wpmath0000016_6

Havel–Hakimi_algorithm.html

  1. S = ( d 1 , , d n ) S=(d_{1},\dots,d_{n})
  2. S S
  3. S = ( d 2 - 1 , d 3 - 1 , , d d 1 + 1 - 1 , d d 1 + 2 , , d n ) S^{\prime}=(d_{2}-1,d_{3}-1,\dots,d_{d_{1}+1}-1,d_{d_{1}+2},\dots,d_{n})
  4. S S
  5. n - 1 n-1
  6. S := S S:=S^{\prime}
  7. S S^{\prime}
  8. v 1 , , v n v_{1},\cdots,v_{n}
  9. S S
  10. S S^{\prime}
  11. { v 1 , v 2 } , { v 1 , v 3 } , , { v 1 , v d 1 + 1 } \{v_{1},v_{2}\},\{v_{1},v_{3}\},\cdots,\{v_{1},v_{d_{1}+1}\}
  12. S S
  13. S S^{\prime}
  14. S S

Heat_rate_(efficiency).html

  1. H e a t R a t e = T h e r m a l E n e r g y I n E l e c t r i c a l E n e r g y O u t HeatRate=\frac{ThermalEnergyIn}{ElectricalEnergyOut}

Held–Karp_algorithm.html

  1. C = ( 0 2 9 10 1 0 6 4 15 7 0 8 6 3 12 0 ) C=\begin{pmatrix}0&2&9&10\\ 1&0&6&4\\ 15&7&0&8\\ 6&3&12&0\end{pmatrix}
  2. N ! 2 π N ( N e ) N N!\approx\sqrt{2\pi N}({N\over e})^{N}
  3. O ( 2 n n 2 ) O(2^{n}n^{2})
  4. O ( 2 n n ) O(2^{n}n)
  5. ( k = 2 n - 1 k ( k - 1 ) ( n - 1 k ) ) + ( n - 1 ) = ( n - 1 ) ( n - 2 ) 2 n - 3 + ( n - 1 ) \left(\sum_{k=2}^{n-1}k(k-1){\left({{n-1}\atop{k}}\right)}\right)+(n-1)=(n-1)(% n-2)2^{n-3}+(n-1)
  6. k = 2 n - 1 k = n ( n - 1 ) 2 - 1 \sum_{k=2}^{n-1}k={n(n-1)\over 2}-1
  7. ( k = 2 n - 1 k ( n - 1 k ) ) + ( n - 1 ) = ( n - 1 ) 2 n - 2 \left(\sum_{k=2}^{n-1}k{\left({{n-1}\atop{k}}\right)}\right)+(n-1)=(n-1)2^{n-2}

Heme_ligase.html

  1. \rightleftharpoons

Hermite_distribution.html

  1. a 1 + 16 a 2 ( a 1 + 4 a 2 ) 2 \frac{a_{1}+16a_{2}}{(a_{1}+4a_{2})^{2}}
  2. exp ( a 1 ( s - 1 ) + a 2 ( s 2 - 1 ) ) \exp(a_{1}(s-1)+a_{2}(s^{2}-1))\,
  3. exp ( a 1 ( e t - 1 ) + a 2 ( e 2 t - 1 ) ) \exp(a_{1}(e^{t}-1)+a_{2}(e^{2t}-1))\,
  4. exp ( a 1 ( e t i - 1 ) + a 2 ( e 2 t i - 1 ) ) \exp(a_{1}(e^{ti}-1)+a_{2}(e^{2ti}-1))\,
  5. f x ( m ) , ( x 1 ) {f_{x}}(m),(x\geq 1)
  6. f x ( m ) {f_{x}}(m)
  7. x x\to\infty
  8. p n = P ( Y = n ) = e [ - a 1 + a 2 ] j = 0 [ n / 2 ] a 1 n - 2 j a 2 j ( n - 2 j ) ! j ! p_{n}=P(Y=n)=e^{[-a_{1}+a_{2}]}\sum_{j=0}^{[n/2]}\frac{a_{1}^{n-2j}a_{2}^{j}}{% (n-2j)!j!}
  9. G Y ( s ) = n = 0 p n s n = exp ( a 1 ( s - 1 ) + a 2 ( s 2 - 1 ) ) G_{Y}(s)=\sum_{n=0}^{\infty}p_{n}s^{n}=\exp(a_{1}(s-1)+a_{2}(s^{2}-1))
  10. Y Herm ( a 1 , a 2 ) Y\ \sim\ \mathrm{Herm}(a_{1},a_{2})\,
  11. M ( t ) = G ( e t ) = exp ( a 1 ( e t - 1 ) + a 2 ( e 2 t - 1 ) ) M(t)=G(e^{t})=\exp(a_{1}(e^{t}-1)+a_{2}(e^{2t}-1))
  12. K ( t ) = log ( M ( t ) ) = a 1 ( e t - 1 ) + a 2 ( e 2 t - 1 ) K(t)=\log(M(t))=a_{1}(e^{t}-1)+a_{2}(e^{2t}-1)
  13. k n = a 1 + 2 n a 2 k_{n}=a_{1}+2^{n}a_{2}
  14. μ 1 = k 1 = a 1 + 2 a 2 \mu_{1}=k_{1}=a_{1}+2a_{2}
  15. μ \mu
  16. μ 2 = k 2 = a 1 + 4 a 2 \mu_{2}=k_{2}=a_{1}+4a_{2}
  17. σ 2 \sigma^{2}
  18. μ 3 = k 3 = a 1 + 8 a 2 \mu_{3}=k_{3}=a_{1}+8a_{2}
  19. k 3 k_{3}
  20. μ 4 = k 4 + 3 k 2 2 = a 1 + 16 a 2 + 3 ( a 1 + 4 a 2 ) 2 \mu_{4}=k_{4}+3k_{2}^{2}=a_{1}+16a_{2}+3(a_{1}+4a_{2})^{2}
  21. k 4 k_{4}
  22. γ 1 = μ 3 μ 2 3 / 2 = ( a 1 + 8 a 2 ) ( a 1 + 4 a 2 ) 3 / 2 \gamma_{1}=\frac{\mu_{3}}{\mu_{2}^{3/2}}=\frac{(a_{1}+8a_{2})}{(a_{1}+4a_{2})^% {3/2}}
  23. γ 1 > 0 \gamma_{1}>0
  24. β 2 = μ 4 μ 2 2 = a 1 + 16 a 2 + 3 ( a 1 + 4 a 2 ) 2 ( a 1 + 4 a 2 ) 2 = a 1 + 16 a 2 ( a 1 + 4 a 2 ) 2 + 3 \beta_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}=\frac{a_{1}+16a_{2}+3(a_{1}+4a_{2})^{2}}% {(a_{1}+4a_{2})^{2}}=\frac{a_{1}+16a_{2}}{(a_{1}+4a_{2})^{2}}+3
  25. γ 2 = μ 4 μ 2 2 - 3 = a 1 + 16 a 2 ( a 1 + 4 a 2 ) 2 \gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3=\frac{a_{1}+16a_{2}}{(a_{1}+4a_{2})^{% 2}}
  26. β 2 > 3 \beta_{2}>3
  27. γ 2 > 0 \gamma_{2}>0
  28. e i t X e^{itX}
  29. ϕ ( t ) = E [ e i t X ] = j = 0 e i j t P [ X = j ] \phi(t)=E[e^{itX}]=\sum_{j=0}^{\infty}e^{ijt}P[X=j]
  30. ϕ x ( t ) = M X ( i t ) \phi_{x}(t)=M_{X}(it)
  31. ϕ x ( t ) = exp ( a 1 ( e i t - 1 ) + a 2 ( e 2 i t - 1 ) ) \phi_{x}(t)=\exp(a_{1}(e^{it}-1)+a_{2}(e^{2it}-1))
  32. F ( x ; a 1 , a 2 ) = P ( X x ) = exp ( - ( a 1 + a 2 ) ) i = 0 x j = 0 [ i / 2 ] a 1 i - 2 j a 2 j ( i - 2 j ) ! j ! \begin{aligned}\displaystyle F(x;a_{1},a_{2})&\displaystyle=P(X\leq x)\\ &\displaystyle=\exp(-(a_{1}+a_{2}))\sum_{i=0}^{\lfloor x\rfloor}\sum_{j=0}^{[i% /2]}\frac{a_{1}^{i-2j}a_{2}^{j}}{(i-2j)!j!}\end{aligned}
  33. a 1 ^ = 0.0135 \hat{a_{1}}=0.0135
  34. a 2 ^ = 0.0932 \hat{a_{2}}=0.0932
  35. X 1 Herm ( a 1 , a 2 ) X_{1}\sim\mathrm{Herm}(a_{1},a_{2})
  36. X 2 Herm ( b 1 , b 2 ) X_{2}\sim\mathrm{Herm}(b_{1},b_{2})
  37. Y Herm ( a 1 + b 1 , a 2 + b 2 ) Y\sim\mathrm{Herm}(a_{1}+b_{1},a_{2}+b_{2})
  38. d = Var ( Y ) E ( y ) = a 1 + 4 a 2 a 1 + 2 a 2 = 1 + 2 a 2 a 1 + 2 a 2 d=\frac{\mathrm{Var}(Y)}{E(y)}=\frac{a_{1}+4a_{2}}{a_{1}+2a_{2}}=1+\frac{2a_{2% }}{a_{1}+2a_{2}}
  39. μ = a 1 + 2 a 2 \mu=a_{1}+2a_{2}
  40. σ 2 = a 1 + 4 a 2 \sigma^{2}=a_{1}+4a_{2}
  41. { x ¯ = a 1 + 2 a 2 σ 2 = a 1 + 4 a 2 \begin{cases}\bar{x}=a_{1}+2a_{2}\\ \sigma^{2}=a_{1}+4a_{2}\end{cases}
  42. a 1 ^ \hat{a_{1}}
  43. a 2 ^ \hat{a_{2}}
  44. a 1 ^ = 2 x ¯ - σ 2 \hat{a_{1}}=2\bar{x}-\sigma^{2}
  45. a 2 ^ = σ 2 - x ^ 2 \hat{a_{2}}=\frac{\sigma^{2}-\hat{x}}{2}
  46. a 1 ^ \hat{a_{1}}
  47. a 2 ^ \hat{a_{2}}
  48. x ¯ < σ 2 < 2 x ¯ \bar{x}<\sigma^{2}<2\bar{x}
  49. a 1 ^ \hat{a_{1}}
  50. a 2 ^ \hat{a_{2}}
  51. μ = a 1 + 2 a 2 \mu=a_{1}+2a_{2}
  52. σ 2 = a 1 + 4 a 2 \sigma^{2}=a_{1}+4a_{2}
  53. { a 1 = μ ( 2 - d ) a 2 = μ ( d - 1 ) 2 \begin{cases}a_{1}=\mu(2-d)\\ a_{2}=\frac{\mu(d-1)}{2}\end{cases}
  54. P ( X = x ) = exp ( - ( μ ( 2 - d ) + μ ( d - 1 ) 2 ) ) j = 0 [ x / 2 ] ( μ ( 2 - d ) ) x - 2 j ( μ ( d - 1 ) 2 ) j ( x - 2 j ) ! j ! P(X=x)=\exp\left(-\left(\mu(2-d)+\frac{\mu(d-1)}{2}\right)\right)\sum_{j=0}^{[% x/2]}\frac{(\mu(2-d))^{x-2j}\left(\frac{\mu(d-1)}{2}\right)^{j}}{(x-2j)!j!}
  55. ( x 1 , , x m ; μ , d ) = log ( ( x 1 , , x m ; μ , d ) ) = m μ ( - 1 + d - 1 2 ) + log ( μ ( 2 - d ) ) i = 1 m x i + i = 1 m log ( q i ( θ ) ) \begin{aligned}\displaystyle\mathcal{L}(x_{1},\ldots,x_{m};\mu,d)&% \displaystyle=\log(\mathcal{L}(x_{1},\ldots,x_{m};\mu,d))\\ &\displaystyle=m\mu\left(-1+\frac{d-1}{2}\right)+\log(\mu(2-d))\sum_{i=1}^{m}x% _{i}+\sum_{i=1}^{m}\log(q_{i}(\theta))\end{aligned}
  56. q i ( θ ) = j = 0 [ x i / 2 ] θ j ( x i - 2 j ) ! j ! q_{i}(\theta)=\sum_{j=0}^{[x_{i}/2]}\frac{\theta^{j}}{(x_{i}-2j)!j!}
  57. θ = d - 1 2 μ ( 2 - d ) 2 \theta=\frac{d-1}{2\mu(2-d)^{2}}
  58. l μ = m ( - 1 + d - 1 2 ) + 1 μ i = 1 m x i - d - 1 2 μ 2 ( 2 - d ) 2 i = 1 m q i ( θ ) q i ( θ ) \frac{\partial l}{\partial\mu}=m\left(-1+\frac{d-1}{2}\right)+\frac{1}{\mu}% \sum_{i=1}^{m}x_{i}-\frac{d-1}{2\mu^{2}(2-d)^{2}}\sum_{i=1}^{m}\frac{q_{i}^{{}% ^{\prime}}(\theta)}{q_{i}(\theta)}
  59. l d = m μ 2 - i = 1 m x i 2 - d - d 2 μ ( 2 - d ) 3 i = 1 m i = 1 m q i ( θ ) q i ( θ ) \frac{\partial l}{\partial d}=m\frac{\mu}{2}-\frac{\sum_{i=1}^{m}x_{i}}{2-d}-% \frac{d}{2\mu(2-d)^{3}}\sum_{i=1}^{m}\sum_{i=1}^{m}\frac{q_{i}^{{}^{\prime}}(% \theta)}{q_{i}(\theta)}
  60. μ = x ¯ \mu=\bar{x}
  61. i = 1 m q i ( θ ~ ) q i ( θ ~ ) = m ( x ¯ ( 2 - d ) ) 2 \sum_{i=1}^{m}\frac{q_{i}^{{}^{\prime}}(\tilde{\theta})}{q_{i}(\tilde{\theta})% }=m(\bar{x}(2-d))^{2}
  62. θ ~ = d - 1 2 x ¯ ( 2 - d ) 2 \tilde{\theta}=\frac{d-1}{2\bar{x}(2-d)^{2}}
  63. μ ^ \hat{\mu}
  64. d ~ \tilde{d}
  65. m ( 2 ) / x ¯ 2 > 1 m^{(2)}/\bar{x}^{2}>1
  66. m ( 2 ) = i = 1 n x i ( x i - 1 ) / n m^{(2)}=\sum_{i=1}^{n}x_{i}(x_{i}-1)/n
  67. m ( 2 ) / x ¯ 2 > 1 m^{(2)}/\bar{x}^{2}>1
  68. d ~ > 1 \tilde{d}>1
  69. d ~ = σ 2 / x ¯ \tilde{d}=\sigma^{2}/\bar{x}
  70. μ ^ = x ¯ \hat{\mu}=\bar{x}
  71. d ~ = 1 \tilde{d}=1
  72. f 0 = exp ( - ( a 1 + a 2 ) ) f_{0}=\exp(-(a_{1}+a_{2}))
  73. μ = a 1 + 2 a 2 \mu=a_{1}+2a_{2}
  74. { x ¯ = a 1 + 2 a 2 f 0 = exp ( - ( a 1 + a 2 ) ) \begin{cases}\bar{x}=a_{1}+2a_{2}\\ f_{0}=\exp(-(a_{1}+a_{2}))\end{cases}
  75. a 1 ^ \hat{a_{1}}
  76. a 2 ^ \hat{a_{2}}
  77. a 1 ^ = - ( x ¯ + 2 log ( f 0 ) ) \hat{a_{1}}=-(\bar{x}+2\log(f_{0}))
  78. a 2 ^ = x ¯ + log ( f 0 ) \hat{a_{2}}=\bar{x}+\log(f_{0})
  79. f 0 = n 0 n f_{0}=\frac{n_{0}}{n}
  80. a 1 ^ \hat{a_{1}}
  81. a 2 ^ \hat{a_{2}}
  82. - log ( n 0 n ) < x ¯ < - 2 log ( n 0 n ) -\log\left(\frac{n_{0}}{n}\right)<\bar{x}<-2\log\left(\frac{n_{0}}{n}\right)
  83. { H 0 : d = 1 H 1 : d > 1 \begin{cases}H_{0}:d=1\\ H_{1}:d>1\end{cases}
  84. W = 2 ( ( X ; μ ^ , d ^ ) - ( X ; μ ^ , 1 ) ) W=2(\mathcal{L}(X;\hat{\mu},\hat{d})-\mathcal{L}(X;\hat{\mu},1))
  85. ( ) \mathcal{L}()
  86. χ 1 2 \chi_{1}^{2}
  87. χ 1 2 \chi_{1}^{2}
  88. χ 1 2 \chi_{1}^{2}
  89. S 2 = 2 m [ m ( 2 ) - x ¯ 2 2 x ¯ ] 2 = m ( d ~ - 1 ) 2 2 S_{2}=2m\left[\frac{m^{(2)}-\bar{x}^{2}}{2\bar{x}}\right]^{2}=\frac{m(\tilde{d% }-1)^{2}}{2}
  90. χ 1 2 \chi_{1}^{2}
  91. sgn ( m ( 2 ) - x ¯ 2 ) S \operatorname{sgn}(m^{(2)}-\bar{x}^{2})\sqrt{S}

Hermite–Minkowski_theorem.html

  1. | d K | n n n ! ( π 4 ) n / 2 \sqrt{|d_{K}|}\geq\frac{n^{n}}{n!}\left(\frac{\pi}{4}\right)^{n/2}

Hertz–Knudsen_equation.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  2. 1 A d N d t ϕ = α p 2 π m k B T = α p N A 2 π M R T , \frac{1}{A}\frac{\mathrm{d}N}{\mathrm{d}t}\equiv\phi=\frac{\alpha p}{\sqrt{2% \pi mk_{B}T}}=\frac{\alpha pN_{A}}{\sqrt{2\pi MRT}},

Hexagonal_sampling.html

  1. M 2 M\geq 2
  2. w ( t ^ ) = w ( V . n ^ ) w(\hat{t})=w(V.\hat{n})
  3. t ^ \hat{t}
  4. n ^ \hat{n}
  5. X a ( Ω ^ ) = - + x a ( t ^ ) e - j Ω ^ T t d t ^ X_{a}(\hat{\Omega})=\int_{-\infty}^{+\infty}\!x_{a}({\hat{t}})e^{-j\hat{\Omega% }^{T}t}\hat{dt}
  6. x a ( t ^ ) = 1 2 π M - + X ( Ω ^ ) e ( j Ω ^ T t ) d Ω ^ x_{a}(\hat{t})=\frac{1}{2\pi^{M}}\int_{-\infty}^{+\infty}\!X(\hat{\Omega})e^{(% j\hat{\Omega}^{T}t)}\,\mathrm{d}\hat{\Omega}
  7. X ( ω ) = 1 | d e t ( V ) | k X a ( Ω ^ - U k ) X(\omega)=\frac{1}{|det(V)|}\sum_{k}\!X_{a}(\hat{\Omega}-Uk)
  8. S . D = 1 | d e t ( V ) | = | d e t ( U ) | 4 π 2 S.D=\frac{1}{|det(V)|}=\frac{|det(U)|}{4\pi^{2}}
  9. V r e c t = [ T 1 0 0 T 2 ] V_{rect}=\begin{bmatrix}T1&0\\ 0&T2\end{bmatrix}
  10. V h e x = [ T 1 T 1 T 2 - T 2 ] V_{hex}=\begin{bmatrix}T1&T1\\ T2&-T2\end{bmatrix}
  11. 2 R ( 3 ) \frac{2R}{\sqrt{(}3)}

Hexagonal_tiling_honeycomb.html

  1. V ¯ 3 {\bar{V}}_{3}
  2. Y ¯ 3 {\bar{Y}}_{3}
  3. Z ¯ 3 {\bar{Z}}_{3}
  4. V P ¯ 3 {\bar{VP}}_{3}
  5. P P ¯ 3 {\bar{PP}}_{3}
  6. V ¯ 3 {\bar{V}}_{3}
  7. V ¯ 3 {\bar{V}}_{3}
  8. V ¯ 3 {\bar{V}}_{3}
  9. P ¯ 3 {\bar{P}}_{3}
  10. V ¯ 3 {\bar{V}}_{3}
  11. V ¯ 3 {\bar{V}}_{3}
  12. V ¯ 3 {\bar{V}}_{3}
  13. V ¯ 3 {\bar{V}}_{3}
  14. V ¯ 3 {\bar{V}}_{3}
  15. V ¯ 3 {\bar{V}}_{3}

Hénon-Heiles_System.html

  1. V ( x , y ) = 1 2 ( x 2 + y 2 ) + λ ( x 2 y - y 3 3 ) V(x,y)=\frac{1}{2}(x^{2}+y^{2})+\lambda(x^{2}y-\frac{y^{3}}{3})
  2. H = 1 2 ( p x 2 + p y 2 ) + 1 2 ( x 2 + y 2 ) + λ ( x 2 y - y 3 3 ) H=\frac{1}{2}(p_{x}^{2}+p_{y}^{2})+\frac{1}{2}(x^{2}+y^{2})+\lambda(x^{2}y-% \frac{y^{3}}{3})
  3. x ˙ = p x \dot{x}=p_{x}
  4. p x ˙ = - x - 2 λ x y \dot{p_{x}}=-x-2\lambda xy
  5. y ˙ = p y \dot{y}=p_{y}
  6. p y ˙ = - y - λ ( x 2 - y 2 ) \dot{p_{y}}=-y-\lambda(x^{2}-y^{2})
  7. λ \lambda
  8. \R 2 \R^{2}
  9. i t Ψ ( x , y ) = [ - 2 2 m 2 + 1 2 ( x 2 + y 2 ) + λ ( x 2 y - 1 3 y 3 ) ] Ψ ( x , y ) i\hbar\frac{\partial}{\partial t}\Psi(x,y)=\left[\frac{-\hbar^{2}}{2m}\nabla^{% 2}+\frac{1}{2}(x^{2}+y^{2})+\lambda(x^{2}y-\frac{1}{3}y^{3})\right]\Psi(x,y)

Hidden-measurements_interpretation.html

  1. 2 2 2\sqrt{2}

Hierarchical_closeness.html

  1. G ( V , A ) G(V,A)
  2. V V
  3. A A
  4. i i
  5. V V
  6. C h c ( i ) C_{hc}(i)
  7. C h c ( i ) = N R ( i ) + C ( c l o - i ) ( i ) C_{hc}(i)=N_{R}(i)+C_{(clo-i)}(i)
  8. N R ( i ) [ 0 , | V | - 1 ] N_{R}(i)\in[0,|V|-1]
  9. i i
  10. N R ( i ) = | { j V : N_{R}(i)=|\{j\in V:\exists
  11. i i
  12. j } | j\}|
  13. C c l o ( i ) C_{clo}(i)
  14. C c l o - i ( i ) = 1 | V | - 1 j V { i } 1 d ( i , j ) C_{clo-i}(i)=\frac{1}{|V|-1}\sum_{j\in V\setminus\{i\}}\frac{1}{d(i,j)}
  15. d ( i , j ) d(i,j)
  16. i i
  17. j j
  18. d ( i , j ) d(i,j)
  19. N R ( i ) N_{R}(i)
  20. V V
  21. i i
  22. N R ( i ) = 0 N_{R}(i)=0
  23. C h c ( i ) = 0 C_{hc}(i)=0
  24. C ( c l o - i ) ( i ) C_{(clo-i)}(i)
  25. 0
  26. N R ( i ) > 0 N_{R}(i)>0
  27. N R ( i ) 1 N_{R}(i)\geq 1
  28. C ( c l o - i ) ( i ) < 1 C_{(clo-i)}(i)<1

High-frequency_vibrating_screens.html

  1. E o = Q m s ( o ) [ 1 - M u ( o ) ] Q m s ( f ) [ 1 - M u ( f ) ] E_{o}=\frac{Q_{ms}(o)\;[1-M_{u}(o)]}{Q_{ms}(f)\;[1-M_{u}(f)]}
  2. E u = Q m s ( u ) M u ( u ) Q m s ( f ) M u ( f ) E_{u}=\frac{Q_{ms}(u)\;M_{u}(u)}{Q_{ms}(f)\;M_{u}(f)}
  3. E = E o E u E=E_{o}\;E_{u}

High_entropy_alloys.html

  1. F = C - P + 2 F=C-P+2
  2. Δ G = Δ H - T Δ S {\Delta}G={\Delta}H-T{\Delta}S
  3. Δ S m i x = - R i = 1 N c i ln c i {\Delta}S_{mix}=-R\sum_{i=1}^{N}c_{i}\ln{c_{i}}
  4. Δ H m i x = i = 1 , i j N 4 Δ H A B m i x c i c j {\Delta}H_{mix}=\sum_{i=1,i{\neq}j}^{N}4{\Delta}H^{mix}_{AB}c_{i}c_{j}
  5. Δ H A B m i x {\Delta}H^{mix}_{AB}
  6. Ω = T m Δ S m i x | Δ H m i x | \Omega=\frac{T_{m}{\Delta}S_{mix}}{\left|{\Delta}H_{mix}\right|}
  7. δ = i = 1 N c i ( 1 - r i r ¯ ) \delta=\sqrt{\sum_{i=1}^{N}c_{i}\left(1-\frac{r_{i}}{\bar{r}}\right)}
  8. r ¯ = i = 1 N c i r i \bar{r}=\sum_{i=1}^{N}c_{i}r_{i}

Hilbert–Kunz_function.html

  1. f ( q ) = length R ( R / m [ q ] ) f(q)=\operatorname{length}_{R}(R/m^{[q]})

Hildreth–Lu_estimation.html

  1. y t - ρ y t - 1 = α ( 1 - ρ ) + β ( X t - ρ X t - 1 ) + e t . y_{t}-\rho y_{t-1}=\alpha(1-\rho)+\beta(X_{t}-\rho X_{t-1})+e_{t}.\,
  2. ρ \rho

Himmelpforten_Convent.html

  1. 72 6000 \tfrac{72}{6000}

Hirota-Satsuma_equation.html

  1. u t - 0.5 * u x x x + 3 u u x - 3 ( v w ) x = 0 u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0
  2. v t + v x x x - 3 u v x = 0 v_{t}+v_{xxx}-3uv_{x}=0
  3. w t + w x x x - 3 u w x = 0 w_{t}+w_{xxx}-3uw_{x}=0

History_of_aerodynamics.html

  1. v 1 2 2 + p 1 ρ = v 2 2 2 + p 2 ρ {v_{1}^{2}\over 2}+{p_{1}\over\rho}={v_{2}^{2}\over 2}+{p_{2}\over\rho}
  2. F = ρ S V 2 sin 2 ( θ ) F=\rho SV^{2}\sin^{2}(\theta)

Hjorth_parameters.html

  1. A c t i v i t y = v a r ( y ( t ) ) . Activity=var(y(t)).
  2. M o b i l i t y = v a r ( y ( t ) d y d t ) v a r ( y ( t ) ) . Mobility=\sqrt{\frac{{var(y(t)\frac{dy}{dt})}}{var(y(t))}}.
  3. C o m p l e x i t y = M o b i l i t y ( y ( t ) d y d t ) M o b i l i t y ( y ( t ) ) . Complexity=\frac{{Mobility(y(t)\frac{dy}{dt})}}{Mobility(y(t))}.

HKA_test.html

  1. X 2 = i = 1 L ( S i A - E ^ ( S i A ) ) 2 V ^ a r ( S i A ) X^{2}=\frac{\sum_{i=1}^{L}(S_{i}^{A}-\widehat{E}(S_{i}^{A}))^{2}}{\widehat{V}% ar(S_{i}^{A})}

Hodge–de_Rham_spectral_sequence.html

  1. H p ( X , Ω q ) H p + q ( X , 𝐂 ) H^{p}(X,\Omega^{q})\Rightarrow H^{p+q}(X,\mathbf{C})
  2. H p + q ( X , 𝐂 ) H^{p+q}(X,\mathbf{C})
  3. E 2 E_{2}
  4. 𝐂 Ω * := [ Ω 0 d Ω 1 d Ω dim X ] , \mathbf{C}\rightarrow\Omega^{*}:=[\Omega^{0}\stackrel{d}{\to}\Omega^{1}% \stackrel{d}{\to}\cdots\to\Omega^{\dim X}],
  5. F p Ω * := [ 0 Ω p Ω p + 1 ] F^{p}\Omega^{*}:=[\cdots\to 0\to\Omega^{p}\to\Omega^{p+1}\to\cdots]
  6. Ω * \Omega^{*}
  7. E 2 E_{2}
  8. p + q = n H p ( X , Ω q ) = H n ( X , 𝐂 ) . \bigoplus_{p+q=n}H^{p}(X,\Omega^{q})=H^{n}(X,\mathbf{C}).

Hoffman_nucleation_theory.html

  1. T m = T c β + ( 1 - 1 β ) T m T\text{m}={T\text{c}\over\beta}+(1-{1\over\beta})T\text{m}^{\circ}
  2. G I = b i L p G\text{I}=biL\text{p}
  3. G I,n = e - ( K g / T Δ T ) G\text{I,n}=e^{-(K\text{g}/T\Delta T)}\,
  4. K g = 4 b σ l σ f T m 0 k Δ h K\text{g}={4b\sigma\text{l}\sigma\text{f}T_{m}^{0}\over k\Delta h}
  5. G II = b i g G\text{II}=b\sqrt{ig}
  6. G II,n = e - ( K g / T Δ T ) G\text{II,n}=e^{-(K\text{g}^{\prime}/T\Delta T)}\,
  7. K g = 2 b σ l σ f T m 0 k Δ h K\text{g}^{\prime}={2b\sigma\text{l}\sigma\text{f}T_{m}^{0}\over k\Delta h}
  8. G III = b i L p = G III e - U * / k ( T - T 0 ) - ( K g / T Δ T ) G\text{III}=biL\text{p}=G\text{III}^{\circ}e^{{-U^{*}/k(T-T\text{0})}-(K\text{% g}/T\Delta T)}

Homography_(computer_vision).html

  1. P i P_{i}
  2. P i P_{i}
  3. p i b {}^{b}p_{i}
  4. p i a {}^{a}p_{i}
  5. p i a = K a H b a K b - 1 p i b {}^{a}p_{i}=K_{a}\cdot H_{ba}\cdot K_{b}^{-1}\cdot{}^{b}p_{i}
  6. H b a H_{ba}
  7. H b a = R - t n T d . H_{ba}=R-\frac{tn^{T}}{d}.
  8. R R
  9. n T P i + d = 0 n^{T}P_{i}+d=0
  10. n T P i n^{T}P_{i}
  11. P i P_{i}
  12. n T n^{T}
  13. - d -d
  14. t = t ( - n T P i d ) t=t\left(-\frac{n^{T}P_{i}}{d}\right)
  15. H b a P i = R P i + t H_{ba}P_{i}=RP_{i}+t
  16. H b a = R - t n T d H_{ba}=R-\frac{tn^{T}}{d}
  17. R a , R b R_{a},R_{b}
  18. t a , t b t_{a},t_{b}
  19. R = R a R b T R=R_{a}R_{b}^{T}
  20. H b a H_{ba}
  21. H b a = R a R b T - R a ( t b - t a ) n T d R b T = R a ( I - ( t b - t a ) n T d ) R b T . H_{ba}=R_{a}R_{b}^{T}-R_{a}\frac{(t_{b}-t_{a})n^{T}}{d}R_{b}^{T}=R_{a}\left(I-% \frac{(t_{b}-t_{a})n^{T}}{d}\right)R_{b}^{T}.
  22. p a = [ x a y a 1 ] , p b = [ w x b w y b w ] , 𝐇 a b = [ h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 ] p_{a}=\begin{bmatrix}x_{a}\\ y_{a}\\ 1\end{bmatrix},p^{\prime}_{b}=\begin{bmatrix}w^{\prime}x_{b}\\ w^{\prime}y_{b}\\ w^{\prime}\end{bmatrix},\mathbf{H}_{ab}=\begin{bmatrix}h_{11}&h_{12}&h_{13}\\ h_{21}&h_{22}&h_{23}\\ h_{31}&h_{32}&h_{33}\end{bmatrix}
  23. p b = 𝐇 a b p a p^{\prime}_{b}=\mathbf{H}_{ab}p_{a}\,
  24. 𝐇 b a = 𝐇 a b - 1 . \mathbf{H}_{ba}=\mathbf{H}_{ab}^{-1}.
  25. p b = p b / w = [ x b y b 1 ] p_{b}=p^{\prime}_{b}/w^{\prime}=\begin{bmatrix}x_{b}\\ y_{b}\\ 1\end{bmatrix}
  26. h 31 = h 32 = 0 , h 33 = 1. h_{31}=h_{32}=0,\;h_{33}=1.

Homotopy_colimit.html

  1. D n S n - 1 p t D^{n}\sqcup_{S^{n-1}}pt
  2. p t S n - 1 p t pt\sqcup_{S^{n-1}}pt
  3. A B C A\leftarrow B\rightarrow C
  4. A B × [ 0 , 1 ] B B × [ 0 , 1 ] C A\sqcup B\times[0,1]\sqcup B\sqcup B\times[0,1]\sqcup_{0}C
  5. X 0 X 0 × X 1 X 1 X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}
  6. X 0 * X 1 X_{0}*X_{1}
  7. X 1 X 2 , X_{1}\to X_{2}\to\cdots,
  8. X : I S p a c e s , X:I\to Spaces,
  9. Δ : S p a c e s S p a c e s I \Delta:Spaces\to Spaces^{I}
  10. Δ : S p a c e s S p a c e s I \Delta:Spaces\to Spaces^{I}
  11. X × | N ( I / i ) | X\times|N(I/i)|
  12. I / i I/i
  13. hocolim X i colim X i . \mathrm{hocolim}X_{i}\to\mathrm{colim}X_{i}.
  14. X 0 X 0 × X 1 X 1 X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}

Homotopy_excision_theorem.html

  1. ( X ; A , B ) (X;A,B)
  2. C = A B C=A\cap B
  3. ( A , C ) (A,C)
  4. m - 1 m-1
  5. m 2 m\geq 2
  6. ( B , C ) (B,C)
  7. n - 1 n-1
  8. n 1 n\geq 1
  9. i : ( A , C ) ( X , B ) i:(A,C)\to(X,B)
  10. i * : π q ( A , C ) π q ( X , B ) i_{*}:\pi_{q}(A,C)\to\pi_{q}(X,B)
  11. q < m + n - 2 q<m+n-2
  12. q = m + n - 2 q=m+n-2

Homotopy_group_with_coefficients.html

  1. i 2 i\geq 2
  2. ( G , i ) (G,i)
  3. π i ( X ; G ) \pi_{i}(X;G)
  4. i 3 i\geq 3
  5. π i ( X ; G ) \pi_{i}(X;G)
  6. π i ( X ; ) \pi_{i}(X;\mathbb{Z})

Hook_length_formula.html

  1. λ = ( λ 1 , , λ m ) \lambda=(\lambda_{1},\ldots,\lambda_{m})
  2. n n
  3. λ \lambda
  4. m m
  5. λ i \lambda_{i}
  6. i i
  7. 1 i m 1\leq i\leq m
  8. λ \lambda
  9. λ \lambda
  10. n n
  11. n n
  12. ( i , j ) (i,j)
  13. i i
  14. j j
  15. H λ ( i , j ) H_{\lambda}(i,j)
  16. ( a , b ) (a,b)
  17. a = i a=i
  18. b j b\geq j
  19. a i a\geq i
  20. b = j b=j
  21. h λ ( i , j ) h_{\lambda}(i,j)
  22. H λ ( i , j ) H_{\lambda}(i,j)
  23. λ \lambda
  24. d λ d_{\lambda}
  25. d λ = n ! h λ ( i , j ) , d_{\lambda}=\frac{n!}{\prod h_{\lambda}(i,j)},
  26. ( i , j ) (i,j)
  27. λ \lambda
  28. λ λ
  29. d λ d_{\lambda}
  30. d λ = 9 ! 7 5 4 3 2 2 1 1 1 = 216. d_{\lambda}=\frac{9!}{7\cdot 5\cdot 4\cdot 3\cdot 2\cdot 2\cdot 1\cdot 1\cdot 1% }=216.
  31. d λ d_{\lambda}
  32. d λ d_{\lambda}
  33. ( i , j ) (i,j)
  34. i i
  35. j j
  36. e λ = n ! ( i , j ) Y ( λ ) h λ ( i , j ) . e_{\lambda}=\frac{n!}{\prod_{(i,j)\in Y(\lambda)}h_{\lambda}(i,j)}.
  37. d λ = e λ d_{\lambda}=e_{\lambda}
  38. d λ d_{\lambda}
  39. d λ = μ λ d μ , d_{\lambda}=\sum_{\mu\uparrow\lambda}d_{\mu},
  40. μ λ \mu\uparrow\lambda
  41. μ \mu
  42. λ \lambda
  43. λ \lambda
  44. μ \mu
  45. d ϕ = 1 d_{\phi}=1
  46. ϕ \phi
  47. λ \lambda
  48. μ \mu
  49. μ \mu
  50. d μ d_{\mu}
  51. μ \mu
  52. e ϕ = 1 e_{\phi}=1
  53. e λ = μ λ e μ , e_{\lambda}=\sum_{\mu\uparrow\lambda}e_{\mu},
  54. d λ = e λ d_{\lambda}=e_{\lambda}
  55. μ λ e μ e λ = 1. \sum_{\mu\uparrow\lambda}\frac{e_{\mu}}{e_{\lambda}}=1.
  56. e μ e λ \frac{e_{\mu}}{e_{\lambda}}
  57. μ \mu
  58. μ λ \mu\uparrow\lambda
  59. λ \lambda
  60. λ \lambda
  61. μ \mu
  62. μ λ \mu\uparrow\lambda
  63. | λ | |\lambda|
  64. ( i , j ) (i,j)
  65. H λ ( i , j ) { ( i , j ) } H_{\lambda}(i,j)\setminus\{(i,j)\}
  66. 𝐜 \,\textbf{c}
  67. ( a , b ) (a,b)
  68. λ \lambda
  69. ( 𝐜 = ( a , b ) ) = e μ e λ , \mathbb{P}\left(\,\textbf{c}=(a,b)\right)=\frac{e_{\mu}}{e_{\lambda}},
  70. μ = λ { ( a , b ) } \mu=\lambda\setminus\{(a,b)\}
  71. 𝐜 = ( a , b ) \,\textbf{c}=(a,b)
  72. μ λ e μ e λ = 1 \sum_{\mu\uparrow\lambda}\frac{e_{\mu}}{e_{\lambda}}=1
  73. S n S_{n}
  74. d λ d_{\lambda}
  75. V λ V_{\lambda}
  76. λ \lambda
  77. dim V λ \dim V_{\lambda}
  78. dim λ \dim\lambda
  79. f λ f^{\lambda}
  80. V λ V_{\lambda}
  81. λ \lambda
  82. n n
  83. χ λ ( w ) = s λ , p τ ( w ) \chi^{\lambda}(w)=\langle s_{\lambda},p_{\tau(w)}\rangle
  84. s λ s_{\lambda}
  85. λ \lambda
  86. p τ ( w ) p_{\tau(w)}
  87. τ ( w ) \tau(w)
  88. w w
  89. w = ( 154 ) ( 238 ) ( 6 ) ( 79 ) w=(154)(238)(6)(79)
  90. τ ( w ) = ( 3 , 3 , 2 , 1 ) \tau(w)=(3,3,2,1)
  91. e = ( 1 ) ( 2 ) ( n ) e=(1)(2)\cdots(n)
  92. τ ( e ) = 1 + 1 + + 1 = 1 ( n ) \tau(e)=1+1+\cdots+1=1^{(n)}
  93. dim V λ = χ λ ( e ) = s λ , p 1 ( n ) \dim V_{\lambda}=\chi^{\lambda}(e)=\langle s_{\lambda},p_{1^{(n)}}\rangle
  94. s λ = μ K λ μ m μ . s_{\lambda}=\sum_{\mu}K_{\lambda\mu}m_{\mu}.
  95. p 1 ( n ) = h 1 ( n ) p_{1^{(n)}}=h_{1^{(n)}}
  96. K λ 1 ( n ) K_{\lambda 1^{(n)}}
  97. m μ , h ν = δ μ ν \langle m_{\mu},h_{\nu}\rangle=\delta_{\mu\nu}
  98. K λ 1 ( n ) K_{\lambda 1^{(n)}}
  99. d λ d_{\lambda}
  100. dim V λ = d λ . \dim V_{\lambda}=d_{\lambda}.
  101. λ n ( f λ ) 2 = n ! \sum_{\lambda\vdash n}\left(f^{\lambda}\right)^{2}=n!
  102. ( x 1 + x 2 + + x k ) n = λ n s λ f λ . (x_{1}+x_{2}+\cdots+x_{k})^{n}=\sum_{\lambda\vdash n}s_{\lambda}f^{\lambda}.
  103. p 1 ( n ) p_{1^{(n)}}
  104. s μ , s ν = δ μ ν \langle s_{\mu},s_{\nu}\rangle=\delta_{\mu\nu}
  105. V n V^{\otimes n}
  106. G L ( V ) GL(V)
  107. p 1 ( n ) = λ n s λ f λ p_{1^{(n)}}=\sum_{\lambda\vdash n}s_{\lambda}f^{\lambda}
  108. Δ ( x ) p 1 ( n ) = λ n Δ ( x ) s λ f λ \Delta(x)p_{1^{(n)}}=\sum_{\lambda\vdash n}\Delta(x)s_{\lambda}f^{\lambda}
  109. Δ ( x ) = i < j ( x i - x j ) \Delta(x)=\prod_{i<j}(x_{i}-x_{j})
  110. λ = ( λ 1 , λ 2 , , λ k ) \lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{k})
  111. l i = λ i + k - 1 l_{i}=\lambda_{i}+k-1
  112. i = 1 , 2 , , k i=1,2,\cdots,k
  113. n n
  114. x 1 , , x n x_{1},\cdots,x_{n}
  115. Δ ( x ) s λ \Delta(x)s_{\lambda}
  116. a ( λ 1 + n - 1 , λ 2 + n - 2 , , λ k ) ( x 1 , x 2 , , x k ) = det [ x 1 l 1 x 2 l 1 x k l 1 x 1 l 2 x 2 l 2 x k l 2 x 1 l k x 2 l k x k l k ] a_{(\lambda_{1}+n-1,\lambda_{2}+n-2,\dots,\lambda_{k})}(x_{1},x_{2},\dots,x_{k% })=\det\left[\begin{matrix}x_{1}^{l_{1}}&x_{2}^{l_{1}}&\dots&x_{k}^{l_{1}}\\ x_{1}^{l_{2}}&x_{2}^{l_{2}}&\dots&x_{k}^{l_{2}}\\ \vdots&\vdots&\ddots&\vdots\\ x_{1}^{l_{k}}&x_{2}^{l_{k}}&\dots&x_{k}^{l_{k}}\end{matrix}\right]
  117. ( l 1 , l 2 , , l k ) (l_{1},l_{2},\cdots,l_{k})
  118. x 1 l 1 x k l k x_{1}^{l_{1}}\cdots x_{k}^{l_{k}}
  119. Δ ( x ) p 1 ( n ) \Delta(x)p_{1^{(n)}}
  120. [ Δ ( x ) p 1 ( n ) ] l 1 , , l k \left[\Delta(x)p_{1^{(n)}}\right]_{l_{1},\cdots,l_{k}}
  121. f λ f^{\lambda}
  122. Δ ( x ) = w S n sgn ( w ) x 1 w ( 1 ) - 1 x 2 w ( 2 ) - 1 x k w ( k ) - 1 \Delta(x)=\sum_{w\in S_{n}}\operatorname{sgn}(w)x_{1}^{w(1)-1}x_{2}^{w(2)-1}% \cdots x_{k}^{w(k)-1}
  123. ( x 1 + x 2 + + x k ) n = n ! d 1 ! d 2 ! d k ! x 1 d 1 x 2 d 2 x k d k (x_{1}+x_{2}+\cdots+x_{k})^{n}=\sum\frac{n!}{d_{1}!d_{2}!\cdots d_{k}!}x_{1}^{% d_{1}}x_{2}^{d_{2}}\cdots x_{k}^{d_{k}}
  124. w S n sgn ( w ) n ! ( l 1 - w ( 1 ) + 1 ) ! ( l 2 - w ( 2 ) + 1 ) ! ( l k - w ( k ) + 1 ) ! \sum_{w\in S_{n}}\operatorname{sgn}(w)\frac{n!}{(l_{1}-w(1)+1)!(l_{2}-w(2)+1)!% \cdots(l_{k}-w(k)+1)!}
  125. n ! l 1 ! l 2 ! l k ! w S n sgn ( w ) [ ( l 1 ) ( l 1 - 1 ) ( l 1 - w ( 1 ) + 2 ) ] [ ( l 2 ) ( l 2 - 1 ) ( l 2 - w ( 2 ) + 2 ) ] [ ( l k ) ( l k - 1 ) ( l k - w ( k ) + 2 ) ] \frac{n!}{l_{1}!l_{2}!\cdots l_{k}!}\sum_{w\in S_{n}}\operatorname{sgn}(w)% \left[(l_{1})(l_{1}-1)\cdots(l_{1}-w(1)+2)\right]\left[(l_{2})(l_{2}-1)\cdots(% l_{2}-w(2)+2)\right]\left[(l_{k})(l_{k}-1)\cdots(l_{k}-w(k)+2)\right]
  126. det [ 1 l 1 l 1 ( l 1 - 1 ) i = 0 k - 2 l 1 - i 1 l 2 l 1 ( l 2 - 1 ) i = 0 k - 2 l 2 - i 1 l k l k ( l k - 1 ) i = 0 k - 2 l k - i ] \det\left[\begin{matrix}1&l_{1}&l_{1}(l_{1}-1)&\dots&\prod_{i=0}^{k-2}l_{1}-i% \\ 1&l_{2}&l_{1}(l_{2}-1)&\dots&\prod_{i=0}^{k-2}l_{2}-i\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&l_{k}&l_{k}(l_{k}-1)&\dots&\prod_{i=0}^{k-2}l_{k}-i\end{matrix}\right]
  127. n ! l 1 ! l 2 ! l k ! i < j ( l i - l j ) \frac{n!}{l_{1}!l_{2}!\cdots l_{k}!}\prod_{i<j}(l_{i}-l_{j})
  128. l i l_{i}
  129. σ n \sigma_{n}
  130. n n
  131. L ( σ n ) L(\sigma_{n})
  132. σ n \sigma_{n}
  133. n \ell_{n}
  134. L ( σ n ) L(\sigma_{n})
  135. n n
  136. n \ell_{n}
  137. 2 n 2\sqrt{n}
  138. d λ d_{\lambda}
  139. d λ d_{\lambda}
  140. n 1 , , n m n_{1},\dots,n_{m}
  141. f ( n 1 , n 2 , , n m ) = n ! Δ ( n m , n m - 1 + 1 , , n 1 + m - 1 ) n m ! ( n m - 1 + 1 ) ! ( n 1 + m - 1 ) ! f(n_{1},n_{2},\ldots,n_{m})=\frac{n!\,\Delta(n_{m},n_{m-1}+1,\ldots,n_{1}+m-1)% }{n_{m}!(n_{m-1}+1)!\cdots(n_{1}+m-1)!}
  142. λ λ
  143. p p
  144. n n
  145. λ λ
  146. n n
  147. h 1 , , h p h_{1},\dots,h_{p}
  148. n n
  149. λ λ
  150. n = 0 π n x n = k = 1 p ( 1 - x h k ) - 1 \sum_{n=0}^{\infty}\pi_{n}x^{n}=\prod_{k=1}^{p}(1-x^{h_{k}})^{-1}
  151. A A
  152. n n
  153. A A
  154. P ( x ) ( 1 - x ) ( 1 - x 2 ) ( 1 - x n ) \frac{P(x)}{(1-x)(1-x^{2})\cdots(1-x^{n})}
  155. P ( x ) P(x)
  156. P ( 1 ) P(1)
  157. A A
  158. λ \lambda
  159. ( i , j ) ( i , j ) i i and j j (i,j)\leq(i^{\prime},j^{\prime})\iff i\leq i^{\prime}\qquad\textrm{and}\qquad j% \leq j^{\prime}
  160. P ( 1 ) = f λ P(1)=f^{\lambda}
  161. P ( x ) ( 1 - x ) ( 1 - x 2 ) ( 1 - x ) n = ( i , j ) λ ( 1 - x h ( i , j ) ) - 1 \frac{P(x)}{(1-x)(1-x^{2})\cdots(1-x)^{n}}=\prod_{(i,j)\in\lambda}(1-x^{h_{(i,% j)}})^{-1}
  162. | x | < 1 |x|<1
  163. P ( x ) = k = 1 n ( 1 - x k ) ( i , j ) λ ( 1 - x h ( i , j ) ) . P(x)=\frac{\prod_{k=1}^{n}(1-x^{k})}{\prod_{(i,j)\in\lambda}(1-x^{h_{(i,j)}})}.
  164. P ( 1 ) = lim x 1 k = 1 n ( 1 - x k ) ( i , j ) λ ( 1 - x h ( i , j ) ) . P(1)=\lim_{x\to 1}\frac{\prod_{k=1}^{n}(1-x^{k})}{\prod_{(i,j)\in\lambda}(1-x^% {h_{(i,j)}})}.
  165. n n
  166. P ( 1 ) = n ! ( i , j ) λ h ( i , j ) . P(1)=\frac{n!}{\prod_{(i,j)\in\lambda}h_{(i,j)}}.
  167. 1 , t , t 2 , t 3 , 1,t,t^{2},t^{3},\cdots
  168. s λ ( 1 , t , t 2 , ) = t n ( λ ) ( i , j ) Y ( λ ) ( 1 - t h λ ( i , j ) ) s_{\lambda}(1,t,t^{2},\cdots)=\frac{t^{n\left(\lambda\right)}}{\prod_{(i,j)\in Y% (\lambda)}(1-t^{h_{\lambda}(i,j)})}
  169. n ( λ ) n(\lambda)
  170. n ( λ ) = i ( i - 1 ) λ i = i ( λ i 2 ) n(\lambda)=\sum_{i}(i-1)\lambda_{i}=\sum_{i}{\left({{\lambda_{i}^{\prime}}% \atop{2}}\right)}
  171. λ \lambda^{\prime}
  172. C n = f ( n , n ) C_{n}=f^{(n,n)}
  173. n n
  174. ( n , n ) (n,n)
  175. i i
  176. i i
  177. C n = ( 2 n ) ! ( n + 1 ) ( n ) ( 3 ) ( 2 ) ( n ) ( n - 1 ) ( 2 ) ( 1 ) = ( 2 n ) ! ( n + 1 ) ! n ! = 1 n + 1 ( 2 n n ) C_{n}=\frac{(2n)!}{(n+1)(n)\cdots(3)(2)(n)(n-1)\cdots(2)(1)}=\frac{(2n)!}{(n+1% )!n!}=\frac{1}{n+1}{\left({{2n}\atop{n}}\right)}

Hoon_Balakram.html

  1. ( 2 n ) ! ( n + 1 ) ! ( n + 1 ) ! {(2n)!\over(n+1)!(n+1)!}
  2. n . n.
  3. n 100 n\leq 100

Hopfield_dielectric.html

  1. N N
  2. H = A = 1 N p A 2 2 m + m ω 2 2 x A 2 - e x A E ( r A ) + λ = 1 2 d 3 k a λ k + a λ k c k H=\sum\limits_{A=1}^{N}{{p_{A}}^{2}\over 2m}+{{m{\omega}^{2}}\over 2}{x_{A}}^{% 2}-e{x_{A}}\cdot E(r_{A})+\sum\limits_{\lambda=1}^{2}\int d^{3}ka_{\lambda k}^% {+}a_{\lambda k}\hbar ck
  3. E ( r A ) = i L 3 λ = 1 2 d 3 k [ c k 2 ϵ 0 ] 1 2 [ e λ ( k ) a λ ( k ) exp ( i k r A ) - H . C . ] E(r_{A})={i\over L^{3}}\sum\limits_{\lambda=1}^{2}\int d^{3}k[{{ck}\over{2% \epsilon_{0}}}]^{1\over 2}[e_{\lambda}(k)a_{\lambda}(k)\exp(ikr_{A})-H.C.]
  4. r A r_{A}
  5. H = A = 1 N ( a A + a A ) ω - e 2 β ( a A + a A + ) E ( r A ) + λ k a λ k + a λ k c k H=\sum\limits_{A=1}^{N}(a_{A}^{+}\cdot a_{A})\hbar\omega-{e\over{{\sqrt{2}}% \beta}}(a_{A}+{a_{A}}^{+})\cdot E(r_{A})+\sum_{\lambda}\sum_{k}a_{\lambda k}^{% +}a_{\lambda k}\hbar ck
  6. B k + = 1 N A = 1 N exp ( i k r A ) a A + , B_{k}^{+}={1\over{\sqrt{N}}}\sum\limits_{A=1}^{N}\exp(ikr_{A})a_{A}^{+},
  7. B k = 1 N A = 1 N exp ( - i k r A ) a A B_{k}={1\over{\sqrt{N}}}\sum\limits_{A=1}^{N}\exp(-ikr_{A})a_{A}
  8. B λ k + = e λ ( k ) B k + B_{\lambda k}^{+}=e_{\lambda}(k)\cdot B_{k}^{+}
  9. B λ k = e λ ( k ) B k , B_{\lambda k}=e_{\lambda}(k)\cdot B_{k},
  10. H = λ k ( B λ k + B λ k + 1 2 ) ω + c k a λ k + a λ k + i e ϵ 0 m ω N V c k [ B λ k a λ - k + B λ k + a λ k - B λ k + a λ - k + - B λ k a λ k + ] H=\sum_{\lambda}\sum_{k}(B_{\lambda k}^{+}B_{\lambda k}+{1\over 2})\hbar\omega% +\hbar cka_{\lambda k}^{+}a_{\lambda k}+{ie\hbar\over{\sqrt{\epsilon_{0}m% \omega}}}\sqrt{N\over V}{\sqrt{ck}}[B_{\lambda k}a_{\lambda-k}+B_{\lambda k}^{% +}a_{\lambda k}-B_{\lambda k}^{+}a_{\lambda-k}^{+}-B_{\lambda k}a_{\lambda k}^% {+}]
  11. H = λ k [ Ω + ( k ) C λ + k + C λ + k + Ω - ( k ) C λ - k + C λ - k ] + c o n s t H=\sum_{\lambda}\sum_{k}\left[\Omega_{+}(k)C_{\lambda+k}^{+}C_{\lambda+k}+% \Omega_{-}(k)C_{\lambda-k}^{+}C_{\lambda-k}\right]+const
  12. [ C λ ± k , H ] = Ω ± ( k ) C λ ± k [C_{\lambda\pm k},H]=\Omega_{\pm}(k)C_{\lambda\pm k}
  13. C λ ± k = c 1 a λ k + c 2 a λ - k + c 3 a λ k + + c 4 a λ - k + + c 5 B λ k + c 6 B λ - k + c 7 B λ k + + c 8 B λ - k + C_{\lambda\pm k}=c_{1}a_{\lambda k}+c_{2}a_{\lambda-k}+c_{3}a_{\lambda k}^{+}+% c_{4}a_{\lambda-k}^{+}+c_{5}B_{\lambda k}+c_{6}B_{\lambda-k}+c_{7}B_{\lambda k% }^{+}+c_{8}B_{\lambda-k}^{+}
  14. Ω - ( k ) 2 = ω 2 + Ω 2 - ( ω 2 - Ω 2 ) 2 + 4 g ω 2 Ω 2 2 , \Omega_{-}(k)^{2}={{\omega^{2}+\Omega^{2}-\sqrt{{(\omega^{2}-\Omega^{2})}^{2}+% 4{g}\omega^{2}\Omega^{2}}\over 2}},
  15. Ω + ( k ) 2 = ω 2 + Ω 2 + ( ω 2 - Ω 2 ) 2 + 4 g ω 2 Ω 2 2 \Omega_{+}(k)^{2}={{\omega^{2}+\Omega^{2}+\sqrt{{(\omega^{2}-\Omega^{2})}^{2}+% 4{g}\omega^{2}\Omega^{2}}\over 2}}
  16. Ω ( k ) = c k , \Omega(k)=ck,
  17. g = N e 2 V m ϵ 0 ω 2 g={{Ne^{2}}\over{Vm\epsilon_{0}\omega^{2}}}
  18. N / V N/V
  19. ω \omega
  20. < a λ k + a λ k > <a_{\lambda k}^{+}a_{\lambda k}>
  21. C k ± | 𝟎 0 C_{k\pm}|\mathbf{0}>=0
  22. Ω - \Omega_{-}
  23. g > 1 g>1

Hopper_cooling.html

  1. ( 2260 J g - 1 / 4.1813 J g - 1 K - 1 ) ( 100 °C - 20 °C ) 7 \frac{(2260\,\text{J}\,\text{g}^{-1}/4.1813\,\text{J}\,\text{g}^{-1}\,\text{K}% ^{-1})}{(100\,\text{°C}-20\,\text{°C})}\approx 7

Hölder_summation.html

  1. a 1 + a 2 + , a_{1}+a_{2}+\cdots,\,
  2. H n 0 = a 1 + a 2 + + a n H^{0}_{n}=a_{1}+a_{2}+\cdots+a_{n}\,
  3. H n k + 1 = H 1 k + + H n k n H^{k+1}_{n}=\frac{H^{k}_{1}+\cdots+H^{k}_{n}}{n}
  4. lim n H n k \lim_{n\rightarrow\infty}H^{k}_{n}\,

Hsu_diffusion.html

  1. D = < Δ v Δ τ Δ a Δ τ > ( c δ E B ) ( k 2 D ) - 1 ( k 2 m v t h 2 c 2 δ E q B 2 ) ( k v t h ) - 1 . D=<\Delta v\Delta\tau\Delta a\Delta\tau>\sim\left(\frac{c\,\delta E_{\perp}}{B% }\right)\left(k_{\perp}^{2}\,D\right)^{-1}\left(\frac{k_{\perp}^{2}mv_{th}^{2}% c^{2}\,\delta E_{\perp}}{qB^{2}}\right)\left(k_{\parallel}\,v_{th}\right)^{-1}.

Hsu–Robbins–Erdős_theorem.html

  1. X 1 , , X n X_{1},\ldots,X_{n}
  2. S n = X 1 + + X n , S_{n}=X_{1}+\cdots+X_{n},\,
  3. n 1 P ( | S n | > ε n ) < \sum\limits_{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty
  4. ε > 0 \varepsilon>0
  5. X X
  6. n 1 P ( | S n | > ε n ) < \sum\limits_{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty

Hub_(network_science_concept).html

  1. m 0 m_{0}
  2. m m 0 m\leq m_{0}
  3. p i p_{i}
  4. i i
  5. p i = k i j k j , p_{i}=\frac{k_{i}}{\sum_{j}k_{j}},
  6. k i k_{i}
  7. i i
  8. j j
  9. ln N ln ln N . \ell\sim\frac{\ln N}{\ln\ln N}.

Huggins_equation.html

  1. [ n i c ] = [ n ] + k H [ n ] 2 c \left[\frac{n_{i}}{c}\right]\quad=[n]+k_{H}[n]^{2}c
  2. n i c \frac{n_{i}}{c}
  3. n n
  4. k H k_{H}
  5. c c
  6. n i n_{i}
  7. 0.3 0.3
  8. 0.5 0.5

HWB_color_model.html

  1. H = H W = ( 1 - S ) V B = 1 - V \begin{aligned}\displaystyle H&\displaystyle=H\\ \displaystyle W&\displaystyle=(1-S)V\\ \displaystyle B&\displaystyle=1-V\end{aligned}
  2. H = H S = 1 - W 1 - B V = 1 - B \begin{aligned}\displaystyle H&\displaystyle=H\\ \displaystyle S&\displaystyle=1-\frac{W}{1-B}\\ \displaystyle V&\displaystyle=1-B\end{aligned}

HY-80.html

  1. E E
  2. ν \nu
  3. G = E / 2 ( 1 + ν ) G=E/2(1+\nu)
  4. K = E / 3 ( 1 - 2 ν ) K=E/3(1-2\nu)
  5. ρ \rho
  6. k k
  7. c p c_{p}
  8. k / ρ c p k/\rho c_{p}
  9. α \alpha
  10. T m e l t T_{melt}

Hydantoin_racemase.html

  1. \rightleftharpoons

Hydrogel_encapsulation_of_quantum_dots.html

  1. D = ( 1.6122 × 10 - 9 ) λ 4 - ( 2.66575 × 10 - 6 ) λ 3 + ( 1.6242 × 10 - 3 ) λ 2 - ( 0.4277 ) λ + 41.57 {D=\ }{(1.6122\times 10^{-9})\lambda^{4}}-{(2.66575\times 10^{-6})\lambda^{3}}% +{(1.6242\times 10^{-3})\lambda^{2}}-{(0.4277)\lambda}+{41.57}
  2. D D
  3. λ \lambda
  4. D = R T 6 π r N A η , {D=\ }{RT\over\ 6\pi rN_{A}\eta},
  5. R R
  6. T T
  7. r r
  8. N A N_{A}
  9. η \eta
  10. C 10 F 21 C_{10}F_{21}
  11. C 8 F 17 C_{8}F_{17}
  12. G ′′ ( ω ) = [ G ( ω ) G 0 - G ( ω ) 2 ] - 1 2 {G^{\prime\prime}(\omega)=\ }[G^{\prime}(\omega)G_{0}-{G^{\prime}(\omega)}^{2}% ]^{-{1\over 2}}
  13. G 0 G_{0}
  14. ω \omega
  15. 10KC10 \ \ \ \ \,\text{10KC10}
  16. 10KC8 \,\text{10KC8}
  17. 6KC8 \,\text{6KC8}
  18. Composition ( w t % ) \,\text{Composition}\ (wt\%)
  19. Relaxation Time ( τ r ) ( s ) \,\text{Relaxation Time}\ (\tau_{r})\ (s)
  20. Plateau Modulus ( G o ) ( k P a ) \,\text{Plateau Modulus}\ (G_{o})\ (kPa)
  21. Viscosity ( η o ) ( k P a s ) \,\text{Viscosity}\ (\eta_{o})\ (kPa\cdot s)
  22. Conversion Percent ( % ) \,\text{Conversion Percent}\ (\%)

Hyper-Erlang_distribution.html

  1. A ( x ) = i = 1 n p i E l i ( x ) A(x)=\sum_{i=1}^{n}p_{i}E_{l_{i}}(x)

Hyper_basis_function_network.html

  1. x n x\in\mathbb{R}^{n}
  2. ϕ : n \phi:\mathbb{R}^{n}\to\mathbb{R}
  3. ϕ ( x ) = j = 1 N a j ρ j ( || x - μ j || ) \phi(x)=\sum_{j=1}^{N}a_{j}\rho_{j}(||x-\mu_{j}||)
  4. N N
  5. μ j \mu_{j}
  6. a j a_{j}
  7. j j
  8. ρ j ( || x - μ j || ) \rho_{j}(||x-\mu_{j}||)
  9. ρ j ( || x - μ j || ) = e ( x - μ j ) T R j ( x - μ j ) \rho_{j}(||x-\mu_{j}||)=e^{(x-\mu_{j})^{T}R_{j}(x-\mu_{j})}
  10. R j R_{j}
  11. d × d d\times d
  12. R j R_{j}
  13. R j = 1 2 σ 2 𝕀 d × d R_{j}=\frac{1}{2\sigma^{2}}\mathbb{I}_{d\times d}
  14. σ > 0 \sigma>0
  15. R j = 1 2 σ j 2 𝕀 d × d R_{j}=\frac{1}{2\sigma_{j}^{2}}\mathbb{I}_{d\times d}
  16. σ j > 0 \sigma_{j}>0
  17. R j = d i a g ( 1 2 σ j 1 2 , , 1 2 σ j z 2 ) 𝕀 d × d R_{j}=diag\left(\frac{1}{2\sigma_{j1}^{2}},...,\frac{1}{2\sigma_{jz}^{2}}% \right)\mathbb{I}_{d\times d}
  18. σ j i > 0 \sigma_{ji}>0
  19. a j a_{j}
  20. R j R_{j}
  21. μ j \mu_{j}
  22. H [ ϕ * ] = i = 1 N ( y i - ϕ * ( x i ) ) 2 H[\phi^{*}]=\sum_{i=1}^{N}(y_{i}-\phi^{*}(x_{i}))^{2}
  23. H ( ϕ * ) a j = 0 \frac{\partial H(\phi^{*})}{\partial a_{j}}=0
  24. H ( ϕ * ) μ j = 0 \frac{\partial H(\phi^{*})}{\partial\mu_{j}}=0
  25. H ( ϕ * ) W = 0 \frac{\partial H(\phi^{*})}{\partial W}=0
  26. R j = W T W R_{j}=W^{T}W
  27. a j , μ j , W a_{j},\mu_{j},W
  28. H [ ϕ * ] H[\phi^{*}]
  29. a j ˙ = - ω H ( ϕ * ) a j \dot{a_{j}}=-\omega\frac{\partial H(\phi^{*})}{\partial a_{j}}
  30. μ j ˙ = - ω H ( ϕ * ) μ j \dot{\mu_{j}}=-\omega\frac{\partial H(\phi^{*})}{\partial\mu_{j}}
  31. W ˙ = - ω H ( ϕ * ) W \dot{W}=-\omega\frac{\partial H(\phi^{*})}{\partial W}
  32. ω \omega

Hyperbolic_geometric_graph.html

  1. G ( V , E ) G(V,E)
  2. N = | V | N=|V|
  3. ζ 2 \mathbb{H}^{2}_{\zeta}
  4. - ζ 2 -\zeta^{2}
  5. R R
  6. i i
  7. ( r i , θ i ) (r_{i},\theta_{i})
  8. 0 r i R 0\leq r_{i}\leq R
  9. 0 θ i < 2 π 0\leq\theta_{i}<2\pi
  10. d i j d_{ij}
  11. i i
  12. j j
  13. cosh ( ζ d i j ) = cosh ( ζ r i ) cosh ( ζ r j ) \cosh(\zeta d_{ij})=\cosh(\zeta r_{i})\cosh(\zeta r_{j})
  14. - sinh ( ζ r i ) sinh ( ζ r j ) cos ( π - | π - | θ i - θ j | | Δ ) . -\sinh(\zeta r_{i})\sinh(\zeta r_{j})\cos\bigg(\underbrace{\pi\!-\!\bigg|\pi-|% \theta_{i}\!-\!\theta_{j}|\bigg|}_{\Delta}\bigg).
  15. Δ \Delta
  16. ( i , j ) (i,j)
  17. r r
  18. d i j r d_{ij}\leq r
  19. d i j d_{ij}
  20. γ ( s ) : + [ 0 , 1 ] \gamma(s):\mathbb{R}^{+}\to[0,1]
  21. s s
  22. ζ = 1 \zeta=1
  23. K = - 1 K=-1

Hyperharmonic_number.html

  1. H n ( r ) H_{n}^{(r)}
  2. H n ( 0 ) = 1 n , H_{n}^{(0)}=\frac{1}{n},
  3. H n ( r ) = k = 1 n H k ( r - 1 ) ( r > 0 ) . H_{n}^{(r)}=\sum_{k=1}^{n}H_{k}^{(r-1)}\quad(r>0).
  4. H n = H n ( 1 ) H_{n}=H_{n}^{(1)}
  5. H n ( r ) = H n - 1 ( r ) + H n ( r - 1 ) . H_{n}^{(r)}=H_{n-1}^{(r)}+H_{n}^{(r-1)}.
  6. H n ( r ) = ( n + r - 1 r - 1 ) ( H n + r - 1 - H r - 1 ) . H_{n}^{(r)}={\left({{n+r-1}\atop{r-1}}\right)}(H_{n+r-1}-H_{r-1}).
  7. H n = 1 n ! [ n + 1 2 ] . H_{n}=\frac{1}{n!}\left[{n+1\atop 2}\right].
  8. H n ( r ) = 1 n ! [ n + r r + 1 ] r , H_{n}^{(r)}=\frac{1}{n!}\left[{n+r\atop r+1}\right]_{r},
  9. [ n r ] r \left[{n\atop r}\right]_{r}
  10. H n ( r ) 1 ( r - 1 ) ! ( n r - 1 ln ( n ) ) , H_{n}^{(r)}\sim\frac{1}{(r-1)!}\left(n^{r-1}\ln(n)\right),
  11. n = 1 H n ( r ) n m < + \sum_{n=1}^{\infty}\frac{H_{n}^{(r)}}{n^{m}}<+\infty
  12. n = 0 H n ( r ) z n = - ln ( 1 - z ) ( 1 - z ) r . \sum_{n=0}^{\infty}H_{n}^{(r)}z^{n}=-\frac{\ln(1-z)}{(1-z)^{r}}.
  13. n = 0 H n ( r ) t n n ! = e t ( n = 1 r - 1 H n ( r - n ) t n n ! + ( r - 1 ) ! ( r ! ) 2 t 2 r F 2 ( 1 , 1 ; r + 1 , r + 1 ; - t ) ) , \sum_{n=0}^{\infty}H_{n}^{(r)}\frac{t^{n}}{n!}=e^{t}\left(\sum_{n=1}^{r-1}H_{n% }^{(r-n)}\frac{t^{n}}{n!}+\frac{(r-1)!}{(r!)^{2}}t^{r}\,_{2}F_{2}\left(1,1;r+1% ,r+1;-t\right)\right),
  14. n = 1 H n ( r ) n m = n = 1 H n ( r - 1 ) ζ ( m , n ) ( r 1 , m r + 1 ) . \sum_{n=1}^{\infty}\frac{H_{n}^{(r)}}{n^{m}}=\sum_{n=1}^{\infty}H_{n}^{(r-1)}% \zeta(m,n)\quad(r\geq 1,m\geq r+1).
  15. H n ( 4 ) H_{n}^{(4)}

HyperLogLog.html

  1. > 10 9 >10^{9}
  2. n n
  3. 2 n 2^{n}

Hypersimplex.html

  1. ( d k ) {\textstyle\left({{d}\atop{k}}\right)}
  2. ( d k ) {\textstyle\left({{d}\atop{k}}\right)}

Hypoelastic_material.html

  1. s y m b o l T ˙ = 𝖣 : s y m b o l F ˙ \dot{symbol{T}}=\mathsf{D}:\dot{symbol{F}}
  2. s y m b o l T symbol{T}
  3. 𝖣 \mathsf{D}
  4. s y m b o l F symbol{F}
  5. 𝖣 \mathsf{D}
  6. ε i j = 1 2 ( u i , j + u j , i ) \varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})
  7. u i u_{i}
  8. x i x_{i}
  9. ( i = 1 , 2 , 3 ) (i=1,2,3)
  10. u i , j = u i / x j u_{i,j}=\partial u_{i}/\partial x_{j}
  11. W ( s y m b o l F ) W(symbol{F})
  12. W W
  13. d i j = ε ˙ i j = 1 2 ( v i , j + v j , i ) d_{ij}=\dot{\varepsilon}_{ij}=\frac{1}{2}(v_{i,j}+v_{j,i})
  14. Δ ε i j = ε ˙ i j Δ t = d i j Δ t \Delta\varepsilon_{ij}=\dot{\varepsilon}_{ij}\Delta t=d_{ij}\Delta t
  15. / t \partial/\partial t
  16. Δ \Delta
  17. t t
  18. v i = u ˙ i v_{i}=\dot{u}_{i}
  19. σ i j \sigma_{ij}
  20. X i X_{i}
  21. σ ^ i j \hat{\sigma}_{ij}
  22. Δ σ i j = σ ^ i j Δ t \Delta\sigma_{ij}=\hat{\sigma}_{ij}\Delta t
  23. σ ^ i j \hat{\sigma}_{ij}
  24. σ ^ i j \hat{\sigma}_{ij}

Icosahedral_number.html

  1. n ( 5 n 2 - 5 n + 2 ) 2 {n(5n^{2}-5n+2)\over 2}

Icosahedron.html

  1. s { 3 3 } s\begin{Bmatrix}3\\ 3\end{Bmatrix}

Ideal_surface.html

  1. γ α θ + γ θ β cos θ + γ α β cos α = 0 \gamma_{\alpha\theta}+\gamma_{\theta\beta}\cos{\theta}+\gamma_{\alpha\beta}% \cos{\alpha}\ =0
  2. γ α θ cos θ + γ θ β + γ α β cos β = 0 \gamma_{\alpha\theta}\cos{\theta}+\gamma_{\theta\beta}+\gamma_{\alpha\beta}% \cos{\beta}\ =0
  3. γ α θ cos α + γ θ β cos β + γ α β = 0 \gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{% \alpha\beta}\ =0
  4. α + β + θ = 2 π \alpha+\beta+\theta=2\pi
  5. γ S G = γ S L + γ L G cos θ \gamma_{SG}\ =\gamma_{SL}+\gamma_{LG}\cos{\theta}
  6. θ A \theta_{\mathrm{A}}
  7. θ R \theta_{\mathrm{R}}
  8. θ c \theta_{\mathrm{c}}
  9. θ A \theta_{\mathrm{A}}
  10. θ R \theta_{\mathrm{R}}
  11. θ c = arccos ( r A cos θ A + r R cos θ R r A + r R ) \theta_{\mathrm{c}}=\arccos\left(\frac{r_{\mathrm{A}}\cos{\theta_{\mathrm{A}}}% +r_{\mathrm{R}}\cos{\theta_{\mathrm{R}}}}{r_{\mathrm{A}}+r_{\mathrm{R}}}\right)
  12. r A = ( sin 3 θ A 2 - 3 cos θ A + cos 3 θ A ) 1 / 3 ; r R = ( sin 3 θ R 2 - 3 cos θ R + cos 3 θ R ) 1 / 3 r_{\mathrm{A}}=\left(\frac{\sin^{3}{\theta_{\mathrm{A}}}}{2-3\cos{\theta_{% \mathrm{A}}}+\cos^{3}{\theta_{\mathrm{A}}}}\right)^{1/3}~{};~{}~{}r_{\mathrm{R% }}=\left(\frac{\sin^{3}{\theta_{\mathrm{R}}}}{2-3\cos{\theta_{\mathrm{R}}}+% \cos^{3}{\theta_{\mathrm{R}}}}\right)^{1/3}
  13. S = γ S G - ( γ S L + γ L G ) S\ =\gamma_{SG}-(\gamma_{SL}+\gamma_{LG})

Igusa_group.html

  1. ( A B C D ) \begin{pmatrix}A&B\\ C&D\end{pmatrix}
  2. ( A B C D ) \begin{pmatrix}A&B\\ C&D\end{pmatrix}

Image_derivatives.html

  1. p u = 𝐝 G p^{\prime}_{u}=\mathbf{d}\ast G
  2. 𝐝 \mathbf{d}
  3. G G
  4. \ast
  5. u u
  6. v v
  7. p u = [ - 1 - 2 - 1 0 0 0 + 1 + 2 + 1 ] * 𝐆 and p v = [ - 1 0 + 1 - 2 0 + 2 - 1 0 + 1 ] * 𝐆 p^{\prime}_{u}=\begin{bmatrix}-1&-2&-1\\ 0&0&0\\ +1&+2&+1\end{bmatrix}*\mathbf{G}\quad\mbox{and}~{}\quad p^{\prime}_{v}=\begin{% bmatrix}-1&0&+1\\ \ \ -2&\ \ 0&\ \ +2\\ -1&0&+1\end{bmatrix}*\mathbf{G}
  8. * *
  9. p v p^{\prime}_{v}
  10. [ - 1 0 + 1 - 2 0 + 2 - 1 0 + 1 ] = [ 1 2 1 ] [ - 1 0 + 1 ] \begin{bmatrix}-1&0&+1\\ -2&0&+2\\ -1&0&+1\end{bmatrix}=\begin{bmatrix}1\\ 2\\ 1\end{bmatrix}\begin{bmatrix}-1&0&+1\end{bmatrix}
  11. p u = 𝐝 ( 𝐤 G ) = ( 𝐝 𝐤 ) G p^{\prime}_{u}=\mathbf{d}\ast(\mathbf{k}\ast G)=(\mathbf{d}\ast\mathbf{k})\ast G
  12. 𝐤 𝐝 \mathbf{k_{d}}
  13. 𝐤 \mathbf{k}
  14. 𝐝 \mathbf{d}
  15. 𝐤 𝐝 = 𝐝 𝐤 \mathbf{k_{d}}=\mathbf{d}\ast\mathbf{k}
  16. p ( u , v ) = 𝐮 T M G M T 𝐯 = M T 𝐮 𝐯 T M G p(u,v)=\mathbf{u}^{T}MGM^{T}\mathbf{v}=M^{T}\mathbf{u}\otimes\mathbf{v}^{T}M\ast G
  17. M M
  18. 𝐮 \mathbf{u}
  19. 𝐯 \mathbf{v}
  20. u u
  21. v v
  22. 𝐮 = [ u 3 , u 2 , u , 1 ] T \mathbf{u}=[u^{3},u^{2},u,1]^{T}
  23. 𝐯 = [ v 3 , v 2 , v , 1 ] T \mathbf{v}=[v^{3},v^{2},v,1]^{T}
  24. 𝐤 = 𝐮 T M = ( M T 𝐯 ) T \mathbf{k}=\mathbf{u}^{T}M=(M^{T}\mathbf{v})^{T}
  25. 𝐝 = 𝐮 T u M = ( M T 𝐯 v ) T \mathbf{d}=\frac{\partial\mathbf{u}^{T}}{\partial u}M=\left(M^{T}\frac{% \partial\mathbf{v}}{\partial v}\right)^{T}
  26. 𝐝 𝟐 = 2 𝐮 T u 2 M = ( M T 2 𝐯 v 2 ) T \mathbf{d^{2}}=\frac{\partial^{2}\mathbf{u}^{T}}{\partial u^{2}}M=\left(M^{T}% \frac{\partial^{2}\mathbf{v}}{\partial v^{2}}\right)^{T}
  27. D u = 𝐮 T u M 𝐮 T M G M T 𝐯 M T 𝐯 = 𝐝 𝐤 ( 𝐤 𝐤 ) T G D_{u}=\frac{\partial\mathbf{u}^{T}}{\partial u}M\ast\mathbf{u}^{T}MGM^{T}% \mathbf{v}\ast M^{T}\mathbf{v}=\mathbf{d}\ast\mathbf{k}\otimes(\mathbf{k}\ast% \mathbf{k})^{T}\ast G
  28. D v = 𝐮 T M 𝐮 T M G M T 𝐯 M T 𝐯 v = 𝐤 𝐤 ( 𝐝 𝐤 ) T G D_{v}=\mathbf{u}^{T}M\ast\mathbf{u}^{T}MGM^{T}\mathbf{v}\ast M^{T}\frac{% \partial\mathbf{v}}{\partial v}=\mathbf{k}\ast\mathbf{k}\otimes(\mathbf{d}\ast% \mathbf{k})^{T}\ast G
  29. D u 2 = 2 𝐮 T u 2 M 𝐮 T M G M T 𝐯 M T 𝐯 = 𝐝 𝟐 𝐤 ( 𝐤 𝐤 ) T G D^{2}_{u}=\frac{\partial^{2}\mathbf{u}^{T}}{\partial u^{2}}M\ast\mathbf{u}^{T}% MGM^{T}\mathbf{v}\ast M^{T}\mathbf{v}=\mathbf{d^{2}}\ast\mathbf{k}\otimes(% \mathbf{k}\ast\mathbf{k})^{T}\ast G
  30. D v 2 = 𝐮 T M 𝐮 T M G M T 𝐯 M T 2 𝐯 v 2 = 𝐤 𝐤 ( 𝐝 𝟐 𝐤 ) T G D^{2}_{v}=\mathbf{u}^{T}M\ast\mathbf{u}^{T}MGM^{T}\mathbf{v}\ast M^{T}\frac{% \partial^{2}\mathbf{v}}{\partial v^{2}}=\mathbf{k}\ast\mathbf{k}\otimes(% \mathbf{d^{2}}\ast\mathbf{k})^{T}\ast G
  31. u = v = 0.5 u=v=0.5
  32. 𝐮 = 𝐯 = [ ( 0.5 ) 3 , ( 0.5 ) 2 , 0.5 , 1 ] T = [ 0.125 , 0.25 , 0.5 , 1 ] T \mathbf{u}=\mathbf{v}=[(0.5)^{3},(0.5)^{2},0.5,1]^{T}=[0.125,0.25,0.5,1]^{T}
  33. 𝐮 ( 0.5 ) u = 𝐯 ( 0.5 ) v = [ 3 ( 0.5 ) 2 , 2 ( 0.5 ) , 1 , 0 ] T = [ 0.75 , 1 , 1 , 0 ] T \frac{\partial\mathbf{u}(0.5)}{\partial u}=\frac{\partial\mathbf{v}(0.5)}{% \partial v}=[3\cdot(0.5)^{2},2\cdot(0.5),1,0]^{T}=[0.75,1,1,0]^{T}
  34. d 2 𝐮 ( 0.5 ) u 2 = d 2 𝐯 ( 0.5 ) d v 2 = [ 6 ( 0.5 ) , 2 , 0 , 0 ] T = [ 3 , 2 , 0 , 0 ] T \frac{d^{2}\mathbf{u}(0.5)}{u^{2}}=\frac{d^{2}\mathbf{v}(0.5)}{dv^{2}}=[6\cdot% (0.5),2,0,0]^{T}=[3,2,0,0]^{T}

Impedance_(accelerator_physics).html

  1. Z 0 | | ( ω ) = - d z c e - i ω z / c W 0 ( z ) Z_{0}^{||}(\omega)=\int_{-\infty}^{\infty}\frac{dz}{c}e^{-i\omega z/c}W_{0}^{^% {\prime}}(z)
  2. Z * | | ( ω ) = Z | | ( - ω ) Z^{*||}(\omega)=Z^{||}(-\omega)
  3. Z | | ( ω ) = R s 1 - i Q ( ω r ω - ω ω r ) 1 + Q 2 ( ω r ω - ω ω r ) 2 Z_{||}(\omega)=R_{s}\frac{1-iQ(\frac{\omega_{r}}{\omega}-\frac{\omega}{\omega_% {r}})}{1+Q^{2}\left(\frac{\omega_{r}}{\omega}-\frac{\omega}{\omega_{r}}\right)% ^{2}}
  4. R s R_{s}
  5. Q Q
  6. ω r \omega_{r}
  7. b b
  8. σ \sigma
  9. Z ( ω ) = 1 - i c b ω 2 π σ Z(\omega)=\frac{1-i}{cb}\sqrt{\frac{\omega}{2\pi\sigma}}
  10. W ( s ) = q 2 π b c σ 1 s 3 / 2 W(s)=\frac{q}{2\pi b}\sqrt{\frac{c}{\sigma}}\frac{1}{s^{3/2}}
  11. W ( s ) 1 s 1 / 2 W(s)\approx\frac{1}{s^{1/2}}

Implication(FCA).html

  1. ( U , V ) (U,V)
  2. U A U\subseteq A
  3. V A V\subseteq A
  4. A A
  5. U V U\rightarrow V
  6. U W or V W U\nsubseteq W\,\text{ or }V\subseteq W
  7. U V U\rightarrow V
  8. U V U^{\prime}\subseteq V^{\prime}
  9. V U ′′ V\subseteq U^{\prime\prime}
  10. U V U\rightarrow V
  11. A A
  12. U V U\rightarrow V
  13. { X A | X respects L } \{X\subseteq A|X\;\,\text{respects}\;L\}
  14. X X ′′ X\;\mapsto\;X^{\prime\prime}
  15. P P ′′ P\;\neq\;P^{\prime\prime}
  16. Q P Q ′′ P Q\subset P\implies Q^{\prime\prime}\subseteq P
  17. { P P ′′ | P pseudo-closed } \{P\mapsto P^{\prime\prime}|P\;\,\text{pseudo-closed}\}
  18. | U | | V | \sum|U||V|
  19. U V U\rightarrow V
  20. V V\neq\emptyset
  21. U V = U\cap V=\emptyset

Imprecise_Dirichlet_process.html

  1. DP ( s , G 0 ) \mathrm{DP}\left(s,G_{0}\right)
  2. G 0 G_{0}
  3. s s
  4. α \alpha
  5. ( s , G 0 ) \left(s,G_{0}\right)
  6. G 0 G_{0}
  7. s 0 s\rightarrow 0
  8. s 0 s\rightarrow 0
  9. s 0 s\rightarrow 0
  10. s > 0 s>0
  11. G 0 G_{0}
  12. IDP : { DP ( s , G 0 ) : G 0 } ~{}~{}\mathrm{IDP}:~{}\left\{\mathrm{DP}\left(s,G_{0}\right):~{}~{}G_{0}\in% \mathbb{P}\right\}
  13. \mathbb{P}
  14. s > 0 s>0
  15. G 0 G_{0}
  16. P P
  17. ( 𝕏 , ) (\mathbb{X},\mathcal{B})
  18. 𝕏 \mathbb{X}
  19. σ \sigma
  20. \mathcal{B}
  21. P DP ( s , G 0 ) P\sim\mathrm{DP}(s,G_{0})
  22. f f
  23. ( 𝕏 , ) (\mathbb{X},\mathcal{B})
  24. E [ f ] E[f]
  25. [ E ( f ) ] = [ f d P ] = f d [ P ] = f d G 0 . \mathcal{E}[E(f)]=\mathcal{E}\left[\int f\,dP\right]=\int f\,d\mathcal{E}[P]=% \int f\,dG_{0}.
  26. P P
  27. X 1 , , X n X_{1},\dots,X_{n}
  28. P P
  29. P D p ( s , G 0 ) P\sim Dp(s,G_{0})
  30. P P
  31. P X 1 , , X n D p ( s + n , G n ) , with G n = s s + n G 0 + 1 s + n i = 1 n δ X i , P\mid X_{1},\dots,X_{n}\sim Dp\left(s+n,G_{n}\right),~{}~{}~{}\,\text{with}~{}% ~{}~{}~{}~{}~{}G_{n}=\frac{s}{s+n}G_{0}+\frac{1}{s+n}\sum\limits_{i=1}^{n}% \delta_{X_{i}},
  32. δ X i \delta_{X_{i}}
  33. X i X_{i}
  34. [ E ( f ) X 1 , , X n ] = f d G n . \mathcal{E}[E(f)\mid X_{1},\dots,X_{n}]=\int f\,dG_{n}.
  35. G 0 G_{0}
  36. G 0 G_{0}
  37. \mathbb{P}
  38. E ( f ) E(f)
  39. G 0 G_{0}
  40. E ( f ) E(f)
  41. G 0 G_{0}\in\mathbb{P}
  42. ¯ [ E ( f ) ] = inf G 0 f d G 0 = inf f , ¯ [ E ( f ) ] = sup G 0 f d G 0 = sup f , \underline{\mathcal{E}}[E(f)]=\inf\limits_{G_{0}\in\mathbb{P}}\int f\,dG_{0}=% \inf f,~{}~{}~{}~{}\overline{\mathcal{E}}[E(f)]=\sup\limits_{G_{0}\in\mathbb{P% }}\int f\,dG_{0}=\sup f,
  43. f f
  44. G 0 = δ X 0 G_{0}=\delta_{X_{0}}
  45. X 0 = arg inf f X_{0}=\arg\inf f
  46. X 0 = arg sup f X_{0}=\arg\sup f
  47. [ E ( f ) ] \mathcal{E}[E(f)]
  48. f f
  49. f f
  50. E ( f ) E(f)
  51. E ( f ) E(f)
  52. ¯ [ E ( f ) X 1 , , X n ] \displaystyle\underline{\mathcal{E}}[E(f)\mid X_{1},\dots,X_{n}]
  53. G 0 G_{0}
  54. s s
  55. n n\rightarrow\infty
  56. ¯ [ E ( f ) X 1 , , X n ] , ¯ [ E ( f ) X 1 , , X n ] S ( f ) , \underline{\mathcal{E}}\left[E(f)\mid X_{1},\dots,X_{n}\right],\quad\overline{% \mathcal{E}}\left[E(f)\mid X_{1},\dots,X_{n}\right]\rightarrow S(f),
  57. S ( f ) = lim n 1 n i = 1 n f ( X i ) S(f)=\lim_{n\rightarrow\infty}\tfrac{1}{n}\sum_{i=1}^{n}f(X_{i})
  58. s s
  59. s s
  60. s s
  61. s s
  62. s s
  63. X 1 , , X n X_{1},\dots,X_{n}
  64. F ( x ) F(x)
  65. F ( x ) = E [ 𝕀 ( , x ] ] F(x)=E[\mathbb{I}_{(\infty,x]}]
  66. 𝕀 ( , x ] \mathbb{I}_{(\infty,x]}
  67. F ( x ) . F(x).
  68. F ( x ) F(x)
  69. ¯ [ F ( x ) X 1 , , X n ] = ¯ [ E ( 𝕀 ( , x ] ) X 1 , , X n ] \displaystyle\underline{\mathcal{E}}\left[F(x)\mid X_{1},\dots,X_{n}\right]=% \underline{\mathcal{E}}[E(\mathbb{I}_{(\infty,x]})\mid X_{1},\dots,X_{n}]
  70. F ^ ( x ) \hat{F}(x)
  71. inf 𝕀 ( , x ] = 0 \inf\mathbb{I}_{(\infty,x]}=0
  72. sup 𝕀 ( , x ] = 1 \sup\mathbb{I}_{(\infty,x]}=1
  73. G 0 G_{0}
  74. 𝒩 ( x ; 0 , 1 ) \mathcal{N}(x;0,1)
  75. F ( x ) F(x)
  76. F ( 0 ) < 0.5 F(0)<0.5
  77. F F
  78. ( - , 0 ] , ( 0 , ) (-\infty,0],(0,\infty)
  79. F ( 0 ) F(0)
  80. F ( 0 ) Beta ( α 0 + n < 0 , β 0 + n - n < 0 ) F(0)\sim\mathrm{Beta}(\alpha_{0}+n_{<0},\beta_{0}+n-n_{<0})
  81. n < 0 n_{<0}
  82. α 0 = s - 0 d G 0 \alpha_{0}=s\int_{-\infty}^{0}dG_{0}
  83. β 0 = s 0 d G 0 . \beta_{0}=s\int_{0}^{\infty}dG_{0}.
  84. 𝒫 ¯ [ F ( 0 ) < 0.5 X 1 , , X n ] = 0 0.5 Beta ( θ ; s + n < 0 , n - n < 0 ) d θ = I 1 / 2 ( s + n < 0 , n - n < 0 ) , \underline{\mathcal{P}}[F(0)<0.5\mid X_{1},\dots,X_{n}]=\int\limits_{0}^{0.5}% \mathrm{Beta}(\theta;s+n_{<0},n-n_{<0})d\theta=I_{1/2}(s+n_{<0},n-n_{<0}),
  85. 𝒫 ¯ [ F ( 0 ) < 0.5 X 1 , , X n ] = 0 0.5 Beta ( θ ; n < 0 , s + n - n < 0 ) d θ = I 1 / 2 ( s + n < 0 , n - n < 0 ) . \overline{\mathcal{P}}[F(0)<0.5\mid X_{1},\dots,X_{n}]=\int\limits_{0}^{0.5}% \mathrm{Beta}(\theta;n_{<0},s+n-n_{<0})d\theta=I_{1/2}(s+n_{<0},n-n_{<0}).
  86. I x ( α , β ) I_{x}(\alpha,\beta)
  87. 𝒫 ¯ [ F ( 0 ) < 0.5 X 1 , , X n ] > 1 - γ , 𝒫 ¯ [ F ( 0 ) < 0.5 X 1 , , X n ] > 1 - γ , \underline{\mathcal{P}}[F(0)<0.5\mid X_{1},\dots,X_{n}]>1-\gamma,~{}~{}% \overline{\mathcal{P}}[F(0)<0.5\mid X_{1},\dots,X_{n}]>1-\gamma,
  88. 1 - γ = 0.95 1-\gamma=0.95
  89. F ( 0 ) < 0.5 F(0)<0.5
  90. 1 - γ 1-\gamma
  91. F ( 0 ) < 0.5 F(0)<0.5
  92. 1 - γ 1-\gamma
  93. G 0 G_{0}
  94. F ( k ; n , p ) = Pr ( Z k ) = I 1 - p ( n - k , k + 1 ) = 1 - I p ( k + 1 , n - k ) , F(k;n,p)=\Pr(Z\leq k)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k),
  95. s 1 s\geq 1
  96. s = 1 s=1
  97. p p
  98. 1 - 𝒫 ¯ [ F ( 0 ) < 0.5 X 1 , , X n ] 1-\underline{\mathcal{P}}[F(0)<0.5\mid X_{1},\dots,X_{n}]
  99. 𝒫 ¯ [ F ( 0 ) < 0.5 X 1 , , X n ] > 0.95 \underline{\mathcal{P}}[F(0)<0.5\mid X_{1},\dots,X_{n}]>0.95
  100. p p
  101. 0.05 0.05
  102. 𝕏 \mathbb{X}

Incidence_(graph).html

  1. ( u , e ) (u,e)
  2. u u
  3. e e
  4. u u
  5. ( u , e ) (u,e)
  6. ( v , f ) (v,f)
  7. u = v u=v
  8. e = f e=f
  9. u v = e uv=e
  10. f f
  11. G G
  12. G G

Incidence_coloring.html

  1. χ i \chi_{i}
  2. A c A_{c}
  3. A c A_{c}
  4. χ i \chi_{i}
  5. χ i \chi_{i}
  6. χ i \chi_{i}
  7. K n K_{n}
  8. χ i \chi_{i}
  9. K m , n K_{m,n}
  10. χ i \chi_{i}
  11. χ i \chi_{i}
  12. H 2 H^{2}
  13. T 2 T^{2}
  14. H 1 H_{1}
  15. H 2 H_{2}
  16. G 1 G_{1}
  17. H 1 H_{1}
  18. G 2 G_{2}
  19. H 2 H_{2}
  20. G 1 , G 2 G_{1},G_{2}
  21. d G d_{G}
  22. d G d_{G}

Inclusion_(Boolean_algebra).html

  1. a b a\leq b
  2. a b = 0 ab^{\prime}=0
  3. a < b a<b
  4. a < b a<b
  5. a b = 0 ab^{\prime}=0
  6. a + b = 1 a^{\prime}+b=1
  7. b < a b^{\prime}<a^{\prime}
  8. a + b = b a+b=b
  9. a b = a ab=a
  10. a a + b a\leq a+b
  11. a b a ab\leq a
  12. a x b a\leq x\leq b

Incomplete_information_network_game.html

  1. σ ( d ) [ 0 , 1 ] \textstyle\sigma_{(d)}\in[0,1]
  2. P ~ \textstyle\tilde{P}
  3. P ~ ( d ) = P ( d ) d d \textstyle\tilde{P}(d)=\frac{P(d)d}{\langle d\rangle}
  4. P ~ \textstyle\tilde{P}
  5. p σ = d σ ( d ) P ~ ( d ) \textstyle p_{\sigma}=\sum_{d}\sigma(d)\tilde{P}(d)
  6. ( d i m ) p σ m ( 1 - p σ ) ( d i - m ) \textstyle\ {d_{i}\choose m}p_{\sigma}^{m}(1-p_{\sigma})^{(d_{i}-m)}
  7. d i \textstyle\ d_{i}
  8. x i \textstyle\ x_{i}
  9. U d i ( x i , p σ ) = m = 0 d i u d i ( x i , m ) ( d i m ) p σ m ( 1 - p σ ) ( d i - m ) \textstyle\ U_{d_{i}}(x_{i},p_{\sigma})=\sum_{m=0}^{d_{i}}u_{d_{i}}(x_{i},m){d% _{i}\choose m}p_{\sigma}^{m}(1-p_{\sigma})^{(d_{i}-m)}
  10. u d i \textstyle\ u_{d_{i}}
  11. u d \textstyle\ u_{d}
  12. P ~ \textstyle\tilde{P}
  13. σ ( d ) \textstyle\sigma(d)
  14. σ ( d ) > 0 \textstyle\sigma(d)>0
  15. U d ( 1 , p σ ) U d ( 0 , p σ ) \textstyle\ U_{d}(1,p_{\sigma})\geq U_{d}(0,p_{\sigma})
  16. σ ( d ) < 1 \textstyle\sigma(d)<1
  17. U d ( 1 , p σ ) U d ( 0 , p σ ) \textstyle\ U_{d}(1,p_{\sigma})\leq U_{d}(0,p_{\sigma})
  18. N = { 1 , , n } \textstyle N=\left\{1,...,n\right\}
  19. g { 0 , 1 } n × n \textstyle\ g\in\left\{0,1\right\}^{n\times n}
  20. g i j = 1 \textstyle\ g_{ij}=1
  21. g i i = 0 \textstyle\ g_{ii}=0
  22. i N \textstyle\ i\in N
  23. i \textstyle\ i
  24. N i ( g ) = { j | g i j = 1 } \textstyle\ N_{i}(g)=\left\{j|g_{ij}=1\right\}
  25. i \textstyle\ i
  26. k i ( g ) = | N i ( g ) | \textstyle\ k_{i}(g)=|N_{i}(g)|
  27. X = { 0 , 1 } \textstyle\ X=\left\{0,1\right\}
  28. y i x i + x ¯ N i \textstyle\ y_{i}\equiv x_{i}+\overline{x}_{Ni}
  29. x i \textstyle\ x_{i}
  30. x ¯ N i j N i x j \textstyle\overline{x}_{Ni}\equiv\sum_{j\in N_{i}}x_{j}
  31. y i 1 \textstyle\ y_{i}\geq 1
  32. 0 < c < 1 \textstyle\ 0<c<1
  33. σ \textstyle\sigma
  34. X = { 0 , 1 } \textstyle\ X=\left\{0,1\right\}
  35. p ( 0 , 1 ) \textstyle\ p\in(0,1)
  36. \Q ( k ; p ) = ( N - 2 k - 1 ) p ( k - 1 ) ( 1 - p ) ( N - k - 1 ) \textstyle\Q(k;p)={N-2\choose k-1}p^{(k-1)}(1-p)^{(N-k-1)}
  37. 1 - [ 1 - k = 1 t Q ( k ; p ) ] t 1 - c \textstyle\ 1-{[1-\sum_{k=1}^{t}Q(k;p)]}^{t}\geq 1-c
  38. σ \textstyle\sigma
  39. σ ( k ) = 1 \textstyle\sigma(k)=1
  40. k < t \textstyle k<t
  41. σ ( k ) = 0 \textstyle\sigma(k)=0
  42. k > t \textstyle k>t
  43. σ ( t ) [ 0 , 1 ] \textstyle\ \sigma(t)\in[0,1]
  44. σ ( k ) \textstyle\sigma(k)

Ind-scheme.html

  1. P = lim P N \mathbb{C}P^{\infty}=\underrightarrow{\lim}\mathbb{C}P^{N}

Indicator_vector.html

  1. x T := ( x s ) s S x_{T}:=(x_{s})_{s\in S}
  2. x s = 1 x_{s}=1
  3. s T s\in T
  4. x s = 0 x_{s}=0
  5. s T . s\notin T.
  6. S = { s 1 , s 2 , , s n } S=\{s_{1},s_{2},\ldots,s_{n}\}
  7. x T = ( x 1 , x 2 , , x n ) x_{T}=(x_{1},x_{2},\ldots,x_{n})
  8. x i = 1 x_{i}=1
  9. s i T s_{i}\in T
  10. x i = 0 x_{i}=0
  11. s i T . s_{i}\notin T.

Indifference_graph.html

  1. u v w uvw
  2. u u
  3. v v
  4. w w
  5. u u
  6. w w
  7. n n
  8. n 2 / 3 n^{2/3}
  9. n 2 / 3 n^{2/3}
  10. G G
  11. G G
  12. O ( n log n ) O(n\log n)

Individual_mobility.html

  1. P ( r ) P(r)
  2. r g ( t ) r_{g}(t)
  3. S ( t ) S(t)
  4. P ( r ) r - ( 1 + β ) P(r)\ \sim r^{-(1+\beta)}
  5. β = 0.6 \beta=0.6
  6. r g ( t ) r_{g}(t)
  7. P ( r ) P(r)
  8. S ( t ) t μ S(t)\ \sim t^{\mu}
  9. μ = 0.6 \mu=0.6

Induction_equation.html

  1. × E = - B t , \vec{\nabla}\times\vec{E}=-{\partial\vec{B}\over\partial t},
  2. × B = μ 0 J , \vec{\nabla}\times\vec{B}=\mu_{0}\vec{J},
  3. E \vec{E}
  4. B \vec{B}
  5. J \vec{J}
  6. E + v × B = J / σ \vec{E}+\vec{v}\times\vec{B}=\vec{J}/\sigma
  7. v \vec{v}
  8. σ \sigma
  9. B t = η 2 B + × ( v × B ) , {\partial\vec{B}\over\partial t}=\eta\nabla^{2}\vec{B}+\vec{\nabla}\times(\vec% {v}\times\vec{B}),
  10. η = 1 / μ 0 σ \eta=1/\mu_{0}\sigma
  11. 1 / σ 1/\sigma
  12. 1 / μ 0 σ 1/\mu_{0}\sigma
  13. V V
  14. L L
  15. η 2 B η B L 2 , × ( v × B ) V B L . \eta\nabla^{2}\vec{B}\sim{\eta B\over L^{2}},\vec{\nabla}\times(\vec{v}\times% \vec{B})\sim{VB\over L}.
  16. R m = L V η R_{m}={LV\over\eta}
  17. η 0 \eta\rightarrow 0
  18. 10 9 10^{9}
  19. B t = × ( v × B ) . {\partial\vec{B}\over\partial t}=\vec{\nabla}\times(\vec{v}\times\vec{B}).
  20. η \eta
  21. B t = η 2 B . {\partial\vec{B}\over\partial t}=\eta\nabla^{2}\vec{B}.
  22. τ d = L 2 / η \tau_{d}=L^{2}/\eta
  23. L L

Induction_motors_modelling_in_ABC_frame_of_reference.html

  1. 2 π / 3 {2π}/{3}
  2. e A = i A r A + d φ A d t e_{A}=i_{A}r_{A}+{{d\varphi_{A}}\over{dt}}
  3. e B = i B r B + d φ B d t e_{B}=i_{B}r_{B}+{{d\varphi_{B}}\over{dt}}
  4. e C = i C r C + d φ C d t e_{C}=i_{C}r_{C}+{{d\varphi_{C}}\over{dt}}
  5. e a = i a r a + d φ a d t e_{a}=i_{a}r_{a}+{{d\varphi_{a}}\over{dt}}
  6. e b = i b r b + d φ b d t e_{b}=i_{b}r_{b}+{{d\varphi_{b}}\over{dt}}
  7. e c = i c r c + d φ c d t e_{c}=i_{c}r_{c}+{{d\varphi_{c}}\over{dt}}
  8. e = i [ R ] + p [ φ ] e=i[R]+p[\varphi]
  9. φ = [ L ] i \varphi=[L]i
  10. φ = [ φ s t a t o r φ r o t o r ] \varphi=\begin{bmatrix}\varphi_{stator}\\ \varphi_{rotor}\end{bmatrix}
  11. φ s t a t o r = [ L s s ] i s t a t o r + [ L s r ] i r o t o r \varphi_{stator}=[L_{ss}]i_{stator}+[L_{sr}]i_{rotor}
  12. φ r o t o r = [ L r s ] i s t a t o r + [ L r r ] i r o t o r \varphi_{rotor}=[L_{rs}]i_{stator}+[L_{rr}]i_{rotor}
  13. [ L s s ] = [ L A A L A B L A C L B A L B B L B C L C A L C B L C C ] [L_{ss}]=\begin{bmatrix}L_{AA}&L_{AB}&L_{AC}\\ L_{BA}&L_{BB}&L_{BC}\\ L_{CA}&L_{CB}&L_{CC}\end{bmatrix}
  14. [ L s r ] = [ L A a L A b L A c L B a L B b L B c L C a L C b L C c ] [L_{sr}]=\begin{bmatrix}L_{Aa}&L_{Ab}&L_{Ac}\\ L_{Ba}&L_{Bb}&L_{Bc}\\ L_{Ca}&L_{Cb}&L_{Cc}\end{bmatrix}
  15. [ L r s ] = [ L a A L a B L a C L b A L b B L b C L c A L c B L c C ] [L_{rs}]=\begin{bmatrix}L_{aA}&L_{aB}&L_{aC}\\ L_{bA}&L_{bB}&L_{bC}\\ L_{cA}&L_{cB}&L_{cC}\end{bmatrix}
  16. [ L r r ] = [ L a a L a b L a c L b a L b b L b c L c a L c b L c c ] [L_{rr}]=\begin{bmatrix}L_{aa}&L_{ab}&L_{ac}\\ L_{ba}&L_{bb}&L_{bc}\\ L_{ca}&L_{cb}&L_{cc}\end{bmatrix}
  17. L A A = L B B = L C C = L l s + L m s L_{AA}=L_{BB}=L_{CC}=L_{ls}+L_{ms}
  18. L A B = L B A = L A C = L C A = L B C = L C B = - 1 2 L m s L_{AB}=L_{BA}=L_{AC}=L_{CA}=L_{BC}=L_{CB}=-\frac{1}{2}L_{ms}
  19. L a a = L b b = L c c = L l r + L m r L_{aa}=L_{bb}=L_{cc}=L_{lr}+L_{mr}
  20. L a b = L b a = L a c = L c b = L b c = L c a = - 1 2 L m r L_{ab}=L_{ba}=L_{ac}=L_{cb}=L_{bc}=L_{ca}=-\frac{1}{2}L_{mr}
  21. T e = [ i A ( i a - i b 2 - i c 2 ) + i B ( i b - i a 2 - i c 2 ) + i C ( i c - i b 2 - i a 2 ) sin θ r + 3 2 i A ( i b - i c ) + i B ( i c - i a ) + i C ( i a - i b ) cos θ r ] T_{e}=[{i_{A}(i_{a}-\frac{i_{b}}{2}-\frac{i_{c}}{2})+i_{B}(i_{b}-\frac{i_{a}}{% 2}-\frac{i_{c}}{2})+i_{C}(i_{c}-\frac{i_{b}}{2}-\frac{i_{a}}{2})}\sin\theta_{r% }+\frac{\sqrt{3}}{2}{i_{A}(i_{b}-i_{c})+i_{B}(i_{c}-i_{a})+i_{C}(i_{a}-i_{b})}% \cos\theta_{r}]
  22. d ω r d t = 1 J ( T e - B m ω r - T L ) \frac{d\omega_{r}}{dt}=\frac{1}{J(T_{e}-B_{m}\omega_{r}-T_{L})}
  23. ω r = d θ r d t \omega_{r}=\frac{d\theta_{r}}{dt}
  24. α β γ \alpha\beta\gamma

Inductive_probability.html

  1. P ( A and B ) = P ( B ) P ( A B ) = P ( A ) P ( B A ) P(A\and B)=P(B)\cdot P(A\mid B)=P(A)\cdot P(B\mid A)
  2. L ( A and B ) = L ( B ) + L ( A B ) = L ( A ) + L ( B A ) L(A\and B)=L(B)+L(A\mid B)=L(A)+L(B\mid A)
  3. L ( K ) L(K)
  4. P ( K ) = 2 - L ( K ) P(K)=2^{-L(K)}
  5. 2 - n 2^{-n}
  6. 2 - k 2^{-k}
  7. P ( x ) = 2 - L ( x ) P(x)=2^{-L(x)}
  8. P ( x ) P(x)
  9. L ( x ) L(x)
  10. 2 - L ( x ) 2^{-L(x)}
  11. L ( A and B ) = L ( A ) + L ( B ) L(A\and B)=L(A)+L(B)
  12. P ( A and B ) = P ( A ) P ( B ) P(A\and B)=P(A)P(B)
  13. b = { x : B ( x ) } b=\{x:B(x)\}
  14. P ( B A ) P(B\mid A)
  15. P ( B ) = P ( B true ) P(B)=P(B\mid\,\text{true})
  16. P ( A ) = | { x : A ( x ) } | | x : t r u e | P(A)=\frac{|\{x:A(x)\}|}{|x:true|}
  17. P ( B A ) = | { x : A ( x ) and B ( X ) } | | x : A ( x ) | P(B\mid A)=\frac{|\{x:A(x)\and B(X)\}|}{|x:A(x)|}
  18. P ( A and B ) P(A\and B)
  19. = | { x : A ( x ) and B ( x ) } | | x : t r u e | =\frac{|\{x:A(x)\and B(x)\}|}{|x:true|}
  20. = | { x : A ( x ) and B ( x ) } | | { x : A ( x ) } | | { x : A ( x ) } | | x : t r u e | =\frac{|\{x:A(x)\and B(x)\}|}{|\{x:A(x)\}|}\frac{|\{x:A(x)\}|}{|x:true|}
  21. = P ( A ) P ( B A ) =P(A)P(B\mid A)
  22. P ( A and B ) = P ( A ) P ( B A ) = P ( B ) P ( A B ) P(A\and B)=P(A)P(B\mid A)=P(B)P(A\mid B)
  23. L ( A and B ) = L ( A ) + L ( B A ) = L ( B ) + L ( A B ) L(A\and B)=L(A)+L(B\mid A)=L(B)+L(A\mid B)
  24. P ( B ) = P ( B A ) P(B)=P(B\mid A)
  25. P ( A and B ) = P ( A ) P ( B ) P(A\and B)=P(A)P(B)
  26. A i A_{i}
  27. i P ( A i B ) = 1 \sum_{i}{P(A_{i}\mid B)}=1
  28. i P ( B A i ) P ( A i ) = i P ( A i B ) P ( B ) \sum_{i}{P(B\mid A_{i})P(A_{i})}=\sum_{i}{P(A_{i}\mid B)P(B)}
  29. P ( B ) = i P ( B A i ) P ( A i ) P(B)=\sum_{i}{P(B\mid A_{i})P(A_{i})}
  30. P ( A i B ) = P ( B A i ) P ( A i ) j P ( B A j ) P ( A j ) P(A_{i}\mid B)=\frac{P(B\mid A_{i})P(A_{i})}{\sum_{j}{P(B\mid A_{j})P(A_{j})}}
  31. A i A_{i}
  32. P ( A B ) = P ( A ) + P ( B ) P(AB)=P(A)+P(B)
  33. P ( A and B ) = 0 P(A\and B)=0
  34. A B = ( A and ¬ ( A and B ) ) ( B and ¬ ( A and B ) ) ( A and B ) AB=(A\and\neg(A\and B))(B\and\neg(A\and B))(A\and B)
  35. A and ¬ ( A and B ) A\and\neg(A\and B)
  36. B and ¬ ( A and B ) B\and\neg(A\and B)
  37. A and B A\and B
  38. ( A and ¬ ( A and B ) ) ( A and B ) = A (A\and\neg(A\and B))(A\and B)=A
  39. P ( A and ¬ ( A and B ) ) + P ( A and B ) = P ( A ) P(A\and\neg(A\and B))+P(A\and B)=P(A)
  40. P ( A and ¬ ( A and B ) ) = P ( A ) - P ( A and B ) P(A\and\neg(A\and B))=P(A)-P(A\and B)
  41. P ( A B ) P(AB)
  42. = P ( ( A and ¬ ( A and B ) ) ( B and ¬ ( A and B ) ) ( A and B ) ) =P((A\and\neg(A\and B))(B\and\neg(A\and B))(A\and B))
  43. = P ( A and ¬ ( A and B ) + P ( B and ¬ ( A and B ) ) + P ( A and B ) =P(A\and\neg(A\and B)+P(B\and\neg(A\and B))+P(A\and B)
  44. = P ( A ) - P ( A and B ) + P ( B ) - P ( A and B ) + P ( A and B ) =P(A)-P(A\and B)+P(B)-P(A\and B)+P(A\and B)
  45. = P ( A ) + P ( B ) - P ( A and B ) =P(A)+P(B)-P(A\and B)
  46. A ¬ A = t r u e A\neg A=true
  47. P ( A ) + P ( ¬ A ) = 1 P(A)+P(\neg A)=1
  48. A B P ( B A ) = 1 A\to B\iff P(B\mid A)=1
  49. A B A\to B
  50. P ( A B ) = 1 \iff P(A\to B)=1
  51. P ( A and B ¬ A ) = 1 \iff P(A\and B\neg A)=1
  52. P ( A and B ) + P ( ¬ A ) = 1 \iff P(A\and B)+P(\neg A)=1
  53. P ( A and B ) = P ( A ) \iff P(A\and B)=P(A)
  54. P ( A ) P ( B A ) = P ( A ) \iff P(A)\cdot P(B\mid A)=P(A)
  55. P ( B A ) = 1 \iff P(B\mid A)=1
  56. P ( H F ) = P ( H ) P ( F H ) P ( F ) P(H\mid F)=\frac{P(H)P(F\mid H)}{P(F)}
  57. P ( H F ) = 2 - ( L ( H ) + L ( F H ) - L ( F ) ) P(H\mid F)=2^{-(L(H)+L(F\mid H)-L(F))}
  58. L ( H ) + L ( F H ) L(H)+L(F\mid H)
  59. P ( F ) P(F)
  60. H i H_{i}
  61. L ( H i ) + L ( F H i ) < L ( F ) L(H_{i})+L(F\mid H_{i})<L(F)
  62. P ( H i F ) = P ( H i ) P ( F H i ) P ( F | R ) + j P ( H j ) P ( F H j ) P(H_{i}\mid F)=\frac{P(H_{i})P(F\mid H_{i})}{P(F|R)+\sum_{j}{P(H_{j})P(F\mid H% _{j})}}
  63. P ( R F ) = P ( F R ) P ( F R ) + j P ( H j ) P ( F H j ) P(R\mid F)=\frac{P(F\mid R)}{P(F\mid R)+\sum_{j}{P(H_{j})P(F\mid H_{j})}}
  64. P ( H i F ) = 2 - ( L ( H i ) + L ( F H i ) ) 2 - L ( F R ) + j 2 - ( L ( H j ) + L ( F | H j ) ) P(H_{i}\mid F)=\frac{2^{-(L(H_{i})+L(F\mid H_{i}))}}{2^{-L(F\mid R)}+\sum_{j}{% 2^{-(L(H_{j})+L(F|\mid H_{j}))}}}
  65. P ( R F ) = 2 - L ( F R ) 2 - L ( F R ) + j 2 - ( L ( H j ) + L ( F H j ) ) P(R\mid F)=\frac{2^{-L(F\mid R)}}{2^{-L(F\mid R)}+\sum_{j}{2^{-(L(H_{j})+L(F% \mid H_{j}))}}}
  66. P ( F R ) = P ( F ) P(F\mid R)=P(F)
  67. P ( H i F ) 2 - ( L ( H i ) + L ( F H i ) ) 2 - L ( F ) + j 2 - ( L ( H j ) + L ( F | H j ) ) P(H_{i}\mid F)\approx\frac{2^{-(L(H_{i})+L(F\mid H_{i}))}}{2^{-L(F)}+\sum_{j}{% 2^{-(L(H_{j})+L(F|H_{j}))}}}
  68. P ( R F ) 2 - L ( F ) 2 - L ( F ) + j 2 - ( L ( H j ) + L ( F H j ) ) P(R\mid F)\approx\frac{2^{-L(F)}}{2^{-L(F)}+\sum_{j}{2^{-(L(H_{j})+L(F\mid H_{% j}))}}}
  69. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  70. F 1 = m 1 a 1 = m 1 k 1 r 2 i 1 F_{1}=m_{1}a_{1}=\frac{m_{1}k_{1}}{r^{2}}i_{1}
  71. F 2 = m 2 a 2 = m 2 k 2 r 2 i 2 F_{2}=m_{2}a_{2}=\frac{m_{2}k_{2}}{r^{2}}i_{2}
  72. i 1 i_{1}
  73. i 2 i_{2}
  74. F 1 = - F 2 F_{1}=-F_{2}
  75. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  76. T F P ( F T ) = 1 T\to F\iff P(F\mid T)=1
  77. P ( F T ) = 1 P(F\mid T)=1
  78. L ( F T ) = 0 L(F\mid T)=0
  79. P ( T F ) = P ( T ) P ( F ) P(T\mid F)=\frac{P(T)}{P(F)}
  80. P ( T F ) = 2 - ( L ( T ) - L ( F ) ) P(T\mid F)=2^{-(L(T)-L(F))}
  81. L ( T i ) < L ( F ) L(T_{i})<L(F)
  82. T i = H i T_{i}=H_{i}
  83. L ( T i ) < L ( F ) L(T_{i})<L(F)
  84. P ( T i F ) = P ( T i ) P ( F R ) + j P ( T j ) P(T_{i}\mid F)=\frac{P(T_{i})}{P(F\mid R)+\sum_{j}{P(T_{j})}}
  85. P ( R F ) = P ( F R ) P ( F R ) + j P ( T j ) P(R\mid F)=\frac{P(F\mid R)}{P(F\mid R)+\sum_{j}{P(T_{j})}}
  86. P ( T i F ) 2 - L ( T i ) 2 - L ( F ) + j 2 - L ( T j ) P(T_{i}\mid F)\approx\frac{2^{-L(T_{i})}}{2^{-L(F)}+\sum_{j}{2^{-L(T_{j})}}}
  87. P ( R F ) 2 - L ( F ) 2 - L ( F ) + j 2 - L ( T j ) P(R\mid F)\approx\frac{2^{-L(F)}}{2^{-L(F)}+\sum_{j}{2^{-L(T_{j})}}}
  88. K i K_{i}
  89. T n ( R ( K i ) ) = x T_{n}(R(K_{i}))=x
  90. R ( K i ) R(K_{i})
  91. K i K_{i}
  92. T n T_{n}
  93. K i K_{i}
  94. P ( s = R ( K i ) T n ( s ) = x ) P(s=R(K_{i})\mid T_{n}(s)=x)
  95. P ( A i B ) = P ( B A i ) P ( A i ) j P ( B A j ) P ( A j ) P(A_{i}\mid B)=\frac{P(B\mid A_{i})\,P(A_{i})}{\sum\limits_{j}P(B\mid A_{j})\,% P(A_{j})}\cdot
  96. B = ( T n ( s ) = x ) B=(T_{n}(s)=x)
  97. A i = ( s = R ( K i ) ) A_{i}=(s=R(K_{i}))
  98. A i A_{i}
  99. K i K_{i}
  100. A i A_{i}
  101. T n ( s ) T_{n}(s)
  102. B B
  103. A i A_{i}
  104. P ( s = R ( K i ) T n ( s ) = x ) = P ( T n ( s ) = x s = R ( K i ) ) P ( s = R ( K i ) ) j P ( T n ( s ) = x s = R ( K j ) ) P ( s = R ( K j ) ) P(s=R(K_{i})\mid T_{n}(s)=x)=\frac{P(T_{n}(s)=x\mid s=R(K_{i}))\,P(s=R(K_{i}))% }{\sum\limits_{j}P(T_{n}(s)=x\mid s=R(K_{j}))\,P(s=R(K_{j}))}\cdot
  105. K i K_{i}
  106. T n ( R ( K i ) ) = x T_{n}(R(K_{i}))=x
  107. P ( T n ( s ) = x s = R ( K i ) ) = 1 P(T_{n}(s)=x\mid s=R(K_{i}))=1
  108. P ( s = R ( K i ) ) = 2 - I ( K i ) P(s=R(K_{i}))=2^{-I(K_{i})}
  109. P ( s = R ( K i ) T n ( s ) = x ) = 2 - I ( K i ) j 2 - I ( K j ) P(s=R(K_{i})\mid T_{n}(s)=x)=\frac{2^{-I(K_{i})}}{\sum\limits_{j}2^{-I(K_{j})}}\cdot
  110. P ( s = R ( K i ) T n ( s ) = x ) = 2 - I ( K i ) j : I ( K j ) < n 2 - I ( K j ) + j : I ( K j ) n 2 - I ( K j ) P(s=R(K_{i})\mid T_{n}(s)=x)=\frac{2^{-I(K_{i})}}{\sum\limits_{j:I(K_{j})<n}2^% {-I(K_{j})}+\sum\limits_{j:I(K_{j})>=n}2^{-I(K_{j})}}\cdot
  111. = j : I ( K j ) < n 2 - I ( K j ) =\sum\limits_{j:I(K_{j})<n}2^{-I(K_{j})}
  112. = j : I ( K j ) n 2 - I ( K j ) =\sum\limits_{j:I(K_{j})>=n}2^{-I(K_{j})}
  113. 2 - n 2^{-n}
  114. P ( s = R ( K i ) T n ( s ) = x ) = 2 - I ( K i ) 2 - n + j : I ( K j ) < n 2 - I ( K j ) P(s=R(K_{i})\mid T_{n}(s)=x)=\frac{2^{-I(K_{i})}}{2^{-n}+\sum\limits_{j:I(K_{j% })<n}2^{-I(K_{j})}}\cdot
  115. P ( random ( s ) T n ( s ) = x ) = 2 - n 2 - n + j : I ( K j ) < n 2 - I ( K j ) P(\operatorname{random}(s)\mid T_{n}(s)=x)=\frac{2^{-n}}{2^{-n}+\sum\limits_{j% :I(K_{j})<n}2^{-I(K_{j})}}\cdot
  116. R ( w ) R(w)
  117. C = R ( c ) C=R(c)
  118. E ( x ) E(x)
  119. x , E ( x ) = { w : R ( w ) x } \forall x,E(x)=\{w:R(w)\equiv x\}
  120. T ( C ) = { t : t C } T(C)=\{t:t\to C\}
  121. P ( A i B ) = P ( B A i ) P ( A i ) j P ( B A j ) P ( A j ) P(A_{i}\mid B)=\frac{P(B\mid A_{i})\,P(A_{i})}{\sum\limits_{j}P(B\mid A_{j})\,% P(A_{j})}\cdot
  122. B = E ( C ) B=E(C)
  123. A i = E ( t ) A_{i}=E(t)
  124. A i A_{i}
  125. T ( C ) T(C)
  126. N T and N M and N M and n E ( N ) and n E ( M ) N\in T\and N\in M\and N\neq M\and n\in E(N)\and n\in E(M)
  127. N M and R ( n ) N and R ( n ) M \implies N\neq M\and R(n)\equiv N\and R(n)\equiv M
  128. false \implies\operatorname{false}
  129. R ( w ) R(w)
  130. t T ( C ) , P ( E ( t ) E ( C ) ) = P ( E ( t ) ) P ( E ( C ) E ( t ) ) j T ( C ) P ( E ( j ) ) P ( E ( C ) E ( j ) ) \forall t\in T(C),P(E(t)\mid E(C))=\frac{P(E(t))\cdot P(E(C)\mid E(t))}{\sum_{% j\in T(C)}P(E(j))\cdot P(E(C)\mid E(j))}
  131. T ( C ) T(C)
  132. t T ( C ) , P ( E ( C ) E ( t ) ) = 1 \forall t\in T(C),P(E(C)\mid E(t))=1
  133. t T ( C ) , P ( E ( t ) ) = n : R ( n ) t 2 - L ( n ) \forall t\in T(C),P(E(t))=\sum_{n:R(n)\equiv t}2^{-L(n)}
  134. t T ( C ) , P ( E ( t ) E ( C ) ) = n : R ( n ) t 2 - L ( n ) j T ( C ) m : R ( m ) j 2 - L ( m ) ) \forall t\in T(C),P(E(t)\mid E(C))=\frac{\sum_{n:R(n)\equiv t}2^{-L(n)}}{\sum_% {j\in T(C)}\sum_{m:R(m)\equiv j}2^{-L(m)})}
  135. t T ( C ) , P ( E ( t ) E ( C ) ) = P ( t C ) \forall t\in T(C),P(E(t)\mid E(C))=P(t\mid C)
  136. t T ( C ) , P ( t C ) = n : R ( n ) t 2 - L ( n ) j T ( C ) m : R ( m ) j 2 - L ( m ) \forall t\in T(C),P(t\mid C)=\frac{\sum_{n:R(n)\equiv t}2^{-L(n)}}{\sum_{j\in T% (C)}\sum_{m:R(m)\equiv j}2^{-L(m)}}
  137. t T ( C ) , P ( t C ) = P ( E ( t ) ) ( j : j T ( C ) and P ( E ( j ) ) > P ( E ( C ) ) P ( E ( j ) ) ) + ( j : j T ( C ) and P ( E ( j ) ) P ( E ( C ) ) P ( j ) ) \forall t\in T(C),P(t\mid C)=\frac{P(E(t))}{(\sum_{j:j\in T(C)\and P(E(j))>P(E% (C))}P(E(j)))+(\sum_{j:j\in T(C)\and P(E(j))\leq P(E(C))}P(j))}
  138. P ( E ( C ) ) = j : j T ( C ) and P ( E ( j ) ) P ( E ( C ) ) P ( j ) P(E(C))=\sum_{j:j\in T(C)\and P(E(j))\leq P(E(C))}P(j)
  139. t T ( C ) , P ( t C ) = P ( E ( t ) ) P ( E ( C ) ) + j : j T ( C ) and P ( E ( j ) ) > P ( E ( C ) ) P ( E ( j ) ) \forall t\in T(C),P(t\mid C)=\frac{P(E(t))}{P(E(C))+\sum_{j:j\in T(C)\and P(E(% j))>P(E(C))}P(E(j))}
  140. random ( C ) \operatorname{random}(C)
  141. P ( random ( C ) C ) = P ( E ( C ) ) P ( E ( C ) ) + j : j T ( C ) and P ( E ( j ) ) > P ( E ( C ) ) P ( E ( j ) ) P(\operatorname{random}(C)\mid C)=\frac{P(E(C))}{P(E(C))+\sum_{j:j\in T(C)\and P% (E(j))>P(E(C))}P(E(j))}
  142. t , P ( E ( t ) ) = n : R ( n ) t 2 - L ( n ) \forall t,P(E(t))=\sum_{n:R(n)\equiv t}2^{-L(n)}
  143. t , P ( F ( t , c ) ) = n : R ( n ) t and L ( n ) < L ( c ) 2 - L ( n ) \forall t,P(F(t,c))=\sum_{n:R(n)\equiv t\and L(n)<L(c)}2^{-L(n)}
  144. t T ( C ) , P ( t C ) = P ( F ( t , c ) ) P ( F ( C , c ) ) + j : j T ( C ) and P ( F ( j , c ) ) > P ( F ( C , c ) ) P ( E ( j , c ) ) \forall t\in T(C),P(t\mid C)=\frac{P(F(t,c))}{P(F(C,c))+\sum_{j:j\in T(C)\and P% (F(j,c))>P(F(C,c))}P(E(j,c))}
  145. P ( random ( C ) C ) = P ( F ( C , c ) ) P ( F ( C , c ) ) + j : j T ( C ) and P ( F ( j , c ) ) > P ( F ( C , c ) ) P ( F ( j , c ) ) P(\operatorname{random}(C)\mid C)=\frac{P(F(C,c))}{P(F(C,c))+\sum_{j:j\in T(C)% \and P(F(j,c))>P(F(C,c))}P(F(j,c))}

Infinite_loop_space_machine.html

  1. X = B S X=BS
  2. B S K ( S ) BS\to K(S)
  3. K ( S ) K(S)

Information_causality.html

  1. n n
  2. n n

Information_distance.html

  1. I D ( x , y ) ID(x,y)
  2. x x
  3. y y
  4. I D ( x , y ) = min { | p | : p ( x ) = y & p ( y ) = x } , ID(x,y)=\min\{|p|:p(x)=y\;\&\;p(y)=x\},
  5. p p
  6. x , y x,y
  7. I D ( x , y ) = E ( x , y ) + O ( log max { K ( x y ) , K ( y x ) } ) ID(x,y)=E(x,y)+O(\log\cdot\max\{K(x\mid y),K(y\mid x)\})
  8. E ( x , y ) = max { K ( x y ) , K ( y x ) } , E(x,y)=\max\{K(x\mid y),K(y\mid x)\},
  9. K ( ) K(\cdot\mid\cdot)
  10. E ( x , y ) E(x,y)
  11. Δ \Delta
  12. D ( x , y ) D(x,y)
  13. x : x y 2 - D ( x , y ) 1 , y : y x 2 - D ( x , y ) 1 , \sum_{x:x\neq y}2^{-D(x,y)}\leq 1,\;\sum_{y:y\neq x}2^{-D(x,y)}\leq 1,
  14. D ( x , y ) = 1 2 D(x,y)=\frac{1}{2}
  15. x y x\neq y
  16. D Δ D\in\Delta
  17. E ( x , y ) D ( x , y ) E(x,y)\leq D(x,y)
  18. E ( x , y ) E(x,y)
  19. O ( log . max { K ( x y ) , K ( y x ) } ) O(\log.\max\{K(x\mid y),K(y\mid x)\})
  20. E ( x , y ) = K ( x y ) E(x,y)=K(x\mid y)
  21. p p
  22. K ( x y ) K(x\mid y)
  23. y y
  24. x x
  25. q q
  26. K ( y x ) - K ( x y ) K(y\mid x)-K(x\mid y)
  27. q p qp
  28. x x
  29. y y
  30. K ( x y ) K ( y x ) K(x\mid y)\leq K(y\mid x)
  31. x x
  32. y y
  33. y y
  34. x x
  35. x x
  36. y y
  37. p p
  38. K ( x y ) K(x\mid y)
  39. O ( log ( max { K ( x y ) , K ( y x ) } ) ) O(\log(\max\{K(x\mid y),K(y\mid x)\}))
  40. y y
  41. x x
  42. x x
  43. K ( p x ) 0 K(p\mid x)\approx 0

Initial_attractiveness.html

  1. Π ( k i ) = k i j k j \Pi\left(k_{i}\right)=\frac{k_{i}}{\sum_{j}k_{j}}
  2. Π ( 0 ) = 0 \Pi(0)=0
  3. Π ( k ) = A + k \Pi(k)=A+k
  4. A A
  5. Π ( k i ) = A i + k i j ( A j + k j ) \Pi(k_{i})=\frac{A_{i}+k_{i}}{\sum\limits_{j}(A_{j}+k_{j})}
  6. Π ( 0 ) A \Pi(0)\sim A
  7. Π ( 0 ) \Pi(0)
  8. p k = C ( k + A ) - γ p_{k}=C\cdot(k+A)^{-\gamma}
  9. γ \gamma
  10. γ = 3 + A m \gamma=3+\frac{A}{m}
  11. m m
  12. γ \gamma
  13. A A

Inserter_category.html

  1. G ( h ) f = g F ( h ) G(h)\circ f=g\circ F(h)

Institutional_complementarity.html

  1. A A
  2. B B
  3. C C
  4. D D
  5. A A
  6. B B
  7. A A
  8. B B
  9. u ( A 1 ; B 1 ) - u ( A 2 ; B 1 ) u ( A 1 ; B 2 ) - u ( A 2 ; B 2 ) u(A^{1};B^{1})-u(A^{2};B^{1})\geq u(A^{1};B^{2})-u(A^{2};B^{2})
  10. u ( B 2 ; A 2 ) - u ( B 1 ; A 2 ) u ( B 2 ; A 1 ) - u ( B 1 ; A 1 ) u(B^{2};A^{2})-u(B^{1};A^{2})\geq u(B^{2};A^{1})-u(B^{1};A^{1})
  11. i i
  12. j j
  13. A A
  14. A < s u p > 1 A<sup>1

Instrumental_magnitude.html

  1. m m
  2. m = - 2.5 log 10 ( f ) m=-2.5\log_{10}(f)
  3. f f
  4. 100 1 / 5 100^{1/5}
  5. 100 1 / 5 = ( 10 2 ) 1 / 5 = 10 2 / 5 = 10 0.4 = 2.51188643 100^{1/5}=(10^{2})^{1/5}=10^{2/5}=10^{0.4}=2.51188643\cdots

Integer-valued_function.html

  1. X X
  2. 𝐙 \mathbf{Z}
  3. f g x : f ( x ) g ( x ) . f\leq g\quad\iff\quad\forall x:f(x)\leq g(x).
  4. 𝐍 < s u p > n \mathbf{N}<sup>n

Integral_closure_of_an_ideal.html

  1. I ¯ \overline{I}
  2. a i I i a_{i}\in I^{i}
  3. r n + a 1 r n - 1 + + a n - 1 r + a n = 0. r^{n}+a_{1}r^{n-1}+\cdots+a_{n-1}r+a_{n}=0.
  4. I ¯ \overline{I}
  5. r M I M rM\subset IM
  6. I ¯ \overline{I}
  7. I = I ¯ I=\overline{I}
  8. [ x , y ] \mathbb{C}[x,y]
  9. x i y d - i x^{i}y^{d-i}
  10. ( x d , y d ) (x^{d},y^{d})
  11. R = k [ X 1 , , X n ] R=k[X_{1},\ldots,X_{n}]
  12. X 1 a 1 X n a n X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}
  13. R [ I t ] = n 0 I n t n R[It]=\oplus_{n\geq 0}I^{n}t^{n}
  14. R [ I t ] R[It]
  15. R [ t ] R[t]
  16. n 0 I n ¯ t n \oplus_{n\geq 0}\overline{I^{n}}t^{n}
  17. I ¯ \overline{I}
  18. I ¯ = I ¯ ¯ \overline{I}=\overline{\overline{I}}
  19. I I
  20. l l
  21. I n + l ¯ I n + 1 \overline{I^{n+l}}\subset I^{n+1}
  22. n 0 n\geq 0
  23. I J I\subset J

Integral_of_inverse_functions.html

  1. f - 1 f^{-1}
  2. f f
  3. f - 1 f^{-1}
  4. f f
  5. I 1 I_{1}
  6. I 2 I_{2}
  7. \mathbb{R}
  8. f : I 1 I 2 f:I_{1}\to I_{2}
  9. f - 1 f^{-1}
  10. I 2 I 1 I_{2}\to I_{1}
  11. f f
  12. f - 1 f^{-1}
  13. F F
  14. f f
  15. f - 1 f^{-1}
  16. f - 1 ( y ) d y = y f - 1 ( y ) - F f - 1 ( y ) + C , \int f^{-1}(y)\,dy=yf^{-1}(y)-F\circ f^{-1}(y)+C,
  17. C C
  18. f - 1 f^{-1}
  19. f ( a ) = c f(a)=c
  20. f ( b ) = d f(b)=d
  21. c d f - 1 ( y ) d y + a b f ( x ) d x = b d - a c . \int_{c}^{d}f^{-1}(y)\,dy+\int_{a}^{b}f(x)\,dx=bd-ac.
  22. f f
  23. f - 1 ( f ( x ) ) = x f^{-1}(f(x))=x
  24. f ( x ) f^{\prime}(x)
  25. x f ( x ) - f ( x ) d x xf(x)-\textstyle\int f(x)\,dx
  26. f f
  27. f - 1 f^{-1}
  28. f - 1 f^{-1}
  29. y y
  30. I 2 I_{2}
  31. y y f - 1 ( y ) - F ( f - 1 ( y ) ) y\to yf^{-1}(y)-F(f^{-1}(y))
  32. f - 1 ( y ) f^{-1}(y)
  33. x I 1 lim h 0 ( x + h ) f ( x + h ) - x f ( x ) - ( F ( x + h ) - F ( x ) ) f ( x + h ) - f ( x ) = x . \forall x\in I_{1}\quad\lim_{h\to 0}\frac{(x+h)f(x+h)-xf(x)-\left(F(x+h)-F(x)% \right)}{f(x+h)-f(x)}=x.
  34. F F
  35. x x
  36. x + h x+h
  37. f f
  38. f ( x ) = exp ( x ) f(x)=\exp(x)
  39. f - 1 ( y ) = ln ( y ) f^{-1}(y)=\ln(y)
  40. ln ( y ) d y = y ln ( y ) - y + C . \int\ln(y)\,dy=y\ln(y)-y+C.
  41. f ( x ) = cos ( x ) f(x)=\cos(x)
  42. f - 1 ( y ) = arccos ( y ) f^{-1}(y)=\arccos(y)
  43. arccos ( y ) d y = y arccos ( y ) - sin ( arccos ( y ) ) + C . \quad\quad\int\arccos(y)\,dy=y\arccos(y)-\sin(\arccos(y))+C.
  44. f ( x ) = tan ( x ) \quad\quad f(x)=\tan(x)
  45. f - 1 ( y ) = arctan ( y ) f^{-1}(y)=\arctan(y)
  46. arctan ( y ) d y = y arctan ( y ) + ln | cos ( arctan ( y ) ) | + C . \quad\quad\int\arctan(y)\,dy=y\arctan(y)+\ln|\cos(\arctan(y))|+C.
  47. f f
  48. f f
  49. U U
  50. V V
  51. \mathbb{C}
  52. f : U V f:U\to V
  53. f f
  54. f - 1 f^{-1}
  55. F F
  56. f f
  57. f - 1 f^{-1}
  58. G ( z ) = z f - 1 ( z ) - F f - 1 ( z ) + C . G(z)=zf^{-1}(z)-F\circ f^{-1}(z)+C.

Integration_along_fibers.html

  1. ( k - m ) (k-m)
  2. π : E B \pi:E\to B
  3. α \alpha
  4. ( π * α ) b ( w 1 , , w k - m ) = π - 1 ( b ) β (\pi_{*}\alpha)_{b}(w_{1},\dots,w_{k-m})=\int_{\pi^{-1}(b)}\beta
  5. β \beta
  6. π - 1 ( b ) \pi^{-1}(b)
  7. m m
  8. β ( v 1 , , v m ) = α ( w 1 ~ , , w k - m ~ , v 1 , , v m ) , w i ~ the lifts of w i . \beta(v_{1},\dots,v_{m})=\alpha(\widetilde{w_{1}},\dots,\widetilde{w_{k-m}},v_% {1},\dots,v_{m}),\quad\widetilde{w_{i}}\,\text{ the lifts of }w_{i}.
  9. b ( π * α ) b b\mapsto(\pi_{*}\alpha)_{b}
  10. π * \pi_{*}
  11. Ω k ( E ) Ω k - m ( B ) \Omega^{k}(E)\to\Omega^{k-m}(B)
  12. π * : H k ( E ) H k - m ( B ) . \pi_{*}:\operatorname{H}^{k}(E)\to\operatorname{H}^{k-m}(B).
  13. π \pi
  14. 0 K Ω * ( E ) π * Ω * ( B ) 0 0\to K\to\Omega^{*}(E)\overset{\pi_{*}}{\to}\Omega^{*}(B)\to 0
  15. H k ( B ) H k + m ( K ) \operatorname{H}^{k}(B)\simeq\operatorname{H}^{k+m}(K)
  16. H k ( B ) 𝛿 H k + m + 1 ( B ) π * H k + m + 1 ( E ) π * H k + 1 ( B ) \dots\rightarrow\operatorname{H}^{k}(B)\overset{\delta}{\to}\operatorname{H}^{% k+m+1}(B)\overset{\pi^{*}}{\rightarrow}\operatorname{H}^{k+m+1}(E)\overset{\pi% _{*}}{\rightarrow}\operatorname{H}^{k+1}(B)\rightarrow\dots
  17. π : M × [ 0 , 1 ] M \pi:M\times[0,1]\to M
  18. M = n M=\mathbb{R}^{n}
  19. x j x_{j}
  20. α = f d x i 1 d x i k + g d t d x j 1 d x j k - 1 . \alpha=f\,dx_{i_{1}}\wedge\dots\wedge dx_{i_{k}}+g\,dt\wedge dx_{j_{1}}\wedge% \dots\wedge dx_{j_{k-1}}.
  21. π * ( α ) = π * ( g d t d x j 1 d x j k - 1 ) = ( 0 1 g ( , t ) d t ) d x j 1 d x j k - 1 . \pi_{*}(\alpha)=\pi_{*}(g\,dt\wedge dx_{j_{1}}\wedge\dots\wedge dx_{j_{k-1}})=% \left(\int_{0}^{1}g(\cdot,t)\,dt\right)\,{dx_{j_{1}}\wedge\dots\wedge dx_{j_{k% -1}}}.
  22. α \alpha
  23. M × I , M\times I,
  24. π * ( d α ) = α 1 - α 0 - d π * ( α ) \pi_{*}(d\alpha)=\alpha_{1}-\alpha_{0}-d\pi_{*}(\alpha)
  25. α i \alpha_{i}
  26. α \alpha
  27. M × { i } M\times\{i\}
  28. f : M × [ 0 , 1 ] N f:M\times[0,1]\to N
  29. h = π * f * h=\pi_{*}\circ f^{*}
  30. d h + h d = f 1 * - f 0 * : Ω k ( N ) Ω k ( M ) , d\circ h+h\circ d=f_{1}^{*}-f_{0}^{*}:\Omega^{k}(N)\to\Omega^{k}(M),
  31. f 1 , f 0 f_{1},f_{0}
  32. f t : U U , x t x f_{t}:U\to U,x\mapsto tx
  33. H k ( U ) = H k ( p t ) \operatorname{H}^{k}(U)=\operatorname{H}^{k}(pt)

Intensity_mapping.html

  1. z 30 z\approx 30
  2. z 6 - 12 z\approx 6-12

Intensity_of_counting_processes.html

  1. λ \lambda
  2. { N ( t ) , t 0 } \{N(t),t\geq 0\}
  3. N ( t ) = M ( t ) + Λ ( t ) N(t)=M(t)+\Lambda(t)
  4. M ( t ) M(t)
  5. Λ ( t ) \Lambda(t)
  6. Λ ( t ) \Lambda(t)
  7. N ( t ) N(t)
  8. λ \lambda
  9. Λ ( t ) = 0 t λ ( s ) d s \Lambda(t)=\int_{0}^{t}\lambda(s)ds
  10. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  11. { N ( t ) , t 0 } \{N(t),t\geq 0\}
  12. { t , t 0 } \{\mathcal{F}_{t},t\geq 0\}
  13. N N
  14. { λ ( t ) , t 0 } \{\lambda(t),t\geq 0\}
  15. λ ( t ) = lim h 0 1 h 𝔼 [ N ( t + h ) - N ( t ) | t ] \lambda(t)=\lim_{h\downarrow 0}\frac{1}{h}\mathbb{E}[N(t+h)-N(t)|\mathcal{F}_{% t}]
  16. λ \lambda
  17. λ ^ \hat{\lambda}
  18. N ( t ) N(t)
  19. t [ 0 , 1 ] t\in[0,1]
  20. n n
  21. N 1 , N 2 , , N n N_{1},N_{2},\ldots,N_{n}
  22. R n ( λ ) = 0 1 λ ( t ) 2 d t - 2 n i = 1 n 0 1 λ ( t ) d N i ( t ) R_{n}(\lambda)=\int_{0}^{1}\lambda(t)^{2}dt-\frac{2}{n}\sum_{i=1}^{n}\int_{0}^% {1}\lambda(t)dN_{i}(t)
  23. λ ( t ) \lambda(t)
  24. [ 0 , 1 ] [0,1]
  25. β = ( β 1 , β 2 , , β m ) \R + m \beta=(\beta_{1},\beta_{2},\ldots,\beta_{m})\in\R_{+}^{m}
  26. λ β = j = 1 m β j λ j , m , λ j , m = m 𝟏 ( j - 1 m , j m ] \lambda_{\beta}=\sum_{j=1}^{m}\beta_{j}\lambda_{j,m},\;\;\;\;\;\;\lambda_{j,m}% =\sqrt{m}\mathbf{1}_{(\frac{j-1}{m},\frac{j}{m}]}
  27. λ j , m \lambda_{j,m}
  28. m \sqrt{m}
  29. L 2 L^{2}
  30. w ^ j \hat{w}_{j}
  31. x > 0 x>0
  32. β w ^ = j = 2 m w ^ j | β j - β j - 1 | \|\beta\|_{\hat{w}}=\sum_{j=2}^{m}\hat{w}_{j}|\beta_{j}-\beta_{j-1}|
  33. β \beta
  34. β ^ = arg min β \R + m { R n ( λ β ) + β w ^ } \hat{\beta}=\arg\min_{\beta\in\R_{+}^{m}}\left\{R_{n}(\lambda_{\beta})+\|\beta% \|_{\hat{w}}\right\}
  35. λ ^ \hat{\lambda}
  36. λ β ^ \lambda_{\hat{\beta}}
  37. L 2 L^{2}
  38. λ ^ - λ \|\hat{\lambda}-\lambda\|
  39. w ^ j ( x ) \hat{w}_{j}(x)
  40. λ ^ - λ 2 inf β \R + m { λ β - λ 2 + 2 β w ^ } \|\hat{\lambda}-\lambda\|^{2}\leq\inf_{\beta\in\R_{+}^{m}}\left\{\|\lambda_{% \beta}-\lambda\|^{2}+2\|\beta\|_{\hat{w}}\right\}
  41. 1 - 12.85 e - x 1-12.85e^{-x}

Interatomic_potential.html

  1. V T O T = i N V 1 ( r i ) + i , j N V 2 ( r i , r j ) + i , j , k N V 3 ( r i , r j , r k ) + V_{TOT}=\sum_{i}^{N}V_{1}(\vec{r}_{i})+\sum_{i,j}^{N}V_{2}(\vec{r}_{i},\vec{r}% _{j})+\sum_{i,j,k}^{N}V_{3}(\vec{r}_{i},\vec{r}_{j},\vec{r}_{k})+\cdots
  2. V 1 \textstyle V_{1}
  3. V 2 \textstyle V_{2}
  4. V 3 \textstyle V_{3}
  5. N \textstyle N
  6. r i \vec{r}_{i}
  7. i < j \textstyle i<j
  8. i < j < k \textstyle i<j<k
  9. r i j = | r i - r j | \textstyle r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|
  10. θ i j k \textstyle\theta_{ijk}
  11. V T O T = i , j N V 2 ( r i j ) + i , j , k N V 3 ( r i j , r i k , θ i j k ) + V_{TOT}=\sum_{i,j}^{N}V_{2}(r_{ij})+\sum_{i,j,k}^{N}V_{3}(r_{ij},r_{ik},\theta% _{ijk})+\cdots
  12. V 3 \textstyle V_{3}
  13. r j k \textstyle r_{jk}
  14. r i j , r i k , θ i j k \textstyle r_{ij},r_{ik},\theta_{ijk}
  15. V ( r ) 0 \textstyle V(r)\equiv 0
  16. r c u t \textstyle r_{cut}
  17. F i = r i V T O T \vec{F}_{i}=\nabla_{\vec{r}_{i}}V_{TOT}
  18. r i j \textstyle r_{ij}
  19. r i \textstyle\vec{r}_{i}
  20. r i \textstyle\nabla_{\vec{r}_{i}}
  21. V L J = 4 ε [ ( σ r ) 12 - ( σ r ) 6 ] V_{LJ}=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma% }{r}\right)^{6}\right]
  22. ε \textstyle\varepsilon
  23. σ \textstyle\sigma
  24. 1 / r 6 \textstyle 1/r^{6}
  25. V ( r ) = D e ( e - 2 a ( r - r e ) - 2 e - a ( r - r e ) ) V(r)=D_{e}(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})})
  26. D e \textstyle D_{e}
  27. r e \textstyle r_{e}
  28. V T O T = i , j N V 2 ( r i j ) + i , j , k N V 3 ( r i j , r i k , θ i j k ) V_{TOT}=\sum_{i,j}^{N}V_{2}(r_{ij})+\sum_{i,j,k}^{N}V_{3}(r_{ij},r_{ik},\theta% _{ijk})
  29. V T O T = i N F i ( j ρ ( r i j ) ) + 1 2 i , j N V 2 ( r i j ) V_{TOT}=\sum_{i}^{N}F_{i}\left(\sum_{j}\rho(r_{ij})\right)+\frac{1}{2}\sum_{i,% j}^{N}V_{2}(r_{ij})
  30. F i \textstyle F_{i}
  31. F i \textstyle\vec{F}_{i}
  32. ρ ( r i j ) \textstyle\rho(r_{ij})
  33. V 2 \textstyle V_{2}
  34. ρ ( r i j ) \textstyle\rho(r_{ij})
  35. V i j ( r i j ) = V r e p u l s i v e ( r i j ) + b i j k V a t t r a c t i v e ( r i j ) V_{ij}(r_{ij})=V_{repulsive}(r_{ij})+b_{ijk}V_{attractive}(r_{ij})
  36. i i
  37. b i j k b_{ijk}
  38. V ( r i j ) = 1 4 π ε 0 Z 1 Z 2 e 2 r i j φ ( r / a ) V(r_{ij})={1\over 4\pi\varepsilon_{0}}{Z_{1}Z_{2}e^{2}\over r_{ij}}\varphi(r/a)
  39. Z 1 Z_{1}
  40. Z 2 Z_{2}

Interchangeability_algorithm.html

  1. O ( n d ( n - l ) * d ) = O ( n 2 d 2 ) O(nd(n-l)*d)=O(n^{2}d^{2})
  2. O ( n k - 1 ) O(n^{k-1})
  3. ( k - 1 ) (k-1)
  4. d k - 1 d^{k-1}
  5. ( k - 1 ) (k-1)
  6. O ( n d n k - l d k - 1 ) = O ( n k d k ) O(ndn^{k-l}d^{k-1})=O(n^{k}d^{k})

Interior_reconstruction.html

  1. x x
  2. [ f g ] = [ A B C D ] [ x y ] . \begin{bmatrix}f\\ g\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}.
  3. X X
  4. Y Y
  5. X X
  6. A A
  7. B B
  8. C C
  9. D D
  10. x x
  11. y y
  12. f f
  13. g g
  14. f f
  15. g g
  16. x x
  17. X X
  18. x X x\in X
  19. y y
  20. Y Y
  21. y Y y\in Y
  22. X X
  23. f f
  24. X X
  25. F F
  26. f F f\in F
  27. g g
  28. F F
  29. Y Y
  30. G G
  31. g G g\in G
  32. C = 0 C=0
  33. A A
  34. B B
  35. C C
  36. D D
  37. [ f g ] = [ A B C D ] [ x y ] \begin{bmatrix}f\\ g\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  38. A A
  39. B B
  40. C C
  41. D D
  42. x x
  43. y y
  44. f f
  45. g g
  46. x x
  47. x x
  48. y y
  49. f f
  50. g g
  51. x x
  52. X X
  53. x X x\in X
  54. X X
  55. y y
  56. X X
  57. Y Y
  58. y Y y\in Y
  59. f f
  60. X X
  61. F F
  62. f F f\in F
  63. g g
  64. F F
  65. Y Y
  66. G G
  67. g G g\in G
  68. C = 0 C=0
  69. x x
  70. A A
  71. B B
  72. C C
  73. D D
  74. G G
  75. g e x g_{ex}
  76. [ x 0 y 0 ] = [ A B C D ] - 1 [ f g e x ] \begin{bmatrix}x_{0}\\ y_{0}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}^{-1}\begin{bmatrix}f\\ g_{ex}\end{bmatrix}
  77. x x
  78. x 0 x_{0}
  79. g e x g_{ex}
  80. g e x | G = f | F g_{ex}|_{\partial G}=f|_{\partial F}
  81. x 0 x_{0}
  82. y 0 y_{0}
  83. g 1 g_{1}
  84. g 1 = C x 0 + D y 0 g_{1}=Cx_{0}+Dy_{0}
  85. [ x 1 y 1 ] = [ A B C D ] - 1 [ f g 1 + g 1 e x ] \begin{bmatrix}x_{1}\\ y_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}^{-1}\begin{bmatrix}f\\ g_{1}+g_{1ex}\end{bmatrix}
  86. f | F = ( g 1 + g 1 e x ) | G f|_{\partial F}=(g_{1}+g_{1ex})|_{\partial G}
  87. x 1 x_{1}
  88. g 1 e x g_{1ex}
  89. x 0 x_{0}
  90. y 0 y_{0}
  91. [ f 1 g 1 ] = [ A B C D ] [ 0 y 0 ] \begin{bmatrix}f_{1}\\ g_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}0\\ y_{0}\end{bmatrix}
  92. f 1 = B y 0 f_{1}=By_{0}
  93. [ x 1 y 1 ] = [ A B C D ] - 1 [ f - f 1 g e x ] \begin{bmatrix}x_{1}\\ y_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}^{-1}\begin{bmatrix}f-f_{1}\\ g_{ex}\end{bmatrix}
  94. g 1 e x g_{1ex}
  95. ( f - f 1 ) | F = g 1 e x | G (f-f_{1})|_{\partial F}=g_{1ex}|_{\partial G}
  96. x 1 x_{1}
  97. f = A x + B y f=Ax+By
  98. B + B^{+}
  99. B B + B = B BB^{+}B=B
  100. Q = [ I - B B + ] Q=[I-BB^{+}]
  101. Q B = 0 QB=0
  102. Q f = Q A x Qf=QAx
  103. x 1 = A + Q + Q f x_{1}=A^{+}Q^{+}Qf
  104. Q Q = Q QQ=Q
  105. Q Q Q = Q QQQ=Q
  106. Q Q
  107. Q Q
  108. Q + = Q Q^{+}=Q
  109. x 1 = A + Q f x_{1}=A^{+}Qf
  110. A + Q = A + [ I - B B + ] A^{+}Q=A^{+}[I-BB^{+}]
  111. [ A B C D ] {\begin{bmatrix}A&B\\ C&D\\ \end{bmatrix}}
  112. A A
  113. min ( R x + S y + T g ) \min(R\|x\|+S\|y\|+T\|g\|)
  114. [ x y ] = [ A B C D ] - 1 [ f g ] \begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}^{-1}\begin{bmatrix}f\\ g\end{bmatrix}
  115. f f
  116. R R
  117. S S
  118. T T
  119. \|\cdot\|
  120. L 0 L_{0}
  121. L 1 L_{1}
  122. L 2 L_{2}
  123. L + L_{+\infty}
  124. x x
  125. | x | > | y | |x|>|y|
  126. | X | > | Y | |X|>|Y|
  127. | x | | y | |x|\sim|y|
  128. | X | | Y | |X|\sim|Y|
  129. | x | | y | |x|\sim|y|
  130. | X | | Y | |X|\sim|Y|
  131. | x | | y | |x|\ll|y|
  132. | X | | Y | |X|\ll|Y|
  133. | y | |y|
  134. | x | | y | |x|\ll|y|
  135. | X | | Y | |X|\ll|Y|
  136. | y | |y|

Interleave_lower_bound.html

  1. I B ( X ) IB(X)
  2. I B ( X ) / 2 - n IB(X)/2-n

Interpolative_decomposition.html

  1. A A
  2. m × n m\times n
  3. r r
  4. A A
  5. A = A ( : , J ) X , A=A_{(:,J)}X,\,
  6. J J
  7. r r
  8. 1 , , n 1,\ldots,n
  9. m × r m\times r
  10. A ( : , J ) A_{(:,J)}
  11. J J
  12. A A
  13. X X
  14. r × n r\times n
  15. X X
  16. r × r r\times r
  17. A A
  18. A A
  19. 3 × 3 3\times 3
  20. A = [ 34 58 52 59 89 80 17 29 26 ] A=\begin{bmatrix}34&58&52\\ 59&89&80\\ 17&29&26\end{bmatrix}
  21. A = [ 58 34 89 59 29 17 ] [ 0 1 0.8788 1 0 0.0303 ] A=\begin{bmatrix}58&34\\ 89&59\\ 29&17\end{bmatrix}\begin{bmatrix}0&1&0.8788\\ 1&0&0.0303\end{bmatrix}

Intersection_(Euclidean_geometry).html

  1. a 1 x + b 1 y = c 1 , a 2 x + b 2 y = c 2 a_{1}x+b_{1}y=c_{1},\ a_{2}x+b_{2}y=c_{2}
  2. ( x s , y s ) (x_{s},y_{s})
  3. x s = c 1 b 2 - c 2 b 1 a 1 b 2 - a 2 b 1 , y s = a 1 c 2 - a 2 c 1 a 1 b 2 - a 2 b 1 . x_{s}=\frac{c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}},\quad y_{s}=\frac{a_% {1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}.
  4. a 1 b 2 - a 2 b 1 = 0 a_{1}b_{2}-a_{2}b_{1}=0
  5. ( x 1 , y 1 ) , ( x 2 , y 2 ) (x_{1},y_{1}),(x_{2},y_{2})
  6. ( x 3 , y 3 ) , ( x 4 , y 4 ) (x_{3},y_{3}),(x_{4},y_{4})
  7. ( x 0 , y 0 ) (x_{0},y_{0})
  8. ( x ( s ) , y ( s ) ) = ( x 1 + s ( x 2 - x 1 ) , y 1 + s ( y 2 - y 1 ) ) , (x(s),y(s))=(x_{1}+s(x_{2}-x_{1}),y_{1}+s(y_{2}-y_{1})),
  9. ( x ( t ) , y ( t ) ) = ( x 3 + t ( x 4 - x 3 ) , y 3 + t ( y 4 - y 3 ) ) . (x(t),y(t))=(x_{3}+t(x_{4}-x_{3}),y_{3}+t(y_{4}-y_{3})).
  10. ( x 0 , y 0 ) (x_{0},y_{0})
  11. s 0 , t 0 s_{0},t_{0}
  12. 0 s 0 , t 0 1 0\leq s_{0},t_{0}\leq 1
  13. s 0 , t 0 s_{0},t_{0}
  14. s ( x 2 - x 1 ) - t ( x 4 - x 3 ) = x 3 - x 1 , s(x_{2}-x_{1})-t(x_{4}-x_{3})=x_{3}-x_{1},
  15. s ( y 2 - y 1 ) - t ( y 4 - y 3 ) = y 3 - y 1 . s(y_{2}-y_{1})-t(y_{4}-y_{3})=y_{3}-y_{1}\ .
  16. 0 s 0 , t 0 1 0\leq s_{0},t_{0}\leq 1
  17. s 0 s_{0}
  18. t 0 t_{0}
  19. ( x 0 , y 0 ) (x_{0},y_{0})
  20. ( 1 , 1 ) , ( 3 , 2 ) (1,1),(3,2)
  21. ( 1 , 4 ) , ( 2 , - 1 ) (1,4),(2,-1)
  22. 2 s - t = 0 2s-t=0
  23. s + 5 t = 3 s+5t=3
  24. s 0 = 3 11 , t 0 = 6 11 s_{0}=\tfrac{3}{11},t_{0}=\tfrac{6}{11}
  25. ( 17 11 , 14 11 ) (\tfrac{17}{11},\tfrac{14}{11})
  26. 0 s 0 , t 0 1 0\leq s_{0},t_{0}\leq 1
  27. a x + b y = c ax+by=c
  28. x 2 + y 2 = r 2 x^{2}+y^{2}=r^{2}
  29. x x
  30. y y
  31. ( x 1 , y 1 ) , ( x 2 , y 2 ) (x_{1},y_{1}),(x_{2},y_{2})
  32. x 1 / 2 = a c ± b r 2 ( a 2 + b 2 ) - c 2 a 2 + b 2 , x_{1/2}=\frac{ac\pm b\sqrt{r^{2}(a^{2}+b^{2})-c^{2}}}{a^{2}+b^{2}}\ ,
  33. y 1 / 2 = b c a r 2 ( a 2 + b 2 ) - c 2 a 2 + b 2 , y_{1/2}=\frac{bc\mp a\sqrt{r^{2}(a^{2}+b^{2})-c^{2}}}{a^{2}+b^{2}}\ ,
  34. r 2 ( a 2 + b 2 ) - c 2 0 . r^{2}(a^{2}+b^{2})-c^{2}\geq 0\ .
  35. r 2 ( a 2 + b 2 ) - c 2 = 0 r^{2}(a^{2}+b^{2})-c^{2}=0
  36. ( x - x 1 ) 2 + ( y - y 1 ) 2 = r 1 2 , ( x - x 2 ) 2 + ( y - y 2 ) 2 = r 2 2 (x-x_{1})^{2}+(y-y_{1})^{2}=r_{1}^{2},\ \quad(x-x_{2})^{2}+(y-y_{2})^{2}=r_{2}% ^{2}
  37. 2 ( x 2 - x 1 ) x + 2 ( y 2 - y 1 ) y = r 1 2 - x 1 2 - y 1 2 - r 2 2 + x 2 2 + y 2 2 . 2(x_{2}-x_{1})x+2(y_{2}-y_{1})y=r_{1}^{2}-x_{1}^{2}-y_{1}^{2}-r_{2}^{2}+x_{2}^% {2}+y_{2}^{2}.
  38. \R 2 \R^{2}
  39. S S
  40. S S
  41. y = f 1 ( x ) , y = f 2 ( x ) y=f_{1}(x),\ y=f_{2}(x)
  42. f 1 ( x ) = f 2 ( x ) . f_{1}(x)=f_{2}(x)\ .
  43. C 1 : ( x 1 ( t ) , y 1 ( t ) ) , C 2 : ( x 2 ( s ) , y 2 ( s ) ) . C_{1}:(x_{1}(t),y_{1}(t)),\ C_{2}:(x_{2}(s),y_{2}(s)).
  44. x 1 ( t ) = x 2 ( s ) , y 1 ( t ) = y 2 ( s ) . x_{1}(t)=x_{2}(s),\ y_{1}(t)=y_{2}(s)\ .
  45. C 1 : ( x 1 ( t ) , y 1 ( t ) ) , C 2 : f ( x , y ) = 0. C_{1}:(x_{1}(t),y_{1}(t)),\ C_{2}:f(x,y)=0.
  46. C 1 C_{1}
  47. f ( x , y ) = 0 f(x,y)=0
  48. C 2 C_{2}
  49. f ( x ( t ) , y ( t ) ) = 0 . f(x(t),y(t))=0\ .
  50. C 1 : f 1 ( x , y ) = 0 , C 2 : f 2 ( x , y ) = 0. C_{1}:f_{1}(x,y)=0,\ C_{2}:f_{2}(x,y)=0.
  51. f 1 ( x , y ) = 0 , f 2 ( x , y ) = 0 . f_{1}(x,y)=0,\ f_{2}(x,y)=0\ .
  52. t t
  53. x x
  54. C 1 : ( t , t 3 ) C_{1}:(t,t^{3})
  55. C 2 : ( x - 1 ) 2 + ( y - 1 ) 2 - 10 = 0 C_{2}:(x-1)^{2}+(y-1)^{2}-10=0
  56. t n + 1 := t n - f ( t n ) f ( t n ) t_{n+1}:=t_{n}-\frac{f(t_{n})}{f^{\prime}(t_{n})}
  57. f ( t ) = ( t - 1 ) 2 + ( t 3 - 1 ) 2 - 10 f(t)=(t-1)^{2}+(t^{3}-1)^{2}-10
  58. C 1 : f 1 ( x , y ) = x 4 + y 4 - 1 = 0 , C_{1}:f_{1}(x,y)=x^{4}+y^{4}-1=0,
  59. C 2 : f 2 ( x , y ) = ( x - 0.5 ) 2 + ( y - 0.5 ) 2 - 1 = 0 C_{2}:f_{2}(x,y)=(x-0.5)^{2}+(y-0.5)^{2}-1=0
  60. ( x n + 1 y n + 1 ) = ( x n + δ x y n + δ y ) {x_{n+1}\choose y_{n+1}}={x_{n}+\delta_{x}\choose y_{n}+\delta_{y}}
  61. ( δ x δ y ) {\delta_{x}\choose\delta_{y}}
  62. ( f 1 x f 1 y f 2 x f 2 y ) ( δ x δ y ) = ( - f 1 - f 2 ) \begin{pmatrix}\frac{\partial f_{1}}{\partial x}&\frac{\partial f_{1}}{% \partial y}\\ \frac{\partial f_{2}}{\partial x}&\frac{\partial f_{2}}{\partial y}\end{% pmatrix}{\delta_{x}\choose\delta_{y}}={-f_{1}\choose-f_{2}}
  63. ( x n , y n ) (x_{n},y_{n})
  64. ( x ( t ) , y ( t ) , z ( t ) ) (x(t),y(t),z(t))
  65. a x + b y + c z = d ax+by+cz=d
  66. a x ( t ) + b y ( t ) + c z ( t ) = d , ax(t)+by(t)+cz(t)=d\ ,
  67. t 0 t_{0}
  68. ( x ( t 0 ) , y ( t 0 ) , z ( t 0 ) ) (x(t_{0}),y(t_{0}),z(t_{0}))
  69. ε i : n i x = d i , i = 1 , 2 \varepsilon_{i}:\ \vec{n}_{i}\cdot\vec{x}=d_{i},\ i=1,2
  70. ε 3 : n 3 x = d 3 \varepsilon_{3}:\ \vec{n}_{3}\cdot\vec{x}=d_{3}
  71. ε i : n i x = d i , i = 1 , 2 , 3 \varepsilon_{i}:\ \vec{n}_{i}\cdot\vec{x}=d_{i},\ i=1,2,3
  72. n 1 , n 2 , n 3 \vec{n}_{1},\vec{n}_{2},\vec{n}_{3}
  73. p 0 = d 1 ( n 2 × n 3 ) + d 2 ( n 3 × n 1 ) + d 3 ( n 1 × n 2 ) n 1 ( n 2 × n 3 ) . \vec{p}_{0}=\frac{d_{1}(\vec{n}_{2}\times\vec{n}_{3})+d_{2}(\vec{n}_{3}\times% \vec{n}_{1})+d_{3}(\vec{n}_{1}\times\vec{n}_{2})}{\vec{n}_{1}\cdot(\vec{n}_{2}% \times\vec{n}_{3})}\ .
  74. n i p 0 = d i , i = 1 , 2 , 3 , \vec{n}_{i}\cdot\vec{p}_{0}=d_{i},\ i=1,2,3,
  75. C : ( x ( t ) , y ( t ) , z ( t ) C:(x(t),y(t),z(t)
  76. S : ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) , S:(x(u,v),y(u,v),z(u,v))\ ,
  77. C : ( x ( t ) , y ( t ) , z ( t ) C:(x(t),y(t),z(t)
  78. S : f ( x , y , z ) = 0 . S:f(x,y,z)=0\ .
  79. C : ( t , t 2 , t 3 ) C:(t,t^{2},t^{3})
  80. S : x 4 + y 4 + z 4 - 1 = 0 S:x^{4}+y^{4}+z^{4}-1=0

Intersection_curve.html

  1. ε i : n i x = d i , i = 1 , 2 , n 1 , n 2 \varepsilon_{i}:\quad\vec{n}_{i}\cdot\vec{x}=d_{i},\quad i=1,2,\quad\vec{n}_{1% },\vec{n}_{2}
  2. x = p + t r \vec{x}=\vec{p}+t\vec{r}
  3. r = n 1 × n 2 \vec{r}=\vec{n}_{1}\times\vec{n}_{2}
  4. P : p P:\vec{p}
  5. ε 1 , ε 2 \varepsilon_{1},\varepsilon_{2}
  6. ε 3 : x = s 1 n 1 + s 2 n 2 \varepsilon_{3}:\vec{x}=s_{1}\vec{n}_{1}+s_{2}\vec{n}_{2}
  7. ε 1 \varepsilon_{1}
  8. ε 2 \varepsilon_{2}
  9. ε 3 \varepsilon_{3}
  10. ε 1 \varepsilon_{1}
  11. ε 2 \varepsilon_{2}
  12. s 1 s_{1}
  13. s 2 s_{2}
  14. P : p = d 1 ( n 2 n 2 ) - d 2 ( n 1 n 2 ) ( n 1 n 1 ) ( n 2 n 2 ) - ( n 1 n 2 ) 2 n 1 + d 2 ( n 1 n 1 ) - d 1 ( n 1 n 2 ) ( n 1 n 1 ) ( n 2 n 2 ) - ( n 1 n 2 ) 2 n 2 . P:\vec{p}=\frac{d_{1}(\vec{n}_{2}\cdot\vec{n}_{2})-d_{2}(\vec{n}_{1}\cdot\vec{% n}_{2})}{(\vec{n}_{1}\cdot\vec{n}_{1})(\vec{n}_{2}\cdot\vec{n}_{2})-(\vec{n}_{% 1}\cdot\vec{n}_{2})^{2}}\vec{n}_{1}+\frac{d_{2}(\vec{n}_{1}\cdot\vec{n}_{1})-d% _{1}(\vec{n}_{1}\cdot\vec{n}_{2})}{(\vec{n}_{1}\cdot\vec{n}_{1})(\vec{n}_{2}% \cdot\vec{n}_{2})-(\vec{n}_{1}\cdot\vec{n}_{2})^{2}}\vec{n}_{2}\ .
  15. ε 1 : x + 2 y + z = 1 , ε 2 : 2 x - 3 y + 2 z = 2 . \varepsilon_{1}:\ x+2y+z=1,\quad\varepsilon_{2}:\ 2x-3y+2z=2\ .
  16. n 1 = ( 1 , 2 , 1 ) , n 2 = ( 2 , - 3 , 2 ) \vec{n}_{1}=(1,2,1)^{\top},\ \vec{n}_{2}=(2,-3,2)^{\top}
  17. r = n 1 × n 2 = ( 7 , 0 , - 7 ) \vec{r}=\vec{n}_{1}\times\vec{n}_{2}=(7,0,-7)^{\top}
  18. P : p P:\vec{p}
  19. p = 1 2 ( 1 , 0 , 1 ) . \vec{p}=\tfrac{1}{2}(1,0,1)^{\top}\ .
  20. x = 1 2 ( 1 , 0 , 1 ) + t ( 7 , 0 , - 7 ) \vec{x}=\tfrac{1}{2}(1,0,1)^{\top}+t(7,0,-7)^{\top}
  21. x = p + s v + t w \vec{x}=\vec{p}+s\vec{v}+t\vec{w}
  22. n = v × w \vec{n}=\vec{v}\times\vec{w}
  23. n x = n p \vec{n}\cdot\vec{x}=\vec{n}\cdot\vec{p}
  24. f 1 ( x , y , z ) = 0 , f 2 ( x , y , z ) = 0 f_{1}(x,y,z)=0,\ f_{2}(x,y,z)=0
  25. x 4 + y 4 + z 4 = 1 x^{4}+y^{4}+z^{4}=1
  26. x 4 + y 4 + z 4 = 1 x^{4}+y^{4}+z^{4}=1
  27. x 4 + y 4 + z 4 = 1 x^{4}+y^{4}+z^{4}=1
  28. x 4 + y 4 + z 4 = 1 x^{4}+y^{4}+z^{4}=1
  29. ( x , y , z ) (x,y,z)
  30. f ( x , y , z ) = 0 f(x,y,z)=0
  31. v \vec{v}
  32. g ( x , y , z ) = f ( x , y , z ) v = 0 g(x,y,z)=\nabla f(x,y,z)\cdot\vec{v}=0
  33. v \vec{v}
  34. f ( x , y , z ) = 0 , g ( x , y , z ) = 0 f(x,y,z)=0,\ g(x,y,z)=0
  35. g g
  36. f ( x , y , z ) = x 4 + y 4 + z 4 - 1 = 0 f(x,y,z)=x^{4}+y^{4}+z^{4}-1=0
  37. x = x ( s , t ) \vec{x}=\vec{x}(s,t)
  38. g ( s , t ) = ( x s ( s , t ) × x t ( s , t ) ) v = 0 g(s,t)=(\vec{x}_{s}(s,t)\times\vec{x}_{t}(s,t))\cdot\vec{v}=0

Intertemporal_budget_constraint.html

  1. t = 0 T x t ( 1 + r ) t t = 0 T w t ( 1 + r ) t , \sum_{t=0}^{T}\frac{x_{t}}{(1+r)^{t}}\leq\sum_{t=0}^{T}\frac{w_{t}}{(1+r)^{t}},
  2. x t x_{t}
  3. w t w_{t}
  4. 1 1 + r \frac{1}{1+r}

Intertemporal_portfolio_choice.html

  1. W T , W_{T},
  2. Utility = ln W T , \,\text{Utility}=\ln W_{T},
  3. W 0 W_{0}
  4. R t . R_{t}.
  5. R t R_{t}
  6. w i t w_{it}
  7. W t - 1 W_{t-1}
  8. W T = W 0 R 1 R 2 R T W_{T}=W_{0}R_{1}R_{2}\cdots R_{T}
  9. R t = w 1 t r 1 t + w 2 t r 2 t + + w n t r n t , R_{t}=w_{1t}r_{1t}+w_{2t}r_{2t}+\cdots+w_{nt}r_{nt},
  10. r i t r_{it}
  11. w i t w_{it}
  12. W T W_{T}
  13. R t R_{t}
  14. W T W_{T}
  15. ln [ W 0 ] + Σ t = 1 T E ln [ w 1 t r 1 t + w 2 t r 2 t + + w n t r n t ] . \ln[W_{0}]+\Sigma_{t=1}^{T}\,\text{E}\ln[w_{1t}r_{1t}+w_{2t}r_{2t}+\cdots+w_{% nt}r_{nt}].
  16. w i t w_{it}
  17. Utility = a W T a \,\text{Utility}=aW_{T}^{a}
  18. R t = w 1 t r 1 t + w 2 t r 2 t + + w n t r n t R_{t}=w_{1t}r_{1t}+w_{2t}r_{2t}+\cdots+w_{nt}r_{nt}
  19. a W 0 a E R 1 a E R 2 a E R T a ; a\cdot W_{0}^{a}\cdot\,\text{E}R_{1}^{a}\cdot\,\text{E}R_{2}^{a}\cdots\,\text{% E}R_{T}^{a};
  20. a W 0 a E [ R 1 a R 2 a R T a ] . a\cdot W_{0}^{a}\cdot\,\text{E}[R_{1}^{a}\cdot R_{2}^{a}\cdots R_{T}^{a}].
  21. E U ( W T ) = E U ( W 0 R 1 R 2 R T ) , \,\text{E}U(W_{T})=\,\text{E}U(W_{0}R_{1}R_{2}\cdots R_{T}),

Inverted_Dirichlet_distribution.html

  1. p ( x 1 , , x k ) = Γ ( ν 1 + + ν k + 1 ) j = 1 k + 1 Γ ( ν j ) x 1 ν 1 - 1 x k ν k - 1 × ( 1 + i = 1 k x i ) - j = 1 k + 1 ν j , x i > 0. p\left(x_{1},\ldots,x_{k}\right)=\frac{\Gamma\left(\nu_{1}+\cdots+\nu_{k+1}% \right)}{\prod_{j=1}^{k+1}\Gamma\left(\nu_{j}\right)}x_{1}^{\nu_{1}-1}\cdots x% _{k}^{\nu_{k}-1}\times\left(1+\sum_{i=1}^{k}x_{i}\right)^{-\sum_{j=1}^{k+1}\nu% _{j}},\qquad x_{i}>0.
  2. E [ i = 1 k x i q i ] = Γ ( ν k + 1 - j = 1 k ν j ) Γ ( ν n + 1 ) j = 1 k Γ ( ν j + q j ) Γ ( ν j ) E\left[\prod_{i=1}^{k}x_{i}^{q_{i}}\right]=\frac{\Gamma\left(\nu_{k+1}-\sum_{j% =1}^{k}\nu_{j}\right)}{\Gamma\left(\nu_{n+1}\right)}\prod_{j=1}^{k}\frac{% \Gamma\left(\nu_{j}+q_{j}\right)}{\Gamma\left(\nu_{j}\right)}
  3. q j > - ν j , 1 j k q_{j}>-\nu_{j},1\leqslant j\leqslant k
  4. ν n + 1 > q 1 + + q k \nu_{n+1}>q_{1}+\ldots+q_{k}

IQ_imbalance.html

  1. y ( t ) = R e { x ( t ) e j 2 π f c t } = x I ( t ) cos ( 2 π f c t ) - x Q ( t ) sin ( 2 π f c t ) w h e r e x ( t ) = x I ( t ) + j x Q ( t ) i s t h e t r a n s m i t t e d b a s e b a n d s i g n a l y(t)=Re\{x(t)e^{j2\pi f_{c}t}\}=x_{I}(t)\cos(2\pi f_{c}t)-x_{Q}(t)\sin(2\pi f_% {c}t)\ \ \ \ \ where\ x(t)=x_{I}(t)+jx_{Q}(t)\ is\ the\ transmitted\ baseband% \ signal
  2. 20 log [ ( 1 + ϵ A ) / ( 1 - ϵ A ) ] 20\log[(1+\epsilon_{A})/(1-\epsilon_{A})]
  3. ε θ \varepsilon_{\theta}
  4. 2 ( 1 + ϵ A ) cos ( 2 π f c t - ε θ / 2 ) , 2(1+\epsilon_{A})\cos(2\pi f_{c}t-\varepsilon_{\theta}/2),
  5. - 2 ( 1 - ϵ A ) cos ( 2 π f c t + ε θ / 2 ) . -2(1-\epsilon_{A})\cos(2\pi f_{c}t+\varepsilon_{\theta}/2).
  6. x I ( t ) = ( 1 + ε A ) [ x I ( t ) cos ( ε θ / 2 ) - x Q ( t ) sin ( ε θ / 2 ) ] , x_{I}(t)=(1+\varepsilon_{A})[x_{I}(t)\cos(\varepsilon_{\theta}/2)-x_{Q}(t)\sin% (\varepsilon_{\theta}/2)],
  7. x Q ( t ) = ( 1 - ε A ) [ x Q ( t ) cos ( ε θ / 2 ) - x I ( t ) sin ( ε θ / 2 ) ] . x_{Q}(t)=(1-\varepsilon_{A})[x_{Q}(t)\cos(\varepsilon_{\theta}/2)-x_{I}(t)\sin% (\varepsilon_{\theta}/2)].
  8. x ( t ) = x I ( t ) + j x Q ( t ) = [ cos ( ε θ / 2 ) + j ε A sin ( ε θ / 2 ) ] x ( t ) + [ ε A cos ( ε θ / 2 ) - j ε A sin ( ε θ / 2 ) ] x * ( t ) = η α x ( t ) + η β x * ( t ) x(t)=x_{I}(t)+jx_{Q}(t)=[\cos(\varepsilon_{\theta}/2)+j\varepsilon_{A}\sin(% \varepsilon_{\theta}/2)]x(t)+[\varepsilon_{A}\cos(\varepsilon_{\theta}/2)-j% \varepsilon_{A}\sin(\varepsilon_{\theta}/2)]x^{*}(t)=\eta_{\alpha}x(t)+\eta_{% \beta}x^{*}(t)
  9. ( . ) * (.)^{*}
  10. X k X_{k}
  11. X k * X^{*}_{k}
  12. ( ( X k , I + j X k , Q ) e j 2 π k f S t ) * = ( X k , I - j X k , Q ) e - j 2 π k f S t = X k * e j 2 π ( - k ) f S t ((X_{k,I+jX_{k,Q}})e^{j2\pi kf_{S}t})^{*}=(X_{k,I}-jX_{k,Q})e^{-j2\pi kf_{S}t}% =X^{*}_{k}e^{j2\pi(-k)f_{S}t}
  13. f S f_{S}
  14. X k = η α X k + η β X - k * X_{k}=\eta_{\alpha}X_{k}+\eta_{\beta}X^{*}_{-k}
  15. z i , n = η α z ( t ) + η β z * ( t ) | t = i ( N + N | g ) T s + N g T s + n T s z_{i,n}=\eta_{\alpha}z(t)+\eta_{\beta}z^{*}(t)|_{t=i(N+N|g)T_{s}+N_{g}T_{s}+nT% _{s}}
  16. η α \eta_{\alpha}
  17. η β \eta_{\beta}
  18. η α \eta_{\alpha}
  19. η β \eta_{\beta}
  20. z i , k = η α H i , k X i , k + η β H i , - k * X i , - k * + V i , k z_{i,k}=\eta_{\alpha}H_{i,k}X_{i,k}+\eta_{\beta}H^{*}_{i,-k}X^{*}_{i,-k}+V_{i,k}
  21. X i , - k X_{i,-k}
  22. { Z i , k ( 0 ) = η α ( 0 ) ( H i , k ( 0 , 0 ) X i , k ( 0 ) + H i , k ( 0 , 1 ) X i , k ( 1 ) ) + η β ( 0 ) ( H i , - k ( 0 , 0 ) X i , - k ( 0 ) + H i , - k ( 0 , 1 ) X i , - k ( 1 ) ) * + V i , k ( 0 ) Z i , k ( 1 ) = η α ( 1 ) ( H i , k ( 1 , 0 ) X i , k ( 0 ) + H i , k ( 1 , 1 ) X i , k ( 1 ) ) + η β ( 1 ) ( H i , - k ( 1 , 0 ) X i , - k ( 0 ) + H i , - k ( 1 , 1 ) X i , - k ( 1 ) ) * + V i , k ( 1 ) \begin{cases}Z^{(0)}_{i,k}=\eta^{(0)}_{\alpha}(H^{(0,0)}_{i,k}X^{(0)}_{i,k}+H^% {(0,1)}_{i,k}X^{(1)}_{i,k})+\eta^{(0)}_{\beta}(H^{(0,0)}_{i,-k}X^{(0)}_{i,-k}+% H^{(0,1)}_{i,-k}X^{(1)}_{i,-k})^{*}+V^{(0)}_{i,k}\\ Z^{(1)}_{i,k}=\eta^{(1)}_{\alpha}(H^{(1,0)}_{i,k}X^{(0)}_{i,k}+H^{(1,1)}_{i,k}% X^{(1)}_{i,k})+\eta^{(1)}_{\beta}(H^{(1,0)}_{i,-k}X^{(0)}_{i,-k}+H^{(1,1)}_{i,% -k}X^{(1)}_{i,-k})^{*}+V^{(1)}_{i,k}\end{cases}
  23. η α ( q ) \eta^{(q)}_{\alpha}
  24. η β ( q ) \eta^{(q)}_{\beta}
  25. η α ( q ) \eta^{(q)}_{\alpha}
  26. η β ( q ) \eta^{(q)}_{\beta}
  27. z i , α ( q ) z^{(q)}_{i,\alpha}
  28. 𝐳 i , α ( 0 ) = [ z i , α 0 ( 0 ) z i , α 1 ( 0 ) z i , α J - 1 ( 0 ) ] = 𝐀 i , α ( 0 ) [ η α ( 0 ) η β ( 0 ) ] + 𝐯 i , α ( 0 ) \mathbf{z}^{(0)}_{i,\alpha}=\begin{bmatrix}z^{(0)}_{i,\alpha 0}\\ z^{(0)}_{i,\alpha 1}\\ \vdots\\ z^{(0)}_{i,\alpha J-1}\end{bmatrix}=\mathbf{A}^{(0)}_{i,\alpha}\begin{bmatrix}% \eta^{(0)}_{\alpha}\\ \eta^{(0)}_{\beta}\end{bmatrix}+\mathbf{v}^{(0)}_{i,\alpha}
  29. 𝐀 i , α ( 0 ) \mathbf{A}^{(0)}_{i,\alpha}
  30. 𝐉 × 2 \mathbf{J}\times 2
  31. 𝐀 i , α ( 0 ) = [ ( H i , α 0 ( 0 , 0 ) X i , α 0 ( 0 ) + H i , α 0 ( 0 , 1 ) X i , α 0 ( 1 ) ) ( H i , α J - 1 ( 0 , 0 ) X i , α J - 1 ( 0 ) + H i , α J - 1 ( 0 , 1 ) X i , α J - 1 ( 1 ) ) * ( H i , α 1 ( 0 , 0 ) X i , α 1 ( 0 ) + H i , α 1 ( 0 , 1 ) X i , α 1 ( 1 ) ) ( H i , α J - 2 ( 0 , 0 ) X i , α J - 2 ( 0 ) + H i , α J - 2 ( 0 , 1 ) X i , α J - 2 ( 1 ) ) * ( H i , α J - 1 ( 0 , 0 ) X i , α J - 1 ( 0 ) + H i , α J - 1 ( 0 , 1 ) X i , α J - 1 ( 1 ) ) ( H i , α 0 ( 0 , 0 ) X i , α 0 ( 0 ) + H i , α 0 ( 0 , 1 ) X i , α 0 ( 1 ) ) * ] \mathbf{A}^{(0)}_{i,\alpha}=\begin{bmatrix}(H^{(0,0)}_{i,\alpha 0}X^{(0)}_{i,% \alpha 0}+H^{(0,1)}_{i,\alpha 0}X^{(1)}_{i,\alpha 0})&(H^{(0,0)}_{i,\alpha_{J-% 1}}X^{(0)}_{i,\alpha_{J-1}}+H^{(0,1)}_{i,\alpha_{J-1}}X^{(1)}_{i,\alpha_{J-1}}% )^{*}\\ (H^{(0,0)}_{i,\alpha 1}X^{(0)}_{i,\alpha 1}+H^{(0,1)}_{i,\alpha 1}X^{(1)}_{i,% \alpha 1})&(H^{(0,0)}_{i,\alpha_{J-2}}X^{(0)}_{i,\alpha_{J-2}}+H^{(0,1)}_{i,% \alpha_{J-2}}X^{(1)}_{i,\alpha_{J-2}})^{*}\\ \vdots&\vdots\\ (H^{(0,0)}_{i,\alpha_{J-1}}X^{(0)}_{i,\alpha_{J-1}}+H^{(0,1)}_{i,\alpha_{J-1}}% X^{(1)}_{i,\alpha_{J-1}})&(H^{(0,0)}_{i,\alpha_{0}}X^{(0)}_{i,\alpha_{0}}+H^{(% 0,1)}_{i,\alpha_{0}}X^{(1)}_{i,\alpha_{0}})^{*}\\ \end{bmatrix}
  32. z ¯ m = η ^ α * z m - η ^ β z m * | η ^ α | 2 - | η ^ β | 2 = η ^ α * | η ^ α | 2 - | η ^ β * | 2 ( z m - η ^ β η ^ α * z m * ) \overline{z}_{m}=\frac{\widehat{\eta}^{*}_{\alpha}z_{m}-\widehat{\eta}_{\beta}% z^{*}_{m}}{|\widehat{\eta}_{\alpha}|^{2}-|\widehat{\eta}_{\beta}|^{2}}=\frac{% \widehat{\eta}^{*}_{\alpha}}{|\widehat{\eta}_{\alpha}|^{2}-|\widehat{\eta}^{*}% _{\beta}|^{2}}(z_{m}-\frac{\widehat{\eta}_{\beta}}{\widehat{\eta}^{*}_{\alpha}% }z^{*}_{m})
  33. η ^ β η ^ α * \frac{\widehat{\eta}_{\beta}}{\widehat{\eta}^{*}_{\alpha}}
  34. η ^ α * / ( | η ^ α | 2 - | η ^ β | 2 ) \widehat{\eta}^{*}_{\alpha}/(|\widehat{\eta}_{\alpha}|^{2}-|\widehat{\eta}_{% \beta}|^{2})
  35. η α \eta_{\alpha}
  36. η β \eta_{\beta}
  37. Z ¯ i , k = η ^ α * Z i , k - η ^ β Z i , k * | η ^ α | 2 - | η ^ β | 2 \overline{Z}_{i,k}=\frac{\widehat{\eta}^{*}_{\alpha}Z_{i,k}-\widehat{\eta}_{% \beta}Z^{*}_{i,k}}{|\widehat{\eta}_{\alpha}|^{2}-|\widehat{\eta}_{\beta}|^{2}}
  38. z i , k = η α H i , k X i , k + η β H i , - k * X i , - k * + V i , k z_{i,k}=\eta_{\alpha}H_{i,k}X_{i,k}+\eta_{\beta}H^{*}_{i,-k}X^{*}_{i,-k}+V_{i,k}
  39. η α \eta_{\alpha}
  40. η β \eta_{\beta}
  41. Z i , k = η α H i , k + V i , k k = 1 , , N / 2 - 1 Z_{i,k}=\eta_{\alpha}H_{i,k}+V_{i,k}\ \ \ \ k=1,\cdots,N/2-1
  42. Z i , - k = η β H i , k * + V i , - k k = 1 , , N / 2 - 1 Z_{i,-k}=\eta_{\beta}H^{*}_{i,k}+V_{i,-k}\ \ \ \ k=1,\cdots,N/2-1
  43. η β η α * \frac{\eta_{\beta}}{\eta^{*}_{\alpha}}
  44. Z i , - k / Z i , k * Z_{i,-k}/Z^{*}_{i,k}
  45. η β η α * \frac{\eta_{\beta}}{\eta^{*}_{\alpha}}
  46. η β η α * \frac{\eta_{\beta}}{\eta^{*}_{\alpha}}

Isentropic_expansion_waves.html

  1. α = s i n - 1 ( 1 M ) \alpha=sin^{-1}(\frac{1}{M})
  2. v v
  3. ( v + d v ) (v+dv)
  4. d θ d\theta
  5. t C V ρ d V + C S ρ v ¯ . d A ¯ = 0 \frac{\partial}{\partial t}\int\limits_{CV}\rho dV+\int\limits_{CS}\rho\bar{v}% .d\bar{A}=0
  6. - ρ v sin α A + ( ρ + d ρ ) ( v + d v ) sin ( a - d θ ) A = 0 {-\rho v\sin\alpha A}+{(\rho+d\rho)(v+dv)\sin(a-d\theta)A}=0
  7. ρ v sin α = ( ρ + d ρ ) ( v + d v ) sin ( α - d θ ) \rho v\sin\alpha=(\rho+d\rho)(v+dv)\sin(\alpha-d\theta)
  8. y y
  9. F S y + F B y = t C V v y ρ d V + C S v y ρ v ¯ . d A ¯ F_{S_{y}}+F_{B_{y}}=\frac{\partial}{\partial t}\int\limits_{CV}v_{y}\rho dV+% \int\limits_{CS}v_{y}\rho\bar{v}.d\bar{A}
  10. 0 = v cos α ( - ρ v sin α A ) + ( v + d v ) cos ( α - d θ ) ( ρ + d ρ ) ( v + d v ) sin ( α - d θ ) A 0=v\cos\alpha(-\rho v\sin\alpha A)+(v+dv)\cos(\alpha-d\theta){(\rho+d\rho)(v+% dv)\sin(\alpha-d\theta)A}
  11. v cos α = ( v + d v ) cos ( α - d θ ) v\cos\alpha=(v+dv)\cos(\alpha-d\theta)
  12. d θ 0 d\theta\rightarrow 0
  13. cos d θ 1 \cos{d\theta}\rightarrow 1
  14. sin d θ d θ \sin{d\theta}\rightarrow d\theta
  15. d θ = - d v v tan α d\theta=\frac{-dv}{v\tan\alpha}
  16. sin α = 1 M \sin\alpha=\frac{1}{M}
  17. tan α = 1 ( M 2 - 1 ) \tan\alpha=\frac{1}{\sqrt{(M^{2}-1)}}
  18. d θ = - ( M 2 - 1 ) d v v d\theta=-\frac{\sqrt{(M^{2}-1)}dv}{v}
  19. x x
  20. Q ˙ - W ˙ s - W ˙ s h e a r - W ˙ o t h e r = t C V e ρ d V + C S h ρ v ¯ . d A ¯ \dot{Q}-\dot{W}_{s}-\dot{W}_{shear}-\dot{W}_{other}=\frac{\partial}{\partial t% }\int\limits_{CV}e\rho dV+\int\limits_{CS}h\rho\bar{v}.d\bar{A}
  21. e = u + v 2 2 + g z e=u+\frac{v^{2}}{2}+gz
  22. 0 = [ h + v 2 2 ] ( - ρ v sin α A ) + [ ( h + d h ) + ( v + d v ) 2 2 ] [ ( ρ + d ρ ) ( v + d v ) sin ( α - d θ ) A ] 0=[{h+\frac{v^{2}}{2}}](-\rho v\sin\alpha A)+[(h+dh)+\frac{(v+dv)^{2}}{2}][(% \rho+d\rho)(v+dv)\sin(\alpha-d\theta)A]
  23. h + v 2 2 = ( h + d h ) + ( v + d v ) 2 2 {h+\frac{v^{2}}{2}}=(h+dh)+\frac{(v+dv)^{2}}{2}
  24. d h = - v d v dh=-vdv
  25. d h = c p d T dh=c_{p}dT
  26. c p d T = - v d v c_{p}dT=-vdv
  27. M M
  28. v v
  29. v = M c = M k R T v=Mc=M\sqrt{kRT}
  30. v v
  31. k R T \sqrt{kRT}
  32. d v v = d M M + d T 2 T \frac{dv}{v}=\frac{dM}{M}+\frac{dT}{2T}
  33. d v v = d M M - v d v 2 c p T = d M M - d v v 2 c p T 2 v = d M M - d v M 2 c 2 c p T 2 v \frac{dv}{v}=\frac{dM}{M}-\frac{vdv}{2c_{p}T}=\frac{dM}{M}-\frac{dv\frac{v^{2}% }{c_{p}T}}{2v}=\frac{dM}{M}-\frac{dv\frac{M^{2}c^{2}}{c_{p}T}}{2v}
  34. d M M - d v M 2 k R T c p T 2 v = d M M - d v [ M 2 ( k - 1 ) ] 2 v \frac{dM}{M}-\frac{dv\frac{M^{2}kRT}{c_{p}T}}{2v}=\frac{dM}{M}-\frac{dv[M^{2}(% k-1)]}{2v}
  35. d v v = 2 2 + M 2 ( k - 1 ) d M M \frac{dv}{v}=\frac{2}{2+M^{2}(k-1)}\frac{dM}{M}
  36. d θ 2 ( M 2 - 1 ) = - 1 2 + M 2 ( k - 1 ) d M M \frac{d\theta}{2\sqrt{(M^{2}-1)}}=-\frac{1}{2+M^{2}(k-1)}\frac{dM}{M}
  37. d θ d\theta
  38. d ω = d θ d\omega=d\theta
  39. M = 1 M=1
  40. M M
  41. ω \omega
  42. M = 1 M=1
  43. 0 ω d ω = 1 M 2 ( M 2 - 1 ) 2 + M 2 ( k - 1 ) d M M \int\limits_{0}^{\omega}d\omega=\int\limits_{1}^{M}\frac{2\sqrt{(M^{2}-1)}}{2+% M^{2}(k-1)}\frac{dM}{M}
  44. ω = ( k + 1 k - 1 ) t a n - 1 [ k - 1 k + 1 ( M 2 - 1 ) ] - t a n - 1 ( M 2 - 1 ) \omega=\sqrt{(\frac{k+1}{k-1})}\ tan^{-1}[\frac{\sqrt{k-1}}{\sqrt{k+1}}(M^{2}-% 1)]-\ tan^{-1}(M^{2}-1)

Isentropic_nozzle_flow.html

  1. ρ \rho
  2. q n e t + h + V 2 2 = w n e t + h o + V o 2 2 q_{net}+h+\frac{\vec{V}^{2}}{2}=w_{net}+h_{o}+\frac{\vec{V_{o}}^{2}}{2}
  3. ρ A V = ( ρ + d ρ ) ( A + d A ) ( V + d V ) \rho AV=(\rho+d\rho)(A+dA)(V+dV)
  4. d V V + d A A + d ρ ρ = 0 \frac{dV}{V}+\frac{dA}{A}+\frac{d\rho}{\rho}=0
  5. V 2 2 + k k - 1 . p ρ = ( V + d V ) 2 2 + k k - 1 . p + d p ρ + d ρ \frac{V^{2}}{2}+\frac{k}{k-1}.\frac{p}{\rho}=\frac{(V+dV)^{2}}{2}+\frac{k}{k-1% }.\frac{p+dp}{\rho+d\rho}
  6. V d V + k k - 1 . ρ d p - p d ρ ρ 2 = 0 VdV+\frac{k}{k-1}.\frac{\rho dp-pd\rho}{\rho^{2}}=0
  7. V d V + k . p ρ 2 . d ρ = 0 VdV+\frac{k.p}{\rho^{2}}.d\rho=0
  8. d V V . ( ρ V 2 k p - 1 ) = d A A \frac{dV}{V}.(\frac{\rho V^{2}}{kp}-1)=\frac{dA}{A}
  9. d V V . ( M 2 - 1 ) = d A A \frac{dV}{V}.(M^{2}-1)=\frac{dA}{A}
  10. c p . T o = V 2 2 + c p . T c_{p}.T_{o}=\frac{V^{2}}{2}+c_{p}.T
  11. T o T = 1 + k - 1 2 M 2 \frac{T_{o}}{T}=1+\frac{k-1}{2}M^{2}
  12. p 0 p = ( 1 + k - 1 2 M 2 ) ( k k - 1 ) \frac{p_{0}}{p}=(1+\frac{k-1}{2}M^{2})^{(}\frac{k}{k-1})
  13. ρ o ρ = ( 1 + k - 1 2 M 2 ) ( 1 k - 1 ) \frac{\rho_{o}}{\rho}=(1+\frac{k-1}{2}M^{2})^{(}\frac{1}{k-1})
  14. T * T o = 2 k + 1 \frac{T^{*}}{T_{o}}=\frac{2}{k+1}
  15. p * p o = ( 2 k + 1 ) ( k k - 1 ) \frac{p^{*}}{p_{o}}=(\frac{2}{k+1})^{(}\frac{k}{k-1})
  16. ρ * ρ o = ( 2 k + 1 ) ( 1 k - 1 ) \frac{\rho^{*}}{\rho_{o}}=(\frac{2}{k+1})^{(}\frac{1}{k-1})
  17. m ˙ = ρ A V = p R T × A × M k R T = p k R T A M \dot{m}=\rho AV=\frac{p}{RT}\times A\times M\sqrt{kRT}=p\sqrt{k\over RT}AM
  18. m ˙ = p o M A k R T o ( 1 + k - 1 2 M 2 ) k + 1 - 2 ( k - 1 ) \dot{m}=p_{o}MA\sqrt{\frac{k}{RT_{o}}}\left(1+\frac{k-1}{2}M^{2}\right)^{\frac% {k+1}{-2(k-1)}}
  19. m ˙ = p o A * k R T o ( 1 + k - 1 2 ) k + 1 - 2 ( k - 1 ) \dot{m}=p_{o}A^{*}\sqrt{\frac{k}{RT_{o}}}\left(1+\frac{k-1}{2}\right)^{\frac{k% +1}{-2(k-1)}}
  20. A A * = 1 M ( 2 + ( k - 1 ) . M 2 k + 1 ) k + 1 2 ( k - 1 ) \frac{A}{A^{*}}=\frac{1}{M}\left(\frac{2+(k-1).M^{2}}{k+1}\right)^{\frac{k+1}{% 2(k-1)}}
  21. p r = 0.5283 p 0 p_{r}=0.5283p_{0}
  22. p r p_{r}
  23. M e = 1 M_{e}=1
  24. p 0 p_{0}
  25. p e p_{e}
  26. p r p_{r}
  27. p e p_{e}
  28. p r p_{r}
  29. A A

Isomultiflorenol_synthase.html

  1. \rightleftharpoons

Isopimara-7,15-diene_synthase.html

  1. \rightleftharpoons

Isotypical_representation.html

  1. π : G ( ) \pi:G\longrightarrow\mathcal{B}(\mathcal{H})
  2. π \pi
  3. π ( G ) ′′ \pi(G)^{{}^{\prime\prime}}
  4. π : G ( ) \pi:G\longrightarrow\mathcal{B}(\mathcal{H})
  5. \mathcal{H}
  6. \mathcal{H}

Isovalent_hybridization.html

  1. λ λ
  2. n n
  3. n = λ 2 \ n=\lambda^{2}
  4. i i 1 + λ i 2 = 1 \sum_{i}\frac{i}{1+\lambda_{i}^{2}}=1
  5. i λ i 2 1 + λ i 2 = 1 , 2 o r 3 \sum_{i}\frac{\lambda_{i}^{2}}{1+\lambda_{i}^{2}}=1,2or3
  6. i i
  7. j j
  8. 1 + λ i λ j cos θ i = 0 \ 1+\lambda_{i}\lambda_{j}\cos\theta_{i}=0
  9. 1 + λ i 2 cos θ i i = 0 \ 1+\lambda_{i}^{2}\cos\theta_{ii}=0
  10. J = 500 1 + λ i 2 \ J=\frac{500}{1+\lambda_{i}^{2}}
  11. J J

István_Fenyő.html

  1. L 2 L^{2}
  2. g ( x ) - 0 K ( x - t ) g ( t ) d t = f ( x ) g(x)-\int_{0}^{\infty}K(x-t)g(t)\,dt=f(x)
  3. f ( x ) f(x)
  4. g ( x ) g(x)
  5. f f
  6. f ( x + y ) ( f ( x ) + f ( y ) - 1 ) = f ( x ) f ( y ) f(x+y)(f(x)+f(y)-1)=f(x)f(y)
  7. E E
  8. f ( x + y ) - f ( x ) - f ( y ) = d ( x , y ) f(x+y)-f(x)-f(y)=d(x,y)
  9. x , y x,y\in\mathbb{R}
  10. d : × E d:\mathbb{R}\times\mathbb{R}\rightarrow E
  11. f ( ψ ( x , y ) ) = f ( x ) + f ( y ) f(\psi(x,y))=f(x)+f(y)
  12. ψ ( x , y ) \psi(x,y)
  13. ψ ( x , y ) = x y ( β γ - α ) + ( x + y ) α β ( 1 - γ ) + α β ( α γ - β ) x y ( γ - 1 ) + ( x + y ) ( β - α γ ) + α 2 γ - β 2 \psi(x,y)=\dfrac{xy(\beta\gamma-\alpha)+(x+y)\alpha\beta(1-\gamma)+\alpha\beta% (\alpha\gamma-\beta)}{xy(\gamma-1)+(x+y)(\beta-\alpha\gamma)+\alpha^{2}\gamma-% \beta^{2}}
  14. α , β , γ \alpha,\beta,\gamma\in\mathbb{R}
  15. γ ( β - α ) 0 \gamma(\beta-\alpha)\neq 0
  16. f ( x ) = c log | γ ( x - α ) / ( x - β ) | f(x)=c\log|\gamma(x-\alpha)/(x-\beta)|
  17. c 0 c\neq 0
  18. ψ ( x , y ) = ( α λ + 1 ) x y - α 2 γ ( x + y ) + α 2 ( α λ - 1 ) λ x y - ( α λ - 1 ) ( x + y ) + α ( α λ - 2 ) \psi(x,y)=\dfrac{(\alpha\lambda+1)xy-\alpha^{2}\gamma(x+y)+\alpha^{2}(\alpha% \lambda-1)}{\lambda xy-(\alpha\lambda-1)(x+y)+\alpha(\alpha\lambda-2)}
  19. α , γ \alpha,\gamma\in\mathbb{R}
  20. f ( x ) = c ( 1 / ( α - x ) - λ ) f(x)=c(1/(\alpha-x)-\lambda)
  21. c 0 c\neq 0
  22. ( ( r y ′′ ) + q y ) - p y = 0 ((ry^{\prime\prime})+qy)^{\prime}-py=0
  23. p , q , r C ( a , ) p,q,r\in C(a,\infty)
  24. r ( x ) > 0 r(x)>0
  25. x x
  26. h n h_{n}
  27. n n
  28. h n ( u ) , s ψ = u , t h n ( ψ ) \langle h_{n}(u),s\psi\rangle=\langle u,th_{n}(\psi)\rangle
  29. u 𝒟 + u\in\mathcal{D}^{+}
  30. ψ H k \psi\in H_{k}
  31. 𝒟 + \mathcal{D}^{+}
  32. H k H_{k}
  33. \mathbb{Z}
  34. H k H_{k}

Iterated_forcing.html

  1. ω 1 \omega_{1}
  2. ω 1 \omega_{1}
  3. ω 1 \omega_{1}
  4. ω 1 \omega_{1}
  5. ω 1 \omega_{1}

Iterative_impedance.html

  1. Z IT = Z + Y Z IT Z_{\mathrm{IT}}=Z+Y\parallel Z_{\mathrm{IT}}
  2. Z IT = Z 2 ± Z 2 4 + Z Y Z_{\mathrm{IT}}={Z\over 2}\pm\sqrt{{Z^{2}\over 4}+{Z\over Y}}
  3. Y IT = Y 2 ± Y 2 4 + Y Z Y_{\mathrm{IT}}={Y\over 2}\pm\sqrt{{Y^{2}\over 4}+{Y\over Z}}
  4. Y IT = 1 Z IT Y_{\mathrm{IT}}={1\over Z_{\mathrm{IT}}}
  5. Z IT = Z 2 + Z IM Z_{\mathrm{IT}}={Z\over 2}+Z_{\mathrm{IM}}

J-multiplicity.html

  1. ( R , 𝔪 ) (R,\mathfrak{m})
  2. d > 0 d>0
  3. j ( I ) = j ( gr I R ) j(I)=j(\operatorname{gr}_{I}R)
  4. j ( gr I R ) j(\operatorname{gr}_{I}R)
  5. Γ 𝔪 ( gr I R ) \Gamma_{\mathfrak{m}}(\operatorname{gr}_{I}R)
  6. Γ 𝔪 \Gamma_{\mathfrak{m}}
  7. 𝔪 \mathfrak{m}

Jackknife_Variance_Estimates_for_Random_Forest.html

  1. V ( x ) = V a r [ θ ^ ( x ) ] V(x)=Var[\hat{\theta}^{\infty}(x)]
  2. V ^ j = n - 1 n i = 1 n ( θ ^ ( - i ) - θ ¯ ) 2 \hat{V}_{j}=\frac{n-1}{n}\sum_{i=1}^{n}(\hat{\theta}_{(-i)}-\overline{\theta})% ^{2}
  3. V ^ j = n - 1 n i = 1 n ( t ¯ ( - i ) ( x ) - t ¯ ( x ) ) 2 \hat{V}_{j}=\frac{n-1}{n}\sum_{i=1}^{n}(\overline{t}^{\star}_{(-i)}(x)-% \overline{t}^{\star}(x))^{2}
  4. t t^{\star}
  5. t ( - i ) t^{\star}_{(-i)}
  6. i t h ith
  7. E r r o r R a t e = 1 N i = 1 N j = 1 M y i j , ErrorRate=\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{M}y_{ij},
  8. y i j y_{ij}
  9. i t h ith
  10. l o g l o s s = 1 N i = 1 N j = 1 M y i j l o g ( p i j ) logloss=\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{M}y_{ij}log(p_{ij})
  11. y i j y_{ij}
  12. i t h ith
  13. p i j p_{ij}
  14. i t h ith
  15. j j
  16. V I J V_{IJ}^{\infty}
  17. V J V_{J}^{\infty}
  18. E [ V ^ I J B ] - V ^ I J n b = 1 B ( t b - t ¯ ) 2 B E[\hat{V}_{IJ}^{B}]-\hat{V}_{IJ}^{\infty}\approx\frac{n\sum_{b=1}^{B}(t_{b}^{% \star}-\bar{t}^{\star})^{2}}{B}
  19. V ^ I J - U B = V ^ I J B - n b = 1 B ( t b - t ¯ ) 2 B \hat{V}_{IJ-U}^{B}=\hat{V}_{IJ}^{B}-\frac{n\sum_{b=1}^{B}(t_{b}^{\star}-\bar{t% }^{\star})^{2}}{B}
  20. V ^ J - U B = V ^ J B - ( e - 1 ) n b = 1 B ( t b - t ¯ ) 2 B \hat{V}_{J-U}^{B}=\hat{V}_{J}^{B}-(e-1)\frac{n\sum_{b=1}^{B}(t_{b}^{\star}-% \bar{t}^{\star})^{2}}{B}

Jade_Mirror_of_the_Four_Unknowns.html

  1. ( a + b - c ) (a+b-c)
  2. a b ab
  3. ( a + b - c ) a b = 24 (a+b-c)ab=24
  4. b + c = 9 b+c=9
  5. x = a x=a
  6. x 5 - 9 x 4 - 81 x 3 + 729 x 2 = 3888 x^{5}-9x^{4}-81x^{3}+729x^{2}=3888
  7. - 2 y 2 - x y 2 + 2 x y + 2 x 2 y + x 3 = 0 -2y^{2}-xy^{2}+2xy+2x^{2}y+x^{3}=0
  8. 2 y 2 - x y 2 + 2 x y + x 3 = 0 2y^{2}-xy^{2}+2xy+x^{3}=0
  9. 8 x + 4 x 2 = 0 8x+4x^{2}=0
  10. 2 x 2 + x 3 = 0 2x^{2}+x^{3}=0
  11. x 2 - 2 x - 8 = 0 x^{2}-2x-8=0
  12. x = 4 x=4
  13. - y - z - y 2 x - x + x y z = 0 -y-z-y^{2}x-x+xyz=0
  14. - y - z + x - x 2 + x z = 0 -y-z+x-x^{2}+xz=0
  15. y 2 - z 2 + x 2 = 0 ; y^{2}-z^{2}+x^{2}=0;
  16. - x - 2 x 2 + y + y 2 + x y - x y 2 + x 2 y -x-2x^{2}+y+y^{2}+xy-xy^{2}+x^{2}y
  17. - 2 x - 2 x 2 + 2 y - 2 y 2 + y 3 + 4 x y - 2 x y 2 + x y 2 -2x-2x^{2}+2y-2y^{2}+y^{3}+4xy-2xy^{2}+xy^{2}
  18. x 4 - 6 x 3 + 4 x 2 + 6 x - 5 = 0 x^{4}-6x^{3}+4x^{2}+6x-5=0
  19. x = 5 x=5
  20. - 2 y + x + z = 0 -2y+x+z=0
  21. - y 2 x + 4 y + 2 x - x 2 + 4 z + x z = 0 -y^{2}x+4y+2x-x^{2}+4z+xz=0
  22. x 2 + y 2 - z 2 = 0 x^{2}+y^{2}-z^{2}=0
  23. 2 y - w + 2 x = 0 2y-w+2x=0
  24. 4 x 2 - 7 x - 686 = 0 4x^{2}-7x-686=0
  25. 16 x 1 0 - 64 x 9 + 160 x 8 - 384 x 7 + 512 x 6 - 544 x 5 + 456 x 4 + 126 x 3 + 3 x 2 - 4 x - 177162 = 0 16x^{1}0-64x^{9}+160x^{8}-384x^{7}+512x^{6}-544x^{5}+456x^{4}+126x^{3}+3x^{2}-% 4x-177162=0
  26. x 4 + 480 * x 3 - 270000 * x 2 + 15552000 * x + 1866240000 = 0 x^{4}+480*x^{3}-270000*x^{2}+15552000*x+1866240000=0
  27. x 4 + 360 * x 3 - 270000 * x 2 + 20736000 * x + 1866240000 = 0 x^{4}+360*x^{3}-270000*x^{2}+20736000*x+1866240000=0
  28. n * a + 1 2 * 1 * n * ( n - 1 ) * b + 1 3 * 2 * 1 * n * ( n - 1 ) * ( n - 2 ) * c n*a+\frac{1}{2*1}*n*(n-1)*b+\frac{1}{3*2*1}*n*(n-1)*(n-2)*c
  29. + 1 4 * 3 * 2 * 1 n * ( n - 1 ) * ( n - 2 ) * ( n - 3 ) * d +\frac{1}{4*3*2*1}n*(n-1)*(n-2)*(n-3)*d
  30. 1 + 3 + 6 + 10 + + 1+3+6+10+...+
  31. 1 2 1\over 2
  32. n ( n + 1 ) n(n+1)
  33. v = 2 + 9 + 24 + 50 + 90 + 147 + 224 + v=2+9+24+50+90+147+224+
  34. 1 2 1\over 2
  35. n ( n + 1 ) 2 n(n+1)^{2}
  36. v = v=
  37. 1 2 * 3 * 4 1\over 2*3*4
  38. ( 3 x + 5 ) * x * ( x + 1 ) * ( x + 2 ) (3x+5)*x*(x+1)*(x+2)
  39. v = 1320 v=1320
  40. 3 * x 4 + 14 x 3 + 21 x 2 + 10 x - 31680 = 0 3*x^{4}+14x^{3}+21x^{2}+10x-31680=0
  41. x = n = 9 x=n=9
  42. v = 2 + 9 + 24 + 50 + 90 + 147 + 224 + 324 + 450 = 1320 v=2+9+24+50+90+147+224+324+450=1320
  43. - 3 * y 2 + 8 * y - 8 * x + 8 * z = 0 -3*y^{2}+8*y-8*x+8*z=0
  44. 4 * y 2 - 8 * x * y + 3 * x 2 - 8 * y * z + 6 * x * z + 3 * z 2 = 0 4*y^{2}-8*x*y+3*x^{2}-8*y*z+6*x*z+3*z^{2}=0
  45. y 2 + x 2 - z 2 = 0 y^{2}+x^{2}-z^{2}=0
  46. 2 * y + 4 * x + 2 * z - w = 0 2*y+4*x+2*z-w=0

Jaimovich–Rebelo_preferences.html

  1. C t C_{t}
  2. N t N_{t}
  3. t t
  4. u ( C t , N t ) = ( C t - ψ N t θ X t ) 1 - σ - 1 1 - σ , u\left({C_{t},N_{t}}\right)=\frac{\left(C_{t}-\psi N_{t}^{\theta}X_{t}\right)^% {1-\sigma}-1}{1-\sigma},
  5. X t = C t γ X t - 1 1 - γ . X_{t}=C_{t}^{\gamma}X_{t-1}^{1-\gamma}.
  6. θ > 1 \theta>1
  7. ψ > 0 \psi>0
  8. σ > 0 \sigma>0
  9. U U
  10. U = E 0 t = 0 β t u ( C t , N t ) , U=E_{0}\sum_{t=0}^{\infty}\beta^{t}u\left({C_{t},N_{t}}\right),
  11. E 0 E_{0}
  12. X t X_{t}
  13. γ = 1 \gamma=1
  14. X t X_{t}
  15. X t = C t , X_{t}=C_{t},
  16. u ( C t , N t ) = ( C t ( 1 - ψ N t θ ) ) 1 - σ - 1 1 - σ , u\left({C_{t},N_{t}}\right)=\frac{\left(C_{t}\left(1-\psi N_{t}^{\theta}\right% )\right)^{1-\sigma}-1}{1-\sigma},
  17. γ 0 \gamma\rightarrow 0
  18. X t X_{t}
  19. X t = X > 0 , X_{t}=X>0,
  20. u ( C t , N t ) = ( C t - ψ X N t θ ) 1 - σ - 1 1 - σ , u\left({C_{t},N_{t}}\right)=\frac{\left(C_{t}-\psi XN_{t}^{\theta}\right)^{1-% \sigma}-1}{1-\sigma},
  21. 0 < γ 1 0<\gamma\leq 1
  22. 0 < γ 1 0<\gamma\leq 1
  23. X t X_{t}
  24. z t z_{t}
  25. C t C_{t}
  26. X t X_{t}
  27. z t z_{t}
  28. γ 0 \gamma\rightarrow 0
  29. X t z t \frac{X_{t}}{z_{t}}
  30. X t z t = X t - 1 z t - 1 z t - 1 z t , \frac{X_{t}}{z_{t}}=\frac{X_{t-1}}{z_{t-1}}\frac{z_{t-1}}{z_{t}},
  31. X t = X z t , X_{t}=Xz_{t},
  32. X > 0 X>0
  33. u ( C t , N t ) = ( C t - z t ψ X N t θ ) 1 - σ - 1 1 - σ , u\left({C_{t},N_{t}}\right)=\frac{\left(C_{t}-z_{t}\psi XN_{t}^{\theta}\right)% ^{1-\sigma}-1}{1-\sigma},

James'_space.html

  1. 𝒫 \mathcal{P}
  2. x = ( x n ) x=(x_{n})
  3. p = ( p 1 , p 2 , , p 2 n + 1 ) 𝒫 p=(p_{1},p_{2},\ldots,p_{2n+1})\in\mathcal{P}
  4. x p := ( x p 2 n + 1 2 + m = 1 n ( x p 2 m - 1 - x p 2 m ) 2 ) 1 / 2 . \|x\|_{p}:=\left(x_{p_{2n+1}}^{2}+\sum_{m=1}^{n}(x_{p_{2m-1}}-x_{p_{2m}})^{2}% \right)^{1/2}.
  5. sup { x p : p 𝒫 } < \sup\{\|x\|_{p}:p\in\mathcal{P}\}<\infty
  6. x := sup { x p : p 𝒫 } ( x 𝐉 ) \|x\|:=\sup\{\|x\|_{p}:p\in\mathcal{P}\}\ (x\in\mathbf{J})

James_E._Humphreys.html

  1. ( 2 , p ) (2,p)

James_Maynard_(mathematician).html

  1. m m
  2. m m
  3. m m
  4. m m
  5. m m
  6. m m
  7. lim inf n ( p n + 1 - p n ) 600 , \liminf_{n\to\infty}\left(p_{n+1}-p_{n}\right)\leq 600,

Janet_basis.html

  1. x x
  2. y y
  3. δ \delta
  4. δ 1 \delta_{1}
  5. δ 2 \delta_{2}
  6. θ \theta
  7. δ θ δ \delta\leq\theta\delta
  8. δ 1 δ 2 δ δ 1 δ δ 2 \delta_{1}\leq\delta_{2}\rightarrow\delta\delta_{1}\leq\delta\delta_{2}
  9. δ 2 \delta_{2}
  10. δ 1 \delta_{1}
  11. δ 2 > δ 1 \delta_{2}>\delta_{1}
  12. z z
  13. x x
  14. y y
  15. x > y x>y
  16. L E X LEX
  17. z x x > z x y > z x > z y y > z y > z z_{xx}>z_{xy}>z_{x}>z_{yy}>z_{y}>z
  18. G R L E X GRLEX
  19. z x x > z x y > z y y > z x > z y > z z_{xx}>z_{xy}>z_{yy}>z_{x}>z_{y}>z
  20. x z = z x , y z = z y , \partial_{x}z=z_{x},\partial_{y}z=z_{y},\ldots
  21. T O P TOP
  22. P O T POT
  23. e 1 e_{1}
  24. e 2 e_{2}
  25. e 1 e_{1}
  26. e 2 e_{2}
  27. e 1 e_{1}
  28. e 1 e_{1}
  29. e 2 e_{2}
  30. S { e 1 , e 2 , } S\equiv\{e_{1},e_{2},\ldots\}
  31. S S
  32. S := Autoreduce ( S ) S:=\operatorname{Autoreduce}(S)
  33. S := CompleteSystem ( S ) S:=\operatorname{CompleteSystem}(S)
  34. v i v_{i}
  35. e i e_{i}
  36. v j v_{j}
  37. e j e_{j}
  38. x i k x_{i_{k}}
  39. x j 1 , , x j l x_{j_{1}},\ldots,x_{j_{l}}
  40. v i x i k = p 1 + + p l v j x j 1 p 1 x j l p l \frac{\partial v_{i}}{\partial x_{i_{k}}}=\frac{\partial^{p_{1}+\cdots+p_{l}}v% _{j}}{\partial x_{j_{1}}^{p_{1}}\cdots\partial x_{j_{l}}^{p_{l}}}
  41. c i , j = Lcoef ( e j ) e i x i k - Lcoef ( e i ) p 1 + + p l e j x j 1 p 1 x j l p l c_{i,j}=\operatorname{Lcoef}(e_{j})\cdot\frac{\partial e_{i}}{\partial x_{i_{k% }}}-\operatorname{Lcoef}(e_{i})\cdot\frac{\partial^{p_{1}+\cdots+p_{l}}e_{j}}{% \partial x_{j_{1}}^{p_{1}}\cdots\partial x_{j_{l}}^{p_{l}}}
  42. c i , j c_{i,j}
  43. c i , j := Reduce ( c i , j , S ) c_{i,j}:=\operatorname{Reduce}(c_{i,j},S)
  44. c i , j c_{i,j}
  45. S S
  46. S := S { c i , j | c i , j 0 } S:=S\cup\{c_{i,j}|c_{i,j}\neq 0\}
  47. S S
  48. A u t o r e d u c e Autoreduce
  49. C o m p l e t i o n Completion
  50. { e 1 z x y - x 2 y 2 z x - x - y y 2 z = 0 , e 2 z x + 1 x z y + x z = 0 } \{e_{1}\equiv z_{xy}-\frac{x^{2}}{y^{2}}z_{x}-\frac{x-y}{y^{2}}z=0,e_{2}\equiv z% _{x}+\frac{1}{x}z_{y}+xz=0\}
  51. G R L E X GRLEX
  52. x > y x>y
  53. { e 3 z y y + 1 y 2 ( x y 3 - x 2 - y ) z y - 1 y ( x 3 - x + y ) z = 0 , e 2 = z x + 1 y z y + x z = 0 } . \{e_{3}\equiv z_{yy}+\frac{1}{y^{2}}(xy^{3}-x^{2}-y)z_{y}-\frac{1}{y}(x^{3}-x+% y)z=0,e_{2}=z_{x}+\frac{1}{y}z_{y}+xz=0\}.
  54. c 3 , 2 e 3 x - 2 e 2 y 2 c_{3,2}\equiv\frac{\partial e_{3}}{\partial x}-\frac{\partial^{2}e_{2}}{% \partial y^{2}}
  55. z = 0 z=0
  56. { z = 0 } \{z=0\}
  57. z = 0 z=0
  58. w w
  59. z z
  60. x x
  61. y y
  62. { f 1 w x x - 2 z x y - 1 2 x w x + 1 2 x 2 w = 0 , f 2 w x y - 1 2 z y y - 1 2 x w y - 6 x 2 z x , \{f_{1}\equiv w_{xx}-2z_{xy}-\frac{1}{2x}w_{x}+\frac{1}{2x^{2}}w=0,f_{2}\equiv w% _{xy}-\frac{1}{2}z_{yy}-\frac{1}{2x}w_{y}-6x^{2}z_{x},
  63. f 3 w y y + 4 x 2 w x - 8 x 2 z y - 8 x w = 0 , f 4 z x x + 1 2 x z x = 0 } f_{3}\equiv w_{yy}+4x^{2}w_{x}-8x^{2}z_{y}-8xw=0,f_{4}\equiv z_{xx}+\frac{1}{2% x}z_{x}=0\}
  64. G R L E X , w > z , x > y GRLEX,w>z,x>y
  65. c 1 , 2 f 1 y - f 2 x and c 2 , 3 f 2 y - f 3 x . c_{1,2}\equiv\frac{\partial f_{1}}{\partial y}-\frac{\partial f_{2}}{\partial x% }\,\text{ and }c_{2,3}\equiv\frac{\partial f_{2}}{\partial y}-\frac{\partial f% _{3}}{\partial x}.
  66. c 1 , 2 = z x y y - 6 x z x = 0 , c 2 , 3 = z y y y + 3 x 2 z x y - 24 x z y - 12 w = 0. c_{1,2}=z_{xyy}-6xz_{x}=0,c_{2,3}=z_{yyy}+3x^{2}z_{xy}-24xz_{y}-12w=0.
  67. { z y + 1 2 x w = 0 , z x = 0 , w y = 0 , w x - 1 x w = 0 } \{z_{y}+\frac{1}{2x}w=0,z_{x}=0,w_{y}=0,w_{x}-\frac{1}{x}w=0\}
  68. z = C 1 - C 2 x , w = 2 C 2 y z=C_{1}-C_{2}x,w=2C_{2}y
  69. C 1 C_{1}
  70. C 2 C_{2}

Jean-Louis_Nicolas.html

  1. φ ( n ) < e - γ n log log n \varphi(n)<e^{-\gamma}\frac{n}{\log\log n}
  2. φ ( n ) \varphi(n)

Jet_mill.html

  1. V = 2 9 ( ρ p - ρ f ) μ g R 2 V=\frac{2}{9}\frac{\left(\rho_{p}-\rho_{f}\right)}{\mu}g\,R^{2}

Jiles-Atherton_model.html

  1. M M
  2. H H
  3. H e H\text{e}
  4. α \alpha
  5. M M
  6. M an M\text{an}
  7. H e H\text{e}
  8. M M
  9. H H
  10. α \alpha
  11. a a
  12. M s M\text{s}
  13. k k
  14. c c
  15. K an K\text{an}
  16. J / m 3 J/m^{3}
  17. ψ \psi
  18. H H
  19. t t
  20. H e H\text{e}
  21. H e = H + α M H\text{e}=H+\alpha M
  22. M an = ( 1 - t ) M aniso + t M ananiso M\text{an}=(1-t)M\text{an}\text{iso}+tM\text{an}\text{aniso}
  23. M aniso M\text{an}\text{iso}
  24. H e H\text{e}
  25. M aniso = M s ( coth ( H e / a ) - ( a / H e ) ) M\text{an}\text{iso}=M\text{s}(\coth(H\text{e}/a)-(a/H\text{e}))
  26. M ananiso M\text{an}\text{aniso}
  27. M ananiso M\text{an}\text{aniso}
  28. M ananiso = M s 0 π e E ( 1 ) + E ( 2 ) sin ( θ ) cos ( θ ) d θ 0 π e E ( 1 ) + E ( 2 ) sin ( θ ) d θ M\text{an}\text{aniso}=M\text{s}\frac{\int_{0}^{\pi}\!e^{E(1)+E(2)}\sin(\theta% )\cos(\theta)\,d\theta}{\int_{0}^{\pi}\!e^{E(1)+E(2)}\sin(\theta)\,d\theta}
  29. E ( 1 ) = H e a cos θ - K an M s μ 0 a sin 2 ( ψ - θ ) E(1)=\frac{H\text{e}}{a}\cos\theta-\frac{K\text{an}}{M\text{s}\mu_{0}a}\sin^{2% }(\psi-\theta)
  30. E ( 2 ) = H e a cos θ - K an M s μ 0 a sin 2 ( ψ + θ ) E(2)=\frac{H\text{e}}{a}\cos\theta-\frac{K\text{an}}{M\text{s}\mu_{0}a}\sin^{2% }(\psi+\theta)
  31. K an = 0 ) K\text{an}=0)
  32. M ananiso M\text{an}\text{aniso}
  33. M aniso M\text{an}\text{iso}
  34. M ananiso M\text{an}\text{aniso}
  35. M ananiso = M s 0 π e 0.5 ( E ( 1 ) + E ( 2 ) ) sin ( θ ) cos ( θ ) d θ 0 π e 0.5 ( E ( 1 ) + E ( 2 ) ) sin ( θ ) d θ M\text{an}\text{aniso}=M\text{s}\frac{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(% \theta)\cos(\theta)\,d\theta}{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(\theta)\,% d\theta}
  36. M ananiso M\text{an}\text{aniso}
  37. d M d H = 1 1 + c M an - M δ k - α ( M an - M ) + c 1 + c d M an d H \frac{dM}{dH}=\frac{1}{1+c}\frac{M\text{an}-M}{\delta k-\alpha(M\text{an}-M)}+% \frac{c}{1+c}\frac{dM\text{an}}{dH}
  38. δ \delta
  39. H H
  40. δ = 1 \delta=1
  41. δ = - 1 \delta=-1
  42. B B
  43. B ( H ) = μ 0 M ( H ) B(H)=\mu_{0}M(H)
  44. μ 0 \mu_{0}
  45. M ananiso M\text{an}\text{aniso}
  46. M ( H ) M(H)
  47. M ananiso M\text{an}\text{aniso}
  48. M ( H ) M(H)

Job_characteristic_theory.html

  1. MPS = Skill Variety + Task Identity + Task Significance 3 x Autonomy x Feedback {\,\text{MPS}}=\frac{\,\text{Skill Variety + Task Identity + Task Significance% }}{\,\text{3}}{\,\text{ x Autonomy x Feedback}}

Joel_Feldman.html

  1. λ Φ 3 4 \lambda\cdot{\Phi}^{4}_{3}
  2. Φ 4 4 \Phi^{4}_{4}

John_Rust.html

  1. x t x_{t}
  2. t t
  3. c ( x t , θ ) c(x_{t},\theta)
  4. θ \theta
  5. R C RC
  6. β \beta
  7. U ( x t , ξ t , d , θ ) = { - c ( x t , θ ) + ξ t , k e e p , - R C - c ( 0 , θ ) + ξ t , r e p l a c e , = u ( x t , d , θ ) + { ξ t , k e e p , if d = keep , ξ t , r e p l a c e , if d = replace , U(x_{t},\xi_{t},d,\theta)=\begin{cases}-c(x_{t},\theta)+\xi_{t,keep},&\\ -RC-c(0,\theta)+\xi_{t,replace},&\end{cases}=u(x_{t},d,\theta)+\begin{cases}% \xi_{t,keep},&\textrm{if}\;\;d=\,\text{keep},\\ \xi_{t,replace},&\textrm{if}\;\;d=\,\text{replace},\end{cases}
  8. d d
  9. ξ t , k e e p \xi_{t,keep}
  10. ξ t , r e p l a c e \xi_{t,replace}
  11. ξ t , k e e p \xi_{t,keep}
  12. ξ t , r e p l a c e \xi_{t,replace}
  13. ξ t , \xi_{t,\bullet}
  14. ξ t - 1 , \xi_{t-1,\bullet}
  15. x t x_{t}
  16. V ( x , ξ , θ ) = max d = keep , replace { u ( x , d , θ ) + ξ d + V ( x , ξ , θ ) q ( d ξ | x , θ ) p ( d x | x , d , θ ) } V(x,\xi,\theta)=\max_{d=\,\text{keep},\,\text{replace}}\left\{u(x,d,\theta)+% \xi_{d}+\int\int V(x^{\prime},\xi^{\prime},\theta)q(d\xi^{\prime}|x^{\prime},% \theta)p(dx^{\prime}|x,d,\theta)\right\}
  17. p ( d x | x , d , θ ) p(dx^{\prime}|x,d,\theta)
  18. q ( d ξ | x , θ ) q(d\xi^{\prime}|x^{\prime},\theta)
  19. q ( d ξ | x , θ ) q(d\xi^{\prime}|x^{\prime},\theta)
  20. d d
  21. P ( d | x , θ ) = exp { u ( x , d , θ ) + β E V ( x , d , θ ) } d D ( x ) exp { u ( x , d , θ ) + β E V ( x , d , θ ) } P(d|x,\theta)={\exp\{u(x,d,\theta)+\beta EV(x,d,\theta)\}\over\sum_{d^{\prime}% \in D(x)}\exp\{u(x,d^{\prime},\theta)+\beta EV(x,d^{\prime},\theta)\}}
  22. E V ( x , d , θ ) EV(x,d,\theta)
  23. E V ( x , d , θ ) = [ log ( d = keep , replace exp { u ( x , d , θ ) + β E V ( x , d , θ ) } ) ] p ( x | x , d , θ ) . EV(x,d,\theta)=\int\left[\log\left(\sum_{d=\,\text{keep},\,\text{replace}}\exp% \{u(x,d^{\prime},\theta)+\beta EV(x^{\prime},d^{\prime},\theta)\}\right)\right% ]p(x^{\prime}|x,d,\theta).
  24. x t x_{t}
  25. E V ( x , d , θ ) EV(x,d,\theta)
  26. θ \theta
  27. E V ( x , d , θ ) EV(x,d,\theta)
  28. θ \theta
  29. ( x , d ) (x,d)
  30. E V ( x , d , θ ) EV(x,d,\theta)
  31. P ( d | x , θ ) P(d|x,\theta)
  32. θ \theta
  33. L ( θ ) = i = 1 N t = 1 T i log ( P ( d i t | x i t , θ ) ) + log ( p ( x i t | x i t - 1 , d i t - 1 , θ ) ) , L(\theta)=\sum_{i=1}^{N}\sum_{t=1}^{T_{i}}\log(P(d_{it}|x_{it},\theta))+\log(p% (x_{it}|x_{it-1},d_{it-1},\theta)),
  34. x i , t x_{i,t}
  35. d i , t d_{i,t}
  36. i = 1 , , N i=1,\dots,N
  37. t = 1 , , T i t=1,\dots,T_{i}
  38. θ \theta
  39. L ( θ ) L(\theta)
  40. θ \theta

Jordan_map.html

  1. a i a^{\dagger}_{i}
  2. a i a_{i}
  3. [ a i , a j ] a i a j - a j a i = δ i j , [a_{i},a^{\dagger}_{j}]\equiv a_{i}a^{\dagger}_{j}-a^{\dagger}_{j}a_{i}=\delta% _{ij},
  4. [ a i , a j ] = [ a i , a j ] = 0 , [a^{\dagger}_{i},a^{\dagger}_{j}]=[a_{i},a_{j}]=0,
  5. [ , ] [\ \ ,\ \ ]
  6. δ i j \delta_{ij}
  7. N = i n i = i a i a i N=\sum_{i}n_{i}=\sum_{i}a^{\dagger}_{i}a_{i}
  8. M M
  9. 𝐌 M i , j a i 𝐌 i j a j , {\mathbf{M}}\qquad\longmapsto\qquad M\equiv\sum_{i,j}a^{\dagger}_{i}{\mathbf{M% }}_{ij}a_{j}~{},
  10. M M
  11. 𝐌 \mathbf{M}
  12. J 𝐚 σ 2 𝐚 , {\vec{J}}\equiv{\mathbf{a}}^{\dagger}\cdot\frac{{\vec{\sigma}}}{2}\cdot{% \mathbf{a}}~{},
  13. J 2 J J = N 2 ( N 2 + 1 ) . J^{2}\equiv{\vec{J}}\cdot{\vec{J}}=\frac{N}{2}\left(\frac{N}{2}+1\right)~{}.
  14. J 2 a 1 k a 2 n | 0 = k + n 2 ( k + n 2 + 1 ) a 1 k a 2 n | 0 , J^{2}~{}a^{\dagger k}_{1}a^{\dagger n}_{2}|0\rangle=\frac{k+n}{2}\left(\frac{k% +n}{2}+1\right)~{}a^{\dagger k}_{1}a^{\dagger n}_{2}|0\rangle~{},
  15. J z a 1 k a 2 n | 0 = 1 2 ( k - n ) a 1 k a 2 n | 0 , J_{z}~{}a^{\dagger k}_{1}a^{\dagger n}_{2}|0\rangle=\frac{1}{2}\left(k-n\right% )~{}a^{\dagger k}_{1}a^{\dagger n}_{2}|0\rangle~{},
  16. j = ( k + n ) / 2 , m = ( k n ) / 2 j=(k+n)/2,m=(k−n)/2
  17. | j , m |j,m〉
  18. b i b^{\dagger}_{i}
  19. b i b_{i}
  20. { , } \{\ \ ,\ \ \}
  21. { b i , b j } b i b j + b j b i = δ i j , \{b_{i},b^{\dagger}_{j}\}\equiv b_{i}b^{\dagger}_{j}+b^{\dagger}_{j}b_{i}=% \delta_{ij},
  22. { b i , b j } = { b i , b j } = 0. \{b^{\dagger}_{i},b^{\dagger}_{j}\}=\{b_{i},b_{j}\}=0.
  23. i j i\neq j

Josef_Kittler.html

  1. h i j ( x ) h^{j}_{i}(x)
  2. x x
  3. c j c_{j}
  4. h i h_{i}
  5. H j ( x ) = m e d i a n ( ( h i j ( x ) ) H^{j}(x)=median((h^{j}_{i}(x))
  6. i i

Joseph_A._Thas.html

  1. M 3 ( K ) M_{3}(K)

Jost_function.html

  1. - ψ ′′ + V ψ = k 2 ψ -\psi^{\prime\prime}+V\psi=k^{2}\psi
  2. ψ ( k , r ) \psi(k,r)
  3. = 0 \ell=0
  4. - ψ ′′ + V ψ = k 2 ψ . -\psi^{\prime\prime}+V\psi=k^{2}\psi.
  5. φ ( k , r ) \varphi(k,r)
  6. φ ( k , 0 ) \displaystyle\varphi(k,0)
  7. 0 r | V ( r ) | < \int_{0}^{\infty}r|V(r)|<\infty
  8. φ ( k , r ) = k - 1 sin ( k r ) + k - 1 0 r d r sin ( k ( r - r ) ) V ( r ) φ ( k , r ) . \varphi(k,r)=k^{-1}\sin(kr)+k^{-1}\int_{0}^{r}dr^{\prime}\sin(k(r-r^{\prime}))% V(r^{\prime})\varphi(k,r^{\prime}).
  9. f ± f_{\pm}
  10. f ± = e ± i k r + o ( 1 ) f_{\pm}=e^{\pm ikr}+o(1)
  11. r r\to\infty
  12. f ± ( k , r ) = e ± i k r - k - 1 r d r sin ( k ( r - r ) ) V ( r ) f ± ( k , r ) . f_{\pm}(k,r)=e^{\pm ikr}-k^{-1}\int_{r}^{\infty}dr^{\prime}\sin(k(r-r^{\prime}% ))V(r^{\prime})f_{\pm}(k,r^{\prime}).
  13. k 0 k\neq 0
  14. f + , f - f_{+},f_{-}
  15. φ \varphi
  16. ω ( k ) := W ( f + , φ ) φ r ( k , r ) f + ( k , r ) - φ ( k , r ) f + , x ( k , r ) \omega(k):=W(f_{+},\varphi)\equiv\varphi_{r}^{\prime}(k,r)f_{+}(k,r)-\varphi(k% ,r)f_{+,x}^{\prime}(k,r)
  17. f + , φ f_{+},\varphi
  18. r = 0 r=0
  19. φ \varphi
  20. ω ( k ) = f + ( k , 0 ) \omega(k)=f_{+}(k,0)
  21. [ - 2 r 2 + V ( r ) - k 2 ] G = - δ ( r - r ) . \left[-\frac{\partial^{2}}{\partial r^{2}}+V(r)-k^{2}\right]G=-\delta(r-r^{% \prime}).
  22. G + ( k ; r , r ) = - φ ( k , r r ) f + ( k , r r ) ω ( k ) , G^{+}(k;r,r^{\prime})=-\frac{\varphi(k,r\wedge r^{\prime})f_{+}(k,r\vee r^{% \prime})}{\omega(k)},
  23. r r min ( r , r ) r\wedge r^{\prime}\equiv\min(r,r^{\prime})
  24. r r max ( r , r ) r\vee r^{\prime}\equiv\max(r,r^{\prime})

K-anonymity.html

  1. 4 \mathcal{E}_{4}
  2. k = 1 k=1

K-convex_function.html

  1. ( s , S ) (s,S)
  2. s s
  3. S S
  4. S s S\leq s
  5. s s
  6. S S
  7. g : g:\mathbb{R}\rightarrow\mathbb{R}
  8. g ( u ) + z [ g ( u ) - g ( u - b ) b ] g ( u + z ) + K g(u)+z\left[\frac{g(u)-g(u-b)}{b}\right]\leq g(u+z)+K
  9. u , z 0 , u,z\geq 0,
  10. b > 0 b>0
  11. g : g:\mathbb{R}\rightarrow\mathbb{R}
  12. g ( λ x + λ ¯ y ) λ g ( x ) + λ ¯ [ g ( y ) + K ] g(\lambda x+\bar{\lambda}y)\leq\lambda g(x)+\bar{\lambda}[g(y)+K]
  13. x y , λ [ 0 , 1 ] x\leq y,\lambda\in[0,1]
  14. λ ¯ = 1 - λ \bar{\lambda}=1-\lambda
  15. a 0 a\geq 0
  16. ( x , f ( x ) ) (x,f(x))
  17. ( y , f ( y ) + a ) (y,f(y)+a)
  18. ( λ x + λ ¯ y , f ( λ x + λ ¯ y ) ) , 0 λ 1 (\lambda x+\bar{\lambda}y,f(\lambda x+\bar{\lambda}y)),0\leq\lambda\leq 1
  19. g g
  20. ( x , g ( x ) ) (x,g(x))
  21. ( y , g ( y ) + K ) (y,g(y)+K)
  22. y x y\geq x
  23. λ = z / ( b + z ) , x = u - b , y = u + z . \lambda=z/(b+z),\quad x=u-b,\quad y=u+z.
  24. g : g:\mathbb{R}\rightarrow\mathbb{R}
  25. L K L\geq K
  26. g g
  27. K 0 K\geq 0
  28. g 1 g_{1}
  29. g 2 g_{2}
  30. α 0 , β 0 , g = α g 1 + β g 2 \alpha\geq 0,\beta\geq 0,\;g=\alpha g_{1}+\beta g_{2}
  31. ( α K + β L ) (\alpha K+\beta L)
  32. g g
  33. ξ \xi
  34. E | g ( x - ξ ) | < E|g(x-\xi)|<\infty
  35. x x
  36. E g ( x - ξ ) Eg(x-\xi)
  37. g : g:\mathbb{R}\rightarrow\mathbb{R}
  38. g ( y ) g(y)\rightarrow\infty
  39. | y | |y|\rightarrow\infty
  40. s s
  41. S S
  42. s S s\leq S
  43. g ( S ) g ( y ) g(S)\leq g(y)
  44. y y\in\mathbb{R}
  45. g ( S ) + K = g ( s ) < g ( y ) g(S)+K=g(s)<g(y)
  46. y < s y<s
  47. g ( y ) g(y)
  48. ( - , s ) (-\infty,s)
  49. g ( y ) g ( z ) + K g(y)\leq g(z)+K
  50. y , z y,z
  51. s y z s\leq y\leq z

K-D-B-tree.html

  1. log 2 ( N / M ) \log_{2}(N/M)
  2. M M

K-epsilon_turbulence_model.html

  1. ϵ \epsilon
  2. ( ρ k ) t + ( ρ k u i ) x i = x j [ μ t σ k k x j ] + 2 μ t E i j E i j - ρ ϵ \frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{i% }}=\frac{\partial}{\partial x_{j}}\left[\frac{\mu_{t}}{\sigma_{k}}\frac{% \partial k}{\partial x_{j}}\right]+2{\mu_{t}}{E_{ij}}{E_{ij}}-\rho\epsilon
  3. ϵ \epsilon
  4. ( ρ ϵ ) t + ( ρ ϵ u i ) x i = x j [ μ t σ ϵ ϵ x j ] + C 1 ϵ ϵ k 2 μ t E i j E i j - C 2 ϵ ρ ϵ 2 k \frac{\partial(\rho\epsilon)}{\partial t}+\frac{\partial(\rho\epsilon u_{i})}{% \partial x_{i}}=\frac{\partial}{\partial x_{j}}\left[\frac{\mu_{t}}{\sigma_{% \epsilon}}\frac{\partial\epsilon}{\partial x_{j}}\right]+C_{1\epsilon}\frac{% \epsilon}{k}2{\mu_{t}}{E_{ij}}{E_{ij}}-C_{2\epsilon}\rho\frac{\epsilon^{2}}{k}
  5. u i u_{i}
  6. E i j E_{ij}
  7. μ t \mu_{t}
  8. μ t = ρ C μ k 2 ϵ \mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon}
  9. σ k \sigma_{k}
  10. σ ϵ \sigma_{\epsilon}
  11. C 1 ϵ C_{1\epsilon}
  12. C 2 ϵ C_{2\epsilon}
  13. C μ = 0.09 C_{\mu}=0.09
  14. σ k = 1.00 \sigma_{k}=1.00
  15. σ ϵ = 1.30 \sigma_{\epsilon}=1.30
  16. C 1 ϵ = 1.44 C_{1\epsilon}=1.44
  17. C 2 ϵ = 1.92 C_{2\epsilon}=1.92

K-space_(functional_analysis).html

  1. V V
  2. 0 X V 0. 0\rightarrow\mathbb{R}\rightarrow X\rightarrow V\rightarrow 0.\,\!
  3. 0 × V V 0. 0\rightarrow\mathbb{R}\rightarrow\mathbb{R}\times V\rightarrow V\rightarrow 0.\,\!
  4. \mathbb{R}
  5. p \ell^{p}
  6. 0 < p < 1 0<p<1
  7. 1 \ell^{1}

K-SVD.html

  1. D n × K D\in\mathbb{R}^{n\times K}
  2. K K
  3. D D
  4. y n y\in\mathbb{R}^{n}
  5. y y
  6. x x
  7. y = D x y=Dx
  8. y D x y\approx Dx
  9. y - D x p ϵ \|y-Dx\|_{p}\leq\epsilon
  10. ε ε
  11. L Lₚ
  12. x K x\in\mathbb{R}^{K}
  13. y y
  14. p p
  15. n < K n<K
  16. ( P 0 ) min x x 0 subject to y = D x (P_{0})\quad\min\limits_{x}\|x\|_{0}\qquad\,\text{subject to }y=Dx
  17. ( P 0 , ϵ ) min x x 0 subject to y - D x 2 ϵ (P_{0,\epsilon})\quad\min\limits_{x}\|x\|_{0}\qquad\,\text{subject to }\|y-Dx% \|_{2}\leq\epsilon
  18. x 0 \|x\|_{0}
  19. x x
  20. { y i } i = 1 M \{y_{i}\}^{M}_{i=1}
  21. min D , X { Y - D X F 2 } subject to i , x i = e k for some k . \quad\min\limits_{D,X}\{\|Y-DX\|^{2}_{F}\}\qquad\,\text{subject to }\forall i,% x_{i}=e_{k}\,\text{ for some }k.
  22. min D , X { Y - D X F 2 } subject to i , x i 0 = 1. \quad\min\limits_{D,X}\{\|Y-DX\|^{2}_{F}\}\qquad\,\text{subject to }\quad% \forall i,\|x_{i}\|_{0}=1.
  23. x i = e k x_{i}=e_{k}
  24. x i x_{i}
  25. T 0 T_{0}
  26. min D , X { Y - D X F 2 } subject to i , x i 0 T 0 . \quad\min\limits_{D,X}\{\|Y-DX\|^{2}_{F}\}\qquad\,\text{subject to }\quad% \forall i\;,\|x_{i}\|_{0}\leq T_{0}.
  27. min D , X i x i 0 subject to i , Y - D X F 2 ϵ . \quad\min\limits_{D,X}\sum_{i}\|x_{i}\|_{0}\qquad\,\text{subject to }\quad% \forall i\;,\|Y-DX\|^{2}_{F}\leq\epsilon.
  28. D D
  29. X X
  30. X X
  31. T 0 T_{0}
  32. D D
  33. D D
  34. X X
  35. k - t h k-th
  36. Y - D X F 2 = | Y - j = 1 K d j x T j | F 2 = | ( Y - j k d j x T j ) - d k x T k | F 2 = E k - d k x T k F 2 \|Y-DX\|^{2}_{F}=\left|Y-\sum_{j=1}^{K}d_{j}x^{j}_{T}\right|^{2}_{F}=\left|% \left(Y-\sum_{j\neq k}d_{j}x^{j}_{T}\right)-d_{k}x^{k}_{T}\right|^{2}_{F}=\|E_% {k}-d_{k}x^{k}_{T}\|^{2}_{F}
  37. x T k x^{k}_{T}
  38. D X DX
  39. K K
  40. K - 1 K-1
  41. k - t h k-th
  42. E k E_{k}
  43. r a n k - 1 rank-1
  44. d k d_{k}
  45. x T k x^{k}_{T}
  46. ω k \omega_{k}
  47. ω k = { i 1 i N , x T k ( i ) 0 } . \omega_{k}=\{i\mid 1\leq i\leq N,x^{k}_{T}(i)\neq 0\}.
  48. { y i } \{y_{i}\}
  49. d k d_{k}
  50. x i x_{i}
  51. Ω k \Omega_{k}
  52. N × | ω k | N\times|\omega_{k}|
  53. ( i -th , ω k ( i ) ) (i\,\text{-th},\omega_{k}(i))
  54. x R k = x T k Ω k x_{R}^{k}=x^{k}_{T}\Omega_{k}
  55. x T k x_{T}^{k}
  56. Y k R = Y Ω k Y^{R}_{k}=Y\Omega_{k}
  57. d k d_{k}
  58. E k R = E k Ω k E^{R}_{k}=E_{k}\Omega_{k}
  59. E k Ω k - d k x T k Ω k F 2 = E k R - d k x R k F 2 \|E_{k}\Omega_{k}-d_{k}x^{k}_{T}\Omega_{k}\|^{2}_{F}=\|E^{R}_{k}-d_{k}x^{k}_{R% }\|^{2}_{F}
  60. E k R E^{R}_{k}
  61. U Δ V T U\Delta V^{T}
  62. d k d_{k}
  63. x R k x^{k}_{R}
  64. V × Δ ( 1 , 1 ) V\times\Delta(1,1)

K-theory_spectrum.html

  1. K R K_{R}
  2. Σ R \Sigma R
  3. ( K R ) n = K 0 ( Σ n R ) × B G L ( Σ n R ) + (K_{R})_{n}=K_{0}(\Sigma^{n}R)\times BGL(\Sigma^{n}R)^{+}
  4. K i ( R ) = π i ( K R ) K_{i}(R)=\pi_{i}(K_{R})

K-trivial_set.html

  1. n K ( A n ) K ( n ) + b \forall nK(A\upharpoonright n)\leq K(n)+b
  2. n C ( A n ) C ( n ) + b \forall nC(A\upharpoonright n)\leq C(n)+b
  3. Δ 2 0 \Delta_{2}^{0}
  4. c : × 0 . c:\mathbb{N}\times\mathbb{N}\to\mathbb{Q}^{\geq 0}.
  5. A s \langle A_{s}\rangle
  6. Δ 2 0 \Delta_{2}^{0}
  7. Δ 2 0 \Delta_{2}^{0}
  8. Δ 2 0 \Delta_{2}^{0}
  9. A s : s ω \langle A_{s}:s\in\omega\rangle
  10. S = Σ x , s c ( x , s ) [ x < s x is the least s.t. A s - 1 ( x ) A s ( x ) ] < . S=\Sigma_{x,s}c(x,s)[x<s\wedge\,\text{x is the least s.t. }A_{s-1}(x)\neq A_{s% }(x)]<\infty.
  11. c K ( x , s ) = Σ x < y s 2 - K s ( x ) c_{K}(x,s)=\Sigma_{x<y\leq s}2^{-K_{s}(x)}
  12. K s ( x ) = min { | σ | : 𝕌 s ( σ ) = x } K_{s}(x)=\min\{|\sigma|:\mathbb{U}_{s}(\sigma)=x\}
  13. 𝕌 s \mathbb{U}_{s}
  14. 𝕌 \mathbb{U}
  15. P S e : | W e | = s x [ x W e , s \ W e , s - 1 x A s ] PS_{e}:|W_{e}|=\infty\Rightarrow\exists s\exists x[x\in W_{e,s}\backslash W_{e% ,s-1}\wedge x\in A_{s}]
  16. lim x sup s > x c ( x , s ) = 0 \lim_{x}\sup_{s>x}c(x,s)=0
  17. A A
  18. A s : s ω \langle A_{s}:s\in\omega\rangle
  19. A 0 = A_{0}=\emptyset
  20. x W e , s \ W e , s - 1 x\in W_{e,s}\backslash W_{e,s-1}
  21. c ( x , s ) 2 - e c(x,s)\leq 2^{-e}
  22. A s A_{s}
  23. P S e PS_{e}
  24. Σ e 2 - e < . \Sigma_{e}2^{-e}<\infty.
  25. W e W_{e}
  26. n K A ( n ) + b K ( n ) . \forall nK^{A}(n)+b\geq K(n).
  27. K A K^{A}
  28. A A

Kadison_transitivity_theorem.html

  1. \mathcal{F}
  2. \mathcal{H}
  3. { 0 } \{0\}
  4. \mathcal{H}
  5. \mathcal{F}
  6. \mathcal{F}
  7. { 0 } \{0\}
  8. \mathcal{H}
  9. \mathcal{H}
  10. \mathcal{F}
  11. 𝔄 \mathfrak{A}
  12. , { y 1 , , y n } \mathcal{H},\{y_{1},\cdots,y_{n}\}
  13. { x 1 , , x n } \{x_{1},\cdots,x_{n}\}
  14. \mathcal{H}
  15. A A
  16. 𝔄 \mathfrak{A}
  17. A x j = y j Ax_{j}=y_{j}
  18. B x j = y j Bx_{j}=y_{j}
  19. B B
  20. A A
  21. 𝔄 \mathfrak{A}
  22. \mathcal{H}

Kadison–Kastler_metric.html

  1. \mathcal{H}
  2. B ( ) B(\mathcal{H})
  3. \mathcal{H}
  4. 𝔄 \mathfrak{A}
  5. 𝔅 \mathfrak{B}
  6. B ( ) B(\mathcal{H})
  7. 𝔄 1 , 𝔅 1 \mathfrak{A}_{1},\mathfrak{B}_{1}
  8. 𝔄 - 𝔅 := sup { A - 𝔅 1 , B - 𝔄 1 : A 𝔄 1 , B 𝔅 1 } . \|\mathfrak{A}-\mathfrak{B}\|:=\sup\{\|A-\mathfrak{B}_{1}\|,\|B-\mathfrak{A}_{% 1}\|:A\in\mathfrak{A}_{1},B\in\mathfrak{B}_{1}\}.

Kai_Wehmeier.html

  1. Δ 1 1 \Delta^{1}_{1}

Kapitza_number.html

  1. K a = σ ρ ( g sin β ) 1 / 3 ν 4 / 3 Ka=\frac{\sigma}{\rho(g\sin\beta)^{1/3}\nu^{4/3}}

Kaplan–Yorke_conjecture.html

  1. λ 1 λ 2 λ n \lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}
  2. i = 1 j λ i > 0 \sum_{i=1}^{j}\lambda_{i}>0
  3. i = 1 j + 1 λ i < 0. \sum_{i=1}^{j+1}\lambda_{i}<0.
  4. D = j + i = 1 j λ i | λ j + 1 | . D=j+\frac{\sum_{i=1}^{j}\lambda_{i}}{|\lambda_{j+1}|}.
  5. λ 1 = 0.603 \lambda_{1}=0.603
  6. λ 2 = - 2.34 \lambda_{2}=-2.34
  7. D = j + λ 1 | λ 2 | = 1 + 0.603 | - 2.34 | = 1.26. D=j+\frac{\lambda_{1}}{|\lambda_{2}|}=1+\frac{0.603}{|-2.34|}=1.26.
  8. σ = 16 \sigma=16
  9. ρ = 45.92 \rho=45.92
  10. β = 4.0 \beta=4.0
  11. D = 2 + 2.16 + 0.00 | - 32.4 | = 2.07. D=2+\frac{2.16+0.00}{|-32.4|}=2.07.

Kardashian_Index.html

  1. F = 43.3 C 0.32 F=43.3C^{0.32}
  2. K - i n d e x = F ( a ) F ( c ) K-index=\frac{F(a)}{F(c)}

KCBS_pentagram.html

  1. 1 - 2 5 1-\frac{2}{\sqrt{5}}
  2. 1 5 \frac{1}{\sqrt{5}}
  3. 1 5 \frac{1}{\sqrt{5}}
  4. 2 5 4.47 > 4 2\sqrt{5}\approx 4.47>4
  5. { | A , | B , | C } \left\{|A\rangle,|B\rangle,|C\rangle\right\}
  6. | C |C\rangle
  7. 1 5 | C + 1 - 1 5 [ cos ( 4 π n 5 ) | A + sin ( 4 π n 5 ) | B ] \frac{1}{\sqrt{\sqrt{5}}}|C\rangle+\sqrt{1-\frac{1}{\sqrt{5}}}\left[\cos\left(% \frac{4\pi n}{5}\right)|A\rangle+\sin\left(\frac{4\pi n}{5}\right)|B\rangle\right]

Kenneth_I._Gross.html

  1. G 2 G_{2}
  2. G 2 G_{2}

Kernel_embedding_of_distributions.html

  1. Ω \Omega
  2. Ω \Omega
  3. d \mathbb{R}^{d}
  4. X X
  5. Ω \Omega
  6. P ( X ) P(X)
  7. k k
  8. Ω × Ω \Omega\times\Omega
  9. \mathcal{H}
  10. f : Ω f:\Omega\mapsto\mathbb{R}
  11. , \langle\cdot,\cdot\rangle_{\mathcal{H}}
  12. | | | | ||\cdot||_{\mathcal{H}}
  13. k ( x , ) \ k(x,\cdot)
  14. f , k ( x , ) = f ( x ) f , x Ω \langle f,k(x,\cdot)\rangle_{\mathcal{H}}=f(x)\ \forall f\in\mathcal{H},% \forall x\in\Omega
  15. k ( x , ) \ k(x,\cdot)
  16. ϕ ( x ) \phi(x)
  17. Ω \Omega
  18. \mathcal{H}
  19. k ( x , x ) = ϕ ( x ) , ϕ ( x ) \ k(x,x^{\prime})=\langle\phi(x),\phi(x^{\prime})\rangle_{\mathcal{H}}
  20. x , x Ω x,x^{\prime}\in\Omega
  21. P ( X ) P(X)
  22. \mathcal{H}
  23. μ X := 𝔼 X [ k ( X , ) ] = 𝔼 X [ ϕ ( X ) ] = Ω ϕ ( x ) d P ( x ) \mu_{X}:=\mathbb{E}_{X}[k(X,\cdot)]=\mathbb{E}_{X}[\phi(X)]=\int_{\Omega}\phi(% x)\ \mathrm{d}P(x)
  24. μ : { family of distributions over Ω } \mu:\{\,\text{family of distributions over }\Omega\}\mapsto\mathcal{H}
  25. n n
  26. { x 1 , , x n } \{x_{1},\dots,x_{n}\}
  27. P P
  28. P P
  29. μ ^ X = 1 n i = 1 n ϕ ( x i ) \widehat{\mu}_{X}=\frac{1}{n}\sum_{i=1}^{n}\phi(x_{i})
  30. Y Y
  31. Y Y
  32. Ω \Omega
  33. k k
  34. ϕ ( x ) ϕ ( y ) , ϕ ( x ) ϕ ( y ) = k ( x , x ) k ( y , y ) \langle\phi(x)\otimes\phi(y),\phi(x^{\prime})\otimes\phi(y^{\prime})\rangle=k(% x,x^{\prime})\otimes k(y,y^{\prime})
  35. P ( X , Y ) P(X,Y)
  36. \mathcal{H}\otimes\mathcal{H}
  37. 𝒞 X Y = 𝔼 X Y [ ϕ ( X ) ϕ ( Y ) ] = Ω × Ω ϕ ( x ) ϕ ( y ) d P ( x , y ) \mathcal{C}_{XY}=\mathbb{E}_{XY}[\phi(X)\otimes\phi(Y)]=\int_{\Omega\times% \Omega}\phi(x)\otimes\phi(y)\ \mathrm{d}P(x,y)
  38. 𝒞 X Y : \mathcal{C}_{XY}:\mathcal{H}\mapsto\mathcal{H}
  39. f , g f,g\in\mathcal{H}
  40. Cov X Y ( f ( X ) , g ( Y ) ) := 𝔼 X Y [ f ( X ) g ( Y ) ] = f , 𝒞 X Y g = f g , 𝒞 X Y \,\text{Cov}_{XY}(f(X),g(Y)):=\mathbb{E}_{XY}[f(X)g(Y)]=\langle f,\mathcal{C}_% {XY}g\rangle_{\mathcal{H}}=\langle f\otimes g,\mathcal{C}_{XY}\rangle_{% \mathcal{H}\otimes\mathcal{H}}
  41. n n
  42. { ( x 1 , y 1 ) , , ( x n , y n ) } \{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  43. P P
  44. 𝒞 ^ X Y = 1 n i = 1 n ϕ ( x i ) ϕ ( y i ) \widehat{\mathcal{C}}_{XY}=\frac{1}{n}\sum_{i=1}^{n}\phi(x_{i})\otimes\phi(y_{% i})
  45. P ( Y X ) P(Y\mid X)
  46. μ Y x = 𝔼 Y x [ ϕ ( Y ) ] = Ω ϕ ( y ) d P ( y x ) \mu_{Y\mid x}=\mathbb{E}_{Y\mid x}[\phi(Y)]=\int_{\Omega}\phi(y)\ \mathrm{d}P(% y\mid x)
  47. P ( Y X ) P(Y\mid X)
  48. x x
  49. X X
  50. X X
  51. \mathcal{H}
  52. 𝒞 Y X : \mathcal{C}_{Y\mid X}:\mathcal{H}\mapsto\mathcal{H}
  53. 𝒞 Y X = 𝒞 Y X 𝒞 X X - 1 \mathcal{C}_{Y\mid X}=\mathcal{C}_{YX}\mathcal{C}_{XX}^{-1}
  54. x x
  55. Y Y
  56. X = x X=x
  57. g : 𝔼 Y X [ g ( Y ) ] g\in\mathcal{H}:\ \mathbb{E}_{Y\mid X}[g(Y)]\in\mathcal{H}
  58. μ Y x = 𝒞 Y X ϕ ( x ) \mu_{Y\mid x}=\mathcal{C}_{Y\mid X}\phi(x)
  59. 𝒞 Y X ϕ ( x ) \mathcal{C}_{Y\mid X}\phi(x)
  60. μ Y x \mu_{Y\mid x}
  61. ( 𝒞 X X + λ 𝐈 ) - 1 (\mathcal{C}_{XX}+\lambda\mathbf{I})^{-1}
  62. 𝐈 \mathbf{I}
  63. { ( x 1 , y 1 ) , , ( x n , y n ) } \{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  64. C ^ Y X = s y m b o l Φ ( 𝐊 + λ 𝐈 ) - 1 s y m b o l Υ T \widehat{C}_{Y\mid X}=symbol{\Phi}(\mathbf{K}+\lambda\mathbf{I})^{-1}symbol{% \Upsilon}^{T}
  65. s y m b o l Φ = ( ϕ ( y i ) , , ( y n ) ) , s y m b o l Υ = ( ϕ ( x i ) , , ( x n ) ) symbol{\Phi}=\left(\phi(y_{i}),\dots,(y_{n})\right),symbol{\Upsilon}=\left(% \phi(x_{i}),\dots,(x_{n})\right)
  66. 𝐊 = s y m b o l Υ T s y m b o l Υ \mathbf{K}=symbol{\Upsilon}^{T}symbol{\Upsilon}
  67. X X
  68. λ \lambda
  69. Y Y
  70. μ ^ Y x = i = 1 n β i ( x ) ϕ ( y i ) = s y m b o l Φ s y m b o l β ( x ) \widehat{\mu}_{Y\mid x}=\sum_{i=1}^{n}\beta_{i}(x)\phi(y_{i})=symbol{\Phi}% symbol{\beta}(x)
  71. s y m b o l β ( x ) = ( 𝐊 + λ 𝐈 ) - 1 𝐊 x symbol{\beta}(x)=(\mathbf{K}+\lambda\mathbf{I})^{-1}\mathbf{K}_{x}
  72. 𝐊 x = ( k ( x 1 , x ) , , k ( x n , x ) ) T \mathbf{K}_{x}=\left(k(x_{1},x),\dots,k(x_{n},x)\right)^{T}
  73. f f
  74. 𝔼 X [ f ( X ) ] = f , μ X \mathbb{E}_{X}[f(X)]=\langle f,\mu_{X}\rangle_{\mathcal{H}}
  75. n × n n\times n
  76. k k
  77. f [ 0 , 1 ] f\in[0,1]
  78. f f\in\mathcal{H}
  79. || f || 1 ||f||_{\mathcal{H}}\leq 1
  80. 1 - δ \ 1-\delta
  81. || μ X - μ ^ X || = sup f ( 0 , 1 ) | 𝔼 X [ f ( X ) ] - 1 n i = 1 n f ( x i ) | 2 n 𝔼 X [ tr K ] + log ( 2 / δ ) 2 n ||\mu_{X}-\widehat{\mu}_{X}||_{\mathcal{H}}=\sup_{f\in\mathcal{B}(0,1)}\left|% \mathbb{E}_{X}[f(X)]-\frac{1}{n}\sum_{i=1}^{n}f(x_{i})\right|\leq\frac{2}{n}% \mathbb{E}_{X}\left[\sqrt{\,\text{tr}K}\right]+\sqrt{\frac{\log(2/\delta)}{2n}}
  82. ( 0 , 1 ) \mathcal{B}(0,1)
  83. \mathcal{H}
  84. 𝐊 \mathbf{K}
  85. i , j i,j
  86. k ( x i , x j ) k(x_{i},x_{j})
  87. O ( n - 1 / 2 ) O(n^{-1/2})
  88. X X
  89. O ( n - 1 / 2 ) O(n^{-1/2})
  90. n n
  91. β i ( x ) \beta_{i}(x)
  92. O ( n - 1 / 4 ) O\left(n^{-1/4}\right)
  93. λ \lambda
  94. O ( n - 1 / 2 ) O\left(n^{-1/2}\right)
  95. C ( 𝒳 ) C(\mathcal{X})
  96. 𝒳 \mathcal{X}
  97. k k
  98. k ( x , ) k(x,\cdot)
  99. x x
  100. k k
  101. C ( 𝒳 ) C(\mathcal{X})
  102. k k
  103. k ( x , x ) = exp ( - 1 2 σ 2 || x - x || 2 ) k(x,x^{\prime})=\exp\left(-\frac{1}{2\sigma^{2}}||x-x^{\prime}||^{2}\right)
  104. d \mathbb{R}^{d}
  105. k k
  106. 𝒞 ^ Y | X \widehat{\mathcal{C}}_{Y|X}
  107. min 𝒞 : i = 1 n || ϕ ( y i ) - 𝒞 ϕ ( x i ) || 2 + λ || 𝒞 || H S 2 \min_{\mathcal{C}:\mathcal{H}\mapsto\mathcal{H}}\sum_{i=1}^{n}||\phi(y_{i})-% \mathcal{C}\phi(x_{i})||_{\mathcal{H}}^{2}+\lambda||\mathcal{C}||_{HS}^{2}
  108. | | | | H S ||\cdot||_{HS}
  109. λ \lambda
  110. P ( X , Y ) = P(X,Y)=
  111. X , Y X,Y
  112. P ( X ) = Ω P ( X , d y ) = P(X)=\int_{\Omega}P(X,\mathrm{d}y)=
  113. X X
  114. P ( Y ) = P(Y)=
  115. Y Y
  116. P ( Y X ) = P ( X , Y ) P ( X ) = P(Y\mid X)=\frac{P(X,Y)}{P(X)}=
  117. Y Y
  118. X X
  119. 𝒞 Y X \mathcal{C}_{Y\mid X}
  120. π ( Y ) = \pi(Y)=
  121. Y Y
  122. Q Q
  123. P P
  124. { ( x 1 , y 1 ) , , ( x n , y n ) } \{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  125. { y ~ 1 , , y ~ n ~ } \{\widetilde{y}_{1},\dots,\widetilde{y}_{\widetilde{n}}\}
  126. π ( Y ) \pi(Y)
  127. X X
  128. Y Y
  129. Y Y
  130. Q ( X ) = Ω P ( X Y ) d π ( Y ) Q(X)=\int_{\Omega}P(X\mid Y)\mathrm{d}\pi(Y)
  131. μ X π \mu_{X}^{\pi}
  132. Q ( X ) Q(X)
  133. μ X π = 𝔼 Y [ 𝒞 X Y ϕ ( Y ) ] = 𝒞 X Y 𝔼 Y [ ϕ ( Y ) ] = 𝒞 X Y μ Y π \mu_{X}^{\pi}=\mathbb{E}_{Y}[\mathcal{C}_{X\mid Y}\phi(Y)]=\mathcal{C}_{X\mid Y% }\mathbb{E}_{Y}[\phi(Y)]=\mathcal{C}_{X\mid Y}\mu_{Y}^{\pi}
  134. μ Y π \mu_{Y}^{\pi}
  135. π ( Y ) \pi(Y)
  136. μ ^ X π = 𝒞 ^ X Y μ ^ Y π = s y m b o l Υ ( G + λ 𝐈 ) - 1 s y m b o l G ~ s y m b o l α \widehat{\mu}_{X}^{\pi}=\widehat{\mathcal{C}}_{X\mid Y}\widehat{\mu}_{Y}^{\pi}% =symbol{\Upsilon}(G+\lambda\mathbf{I})^{-1}\widetilde{symbol{G}}symbol{\alpha}
  137. μ Y π = i = 1 n ~ α i ϕ ( y ~ i ) \mu_{Y}^{\pi}=\sum_{i=1}^{\widetilde{n}}\alpha_{i}\phi(\widetilde{y}_{i})
  138. s y m b o l α = ( α 1 , , α n ~ ) T symbol{\alpha}=(\alpha_{1},\dots,\alpha_{\widetilde{n}})^{T}
  139. s y m b o l Υ = ( ϕ ( x 1 ) , , ϕ ( x n ) ) symbol{\Upsilon}=\left(\phi(x_{1}),\dots,\phi(x_{n})\right)
  140. 𝐆 , 𝐆 ~ \mathbf{G},\widetilde{\mathbf{G}}
  141. 𝐆 i j = k ( y i , y j ) , 𝐆 ~ i j = k ( y i , y ~ j ) \mathbf{G}_{ij}=k(y_{i},y_{j}),\widetilde{\mathbf{G}}_{ij}=k(y_{i},\widetilde{% y}_{j})
  142. Q ( X , Y ) = P ( X Y ) π ( Y ) Q(X,Y)=P(X\mid Y)\pi(Y)
  143. 𝒞 X Y π \mathcal{C}_{XY}^{\pi}
  144. Q ( X , Y ) Q(X,Y)
  145. π ( Y ) \pi(Y)
  146. 𝒞 X Y π = 𝒞 X Y 𝒞 Y Y π \mathcal{C}_{XY}^{\pi}=\mathcal{C}_{X\mid Y}\mathcal{C}_{YY}^{\pi}
  147. 𝒞 X Y π = 𝔼 X Y [ ϕ ( X ) ϕ ( Y ) ] \mathcal{C}_{XY}^{\pi}=\mathbb{E}_{XY}[\phi(X)\otimes\phi(Y)]
  148. 𝒞 Y Y π = 𝔼 Y [ ϕ ( Y ) ϕ ( Y ) ] \mathcal{C}_{YY}^{\pi}=\mathbb{E}_{Y}[\phi(Y)\otimes\phi(Y)]
  149. 𝒞 ^ X Y π = 𝒞 ^ X Y 𝒞 ^ Y Y π = s y m b o l Υ ( 𝐆 + λ 𝐈 ) - 1 𝐆 ~ diag ( s y m b o l α ) s y m b o l Φ T \widehat{\mathcal{C}}_{XY}^{\pi}=\widehat{\mathcal{C}}_{X\mid Y}\widehat{% \mathcal{C}}_{YY}^{\pi}=symbol{\Upsilon}(\mathbf{G}+\lambda\mathbf{I})^{-1}% \widetilde{\mathbf{G}}\,\text{diag}(symbol{\alpha})symbol{\Phi}^{T}
  150. Q ( Y x ) = P ( x Y ) π ( Y ) Q ( x ) Q(Y\mid x)=\frac{P(x\mid Y)\pi(Y)}{Q(x)}
  151. Q ( x ) = Ω P ( x y ) d π ( y ) Q(x)=\int_{\Omega}P(x\mid y)\mathrm{d}\pi(y)
  152. μ Y x π = 𝒞 Y X π ϕ ( x ) = 𝒞 Y X π ( 𝒞 X X π ) - 1 ϕ ( x ) \mu_{Y\mid x}^{\pi}=\mathcal{C}_{Y\mid X}^{\pi}\phi(x)=\mathcal{C}_{YX}^{\pi}(% \mathcal{C}_{XX}^{\pi})^{-1}\phi(x)
  153. 𝒞 Y X π = ( 𝒞 X Y 𝒞 Y Y π ) T \mathcal{C}_{YX}^{\pi}=\left(\mathcal{C}_{X\mid Y}\mathcal{C}_{YY}^{\pi}\right% )^{T}
  154. μ ^ Y x π = 𝒞 ^ Y X π ( ( 𝒞 ^ X X ) 2 + λ ~ 𝐈 ) - 1 𝒞 ^ X X π ϕ ( x ) = s y m b o l Φ ~ s y m b o l Λ T ( ( 𝐃𝐊 ) 2 + λ ~ 𝐈 ) - 1 𝐊𝐃𝐊 x \widehat{\mu}_{Y\mid x}^{\pi}=\widehat{\mathcal{C}}_{YX}^{\pi}\left((\widehat{% \mathcal{C}}_{XX})^{2}+\widetilde{\lambda}\mathbf{I}\right)^{-1}\widehat{% \mathcal{C}}_{XX}^{\pi}\phi(x)=\widetilde{symbol{\Phi}}symbol{\Lambda}^{T}% \left((\mathbf{D}\mathbf{K})^{2}+\widetilde{\lambda}\mathbf{I}\right)^{-1}% \mathbf{K}\mathbf{D}\mathbf{K}_{x}
  155. s y m b o l Λ = ( 𝐆 + λ ~ 𝐈 ) - 1 𝐆 ~ diag ( s y m b o l α ) , 𝐃 = diag ( ( 𝐆 + λ ~ 𝐈 ) - 1 𝐆 ~ s y m b o l α ) symbol{\Lambda}=\left(\mathbf{G}+\widetilde{\lambda}\mathbf{I}\right)^{-1}% \widetilde{\mathbf{G}}\,\text{diag}(symbol{\alpha}),\mathbf{D}=\,\text{diag}% \left(\left(\mathbf{G}+\widetilde{\lambda}\mathbf{I}\right)^{-1}\widetilde{% \mathbf{G}}symbol{\alpha}\right)
  156. λ \lambda
  157. 𝒞 ^ Y X π , 𝒞 ^ X X π = s y m b o l Υ 𝐃 s y m b o l Υ T \widehat{\mathcal{C}}_{YX}^{\pi},\widehat{\mathcal{C}}_{XX}^{\pi}=symbol{% \Upsilon}\mathbf{D}symbol{\Upsilon}^{T}
  158. λ ~ \widetilde{\lambda}
  159. 𝒞 ^ Y X π = 𝒞 ^ Y X π ( ( 𝒞 ^ X X π ) 2 + λ ~ 𝐈 ) - 1 𝒞 ^ X X π \widehat{\mathcal{C}}_{Y\mid X}^{\pi}=\widehat{\mathcal{C}}_{YX}^{\pi}\left((% \widehat{\mathcal{C}}_{XX}^{\pi})^{2}+\widetilde{\lambda}\mathbf{I}\right)^{-1% }\widehat{\mathcal{C}}_{XX}^{\pi}
  160. 𝒞 ^ X X π \widehat{\mathcal{C}}_{XX}^{\pi}
  161. D D
  162. P ( X ) P(X)
  163. Q ( Y ) Q(Y)
  164. MMD ( P , Q ) = || μ X - μ Y || 2 \,\text{MMD}(P,Q)=\left|\left|\mu_{X}-\mu_{Y}\right|\right|_{\mathcal{H}}^{2}
  165. MMD ( P , Q ) = sup || f || 1 ( 𝔼 X [ f ( X ) ] - 𝔼 Y [ f ( Y ) ] ) \,\text{MMD}(P,Q)=\sup_{||f||_{\mathcal{H}}\leq 1}\left(\mathbb{E}_{X}[f(X)]-% \mathbb{E}_{Y}[f(Y)]\right)
  166. P ( X ) P(X)
  167. Q ( Y ) Q(Y)
  168. MMD ^ ( P , Q ) = || 1 n i = 1 n ϕ ( x i ) - 1 m i = 1 m ϕ ( y i ) || 2 = 1 n m i = 1 n j = 1 m [ k ( x i , x j ) + k ( y i , y j ) - 2 k ( x i , y j ) ] \widehat{\,\text{MMD}}(P,Q)=\left|\left|\frac{1}{n}\sum_{i=1}^{n}\phi(x_{i})-% \frac{1}{m}\sum_{i=1}^{m}\phi(y_{i})\right|\right|_{\mathcal{H}}^{2}=\frac{1}{% nm}\sum_{i=1}^{n}\sum_{j=1}^{m}\left[k(x_{i},x_{j})+k(y_{i},y_{j})-2k(x_{i},y_% {j})\right]
  169. P = Q P=Q
  170. P Q P\neq Q
  171. P X * P_{X}^{*}
  172. max P X H ( P X ) \max_{P_{X}}H(P_{X})
  173. || μ ^ X - μ X [ P X ] || ϵ ||\widehat{\mu}_{X}-\mu_{X}[P_{X}]||_{\mathcal{H}}\leq\epsilon
  174. Ω \Omega
  175. μ X [ P X ] \mu_{X}[P_{X}]
  176. P X P_{X}
  177. H H
  178. X X
  179. Y Y
  180. HSIC ( X , Y ) = || 𝒞 X Y - μ X μ Y || 2 \,\text{HSIC}(X,Y)=\left|\left|\mathcal{C}_{XY}-\mu_{X}\otimes\mu_{Y}\right|% \right|_{\mathcal{H}\otimes\mathcal{H}}^{2}
  181. O ( n ( d f 2 + d g 2 ) ) O(n(d_{f}^{2}+d_{g}^{2}))
  182. 𝐀𝐀 T , 𝐁𝐁 T \mathbf{A}\mathbf{A}^{T},\mathbf{B}\mathbf{B}^{T}
  183. 𝐀 n × d f , 𝐁 n × d g \mathbf{A}\in\mathbb{R}^{n\times d_{f}},\mathbf{B}\in\mathbb{R}^{n\times d_{g}}
  184. m u t ( ) = i = 1 n β u t i ϕ ( x t i ) m_{ut}(\cdot)=\sum_{i=1}^{n}\beta_{ut}^{i}\phi(x_{t}^{i})
  185. m ^ t s = ( u N ( t ) \ s 𝐊 t s y m b o l β u t ) T ( 𝐊 s + λ 𝐈 ) - 1 s y m b o l Υ s T ϕ ( x s ) \widehat{m}_{ts}=\left(\odot_{u\in N(t)\backslash s}\mathbf{K}_{t}symbol{\beta% }_{ut}\right)^{T}(\mathbf{K}_{s}+\lambda\mathbf{I})^{-1}symbol{\Upsilon}_{s}^{% T}\phi(x_{s})
  186. \odot
  187. N ( t ) \ s N(t)\backslash s
  188. s y m b o l β u t = ( β u t 1 , , β u t n ) symbol{\beta}_{ut}=\left(\beta_{ut}^{1},\dots,\beta_{ut}^{n}\right)
  189. 𝐊 t , 𝐊 s \mathbf{K}_{t},\mathbf{K}_{s}
  190. X t , X s X_{t},X_{s}
  191. s y m b o l Υ s = ( ϕ ( x s 1 ) , , ϕ ( x s n ) ) symbol{\Upsilon}_{s}=\left(\phi(x_{s}^{1}),\dots,\phi(x_{s}^{n})\right)
  192. X s X_{s}
  193. X t X_{t}
  194. X s X_{s}
  195. P ( S t S t - 1 ) P(S^{t}\mid S^{t-1})
  196. P ( O t S t ) P(O^{t}\mid S^{t})
  197. s t s^{t}
  198. h t = ( o 1 , , o t ) h^{t}=(o^{1},\dots,o^{t})
  199. P ( S t + 1 h t + 1 ) P(S^{t+1}\mid h^{t+1})
  200. P ( S t + 1 h t ) = 𝔼 S t h t [ P ( S t + 1 S t ) ] P(S^{t+1}\mid h^{t})=\mathbb{E}_{S^{t}\mid h^{t}}[P(S^{t+1}\mid S^{t})]
  201. P ( S t + 1 h t , o t + 1 ) P ( o t + 1 S t + 1 ) P ( S t + 1 h t ) P(S^{t+1}\mid h^{t},o^{t+1})\propto P(o^{t+1}\mid S^{t+1})P(S^{t+1}\mid h^{t})
  202. μ S t + 1 h t + 1 = 𝒞 S t + 1 O t + 1 π ( 𝒞 O t + 1 O t + 1 π ) - 1 ϕ ( o t + 1 ) \mu_{S^{t+1}\mid h^{t+1}}=\mathcal{C}_{S^{t+1}O^{t+1}}^{\pi}\left(\mathcal{C}_% {O^{t+1}O^{t+1}}^{\pi}\right)^{-1}\phi(o^{t+1})
  203. ( s ~ 1 , , s ~ T , o ~ 1 , , o ~ T ) (\widetilde{s}^{1},\dots,\widetilde{s}^{T},\widetilde{o}^{1},\dots,\widetilde{% o}^{T})
  204. μ ^ S t + 1 h t + 1 = i = 1 T α i t ϕ ( s ~ t ) \widehat{\mu}_{S^{t+1}\mid h^{t+1}}=\sum_{i=1}^{T}\alpha_{i}^{t}\phi(% \widetilde{s}^{t})
  205. s y m b o l α = ( α 1 , , α T ) symbol{\alpha}=(\alpha_{1},\dots,\alpha_{T})
  206. 𝐃 t + 1 = diag ( ( G + λ 𝐈 ) - 1 G ~ s y m b o l α t ) \mathbf{D}^{t+1}=\,\text{diag}\left((G+\lambda\mathbf{I})^{-1}\widetilde{G}% symbol{\alpha}^{t}\right)
  207. s y m b o l α t + 1 = 𝐃 t + 1 𝐊 ( ( 𝐃 t + 1 K ) 2 + λ ~ 𝐈 ) - 1 𝐃 t + 1 𝐊 o t + 1 symbol{\alpha}^{t+1}=\mathbf{D}^{t+1}\mathbf{K}\left((\mathbf{D}^{t+1}K)^{2}+% \widetilde{\lambda}\mathbf{I}\right)^{-1}\mathbf{D}^{t+1}\mathbf{K}_{o^{t+1}}
  208. 𝐆 , 𝐊 \mathbf{G},\mathbf{K}
  209. s ~ 1 , , s ~ T \widetilde{s}^{1},\dots,\widetilde{s}^{T}
  210. o ~ 1 , , o ~ T \widetilde{o}^{1},\dots,\widetilde{o}^{T}
  211. 𝐆 ~ \widetilde{\mathbf{G}}
  212. 𝐆 ~ i j = k ( s ~ i , s ~ j + 1 ) \widetilde{\mathbf{G}}_{ij}=k(\widetilde{s}_{i},\widetilde{s}_{j+1})
  213. 𝐊 o t + 1 = ( k ( o ~ 1 , o t + 1 ) , , k ( o ~ T , o t + 1 ) ) T \mathbf{K}_{o^{t+1}}=(k(\widetilde{o}^{1},o^{t+1}),\dots,k(\widetilde{o}^{T},o% ^{t+1}))^{T}
  214. { P i , y i } i = 1 n , y i { + 1 , - 1 } \{P_{i},y_{i}\}_{i=1}^{n},\ y_{i}\in\{+1,-1\}
  215. K ( P ( X ) , Q ( Z ) ) = μ X , μ Z = 𝔼 X Z [ k ( x , z ) ] K\left(P(X),Q(Z)\right)=\langle\mu_{X},\mu_{Z}\rangle_{\mathcal{H}}=\mathbb{E}% _{XZ}[k(x,z)]
  216. P i P_{i}
  217. k k
  218. { x i } i = 1 n P ( X ) , { z j } j = 1 m Q ( Z ) \{x_{i}\}_{i=1}^{n}\sim P(X),\{z_{j}\}_{j=1}^{m}\sim Q(Z)
  219. K ^ ( X , Z ) = 1 n m i = 1 n j = 1 m k ( x i , z j ) \widehat{K}\left(X,Z\right)=\frac{1}{nm}\sum_{i=1}^{n}\sum_{j=1}^{m}k(x_{i},z_% {j})
  220. k k
  221. { P i , y i } i = 1 n \{P_{i},y_{i}\}_{i=1}^{n}
  222. { x i , y i } i = 1 n \{x_{i},y_{i}\}_{i=1}^{n}
  223. P i P_{i}
  224. { ( x i t r , y i t r ) } i = 1 n \{(x_{i}^{tr},y_{i}^{tr})\}_{i=1}^{n}
  225. { ( x j t e , y j t e ) } j = 1 m \{(x_{j}^{te},y_{j}^{te})\}_{j=1}^{m}
  226. y j t e y_{j}^{te}
  227. P t r ( X , Y ) P^{tr}(X,Y)
  228. P t e ( X , Y ) P^{te}(X,Y)
  229. P t r ( X ) P t e ( X ) P^{tr}(X)\neq P^{te}(X)
  230. P t r ( Y ) P t e ( Y ) P^{tr}(Y)\neq P^{te}(Y)
  231. P ( Y ) P(Y)
  232. P t r ( X Y ) P t e ( X Y ) P^{tr}(X\mid Y)\neq P^{te}(X\mid Y)
  233. P ( X Y ) P(X\mid Y)
  234. X X
  235. P t e ( X ) / P t r ( X ) P^{te}(X)/P^{tr}(X)
  236. X X
  237. Y Y
  238. s y m b o l β * ( 𝐲 t r ) symbol{\beta}^{*}(\mathbf{y}^{tr})
  239. min s y m b o l β ( y ) || 𝒞 ( X Y ) t r 𝔼 Y t r [ s y m b o l β ( y ) ϕ ( y ) ] - μ X t e || 2 \min_{symbol{\beta}(y)}\left|\left|\mathcal{C}_{{(X\mid Y)}^{tr}}\mathbb{E}_{Y% ^{tr}}[symbol{\beta}(y)\phi(y)]-\mu_{X^{te}}\right|\right|_{\mathcal{H}}^{2}
  240. s y m b o l β ( y ) 0 , 𝔼 Y t r [ s y m b o l β ( y ) ] = 1 symbol{\beta}(y)\geq 0,\mathbb{E}_{Y^{tr}}[symbol{\beta}(y)]=1
  241. 𝐗 n e w = 𝐗 t r 𝐖 + 𝐁 \mathbf{X}^{new}=\mathbf{X}^{tr}\odot\mathbf{W}+\mathbf{B}
  242. \odot
  243. 𝐖 , 𝐁 \mathbf{W},\mathbf{B}
  244. || μ ^ X n e w - μ ^ X t e || 2 = || 𝒞 ^ ( X Y ) n e w μ ^ Y t r - μ ^ X t e || 2 \left|\left|\widehat{\mu}_{X^{new}}-\widehat{\mu}_{X^{te}}\right|\right|_{% \mathcal{H}}^{2}=\left|\left|\widehat{\mathcal{C}}_{(X\mid Y)^{new}}\widehat{% \mu}_{Y^{tr}}-\widehat{\mu}_{X^{te}}\right|\right|_{\mathcal{H}}^{2}
  245. P ( 1 ) ( X , Y ) , P ( 2 ) ( X , Y ) , , P ( N ) ( X , Y ) P^{(1)}(X,Y),P^{(2)}(X,Y),\dots,P^{(N)}(X,Y)
  246. P * ( X , Y ) P^{*}(X,Y)
  247. P ( Y X ) P(Y\mid X)
  248. P ( X ) P(X)
  249. 𝒫 \mathcal{P}
  250. \mathcal{H}
  251. 𝒫 ( μ X ( i ) Y ( i ) ) = 1 / N for i = 1 , , N \mathcal{P}(\mu_{X^{(i)}Y^{(i)}})=1/N\,\text{ for }i=1,\dots,N
  252. V ( 𝒫 ) = 1 N tr ( 𝐆 ) - 1 N 2 i , j = 1 N 𝐆 i j V_{\mathcal{H}}(\mathcal{P})=\frac{1}{N}\,\text{tr}(\mathbf{G})-\frac{1}{N^{2}% }\sum_{i,j=1}^{N}\mathbf{G}_{ij}
  253. 𝐆 i j = μ X ( i ) , μ X ( j ) \mathbf{G}_{ij}=\langle\mu_{X^{(i)}},\mu_{X^{(j)}}\rangle_{\mathcal{H}}
  254. 𝐆 \mathbf{G}
  255. N × N N\times N
  256. Y Y
  257. X X
  258. C T X C^{T}X
  259. Y Y
  260. X X
  261. X , Y X,Y
  262. { 1 , , K } \{1,\dots,K\}
  263. k ( x , x ) = δ ( x , x ) k(x,x^{\prime})=\delta(x,x^{\prime})
  264. ϕ ( x ) = 𝐞 x \phi(x)=\mathbf{e}_{x}
  265. K × K K\times K
  266. μ X = 𝔼 X [ 𝐞 x ] = ( P ( X = 1 ) P ( X = K ) ) \mu_{X}=\mathbb{E}_{X}[\mathbf{e}_{x}]=\left(\begin{array}[]{c}P(X=1)\\ \vdots\\ P(X=K)\\ \end{array}\right)
  267. 𝒞 X Y = 𝔼 X Y [ 𝐞 X e Y ] = ( P ( X = s , Y = t ) ) s , t { 1 , , K } \mathcal{C}_{XY}=\mathbb{E}_{XY}[\mathbf{e}_{X}\otimes e_{Y}]=\bigg(P(X=s,Y=t)% \bigg)_{s,t\in\{1,\dots,K\}}
  268. 𝒞 Y X = 𝒞 Y X 𝒞 X X - 1 \mathcal{C}_{Y\mid X}=\mathcal{C}_{YX}\mathcal{C}_{XX}^{-1}
  269. 𝒞 Y X = ( P ( Y = s X = t ) ) s , t { 1 , , K } \mathcal{C}_{Y\mid X}=\bigg(P(Y=s\mid X=t)\bigg)_{s,t\in\{1,\dots,K\}}
  270. 𝒞 X X = ( P ( X = 1 ) 0 0 P ( X = K ) ) \mathcal{C}_{XX}=\left(\begin{array}[]{c c c}P(X=1)&\dots&0\\ \vdots&\ddots&\vdots\\ 0&\dots&P(X=K)\\ \end{array}\right)
  271. X X
  272. μ Y x = 𝒞 Y X ϕ ( x ) = ( P ( Y = 1 X = x ) P ( Y = K X = x ) ) \mu_{Y\mid x}=\mathcal{C}_{Y\mid X}\phi(x)=\left(\begin{array}[]{c}P(Y=1\mid X% =x)\\ \vdots\\ P(Y=K\mid X=x)\\ \end{array}\right)
  273. ( Q ( X = 1 ) P ( X = N ) ) μ Y π = ( P ( X = s Y = t ) ) 𝒞 X Y ( π ( Y = 1 ) p i ( Y = N ) ) μ Y π \underbrace{\left(\begin{array}[]{c}Q(X=1)\\ \vdots\\ P(X=N)\\ \end{array}\right)}_{\mu_{Y}^{\pi}}=\underbrace{\left(\begin{array}[]{c}\\ P(X=s\mid Y=t)\\ \\ \end{array}\right)}_{\mathcal{C}_{X\mid Y}}\underbrace{\left(\begin{array}[]{c% }\pi(Y=1)\\ \vdots\\ pi(Y=N)\\ \end{array}\right)}_{\mu_{Y}^{\pi}}
  274. ( Q ( X = s , Y = t ) ) 𝒞 X Y π = ( P ( X = s Y = t ) ) 𝒞 X Y ( π ( Y = 1 ) 0 0 π ( Y = K ) ) 𝒞 Y Y π \underbrace{\left(\begin{array}[]{c}\\ Q(X=s,Y=t)\\ \\ \end{array}\right)}_{\mathcal{C}_{XY}^{\pi}}=\underbrace{\left(\begin{array}[]% {c}\\ P(X=s\mid Y=t)\\ \\ \end{array}\right)}_{\mathcal{C}_{X\mid Y}}\underbrace{\left(\begin{array}[]{c% c c}\pi(Y=1)&\dots&0\\ \vdots&\ddots&\vdots\\ 0&\dots&\pi(Y=K)\\ \end{array}\right)}_{\mathcal{C}_{YY}^{\pi}}