w M_{\mathrm{w}}
  451. M w M_{\mathrm{w}}
  452. M w M_{\mathrm{w}}
  453. M w M_{\mathrm{w}}
  454. M w M_{\mathrm{w}}
  455. M w M_{\mathrm{w}}
  456. m b m_{b}
  457. M w M_{\mathrm{w}}
  458. M w M_{\mathrm{w}}
  459. M L M\text{L}
  460. M w M_{\mathrm{w}}
  461. M w M_{\mathrm{w}}
  462. M w M_{\mathrm{w}}
  463. M w M_{\mathrm{w}}
  464. M L M\text{L}
  465. m b m_{b}
  466. M w M_{\mathrm{w}}
  467. M w M_{\mathrm{w}}
  468. M w M_{\mathrm{w}}
  469. M w M_{\mathrm{w}}
  470. M w M_{\mathrm{w}}
  471. M w M_{\mathrm{w}}
  472. M w M_{\mathrm{w}}
  473. M w M_{\mathrm{w}}
  474. M w M_{\mathrm{w}}
  475. M w M_{\mathrm{w}}
  476. M w M_{\mathrm{w}}
  477. M w M_{\mathrm{w}}
  478. M w M_{\mathrm{w}}
  479. M w M_{\mathrm{w}}
  480. M w M_{\mathrm{w}}
  481. m b m_{b}
  482. M w M_{\mathrm{w}}
  483. m b m_{b}
  484. M w M_{\mathrm{w}}
  485. m b m_{b}
  486. M w M_{\mathrm{w}}
  487. M w M_{\mathrm{w}}
  488. M w M_{\mathrm{w}}
  489. M w M_{\mathrm{w}}
  490. M w M_{\mathrm{w}}
  491. M w M_{\mathrm{w}}
  492. M w M_{\mathrm{w}}
  493. M w M_{\mathrm{w}}
  494. M w M_{\mathrm{w}}
  495. M w M_{\mathrm{w}}
  496. m b m_{b}
  497. M w M_{\mathrm{w}}
  498. M w M_{\mathrm{w}}
  499. M w M_{\mathrm{w}}
  500. M w M_{\mathrm{w}}
  501. M w M_{\mathrm{w}}
  502. M w M_{\mathrm{w}}
  503. M w M_{\mathrm{w}}
  504. M w M_{\mathrm{w}}
  505. M w M_{\mathrm{w}}
  506. M w M_{\mathrm{w}}
  507. m b m_{b}
  508. m b m_{b}
  509. M L M\text{L}
  510. M w M_{\mathrm{w}}
  511. M w M_{\mathrm{w}}
  512. M w M_{\mathrm{w}}
  513. M w M_{\mathrm{w}}
  514. M w M_{\mathrm{w}}
  515. M w M_{\mathrm{w}}
  516. M w M_{\mathrm{w}}
  517. M L M\text{L}
  518. M w M_{\mathrm{w}}
  519. M w M_{\mathrm{w}}
  520. M w M_{\mathrm{w}}
  521. M w M_{\mathrm{w}}
  522. M w M_{\mathrm{w}}
  523. M L M\text{L}
  524. M w M_{\mathrm{w}}
  525. M w M_{\mathrm{w}}
  526. M w M_{\mathrm{w}}
  527. M L M\text{L}
  528. M w M_{\mathrm{w}}
  529. M w M_{\mathrm{w}}
  530. M w M_{\mathrm{w}}
  531. M w M_{\mathrm{w}}
  532. M w M_{\mathrm{w}}
  533. M w M_{\mathrm{w}}
  534. M w M_{\mathrm{w}}
  535. M L M\text{L}
  536. M w M_{\mathrm{w}}
  537. M w M_{\mathrm{w}}
  538. M w M_{\mathrm{w}}
  539. M w M_{\mathrm{w}}
  540. M w M_{\mathrm{w}}
  541. M w M_{\mathrm{w}}
  542. M L M\text{L}
  543. M w M_{\mathrm{w}}
  544. M w M_{\mathrm{w}}
  545. M w M_{\mathrm{w}}
  546. m b m_{b}
  547. M w M_{\mathrm{w}}
  548. M w M_{\mathrm{w}}
  549. M w M_{\mathrm{w}}
  550. M w M_{\mathrm{w}}
  551. M w M_{\mathrm{w}}
  552. M w M_{\mathrm{w}}
  553. M w M_{\mathrm{w}}
  554. M w M_{\mathrm{w}}
  555. M w M_{\mathrm{w}}
  556. M w M_{\mathrm{w}}
  557. M w M_{\mathrm{w}}
  558. M w M_{\mathrm{w}}
  559. m b m_{b}
  560. M w M_{\mathrm{w}}
  561. M L M\text{L}
  562. M w M_{\mathrm{w}}
  563. M w M_{\mathrm{w}}
  564. M w M_{\mathrm{w}}
  565. M w M_{\mathrm{w}}
  566. M w M_{\mathrm{w}}
  567. M w M_{\mathrm{w}}
  568. M w M_{\mathrm{w}}
  569. M w M_{\mathrm{w}}
  570. M w M_{\mathrm{w}}
  571. M w M_{\mathrm{w}}
  572. M w M_{\mathrm{w}}
  573. M w M_{\mathrm{w}}
  574. M w M_{\mathrm{w}}
  575. M w M_{\mathrm{w}}
  576. M w M_{\mathrm{w}}
  577. M w M_{\mathrm{w}}
  578. M L M\text{L}
  579. M L M\text{L}
  580. m b m_{b}
  581. M w M_{\mathrm{w}}
  582. M w M_{\mathrm{w}}
  583. M w M_{\mathrm{w}}
  584. M w M_{\mathrm{w}}
  585. M w M_{\mathrm{w}}
  586. M L M\text{L}
  587. M w M_{\mathrm{w}}
  588. M w M_{\mathrm{w}}
  589. M L M\text{L}
  590. M w M_{\mathrm{w}}
  591. M w M_{\mathrm{w}}
  592. M w M_{\mathrm{w}}
  593. M w M_{\mathrm{w}}
  594. m b m_{b}
  595. M w M_{\mathrm{w}}
  596. M w M_{\mathrm{w}}
  597. M w M_{\mathrm{w}}
  598. M w M_{\mathrm{w}}
  599. M w M_{\mathrm{w}}
  600. M w M_{\mathrm{w}}
  601. M w M_{\mathrm{w}}
  602. M w M_{\mathrm{w}}
  603. M w M_{\mathrm{w}}
  604. m b m_{b}
  605. M w M_{\mathrm{w}}
  606. M L M\text{L}
  607. M w M_{\mathrm{w}}
  608. M w M_{\mathrm{w}}
  609. M w M_{\mathrm{w}}
  610. M w M_{\mathrm{w}}
  611. M w M_{\mathrm{w}}
  612. M w M_{\mathrm{w}}
  613. M w M_{\mathrm{w}}
  614. M L M\text{L}
  615. M L M\text{L}
  616. M w M_{\mathrm{w}}
  617. M w M_{\mathrm{w}}
  618. m b m_{b}
  619. M w M_{\mathrm{w}}
  620. m b m_{b}
  621. M w M_{\mathrm{w}}
  622. M w M_{\mathrm{w}}
  623. M w M_{\mathrm{w}}
  624. M w M_{\mathrm{w}}
  625. M w M_{\mathrm{w}}
  626. M w M_{\mathrm{w}}
  627. M w M_{\mathrm{w}}
  628. M w M_{\mathrm{w}}
  629. M w M_{\mathrm{w}}
  630. M w M_{\mathrm{w}}
  631. M w M_{\mathrm{w}}
  632. M w M_{\mathrm{w}}
  633. M w M_{\mathrm{w}}
  634. M L M\text{L}
  635. M w M_{\mathrm{w}}
  636. M w M_{\mathrm{w}}
  637. M w M_{\mathrm{w}}
  638. M w M_{\mathrm{w}}
  639. M w M_{\mathrm{w}}
  640. M w M_{\mathrm{w}}
  641. M s M_{s}
  642. m b m_{b}
  643. M w M_{\mathrm{w}}
  644. M w M_{\mathrm{w}}
  645. M d Md
  646. M w M_{\mathrm{w}}
  647. m b m_{b}
  648. M w M_{\mathrm{w}}
  649. M w M_{\mathrm{w}}
  650. M w M_{\mathrm{w}}
  651. M w M_{\mathrm{w}}
  652. M w M_{\mathrm{w}}
  653. M w M_{\mathrm{w}}
  654. M w M_{\mathrm{w}}
  655. M w M_{\mathrm{w}}
  656. M w M_{\mathrm{w}}
  657. M w M_{\mathrm{w}}
  658. M w M_{\mathrm{w}}
  659. M w M_{\mathrm{w}}
  660. M w M_{\mathrm{w}}
  661. M w M_{\mathrm{w}}
  662. M w M_{\mathrm{w}}
  663. M w M_{\mathrm{w}}
  664. M w M_{\mathrm{w}}
  665. M w M_{\mathrm{w}}
  666. M w M_{\mathrm{w}}
  667. M w M_{\mathrm{w}}
  668. M w M_{\mathrm{w}}
  669. M w M_{\mathrm{w}}
  670. M w M_{\mathrm{w}}
  671. M w M_{\mathrm{w}}
  672. M w M_{\mathrm{w}}
  673. M w M_{\mathrm{w}}
  674. M w M_{\mathrm{w}}
  675. M w M_{\mathrm{w}}
  676. M w M_{\mathrm{w}}
  677. M L M\text{L}
  678. M w M_{\mathrm{w}}
  679. M w M_{\mathrm{w}}
  680. M w M_{\mathrm{w}}
  681. M w M_{\mathrm{w}}
  682. m b m_{b}
  683. M w M_{\mathrm{w}}
  684. M w M_{\mathrm{w}}
  685. M w M_{\mathrm{w}}
  686. m b m_{b}
  687. M w M_{\mathrm{w}}
  688. M w M_{\mathrm{w}}
  689. M w M_{\mathrm{w}}
  690. M w M_{\mathrm{w}}
  691. M w M_{\mathrm{w}}
  692. M w M_{\mathrm{w}}
  693. M w M_{\mathrm{w}}
  694. M w M_{\mathrm{w}}
  695. M w M_{\mathrm{w}}
  696. M w M_{\mathrm{w}}
  697. M w M_{\mathrm{w}}
  698. m b m_{b}
  699. M w M_{\mathrm{w}}
  700. M w M_{\mathrm{w}}
  701. M w M_{\mathrm{w}}
  702. M w M_{\mathrm{w}}
  703. M w M_{\mathrm{w}}
  704. M w M_{\mathrm{w}}
  705. M s M_{s}
  706. M w M_{\mathrm{w}}
  707. M w M_{\mathrm{w}}
  708. M w M_{\mathrm{w}}
  709. M w M_{\mathrm{w}}
  710. M w M_{\mathrm{w}}
  711. m b m_{b}
  712. M w M_{\mathrm{w}}
  713. M w M_{\mathrm{w}}
  714. M w M_{\mathrm{w}}
  715. M d Md
  716. M w M_{\mathrm{w}}
  717. M w M_{\mathrm{w}}
  718. M w M_{\mathrm{w}}
  719. m b m_{b}
  720. M L M\text{L}
  721. M w M_{\mathrm{w}}
  722. M w M_{\mathrm{w}}
  723. M w M_{\mathrm{w}}
  724. M L M\text{L}
  725. m b m_{b}
  726. M w M_{\mathrm{w}}
  727. M w M_{\mathrm{w}}
  728. M w M_{\mathrm{w}}
  729. M w M_{\mathrm{w}}
  730. M w M_{\mathrm{w}}
  731. M w M_{\mathrm{w}}
  732. M w M_{\mathrm{w}}
  733. M w M_{\mathrm{w}}
  734. M w M_{\mathrm{w}}
  735. M w M_{\mathrm{w}}
  736. M w M_{\mathrm{w}}
  737. M w M_{\mathrm{w}}
  738. M w M_{\mathrm{w}}
  739. M w M_{\mathrm{w}}
  740. M w M_{\mathrm{w}}
  741. M w M_{\mathrm{w}}
  742. M w M_{\mathrm{w}}
  743. M w M_{\mathrm{w}}
  744. M w M_{\mathrm{w}}
  745. M w M_{\mathrm{w}}
  746. M w M_{\mathrm{w}}
  747. M w M_{\mathrm{w}}
  748. M w M_{\mathrm{w}}
  749. M w M_{\mathrm{w}}
  750. M w M_{\mathrm{w}}
  751. M w M_{\mathrm{w}}
  752. M w M_{\mathrm{w}}
  753. M w M_{\mathrm{w}}
  754. M w M_{\mathrm{w}}
  755. M w M_{\mathrm{w}}
  756. M w M_{\mathrm{w}}
  757. M w M_{\mathrm{w}}
  758. M w M_{\mathrm{w}}
  759. M L M\text{L}
  760. M w M_{\mathrm{w}}
  761. M w M_{\mathrm{w}}
  762. M w M_{\mathrm{w}}
  763. M w M_{\mathrm{w}}
  764. M w M_{\mathrm{w}}
  765. M w M_{\mathrm{w}}
  766. M w M_{\mathrm{w}}
  767. m b m_{b}
  768. M w M_{\mathrm{w}}
  769. M w M_{\mathrm{w}}
  770. M w M_{\mathrm{w}}
  771. m b m_{b}
  772. M w M_{\mathrm{w}}
  773. M w M_{\mathrm{w}}
  774. M w M_{\mathrm{w}}
  775. M w M_{\mathrm{w}}
  776. M w M_{\mathrm{w}}
  777. M w M_{\mathrm{w}}
  778. M w M_{\mathrm{w}}
  779. M w M_{\mathrm{w}}
  780. M w M_{\mathrm{w}}
  781. M w M_{\mathrm{w}}
  782. M w M_{\mathrm{w}}
  783. M w M_{\mathrm{w}}
  784. M w M_{\mathrm{w}}
  785. M w M_{\mathrm{w}}
  786. M w M_{\mathrm{w}}
  787. M w M_{\mathrm{w}}
  788. M w M_{\mathrm{w}}
  789. M w M_{\mathrm{w}}
  790. M w M_{\mathrm{w}}
  791. M w M_{\mathrm{w}}
  792. M w M_{\mathrm{w}}
  793. M w M_{\mathrm{w}}
  794. M w M_{\mathrm{w}}
  795. M w M_{\mathrm{w}}
  796. M w M_{\mathrm{w}}
  797. M w M_{\mathrm{w}}
  798. M w M_{\mathrm{w}}
  799. M w M_{\mathrm{w}}
  800. M w M_{\mathrm{w}}
  801. M w M_{\mathrm{w}}
  802. M w M_{\mathrm{w}}
  803. M w M_{\mathrm{w}}
  804. M w M_{\mathrm{w}}
  805. M w M_{\mathrm{w}}
  806. M w M_{\mathrm{w}}
  807. M w M_{\mathrm{w}}
  808. M w M_{\mathrm{w}}
  809. M w M_{\mathrm{w}}
  810. m b m_{b}
  811. M w M_{\mathrm{w}}
  812. M w M_{\mathrm{w}}
  813. M w M_{\mathrm{w}}
  814. m b m_{b}
  815. M w M_{\mathrm{w}}
  816. M w M_{\mathrm{w}}
  817. M w M_{\mathrm{w}}
  818. M w M_{\mathrm{w}}
  819. m b m_{b}
  820. M w M_{\mathrm{w}}
  821. M w M_{\mathrm{w}}
  822. M w M_{\mathrm{w}}
  823. M w M_{\mathrm{w}}
  824. M w M_{\mathrm{w}}
  825. M w M_{\mathrm{w}}
  826. M w M_{\mathrm{w}}
  827. M w M_{\mathrm{w}}
  828. M w M_{\mathrm{w}}
  829. M w M_{\mathrm{w}}
  830. M w M_{\mathrm{w}}
  831. M w M_{\mathrm{w}}
  832. M w M_{\mathrm{w}}
  833. M w M_{\mathrm{w}}
  834. M w M_{\mathrm{w}}
  835. M w M_{\mathrm{w}}
  836. M w M_{\mathrm{w}}
  837. M w M_{\mathrm{w}}
  838. M w M_{\mathrm{w}}
  839. M w M_{\mathrm{w}}
  840. M w M_{\mathrm{w}}
  841. M w M_{\mathrm{w}}
  842. M w M_{\mathrm{w}}
  843. M w M_{\mathrm{w}}
  844. M w M_{\mathrm{w}}
  845. M w M_{\mathrm{w}}
  846. M w M_{\mathrm{w}}
  847. M w M_{\mathrm{w}}
  848. M w M_{\mathrm{w}}
  849. M w M_{\mathrm{w}}
  850. M w M_{\mathrm{w}}
  851. M w M_{\mathrm{w}}
  852. M w M_{\mathrm{w}}
  853. M w M_{\mathrm{w}}
  854. M w M_{\mathrm{w}}
  855. m b m_{b}
  856. M w M_{\mathrm{w}}
  857. M w M_{\mathrm{w}}
  858. M w M_{\mathrm{w}}
  859. M w M_{\mathrm{w}}
  860. m b m_{b}
  861. m b m_{b}
  862. M w M_{\mathrm{w}}
  863. M w M_{\mathrm{w}}
  864. M w M_{\mathrm{w}}
  865. M w M_{\mathrm{w}}
  866. M w M_{\mathrm{w}}
  867. M w M_{\mathrm{w}}
  868. M w M_{\mathrm{w}}
  869. M w M_{\mathrm{w}}
  870. M w M_{\mathrm{w}}
  871. M w M_{\mathrm{w}}
  872. M w M_{\mathrm{w}}
  873. M w M_{\mathrm{w}}
  874. M L M\text{L}
  875. M w M_{\mathrm{w}}
  876. M w M_{\mathrm{w}}
  877. M w M_{\mathrm{w}}
  878. m b m_{b}
  879. M w M_{\mathrm{w}}
  880. M w M_{\mathrm{w}}
  881. M w M_{\mathrm{w}}
  882. M w M_{\mathrm{w}}
  883. m b m_{b}
  884. M w M_{\mathrm{w}}
  885. M w M_{\mathrm{w}}
  886. M d Md
  887. M w M_{\mathrm{w}}
  888. M w M_{\mathrm{w}}
  889. M w M_{\mathrm{w}}
  890. M w M_{\mathrm{w}}
  891. M w M_{\mathrm{w}}
  892. M w M_{\mathrm{w}}
  893. M w M_{\mathrm{w}}
  894. M w M_{\mathrm{w}}
  895. M w M_{\mathrm{w}}
  896. M d Md
  897. M w M_{\mathrm{w}}
  898. m b m_{b}
  899. M w M_{\mathrm{w}}
  900. M w M_{\mathrm{w}}
  901. M w M_{\mathrm{w}}
  902. M w M_{\mathrm{w}}
  903. M w M_{\mathrm{w}}
  904. M w M_{\mathrm{w}}
  905. M w M_{\mathrm{w}}
  906. M w M_{\mathrm{w}}
  907. M w M_{\mathrm{w}}
  908. M w M_{\mathrm{w}}
  909. M w M_{\mathrm{w}}
  910. M w M_{\mathrm{w}}
  911. m b m_{b}
  912. M w M_{\mathrm{w}}
  913. M w M_{\mathrm{w}}
  914. M w M_{\mathrm{w}}
  915. M w M_{\mathrm{w}}
  916. M w M_{\mathrm{w}}
  917. M w M_{\mathrm{w}}
  918. M L M\text{L}
  919. M w M_{\mathrm{w}}
  920. M w M_{\mathrm{w}}
  921. M w M_{\mathrm{w}}
  922. M w M_{\mathrm{w}}
  923. M w M_{\mathrm{w}}
  924. M w M_{\mathrm{w}}
  925. M w M_{\mathrm{w}}
  926. M w M_{\mathrm{w}}
  927. M w M_{\mathrm{w}}
  928. M w M_{\mathrm{w}}
  929. M L M\text{L}
  930. M w M_{\mathrm{w}}
  931. M w M_{\mathrm{w}}
  932. M w M_{\mathrm{w}}
  933. M w M_{\mathrm{w}}
  934. M w M_{\mathrm{w}}
  935. M w M_{\mathrm{w}}
  936. M w M_{\mathrm{w}}
  937. M w M_{\mathrm{w}}
  938. M w M_{\mathrm{w}}
  939. M w M_{\mathrm{w}}
  940. m b m_{b}
  941. M w M_{\mathrm{w}}
  942. M w M_{\mathrm{w}}
  943. m b m_{b}
  944. M w M_{\mathrm{w}}
  945. M w M_{\mathrm{w}}
  946. M w M_{\mathrm{w}}
  947. M w M_{\mathrm{w}}
  948. M w M_{\mathrm{w}}
  949. M w M_{\mathrm{w}}
  950. M w M_{\mathrm{w}}
  951. M w M_{\mathrm{w}}
  952. M w M_{\mathrm{w}}
  953. M w M_{\mathrm{w}}
  954. M w M_{\mathrm{w}}
  955. M w M_{\mathrm{w}}
  956. M w M_{\mathrm{w}}
  957. M w M_{\mathrm{w}}
  958. M w M_{\mathrm{w}}
  959. M w M_{\mathrm{w}}
  960. M w M_{\mathrm{w}}
  961. M w M_{\mathrm{w}}
  962. M w M_{\mathrm{w}}
  963. M w M_{\mathrm{w}}
  964. M w M_{\mathrm{w}}
  965. M w M_{\mathrm{w}}
  966. M w M_{\mathrm{w}}
  967. M w M_{\mathrm{w}}
  968. M w M_{\mathrm{w}}
  969. M w M_{\mathrm{w}}
  970. M w M_{\mathrm{w}}
  971. m b m_{b}
  972. M w M_{\mathrm{w}}
  973. M w M_{\mathrm{w}}
  974. M w M_{\mathrm{w}}
  975. m b m_{b}
  976. M w M_{\mathrm{w}}
  977. M w M_{\mathrm{w}}
  978. M w M_{\mathrm{w}}
  979. M w M_{\mathrm{w}}
  980. M w M_{\mathrm{w}}
  981. M w M_{\mathrm{w}}
  982. M w M_{\mathrm{w}}
  983. M w M_{\mathrm{w}}
  984. M w M_{\mathrm{w}}
  985. M w M_{\mathrm{w}}
  986. M w M_{\mathrm{w}}
  987. M w M_{\mathrm{w}}
  988. M w M_{\mathrm{w}}
  989. M w M_{\mathrm{w}}
  990. M w M_{\mathrm{w}}
  991. M w M_{\mathrm{w}}
  992. M w M_{\mathrm{w}}
  993. M w M_{\mathrm{w}}
  994. M w M_{\mathrm{w}}
  995. M w M_{\mathrm{w}}
  996. m b m_{b}
  997. M w M_{\mathrm{w}}
  998. M w M_{\mathrm{w}}
  999. M w M_{\mathrm{w}}
  1000. M d Md
  1001. M w M_{\mathrm{w}}
  1002. M w M_{\mathrm{w}}
  1003. M w M_{\mathrm{w}}
  1004. M w M_{\mathrm{w}}
  1005. m b m_{b}
  1006. M w M_{\mathrm{w}}
  1007. M w M_{\mathrm{w}}
  1008. M w M_{\mathrm{w}}
  1009. M w M_{\mathrm{w}}
  1010. M w M_{\mathrm{w}}
  1011. m b m_{b}
  1012. M w M_{\mathrm{w}}
  1013. M w M_{\mathrm{w}}
  1014. M w M_{\mathrm{w}}
  1015. M w M_{\mathrm{w}}
  1016. M w M_{\mathrm{w}}
  1017. M w M_{\mathrm{w}}
  1018. M w M_{\mathrm{w}}
  1019. M w M_{\mathrm{w}}
  1020. M w M_{\mathrm{w}}
  1021. M w M_{\mathrm{w}}
  1022. M w M_{\mathrm{w}}
  1023. M w M_{\mathrm{w}}
  1024. M w M_{\mathrm{w}}
  1025. m b m_{b}
  1026. M w M_{\mathrm{w}}
  1027. M w M_{\mathrm{w}}
  1028. m b m_{b}
  1029. M w M_{\mathrm{w}}
  1030. M w M_{\mathrm{w}}
  1031. M w M_{\mathrm{w}}
  1032. M w M_{\mathrm{w}}
  1033. M w M_{\mathrm{w}}
  1034. M L M\text{L}
  1035. M w M_{\mathrm{w}}
  1036. M w M_{\mathrm{w}}
  1037. M w M_{\mathrm{w}}
  1038. m b m_{b}
  1039. M w M_{\mathrm{w}}
  1040. m b m_{b}
  1041. M w M_{\mathrm{w}}
  1042. M w M_{\mathrm{w}}
  1043. M w M_{\mathrm{w}}
  1044. M w M_{\mathrm{w}}
  1045. M w M_{\mathrm{w}}
  1046. M w M_{\mathrm{w}}
  1047. M w M_{\mathrm{w}}
  1048. M w M_{\mathrm{w}}
  1049. M w M_{\mathrm{w}}
  1050. M w M_{\mathrm{w}}
  1051. M w M_{\mathrm{w}}
  1052. M w M_{\mathrm{w}}
  1053. M w M_{\mathrm{w}}
  1054. M L M\text{L}
  1055. M w M_{\mathrm{w}}
  1056. M w M_{\mathrm{w}}
  1057. M w M_{\mathrm{w}}
  1058. M w M_{\mathrm{w}}
  1059. M w M_{\mathrm{w}}
  1060. M w M_{\mathrm{w}}
  1061. M w M_{\mathrm{w}}
  1062. M w M_{\mathrm{w}}
  1063. M w M_{\mathrm{w}}
  1064. M w M_{\mathrm{w}}
  1065. M w M_{\mathrm{w}}
  1066. M w M_{\mathrm{w}}
  1067. M w M_{\mathrm{w}}
  1068. M w M_{\mathrm{w}}
  1069. M w M_{\mathrm{w}}
  1070. M w M_{\mathrm{w}}
  1071. M w M_{\mathrm{w}}
  1072. M w M_{\mathrm{w}}
  1073. M w M_{\mathrm{w}}
  1074. M w M_{\mathrm{w}}
  1075. M w M_{\mathrm{w}}
  1076. M w M_{\mathrm{w}}
  1077. M w M_{\mathrm{w}}
  1078. M w M_{\mathrm{w}}
  1079. M w M_{\mathrm{w}}
  1080. M w M_{\mathrm{w}}
  1081. M w M_{\mathrm{w}}
  1082. M w M_{\mathrm{w}}
  1083. M w M_{\mathrm{w}}
  1084. m b m_{b}
  1085. M w M_{\mathrm{w}}
  1086. M w M_{\mathrm{w}}
  1087. M w M_{\mathrm{w}}
  1088. m b m_{b}
  1089. M w M_{\mathrm{w}}
  1090. M w M_{\mathrm{w}}
  1091. M w M_{\mathrm{w}}
  1092. M w M_{\mathrm{w}}
  1093. M w M_{\mathrm{w}}
  1094. M w M_{\mathrm{w}}
  1095. M w M_{\mathrm{w}}
  1096. M w M_{\mathrm{w}}
  1097. m b m_{b}
  1098. M w M_{\mathrm{w}}
  1099. M s M_{s}
  1100. M w M_{\mathrm{w}}
  1101. M w M_{\mathrm{w}}
  1102. M w M_{\mathrm{w}}
  1103. M w M_{\mathrm{w}}
  1104. M w M_{\mathrm{w}}
  1105. M w M_{\mathrm{w}}
  1106. M w M_{\mathrm{w}}
  1107. M w M_{\mathrm{w}}
  1108. M w M_{\mathrm{w}}
  1109. M w M_{\mathrm{w}}
  1110. M w M_{\mathrm{w}}
  1111. M w M_{\mathrm{w}}
  1112. M w M_{\mathrm{w}}
  1113. M w M_{\mathrm{w}}
  1114. M w M_{\mathrm{w}}
  1115. M w M_{\mathrm{w}}
  1116. M w M_{\mathrm{w}}
  1117. M w M_{\mathrm{w}}
  1118. M w M_{\mathrm{w}}
  1119. M w M_{\mathrm{w}}
  1120. M w M_{\mathrm{w}}
  1121. M w M_{\mathrm{w}}
  1122. M w M_{\mathrm{w}}