wpmath0000005_8

Matrix_norm.html

  1. K K
  2. K m × n K^{m\times n}
  3. m m
  4. n n
  5. K K
  6. A * A^{*}
  7. A A
  8. K m × n K^{m\times n}
  9. A \|A\|
  10. A A
  11. A 0 \|A\|\geq 0
  12. A = 0 \|A\|=0
  13. A = 0 A=0
  14. α A = | α | A \|\alpha A\|=|\alpha|\|A\|
  15. α \alpha
  16. K K
  17. A A
  18. K m × n K^{m\times n}
  19. A + B A + B \|A+B\|\leq\|A\|+\|B\|
  20. A A
  21. B B
  22. K m × n . K^{m\times n}.
  23. A B A B \|AB\|\leq\|A\|\|B\|
  24. A A
  25. B B
  26. K n × n . K^{n\times n}.
  27. A \displaystyle\|A\|
  28. A p = sup x 0 A x p x p . \left\|A\right\|_{p}=\sup\limits_{x\neq 0}\frac{\left\|Ax\right\|_{p}}{\left\|% x\right\|_{p}}.
  29. A p . \left\|A\right\|_{p}.
  30. p = 1 p=1
  31. p = p=\infty
  32. A 1 = max 1 j n i = 1 m | a i j | , \left\|A\right\|_{1}=\max\limits_{1\leq j\leq n}\sum_{i=1}^{m}|a_{ij}|,
  33. A = max 1 i m j = 1 n | a i j | , \left\|A\right\|_{\infty}=\max\limits_{1\leq i\leq m}\sum_{j=1}^{n}|a_{ij}|,
  34. A = [ - 3 5 7 2 6 4 0 2 8 ] , A=\begin{bmatrix}-3&5&7\\ 2&6&4\\ 0&2&8\\ \end{bmatrix},
  35. A 2 = λ max ( A * A ) = σ max ( A ) \left\|A\right\|_{2}=\sqrt{\lambda_{\,\text{max}}(A^{{}^{*}}A)}=\sigma_{\,% \text{max}}(A)
  36. K m × n K^{m\times n}
  37. α \|\cdot\|_{\alpha}
  38. K n K^{n}
  39. β \|\cdot\|_{\beta}
  40. K m K^{m}
  41. A α , β = max x 0 A x β x α . \left\|A\right\|_{\alpha,\beta}=\max\limits_{x\neq 0}\frac{\left\|Ax\right\|_{% \beta}}{\left\|x\right\|_{\alpha}}.
  42. A x β A α , β x α . \|Ax\|_{\beta}\leq\|A\|_{\alpha,\beta}\|x\|_{\alpha}.
  43. α = β \alpha=\beta
  44. A B x A B x A B x \|ABx\|\leq\|A\|\|Bx\|\leq\|A\|\|B\|\|x\|
  45. max x = 1 A B x = A B . \max\limits_{\|x\|=1}\|ABx\|=\|AB\|.
  46. A r 1 / r ρ ( A ) , \left\|A^{r}\right\|^{1/r}\geq\rho(A),
  47. A A
  48. A A
  49. A = [ 0 1 0 0 ] , A=\begin{bmatrix}0&1\\ 0&0\\ \end{bmatrix},
  50. A A
  51. A A
  52. A A
  53. lim r A r 1 / r = ρ ( A ) . \lim_{r\rightarrow\infty}\|A^{r}\|^{1/r}=\rho(A).
  54. m × n m\times n
  55. m n mn
  56. A p = vec ( A ) p = ( i = 1 m j = 1 n | a i j | p ) 1 / p \|A\|_{p}=\|\mathrm{vec}(A)\|_{p}=\left(\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^{% p}\right)^{1/p}
  57. ( a 1 , , a n ) (a_{1},\cdots,a_{n})
  58. A A
  59. L 2 , 1 L_{2,1}
  60. A 2 , 1 = j = 1 n a j 2 = j = 1 n ( i = 1 m | a i j | 2 ) 1 / 2 \|A\|_{2,1}=\sum_{j=1}^{n}\|a_{j}\|_{2}=\sum_{j=1}^{n}\left(\sum_{i=1}^{m}|a_{% ij}|^{2}\right)^{1/2}
  61. i i
  62. j j
  63. A i , j A_{i,j}
  64. L 2 , 1 L_{2,1}
  65. L 2 , 1 L_{2,1}
  66. L 2 , 1 L_{2,1}
  67. L p , q L_{p,q}
  68. A p , q = [ j = 1 n ( i = 1 m | a i j | p ) q / p ] 1 / q \|A\|_{p,q}=\left[\sum_{j=1}^{n}\left(\sum_{i=1}^{m}|a_{ij}|^{p}\right)^{q/p}% \right]^{1/q}
  69. A F = i = 1 m j = 1 n | a i j | 2 = trace ( A * A ) = i = 1 min { m , n } σ i 2 \|A\|_{F}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^{2}}=\sqrt{\operatorname{% trace}(A^{{}^{*}}A)}=\sqrt{\sum_{i=1}^{\min\{m,\,n\}}\sigma_{i}^{2}}
  70. A F 2 = A R F 2 = R A F 2 \|A\|_{F}^{2}=\|AR\|_{F}^{2}=\|RA\|_{F}^{2}
  71. R R
  72. A R F 2 = trace ( R T A T A R ) = trace ( R R T A T A ) = trace ( A T A ) = A F 2 \|AR\|_{F}^{2}=\operatorname{trace}\left(R^{\rm T}A^{\rm T}AR\right)=% \operatorname{trace}\left(RR^{\rm T}A^{\rm T}A\right)=\operatorname{trace}% \left(A^{\rm T}A\right)=\|A\|_{F}^{2}
  73. R A F 2 = trace ( A T R T R A ) = trace ( A T A ) = A F 2 \|RA\|_{F}^{2}=\operatorname{trace}\left(A^{\rm T}R^{\rm T}RA\right)=% \operatorname{trace}\left(A^{\rm T}A\right)=\|A\|_{F}^{2}
  74. R R
  75. R T R = R R T = 𝐈 R^{\rm T}R=RR^{\rm T}=\mathbf{I}
  76. trace ( X Y Z ) = trace ( Z X Y ) \operatorname{trace}(XYZ)=\operatorname{trace}(ZXY)
  77. A max = max { | a i j | } . \|A\|_{\,\text{max}}=\max\{|a_{ij}|\}.
  78. A p = ( i = 1 min { m , n } σ i p ) 1 / p . \|A\|_{p}=\left(\sum_{i=1}^{\min\{m,\,n\}}\sigma_{i}^{p}\right)^{1/p}.\,
  79. A * = trace ( A * A ) = i = 1 min { m , n } σ i . \|A\|_{*}=\operatorname{trace}\left(\sqrt{A^{*}A}\right)=\sum_{i=1}^{\min\{m,% \,n\}}\sigma_{i}.
  80. A * A \sqrt{A^{*}A}
  81. B B
  82. B B = A * A BB=A^{*}A
  83. A * A A^{*}A
  84. a b \|\cdot\|_{ab}
  85. K m × n K^{m\times n}
  86. a \|\cdot\|_{a}
  87. K n K^{n}
  88. b \|\cdot\|_{b}
  89. K m K^{m}
  90. A x b A a b x a \|Ax\|_{b}\leq\|A\|_{ab}\|x\|_{a}
  91. A K m × n , x K n A\in K^{m\times n},x\in K^{n}
  92. b \|\cdot\|_{b}
  93. K n × n K^{n\times n}
  94. a \|\cdot\|_{a}
  95. K n K^{n}
  96. A x a A b x a \|Ax\|_{a}\leq\|A\|_{b}\|x\|_{a}
  97. A K n × n , x K n A\in K^{n\times n},x\in K^{n}
  98. α \|\cdot\|_{\alpha}
  99. β \|\cdot\|_{\beta}
  100. r A α A β s A α r\left\|A\right\|_{\alpha}\leq\left\|A\right\|_{\beta}\leq s\left\|A\right\|_{\alpha}
  101. K m × n K^{m\times n}
  102. K m × n K^{m\times n}
  103. K m × n K^{m\times n}
  104. K m × n K^{m\times n}
  105. m × n m\times n
  106. \|\cdot\|
  107. n × n \mathbb{R}^{n\times n}
  108. k k
  109. l l\|\cdot\|
  110. l k l\geq k
  111. α \|\cdot\|_{\alpha}
  112. β \|\cdot\|_{\beta}
  113. β < α \|\cdot\|_{\beta}<\|\cdot\|_{\alpha}
  114. A m × n A\in\mathbb{R}^{m\times n}
  115. r r
  116. A 2 A F r A 2 \|A\|_{2}\leq\|A\|_{F}\leq\sqrt{r}\|A\|_{2}
  117. A F A * r A F \|A\|_{F}\leq\|A\|_{*}\leq\sqrt{r}\|A\|_{F}
  118. A max A 2 m n A max \|A\|_{\,\text{max}}\leq\|A\|_{2}\leq\sqrt{mn}\|A\|_{\,\text{max}}
  119. 1 n A A 2 m A \frac{1}{\sqrt{n}}\|A\|_{\infty}\leq\|A\|_{2}\leq\sqrt{m}\|A\|_{\infty}
  120. 1 m A 1 A 2 n A 1 . \frac{1}{\sqrt{m}}\|A\|_{1}\leq\|A\|_{2}\leq\sqrt{n}\|A\|_{1}.
  121. A p \|A\|_{p}
  122. A 2 A 1 A , \|A\|_{2}\leq\sqrt{\|A\|_{1}\|A\|_{\infty}},

Matrix_unit.html

  1. [ ] [ ] [ ] [ ] = [ ] \begin{bmatrix}\cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot\\ \end{bmatrix}\begin{bmatrix}\cdot&\cdot\\ \cdot&\cdot\\ \cdot&\cdot\\ \end{bmatrix}\begin{bmatrix}\cdot\\ \cdot\\ \end{bmatrix}\begin{bmatrix}\cdot&\cdot&\cdot&\cdot\\ \end{bmatrix}=\begin{bmatrix}\cdot&\cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot&\\ \end{bmatrix}

Maximal_compact_subgroup.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 = 𝔨 𝔭 , \displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p},}
  3. 𝔨 \mathfrak{k}
  4. G = K exp 𝔭 = K P = P K . \displaystyle{G=K\cdot\exp\mathfrak{p}=K\cdot P=P\cdot K.}
  5. 𝔤 \mathfrak{g}
  6. ( X , Y ) σ = - B ( X , σ ( Y ) ) \displaystyle{(X,Y)_{\sigma}=-B(X,\sigma(Y))}
  7. 𝔤 \mathfrak{g}
  8. ( S X , Y ) σ , (S\cdot X,Y)_{\sigma},
  9. 𝔤 \mathfrak{g}
  10. Ad σ ( x ) = ( Ad ( x ) - 1 ) t . \displaystyle{\mathrm{Ad}\,\sigma(x)=(\mathrm{Ad}\,(x)^{-1})^{t}.}
  11. S Ad ( σ ( h ) ) = Ad ( h ) S . \displaystyle{S\circ\mathrm{Ad}(\sigma(h))=\mathrm{Ad}(h)\circ S.}
  12. 𝔭 \mathfrak{p}
  13. f ( e X ) = Tr Ad ( e X ) S . \displaystyle{f(e^{X})=\mathrm{Tr}\,\mathrm{Ad}(e^{X})S.}
  14. f ( e X ) = λ i ( Ad ( e X ) e i , e i ) σ ( min λ i ) Tr e ad X , \displaystyle{f(e^{X})=\sum\lambda_{i}(\mathrm{Ad}(e^{X})e_{i},e_{i})_{\sigma}% \geq(\min\lambda_{i})\cdot\mathrm{Tr}\,e^{\mathrm{ad}\,X},}
  15. 𝔨 i 𝔭 \mathfrak{k}\oplus i\mathfrak{p}
  16. e Z = e Y / 2 e X e Y / 2 , \displaystyle{e^{Z}=e^{Y/2}e^{X}e^{Y/2},}
  17. 𝔭 \mathfrak{p}
  18. e Z / 2 e - Y / 2 = k e X / 2 . \displaystyle{e^{Z/2}e^{-Y/2}=k\cdot e^{X/2}.}
  19. g ( t ) = f ( e Y / 2 e t X e Y / 2 ) = e μ i t A d ( e Y / 2 ) f i σ 2 . \displaystyle{g(t)=f(e^{Y/2}e^{tX}e^{Y/2})=\sum e^{\mu_{i}t}\|Ad(e^{Y/2})f_{i}% \|^{2}_{\sigma}.}
  20. K G K\hookrightarrow G
  21. G K G\twoheadrightarrow K

Maximal_independent_set.html

  1. n / k k - ( n mod k ) n / k + 1 n mod k . \lfloor n/k\rfloor^{k-(n\bmod k)}\lfloor n/k+1\rfloor^{n\bmod k}.
  2. O ( log ( n ) ) O(\log(n))
  3. Θ ( log ( n ) ) \Theta(\log(n))
  4. log 4 ( n ) \log_{4}(n)
  5. 3 log 4 / 3 ( m ) + 1 3\log_{4/3}(m)+1
  6. O ( log ( n ) ) O(\log(n))
  7. O ( log 2 ( n ) ) O(\log^{2}(n))
  8. O ( log 2 ( n ) ) O(\log^{2}(n))
  9. δ = 2 i log ( n ) / D \delta=2^{i}\log{(n)}/D
  10. D / 2 i D/2^{i}
  11. δ = C log ( n ) / d \delta=C\log{(n)}/d
  12. O ( log ( n ) ) O(\log(n))
  13. O ( log ( n ) ) O(\log(n))
  14. δ = 2 i log ( n ) / D \delta=2^{i}\log{(n)}/D
  15. O ( log ( D ) log ( n ) ) = O ( log 2 ( n ) ) O(\log(D)\log(n))=O(\log^{2}(n))

Maximum_a_posteriori_estimation.html

  1. θ \theta
  2. x x
  3. f f
  4. x x
  5. f ( x | θ ) f(x|\theta)
  6. x x
  7. θ \theta
  8. θ f ( x | θ ) \theta\mapsto f(x|\theta)\!
  9. θ ^ ML ( x ) = arg max 𝜃 f ( x | θ ) \hat{\theta}_{\mathrm{ML}}(x)=\underset{\theta}{\operatorname{arg\,max}}\ f(x|% \theta)\!
  10. θ \theta
  11. g g
  12. θ \theta
  13. θ \theta
  14. θ \theta
  15. θ f ( θ | x ) = f ( x | θ ) g ( θ ) ϑ Θ f ( x | ϑ ) g ( ϑ ) d ϑ \theta\mapsto f(\theta|x)=\frac{f(x|\theta)\,g(\theta)}{\displaystyle\int_{% \vartheta\in\Theta}f(x|\vartheta)\,g(\vartheta)\,d\vartheta}\!
  16. g g
  17. θ \theta
  18. Θ \Theta
  19. g g
  20. θ \theta
  21. θ ^ MAP ( x ) = arg max 𝜃 f ( x | θ ) g ( θ ) ϑ f ( x | ϑ ) g ( ϑ ) d ϑ = arg max 𝜃 f ( x | θ ) g ( θ ) . \hat{\theta}_{\mathrm{MAP}}(x)=\underset{\theta}{\operatorname{arg\,max}}\ % \frac{f(x|\theta)\,g(\theta)}{\displaystyle\int_{\vartheta}f(x|\vartheta)\,g(% \vartheta)\,d\vartheta}=\underset{\theta}{\operatorname{arg\,max}}\ f(x|\theta% )\,g(\theta).\!
  22. θ \theta
  23. θ \theta
  24. g g
  25. L ( θ , a ) = { 0 , if | a - θ | < c 1 , otherwise L(\theta,a)=\begin{cases}0&\mbox{, if }~{}|a-\theta|<c\\ 1&\mbox{, otherwise}\\ \end{cases}\!
  26. c c
  27. θ \theta
  28. θ \theta
  29. x x
  30. h 1 h_{1}
  31. h 2 h_{2}
  32. h 3 h_{3}
  33. x x
  34. h 1 h_{1}
  35. h 1 h_{1}
  36. x x
  37. x x
  38. ( x 1 , , x n ) (x_{1},\dots,x_{n})
  39. N ( μ , σ v 2 ) N(\mu,\sigma_{v}^{2})
  40. μ \mu
  41. N ( μ 0 , σ m 2 ) N(\mu_{0},\sigma_{m}^{2})
  42. μ \mu
  43. f ( μ ) f ( x | μ ) = π ( μ ) L ( μ ) = 1 2 π σ m exp ( - 1 2 ( μ - μ 0 σ m ) 2 ) j = 1 n 1 2 π σ v exp ( - 1 2 ( x j - μ σ v ) 2 ) , f(\mu)f(x|\mu)=\pi(\mu)L(\mu)=\frac{1}{\sqrt{2\pi}\sigma_{m}}\exp\left(-\frac{% 1}{2}\left(\frac{\mu-\mu_{0}}{\sigma_{m}}\right)^{2}\right)\prod_{j=1}^{n}% \frac{1}{\sqrt{2\pi}\sigma_{v}}\exp\left(-\frac{1}{2}\left(\frac{x_{j}-\mu}{% \sigma_{v}}\right)^{2}\right),
  44. μ \mu
  45. j = 1 n ( x j - μ σ v ) 2 + ( μ - μ 0 σ m ) 2 . \sum_{j=1}^{n}\left(\frac{x_{j}-\mu}{\sigma_{v}}\right)^{2}+\left(\frac{\mu-% \mu_{0}}{\sigma_{m}}\right)^{2}.
  46. μ ^ M A P = n σ m 2 n σ m 2 + σ v 2 ( 1 n j = 1 n x j ) + σ v 2 n σ m 2 + σ v 2 μ 0 . \hat{\mu}_{MAP}=\frac{n\sigma_{m}^{2}}{n\sigma_{m}^{2}+\sigma_{v}^{2}}\left(% \frac{1}{n}\sum_{j=1}^{n}x_{j}\right)+\frac{\sigma_{v}^{2}}{n\sigma_{m}^{2}+% \sigma_{v}^{2}}\mu_{0}.
  47. σ m \sigma_{m}\to\infty
  48. μ ^ M A P μ ^ M L . \hat{\mu}_{MAP}\to\hat{\mu}_{ML}.

Maximum_entropy_probability_distribution.html

  1. Pr ( X = x k ) = p k for k = 1 , 2 , \operatorname{Pr}(X=x_{k})=p_{k}\quad\mbox{ for }~{}k=1,2,\ldots
  2. H ( X ) = - k 1 p k log p k . H(X)=-\sum_{k\geq 1}p_{k}\log p_{k}\;.
  3. H ( X ) = - - p ( x ) log p ( x ) d x . H(X)=-\int_{-\infty}^{\infty}p(x)\log p(x)dx\;.
  4. H ( X ) H(X)
  5. E ( f j ( X ) ) = a j for j = 1 , , n \operatorname{E}(f_{j}(X))=a_{j}\quad\mbox{ for }~{}j=1,\ldots,n
  6. p ( x ) = c exp ( j = 1 n λ j f j ( x ) ) for all x S p(x)=c\exp\left(\sum_{j=1}^{n}\lambda_{j}f_{j}(x)\right)\quad\mbox{ for all }~% {}x\in S
  7. E ( f j ( X ) ) = a j for j = 1 , , n \operatorname{E}(f_{j}(X))=a_{j}\quad\mbox{ for }~{}j=1,\ldots,n
  8. Pr ( X = x k ) = c exp ( j = 1 n λ j f j ( x k ) ) for k = 1 , 2 , \operatorname{Pr}(X=x_{k})=c\exp\left(\sum_{j=1}^{n}\lambda_{j}f_{j}(x_{k})% \right)\quad\mbox{ for }~{}k=1,2,\ldots
  9. - f j ( x ) p ( x ) d x = a j \int_{-\infty}^{\infty}f_{j}(x)p(x)dx=a_{j}
  10. J ( p ( x ) ) = - - p ( x ) ln p ( x ) d x + λ 0 ( - p ( x ) d x - 1 ) + j = 1 n λ j ( - f j ( x ) p ( x ) d x - a j ) J(p(x))=-\int_{-\infty}^{\infty}p(x)\ln{p(x)}dx+\lambda_{0}\left(\int_{-\infty% }^{\infty}p(x)dx-1\right)+\sum_{j=1}^{n}\lambda_{j}\left(\int_{-\infty}^{% \infty}f_{j}(x)p(x)dx-a_{j}\right)
  11. λ j \lambda_{j}
  12. n n
  13. δ J ( p ( x ) ) δ p ( x ) = - ln p ( x ) - 1 + λ 0 + j = 1 n λ j f j ( x ) = 0 \frac{\delta{J(p(x))}}{\delta{p(x)}}=-\ln{p(x)}-1+\lambda_{0}+\sum_{j=1}^{n}% \lambda_{j}f_{j}(x)=0
  14. p ( x ) = e - 1 + λ 0 e j = 1 n λ j f j ( x ) = c exp ( j = 1 n λ j f j ( x ) ) . p(x)=e^{-1+\lambda_{0}}\cdot e^{\sum_{j=1}^{n}\lambda_{j}f_{j}(x)}=c\cdot\exp% \left(\sum_{j=1}^{n}\lambda_{j}f_{j}(x)\right)\;.
  15. p ( x ) = exp ( ln p ( x ) ) p(x)=\exp{(\ln{p(x)})}
  16. ln p ( x ) f ( x ) \ln{p(x)}\rightarrow f(x)
  17. exp ( f ( x ) ) f ( x ) d x = - H \int\exp{(f(x))}f(x)dx=-H
  18. p ( x ) p(x)
  19. p ( x ) f ( x ) d x = - H \int p(x)f(x)dx=-H
  20. ln p ( x ) f ( x ) \ln{p(x)}\rightarrow f(x)
  21. f ( x ) f(x)
  22. Pr ( a j - 1 X < a j ) = p j for j = 1 , , k \operatorname{Pr}(a_{j-1}\leq X<a_{j})=p_{j}\quad\mbox{ for }~{}j=1,\ldots,k
  23. p ( x | λ ) = { λ e - λ x x 0 , 0 x < 0 , p(x|\lambda)=\begin{cases}\lambda e^{-\lambda x}&x\geq 0,\\ 0&x<0,\end{cases}
  24. p ( x | μ , σ ) = 1 σ 2 π e - ( x - μ ) 2 2 σ 2 , p(x|\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}},
  25. Pr ( X = x k ) = C r x k for k = 1 , , n \operatorname{Pr}(X=x_{k})=Cr^{x_{k}}\quad\mbox{ for }~{}k=1,\ldots,n
  26. Pr ( X = x k ) = C r x k for k = 1 , 2 , , \operatorname{Pr}(X=x_{k})=Cr^{x_{k}}\quad\mbox{ for }~{}k=1,2,\ldots,
  27. C = 1 μ - 1 , r = μ - 1 μ , C=\frac{1}{\mu-1},\quad\quad r=\frac{\mu-1}{\mu},
  28. θ i \theta_{i}
  29. θ i \theta_{i}
  30. 2 π 2\pi
  31. \mathbb{R}
  32. p ( x ) = c exp ( λ 1 x + λ 2 x 2 + λ 3 x 3 ) p(x)=c\exp{(\lambda_{1}x+\lambda_{2}x^{2}+\lambda_{3}x^{3})}
  33. λ 3 = 0 \lambda_{3}=0
  34. ϵ \epsilon
  35. σ \sigma
  36. Γ ( x ) = 0 e - t t x - 1 d t \Gamma(x)=\int_{0}^{\infty}e^{-t}t^{x-1}dt
  37. ψ ( x ) = d d x ln Γ ( x ) = Γ ( x ) Γ ( x ) \psi(x)=\frac{d}{dx}\ln\Gamma(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}
  38. B ( p , q ) = Γ ( p ) Γ ( q ) Γ ( p + q ) B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}
  39. f ( k ) = 1 b - a + 1 f(k)=\frac{1}{b-a+1}
  40. { a , a + 1 , , b - 1 , b } \{a,a+1,...,b-1,b\}\,
  41. f ( x ) = 1 b - a f(x)=\frac{1}{b-a}
  42. [ a , b ] [a,b]\,
  43. f ( k ) = p k ( 1 - p ) 1 - k f(k)=p^{k}(1-p)^{1-k}
  44. E ( k ) = p E(k)=p\,
  45. { 0 , 1 } \{0,1\}\,
  46. f ( k ) = ( 1 - p ) k - 1 p f(k)=(1-p)^{k-1}\,p
  47. E ( k ) = 1 p E(k)=\frac{1}{p}\,
  48. { 1 , 2 , 3 , } \{1,2,3,...\}\,
  49. f ( x ) = λ exp ( - λ x ) f(x)=\lambda\exp\left(-\lambda x\right)
  50. E ( x ) = 1 λ E(x)=\frac{1}{\lambda}\,
  51. [ 0 , ) [0,\infty)\,
  52. f ( x ) = 1 2 b exp ( - | x - μ | b ) f(x)=\frac{1}{2b}\exp\left(-\frac{|x-\mu|}{b}\right)
  53. E ( | x - μ | ) = b E(|x-\mu|)=b\,
  54. ( - , ) (-\infty,\infty)\,
  55. f ( x ) = α x m α x α + 1 f(x)=\frac{\alpha x_{m}^{\alpha}}{x^{\alpha+1}}
  56. E ( ln ( x ) ) = 1 α + ln ( x m ) E(\ln(x))=\frac{1}{\alpha}+\ln(x_{m})\,
  57. [ x m , ) [x_{m},\infty)\,
  58. f ( x ) = 1 2 π σ 2 exp ( - ( x - μ ) 2 2 σ 2 ) f(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)
  59. E ( x ) = μ , E ( ( x - μ ) 2 ) = σ 2 E(x)=\mu,\,E((x-\mu)^{2})=\sigma^{2}
  60. ( - , ) (-\infty,\infty)\,
  61. f ( θ ) = 1 2 π I 0 ( κ ) exp ( κ cos ( θ - μ ) ) f(\theta)=\frac{1}{2\pi I_{0}(\kappa)}\exp{(\kappa\cos{(\theta-\mu)})}
  62. E ( cos θ ) = I 1 ( κ ) I 0 ( κ ) cos μ , E ( sin θ ) = I 1 ( κ ) I 0 ( κ ) sin μ E(\cos\theta)=\frac{I_{1}(\kappa)}{I_{0}(\kappa)}\cos\mu,\,E(\sin\theta)=\frac% {I_{1}(\kappa)}{I_{0}(\kappa)}\sin\mu
  63. [ 0 , 2 π ) [0,2\pi)\,
  64. f ( x ) = x σ 2 exp ( - x 2 2 σ 2 ) f(x)=\frac{x}{\sigma^{2}}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)
  65. E ( x 2 ) = 2 σ 2 , E ( ln ( x ) ) = ln ( 2 σ 2 ) - γ E 2 E(x^{2})=2\sigma^{2},E(\ln(x))=\frac{\ln(2\sigma^{2})-\gamma_{E}}{2}\,
  66. [ 0 , ) [0,\infty)\,
  67. f ( x ) = x α - 1 ( 1 - x ) β - 1 B ( α , β ) f(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}
  68. 0 x 1 0\leq x\leq 1
  69. E ( ln ( x ) ) = ψ ( α ) - ψ ( α + β ) E(\ln(x))=\psi(\alpha)-\psi(\alpha+\beta)\,
  70. E ( ln ( 1 - x ) ) = ψ ( β ) - ψ ( α + β ) E(\ln(1-x))=\psi(\beta)-\psi(\alpha+\beta)\,
  71. [ 0 , 1 ] [0,1]\,
  72. f ( x ) = 1 π ( 1 + x 2 ) f(x)=\frac{1}{\pi(1+x^{2})}
  73. E ( ln ( 1 + x 2 ) ) = 2 ln 2 E(\ln(1+x^{2}))=2\ln 2
  74. ( - , ) (-\infty,\infty)\,
  75. f ( x ) = 2 2 k / 2 Γ ( k / 2 ) x k - 1 exp ( - x 2 2 ) f(x)=\frac{2}{2^{k/2}\Gamma(k/2)}x^{k-1}\exp\left(-\frac{x^{2}}{2}\right)
  76. E ( x 2 ) = k , E ( ln ( x ) ) = 1 2 [ ψ ( k 2 ) + ln ( 2 ) ] E(x^{2})=k,\,E(\ln(x))=\frac{1}{2}\left[\psi\left(\frac{k}{2}\right)\!+\!\ln(2% )\right]
  77. [ 0 , ) [0,\infty)\,
  78. f ( x ) = 1 2 k / 2 Γ ( k / 2 ) x k 2 - 1 exp ( - x 2 ) f(x)=\frac{1}{2^{k/2}\Gamma(k/2)}x^{\frac{k}{2}\!-\!1}\exp\left(-\frac{x}{2}\right)
  79. E ( x ) = k , E ( ln ( x ) ) = ψ ( k 2 ) + ln ( 2 ) E(x)=k,\,E(\ln(x))=\psi\left(\frac{k}{2}\right)+\ln(2)
  80. [ 0 , ) [0,\infty)\,
  81. f ( x ) = λ k ( k - 1 ) ! x k - 1 exp ( - λ x ) f(x)=\frac{\lambda^{k}}{(k-1)!}x^{k-1}\exp(-\lambda x)
  82. E ( x ) = k / λ , E ( ln ( x ) ) = ψ ( k ) - ln ( λ ) E(x)=k/\lambda,\,E(\ln(x))=\psi(k)-\ln(\lambda)
  83. [ 0 , ) [0,\infty)\,
  84. f ( x ) = x k - 1 exp ( - x θ ) θ k Γ ( k ) f(x)=\frac{x^{k-1}\exp(-\frac{x}{\theta})}{\theta^{k}\Gamma(k)}
  85. E ( x ) = k θ , E ( ln ( x ) ) = ψ ( k ) + ln ( θ ) E(x)=k\theta,\,E(\ln(x))=\psi(k)+\ln(\theta)
  86. [ 0 , ) [0,\infty)\,
  87. f ( x ) = 1 σ x 2 π exp ( - ( ln x - μ ) 2 2 σ 2 ) f(x)=\frac{1}{\sigma x\sqrt{2\pi}}\exp\left(-\frac{(\ln x-\mu)^{2}}{2\sigma^{2% }}\right)
  88. E ( ln ( x ) ) = μ , E ( ( ln ( x ) - μ ) 2 ) = σ 2 E(\ln(x))=\mu,E((\ln(x)-\mu)^{2})=\sigma^{2}\,
  89. [ 0 , ) [0,\infty)\,
  90. f ( x ) = 1 a 3 2 π x 2 exp ( - x 2 2 a 2 ) f(x)=\frac{1}{a^{3}}\sqrt{\frac{2}{\pi}}\,x^{2}\exp\left(-\frac{x^{2}}{2a^{2}}\right)
  91. E ( x 2 ) = 3 a 2 , E ( ln ( x ) ) = 1 + ln ( a 2 ) - γ E 2 E(x^{2})=3a^{2},\,E(\ln(x))\!=\!1\!+\!\ln\left(\frac{a}{\sqrt{2}}\right)\!-\!% \frac{\gamma_{E}}{2}
  92. [ 0 , ) [0,\infty)\,
  93. f ( x ) = k λ k x k - 1 exp ( - x k λ k ) f(x)=\frac{k}{\lambda^{k}}x^{k-1}\exp\left(-\frac{x^{k}}{\lambda^{k}}\right)
  94. E ( x k ) = λ k , E ( ln ( x ) ) = ln ( λ ) - γ E k E(x^{k})=\lambda^{k},E(\ln(x))=\ln(\lambda)-\frac{\gamma_{E}}{k}\,
  95. [ 0 , ) [0,\infty)\,
  96. f X ( x ) = f_{X}(\vec{x})=
  97. exp ( - 1 2 ( x - μ ) Σ - 1 ( x - μ ) ) ( 2 π ) N / 2 | Σ | 1 / 2 \frac{\exp\left(-\frac{1}{2}(\vec{x}-\vec{\mu})^{\top}\Sigma^{-1}\cdot(\vec{x}% -\vec{\mu})\right)}{(2\pi)^{N/2}\left|\Sigma\right|^{1/2}}
  98. E ( x ) = μ , E ( ( x - μ ) ( x - μ ) T ) = Σ E(\vec{x})=\vec{\mu},\,E((\vec{x}-\vec{\mu})(\vec{x}-\vec{\mu})^{T})=\Sigma\,
  99. ( - , ) (-\vec{\infty},\vec{\infty})\,
  100. f ( k ) = ( n k ) p k ( 1 - p ) n - k f(k)={n\choose k}p^{k}(1-p)^{n-k}
  101. E ( x ) = μ , f n-generalized binomial distribution E(x)=\mu,f\in\,\text{n-generalized binomial distribution}
  102. [ 0 , n ] [0,n]
  103. f ( k ) = exp - λ λ k k ! f(k)=\frac{\exp^{-\lambda}\lambda^{k}}{k!}
  104. E ( x ) = μ , f -generalized binomial distribution E(x)=\mu,f\in{\infty}\,\text{-generalized binomial distribution}
  105. [ 0 , n ] [0,n]

Maximum_length_sequence.html

  1. { a 3 [ n + 1 ] = a 0 [ n ] + a 1 [ n ] a 2 [ n + 1 ] = a 3 [ n ] a 1 [ n + 1 ] = a 2 [ n ] a 0 [ n + 1 ] = a 1 [ n ] \begin{cases}a_{3}[n+1]=a_{0}[n]+a_{1}[n]\\ a_{2}[n+1]=a_{3}[n]\\ a_{1}[n+1]=a_{2}[n]\\ a_{0}[n+1]=a_{1}[n]\\ \end{cases}
  2. + +
  3. 2 n - 1 2^{n}-1
  4. 2 n - 1 2^{n-1}
  5. 2 n - 1 - 1 2^{n-1}-1
  6. y [ n ] = ( h * s ) [ n ] . y[n]=(h*s)[n].\,
  7. ϕ s y = h [ n ] * ϕ s s {\phi}_{sy}=h[n]*{\phi}_{ss}\,
  8. h [ n ] = ϕ s y . h[n]={\phi}_{sy}.\,

Maximum_principle.html

  1. x 0 x_{0}\,
  2. x 0 x_{0}\,
  3. x 0 x_{0}\,
  4. f ( x 0 ) f ( x ) f(x_{0})\geq f(x)\,
  5. x 0 x_{0}\,
  6. x 0 x_{0}\,
  7. x 0 x_{0}\,

Maxwell_coil.html

  1. 4 7 R \sqrt{\frac{4}{7}}R
  2. 3 7 R \sqrt{\frac{3}{7}}R
  3. R R
  4. 49 64 \frac{49}{64}

Mean_motion.html

  1. n n\,\!
  2. n = G ( M + m ) a 3 n=\sqrt{\frac{G(M\!+\!m)}{a^{3}}}\,\!
  3. n = d G ( M + m ) 4 π 2 a 3 n=d\sqrt{\frac{G(M\!+\!m)}{4\pi^{2}a^{3}}}\,\!
  4. d d\!
  5. G G\!
  6. M M\!
  7. m m\!
  8. a a\!
  9. n = 2 π P n=\frac{2\pi}{P}
  10. n = d P n=\frac{d}{P}
  11. n = M 1 - M 0 t n=\frac{M_{1}-M_{0}}{t}

Mean_radiant_temperature.html

  1. M R T 4 = T 1 4 F p - 1 + T 2 4 F p - 2 + + T n 4 F p - n MRT^{4}=T_{1}^{4}F_{p-1}+T_{2}^{4}F_{p-2}+...+T_{n}^{4}F_{p-n}
  2. M R T MRT
  3. T n T_{n}
  4. F p - n F_{p-n}
  5. M R T = T 1 F p - 1 + T 2 F p - 2 + + T n F p - n MRT=T_{1}F_{p-1}+T_{2}F_{p-2}+...+T_{n}F_{p-n}
  6. M R T = [ ( G T + 273 ) 4 + 1 , 1 10 8 v a 0 , 6 ε D 0 , 4 ( G T - T a ) ] 1 / 4 - 273 MRT=\left[\left(GT+273\right)^{4}+\frac{1,1\cdot 10^{8}\cdot v_{a}^{0,6}}{% \varepsilon\cdot D^{0,4}}(GT-T_{a})\right]^{1/4}-273
  7. M R T MRT
  8. G T GT
  9. v a v_{a}
  10. ε \varepsilon
  11. D D
  12. T a T_{a}
  13. ε \varepsilon
  14. M R T = [ ( G T + 273 ) 4 + 2 , 5 10 8 v a 0 , 6 ( G T - T a ) ] 1 / 4 - 273 MRT=\left[\left(GT+273\right)^{4}+2,5\cdot 10^{8}\cdot v_{a}^{0,6}(GT-T_{a})% \right]^{1/4}-273

Mediant_(mathematics).html

  1. a c and b d \frac{a}{c}\,\text{ and }\frac{b}{d}
  2. a + b c + d . \frac{a+b}{c+d}.
  3. a / c < b / d a/c<b/d
  4. a , b , c , d 0 a,b,c,d\geq 0
  5. a c < a + b c + d < b d . \frac{a}{c}<\frac{a+b}{c+d}<\frac{b}{d}.
  6. a + b c + d - a c = b c - a d c ( c + d ) = d c + d ( b d - a c ) \frac{a+b}{c+d}-\frac{a}{c}={{bc-ad}\over{c(c+d)}}={d\over{c+d}}\left(\frac{b}% {d}-\frac{a}{c}\right)
  7. b d - a + b c + d = b c - a d d ( c + d ) = c c + d ( b d - a c ) . \frac{b}{d}-\frac{a+b}{c+d}={{bc-ad}\over{d(c+d)}}={c\over{c+d}}\left(\frac{b}% {d}-\frac{a}{c}\right).
  8. b c - a d = 1 bc-ad=1
  9. a / c a^{\prime}/c^{\prime}
  10. a = λ 1 a + λ 2 b \,a^{\prime}=\lambda_{1}a+\lambda_{2}b
  11. c = λ 1 c + λ 2 d \,c^{\prime}=\lambda_{1}c+\lambda_{2}d
  12. λ 1 , λ 2 \lambda_{1},\,\lambda_{2}
  13. λ i \lambda_{i}
  14. λ 1 a + λ 2 b λ 1 c + λ 2 d - a c = λ 2 b c - a d c ( λ 1 c + λ 2 d ) \frac{\lambda_{1}a+\lambda_{2}b}{\lambda_{1}c+\lambda_{2}d}-\frac{a}{c}=% \lambda_{2}{{bc-ad}\over{c(\lambda_{1}c+\lambda_{2}d)}}
  15. b d - λ 1 a + λ 2 b λ 1 c + λ 2 d = λ 1 b c - a d d ( λ 1 c + λ 2 d ) \frac{b}{d}-\frac{\lambda_{1}a+\lambda_{2}b}{\lambda_{1}c+\lambda_{2}d}=% \lambda_{1}{{bc-ad}\over{d(\lambda_{1}c+\lambda_{2}d)}}
  16. b c - a d = 1 bc-ad=1\,
  17. λ 1 , λ 2 \lambda_{1},\,\lambda_{2}
  18. a = λ 1 a + λ 2 b \,a^{\prime}=\lambda_{1}a+\lambda_{2}b
  19. c = λ 1 c + λ 2 d \,c^{\prime}=\lambda_{1}c+\lambda_{2}d
  20. λ 1 , λ 2 \lambda_{1},\lambda_{2}
  21. c c + d . c^{\prime}\geq c+d.
  22. Δ ( v 1 , v 2 , v 3 ) \Delta(v_{1},v_{2},v_{3})
  23. area ( Δ ) = b c - a d 2 . \,\text{area}(\Delta)={{bc-ad}\over 2}\ .
  24. p = ( p 1 , p 2 ) p=(p_{1},p_{2})
  25. p 1 = λ 1 a + λ 2 b , p 2 = λ 1 c + λ 2 d , p_{1}=\lambda_{1}a+\lambda_{2}b,\;p_{2}=\lambda_{1}c+\lambda_{2}d,
  26. λ 1 0 , λ 2 0 , λ 1 + λ 2 1. \lambda_{1}\geq 0,\,\lambda_{2}\geq 0,\,\lambda_{1}+\lambda_{2}\leq 1.\,
  27. area ( Δ ) = v interior + v boundary 2 - 1 \,\text{area}(\Delta)=v_{\mathrm{interior}}+{v_{\mathrm{boundary}}\over 2}-1
  28. 1 \geq 1
  29. q 2 = λ 1 c + λ 2 d max ( c , d ) < c + d q_{2}=\lambda_{1}c+\lambda_{2}d\leq\max(c,d)<c+d
  30. λ 1 + λ 2 1. \lambda_{1}+\lambda_{2}\leq 1.\,
  31. ? ( p + r q + s ) = 1 2 ( ? ( p q ) + ? ( r s ) ) ?\left(\frac{p+r}{q+s}\right)=\frac{1}{2}\left(?\bigg(\frac{p}{q}\bigg)+{}?% \bigg(\frac{r}{s}\bigg)\right)
  32. b c - a d = 1 bc-ad=1
  33. m w m_{w}
  34. a 1 / b 2 , , a n / b n a_{1}/b_{2},\ldots,a_{n}/b_{n}
  35. i w i a i i w i b i \frac{\sum_{i}w_{i}a_{i}}{\sum_{i}w_{i}b_{i}}
  36. w i > 0 w_{i}>0
  37. m w m_{w}
  38. a i / b i a_{i}/b_{i}

Megaminx.html

  1. 20 ! 2 × 3 19 × 30 ! 2 × 2 29 1.01 × 10 68 \frac{20!}{2}\times 3^{19}\times\frac{30!}{2}\times 2^{29}\approx 1.01\times 1% 0^{68}
  2. 20 ! 2 × 3 19 × 30 ! 2 × 2 29 2 14 6.14 × 10 63 \frac{20!}{2}\times 3^{19}\times\frac{30!}{2}\times\frac{2^{29}}{2^{14}}% \approx 6.14\times 10^{63}

Megathrust_earthquake.html

  1. M w M_{w}
  2. M W M_{W}

Mellin_inversion_theorem.html

  1. φ ( s ) \varphi(s)
  2. a < ( s ) < b a<\Re(s)<b
  3. ( s ) ± \Im(s)\to\pm\infty
  4. f ( x ) = { - 1 φ } = 1 2 π i c - i c + i x - s φ ( s ) d s f(x)=\{\mathcal{M}^{-1}\varphi\}=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x% ^{-s}\varphi(s)\,ds
  5. φ ( s ) = { f } = 0 x s f ( x ) d x x . \varphi(s)=\{\mathcal{M}f\}=\int_{0}^{\infty}x^{s}f(x)\,\frac{dx}{x}.
  6. φ ( s ) = 0 x s f ( x ) d x x \varphi(s)=\int_{0}^{\infty}x^{s}f(x)\,\frac{dx}{x}
  7. a < ( s ) < b a<\Re(s)<b
  8. φ \varphi
  9. φ ( s ) \varphi(s)
  10. φ ( s ) \varphi(s)
  11. a < ( s ) < b a<\Re(s)<b
  12. | φ ( s ) | < K | s | - 2 |\varphi(s)|<K|s|^{-2}
  13. φ \varphi
  14. a < ( s ) < b a<\Re(s)<b
  15. φ \varphi
  16. a < ( s ) < b a<\Re(s)<b
  17. L ν , p ( R + ) L_{\nu,p}(R^{+})
  18. f = ( 0 | x ν f ( x ) | p d x x ) 1 / p < \|f\|=\left(\int_{0}^{\infty}|x^{\nu}f(x)|^{p}\,\frac{dx}{x}\right)^{1/p}<\infty
  19. L ν , p ( R + ) L_{\nu,p}(R^{+})
  20. 1 < p 2 1<p\leq 2
  21. φ ( s ) \varphi(s)
  22. L ν , q ( R + ) L_{\nu,q}(R^{+})
  23. q = p / ( p - 1 ) q=p/(p-1)
  24. f ( x ) = 1 2 π i ν - i ν + i x - s φ ( s ) d s . f(x)=\frac{1}{2\pi i}\int_{\nu-i\infty}^{\nu+i\infty}x^{-s}\varphi(s)\,ds.
  25. { f } ( s ) = { f ( - ln x ) } ( s ) \left\{\mathcal{B}f\right\}(s)=\left\{\mathcal{M}f(-\ln x)\right\}(s)

Membership_function_(mathematics).html

  1. X X
  2. X X
  3. X X
  4. [ 0 , 1 ] [0,1]
  5. X X
  6. X X
  7. A ~ \tilde{A}
  8. μ A . \mu_{A}.
  9. x x
  10. X X
  11. μ A ( x ) \mu_{A}(x)
  12. x x
  13. A ~ . \tilde{A}.
  14. μ A ( x ) \mu_{A}(x)
  15. x x
  16. A ~ . \tilde{A}.
  17. x x
  18. x x
  19. L L
  20. L L
  21. ν \nu
  22. [ 0 , 1 ] [0,1]
  23. ν \nu
  24. ν ( ) = 0 , ν ( Ω ) = 1 ) . \nu()=0,\nu(\Omega)=1).

Menelaus'_theorem.html

  1. A F F B × B D D C × C E E A = - 1. \frac{AF}{FB}\times\frac{BD}{DC}\times\frac{CE}{EA}=-1.
  2. A F × B D × C E = - F B × D C × E A . AF\times BD\times CE=-FB\times DC\times EA.
  3. A F F B × B D D C × C E E A = - 1 , \frac{AF}{FB}\times\frac{BD}{DC}\times\frac{CE}{EA}=-1,
  4. | A F F B | | B D D C | | C E E A | = | a b b c c a | = 1. \left|\frac{AF}{FB}\right|\cdot\left|\frac{BD}{DC}\right|\cdot\left|\frac{CE}{% EA}\right|=\left|\frac{a}{b}\cdot\frac{b}{c}\cdot\frac{c}{a}\right|=1.
  5. | B D D C | = | B F C K | , | A E E C | = | A F C K | \left|\frac{BD}{DC}\right|=\left|\frac{BF}{CK}\right|,\,\left|\frac{AE}{EC}% \right|=\left|\frac{AF}{CK}\right|
  6. A F F B = A F F B . \frac{AF}{FB}=\frac{AF^{\prime}}{F^{\prime}B}.
  7. D C D B × E A E C × F B F A = 1 , \frac{\overrightarrow{DC}}{\overrightarrow{DB}}\times\frac{\overrightarrow{EA}% }{\overrightarrow{EC}}\times\frac{\overrightarrow{FB}}{\overrightarrow{FA}}=1,

Mercury_coulometer.html

  1. Q = 2 Δ m F M Hg Q=\frac{2\Delta mF}{M_{\rm Hg}}
  2. Δ m \Delta\ m
  3. M Hg M_{\rm Hg}

Mereotopology.html

  1. C x x . \ Cxx.
  2. C x y C y x . Cxy\rightarrow Cyx.
  3. E x y [ C z x C z y ] . Exy\leftrightarrow[Czx\rightarrow Czy].
  4. ( E x a E x b ) ( a = b ) , (Exa\leftrightarrow Exb)\leftrightarrow(a=b),
  5. P x y E x y . \ Pxy\rightarrow Exy.
  6. O x y \exist z [ P z x and P z y ] . Oxy\leftrightarrow\exist z[Pzx\and\ Pzy].
  7. O x y C x y . Oxy\rightarrow Cxy.
  8. I P x y ( P x y and ( C z x O z y ) ) . IPxy\leftrightarrow(Pxy\and(Czx\rightarrow Ozy)).
  9. 𝐢 x , \mathbf{i}x,
  10. 𝐢 x = \mathbf{i}x=
  11. σ z [ I P z x ] . \sigma z[IPzx].
  12. 𝐢 W = W , \mathbf{i}W=W,
  13. P ( 𝐢 x ) x . \ P(\mathbf{i}x)x.
  14. 𝐢 ( 𝐢 x ) = 𝐢 x . \mathbf{i}(\mathbf{i}x)=\mathbf{i}x.
  15. 𝐢 ( x × y ) = 𝐢 x × 𝐢 y , \mathbf{i}(x\times y)=\mathbf{i}x\times\mathbf{i}y,
  16. S C x ( ( O w x ( O w y O w z ) ) C y z ) . SCx\leftrightarrow((Owx\leftrightarrow(OwyOwz))\rightarrow Cyz).
  17. C x y \exist z [ S C z and O z x and ( P w z ( O w x O w y ) ) . Cxy\rightarrow\exist z[SCz\and Ozx\and(Pwz\rightarrow(OwxOwy)).
  18. x \exist y [ P y x and ( C z y O z x ) and ¬ ( P x y and ( C z x O z y ) ) ] . \forall x\exist y[Pyx\and(Czy\rightarrow Ozx)\and\lnot(Pxy\and(Czx\rightarrow Ozy% ))].

Metabelian_group.html

  1. x a x + b x\mapsto ax+b
  2. x x + b x\mapsto x+b
  3. x a x x\mapsto ax

Metabolic_acidosis.html

  1. H + + HCO 3 - H 2 CO 3 CO 2 + H 2 O H^{+}+\,\text{HCO}_{3}{}^{-}\Longleftrightarrow H_{2}\,\text{CO}_{3}% \Longleftrightarrow\,\text{CO}_{2}+H_{2}O
  2. pH = pK a + Log [ HCO 3 - ] [ CO 2 ] \,\text{pH}=\,\text{pK}_{a}+\,\text{Log}\frac{\left[\,\text{HCO}_{3}^{-}\right% ]}{\left[\,\text{CO}_{2}\right]}
  3. pH = 6.1 + Log [ 24 0.03 × 40 ] \,\text{pH}=6.1+\,\text{Log}\left[\frac{24}{0.03\times 40}\right]
  4. = 6.1 + 1.3 =6.1+1.3
  5. = 7.4 =7.4

Metabolic_engineering.html

  1. V x = - ( G x ) - 1 * ( G m * V m ) V_{x}=-(G_{x})^{-1}*(G_{m}*V_{m})

Metabolic_equivalent.html

  1. 1 MET 1 kcal kg * h 4.184 kJ kg * h \,\text{1 MET}\ \equiv\ 1\dfrac{\,\text{kcal}}{\,\text{kg}*{h}}\ \equiv\ 4.184% \dfrac{\,\text{kJ}}{\,\text{kg}*{h}}

Metapopulation.html

  1. d N d t = c N ( 1 - N ) - e N . \frac{dN}{dt}=cN(1-N)-eN.\,
  2. K = 1 - e c K=1-\frac{e}{c}\,
  3. r = c - e . r=c-e.\,

Method_of_distinguished_element.html

  1. 𝒜 \mathcal{A}
  2. A A
  3. x A x\in A
  4. A A
  5. P ( X , x ) P(X,x)
  6. X A X\subseteq A
  7. x x
  8. 𝒜 ( x ) \mathcal{A}(x)
  9. X X
  10. 𝒜 \mathcal{A}
  11. P ( X , x ) P(X,x)
  12. 𝒜 - x \mathcal{A}-x
  13. X X
  14. 𝒜 \mathcal{A}
  15. P ( X , x ) P(X,x)
  16. 𝒜 ( x ) \mathcal{A}(x)
  17. 𝒜 - x \mathcal{A}-x
  18. | 𝒜 | = | 𝒜 ( x ) | + | 𝒜 - x | |\mathcal{A}|=|\mathcal{A}(x)|+|\mathcal{A}-x|
  19. ( n k ) {n\choose k}
  20. ( n k - 1 ) + ( n k ) = ( n + 1 k ) . {n\choose k-1}+{n\choose k}={n+1\choose k}.
  21. ( n k - 1 ) {n\choose k-1}
  22. ( n k ) {n\choose k}
  23. a b c a / b c b / a c c / a b a / b / c \begin{matrix}abc\\ a/bc\\ b/ac\\ c/ab\\ a/b/c\end{matrix}
  24. B n + C n = B n + 1 . B_{n}+C_{n}=B_{n+1}.\,

Method_of_undetermined_coefficients.html

  1. i = 0 n c i y ( i ) + y ( n + 1 ) = g ( x ) \sum_{i=0}^{n}c_{i}y^{(i)}+y^{(n+1)}=g(x)
  2. y ( i ) y^{(i)}
  3. c i c_{i}
  4. y c y_{c}
  5. i = 0 n c i y ( i ) + y ( n + 1 ) = 0 , \sum_{i=0}^{n}c_{i}y^{(i)}+y^{(n+1)}=0,
  6. y p y_{p}
  7. g ( x ) g(x)
  8. y y
  9. y = y c + y p . y=y_{c}+y_{p}.
  10. g ( x ) g(x)
  11. h ( x ) + w ( x ) h(x)+w(x)
  12. y p 1 y_{p_{1}}
  13. h ( x ) h(x)
  14. y p 2 y_{p_{2}}
  15. w ( x ) w(x)
  16. y p y_{p}
  17. y p = y p 1 + y p 2 . y_{p}=y_{p_{1}}+y_{p_{2}}.
  18. k e a x ke^{ax}\!
  19. C e a x Ce^{ax}\!
  20. k x n , n = 0 , 1 , 2 , kx^{n},\;n=0,1,2,\ldots\!
  21. i = 0 n K i x i \sum_{i=0}^{n}K_{i}x^{i}\!
  22. k cos ( a x ) or k sin ( a x ) k\cos(ax)\,\text{ or }k\sin(ax)\!
  23. K cos ( a x ) + M sin ( a x ) K\cos(ax)+M\sin(ax)\!
  24. k e a x cos ( b x ) or k e a x sin ( b x ) ke^{ax}\cos(bx)\,\text{ or }ke^{ax}\sin(bx)\!
  25. e a x ( K cos ( b x ) + M sin ( b x ) ) e^{ax}(K\cos(bx)+M\sin(bx))\!
  26. ( i = 1 n k i x i ) cos ( b x ) or ( i = 1 n k i x i ) sin ( b x ) \left(\sum_{i=1}^{n}k_{i}x^{i}\right)\cos(bx)\,\text{ or }\ \left(\sum_{i=1}^{% n}k_{i}x^{i}\right)\sin(bx)\!
  27. ( i = 1 n k i x i ) ( K cos ( b x ) + M sin ( b x ) ) \left(\sum_{i=1}^{n}k_{i}x^{i}\right)(K\cos(bx)+M\sin(bx))\!
  28. ( i = 1 n k i x i ) e a x cos ( b x ) or ( i = 1 n k i x i ) e a x sin ( b x ) \left(\sum_{i=1}^{n}k_{i}x^{i}\right)e^{ax}\cos(bx)\,\text{ or }\left(\sum_{i=% 1}^{n}k_{i}x^{i}\right)e^{ax}\sin(bx)\!
  29. e a x ( ( i = 1 n Q i x i ) K cos ( b x ) + ( i = 1 n R i x i ) M sin ( b x ) ) e^{ax}\left(\left(\sum_{i=1}^{n}Q_{i}x^{i}\right)K\cos(bx)+\left(\sum_{i=1}^{n% }R_{i}x^{i}\right)M\sin(bx)\right)
  30. y ′′ + y = t cos t . y^{\prime\prime}+y=t\cos{t}.\!
  31. P n e α t cos β t P_{n}e^{\alpha t}\cos{\beta t}\!
  32. λ 2 + 1 = 0 \lambda^{2}+1=0\!
  33. y p = t [ F 1 ( t ) e α t cos β t + G 1 ( t ) e α t sin β t ] = t [ F 1 ( t ) cos t + G 1 ( t ) sin t ] = t [ ( A 0 t + A 1 ) cos t + ( B 0 t + B 1 ) sin t ] = ( A 0 t 2 + A 1 t ) cos t + ( B 0 t 2 + B 1 t ) sin t . \begin{aligned}\displaystyle y_{p}&\displaystyle=t[F_{1}(t)e^{\alpha t}\cos{% \beta t}+G_{1}(t)e^{\alpha t}\sin{\beta t}]\\ &\displaystyle=t[F_{1}(t)\cos t+G_{1}(t)\sin t]\\ &\displaystyle=t[(A_{0}t+A_{1})\cos t+(B_{0}t+B_{1})\sin t]\\ &\displaystyle=(A_{0}t^{2}+A_{1}t)\cos t+(B_{0}t^{2}+B_{1}t)\sin t.\end{aligned}
  34. t cos t = y p ′′ + y p = [ ( A 0 t 2 + A 1 t ) cos t + ( B 0 t 2 + B 1 t ) sin t ] ′′ + [ ( A 0 t 2 + A 1 t ) cos t + ( B 0 t 2 + B 1 t ) sin t ] = [ 2 A 0 cos t + 2 ( 2 A 0 t + A 1 ) ( - sin t ) + ( A 0 t 2 + A 1 t ) ( - cos t ) ] + [ 2 B 0 sin t + 2 ( 2 B 0 t + B 1 ) cos t + ( B 0 t 2 + B 1 t ) ( - sin t ) ] + [ ( A 0 t 2 + A 1 t ) cos t + ( B 0 t 2 + B 1 t ) sin t ] = [ 4 B 0 t + ( 2 A 0 + 2 B 1 ) ] cos t + [ - 4 A 0 t + ( - 2 A 1 + 2 B 0 ) ] sin t . \begin{aligned}\displaystyle t\cos t&\displaystyle=y_{p}^{\prime\prime}+y_{p}% \\ &\displaystyle=[(A_{0}t^{2}+A_{1}t)\cos t+(B_{0}t^{2}+B_{1}t)\sin t]^{\prime% \prime}\\ &\displaystyle\quad+[(A_{0}t^{2}+A_{1}t)\cos t+(B_{0}t^{2}+B_{1}t)\sin t]\\ &\displaystyle=[2A_{0}\cos t+2(2A_{0}t+A_{1})(-\sin t)+(A_{0}t^{2}+A_{1}t)(-% \cos t)]\\ &\displaystyle\quad+[2B_{0}\sin t+2(2B_{0}t+B_{1})\cos t+(B_{0}t^{2}+B_{1}t)(-% \sin t)]\\ &\displaystyle\quad+[(A_{0}t^{2}+A_{1}t)\cos t+(B_{0}t^{2}+B_{1}t)\sin t]\\ &\displaystyle=[4B_{0}t+(2A_{0}+2B_{1})]\cos t+[-4A_{0}t+(-2A_{1}+2B_{0})]\sin t% .\end{aligned}
  35. 4 B 0 = 1 2 A 0 + 2 B 1 = 0 - 4 A 0 = 0 - 2 A 1 + 2 B 0 = 0 \begin{array}[]{rrrrl}&&4B_{0}&&=1\\ 2A_{0}&&&+2B_{1}&=0\\ -4A_{0}&&&&=0\\ &-2A_{1}&+2B_{0}&&=0\end{array}
  36. A 0 = 0 A_{0}=0
  37. A 1 = 1 / 4 A_{1}=1/4
  38. B 0 = 1 / 4 B_{0}=1/4
  39. B 1 = 0 B_{1}=0
  40. y p = 1 4 t cos t + 1 4 t 2 sin t . y_{p}=\frac{1}{4}t\cos t+\frac{1}{4}t^{2}\sin t.
  41. d y d x = y + e x . \frac{dy}{dx}=y+e^{x}.
  42. e x e^{x}
  43. c 1 e x c_{1}e^{x}
  44. y p = A x e x . y_{p}=Axe^{x}.
  45. d d x ( A x e x ) = A x e x + e x \frac{d}{dx}\left(Axe^{x}\right)=Axe^{x}+e^{x}
  46. A x e x + A e x = A x e x + e x Axe^{x}+Ae^{x}=Axe^{x}+e^{x}
  47. A = 1. A=1.
  48. y = c 1 e x + x e x . y=c_{1}e^{x}+xe^{x}.
  49. d y d t = t 2 - y \frac{dy}{dt}=t^{2}-y
  50. t 2 t^{2}
  51. y p = A t 2 + B t + C y_{p}=At^{2}+Bt+C
  52. d y p d t = 2 A t + B \frac{dy_{p}}{dt}=2At+B
  53. 2 A t + B = t 2 - ( A t 2 + B t + C ) 2At+B=t^{2}-(At^{2}+Bt+C)
  54. t 2 - A t 2 = 0 t^{2}-At^{2}=0
  55. - B t = 2 A t -Bt=2At
  56. - C = B -C=B
  57. y p = t 2 - 2 t + 2 y_{p}=t^{2}-2t+2
  58. y = y p + y c y=y_{p}+y_{c}
  59. y c y_{c}
  60. y c = c 1 e - t y_{c}=c_{1}e^{-t}
  61. y = t 2 - 2 t + 2 + c 1 e - t y=t^{2}-2t+2+c_{1}e^{-t}

Metric_(mathematics).html

  1. 6 2 8.49 6\sqrt{2}\approx 8.49
  2. ( p n ) n N (p_{n})_{n\in N}
  3. d ( x , y ) = n = 1 1 2 n p n ( x - y ) 1 + p n ( x - y ) d(x,y)=\sum_{n=1}^{\infty}\frac{1}{2^{n}}\frac{p_{n}(x-y)}{1+p_{n}(x-y)}
  4. 1 2 n \frac{1}{2^{n}}
  5. ( a n ) (a_{n})
  6. d d
  7. min ( d , 1 ) \min(d,1)
  8. d 1 + d {d\over 1+d}
  9. d . d.
  10. ( X , ) (X,\|\cdot\|)
  11. d ( x , y ) := x - y d(x,y):=\|x-y\|
  12. \|\cdot\|
  13. d ( x , y ) = d ( x + a , y + a ) d(x,y)=d(x+a,y+a)
  14. d ( α x , α y ) = | α | d ( x , y ) d(\alpha x,\alpha y)=|\alpha|d(x,y)
  15. x := d ( x , 0 ) \|x\|:=d(x,0)
  16. Z = X Y Z=XY
  17. Z Z
  18. X X
  19. Y Y
  20. x x
  21. X X
  22. Y Y
  23. Z Z
  24. d d
  25. d ( X ) = 0 d(X)=0
  26. X X
  27. d ( X ) > 0 d(X)>0
  28. d ( X ) d(X)
  29. X X
  30. d ( X Y ) d ( X Z ) + d ( Z Y ) d(XY)\leq d(XZ)+d(ZY)
  31. X X
  32. X , Y , Z X,Y,Z
  33. X X
  34. x x
  35. d ( X ) = 0 d(X)=0
  36. X X
  37. d ( X ) = max { x : x X } - min { x : x X } d(X)=\max\{x:x\in X\}-\min\{x:x\in X\}
  38. d ( x , y ) = 0 d(x,y)=0
  39. x y x\neq y
  40. 1 / 2 {1}/{2}

Metric_dimension_(graph_theory).html

  1. W = { w 1 , w 2 , , w k } W=\{w_{1},w_{2},\dots,w_{k}\}
  2. r ( v | W ) = ( d ( v , w 1 ) , d ( v , w 2 ) , , d ( v , w k ) ) r(v|W)=(d(v,w_{1}),d(v,w_{2}),\dots,d(v,w_{k}))
  3. G G
  4. G G
  5. n n
  6. n 1 n−1
  7. n n
  8. n 2 n−2
  9. K s + K t ¯ ( s 1 , t 2 ) K_{s}+\overline{K_{t}}(s\geq 1,t\geq 2)
  10. K s + ( K 1 K t ) ( s , t 1 ) K_{s}+(K_{1}\cup K_{t})(s,t\geq 1)
  11. n D β - 1 + β n\leq D^{\beta-1}+\beta
  12. n n
  13. D D
  14. 2 log n 2\log n
  15. ( n 2 ) {\textstyle\left({{n}\atop{2}}\right)}
  16. log n + log log 2 n + 1 \log n+\log\log_{2}n+1
  17. ( 1 - ϵ ) log n (1-\epsilon)\log n
  18. ϵ > 0 \epsilon>0

Micellar_electrokinetic_chromatography.html

  1. u o u_{o}
  2. t M t_{M}
  3. u c u_{c}
  4. t c t_{c}
  5. u c = u p + u o u_{c}=u_{p}+u_{o}
  6. u p u_{p}
  7. k 1 k^{1}
  8. k 1 = n c n w k^{1}=\frac{n_{c}}{n_{w}}
  9. n c n_{c}
  10. n w n_{w}
  11. t M t r t c t_{M}\leq t_{r}\leq t_{c}
  12. R R
  13. R = u s - u c u o - u c R=\frac{u_{s}-u_{c}}{u_{o}-u_{c}}
  14. u s u_{s}
  15. R R
  16. R = 1 1 + k 1 R=\frac{1}{1+k^{1}}
  17. L L
  18. u o = L / t M u_{o}=L/t_{M}
  19. u c = L / t c u_{c}=L/t_{c}
  20. u s = L / t r u_{s}=L/t_{r}
  21. k 1 = t r - t M t M ( 1 - ( t r / t c ) ) k^{1}=\frac{t_{r}-t_{M}}{t_{M}(1-(t_{r}/t_{c}))}
  22. k 1 k^{1}
  23. k = t r - t M t M k=\frac{t_{r}-t_{M}}{t_{M}}
  24. t r = ( 1 + k 1 1 + ( t M / t c ) k 1 ) t M t_{r}=\left(\frac{1+k^{1}}{1+(t_{M}/t_{c})k^{1}}\right)t_{M}
  25. k 1 k^{1}
  26. t c t_{c}
  27. t M t_{M}
  28. k 1 k^{1}

Microstrip.html

  1. Z microstrip = Z 0 2 π 2 ( 1 + ε r ) ln ( 1 + 4 h w eff ( 14 + 8 ε r 11 4 h w eff + ( 14 + 8 ε r 11 4 h w eff ) 2 + π 2 1 + 1 ε r 2 ) ) , Z_{\textrm{microstrip}}=\frac{Z_{0}}{2\pi\sqrt{2(1+\varepsilon_{r})}}\mathrm{% ln}\left(1+\frac{4h}{w_{\textrm{eff}}}\left(\frac{14+\frac{8}{\varepsilon_{r}}% }{11}\frac{4h}{w_{\textrm{eff}}}+\sqrt{\left(\frac{14+\frac{8}{\varepsilon_{r}% }}{11}\frac{4h}{w_{\textrm{eff}}}\right)^{2}+\pi^{2}\frac{1+\frac{1}{% \varepsilon_{r}}}{2}}\right)\right),
  2. w eff w_{\mathrm{eff}}
  3. w eff = w + t 1 + 1 ε r 2 π ln ( 4 e ( t h ) 2 + ( 1 π 1 w t + 11 10 ) 2 ) . w_{\textrm{eff}}=w+t\frac{1+\frac{1}{\varepsilon_{r}}}{2\pi}\mathrm{ln}\left(% \frac{4e}{\sqrt{\left(\frac{t}{h}\right)^{2}+\left(\frac{1}{\pi}\frac{1}{\frac% {w}{t}+\frac{11}{10}}\right)^{2}}}\right).
  4. w h w\gg h
  5. ε r \varepsilon_{r}
  6. w h w\ll h
  7. ε r = 1 \varepsilon_{r}=1
  8. w h w\ll h
  9. ε r 1. \varepsilon_{r}\gg 1.
  10. w / h > 3.3 w/h>3.3
  11. w / h 3.3 w/h\leq 3.3
  12. w / h = 3.3 w/h=3.3
  13. M = 100 x d % = ( 52 + 65 e - 27 20 w h ) % M=100\frac{x}{d}\%=(52+65e^{-\frac{27}{20}\frac{w}{h}})\%
  14. w / h 0.25 w/h\geq 0.25
  15. ε r 25 \varepsilon_{r}\leq 25
  16. ε r \varepsilon_{r}
  17. 0.25 w / h 2.75 0.25\leq w/h\leq 2.75
  18. 2.5 ε r 25 2.5\leq\varepsilon_{r}\leq 25
  19. d d
  20. w / h w/h

Midparent.html

  1. h e i g h t m i d p a r e n t = 1 2 ( h e i g h t f a t h e r + ( 1.08 × h e i g h t m o t h e r ) ) height_{midparent}=\frac{1}{2}(height_{father}+(1.08\times height_{mother}))

Midpoint_method.html

  1. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 y^{\prime}(t)=f(t,y(t)),\quad y(t_{0})=y_{0}
  2. y n + 1 = y n + h f ( t n + h 2 , y n + h 2 f ( t n , y n ) ) , ( 1 e ) y_{n+1}=y_{n}+hf\left(t_{n}+\frac{h}{2},y_{n}+\frac{h}{2}f(t_{n},y_{n})\right)% ,\qquad\qquad(1e)
  3. y n + 1 = y n + h f ( t n + h 2 , 1 2 ( y n + y n + 1 ) ) , ( 1 i ) y_{n+1}=y_{n}+hf\left(t_{n}+\frac{h}{2},\frac{1}{2}(y_{n}+y_{n+1})\right),% \qquad\qquad(1i)
  4. n = 0 , 1 , 2 , n=0,1,2,\dots
  5. h h
  6. t n = t 0 + n h , t_{n}=t_{0}+nh,
  7. y n y_{n}
  8. y ( t n ) . y(t_{n}).
  9. f f
  10. t = t n + h / 2 , t=t_{n}+h/2,
  11. t n t_{n}
  12. t n + 1 t_{n+1}
  13. O ( h 3 ) O\left(h^{3}\right)
  14. O ( h 2 ) O\left(h^{2}\right)
  15. y n + 1 = y n + h f ( t n , y n ) , y_{n+1}=y_{n}+hf(t_{n},y_{n}),\,
  16. y ( t + h ) y ( t ) + h f ( t , y ( t ) ) ( 2 ) y(t+h)\approx y(t)+hf(t,y(t))\qquad\qquad(2)
  17. y ( t ) y ( t + h ) - y ( t ) h ( 3 ) y^{\prime}(t)\approx\frac{y(t+h)-y(t)}{h}\qquad\qquad(3)
  18. y = f ( t , y ) . y^{\prime}=f(t,y).
  19. y ( t + h 2 ) y ( t + h ) - y ( t ) h y^{\prime}\left(t+\frac{h}{2}\right)\approx\frac{y(t+h)-y(t)}{h}
  20. y ( t + h ) y ( t ) + h f ( t + h 2 , y ( t + h 2 ) ) . ( 4 ) y(t+h)\approx y(t)+hf\left(t+\frac{h}{2},y\left(t+\frac{h}{2}\right)\right).% \qquad\qquad(4)
  21. y ( t + h ) y(t+h)
  22. y y
  23. t + h / 2 t+h/2
  24. y ( t + h / 2 ) y(t+h/2)
  25. y ( t + h 2 ) y ( t ) + h 2 y ( t ) = y ( t ) + h 2 f ( t , y ( t ) ) , y\left(t+\frac{h}{2}\right)\approx y(t)+\frac{h}{2}y^{\prime}(t)=y(t)+\frac{h}% {2}f(t,y(t)),
  26. y ( t + h ) y ( t ) + h f ( t + h 2 , y ( t ) + h 2 f ( t , y ( t ) ) ) y(t+h)\approx y(t)+hf\left(t+\frac{h}{2},y(t)+\frac{h}{2}f(t,y(t))\right)
  27. t + h / 2 t+h/2
  28. y ( t ) y(t)
  29. y ( t + h ) y(t+h)
  30. y ( t + h 2 ) 1 2 ( y ( t ) + y ( t + h ) ) y\left(t+\frac{h}{2}\right)\approx\frac{1}{2}\bigl(y(t)+y(t+h)\bigr)
  31. y ( t + h ) - y ( t ) h y ( t + h 2 ) k = f ( t + h 2 , 1 2 ( y ( t ) + y ( t + h ) ) ) \frac{y(t+h)-y(t)}{h}\approx y^{\prime}\left(t+\frac{h}{2}\right)\approx k=f% \left(t+\frac{h}{2},\frac{1}{2}\bigl(y(t)+y(t+h)\bigr)\right)
  32. y n + h k y_{n}+h\,k
  33. y ( t n + h ) y(t_{n}+h)
  34. k = f ( t n + h 2 , y n + h 2 k ) y n + 1 = y n + h k \begin{aligned}\displaystyle k&\displaystyle=f\left(t_{n}+\frac{h}{2},y_{n}+% \frac{h}{2}k\right)\\ \displaystyle y_{n+1}&\displaystyle=y_{n}+h\,k\end{aligned}
  35. h / 2 h/2
  36. h h
  37. 𝒪 ( h 3 ) \mathcal{O}(h^{3})
  38. k k

Mills'_constant.html

  1. A 3 n \lfloor A^{3^{n}}\rfloor
  2. a i - 1 3 a_{i-1}^{3}
  3. A 3 n A^{3^{n}}
  4. a i < ( a i - 1 + 1 ) 3 a_{i}<(a_{i-1}+1)^{3}
  5. a 1 a_{1}
  6. ( ( ( ( ( ( ( ( ( 2 3 + 3 ) 3 + 30 ) 3 + 6 ) 3 + 80 ) 3 + 12 ) 3 + 450 ) 3 + 894 ) 3 + 3636 ) 3 + 70756 ) 3 + 97220 , \displaystyle(((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894% )^{3}+3636)^{3}+70756)^{3}+97220,
  7. A a ( n ) 1 / 3 n . A\approx a(n)^{1/3^{n}}.

Mineral_processing.html

  1. C C CC
  2. S G SG
  3. C C = S G ( h e a v y m i n e r a l ) - S G ( f l u i d ) S G ( l i g h t m i n e r a l ) - S G ( f l u i d ) CC=\frac{SG(heavy\ mineral)-SG(fluid)}{SG(light\ mineral)-SG(fluid)}

Minimum-variance_unbiased_estimator.html

  1. g ( θ ) g(\theta)
  2. X 1 , X 2 , , X n X_{1},X_{2},\ldots,X_{n}
  3. p θ , θ Ω p_{\theta},\theta\in\Omega
  4. Ω \Omega
  5. δ ( X 1 , X 2 , , X n ) \delta(X_{1},X_{2},\ldots,X_{n})
  6. g ( θ ) g(\theta)
  7. θ Ω \forall\theta\in\Omega
  8. var ( δ ( X 1 , X 2 , , X n ) ) var ( δ ~ ( X 1 , X 2 , , X n ) ) \mathrm{var}(\delta(X_{1},X_{2},\ldots,X_{n}))\leq\mathrm{var}(\tilde{\delta}(% X_{1},X_{2},\ldots,X_{n}))
  9. δ ~ . \tilde{\delta}.
  10. g ( θ ) g(\theta)
  11. p θ , θ Ω p_{\theta},\theta\in\Omega
  12. δ ( X 1 , X 2 , , X n ) \delta(X_{1},X_{2},\ldots,X_{n})
  13. g ( θ ) g(\theta)
  14. T T
  15. η ( X 1 , X 2 , , X n ) = E ( δ ( X 1 , X 2 , , X n ) | T ) \eta(X_{1},X_{2},\ldots,X_{n})=\mathrm{E}(\delta(X_{1},X_{2},\ldots,X_{n})|T)\,
  16. g ( θ ) . g(\theta).
  17. MSE ( δ ) = var ( δ ) + [ bias ( δ ) ] 2 \operatorname{MSE}(\delta)=\mathrm{var}(\delta)+[\mathrm{bias}(\delta)]^{2}
  18. \mathbb{R}
  19. p θ ( x ) = θ e - x ( 1 + e - x ) θ + 1 p_{\theta}(x)=\frac{\theta e^{-x}}{(1+e^{-x})^{\theta+1}}
  20. g ( θ ) = 1 θ 2 g(\theta)=\frac{1}{\theta^{2}}
  21. e - x 1 + e - x exp ( - θ log ( 1 + e - x ) + log ( θ ) ) \frac{e^{-x}}{1+e^{-x}}\exp(-\theta\log(1+e^{-x})+\log(\theta))
  22. T = log ( 1 + e - x ) T=\mathrm{log}(1+e^{-x})
  23. T T
  24. E ( T ) = - 1 θ , var ( T ) = 1 θ 2 \mathrm{E}(T)=-\frac{1}{\theta},\quad\mathrm{var}(T)=\frac{1}{\theta^{2}}
  25. E ( T 2 ) = 2 θ 2 \mathrm{E}(T^{2})=\frac{2}{\theta^{2}}
  26. δ ( X ) = T 2 2 \delta(X)=\frac{T^{2}}{2}
  27. η ( X ) = E ( δ ( X ) | T ) = E ( T 2 2 | T ) = T 2 2 = log ( 1 + e - X ) 2 2 \eta(X)=\mathrm{E}(\delta(X)|T)=\mathrm{E}\left(\left.\frac{T^{2}}{2}\,\right|% \,T\right)=\frac{T^{2}}{2}=\frac{\log(1+e^{-X})^{2}}{2}
  28. k + 1 k m - 1 , \frac{k+1}{k}m-1,

Minimum_mean_square_error.html

  1. x x
  2. n × 1 n\times 1
  3. y y
  4. m × 1 m\times 1
  5. x ^ ( y ) \hat{x}(y)
  6. x x
  7. y y
  8. e = x ^ - x e=\hat{x}-x
  9. MSE = tr { E { ( x ^ - x ) ( x ^ - x ) T } } \mathrm{MSE}=\mathrm{tr}\left\{\mathrm{E}\{(\hat{x}-x)(\hat{x}-x)^{T}\}\right\}
  10. E \mathrm{E}
  11. x x
  12. y y
  13. x x
  14. E { ( x ^ - x ) 2 } \mathrm{E}\left\{(\hat{x}-x)^{2}\right\}
  15. tr { E { e e T } } = E { tr { e e T } } = E { e T e } = i = 1 n E { e i 2 } . \mathrm{tr}\left\{\mathrm{E}\{ee^{T}\}\right\}=\mathrm{E}\left\{\mathrm{tr}\{% ee^{T}\}\right\}=\mathrm{E}\{e^{T}e\}=\sum_{i=1}^{n}\mathrm{E}\{e_{i}^{2}\}.
  16. x ^ MMSE ( y ) = E { x | y } . \hat{x}_{\mathrm{MMSE}}(y)=\mathrm{E}\left\{x|y\right\}.
  17. x x
  18. E { x ^ MMSE ( y ) } = E { E { x | y } } = E { x } . \mathrm{E}\{\hat{x}_{\mathrm{MMSE}}(y)\}=\mathrm{E}\{\mathrm{E}\{x|y\}\}=% \mathrm{E}\{x\}.
  19. n ( x ^ - x ) 𝑑 𝒩 ( 0 , I - 1 ( x ) ) , \sqrt{n}(\hat{x}-x)\xrightarrow{d}\mathcal{N}\left(0,I^{-1}(x)\right),
  20. I ( x ) I(x)
  21. x x
  22. x x
  23. x ^ = g ( y ) \hat{x}=g(y)
  24. x ^ MMSE = g * ( y ) , \hat{x}_{\mathrm{MMSE}}=g^{*}(y),
  25. E { ( x ^ MMSE - x ) g ( y ) } = 0 \mathrm{E}\{(\hat{x}_{\mathrm{MMSE}}-x)g(y)\}=0
  26. g ( y ) g(y)
  27. 𝒱 = { g ( y ) | g : m , E { g ( y ) 2 } < + } \mathcal{V}=\{g(y)|g:\mathbb{R}^{m}\rightarrow\mathbb{R},\mathrm{E}\{g(y)^{2}% \}<+\infty\}
  28. E { ( g i * ( y ) - x i ) g j ( y ) } = 0 , \mathrm{E}\{(g_{i}^{*}(y)-x_{i})g_{j}(y)\}=0,
  29. x ^ MMSE - x \hat{x}_{\mathrm{MMSE}}-x
  30. x ^ \hat{x}
  31. E { ( x ^ MMSE - x ) x ^ T } = 0. \mathrm{E}\{(\hat{x}_{\mathrm{MMSE}}-x)\hat{x}^{T}\}=0.
  32. x x
  33. y y
  34. W y + b Wy+b
  35. W W
  36. b b
  37. E { x | y } \mathrm{E}\{x|y\}
  38. x x
  39. y y
  40. y y
  41. E { x | y } = W y + b \mathrm{E}\{x|y\}=Wy+b
  42. y y
  43. W W
  44. b b
  45. x x
  46. x x
  47. y y
  48. b b
  49. W W
  50. b = x ¯ - W y ¯ , b=\bar{x}-W\bar{y},
  51. W = C X Y C Y - 1 . W=C_{XY}C^{-1}_{Y}.
  52. x ^ = W ( y - y ¯ ) + x ¯ , \hat{x}=W(y-\bar{y})+\bar{x},
  53. E { x ^ } = x ¯ , \mathrm{E}\{\hat{x}\}=\bar{x},
  54. C X ^ = C X Y C Y - 1 C Y X , C_{\hat{X}}=C_{XY}C^{-1}_{Y}C_{YX},
  55. x ¯ = E { x } \bar{x}=\mathrm{E}\{x\}
  56. y ¯ = E { y } , \bar{y}=\mathrm{E}\{y\},
  57. C X Y C_{XY}
  58. x x
  59. y y
  60. C Y C_{Y}
  61. y y
  62. C Y X C_{YX}
  63. y y
  64. x x
  65. C e = C X - C X ^ = C X - C X Y C Y - 1 C Y X , C_{e}=C_{X}-C_{\hat{X}}=C_{X}-C_{XY}C^{-1}_{Y}C_{YX},
  66. LMMSE = tr { C e } . \mathrm{LMMSE}=\mathrm{tr}\{C_{e}\}.
  67. x x
  68. y y
  69. x ^ = σ X Y σ Y 2 ( y - y ¯ ) + x ¯ , \hat{x}=\frac{\sigma_{XY}}{\sigma_{Y}^{2}}(y-\bar{y})+\bar{x},
  70. σ e 2 = σ X 2 - σ X Y 2 σ Y 2 . \sigma^{2}_{e}=\sigma_{X}^{2}-\frac{\sigma_{XY}^{2}}{\sigma_{Y}^{2}}.
  71. x ^ = W y + b \hat{x}=Wy+b
  72. W W
  73. b b
  74. E { x ^ } = E { x } . \mathrm{E}\{\hat{x}\}=\mathrm{E}\{x\}.
  75. x ^ \hat{x}
  76. b = x ¯ - W y ¯ , b=\bar{x}-W\bar{y},
  77. x ¯ = E { x } \bar{x}=\mathrm{E}\{x\}
  78. y ¯ = E { y } \bar{y}=\mathrm{E}\{y\}
  79. x ^ = W ( y - y ¯ ) + x ¯ \hat{x}=W(y-\bar{y})+\bar{x}
  80. x ^ - x = W ( y - y ¯ ) - ( x - x ¯ ) . \hat{x}-x=W(y-\bar{y})-(x-\bar{x}).
  81. E { ( x ^ - x ) ( y - y ¯ ) T } = 0 \mathrm{E}\{(\hat{x}-x)(y-\bar{y})^{T}\}=0
  82. g ( y ) = y - y ¯ g(y)=y-\bar{y}
  83. E { ( x ^ - x ) ( y - y ¯ ) T } = E { ( W ( y - y ¯ ) - ( x - x ¯ ) ) ( y - y ¯ ) T } = W E { ( y - y ¯ ) ( y - y ¯ ) T } - E { ( x - x ¯ ) ( y - y ¯ ) T } = W C Y - C X Y . \begin{array}[]{ll}\mathrm{E}\{(\hat{x}-x)(y-\bar{y})^{T}\}&=\mathrm{E}\{(W(y-% \bar{y})-(x-\bar{x}))(y-\bar{y})^{T}\}\\ &=W\mathrm{E}\{(y-\bar{y})(y-\bar{y})^{T}\}-\mathrm{E}\{(x-\bar{x})(y-\bar{y})% ^{T}\}\\ &=WC_{Y}-C_{XY}.\end{array}
  84. W W
  85. W = C X Y C Y - 1 . W=C_{XY}C^{-1}_{Y}.
  86. C X Y C_{XY}
  87. C Y C_{Y}
  88. C X Y = C Y X T C_{XY}=C^{T}_{YX}
  89. C Y X C_{YX}
  90. W T = C Y - 1 C Y X . W^{T}=C^{-1}_{Y}C_{YX}.
  91. x ^ = C X Y C Y - 1 ( y - y ¯ ) + x ¯ . \hat{x}=C_{XY}C^{-1}_{Y}(y-\bar{y})+\bar{x}.
  92. x ^ \hat{x}
  93. E { x ^ } = x ¯ \mathrm{E}\{\hat{x}\}=\bar{x}
  94. C X ^ = E { ( x ^ - x ¯ ) ( x ^ - x ¯ ) T } = W E { ( y - y ¯ ) ( y - y ¯ ) T } W T = W C Y W T . \begin{array}[]{ll}C_{\hat{X}}&=\mathrm{E}\{(\hat{x}-\bar{x})(\hat{x}-\bar{x})% ^{T}\}\\ &=W\mathrm{E}\{(y-\bar{y})(y-\bar{y})^{T}\}W^{T}\\ &=WC_{Y}W^{T}.\\ \end{array}
  95. W W
  96. W T W^{T}
  97. C X ^ = C X Y C Y - 1 C Y X . C_{\hat{X}}=C_{XY}C^{-1}_{Y}C_{YX}.
  98. C e = E { ( x ^ - x ) ( x ^ - x ) T } = E { ( x ^ - x ) ( W ( y - y ¯ ) - ( x - x ¯ ) ) T } = E { ( x ^ - x ) ( y - y ¯ ) T } 0 W T - E { ( x ^ - x ) ( x - x ¯ ) T } = - E { ( W ( y - y ¯ ) - ( x - x ¯ ) ) ( x - x ¯ ) T } = E { ( x - x ¯ ) ( x - x ¯ ) T } - W E { ( y - y ¯ ) ( x - x ¯ ) T } = C X - W C Y X . \begin{array}[]{ll}C_{e}&=\mathrm{E}\{(\hat{x}-x)(\hat{x}-x)^{T}\}\\ &=\mathrm{E}\{(\hat{x}-x)(W(y-\bar{y})-(x-\bar{x}))^{T}\}\\ &=\underbrace{\mathrm{E}\{(\hat{x}-x)(y-\bar{y})^{T}\}}_{0}W^{T}-\mathrm{E}\{(% \hat{x}-x)(x-\bar{x})^{T}\}\\ &=-\mathrm{E}\{(W(y-\bar{y})-(x-\bar{x}))(x-\bar{x})^{T}\}\\ &=\mathrm{E}\{(x-\bar{x})(x-\bar{x})^{T}\}-W\mathrm{E}\{(y-\bar{y})(x-\bar{x})% ^{T}\}\\ &=C_{X}-WC_{YX}.\\ \end{array}
  99. W = C X Y C Y - 1 W=C_{XY}C^{-1}_{Y}
  100. C e C_{e}
  101. C e = C X - C X Y C Y - 1 C Y X . C_{e}=C_{X}-C_{XY}C^{-1}_{Y}C_{YX}.
  102. C e = C X - C X ^ . C_{e}=C_{X}-C_{\hat{X}}.
  103. LMMSE = tr { C e } \mathrm{LMMSE}=\mathrm{tr}\{C_{e}\}
  104. W W
  105. C Y C_{Y}
  106. W W
  107. C Y C_{Y}
  108. y y
  109. y = A x + z y=Ax+z
  110. A A
  111. z z
  112. E { z } = 0 \mathrm{E}\{z\}=0
  113. C X Z = 0 C_{XZ}=0
  114. E { y } = A x ¯ , \mathrm{E}\{y\}=A\bar{x},
  115. C Y = A C X A T + C Z , C_{Y}=AC_{X}A^{T}+C_{Z},
  116. C X Y = C X A T . C_{XY}=C_{X}A^{T}.
  117. W W
  118. W = C X A T ( A C X A T + C Z ) - 1 . W=C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}.
  119. x ^ \hat{x}
  120. x ^ = C X A T ( A C X A T + C Z ) - 1 ( y - A x ¯ ) + x ¯ . \hat{x}=C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}(y-A\bar{x})+\bar{x}.
  121. C e = C X - C X ^ = C X - C X A T ( A C X A T + C Z ) - 1 A C X . C_{e}=C_{X}-C_{\hat{X}}=C_{X}-C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}AC_{X}.
  122. y y
  123. x x
  124. ( A C X A T + C Z ) - 1 (AC_{X}A^{T}+C_{Z})^{-1}
  125. C Z C_{Z}
  126. x x
  127. A C X A T AC_{X}A^{T}
  128. C X A T ( A C X A T + C Z ) - 1 = ( A T C Z - 1 A + C X - 1 ) - 1 A T C Z - 1 , C_{X}A^{T}(AC_{X}A^{T}+C_{Z})^{-1}=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}A^{T}C_{Z% }^{-1},
  129. ( A C X A T + C Z ) (AC_{X}A^{T}+C_{Z})
  130. ( A T C Z - 1 A + C X - 1 ) , (A^{T}C_{Z}^{-1}A+C_{X}^{-1}),
  131. W = ( A T C Z - 1 A + C X - 1 ) - 1 A T C Z - 1 , W=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}A^{T}C_{Z}^{-1},
  132. C e = ( A T C Z - 1 A + C X - 1 ) - 1 . C_{e}=(A^{T}C_{Z}^{-1}A+C_{X}^{-1})^{-1}.
  133. W W
  134. C e C_{e}
  135. W = C e A T C Z - 1 W=C_{e}A^{T}C_{Z}^{-1}
  136. x ^ \hat{x}
  137. x ^ = C e A T C Z - 1 ( y - A x ¯ ) + x ¯ . \hat{x}=C_{e}A^{T}C_{Z}^{-1}(y-A\bar{x})+\bar{x}.
  138. C X - 1 = 0 C_{X}^{-1}=0
  139. x x
  140. W = ( A T C Z - 1 A ) - 1 A T C Z - 1 W=(A^{T}C_{Z}^{-1}A)^{-1}A^{T}C_{Z}^{-1}
  141. C Z - 1 C_{Z}^{-1}
  142. z z
  143. C Z = σ 2 I , C_{Z}=\sigma^{2}I,
  144. I I
  145. W = ( A T A ) - 1 A T W=(A^{T}A)^{-1}A^{T}
  146. x x
  147. x x
  148. x ^ 1 \hat{x}_{1}
  149. Y 1 Y_{1}
  150. x ^ 1 \hat{x}_{1}
  151. C e 1 C_{e_{1}}
  152. y y
  153. x ^ 1 \hat{x}_{1}
  154. y ^ = A x ^ 1 \hat{y}=A\hat{x}_{1}
  155. y ^ \hat{y}
  156. y y
  157. y ~ = y - y ^ = A ( x - x ^ 1 ) + z = A e 1 + z \tilde{y}=y-\hat{y}=A(x-\hat{x}_{1})+z=Ae_{1}+z
  158. x ^ 2 = x ^ 1 + C X Y ~ C Y ~ - 1 y ~ , \hat{x}_{2}=\hat{x}_{1}+C_{X\tilde{Y}}C_{\tilde{Y}}^{-1}\tilde{y},
  159. C X Y ~ C_{X\tilde{Y}}
  160. x x
  161. y ~ \tilde{y}
  162. C Y ~ C_{\tilde{Y}}
  163. y ~ . \tilde{y}.
  164. E { y ~ } = 0 \mathrm{E}\{\tilde{y}\}=0
  165. x = x ^ 1 + e 1 x=\hat{x}_{1}+e_{1}
  166. C Y ~ = A C e 1 A T + C Z , C_{\tilde{Y}}=AC_{e_{1}}A^{T}+C_{Z},
  167. C X Y ~ = E { ( x ^ 1 + e 1 - x ¯ ) ( A e 1 + z ) T } = C e 1 A T . C_{X\tilde{Y}}=\mathrm{E}\{(\hat{x}_{1}+e_{1}-\bar{x})(Ae_{1}+z)^{T}\}=C_{e_{1% }}A^{T}.
  168. x ^ 2 = x ^ 1 + C e 1 A T ( A C e 1 A T + C Z ) - 1 ( y - A x ^ 1 ) , \hat{x}_{2}=\hat{x}_{1}+C_{e_{1}}A^{T}(AC_{e_{1}}A^{T}+C_{Z})^{-1}(y-A\hat{x}_% {1}),
  169. C e 2 = C e 1 - C e 1 A T ( A C e 1 A T + C Z ) - 1 A C e 1 . C_{e_{2}}=C_{e_{1}}-C_{e_{1}}A^{T}(AC_{e_{1}}A^{T}+C_{Z})^{-1}AC_{e_{1}}.
  170. K 2 = C e 1 A T ( A C e 1 A T + C Z ) - 1 , K_{2}=C_{e_{1}}A^{T}(AC_{e_{1}}A^{T}+C_{Z})^{-1},
  171. x ^ 2 = x ^ 1 + K 2 ( y - A x ^ 1 ) , \hat{x}_{2}=\hat{x}_{1}+K_{2}(y-A\hat{x}_{1}),
  172. C e 2 = ( I - K 2 A ) C e 1 . C_{e_{2}}=(I-K_{2}A)C_{e_{1}}.
  173. K K
  174. y m = a m T x m + z m y_{m}=a_{m}^{T}x_{m}+z_{m}
  175. a m T a_{m}^{T}
  176. x m x_{m}
  177. z m z_{m}
  178. σ m 2 \sigma_{m}^{2}
  179. x ^ m + 1 \hat{x}_{m+1}
  180. x ^ m + 1 = x ^ m + k m + 1 ( y m + 1 - a m + 1 T x ^ m ) \hat{x}_{m+1}=\hat{x}_{m}+k_{m+1}(y_{m+1}-a^{T}_{m+1}\hat{x}_{m})
  181. y m + 1 y_{m+1}
  182. k m + 1 k_{m+1}
  183. k m + 1 = ( C e ) m a m + 1 σ m + 1 2 + a m + 1 T ( C e ) m a m + 1 . k_{m+1}=\frac{(C_{e})_{m}a_{m+1}}{\sigma^{2}_{m+1}+a^{T}_{m+1}(C_{e})_{m}a_{m+% 1}}.
  184. ( C e ) m + 1 (C_{e})_{m+1}
  185. ( C e ) m + 1 = ( I - k m + 1 a m + 1 T ) ( C e ) m . (C_{e})_{m+1}=(I-k_{m+1}a^{T}_{m+1})(C_{e})_{m}.
  186. k m + 1 k_{m+1}
  187. x ^ \hat{x}
  188. C e C_{e}
  189. x x
  190. x 1 , x 2 x_{1},x_{2}
  191. x 3 x_{3}
  192. x 4 x_{4}
  193. x ^ 4 = i = 1 3 w i x i \hat{x}_{4}=\sum_{i=1}^{3}w_{i}x_{i}
  194. x = [ x 1 , x 2 , x 3 , x 4 ] T x=[x_{1},x_{2},x_{3},x_{4}]^{T}
  195. cov ( X ) = E [ x x T ] = [ 1 2 3 4 2 5 8 9 3 8 6 10 4 9 10 15 ] , \operatorname{cov}(X)=\mathrm{E}[xx^{T}]=\left[\begin{array}[]{cccc}1&2&3&4\\ 2&5&8&9\\ 3&8&6&10\\ 4&9&10&15\end{array}\right],
  196. w i w_{i}
  197. x ^ 4 \hat{x}_{4}
  198. y = [ x 1 , x 2 , x 3 ] T y=[x_{1},x_{2},x_{3}]^{T}
  199. W = [ w 1 , w 2 , w 3 ] W=[w_{1},w_{2},w_{3}]
  200. x = x 4 x=x_{4}
  201. C Y C_{Y}
  202. C Y = [ E [ x 1 , x 1 ] E [ x 2 , x 1 ] E [ x 3 , x 1 ] E [ x 1 , x 2 ] E [ x 2 , x 2 ] E [ x 3 , x 2 ] E [ x 1 , x 3 ] E [ x 2 , x 3 ] E [ x 3 , x 3 ] ] = [ 1 2 3 2 5 8 3 8 6 ] . C_{Y}=\left[\begin{array}[]{ccc}E[x_{1},x_{1}]&E[x_{2},x_{1}]&E[x_{3},x_{1}]\\ E[x_{1},x_{2}]&E[x_{2},x_{2}]&E[x_{3},x_{2}]\\ E[x_{1},x_{3}]&E[x_{2},x_{3}]&E[x_{3},x_{3}]\end{array}\right]=\left[\begin{% array}[]{ccc}1&2&3\\ 2&5&8\\ 3&8&6\end{array}\right].
  203. C Y X C_{YX}
  204. C Y X = [ E [ x 4 , x 1 ] E [ x 4 , x 2 ] E [ x 4 , x 3 ] ] = [ 4 9 10 ] . C_{YX}=\left[\begin{array}[]{c}E[x_{4},x_{1}]\\ E[x_{4},x_{2}]\\ E[x_{4},x_{3}]\end{array}\right]=\left[\begin{array}[]{c}4\\ 9\\ 10\end{array}\right].
  205. C Y W T = C Y X C_{Y}W^{T}=C_{YX}
  206. C Y C_{Y}
  207. C Y - 1 C Y X = [ 4.85 - 1.71 - .142 - 1.71 .428 .2857 - .142 .2857 - .1429 ] [ 4 9 10 ] = [ 2.57 - .142 .5714 ] = W T . C_{Y}^{-1}C_{YX}=\left[\begin{array}[]{ccc}4.85&-1.71&-.142\\ -1.71&.428&.2857\\ -.142&.2857&-.1429\end{array}\right]\left[\begin{array}[]{c}4\\ 9\\ 10\end{array}\right]=\left[\begin{array}[]{c}2.57\\ -.142\\ .5714\end{array}\right]=W^{T}.
  208. w 1 = 2.57 , w_{1}=2.57,
  209. w 2 = - .142 , w_{2}=-.142,
  210. w 3 = .5714 w_{3}=.5714
  211. x ^ 4 \hat{x}_{4}
  212. e min 2 = E [ x 4 x 4 ] - W C Y X = 15 - W C Y X = .2857 \left\|e\right\|_{\min}^{2}=\mathrm{E}[x_{4}x_{4}]-WC_{YX}=15-WC_{YX}=.2857
  213. C Y C_{Y}
  214. W W
  215. y y
  216. N N
  217. x x
  218. y = 1 x + z y=1x+z
  219. 1 = [ 1 , 1 , , 1 ] T 1=[1,1,\ldots,1]^{T}
  220. 1 1
  221. [ - x 0 , x 0 ] [-x_{0},x_{0}]
  222. x x
  223. x x
  224. [ - x 0 , x 0 ] [-x_{0},x_{0}]
  225. x x
  226. σ X 2 = x 0 2 / 3. \sigma_{X}^{2}=x_{0}^{2}/3.
  227. z z
  228. N ( 0 , σ Z 2 I ) N(0,\sigma_{Z}^{2}I)
  229. I I
  230. x x
  231. z z
  232. C X Z = 0 C_{XZ}=0
  233. E { y } = 0 , C Y = E { y y T } = σ X 2 11 T + σ Z 2 I , C X Y = E { x y T } = σ X 2 1 T . \begin{aligned}&\displaystyle\mathrm{E}\{y\}=0,\\ &\displaystyle C_{Y}=\mathrm{E}\{yy^{T}\}=\sigma_{X}^{2}11^{T}+\sigma_{Z}^{2}I% ,\\ &\displaystyle C_{XY}=\mathrm{E}\{xy^{T}\}=\sigma_{X}^{2}1^{T}.\end{aligned}
  234. x ^ = C X Y C Y - 1 y = σ X 2 1 T ( σ X 2 11 T + σ Z 2 I ) - 1 y . \begin{aligned}\displaystyle\hat{x}&\displaystyle=C_{XY}C_{Y}^{-1}y\\ &\displaystyle=\sigma_{X}^{2}1^{T}(\sigma_{X}^{2}11^{T}+\sigma_{Z}^{2}I)^{-1}y% .\end{aligned}
  235. W W
  236. x ^ = ( 1 T 1 σ Z 2 I 1 + 1 σ X 2 ) - 1 1 T 1 σ Z 2 I y = 1 σ Z 2 ( N σ Z 2 + 1 σ X 2 ) - 1 1 T y = σ X 2 σ X 2 + σ Z 2 / N y ¯ , \begin{aligned}\displaystyle\hat{x}&\displaystyle=(1^{T}\frac{1}{\sigma_{Z}^{2% }}I1+\frac{1}{\sigma_{X}^{2}})^{-1}1^{T}\frac{1}{\sigma_{Z}^{2}}Iy\\ &\displaystyle=\frac{1}{\sigma_{Z}^{2}}(\frac{N}{\sigma_{Z}^{2}}+\frac{1}{% \sigma_{X}^{2}})^{-1}1^{T}y\\ &\displaystyle=\frac{\sigma_{X}^{2}}{\sigma_{X}^{2}+\sigma_{Z}^{2}/N}\bar{y},% \end{aligned}
  237. y = [ y 1 , y 2 , , y N ] T y=[y_{1},y_{2},\ldots,y_{N}]^{T}
  238. y ¯ = 1 T y N = i = 1 N y i N . \bar{y}=\frac{1^{T}y}{N}=\frac{\sum_{i=1}^{N}y_{i}}{N}.
  239. σ X ^ 2 = C X Y C Y - 1 C Y X = ( σ X 2 σ X 2 + σ Z 2 / N ) σ X 2 . \sigma_{\hat{X}}^{2}=C_{XY}C_{Y}^{-1}C_{YX}=\Big(\frac{\sigma_{X}^{2}}{\sigma_% {X}^{2}+\sigma_{Z}^{2}/N}\Big)\sigma_{X}^{2}.
  240. LMMSE = σ X 2 - σ X ^ 2 = ( σ Z 2 σ X 2 + σ Z 2 / N ) σ X 2 N . \mathrm{LMMSE}=\sigma_{X}^{2}-\sigma_{\hat{X}}^{2}=\Big(\frac{\sigma_{Z}^{2}}{% \sigma_{X}^{2}+\sigma_{Z}^{2}/N}\Big)\frac{\sigma_{X}^{2}}{N}.
  241. N N
  242. x ^ = 1 N i = 1 N y i , \hat{x}=\frac{1}{N}\sum_{i=1}^{N}y_{i},
  243. σ X ^ 2 = σ X 2 , \sigma_{\hat{X}}^{2}=\sigma_{X}^{2},
  244. x x
  245. x x
  246. x [ 0 , 1 ] . x\in[0,1].
  247. 1 - x . 1-x.
  248. x x
  249. [ 0 , 1 ] [0,1]
  250. x ¯ = 1 / 2 \bar{x}=1/2
  251. σ X 2 = 1 / 12. \sigma_{X}^{2}=1/12.
  252. y 1 y_{1}
  253. z 1 z_{1}
  254. σ Z 1 2 . \sigma_{Z_{1}}^{2}.
  255. y 2 y_{2}
  256. z 2 z_{2}
  257. σ Z 2 2 . \sigma_{Z_{2}}^{2}.
  258. y 1 = x + z 1 y 2 = x + z 2 . \begin{aligned}\displaystyle y_{1}&\displaystyle=x+z_{1}\\ \displaystyle y_{2}&\displaystyle=x+z_{2}.\end{aligned}
  259. E { y 1 } = E { y 2 } = x ¯ = 1 / 2 \mathrm{E}\{y_{1}\}=\mathrm{E}\{y_{2}\}=\bar{x}=1/2
  260. y 1 y_{1}
  261. y 2 y_{2}
  262. x ^ = w 1 ( y 1 - x ¯ ) + w 2 ( y 2 - x ¯ ) + x ¯ , \hat{x}=w_{1}(y_{1}-\bar{x})+w_{2}(y_{2}-\bar{x})+\bar{x},
  263. w 1 = 1 / σ Z 1 2 1 / σ Z 1 2 + 1 / σ Z 2 2 + 1 / σ X 2 , w 2 = 1 / σ Z 2 2 1 / σ Z 1 2 + 1 / σ Z 2 2 + 1 / σ X 2 . \begin{aligned}\displaystyle w_{1}&\displaystyle=\frac{1/\sigma_{Z_{1}}^{2}}{1% /\sigma_{Z_{1}}^{2}+1/\sigma_{Z_{2}}^{2}+1/\sigma_{X}^{2}},\\ \displaystyle w_{2}&\displaystyle=\frac{1/\sigma_{Z_{2}}^{2}}{1/\sigma_{Z_{1}}% ^{2}+1/\sigma_{Z_{2}}^{2}+1/\sigma_{X}^{2}}.\end{aligned}
  264. σ X ^ 2 = 1 / σ Z 1 2 + 1 / σ Z 2 2 1 / σ Z 1 2 + 1 / σ Z 2 2 + 1 / σ X 2 σ X 2 , \sigma_{\hat{X}}^{2}=\frac{1/\sigma_{Z_{1}}^{2}+1/\sigma_{Z_{2}}^{2}}{1/\sigma% _{Z_{1}}^{2}+1/\sigma_{Z_{2}}^{2}+1/\sigma_{X}^{2}}\sigma_{X}^{2},
  265. σ X ^ 2 \sigma_{\hat{X}}^{2}
  266. σ X 2 . \sigma_{X}^{2}.
  267. N N
  268. w i = 1 / σ Z i 2 i = 1 N 1 / σ Z i 2 + 1 / σ X 2 w_{i}=\frac{1/\sigma_{Z_{i}}^{2}}{\sum_{i=1}^{N}1/\sigma_{Z_{i}}^{2}+1/\sigma_% {X}^{2}}
  269. a 1 a_{1}
  270. a 2 a_{2}
  271. z 1 z_{1}
  272. z 2 z_{2}
  273. σ Z 1 2 \sigma_{Z_{1}}^{2}
  274. σ Z 2 2 \sigma_{Z_{2}}^{2}
  275. x x
  276. σ X 2 . \sigma_{X}^{2}.
  277. y 1 = a 1 x + z 1 y 2 = a 2 x + z 2 . \begin{aligned}\displaystyle y_{1}&\displaystyle=a_{1}x+z_{1}\\ \displaystyle y_{2}&\displaystyle=a_{2}x+z_{2}.\end{aligned}
  278. E { y 1 } = E { y 2 } = 0 \mathrm{E}\{y_{1}\}=\mathrm{E}\{y_{2}\}=0
  279. y = w 1 y 1 + w 2 y 2 y=w_{1}y_{1}+w_{2}y_{2}
  280. w i = a i / σ Z i 2 i a i 2 / σ Z i 2 + 1 / σ X 2 . w_{i}=\frac{a_{i}/\sigma_{Z_{i}}^{2}}{\sum_{i}a_{i}^{2}/\sigma_{Z_{i}}^{2}+1/% \sigma_{X}^{2}}.

Minkowski's_question_mark_function.html

  1. s i z e = 120 % ? ( x ) size=120\%?(x)
  2. [ a 0 ; a 1 , a 2 , ] [a_{0};a_{1},a_{2},\ldots]
  3. x x
  4. ? ( x ) = a 0 + 2 n = 1 ( - 1 ) n + 1 2 a 1 + + a n {\rm?}(x)=a_{0}+2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{2^{a_{1}+\cdots+a_{n}}}
  5. [ a 0 ; a 1 , a 2 , , a m ] [a_{0};a_{1},a_{2},\ldots,a_{m}]
  6. x x
  7. ? ( x ) = a 0 + 2 n = 1 m ( - 1 ) n + 1 2 a 1 + + a n {\rm?}(x)=a_{0}+2\sum_{n=1}^{m}\frac{(-1)^{n+1}}{2^{a_{1}+\cdots+a_{n}}}
  8. [ s i z e = 120 % 0 , 1 ] [size=120\%0,1]
  9. 2 / 7 2/7
  10. s i z e = 120 % 0 ; 2 , 1 , 2 , 1 , 2 , 1 , = size=120\%0; 2, 1, 2, 1, 2, 1, ……=
  11. 3 1 2 \sqrt{3}\frac{−1}{2}
  12. [ s i z e = 120 % 0 , 1 ] [size=120\%0,1]
  13. ? ( 3 - 1 2 ) = 2 7 . ?\left(\frac{\sqrt{3}-1}{2}\right)=\frac{2}{7}.
  14. p / q p/q
  15. r / s r/s
  16. | p s r q | = 1 |ps−rq|=1
  17. ? ( p + r q + s ) = 1 2 ( ? ( p q ) + ? ( r s ) ) ?\left(\frac{p+r}{q+s}\right)=\frac{1}{2}\left(?\bigg(\frac{p}{q}\bigg)+{}?% \bigg(\frac{r}{s}\bigg)\right)
  18. ? ( 0 1 ) = 0 and ? ( 1 1 ) = 1 ?\left(\frac{0}{1}\right)=0\quad\mbox{ and }~{}\quad?\left(\frac{1}{1}\right)=1
  19. ? ( x ) ?(x)
  20. x x
  21. p n - 1 / q n - 1 p_{n-1}/q_{n-1}
  22. p n / q n p_{n}/q_{n}
  23. ( p n - 1 p n q n - 1 q n ) \begin{pmatrix}p_{n-1}&p_{n}\\ q_{n-1}&q_{n}\end{pmatrix}
  24. x x
  25. q r - p s = 1 qr-ps=1
  26. m n = p + r q + s , \frac{m}{n}=\frac{p+r}{q+s},
  27. p q x < r s . \frac{p}{q}\leq x<\frac{r}{s}.
  28. y ? ( x ) < y + d , y\leq\;?(x)<y+d,
  29. d d
  30. y ? ( x ) < y + 2 d y\leq\;?(x)<y+2d
  31. q + s q+s

Missionaries_and_cannibals_problem.html

  1. α \alpha
  2. β \beta
  3. γ \gamma
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  21. γ \gamma
  22. α \alpha
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  24. γ \gamma
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  27. γ \gamma
  28. α \alpha
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  43. γ \gamma
  44. α \alpha
  45. β \beta
  46. γ \gamma
  47. β \beta
  48. γ \gamma
  49. α \alpha
  50. β \beta
  51. γ \gamma
  52. α \alpha
  53. β \beta
  54. γ \gamma
  55. α \alpha
  56. β \beta
  57. γ \gamma
  58. α \alpha
  59. γ \gamma
  60. β \beta
  61. α \alpha
  62. γ \gamma
  63. α \alpha
  64. β \beta
  65. γ \gamma
  66. α \alpha
  67. β \beta
  68. γ \gamma
  69. α \alpha
  70. β \beta
  71. γ \gamma
  72. α \alpha
  73. β \beta
  74. α \alpha
  75. β \beta
  76. γ \gamma

Misuse_of_statistics.html

  1. x x
  2. 1.32 x 1.32x
  3. H 0 H_{0}
  4. H 0 H_{0}
  5. H A H_{A}
  6. H 0 H_{0}
  7. H 0 H_{0}
  8. H A H_{A}
  9. H 0 H_{0}
  10. H 0 H_{0}
  11. H 0 H_{0}
  12. H 0 H_{0}
  13. H 0 H_{0}
  14. H 0 H_{0}
  15. H 0 H_{0}

Mittag-Leffler_function.html

  1. E α , β ( z ) = k = 0 z k Γ ( α k + β ) . E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)}.
  2. α = 0 , 1 / 2 , 1 , 2 \alpha=0,1/2,1,2
  3. E 0 , 1 ( z ) = k = 0 z k = 1 1 - z . E_{0,1}(z)=\sum_{k=0}^{\infty}z^{k}=\frac{1}{1-z}.
  4. E 1 , 1 ( z ) = k = 0 z k Γ ( k + 1 ) = k = 0 z k k ! = exp ( z ) . E_{1,1}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(k+1)}=\sum_{k=0}^{\infty}% \frac{z^{k}}{k!}=\exp(z).
  5. E 1 / 2 , 1 ( z ) = exp ( z 2 ) erfc ( - z ) . E_{1/2,1}(z)=\exp(z^{2})\operatorname{erfc}(-z).
  6. E 2 , 1 ( z ) = cosh ( z ) . E_{2,1}(z)=\cosh(\sqrt{z}).
  7. α = 0 , 1 , 2 \alpha=0,1,2
  8. 0 z E α , 1 ( - s 2 ) d s \int_{0}^{z}E_{\alpha,1}(-s^{2}){\mathrm{d}}s
  9. arctan ( z ) \arctan(z)
  10. π 2 erf ( z ) \tfrac{\sqrt{\pi}}{2}\operatorname{erf}(z)
  11. sin ( z ) \sin(z)
  12. E α , β ( z ) = 1 2 π i C t α - β e t t α - z d t E_{\alpha,\beta}(z)=\frac{1}{2\pi i}\int_{C}\frac{t^{\alpha-\beta}e^{t}}{t^{% \alpha}-z}\,dt

Mixed-data_sampling.html

  1. y t = β 0 + β 1 B ( L 1 / m ; θ ) x t ( m ) + ε t ( m ) , y_{t}=\beta_{0}+\beta_{1}B(L^{1/m};\theta)x_{t}^{(m)}+\varepsilon_{t}^{(m)},\,
  2. x t ( 4 ) x_{t}^{(4)}
  3. ε \varepsilon
  4. B ( L 1 / m ; θ ) B(L^{1/m};\theta)

Mixed_anomaly.html

  1. n = 1 + D / 2 n=1+D/2
  2. D D

Mixing_(mathematics).html

  1. X t = { , X t - 1 , X t , X t + 1 , } \langle X_{t}\rangle=\{\ldots,X_{t-1},X_{t},X_{t+1},\ldots\}
  2. α ( s ) \alpha(s)
  3. α ( s ) sup { | P ( A B ) - P ( A ) P ( B ) | : - < t < , A X - t , B X t + s } . \alpha(s)\equiv\sup\left\{\,|P(A\cap B)-P(A)P(B)|:-\infty<t<\infty,A\in X_{-% \infty}^{t},B\in X_{t+s}^{\infty}\,\right\}.
  4. X a b X_{a}^{b}
  5. - a b -\infty\leq a\leq b\leq\infty
  6. X a X_{a}
  7. X a + 1 X_{a+1}
  8. a a
  9. a + 1 a+1
  10. { X a , X a + 1 , , X b } . \{X_{a},X_{a+1},\ldots,X_{b}\}.
  11. X t \langle X_{t}\rangle
  12. α ( s ) 0 \alpha(s)\rightarrow 0
  13. s s\rightarrow\infty
  14. ρ t = sup ϕ Z : ϕ 2 = 1 t ϕ 2 . \rho_{t}=\sup_{\phi\in Z:\,\|\phi\|_{2}=1}\|\mathcal{E}_{t}\phi\|_{2}.
  15. α t = sup ϕ Z : ϕ = 1 t ϕ 1 . \alpha_{t}=\sup_{\phi\in Z:\,\|\phi\|_{\infty}=1}\|\mathcal{E}_{t}\phi\|_{1}.
  16. β t = sup 0 ϕ 1 | t ϕ ( x ) - ϕ d Q | d Q . \beta_{t}=\int\sup_{0\leq\phi\leq 1}\Big|\mathcal{E}_{t}\phi(x)-\int\phi dQ% \Big|dQ.
  17. ( X , 𝒜 , μ , T ) (X,\mathcal{A},\mu,T)
  18. A , B 𝒜 A,B\in\mathcal{A}
  19. lim n μ ( A T - n B ) = μ ( A ) μ ( B ) \lim_{n\to\infty}\mu(A\cap T^{-n}B)=\mu(A)\mu(B)
  20. T - n T^{-n}
  21. T g T_{g}
  22. M M
  23. A A
  24. B B
  25. B B
  26. μ ( T n A B ) μ ( B ) . \frac{\mu\left(T^{n}A\cap B\right)}{\mu\left(B\right)}.
  27. n n\rightarrow\infty
  28. B B
  29. lim n μ ( T n A B ) μ ( B ) = μ ( A ) μ ( M ) = μ ( A ) \lim_{n\rightarrow\infty}\frac{\mu\left(T^{n}A\cap B\right)}{\mu\left(B\right)% }=\frac{\mu\left(A\right)}{\mu\left(M\right)}=\mu\left(A\right)
  30. μ ( M ) = 1 \mu(M)=1
  31. lim n 1 n k = 0 n - 1 | μ ( A T - k B ) - μ ( A ) μ ( B ) | = 0. \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}|\mu(A\cap T^{-k}B)-\mu(A)\mu(B)|=0.
  32. T T
  33. μ ( A T - n B ) - μ ( A ) μ ( B ) \mu(A\cap T^{-n}B)-\mu(A)\mu(B)
  34. 0
  35. | μ ( A T - n B ) - μ ( A ) μ ( B ) | |\mu(A\cap T^{-n}B)-\mu(A)\mu(B)|
  36. 0
  37. μ ( A T - n B ) - μ ( A ) μ ( B ) \mu(A\cap T^{-n}B)-\mu(A)\mu(B)
  38. 0
  39. L 2 L^{2}
  40. ( X , 𝒜 , μ , T ) (X,\mathcal{A},\mu,T)
  41. f L 2 ( X , μ ) f\in L^{2}(X,\mu)
  42. ( f T n ) n 0 (f\circ T^{n})_{n\geq 0}
  43. X f d μ \int_{X}fd\mu
  44. lim N 1 N n = 0 N - 1 f T n - X f d μ L 2 ( X , μ ) = 0. \lim_{N\to\infty}\|{1\over N}\sum_{n=0}^{N-1}f\circ T^{n}-\int_{X}fd\mu\|_{L^{% 2}(X,\mu)}=0.
  45. ( X , 𝒜 , μ , T ) (X,\mathcal{A},\mu,T)
  46. f f
  47. g L 2 ( X , μ ) g\in L^{2}(X,\mu)
  48. lim N 1 N n = 0 N - 1 | X f T n g d μ - X f d μ X g d μ | = 0. \lim_{N\to\infty}{1\over N}\sum_{n=0}^{N-1}|\int_{X}f\circ T^{n}\cdot gd\mu-% \int_{X}fd\mu\cdot\int_{X}gd\mu|=0.
  49. ( X , 𝒜 , μ , T ) (X,\mathcal{A},\mu,T)
  50. f L 2 ( X , μ ) f\in L^{2}(X,\mu)
  51. ( f T n ) n 0 (f\circ T^{n})_{n\geq 0}
  52. X f d μ \int_{X}fd\mu
  53. g L 2 ( X , μ ) g\in L^{2}(X,\mu)
  54. lim n X f T n g d μ = X f d μ X g d μ . \lim_{n\to\infty}\int_{X}f\circ T^{n}\cdot gd\mu=\int_{X}fd\mu\cdot\int_{X}gd\mu.
  55. lim n C o v ( f T n , g ) = 0 \lim_{n\to\infty}Cov(f\circ T^{n},g)=0
  56. f T n f\circ T^{n}
  57. g g
  58. n n
  59. g g
  60. f T n f\circ T^{n}
  61. g g
  62. n n
  63. ( X , μ , T ) (X,\mu,T)
  64. ( Y , μ , S ) (Y,\mu,S)
  65. ( X × Y , μ ν , T × S ) (X\times Y,\mu\otimes\nu,T\times S)
  66. ( T × S ) ( x , y ) = ( T ( x ) , S ( y ) ) (T\times S)(x,y)=(T(x),S(y))
  67. ( X , μ , T ) (X,\mu,T)
  68. ( Y , μ , S ) (Y,\mu,S)
  69. ( X × Y , μ ν , T × S ) (X\times Y,\mu\otimes\nu,T\times S)
  70. ( X , μ , T ) (X,\mu,T)
  71. ( X 2 , μ μ , T × T ) (X^{2},\mu\otimes\mu,T\times T)
  72. ( X 2 , μ μ , T × T ) (X^{2},\mu\otimes\mu,T\times T)
  73. lim m , n μ ( A T - m B T - m - n C ) = μ ( A ) μ ( B ) μ ( C ) \lim_{m,n\to\infty}\mu(A\cap T^{-m}B\cap T^{-m-n}C)=\mu(A)\mu(B)\mu(C)
  74. f : X X f:X\to X
  75. A , B X A,B\subset X
  76. f n ( A ) B f^{n}(A)\cap B\neq\varnothing
  77. f n f^{n}
  78. x X x\in X
  79. { f n ( x ) : n } \{f^{n}(x):n\in\mathbb{N}\}
  80. A A
  81. B B
  82. n > N n>N
  83. f n ( A ) B f^{n}(A)\cap B\neq\varnothing
  84. f n f^{n}
  85. ϕ g \phi_{g}
  86. g > N \|g\|>N

Mode_(statistics).html

  1. x ¯ = 1 n i = 1 n x i \scriptstyle\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}
  2. 3 \sqrt{3}
  3. mean = e μ + σ 2 / 2 = e 0 + 0.25 2 / 2 1.032 mode = e μ - σ 2 = e 0 - 0.25 2 0.939 median = e μ = e 0 = 1 \begin{array}[]{rlll}\,\text{mean}&=e^{\mu+\sigma^{2}/2}&=e^{0+0.25^{2}/2}&% \approx 1.032\\ \,\text{mode}&=e^{\mu-\sigma^{2}}&=e^{0-0.25^{2}}&\approx 0.939\\ \,\text{median}&=e^{\mu}&=e^{0}&=1\end{array}
  4. mean = e μ + σ 2 / 2 = e 0 + 1 2 / 2 1.649 mode = e μ - σ 2 = e 0 - 1 2 0.368 median = e μ = e 0 = 1 \begin{array}[]{rlll}\,\text{mean}&=e^{\mu+\sigma^{2}/2}&=e^{0+1^{2}/2}&% \approx 1.649\\ \,\text{mode}&=e^{\mu-\sigma^{2}}&=e^{0-1^{2}}&\approx 0.368\\ \,\text{median}&=e^{\mu}&=e^{0}&=1\end{array}
  5. X ~ \tilde{X}
  6. X ¯ \bar{X}
  7. | X ~ - X ¯ | σ ( 3 / 5 ) 1 / 2 \frac{\left|\tilde{X}-\bar{X}\right|}{\sigma}\leq(3/5)^{1/2}
  8. | X ~ - mode | σ 3 1 / 2 . \frac{\left|\tilde{X}-\mathrm{mode}\right|}{\sigma}\leq 3^{1/2}.
  9. X ± ( 2 α - 1 ) | X - θ | X\pm\left(\frac{2}{\alpha}-1\right)|X-\theta|
  10. X ± ( 1 α - 1 ) | X - θ | X\pm\left(\frac{1}{\alpha}-1\right)|X-\theta|
  11. X ± ( 0.484 α - 1 ) | X - θ | X\pm\left(\frac{0.484}{\alpha}-1\right)|X-\theta|
  12. x ± 5 | x - ν | x\pm 5|x-\nu|
  13. P ( X - k | X - a | μ X + k | X - a | ) 1 - 1 1 + k P(X-k|X-a|\leq\mu\leq X+k|X-a|)\geq 1-\frac{1}{1+k}
  14. X - 23.3 | X | μ X + 23.3 | X | X-23.3|X|\leq\mu\leq X+23.3|X|
  15. σ 10 | X | \sigma\leq 10|X|
  16. μ 0 < x + m 2 ± k | x - m | \mu_{0}<\frac{x+m}{2}\pm k|x-m|
  17. μ 0 > x + m 2 ± k | x - m | \mu_{0}>\frac{x+m}{2}\pm k|x-m|
  18. μ 0 < x + m 2 ± 9.66 | x - m | \mu_{0}<\frac{x+m}{2}\pm 9.66|x-m|

Mode_7.html

  1. [ x y ] = [ a b x 0 c d y 0 ] [ x - x 0 y - y 0 1 ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}a&b&x_{0}\\ c&d&y_{0}\end{bmatrix}\begin{bmatrix}x-x_{0}\\ y-y_{0}\\ 1\end{bmatrix}

Mode_choice.html

  1. c transit c auto = R \frac{c\text{transit}}{c\text{auto}}=R
  2. log ( P i 1 - P i ) = v ( x i ) \log\left(\frac{P_{i}}{1-P_{i}}\right)=v(x_{i})
  3. log ( P A 1 - P A ) = β 0 + β 1 ( c A - c T ) + β 2 ( t A - t T ) + β 3 I + β 4 N = v A \log\left(\frac{P_{A}}{1-P_{A}}\right)=\beta_{0}+\beta_{1}\left(c_{A}-c_{T}% \right)+\beta_{2}\left(t_{A}-t_{T}\right)+\beta_{3}I+\beta_{4}N=v_{A}
  4. P A 1 - P A = e v A \frac{P_{A}}{1-P_{A}}=e^{v_{A}}
  5. P A = e v A - P A e v A P_{A}=e^{v_{A}}-P_{A}e^{v_{A}}
  6. P A ( 1 + e v A ) = e v A P_{A}\left(1+e^{v_{A}}\right)=e^{v_{A}}
  7. P A = e v A 1 + e v A P_{A}=\frac{e^{v_{A}}}{1+e^{v_{A}}}
  8. v A = β 0 + β 1 ( c A - c T ) + β 2 ( t A - t T ) + β 3 I + β 4 N v_{A}=\beta_{0}+\beta_{1}\left(c_{A}-c_{T}\right)+\beta_{2}\left(t_{A}-t_{T}% \right)+\beta_{3}I+\beta_{4}N
  9. L * = n = 1 N f ( y n | x n , θ ) L^{*}=\prod_{n=1}^{N}{f\left({y_{n}\left|{x_{n},\theta}\right.}\right)}
  10. θ ^ \hat{\theta}\,
  11. L θ ^ N = 0 \frac{\partial L}{\partial\hat{\theta}_{N}}=0
  12. ln L * = n = 1 N ln f ( y n | x n , θ ) \ln L^{*}=\sum_{n=1}^{N}\ln f\left(y_{n}\left|x_{n},\theta\right.\right)
  13. L * = n = 1 N P i Y i ( 1 - P i ) 1 - Y i L^{*}=\prod_{n=1}^{N}{P_{i}^{Y_{i}}}\left(1-P_{i}\right)^{1-Y_{i}}
  14. = ln L * = i = 1 n [ Y i ln P i + ( 1 - Y i ) ln ( 1 - P i ) ] \ell=\ln L^{*}=\sum_{i=1}^{n}\left[Y_{i}\ln P_{i}+\left(1-Y_{i}\right)\ln\left% (1-P_{i}\right)\right]
  15. P auto = e v ( x auto ) 1 + e v ( x auto ) P\text{auto}=\frac{e^{v(x\text{auto})}}{1+e^{v(x\text{auto})}}
  16. = ln L * = i = 1 n [ Y i v ( x auto ) - ln ( 1 + e v ( x auto ) ) ] \ell=\ln L^{*}=\sum_{i=1}^{n}\left[Y_{i}v(x\text{auto})-\ln\left(1+e^{v(x\text% {auto})}\right)\right]
  17. β = i = 1 n ( Y i - P ^ i ) = 0 \frac{\partial\ell}{\partial\beta}=\sum_{i=1}^{n}\left(Y_{i}-\hat{P}_{i}\right% )=0

Modelica.html

  1. x ˙ = - c x \dot{x}=-c\cdot x

Modulo_(jargon).html

  1. 1 4 2 8 5 7 7 1 4 2 8 5 \begin{array}[]{ccccccccccccc}&1&&4&&2&&8&&5&&7\\ \searrow&&\searrow&&\searrow&&\searrow&&\searrow&&\searrow&&\searrow\\ &7&&1&&4&&2&&8&&5\end{array}

Modulo_operation.html

  1. a a
  2. n n
  3. a a
  4. n n
  5. q q
  6. r r
  7. a a
  8. n n
  9. q \displaystyle q
  10. ( a mod n ) mod n = a mod n (a\,\bmod\,n)\,\bmod\,n=a\,\bmod\,n
  11. n x mod n = 0 n^{x}\,\bmod\,n=0
  12. x x
  13. n n
  14. b b
  15. a b n - 1 mod n = a mod n ab^{n-1}\,\bmod\,n=a\,\bmod\,n
  16. ( ( - a mod n ) + ( a mod n ) ) mod n = 0 ((-a\,\bmod\,n)+(a\,\bmod\,n))\,\bmod\,n=0
  17. b - 1 mod n b^{-1}\,\bmod\,n
  18. b b
  19. n n
  20. ( ( b - 1 mod n ) ( b mod n ) ) mod n = 1 ((b^{-1}\,\bmod\,n)\,(b\,\bmod\,n))\,\bmod\,n=1
  21. ( a + b ) mod n = ( ( a mod n ) + ( b mod n ) ) mod n (a+b)\,\bmod\,n=((a\,\bmod\,n)+(b\,\bmod\,n))\,\bmod\,n
  22. a b mod n = ( ( a mod n ) ( b mod n ) ) mod n ab\,\bmod\,n=((a\,\bmod\,n)\,(b\,\bmod\,n))\,\bmod\,n
  23. a b mod n = ( ( a mod n ) ( b - 1 mod n ) ) mod n \frac{a}{b}\,\bmod\,n=((a\,\bmod\,n)(b^{-1}\,\bmod\,n))\,\bmod\,n
  24. ( ( a b mod n ) ( b - 1 mod n ) ) mod n = a mod n ((ab\,\bmod\,n)\,(b^{-1}\,\bmod\,n))\,\bmod\,n=a\,\bmod\,n

Mohr's_circle.html

  1. σ n \sigma_{\mathrm{n}}
  2. τ n \tau_{\mathrm{n}}
  3. σ i j \sigma_{ij}
  4. s y m b o l σ symbol\sigma
  5. s y m b o l σ = [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] [ σ x τ x y τ x z τ y x σ y τ y z τ z x τ z y σ z ] symbol{\sigma}=\left[{\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\\ \end{matrix}}\right]\equiv\left[{\begin{matrix}\sigma_{xx}&\sigma_{xy}&\sigma_% {xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\ \end{matrix}}\right]\equiv\left[{\begin{matrix}\sigma_{x}&\tau_{xy}&\tau_{xz}% \\ \tau_{yx}&\sigma_{y}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{z}\\ \end{matrix}}\right]
  6. ( x , y ) (x,y)
  7. P P
  8. ( x , y ) (x^{\prime},y^{\prime})
  9. ( x , y ) (x,y)
  10. P P
  11. ( x , y ) (x,y)
  12. σ x \sigma_{x}
  13. σ y \sigma_{y}
  14. τ x y \tau_{xy}
  15. τ x y = τ y x \tau_{xy}=\tau_{yx}
  16. s y m b o l σ = [ σ x τ x y 0 τ x y σ y 0 0 0 0 ] [ σ x τ x y τ x y σ y ] symbol{\sigma}=\left[{\begin{matrix}\sigma_{x}&\tau_{xy}&0\\ \tau_{xy}&\sigma_{y}&0\\ 0&0&0\\ \end{matrix}}\right]\equiv\left[{\begin{matrix}\sigma_{x}&\tau_{xy}\\ \tau_{xy}&\sigma_{y}\\ \end{matrix}}\right]
  17. σ n \sigma_{\mathrm{n}}
  18. τ n \tau_{\mathrm{n}}
  19. ( x , y ) (x^{\prime},y^{\prime})
  20. P P
  21. x x
  22. y y
  23. ( x , y ) (x^{\prime},y^{\prime})
  24. θ \theta
  25. ( x , y ) (x,y)
  26. P P
  27. y y
  28. z z
  29. σ n \sigma_{\mathrm{n}}
  30. τ n \tau_{\mathrm{n}}
  31. σ n = 1 2 ( σ x + σ y ) + 1 2 ( σ x - σ y ) cos 2 θ + τ x y sin 2 θ \sigma_{\mathrm{n}}=\frac{1}{2}(\sigma_{x}+\sigma_{y})+\frac{1}{2}(\sigma_{x}-% \sigma_{y})\cos 2\theta+\tau_{xy}\sin 2\theta
  32. τ n = - 1 2 ( σ x - σ y ) sin 2 θ + τ x y cos 2 θ \tau_{\mathrm{n}}=-\frac{1}{2}(\sigma_{x}-\sigma_{y})\sin 2\theta+\tau_{xy}% \cos 2\theta
  33. σ n \sigma_{\mathrm{n}}
  34. x x^{\prime}
  35. σ n \sigma_{\mathrm{n}}
  36. d A dA
  37. F x = σ n d A - σ x d A cos 2 θ - σ y d A sin 2 θ - τ x y d A cos θ sin θ - τ x y d A sin θ cos θ = 0 σ n = σ x cos 2 θ + σ y sin 2 θ + 2 τ x y sin θ cos θ \ \begin{aligned}\displaystyle\sum F_{x^{\prime}}&\displaystyle=\sigma_{% \mathrm{n}}dA-\sigma_{x}dA\cos^{2}\theta-\sigma_{y}dA\sin^{2}\theta-\tau_{xy}% dA\cos\theta\sin\theta-\tau_{xy}dA\sin\theta\cos\theta=0\\ \displaystyle\sigma_{\mathrm{n}}&\displaystyle=\sigma_{x}\cos^{2}\theta+\sigma% _{y}\sin^{2}\theta+2\tau_{xy}\sin\theta\cos\theta\\ \end{aligned}
  38. cos 2 θ = 1 + cos 2 θ 2 , sin 2 θ = 1 - cos 2 θ 2 and sin 2 θ = 2 sin θ cos θ \cos^{2}\theta=\frac{1+\cos 2\theta}{2},\qquad\sin^{2}\theta=\frac{1-\cos 2% \theta}{2}\qquad\,\text{and}\qquad\sin 2\theta=2\sin\theta\cos\theta
  39. σ n = 1 2 ( σ x + σ y ) + 1 2 ( σ x - σ y ) cos 2 θ + τ x y sin 2 θ \sigma_{\mathrm{n}}=\frac{1}{2}(\sigma_{x}+\sigma_{y})+\frac{1}{2}(\sigma_{x}-% \sigma_{y})\cos 2\theta+\tau_{xy}\sin 2\theta
  40. τ n \tau_{\mathrm{n}}
  41. y y^{\prime}
  42. τ n \tau_{\mathrm{n}}
  43. d A dA
  44. F y = τ n d A + σ x d A cos θ sin θ - σ y d A sin θ cos θ - τ x y d A cos 2 θ + τ x y d A sin 2 θ = 0 τ n = - ( σ x - σ y ) sin θ cos θ + τ x y ( cos 2 θ - sin 2 θ ) \ \begin{aligned}\displaystyle\sum F_{y^{\prime}}&\displaystyle=\tau_{\mathrm{% n}}dA+\sigma_{x}dA\cos\theta\sin\theta-\sigma_{y}dA\sin\theta\cos\theta-\tau_{% xy}dA\cos^{2}\theta+\tau_{xy}dA\sin^{2}\theta=0\\ \displaystyle\tau_{\mathrm{n}}&\displaystyle=-(\sigma_{x}-\sigma_{y})\sin% \theta\cos\theta+\tau_{xy}\left(\cos^{2}\theta-\sin^{2}\theta\right)\\ \end{aligned}
  45. cos 2 θ - sin 2 θ = cos 2 θ and sin 2 θ = 2 sin θ cos θ \cos^{2}\theta-\sin^{2}\theta=\cos 2\theta\qquad\,\text{and}\qquad\sin 2\theta% =2\sin\theta\cos\theta
  46. τ n = - 1 2 ( σ x - σ y ) sin 2 θ + τ x y cos 2 θ \tau_{\mathrm{n}}=-\frac{1}{2}(\sigma_{x}-\sigma_{y})\sin 2\theta+\tau_{xy}% \cos 2\theta
  47. σ n \sigma_{\mathrm{n}}
  48. τ n \tau_{\mathrm{n}}
  49. s y m b o l σ = 𝐀 s y m b o l σ 𝐀 T [ σ x τ x y τ y x σ y ] = [ a x a x y a y x a y ] [ σ x τ x y τ y x σ y ] [ a x a y x a x y a y ] = [ cos θ sin θ - sin θ cos θ ] [ σ x τ x y τ y x σ y ] [ cos θ - sin θ sin θ cos θ ] \begin{aligned}\displaystyle symbol{\sigma}^{\prime}&\displaystyle=\mathbf{A}% symbol{\sigma}\mathbf{A}^{T}\\ \displaystyle\left[{\begin{matrix}\sigma_{x^{\prime}}&\tau_{x^{\prime}y^{% \prime}}\\ \tau_{y^{\prime}x^{\prime}}&\sigma_{y^{\prime}}\\ \end{matrix}}\right]&\displaystyle=\left[{\begin{matrix}a_{x}&a_{xy}\\ a_{yx}&a_{y}\\ \end{matrix}}\right]\left[{\begin{matrix}\sigma_{x}&\tau_{xy}\\ \tau_{yx}&\sigma_{y}\\ \end{matrix}}\right]\left[{\begin{matrix}a_{x}&a_{yx}\\ a_{xy}&a_{y}\\ \end{matrix}}\right]\\ &\displaystyle=\left[{\begin{matrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{matrix}}\right]\left[{\begin{matrix}\sigma_{x}&\tau_{xy}\\ \tau_{yx}&\sigma_{y}\\ \end{matrix}}\right]\left[{\begin{matrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{matrix}}\right]\end{aligned}
  50. σ x = σ n \sigma_{x^{\prime}}=\sigma_{\mathrm{n}}
  51. τ x y = τ n \tau_{x^{\prime}y^{\prime}}=\tau_{\mathrm{n}}
  52. σ n = σ x cos 2 θ + σ y sin 2 θ + 2 τ x y sin θ cos θ \sigma_{\mathrm{n}}=\sigma_{x}\cos^{2}\theta+\sigma_{y}\sin^{2}\theta+2\tau_{% xy}\sin\theta\cos\theta
  53. cos 2 θ = 1 + cos 2 θ 2 , sin 2 θ = 1 - cos 2 θ 2 and sin 2 θ = 2 sin θ cos θ \cos^{2}\theta=\frac{1+\cos 2\theta}{2},\qquad\sin^{2}\theta=\frac{1-\cos 2% \theta}{2}\qquad\,\text{and}\qquad\sin 2\theta=2\sin\theta\cos\theta
  54. σ n = 1 2 ( σ x + σ y ) + 1 2 ( σ x - σ y ) cos 2 θ + τ x y sin 2 θ \sigma_{\mathrm{n}}=\frac{1}{2}(\sigma_{x}+\sigma_{y})+\frac{1}{2}(\sigma_{x}-% \sigma_{y})\cos 2\theta+\tau_{xy}\sin 2\theta
  55. τ n = - ( σ x - σ y ) sin θ cos θ + τ x y ( cos 2 θ - sin 2 θ ) \tau_{\mathrm{n}}=-(\sigma_{x}-\sigma_{y})\sin\theta\cos\theta+\tau_{xy}\left(% \cos^{2}\theta-\sin^{2}\theta\right)
  56. cos 2 θ - sin 2 θ = cos 2 θ and sin 2 θ = 2 sin θ cos θ \cos^{2}\theta-\sin^{2}\theta=\cos 2\theta\qquad\,\text{and}\qquad\sin 2\theta% =2\sin\theta\cos\theta
  57. τ n = - 1 2 ( σ x - σ y ) sin 2 θ + τ x y cos 2 θ \tau_{\mathrm{n}}=-\frac{1}{2}(\sigma_{x}-\sigma_{y})\sin 2\theta+\tau_{xy}% \cos 2\theta
  58. σ y \sigma_{y^{\prime}}
  59. σ x \sigma_{x^{\prime}}
  60. 2 θ 2\theta
  61. σ n \sigma_{\mathrm{n}}
  62. τ n \tau_{\mathrm{n}}
  63. σ n \sigma_{\mathrm{n}}
  64. τ n \tau_{\mathrm{n}}
  65. θ \theta
  66. 2 θ 2\theta
  67. σ n \sigma_{\mathrm{n}}
  68. τ n \tau_{\mathrm{n}}
  69. [ σ n - 1 2 ( σ x + σ y ) ] 2 + τ n 2 = [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 ( σ n - σ avg ) 2 + τ n 2 = R 2 \begin{aligned}\displaystyle\left[\sigma_{\mathrm{n}}-\tfrac{1}{2}(\sigma_{x}+% \sigma_{y})\right]^{2}+\tau_{\mathrm{n}}^{2}&\displaystyle=\left[\tfrac{1}{2}(% \sigma_{x}-\sigma_{y})\right]^{2}+\tau_{xy}^{2}\\ \displaystyle(\sigma_{\mathrm{n}}-\sigma_{\mathrm{avg}})^{2}+\tau_{\mathrm{n}}% ^{2}&\displaystyle=R^{2}\end{aligned}
  70. R = [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 and σ avg = 1 2 ( σ x + σ y ) R=\sqrt{\left[\tfrac{1}{2}(\sigma_{x}-\sigma_{y})\right]^{2}+\tau_{xy}^{2}}% \quad\,\text{and}\quad\sigma_{\mathrm{avg}}=\tfrac{1}{2}(\sigma_{x}+\sigma_{y})
  71. ( x - a ) 2 + ( y - b ) 2 = r 2 (x-a)^{2}+(y-b)^{2}=r^{2}
  72. r = R r=R
  73. ( a , b ) = ( σ avg , 0 ) (a,b)=(\sigma_{\mathrm{avg}},0)
  74. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  75. τ x y \tau_{xy}
  76. x x
  77. y y
  78. τ x y \tau_{xy}
  79. τ y x \tau_{yx}
  80. y y
  81. x x
  82. τ x y \tau_{xy}
  83. τ y x \tau_{yx}
  84. x x
  85. y y
  86. τ x y \tau_{xy}
  87. τ y x \tau_{yx}
  88. τ n \tau_{\mathrm{n}}
  89. 2 θ 2\theta
  90. 2 θ 2\theta
  91. 2 θ 2\theta
  92. τ n \tau_{\mathrm{n}}
  93. τ n \tau_{\mathrm{n}}
  94. σ x \sigma_{x}
  95. σ y \sigma_{y}
  96. τ x y \tau_{xy}
  97. P P
  98. A A
  99. B B
  100. P P
  101. A A
  102. B B
  103. A A
  104. B B
  105. σ n \sigma_{\mathrm{n}}
  106. τ n \tau_{\mathrm{n}}
  107. D D
  108. D D
  109. P P
  110. O B ¯ \overline{OB}
  111. O D ¯ \overline{OD}
  112. θ \theta
  113. B B
  114. D D
  115. P P
  116. σ x \sigma_{x}
  117. σ y \sigma_{y}
  118. τ x y \tau_{xy}
  119. P P
  120. P P
  121. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  122. σ n \sigma_{\mathrm{n}}
  123. τ n \tau_{\mathrm{n}}
  124. A ( σ y , τ x y ) A(\sigma_{y},\tau_{xy})
  125. B ( σ x , - τ x y ) B(\sigma_{x},-\tau_{xy})
  126. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  127. A A
  128. B B
  129. A A
  130. B B
  131. A B ¯ \overline{AB}
  132. O O
  133. A B ¯ \overline{AB}
  134. σ n \sigma_{\mathrm{n}}
  135. C C
  136. E E
  137. σ n \sigma_{\mathrm{n}}
  138. σ 1 \sigma_{1}
  139. σ 2 \sigma_{2}
  140. σ 1 = σ max = σ avg + R \sigma_{1}=\sigma_{\max}=\sigma\text{avg}+R
  141. σ 2 = σ min = σ avg - R \sigma_{2}=\sigma_{\min}=\sigma\text{avg}-R
  142. σ avg \sigma\text{avg}
  143. O O
  144. σ avg = 1 2 ( σ x + σ y ) \sigma\text{avg}=\tfrac{1}{2}(\sigma_{x}+\sigma_{y})
  145. R R
  146. R = [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 R=\sqrt{\left[\tfrac{1}{2}(\sigma_{x}-\sigma_{y})\right]^{2}+\tau_{xy}^{2}}
  147. O O
  148. R R
  149. τ max , min = ± R \tau_{\max,\min}=\pm R
  150. σ x \sigma_{x}
  151. σ y \sigma_{y}
  152. τ x y \tau_{xy}
  153. P P
  154. A A
  155. B B
  156. P P
  157. σ n \sigma_{\mathrm{n}}
  158. τ n \tau_{\mathrm{n}}
  159. D D
  160. D D
  161. P P
  162. θ \theta
  163. B B
  164. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  165. D D
  166. θ \theta
  167. B B
  168. σ x \sigma_{x}
  169. 2 θ 2\theta
  170. B ( σ x , - τ x y ) B(\sigma_{x},-\tau_{xy})
  171. D ( σ n , τ n ) D(\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  172. 2 θ 2\theta
  173. O B ¯ \overline{OB}
  174. O D ¯ \overline{OD}
  175. θ \theta
  176. P P
  177. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  178. 2 θ 2\theta
  179. A A
  180. B B
  181. P P
  182. θ = 90 \theta=90^{\circ}
  183. 180 180^{\circ}
  184. σ \sigma
  185. τ \tau
  186. σ n \sigma_{\mathrm{n}}
  187. τ n \tau_{\mathrm{n}}
  188. σ x , \sigma_{x},\!
  189. σ y , \sigma_{y},\!
  190. τ x y , \tau_{xy},\!
  191. B B
  192. σ x \sigma_{x}
  193. A A
  194. σ y \sigma_{y}
  195. θ \theta
  196. θ \theta
  197. O B ¯ \overline{OB}
  198. O C ¯ \overline{OC}
  199. θ p \theta_{p}
  200. B B
  201. θ p 1 \theta_{p1}
  202. θ p 2 \theta_{p2}
  203. tan 2 θ p = 2 τ x y σ x - σ y \tan 2\theta_{\mathrm{p}}=\frac{2\tau_{xy}}{\sigma_{x}-\sigma_{y}}
  204. θ p \theta_{\mathrm{p}}
  205. 90 90^{\circ}
  206. τ n \tau_{\mathrm{n}}
  207. σ x = - 10 MPa \sigma_{x^{\prime}}=-10\textrm{ MPa}
  208. σ y = 50 MPa \sigma_{y^{\prime}}=50\textrm{ MPa}
  209. τ x y = 40 MPa \tau_{x^{\prime}y^{\prime}}=40\textrm{ MPa}
  210. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  211. τ n \tau_{\mathrm{n}}
  212. σ n \sigma_{\mathrm{n}}
  213. σ n \sigma_{\mathrm{n}}
  214. R = [ 1 2 ( σ x - σ y ) ] 2 + τ x y 2 = [ 1 2 ( - 10 - 50 ) ] 2 + 40 2 = 50 MPa \begin{aligned}\displaystyle R&\displaystyle=\sqrt{\left[\tfrac{1}{2}(\sigma_{% x}-\sigma_{y})\right]^{2}+\tau_{xy}^{2}}\\ &\displaystyle=\sqrt{\left[\tfrac{1}{2}(-10-50)\right]^{2}+40^{2}}\\ &\displaystyle=50\textrm{ MPa}\\ \end{aligned}
  215. σ avg = 1 2 ( σ x + σ y ) = 1 2 ( - 10 + 50 ) = 20 MPa \begin{aligned}\displaystyle\sigma_{\mathrm{avg}}&\displaystyle=\tfrac{1}{2}(% \sigma_{x}+\sigma_{y})\\ &\displaystyle=\tfrac{1}{2}(-10+50)\\ &\displaystyle=20\textrm{ MPa}\\ \end{aligned}
  216. σ 1 = σ avg + R = 70 MPa \begin{aligned}\displaystyle\sigma_{1}&\displaystyle=\sigma_{\mathrm{avg}}+R\\ &\displaystyle=70\textrm{ MPa}\\ \end{aligned}
  217. σ 2 = σ avg - R = - 30 MPa \begin{aligned}\displaystyle\sigma_{2}&\displaystyle=\sigma_{\mathrm{avg}}-R\\ &\displaystyle=-30\textrm{ MPa}\\ \end{aligned}
  218. τ max , min = ± R = ± 50 MPa \tau_{\max,\min}=\pm R=\pm 50\textrm{ MPa}
  219. σ avg \sigma_{\mathrm{avg}}
  220. tan 2 θ p = 2 τ x y σ x - σ y = 2 * 40 ( - 10 - 50 ) = - 53.13 \begin{aligned}\displaystyle\tan 2\theta_{\mathrm{p}}&\displaystyle=\frac{2% \tau_{xy}}{\sigma_{x}-\sigma_{y}}\\ &\displaystyle=\frac{2*40}{(-10-50)}=-53.13^{\circ}\end{aligned}
  221. θ p 2 = - 26.565 \theta_{p2}=-26.565^{\circ}
  222. 2 θ p 1 = 180 - 53.13 = 126.87 θ p 1 = 63.435 \begin{aligned}\displaystyle 2\theta_{p1}&\displaystyle=180-53.13^{\circ}=126.% 87^{\circ}\\ \displaystyle\theta_{p1}&\displaystyle=63.435^{\circ}\\ \end{aligned}
  223. θ p 1 \theta_{p1}
  224. θ p 2 \theta_{p2}
  225. σ x \sigma_{x^{\prime}}
  226. x x^{\prime}
  227. σ x \sigma_{x}
  228. x x
  229. σ y \sigma_{y^{\prime}}
  230. σ x \sigma_{x^{\prime}}
  231. σ 1 \sigma_{1}
  232. ( σ 1 , σ 2 , σ 3 ) \left(\sigma_{1},\sigma_{2},\sigma_{3}\right)
  233. ( n 1 , n 2 , n 3 ) \left(n_{1},n_{2},n_{3}\right)
  234. x 1 x_{1}
  235. x 2 x_{2}
  236. x 3 x_{3}
  237. σ 1 > σ 2 > σ 3 \sigma_{1}>\sigma_{2}>\sigma_{3}
  238. 𝐓 ( 𝐧 ) \mathbf{T}^{(\mathbf{n})}
  239. 𝐧 \mathbf{n}
  240. ( T ( n ) ) 2 = σ i j σ i k n j n k σ n 2 + τ n 2 = σ 1 2 n 1 2 + σ 2 2 n 2 2 + σ 3 2 n 3 2 \begin{aligned}\displaystyle\left(T^{(n)}\right)^{2}&\displaystyle=\sigma_{ij}% \sigma_{ik}n_{j}n_{k}\\ \displaystyle\sigma_{\mathrm{n}}^{2}+\tau_{\mathrm{n}}^{2}&\displaystyle=% \sigma_{1}^{2}n_{1}^{2}+\sigma_{2}^{2}n_{2}^{2}+\sigma_{3}^{2}n_{3}^{2}\end{aligned}
  241. σ n = σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 . \sigma_{\mathrm{n}}=\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}.
  242. n i n i = n 1 2 + n 2 2 + n 3 2 = 1 n_{i}n_{i}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1
  243. n 1 2 n_{1}^{2}
  244. n 2 2 n_{2}^{2}
  245. n 3 2 n_{3}^{2}
  246. n 1 2 = τ n 2 + ( σ n - σ 2 ) ( σ n - σ 3 ) ( σ 1 - σ 2 ) ( σ 1 - σ 3 ) 0 n 2 2 = τ n 2 + ( σ n - σ 3 ) ( σ n - σ 1 ) ( σ 2 - σ 3 ) ( σ 2 - σ 1 ) 0 n 3 2 = τ n 2 + ( σ n - σ 1 ) ( σ n - σ 2 ) ( σ 3 - σ 1 ) ( σ 3 - σ 2 ) 0. \begin{aligned}\displaystyle n_{1}^{2}&\displaystyle=\frac{\tau_{\mathrm{n}}^{% 2}+(\sigma_{\mathrm{n}}-\sigma_{2})(\sigma_{\mathrm{n}}-\sigma_{3})}{(\sigma_{% 1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}\geq 0\\ \displaystyle n_{2}^{2}&\displaystyle=\frac{\tau_{\mathrm{n}}^{2}+(\sigma_{% \mathrm{n}}-\sigma_{3})(\sigma_{\mathrm{n}}-\sigma_{1})}{(\sigma_{2}-\sigma_{3% })(\sigma_{2}-\sigma_{1})}\geq 0\\ \displaystyle n_{3}^{2}&\displaystyle=\frac{\tau_{\mathrm{n}}^{2}+(\sigma_{% \mathrm{n}}-\sigma_{1})(\sigma_{\mathrm{n}}-\sigma_{2})}{(\sigma_{3}-\sigma_{1% })(\sigma_{3}-\sigma_{2})}\geq 0.\end{aligned}
  247. σ 1 > σ 2 > σ 3 \sigma_{1}>\sigma_{2}>\sigma_{3}
  248. ( n i ) 2 (n_{i})^{2}
  249. τ n 2 + ( σ n - σ 2 ) ( σ n - σ 3 ) 0 \tau_{\mathrm{n}}^{2}+(\sigma_{\mathrm{n}}-\sigma_{2})(\sigma_{\mathrm{n}}-% \sigma_{3})\geq 0
  250. σ 1 - σ 2 > 0 \sigma_{1}-\sigma_{2}>0
  251. σ 1 - σ 3 > 0 \sigma_{1}-\sigma_{3}>0
  252. τ n 2 + ( σ n - σ 3 ) ( σ n - σ 1 ) 0 \tau_{\mathrm{n}}^{2}+(\sigma_{\mathrm{n}}-\sigma_{3})(\sigma_{\mathrm{n}}-% \sigma_{1})\leq 0
  253. σ 2 - σ 3 > 0 \sigma_{2}-\sigma_{3}>0
  254. σ 2 - σ 1 < 0 \sigma_{2}-\sigma_{1}<0
  255. τ n 2 + ( σ n - σ 1 ) ( σ n - σ 2 ) 0 \tau_{\mathrm{n}}^{2}+(\sigma_{\mathrm{n}}-\sigma_{1})(\sigma_{\mathrm{n}}-% \sigma_{2})\geq 0
  256. σ 3 - σ 1 < 0 \sigma_{3}-\sigma_{1}<0
  257. σ 3 - σ 2 < 0. \sigma_{3}-\sigma_{2}<0.
  258. τ n 2 + [ σ n - 1 2 ( σ 2 + σ 3 ) ] 2 ( 1 2 ( σ 2 - σ 3 ) ) 2 τ n 2 + [ σ n - 1 2 ( σ 1 + σ 3 ) ] 2 ( 1 2 ( σ 1 - σ 3 ) ) 2 τ n 2 + [ σ n - 1 2 ( σ 1 + σ 2 ) ] 2 ( 1 2 ( σ 1 - σ 2 ) ) 2 \begin{aligned}\displaystyle\tau_{\mathrm{n}}^{2}+\left[\sigma_{\mathrm{n}}-% \tfrac{1}{2}(\sigma_{2}+\sigma_{3})\right]^{2}\geq\left(\tfrac{1}{2}(\sigma_{2% }-\sigma_{3})\right)^{2}\\ \displaystyle\tau_{\mathrm{n}}^{2}+\left[\sigma_{\mathrm{n}}-\tfrac{1}{2}(% \sigma_{1}+\sigma_{3})\right]^{2}\leq\left(\tfrac{1}{2}(\sigma_{1}-\sigma_{3})% \right)^{2}\\ \displaystyle\tau_{\mathrm{n}}^{2}+\left[\sigma_{\mathrm{n}}-\tfrac{1}{2}(% \sigma_{1}+\sigma_{2})\right]^{2}\geq\left(\tfrac{1}{2}(\sigma_{1}-\sigma_{2})% \right)^{2}\\ \end{aligned}
  259. C 1 C_{1}
  260. C 2 C_{2}
  261. C 3 C_{3}
  262. R 1 = 1 2 ( σ 2 - σ 3 ) R_{1}=\tfrac{1}{2}(\sigma_{2}-\sigma_{3})
  263. R 2 = 1 2 ( σ 1 - σ 3 ) R_{2}=\tfrac{1}{2}(\sigma_{1}-\sigma_{3})
  264. R 3 = 1 2 ( σ 1 - σ 2 ) R_{3}=\tfrac{1}{2}(\sigma_{1}-\sigma_{2})
  265. [ 1 2 ( σ 2 + σ 3 ) , 0 ] \left[\tfrac{1}{2}(\sigma_{2}+\sigma_{3}),0\right]
  266. [ 1 2 ( σ 1 + σ 3 ) , 0 ] \left[\tfrac{1}{2}(\sigma_{1}+\sigma_{3}),0\right]
  267. [ 1 2 ( σ 1 + σ 2 ) , 0 ] \left[\tfrac{1}{2}(\sigma_{1}+\sigma_{2}),0\right]
  268. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  269. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  270. C 1 C_{1}
  271. C 1 C_{1}
  272. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  273. C 2 C_{2}
  274. C 2 C_{2}
  275. ( σ n , τ n ) (\sigma_{\mathrm{n}},\tau_{\mathrm{n}})
  276. C 3 C_{3}
  277. C 3 C_{3}

Molar_mass_distribution.html

  1. M n = M i N i N i , M w = M i 2 N i M i N i , M z = M i 3 N i M i 2 N i , M v = [ M i 1 + a N i M i N i ] 1 a M_{n}=\frac{\sum M_{i}N_{i}}{\sum N_{i}},\quad M_{w}=\frac{\sum M_{i}^{2}N_{i}% }{\sum M_{i}N_{i}},\quad M_{z}=\frac{\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}% ,\quad M_{v}=\left[\frac{\sum M_{i}^{1+a}N_{i}}{\sum M_{i}N_{i}}\right]^{\frac% {1}{a}}
  2. M ¯ n = i N i M i i N i \bar{M}_{n}=\frac{\sum_{i}N_{i}M_{i}}{\sum_{i}N_{i}}
  3. M ¯ w = i N i M i 2 i N i M i \bar{M}_{w}=\frac{\sum_{i}N_{i}M_{i}^{2}}{\sum_{i}N_{i}M_{i}}
  4. N i N_{i}
  5. M i M_{i}
  6. X ¯ w = 1 + p 1 - p M ¯ w = M o ( 1 + p ) 1 - p \bar{X}_{w}=\frac{1+p}{1-p}\quad\bar{M}_{w}=\frac{M_{o}\left(1+p\right)}{1-p}
  7. M ¯ z = M i 3 N i M i 2 N i \bar{M}_{z}=\frac{\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}\quad

Molecular_term_symbol.html

  1. Λ Ω , ( g / u ) ( + / - ) 2 S + 1 {}^{2S+1}\!\Lambda^{(+/-)}_{\Omega,(g/u)}
  2. [ 𝐋 ^ 2 , H ^ ] = 0 [\hat{\mathbf{L}}^{2},\hat{H}]=0
  3. { 𝐒 ^ 2 , 𝐒 ^ z , 𝐋 ^ z , 𝐉 ^ z = 𝐒 ^ z + 𝐋 ^ z } \{\hat{\mathbf{S}}^{2},\hat{\mathbf{S}}_{z},\hat{\mathbf{L}}_{z},\hat{\mathbf{% J}}_{z}=\hat{\mathbf{S}}_{z}+\hat{\mathbf{L}}_{z}\}
  4. M L = i m l i . M_{L}=\sum_{i}{m_{l}}_{i}.
  5. Λ Ω 2 S + 1 {}^{2S+1}\Lambda_{\Omega}
  6. X ~ \tilde{X}
  7. a ~ \tilde{a}

Molecularity.html

  1. A + B 𝑀 C A+B\overset{M}{\to}C
  2. A + B A B * A+B\to AB^{*}
  3. A B * + M C + M AB^{*}+M\to C+M

Mollifier.html

  1. φ \varphi
  2. n φ ( x ) d x = 1 \int_{\mathbb{R}^{n}}\!\varphi(x)\mathrm{d}x=1
  3. lim ϵ 0 φ ϵ ( x ) = lim ϵ 0 ϵ - n φ ( x / ϵ ) = δ ( x ) \lim_{\epsilon\to 0}\varphi_{\epsilon}(x)=\lim_{\epsilon\to 0}\epsilon^{-n}% \varphi(x/\epsilon)=\delta(x)
  4. δ ( x ) \delta(x)
  5. φ \varphi
  6. φ \varphi
  7. φ \varphi
  8. ( x ) (x)
  9. φ \varphi
  10. ( x ) (x)
  11. μ \mu
  12. ( | x | ) (|x|)
  13. μ \mu
  14. φ ϵ \scriptstyle\varphi_{\epsilon}
  15. Φ ϵ ( f ) ( x ) = n φ ϵ ( x - y ) f ( y ) d y \Phi_{\epsilon}(f)(x)=\int_{\mathbb{R}^{n}}\varphi_{\epsilon}(x-y)f(y)\mathrm{% d}y
  16. φ ϵ ( x ) = ϵ - n φ ( x / ϵ ) \scriptstyle\varphi_{\epsilon}(x)=\epsilon^{-n}\varphi(x/\epsilon)
  17. φ \varphi
  18. φ \varphi
  19. ( x ) (x)
  20. φ ( x ) = { e - 1 / ( 1 - | x | 2 ) / I n if | x | < 1 0 if | x | 1 \varphi(x)=\begin{cases}e^{-1/(1-|x|^{2})}/I_{n}&\,\text{ if }|x|<1\\ 0&\,\text{ if }|x|\geq 1\end{cases}
  21. I n I_{n}
  22. φ \varphi
  23. φ \varphi
  24. ( x ) (x)
  25. T T
  26. ϵ \epsilon
  27. T ϵ = T φ ϵ T_{\epsilon}=T\ast\varphi_{\epsilon}
  28. \ast
  29. T T
  30. ϵ \epsilon
  31. T T
  32. lim ϵ 0 T ϵ = lim ϵ 0 T φ ϵ = T D ( n ) \lim_{\epsilon\to 0}T_{\epsilon}=\lim_{\epsilon\to 0}T\ast\varphi_{\epsilon}=T% \in D^{\prime}(\mathbb{R}^{n})
  33. T T
  34. supp T ϵ = supp ( T φ ϵ ) supp T + supp φ ϵ \mathrm{supp}T_{\epsilon}=\mathrm{supp}(T\ast\varphi_{\epsilon})\subset\mathrm% {supp}T+\mathrm{supp}\varphi_{\epsilon}
  35. supp \mathrm{supp}
  36. + +
  37. S S
  38. T T
  39. lim ϵ 0 S ϵ T = lim ϵ 0 S T ϵ = def S T \lim_{\epsilon\to 0}S_{\epsilon}\cdot T=\lim_{\epsilon\to 0}S\cdot T_{\epsilon% }\overset{\mathrm{def}}{=}S\cdot T
  40. B 1 = { x : | x | < 1 } B_{1}=\{x:|x|<1\}
  41. φ 2 \varphi_{2}
  42. ϵ = 1 / 2 \scriptstyle\epsilon=1/2
  43. χ B 1 , 1 / 2 ( x ) = χ B 1 φ 1 / 2 ( x ) = n χ B 1 ( x - y ) φ 1 / 2 ( y ) d y = B 1 / 2 χ B 1 ( x - y ) φ 1 / 2 ( y ) d y ( s u p p ( φ 1 / 2 ) = B 1 / 2 ) \chi_{B_{1},1/2}(x)=\chi_{B_{1}}\ast\varphi_{1/2}(x)=\int_{\mathbb{R}^{n}}\!\!% \!\chi_{B_{1}}(x-y)\varphi_{1/2}(y)\mathrm{d}y=\int_{B_{1/2}}\!\!\!\chi_{B_{1}% }(x-y)\varphi_{1/2}(y)\mathrm{d}y\ \ \ (\because supp(\varphi_{1/2})=B_{1/2})
  44. 1 1
  45. B 1 / 2 = { x : | x | < 1 / 2 } B_{1/2}=\{x:|x|<1/2\}
  46. B 3 / 2 = { x : | x | < 3 / 2 } B_{3/2}=\{x:|x|<3/2\}
  47. | x | |x|
  48. 1 / 2 1/2
  49. | y | |y|
  50. 1 / 2 1/2
  51. | x - y | |x-y|
  52. 1 1
  53. | x | |x|
  54. 1 / 2 1/2
  55. B 1 / 2 χ B 1 ( x - y ) φ 1 / 2 ( y ) d y = B 1 / 2 φ 1 / 2 ( y ) d y = 1 \int_{B_{1/2}}\!\!\!\chi_{B_{1}}(x-y)\varphi_{1/2}(y)\mathrm{d}y=\int_{B_{1/2}% }\!\!\!\varphi_{1/2}(y)\mathrm{d}y=1
  56. ϵ \scriptstyle\epsilon
  57. f ( t ) = e x p ( - 1 / t ) f(t)=exp(-1/t)
  58. t t
  59. f ( x ) = f ( 1 - < n o w i k i > x 2 ) = e x p ( - 1 / ( 1 - x 2 ) ) f(x)=f(1-<nowiki>x^{2})=exp(-1/(1-x^{2}))

Moment_problem.html

  1. m n = - x n d μ ( x ) . m_{n}=\int_{-\infty}^{\infty}x^{n}\,d\mu(x)\,.\,\!
  2. m n = - M n ( x ) d μ ( x ) . m_{n}=\int_{-\infty}^{\infty}M_{n}(x)\,d\mu(x)\,.\,\!
  3. ( H n ) i j = m i + j , (H_{n})_{ij}=m_{i+j}\,,\,\!
  4. μ \mu
  5. φ \scriptstyle\varphi
  6. P ( x ) = k a k x k P(x)=\sum_{k}a_{k}x^{k}\,\!
  7. k a k m k . \sum_{k}a_{k}m_{k}.\,\!
  8. ϕ \phi
  9. φ ( f ) 0 for any f C 0 ( [ a , b ] ) \qquad\varphi(f)\geq 0\,\text{ for any }f\in C_{0}([a,b])
  10. ϕ ( f ) = f d μ \phi(f)=\int f\,d\mu\,\!
  11. μ \mu

Momentum_(technical_analysis).html

  1. 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑐𝑙𝑜𝑠𝑒 𝑡𝑜𝑑𝑎𝑦 - 𝑐𝑙𝑜𝑠𝑒 N 𝑑𝑎𝑦𝑠 𝑎𝑔𝑜 \mathit{momentum}=\mathit{close}_{\mathit{today}}-\mathit{close}_{N\,\mathit{% days\,ago}}
  2. 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 = 𝑐𝑙𝑜𝑠𝑒 𝑡𝑜𝑑𝑎𝑦 - 𝑐𝑙𝑜𝑠𝑒 N 𝑑𝑎𝑦𝑠 𝑎𝑔𝑜 𝑐𝑙𝑜𝑠𝑒 N 𝑑𝑎𝑦𝑠 𝑎𝑔𝑜 \mathit{rate\,of\,change}={\mathit{close}_{\mathit{today}}-\mathit{close}_{N\,% \mathit{days\,ago}}\over\mathit{close}_{N\,\mathit{days\,ago}}}
  3. 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 N + 1 = 𝑆𝑀𝐴 𝑡𝑜𝑑𝑎𝑦 - 𝑆𝑀𝐴 𝑦𝑒𝑠𝑡𝑒𝑟𝑑𝑎𝑦 {\mathit{momentum}\over N+1}=\mathit{SMA}_{\mathit{today}}-\mathit{SMA}_{% \mathit{yesterday}}

Momentum_theory.html

  1. P = T 3 2 ρ A P=\sqrt{\frac{T^{3}}{2\rho A}}
  2. ρ \rho

Money_creation.html

  1. M = 1 R M=\frac{1}{R}
  2. M M
  3. R R

Money_flow_index.html

  1. t y p i c a l p r i c e = h i g h + l o w + c l o s e 3 typical\ price={high+low+close\over 3}
  2. m o n e y f l o w = t y p i c a l p r i c e × v o l u m e money\ flow=typical\ price\times volume
  3. m o n e y r a t i o = p o s i t i v e m o n e y f l o w n e g a t i v e m o n e y f l o w money\ ratio={positive\ money\ flow\over negative\ money\ flow}
  4. M F I = 100 - 100 1 + m o n e y r a t i o MFI=100-{100\over 1+money\ ratio}
  5. M F I = 100 × p o s i t i v e m o n e y f l o w p o s i t i v e m o n e y f l o w + n e g a t i v e m o n e y f l o w MFI=100\times{positive\ money\ flow\over positive\ moneyflow+negative\ money\ flow}

Money_multiplier.html

  1. R / M R R ; R/M\geq RR;
  2. M / R 1 / R R , M/R\leq 1/RR,
  3. M R × ( 1 / R R ) , M\leq R\times(1/RR),
  4. ( 1 / R R ) , (1/RR),
  5. 1 / R R . 1/RR.
  6. ( 1 + C D ) / ( R R + C D ) . (1+CD)/(RR+CD).
  7. m = 1 R R m=\frac{1}{RR}
  8. 1 - R R . 1-RR.
  9. m = ( 1 + C u r r e n c y D r a i n R a t i o ) ( C u r r e n c y D r a i n R a t i o + D e s i r e d R e s e r v e R a t i o ) m=\frac{(1+CurrencyDrainRatio)}{(CurrencyDrainRatio+DesiredReserveRatio)}
  10. α ( 0 , 1 ) \alpha\in\left(0,1\right)\;
  11. β ( 0 , 1 ) \beta\in\left(0,1\right)\;
  12. γ ( 0 , 1 ) \gamma\in\left(0,1\right)\;
  13. D e p o s i t s = n = 0 [ ( 1 - α - β - γ ) ] n = 1 α + β + γ Deposits=\sum_{n=0}^{\infty}\left[\left(1-\alpha-\beta-\gamma\right)\right]^{n% }=\frac{1}{\alpha+\beta+\gamma}
  14. P u b l i c l y H e l d C u r r e n c y = γ D e p o s i t s = γ α + β + γ PubliclyHeldCurrency=\gamma\cdot Deposits=\frac{\gamma}{\alpha+\beta+\gamma}
  15. L o a n s = ( 1 - α - β ) D e p o s i t s = 1 - α - β α + β + γ Loans=\left(1-\alpha-\beta\right)\cdot Deposits=\frac{1-\alpha-\beta}{\alpha+% \beta+\gamma}
  16. m = M o n e y S t o c k M o n e t a r y B a s e = D e p o s i t s + P u b l i c l y H e l d C u r r e n c y M o n e t a r y B a s e = 1 + γ α + β + γ m=\frac{MoneyStock}{MonetaryBase}=\frac{Deposits+PubliclyHeldCurrency}{% MonetaryBase}=\frac{1+\gamma}{\alpha+\beta+\gamma}
  17. α + β = D e s i r e d R e s e r v e R a t i o \alpha+\beta=DesiredReserveRatio
  18. γ = C u r r e n c y D r a i n R a t i o \gamma=CurrencyDrainRatio
  19. k k\;
  20. L k = ( 1 - α - β ) D k - 1 L_{k}=\left(1-\alpha-\beta\right)\cdot D_{k-1}
  21. k k\;
  22. P H M k = γ D k - 1 PHM_{k}=\gamma\cdot D_{k-1}
  23. k k\;
  24. D k = L k - P H M k D_{k}=L_{k}-PHM_{k}\;
  25. n n\;
  26. n = 0 n=0\;
  27. D 0 = 1 D_{0}=1\;
  28. n = 1 n=1\;
  29. D 1 = ( 1 - α - β - γ ) D_{1}=\left(1-\alpha-\beta-\gamma\right)
  30. L 1 = ( 1 - α - β ) L_{1}=\left(1-\alpha-\beta\right)
  31. P H M 1 = γ PHM_{1}=\gamma\;
  32. n = 2 n=2\;
  33. D 2 = ( 1 - α - β - γ ) 2 D_{2}=\left(1-\alpha-\beta-\gamma\right)^{2}
  34. L 2 = ( 1 - α - β ) ( 1 - α - β - γ ) L_{2}=\left(1-\alpha-\beta\right)\left(1-\alpha-\beta-\gamma\right)
  35. P H M 2 = γ ( 1 - α - β - γ ) PHM_{2}=\gamma\left(1-\alpha-\beta-\gamma\right)
  36. n = 3 n=3\;
  37. D 3 = ( 1 - α - β - γ ) 3 D_{3}=\left(1-\alpha-\beta-\gamma\right)^{3}
  38. L 3 = ( 1 - α - β ) ( 1 - α - β - γ ) 2 L_{3}=\left(1-\alpha-\beta\right)\left(1-\alpha-\beta-\gamma\right)^{2}
  39. P H M 3 = γ ( 1 - α - β - γ ) 2 PHM_{3}=\gamma\left(1-\alpha-\beta-\gamma\right)^{2}
  40. n = k n=k\;
  41. D k = ( 1 - α - β - γ ) k D_{k}=\left(1-\alpha-\beta-\gamma\right)^{k}
  42. L k = ( 1 - α - β ) ( 1 - α - β - γ ) k - 1 L_{k}=\left(1-\alpha-\beta\right)\left(1-\alpha-\beta-\gamma\right)^{k-1}
  43. P H M k = γ ( 1 - α - β - γ ) k - 1 PHM_{k}=\gamma\left(1-\alpha-\beta-\gamma\right)^{k-1}
  44. n n\rightarrow\infty
  45. D = 0 D_{\infty}=0
  46. L = 0 L_{\infty}=0
  47. P H M = 0 PHM_{\infty}=0
  48. D = 1 α + β + γ D=\frac{1}{\alpha+\beta+\gamma}
  49. L = 1 - α - β α + β + γ L=\frac{1-\alpha-\beta}{\alpha+\beta+\gamma}
  50. P H M = γ α + β + γ PHM=\frac{\gamma}{\alpha+\beta+\gamma}
  51. R R = 1 5 RR=\tfrac{1}{5}
  52. m = 1 / 1 5 = 5 m=1/\tfrac{1}{5}=5
  53. M = C + D M=C+D
  54. B = C + R B=C+R
  55. M B = C + D C + R = C + D C + R . D / ( C R ) D / ( C R ) M = B . C + D C + R . D / ( C R ) D / ( C R ) = B . D R ( 1 + D C ) D R + D C multiplier \begin{aligned}\displaystyle\frac{M}{B}&\displaystyle=\frac{C+D}{C+R}=\frac{C+% D}{C+R}.\frac{D/(CR)}{D/(CR)}\\ \displaystyle M&\displaystyle=B.\frac{C+D}{C+R}.\frac{D/(CR)}{D/(CR)}\\ &\displaystyle=B.\underbrace{\frac{\tfrac{D}{R}(1+\tfrac{D}{C})}{\tfrac{D}{R}+% \tfrac{D}{C}}}_{\textrm{multiplier}}\end{aligned}
  56. D R ( 1 + D C ) D R + D C \frac{\tfrac{D}{R}(1+\tfrac{D}{C})}{\tfrac{D}{R}+\tfrac{D}{C}}
  57. x = D / R x=D/R
  58. y = D / C y=D/C
  59. M ( x , y ) = B x ( 1 + y ) x + y M(x,y)=B\frac{x(1+y)}{x+y}
  60. { x 1 , y 1 } \{x\geq 1,y\geq 1\}
  61. M M

Monitor_(synchronization).html

  1. P P
  2. P P
  3. c c
  4. P c P_{c}
  5. c c
  6. P c P_{c}
  7. P c P_{c}
  8. P c P_{c}
  9. P c P_{c}
  10. c f u l l c_{full}
  11. c e m p t y c_{empty}
  12. P s I s P o s i t i v e = ( s > 0 ) P_{sIsPositive}=(s>0)
  13. c c
  14. c c
  15. c c
  16. c c
  17. c c
  18. c c
  19. c c
  20. c c
  21. c c
  22. c c
  23. P c P_{c}
  24. c c
  25. c c
  26. I I
  27. I I
  28. I I
  29. c c
  30. I I
  31. P c P_{c}
  32. I I
  33. c c
  34. P c P_{c}
  35. I I
  36. I I
  37. c c
  38. P c P_{c}
  39. I I
  40. I I
  41. P c P_{c}
  42. c c
  43. I I
  44. P c P_{c}
  45. c c
  46. c c
  47. P c P_{c}
  48. c c
  49. I I
  50. I I
  51. P c P_{c}
  52. c c
  53. c c
  54. c c
  55. c c
  56. c c
  57. c c
  58. c c
  59. P c P_{c}
  60. c c
  61. P c P_{c}
  62. P c P_{c}
  63. P c P_{c}
  64. P P
  65. P P
  66. P c P_{c}
  67. P P
  68. P P
  69. I I
  70. P P
  71. I I

Monster_Lie_algebra.html

  1. j ( q ) - 744 = 1 q + 196884 q + 21493760 q 2 + . j(q)-744={1\over q}+196884q+21493760q^{2}+\cdots.
  2. j ( p ) - j ( q ) = ( 1 p - 1 q ) n , m = 1 ( 1 - p n q m ) c n m . j(p)-j(q)=\left({1\over p}-{1\over q}\right)\prod_{n,m=1}^{\infty}(1-p^{n}q^{m% })^{c_{nm}}.

Monte_Carlo_methods_for_option_pricing.html

  1. S t \ S_{t}\,
  2. μ \mu\,
  3. σ \sigma\,
  4. d S t = μ S t d t + σ S t d W t dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}\,
  5. d W t dW_{t}\,

Monte_Carlo_methods_in_finance.html

  1. \mathbb{P}
  2. S 1 , , S n S_{1},...,S_{n}
  3. ω \omega
  4. H ( S 1 ( ω ) , S 2 ( ω ) , , S n ( ω ) ) = : H ( ω ) H(S_{1}(\omega),S_{2}(\omega),\dots,S_{n}(\omega))=:H(\omega)
  5. H 0 = D F T ω H ( ω ) d ( ω ) H_{0}={DF}_{T}\int_{\omega}H(\omega)\,d\mathbb{P}(\omega)
  6. D F T {DF}_{T}
  7. H 0 D F T 1 N ω sample set H ( ω ) H_{0}\approx{DF}_{T}\frac{1}{N}\sum_{\omega\in\,\text{sample set}}H(\omega)
  8. d S = μ S d t + σ S d W t . dS=\mu S\,dt+\sigma S\,dW_{t}.
  9. δ t \delta t
  10. d t dt
  11. δ t \delta t
  12. S ( k δ t ) = S ( 0 ) exp ( i = 1 k [ ( μ - σ 2 2 ) δ t + σ ε i δ t ] ) S(k\delta t)=S(0)\exp\left(\sum_{i=1}^{k}\left[\left(\mu-\frac{\sigma^{2}}{2}% \right)\delta t+\sigma\varepsilon_{i}\sqrt{\delta t}\right]\right)
  13. ε i \varepsilon_{i}
  14. ω \omega
  15. { ε 1 , , ε M } \{\varepsilon_{1},\dots,\varepsilon_{M}\}
  16. H ( ω ) = 1 M k = 1 M S ( k δ t ) . H(\omega)=\frac{1}{M}\sum_{k=1}^{M}S(k\delta t).
  17. ϵ = 𝒪 ( N - 1 / 2 ) \epsilon=\mathcal{O}\left(N^{-1/2}\right)
  18. { ε 1 , , ε M } \{\varepsilon_{1},\dots,\varepsilon_{M}\}
  19. { - ε 1 , , - ε M } \{-\varepsilon_{1},\dots,-\varepsilon_{M}\}

Montel's_theorem.html

  1. \mathcal{F}
  2. D D
  3. z 0 D z_{0}\in D
  4. \mathcal{F}
  5. z 0 z_{0}
  6. U D U\subset D
  7. z 0 z_{0}
  8. f f ( U ) \bigcup_{f\in\mathcal{F}}f(U)
  9. a , b a,b\in\mathbb{C}
  10. { z z + a : a \C } \{z\mapsto z+a:a\in\C\}
  11. \mathcal{F}
  12. z 1 z_{1}
  13. z 1 z - z 1 z\mapsto\frac{1}{z-z_{1}}
  14. { a , b } \mathbb{C}\setminus\{a,b\}

Montel_space.html

  1. f K , n = sup | α | n sup x K | α f ( x ) | \|f\|_{K,n}=\sup_{|\alpha|\leq n}\sup_{x\in K}\left|\partial^{\alpha}f(x)\right|
  2. C 0 ( K ) C 0 ( Ω ) \scriptstyle{C^{\infty}_{0}(K)\subset C^{\infty}_{0}(\Omega)}

Moseley's_law.html

  1. f = k 1 ( Z - k 2 ) \sqrt{f}=k_{1}\cdot\left(Z-k_{2}\right)
  2. f f
  3. k 1 k_{1}
  4. k 2 k_{2}
  5. k 1 k_{1}
  6. k 2 k_{2}
  7. K α K_{\alpha}
  8. f = ( 2.47 × 10 15 ) × ( Z - 1 ) 2 f=\left(2.47\times 10^{15}\right)\times\left(Z-1\right)^{2}
  9. k 1 k_{1}
  10. k 1 k_{1}
  11. k 2 k_{2}
  12. k 2 k_{2}
  13. k 2 k_{2}
  14. f ( K α ) = ( 3.29 × 10 15 ) × 3 / 4 × ( Z - 1 ) 2 f\left(K_{\alpha}\right)=\left(3.29\times 10^{15}\right)\times 3/4\times\left(% Z-1\right)^{2}
  15. f ( L α ) = ( 3.29 × 10 15 ) × 5 / 36 × ( Z - 7.4 ) 2 f\left(L_{\alpha}\right)=\left(3.29\times 10^{15}\right)\times 5/36\times\left% (Z-7.4\right)^{2}
  16. E = h ν = E i - E f = m e q e 2 q Z 2 8 h 2 ϵ 0 2 ( 1 n f 2 - 1 n i 2 ) E=h\nu=E_{i}-E_{f}=\frac{m_{e}q_{e}^{2}q_{Z}^{2}}{8h^{2}\epsilon_{0}^{2}}\left% (\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right)\,
  17. ϵ 0 = 1 / 4 π \scriptstyle\epsilon_{0}=1/4\pi
  18. m e \scriptstyle m_{e}\,
  19. q e \scriptstyle q_{e}\,
  20. n f \scriptstyle n_{f}\,
  21. n i \scriptstyle n_{i}\,
  22. q e 2 q Z 2 = q e 4 \scriptstyle q_{e}^{2}q_{Z}^{2}=q_{e}^{4}\,
  23. q e \scriptstyle q_{e}
  24. q e 4 \scriptstyle q_{e}^{4}
  25. E = h ν = E i - E f = m e q e 4 ( Z - 1 ) 2 8 h 2 ϵ 0 2 ( 1 1 2 - 1 2 2 ) E=h\nu=E_{i}-E_{f}=\frac{m_{e}q_{e}^{4}(Z-1)^{2}}{8h^{2}\epsilon_{0}^{2}}\left% (\frac{1}{1^{2}}-\frac{1}{2^{2}}\right)\,
  26. f = ν = m e q e 4 8 h 3 ϵ 0 2 ( 3 4 ) ( Z - 1 ) 2 = ( 2.48 * 10 15 Hz ) ( Z - 1 ) 2 f=\nu=\frac{m_{e}q_{e}^{4}}{8h^{3}\epsilon_{0}^{2}}\left(\frac{3}{4}\right)(Z-% 1)^{2}=(2.48*10^{15}\ \mathrm{Hz})(Z-1)^{2}\,
  27. K 2 K_{2}

Motion_simulator.html

  1. y ( s ) y(s)
  2. f ( s ) f(s)
  3. y ( s ) f ( s ) = 2.02 ( s + 0.1 ) s + 0.2 \frac{y(s)}{f(s)}=\frac{2.02(s+0.1)}{s+0.2}
  4. y ( s ) y(s)
  5. p h i ( s ) phi(s)
  6. y ( s ) p h i ( s ) = 0.07 s 3 ( s + 50 ) ( s + 0.05 ) ( s + 0.03 ) \frac{y(s)}{phi(s)}=\frac{0.07s^{3}(s+50)}{(s+0.05)(s+0.03)}
  7. θ \theta
  8. θ ¨ + 2 ζ ω n θ ˙ + ω n θ = u ( t ) \ddot{\theta}+2\zeta\omega_{n}\dot{\theta}+\omega_{n}\theta=u(t)
  9. ζ \zeta
  10. ω n \omega_{n}
  11. u ( t ) u(t)
  12. ζ \zeta
  13. ω n \omega_{n}
  14. 0.5 / sec 2 0.5^{\circ}/\sec^{2}
  15. β = sin - 1 ( f g ) \beta=\sin^{-1}(\frac{f}{g})
  16. L P ( s ) = 1 1 + s T LP(s)=\frac{1}{1+sT}
  17. U ( s ) = ( β + L P ( s ) ) X ( s ) U(s)=(\beta+LP(s))X(s)
  18. X ( s ) X(s)
  19. H P ( s ) = Y ( s ) U ( s ) = s 2 T 2 2 1 + s 2 d T 2 + s 2 T 2 2 ( 1 s ) s T 1 1 + s T 1 ( 1 s ) HP(s)=\frac{Y(s)}{U(s)}=\frac{s^{2}T_{2}^{2}}{1+s2dT_{2}+s^{2}T_{2}^{2}}(\frac% {1}{s})\frac{sT_{1}}{1+sT_{1}}(\frac{1}{s})
  20. T , T,
  21. T 1 T_{1}
  22. T 2 T_{2}

Mousetrap_car.html

  1. τ = - κ θ \tau=-\kappa\theta\,
  2. τ \tau\,
  3. θ \theta\,
  4. κ \kappa\,
  5. U = 1 2 κ θ 2 U=\frac{1}{2}\kappa\theta^{2}

Moyal_product.html

  1. f g = f g + n = 1 n C n ( f , g ) f\star g=fg+\sum_{n=1}^{\infty}\hbar^{n}C_{n}(f,g)
  2. 1. f g = f g + 𝒪 ( ) 1.\quad f\star g=fg+\mathcal{O}(\hbar)
  3. 2. f g - g f = i { f , g } + 𝒪 ( 3 ) i { { f , g } } 2.\quad f\star g-g\star f=\mathrm{i}\hbar\{f,g\}+\mathcal{O}(\hbar^{3})\equiv% \mathrm{i}\hbar\{\{f,g\}\}
  4. 3. f 1 = 1 f = f 3.\quad f\star 1=1\star f=f
  5. 4. f g ¯ = g ¯ f ¯ 4.\quad\overline{f\star g}=\overline{g}\star\overline{f}
  6. i \mathrm{i}
  7. n n
  8. Π = i , j Π i j i j , \Pi=\sum_{i,j}\Pi^{ij}\partial_{i}\wedge\partial_{j},
  9. f f
  10. g g
  11. f g = f g + i 2 i , j Π i j ( i f ) ( j g ) - 2 8 i , j , k , m Π i j Π k m ( i k f ) ( j m g ) + f\star g=fg+\frac{i\hbar}{2}\sum_{i,j}\Pi^{ij}(\partial_{i}f)(\partial_{j}g)-% \frac{\hbar^{2}}{8}\sum_{i,j,k,m}\Pi^{ij}\Pi^{km}(\partial_{i}\partial_{k}f)(% \partial_{j}\partial_{m}g)+\ldots
  12. f g = m e i 2 Π ( f g ) , f\star g=m\circ e^{\frac{i\hbar}{2}\Pi}(f\otimes g),
  13. m m
  14. m ( a b ) = a b m(a\otimes b)=ab
  15. e A := 1 + n = 1 1 n ! A n \textstyle e^{A}:=1+\sum_{n=1}^{\infty}\frac{1}{n!}A^{n}
  16. C n C_{n}
  17. C n = i n 2 n n ! m Π n . C_{n}=\frac{i^{n}}{2^{n}n!}m\circ\Pi^{n}.
  18. i \mathrm{i}
  19. exp ( - a ( x 2 + p 2 ) ) exp ( - b ( x 2 + p 2 ) ) = 1 1 + 2 a b exp ( - a + b 1 + 2 a b ( x 2 + p 2 ) ) . \exp\left(-a(x^{2}+p^{2})\right)~{}\star~{}\exp\left(-b(x^{2}+p^{2})\right)={1% \over 1+\hbar^{2}ab}\exp\left(-{a+b\over 1+\hbar^{2}ab}(x^{2}+p^{2})\right).
  20. ħ 0 ħ→0

MQV.html

  1. R ¯ \bar{R}
  2. R = ( x , y ) R=(x,y)
  3. R ¯ = ( x mod 2 L ) + 2 L \bar{R}=(x\,\bmod\,2^{L})+2^{L}
  4. L = log 2 n + 1 2 L=\left\lceil\frac{\lfloor\log_{2}n\rfloor+1}{2}\right\rceil
  5. R ¯ \bar{R}
  6. S a = x + X ¯ a S_{a}=x+\bar{X}a
  7. S b = y + Y ¯ b S_{b}=y+\bar{Y}b
  8. K = h S a ( Y + Y ¯ B ) K=h\cdot S_{a}(Y+\bar{Y}B)
  9. K = h S b ( X + X ¯ A ) K=h\cdot S_{b}(X+\bar{X}A)
  10. K K
  11. K = h S b ( X + X ¯ A ) = h S b ( x P + X ¯ a P ) = h S b ( x + X ¯ a ) P = h S b S a P K=h\cdot S_{b}(X+\bar{X}A)=h\cdot S_{b}(xP+\bar{X}aP)=h\cdot S_{b}(x+\bar{X}a)% P=h\cdot S_{b}S_{a}P
  12. K = h S a ( Y + Y ¯ B ) = h S a ( y P + Y ¯ b P ) = h S a ( y + Y ¯ b ) P = h S b S a P K=h\cdot S_{a}(Y+\bar{Y}B)=h\cdot S_{a}(yP+\bar{Y}bP)=h\cdot S_{a}(y+\bar{Y}b)% P=h\cdot S_{b}S_{a}P
  13. K = h S b S a P K=h\cdot S_{b}S_{a}P

Muller's_method.html

  1. y k ( x ) = f ( x k - 1 ) + ( x - x k - 1 ) f [ x k - 1 , x k - 2 ] + ( x - x k - 1 ) ( x - x k - 2 ) f [ x k - 1 , x k - 2 , x k - 3 ] , y_{k}(x)=f(x_{k-1})+(x-x_{k-1})f[x_{k-1},x_{k-2}]+(x-x_{k-1})(x-x_{k-2})f[x_{k% -1},x_{k-2},x_{k-3}],\,
  2. y k ( x ) = f ( x k - 1 ) + w ( x - x k - 1 ) + f [ x k - 1 , x k - 2 , x k - 3 ] ( x - x k - 1 ) 2 y_{k}(x)=f(x_{k-1})+w(x-x_{k-1})+f[x_{k-1},x_{k-2},x_{k-3}]\,(x-x_{k-1})^{2}\,
  3. w = f [ x k - 1 , x k - 2 ] + f [ x k - 1 , x k - 3 ] - f [ x k - 2 , x k - 3 ] . w=f[x_{k-1},x_{k-2}]+f[x_{k-1},x_{k-3}]-f[x_{k-2},x_{k-3}].\,
  4. x k = x k - 1 - 2 f ( x k - 1 ) w ± w 2 - 4 f ( x k - 1 ) f [ x k - 1 , x k - 2 , x k - 3 ] . x_{k}=x_{k-1}-\frac{2f(x_{k-1})}{w\pm\sqrt{w^{2}-4f(x_{k-1})f[x_{k-1},x_{k-2},% x_{k-3}]}}.
  5. lim k | x k - ξ | | x k - 1 - ξ | μ = | f ′′′ ( ξ ) 6 f ( ξ ) | ( μ - 1 ) / 2 , \lim_{k\to\infty}\frac{|x_{k}-\xi|}{|x_{k-1}-\xi|^{\mu}}=\left|\frac{f^{\prime% \prime\prime}(\xi)}{6f^{\prime}(\xi)}\right|^{(\mu-1)/2},
  6. x 3 - x 2 - x - 1 = 0 x^{3}-x^{2}-x-1=0
  7. x m + 1 - x m - x m - 1 - - x - 1 = 0 x^{m+1}-x^{m}-x^{m-1}-\dots-x-1=0

Multicollinearity.html

  1. y = X β + ϵ y=X\beta+\epsilon
  2. β ^ O L S = ( X X ) - 1 X y \hat{\beta}_{OLS}=(X^{\prime}X)^{-1}X^{\prime}y
  3. X 1 X_{1}
  4. X 2 X_{2}
  5. λ 0 \lambda_{0}
  6. λ 1 \lambda_{1}
  7. X 2 i = λ 0 + λ 1 X 1 i . X_{2i}=\lambda_{0}+\lambda_{1}X_{1i}.
  8. λ 0 + λ 1 X 1 i + λ 2 X 2 i + + λ k X k i = 0 \lambda_{0}+\lambda_{1}X_{1i}+\lambda_{2}X_{2i}+\cdots+\lambda_{k}X_{ki}=0
  9. λ j \lambda_{j}
  10. X j i X_{ji}
  11. Y i = β 0 + β 1 X 1 i + + β k X k i + ε i . Y_{i}=\beta_{0}+\beta_{1}X_{1i}+\cdots+\beta_{k}X_{ki}+\varepsilon_{i}.
  12. X T X X^{T}X
  13. X = [ 1 X 11 X k 1 1 X 1 N X k N ] . X=\begin{bmatrix}1&X_{11}&\cdots&X_{k1}\\ \vdots&\vdots&&\vdots\\ 1&X_{1N}&\cdots&X_{kN}\end{bmatrix}.
  14. v i v_{i}
  15. λ 0 + λ 1 X 1 i + λ 2 X 2 i + + λ k X k i + v i = 0. \lambda_{0}+\lambda_{1}X_{1i}+\lambda_{2}X_{2i}+\cdots+\lambda_{k}X_{ki}+v_{i}% =0.
  16. X j X_{j}
  17. v i v_{i}
  18. λ \lambda
  19. tolerance = 1 - R j 2 , VIF = 1 tolerance , \mathrm{tolerance}=1-R_{j}^{2},\quad\mathrm{VIF}=\frac{1}{\mathrm{tolerance}},
  20. R j 2 R_{j}^{2}
  21. Y Y
  22. X 1 X_{1}
  23. X 1 X_{1}
  24. X 2 X_{2}
  25. X 1 X_{1}
  26. X 2 X_{2}
  27. X 1 X_{1}
  28. X 2 X_{2}
  29. X 1 X_{1}
  30. x 1 x_{1}
  31. x 1 2 x_{1}^{2}
  32. x 1 3 x_{1}^{3}

Multigraph.html

  1. s : A V s:A\rightarrow V
  2. t : A V t:A\rightarrow V
  3. G = ( Σ V , Σ A , V , A , s , t , V , A ) G=(\Sigma_{V},\Sigma_{A},V,A,s,t,\ell_{V},\ell_{A})
  4. Σ V \Sigma_{V}
  5. Σ A \Sigma_{A}
  6. s : A V s\colon A\rightarrow\ V
  7. t : A V t\colon A\rightarrow\ V
  8. V : V Σ V \ell_{V}\colon V\rightarrow\Sigma_{V}
  9. A : A Σ A \ell_{A}\colon A\rightarrow\Sigma_{A}

Multigrid_method.html

  1. i i
  2. N i N_{i}
  3. N i N_{i}
  4. W i = ρ K N i W_{i}=\rho KN_{i}
  5. i + 1 i+1
  6. ρ = N i + 1 / N i < 1 \rho=N_{i+1}/N_{i}<1
  7. K K
  8. k k
  9. W k = W k + 1 + ρ K N k W_{k}=W_{k+1}+\rho KN_{k}
  10. N 1 N_{1}
  11. W 1 = W 2 + ρ K N 1 W_{1}=W_{2}+\rho KN_{1}
  12. N k = ρ k - 1 N 1 N_{k}=\rho^{k-1}N_{1}
  13. W 1 = K N 1 p = 0 n ρ p W_{1}=KN_{1}\sum_{p=0}^{n}\rho^{p}
  14. n n
  15. W 1 < K N 1 1 1 - ρ W_{1}<KN_{1}\frac{1}{1-\rho}
  16. O ( N ) O(N)
  17. O ( N ) O(N)

Multiple_correlation.html

  1. 𝐜 = ( r x 1 y , r x 2 y , , r x N y ) \mathbf{c}={(r_{x_{1}y},r_{x_{2}y},\dots,r_{x_{N}y})}^{\top}
  2. r x n y r_{x_{n}y}
  3. x n x_{n}
  4. y y
  5. R x x R_{xx}
  6. R 2 = 𝐜 R x x - 1 𝐜 R^{2}=\mathbf{c}^{\top}R_{xx}^{-1}\,\mathbf{c}
  7. 𝐜 \mathbf{c}^{\top}
  8. 𝐜 \mathbf{c}
  9. R x x - 1 R_{xx}^{-1}
  10. R x x = ( r x 1 x 1 r x 1 x 2 r x 1 x N r x 2 x 1 r x N x 1 r x N x N ) R_{xx}=\left(\begin{array}[]{cccc}r_{x_{1}x_{1}}&r_{x_{1}x_{2}}&\dots&r_{x_{1}% x_{N}}\\ r_{x_{2}x_{1}}&\ddots&&\vdots\\ \vdots&&\ddots&\\ r_{x_{N}x_{1}}&\dots&&r_{x_{N}x_{N}}\end{array}\right)
  11. R x x R_{xx}
  12. R 2 R^{2}
  13. 𝐜 𝐜 \mathbf{c}^{\top}\,\mathbf{c}
  14. R x x R_{xx}
  15. y y
  16. x x
  17. z z
  18. R R
  19. z z
  20. x x
  21. y y
  22. z z
  23. x x
  24. y y
  25. x x
  26. y y
  27. z z
  28. y y
  29. x x
  30. R R
  31. y y
  32. x x
  33. z z
  34. R R
  35. y y
  36. x x
  37. z z
  38. y y
  39. x x
  40. z z

Multipole_expansion.html

  1. f ( θ , ϕ ) f(\theta,\phi)
  2. f ( θ , ϕ ) = l = 0 m = - l l C l m Y l m ( θ , ϕ ) . f(\theta,\phi)=\sum_{l=0}^{\infty}\,\sum_{m=-l}^{l}\,C^{m}_{l}\,Y^{m}_{l}(% \theta,\phi).
  3. Y l m ( θ , ϕ ) Y^{m}_{l}(\theta,\phi)
  4. C l m C^{m}_{l}
  5. C 0 0 C^{0}_{0}
  6. C 1 - 1 , C 1 0 , C 1 1 C^{-1}_{1},C^{0}_{1},C^{1}_{1}
  7. f ( θ , ϕ ) = C + C i n i + C i j n i n j + C i j k n i n j n k + C i j k l n i n j n k n l + . f(\theta,\phi)=C+C_{i}n^{i}+C_{ij}n^{i}n^{j}+C_{ijk}n^{i}n^{j}n^{k}+C_{ijkl}n^% {i}n^{j}n^{k}n^{l}+\cdots.
  8. n i n^{i}
  9. θ \theta
  10. ϕ \phi
  11. C C
  12. C i C_{i}
  13. C l - m = ( - 1 ) m C l m . C_{l}^{-m}=(-1)^{m}C^{m\ast}_{l}\ .
  14. C = C ; C i = C i ; C i j = C i j ; C i j k = C i j k ; C=C^{\ast};\ C_{i}=C_{i}^{\ast};\ C_{ij}=C_{ij}^{\ast};\ C_{ijk}=C_{ijk}^{\ast% };\ \ldots
  15. r r
  16. r r
  17. V V
  18. V ( r , θ , ϕ ) = l = 0 m = - l l C l m ( r ) Y l m ( θ , ϕ ) = j = 1 l = 0 m = - l l D l , j m r j Y l m ( θ , ϕ ) . V(r,\theta,\phi)=\sum_{l=0}^{\infty}\,\sum_{m=-l}^{l}\,C^{m}_{l}(r)\,Y^{m}_{l}% (\theta,\phi)=\sum_{j=1}^{\infty}\,\sum_{l=0}^{\infty}\,\sum_{m=-l}^{l}\,\frac% {D^{m}_{l,j}}{r^{j}}\,Y^{m}_{l}(\theta,\phi).
  19. v ( 𝐑 - 𝐫 ) = v ( 𝐑 ) - α = x , y , z r α v α ( 𝐑 ) + 1 2 α = x , y , z β = x , y , z r α r β v α β ( 𝐑 ) - + v(\mathbf{R}-\mathbf{r})=v(\mathbf{R})-\sum_{\alpha=x,y,z}r_{\alpha}v_{\alpha}% (\mathbf{R})+\frac{1}{2}\sum_{\alpha=x,y,z}\sum_{\beta=x,y,z}r_{\alpha}r_{% \beta}v_{\alpha\beta}(\mathbf{R})-\cdots+\cdots
  20. v α ( 𝐑 ) ( v ( 𝐫 - 𝐑 ) r α ) 𝐫 = 𝟎 and v α β ( 𝐑 ) ( 2 v ( 𝐫 - 𝐑 ) r α r β ) 𝐫 = 𝟎 . v_{\alpha}(\mathbf{R})\equiv\left(\frac{\partial v(\mathbf{r}-\mathbf{R})}{% \partial r_{\alpha}}\right)_{\mathbf{r}=\mathbf{0}}\quad\hbox{and}\quad v_{% \alpha\beta}(\mathbf{R})\equiv\left(\frac{\partial^{2}v(\mathbf{r}-\mathbf{R})% }{\partial r_{\alpha}\partial r_{\beta}}\right)_{\mathbf{r}=\mathbf{0}}.
  21. ( 2 v ( 𝐫 - 𝐑 ) ) 𝐫 = 𝟎 = α = x , y , z v α α ( 𝐑 ) = 0 \left(\nabla^{2}v(\mathbf{r}-\mathbf{R})\right)_{\mathbf{r}=\mathbf{0}}=\sum_{% \alpha=x,y,z}v_{\alpha\alpha}(\mathbf{R})=0
  22. α = x , y , z β = x , y , z r α r β v α β ( 𝐑 ) = 1 3 α = x , y , z β = x , y , z ( 3 r α r β - δ α β r 2 ) v α β ( 𝐑 ) , \sum_{\alpha=x,y,z}\sum_{\beta=x,y,z}r_{\alpha}r_{\beta}v_{\alpha\beta}(% \mathbf{R})=\frac{1}{3}\sum_{\alpha=x,y,z}\sum_{\beta=x,y,z}(3r_{\alpha}r_{% \beta}-\delta_{\alpha\beta}r^{2})v_{\alpha\beta}(\mathbf{R}),
  23. v ( 𝐫 - 𝐑 ) 1 | 𝐫 - 𝐑 | . v(\mathbf{r}-\mathbf{R})\equiv\frac{1}{|\mathbf{r}-\mathbf{R}|}.
  24. v ( 𝐑 ) = 1 R , v α ( 𝐑 ) = - R α R 3 , and v α β ( 𝐑 ) = 3 R α R β - δ α β R 2 R 5 . v(\mathbf{R})=\frac{1}{R},\quad v_{\alpha}(\mathbf{R})=-\frac{R_{\alpha}}{R^{3% }},\quad\hbox{and}\quad v_{\alpha\beta}(\mathbf{R})=\frac{3R_{\alpha}R_{\beta}% -\delta_{\alpha\beta}R^{2}}{R^{5}}.
  25. q tot i = 1 N q i , P α i = 1 N q i r i α , and Q α β i = 1 N q i ( 3 r i α r i β - δ α β r i 2 ) , q_{\mathrm{tot}}\equiv\sum_{i=1}^{N}q_{i},\quad P_{\alpha}\equiv\sum_{i=1}^{N}% q_{i}r_{i\alpha},\quad\hbox{and}\quad Q_{\alpha\beta}\equiv\sum_{i=1}^{N}q_{i}% (3r_{i\alpha}r_{i\beta}-\delta_{\alpha\beta}r_{i}^{2}),
  26. 4 π ε 0 V ( 𝐑 ) i = 1 N q i v ( 𝐫 i - 𝐑 ) 4\pi\varepsilon_{0}V(\mathbf{R})\equiv\sum_{i=1}^{N}q_{i}v(\mathbf{r}_{i}-% \mathbf{R})
  27. = q tot R + 1 R 3 α = x , y , z P α R α + 1 6 R 5 α , β = x , y , z Q α β ( 3 R α R β - δ α β R 2 ) + =\frac{q_{\mathrm{tot}}}{R}+\frac{1}{R^{3}}\sum_{\alpha=x,y,z}P_{\alpha}R_{% \alpha}+\frac{1}{6R^{5}}\sum_{\alpha,\beta=x,y,z}Q_{\alpha\beta}(3R_{\alpha}R_% {\beta}-\delta_{\alpha\beta}R^{2})+\cdots
  28. α v α α = 0 and α Q α α = 0. \sum_{\alpha}v_{\alpha\alpha}=0\quad\hbox{and}\quad\sum_{\alpha}Q_{\alpha% \alpha}=0.
  29. V ( 𝐑 ) = 1 4 π ε 0 R 3 ( 𝐏 𝐑 ) , V(\mathbf{R})=\frac{1}{4\pi\varepsilon_{0}R^{3}}(\mathbf{P}\cdot\mathbf{R}),
  30. V ( 𝐑 ) i = 1 N q i 4 π ε 0 | 𝐫 i - 𝐑 | = 1 4 π ε 0 = 0 m = - ( - 1 ) m I - m ( 𝐑 ) i = 1 N q i R m ( 𝐫 i ) , V(\mathbf{R})\equiv\sum_{i=1}^{N}\frac{q_{i}}{4\pi\varepsilon_{0}|\mathbf{r}_{% i}-\mathbf{R}|}=\frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}\sum_{m=-% \ell}^{\ell}(-1)^{m}I^{-m}_{\ell}(\mathbf{R})\sum_{i=1}^{N}q_{i}R^{m}_{\ell}(% \mathbf{r}_{i}),
  31. I - m ( 𝐑 ) I^{-m}_{\ell}(\mathbf{R})
  32. R + 1 R^{\ell+1}
  33. R m ( 𝐫 ) R^{m}_{\ell}(\mathbf{r})
  34. Q m i = 1 N q i R m ( 𝐫 i ) , - m . Q^{m}_{\ell}\equiv\sum_{i=1}^{N}q_{i}R^{m}_{\ell}(\mathbf{r}_{i}),\qquad-\ell% \leq m\leq\ell.
  35. R ^ \hat{R}
  36. I m ( 𝐑 ) 4 π 2 + 1 Y m ( R ^ ) R + 1 I^{m}_{\ell}(\mathbf{R})\equiv\sqrt{\frac{4\pi}{2\ell+1}}\frac{Y^{m}_{\ell}(% \hat{R})}{R^{\ell+1}}
  37. V ( 𝐑 ) = 1 4 π ε 0 = 0 m = - ( - 1 ) m I - m ( 𝐑 ) Q m V(\mathbf{R})=\frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}\sum_{m=-\ell% }^{\ell}(-1)^{m}I^{-m}_{\ell}(\mathbf{R})Q^{m}_{\ell}
  38. = 1 4 π ε 0 = 0 [ 4 π 2 + 1 ] 1 / 2 1 R + 1 m = - ( - 1 ) m Y - m ( R ^ ) Q m , R > r max . =\frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}\left[\frac{4\pi}{2\ell+1}% \right]^{1/2}\;\frac{1}{R^{\ell+1}}\;\sum_{m=-\ell}^{\ell}(-1)^{m}Y^{-m}_{\ell% }(\hat{R})Q^{m}_{\ell},\qquad R>r_{\mathrm{max}}.
  39. V = 0 ( 𝐑 ) = q tot 4 π ε 0 R with q tot i = 1 N q i . V_{\ell=0}(\mathbf{R})=\frac{q_{\mathrm{tot}}}{4\pi\varepsilon_{0}R}\qquad% \hbox{with}\quad q_{\mathrm{tot}}\equiv\sum_{i=1}^{N}q_{i}.
  40. 𝐑 = ( R x , R y , R z ) , 𝐏 = ( P x , P y , P z ) with P α i = 1 N q i r i α , α = x , y , z . \mathbf{R}=(R_{x},R_{y},R_{z}),\quad\mathbf{P}=(P_{x},P_{y},P_{z})\quad\hbox{% with}\quad P_{\alpha}\equiv\sum_{i=1}^{N}q_{i}r_{i\alpha},\quad\alpha=x,y,z.
  41. V = 1 ( 𝐑 ) = 1 4 π ε 0 R 3 ( R x P x + R y P y + R z P z ) = 𝐑 𝐏 4 π ε 0 R 3 = R ^ 𝐏 4 π ε 0 R 2 . V_{\ell=1}(\mathbf{R})=\frac{1}{4\pi\varepsilon_{0}R^{3}}(R_{x}P_{x}+R_{y}P_{y% }+R_{z}P_{z})=\frac{\mathbf{R}\cdot\mathbf{P}}{4\pi\varepsilon_{0}R^{3}}=\frac% {\hat{R}\cdot\mathbf{P}}{4\pi\varepsilon_{0}R^{2}}.
  42. Q z 2 i = 1 N q i 1 2 ( 3 z i 2 - r i 2 ) , Q_{z^{2}}\equiv\sum_{i=1}^{N}q_{i}\;\frac{1}{2}(3z_{i}^{2}-r_{i}^{2}),
  43. U A B = i A j B q i q j 4 π ε 0 r i j . U_{AB}=\sum_{i\in A}\sum_{j\in B}\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}r_{ij}}.
  44. 𝐑 A B + 𝐫 B j + 𝐫 j i + 𝐫 i A = 0 𝐫 i j = 𝐑 A B - 𝐫 A i + 𝐫 B j . \mathbf{R}_{AB}+\mathbf{r}_{Bj}+\mathbf{r}_{ji}+\mathbf{r}_{iA}=0\quad% \Leftrightarrow\quad\mathbf{r}_{ij}=\mathbf{R}_{AB}-\mathbf{r}_{Ai}+\mathbf{r}% _{Bj}.
  45. | 𝐑 A B | > | 𝐫 B j - 𝐫 A i | for all i , j . |\mathbf{R}_{AB}|>|\mathbf{r}_{Bj}-\mathbf{r}_{Ai}|\quad\hbox{for all}\quad i,j.
  46. 1 | 𝐫 j - 𝐫 i | = 1 | 𝐑 A B - ( 𝐫 A i - 𝐫 B j ) | = L = 0 M = - L L ( - 1 ) M I L - M ( 𝐑 A B ) R L M ( 𝐫 A i - 𝐫 B j ) , \frac{1}{|\mathbf{r}_{j}-\mathbf{r}_{i}|}=\frac{1}{|\mathbf{R}_{AB}-(\mathbf{r% }_{Ai}-\mathbf{r}_{Bj})|}=\sum_{L=0}^{\infty}\sum_{M=-L}^{L}\,(-1)^{M}I_{L}^{-% M}(\mathbf{R}_{AB})\;R^{M}_{L}(\mathbf{r}_{Ai}-\mathbf{r}_{Bj}),
  47. I L M I^{M}_{L}
  48. R L M R^{M}_{L}
  49. R L M ( 𝐫 A i - 𝐫 B j ) = A = 0 L ( - 1 ) L - A ( 2 L 2 A ) 1 / 2 R^{M}_{L}(\mathbf{r}_{Ai}-\mathbf{r}_{Bj})=\sum_{\ell_{A}=0}^{L}(-1)^{L-\ell_{% A}}{\left({{2L}\atop{2\ell_{A}}}\right)}^{1/2}
  50. × m A = - A A R A m A ( 𝐫 A i ) R L - A M - m A ( 𝐫 B j ) A , m A ; L - A , M - m A | L M , \times\sum_{m_{A}=-\ell_{A}}^{\ell_{A}}R^{m_{A}}_{\ell_{A}}(\mathbf{r}_{Ai})R^% {M-m_{A}}_{L-\ell_{A}}(\mathbf{r}_{Bj})\;\langle\ell_{A},m_{A};L-\ell_{A},M-m_% {A}|LM\rangle,
  51. R m ( - 𝐫 ) = ( - 1 ) R m ( 𝐫 ) . R^{m}_{\ell}(-\mathbf{r})=(-1)^{\ell}R^{m}_{\ell}(\mathbf{r}).
  52. U A B = 1 4 π ε 0 A = 0 B = 0 ( - 1 ) B ( 2 A + 2 B 2 A ) 1 / 2 U_{AB}=\frac{1}{4\pi\varepsilon_{0}}\sum_{\ell_{A}=0}^{\infty}\sum_{\ell_{B}=0% }^{\infty}(-1)^{\ell_{B}}{\left({{2\ell_{A}+2\ell_{B}}\atop{2\ell_{A}}}\right)% }^{1/2}\,
  53. × m A = - A A m B = - B B ( - 1 ) m A + m B I A + B - m A - m B ( 𝐑 A B ) Q A m A Q B m B A , m A ; B , m B | A + B , m A + m B . \times\sum_{m_{A}=-\ell_{A}}^{\ell_{A}}\sum_{m_{B}=-\ell_{B}}^{\ell_{B}}(-1)^{% m_{A}+m_{B}}I_{\ell_{A}+\ell_{B}}^{-m_{A}-m_{B}}(\mathbf{R}_{AB})\;Q^{m_{A}}_{% \ell_{A}}Q^{m_{B}}_{\ell_{B}}\;\langle\ell_{A},m_{A};\ell_{B},m_{B}|\ell_{A}+% \ell_{B},m_{A}+m_{B}\rangle.
  54. I A + B - ( m A + m B ) ( 𝐑 A B ) [ 4 π 2 A + 2 B + 1 ] 1 / 2 Y A + B - ( m A + m B ) ( 𝐑 ^ A B ) R A B A + B + 1 I_{\ell_{A}+\ell_{B}}^{-(m_{A}+m_{B})}(\mathbf{R}_{AB})\equiv\left[\frac{4\pi}% {2\ell_{A}+2\ell_{B}+1}\right]^{1/2}\;\frac{Y^{-(m_{A}+m_{B})}_{\ell_{A}+\ell_% {B}}(\widehat{\mathbf{R}}_{AB})}{R^{\ell_{A}+\ell_{B}+1}_{AB}}
  55. Q m i = 1 N e Z i R m ( 𝐫 i ) , Q^{m}_{\ell}\equiv\sum_{i=1}^{N}eZ_{i}\;R^{m}_{\ell}(\mathbf{r}_{i}),
  56. R m ( 𝐫 i ) R^{m}_{\ell}(\mathbf{r}_{i})
  57. \ell
  58. M m Ψ | Q m | Ψ . M^{m}_{\ell}\equiv\langle\Psi|Q^{m}_{\ell}|\Psi\rangle.
  59. Q 1 m Q^{m}_{1}
  60. M 0 0 = i = 1 N e Z i , M^{0}_{0}=\sum_{i=1}^{N}eZ_{i},
  61. M 1 1 = - 1 2 i = 1 N e Z i Ψ | x i + i y i | Ψ and M 1 - 1 = 1 2 i = 1 N e Z i Ψ | x i - i y i | Ψ . M^{1}_{1}=-\sqrt{\tfrac{1}{2}}\sum_{i=1}^{N}eZ_{i}\langle\Psi|x_{i}+iy_{i}|% \Psi\rangle\quad\hbox{and}\quad M^{-1}_{1}=\sqrt{\tfrac{1}{2}}\sum_{i=1}^{N}eZ% _{i}\langle\Psi|x_{i}-iy_{i}|\Psi\rangle.
  62. M 1 0 = i = 1 N e Z i Ψ | z i | Ψ . M^{0}_{1}=\sum_{i=1}^{N}eZ_{i}\langle\Psi|z_{i}|\Psi\rangle.
  63. C m C^{m}_{\ell}
  64. S m S^{m}_{\ell}
  65. C 1 0 \displaystyle C^{0}_{1}

Multiresolution_analysis.html

  1. L 2 ( ) L^{2}(\mathbb{R})
  2. { 0 } V 1 V 0 V - 1 V - n V - ( n + 1 ) L 2 ( \R ) \{0\}\dots\subset V_{1}\subset V_{0}\subset V_{-1}\subset\dots\subset V_{-n}% \subset V_{-(n+1)}\subset\dots\subset L^{2}(\R)
  3. f V k , m \Z f\in V_{k},\;m\in\Z
  4. g ( x ) = f ( x - m 2 k ) g(x)=f(x-m2^{k})
  5. V k V_{k}
  6. V k V l , k > l , V_{k}\subset V_{l},\;k>l,
  7. f V k f\in V_{k}
  8. g V l g\in V_{l}
  9. x \R : g ( x ) = f ( 2 k - l x ) \forall x\in\R:\;g(x)=f(2^{k-l}x)
  10. ϕ \phi
  11. ϕ 1 , , ϕ r \phi_{1},\dots,\phi_{r}
  12. V 0 L 2 ( \R ) V_{0}\subset L^{2}(\R)
  13. L 2 ( \R ) L^{2}(\R)
  14. V 0 V - 1 V_{0}\subset V_{-1}
  15. a k = 2 ϕ ( x ) , ϕ ( 2 x - k ) a_{k}=2\langle\phi(x),\phi(2x-k)\rangle
  16. | k | N |k|\leq N
  17. a k = 0 a_{k}=0
  18. | k | > N |k|>N
  19. ϕ ( x ) = k = - N N a k ϕ ( 2 x - k ) . \phi(x)=\sum_{k=-N}^{N}a_{k}\phi(2x-k).
  20. ψ ( x ) := k = - N N ( - 1 ) k a 1 - k ϕ ( 2 x - k ) , \psi(x):=\sum_{k=-N}^{N}(-1)^{k}a_{1-k}\phi(2x-k),
  21. W 0 V - 1 W_{0}\subset V_{-1}
  22. V 0 V_{0}
  23. V - 1 V_{-1}
  24. V - 1 V_{-1}
  25. \oplus
  26. W 0 W_{0}
  27. V 0 V_{0}
  28. W k W_{k}
  29. W 0 W_{0}
  30. L 2 ( \R ) = closure of k \Z W k , L^{2}(\R)=\mbox{closure of }~{}\bigoplus_{k\in\Z}W_{k},
  31. { ψ k , n ( x ) = 2 - k ψ ( 2 - k x - n ) : k , n \Z } \{\psi_{k,n}(x)=\sqrt{2}^{-k}\psi(2^{-k}x-n):\;k,n\in\Z\}
  32. L 2 ( \R ) L^{2}(\R)

Møller–Plesset_perturbation_theory.html

  1. H ^ 0 \hat{H}_{0}
  2. V ^ \hat{V}
  3. H ^ = H ^ 0 + λ V ^ . \hat{H}=\hat{H}_{0}+\lambda\hat{V}.
  4. Ψ = lim m i = 0 m λ i Ψ ( i ) , \Psi=\lim_{m\to\infty}\sum_{i=0}^{m}\lambda^{i}\Psi^{(i)},
  5. E = lim m i = 0 m λ i E ( i ) . E=\lim_{m\to\infty}\sum_{i=0}^{m}\lambda^{i}E^{(i)}.
  6. m m\to\infty
  7. ( H ^ 0 + λ V ) ( i = 0 m λ i Ψ ( i ) ) = ( i = 0 m λ i E ( i ) ) ( i = 0 m λ i Ψ ( i ) ) . \left(\hat{H}_{0}+\lambda V\right)\left(\sum_{i=0}^{m}\lambda^{i}\Psi^{(i)}% \right)=\left(\sum_{i=0}^{m}\lambda^{i}E^{(i)}\right)\left(\sum_{i=0}^{m}% \lambda^{i}\Psi^{(i)}\right).
  8. λ k \lambda^{k}
  9. H ^ 0 F ^ + Φ 0 | ( H ^ - F ^ ) | Φ 0 \hat{H}_{0}\equiv\hat{F}+\langle\Phi_{0}|(\hat{H}-\hat{F})|\Phi_{0}\rangle
  10. V ^ H ^ - H ^ 0 = H ^ - ( F ^ + Φ 0 | ( H ^ - F ^ ) | Φ 0 ) , \hat{V}\equiv\hat{H}-\hat{H}_{0}=\hat{H}-\left(\hat{F}+\langle\Phi_{0}|(\hat{H% }-\hat{F})|\Phi_{0}\rangle\right),
  11. F ^ Φ 0 k = 1 N f ^ ( k ) Φ 0 = 2 i = 1 N / 2 ε i Φ 0 . \hat{F}\Phi_{0}\equiv\sum_{k=1}^{N}\hat{f}(k)\Phi_{0}=2\sum_{i=1}^{N/2}% \varepsilon_{i}\Phi_{0}.
  12. H ^ \hat{H}
  13. f ^ ( k ) \hat{f}(k)
  14. F ^ \hat{F}
  15. F ^ Φ 0 - Φ 0 | F ^ | Φ 0 Φ 0 = 0 H ^ 0 Φ 0 = Φ 0 | H ^ | Φ 0 Φ 0 , \hat{F}\Phi_{0}-\langle\Phi_{0}|\hat{F}|\Phi_{0}\rangle\Phi_{0}=0\implies\hat{% H}_{0}\Phi_{0}=\langle\Phi_{0}|\hat{H}|\Phi_{0}\rangle\Phi_{0},
  16. H ^ \hat{H}
  17. E MP1 Φ 0 | V ^ | Φ 0 = 0 E_{\,\text{MP1}}\equiv\langle\Phi_{0}|\hat{V}|\Phi_{0}\rangle=0
  18. E MP2 = 1 4 i , j , a , b φ i φ j | v ~ ^ | φ a φ b φ a φ b | v ~ ^ | φ i φ j ε i + ε j - ε a - ε b \begin{aligned}\displaystyle E_{\,\text{MP2}}&\displaystyle=\frac{1}{4}\sum_{i% ,j,a,b}\frac{\langle\varphi_{i}\varphi_{j}|\hat{\tilde{v}}|\varphi_{a}\varphi_% {b}\rangle\langle\varphi_{a}\varphi_{b}|\hat{\tilde{v}}|\varphi_{i}\varphi_{j}% \rangle}{\varepsilon_{i}+\varepsilon_{j}-\varepsilon_{a}-\varepsilon_{b}}\\ \end{aligned}
  19. Φ 0 | ( H ^ - F ^ ) | Φ 0 0 E HF 2 i = 1 N / 2 ε i . \langle\Phi_{0}|(\hat{H}-\hat{F})|\Phi_{0}\rangle\neq 0\qquad% \Longleftrightarrow\qquad E_{\,\text{HF}}\neq 2\sum_{i=1}^{N/2}\varepsilon_{i}.
  20. H ^ 0 F ^ , V ^ H ^ - F ^ . \hat{H}_{0}\equiv\hat{F},\qquad\hat{V}\equiv\hat{H}-\hat{F}.
  21. E MP0 = 2 i = 1 N / 2 ε i , E MP1 = E HF - 2 i = 1 N / 2 ε i . E_{\,\text{MP0}}=2\sum_{i=1}^{N/2}\varepsilon_{i},\qquad E_{\,\text{MP1}}=E_{% \,\text{HF}}-2\sum_{i=1}^{N/2}\varepsilon_{i}.

Nagata–Smirnov_metrization_theorem.html

  1. X X

Naimark's_problem.html

  1. * *
  2. * *
  3. \diamondsuit
  4. 1 \aleph_{1}
  5. 1 \aleph_{1}
  6. 𝖹𝖥𝖢 \mathsf{ZFC}
  7. 𝖹𝖥𝖢 \mathsf{ZFC}
  8. 𝖹𝖥𝖢 \mathsf{ZFC}

Nakayama_lemma.html

  1. k = 1 I k M = 0. \textstyle{\bigcap_{k=1}^{\infty}I^{k}M=0.}
  2. Tor i + 1 R ( k , M ) = 0. \operatorname{Tor}_{i+1}^{R}(k,M)=0.
  3. p ( x ) = x n + p 1 x n - 1 + + p n p(x)=x^{n}+p_{1}x^{n-1}+\cdots+p_{n}
  4. p ( φ ) = 0 p(\varphi)=0
  5. φ ( x i ) = j = 1 n a i j x j \varphi(x_{i})=\sum_{j=1}^{n}a_{ij}x_{j}
  6. j = 1 n ( φ δ i j - a i j ) x j = 0. \sum_{j=1}^{n}\left(\varphi\delta_{ij}-a_{ij}\right)x_{j}=0.
  7. p ( t ) = det ( t δ i j - a i j ) . p(t)=\det(t\delta_{ij}-a_{ij}).
  8. r = p ( 1 ) = 1 + p 1 + p 2 + + p n r=p(1)=1+p_{1}+p_{2}+\cdots+p_{n}
  9. i { 1 , , n } i\in\{1,\ldots,n\}
  10. i = 1 n x i R = U \sum_{i=1}^{n}x_{i}R=U
  11. U J ( R ) = U U\cdot J(R)=U
  12. i = 1 n x i U \sum_{i=1}^{n}x_{i}\in U
  13. i = 1 n ( x i r i ) j i = i = 1 n x i \sum_{i=1}^{n}(x_{i}r_{i})j_{i}=\sum_{i=1}^{n}x_{i}
  14. r i R r_{i}\in R
  15. j i J ( R ) j_{i}\in J(R)
  16. i = 1 n x i ( r i j i ) = i = 1 n x i \sum_{i=1}^{n}x_{i}(r_{i}j_{i})=\sum_{i=1}^{n}x_{i}
  17. r i j i J ( R ) r_{i}j_{i}\in J\left(R\right)
  18. i = 1 n x i k i = i = 1 n x i \sum_{i=1}^{n}x_{i}k_{i}=\sum_{i=1}^{n}x_{i}
  19. k i J ( R ) k_{i}\in J(R)
  20. i = 1 n x i ( 1 - k i ) = 0 \sum_{i=1}^{n}x_{i}(1-k_{i})=0
  21. k i J ( R ) k_{i}\in J(R)
  22. 1 - k i U ( R ) 1-k_{i}\in U(R)
  23. i = 1 n x i ( 1 - k i ) ( 1 - k j ) - 1 = 0 \sum_{i=1}^{n}x_{i}(1-k_{i})(1-k_{j})^{-1}=0
  24. i j x i ( 1 - k i ) ( 1 - k j ) - 1 = - x j \sum_{i\neq j}x_{i}(1-k_{i})(1-k_{j})^{-1}=-x_{j}
  25. R + R_{+}
  26. M i = 0 M_{i}=0
  27. R + M = M R_{+}M=M
  28. M = 0 M=0
  29. M i 0 M_{i}\neq 0
  30. M i M_{i}
  31. R + M R_{+}M
  32. M R + M M\neq R_{+}M
  33. M = 0 M=0

Napierian_logarithm.html

  1. NapLog ( x ) = log 10 7 x log 10 7 10 7 - 1 . \mathrm{NapLog}(x)=\frac{\log\frac{10^{7}}{x}}{\log\frac{10^{7}}{10^{7}-1}}.
  2. NapLog ( x ) = log 10 7 10 7 - 1 10 7 - log 10 7 10 7 - 1 x \mathrm{NapLog}(x)=\log_{\frac{10^{7}}{10^{7}-1}}10^{7}-\log_{\frac{10^{7}}{10% ^{7}-1}}x
  3. NapLog ( x y ) = NapLog ( x ) + NapLog ( y ) - 161180950 \mathrm{NapLog}(xy)=\mathrm{NapLog}(x)+\mathrm{NapLog}(y)-161180950
  4. NapLog ( x ) 9999999.5 ( 16.11809565 - ln x ) \mathrm{NapLog}(x)\approx 9999999.5(16.11809565-\ln x)
  5. NapLog ( x ) 23025850 ( 7 - log 10 x ) . \mathrm{NapLog}(x)\approx 23025850(7-\log_{10}x).
  6. 16.11809565 7 ln ( 10 ) 16.11809565\approx 7\ln\left(10\right)
  7. 23025850 10 7 ln ( 10 ) . 23025850\approx 10^{7}\ln(10).

Natural_density.html

  1. \mathbb{N}
  2. A A
  3. = { 1 , 2 , } . \mathbb{N}=\{1,2,\ldots\}.
  4. n n\in\mathbb{N}
  5. A ( n ) = { 1 , 2 , , n } A . A(n)=\{1,2,\ldots,n\}\cap A.
  6. a ( n ) = | A ( n ) | a(n)=|A(n)|
  7. d ¯ ( A ) \overline{d}(A)
  8. A A
  9. d ¯ ( A ) = lim sup n a ( n ) n \overline{d}(A)=\limsup_{n\rightarrow\infty}\frac{a(n)}{n}
  10. d ¯ ( A ) \overline{d}(A)
  11. A . A.
  12. d ¯ ( A ) \underline{d}(A)
  13. A A
  14. d ¯ ( A ) = lim inf n a ( n ) n \underline{d}(A)=\liminf_{n\rightarrow\infty}\frac{a(n)}{n}
  15. A A
  16. d ( A ) d(A)
  17. d ¯ ( A ) = d ¯ ( A ) \underline{d}(A)=\overline{d}(A)
  18. d ( A ) d(A)
  19. d ( A ) = lim n a ( n ) n d(A)=\lim_{n\rightarrow\infty}\frac{a(n)}{n}
  20. \mathbb{N}
  21. A = { a 1 < a 2 < < a n < ; n } A=\{a_{1}<a_{2}<\ldots<a_{n}<\ldots;n\in\mathbb{N}\}
  22. d ¯ ( A ) = lim inf n n a n , \underline{d}(A)=\liminf_{n\rightarrow\infty}\frac{n}{a_{n}},
  23. d ¯ ( A ) = lim sup n n a n \overline{d}(A)=\limsup_{n\rightarrow\infty}\frac{n}{a_{n}}
  24. d ( A ) = lim n n a n d(A)=\lim_{n\rightarrow\infty}\frac{n}{a_{n}}
  25. A A\subseteq\mathbb{N}
  26. d * ( A ) d^{*}(A)
  27. d * ( A ) = lim sup N - M | A { M , M + 1 , , N } | N - M + 1 d^{*}(A)=\limsup_{N-M\rightarrow\infty}\frac{|A\bigcap\{M,M+1,\ldots,N\}|}{N-M% +1}
  28. d ( A ) d(A)
  29. d ( B ) d(B)
  30. d ( A B ) d(A\cup B)
  31. max { d ( A ) , d ( B ) } d ( A B ) min { d ( A ) + d ( B ) , 1 } \max\{d(A),d(B)\}\leq d(A\cup B)\leq\min\{d(A)+d(B),1\}
  32. d ¯ ( A ) + d ¯ ( B ) d ¯ ( A B ) d ¯ ( A ) + d ¯ ( B ) \underline{d}(A)+\overline{d}(B)\leq\overline{d}(A\cup B)\leq\overline{d}(A)+% \overline{d}(B)
  33. A = { n 2 ; n } A=\{n^{2};n\in\mathbb{N}\}
  34. A = { 2 n ; n } A=\{2n;n\in\mathbb{N}\}
  35. A = { a n + b ; n } A=\{an+b;n\in\mathbb{N}\}
  36. 6 π 2 \tfrac{6}{\pi^{2}}
  37. A = n = 0 { 2 2 n , , 2 2 n + 1 - 1 } A=\bigcup\limits_{n=0}^{\infty}\{2^{2n},\ldots,2^{2n+1}-1\}
  38. d ¯ ( A ) = lim m 1 + 2 2 + + 2 2 m 2 2 m + 1 - 1 = lim m 2 2 m + 2 - 1 3 ( 2 2 m + 1 - 1 ) = 2 3 , \overline{d}(A)=\lim_{m\rightarrow\infty}\frac{1+2^{2}+\cdots+2^{2m}}{2^{2m+1}% -1}=\lim_{m\rightarrow\infty}\frac{2^{2m+2}-1}{3(2^{2m+1}-1)}=\frac{2}{3}\,,
  39. d ¯ ( A ) = lim m 1 + 2 2 + + 2 2 m 2 2 m + 2 - 1 = lim m 2 2 m + 2 - 1 3 ( 2 2 m + 2 - 1 ) = 1 3 . \underline{d}(A)=\lim_{m\rightarrow\infty}\frac{1+2^{2}+\cdots+2^{2m}}{2^{2m+2% }-1}=\lim_{m\rightarrow\infty}\frac{2^{2m+2}-1}{3(2^{2m+2}-1)}=\frac{1}{3}\,.
  40. { α n } n 𝒩 \{\alpha_{n}\}_{n\in\mathcal{N}}
  41. [ 0 , 1 ] [0,1]
  42. { A x } x [ 0 , 1 ] \{A_{x}\}_{x\in[0,1]}
  43. A x := { n : α n < x } . A_{x}:=\{n\in\mathbb{N}\,:\,\alpha_{n}<x\}\,.
  44. d ( A x ) = x d(A_{x})=x
  45. x x
  46. δ ( A ) = lim x 1 log x n A , n x 1 n . \mathbf{\delta}(A)=\lim_{x\rightarrow\infty}\frac{1}{\log x}\sum_{n\in A,n\leq x% }\frac{1}{n}\ .

Need_for_Speed:_Underground_2.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Neighbourhood_(mathematics).html

  1. X X
  2. p p
  3. X X
  4. p p
  5. V V
  6. X X
  7. U U
  8. p p
  9. p U V . p\in U\subseteq V.
  10. p X p\in X
  11. V V
  12. V V
  13. V V
  14. S S
  15. X X
  16. S S
  17. V V
  18. U U
  19. S S
  20. V V
  21. S S
  22. S S
  23. V V
  24. S S
  25. S S
  26. V V
  27. M = ( X , d ) M=(X,d)
  28. V V
  29. p p
  30. p p
  31. r > 0 r>0
  32. B r ( p ) = B ( p ; r ) = { x X d ( x , p ) < r } B_{r}(p)=B(p;r)=\{x\in X\mid d(x,p)<r\}
  33. V V
  34. V V
  35. S S
  36. r r
  37. p p
  38. S S
  39. B r ( p ) = { x X d ( x , p ) < r } B_{r}(p)=\{x\in X\mid d(x,p)<r\}
  40. V V
  41. r > 0 r>0
  42. r r
  43. S r S_{r}
  44. S S
  45. X X
  46. r r
  47. S S
  48. S S
  49. r r
  50. r r
  51. S S
  52. r r
  53. r r
  54. r r
  55. \mathbb{R}
  56. V V
  57. V := n B ( n ; 1 / n ) , V:=\bigcup_{n\in\mathbb{N}}B\left(n\,;\,1/n\right),
  58. V V
  59. \mathbb{N}
  60. X X
  61. N ( x ) N(x)
  62. X X
  63. x x
  64. X X
  65. x x
  66. U U
  67. N ( x ) N(x)
  68. U U
  69. N ( x ) N(x)
  70. V V
  71. N ( x ) N(x)
  72. y y
  73. V V
  74. U U
  75. N ( y ) N(y)
  76. S = ( X , δ ) S=(X,\delta)
  77. V V
  78. P P
  79. P P
  80. X V X\setminus V
  81. P P
  82. X V X\setminus V
  83. p p
  84. p p
  85. { p } \{p\}
  86. ( - 1 , 1 ) = { y : - 1 < y < 1 } (-1,1)=\{y:-1<y<1\}
  87. p = 0 p=0
  88. ( - 1 , 0 ) ( 0 , 1 ) = ( - 1 , 1 ) { 0 } (-1,0)\cup(0,1)=(-1,1)\setminus\{0\}
  89. 0

Neighbourhood_system.html

  1. 𝒱 ( x ) \mathcal{V}(x)
  2. ( x ) 𝒱 ( x ) \mathcal{B}(x)\subset\mathcal{V}(x)
  3. V 𝒱 ( x ) B ( x ) with B V \forall V\in\mathcal{V}(x)\quad\exists B\in\mathcal{B}(x)\mbox{ with }~{}B\subset V
  4. V V
  5. B B
  6. V V
  7. 𝒱 ( x ) = { V B : B ( x ) } \mathcal{V}(x)=\left\{V\supset B~{}:~{}B\in\mathcal{B}(x)\right\}
  8. 𝒱 ( x ) = { X } \mathcal{V}(x)=\{X\}
  9. ( x ) = { B 1 / n ( x ) ; n * } \mathcal{B}(x)=\{B_{1/n}(x);n\in\mathbb{N}^{*}\}
  10. ν \nu
  11. { μ ( E ) : | μ f i - ν f i | < ε i , i = 1 , , n } \left\{\mu\in\mathcal{M}(E):|\mu f_{i}-\nu f_{i}|<\varepsilon_{i},i=1,\ldots,n\right\}
  12. f i f_{i}
  13. 𝒱 ( x ) = 𝒱 ( 0 ) + x . \mathcal{V}(x)=\mathcal{V}(0)+x.

Nernst_effect.html

  1. | N | = E Y / B Z d T / d x |N|=\frac{E_{Y}/B_{Z}}{dT/dx}
  2. E Y E_{Y}
  3. B Z B_{Z}
  4. d T / d x dT/dx

Nerve_conduction_study.html

  1. 2 D F - M - 1 \frac{2D}{F-M-1}

Nesbitt's_inequality.html

  1. a b + c + b a + c + c a + b 3 2 . \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}.
  2. ( a + b ) , ( b + c ) , ( c + a ) (a+b),(b+c),(c+a)
  3. ( a + b ) + ( a + c ) + ( b + c ) 3 3 1 a + b + 1 a + c + 1 b + c . \frac{(a+b)+(a+c)+(b+c)}{3}\geq\frac{3}{\displaystyle\frac{1}{a+b}+\frac{1}{a+% c}+\frac{1}{b+c}}.
  4. ( ( a + b ) + ( a + c ) + ( b + c ) ) ( 1 a + b + 1 a + c + 1 b + c ) 9 , ((a+b)+(a+c)+(b+c))\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)\geq 9,
  5. 2 a + b + c b + c + 2 a + b + c a + c + 2 a + b + c a + b 9 2\frac{a+b+c}{b+c}+2\frac{a+b+c}{a+c}+2\frac{a+b+c}{a+b}\geq 9
  6. a b c a\geq b\geq c
  7. 1 b + c 1 a + c 1 a + b \frac{1}{b+c}\geq\frac{1}{a+c}\geq\frac{1}{a+b}
  8. x = ( a , b , c ) \vec{x}=(a,b,c)
  9. y = ( 1 b + c , 1 a + c , 1 a + b ) \vec{y}=\left(\frac{1}{b+c},\frac{1}{a+c},\frac{1}{a+b}\right)
  10. y 1 \vec{y}_{1}
  11. y 2 \vec{y}_{2}
  12. y \vec{y}
  13. x y x y 1 \vec{x}\cdot\vec{y}\geq\vec{x}\cdot\vec{y}_{1}
  14. x y x y 2 \vec{x}\cdot\vec{y}\geq\vec{x}\cdot\vec{y}_{2}
  15. a , b , c : a,b,c:
  16. a b + c + b a + c + c a + b = 3 2 + 1 2 ( ( a - b ) 2 ( a + c ) ( b + c ) + ( a - c ) 2 ( a + b ) ( b + c ) + ( b - c ) 2 ( a + b ) ( a + c ) ) \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=\frac{3}{2}+\frac{1}{2}\left(\frac{(% a-b)^{2}}{(a+c)(b+c)}+\frac{(a-c)^{2}}{(a+b)(b+c)}+\frac{(b-c)^{2}}{(a+b)(a+c)% }\right)
  17. 3 2 \frac{3}{2}
  18. a + b , b + c , c + a , 1 a + b , 1 b + c , 1 c + a \displaystyle\left\langle\sqrt{a+b},\sqrt{b+c},\sqrt{c+a}\right\rangle,\left% \langle\frac{1}{\sqrt{a+b}},\frac{1}{\sqrt{b+c}},\frac{1}{\sqrt{c+a}}\right\rangle
  19. ( ( b + c ) + ( a + c ) + ( a + b ) ) ( 1 b + c + 1 a + c + 1 a + b ) 9 , ((b+c)+(a+c)+(a+b))\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)\geq 9,
  20. x = a + b , y = b + c , z = c + a x=a+b,y=b+c,z=c+a
  21. { x 2 z , z 2 x , y 2 z , z 2 y , x 2 y , y 2 x } \left\{x^{2}z,z^{2}x,y^{2}z,z^{2}y,x^{2}y,y^{2}x\right\}
  22. ( x 2 z + z 2 x ) + ( y 2 z + z 2 y ) + ( x 2 y + y 2 x ) 6 x 2 z z 2 x y 2 z z 2 y x 2 y y 2 x 6 = x y z . \frac{\left(x^{2}z+z^{2}x\right)+\left(y^{2}z+z^{2}y\right)+\left(x^{2}y+y^{2}% x\right)}{6}\geq\sqrt[6]{x^{2}z\cdot z^{2}x\cdot y^{2}z\cdot z^{2}y\cdot x^{2}% y\cdot y^{2}x}=xyz.
  23. x y z / 6 xyz/6
  24. x + z y + y + z x + x + y z 6. \frac{x+z}{y}+\frac{y+z}{x}+\frac{x+y}{z}\geq 6.
  25. x , y , z x,y,z
  26. a , b , c a,b,c
  27. 2 a + b + c b + c + a + b + 2 c a + b + a + 2 b + c c + a 6 , \frac{2a+b+c}{b+c}+\frac{a+b+2c}{a+b}+\frac{a+2b+c}{c+a}\geq 6,
  28. n n
  29. ( x k ) (x_{k})
  30. n n
  31. ( a k ) (a_{k})
  32. k = 1 n x k 2 a k k = 1 n x k 2 k = 1 n a k \displaystyle\sum_{k=1}^{n}\frac{x_{k}^{2}}{a_{k}}\geq\frac{\sum_{k=1}^{n}x_{k% }^{2}}{\sum_{k=1}^{n}a_{k}}
  33. x x
  34. a , b , c a,b,c
  35. a a
  36. a ( b + c ) , b ( c + a ) , c ( a + b ) a(b+c),b(c+a),c(a+b)
  37. a 2 a ( b + c ) + b 2 b ( c + a ) + c 2 c ( a + b ) ( a + b + c ) 2 a ( b + c ) + b ( c + a ) + c ( a + b ) \frac{a^{2}}{a(b+c)}+\frac{b^{2}}{b(c+a)}+\frac{c^{2}}{c(a+b)}\geq\frac{(a+b+c% )^{2}}{a(b+c)+b(c+a)+c(a+b)}
  38. a 2 a ( b + c ) + b 2 b ( c + a ) + c 2 c ( a + b ) a 2 + b 2 + c 2 + 2 ( a b + b c + c a ) 2 ( a b + b c + c a ) , \frac{a^{2}}{a(b+c)}+\frac{b^{2}}{b(c+a)}+\frac{c^{2}}{c(a+b)}\geq\frac{a^{2}+% b^{2}+c^{2}+2(ab+bc+ca)}{2(ab+bc+ca)},
  39. a b + c + b c + a + c a + b a 2 + b 2 + c 2 2 ( a b + b c + c a ) + 1. \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq\frac{a^{2}+b^{2}+c^{2}}{2(ab+bc+% ca)}+1.
  40. a 2 + b 2 + c 2 a b + b c + c a a^{2}+b^{2}+c^{2}\geq ab+bc+ca
  41. 1 2 \displaystyle\frac{1}{2}
  42. a b + c + b c + a + c a + b 3 2 . \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq\frac{3}{2}.

Net_energy_gain.html

  1. N E G = E n e r g y Consumable - E n e r g y Expended . NEG=Energy_{\hbox{Consumable}}-Energy_{\hbox{Expended}}.

Net_migration_rate.html

  1. 100 , 000 ÷ 1 , 100 , 000 = 0.09091 100,000\div 1,100,000=0.09091
  2. 0.09091 × 1 , 000 = 90.91 0.09091\times 1,000=90.91

Net_national_product.html

  1. N N P = G N P - D e p r e c i a t i o n NNP=GNP-Depreciation
  2. N D P = G D P - D e p r e c i a t i o n NDP=GDP-Depreciation

Network_performance.html

  1. t = s / c m t=s/c_{m}

Neutrino_oscillation.html

  1. | ν α = i U α i * | ν i \left|\nu_{\alpha}\right\rangle=\sum_{i}U^{*}_{\alpha i}\left|\nu_{i}\right\rangle\,
  2. | ν i = α U α i | ν α \left|\nu_{i}\right\rangle=\sum_{\alpha}U_{\alpha i}\left|\nu_{\alpha}\right\rangle
  3. | ν α \left|\nu_{\alpha}\right\rangle
  4. | ν i \left|\nu_{i}\right\rangle
  5. m i m_{i}
  6. i = i=
  7. * {}^{*}
  8. U α i U_{\alpha i}
  9. U = [ U e 1 U e 2 U e 3 U μ 1 U μ 2 U μ 3 U τ 1 U τ 2 U τ 3 ] = [ 1 0 0 0 c 23 s 23 0 - s 23 c 23 ] [ c 13 0 s 13 e - i δ 0 1 0 - s 13 e i δ 0 c 13 ] [ c 12 s 12 0 - s 12 c 12 0 0 0 1 ] [ 1 0 0 0 e i α 1 / 2 0 0 0 e i α 2 / 2 ] = [ c 12 c 13 s 12 c 13 s 13 e - i δ - s 12 c 23 - c 12 s 23 s 13 e i δ c 12 c 23 - s 12 s 23 s 13 e i δ s 23 c 13 s 12 s 23 - c 12 c 23 s 13 e i δ - c 12 s 23 - s 12 c 23 s 13 e i δ c 23 c 13 ] [ 1 0 0 0 e i α 1 / 2 0 0 0 e i α 2 / 2 ] \begin{aligned}\displaystyle U&\displaystyle=\begin{bmatrix}U_{e1}&U_{e2}&U_{e% 3}\\ U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\ U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}1&0&0\\ 0&c_{23}&s_{23}\\ 0&-s_{23}&c_{23}\end{bmatrix}\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta}\\ 0&1&0\\ -s_{13}e^{i\delta}&0&c_{13}\end{bmatrix}\begin{bmatrix}c_{12}&s_{12}&0\\ -s_{12}&c_{12}&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&e^{i\alpha_{1}/2}&0\\ 0&0&e^{i\alpha_{2}/2}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{% i\delta}&s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{% i\delta}&c_{23}c_{13}\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&e^{i\alpha_{1}/2}&0\\ 0&0&e^{i\alpha_{2}/2}\end{bmatrix}\\ \end{aligned}
  10. | ν i \left|\nu_{i}\right\rangle
  11. | ν i ( t ) = e - i ( E i t - p i x ) | ν i ( 0 ) , |\nu_{i}(t)\rangle=e^{-i(E_{i}t-\vec{p}_{i}\cdot\vec{x})}|\nu_{i}(0)\rangle,
  12. ( c = 1 , = 1 ) (c=1,\hbar=1)
  13. E i E_{i}
  14. i i
  15. t t
  16. p i \vec{p}_{i}
  17. x \vec{x}
  18. | p i | = p i m i |\vec{p}_{i}|=p_{i}\gg m_{i}
  19. E i = p i 2 + m i 2 p i + m i 2 2 p i E + m i 2 2 E , E_{i}=\sqrt{p_{i}^{2}+m_{i}^{2}}\simeq p_{i}+\frac{m_{i}^{2}}{2p_{i}}\approx E% +\frac{m_{i}^{2}}{2E},
  20. | ν i ( L ) = e - i m i 2 L / 2 E | ν i ( 0 ) . |\nu_{i}(L)\rangle=e^{-im_{i}^{2}L/2E}|\nu_{i}(0)\rangle.
  21. P α β = | ν β | ν α ( t ) | 2 = | i U α i * U β i e - i m i 2 L / 2 E | 2 . P_{\alpha\rightarrow\beta}=\left|\left\langle\nu_{\beta}|\nu_{\alpha}(t)\right% \rangle\right|^{2}=\left|\sum_{i}U_{\alpha i}^{*}U_{\beta i}e^{-im_{i}^{2}L/2E% }\right|^{2}.
  22. P α β = δ α β - 4 i > j Re ( U α i * U β i U α j U β j * ) sin 2 ( Δ m i j 2 L 4 E ) + 2 i > j Im ( U α i * U β i U α j U β j * ) sin ( Δ m i j 2 L 2 E ) , \begin{matrix}P_{\alpha\rightarrow\beta}=\delta_{\alpha\beta}&-&4{% \displaystyle\sum_{i>j}{\rm Re}(U_{\alpha i}^{*}U_{\beta i}U_{\alpha j}U_{% \beta j}^{*}})\sin^{2}(\frac{\Delta m_{ij}^{2}L}{4E})\\ &+&{\displaystyle 2\sum_{i>j}{\rm Im}(U_{\alpha i}^{*}U_{\beta i}U_{\alpha j}U% _{\beta j}^{*})\sin(}\frac{\Delta m_{ij}^{2}L}{2E}),\end{matrix}
  23. Δ m i j 2 m i 2 - m j 2 \Delta m_{ij}^{2}\ \equiv m_{i}^{2}-m_{j}^{2}
  24. \hbar
  25. Δ m 2 c 3 L 4 E = GeV fm 4 c × Δ m 2 eV 2 L km GeV E 1.27 × Δ m 2 eV 2 L km GeV E , \frac{\Delta m^{2}\,c^{3}\,L}{4\hbar E}=\frac{{\rm GeV}\,{\rm fm}}{4\hbar c}% \times\frac{\Delta m^{2}}{{\rm eV}^{2}}\frac{L}{\rm km}\frac{\rm GeV}{E}% \approx 1.27\times\frac{\Delta m^{2}}{{\rm eV}^{2}}\frac{L}{\rm km}\frac{\rm GeV% }{E},
  26. A CP ( α β ) = P ( ν α ν β ) - P ( ν ¯ α ν ¯ β ) = 4 i > j Im ( U α i * U β i U α j U β j * ) sin ( Δ m i j 2 L 2 E ) A^{(\alpha\beta)}_{\,\text{CP}}=P(\nu_{\alpha}\rightarrow\nu_{\beta})-P(\bar{% \nu}_{\alpha}\rightarrow\bar{\nu}_{\beta})=4\sum_{i>j}\mathrm{Im}\big(U^{*}_{% \alpha i}U_{\beta i}U_{\alpha j}U^{*}_{\beta j}\big)\sin\Big(\tfrac{\Delta m^{% 2}_{ij}L}{2E}\Big)
  27. Im ( U α i U β i * U α j * U β j ) = J γ , k ε α β γ ε i j k \mathrm{Im}\big(U_{\alpha i}U^{*}_{\beta i}U^{*}_{\alpha j}U_{\beta j}\big)=J% \sum_{\gamma,k}\varepsilon_{\alpha\beta\gamma}\varepsilon_{ijk}
  28. A CP ( α β ) = 16 J γ ε α β γ sin ( Δ m 21 2 L 4 E ) sin ( Δ m 32 2 L 4 E ) sin ( Δ m 31 2 L 4 E ) A^{(\alpha\beta)}_{\,\text{CP}}=16J\sum_{\gamma}\varepsilon_{\alpha\beta\gamma% }\sin\Big(\tfrac{\Delta m^{2}_{21}L}{4E}\Big)\sin\Big(\tfrac{\Delta m^{2}_{32}% L}{4E}\Big)\sin\Big(\tfrac{\Delta m^{2}_{31}L}{4E}\Big)
  29. U = ( cos θ sin θ - sin θ cos θ ) . U=\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}.
  30. P α β , α β = sin 2 ( 2 θ ) sin 2 ( Δ m 2 L 4 E ) ( natural units ) . P_{\alpha\rightarrow\beta,\alpha\neq\beta}=\sin^{2}(2\theta)\,\sin^{2}\left(% \frac{\Delta m^{2}L}{4E}\right)\,\mathrm{(natural\,units)}.
  31. P α β , α β = sin 2 ( 2 θ ) sin 2 ( 1.27 Δ m 2 L E [ eV 2 ] [ km ] [ GeV ] ) . P_{\alpha\rightarrow\beta,\alpha\neq\beta}=\sin^{2}(2\theta)\,\sin^{2}\left(1.% 27\frac{\Delta m^{2}L}{E}\frac{\rm[eV^{2}]\,[km]}{\rm[GeV]}\right).
  32. 1 2 m g L x 2 \tfrac{1}{2}\tfrac{mg}{L}x^{2}
  33. g / L \sqrt{g/L}
  34. 1 2 k x 2 \tfrac{1}{2}kx^{2}
  35. k / m \sqrt{k/m}
  36. V = m 2 ( g L a x a 2 + g L b x b 2 + k m ( x b - x a ) 2 ) . V=\frac{m}{2}\left(\frac{g}{L_{a}}x_{a}^{2}+\frac{g}{L_{b}}x_{b}^{2}+\frac{k}{% m}(x_{b}-x_{a})^{2}\right).
  37. V = m 2 ( x a x b ) ( g L a + k m - k m - k m g L b + k m ) ( x a x b ) . V=\frac{m}{2}\begin{pmatrix}x_{a}\ x_{b}\end{pmatrix}\begin{pmatrix}\tfrac{g}{% L_{a}}+\tfrac{k}{m}&-\tfrac{k}{m}\\ -\tfrac{k}{m}&\tfrac{g}{L_{b}}+\tfrac{k}{m}\end{pmatrix}\begin{pmatrix}x_{a}\\ x_{b}\end{pmatrix}.
  38. ( x a x b ) = ( cos θ sin θ - sin θ cos θ ) ( x 1 x 2 ) \begin{pmatrix}x_{a}\\ x_{b}\end{pmatrix}=\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}
  39. V = m 2 ( x 1 x 2 ) ( λ 1 0 0 λ 2 ) ( x 1 x 2 ) V=\frac{m}{2}\begin{pmatrix}x_{1}\ x_{2}\end{pmatrix}\begin{pmatrix}\lambda_{1% }&0\\ 0&\lambda_{2}\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}
  40. λ 1 \sqrt{\lambda_{1}}
  41. λ 2 \sqrt{\lambda_{2}}

Neutron_star_spin-up.html

  1. P / P ˙ P/\dot{P}
  2. P P
  3. P ˙ \dot{P}

New_York_Yacht_Club.html

  1. Rating = 2 Load Waterline Length + Sail Area 3 \,\text{Rating}=\frac{2\cdot\,\text{Load Waterline Length}+\sqrt{\,\text{Sail % Area}}}{3}

Newton's_method_in_optimization.html

  1. f f
  2. f ( x ) = 0 f(x)=0
  3. f f^{\prime}
  4. f f
  5. f ( x ) = 0 f^{\prime}(x)=0
  6. f f
  7. f T ( x ) = f T ( x n + Δ x ) f ( x n ) + f ( x n ) Δ x + 1 2 f ′′ ( x n ) Δ x 2 \displaystyle f_{T}(x)=f_{T}(x_{n}+\Delta x)\approx f(x_{n})+f^{\prime}(x_{n})% \Delta x+\frac{1}{2}f^{\prime\prime}(x_{n})\Delta x^{2}
  8. 0 = d d Δ x ( f ( x n ) + f ( x n ) Δ x + 1 2 f ′′ ( x n ) Δ x 2 ) = f ( x n ) + f ′′ ( x n ) Δ x \displaystyle 0=\frac{d}{d\Delta x}\left(f(x_{n})+f^{\prime}(x_{n})\Delta x+% \frac{1}{2}f^{\prime\prime}(x_{n})\Delta x^{2}\right)=f^{\prime}(x_{n})+f^{% \prime\prime}(x_{n})\Delta x
  9. Δ x = - f ( x n ) f ′′ ( x n ) \Delta x=-\frac{f^{\prime}(x_{n})}{f^{\prime\prime}(x_{n})}
  10. x n + 1 = x n + Δ x = x n - f ( x n ) f ′′ ( x n ) x_{n+1}=x_{n}+\Delta x=x_{n}-\frac{f^{\prime}(x_{n})}{f^{\prime\prime}(x_{n})}
  11. f ( 𝐱 ) f(\mathbf{x})
  12. 𝐱 n \mathbf{x}_{n}
  13. f ( 𝐱 ) f(\mathbf{x})
  14. f ( 𝐱 ) \nabla f(\mathbf{x})
  15. H f ( 𝐱 ) Hf(\mathbf{x})
  16. 𝐱 n + 1 = 𝐱 n - [ H f ( 𝐱 n ) ] - 1 f ( 𝐱 n ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-[Hf(\mathbf{x}_{n})]^{-1}\nabla f(\mathbf{x}_{% n}),\ n\geq 0.
  17. γ ( 0 , 1 ) \gamma\in(0,1)
  18. γ = 1 \gamma=1
  19. 𝐱 n + 1 = 𝐱 n - γ [ H f ( 𝐱 n ) ] - 1 f ( 𝐱 n ) . \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma[Hf(\mathbf{x}_{n})]^{-1}\nabla f(% \mathbf{x}_{n}).
  20. 𝐱 n 𝐱 n + 1 \mathbf{x}_{n}\to\mathbf{x}_{n+1}
  21. N N
  22. 𝐱 0 N , \mathbf{x}_{0}\in N,
  23. γ = 1 \gamma=1
  24. 𝐱 \mathbf{x}
  25. 𝐩 n = [ H f ( 𝐱 n ) ] - 1 f ( 𝐱 n ) \mathbf{p}_{n}=[Hf(\mathbf{x}_{n})]^{-1}\nabla f(\mathbf{x}_{n})
  26. [ H f ( 𝐱 n ) ] 𝐩 n = f ( 𝐱 n ) [Hf(\mathbf{x}_{n})]\mathbf{p}_{n}=\nabla f(\mathbf{x}_{n})
  27. [ H f ( 𝐱 n ) ] [Hf(\mathbf{x}_{n})]
  28. [ H f ( 𝐱 n ) ] [Hf(\mathbf{x}_{n})]
  29. 𝐩 n \mathbf{p}_{n}
  30. B n B_{n}
  31. H f ( 𝐱 n ) + B n H_{f}(\mathbf{x}_{n})+B_{n}
  32. H f H_{f}
  33. B n B_{n}
  34. H f ( 𝐱 n ) + B n H_{f}(\mathbf{x}_{n})+B_{n}
  35. H f H_{f}
  36. ϵ > 0. \epsilon>0.
  37. μ 𝐈 \mu\mathbf{I}
  38. μ \mu
  39. 1 μ \frac{1}{\mu}