wpmath0000014_8

Mineral_physics.html

  1. ( d P d T ) = γ D V C V , \left(\frac{dP}{dT}\right)=\frac{\gamma_{D}}{V}C_{V},
  2. C V C_{V}
  3. γ D \gamma_{D}
  4. v p v_{p}
  5. M ¯ \overline{M}
  6. ρ \rho
  7. v p = a M ¯ + b ρ v_{p}=a\overline{M}+b\rho

Minerals_Resource_Rent_Tax.html

  1. M R R T = A ( B - C ) - D - E MRRT=A(B-C)-D-E
  2. MRRT rate = 30 % ( 1 - Extraction factor ) \,\text{MRRT rate}=30\%(1-\,\text{Extraction factor})

MinHash.html

  1. A A
  2. B B
  3. J ( A , B ) = | A B | | A B | . J(A,B)={{|A\cap B|}\over{|A\cup B|}}.
  4. J ( A , B ) J(A,B)
  5. h h
  6. A A
  7. B B
  8. S S
  9. S S
  10. h h
  11. x x
  12. S S
  13. h ( x ) h(x)
  14. A A
  15. B B
  16. A B A∪B
  17. A B A∩B
  18. J ( A , B ) J(A,B)
  19. A A
  20. B B
  21. r r
  22. r r
  23. J ( A , B ) J(A,B)
  24. r r
  25. k k
  26. k k
  27. S S
  28. k k
  29. k k
  30. J ( A , B ) J(A,B)
  31. y y
  32. y / k y/k
  33. k k
  34. J ( A , B ) J(A,B)
  35. O ( 1 / k ) O(1/√k)
  36. ε > 0 ε>0
  37. ε ε
  38. J ( A , B ) J(A,B)
  39. h h
  40. k k
  41. S S
  42. k k
  43. h h
  44. k k
  45. S S
  46. h h
  47. S S
  48. A B A∪B
  49. X X
  50. A B A∪B
  51. X X
  52. A B A∩B
  53. Y Y
  54. k k
  55. J ( A , B ) J(A,B)
  56. X X
  57. k k
  58. k k
  59. O ( 1 / k ) O(1/√k)
  60. Y / k Y/k
  61. O ( k ) O(k)
  62. ε ε
  63. k k
  64. n n
  65. O ( n k ) O(nk)
  66. O ( n l o g k ) O(nlogk)
  67. h h
  68. n n
  69. n n
  70. n ! n!
  71. Ω ( n l o g n ) Ω(nlogn)
  72. n n
  73. l c m ( 1 , 2 , , n ) e n - o ( n ) lcm(1,2,...,n)\geq e^{n-o(n)}
  74. Ω ( n ) Ω(n)
  75. k k
  76. ε ε

Minimal_coupling.html

  1. q q
  2. A μ A_{\mu}
  3. p μ p_{\mu}
  4. p μ p μ - q A μ p_{\mu}\mapsto p_{\mu}-q\ A_{\mu}
  5. φ \varphi
  6. - g d 4 x \sqrt{-g}\,d^{4}x
  7. S = d 4 x g ( - 1 2 R + 1 2 μ φ μ φ - V ( φ ) ) S=\int d^{4}x\,\sqrt{g}\,\left(-\frac{1}{2}R+\frac{1}{2}\nabla_{\mu}\varphi% \nabla^{\mu}\varphi-V(\varphi)\right)
  8. g := det g μ ν g:=\det g_{\mu\nu}

Minimal_ideal.html

  1. soc ( R R ) \mathrm{soc}(R_{R})

Mironenko_reflecting_function.html

  1. F ( t , x ) \,F(t,x)
  2. x ( - t ) \,x(-t)
  3. x ( t ) \,x(t)
  4. x ( - t ) = F ( t , x ( t ) ) . \,x(-t)=F(t,x(t)).
  5. x ˙ = X ( t , x ) \dot{x}=X(t,x)
  6. φ ( t ; t 0 , x ) \varphi(t;t_{0},x)
  7. F ( t , x ) = φ ( - t ; t , x ) . F(t,x)=\varphi(-t;t,x).
  8. x ( - t ) \,x(-t)
  9. 2 ω \,2\omega
  10. t \,t
  11. F ( - ω , x ) \,F(-\omega,x)
  12. [ - ω ; ω ] \,[-\omega;\omega]
  13. x ˙ = X ( t , x ) . \dot{x}=X(t,x).
  14. ( ω , x 0 ) \,(\omega,x_{0})
  15. F ( t , x ) \,F(t,x)
  16. x ˙ = X ( t , x ) \dot{x}=X(t,x)
  17. F t + F x X + X ( - t , F ) = 0 , F ( 0 , x ) = x . \,F_{t}+F_{x}X+X(-t,F)=0,\qquad F(0,x)=x.

Mirsky's_theorem.html

  1. 1 + log 2 N 1+\lfloor\log_{2}N\rfloor
  2. 1 + log 2 N 1+\lfloor\log_{2}N\rfloor

Mittag-Leffler_polynomials.html

  1. ( 1 + t ) z ( 1 - t ) - z = n g n ( z ) t n . \displaystyle(1+t)^{z}(1-t)^{-z}=\sum_{n}g_{n}(z)t^{n}.

Mixed_volume.html

  1. f ( λ 1 , , λ r ) = Vol n ( λ 1 K 1 + + λ r K r ) , λ i 0 , f(\lambda_{1},\ldots,\lambda_{r})=\mathrm{Vol}_{n}(\lambda_{1}K_{1}+\cdots+% \lambda_{r}K_{r}),\qquad\lambda_{i}\geq 0,
  2. f ( λ 1 , , λ r ) = j 1 , , j n = 1 r V ( K j 1 , , K j n ) λ j 1 λ j n , f(\lambda_{1},\ldots,\lambda_{r})=\sum_{j_{1},\ldots,j_{n}=1}^{r}V(K_{j_{1}},% \ldots,K_{j_{n}})\lambda_{j_{1}}\cdots\lambda_{j_{n}},
  3. V ( T 1 , , T n ) = n λ 1 λ n | λ 1 = = λ n = + 0 Vol n ( λ 1 T 1 + + λ n T n ) . V(T_{1},\ldots,T_{n})=\left.\frac{\partial^{n}}{\partial\lambda_{1}\cdots% \partial\lambda_{n}}\right|_{\lambda_{1}=\cdots=\lambda_{n}=+0}\mathrm{Vol}_{n% }(\lambda_{1}T_{1}+\cdots+\lambda_{n}T_{n}).
  4. V ( T 1 , T 2 , T 3 , , T n ) V ( T 1 , T 1 , T 3 , , T n ) V ( T 2 , T 2 , T 3 , , T n ) . V(T_{1},T_{2},T_{3},\ldots,T_{n})\geq\sqrt{V(T_{1},T_{1},T_{3},\ldots,T_{n})V(% T_{2},T_{2},T_{3},\ldots,T_{n})}.
  5. W j ( K ) = V ( K , K , , K n - j times , B , B , , B j times ) W_{j}(K)=V(\overset{n-j\,\text{ times}}{\overbrace{K,K,\ldots,K}},\overset{j\,% \text{ times}}{\overbrace{B,B,\ldots,B}})
  6. Vol n ( K + t B ) = j = 0 n ( n j ) W j ( K ) t j . \mathrm{Vol}_{n}(K+tB)=\sum_{j=0}^{n}{\left({{n}\atop{j}}\right)}W_{j}(K)t^{j}.
  7. V j ( K ) = ( n j ) W n - j ( K ) κ n - j , V_{j}(K)={\left({{n}\atop{j}}\right)}\frac{W_{n-j}(K)}{\kappa_{n-j}},
  8. Vol n ( K + t B ) = j = 0 n V j ( K ) Vol n - j ( t B n - j ) . \mathrm{Vol}_{n}(K+tB)=\sum_{j=0}^{n}V_{j}(K)\,\mathrm{Vol}_{n-j}(tB_{n-j}).

MM_algorithm.html

  1. f ( θ ) f(\theta)
  2. m m
  3. m = 0 , 1... m=0,1...
  4. g ( θ | θ m ) g(\theta|\theta_{m})
  5. θ m \theta_{m}
  6. g ( θ | θ m ) g(\theta|\theta_{m})
  7. f ( θ ) f(\theta)
  8. θ \theta
  9. g ( θ m | θ m ) = f ( θ m ) g(\theta_{m}|\theta_{m})=f(\theta_{m})
  10. g ( θ | θ m ) g(\theta|\theta_{m})
  11. f ( θ ) f(\theta)
  12. θ m + 1 = arg max θ g ( θ | θ m ) \theta_{m+1}=\arg\max_{\theta}g(\theta|\theta_{m})
  13. f ( θ m ) f(\theta_{m})
  14. m m
  15. f ( θ m + 1 ) f(\theta_{m+1})
  16. g ( θ m + 1 | θ m ) g(\theta_{m+1}|\theta_{m})
  17. g ( θ m | θ m ) = f ( θ m ) g(\theta_{m}|\theta_{m})=f(\theta_{m})
  18. θ m \theta_{m}

MMH-Badger_MAC.html

  1. 1 / ( 2 32 - 5 ) 1/(2^{32}-5)
  2. 2 64 - 1 2^{64}-1
  3. u 32 u\cdot 32
  4. 1 u 5 1\leq u\leq 5
  5. u u
  6. 2 64 2^{64}
  7. H H^{\triangle}
  8. ( m , m b ) A × B (m,m_{b})\in A\times B
  9. h ( m , m b ) = H ( m ) + m b h(m,m_{b})=H^{\triangle}(m)+m_{b}
  10. m m m\neq m^{\prime}
  11. h ( m , m b ) = h ( m , m b ) h(m,m_{b})=h(m^{\prime},m^{\prime}_{b})
  12. H H^{\triangle}
  13. m = m m=m^{\prime}
  14. m b m b m_{b}\neq m_{b}^{\prime}
  15. N H K ( M ) = i = 1 2 ( k ( 2 i - 1 ) + w m ( 2 i - 1 ) ) × ( k 2 i + w m 2 i ) mod 2 2 w NH_{K}(M)=\sum_{i=1}^{\frac{\ell}{2}}(k_{(2i-1)}+_{w}m_{(2i-1)})\times(k_{2i}+% _{w}m_{2i})\mod 2^{2w}
  16. + w +_{w}
  17. 2 w 2^{w}
  18. m i , k i { 0 , , 2 w - 1 } m_{i},k_{i}\in\big\{0,\ldots,2^{w}-1\big\}
  19. 2 - w 2^{-w}
  20. 2 - w 2^{-w}
  21. N H K ( M ) = ( k 1 + w m 1 ) × ( k 2 + w m 2 ) mod 2 2 w NH_{K}(M)=(k_{1}+_{w}m_{1})\times(k_{2}+_{w}m_{2})\mod 2^{2w}
  22. 2 - 32 2^{-32}
  23. E N H k 1 , k 2 ( m 1 , m 2 , m 3 , m 4 ) = ( m 1 + 32 k 1 ) ( m 2 + 32 k 2 ) + 64 m 3 + 64 2 32 m 4 ENH_{k_{1},k_{2}}(m_{1},m_{2},m_{3},m_{4})=(m_{1}+_{32}k_{1})(m_{2}+_{32}k_{2}% )+_{64}m_{3}+_{64}2^{32}m_{4}
  24. = H * × F , \mathcal{H}=H^{*}\times F,
  25. ϵ H * \epsilon_{H^{*}}
  26. ϵ F \epsilon_{F}
  27. 2 64 - 1 2^{64}-1
  28. 2 - 32 2^{-32}
  29. 2 - 26.14 2^{-26.14}
  30. h h
  31. { 0 , 1 } 64 × { 0 , 1 } 128 { 0 , 1 } 64 \big\{0,1\big\}^{64}\times\big\{0,1\big\}^{128}\to\big\{0,1\big\}^{64}
  32. m 2 m 1 m_{2}\parallel m_{1}
  33. h ( k , m 2 , m 1 ) h(k,m_{2},m_{1})
  34. h ( k , m 2 , m 1 ) = ( L ( m 1 ) + 32 L ( k ) ) ( U ( m 1 ) + 32 U ( k ) ) + 64 m 2 h(k,m_{2},m_{1})=(L(m_{1})+_{32}L(k))\cdot(U(m_{1})+_{32}U(k))+_{64}m_{2}\,
  35. + n +_{n}
  36. 2 n 2^{n}
  37. k j i k_{j}^{i}
  38. k j i k_{j}^{i}
  39. 2 11 2^{11}
  40. 2 - 57.7 2^{-57.7}
  41. 2 15 2^{15}
  42. 2 - 56.6 2^{-56.6}
  43. 2 32 2^{32}
  44. 2 - 54.2 2^{-54.2}
  45. 2 61 - 1 2^{61}-1
  46. 2 - 52.2 2^{-52.2}
  47. p p
  48. F p F_{p}
  49. p p
  50. F p k F_{p}^{k}
  51. k k
  52. F p k F_{p}^{k}
  53. F p F_{p}
  54. MMH * = { g x : F p k F p | x F p k } \mathrm{MMH}^{*}=\{g_{x}:F_{p}^{k}\rightarrow F_{p}|x\in F_{p}^{k}\}\,
  55. g x g_{x}
  56. g x ( m ) \!g_{x}(m)
  57. m x mod p mx\mod p
  58. i = 1 n m i x i mod p \sum_{i=1}^{n}m_{i}\,x_{i}\mod p
  59. m m
  60. x x
  61. m = ( m 1 , , m k ) m=(m_{1},\ldots,m_{k})
  62. x = ( x 1 , , x k ) , x i , m i F p x=(x_{1},\ldots,x_{k}),x_{i},m_{i}\in\!F_{p}
  63. x x
  64. m m^{\prime}
  65. m m
  66. m 1 m 1 m_{1}\neq m^{\prime}_{1}
  67. g x ( m ) g_{x}(m)
  68. m m
  69. a F p a\in F_{p}
  70. m , m m,m^{\prime}
  71. m 1 m 1 m_{1}\neq m^{\prime}_{1}
  72. x 2 , x 3 , , x s x_{2},x_{3},\ldots,x_{s}
  73. Pr x 1 [ g x ( m ) - g x ( m ) a mod p ] \displaystyle{\Pr}_{x_{1}}[g_{x}(m)-g_{x}(m^{\prime})\equiv a\mod p]
  74. F p F_{p}
  75. p p
  76. F p = { 0 , 1 , , p - 1 } p F_{p}=\underbrace{\big\{0,1,\ldots,p-1\big\}}_{p}
  77. F p F_{p}
  78. 0 F p 0\in F_{p}
  79. x 1 = 0 x_{1}=0
  80. Pr x 1 F p ( x 1 = 0 ) = 1 p {\Pr}_{x_{1}\in\!{F_{p}}}(x_{1}=0)=\frac{1}{p}
  81. Pr ( x 1 , , x k ) F p k ( g x ( m ) g x ( m ) mod p ) {\Pr}_{(x_{1},\ldots,x_{k})\in\!{F_{p}^{k}}}(g_{x}(m)\equiv g_{x}(m^{\prime})% \mod p)
  82. Pr ( x 1 , , x k ) F p k ( g x ( m ) g x ( m ) mod p ) = ( x 2 , , x k ) F p k - 1 Pr ( x 2 , x k ) F p k - 1 ( x 2 = x 2 , , x k = x k ) Pr x 1 F p ( g x ( m ) g x ( m ) mod p ) = ( x 2 , , x k ) F p k - 1 1 p k - 1 1 p = p k - 1 1 p k - 1 1 p = 1 p \begin{aligned}\displaystyle{\Pr}_{(x_{1},\ldots,x_{k})\in\!{F_{p}^{k}}}(g_{x}% (m)\equiv g_{x}(m^{\prime})\mod p)&\displaystyle=\sum_{(x_{2},\ldots,x_{k})\in% \!{F_{p}^{k-1}}}{\Pr}_{(x_{2}^{^{\prime}}\cdots,x_{k}^{^{\prime}})\in\!{F_{p}^% {k-1}}}({x_{2}=x_{2}^{^{\prime}}},\ldots,{x_{k}=x_{k}^{^{\prime}}})\cdot{\Pr}_% {x_{1}\in\!F_{p}}(g_{x}(m)\equiv g_{x}(m^{\prime})\mod p)\\ &\displaystyle=\sum_{(x_{2},\ldots,x_{k})\in\!{F_{p}^{k-1}}}\frac{1}{p^{k-1}}% \cdot\frac{1}{p}\\ &\displaystyle=p^{k-1}\cdot\frac{1}{p^{k-1}}\cdot\frac{1}{p}\\ &\displaystyle=\frac{1}{p}\end{aligned}
  83. 1 p \frac{1}{p}
  84. p p
  85. n n
  86. n n
  87. 1 p n \frac{1}{p^{n}}
  88. n n
  89. n n
  90. M M H 32 * MMH^{*}_{32}
  91. p = 2 32 + 15 p=2^{32}+15
  92. 2 32 < p < 2 32 + 2 16 2^{32}<p<2^{32}+2^{16}
  93. 2 16 + 1 2^{16}+1
  94. 2 31 - 1 2^{31}-1
  95. MMH 32 * \mathrm{MMH}^{*}_{32}
  96. M M H 32 * = { g x ( { 0 , 1 } 32 ) k } F p , MMH^{*}_{32}=\big\{g_{x}(\big\{0,1\big\}^{32})^{k}\big\}\to F_{p},
  97. { 0 , 1 } 32 \big\{0,1\big\}^{32}
  98. { 0 , 1 , , 2 32 - 1 } \big\{0,1,\ldots,2^{32}-1\big\}
  99. g x g_{x}
  100. g x ( m ) \displaystyle g_{x}(m)
  101. x = ( x 1 , , x k ) x=(x_{1},\ldots,x_{k})
  102. m = ( m , , m k ) m=(m,\ldots,m_{k})
  103. 2 - 32 2^{-32}
  104. M M H 32 * MMH^{*}_{32}
  105. 2 - 32 2^{-32}
  106. 1 / p 1/p
  107. p 2 k p\approx 2^{k}

Mnemonics_in_trigonometry.html

  1. 1 sin A = csc A or 1 csc A = sin A 1 tan A = cot A or 1 cot A = tan A 1 sec A = cos A or 1 cos A = sec A \begin{array}[]{ccc}{1\over\sin A}=\csc A&\,\text{or}&{1\over\csc A}=\sin A\\ \\ {1\over\tan A}=\cot A&\,\text{or}&{1\over\cot A}=\tan A\\ \\ {1\over\sec A}=\cos A&\,\text{or}&{1\over\cos A}=\sec A\end{array}
  2. sin 2 A + cos 2 A = 1 \sin^{2}A+\cos^{2}A=1
  3. 1 + cot 2 A = csc 2 A 1+\cot^{2}A=\csc^{2}A
  4. tan 2 A + 1 = sec 2 A \tan^{2}A+1=\sec^{2}A
  5. tan A = sin A cos A \tan A={\sin A\over\cos A}
  6. sin A = cos A cot A \sin A={\cos A\over\cot A}
  7. cos A = cot A csc A \cos A={\cot A\over\csc A}
  8. cot A = csc A sec A \cot A={\csc A\over\sec A}
  9. csc A = sec A tan A \csc A={\sec A\over\tan A}
  10. sec A = tan A sin A \sec A={\tan A\over\sin A}
  11. tan A = sec A csc A \tan A={\sec A\over\csc A}
  12. sec A = csc A cot A \sec A={\csc A\over\cot A}
  13. csc A = cot A cos A \csc A={\cot A\over\cos A}
  14. cot A = cos A sin A \cot A={\cos A\over\sin A}
  15. cos A = sin A tan A \cos A={\sin A\over\tan A}
  16. sin A = tan A sec A \sin A={\tan A\over\sec A}
  17. tan A = sin A sec A \tan A=\sin A\cdot\sec A
  18. sin A = cos A tan A \sin A=\cos A\cdot\tan A
  19. cos A = sin A cot A \cos A=\sin A\cdot\cot A
  20. cot A = cos A csc A \cot A=\cos A\cdot\csc A
  21. csc A = cot A sec A \csc A=\cot A\cdot\sec A
  22. sec A = csc A tan A \sec A=\csc A\cdot\tan A

Modern_searches_for_Lorentz_violation.html

  1. κ ~ e - \tilde{\kappa}_{e-}
  2. κ ~ o + \tilde{\kappa}_{o+}
  3. κ ~ t r \tilde{\kappa}_{tr}
  4. κ ~ e - \tilde{\kappa}_{e-}
  5. κ ~ o + \tilde{\kappa}_{o+}
  6. κ ~ t r \tilde{\kappa}_{tr}
  7. κ ~ o + \tilde{\kappa}_{o+}
  8. κ ~ e - \tilde{\kappa}_{e-}
  9. κ ~ o + \tilde{\kappa}_{o+}
  10. κ ~ t r \tilde{\kappa}_{tr}
  11. 0.7 ± 1 × 10 - 14 \scriptstyle 0.7\pm 1\times 10^{-14}
  12. - 0.4 ± 0.9 × 10 - 10 \scriptstyle-0.4\pm 0.9\times 10^{-10}
  13. 3 ± 11 × 10 - 10 \scriptstyle 3\pm 11\times 10^{-10}
  14. 0.7 ± 1.4 × 10 - 12 \scriptstyle 0.7\pm 1.4\times 10^{-12}
  15. 3.4 ± 6.2 × 10 - 9 \scriptstyle 3.4\pm 6.2\times 10^{-9}
  16. 0.8 ( 0.6 ) × 10 - 16 \scriptstyle 0.8(0.6)\times 10^{-16}
  17. - 1.5 ( 1.2 ) × 10 - 12 \scriptstyle-1.5(1.2)\times 10^{-12}
  18. - 1.5 ( 0.74 ) × 10 - 8 \scriptstyle-1.5(0.74)\times 10^{-8}
  19. 1.6 × 10 - 14 \scriptstyle\leq 1.6\times 10^{-14}
  20. ( 4 ± 8 ) × 10 - 12 \scriptstyle(4\pm 8)\times 10^{-12}
  21. ( - 0.31 ± 0.73 ) × 10 - 17 \scriptstyle(-0.31\pm 0.73)\times 10^{-17}
  22. ( - 0.14 ± 0.78 ) × 10 - 13 \scriptstyle(-0.14\pm 0.78)\times 10^{-13}
  23. ( - 1.6 ± 6 ± 1.2 ) × 10 - 12 \scriptstyle(-1.6\pm 6\pm 1.2)\times 10^{-12}
  24. ( 0.0 ± 1.0 ± 0.3 ) × 10 - 17 \scriptstyle(0.0\pm 1.0\pm 0.3)\times 10^{-17}
  25. ( 1.5 ± 1.5 ± 0.2 ) × 10 - 13 \scriptstyle(1.5\pm 1.5\pm 0.2)\times 10^{-13}
  26. - 4 , 8 ( 3 , 7 ) × 10 - 8 \scriptstyle-4,8(3,7)\times 10^{-8}
  27. - 0.3 ± 3 × 10 - 7 \scriptstyle-0.3\pm 3\times 10^{-7}
  28. ( 7.7 ( 4.0 ) ) × 10 - 16 \scriptstyle(7.7(4.0))\times 10^{-16}
  29. ( 1.7 ( 2.0 ) ) × 10 - 12 \scriptstyle(1.7(2.0))\times 10^{-12}
  30. 3 × 10 - 8 \scriptstyle\lesssim 3\times 10^{-8}
  31. 9.4 ( 8.1 ) × 10 - 11 \scriptstyle 9.4(8.1)\times 10^{-11}
  32. ( - 6.9 ( 2.2 ) ) × 10 - 16 \scriptstyle(-6.9(2.2))\times 10^{-16}
  33. ( - 0.9 ( 2.6 ) ) × 10 - 12 \scriptstyle(-0.9(2.6))\times 10^{-12}
  34. ( - 2.1 ± 1.9 ) × 10 - 10 \scriptstyle(-2.1\pm 1.9)\times 10^{-10}
  35. ( - 3.1 ( 2.5 ) ) × 10 - 16 \scriptstyle(-3.1(2.5))\times 10^{-16}
  36. ( - 2.5 ( 5.1 ) ) × 10 - 12 \scriptstyle(-2.5(5.1))\times 10^{-12}
  37. - 0.9 ( 2.0 ) × 10 - 10 \scriptstyle-0.9(2.0)\times 10^{-10}
  38. ( - 0.63 ( 0.43 ) ) × 10 - 15 \scriptstyle(-0.63(0.43))\times 10^{-15}
  39. ( 0.20 ( 0.21 ) ) × 10 - 11 \scriptstyle(0.20(0.21))\times 10^{-11}
  40. ( + 0.5 ± 3 ± 0.7 ) × 10 - 10 \scriptstyle(+0.5\pm 3\pm 0.7)\times 10^{-10}
  41. ( - 2 ± 0 , 2 ) × 10 - 14 \scriptstyle(-2\pm 0{,}2)\times 10^{-14}
  42. ( - 5.7 ± 2.3 ) × 10 - 15 \scriptstyle(-5.7\pm 2.3)\times 10^{-15}
  43. ( - 1.8 ± 1.5 ) × 10 - 11 \scriptstyle(-1.8\pm 1.5)\times 10^{-11}
  44. ( + 1.2 ± 2.2 ) × 10 - 9 \scriptstyle(+1.2\pm 2.2)\times 10^{-9}
  45. ( 3.7 ± 3.0 ) × 10 - 7 \scriptstyle(3.7\pm 3.0)\times 10^{-7}
  46. ( + 2.2 ± 1.5 ) × 10 - 9 \scriptstyle(+2.2\pm 1.5)\times 10^{-9}
  47. ( 1.7 ± 2.6 ) × 10 - 15 \scriptstyle(1.7\pm 2.6)\times 10^{-15}
  48. ( 14 ± 14 ) × 10 - 11 \scriptstyle(14\pm 14)\times 10^{-11}
  49. ( 1.4 ± 1.4 × 10 - 13 \scriptstyle(1.4\pm 1.4\times 10^{-13}
  50. 10 - 9 \scriptstyle\leq 10^{-9}
  51. ( + 1.5 ± 4.2 ) × 10 - 9 \scriptstyle(+1.5\pm 4.2)\times 10^{-9}
  52. ( 1.9 ± 2.1 ) × 10 - 5 \scriptstyle(1.9\pm 2.1)\times 10^{-5}
  53. 6.6 × 10 - 5 \scriptstyle 6.6\times 10^{-5}
  54. 5 × 10 - 9 \scriptstyle\lesssim 5\times 10^{-9}
  55. 10 - 15 \scriptstyle\lesssim 10^{-15}
  56. ( - 5 ± 12 ) × 10 - 5 \scriptstyle(-5\pm 12)\times 10^{-5}
  57. E Pl 1.22 × 10 19 \scriptstyle E_{\mathrm{Pl}}\sim 1.22\times 10^{19}
  58. > 7.6 × E Pl \scriptstyle>7.6\times E_{\mathrm{Pl}}
  59. > 3.42 × E Pl \scriptstyle>3.42\times E_{\mathrm{Pl}}
  60. > 1.19 × E Pl \scriptstyle>1.19\times E_{\mathrm{Pl}}
  61. 7.2 × 10 17 \scriptstyle\geq 7.2\times 10^{17}
  62. 0.21 × 10 18 \scriptstyle\geq 0.21\times 10^{18}
  63. 1.4 × 10 16 \scriptstyle\geq 1.4\times 10^{16}
  64. 3.2 × 10 11 \scriptstyle\geq 3.2\times 10^{11}
  65. 0.66 × 10 17 \scriptstyle\geq 0.66\times 10^{17}
  66. 1.8 × 10 17 \scriptstyle\geq 1.8\times 10^{17}
  67. 6.9 × 10 15 \scriptstyle\geq 6.9\times 10^{15}
  68. 10 15 \scriptstyle\geq 10^{15}
  69. > 1.8 × 10 15 \scriptstyle>1.8\times 10^{15}
  70. 2.7 × 10 16 \scriptstyle\geq 2.7\times 10^{16}
  71. > 4 × 10 16 \scriptstyle>4\times 10^{16}
  72. k ( V ) 00 ( 3 ) \scriptstyle k_{(V)00}^{(3)}
  73. k ( V ) 00 ( 5 ) \scriptstyle k_{(V)00}^{(5)}
  74. ξ \xi
  75. k ( V ) 00 ( 5 ) = 3 4 π ξ 5 m P {\scriptstyle k_{(V)00}^{(5)}=\frac{3\sqrt{4\pi}\xi}{5m_{\mathrm{P}}}}
  76. m P m_{P}
  77. ξ \xi
  78. k ( V ) 00 ( 3 ) \scriptstyle k_{(V)00}^{(3)}
  79. k ( V ) 00 ( 5 ) \scriptstyle k_{(V)00}^{(5)}
  80. 5.9 × 10 - 35 \scriptstyle\leq 5.9\times 10^{-35}
  81. 3.4 × 10 - 16 \scriptstyle\leq 3.4\times 10^{-16}
  82. 1.4 × 10 - 34 \scriptstyle\leq 1.4\times 10^{-34}
  83. 8 × 10 - 16 \scriptstyle\leq 8\times 10^{-16}
  84. 1.9 × 10 - 33 \scriptstyle\leq 1.9\times 10^{-33}
  85. 1.1 × 10 - 14 \scriptstyle\leq 1.1\times 10^{-14}
  86. 4.2 × 10 - 34 \scriptstyle\leq 4.2\times 10^{-34}
  87. 2.4 × 10 - 15 \scriptstyle\leq 2.4\times 10^{-15}
  88. 1 × 10 - 32 \scriptstyle\leq 1\times 10^{-32}
  89. 9 × 10 - 14 \scriptstyle\leq 9\times 10^{-14}
  90. 2 × 10 - 43 \scriptstyle\leq 2\times 10^{-43}
  91. = ( 2.3 ± 5.4 ) × 10 - 43 \scriptstyle=(2.3\pm 5.4)\times 10^{-43}
  92. 1.5 × 10 - 28 \scriptstyle\leq 1.5\times 10^{-28}
  93. 9 × 10 - 10 \scriptstyle\leq 9\times 10^{-10}
  94. = ( 1.2 ± 2.2 ) × 10 - 43 \scriptstyle=(1.2\pm 2.2)\times 10^{-43}
  95. 2.5 × 10 - 43 \scriptstyle\leq 2.5\times 10^{-43}
  96. = ( 2.6 ± 1.9 ) × 10 - 43 \scriptstyle=(2.6\pm 1.9)\times 10^{-43}
  97. = ( 2.5 ± 3.0 ) × 10 - 43 \scriptstyle=(2.5\pm 3.0)\times 10^{-43}
  98. 3.4 × 10 - 26 \scriptstyle\leq 3.4\times 10^{-26}
  99. 2 × 10 - 7 \scriptstyle\leq 2\times 10^{-7}
  100. = ( 6.0 ± 4.0 ) × 10 - 43 \scriptstyle=(6.0\pm 4.0)\times 10^{-43}
  101. 8.7 × 10 - 23 \scriptstyle\leq 8.7\times 10^{-23}
  102. 4 × 10 - 4 \scriptstyle\leq 4\times 10^{-4}
  103. 2 × 10 - 42 \scriptstyle\leq 2\times 10^{-42}
  104. 5 × 10 - 21 \scriptstyle\leq 5\times 10^{-21}
  105. 4.5 × 10 - 23 \scriptstyle\leq 4.5\times 10^{-23}
  106. 5 × 10 - 15 \scriptstyle\leq 5\times 10^{-15}
  107. - 5.8 × 10 - 12 \scriptstyle\leq-5.8\times 10^{-12}
  108. 1.2 × 10 - 11 \scriptstyle\leq 1.2\times 10^{-11}
  109. 3 × 10 - 23 \scriptstyle\leq 3\times 10^{-23}
  110. - 9 × 10 - 16 \scriptstyle\leq-9\times 10^{-16}
  111. 6 × 10 - 20 \scriptstyle\leq 6\times 10^{-20}
  112. 2 × 10 - 19 \scriptstyle\leq 2\times 10^{-19}
  113. 10 - 17 \scriptstyle\leq 10^{-17}
  114. 6 × 10 - 20 \scriptstyle\leq 6\times 10^{-20}
  115. - 2 × 10 - 21 \scriptstyle\leq-2\times 10^{-21}
  116. 5 × 10 - 24 \scriptstyle\leq 5\times 10^{-24}
  117. - 2 × 10 - 16 \scriptstyle\leq-2\times 10^{-16}
  118. 5 × 10 - 20 \scriptstyle\leq 5\times 10^{-20}
  119. - 1.5 × 10 - 15 \scriptstyle\leq-1.5\times 10^{-15}
  120. 5 × 10 - 23 \scriptstyle\leq 5\times 10^{-23}
  121. < 6.7 × 10 - 34 \scriptstyle<6.7\times 10^{-34}
  122. ( - 9.0 ± 11 ) × 10 - 17 \scriptstyle(-9.0\pm 11)\times 10^{-17}
  123. < 4 × 10 - 30 \scriptstyle<4\times 10^{-30}
  124. < 3.7 × 10 - 31 \scriptstyle<3.7\times 10^{-31}
  125. ( 4.8 ± 4.4 ) × 10 - 32 \scriptstyle(4.8\pm 4.4)\times 10^{-32}
  126. < 3.7 × 10 - 32 \scriptstyle<3.7\times 10^{-32}
  127. < 6 × 10 - 32 \scriptstyle<6\times 10^{-32}
  128. < 3.7 × 10 - 33 \scriptstyle<3.7\times 10^{-33}
  129. < 2 × 10 - 29 \scriptstyle<2\times 10^{-29}
  130. ( 4.0 ± 3.3 ) × 10 - 31 \scriptstyle(4.0\pm 3.3)\times 10^{-31}
  131. ( - 1.8 ( 2.8 ) ) × 10 - 25 \scriptstyle(-1.8(2.8))\times 10^{-25}
  132. ( 8.0 ± 9.5 ) × 10 - 32 \scriptstyle(8.0\pm 9.5)\times 10^{-32}
  133. < 5 × 10 - 30 \scriptstyle<5\times 10^{-30}
  134. < 2 × 10 - 27 \scriptstyle<2\times 10^{-27}
  135. ( 1.8 ± 5.3 ) × 10 - 30 \scriptstyle(1.8\pm 5.3)\times 10^{-30}
  136. < 2 × 10 - 27 \scriptstyle<2\times 10^{-27}
  137. ( 4.0 ± 3.3 ) × 10 - 31 \scriptstyle(4.0\pm 3.3)\times 10^{-31}
  138. κ ~ t r \tilde{\kappa}_{tr}
  139. 1.3 × 10 - 6 \scriptstyle\leq 1.3\times 10^{-6}
  140. 1.2 × 10 - 5 \scriptstyle\leq 1.2\times 10^{-5}
  141. 8.4 × 10 - 8 \scriptstyle\leq 8.4\times 10^{-8}
  142. 2.2 × 10 - 7 \scriptstyle\leq 2.2\times 10^{-7}
  143. 1 × 10 - 6 \scriptstyle\leq 1\times 10^{-6}
  144. 2.7 × 10 - 4 \scriptstyle\leq 2.7\times 10^{-4}
  145. ( - 3.0 ± 2.4 ) × 10 - 15 \scriptstyle\leq(-3.0\pm 2.4)\times 10^{-15}
  146. 10 14 ( eV ) - 1 \scriptstyle\leq 10^{14}(\,\text{eV})^{-1}
  147. α 1 , α 2 , α 3 \alpha_{1},\alpha_{2},\alpha_{3}
  148. 10 - 9 \scriptstyle 10^{-9}
  149. 10 - 13 \scriptstyle 10^{-13}
  150. ( 6.9 ± 4.5 ) × 10 - 11 \scriptstyle(6.9\pm 4.5)\times 10^{-11}
  151. 10 - 9 \scriptstyle 10^{-9}
  152. 10 - 10 \scriptstyle 10^{-10}
  153. 10 - 20 \scriptstyle\leq 10^{-20}
  154. 10 - 23 \scriptstyle\leq 10^{-23}
  155. 10 - 20 \scriptstyle\leq 10^{-20}
  156. 10 - 23 \scriptstyle\leq 10^{-23}
  157. 10 - 23 \scriptstyle\leq 10^{-23}
  158. 10 - 20 \scriptstyle\leq 10^{-20}
  159. 10 - 19 \scriptstyle\leq 10^{-19}
  160. | v - c | / c < 10 - 9 \scriptstyle|v-c|/c<10^{-9}
  161. < 6 × 10 - 22 \scriptstyle<6\times 10^{-22}
  162. < 5.6 × 10 - 19 \scriptstyle<5.6\times 10^{-19}
  163. < 10 - 18 \scriptstyle<10^{-18}
  164. < 10 - 13 \scriptstyle<10^{-13}
  165. < 7.8 × 10 - 12 \scriptstyle<7.8\times 10^{-12}
  166. < 2.5 × 10 - 8 \scriptstyle<2.5\times 10^{-8}
  167. < 10 - 12 \scriptstyle<10^{-12}
  168. < 10 - 12 \scriptstyle<10^{-12}
  169. < 1.7 × 10 - 11 \scriptstyle<1.7\times 10^{-11}
  170. κ ~ o + \scriptstyle{\tilde{\kappa}_{o+}}

Modigliani_risk-adjusted_performance.html

  1. D t D_{t}
  2. t t
  3. D t R P t - R F t D_{t}\equiv R_{P_{t}}-R_{F_{t}}
  4. R P t R_{P_{t}}
  5. t t
  6. R F t R_{F_{t}}
  7. t t
  8. S S
  9. S D ¯ σ D S\equiv\frac{\overline{D}}{\sigma_{D}}
  10. D ¯ \overline{D}
  11. σ D \sigma_{D}
  12. M 2 S × σ B + R F ¯ M^{2}\equiv S\times\sigma_{B}+\overline{R_{F}}
  13. S S
  14. σ B \sigma_{B}
  15. R F ¯ \overline{R_{F}}
  16. S S
  17. M 2 D ¯ × σ B σ D + R F ¯ M^{2}\equiv\overline{D}\times\frac{\sigma_{B}}{\sigma_{D}}+\overline{R_{F}}
  18. M 2 M^{2}
  19. M 2 a l p h a S × σ B M^{2}alpha\equiv S\times\sigma_{B}
  20. M 2 a l p h a D ¯ × σ B σ D M^{2}alpha\equiv\overline{D}\times\frac{\sigma_{B}}{\sigma_{D}}
  21. σ B σ D \frac{\sigma_{B}}{\sigma_{D}}
  22. σ D \sigma_{D}
  23. σ B \sigma_{B}
  24. M β 2 D ¯ × β B β D + R F ¯ M^{2}_{\beta}\equiv\overline{D}\times\frac{\beta_{B}}{\beta_{D}}+\overline{R_{% F}}

Modular_decomposition.html

  1. v X v\not\in X
  2. G = ( V , E ) G=(V,E)
  3. M V M\subseteq V
  4. G G
  5. M M
  6. V \ M V\backslash M
  7. u , v M , x V \ M \forall u,v\in M,\forall x\in V\backslash M
  8. x x
  9. u u
  10. v v
  11. x x
  12. u u
  13. v v
  14. M M
  15. u , v M , N ( u ) M = N ( v ) M \forall u,v\in M,N(u)\setminus M=N(v)\setminus M
  16. \emptyset
  17. V V
  18. { v } \{v\}
  19. v V v\in V
  20. G G
  21. G G
  22. M M
  23. G G
  24. G G
  25. M \forall M^{\prime}
  26. G G
  27. M M = M\cap M^{\prime}=\emptyset
  28. M M M\subseteq M^{\prime}
  29. M M M^{\prime}\subseteq M
  30. X X
  31. Y Y
  32. X X
  33. Y Y
  34. X X
  35. Y Y
  36. V V
  37. P P
  38. G / P G/P
  39. P P
  40. G / P G/P
  41. G / P G/P
  42. { { x } | x V } \{\{x\}|x\in V\}
  43. G / { V } G/\{V\}
  44. G / { { x } | x V } G/\{\{x\}|x\in V\}
  45. X X
  46. X X
  47. V \ X V\backslash X
  48. V V
  49. P P
  50. Θ ( n 2 ) \Theta(n^{2})
  51. u Y u\in Y
  52. v Z v\in Z

Modular_invariant_theory.html

  1. | x 1 x 1 q x 1 q 2 x 2 x 2 q x 2 q 2 x 3 x 3 q x 3 q 2 | \begin{vmatrix}x_{1}&x_{1}^{q}&x_{1}^{q^{2}}\\ x_{2}&x_{2}^{q}&x_{2}^{q^{2}}\\ x_{3}&x_{3}^{q}&x_{3}^{q^{2}}\end{vmatrix}

Modular_lambda_function.html

  1. / 1 , τ \mathbb{C}/\langle 1,\tau\rangle
  2. q = e π i τ q=e^{\pi i\tau}
  3. λ ( τ ) = 16 q - 128 q 2 + 704 q 3 - 3072 q 4 + 11488 q 5 - 38400 q 6 + \lambda(\tau)=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots
  4. S L 2 ( ) SL_{2}(\mathbb{Z})
  5. λ ( τ ) \lambda(\tau)
  6. τ τ + 2 ; τ τ 1 - 2 τ . \tau\mapsto\tau+2\ ;\ \tau\mapsto\frac{\tau}{1-2\tau}\ .
  7. τ τ + 1 : λ λ λ - 1 ; \tau\mapsto\tau+1\ :\ \lambda\mapsto\frac{\lambda}{\lambda-1}\,;
  8. τ - 1 τ : λ 1 - λ . \tau\mapsto-\frac{1}{\tau}\ :\ \lambda\mapsto 1-\lambda\ .
  9. λ ( τ ) \lambda(\tau)
  10. { λ , 1 1 - λ , λ - 1 λ , 1 λ , λ λ - 1 , 1 - λ } . \left\{{\lambda,\frac{1}{1-\lambda},\frac{\lambda-1}{\lambda},\frac{1}{\lambda% },\frac{\lambda}{\lambda-1},1-\lambda}\right\}\ .
  11. λ ( τ ) = k 2 ( τ ) \lambda(\tau)=k^{2}(\tau)
  12. η ( τ ) \eta(\tau)
  13. λ ( τ ) = θ 2 4 ( 0 , τ ) θ 3 4 ( 0 , τ ) = [ 2 η ( τ 2 ) η 2 ( 2 τ ) η 3 ( τ ) ] 8 \lambda(\tau)=\frac{\theta_{2}^{4}(0,\tau)}{\theta_{3}^{4}(0,\tau)}=\left[% \frac{\sqrt{2}\,\eta(\tfrac{\tau}{2})\eta^{2}(2\tau)}{\eta^{3}(\tau)}\right]^{8}
  14. θ 2 ( 0 , τ ) = n = - q ( n + 1 2 ) 2 and θ 3 ( 0 , τ ) = n = - q n 2 \theta_{2}(0,\tau)=\sum_{n=-\infty}^{\infty}q^{\left({n+\frac{1}{2}}\right)^{2% }}\mathrm{and}\ \theta_{3}(0,\tau)=\sum_{n=-\infty}^{\infty}q^{n^{2}}
  15. q = e π i τ q=e^{\pi i\tau}
  16. [ ω 1 , ω 2 ] [\omega_{1},\omega_{2}]
  17. τ = ω 2 ω 1 \tau=\frac{\omega_{2}}{\omega_{1}}
  18. e 1 = ( ω 1 2 ) , e 2 = ( ω 2 2 ) , e 3 = ( ω 1 + ω 2 2 ) e_{1}=\wp\left(\frac{\omega_{1}}{2}\right),e_{2}=\wp\left(\frac{\omega_{2}}{2}% \right),e_{3}=\wp\left(\frac{\omega_{1}+\omega_{2}}{2}\right)
  19. λ = e 3 - e 2 e 1 - e 2 . \lambda=\frac{e_{3}-e_{2}}{e_{1}-e_{2}}\,.
  20. j ( τ ) = 256 ( 1 - λ + λ 2 ) 3 λ 2 ( 1 - λ ) 2 . j(\tau)=\frac{256(1-\lambda+\lambda^{2})^{3}}{\lambda^{2}(1-\lambda)^{2}}\ .
  21. y 2 = x ( x - 1 ) ( x - λ ) y^{2}=x(x-1)(x-\lambda)
  22. 16 λ ( 2 τ ) - 8 \frac{16}{\lambda(2\tau)}-8
  23. Γ 0 ( 4 ) \Gamma_{0}(4)
  24. q - 1 + 20 q - 62 q 3 + q^{-1}+20q-62q^{3}+\dots

Modulation_space.html

  1. 1 p , q 1\leq p,q\leq\infty
  2. m ( x , ω ) m(x,\omega)
  3. 2 d \mathbb{R}^{2d}
  4. g 𝒮 ( d ) g\in\mathcal{S}(\mathbb{R}^{d})
  5. M m p , q ( d ) M^{p,q}_{m}(\mathbb{R}^{d})
  6. M m p , q ( d ) = { f 𝒮 ( d ) : ( d ( d | V g f ( x , ω ) | p m ( x , ω ) p d x ) q / p d ω ) 1 / q < } . M^{p,q}_{m}(\mathbb{R}^{d})=\left\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})\ :% \ \left(\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}|V_{g}f(x,\omega)|^{p}% m(x,\omega)^{p}dx\right)^{q/p}d\omega\right)^{1/q}<\infty\right\}.
  7. V g f V_{g}f
  8. f f
  9. g g
  10. ( x , ω ) (x,\omega)
  11. V g f ( x , ω ) = d f ( t ) g ( t - x ) ¯ e - 2 π i t ω d t = ξ - 1 ( g ^ ( ξ ) ¯ f ^ ( ξ + ω ) ) ( x ) . V_{g}f(x,\omega)=\int_{\mathbb{R}^{d}}f(t)\overline{g(t-x)}e^{-2\pi it\cdot% \omega}dt=\mathcal{F}^{-1}_{\xi}(\overline{\hat{g}(\xi)}\hat{f}(\xi+\omega))(x).
  12. f M m p , q ( d ) f\in M^{p,q}_{m}(\mathbb{R}^{d})
  13. V g f L m p , q ( 2 d ) V_{g}f\in L^{p,q}_{m}(\mathbb{R}^{2d})
  14. M m p , q ( d ) M^{p,q}_{m}(\mathbb{R}^{d})
  15. g 𝒮 ( d ) g\in\mathcal{S}(\mathbb{R}^{d})
  16. M p , q s ( d ) = { f 𝒮 ( d ) : ( k d k s q ψ k ( D ) f p q ) 1 / q < } , x := | x | + 1 M^{s}_{p,q}(\mathbb{R}^{d})=\left\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})\ :% \ \left(\sum_{k\in\mathbb{Z}^{d}}\langle k\rangle^{sq}\|\psi_{k}(D)f\|_{p}^{q}% \right)^{1/q}<\infty\right\},\langle x\rangle:=|x|+1
  17. { ψ k } \{\psi_{k}\}
  18. m ( x , ω ) = ω s m(x,\omega)=\langle\omega\rangle^{s}
  19. M p , q s = M m p , q M^{s}_{p,q}=M^{p,q}_{m}
  20. p = q = 1 p=q=1
  21. m ( x , ω ) = 1 m(x,\omega)=1
  22. M m 1 , 1 ( d ) = M 1 ( d ) M^{1,1}_{m}(\mathbb{R}^{d})=M^{1}(\mathbb{R}^{d})
  23. S 0 S_{0}
  24. M 1 ( d ) M^{1}(\mathbb{R}^{d})
  25. L 1 ( d ) C 0 ( d ) L^{1}(\mathbb{R}^{d})\cap C_{0}(\mathbb{R}^{d})
  26. M 1 ( d ) M^{1}(\mathbb{R}^{d})
  27. \mathcal{F}
  28. M 1 , 1 M^{1,1}

Moens–Korteweg_equation.html

  1. P W V = E inc h 2 r ρ PWV=\sqrt{\dfrac{E\text{inc}\cdot h}{2r\rho}}

Moist_static_energy.html

  1. S = C p T + g z + L v q S=C_{p}\cdot T+g\cdot z+L_{v}\cdot q

Mojette_Transform.html

  1. p r o j ( b , p , q ) = k l f ( k , l ) Δ ( b + k q - p l ) proj(b,p,q)=\sum_{k}\sum_{l}f(k,l)\Delta(b+kq-pl)
  2. Δ ( b ) = 1 \Delta(b)=1
  3. i f b = 0 if\,\ b=0
  4. Δ ( b ) = 0 \Delta(b)=0
  5. o t h e r w i s e otherwise

Moment_curve.html

  1. ( x , x 2 , x 3 , , x d ) . \left(x,x^{2},x^{3},\dots,x^{d}\right).
  2. Ω ( n d / 2 ) \Omega(n^{\lceil d/2\rceil})

Momentum-depth_relationship_in_a_rectangular_channel.html

  1. M 1 + M 2 = F w + F f + F P 1 + F P 2 M_{1}+M_{2}=F_{w}+F_{f}+F_{P1}+F_{P2}
  2. M 1 x + M 2 x = F P 1 x + F P 2 x M_{1x}+M_{2x}=F_{P1x}+F_{P2x}
  3. M 1 x = m ˙ V 1 x = - ρ Q V 1 a n d F P 1 x = P ¯ 1 A 1 M_{1x}=\dot{m}V_{1x}=-\rho QV_{1}\quad and\quad F_{P1x}=\overline{P}_{1}A_{1}
  4. M 2 x = m ˙ V 2 x = - ρ Q V 2 a n d F P 2 x = P ¯ 2 A 2 M_{2x}=\dot{m}V_{2x}=-\rho QV_{2}\quad and\quad F_{P2x}=\overline{P}_{2}A_{2}
  5. - ρ Q V 1 + ρ Q V 2 = P ¯ 1 A 1 - P ¯ 2 A 2 -\rho QV_{1}+\rho QV_{2}=\overline{P}_{1}A_{1}-\overline{P}_{2}A_{2}
  6. m ˙ \dot{m}
  7. P ¯ \overline{P}
  8. P ¯ = 1 2 ρ g y \overline{P}={1\over 2}\rho gy
  9. Q = V 1 A 1 = V 2 A 2 Q=V_{1}A_{1}=V_{2}A_{2}
  10. q = V 1 y 1 = V 2 y 2 q=V_{1}y_{1}=V_{2}y_{2}
  11. V 1 = q / y 1 a n d V 2 = q / y 2 V_{1}=q/y_{1}\quad and\quad V_{2}=q/y_{2}
  12. - ρ q ( q y 1 ) + ρ q ( q y 2 ) = ( 1 2 ρ g y 1 ) y 1 - ( 1 2 ρ g y 2 ) y 2 -{\rho q\left({q\over{y_{1}}}\right)}+{\rho q\left({q\over{y_{2}}}\right)}=% \left({1\over 2}\rho g{y_{1}}\right){y_{1}}-{\left({1\over 2}\rho g{y_{2}}% \right){y_{2}}}
  13. - q 2 g y 1 + q 2 g y 2 = y 1 2 2 - y 2 2 2 -{{q^{2}\over g{y_{1}}}}+{q^{2}\over g{y_{2}}}={{y_{1}}^{2}\over 2}-{{y_{2}}^{% 2}\over 2}
  14. y 1 2 2 + q 2 g y 1 = y 2 2 2 + q 2 g y 2 {{y_{1}}^{2}\over 2}+{q^{2}\over g{y_{1}}}={{y_{2}}^{2}\over 2}+{q^{2}\over g{% y_{2}}}
  15. M u n i t = y 2 2 + q 2 g y M_{unit}={y^{2}\over 2}+{q^{2}\over gy}
  16. M u n i t = q 2 g y + y 2 2 M_{unit}={q^{2}\over gy}+{y^{2}\over 2}
  17. V V
  18. g y \sqrt{gy}
  19. y c = ( q 2 g ) 1 3 y_{c}=\left({q^{2}\over g}\right)^{1\over 3}
  20. ( g y ) \left({\sqrt{gy}}\right)
  21. F r = V g ( A B ) = V g y F_{r}={{V}\over{\sqrt{g\left({A\over B}\right)}}}={{V}\over{\sqrt{gy}}}
  22. y 2 = y 1 2 ( - 1 + ( 1 + 8 F r 1 2 ) ) o r y 1 = y 2 2 ( - 1 + ( 1 + 8 F r 2 2 ) ) y_{2}={y_{1}\over 2}\left(-1+\sqrt{(1+8{F_{r_{1}}}^{2})}\right)\quad or\quad y% _{1}={y_{2}\over 2}\left(-1+\sqrt{(1+8{F_{r_{2}}}^{2})}\right)
  23. M 1 = M 2 M_{1}=M_{2}
  24. ( q 2 g y 1 + y 1 2 2 ) = ( q 2 g y 2 + y 2 2 2 ) \left({q^{2}\over gy_{1}}+{y_{1}^{2}\over 2}\right)=\left({q^{2}\over gy_{2}}+% {y_{2}^{2}\over 2}\right)
  25. y 2 y^{2}
  26. q 2 g y 1 - q 2 g y 2 = - y 1 2 2 + y 2 2 2 {q^{2}\over gy_{1}}-{q^{2}\over gy_{2}}=-{y_{1}^{2}\over 2}+{y_{2}^{2}\over 2}
  27. q 2 g ( 1 y 1 - 1 y 2 ) = 1 2 ( y 2 2 - y 1 2 ) {q^{2}\over g}\left({1\over y_{1}}-{1\over y_{2}}\right)={1\over 2}({y_{2}^{2}% }-{y_{1}^{2}})
  28. q 2 g ( y 2 - y 1 y 1 y 2 ) = 1 2 ( y 2 - y 1 ) ( y 2 + y 1 ) {q^{2}\over g}\left({{y_{2}-y_{1}}\over{y_{1}y_{2}}}\right)={1\over 2}({y_{2}-% y_{1}})({y_{2}+y_{1}})
  29. ( y 2 - y 1 ) (y_{2}-y_{1})
  30. q 2 g ( 1 y 1 y 2 ) = 1 2 ( y 2 + y 1 ) {q^{2}\over g}\left({1\over{y_{1}y_{2}}}\right)={1\over 2}({y_{2}+y_{1}})
  31. q = V 1 y 1 = V 2 y 2 q=V_{1}y_{1}=V_{2}y_{2}
  32. V 1 y 1 V_{1}y_{1}
  33. V 1 2 y 1 2 g y 1 y 2 = V 1 2 y 1 g y 2 = y 2 + y 1 2 {{V_{1}^{2}y_{1}^{2}}\over{gy_{1}y_{2}}}={{V_{1}^{2}y_{1}}\over{gy_{2}}}={{y_{% 2}+y_{1}}\over 2}
  34. y 1 / y 2 {y_{1}/y_{2}}
  35. V 1 2 g = y 2 ( y 2 + y 1 ) 2 y 1 {{V_{1}^{2}}\over{g}}={{y_{2}({y_{2}+y_{1}})}\over{2y_{1}}}
  36. y 1 y_{1}
  37. V 1 2 g y 1 = y 2 ( y 2 + y 1 ) 2 y 1 2 V 1 2 g y 1 = F r 1 2 = y 2 2 2 y 1 2 + y 2 y 1 2 y 1 2 = y 2 2 2 y 1 2 + y 2 2 y 1 {{V_{1}^{2}}\over{gy_{1}}}={{y_{2}({y_{2}+y_{1}})}\over{2y_{1}^{2}}}% \Rightarrow{V_{1}^{2}\over gy_{1}}=F_{r_{1}}^{2}={y_{2}^{2}\over{2y_{1}^{2}}}+% {{y_{2}y_{1}}\over 2y_{1}^{2}}={y_{2}^{2}\over{2y_{1}^{2}}}+{{y_{2}}\over 2y_{% 1}}
  38. 0 = y 2 2 2 y 1 2 + y 2 2 y 1 - F r 1 2 0={y_{2}^{2}\over{2y_{1}^{2}}}+{{y_{2}}\over 2y_{1}}-F_{r_{1}}^{2}
  39. x = y 2 / y 1 x={y_{2}/y_{1}}
  40. 0 = 1 2 x 2 + 1 2 x - F r 1 2 0={1\over{2}}x^{2}+{1\over{2}}x-F_{r_{1}}^{2}
  41. x x
  42. a = 1 / 2 a={1/2}
  43. b = 1 / 2 b={1/2}
  44. c = c=
  45. x = - b ± b 2 - 4 a c 2 a x = y 2 y 1 = - 1 2 ± 1 2 2 - 4 ( 1 2 ) ( - F r 1 2 ) 2 ( 1 2 ) = - 1 2 ± 1 4 + 2 F r 1 2 1 x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\quad\Rightarrow\quad x={y_{2}\over y_{1}}=% \frac{-{1\over 2}\pm\sqrt{{1\over 2}^{2}-4\left({1\over 2}\right)\left(-F_{r_{% 1}}^{2}\right)}}{2\left({1\over 2}\right)}=\frac{-{1\over 2}\pm\sqrt{{1\over 4% }+2{F_{r_{1}}^{2}}}}{1}
  46. y 2 y 1 = - 1 2 ± 1 4 ( 1 + 8 F r 1 2 ) = - 1 2 ± 1 2 ( 1 + 8 F r 1 2 ) {y_{2}\over y_{1}}=-{1\over 2}\pm\sqrt{{1\over 4}\left(1+8{F_{r_{1}}^{2}}% \right)}=-{1\over 2}\pm{1\over 2}\sqrt{\left(1+8{F_{r_{1}}^{2}}\right)}
  47. y 2 y 1 = - 1 2 + 1 2 ( 1 + 8 F r 1 2 ) y 2 = - y 1 2 + y 1 2 ( 1 + 8 F r 1 2 ) {y_{2}\over y_{1}}=-{1\over 2}+{1\over 2}\sqrt{\left(1+8{F_{r_{1}}^{2}}\right)% }\quad\Rightarrow\quad y_{2}=-{y_{1}\over 2}+{y_{1}\over 2}\sqrt{\left(1+8{F_{% r_{1}}^{2}}\right)}
  48. y 2 = y 1 2 ( - 1 + ( 1 + 8 F r 1 2 ) ) y_{2}={y_{1}\over 2}\left(-1+\sqrt{\left(1+8{F_{r_{1}}^{2}}\right)}\right)
  49. F t h r u s t - g a t e = γ b ( Δ M ) = γ b ( M u n i t , 1 - M u n i t , 2 ) F_{thrust-gate}=\gamma b(\Delta M)=\gamma b(M_{unit,1}-M_{unit,2})
  50. M u n i t = y 2 2 + q 2 g y M_{unit}={{y^{2}}\over 2}+{{q^{2}}\over gy}
  51. M u n i t , 1 = y 1 2 2 + q 2 g y 1 = 16.3 2 2 + ( 100 10 ) 2 ( 32.2 ) ( 16.3 ) M_{unit,1}={{{y_{1}}^{2}}\over 2}+{{q^{2}}\over gy_{1}}={{{16.3}^{2}}\over 2}+% {{{\left({100\over 10}\right)}^{2}}\over{(32.2)(16.3)}}
  52. M u n i t , 1 = 132 f t 2 M_{unit,1}=132\quad ft^{2}
  53. M u n i t , 2 = 10 f t 2 M_{unit,2}=10\quad ft^{2}
  54. l b / f t 3 {lb/ft^{3}}
  55. F t h r u s t - g a t e = γ b ( M u n i t , 1 - M u n i t , 2 ) F_{thrust-gate}=\gamma b(M_{unit,1}-M_{unit,2})
  56. F t h r u s t - g a t e = ( 62.30 ) ( 10 ) ( 132 - 10 ) F_{thrust-gate}=(62.30)(10)(132-10)
  57. F t h r u s t - g a t e = 76 , 142 l b s F_{thrust-gate}=76,142\quad lbs

Monitoring_(medicine).html

  1. C D = K × C V a 2 + C V i 2 CD=K\times\sqrt{CV_{a}^{2}+CV_{i}^{2}}

Monod_equation.html

  1. μ = μ max S K s + S \mu=\mu_{\max}{S\over K_{s}+S}

Monotonic_query.html

  1. I J q ( I ) q ( J ) I\subseteq J\Rightarrow q(I)\subseteq q(J)
  2. A ( Q , τ ) = t = 1 τ ( A ( Q , t ) - A ( Q , t - 1 ) ) A ( Q , 0 ) A(Q,\tau)=\bigcup_{t=1}^{\tau}(A(Q,t)-A(Q,t-1))\cup A(Q,0)
  3. A ( Q , τ ) = t = 0 τ A ( Q , t ) A(Q,\tau)=\bigcup_{t=0}^{\tau}A(Q,t)

Monte_Carlo_methods_for_electron_transport.html

  1. f t + 1 k E ( k ) r f + q F ( r ) k f = [ f t ] collision \frac{\partial f}{\partial t}+\frac{1}{\hbar}\nabla_{k}E(k)\nabla_{r}f+\frac{% qF(r)}{\hbar}\nabla_{k}f=\left[\frac{\partial f}{\partial t}\right]_{\mathrm{% collision}}
  2. v = 1 k E ( k ) v=\frac{1}{\hbar}\nabla_{k}E(k)
  3. d r d t = 1 k E ( k ) \frac{dr}{dt}=\frac{1}{\hbar}\nabla_{k}E(k)
  4. d k d t = q F ( r ) \frac{dk}{dt}=\frac{qF(r)}{\hbar}
  5. E ( k ) = E ( 0 ) + E ( k ) k | k = 0 k + 1 2 2 E ( k ) k 2 k 2 E(k)=E(0)+\left.\frac{\partial E(k)}{\partial k}\right|_{\mathrm{k=0}}\cdot k+% \frac{1}{2}\frac{\partial^{2}E(k)}{\partial k^{2}}\cdot k^{2}
  6. E ( k ) = 2 k 2 2 m * E(k)=\frac{\hbar^{2}k^{2}}{2m^{*}}
  7. 1 m * = 1 2 2 E ( k ) k 2 \frac{1}{m^{*}}=\frac{1}{\hbar^{2}}\frac{\partial^{2}E(k)}{\partial k^{2}}
  8. E ( k ) = 2 ( k l 2 m l * + 2 k t 2 m t * ) E(k)=\frac{\hbar}{2}\left(\frac{k^{2}_{l}}{m^{*}_{l}}+\frac{2k^{2}_{t}}{m^{*}_% {t}}\right)
  9. m l * , m t * m^{*}_{l},m^{*}_{t}
  10. E ( 1 + α E ) = 2 k 2 2 m * E(1+\alpha E)=\frac{\hbar^{2}k^{2}}{2m^{*}}
  11. α \alpha
  12. α = ( 1 - m * / m 0 ) 2 E g \alpha=\frac{(1-m^{*}/m_{0})^{2}}{E_{g}}
  13. m 0 m_{0}
  14. p ( t ) d t = P [ k ( t ) ] exp [ - 0 t P [ k ( t ) ] d t ] d t p(t)\,dt=P[k(t)]\exp[-\int^{t}_{0}P[k(t^{\prime})]\,dt^{\prime}]\,dt
  15. Γ \Gamma
  16. p ( k ) = τ 0 - 1 p(k)=\tau_{0}^{-1}
  17. p ( t ) = 1 τ 0 exp ( - t / τ 0 ) . p(t)=\frac{1}{\tau_{0}}\exp(-t/\tau_{0}).
  18. t r = - τ 0 ln ( r ) t_{r}=-\tau_{0}\ln(r)
  19. | k |k\rangle
  20. | k |k^{\prime}\rangle
  21. S ( k , k ) = 2 π | k | H | k | 2 δ ( E - E ) S(k,k^{\prime})=\frac{2\pi}{\hbar}\left|\langle k|H^{\prime}|k^{\prime}\rangle% \right|^{2}\cdot\delta(E-E^{\prime})
  22. δ \delta
  23. k | H | k \langle k|H^{\prime}|k^{\prime}\rangle
  24. k | H | k = 1 V o l Vol ψ k ( r ) H ψ k * ( r ) d r \langle k|H^{\prime}|k^{\prime}\rangle=\frac{1}{Vol}\int_{\mathrm{Vol}}\psi_{k% }(r)H^{\prime}\psi^{*}_{k^{\prime}}(r)\,dr
  25. ψ k ( r ) \psi_{k}(r)
  26. ψ k ( r ) \psi_{k^{\prime}}(r)
  27. ω q \omega_{q}
  28. E - E = E ( k ) - E ( k ) ± ω q E^{\prime}-E=E(k^{\prime})-E(k)\pm\hbar\omega_{q}\,
  29. k - k ± q = { 0 R Umklapp-process k^{\prime}-k\pm q=\begin{cases}0&\,\text{ }\\ R&\,\text{Umklapp-process}\end{cases}
  30. λ ( k ) = k S ( k , k ) \lambda(k)=\sum_{k^{\prime}}S(k,k^{\prime})
  31. λ 1 , λ 2 , , λ n \lambda_{1},\lambda_{2},...,\lambda_{n}
  32. λ t o t ( t s c ) = i λ i . \lambda_{tot}(t_{sc})=\sum_{i}\lambda_{i}.
  33. λ tot \lambda_{\mathrm{tot}}
  34. E ( k ) = E ( k ) ± ω q ± Δ E C E(k^{\prime})=E(k)\pm\hbar\omega_{q}\pm\Delta E_{C}\,
  35. ω q \hbar\omega_{q}
  36. Δ E C \Delta E_{C}
  37. θ \theta
  38. ψ \psi
  39. p ( θ , ψ ) p(\theta,\psi)
  40. p ( θ , ψ ) d θ d ψ = sin θ d θ d ψ 4 π p(\theta,\psi)\,d\theta d\psi=\frac{\sin\theta\,d\theta\,d\psi}{4\pi}
  41. ψ \psi
  42. θ \theta
  43. p ( θ ) = sin θ 2 p(\theta)=\frac{\sin\theta}{2}
  44. p ( ψ ) = 1 2 π p(\psi)=\frac{1}{2\pi}
  45. r 2 = 0 θ p ( θ ) d θ = 1 - cos θ 2 r_{2}=\int_{0}^{\theta}p(\theta^{\prime})\,d\theta^{\prime}=\frac{1-\cos\theta% }{2}
  46. f t + r r f - 1 r V k f + α = 1 ( - 1 ) α + 1 4 α ( 2 α + 1 ) ! × ( r k ) 2 α + 1 V f = ( f t ) c \frac{\partial f}{\partial t}+r\cdot\nabla_{r}f-\frac{1}{\hbar}\nabla_{r}V% \cdot\nabla_{k}f+\sum_{\alpha=1}^{\infty}\frac{(-1)^{\alpha+1}}{\hbar 4^{% \alpha}(2\alpha+1)!}\times(\nabla_{r}\nabla_{k})^{2\alpha+1}Vf=\left(\frac{% \partial f}{\partial t}\right)_{c}
  47. α = 0 \alpha=0
  48. f t + r r f - 1 r V k f = ( f t ) c \frac{\partial f}{\partial t}+r\cdot\nabla_{r}f-\frac{1}{\hbar}\nabla_{r}V% \cdot\nabla_{k}f=\left(\frac{\partial f}{\partial t}\right)_{c}
  49. V ω V_{\omega}
  50. V ω V_{\omega}
  51. V eff ( x ) = 1 2 π a - V ( x ) e - ( x - x ) 2 2 a 2 d x V_{\mathrm{eff}}(x)=\frac{1}{\sqrt{2\pi a}}\int^{\infty}_{-\infty}V(x^{\prime}% )e^{-\frac{(x^{\prime}-x)^{2}}{2a^{2}}}dx^{\prime}
  52. a 2 = 2 12 m * k B T a^{2}=\frac{\hbar^{2}}{12m^{*}k_{B}T}
  53. V schr ( z ) = - k B T log ( n q ( z ) ) - V p ( z ) + V 0 V_{\mathrm{schr}}(z)=-k_{B}T\cdot\log(n_{q}(z))-V_{p}(z)+V_{0}

Morrey–Campanato_space.html

  1. L λ , p ( Ω ) L^{\lambda,p}(\Omega)
  2. λ \lambda
  3. L λ , p ( Ω ) L^{\lambda,p}(\Omega)
  4. Ω \Omega
  5. [ u ] λ , p p = sup 0 < r < diam ( Ω ) , x 0 Ω 1 r λ B r ( x 0 ) Ω | u ( y ) | p d y . \left[u\right]_{\lambda,p}^{p}=\sup_{0<r<\operatorname{diam}(\Omega),x_{0}\in% \Omega}\frac{1}{r^{\lambda}}\int_{B_{r}(x_{0})\cap\Omega}|u(y)|^{p}dy.
  6. λ = 0 \lambda=0
  7. L p L^{p}
  8. λ = n \lambda=n
  9. L L^{\infty}
  10. λ > n \lambda>n
  11. [ u ] λ , p p = sup 0 < r < diam ( Ω ) , x 0 Ω 1 r λ B r ( x 0 ) Ω | u ( y ) - u r , x 0 | p d y \left[u\right]_{\lambda,p}^{p}=\sup_{0<r<\operatorname{diam}(\Omega),x_{0}\in% \Omega}\frac{1}{r^{\lambda}}\int_{B_{r}(x_{0})\cap\Omega}|u(y)-u_{r,x_{0}}|^{p% }dy
  12. u r , x 0 = 1 | B r ( x 0 ) Ω | B r ( x 0 ) Ω u ( y ) d y . u_{r,x_{0}}=\frac{1}{|B_{r}(x_{0})\cap\Omega|}\int_{B_{r}(x_{0})\cap\Omega}u(y% )dy.
  13. 0 λ < n 0\leq\lambda<n
  14. λ \lambda
  15. Ω \Omega
  16. | Ω B r ( x 0 ) | > A r n |\Omega\cap B_{r}(x_{0})|>Ar^{n}
  17. x 0 Ω x_{0}\in\Omega
  18. r < diam ( Ω ) r<\operatorname{diam}(\Omega)
  19. n = λ n=\lambda
  20. n < λ n + p n<\lambda\leq n+p
  21. C α ( Ω ) C^{\alpha}(\Omega)
  22. α = λ - n p \alpha=\frac{\lambda-n}{p}
  23. λ > n + p \lambda>n+p

Morris–Lecar_model.html

  1. C d V d t \displaystyle C\frac{dV}{dt}
  2. M ss = 1 2 ( 1 + tanh [ V - V 1 V 2 ] ) N ss = 1 2 ( 1 + tanh [ V - V 3 V 4 ] ) τ N = 1 / ( ϕ cosh [ V - V 3 2 V 4 ] ) \begin{aligned}\displaystyle M_{\mathrm{ss}}&\displaystyle~{}=~{}\tfrac{1}{2}% \cdot(1+\tanh[\tfrac{V-V_{1}}{V_{2}}])\\ \displaystyle N_{\mathrm{ss}}&\displaystyle~{}=~{}\tfrac{1}{2}\cdot(1+\tanh[% \tfrac{V-V_{3}}{V_{4}}])\\ \displaystyle\tau_{N}&\displaystyle~{}=~{}1/(\phi\cosh[\tfrac{V-V_{3}}{2V_{4}}% ])\end{aligned}
  3. M < s u b > s s M<sub>ss

MOSCED.html

  1. ln γ 2 = ν 2 R T [ ( λ 1 - λ 2 ) 2 + q 1 2 q 2 2 ( τ 1 T - τ 2 T ) 2 ψ 1 + ( α 1 T - α 2 T ) ( β 1 T - β 2 T ) ξ 1 ] + d 12 \ln\gamma_{2}^{\infty}=\frac{\nu_{2}}{RT}\left[\left(\lambda_{1}-\lambda_{2}% \right)^{2}+\frac{q_{1}^{2}q_{2}^{2}\left(\tau_{1}^{T}-\tau_{2}^{T}\right)^{2}% }{\psi_{1}}+\frac{\left(\alpha_{1}^{T}-\alpha_{2}^{T}\right)\left(\beta_{1}^{T% }-\beta_{2}^{T}\right)}{\xi_{1}}\right]+d_{12}
  2. d 12 = ln ( ν 2 ν 1 ) a a + 1 + ( ν 2 ν 1 ) a a d_{12}=\ln\left(\frac{\nu_{2}}{\nu_{1}}\right)^{aa}+1+\left(\frac{\nu_{2}}{\nu% _{1}}\right)^{aa}
  3. a a = 0.953 - 0.002314 ( ( τ 2 T ) 2 + α 2 T β 2 T ) aa=0.953-0.002314\left(\left(\tau_{2}^{T}\right)^{2}+\alpha_{2}^{T}\beta_{2}^{% T}\right)
  4. α T = α ( 293 K T ) 0.8 \alpha^{T}=\alpha\left(\frac{293K}{T}\right)^{0.8}
  5. β T = β ( 293 K T ) 0.8 \beta^{T}=\beta\left(\frac{293K}{T}\right)^{0.8}
  6. τ T = τ ( 293 K T ) 0.4 \tau^{T}=\tau\left(\frac{293K}{T}\right)^{0.4}
  7. ψ 1 = P O L + 0.002629 α 1 T β 1 T \psi_{1}=POL+0.002629\alpha_{1}^{T}\beta_{1}^{T}
  8. ξ 1 = 0.68 ( P O L - 1 ) + [ 3.4 - 2.4 exp ( - 0.002687 ( α 1 β 1 ) 1.5 ) ] ( 293 K / T ) 2 \xi_{1}=0.68\left(POL-1\right)+\left[3.4-2.4\exp\left(-0.002687\left(\alpha_{1% }\beta_{1}\right)^{1.5}\right)\right]^{\left(293K/T\right)^{2}}
  9. P O L = q 1 4 [ 1.15 - 1.15 exp ( - 0.002337 ( τ 1 T ) 3 ) ] + 1 POL=q_{1}^{4}\left[1.15-1.15\exp\left(-0.002337\left(\tau_{1}^{T}\right)^{3}% \right)\right]+1
  10. ln γ 2 = ( ln γ 2 + 2 ( ln γ 1 - ln γ 2 ) Φ 2 ) Φ 1 2 \ln\gamma_{2}=\left(\ln\gamma_{2}^{\infty}+2\left(\ln\gamma_{1}^{\infty}-\ln% \gamma_{2}^{\infty}\right)\Phi_{2}\right)\Phi_{1}^{2}
  11. ln γ 1 = ( ln γ 1 + 2 ( ln γ 2 - ln γ 1 ) Φ 1 ) Φ 2 2 \ln\gamma_{1}=\left(\ln\gamma_{1}^{\infty}+2\left(\ln\gamma_{2}^{\infty}-\ln% \gamma_{1}^{\infty}\right)\Phi_{1}\right)\Phi_{2}^{2}
  12. Φ i = x i ν i j ν j x j \Phi_{i}=\frac{x_{i}\nu_{i}}{\sum_{j}\nu_{j}x_{j}}
  13. x i x_{i}

Mott_polynomials.html

  1. e x ( 1 - t 2 - 1 ) / t = n s n ( x ) t n / n ! . e^{x(\sqrt{1-t^{2}}-1)/t}=\sum_{n}s_{n}(x)t^{n}/n!.
  2. s 0 ( x ) = 1 ; s_{0}(x)=1;
  3. s 1 ( x ) = - 1 2 x ; s_{1}(x)=-\frac{1}{2}x;
  4. s 2 ( x ) = 1 4 x 2 ; s_{2}(x)=\frac{1}{4}x^{2};
  5. s 3 ( x ) = - 3 4 x - 1 8 x 3 ; s_{3}(x)=-\frac{3}{4}x-\frac{1}{8}x^{3};
  6. s 4 ( x ) = 3 2 x 2 + 1 16 x 4 ; s_{4}(x)=\frac{3}{2}x^{2}+\frac{1}{16}x^{4};
  7. s 5 ( x ) = - 15 2 x - 15 8 x 3 - 1 32 x 5 ; s_{5}(x)=-\frac{15}{2}x-\frac{15}{8}x^{3}-\frac{1}{32}x^{5};
  8. s 6 ( x ) = 225 8 x 2 + 15 8 x 4 + 1 64 x 6 ; s_{6}(x)=\frac{225}{8}x^{2}+\frac{15}{8}x^{4}+\frac{1}{64}x^{6};

Mott–Bethe_formula.html

  1. f B ( q , Z ) = m e 2 2 π 2 ϵ 0 ( Z - f x ( q , Z ) q 2 ) f^{B}(q,Z)=\frac{me^{2}}{2\pi\hbar^{2}\epsilon_{0}}\Bigg(\frac{Z-f_{x}(q,Z)}{q% ^{2}}\Bigg)
  2. f e ( q , Z ) f_{e}(q,Z)
  3. f x ( q , Z ) f_{x}(q,Z)
  4. q q
  5. \hbar
  6. ϵ 0 \epsilon_{0}
  7. Potential electric / distance 3 \mathrm{Potential_{electric}}/\mathrm{distance}^{3}

Mountain_Car.html

  1. V e l o c i t y = ( - 0.07 , 0.07 ) Velocity=(-0.07,0.07)
  2. P o s i t i o n = ( - 1.2 , 0.6 ) Position=(-1.2,0.6)
  3. m o t o r = ( l e f t , n e u t r a l , r i g h t ) motor=(left,neutral,right)
  4. r e w a r d = - 1 + h e i g h t reward=-1+height
  5. A c t i o n = [ - 1 , 0 , 1 ] Action=[-1,0,1]
  6. V e l o c i t y = V e l o c i t y + ( A c t i o n ) * 0.001 + cos ( 3 * P o s i t i o n ) * ( - 0.0025 ) Velocity=Velocity+(Action)*0.001+\cos(3*Position)*(-0.0025)
  7. P o s i t i o n = P o s i t i o n + V e l o c i t y Position=Position+Velocity
  8. P o s i t i o n = - 0.5 Position=-0.5
  9. V e l o c i t y = 0.0 Velocity=0.0
  10. P o s i t i o n 0.6 Position\geq 0.6

Multicriteria_classification.html

  1. X = { 𝐱 1 , 𝐱 2 , , 𝐱 m } X=\{\mathbf{x}_{1},\mathbf{x}_{2},...,\mathbf{x}_{m}\}
  2. t r + 1 < V ( 𝐱 i ) < t r t_{r+1}<V(\mathbf{x}_{i})<t_{r}
  3. f f
  4. β * = arg min β B L [ D ( X ) , D ( X , f β ) ] \beta^{*}=\arg\min_{\beta\in B}L[D(X),D^{^{\prime}}(X,f_{\beta})]
  5. minimize \displaystyle\,\text{minimize}

Multidimensional_Poverty_Index.html

  1. M P I = H × A MPI=H\times A
  2. 1 + 1 + 0 3 = 0.667 \frac{1+1+0}{3}=0.667
  3. 33.33 % + 50.00 % 2 = 0.417 \frac{33.33\%+50.00\%}{2}=0.417
  4. 0.667 × 0.417 = 0.278 0.667\times 0.417=0.278

Multiparty_communication_complexity.html

  1. f ( x 1 , x 2 ) : { 0 , 1 } n { 0 , 1 } , x 1 , x 2 { 0 , 1 } n , 2 n = n f(x_{1},x_{2}):\{0,1\}^{n}\to\{0,1\},\ x_{1},x_{2}\in\{0,1\}^{n^{\prime}},\ 2n% ^{\prime}=n
  2. f ( x 1 , x 2 , , x n ) : { 0 , 1 } n { 0 , 1 } f(x_{1},x_{2},\ldots,x_{n}):\{0,1\}^{n}\to\{0,1\}
  3. C ( k ) ( f ) = max A C A ( k ) ( f ) C^{(k)}(f)=\max_{A}C^{(k)}_{A}(f)\,
  4. C ( k ) ( f ) n k + 1. C^{(k)}(f)\leq\bigg\lfloor{n\over k}\bigg\rfloor+1.
  5. C ( k ) ( G I P ) c n 4 k , C^{(k)}(GIP)\geq c{n\over 4^{k}},
  6. C ( k ) ( G I P ) c n 2 k , C^{(k)}(GIP)\leq c{n\over 2^{k}},
  7. O ( k 2 log ( n L 1 ( f ) ) n L 1 2 ( f ) 2 k ) O\Bigg(k^{2}\log(nL_{1}(f))\Bigg\lceil{nL_{1}^{2}(f)\over 2^{k}}\Bigg\rceil\Bigg)

Multiphase_particle-in-cell_method.html

  1. θ f t + ( θ f u f ) = 0 , \frac{\partial\theta_{f}}{\partial t}+\nabla\cdot(\theta_{f}{u}_{f})=0,
  2. θ f \theta_{f}\;
  3. u f {u}_{f}\;
  4. ρ f \rho_{f}\;
  5. p p\;
  6. g {g}\;
  7. θ f u f t + ( θ f u f u f ) = - p ρ f - F ρ f + θ f g \frac{\partial\theta_{f}{u}_{f}}{\partial t}+\nabla\cdot(\theta_{f}{u}_{f}{u}_% {f})=-\frac{\nabla p}{\rho_{f}}-\frac{{F}}{\rho_{f}}+\theta_{f}{g}
  8. F {F}\;
  9. ϕ ( x , u f , ρ p , Ω p , t ) ; \phi\left({x},{u}_{f},\rho_{p},\Omega_{p},t\right);
  10. u f {u}_{f}\;
  11. ρ p \rho_{p}\;
  12. Ω p \Omega_{p}\;
  13. x {x}\;
  14. t t\;
  15. ϕ t + ( ϕ u p ) + u p ( ϕ A ) = 0 \frac{\partial\phi}{\partial t}+\nabla\cdot(\phi{u}_{p})+\nabla_{{u}_{p}}\cdot% \left(\phi{A}\right)=0
  16. A {A}\;
  17. θ p = ϕ Ω p d Ω p d ρ p d u p \theta_{p}=\int\!\!\!\int\!\!\!\int\phi\Omega_{p}\;d\Omega_{p}d\rho_{p}d{u}_{p}
  18. θ p ρ p ¯ = ϕ Ω p ρ p d Ω p d ρ p d u p \overline{\theta_{p}\rho_{p}}=\int\!\!\!\int\!\!\!\int\phi\Omega_{p}\rho_{p}\;% d\Omega_{p}d\rho_{p}d{u}_{p}
  19. u ¯ p = 1 θ p ρ p ¯ ϕ Ω p ρ p u p d Ω p d ρ p d u p \overline{{u}}_{p}=\frac{1}{\overline{\theta_{p}\rho_{p}}}\int\!\!\!\int\!\!\!% \int\phi\Omega_{p}\rho_{p}{u}_{p}\;d\Omega_{p}d\rho_{p}d{u}_{p}
  20. A {A}\;
  21. A = D p ( u f - u p ) - p ρ p + g - τ θ p ρ p . {A}=D_{p}\left({u}_{f}-{u}_{p}\right)-\frac{\nabla p}{\rho_{p}}+{g}-\frac{% \nabla\tau}{\theta_{p}\rho_{p}}.
  22. D p D_{p}\;
  23. τ \tau\;
  24. F {F}\;
  25. F = ϕ Ω p ρ p [ D p ( u f - u p ) - p ρ p ] d Ω p d ρ p d u p {F}=\int\!\!\!\int\!\!\!\int\phi\Omega_{p}\rho_{p}\left[D_{p}\left({u}_{f}-{u}% _{p}\right)-\frac{\nabla p}{\rho_{p}}\right]\;d\Omega_{p}d\rho_{p}d{u}_{p}
  26. τ \tau\;
  27. τ = P s θ P β max [ θ c p - θ p , ϵ ( 1 - θ p ) ] \tau=\frac{P_{s}{\theta_{P}}^{\beta}}{\max\left[\theta_{cp}-\theta_{p},% \epsilon\left(1-\theta_{p}\right)\right]}
  28. θ c p \theta_{cp}\;
  29. β \beta\;
  30. P s P_{s}\;
  31. ϵ \epsilon\;

Multiple-prism_grating_laser_oscillator.html

  1. Δ λ Δ θ ( M θ λ ) - 1 \Delta\lambda\approx\Delta\theta\left(M{\partial\theta\over\partial\lambda}% \right)^{-1}
  2. Δ θ \Delta\theta
  3. Δ λ Δ θ ( M θ λ + ϕ 2 , m λ ) - 1 \Delta\lambda\approx\Delta\theta\left(M{\partial\theta\over\partial\lambda}+{% \partial\phi_{2,m}\over\partial\lambda}\right)^{-1}
  4. Δ ν \Delta\nu
  5. Δ λ \Delta\lambda

Multiplier-accelerator_model.html

  1. Y t = g t + C t + I t Y_{t}=g_{t}+C_{t}+I_{t}
  2. C t = α Y t - 1 C_{t}=\alpha Y_{t-1}
  3. I t = β C t - C t - 1 I_{t}=\beta\mid C_{t}-C_{t-1}\mid
  4. g t = 1 g_{t}=1
  5. Y t Y_{t}
  6. g t g_{t}
  7. C t C_{t}
  8. I t I_{t}
  9. t t
  10. Y t = 1 + α ( 1 + β ) Y t - 1 - α β Y t - 2 Y_{t}=1+\alpha(1+\beta)Y_{t-1}-\alpha\beta Y_{t-2}
  11. α \alpha
  12. β \beta

Multiplier_ideal.html

  1. | h | 2 | f i 2 | c \frac{|h|^{2}}{\sum|f_{i}^{2}|^{c}}
  2. \mathbb{Q}
  3. \mathbb{Q}
  4. μ : X X \mu:X^{\prime}\to X
  5. J ( D ) = μ * 𝒪 ( K X / X - [ μ * D ] ) J(D)=\mu_{*}\mathcal{O}(K_{X^{\prime}/X}-[\mu^{*}D])
  6. K X / X K_{X^{\prime}/X}
  7. K X / X = K X - μ * K X K_{X^{\prime}/X}=K_{X^{\prime}}-\mu^{*}K_{X}
  8. 𝒪 X \mathcal{O}_{X}
  9. J ( D ) = 𝒪 X ( - D ) J(D)=\mathcal{O}_{X}(-D)

Multiplier_uncertainty.html

  1. σ a 2 \sigma^{2}_{a}
  2. σ u 2 \sigma^{2}_{u}
  3. y = a P + u . y=aP+u.
  4. y d y_{d}
  5. E L = E ( y - y d ) 2 = E ( a P + u - y d ) 2 = [ E ( a P + u - y d ) ] 2 + var ( a P + u - y d ) = [ ( E a ) P + E u - y d ] 2 + P 2 σ a 2 + σ u 2 . \,\text{E}L=\,\text{E}(y-y_{d})^{2}=\,\text{E}(aP+u-y_{d})^{2}=[\,\text{E}(aP+% u-y_{d})]^{2}+\,\text{var}(aP+u-y_{d})=[(\,\text{E}a)P+\,\text{E}u-y_{d}]^{2}+% P^{2}\sigma^{2}_{a}+\sigma^{2}_{u}.
  6. P o p t = ( E a ) ( y d - E u ) ( E a ) 2 + σ a 2 . P^{opt}=\frac{(\,\text{E}a)(y_{d}-\,\text{E}u)}{(\,\text{E}a)^{2}+\sigma^{2}_{% a}}.
  7. σ a 2 \sigma^{2}_{a}
  8. σ a 2 > 0 \sigma^{2}_{a}>0

Multipole_radiation.html

  1. 2 2^{\ell}
  2. 1 / r + 2 1/r^{\ell+2}
  3. r r
  4. 1 / r 2 1/r^{2}
  5. 4 π r 2 4\pi r^{2}
  6. 1 / r 2 1/r^{2}
  7. 1 / ( 2 + 1 ) ! ! 1/(2\ell+1)!!
  8. ρ ( 𝐱 , t ) = - ρ ^ ( 𝐱 , ω ) e - i ω t \rho(\mathbf{x},t)=\int_{-\infty}^{\infty}\hat{\rho}(\mathbf{x},\omega)e^{-i% \omega t}
  9. 𝐉 ( 𝐱 , t ) = - 𝐉 ^ ( 𝐱 , ω ) e - i ω t \mathbf{J}(\mathbf{x},t)=\int_{-\infty}^{\infty}\hat{\mathbf{J}}(\mathbf{x},% \omega)e^{-i\omega t}
  10. ρ ( 𝐱 , t ) = ρ ( 𝐱 ) e - i ω t \rho(\mathbf{x},t)=\rho(\mathbf{x})e^{-i\omega t}
  11. 𝐉 ( 𝐱 , t ) = 𝐉 ( 𝐱 ) e - i ω t \mathbf{J}(\mathbf{x},t)=\mathbf{J}(\mathbf{x})e^{-i\omega t}
  12. 𝐌 ( 𝐱 , t ) \mathbf{M}(\mathbf{x},t)
  13. ϕ ( 𝐱 , t ) = 1 4 π ϵ 0 d 3 𝐱 d t ρ ( 𝐱 , t ) 𝐱 - 𝐱 2 δ ( t - ( t - 𝐱 - 𝐱 2 c ) ) \phi(\mathbf{x},t)=\frac{1}{4\pi\epsilon_{0}}\int d^{3}\mathbf{x^{\prime}}\int dt% ^{\prime}\frac{\rho(\mathbf{x^{\prime}},t^{\prime})}{\|\mathbf{x}-\mathbf{x^{% \prime}}\|_{2}}\delta\left(t^{\prime}-(t-\frac{\|\mathbf{x}-\mathbf{x^{\prime}% }\|_{2}}{c})\right)
  14. 𝐀 ( 𝐱 , t ) = μ 0 4 π d 3 𝐱 d t 𝐉 ( 𝐱 , t ) 𝐱 - 𝐱 2 δ ( t - ( t - 𝐱 - 𝐱 2 c ) ) \mathbf{A}(\mathbf{x},t)=\frac{\mu_{0}}{4\pi}\int d^{3}\mathbf{x^{\prime}}\int dt% ^{\prime}\frac{\mathbf{J}(\mathbf{x^{\prime}},t^{\prime})}{\|\mathbf{x}-% \mathbf{x^{\prime}}\|_{2}}\delta\left(t^{\prime}-(t-\frac{\|\mathbf{x}-\mathbf% {x^{\prime}}\|_{2}}{c})\right)
  15. δ \delta
  16. 𝐱 - 𝐱 2 \|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}
  17. ϕ ( 𝐱 , t ) = 1 4 π ϵ 0 e - i ω t d 3 𝐱 ρ ( 𝐱 ) e i k 𝐱 - 𝐱 2 𝐱 - 𝐱 2 \phi(\mathbf{x},t)=\frac{1}{4\pi\epsilon_{0}}e^{-i\omega t}\int d^{3}\mathbf{x% ^{\prime}}\rho(\mathbf{x^{\prime}})\frac{e^{ik\|\mathbf{x}-\mathbf{x^{\prime}}% \|_{2}}}{\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}
  18. 𝐀 ( 𝐱 , t ) = μ 0 4 π e - i ω t d 3 𝐱 𝐉 ( 𝐱 ) e i k 𝐱 - 𝐱 2 𝐱 - 𝐱 2 \mathbf{A}(\mathbf{x},t)=\frac{\mu_{0}}{4\pi}e^{-i\omega t}\int d^{3}\mathbf{x% ^{\prime}}\mathbf{J}(\mathbf{x^{\prime}})\frac{e^{ik\|\mathbf{x}-\mathbf{x^{% \prime}}\|_{2}}}{\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}
  19. r = 𝐱 2 r=\|\mathbf{x}\|_{2}
  20. λ = 2 π / k \lambda=2\pi/k
  21. k r 1 kr\ll 1
  22. e i k 𝐱 - 𝐱 2 = 1 + O ( k r ) e^{ik\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}=1+O(kr)
  23. 1 / 𝐱 - 𝐱 2 1/\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}
  24. λ r \lambda\ll r
  25. 1 𝐱 - 𝐱 2 = 1 r + O ( 1 / r 2 ) \frac{1}{\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}=\frac{1}{r}+O(1/r^{2})
  26. e i k 𝐱 - 𝐱 2 = e i k ( r - 𝐧 𝐱 + O ( 1 / r ) ) = e i k r - i k 𝐧 𝐱 ( 1 + O ( 1 / r ) ) e^{ik\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}=e^{ik(r-\mathbf{n}\cdot\mathbf{x^% {\prime}}+O(1/r))}=e^{ikr-ik\mathbf{n}\cdot\mathbf{x^{\prime}}}(1+O(1/r))
  27. 1 / r 1/r
  28. e i k 𝐱 - 𝐱 2 𝐱 - 𝐱 2 = e i k r r ( 1 - i k ( 𝐧 𝐱 ) + ( - i k ) 2 2 ( 𝐧 𝐱 ) 2 + ) + O ( 1 / r 2 ) \frac{e^{ik\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}}{\|\mathbf{x}-\mathbf{x^{% \prime}}\|_{2}}=\frac{e^{ikr}}{r}(1-ik(\mathbf{n}\cdot\mathbf{x^{\prime}})+% \frac{(-ik)^{2}}{2}(\mathbf{n}\cdot\mathbf{x^{\prime}})^{2}+...)+O(1/r^{2})
  29. 𝐧 𝐱 \mathbf{n}\cdot\mathbf{x^{\prime}}
  30. e i k 𝐱 - 𝐱 2 𝐱 - 𝐱 2 e i k r r \frac{e^{ik\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}}{\|\mathbf{x}-\mathbf{x^{% \prime}}\|_{2}}\rightarrow\frac{e^{ikr}}{r}
  31. ϕ Electric monopole ( 𝐱 , t ) = 1 4 π ϵ 0 e i k r - i ω t r d 3 𝐱 ρ ( 𝐱 ) = e i k r - i ω t 4 π ϵ 0 r q \phi_{\,\text{Electric monopole}}(\mathbf{x},t)=\frac{1}{4\pi\epsilon_{0}}% \frac{e^{ikr-i\omega t}}{r}\int d^{3}\mathbf{x^{\prime}}\rho(\mathbf{x^{\prime% }})=\frac{e^{ikr-i\omega t}}{4\pi\epsilon_{0}r}q
  32. q = d 3 𝐱 ρ ( 𝐱 ) q=\int d^{3}\mathbf{x^{\prime}}\rho(\mathbf{x^{\prime}})
  33. q ( t ) = d 3 𝐱 ρ ( 𝐱 , t ) = d 3 𝐱 ρ ( 𝐱 ) e - i ω t = q e - i ω t q(t)=\int d^{3}\mathbf{x^{\prime}}\rho(\mathbf{x^{\prime}},t)=\int d^{3}% \mathbf{x^{\prime}}\rho(\mathbf{x^{\prime}})e^{-i\omega t}=qe^{-i\omega t}
  34. ϕ Electric monopole ( 𝐱 , t ) = 0 \phi_{\,\text{Electric monopole}}(\mathbf{x},t)=0
  35. 𝐀 Electric dipole ( 𝐱 , t ) = μ 0 4 π e i k r - i ω t r d 3 𝐱 𝐉 ( 𝐱 ) \mathbf{A}_{\,\text{Electric dipole}}(\mathbf{x},t)=\frac{\mu_{0}}{4\pi}\frac{% e^{ikr-i\omega t}}{r}\int d^{3}\mathbf{x^{\prime}}\mathbf{J}(\mathbf{x^{\prime% }})
  36. d 3 𝐱 𝐉 ( 𝐱 ) = - d 3 𝐱 𝐱 ( 𝐉 ( 𝐱 ) ) \int d^{3}\mathbf{x^{\prime}}\mathbf{J}(\mathbf{x^{\prime}})=-\int d^{3}% \mathbf{x^{\prime}}\mathbf{x^{\prime}}(\mathbf{\nabla}\cdot\mathbf{J}(\mathbf{% x^{\prime}}))
  37. ρ ( 𝐱 , t ) t + 𝐉 ( 𝐱 , t ) = ( - i ω ρ ( 𝐱 ) + 𝐉 ( 𝐱 ) ) e - i ω t = 0 \frac{\partial\rho(\mathbf{x},t)}{\partial t}+\mathbf{\nabla}\cdot\mathbf{J}(% \mathbf{x},t)=\left(-i\omega\rho(\mathbf{x})+\mathbf{\nabla}\cdot\mathbf{J}(% \mathbf{x})\right)e^{-i\omega t}=0
  38. 𝐀 Electric dipole ( 𝐱 , t ) = - i ω μ 0 4 π e i k r - i ω t r d 3 𝐱 𝐱 ρ ( 𝐱 ) \mathbf{A}_{\,\text{Electric dipole}}(\mathbf{x},t)=\frac{-i\omega\mu_{0}}{4% \pi}\frac{e^{ikr-i\omega t}}{r}\int d^{3}\mathbf{x^{\prime}}\mathbf{x^{\prime}% }\rho(\mathbf{x^{\prime}})
  39. e i k 𝐱 - 𝐱 2 𝐱 - 𝐱 2 e i k r r ( - i k ) ( 𝐧 𝐱 ) \frac{e^{ik\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}}{\|\mathbf{x}-\mathbf{x^{% \prime}}\|_{2}}\rightarrow\frac{e^{ikr}}{r}(-ik)(\mathbf{n}\cdot\mathbf{x^{% \prime}})
  40. 𝐩 = d 3 𝐱 𝐱 ρ ( 𝐱 ) \mathbf{p}=\int d^{3}\mathbf{x^{\prime}}\mathbf{x^{\prime}}\rho(\mathbf{x^{% \prime}})
  41. ρ Electric dipole ( 𝐱 , t ) = - i k 4 π ϵ 0 e i k r - i ω t r 𝐧 𝐩 \rho_{\,\text{Electric dipole}}(\mathbf{x},t)=\frac{-ik}{4\pi\epsilon_{0}}% \frac{e^{ikr-i\omega t}}{r}\mathbf{n}\cdot\mathbf{p}
  42. 𝐀 Electric dipole ( 𝐱 , t ) = - i ω μ 0 4 π e i k r - i ω t r 𝐩 \mathbf{A}_{\,\text{Electric dipole}}(\mathbf{x},t)=\frac{-i\omega\mu_{0}}{4% \pi}\frac{e^{ikr-i\omega t}}{r}\mathbf{p}
  43. 𝐄 ( 𝐱 , t ) = - ϕ ( 𝐱 , t ) - 𝐀 ( 𝐱 , t ) t \mathbf{E}(\mathbf{x},t)=-\mathbf{\nabla}\phi(\mathbf{x},t)-\frac{\partial% \mathbf{A}(\mathbf{x},t)}{\partial t}
  44. 𝐁 ( 𝐱 , t ) = × 𝐀 ( 𝐱 , t ) \mathbf{B}(\mathbf{x},t)=\mathbf{\nabla}\times\mathbf{A}(\mathbf{x},t)
  45. 𝐇 ( 𝐱 , t ) = 1 μ 0 × 𝐀 ( 𝐱 , t ) \mathbf{H}(\mathbf{x},t)=\frac{1}{\mu_{0}}\mathbf{\nabla}\times\mathbf{A}(% \mathbf{x},t)
  46. 𝐄 ( 𝐱 , t ) = i Z 0 k × 𝐇 ( 𝐱 , t ) \mathbf{E}(\mathbf{x},t)=\frac{iZ_{0}}{k}\mathbf{\nabla}\times\mathbf{H}(% \mathbf{x},t)
  47. Z 0 = μ 0 / ϵ 0 Z_{0}=\sqrt{\mu_{0}/\epsilon_{0}}
  48. 𝐇 Electric dipole ( 𝐱 , t ) = c k 2 4 π ( 𝐧 × 𝐩 ) e i k r - i ω t r \mathbf{H}_{\,\text{Electric dipole}}(\mathbf{x},t)=\frac{ck^{2}}{4\pi}(% \mathbf{n}\times\mathbf{p})\frac{e^{ikr-i\omega t}}{r}
  49. 𝐄 Electric dipole ( 𝐱 , t ) = Z 0 ( 𝐇 Electric dipole × 𝐧 ) \mathbf{E}_{\,\text{Electric dipole}}(\mathbf{x},t)=Z_{0}(\mathbf{H}_{\,\text{% Electric dipole}}\times\mathbf{n})
  50. 𝐒 = 𝐄 × 𝐇 \mathbf{S}=\mathbf{E}\times\mathbf{H}
  51. d P ( 𝐱 ) d Ω = r 2 2 ( 𝐧 𝐄 × 𝐇 ) \frac{dP(\mathbf{x})}{d\Omega}=\frac{r^{2}}{2}\Re(\mathbf{n}\cdot\mathbf{E}% \times\mathbf{H})
  52. 𝐧 \mathbf{n}
  53. r 2 r^{2}
  54. d P Electric dipole ( 𝐱 ) d Ω = c 2 Z 0 32 π 2 k 4 𝐩 2 2 sin 2 θ \frac{dP_{\,\text{Electric dipole}}(\mathbf{x})}{d\Omega}=\frac{c^{2}Z_{0}}{32% \pi^{2}}k^{4}\|\mathbf{p}\|_{2}^{2}\sin^{2}\theta
  55. 𝐩 \mathbf{p}
  56. P Electric dipole = c 2 Z 0 12 π k 4 𝐩 2 2 P_{\,\text{Electric dipole}}=\frac{c^{2}Z_{0}}{12\pi}k^{4}\|\mathbf{p}\|_{2}^{2}
  57. e i k 𝐱 - 𝐱 2 𝐱 - 𝐱 2 e i k r r ( - i k ) ( 𝐧 𝐱 ) \frac{e^{ik\|\mathbf{x}-\mathbf{x^{\prime}}\|_{2}}}{\|\mathbf{x}-\mathbf{x^{% \prime}}\|_{2}}\rightarrow\frac{e^{ikr}}{r}(-ik)(\mathbf{n}\cdot\mathbf{x^{% \prime}})
  58. 𝐀 Magnetic dipole / Electric quadrupole ( 𝐱 , t ) = μ 0 4 π e i k r - i ω t r ( - i k ) d 3 𝐱 ( 𝐧 𝐱 ) 𝐉 ( 𝐱 ) \mathbf{A}_{\,\text{Magnetic dipole / Electric quadrupole}}(\mathbf{x},t)=% \frac{\mu_{0}}{4\pi}\frac{e^{ikr-i\omega t}}{r}(-ik)\int d^{3}\mathbf{x^{% \prime}}(\mathbf{n}\cdot\mathbf{x^{\prime}})\mathbf{J}(\mathbf{x^{\prime}})
  59. ( 𝐧 𝐱 ) 𝐉 ( 𝐱 ) = 1 2 ( ( 𝐧 𝐱 ) 𝐉 ( 𝐱 ) + ( 𝐧 𝐉 ( 𝐱 ) ) 𝐱 ) + 1 2 ( 𝐱 × 𝐉 ( 𝐱 ) ) × 𝐧 (\mathbf{n}\cdot\mathbf{x^{\prime}})\mathbf{J}(\mathbf{x^{\prime}})=\frac{1}{2% }\left((\mathbf{n}\cdot\mathbf{x^{\prime}})\mathbf{J}(\mathbf{x^{\prime}})+(% \mathbf{n}\cdot\mathbf{J}(\mathbf{x^{\prime}}))\mathbf{x^{\prime}}\right)+% \frac{1}{2}(\mathbf{x^{\prime}}\times\mathbf{J}(\mathbf{x^{\prime}}))\times% \mathbf{n}
  60. 𝐌 effective ( 𝐱 ) = 1 / 2 ( 𝐱 × 𝐉 ( 𝐱 ) ) \mathbf{M}_{\,\text{effective}}(\mathbf{x^{\prime}})=1/2(\mathbf{x^{\prime}}% \times\mathbf{J}(\mathbf{x^{\prime}}))
  61. d 3 𝐱 𝐌 effective ( 𝐱 ) = 𝐦 \int d^{3}\mathbf{x^{\prime}}\mathbf{M}_{\,\text{effective}}(\mathbf{x^{\prime% }})=\mathbf{m}
  62. 𝐀 Magnetic dipole ( 𝐱 , t ) = - i k μ 0 4 π e i k r - i ω t r 𝐦 × 𝐧 \mathbf{A}_{\,\text{Magnetic dipole}}(\mathbf{x},t)=\frac{-ik\mu_{0}}{4\pi}% \frac{e^{ikr-i\omega t}}{r}\mathbf{m}\times\mathbf{n}
  63. 𝐀 Magnetic dipole \mathbf{A}_{\,\text{Magnetic dipole}}
  64. 𝐇 Electric dipole \mathbf{H}_{\,\text{Electric dipole}}
  65. 𝐄 Electric dipole Z 0 𝐇 Magnetic dipole \mathbf{E}_{\,\text{Electric dipole}}\rightarrow Z_{0}\mathbf{H}_{\,\text{% Magnetic dipole}}
  66. 𝐇 Electric dipole - 1 Z 0 𝐄 Magnetic dipole \mathbf{H}_{\,\text{Electric dipole}}\rightarrow\frac{-1}{Z_{0}}\mathbf{E}_{\,% \text{Magnetic dipole}}
  67. 𝐩 𝐦 / c \mathbf{p}\rightarrow\mathbf{m}/c
  68. 𝐄 Magnetic dipole ( 𝐱 , t ) = - k 2 Z 0 4 π ( 𝐧 × 𝐦 ) e i k r - i ω t r \mathbf{E}_{\,\text{Magnetic dipole}}(\mathbf{x},t)=\frac{-k^{2}Z_{0}}{4\pi}(% \mathbf{n}\times\mathbf{m})\frac{e^{ikr-i\omega t}}{r}
  69. 𝐇 Magnetic dipole ( 𝐱 , t ) = - 1 Z 0 ( 𝐄 Magnetic dipole × 𝐧 ) \mathbf{H}_{\,\text{Magnetic dipole}}(\mathbf{x},t)=\frac{-1}{Z_{0}}(\mathbf{E% }_{\,\text{Magnetic dipole}}\times\mathbf{n})
  70. d P Magnetic dipole ( 𝐱 ) d Ω = Z 0 32 π 2 k 4 𝐦 2 2 sin 2 θ \frac{dP_{\,\text{Magnetic dipole}}(\mathbf{x})}{d\Omega}=\frac{Z_{0}}{32\pi^{% 2}}k^{4}\|\mathbf{m}\|_{2}^{2}\sin^{2}\theta
  71. 𝐦 \mathbf{m}
  72. P Magnetic dipole = Z 0 12 π k 4 𝐦 2 2 P_{\,\text{Magnetic dipole}}=\frac{Z_{0}}{12\pi}k^{4}\|\mathbf{m}\|_{2}^{2}
  73. 1 2 d 3 𝐱 ( ( 𝐧 𝐱 ) 𝐉 ( 𝐱 ) + ( 𝐧 𝐉 ( 𝐱 ) ) 𝐱 ) = - i ω 2 d 3 𝐱 𝐱 ( 𝐧 𝐱 ) ρ ( 𝐱 ) \frac{1}{2}\int d^{3}\mathbf{x}\left((\mathbf{n}\cdot\mathbf{x^{\prime}})% \mathbf{J}(\mathbf{x^{\prime}})+(\mathbf{n}\cdot\mathbf{J}(\mathbf{x^{\prime}}% ))\mathbf{x^{\prime}}\right)=\frac{-i\omega}{2}\int d^{3}\mathbf{x^{\prime}}% \mathbf{x^{\prime}}(\mathbf{n}\cdot\mathbf{x^{\prime}})\rho(\mathbf{x^{\prime}})
  74. 𝐀 Electric quadrupole ( 𝐱 , t ) = - k ω μ 0 8 π e i k r - i ω t r d 3 𝐱 𝐱 ( 𝐧 𝐱 ) ρ ( 𝐱 ) \mathbf{A}_{\,\text{Electric quadrupole}}(\mathbf{x},t)=\frac{-k\omega\mu_{0}}% {8\pi}\frac{e^{ikr-i\omega t}}{r}\int d^{3}\mathbf{x^{\prime}}\mathbf{x^{% \prime}}(\mathbf{n}\cdot\mathbf{x^{\prime}})\rho(\mathbf{x^{\prime}})
  75. Q α β = d 3 𝐱 ( 3 x α x β - 𝐱 2 2 δ α β ) Q_{\alpha\beta}=\int d^{3}\mathbf{x^{\prime}}(3x^{\prime}_{\alpha}x^{\prime}_{% \beta}-\|\mathbf{x^{\prime}}\|_{2}^{2}\delta_{\alpha\beta})
  76. [ Q ( 𝐧 ) ] α = β Q α β n β [Q(\mathbf{n})]_{\alpha}=\sum_{\beta}Q_{\alpha\beta}n_{\beta}
  77. 𝐀 Electric quadrupole ( 𝐱 , t ) = - k ω μ 0 8 π e i k r - i ω t r 1 3 𝐐 ( 𝐧 ) \mathbf{A}_{\,\text{Electric quadrupole}}(\mathbf{x},t)=\frac{-k\omega\mu_{0}}% {8\pi}\frac{e^{ikr-i\omega t}}{r}\frac{1}{3}\mathbf{Q(n)}
  78. 𝐇 Electric quadrupole ( 𝐱 , t ) = - i c k 3 24 π e i k r - i ω t r 𝐧 × 𝐐 ( 𝐧 ) \mathbf{H}_{\,\text{Electric quadrupole}}(\mathbf{x},t)=\frac{-ick^{3}}{24\pi}% \frac{e^{ikr-i\omega t}}{r}\mathbf{n}\times\mathbf{Q(n)}
  79. 𝐄 Electric quadrupole ( 𝐱 , t ) = Z 0 ( 𝐇 Electric quadrupole × 𝐧 ) \mathbf{E}_{\,\text{Electric quadrupole}}(\mathbf{x},t)=Z_{0}(\mathbf{H}_{\,% \text{Electric quadrupole}}\times\mathbf{n})
  80. d P Electric quadrupole ( 𝐱 ) d Ω = c 2 Z 0 1152 π 2 k 6 ( 𝐧 × 𝐐 ( 𝐧 ) ) × 𝐧 2 2 \frac{dP_{\,\text{Electric quadrupole}}(\mathbf{x})}{d\Omega}=\frac{c^{2}Z_{0}% }{1152\pi^{2}}k^{6}\|(\mathbf{n}\times\mathbf{Q(n)})\times\mathbf{n}\|_{2}^{2}
  81. 𝐦 \mathbf{m}
  82. P Electric quadrupole = c 2 Z 0 1440 π k 6 α , β Q α β 2 P_{\,\text{Electric quadrupole}}=\frac{c^{2}Z_{0}}{1440\pi}k^{6}\sum_{\alpha,% \beta}Q_{\alpha\beta}^{2}
  83. ω \omega
  84. ρ ( 𝐱 , t ) = ρ ( 𝐱 ) e - i ω t \rho(\mathbf{x},t)=\rho(\mathbf{x})e^{-i\omega t}
  85. 𝐉 ( 𝐱 , t ) = 𝐉 ( 𝐱 ) e - i ω t \mathbf{J}(\mathbf{x},t)=\mathbf{J}(\mathbf{x})e^{-i\omega t}
  86. 𝐌 ( 𝐱 , t ) = 𝐌 ( 𝐱 ) e - i ω t \mathbf{M}(\mathbf{x},t)=\mathbf{M}(\mathbf{x})e^{-i\omega t}
  87. 𝐄 ( 𝐱 , t ) = 𝐄 ( 𝐱 ) e - i ω t \mathbf{E}(\mathbf{x},t)=\mathbf{E}(\mathbf{x})e^{-i\omega t}
  88. 𝐇 ( 𝐱 , t ) = 𝐇 ( 𝐱 ) e - i ω t \mathbf{H}(\mathbf{x},t)=\mathbf{H}(\mathbf{x})e^{-i\omega t}
  89. 𝐄 ( 𝐱 ) = - i Z 0 k 𝐉 ( 𝐱 ) \mathbf{\nabla}\cdot\mathbf{E}(\mathbf{x})=-\frac{iZ_{0}}{k}\mathbf{\nabla}% \cdot\mathbf{J}(\mathbf{x})
  90. 𝐇 ( 𝐱 ) = - 𝐌 ( 𝐱 ) \mathbf{\nabla}\cdot\mathbf{H}(\mathbf{x})=-\mathbf{\nabla}\cdot\mathbf{M}(% \mathbf{x})
  91. × 𝐄 ( 𝐱 ) = i k Z 0 ( 𝐇 ( 𝐱 ) + 𝐌 ( 𝐱 ) ) \mathbf{\nabla}\times\mathbf{E}(\mathbf{x})=ikZ_{0}\left(\mathbf{H}(\mathbf{x}% )+\mathbf{M}(\mathbf{x})\right)
  92. × 𝐇 ( 𝐱 ) = - i k Z 0 𝐄 ( 𝐱 ) + 𝐉 ( 𝐱 ) \mathbf{\nabla}\times\mathbf{H}(\mathbf{x})=-\frac{ik}{Z_{0}}\mathbf{E}(% \mathbf{x})+\mathbf{J}(\mathbf{x})
  93. × ( × 𝐕 ) = ( 𝐕 ) - 2 𝐕 \mathbf{\nabla}\times(\mathbf{\nabla}\times\mathbf{V})=\mathbf{\nabla}(\mathbf% {\nabla}\cdot\mathbf{V})-\mathbf{\nabla}^{2}\mathbf{V}
  94. ( 2 + k 2 ) 𝐄 ( 𝐱 ) = - [ i k Z 0 𝐉 ( 𝐱 ) + i k Z 0 × 𝐌 ( 𝐱 ) ) + i Z 0 k ( 𝐉 ( 𝐱 ) ) ] (\nabla^{2}+k^{2})\mathbf{E}(\mathbf{x})=-\left[ikZ_{0}\mathbf{J}(\mathbf{x})+% ikZ_{0}\mathbf{\nabla}\times\mathbf{M}(\mathbf{x}))+\frac{iZ_{0}}{k}\mathbf{% \nabla}(\mathbf{\nabla}\cdot\mathbf{J}(\mathbf{x}))\right]
  95. ( 2 + k 2 ) 𝐇 ( 𝐱 ) = - [ k 2 𝐌 ( 𝐱 ) + × 𝐉 ( 𝐱 ) + ( 𝐌 ( 𝐱 ) ) ] (\nabla^{2}+k^{2})\mathbf{H}(\mathbf{x})=-\left[k^{2}\mathbf{M}(\mathbf{x})+% \mathbf{\nabla}\times\mathbf{J}(\mathbf{x})+\mathbf{\nabla}(\mathbf{\nabla}% \cdot\mathbf{M}(\mathbf{x}))\right]
  96. ω \omega
  97. ( 2 + k 2 ) 𝚿 ( 𝐱 ) = 0 (\mathbf{\nabla}^{2}+k^{2})\mathbf{\Psi}(\mathbf{x})=0
  98. 𝚿 ( 𝐱 ) \mathbf{\Psi}(\mathbf{x})
  99. 𝚿 ( 𝐱 ) = = 0 m = - f m ( k r ) 𝐗 m ( θ , ϕ ) \mathbf{\Psi}(\mathbf{x})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}f_{\ell m% }(kr)\mathbf{X}_{\ell m}(\theta,\phi)
  100. f m ( k r ) = A m ( 1 ) h ( 1 ) ( k r ) + A m ( 2 ) h ( 2 ) ( k r ) f_{\ell m}(kr)=A_{\ell m}^{(1)}h_{\ell}^{(1)}(kr)+A_{\ell m}^{(2)}h_{\ell}^{(2% )}(kr)
  101. 𝐗 m ( θ , ϕ ) = 𝐋 Y m ( θ , ϕ ) / ( + 1 ) \mathbf{X}_{\ell m}(\theta,\phi)=\mathbf{L}Y_{\ell m}(\theta,\phi)/\sqrt{\ell(% \ell+1)}
  102. h ( 1 ) h_{\ell}^{(1)}
  103. h ( 2 ) h_{\ell}^{(2)}
  104. 𝐋 = - i 𝐱 × \mathbf{L}=-i\mathbf{x}\times\mathbf{\nabla}
  105. L 2 Y m = ( + 1 ) Y m L^{2}Y_{\ell m}=\ell(\ell+1)Y_{\ell m}
  106. A m ( 1 ) A_{\ell m}^{(1)}
  107. A m ( 2 ) A_{\ell m}^{(2)}
  108. A m ( 2 ) = 0 A_{\ell m}^{(2)}=0
  109. ( 2 + k 2 ) 𝚿 ( 𝐱 ) = - 𝐕 ( 𝐱 ) (\mathbf{\nabla}^{2}+k^{2})\mathbf{\Psi}(\mathbf{x})=-\mathbf{V}(\mathbf{x})
  110. Ψ α ( 𝐱 ) = β d 3 𝐱 G α β ( 𝐱 , 𝐱 ) V β ( 𝐱 ) \Psi_{\alpha}(\mathbf{x})=\sum_{\beta}\int d^{3}\mathbf{x^{\prime}}G_{\alpha% \beta}(\mathbf{x},\mathbf{x^{\prime}})V_{\beta}(\mathbf{x^{\prime}})
  111. G α β ( 𝐱 , 𝐱 ) = = 0 m = - i k h ( 1 ) ( k r ) j ( k r ) X m α ( θ , ϕ ) X m β * ( θ , ϕ ) G_{\alpha\beta}(\mathbf{x},\mathbf{x^{\prime}})=\sum_{\ell=0}^{\infty}\sum_{m=% -\ell}^{\ell}ikh_{\ell}^{(1)}(kr)j_{\ell}(kr^{\prime})X_{\ell m\alpha}(\theta,% \phi)X_{\ell m\beta}^{*}(\theta^{\prime},\phi^{\prime})
  112. 𝐗 m * = Y m * 𝐋 / ( + 1 ) \mathbf{X}_{\ell m}^{*}=Y_{\ell m}^{*}\mathbf{L}/\sqrt{\ell(\ell+1)}
  113. 𝐕 \mathbf{V}
  114. 𝚿 ( 𝐱 ) = = 0 m = - i k ( + 1 ) h ( 1 ) ( k r ) 𝐗 m ( θ , ϕ ) d 3 𝐱 j ( k r ) Y m * ( θ , ϕ ) 𝐋 𝐕 ( 𝐱 ) \mathbf{\Psi}(\mathbf{x})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\frac{ik}% {\sqrt{\ell(\ell+1)}}h_{\ell}^{(1)}(kr)\mathbf{X}_{\ell m}(\theta,\phi)\int d^% {3}\mathbf{x^{\prime}}j_{\ell}(kr^{\prime})Y_{\ell m}^{*}(\theta^{\prime},\phi% ^{\prime})\mathbf{L^{\prime}}\cdot\mathbf{V}(\mathbf{x^{\prime}})
  115. ( 2 + k 2 ) 𝐇 ( 𝐱 ) = - [ k 2 𝐌 ( 𝐱 ) + × 𝐉 ( 𝐱 ) + ( 𝐌 ( 𝐱 ) ) ] (\nabla^{2}+k^{2})\mathbf{H}(\mathbf{x})=-\left[k^{2}\mathbf{M}(\mathbf{x})+% \mathbf{\nabla}\times\mathbf{J}(\mathbf{x})+\mathbf{\nabla}(\mathbf{\nabla}% \cdot\mathbf{M}(\mathbf{x}))\right]
  116. 𝐇 ( E ) ( 𝐱 ) = = 0 m = - a m ( E ) h ( 1 ) ( k r ) 𝐗 m ( θ , ϕ ) \mathbf{H}^{(E)}(\mathbf{x})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a_{% \ell m}^{(E)}h_{\ell}^{(1)}(kr)\mathbf{X}_{\ell m}(\theta,\phi)
  117. a m ( E ) = i k ( + 1 ) d 3 𝐱 j ( k r ) Y m * ( θ , ϕ ) 𝐋 [ k 2 𝐌 ( 𝐱 ) + × 𝐉 ( 𝐱 ) + ( 𝐌 ( 𝐱 ) ) ] a_{\ell m}^{(E)}=\frac{ik}{\sqrt{\ell(\ell+1)}}\int d^{3}\mathbf{x^{\prime}}j_% {\ell}(kr^{\prime})Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})\mathbf{L^{% \prime}}\cdot\left[k^{2}\mathbf{M}(\mathbf{x^{\prime}})+\mathbf{\nabla^{\prime% }}\times\mathbf{J}(\mathbf{x^{\prime}})+\mathbf{\nabla^{\prime}}(\mathbf{% \nabla^{\prime}}\cdot\mathbf{M}(\mathbf{x^{\prime}}))\right]
  118. 𝐄 ( E ) ( 𝐱 ) = i Z 0 k × 𝐇 ( E ) ( 𝐱 ) \mathbf{E}^{(E)}(\mathbf{x})=\frac{iZ_{0}}{k}\mathbf{\nabla}\times\mathbf{H}^{% (E)}(\mathbf{x})
  119. 𝐋 𝐕 ( 𝐱 ) = i ( 𝐱 × 𝐕 ( 𝐱 ) ) \mathbf{L}\cdot\mathbf{V}(\mathbf{x})=i\mathbf{\nabla}\cdot(\mathbf{x}\times% \mathbf{V}(\mathbf{x}))
  120. 𝐋 ( × 𝐕 ( 𝐱 ) ) = i 2 ( 𝐱 𝐕 ( 𝐱 ) ) - i r r ( r 2 𝐕 ( 𝐱 ) ) \mathbf{L}\cdot(\mathbf{\nabla}\times\mathbf{V}(\mathbf{x}))=i\nabla^{2}(% \mathbf{x}\cdot\mathbf{V}(\mathbf{x}))-\frac{i\partial}{r\partial r}(r^{2}% \mathbf{\nabla}\cdot\mathbf{V}(\mathbf{x}))
  121. 𝐋 s ( 𝐱 ) = 0 \mathbf{L}\cdot\mathbf{\nabla}s(\mathbf{x})=0
  122. a m ( E ) = - i k 2 ( + 1 ) d 3 𝐱 j ( k r ) Y m * ( θ , ϕ ) [ - i k ( 𝐱 × 𝐌 ( 𝐱 ) ) - i k 2 ( 𝐱 𝐉 ( 𝐱 ) ) - c r r ( r 2 ρ ( 𝐱 ) ) ] a_{\ell m}^{(E)}=\frac{-ik^{2}}{\sqrt{\ell(\ell+1)}}\int d^{3}\mathbf{x^{% \prime}}j_{\ell}(kr^{\prime})Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})% \left[-ik\mathbf{\nabla}\cdot(\mathbf{x^{\prime}}\times\mathbf{M}(\mathbf{x^{% \prime}}))-\frac{i}{k}\nabla^{2}(\mathbf{x^{\prime}}\cdot\mathbf{J}(\mathbf{x^% {\prime}}))-\frac{c\partial}{r^{\prime}\partial r^{\prime}}(r^{\prime 2}\rho(% \mathbf{x^{\prime}}))\right]
  123. a m ( E ) = - i k 2 ( + 1 ) d 3 𝐱 j ( k r ) Y m * ( θ , ϕ ) [ - i k ( 𝐱 × 𝐌 ( 𝐱 ) ) + i k 𝐱 𝐉 ( 𝐱 ) ] + c Y m * ( θ , ϕ ) ρ ( 𝐱 ) r ( r j ( k r ) ) a_{\ell m}^{(E)}=\frac{-ik^{2}}{\sqrt{\ell(\ell+1)}}\int d^{3}\mathbf{x^{% \prime}}j_{\ell}(kr^{\prime})Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})% \left[-ik\mathbf{\nabla}\cdot(\mathbf{x^{\prime}}\times\mathbf{M}(\mathbf{x^{% \prime}}))+ik\mathbf{x^{\prime}}\cdot\mathbf{J}(\mathbf{x^{\prime}})\right]+cY% _{\ell m}^{*}(\theta^{\prime},\phi^{\prime})\rho(\mathbf{x^{\prime}})\frac{% \partial}{\partial r^{\prime}}(r^{\prime}j_{\ell}(kr^{\prime}))
  124. j ( k r ) j_{\ell}(kr^{\prime})
  125. j ( k r ) = ( k r ) ( 2 + 1 ) ! ! + O ( ( k r ) + 2 ) j_{\ell}(kr^{\prime})=\frac{(kr^{\prime})^{\ell}}{(2\ell+1)!!}+O((kr^{\prime})% ^{\ell+2})
  126. a m ( E ) = - i c k + 2 ( 2 + 1 ) ! ! ( + 1 ) 1 / 2 [ Q m + Q m ] a_{\ell m}^{(E)}=\frac{-ick^{\ell+2}}{(2\ell+1)!!}\left(\frac{\ell+1}{\ell}% \right)^{1/2}[Q_{\ell m}+Q_{\ell m}^{\prime}]
  127. Q m = d 3 𝐱 r Y m * ( θ , ϕ ) ρ ( 𝐱 ) Q_{\ell m}=\int d^{3}\mathbf{x^{\prime}}r^{\prime\ell}Y_{\ell m}^{*}(\theta^{% \prime},\phi^{\prime})\rho(\mathbf{x^{\prime}})
  128. Q m = - i k c ( + 1 ) d 3 𝐱 r Y m * ( θ , ϕ ) ( 𝐱 × 𝐌 ( 𝐱 ) ) Q_{\ell m}^{\prime}=-\frac{ik}{c(\ell+1)}\int d^{3}\mathbf{x^{\prime}}r^{% \prime\ell}Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})\mathbf{\nabla}\cdot(% \mathbf{x^{\prime}}\times\mathbf{M}(\mathbf{x^{\prime}}))
  129. Q m Q_{\ell m}
  130. ρ ( 𝐱 ) \rho(\mathbf{x})
  131. Q m Q_{\ell m}^{\prime}
  132. ( 2 + k 2 ) 𝐄 ( 𝐱 ) = - [ i k Z 0 𝐉 ( 𝐱 ) + i k Z 0 × 𝐌 ( 𝐱 ) ) + i Z 0 k ( 𝐉 ( 𝐱 ) ) ] (\nabla^{2}+k^{2})\mathbf{E}(\mathbf{x})=-\left[ikZ_{0}\mathbf{J}(\mathbf{x})+% ikZ_{0}\mathbf{\nabla}\times\mathbf{M}(\mathbf{x}))+\frac{iZ_{0}}{k}\mathbf{% \nabla}(\mathbf{\nabla}\cdot\mathbf{J}(\mathbf{x}))\right]
  133. 𝐄 ( M ) ( 𝐱 ) = = 0 m = - a m ( M ) h ( 1 ) ( k r ) 𝐗 m ( θ , ϕ ) \mathbf{E}^{(M)}(\mathbf{x})=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a_{% \ell m}^{(M)}h_{\ell}^{(1)}(kr)\mathbf{X}_{\ell m}(\theta,\phi)
  134. a m ( M ) = i k ( + 1 ) d 3 𝐱 j ( k r ) Y m * ( θ , ϕ ) 𝐋 [ i k Z 0 𝐉 ( 𝐱 ) + i k Z 0 × 𝐌 ( 𝐱 ) ) + i Z 0 k ( 𝐉 ( 𝐱 ) ) ] a_{\ell m}^{(M)}=\frac{ik}{\sqrt{\ell(\ell+1)}}\int d^{3}\mathbf{x^{\prime}}j_% {\ell}(kr^{\prime})Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})\mathbf{L^{% \prime}}\cdot\left[ikZ_{0}\mathbf{J}(\mathbf{x})+ikZ_{0}\mathbf{\nabla}\times% \mathbf{M}(\mathbf{x}))+\frac{iZ_{0}}{k}\mathbf{\nabla}(\mathbf{\nabla}\cdot% \mathbf{J}(\mathbf{x}))\right]
  135. 𝐇 ( M ) ( 𝐱 ) = - i k Z 0 × 𝐄 ( M ) ( 𝐱 ) \mathbf{H}^{(M)}(\mathbf{x})=-\frac{i}{kZ_{0}}\mathbf{\nabla}\times\mathbf{E}^% {(M)}(\mathbf{x})
  136. a m ( M ) = - i k 2 ( + 1 ) d 3 𝐱 j ( k r ) Y m * ( θ , ϕ ) [ ( 𝐱 × 𝐉 ( 𝐱 ) ) - k 2 𝐱 𝐌 ( 𝐱 ) ] + Y m * ( θ , ϕ ) 𝐌 ( 𝐱 ) r ( r j ( k r ) ) a_{\ell m}^{(M)}=\frac{-ik^{2}}{\sqrt{\ell(\ell+1)}}\int d^{3}\mathbf{x^{% \prime}}j_{\ell}(kr^{\prime})Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})% \left[\mathbf{\nabla}\cdot(\mathbf{x^{\prime}}\times\mathbf{J}(\mathbf{x^{% \prime}}))-k^{2}\mathbf{x^{\prime}}\cdot\mathbf{M}(\mathbf{x^{\prime}})\right]% +Y_{\ell m}^{*}(\theta^{\prime},\phi^{\prime})\mathbf{\nabla}\cdot\mathbf{M}(% \mathbf{x^{\prime}})\frac{\partial}{\partial r^{\prime}}(r^{\prime}j_{\ell}(kr% ^{\prime}))
  137. a m ( M ) = - i k + 2 ( 2 + 1 ) ! ! ( + 1 ) 1 / 2 [ M m + M m ] a_{\ell m}^{(M)}=\frac{-ik^{\ell+2}}{(2\ell+1)!!}\left(\frac{\ell+1}{\ell}% \right)^{1/2}[M_{\ell m}+M_{\ell m}^{\prime}]
  138. M m = 1 + 1 d 3 𝐱 r Y m * ( θ , ϕ ) ( 𝐱 × 𝐉 ( 𝐱 ) ) M_{\ell m}=\frac{1}{\ell+1}\int d^{3}\mathbf{x^{\prime}}r^{\prime\ell}Y_{\ell m% }^{*}(\theta^{\prime},\phi^{\prime})\mathbf{\nabla}\cdot(\mathbf{x^{\prime}}% \times\mathbf{J}(\mathbf{x^{\prime}}))
  139. M m = d 3 𝐱 r Y m * ( θ , ϕ ) 𝐌 ( 𝐱 ) M_{\ell m}^{\prime}=\int d^{3}\mathbf{x^{\prime}}r^{\prime\ell}Y_{\ell m}^{*}(% \theta^{\prime},\phi^{\prime})\mathbf{\nabla}\cdot\mathbf{M}(\mathbf{x^{\prime% }})
  140. M m M_{\ell m}
  141. 𝐱 × 𝐉 ( 𝐱 ) / 2 \mathbf{x}\times\mathbf{J}(\mathbf{x})/2
  142. M m M_{\ell m}^{\prime}
  143. 𝐌 ( 𝐱 ) \mathbf{M}(\mathbf{x})
  144. 𝐄 ( 𝐱 , t ) = ( = 0 m = - [ a m ( M ) h ( 1 ) ( k r ) 𝐗 m ( θ , ϕ ) + i Z 0 k a m ( E ) × ( h ( 1 ) ( k r ) 𝐗 m ( θ , ϕ ) ) ] e - i ω t ) \mathbf{E}(\mathbf{x},t)=\Re\left(\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}% \left[a_{\ell m}^{(M)}h_{\ell}^{(1)}(kr)\mathbf{X}_{\ell m}(\theta,\phi)+\frac% {iZ_{0}}{k}a_{\ell m}^{(E)}\mathbf{\nabla}\times(h_{\ell}^{(1)}(kr)\mathbf{X}_% {\ell m}(\theta,\phi))\right]e^{-i\omega t}\right)
  145. 𝐇 ( 𝐱 , t ) = ( = 0 m = - [ a m ( E ) h ( 1 ) ( k r ) 𝐗 m ( θ , ϕ ) - i k Z 0 a m ( M ) × ( h ( 1 ) ( k r ) 𝐗 m ( θ , ϕ ) ) ] e - i ω t ) \mathbf{H}(\mathbf{x},t)=\Re\left(\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}% \left[a_{\ell m}^{(E)}h_{\ell}^{(1)}(kr)\mathbf{X}_{\ell m}(\theta,\phi)-\frac% {i}{kZ_{0}}a_{\ell m}^{(M)}\mathbf{\nabla}\times(h_{\ell}^{(1)}(kr)\mathbf{X}_% {\ell m}(\theta,\phi))\right]e^{-i\omega t}\right)
  146. h ( 1 ) ( k r ) h_{\ell}^{(1)}(kr)
  147. 1 / r 1 1/r\ll 1
  148. h ( 1 ) ( k r ) = ( - i ) + 1 e i k r k r + O ( 1 / r 2 ) h_{\ell}^{(1)}(kr)=(-i)^{\ell+1}\frac{e^{ikr}}{kr}+O(1/r^{2})

Multiscroll_attractor.html

  1. d x ( t ) d t = a * ( y ( t ) - x ( t ) ) \frac{dx(t)}{dt}=a*(y(t)-x(t))
  2. d y ( t ) d t = ( c - a ) * x ( t ) - x ( t ) * z ( t ) + c * y ( t ) \frac{dy(t)}{dt}=(c-a)*x(t)-x(t)*z(t)+c*y(t)
  3. d z ( t ) d t = x ( t ) * y ( t ) - b * z ( t ) \frac{dz(t)}{dt}=x(t)*y(t)-b*z(t)
  4. d x ( t ) d t = a * ( y ( t ) - x ( t ) ) \frac{dx(t)}{dt}=a*(y(t)-x(t))
  5. d y ( t ) d t = x ( t ) - x ( t ) * z ( t ) + c * y ( t ) + u \frac{dy(t)}{dt}=x(t)-x(t)*z(t)+c*y(t)+u
  6. d z ( t ) d t = x ( t ) * y ( t ) - b * z ( t ) \frac{dz(t)}{dt}=x(t)*y(t)-b*z(t)
  7. d x ( t ) d t = a * ( y ( t ) - x ( t ) ) , \frac{dx(t)}{dt}=a*(y(t)-x(t)),
  8. d y ( t ) d t = ( c - a ) * x ( t ) - x ( t ) * f + c * y ( t ) , \frac{dy(t)}{dt}=(c-a)*x(t)-x(t)*f+c*y(t),
  9. d z ( t ) d t = x ( t ) * y ( t ) - b * z ( t ) \frac{dz(t)}{dt}=x(t)*y(t)-b*z(t)
  10. f = d 0 * z ( t ) + d 1 * z ( t - τ ) - d 2 * sin ( z ( t - τ ) ) f=d0*z(t)+d1*z(t-\tau)-d2*\sin(z(t-\tau))
  11. d x ( t ) d t = α * ( y ( t ) - h ) \frac{dx(t)}{dt}=\alpha*(y(t)-h)
  12. d y ( t ) d t = x ( t ) - y ( t ) + z ( t ) \frac{dy(t)}{dt}=x(t)-y(t)+z(t)
  13. d z ( t ) d t = - β * y ( t ) \frac{dz(t)}{dt}=-\beta*y(t)
  14. h := - b * s i n ( π * x ( t ) 2 * a + d ) h:=-b*sin(\frac{\pi*x(t)}{2*a}+d)
  15. d x ( t ) d t = y ( t ) \frac{dx(t)}{dt}=y(t)
  16. d y ( t ) d t = - m 1 * x ( t ) - ( 1 / 2 * ( m 0 - m 1 ) ) * ( | x ( t ) + 1 | - | x ( t ) - 1 | ) - e * y ( t ) + γ * c o s ( ω * t ) \frac{dy(t)}{dt}=-m1*x(t)-(1/2*(m0-m1))*(|x(t)+1|-|x(t)-1|)-e*y(t)+\gamma*cos(% \omega*t)
  17. d x ( t ) d t = 1 / 3 * ( - ( a + 1 ) * x ( t ) + a - c + z ( t ) * y ( t ) ) + ( ( 1 - a ) * ( x ( t ) 2 - y ( t ) 2 ) + ( 2 * ( a + c - z ( t ) ) ) * x ( t ) * y ( t ) ) \frac{dx(t)}{dt}=1/3*(-(a+1)*x(t)+a-c+z(t)*y(t))+((1-a)*(x(t)^{2}-y(t)^{2})+(2% *(a+c-z(t)))*x(t)*y(t))
  18. * 1 3 * x ( t ) 2 + y ( t ) 2 *\frac{1}{3*\sqrt{x(t)^{2}+y(t)^{2}}}
  19. d y ( t ) d t = 1 / 3 * ( ( c - a - z ( t ) ) * x ( t ) - ( a + 1 ) * y ( t ) ) + ( ( 2 * ( a - 1 ) ) * x ( t ) * y ( t ) + ( a + c - z ( t ) ) * ( x ( t ) 2 - y ( t ) 2 ) ) \frac{dy(t)}{dt}=1/3*((c-a-z(t))*x(t)-(a+1)*y(t))+((2*(a-1))*x(t)*y(t)+(a+c-z(% t))*(x(t)^{2}-y(t)^{2}))
  20. * 1 3 * x ( t ) 2 + y ( t ) 2 *\frac{1}{3*\sqrt{x(t)^{2}+y(t)^{2}}}
  21. d z ( t d t = 1 / 2 * ( 3 * x ( t ) 2 * y ( t ) - y ( t ) 3 ) - b * z ( t ) \frac{dz(t}{dt}=1/2*(3*x(t)^{2}*y(t)-y(t)^{3})-b*z(t)

Multivariate_kernel_density_estimation.html

  1. f ^ H ( x ) = 1 n i = 1 n K H ( x - x i ) \hat{f}_{H}({x})=\frac{1}{n}\sum_{i=1}^{n}K_{H}({x}-{x}_{i})
  2. MISE ( H ) = E [ ( f ^ H ( x ) - f ( x ) ) 2 d x ] . \operatorname{MISE}({H})=\operatorname{E}\!\left[\,\int(\hat{f}_{H}({x})-f({x}% ))^{2}\,d{x}\;\right].
  3. AMISE ( H ) = n - 1 | H | - 1 / 2 R ( K ) + 1 4 m 2 ( K ) 2 ( vec T H ) Ψ 4 ( vec H ) \operatorname{AMISE}({H})=n^{-1}|{H}|^{-1/2}R(K)+\tfrac{1}{4}m_{2}(K)^{2}(% \operatorname{vec}^{T}{H}){\Psi}_{4}(\operatorname{vec}{H})
  4. R ( K ) = K ( x ) 2 d x R(K)=\int K({x})^{2}\,d{x}
  5. x x T K ( x ) d x = m 2 ( K ) I d \int{x}{x}^{T}K({x})\,d{x}=m_{2}(K){I}_{d}
  6. Ψ 4 = ( vec D 2 f ( x ) ) ( vec T D 2 f ( x ) ) d x {\Psi}_{4}=\int(\operatorname{vec}\,\operatorname{D}^{2}f({x}))(\operatorname{% vec}^{T}\operatorname{D}^{2}f({x}))\,d{x}
  7. vec [ a c b d ] = [ a b c d ] T . \operatorname{vec}\begin{bmatrix}a&c\\ b&d\end{bmatrix}=\begin{bmatrix}a&b&c&d\end{bmatrix}^{T}.
  8. MISE ( H ) = AMISE ( H ) + o ( n - 1 | H | - 1 / 2 + tr H 2 ) \operatorname{MISE}({H})=\operatorname{AMISE}({H})+o(n^{-1}|{H}|^{-1/2}+% \operatorname{tr}\,{H}^{2})
  9. H AMISE = argmin H F AMISE ( H ) . {H}_{\operatorname{AMISE}}=\operatorname{argmin}_{{H}\in F}\,\operatorname{% AMISE}({H}).
  10. Ψ ^ 4 \hat{{\Psi}}_{4}
  11. PI ( H ) = n - 1 | H | - 1 / 2 R ( K ) + 1 4 m 2 ( K ) 2 ( vec T H ) Ψ ^ 4 ( G ) ( vec H ) \operatorname{PI}({H})=n^{-1}|{H}|^{-1/2}R(K)+\tfrac{1}{4}m_{2}(K)^{2}(% \operatorname{vec}^{T}{H})\hat{{\Psi}}_{4}({G})(\operatorname{vec}\,{H})
  12. Ψ ^ 4 ( G ) = n - 2 i = 1 n j = 1 n [ ( vec D 2 ) ( vec T D 2 ) ] K G ( X i - X j ) \hat{{\Psi}}_{4}({G})=n^{-2}\sum_{i=1}^{n}\sum_{j=1}^{n}[(\operatorname{vec}\,% \operatorname{D}^{2})(\operatorname{vec}^{T}\operatorname{D}^{2})]K_{G}({X}_{i% }-{X}_{j})
  13. H ^ PI = argmin H F PI ( H ) \hat{{H}}_{\operatorname{PI}}=\operatorname{argmin}_{{H}\in F}\,\operatorname{% PI}({H})
  14. H ^ PI \hat{{H}}_{\operatorname{PI}}
  15. SCV ( H ) = n - 1 | H | - 1 / 2 R ( K ) + n - 2 i = 1 n j = 1 n ( K 2 H + 2 G - 2 K H + 2 G + K 2 G ) ( X i - X j ) \operatorname{SCV}({H})=n^{-1}|{H}|^{-1/2}R(K)+n^{-2}\sum_{i=1}^{n}\sum_{j=1}^% {n}(K_{2{H}+2{G}}-2K_{{H}+2{G}}+K_{2{G}})({X}_{i}-{X}_{j})
  16. H ^ SCV = argmin H F SCV ( H ) \hat{{H}}_{\operatorname{SCV}}=\operatorname{argmin}_{{H}\in F}\,\operatorname% {SCV}({H})
  17. H ^ SCV \hat{{H}}_{\operatorname{SCV}}
  18. 𝐇 i i = ( 4 d + 2 ) 1 d + 4 n - 1 d + 4 σ i \sqrt{\mathbf{H}_{ii}}=\left(\frac{4}{d+2}\right)^{\frac{1}{d+4}}n^{\frac{-1}{% d+4}}\sigma_{i}
  19. σ i \sigma_{i}
  20. 𝐇 i j = 0 , i j \mathbf{H}_{ij}=0,i\neq j
  21. 𝐇 i i = n - 1 d + 4 σ i \sqrt{\mathbf{H}_{ii}}=n^{\frac{-1}{d+4}}\sigma_{i}
  22. E f ^ ( x ; H ) = K H * f ( x ) = f ( x ) + 1 2 m 2 ( K ) tr ( H D 2 f ( x ) ) d x + O ( tr H 2 ) \operatorname{E}\hat{f}({x};{H})=K_{H}*f({x})=f({x})+\frac{1}{2}m_{2}(K)\int% \operatorname{tr}({H}\operatorname{D}^{2}f({x}))\,d{x}+O(\operatorname{tr}\,{H% }^{2})
  23. Var f ^ ( x ; H ) = n - 1 | H | - 1 / 2 R ( K ) + o ( n - 1 | H | - 1 / 2 ) . \operatorname{Var}\hat{f}({x};{H})=n^{-1}|{H}|^{-1/2}R(K)+o(n^{-1}|{H}|^{-1/2}).
  24. MSE f ^ ( x ; H ) = Var f ^ ( x ; H ) + [ E f ^ ( x ; H ) - f ( x ) ] 2 \operatorname{MSE}\,\hat{f}({x};{H})=\operatorname{Var}\hat{f}({x};{H})+[% \operatorname{E}\hat{f}({x};{H})-f({x})]^{2}
  25. vec ( H ^ - H AMISE ) = O ( n - 2 α ) vec H AMISE . \operatorname{vec}(\hat{{H}}-{H}_{\operatorname{AMISE}})=O(n^{-2\alpha})% \operatorname{vec}{H}_{\operatorname{AMISE}}.
  26. H ^ PI = [ 0.052 0.510 0.510 8.882 ] . \hat{{H}}_{\operatorname{PI}}=\begin{bmatrix}0.052&0.510\\ 0.510&8.882\end{bmatrix}.
  27. MIAE ( H ) = E | f ^ H ( x ) - f ( x ) | d x . \operatorname{MIAE}({H})=\operatorname{E}\,\int|\hat{f}_{H}({x})-f({x})|\,d{x}.
  28. MUAE ( H ) = E sup x | f ^ H ( x ) - f ( x ) | . \operatorname{MUAE}({H})=\operatorname{E}\,\operatorname{sup}_{{x}}|\hat{f}_{H% }({x})-f({x})|.
  29. MKL ( H ) = f ( x ) log [ f ( x ) ] d x - E f ( x ) log [ f ^ ( x ; H ) ] d x \operatorname{MKL}({H})=\int f({x})\,\operatorname{log}[f({x})]\,d{x}-% \operatorname{E}\int f({x})\,\operatorname{log}[\hat{f}({x};{H})]\,d{x}
  30. MH ( H ) = E ( f ^ H ( x ) 1 / 2 - f ( x ) 1 / 2 ) 2 d x . \operatorname{MH}({H})=\operatorname{E}\int(\hat{f}_{H}({x})^{1/2}-f({x})^{1/2% })^{2}\,d{x}.

Multivariate_Pareto_distribution.html

  1. F ( x 1 , x 2 ) = 1 - i = 1 2 ( x i θ i ) - a + ( i = 1 2 x i θ i - 1 ) - a , x i > θ i > 0 , i = 1 , 2 ; a > 0 , F(x_{1},x_{2})=1-\sum_{i=1}^{2}\left(\frac{x_{i}}{\theta_{i}}\right)^{-a}+% \left(\sum_{i=1}^{2}\frac{x_{i}}{\theta_{i}}-1\right)^{-a},\qquad x_{i}>\theta% _{i}>0,i=1,2;a>0,
  2. f ( x 1 , x 2 ) = ( a + 1 ) a ( θ 1 θ 2 ) a + 1 ( θ 2 x 1 + θ 1 x 2 - θ 1 θ 2 ) - ( a + 2 ) , x i θ i > 0 , i = 1 , 2 ; a > 0. f(x_{1},x_{2})=(a+1)a(\theta_{1}\theta_{2})^{a+1}(\theta_{2}x_{1}+\theta_{1}x_% {2}-\theta_{1}\theta_{2})^{-(a+2)},\qquad x_{i}\geq\theta_{i}>0,i=1,2;a>0.
  3. f ( x i ) = a θ i a x i - ( a + 1 ) , x i θ i > 0 , i = 1 , 2. f(x_{i})=a\theta_{i}^{a}x_{i}^{-(a+1)},\qquad x_{i}\geq\theta_{i}>0,i=1,2.
  4. E [ X i ] = a θ i a - 1 , a > 1 ; V a r ( X i ) = a θ i 2 ( a - 1 ) 2 ( a - 2 ) , a > 2 ; i = 1 , 2 , E[X_{i}]=\frac{a\theta_{i}}{a-1},a>1;\quad Var(X_{i})=\frac{a\theta_{i}^{2}}{(% a-1)^{2}(a-2)},a>2;\quad i=1,2,
  5. cov ( X 1 , X 2 ) = θ 1 θ 2 ( a - 1 ) 2 ( a - 2 ) , and cor ( X 1 , X 2 ) = 1 a . \operatorname{cov}(X_{1},X_{2})=\frac{\theta_{1}\theta_{2}}{(a-1)^{2}(a-2)},\,% \text{ and }\operatorname{cor}(X_{1},X_{2})=\frac{1}{a}.
  6. F ¯ ( x 1 , x 2 ) = ( 1 + i = 1 2 x i - θ i θ i ) - a , x i > θ i , i = 1 , 2. \overline{F}(x_{1},x_{2})=\left(1+\sum_{i=1}^{2}\frac{x_{i}-\theta_{i}}{\theta% _{i}}\right)^{-a},\qquad x_{i}>\theta_{i},i=1,2.
  7. F ¯ ( x 1 , x 2 ) = ( 1 + i = 1 2 x i - μ i σ i ) - a , x i > μ i , i = 1 , 2 , \overline{F}(x_{1},x_{2})=\left(1+\sum_{i=1}^{2}\frac{x_{i}-\mu_{i}}{\sigma_{i% }}\right)^{-a},\qquad x_{i}>\mu_{i},i=1,2,
  8. E [ X i ] = μ i + σ i a - 1 , i = 1 , 2 , E[X_{i}]=\mu_{i}+\frac{\sigma_{i}}{a-1},\qquad i=1,2,
  9. f ( x 1 , , x k ) = a ( a + 1 ) ( a + k - 1 ) ( i = 1 k θ i ) - 1 ( i = 1 k x i θ i - k + 1 ) - ( a + k ) , x i > θ i > 0 , a > 0 , ( 1 ) f(x_{1},\dots,x_{k})=a(a+1)\cdots(a+k-1)\left(\prod_{i=1}^{k}\theta_{i}\right)% ^{-1}\left(\sum_{i=1}^{k}\frac{x_{i}}{\theta_{i}}-k+1\right)^{-(a+k)},\qquad x% _{i}>\theta_{i}>0,a>0,\qquad(1)
  10. F ¯ ( x 1 , , x k ) = ( i = 1 k x i θ i - k + 1 ) - a , x i > θ i > 0 , i = 1 , , k ; a > 0. ( 2 ) \overline{F}(x_{1},\dots,x_{k})=\left(\sum_{i=1}^{k}\frac{x_{i}}{\theta_{i}}-k% +1\right)^{-a},\qquad x_{i}>\theta_{i}>0,i=1,\dots,k;a>0.\quad(2)
  11. E [ X i ] = a θ i a - 1 , for a > 1 , and V a r ( X i ) = a θ i 2 ( a - 1 ) 2 ( a - 2 ) , for a > 2. E[X_{i}]=\frac{a\theta_{i}}{a-1},\,\text{ for }a>1,\,\text{ and }Var(X_{i})=% \frac{a\theta_{i}^{2}}{(a-1)^{2}(a-2)},\,\text{ for }a>2.
  12. cov ( X i , X j ) = θ i θ j ( a - 1 ) 2 ( a - 2 ) , cor ( X i , X j ) = 1 a , i j . \operatorname{cov}(X_{i},X_{j})=\frac{\theta_{i}\theta_{j}}{(a-1)^{2}(a-2)},% \qquad\operatorname{cor}(X_{i},X_{j})=\frac{1}{a},\qquad i\neq j.
  13. F ¯ ( x 1 , , x k ) = ( 1 + i = 1 k x i - θ i θ i ) - a , x i > θ i > 0 , i = 1 , , k . \overline{F}(x_{1},\dots,x_{k})=\left(1+\sum_{i=1}^{k}\frac{x_{i}-\theta_{i}}{% \theta_{i}}\right)^{-a},\qquad x_{i}>\theta_{i}>0,\quad i=1,\dots,k.
  14. F ¯ ( x 1 , , x k ) = ( 1 + i = 1 k x i - μ i σ i ) - a , x i > μ i , i = 1 , , k , ( 3 ) \overline{F}(x_{1},\dots,x_{k})=\left(1+\sum_{i=1}^{k}\frac{x_{i}-\mu_{i}}{% \sigma_{i}}\right)^{-a},\qquad x_{i}>\mu_{i},\quad i=1,\dots,k,\qquad(3)
  15. E [ X i ] = μ i + σ i a - 1 , i = 1 , , k , E[X_{i}]=\mu_{i}+\frac{\sigma_{i}}{a-1},\qquad i=1,\dots,k,
  16. F ¯ ( x 1 , , x k ) = ( 1 + i = 1 k ( x i - μ i σ i ) 1 / γ i ) - a , x i > μ i , σ i > 0 , i = 1 , , k ; a > 0. ( 4 ) \overline{F}(x_{1},\dots,x_{k})=\left(1+\sum_{i=1}^{k}\left(\frac{x_{i}-\mu_{i% }}{\sigma_{i}}\right)^{1/\gamma_{i}}\right)^{-a},\qquad x_{i}>\mu_{i},\sigma_{% i}>0,i=1,\dots,k;a>0.\qquad(4)
  17. W i Γ ( β i , 1 ) , i = 1 , , k , Z Γ ( α , 1 ) , W_{i}\sim\Gamma(\beta_{i},1),\quad i=1,\dots,k,\qquad Z\sim\Gamma(\alpha,1),

Multivariate_stable_distribution.html

  1. d : 𝕊 = { u d : | u | = 1 } \mathbb{R}^{d}:\mathbb{S}=\{u\in\mathbb{R}^{d}:|u|=1\}
  2. X X
  3. X S ( α , Λ , δ ) X\sim S(\alpha,\Lambda,\delta)
  4. X X
  5. E exp ( u T X ) = exp { - s 𝕊 { | u T s | α + i ν ( u T s , α ) } Λ ( d s ) + i u T δ } \operatorname{E}\exp(u^{T}X)=\exp\left\{-\int\limits_{s\in\mathbb{S}}\left\{|u% ^{T}s|^{\alpha}+i\nu(u^{T}s,\alpha)\right\}\,\Lambda(ds)+iu^{T}\delta\right\}
  6. ν ( y , α ) = { - 𝐬𝐢𝐠𝐧 ( y ) tan ( π α / 2 ) | y | α α 1 , ( 2 / π ) y ln | y | α = 1. \nu(y,\alpha)=\begin{cases}-\mathbf{sign}(y)\tan(\pi\alpha/2)|y|^{\alpha}&% \alpha\neq 1,\\ (2/\pi)y\ln|y|&\alpha=1.\end{cases}
  7. Λ \Lambda
  8. 𝕊 \mathbb{S}
  9. δ d \delta\in\mathbb{R}^{d}
  10. u u
  11. u T X u^{T}X
  12. α - \alpha-
  13. β ( u ) \beta(u)
  14. γ ( u ) \gamma(u)
  15. δ ( u ) \delta(u)
  16. X S ( α , β ( ) , γ ( ) , δ ( ) ) X\sim S(\alpha,\beta(\cdot),\gamma(\cdot),\delta(\cdot))
  17. u T X s ( α , β ( ) , γ ( ) , δ ( ) ) u^{T}X\sim s(\alpha,\beta(\cdot),\gamma(\cdot),\delta(\cdot))
  18. u d u\in\mathbb{R}^{d}
  19. γ ( u ) = s 𝕊 | u T s | α Λ ( d s ) \gamma(u)=\int_{s\in\mathbb{S}}|u^{T}s|^{\alpha}\Lambda(ds)
  20. β ( u ) = s 𝕊 | u T s | α 𝐬𝐢𝐠𝐧 ( u T s ) Λ ( d s ) \beta(u)=\int_{s\in\mathbb{S}}|u^{T}s|^{\alpha}\mathbf{sign}(u^{T}s)\Lambda(ds)
  21. δ ( u ) = { u T δ α 1 u T δ - s 𝕊 π 2 u T s ln | u T s | Λ ( d s ) α = 1 \delta(u)=\begin{cases}u^{T}\delta&\alpha\neq 1\\ u^{T}\delta-\int_{s\in\mathbb{S}}\tfrac{\pi}{2}u^{T}s\ln|u^{T}s|\Lambda(ds)&% \alpha=1\end{cases}
  22. ω ( y | α , β ) = { | y | α [ 1 - i β ( tan π α 2 ) 𝐬𝐢𝐠𝐧 ( y ) ] α 1 | y | [ 1 + i β 2 π 𝐬𝐢𝐠𝐧 ( y ) ln | y | ] α = 1 \omega(y|\alpha,\beta)=\begin{cases}|y|^{\alpha}\left[1-i\beta(\tan\tfrac{\pi% \alpha}{2})\mathbf{sign}(y)\right]&\alpha\neq 1\\ |y|\left[1+i\beta\tfrac{2}{\pi}\mathbf{sign}(y)\ln|y|\right]&\alpha=1\end{cases}
  23. E exp ( i u T X ) = exp { - γ 0 | u | α + i u T δ ) } E\exp(iu^{T}X)=\exp\{-\gamma_{0}|u|^{\alpha}+iu^{T}\delta)\}
  24. α = 2 \alpha=2
  25. α = 2 \alpha=2
  26. Σ \Sigma
  27. E exp ( i u T X ) = exp { - ( u T Σ u ) α / 2 + i u T δ ) } E\exp(iu^{T}X)=\exp\{-(u^{T}\Sigma u)^{\alpha/2}+iu^{T}\delta)\}
  28. δ R d \delta\in R^{d}
  29. Σ \Sigma
  30. E exp ( i u T X ) = exp { - ( u T Σ u ) + i u T δ ) } E\exp(iu^{T}X)=\exp\{-(u^{T}\Sigma u)+iu^{T}\delta)\}
  31. X j S ( α , β j , γ j , δ j ) X_{j}\sim S(\alpha,\beta_{j},\gamma_{j},\delta_{j})
  32. E exp ( i u T X ) = exp { - j = 1 m ω ( u j | α , β j ) γ j α + i u T δ ) } E\exp(iu^{T}X)=\exp\left\{-\sum_{j=1}^{m}\omega(u_{j}|\alpha,\beta_{j})\gamma_% {j}^{\alpha}+iu^{T}\delta)\right\}
  33. λ j \lambda_{j}
  34. s j 𝕊 , j = 1 , , m s_{j}\in\mathbb{S},j=1,\ldots,m
  35. E exp ( i u T X ) = exp { - j = 1 m ω ( u T s j | α , 1 ) γ j α + i u T δ ) } E\exp(iu^{T}X)=\exp\left\{-\sum_{j=1}^{m}\omega(u^{T}s_{j}|\alpha,1)\gamma_{j}% ^{\alpha}+iu^{T}\delta)\right\}
  36. X S ( α , β ( ) , γ ( ) , δ ( ) ) X\sim S(\alpha,\beta(\cdot),\gamma(\cdot),\delta(\cdot))
  37. b m b\in\mathbb{R}^{m}
  38. α \alpha
  39. γ ( A T ) \gamma(A^{T}\cdot)
  40. β ( A T ) \beta(A^{T}\cdot)
  41. δ ( A T ) + b T \delta(A^{T}\cdot)+b^{T}\cdot
  42. X i S ( α , β x i , γ x i , δ x i ) , i = 1 , , n X_{i}\sim S(\alpha,\beta_{x_{i}},\gamma_{x_{i}},\delta_{x_{i}}),i=1,\ldots,n
  43. n × n n\times n
  44. Y i = i = 1 n A i j X j Y_{i}=\sum_{i=1}^{n}A_{ij}X_{j}
  45. X i X_{i}
  46. Y i = S ( α , β y i , γ y i , δ y i ) Y_{i}=S(\alpha,\beta_{y_{i}},\gamma_{y_{i}},\delta_{y_{i}})
  47. X i X_{i}
  48. Y i Y_{i}

Munn_semigroup.html

  1. E E
  2. e , f E \bigcup_{e,f\in E}
  3. E E
  4. T E T_{E}
  5. E = { 0 , 1 , 2 , } E=\{0,1,2,...\}
  6. E E
  7. 0 < 1 < 2 < 0<1<2<...
  8. E E
  9. E n = { 0 , 1 , 2 , , n } En=\{0,1,2,...,n\}
  10. n n
  11. E m Em
  12. E n En
  13. m = n m=n
  14. T n , n T_{n,n}
  15. 1 E n 1_{En}
  16. 1 E n 1_{En}
  17. T m , n = T_{m,n}=\emptyset
  18. m n m\not=n
  19. T E = { 1 E 0 , 1 E 1 , 1 E 2 , } E . T_{E}=\{1_{E0},1_{E1},1_{E2},\ldots\}\cong E.

Murnaghan_equation_of_state.html

  1. P ( V ) = K 0 K 0 [ ( V V 0 ) - K 0 - 1 ] . P(V)=\frac{K_{0}}{K_{0}^{\prime}}\left[\left(\frac{V}{V_{0}}\right)^{-K_{0}^{% \prime}}-1\right]\,.
  2. K = - V ( P V ) T K=-V\left(\frac{\partial P}{\partial V}\right)_{T}
  3. K = K 0 + P K 0 K=K_{0}+P\ K_{0}^{\prime}
  4. P ( V ) = K 0 K 0 [ ( V V 0 ) - K 0 - 1 ] P(V)=\frac{K_{0}}{K_{0}^{\prime}}\left[\left(\frac{V}{V_{0}}\right)^{-K_{0}^{% \prime}}-1\right]
  5. V ( P ) = V 0 [ 1 + P ( K 0 K 0 ) ] - 1 / K 0 V(P)=V_{0}\left[1+P\left(\frac{K^{\prime}_{0}}{K_{0}}\right)\right]^{-1/K^{% \prime}_{0}}
  6. E ( V ) = E 0 + K 0 V 0 [ 1 K 0 ( K 0 - 1 ) ( V V 0 ) 1 - K 0 + 1 K 0 V V 0 - 1 K 0 - 1 ] . E(V)=E_{0}+K_{0}\,V_{0}\left[\frac{1}{K_{0}^{\prime}(K_{0}^{\prime}-1)}\left(% \frac{V}{V_{0}}\right)^{1-K_{0}^{\prime}}+\frac{1}{K_{0}^{\prime}}\frac{V}{V_{% 0}}-\frac{1}{K_{0}^{\prime}-1}\right].
  7. V 0 V_{0}
  8. E = E 0 + 1 2 K 0 ( V - V 0 ) 2 V 0 . E=E_{0}+\frac{1}{2}K_{0}\frac{(V-V_{0})^{2}}{V_{0}}.
  9. K 0 = - V ( P V ) T . K_{0}=-V\left(\frac{\partial P}{\partial V}\right)_{T}.
  10. P = K 0 ln ( V 0 / V ) . P=K_{0}\ln(V_{0}/V).\,
  11. V = V 0 exp ( - P / K 0 ) . V=V_{0}\exp(-P/K_{0}).\,
  12. E = E 0 + K 0 ( V 0 - V + V ln ( V / V 0 ) ) . E=E_{0}+K_{0}\left(V_{0}-V+V\ln(V/V_{0})\right).\,
  13. P = - ( E V ) S ( 1 ) P=-\left(\frac{\partial E}{\partial V}\right)_{S}\qquad(1)
  14. K = - V ( P V ) T . ( 2 ) K=-V\left(\frac{\partial P}{\partial V}\right)_{T}.\qquad(2)
  15. K = ( K P ) T ( 3 ) K^{\prime}=\left(\frac{\partial K}{\partial P}\right)_{T}\qquad(3)
  16. K = K 0 K^{\prime}=K^{\prime}_{0}
  17. K = K 0 + K 0 P ( 4 ) K=K_{0}+K^{\prime}_{0}P\qquad(4)
  18. K 0 K_{0}
  19. K K
  20. P = 0. P=0.
  21. d V V = - d P K 0 + K 0 P . ( 5 ) \frac{dV}{V}=-\frac{dP}{K_{0}+K^{\prime}_{0}P}.\qquad(5)
  22. P ( V ) = K 0 K 0 ( ( V 0 V ) K 0 - 1 ) ( 6 ) P(V)=\frac{K_{0}}{K^{\prime}_{0}}\left(\left(\frac{V_{0}}{V}\right)^{K^{\prime% }_{0}}-1\right)\qquad(6)
  23. V ( P ) = V 0 ( 1 + K 0 P K 0 ) - 1 / K 0 . ( 7 ) V(P)=V_{0}\left(1+K^{\prime}_{0}\frac{P}{K_{0}}\right)^{-1/K^{\prime}_{0}}.% \qquad(7)
  24. E = E 0 - P d V E=E_{0}-\int P\,dV
  25. T = 0 T=0
  26. E ( V ) = E 0 + K 0 V K 0 ( ( V 0 / V ) K 0 K 0 - 1 + 1 ) - K 0 V 0 K 0 - 1 . ( 8 ) E(V)=E_{0}+\frac{K_{0}V}{K_{0}^{\prime}}\left(\frac{(V_{0}/V)^{K_{0}^{\prime}}% }{K_{0}^{\prime}-1}+1\right)-\frac{K_{0}V_{0}}{K_{0}^{\prime}-1}.\qquad(8)
  27. K 0 K^{\prime}_{0}
  28. K 0 K_{0}
  29. K 0 K^{\prime}_{0}
  30. K = K 0 + P K 0 + P 2 K 0 ′′ K=K_{0}+PK_{0}^{\prime}+P^{2}K_{0}^{\prime\prime}
  31. P ( V ) = 2 K 0 K 0 [ Γ K 0 ( V 0 V ) Γ + 1 ( V 0 V ) Γ - 1 - 1 ] - 1 P(V)=2\frac{K_{0}}{K_{0}^{\prime}}\left[\frac{\Gamma}{K_{0}^{\prime}}\,\frac{(% \frac{V_{0}}{V})^{\Gamma}+1}{(\frac{V_{0}}{V})^{\Gamma}-1}-1\right]^{-1}
  32. Γ 2 = K 0 2 - 2 K 0 K 0 ′′ > 0 \Gamma^{2}=K_{0}^{\prime 2}-2K_{0}K_{0}^{\prime\prime}>0
  33. K 0 ′′ = 0 K_{0}^{\prime\prime}=0

Mutual_fund_separation_theorem.html

  1. σ 2 \sigma^{2}
  2. μ \mu
  3. r r
  4. X X
  5. W W
  6. 1 1
  7. σ 2 \sigma^{2}
  8. X T r = μ X^{T}r=\mu
  9. X T 1 = W X^{T}1=W
  10. T {}^{T}
  11. σ 2 = X T V X , \sigma^{2}=X^{T}VX,
  12. V V
  13. L = X T V X + 2 λ ( μ - X T r ) + 2 η ( W - X T 1 ) , L=X^{T}VX+2\lambda(\mu-X^{T}r)+2\eta(W-X^{T}1),
  14. λ \lambda
  15. η \eta
  16. X X
  17. X X
  18. λ \lambda
  19. η \eta
  20. X X
  21. λ \lambda
  22. η \eta
  23. λ \lambda
  24. η \eta
  25. X X
  26. X opt = W Δ [ ( r T V - 1 r ) V - 1 1 - ( 1 T V - 1 r ) V - 1 r ] + μ Δ [ ( 1 T V - 1 1 ) V - 1 r - ( r T V - 1 1 ) V - 1 1 ] X^{\mathrm{opt}}=\frac{W}{\Delta}[(r^{T}V^{-1}r)V^{-1}1-(1^{T}V^{-1}r)V^{-1}r]% +\frac{\mu}{\Delta}[(1^{T}V^{-1}1)V^{-1}r-(r^{T}V^{-1}1)V^{-1}1]
  27. Δ = ( r T V - 1 r ) ( 1 T V - 1 1 ) - ( r T V - 1 1 ) 2 > 0. \Delta=(r^{T}V^{-1}r)(1^{T}V^{-1}1)-(r^{T}V^{-1}1)^{2}>0.
  28. X opt = α W + β μ X^{\mathrm{opt}}=\alpha W+\beta\mu
  29. α \alpha
  30. β \beta
  31. μ 1 \mu_{1}
  32. μ 2 \mu_{2}
  33. X 1 opt = α W + β μ 1 X_{1}^{\mathrm{opt}}=\alpha W+\beta\mu_{1}
  34. X 2 opt = α W + β μ 2 . X_{2}^{\mathrm{opt}}=\alpha W+\beta\mu_{2}.
  35. μ 3 \mu_{3}
  36. X 1 opt X_{1}^{\mathrm{opt}}
  37. X 2 opt X_{2}^{\mathrm{opt}}
  38. X 3 opt = α W + β μ 3 = μ 3 - μ 2 μ 1 - μ 2 X 1 opt + μ 1 - μ 3 μ 1 - μ 2 X 2 opt . X_{3}^{\mathrm{opt}}=\alpha W+\beta\mu_{3}=\frac{\mu_{3}-\mu_{2}}{\mu_{1}-\mu_% {2}}X_{1}^{\mathrm{opt}}+\frac{\mu_{1}-\mu_{3}}{\mu_{1}-\mu_{2}}X_{2}^{\mathrm% {opt}}.
  39. V V
  40. σ 2 \sigma^{2}
  41. ( W - X T 1 ) r f + X T r = μ , (W-X^{T}1)r_{f}+X^{T}r=\mu,
  42. r f r_{f}
  43. X X
  44. r r
  45. ( W - X T 1 ) (W-X^{T}1)
  46. σ 2 = X T V X \sigma^{2}=X^{T}VX
  47. V V
  48. X opt = ( μ - W r f ) ( r - 1 r f ) T V - 1 ( r - 1 r f ) V - 1 ( r - 1 r f ) . X^{\mathrm{opt}}=\frac{(\mu-Wr_{f})}{(r-1r_{f})^{T}V^{-1}(r-1r_{f})}V^{-1}(r-1% r_{f}).
  49. μ = W r f \mu=Wr_{f}
  50. μ = W r T V - 1 ( r - 1 r f ) 1 T V - 1 ( r - 1 r f ) \mu=\tfrac{Wr^{T}V^{-1}(r-1r_{f})}{1^{T}V^{-1}(r-1r_{f})}
  51. X * = W 1 T V - 1 ( r - 1 r f ) V - 1 ( r - 1 r f ) . X^{*}=\frac{W}{1^{T}V^{-1}(r-1r_{f})}V^{-1}(r-1r_{f}).
  52. X opt X^{\mathrm{opt}}
  53. μ \mu

N-ellipse.html

  1. { ( x , y ) R 2 : i = 1 n ( x - u i ) 2 + ( y - v i ) 2 = d } \left\{(x,y)\in R^{2}:\sum_{i=1}^{n}\sqrt{(x-u_{i})^{2}+(y-v_{i})^{2}}=d\right\}

N-flake.html

  1. r = 1 2 ( 1 + k = 1 n / 4 cos 2 π k n ) r=\frac{1}{2\left(1+\displaystyle\sum_{k=1}^{\lfloor n/4\rfloor}{\cos\frac{2% \pi k}{n}}\right)}
  2. log m log r \textstyle\frac{\log m}{\log r}
  3. log ( 3 ) log ( 2 ) \textstyle{\frac{\log(3)}{\log(2)}}
  4. log ( 3 ) log ( 2 ) \textstyle{\frac{\log(3)}{\log(2)}}
  5. log ( 5 ) log ( 3 ) \textstyle{\frac{\log(5)}{\log(3)}}
  6. log ( 5 ) log ( 3 ) \textstyle{\frac{\log(5)}{\log(3)}}
  7. log ( 6 ) log ( 1 + φ ) \textstyle{\frac{\log(6)}{\log(1+\varphi)}}
  8. φ = 1 + 5 2 \textstyle{\varphi=\frac{1+\sqrt{5}}{2}}
  9. log ( 6 ) log ( 1 + φ ) \textstyle{\frac{\log(6)}{\log(1+\varphi)}}
  10. 1 1 + φ \textstyle{\frac{1}{1+\varphi}}
  11. log ( 5 ) log ( 1 + φ ) \textstyle{\frac{\log(5)}{\log(1+\varphi)}}
  12. log ( 7 ) log ( 3 ) \textstyle{\frac{\log(7)}{\log(3)}}
  13. log ( 7 ) log ( 3 ) \textstyle{\frac{\log(7)}{\log(3)}}
  14. log ( 6 ) log ( 3 ) \textstyle{\frac{\log(6)}{\log(3)}}
  15. log ( 4 ) log ( 2 ) \textstyle{\frac{\log(4)}{\log(2)}}
  16. log ( 20 ) log ( 3 ) \textstyle{\frac{\log(20)}{\log(3)}}
  17. log ( 7 ) log ( 3 ) \textstyle{\frac{\log(7)}{\log(3)}}
  18. log ( 6 ) log ( 2 ) \textstyle{\frac{\log(6)}{\log(2)}}
  19. 1 2 + φ \textstyle{\frac{1}{2+\varphi}}
  20. log ( 20 ) log ( 2 + φ ) \textstyle{\frac{\log(20)}{\log(2+\varphi)}}
  21. 1 1 + φ \textstyle{\frac{1}{1+\varphi}}
  22. log ( 12 ) log ( 1 + φ ) \textstyle{\frac{\log(12)}{\log(1+\varphi)}}

N_=_2_superconformal_algebra.html

  1. r r\in{\mathbb{Z}}
  2. r 1 2 + r\in{1\over 2}+{\mathbb{Z}}
  3. [ L m , L n ] = ( m - n ) L m + n + c 12 ( m 3 - m ) δ m + n , 0 \displaystyle{[L_{m},L_{n}]=(m-n)L_{m+n}+{c\over 12}(m^{3}-m)\delta_{m+n,0}}
  4. [ L m , J n ] = - n J m + n \displaystyle{[L_{m},\,J_{n}]=-nJ_{m+n}}
  5. [ J m , J n ] = c 3 m δ m + n , 0 \displaystyle{[J_{m},J_{n}]={c\over 3}m\delta_{m+n,0}}
  6. { G r + , G s - } = L r + s + 1 2 ( r - s ) J r + s + c 6 ( r 2 - 1 4 ) δ r + s , 0 \displaystyle{\{G_{r}^{+},G_{s}^{-}\}=L_{r+s}+{1\over 2}(r-s)J_{r+s}+{c\over 6% }(r^{2}-{1\over 4})\delta_{r+s,0}}
  7. { G r + , G s + } = 0 = { G r - , G s - } \displaystyle{\{G_{r}^{+},G_{s}^{+}\}=0=\{G_{r}^{-},G_{s}^{-}\}}
  8. [ L m , G r ± ] = ( m 2 - r ) G r + m ± \displaystyle{[L_{m},G_{r}^{\pm}]=({m\over 2}-r)G^{\pm}_{r+m}}
  9. [ J m , G r ± ] = ± G m + r ± \displaystyle{[J_{m},G_{r}^{\pm}]=\pm G_{m+r}^{\pm}}
  10. r , s r,s\in{\mathbb{Z}}
  11. r , s 1 2 + r,s\in{1\over 2}+{\mathbb{Z}}
  12. L n L_{n}
  13. G r = G r + + G r - G_{r}=G_{r}^{+}+G_{r}^{-}
  14. r , s r,s
  15. c c
  16. L n * = L - n , J m * = J - m , ( G r ± ) * = G - r , c * = c \displaystyle{L_{n}^{*}=L_{-n},\,\,J_{m}^{*}=J_{-m},\,\,(G_{r}^{\pm})^{*}=G_{-% r}^{\mp},\,\,c^{*}=c}
  17. α \alpha
  18. α ( L n ) = L n + 1 2 J n + c 24 δ n , 0 \alpha(L_{n})=L_{n}+{1\over 2}J_{n}+{c\over 24}\delta_{n,0}
  19. α ( J n ) = J n + c 6 δ n , 0 \alpha(J_{n})=J_{n}+{c\over 6}\delta_{n,0}
  20. α ( G r ± ) = G r ± 1 2 ± \alpha(G_{r}^{\pm})=G_{r\pm{1\over 2}}^{\pm}
  21. α - 1 ( L n ) = L n - 1 2 J n + c 24 δ n , 0 \alpha^{-1}(L_{n})=L_{n}-{1\over 2}J_{n}+{c\over 24}\delta_{n,0}
  22. α - 1 ( J n ) = J n - c 6 δ n , 0 \alpha^{-1}(J_{n})=J_{n}-{c\over 6}\delta_{n,0}
  23. α - 1 ( G r ± ) = G r 1 2 ± \alpha^{-1}(G_{r}^{\pm})=G_{r\mp{1\over 2}}^{\pm}
  24. L 0 L_{0}
  25. J 0 J_{0}
  26. G 0 ± G_{0}^{\pm}
  27. L 0 L_{0}
  28. J 0 J_{0}
  29. G 0 ± G_{0}^{\pm}
  30. \partial
  31. ¯ \overline{\partial}
  32. β \beta
  33. β ( L m ) = L m , \displaystyle{\beta(L_{m})=L_{m}},
  34. β ( J m ) = - J m - c 3 δ m , 0 , \beta(J_{m})=-J_{m}-{c\over 3}\delta_{m,0},
  35. β ( G r ± ) = G r \beta(G_{r}^{\pm})=G_{r}^{\mp}
  36. β \beta
  37. β α β - 1 = α - 1 \beta\alpha\beta^{-1}=\alpha^{-1}
  38. α 2 \alpha^{2}
  39. β \beta
  40. 2 {\mathbb{Z}}\rtimes{\mathbb{Z}}_{2}
  41. n = L n + 1 2 ( n + 1 ) J n {\mathcal{L}}_{n}=L_{n}+{1\over 2}(n+1)J_{n}
  42. [ m , n ] = ( m - n ) m + n [{\mathcal{L}}_{m},{\mathcal{L}}_{n}]=(m-n){\mathcal{L}}_{m+n}
  43. c c
  44. J m J_{m}
  45. [ m , J n ] = - n J m + n + c 6 ( m 2 + m ) δ m + n , 0 \displaystyle{[{\mathcal{L}}_{m},J_{n}]=-nJ_{m+n}+{c\over 6}(m^{2}+m)\delta_{m% +n,0}}
  46. { G r + , G s - } = 2 r + s - 2 s J r + s + c 3 ( m 2 + m ) δ m + n , 0 \displaystyle{\{G_{r}^{+},G_{s}^{-}\}=2{\mathcal{L}}_{r+s}-2sJ_{r+s}+{c\over 3% }(m^{2}+m)\delta_{m+n,0}}
  47. ( a n ) (a_{n})
  48. ( b n ) (b_{n})
  49. [ a m , a n ] = m 2 δ m + n , 0 , [ b m , b n ] = m 2 δ m + n , 0 , a n * = a - n , b n * = b - n \displaystyle{[a_{m},a_{n}]={m\over 2}\delta_{m+n,0},\,\,\,\,[b_{m},b_{n}]={m% \over 2}\delta_{m+n,0}},\,\,\,\,a_{n}^{*}=a_{-n},\,\,\,\,b_{n}^{*}=b_{-n}
  50. ( e r ) (e_{r})
  51. { e r , e s * } = δ r , s , { e r , e s } = 0. \displaystyle{\{e_{r},e^{*}_{s}\}=\delta_{r,s},\,\,\,\,\{e_{r},e_{s}\}=0.}
  52. L n L_{n}
  53. L n = m : a - m + n a m : + m : b - m + n b m : + r ( r + n 2 ) : e r * e n + r : L_{n}=\sum_{m}:a_{-m+n}a_{m}:+\sum_{m}:b_{-m+n}b_{m}:+\sum_{r}(r+{n\over 2}):e% ^{*}_{r}e_{n+r}:
  54. J n J_{n}
  55. J n = r : e r * e n + r : J_{n}=\sum_{r}:e_{r}^{*}e_{n+r}:
  56. G r ± G_{r}^{\pm}
  57. G r + = ( a - m + i b - m ) e r + m , G r - = ( a r + m - i b r + m ) e m * G^{+}_{r}=\sum(a_{-m}+ib_{-m})\cdot e_{r+m},\,\,\,\,G_{r}^{-}=\sum(a_{r+m}-ib_% {r+m})\cdot e^{*}_{m}
  58. \ell
  59. E n , F n , H n E_{n},F_{n},H_{n}
  60. [ H m , H n ] = 2 m δ n + m , 0 , [H_{m},H_{n}]=2m\ell\delta_{n+m,0},
  61. [ E m , F n ] = H m + n + m δ m + n , 0 , [E_{m},F_{n}]=H_{m+n}+m\ell\delta_{m+n,0},
  62. [ H m , E n ] = 2 E m + n , \displaystyle{[H_{m},E_{n}]=2E_{m+n},}
  63. [ H m , F n ] = - 2 F m + n , \displaystyle{[H_{m},F_{n}]=-2F_{m+n},}
  64. G r + = ( / 2 + 1 ) - 1 / 2 E - m e m + r , G r - = ( / 2 + 1 ) - 1 / 2 F r + m e m * . \displaystyle{G^{+}_{r}=(\ell/2+1)^{-1/2}\sum E_{-m}\cdot e_{m+r},\,\,\,G^{-}_% {r}=(\ell/2+1)^{-1/2}\sum F_{r+m}\cdot e_{m}^{*}.}
  65. c = 3 / ( + 2 ) c=3\ell/(\ell+2)
  66. X n = H n - 2 r : e r * e n + r : . X_{n}=H_{n}-2\sum_{r}:e_{r}^{*}e_{n+r}:.
  67. X 0 X_{0}
  68. X n X_{n}
  69. n n
  70. Q = G 1 / 2 + + G - 1 / 2 - Q=G_{1/2}^{+}+G_{-1/2}^{-}
  71. Q = G 0 + + G 0 - . Q=G_{0}^{+}+G_{0}^{-}.
  72. G G
  73. H H
  74. T T
  75. G G
  76. H H
  77. G / H G/H
  78. H = T H=T
  79. G G

Nadal_formula.html

  1. ( L V ) < m t p l ( tan ( δ ) - μ 1 + μ * tan ( δ ) ) \left(\frac{L}{V}\right)<mtpl>{{=}}\left(\frac{\tan(\delta)-\mu}{1+\mu*\tan(% \delta)}\right)
  2. ( L V ) < m t p l ( tan ( δ ) - μ * cos β ( 1 + μ * tan ( δ ) ) * cos β ) ) \left(\frac{L}{V}\right)<mtpl>{{=}}\left(\frac{\tan(\delta)-\mu*\cos\beta}{(1+% \mu*\tan(\delta))*\cos\beta)}\right)

Nagata's_conjecture.html

  1. ( x , y , z ) ( x - 2 ( x z + y 2 ) y - ( x z + y 2 ) 2 z , y + ( x z + y 2 ) z , z ) . (x,y,z)\mapsto(x-2(xz+y^{2})y-(xz+y^{2})^{2}z,y+(xz+y^{2})z,z).

Nakamura_number.html

  1. a a
  2. b b
  3. b b
  4. c c
  5. c c
  6. a a
  7. { 1 , 2 , 3 } \{1,2,3\}
  8. { 4 , 5 , 1 } \{4,5,1\}
  9. { 2 , 3 , 4 } \{2,3,4\}
  10. x x
  11. y y
  12. x x
  13. S S
  14. S S
  15. x x
  16. y y
  17. x x
  18. y y
  19. x x
  20. y y
  21. N N
  22. N N
  23. W W
  24. W W
  25. W W
  26. W W
  27. S W S\in W
  28. S T S\subseteq T
  29. T W T\in W
  30. S W S\in W
  31. N S W N\setminus S\notin W
  32. S W S\notin W
  33. N S W N\setminus S\in W
  34. T N T\subseteq N
  35. S S
  36. S W S\in W
  37. S T W S\cap T\in W
  38. X X
  39. # X \#X
  40. \succ
  41. X X
  42. x y x\succ y
  43. x x
  44. y y
  45. y x y\not\succ x
  46. \succ
  47. x 1 , , x m x_{1},\ldots,x_{m}
  48. x 1 x 2 x_{1}\succ x_{2}
  49. x 2 x 3 x_{2}\succ x_{3}
  50. x m - 1 x m x_{m-1}\succ x_{m}
  51. x m x 1 x_{m}\not\succ x_{1}
  52. p = ( i p ) i N p=(\succ_{i}^{p})_{i\in N}
  53. i p \succ_{i}^{p}
  54. x i p y x\succ_{i}^{p}y
  55. i i
  56. x x
  57. y y
  58. p p
  59. ( W , p ) (W,p)
  60. W W
  61. p p
  62. ( W , p ) (W,p)
  63. W p \succ^{p}_{W}
  64. X X
  65. x W p y x\succ^{p}_{W}y
  66. S W S\in W
  67. x i p y x\succ_{i}^{p}y
  68. i S i\in S
  69. C ( W , p ) C(W,p)
  70. ( W , p ) (W,p)
  71. W p \succ^{p}_{W}
  72. X X
  73. W p \succ^{p}_{W}
  74. x C ( W , p ) x\in C(W,p)
  75. y X y\in X
  76. y W p x y\succ^{p}_{W}x
  77. ν ( W ) \nu(W)
  78. W W
  79. ν ( W ) = min { # W : W W ; W = } \nu(W)=\min\{\#W^{\prime}:W^{\prime}\subseteq W;\cap W^{\prime}=\emptyset\}
  80. W = S W S = \cap W=\cap_{S\in W}S=\emptyset
  81. ν ( W ) = + \nu(W)=+\infty
  82. W W
  83. 2 ν ( W ) # N 2\leq\nu(W)\leq\#N
  84. N = { 1 , , n } N=\{1,\ldots,n\}
  85. W W
  86. W W
  87. ν ( W ) = 3 \nu(W)=3
  88. W W
  89. ν ( W ) = 3 \nu(W)=3
  90. n 4 n\neq 4
  91. ν ( W ) = 4 \nu(W)=4
  92. n = 4 n=4
  93. W W
  94. q q
  95. q q
  96. n / 2 < q < n n/2<q<n
  97. ν ( W ) = [ n / ( n - q ) ] \nu(W)=[n/(n-q)]
  98. [ x ] [x]
  99. x x
  100. N = { 1 , 2 , } N=\{1,2,\ldots\}
  101. + +\infty
  102. k 3 k\geq 3
  103. k < 3 k<3
  104. W W
  105. C ( W , p ) C(W,p)
  106. p p
  107. X X
  108. # X < ν ( W ) \#X<\nu(W)
  109. W p \succ_{W}^{p}
  110. p p
  111. # B < ν ( W ) \#B<\nu(W)
  112. B X B\subseteq X
  113. p p
  114. p p
  115. p p
  116. \succeq
  117. x y y x x\succeq y\iff y\not\succ x
  118. \succ
  119. \succeq
  120. \succ
  121. \succeq
  122. \succ
  123. x y x\neq y
  124. x y x\succ y
  125. y x y\succ x
  126. \mathcal{B}
  127. \mathcal{B}
  128. N N
  129. σ \sigma
  130. \mathcal{B}
  131. \mathcal{B}
  132. p p
  133. x , y X x,y\in X
  134. { i : x i p y } \{i:x\succ_{i}^{p}y\}\in\mathcal{B}
  135. X X
  136. x X x\in X
  137. i p \succ_{i}^{p}
  138. i p \succ_{i}^{p}
  139. x x
  140. y X y\in X
  141. y i p x y\succ_{i}^{p}x
  142. X X
  143. x x
  144. C ( W , p ) C(W,p)
  145. i i
  146. x x
  147. i i
  148. y i y_{i}
  149. x x
  150. x x
  151. x X x\in X
  152. C + ( W , p ) C^{+}(W,p)
  153. S W S\in W
  154. i S i\in S
  155. x x
  156. y i X y_{i}\in X
  157. y i i p x y_{i}\succ_{i}^{p}x
  158. C + ( W , p ) C^{+}(W,p)
  159. p p
  160. C + ( W , p ) C ( W , p ) C^{+}(W,p)\subseteq C(W,p)
  161. W W
  162. # X < ν ( W ) \#X<\nu(W)
  163. C + ( W , p ) C^{+}(W,p)
  164. p p
  165. C ( W , p ) C(W,p)
  166. p p
  167. X X
  168. C + ( W , p ) C^{+}(W,p)
  169. C ( W , p ) C(W,p)
  170. p p
  171. X X
  172. # X < ν ( W ) \#X<\nu(W)
  173. p p
  174. p p
  175. p p
  176. \mathcal{B}
  177. W W^{\prime}\subseteq\mathcal{B}^{\prime}
  178. \mathcal{B}
  179. \mathcal{B}^{\prime}
  180. \mathcal{B}
  181. \mathcal{B}^{\prime}

Nanogenerator.html

  1. V max = ± 3 4 ( κ 0 + κ ) [ e 33 - 2 ( 1 + ν ) e 15 - 2 ν e 31 ] a 3 l 3 ν max V_{\,\text{max}}=\pm\frac{3}{4(\kappa_{0}+\kappa)}[e_{\,\text{33}}-2(1+\nu)e_{% \,\text{15}}-2\nu e_{\,\text{31}}]\frac{a^{3}}{l^{3}}\nu_{\,\text{max}}

Naor-Reingold_Pseudorandom_Function.html

  1. 𝔽 p * {\mathbb{F}_{p}}^{*}
  2. ( 𝔽 l ) n (\mathbb{F}_{l})^{n}
  3. f a ( x ) = g a 1 x 1 a 2 x 2 a n x n 𝔽 p f_{a}(x)=g^{a_{1}^{x_{1}}a_{2}^{x_{2}}...a_{n}^{x_{n}}}\in\mathbb{F}_{p}
  4. 𝔽 7 * {\mathbb{F}_{7}}^{*}
  5. f a ( 5 ) f_{a}\;(5)
  6. f a ( x ) = g a 1 x 1 a 2 x 2 a n x n f_{a}(x)=g^{a_{1}^{x_{1}}a_{2}^{x_{2}}...a_{n}^{x_{n}}}
  7. f a ( 5 ) = 4 1 1 2 0 1 1 = 4 1 = 4 𝔽 7 f_{a}(5)=4^{1^{1}2^{0}1^{1}}=4^{1}=4\in\mathbb{F}_{7}
  8. f a ( x ) f_{a}\;(x)
  9. f a ( x ) f_{a}\;(x)
  10. f a ( 1 ) = g a 1 , f a ( 2 ) = g a 2 , f a ( 3 ) = g a 1 a 2 f_{a}(1)=g^{a_{1}},f_{a}(2)=g^{a_{2}},f_{a}(3)=g^{a_{1}a_{2}}
  11. f a ( k ) = g a 1 x 1 a 2 x 2 a n x n f_{a}(k)=g^{a_{1}^{x_{1}}a_{2}^{x_{2}}...a_{n}^{x_{n}}}
  12. f a ( k + 1 ) f_{a}\;(k+1)
  13. f a ( 1 ) = g a 1 f_{a}(1)=g^{a_{1}}
  14. f a ( k ) = g a 2 x 2 a n x n f_{a}(k)=g^{a_{2}^{x_{2}}...a_{n}^{x_{n}}}
  15. f a ( k + 1 ) = g a 1 a 2 x 2 a n x n f_{a}(k+1)=g^{a_{1}a_{2}^{x_{2}}\dots a_{n}^{x_{n}}}
  16. f a ( k + 1 ) f_{a}\;(k+1)
  17. f a ( 5 ) = 4 1 1 2 0 1 1 = 4 1 = 4 f_{a}(5)=4^{1^{1}2^{0}1^{1}}=4^{1}=4
  18. f a ( 1 ) = 4 1 0 2 0 1 1 = 4 1 = 4 f_{a}(1)=4^{1^{0}2^{0}1^{1}}=4^{1}=4
  19. f a ( 6 ) f_{a}\;(6)
  20. f a ( 6 ) f_{a}\;(6)
  21. f a ( 1 ) f_{a}\;(1)
  22. f a ( 5 ) f_{a}\;(5)
  23. 𝒜 f \mathcal{A}^{f}
  24. 𝒜 \mathcal{A}
  25. f a ( x ) f_{a}\;(x)
  26. 𝔽 p \mathbb{F}_{p}
  27. 𝒜 \mathcal{A}
  28. Pr [ 𝒜 f a ( x ) ( p , g ) 1 ] - Pr [ 𝒜 R ( p , g ) 1 ] \,\text{Pr }[\mathcal{A}^{f_{a}(x)}(p,g)\to 1]-\,\text{Pr }[\mathcal{A}^{R}(p,% g)\to 1]
  29. 𝒢 ( n ) \mathcal{I}\mathcal{G}(n)
  30. R a ( x ) R_{a}\;(x)
  31. { 0 , 1 } n 𝔽 p \{0,1\}^{n}\to\mathbb{F}_{p}
  32. \mathcal{R}
  33. \mathcal{R}
  34. γ \gamma
  35. γ \gamma
  36. log l \log l
  37. δ > 0 \delta>0
  38. f a ( x ) f_{a}\;(x)
  39. L a L_{a}
  40. L a { l 1 - δ , if γ 2 l ( γ 2 - δ ) , if γ < 2 L_{a}\geqslant\begin{cases}l^{1-\ \delta\,\!}&\,\text{, if }\gamma\,\!% \geqslant 2\\ l^{\left(\tfrac{\ \gamma\,\!}{2-\ \delta\,\!}\right)}&\,\text{, if }\gamma\,\!% <2\end{cases}
  41. 3 ( l - 1 ) n - δ 3(l-1)^{n-\delta}
  42. ( 𝔽 l ) n (\mathbb{F}_{l})^{n}
  43. log p log n n . \log p\approx\log n\approx{n.}
  44. f a ( x ) f_{a}\;(x)
  45. ( 𝔽 l ) n (\mathbb{F}_{l})^{n}
  46. 𝐃 a {\mathbf{D}}_{a}
  47. { f a ( x ) | 0 x 2 n - 1 } \{f_{a}(x)|0\leq x\leq 2^{n-1}\}
  48. n = log p n=\log p
  49. ( 𝔽 l ) n (\mathbb{F}_{l})^{n}
  50. 𝐃 a Δ ( l , p ) {\mathbf{D}}_{a}\leq\Delta(l,p)
  51. Δ ( l , p ) = { p ( 1 - γ 2 ) l ( - 1 2 ) log 2 p , if l p γ p ( 1 2 ) l - 1 log 2 p , if p γ > l p ( 2 3 ) p ( 1 4 ) l ( - 5 8 ) log 2 p , if p ( 2 3 ) > l p ( 1 2 ) p ( 1 8 ) l ( - 3 8 ) log 2 p , if p ( 1 2 ) > l p ( 1 3 ) \Delta(l,p)=\begin{cases}p^{\left(\tfrac{1-\ \gamma\,\!}{2}\right)}l^{\left(% \tfrac{-1}{2}\right)}\log^{2}p&\,\text{, if }l\geqslant p^{\gamma\,\!}\\ p^{\left(\tfrac{1}{2}\right)}l^{-1}\log^{2}p&\,\text{, if }p^{\gamma\,\!}>l% \geqslant p^{\left(\tfrac{2}{3}\right)}\\ p^{\left(\tfrac{1}{4}\right)}l^{\left(\tfrac{-5}{8}\right)}\log^{2}p&\,\text{,% if }p^{\left(\tfrac{2}{3}\right)}>l\geqslant p^{\left(\tfrac{1}{2}\right)}\\ p^{\left(\tfrac{1}{8}\right)}l^{\left(\tfrac{-3}{8}\right)}\log^{2}p&\,\text{,% if }p^{\left(\tfrac{1}{2}\right)}>l\geqslant p^{\left(\tfrac{1}{3}\right)}\\ \end{cases}
  52. γ \gamma
  53. log 3 \log 3
  54. 𝔽 p \mathbb{F}_{p}
  55. G \langle G\rangle
  56. F a ( x ) = ( a 1 x 1 a 2 x 2 a n x n ) G F_{a}(x)=(a_{1}^{x_{1}}a_{2}^{x_{2}}\dots a_{n}^{x_{n}})G
  57. x = x 1 x n x=x_{1}\dots x_{n}
  58. x , 0 x 2 n - 1 x,0\leq x\leq 2^{n-1}
  59. u k = X ( f a ( k ) ) where X ( P ) is the abscissa of P E . u_{k}=X(f_{a}(k))\;\mbox{where }~{}X(P)\mbox{ is the abscissa of}~{}\;P\in E.
  60. u k u_{k}

Narumi_polynomials.html

  1. s n ( x ) t n / n ! = ( t log ( 1 + t ) ) a ( 1 + t ) x \displaystyle\sum s_{n}(x)t^{n}/n!=\left(\frac{t}{\log(1+t)}\right)^{a}(1+t)^{x}

Nasir_al-Din_al-Tusi.html

  1. a sin A = b sin B = c sin C \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

Natural_borrowing_limit.html

  1. { y t } \{y_{t}\}
  2. b t + 1 b_{t+1}
  3. r r
  4. max E 0 t = 0 β t u ( c t ) \max E_{0}\sum_{t=0}^{\infty}\beta^{t}u(c_{t})
  5. c t + b t + 1 y t + ( 1 + r ) b t c_{t}+b_{t+1}\leq y_{t}+(1+r)b_{t}
  6. t = τ , τ + 1 , τ + 2 , t=\tau,\,\tau+1,\,\tau+2,...
  7. b t + 1 η b_{t+1}\geq\eta
  8. t = τ , τ + 1 , τ + 2 , t=\tau,\,\tau+1,\,\tau+2,...
  9. η \eta
  10. η \eta
  11. c t = 0 c_{t}=0
  12. t t
  13. η = - 1 r y m i n \eta=-\frac{1}{r}y_{min}
  14. β ( 1 + r ) = 1 \beta(1+r)=1
  15. u ( c t ) = E u ( c t + 1 ) u^{\prime}(c_{t})=Eu^{\prime}(c_{t+1})
  16. t = τ , τ + 1 , τ + 2 , t=\tau,\,\tau+1,\,\tau+2,...
  17. η = 0 \eta=0
  18. β ( 1 + r ) = 1 \beta(1+r)=1
  19. u ( c t ) = E u ( c t + 1 ) u^{\prime}(c_{t})=Eu^{\prime}(c_{t+1})
  20. b t + 1 > η b_{t+1}>\eta
  21. c t = y t + ( 1 + r ) b t c_{t}=y_{t}+(1+r)b_{t}
  22. b t + 1 = η b_{t+1}=\eta

Natural_evolution_strategy.html

  1. θ \theta
  2. π ( x | θ ) \pi(x\,|\,\theta)
  3. f ( x ) f(x)
  4. x x
  5. J ( θ ) = E θ [ f ( x ) ] = f ( x ) π ( x | θ ) d x J(\theta)=\operatorname{E}_{\theta}[f(x)]=\int f(x)\;\pi(x\,|\,\theta)\;dx
  6. θ J ( θ ) = θ f ( x ) π ( x | θ ) d x \nabla_{\theta}J(\theta)=\nabla_{\theta}\int f(x)\;\pi(x\,|\,\theta)\;dx
  7. = f ( x ) θ π ( x | θ ) d x =\int f(x)\;\nabla_{\theta}\pi(x\,|\,\theta)\;dx
  8. = f ( x ) θ π ( x | θ ) π ( x | θ ) π ( x | θ ) d x =\int f(x)\;\nabla_{\theta}\pi(x\,|\,\theta)\;\frac{\pi(x\,|\,\theta)}{\pi(x\,% |\,\theta)}\;dx
  9. = [ f ( x ) θ log π ( x | θ ) ] π ( x | θ ) d x =\int\Big[f(x)\;\nabla_{\theta}\log\pi(x\,|\,\theta)\Big]\;\pi(x\,|\,\theta)\;dx
  10. = E θ [ f ( x ) θ log π ( x | θ ) ] =\operatorname{E}_{\theta}\left[f(x)\;\nabla_{\theta}\log\pi(x\,|\,\theta)\right]
  11. f ( x ) f(x)
  12. x x
  13. λ \lambda
  14. θ J ( θ ) 1 λ k = 1 λ f ( x k ) θ log π ( x k | θ ) \nabla_{\theta}J(\theta)\approx\frac{1}{\lambda}\sum_{k=1}^{\lambda}f(x_{k})\;% \nabla_{\theta}\log\pi(x_{k}\,|\,\theta)
  15. θ θ + η θ J ( θ ) \theta\leftarrow\theta+\eta\nabla_{\theta}J(\theta)
  16. θ θ + η 𝐅 - 1 θ J ( θ ) \theta\leftarrow\theta+\eta\mathbf{F}^{-1}\nabla_{\theta}J(\theta)
  17. 𝐅 \mathbf{F}
  18. θ log π ( x | θ ) \nabla_{\theta}\log\pi(x|\theta)
  19. u 1 u λ u_{1}\geq\dots\geq u_{\lambda}
  20. x i x_{i}
  21. θ J ( θ ) = k = 1 λ u k θ log π ( x k | θ ) \nabla_{\theta}J(\theta)=\sum_{k=1}^{\lambda}u_{k}\;\nabla_{\theta}\log\pi(x_{% k}\,|\,\theta)
  22. f , θ i n i t f,\;\;\theta_{init}
  23. k = 1 λ k=1\ldots\lambda
  24. λ \lambda
  25. x k π ( | θ ) x_{k}\sim\pi(\cdot|\theta)
  26. f ( x k ) f(x_{k})
  27. θ log π ( x k | θ ) \nabla_{\theta}\log\pi(x_{k}|\theta)
  28. u k u_{k}
  29. θ J 1 λ k = 1 λ u k θ log π ( x k | θ ) \nabla_{\theta}J\leftarrow\frac{1}{\lambda}\sum_{k=1}^{\lambda}u_{k}\cdot% \nabla_{\theta}\log\pi(x_{k}|\theta)
  30. 𝐅 1 λ k = 1 λ θ log π ( x k | θ ) θ log π ( x k | θ ) \mathbf{F}\leftarrow\frac{1}{\lambda}\sum_{k=1}^{\lambda}\nabla_{\theta}\log% \pi(x_{k}|\theta)\nabla_{\theta}\log\pi(x_{k}|\theta)^{\top}
  31. θ θ + η 𝐅 - 1 θ J \theta\leftarrow\theta+\eta\cdot\mathbf{F}^{-1}\nabla_{\theta}J
  32. η \eta

Natural_units.html

  1. 1 1
  2. c = 1 c=1
  3. c c
  4. v v
  5. v = c / 2 v={c}/{2}
  6. c = 1 c=1
  7. v = 1 / 2 v={1}/{2}
  8. v = 1 / 2 v={1}/{2}
  9. v v
  10. v v
  11. c = 1 c=1
  12. E = m E=m
  13. 1 1
  14. G G
  15. 1 1
  16. 1 1
  17. α 1 / 137 α≈1/137
  18. 1 1
  19. α = k e e 2 c , \alpha=\frac{k\text{e}e^{2}}{\hbar c},
  20. e e
  21. c c
  22. e = 4 π α c e=\sqrt{4\pi\alpha\hbar c}
  23. e = α c e=\sqrt{\alpha\hbar c}
  24. c c
  25. α 1 / 137 α≈1/137
  26. c = 1 c=1
  27. 4 π
  28. l P = G c 3 l\text{P}=\sqrt{\hbar G\over c^{3}}
  29. m P = c G m\text{P}=\sqrt{\hbar c\over G}
  30. t P = G c 5 t\text{P}=\sqrt{\hbar G\over c^{5}}
  31. T P = c 5 G k B 2 T\text{P}=\sqrt{\frac{\hbar c^{5}}{G{k\text{B}}^{2}}}
  32. q P = c k e q\text{P}=\sqrt{\hbar c\over k\text{e}}
  33. c = = G = k e = k B = 1 , c=\hbar=G=k\text{e}=k\text{B}=1,
  34. c c
  35. G G
  36. c c
  37. G G
  38. l S = G k e e 2 c 4 l\text{S}=\sqrt{\frac{Gk\text{e}e^{2}}{c^{4}}}
  39. m S = k e e 2 G m\text{S}=\sqrt{\frac{k\text{e}e^{2}}{G}}
  40. t S = G k e e 2 c 6 t\text{S}=\sqrt{\frac{Gk\text{e}e^{2}}{c^{6}}}
  41. T S = c 4 k e e 2 G k B 2 T\text{S}=\sqrt{\frac{c^{4}k\text{e}e^{2}}{G{k\text{B}}^{2}}}
  42. q S = e q\text{S}=e
  43. c = G = k e = e = k B = 1 , c=G=k\text{e}=e=k\text{B}=1,
  44. c c
  45. G G
  46. e e
  47. l A = 2 ( 4 π ϵ 0 ) m e e 2 l\text{A}=\frac{\hbar^{2}(4\pi\epsilon_{0})}{m\text{e}e^{2}}
  48. m A = m e m\text{A}=m\text{e}
  49. t A = 3 ( 4 π ϵ 0 ) 2 m e e 4 t\text{A}=\frac{\hbar^{3}(4\pi\epsilon_{0})^{2}}{m\text{e}e^{4}}
  50. q A = e q\text{A}=e
  51. T A = m e e 4 2 ( 4 π ϵ 0 ) 2 k B T\text{A}=\frac{m\text{e}e^{4}}{\hbar^{2}(4\pi\epsilon_{0})^{2}k\text{B}}
  52. e = m e = = k e = k B = 1 e=m\text{e}=\hbar=k\text{e}=k\text{B}=1
  53. c = 1 α c=\frac{1}{\alpha}
  54. e 2 = 2 m e = = k e = k B = 1 \frac{e}{\sqrt{2}}=2m\text{e}=\hbar=k\text{e}=k\text{B}=1
  55. c = 2 α c=\frac{2}{\alpha}
  56. k e = 1 4 π ε 0 . k\text{e}=\frac{1}{4\pi\varepsilon_{0}}.
  57. o r b i t a l v e l o c i t y = 1 orbitalvelocity=1
  58. o r b i t a l r a d i u s = 1 orbitalradius=1
  59. a n g u l a r m o m e n t u m = 1 angularmomentum=1
  60. 1 / 2 {1}/{2}
  61. 2 2
  62. 137 137
  63. 274 274
  64. c c
  65. e e
  66. l QCD = m p c l_{\mathrm{QCD}}=\frac{\hbar}{m\text{p}c}
  67. m QCD = m p m_{\mathrm{QCD}}=m\text{p}
  68. t QCD = m p c 2 t_{\mathrm{QCD}}=\frac{\hbar}{m\text{p}c^{2}}
  69. T QCD = m p c 2 k B T_{\mathrm{QCD}}=\frac{m\text{p}c^{2}}{k\text{B}}
  70. q QCD = e / 4 π α q_{\mathrm{QCD}}=e/\sqrt{4\pi\alpha}
  71. q QCD = e / α q_{\mathrm{QCD}}=e/\sqrt{\alpha}
  72. c = m p = = k B = 1 c=m\text{p}=\hbar=k\text{B}=1
  73. = ( 1 eV - 1 ) c =(1\,\,\text{eV}^{-1})\hbar c
  74. = ( 1 eV ) / c 2 =(1\,\,\text{eV})/c^{2}
  75. = ( 1 eV - 1 ) =(1\,\,\text{eV}^{-1})\hbar
  76. = 1 eV / k B =1\,\,\text{eV}/k\text{B}
  77. = e / 4 π α =e/\sqrt{4\pi\alpha}
  78. = e / α =e/\sqrt{\alpha}
  79. = c = k B = 1. \hbar=c=k\text{B}=1.
  80. c c
  81. 1.0 cm = 1.0 cm c 51000 eV - 1 1.0\,\,\text{cm}=\frac{1.0\,\,\text{cm}}{\hbar c}\approx 51000\,\,\text{eV}^{-1}
  82. c = G = 1 c=G=1
  83. c c\,
  84. 1 1\,
  85. 1 1\,
  86. 1 α \frac{1}{\alpha}
  87. 2 α \frac{2}{\alpha}
  88. 1 1\,
  89. 1 1\,
  90. = h 2 π \hbar=\frac{h}{2\pi}
  91. 1 1\,
  92. 1 α \frac{1}{\alpha}
  93. 1 1\,
  94. 1 1\,
  95. 1 1\,
  96. 1 1\,
  97. e e\,
  98. α \sqrt{\alpha}\,
  99. 1 1\,
  100. 1 1\,
  101. 2 \sqrt{2}\,
  102. 4 π α \sqrt{4\pi\alpha}
  103. α \sqrt{\alpha}
  104. K J = e π K\text{J}=\frac{e}{\pi\hbar}\,
  105. α π \frac{\sqrt{\alpha}}{\pi}\,
  106. α π \frac{\alpha}{\pi}\,
  107. 1 π \frac{1}{\pi}\,
  108. 2 π \frac{\sqrt{2}}{\pi}\,
  109. 4 α π \sqrt{\frac{4\alpha}{\pi}}\,
  110. α π \frac{\sqrt{\alpha}}{\pi}\,
  111. R K = 2 π e 2 R\text{K}=\frac{2\pi\hbar}{e^{2}}\,
  112. 2 π α \frac{2\pi}{\alpha}\,
  113. 2 π α \frac{2\pi}{\alpha}\,
  114. 2 π 2\pi\,
  115. π \pi\,
  116. 1 2 α \frac{1}{2\alpha}
  117. 2 π α \frac{2\pi}{\alpha}
  118. G G\,
  119. 1 1\,
  120. 1 1\,
  121. α G α \frac{\alpha\text{G}}{\alpha}\,
  122. 8 α G α \frac{8\alpha\text{G}}{\alpha}\,
  123. α G m e 2 \frac{\alpha\text{G}}{{m\text{e}}^{2}}\,
  124. α G m e 2 \frac{\alpha\text{G}}{{m\text{e}}^{2}}\,
  125. k B k\text{B}\,
  126. 1 1\,
  127. 1 1\,
  128. 1 1\,
  129. 1 1\,
  130. 1 1\,
  131. 1 1\,
  132. m e m\text{e}\,
  133. α G \sqrt{\alpha\text{G}}\,
  134. α G α \sqrt{\frac{\alpha\text{G}}{\alpha}}\,
  135. 1 1\,
  136. 1 2 \frac{1}{2}\,
  137. 511 keV 511\,\text{ keV}
  138. 511 keV 511\,\text{ keV}

Nbn.html

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

ND_experiment.html

  1. ρ 0 \rho^{0}
  2. ω \omega
  3. ϕ \phi
  4. π 0 \pi^{0}
  5. η \eta
  6. e + e - ρ , ω , ϕ π 0 γ , η γ e^{+}e^{-}\to\rho,\omega,\phi\to\pi^{0}\gamma,\eta\gamma
  7. ϕ η γ \phi\to\eta^{\prime}\gamma
  8. ρ 0 \rho^{0}
  9. ω \omega
  10. ϕ \phi
  11. ω , ϕ π 0 e + e - \omega,\phi\to\pi^{0}e^{+}e^{-}
  12. ϕ π + π - \phi\to\pi^{+}\pi^{-}
  13. ρ π + π - π 0 \rho\to\pi^{+}\pi^{-}\pi^{0}
  14. ω , ϕ π 0 π 0 γ \omega,\phi\to\pi^{0}\pi^{0}\gamma
  15. ϕ π 0 η γ \phi\to\pi^{0}\eta\gamma
  16. a 0 ( 980 ) a_{0}(980)
  17. f 0 ( 975 ) f_{0}(975)
  18. ϕ \phi
  19. e + e - ω π 0 e^{+}e^{-}\to\omega\pi^{0}
  20. e + e - π + π - π 0 , π + π - η e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0},~{}~{}\pi^{+}\pi^{-}\eta
  21. e + e - π + π - π + π - , π + π - π 0 π 0 e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-},~{}~{}\pi^{+}\pi^{-}\pi^{0}\pi^{0}
  22. e + e - γ γ γ γ e^{+}e^{-}\to\gamma\gamma\gamma\gamma
  23. e + e - e + e - γ γ e^{+}e^{-}\to e^{+}e^{-}\gamma\gamma
  24. e + e - e + e - e + e - e^{+}e^{-}\to e^{+}e^{-}e^{+}e^{-}
  25. e + e - e + e - γ e^{+}e^{-}\to e^{+}e^{-}\gamma
  26. f 0 ( 975 ) f_{0}(975)
  27. f 2 ( 1270 ) f_{2}(1270)
  28. f 0 ( 1300 ) f_{0}(1300)
  29. a 0 ( 980 ) a_{0}(980)
  30. a 2 ( 1320 ) a_{2}(1320)
  31. ϕ π + π - π 0 , K S 0 K L 0 \phi\to\pi^{+}\pi^{-}\pi^{0},~{}~{}K^{0}_{S}K^{0}_{L}
  32. e + e - π 0 π 0 γ e^{+}e^{-}\to\pi^{0}\pi^{0}\gamma
  33. K S 0 γ γ K^{0}_{S}\to\gamma\gamma
  34. e + e - C ( 1480 ) ϕ π 0 e^{+}e^{-}\to C(1480)\to\phi\pi^{0}

Nearest-neighbor_chain_algorithm.html

  1. A B C A→B→C→...
  2. n n
  3. S S
  4. S S
  5. S S
  6. C C
  7. S S
  8. C C
  9. D D
  10. D D
  11. S S
  12. C C
  13. S S
  14. S S
  15. D D
  16. S S
  17. S S
  18. C C
  19. 2 n 2 2n−2
  20. n n
  21. n 2 n−2
  22. 2 n 2 2n−2
  23. n 1 n−1
  24. n 1 n−1
  25. d d
  26. A A
  27. B B
  28. C C
  29. A A
  30. B B
  31. d ( A B , C ) m i n ( d ( A , C ) , d ( B , C ) ) d(A∪B,C)≥min(d(A,C),d(B,C))
  32. C C
  33. D D
  34. E E
  35. C C
  36. D D
  37. S S
  38. C C
  39. D D
  40. A A
  41. B B
  42. d ( A , B ) = x A , y B d 2 ( x , y ) | A | + | B | - x , y A d 2 ( x , y ) | A | - x , y B d 2 ( x , y ) | B | . d(A,B)=\sum_{x\in A,y\in B}\frac{d^{2}(x,y)}{|A|+|B|}-\sum_{x,y\in A}\frac{d^{% 2}(x,y)}{|A|}-\sum_{x,y\in B}\frac{d^{2}(x,y)}{|B|}.
  43. c A c_{A}
  44. n A n_{A}
  45. d ( A , B ) = d 2 ( c a , c b ) 1 / n A + 1 / n B , d(A,B)=\frac{d^{2}(c_{a},c_{b})}{1/n_{A}+1/n_{B}},
  46. d ( A B , C ) = n A + n C n A + n B + n C d ( A , C ) + n B + n C n A + n B + n C d ( B , C ) - n C n A + n B + n C d ( A , B ) . d(A\cup B,C)=\frac{n_{A}+n_{C}}{n_{A}+n_{B}+n_{C}}d(A,C)+\frac{n_{B}+n_{C}}{n_% {A}+n_{B}+n_{C}}d(B,C)-\frac{n_{C}}{n_{A}+n_{B}+n_{C}}d(A,B).
  47. d ( A , B ) d(A,B)
  48. A A
  49. B B
  50. n C n_{C}
  51. d ( A , C ) d(A,C)
  52. d ( B , C ) d(B,C)
  53. n n
  54. O ( n ) O(n)
  55. d ( A B , C ) d(A\cup B,C)
  56. d ( A , C ) d(A,C)
  57. d ( B , C ) d(B,C)
  58. d ( A , C ) d(A,C)
  59. d ( B , C ) d(B,C)
  60. d ( A B , C ) = min ( d ( A , C ) , d ( B , C ) ) , d(A\cup B,C)=\min(d(A,C),d(B,C)),
  61. O ( n < s u p > 2 ) O(n<sup>2)

Nearly_completely_decomposable_Markov_chain.html

  1. P = ( 1 2 1 2 0 0 1 2 1 2 0 0 0 0 1 2 1 2 0 0 1 2 1 2 ) + ϵ ( - 1 2 0 1 2 0 0 - 1 2 0 1 2 1 2 0 - 1 2 0 0 1 2 0 - 1 2 ) P=\begin{pmatrix}\frac{1}{2}&\frac{1}{2}&0&0\\ \frac{1}{2}&\frac{1}{2}&0&0\\ 0&0&\frac{1}{2}&\frac{1}{2}\\ 0&0&\frac{1}{2}&\frac{1}{2}\\ \end{pmatrix}+\epsilon\begin{pmatrix}-\frac{1}{2}&0&\frac{1}{2}&0\\ 0&-\frac{1}{2}&0&\frac{1}{2}\\ \frac{1}{2}&0&-\frac{1}{2}&0\\ 0&\frac{1}{2}&0&-\frac{1}{2}\\ \end{pmatrix}

Nearly_Kähler_manifold.html

  1. M M
  2. J J
  3. J \nabla J
  4. ( X J ) X = 0 (\nabla_{X}J)X=0\,
  5. X X
  6. M M
  7. S 6 S^{6}
  8. S 6 S^{6}
  9. S 6 = G 2 / S U ( 3 ) , S p ( 2 ) / S U ( 2 ) × U ( 1 ) , S U ( 3 ) / U ( 1 ) × U ( 1 ) , S 3 × S 3 S^{6}=G_{2}/SU(3),Sp(2)/SU(2)\times U(1),SU(3)/U(1)\times U(1),S^{3}\times S^{3}
  10. M M
  11. d ω = 0 d\omega=0
  12. ω \omega
  13. ω ( X , Y ) = g ( J X , Y ) , \omega(X,Y)=g(JX,Y),\,
  14. g g
  15. M M

Neat_submanifold.html

  1. M M
  2. A A
  3. M M
  4. M M
  5. A = A M \partial A=A\cap\partial M
  6. A A
  7. ( U , ϕ ) (U,\phi)
  8. M M
  9. A U = ϕ - 1 ( m ) A\cap U=\phi^{-1}(\mathbb{R}^{m})
  10. m m
  11. A A

Necklace_ring.html

  1. c n = [ i , j ] = n ( i , j ) a i b j \displaystyle c_{n}=\sum_{[i,j]=n}(i,j)a_{i}b_{j}

Neighbourhood_space.html

  1. x X x\in X
  2. x \mathfrak{R}_{x}
  3. x O x\in O

NERD_(sabermetrics).html

  1. p NERD = ( x FIP z × 2 ) + ( SwStrk % z / 2 ) + ( Strike % z / 2 ) + LUCK + 4.69 p\,\text{NERD}=(x\,\text{FIP}z\times 2)+(\,\text{SwStrk}\%z/2)+(\,\text{Strike% }\%z/2)+\,\text{LUCK}+4.69
  2. AGE z × 2 + BAT z + HR / FB z + ( SBA z + SBR z + XBT z ) × .33 + BL z + UZR z + PAY z + LUCK \,\text{AGE}z\times 2+\,\text{BAT}z+\,\text{HR}/\,\text{FB}z+(\,\text{SBA}z+\,% \text{SBR}z+\,\text{XBT}z)\times.33+\,\text{BL}z+\,\text{UZR}z+\,\text{PAY}z+% \,\text{LUCK}

Network_controllability.html

  1. 𝐗 ˙ ( t ) = 𝐀 𝐗 ( t ) + 𝐁 𝐮 ( t ) \dot{\mathbf{X}}(t)=\mathbf{A}\cdot\mathbf{X}(t)+\mathbf{B}\cdot\mathbf{u}(t)
  2. 𝐗 ( t ) = ( x 1 ( t ) , , x N ( t ) ) T \mathbf{X}(t)=(x_{1}(t),\cdots,x_{N}(t))^{\mathrm{T}}
  3. N N
  4. t t
  5. N × N N\times N
  6. 𝐀 \mathbf{A}
  7. N × M N\times M
  8. 𝐁 \mathbf{B}
  9. 𝐮 ( t ) = ( u 1 ( t ) , , u M ( t ) ) T \mathbf{u}(t)=(u_{1}(t),\cdots,u_{M}(t))^{\mathrm{T}}
  10. N D N_{\mathrm{D}}
  11. n D = N D / N n_{\mathrm{D}}=N_{\mathrm{D}}/N
  12. n D {n_{\mathrm{D}}}
  13. n D r e a l {n_{\mathrm{D}}}^{real}
  14. n D rand _ degree {n_{\mathrm{D}}}^{\mathrm{rand\_degree}}
  15. n D r e a l {n_{\mathrm{D}}}^{real}
  16. n D rand _ degree {n_{\mathrm{D}}}^{\mathrm{rand\_degree}}
  17. P ( k in , k out ) P(k_{\mathrm{in}},k_{\mathrm{out}})
  18. n D {n_{\mathrm{D}}}
  19. n D r e a l {n_{\mathrm{D}}}^{real}
  20. n D rand _ degree {n_{\mathrm{D}}}^{\mathrm{rand\_degree}}
  21. n D r e a l {n_{\mathrm{D}}}^{real}
  22. n D rand _ degree {n_{\mathrm{D}}}^{\mathrm{rand\_degree}}

Network_operator_matrix.html

  1. y = x 1 + sin ( x 1 ) + q 1 x 1 e - x 2 y=x_{1}+\sin(x_{1})+q_{1}x_{1}e^{-x_{2}}
  2. ρ 1 ( z ) = z ; \rho_{1}(z)=z\,;
  3. ρ 3 ( z ) = - z ; \rho_{3}(z)=-z\,;
  4. ρ 6 ( n ) = { ε - 1 , if z > ln ε e z , otherwise ; \rho_{6}(n)=\begin{cases}\varepsilon\!^{-1},&\,\text{if }z>\ln\varepsilon\\ e^{z},&\,\text{ otherwise}\end{cases};
  5. ρ 12 ( z ) = sin z ; \rho_{12}(z)=\sin z\,;
  6. χ 0 ( z , z ′′ ) = z + z ′′ ; \chi_{0}(z^{\prime},z^{\prime\prime})=z^{\prime}+z^{\prime\prime}\,;
  7. χ 1 ( z , z ′′ ) = z × z ′′ ; \chi_{1}(z^{\prime},z^{\prime\prime})=z^{\prime}\times z^{\prime\prime}\,;
  8. Ψ = [ 0 0 0 1 1 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 ] \Psi=\begin{bmatrix}0&0&0&1&1&0&0&12\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&3&0&0\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&1&0&1&0\\ 0&0&0&0&0&0&6&0\\ 0&0&0&0&0&0&1&1\\ 0&0&0&0&0&0&0&0\end{bmatrix}
  9. 𝐳 = [ x 1 q 1 x 2 0 1 0 1 0 ] T \mathbf{z}=[x_{1}\ q_{1}\ x_{2}\ 0\ 1\ 0\ 1\ 0]^{T}
  10. Ψ \Psi
  11. ψ ( i , j ) 0 \psi_{(i,j)}\neq 0
  12. z j = χ ψ j , j ( z j , ρ ψ i , j ( z i ) ) z_{j}=\chi_{\psi_{j,j}}(z_{j},\rho_{\psi_{i,j}}(z_{i}))
  13. ψ 1 , 4 = 1 \psi_{1,4}=1
  14. i = 1 i=1
  15. j = 4 j=4
  16. ψ j , j = ψ 4 , 4 = 0 \psi_{j,j}=\psi_{4,4}=0
  17. 𝐳 = [ x 1 q 1 x 2 0 1 0 1 0 ] T \mathbf{z}=[x_{1}\ q_{1}\ x_{2}\ 0\ 1\ 0\ 1\ 0]^{T}
  18. z 1 = x 1 z_{1}=x_{1}
  19. z 4 = 0 z_{4}=0
  20. z 4 = χ ψ 4 , 4 ( z 4 , ρ ψ 1 , 4 ( z 1 ) ) = χ 0 ( z 4 , ρ 1 ( z 1 ) ) = 0 + x 1 = x 1 z_{4}=\chi_{\psi_{4,4}}(z_{4},\rho_{\psi_{1,4}}(z_{1}))=\chi_{0}(z_{4},\rho_{1% }(z_{1}))=0+x_{1}=x_{1}
  21. ψ 1 , 5 = 1 \psi_{1,5}=1
  22. z 5 = χ ψ 5 , 5 ( z 5 , ρ ψ 1 , 5 ( z 1 ) ) = χ 1 ( z 5 , ρ 1 ( z 1 ) ) = 1 × x 1 = x 1 z_{5}=\chi_{\psi_{5,5}}(z_{5},\rho_{\psi_{1,5}}(z_{1}))=\chi_{1}(z_{5},\rho_{1% }(z_{1}))=1\times x_{1}=x_{1}
  23. ψ 1 , 8 = 12 \psi_{1,8}=12
  24. z 8 = χ ψ 8 , 8 ( z 8 , ρ ψ 1 , 8 ( z 1 ) ) = χ 0 ( z 8 , ρ 12 ( z 1 ) ) = 0 + sin x 1 = sin x 1 z_{8}=\chi_{\psi_{8,8}}(z_{8},\rho_{\psi_{1,8}}(z_{1}))=\chi_{0}(z_{8},\rho_{1% 2}(z_{1}))=0+\sin x_{1}=\sin x_{1}
  25. 𝐳 = [ x 1 q 1 x 2 x 1 x 1 0 1 sin x 1 ] T \mathbf{z}=[x_{1}\ q_{1}\ x_{2}\ x_{1}\ x_{1}\ 0\ 1\ \sin x_{1}]^{T}
  26. ψ 2 , 5 = 1 \psi_{2,5}=1
  27. z 5 = χ ψ 5 , 5 ( z 5 , ρ ψ 2 , 5 ( z 2 ) ) = χ 1 ( z 5 , ρ 1 ( z 1 ) ) = x 1 q 1 z_{5}=\chi_{\psi_{5,5}}(z_{5},\rho_{\psi_{2,5}}(z_{2}))=\chi_{1}(z_{5},\rho_{1% }(z_{1}))=x_{1}q_{1}
  28. ψ 3 , 6 = 3 \psi_{3,6}=3
  29. z 6 = χ ψ 6 , 6 ( z 6 , ρ ψ 3 , 6 ( z 3 ) ) = χ 0 ( z 6 , ρ 3 ( z 3 ) ) = 0 + ( - x 2 ) = - x 2 z_{6}=\chi_{\psi_{6,6}}(z_{6},\rho_{\psi_{3,6}}(z_{3}))=\chi_{0}(z_{6},\rho_{3% }(z_{3}))=0+(-x_{2})=-x_{2}
  30. ψ 4 , 8 = 1 \psi_{4,8}=1
  31. z 8 = χ ψ 8 , 8 ( z 8 , ρ ψ 4 , 8 ( z 4 ) ) = χ 0 ( z 8 , ρ 1 ( z 4 ) ) = sin x 1 + x 1 z_{8}=\chi_{\psi_{8,8}}(z_{8},\rho_{\psi_{4,8}}(z_{4}))=\chi_{0}(z_{8},\rho_{1% }(z_{4}))=\sin x_{1}+x_{1}
  32. ψ 5 , 7 = 1 \psi_{5,7}=1
  33. z 7 = χ ψ 7 , 7 ( z 7 , ρ ψ 5 , 7 ( z 5 ) ) = χ 1 ( z 7 , ρ 1 ( z 5 ) ) = 1 x 1 q 1 = x 1 q 1 z_{7}=\chi_{\psi_{7,7}}(z_{7},\rho_{\psi_{5,7}}(z_{5}))=\chi_{1}(z_{7},\rho_{1% }(z_{5}))=1x_{1}q_{1}=x_{1}q_{1}
  34. ψ 6 , 7 = 6 \psi_{6,7}=6
  35. z 7 = χ ψ 7 , 7 ( z 7 , ρ ψ 6 , 7 ( z 6 ) ) = χ 1 ( z 7 , ρ 6 ( z 6 ) ) = x 1 q 1 e - x 2 z_{7}=\chi_{\psi_{7,7}}(z_{7},\rho_{\psi_{6,7}}(z_{6}))=\chi_{1}(z_{7},\rho_{6% }(z_{6}))=x_{1}q_{1}e^{-x_{2}}
  36. ψ 7 , 8 = 1 \psi_{7,8}=1
  37. z 8 = χ ψ 8 , 8 ( z 8 , ρ ψ 7 , 8 ( z 7 ) ) = χ 0 ( z 8 , ρ 1 ( z 7 ) ) = sin x 1 + x 1 + x 1 q 1 e - x 2 z_{8}=\chi_{\psi_{8,8}}(z_{8},\rho_{\psi_{7,8}}(z_{7}))=\chi_{0}(z_{8},\rho_{1% }(z_{7}))=\sin x_{1}+x_{1}+x_{1}q_{1}e^{-x_{2}}

Neural_decoding.html

  1. t 1 , t 2 , , t n = { t i } t_{1},\,\text{ }t_{2},\,\text{ }...,\,\text{ }t_{n}\,\text{ }=\,\text{ }\{t_{i}\}
  2. P [ s ( t ) ] P[s(t)]
  3. P [ s ( t ) ] P[s(t)]
  4. P [ { t i } ] P[\{t_{i}\}]
  5. P [ s ( t ) | { t i } ] P[s(t)|\{t_{i}\}]
  6. P [ { t i } | s ( t ) ] P[\{t_{i}\}|s(t)]
  7. P [ s ( t ) | { t i } ] = P [ { t i } | s ( t ) ] * ( P [ s ( t ) ] / P [ { t i } ] ) P[s(t)|\{t_{i}\}]=P[\{t_{i}\}|s(t)]*(P[s(t)]/P[\{t_{i}\}])
  8. P [ { t i } | s ( t ) ] P[\{t_{i}\}|s(t)]
  9. P ( r | s ) = P ( n i j | s ) P(r|s)=\prod P(n_{ij}|s)
  10. r = n = r=n=
  11. n i j n_{ij}
  12. i i
  13. j j
  14. P ( r | s ) = l [ i , j v i ( t i j l | s ) d t ] e x p [ - i 0 T d t v i ( t | s ) ] P(r|s)=\prod_{l}\left[\prod_{i,j}v_{i}(t_{ijl}|s)dt\right]exp\left[-\sum_{i}% \int_{0}^{T}dtv_{i}(t|s)\right]
  15. t i j l t_{ijl}
  16. v i ( t | s ) v_{i}(t|s)
  17. t i t_{i}
  18. t i - 1 t_{i-1}
  19. P ( r | s ) = l [ i , j v i ( t i j l , τ ( t i j l ) | s ) d t ] e x p [ - i 0 T d t v i ( t , τ ( t ) | s ) ] P(r|s)=\prod_{l}\left[\prod_{i,j}v_{i}(t_{ijl},\tau(t_{ijl})|s)dt\right]exp% \left[-\sum_{i}\int_{0}^{T}dtv_{i}(t,\tau(t)|s)\right]
  20. τ ( t ) \tau(t)
  21. P ( r | s ) = 1 \Zeta ( s ) e x p ( i h i ( s ) r i + 1 2 i j J i j ( s ) r i r j ) P(r|s)=\frac{1}{\Zeta(s)}exp\left(\sum_{i}h_{i}(s)r_{i}+\frac{1}{2}\sum_{i\neq j% }J_{ij}(s)r_{i}r_{j}\right)
  22. r = ( r 1 , r 2 , , r n ) T r=(r_{1},r_{2},...,r_{n})^{T}
  23. h i h_{i}
  24. J i j J_{ij}
  25. \Zeta ( s ) \Zeta(s)

Neutrino_theory_of_light.html

  1. γ μ p μ Ψ = 0. \gamma^{\mu}p_{\mu}\Psi=0.
  2. γ 0 = ( 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ) , γ 1 = ( 0 0 0 1 0 0 1 0 0 - 1 0 0 - 1 0 0 0 ) , \gamma^{0}=\left(\begin{array}[]{cccc}0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\end{array}\right),\;\;\;\;\gamma^{1}=\left(\begin{array}[]{cccc}0&0&0&% 1\\ 0&0&1&0\\ 0&-1&0&0\\ -1&0&0&0\end{array}\right),
  3. γ 2 = ( 0 0 0 - i 0 0 i 0 0 i 0 0 - i 0 0 0 ) , γ 3 = ( 0 0 1 0 0 0 0 - 1 - 1 0 0 0 0 1 0 0 ) . \gamma^{2}=\left(\begin{array}[]{cccc}0&0&0&-i\\ 0&0&i&0\\ 0&i&0&0\\ -i&0&0&0\end{array}\right),\;\;\;\;\gamma^{3}=\left(\begin{array}[]{cccc}0&0&1% &0\\ 0&0&0&-1\\ -1&0&0&0\\ 0&1&0&0\end{array}\right).
  4. γ 0 \gamma^{0}
  5. γ k \gamma^{k}
  6. γ μ γ ν + γ ν γ μ = 2 η μ ν I \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\eta^{\mu\nu}I
  7. η μ ν \eta^{\mu\nu}
  8. ( + - - - ) (+---)
  9. I I
  10. Ψ ( x ) = 1 V 𝐤 { [ a 1 ( 𝐤 ) u + 1 + 1 ( 𝐤 ) + a 2 ( 𝐤 ) u - 1 + 1 ( 𝐤 ) ] e i k x \Psi(x)={1\over\sqrt{V}}\sum_{\mathbf{k}}\left\{\left[a_{1}(\mathbf{k})u^{+1}_% {+1}(\mathbf{k})+a_{2}(\mathbf{k})u^{+1}_{-1}(\mathbf{k})\right]e^{ikx}\right.
  11. + [ c 1 ( 𝐤 ) u - 1 - 1 ( - 𝐤 ) + c 2 ( 𝐤 ) u + 1 - 1 ( - 𝐤 ) ] e - i k x } , \left.+\left[c_{1}^{\dagger}(\mathbf{k})u^{-1}_{-1}(\mathbf{-k})+c_{2}^{% \dagger}(\mathbf{k})u^{-1}_{+1}(\mathbf{-k})\right]e^{-ikx}\right\},
  12. k x kx
  13. 𝐤 𝐱 - k 0 t \mathbf{k}\cdot\mathbf{x}-k_{0}t
  14. a 1 a_{1}
  15. c 1 c_{1}
  16. ν 1 \nu_{1}
  17. ν ¯ 1 \overline{\nu}_{1}
  18. a 2 a_{2}
  19. c 2 c_{2}
  20. ν 2 \nu_{2}
  21. ν ¯ 2 \overline{\nu}_{2}
  22. ν 1 \nu_{1}
  23. ν 2 \nu_{2}
  24. u u
  25. u + 1 + 1 ( 𝐩 ) = E + p 3 2 E ( 1 p 1 + i p 2 E + p 3 0 0 ) , u^{+1}_{+1}(\mathbf{p})=\sqrt{{E+p_{3}}\over 2E}\left(\begin{array}[]{c}1\\ {{p_{1}+ip_{2}}\over{E+p_{3}}}\\ 0\\ 0\end{array}\right),
  26. u - 1 - 1 ( 𝐩 ) = E + p 3 2 E ( - p 1 + i p 2 E + p 3 1 0 0 ) , u^{-1}_{-1}(\mathbf{p})=\sqrt{{E+p_{3}}\over 2E}\left(\begin{array}[]{c}{{-p_{% 1}+ip_{2}}\over{E+p_{3}}}\\ 1\\ 0\\ 0\end{array}\right),
  27. u + 1 - 1 ( 𝐩 ) = E + p 3 2 E ( 0 0 1 p 1 + i p 2 E + p 3 ) , u^{-1}_{+1}(\mathbf{p})=\sqrt{{E+p_{3}}\over 2E}\left(\begin{array}[]{c}0\\ 0\\ 1\\ {{p_{1}+ip_{2}}\over{E+p_{3}}}\end{array}\right),
  28. u - 1 + 1 ( 𝐩 ) = E + p 3 2 E ( 0 0 - p 1 + i p 2 E + p 3 1 ) . u^{+1}_{-1}(\mathbf{p})=\sqrt{{E+p_{3}}\over 2E}\left(\begin{array}[]{c}0\\ 0\\ {{-p_{1}+ip_{2}}\over{E+p_{3}}}\\ 1\end{array}\right).
  29. u + 1 + 1 ( - 𝐩 ) = u - 1 - 1 ( 𝐩 ) , u^{+1}_{+1}(\mathbf{-p})=u^{-1}_{-1}(\mathbf{p}),
  30. u - 1 - 1 ( - 𝐩 ) = u + 1 + 1 ( 𝐩 ) , u^{-1}_{-1}(\mathbf{-p})=u^{+1}_{+1}(\mathbf{p}),
  31. u - 1 + 1 ( - 𝐩 ) = u + 1 - 1 ( 𝐩 ) , u^{+1}_{-1}(\mathbf{-p})=u^{-1}_{+1}(\mathbf{p}),
  32. u + 1 - 1 ( - 𝐩 ) = u - 1 + 1 ( 𝐩 ) . u^{-1}_{+1}(\mathbf{-p})=u^{+1}_{-1}(\mathbf{p}).
  33. Ψ γ 0 γ μ Ψ . \Psi^{\dagger}\gamma_{0}\gamma_{\mu}\Psi.
  34. A μ ( x ) = 𝐩 - 1 2 V p 0 { [ Q R ( 𝐩 ) u - 1 - 1 ( 𝐩 ) γ 0 γ μ u + 1 + 1 ( 𝐩 ) + Q L ( 𝐩 ) u + 1 + 1 ( 𝐩 ) γ 0 γ μ u - 1 - 1 ( 𝐩 ) ] e i p x A_{\mu}(x)=\sum_{\mathbf{p}}{-1\over 2\sqrt{Vp_{0}}}\left\{\left[Q_{R}(\mathbf% {p})u^{-1}_{-1}(\mathbf{p})^{\dagger}\gamma_{0}\gamma_{\mu}u^{+1}_{+1}(\mathbf% {p})+Q_{L}(\mathbf{p})u^{+1}_{+1}(\mathbf{p})^{\dagger}\gamma_{0}\gamma_{\mu}u% ^{-1}_{-1}(\mathbf{p})\right]e^{ipx}\right.
  35. + [ Q R ( 𝐩 ) u + 1 + 1 ( 𝐩 ) γ 0 γ μ u - 1 - 1 ( 𝐩 ) + Q L ( 𝐩 ) u - 1 - 1 ( 𝐩 ) γ 0 γ μ u + 1 + 1 ( 𝐩 ) ] e - i p x } , ( 1 ) \left.+\left[Q_{R}^{\dagger}(\mathbf{p})u^{+1}_{+1}(\mathbf{p})^{\dagger}% \gamma_{0}\gamma_{\mu}u^{-1}_{-1}(\mathbf{p})+Q_{L}^{\dagger}(\mathbf{p})u^{-1% }_{-1}(\mathbf{p})^{\dagger}\gamma_{0}\gamma_{\mu}u^{+1}_{+1}(\mathbf{p})% \right]e^{-ipx}\right\},\quad\quad(1)
  36. p x = 𝐩 𝐱 - p 0 t = 𝐩 𝐱 - E t px=\mathbf{p}\cdot\mathbf{x}-p_{0}t=\mathbf{p}\cdot\mathbf{x}-Et
  37. Q R ( 𝐩 ) = 𝐤 F ( 𝐤 ) [ c 1 ( 𝐩 / 2 - 𝐤 ) a 1 ( 𝐩 / 2 + 𝐤 ) + c 2 ( 𝐩 / 2 + 𝐤 ) a 2 ( 𝐩 / 2 - 𝐤 ) ] Q_{R}(\mathbf{p})=\sum_{\mathbf{k}}F^{\dagger}(\mathbf{k})\left[c_{1}(\mathbf{% p}/2-\mathbf{k})a_{1}(\mathbf{p}/2+\mathbf{k})+c_{2}(\mathbf{p}/2+\mathbf{k})a% _{2}(\mathbf{p}/2-\mathbf{k})\right]
  38. Q L ( 𝐩 ) = 𝐤 F ( 𝐤 ) [ c 2 ( 𝐩 / 2 - 𝐤 ) a 2 ( 𝐩 / 2 + 𝐤 ) + c 1 ( 𝐩 / 2 + 𝐤 ) a 1 ( 𝐩 / 2 - 𝐤 ) , ] . Q_{L}(\mathbf{p})=\sum_{\mathbf{k}}F^{\dagger}(\mathbf{k})\left[c_{2}(\mathbf{% p}/2-\mathbf{k})a_{2}(\mathbf{p}/2+\mathbf{k})+c_{1}(\mathbf{p}/2+\mathbf{k})a% _{1}(\mathbf{p}/2-\mathbf{k}),\right].
  39. F ( 𝐤 ) F(\mathbf{k})
  40. 𝐤 | F ( 𝐤 ) | 2 = 1. \sum_{\mathbf{k}}\left|F(\mathbf{k})\right|^{2}=1.
  41. ϵ μ 1 ( p ) = - 1 2 [ u - 1 - 1 ( 𝐩 ) ] γ 0 γ μ u + 1 + 1 ( 𝐩 ) , \epsilon_{\mu}^{1}(p)={-1\over\sqrt{2}}[u^{-1}_{-1}(\mathbf{p})]^{\dagger}% \gamma_{0}\gamma_{\mu}u^{+1}_{+1}(\mathbf{p}),
  42. ϵ μ 2 ( p ) = - 1 2 [ u + 1 + 1 ( 𝐩 ) ] γ 0 γ μ u - 1 - 1 ( 𝐩 ) . \epsilon_{\mu}^{2}(p)={-1\over\sqrt{2}}[u^{+1}_{+1}(\mathbf{p})]^{\dagger}% \gamma_{0}\gamma_{\mu}u^{-1}_{-1}(\mathbf{p}).
  43. ϵ μ 1 ( p ) = 1 2 ( - i p 1 p 2 + E 2 + p 3 E - p 1 2 E ( E + p 3 ) , - p 1 p 2 + i E 2 + i p 3 E - i p 2 2 E ( E + p 3 ) , - p 1 - i p 2 E , 0 ) , \epsilon_{\mu}^{1}(p)\!=\!{1\over\sqrt{2}}\left({{-ip_{1}p_{2}\!+\!E^{2}\!+\!p% _{3}E\!-\!p_{1}^{2}}\over{E(E+p_{3})}},{{-p_{1}p_{2}\!+\!iE^{2}\!+\!ip_{3}E\!-% \!ip_{2}^{2}}\over{E(E+p_{3})}},{{\!-p_{1}\!-\!ip_{2}}\over E},0\right),
  44. ϵ μ 2 ( p ) = 1 2 ( i p 1 p 2 + E 2 + p 3 E - p 1 2 E ( E + p 3 ) , - p 1 p 2 - i E 2 - i p 3 E + i p 2 2 E ( E + p 3 ) , - p 1 + i p 2 E , 0 ) , ( 2 ) \epsilon_{\mu}^{2}(p)\!=\!{1\over\sqrt{2}}\left({{ip_{1}p_{2}\!+\!E^{2}\!+\!p_% {3}E\!-\!p_{1}^{2}}\over{E(E+p_{3})}},{{-p_{1}p_{2}\!-\!iE^{2}\!-\!ip_{3}E\!+% \!ip_{2}^{2}}\over{E(E+p_{3})}},{{\!-p_{1}\!+\!ip_{2}}\over E},0\right),\quad(2)
  45. ϵ 0 1 ( p ) \epsilon_{0}^{1}(p)
  46. ϵ 0 2 ( p ) \epsilon_{0}^{2}(p)
  47. 𝐩 \mathbf{p}
  48. 𝐧 = 𝐩 / | 𝐩 | \mathbf{n}=\mathbf{p}/|\mathbf{p}|
  49. ϵ μ 1 ( n ) = 1 2 ( - i n 1 n 2 + 1 + n 3 - n 1 2 1 + n 3 , - n 1 n 2 + i n 1 2 + i n 3 2 + i n 3 1 + n 3 , - n 1 - i n 2 , 0 ) , \epsilon_{\mu}^{1}(n)\!=\!{1\over\sqrt{2}}\left({{-in_{1}n_{2}\!+\!1\!+\!n_{3}% \!-\!n_{1}^{2}}\over{1+n_{3}}},{{-n_{1}n_{2}\!+\!in_{1}^{2}\!+\!in_{3}^{2}\!+% \!in_{3}}\over{1+n_{3}}},\!-n_{1}\!-\!in_{2},0\right),
  50. ϵ μ 2 ( n ) = 1 2 ( i n 1 n 2 + 1 + n 3 - n 1 2 1 + n 3 , - n 1 n 2 - i n 1 2 - i n 3 2 - i n 3 1 + n 3 , - n 1 + i n 2 , 0 ) . \epsilon_{\mu}^{2}(n)\!=\!{1\over\sqrt{2}}\left({{in_{1}n_{2}\!+\!1\!+\!n_{3}% \!-\!n_{1}^{2}}\over{1+n_{3}}},{{-n_{1}n_{2}\!-\!in_{1}^{2}\!-\!in_{3}^{2}\!-% \!in_{3}}\over{1+n_{3}}},\!-n_{1}\!+\!in_{2},0\right).
  51. ϵ μ j ( p ) ϵ μ j * ( p ) = 1 , \epsilon_{\mu}^{j}(p)\cdot\epsilon_{\mu}^{j*}(p)=1,
  52. ϵ μ j ( p ) ϵ μ k * ( p ) = 0 for k j . \epsilon_{\mu}^{j}(p)\cdot\epsilon_{\mu}^{k*}(p)=0\;\;\,\text{for}\;\;k\neq j.
  53. p μ p_{\mu}
  54. p μ ϵ μ 1 ( p ) = 0 , p_{\mu}\epsilon_{\mu}^{1}(p)=0,
  55. p μ ϵ μ 2 ( p ) = 0. ( 3 ) p_{\mu}\epsilon_{\mu}^{2}(p)=0.\quad\quad\quad\quad(3)
  56. 𝐩 ϵ 𝟏 ( 𝐩 ) = 𝐩 ϵ 𝟐 ( 𝐩 ) = 0 , \mathbf{p}\cdot\mathbf{\epsilon^{1}}(\mathbf{p})=\mathbf{p}\cdot\mathbf{% \epsilon^{2}}(\mathbf{p})=0,
  57. ϵ 𝟏 ( 𝐩 ) × ϵ 𝟐 ( 𝐩 ) = - i 𝐩 / p 0 , \mathbf{\epsilon^{1}}(\mathbf{p})\times\mathbf{\epsilon^{2}}(\mathbf{p})=-i% \mathbf{p}/p_{0},
  58. 𝐩 × ϵ 𝟏 ( 𝐩 ) = - i p 0 ϵ 𝟏 ( 𝐩 ) , \mathbf{p}\times\mathbf{\epsilon^{1}}(\mathbf{p})=-ip_{0}\mathbf{\epsilon^{1}}% (\mathbf{p}),
  59. 𝐩 × ϵ 𝟐 ( 𝐩 ) = i p 0 ϵ 𝟐 ( 𝐩 ) . ( 4 ) \mathbf{p}\times\mathbf{\epsilon^{2}}(\mathbf{p})=ip_{0}\mathbf{\epsilon^{2}}(% \mathbf{p}).\quad\quad\quad\quad(4)
  60. A μ ( x ) A_{\mu}(x)
  61. A μ ( x ) = 𝐩 1 2 V p 0 { [ Q R ( 𝐩 ) ϵ μ 1 ( 𝐩 ) + Q L ( 𝐩 ) ϵ μ 2 ( 𝐩 ) ] e i p x A_{\mu}(x)=\sum_{\mathbf{p}}{1\over\sqrt{2Vp_{0}}}\left\{\left[Q_{R}(\mathbf{p% })\epsilon_{\mu}^{1}(\mathbf{p})+Q_{L}(\mathbf{p})\epsilon_{\mu}^{2}(\mathbf{p% })\right]e^{ipx}\right.
  62. + [ Q R ( 𝐩 ) ϵ μ 1 * ( 𝐩 ) + Q L ( 𝐩 ) ϵ μ 2 * ( 𝐩 ) ] e - i p x } . ( 5 ) \left.+\left[Q_{R}^{\dagger}(\mathbf{p})\epsilon_{\mu}^{1*}(\mathbf{p})+Q_{L}^% {\dagger}(\mathbf{p})\epsilon_{\mu}^{2*}(\mathbf{p})\right]e^{-ipx}\right\}.% \quad\quad\quad(5)
  63. 𝐄 \mathbf{E}~{}
  64. 𝐇 \mathbf{H}~{}
  65. 𝐄 ( x ) = - 𝐀 ( x ) t , \mathbf{E}(x)=-{\partial\mathbf{A}(x)\over\partial t},
  66. 𝐇 ( x ) = × 𝐀 ( x ) . ( 6 ) \mathbf{H}(x)=\nabla\times\mathbf{A}(x).\quad\quad\quad\quad(6)
  67. E μ ( x ) = i 𝐩 p 0 2 V { [ Q R ( 𝐩 ) ϵ μ 1 ( 𝐩 ) + Q L ( 𝐩 ) ϵ μ 2 ( 𝐩 ) ] e i p x E_{\mu}(x)=i\sum_{\mathbf{p}}{\sqrt{p_{0}}\over\sqrt{2V}}\left\{\left[Q_{R}(% \mathbf{p})\epsilon_{\mu}^{1}(\mathbf{p})+Q_{L}(\mathbf{p})\epsilon_{\mu}^{2}(% \mathbf{p})\right]e^{ipx}\right.
  68. - [ Q R ( 𝐩 ) ϵ μ 1 * ( 𝐩 ) + Q L ( 𝐩 ) ϵ μ 2 * ( 𝐩 ) , ] e - i p x } . \left.-\left[Q_{R}^{\dagger}(\mathbf{p})\epsilon_{\mu}^{1*}(\mathbf{p})+Q_{L}^% {\dagger}(\mathbf{p})\epsilon_{\mu}^{2*}(\mathbf{p}),\right]e^{-ipx}\right\}.
  69. H μ ( x ) = 𝐩 p 0 2 V { [ Q R ( 𝐩 ) ϵ μ 1 ( 𝐩 ) - Q L ( 𝐩 ) ϵ μ 2 ( 𝐩 ) ] e i p x H_{\mu}(x)=\sum_{\mathbf{p}}{\sqrt{p_{0}}\over\sqrt{2V}}\left\{\left[Q_{R}(% \mathbf{p})\epsilon_{\mu}^{1}(\mathbf{p})-Q_{L}(\mathbf{p})\epsilon_{\mu}^{2}(% \mathbf{p})\right]e^{ipx}\right.
  70. + [ Q R ( 𝐩 ) ϵ μ 1 * ( 𝐩 ) - Q L ( 𝐩 ) ϵ μ 2 * ( 𝐩 ) , ] e - i p x } . \left.+\left[Q_{R}^{\dagger}(\mathbf{p})\epsilon_{\mu}^{1*}(\mathbf{p})-Q_{L}^% {\dagger}(\mathbf{p})\epsilon_{\mu}^{2*}(\mathbf{p}),\right]e^{-ipx}\right\}.
  71. E 1 ( x ) / x 1 = i 𝐩 p 0 2 V { [ Q R ( 𝐩 ) p 1 ϵ 1 1 ( 𝐩 ) + Q L ( 𝐩 ) p 1 ϵ 1 2 ( 𝐩 ) ] e i p x \partial E_{1}(x)/\partial x_{1}=i\sum_{\mathbf{p}}{\sqrt{p_{0}}\over\sqrt{2V}% }\left\{\left[Q_{R}(\mathbf{p})p_{1}\epsilon_{1}^{1}(\mathbf{p})+Q_{L}(\mathbf% {p})p_{1}\epsilon_{1}^{2}(\mathbf{p})\right]e^{ipx}\right.
  72. + [ Q R ( 𝐩 ) p 1 ϵ 1 1 * ( 𝐩 ) + Q L ( 𝐩 ) p 1 ϵ 1 2 * ( 𝐩 ) ] e - i p x } . \left.+\left[Q_{R}^{\dagger}(\mathbf{p})p_{1}\epsilon_{1}^{1*}(\mathbf{p})+Q_{% L}^{\dagger}(\mathbf{p})p_{1}\epsilon_{1}^{2*}(\mathbf{p})\right]e^{-ipx}% \right\}.
  73. E 1 ( x ) / x 1 + E 2 ( x ) / x 2 + E 3 ( x ) / x 3 \partial E_{1}(x)/\partial x_{1}+\partial E_{2}(x)/\partial x_{2}+\partial E_{% 3}(x)/\partial x_{3}
  74. p 1 ϵ 1 1 ( 𝐩 ) + p 2 ϵ 2 1 ( 𝐩 ) + p 3 ϵ 3 1 ( 𝐩 ) p_{1}\epsilon_{1}^{1}(\mathbf{p})+p_{2}\epsilon_{2}^{1}(\mathbf{p})+p_{3}% \epsilon_{3}^{1}(\mathbf{p})
  75. 𝐄 ( x ) = 0 , \nabla\cdot\mathbf{E}(x)=0,
  76. 𝐇 ( x ) = 0. \nabla\cdot\mathbf{H}(x)=0.
  77. 𝐇 \mathbf{H}
  78. × 𝐄 ( x ) \nabla\times\mathbf{E}(x)
  79. 𝐩 × ϵ 𝟏 ( 𝐩 ) \mathbf{p}\times\mathbf{\epsilon^{1}}(\mathbf{p})
  80. 𝐇 ( x ) / t \partial\mathbf{H}(x)/\partial t
  81. i p 0 ϵ 𝟏 ( 𝐩 ) ip_{0}\mathbf{\epsilon^{1}}(\mathbf{p})
  82. × 𝐄 ( x ) = - 𝐇 ( x ) / t , \nabla\times\mathbf{E}(x)=-\partial\mathbf{H}(x)/\partial t,
  83. × 𝐇 ( x ) = 𝐄 ( x ) / t . \nabla\times\mathbf{H}(x)=\partial\mathbf{E}(x)/\partial t.
  84. 𝐄 \mathbf{E}
  85. 𝐇 \mathbf{H}
  86. P 𝐄 ( 𝐱 , t ) P - 1 = - 𝐄 ( - 𝐱 , t ) , P\mathbf{E}(\mathbf{x},t)P^{-}1=-\mathbf{E}(\mathbf{-x},t),
  87. P 𝐇 ( 𝐱 , t ) P - 1 = 𝐇 ( - 𝐱 , t ) , P\mathbf{H}(\mathbf{x},t)P^{-}1=\mathbf{H}(\mathbf{-x},t),
  88. C 𝐄 ( 𝐱 , t ) C - 1 = - 𝐄 ( 𝐱 , t ) , C\mathbf{E}(\mathbf{x},t)C^{-}1=-\mathbf{E}(\mathbf{x},t),
  89. C 𝐇 ( 𝐱 , t ) C - 1 = - 𝐇 ( 𝐱 , t ) . C\mathbf{H}(\mathbf{x},t)C^{-}1=-\mathbf{H}(\mathbf{x},t).
  90. A μ A_{\mu}
  91. A μ / x μ = 0 \partial A_{\mu}/\partial x_{\mu}=0
  92. 𝐄 \mathbf{E}
  93. 𝐇 \mathbf{H}
  94. j = 1 2 ϵ μ j ( 𝐩 ) ϵ ν j * ( 𝐩 ) = = j = = 1 2 ϵ μ j * ( 𝐩 ) ϵ ν j ( 𝐩 ) = δ μ ν - p μ p ν E 2 . \sum_{j=1}^{2}\epsilon_{\mu}^{j}(\mathbf{p})\epsilon_{\nu}^{j*}(\mathbf{p})==% \sum_{j==1}^{2}\epsilon_{\mu}^{j*}(\mathbf{p})\epsilon_{\nu}^{j}(\mathbf{p})=% \delta_{\mu\nu}-{p_{\mu}p_{\nu}\over E^{2}}.
  95. ϵ μ 1 ( n ) \epsilon_{\mu}^{1}(n)
  96. ϵ μ 2 ( n ) \epsilon_{\mu}^{2}(n)
  97. ϵ μ 1 ( n ) = 1 2 ( 1 , i , 0 , 0 ) , \epsilon_{\mu}^{1}(n)={1\over\sqrt{2}}(1,i,0,0),
  98. ϵ μ 2 ( n ) = 1 2 ( 1 , - i , 0 , 0 ) . \epsilon_{\mu}^{2}(n)={1\over\sqrt{2}}(1,-i,0,0).
  99. { a ( 𝐤 ) , a ( 𝐥 ) } = 0 , \{a(\mathbf{k}),a(\mathbf{l})\}=0,
  100. { a ( 𝐤 ) , a ( 𝐥 ) } = 0 , \{a^{\dagger}(\mathbf{k}),a^{\dagger}(\mathbf{l})\}=0,
  101. { a ( 𝐤 ) , a ( 𝐥 ) } = δ ( 𝐤 - 𝐥 ) , \{a(\mathbf{k}),a^{\dagger}(\mathbf{l})\}=\delta(\mathbf{k}-\mathbf{l}),
  102. [ b ( 𝐤 ) , b ( 𝐥 ) ] = 0 , \left[b(\mathbf{k}),b(\mathbf{l})\right]=0,
  103. [ b ( 𝐤 ) , b ( 𝐥 ) ] = 0 , \left[b^{\dagger}(\mathbf{k}),b^{\dagger}(\mathbf{l})\right]=0,
  104. [ b ( 𝐤 ) , b ( 𝐥 ) ] = δ ( 𝐤 - 𝐥 ) . ( 7 ) \left[b(\mathbf{k}),b^{\dagger}(\mathbf{l})\right]=\delta(\mathbf{k}-\mathbf{l% }).\quad\quad(7)
  105. [ Q ( 𝐤 ) , Q ( 𝐥 ) ] = 0 , \left[Q(\mathbf{k}),Q(\mathbf{l})\right]=0,
  106. [ Q ( 𝐤 ) , Q ( 𝐥 ) ] = 0 , \left[Q^{\dagger}(\mathbf{k}),Q^{\dagger}(\mathbf{l})\right]=0,
  107. [ Q ( 𝐤 ) , Q ( 𝐥 ) ] = δ ( 𝐤 - 𝐥 ) - Δ ( 𝐤 , 𝐥 ) . ( 8 ) \left[Q(\mathbf{k}),Q^{\dagger}(\mathbf{l})\right]=\delta(\mathbf{k}-\mathbf{l% })-\Delta(\mathbf{k},\mathbf{l}).\quad\quad(8)
  108. Δ ( 𝐩 , 𝐩 ) = 𝐤 F ( 𝐤 ) [ F ( 𝐩 / 2 - 𝐩 / 2 + 𝐤 ) a ( 𝐩 - 𝐩 / 2 - 𝐤 ) a ( 𝐩 / 2 - 𝐤 ) \Delta(\mathbf{p}^{\prime},\mathbf{p})=\sum_{\mathbf{k}}F^{\dagger}(\mathbf{k}% )\left[F(\mathbf{p}^{\prime}/2-\mathbf{p}/2+\mathbf{k})a^{\dagger}(\mathbf{p}-% \mathbf{p}^{\prime}/2-\mathbf{k})a(\mathbf{p}^{\prime}/2-\mathbf{k})\right.
  109. + F ( 𝐩 / 2 - 𝐩 / 2 + 𝐤 ) c ( 𝐩 - 𝐩 / 2 + 𝐤 ) c ( 𝐩 / 2 + 𝐤 ) ] . ( 9 ) \left.+F(\mathbf{p}/2-\mathbf{p}^{\prime}/2+\mathbf{k})c^{\dagger}(\mathbf{p}-% \mathbf{p}^{\prime}/2+\mathbf{k})c(\mathbf{p}^{\prime}/2+\mathbf{k})\right].% \quad\quad(9)
  110. Δ ( 𝐩 , 𝐩 ) , \Delta(\mathbf{p}^{\prime},\mathbf{p}),
  111. | Φ = a ( 𝐤 𝟏 ) a ( 𝐤 𝟐 ) a ( 𝐤 𝐧 ) c ( 𝐪 𝟏 ) c ( 𝐪 𝟐 ) c ( 𝐪 𝐦 ) | 0 |\Phi\rangle=a^{\dagger}(\mathbf{k_{1}})a^{\dagger}(\mathbf{k_{2}})...a^{% \dagger}(\mathbf{k_{n}})c^{\dagger}(\mathbf{q_{1}})c^{\dagger}(\mathbf{q_{2}})% ...c^{\dagger}(\mathbf{q_{m}})|0\rangle
  112. 𝐩 𝐩 \mathbf{p}^{\prime}\neq\mathbf{p}
  113. Δ ( 𝐩 , 𝐩 ) \Delta(\mathbf{p}^{\prime},\mathbf{p})
  114. Δ ( 𝐩 , 𝐩 ) = δ ( 𝐩 - 𝐩 ) 𝐤 | F ( 𝐤 ) | 2 [ a ( 𝐩 / 2 - 𝐤 ) a ( 𝐩 / 2 - 𝐤 ) \Delta(\mathbf{p}^{\prime},\mathbf{p})=\delta(\mathbf{p}^{\prime}-\mathbf{p})% \sum_{\mathbf{k}}\left|F(\mathbf{k})\right|^{2}\left[a^{\dagger}(\mathbf{p}/2-% \mathbf{k})a(\mathbf{p}/2-\mathbf{k})\right.
  115. + c ( 𝐩 / 2 + 𝐤 ) c ( 𝐩 / 2 + 𝐤 ) ] . \left.+c^{\dagger}(\mathbf{p}/2+\mathbf{k})c(\mathbf{p}/2+\mathbf{k})\right].
  116. n a ( 𝐤 ) n_{a}(\mathbf{k})
  117. n c ( 𝐤 ) n_{c}(\mathbf{k})
  118. Δ ( 𝐩 , 𝐩 ) = δ ( 𝐩 - 𝐩 ) 𝐤 | F ( 𝐤 ) | 2 [ n a ( 𝐩 / 2 - 𝐤 ) + n c ( 𝐩 / 2 + 𝐤 ) ] \Delta(\mathbf{p}^{\prime},\mathbf{p})=\delta(\mathbf{p}^{\prime}-\mathbf{p})% \sum_{\mathbf{k}}\left|F(\mathbf{k})\right|^{2}\left[n_{a}(\mathbf{p}/2-% \mathbf{k})+n_{c}(\mathbf{p}/2+\mathbf{k})\right]
  119. = δ ( 𝐩 - 𝐩 ) 𝐤 [ | F ( 𝐩 / 2 - 𝐤 ) | 2 n a ( 𝐤 ) + | F ( 𝐤 - 𝐩 / 2 ) | 2 n c ( 𝐤 ) ] =\delta(\mathbf{p}^{\prime}-\mathbf{p})\sum_{\mathbf{k}}\left[\left|F(\mathbf{% p}/2-\mathbf{k})\right|^{2}n_{a}(\mathbf{k})+\left|F(\mathbf{k}-\mathbf{p}/2)% \right|^{2}n_{c}(\mathbf{k})\right]
  120. = δ ( 𝐩 - 𝐩 ) Δ ¯ ( 𝐩 , 𝐩 ) =\delta(\mathbf{p}^{\prime}-\mathbf{p})\overline{\Delta}(\mathbf{p},\mathbf{p})
  121. 𝐤 \mathbf{k}
  122. F ( 𝐩 / 2 - 𝐤 ) F(\mathbf{p}/2-\mathbf{k})
  123. F ( 𝐤 - 𝐩 / 2 ) F(\mathbf{k}-\mathbf{p}/2)
  124. 𝐩 \mathbf{p}
  125. 𝐩 + 𝐤 \mathbf{p+k}
  126. 𝐤 \mathbf{k}
  127. [ Q R ( 𝐩 ) , Q R ( 𝐩 ) ] = 0 , [ Q L ( 𝐩 ) , Q L ( 𝐩 ) ] = 0 , \left[Q_{R}(\mathbf{p}^{\prime}),Q_{R}(\mathbf{p})\right]=0,\;\left[Q_{L}(% \mathbf{p}^{\prime}),Q_{L}(\mathbf{p})\right]=0,
  128. [ Q R ( 𝐩 ) , Q R ( 𝐩 ) ] = δ ( 𝐩 - 𝐩 ) ( 1 - Δ ¯ 12 ( 𝐩 , 𝐩 ) ) , \left[Q_{R}(\mathbf{p}^{\prime}),Q_{R}^{\dagger}(\mathbf{p})\right]=\delta(% \mathbf{p}^{\prime}-\mathbf{p})(1-{\overline{\Delta}_{12}}(\mathbf{p},\mathbf{% p})),
  129. [ Q L ( 𝐩 ) , Q L ( 𝐩 ) ] = δ ( 𝐩 - 𝐩 ) ( 1 - Δ ¯ 21 ( 𝐩 , 𝐩 ) ) , \left[Q_{L}(\mathbf{p}^{\prime}),Q_{L}^{\dagger}(\mathbf{p})\right]=\delta(% \mathbf{p}^{\prime}-\mathbf{p})(1-{\overline{\Delta}_{21}}(\mathbf{p},\mathbf{% p})),
  130. [ Q R ( 𝐩 ) , Q L ( 𝐩 ) ] = 0 , [ Q R ( 𝐩 ) , Q L ( 𝐩 ) ] = 0 , ( 10 ) \left[Q_{R}(\mathbf{p}^{\prime}),Q_{L}(\mathbf{p})\right]=0,\;\left[Q_{R}(% \mathbf{p}^{\prime}),Q_{L}^{\dagger}(\mathbf{p})\right]=0,\quad\quad\quad\quad% (10)
  131. Δ ¯ 12 ( 𝐩 , 𝐩 ) = 𝐤 [ | F ( 𝐤 - 𝐩 / 2 ) | 2 ( n a 1 ( 𝐤 ) + n c 2 ( 𝐤 ) ) {\overline{\Delta}_{12}}(\mathbf{p},\mathbf{p})=\sum_{\mathbf{k}}\left[\left|F% (\mathbf{k}-\mathbf{p}/2)\right|^{2}(n_{a1}(\mathbf{k})+n_{c2}(\mathbf{k}))\right.
  132. + | F ( 𝐩 / 2 - 𝐤 ) | 2 ( n c 1 ( 𝐤 ) + n a 2 ( 𝐤 ) ) ] . ( 11 ) \left.+\left|F(\mathbf{p}/2-\mathbf{k})\right|^{2}(n_{c1}(\mathbf{k})+n_{a2}(% \mathbf{k}))\right].\quad\quad\quad\quad(11)
  133. Δ ¯ 12 ( 𝐩 , 𝐩 ) \overline{\Delta}_{12}(\mathbf{p},\mathbf{p})
  134. Δ ¯ 21 ( 𝐩 , 𝐩 ) \overline{\Delta}_{21}(\mathbf{p},\mathbf{p})
  135. ξ ( 𝐩 ) = 1 2 [ Q L ( 𝐩 ) + Q R ( 𝐩 ) ] , \xi(\mathbf{p})={1\over\sqrt{2}}\left[Q_{L}(\mathbf{p})+Q_{R}(\mathbf{p})% \right],
  136. η ( 𝐩 ) = i 2 [ Q L ( 𝐩 ) - Q R ( 𝐩 ) ] . ( 12 ) \eta(\mathbf{p})={i\over\sqrt{2}}\left[Q_{L}(\mathbf{p})-Q_{R}(\mathbf{p})% \right].\quad\quad\quad\quad(12)
  137. [ ξ ( 𝐩 ) , η ( 𝐩 ) ] = i 2 δ ( 𝐩 - 𝐩 ) [ Δ ¯ 21 ( 𝐩 , 𝐩 ) - Δ ¯ 12 ( 𝐩 , 𝐩 ) ] , ( 13 ) [\xi(\mathbf{p}^{\prime}),\eta^{\dagger}(\mathbf{p})]={i\over 2}\delta(\mathbf% {p}^{\prime}-\mathbf{p})[\overline{\Delta}_{21}(\mathbf{p},\mathbf{p})-% \overline{\Delta}_{12}(\mathbf{p},\mathbf{p})],\quad\quad(13)
  138. [ ξ ( 𝐩 ) , η ( 𝐩 ) ] = 0 ( 14 ) [\xi(\mathbf{p}^{\prime}),\eta^{\dagger}(\mathbf{p})]=0\quad\quad\quad\quad(14)
  139. 𝐤 [ | F ( 𝐤 - 𝐩 / 2 ) | 2 ( n a 1 ( 𝐤 ) + n c 2 ( 𝐤 ) - n a 2 ( 𝐤 ) - n c 1 ( 𝐤 ) ) \sum_{\mathbf{k}}\left[\left|F(\mathbf{k}-\mathbf{p}/2)\right|^{2}(n_{a1}(% \mathbf{k})+n_{c2}(\mathbf{k})-n_{a2}(\mathbf{k})-n_{c1}(\mathbf{k}))\right.
  140. + | F ( 𝐩 / 2 - 𝐤 ) | 2 ( n c 1 ( 𝐤 ) + n a 2 ( 𝐤 ) - n c 2 ( 𝐤 ) - n a 1 ( 𝐤 ) ) ] \left.+\left|F(\mathbf{p}/2-\mathbf{k})\right|^{2}(n_{c1}(\mathbf{k})+n_{a2}(% \mathbf{k})-n_{c2}(\mathbf{k})-n_{a1}(\mathbf{k}))\right]
  141. n a 1 ( 𝐤 ) n_{a1}(\mathbf{k})
  142. n c 1 ( 𝐤 ) n_{c1}(\mathbf{k})
  143. F ( 𝐤 ) = 0 F(\mathbf{k})=0
  144. N ( 𝐩 ) = Q ( 𝐩 ) Q ( 𝐩 ) . N(\mathbf{p})=Q^{\dagger}(\mathbf{p})Q(\mathbf{p}).
  145. F ( 𝐤 ) = 1 / Ω F(\mathbf{k})=1/\Omega
  146. Ω \Omega
  147. m m
  148. N ( 𝐩 ) ( Q ( 𝐩 ) ) m | 0 = ( m - m ( m - 1 ) Ω ) ( Q ( 𝐩 ) ) m | 0 , N(\mathbf{p})(Q^{\dagger}(\mathbf{p}))^{m}|0\rangle\;=\left(m-{m(m-1)\over% \Omega}\right)(Q^{\dagger}(\mathbf{p}))^{m}|0\rangle,
  149. N ( 𝐩 ) | 0 = 0 N(\mathbf{p})|0\rangle=0
  150. Ω \Omega
  151. Q ( 𝐩 ) | n 𝐩 = ( n 𝐩 + 1 ) ( 1 - n 𝐩 Ω ) | n 𝐩 + 1 , Q^{\dagger}(\mathbf{p})|n_{\mathbf{p}}\rangle\;=\sqrt{(n_{\mathbf{p}}+1)\left(% 1-{n_{\mathbf{p}}\over\Omega}\right)}|n_{\mathbf{p}}+1\rangle,
  152. Q ( 𝐩 ) | n 𝐩 = n 𝐩 ( 1 - ( n 𝐩 - 1 ) Ω ) | n 𝐩 - 1 , ( 15 ) Q(\mathbf{p})|n_{\mathbf{p}}\rangle\;=\sqrt{n_{\mathbf{p}}\left(1-{(n_{\mathbf% {p}}-1)\over\Omega}\right)}|n_{\mathbf{p}}-1\rangle,\quad\quad\quad\quad(15)
  153. | n 𝐩 |n_{\mathbf{p}}\rangle
  154. n 𝐩 n_{\mathbf{p}}
  155. 𝐩 \mathbf{p}
  156. Q ( 𝐩 ) Q^{\dagger}(\mathbf{p})
  157. n 𝐩 n_{\mathbf{p}}
  158. Q ( 𝐩 ) | 0 = | 1 𝐩 , Q^{\dagger}(\mathbf{p})|0\rangle=|1_{\mathbf{p}}\rangle,
  159. Q ( 𝐩 ) | 1 𝐩 = | 0 , Q(\mathbf{p})|1_{\mathbf{p}}\rangle=|0\rangle,
  160. Ω \Omega
  161. w α β ( n 𝐩 + 1 n 𝐩 ) = ( n 𝐩 + 1 ) ( 1 - n 𝐩 Ω ) w α β ( 1 𝐩 0 ) , ( 16 ) w_{\alpha\beta}(n_{\mathbf{p}}+1\leftarrow n_{\mathbf{p}})=(n_{\mathbf{p}}+1)% \left(1-{n_{\mathbf{p}}\over\Omega}\right)w_{\alpha\beta}(1_{\mathbf{p}}% \leftarrow 0),\quad(16)
  162. w β α ( n 𝐩 - 1 n 𝐩 ) = n 𝐩 ( 1 - n 𝐩 - 1 Ω ) w β α ( 0 1 𝐩 ) . ( 17 ) w_{\beta\alpha}(n_{\mathbf{p}}-1\leftarrow n_{\mathbf{p}})=n_{\mathbf{p}}\left% (1-{n_{\mathbf{p}}-1\over\Omega}\right)w_{\beta\alpha}(0\leftarrow 1_{\mathbf{% p}}).\quad(17)
  163. w β α ( 0 1 𝐩 ) = w α β ( 1 𝐩 0 ) , w_{\beta\alpha}(0\leftarrow 1_{\mathbf{p}})=w_{\alpha\beta}(1_{\mathbf{p}}% \leftarrow 0),
  164. w α β ( n 𝐩 + 1 n 𝐩 ) w β α ( n 𝐩 - 1 n 𝐩 ) = ( n 𝐩 + 1 ) ( 1 - n 𝐩 Ω ) n 𝐩 ( 1 - ( n 𝐩 - 1 ) Ω ) . {w_{\alpha\beta}(n_{\mathbf{p}}+1\leftarrow n_{\mathbf{p}})\over w_{\beta% \alpha}(n_{\mathbf{p}}-1\leftarrow n_{\mathbf{p}})}={(n_{\mathbf{p}}+1)\left(1% -{n_{\mathbf{p}}\over\Omega}\right)\over n_{\mathbf{p}}\left(1-{(n_{\mathbf{p}% }-1)\over\Omega}\right)}.
  165. w α β ( n 𝐩 + 1 n 𝐩 ) e - E β / k T = w β α ( n 𝐩 - 1 n 𝐩 ) e - E α / k T , w_{\alpha\beta}(n_{\mathbf{p}}+1\leftarrow n_{\mathbf{p}})e^{-E_{\beta}/kT}=w_% {\beta\alpha}(n_{\mathbf{p}}-1\leftarrow n_{\mathbf{p}})e^{-E_{\alpha}/kT},
  166. ω p = E β - E α \omega_{p}=E_{\beta}-E_{\alpha}
  167. n 𝐩 = 2 u + ( u + 2 ) / Ω + u 2 ( 1 + 2 / Ω ) + ( u + 2 ) 2 / Ω 2 , n_{\mathbf{p}}={2\over u+(u+2)/\Omega+\sqrt{u^{2}(1+2/\Omega)+(u+2)^{2}/\Omega% ^{2}}},
  168. u = e ω p / k T - 1 u=e^{\omega_{p}/kT}-1
  169. Ω ( ω p / k T ) 1 \Omega(\omega_{p}/kT)>>1
  170. n 𝐩 = 1 e ω p / k T ( 1 + 1 Ω ) - 1 . n_{\mathbf{p}}={1\over e^{\omega_{p}/kT}\left(1+{1\over\Omega}\right)-1}.
  171. 1 / Ω 1/\Omega

Nevanlinna's_criterion.html

  1. z h ( z ) / h ( z ) zh^{\prime}(z)/h(z)
  2. f t ( z ) = h - 1 ( e - t h ( z ) ) , f_{t}(z)=h^{-1}(e^{-t}h(z)),\,
  3. t | f t ( z ) | 2 0. \partial_{t}|f_{t}(z)|^{2}\leq 0.
  4. t | f t ( z ) | 2 = 2 f t ( z ) ¯ t f t ( z ) = 2 w ¯ v ( w ) , \partial_{t}|f_{t}(z)|^{2}=2\Re\,\overline{f_{t}(z)}\partial_{t}f_{t}(z)=2\Re% \,\overline{w}v(w),
  5. v ( w ) = - h ( w ) h ( w ) . v(w)=-{h(w)\over h^{\prime}(w)}.
  6. w ¯ h ( w ) h ( w ) 0. \Re\,\overline{w}{h(w)\over h^{\prime}(w)}\geq 0.
  7. h ( w ) w h ( w ) 0. \Re\,{h(w)\over wh^{\prime}(w)}\geq 0.
  8. z h ( z ) h ( z ) 0 \Re\,z{h^{\prime}(z)\over h(z)}\geq 0
  9. θ arg h ( r e i θ ) = θ log h ( z ) = θ log h ( z ) = z θ z log h ( z ) = z h ( z ) h ( z ) . \partial_{\theta}\arg h(re^{i\theta})=\partial_{\theta}\Im\,\log h(z)=\Im\,% \partial_{\theta}\log h(z)=\Im\,{\partial z\over\partial\theta}\cdot\partial_{% z}\log h(z)=\Re\,z{h^{\prime}(z)\over h(z)}.
  10. z = r e i θ z=re^{i\theta}
  11. h ( r e i θ ) h(re^{i\theta})
  12. h h
  13. | b n | 2. |b_{n}|\leq 2.
  14. e i θ + z e i θ - z = 1 + 2 n 1 e - i n θ z n , {e^{i\theta}+z\over e^{i\theta}-z}=1+2\sum_{n\geq 1}e^{-in\theta}z^{n},
  15. 0 2 π g ( e i θ ) d θ = 1 \int_{0}^{2\pi}\Re g(e^{i\theta})\,d\theta=1
  16. b n = 2 0 2 π e - i n t g ( e i θ ) d θ . b_{n}=2\int_{0}^{2\pi}e^{-int}\Re g(e^{i\theta})\,d\theta.
  17. | b n | 2 0 2 π g ( e i θ ) d θ = 2. |b_{n}|\leq 2\int_{0}^{2\pi}\Re g(e^{i\theta})\,d\theta=2.
  18. f ( z ) = z + a 2 z 2 + a 3 z 3 + f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots
  19. | a n | n . |a_{n}|\leq n.
  20. g ( z ) = z f ( z ) f ( z ) = 1 + b 1 z + b 2 z 2 + g(z)=z{f^{\prime}(z)\over f(z)}=1+b_{1}z+b_{2}z^{2}+\cdots
  21. z f ( z ) = g ( z ) f ( z ) zf^{\prime}(z)=g(z)f(z)
  22. ( n - 1 ) a n = k = 1 n - 1 b n - k a k . (n-1)a_{n}=\sum_{k=1}^{n-1}b_{n-k}a_{k}.
  23. | a n | 2 n - 1 k = 1 n - 1 | a k | , |a_{n}|\leq{2\over n-1}\sum_{k=1}^{n-1}|a_{k}|,
  24. | a n | n . |a_{n}|\leq n.

Nevanlinna_function.html

  1. N ( z ) = C + D z + ( 1 λ - z - λ 1 + λ 2 ) d μ ( λ ) , z , N(z)=C+Dz+\int_{\mathbb{R}}\left(\frac{1}{\lambda-z}-\frac{\lambda}{1+\lambda^% {2}}\right)d\mu(\lambda),\quad z\in\mathbb{H},
  2. d μ ( λ ) 1 + λ 2 < . \int_{\mathbb{R}}\frac{d\mu(\lambda)}{1+\lambda^{2}}<\infty.
  3. C = Re ( N ( i ) ) and D = lim y N ( i y ) i y C=\mathrm{Re}(N(i))\qquad\,\text{and}\qquad D=\lim_{y\rightarrow\infty}\frac{N% (iy)}{iy}
  4. μ ( ( λ 1 , λ 2 ] ) = lim δ 0 lim ε 0 1 π λ 1 + δ λ 2 + δ Im ( N ( λ + i ε ) ) d λ . \mu((\lambda_{1},\lambda_{2}])=\lim_{\delta\rightarrow 0}\lim_{\varepsilon% \rightarrow 0}\frac{1}{\pi}\int_{\lambda_{1}+\delta}^{\lambda_{2}+\delta}% \mathrm{Im}(N(\lambda+i\varepsilon))d\lambda.
  5. z z
  6. z - a z-a
  7. a . a.
  8. z p with 0 p 1 z^{p}\,\text{ with }0\leq p\leq 1
  9. - z p with - 1 p 0 -z^{p}\,\text{ with }-1\leq p\leq 0
  10. i ( z / i ) p with - 1 p 1. i(z/i)^{p}\,\text{ with }-1\leq p\leq 1.
  11. ln ( z ) \ln(z)
  12. tan ( z ) \tan(z)
  13. z a z + b c z + d z\mapsto\frac{az+b}{cz+d}
  14. a d - b c a^{\ast}d-bc^{\ast}
  15. Im ( b d ) = Im ( a c ) = 0. \mathrm{Im}(b^{\ast}d)=\mathrm{Im}(a^{\ast}c)=0.
  16. i z + i - 2 z + 1 + i \frac{iz+i-2}{z+1+i}
  17. ( S - z ) - 1 f , f \langle(S-z)^{-1}f,f\rangle

Néel_relaxation_theory.html

  1. τ N = τ 0 exp ( K V k B T ) \tau\text{N}=\tau_{0}\exp\left(\frac{KV}{k\text{B}T}\right)
  2. K K
  3. V V
  4. T T
  5. T B = K V k B ln ( τ m / τ 0 ) T\text{B}=\frac{KV}{k\text{B}\ln\left(\tau\text{m}/\tau_{0}\right)}

NGC_3114.html

  1. E ( B - V ) = 0.27 E(B-V)=0.27
  2. 6 × 10 7 6\times 10^{7}
  3. 2 × 10 8 2\times 10^{8}
  4. E ( B - V ) = 0.03 E(B-V)=0.03
  5. 940 ± 60 940\pm 60
  6. 1 - 2 × 10 8 1-2\times 10^{8}
  7. [ F e / H ] = - 0.04 ± 0.04 [Fe/H]=-0.04\pm 0.04

Nichols_algebra.html

  1. 𝔅 ( V ) \mathfrak{B}(V)
  2. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}
  3. V 𝒴 H H 𝒟 V\in{}^{H}_{H}\mathcal{YD}
  4. Δ ( v ) = 1 v + v 1 \Delta(v)=1\otimes v+v\otimes 1
  5. V = 𝔤 V=\mathfrak{g}
  6. V 𝒴 H H 𝒟 V\in{}^{H}_{H}\mathcal{YD}
  7. I n = 2 T n V , I\subset\bigoplus_{n=2}^{\infty}T^{n}V,
  8. Δ ( I ) I T V + T V I . \Delta(I)\subset I\otimes TV+TV\otimes I.
  9. I 𝒴 H H 𝒟 I\in{}^{H}_{H}\mathcal{YD}
  10. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}
  11. v i V . v_{i}\in V.
  12. v i g i v i v_{i}\mapsto g_{i}\otimes v_{i}
  13. i \partial_{i}
  14. i ( 1 ) = 0 i ( v j ) = δ i j \partial_{i}(1)=0\quad\partial_{i}(v_{j})=\delta_{ij}
  15. i ( a b ) = a i ( b ) + i ( a ) ( g i . b ) \partial_{i}(ab)=a\partial_{i}(b)+\partial_{i}(a)(g_{i}.b)
  16. i \partial_{i}
  17. V ± = k x V_{\pm}=kx
  18. V ± V_{\pm}
  19. x g x x\mapsto g\otimes x
  20. V ± V_{\pm}
  21. g x ± x g\otimes x\mapsto\pm x
  22. x x ± x x x\otimes x\rightarrow\pm x\otimes x
  23. 𝔅 ( V + ) = k [ x ] 𝔅 ( V - ) = k [ x ] / ( x 2 ) \mathfrak{B}(V_{+})=k[x]\qquad\mathfrak{B}(V_{-})=k[x]/(x^{2})
  24. 𝔅 ( V q ) = k [ x ] / ( x n ) \mathfrak{B}(V_{q})=k[x]/(x^{n})
  25. 𝔅 ( V q ) = k [ x ] \mathfrak{B}(V_{q})=k[x]
  26. x g x x\mapsto g\otimes x
  27. x g x x\mapsto g\otimes x
  28. g x - x g\otimes x\mapsto-x
  29. g y + y g\otimes y\mapsto+y
  30. h y - y h\otimes y\mapsto-y
  31. h x ± x h\otimes x\mapsto\pm x
  32. x x - x x x\otimes x\rightarrow-x\otimes x
  33. y y - y y y\otimes y\rightarrow-y\otimes y
  34. x y y x x\otimes y\rightarrow y\otimes x
  35. y x ± x y y\otimes x\rightarrow\pm x\otimes y
  36. q 12 q 21 = ± 1 q_{12}q_{21}=\pm 1
  37. 𝔅 ( V 0 ) = k [ x , y ] / ( x 2 , y 2 , x y + y x ) , \mathfrak{B}(V_{0})=k[x,y]/(x^{2},y^{2},xy+yx),
  38. 𝔅 ( V 1 ) = k [ x ] / ( x 2 , y 2 , x y x y + y x y x ) \mathfrak{B}(V_{1})=k[x]/(x^{2},y^{2},xyxy+yxyx)
  39. A 1 A 1 A_{1}\cup A_{1}
  40. 𝒪 [ g ] χ \mathcal{O}_{[g]}^{\chi}
  41. 𝒪 [ g ] χ \mathcal{O}_{[g]}^{\chi}
  42. 𝔅 ( V ) \mathfrak{B}(V)
  43. 𝕊 3 \mathbb{S}_{3}\;
  44. 𝒪 [ ( 12 ) ] \mathcal{O}_{[(12)]}\;
  45. 𝕊 4 \mathbb{S}_{4}\;
  46. 𝒪 [ ( 12 ) ] \qquad\mathcal{O}_{[(12)]}\;
  47. 𝕊 4 \mathbb{S}_{4}\;
  48. 𝒪 [ ( 1234 ) ] \qquad\mathcal{O}_{[(1234)]}\;
  49. 𝕊 5 \mathbb{S}_{5}\;
  50. 𝒪 [ ( 12 ) ] \qquad\mathcal{O}_{[(12)]}\;
  51. 𝔻 4 \mathbb{D}_{4}\;
  52. 𝒪 [ a b ] 𝒪 [ b ] \qquad\mathcal{O}_{[ab]}\oplus\mathcal{O}_{[b]}
  53. 𝔻 4 \mathbb{D}_{4}\;
  54. x i x j q i j x j x i x_{i}\otimes x_{j}\mapsto q_{ij}x_{j}\otimes x_{i}
  55. x x τ q ( x x ) g . x = q x x\otimes x\;\stackrel{\tau}{\longmapsto}\;q(x\otimes x)\;\;\Longleftrightarrow% \;\;g.x=qx
  56. g ( j 2 ) = g g^{(j^{2})}=g
  57. 𝒪 [ g ] χ \mathcal{O}_{[g]}^{\chi}
  58. 𝔅 ( 𝒪 [ g ] 𝒪 [ h ] ) \mathfrak{B}\left(\mathcal{O}_{[g]}\oplus\mathcal{O}_{[h]}\right)
  59. x 𝒪 [ g ] y 𝒪 [ h ] x\in\mathcal{O}_{[g]}\;y\in\mathcal{O}_{[h]}
  60. a d 𝒪 [ g ] 𝒪 [ h ] = [ 𝒪 [ g ] , 𝒪 [ h ] ] ad_{\mathcal{O}_{[g]}}\mathcal{O}_{[h]}=[\mathcal{O}_{[g]},\mathcal{O}_{[h]}]
  61. 𝒪 [ g h ] \mathcal{O}_{[gh]}
  62. ( g h ) 2 = ( h g ) 2 \;(gh)^{2}=(hg)^{2}
  63. r , s [ g ] r,s\in[g]\;
  64. r , s \langle r,s\rangle\;
  65. ( r s ) 2 ( s r ) 2 (rs)^{2}\neq(sr)^{2}\;
  66. 𝔸 n 5 \mathbb{A}_{n\geq 5}
  67. 𝕊 n 6 \mathbb{S}_{n\geq 6}
  68. F i 22 Fi_{22}\;
  69. 22 A , 22 B 22A,22B\;
  70. 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\;
  71. 32 A , 32 B , 46 A , 46 B , 92 A , 92 B , 94 A , 94 B 32A,\;32B,\;46A,\;46B,\;92A,\;92B,\;94A,\;94B\;
  72. U q ( 𝔤 ) U_{q}(\mathfrak{g})
  73. U q ( 𝔤 ) ( 𝔅 ( V ) k [ n ] 𝔅 ( V * ) ) σ U_{q}(\mathfrak{g})\cong\left(\mathfrak{B}(V)\otimes k[\mathbb{Z}^{n}]\otimes% \mathfrak{B}(V^{*})\right)^{\sigma}
  74. 𝔤 \mathfrak{g}

Niobium(IV)_fluoride.html

  1. 2 NbF 4 NbF 5 + NbF 3 \mathrm{\ 2NbF_{4}\rightarrow NbF_{5}+NbF_{3}}

NIP_(model_theory).html

  1. M φ ( s y m b o l a , s y m b o l b i ) i X . M\models\varphi(symbol{a},symbol{b}_{i})\quad\Leftrightarrow\quad i\in X.
  2. ( k ) ( y k = x ) . (\exists k)(y\cdot k=x).

Niven's_theorem.html

  1. sin 0 = 0 , sin 30 = 1 2 , sin 90 = 1. \begin{aligned}\displaystyle\sin 0^{\circ}&\displaystyle=0,\\ \displaystyle\sin 30^{\circ}&\displaystyle=\frac{1}{2},\\ \displaystyle\sin 90^{\circ}&\displaystyle=1.\end{aligned}
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi

NK_model.html

  1. N N
  2. K K
  3. K = 0 K=0
  4. K = N K=N
  5. N N
  6. S S
  7. S i S_{i}
  8. F ( S ) = i f ( S i ) , F(S)=\sum_{i}f(S_{i}),
  9. K K
  10. f ( S i ) = f ( S i , S 1 i , , S K i ) , f(S_{i})=f(S_{i},S^{i}_{1},\dots,S^{i}_{K}),\,
  11. S j i S^{i}_{j}
  12. S i S_{i}
  13. f ( S i , S 1 i , , S K i ) f(S_{i},S^{i}_{1},\dots,S^{i}_{K})
  14. 1 k N 1<<k\leq N
  15. μ + σ 2 ln ( k + 1 ) k + 1 \mu+\sigma\sqrt{{2\ln(k+1)}\over{k+1}}
  16. ( k + 1 ) σ 2 N [ k + 1 + 2 ( k + 2 ) ln ( k + 1 ) ] {{(k+1)\sigma^{2}}\over{N[k+1+2(k+2)\ln(k+1)]}}
  17. f ( S i ) = f ( S i , S i + 1 ) f(S_{i})=f(S_{i},S_{i+1})
  18. f ( S 5 ) = f ( S 5 , S 1 ) f(S_{5})=f(S_{5},S_{1})
  19. F ( 00101 ) = f ( 0 , 0 ) + f ( 0 , 1 ) + f ( 1 , 0 ) + f ( 0 , 1 ) + f ( 1 , 0 ) = 0 + 1 + 2 + 1 + 2 = 6. F(00101)=f(0,0)+f(0,1)+f(1,0)+f(0,1)+f(1,0)=0+1+2+1+2=6.\,
  20. F ( 11100 ) = f ( 1 , 1 ) + f ( 1 , 1 ) + f ( 1 , 0 ) + f ( 0 , 0 ) + f ( 0 , 1 ) = 0 + 0 + 2 + 0 + 1 = 3. F(11100)=f(1,1)+f(1,1)+f(1,0)+f(0,0)+f(0,1)=0+0+2+0+1=3.\,
  21. P P
  22. P P
  23. 2 K 2^{K}
  24. Q Q
  25. Q Q
  26. P = 0 P=0
  27. Q = Q=\infty

No_free_lunch_with_vanishing_risk.html

  1. K = { ( H S ) : H admissible , ( H S ) = lim t ( H S ) t exists a.s. } K=\{(H\cdot S)_{\infty}:H\,\text{ admissible},(H\cdot S)_{\infty}=\lim_{t\to% \infty}(H\cdot S)_{t}\,\text{ exists a.s.}\}
  2. C = { g L ( P ) : g f f K } C=\{g\in L^{\infty}(P):g\leq f\;\forall f\in K\}
  3. C ¯ L + ( P ) = { 0 } \bar{C}\cap L^{\infty}_{+}(P)=\{0\}
  4. C ¯ \bar{C}
  5. L + ( P ) L^{\infty}_{+}(P)
  6. S = ( S t ) t = 0 T S=(S_{t})_{t=0}^{T}
  7. d \mathbb{R}^{d}
  8. \mathbb{Q}
  9. \mathbb{Q}

Noisy-storage_model.html

  1. n n
  2. O ( n 2 ) O(n^{2})
  3. Δ t \Delta t
  4. : 𝒮 ( in ) 𝒮 ( out ) \mathcal{F}:\mathcal{S}(\mathcal{H}_{\rm in})\rightarrow\mathcal{S}(\mathcal{H% }_{\rm out})
  5. ρ in 𝒮 ( in ) \rho_{\rm in}\in\mathcal{S}(\mathcal{H}_{\rm in})
  6. ρ out 𝒮 ( out ) \rho_{\rm out}\in\mathcal{S}(\mathcal{H}_{\rm out})
  7. \mathcal{E}
  8. 𝒟 \mathcal{D}
  9. \mathcal{E}
  10. \mathcal{F}
  11. 𝒟 \mathcal{D}
  12. N N
  13. = 𝒩 N \mathcal{F}=\mathcal{N}^{\otimes N}
  14. 𝒩 : S ( \Complex d ) S ( \Complex d ) \mathcal{N}:S(\Complex^{d})\rightarrow S(\Complex^{d})
  15. d d
  16. N N
  17. = 𝒩 N \mathcal{F}=\mathcal{N}^{\otimes N}
  18. 𝒩 ( ρ ) = λ ρ + ( 1 - λ ) 𝗂𝖽 / 2 \mathcal{N}(\rho)=\lambda\rho+(1-\lambda)\mathsf{id}/2
  19. N N
  20. = 𝗂𝖽 N \mathcal{F}=\mathsf{id}^{\otimes N}
  21. 𝗂𝖽 \mathsf{id}
  22. n n
  23. = 𝒩 N \mathcal{F}=\mathcal{N}^{\otimes N}
  24. 𝒩 = 𝗂𝖽 \mathcal{N}=\mathsf{id}
  25. n > 2 N n>2N
  26. N < n / 2 N<n/2
  27. n n
  28. n N n\gtrapprox N
  29. n = N n=N
  30. = 𝒩 N \mathcal{F}=\mathcal{N}^{\otimes N}
  31. n n
  32. n > 2 N C ( 𝒩 ) n>2\cdot N\cdot C(\mathcal{N})
  33. C ( 𝒩 ) C(\mathcal{N})
  34. 𝒩 \mathcal{N}
  35. 𝒩 \mathcal{N}
  36. n > 2 N E C ( 𝒩 ) n>2\cdot N\cdot E_{C}(\mathcal{N})
  37. E C ( 𝒩 ) E_{C}(\mathcal{N})
  38. 𝒩 \mathcal{N}
  39. E C ( 𝒩 ) E_{C}(\mathcal{N})
  40. = 𝒩 N \mathcal{F}=\mathcal{N}^{\otimes N}
  41. n n
  42. n > Q ( 𝒩 ) N n>Q(\mathcal{N})N
  43. Q Q
  44. 𝒩 \mathcal{N}
  45. 𝒩 \mathcal{N}

Nominal_income_target.html

  1. L t d L^{d}_{t}
  2. L t s L^{s}_{t}
  3. ln L t d = 1 a [ - ln W t + ln P t + b + s t ] \ln{L^{d}_{t}}=\frac{1}{a}\left[-\ln{W_{t}}+\ln{P_{t}}+b+s_{t}\right]
  4. ln L t s = 1 d [ ln W t - ln P t - c ] , \ln{L^{s}_{t}}=\frac{1}{d}\left[\ln{W_{t}}-\ln{P_{t}}-c\right]\;,
  5. W t W_{t}
  6. P t P_{t}
  7. s t s_{t}
  8. b = ln ( 1 - a ) , ( 0 < a < 1 ) b=\ln(1-a),\;\;(0<a<1)
  9. ln W t * = ln P t + a c + b d a + d + d s t a + d , \ln{W^{*}_{t}}=\ln{P_{t}}+\frac{ac+bd}{a+d}+\frac{ds_{t}}{a+d}\;,
  10. W t * W^{*}_{t}
  11. E t - 1 [ ln W ] = E t - 1 [ ln P t ] + a c + b d a + d + d a + d E t - 1 [ s t ] . E_{t-1}[\ln{W}]=E_{t-1}[\ln{P_{t}}]+\frac{ac+bd}{a+d}+\frac{d}{a+d}E_{t-1}[s_{% t}]\;.
  12. ln L t d = 1 a ( ln P t - E t - 1 [ ln P t ] + s t - d a + d E t - 1 [ s t ] + b - a c + b d a + d ) . \ln{L^{d}_{t}}=\frac{1}{a}\left(\ln{P_{t}}-E_{t-1}[\ln{P_{t}}]+s_{t}-\frac{d}{% a+d}E_{t-1}[s_{t}]+b-\frac{ac+bd}{a+d}\right)\;.
  13. Y t Y_{t}
  14. ln Y t = ( 1 - a ) ln L t + s t , \ln{Y_{t}}=(1-a)\ln{L_{t}}+s_{t}\;,
  15. ln Y t = 1 - a a ( ln P t - E t - 1 [ ln P t ] - d a + d E t - 1 [ s t ] + b - a c + b d a + d ) + 1 a s t . \ln{Y_{t}}=\frac{1-a}{a}\left(\ln{P_{t}}-E_{t-1}[\ln{P_{t}}]-\frac{d}{a+d}E_{t% -1}[s_{t}]+b-\frac{ac+bd}{a+d}\right)+\frac{1}{a}s_{t}\;.
  16. E t - 1 [ ln Y t ] = 1 - a a ( - d a + d E t - 1 [ s t ] + b - a c + b d a + d ) + 1 a E t - 1 [ s t ] , E_{t-1}[\ln{Y_{t}}]=\frac{1-a}{a}\left(-\frac{d}{a+d}E_{t-1}[s_{t}]+b-\frac{ac% +bd}{a+d}\right)+\frac{1}{a}E_{t-1}[s_{t}]\;,
  17. ln Y t - E t - 1 [ ln Y t ] = 1 - a a ( ln P t - E t - 1 [ ln P t ] ) + 1 a ( s t - E t - 1 [ s t ] ) . \ln{Y_{t}}-E_{t-1}[\ln{Y_{t}}]=\frac{1-a}{a}(\ln{P_{t}}-E_{t-1}[\ln{P_{t}}])+% \frac{1}{a}(s_{t}-E_{t-1}[s_{t}])\;.
  18. Y t * Y^{*}_{t}
  19. ln W t = ln W t * \ln{W_{t}}=\ln{W^{*}_{t}}
  20. ln Y t * = 1 - a a ( b - a c + b d a + d + s t - d a + d s t ) + s t . \ln{Y^{*}_{t}}=\frac{1-a}{a}\left(b-\frac{ac+bd}{a+d}+s_{t}-\frac{d}{a+d}s_{t}% \right)+s_{t}\;\;.
  21. ln Y t - ln Y t * = 1 - a a ( ln P t - E t - 1 [ ln P t ] ) + 1 - a a d a + d ( s t - E t - 1 [ s t ] ) . \ln{Y_{t}}-\ln{Y^{*}_{t}}=\frac{1-a}{a}(\ln{P_{t}}-E_{t-1}[\ln{P_{t}}])+\frac{% 1-a}{a}\cdot\frac{d}{a+d}(s_{t}-E_{t-1}[s_{t}])\;.
  22. X t X_{t}
  23. X t = Y t P t X_{t}=Y_{t}P_{t}
  24. ln Y t - ln Y t * = ( 1 - a ) d a + d ( ln X t - E t - 1 [ ln X t ] ) + 1 - a a a a + d ( ln P t - E t - 1 [ ln P t ] ) . \ln{Y_{t}}-\ln{Y^{*}_{t}}=\frac{(1-a)d}{a+d}(\ln{X_{t}}-E_{t-1}[\ln{X_{t}}])+% \frac{1-a}{a}\cdot\frac{a}{a+d}(\ln{P_{t}}-E_{t-1}[\ln{P_{t}}])\;\;.
  25. ln Y t - ln Y t * \ln{Y_{t}}-\ln{Y^{*}_{t}}
  26. a a + d = 0 \frac{a}{a+d}=0
  27. d = d=\infty

Non-Archimedean_ordered_field.html

  1. x / y x/y
  2. π / 2 π/2
  3. \R \R
  4. \R \R

Non-linear_effects.html

  1. e e p r o d u c t = e e m a x e e c a t a l y s t ee_{product}=ee_{max}ee_{catalyst}
  2. e e p r o d u c t = f e e m a x e e c a t a l y s t ee_{product}=f\,ee_{max}ee_{catalyst}
  3. e e product = e e max e e auxiliary ( 1 + β ) / ( 1 + g β ) ee\text{product}=ee\text{max}ee\text{auxiliary}(1+\beta)/(1+g\,\beta)
  4. β = z / x + y \beta=z/x+y
  5. g = r r s / r r r g=r_{rs}/r_{rr}
  6. r = ( x + y + g z ) r R R r=(x+y+gz)r_{RR}
  7. E E prod = E E 0 e e aux 3 + 3 g E E 0 / E E 0 + ( 1 - 3 g E E 0 / E E 0 ) e e aux 2 1 + 3 g + 3 ( 1 - g ) e e aux 2 EE\text{prod}=EE_{0}ee\text{aux}\frac{3+3\,g\,EE^{\prime}_{0}/EE_{0}+(1-3\,g\,% EE^{\prime}_{0}/EE_{0})ee\text{aux}^{2}}{1+3\,g+3(1-g)ee\text{aux}^{2}}

Non-local_means.html

  1. Ω \Omega
  2. p p
  3. q q
  4. u ( p ) = 1 C ( p ) Ω v ( q ) f ( p , q ) d q . u(p)={1\over C(p)}\int_{\Omega}v(q)f(p,q)dq.
  5. u ( p ) u(p)
  6. p p
  7. v ( q ) v(q)
  8. q q
  9. f ( p , q ) f(p,q)
  10. q Ω \forall q\in\Omega
  11. C ( p ) C(p)
  12. C ( p ) = Ω f ( p , q ) d q . C(p)=\int_{\Omega}f(p,q)dq.
  13. f ( p , q ) f(p,q)
  14. p p
  15. q q
  16. μ = B ( p ) \mu=B(p)
  17. f ( p , q ) = e - | B ( q ) - B ( p ) | 2 h 2 f(p,q)=e^{-{{\left|B(q)-B(p)\right|^{2}}\over h^{2}}}
  18. h h
  19. B ( p ) B(p)
  20. p p
  21. Ω \Omega
  22. u ( p ) = 1 C ( p ) q Ω v ( q ) f ( p , q ) u(p)={1\over C(p)}\sum_{q\in\Omega}v(q)f(p,q)
  23. C ( p ) C(p)
  24. C ( p ) = q Ω f ( p , q ) C(p)=\sum_{q\in\Omega}f(p,q)
  25. f ( p , q ) = e - | B ( q ) - B ( p ) | 2 h 2 f(p,q)=e^{-{{\left|B(q)-B(p)\right|^{2}}\over h^{2}}}
  26. B ( p ) B(p)
  27. B ( p ) = 1 | R ( p ) | i R ( p ) v ( i ) B(p)={1\over|R(p)|}\sum_{i\in R(p)}v(i)
  28. R ( p ) Ω R(p)\subseteq\Omega
  29. p p
  30. | R ( p ) | |R(p)|
  31. R R

Non-radiative_dielectric_waveguide.html

  1. k z = β - j α k_{z}=\beta-j\alpha
  2. k y = m π a k_{y}=\frac{m\pi}{a}
  3. k o 2 = ( 2 π λ o ) 2 = k x o 2 + k y 2 + k z 2 = - | k x o | 2 + k y 2 + β 2 ( 1 ) k_{o}^{2}=\left(\frac{2\pi}{\lambda_{o}}\right)^{2}=k_{xo}^{2}+k_{y}^{2}+k_{z}% ^{2}=-\left|k_{xo}\right|^{2}+k_{y}^{2}+\beta^{2}\ \ \ \ (1)
  4. k 2 = k o 2 ε r = ( 2 π λ ) 2 = k x ε 2 + k y 2 + k z 2 = k x ε 2 + k y 2 + β 2 ( 2 ) k^{2}=k_{o}^{2}\varepsilon_{r}=\left(\frac{2\pi}{\lambda}\right)^{2}=k_{x% \varepsilon}^{2}+k_{y}^{2}+k_{z}^{2}=k_{x\varepsilon}^{2}+k_{y}^{2}+\beta^{2}% \ \ \ \ \ \ (2)
  5. ε r \varepsilon_{r}
  6. a < λ o 2 a<\frac{\lambda_{o}}{2}
  7. λ = λ o ε r \lambda=\frac{\lambda_{o}}{\sqrt{\varepsilon_{r}}}
  8. k x o = - j | k x o | k_{xo}=-j\left|k_{xo}\right|
  9. Y + Y = 0 \overleftarrow{Y}+\overrightarrow{Y}=0\ \ \ \ \
  10. Y a n d Y \ \ \ \ \ \ \ \overleftarrow{Y}\ \ \ and\ \ \ \overrightarrow{Y}\
  11. Y = Y o \overrightarrow{Y}=Y_{o}
  12. Y = - j Y ε c o t ( k x ε w ) \overleftarrow{Y}=-jY_{\varepsilon}cot(k_{x\varepsilon}w)
  13. - j Y ε c o t ( k x ε w ) + Y o = 0 -jY_{\varepsilon}cot(k_{x\varepsilon}w)+Y_{o}=0
  14. - j ε r k x o c o t ( k x ε w ) + k x ε = 0 ( 4 ) -j\varepsilon_{r}k_{xo}cot(k_{x\varepsilon}w)+k_{x\varepsilon}=0\ \ \ \ (4)
  15. k x o = k o 2 - ( m π a ) 2 - β 2 = k o 1 - ( m π a k o ) 2 - β 2 k o k_{xo}=\sqrt{{k_{o}^{2}}-(\frac{m\pi}{a})^{2}-\beta^{2}}=k_{o}\sqrt{1-(\frac{m% \pi}{ak_{o}})^{2}-\frac{\beta^{2}}{k_{o}}}
  16. k x o = k o 2 ε r - ( m π a ) 2 - β 2 = k o ε r - ( m π a k o ) 2 - β 2 k o k_{xo}=\sqrt{{k_{o}^{2}\varepsilon_{r}}-(\frac{m\pi}{a})^{2}-\beta^{2}}=k_{o}% \sqrt{\varepsilon_{r}-(\frac{m\pi}{ak_{o}})^{2}-\frac{\beta^{2}}{k_{o}}}
  17. β k o = ε e f f \frac{\beta}{k_{o}}=\sqrt{\varepsilon_{eff}}
  18. ε e f f \varepsilon_{eff}
  19. β = k 2 - k t 2 = ω 2 μ ε - ω c 2 μ ε \beta=\sqrt{k^{2}-k_{t}^{2}}=\sqrt{\omega^{2}\mu\varepsilon-\omega_{c}^{2}\mu\varepsilon}
  20. Y o = k x o μ o ω , Y ε = k x ε μ o ω ( 5 ) Y_{o}=\frac{k_{xo}}{\mu_{o}\omega}\ \ \ ,\ \ \ \ Y_{\varepsilon}=\frac{k_{x% \varepsilon}}{\mu_{o}\omega}\ \ \ \ \ \ \ (5)
  21. Y = Y o , Y = j Y ε t a n ( k x ε w ) \overrightarrow{Y}=Y_{o}\ \ \ ,\ \ \ \overleftarrow{Y}=jY_{\varepsilon}tan(k_{% x\varepsilon}w)
  22. j k x ε t a n ( k x ε w ) + k x o = 0 ( 6 ) jk_{x\varepsilon}tan(k_{x\varepsilon}w)+k_{xo}=0\ \ \ (6)
  23. Y = Y o Y = Y ε Y o + j Y ε t a n ( k x ε b ) Y ε + j Y o t a n ( k x ε b ) \overrightarrow{Y}=Y_{o}\ \ \ \ \ \ \ \ \overleftarrow{Y}=Y_{\varepsilon}\frac% {Y_{o}+jY_{\varepsilon}tan(k_{x\varepsilon}b)}{Y_{\varepsilon}+jY_{o}tan(k_{x% \varepsilon}b)}
  24. Y o + Y ε Y o + j Y ε t a n ( k x ε b ) Y ε + j Y o t a n ( k x ε b ) = 0 ( 7 ) Y_{o}+Y_{\varepsilon}\frac{Y_{o}+jY_{\varepsilon}tan(k_{x\varepsilon}b)}{Y_{% \varepsilon}+jY_{o}tan(k_{x\varepsilon}b)}=0\ \ \ \ \ \ \ \ \ (7)
  25. A ¯ \underline{A}
  26. H ¯ = × A ¯ , E ¯ = - j ω μ A ¯ + A ¯ j ω ε ( 8 ) \underline{H}=\triangledown\times\underline{A}\ \ \ \ \ ,\ \ \ \ \ \ % \underline{E}=-j\omega\mu\underline{A}+\frac{\triangledown\triangledown\cdot% \underline{A}}{j\omega\varepsilon}\ \ \ \ \ \ (8)
  27. F ¯ \underline{F}
  28. E ¯ = - × F ¯ , H ¯ = - j ω F ¯ + F ¯ j ω μ ( 9 ) \underline{E}=-\triangledown\times\underline{F}\ \ \ \ \ \ ,\ \ \ \ \ \ % \underline{H}=-j\omega\underline{F}+\frac{\triangledown\triangledown\cdot% \underline{F}}{j\omega\mu}\ \ \ \ \ \ \ (9)
  29. A ¯ = A z z o ¯ = L T M ( z ) T T M ( x , y ) z o ¯ ( 10 ) \underline{A}=A_{z}\underline{z_{o}}=L^{TM}(z)T^{TM}(x,y)\underline{z_{o}}\ \ % \ \ \ \ \ (10)
  30. F ¯ = F z z o ¯ = L T M ( z ) T T M ( x , y ) z o ¯ ( 11 ) \underline{F}=F_{z}\underline{z_{o}}=L^{TM}(z)T^{TM}(x,y)\underline{z_{o}}\ \ % \ \ \ \ (11)
  31. 2 A z + k 2 A z = 0 ( 12 ) \triangledown^{2}A_{z}+k^{2}A_{z}=0\ \ \ \ \ \ (12)
  32. 2 F z + k 2 F z = 0 ( 13 ) \triangledown^{2}F_{z}+k^{2}F_{z}=0\ \ \ \ \ \ (13)
  33. d 2 L d z 2 + k z 2 L = 0 ( 14 ) \frac{d^{2}L}{dz^{2}}+k_{z}^{2}L=0\ \ \ \ \ \ (14)
  34. t 2 T + k t 2 T = 0 ( 15 ) \triangledown_{t}^{2}T+k_{t}^{2}T=0\ \ \ \ \ \ (15)
  35. t 2 = 2 - 2 z 2 a n d k t 2 = k 2 - k z 2 \triangledown_{t}^{2}=\triangledown^{2}-\frac{\partial^{2}}{\partial z^{2}}\ % \ \ and\ \ \ k_{t}^{2}=k^{2}-k_{z}^{2}
  36. L ( z ) = L o e - j k z z + L o - e j k z z ( 16 ) L(z)=L_{o}^{\dotplus}e^{-jk_{z}z}+L_{o}^{-}e^{jk_{z}z}\ \ \ \ \ \ \ (16)
  37. d 2 X d x 2 + k x 2 X = 0 ( 17 ) \frac{d^{2}X}{dx^{2}}+k_{x}^{2}X=0\ \ \ \ \ \ \ (17)
  38. d 2 Y d y 2 + k y 2 Y = 0 ( 18 ) \frac{d^{2}Y}{dy^{2}}+k_{y}^{2}Y=0\ \ \ \ \ \ \ \ (18)
  39. Y ( y ) = C 1 cos ( k y y ) + C 2 s i n ( k y y ) Y(y)=C_{1}\cos(k_{y}y)+C_{2}sin(k_{y}y)
  40. s i n ( m π a y ) ( m = 1 , 2 , 3 , ) sin(\frac{m\pi}{a}y)\ \ \ \ \ (m=1,2,3,...)
  41. c o s ( m π a y ) ( m = 1 , 2 , 3 , ) cos(\frac{m\pi}{a}y)\ \ \ \ \ \ (m=1,2,3,...)
  42. X ( x ) = C 3 e - j k x x + C 4 e j k x x X(x)=C_{3}e^{-jk_{x}x}+C_{4}e^{jk_{x}x}
  43. T ε T E = c o s ( m π a y ) ( C e - j k x ε x + D e j k x ε x ) m = 1 , 2 , 3 , ( 20 ) T_{\varepsilon}^{TE}=cos(\frac{m\pi}{a}y)\cdot(Ce^{-jk_{x\varepsilon}x}+De^{jk% _{x\varepsilon}x})\ \ \ \ \ \ m=1,2,3,...\ \ \ \ (20)
  44. k x ε = k o ε r - ( m π k o a ) 2 - ( k z k o ) 2 ( 21 ) k_{x\varepsilon}=k_{o}\sqrt{\varepsilon_{r}-(\frac{m\pi}{k_{o}a})^{2}-(\frac{k% _{z}}{k_{o}})^{2}}\ \ \ \ (21)
  45. T o + T M = E s i n ( m π a y ) e - j k x o ( x - w ) ( 22 ) T_{o+}^{TM}=Esin(\frac{m\pi}{a}y)e^{-jk_{xo}(x-w)}\ \ \ \ (22)
  46. T o + T E = F c o s ( m π a y ) e - j k x o ( x - w ) ( 23 ) T_{o+}^{TE}=Fcos(\frac{m\pi}{a}y)e^{-jk_{xo}(x-w)}\ \ \ \ \ \ \ \ \ (23)
  47. T o - T E = H c o s ( m π a y ) e j k x o ( x - w ) ( 25 ) T_{o-}^{TE}=Hcos(\frac{m\pi}{a}y)e^{jk_{xo}(x-w)}\ \ \ \ \ \ \ \ (25)
  48. k x o = k o 1 - ( m π k o a ) 2 - ( k z k o ) 2 ( 26 ) k_{xo}=k_{o}\sqrt{1-(\frac{m\pi}{k_{o}a})^{2}-(\frac{k_{z}}{k_{o}})^{2}}\ \ \ % \ \ \ \ (26)
  49. E x = 1 j ω ε dL d z T x T M - L T y T E = - k z ω ε L T x T M - L T y T E ( 27 ) E_{x}=\frac{1}{j\omega\varepsilon}\frac{\mathrm{dL}}{\mathrm{d}z}\frac{% \partial T}{\partial x}^{TM}-L\frac{\partial T}{\partial y}^{TE}=\frac{-k_{z}}% {\omega\varepsilon}L\frac{\partial T}{\partial x}^{TM}-L\frac{\partial T}{% \partial y}^{TE}\ \ \ \ \ \ \ (27)
  50. E y = 1 j ω ε dL d z T y T M - L T x T E = - k z ω ε L T y T M - L T x T E ( 28 ) E_{y}=\frac{1}{j\omega\varepsilon}\frac{\mathrm{dL}}{\mathrm{d}z}\frac{% \partial T}{\partial y}^{TM}-L\frac{\partial T}{\partial x}^{TE}=\frac{-k_{z}}% {\omega\varepsilon}L\frac{\partial T}{\partial y}^{TM}-L\frac{\partial T}{% \partial x}^{TE}\ \ \ \ \ \ (28)
  51. E z = k t 2 j ω ε L T T M = k 2 - k z 2 j ω ε L T T M ( 29 ) E_{z}=\frac{k_{t}^{2}}{j\omega\varepsilon}L\ T^{TM}=\frac{k^{2}-k_{z}^{2}}{j% \omega\varepsilon}L\ T^{TM}\ \ \ \ \ \ \ (29)
  52. H x = L T y T M + 1 j ω μ dL d z T x T E = L T y T M - k z ω μ L T x T E ( 30 ) H_{x}=L\frac{\partial T}{\partial y}^{TM}+\frac{1}{j\omega\mu}\frac{\mathrm{dL% }}{\mathrm{d}z}\frac{\partial T}{\partial x}^{TE}=L\frac{\partial T}{\partial y% }^{TM}-\frac{k_{z}}{\omega\mu}L\frac{\partial T}{\partial x}^{TE}\ \ \ \ \ (30)
  53. H y = - L T x T M + 1 j ω μ dL d z T y T E = - L T x T M - k z ω μ L T y T E ( 31 ) H_{y}=-L\frac{\partial T}{\partial x}^{TM}+\frac{1}{j\omega\mu}\frac{\mathrm{% dL}}{\mathrm{d}z}\frac{\partial T}{\partial y}^{TE}=-L\frac{\partial T}{% \partial x}^{TM}-\frac{k_{z}}{\omega\mu}L\frac{\partial T}{\partial y}^{TE}\ % \ \ \ (31)
  54. H z = k t 2 j ω μ L T T E = k 2 - k z 2 j ω μ L T T E ( 32 ) H_{z}=\frac{k_{t}^{2}}{j\omega\mu}L\ T^{TE}=\frac{k^{2}-k_{z}^{2}}{j\omega\mu}% L\ T^{TE}\ \ \ \ (32)
  55. - k z ω ε o T o y T M + T o x T E = - k z ω ε o ε r T ε y T M + T ε x T E -\frac{k_{z}}{\omega\varepsilon_{o}}\frac{\partial T_{o}}{\partial y}^{TM}+% \frac{\partial T_{o}}{\partial x}^{TE}=-\frac{k_{z}}{\omega\varepsilon_{o}% \varepsilon_{r}}\frac{\partial T_{\varepsilon}}{\partial y}^{TM}+\frac{% \partial T_{\varepsilon}}{\partial x}^{TE}
  56. k o 2 - k z 2 j ω ε o T o T M = k o 2 ε r - k z 2 j ω ε r ε o T ε T M \frac{k_{o}^{2}-k_{z}^{2}}{j\omega\varepsilon_{o}}\ T_{o}^{TM}=\frac{k_{o}^{2}% \varepsilon_{r}-k_{z}^{2}}{j\omega\varepsilon_{r}\varepsilon_{o}}T_{% \varepsilon}^{TM}
  57. - T o x T M - k z ω μ T o y T E = - T ε x T M - k z ω μ T ε y T E -\frac{\partial T_{o}}{\partial x}^{TM}-\frac{k_{z}}{\omega\mu}\frac{\partial T% _{o}}{\partial y}^{TE}=-\frac{\partial T_{\varepsilon}}{\partial x}^{TM}-\frac% {k_{z}}{\omega\mu}\frac{\partial T_{\varepsilon}}{\partial y}^{TE}
  58. k o 2 - k z 2 j ω μ T o T E = k o 2 ε r - k z 2 j ω μ T E \frac{k_{o}^{2}-k_{z}^{2}}{j\omega\mu}\ T_{o}^{TE}=\frac{k_{o}^{2}\varepsilon_% {r}-k_{z}^{2}}{j\omega\mu}^{TE}
  59. E y = [ - k z ω ε o ε r m π a ( A e - j k x ε x + B e j k x ε x ) - j k x ε ( C e - j k x ε x - D e j k x ε x ) ] c o s ( m π a y ) e - j k z z ( 34 ) E_{y}=\left[-\frac{k_{z}}{\omega\varepsilon_{o}\varepsilon_{r}}\frac{m\pi}{a}(% Ae^{-jk_{x\varepsilon}x}+Be^{jk_{x\varepsilon}x})-jk_{x\varepsilon}(Ce^{-jk_{x% \varepsilon}x}-De^{jk_{x\varepsilon}x})\right]cos(\frac{m\pi}{a}y)e^{-jk_{z}z}% \ \ \ \ \ \ \ (34)
  60. E z = k o 2 ε r - k z 2 j ω ε o ε r ( A e - j k x ε x - B e j k x ε x ) s i n ( m π a y ) e - j k z z ( 35 ) E_{z}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{j\omega\varepsilon_{o}% \varepsilon_{r}}(Ae^{-jk_{x\varepsilon}x}-Be^{jk_{x\varepsilon}x})sin(\frac{m% \pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ \ (35)
  61. H x = [ m π a ( A e - j k x ε x + B e j k x ε x ) + j k x ε k z ω μ ( C e - j k x ε x - D e j k x ε x ) ] c o s ( m π a y ) e - j k z z ( 36 ) H_{x}=\left[\frac{m\pi}{a}(Ae^{-jk_{x\varepsilon}x}+Be^{jk_{x\varepsilon}x})+j% \frac{k_{x\varepsilon}k_{z}}{\omega\mu}(Ce^{-jk_{x\varepsilon}x}-De^{jk_{x% \varepsilon}x})\right]cos(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ \ \ (36)
  62. H y = [ j k x ε ( A e - j k x ε x - B e j k x ε x ) + k z ω μ m π a ( C e - j k x ε x + D e j k x ε x ) ] s i n ( m π a y ) e - j k z z ( 37 ) H_{y}=\left[jk_{x\varepsilon}(Ae^{-jk_{x\varepsilon}x}-Be^{jk_{x\varepsilon}x}% )+\frac{k_{z}}{\omega\mu}\frac{m\pi}{a}(Ce^{-jk_{x\varepsilon}x}+De^{jk_{x% \varepsilon}x})\right]sin(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ \ \ (37)
  63. H z = k o 2 ε r - k z 2 j ω μ ( C e - j k x ε x + D e j k x ε x ) c o s ( m π a y ) e - j k z z ( 38 ) H_{z}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{j\omega\mu}(Ce^{-jk_{x% \varepsilon}x}+De^{jk_{x\varepsilon}x})cos(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ % \ \ \ \ \ \ \ \ (38)
  64. E x = k o 2 ε r - k z 2 k o 2 - k z 2 [ j k x o k z ω ε o ε r ( A e - j k x ε w + B e j k x ε w ) + m π a ( C e - j k x ε w + D e j k x ε w ) ] e - j k x o ( x - w ) s i n ( m π a y ) e - j k z z ( 39 ) E_{x}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[\frac{jk_% {xo}k_{z}}{\omega\varepsilon_{o}\varepsilon_{r}}(A\ e^{-jk_{x\varepsilon}w}+B% \ e^{jk_{x\varepsilon}w})+\frac{m\pi}{a}(C\ e^{-jk_{x\varepsilon}w}+D\ e^{jk_{% x\varepsilon}w})]e^{-jk_{xo}(x-w)}sin(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ \ % \ \ (39)
  65. E y = k o 2 ε r - k z 2 k o 2 - k z 2 [ - k z ω ε o ε r m π a ( A e - j k x ε w + B e j k x ε w ) - j k x o ( C e - j k x ε w + D e j k x ε w ) ] e - j k x o ( x - w ) c o s ( m π a y ) e - j k z z ( 40 ) E_{y}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[-\frac{k_% {z}}{\omega\varepsilon_{o}\varepsilon_{r}}\frac{m\pi}{a}(A\ e^{-jk_{x% \varepsilon}w}+B\ e^{jk_{x\varepsilon}w})-jk_{xo}(C\ e^{-jk_{x\varepsilon}w}+D% \ e^{jk_{x\varepsilon}w})]e^{-jk_{xo}(x-w)}cos(\frac{m\pi}{a}y)e^{-jk_{z}z}\ % \ \ \ \ \ (40)
  66. E z = k o 2 ε r - k z 2 j ω ε o ε r ( A e - j k x ε w + B e j k x ε w ) e - j k x o ( x - w ) s i n ( m π a y ) e - j k z z ( 41 ) E_{z}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{j\omega\varepsilon_{o}% \varepsilon_{r}}(A\ e^{-jk_{x\varepsilon}w}+B\ e^{jk_{x\varepsilon}w})e^{-jk_{% xo}(x-w)}sin(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ (41)
  67. H x = k o 2 ε r - k z 2 k o 2 - k z 2 [ m π a 1 ε r ( A e - j k x ε w + B e j k x ε w ) + j k x o k z ω μ ( C e - j k x ε w + D e j k x ε w ) ] e - j k x o ( x - w ) c o s ( m π a y ) e - j k z z ( 42 ) H_{x}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[\frac{m% \pi}{a}\frac{1}{\varepsilon_{r}}(A\ e^{-jk_{x\varepsilon}w}+B\ e^{jk_{x% \varepsilon}w})+\frac{jk_{xo}k_{z}}{\omega\mu}(C\ e^{-jk_{x\varepsilon}w}+D\ e% ^{jk_{x\varepsilon}w})]e^{-jk_{xo}(x-w)}cos(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ % \ \ \ \ (42)
  68. H y = k o 2 ε r - k z 2 k o 2 - k z 2 [ j k x o ε r ( A e - j k x ε w + B e j k x ε w ) + k z ω μ m π a ( C e - j k x ε w + D e j k x ε w ) ] e - j k x 0 ( x - w ) s i n ( m π a y ) e - j k z z ( 43 ) H_{y}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[j\frac{k_% {xo}}{\varepsilon_{r}}(A\ e^{-jk_{x\varepsilon}w}+B\ e^{jk_{x\varepsilon}w})+% \frac{k_{z}}{\omega\mu}\frac{m\pi}{a}(C\ e^{-jk_{x\varepsilon}w}+D\ e^{jk_{x% \varepsilon}w})]e^{-jk_{x0}(x-w)}sin(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ \ (% 43)
  69. H z = k o 2 ε r - k z 2 j ω μ ( C e - j k x ε w + D e j k x ε w ) e - j k x o ( x - w ) c o s ( m π a y ) e - j k z z ( 44 ) H_{z}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{j\omega\mu}(C\ e^{-jk_{x% \varepsilon}w}+D\ e^{jk_{x\varepsilon}w})e^{-jk_{xo}(x-w)}cos(\frac{m\pi}{a}y)% e^{-jk_{z}z}\ \ \ \ \ (44)
  70. E y = k o 2 ε r - k z 2 k o 2 - k z 2 [ - k z ω ε o ε r m π a ( A e j k x ε w + B e - j k x ε w ) - j k x o ( C e j k x ε w + D e - j k x ε w ) ] e j k x o ( x + w ) c o s ( m π a y ) e - j k z z ( 46 ) E_{y}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[-\frac{k_% {z}}{\omega\varepsilon_{o}\varepsilon_{r}}\frac{m\pi}{a}(A\ e^{jk_{x% \varepsilon}w}+B\ e^{-jk_{x\varepsilon}w})-jk_{xo}(C\ e^{jk_{x\varepsilon}w}+D% \ e^{-jk_{x\varepsilon}w})]e^{jk_{xo}(x+w)}cos(\frac{m\pi}{a}y)e^{-jk_{z}z}\ % \ \ \ \ \ (46)
  71. E z = k o 2 ε r - k z 2 j ω ε o ε r ( A e j k x ε w + B e - j k x ε w ) e - j k x o ( x + w ) s i n ( m π a y ) e - j k z z ( 47 ) E_{z}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{j\omega\varepsilon_{o}% \varepsilon_{r}}(A\ e^{jk_{x\varepsilon}w}+B\ e^{-jk_{x\varepsilon}w})e^{-jk_{% xo}(x+w)}sin(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ (47)
  72. H x = k o 2 ε r - k z 2 k o 2 - k z 2 [ m π a 1 ε r ( A e j k x ε w + B e - j k x ε w ) - j k x o k z ω μ ( C e j k x ε w + D e - j k x ε w ) ] e j k x o ( x + w ) c o s ( m π a y ) e - j k z z ( 48 ) H_{x}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[\frac{m% \pi}{a}\frac{1}{\varepsilon_{r}}(A\ e^{jk_{x\varepsilon}w}+B\ e^{-jk_{x% \varepsilon}w})-\frac{jk_{xo}k_{z}}{\omega\mu}(C\ e^{jk_{x\varepsilon}w}+D\ e^% {-jk_{x\varepsilon}w})]e^{jk_{xo}(x+w)}cos(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ % \ \ \ \ (48)
  73. H y = k o 2 ε r - k z 2 k o 2 - k z 2 [ - j k x o ε r ( A e j k x ε w + B e - j k x ε w ) + k z ω μ m π a ( C e j k x ε w + D e - j k x ε w ) ] e j k x o ( x + w ) s i n ( m π a y ) e - j k z z ( 49 ) H_{y}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[-j\frac{k% _{xo}}{\varepsilon_{r}}(A\ e^{jk_{x\varepsilon}w}+B\ e^{-jk_{x\varepsilon}w})+% \frac{k_{z}}{\omega\mu}\frac{m\pi}{a}(C\ e^{jk_{x\varepsilon}w}+D\ e^{-jk_{x% \varepsilon}w})]e^{jk_{xo}(x+w)}sin(\frac{m\pi}{a}y)e^{-jk_{z}z}\ \ \ \ \ \ (49)
  74. H z = k o 2 ε r - k z 2 j ω μ ( C e j k x ε w + D e - j k x ε w ) e j k x o ( x + w ) c o s ( m π a y ) e - j k z z ( 50 ) H_{z}=\frac{k_{o}^{2}\varepsilon_{r}-k_{z}^{2}}{j\omega\mu}(C\ e^{jk_{x% \varepsilon}w}+D\ e^{-jk_{x\varepsilon}w})e^{jk_{xo}(x+w)}cos(\frac{m\pi}{a}y)% e^{-jk_{z}z}\ \ \ \ \ (50)

Noncentral_beta_distribution.html

  1. j = 0 e - λ / 2 ( λ 2 ) j j ! I x ( α + j , β ) \sum_{j=0}^{\infty}e^{-\lambda/2}\frac{\left(\frac{\lambda}{2}\right)^{j}}{j!}% I_{x}\left(\alpha+j,\beta\right)
  2. e - λ 2 Γ ( α + 1 ) Γ ( α ) Γ ( α + β ) Γ ( α + β + 1 ) F 2 2 ( α + β , α + 1 ; α , α + β + 1 ; λ 2 ) e^{-\frac{\lambda}{2}}\frac{\Gamma\left(\alpha+1\right)}{\Gamma\left(\alpha% \right)}\frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha+\beta+1% \right)}{}_{2}F_{2}\left(\alpha+\beta,\alpha+1;\alpha,\alpha+\beta+1;\frac{% \lambda}{2}\right)
  3. e - λ 2 Γ ( α + 2 ) Γ ( α ) Γ ( α + β ) Γ ( α + β + 2 ) F 2 2 ( α + β , α + 2 ; α , α + β + 2 ; λ 2 ) - μ 2 e^{-\frac{\lambda}{2}}\frac{\Gamma\left(\alpha+2\right)}{\Gamma\left(\alpha% \right)}\frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha+\beta+2% \right)}{}_{2}F_{2}\left(\alpha+\beta,\alpha+2;\alpha,\alpha+\beta+2;\frac{% \lambda}{2}\right)-\mu^{2}
  4. μ \mu
  5. X = χ m 2 ( λ ) χ m 2 ( λ ) + χ n 2 , X=\frac{\chi^{2}_{m}(\lambda)}{\chi^{2}_{m}(\lambda)+\chi^{2}_{n}},
  6. χ m 2 ( λ ) \chi^{2}_{m}(\lambda)
  7. λ \lambda
  8. χ n 2 \chi^{2}_{n}
  9. χ m 2 ( λ ) \chi^{2}_{m}(\lambda)
  10. X Beta ( m 2 , n 2 , λ ) X\sim\mbox{Beta}~{}\left(\frac{m}{2},\frac{n}{2},\lambda\right)
  11. Y = χ n 2 χ n 2 + χ m 2 ( λ ) , Y=\frac{\chi^{2}_{n}}{\chi^{2}_{n}+\chi^{2}_{m}(\lambda)},
  12. Y Y
  13. X = 1 - Y X=1-Y
  14. F ( x ) = j = 0 P ( j ) I x ( α + j , β ) , F(x)=\sum_{j=0}^{\infty}P(j)I_{x}(\alpha+j,\beta),
  15. I x ( a , b ) I_{x}(a,b)
  16. F ( x ) = j = 0 1 j ! ( λ 2 ) j e - λ / 2 I x ( α + j , β ) . F(x)=\sum_{j=0}^{\infty}\frac{1}{j!}\left(\frac{\lambda}{2}\right)^{j}e^{-% \lambda/2}I_{x}(\alpha+j,\beta).
  17. F ( x ) = j = 0 P ( j ) I x ( α , β + j ) . F(x)=\sum_{j=0}^{\infty}P(j)I_{x}(\alpha,\beta+j).
  18. f ( x ) = j = 0 1 j ! ( λ 2 ) j e - λ / 2 x α + j - 1 ( 1 - x ) β - 1 B ( α + j , β ) . f(x)=\sum_{j=0}^{\infty}\frac{1}{j!}\left(\frac{\lambda}{2}\right)^{j}e^{-% \lambda/2}\frac{x^{\alpha+j-1}(1-x)^{\beta-1}}{B(\alpha+j,\beta)}.
  19. B B
  20. α \alpha
  21. β \beta
  22. λ \lambda
  23. X Beta ( α , β , λ ) X\sim\mbox{Beta}~{}\left(\alpha,\beta,\lambda\right)
  24. β X α ( 1 - X ) \frac{\beta X}{\alpha(1-X)}
  25. 2 α , 2 β 2\alpha,2\beta
  26. λ \lambda
  27. X X
  28. F μ 1 , μ 2 ( λ ) F_{\mu_{1},\mu_{2}}\left(\lambda\right)
  29. μ 1 \mu_{1}
  30. μ 2 \mu_{2}
  31. Z = μ 2 μ 1 μ 2 μ 1 + X - 1 Z=\cfrac{\cfrac{\mu_{2}}{\mu_{1}}}{\cfrac{\mu_{2}}{\mu_{1}}+X^{-1}}
  32. Z Beta ( 1 2 μ 1 , 1 2 μ 2 , λ ) Z\sim\mbox{Beta}~{}\left(\frac{1}{2}\mu_{1},\frac{1}{2}\mu_{2},\lambda\right)
  33. λ = 0 \lambda=0