wpmath0000016_9

Mathisson–Papapetrou–Dixon_equations.html

  1. D S μ ν d s + u μ u σ D S ν σ d s - u ν u σ D S μ σ d s = 0 \frac{DS^{\mu\nu}}{ds}+u^{\mu}u_{\sigma}\frac{DS^{\nu\sigma}}{ds}-u^{\nu}u_{% \sigma}\frac{DS^{\mu\sigma}}{ds}=0

Matrix-exponential_distribution.html

  1. f ( x ) = α e x T 𝐬 for x 0 f(x)=\mathbf{\alpha}e^{x\,T}\mathbf{s}\,\text{ for }x\geq 0
  2. 𝔼 ( X k ) = ( - 1 ) k + 1 k ! α T - ( k + 1 ) 𝐬 . \mathbb{E}(X^{k})=(-1)^{k+1}k!\mathbf{\alpha}T^{-(k+1)}\mathbf{s}.

Matrix_analysis.html

  1. 𝐀 , 𝐁 M m n ( F ) , 𝐀 + 𝐁 M m n ( F ) \mathbf{A},\mathbf{B}\in M_{mn}(F)\,,\quad\mathbf{A}+\mathbf{B}\in M_{mn}(F)
  2. α F , α 𝐀 M m n ( F ) \alpha\in F\,,\quad\alpha\mathbf{A}\in M_{mn}(F)
  3. α 𝐀 + β 𝐁 M m n ( F ) \alpha\mathbf{A}+\beta\mathbf{B}\in M_{mn}(F)
  4. ( 1 0 0 0 ) , ( 0 1 0 0 ) , ( 0 0 1 0 ) , ( 0 0 0 1 ) , \begin{pmatrix}1&0\\ 0&0\end{pmatrix}\,,\quad\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\,,\quad\begin{pmatrix}0&0\\ 1&0\end{pmatrix}\,,\quad\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\,,
  5. ( a b c d ) = a ( 1 0 0 0 ) + b ( 0 1 0 0 ) + c ( 0 0 1 0 ) + d ( 0 0 0 1 ) , \begin{pmatrix}a&b\\ c&d\end{pmatrix}=a\begin{pmatrix}1&0\\ 0&0\end{pmatrix}+b\begin{pmatrix}0&1\\ 0&0\end{pmatrix}+c\begin{pmatrix}0&0\\ 1&0\end{pmatrix}+d\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\,,
  6. 𝐀𝐱 = λ 𝐱 \mathbf{A}\mathbf{x}=\lambda\mathbf{x}
  7. p 𝐀 ( λ ) = det ( 𝐀 - λ 𝐈 ) = 0 p_{\mathbf{A}}(\lambda)=\det(\mathbf{A}-\lambda\mathbf{I})=0
  8. 𝐁 = 𝐏𝐀𝐏 - 1 \mathbf{B}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}
  9. 𝐀 0 \|\mathbf{A}\|\geq 0
  10. α 𝐀 = | α | 𝐀 \|\alpha\mathbf{A}\|=|\alpha|\|\mathbf{A}\|
  11. 𝐀 + 𝐁 𝐀 + 𝐁 \|\mathbf{A}+\mathbf{B}\|\leq\|\mathbf{A}\|+\|\mathbf{B}\|
  12. 𝐀 = 𝐀 : 𝐀 = i = 1 m j = 1 n ( A i j ) 2 \|\mathbf{A}\|=\sqrt{\mathbf{A}:\mathbf{A}}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}% (A_{ij})^{2}}

Matrix_geometric_method.html

  1. Q = ( B 00 B 01 B 10 A 1 A 2 A 0 A 1 A 2 A 0 A 1 A 2 A 0 A 1 A 2 ) Q=\begin{pmatrix}B_{00}&B_{01}\\ B_{10}&A_{1}&A_{2}\\ &A_{0}&A_{1}&A_{2}\\ &&A_{0}&A_{1}&A_{2}\\ &&&A_{0}&A_{1}&A_{2}\\ &&&&\ddots&\ddots&\ddots\end{pmatrix}
  2. π 0 B 00 + π 1 B 10 \displaystyle\pi_{0}B_{00}+\pi_{1}B_{10}
  3. π i = π 1 R i - 1 \pi_{i}=\pi_{1}R^{i-1}
  4. ( π 0 π 1 ) ( B 00 B 01 B 10 A 1 + R A 0 ) = ( 0 0 ) \displaystyle\begin{pmatrix}\pi_{0}&\pi_{1}\end{pmatrix}\begin{pmatrix}B_{00}&% B_{01}\\ B_{10}&A_{1}+RA_{0}\end{pmatrix}=\begin{pmatrix}0&0\end{pmatrix}

Matrix_product_state.html

  1. | Ψ = { s } Tr [ A 1 ( s 1 ) A 2 ( s 2 ) A N ( s N ) ] | s 1 s 2 s N , |\Psi\rangle=\sum_{\{s\}}\,\text{Tr}[A_{1}^{(s_{1})}A_{2}^{(s_{2})}\cdots A_{N% }^{(s_{N})}]|s_{1}s_{2}\ldots s_{N}\rangle,
  2. A i ( s ) A_{i}^{(s)}
  3. χ \chi
  4. s i s_{i}
  5. s i { 0 , 1 } s_{i}\in\{0,1\}
  6. s i { 0 , 1 , , d - 1 } s_{i}\in\{0,1,\ldots,d-1\}
  7. χ \chi
  8. χ = 1 \chi=1
  9. A 1 ( s ) = A 2 ( s ) = = A N ( s ) A ( s ) . A_{1}^{(s)}=A_{2}^{(s)}=\cdots=A_{N}^{(s)}\equiv A^{(s)}.
  10. χ \chi
  11. χ \chi
  12. | GHZ = | 0 N + | 1 N 2 |\mathrm{GHZ}\rangle=\frac{|0\rangle^{\otimes N}+|1\rangle^{\otimes N}}{\sqrt{% 2}}
  13. A ( 0 ) = [ 1 0 0 0 ] A ( 1 ) = [ 0 0 0 1 ] , A^{(0)}=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}\quad A^{(1)}=\begin{bmatrix}0&0\\ 0&1\end{bmatrix},
  14. A = [ | 0 0 0 | 1 ] . A=\begin{bmatrix}|0\rangle&0\\ 0&|1\rangle\end{bmatrix}.
  15. A | 0 A ( 0 ) + | 1 A ( 1 ) + + | d - 1 A ( d - 1 ) . A\equiv|0\rangle A^{(0)}+|1\rangle A^{(1)}+\ldots+|d-1\rangle A^{(d-1)}.
  16. A A = [ | 00 0 0 | 11 ] . AA=\begin{bmatrix}|00\rangle&0\\ 0&|11\rangle\end{bmatrix}.
  17. A 1 = [ | 0 0 | 0 | 1 ] A 2 = [ | 0 | 1 0 | 0 ] A 3 = [ | 1 0 0 | 0 ] . A_{1}=\begin{bmatrix}|0\rangle&0\\ |0\rangle&|1\rangle\end{bmatrix}\quad A_{2}=\begin{bmatrix}|0\rangle&|1\rangle% \\ 0&|0\rangle\end{bmatrix}\quad A_{3}=\begin{bmatrix}|1\rangle&0\\ 0&|0\rangle\end{bmatrix}.
  18. A + = - 2 3 σ + = [ 0 0 - 2 / 3 0 ] A^{+}=-\sqrt{\frac{2}{3}}\ \sigma^{+}=\begin{bmatrix}0&0\\ -\sqrt{2/3}&0\end{bmatrix}
  19. A 0 = - 1 3 σ z = [ - 1 / 3 0 0 1 / 3 ] A^{0}=\frac{-1}{\sqrt{3}}\ \sigma^{z}=\begin{bmatrix}-1/\sqrt{3}&0\\ 0&1/\sqrt{3}\end{bmatrix}
  20. A - = - 2 3 σ - = [ 0 2 / 3 0 0 ] A^{-}=-\sqrt{\frac{2}{3}}\ \sigma^{-}=\begin{bmatrix}0&\sqrt{2/3}\\ 0&0\end{bmatrix}
  21. σ ’s \sigma\,\text{'s}
  22. A = 1 3 [ - | 0 2 | - - 2 | + | 0 ] . A=\frac{1}{\sqrt{3}}\begin{bmatrix}-|0\rangle&\sqrt{2}|-\rangle\\ -\sqrt{2}|+\rangle&|0\rangle\end{bmatrix}.
  23. A = [ 0 | | - 1 2 | 0 0 1 2 | 0 0 ] . A=\begin{bmatrix}0&|\uparrow\rangle&|\downarrow\rangle\\ \frac{-1}{\sqrt{2}}|\downarrow\rangle&0&0\\ \frac{1}{\sqrt{2}}|\uparrow\rangle&0&0\end{bmatrix}.

Matrix_regularization.html

  1. min x A x - y 2 + λ x 2 \min_{x}\|Ax-y\|^{2}+\lambda\|x\|^{2}
  2. x x
  3. min X A X - Y 2 + λ X 2 \min_{X}\|AX-Y\|^{2}+\lambda\|X\|^{2}
  4. x x
  5. X X
  6. W W
  7. S = ( X i t , y i t ) S=(X_{i}^{t},y_{i}^{t})
  8. i i
  9. 1 1
  10. n n
  11. t t
  12. 1 1
  13. T T
  14. X i X_{i}
  15. D T \in\mathbb{R}^{DT}
  16. W W
  17. D × T D\times T
  18. y y
  19. y i t = W , X i t F y_{i}^{t}=\langle W,X_{i}^{t}\rangle_{F}
  20. X i X_{i}
  21. W W
  22. min W E ( W ) + R ( W ) \min_{W\in\mathcal{H}}E(W)+R(W)
  23. E E
  24. W W
  25. R ( W ) R(W)
  26. R ( W ) R(W)
  27. 1 \ell^{1}
  28. 2 \ell^{2}
  29. W W
  30. \mathcal{H}
  31. X i t X_{i}^{t}
  32. X i t = e t e i X_{i}^{t}=e_{t}\otimes e_{i}^{\prime}
  33. ( e t ) t (e_{t})_{t}
  34. ( e i ) i (e_{i}^{\prime})_{i}
  35. T \mathbb{R}^{T}
  36. D \mathbb{R}^{D}
  37. w i t w_{i}^{t}
  38. W W
  39. y y
  40. W W
  41. W W
  42. W W
  43. R ( W ) = λ W * = λ | σ i | R(W)=\lambda\|W\|_{*}=\lambda\sum|\sigma_{i}|
  44. σ i \sigma_{i}
  45. i i
  46. 1 1
  47. min D , T \min D,T
  48. W W
  49. X X
  50. X i t = e t x i X_{i}^{t}=e_{t}\otimes x_{i}\,
  51. Y = X W + b Y=XW+b\,
  52. 2 \ell^{2}
  53. R ( W ) = λ W F 2 = λ | w i j | 2 = λ Tr ( W * W ) = λ σ i 2 . R(W)=\lambda\|W\|_{F}^{2}=\lambda\sum\sum|w_{ij}|^{2}=\lambda\operatorname{Tr}% (W^{*}W)=\lambda\sum\sigma_{i}^{2}.
  54. Y Y
  55. X i t = e t x i t . X_{i}^{t}=e_{t}\otimes x_{i}^{t}.
  56. min W X W - Y 2 2 + λ W 2 2 \min_{W}\|XW-Y\|_{2}^{2}+\lambda\|W\|_{2}^{2}
  57. Y Y
  58. min W , Ω X W - Y 2 2 + λ 1 W 2 2 + λ 2 Tr ( W T Ω - 1 W ) \min_{W,\Omega}\|XW-Y\|_{2}^{2}+\lambda_{1}\|W\|_{2}^{2}+\lambda_{2}% \operatorname{Tr}(W^{T}\Omega^{-1}W)
  59. Ω \Omega
  60. W W
  61. Ω \Omega
  62. min W * \min\|W\|_{*}
  63. W i , j = Y i j . W_{i,j}=Y_{ij}.
  64. n n
  65. 0 \ell^{0}
  66. 0 \ell^{0}
  67. 1 \ell^{1}
  68. 1 \ell^{1}
  69. 1 \ell^{1}
  70. p , q \ell_{p,q}
  71. p = 2 p=2
  72. q = 1 q=1
  73. W 2 , 1 = w i 2 . \|W\|_{2,1}=\sum\|w_{i}\|_{2}.
  74. 2 , 1 \ell_{2,1}
  75. 2 \ell^{2}
  76. 2 \ell^{2}
  77. 2 , 1 \ell_{2,1}
  78. R ( W ) = λ g G j | G g | | w g j | 2 = λ g G w g g R(W)=\lambda\sum_{g}^{G}\sqrt{\sum_{j}^{|G_{g}|}|w_{g}^{j}|^{2}}=\lambda\sum_{% g}^{G}\|w_{g}\|_{g}
  79. g g
  80. | G g | |G_{g}|
  81. g g
  82. w g i w_{g}^{i}
  83. prox λ , R g ( w g ) i = ( w g i - λ w g i w g g ) 𝟏 w g g λ . \operatorname{prox}_{\lambda,R_{g}}(w_{g})^{i}=\left(w_{g}^{i}-\lambda\frac{w_% {g}^{i}}{\|w_{g}\|_{g}}\right)\mathbf{1}_{\|w_{g}\|_{g}\geq\lambda}.
  84. 𝟏 w g g λ \mathbf{1}_{\|w_{g}\|_{g}\geq\lambda}
  85. λ \geq\lambda
  86. 2 , 1 \ell_{2,1}
  87. Y Y
  88. A A
  89. B B
  90. 𝒜 , \mathcal{H_{A}},\mathcal{H_{B}}
  91. 𝒟 \mathcal{H_{D}}
  92. 𝒟 : f = h + h ; h 𝒜 , h \mathcal{H_{D}}:f=h+h^{\prime};h\in\mathcal{H_{A}},h^{\prime}\in\mathcal{H_{B}}
  93. A A
  94. B B
  95. 2 , 1 \ell_{2,1}
  96. f 𝒟 , 1 = h 𝒜 + h \|f\|_{\mathcal{H_{D}},1}=\|h\|_{\mathcal{H_{A}}}+\|h^{\prime}\|_{\mathcal{H_{% B}}}

Max::min_CSP::Ones_classification_theorems.html

  1. x 1 ¬ x 2 x 3 = 1 x_{1}\oplus\lnot x_{2}\oplus x_{3}=1
  2. 2 log 1 - ϵ ( n ) 2^{\log^{1-\epsilon}(n)}
  3. ϵ \epsilon
  4. 2 log 1 - ϵ ( n ) 2^{\log^{1-\epsilon}(n)}
  5. ϵ \epsilon

Maximally_informative_dimensions.html

  1. 𝐬 \mathbf{s}
  2. 𝐬 \mathbf{s}
  3. D D
  4. K K
  5. K D K\ll D
  6. { 𝐯 K } \{\mathbf{v}^{K}\}
  7. 𝐬 K \mathbf{s}^{K}
  8. 𝐬 \mathbf{s}
  9. { 𝐯 K } \{\mathbf{v}^{K}\}
  10. P ( s p i k e | 𝐬 K ) = P ( s p i k e ) f ( 𝐬 K ) P(spike|\mathbf{s}^{K})=P(spike)f(\mathbf{s}^{K})
  11. f ( 𝐬 K ) = P ( 𝐬 K | s p i k e ) P ( 𝐬 K ) f(\mathbf{s}^{K})=\frac{P(\mathbf{s}^{K}|spike)}{P(\mathbf{s}^{K})}
  12. { 𝐯 K } \{\mathbf{v}^{K}\}
  13. P ( 𝐬 ) P(\mathbf{s})
  14. P ( 𝐬 | s p i k e ) P(\mathbf{s}|spike)
  15. I s p i k e = 𝐬 P ( 𝐬 | s p i k e ) l o g 2 [ P ( 𝐬 | s p i k e ) / P ( 𝐬 ) ] I_{spike}=\sum_{\mathbf{s}}P(\mathbf{s}|spike)log_{2}[P(\mathbf{s}|spike)/P(% \mathbf{s})]
  16. K = 1 K=1
  17. 𝐯 \mathbf{v}
  18. x = 𝐬 𝐯 x=\mathbf{s}\cdot\mathbf{v}
  19. I ( 𝐯 ) = d x P 𝐯 ( x | s p i k e ) l o g 2 [ P 𝐯 ( x | s p i k e ) / P 𝐯 ( x ) ] I(\mathbf{v})=\int dxP_{\mathbf{v}}(x|spike)log2[P_{\mathbf{v}}(x|spike)/P_{% \mathbf{v}}(x)]
  20. P 𝐯 ( x | s p i k e ) = δ ( x - 𝐬 𝐯 ) | s p i k e 𝐬 P_{\mathbf{v}}(x|spike)=\langle\delta(x-\mathbf{s}\cdot\mathbf{v})|spike% \rangle_{\mathbf{s}}
  21. P 𝐯 ( x ) = δ ( x - 𝐬 𝐯 ) 𝐬 P_{\mathbf{v}}(x)=\langle\delta(x-\mathbf{s}\cdot\mathbf{v})\rangle_{\mathbf{s}}
  22. 𝐯 \mathbf{v}
  23. K = 1 K=1
  24. 𝐯 \mathbf{v}
  25. I ( 𝐯 ) I(\mathbf{v})
  26. K > 1 K>1
  27. P 𝐯 K ( 𝐱 | s p i k e ) = i = 1 K δ ( x i - 𝐬 𝐯 i ) | s p i k e 𝐬 P_{\mathbf{v}^{K}}(\mathbf{x}|spike)=\langle\prod_{i=1}^{K}\delta(x_{i}-% \mathbf{s}\cdot\mathbf{v}_{i})|spike\rangle_{\mathbf{s}}
  28. P 𝐯 K ( 𝐱 ) = i = 1 K δ ( x i - 𝐬 𝐯 i ) 𝐬 P_{\mathbf{v}^{K}}(\mathbf{x})=\langle\prod_{i=1}^{K}\delta(x_{i}-\mathbf{s}% \cdot\mathbf{v}_{i})\rangle_{\mathbf{s}}
  29. I ( 𝐯 K ) I({\mathbf{v}^{K}})

Maximum_disjoint_set.html

  1. | S | | M D S ( C ) | M |S|\geq\frac{|MDS(C)|}{M}
  2. | M D S ( C ) | log n \frac{|MDS(C)|}{\log{n}}
  3. O ( n log n ) O(n\log{n})
  4. x med x_{\mathrm{med}}
  5. x = x med x=x_{\mathrm{med}}
  6. R left R_{\mathrm{left}}
  7. R right R_{\mathrm{right}}
  8. R int R_{\mathrm{int}}
  9. R left R_{\mathrm{left}}
  10. R right R_{\mathrm{right}}
  11. R left R_{\mathrm{left}}
  12. M left M_{\mathrm{left}}
  13. R right R_{\mathrm{right}}
  14. M right M_{\mathrm{right}}
  15. M left M_{\mathrm{left}}
  16. M right M_{\mathrm{right}}
  17. M left M right M_{\mathrm{left}}\cup M_{\mathrm{right}}
  18. R int R_{\mathrm{int}}
  19. M int M_{\mathrm{int}}
  20. R int R_{\mathrm{int}}
  21. x = x med x=x_{\mathrm{med}}
  22. M left M right M_{\mathrm{left}}\cup M_{\mathrm{right}}
  23. M int M_{\mathrm{int}}
  24. M left M right M_{\mathrm{left}}\cup M_{\mathrm{right}}
  25. M int M_{\mathrm{int}}
  26. | M D S ( C ) | log n \frac{|MDS(C)|}{\log{n}}
  27. O ( log log n ) O(\log{\log{n}})
  28. R i , , R m R_{i},\ldots,R_{m}
  29. R i R_{i}
  30. M i M_{i}
  31. R i R_{i}
  32. R i + 1 R_{i+1}
  33. R i - 1 R_{i-1}
  34. M 1 M 3 M_{1}\cup M_{3}\cup\cdots
  35. M 2 M 4 M_{2}\cup M_{4}\cup\cdots
  36. | MDS ( D ( r , s ) ) | ( 1 - 1 k ) 2 | MDS | |\mathrm{MDS}(D(r,s))|\geq(1-\frac{1}{k})^{2}\cdot|\mathrm{MDS}|
  37. j = 0 , , k - 1 | A ( j ) | ( k - 2 ) | A * | \sum_{j=0,\ldots,k-1}{|A(j)|}\geq(k-2)|A*|
  38. E ( m ) = 0 if m b E(m)=0\ \ \ \ \,\text{ if }m\leq b
  39. E ( m ) = E ( a m ) + E ( ( 1 - a ) m ) + c m if m > b E(m)=E(a\cdot m)+E((1-a)\cdot m)+c\cdot\sqrt{m}\,\text{ if }m>b
  40. E ( m ) = ( 0 b + c b ( a + 1 - a - 1 ) ) m - c a + 1 - a - 1 m . E(m)=(\frac{0}{b}+\frac{c}{\sqrt{b}(\sqrt{a}+\sqrt{1-a}-1)})\cdot m-\frac{c}{% \sqrt{a}+\sqrt{1-a}-1}\cdot\sqrt{m}.
  41. E ( m ) = O ( m / b ) E(m)=O(m/\sqrt{b})
  42. 2 O ( r n ) 2^{O(r\cdot\sqrt{n})}
  43. 2 O ( r n ) 2^{O(r\cdot\sqrt{n})}
  44. T ( 1 ) = 1 T(1)=1
  45. T ( n ) = 2 O ( r n ) T ( 2 n 3 ) if n > 1 T(n)=2^{O(r\cdot\sqrt{n})}T\left(\frac{2n}{3}\right)\,\text{ if }n>1
  46. T ( n ) = 2 O ( r n ) T(n)=2^{O(r\cdot\sqrt{n})}
  47. ( 1 - O ( 1 b ) ) | M D S ( C ) | (1-O(\frac{1}{\sqrt{b}}))\cdot|MDS(C)|
  48. O ( n b + 3 ) O(n^{b+3})
  49. b 0 b\geq 0
  50. S S
  51. C - S C-S
  52. b + 1 b+1
  53. | Y | < | X | |Y|<|X|
  54. S := S - Y + X S:=S-Y+X
  55. n u | M D S ( C ) | \frac{n}{u}\cdot|MDS(C)|
  56. O ( n 3 ) O(n^{3})

Maxwell–Jüttner_distribution.html

  1. γ = 1 / 1 - v 2 / c 2 \gamma=1/\sqrt{1-v^{2}/c^{2}}
  2. f ( γ ) = γ 2 β θ K 2 ( 1 / θ ) exp ( - γ θ ) f(\gamma)=\frac{\gamma^{2}\beta}{\theta K_{2}(1/\theta)}\exp\left(-\frac{% \gamma}{\theta}\right)
  3. β = v c = 1 - 1 / γ 2 , \beta=\frac{v}{c}=\sqrt{1-1/\gamma^{2}},
  4. θ = k T m c 2 , \theta=\frac{kT}{mc^{2}},
  5. K 2 K_{2}
  6. f ( 𝐩 ) = 1 4 π m 3 c 3 θ K 2 ( 1 / θ ) exp ( - γ ( p ) θ ) f(\mathbf{p})=\frac{1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta)}\exp\left(-\frac{% \gamma(p)}{\theta}\right)
  7. γ ( p ) = 1 + ( p m c ) 2 \gamma(p)=\sqrt{1+\left(\frac{p}{mc}\right)^{2}}

McKay's_approximation_for_the_coefficient_of_variation.html

  1. x i x_{i}
  2. i = 1 , 2 , , n i=1,2,\ldots,n
  3. n n
  4. N ( μ , σ 2 ) N(\mu,\sigma^{2})
  5. c v = σ / μ c_{v}=\sigma/\mu
  6. x ¯ \bar{x}
  7. s s\,
  8. c ^ v = s / x ¯ \hat{c}_{v}=s/\bar{x}
  9. K = ( 1 + 1 c v 2 ) ( n - 1 ) c ^ v 2 1 + ( n - 1 ) c ^ v 2 / n K=\left(1+\frac{1}{c_{v}^{2}}\right)\ \frac{(n-1)\ \hat{c}_{v}^{2}}{1+(n-1)\ % \hat{c}_{v}^{2}/n}
  10. c v c_{v}
  11. K K
  12. n - 1 n-1
  13. K K
  14. σ 2 \sigma^{2}
  15. n n
  16. n - 1 n-1
  17. K K

McKelvey-Schofield_chaos_theorem.html

  1. R N R^{N}

Mean_field_particle_methods.html

  1. ( η n ) n = 0 , 1 , \textstyle\left(\eta_{n}\right)_{n=0,1,\ldots}
  2. S \textstyle S
  3. η n + 1 = Φ ( η n ) \textstyle\eta_{n+1}=\Phi(\eta_{n})\quad
  4. Φ \textstyle\Phi
  5. P ( S ) \textstyle P(S)
  6. S \textstyle S
  7. η n = ( η n ( x ) ) x S \textstyle\eta_{n}=(\eta_{n}(x))_{x\in S}
  8. η n ( x ) [ 0 , 1 ] \textstyle\eta_{n}(x)\in[0,1]
  9. x S \textstyle x\in S
  10. x S η n ( x ) = 1 \textstyle\sum_{x\in S}\eta_{n}(x)=1
  11. Φ \textstyle\Phi
  12. ( s - 1 ) \textstyle(s-1)
  13. s \textstyle s
  14. S \textstyle S
  15. s \textstyle s
  16. ξ n ( N ) = ( ξ n ( N , i ) ) 1 i N \textstyle\xi^{(N)}_{n}=\left(\xi^{(N,i)}_{n}\right)_{1\leq i\leq N}
  17. S N \textstyle S^{N}
  18. N \textstyle N
  19. η 0 \textstyle\eta_{0}
  20. P r ( ξ n + 1 ( N , 1 ) = y 1 , ξ n + 1 ( N , 2 ) = y 2 , , ξ n + 1 ( N , N ) = y N ξ n ( N ) ) = 1 i N Φ ( η n N ) ( y i ) Pr\left(\xi^{(N,1)}_{n+1}=y^{1},\xi^{(N,2)}_{n+1}=y^{2},\ldots,\xi^{(N,N)}_{n+% 1}=y^{N}\mid\xi^{(N)}_{n}\right)=\prod_{1\leq i\leq N}\Phi\left(\eta_{n}^{N}% \right)\left(y^{i}\right)\quad
  21. η n N = 1 N 1 j N 1 ξ n ( N , j ) \quad\eta^{N}_{n}=\frac{1}{N}\sum_{1\leq j\leq N}1_{\xi^{(N,j)}_{n}}
  22. y 1 x ( y ) \textstyle\quad y\mapsto 1_{x}(y)
  23. x x
  24. ξ n ( N ) \textstyle\xi^{(N)}_{n}
  25. ξ n + 1 ( N ) \textstyle\xi^{(N)}_{n+1}
  26. Φ ( η n N ) \textstyle\Phi\left(\eta_{n}^{N}\right)
  27. η n N \textstyle\eta_{n}^{N}
  28. η n \textstyle\eta_{n}
  29. Φ ( η n N ) \textstyle\Phi\left(\eta_{n}^{N}\right)
  30. Φ ( η n ) = η n + 1 \textstyle\Phi\left(\eta_{n}\right)=\eta_{n+1}
  31. η n + 1 N \textstyle\eta_{n+1}^{N}
  32. N \textstyle N
  33. Φ ( η n N ) \textstyle\Phi\left(\eta_{n}^{N}\right)
  34. η n + 1 N \textstyle\eta_{n+1}^{N}
  35. η n + 1 \textstyle\eta_{n+1}
  36. K η n = ( K η n ( x , y ) ) x , y S \textstyle K_{\eta_{n}}=\left(K_{\eta_{n}}(x,y)\right)_{x,y\in S}
  37. η n P ( S ) \eta_{n}\in\textstyle P(S)
  38. x S η n ( x ) K η n ( x , y ) = Φ ( η n ) ( y ) = η n + 1 ( y ) \displaystyle\sum_{x\in S}\eta_{n}(x)K_{\eta_{n}}(x,y)=\Phi(\eta_{n})(y)=\eta_% {n+1}(y)
  39. ( η n ) n 0 \textstyle\left(\eta_{n}\right)_{n\geq 0}
  40. ( X ¯ n ) n 0 \textstyle\left(\overline{X}_{n}\right)_{n\geq 0}
  41. P r ( X ¯ n + 1 = y X ¯ n = x ) = K η n ( x , y ) \textstyle Pr(\overline{X}_{n+1}=y\mid\overline{X}_{n}=x)=K_{\eta_{n}}(x,y)
  42. L a w ( X ¯ n ) = η n \textstyle Law(\overline{X}_{n})=\eta_{n}
  43. K η n K_{\eta_{n}}
  44. η n \textstyle\eta_{n}
  45. ξ n ( N ) = ( ξ n ( N , i ) ) 1 i N \textstyle\xi^{(N)}_{n}=\left(\xi^{(N,i)}_{n}\right)_{1\leq i\leq N}
  46. S N \textstyle S^{N}
  47. N \textstyle N
  48. X 0 \textstyle X_{0}
  49. P r ( ξ n + 1 ( N , 1 ) = y 1 , ξ n + 1 ( N , 2 ) = y 2 , , ξ n + 1 ( N , N ) = y N ξ n ( N ) ) = 1 i N K n + 1 , η n N ( ξ n ( N , i ) , y i ) Pr\left(\xi^{(N,1)}_{n+1}=y^{1},\xi^{(N,2)}_{n+1}=y^{2},\ldots,\xi^{(N,N)}_{n+% 1}=y^{N}\mid\xi^{(N)}_{n}\right)=\prod_{1\leq i\leq N}K_{n+1,\eta_{n}^{N}}% \left(\xi^{(N,i)}_{n},y^{i}\right)\quad
  50. η n N = 1 N 1 j N 1 ξ n ( N , j ) \quad\eta^{N}_{n}=\frac{1}{N}\sum_{1\leq j\leq N}1_{\xi^{(N,j)}_{n}}
  51. Φ \textstyle\Phi
  52. f : x S f ( x ) \textstyle f:x\in S\mapsto f(x)\in\mathbb{R}
  53. 1 N 1 j N f ( ξ n ( N , j ) ) N E ( f ( X ¯ n ) ) = x S η n ( x ) f ( x ) \frac{1}{N}\sum_{1\leq j\leq N}f\left(\xi^{(N,j)}_{n}\right)\longrightarrow_{N% \uparrow\infty}\quad E\left(f(\overline{X}_{n})\right)=\sum_{x\in S}\eta_{n}(x% )f(x)
  54. M = ( M ( x , y ) ) x , y S \textstyle M=(M(x,y))_{x,y\in S}
  55. G : x S G ( x ) ] 0 , 1 ] \textstyle G:x\in S~{}\mapsto~{}G(x)\in]0,1]
  56. Φ : η n = ( η n ( x ) ) x S P ( S ) Φ ( η n ) = ( Φ ( η n ) ( y ) ) y S P ( S ) \Phi~{}:~{}\eta_{n}=(\eta_{n}(x))_{x\in S}\in P(S)~{}\mapsto~{}\Phi(\eta_{n})=% \left(\Phi(\eta_{n})(y)\right)_{y\in S}\in P(S)
  57. Φ ( η n ) ( y ) \Phi(\eta_{n})(y)
  58. y S y\in S
  59. Φ ( η n ) ( y ) = x S Ψ G ( η n ) ( x ) M ( x , y ) \displaystyle\Phi(\eta_{n})(y)=\sum_{x\in S}~{}\Psi_{G}(\eta_{n})(x)M(x,y)~{}
  60. Ψ G ( η n ) ( x ) \textstyle\Psi_{G}(\eta_{n})(x)
  61. Ψ G ( η n ) ( x ) = η n ( x ) G ( x ) z S η n ( z ) G ( z ) \displaystyle\Psi_{G}(\eta_{n})(x)=\frac{\eta_{n}(x)G(x)}{\sum_{z\in S}\eta_{n% }(z)G(z)}
  62. K η n = ( K η n ( x , y ) ) x , y S \textstyle K_{\eta_{n}}=\left(K_{\eta_{n}}(x,y)\right)_{x,y\in S}
  63. η n P ( S ) \eta_{n}\in\textstyle P(S)
  64. K η n ( x , y ) = ϵ G ( x ) M ( x , y ) + ( 1 - ϵ G ( x ) ) Φ ( η n ) ( y ) K_{\eta_{n}}(x,y)=\epsilon G(x)~{}M(x,y)+(1-\epsilon G(x))~{}\Phi(\eta_{n})(y)
  65. ϵ [ 0 , 1 ] \textstyle\epsilon\in[0,1]
  66. η n ( x ) = E ( 1 x ( X n ) 0 p < n G ( X p ) ) E ( 0 p < n G ( X p ) ) \textstyle\displaystyle\eta_{n}(x)=\frac{E\left(1_{x}(X_{n})\prod_{0\leq p<n}G% (X_{p})\right)}{E\left(\prod_{0\leq p<n}G(X_{p})\right)}~{}\;
  67. X n \textstyle X_{n}
  68. η 0 \textstyle\eta_{0}
  69. M \textstyle M
  70. f : x S \textstyle f~{}:~{}x\in S\mapsto\mathbb{R}
  71. η n ( f ) := x S η n ( x ) f ( x ) = E ( f ( X n ) 0 p < n G ( X p ) ) E ( 0 p < n G ( X p ) ) \textstyle\displaystyle\eta_{n}(f):=\sum_{x\in S}\eta_{n}(x)f(x)=\frac{E\left(% f(X_{n})\prod_{0\leq p<n}G(X_{p})\right)}{E\left(\prod_{0\leq p<n}G(X_{p})% \right)}~{}\;
  72. G ( x ) = 1 \textstyle G(x)=1
  73. ϵ = 1 \epsilon=1
  74. K η n ( x , y ) = M ( x , y ) = P r ( X n + 1 = y | X n = x ) K_{\eta_{n}}(x,y)=M(x,y)=Pr\left(X_{n+1}=y|X_{n}=x\right)
  75. η n ( x ) = E ( 1 x ( X n ) ) = P r ( X n = x ) \textstyle\displaystyle\eta_{n}(x)=E\left(1_{x}(X_{n})\right)=Pr(X_{n}=x)
  76. η n + 1 ( y ) = x S η n ( x ) M ( x , y ) P r ( X n + 1 = y ) = x S P r ( X n + 1 = y | X n = x ) P r ( X n = x ) \displaystyle\eta_{n+1}(y)=\sum_{x\in S}\eta_{n}(x)M(x,y)\quad\Leftrightarrow% \quad Pr\left(X_{n+1}=y\right)=\sum_{x\in S}~{}Pr\left(X_{n+1}=y|X_{n}=x\right% )Pr\left(X_{n}=x\right)
  77. N \textstyle N
  78. ξ n + 1 ( N , i ) \textstyle\xi^{(N,i)}_{n+1}
  79. K n + 1 , η n N ( ξ n ( N , i ) , y ) = ϵ G ( ξ n ( N , i ) ) M ( ξ n ( N , i ) , y ) + ( 1 - ϵ G ( ξ n ( N , i ) ) ) 1 j N G ( ξ n ( N , j ) ) 1 k N G ( ξ n ( N , k ) ) M ( ξ n ( N , j ) , y ) K_{n+1,\eta_{n}^{N}}\left(\xi^{(N,i)}_{n},y\right)=\epsilon G\left(\xi^{(N,i)}% _{n}\right)M\left(\xi^{(N,i)}_{n},y\right)+\left(1-\epsilon G\left(\xi^{(N,i)}% _{n}\right)\right)~{}\sum_{1\leq j\leq N}\frac{G\left(\xi^{(N,j)}_{n}\right)}{% \sum_{1\leq k\leq N}G\left(\xi^{(N,k)}_{n}\right)}~{}M\left(\xi^{(N,j)}_{n},y\right)
  80. ϵ G ( ξ n ( N , i ) ) \textstyle\epsilon G\left(\xi^{(N,i)}_{n}\right)
  81. ξ n ( N , i ) \textstyle\xi^{(N,i)}_{n}
  82. ξ n + 1 ( N , i ) = y \textstyle\xi^{(N,i)}_{n+1}=y
  83. M ( ξ n ( N , i ) , y ) \textstyle M\left(\xi^{(N,i)}_{n},y\right)
  84. ξ n ( N , i ) \textstyle\xi^{(N,i)}_{n}
  85. ξ n ( N , j ) \textstyle\xi^{(N,j)}_{n}
  86. G ( ξ n ( N , j ) ) \textstyle G\left(\xi^{(N,j)}_{n}\right)
  87. ξ n + 1 ( N , i ) = y \textstyle\xi^{(N,i)}_{n+1}=y
  88. M ( ξ n ( N , j ) , y ) \textstyle M\left(\textstyle\xi^{(N,j)}_{n},y\right)
  89. G ( x ) = 1 \textstyle G(x)=1
  90. ϵ = 1 \epsilon=1
  91. X n \textstyle X_{n}
  92. ϵ = 0 \textstyle\epsilon=0
  93. G \textstyle G
  94. M \textstyle M
  95. X ¯ n \textstyle\overline{X}_{n}
  96. n = 0 , 1 , \textstyle n=0,1,\ldots
  97. X ¯ n + 1 = E ( a ( X ¯ n ) ) b ( X ¯ n ) + c ( X ¯ n ) + σ W n \overline{X}_{n+1}=E\left(a\left(\overline{X}_{n}\right)\right)~{}b(\overline{% X}_{n})+c(\overline{X}_{n})+\sigma~{}W_{n}
  98. W n \textstyle W_{n}
  99. σ > 0 \textstyle\sigma>0
  100. a , b , c \textstyle a,b,c
  101. \mathbb{R}
  102. X ¯ 0 \textstyle\overline{X}_{0}
  103. η n \textstyle\eta_{n}
  104. X ¯ n \textstyle\overline{X}_{n}
  105. f f
  106. E ( f ( X ¯ n ) ) = - + f ( x ) η n ( d x ) \textstyle E\left(f(\overline{X}_{n})\right)=\displaystyle\int_{-\infty}^{+% \infty}f(x)~{}\eta_{n}(dx)\quad
  107. P r ( X ¯ n d x ) = η n ( d x ) \quad\textstyle Pr(\overline{X}_{n}\in dx)=\eta_{n}(dx)
  108. d x \textstyle dx
  109. x \textstyle x
  110. f \textstyle f
  111. E ( f ( X ¯ n + 1 ) | X ¯ n = x ) = - + K η n ( x , d y ) f ( y ) E\left(f(\overline{X}_{n+1})~{}|~{}\overline{X}_{n}=x\right)=\int_{-\infty}^{+% \infty}K_{\eta_{n}}(x,dy)~{}f(y)\quad
  112. K η n ( x , d y ) = P r ( X ¯ n + 1 d y X ¯ n = x ) = 1 2 π σ exp { - 1 2 σ 2 ( y - [ b ( x ) - + a ( z ) η n ( d z ) + c ( x ) ] ) 2 } d y \quad K_{\eta_{n}}(x,dy)=Pr(\overline{X}_{n+1}\in dy\mid\overline{X}_{n}=x)=% \frac{1}{\sqrt{2\pi}\sigma}~{}\exp{\left\{-\frac{1}{2\sigma^{2}}\left(y-\left[% b(x)\int_{-\infty}^{+\infty}a(z)~{}\eta_{n}(dz)+c(x)\right]\right)^{2}\right\}% }~{}dy
  113. η n \textstyle\eta_{n}
  114. - + η n + 1 ( d y ) f ( y ) = - + [ - + η n ( d x ) K η n ( x , d y ) ] f ( y ) \int_{-\infty}^{+\infty}\eta_{n+1}(dy)~{}f(y)=\int_{-\infty}^{+\infty}\left[% \int_{-\infty}^{+\infty}\eta_{n}(dx)K_{\eta_{n}}(x,dy)\right]~{}f(y)\quad
  115. f f
  116. η n + 1 = Φ ( η n ) = η n K η n η n + 1 ( d y ) = ( η n K η n ) ( d y ) = x η n ( d x ) K η n ( x , d y ) \eta_{n+1}~{}=\Phi\left(\eta_{n}\right)=~{}\eta_{n}K_{\eta_{n}}\quad% \Leftrightarrow\quad\eta_{n+1}(dy)~{}=~{}\left(\eta_{n}K_{\eta_{n}}\right)(dy)% ~{}=~{}\int_{x\in\mathbb{R}}\eta_{n}(dx)K_{\eta_{n}}(x,dy)
  117. ξ n ( N ) = ( ξ n ( N , i ) ) 1 i N \textstyle\xi^{(N)}_{n}=\left(\xi^{(N,i)}_{n}\right)_{1\leq i\leq N}
  118. N \textstyle\mathbb{R}^{N}
  119. ξ n + 1 ( N , i ) = ( 1 N 1 j N a ( ξ n ( N , i ) ) ) b ( ξ n ( N , i ) ) + c ( ξ n ( N , i ) ) + σ W n i 1 i N \xi^{(N,i)}_{n+1}=\left(\displaystyle\frac{1}{N}\sum_{1\leq j\leq N}a\left(\xi% ^{(N,i)}_{n}\right)\right)~{}b\left(\xi^{(N,i)}_{n}\right)+c\left(\xi^{(N,i)}_% {n}\right)+\sigma~{}W^{i}_{n}\quad 1\leq i\leq N
  120. ξ 0 ( N ) = ( ξ 0 ( N , i ) ) 1 i N \textstyle\xi^{(N)}_{0}=\left(\xi^{(N,i)}_{0}\right)_{1\leq i\leq N}
  121. ( W n i ) 1 i N \textstyle\left(W^{i}_{n}\right)_{1\leq i\leq N}
  122. N \textstyle N
  123. X ¯ 0 \textstyle\overline{X}_{0}
  124. W n \textstyle W_{n}
  125. n = 1 , 2 , \textstyle n=1,2,\ldots
  126. a , b , c \textstyle a,b,c
  127. 1 N 1 j N f ( ξ n ( N , i ) ) = f ( y ) η n N ( d y ) N E ( f ( X ¯ n ) ) = f ( y ) η n ( d y ) \frac{1}{N}\sum_{1\leq j\leq N}f\left(\xi^{(N,i)}_{n}\right)=\int_{\mathbb{R}}% ~{}f(y)~{}\eta^{N}_{n}(dy)\longrightarrow_{N\uparrow\infty}\quad E\left(f(% \overline{X}_{n})\right)=\int_{\mathbb{R}}~{}f(y)~{}\eta_{n}(dy)
  128. η n N = 1 N 1 j N δ ξ n ( N , i ) \eta^{N}_{n}=\frac{1}{N}\sum_{1\leq j\leq N}\delta_{\xi^{(N,i)}_{n}}
  129. f \textstyle f
  130. δ x \textstyle\delta_{x}
  131. x \textstyle x
  132. W ¯ t n \textstyle\overline{W}_{t_{n}}
  133. t 0 = 0 < t 1 < < t n < \textstyle t_{0}=0<t_{1}<\ldots<t_{n}<\ldots
  134. t n - t n - 1 = h \textstyle t_{n}-t_{n-1}=h
  135. c ( x ) = x \textstyle c(x)=x
  136. b ( x ) \textstyle b(x)
  137. σ \textstyle\sigma
  138. b ( x ) × h \textstyle b(x)\times h
  139. σ × h \textstyle\sigma\times\sqrt{h}
  140. X ¯ t n \textstyle\overline{X}_{t_{n}}
  141. X ¯ n \textstyle\overline{X}_{n}
  142. t n \textstyle t_{n}
  143. ( W ¯ t n + 1 - W ¯ t n ) \textstyle\left(\overline{W}_{t_{n+1}}-\overline{W}_{t_{n}}\right)
  144. t n - t n - 1 = h \textstyle t_{n}-t_{n-1}=h
  145. X ¯ t n + 1 - X ¯ t n = E ( a ( X ¯ t n ) ) b ( X ¯ t n ) h + σ ( W ¯ t n + 1 - W ¯ t n ) \overline{X}_{t_{n+1}}-\overline{X}_{t_{n}}=E\left(a\left(\overline{X}_{t_{n}}% \right)\right)~{}b(\overline{X}_{t_{n}})~{}h+\sigma~{}\left(\overline{W}_{t_{n% +1}}-\overline{W}_{t_{n}}\right)
  146. h \textstyle h
  147. 0 \textstyle 0
  148. d X ¯ t = E ( a ( X ¯ t ) ) b ( X ¯ t ) d t + σ d W ¯ t d\overline{X}_{t}=E\left(a\left(\overline{X}_{t}\right)\right)~{}b(\overline{X% }_{t})~{}dt+\sigma~{}d\overline{W}_{t}
  149. ξ t ( N ) = ( ξ t ( N , i ) ) 1 i N \textstyle\xi^{(N)}_{t}=\left(\xi^{(N,i)}_{t}\right)_{1\leq i\leq N}
  150. N \textstyle\mathbb{R}^{N}
  151. d ξ t ( N , i ) = ( 1 N 1 j N a ( ξ t ( N , i ) ) ) b ( ξ t ( N , i ) ) + σ d W ¯ t i 1 i N d\xi^{(N,i)}_{t}=\left(\displaystyle\frac{1}{N}\sum_{1\leq j\leq N}a\left(\xi^% {(N,i)}_{t}\right)\right)~{}b\left(\xi^{(N,i)}_{t}\right)+\sigma~{}d\overline{% W}_{t}^{i}\quad 1\leq i\leq N
  152. ξ 0 ( N ) = ( ξ 0 ( N , i ) ) 1 i N \textstyle\xi^{(N)}_{0}=\left(\xi^{(N,i)}_{0}\right)_{1\leq i\leq N}
  153. ( W ¯ t i ) 1 i N \textstyle\left(\overline{W}_{t}^{i}\right)_{1\leq i\leq N}
  154. N \textstyle N
  155. X ¯ 0 \textstyle\overline{X}_{0}
  156. W ¯ t \textstyle\overline{W}_{t}
  157. a , b \textstyle a,b
  158. 1 N 1 j N f ( ξ t ( N , i ) ) = f ( y ) η t N ( d y ) N E ( f ( X ¯ t ) ) = f ( y ) η t ( d y ) \frac{1}{N}\sum_{1\leq j\leq N}f\left(\xi^{(N,i)}_{t}\right)=\int_{\mathbb{R}}% ~{}f(y)~{}\eta^{N}_{t}(dy)\longrightarrow_{N\uparrow\infty}\quad E\left(f(% \overline{X}_{t})\right)=\int_{\mathbb{R}}~{}f(y)~{}\eta_{t}(dy)
  159. η t = L a w ( X ¯ t ) \textstyle\eta_{t}=Law\left(\overline{X}_{t}\right)
  160. η t N = 1 N 1 j N δ ξ t ( N , i ) \eta^{N}_{t}=\frac{1}{N}\sum_{1\leq j\leq N}\delta_{\xi^{(N,i)}_{t}}
  161. f \textstyle f

Mean_log_deviation.html

  1. MLD = 1 N i = 1 N ln x ¯ x i \mathrm{MLD}=\frac{1}{N}\sum_{i=1}^{N}\ln\frac{\overline{x}}{x_{i}}
  2. x i x_{i}
  3. x ¯ \overline{x}
  4. x i x_{i}
  5. MLD = 1 N i = 1 N ( ln x ¯ - ln x i ) = ln x ¯ - ln x ¯ \mathrm{MLD}=\frac{1}{N}\sum_{i=1}^{N}(\ln\overline{x}-\ln x_{i})=\ln\overline% {x}-\overline{\ln x}
  6. ln x ¯ \overline{\ln x}
  7. ln x ¯ ln x ¯ \ln{\overline{x}}\geq\overline{\ln x}
  8. SDL = 1 N i = 1 N ( ln x i - ln x ¯ ) 2 \mathrm{SDL}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\ln x_{i}-\overline{\ln x})^{2}}

Mean_of_a_function.html

  1. f ¯ = 1 b - a a b f ( x ) d x . \bar{f}=\frac{1}{b-a}\int_{a}^{b}f(x)\,dx.
  2. y ¯ \bar{y}
  3. y 1 , y 2 , , y n y_{1},y_{2},\dots,y_{n}
  4. n y ¯ = y 1 + y 2 + + y n n\bar{y}=y_{1}+y_{2}+\cdots+y_{n}
  5. y ¯ \bar{y}
  6. n n
  7. n n
  8. y i y_{i}
  9. f ¯ \bar{f}
  10. [ a , b ] [a,b]
  11. a b f ¯ d x = a b f ( x ) d x \int_{a}^{b}\bar{f}\,dx=\int_{a}^{b}f(x)\,dx
  12. f ¯ \bar{f}
  13. [ a , b ] [a,b]
  14. f ( x ) f(x)
  15. [ a , b ] [a,b]
  16. f ¯ \bar{f}
  17. a b f ¯ d x = f ¯ x | a b = f ¯ b - f ¯ a = ( b - a ) f ¯ \int_{a}^{b}\bar{f}\,dx=\bar{f}x\bigr|_{a}^{b}=\bar{f}b-\bar{f}a=(b-a)\bar{f}
  18. f f
  19. c ( a , b ) c\in(a,b)
  20. a b f ( x ) d x = f ( c ) ( b - a ) \int_{a}^{b}f(x)\,dx=f(c)(b-a)
  21. f ( c ) f(c)
  22. f ( x ) f(x)
  23. [ a , b ] [a,b]
  24. f ¯ = f ( c ) \bar{f}=f(c)
  25. f ¯ = 1 Vol ( U ) U f . \bar{f}=\frac{1}{\hbox{Vol}(U)}\int_{U}f.
  26. exp ( 1 Vol ( U ) U log f ) . \exp\left(\frac{1}{\hbox{Vol}(U)}\int_{U}\log f\right).

Measure_problem_(cosmology).html

  1. P ( ϕ , t ) P(\phi,t)
  2. ϕ \phi
  3. t t
  4. e 3 H Δ t e^{3H\Delta t}
  5. Δ t \Delta t
  6. t t
  7. a a
  8. η log a \eta\sim\log a
  9. e 3 Δ η e^{3\Delta\eta}
  10. H H
  11. e 3 H β Δ t β e^{3H^{\beta}\Delta t_{\beta}}
  12. β \beta
  13. t β t_{\beta}
  14. β \beta
  15. β = 0 \beta=0
  16. β \beta
  17. β > 0 \beta>0
  18. β < 0 \beta<0
  19. P ( ϕ , t ) P(\phi,t)
  20. Q Q
  21. G G

Mechanical-electrical_analogies.html

  1. d λ d t = v \frac{d\lambda}{dt}=v
  2. d q d t = i \frac{dq}{dt}=i
  3. d p d t = F \frac{dp}{dt}=F
  4. d x d t = u \frac{dx}{dt}=u
  5. F d x \int Fdx
  6. v d p \int vdp

Medcouple.html

  1. X := x 0 x 1 x n - 1 X:=x_{0}\geq x_{1}\geq\ldots\geq x_{n-1}
  2. n n
  3. x m x_{m}
  4. X X
  5. X + := { x i | x i x m } X^{+}:=\{x_{i}~{}|~{}x_{i}\geq x_{m}\}
  6. X - := { x j | x j x m } X^{-}:=\{x_{j}~{}|~{}x_{j}\leq x_{m}\}
  7. p := | X + | p:=|X^{+}|
  8. q := | X - | q:=|X^{-}|
  9. x i + X + x_{i}^{+}\in X^{+}
  10. x j - X - x_{j}^{-}\in X^{-}
  11. h ( x i + , x j - ) := { ( x i + - x m ) - ( x m - x j - ) x i + - x j - if x i + > x j - , signum ( p - 1 - i - j ) if x i + = x m = x j - , h(x_{i}^{+},x_{j}^{-}):=\begin{cases}\displaystyle\frac{(x_{i}^{+}-x_{m})-(x_{% m}-x_{j}^{-})}{x_{i}^{+}-x_{j}^{-}}&\text{ if }x_{i}^{+}>x_{j}^{-},\\ \operatorname{signum}(p-1-i-j)&\,\text{ if }x_{i}^{+}=x_{m}=x_{j}^{-},\end{cases}
  12. signum \operatorname{signum}
  13. { h ( x i + , x j - ) | x i + X + and x j - X - } \{h(x_{i}^{+},x_{j}^{-})~{}|~{}x_{i}^{+}\in X^{+}\,\text{ and }x_{j}^{-}\in X^% {-}\}
  14. p p
  15. q q
  16. p q pq
  17. h ( x i + , x j - ) h(x_{i}^{+},x_{j}^{-})
  18. ( x i , x j ) (x_{i},x_{j})
  19. x i + x m x j - x_{i}^{+}\geq x_{m}\geq x_{j}^{-}
  20. h ( x i + , x j - ) h(x_{i}^{+},x_{j}^{-})
  21. X X
  22. X X
  23. Z + := { x i + - x m r | x i + X + } Z^{+}:=\left.\left\{\frac{x_{i}^{+}-x_{m}}{r}~{}\right|~{}x_{i}^{+}\in X^{+}\right\}
  24. Z - := { x j - - x m r | x j - X - } Z^{-}:=\left.\left\{\frac{x_{j}^{-}-x_{m}}{r}~{}\right|~{}x_{j}^{-}\in X^{-}\right\}
  25. r = 2 max 0 i n - 1 | x i | r=2\max_{0\leq i\leq n-1}|x_{i}|
  26. Z := Z + Z - Z:=Z^{+}\cup Z^{-}
  27. X X
  28. Z Z
  29. h ( z i + , z j - ) := { z i + + z j - z i + - z j - if z i + > z j - signum ( p - 1 - i - j ) if z i + = 0 = z j - h(z_{i}^{+},z_{j}^{-}):=\begin{cases}\displaystyle\frac{z_{i}^{+}+z_{j}^{-}}{z% _{i}^{+}-z_{j}^{-}}&\text{ if }z_{i}^{+}>z_{j}^{-}\\ \operatorname{signum}(p-1-i-j)&\,\text{ if }z_{i}^{+}=0=z_{j}^{-}\end{cases}
  30. Z Z
  31. | h ( z i + , z j - ) | 1 |h(z_{i}^{+},z_{j}^{-})|\leq 1
  32. | a | - | b | | a - b | |a|-|b|\leq|a-b|
  33. a = z i + a=z_{i}^{+}
  34. b = z j - b=z_{j}^{-}
  35. z i + 0 z j - z_{i}^{+}\geq 0\geq z_{j}^{-}
  36. h ( z i + , z j - ) h(z_{i}^{+},z_{j}^{-})
  37. h z i + \frac{\partial h}{\partial z_{i}^{+}}
  38. h z j - \frac{\partial h}{\partial z_{j}^{-}}
  39. z i + 0 z j - z_{i}^{+}\geq 0\geq z_{j}^{-}
  40. H := ( h i j ) = ( h ( z i + , z j - ) ) = ( h ( z 0 + , z 0 - ) h ( z 0 + , z q - 1 - ) h ( z p - 1 + , z 0 - ) h ( z p - 1 + , z q - 1 - ) ) . H:=(h_{ij})=(h(z_{i}^{+},z_{j}^{-}))=\begin{pmatrix}h(z_{0}^{+},z_{0}^{-})&% \cdots&h(z_{0}^{+},z_{q-1}^{-})\\ \vdots&\ddots&\vdots\\ h(z_{p-1}^{+},z_{0}^{-})&\cdots&h(z_{p-1}^{+},z_{q-1}^{-})\end{pmatrix}.
  41. Z + Z^{+}
  42. Z - Z^{-}
  43. H H
  44. H = ( h ( z 0 + , z 0 - ) h ( z 0 + , z q - 1 - ) h ( z p - 1 + , z 0 - ) h ( z p - 1 + , z q - 1 - ) ) . H=\begin{pmatrix}h(z_{0}^{+},z_{0}^{-})&\geq&\cdots&\geq&h(z_{0}^{+},z_{q-1}^{% -})\\ \geq&&&&\geq\\ \vdots&&\ddots&&\vdots\\ \geq&&&&\geq\\ h(z_{p-1}^{+},z_{0}^{-})&\geq&\cdots&\geq&h(z_{p-1}^{+},z_{q-1}^{-})\end{% pmatrix}.
  45. X X
  46. ( x i , x j ) (x_{i},x_{j})
  47. x i x m x j x_{i}\geq x_{m}\geq x_{j}
  48. O ( n ) O(n)
  49. H H
  50. p q n 2 4 pq\approx\frac{n^{2}}{4}
  51. X X
  52. O ( n 2 ) O(n^{2})
  53. O ( n 2 ) O(n^{2})
  54. O ( n 2 ) O(n^{2})
  55. H H
  56. H = ( h i j ) H=(h_{ij})
  57. h i j h_{ij}
  58. u u
  59. h i j h_{ij}
  60. H H
  61. O ( n ) O(n)
  62. i i
  63. j j
  64. h i j > u h_{ij}>u
  65. P P
  66. u u
  67. u u
  68. H = ( h i j ) H=(h_{ij})
  69. p + q p+q
  70. h i j h_{ij}
  71. O ( n ) O(n)
  72. P P
  73. u u
  74. h i j h_{ij}
  75. u u
  76. H H
  77. i i
  78. P i Q i P_{i}\geq Q_{i}
  79. u u
  80. i = 0 p - 1 ( P i + 1 ) i = 0 p - 1 Q i \sum_{i=0}^{p-1}(P_{i}+1)~{}\qquad~{}\sum_{i=0}^{p-1}Q_{i}
  81. H H
  82. u u
  83. u u
  84. u u
  85. h i j h_{ij}
  86. H H
  87. P P
  88. Q Q
  89. O ( 1 ) O(1)
  90. O ( n ) O(n)
  91. h ( i , j ) h(i,j)
  92. p p
  93. q q
  94. P P
  95. Q Q
  96. p q / 2 pq/2
  97. P P
  98. Q Q
  99. p q / 2 pq/2
  100. p p
  101. p q / 2 pq/2
  102. h ( i , j ) h(i,j)
  103. O ( n log n ) O(n\log n)
  104. O ( n ) O(n)
  105. P P
  106. Q Q
  107. O ( log n ) O(\log n)
  108. O ( n log n ) O(n\log n)

Median_of_medians.html

  1. n n
  2. n 5 \frac{n}{5}
  3. T ( n ) T ( n 2 / 10 ) + T ( n 7 / 10 ) + c n . T(n)\leq T(n\cdot 2/10)+T(n\cdot 7/10)+c\cdot n.
  4. T ( n ) 10 c n O ( n ) . T(n)\leq 10\cdot c\cdot n\in O(n).
  5. 2 3 n \frac{2}{3}n
  6. n \sqrt{n}
  7. n \sqrt{n}

Membrane_osmometer.html

  1. M n M_{n}
  2. M n M_{n}
  3. M n M_{n}
  4. M n M_{n}
  5. M n M_{n}
  6. lim c 0 ( Π c ) = R T M n \lim_{c\to 0}\Big({\Pi\over c}\Big)={{RT}\over M_{n}}
  7. M n M_{n}
  8. R R
  9. T T
  10. c c
  11. Π \Pi
  12. M n M_{n}
  13. M n M_{n}
  14. Π c = R T ( 1 M n + A 1 c + A 2 c 2 + A 3 c 3 + ) {\Pi\over c}=RT({1\over M_{n}}+A_{1}c+A_{2}c^{2}+A_{3}c^{3}+\dots)
  15. A n A_{n}
  16. c c
  17. Π c = A M n + B c + C c 2 + D c 3 + {\Pi\over c}={A\over M_{n}}+Bc+Cc^{2}+Dc^{3}+\dots
  18. Π c = M n ( Γ 1 + Γ 2 c + Γ 3 c 2 + Γ 4 c 3 + ) {\Pi\over c}=M_{n}(\Gamma_{1}+\Gamma_{2}c+\Gamma_{3}c^{2}+\Gamma_{4}c^{3}+\dots)
  19. B B
  20. Γ \Gamma
  21. R T A 2 = B = R T M n Γ 2 RTA_{2}=B={RT\over M_{n}}\Gamma_{2}
  22. Π = Δ H ρ g \Pi=\Delta H\rho g
  23. Π \Pi
  24. Δ H \Delta H
  25. ρ \rho
  26. g g
  27. < M n < <M_{n}<
  28. M n M_{n}
  29. M n M_{n}

Memory_cell_(binary).html

  1. Q \scriptstyle Q
  2. Q ¯ \scriptstyle\overline{Q}
  3. Q \scriptstyle Q
  4. Q ¯ \scriptstyle\overline{Q}
  5. W L \scriptstyle WL
  6. Q \scriptstyle Q
  7. Q ¯ \scriptstyle\overline{Q}
  8. B L \scriptstyle BL
  9. B L ¯ \scriptstyle\overline{BL}
  10. B L \scriptstyle BL
  11. B L ¯ \scriptstyle\overline{BL}
  12. W L \scriptstyle WL

Meshedness_coefficient.html

  1. α = m - n + 1 2 n - 5 . \alpha=\frac{m-n+1}{2n-5}.
  2. < k 2 m / n <k>=2m/n
  3. n 1 n\gg 1
  4. α = < k > - 2 4 \alpha=\frac{<k>-2}{4}

Metallic_mean.html

  1. n + 1 n + 1 n + 1 n + 1 n + = [ n ; n , n , n , n , ] = 1 2 ( n + n 2 + 4 ) n+\cfrac{1}{n+\cfrac{1}{n+\cfrac{1}{n+\cfrac{1}{n+\ddots\,}}}}=[n;n,n,n,n,% \dots]=\frac{1}{2}\left(n+\sqrt{n^{2}+4}\right)\,
  2. 0 + 4 2 0+\frac{\sqrt{4}}{2}
  3. 1 + 5 2 1+\frac{\sqrt{5}}{2}
  4. 2 + 8 2 2+\frac{\sqrt{8}}{2}
  5. 3 + 13 2 3+\frac{\sqrt{13}}{2}
  6. 4 + 20 2 4+\frac{\sqrt{20}}{2}
  7. 5 + 29 2 5+\frac{\sqrt{29}}{2}
  8. 6 + 40 2 6+\frac{\sqrt{40}}{2}
  9. 7 + 53 2 7+\frac{\sqrt{53}}{2}
  10. 8 + 68 2 8+\frac{\sqrt{68}}{2}
  11. 9 + 85 2 9+\frac{\sqrt{85}}{2}
  12. n + 4 + n < s u p > 2 n\frac{+}{\sqrt{4+}{n}{<sup>2}}
  13. x 2 - n x = 1 , x^{2}-nx=1,\,
  14. S m n = K n S m + K ( n - 1 ) \!\ S_{m}^{n}=K_{n}S_{m}+K_{(n-1)}
  15. K n = m K ( n - 1 ) + K ( n - 2 ) . \!\ K_{n}=mK_{(n-1)}+K_{(n-2)}.
  16. K n = 1 m 2 + 4 ( S m n + 1 - ( m - S m ) n + 1 ) . \!\ K_{n}=\frac{1}{\sqrt{m^{2}+4}}{(S_{m}^{n+1}-{(m-S_{m})}^{n+1})}.
  17. S m n - S m n = 1 - S m - n . \!\ {S_{m}^{n}-\lfloor S_{m}^{n}\rfloor}=1-S_{m}^{-n}.
  18. 1 S m 4 - S m 4 + S m 4 - 1 = S ( m 4 + 4 m 2 + 1 ) \!\ {1\over{S_{m}^{4}-\lfloor S_{m}^{4}\rfloor}}+\lfloor S_{m}^{4}-1\rfloor=S_% {(m^{4}+4m^{2}+1)}
  19. 1 S m 6 - S m 6 + S m 6 - 1 = S ( m 6 + 6 m 4 + 9 m 2 + 1 ) . \!\ {1\over{S_{m}^{6}-\lfloor S_{m}^{6}\rfloor}}+\lfloor S_{m}^{6}-1\rfloor=S_% {(m^{6}+6m^{4}+9m^{2}+1)}.
  20. S m 3 = S ( m 3 + 3 m ) \!\ S_{m}^{3}=S_{(m^{3}+3m)}
  21. S m 5 = S ( m 5 + 5 m 3 + 5 m ) \!\ S_{m}^{5}=S_{(m^{5}+5m^{3}+5m)}
  22. S m 7 = S ( m 7 + 7 m 5 + 14 m 3 + 7 m ) \!\ S_{m}^{7}=S_{(m^{7}+7m^{5}+14m^{3}+7m)}
  23. S m 9 = S ( m 9 + 9 m 7 + 27 m 5 + 30 m 3 + 9 m ) \!\ S_{m}^{9}=S_{(m^{9}+9m^{7}+27m^{5}+30m^{3}+9m)}
  24. S m 11 = S ( m 11 + 11 m 9 + 44 m 7 + 77 m 5 + 55 m 3 + 11 m ) . \!\ S_{m}^{11}=S_{(m^{11}+11m^{9}+44m^{7}+77m^{5}+55m^{3}+11m)}.
  25. S m 2 n + 1 = S k = 0 n 2 n + 1 2 k + 1 ( n + k 2 k ) m 2 k + 1 . \!\ S_{m}^{2n+1}=S_{\sum_{k=0}^{n}{{2n+1}\over{2k+1}}{{n+k}\choose{2k}}m^{2k+1% }}.
  26. 1 / S m = S m - m \!\ 1/S_{m}=S_{m}-m
  27. S m = a + b \!\ S_{m}=a+b
  28. S m 2 = a 2 + m b + 1. \!\ S_{m}^{2}=a^{2}+mb+1.
  29. S m 2 = m a + m b + 1 \!\ S_{m}^{2}=ma+mb+1
  30. S m 2 = m ( a + b ) + 1 \!\ S_{m}^{2}=m(a+b)+1
  31. S m 2 = m ( S m ) + 1. \!\ S_{m}^{2}=m(S_{m})+1.
  32. x 2 - m x - 1 = 0. \!\ x^{2}-mx-1=0.
  33. 1 / S m = S ( - m ) = S m - m . \!\ 1/S_{m}=S_{(-m)}=S_{m}-m.
  34. 1 2 ( n + n 2 + 4 c ) = R \!\ \frac{1}{2}\left(n+\sqrt{n^{2}+4c}\right)=R
  35. R - R = c / R \!\ R-\lfloor R\rfloor=c/R
  36. ( 1 R ) c = R - Re ( R ) \!\ \left({1\over R}\right)c=R-\lfloor\operatorname{Re}(R)\rfloor
  37. S m = 0 m ( x 2 x 2 + 4 + m + 2 2 m ) d x . S_{m}=\int_{0}^{m}{\left({x\over{2\sqrt{x^{2}+4}}}+{{m+2}\over{2m}}\right)}\,dx.

Metallocarbohedryne.html

  1. T h T_{h}
  2. T d T_{d}
  3. T h T_{h}
  4. D 2 d D_{2d}
  5. T d T_{d}
  6. T h T_{h}
  7. T d T_{d}
  8. T h T_{h}

Method_of_continued_fractions.html

  1. | ψ = | ϕ + G 0 V | ψ |\psi\rangle=|\phi\rangle+G_{0}V|\psi\rangle
  2. T = ϕ | V | ψ . T=\langle\phi|V|\psi\rangle.
  3. V V
  4. | ϕ |\phi\rangle
  5. A = G 0 V A=G_{0}V
  6. V = V | ϕ ϕ | V ϕ | V | ϕ + V 1 . V=\frac{V|\phi\rangle\langle\phi|V}{\langle\phi|V|\phi\rangle}+V_{1}.
  7. | ψ = | ϕ + T ϕ | V | ϕ | ψ 1 , T = ϕ | V | ϕ 2 ϕ | V | ϕ - ϕ | V | ψ 1 , |\psi\rangle=|\phi\rangle+\frac{T}{\langle\phi|V|\phi\rangle}|\psi_{1}\rangle,% \qquad T=\frac{\langle\phi|V|\phi\rangle^{2}}{\langle\phi|V|\phi\rangle-% \langle\phi|V|\psi_{1}\rangle},
  8. | ψ 1 |\psi_{1}\rangle
  9. | ψ 1 = | ϕ 1 + G 0 V 1 | ψ 1 , |\psi_{1}\rangle=|\phi_{1}\rangle+G_{0}V_{1}|\psi_{1}\rangle,
  10. | ϕ 1 = G 0 V | ϕ . |\phi_{1}\rangle=G_{0}V|\phi\rangle.
  11. V 1 V_{1}
  12. V 1 | ϕ = ϕ | V 1 = 0 , V_{1}|\phi\rangle=\langle\phi|V_{1}=0,
  13. | ψ 1 |\psi_{1}\rangle
  14. V i = V i - 1 - V i - 1 | ϕ i - 1 ϕ i - 1 | V i - 1 ϕ i - 1 | V i - 1 | ϕ i - 1 V_{i}=V_{i-1}-\frac{V_{i-1}|\phi_{i-1}\rangle\langle\phi_{i-1}|V_{i-1}}{% \langle\phi_{i-1}|V_{i-1}|\phi_{i-1}\rangle}
  15. | ϕ i = G 0 V i - 1 | ϕ i - 1 . |\phi_{i}\rangle=G_{0}V_{i-1}|\phi_{i-1}\rangle.
  16. T = β 0 2 β 0 - γ 1 - β 1 2 β 1 - γ 2 - β 2 2 β 2 - γ 3 - , T=\cfrac{\beta_{0}^{2}}{\beta_{0}-\gamma_{1}-\cfrac{\beta_{1}^{2}}{\beta_{1}-% \gamma_{2}-\cfrac{\beta_{2}^{2}}{\beta_{2}-\gamma_{3}-\ddots}}},
  17. β i = ϕ i - 1 | V i - 1 | ϕ i - 1 , γ i = ϕ i - 1 | V i - 1 | ϕ i . \beta_{i}=\langle\phi_{i-1}|V_{i-1}|\phi_{i-1}\rangle,\qquad\gamma_{i}=\langle% \phi_{i-1}|V_{i-1}|\phi_{i}\rangle.
  18. β N = β N + 1 = = 0 , γ N = γ N + 1 = = 0. \beta_{N}=\beta_{N+1}=\dots=0,\qquad\gamma_{N}=\gamma_{N+1}=\dots=0.
  19. | ψ N = | ϕ N + G 0 V N | ψ N , |\psi_{N}\rangle=|\phi_{N}\rangle+G_{0}V_{N}|\psi_{N}\rangle,
  20. V N V_{N}
  21. | ϕ i , i = 0 , 1 , , N - 1 |\phi_{i}\rangle,i=0,1,...,N-1
  22. G i + 1 = G i - | ϕ i + 1 ϕ i + 1 | ϕ i | V | ϕ i + 1 , G_{i+1}=G_{i}-\frac{|\phi_{i+1}\rangle\langle\phi_{i+1}|}{\langle\phi_{i}|V|% \phi_{i+1}\rangle},
  23. | ϕ i + 1 = G i V | ϕ i |\phi_{i+1}\rangle=G_{i}V|\phi_{i}\rangle
  24. β i , γ i \beta_{i},\gamma_{i}
  25. | ψ = | ϕ |\psi\rangle=|\phi\rangle

Methylornithine_synthase.html

  1. \rightleftharpoons

Metric-affine_gravitation_theory.html

  1. X X
  2. T X TX
  3. X X
  4. ( x μ , x ˙ μ ) (x^{\mu},\dot{x}^{\mu})
  5. T X TX
  6. Γ = d x λ ( λ + Γ λ x ˙ ν μ ν ˙ μ ) . \Gamma=dx^{\lambda}\otimes(\partial_{\lambda}+\Gamma_{\lambda}{}^{\mu}{}_{\nu}% \dot{x}^{\nu}\dot{\partial}_{\mu}).
  7. F X FX
  8. X X
  9. G L ( 4 , ) GL(4,\mathbb{R})
  10. g = g μ ν d x μ d x ν g=g_{\mu\nu}dx^{\mu}\otimes dx^{\nu}
  11. T X TX
  12. F X / S O ( 1 , 3 ) X FX/SO(1,3)\to X
  13. S O ( 1 , 3 ) SO(1,3)
  14. g g
  15. Γ \Gamma
  16. T X TX
  17. Γ μ ν α = { μ ν α } + S μ ν α + 1 2 C μ ν α \Gamma_{\mu\nu\alpha}=\{_{\mu\nu\alpha}\}+S_{\mu\nu\alpha}+\frac{1}{2}C_{\mu% \nu\alpha}
  18. { μ ν α } = - 1 2 ( μ g ν α + α g ν μ - ν g μ α ) , \{_{\mu\nu\alpha}\}=-\frac{1}{2}(\partial_{\mu}g_{\nu\alpha}+\partial_{\alpha}% g_{\nu\mu}-\partial_{\nu}g_{\mu\alpha}),
  19. C μ ν α = C μ α ν = μ Γ g ν α = μ g ν α + Γ μ ν α + Γ μ α ν C_{\mu\nu\alpha}=C_{\mu\alpha\nu}=\nabla^{\Gamma}_{\mu}g_{\nu\alpha}=\partial_% {\mu}g_{\nu\alpha}+\Gamma_{\mu\nu\alpha}+\Gamma_{\mu\alpha\nu}
  20. S μ ν α = - S μ α ν = 1 2 ( T ν μ α + T ν α μ + T μ ν α + C α ν μ - C ν α μ ) , S_{\mu\nu\alpha}=-S_{\mu\alpha\nu}=\frac{1}{2}(T_{\nu\mu\alpha}+T_{\nu\alpha% \mu}+T_{\mu\nu\alpha}+C_{\alpha\nu\mu}-C_{\nu\alpha\mu}),
  21. T μ ν α = 1 2 ( Γ μ ν α - Γ α ν μ ) T_{\mu\nu\alpha}=\frac{1}{2}(\Gamma_{\mu\nu\alpha}-\Gamma_{\alpha\nu\mu})
  22. Γ \Gamma
  23. Γ \Gamma
  24. R R
  25. Γ \Gamma
  26. Γ \Gamma
  27. g g
  28. g g
  29. μ Γ g ν α = 0 \nabla^{\Gamma}_{\mu}g_{\nu\alpha}=0
  30. Γ μ ν α = { μ ν α } + 1 2 ( T ν μ α + T ν α μ + T μ ν α ) . \Gamma_{\mu\nu\alpha}=\{_{\mu\nu\alpha}\}+\frac{1}{2}(T_{\nu\mu\alpha}+T_{\nu% \alpha\mu}+T_{\mu\nu\alpha}).
  31. F g X F^{g}X
  32. F X FX
  33. g g
  34. F X / S O ( 1 , 3 ) X FX/SO(1,3)\to X
  35. Γ \Gamma
  36. Γ g \Gamma^{g}
  37. F g X F^{g}X
  38. G L ( 4 , ) GL(4,\mathbb{R})
  39. Γ \Gamma
  40. Γ g \Gamma^{g}

Michael_Shub.html

  1. C C^{\infty}

Michel_Lazard.html

  1. G L n ( p ) GL_{n}(\mathbb{Z}_{p})

Micromotor.html

  1. M g + 2 H 2 O M g ( O H ) 2 + H 2 Mg+2H_{2}O\to Mg(OH)_{2}+H_{2}

Microscale_and_macroscale_models.html

  1. N ( t ) N(t)
  2. t t
  3. r = β - δ r=\beta-\delta
  4. β \beta
  5. δ \delta
  6. t t
  7. σ > 0 \sigma>0
  8. N ( t ) N(t)
  9. t t
  10. d N ( t ) / d t dN(t)/dt
  11. N N
  12. N ( 0 ) N(0)
  13. t = 0 t=0
  14. β \beta
  15. δ \delta
  16. Microscale ( ) \operatorname{Microscale}()
  17. Mutation ( v ) \operatorname{Mutation}(v)
  18. s i g m a sigma
  19. Rand ( ) \operatorname{Rand}()
  20. 0 R a n d ( ) < 1 0\leq Rand()<1
  21. Microscale ( ) \operatorname{Microscale}()
  22. β = 1 / 5 \beta=1/5
  23. δ = 1 / 10 \delta=1/10
  24. N 0 = 1000 N_{0}=1000
  25. α \alpha
  26. β \beta
  27. N 0 N_{0}
  28. ( σ = 0 ) (\sigma=0)
  29. σ = 0.006 \sigma=0.006
  30. σ = 0.010 \sigma=0.010

Mildly_context-sensitive_grammar_formalism.html

  1. 𝐶𝑂𝑃𝑌 = { w w w { a , b } * } \mathit{COPY}=\{\,ww\mid w\in\{a,b\}^{*}\,\}

Milnor–Moore_theorem.html

  1. dim A n < \dim A_{n}<\infty
  2. U ( P ( A ) ) A U(P(A))\to A
  3. P ( A ) P(A)

Miltiradiene_synthase.html

  1. \rightleftharpoons

Min_entropy.html

  1. ρ A B \rho_{AB}
  2. ρ A B \rho_{AB}
  3. A B \mathcal{H}_{A}\otimes\mathcal{H}_{B}
  4. H min ( A | B ) ρ - inf σ B D max ( ρ A B | | I A σ B ) H_{\min}(A|B)_{\rho}\equiv-\inf_{\sigma_{B}}D_{\max}(\rho_{AB}||I_{A}\otimes% \sigma_{B})
  5. σ B \sigma_{B}
  6. B \mathcal{H}_{B}
  7. D max D_{\max}
  8. D max ( ρ | | σ ) = inf λ { λ : ρ 2 λ σ } D_{\max}(\rho||\sigma)=\inf_{\lambda}\{\lambda:\rho\leq 2^{\lambda}\sigma\}
  9. H min ϵ ( A | B ) ρ = sup ρ H min ( A | B ) ρ H_{\min}^{\epsilon}(A|B)_{\rho}=\sup_{\rho^{\prime}}H_{\min}(A|B)_{\rho^{% \prime}}
  10. ρ A B \rho^{\prime}_{AB}
  11. ϵ \epsilon
  12. ρ A B \rho_{AB}
  13. ϵ \epsilon
  14. P ( ρ , σ ) = 1 - F ( ρ , σ ) 2 P(\rho,\sigma)=\sqrt{1-F(\rho,\sigma)^{2}}
  15. F ( ρ , σ ) F(\rho,\sigma)
  16. S ( A | B ) ρ = lim ϵ 0 lim n 1 n H min ϵ ( A n | B n ) ρ n . S(A|B)_{\rho}=\lim_{\epsilon\rightarrow 0}\lim_{n\rightarrow\infty}\frac{1}{n}% H_{\min}^{\epsilon}(A^{n}|B^{n})_{\rho^{\otimes n}}~{}.
  17. H min ϵ ( A | B ) ρ H min ϵ ( A | B C ) ρ . H_{\min}^{\epsilon}(A|B)_{\rho}\geq H_{\min}^{\epsilon}(A|BC)_{\rho}~{}.
  18. ρ \rho
  19. ρ B x \rho_{B}^{x}
  20. x x
  21. P X ( x ) P_{X}(x)
  22. ρ X B = x P X ( x ) | x x | ρ B x \rho_{XB}=\sum_{x}P_{X}(x)|x\rangle\langle x|\otimes\rho_{B}^{x}
  23. { | x } \{|x\rangle\}
  24. x x
  25. p g ( X | B ) p_{g}(X|B)
  26. p g ( X | B ) = x P X ( x ) t r ( E x ρ B x ) p_{g}(X|B)=\sum_{x}P_{X}(x)tr(E_{x}\rho_{B}^{x})
  27. E x E_{x}
  28. p g ( X | B ) = 2 - H min ( X | B ) . p_{g}(X|B)=2^{-H_{\min}(X|B)}~{}.
  29. ρ X B \rho_{XB}
  30. ρ X B = σ X τ B \rho_{XB}=\sigma_{X}\otimes\tau_{B}
  31. σ X \sigma_{X}
  32. τ B \tau_{B}
  33. 2 - H min ( X | B ) = max x P X ( x ) . 2^{-H_{\min}(X|B)}=\max_{x}P_{X}(x)~{}.
  34. | ϕ + |\phi^{+}\rangle
  35. A B \mathcal{H}_{A}\otimes\mathcal{H}_{B}
  36. | ϕ + A B = 1 d x A , x B | x A | x B |\phi^{+}\rangle_{AB}=\frac{1}{\sqrt{d}}\sum_{x_{A},x_{B}}|x_{A}\rangle|x_{B}\rangle
  37. { | x A } \{|x_{A}\rangle\}
  38. { | x B } \{|x_{B}\rangle\}
  39. ρ A B \rho_{AB}
  40. q c ( A | B ) = d max F ( ( I A ) ρ A B , | ϕ + ϕ + | ) q_{c}(A|B)=d\max_{\mathcal{E}}F\left((I_{A}\otimes\mathcal{E})\rho_{AB},|\phi^% {+}\rangle\langle\phi^{+}|\right)
  41. \mathcal{E}
  42. ρ A B \rho_{AB}
  43. q c ( A | B ) = 2 - H min ( A | B ) q_{c}(A|B)=2^{-H_{\min}(A|B)}
  44. ρ A B \rho_{AB}
  45. H min ( A | B ) = - inf σ B inf λ { λ | ρ A B 2 λ ( I A σ B ) } . H_{\min}(A|B)=-\inf_{\sigma_{B}}\inf_{\lambda}\{\lambda|\rho_{AB}\leq 2^{% \lambda}(I_{A}\otimes\sigma_{B})\}~{}.
  46. log inf σ B Tr ( σ B ) \log\inf_{\sigma_{B}}\operatorname{Tr}(\sigma_{B})
  47. σ B 0 \sigma_{B}\geq 0
  48. I A σ B ρ A B . I_{A}\otimes\sigma_{B}\geq\rho_{AB}~{}.
  49. min: Tr ( σ B ) \,\text{min:}\operatorname{Tr}(\sigma_{B})
  50. subject to: I A σ B ρ A B \,\text{subject to: }I_{A}\otimes\sigma_{B}\geq\rho_{AB}
  51. σ B 0 . \sigma_{B}\geq 0~{}.
  52. ( ρ A B , I B , Tr * ) (\rho_{AB},I_{B},\operatorname{Tr}^{*})
  53. Tr * \operatorname{Tr}^{*}
  54. Tr * \operatorname{Tr}^{*}
  55. Tr * ( X ) = I A X . \operatorname{Tr}^{*}(X)=I_{A}\otimes X~{}.
  56. E A B E_{AB}
  57. max: Tr ( ρ A B E A B ) \,\text{max:}\operatorname{Tr}(\rho_{AB}E_{AB})
  58. subject to: Tr A ( E A B ) = I B \,\text{subject to: }\operatorname{Tr}_{A}(E_{AB})=I_{B}
  59. E A B 0 . E_{AB}\geq 0~{}.
  60. \mathcal{E}
  61. I A ( | ϕ + ϕ + | ) = E A B I_{A}\otimes\mathcal{E}^{\dagger}(|\phi^{+}\rangle\langle\phi^{+}|)=E_{AB}
  62. ρ A B , E A B = ρ A B , I A ( | ϕ + ϕ + | ) \langle\rho_{AB},E_{AB}\rangle=\langle\rho_{AB},I_{A}\otimes\mathcal{E}^{% \dagger}(|\phi^{+}\rangle\langle\phi^{+}|)\rangle
  63. = I A ( ρ A B ) , | ϕ + ϕ + | ) =\langle I_{A}\otimes\mathcal{E}(\rho_{AB}),|\phi^{+}\rangle\langle\phi^{+}|)\rangle
  64. max P X ( x ) x | ( ρ B x ) | x . \max P_{X}(x)\langle x|\mathcal{E}(\rho_{B}^{x})|x\rangle~{}.
  65. \mathcal{E}
  66. x x

Ming_Antu's_infinite_series_expansion_of_trigonometric_functions.html

  1. π = 3 ( 1 + 1 4 3 ! + 3 2 4 2 5 ! + 3 2 5 2 4 3 7 ! + ) \pi=3\left(1+\frac{1}{4\cdot 3!}+\frac{3^{2}}{4^{2}\cdot 5!}+\frac{3^{2}\cdot 5% ^{2}}{4^{3}\cdot 7!}+\cdots\right)
  2. sin x = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + \sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots
  3. vers x = x 2 2 ! - x 4 4 ! + x 6 6 ! + . \operatorname{vers}x=\frac{x^{2}}{2!}-\frac{x^{4}}{4!}+\frac{x^{6}}{6!}+\cdots.
  4. π \pi
  5. B D = 2 x - G H BD=2x-GH
  6. A B : B C : C I = 1 : x : x 2 AB:BC:CI=1:x:x^{2}
  7. \therefore
  8. A B : B E : E F : F J : J K = 1 : p : p 2 : p 3 : p 4 AB:BE:EF:FJ:JK=1:p:p^{2}:p^{3}:p^{4}
  9. 1 : B E = B E : E F ; 1:BE=BE:EF;
  10. E F = B E 2 EF=BE^{2}
  11. 1 : B E 2 = x : G H 1:BE^{2}=x:GH
  12. G H = x B E 2 = x p 2 GH=x\cdot BE^{2}=xp^{2}
  13. B D = 2 x - x p 2 BD=2x-xp^{2}
  14. E F = L C = C M = M G = N G = I N EF=LC=CM=MG=NG=IN
  15. L M + M N = C M + M N + I N = C I + O P = J K + C I LM+MN=CM+MN+IN=CI+OP=JK+CI
  16. A B : ( B E + E C ) = B L : ( L M + M N ) \therefore AB:(BE+EC)=BL:(LM+MN)
  17. A B : B L = B L : ( C I + J K ) AB:BL=BL:(CI+JK)
  18. B L = q BL=q
  19. A B : B L : ( C I + J K ) = 1 : q : q 2 AB:BL:(CI+JK)=1:q:q^{2}
  20. J K = p 4 JK=p^{4}
  21. C I = y 2 CI=y^{2}
  22. C I + J K = q 2 = B L 2 = ( 2 B E ) 2 = ( 2 p ) 2 = 4 p 2 CI+JK=q^{2}=BL^{2}=(2BE)^{2}=(2p)^{2}=4p^{2}
  23. q 2 = 4 p 2 q^{2}=4p^{2}
  24. p = q 2 p=\frac{q}{2}
  25. C I + J K = x 2 + p 4 = q 2 CI+JK=x^{2}+p^{4}=q^{2}
  26. x 2 + q 4 16 = q 2 , x^{2}+\frac{q^{4}}{16}=q^{2},
  27. x 2 = q 2 - q 4 16 x^{2}=q^{2}-\frac{q^{4}}{16}
  28. ( x 2 ) 2 16 = ( q 2 - q 4 16 ) 2 16 = j = 0 2 ( - 1 ) j ( 2 j ) q 2 ( 2 + j ) 16 j \frac{(x^{2})^{2}}{16}=\frac{(q^{2}-\frac{q^{4}}{16})^{2}}{16}=\sum_{j=0}^{2}(% -1)^{j}{2\choose j}\frac{q^{2(2+j)}}{16^{j}}
  29. x 4 16 = q 4 16 - q 6 128 + q 8 4096 16 \frac{x^{4}}{16}=\frac{q^{4}}{16}-\frac{q^{6}}{128}+\frac{q^{8}}{4096}{16}
  30. x 2 n 16 n - 1 = j = 0 n ( - 1 ) j ( n j ) q 2 ( n + j ) 16 n + j - 1 \frac{x^{2n}}{16^{n-1}}=\sum_{j=0}^{n}(-1)^{j}{n\choose j}\frac{q^{2(n+j)}}{16% ^{n+j-1}}
  31. q 4 q^{4}
  32. x 2 = q 2 - q 4 16 x^{2}=q^{2}-\frac{q^{4}}{16}
  33. x 4 16 = q 4 16 - 2 q 6 16 2 + q 8 4096 16 \frac{x^{4}}{16}={\frac{q^{4}}{16}-\frac{2q^{6}}{16^{2}}+\frac{q^{8}}{4096}}{16}
  34. x 2 + x 4 16 = q 2 - q 6 128 + q 8 4096 x^{2}+\frac{x^{4}}{16}=q^{2}-\frac{q^{6}}{128}+\frac{q^{8}}{4096}
  35. x 2 + x 4 16 + 2 x 6 16 2 = q 2 - 5 q 8 4096 + 3 q 10 32768 - q 12 524288 , x^{2}+\frac{x^{4}}{16}+\frac{2x^{6}}{16^{2}}=q^{2}-\frac{5q^{8}}{4096}+\frac{3% q^{10}}{32768}-\frac{q^{12}}{524288},
  36. q 6 q^{6}
  37. x 2 + x 4 16 + 2 x 6 16 2 + 5 x 8 16 3 + 14 x 10 16 4 + 42 x 12 16 5 \displaystyle x^{2}+\frac{x^{4}}{16}+\frac{2x^{6}}{16^{2}}+\frac{5x^{8}}{16^{3% }}+\frac{14x^{10}}{16^{4}}+\frac{42x^{12}}{16^{5}}
  38. q 2 = n = 1 C n x 2 n 4 2 n - 2 q^{2}=\sum_{n=1}^{\infty}C_{n}\frac{x^{2n}}{4^{2n-2}}
  39. C n = 1 n + 1 ( 2 n n ) C_{n}=\frac{1}{n+1}{2n\choose n}
  40. C n = k ( - 1 ) k ( n k k + 1 ) C n k C_{n}=\sum_{k}(-1)^{k}{nk\choose k+1}C_{nk}
  41. B C : C G : G H = A B : B E : E F = 1 : p : p 2 = x : p x : p 2 x \because BC:CG:GH=AB:BE:EF=1:p:p^{2}=x:px:p^{2}x
  42. G H := p 2 x = ( q 2 ) 2 x = q 2 x 4 \therefore GH:=p^{2}x=(\frac{q}{2})^{2}x=\frac{q^{2}x}{4}
  43. B D = 2 x - G H BD=2x-GH
  44. B D = 2 x - x 4 q 2 BD=2x-\frac{x}{4}q^{2}
  45. = 2 x - n = 1 C n x 2 n + 1 4 2 n - 1 =2x-\sum_{n=1}^{\infty}C_{n}\frac{x^{2n+1}}{4^{2n-1}}
  46. B D = 2 x - x B E 2 BD=2x-x\cdot BE^{2}
  47. sin ( 2 α ) = 2 sin α - n = 1 C n ( sin α ) 2 n + 1 4 n - 1 \sin(2\alpha)=2\sin\alpha-\sum_{n=1}^{\infty}C_{n}\frac{(\sin\alpha)^{2n+1}}{4% ^{n-1}}
  48. = 2 sin ( α ) - 2 sin ( α ) 3 1 + cos ( α ) =2\sin(\alpha)-\frac{2\sin(\alpha)^{3}}{1+\cos(\alpha)}
  49. q 2 = B L 2 = n = 1 C n x 2 n 4 2 n - 2 q^{2}=BL^{2}=\sum_{n=1}^{\infty}C_{n}\frac{x^{2n}}{4^{2n-2}}
  50. sin ( α 2 ) 2 = n = 1 C n ( s i n α ) 2 n 4 2 n \sin(\frac{\alpha}{2})^{2}=\sum_{n=1}^{\infty}C_{n}\frac{(sin\alpha)^{2n}}{4^{% 2n}}
  51. A B : B C = B C : C G = C G : G F AB:BC=BC:CG=CG:GF
  52. B C : F G = B D : δ γ BC:FG=BD:\delta\gamma
  53. 2 B D = B E + δ α 2BD=BE+\delta\alpha
  54. 2 B D - δ γ = B E + B C 2BD-\delta\gamma=BE+BC
  55. 2 * B D - δ γ - B C = B E \therefore 2*BD-\delta\gamma-BC=BE
  56. B E = 3 * a - a 3 BE=3*a-a^{3}
  57. y 4 y_{4}
  58. y 4 = 4 * a - 10 * a 3 4 + 14 * a 5 4 3 - 12 * a 7 4 5 y_{4}=4*a-\frac{10*a^{3}}{4}+\frac{14*a^{5}}{4^{3}}-\frac{12*a^{7}}{4^{5}}
  59. 4 a - 10 * a 3 / 4 + n = 1 ( 16 C n - 2 C n + 1 ) * a 2 n + 1 4 2 n - 1 4a-10*a^{3}/4+\sum_{n=1}^{\infty}(16C_{n}-2C_{n+1})*\frac{a^{2n+1}}{4^{2n-1}}
  60. sin ( 4 * α ) = 4 * s i n ( α ) - 10 * s i n 3 α \sin(4*\alpha)=4*sin(\alpha)-10*sin^{3}\alpha
  61. + n = 1 ( 16 * C n - 2 C n + 1 ) * sin 2 n + 3 ( α ) 4 n +\sum_{n=1}^{\infty}(16*C_{n}-2C_{n+1})*\frac{\sin^{2n+3}(\alpha)}{4^{n}}
  62. y 5 = 5 a - 5 a 3 + a 5 y_{5}=5a-5a^{3}+a^{5}
  63. sin ( 5 α ) = 5 sin ( α ) - 20 sin 3 ( α ) + 16 sin 5 ( α ) \sin(5\alpha)=5\sin(\alpha)-20\sin^{3}(\alpha)+16\sin^{5}(\alpha)
  64. y 10 = y 5 ( y 2 ) \therefore y_{10}=y_{5}(y_{2})
  65. y 10 ( a ) = 5 * y 2 - 5 * ( y 2 ) 3 + ( y 2 ) 5 y_{10}(a)=5*y_{2}-5*(y_{2})^{3}+(y_{2})^{5}
  66. y 2 y_{2}
  67. y 10 ( a ) = 10 * a - 165 * a 3 4 + 3003 * a 5 4 3 - 21450 * a 7 4 5 y_{10}(a)=10*a-\frac{165*a^{3}}{4}+\frac{3003*a^{5}}{4^{3}}-\frac{21450*a^{7}}% {4^{5}}
  68. a = y 10 a=y_{10}
  69. y 10 y_{10}
  70. y 100 = y 10 ( a = y 10 ) y_{100}=y_{10}(a=y_{10})
  71. y 100 ( a ) = 100 * a - 166650 * a 3 4 + 333000030 * a 5 4 * 16 - 316350028500 * a 7 4 * 16 2 + 17488840755750 * a 9 4 * 16 3 + y_{100}(a)=100*a-166650*\frac{a^{3}}{4}+333000030*\frac{a^{5}}{4*16}-316350028% 500*\frac{a^{7}}{4*16^{2}}+17488840755750*\frac{a^{9}}{4*16^{3}}+
  72. y 1000 = y 100 ( y 10 ) y_{1000}=y_{100}(y_{10})
  73. y 1000 ( a ) = 1000 * a - 1666666500 * a 3 4 + 33333000000300 * a 5 4 * 16 - 3174492064314285000 a 7 4 * 16 2 + y_{1000}(a)=1000*a-1666666500*\frac{a^{3}}{4}+33333000000300*\frac{a^{5}}{4*16% }-3174492064314285000\frac{a^{7}}{4*16^{2}}+
  74. y 10000 = 10000 * a - 166666665000 * a 3 4 + 33333330000000300 * a 5 4 3 + y_{10000}=10000*a-\frac{166666665000*a^{3}}{4}+\frac{33333330000000300*a^{5}}{% 4^{3}}+
  75. y 100 = 100 a - ( 100 a ) 3 24.002400240024002400 + ( 100 a ) 5 24.024021859697730358 * 80 + y100=100a-\frac{(100a)^{3}}{24.002400240024002400}+\frac{(100a)^{5}}{24.024021% 859697730358*80}+
  76. y 1000 := 1000 a - ( 1000 a ) 3 24.000024000024000024 + ( 1000 a ) 5 24.000240002184019680 * 80 + y1000:=1000a-\frac{(1000a)^{3}}{24.000024000024000024}+\frac{(1000a)^{5}}{24.0% 00240002184019680*80}+
  77. y 10000 := 10000 a - ( 10000 a ) 3 24.000000240000002400 + ( 10000 a ) 5 24.000002400000218400 * 80 + y10000:=10000a-\frac{(10000a)^{3}}{24.000000240000002400}+\frac{(10000a)^{5}}{% 24.000002400000218400*80}+
  78. c h o r d = a r c - a r c 3 4 * 3 ! + a r c 5 4 2 * 5 ! - a r c 7 4 3 * 7 ! + chord=arc-\frac{arc^{3}}{4*3!}+\frac{arc^{5}}{4^{2}*5!}-\frac{arc^{7}}{4^{3}*7% !}+
  79. = n = 1 ( - 1 ) n - 1 * a r c 2 * n - 1 ( 4 n - 1 * ( 2 * n - 1 ) ! ) =\sum_{n=1}^{\infty}\frac{(-1)^{n-1}*arc^{2*n-1}}{(4^{n-1}*(2*n-1)!)}
  80. a r c := c h o r d + c h o r d 3 24 + 3 * c h o r d 5 640 + 5 * c h o r d 7 7168 + arc:=chord+\frac{chord^{3}}{24}+\frac{3*chord^{5}}{640}+\frac{5*chord^{7}}{716% 8}+

Minimal_generating_set.html

  1. dim k M / m M = dim k M R k \dim_{k}M/mM=\dim_{k}M\otimes_{R}k

Minimum_overlap_problem.html

  1. n n
  2. k k
  3. 2 n a n d 2 n −2nand2n
  4. M ( n ) M(n)
  5. M ( n ) := min A , B max k M k . M(n):=\min_{A,B}\max_{k}M_{k}.\,\!
  6. M ( n ) M(n)
  7. n n
  8. M ( n ) M(n)
  9. M ( n ) > n / 4 M(n)>n/4
  10. M ( n ) > ( 1 - 2 - 1 / 2 ) n M(n)>(1-2^{-1/2})\,n
  11. M ( n ) > ( 4 - 6 - 1 / 2 ) n / 5 M(n)>(4-6^{-1/2})\,n/5
  12. M ( n ) > ( 4 - 15 1 / 2 ) 1 / 2 ( n - 1 ) M(n)>(4-15^{1/2})^{1/2}\,(n-1)
  13. M ( n ) > ( 4 - 15 1 / 2 ) 1 / 2 n M(n)>(4-15^{1/2})^{1/2}\,n
  14. M ( n ) = n / 2 M(n)=n/2
  15. M ( n ) < 2 n / 5 M(n)<2n/5
  16. M ( n ) < 0.38201 n M(n)<0.38201n
  17. M ( n ) / n M(n) /n
  18. M ( n ) / n M(n) /n
  19. M ( n ) M(n)
  20. M ( n ) M(n)
  21. n n\,\!
  22. M ( n ) M(n)\,\!
  23. 5 ( n + 3 ) / 13 \textstyle\lfloor 5(n+3)/13\rfloor

Mining_pool.html

  1. B B
  2. p p
  3. p = 1 / D p=1/D
  4. p p
  5. R = B p R=B\cdot p
  6. R = B n N R=B\cdot\frac{n}{N}
  7. n n
  8. N N
  9. f f
  10. c c
  11. s = 1 s=1
  12. k k
  13. S k S_{k}
  14. S k = 0 S_{k}=0
  15. r = 1 - p + p c r=1-p+\frac{p}{c}
  16. p = 1 / D p=1/D
  17. r r
  18. k k
  19. S k = S k + s p B S_{k}=S_{k}+spB
  20. s = s r s=sr
  21. k k
  22. ( 1 - f ) ( r - 1 ) S k s p \frac{(1-f)(r-1)S_{k}}{sp}
  23. o o
  24. o = 0 o=0
  25. o = 1 o=1
  26. f f
  27. c c
  28. o o
  29. s = 1 s=1
  30. k k
  31. S k S_{k}
  32. S k = 0 S_{k}=0
  33. r = 1 + p ( 1 - c ) ( 1 - o ) c r=1+\frac{p(1-c)(1-o)}{c}
  34. r r
  35. k k
  36. S k = S k + ( 1 - f ) ( 1 - c ) s p B S_{k}=S_{k}+(1-f)(1-c)spB
  37. B B
  38. s = s r s=sr
  39. k k
  40. ( 1 - o ) S k c s \frac{(1-o)S_{k}}{c_{s}}
  41. S k = S k o S_{k}=S_{k}\cdot o

Mitochondrial_tRNA_pseudouridine27::28_synthase.html

  1. \rightleftharpoons

Mittag-Leffler_distribution.html

  1. [ 0 , ) [0,\infty)
  2. α ( 0 , 1 ] \alpha\in(0,1]
  3. α [ 0 , 1 ] \alpha\in[0,1]
  4. α \alpha
  5. E α ( z ) := n = 0 z n Γ ( 1 + α n ) E_{\alpha}(z):=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(1+\alpha n)}
  6. α = 0 \alpha=0
  7. - { 1 } \mathbb{C}-\{1\}
  8. α ( 0 , 1 ] \alpha\in(0,1]
  9. E α E_{\alpha}
  10. 0
  11. - -\infty
  12. E α ( 0 ) = 1 E_{\alpha}(0)=1
  13. x 1 - E α ( - x α ) x\mapsto 1-E_{\alpha}(-x^{\alpha})
  14. α \alpha
  15. E 1 E_{1}
  16. 1 1
  17. α ( 0 , 1 ) \alpha\in(0,1)
  18. 𝔼 ( e - λ X α ) = 1 1 + λ α , \mathbb{E}(e^{-\lambda X_{\alpha}})=\frac{1}{1+\lambda^{\alpha}},
  19. α ( 0 , 1 ) \alpha\in(0,1)
  20. α [ 0 , 1 ] \alpha\in[0,1]
  21. X α X_{\alpha}
  22. α \alpha
  23. C > 0 C>0
  24. 𝔼 ( e z X α ) = E α ( C z ) , \mathbb{E}(e^{zX_{\alpha}})=E_{\alpha}(Cz),
  25. z z
  26. α ( 0 , 1 ] \alpha\in(0,1]
  27. z z
  28. 1 / C 1/C
  29. α = 0 \alpha=0
  30. 0
  31. 1 / 2 1/2
  32. 1 1

Mixed-order_Ambisonics.html

  1. ( # V + 1 ) (\#V+1)

Mobility_analogy.html

  1. i = v G i=vG
  2. F = u R m F=uR_{\mathrm{m}}
  3. v = L d i d t v=L\frac{di}{dt}
  4. u = C m d F d t u=C_{\mathrm{m}}\frac{dF}{dt}
  5. Z = j ω L Z=j\omega L
  6. Y m = j ω C m Y_{\mathrm{m}}=j\omega C_{\mathrm{m}}
  7. i = C d v d t i=C\frac{dv}{dt}
  8. F = M d u d t F=M\frac{du}{dt}
  9. Z = 1 j ω C Z={1\over j\omega C}
  10. Y m = 1 j ω M Y_{\mathrm{m}}={1\over j\omega M}
  11. F = B ( d u 2 d t - d u 1 d t ) = B d Δ u d t F=B\left(\frac{du_{\mathrm{2}}}{dt}-\frac{du_{\mathrm{1}}}{dt}\right)=B\frac{d% \Delta u}{dt}
  12. [ i u ] = [ y 11 y 12 y 21 y 22 ] [ v F ] \begin{bmatrix}i\\ u\end{bmatrix}=\begin{bmatrix}y_{11}&y_{12}\\ y_{21}&y_{22}\end{bmatrix}\begin{bmatrix}v\\ F\end{bmatrix}
  13. y 22 y_{22}\,
  14. y 11 y_{11}\,
  15. y 21 y_{21}\,
  16. y 12 y_{12}\,
  17. E = v i d t E=\int vi\ dt
  18. E = F u d t E=\int Fu\ dt
  19. P = v i P=vi
  20. P = F u P=Fu
  21. P = i 2 R = v 2 R P=i^{2}R={v^{2}\over R}
  22. P = u 2 R m = F 2 R m P=u^{2}R_{\mathrm{m}}={F^{2}\over R_{\mathrm{m}}}
  23. E = 1 2 L i 2 E=\tfrac{1}{2}Li^{2}
  24. E = 1 2 C m F 2 E=\tfrac{1}{2}C_{\mathrm{m}}F^{2}
  25. E = 1 2 C v 2 E=\tfrac{1}{2}Cv^{2}
  26. E = 1 2 M u 2 E=\tfrac{1}{2}Mu^{2}

Modified_hyperbolic_tangent.html

  1. mtanh x = e a x - e - b x e c x + e - d x \operatorname{mtanh}x=\frac{e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}

Modified_KdV-Burgers_equation.html

  1. u t + u x x x - α u 2 u x - β u x x = 0. u_{t}+u_{xxx}-\alpha u^{2}\,u_{x}-\beta u_{xx}=0.

Module:Buffer::doc.html

  1. y < m t p l 0.5 / x y<mtpl>{{=}}0.5/\sqrt{x}
  2. y y
  3. x x
  4. = =

Module:Random::doc.html

  1. [ 0 , 1 ) [0,1)
  2. [ 1 , m ] [1,m]
  3. [ m , n ] [m,n]
  4. [ n , m ] [n,m]

Module:Rfx::doc.html

  1. supports supports + opposes × 100 \frac{\,\text{supports}}{\,\text{supports}+\,\text{opposes}}\times 100

Module:User:Lesser_Cartographies::ZBL.html

  1. p p
  2. a = b a=b

Module_of_covariants.html

  1. A G A^{G}
  2. ( M k A ) G . (M\otimes_{k}A)^{G}.

Moduli_stack_of_formal_group_laws.html

  1. FG \mathcal{M}_{\,\text{FG}}
  2. FG \mathcal{M}_{\,\text{FG}}
  3. FG n \mathcal{M}^{n}_{\,\text{FG}}
  4. FG n ( R ) \mathcal{M}^{n}_{\,\text{FG}}(R)
  5. FG \mathcal{M}_{\,\text{FG}}
  6. Spec 𝔽 p ¯ FG n \operatorname{Spec}\overline{\mathbb{F}_{p}}\to\mathcal{M}^{n}_{\,\text{FG}}
  7. FG n \mathcal{M}^{n}_{\,\text{FG}}
  8. Spec 𝔽 p ¯ / Aut ( 𝔽 p ¯ , f ) \operatorname{Spec}\overline{\mathbb{F}_{p}}/\operatorname{Aut}(\overline{% \mathbb{F}_{p}},f)
  9. Aut ( 𝔽 p ¯ , f ) \operatorname{Aut}(\overline{\mathbb{F}_{p}},f)
  10. FG n \mathcal{M}^{n}_{\,\text{FG}}

Moduli_stack_of_principal_bundles.html

  1. 𝐅 q \mathbf{F}_{q}
  2. Bun G ( X ) \operatorname{Bun}_{G}(X)
  3. 𝐅 q \mathbf{F}_{q}
  4. Bun G ( X ) ( R ) = \operatorname{Bun}_{G}(X)(R)=
  5. X × 𝐅 q Spec R X\times_{\mathbf{F}_{q}}\operatorname{Spec}R
  6. 𝐅 q \mathbf{F}_{q}
  7. Bun G ( X ) \operatorname{Bun}_{G}(X)
  8. Bun G ( X ) ( 𝐅 q ) \operatorname{Bun}_{G}(X)(\mathbf{F}_{q})
  9. Bun G ( X ) \operatorname{Bun}_{G}(X)
  10. Bun G ( X ) \operatorname{Bun}_{G}(X)
  11. Bun G ( X ) \operatorname{Bun}_{G}(X)
  12. Bun G ( X ) \operatorname{Bun}_{G}(X)
  13. Bun G ( X ) \operatorname{Bun}_{G}(X)
  14. Bun G ( X ) \operatorname{Bun}_{G}(X)
  15. ( g ( X ) - 1 ) dim G (g(X)-1)\dim G
  16. g ( X ) g(X)
  17. π 0 ( Bun G ( X ) ) \pi_{0}(\operatorname{Bun}_{G}(X))
  18. π 1 ( G ) \pi_{1}(G)
  19. Bun G ( X ) \operatorname{Bun}_{G}(X)
  20. # Bun G ( X ) ( 𝐅 q ) = q dim Bun G ( X ) tr ( ϕ - 1 | H * ( Bun G ( X ) ; l ) ) \#\operatorname{Bun}_{G}(X)(\mathbf{F}_{q})=q^{\dim\operatorname{Bun}_{G}(X)}% \operatorname{tr}(\phi^{-1}|H^{*}(\operatorname{Bun}_{G}(X);\mathbb{Z}_{l}))
  21. l \mathbb{Z}_{l}
  22. \mathbb{C}
  23. ϕ \phi
  24. # Bun G ( X ) ( 𝐅 q ) = P 1 # Aut ( P ) \#\operatorname{Bun}_{G}(X)(\mathbf{F}_{q})=\sum_{P}{1\over\#\operatorname{Aut% }(P)}
  25. tr ( ϕ - 1 | V * ) = i = 0 ( - 1 ) i tr ( ϕ - 1 | V i ) \operatorname{tr}(\phi^{-1}|V_{*})=\sum_{i=0}^{\infty}(-1)^{i}\operatorname{tr% }(\phi^{-1}|V_{i})
  26. V * V_{*}

Modulus_of_smoothness.html

  1. n n
  2. f C [ a , b ] f\in C[a,b]
  3. ω n : [ 0 , ) \omega_{n}:[0,\infty)\to\mathbb{R}
  4. ω n ( t , f , [ a , b ] ) = sup h [ 0 , t ] sup x [ a , b - n h ] | Δ h n ( f , x ) | for 0 t b - a n , \omega_{n}(t,f,[a,b])=\sup_{h\in[0,t]}\sup_{x\in[a,b-nh]}|\Delta_{h}^{n}(f,x)|% \,\text{ for }0\leq t\leq\frac{b-a}{n},
  5. ω n ( t , f , [ a , b ] ) = ω n ( b - a n , f , [ a , b ] ) , for t > b - a n , \omega_{n}(t,f,[a,b])=\omega_{n}\left(\frac{b-a}{n},f,[a,b]\right),\,\text{ % for }t>\frac{b-a}{n},
  6. Δ h n ( f , x 0 ) = i = 1 n ( - 1 ) n - i ( n i ) f ( x 0 + i h ) . \Delta_{h}^{n}(f,x_{0})=\sum_{i=1}^{n}(-1)^{n-i}{\left({{n}\atop{i}}\right)}f(% x_{0}+ih).
  7. ω n ( 0 ) = 0 , \omega_{n}(0)=0,
  8. ω n ( 0 + ) = 0. \omega_{n}(0+)=0.
  9. ω n \omega_{n}
  10. [ 0 , ) . [0,\infty).
  11. ω n \omega_{n}
  12. [ 0 , ) . [0,\infty).
  13. ω n ( m t ) m n ω n ( t ) \omega_{n}(mt)\leq m^{n}\omega_{n}(t)
  14. m m\in\mathbb{N}
  15. t 0. t\geq 0.
  16. ω n ( f , λ t ) ( λ + 1 ) n ω n ( f , λ t ) \omega_{n}(f,\lambda t)\leq(\lambda+1)^{n}\omega_{n}(f,\lambda t)
  17. λ > 0. \lambda>0.
  18. r N r\in N
  19. W r W^{r}
  20. [ - 1 , 1 ] [-1,1]
  21. ( r - 1 ) (r-1)
  22. [ - 1 , 1 ] [-1,1]
  23. f ( r ) L [ - 1 , 1 ] < + . \|f^{(r)}\|_{L_{\infty}[-1,1]}<+\infty.
  24. f W r f\in W^{r}
  25. ω r ( t , f , [ - 1 , 1 ] ) t r f ( r ) L [ - 1 , 1 ] , t 0 , \omega_{r}(t,f,[-1,1])\leq t^{r}\|f^{(r)}\|_{L_{\infty}[-1,1]},t\geq 0,
  26. g ( x ) L [ - 1 , 1 ] = ess sup x [ - 1 , 1 ] | g ( x ) | . \|g(x)\|_{L_{\infty}[-1,1]}={\mathrm{ess}\sup}_{x\in[-1,1]}|g(x)|.
  27. n n
  28. f f
  29. 2 π 2\pi
  30. T n T_{n}
  31. n \leq n
  32. | f ( x ) - T n ( x ) | c ( k ) ω k ( 1 n , f ) , x [ 0 , 2 π ] , |f(x)-T_{n}(x)|\leq c(k)\omega_{k}\left(\frac{1}{n},f\right),\quad x\in[0,2\pi],
  33. c ( k ) c(k)
  34. k . k\in\mathbb{N}.

Molar_attenuation_coefficient.html

  1. A = ε c , A=\varepsilon c\ell,
  2. A = i = 1 N A i = i = 1 N ε i c i . A=\sum_{i=1}^{N}A_{i}=\ell\sum_{i=1}^{N}\varepsilon_{i}c_{i}.
  3. { A ( λ 1 ) = i = 1 N ε i ( λ 1 ) c i , A ( λ N ) = i = 1 N ε i ( λ N ) c i . \begin{cases}A(\lambda_{1})=\ell\sum_{i=1}^{N}\varepsilon_{i}(\lambda_{1})c_{i% },\\ \ldots\\ A(\lambda_{N})=\ell\sum_{i=1}^{N}\varepsilon_{i}(\lambda_{N})c_{i}.\\ \end{cases}
  4. σ = ln ( 10 ) 10 3 N A ε = 3.823 , 532 , 16 × 10 - 21 ε . \sigma=\ln(10)\frac{10^{3}}{N_{A}}\varepsilon=3.823,532,16\times 10^{-21}\,\varepsilon.

Moment_measure.html

  1. 𝐑 d \textstyle\,\textbf{R}^{d}
  2. x \textstyle x
  3. N \textstyle{N}
  4. x N , \textstyle x\in{N},
  5. N \textstyle{N}
  6. B \textstyle B
  7. N ( B ) , \textstyle{N}(B),
  8. n = 1 , 2 , \textstyle n=1,2,\dots
  9. n \textstyle n
  10. N \textstyle{N}
  11. N n ( B 1 × × B n ) = i = 1 n N ( B i ) {N}^{n}(B_{1}\times\cdots\times B_{n})=\prod_{i=1}^{n}{N}(B_{i})
  12. B 1 , , B n \textstyle B_{1},...,B_{n}
  13. 𝐑 d \textstyle\,\textbf{R}^{d}
  14. n \textstyle n
  15. B 1 × , , × B n B_{1}\times,\dots,\times B_{n}
  16. Π \textstyle\Pi
  17. N ( B i ) \textstyle{N}(B_{i})
  18. N \textstyle{N}
  19. n \textstyle n
  20. N \textstyle{N}
  21. N n ( B 1 × × B n ) = ( x 1 , , x n ) N i = 1 n 𝟏 B i ( x i ) {N}^{n}(B_{1}\times\cdots\times B_{n})=\sum_{(x_{1},\dots,x_{n})\in{N}}\prod_{% i=1}^{n}\mathbf{1}_{B_{i}}(x_{i})
  22. n \textstyle n
  23. 𝟏 \textstyle\mathbf{1}
  24. 𝟏 B 1 \textstyle\mathbf{1}_{B_{1}}
  25. n \textstyle n
  26. M n ( B 1 × × B n ) = E [ N n ( B 1 × × B n ) ] , M^{n}(B_{1}\times\ldots\times B_{n})=E[{N}^{n}(B_{1}\times\ldots\times B_{n})],
  27. N \textstyle{N}
  28. n \textstyle n\,
  29. N \textstyle{N}
  30. 𝐑 n d f ( x 1 , , x n ) M n ( d x 1 , , d x n ) = E [ ( x 1 , , x n ) N f ( x 1 , , x n ) ] , \int_{\,\textbf{R}^{nd}}f(x_{1},\dots,x_{n})M^{n}(dx_{1},\dots,dx_{n})=E\left[% \sum_{(x_{1},\dots,x_{n})\in{N}}f(x_{1},\dots,x_{n})\right],
  31. f \textstyle f
  32. 𝐑 n d \textstyle\,\textbf{R}^{nd}
  33. n \textstyle n
  34. M 1 ( B ) = E [ N ( B ) ] , M^{1}(B)=E[{N}(B)],
  35. M 1 \textstyle M^{1}
  36. N \textstyle{N}
  37. B \textstyle B
  38. A \textstyle A
  39. B \textstyle B
  40. M 2 ( A × B ) = E [ N ( A ) N ( B ) ] , M^{2}(A\times B)=E[{N}(A){N}(B)],
  41. B \textstyle B
  42. M 2 ( B ) = ( E [ N ( B ) ] ) 2 + Var [ N ( B ) ] , M^{2}(B)=(E[{N}(B)])^{2}+\,\text{Var}[{N}(B)],
  43. Var [ N ( B ) ] \textstyle\,\text{Var}[{N}(B)]
  44. N ( B ) \textstyle{N}(B)
  45. N \textstyle{N}
  46. A \textstyle A
  47. B \textstyle B
  48. Cov [ N ( A ) , N ( B ) ] = M 2 ( A × B ) - M 1 ( A ) M 1 ( B ) \,\text{Cov}[{N}(A),{N}(B)]=M^{2}(A\times B)-M^{1}(A)M^{1}(B)
  49. Λ \textstyle\Lambda
  50. M 1 ( B ) = Λ ( B ) , M^{1}(B)=\Lambda(B),
  51. λ \textstyle\lambda
  52. M 1 ( B ) = λ | B | , M^{1}(B)=\lambda|B|,
  53. | B | \textstyle|B|
  54. B \textstyle B
  55. Λ \textstyle\Lambda
  56. M 2 ( B ) = Λ ( B ) + Λ ( B ) 2 . M^{2}(B)=\Lambda(B)+\Lambda(B)^{2}.
  57. M 2 ( B ) = λ | B | + ( λ | B | ) 2 . M^{2}(B)=\lambda|B|+(\lambda|B|)^{2}.

Moment_of_inertia_factor.html

  1. C M R 2 \frac{C}{MR^{2}}

Monin–Obukhov_similarity_theory.html

  1. L L
  2. L L
  3. L = - u * 3 κ g T Q ρ c p L=-\dfrac{u_{*}^{3}}{\kappa\dfrac{g}{T}\dfrac{Q}{\rho c_{p}}}
  4. κ 0.40 \kappa\approx 0.40
  5. u * u_{*}
  6. Q Q
  7. c p c_{p}
  8. θ v \theta_{v}
  9. T T
  10. Q Q
  11. Q = ρ c p w θ v ¯ Q=\rho c_{p}\overline{w^{\prime}\theta_{v}^{\prime}}
  12. w w^{\prime}
  13. θ v \theta_{v}^{\prime}
  14. L = - u * 3 κ g θ v ¯ w θ v ¯ L=-\dfrac{u_{*}^{3}}{\kappa\dfrac{g}{\overline{\theta_{v}}}\overline{w^{\prime% }\theta_{v}^{\prime}}}
  15. L < 0 L<0
  16. L > 0 L>0
  17. L L
  18. | L | |L|
  19. | L | |L|
  20. L L\rightarrow\infty
  21. L L
  22. z z
  23. ζ = z / L \zeta=z/L
  24. { u * , g / θ v ¯ , u ¯ / z , z , w θ v ¯ } \{u_{*},g/\overline{\theta_{v}},\partial\overline{u}/\partial z,z,\overline{w^% {\prime}\theta_{v}^{\prime}}\}
  25. κ z u * u ¯ z \dfrac{\kappa z}{u_{*}}\dfrac{\partial\overline{u}}{\partial z}
  26. ζ = z L \zeta=\dfrac{z}{L}
  27. φ M ( ζ ) \varphi_{M}(\zeta)
  28. φ H ( ζ ) \varphi_{H}(\zeta)
  29. u ¯ z = u * κ z φ M ( ζ ) \dfrac{\partial\overline{u}}{\partial z}=\dfrac{u_{*}}{\kappa z}\varphi_{M}(\zeta)
  30. θ v ¯ z = θ * κ z φ H ( ζ ) \dfrac{\partial\overline{\theta_{v}}}{\partial z}=\dfrac{\theta_{*}}{\kappa z}% \varphi_{H}(\zeta)
  31. θ * = - w θ v ¯ u * \theta_{*}=-\dfrac{\overline{w^{\prime}\theta_{v}^{\prime}}}{u_{*}}
  32. φ M \varphi_{M}
  33. φ H \varphi_{H}
  34. K M = κ z u * φ M ( ζ ) , K H = κ z u * φ H ( ζ ) K_{M}=\kappa z\dfrac{u_{*}}{\varphi_{M}(\zeta)},\ K_{H}=\kappa z\dfrac{u_{*}}{% \varphi_{H}(\zeta)}
  35. K M K_{M}
  36. K H K_{H}
  37. P r t Pr_{t}
  38. K H K M = 1 P r t > 1 \dfrac{K_{H}}{K_{M}}=\dfrac{1}{Pr_{t}}>1
  39. L ( R i z ) z = 0 = 1 L\Big(\dfrac{\partial Ri}{\partial z}\Big)_{z=0}=1
  40. φ ( 0 ) = 1 \varphi(0)=1
  41. φ M ( ζ ) = 1 + β ζ \varphi_{M}(\zeta)=1+\beta\zeta
  42. β 6 \beta\approx 6
  43. 0 < ζ < 1 0<\zeta<1
  44. ζ < 0 \zeta<0
  45. φ M 4 - γ ζ φ M 3 = 1 \varphi_{M}^{4}-\gamma\zeta\varphi_{M}^{3}=1
  46. γ \gamma
  47. φ M = ( 1 + γ ζ ) - 1 / 4 \varphi_{M}=(1+\gamma\zeta)^{-1/4}
  48. - 2 < ζ < 0 -2<\zeta<0
  49. φ M ( ζ ) = ( 1 - 15 ζ ) - 1 / 4 - 2 < ζ < 0 \varphi_{M}(\zeta)=(1-15\zeta)^{-1/4}\quad-2<\zeta<0
  50. φ M ( ζ ) = 1 + 4.7 ζ 0 < ζ < 1 \varphi_{M}(\zeta)=1+4.7\zeta\quad 0<\zeta<1
  51. φ H ( ζ ) = 0.74 ( 1 - 9 ζ ) - 1 / 2 - 2 < ζ < 0 \varphi_{H}(\zeta)=0.74(1-9\zeta)^{-1/2}\quad-2<\zeta<0
  52. φ H ( ζ ) = 0.74 + 4.7 ζ 0 < ζ < 1 \varphi_{H}(\zeta)=0.74+4.7\zeta\quad 0<\zeta<1
  53. ζ \zeta
  54. R i Ri
  55. g Q T \dfrac{gQ}{T}
  56. ϵ 0 \epsilon_{0}

Monoidal-category_action.html

  1. : S × X X \cdot:S\times X\to X
  2. s ( t x ) ( s t ) x s\cdot(t\cdot x)\simeq(s\cdot t)\cdot x
  3. e x x e\cdot x\simeq x

Monopoly_price.html

  1. M R = P + P ( Q ) * Q MR=P+P^{\prime}(Q)*Q
  2. 0 > P ( Q ) 0>P^{\prime}(Q)
  3. M C = C ( Q ) MC=C^{\prime}(Q)
  4. 0 < C ( Q ) 0<C^{\prime}(Q)
  5. π = P ( Q ) * Q - C ( Q ) \pi=P(Q)*Q-C(Q)
  6. π \pi
  7. π \pi
  8. P ( Q ) Q + P - C ( Q ) = 0 P^{\prime}(Q)Q+P-C^{\prime}(Q)=0
  9. P ( Q ) * Q + P = C ( Q ) P^{\prime}(Q)*Q+P=C^{\prime}(Q)
  10. P ( P ( Q / P ) + 1 ) = M C P(P^{\prime}(Q/P)+1)=MC
  11. P ( Q / P ) P^{\prime}(Q/P)
  12. 1 / ϵ 1/\epsilon
  13. P ( 1 + 1 / ϵ ) = M C P(1+1/{\epsilon})=MC
  14. P = ϵ ϵ + 1 M C P=\frac{\epsilon}{\epsilon+1}MC
  15. η \eta
  16. P = 1 1 + η M C P=\frac{1}{1+\eta}MC
  17. η < 0 \eta<0
  18. η = 0 \eta=0
  19. 1 / η < 1 1/{\eta}<1

Monotone_comparative_statics.html

  1. X X\subseteq\mathbb{R}
  2. f ( ; s ) : X f(\cdot;s):X\rightarrow\mathbb{R}
  3. s S s\in S
  4. ( S , S ) (S,\geq_{S})
  5. arg max x X f ( x ; s ) \arg\max\limits_{x\in X}f(x;s)
  6. s s
  7. X X
  8. f ( ; s ) f(\cdot;s)
  9. x x
  10. x ¯ ( s ) \bar{x}(s)
  11. f ( ; s ) f(\cdot;s)
  12. f ( x ¯ ( s ) ; s ) 0 f^{\prime}(\bar{x}(s);s^{\prime})\geq 0
  13. s > s s^{\prime}>s
  14. x ¯ ( s ) \bar{x}(s)
  15. s s
  16. x ¯ ( s ) x ¯ ( s ) \bar{x}(s^{\prime})\geq\bar{x}(s)
  17. f ( ; s ) f(\cdot;s)
  18. arg max x X f ( x ; s ) \arg\max_{x\in X}f(x;s)
  19. s s
  20. Y Y
  21. Y Y^{\prime}
  22. \mathbb{R}
  23. Y Y^{\prime}
  24. Y Y
  25. Y S S O Y Y^{\prime}\geq_{SSO}Y
  26. x x^{\prime}
  27. Y Y^{\prime}
  28. x x
  29. Y Y
  30. max { x , x } \max\{x^{\prime},x\}
  31. Y Y^{\prime}
  32. min { x , x } \min\{x^{\prime},x\}
  33. Y Y
  34. Y := { x } Y:=\{x\}
  35. Y := { x } Y^{\prime}:=\{x^{\prime}\}
  36. Y S S O Y Y^{\prime}\geq_{SSO}Y
  37. x x x^{\prime}\geq x
  38. arg max x X f ( x ; s ) \arg\max_{x\in X}f(x;s)
  39. arg max x X f ( x ; s ) S S O arg max x X f ( x ; s ) \arg\max_{x\in X}f(x;s^{\prime})\geq_{SSO}\arg\max_{x\in X}f(x;s)
  40. s > S s s^{\prime}>_{S}s
  41. ϕ : S \phi:S\rightarrow\mathbb{R}
  42. ϕ \phi
  43. s S s s^{\prime}\geq_{S}s
  44. ϕ ( s ) ( > ) 0 ϕ ( s ) ( > ) 0 \phi(s)\geq(>)\ 0\ \Rightarrow\ \phi(s^{\prime})\geq(>)\ 0
  45. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  46. f : X × S f:X\times S\to\mathbb{R}
  47. x x x^{\prime}\geq x
  48. Δ ( s ) = f ( x ; s ) - f ( x ; s ) \Delta(s)=f(x^{\prime};s)-f(x;s)
  49. Δ ( s ) \Delta(s)
  50. s s
  51. x > x x^{\prime}>x
  52. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  53. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  54. { g ( ; s ) } s S \{g(\cdot;s)\}_{s\in S}
  55. g ( x ; s ) = H ( f ( x ; s ) ; s ) g(x;s)=H(f(x;s);s)
  56. H ( ; s ) H(\cdot;s)
  57. x x
  58. F Y ( s ) := arg max x Y f ( x ; s ) F_{Y}(s):=\arg\max_{x\in Y}f(x;s)
  59. f ( ; s ) } s S f(\cdot;s)\}_{s\in S}
  60. Y X Y\subseteq X
  61. F Y ( s ) S S O F Y ( s ) F_{Y}(s^{\prime})\geq_{SSO}F_{Y}(s)
  62. s S s s^{\prime}\geq_{S}s
  63. s S s s^{\prime}\geq_{S}s
  64. x F Y ( s ) x\in F_{Y}(s)
  65. x F Y ( s ) x^{\prime}\in F_{Y}(s^{\prime})
  66. max { x , x } F Y ( s ) \max\{x^{\prime},x\}\in F_{Y}(s^{\prime})
  67. min { x , x } F Y ( s ) \min\{x^{\prime},x\}\in F_{Y}(s)
  68. x > x x>x^{\prime}
  69. x F Y ( s ) x\in F_{Y}(s)
  70. f ( x ; s ) f ( x ; s ) f(x;s)\geq f(x^{\prime};s)
  71. x F Y ( s ) x\in F_{Y^{\prime}}(s^{\prime})
  72. f ( x ; s ) = f ( x ; s ) f(x;s)=f(x^{\prime};s)
  73. x F Y ( s ) x^{\prime}\in F_{Y}(s)
  74. f ( x ; s ) > f ( x ; s ) f(x;s)>f(x^{\prime};s)
  75. f ( x ; s ) > f ( x ; s ) f(x;s^{\prime})>f(x^{\prime};s^{\prime})
  76. x x^{\prime}
  77. s s^{\prime}
  78. Y := { x , x ¯ } Y:=\{x,\bar{x}\}
  79. x ¯ x \bar{x}\geq x
  80. F Y ( s ) S S O F Y ( s ) F_{Y}(s^{\prime})\geq_{SSO}F_{Y}(s)
  81. s S s s^{\prime}\geq_{S}s
  82. f ( x ¯ ; s ) ( > ) f ( x ; s ) f(\bar{x};s)\geq(>)\ f(x;s)
  83. f ( x ¯ ; s ) ( > ) f ( x ; s ) f(\bar{x};s^{\prime})\geq(>)\ f(x;s^{\prime})
  84. x X + x\in X\subseteq\mathbb{R}_{+}
  85. Π ( x ; - c ) = x P ( x ) - c x \Pi(x;-c)=xP(x)-cx
  86. P : + + P:\mathbb{R}_{+}\to\mathbb{R}_{+}
  87. c 0 c\geq 0
  88. { Π ( , - c ) } ( - c ) - \{\Pi(\cdot,-c)\}_{(-c)\in\mathbb{R}_{-}}
  89. x x x^{\prime}\geq x
  90. x P ( x ) - c x ( > ) x P ( x ) - c x x^{\prime}P(x^{\prime})-cx^{\prime}\geq(>)\ xP(x)-cx
  91. c c^{\prime}
  92. ( - c ) ( - c ) (-c^{\prime})\geq(-c)
  93. x P ( x ) - c x ( > ) x P ( x ) - c x x^{\prime}P(x^{\prime})-c^{\prime}x^{\prime}\geq(>)\ xP(x)-c^{\prime}x
  94. ( - c ) (-c)
  95. arg max x Y f ( x ; s ) \arg\max_{x\in Y}f(x;s)
  96. s s
  97. Y X Y\subset X
  98. X X
  99. X X\subseteq\mathbb{R}
  100. Y X Y\subseteq X
  101. X X
  102. x * x^{*}
  103. x * * x^{**}
  104. Y Y
  105. x X x\in X
  106. x * x x * * x^{*}\leq x\leq x^{**}
  107. Y Y
  108. X = X=\mathbb{N}
  109. { 1 , 2 , 3 , 4 } \{1,2,3,4\}
  110. X X
  111. { 1 , 2 , 4 } \{1,2,4\}
  112. [ x * , x * * ] = { x X | x * x x * * } [x^{*},x^{**}]=\{x\in X\ |\ x^{*}\leq x\leq x^{**}\}
  113. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  114. x ′′ > x x^{\prime\prime}>x^{\prime}
  115. s S s s^{\prime}\geq_{S}s
  116. f ( x ′′ ; s ) f ( x ; s ) f(x^{\prime\prime};s)\geq f(x;s)
  117. x [ x , x ′′ ] x\in[x^{\prime},x^{\prime\prime}]
  118. f ( x ′′ ; s ) ( > ) f ( x ; s ) f ( x ′′ ; s ) ( > ) f ( x ; s ) f(x^{\prime\prime};s)\geq(>)\ f(x^{\prime};s)\ \Rightarrow\ f(x^{\prime\prime}% ;s^{\prime})\geq(>)\ f(x^{\prime};s^{\prime})
  119. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  120. arg max x X f ( x , s ) \arg\max_{x\in X}f(x,s)
  121. s s
  122. f : X × S f:X\times S\to\mathbb{R}
  123. arg max x [ x * , x * * ] f ( x ; s ) \arg\max_{x\in[x^{*},x^{**}]}f(x;s)
  124. x * * x * x^{**}\geq x^{*}
  125. [ x * , x * * ] [x^{*},x^{**}]
  126. { x X | x * x x * * } \{x\in X\ |\ x^{*}\leq x\leq x^{**}\}
  127. F Y ( s ) := arg max x Y f ( x ; s ) F_{Y}(s):=\arg\max_{x\in Y}f(x;s)
  128. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  129. F Y ( s ) F_{Y}(s)
  130. s s
  131. Y X Y\subseteq X
  132. s S s s^{\prime}\geq_{S}s
  133. x F Y ( s ) x^{\prime}\in F_{Y}(s)
  134. x ′′ F Y ( s ) x^{\prime\prime}\in F_{Y}(s^{\prime})
  135. x > x ′′ x^{\prime}>x^{\prime\prime}
  136. f ( x ; s ) f ( x ; s ) f(x^{\prime};s)\geq f(x;s)
  137. x [ x ′′ , x ] Y x\in[x^{\prime\prime},x^{\prime}]\subset Y
  138. f ( x ; s ) f ( x ′′ ; s ) f(x^{\prime};s^{\prime})\geq f(x^{\prime\prime};s^{\prime})
  139. x F Y ( s ) x^{\prime}\in F_{Y}(s^{\prime})
  140. f ( x ; s ) = f ( x ′′ ; s ) f(x^{\prime};s)=f(x^{\prime\prime};s)
  141. f ( x ; s ) > f ( x ′′ ; s ) f(x^{\prime};s)>f(x^{\prime\prime};s)
  142. f ( x ; s ) > f ( x ′′ ; s ) f(x^{\prime};s^{\prime})>f(x^{\prime\prime};s^{\prime})
  143. x ′′ F Y ( s ) x^{\prime\prime}\in F_{Y}(s^{\prime})
  144. [ x ′′ , x ] [x^{\prime\prime},x^{\prime}]
  145. f ( x ; s ) f ( x ; s ) f(x^{\prime};s)\geq f(x;s)
  146. x [ x ′′ , x ] x\in[x^{\prime\prime},x^{\prime}]
  147. x arg max x [ x ′′ , x ] f ( x ; s ) x^{\prime}\in\arg\max_{x\in[x^{\prime\prime},x^{\prime}]}f(x;s)
  148. f ( x ′′ ; s ) > f ( x ; s ) f(x^{\prime\prime};s^{\prime})>f(x^{\prime};s^{\prime})
  149. f ( ; s ) f(\cdot;s^{\prime})
  150. arg max x [ x ′′ , x ] f ( x ; s ) \arg\max_{x\in[x^{\prime\prime},x^{\prime}]}f(x;s^{\prime})
  151. x x^{\prime}
  152. arg max x [ x ′′ , x ] f ( x ; s ) \arg\max_{x\in[x^{\prime\prime},x^{\prime}]}f(x;s)
  153. s s
  154. f ( x ′′ ; s ) = f ( x ; s ) f(x^{\prime\prime};s^{\prime})=f(x^{\prime};s^{\prime})
  155. f ( x ′′ ; s ) < f ( x ; s ) f(x^{\prime\prime};s)<f(x^{\prime};s)
  156. arg max x [ x ′′ , x ] f ( x ; s ) \arg\max_{x\in[x^{\prime\prime},x^{\prime}]}f(x;s^{\prime})
  157. x ′′ x^{\prime\prime}
  158. arg max x [ x ′′ , x ] f ( x ; s ) \arg\max_{x\in[x^{\prime\prime},x^{\prime}]}f(x;s)
  159. s s
  160. x ′′ arg max x [ x ′′ , x ] f ( x ; s ) x^{\prime\prime}\not\in\arg\max_{x\in[x^{\prime\prime},x^{\prime}]}f(x;s^{% \prime})
  161. x x^{\prime}
  162. arg max x [ x ′′ , x ] f ( x ; s ) \arg\max_{x\in[x^{\prime\prime},x^{\prime}]}f(x;s)
  163. X X
  164. \mathbb{R}
  165. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  166. s S s s^{\prime}\geq_{S}s
  167. α > 0 \alpha>0
  168. f ( x ; s ) α f ( x ; s ) f^{\prime}(x;s^{\prime})\geq\alpha f^{\prime}(x;s)
  169. x X x\in X
  170. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  171. s S s s^{\prime}\geq_{S}s
  172. α : X \alpha:X\rightarrow\mathbb{R}
  173. f ( x ; s ) α ( x ) f ( x ; s ) f^{\prime}(x;s^{\prime})\geq\alpha(x)f^{\prime}(x;s)
  174. x X x\in X
  175. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  176. π ( t ) \pi(t)
  177. x x
  178. V ( x ; - r ) = 0 x e - r t π ( t ) d t , V(x;-r)=\int_{0}^{x}e^{-rt}\pi(t)dt,
  179. r > 0 r>0
  180. V ( x ; - r ) = e - r x π ( x ) V^{\prime}(x;-r)=e^{-rx}\pi(x)
  181. V V
  182. r r
  183. r > r > 0 r^{\prime}>r>0
  184. arg max x 0 V ( x ; - r ) S S O arg max x 0 V ( x ; - r ) \arg\max_{x\geq 0}V(x;-r)\geq_{SSO}\arg\max_{x\geq 0}V(x;-r^{\prime})
  185. r < r r^{\prime}<r
  186. V ( x ; - r ) = e - r x π ( x ) = e ( r - r ) x V ( x ; - r ) . V^{\prime}(x;-r)=e^{-rx}\pi(x)=e^{(r^{\prime}-r)x}V^{\prime}(x;-r^{\prime}).
  187. α ( x ) = e ( r - r ) x \alpha(x)=e^{(r^{\prime}-r)x}
  188. { V ( ; - r ) } ( - r ) < 0 \{V(\cdot;-r)\}_{(-r)<0}
  189. ( X , X ) (X,\geq_{X})
  190. x x
  191. x x^{\prime}
  192. X X
  193. x x x^{\prime}\vee x
  194. x x x^{\prime}\wedge x
  195. ( X , X ) (X,\geq_{X})
  196. Y Y
  197. Y Y^{\prime}
  198. X X
  199. Y Y^{\prime}
  200. Y Y
  201. Y S S O Y Y^{\prime}\geq_{SSO}Y
  202. x x^{\prime}
  203. Y Y^{\prime}
  204. x x
  205. Y Y
  206. x x x\vee x^{\prime}
  207. Y Y^{\prime}
  208. x x x\wedge x^{\prime}
  209. Y Y
  210. X = X=\mathbb{R}
  211. Y := [ a , b ] Y:=[a,b]
  212. Y := [ a , b ] Y^{\prime}:=[a^{\prime},b^{\prime}]
  213. X X
  214. ( X , ) (X,\geq)
  215. \geq
  216. \mathbb{R}
  217. Y S S O Y Y^{\prime}\geq_{SSO}Y
  218. a a a^{\prime}\geq a
  219. b b b^{\prime}\geq b
  220. X = n X=\mathbb{R}^{n}
  221. Y Y
  222. Y X Y^{\prime}\subset X
  223. a a
  224. b b
  225. a a^{\prime}
  226. b b^{\prime}
  227. X X
  228. Y := { x X | a x b } Y:=\{x\in X\ |\ a\leq x\leq b\}
  229. Y := { x X | a x b } Y^{\prime}:=\{x\in X\ |\ a^{\prime}\leq x\leq b^{\prime}\}
  230. \geq
  231. n \mathbb{R}^{n}
  232. ( X , ) (X,\geq)
  233. Y S S O Y Y^{\prime}\geq_{SSO}Y
  234. a a a^{\prime}\geq a
  235. b b b^{\prime}\geq b
  236. ( X , X ) (X,\geq_{X})
  237. \mathbb{R}
  238. X \geq_{X}
  239. ( X , X ) (X,\geq_{X})
  240. Y := Δ ( [ a , b ] ) Y:=\Delta([a,b])
  241. Y := Δ ( [ a , b ] ) Y^{\prime}:=\Delta([a^{\prime},b^{\prime}])
  242. [ a , b ] [a,b]
  243. [ a , b ] [a^{\prime},b^{\prime}]
  244. Y S S O Y Y^{\prime}\geq_{SSO}Y
  245. X \geq_{X}
  246. a a a^{\prime}\geq a
  247. b b b^{\prime}\geq b
  248. ( X , X ) (X,\geq_{X})
  249. f : X f:X\to\mathbb{R}
  250. f ( x ) ( > ) f ( x x ) f ( x x ) ( > ) f ( x ) . f(x)\geq(>)\ f(x\wedge x^{\prime})\ \Rightarrow\ f(x\vee x^{\prime})\geq(>)\ f% (x^{\prime}).
  251. f f
  252. f ( x x ) - f ( x ) f ( x ) - f ( x x ) . f(x\vee x^{\prime})-f(x^{\prime})\geq f(x)-f(x\wedge x^{\prime}).
  253. f f
  254. g := H f g:=H\circ f
  255. H H
  256. ( X , X ) (X,\geq_{X})
  257. ( S , S ) (S,\geq_{S})
  258. Y Y
  259. Y Y^{\prime}
  260. X X
  261. f : X × S f:X\times S\to\mathbb{R}
  262. arg max x Y f ( x ; s ) \arg\max_{x\in Y}f(x;s)
  263. F Y ( s ) F_{Y}(s)
  264. F Y ( s ) S S O F Y ( s ) F_{Y^{\prime}}(s^{\prime})\geq_{SSO}F_{Y}(s)
  265. s S s s^{\prime}\geq_{S}s
  266. Y S S O Y Y^{\prime}\geq_{SSO}Y
  267. ( ) (\Leftarrow)
  268. Y S S O Y Y^{\prime}\geq_{SSO}Y
  269. s S s s^{\prime}\geq_{S}s
  270. x F Y ( s ) x^{\prime}\in F_{Y^{\prime}}(s^{\prime})
  271. x F Y ( s ) x\in F_{Y}(s)
  272. x F Y ( s ) x\in F_{Y}(s)
  273. Y S S O Y Y^{\prime}\geq_{SSO}Y
  274. f ( x ; s ) f ( x x ; s ) f(x;s)\geq f(x^{\prime}\wedge x;s)
  275. f ( x x ; s ) f ( x ; s ) f(x^{\prime}\vee x;s)\geq f(x^{\prime};s)
  276. f ( x x ; s ) f ( x ; s ) f(x^{\prime}\vee x;s^{\prime})\geq f(x^{\prime};s^{\prime})
  277. x x F Y ( s ) x^{\prime}\vee x\in F_{Y^{\prime}}(s^{\prime})
  278. x x F Y ( s ) x^{\prime}\wedge x\not\in F_{Y}(s)
  279. f ( x ; s ) > f ( x x ; s ) f(x;s)>f(x^{\prime}\wedge x;s)
  280. f ( x x ; s ) > f ( x ; s ) f(x^{\prime}\vee x;s)>f(x^{\prime};s)
  281. f ( x x ; s ) > f ( x ; s ) f(x^{\prime}\vee x;s^{\prime})>f(x^{\prime};s^{\prime})
  282. x F Y ( s ) x^{\prime}\in F_{Y^{\prime}}(s^{\prime})
  283. x x F Y ( s ) x^{\prime}\wedge x\in F_{Y}(s)
  284. ( ) (\Rightarrow)
  285. Y := { x , x x } Y^{\prime}:=\{x^{\prime},x^{\prime}\vee x\}
  286. Y := { x , x x } Y:=\{x,x^{\prime}\wedge x\}
  287. Y S S O Y Y^{\prime}\geq_{SSO}Y
  288. F Y ( s ) S S O F Y ( s ) F_{Y^{\prime}}(s)\geq_{SSO}F_{Y}(s)
  289. f ( x ; s ) ( > ) f ( x x ; s ) f(x;s)\geq(>)\ f(x^{\prime}\wedge x;s)
  290. f ( x x ; s ) ( > ) f ( x ; s ) f(x^{\prime}\vee x;s)\geq(>)\ f(x^{\prime};s)
  291. Y := { x , x ¯ } Y:=\{x,\bar{x}\}
  292. x ¯ x \bar{x}\geq x
  293. F Y ( s ) S S O F Y ( s ) F_{Y}(s^{\prime})\geq_{SSO}F_{Y}(s)
  294. s S s s^{\prime}\geq_{S}s
  295. f ( x ¯ ; s ) ( > ) f ( x ; s ) f(\bar{x};s)\geq(>)\ f(x;s)
  296. f ( x ¯ ; s ) ( > ) f ( x ; s ) f(\bar{x};s^{\prime})\geq(>)\ f(x;s^{\prime})
  297. x x
  298. X X
  299. + l \mathbb{R}^{l}_{+}
  300. p + + l p\in\mathbb{R}^{l}_{++}
  301. V V
  302. x x
  303. \mathbb{R}
  304. Π ( x ; p ) = V ( x ) - p x \Pi(x;p)=V(x)-p\cdot x
  305. x x^{\prime}
  306. x X x\in X
  307. x x x^{\prime}\geq x
  308. V ( x ) - V ( x ) + ( - p ) ( x - x ) V(x^{\prime})-V(x)+(-p)(x^{\prime}-x)
  309. ( - p ) (-p)
  310. { Π ( ; p ) } p + + l \{\Pi(\cdot;p)\}_{p\in\mathbb{R}_{++}^{l}}
  311. V V
  312. Π ( ; p ) \Pi(\cdot;p)
  313. arg max x X Π ( x ; p ) S S O arg max x X Π ( x ; p ) \arg\max_{x\in X}\Pi(x;p)\geq_{SSO}\arg\max_{x\in X}\Pi(x;p^{\prime})
  314. p p p^{\prime}\geq p
  315. u : X u:X\to\mathbb{R}
  316. p p
  317. + + n \mathbb{R}_{++}^{n}
  318. w > 0 w>0
  319. B ( p , w ) = { x X | p x w } B(p,w)=\{x\in X\ |\ p\cdot x\leq w\}
  320. ( p , w ) (p,w)
  321. D ( p , w ) = arg max x B ( p , w ) u ( x ) D(p,w)=\arg\max_{x\in B(p,w)}u(x)
  322. B ( p , w ) S S O B ( p , w ) B(p,w^{\prime})\not\geq_{SSO}B(p,w)
  323. w > w w^{\prime}>w
  324. S S O \geq_{SSO}
  325. u : + + n u:\mathbb{R}_{++}^{n}\rightarrow\mathbb{R}
  326. w ′′ > w w^{\prime\prime}>w^{\prime}
  327. x ′′ D ( p , w ′′ ) x^{\prime\prime}\in D(p,w^{\prime\prime})
  328. x D ( p , w ) x^{\prime}\in D(p,w)
  329. z ′′ D ( p , w ′′ ) z^{\prime\prime}\in D(p,w^{\prime\prime})
  330. z D ( p , w ) z^{\prime}\in D(p,w^{\prime})
  331. z ′′ x z^{\prime\prime}\geq x^{\prime}
  332. x ′′ z x^{\prime\prime}\geq z^{\prime}
  333. u u
  334. x x
  335. y y
  336. u ( x y ) - u ( y ) u ( x ) - u ( x y ) u(x\wedge y)-u(y)\geq u(x)-u(x\vee y)
  337. x x
  338. y y
  339. x y x\wedge y
  340. x y x\vee y
  341. x y - x = y - x y x\wedge y-x=y-x\vee y
  342. x - x y = x y - y x-x\vee y=x\wedge y-y
  343. x y - x x\wedge y-x
  344. x - x y x-x\vee y
  345. u ( x y - λ v ) - u ( y ) u ( x ) - u ( x y + λ v ) . u(x\vee y-\lambda v)-u(y)\geq u(x)-u(x\wedge y+\lambda v).
  346. λ [ 0 , 1 ] \lambda\in[0,1]
  347. v = y - x y = x y - x v=y-x\wedge y=x\vee y-x
  348. x x
  349. y y
  350. x y - λ v x\vee y-\lambda v
  351. x y + λ v x\wedge y+\lambda v
  352. X X\subset\mathbb{R}
  353. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  354. X X
  355. arg max x X f ( x , ; s ) \arg\max_{x\in X}f(x,;s)
  356. s s
  357. s s
  358. x x
  359. s s
  360. { λ ( ; t ) } t T \{\lambda(\cdot;t)\}_{t\in T}
  361. t t
  362. ( T , T ) (T,\geq_{T})
  363. t t
  364. F ( x ; t ) = S f ( x ; s ) λ ( s ; t ) d s . F(x;t)=\int_{S}f(x;s)\,\lambda(s;t)\,ds.
  365. arg max x X F ( x ; t ) \arg\max_{x\in X}F(x;t)
  366. t t
  367. { F ( ; t ) } t T \{F(\cdot;t)\}_{t\in T}
  368. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  369. ( S ) (S\subseteq\mathbb{R})
  370. { λ ( ; t ) } t T \{\lambda(\cdot;t)\}_{t\in T}
  371. { F ( ; t ) } t T \{F(\cdot;t)\}_{t\in T}
  372. x , x X x^{\prime},x\in X
  373. ϕ ( s ) := f ( x ; s ) - f ( x ; s ) \phi(s):=f(x^{\prime};s)-f(x;s)
  374. F ( x ; t ) - F ( x ; t ) = S [ f ( x ; s ) - f ( x ; s ) ] λ ( s ; t ) d s F(x^{\prime};t)-F(x;t)=\int_{S}[f(x^{\prime};s)-f(x;s)]\lambda(s;t)ds
  375. F ( x ; t ) - F ( x ; t ) = S ϕ ( s ) λ ( s ; t ) d s F(x^{\prime};t)-F(x;t)=\int_{S}\phi(s)\lambda(s;t)ds
  376. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  377. ϕ \phi
  378. s s
  379. F ( x ; t ) - F ( x ; t ) F(x^{\prime};t)-F(x;t)
  380. t t
  381. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  382. S S\subseteq\mathbb{R}
  383. { F ( ; t ) } t T \{F(\cdot;t)\}_{t\in T}
  384. { λ ( ; t ) } t T \{\lambda(\cdot;t)\}_{t\in T}
  385. λ ( ; t ) \lambda(\cdot;t^{\prime})
  386. λ ( ; t ) \lambda(\cdot;t)
  387. S := { 1 , 2 , , N } S:=\{1,2,\ldots,N\}
  388. λ ( ; t ′′ ) \lambda(\cdot;t^{\prime\prime})
  389. λ ( ; t ) \lambda(\cdot;t^{\prime})
  390. { f ( ; s ) } s S \{f(\cdot;s)\}_{s\in S}
  391. X X\subset\mathbb{R}
  392. arg max x X F ( x ; t ′′ ) < arg max x X F ( x ; t ) \arg\max_{x\in X}F(x;t^{\prime\prime})<\arg\max_{x\in X}F(x;t^{\prime})
  393. F ( x ; t ) = s S λ ( s , t ) f ( x , s ) F(x;t)=\sum_{s\in S}\lambda(s,t)f(x,s)
  394. t = t , t ′′ t=t^{\prime},\,t^{\prime\prime}
  395. u : + u:\mathbb{R}_{+}\to\mathbb{R}
  396. w > 0 w>0
  397. R 0 R\geq 0
  398. s s
  399. λ ( s ; t ) \lambda(s;t)
  400. x x
  401. s s
  402. ( w - x ) R + x s (w-x)R+xs
  403. x x
  404. V ( x ; t ) := S u ( ( w - x ) R + x s ) λ ( s ; t ) d a . V(x;t):=\int_{S}u((w-x)R+xs)\lambda(s;t)\,da.
  405. { u ^ ( ; s ) } s S \{\hat{u}(\cdot;s)\}_{s\in S}
  406. u ^ ( x ; s ) := u ( w R + x ( s - R ) ) \hat{u}(x;s):=u(wR+x(s-R))
  407. { V ( ; t ) } t T \{V(\cdot;t)\}_{t\in T}
  408. arg max x 0 V ( x ; t ) \arg\max_{x\geq 0}V(x;t)
  409. t t
  410. λ ( ; t ) } t T \lambda(\cdot;t)\}_{t\in T}
  411. ( S , S ) (S,\geq_{S})
  412. f , g : S f,g:S\to\mathbb{R}
  413. s s s^{\prime}\geq s
  414. f ( s ) > 0 f(s)>0
  415. g ( s ) < 0 g(s)<0
  416. - g ( s ) f ( s ) - g ( s ) f ( s ) ; -\frac{g(s)}{f(s)}\geq-\frac{g(s^{\prime})}{f(s^{\prime})};
  417. f ( s ) < 0 f(s)<0
  418. g ( s ) > 0 g(s)>0
  419. - f ( s ) g ( s ) - f ( s ) g ( s ) . -\frac{f(s)}{g(s)}\geq-\frac{f(s^{\prime})}{g(s^{\prime})}.
  420. f f
  421. g g
  422. α f + β g \alpha f+\beta g
  423. α \alpha
  424. β \beta
  425. f f
  426. g g
  427. f ( s ) > 0 f(s)>0
  428. g ( s ) < 0 g(s)<0
  429. α * = - g ( s ) / f ( s ) \alpha^{*}=-g(s)/f(s)
  430. α * f ( s ) + g ( s ) = 0 \alpha^{*}f(s)+g(s)=0
  431. α * f ( s ) + g ( s ) \alpha^{*}f(s)+g(s)
  432. α * f ( s ) + g ( s ) 0 \alpha^{*}f(s^{\prime})+g(s^{\prime})\geq 0
  433. s s s^{\prime}\geq s
  434. f f
  435. f ( s ) > 0 f(s^{\prime})>0
  436. α * = - g ( s ) f ( s ) - g ( s ) f ( s ) . \alpha^{*}=-\frac{g(s)}{f(s)}\geq-\frac{g(s^{\prime})}{f(s^{\prime})}.
  437. β = 1 \beta=1
  438. α f ( s ) + g ( s ) ( > ) 0. \alpha f(s)+g(s)\geq(>)0.
  439. f ( s ) 0 f(s)\geq 0
  440. g ( s ) 0 g(s)\geq 0
  441. f ( s ) 0 f(s^{\prime})\geq 0
  442. g ( s ) 0 g(s^{\prime})\geq 0
  443. α f ( s ) + g ( s ) ( > ) 0 \alpha f(s^{\prime})+g(s^{\prime})\geq(>)0
  444. g ( s ) < 0 g(s)<0
  445. f ( s ) > 0 f(s)>0
  446. f f
  447. g g
  448. α ( > ) - g ( s ) f ( s ) - g ( s ) f ( s ) . \alpha\geq(>)-\frac{g(s)}{f(s)}\geq-\frac{g(s^{\prime})}{f(s^{\prime})}.
  449. f f
  450. f ( s ) > 0 f(s^{\prime})>0
  451. α f ( s ) + g ( s ) ( > ) 0. \alpha f(s^{\prime})+g(s^{\prime})\geq(>)\ 0.
  452. ( T , 𝒯 , μ ) (T,\mathcal{T},\mu)
  453. s S s\in S
  454. f ( s ; t ) f(s;t)
  455. t T t\in T
  456. F ( s ) = T f ( s ; t ) d μ ( t ) F(s)=\int_{T}f(s;t)d\mu(t)
  457. t t
  458. t T t^{\prime}\in T
  459. f ( s ; t ) f(s;t)
  460. f ( s ; t ) f(s;t^{\prime})
  461. s S s\in S
  462. 𝒯 \mathcal{T}
  463. F F
  464. μ \mu
  465. x x
  466. t T t\in T\subset\mathbb{R}
  467. Π ( x ; - c , t ) = x P ( x ; t ) - c x \Pi(x;-c,t)=xP(x;t)-cx
  468. c c
  469. P ( x , t ) P(x,t)
  470. t t
  471. V ( x ; - c ) = T u ( Π ( x ; - c , t ) ) d λ ( t ) , V(x;-c)=\int_{T}u(\Pi(x;-c,t))d\lambda(t),
  472. λ \lambda
  473. t t
  474. u : u:\mathbb{R}\to\mathbb{R}
  475. arg max x 0 V ( x ; - c ) \arg\max_{x\geq 0}V(x;-c)
  476. - c -c
  477. { V ( x ; - c ) } c + \{V(x;-c)\}_{c\in\mathbb{R}_{+}}
  478. x x x^{\prime}\geq x
  479. Δ ( - c ) = T [ u ( Π ( x ; - c , t ) ) - u ( Π ( x ; - c , t ) ) ] d λ ( t ) , \Delta(-c)=\int_{T}[u(\Pi(x^{\prime};-c,t))-u(\Pi(x;-c,t))]\,d\lambda(t),
  480. t t
  481. δ ( - c , t ) = u ( Π ( x ; - c , t ) ) - u ( Π ( x ; - c , t ) ) \delta(-c,t)=u(\Pi(x^{\prime};-c,t))-u(\Pi(x;-c,t))
  482. - c -c
  483. u u
  484. δ \delta
  485. - c -c
  486. Δ \Delta
  487. t , t T t^{\prime},t\in T
  488. δ ( - c , t ) \delta(-c,t)
  489. δ ( - c , t ) \delta(-c,t^{\prime})
  490. - c -c
  491. P P
  492. x x
  493. t t
  494. { log ( P ( , t ) ) } t T \{\log(P(\cdot,t))\}_{t\in T}
  495. u : u:\mathbb{R}\to\mathbb{R}
  496. u > 0 u^{\prime}>0

Monte_Carlo_tree_search.html

  1. R R
  2. L L
  3. L L
  4. C C
  5. C C
  6. C C
  7. R R
  8. w i n i + c ln t n i \frac{w_{i}}{n_{i}}+c\sqrt{\frac{\ln t}{n_{i}}}
  9. w i w_{i}
  10. i i
  11. n i n_{i}
  12. i i
  13. c c
  14. 2 \sqrt{2}
  15. t t
  16. n i n_{i}
  17. b i n i \frac{b_{i}}{n_{i}}
  18. b i b_{i}
  19. i i
  20. N N
  21. C i C_{i}
  22. N N
  23. N N
  24. i i
  25. N N
  26. ( 1 - β ( n i , n ~ i ) ) w i n i + β ( n i , n ~ i ) w ~ i n ~ i + c ln t n i (1-\beta(n_{i},\tilde{n}_{i}))\frac{w_{i}}{n_{i}}+\beta(n_{i},\tilde{n}_{i})% \frac{\tilde{w}_{i}}{\tilde{n}_{i}}+c\sqrt{\frac{\ln t}{n_{i}}}
  27. w ~ i \tilde{w}_{i}
  28. n ~ i \tilde{n}_{i}
  29. i i
  30. i i
  31. β ( n i , n ~ i ) \beta(n_{i},\tilde{n}_{i})
  32. n i n_{i}
  33. n ~ i \tilde{n}_{i}
  34. β ( n i , n ~ i ) \beta(n_{i},\tilde{n}_{i})
  35. β ( n i , n ~ i ) = n ~ i n i + n ~ i + 4 b 2 n i n ~ i \beta(n_{i},\tilde{n}_{i})=\frac{\tilde{n}_{i}}{n_{i}+\tilde{n}_{i}+4b^{2}n_{i% }\tilde{n}_{i}}
  36. b b

Montesinos_link.html

  1. K ( e ; α 1 / β 1 , α 2 / β 2 , , α n / β n ) K(e;\alpha_{1}/\beta_{1},\alpha_{2}/\beta_{2},\ldots,\alpha_{n}/\beta_{n})
  2. e e
  3. α i \alpha_{i}
  4. β i \beta_{i}
  5. e e
  6. α 1 / β 1 , α 2 / β 2 , , α n / β n \alpha_{1}/\beta_{1},\alpha_{2}/\beta_{2},\ldots,\alpha_{n}/\beta_{n}

Mosaicity.html

  1. 𝐤 ^ ± I ± = μ I - ( μ + σ ) I ± , \mathbf{\hat{k}}_{\pm}\mathbf{\nabla}I_{\pm}=\mu I_{\mp}-(\mu+\sigma)I_{\pm},
  2. 𝐤 ^ \mathbf{\hat{k}}
  3. I ± I_{\pm}

Moufang_set.html

  1. ( X ; { U x } x X ) \left({X;\{U_{x}\}_{x\in X}}\right)
  2. { U x } x X \{U_{x}\}_{x\in X}
  3. Σ X \Sigma_{X}
  4. U y U_{y}
  5. X { y } X\setminus\{y\}
  6. U y U_{y}
  7. { U x } x X \{U_{x}\}_{x\in X}
  8. x - x - 1 = - U x - 1 ( x ) . x\mapsto-x^{-1}=-U_{x}^{-1}(x)\ .

Mountains_of_Io.html

  1. σ h \sigma_{h}
  2. σ v \sigma_{v}
  3. ν \nu
  4. ν \nu
  5. ν \nu
  6. σ h \sigma_{h}
  7. σ v \sigma_{v}

Multi-fractional_order_estimator.html

  1. y n = j = 1 J c j n j - 1 + η n = x n + η n y_{n}=\sum_{j=1}^{J}c_{j}n^{j-1}+\eta_{n}=x_{n}+\eta_{n}
  2. η n \eta_{n}
  3. σ n 2 \sigma_{n}^{2}
  4. τ \tau
  5. x ^ ( τ ) = n = 1 N y n w n ( τ ) \hat{x}(\tau)=\sum_{n=1}^{N}y_{n}w_{n}(\tau)
  6. τ \tau
  7. w n ( τ ) = m U m n T m ( τ ) f m w_{n}(\tau)=\sum_{m}U_{mn}T_{m}(\tau)f_{m}
  8. U m n U_{mn}
  9. T m ( τ ) T_{m}(\tau)
  10. c m c_{m}
  11. τ \tau
  12. f m f_{m}
  13. U m n T m U_{mn}T_{m}
  14. f m f_{m}
  15. τ \tau
  16. f m f_{m}
  17. f m = 1 f_{m}=1
  18. f m f_{m}
  19. f m f_{m}
  20. f 1 = 1 f_{1}=1
  21. f 5 0 f_{5}\gtrsim 0
  22. f m = 0 f_{m}=0
  23. f m s = 1 f_{m}s=1
  24. x ^ = < ψ , ω m > \hat{x}=<\psi,\omega_{m}>
  25. w n ( τ ) w_{n}(\tau)
  26. ω m ( τ ) \omega_{m}(\tau)
  27. x ^ x ^ ( τ ) \hat{x}\doteq\hat{x}(\tau)
  28. ω m ω m ( τ ) \omega_{m}\doteq\omega_{m}(\tau)
  29. < , > <,>
  30. ψ \psi
  31. y n y_{n}
  32. f 1 = f 2 = 1 f_{1}=f_{2}=1
  33. f m = 0 f_{m}=0
  34. 0 f 3 1 0\leq f_{3}\leq 1
  35. 2 + f 3 2+f_{3}
  36. w 2 + f 3 = ( 1 - f 3 ) ω 2 + f 3 ω 3 = ω 2 + f 3 ( ω 3 - ω 2 ) = ω 2 + f 3 ν 3 w_{2+f_{3}}=(1-f_{3})\omega_{2}+f_{3}\omega_{3}=\omega_{2}+f_{3}(\omega_{3}-% \omega_{2})=\omega_{2}+f_{3}\nu_{3}
  37. f 3 f_{3}
  38. ν 3 = ω 3 - ω 2 = υ 3 T 3 \nu_{3}=\omega_{3}-\omega_{2}=\upsilon_{3}T_{3}
  39. υ 3 \upsilon_{3}
  40. U 3 n U_{3n}
  41. c 3 a Δ 2 2 c_{3}\equiv\tfrac{a\Delta^{2}}{2}
  42. ν 3 \nu_{3}
  43. ω 3 \omega_{3}
  44. M S E = σ η 2 ( | ω 2 | 2 + f 3 2 | ν 3 | 2 ) + [ c 3 T 3 ( 1 - f 3 ) ] 2 MSE=\sigma_{\eta}^{2}(|\omega_{2}|^{2}+f_{3}^{2}|\nu_{3}|^{2})+[c_{3}T_{3}(1-f% _{3})]^{2}
  45. θ \theta
  46. | θ | 2 < θ , θ Align g t ; |\theta|^{2}\doteq<\theta,\theta&gt;
  47. σ η 2 ( | ω 2 | 2 + f 3 2 | ν 3 | 2 ) \sigma_{\eta}^{2}(|\omega_{2}|^{2}+f_{3}^{2}|\nu_{3}|^{2})
  48. f 3 2 f_{3}^{2}
  49. [ c 3 T 3 ( 1 - f 3 ) ] 2 [c_{3}T_{3}(1-f_{3})]^{2}
  50. c 3 c_{3}
  51. f 3 f_{3}
  52. f 3 f_{3}
  53. f 3 , o p t f 3 , o p t ( τ ) = ( c 3 T 3 ) 2 ( c 3 T 3 ) 2 + σ η 2 | ν 3 | 2 = c 3 2 c 3 2 + σ η 2 | υ 3 | 2 = ρ 3 2 ρ 3 2 + | υ 3 | 2 f_{3,opt}\doteq f_{3,opt}(\tau)=\frac{(c_{3}T_{3})^{2}}{(c_{3}T_{3})^{2}+% \sigma_{\eta}^{2}|\nu_{3}|^{2}}=\frac{c_{3}^{2}}{c_{3}^{2}+\sigma_{\eta}^{2}|% \upsilon_{3}|^{2}}=\frac{\rho_{3}^{2}}{\rho_{3}^{2}+|\upsilon_{3}|^{2}}
  54. ρ 3 c 3 σ η = a Δ 2 2 σ η \rho_{3}\equiv\frac{c_{3}}{\sigma_{\eta}}=\frac{a\Delta^{2}}{2\sigma_{\eta}}
  55. w 2 + f 3 , o p t = ω 2 + f 3 , o p t ν 3 = ω 2 + υ 3 T 3 f 3 , o p t = ω 2 + υ 3 T 3 ρ 3 2 ρ 3 2 + | υ 3 | 2 w_{2+f_{3,opt}}=\omega_{2}+f_{3,opt}\nu_{3}=\omega_{2}+\upsilon_{3}T_{3}f_{3,% opt}=\omega_{2}+\upsilon_{3}T_{3}\frac{\rho_{3}^{2}}{\rho_{3}^{2}+|\upsilon_{3% }|^{2}}
  56. M S E m i n = σ η 2 ( | ω 2 | 2 + f 3 , o p t | ν 3 | 2 ) MSE_{min}=\sigma_{\eta}^{2}(|\omega_{2}|^{2}+f_{3,opt}|\nu_{3}|^{2})
  57. M S E m i n MSE_{min}
  58. f 3 , o p t 2 f_{3,opt}^{2}
  59. M S E m i n MSE_{min}
  60. f 3 , o p t f_{3,opt}
  61. M S E = 1 N + 3 ( N + 1 ) N ( N - 1 ) + f 3 2 5 ( N + 1 ) ( N + 2 ) N ( ( N - 1 ) ( N - 2 ) + ρ 3 2 [ ( N + 1 ) ( N + 2 ) 6 ] 2 ( 1 - f 3 ) 2 MSE=\frac{1}{N}+\frac{3(N+1)}{N(N-1)}+f_{3}^{2}\frac{5(N+1)(N+2)}{N((N-1)(N-2)% }+\rho_{3}^{2}\left[\frac{(N+1)(N+2)}{6}\right]^{2}(1-f_{3})^{2}
  62. f 3 = 1 f_{3}=1
  63. 4 N + 2 N ( N - 1 ) \frac{4N+2}{N(N-1)}
  64. 9 N 2 + 9 N + 6 N ( N - 1 ) ( N - 2 ) \frac{9N^{2}+9N+6}{N(N-1)(N-2)}
  65. f 3 = ρ 3 = 0 f_{3}=\rho_{3}=0
  66. ρ 3 \rho_{3}
  67. 20 m / s 2 20m/s^{2}
  68. σ η = 25 m \sigma_{\eta}=25m
  69. Δ \Delta
  70. ρ 3 = a Δ 2 2 σ η = 20 / 2 / 25 = 0.4 \rho_{3}=\frac{a\Delta^{2}}{2\sigma_{\eta}}=20/2/25=0.4
  71. f 3 = 0 f_{3}=0
  72. M S E = 4 N + 2 N ( N - 1 ) + ρ 3 2 [ ( N + 1 ) ( N + 2 ) 6 ] 2 MSE=\frac{4N+2}{N(N-1)}+\rho_{3}^{2}\left[\frac{(N+1)(N+2)}{6}\right]^{2}
  73. α 0.85 \alpha\approx 0.85
  74. β 0.53 \beta\approx 0.53
  75. α = 4 N - 2 N ( N + 1 ) \alpha=\frac{4N-2}{N(N+1)}
  76. β = 6 N ( N + 1 ) \beta=\frac{6}{N(N+1)}
  77. τ \tau
  78. 0 f 3 1 0\leq f_{3}\leq 1
  79. M S E = 1 N + 3 ( N - 1 ) N ( N + 1 ) + f 3 2 5 ( N - 1 ) ( N - 2 ) N ( ( N + 1 ) ( N + 2 ) + ρ 3 2 [ ( N - 1 ) ( N - 2 ) 6 ] 2 ( 1 - f 3 ) 2 MSE=\frac{1}{N}+\frac{3(N-1)}{N(N+1)}+f_{3}^{2}\frac{5(N-1)(N-2)}{N((N+1)(N+2)% }+\rho_{3}^{2}\left[\frac{(N-1)(N-2)}{6}\right]^{2}(1-f_{3})^{2}
  80. M S E m i n = 4 N - 2 N ( N + 1 ) + f 3 , o p t 5 ( N - 1 ) ( N - 2 ) N ( ( N + 1 ) ( N + 2 ) MSE_{min}=\frac{4N-2}{N(N+1)}+f_{3,opt}\frac{5(N-1)(N-2)}{N((N+1)(N+2)}
  81. | υ 3 | 2 = 180 N ( N 2 - 1 ) ( N 2 - 4 ) |\upsilon_{3}|^{2}=\frac{180}{N(N^{2}-1)(N^{2}-4)}
  82. f 3 , o p t = ρ 3 2 ρ 3 2 + | υ 3 | 2 f_{3,opt}=\frac{\rho_{3}^{2}}{\rho_{3}^{2}+|\upsilon_{3}|^{2}}
  83. M S E m i n MSE_{min}
  84. ρ 3 \rho_{3}
  85. M S E m i n MSE_{min}
  86. M S E m i n MSE_{min}
  87. 20 m / s 2 20m/s^{2}
  88. ρ 3 = 0.4 \rho_{3}=0.4
  89. M S E m i n MSE_{min}
  90. M S E m i n MSE_{min}
  91. M S E m i n MSE_{min}
  92. N N\to\infty
  93. ρ 3 = 0 \rho_{3}=0
  94. M S E m i n MSE_{min}
  95. ρ 3 = 0.4 \rho_{3}=0.4
  96. ρ 3 \rho_{3}
  97. M S E = 4 N - 2 N ( N + 1 ) + ρ 3 2 [ ( N - 1 ) ( N - 2 ) 6 ] 2 MSE=\frac{4N-2}{N(N+1)}+\rho_{3}^{2}\left[\frac{(N-1)(N-2)}{6}\right]^{2}
  98. M S E m i n MSE_{min}
  99. M S E m i n MSE_{min}
  100. μ 1 ( k ) \mu_{1}(k)
  101. μ 2 ( k ) \mu_{2}(k)
  102. X ^ ( k | k ) = X ^ 1 ( k | k ) μ 1 ( k ) + X ^ 2 ( k | k ) μ 2 ( k ) \hat{X}(k|k)=\hat{X}_{1}(k|k)\mu_{1}(k)+\hat{X}_{2}(k|k)\mu_{2}(k)
  103. X ^ 1 ( k | k ) \hat{X}_{1}(k|k)
  104. X ^ 2 ( | k | k ) \hat{X}_{2}(|k|k)
  105. μ 1 ( k ) + μ 2 ( k ) = 1 \mu_{1}(k)+\mu_{2}(k)=1
  106. μ 1 ( k ) \mu_{1}(k)
  107. ( 1 - f 3 ) (1-f_{3})
  108. μ 2 ( k ) \mu_{2}(k)
  109. f 3 f_{3}
  110. μ 2 ( k ) \mu_{2}(k)
  111. X ^ ( k | k ) = X ^ 1 ( k | k ) + [ X ^ 2 ( k | k ) - X ^ 1 ( k | k ] μ 2 ( k ) \hat{X}(k|k)=\hat{X}_{1}(k|k)+[\hat{X}_{2}(k|k)-\hat{X}_{1}(k|k]\mu_{2}(k)
  112. μ 2 ( k ) \mu_{2}(k)
  113. f 3 f_{3}
  114. μ 2 ( k ) \mu_{2}(k)
  115. f 3 , o p t f_{3,opt}
  116. μ 2 ( k ) \mu_{2}(k)
  117. μ 2 ( k ) \mu_{2}(k)
  118. μ 2 \mu_{2}
  119. ρ 3 \rho_{3}
  120. f 3 , o p t f_{3,opt}
  121. μ 2 \mu_{2}
  122. f 3 , o p t f_{3,opt}
  123. ρ 3 = 0.7 \rho_{3}=0.7
  124. ρ 3 = 1.4 \rho_{3}=1.4
  125. μ 2 \mu_{2}
  126. M S E m i n MSE_{min}
  127. f 3 f_{3}
  128. μ 2 \mu_{2}
  129. μ 2 \mu_{2}
  130. f 3 , o p t f_{3,opt}
  131. ρ 3 = 0.7 \rho_{3}=0.7
  132. μ 2 \mu_{2}
  133. f 3 , o p t f_{3,opt}
  134. ρ 3 = 0.85 \rho_{3}=0.85
  135. ρ 3 = 1.4 \rho_{3}=1.4
  136. μ 2 \mu_{2}
  137. f 3 , o p t f_{3,opt}
  138. μ 2 \mu_{2}
  139. μ 2 \mu_{2}
  140. M S E m i n MSE_{min}
  141. ρ 3 < 0.5 \rho_{3}<0.5
  142. ( a Δ 2 ) / σ η < 1 (a\Delta^{2})/\sigma_{\eta}<1
  143. μ 2 \mu_{2}

Multi-state_modeling_of_biomolecules.html

  1. 10 23 \sim 10^{23}
  2. 10 100 10^{100}
  3. 2 8 = 256 2^{8}=256

Multi-surface_method.html

  1. 𝒜 , n \mathcal{A,B}\in\mathbb{R}^{n}
  2. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  3. f ( 𝒜 ) > 0 , f ( ) 0 f(\mathcal{A})>0,f(\mathcal{B})\leq 0
  4. f ( x ) = c x + γ f(x)=cx+\gamma

Multidimensional_Filter_Design.html

  1. M M
  2. M M
  3. x ( n ¯ ) x\left(\underline{n}\right)
  4. y ( n ¯ ) y\left(\underline{n}\right)
  5. m m
  6. y y
  7. y ( n 1 , n 2 , , n m ) = l 1 = 0 L 1 - 1 l 2 = 0 L 2 - 1 l m = 0 L m - 1 b ( l 1 , l 2 , , l m ) x ( n 1 - l 1 , n 2 - l 2 , , n m - l m ) y\left(n_{1},n_{2},...,n_{m}\right)=\sum_{l_{1}=0}^{L_{1}-1}\sum_{l_{2}=0}^{L_% {2}-1}...\sum_{l_{m}=0}^{L_{m}-1}b(l_{1},l_{2},...,l_{m})x(n_{1}-l_{1},n_{2}-l% _{2},...,n_{m}-l_{m})
  8. m m
  9. y y
  10. y ( n 1 , n 2 , , n m ) = l 1 = 0 L 1 - 1 l 2 = 0 L 2 - 1 l m = 0 L m - 1 b ( l 1 , l 2 , , l m ) x ( n 1 - l 1 , n 2 - l 2 , , n m - l m ) - m 1 = 0 M 1 - 1 m 2 = 0 M 2 - 1 m m = 0 M m - 1 a ( m 1 , m 2 , , m m ) y ( n 1 - m 1 , n 2 - m 2 , , n m - m m ) y\left(n_{1},n_{2},...,n_{m}\right)=\sum_{l_{1}=0}^{L_{1}-1}\sum_{l_{2}=0}^{L_% {2}-1}...\sum_{l_{m}=0}^{L_{m}-1}b(l_{1},l_{2},...,l_{m})x(n_{1}-l_{1},n_{2}-l% _{2},...,n_{m}-l_{m})-\sum_{m_{1}=0}^{M_{1}-1}\sum_{m_{2}=0}^{M_{2}-1}...\sum_% {m_{m}=0}^{M_{m}-1}a(m_{1},m_{2},...,m_{m})y(n_{1}-m_{1},n_{2}-m_{2},...,n_{m}% -m_{m})
  11. n 1 n_{1}
  12. n 2 n_{2}
  13. n m n_{m}
  14. 0
  15. n 1 n_{1}
  16. 0
  17. n 2 n_{2}
  18. 0
  19. n m n_{m}
  20. y ( n 1 , n 2 , , n m ) = k 1 = 0 N 1 - 1 k 2 = 0 N 2 - 1 k m = 0 N m - 1 h ( k 1 , k 2 , , k m ) x ( n 1 - k 1 , n 2 - k 2 , , n m - k m ) y\left(n_{1},n_{2},...,n_{m}\right)=\sum_{k_{1}=0}^{N_{1}-1}\sum_{k_{2}=0}^{N_% {2}-1}...\sum_{k_{m}=0}^{N_{m}-1}h(k_{1},k_{2},...,k_{m})x(n_{1}-k_{1},n_{2}-k% _{2},...,n_{m}-k_{m})
  21. N 1 N_{1}
  22. N 2 N_{2}
  23. N m N_{m}
  24. N 1 N_{1}
  25. N 2 N_{2}
  26. N m N_{m}
  27. 1 1
  28. y ( n 1 , n 2 ) y\left(n_{1},n_{2}\right)
  29. N 1 N_{1}
  30. 1 1
  31. N 2 N_{2}
  32. 1 1
  33. N 2 N_{2}
  34. N 1 N_{1}
  35. h ( n 1 , n 2 ) h\left(n_{1},n_{2}\right)
  36. h ( - n 1 , - n 2 ) h\left(-n_{1},-n_{2}\right)
  37. y ( n 1 , n 2 ) = k 1 = - N 1 N 1 k 2 = - N 2 N 2 h ( k 1 , k 2 ) x ( n 1 - k 1 , n 2 - k 2 ) y\left(n_{1},n_{2}\right)=\sum_{k_{1}=-N_{1}}^{N_{1}}\sum_{k_{2}=-N_{2}}^{N_{2% }}h(k_{1},k_{2})x(n_{1}-k_{1},n_{2}-k_{2})
  38. y ( n 1 , n 2 ) = k 1 = - N 1 N 1 k 2 = 1 N 2 h ( k 1 , k 2 ) [ x ( n 1 - k 1 , n 2 - k 2 ) + x ( n 1 + k 1 , n 2 + k 2 ) ] + k 1 = 1 N 1 h ( k 1 , 0 ) [ x ( n 1 - k 1 , n 2 ) + x ( n 1 + k 1 , n 2 ) ] + h ( 0 , 0 ) x ( n 1 , n 2 ) y\left(n_{1},n_{2}\right)=\sum_{k_{1}=-N_{1}}^{N_{1}}\sum_{k_{2}=1}^{N_{2}}h(k% _{1},k_{2})[x(n_{1}-k_{1},n_{2}-k_{2})+x(n_{1}+k_{1},n_{2}+k_{2})]+\sum_{k_{1}% =1}^{N_{1}}h(k_{1},0)[x(n_{1}-k_{1},n_{2})+x(n_{1}+k_{1},n_{2})]+h(0,0)x(n_{1}% ,n_{2})
  39. N N
  40. x ( n ¯ ) x(\underline{n})
  41. X ( k ¯ ) X(\underline{k})
  42. x ( n ¯ ) x(\underline{n})
  43. I N I_{N}
  44. d e t N detN
  45. n n
  46. J N J_{N}
  47. d e t N T detN^{T}
  48. = d e t N =detN
  49. X ( k ) = n ϵ I n x ( n ) e - j k T ( 2 π N - 1 ) n X\left(k\right)=\sum_{n\epsilon I_{n}}x\left(n\right)e^{-jk^{T}\left(2\pi N^{-% 1}\right)n}
  50. y ( n 1 , n 2 , , n m ) y\left(n_{1},n_{2},...,n_{m}\right)
  51. x ( n 1 , n 2 , , n m ) x\left(n_{1},n_{2},...,n_{m}\right)
  52. h ( n 1 , n 2 , , n m ) h\left(n_{1},n_{2},...,n_{m}\right)
  53. y ( n 1 , n 2 , , n m ) = x ( n 1 , n 2 , , n m ) * h ( n 1 , n 2 , , n m ) y\left(n_{1},n_{2},...,n_{m}\right)=x\left(n_{1},n_{2},...,n_{m}\right)*h\left% (n_{1},n_{2},...,n_{m}\right)
  54. Y ( w 1 , w 2 , , w m ) = X ( w 1 , w 2 , , w m ) Y\left(w_{1},w_{2},...,w_{m}\right)=X\left(w_{1},w_{2},...,w_{m}\right)
  55. H ( w 1 , w 2 , , w m ) H\left(w_{1},w_{2},...,w_{m}\right)
  56. y ( n 1 , n 2 , , n m ) y\left(n_{1},n_{2},...,n_{m}\right)
  57. Y ( w 1 , w 2 , , w m ) Y\left(w_{1},w_{2},...,w_{m}\right)
  58. N 1 N_{1}
  59. N 2 N_{2}
  60. N m N_{m}
  61. Y ( k 1 , k 2 , , k m ) = Y ( w 1 , w 2 , , w m ) | w 1 = 2 π k 1 N 1 ; w 2 = 2 π k 2 N 2 ; ; w m = 2 π k m N m Y\left(k_{1},k_{2},...,k_{m}\right)=Y\left(w_{1},w_{2},...,w_{m}\right)|w_{1}=% \frac{2\pi k_{1}}{N_{1}};w_{2}=\frac{2\pi k_{2}}{N_{2}};...;w_{m}=\frac{2\pi k% _{m}}{N_{m}}
  62. Y ( k 1 , k 2 , , k m ) = X ( k 1 , k 2 , , k m ) Y\left(k_{1},k_{2},...,k_{m}\right)=X\left(k_{1},k_{2},...,k_{m}\right)
  63. H ( k 1 , k 2 , , k m ) H\left(k_{1},k_{2},...,k_{m}\right)
  64. N 1 N_{1}
  65. N 2 N_{2}
  66. N m N_{m}
  67. x x
  68. h h
  69. y ^ ( n 1 , n 2 , , n m ) \hat{y}\left(n_{1},n_{2},...,n_{m}\right)
  70. X ( k 1 , k 2 , , k m ) X\left(k_{1},k_{2},...,k_{m}\right)
  71. H ( k 1 , k 2 , , k m ) H\left(k_{1},k_{2},...,k_{m}\right)
  72. y ^ ( n 1 , n 2 , , n m ) \hat{y}\left(n_{1},n_{2},...,n_{m}\right)
  73. h ( n 1 , n 2 , , n m ) h\left(n_{1},n_{2},...,n_{m}\right)
  74. x ( n 1 , n 2 , , n m ) x\left(n_{1},n_{2},...,n_{m}\right)
  75. N 1 N_{1}
  76. N 2 N_{2}
  77. N m N_{m}
  78. y ( n 1 , n 2 , , n m ) y\left(n_{1},n_{2},...,n_{m}\right)
  79. y ^ ( n 1 , n 2 , , n m ) = y ( n 1 , n 2 , , n m ) \hat{y}\left(n_{1},n_{2},...,n_{m}\right)=y\left(n_{1},n_{2},...,n_{m}\right)
  80. N 1 N_{1}
  81. N 2 N_{2}
  82. N m N_{m}
  83. x x
  84. h h
  85. N 1 N_{1}
  86. N 2 N_{2}
  87. N m N_{m}
  88. x ( n 1 , n 2 , , n m ) x\left(n_{1},n_{2},...,n_{m}\right)
  89. H ( k 1 , k 2 , , k m ) H\left(k_{1},k_{2},...,k_{m}\right)
  90. H ( k 1 , k 2 ) H\left(k_{1},k_{2}\right)
  91. y ( n 1 , n 2 ) y\left(n_{1},n_{2}\right)
  92. 2 2
  93. N 1 N_{1}
  94. N 2 N_{2}
  95. log 2 \log_{2}
  96. N 1 N_{1}
  97. N 2 N_{2}
  98. + +
  99. 2 2
  100. N 1 N_{1}
  101. N 2 N_{2}
  102. N 1 N_{1}
  103. N 2 N_{2}
  104. Z d Z_{d}
  105. N d N_{d}
  106. x x
  107. d i a g ( x ) diag\left(x\right)
  108. x i x_{i}
  109. X X
  110. d i a g ( X ) diag\left(X\right)
  111. i t h i^{th}
  112. X i i Xii
  113. A > B A>\geq B
  114. A - B A-B
  115. T r ( A ) Tr\left(A\right)
  116. A A
  117. R R
  118. d d
  119. n ϵ Z d n\epsilon Z_{d}
  120. x k x_{k}
  121. x - k x_{-k}
  122. R ( w ) = k = - n n x k e - j k T w R\left(w\right)=\sum_{k=-n}^{n}x_{k}e^{-jk^{T}w}
  123. k k
  124. - n k n -n\leq k\leq n
  125. R R
  126. [ - π , π ] d [-\pi,\pi]^{d}
  127. H l ( w ) = l = 1 r | H l ( w ) | 2 H_{l}\left(w\right)=\sum_{l=1}^{r}|H_{l}\left(w\right)|^{2}
  128. H l ( w ) = k = 0 n l h l , k e - j k T w H_{l}\left(w\right)=\sum_{k=0}^{n_{l}}h_{l,k}e^{-jk^{T}w}
  129. H H
  130. n = ( n 1 , n 2 ) n=\left(n_{1},n_{2}\right)
  131. h k h_{k}
  132. h - k h_{-k}
  133. H ( w ) = k = - n n h k e - j k T w H\left(w\right)=\sum_{k=-n}^{n}h_{k}e^{-jk^{T}w}
  134. h k h_{k}
  135. δ s \delta_{s}
  136. D s D_{s}
  137. δ p \delta_{p}
  138. D p D_{p}
  139. δ s \delta_{s}
  140. | 1 - H ( w ) | δ p , w ϵ D p |1-H\left(w\right)|\leq\delta_{p},\,\,w\epsilon D_{p}
  141. | H ( w ) | δ s , w ϵ D s |H\left(w\right)|\leq\delta_{s},\,\,w\epsilon D_{s}
  142. δ s \delta_{s}
  143. h k h_{k}
  144. R 1 ( w ) = H ( w ) - 1 + δ p 0 , w ϵ D p R_{1}\left(w\right)=H\left(w\right)-1+\delta_{p}\geq 0,\,\,w\epsilon D_{p}
  145. R 2 ( w ) = 1 - H ( w ) + δ p 0 , w ϵ D p R_{2}\left(w\right)=1-H\left(w\right)+\delta_{p}\geq 0,\,\,w\epsilon D_{p}
  146. R 3 ( w ) = H ( w ) + δ s 0 , w ϵ D s R_{3}\left(w\right)=H\left(w\right)+\delta_{s}\geq 0,\,\,w\epsilon D_{s}
  147. R 4 ( w ) = H ( w ) - δ s 0 , w ϵ D s R_{4}\left(w\right)=H\left(w\right)-\delta_{s}\geq 0,\,\,w\epsilon D_{s}
  148. R i R_{i}
  149. n n
  150. H ( w 1 , w 2 , , w m ) = A ( w 1 , w 2 , , w m ) B ( w 1 , w 2 , , w m ) = l 1 l 2 l m a ( l 1 , l 2 , , l m ) e - j ( w 1 l 1 + w 2 l 2 + + w m l m ) k 1 k 2 k m b ( k 1 , k 2 , , k m ) e - j ( w 1 k 1 + w 2 k 2 + + w m k m ) H\left(w_{1},w_{2},...,w_{m}\right)=\frac{A\left(w_{1},w_{2},...,w_{m}\right)}% {B\left(w_{1},w_{2},...,w_{m}\right)}=\frac{\sum_{l_{1}}\sum_{l_{2}}...\sum_{l% _{m}}a\left(l_{1},l_{2},...,l_{m}\right)e^{-j\left(w_{1}l_{1}+w_{2}l_{2}+...+w% _{m}l_{m}\right)}}{\sum_{k_{1}}\sum_{k_{2}}...\sum_{k_{m}}b\left(k_{1},k_{2},.% ..,k_{m}\right)e^{-j\left(w_{1}k_{1}+w_{2}k_{2}+...+w_{m}k_{m}\right)}}
  151. a ( l 1 , l 2 , , l m ) a\left(l_{1},l_{2},...,l_{m}\right)
  152. b ( k 1 , k 2 , , k m ) b\left(k_{1},k_{2},...,k_{m}\right)
  153. b ( 0 , 0 , , 0 ) = 1 b\left(0,0,...,0\right)=1
  154. X ( w 1 , w 2 , , w m ) X\left(w_{1},w_{2},...,w_{m}\right)
  155. x ( n 1 , n 2 , , n m ) x\left(n_{1},n_{2},...,n_{m}\right)
  156. Y ( w 1 , w 2 , , w m ) Y\left(w_{1},w_{2},...,w_{m}\right)
  157. y ( n 1 , n 2 , , n m ) y\left(n_{1},n_{2},...,n_{m}\right)
  158. Y ( w 1 , w 2 , , w m ) = A ( w 1 , w 2 , , w m ) X ( w 1 , w 2 , , w m ) + C ( w 1 , w 2 , , w m ) Y ( w 1 , w 2 , , w m ) Y\left(w_{1},w_{2},...,w_{m}\right)=A\left(w_{1},w_{2},...,w_{m}\right)X\left(% w_{1},w_{2},...,w_{m}\right)+C\left(w_{1},w_{2},...,w_{m}\right)Y\left(w_{1},w% _{2},...,w_{m}\right)
  159. C ( w 1 , w 2 , , w m ) C\left(w_{1},w_{2},...,w_{m}\right)
  160. C ( w 1 , w 2 , , w m ) = 1 - B ( w 1 , w 2 , , w m ) C\left(w_{1},w_{2},...,w_{m}\right)=1-B\left(w_{1},w_{2},...,w_{m}\right)
  161. y ( n 1 , n 2 , , n m ) = a ( n 1 , n 2 , , n m ) * x ( n 1 , n 2 , , n m ) + c ( n 1 , n 2 , , n m ) * y ( n 1 , n 2 , , n m ) y\left(n_{1},n_{2},...,n_{m}\right)=a\left(n_{1},n_{2},...,n_{m}\right)*x\left% (n_{1},n_{2},...,n_{m}\right)+c\left(n_{1},n_{2},...,n_{m}\right)*y\left(n_{1}% ,n_{2},...,n_{m}\right)
  162. y ( n 1 , n 2 , , n m ) y\left(n_{1},n_{2},...,n_{m}\right)
  163. y i ( n 1 , n 2 , , n m ) = a ( n 1 , n 2 , , n m ) * x ( n 1 , n 2 , , n m ) + c ( n 1 , n 2 , , n m ) * y i - 1 ( n 1 , n 2 , , n m ) y_{i}\left(n_{1},n_{2},...,n_{m}\right)=a\left(n_{1},n_{2},...,n_{m}\right)*x% \left(n_{1},n_{2},...,n_{m}\right)+c\left(n_{1},n_{2},...,n_{m}\right)*y_{i-1}% \left(n_{1},n_{2},...,n_{m}\right)
  164. i i
  165. Y i ( w 1 , w 2 , , w m ) = A ( w 1 , w 2 , , w m ) X ( w 1 , w 2 , , w m ) + C ( w 1 , w 2 , , w m ) Y i - 1 ( w 1 , w 2 , , w m ) Y_{i}\left(w_{1},w_{2},...,w_{m}\right)=A\left(w_{1},w_{2},...,w_{m}\right)X% \left(w_{1},w_{2},...,w_{m}\right)+C\left(w_{1},w_{2},...,w_{m}\right)Y_{i-1}% \left(w_{1},w_{2},...,w_{m}\right)
  166. C ( w 1 , w 2 , , w m ) 0 C\left(w_{1},w_{2},...,w_{m}\right)\neq 0
  167. | C ( w 1 , w 2 , , w m ) | < 1 |C\left(w_{1},w_{2},...,w_{m}\right)|<1
  168. lim I Y I ( w 1 , w 2 , , w m ) = A ( w 1 , w 2 , , w m ) X ( w 1 , w 2 , , w m ) 1 - C ( w 1 , w 2 , , w m ) = Y ( w 1 , w 2 , , w m ) \lim_{I\to\infty}Y_{I}\left(w_{1},w_{2},...,w_{m}\right)=\frac{A\left(w_{1},w_% {2},...,w_{m}\right)X\left(w_{1},w_{2},...,w_{m}\right)}{1-C\left(w_{1},w_{2},% ...,w_{m}\right)}=Y\left(w_{1},w_{2},...,w_{m}\right)
  169. H ( w 1 , w 2 , , w m ) H\left(w_{1},w_{2},...,w_{m}\right)
  170. a ( n 1 , n 2 , , n m ) a\left(n_{1},n_{2},...,n_{m}\right)
  171. b ( n 1 , n 2 , , n m ) b\left(n_{1},n_{2},...,n_{m}\right)
  172. y ( n 1 , n 2 ) y\left(n_{1},n_{2}\right)
  173. d ( n 1 , n 2 , , n m ) d\left(n_{1},n_{2},...,n_{m}\right)
  174. e ( n 1 , n 2 , , n m ) = y ( n 1 , n 2 , , n m ) - d ( n 1 , n 2 , , n m ) e\left(n_{1},n_{2},...,n_{m}\right)=y\left(n_{1},n_{2},...,n_{m}\right)-d\left% (n_{1},n_{2},...,n_{m}\right)
  175. E ( w 1 , w 2 , , w m ) E\left(w_{1},w_{2},...,w_{m}\right)
  176. E ( w 1 , w 2 , , w m ) = A ( w 1 , w 2 , , w m ) X ( w 1 , w 2 , , w m ) B ( w 1 , w 2 , , w m ) - D ( w 1 , w 2 , , w m ) E\left(w_{1},w_{2},...,w_{m}\right)=\frac{A\left(w_{1},w_{2},...,w_{m}\right)X% \left(w_{1},w_{2},...,w_{m}\right)}{B\left(w_{1},w_{2},...,w_{m}\right)}-D% \left(w_{1},w_{2},...,w_{m}\right)
  177. B ( w 1 , w 2 , , w m ) B\left(w_{1},w_{2},...,w_{m}\right)
  178. e ( n 1 , n 2 , , n m ) = a ( n 1 , n 2 , , n m ) * x ( n 1 , n 2 , , n m ) - b ( n 1 , n 2 , , n m ) * d ( n 1 , n 2 , , n m ) e^{\prime}\left(n_{1},n_{2},...,n_{m}\right)=a\left(n_{1},n_{2},...,n_{m}% \right)*x\left(n_{1},n_{2},...,n_{m}\right)-b\left(n_{1},n_{2},...,n_{m}\right% )*d\left(n_{1},n_{2},...,n_{m}\right)
  179. e 2 ( n 1 , n 2 , , n m ) = n 1 n 2 n m [ e ( n 1 , n 2 , , n m ) ] 2 e^{\prime}_{2}\left(n_{1},n_{2},...,n_{m}\right)=\sum_{n_{1}}\sum_{n_{2}}...% \sum_{n_{m}}[e^{\prime}\left(n_{1},n_{2},...,n_{m}\right)]^{2}
  180. δ ( n 1 , n 2 , , n m ) \delta\left(n_{1},n_{2},...,n_{m}\right)
  181. a ( n 1 , n 2 , , n m ) a\left(n_{1},n_{2},...,n_{m}\right)
  182. 0 n 1 N 1 - 1 & 0 n 2 N 2 - 1... , 0 n m N m - 1 0\leq n_{1}\leq N_{1}-1\,\&\,0\leq n_{2}\leq N_{2}-1...,0\leq n_{m}\leq N_{m}-1
  183. e ( n 1 , n 2 , , n m ) = - q 1 = 0 M 1 - 1 q 2 = 0 M 2 - 1 q m = 0 M m - 1 b ( q 1 , q 2 , . . , q m ) d ( n 1 - q 1 , n 2 - q 2 ) ; e^{\prime}\left(n_{1},n_{2},...,n_{m}\right)=-\sum_{q_{1}=0}^{M_{1}-1}\sum_{q_% {2}=0}^{M_{2}-1}...\sum_{q_{m}=0}^{M_{m}-1}b\left(q_{1},q_{2},..,q_{m}\right)d% \left(n_{1}-q_{1},n_{2}-q_{2}\right)\,;\,
  184. n 1 N 1 - 1 n_{1}\geq N_{1}-1
  185. n 2 N 2 - 1 n_{2}\geq N_{2}-1
  186. n m N m - 1 n_{m}\geq N_{m}-1
  187. e e^{\prime}
  188. e 2 e_{2}^{\prime}
  189. e 2 e^{\prime}_{2}
  190. b ( q 1 , q 2 , , q n ) b\left(q_{1},q_{2},...,q_{n}\right)
  191. m 1 = 0 M 1 - 1 m 2 = 0 M 2 - 1 m m = 0 M m - 1 b ( m 1 , m 2 , , m m ) r ( m 1 , m 2 , , m m ; q 1 , q 2 , , q m ) = 0 ; \sum_{m_{1}=0}^{M_{1}-1}\sum_{m_{2}=0}^{M_{2}-1}...\sum_{m_{m}=0}^{M_{m}-1}b% \left(m_{1},m_{2},...,m_{m}\right)r\left(m_{1},m_{2},...,m_{m};q_{1},q_{2},...% ,q_{m}\right)=0\,;\,
  192. 0 m 1 < M 1 ; 0 m 2 < M 2 ; ; 0 m m < M m 0\leq m_{1}<M_{1};0\leq m_{2}<M_{2};...;0\leq m_{m}<M_{m}
  193. n 1 N 1 - 1 n_{1}\geq N_{1}-1
  194. n 2 N 2 - 1 n_{2}\geq N_{2}-1
  195. n m N m - 1 n_{m}\geq N_{m}-1
  196. b ( n 1 , n 2 , , n m ) b\left(n_{1},n_{2},...,n_{m}\right)
  197. a ( n 1 , n 2 , , n m ) a\left(n_{1},n_{2},...,n_{m}\right)
  198. a ( n 1 , n 2 , , n m ) b ( n 1 , n 2 , , n m ) * d ( n 1 , n 2 , , n m ) a\left(n_{1},n_{2},...,n_{m}\right)\approx b\left(n_{1},n_{2},...,n_{m}\right)% *d\left(n_{1},n_{2},...,n_{m}\right)
  199. y y
  200. d d
  201. 1 ( 2 π ) m - π π - π π - π π | Y ( w 1 , w 2 , , w m ) - D ( w 1 , w 2 , , w m ) | 2 d w 1 d w 2 d w m \frac{1}{\left(2\pi\right)^{m}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}...\int_{-\pi% }^{\pi}|Y\left(w_{1},w_{2},...,w_{m}\right)-D\left(w_{1},w_{2},...,w_{m}\right% )|^{2}dw_{1}dw_{2}...dw_{m}
  202. J a J_{a}
  203. D ( w 1 , w 2 , , w m ) D\left(w_{1},w_{2},...,w_{m}\right)
  204. Y ( w 1 , w 2 , , w m ) Y\left(w_{1},w_{2},...,w_{m}\right)
  205. f f
  206. Y Y
  207. D D
  208. f / l e f t ( Y ) - D f/left(Y)-D
  209. J a = k W ( w 1 k , w 2 k , , w m k ) [ f ( A ( w 1 k , w 2 k , , w m k ) B ( w 1 k , w 2 k , , w m k ) ) - D ( w 1 k , w 2 k , , w m k ) ] 2 J_{a}=\sum_{k}W\left(w_{1k},w_{2k},...,w_{mk}\right)[f\left(\frac{A\left(w_{1k% },w_{2k},...,w_{mk}\right)}{B\left(w_{1k},w_{2k},...,w_{mk}\right)}\right)-D% \left(w_{1k},w_{2k},...,w_{mk}\right)]^{2}
  210. W ( w 1 , w 2 , , w m ) W\left(w_{1},w_{2},...,w_{m}\right)
  211. ( w 1 k , w 2 k , w m k ) \left(w_{1k},w_{2k},...w_{mk}\right)
  212. { a ( n 1 , n 2 , , w m ) b ( n 1 , n 2 , , w m ) } \{a\left(n_{1},n_{2},...,w_{m}\right)\,b\left(n_{1},n_{2},...,w_{m}\right)\}
  213. J a J_{a}
  214. f f
  215. Y ( w 1 , w 2 , , w m ) Y\left(w_{1},w_{2},...,w_{m}\right)

Multidimensional_Multirate_Systems.html

  1. Y [ w ] = 1 J ( M ) k S X ( M ( w - 2 π k ) ) Y[w]=\frac{1}{J(M)}\sum_{k\mathop{\in}S}X(M(w-2\cdot\pi\cdot k))
  2. M [ 0 , 0 ] w 0 + M [ 1 , 0 ] w 1 M[0,0]\cdot w_{0}+M[1,0]\cdot w_{1}
  3. [ - π , π ) [-\pi,\pi)
  4. M [ 0 , 1 ] w 0 + M [ 1 , 1 ] w 1 M[0,1]\cdot w_{0}+M[1,1]\cdot w_{1}
  5. [ - π , π ) [-\pi,\pi)
  6. w = π ( M - T ) x w=\pi\cdot(M^{-T})\cdot x
  7. P ( w ) P(w)
  8. h ( s ) ( n ) h^{(s)}(n)
  9. p ( n ) p(n)
  10. P ( w ) P(w)
  11. h ( s ) ( n ) h^{(s)}(n)
  12. h ( n ) h(n)
  13. P ( w ) P(w)
  14. H s ( w ) = P ( w 0 ) P ( w 1 ) P ( w k ) H_{s}(w)=P(w_{0})\cdot P(w_{1})...P(w_{k})
  15. h s ( n ) = p ( n 0 ) p ( n 1 ) p ( n k ) = i = 0 k s i n π n i J ( M ) ( π n i ) h_{s}(n)=p(n_{0})\cdot p(n_{1})\dots p(n_{k})=\prod_{i=0}^{k}\frac{sin\frac{% \pi\cdot n_{i}}{J(M)}}{(\pi\cdot n_{i})}
  16. h ( s ) ( M n ) h^{(s)}(Mn)
  17. h ( s ) ( n ) h^{(s)}(n)
  18. M n = J ( M ) M - 1 n = J ( M ) m M\cdot n=J(M)\cdot M^{-1}\cdot n=J(M)\cdot m
  19. = 1 J ( M ) D i = 0 k s i n ( π m i ) π m i =\frac{1}{J(M)^{D}}\prod_{i=0}^{k}\frac{sin(\pi\cdot m_{i})}{\pi\cdot m_{i}}
  20. r i ( n ) = x ( M n - k i ) r_{i}(n)=x(Mn-k_{i})
  21. X ( z ) X(z)
  22. X ( z ) = k S z - k i R i ( z M ) X(z)=\sum_{k\mathop{\in}S}z^{-k_{i}}\cdot R_{i}\cdot(z^{M})
  23. R i R_{i}
  24. X ( z ) X(z)
  25. H l ( z ) = k S z - k i E i , l ( z M ) H_{l}(z)=\sum_{k\mathop{\in}S}z^{-k_{i}}\cdot E_{i,l}\cdot(z^{M})
  26. l = 0 , 1 , , J ( M ) - 1 l=0,1,\dots,J(M)-1
  27. E i , l E_{i,l}

Multidimensional_network.html

  1. G = ( V , E ) G=(V,E)
  2. V V
  3. E E
  4. u , v V u,v\in V
  5. G = ( V , E , D ) G=(V,E,D)
  6. D D
  7. E E
  8. ( u , v , d ) (u,v,d)
  9. u , v V u,v\in V
  10. d D d\in D
  11. e = ( u , v , d , w ) e=(u,v,d,w)
  12. w w
  13. u u
  14. v v
  15. d d
  16. G = ( V , E , D ) G=(V,E,D)
  17. D = { - 1 , 0 , 1 } D=\{-1,0,1\}
  18. E = { ( u , v , d ) ; u , v V , d D } E=\{(u,v,d);u,v\in V,d\in D\}
  19. e = ( u , v , d 1 d n - 2 ) e=(u,v,d_{1}\dots d_{n-2})
  20. ( u , v , d ) (u,v,d)
  21. ( v , u , d ) (v,u,d)
  22. | D | |D|
  23. | D | : k = ( k i 1 , k i | D | ) |D|:{k}=(k^{1}_{i},\dots k^{|D|}_{i})
  24. α = 1 | D | k i α \sum_{\alpha=1}^{|D|}k^{\alpha}_{i}
  25. v v
  26. v v
  27. V × V V\times V
  28. D D
  29. V × V × D V\times V\times D
  30. = ( r 1 , r | D | ) =(r_{1},\dots r_{|D|})
  31. i i
  32. i i
  33. G G
  34. d 1 , | D | , r l s l \forall d\in\langle 1,|D|\rangle,r_{l}\leq s_{l}
  35. i \exists i
  36. r l < s l r_{l}<s_{l}
  37. M P ( u , v ) MP(u,v)
  38. u u
  39. v v
  40. u u
  41. v v
  42. P M P P\subseteq MP
  43. p P , p M P \forall p\in P,\nexists p^{\prime}\in MP
  44. p p^{\prime}
  45. p p
  46. G = ( V , E , D ) G=(V,E,D)
  47. D D^{\prime}
  48. Neighbors ( v , D ) Neighbors ( v , D ) \frac{\,\text{Neighbors}(v,D^{\prime})}{\,\text{Neighbors}(v,D)}
  49. | { ( u , v , d ) E | u , v V } | | E | \frac{|\{(u,v,d)\in E|u,v\in V\}|}{|E|}
  50. | { ( u , v , d ) E | u , v V j D , j d : ( u , v , j ) E } | | { ( u , v , d ) E | u , v V } | \frac{|\{(u,v,d)\in E|u,v\in V\wedge\forall j\in D,j\neq d:(u,v,j)\notin E\}|}% {|\{(u,v,d)\in E|u,v\in V\}|}

Multidimensional_signal_processing.html

  1. X ( k 1 , k 2 , , k m ) = n 1 = - n 2 = - n m = - x ( n 1 , n 2 , , n m ) e - j 2 π k 1 n 1 e - j 2 π k 2 n 2 e - j 2 π k m n m X(k_{1},k_{2},\dots,k_{m})=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=-\infty}^{% \infty}\cdots\sum_{n_{m}=-\infty}^{\infty}x(n_{1},n_{2},\dots,n_{m})e^{-j2\pi k% _{1}n_{1}}e^{-j2\pi k_{2}n_{2}}\cdots e^{-j2\pi k_{m}n_{m}}

Multidimensional_transform.html

  1. F ( w 1 , w 2 , , w m ) = n 1 = - n 2 = - n m = - f ( n 1 , n 2 , , n m ) e - j w 1 n 1 - j w 2 n 2 - j w m n m F(w_{1},w_{2},\dots,w_{m})=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=-\infty}^{% \infty}\cdots\sum_{n_{m}=-\infty}^{\infty}f(n_{1},n_{2},\dots,n_{m})e^{-jw_{1}% n_{1}-jw_{2}n_{2}\cdots-jw_{m}n_{m}}
  2. f ( n 1 , n 2 , , n m ) = ( 1 2 π ) m - π π - π π F ( w 1 , w 2 , , w m ) e j w 1 n 1 + j w 2 n 2 + + j w m n m d w 1 d w m f(n_{1},n_{2},\dots,n_{m})=\left(\frac{1}{2\pi}\right)^{m}\int_{-\pi}^{\pi}% \cdots\int_{-\pi}^{\pi}F(w_{1},w_{2},\ldots,w_{m})e^{jw_{1}n_{1}+jw_{2}n_{2}+% \cdots+jw_{m}n_{m}}\,dw_{1}\cdots\,dw_{m}
  3. F ( Ω 1 , Ω 2 , , Ω m ) = - - f ( t 1 , t 2 , , t m ) e - j Ω 1 t 1 - j Ω 2 t 2 - j Ω m t m d t 1 d t m F(\Omega_{1},\Omega_{2},\ldots,\Omega_{m})=\int_{-\infty}^{\infty}\cdots\int_{% -\infty}^{\infty}f(t_{1},t_{2},\ldots,t_{m})e^{-j\Omega_{1}t_{1}-j\Omega_{2}t_% {2}\cdots-j\Omega_{m}t_{m}}\,dt_{1}\cdots\,dt_{m}
  4. F x ( K 1 , K 2 , , K n ) = n 1 = 0 N 1 - 1 n m N m - 1 f x ( n 1 , n 2 , , n N ) e - j 2 π N 1 n 1 K 1 - j 2 π N 2 n 2 K 2 - j 2 π N m n m K m Fx(K_{1},K_{2},\ldots,K_{n})=\sum_{n_{1}=0}^{N_{1}-1}\cdots\sum_{n_{m}}^{N_{m}% -1}fx(n_{1},n_{2},\ldots,n_{N})e^{-j\frac{2\pi}{N_{1}}n_{1}K_{1}-j\frac{2\pi}{% N_{2}}n_{2}K_{2}\cdots-j\frac{2\pi}{N_{m}}n_{m}K_{m}}
  5. f x ( n 1 , n 2 , , n m ) = 1 N 1 N m K 1 = 0 N 1 - 1 K m N m - 1 F x ( K 1 , K 2 , , K m ) e j 2 π N 1 n 1 K 1 + j 2 π N 2 n 2 K 2 + j 2 π N m n m K m fx(n_{1},n_{2},\ldots,n_{m})=\frac{1}{N_{1}\cdots N_{m}}\sum_{K_{1}=0}^{N_{1}-% 1}\cdots\sum_{K_{m}}^{N_{m}-1}Fx(K_{1},K_{2},\ldots,K_{m})e^{j\frac{2\pi}{N_{1% }}n_{1}K_{1}+j\frac{2\pi}{N_{2}}n_{2}K_{2}\cdots+j\frac{2\pi}{N_{m}}n_{m}K_{m}}
  6. F x ( K 1 , K 2 , , K r ) = n 1 = 0 N 1 - 1 n 2 = 0 N 2 - 1 n r = 0 N r - 1 f x ( n 1 , n 2 , , n r ) cos π ( 2 n 1 + 1 ) K 1 2 N 1 cos π ( 2 n r + 1 ) K r 2 N r Fx(K_{1},K_{2},\ldots,K_{r})=\sum_{n_{1}=0}^{N_{1}-1}\sum_{n_{2}=0}^{N_{2}-1}% \cdots\sum_{n_{r}=0}^{N_{r}-1}fx(n_{1},n_{2},\ldots,n_{r})\cos{\frac{\pi(2n_{1% }+1)K_{1}}{2N_{1}}}\cdots\cos{\frac{\pi(2n_{r}+1)K_{r}}{2N_{r}}}
  7. w ( x , y ) = 1 4 ( 1 + cos x π N ) ( 1 + cos y π N ) w(x,y)=\frac{1}{4}\left(1+\cos{\frac{x\pi}{N}}\right)\left(1+\cos{\frac{y\pi}{% N}}\right)
  8. A m ( f ) 2 = [ i = - f f FFT ( - f , i ) 2 + i = - f f FFT ( f , i ) 2 + i = - f + 1 f - 1 FFT ( i , - f ) 2 + i = - f + 1 f - 1 FFT ( i , f ) 2 ] A_{m}{(f)}^{2}=\left[\sum_{i=-f}^{f}\operatorname{FFT}(-f,i)^{2}+\sum_{i=-f}^{% f}\operatorname{FFT}(f,i)^{2}+\sum_{i=-f+1}^{f-1}\operatorname{FFT}(i,-f)^{2}+% \sum_{i=-f+1}^{f-1}\operatorname{FFT}(i,f)^{2}\right]

Multimodal_learning.html

  1. E ( 𝐯 , 𝐡 ; θ ) = - i = 1 D j = 1 F W i j v i h j - i = 1 D b i v i - j = 1 F a j h j E(\mathbf{v},\mathbf{h};\theta)=-\sum_{i=1}^{D}\sum_{j=1}^{F}W_{ij}v_{i}h_{j}-% \sum_{i=1}^{D}b_{i}v_{i}-\sum_{j=1}^{F}a_{j}h_{j}
  2. θ = { 𝐯 , 𝐡 ; θ } \theta=\{\mathbf{v},\mathbf{h};\theta\}
  3. W i j W_{ij}
  4. i i
  5. j j
  6. b i b_{i}
  7. a j a_{j}
  8. P ( 𝐯 ; θ ) = 1 𝒵 ( θ ) 𝐡 exp ( - E ( 𝐯 , 𝐡 ; θ ) ) P(\mathbf{v};\theta)=\frac{1}{\mathcal{Z}(\theta)}\sum_{\mathbf{h}}\mathrm{exp% }(-E(\mathbf{v},\mathbf{h};\theta))
  9. 𝒵 ( θ ) \mathcal{Z}(\theta)
  10. 𝐡 \mathbf{h}
  11. 𝐯 \mathbf{v}
  12. P ( 𝐡 | 𝐯 ; θ ) = j = 1 F p ( h j | 𝐯 ) P(\mathbf{h}|\mathbf{v};\theta)=\prod_{j=1}^{F}p(h_{j}|\mathbf{v})
  13. p ( h j = 1 | 𝐯 ) = g ( i = 1 D W i j v i + a j ) p(h_{j}=1|\mathbf{v})=g(\sum_{i=1}^{D}W_{ij}v_{i}+a_{j})
  14. P ( 𝐯 | 𝐡 ; θ ) = i = 1 D p ( v i | 𝐯 ) P(\mathbf{v}|\mathbf{h};\theta)=\prod_{i=1}^{D}p(v_{i}|\mathbf{v})
  15. p ( v i = 1 | 𝐡 ) = g ( j = 1 F W i j h j + b i ) p(v_{i}=1|\mathbf{h})=g(\sum_{j=1}^{F}W_{ij}h_{j}+b_{i})
  16. g ( x ) = 1 ( 1 + exp ( - x ) ) g(x)=\frac{1}{(1+\mathrm{exp}(-x))}
  17. E ( 𝐯 , 𝐡 ; θ ) = i = 1 D ( v i - b i ) 2 2 σ i 2 - i = 1 D j = 1 F v i σ i W i j v i h j - i = 1 D b i v i - j = 1 F a j h j E(\mathbf{v},\mathbf{h};\theta)=\sum_{i=1}^{D}\frac{(v_{i}-b_{i})^{2}}{2\sigma% _{i}^{2}}-\sum_{i=1}^{D}\sum_{j=1}^{F}\frac{v_{i}}{\sigma_{i}}W_{ij}v_{i}h_{j}% -\sum_{i=1}^{D}b_{i}v_{i}-\sum_{j=1}^{F}a_{j}h_{j}
  18. θ = { 𝐚 , 𝐛 , 𝐰 , σ } \theta=\{\mathbf{a},\mathbf{b},\mathbf{w},\mathbf{\sigma}\}
  19. P ( 𝐡 | 𝐯 ; θ ) = j = 1 F p ( h j | 𝐯 ) P(\mathbf{h}|\mathbf{v};\theta)=\prod_{j=1}^{F}p(h_{j}|\mathbf{v})
  20. p ( h j = 1 | 𝐯 ) = g ( i = 1 D W i j v i σ i + a j ) p(h_{j}=1|\mathbf{v})=g(\sum_{i=1}^{D}W_{ij}\frac{v_{i}}{\sigma_{i}}+a_{j})
  21. P ( 𝐯 | 𝐡 ; θ ) = i = 1 D p ( v i | 𝐡 ) P(\mathbf{v}|\mathbf{h};\theta)=\prod_{i=1}^{D}p(v_{i}|\mathbf{h})
  22. p ( v i | 𝐡 ) 𝒩 ( σ i j = 1 F W i j h j + b i , σ i 2 ) p(v_{i}|\mathbf{h})\sim\mathcal{N}(\sigma_{i}\sum_{j=1}^{F}W_{ij}h_{j}+b_{i},% \sigma_{i}^{2})
  23. K K
  24. M M
  25. 𝐕 \mathbf{V}
  26. M × K M\times K
  27. v i k = 1 v_{ik}=1
  28. i t h i^{th}
  29. k t h k^{th}
  30. v ^ k \hat{v}_{k}
  31. k t h k^{th}
  32. { 𝐕 , 𝐡 } \{\mathbf{V},\mathbf{h}\}
  33. M M
  34. E ( 𝐕 , 𝐡 ) = - j = 1 F k = 1 K W j k v ^ k h j - k = 1 K b k v ^ k - M j = 1 F a j h j E(\mathbf{V},\mathbf{h})=-\sum_{j=1}^{F}\sum_{k=1}^{K}W_{jk}\hat{v}_{k}h_{j}-% \sum_{k=1}^{K}b_{k}\hat{v}_{k}-M\sum_{j=1}^{F}a_{j}h_{j}
  35. p ( h j = 1 | 𝐕 ) = g ( M a j + k = 1 K v ^ k W j k ) p(h_{j}=1|\mathbf{V})=g(Ma_{j}+\sum_{k=1}^{K}\hat{v}_{k}W_{jk})
  36. p ( v i k = 1 | 𝐡 ) = exp ( b k + j = 1 F h j W j k q = 1 K exp ( b q + j = 1 F h j W j q ) p(v_{ik}=1|\mathbf{h})=\frac{\mathrm{exp}(b_{k}+\sum_{j=1}^{F}h_{j}W_{jk}}{% \sum_{q=1}^{K}\mathrm{exp}(b_{q}+\sum_{j=1}^{F}h_{j}W_{jq}})
  37. E ( 𝐯 , 𝐡 ; θ ) = - i = 1 D j = 1 F 1 W i j ( 1 ) v i h j ( 1 ) - j = 1 F 1 l = 1 F 2 W j l ( 2 ) h j ( 1 ) h l ( 2 ) - l = 1 F 2 p = 1 F 3 W l p ( 3 ) h l ( 2 ) h p ( 3 ) - i = 1 D b i v i - j = 1 F 1 b j ( 1 ) h j ( 1 ) - l = 1 F 2 b l ( 2 ) h l ( 2 ) - p = 1 F 3 b p ( 3 ) h p ( 3 ) \begin{aligned}\displaystyle E({\mathbf{v},\mathbf{h};\theta})=&\displaystyle-% \sum_{i=1}^{D}\sum_{j=1}^{F_{1}}W_{ij}^{(1)}v_{i}h_{j}^{(1)}-\sum_{j=1}^{F_{1}% }\sum_{l=1}^{F_{2}}W_{jl}^{(2)}h_{j}^{(1)}h_{l}^{(2)}\\ &\displaystyle-\sum_{l=1}^{F_{2}}\sum_{p=1}^{F_{3}}W_{lp}^{(3)}h_{l}^{(2)}h_{p% }^{(3)}-\sum_{i=1}^{D}b_{i}v_{i}-\sum_{j=1}^{F_{1}}b_{j}^{(1)}h_{j}^{(1)}-\sum% _{l=1}^{F_{2}}b_{l}^{(2)}h_{l}^{(2)}-\sum_{p=1}^{F_{3}}b_{p}^{(3)}h_{p}^{(3)}% \end{aligned}
  38. P ( 𝐯 ; θ ) = 1 𝒵 ( θ ) 𝐡 exp ( - E ( 𝐯 , 𝐡 ( 1 ) , 𝐡 ( 2 ) , 𝐡 ( 3 ) ; θ ) ) P(\mathbf{v};\theta)=\frac{1}{\mathcal{Z}(\theta)}\sum_{\mathbf{h}}\mathrm{exp% }(-E(\mathbf{v},\mathbf{h}^{(1)},\mathbf{h}^{(2)},\mathbf{h}^{(3)};\theta))
  39. P ( 𝐯 m , 𝐯 t ; θ ) = 𝐡 ( 2 m ) , 𝐡 ( 2 t ) , 𝐡 ( 3 ) P ( 𝐡 ( 2 m ) , 𝐡 ( 2 t ) , 𝐡 ( 3 ) ) ( 𝐡 ( 1 m ) P ( 𝐯 m , 𝐡 ( 1 m ) | 𝐡 ( 2 m ) ) ) ( 𝐡 ( 1 t ) P ( 𝐯 t , 𝐡 ( 1 t ) | 𝐡 ( 2 t ) ) ) = 1 𝒵 M ( θ ) 𝐡 exp ( k j W k j ( 1 t ) v k t h j ( 1 t ) + j l W j l ( 2 t ) h j ( 1 t ) h l ( 2 t ) + k b k t v k t + M j b j ( 1 t ) h j ( 1 t ) + l b l ( 2 t ) h l ( 2 t ) - i ( v i m - b i m ) 2 2 σ 2 + i j v i m σ i W i j ( 1 m ) h j ( 1 m ) + j l W j l ( 2 m ) h j ( 1 m ) h l ( 2 m ) + j b j ( 1 m ) h j ( 1 m ) + l b l ( 2 m ) h l ( 2 m ) + l p W ( 3 t ) h l ( 2 t ) h p ( 3 ) + l p W ( 3 m ) h l ( 2 m ) h p ( 3 ) + p b p ( 3 ) h p ( 3 ) \begin{aligned}\displaystyle P(\mathbf{v}^{m},\mathbf{v}^{t};\theta)&% \displaystyle=\sum_{\mathbf{h}^{(2m)},\mathbf{h}^{(2t)},\mathbf{h}^{(3)}}P(% \mathbf{h}^{(2m)},\mathbf{h}^{(2t)},\mathbf{h}^{(3)})(\sum_{\mathbf{h}^{(1m)}}% P(\mathbf{v}_{m},\mathbf{h}^{(1m)}|\mathbf{h}^{(2m)}))(\sum_{\mathbf{h}^{(1t)}% }P(\mathbf{v}^{t},\mathbf{h}^{(1t)}|\mathbf{h}^{(2t)}))\\ &\displaystyle=\frac{1}{\mathcal{Z}_{M}(\theta)}\sum_{\mathbf{h}}\mathrm{exp}(% \sum_{kj}W_{kj}^{(1t)}v_{k}^{t}h_{j}^{(1t)}\\ &\displaystyle+\sum_{jl}W_{jl}^{(2t)}h_{j}^{(1t)}h_{l}^{(2t)}+\sum_{k}b_{k}^{t% }v_{k}^{t}+M\sum_{j}b_{j}^{(1t)}h_{j}^{(1t)}+\sum_{l}b_{l}^{(2t)}h_{l}^{(2t)}% \\ &\displaystyle-\sum_{i}\frac{(v_{i}^{m}-b_{i}^{m})^{2}}{2\sigma^{2}}+\sum_{ij}% \frac{v_{i}^{m}}{\sigma_{i}}W_{ij}^{(1m)}h_{j}^{(1m)}\\ &\displaystyle+\sum_{jl}W_{jl}^{(2m)}h_{j}^{(1m)}h_{l}^{(2m)}+\sum_{j}b_{j}^{(% 1m)}h_{j}^{(1m)}+\sum_{l}b_{l}^{(2m)}h_{l}{(2m)}\\ &\displaystyle+\sum_{lp}W^{(3t)}h_{l}^{(2t)}h_{p}^{(3)}+\sum_{lp}W^{(3m)}h_{l}% ^{(2m)}h_{p}^{(3)}+\sum_{p}b_{p}^{(3)}h_{p}^{(3)}\end{aligned}
  40. p ( h j ( 1 m ) = 1 | 𝐯 m , 𝐡 ( 2 m ) ) = g ( i = 1 D W i j ( 1 m ) v i m σ i + l = 1 F 2 m W j l ( 2 m ) h l ( 2 m ) + b j ( 1 m ) ) p(h_{j}^{(1m)}=1|\mathbf{v}^{m},\mathbf{h}^{(2m)})=g(\sum_{i=1}^{D}W_{ij}^{(1m% )}\frac{v_{i}^{m}}{\sigma_{i}}+\sum_{l=1}^{F_{2}^{m}}W_{jl}^{(2m)}h_{l}^{(2m)}% +b_{j}^{(1m)})
  41. p ( h l ( 2 m ) = 1 | 𝐡 ( 1 m ) , 𝐡 ( 3 ) ) = g ( j = 1 F 1 m W j l ( 2 m ) h j ( 1 m ) + p = 1 F 3 W l p ( 3 m ) h p ( 3 ) + b l ( 2 m ) ) p(h_{l}^{(2m)}=1|\mathbf{h}^{(1m)},\mathbf{h}^{(3)})=g(\sum_{j=1}^{F_{1}^{m}}W% _{jl}^{(2m)}h_{j}^{(1m)}+\sum_{p=1}^{F_{3}}W_{lp}^{(3m)}h_{p}^{(3)}+b_{l}^{(2m% )})
  42. p ( h j ( 1 t ) = 1 | 𝐯 t , 𝐡 ( 2 t ) ) = g ( k = 1 K W k l ( 1 t ) v k ( t ) + l = 1 F 2 t W j l ( 2 t ) h l ( 2 t ) + M b j ( 1 t ) ) p(h_{j}^{(1t)}=1|\mathbf{v}^{t},\mathbf{h}^{(2t)})=g(\sum_{k=1}^{K}W_{kl}^{(1t% )}v_{k}^{(t)}+\sum_{l=1}^{F_{2}^{t}}W_{jl}^{(2t)}h_{l}^{(2t)}+Mb_{j}^{(1t)})
  43. p ( h l ( 2 t ) = 1 | 𝐡 ( 1 t ) , 𝐡 ( 3 ) ) = g ( j = 1 F 1 t W j l ( 2 t ) h j ( 1 t ) + p = 1 F 3 W l p ( 3 t ) h p ( 3 ) + b l ( 2 t ) ) p(h_{l}^{(2t)}=1|\mathbf{h}^{(1t)},\mathbf{h}^{(3)})=g(\sum_{j=1}^{F_{1}^{t}}W% _{jl}^{(2t)}h_{j}^{(1t)}+\sum_{p=1}^{F_{3}}W_{lp}^{(3t)}h_{p}^{(3)}+b_{l}^{(2t% )})
  44. p ( h p 3 ) = 1 | 𝐡 ( 2 ) ) = g ( l = 1 F 2 m W l p ( 3 m ) h l ( 2 m ) + l = 1 F 2 t W l p ( 3 t ) h l ( 2 t ) + b p ( 3 ) ) p(h_{p}^{3)}=1|\mathbf{h}^{(2)})=g(\sum_{l=1}^{F_{2}^{m}}W_{lp}^{(3m)}h_{l}^{(% 2m)}+\sum_{l=1}^{F_{2}^{t}}W_{lp}^{(3t)}h_{l}^{(2t)}+b_{p}^{(3)})
  45. p ( v i k t = 1 | 𝐡 ( 1 t ) ) = exp ( j = 1 F 1 t h j ( 1 t ) W j k ( 1 t ) + b k t ) q = 1 K exp ( j = 1 F 1 t h j ( 1 t ) W j q ( 1 t ) + b k t ) p(v_{ik}^{t}=1|\mathbf{h}^{(1t)})=\frac{\mathrm{exp}(\sum_{j=1}^{F_{1}^{t}}h_{% j}^{(1t)}W_{jk}^{(1t)}+b_{k}^{t})}{\sum_{q=1}^{K}\mathrm{exp}(\sum_{j=1}^{F_{1% }^{t}}h_{j}^{(1t)}W_{jq}^{(1t)}+b_{k}^{t})}
  46. p ( v i m | 𝐡 ( 1 m ) ) 𝒩 ( σ i j = 1 F 1 m W i j ( 1 m ) h j ( 1 m ) + b i m , σ i 2 ) p(v_{i}^{m}|\mathbf{h}^{(1m)})\sim\mathcal{N}(\sigma_{i}\sum_{j=1}^{F_{1}^{m}}% W_{ij}^{(1m)}h_{j}^{(1m)}+b_{i}^{m},\sigma_{i}^{2})

Multiple_factor_analysis.html

  1. j j
  2. j j
  3. λ 1 j \lambda_{1}^{j}
  4. j j
  5. 1 / λ 1 j 1/\lambda_{1}^{j}
  6. j j
  7. A A
  8. B B
  9. C 1 C_{1}
  10. C 2 C_{2}
  11. 1 1
  12. 2 2
  13. 3 3
  14. 4 4
  15. 5 5
  16. 6 6
  17. F 1 F_{1}
  18. F 2 F_{2}
  19. F 1 F_{1}
  20. F 1 F_{1}
  21. i i
  22. k k
  23. i i
  24. k k
  25. K K
  26. I I
  27. J J
  28. I I
  29. J J
  30. K K
  31. N i j N_{i}^{j}
  32. I I
  33. j j
  34. i j , j = 1 , J {i^{j},j=1,J}
  35. j j
  36. N i j N_{i}^{j}
  37. N i j N_{i}^{j}
  38. j j
  39. s s
  40. j j
  41. s s
  42. j j
  43. s s

Multiple_kernel_learning.html

  1. n n
  2. K K
  3. K = i = 1 n β i K i K^{\prime}=\sum_{i=1}^{n}\beta_{i}K_{i}
  4. β i \beta_{i}
  5. X X
  6. Y Y
  7. min β , c \Epsilon ( Y , K c ) + R ( K , c ) \min_{\beta,c}\Epsilon(Y,K^{\prime}c)+R(K,c)
  8. \Epsilon \Epsilon
  9. R R
  10. \Epsilon \Epsilon
  11. R R
  12. n \ell_{n}
  13. k ( ( x 1 i , x 1 j ) , ( x 2 i , x 2 j ) ) = k ( x 1 i , x 2 i ) k ( x 1 j , x 2 j ) + k ( x 1 i , x 2 j ) k ( x 1 j , x 2 i ) k((x_{1i},x_{1j}),(x_{2i},x_{2j}))=k(x_{1i},x_{2i})k(x_{1j},x_{2j})+k(x_{1i},x% _{2j})k(x_{1j},x_{2i})
  14. π m \pi_{m}
  15. K m K_{m}
  16. δ \delta
  17. β m = π m - δ h = 1 n ( π h - δ ) \beta_{m}=\frac{\pi_{m}-\delta}{\sum_{h=1}^{n}(\pi_{h}-\delta)}
  18. A ( K 1 , K 2 ) = < K 1 , K 2 > < K 1 , K 1 > < K 2 , K 2 > A(K_{1},K_{2})=\frac{<K_{1},K_{2}>}{\sqrt{<K_{1},K_{1}><K_{2},K_{2}>}}
  19. β m = A ( K m , Y Y T ) h = 1 n A ( K h , Y Y T ) \beta_{m}=\frac{A(K_{m},YY^{T})}{\sum_{h=1}^{n}A(K_{h},YY^{T})}
  20. max β , tr ( K t r a ) = 1 , K 0 A ( K t r a , Y Y T ) . \max_{\beta,\operatorname{tr}(K^{\prime}_{tra})=1,K^{\prime}\geq 0}A(K^{\prime% }_{tra},YY^{T}).
  21. K t r a K^{\prime}_{tra}
  22. ω ( K ) \omega(K)
  23. min tr ( K t r a ) = c ω ( K t r a ) \min_{\operatorname{tr}(K^{\prime}_{tra})=c}\omega(K^{\prime}_{tra})
  24. c c
  25. f ( x ) = i = 0 n α i m = 1 p η m K m ( x i m , x m ) f(x)=\sum^{n}_{i=0}\alpha_{i}\sum^{p}_{m=1}\eta_{m}K_{m}(x_{i}^{m},x^{m})
  26. η \eta
  27. α \alpha
  28. f ( x ) = i = 1 N m = 1 P α i m K m ( x i m , x m ) + b f(x)=\sum_{i=1}^{N}\sum_{m=1}^{P}\alpha_{i}^{m}K_{m}(x_{i}^{m},x^{m})+b
  29. α i m \alpha_{i}^{m}
  30. b b
  31. α i \alpha_{i}
  32. b b
  33. L = ( x i , y i ) L={(x_{i},y_{i})}
  34. U = x i U={x_{i}}
  35. f ( x ) = α 0 + i = 1 | L | α i K i ( x ) f(x)=\alpha_{0}+\sum_{i=1}^{|L|}\alpha_{i}K_{i}(x)
  36. min f L ( f ) + λ R ( f ) + γ Θ ( f ) \min_{f}L(f)+\lambda R(f)+\gamma\Theta(f)
  37. L L
  38. R R
  39. Θ \Theta
  40. g m π ( x ) = < ϕ m π , ψ m ( x ) Align g t ; g^{\pi}_{m}(x)=<\phi^{\pi}_{m},\psi_{m}(x)&gt;
  41. ψ m ( x ) = [ K m ( x 1 , x ) , , K m ( x L , x ) ] T \psi_{m}(x)=[K_{m}(x_{1},x),\ldots,K_{m}(x_{L},x)]^{T}
  42. ϕ m π \phi^{\pi}_{m}
  43. Π \Pi
  44. q m p i ( y | g m π ( x ) ) q^{pi}_{m}(y|g^{\pi}_{m}(x))
  45. p m π ( f ( x ) | g m π ( x ) ) p^{\pi}_{m}(f(x)|g^{\pi}_{m}(x))
  46. Θ = 1 Π π = 1 Π m = 1 M D ( q m p i ( y | g m π ( x ) ) | | p m π ( f ( x ) | g m π ( x ) ) ) \Theta=\frac{1}{\Pi}\sum^{\Pi}_{\pi=1}\sum^{M}_{m=1}D(q^{pi}_{m}(y|g^{\pi}_{m}% (x))||p^{\pi}_{m}(f(x)|g^{\pi}_{m}(x)))
  47. D ( Q | | P ) = i Q ( i ) ln Q ( i ) P ( i ) D(Q||P)=\sum_{i}Q(i)\ln\frac{Q(i)}{P(i)}
  48. U = x i U={x_{i}}
  49. K = i = 1 M β i K m K^{\prime}=\sum_{i=1}^{M}\beta_{i}K_{m}
  50. B i B_{i}
  51. x i x_{i}
  52. i = 1 n x i - x j B i K ( x i , x j ) x j 2 \sum^{n}_{i=1}\left\|x_{i}-\sum_{x_{j}\in B_{i}}K(x_{i},x_{j})x_{j}\right\|^{2}
  53. i = 1 n x j B i K ( x i , x j ) x i - x j 2 \sum_{i=1}^{n}\sum_{x_{j}\in B_{i}}K(x_{i},x_{j})\left\|x_{i}-x_{j}\right\|^{2}
  54. min β , B i = 1 n x i - x j B i K ( x i , x j ) x j 2 + γ 1 i = 1 n x j B i K ( x i , x j ) x i - x j 2 + γ 2 i | B i | \min_{\beta,B}\sum^{n}_{i=1}\left\|x_{i}-\sum_{x_{j}\in B_{i}}K(x_{i},x_{j})x_% {j}\right\|^{2}+\gamma_{1}\sum_{i=1}^{n}\sum_{x_{j}\in B_{i}}K(x_{i},x_{j})% \left\|x_{i}-x_{j}\right\|^{2}+\gamma_{2}\sum_{i}|B_{i}|
  55. D 0 , 1 n × n D\in{0,1}^{n\times n}
  56. D i j = 1 D_{ij}=1
  57. x i x_{i}
  58. x j x_{j}
  59. B i = x j : D i j = 1 B_{i}={x_{j}:D_{ij}=1}
  60. K K
  61. B i B_{i}
  62. 1 \ell_{1}
  63. 2 \ell_{2}
  64. p p

Multiplicative_sequence.html

  1. i p i z i = i p i z i i p i ′′ z i \sum_{i}p_{i}z^{i}=\sum_{i}p^{\prime}_{i}z^{i}\cdot\sum_{i}p^{\prime\prime}_{i% }z^{i}
  2. i K i ( p 1 , , p i ) z i = j K j ( p 1 , , p j ) z j k K k ( p 1 ′′ , , p k ′′ ) z k \sum_{i}K_{i}(p_{1},\ldots,p_{i})z^{i}=\sum_{j}K_{j}(p^{\prime}_{1},\ldots,p^{% \prime}_{j})z^{j}\cdot\sum_{k}K_{k}(p^{\prime\prime}_{1},\ldots,p^{\prime% \prime}_{k})z^{k}
  3. K n ( 1 , 0 , , 0 ) z n \sum K_{n}(1,0,\ldots,0)z^{n}
  4. i = 1 m Q ( β i z ) \prod_{i=1}^{m}Q(\beta_{i}z)
  5. Q ( z ) = z tanh z = 1 - k = 1 ( - 1 ) k 2 2 k ( 2 k ) ! B k z k Q(z)=\frac{\sqrt{z}}{\tanh\sqrt{z}}=1-\sum_{k=1}^{\infty}(-1)^{k}\frac{2^{2k}}% {(2k)!}B_{k}z^{k}
  6. Q ( z ) = 2 z sinh 2 z Q(z)=\frac{2\sqrt{z}}{\sinh 2\sqrt{z}}
  7. Q ( z ) = z 1 - exp ( - z ) = 1 + x 2 - k = 1 ( - 1 ) k B k ( 2 k ) ! z 2 k Q(z)=\frac{z}{1-\exp(-z)}=1+\frac{x}{2}-\sum_{k=1}^{\infty}(-1)^{k}\frac{B_{k}% }{(2k)!}z^{2k}
  8. z 1 - exp ( - z ) \frac{z}{1-\exp(-z)}

Multirate_Filter_Bank_and_Multidimensional_Directional_Filter_Banks.html

  1. M M
  2. M t h M^{th}
  3. M M
  4. x ( n ) M = x ( M . n ) {x(n)}_{\downarrow{}M}=x(M.n)
  5. X ( z ) M = 1 M m = 0 M - 1 X ( z 1 M ) X(z)_{\downarrow M}=\frac{1}{M}\sum_{m=0}^{M-1}X(z^{\frac{1}{M}})
  6. x ( n ) M = { x ( n M ) 0 n M o t h e r w i s e x(n)_{\uparrow M}=\begin{cases}\begin{array}[]{c}x(\frac{n}{M})\\ 0\end{array}&\begin{array}[]{c}\frac{n}{M}\\ otherwise\end{array}\end{cases}
  7. X ( z ) M = X ( z M ) {X(z)}_{\uparrow{}M}=X(z^{M})
  8. x ( n ) x\left(n\right)
  9. x 1 ( n ) , x 2 ( n ) , x 3 ( n ) , x_{1}(n),x_{2}(n),x_{3}(n),...
  10. x ( n ) x\left(n\right)
  11. x 1 ( n ) , x 2 ( n ) , x 3 ( n ) , x_{1}(n),x_{2}(n),x_{3}(n),...
  12. B W 1 , B W 2 , B W 3 , BW_{1},BW_{2},BW_{3},...
  13. f c 1 , f c 2 , f c 3 , f_{c1},f_{c2},f_{c3},...
  14. H k ( z ) H_{k}(z)
  15. F k ( z ) F_{k}(z)
  16. { h k [ n ] } k = 1 K \left\{h_{k}[n]\right\}_{k=1}^{K}
  17. { g k [ n ] } k = 1 K \left\{g_{k}[n]\right\}_{k=1}^{K}
  18. { M k [ n ] } k = 1 K \left\{M_{k}[n]\right\}_{k=1}^{K}
  19. l 2 ( Z d ) l^{2}(Z^{d})
  20. φ k , m [ n ] = d e f h k * [ M k m - n ] \varphi_{k,m}[n]\stackrel{def}{=}h_{k}^{*}[M_{k}m-n]
  21. 1 k K 1\leq k\leq K
  22. m Z 2 m\in Z^{2}
  23. g k [ n ] g_{k}[n]
  24. ψ k , m [ n ] = d e f h k * [ M k m - n ] \psi_{k,m}[n]\stackrel{def}{=}h_{k}^{*}[M_{k}m-n]
  25. c k [ m ] = < x [ n ] , φ k , m [ n ] Align g t ; c_{k}[m]=<x[n],\varphi_{k,m}[n]&gt;
  26. x ^ [ n ] = 1 k K , m Z 2 c k [ m ] ψ k , m [ n ] \hat{x}[n]=\sum_{1\leq k\leq K,m\in Z^{2}}c_{k}[m]\psi_{k,m}[n]
  27. x ^ [ n ] = 1 k K , m Z 2 < x [ n ] , φ k , m [ n ] > ψ k , m [ n ] \hat{x}[n]=\sum_{1\leq k\leq K,m\in Z^{2}}<x[n],\varphi_{k,m}[n]>\psi_{k,m}[n]
  28. x [ n ] = x [ n ] ^ x[n]=\hat{x[n]}
  29. x [ n ] x[n]
  30. y j [ n ] , y_{j}[n],
  31. j = 0 , 1 , , N - 1 j=0,1,...,N-1
  32. y j [ n ] y_{j}[n]
  33. H ( z ) H(z)
  34. G ( z ) G(z)
  35. N × M N\times M
  36. M × N M\times N
  37. M = d e f | M | M\stackrel{def}{=}|M|
  38. H ( z ) H(z)
  39. G ( z ) G(z)
  40. F ( z ) = k Z d f [ k ] z k = k Z d f [ k 1 , , k d ] z 1 k 1 z d k d F(z)=\sum_{k\in Z^{d}}f[k]z^{k}=\sum_{k\in Z^{d}}f[k_{1},...,k_{d}]z_{1}^{k_{1% }}...z_{d}^{k_{d}}
  41. G ( z ) H ( z ) = I | M | G(z)H(z)=I_{|M|}
  42. G ( z ) H ( z ) = I | M | G(z)H(z)=I_{|M|}
  43. M o d u l e { h 1 ( z ) , , h N ( z ) } = d e f { c 1 ( z ) h 1 ( z ) + + c N ( z ) h N ( z ) } Module\left\{h_{1}(z),...,h_{N}(z)\right\}\stackrel{def}{=}\{c_{1}(z)h_{1}(z)+% ...+c_{N}(z)h_{N}(z)\}
  44. : c 1 ( z ) , , c N ( z ) :c_{1}(z),...,c_{N}(z)
  45. { b 1 ( z ) , , b N ( z ) } \left\{b_{1}(z),...,b_{N}(z)\right\}
  46. { h 1 ( z ) , , h N ( z ) } \left\{h_{1}(z),...,h_{N}(z)\right\}
  47. b i ( z ) b_{i}(z)
  48. h j ( z ) h_{j}(z)
  49. K × N K\times N
  50. W i j ( z ) W_{ij}(z)
  51. b i ( z ) = j = 1 N W i j ( z ) h j ( z ) , i = 1 , , K b_{i}(z)=\sum_{j=1}^{N}W_{ij}(z)h_{j}(z),i=1,...,K
  52. 2 l 2^{l}

Multiscale_turbulence.html

  1. C ϵ C_{\epsilon}
  2. ε = C ε 𝒰 3 \varepsilon=C_{\varepsilon}\frac{\mathcal{U}^{3}}{\mathcal{L}}
  3. C ϵ R e I m R e L n C_{\epsilon}\propto\frac{Re_{I}^{m}}{Re_{L}^{n}}
  4. m 1 n m\approx 1\approx n
  5. R e I Re_{I}
  6. R e L Re_{L}