wpmath0000016_8

Linalool_dehydratase.html

  1. \rightleftharpoons

LINCOA.html

  1. m m
  2. n + 2 n+2
  3. ( n + 1 ) ( n + 2 ) / 2 (n+1)(n+2)/2
  4. m m
  5. n + 6 n+6
  6. 2 n + 1 2n+1

Linear_equation_over_a_ring.html

  1. a 1 x 1 + + a k x k = b a_{1}x_{1}+\cdots+a_{k}x_{k}=b
  2. a 1 , , a k a_{1},\ldots,a_{k}
  3. b b
  4. R R
  5. x 1 , , x k x_{1},\ldots,x_{k}
  6. R R
  7. b b
  8. b = 1 b=1
  9. a a
  10. R R
  11. a 1 , , a k a_{1},\ldots,a_{k}
  12. R R
  13. ( a 1 , , a k ) , (a_{1},\ldots,a_{k}),
  14. ( x 1 , , x k ) (x_{1},\ldots,x_{k})
  15. R R
  16. a 1 x 1 + + a k x k = 0. a_{1}x_{1}+\cdots+a_{k}x_{k}=0.
  17. a x = 0 ax=0
  18. a x = 1 ax=1
  19. [ s t u v ] \begin{bmatrix}s&t\\ u&v\end{bmatrix}
  20. [ s t u v ] [ a b ] = [ gcd ( a , b ) 0 ] . \begin{bmatrix}s&t\\ u&v\end{bmatrix}\begin{bmatrix}a\\ b\end{bmatrix}=\begin{bmatrix}\gcd(a,b)\\ 0\end{bmatrix}.

Linear_function_(calculus).html

  1. x x
  2. f ( x ) = a x + b s i z e = 120 % f(x)=ax+bsize=120\%
  3. x x
  4. ( x , f ( x ) ) (x,f(x))
  5. a a
  6. a x + b ax+b
  7. f ( x ) = b f(x)=b
  8. x x
  9. 𝐑 \mathbf{R}
  10. x x
  11. x x
  12. f ( x ) f(x)
  13. f f
  14. a a
  15. b b
  16. y y
  17. ( 0 , b ) (0,b)
  18. x x
  19. ( b a , 0 ) (−\frac{b}{a}, 0)
  20. b a −\frac{b}{a}
  21. f ( x ) = a x + b f(x)=ax+b
  22. a a
  23. f ( x ) = a x + b f(x)=ax+b
  24. a a
  25. x x
  26. a a
  27. a a
  28. a a
  29. c c
  30. c a ca
  31. y = a x + b y=ax+b
  32. x x
  33. y y
  34. a a
  35. b b
  36. y y
  37. x x
  38. y y
  39. f ( x ) = a x + b f(x)=ax+b
  40. y = f ( x ) y=f(x)
  41. f f
  42. f f
  43. y y
  44. f ( x ) f(x)
  45. y = f ( x ) = a x + b y=f(x)=ax+b
  46. y y
  47. b b
  48. y y
  49. y y
  50. y y
  51. a a
  52. a a
  53. x x
  54. y y
  55. a a
  56. f ( x + 1 ) = f ( x ) + a f(x+1)=f(x)+a
  57. f ( x ) = a x + b f(x)=ax+b
  58. y = 2 x + 4 y=−2x+4
  59. a = 2 a=−2
  60. b = 4 b=4
  61. ( 0 , b ) = ( 0 , 4 ) (0,b)=(0, 4)
  62. y y
  63. ( b a , 0 ) = (−\frac{b}{a}, 0)=
  64. ( 4 2 , 0 ) = ( 2 , 0 ) (\frac{−4}{−2}, 0)=(2, 0)
  65. x x
  66. a = 2 a=−2
  67. x x
  68. y y
  69. A x + B y = C s i z e = 120 % Ax+By=Csize=120\%
  70. B 0 B≠0
  71. y y
  72. y = ( A B ) x + ( C B ) = f ( x ) y=−(\frac{A}{B})x+(\frac{C}{B})=f(x)
  73. a 0 a≠0
  74. l o g ( g ( x ) ) log(g(x))
  75. x x
  76. g g
  77. l o g ( y ) log(y)
  78. l o g ( x ) log(x)
  79. log r y = a log r x + b y = r b x a \log_{r}y=a\log_{r}x+b\quad\Rightarrow\quad y=r^{b}\cdot x^{a}
  80. r = f ( φ ) = a φ + b r=f(\varphi)=a\varphi+b
  81. a 0 a\neq 0

Linear_optical_quantum_computing.html

  1. N × N N\times N
  2. U ( N ) U(N)
  3. U ( N ) U(N)
  4. N N
  5. N N
  6. 𝒪 ( N 2 ) \mathcal{O}(N^{2})
  7. | 000 |000\cdots\rangle
  8. α | 0 + β | 1 \alpha|0\rangle+\beta|1\rangle
  9. | α | 2 |\alpha|^{2}
  10. | 0 |0\rangle
  11. | β | 2 |\beta|^{2}
  12. | 1 |1\rangle
  13. | α | 2 + | β | 2 = 1 |\alpha|^{2}+|\beta|^{2}=1
  14. | 0 |0\rangle
  15. | 1 |1\rangle
  16. | n , n = 0 , 1 , 2 , |n\rangle,n=0,1,2,\cdots
  17. n n
  18. | 01 V H | 0 V | 1 H |01\rangle_{VH}\equiv|0\rangle_{V}|1\rangle_{H}
  19. V V
  20. H H
  21. | 1 |1\rangle
  22. t t
  23. S U ( 2 ) SU(2)
  24. 𝐁 θ , ϕ \mathbf{B}_{\theta,\phi}
  25. U ( 𝐁 θ , ϕ ) = [ cos θ - e i ϕ sin θ e - i ϕ sin θ cos θ ] , U(\mathbf{B}_{\theta,\phi})=\begin{bmatrix}\cos\theta&-e^{i\phi}\sin\theta\\ e^{-i\phi}\sin\theta&\cos\theta\end{bmatrix}\,,
  26. θ \theta
  27. ϕ \phi
  28. r r
  29. t t
  30. ϕ = π 2 \phi=\frac{\pi}{2}
  31. | t | 2 + | r | 2 = 1 |t|^{2}+|r|^{2}=1
  32. t * r + t r * = 0 t^{*}r+tr^{*}=0
  33. U ( 𝐁 θ , ϕ = π 2 ) = [ t r r t ] = [ cos θ - i sin θ - i sin θ cos θ ] = cos θ I ^ - i sin θ σ ^ x = e - i θ σ ^ x , U(\mathbf{B}_{\theta,\phi=\frac{\pi}{2}})=\begin{bmatrix}t&r\\ r&t\end{bmatrix}=\begin{bmatrix}\cos\theta&-i\sin\theta\\ -i\sin\theta&\cos\theta\end{bmatrix}=\cos\theta\hat{I}-i\sin\theta\hat{\sigma}% _{x}=e^{-i\theta\hat{\sigma}_{x}}\,,
  34. x x
  35. 2 θ = 2 cos - 1 ( | t | ) 2\theta=2\cos^{-1}(|t|)
  36. σ ^ x \hat{\sigma}_{x}
  37. R ( θ ) = [ cos θ - sin θ sin θ cos θ ] . R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}\,.
  38. θ = 45 \theta=45^{\circ}
  39. 𝐏 ϕ \mathbf{P}_{\phi}
  40. U ( 𝐏 ϕ ) = e i ϕ U(\mathbf{P}_{\phi})=e^{i\phi}
  41. U ( 𝐏 ϕ ) = [ e i ϕ 0 0 1 ] = [ e i ϕ / 2 0 0 e - i ϕ / 2 ] (global phase ignored) = e i ϕ 2 σ ^ z U(\mathbf{P}_{\phi})=\begin{bmatrix}e^{i\phi}&0\\ 0&1\end{bmatrix}=\begin{bmatrix}e^{i\phi/2}&0\\ 0&e^{-i\phi/2}\end{bmatrix}\,\text{(global phase ignored)}=e^{i\frac{\phi}{2}% \hat{\sigma}_{z}}
  42. - ϕ -\phi
  43. z z
  44. S U ( 2 ) SU(2)
  45. S U ( 2 ) SU(2)
  46. θ \theta
  47. ϕ \phi
  48. ϕ \phi
  49. | 0 \left|0\right\rangle
  50. | 1 \left|1\right\rangle
  51. | 10 2 , 3 |10\rangle_{2,3}
  52. x x
  53. x = - 1 x=-1
  54. θ 1 = 22.5 \theta_{1}=22.5^{\circ}
  55. ϕ 1 = 0 \phi_{1}=0^{\circ}
  56. θ 2 = 65.5302 \theta_{2}=65.5302^{\circ}
  57. ϕ 2 = 0 \phi_{2}=0^{\circ}
  58. θ 3 = - 22.5 \theta_{3}=-22.5^{\circ}
  59. ϕ 3 = 0 \phi_{3}=0^{\circ}
  60. ϕ 4 = 180 \phi_{4}=180^{\circ}
  61. x = e i π / 2 x=e^{i\pi/2}
  62. θ 1 = 36.53 \theta_{1}=36.53^{\circ}
  63. ϕ 1 = 88.24 \phi_{1}=88.24^{\circ}
  64. θ 2 = 62.25 \theta_{2}=62.25^{\circ}
  65. ϕ 2 = - 66.53 \phi_{2}=-66.53^{\circ}
  66. θ 3 = - 36.53 \theta_{3}=-36.53^{\circ}
  67. ϕ 3 = - 11.25 \phi_{3}=-11.25^{\circ}
  68. ϕ 4 = 102.24 \phi_{4}=102.24^{\circ}
  69. x = - 1 x=-1
  70. 1 / 4 1/4
  71. p N p^{N}
  72. N N
  73. p p
  74. p - N p^{-N}
  75. p - N p^{-N}
  76. | Φ + |\Phi^{+}\rangle
  77. n n
  78. n n
  79. n 2 ( n + 1 ) 2 \frac{n^{2}}{(n+1)^{2}}
  80. n n
  81. F = 0.82 ± 0.01 F=0.82\pm 0.01
  82. n n
  83. n + 1 n+1
  84. 10 4 10^{4}

Link-centric_preferential_attachment.html

  1. n n
  2. P ( n ) = c n + c P(n)={c\over n+c}\,
  3. 1 - P ( n ) = n n + c . 1-P(n)={n\over n+c}.
  4. n n

Liouvillian_function.html

  1. 1 / x 1/x
  2. erf ( x ) = 2 π 0 x e - t 2 d t , \mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\,dt,

Liquid_droplet_radiator.html

  1. q ˙ = ( 4 π a 2 ) σ F T 4 \dot{q}=(4\pi a^{2})\sigma FT^{4}
  2. σ \sigma
  3. q ˙ \dot{q}
  4. a a
  5. F F
  6. T T
  7. ( 4 π a 2 ) σ F T 4 = - c ρ 4 π a 4 3 d T d t (4\pi a^{2})\sigma FT^{4}=-c\rho\frac{4\pi a^{4}}{3}\frac{dT}{dt}
  8. c c
  9. ρ \rho
  10. t t

List_of_countries_by_proportions_of_national_flags.html

  1. 3 2 \frac{3}{2}
  2. 75 28 ( 7 5 15 ) 1.7477 \frac{75}{28}⋅(7\sqrt{5}−15)≈1.7477
  3. φ = 1 + 5 2 1.618034 φ=1+\frac{\sqrt{5}}{2}≈1.618034
  4. 3 1 2 = 3 2 3⋅\frac{1}{2}=\frac{3}{2}
  5. 2 + 78 + 2 2 + 40 + 2 = 82 44 = 41 22 \frac{2+78+2}{2+40+2}=\frac{82}{44}=\frac{41}{22}

List_of_earthquakes_in_South_Africa.html

  1. M L M_{L}
  2. M L M_{L}

List_of_finite-dimensional_Nichols_algebras.html

  1. U q ( 𝔤 ) + U_{q}(\mathfrak{g})^{+}
  2. u q ( 𝔤 ) + u_{q}(\mathfrak{g})^{+}
  3. 𝔅 ( V ) \mathfrak{B}(V)
  4. V V
  5. G G
  6. V V
  7. G G
  8. V V
  9. x i x j q i j x j x i x_{i}\otimes x_{j}\mapsto q_{ij}x_{j}\otimes x_{i}
  10. G G
  11. V = i I V i V=\bigoplus_{i\in I}V_{i}
  12. V V
  13. V i = 𝒪 [ g ] χ V_{i}=\mathcal{O}_{[g]}^{\chi}
  14. [ g ] G [g]\subset G
  15. χ \chi
  16. Cent ( g ) \operatorname{Cent}(g)
  17. V i V_{i}
  18. 𝔅 ( V ) \mathfrak{B}(V)
  19. ( n ) t := 1 + t + t 2 + + t n - 1 (n)_{t}:=1+t+t^{2}+\cdots+t^{n-1}
  20. 0
  21. V V
  22. V V
  23. 𝔸 n 5 \mathbb{A}_{n\geq 5}
  24. 𝕊 n 6 \mathbb{S}_{n\geq 6}
  25. P S L n ( 𝔽 q ) PSL_{n}(\mathbb{F}_{q})
  26. S p 2 n ( 𝔽 q ) Sp_{2n}(\mathbb{F}_{q})
  27. F i 22 Fi_{22}\;
  28. 22 A , 22 B 22A,22B\;
  29. 16 C , 16 D , 32 A , 32 B , 32 C , 32 D , 34 A , 46 A , 46 B 16C,\;16D,\;32A,\;32B,\;32C,\;32D,\;34A,\;46A,\;46B\;
  30. 32 A , 32 B , 46 A , 46 B , 92 A , 92 B , 94 A , 94 B 32A,\;32B,\;46A,\;46B,\;92A,\;92B,\;94A,\;94B\;
  31. q i j q_{ij}
  32. q i i , q i j q j i q_{ii},q_{ij}q_{ji}
  33. u q ( 𝔤 ) + u_{q}(\mathfrak{g})^{+}
  34. q i j = q ( α i , α j ) q_{ij}=q^{(\alpha_{i},\alpha_{j})}
  35. ( W , S ) (W,S)
  36. 𝔅 ( V ) \mathfrak{B}(V)
  37. A 1 A_{1}
  38. A 1 A_{1}
  39. A 1 A_{1}
  40. A 2 A_{2}
  41. V V
  42. 3 3
  43. 6 6
  44. 10 10
  45. 2 + 2 2+2
  46. 12 12
  47. 576 = 24 2 576\;=24^{2}
  48. 8294400 8294400
  49. 64 64
  50. ( 2 ) t 2 ( 3 ) t (2)_{t}^{2}(3)_{t}
  51. ( 2 ) t 2 ( 3 ) t 2 ( 4 ) t 2 (2)^{2}_{t}(3)^{2}_{t}(4)^{2}_{t}
  52. ( 4 ) t 4 ( 5 ) t 2 ( 6 ) t 4 (4)_{t}^{4}(5)_{t}^{2}(6)_{t}^{4}
  53. ( 2 ) t 4 ( 2 ) t 2 2 (2)^{4}_{t}(2)^{2}_{t^{2}}
  54. 𝕊 3 \;\mathbb{S}_{3}
  55. 𝕊 4 \;\mathbb{S}_{4}
  56. 𝕊 5 \;\mathbb{S}_{5}
  57. 𝔻 4 \;\mathbb{D}_{4}
  58. 𝒪 ( 12 ) - 1 \mathcal{O}_{(12)}^{-1}
  59. 𝒪 ( 1234 ) - 1 , 𝒪 ( 12 ) - 1 , 𝒪 ( 12 ) - 1 sgn \mathcal{O}_{(1234)}^{-1},\quad\mathcal{O}_{(12)}^{-1},\quad\mathcal{O}_{(12)}% ^{-1\otimes\operatorname{sgn}}
  60. 𝒪 ( 12 ) - 1 , 𝒪 ( 12 ) - 1 sgn \mathcal{O}_{(12)}^{-1},\quad\mathcal{O}_{(12)}^{-1\otimes\operatorname{sgn}}
  61. 𝒪 b ϵ 𝒪 a 2 b ϵ , 𝒪 b s g n 𝒪 a 2 b s g n \mathcal{O}_{b}^{\epsilon}\oplus\mathcal{O}^{\epsilon}_{a^{2}b},\quad\mathcal{% O}_{b}^{sgn}\oplus\mathcal{O}^{sgn}_{a^{2}b}
  62. Γ 2 \Gamma_{2}
  63. A n A_{n}
  64. A n A n A_{n}\cup A_{n}
  65. 𝕊 2 2 \mathbb{S}_{2}\cong\mathbb{Z}_{2}
  66. u i ( A 1 ) + u_{i}(A_{1})^{+}
  67. 𝔅 ( V ) \mathfrak{B}(V)
  68. A 1 A_{1}
  69. A 1 A_{1}
  70. A 1 A_{1}
  71. A 1 A_{1}
  72. V V
  73. 4 4
  74. 4 4
  75. 5 5
  76. 7 7
  77. 72 72
  78. 5 , 184 5,184
  79. 1 , 280 1,280
  80. 326 , 592 326,592
  81. ( 2 ) t 2 ( 3 ) t ( 6 ) t (2)^{2}_{t}(3)_{t}(6)_{t}
  82. ( 6 ) t 4 ( 2 ) t 2 2 (6)^{4}_{t}(2)^{2}_{t^{2}}
  83. ( 4 ) t 4 ( 5 ) t (4)^{4}_{t}(5)_{t}
  84. ( 6 ) t 6 ( 7 ) t (6)^{6}_{t}(7)_{t}
  85. S L 2 ( 3 ) SL_{2}(3)
  86. 𝔸 4 \mathbb{A}_{4}
  87. 5 5 × \mathbb{Z}_{5}\rtimes\mathbb{Z}_{5}^{\times}
  88. 7 7 × \mathbb{Z}_{7}\rtimes\mathbb{Z}_{7}^{\times}
  89. 𝒪 ( - 1 1 0 - 1 ) - 1 \mathcal{O}_{\begin{pmatrix}-1&1\\ 0&-1\end{pmatrix}}^{-1}
  90. 𝒪 ( 1 1 0 1 ) ζ 6 \mathcal{O}_{\begin{pmatrix}1&1\\ 0&1\end{pmatrix}}^{\zeta_{6}}
  91. 𝒪 i 2 - 1 , 𝒪 i 3 - 1 \mathcal{O}_{i\rtimes 2}^{-1},\quad\mathcal{O}_{i\rtimes 3}^{-1}
  92. 𝒪 i 3 - 1 , 𝒪 i 5 - 1 \mathcal{O}_{i\rtimes 3}^{-1},\quad\mathcal{O}_{i\rtimes 5}^{-1}
  93. 𝔅 ( V ) \mathfrak{B}(V)
  94. B 2 , B 2 B_{2},B_{2}
  95. B 2 , B 2 B_{2},B_{2}
  96. ( 2 - 2 - 2 2 ) \begin{pmatrix}2&-2\\ -2&2\end{pmatrix}
  97. ( 2 - 2 - 1 2 ) \begin{pmatrix}2&-2\\ -1&2\end{pmatrix}
  98. ( 2 - 4 - 1 2 ) \begin{pmatrix}2&-4\\ -1&2\end{pmatrix}
  99. V V
  100. 3 + 2 3+2
  101. 3 + 1 3+1
  102. 3 + 2 3+2
  103. 3 + 1 3+1
  104. 3 + 2 3+2
  105. 2 , 304 2,304
  106. 10 , 368 10,368
  107. 2 , 239 , 488 2,239,488
  108. ( 2 ) t 2 ( 3 ) t ( 2 ) t 2 (2)_{t}^{2}(3)_{t}\cdot(2)_{t}^{2}
  109. ( 2 ) t 2 ( 3 ) t ( 6 ) t (2)_{t}^{2}(3)_{t}\cdot(6)_{t}
  110. ( 2 ) t 2 2 ( 3 ) t 2 ( 2 ) t 3 ( 6 ) t 3 \cdot(2)_{t^{2}}^{2}(3)_{t^{2}}\cdot(2)_{t^{3}}(6)_{t^{3}}
  111. 𝕊 3 × 2 \mathbb{S}_{3}\times\mathbb{Z}_{2}
  112. 𝕊 3 × 6 \mathbb{S}_{3}\times\mathbb{Z}_{6}
  113. 𝒪 ( 12 ) - 1 , 1 𝒪 { z } 1 , σ 2 \mathcal{O}_{(12)}^{-1,1}\oplus\mathcal{O}_{\{z\}}^{1,\sigma_{2}}
  114. 𝒪 ( 12 ) - 1 , ζ 6 𝒪 { z } 1 , ζ 6 - 1 \mathcal{O}_{(12)}^{-1,\zeta_{6}}\oplus\mathcal{O}_{\{z\}}^{1,\zeta_{6}^{-1}}
  115. σ \sigma
  116. B 2 B_{2}
  117. V V
  118. 2 + 4 2+4
  119. 262 , 144 = 2 18 262,144\;=2^{18}
  120. ( 2 ) t 2 ( 2 ) t 2 ( 2 ) t 4 ( 2 ) t 2 2 ( 2 ) t 2 4 ( 2 2 ) t 4 2 ( 2 ) t 3 2 ( 2 ) t 6 (2)_{t}^{2}(2)_{t^{2}}\cdot(2)_{t}^{4}(2)_{t^{2}}^{2}\cdot(2)_{t^{2}}^{4}(2^{2% })_{t^{4}}^{2}\cdot(2)_{t^{3}}^{2}(2)_{t^{6}}
  121. 𝔻 ~ 8 \tilde{\mathbb{D}}_{8}
  122. 𝒪 [ h ] - 1 𝒪 [ g ] ζ 8 \mathcal{O}_{[h]}^{-1}\oplus\mathcal{O}_{[g]}^{\zeta_{8}}
  123. 𝔻 4 \mathbb{D}_{4}
  124. A 2 A_{2}
  125. G 2 G_{2}
  126. V V
  127. 1 + 4 1+4
  128. 80 , 621 , 568 80,621,568
  129. ( 6 ) t ( 2 ) t 2 ( 3 ) t ( 6 ) t ( 2 ) t 2 2 ( 3 ) t 2 ( 6 ) t 2 ( 2 ) t 3 2 ( 3 ) t 3 ( 6 ) t 3 ( 6 ) t 4 ( 6 ) t 5 (6)_{t}\cdot(2)_{t}^{2}(3)_{t}(6)_{t}\cdot(2)_{t^{2}}^{2}(3)_{t^{2}}(6)_{t^{2}% }\cdot(2)_{t^{3}}^{2}(3)_{t^{3}}(6)_{t^{3}}\cdot(6)_{t^{4}}\cdot(6)_{t^{5}}
  130. \Z 6 × S L 2 ( 3 ) \Z_{6}\times SL_{2}(3)
  131. 𝒪 { z } ζ 6 , ζ 6 - 1 𝒪 ( - 1 1 0 - 1 ) 1 , - 1 \mathcal{O}_{\{z\}}^{\zeta_{6},\zeta_{6}^{-1}}\oplus\mathcal{O}_{\begin{% pmatrix}-1&1\\ 0&-1\end{pmatrix}}^{1,-1}
  132. S L 2 ( 3 ) SL_{2}(3)
  133. V V
  134. 3 + 2 + 1 3+2+1
  135. 3 + 1 + 1 3+1+1
  136. 1 , 671 , 768 , 834 , 048 1,671,768,834,048
  137. ( 2 ) t 2 ( 3 ) t ( 6 ) t ( 6 ) t ( 2 ) t 2 2 ( 3 ) t 2 ( 6 ) t 2 ( 2 ) t 3 ( 6 ) t 3 ( 2 ) t 3 2 ( 3 ) t 3 (2)_{t}^{2}(3)_{t}\cdot(6)_{t}\cdot(6)_{t}\cdot(2)_{t^{2}}^{2}(3)_{t^{2}}\cdot% (6)_{t^{2}}\cdot(2)_{t^{3}}(6)_{t^{3}}\cdot(2)_{t^{3}}^{2}(3)_{t^{3}}
  138. ( 2 ) t 4 ( 6 ) t 4 ( 2 ) t 5 ( 6 ) t 5 ( 2 ) t 6 2 ( 3 ) t 6 ( 6 ) t 7 ( 6 ) t 8 ( 6 ) t 9 \cdot(2)_{t^{4}}(6)_{t^{4}}\cdot(2)_{t^{5}}(6)_{t^{5}}\cdot(2)_{t^{6}}^{2}(3)_% {t^{6}}\cdot(6)_{t^{7}}\cdot(6)_{t^{8}}\cdot(6)_{t^{9}}
  139. 𝕊 3 × 6 × 6 \mathbb{S}_{3}\times\mathbb{Z}_{6}\times\mathbb{Z}_{6}
  140. 𝒪 ( 12 ) - 1 , ζ 6 , 1 𝒪 { z } 1 , ζ 6 - 1 , 1 𝒪 { w } 1 , ζ 6 , ζ 6 - 1 \mathcal{O}_{(12)}^{-1,\zeta_{6},1}\oplus\mathcal{O}_{\{z\}}^{1,\zeta_{6}^{-1}% ,1}\oplus\mathcal{O}_{\{w\}}^{1,\zeta_{6},\zeta_{6}^{-1}}
  141. Γ 3 \Gamma_{3}
  142. A 2 A_{2}
  143. A 3 A_{3}
  144. 𝔅 ( V ) \mathfrak{B}(V)
  145. A n , n 2 A_{n},\;n\geq 2
  146. D n , n 4 D_{n},\;n\geq 4
  147. E 6 , E 7 , E 8 E_{6},E_{7},E_{8}
  148. B n , n 2 B_{n},\;n\geq 2
  149. q i j = ± 1 q_{ij}=\pm 1
  150. A n × A n A_{n}\times A_{n}
  151. D n × D n D_{n}\times D_{n}
  152. E n × E n E_{n}\times E_{n}
  153. B n × B n B_{n}\times B_{n}
  154. V V
  155. 2 + 2 + 2+2+\cdots
  156. 2 + 2 + 2+2+\cdots
  157. 2 + 2 + 2+2+\cdots
  158. 2 + 2 + 2+2+\cdots
  159. ( 2 ( n + 1 2 ) ) 2 \left(2^{n+1\choose 2}\right)^{2}
  160. ( 2 n ( n - 1 ) ) 2 \left(2^{n(n-1)}\right)^{2}
  161. ( 2 36 ) 2 , ( 2 63 ) 2 , ( 2 120 ) 2 \left(2^{36}\right)^{2},\;\left(2^{63}\right)^{2},\;\left(2^{120}\right)^{2}
  162. ( 3 n ( n - 1 ) 2 n ) 2 \left(3^{n(n-1)}2^{n}\right)^{2}
  163. 2 n + 1 2^{n+1}
  164. D n , 2 | n D_{n},\;2|n
  165. 2 3 2^{3}
  166. B n B_{n}
  167. q = ± 1 q=\pm 1
  168. A 2 n - 1 2 , E 6 2 {{}^{2}}A_{2n-1},{{}^{2}}E_{6}
  169. 𝔅 ( V ) \mathfrak{B}(V)
  170. C n , n 3 C_{n},\;n\geq 3
  171. F 4 F_{4}
  172. q i j = ± 1 q_{ij}=\pm 1
  173. A 2 n - 1 A_{2n-1}
  174. E 6 E_{6}
  175. V V
  176. 1 + 2 + 1+2+\cdots
  177. 1 + 1 + 2 + 2 1+1+2+2
  178. 2 < m t p l > ( 2 n 2 ) 2^{<}mtpl>{{2n\choose 2}}
  179. 2 36 = 68 , 719 , 476 , 736 2^{36}=68,719,476,736
  180. ( 2 ) t 6 ( 2 ) t 2 5 ( 2 ) t 3 5 ( 2 ) t 4 5 ( 2 ) t 5 4 ( 2 ) t 6 3 (2)^{6}_{t}(2)^{5}_{t^{2}}(2)^{5}_{t^{3}}(2)^{5}_{t^{4}}(2)^{4}_{t^{5}}(2)^{3}% _{t^{6}}
  181. ( 2 ) t 7 3 ( 2 ) t 8 2 ( 2 ) t 9 ( 2 ) t 10 ( 2 ) t 11 \cdot(2)^{3}_{t^{7}}(2)^{2}_{t^{8}}(2)_{t^{9}}(2)_{t^{10}}(2)_{t^{11}}
  182. 2 n + 1 2^{n+1}
  183. 2 4 + 1 2^{4+1}
  184. 2 3 2^{3}
  185. 2 × 2 × 𝔻 4 \mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{D}_{4}
  186. 𝒪 { z 1 } 𝒪 { z 2 } 𝒪 [ x 1 ] 𝒪 [ y 1 ] \mathcal{O}_{\{z_{1}\}}\oplus\mathcal{O}_{\{z_{2}\}}\oplus\mathcal{O}_{[x_{1}]% }\oplus\mathcal{O}_{[y_{1}]}
  187. D 4 3 , D n 2 {{}^{3}}D_{4},{{}^{2}}D_{n}

List_of_permanent_Ambisonic_playback_systems.html

  1. \ell

List_of_shapes_with_known_packing_constant.html

  1. π / 12 0.906900 π/\sqrt{12}≈0.906900
  2. η s o = 8 - 4 2 - ln 2 2 2 - 1 0.902414 . \eta_{so}=\frac{8-4\sqrt{2}-\ln{2}}{2\sqrt{2}-1}\approx 0.902414\,.
  3. π / 18 0.7404805 π/\sqrt{18}≈0.7404805
  4. π / 12 0.906900 π/\sqrt{12}≈0.906900

List_of_sums_of_reciprocals.html

  1. 2 p - 1 - 1 p \frac{2^{p-1}-1}{p}
  2. π p , \frac{\pi}{p},
  3. π q , \frac{\pi}{q},
  4. π r . \frac{\pi}{r}.
  5. π 2 6 . \frac{\pi^{2}}{6}.
  6. n = 1 1 n s . \sum_{n=1}^{\infty}\frac{1}{n^{s}}.
  7. γ , \gamma,

List_of_things_named_after_Joseph_Fourier.html

  1. 𝐹𝑜 \mathit{Fo}
  2. α t / d 2 \alpha t/d^{2}
  3. α t \alpha t
  4. d 2 d^{2}

Littlewood's_Tauberian_theorem.html

  1. a n x n s , \sum a_{n}x^{n}\to s,
  2. a n = s . \sum a_{n}=s.

Local_case-control_sampling.html

  1. ( x i , y i ) i = 1 N (x_{i},y_{i})_{i=1}^{N}
  2. x i d x_{i}\in\mathbb{R}^{d}
  3. y i { 0 , 1 } y_{i}\in\{0,1\}
  4. ( Y = 1 ) 0 \mathbb{P}(Y=1)\approx 0
  5. X { 0 , 1 } X\in\{0,1\}
  6. ( Y = 1 X = 0 ) 0 \mathbb{P}(Y=1\mid X=0)\approx 0
  7. ( Y = 1 X = 1 ) 1 \mathbb{P}(Y=1\mid X=1)\approx 1
  8. θ = ( α , β ) \theta=(\alpha,\beta)
  9. ( Y = 1 X ; θ ) = p ~ θ ( x ) = exp ( α + β T x ) 1 + exp ( α + β T x ) \mathbb{P}(Y=1\mid X;\theta)=\tilde{p}_{\theta}(x)=\frac{\exp(\alpha+\beta^{T}% x)}{1+\exp(\alpha+\beta^{T}x)}
  10. θ ~ = ( α ~ , β ~ ) \tilde{\theta}=(\tilde{\alpha},\tilde{\beta})
  11. ( x , y ) (x,y)
  12. a ( x , y ) = | y - p ~ θ ~ ( x ) | a(x,y)=|y-\tilde{p}_{\tilde{\theta}}(x)|
  13. z i Bernoulli ( a ( x i , y i ) ) z_{i}\sim\,\text{Bernoulli}(a(x_{i},y_{i}))
  14. i { 1 , , N } i\in\{1,\ldots,N\}
  15. S = { ( x i , y i ) : z i = 1 } S=\{(x_{i},y_{i}):z_{i}=1\}
  16. θ ^ S = ( α ^ S , β ^ S ) \hat{\theta}_{S}=(\hat{\alpha}_{S},\hat{\beta}_{S})
  17. θ ^ = ( α ^ , β ^ ) \hat{\theta}=(\hat{\alpha},\hat{\beta})
  18. α ^ α ^ S + α ~ \hat{\alpha}\leftarrow\hat{\alpha}_{S}+\tilde{\alpha}
  19. β ^ β ^ S + β ~ \hat{\beta}\leftarrow\hat{\beta}_{S}+\tilde{\beta}
  20. N = 1000 N=1000
  21. θ ~ \tilde{\theta}
  22. N h = 500 N_{h}=500
  23. N h = 500 N_{h}=500
  24. c c
  25. c > 1 c>1
  26. min ( c a ( x i , y i ) , 1 ) \min(ca(x_{i},y_{i}),1)
  27. c > 1 c>1
  28. 1 + 1 c 1+\frac{1}{c}

Local_inverse.html

  1. [ f g ] = [ A B C D ] [ x y ] \begin{bmatrix}f\\ g\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  2. E E
  3. F F
  4. G G
  5. H H
  6. [ E F G H ] [ A B C D ] = J \begin{bmatrix}E&F\\ G&H\end{bmatrix}\begin{bmatrix}A&B\\ C&D\end{bmatrix}=J
  7. J J
  8. I I
  9. J J
  10. I I
  11. I I
  12. [ E F G H ] \begin{bmatrix}E&F\\ G&H\end{bmatrix}
  13. [ x 0 y 0 ] = [ E F G H ] [ f g ] \begin{bmatrix}x_{0}\\ y_{0}\end{bmatrix}=\begin{bmatrix}E&F\\ G&H\end{bmatrix}\begin{bmatrix}f\\ g\end{bmatrix}
  14. x 1 x_{1}
  15. [ x 1 y 1 ] = [ E F G H ] [ f - B y 0 g - D y 0 ] \begin{bmatrix}x_{1}\\ y_{1}\end{bmatrix}=\begin{bmatrix}E&F\\ G&H\end{bmatrix}\begin{bmatrix}f-By_{0}\\ g-Dy_{0}\end{bmatrix}
  16. y 1 y_{1}
  17. x 1 = E ( f - B y 0 ) + F ( g - D y 0 ) x_{1}=E(f-By_{0})+F(g-Dy_{0})
  18. y 1 = G ( f - A x 0 ) + H ( g - C x 0 ) y_{1}=G(f-Ax_{0})+H(g-Cx_{0})
  19. x x
  20. y y
  21. [ f g ] = [ A B C D ] [ x y ] \begin{bmatrix}f\\ g\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  22. A A
  23. B B
  24. C C
  25. D D
  26. x x
  27. y y
  28. f f
  29. g g
  30. [ E F G H ] \begin{bmatrix}E&F\\ G&H\end{bmatrix}
  31. [ A B C D ] [ E F G H ] = J \begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}E&F\\ G&H\end{bmatrix}=J
  32. J = I J=I
  33. J J
  34. I I
  35. g e x g_{ex}
  36. g g
  37. g e x | G = f | F g_{ex}|_{\partial G}=f|_{\partial F}
  38. y 0 y_{0}
  39. y y
  40. y 0 = H g e x y_{0}=Hg_{ex}
  41. y y^{\prime}
  42. y y
  43. y = y 0 + y c o y^{\prime}=y_{0}+y_{co}
  44. f f^{\prime}
  45. f f
  46. f = f - B y f^{\prime}=f-By^{\prime}
  47. g 1 e x g_{1ex}
  48. g g
  49. g 1 e x | G = f | F g_{1ex}|_{\partial G}=f^{\prime}|_{\partial F}
  50. x 1 x_{1}
  51. x 1 = E f + F g 1 e x x_{1}=Ef^{\prime}+Fg_{1ex}
  52. g g
  53. y y
  54. y y
  55. y y
  56. y = y 0 + y c o = k y 0 y^{\prime}=y_{0}+y_{co}=ky_{0}
  57. k = 1.04 k=1.04
  58. A + B A^{+}B
  59. 0
  60. f = A x + B y f=Ax+By
  61. A x = f - B y = f Ax=f-By=f^{\prime}
  62. f = f - B y f^{\prime}=f-By
  63. x = A - 1 f x^{\prime}=A^{-1}f^{\prime}
  64. x x^{\prime}
  65. x x
  66. x = x x^{\prime}=x
  67. A A
  68. x = A + ( f - B y ) = A + f x^{\prime}=A^{+}(f-By)=A^{+}f^{\prime}
  69. y y
  70. 0
  71. x 0 = A + f x_{0}=A^{+}f
  72. x 0 x_{0}
  73. y y
  74. y y
  75. x 0 x_{0}
  76. error 0 = | x 0 - x | = | A + B y | \mathrm{error}_{0}=|x_{0}-x^{\prime}|=|A^{+}By|
  77. Q Q
  78. Q B = 0 QB=0
  79. Q A x = Q f - Q B y = Q f QAx=Qf-QBy=Qf
  80. x 1 = [ Q A ] + Q f = [ A ] + Q + Q f x_{1}=[QA]^{+}Qf=[A]^{+}Q^{+}Qf
  81. Q + Q^{+}
  82. Q Q
  83. x 1 x_{1}
  84. x x
  85. Q B = 0 QB=0
  86. Q Q
  87. Q = I - B B + Q=I-BB^{+}
  88. Q Q
  89. B B
  90. B + B^{+}
  91. B B
  92. B + B^{+}
  93. B B + B = B BB^{+}B=B
  94. Q B = [ I - B B + ] B = B - B B + B = B - B = 0 QB=[I-BB^{+}]B=B-BB^{+}B=B-B=0
  95. Q Q = Q QQ=Q
  96. Q Q = [ I - B B + ] [ I - B B + ] = I - 2 B B + + B B + B B + = I - 2 B B + + B B + = I - B B + = Q \begin{aligned}\displaystyle QQ&\displaystyle=[I-BB^{+}][I-BB^{+}]=I-2BB^{+}+% BB^{+}BB^{+}\\ &\displaystyle=I-2BB^{+}+BB^{+}=I-BB^{+}=Q\end{aligned}
  97. Q Q Q = ( Q Q ) Q = Q Q = Q QQQ=(QQ)Q=QQ=Q
  98. Q + Q = Q Q = Q Q^{+}Q=QQ=Q
  99. x 1 = A + [ Q ] + Q f = A + Q f x_{1}=A^{+}[Q]^{+}Qf=A^{+}Qf
  100. x 1 = [ A ] + [ I - B B + ] f x_{1}=[A]^{+}[I-BB^{+}]f
  101. A L = [ A ] + [ I - B B + ] A^{L}=[A]^{+}[I-BB^{+}]
  102. A L A^{L}
  103. [ A B C D ] \begin{bmatrix}A&B\\ C&D\end{bmatrix}
  104. [ A ] + [ I - B B + ] f = [ A ] + [ I - B B + ] ( f - B y ) = [ A ] + [ I - B B + ] f [A]^{+}[I-BB^{+}]f^{\prime}=[A]^{+}[I-BB^{+}](f-By)=[A]^{+}[I-BB^{+}]f
  105. x 1 = [ A ] + [ I - B B + ] f x_{1}=[A]^{+}[I-BB^{+}]f^{\prime}
  106. x 1 x_{1}
  107. f f^{\prime}
  108. error 1 = | x 1 - x | = | [ A ] + B B + f | \mathrm{error}_{1}=|x_{1}-x^{\prime}|=|[A]^{+}BB^{+}f^{\prime}|
  109. y y
  110. f f^{\prime}
  111. [ A ] + B B + f [A]^{+}BB^{+}f^{\prime}
  112. x x
  113. [ A ] + B y [A]^{+}By
  114. error 1 error 0 \mathrm{error}_{1}\ll\mathrm{error}_{0}
  115. x 1 x_{1}
  116. x 0 x_{0}
  117. x 1 x_{1}
  118. x 0 x_{0}
  119. y y
  120. 0

Local_pixel_grouping.html

  1. I I
  2. v v
  3. I v = I + v I_{v}=I+v
  4. I v I_{v}
  5. X v X_{v}
  6. X v X_{v}
  7. p x px
  8. K × K K\times K
  9. p x px
  10. x = [ x 1 x m ] T , m = K 2 x=[x_{1}...x_{m}]^{\rm T},m=K^{2}
  11. p x px
  12. L × L , L > K L\times L,L>K
  13. K × K K\times K
  14. L × L L\times L
  15. ( L - K + 1 ) 2 (L-K+1)^{2}
  16. x i v \overrightarrow{x_{i}}^{\rm v}
  17. x 0 v \overrightarrow{x_{0}}^{\rm v}
  18. X v X_{v}
  19. X v X_{v}
  20. X v ¯ \overline{X_{v}}
  21. X v ¯ \overline{X_{v}}
  22. Ω x ¯ \Omega_{\overline{x}}
  23. P x ¯ P_{\overline{x}}
  24. P x ¯ P_{\overline{x}}
  25. X v ¯ \overline{X_{v}}
  26. Y v ¯ = P x ¯ X v ¯ \overline{Y_{v}}=P_{\overline{x}}\overline{X_{v}}
  27. Y v ¯ \overline{Y_{v}}
  28. Ω y v ¯ = 1 n Y v ¯ Y v ¯ T \Omega_{\overline{y_{v}}}=\frac{1}{n}\overline{Y_{v}}\ \overline{Y_{v}}^{\rm T}
  29. Ω y ¯ v \Omega_{\overline{y}_{v}}
  30. Y ¯ ^ k = w k Y ¯ v k \hat{\overrightarrow{\overline{Y}}}_{k}=w_{k}\ \overrightarrow{\overline{Y}}_{% v}^{k}
  31. w k = Ω y ¯ ( k , k ) Ω y ¯ ( k , k ) + Ω v y ( k , k ) w_{k}=\frac{\Omega_{\overline{y}}(k,k)}{\Omega_{\overline{y}}(k,k)+\Omega_{v_{% y}}(k,k)}
  32. X ¯ ^ \hat{\overline{X}}

Local_World_Evolving_Network_Models.html

  1. m 0 m_{0}
  2. e 0 e_{0}
  3. P l o c a l ( k i ) = P ( i L o c a l - W o r l d ) k i j L o c a l k i P_{local}^{{}^{\prime}}(k_{i})=P^{{}^{\prime}}(i\in Local-World)\frac{k_{i}}{% \sum_{j\in Local}k_{i}}
  4. P ( i L o c a l - W o r l d ) = M m 0 + t P^{{}^{\prime}}(i\in Local-World)=\frac{M}{m_{0}+t}
  5. P l o c a l ( k i ) = M m 0 + t k i j L o c a l k i P_{local}^{{}^{\prime}}(k_{i})=\frac{M}{m_{0}+t}\frac{k_{i}}{\sum_{j\in Local}% k_{i}}
  6. k i t = m M m 0 + t k i j L o c a l k i \frac{\partial k_{i}}{\partial t}=\frac{mM}{m_{0}+t}\frac{k_{i}}{\sum_{j\in Local% }k_{i}}
  7. m M m 0 + t m\leqslant M\leqslant m_{0}+t
  8. M = m M=m
  9. M = m 0 + t M=m_{0}+t
  10. M = m M=m
  11. k i t = m m 0 + t k i j L o c a l k i \frac{\partial k_{i}}{\partial t}=\frac{m}{m_{0}+t}\frac{k_{i}}{\sum_{j\in Local% }k_{i}}
  12. P ( k ) e - k / m P(k)\sim e^{-k/m}
  13. M = m 0 + t M=m_{0}+t
  14. k i t = k i 2 t \frac{\partial k_{i}}{\partial t}=\frac{k_{i}}{2t}
  15. P ( k ) 2 m 2 / k 3 P(k)\sim 2m^{2}/k^{3}
  16. δ \delta
  17. δ \delta
  18. δ \delta
  19. δ [ 0 , 1 ] \delta\in\left[0,1\right]
  20. δ = 0 \delta=0
  21. δ = 1 \delta=1
  22. e 0 e_{0}
  23. P g l o b a l ( k i ) = k i j k j P_{global}(k_{i})=\frac{k_{i}}{\sum_{j}k_{j}}
  24. P l o c a l ( k i ) = P ( i L o c a l - W o r l d ) k j j L o c a l k j P_{local}^{{}^{\prime}}(k_{i})=P^{{}^{\prime}}(i\in Local-World)\frac{k_{j}}{% \sum_{j\in Local}k_{j}}
  25. P a l l ( k i ) = = δ k i j k j + ( 1 - δ ) P ( i L o c a l - W o r l d ) k i j k j P_{all}^{{}^{\prime}}(k_{i})==\delta\frac{k_{i}}{\sum_{j}k_{j}}+(1-\delta)P^{{% }^{\prime}}(i\in Local-World)\frac{k_{i}}{\sum_{j}k_{j}}
  26. δ \delta
  27. m = M m=M
  28. m M m 0 + t m\ll M\ll m_{0}+t
  29. M = m 0 + t M=m_{0}+t
  30. M = m M=m
  31. k i t = δ k i + 2 ( 1 - δ ) m 2 t \frac{\partial k_{i}}{\partial t}=\frac{\delta k_{i}+2(1-\delta)m}{2t}
  32. j l o c a l k j = M k i = m k i m k i \sum_{j\in local}k_{j}=M\left\langle k_{i}\right\rangle=m\left\langle k_{i}% \right\rangle\approx mk_{i}
  33. ( σ ) (\sigma)
  34. 1 + δ 2 1+\frac{\delta}{2}
  35. δ \delta
  36. m M m 0 + t m\ll M\ll m_{0}+t
  37. m 0 + t m_{0}+t
  38. M = m 0 + t M=m_{0}+t
  39. k i t = [ δ 2 - 1 + δ 1 - δ ] k i t \frac{\partial k_{i}}{\partial t}=\left[\frac{\delta}{2}-\frac{1+\delta}{1-% \delta}\right]\frac{k_{i}}{t}
  40. j l o c a l k j = j l o c a l k j = [ δ k i + ( 1 - δ ) m ] M \sum_{j\in local}k{j}=\sum_{j\in local}k_{j}=\left[\delta\left\langle k_{i}% \right\rangle+(1-\delta)m\right]M
  41. 4 + δ + δ 2 2 - δ - δ 2 \frac{4+\delta+\delta^{2}}{2-\delta-\delta^{2}}
  42. δ \delta
  43. δ \delta

Localization_formula_for_equivariant_cohomology.html

  1. α \alpha
  2. ξ \xi
  3. 1 d M M α ( ξ ) = F 1 d F F α ( ξ ) e T ( F ) ( ξ ) {1\over d_{M}}\int_{M}\alpha(\xi)=\sum_{F}{1\over d_{F}}\int_{F}{\alpha(\xi)% \over e_{T}(F)(\xi)}
  4. M T M^{T}
  5. d M d_{M}
  6. e T ( F ) e_{T}(F)
  7. M e - t H ω n / n ! = p e - t H ( p ) t n α j ( p ) . \int_{M}e^{-tH}\omega^{n}/n!=\sum_{p}{e^{-tH(p)}\over t^{n}\prod\alpha_{j}(p)}.
  8. α j ( p ) \alpha_{j}(p)

Localizing_subcategory.html

  1. 𝒜 \mathcal{A}
  2. 𝒞 \mathcal{C}
  3. 0 A A A ′′ 0 0\rightarrow A^{\prime}\rightarrow A\rightarrow A^{\prime\prime}\rightarrow 0
  4. 𝒜 \mathcal{A}
  5. A A
  6. 𝒞 \mathcal{C}
  7. A A^{\prime}
  8. A ′′ A^{\prime\prime}
  9. 𝒞 \mathcal{C}
  10. 𝒞 \mathcal{C}
  11. 𝒜 \mathcal{A}
  12. 𝒜 / 𝒞 \mathcal{A}/\mathcal{C}
  13. 𝒜 \mathcal{A}
  14. T : 𝒜 𝒜 / 𝒞 T\colon\mathcal{A}\rightarrow\mathcal{A}/\mathcal{C}
  15. 𝒞 \mathcal{C}
  16. 𝒜 \mathcal{A}
  17. 𝒞 \mathcal{C}
  18. T : 𝒜 𝒜 / 𝒞 T\colon\mathcal{A}\rightarrow\mathcal{A}/\mathcal{C}
  19. S : 𝒜 / 𝒞 𝒜 S\colon\mathcal{A}/\mathcal{C}\rightarrow\mathcal{A}
  20. T T
  21. T T
  22. S T ST
  23. S S
  24. 𝒜 \mathcal{A}
  25. 𝒞 \mathcal{C}
  26. 𝒞 \mathcal{C}
  27. 𝒜 \mathcal{A}
  28. 𝒞 \mathcal{C}
  29. 𝒜 / 𝒞 \mathcal{A}/\mathcal{C}
  30. Mod ( R ) \operatorname{Mod}(R)
  31. R R

Lode_Coordinates.html

  1. ξ \xi
  2. ρ \rho
  3. θ \theta
  4. ( z , r , θ ) (z,r,\theta)
  5. ( ξ , ρ , θ ) (\xi,\rho,\theta)
  6. ( σ 1 , σ 2 , σ 3 ) (\sigma_{1},\sigma_{2},\sigma_{3})
  7. ( I 1 , J 2 , J 3 ) (I_{1},J_{2},J_{3})
  8. ( σ 1 , σ 2 , σ 3 ) = ( 1 , 1 , 1 ) (\sigma_{1},\sigma_{2},\sigma_{3})=(1,1,1)
  9. s y m b o l σ symbol{\sigma}
  10. s y m b o l s symbol{s}
  11. I 1 = tr ( s y m b o l σ ) I_{1}=\mathrm{tr}(symbol{\sigma})
  12. J 2 = 1 2 ( ( tr ( s y m b o l σ ) ) 2 - tr ( s y m b o l σ \cdotsymbol σ ) ) = 1 2 tr ( s y m b o l s \cdotsymbol s ) = 1 2 s y m b o l s 2 J_{2}=\frac{1}{2}\left((\mathrm{tr}(symbol{\sigma}))^{2}-\mathrm{tr}(symbol{% \sigma}\cdotsymbol{\sigma})\right)=\frac{1}{2}\mathrm{tr}\left(symbol{s}% \cdotsymbol{s}\right)=\frac{1}{2}\lVert symbol{s}\rVert^{2}
  13. J 3 = det ( s y m b o l s ) = 1 3 tr ( s y m b o l s \cdotsymbol s \cdotsymbol s ) J_{3}=\mathrm{det}(symbol{s})=\frac{1}{3}\mathrm{tr}\left(symbol{s}\cdotsymbol% {s}\cdotsymbol{s}\right)
  14. I 1 = σ k k I_{1}=\sigma_{kk}
  15. J 2 = 1 2 ( ( σ k k ) 2 - σ i j σ j i ) = 1 2 s i j s j i = 1 2 s i j s i j J_{2}=\frac{1}{2}\left((\sigma_{kk})^{2}-\sigma_{ij}\sigma_{ji}\right)=\frac{1% }{2}s_{ij}s_{ji}=\frac{1}{2}s_{ij}s_{ij}
  16. J 3 = 1 6 ϵ i j k ϵ p q r σ i p σ j q σ k r = 1 3 s i j s j k s k i J_{3}=\frac{1}{6}\epsilon_{ijk}\epsilon_{pqr}\sigma_{ip}\sigma_{jq}\sigma_{kr}% =\frac{1}{3}s_{ij}s_{jk}s_{ki}
  17. ϵ \epsilon
  18. J 2 J_{2}
  19. s y m b o l s symbol{s}
  20. s i j = s j i s_{ij}=s_{ji}
  21. I 1 s y m b o l σ = s y m b o l I \frac{\partial I_{1}}{\partial symbol{\sigma}}=symbol{I}
  22. J 2 s y m b o l σ = s y m b o l s = s y m b o l σ - tr ( s y m b o l σ ) 3 s y m b o l I \frac{\partial J_{2}}{\partial symbol{\sigma}}=symbol{s}=symbol{\sigma}-\frac{% \mathrm{tr}\left(symbol{\sigma}\right)}{3}symbol{I}
  23. J 3 s y m b o l σ = s y m b o l T = s y m b o l s \cdotsymbol s - 2 J 2 3 s y m b o l I \frac{\partial J_{3}}{\partial symbol{\sigma}}=symbol{T}=symbol{s}\cdotsymbol{% s}-\frac{2J_{2}}{3}symbol{I}
  24. s y m b o l I symbol{I}
  25. s y m b o l T symbol{T}
  26. ( z ) (z)
  27. z z
  28. z = s y m b o l E z : s y m b o l σ = tr ( s y m b o l σ ) 3 = I 1 3 z=symbol{E_{z}}\colon symbol{\sigma}=\frac{\mathrm{tr}(symbol{\sigma})}{\sqrt{% 3}}=\frac{I_{1}}{\sqrt{3}}
  29. s y m b o l E z = s y m b o l I s y m b o l I = s y m b o l I 3 symbol{E_{z}}=\frac{symbol{I}}{\lVert symbol{I}\rVert}=\frac{symbol{I}}{\sqrt{% 3}}
  30. ( r ) (r)
  31. r r
  32. r = s y m b o l E r : s y m b o l σ = s y m b o l s = 2 J 2 r=symbol{E_{r}}\colon symbol{\sigma}=\lVert symbol{s}\rVert=\sqrt{2J_{2}}
  33. s y m b o l E r = s y m b o l s s y m b o l s symbol{E_{r}}=\frac{symbol{s}}{\lVert symbol{s}\rVert}
  34. r = 2 J 2 r=\sqrt{2J_{2}}
  35. r = s y m b o l s s y m b o l s \colonsymbol σ r=\frac{symbol{s}}{\lVert symbol{s}\rVert}\colonsymbol{\sigma}
  36. σ \sigma
  37. s y m b o l s symbol{s}
  38. r = s y m b o l s s y m b o l s \colonsymbol s : ( s y m b o l s + z s y m b o l E z ) = 1 s y m b o l s \colonsymbol s ( s y m b o l s \colonsymbol s + z s y m b o l s \colonsymbol E z ) r=\frac{symbol{s}}{\sqrt{symbol{s}\colonsymbol{s}}}\colon\left(symbol{s}+% zsymbol{E_{z}}\right)=\frac{1}{\sqrt{symbol{s}\colonsymbol{s}}}\left(symbol{s}% \colonsymbol{s}+zsymbol{s}\colonsymbol{E_{z}}\right)
  39. s y m b o l E z symbol{E_{z}}
  40. s y m b o l s symbol{s}
  41. r = s y m b o l s \colonsymbol s s y m b o l s \colonsymbol s = s y m b o l s \colonsymbol s r=\frac{symbol{s}\colonsymbol{s}}{\sqrt{symbol{s}\colonsymbol{s}}}=\sqrt{% symbol{s}\colonsymbol{s}}
  42. s y m b o l A \colonsymbol B = tr ( s y m b o l A T \cdotsymbol B ) symbol{A}\colonsymbol{B}=\mathrm{tr}\left(symbol{A}^{T}\cdotsymbol{B}\right)
  43. J 2 = 1 2 tr ( s y m b o l s \cdotsymbol s ) J_{2}=\frac{1}{2}\mathrm{tr}\left(symbol{s}\cdotsymbol{s}\right)
  44. r = tr ( s y m b o l s \cdotsymbol s ) = 2 J 2 r=\sqrt{\mathrm{tr}\left(symbol{s}\cdotsymbol{s}\right)}=\sqrt{2J_{2}}
  45. ( θ ) (\theta)
  46. λ m \lambda_{m}
  47. θ s \theta_{s}
  48. θ ¯ s \bar{\theta}_{s}
  49. θ c \theta_{c}
  50. sin ( 3 θ s ) = - sin ( 3 θ ¯ s ) = cos ( 3 θ c ) = J 3 2 ( 3 J 2 ) 3 / 2 \sin(3\theta_{s})=-\sin(3\bar{\theta}_{s})=\cos(3\theta_{c})=\frac{J_{3}}{2}% \left(\frac{3}{J_{2}}\right)^{3/2}
  51. θ s = π 6 - θ c θ s = - θ ¯ s \theta_{s}=\frac{\pi}{6}-\theta_{c}\qquad\qquad\theta_{s}=-\bar{\theta}_{s}
  52. θ c \theta_{c}
  53. θ s \theta_{s}
  54. cos ( 3 θ c ) = sin ( 3 θ s ) \cos(3\theta_{c})=\sin(3\theta_{s})
  55. cos ( 3 θ c ) = sin ( ( 3 θ s - π 2 ) + π 2 ) \cos(3\theta_{c})=\sin\left(\left(3\theta_{s}-\frac{\pi}{2}\right)+\frac{\pi}{% 2}\right)
  56. cos ( 3 θ c ) = cos ( 3 θ s - π 2 ) \cos(3\theta_{c})=\cos\left(3\theta_{s}-\frac{\pi}{2}\right)
  57. 0 cos - 1 ( θ ) 1 0\leq\cos^{-1}(\theta)\leq 1
  58. θ s \theta_{s}
  59. θ c \theta_{c}
  60. 3 θ c = ± ( 3 θ s - π 2 ) 3\theta_{c}=\pm\left(3\theta_{s}-\frac{\pi}{2}\right)
  61. θ c = π 6 - θ s \theta_{c}=\frac{\pi}{6}-\theta_{s}
  62. π / 3 \pi/3
  63. σ 1 σ 2 σ 3 \sigma_{1}\geq\sigma_{2}\geq\sigma_{3}
  64. θ s \theta_{s}
  65. θ ¯ s \bar{\theta}_{s}
  66. θ c \theta_{c}
  67. - π 6 θ s π 6 -\frac{\pi}{6}\leq\theta_{s}\leq\frac{\pi}{6}
  68. - π 6 θ s π 6 -\frac{\pi}{6}\leq\theta_{s}\leq\frac{\pi}{6}
  69. 0 θ c π 3 0\leq\theta_{c}\leq\frac{\pi}{3}
  70. σ 1 = σ 2 σ 3 \sigma_{1}=\sigma_{2}\geq\sigma_{3}
  71. - π 6 -\frac{\pi}{6}
  72. π 6 \frac{\pi}{6}
  73. π 3 \frac{\pi}{3}
  74. σ 2 = ( σ 1 + σ 3 ) / 2 \sigma_{2}=(\sigma_{1}+\sigma_{3})/2
  75. 0
  76. 0
  77. π 6 \frac{\pi}{6}
  78. σ 1 σ 2 = σ 3 \sigma_{1}\geq\sigma_{2}=\sigma_{3}
  79. π 6 \frac{\pi}{6}
  80. - π 6 -\frac{\pi}{6}
  81. 0
  82. θ s \theta_{s}
  83. θ c \theta_{c}
  84. s y m b o l E θ s = s y m b o l T / \lVertsymbol T - sin ( 3 θ s ) s y m b o l E z cos ( 3 θ s ) s y m b o l E θ c = s y m b o l T / \lVertsymbol T - cos ( 3 θ c ) s y m b o l E z sin ( 3 θ s ) symbol{E_{\theta s}}=\frac{symbol{T}/\lVertsymbol{T}\rVert-\sin(3\theta_{s})% symbol{E_{z}}}{\cos(3\theta_{s})}\qquad symbol{E_{\theta c}}=\frac{symbol{T}/% \lVertsymbol{T}\rVert-\cos(3\theta_{c})symbol{E_{z}}}{\sin(3\theta_{s})}
  85. ( z , r ) (z,r)
  86. θ \theta
  87. ( z , r ) (z,r)
  88. r r
  89. r r
  90. ( z , r ) (z,r)
  91. ( z , r ) (z,r)
  92. p = - I 1 / 3 p=-I1/3
  93. σ v = 3 J 2 \sigma_{v}=\sqrt{3J_{2}}
  94. p = - 1 3 z p=-\frac{1}{\sqrt{3}}z
  95. σ v = 3 2 r \sigma_{v}=\sqrt{\frac{3}{2}}r
  96. | - 1 / 3 | | 3 / 2 | |-1/\sqrt{3}|\neq|\sqrt{3/2}|
  97. r = 2 J 2 r=\sqrt{2J_{2}}
  98. ( r , θ ) (r,\theta)
  99. z z

Loewner_order.html

  1. A = [ 1 0 0 0 ] A=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}
  2. B = [ 0 0 0 1 ] B=\begin{bmatrix}0&0\\ 0&1\end{bmatrix}

Loewy_decomposition.html

  1. D d d x D\equiv\frac{d}{dx}
  2. x x
  3. n n
  4. L D n + a 1 D n - 1 + + a n - 1 D + a n L\equiv D^{n}+a_{1}D^{n-1}+\cdots+a_{n-1}D+a_{n}
  5. a i a_{i}
  6. i = 1 , , n i=1,\ldots,n
  7. L L
  8. x x
  9. a i ( x ) a_{i}\in{\mathbb{Q}}(x)
  10. y y
  11. d y d x 0 \frac{dy}{dx}\neq 0
  12. L y Ly
  13. L y = 0 Ly=0
  14. L L
  15. L L
  16. n n
  17. L 1 L_{1}
  18. L 2 L_{2}
  19. n n
  20. L = L 1 L 2 L=L_{1}L_{2}
  21. D a i = a i D + a i Da_{i}=a_{i}D+a_{i}^{\prime}
  22. L 1 L_{1}
  23. L L
  24. L 2 L_{2}
  25. L L
  26. ¯ ( x ) {\bar{\mathbb{Q}}}(x)
  27. L 1 L_{1}
  28. L 2 L_{2}
  29. L c l m ( L 1 , L 2 ) Lclm(L_{1},L_{2})
  30. L 1 L_{1}
  31. L 2 L_{2}
  32. G c r d ( L 1 , L 2 ) Gcrd(L_{1},L_{2})
  33. L 1 L_{1}
  34. L 2 L_{2}
  35. L c l m Lclm
  36. L c l m Lclm
  37. D = d d x D=\frac{d}{dx}
  38. a i ( x ) a_{i}\in{\mathbb{Q}(x)}
  39. L D n + a 1 D n - 1 + + a n - 1 D + a n L\equiv D^{n}+a_{1}D^{n-1}+\cdots+a_{n-1}D+a_{n}
  40. n n
  41. L k ( d k ) L^{(d_{k})}_{k}
  42. d k d_{k}
  43. ( x ) {\mathbb{Q}}(x)
  44. L = L m ( d m ) L m - 1 ( d m - 1 ) L 1 ( d 1 ) L=L_{m}^{(d_{m})}L_{m-1}^{(d_{m-1})}\ldots L_{1}^{(d_{1})}
  45. d 1 + + d m = n d_{1}+\ldots+d_{m}=n
  46. L k ( d k ) L^{(d_{k})}_{k}
  47. L k ( d k ) L^{(d_{k})}_{k}
  48. k = 1 , , m k=1,\ldots,m
  49. L k ( d k ) = L c l m ( l j 1 ( e 1 ) , l j 2 ( e 2 ) , , l j k ( e k ) ) L^{(d_{k})}_{k}=Lclm(l^{(e_{1})}_{j_{1}},l^{(e_{2})}_{j_{2}},\ldots,l^{(e_{k})% }_{j_{k}})
  50. e 1 + e 2 + + e k = d k e_{1}+e_{2}+\ldots+e_{k}=d_{k}
  51. l j i ( e i ) l^{(e_{i})}_{j_{i}}
  52. i = 1 , , k i=1,\ldots,k
  53. e i e_{i}
  54. ( x ) {\mathbb{Q}}(x)
  55. L L
  56. L y = 0 Ly=0
  57. n = 2 n=2
  58. L L
  59. 0 2 , 3 2 {\mathcal{L}}^{2}_{0},\ldots{\mathcal{L}}^{2}_{3}
  60. l ( i ) l^{(i)}
  61. l j ( i ) l^{(i)}_{j}
  62. i i
  63. C C
  64. 1 2 : L = l 2 ( 1 ) l 1 ( 1 ) ; {\mathcal{L}}^{2}_{1}:L=l^{(1)}_{2}l^{(1)}_{1};
  65. 2 2 : L = L c l m ( l 2 ( 1 ) , l 1 ( 1 ) ) ; {\mathcal{L}}^{2}_{2}:L=Lclm(l^{(1)}_{2},l^{(1)}_{1});
  66. 3 2 : L = L c l m ( l ( 1 ) ( C ) ) . {\mathcal{L}}^{2}_{3}:L=Lclm(l^{(1)}(C)).
  67. i 2 {\mathcal{L}}^{2}_{i}
  68. i i
  69. 0 2 {\mathcal{L}}^{2}_{0}
  70. 0 2 {\mathcal{L}}^{2}_{0}
  71. 2 2 {\mathcal{L}}^{2}_{2}
  72. 3 2 {\mathcal{L}}^{2}_{3}
  73. i 2 {\mathcal{L}}^{2}_{i}
  74. i = 1 , 2 i=1,2
  75. 3 3
  76. L y = 0 Ly=0
  77. L L
  78. D d d x D\equiv\frac{d}{dx}
  79. y y
  80. a i ( x ) a_{i}\in{\mathbb{Q}}(x)
  81. ε i ( x ) exp ( - a i d x ) \varepsilon_{i}(x)\equiv\exp{(-\int a_{i}dx)}
  82. i = 1 , 2 i=1,2
  83. ε ( x , C ) exp ( - a ( C ) d x ) \varepsilon(x,C)\equiv\exp{(-\int a(C)dx)}
  84. C C
  85. C ¯ \bar{C}
  86. C ¯ ¯ \bar{\bar{C}}
  87. C ¯ C ¯ ¯ \bar{C}\neq\bar{\bar{C}}
  88. y 1 y_{1}
  89. y 2 y_{2}
  90. 1 2 {\mathcal{L}}^{2}_{1}
  91. L y = ( D + a 2 ) ( D + a 1 ) y = 0 Ly=(D+a_{2})(D+a_{1})y=0
  92. y 1 = ε 1 ( x ) , y_{1}=\varepsilon_{1}(x),
  93. y 2 = ε 1 ( x ) ε 2 ( x ) ε 1 ( x ) d x . y_{2}=\varepsilon_{1}(x)\int\frac{\varepsilon_{2}(x)}{\varepsilon_{1}(x)}\,dx.
  94. 2 2 {\mathcal{L}}^{2}_{2}
  95. L y = L c l m ( D + a 2 , D + a 1 ) y = 0 ; Ly=Lclm(D+a_{2},D+a_{1})y=0;
  96. y i = ε i ( x ) ; y_{i}=\varepsilon_{i}(x);
  97. a 1 a_{1}
  98. a 2 a_{2}
  99. 3 2 {\mathcal{L}}^{2}_{3}
  100. L y = L c l m ( D + a ( C ) ) y = 0 ; Ly=Lclm(D+a(C))y=0;
  101. y 1 = ε ( x , C ¯ ) y_{1}=\varepsilon(x,\bar{C})
  102. y 2 = ε ( x , C ¯ ¯ ) . y_{2}=\varepsilon(x,\bar{\bar{C}}).
  103. p , q ( x ) p,q\in{\mathbb{Q}}(x)
  104. r ( x ) r\in{\mathbb{Q}}(x)
  105. p - q = r r p-q=\frac{r^{\prime}}{r}
  106. 2 2 {\mathcal{L}}^{2}_{2}
  107. y ′′ + ( 2 + 1 x ) y - 4 x 2 y = L c l m ( D + 2 x - 2 x - 2 x 2 - 2 x + 3 2 , D + 2 + 2 x - 1 x + 3 2 ) y = 0. y^{\prime\prime}+(2+\frac{1}{x})y^{\prime}-\frac{4}{x^{2}}y=Lclm\Big(D+\frac{2% }{x}-\frac{2x-2}{x^{2}-2x+{\frac{3}{2}}},D+2+\frac{2}{x}-\frac{1}{x+{\frac{3}{% 2}}}\Big)y=0.
  108. a 1 = 2 + 2 x - 1 x + 3 2 a_{1}=2+\frac{2}{x}-\frac{1}{x+\frac{3}{2}}
  109. a 2 = 2 x - 2 x - 2 x 2 - 2 x + 3 2 a_{2}=\frac{2}{x}-\frac{2x-2}{x^{2}-2x+\frac{3}{2}}
  110. a - a 2 + ( 2 + 1 x ) + 4 x 2 = 0 a^{\prime}-a^{2}+\big(2+\frac{1}{x}\big)+\frac{4}{x^{2}}=0
  111. y 1 = 2 3 - 4 3 x + 1 x 2 , y_{1}=\frac{2}{3}-\frac{4}{3x}+\frac{1}{x^{2}},
  112. y 2 = 2 x + 3 x 2 e - 2 x . y_{2}=\frac{2}{x}+\frac{3}{x^{2}}e^{-2x}.
  113. 3 2 {\mathcal{L}}^{2}_{3}
  114. y ′′ - 6 x 2 y = L c l m ( D + 2 x - 5 x 4 x 5 + C ) y = 0. y^{\prime\prime}-\frac{6}{x^{2}}y=Lclm\big(D+\frac{2}{x}-\frac{5x^{4}}{x^{5}+C% }\big)y=0.
  115. a - a 2 + 6 x 2 = 0 a^{\prime}-a^{2}+\frac{6}{x^{2}}=0
  116. y 1 = x 3 y_{1}=x^{3}
  117. y 2 = 1 x 2 y_{2}=\frac{1}{x^{2}}
  118. C = 0 C=0
  119. C C\rightarrow\infty
  120. n = 3 n=3
  121. {\mathcal{F}}
  122. δ \delta
  123. {\mathcal{F}}
  124. δ ( a + b ) = δ ( a ) + δ ( b ) \delta(a+b)=\delta(a)+\delta(b)
  125. δ ( a b ) = δ ( a ) b + a δ ( b ) \delta(ab)=\delta(a)b+a\delta(b)
  126. a , b a,b\in{\mathcal{F}}
  127. x = x \partial_{x}=\frac{\partial}{\partial x}
  128. y = y \partial_{y}=\frac{\partial}{\partial y}
  129. i , j r i , j ( x , y ) x i y j \sum_{i,j}r_{i,j}(x,y)\partial_{x}^{i}\partial_{y}^{j}
  130. r i , j r_{i,j}
  131. ( x , y ) {\mathbb{Q}}(x,y)
  132. 𝒟 = ( x , y ) [ x , y ] {\mathcal{D}}={\mathbb{Q}}(x,y)[\partial_{x},\partial_{y}]
  133. 𝒟 = [ x , y ] {\mathcal{D}}={\mathcal{F}}[\partial_{x},\partial_{y}]
  134. 𝒟 {\mathcal{D}}
  135. x a = a x + a x \partial_{x}a=a\partial_{x}+\frac{\partial a}{\partial x}
  136. a a
  137. L = i + j n r i , j ( x , y ) x i y j L=\sum_{i+j\leq n}r_{i,j}(x,y)\partial_{x}^{i}\partial_{y}^{j}
  138. n n
  139. s y m b ( L ) i + j = n r i , j ( x , y ) X i Y j symb(L)\equiv\sum_{i+j=n}r_{i,j}(x,y)X^{i}Y^{j}
  140. X X
  141. Y Y
  142. I I
  143. l i 𝒟 l_{i}\in{\mathcal{D}}
  144. i = 1 , , p i=1,\ldots,p
  145. I = l 1 , , l p I=\langle l_{1},\ldots,l_{p}\rangle
  146. I I
  147. 𝒟 {\mathcal{D}}
  148. l i 𝒟 l_{i}\in{\mathcal{D}}
  149. z z
  150. I = l 1 , l 2 , I=\langle l_{1},l_{2},\ldots\rangle
  151. l 1 z = 0 l_{1}z=0
  152. l 2 z = 0 , l_{2}z=0,\ldots
  153. z z
  154. δ \delta
  155. δ 1 \delta_{1}
  156. δ 2 \delta_{2}
  157. θ \theta
  158. δ θ δ \delta\preceq\theta\delta
  159. δ 1 δ 2 δ δ 1 δ δ 2 \delta_{1}\preceq\delta_{2}\rightarrow\delta\delta_{1}\preceq\delta\delta_{2}
  160. g r l e x grlex
  161. l 1 , , l p l_{1},\ldots,l_{p}
  162. I I
  163. I = l 1 , , l p I={\big\langle\big\langle}l_{1},\ldots,l_{p}{\big\rangle\big\rangle}
  164. I = l 1 x x - 1 x x - y x ( x + y ) y , I=\Big\langle l_{1}\equiv\partial_{xx}-\frac{1}{x}\partial_{x}-\frac{y}{x(x+y)% }\partial_{y},
  165. l 2 x y + 1 x + y y , l_{2}\equiv\partial_{xy}+\frac{1}{x+y}\partial_{y},
  166. l 3 y y + 1 x + y y l_{3}\equiv\partial_{yy}+\frac{1}{x+y}\partial_{y}\Big\rangle
  167. g r l e x grlex
  168. x y x\succ y
  169. l 1 , y = l 2 , x - l 2 , y = y + 2 x x ( x + y ) x y + y x ( x + y ) y y l_{1,y}=l_{2,x}-l_{2,y}=\frac{y+2x}{x(x+y)}\partial_{xy}+\frac{y}{x(x+y)}% \partial_{yy}
  170. I I
  171. y \partial_{y}
  172. I = x x - 1 x x , y I={\Big\langle\Big\langle}\partial_{xx}-\frac{1}{x}\partial_{x},\partial_{y}{% \Big\rangle\Big\rangle}
  173. I I
  174. J J
  175. I I
  176. J J
  177. I I
  178. I I
  179. J J
  180. I I
  181. J J
  182. I f 1 , , f p I\equiv\langle f_{1},\ldots,f_{p}\rangle
  183. J g 1 , , g q , J\equiv\langle g_{1},\ldots,g_{q}\rangle,
  184. f i f_{i}
  185. g j 𝒟 g_{j}\in{\mathcal{D}}
  186. i i
  187. j j
  188. I I
  189. J J
  190. G c r d ( I , J ) Gcrd(I,J)
  191. I I
  192. J J
  193. I I
  194. J J
  195. L c l m ( I , J ) z = 0 Lclm(I,J)z=0
  196. L L
  197. L L
  198. 𝔩 m x m + a m - 1 x m - 1 + + a 1 x + a 0 {\mathfrak{l}}_{m}\equiv\partial_{x^{m}}+a_{m-1}\partial_{x^{m-1}}+\ldots+a_{1% }\partial_{x}+a_{0}
  199. 𝔨 n y n + b n - 1 y n - 1 + + b 1 y + b 0 {\mathfrak{k}}_{n}\equiv\partial_{y^{n}}+b_{n-1}\partial_{y^{n-1}}+\ldots+b_{1% }\partial_{y}+b_{0}
  200. x x
  201. y y
  202. a i , b i ( x , y ) a_{i},b_{i}\in{\mathbb{Q}}(x,y)
  203. m m
  204. n n
  205. a i a_{i}
  206. i = 0 , , m - 1 i=0,\ldots,m-1
  207. L L
  208. 𝔩 m {\mathfrak{l}}_{m}
  209. m m
  210. 𝕃 x m ( L ) L , 𝔩 m {\mathbb{L}}_{x^{m}}(L)\equiv{\langle\langle}L,{\mathfrak{l}}_{m}{\rangle\rangle}
  211. L L
  212. b j b_{j}
  213. j = 0 , , n - 1 j=0,\ldots,n-1
  214. L L
  215. 𝔨 n {\mathfrak{k}}_{n}
  216. n n
  217. 𝕃 y n ( L ) L , 𝔨 n {\mathbb{L}}_{y^{n}}(L)\equiv{\langle\langle}L,{\mathfrak{k}}_{n}{\rangle\rangle}
  218. L L
  219. L L
  220. x x
  221. y y
  222. x x \partial_{xx}
  223. x y \partial_{xy}
  224. x x \partial_{xx}
  225. L L
  226. L x x + A 1 x y + A 2 y y + A 3 x + A 4 y + A 5 L\equiv\partial_{xx}+A_{1}\partial_{xy}+A_{2}\partial_{yy}+A_{3}\partial_{x}+A% _{4}\partial_{y}+A_{5}
  227. A i ( x , y ) A_{i}\in{\mathbb{Q}}(x,y)
  228. i i
  229. l i x + a i y + b i l_{i}\equiv\partial_{x}+a_{i}\partial_{y}+b_{i}
  230. i = 1 i=1
  231. i = 2 i=2
  232. l ( Φ ) x + a y + b ( Φ ) l(\Phi)\equiv\partial_{x}+a\partial_{y}+b(\Phi)
  233. a i , b i , a ( x , y ) a_{i},b_{i},a\in{\mathbb{Q}}(x,y)
  234. Φ \Phi
  235. L L
  236. x x 1 : L = l 2 l 1 ; {\mathcal{L}}_{xx}^{1}:L=l_{2}l_{1};
  237. x x 2 : L = L c l m ( l 2 , l 1 ) ; {\mathcal{L}}_{xx}^{2}:L=Lclm(l_{2},l_{1});
  238. x x 3 : L = L c l m ( l ( Φ ) ) . {\mathcal{L}}_{xx}^{3}:L=Lclm(l(\Phi)).
  239. L L
  240. x x i {\mathcal{L}}_{xx}^{i}
  241. i i
  242. L L
  243. x x 0 {\mathcal{L}}_{xx}^{0}
  244. x x 0 {\mathcal{L}}_{xx}^{0}
  245. x x 2 {\mathcal{L}}_{xx}^{2}
  246. x x 3 {\mathcal{L}}_{xx}^{3}
  247. L L
  248. L z z x x + A 1 z x y + A 2 z y y + A 3 z x + A 4 z y + A 5 z = 0 Lz\equiv z_{xx}+A_{1}z_{xy}+A_{2}z_{yy}+A_{3}z_{x}+A_{4}z_{y}+A_{5}z=0
  249. A 1 , , A 5 ( x , y ) A_{1},\ldots,A_{5}\in{\mathbb{Q}}(x,y)
  250. l i x + a i y + b i l_{i}\equiv\partial_{x}+a_{i}\partial_{y}+b_{i}
  251. a i , b i ( x , y ) a_{i},b_{i}\in{\mathbb{Q}}(x,y)
  252. i = 1 , 2 i=1,2
  253. φ i ( x , y ) = c o n s t \varphi_{i}(x,y)=const
  254. d y d x = a i ( x , y ) \frac{dy}{dx}=a_{i}(x,y)
  255. y ¯ φ i ( x , y ) \bar{y}\equiv\varphi_{i}(x,y)
  256. y = ψ i ( x , y ¯ ) y=\psi_{i}(x,\bar{y})
  257. φ i \varphi_{i}
  258. ψ i \psi_{i}
  259. i ( x , y ) exp ( - b i ( x , y ) | y = ψ i ( x , y ¯ ) d x ) | y ¯ = φ i ( x , y ) {\mathcal{E}}_{i}(x,y)\equiv\exp\Big(-{\displaystyle\int}b_{i}(x,y)\big|_{y=% \psi_{i}(x,\bar{y})}dx\Big)\Big|_{\bar{y}=\varphi_{i}(x,y)}
  260. i = 1 , 2 i=1,2
  261. x x 1 : z 1 ( x , y ) = 1 ( x , y ) F 1 ( φ 1 ) {\mathcal{L}}^{1}_{xx}:z_{1}(x,y)={\mathcal{E}}_{1}(x,y)F_{1}(\varphi_{1})
  262. z 2 ( x , y ) = 1 ( x , y ) 2 ( x , y ) 1 ( x , y ) F 2 ( φ 2 ( x , y ) ) | y = ψ 1 ( x , y ¯ ) d x | y ¯ = φ 1 ( x , y ) ; z_{2}(x,y)={\mathcal{E}}_{1}(x,y){\displaystyle\int}\frac{{\mathcal{E}}_{2}(x,% y)}{{\mathcal{E}}_{1}(x,y)}F_{2}\big(\varphi_{2}(x,y)\big)\big|_{y=\psi_{1}(x,% \bar{y})}dx\Big|_{\bar{y}=\varphi_{1}(x,y)};
  263. x x 2 : z i ( x , y ) = i ( x , y ) F i ( φ i ( x , y ) ) , i = 1 , 2 ; {\mathcal{L}}^{2}_{xx}:z_{i}(x,y)={\mathcal{E}}_{i}(x,y)F_{i}\big(\varphi_{i}(% x,y)\big),i=1,2;
  264. x x 3 : z i ( x , y ) = i ( x , y ) F i ( φ ( x , y ) ) , i = 1 , 2. {\mathcal{L}}^{3}_{xx}:z_{i}(x,y)={\mathcal{E}}_{i}(x,y)F_{i}\big(\varphi(x,y)% \big),i=1,2.
  265. F i F_{i}
  266. φ \varphi
  267. φ 1 \varphi_{1}
  268. φ 2 \varphi_{2}
  269. ψ 1 \psi_{1}
  270. φ 1 φ 2 \varphi_{1}\neq\varphi_{2}
  271. A 1 A_{1}
  272. A 2 A_{2}
  273. A 3 A_{3}
  274. z x x - z y y + 4 x + y z x = 0 z_{xx}-z_{yy}+\frac{4}{x+y}z_{x}=0
  275. L z l 2 l 1 z = ( x + y + 2 x + y ) ( x - y + 2 x + y ) z = 0 Lz\equiv l_{2}l_{1}z=\Big(\partial_{x}+\partial_{y}+\frac{2}{x+y}\Big)\Big(% \partial_{x}-\partial_{y}+\frac{2}{x+y}\Big)z=0
  276. φ 1 ( x , y ) = x + y , ψ 1 ( x , y ) = y ¯ - x , 1 ( x , y ) = exp ( 2 y x + y ) \varphi_{1}(x,y)=x+y,\psi_{1}(x,y)=\bar{y}-x,{\mathcal{E}}_{1}(x,y)=\exp{\Big(% \frac{2y}{x+y}\Big)}
  277. φ 2 ( x , y ) = x - y , ψ 2 ( x , y ) = x - y ¯ , 2 ( x , y ) = - 1 x + y . \varphi_{2}(x,y)=x-y,\psi_{2}(x,y)=x-\bar{y},{\mathcal{E}}_{2}(x,y)=-\frac{1}{% x+y}.
  278. z 1 ( x , y ) = exp ( 2 y x + y ) F ( x + y ) , z_{1}(x,y)=\exp{\Big(\frac{2y}{x+y}\Big)}F(x+y),
  279. z 2 ( x , y ) = 1 x + y exp ( 2 y x + y ) exp ( 2 x - y ¯ y ¯ ) G ( 2 x - y ¯ ) d x | y ¯ = x + y . z_{2}(x,y)=\frac{1}{x+y}\exp{\Big(\frac{2y}{x+y}\Big)}{\displaystyle\int}\exp{% \Big(\frac{2x-\bar{y}}{\bar{y}}\Big)}G(2x-\bar{y})dx\Big|_{\bar{y}=x+y}.
  280. F F
  281. G G
  282. x y \partial_{xy}
  283. L L
  284. L x y + A 1 x + A 2 y + A 3 L\equiv\partial_{xy}+A_{1}\partial_{x}+A_{2}\partial_{y}+A_{3}
  285. A i ( x , y ) A_{i}\in{\mathbb{Q}}(x,y)
  286. i i
  287. l x + A 2 l\equiv\partial_{x}+A_{2}
  288. k y + A 1 k\equiv\partial_{y}+A_{1}
  289. L L
  290. x y 1 : L = k l ; {\mathcal{L}}_{xy}^{1}:L=kl;
  291. x y 2 : L = l k ; {\mathcal{L}}_{xy}^{2}:L=lk;
  292. x y 3 : L = L c l m ( k , l ) . {\mathcal{L}}_{xy}^{3}:L=Lclm(k,l).
  293. L L
  294. x y i {\mathcal{L}}_{xy}^{i}
  295. i i
  296. x y 3 {\mathcal{L}}_{xy}^{3}
  297. L L
  298. L x y + A 1 x + A 2 y + A 3 L\equiv\partial_{xy}+A_{1}\partial_{x}+A_{2}\partial_{y}+A_{3}
  299. A i ( x , y ) A_{i}\in{\mathbb{Q}}(x,y)
  300. i i
  301. 𝕃 x m ( L ) \mathbb{L}_{x^{m}}(L)
  302. 𝕃 y n ( L ) \mathbb{L}_{y^{n}}(L)
  303. 𝔩 m {\mathfrak{l}}_{m}
  304. 𝔨 n {\mathfrak{k}}_{n}
  305. l x + a l\equiv\partial_{x}+a
  306. k y + b k\equiv\partial_{y}+b
  307. a , b ( x , y ) a,b\in{\mathbb{Q}}(x,y)
  308. L L
  309. m m
  310. n n
  311. m , n 2 m,n\geq 2
  312. x y 4 : L = L c l m ( 𝕃 x m ( L ) , 𝕃 y n ( L ) ) ; {\mathcal{L}}_{xy}^{4}:L=Lclm\big(\mathbb{L}_{x^{m}}(L),\mathbb{L}_{y^{n}}(L)% \big);
  313. x y 5 : L = E x q u o ( L , 𝕃 x m ( L ) ) 𝕃 x m ( L ) = ( 1 0 0 y + A 1 ) ( L 𝔩 m ) ; {\mathcal{L}}_{xy}^{5}:L=Exquo\big(L,\mathbb{L}_{x^{m}}(L)\big)\mathbb{L}_{x^{% m}}(L)=\left(\begin{array}[]{cc}1&0\\ 0&\partial_{y}+A_{1}\end{array}\right)\left(\begin{array}[]{c}L\\ {\mathfrak{l}}_{m}\end{array}\right);
  314. x y 6 : L = E x q u o ( L , 𝕃 y n ( L ) ) 𝕃 y n ( L ) = ( 1 0 0 x + A 2 ) ( L 𝔨 n ) ; {\mathcal{L}}_{xy}^{6}:L=Exquo\big(L,\mathbb{L}_{y^{n}}(L)\big)\mathbb{L}_{y^{% n}}(L)=\left(\begin{array}[]{cc}1&0\\ 0&\partial_{x}+A_{2}\end{array}\right)\left(\begin{array}[]{c}L\\ {\mathfrak{k}}_{n}\end{array}\right);
  315. x y 7 : L = L c l m ( k , 𝕃 x m ( L ) ) ; {\mathcal{L}}_{xy}^{7}:L=Lclm\big(k,\mathbb{L}_{x^{m}}(L)\big);
  316. x y 8 : L = L c l m ( l , 𝕃 y n ( L ) ) . {\mathcal{L}}_{xy}^{8}:L=Lclm\big(l,\mathbb{L}_{y^{n}}(L)\big).
  317. L L
  318. x y 0 {\mathcal{L}}_{xy}^{0}
  319. x y 0 {\mathcal{L}}_{xy}^{0}
  320. x y 4 {\mathcal{L}}_{xy}^{4}
  321. x y 7 {\mathcal{L}}_{xy}^{7}
  322. x y 8 {\mathcal{L}}_{xy}^{8}
  323. x y 4 {\mathcal{L}}_{xy}^{4}
  324. L x y + 2 x - y x - 2 x - y y - 4 ( x - y ) 2 L\equiv\partial_{xy}+\frac{2}{x-y}\partial_{x}-\frac{2}{x-y}\partial_{y}-\frac% {4}{(x-y)^{2}}
  325. L \langle L\rangle
  326. 𝕃 x 2 ( L ) x x - 2 x - y x + 2 ( x - y ) 2 , L {\mathbb{L}}_{x^{2}}(L)\equiv{\Big\langle\Big\langle}\partial_{xx}-\frac{2}{x-% y}\partial_{x}+\frac{2}{(x-y)^{2}},L{\Big\rangle\Big\rangle}
  327. 𝕃 y 2 ( L ) L , y y + 2 x - y y + 2 ( x - y ) 2 . {\mathbb{L}}_{y^{2}}(L)\equiv{\Big\langle\Big\langle}L,\partial_{yy}+\frac{2}{% x-y}\partial_{y}+\frac{2}{(x-y)^{2}}{\Big\rangle\Big\rangle}.
  328. L L
  329. L = L c l m ( 𝕃 x 2 ( L ) , 𝕃 y 2 ( L ) ) \langle L\rangle=Lclm\big({\mathbb{L}}_{x^{2}}(L),{\mathbb{L}}_{y^{2}}(L)\big)
  330. x y 4 {\mathcal{L}}^{4}_{xy}
  331. L z = 0 Lz=0
  332. z 1 ( x , y ) = 2 ( x - y ) F ( y ) + ( x - y ) 2 F ( y ) z_{1}(x,y)=2(x-y)F(y)+(x-y)^{2}F^{\prime}(y)
  333. z 2 ( x , y ) = 2 ( y - x ) G ( x ) + ( y - x ) 2 G ( x ) z_{2}(x,y)=2(y-x)G(x)+(y-x)^{2}G^{\prime}(x)
  334. L x x x + x x x y + 2 x x + 2 ( x + 1 ) x y + x + ( x + 2 ) y L\equiv\partial_{xxx}+x\partial_{xxy}+2\partial_{xx}+2(x+1)\partial_{xy}+% \partial_{x}+(x+2)\partial_{y}
  335. l 1 x + 1 l_{1}\equiv\partial_{x}+1
  336. l 2 x + x y l_{2}\equiv\partial_{x}+x\partial_{y}
  337. L 1 x x x - x 2 x y y + 3 x x + ( 2 x + 3 ) x y - x 2 y y + 2 x + ( 2 x + 3 ) y L_{1}\equiv\partial_{xxx}-x^{2}\partial_{xyy}+3\partial_{xx}+(2x+3)\partial_{% xy}-x^{2}\partial_{yy}+2\partial_{x}+(2x+3)\partial_{y}
  338. L 2 x x y + x x y y - 1 x x x - 1 x x y + x y y - 1 x x - ( 1 + 1 x ) y . L_{2}\equiv\partial_{xxy}+x\partial_{xyy}-\frac{1}{x}\partial_{xx}-\frac{1}{x}% \partial_{xy}+x\partial_{y}y-\frac{1}{x}\partial_{x}-\big(1+\frac{1}{x}\big)% \partial_{y}{\big\rangle\big\rangle}.
  339. L c l m ( l 2 , l 1 ) = L 1 , L 2 Lclm(l_{2},l_{1})={\langle\langle}L_{1},L_{2}{\rangle\rangle}
  340. L = ( 1 x 0 x + 1 + 1 x ) ( L 1 L 2 ) . L=\left(\begin{array}[]{cc}1&x\\ 0&\partial_{x}+1+\frac{1}{x}\end{array}\right)\left(\begin{array}[]{c}L_{1}\\ L_{2}\end{array}\right).
  341. L z = 0 Lz=0
  342. z 1 ( x , y ) = F ( y - 1 2 x 2 ) z_{1}(x,y)=F(y-\frac{1}{2}x^{2})
  343. z 2 ( x , y ) = G ( y ) e - x z_{2}(x,y)=G(y)e^{-x}
  344. z 3 ( x , y ) = x e - x H ( y ¯ + 1 2 x 2 ) d x | y ¯ = y - 1 2 x 2 z_{3}(x,y)={\displaystyle\int}xe^{-x}H\big(\bar{y}+\frac{1}{2}x^{2}\big)dx\Big% |_{\bar{y}=y-\frac{1}{2}x^{2}}
  345. F , G F,G
  346. H H

Log-space_computable_function.html

  1. f : Σ Σ f\colon\Sigma^{\ast}\rightarrow\Sigma^{\ast}
  2. O ( log n ) O(\log n)

Log_Gabor_filter.html

  1. G ( f ) = exp ( - ( log ( f / f 0 ) ) 2 2 ( log ( σ / f 0 ) ) 2 ) G(f)=\exp\left(\frac{-\left(\log(f/f_{0})\right)^{2}}{2\left(\log(\sigma/f_{0}% )\right)^{2}}\right)
  2. f 0 f_{0}
  3. σ \sigma
  4. f 0 f_{0}
  5. σ \sigma
  6. σ / f 0 \sigma/f_{0}
  7. G ( f , θ ) = exp ( - ( log ( f / f 0 ) ) 2 2 ( log ( σ f / f 0 ) ) 2 ) exp ( - ( θ - θ 0 ) 2 2 σ θ 2 ) G(f,\theta)=\exp\left(\frac{-(\log(f/f_{0}))^{2}}{2(\log(\sigma_{f}/f_{0}))^{2% }}\right)\exp\left(\frac{-(\theta-\theta_{0})^{2}}{2\sigma_{\theta}^{2}}\right)
  8. f 0 f_{0}
  9. σ f \sigma_{f}
  10. θ 0 \theta_{0}
  11. σ θ \sigma_{\theta}
  12. B = 2 2 / log ( 2 ) ( log ( σ f / f 0 ) ) B=2\sqrt{2/\log(2)}\left(\|\log(\sigma_{f}/f_{0})\|\right)
  13. B θ = 2 σ θ 2 log 2 B_{\theta}=2\sigma_{\theta}\sqrt{2\log 2}

Logarithmically_convex_set.html

  1. D D
  2. D * = { z = ( z 1 , , z n ) D / z 1 z n 0 } D^{*}=\{z=(z_{1},\cdots,z_{n})\in D/z_{1}\cdots z_{n}\neq 0\}
  3. λ : z λ ( z ) = ( ln ( | z 1 | ) , , ln ( | z n | ) ) \lambda:z\rightarrow\lambda(z)=(\ln(|z_{1}|),\cdots,\ln(|z_{n}|))
  4. n \mathbb{R}^{n}

Logic.html

  1. m ( x ) m(x)
  2. d ( x ) d(x)
  3. x . ( m ( x ) d ( x ) ) \forall x.(m(x)\rightarrow d(x))
  4. a a
  5. b b
  6. a a
  7. b b
  8. a a
  9. b b
  10. a a
  11. b b

Logic_learning_machine.html

  1. y y
  2. if p r e m i s e then c o n s e q u e n c e \,\textbf{if }\,\text{ }premise\,\text{ }\,\textbf{ then }\,\text{ }consequence
  3. x 1 { A , B , C , } x_{1}\in\{A,B,C,...\}
  4. x 2 α x_{2}\leq\alpha
  5. β x 3 γ \beta\leq x_{3}\leq\gamma
  6. if x 1 { A , B , C , } AND x 2 α AND β x 3 γ then y = y ¯ \,\textbf{if }\,\text{ }x_{1}\in\{A,B,C,...\}\,\text{ AND }x_{2}\leq\alpha\,% \text{ AND }\beta\leq x_{3}\leq\gamma\,\text{ }\,\textbf{ then }\,\text{ }y=% \bar{y}

LogSumExp.html

  1. L S E ( x 1 , , x n ) = log ( exp ( x 1 ) + + exp ( x n ) ) LSE(x_{1},\dots,x_{n})=\log\left(\exp(x_{1})+\cdots+\exp(x_{n})\right)
  2. \R n \R^{n}
  3. \R \R
  4. x k x_{k}

London_dial.html

  1. S H = ϕ SH=\phi
  2. tan H H = sin ϕ tan ( 15 × t ) \tan H_{H}=\sin\phi\tan(15^{\circ}\times t)
  3. H H H_{H}
  4. ϕ \phi
  5. sin ϕ \sin\phi
  6. tan h sin ϕ \tan h\sin\phi
  7. sin ϕ \sin\phi
  8. tan h sin ϕ \tan h\sin\phi

Longifolene_synthase.html

  1. \rightleftharpoons

Loop_representation_in_gauge_theories_and_quantum_gravity.html

  1. E = ρ ϵ 0 × B - ϵ 0 μ 0 E t = μ 0 J × E + B t = 0 B = 0 \nabla\cdot\vec{E}={\rho\over\epsilon_{0}}\qquad\nabla\times\vec{B}-\epsilon_{% 0}\mu_{0}{\partial\vec{E}\over\partial t}=\mu_{0}\vec{J}\qquad\nabla\times\vec% {E}+{\partial\vec{B}\over\partial t}=0\qquad\nabla\cdot\vec{B}=0
  2. ρ \rho
  3. J \vec{J}
  4. ϕ \phi
  5. A \vec{A}
  6. E = - ϕ - A t B = × A \vec{E}=-\nabla\phi-{\partial\vec{A}\over\partial t}\qquad\vec{B}=\nabla\times% \vec{A}
  7. ϕ = ϕ + Λ t A = A - Λ \phi^{\prime}=\phi+{\partial\Lambda\over\partial t}\qquad\vec{A}^{\prime}=\vec% {A}-\nabla\Lambda
  8. Λ ( x , t ) \Lambda(\vec{x},t)
  9. A μ = ( ϕ , A ) A^{\mu}=(\phi,\vec{A})
  10. A μ = A μ + μ Λ {A^{\mu}}^{\prime}=A^{\mu}+\partial^{\mu}\Lambda
  11. F μ ν = μ A ν - ν A μ F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}
  12. F 0 i = E i , ϵ i j k F j k = B i F^{0i}=E^{i},\qquad\epsilon^{ijk}F^{jk}=B^{i}
  13. S = - 1 2 d 4 x ( F μ ν F μ ν ) S=-{1\over 2}\int d^{4}x\Big(F_{\mu\nu}F^{\mu\nu}\Big)
  14. U ( 1 ) U(1)
  15. ψ ( x ) \psi(x)
  16. ψ ( x ) = exp ( i θ ) ψ ( x ) \psi^{\prime}(x)=\exp(i\theta)\psi(x)
  17. ψ ( x ) = Ω ψ ( x ) = exp ( i θ ( x ) ) ψ ( x ) \psi^{\prime}(x)=\Omega\psi(x)=\exp(i\theta(x))\psi(x)
  18. ψ ( x ) \psi(x)
  19. μ ( exp ( i θ ( x ) ) ψ ( x ) ) = Ω μ ψ ( x ) + μ Ω ψ ( x ) \partial_{\mu}(\exp(i\theta(x))\psi(x))=\Omega\partial_{\mu}\psi(x)+\partial_{% \mu}\Omega\psi(x)
  20. 𝒟 μ \mathcal{D}_{\mu}
  21. 𝒟 μ \mathcal{D}_{\mu}
  22. A μ A_{\mu}
  23. 𝒟 μ = μ + i g A μ ( x ) \mathcal{D}_{\mu}=\partial_{\mu}+igA_{\mu}(x)
  24. ( 𝒟 μ ψ ) = μ ψ + i g A μ ψ = Ω μ ψ + ( Ω ) ψ + i g A μ Ω ψ (\mathcal{D}_{\mu}\psi)^{\prime}=\partial_{\mu}\psi^{\prime}+igA_{\mu}^{\prime% }\psi^{\prime}=\Omega\partial_{\mu}\psi+(\partial\Omega)\psi+igA_{\mu}^{\prime% }\Omega\psi
  25. μ Ω \partial_{\mu}\Omega
  26. A μ ( x ) = A μ ( x ) + i g [ μ Ω ( x ) ] Ω - 1 ( x ) E q 1. A_{\mu}^{\prime}(x)=A_{\mu}(x)+{i\over g}[\partial_{\mu}\Omega(x)]\Omega^{-1}(% x)\quad Eq1.
  27. ( 𝒟 μ ψ ) = Ω 𝒟 μ ψ (\mathcal{D}_{\mu}\psi)^{\prime}=\Omega\mathcal{D}_{\mu}\psi
  28. E q 1 Eq1
  29. A μ ( x ) = A μ ( x ) + 1 g μ θ ( x ) A_{\mu}^{\prime}(x)=A_{\mu}(x)+{1\over g}\partial_{\mu}\theta(x)
  30. F μ ν = - i g [ 𝒟 μ , 𝒟 μ ] = - i g [ μ + i g A μ ( x ) , ν + i g A ν ( x ) ] F_{\mu\nu}={-i\over g}[\mathcal{D}_{\mu},\mathcal{D}_{\mu}]={-i\over g}[% \partial_{\mu}+igA_{\mu}(x),\partial_{\nu}+igA_{\nu}(x)]
  31. = - i g ( [ μ , ν ] + i g ( μ A ν - ν A μ ) - g 2 [ A μ , A ν ] ) ={-i\over g}\Big([\partial_{\mu},\partial_{\nu}]+ig(\partial_{\mu}A_{\nu}-% \partial_{\nu}A_{\mu})-g^{2}[A_{\mu},A_{\nu}]\Big)
  32. = μ A ν - ν A ν =\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\nu}
  33. S = - 1 2 d 4 x ( F μ ν F μ ν ) S=-{1\over 2}\int d^{4}x\Big(F_{\mu\nu}F^{\mu\nu}\Big)
  34. [ T i , T j ] = i f i j k T k [T_{i},T_{j}]=if^{ijk}T^{k}
  35. 𝚿 Ω ^ ( x ) 𝚿 ( x ) = exp ( i θ i ( x ) T i ) 𝚿 ( x ) \mathbf{\Psi}^{\prime}\mapsto\hat{\Omega}(x)\mathbf{\Psi}(x)=\exp(i\theta^{i}(% x)T^{i})\mathbf{\Psi}(x)
  36. 𝚿 ( x ) \mathbf{\Psi}(x)
  37. 𝒟 μ = 𝐈 μ + i g 𝐀 μ ( x ) \mathbf{\mathcal{D}}_{\mu}=\mathbf{I}\partial_{\mu}+ig\mathbf{A}_{\mu}(x)
  38. 𝐀 μ ( x ) = A μ i ( x ) T i \mathbf{A}_{\mu}(x)=A^{i}_{\mu}(x)T^{i}
  39. 𝐀 μ ( x ) \mathbf{A}_{\mu}(x)
  40. 𝐀 μ ( x ) = Ω ^ 𝐀 μ ( x ) Ω ^ - 1 + i g Ω ^ ( μ Ω ^ - 1 ) \mathbf{A}_{\mu}^{\prime}(x)=\hat{\Omega}\mathbf{A}_{\mu}(x)\hat{\Omega}^{-1}+% {i\over g}\hat{\Omega}(\partial_{\mu}\hat{\Omega}^{-1})
  41. 𝐅 μ ν = - i g [ 𝒟 μ , 𝒟 ν ] = μ 𝐀 ν - ν 𝐀 μ + i g [ 𝐀 μ , 𝐀 ν ] = ( μ A ν i - ν A μ i + g f i j k A μ j A ν k ) T i \mathbf{F}_{\mu\nu}=-{i\over g}[\mathbf{\mathcal{D}}_{\mu},\mathbf{\mathcal{D}% }_{\nu}]=\partial_{\mu}\mathbf{A}_{\nu}-\partial_{\nu}\mathbf{A}_{\mu}+ig[% \mathbf{A}_{\mu},\mathbf{A}_{\nu}]=(\partial_{\mu}A_{\nu}^{i}-\partial_{\nu}A_% {\mu}^{i}+gf^{ijk}A_{\mu}^{j}A_{\nu}^{k})T^{i}
  42. 𝒟 μ \mathbf{\mathcal{D}}_{\mu}
  43. F μ ν i F^{i}_{\mu\nu}
  44. 𝐅 μ ν 𝐅 μ ν = Ω ^ 𝐅 μ ν Ω ^ - 1 \mathbf{F}_{\mu\nu}\mapsto\mathbf{F}_{\mu\nu}^{\prime}=\hat{\Omega}\mathbf{F}_% {\mu\nu}\hat{\Omega}^{-1}
  45. 𝐅 μ ν \mathbf{F}_{\mu\nu}
  46. Ω ^ \hat{\Omega}
  47. T r ( Ω ^ 𝐅 μ ν Ω ^ - 1 Ω ^ 𝐅 μ ν Ω ^ - 1 ) = T r ( 𝐅 μ ν 𝐅 μ ν ) Tr(\hat{\Omega}\mathbf{F}_{\mu\nu}\hat{\Omega}^{-1}\hat{\Omega}\mathbf{F}^{\mu% \nu}\hat{\Omega}^{-1})=Tr(\mathbf{F}_{\mu\nu}\mathbf{F}^{\mu\nu})
  48. S = - 1 2 d 4 x T r ( 𝐅 μ ν 𝐅 μ ν ) = - 1 2 d 4 x T r ( F μ ν i T j F j μ ν T j ) S=-{1\over 2}\int d^{4}xTr(\mathbf{F}_{\mu\nu}\mathbf{F}^{\mu\nu})=-{1\over 2}% \int d^{4}xTr\Big(F_{\mu\nu}^{i}T^{j}F^{\mu\nu}_{j}T^{j}\Big)
  49. γ \mid\gamma\rangle
  50. A γ = W ( γ ) = exp [ i e γ d y α A α ( y ) ] \langle A\mid\gamma\rangle=W(\gamma)=\exp\left[ie\int_{\gamma}dy^{\alpha}A_{% \alpha}(y)\right]
  51. W ( γ ) W(\gamma)
  52. U ( 1 ) U(1)
  53. S U ( 2 ) SU(2)
  54. S U ( 2 ) SU(2)
  55. A μ i ( x ) A^{i}_{\mu}(x)
  56. i i
  57. S U ( 2 ) SU(2)
  58. 𝐀 μ ( x ) = A μ i ( x ) τ i \mathbf{A}_{\mu}(x)=A^{i}_{\mu}(x)\tau_{i}
  59. τ i \tau_{i}
  60. s u ( 2 ) su(2)
  61. i / 2 i/2
  62. 𝐀 μ ( x ) \mathbf{A}_{\mu}(x)
  63. S U ( 2 ) SU(2)
  64. A a i A_{a}^{i}
  65. E ~ i a \tilde{E}_{i}^{a}
  66. Ψ ( A a i ) \Psi(A_{a}^{i})
  67. A ^ a i Ψ [ A ] = A a i Ψ [ A ] \hat{A}_{a}^{i}\Psi[A]=A_{a}^{i}\Psi[A]
  68. q ^ ψ ( q ) = q ψ ( q ) \hat{q}\psi(q)=q\psi(q)
  69. E ~ ^ a i Ψ [ A ] = - i δ Ψ [ A ] δ A a i \hat{\tilde{E}}_{a}^{i}\Psi[A]=-i{\delta\Psi[A]\over\delta A_{a}^{i}}
  70. p ^ ψ ( q ) = - i d ψ ( q ) d q \hat{p}\psi(q)=-i{d\psi(q)\over dq}
  71. h γ [ A ] h_{\gamma}[A]
  72. ( h e ) α β = U α γ - 1 ( x ) ( h e ) γ σ U σ β ( y ) . (h^{\prime}_{e})_{\alpha\beta}=U_{\alpha\gamma}^{-1}(x)(h_{e})_{\gamma\sigma}U% _{\sigma\beta}(y).
  73. x = y x=y
  74. α = β \alpha=\beta
  75. ( h e ) α α = U α γ - 1 ( x ) ( h e ) γ σ U σ α ( x ) = [ U σ α ( x ) U α γ - 1 ( x ) ] ( h e ) γ σ = δ σ γ ( h e ) γ σ = ( h e ) γ γ (h^{\prime}_{e})_{\alpha\alpha}=U_{\alpha\gamma}^{-1}(x)(h_{e})_{\gamma\sigma}% U_{\sigma\alpha}(x)=[U_{\sigma\alpha}(x)U_{\alpha\gamma}^{-1}(x)](h_{e})_{% \gamma\sigma}=\delta_{\sigma\gamma}(h_{e})_{\gamma\sigma}=(h_{e})_{\gamma\gamma}
  76. Tr h γ = Tr h γ . \operatorname{Tr}h^{\prime}_{\gamma}=\operatorname{Tr}h_{\gamma}.
  77. W γ [ A ] W_{\gamma}[A]
  78. h γ [ A ] = 𝒫 exp { - γ 0 γ 1 d s γ ˙ a A a i ( γ ( s ) ) T i } h_{\gamma}[A]=\mathcal{P}\exp\Big\{-\int_{\gamma_{0}}^{\gamma_{1}}\,ds\dot{% \gamma}^{a}A_{a}^{i}(\gamma(s))T_{i}\Big\}
  79. γ \gamma
  80. s s
  81. 𝒫 \mathcal{P}
  82. s s
  83. T i T_{i}
  84. s u ( 2 ) su(2)
  85. [ T i , T j ] = 2 i ϵ i j k T k . [T^{i},T^{j}]=2i\epsilon^{ijk}T^{k}.\,
  86. ( N + 1 ) × ( N + 1 ) (N+1)\times(N+1)
  87. N = 1 , 2 , 3 , N=1,2,3,\dots
  88. s u ( 2 ) su(2)
  89. N / 2 N/2
  90. Ψ [ A ] = γ Ψ [ γ ] W γ [ A ] \Psi[A]=\sum_{\gamma}\Psi[\gamma]W_{\gamma}[A]
  91. exp ( i k x ) \exp(ikx)
  92. k k
  93. ψ [ x ] = d k ψ ( k ) exp ( i k x ) . \psi[x]=\int dk\psi(k)\exp(ikx).
  94. ψ ( k ) \psi(k)
  95. Ψ [ γ ] = [ d A ] Ψ [ A ] W γ [ A ] . \Psi[\gamma]=\int[dA]\Psi[A]W_{\gamma}[A].
  96. O ^ \hat{O}
  97. Φ [ A ] = O ^ Ψ [ A ] , Eq 1 \Phi[A]=\hat{O}\Psi[A],\qquad\,\text{Eq 1}
  98. O ^ \hat{O}^{\prime}
  99. Ψ [ γ ] \Psi[\gamma]
  100. Φ [ γ ] = O ^ Ψ [ γ ] , Eq 2 \Phi[\gamma]=\hat{O}^{\prime}\Psi[\gamma],\qquad\,\text{Eq 2}
  101. Φ [ γ ] \Phi[\gamma]
  102. Φ [ γ ] = [ d A ] Φ [ A ] W γ [ A ] . Eq 3 \Phi[\gamma]=\int[dA]\Phi[A]W_{\gamma}[A].\qquad\,\text{Eq 3}
  103. O ^ \hat{O}^{\prime}
  104. Ψ [ γ ] \Psi[\gamma]
  105. O ^ \hat{O}
  106. Ψ [ A ] \Psi[A]
  107. E q 2 Eq\;2
  108. E q 3 Eq\;3
  109. E q 1 Eq\;1
  110. E q 3 Eq\;3
  111. O ^ Ψ [ γ ] = [ d A ] W γ [ A ] O ^ Ψ [ A ] , \hat{O}^{\prime}\Psi[\gamma]=\int[dA]W_{\gamma}[A]\hat{O}\Psi[A],
  112. O ^ Ψ [ γ ] = [ d A ] ( O ^ W γ [ A ] ) Ψ [ A ] , \hat{O}^{\prime}\Psi[\gamma]=\int[dA](\hat{O}^{\dagger}W_{\gamma}[A])\Psi[A],
  113. O ^ \hat{O}^{\dagger}
  114. O ^ \hat{O}
  115. Ψ [ A ] \Psi[A]
  116. O ^ \hat{O}^{\prime}
  117. O ^ \hat{O}^{\dagger}
  118. A A
  119. W γ [ A ] W_{\gamma}[A]
  120. γ \gamma
  121. O ^ \hat{O}^{\prime}
  122. γ \gamma
  123. Ψ [ γ ] \Psi[\gamma]
  124. Ψ [ γ ] \Psi[\gamma]
  125. γ \gamma
  126. Σ \Sigma
  127. x 3 = 0 x^{3}=0
  128. Σ \Sigma
  129. sin θ \sin\theta
  130. θ \theta
  131. u \vec{u}
  132. v \vec{v}
  133. A \displaystyle A
  134. Σ \Sigma
  135. A Σ = Σ d x 1 d x 2 det q ( 2 ) A_{\Sigma}=\int_{\Sigma}\,dx^{1}\,dx^{2}\sqrt{\det\;q^{(2)}}
  136. det q ( 2 ) = q 11 q 22 - q 12 2 \det q^{(2)}=q_{11}q_{22}-q_{12}^{2}
  137. Σ \Sigma
  138. det q ( 2 ) = ϵ 3 a b ϵ 3 c d q a c q b c 2 . \det\;q^{(2)}={\epsilon^{3ab}\epsilon^{3cd}q_{ac}q_{bc}\over 2}.
  139. q a b = ϵ a c d ϵ b e f q c e q d f 3 ! det ( q ) q^{ab}={\epsilon^{acd}\epsilon^{bef}q_{ce}q_{df}\over 3!\det(q)}
  140. det q ( 2 ) \det q^{(2)}
  141. E ~ i a E ~ b i = det ( q ) q a b \tilde{E}^{a}_{i}\tilde{E}^{bi}=\det(q)q^{ab}
  142. A Σ = Σ d x 1 d x 2 E ~ i 3 E ~ 3 i . A_{\Sigma}=\int_{\Sigma}\,dx^{1}\,dx^{2}\sqrt{\tilde{E}^{3}_{i}\tilde{E}^{3i}}.
  143. E ~ i 3 \tilde{E}^{3}_{i}
  144. E ~ ^ i 3 δ δ A 3 i . \hat{\tilde{E}}^{3}_{i}\sim{\delta\over\delta A_{3}^{i}}.
  145. A Σ A_{\Sigma}
  146. N = 2 J N=2J
  147. i T i T i = J ( J + 1 ) 1 \sum_{i}T^{i}T^{i}=J(J+1)1
  148. A ^ Σ W γ [ A ] = 8 π P l a n c k 2 β I j I ( j I + 1 ) W γ [ A ] \hat{A}_{\Sigma}W_{\gamma}[A]=8\pi\ell_{Planck}^{2}\beta\sum_{I}\sqrt{j_{I}(j_% {I}+1)}W_{\gamma}[A]
  149. I I
  150. Σ \Sigma
  151. R R
  152. V = R d 3 x det ( q ) = 1 6 R d x 3 ϵ a b c ϵ i j k E ~ i a E ~ j b E ~ k c . V=\int_{R}d^{3}x\sqrt{\det(q)}={1\over 6}\int_{R}dx^{3}\sqrt{\epsilon_{abc}% \epsilon^{ijk}\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}\tilde{E}^{c}_{k}}.
  153. γ ˙ a \dot{\gamma}^{a}
  154. S U ( 2 ) SU(2)
  155. S U ( 2 ) SU(2)
  156. 𝔸 \mathbb{A}
  157. 𝔹 \mathbb{B}
  158. Tr ( 𝔸 ) Tr ( 𝔹 ) = Tr ( 𝔸 𝔹 ) + Tr ( 𝔸 𝔹 - 1 ) . \operatorname{Tr}(\mathbb{A})\operatorname{Tr}(\mathbb{B})=\operatorname{Tr}(% \mathbb{A}\mathbb{B})+\operatorname{Tr}(\mathbb{A}\mathbb{B}^{-1}).
  159. γ \gamma
  160. η \eta
  161. W γ [ A ] W η [ A ] = W γ η [ A ] + W γ η - 1 [ A ] W_{\gamma}[A]W_{\eta}[A]=W_{\gamma\circ\eta}[A]+W_{\gamma\circ\eta^{-1}}[A]
  162. η - 1 \eta^{-1}
  163. η \eta
  164. γ η \gamma\circ\eta
  165. γ \gamma
  166. η \eta
  167. W ( γ 1 γ 2 ) = W ( γ 2 γ 1 ) W(\gamma_{1}\circ\gamma_{2})=W(\gamma_{2}\circ\gamma_{1})
  168. γ \gamma

Lorentz-violating_electrodynamics.html

  1. d d
  2. d = 3 , 4 d=3,4
  3. = - 1 4 F μ ν F μ ν + 1 2 ( k A F ) κ ϵ κ λ μ ν A λ F μ ν - 1 4 ( k F ) κ λ μ ν F κ λ F μ ν . \mathcal{L}=-\textstyle{{1}\over{4}}\,F_{\mu\nu}F^{\mu\nu}+\textstyle{{1}\over% {2}}\,(k_{AF})^{\kappa}\,\epsilon_{\kappa\lambda\mu\nu}A^{\lambda}F^{\mu\nu}-% \textstyle{{1}\over{4}}\,(k_{F})_{\kappa\lambda\mu\nu}F^{\kappa\lambda}F^{\mu% \nu}.
  4. d = 3 d=3
  5. ( k A F ) κ (k_{AF})^{\kappa}
  6. d = 4 d=4
  7. ( k F ) κ λ μ ν (k_{F})_{\kappa\lambda\mu\nu}
  8. ( k A F ) κ (k_{AF})^{\kappa}
  9. ( k F ) κ λ μ ν (k_{F})_{\kappa\lambda\mu\nu}
  10. ( k A F ) κ (k_{AF})^{\kappa}
  11. ( k F ) κ λ μ ν (k_{F})_{\kappa\lambda\mu\nu}
  12. d 5 d\geq 5
  13. = - 1 4 F μ ν F μ ν + 1 2 ϵ κ λ μ ν A λ ( k ^ A F ) κ F μ ν - 1 4 F κ λ ( k ^ F ) κ λ μ ν F μ ν , \mathcal{L}=-\textstyle{1\over 4}F_{\mu\nu}F^{\mu\nu}+\textstyle{1\over 2}% \epsilon^{\kappa\lambda\mu\nu}A_{\lambda}{(\hat{k}_{AF})}_{\kappa}F_{\mu\nu}-% \textstyle{1\over 4}F_{\kappa\lambda}{(\hat{k}_{F})}^{\kappa\lambda\mu\nu}F_{% \mu\nu}\ ,
  14. ( k ^ A F ) κ {(\hat{k}_{AF})}_{\kappa}
  15. ( k ^ F ) κ λ μ ν {(\hat{k}_{F})}^{\kappa\lambda\mu\nu}
  16. ( k ^ A F ) κ {(\hat{k}_{AF})}_{\kappa}
  17. d = 3 , 5 , 7 , d=3,5,7,\ldots
  18. d = 4 , 6 , 8 , d=4,6,8,\ldots
  19. ( k ^ F ) κ λ μ ν {(\hat{k}_{F})}^{\kappa\lambda\mu\nu}
  20. d = 3 , 4 d=3,4
  21. Δ ϕ = 2 π Δ v t / λ \Delta\phi=2\pi\Delta v\,t/\lambda
  22. Δ v \Delta v
  23. t t
  24. λ \lambda
  25. d > 3 d>3
  26. t / λ t/\lambda
  27. Δ v \Delta v
  28. d > 3 d>3
  29. d = 4 d=4
  30. d = 3 d=3
  31. Δ v \Delta v
  32. λ \lambda
  33. d = 3 d=3
  34. d 4 d\neq 4
  35. Δ t = Δ v L / c 2 \Delta t=\Delta vL/c^{2}
  36. Δ v \Delta v
  37. L L
  38. Δ v \Delta v
  39. E d - 4 E^{d-4}
  40. d > 4 d>4
  41. d d
  42. E 2 E^{2}
  43. E 4 E^{4}
  44. E E
  45. E 3 E^{3}
  46. d = 4 d=4
  47. d = 6 d=6
  48. d = 8 d=8
  49. d > 4 d>4

Loss_free_resistor.html

  1. 𝐱 ( 0 ) = [ v i ( t ) i i ( t ) ] = [ k ( t ) 0 0 k - 1 ( t ) ] [ v 0 ( t ) - i 0 ( t ) ] \mathbf{x}^{(0)}=\begin{bmatrix}v_{i}(t)\\ i_{i}(t)\\ \end{bmatrix}=\begin{bmatrix}k(t)&0\\ 0&k^{-1}(t)\\ \end{bmatrix}\begin{bmatrix}v_{0}(t)\\ -i_{0}(t)\\ \end{bmatrix}
  2. v i = k ( r ) U = v a b ( t ) v_{i}=k(r)U=v_{ab}(t)
  3. v a b ( t ) = i a b ( t ) R v_{ab}(t)=i_{ab}(t)R
  4. k ( t ) = i a b ( t ) R U k(t)=\frac{i_{ab}(t)R}{U}
  5. 𝐱 ( 0 ) = [ V i i i ] = [ 0 g - 1 ( t ) g ( t ) 0 ] [ V 0 ( t ) - i 0 ( t ) ] \mathbf{x}^{(0)}=\begin{bmatrix}V_{i}\\ i_{i}\\ \end{bmatrix}=\begin{bmatrix}0&g^{-1}(t)\\ g(t)&0\\ \end{bmatrix}\begin{bmatrix}V_{0}(t)\\ -i_{0}(t)\\ \end{bmatrix}
  6. g ( t ) = v a b ( t ) U R g(t)=\frac{v_{ab}(t)}{UR}

Loss_functions_for_classification.html

  1. X X
  2. f : X f:X\mapsto\mathbb{R}
  3. f ( x ) f(\vec{x})
  4. y y
  5. x \vec{x}
  6. y y
  7. I [ f ] = X Y V ( f ( x ) , y ) p ( x , y ) d x d y I[f]=\displaystyle\int_{X\otimes Y}V(f(\vec{x}),y)p(\vec{x},y)\,d\vec{x}\,dy
  8. V ( f ( x ) , y ) V(f(\vec{x}),y)
  9. p ( x , y ) p(\vec{x},y)
  10. p ( x , y ) = p ( y x ) p ( x ) . p(\vec{x},y)=p(y\mid\vec{x})p(\vec{x}).
  11. p ( x , y ) p(\vec{x},y)
  12. n n
  13. S = { ( x 1 , y 1 ) , , ( x n , y n ) } S=\{(\vec{x}_{1},y_{1}),\dots,(\vec{x}_{n},y_{n})\}
  14. I S [ f ] = 1 n i = 1 n V ( f ( x i ) , y i ) . I_{S}[f]=\frac{1}{n}\sum_{i=1}^{n}V(f(\vec{x}_{i}),y_{i}).
  15. y y
  16. f ( x ) f(\vec{x})
  17. V ( f ( x ) , y ) = ϕ ( - y f ( x ) ) V(f(\vec{x}),y)=\phi(-yf(\vec{x}))
  18. f S * f^{*}_{S}
  19. V ( f ( x ) , y ) = H ( - y f ( x ) ) V(f(\vec{x}),y)=H(-yf(\vec{x}))
  20. H H
  21. f * f^{*}
  22. f * ( x ) = { 1 if p ( 1 x ) > p ( - 1 x ) - 1 if p ( 1 x ) < p ( - 1 x ) f^{*}(\vec{x})\;=\;\begin{cases}1&\,\text{if }p(1\mid\vec{x})>p(-1\mid\vec{x})% \\ -1&\,\text{if }p(1\mid\vec{x})<p(-1\mid\vec{x})\end{cases}
  23. p ( 1 x ) p ( - 1 x ) p(1\mid\vec{x})\neq p(-1\mid\vec{x})
  24. V ( y f 0 ( x ) ) V(yf_{0}(\vec{x}))
  25. f 0 ( x ) 0 f_{0}(\vec{x})\neq 0
  26. f * ( x ) = sgn ( f 0 ( x ) ) f^{*}(\vec{x})=\operatorname{sgn}(f_{0}(\vec{x}))
  27. sgn \operatorname{sgn}
  28. f 0 ( x ) 0 f_{0}(\vec{x})\neq 0
  29. I [ f ] = X Y V ( f ( x ) , y ) p ( x , y ) d x d y = X Y V ( - y f ( x ) ) p ( y x ) p ( x ) d y d x = Y V ( - y f ( x ) ) p ( y x ) d y = V ( - f ( x ) ) p ( 1 x ) + V ( f ( x ) ) p ( - 1 x ) = V ( - f ( x ) ) p ( 1 x ) + V ( f ( x ) ) ( 1 - p ( 1 x ) ) \begin{aligned}\displaystyle I[f]&\displaystyle=\int_{X\otimes Y}V(f(\vec{x}),% y)p(\vec{x},y)\,d\vec{x}\,dy\\ &\displaystyle=\int_{X}\int_{Y}V(-yf(\vec{x}))p(y\mid\vec{x})p(\vec{x})\,dy\,d% \vec{x}\\ &\displaystyle=\int_{Y}V(-yf(\vec{x}))p(y\mid\vec{x})\,dy\\ &\displaystyle=V(-f(\vec{x}))p(1\mid\vec{x})+V(f(\vec{x}))p(-1\mid\vec{x})\\ &\displaystyle=V(-f(\vec{x}))p(1\mid\vec{x})+V(f(\vec{x}))(1-p(1\mid\vec{x}))% \end{aligned}
  30. x \vec{x}
  31. X p ( x ) d x = 1 \int_{X}p(x)\,dx=1
  32. y y
  33. p ( - 1 x ) = 1 - p ( 1 x ) p(-1\mid x)=1-p(1\mid x)
  34. I [ f ] I[f]
  35. f f
  36. f ( x ) f(\vec{x})
  37. p ( 1 x ) p(1\mid x)
  38. ϕ ( y f ( x ) ) \phi(yf(\vec{x}))
  39. V ( f ( x ) , y ) = ( 1 - y f ( x ) ) 2 V(f(\vec{x}),y)=(1-yf(\vec{x}))^{2}
  40. y f ( x ) = 0 yf(\vec{x})=0
  41. y f ( x ) = 1 yf(\vec{x})=1
  42. f ( x ) f(\vec{x})
  43. x X x\in X
  44. y f ( x ) yf(\vec{x})
  45. y y
  46. f ( x ) f(\vec{x})
  47. I [ f ] I[f]
  48. f * Square = 2 p ( 1 x ) - 1 f^{*}\text{Square}=2p(1\mid x)-1
  49. f * f^{*}
  50. p ( 1 x ) = 1 p(1\mid x)=1
  51. p ( 1 x ) = 0 p(1\mid x)=0
  52. x \vec{x}
  53. V ( f ( x ) , y ) = max ( 0 , 1 - y f ( x ) ) = | 1 - y f ( x ) | + . V(f(\vec{x}),y)=\max(0,1-yf(\vec{x}))=|1-yf(\vec{x})|_{+}.
  54. sgn ( f ( x ) ) = y \operatorname{sgn}(f(\vec{x}))=y
  55. | y f ( x ) | 1 |yf(\vec{x})|\geq 1
  56. y f ( x ) = 1 yf(\vec{x})=1
  57. y f ( x ) = 1 yf(\vec{x})=1
  58. I [ f ] I[f]
  59. f * Hinge ( x ) = { 1 if p ( 1 x ) > p ( - 1 x ) - 1 if p ( 1 x ) < p ( - 1 x ) f^{*}\text{Hinge}(\vec{x})\;=\;\begin{cases}1&\,\text{if }p(1\mid\vec{x})>p(-1% \mid\vec{x})\\ -1&\,\text{if }p(1\mid\vec{x})<p(-1\mid\vec{x})\end{cases}
  60. p ( 1 x ) 0.5 p(1\mid x)\neq 0.5
  61. V ( f ( x ) , y ) = 1 ln 2 ln ( 1 + e - y f ( x ) ) V(f(\vec{x}),y)=\frac{1}{\ln 2}\ln(1+e^{-yf(\vec{x})})
  62. | f ( x ) | |f(\vec{x})|
  63. I [ f ] I[f]
  64. f * Logistic = ln ( p ( 1 x ) 1 - p ( 1 x ) ) . f^{*}\text{Logistic}=\ln\left(\frac{p(1\mid x)}{1-p(1\mid x)}\right).
  65. p ( 1 x ) = 1 p(1\mid x)=1
  66. p ( 1 x ) = 0 p(1\mid x)=0
  67. p ( 1 x ) p(1\mid x)
  68. p ( 1 x ) = 0.5 p(1\mid x)=0.5

Louis_Miles_Muggleton.html

  1. N m ( E ) N_{m}(E)
  2. R R
  3. N M ( E ) = k 1 ( f 0 E ) 2 = k 2 ( 1 + 0.00334 R ) N_{M}(E)=k_{1}(f_{0}E)^{2}=k_{2}(1+0.00334R)
  4. f 0 E f_{0}E
  5. k 1 , k 2 k_{1},k_{2}
  6. f 0 E f_{0}E
  7. p p
  8. ( f 0 E ) 4 = A ( c o s χ ) p (f_{0}E)^{4}=A(cos\chi)^{p}
  9. χ \chi
  10. p p
  11. p p
  12. f 0 E f_{0}E
  13. f 0 E f_{0}E
  14. f 0 E f_{0}E

Louvain_Modularity.html

  1. Q = 1 2 m Σ i j [ A i j - k i k j 2 m ] δ ( c i , c j ) Q=\frac{1}{2m}\Sigma_{ij}\bigg[A_{ij}-\frac{k_{i}k_{j}}{2m}\bigg]\delta(c_{i},% c_{j})
  2. A i j A_{ij}
  3. i i
  4. j j
  5. k i k_{i}
  6. k j k_{j}
  7. i i
  8. j j
  9. m m
  10. c i c_{i}
  11. c j c_{j}
  12. δ \delta
  13. i i
  14. i i
  15. j j
  16. i i
  17. Δ Q = [ Σ i n + k i , i n 2 m - ( Σ t o t + k i 2 m ) 2 ] - [ Σ i n 2 m - ( Σ t o t 2 m ) 2 - ( k i 2 m ) 2 ] \Delta Q=\bigg[\frac{\Sigma_{in}+k_{i,in}}{2m}-\bigg(\frac{\Sigma_{tot}+k_{i}}% {2m}\bigg)^{2}\bigg]-\bigg[\frac{\Sigma_{in}}{2m}-\bigg(\frac{\Sigma_{tot}}{2m% }\bigg)^{2}-\bigg(\frac{k_{i}}{2m}\bigg)^{2}\bigg]
  18. Σ i n \Sigma_{in}
  19. i i
  20. Σ t o t \Sigma_{tot}
  21. k i k_{i}
  22. i i
  23. k i , i n k_{i,in}
  24. i i
  25. m m
  26. i i
  27. i i
  28. i i

Low-degree_saturation.html

  1. Π ( k i ) = k i j k j \Pi\left(k_{i}\right)=\frac{k_{i}}{\sum_{j}k_{j}}
  2. Π ( k i ) \Pi\left(k_{i}\right)
  3. k k
  4. A A
  5. Π ( k i ) = A + k i A + j k j \Pi\left(k_{i}\right)=\frac{A+k_{i}}{A+\sum\limits_{j}k_{j}}
  6. k k
  7. p k = C ( k + A ) - γ p_{k}=C\left(k+A\right)^{-\gamma}
  8. k k
  9. ( A ) (A)

Low_Power_FSM_Synthesis.html

  1. P = V D D 2 f i α i C i P=V_{DD}^{2}f\sum_{i}{\alpha_{i}}{C_{i}}
  2. α i \alpha_{i}
  3. i i
  4. C i C_{i}
  5. V D D V_{DD}
  6. f f

LowerUnits.html

  1. φ \varphi
  2. η \eta
  3. η \eta
  4. φ \varphi
  5. η \eta
  6. η \eta^{\prime}
  7. η \eta
  8. η \eta
  9. η \eta^{\prime}
  10. \ell
  11. η \eta^{\prime}
  12. ¯ \overline{\ell}
  13. η \eta
  14. η \eta^{\prime}
  15. ¯ \overline{\ell}
  16. η \eta
  17. η \eta^{\prime}
  18. η \eta
  19. η \eta^{\prime}
  20. η \eta
  21. ¯ \overline{\ell}
  22. η \eta
  23. κ \kappa
  24. κ \kappa^{\prime}
  25. κ \kappa
  26. κ \kappa^{\prime}
  27. κ \kappa
  28. ψ \psi
  29. ψ \psi^{\prime}
  30. ψ b \psi_{b}
  31. ψ \psi
  32. ψ f \psi_{f}
  33. ψ b \psi_{b}
  34. ψ \psi^{\prime}
  35. ψ f \psi_{f}
  36. ψ \psi^{\prime}
  37. ψ \psi
  38. ψ \psi
  39. ψ b \psi_{b}
  40. ψ b \psi_{b}
  41. ψ \psi
  42. ψ b \psi_{b}
  43. η \eta
  44. ψ b \psi_{b}
  45. η \eta
  46. η \eta
  47. η \eta
  48. η \eta
  49. ψ b \psi_{b}
  50. ψ b \psi_{b}
  51. ψ f \psi_{f}
  52. q q
  53. ψ \psi^{\prime}
  54. ψ \psi^{\prime}
  55. ψ f \psi_{f}
  56. q q\neq\emptyset
  57. η \eta
  58. q q
  59. q q
  60. q q
  61. η \eta
  62. ψ \psi^{\prime}
  63. ψ \psi^{\prime}
  64. η \eta
  65. ψ \psi^{\prime}
  66. ψ \psi^{\prime}

Luigi_Amerio.html

  1. Δ 2 u - u = f \Delta_{2}u-u=f

Lupan-3beta,20-diol_synthase.html

  1. \rightleftharpoons

Lupeol_synthase.html

  1. \rightleftharpoons

Lycopene_beta-cyclase.html

  1. \rightleftharpoons

Lycopene_epsilon-cyclase.html

  1. \rightleftharpoons

M-Theory_(learning_framework).html

  1. I I
  2. t t
  3. g g
  4. g I , t = I , g - 1 t ( 1 ) \langle gI,t\rangle=\langle I,g^{-1}t\rangle(1)
  5. I I
  6. { I , g t | g G } \{\langle I,g^{\prime}t\rangle|g^{\prime}\in G\}
  7. g g
  8. I I
  9. { g I , g t | g G } \{\langle gI,g^{\prime}t\rangle|g^{\prime}\in G\}
  10. { I , g - 1 g t | g G } \{\langle I,g^{-1}g^{\prime}t\rangle|g^{\prime}\in G\}
  11. { g - 1 g | g G } \{g^{-1}g^{\prime}|g^{\prime}\in G\}
  12. G G
  13. g - 1 g g^{-1}g^{\prime}
  14. G G
  15. g ′′ g^{\prime\prime}
  16. g g^{\prime}
  17. g ′′ = g - 1 g g^{\prime\prime}=g^{-1}g^{\prime}
  18. g = g g ′′ g^{\prime}=gg^{\prime\prime}
  19. { I , g - 1 g t | g G } = { I , g ′′ t | g ′′ G } \{\langle I,g^{-1}g^{\prime}t\rangle|g^{\prime}\in G\}=\{\langle I,g^{\prime% \prime}t\rangle|g^{\prime\prime}\in G\}
  20. O I O_{I}
  21. g I gI
  22. I I
  23. G , g G G,\forall g\in G
  24. O I O_{I}
  25. I I I\sim I^{\prime}
  26. g G \exists g\in G
  27. I = g I I^{\prime}=gI
  28. P I P_{I}
  29. I I
  30. g I gI
  31. P I P_{I}
  32. K K
  33. P I , t k P_{\langle I,t^{k}\rangle}
  34. I , t k \langle I,t^{k}\rangle
  35. t k , k = 1 , , K t^{k},k=1,...,K
  36. n n
  37. X n X X_{n}\in X
  38. K 2 c ϵ 2 log n δ K\geq\frac{2}{c\epsilon^{2}}\log\frac{n}{\delta}
  39. c c
  40. | d ( P I , P I ) - d K ( P I , P I ) | ϵ , |d(P_{I},P_{I}^{\prime})-dK(P_{I},P_{I}^{\prime})|\leq\epsilon,
  41. 1 - δ 2 1-\delta^{2}
  42. I , I I,I^{\prime}
  43. \in
  44. X n X_{n}
  45. I I
  46. K K
  47. P I , t k P_{\langle I,t^{k}\rangle}
  48. k = 1 , , K k=1,...,K
  49. K K
  50. n n
  51. n n
  52. ϵ \epsilon
  53. 1 - δ 2 1-\delta^{2}
  54. K 2 c ϵ 2 log n δ K\geq\frac{2}{c\epsilon^{2}}\log\frac{n}{\delta}
  55. c c
  56. P I , t k P_{\langle I,t^{k}\rangle}
  57. μ n k ( I ) = 1 / | G | i = 1 | G | η n ( I , g i t k ) \mu^{k}_{n}(I)=1/\left|G\right|\sum_{i=1}^{\left|G\right|}\eta_{n}(\langle I,g% _{i}t^{k}\rangle)
  58. η n , n = 1 , , N \eta_{n},n=1,...,N
  59. G 0 G_{0}
  60. G G
  61. I I
  62. t k t_{k}
  63. I , g - 1 t k \langle I,g^{-1}t_{k}\rangle
  64. G 0 G_{0}
  65. I , g - 1 t k \langle I,g^{-1}t_{k}\rangle
  66. s u p p ( I , g - 1 t k ) supp(\langle I,g^{-1}t_{k}\rangle)
  67. g g^{\prime}
  68. g G 0 g^{\prime}G_{0}
  69. I I
  70. g g^{\prime}
  71. s u p p ( I , g - 1 t k ) supp(\langle I,g^{-1}t_{k}\rangle)
  72. s u p p ( g I , t k ) supp(\langle gI,t_{k}\rangle)
  73. s u p p ( g I , t k ) supp(\langle gI,t_{k}\rangle)
  74. d i s t r ( μ l ( g I ) , μ l ( t ) ) = d i s t r ( μ l ( I ) , μ l ( g - 1 t ) ) distr(\langle\mu_{l}(gI),\mu_{l}(t)\rangle)=distr(\langle\mu_{l}(I),\mu_{l}(g^% {-1}t)\rangle)
  75. l l
  76. μ l ( I ) \mu_{l}(I)
  77. d i s t r distr
  78. g G g\in G
  79. I , g i t k \langle I,g_{i}t^{k}\rangle
  80. i = 1 , , | G | i=1,...,|G|
  81. | G | |G|
  82. 1 | G | i = 1 | G | σ ( I , g i t k + n Δ ) , \frac{1}{|G|}\sum_{i=1}^{|G|}\sigma(\langle I,g_{i}t^{k}\rangle+n\Delta),
  83. σ \sigma
  84. Δ \Delta
  85. n n

M::D::c_queue.html

  1. F ( y ) = 0 F ( x + y - D ) λ c x c - 1 ( c - 1 ) ! e - λ x d x , y 0 c . F(y)=\int_{0}^{\infty}F(x+y-D)\frac{\lambda^{c}x^{c-1}}{(c-1)!}e^{-\lambda x}% \,\text{d}x,\quad y\geq 0\quad c\in\mathbb{N}.
  2. ( W x ) = n = 0 c - 1 P n k = 1 m ( - λ ( x - k D ) ) ( k + 1 ) c - 1 - n ( ( K + 1 ) c - 1 - n ) ! e λ ( x - k D ) , m D x < ( m + 1 ) D . \mathbb{P}(W\leq x)=\sum_{n=0}^{c-1}P_{n}\sum_{k=1}^{m}\frac{(-\lambda(x-kD))^% {(k+1)c-1-n}}{((K+1)c-1-n)!}e^{\lambda(x-kD)},\quad mD\leq x<(m+1)D.

M2-brane.html

  1. ( P o i n c a r e ) 3 × S O ( 8 ) (Poincare)_{3}\times SO(8)
  2. d s M 2 2 \displaystyle ds^{2}_{M2}
  3. η \eta
  4. x μ x^{\mu}
  5. x i x^{i}
  6. q q
  7. F F

Magnetic_resonance_(quantum_mechanics).html

  1. E / < m t p l > h E/<mtpl>{{h}}\,
  2. 1 2 \tfrac{1}{2}
  3. | Ψ ( t ) |\Psi(t)\rangle
  4. e - i H ^ / t e^{-i{\hat{H}\hbar}/t}
  5. i t Ψ = H ^ Ψ i\hbar\frac{\partial}{\partial t}\Psi=\hat{H}\Psi
  6. e - i E t / e^{-iEt/\hbar}
  7. | Ψ ( t ) = | Ψ ( 0 ) e - i E t / |\Psi(t)\rangle=|\Psi(0)\rangle e^{-iEt/\hbar}
  8. | x | Ψ ( t ) | 2 |\langle x|\Psi(t)\rangle|^{2}
  9. | x | Ψ ( 0 ) | 2 |\langle x|\Psi(0)\rangle|^{2}
  10. 1 2 \tfrac{1}{2}
  11. m {m}
  12. 1 2 \tfrac{1}{2}
  13. B 0 z ^ {B_{0}}\hat{z}
  14. H ^ = - m . B 0 \hat{H}=-{m}.{B_{0}}
  15. - 2 γ . σ z . B 0 -\tfrac{\hbar}{2}\gamma.\sigma_{z}.{B_{0}}
  16. - 2 . ω 0 . [ 1 0 0 - 1 ] -\tfrac{\hbar}{2}.\omega_{0}.\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  17. ω 0 \omega_{0}
  18. γ B 0 \gamma B_{0}
  19. B 0 {B_{0}}
  20. σ z \sigma_{z}
  21. H ^ \hat{H}
  22. - 2 ω 0 -\tfrac{\hbar}{2}\omega_{0}
  23. 2 ω 0 \tfrac{\hbar}{2}\omega_{0}
  24. B 1 {B_{1}}
  25. B 0 {B_{0}}
  26. ω . \omega.
  27. B 1 = i ^ B 1 c o s ω t - j ^ B 1 s i n ω t {B_{1}}=\hat{i}B_{1}cos{\omega t}-\hat{j}B_{1}sin{\omega t}
  28. [ 1 0 ] \begin{bmatrix}1\\ 0\end{bmatrix}
  29. [ 0 1 ] \begin{bmatrix}0\\ 1\end{bmatrix}
  30. H ^ = ( B 0 B 1 e ω i t B 1 e - ω i t - B 0 ) \hat{H}=\begin{pmatrix}{B_{0}}&{B_{1}}\ e^{\omega it}\\ {B_{1}}e^{-\omega it}&{-B_{0}}\end{pmatrix}
  31. H ^ \hat{H}
  32. B 1 {B_{1}}
  33. R ^ ( t ) \hat{R}(t)
  34. | Ψ ( t ) |\Psi(t)\rangle
  35. H ^ = R ( t ) H ^ R ( t ) + 2 ω σ z \hat{H^{\prime}}=R(t)\hat{H}R(t)^{\dagger}+\tfrac{\hbar}{2}\omega\sigma_{z}
  36. R ( t ) ^ \hat{R(t)}
  37. σ z \sigma_{z}
  38. R ^ ( t ) = ( e - i ω t / 2 0 0 e i ω t / 2 ) \hat{R}(t)=\begin{pmatrix}e^{-i{\omega}t/2}&0\\ 0&e^{i{\omega}t/2}\end{pmatrix}
  39. H ^ = 2 ( Δ ω - ω 1 - ω 1 - Δ ω ) \hat{H^{\prime}}=\tfrac{\hbar}{2}\begin{pmatrix}\Delta\omega&-\omega_{1}\\ -\omega_{1}&-\Delta\omega\end{pmatrix}
  40. Δ ω = ω - ω 0 \Delta\omega=\omega-\omega_{0}
  41. ω 1 = γ B 1 \omega_{1}=\gamma B_{1}
  42. 2 Δ ω \tfrac{\hbar}{2}\Delta\omega
  43. - 2 Δ ω -\tfrac{\hbar}{2}\Delta\omega
  44. 2 ( 0 - ω 1 - ω 1 0 ) \tfrac{\hbar}{2}\begin{pmatrix}0&-\omega_{1}\\ -\omega_{1}&0\end{pmatrix}
  45. [ 1 0 ] \begin{bmatrix}1\\ 0\end{bmatrix}
  46. B 0 {B_{0}}
  47. - 2 ω 0 -\tfrac{\hbar}{2}\omega_{0}
  48. ( 0 1 ) . \begin{pmatrix}0\\ 1\end{pmatrix}.
  49. ω \omega
  50. ω 0 \omega_{0}
  51. B 1 {B_{1}}
  52. B 0 {B_{0}}
  53. ( 2 n + 1 ) Π ) ω 2 + Δ ω 2 \frac{(2n+1)\Pi)}{\sqrt{\omega^{2}+\Delta\omega^{2}}}
  54. [ 0 1 ] \begin{bmatrix}0\\ 1\end{bmatrix}
  55. ω ω 0 \omega\not=\omega_{0}
  56. τ \tau
  57. P 1 2 P_{1}2
  58. n ( 1 - e - t / τ ) P 12 d t n(1-e^{-t/\tau})P_{12}dt
  59. d N = n . e - t / τ . ( 1 - e - t / τ ) P 12 d t dN=n.e^{-t/\tau}.(1-e^{-t/\tau})P_{12}dt
  60. n . e - t / τ . P 12 d t n.e^{-t/\tau}.P_{12}dt
  61. - 0 n e - t / τ P 12 d t \int_{-\infty}^{0}ne^{-t/\tau}P_{12}\ dt
  62. λ \lambda
  63. ( n / 2 ) ω 2 / ( δ ω 2 + ω 1 2 + 1 / τ 2 ) (n/2)\omega^{2}/(\delta\omega^{2}+\omega_{1}^{2}+1/\tau^{2})
  64. B 0 B_{0}
  65. ω 0 \omega_{0}
  66. δ ω \delta\omega
  67. ω 0 \omega_{0}
  68. ω \omega
  69. B 1 {B}_{1}
  70. γ \gamma
  71. ( B 0 ) m a x (B_{0})_{max}
  72. ω \omega
  73. ( B 0 ) m a x (B_{0})_{max}
  74. γ \gamma
  75. γ \gamma
  76. ω \omega
  77. B 0 B_{0}
  78. ω 1 2 + 1 / τ 2 \sqrt{\omega_{1}^{2}+1/\tau^{2}}
  79. ω 1 \omega_{1}
  80. B 1 B_{1}
  81. ω 1 \omega_{1}
  82. ω 1 \omega_{1}
  83. / 2 \hbar/2
  84. B z \frac{\partial B}{\partial z}
  85. S z = / 2 S_{z}=\hbar/2
  86. S z = - / 2 S_{z}=-\hbar/2
  87. B 1 B_{1}
  88. ω 1 \omega_{1}
  89. S z = + / 2 - > - / 2 S_{z}=+\hbar/2->-\hbar/2
  90. ω 1 \omega_{1}
  91. ω p \omega_{p}
  92. ω 1 \omega_{1}
  93. B 1 B_{1}
  94. ω 1 \omega_{1}
  95. ω p \omega_{p}
  96. ω 1 \omega_{1}
  97. = g e B / 2 =geB/{2\hbar}
  98. μ ( = g q / 4 m ) \mu(=gq\hbar/{4m})
  99. ( Δ ω z ^ - ω 1 X ^ ) / γ (\Delta\omega\hat{z}-\omega_{1}\hat{X})/\gamma
  100. z ^ \hat{z}
  101. B 0 B_{0}
  102. X ^ \hat{X}
  103. B 1 B_{1}
  104. δ ω = ω - ω 0 \delta\omega=\omega-\omega_{0}
  105. m {m}
  106. m {m}
  107. B {B}
  108. d L d t = \frac{d{L}}{dt}=
  109. m {m}
  110. B {B}
  111. L {L}
  112. d m d t = γ \frac{d{m}}{dt}=\gamma
  113. m {m}
  114. B {B}
  115. B {B}
  116. B 1 {B_{1}}
  117. d m d t = γ \frac{d{m}}{dt}=\gamma
  118. m {m}
  119. B + B 1 {B}+{B_{1}}
  120. B 1 {B_{1}}
  121. d m d t = d m d t + \frac{d{m^{^{\prime}}}}{dt}=\frac{d{m}}{dt}+
  122. m {m}
  123. ω {\omega}
  124. ω = ω z ^ {\omega}=\omega\hat{z}
  125. ω 0 = γ B \omega_{0}=\gamma B
  126. ω 1 = γ B 1 \omega_{1}=\gamma B_{1}
  127. d m d t = \frac{d{m^{^{\prime}}}}{dt}=
  128. m {m}
  129. ( ω 0 + ω 1 - ω ) {({\omega_{0}}+{\omega_{1}}-{\omega})}
  130. = =
  131. m {m}
  132. Δ ω {\Delta\omega}
  133. + +
  134. ω 1 {\omega_{1}}
  135. B e f f e c t i v e = Δ ω - ω 1 = Δ ω z ^ - ω 1 X ^ B_{effective}=\Delta{\omega}-{\omega_{1}}={\Delta\omega}{\hat{z}}-\omega_{1}{% \hat{X}}
  136. ω = ω 0 \omega=\omega_{0}
  137. < m γ < S > <{m}>=\gamma<{S}>
  138. < m Align g t ; <{m}&gt;
  139. i d d t m = [ m , H ^ ] i\hbar\frac{d}{dt}\langle{m}\rangle=\langle[{m},\hat{H}]\rangle
  140. H ^ = - m B ( t ) \hat{H}=-{m}\cdot{B(t)}
  141. [ m i , H ^ ] = [ m i , - m j B j ] = [ γ S i , - γ S j B j ] = - γ 2 [ S i , S j B j ] = - γ 2 i [ S k B j - S k B j ] , ( i j , k ) [m_{i},\hat{H}]=[m_{i},-m_{j}B_{j}]=[\gamma{S}_{i},-\gamma{S}_{j}{B}_{j}]=-% \gamma^{2}[{S}_{i},{S}_{j}{B}_{j}]=-\gamma^{2}i\hbar[{{S}_{k}{B}_{j}-{S}_{k}{B% }_{j}}],(i\neq j,k)
  142. [ m i , H ^ ] = i γ [ B j m k - B j m k ] [m_{i},\hat{H}]=i\hbar\gamma[{B}_{j}{m}_{k}-{B}_{j}{m}_{k}]
  143. m {m}
  144. d d t m ( t ) = γ m ( t ) × B ( t ) \frac{d}{dt}{m}(t)=\gamma{m}(t)\times{B}(t)
  145. B B
  146. + γ B / 2 +\gamma\hbar B/2
  147. - γ B / 2 -\gamma\hbar B/2
  148. E = - μ . B E=-{\mu}.{B}
  149. E E
  150. N 0 N_{0}
  151. T T
  152. N 0 e - E / k T N_{0}e^{-E/kT}
  153. / 2 \hbar/2
  154. S z = + / 2 S_{z}=+\hbar/2
  155. S z = - / 2 S_{z}=-\hbar/2

Magnetic_skyrmion.html

  1. 𝐧 = 1 4 π 𝐌 ( 𝐌 x × 𝐌 y ) d x d y \mathbf{n}={\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial\mathbf{M}}% {\partial x}\times\frac{\partial\mathbf{M}}{\partial y}\right)dxdy
  2. 𝐧 \mathbf{n}
  3. 𝐌 \mathbf{M}
  4. 1 4 π 𝐌 ( 𝐌 x × 𝐌 y ) d x d y = 𝐧 {\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial\mathbf{M}}{\partial x% }\times\frac{\partial\mathbf{M}}{\partial y}\right)dxdy=\mathbf{n}
  5. 1 4 π 𝐌 ( 𝐌 x × 𝐌 y ) d x d y = 𝐧 {\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial\mathbf{M}}{\partial x% }\times\frac{\partial\mathbf{M}}{\partial y}\right)dxdy=\mathbf{n}
  6. H = - J 𝐫 𝐌 𝐫 ( 𝐌 𝐫 + 𝐞 𝐱 + 𝐌 𝐫 + 𝐞 𝐲 ) - D 𝐫 ( 𝐌 𝐫 × 𝐌 𝐫 + 𝐞 𝐱 𝐞 𝐱 + 𝐌 𝐫 × 𝐌 𝐫 + 𝐞 𝐲 𝐞 𝐲 ) - 𝐁 𝐫 𝐌 𝐫 - A 𝐫𝐈 M z 𝐫 2 H=-J\sum_{\mathbf{r}}\mathbf{M_{r}}\cdot\left(\mathbf{M_{r+e_{x}}}+\mathbf{M_{% r+e_{y}}}\right)-D\sum_{\mathbf{r}}\left(\mathbf{M_{r}}\times\mathbf{M_{r+e_{x% }}}\cdot\mathbf{e_{x}}+\mathbf{M_{r}}\times\mathbf{M_{r+e_{y}}}\cdot\mathbf{e_% {y}}\right)-\mathbf{B}\cdot\sum_{\mathbf{r}}\mathbf{M_{r}}-A\sum_{\mathbf{rI}}% M_{z\mathbf{r}}^{2}

Magnetoquasistatic_field.html

  1. θ \theta
  2. ϕ \phi

Malmquist's_theorem.html

  1. d w d z = R ( z , w ) \frac{dw}{dz}=R(z,w)

Mandel_Q_parameter.html

  1. Q = ( Δ n ^ ) 2 - n ^ n ^ = n ^ 2 - n ^ 2 n ^ - 1 = n ^ ( g ( 2 ) ( 0 ) - 1 ) Q=\frac{\left\langle(\Delta\hat{n})^{2}\right\rangle-\langle\hat{n}\rangle}{% \langle\hat{n}\rangle}=\frac{\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{% 2}}{\langle\hat{n}\rangle}-1=\langle\hat{n}\rangle\left(g^{(2)}(0)-1\right)
  2. n ^ \hat{n}
  3. g ( 2 ) g^{(2)}
  4. - 1 Q < 0 0 ( Δ n ^ ) 2 n ^ -1\leq Q<0\Leftrightarrow 0\leq\langle(\Delta\hat{n})^{2}\rangle\leq\langle% \hat{n}\rangle
  5. Q = - 1 Q=-1
  6. Δ n = 0 \Delta n=0
  7. Q = n Q=\langle n\rangle
  8. Q = 0 Q=0

Manin_conjecture.html

  1. V V
  2. K K
  3. H H
  4. V ( K ) V(K)
  5. V V
  6. U V U\subset V
  7. K K
  8. N U , H ( B ) = # { x U ( K ) : H ( x ) B } N_{U,H}(B)=\#\{x\in U(K):H(x)\leq B\}
  9. B 1 B\geq 1
  10. N U , H ( B ) c B ( log B ) ρ - 1 , N_{U,H}(B)\sim cB(\log B)^{\rho-1},
  11. B . B\to\infty.
  12. ρ \rho
  13. V V
  14. c c

Mapping_spectrum.html

  1. F ( X , Y ) F(X,Y)
  2. [ X Y , Z ] = [ X , F ( Y , Z ) ] . [X\wedge Y,Z]=[X,F(Y,Z)].

Marcatili's_method.html

  1. k x k_{x}
  2. k y k_{y}
  3. β \beta
  4. k y k_{y}
  5. k x k_{x}
  6. k x k_{x}
  7. k y k_{y}
  8. β 2 = k 2 - k x 2 - k y 2 \beta_{2}=k^{2}-k_{x}^{2}-k_{y}^{2}
  9. k 2 k^{2}
  10. k x k_{x}
  11. k y k_{y}
  12. k β k\approx\beta
  13. k x k y 1 k_{x}\approx k_{y}\ll 1

Markushevich_basis.html

  1. X X
  2. { x α ; f α } x \isin α \{x_{\alpha};f_{\alpha}\}_{x\isin\alpha}
  3. X X
  4. span ¯ { x α } = X \overline{\,\text{span}}\{x_{\alpha}\}=X
  5. { f α } x \isin α \{f_{\alpha}\}_{x\isin\alpha}
  6. X X
  7. { e 2 i π n t } n \isin \{e^{2i\pi nt}\}_{n\isin\mathbb{Z}}
  8. C ~ [ 0 , 1 ] \tilde{C}[0,1]
  9. x α = f α = 1 \|x_{\alpha}\|=\|f_{\alpha}\|=1
  10. α \alpha

Marneral_synthase.html

  1. \rightleftharpoons

Mars_Oxygen_ISRU_Experiment.html

  1. \longrightarrow

Marxian_economics.html

  1. v a l u e = m p + l t value=mp+lt
  2. v a l u e value
  3. m p mp
  4. l t lt

Mass_estimation.html

  1. \Tau ( \Tau(
  2. ) )
  3. x x
  4. \Tau ( x ) \Tau(x)
  5. m ( x ) = | \Tau ( x ) | m(x)=|\Tau(x)|
  6. m ( m(
  7. ) )
  8. \Tau ( \Tau(
  9. ) )
  10. D D
  11. n n
  12. R R
  13. s s
  14. x x
  15. m i ( x ) m_{i}(x)
  16. p ( s i ) p(s_{i})
  17. n - 1 n-1
  18. s i s_{i}
  19. m a s s ( x | D ) = i = 1 n - 1 m i ( x ) × p ( s i ) mass(x|D)=\sum_{i=1}^{n-1}m_{i}(x)\times p(s_{i})
  20. p ( s i ) p(s_{i})
  21. s i s_{i}
  22. x 1 < x 2 < < x n - 1 < x n x_{1}<x_{2}<...<x_{n-1}<x_{n}
  23. D D
  24. p ( s i ) = x i + 1 - x i x n - x 1 > 0 p(s_{i})=\frac{x_{i+1}-x_{i}}{x_{n}-x_{1}}>0
  25. m a s s ( x 1 | D ) = 1 p ( s 1 ) + 2 p ( s 2 ) + 3 p ( s 3 ) + 4 p ( s 4 ) mass(x_{1}|D)=1p(s_{1})+2p(s_{2})+3p(s_{3})+4p(s_{4})
  26. m a s s ( x 2 | D ) = 4 p ( s 1 ) + 2 p ( s 2 ) + 3 p ( s 3 ) + 4 p ( s 4 ) mass(x_{2}|D)=4p(s_{1})+2p(s_{2})+3p(s_{3})+4p(s_{4})
  27. m a s s ( x 3 | D ) = 4 p ( s 1 ) + 3 p ( s 2 ) + 3 p ( s 3 ) + 4 p ( s 4 ) mass(x_{3}|D)=4p(s_{1})+3p(s_{2})+3p(s_{3})+4p(s_{4})
  28. m a s s ( x 4 | D ) = 4 p ( s 1 ) + 3 p ( s 2 ) + 2 p ( s 3 ) + 4 p ( s 4 ) mass(x_{4}|D)=4p(s_{1})+3p(s_{2})+2p(s_{3})+4p(s_{4})
  29. m a s s ( x 5 | D ) = 4 p ( s 1 ) + 3 p ( s 2 ) + 2 p ( s 3 ) + 1 p ( s 4 ) mass(x_{5}|D)=4p(s_{1})+3p(s_{2})+2p(s_{3})+1p(s_{4})
  30. h > 1 h>1
  31. m a s s ( x , h | D ) = i = 1 | D | - 1 m a s s ( x , h - 1 | { y | y \Tau i ( x ) } ) × p ( s i ) mass(x,h|D)=\sum_{i=1}^{|D|-1}mass(x,h-1|\{y|y\in\Tau_{i}(x)\})\times p(s_{i})
  32. R d R^{d}
  33. \Tau i ( \Tau_{i}(
  34. ) ( i = 1 , 2 , , t ) )(i=1,2,\cdots,t)
  35. \Tau i ( \Tau_{i}(
  36. ) )
  37. S i D ( | S i | = ψ < n ) S_{i}\subset D(|S_{i}|=\psi<n)
  38. x x
  39. m a s s ( mass(
  40. x x
  41. ) )
  42. m i ( x ) m_{i}(x)
  43. t t
  44. \Tau i ( x ) \Tau_{i}(x)
  45. x x
  46. t t
  47. \Tau i ( \Tau_{i}(
  48. ) )
  49. \Tau i ( \Tau_{i}(
  50. ) )
  51. S i S_{i}
  52. h h
  53. \Tau i ( x ) \Tau_{i}(x)
  54. x x
  55. \Tau i ( \Tau_{i}(
  56. ) )
  57. m a s s ( x ) = 1 t i = 1 t m i ( x ) × 2 i ( x ) mass(x)=\frac{1}{t}\sum_{i=1}^{t}m_{i}(x)\times 2^{\ell_{i}(x)}
  58. i ( x ) \ell_{i}(x)
  59. x x
  60. \Tau i ( \Tau_{i}(
  61. ) )
  62. m a s s ( x ) = 1 t i = 1 t m i ( x ) mass(x)=\frac{1}{t}\sum_{i=1}^{t}m_{i}(x)
  63. m a s s ( x ) = 1 t i = 1 t 2 i ( x ) mass(x)=\frac{1}{t}\sum_{i=1}^{t}2^{\ell_{i}(x)}
  64. S i D S_{i}\subset D
  65. ψ \psi
  66. ( Π i ) (\Pi_{i})
  67. S i S_{i}
  68. H p H_{p}
  69. p D p\in D
  70. l p = || p - q || 2 l_{p}=\frac{||p-q||_{\infty}}{2}
  71. q = m i n o S i ( || q - o || ) q=min_{o\in S_{i}}(||q-o||_{\infty})
  72. p p
  73. S i S_{i}
  74. Π i = o S i H o \Pi_{i}=\cup_{o\in S_{i}}H_{o}
  75. Π i \Pi_{i}
  76. M i D ( | M i | = Ψ ; ψ < Ψ < n ) M_{i}\subset D(|M_{i}|=\Psi;\psi<\Psi<n)
  77. O ( n ) O(n)
  78. O ( n 2 ) O(n^{2})
  79. O ( 1 ) O(1)
  80. O ( n ) O(n)
  81. O ( n ) O(n)
  82. O ( n 2 ) O(n^{2})
  83. O ( 1 ) O(1)
  84. O ( n ) O(n)
  85. O ( 1 ) O(1)
  86. O ( n ) O(n)
  87. O ( 1 ) O(1)
  88. O ( n ) O(n)
  89. x R d x\in R^{d}
  90. f ¯ m ( x ) = 1 t i = 1 t m i ( x ) ψ × v ( \Tau i ( x ) ) \bar{f}_{m}(x)=\frac{1}{t}\sum_{i=1}^{t}\frac{m_{i}(x)}{\psi\times v(\Tau_{i}(% x))}
  91. v ( v(
  92. ) )
  93. f ¯ l ( x ) = 1 t i = 1 t ρ ( x | Π i , M i ) \bar{f}_{l}(x)=\frac{1}{t}\sum_{i=1}^{t}\rho(x|\Pi_{i},M_{i})
  94. ρ ( x | Π i , M i ) = m ( H ( x | Π i ) | M i ) | M i | × l ( H ( x | Π i ) ) \rho(x|\Pi_{i},M_{i})=\frac{m(H_{(x|\Pi_{i})}|M_{i})}{|M_{i}|\times l(H_{(x|% \Pi_{i})})}
  95. H ( x | Π i ) H_{(x|\Pi_{i})}
  96. Π i \Pi_{i}
  97. x x
  98. m ( H | M ) m(H|M)
  99. M M
  100. H H
  101. l ( H ) l(H)
  102. H H
  103. O ( n 2 ) O(n^{2})
  104. O ( n ) O(n)
  105. O ( n ) O(n)
  106. O ( 1 ) O(1)
  107. t t
  108. R d R^{d}
  109. R t R^{t}
  110. x R d x\in R^{d}
  111. y R t y\in R^{t}
  112. y = m 1 ( x ) , m 2 ( x ) , , m t ( x ) y=\langle m_{1}(x),m_{2}(x),\cdots,m_{t}(x)\rangle
  113. h = 1 h=1
  114. h > 1 h>1
  115. ( τ ) (\tau)
  116. τ \tau
  117. h h
  118. f ^ ( x ) \hat{f}(x)
  119. D = { x ( 1 ) , x ( 2 ) , , x ( n ) } D=\{x^{(1)},x^{(2)},\cdots,x^{(n)}\}
  120. x ( i ) R d x^{(i)}\in R^{d}
  121. f ^ K D E ( x ) = 1 n × b d i = 1 n K ( d i s t ( x , x ( i ) ) b ) \hat{f}_{KDE}(x)=\frac{1}{n\times b^{d}}\sum_{i=1}^{n}K\bigg(\frac{dist(x,x^{(% i)})}{b}\bigg)
  122. K ( ) K(\cdot)
  123. b b
  124. f ^ k N N ( x ) = | N ( x , k ) | n × V N ( x , k ) \hat{f}_{kNN}(x)=\frac{|N(x,k)|}{n\times V_{N(x,k)}}
  125. N ( x , k ) N(x,k)
  126. k k
  127. x x
  128. D D
  129. V N ( x , k ) V_{N(x,k)}
  130. N ( x , k ) N(x,k)
  131. f ^ k N N D i s t ( x ) = | N ( x , k ) | n × y N ( x , k ) d i s t ( x , y ) \hat{f}_{kNNDist}(x)=\frac{|N(x,k)|}{n\times\sum_{y\in N(x,k)}dist(x,y)}
  132. ϵ \epsilon
  133. f ^ ϵ ( x ) = | N ϵ ( x ) | n × ϵ \hat{f}_{\epsilon}(x)=\frac{|N_{\epsilon}(x)|}{n\times\epsilon}
  134. N ϵ ( x ) = { y D | d i s t ( x , y ) ϵ } N_{\epsilon}(x)=\{y\in D|dist(x,y)\leq\epsilon\}
  135. ( O ( n 2 d ) ) \big(O(n^{2}d)\big)
  136. ( O ( n d ) ) \big(O(nd)\big)
  137. ( O ( n ) ) \big(O(n)\big)
  138. ( O ( 1 ) ) \big(O(1)\big)

Master_stability_function.html

  1. N N
  2. x ˙ i = f ( x i ) \dot{x}_{i}=f(x_{i})
  3. x i x_{i}
  4. i i
  5. σ \sigma
  6. A i j A_{ij}
  7. g g
  8. x ˙ i = f ( x i ) + σ j = 1 N A i j g ( x j ) . \dot{x}_{i}=f(x_{i})+\sigma\sum_{j=1}^{N}A_{ij}g(x_{j}).
  9. j A i j \sum_{j}A_{ij}
  10. z z
  11. y ˙ = ( D f + γ D g ) y . \dot{y}=(Df+\gamma Dg)y.
  12. σ λ k \sigma\lambda_{k}
  13. λ k \lambda_{k}
  14. A A

Mathematical_constants_and_functions.html

  1. σ 10 {\sigma_{{}_{10}}}
  2. x 10 + x 9 - x 7 - x 6 - x 5 - x 4 - x 3 + x + 1 x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1
  3. C A r t i n {C}_{Artin}
  4. n = 1 ( 1 - 1 p n ( p n - 1 ) ) p n = prime \prod_{n=1}^{\infty}\left(1-\frac{1}{p_{n}(p_{n}-1)}\right)\quad p_{n}% \scriptstyle\,\text{ = prime}
  5. V R {V_{{}_{R}}}
  6. s 3 12 ( 3 2 - 49 π + 162 arctan 2 ) \frac{s^{3}}{12}(3\sqrt{2}-49\,\pi+162\,\arctan\sqrt{2})
  7. C m {C_{m}}
  8. n = 1 ( 1 + 1 ( p n - 1 ) 2 ) p n : p r i m e \prod_{n=1}^{\infty}\underset{p_{n}:\,{prime}}{\Big(1+\frac{1}{(p_{n}-1)^{2}}% \Big)}
  9. C P a C_{Pa}
  10. n = 2 2 φ φ + φ n , φ = F i \prod_{n=2}^{\infty}\frac{2\varphi}{\varphi+\varphi_{n}}\;,\;\varphi{=}{Fi}
  11. φ n = 1 + φ n - 1 \varphi_{n}{=}\sqrt{1{+}\varphi_{n{-}1}}
  12. φ 1 = 1 \varphi_{1}{=}1
  13. b {b}
  14. ( 4 - 2 Φ ) π = ( 3 - 5 ) π (4-2\,\Phi)\,\pi=(3-\sqrt{5})\,\pi
  15. L 2 {L2}
  16. 1 5 + 25 - 2 5 π = 1 π 0 π | sin ( 5 t 2 ) | sin ( t 2 ) d t \frac{1}{5}+\frac{\sqrt{25-2\sqrt{5}}}{\pi}=\frac{1}{\pi}\int_{0}^{\pi}\frac{% \left|\sin(\frac{5t}{2})\right|}{\sin(\frac{t}{2})}\,dt
  17. V c {V_{c}}
  18. 3 2 n 1 ( 1 + 1 ( 2 e n - 1 ) 2 ) 1 / 2 n + 1 \frac{\sqrt{3}}{\sqrt{2}}\prod_{n\geq 1}\left(1+{1\over(2e_{n}-1)^{2}}\right)^% {\!1/2^{n+1}}
  19. γ \gamma
  20. 6 π - 1 - e \sqrt{6\pi-1}-e
  21. S E f {S_{Ef}}
  22. n = 1 1 n ( n - 1 ) 2 1 = 1 + 1 2 1 + 1 3 2 1 + 1 4 3 2 1 + 1 5 4 3 2 1 + \sum_{n=1}^{\infty}\frac{1}{n^{(n{-}1)^{\cdot^{\cdot^{\cdot^{2^{1}}}}}}}=1{+}% \frac{1}{2^{1}}{+}\frac{1}{3^{2^{1}}}+\frac{1}{4^{3^{2^{1}}}}+\frac{1}{5^{4^{3% ^{2^{1}}}}}{+}\cdots
  23. C G S C_{GS}
  24. 0 log ( s + 1 ) e s - 1 d s = - n = 1 e n n E i ( - n ) E i : I n t e g r a l E x p o n e n t i a l \int^{\infty}_{0}\frac{\log(s+1)}{e^{s}-1}\ ds=\!-\!\sum_{n=1}^{\infty}\frac{e% ^{n}}{n}Ei(-n)\overset{Ei:}{\underset{}{}}{Integral}{\scriptstyle Exponential}
  25. - W ( - 1 ) {-W(-1)}
  26. lim n \lim_{n\rightarrow\infty}
  27. f ( x ) = log ( log ( log ( log ( log ( log ( x ) ) ) ) ) ) log s n times f(x)=\underbrace{\log(\log(\log(\log(\cdots\log(\log(x))))))\,\!}\atop{\log_{s% }\,\text{ n times}}
  28. β {\beta}
  29. 1 2 π \approx\frac{1}{2\sqrt{\pi}}
  30. C 2 {C}_{2}
  31. p = 3 p ( p - 2 ) ( p - 1 ) 2 \prod_{p=3}^{\infty}\frac{p(p-2)}{(p-1)^{2}}
  32. F F
  33. n = 1 ( 1 - ( - 1 φ 2 ) n ) = n = 1 ( 1 - ( 5 - 3 2 ) n ) \prod_{n=1}^{\infty}\left(1-\left(-\frac{1}{{\varphi}^{2}}\right)^{n}\right)=% \prod_{n=1}^{\infty}\left(1-\left(\frac{\sqrt{5}-3}{2}\right)^{n}\right)
  34. ρ {\rho}
  35. n = 3 cos ( π n ) = cos ( π 3 ) cos ( π 4 ) cos ( π 5 ) \prod_{n=3}^{\infty}\cos\left(\frac{\pi}{n}\right)=\cos\left(\frac{\pi}{3}% \right)\cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{5}\right)...
  36. q {q}
  37. 1 = n = 1 t k q k Raiz real de n = 0 ( 1 - 1 q 2 n ) + q - 2 q - 1 = 0 1=\!\sum_{n=1}^{\infty}\frac{t_{k}}{q^{k}}\qquad\scriptstyle\,\text{Raiz real % de}\displaystyle\prod_{n=0}^{\infty}\!\left(\!1{-}\frac{1}{q^{2^{n}}}\!\right)% \!{+}\frac{q{-}2}{q{-}1}=0
  38. σ R r {\sigma}_{\,Rr}
  39. 3 + 13 2 = 1 + 3 + 3 + 3 + 3 + \frac{3+\sqrt{13}}{2}=1+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}}
  40. C 5 {C_{5}}
  41. 1 n = 0 1 ( 3 n + 2 2 ) = 3 3 2 π {\frac{1}{{\sum_{n=0}^{\infty}\frac{1}{{\left({{3n+2}\atop{2}}\right)}}}}}=% \frac{3\sqrt{3}}{2\pi}
  42. G G S G_{\,GS}
  43. 2 2 2^{\sqrt{2}}
  44. γ \gamma
  45. e π 2 / ( 12 ln 2 ) e^{\pi^{2}/(12\ln 2)}
  46. Chi ( ) {\operatorname{Chi()}}
  47. γ + 0 x cosh t - 1 t d t \gamma+\int_{0}^{x}\frac{\cosh t-1}{t}\,dt
  48. γ = Euler–Mascheroni constant= 0.5772156649… \scriptstyle\gamma\,\,\text{= Euler–Mascheroni constant= 0.5772156649...}
  49. C V i {C}_{Vi}
  50. lim n | a n | 1 n \lim_{n\to\infty}|a_{n}|^{\frac{1}{n}}
  51. 3 4 ζ ( 2 ) \tfrac{3}{4}\zeta(2)
  52. π 2 8 = n = 0 1 ( 2 n - 1 ) 2 = 1 1 2 + 1 3 2 + 1 5 2 + 1 7 2 + \frac{\pi^{2}}{8}=\sum_{n=0}^{\infty}\frac{1}{(2n-1)^{2}}=\frac{1}{1^{2}}+% \frac{1}{3^{2}}+\frac{1}{5^{2}}+\frac{1}{7^{2}}+\cdots
  53. 2 π \sqrt{2\pi}
  54. 2 π = lim n n ! e n n n n \color w h i t e . \color b l a c k \sqrt{2\pi}=\lim_{n\to\infty}\frac{n!\;e^{n}}{n^{n}\sqrt{n}}{\color{white}....% \color{black}}
  55. τ e \sqrt{\tau e}
  56. 2 π e \sqrt{2\pi e}
  57. £ L o {\,\text{£}_{{}_{Lo}}}
  58. 6 ln 2 ln 10 π 2 \frac{6\ln 2\ln 10}{\pi^{2}}
  59. 𝒯 R \mathcal{T}_{R}
  60. a 2 ( 2 3 + π 6 - 3 ) a^{2}\cdot\left(2\sqrt{3}+{\frac{\pi}{6}}-3\right)
  61. 𝒞 2 \mathcal{C}_{2}
  62. n = 1 ( 1 - 1 p n ( p n + 1 ) ) p n : p r i m e \underset{p_{n}:\,{prime}}{\prod_{n=1}^{\infty}\left(1-\frac{1}{p_{n}(p_{n}+1)% }\right)}
  63. μ {\mu}
  64. 2 + 2 = lim n c n 1 / n \sqrt{2+\sqrt{2}}\;=\lim_{n\rightarrow\infty}c_{n}^{1/n}
  65. : x 4 - 4 x 2 + 2 = 0 :\;x^{4}-4x^{2}+2=0
  66. λ 2 {\lambda}_{2}
  67. lim n F n ( x ) - ln ( 1 - x ) ( - λ ) n = Ψ ( x ) , \lim_{n\to\infty}\frac{F_{n}(x)-\ln(1-x)}{(-\lambda)^{n}}=\Psi(x),
  68. Ψ ( x ) \Psi(x)
  69. Ψ ( 0 ) = Ψ ( 1 ) = 0 \Psi(0)\!=\!\Psi(1)\!=\!0
  70. π 2 {\frac{\pi}{2}}
  71. n = 1 ( 4 n 2 4 n 2 - 1 ) = 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8 9 \prod_{n=1}^{\infty}\left(\frac{4n^{2}}{4n^{2}-1}\right)=\frac{2}{1}\cdot\frac% {2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot% \frac{8}{7}\cdot\frac{8}{9}\cdots
  72. E B {E}_{\,B}
  73. m = 1 n = 1 1 2 m n = n = 1 1 2 n - 1 = 1 1 + 1 3 + 1 7 + 1 15 + \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{2^{mn}}=\sum_{n=1}^{\infty}% \frac{1}{2^{n}-1}=\frac{1}{1}\!+\!\frac{1}{3}\!+\!\frac{1}{7}\!+\!\frac{1}{15}% \!+\!...
  74. φ {\varphi}
  75. 1 + 5 2 = 1 + 1 + 1 + 1 + \frac{1+\sqrt{5}}{2}=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}
  76. ζ ( 2 ) {\zeta}(\,2)
  77. π 2 6 = n = 1 1 n 2 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + \frac{\pi^{2}}{6}=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{1}{1^{2}}+\frac{1}{% 2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots
  78. 3 \sqrt{3}
  79. 3 3 3 3 3 3 3 3 3 3 \sqrt[3]{3\,\sqrt[3]{3\,\sqrt[3]{3\,\sqrt[3]{3\,\sqrt[3]{3\,\cdots}}}}}
  80. R {R}
  81. 1 + 2 + 3 + 4 + \sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\cdots}}}}
  82. P 2 {P}_{\,2}
  83. ln ( 1 + 2 ) + 2 = arcsinh ( 1 ) + 2 \ln(1+\sqrt{2})+\sqrt{2}\;=\;\operatorname{arcsinh}(1)+\sqrt{2}
  84. 𝒮 m {\mathcal{S}_{{}_{m}}}
  85. n = 1 1 ϕ ( n ) σ 1 ( n ) = n = 1 ( 1 + k = 1 1 p n 2 k - p n k - 1 ) p n : p r i m e \sum_{n=1}^{\infty}\frac{1}{\phi(n)\sigma_{1}(n)}=\underset{p_{n}:\,{prime}}{% \prod_{n=1}^{\infty}\left(1+\sum_{k=1}^{\infty}\frac{1}{p_{n}^{2k}-p_{n}^{k-1}% }\right)}
  86. 𝒜 6 \mathcal{A}_{6}
  87. 3 3 2 l 2 \frac{3\sqrt{3}}{2}\,l^{2}
  88. 𝒞 F T {\mathcal{C}_{{}_{FT}}}
  89. 1 2 n = 1 ( 1 - 2 p n 2 ) + 1 2 p n : p r i m e = 3 π 2 n = 1 ( 1 - 1 p n 2 - 1 ) + 1 2 \underset{p_{n}:\,{prime}}{\frac{1}{2}\prod_{n=1}^{\infty}\left(1-\frac{2}{p_{% n}^{2}}\right){+}\frac{1}{2}}=\frac{3}{\pi^{2}}\prod_{n=1}^{\infty}\left(1-% \frac{1}{p_{n}^{2}-1}\right){+}\frac{1}{2}
  90. 𝒞 2 \mathcal{C}^{2}
  91. n = 1 ( 3 n - 1 ) 2 ( 3 n - 2 ) ( 3 n ) = 3 4 π 2 Γ ( 1 3 ) 3 \prod_{n=1}^{\infty}\frac{(3n-1)^{2}}{(3n-2)(3n)}=\frac{3}{4\pi^{2}}\,\Gamma% \left(\frac{1}{3}\right)^{3}
  92. 𝒯 \mathcal{T}
  93. : x 4 - x 3 - x 2 - x - 1 = 0 :\;\;x^{4}-x^{3}-x^{2}-x-1=0
  94. f ( 3 , 4 ) {f_{(3,4)}}
  95. : 4 x 4 - 28 x 3 - 7 x 2 + 16 x + 16 = 0 :\;\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0
  96. C {C}
  97. 1 + n = 2 ( 1 - 1 ζ ( n ) ) 1+\sum_{n=2}^{\infty}\left(1-\frac{1}{\zeta(n)}\right)
  98. π 3 3 \frac{\pi}{3\sqrt{3}}
  99. n = 1 1 n ( 2 n n ) = 1 - 1 2 + 1 4 - 1 5 + 1 7 - 1 8 + \sum_{n=1}^{\infty}\frac{1}{n{2n\choose n}}=1-\frac{1}{2}+\frac{1}{4}-\frac{1}% {5}+\frac{1}{7}-\frac{1}{8}+\cdots
  100. γ 2 \gamma_{{}_{2}}
  101. 2 3 = 1 cos ( π 6 ) \frac{2}{\sqrt{3}}=\frac{1}{\cos\,(\frac{\pi}{6})}
  102. τ \tau
  103. n = 0 t n 2 n + 1 \sum_{n=0}^{\infty}\frac{t_{n}}{2^{n+1}}
  104. t n {t_{n}}
  105. τ ( x ) = n = 0 ( - 1 ) t n x n = n = 0 ( 1 - x 2 n ) \tau(x)=\sum_{n=0}^{\infty}(-1)^{t_{n}}\,x^{n}=\prod_{n=0}^{\infty}(1-x^{2^{n}})
  106. 𝒫 P e l l {\mathcal{P}_{{}_{Pell}}}
  107. 1 - n = 0 ( 1 - 1 2 2 n + 1 ) 1-\prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2n+1}}\right)
  108. λ {\lambda}
  109. x e x 2 + 1 x 2 + 1 + 1 = 1 \frac{x\;e^{\sqrt{x^{2}+1}}}{\sqrt{x^{2}+1}+1}=1
  110. δ 0 {{\delta}_{{}_{0}}}
  111. 1 + π 2 6 + 2 Li 2 ( - e ) Li 2 = Dilogarithm integral 1+\frac{\pi^{2}}{6}+2\;\mathrm{Li}_{2}\left(-\sqrt{e}\;\right)\quad\mathrm{Li}% _{2}\,\scriptstyle\,\text{= Dilogarithm integral}
  112. C C R {C_{{}_{CR}}}
  113. 1 3 + ( - 23 + 3 i 237 ) 1 3 3 2 2 3 + 11 3 ( 2 ( - 23 + 3 i 237 ) ) 1 3 {1\over 3}+{(-23+3i\sqrt{237})^{\tfrac{1}{3}}\over 3\cdot 2^{\tfrac{2}{3}}}+{1% 1\over 3(2(-23+3i\sqrt{237}))^{\tfrac{1}{3}}}
  114. Γ ( 3 4 ) \Gamma(\tfrac{3}{4})
  115. ( - 1 + 3 4 ) ! = ( - 1 4 ) ! \left(-1+\frac{3}{4}\right)!=\left(-\frac{1}{4}\right)!
  116. ζ ( 3 ) \zeta(3)
  117. n = 1 1 n 3 = 1 1 3 + 1 2 3 + 1 3 3 + 1 4 3 + 1 5 3 + = \sum_{n=1}^{\infty}\frac{1}{n^{3}}=\frac{1}{1^{3}}+\frac{1}{2^{3}}+\frac{1}{3^% {3}}+\frac{1}{4^{3}}+\frac{1}{5^{3}}+\cdots=
  118. 1 2 n = 1 H n n 2 = 1 2 i = 1 j = 1 1 i j ( i + j ) = 0 1 0 1 0 1 d x d y d z 1 - x y z \frac{1}{2}\sum_{n=1}^{\infty}\frac{H_{n}}{n^{2}}=\frac{1}{2}\sum_{i=1}^{% \infty}\sum_{j=1}^{\infty}\frac{1}{ij(i{+}j)}=\!\!\int\limits_{0}^{1}\!\!\int% \limits_{0}^{1}\!\!\int\limits_{0}^{1}\frac{\mathrm{d}x\mathrm{d}y\mathrm{d}z}% {1-xyz}
  119. C {C}
  120. 0 1 0 1 1 1 + x 2 y 2 d x d y = n = 0 ( - 1 ) n ( 2 n + 1 ) 2 = 1 1 2 - 1 3 2 + \int_{0}^{1}\!\!\int_{0}^{1}\!\!\frac{1}{1{+}x^{2}y^{2}}\,dx\,dy=\!\sum_{n=0}^% {\infty}\!\frac{(-1)^{n}}{(2n{+}1)^{2}}\!=\!\frac{1}{1^{2}}{-}\frac{1}{3^{2}}{% +}{\cdots}
  121. β ( 1 ) {\beta}(1)
  122. π 4 = n = 0 ( - 1 ) n 2 n + 1 = 1 1 - 1 3 + 1 5 - 1 7 + 1 9 - \frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}=\frac{1}{1}-\frac{1}{3}% +\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots
  123. C H B M {C_{{}_{HBM}}}
  124. n = 1 ( 1 - 1 p n ) 7 ( 1 + 7 p n + 1 p n 2 ) p n : p r i m e \underset{p_{n}:\,{prime}}{\prod_{n=1}^{\infty}\left(1-\frac{1}{p_{n}}\right)^% {7}\left(1+\frac{7p_{n}+1}{p_{n}^{2}}\right)}
  125. | i | |{}^{\infty}{i}|
  126. lim n | i n | = | lim n i i i n | \lim_{n\to\infty}\left|{}^{n}i\right|=\left|\lim_{n\to\infty}\underbrace{i^{i^% {\cdot^{\cdot^{i}}}}}_{n}\right|
  127. I 1 {I}_{1}
  128. 0 1 x x d x = n = 1 ( - 1 ) n + 1 n n = 1 1 1 - 1 2 2 + 1 3 3 - \int_{0}^{1}\!x^{x}\,dx=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{n}}=\frac{1}{1% ^{1}}-\frac{1}{2^{2}}+\frac{1}{3^{3}}-{\cdots}
  129. I 2 {I}_{2}
  130. 0 1 1 x x d x = n = 1 1 n n = 1 1 1 + 1 2 2 + 1 3 3 + 1 4 4 + \int_{0}^{1}\!\frac{1}{x^{x}}\,dx=\sum_{n=1}^{\infty}\frac{1}{n^{n}}=\frac{1}{% 1^{1}}+\frac{1}{2^{2}}+\frac{1}{3^{3}}+\frac{1}{4^{4}}+\cdots
  131. P # {P_{\#}}
  132. n = 1 1 p n # = 1 2 + 1 6 + 1 30 + 1 210 + = k = 1 n = 1 k 1 p n p n : p r i m e \underset{p_{n}:\,{prime}}{\sum_{n=1}^{\infty}\frac{1}{p_{n}\#}=\frac{1}{2}+% \frac{1}{6}+\frac{1}{30}+\frac{1}{210}+...=\sum_{k=1}^{\infty}\prod_{n=1}^{k}% \frac{1}{p_{n}}}
  133. < m t p l > C <mtpl>{{C}}
  134. 1 π arctan 1 2 = 1 π n = 0 ( - 1 ) n ( 2 2 n + 1 ) ( 2 n + 1 ) \frac{1}{\pi}\arctan{\frac{1}{2}}=\frac{1}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{% n}}{(2^{2n+1})(2n+1)}
  135. = 1 π ( 1 2 - 1 3 2 3 + 1 5 2 5 - 1 7 2 7 + ) =\frac{1}{\pi}\left(\frac{1}{2}-\frac{1}{3\cdot 2^{3}}+\frac{1}{5\cdot 2^{5}}-% \frac{1}{7\cdot 2^{7}}+\cdots\right)
  136. A {A}
  137. 1 2 π \frac{1}{2\pi}
  138. C π {\frac{C}{\pi}}
  139. - π π cosh - 1 ( cos ( t ) + 3 2 ) 4 π d t \int\limits_{-\pi}^{\pi}\frac{\cosh^{-1}\left(\frac{\sqrt{\cos(t)+3}}{\sqrt{2}% }\right)}{4\pi}\,dt
  140. i ! {i}\,!
  141. Γ ( 1 + i ) = i Γ ( i ) = 0 t i e t d t \Gamma(1+i)=i\,\Gamma(i)=\int\limits_{0}^{\infty}\frac{t^{i}}{e^{t}}\mathrm{d}t
  142. W W
  143. 45 - 1929 18 3 + 45 + 1929 18 3 \sqrt[3]{\frac{45-\sqrt{1929}}{18}}+\sqrt[3]{\frac{45+\sqrt{1929}}{18}}
  144. C s a {C_{sa}}
  145. p > 2 ( 1 - p + 2 p 3 ) \prod_{p>2}\Big(1-\frac{p+2}{p^{3}}\Big)
  146. τ {\tau}
  147. lim n 1 - ! n n ! = lim n P ( n ) = 0 1 e - x d x = 1 - 1 e = \lim_{n\to\infty}1-\frac{!n}{n!}=\lim_{n\to\infty}P(n)=\int_{0}^{1}e^{-x}dx=1{% -}\frac{1}{e}=
  148. n = 1 ( - 1 ) n + 1 n ! = 1 1 ! - 1 2 ! + 1 3 ! - 1 4 ! + 1 5 ! - 1 6 ! + \sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}=\frac{1}{1!}{-}\frac{1}{2!}{+}% \frac{1}{3!}{-}\frac{1}{4!}{+}\frac{1}{5!}{-}\frac{1}{6!}{+}\cdots
  149. F 1 F_{1}
  150. n = 1 n ! 2 π n ( n e ) n 1 + 1 n 12 \prod_{n=1}^{\infty}\frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}\sqrt[1% 2]{1+\tfrac{1}{n}}}
  151. 2 ϖ 2\varpi
  152. [ Γ ( 1 4 ) ] 2 2 π = 4 0 1 d x ( 1 - x 2 ) ( 2 - x 2 ) \frac{[\Gamma(\tfrac{1}{4})]^{2}}{\sqrt{2\pi}}=4\int^{1}_{0}\frac{dx}{\sqrt{(1% -x^{2})(2-x^{2})}}
  153. Δ ( 3 ) \Delta(3)
  154. 4 + 17 2 - 6 3 - 7 π 105 + ln ( 1 + 2 ) 5 + 2 ln ( 2 + 3 ) 5 \frac{4\!+\!17\sqrt{2}\!-6\sqrt{3}\!-7\pi}{105}\!+\!\frac{\ln(1\!+\!\sqrt{2})}% {5}\!+\!\frac{2\ln(2\!+\!\sqrt{3})}{5}
  155. λ {\lambda}
  156. x 71 - x 69 - 2 x 68 - x 67 + 2 x 66 + 2 x 65 + x 64 - x 63 - x 62 - x 61 - x 60 - x 59 + 2 x 58 + 5 x 57 + 3 x 56 - 2 x 55 - 10 x 54 - 3 x 53 - 2 x 52 + 6 x 51 + 6 x 50 + x 49 + 9 x 48 - 3 x 47 - 7 x 46 - 8 x 45 - 8 x 44 + 10 x 43 + 6 x 42 + 8 x 41 - 5 x 40 - 12 x 39 + 7 x 38 - 7 x 37 + 7 x 36 + x 35 - 3 x 34 + 10 x 33 + x 32 - 6 x 31 - 2 x 30 - 10 x 29 - 3 x 28 + 2 x 27 + 9 x 26 - 3 x 25 + 14 x 24 - 8 x 23 - 7 x 21 + 9 x 20 + 3 x 19 - 4 x 18 - 10 x 17 - 7 x 16 + 12 x 15 + 7 x 14 + 2 x 13 - 12 x 12 - 4 x 11 - 2 x 10 + 5 x 9 + x 7 - 7 x 6 + 7 x 5 - 4 x 4 + 12 x 3 - 6 x 2 + 3 x - 6 = 0 \begin{smallmatrix}x^{71}\quad\ -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-% x^{63}-x^{62}-x^{61}-x^{60}\\ -x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{5% 0}\\ +x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{4% 0}\\ -12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{3% 0}\\ -10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}\quad\ -7x^{21}+9x^{% 20}\\ +3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{% 12}\!-4x^{11}\!-2x^{10}\\ +5x^{9}+x^{7}\quad\ -7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\end{smallmatrix}
  157. β {\beta}
  158. π 2 12 ln 2 \frac{\pi^{2}}{12\,\ln 2}
  159. β 3 \beta_{3}
  160. 0 1 d t 1 + t 3 = n = 0 ( - 1 ) n 3 n + 1 = 1 3 ( ln 2 + π 3 ) \int^{1}_{0}\frac{{\mathrm{d}t}}{1+t^{3}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{3% n+1}=\frac{1}{3}\left(\ln 2+\frac{\pi}{\sqrt{3}}\right)
  161. K 0 {K_{0}}
  162. 1 + 1 2 + 1 3 + + 1 9 + 1 11 + + 1 19 + 1 21 + + etc. 1{+}\frac{1}{2}{+}\frac{1}{3}{+}\cdots{+}\frac{1}{9}{+}\frac{1}{11}{+}\cdots{+% }\frac{1}{19}{+}\frac{1}{21}{+}\cdots{+}\,\,\text{etc.}
  163. + 1 99 + 1 111 + + 1 119 + 1 121 + d e n o m i n a t o r s c o n t a i n i n g 0. E x c l u d i n g a l l {+}\frac{1}{99}{+}\frac{1}{111}{+}\cdots{+}\frac{1}{119}{+}\frac{1}{121}{+}% \cdots\;\;\overset{Excluding\;all}{\underset{containing\;0.}{\scriptstyle denominators}}
  164. C 1 {C_{1}}
  165. lim n ( L n - 4 π 2 ln ( 2 n + 1 ) ) = 4 π 2 ( k = 1 2 ln k 4 k 2 - 1 - Γ ( 1 2 ) Γ ( 1 2 ) ) \lim_{n\to\infty}\!\!\left(\!{L_{n}{-}\frac{4}{\pi^{2}}\ln(2n{+}1)}\!\!\right)% \!{=}\frac{4}{\pi^{2}}\!\left({\sum_{k=1}^{\infty}\!\frac{2\ln k}{4k^{2}{-}1}}% {-}\frac{\Gamma^{\prime}(\tfrac{1}{2})}{\Gamma(\tfrac{1}{2})}\!\!\right)
  166. C 2 {C_{2}}
  167. e 2 - 7 2 = 0 | d d t ( sin t t ) n | d t - 1 \frac{e^{2}-7}{2}=\int_{0}^{\infty}\left|{\frac{d}{dt}\left(\frac{\sin t}{t}% \right)^{n}}\right|\,dt-1
  168. C L C_{L}
  169. n = 2 ln ( n n - 1 ) n \sum_{n=2}^{\infty}\frac{\ln\left(\frac{n}{n-1}\right)}{n}
  170. F α F_{\alpha}
  171. x n + 1 = ( 1 + 1 x n ) n for n = 1 , 2 , 3 , x_{n+1}=\left(1+\frac{1}{x_{n}}\right)^{n}\,\text{ for }n=1,2,3,\ldots
  172. lim n x n log n n = 1 \,\lim_{n\to\infty}x_{n}\tfrac{\log n}{n}=1
  173. F β F_{\beta}
  174. x x + 1 = ( x + 1 ) x x^{x+1}=(x+1)^{x}
  175. ζ ( 2 ) 2 \frac{{\zeta}(2)}{2}
  176. π 2 12 = n = 1 ( - 1 ) n + 1 n 2 = 1 1 2 - 1 2 2 + 1 3 2 - 1 4 2 + 1 5 2 - \frac{\pi^{2}}{12}=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\frac{1}{1^{2}}% {-}\frac{1}{2^{2}}{+}\frac{1}{3^{2}}{-}\frac{1}{4^{2}}{+}\frac{1}{5^{2}}{-}\cdots
  177. L n ( 2 ) Ln(2)
  178. n = 1 1 n 2 n = n = 1 ( - 1 ) n + 1 n = 1 1 - 1 2 + 1 3 - 1 4 + \sum_{n=1}^{\infty}\frac{1}{n2^{n}}=\sum_{n=1}^{\infty}\frac{({-}1)^{n+1}}{n}=% \frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+{\cdots}
  179. σ ( 1 2 ) \sigma(\tfrac{1}{2})
  180. e π 8 π 4 * 2 3 / 4 ( 1 4 ! ) 2 \frac{e^{\frac{\pi}{8}}\sqrt{\pi}}{4*2^{3/4}{(\frac{1}{4}!)^{2}}}
  181. γ {\gamma}
  182. n = 1 k = 0 ( - 1 ) k 2 n + k = n = 1 ( 1 n - ln ( 1 + 1 n ) ) \sum_{n=1}^{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2^{n}+k}=\sum_{n=1}^{% \infty}\left(\frac{1}{n}-\ln\left(1+\frac{1}{n}\right)\right)
  183. = 0 1 - ln ( ln 1 x ) d x = - Γ ( 1 ) = - Ψ ( 1 ) =\int_{0}^{1}-\ln\left(\ln\frac{1}{x}\right)\,dx=-\Gamma^{\prime}(1)=-\Psi(1)
  184. β \beta
  185. e 2 π 0 π 3 t tan t d t = e - 1 3 1 3 ln 1 + e 2 π i t d t e^{{}^{\textstyle{\frac{2}{\pi}}\displaystyle{\int_{0}^{\frac{\pi}{3}}}% \textstyle{t\tan t\ dt}}}=e^{{}^{\displaystyle{\,\int_{\frac{-1}{3}}^{\frac{1}% {3}}}\textstyle{\,\ln\lfloor 1+e^{2\pi it}}\rfloor dt}}
  186. c c
  187. φ 2 π = ( 1 + 5 2 ) 2 π \varphi^{\frac{2}{\pi}}=\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{2}{\pi}}
  188. C S C_{S}
  189. n = 1 ( 1 - p p 3 - 1 ) \prod_{n=1}^{\infty}\left(1-\frac{p}{p^{3}-1}\right)
  190. d d
  191. lim x cos [ x ] ( c ) = lim x cos ( cos ( cos ( ( cos ( c ) ) ) ) ) x \lim_{x\to\infty}\cos^{[x]}(c)=\lim_{x\to\infty}\underbrace{\cos(\cos(\cos(% \cdots(\cos(c)))))}_{x}
  192. C T C_{T}
  193. n = 1 ( 1 - 3 p n 3 + 2 p n 4 + 1 p n 5 - 1 p n 6 ) \prod_{n=1}^{\infty}\left(1-\frac{3}{{p_{n}}^{3}}+\frac{2}{{p_{n}}^{4}}+\frac{% 1}{{p_{n}}^{5}}-\frac{1}{{p_{n}}^{6}}\right)
  194. p n = prime \scriptstyle p_{n}=\,\,\text{prime}
  195. L / 2 L\,\text{/}\sqrt{2}
  196. 0 d x 1 + x 4 = 1 4 π Γ ( 1 4 ) 2 = 4 ( 1 4 ! ) 2 π \int\limits_{0}^{\infty}\frac{{\mathrm{d}x}}{\sqrt{1+x^{4}}}=\frac{1}{4\sqrt{% \pi}}\,\Gamma\left(\frac{1}{4}\right)^{2}=\frac{4\left(\frac{1}{4}!\right)^{2}% }{\sqrt{\pi}}
  197. Γ ( ) = Gamma function \scriptstyle\Gamma()\,\text{= Gamma function}
  198. P r 1 Pr_{1}
  199. n = 2 ( 1 + 1 n ) 1 n \prod_{n=2}^{\infty}\Big(1+\frac{1}{n}\Big)^{\frac{1}{n}}
  200. \partial
  201. n = 1 1 n 3 + n = n = 1 1 n ( n + 1 ) \sum_{n=1}^{\infty}\frac{1}{\sqrt{n^{3}}+\sqrt{n}}=\sum_{n=1}^{\infty}\frac{1}% {\sqrt{n}(n+1)}
  202. S 5 S_{5}
  203. 21 + 1 2 = 5 + 5 + 5 + 5 + 5 + \displaystyle\frac{\sqrt{21}+1}{2}=\scriptstyle\,\sqrt{5+\sqrt{5+\sqrt{5+\sqrt% {5+\sqrt{5+\cdots}}}}}\;
  204. = 1 + 5 - 5 - 5 - 5 - 5 - =1+\,\scriptstyle\sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5-\cdots}}}}}\;
  205. i \sqrt{i}
  206. - 1 4 = 1 + i 2 = e i π 4 = cos ( π 4 ) + i sin ( π 4 ) \sqrt[4]{-1}=\frac{1+i}{\sqrt{2}}=e^{\frac{i\pi}{4}}=\cos\left(\frac{\pi}{4}% \right)+i\sin\left(\frac{\pi}{4}\right)
  207. 𝒜 A G {\mathcal{A}_{AG}}
  208. e - 1 + k = 2 n = 1 1 n k n + 1 = e - 1 - k = 2 1 k ln ( 1 - 1 k ) e^{-1+\sum\limits_{k=2}^{\infty}\sum\limits_{n=1}^{\infty}\frac{1}{nk^{n+1}}}=% e^{-1-\sum\limits_{k=2}^{\infty}\frac{1}{k}\ln\left(1-\frac{1}{k}\right)}
  209. K {K}
  210. π ( 2 γ + ln 4 π 3 Γ ( 1 4 ) 4 ) = π ( 2 γ + 4 ln Γ ( 3 4 ) - ln π ) \pi\left(2\gamma+\ln\frac{4\pi^{3}}{\Gamma(\tfrac{1}{4})^{4}}\right)=\pi(2% \gamma+4\ln\Gamma(\tfrac{3}{4})-\ln\pi)
  211. = π ( 2 ln 2 + 3 ln π + 2 γ - 4 ln Γ ( 1 4 ) ) =\pi\left(2\ln 2+3\ln\pi+2\gamma-4\ln\Gamma(\tfrac{1}{4})\right)
  212. 1 γ \frac{1}{\gamma}
  213. ( 0 1 - log ( log 1 x ) d x ) - 1 = n = 1 ( - 1 ) n ( - 1 + γ ) n \left(\int_{0}^{1}-\log\left(\log\frac{1}{x}\right)\,dx\right)^{-1}=\sum_{n=1}% ^{\infty}(-1)^{n}(-1+\gamma)^{n}
  214. L 1 {L_{1}}
  215. i = 0 j i n x - x i x j - x i = 1 π 0 π sin 3 t 2 sin t 2 d t = 1 3 + 2 3 π \prod_{\begin{smallmatrix}i=0\\ j\neq i\end{smallmatrix}}^{n}\frac{x-x_{i}}{x_{j}-x_{i}}=\frac{1}{\pi}\int_{0}% ^{\pi}\frac{\lfloor\sin{\frac{3t}{2}}\rfloor}{\sin{\frac{t}{2}}}\,dt=\frac{1}{% 3}+\frac{2\sqrt{3}}{\pi}
  216. ς \varsigma
  217. 2 + 2 cos 2 π 7 = 2 + 2 + 7 + 7 7 + 7 7 + 3 3 3 1 + 7 + 7 7 + 7 7 + 3 3 3 2+2\cos\frac{2\pi}{7}=\textstyle 2+\frac{2+\sqrt[3]{7+7\sqrt[3]{7+7\sqrt[3]{\,% 7+\cdots}}}}{1+\sqrt[3]{7+7\sqrt[3]{7+7\sqrt[3]{\,7+\cdots}}}}
  218. E T ET
  219. p ( 1 + 1 p ( p - 1 ) ) p = primes = ζ ( 2 ) ζ ( 3 ) ζ ( 6 ) = 315 ζ ( 3 ) 2 π 4 \underset{p\,\text{= primes}}{\prod_{p}\Big(1+\frac{1}{p(p-1)}\Big)}=\frac{% \zeta(2)\zeta(3)}{\zeta(6)}=\frac{315\zeta(3)}{2\pi^{4}}
  220. 5 4 \sqrt[4]{5}
  221. 5 5 5 5 5 5 5 5 5 5 \sqrt[5]{5\,\sqrt[5]{5\,\sqrt[5]{5\,\sqrt[5]{5\,\sqrt[5]{5\,\cdots}}}}}
  222. A C F A_{CF}
  223. q 1 ( p , q ) = 1 1 p < q π ( 1 2 q 2 ) 2 = π 4 ζ ( 3 ) ζ ( 4 ) = 45 2 ζ ( 3 ) π 3 ζ ( ) = Riemann Zeta Function \sum_{q\geq 1}\sum_{(p,q)=1\atop 1\leq p<q}\pi\left(\frac{1}{2q^{2}}\right)^{2% }\underset{\zeta()\,\text{= Riemann Zeta Function}}{=\frac{\pi}{4}\frac{\zeta(% 3)}{\zeta(4)}=\frac{45}{2}\frac{\zeta(3)}{\pi^{3}}}
  224. ζ ( 4 ) \zeta(4)
  225. π 4 90 = n = 1 < m t p l > 1 n 4 = 1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 + 1 5 4 + \frac{\pi^{4}}{90}=\sum_{n=1}^{\infty}\frac{<}{m}tpl>{{1}}{n^{4}}=\frac{1}{1^{% 4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\frac{1}{4^{4}}+\frac{1}{5^{4}}+...
  226. R 2 {R_{2}}
  227. 17 - 1 2 = 4 + 4 + 4 + 4 + 4 + 4 + - 1 \frac{\sqrt{17}-1}{2}=\,\scriptstyle\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+% \sqrt{4+\cdots}}}}}}\,\,-1
  228. = 4 - 4 - 4 - 4 - 4 - 4 - =\,\scriptstyle\sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\cdots}}}}}}\textstyle
  229. π 2 {\pi}^{2}
  230. 6 ζ ( 2 ) = 6 n = 1 1 n 2 = 6 1 2 + 6 2 2 + 6 3 2 + 6 4 2 + 6\,\zeta(2)=6\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{6}{1^{2}}+\frac{6}{2^{2}% }+\frac{6}{3^{2}}+\frac{6}{4^{2}}+\cdots
  231. ρ {\rho}
  232. 1 + 1 + 1 + 3 3 3 = 1 2 + 23 108 3 + 1 2 - 23 108 3 \sqrt[3]{1+\!\sqrt[3]{1+\!\sqrt[3]{1+\cdots}}}=\textstyle\sqrt[3]{\frac{1}{2}+% \!\sqrt{\frac{23}{108}}}+\!\sqrt[3]{\frac{1}{2}-\!\sqrt{\frac{23}{108}}}
  233. 2 l n γ 2\,ln\,\gamma
  234. π 2 6 l n ( 2 ) \frac{\pi^{2}}{6ln(2)}
  235. P f {P_{f}}
  236. n = 0 8 2 n 2 2 n + 2 - 1 = n = 0 1 2 2 n 1 - 1 2 2 n + 2 \sum_{n=0}^{\infty}\frac{8^{2^{n}}}{2^{2^{n+2}}-1}=\sum_{n=0}^{\infty}\cfrac{% \tfrac{1}{2^{2^{n}}}}{1-\tfrac{1}{2^{2^{n+2}}}}
  237. σ 3 {\sigma_{3}}
  238. n = 1 n 3 - n = 1 2 3 3 3 3 = 1 1 / 3 2 1 / 9 3 1 / 27 \prod_{n=1}^{\infty}n^{{3}^{-n}}=\sqrt[3]{1\sqrt[3]{2\sqrt[3]{3\cdots}}}=1^{1/% 3}\;2^{1/9}\;3^{1/27}\cdots
  239. C k {C_{k}}
  240. log 4 log 3 \frac{\log 4}{\log 3}
  241. 2 e 2^{\,e}
  242. 2 e 2^{e}
  243. M {M}
  244. lim n ( p n 1 p - ln ( ln ( n ) ) ) = γ + p ( ln ( 1 - 1 p ) + 1 p ) γ : Euler constant , p : prime \lim_{n\rightarrow\infty}\!\!\left(\sum_{p\leq n}\frac{1}{p}\!-\ln(\ln(n))\!% \right)\!\!=\underset{\!\!\!\!\gamma:\,\,\text{Euler constant},\,\,p:\,\,\text% {prime}}{\!\gamma\!+\!\!\sum_{p}\!\left(\!\ln\!\left(\!1\!-\!\frac{1}{p}\!% \right)\!\!+\!\frac{1}{p}\!\right)}
  245. γ \gamma
  246. i i = i - i = ( i i ) - 1 = ( ( ( i ) i ) i ) i = e π 2 = n = 0 π n n ! \sqrt[i]{i}=i^{-i}=(i^{i})^{-1}=(((i)^{i})^{i})^{i}=e^{\frac{\pi}{2}}=\sqrt{% \sum_{n=0}^{\infty}\frac{\pi^{n}}{n!}}
  247. 1 3 \sqrt[3]{1}
  248. { 1 - 1 2 + 3 2 i - 1 2 - 3 2 i . \begin{cases}\ \ 1\\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i\\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i.\end{cases}
  249. £ L i \,\text{£}_{Li}
  250. n = 1 1 10 n ! = 1 10 1 ! + 1 10 2 ! + 1 10 3 ! + 1 10 4 ! + \sum_{n=1}^{\infty}\frac{1}{10^{n!}}=\frac{1}{10^{1!}}+\frac{1}{10^{2!}}+\frac% {1}{10^{3!}}+\frac{1}{10^{4!}}+\cdots
  251. e - e {e}^{-e}
  252. ( 1 e ) e \left(\frac{1}{e}\right)^{e}
  253. ϕ 3 {\phi}_{3}
  254. 1 + 19 + 3 33 3 + 19 - 3 33 3 3 = 1 + ( 1 2 + 1 2 + 1 2 + 3 3 3 ) - 1 \textstyle\frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}=% \scriptstyle\,1+\left(\sqrt[3]{\tfrac{1}{2}+\sqrt[3]{\tfrac{1}{2}+\sqrt[3]{% \tfrac{1}{2}+...}}}\right)^{-1}
  255. l l 2 {ll_{2}}
  256. - ln ( ln ( 2 ) ) -\ln(\ln(2))
  257. π π \pi^{\pi}
  258. π π \pi^{\pi}
  259. 2 + ζ ( 1 2 ) 2+\zeta(\tfrac{1}{2})
  260. 2 - ( 1 + 2 ) n = 1 ( - 1 ) n + 1 n = γ + n = 1 ( - 1 ) 2 n γ n 2 n n ! {2{-}(1{+}\sqrt{2})\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}}}=\gamma+\sum% _{n=1}^{\infty}\frac{(-1)^{2n}\;\gamma_{n}}{2^{n}n!}
  261. e e e^{e}
  262. n = 0 e n n ! = lim n ( 1 + n n ) n - n ( 1 + n ) 1 + n \sum_{n=0}^{\infty}\frac{e^{n}}{n!}=\lim_{n\to\infty}\left(\frac{1+n}{n}\right% )^{n^{-n}(1+n)^{1+n}}
  263. C {C}
  264. γ β ( 1 ) + β ( 1 ) = π ( - ln Γ ( 1 4 ) + 3 4 π + 1 2 ln 2 + 1 2 γ ) \gamma{\beta}(1)\!+\!{\beta}^{\prime}(1)\!=\pi\!\left(-\!\ln\Gamma(\tfrac{1}{4% })+\tfrac{3}{4}\pi+\tfrac{1}{2}\ln 2+\tfrac{1}{2}\gamma\right)
  265. = π ( - ln ( 1 4 ! ) + 3 4 ln π - 3 2 ln 2 + 1 2 γ ) =\pi\!\left(-\!\ln(\tfrac{1}{4}!)+\tfrac{3}{4}\ln\pi-\tfrac{3}{2}\ln 2+\tfrac{% 1}{2}\,\gamma\right)
  266. γ = Euler–Mascheroni constant = 0.5772156649 \scriptstyle\gamma=\,\text{Euler–Mascheroni constant}=0.5772156649\ldots
  267. β ( ) = Beta function , Γ ( ) = Gamma function \scriptstyle\beta()=\,\text{Beta function},\quad\scriptstyle\Gamma()=\,\text{% Gamma function}
  268. π 2 2 \frac{\pi}{2\sqrt{2}}
  269. n = 1 ( - 1 ) n - 1 2 2 n + 1 = 1 1 + 1 3 - 1 5 - 1 7 + 1 9 + 1 11 - \sum_{n=1}^{\infty}\frac{({-}1)^{\lfloor\frac{n-1}{2}\rfloor}}{2n+1}=\frac{1}{% 1}+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-{\cdots}
  270. B {B}
  271. lim k | q k + 1 q k | where: Q ( x ) = 1 P ( x ) = k = 1 q k x k \lim_{k\to\infty}\left|\frac{q_{k+1}}{q_{k}}\right|\quad\scriptstyle\text{% where:}\displaystyle\;\;Q(x)=\frac{1}{P(x)}=\!\sum_{k=1}^{\infty}q_{k}x^{k}
  272. P ( x ) = k = 1 p k x k p k prime = 1 + 2 x + 3 x 2 + 5 x 3 + P(x)=\sum_{k=1}^{\infty}\underset{p_{k}\,\text{ prime}}{p_{k}x^{k}}=1+2x+3x^{2% }+5x^{3}+\cdots
  273. S i ( π ) {Si(\pi)}
  274. 0 π sin t t d t = n = 1 ( - 1 ) n - 1 π 2 n - 1 ( 2 n - 1 ) ( 2 n - 1 ) ! \int_{0}^{\pi}\frac{\sin t}{t}\,dt=\sum\limits_{n=1}^{\infty}(-1)^{n-1}\frac{% \pi^{2n-1}}{(2n-1)(2n-1)!}
  275. = π - π 3 3 3 ! + π 5 5 5 ! - π 7 7 7 ! + =\pi-\frac{\pi^{3}}{3\cdot 3!}+\frac{\pi^{5}}{5\cdot 5!}-\frac{\pi^{7}}{7\cdot 7% !}+\cdots
  276. 𝒞 C E {\mathcal{C}_{CE}}
  277. n = 1 p n 10 n + k = 1 n log 10 p k \sum_{n=1}^{\infty}\frac{p_{n}}{10^{n+\sum\limits_{k=1}^{n}\lfloor\log_{10}{p_% {k}}\rfloor}}
  278. C d {C_{d}}
  279. log ( 1 + 73 - 6 87 3 + 73 + 6 87 3 3 ) log ( 2 ) \frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}% \right)}{\log(2)}
  280. K R {K_{R}}
  281. π 2 log ( 1 + 2 ) \frac{\pi}{2\log(1+\sqrt{2})}
  282. l o g 2 3 {log_{2}3}
  283. log 3 log 2 = n = 0 1 2 2 n + 1 ( 2 n + 1 ) n = 0 1 3 2 n + 1 ( 2 n + 1 ) = 1 2 + 1 24 + 1 160 + 1 3 + 1 81 + 1 1215 + \frac{\log 3}{\log 2}=\frac{\sum_{n=0}^{\infty}\frac{1}{2^{2n+1}(2n+1)}}{\sum_% {n=0}^{\infty}\frac{1}{3^{2n+1}(2n+1)}}=\frac{\frac{1}{2}+\frac{1}{24}+\frac{1% }{160}+\cdots}{\frac{1}{3}+\frac{1}{81}+\frac{1}{1215}+\cdots}
  284. θ {\theta}
  285. θ 3 n \lfloor\theta^{3^{n}}\rfloor
  286. V 8 {V_{8}}
  287. 2 3 n = 1 1 n ( 2 n n ) k = n 2 n - 1 1 k = 6 0 π / 3 log ( 1 2 sin t ) d t = 2\sqrt{3}\,\sum_{n=1}^{\infty}\frac{1}{n{2n\choose n}}\sum_{k=n}^{2n-1}\frac{1% }{k}=6\int\limits_{0}^{\pi/3}\log\left(\frac{1}{2\sin t}\right)\,dt=
  288. 3 < m t p l > 9 n = 0 ( - 1 ) n 27 n { 18 ( 6 n + 1 ) 2 - 18 ( 6 n + 2 ) 2 - 24 ( 6 n + 3 ) 2 - 6 ( 6 n + 4 ) 2 + 2 ( 6 n + 5 ) 2 } \scriptstyle\frac{\sqrt{3}}{<}mtpl>{{9}}\,\sum\limits_{n=0}^{\infty}\frac{(-1)% ^{n}}{27^{n}}\,\left\{\!\frac{{18}}{(6n+1)^{2}}-\frac{{18}}{(6n+2)^{2}}-\frac{% {24}}{(6n+3)^{2}}-\frac{{6}}{(6n+4)^{2}}+\frac{{2}}{(6n+5)^{2}}\!\right\}
  289. R {R}
  290. e π 163 e^{\pi\sqrt{163}}
  291. K - 1 {K_{-1}}
  292. log 2 n = 1 1 n log ( 1 + 1 n ( n + 2 ) ) = lim n n 1 a 1 + 1 a 2 + + 1 a n \frac{\log 2}{\sum\limits_{n=1}^{\infty}\frac{1}{n}\log\bigl(1{+}\frac{1}{n(n+% 2)}\bigr)}=\lim_{n\to\infty}\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+% \frac{1}{a_{n}}}
  293. e \sqrt{e}
  294. n = 0 1 2 n n ! = n = 0 1 ( 2 n ) ! ! = 1 1 + 1 2 + 1 8 + 1 48 + \sum_{n=0}^{\infty}\frac{1}{2^{n}n!}=\sum_{n=0}^{\infty}\frac{1}{(2n)!!}=\frac% {1}{1}+\frac{1}{2}+\frac{1}{8}+\frac{1}{48}+\cdots
  295. ζ ( 6 ) \zeta(6)
  296. π 6 945 = n = 1 1 1 - p n - 6 p n : prime = 1 1 - 2 - 6 1 1 - 3 - 6 1 1 - 5 - 6 \frac{\pi^{6}}{945}\!=\!\prod_{n=1}^{\infty}\!\underset{p_{n}:\,\text{ prime}}% {\frac{1}{{1-p_{n}}^{-6}}}=\frac{1}{1\!-\!2^{-6}}\!\cdot\!\frac{1}{1\!-\!3^{-6% }}\!\cdot\!\frac{1}{1\!-\!5^{-6}}\cdots
  297. T 1 \mathrm{T}_{1}
  298. [ 1 , 0 , 8 , 4 , 1 , 0 , 1 , 5 , 1 , 2 , 2 , 3 , 1 , 1 , 1 , 3 , 6 , ] \textstyle[1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,...]
  299. 1 1 + 1 0 + 1 8 + 1 4 + 1 1 + 1 0 + 1 / \tfrac{1}{1+\tfrac{1}{0+\tfrac{1}{8+\tfrac{1}{4+\tfrac{1}{1+\tfrac{1}{0+1{/% \cdots}}}}}}}
  300. Ω \Omega
  301. p P 2 - | p | | p | : Size in bits of program p P : Domain of all programs that stop. p : Halted program \sum_{p\in P}2^{-|p|}\overset{p:\,\text{ Halted program}}{\underset{P:\,\text{% Domain of all programs that stop.}}{\scriptstyle{|p|}:\,\text{Size in bits of% program }p}}
  302. G {G}
  303. 1 agm ( 1 , 2 ) = 4 2 ( 1 4 ! ) 2 π 3 / 2 = 2 π 0 1 d x 1 - x 4 \frac{1}{\mathrm{agm}(1,\sqrt{2})}=\frac{4\sqrt{2}\,(\tfrac{1}{4}!)^{2}}{\pi^{% 3/2}}=\frac{2}{\pi}\int_{0}^{1}\frac{dx}{\sqrt{1-x^{4}}}
  304. μ {\mu}
  305. li ( x ) = 0 x d t ln t = 0 \color W h i t e \mathrm{li}(x)=\int\limits_{0}^{x}\frac{dt}{\ln t}=0{\color{White}{......}}
  306. li ( x ) = Ei ( ln x ) \color W h i t e . . \mathrm{li}(x)\;=\;\mathrm{Ei}(\ln{x}){\color{White}{........}}
  307. ξ 2 \xi_{2}
  308. k = 1 ( - 1 ) k s k - 1 = 1 1 - 1 2 + 1 6 - 1 42 + 1 1806 ± \sum_{k=1}^{\infty}\frac{(-1)^{k}}{s_{k}-1}=\frac{1}{1}-\frac{1}{2}+\frac{1}{6% }-\frac{1}{42}+\frac{1}{1806}{\,\pm\cdots}
  309. S 0 = 2 , S k = 1 + n = 0 k - 1 S n for k > 0 \,\,S_{0}=\,2,\,\,S_{k}=\,1+\prod\limits_{n=0}^{k-1}S_{n}\,\text{ for}\;k>0
  310. 2 \sqrt{2}
  311. n = 1 ( 1 + ( - 1 ) n + 1 2 n - 1 ) = ( 1 + 1 1 ) ( 1 - 1 3 ) ( 1 + 1 5 ) \!\prod_{n=1}^{\infty}\!\left(1\!+\!\frac{(-1)^{n+1}}{2n-1}\right)\!=\!\left(1% \!+\!\frac{1}{1}\right)\!\left(1\!-\!\frac{1}{3}\right)\!\left(1\!+\!\frac{1}{% 5}\right)\cdots
  312. Γ ( 1 2 ) {\Gamma}(\tfrac{1}{2})
  313. π = ( - 1 2 ) ! = - 1 e x 2 d x = 0 1 1 - ln x d x \sqrt{\pi}=\left(-\frac{1}{2}\right)!=\int_{-\infty}^{\infty}\frac{1}{e^{x^{2}% }}\,dx=\int_{0}^{1}\frac{1}{\sqrt{-\ln x}}\,dx
  314. 2 12 \sqrt[12]{2}
  315. 440 H z .2 1 12 2 2 12 2 3 12 2 4 12 2 5 12 2 6 12 2 7 12 2 8 12 2 9 12 2 10 12 2 11 12 2 \scriptstyle 440\,Hz.\textstyle 2^{\frac{1}{12}}\,2^{\frac{2}{12}}\,2^{\frac{3% }{12}}\,2^{\frac{4}{12}}\,2^{\frac{5}{12}}\,2^{\frac{6}{12}}\,2^{\frac{7}{12}}% \,2^{\frac{8}{12}}\,2^{\frac{9}{12}}\,2^{\frac{10}{12}}\,2^{\frac{11}{12}}\,2
  316. \color w h i t e \color b l a c k D o 1 D o # R e R e # M i F a F a # S o l S o l # L a L a # S i D o 2 \scriptstyle{\color{white}...\color{black}Do_{1}\;\;Do\#\;\,Re\;\,Re\#\;\,Mi\;% \;Fa\;\;Fa\#\;Sol\;\,Sol\#\,La\;\;La\#\;\;Si\;\,Do_{2}}
  317. \color w h i t e . \color b l a c k C 1 C # D D # E F F # G G # A A # B C 2 \scriptstyle{\color{white}....\color{black}C_{1}\;\;\;\;C\#\;\;\;\,D\;\;\;D\#% \;\;\,E\;\;\;\;\,F\;\;\;\,F\#\;\;\;G\;\;\;\;G\#\;\;\;A\;\;\;\,A\#\;\;\;\,B\;\;% \;C_{2}}
  318. π ln β {\pi\ln\beta}
  319. 3 3 4 ( 1 - n = 0 1 ( 3 n + 2 ) 2 + n = 1 1 ( 3 n + 1 ) 2 ) = \frac{3\sqrt{3}}{4}\left(1-\sum_{n=0}^{\infty}\frac{1}{(3n+2)^{2}}+\sum_{n=1}^% {\infty}\frac{1}{(3n+1)^{2}}\right)=
  320. 3 3 4 ( 1 - 1 2 2 + 1 4 2 - 1 5 2 + 1 7 2 - 1 8 2 + 1 10 2 ± ) \textstyle\frac{3\sqrt{3}}{4}\left(1-\frac{1}{2^{2}}+\frac{1}{4^{2}}-\frac{1}{% 5^{2}}+\frac{1}{7^{2}}-\frac{1}{8^{2}}+\frac{1}{10^{2}}\pm\cdots\right)
  321. ϖ {\varpi}
  322. π G = 4 2 π Γ ( 5 4 ) 2 = 1 4 2 π Γ ( 1 4 ) 2 = 4 2 π ( 1 4 ! ) 2 \pi\,{G}=4\sqrt{\tfrac{2}{\pi}}\,\Gamma{\left(\tfrac{5}{4}\right)^{2}}=\tfrac{% 1}{4}\sqrt{\tfrac{2}{\pi}}\,\Gamma{\left(\tfrac{1}{4}\right)^{2}}=4\sqrt{% \tfrac{2}{\pi}}\left(\tfrac{1}{4}!\right)^{2}
  323. A {A}
  324. e 1 12 - ζ ( - 1 ) = e 1 8 - 1 2 n = 0 1 n + 1 k = 0 n ( - 1 ) k ( n k ) ( k + 1 ) 2 ln ( k + 1 ) e^{\frac{1}{12}-\zeta^{\prime}(-1)}=e^{\frac{1}{8}-\frac{1}{2}\sum\limits_{n=0% }^{\infty}\frac{1}{n+1}\sum\limits_{k=0}^{n}\left(-1\right)^{k}{\left({{n}% \atop{k}}\right)}\left(k+1\right)^{2}\ln(k+1)}
  325. ψ ( 1 4 ) {\psi}(\tfrac{1}{4})
  326. - γ - π 2 - 3 ln 2 = - γ + n = 0 ( 1 n + 1 - 1 n + 1 4 ) -\gamma-\frac{\pi}{2}-3\ln{2}=-\gamma+\sum_{n=0}^{\infty}\left(\frac{1}{n+1}-% \frac{1}{n+\tfrac{1}{4}}\right)
  327. K 2 K_{2}
  328. n = 1 ( 1 - 3 p n - 2 p n 3 ) p n : prime = 6 π 2 n = 1 ( 1 - 1 < m t p l > p n ( p n + 1 ) ) p n : prime \prod_{n=1}^{\infty}\underset{p_{n}:\,\text{ prime}}{\left(1-\frac{3p_{n}-2}{{% p_{n}}^{3}}\right)}=\frac{6}{\pi^{2}}\prod_{n=1}^{\infty}\underset{p_{n}:\,% \text{ prime}}{\left(1-\frac{1}{<}mtpl>{{p_{n}(p_{n}+1)}}\right)}
  329. e γ e^{\gamma}
  330. n = 1 e 1 n 1 + 1 n = n = 0 ( k = 0 n ( k + 1 ) ( - 1 ) k + 1 ( n k ) ) 1 n + 1 = \prod_{n=1}^{\infty}\frac{e^{\frac{1}{n}}}{1+\tfrac{1}{n}}=\prod_{n=0}^{\infty% }\left(\prod_{k=0}^{n}(k+1)^{(-1)^{k+1}{n\choose k}}\right)^{\frac{1}{n+1}}=
  331. ( 2 1 ) 1 / 2 ( 2 2 1 3 ) 1 / 3 ( 2 3 4 1 3 3 ) 1 / 4 ( 2 4 4 4 1 3 6 5 ) 1 / 5 \textstyle\left(\frac{2}{1}\right)^{1/2}\left(\frac{2^{2}}{1\cdot 3}\right)^{1% /3}\left(\frac{2^{3}\cdot 4}{1\cdot 3^{3}}\right)^{1/4}\left(\frac{2^{4}\cdot 4% ^{4}}{1\cdot 3^{6}\cdot 5}\right)^{1/5}\cdots
  332. Γ ( 1 4 ) \Gamma(\tfrac{1}{4})
  333. 4 ( 1 4 ) ! = ( - 3 4 ) ! 4\left(\frac{1}{4}\right)!=\left(-\frac{3}{4}\right)!
  334. σ {\sigma}
  335. n = 1 n 1 / 2 n = 1 2 3 = 1 1 / 2 2 1 / 4 3 1 / 8 \prod_{n=1}^{\infty}n^{{1/2}^{n}}=\sqrt{1\sqrt{2\sqrt{3\cdots}}}=1^{1/2}\;2^{1% /4}\;3^{1/8}\cdots
  336. θ m {\theta_{m}}
  337. arctan ( 2 ) = arccos ( 1 3 ) 54.7356 \arctan\left(\sqrt{2}\right)=\arccos\left(\sqrt{\tfrac{1}{3}}\right)\approx% \textstyle{54.7356}^{\circ}
  338. m {m}
  339. 0 e x p ( - 2 0 x 1 - e - y y d y ) d x = e - 2 γ 0 e - 2 Γ ( 0 , n ) n 2 \int\limits_{0}^{\infty}exp\left(\!-2\int\limits_{0}^{x}\frac{1-e^{-y}}{y}dy% \right)\!dx={e^{-2\gamma}}\int\limits_{0}^{\infty}\frac{e^{-2\Gamma(0,n)}}{n^{% 2}}
  340. e e \sqrt[e]{e}
  341. e 1 e \color W h i t e . . e^{\frac{1}{e}}{\color{White}{...........}}
  342. ( 1 e ) 1 e {\left(\frac{1}{e}\right)}^{\frac{1}{e}}
  343. e - 1 e \color W h i t e . {e}^{-\frac{1}{e}}{\color{White}{..........}}
  344. p ( 3 ) {p(3)}
  345. 1 - ( 3 ( 2 π ) 3 - π π - π π - π π d x d y d z 3 - cos x - cos y - cos z ) - 1 1-\!\!\left({3\over(2\pi)^{3}}\int\limits_{-\pi}^{\pi}\int\limits_{-\pi}^{\pi}% \int\limits_{-\pi}^{\pi}{dx\,dy\,dz\over 3-\!\cos x-\!\cos y-\!\cos z}\right)^% {\!-1}
  346. = 1 - 16 2 3 π 3 ( Γ ( 1 24 ) Γ ( 5 24 ) Γ ( 7 24 ) Γ ( 11 24 ) ) - 1 =1-16\sqrt{\tfrac{2}{3}}\;\pi^{3}\left(\Gamma(\tfrac{1}{24})\Gamma(\tfrac{5}{2% 4})\Gamma(\tfrac{7}{24})\Gamma(\tfrac{11}{24})\right)^{-1}
  347. L {L}
  348. = Γ ( 1 3 ) Γ ( 5 6 ) Γ ( 1 6 ) = ( - 2 3 ) ! ( - 1 + 5 6 ) ! ( - 1 + 1 6 ) ! =\frac{\Gamma(\tfrac{1}{3})\;\Gamma(\tfrac{5}{6})}{\Gamma(\tfrac{1}{6})}=\frac% {(-\tfrac{2}{3})!\;(-1+\tfrac{5}{6})!}{(-1+\tfrac{1}{6})!}
  349. C M R B C_{{}_{MRB}}
  350. n = 1 ( - 1 ) n ( n 1 / n - 1 ) = - 1 1 + 2 2 - 3 3 + \sum_{n=1}^{\infty}(-1)^{n}(n^{1/n}-1)=-\sqrt[1]{1}+\sqrt[2]{2}-\sqrt[3]{3}+\cdots
  351. 4 π \frac{4}{\pi}
  352. n = 0 ( ( 2 n - 3 ) ! ! ( 2 n ) ! ! ) 2 = 1 + ( 1 2 ) 2 + ( 1 2 4 ) 2 + ( 1 3 2 4 6 ) 2 + \displaystyle\sum\limits_{n=0}^{\infty}\textstyle\left(\frac{(2n-3)!!}{(2n)!!}% \right)^{2}={1\!+\!\left(\frac{1}{2}\right)^{2}\!+\!\left(\frac{1}{2\cdot 4}% \right)^{2}\!+\!\left(\frac{1\cdot 3}{2\cdot 4\cdot 6}\right)^{2}+\cdots}
  353. C {C}
  354. 6 ln 2 π 2 ( 3 ln 2 + 4 γ - 24 π 2 ζ ( 2 ) - 2 ) - 1 2 \frac{6\ln 2}{\pi^{2}}\left(3\ln 2+4\,\gamma-\frac{24}{\pi^{2}}\,\zeta^{\prime% }(2)-2\right)-\frac{1}{2}
  355. γ = Euler–Mascheroni Constant = 0.5772156649 \scriptstyle\gamma\,\,\text{= Euler–Mascheroni Constant}=0.5772156649\ldots
  356. ζ ( 2 ) = Derivative of ζ ( 2 ) = - n = 2 ln n n 2 = - 0.9375482543 \scriptstyle\zeta^{\prime}(2)\,\,\text{= Derivative of }\zeta(2)=-\sum\limits_% {n=2}^{\infty}\frac{\ln n}{n^{2}}=-0.9375482543\ldots
  357. δ {\delta}
  358. lim n x n + 1 - x n x n + 2 - x n + 1 x ( 3.8284 ; 3.8495 ) \lim_{n\to\infty}\frac{x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}\qquad\scriptstyle x\in(% 3.8284;\,3.8495)
  359. x n + 1 = a x n ( 1 - x n ) or x n + 1 = a sin ( x n ) \scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad\,\text{or}\quad x_{n+1}=\,a\sin(x_% {n})
  360. α \alpha
  361. lim n d n d n + 1 \lim_{n\to\infty}\frac{d_{n}}{d_{n+1}}
  362. λ {\lambda}
  363. 0 f ( x ) x 2 d x Para x > 2 = 0 1 e Li ( n ) d n Li: Logarithmic integral \int\limits_{0}^{\infty}\underset{\,\text{Para }x>2}{\frac{f(x)}{x^{2}}\,dx}=% \int\limits_{0}^{1}e^{\operatorname{Li}(n)}dn\quad\scriptstyle\,\text{Li: % Logarithmic integral}
  364. e π {e}^{\pi}
  365. ( - 1 ) - i = i - 2 i = n = 0 π n n ! = π 1 1 + π 2 2 ! + π 3 3 ! + (-1)^{-i}=i^{-2i}=\sum_{n=0}^{\infty}\frac{\pi^{n}}{n!}=\frac{\pi^{1}}{1}+% \frac{\pi^{2}}{2!}+\frac{\pi^{3}}{3!}+\cdots
  366. e 2 e^{2}
  367. n = 0 2 n n ! = 1 + 2 + 2 2 2 ! + 2 3 3 ! + 2 4 4 ! + 2 5 5 ! + \sum_{n=0}^{\infty}\frac{2^{n}}{n!}=1+2+\frac{2^{2}}{2!}+\frac{2^{3}}{3!}+% \frac{2^{4}}{4!}+\frac{2^{5}}{5!}+\cdots
  368. σ {\sigma}
  369. k = 1 { 1 - [ 1 - j = 1 n ( 1 - p k - j ) ] 2 p k : prime } \prod_{k=1}^{\infty}\left\{1-[1-\prod_{j=1}^{n}\underset{p_{k}:\,\text{ prime}% }{(1-p_{k}^{-j})]^{2}}\right\}
  370. 1 ζ ( 2 ) \frac{1}{\zeta(2)}
  371. 6 π 2 = n = 0 ( 1 - 1 p n 2 ) p n : prime = ( 1 - 1 2 2 ) ( 1 - 1 3 2 ) ( 1 - 1 5 2 ) \frac{6}{\pi^{2}}=\prod_{n=0}^{\infty}\underset{p_{n}:\,\text{ prime}}{\!\left% (\!1-\frac{1}{{p_{n}}^{2}}\!\right)}\!=\!\textstyle\left(1\!-\!\frac{1}{2^{2}}% \right)\!\left(1\!-\!\frac{1}{3^{2}}\right)\!\left(1\!-\!\frac{1}{5^{2}}\right)\cdots
  372. C 10 C_{10}
  373. n = 1 k = 10 n - 1 10 n - 1 k 10 k n - 9 j = 0 n - 1 10 j ( n - j - 1 ) \sum_{n=1}^{\infty}\;\sum_{k=10^{n-1}}^{10^{n}-1}\frac{k}{10^{kn-9\sum_{j=0}^{% n-1}10^{j}(n-j-1)}}
  374. K K
  375. 1 2 p 3 mod 4 ( 1 - 1 p 2 ) - 1 2 p : prime = π 4 p 1 mod 4 ( 1 - 1 p 2 ) 1 2 p : prime \frac{1}{\sqrt{2}}\prod_{p\equiv 3\!\!\!\!\!\mod\!4}\!\!\underset{\!\!\!\!\!\!% \!\!p:\,\text{ prime}}{\left(1-\frac{1}{p^{2}}\right)^{-\frac{1}{2}}}\!\!=% \frac{\pi}{4}\prod_{p\equiv 1\!\!\!\!\!\mod\!4}\!\!\underset{\!\!\!\!p:\,\text% { prime}}{\left(1-\frac{1}{p^{2}}\right)^{\frac{1}{2}}}
  376. ω {\omega}
  377. 2 2 2 2 ω = primes: 2 ω =3, 2 2 ω =13, 2 2 2 ω = 16381 , \left\lfloor 2^{2^{2^{\cdot^{\cdot^{2^{\omega}}}}}}\!\right\rfloor\scriptstyle% \,\text{= primes:}\displaystyle\left\lfloor 2^{\omega}\right\rfloor% \scriptstyle\,\text{=3,}\displaystyle\left\lfloor 2^{2^{\omega}}\right\rfloor% \scriptstyle\,\text{=13,}\displaystyle\left\lfloor 2^{2^{2^{\omega}}}\right% \rfloor\scriptstyle=16381,\ldots
  378. e {e}
  379. lim n ( 1 + 1 n ) n = n = 0 1 n ! = 1 0 ! + 1 1 + 1 2 ! + 1 3 ! + \!\lim_{n\to\infty}\!\left(\!1\!+\!\frac{1}{n}\right)^{n}\!=\!\sum_{n=0}^{% \infty}\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1}+\frac{1}{2!}+\frac{1}{3!}+\textstyle\cdots
  380. 1 e \frac{1}{e}
  381. n = 0 ( - 1 ) n n ! = 1 0 ! - 1 1 ! + 1 2 ! - 1 3 ! + 1 4 ! - 1 5 ! + \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}=\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-% \frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\cdots
  382. H 2 n + 1 {H}_{2n+1}
  383. lim n H 2 n + 1 = ( 1 2 ) ( 1 3 ) ( 1 4 ) ( 1 2 n + 1 ) = 2 - 3 - 4 < m t p l > - 2 n - 1 \lim_{n\to\infty}{H}_{2n+1}=\textstyle\left(\frac{1}{2}\right)^{\left(\frac{1}% {3}\right)^{\left(\frac{1}{4}\right)^{\cdot^{\cdot^{\left(\frac{1}{2n+1}\right% )}}}}}={2}^{-3^{-4^{\cdot^{\cdot^{<}mtpl>{{-2n-1}}}}}}
  384. H 2 n {H}_{2n}
  385. lim n H 2 n = ( 1 2 ) ( 1 3 ) ( 1 4 ) ( 1 2 n ) = 2 - 3 - 4 < m t p l > - 2 n \lim_{n\to\infty}{H}_{2n}=\textstyle\left(\frac{1}{2}\right)^{\left(\frac{1}{3% }\right)^{\left(\frac{1}{4}\right)^{\cdot^{\cdot^{\left(\frac{1}{2n}\right)}}}% }}={2}^{-3^{-4^{\cdot^{\cdot^{<}mtpl>{{-2n}}}}}}
  386. π \pi
  387. lim n 2 n 2 - 2 + 2 + + 2 n \lim_{n\to\infty}\,2^{n}\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}% }_{n}
  388. arctan 1 2 \arctan\frac{1}{2}
  389. n = 0 ( - 1 ) n x 2 n + 1 2 n + 1 = 1 2 - 1 3 2 3 + 1 5 2 5 - 1 7 2 7 + For x = 1 / 2 \underset{\,\text{For }x=1/2\qquad\qquad}{\sum_{n=0}^{\infty}\frac{(\!-1\!)^{n% }\,x^{2n+1}}{2n+1}=\frac{1}{2}{-}\frac{1}{3\!\cdot\!2^{3}}{+}\frac{1}{5\!\cdot% \!2^{5}}{-}\frac{1}{7\!\cdot\!2^{7}}{+}\cdots}
  390. B 2 {B}_{\,2}
  391. ( 1 p + 1 p + 2 ) p , p + 2 : prime = ( 1 3 + 1 5 ) + ( 1 5 + 1 7 ) + ( 1 11 + 1 13 ) + \textstyle\underset{p,\,p+2:\,\text{ prime}}{\sum(\frac{1}{p}+\frac{1}{p+2})}=% (\frac{1}{3}\!+\!\frac{1}{5})+(\tfrac{1}{5}\!+\!\tfrac{1}{7})+(\tfrac{1}{11}\!% +\!\tfrac{1}{13})+\cdots
  392. B 4 {B}_{\,4}
  393. ( 1 p + 1 p + 2 + 1 p + 4 + 1 p + 6 ) p , p + 2 , p + 4 , p + 6 : prime \textstyle{\sum(\frac{1}{p}+\frac{1}{p+2}+\frac{1}{p+4}+\frac{1}{p+6})}% \scriptstyle\quad{p,\;p+2,\;p+4,\;p+6:\,\text{ prime}}
  394. ( 1 5 + 1 7 + 1 11 + 1 13 ) + ( 1 11 + 1 13 + 1 17 + 1 19 ) + \textstyle{\left(\tfrac{1}{5}+\tfrac{1}{7}+\tfrac{1}{11}+\tfrac{1}{13}\right)}% +\left(\tfrac{1}{11}+\tfrac{1}{13}+\tfrac{1}{17}+\tfrac{1}{19}\right)+\dots
  395. 2 π \frac{2}{\pi}
  396. 2 2 2 + 2 2 2 + 2 + 2 2 \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+% \sqrt{2}}}}{2}\cdots
  397. G {G}
  398. 0 e - n 1 + n d n = 0 1 1 1 - ln n d n = 1 1 + 1 1 + 1 1 + 2 1 + 2 1 + 3 1 + 3 / \!\int\limits_{0}^{\infty}\!\!\frac{e^{-n}}{1{+}n}\,dn=\!\!\int\limits_{0}^{1}% \!\!\frac{1}{1{-}\ln n}\,dn=\textstyle{\tfrac{1}{1+\tfrac{1}{1+\tfrac{1}{1+% \tfrac{2}{1+\tfrac{2}{1+\tfrac{3}{1+3{/\cdots}}}}}}}}
  399. i {i}
  400. - 1 = ln ( - 1 ) π e i π = - 1 \sqrt{-1}=\frac{\ln(-1)}{\pi}\qquad\qquad\mathrm{e}^{i\,\pi}=-1
  401. C C F {C}_{CF}
  402. I 1 ( 2 ) I 0 ( 2 ) = n = 0 n n ! n ! n = 0 1 n ! n ! = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 / \frac{I_{1}(2)}{I_{0}(2)}=\frac{\sum\limits_{n=0}^{\infty}\frac{n}{n!n!}}{{% \sum\limits_{n=0}^{\infty}\frac{1}{n!n!}}}=\textstyle\tfrac{1}{1+\tfrac{1}{2+% \tfrac{1}{3+\tfrac{1}{4+\tfrac{1}{5+\tfrac{1}{6+1{/\cdots}}}}}}}
  403. R 5 R_{5}
  404. 5 + 5 + 5 - 5 + 5 + 5 + 5 - = 2 + 5 + 15 - 6 5 2 \scriptstyle\sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}% }}}\;=\textstyle\frac{2+\sqrt{5}+\sqrt{15-6\sqrt{5}}}{2}
  405. Ω {\Omega}
  406. n = 1 ( - n ) n - 1 n ! = ( 1 e ) ( 1 e ) ( 1 e ) = e - Ω = e - e - e < m t p l > - e \sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}=\,\left(\frac{1}{e}\right)^{\left(% \frac{1}{e}\right)^{\cdot^{\cdot^{\left(\frac{1}{e}\right)}}}}=e^{-\Omega}=e^{% -e^{-e^{\cdot^{\cdot^{<}mtpl>{{-e}}}}}}
  407. β ( 3 ) {\beta}(3)
  408. π 3 32 = n = 1 - 1 n + 1 ( - 1 + 2 n ) 3 = 1 1 3 - 1 3 3 + 1 5 3 - 1 7 3 + \frac{\pi^{3}}{32}=\sum_{n=1}^{\infty}\frac{-1^{n+1}}{(-1+2n)^{3}}=\frac{1}{1^% {3}}{-}\frac{1}{3^{3}}{+}\frac{1}{5^{3}}{-}\frac{1}{7^{3}}{+}\cdots
  409. 5 \sqrt{5}
  410. ( n = 5 ) k = 0 n - 1 e 2 k 2 π i n = 1 + e 2 π i 5 + e 8 π i 5 + e 18 π i 5 + e 32 π i 5 \scriptstyle(n=5)\displaystyle\sum_{k=0}^{n-1}e^{\frac{2k^{2}\pi i}{n}}=1+e^{% \frac{2\pi i}{5}}+e^{\frac{8\pi i}{5}}+e^{\frac{18\pi i}{5}}+e^{\frac{32\pi i}% {5}}
  411. Ψ \Psi
  412. n = 1 1 F n = 1 1 + 1 1 + 1 2 + 1 3 + 1 5 + 1 8 + 1 13 + \sum_{n=1}^{\infty}\frac{1}{F_{n}}=\frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1% }{3}+\frac{1}{5}+\frac{1}{8}+\frac{1}{13}+\cdots
  413. K 0 K_{\,0}
  414. n = 1 [ 1 + 1 n ( n + 2 ) ] ln n / ln 2 \prod_{n=1}^{\infty}\left[{1+{1\over n(n+2)}}\right]^{\ln n/\ln 2}

Mathematical_modeling_of_electrophysiological_activity_in_epilepsy.html

  1. ( u , I , v ) (u,I,v)
  2. u t = u - v + R 2 ω ( x - x , y - y ) f ( u - θ ) d x d y + ζ ( x , y , t ) , {\partial u\over\partial t}=u-v+\int_{R^{2}}\omega(x-x^{\prime},y-y^{\prime})f% (u-\theta)\,dxdy+\zeta(x,y,t),
  3. v t = ϵ ( β u - v ) . {\partial v\over\partial t}=\epsilon(\beta u-v).
  4. ϵ > 0 \epsilon>0
  5. β > 0 \beta>0
  6. ω ( x , y ) \omega(x,y)
  7. ω = A e - λ - ( x 2 + y 2 ) \omega=Ae^{-\lambda\sqrt{-(x^{2}+y^{2})}}
  8. ( A , λ ) = ( 2.1 , 1 ) . (A,\lambda)=(2.1,1).
  9. f ( u - θ ) = H ( u - θ ) f(u-\theta)=H(u-\theta)
  10. ζ ( x , y , t ) \zeta(x,y,t)
  11. ( u , v ) (u,v)
  12. ( u , v ) = ( 0 , 0 ) (u,v)=(0,0)
  13. ( u , v ) (u,v)
  14. ( 0 , 0 ) (0,0)
  15. ϵ > 0 \epsilon>0
  16. β \beta
  17. β * \beta^{*}
  18. ( u , v ) (u,v)
  19. β β * \beta\geq\beta^{*}
  20. Δ t , \Delta t,
  21. β β * \beta\geq\beta^{*}
  22. R a t e = ( Δ t / T ) * c ( 1 ) Rate=(\Delta t/T)*c~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1)
  23. Δ t \Delta t
  24. Δ t / T ? \Delta t/T?
  25. Δ t / T \Delta t/T
  26. d u d t = u - v + H ( u - θ ) , {{du}\over{dt}}=u-v+H(u-\theta),
  27. d v d t = ϵ ( β u - v ) . {{dv}\over{dt}}=\epsilon(\beta u-v).
  28. ω = 2.1 e - λ - ( x 2 + y 2 ) \omega=2.1e^{-\lambda\sqrt{-(x^{2}+y^{2})}}
  29. ϵ = 0.1 \epsilon=0.1
  30. θ = 0.1 \theta=0.1
  31. β β * = 12.61 \beta\geq\beta^{*}=12.61
  32. 12.61 β 17 12.61\leq\beta\leq 17
  33. 0.136 Δ t / T 0.238 0.136\leq\Delta t/T\leq 0.238
  34. 3.046 m m / s R a t e 5.331 m m / s ( 2 ) 3.046mm/s\leq Rate\leq 5.331mm/s~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2)
  35. 0.136 Δ t / T 0.238 0.136\leq\Delta t/T\leq 0.238
  36. R a t e 4 m m / s Rate\approx 4mm/s