wpmath0000013_7

Lectionary_147.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Lectionary_184.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Lectionary_60.html

  1. 𝔓 \mathfrak{P}

Lefschetz_theorem_on_(1,1)-classes.html

  1. 𝒥 \mathcal{J}
  2. 0 𝐙 ¯ 2 π i 𝒪 X exp 𝒪 X × 0. 0\to\underline{\mathbf{Z}}\stackrel{2\pi i}{\longrightarrow}\mathcal{O}_{X}% \stackrel{\operatorname{exp}}{\longrightarrow}\mathcal{O}_{X}^{\times}\to 0.
  3. H 1 ( X , 𝒪 X × ) c 1 H 2 ( X , 𝐙 ) i * H 2 ( X , 𝒪 X ) . H^{1}(X,\mathcal{O}_{X}^{\times})\stackrel{c_{1}}{\to}H^{2}(X,\mathbf{Z})% \stackrel{i_{*}}{\to}H^{2}(X,\mathcal{O}_{X}).
  4. H 1 ( X , 𝒪 X × ) H^{1}(X,\mathcal{O}_{X}^{\times})
  5. H 2 ( X , 𝒪 X ) H 0 , 2 ( X ) H^{2}(X,\mathcal{O}_{X})\cong H^{0,2}(X)

Lehmer_code.html

  1. n ! = n × ( n - 1 ) × × 2 × 1 n!=n\times(n-1)\times\cdots\times 2\times 1
  2. n 1 n−1
  3. L ( σ ) = ( L ( σ ) 1 , , L ( σ ) n ) where L ( σ ) i = # { j > i : σ j < σ i } , L(\sigma)=(L(\sigma)_{1},\ldots,L(\sigma)_{n})\quad\,\text{where}\quad L(% \sigma)_{i}=\#\{j>i:\sigma_{j}<\sigma_{i}\},
  4. n i n−i
  5. n + 1 i n+1−i
  6. i Align l t ; j i&lt;j
  7. n i n−i
  8. n n
  9. n 1 n−1
  10. n 2 n−2
  11. j > i j>i
  12. n 1 n−1
  13. 𝟏 5 0 6 3 4 2 1 𝟒 0 5 2 3 1 1 4 𝟎 4 2 3 1 1 4 0 𝟑 1 2 0 1 4 0 3 𝟏 2 0 1 4 0 3 1 𝟏 0 1 4 0 3 1 1 𝟎 \begin{matrix}\mathbf{1}&5&0&6&3&4&2\\ 1&\mathbf{4}&0&5&2&3&1\\ 1&4&\mathbf{0}&4&2&3&1\\ 1&4&0&\mathbf{3}&1&2&0\\ 1&4&0&3&\mathbf{1}&2&0\\ 1&4&0&3&1&\mathbf{1}&0\\ 1&4&0&3&1&1&\mathbf{0}\\ \end{matrix}
  14. n 1 n−1
  15. [ n ] × [ n - 1 ] × × [ 2 ] × [ 1 ] [n]\times[n-1]\times\cdots\times[2]\times[1]
  16. { 0 , 1 , , k - 1 } \{0,1,\ldots,k-1\}
  17. n n + 1 i nn+1−i
  18. = =
  19. 𝔖 n \scriptstyle\ \mathfrak{S}_{n}
  20. { ω B ( k ) } { L ( k , ω ) = 1 } and { ω H ( k ) } { L ( k , ω ) = k } . \{\omega\in B(k)\}\Leftrightarrow\{L(k,\omega)=1\}\quad\,\text{and}\quad\{% \omega\in H(k)\}\Leftrightarrow\{L(k,\omega)=k\}.
  21. N b ( ω ) = 1 k n 1 1 B ( k ) and N b ( ω ) = 1 k n 1 1 H ( k ) . N_{b}(\omega)=\sum_{1\leq k\leq n}\ 1\!\!1_{B(k)}\quad\,\text{and}\quad N_{b}(% \omega)=\sum_{1\leq k\leq n}\ 1\!\!1_{H(k)}.
  22. [ [ 1 , k ] ] , \scriptstyle\ [\![1,k]\!],
  23. ( B ( k ) ) = ( L ( k ) = 1 ) = ( H ( k ) ) = ( L ( k ) = k ) = 1 k . \mathbb{P}(B(k))=\mathbb{P}(L(k)=1)=\mathbb{P}(H(k))=\mathbb{P}(L(k)=k)=\tfrac% {1}{k}.
  24. 1 1 B ( k ) 1\!\!1_{B(k)}
  25. G k ( s ) = k - 1 + s k , G_{k}(s)=\frac{k-1+s}{k},
  26. G ( s ) = k = 1 n G k ( s ) = ( s ) n n ! , G(s)=\prod_{k=1}^{n}G_{k}(s)\ =\ \frac{(s)_{\uparrow n}}{n!},

Leontief_utilities.html

  1. u ( x 1 , , x C ) = min { x 1 a 1 , , x C a C } u(x_{1},\ldots,x_{C})=\min\left\{\frac{x_{1}}{a_{1}},\ldots,\frac{x_{C}}{a_{C}% }\right\}

Leopoldt's_conjecture.html

  1. U 1 = P | p U 1 , P . U_{1}=\prod_{P|p}U_{1,P}.
  2. E 1 E_{1}
  3. r 1 + r 2 - 1 r_{1}+r_{2}-1
  4. r 1 r_{1}
  5. K K
  6. r 2 r_{2}
  7. p \mathbb{Z}_{p}
  8. E 1 E_{1}
  9. U 1 U_{1}
  10. r 1 + r 2 - 1. r_{1}+r_{2}-1.
  11. K K
  12. \mathbb{Q}
  13. \mathbb{Q}

Levi-Civita_field.html

  1. q a q ε q , \sum_{q\in\mathbb{Q}}a_{q}\varepsilon^{q},
  2. a q a_{q}\,
  3. \mathbb{Q}
  4. ε \varepsilon
  5. { q : a q 0 } , \{q\in\mathbb{Q}:a_{q}\neq 0\},
  6. \mathbb{Q}
  7. ε \varepsilon
  8. a 0 a_{0}\,
  9. 7 ε 7\varepsilon
  10. ε \varepsilon
  11. ε 2 \varepsilon^{2}
  12. ε \varepsilon
  13. r ε r\varepsilon
  14. r r
  15. 1 + ε 1+\varepsilon
  16. ε 1 2 \varepsilon^{\frac{1}{2}}
  17. ε \varepsilon
  18. 1 / ε 1/\varepsilon
  19. 1 + ε + 1 2 ε 2 + + 1 n ! ε n + 1+\varepsilon+\frac{1}{2}\varepsilon^{2}+\cdots+\frac{1}{n!}\varepsilon^{n}+\cdots
  20. e ε e^{\varepsilon}
  21. 1 + ε + 2 ε 2 + + n ! ε n + 1+\varepsilon+2\varepsilon^{2}+\cdots+n!\varepsilon^{n}+\cdots
  22. \mathbb{Q}
  23. { q : a q 0 } \{q\in\mathbb{Q}:a_{q}\neq 0\}
  24. 1 + ε 1 / 2 + ε 2 / 3 + ε 3 / 4 + ε 4 / 5 + 1+\varepsilon^{1/2}+\varepsilon^{2/3}+\varepsilon^{3/4}+\varepsilon^{4/5}+\cdots

Levitzky's_theorem.html

  1. { r R a r = 0 } \{r\in R\mid ar=0\}

Lévy_hierarchy.html

  1. Δ 0 = Σ 0 = Π 0 \Delta_{0}=\Sigma_{0}=\Pi_{0}
  2. A A
  3. Σ i + 1 \Sigma_{i+1}
  4. A A
  5. x 1 x n B \exists x_{1}...\exists x_{n}B
  6. B B
  7. Π i \Pi_{i}
  8. Π i + 1 \Pi_{i+1}
  9. A A
  10. x 1 x n B \forall x_{1}...\forall x_{n}B
  11. B B
  12. Σ i \Sigma_{i}
  13. Σ i \Sigma_{i}
  14. Π i \Pi_{i}
  15. Δ i \Delta_{i}
  16. Σ i \Sigma_{i}
  17. Π i \Pi_{i}
  18. Σ i S \Sigma_{i}^{S}
  19. Π i S \Pi_{i}^{S}
  20. Σ i \Sigma_{i}
  21. Π i \Pi_{i}
  22. Σ i \Sigma_{i}
  23. Π i \Pi_{i}
  24. Σ i Z F C \Sigma^{ZFC}_{i}
  25. Π i Z F C \Pi^{ZFC}_{i}

Liability_(financial_accounting).html

  1. Assets = Liabilities + Owner’s Equity \,\text{Assets}=\,\text{Liabilities}+\,\text{Owner's Equity}

Liberman's_lemma.html

  1. γ \gamma
  2. t dist 2 γ ( t ) - t 2 t\mapsto\operatorname{dist}^{2}\circ\gamma(t)-t^{2}\,

Lie_point_symmetry.html

  1. Z Z
  2. Z = ( z 1 , , z n ) Z=(z_{1},\dots,z_{n})
  3. n n
  4. Z Z
  5. δ \delta
  6. ( Z ) \mathbb{R}(Z)
  7. δ : ( Z ) ( Z ) \delta:\mathbb{R}(Z)\rightarrow\mathbb{R}(Z)
  8. \mathbb{R}
  9. ( f 1 , f 2 ) ( Z ) 2 , δ f 1 f 2 = f 1 δ f 2 + f 2 δ f 1 \forall(f_{1},f_{2})\in\mathbb{R}(Z)^{2},\delta f_{1}f_{2}=f_{1}\delta f_{2}+f% _{2}\delta f_{1}
  10. { z 1 , , z n } \left\{\frac{\partial}{\partial z_{1}},\dots,\frac{\partial}{\partial z_{n}}\right\}
  11. δ = i = 1 n ξ z i ( Z ) z i \delta=\sum_{i=1}^{n}\xi_{z_{i}}(Z)\frac{\partial}{\partial z_{i}}
  12. ξ z i \xi_{z_{i}}
  13. ( Z ) \mathbb{R}(Z)
  14. i i
  15. { 1 , , n } \left\{1,\dots,n\right\}
  16. 𝔤 \mathfrak{g}
  17. 𝒟 \mathcal{D}
  18. ( G , + ) (G,+)
  19. M M
  20. 𝒟 : G × M M ν × Z 𝒟 ( ν , Z ) \begin{array}[]{rccc}\mathcal{D}:&G\times M&\rightarrow&M\\ &\nu\times Z&\rightarrow&\mathcal{D}(\nu,Z)\end{array}
  21. Z Z
  22. M M
  23. 𝒟 ( e , Z ) = Z \mathcal{D}(e,Z)=Z
  24. e e
  25. G G
  26. ( ν , ν ^ ) (\nu,\hat{\nu})
  27. G 2 G^{2}
  28. 𝒟 ( ν , 𝒟 ( ν ^ , Z ) ) = 𝒟 ( ν + ν ^ , Z ) \mathcal{D}(\nu,\mathcal{D}(\hat{\nu},Z))=\mathcal{D}(\nu+\hat{\nu},Z)
  29. G G
  30. \mathbb{R}
  31. G G
  32. M M
  33. M = n M=\mathbb{R}^{n}
  34. n n
  35. F = ( f 1 , , f k ) = ( p 1 / q 1 , , p k / q k ) F=(f_{1},\dots,f_{k})=(p_{1}/q_{1},\dots,p_{k}/q_{k})
  36. \mathbb{R}
  37. p i p_{i}
  38. q i q_{i}
  39. [ Z ] \mathbb{R}[Z]
  40. Z = ( z 1 , , z n ) Z=(z_{1},\dots,z_{n})
  41. \mathbb{R}
  42. F F
  43. { p 1 ( Z ) = 0 , p k ( Z ) = 0 and { q 1 ( Z ) 0 , q k ( Z ) 0. \begin{array}[]{ccc}\left\{\begin{array}[]{l}p_{1}(Z)=0,\\ \vdots\\ p_{k}(Z)=0\end{array}\right.&\mbox{and}&\left\{\begin{array}[]{l}q_{1}(Z)\neq 0% ,\\ \vdots\\ q_{k}(Z)\neq 0.\end{array}\right.\end{array}
  44. F = ( f 1 , , f k ) F=(f_{1},\dots,f_{k})
  45. F F
  46. k k
  47. ( f i / z j ) (\partial f_{i}/\partial z_{j})
  48. k k
  49. Z Z
  50. G G
  51. G G
  52. n \mathbb{R}^{n}
  53. F : n k F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}
  54. k n k\leq n
  55. f i ( Z ) = 0 i { 1 , , k } . f_{i}(Z)=0\quad\forall i\in\{1,\dots,k\}.
  56. G G
  57. δ f i ( Z ) = 0 i { 1 , , k } whenever f 1 ( Z ) = = f k ( Z ) = 0 \delta f_{i}(Z)=0\quad\forall i\in\{1,\dots,k\}\mbox{ whenever }~{}f_{1}(Z)=% \dots=f_{k}(Z)=0
  58. δ \delta
  59. 𝔤 \mathfrak{g}
  60. G G
  61. Z = ( P , Q , a , b , c , l ) Z=(P,Q,a,b,c,l)
  62. { f 1 ( Z ) = ( 1 - c P ) + b Q + 1 , f 2 ( Z ) = a P - l Q + 1. \left\{\begin{array}[]{l}f_{1}(Z)=(1-cP)+bQ+1,\\ f_{2}(Z)=aP-lQ+1.\end{array}\right.
  63. δ = a ( a - 1 ) a + ( l + b ) b + ( 2 a c - c ) c + ( - a P + P ) P \delta=a(a-1)\dfrac{\partial}{\partial a}+(l+b)\dfrac{\partial}{\partial b}+(2% ac-c)\dfrac{\partial}{\partial c}+(-aP+P)\dfrac{\partial}{\partial P}
  64. a , b , c a,b,c
  65. P P
  66. δ f 1 = f 1 - f 2 \delta f_{1}=f_{1}-f_{2}
  67. δ f 2 = 0 \delta f_{2}=0
  68. δ f 1 = δ f 2 = 0 \delta f_{1}=\delta f_{2}=0
  69. Z Z
  70. 6 \mathbb{R}^{6}
  71. d / d t d\cdot/dt
  72. t t
  73. X = ( x 1 , , x k ) X=(x_{1},\dots,x_{k})
  74. Θ = ( θ 1 , , θ l ) \Theta=(\theta_{1},\dots,\theta_{l})
  75. Z = ( z 1 , , z n ) = ( t , x 1 , , x k , θ 1 , , θ l ) Z=(z_{1},\dots,z_{n})=(t,x_{1},\dots,x_{k},\theta_{1},\dots,\theta_{l})
  76. n = 1 + k + l n=1+k+l
  77. { d x i d t = f i ( Z ) with f i ( Z ) i { 1 , , k } , d θ j d t = 0 j { 1 , , l } \left\{\begin{array}[]{l}\dfrac{dx_{i}}{dt}=f_{i}(Z)\mbox{ with }~{}f_{i}\in% \mathbb{R}(Z)\quad\forall i\in\{1,\dots,k\},\\ \dfrac{d\theta_{j}}{dt}=0\quad\forall j\in\{1,\dots,l\}\end{array}\right.
  78. F = ( f 1 , , f k ) F=(f_{1},\dots,f_{k})
  79. X X
  80. Θ \Theta
  81. δ \delta
  82. δ = t + i = 1 k f i ( Z ) x i \delta=\dfrac{\partial}{\partial t}+\sum_{i=1}^{k}f_{i}(Z)\dfrac{\partial}{% \partial x_{i}}\cdot
  83. 𝒟 \mathcal{D}
  84. δ 𝒟 \delta_{\mathcal{D}}
  85. 𝒮 \mathcal{S}
  86. 𝒟 \mathcal{D}
  87. 𝒮 \mathcal{S}
  88. 𝒟 \mathcal{D}
  89. δ 𝒮 \delta_{\mathcal{S}}
  90. [ δ 𝒟 , δ 𝒮 ] = λ δ 𝒟 [\delta_{\mathcal{D}},\delta_{\mathcal{S}}]=\lambda\delta_{\mathcal{D}}
  91. λ \lambda
  92. δ 𝒟 \delta_{\mathcal{D}}
  93. δ 𝒮 \delta_{\mathcal{S}}
  94. δ 𝒟 λ = δ 𝒮 λ = 0 \delta_{\mathcal{D}}\lambda=\delta_{\mathcal{S}}\lambda=0
  95. 𝒟 \mathcal{D}
  96. x x
  97. a a
  98. b b
  99. d x d t = ( a - b x ) x , d a d t = d b d t = 0. \dfrac{dx}{dt}=(a-bx)x,\dfrac{da}{dt}=\dfrac{db}{dt}=0.
  100. 𝒟 : ( , + ) × 4 4 ( t ^ , ( t , x , a , b ) ) ( t + t ^ , a x e a t ^ a - ( 1 - e a t ^ ) b x , a , b ) . \begin{array}[]{rccc}\mathcal{D}:&(\mathbb{R},+)\times\mathbb{R}^{4}&% \rightarrow&\mathbb{R}^{4}\\ &(\hat{t},(t,x,a,b))&\rightarrow&\left(t+\hat{t},\frac{axe^{a\hat{t}}}{a-(1-e^% {a\hat{t}})bx},a,b\right).\end{array}
  101. t ^ \hat{t}
  102. \mathbb{R}
  103. δ 𝒟 = t + ( ( a - b x ) x ) x \delta_{\mathcal{D}}=\dfrac{\partial}{\partial t}+((a-bx)x)\dfrac{\partial}{% \partial x}\cdot
  104. 𝒟 \mathcal{D}
  105. δ 𝒮 1 = - x x + b b , δ 𝒮 2 = t t - x x - a a \delta_{\mathcal{S}_{1}}=-x\dfrac{\partial}{\partial x}+b\dfrac{\partial}{% \partial b},\quad\delta_{\mathcal{S}_{2}}=t\dfrac{\partial}{\partial t}-x% \dfrac{\partial}{\partial x}-a\dfrac{\partial}{\partial a}\cdot

Lieb–Liniger_model.html

  1. N N
  2. x x
  3. [ 0 , L ] [0,L]
  4. ψ ( x 1 , x 2 , , x j , , x N ) \psi(x_{1},x_{2},\dots,x_{j},\dots,x_{N})
  5. ψ ( , x i , , x j , ) = ψ ( , x j , , x i , ) \psi(\dots,x_{i},\dots,x_{j},\dots)=\psi(\dots,x_{j},\dots,x_{i},\dots)
  6. i j i\neq j
  7. ψ \psi
  8. ψ ( , x j = 0 , ) = ψ ( , x j = L , ) \psi(\dots,x_{j}=0,\dots)=\psi(\dots,x_{j}=L,\dots)
  9. j j
  10. H = - j = 1 N 2 / x j 2 + 2 c 1 i < j N δ ( x i - x j ) , H=-\sum\nolimits_{j=1}^{N}\partial^{2}/\partial x_{j}^{2}+2c\sum\nolimits_{1% \leq i<j\leq N}\delta(x_{i}-x_{j})\ ,
  11. δ \delta
  12. c 0 c\geq 0
  13. x 1 x_{1}
  14. x 2 x_{2}
  15. x 2 x 1 x_{2}\searrow x_{1}
  16. ( x 2 - x 1 ) ψ ( x 1 , x 2 ) | x 2 = x 1 + = c ψ ( x 1 = x 2 ) (\frac{\partial}{\partial x_{2}}-\frac{\partial}{\partial x_{1}})\psi(x_{1},x_% {2})|_{x_{2}=x_{1}+}=c\psi(x_{1}=x_{2})
  17. c = c=\infty
  18. H ψ = E ψ H\psi=E\psi
  19. ψ \psi
  20. ψ \psi
  21. \mathcal{R}
  22. 0 x 1 x 2 , x N L 0\leq x_{1}\leq x_{2}\leq\dots,\leq x_{N}\leq L
  23. ψ \psi
  24. k 1 < k 2 < < k N k_{1}<k_{2}<\cdots<k_{N}
  25. ψ ( x 1 , , x N ) = P a ( P ) exp ( i j = 1 N k P j x j ) \psi(x_{1},\dots,x_{N})=\sum_{P}a(P)\exp\left(i\sum_{j=1}^{N}k_{Pj}x_{j}\right)
  26. N ! N!
  27. P P
  28. 1 , 2 , , N 1,2,\dots,N
  29. P P
  30. 1 , 2 , , N 1,2,\dots,N
  31. P 1 , P 2 , , P N P1,P2,\dots,PN
  32. a ( P ) a(P)
  33. k k
  34. H ψ = E ψ H\psi=E\psi
  35. E = j = 1 N k j 2 E=\sum\nolimits_{j=1}^{N}\,k_{j}^{2}
  36. a ( P ) = 1 i < j N ( 1 + i c k P i - k P j ) . a(P)=\prod\nolimits_{1\leq i<j\leq N}\left(1+\frac{ic}{k_{Pi}-k_{Pj}}\right)\ .
  37. H H
  38. ψ \psi
  39. k k
  40. N N
  41. L k j = 2 π I j - 2 i = 1 N arctan ( k j - k i c ) for j = 1 , , N , L\,k_{j}=2\pi I_{j}\ -2\sum\nolimits_{i=1}^{N}\arctan\left(\frac{k_{j}-k_{i}}{% c}\right)\qquad\qquad\,\text{for }j=1,\,\dots,\,N\ ,
  42. I 1 < I 2 < < I N I_{1}<I_{2}<\cdots<I_{N}
  43. N N
  44. N N
  45. ± 1 2 , ± 3 2 , \pm\frac{1}{2},\pm\frac{3}{2},\dots
  46. I I
  47. I j + 1 - I j = 1 , for 1 j < N and I 1 = - I N . I_{j+1}-I_{j}=1,\quad{\rm for}\ 1\leq j<N\qquad\,\text{and }I_{1}=-I_{N}.\,
  48. I 1 , , I N - 1 I_{1},\dots,I_{N-1}
  49. I N I_{N}
  50. n > 0 n>0
  51. I 1 I_{1}
  52. n n
  53. p = 2 π n / L p=2\pi n/L
  54. - 2 π n / L -2\pi n/L
  55. 0 < n N / 2 0<n\leq N/2
  56. I i I i + 1 I_{i}\to I_{i}+1
  57. i n i\geq n
  58. p = π - 2 π n / L p=\pi-2\pi n/L
  59. p = - π + 2 π n / L p=-\pi+2\pi n/L
  60. | p | π . |p|\leq\pi.
  61. E 0 E_{0}
  62. E 1 , 2 ( p ) E_{1,2}(p)
  63. ϵ 1 ( p ) = E 1 ( p ) - E 0 \epsilon_{1}(p)=E_{1}(p)-E_{0}
  64. ϵ 2 ( p ) = E 2 ( p ) - E 0 \epsilon_{2}(p)=E_{2}(p)-E_{0}
  65. N N
  66. L L
  67. ρ = N / L \rho=N/L
  68. e = E 0 / N e=E_{0}/N
  69. ϵ 1 , 2 ( p ) \epsilon_{1,2}(p)
  70. N N\to\infty
  71. ρ \rho
  72. c c
  73. x ρ x x\to\rho x
  74. γ = c / ρ \gamma=c/\rho
  75. E 0 E_{0}
  76. k k
  77. K K
  78. L f ( k ) L\,f(k)
  79. f f
  80. - K k K -K\leq k\leq K
  81. 2 c - K K f ( p ) c 2 + ( p - k ) 2 d p = 2 π f ( k ) - 1 and - K K f ( p ) d p = ρ , 2c\int\nolimits_{-K}^{K}\frac{f(p)}{c^{2}+(p-k)^{2}}dp=2\pi f(k)-1\quad{\rm and% }\quad\int\nolimits_{-K}^{K}f(p)dp=\rho\ ,
  82. f f
  83. e = 1 ρ - K K k 2 f ( k ) d k . e=\frac{1}{\rho}\int\nolimits_{-K}^{K}k^{2}f(k)dk.
  84. e e
  85. γ \gamma
  86. e e
  87. γ \gamma
  88. e γ - 4 γ 3 / 2 / π e\approx\gamma-4\gamma^{3/2}/\pi
  89. γ = \gamma=\infty
  90. e = π 2 / 3 e=\pi^{2}/3
  91. ϵ 1 ( p ) \epsilon_{1}(p)
  92. ϵ 2 ( p ) \epsilon_{2}(p)
  93. γ = 0.787 \gamma=0.787
  94. γ > 0 \gamma>0
  95. γ \gamma
  96. e e

Lifting_theory.html

  1. T : L ( X , Σ , μ ) ( X , Σ , μ ) T:L^{\infty}(X,\Sigma,\mu)\to\mathcal{L}^{\infty}(X,\Sigma,\mu)
  2. { ( X , Σ , μ ) L ( X , Σ , μ ) f [ f ] \begin{cases}\mathcal{L}^{\infty}(X,\Sigma,\mu)\to L^{\infty}(X,\Sigma,\mu)\\ f\mapsto[f]\end{cases}
  3. T ( r [ f ] + s [ g ] ) ( p ) = r T [ f ] ( p ) + s T [ g ] ( p ) , p X , r , s 𝐑 ; T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),\qquad\forall p\in X,r,s\in\mathbf{R};
  4. T ( [ f ] × [ g ] ) ( p ) = T [ f ] ( p ) × T [ g ] ( p ) , p X ; T([f]\times[g])(p)=T[f](p)\times T[g](p),\qquad\forall p\in X;
  5. T [ 1 ] = 1. T[1]=1.
  6. ( X , Σ , μ ) \mathcal{L}^{\infty}(X,\Sigma,\mu)
  7. T : L ( X , Σ , μ ) ( X , Σ , μ ) T:L^{\infty}(X,\Sigma,\mu)\to\mathcal{L}^{\infty}(X,\Sigma,\mu)
  8. N := n { p Supp ( μ ) : ( T 0 U n ) ( p ) < U n ( p ) } N:=\bigcup\nolimits_{n}\left\{p\in\mathrm{Supp}(\mu):(T_{0}U_{n})(p)<U_{n}(p)\right\}
  9. ( T [ f ] ) ( p ) := { ( T 0 [ f ] ) ( p ) p N T p [ f ] p N . (T[f])(p):=\begin{cases}(T_{0}[f])(p)&p\notin N\\ T_{p}[f]&p\in N.\end{cases}
  10. Y y λ y Y\ni y\mapsto\lambda_{y}
  11. π - 1 ( { y } ) \pi^{-1}(\{y\})
  12. { y } Φ and λ y ( ( X π - 1 ( { y } ) ) = 0 y Y \{y\}\in\Phi\;\;\mathrm{and}\;\;\lambda_{y}\left((X\setminus\pi^{-1}(\{y\})% \right)=0\qquad\forall y\in Y
  13. X f ( p ) μ ( d p ) = Y ( π - 1 ( { y } ) f ( p ) λ y ( d p ) ) ν ( d y ) ( * ) \int_{X}f(p)\;\mu(dp)=\int_{Y}\left(\int_{\pi^{-1}(\{y\})}f(p)\,\lambda_{y}(dp% )\right)\nu(dy)\qquad(*)
  14. y π - 1 ( { y } ) f ( p ) λ y ( d p ) y\mapsto\int_{\pi^{-1}(\{y\})}f(p)\,\lambda_{y}(dp)
  15. f \lfloor f\rfloor
  16. λ y ( f ) := T ( f ) ( y ) . \lambda_{y}(f):=T(\lfloor f\rfloor)(y).
  17. λ y ( f φ π ) = φ ( y ) λ y ( f ) y Y , φ C b ( Y ) , f L ( X , Σ , μ ) \lambda_{y}(f\cdot\varphi\circ\pi)=\varphi(y)\lambda_{y}(f)\qquad\forall y\in Y% ,\varphi\in C_{b}(Y),f\in L^{\infty}(X,\Sigma,\mu)
  18. ν ( A ) := μ ( π - 1 ( A ) ) \nu(A):=\mu\left(\pi^{-1}(A)\right)

Ligand_K-edge.html

  1. D 0 ( L 1 s ψ * ) = c o n s t | L 1 s | 𝐫 | ψ * | 2 = α 2 c o n s t | L 1 s | 𝐫 | L n p | 2 D_{0}(L\ 1s\rightarrow\psi^{*})=const\ |\langle L\ 1s|\mathbf{r}|\psi^{*}% \rangle|^{2}=\alpha^{2}\ const\ |\langle L\ 1s|\mathbf{r}|L\ np\rangle|^{2}
  2. ψ * \psi^{*}
  3. α 2 \alpha^{2}
  4. ψ * = 1 - α 2 | M d - α | L n p \psi^{*}=\sqrt{1-\alpha^{2}}|M_{d}\rangle-\alpha|L_{np}\rangle
  5. D 0 = α 2 h 3 n I s D_{0}=\frac{\alpha^{2}h}{3n}I_{s}

Light's_associativity_test.html

  1. \star
  2. \circ
  3. \star
  4. \circ
  5. \star
  6. \circ
  7. \star
  8. \circ
  9. \star
  10. \circ
  11. \star
  12. \circ
  13. \star
  14. \circ
  15. \star
  16. \star
  17. \circ
  18. \circ
  19. \star
  20. \circ
  21. \star
  22. \circ
  23. \ldots
  24. \ldots
  25. \mapsto
  26. \ldots
  27. \ldots
  28. \ldots
  29. \ldots
  30. O ( n 2 log 1 δ ) O(n^{2}\log\frac{1}{\delta})
  31. n × n n\times n
  32. δ \delta
  33. a , b , c \langle a,b,c\rangle
  34. ( a b ) c a ( b c ) (ab)c\neq a(bc)
  35. O ( n 2 log n log 1 δ ) O(n^{2}\log n\cdot\log\frac{1}{\delta})

Limit_(mathematics).html

  1. lim n c f ( n ) = L \lim_{n\to c}f(n)=L
  2. f ( n ) L . f(n)\to L\ .
  3. f f
  4. c c
  5. lim x c f ( x ) = L \lim_{x\to c}f(x)=L
  6. f ( x ) f(x)
  7. L L
  8. x x
  9. c c
  10. f f
  11. x x
  12. x x
  13. c c
  14. L L
  15. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  16. f ( x ) f(x)
  17. L L
  18. f ( x ) f(x)
  19. ( L ε , L + ε ) (L−ε,L+ε)
  20. x x
  21. c c
  22. x x
  23. c c
  24. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  25. x x
  26. ( c δ , c ) (c−δ,c)
  27. ( c , c + δ ) (c,c+δ)
  28. c c
  29. 0
  30. x c x≠c
  31. x x
  32. δ δ
  33. c c
  34. f ( c ) L f(c)≠L
  35. f f
  36. c c
  37. f ( x ) = x 2 - 1 x - 1 f(x)=\frac{x^{2}-1}{x-1}
  38. f ( 1 ) f(1)
  39. x x
  40. f ( x ) f(x)
  41. f ( x ) f(x)
  42. x x
  43. 1 1
  44. lim x 1 x 2 - 1 x - 1 = 2 \lim_{x\to 1}\frac{x^{2}-1}{x-1}=2
  45. x 2 - 1 x - 1 = ( x + 1 ) ( x - 1 ) x - 1 = x + 1 \frac{x^{2}-1}{x-1}=\frac{(x+1)(x-1)}{x-1}=x+1
  46. x 1 x≠1
  47. x + 1 x+1
  48. x x
  49. x x
  50. lim x 1 x 2 - 1 x - 1 = 1 + 1 = 2 \lim_{x\to 1}\frac{x^{2}-1}{x-1}=1+1=2
  51. f ( x ) = 2 x - 1 x f(x)={2x-1\over x}
  52. x x
  53. f ( x ) f(x)
  54. f ( x ) f(x)
  55. x x
  56. f ( x ) f(x)
  57. x x
  58. lim x 2 x - 1 x = 2. \lim_{x\to\infty}\frac{2x-1}{x}=2.
  59. L L
  60. lim n a n = L \lim_{n\to\infty}a_{n}=L
  61. ε > 0 ε>0
  62. N N
  63. n > N n>N
  64. L L
  65. n n
  66. a ( n ) a(n)
  67. n n
  68. L L
  69. f ( x ) f(x)
  70. x x
  71. L L
  72. L L
  73. L + 1 / n L+1/n
  74. ( a n ) (a_{n})
  75. a H a_{H}
  76. lim n a n = st ( a H ) \lim_{n\to\infty}a_{n}=\operatorname{st}(a_{H})
  77. a = [ a n ] a=[a_{n}]
  78. ( a n ) (a_{n})
  79. st ( a ) = lim n a n \operatorname{st}(a)=\lim_{n\to\infty}a_{n}
  80. p n {{p}_{n}}
  81. n n
  82. 0
  83. \infty
  84. p p
  85. p n p {p}_{n}\neq p
  86. n n
  87. λ \lambda
  88. α \alpha
  89. lim n | p n + 1 - p | | p n - p | α = λ \lim_{n\rightarrow\infty}\frac{\left|{p}_{n+1}-p\right|}{{\left|{p}_{n}-p% \right|}^{\alpha}}=\lambda
  90. p n {{p}_{n}}
  91. n n
  92. 0
  93. \infty
  94. p p
  95. α \alpha
  96. λ \lambda
  97. f f
  98. p p
  99. p n p_{n}
  100. f ( p ) = p f(p)=p
  101. | f ( p ) | \left|f^{\prime}(p)\right|
  102. | f ( p ) | ( 0 , 1 ) \left|f^{\prime}(p)\right|\in(0,1)
  103. | f ( p ) | > 1 \left|f^{\prime}(p)\right|>1
  104. | f ( p ) | = 0 \left|f^{\prime}(p)\right|=0
  105. | f ′′ ( p ) | \left|f^{\prime\prime}(p)\right|
  106. | f ′′ ( p ) | 0 \left|f^{\prime\prime}(p)\right|\neq 0
  107. f ′′ ( p ) f^{\prime\prime}(p)
  108. | f ′′ ( p ) | = 0 \left|f^{\prime\prime}(p)\right|=0
  109. | f ′′ ( p ) | \left|f^{\prime\prime}(p)\right|

Limiting_similarity.html

  1. d x 1 d t = r 1 x 1 ( K 1 - x 1 - α 12 x 2 K 1 ) {dx_{1}\over dt}=r_{1}x_{1}\left({K_{1}-x_{1}-\alpha_{12}x_{2}\over K_{1}}\right)
  2. d x 2 d t = r 2 x 2 ( K 2 - x 2 - α 21 x 1 K 2 ) . {dx_{2}\over dt}=r_{2}x_{2}\left({K_{2}-x_{2}-\alpha_{21}x_{1}\over K_{2}}% \right).

Lindhard_theory.html

  1. ϵ ( q , ω ) = 1 - V q k f k - q - f k ( ω + i δ ) + E k - q - E k . \epsilon(q,\omega)=1-V_{q}\sum_{k}{\frac{f_{k-q}-f_{k}}{\hbar(\omega+i\delta)+% E_{k-q}-E_{k}}}.
  2. V q V_{q}
  3. V e f f ( q ) - V i n d ( q ) V_{eff}(q)-V_{ind}(q)
  4. f k f_{k}
  5. q 0 q\to 0
  6. E k - q - E k = 2 2 m ( k 2 - 2 k q + q 2 ) - 2 k 2 2 m - 2 k q m E_{k-q}-E_{k}=\frac{\hbar^{2}}{2m}(k^{2}-2\vec{k}\cdot\vec{q}+q^{2})-\frac{% \hbar^{2}k^{2}}{2m}\simeq-\frac{\hbar^{2}\vec{k}\cdot\vec{q}}{m}
  7. f k - q - f k = f k - q k f k + - f k - q k f k f_{k-q}-f_{k}=f_{k}-\vec{q}\cdot\nabla_{k}f_{k}+\cdots-f_{k}\simeq-\vec{q}% \cdot\nabla_{k}f_{k}
  8. δ 0 \delta\to 0
  9. ϵ ( 0 , ω ) \displaystyle\epsilon(0,\omega)
  10. E k = ω k E_{k}=\hbar\omega_{k}
  11. V q = 4 π e 2 ϵ q 2 L 3 V_{q}=\frac{4\pi e^{2}}{\epsilon q^{2}L^{3}}
  12. ω p l 2 = 4 π e 2 N ϵ L 3 m \omega_{pl}^{2}=\frac{4\pi e^{2}N}{\epsilon L^{3}m}
  13. 4 π 4\pi
  14. 1 / ϵ 0 1/\epsilon_{0}
  15. ω + i δ 0 \omega+i\delta\to 0
  16. ϵ ( q , 0 ) = 1 - V q k f k - q - f k E k - q - E k \epsilon(q,0)=1-V_{q}\sum_{k}{\frac{f_{k-q}-f_{k}}{E_{k-q}-E_{k}}}
  17. ϵ ( q , 0 ) = 1 - V q k , i - q i f k i - 2 k q m = 1 - V q k , i q i f k i 2 k q m \epsilon(q,0)=1-V_{q}\sum_{k,i}{\frac{-q_{i}\frac{\partial f}{\partial k_{i}}}% {-\frac{\hbar^{2}\vec{k}\cdot\vec{q}}{m}}}=1-V_{q}\sum_{k,i}{\frac{q_{i}\frac{% \partial f}{\partial k_{i}}}{\frac{\hbar^{2}\vec{k}\cdot\vec{q}}{m}}}
  18. i q i f k k i = - i q i f k μ ϵ k k i = - i q i k i 2 m f k μ \sum_{i}{q_{i}\frac{\partial f_{k}}{\partial k_{i}}}=-\sum_{i}{q_{i}\frac{% \partial f_{k}}{\partial\mu}\frac{\partial\epsilon_{k}}{\partial k_{i}}}=-\sum% _{i}{q_{i}k_{i}\frac{\hbar^{2}}{m}\frac{\partial f_{k}}{\partial\mu}}
  19. ϵ k = 2 k 2 2 m \epsilon_{k}=\frac{\hbar^{2}k^{2}}{2m}
  20. ϵ k k i = 2 k i m \frac{\partial\epsilon_{k}}{\partial k_{i}}=\frac{\hbar^{2}k_{i}}{m}
  21. ϵ ( q , 0 ) \displaystyle\epsilon(q,0)
  22. κ \kappa
  23. κ = 4 π e 2 ϵ n μ \kappa=\sqrt{\frac{4\pi e^{2}}{\epsilon}\frac{\partial n}{\partial\mu}}
  24. V s ( q , ω = 0 ) V q ϵ ( q , ω = 0 ) = 4 π e 2 ϵ q 2 L 3 q 2 + κ 2 q 2 = 4 π e 2 ϵ L 3 1 q 2 + κ 2 V_{s}(q,\omega=0)\equiv\frac{V_{q}}{\epsilon(q,\omega=0)}=\frac{\frac{4\pi e^{% 2}}{\epsilon q^{2}L^{3}}}{\frac{q^{2}+\kappa^{2}}{q^{2}}}=\frac{4\pi e^{2}}{% \epsilon L^{3}}\frac{1}{q^{2}+\kappa^{2}}
  25. V s ( r ) = q 4 π e 2 ϵ L 3 ( q 2 + κ 2 ) e i q r = e 2 ϵ r e - κ r V_{s}(r)=\sum_{q}{\frac{4\pi e^{2}}{\epsilon L^{3}(q^{2}+\kappa^{2})}e^{i\vec{% q}\cdot\vec{r}}}=\frac{e^{2}}{\epsilon r}e^{-\kappa r}
  26. E f = 2 2 m ( 3 π 2 n ) 2 3 E_{f}=\frac{\hbar^{2}}{2m}(3\pi^{2}n)^{\frac{2}{3}}
  27. n = 1 3 π 2 ( 2 m 2 E f ) 3 2 n=\frac{1}{3\pi^{2}}\left(\frac{2m}{\hbar^{2}}E_{f}\right)^{\frac{3}{2}}
  28. E f μ E_{f}\equiv\mu
  29. n μ = 3 2 n E f \frac{\partial n}{\partial\mu}=\frac{3}{2}\frac{n}{E_{f}}
  30. κ = 4 π e 2 ϵ n μ = 6 π e 2 n ϵ E f \kappa=\sqrt{\frac{4\pi e^{2}}{\epsilon}\frac{\partial n}{\partial\mu}}=\sqrt{% \frac{6\pi e^{2}n}{\epsilon E_{f}}}
  31. κ = 4 π e 2 n β ϵ \kappa=\sqrt{\frac{4\pi e^{2}n\beta}{\epsilon}}
  32. q 0 q\to 0
  33. E k - q - E k = 2 2 m ( k 2 - 2 k q + q 2 ) - 2 k 2 2 m - 2 k q m E_{k-q}-E_{k}=\frac{\hbar^{2}}{2m}(k^{2}-2\vec{k}\cdot\vec{q}+q^{2})-\frac{% \hbar^{2}k^{2}}{2m}\simeq-\frac{\hbar^{2}\vec{k}\cdot\vec{q}}{m}
  34. f k - q - f k = f k - q k f k + - k k - q k f k f_{k-q}-f_{k}=f_{k}-\vec{q}\cdot\nabla_{k}f_{k}+\cdots-k_{k}\simeq-\vec{q}% \cdot\nabla_{k}f_{k}
  35. δ 0 \delta\to 0
  36. ϵ ( 0 , ω ) \displaystyle\epsilon(0,\omega)
  37. E k = ϵ k E_{k}=\hbar\epsilon_{k}
  38. V q = 2 π e 2 ϵ q L 2 V_{q}=\frac{2\pi e^{2}}{\epsilon qL^{2}}
  39. ω p l 2 ( q ) = 2 π e 2 n q ϵ m \omega_{pl}^{2}(q)=\frac{2\pi e^{2}nq}{\epsilon m}
  40. ω + i δ 0 \omega+i\delta\to 0
  41. ϵ ( q , 0 ) = 1 - V q k f k - q - f k E k - q - E k \epsilon(q,0)=1-V_{q}\sum_{k}{\frac{f_{k-q}-f_{k}}{E_{k-q}-E_{k}}}
  42. ϵ ( q , 0 ) = 1 - V q k , i - q i f k i - 2 k q m = 1 - V q k , i q i f k i 2 k q m \epsilon(q,0)=1-V_{q}\sum_{k,i}{\frac{-q_{i}\frac{\partial f}{\partial k_{i}}}% {-\frac{\hbar^{2}\vec{k}\cdot\vec{q}}{m}}}=1-V_{q}\sum_{k,i}{\frac{q_{i}\frac{% \partial f}{\partial k_{i}}}{\frac{\hbar^{2}\vec{k}\cdot\vec{q}}{m}}}
  43. i q i f k k i = - i q i f k μ ϵ k k i = - i q i k i 2 m f k μ \sum_{i}{q_{i}\frac{\partial f_{k}}{\partial k_{i}}}=-\sum_{i}{q_{i}\frac{% \partial f_{k}}{\partial\mu}\frac{\partial\epsilon_{k}}{\partial k_{i}}}=-\sum% _{i}{q_{i}k_{i}\frac{\hbar^{2}}{m}\frac{\partial f_{k}}{\partial\mu}}
  44. ϵ k = 2 k 2 2 m \epsilon_{k}=\frac{\hbar^{2}k^{2}}{2m}
  45. ϵ k k i = 2 k i m \frac{\partial\epsilon_{k}}{\partial k_{i}}=\frac{\hbar^{2}k_{i}}{m}
  46. ϵ ( q , 0 ) \displaystyle\epsilon(q,0)
  47. κ \kappa
  48. κ = 2 π e 2 ϵ n μ \kappa=\frac{2\pi e^{2}}{\epsilon}\frac{\partial n}{\partial\mu}
  49. V s ( q , ω = 0 ) V q ϵ ( q , ω = 0 ) = 2 π e 2 ϵ q L 2 q q + κ = 2 π e 2 ϵ L 2 1 q + κ V_{s}(q,\omega=0)\equiv\frac{V_{q}}{\epsilon(q,\omega=0)}=\frac{2\pi e^{2}}{% \epsilon qL^{2}}\frac{q}{q+\kappa}=\frac{2\pi e^{2}}{\epsilon L^{2}}\frac{1}{q% +\kappa}
  50. μ ( n , T ) = 1 β ln ( e 2 β π n / m - 1 ) \mu(n,T)=\frac{1}{\beta}\ln{(e^{\hbar^{2}\beta\pi n/m}-1)}
  51. μ n = 2 π m 1 1 - e - 2 β π n / m \frac{\partial\mu}{\partial n}=\frac{\hbar^{2}\pi}{m}\frac{1}{1-e^{-\hbar^{2}% \beta\pi n/m}}
  52. κ = 2 π e 2 ϵ n μ = 2 π e 2 ϵ m 2 π ( 1 - e - 2 β π n / m ) = 2 m e 2 2 ϵ f k = 0 . \kappa=\frac{2\pi e^{2}}{\epsilon}\frac{\partial n}{\partial\mu}=\frac{2\pi e^% {2}}{\epsilon}\frac{m}{\hbar^{2}\pi}(1-e^{-\hbar^{2}\beta\pi n/m})=\frac{2me^{% 2}}{\hbar^{2}\epsilon}f_{k=0}.
  53. e - k e f f r / r e^{-k_{eff}r}/r

Line_segment.html

  1. \mathbb{R}
  2. \mathbb{C}
  3. L = { 𝐮 + t 𝐯 t [ 0 , 1 ] } L=\{\mathbf{u}+t\mathbf{v}\mid t\in[0,1]\}
  4. 𝐮 , 𝐯 V \mathbf{u},\mathbf{v}\in V\,\!
  5. L = { 𝐮 + t 𝐯 t ( 0 , 1 ) } L=\{\mathbf{u}+t\mathbf{v}\mid t\in(0,1)\}
  6. 𝐮 , 𝐯 V \mathbf{u},\mathbf{v}\in V\,\!
  7. 2 \mathbb{R}^{2}
  8. { ( x , y ) | ( x - c x ) 2 + ( y - c y ) 2 + ( x - a x ) 2 + ( y - a y ) 2 = ( c x - a x ) 2 + ( c y - a y ) 2 } \{(x,y)|\sqrt{(x-c_{x})^{2}+(y-c_{y})^{2}}+\sqrt{(x-a_{x})^{2}+(y-a_{y})^{2}}=% \sqrt{(c_{x}-a_{x})^{2}+(c_{y}-a_{y})^{2}}\}

Linear-fractional_programming.html

  1. maximize 𝐜 T 𝐱 + α 𝐝 T 𝐱 + β subject to A 𝐱 𝐛 , \begin{aligned}\displaystyle\,\text{maximize}&\displaystyle\frac{\mathbf{c}^{T% }\mathbf{x}+\alpha}{\mathbf{d}^{T}\mathbf{x}+\beta}\\ \displaystyle\,\text{subject to}&\displaystyle A\mathbf{x}\leq\mathbf{b},\end{aligned}
  2. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  3. 𝐜 , 𝐝 n \mathbf{c},\mathbf{d}\in\mathbb{R}^{n}
  4. 𝐛 m \mathbf{b}\in\mathbb{R}^{m}
  5. A m × n A\in\mathbb{R}^{m\times n}
  6. α , β \alpha,\beta\in\mathbb{R}
  7. { 𝐱 | 𝐝 T 𝐱 + β > 0 } \{\mathbf{x}|\mathbf{d}^{T}\mathbf{x}+\beta>0\}
  8. 𝐲 = 1 𝐝 T 𝐱 + β 𝐱 ; t = 1 𝐝 T 𝐱 + β \mathbf{y}=\frac{1}{\mathbf{d}^{T}\mathbf{x}+\beta}\cdot\mathbf{x}\;;\;\;t=% \frac{1}{\mathbf{d}^{T}\mathbf{x}+\beta}
  9. maximize \displaystyle\,\text{maximize}
  10. A 𝐲 - 𝐛 t 𝟎 A\mathbf{y}-\mathbf{b}t\leq\mathbf{0}
  11. 𝐝 T 𝐲 + β t - 1 = 0 \mathbf{d}^{T}\mathbf{y}+\beta t-1=0
  12. 𝐮 \mathbf{u}
  13. λ \lambda
  14. minimize λ subject to A T 𝐮 + λ 𝐝 = 𝐜 - 𝐛 T 𝐮 + λ β α 𝐮 + n , λ , \begin{aligned}\displaystyle\,\text{minimize}&\displaystyle\lambda\\ \displaystyle\,\text{subject to}&\displaystyle A^{T}\mathbf{u}+\lambda\mathbf{% d}=\mathbf{c}\\ &\displaystyle-\mathbf{b}^{T}\mathbf{u}+\lambda\beta\geq\alpha\\ &\displaystyle\mathbf{u}\in\mathbb{R}_{+}^{n},\lambda\in\mathbb{R},\end{aligned}

Linear_belief_function.html

  1. M ( X ) = ( μ Σ ) M(X)=\left({\begin{array}[]{*{20}c}\mu\\ \Sigma\\ \end{array}}\right)
  2. M ( X ) = ( μ Σ - 1 - Σ - 1 ) M(\vec{X})=\left({\begin{array}[]{*{20}c}{\mu\Sigma^{-1}}\\ {-\Sigma^{-1}}\\ \end{array}}\right)
  3. M ( X ) M(\vec{X})
  4. M ( X ) = ( μ 0 ) M(X)=\left({\begin{array}[]{*{20}c}\mu\\ 0\\ \end{array}}\right)
  5. M ( X ) = [ 0 0 ] M(\vec{X})=\left[{\begin{array}[]{*{20}c}0\\ 0\\ \end{array}}\right]
  6. M ( X ) M(\vec{X})
  7. M ( X , Y ) = [ μ 1 Σ 11 Σ 21 μ 2 Σ 12 Σ 22 ] M(X,Y)=\left[{\begin{array}[]{*{20}c}{\begin{array}[]{*{20}c}{\mu_{1}}\\ {\Sigma_{11}}\\ {\Sigma_{21}}\\ \end{array}}&{\begin{array}[]{*{20}c}{\mu_{2}}\\ {\Sigma_{12}}\\ {\Sigma_{22}}\\ \end{array}}\\ \end{array}}\right]
  8. M ( X , Y ) = [ μ 1 ( Σ 11 ) - 1 - ( Σ 11 ) - 1 Σ 21 ( Σ 11 ) - 1 μ 2 - μ 1 ( Σ 11 ) - 1 Σ 12 ( Σ 11 ) - 1 Σ 12 Σ 22 - Σ 21 ( Σ 11 ) - 1 Σ 12 ] M(\vec{X},Y)=\left[{\begin{array}[]{*{20}c}{\begin{array}[]{*{20}c}{\mu_{1}(% \Sigma_{11})^{-1}}\\ {-(\Sigma_{11})^{-1}}\\ {\Sigma_{21}(\Sigma_{11})^{-1}}\\ \end{array}}&{\begin{array}[]{*{20}c}{\mu_{2}-\mu_{1}(\Sigma_{11})^{-1}\Sigma_% {12}}\\ {(\Sigma_{11})^{-1}\Sigma_{12}}\\ {\Sigma_{22}-\Sigma_{21}(\Sigma_{11})^{-1}\Sigma_{12}}\\ \end{array}}\\ \end{array}}\right]
  9. μ 1 ( Σ 11 ) - 1 \mu_{1}(\Sigma_{11})^{-1}
  10. - ( Σ 11 ) - 1 -(\Sigma_{11})^{-1}
  11. μ 2 - μ 1 ( Σ 11 ) - 1 Σ 12 \mu_{2}-\mu_{1}(\Sigma_{11})^{-1}\Sigma_{12}
  12. Σ 22 - Σ 21 ( Σ 11 ) - 1 Σ 12 \Sigma_{22}-\Sigma_{21}(\Sigma_{11})^{-1}\Sigma_{12}
  13. ( Σ 11 ) - 1 Σ 12 (\Sigma_{11})^{-1}\Sigma_{12}
  14. M ( X , Y ) M(\vec{X},Y)
  15. M ( X , Y ) = [ 0 0 0 μ 2 0 Σ 22 ] M(\vec{X},Y)=\left[{\begin{array}[]{*{20}c}{\begin{array}[]{*{20}c}0\\ 0\\ 0\\ \end{array}}&{\begin{array}[]{*{20}c}{\mu_{2}}\\ 0\\ {\Sigma_{22}}\\ \end{array}}\\ \end{array}}\right]
  16. μ 1 ( Σ 11 ) - 1 \mu_{1}(\Sigma_{11})^{-1}
  17. - ( Σ 11 ) - 1 -(\Sigma_{11})^{-1}
  18. ( Σ 11 ) - 1 Σ 12 (\Sigma_{11})^{-1}\Sigma_{12}
  19. μ 1 ( Σ 11 ) - 1 Σ 12 \mu_{1}(\Sigma_{11})^{-1}\Sigma_{12}
  20. Σ 21 ( Σ 11 ) - 1 Σ 12 \Sigma_{21}(\Sigma_{11})^{-1}\Sigma_{12}
  21. M ( X , Y ) = [ 0 0 A T b A 0 ] M(\vec{X},Y)=\left[{\begin{array}[]{*{20}c}{\begin{array}[]{*{20}c}0\\ 0\\ {A^{T}}\\ \end{array}}&{\begin{array}[]{*{20}c}b\\ A\\ 0\\ \end{array}}\\ \end{array}}\right]
  22. μ 1 ( Σ 11 ) - 1 \mu_{1}(\Sigma_{11})^{-1}
  23. - ( Σ 11 ) - 1 -(\Sigma_{11})^{-1}
  24. M ( X , Y ) = [ 0 0 A T b A Σ ] M(\vec{X},Y)=\left[{\begin{array}[]{*{20}c}{\begin{array}[]{*{20}c}0\\ 0\\ {A^{T}}\\ \end{array}}&{\begin{array}[]{*{20}c}b\\ A\\ \Sigma\\ \end{array}}\\ \end{array}}\right]
  25. M Y ( X , Y ) = [ μ 2 Σ 22 ] M^{\downarrow Y}(X,Y)=\left[{\begin{array}[]{*{20}c}{\mu_{2}}\\ {\Sigma_{22}}\\ \end{array}}\right]
  26. M ( X , Y ) M(\vec{X},Y)
  27. M Y ( X , Y ) = [ μ 2 - μ 1 ( Σ 11 ) - 1 Σ 12 Σ 22 - Σ 21 ( Σ 11 ) - 1 Σ 12 ] M^{\downarrow Y}(\vec{X},Y)=\left[{\begin{array}[]{*{20}c}{\mu_{2}-\mu_{1}(% \Sigma_{11})^{-1}\Sigma_{12}}\\ {\Sigma_{22}-\Sigma_{21}(\Sigma_{11})^{-1}\Sigma_{12}}\\ \end{array}}\right]
  28. M ( X ) M(\vec{X})
  29. M ( X ) = ( μ ¯ Σ ¯ ) M(\vec{X})=\left({\begin{array}[]{*{20}c}{\bar{\mu}}\\ {\bar{\Sigma}}\\ \end{array}}\right)
  30. M ( X ) M(\vec{X})
  31. M ( X ) = ( - μ ¯ Σ ¯ - 1 - Σ ¯ - 1 ) M(X)=\left({\begin{array}[]{*{20}c}{-\bar{\mu}\bar{\Sigma}^{-1}}\\ {-\bar{\Sigma}^{-1}}\\ \end{array}}\right)
  32. M ( X , Y ) = [ μ ¯ 1 Σ ¯ 11 Σ ¯ 21 μ ¯ 2 Σ ¯ 12 Σ ¯ 22 ] M(\vec{X},Y)=\left[{\begin{array}[]{*{20}c}{\begin{array}[]{*{20}c}{\bar{\mu}_% {1}}\\ {\bar{\Sigma}_{11}}\\ {\bar{\Sigma}_{21}}\\ \end{array}}&{\begin{array}[]{*{20}c}{\bar{\mu}_{2}}\\ {\bar{\Sigma}_{12}}\\ {\bar{\Sigma}_{22}}\\ \end{array}}\\ \end{array}}\right]
  33. M ( X , Y ) = [ - μ ¯ 1 ( Σ ¯ 11 ) - 1 - ( Σ ¯ 11 ) - 1 - Σ ¯ 21 ( Σ ¯ 11 ) - 1 μ ¯ 2 - μ ¯ 1 ( Σ ¯ 11 ) - 1 Σ ¯ 12 - ( Σ ¯ 11 ) - 1 Σ ¯ 12 Σ ¯ 22 - Σ ¯ 21 ( Σ ¯ 11 ) - 1 Σ ¯ 12 ] M(X,Y)=\left[{\begin{array}[]{*{20}c}{\begin{array}[]{*{20}c}{-\bar{\mu}_{1}(% \bar{\Sigma}_{11})^{-1}}\\ {-(\bar{\Sigma}_{11})^{-1}}\\ {-\bar{\Sigma}_{21}(\bar{\Sigma}_{11})^{-1}}\\ \end{array}}&{\begin{array}[]{*{20}c}{\bar{\mu}_{2}-\bar{\mu}_{1}(\bar{\Sigma}% _{11})^{-1}\bar{\Sigma}_{12}}\\ {-(\bar{\Sigma}_{11})^{-1}\bar{\Sigma}_{12}}\\ {\bar{\Sigma}_{22}-\bar{\Sigma}_{21}(\bar{\Sigma}_{11})^{-1}\bar{\Sigma}_{12}}% \\ \end{array}}\\ \end{array}}\right]
  34. M ( X ) M(\vec{X})
  35. M ( X , Y ) M(\vec{X},Y)
  36. M 1 ( X ) = ( μ ¯ 1 Σ ¯ 1 ) M_{1}(\vec{X})=\left({\begin{array}[]{*{20}c}{\bar{\mu}_{1}}\\ {\bar{\Sigma}_{1}}\\ \end{array}}\right)
  37. M 2 ( X ) = ( μ ¯ 2 Σ ¯ 2 ) M_{2}(\vec{X})=\left({\begin{array}[]{*{20}c}{\bar{\mu}_{2}}\\ {\bar{\Sigma}_{2}}\\ \end{array}}\right)
  38. M ( X ) = ( μ ¯ 1 + μ ¯ 2 Σ ¯ 1 + Σ ¯ 2 ) M(\vec{X})=\left({\begin{array}[]{*{20}c}{\bar{\mu}_{1}+\bar{\mu}_{2}}\\ {\bar{\Sigma}_{1}+\bar{\Sigma}_{2}}\\ \end{array}}\right)
  39. M 1 ( X , Y , Z ) M_{1}(X,Y,\vec{Z})
  40. M 2 ( X , Y , Z ) M_{2}(X,\vec{Y},Z)
  41. M 1 ( X , Y , Z ) M_{1}(X,Y,\vec{Z})
  42. M 2 ( X , Y , Z ) M_{2}(X,\vec{Y},Z)
  43. M 1 ( X , E , Y ) = [ 0 0 b 0 0 A 0 0 I A T I 0 ] M_{1}(\vec{X},\vec{\rm E},Y)=\left[{\begin{array}[]{*{20}c}0&0&b\\ 0&0&A\\ 0&0&I\\ {A^{T}}&I&0\\ \end{array}}\right]
  44. M 2 ( E ) = [ 0 - Σ - 1 ] M_{2}(\vec{\rm E})=\left[{\begin{array}[]{*{20}c}0\\ {-\Sigma^{-1}}\\ \end{array}}\right]
  45. M 1 ( X , E , Y ) M_{1}(\vec{X},\vec{\rm E},Y)
  46. M ( X , E , Y ) = [ 0 0 b 0 0 A 0 - Σ - 1 I A T I 0 ] M(\vec{X},\vec{\rm E},Y)=\left[{\begin{array}[]{*{20}c}0&0&b\\ 0&0&A\\ 0&{-\Sigma^{-1}}&I\\ {A^{T}}&I&0\\ \end{array}}\right]
  47. E = B + S - C + R E=B+S-C+R

Linear_bottleneck_assignment_problem.html

  1. max a A C ( a , f ( a ) ) \max_{a\in A}C(a,f(a))
  2. max a A C a , f ( a ) \max_{a\in A}C_{a,f(a)}
  3. min max i , j c i j x i j \min\,\max_{i,j}c_{ij}x_{ij}
  4. j = 1 n x i j = 1 ( i = 1 , 2 , , n ) , \sum^{n}_{j=1}x_{ij}=1(i=1,2,\dots,n),
  5. i = 1 n x i j = 1 ( j = 1 , 2 , , n ) , \sum^{n}_{i=1}x_{ij}=1(j=1,2,\dots,n),
  6. x i j { 0 , 1 } ( i , j = 1 , 2 , , n ) x_{ij}\in\{0,1\}(i,j=1,2,\dots,n)
  7. c n * c^{*}_{n}
  8. c i j c_{ij}
  9. E [ c n * ] = log n + log 2 + γ n + O ( ( log n ) 2 n 7 / 5 ) E[c^{*}_{n}]=\frac{\log n+\log 2+\gamma}{n}+O\left(\frac{(\log n)^{2}}{n^{7/5}% }\right)
  10. V a r [ c n * ] = ζ ( 2 ) - 2 ( log 2 ) 2 n 2 + O ( ( log n ) 2 n 7 / 3 ) . Var[c_{n}^{*}]=\frac{\zeta(2)-2(\log 2)^{2}}{n^{2}}+O\left(\frac{(\log n)^{2}}% {n^{7/3}}\right).

Linear_code_sequence_and_jump.html

  1. number of statements executed by the test data total number of executable statements \frac{\,\text{number of statements executed by the test data}}{\,\text{total % number of executable statements}}
  2. number of control-flow branches executed by the test data total number of control-flow branches \frac{\,\text{number of control-flow branches executed by the test data}}{\,% \text{total number of control-flow branches}}
  3. number of LCSAJs executed by the test data total number of LCSAJs \frac{\,\text{number of LCSAJs executed by the test data}}{\,\text{total % number of LCSAJs}}

Linear_entropy.html

  1. S L = ˙ 1 - Tr ( ρ 2 ) S_{L}\,\dot{=}\,1-\mbox{Tr}~{}(\rho^{2})\,
  2. γ \gamma\,
  3. S L = 1 - γ . S_{L}\,=\,1-\gamma\,.
  4. S = ˙ - Tr ( ρ ln ρ ) = - ln ρ . S\,\dot{=}\,-\mbox{Tr}~{}(\rho\ln\rho)=-\langle\ln\rho\rangle\,.
  5. - ln ρ = 1 - ρ + ( 1 - ρ ) 2 / 2 + ( 1 - ρ ) 3 / 3 + , -\langle\ln\rho\rangle=\langle 1-\rho\rangle+\langle(1-\rho)^{2}\rangle/2+% \langle(1-\rho)^{3}\rangle/3+...~{},
  6. S L = ˙ d d - 1 ( 1 - Tr ( ρ 2 ) ) , S_{L}\,\dot{=}\,\tfrac{d}{d-1}(1-\mbox{Tr}~{}(\rho^{2}))\,,

Linguistics.html

  1. Σ \Sigma

Link_group.html

  1. F n F_{n}
  2. 𝐙 2 . \mathbf{Z}^{2}.
  3. μ ¯ , \bar{\mu},
  4. μ ¯ , \bar{\mu},
  5. L = L 1 L 2 L 3 L=L_{1}\cup L_{2}\cup L_{3}
  6. l k ( L i , L j ) = 0 ; i , j = 1 , 2 , 3 ; i < j lk(L_{i},L_{j})=0;i,j=1,2,3;i<j
  7. F 1 , F 2 , F 3 F_{1},F_{2},F_{3}
  8. F i L j = , i j F_{i}\cap L_{j}=,i\neq j
  9. F 1 F 2 , F 3 F_{1}\cap F_{2},\cap F_{3}
  10. m N k - N k + 1 , mN_{k}-N_{k+1},
  11. N k N_{k}
  12. N k = 1 k d | m ϕ ( d ) ( m k / d ) , N_{k}=\frac{1}{k}\sum_{d|m}\phi(d)\left(m^{k/d}\right),
  13. ϕ \phi
  14. m k + 1 / k 2 . m^{k+1}/k^{2}.

Linked_timestamping.html

  1. N N
  2. N N
  3. N \sqrt{N}
  4. t t
  5. t + 1 t+1
  6. t + 2 t+2
  7. N N
  8. N = 2 l N=2^{l}
  9. l l

Lipid_bilayer_mechanics.html

  1. Λ \Lambda
  2. K b = K a * t K_{b}=K_{a}*t
  3. Λ \Lambda

List_of_box_office_bombs.html

  1. T W G 2 - P B {TWG\over 2}-{PB}

List_of_derivatives_and_integrals_in_alternative_calculi.html

  1. ψ ( x ) = Γ ( x ) Γ ( x ) \psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}\,
  2. K ( x ) = e ζ ( - 1 , x ) - ζ ( - 1 ) = e z - z 2 2 + z 2 ln ( 2 π ) - ψ ( - 2 ) ( z ) K(x)=e^{\zeta^{\prime}(-1,x)-\zeta^{\prime}(-1)}=e^{\frac{z-z^{2}}{2}+\frac{z}% {2}\ln(2\pi)-\psi^{(-2)}(z)}
  3. ( ! x ) = Γ ( x + 1 , - 1 ) e (!x)=\frac{\Gamma(x+1,-1)}{e}
  4. B a ( x ) = - a ζ ( - a + 1 , x ) B_{a}(x)=-a\zeta(-a+1,x)\,
  5. f ( x ) f(x)\,
  6. f ( x ) f^{\prime}(x)\,
  7. f ( x ) d x \int f(x)dx
  8. f * ( x ) f^{*}(x)\,
  9. f ( x ) d x \int f(x)^{dx}
  10. Δ f ( x ) \Delta f(x)\,
  11. Δ - 1 f ( x ) \Delta^{-1}f(x)\,
  12. x f ( x ) \prod_{x}f(x)\,
  13. a a\,
  14. 0 0\,
  15. a x ax\,
  16. 1 1\,
  17. a x a^{x}\,
  18. 0 0\,
  19. a x ax\,
  20. 1 1\,
  21. a x a^{x}\,
  22. x x\,
  23. 1 1\,
  24. x 2 2 \frac{x^{2}}{2}\,
  25. e x \sqrt[x]{e}\,
  26. x x e x \frac{x^{x}}{e^{x}}\,
  27. 1 1\,
  28. x 2 2 - x 2 \frac{x^{2}}{2}-\frac{x}{2}\,
  29. 1 + 1 x 1+\frac{1}{x}\,
  30. Γ ( x ) \Gamma(x)\,
  31. a x + b ax+b\,
  32. a a\,
  33. a x 2 + 2 b x 2 \frac{ax^{2}+2bx}{2}\,
  34. exp ( a a x + b ) \exp\left(\frac{a}{ax+b}\right)\,
  35. ( b + a x ) b a + x e x \frac{(b+ax)^{\frac{b}{a}+x}}{e^{x}}\,
  36. a a\,
  37. a x 2 + 2 b x - a x 2 \frac{ax^{2}+2bx-ax}{2}\,
  38. 1 + a a x + b 1+\frac{a}{ax+b}\,
  39. a x Γ ( a x + b a ) Γ ( a + b a ) \frac{a^{x}\Gamma(\frac{ax+b}{a})}{\Gamma(\frac{a+b}{a})}\,
  40. 1 x \frac{1}{x}\,
  41. - 1 x 2 -\frac{1}{x^{2}}\,
  42. ln | x | \ln|x|\,
  43. 1 e x \frac{1}{\sqrt[x]{e}}\,
  44. e x x x \frac{e^{x}}{x^{x}}\,
  45. - 1 x + x 2 -\frac{1}{x+x^{2}}\,
  46. ψ ( x ) \psi(x)\,
  47. x x + 1 \frac{x}{x+1}\,
  48. 1 Γ ( x ) \frac{1}{\Gamma(x)}\,
  49. x a x^{a}\,
  50. a x a - 1 ax^{a-1}\,
  51. x a + 1 a + 1 \frac{x^{a+1}}{a+1}\,
  52. e a x e^{\frac{a}{x}}\,
  53. e - a x x a x e^{-ax}x^{ax}\,
  54. ( x + 1 ) a - x a (x+1)^{a}-x^{a}\,
  55. a - ; a\notin\mathbb{Z}^{-}\,;
  56. B a + 1 ( x ) a + 1 , \frac{B_{a+1}(x)}{a+1},\,
  57. a - ; a\in\mathbb{Z}^{-}\,;
  58. ( - 1 ) a - 1 ψ ( - a - 1 ) ( x ) Γ ( - a ) , \frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)},\,
  59. ( 1 + 1 x ) a \left(1+\frac{1}{x}\right)^{a}\,
  60. Γ ( x ) a \Gamma(x)^{a}\,
  61. a x a^{x}\,
  62. a x ln a a^{x}\ln a\,
  63. a x ln a \frac{a^{x}}{\ln a}\,
  64. a a\,
  65. a x 2 2 a^{\frac{x^{2}}{2}}\,
  66. ( a - 1 ) a x (a-1)a^{x}\,
  67. a x a - 1 \frac{a^{x}}{a-1}\,
  68. a a\,
  69. a x 2 - x 2 a^{\frac{x^{2}-x}{2}}\,
  70. a x \sqrt[x]{a}\,
  71. - a x ln a x 2 -\frac{\sqrt[x]{a}\ln a}{x^{2}}\,
  72. x a x - Ei ( ln a x ) ln a x\sqrt[x]{a}-\operatorname{Ei}\left(\frac{\ln a}{x}\right)\ln a\,
  73. a - 1 x 2 a^{-\frac{1}{x^{2}}}\,
  74. a ln x a^{\ln x}\,
  75. a 1 1 + x - a 1 x a^{\frac{1}{1+x}}-a^{\frac{1}{x}}\,
  76. ? ?\,
  77. a - 1 x + x 2 a^{-\frac{1}{x+x^{2}}}\,
  78. a ψ ( x ) a^{\psi(x)}\,
  79. log a x \log_{a}x\,
  80. 1 x ln a \frac{1}{x\ln a}\,
  81. log a x x - x ln a \log_{a}x^{x}-\frac{x}{\ln a}
  82. exp ( 1 x ln x ) \exp\left(\frac{1}{x\ln x}\right)\,
  83. ( log a x ) x e li ( x ) \frac{(\log_{a}x)^{x}}{e^{\operatorname{li}(x)}}\,
  84. log a ( 1 x - 1 ) \log_{a}\left(\frac{1}{x}-1\right)\,
  85. log a Γ ( x ) \log_{a}\Gamma(x)\,
  86. log x ( x + 1 ) \log_{x}(x+1)\,
  87. ? ?\,
  88. x x x^{x}\,
  89. x x ( 1 + ln x ) x^{x}(1+\ln x)\,
  90. ? ?\,
  91. e x ex\,
  92. e - 1 4 x 2 ( 1 - 2 ln x ) e^{-\frac{1}{4}x^{2}(1-2\ln x)}\,
  93. ( x + 1 ) x + 1 - x x (x+1)^{x+1}-x^{x}\,
  94. ? ?\,
  95. ( x + 1 ) x + 1 x x \frac{(x+1)^{x+1}}{x^{x}}\,
  96. K ( x ) \operatorname{K}(x)\,
  97. Γ ( x ) \Gamma(x)\,
  98. Γ ( x ) ψ ( x ) \Gamma(x)\psi(x)\,
  99. ? ?\,
  100. e ψ ( x ) e^{\psi(x)}\,
  101. e ψ ( - 2 ) ( x ) e^{\psi^{(-2)}(x)}\,
  102. ( x - 1 ) Γ ( x ) (x-1)\Gamma(x)\,
  103. ( - 1 ) x + 1 Γ ( x ) ( ! ( - x ) ) (-1)^{x+1}\Gamma(x)(!(-x))\,
  104. x x\,
  105. Γ ( x ) x - 1 K ( x ) \frac{\Gamma(x)^{x-1}}{\operatorname{K}(x)}\,

List_of_earthquakes_in_Cuba.html

  1. M i M_{i}
  2. M i M_{i}
  3. M s M_{s}
  4. M i M_{i}
  5. M i M_{i}
  6. M s M_{s}
  7. M s M_{s}
  8. M w M_{w}
  9. M w M_{w}

List_of_map_projections.html

  1. 3 / 2 \sqrt{3}/{2}
  2. 3 / 2 \sqrt{3}/{2}

List_of_mathematical_abbreviations.html

  1. \nabla
  2. \Im
  3. \mathbb{P}
  4. \Re

List_of_poker_hands.html

  1. n k ¯ = n ! ( n - k ) ! n^{\underline{k}}=\frac{n!}{(n-k)!}
  2. 52 ! ( 52 - 5 ) ! = 52 ! 47 ! = 52 × 51 × 50 × 49 × 48 = 311 , 875 , 200 \frac{52!}{(52-5)!}=\frac{52!}{47!}=52\times 51\times 50\times 49\times 48=311% ,875,200
  3. ( 52 5 ) \begin{matrix}{52\choose 5}\end{matrix}
  4. ( 52 7 ) \begin{matrix}{52\choose 7}\end{matrix}
  5. 4 10 2 , 598 , 960 0.0015 % \frac{4\cdot 10}{2{,}598{,}960}\approx 0.0015\%
  6. C 13 1 C 4 4 C 12 1 C 4 1 C 52 5 = 13 1 12 4 2 , 598 , 960 0.024 % \frac{C_{13}^{1}C_{4}^{4}\cdot C_{12}^{1}C_{4}^{1}}{C_{52}^{5}}=\frac{13\cdot 1% \cdot 12\cdot 4}{2{,}598{,}960}\approx 0.024\%
  7. C 13 1 C 4 3 C 12 1 C 4 2 C 52 5 = 13 4 × 12 6 2 , 598 , 960 0.1441 % \frac{C_{13}^{1}C_{4}^{3}\cdot C_{12}^{1}C_{4}^{2}}{C_{52}^{5}}=\frac{13\cdot 4% \times 12\cdot 6}{2{,}598{,}960}\approx 0.1441\%
  8. 4 C 13 5 - 40 C 52 5 = 4 1 , 287 - 40 2 , 598 , 960 = 5 , 108 2 , 598 , 960 0.196 % \frac{4\cdot C_{13}^{5}-40}{C_{52}^{5}}=\frac{4\cdot 1{,}287-40}{2{,}598{,}960% }=\frac{5{,}108}{2{,}598{,}960}\approx 0.196\%
  9. 10 4 5 - 40 2 , 598 , 960 = 10 , 200 2 , 598 , 960 0.39 % \frac{10\cdot 4^{5}-40}{2{,}598{,}960}=\frac{10{,}200}{2{,}598{,}960}\approx 0% .39\%
  10. C 13 1 C 4 3 C 12 2 C 4 1 C 4 1 C 52 5 = 13 4 66 4 4 2 , 598 , 960 = 54 , 912 2 , 598 , 960 2.11 % \frac{C_{13}^{1}C_{4}^{3}\cdot C_{12}^{2}C_{4}^{1}C_{4}^{1}}{C_{52}^{5}}=\frac% {13\cdot 4\cdot 66\cdot 4\cdot 4}{2{,}598{,}960}=\frac{54{,}912}{2{,}598{,}960% }\approx 2.11\%
  11. C 13 2 C 4 2 C 4 2 C 11 1 C 4 1 C 52 5 = 78 6 6 11 4 2 , 598 , 960 = 123 , 552 2 , 598 , 960 4.75 % \frac{C_{13}^{2}C_{4}^{2}C_{4}^{2}\cdot C_{11}^{1}C_{4}^{1}}{C_{52}^{5}}=\frac% {78\cdot 6\cdot 6\cdot 11\cdot 4}{2{,}598{,}960}=\frac{123{,}552}{2{,}598{,}96% 0}\approx 4.75\%
  12. C 13 1 C 4 2 ( C 12 3 4 3 ) C 52 5 = 13 6 ( 220 64 ) 2 , 598 , 960 = 1 , 098 , 240 2 , 598 , 960 42.26 % \frac{C_{13}^{1}C_{4}^{2}\cdot(C_{12}^{3}\cdot 4^{3})}{C_{52}^{5}}=\frac{13% \cdot 6\cdot(220\cdot 64)}{2{,}598{,}960}=\frac{1{,}098{,}240}{2{,}598{,}960}% \approx 42.26\%
  13. ( C 13 5 - 10 ) ( 4 5 - 4 ) = 1 , 302 , 540 (C^{5}_{13}-10)(4^{5}-4)=1{,}302{,}540
  14. 1 , 302 , 540 2 , 598 , 960 50.11 % \frac{1{,}302{,}540}{2{,}598{,}960}\approx 50.11\%
  15. ( 52 5 ) = 52 ! 5 ! ( 52 - 5 ) ! = 52 ! 5 ! 47 ! = 52 × 51 × 50 × 49 × 48 5 × 4 × 3 × 2 × 1 = 2 , 598 , 960 {52\choose 5}=\frac{52!}{5!(52-5)!}=\frac{52!}{5!47!}=\frac{52\times 51\times 5% 0\times 49\times 48}{5\times 4\times 3\times 2\times 1}=2{,}598{,}960
  16. ( 52 7 ) = 52 ! 7 ! ( 52 - 7 ) ! = 52 ! 7 ! 45 ! = 52 × 51 × 50 × 49 × 48 × 47 × 46 7 × 6 × 5 × 4 × 3 × 2 × 1 = 133 , 784 , 560 {52\choose 7}=\frac{52!}{7!(52-7)!}=\frac{52!}{7!45!}=\frac{52\times 51\times 5% 0\times 49\times 48\times 47\times 46}{7\times 6\times 5\times 4\times 3\times 2% \times 1}=133{,}784{,}560

List_of_production_functions.html

  1. Y = A F ( K , L ) \ Y=AF(K,L)
  2. Y = F ( K , A L ) \ Y=F(K,AL)
  3. Y = F ( A K , L ) \ Y=F(AK,L)
  4. \Epsilon = [ ( s l o p e ) ( L / K ) L / K s l o p e ] - 1 \ \Epsilon=[\frac{\partial(slope)}{\partial(L/K)}\frac{L/K}{slope}]^{-1}
  5. s l o p e = - F ( K , L ) / K F ( K , L ) / L \ slope=-\frac{\partial F(K,L)/\partial K}{\partial F(K,L)/\partial L}
  6. Y = A [ α K γ + ( 1 - α ) L γ ] 1 γ Y=A[\alpha K^{\gamma}+(1-\alpha)L^{\gamma}]^{\frac{1}{\gamma}}
  7. γ \isin [ - , 1 ] \gamma\isin[-\infty,1]
  8. Y = A [ α K + ( 1 - α ) L ] \ Y=A[\alpha K+(1-\alpha)L]
  9. γ = 1 \ \gamma=1
  10. Y = A K α L 1 - α \ Y=AK^{\alpha}L^{1-\alpha}
  11. γ 0 \gamma\to 0
  12. Y = M i n [ K , L ] \ Y=Min[K,L]
  13. γ - \gamma\to-\infty
  14. l n ( Y ) = l n ( A ) + a L l n ( L ) + a K l n ( K ) + b L L l n 2 ( L ) + b L K l n ( L ) l n ( K ) + b K K l n 2 ( K ) \ ln(Y)=ln(A)+a_{L}ln(L)+a_{K}ln(K)+b_{LL}ln^{2}(L)+b_{LK}ln(L)ln(K)+b_{KK}ln^% {2}(K)
  15. Y = A F ( K f , K w , L w , L m , b u s i n e s s c o n f i d e n c e , c o n s u m e r c o n f i d e n c e ) Y=A\,F(Kf,Kw,Lw,Lm,business\,confidence,\,consumer\,confidence)
  16. K f Kf
  17. K w Kw
  18. L w Lw
  19. L m Lm

List_of_transitive_finite_linear_groups.html

  1. p p
  2. G G
  3. G L ( d , p ) GL(d,p)
  4. ( F p ) d (F_{p})^{d}
  5. F p F_{p}
  6. G Γ L ( 1 , p d ) ; G\leq\Gamma{}L(1,p^{d});
  7. G S L ( a , q ) and p d = q a ; G\triangleright SL(a,q)\,\text{ and }p^{d}=q^{a};
  8. G S p ( 2 a , q ) and p d = q 2 a ; G\triangleright Sp(2a,q)\,\text{ and }p^{d}=q^{2a};
  9. G G 2 ( q ) , p d = q 6 and p = 2. G\triangleright G_{2}(q)^{\prime},\ p^{d}=q^{6}\,\text{ and }p=2.
  10. G G
  11. p d : G p^{d}:G
  12. p d : G p^{d}:G
  13. p p
  14. d d
  15. G 0 G_{0}
  16. p d : G p^{d}:G
  17. p = 5 p=5
  18. d = 2 d=2
  19. S L ( 2 , 3 ) SL(2,3)
  20. p = 7 p=7
  21. d = 2 d=2
  22. S L ( 2 , 3 ) SL(2,3)
  23. p = 11 p=11
  24. d = 2 d=2
  25. S L ( 2 , 3 ) SL(2,3)
  26. p = 23 p=23
  27. d = 2 d=2
  28. S L ( 2 , 3 ) SL(2,3)
  29. p = 11 p=11
  30. d = 2 d=2
  31. S L ( 2 , 5 ) SL(2,5)
  32. p = 19 p=19
  33. d = 2 d=2
  34. S L ( 2 , 5 ) SL(2,5)
  35. p = 29 p=29
  36. d = 2 d=2
  37. S L ( 2 , 5 ) SL(2,5)
  38. p = 59 p=59
  39. d = 2 d=2
  40. S L ( 2 , 5 ) SL(2,5)
  41. p = 2 p=2
  42. d = 4 d=4
  43. A 6 A_{6}
  44. p = 2 p=2
  45. d = 4 d=4
  46. A 7 A_{7}
  47. p = 3 p=3
  48. d = 4 d=4
  49. S L ( 2 , 5 ) SL(2,5)
  50. p = 3 p=3
  51. d = 4 d=4
  52. 2 - 1 + 4 2^{1+4}_{-}
  53. p = 2 p=2
  54. d = 6 d=6
  55. P S U ( 3 , 3 ) PSU(3,3)
  56. p = 3 p=3
  57. d = 6 d=6
  58. S L ( 2 , 13 ) SL(2,13)

List_of_trigonometric_identities.html

  1. π \pi
  2. π 6 \frac{\pi}{6}\!
  3. π 3 \frac{\pi}{3}\!
  4. 2 π 3 \frac{2\pi}{3}\!
  5. 5 π 6 \frac{5\pi}{6}\!
  6. 7 π 6 \frac{7\pi}{6}\!
  7. 4 π 3 \frac{4\pi}{3}\!
  8. 5 π 3 \frac{5\pi}{3}\!
  9. 11 π 6 \frac{11\pi}{6}\!
  10. π 4 \frac{\pi}{4}\!
  11. π 2 \frac{\pi}{2}\!
  12. 3 π 4 \frac{3\pi}{4}\!
  13. π \pi\!
  14. 5 π 4 \frac{5\pi}{4}\!
  15. 3 π 2 \frac{3\pi}{2}\!
  16. 7 π 4 \frac{7\pi}{4}\!
  17. 2 π 2\pi\!
  18. tan θ = sin θ cos θ . \tan\theta=\frac{\sin\theta}{\cos\theta}.
  19. sec θ = 1 cos θ , csc θ = 1 sin θ , cot θ = 1 tan θ = cos θ sin θ . \sec\theta=\frac{1}{\cos\theta},\quad\csc\theta=\frac{1}{\sin\theta},\quad\cot% \theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}.
  20. sin ( arcsin x ) = x for | x | 1 \sin(\arcsin x)=x\quad\,\text{for}\quad|x|\leq 1
  21. arcsin ( sin x ) = x for | x | π / 2. \arcsin(\sin x)=x\quad\,\text{for}\quad|x|\leq\pi/2.
  22. sin 2 θ + cos 2 θ = 1 \sin^{2}\theta+\cos^{2}\theta=1\!
  23. sin θ \displaystyle\sin\theta
  24. θ θ
  25. 1 + tan 2 θ = sec 2 θ and 1 + cot 2 θ = csc 2 θ . 1+\tan^{2}\theta=\sec^{2}\theta\quad\,\text{and}\quad 1+\cot^{2}\theta=\csc^{2% }\theta.\!
  26. sin θ \sin\theta\!
  27. cos θ \cos\theta\!
  28. tan θ \tan\theta\!
  29. csc θ \csc\theta\!
  30. sec θ \sec\theta\!
  31. cot θ \cot\theta\!
  32. sin θ = \sin\theta=\!
  33. sin θ \sin\theta
  34. ± 1 - cos 2 θ \pm\sqrt{1-\cos^{2}\theta}\!
  35. ± tan θ 1 + tan 2 θ \pm\frac{\tan\theta}{\sqrt{1+\tan^{2}\theta}}\!
  36. 1 csc θ \frac{1}{\csc\theta}\!
  37. ± sec 2 θ - 1 sec θ \pm\frac{\sqrt{\sec^{2}\theta-1}}{\sec\theta}\!
  38. ± 1 1 + cot 2 θ \pm\frac{1}{\sqrt{1+\cot^{2}\theta}}\!
  39. cos θ = \cos\theta=\!
  40. ± 1 - sin 2 θ \pm\sqrt{1-\sin^{2}\theta}\!
  41. cos θ \cos\theta\!
  42. ± 1 1 + tan 2 θ \pm\frac{1}{\sqrt{1+\tan^{2}\theta}}\!
  43. ± csc 2 θ - 1 csc θ \pm\frac{\sqrt{\csc^{2}\theta-1}}{\csc\theta}\!
  44. 1 sec θ \frac{1}{\sec\theta}\!
  45. ± cot θ 1 + cot 2 θ \pm\frac{\cot\theta}{\sqrt{1+\cot^{2}\theta}}\!
  46. tan θ = \tan\theta=\!
  47. ± sin θ 1 - sin 2 θ \pm\frac{\sin\theta}{\sqrt{1-\sin^{2}\theta}}\!
  48. ± 1 - cos 2 θ cos θ \pm\frac{\sqrt{1-\cos^{2}\theta}}{\cos\theta}\!
  49. tan θ \tan\theta\!
  50. ± 1 csc 2 θ - 1 \pm\frac{1}{\sqrt{\csc^{2}\theta-1}}\!
  51. ± sec 2 θ - 1 \pm\sqrt{\sec^{2}\theta-1}\!
  52. 1 cot θ \frac{1}{\cot\theta}\!
  53. csc θ = \csc\theta=\!
  54. 1 sin θ \frac{1}{\sin\theta}\!
  55. ± 1 1 - cos 2 θ \pm\frac{1}{\sqrt{1-\cos^{2}\theta}}\!
  56. ± 1 + tan 2 θ tan θ \pm\frac{\sqrt{1+\tan^{2}\theta}}{\tan\theta}\!
  57. csc θ \csc\theta\!
  58. ± sec θ sec 2 θ - 1 \pm\frac{\sec\theta}{\sqrt{\sec^{2}\theta-1}}\!
  59. ± 1 + cot 2 θ \pm\sqrt{1+\cot^{2}\theta}\!
  60. sec θ = \sec\theta=\!
  61. ± 1 1 - sin 2 θ \pm\frac{1}{\sqrt{1-\sin^{2}\theta}}\!
  62. 1 cos θ \frac{1}{\cos\theta}\!
  63. ± 1 + tan 2 θ \pm\sqrt{1+\tan^{2}\theta}\!
  64. ± csc θ csc 2 θ - 1 \pm\frac{\csc\theta}{\sqrt{\csc^{2}\theta-1}}\!
  65. sec θ \sec\theta\!
  66. ± 1 + cot 2 θ cot θ \pm\frac{\sqrt{1+\cot^{2}\theta}}{\cot\theta}\!
  67. cot θ = \cot\theta=\!
  68. ± 1 - sin 2 θ sin θ \pm\frac{\sqrt{1-\sin^{2}\theta}}{\sin\theta}\!
  69. ± cos θ 1 - cos 2 θ \pm\frac{\cos\theta}{\sqrt{1-\cos^{2}\theta}}\!
  70. 1 tan θ \frac{1}{\tan\theta}\!
  71. ± csc 2 θ - 1 \pm\sqrt{\csc^{2}\theta-1}\!
  72. ± 1 sec 2 θ - 1 \pm\frac{1}{\sqrt{\sec^{2}\theta-1}}\!
  73. cot θ \cot\theta\!
  74. versin ( θ ) \operatorname{versin}(\theta)
  75. vers ( θ ) \operatorname{vers}(\theta)
  76. ver ( θ ) \operatorname{ver}(\theta)
  77. 1 - cos ( θ ) 1-\cos(\theta)
  78. vercosin ( θ ) \operatorname{vercosin}(\theta)
  79. 1 + cos ( θ ) 1+\cos(\theta)
  80. coversin ( θ ) \operatorname{coversin}(\theta)
  81. cvs ( θ ) \operatorname{cvs}(\theta)
  82. 1 - sin ( θ ) 1-\sin(\theta)
  83. covercosin ( θ ) \operatorname{covercosin}(\theta)
  84. 1 + sin ( θ ) 1+\sin(\theta)
  85. haversin ( θ ) \operatorname{haversin}(\theta)
  86. 1 - cos ( θ ) 2 \frac{1-\cos(\theta)}{2}
  87. havercosin ( θ ) \operatorname{havercosin}(\theta)
  88. 1 + cos ( θ ) 2 \frac{1+\cos(\theta)}{2}
  89. hacoversin ( θ ) \operatorname{hacoversin}(\theta)
  90. 1 - sin ( θ ) 2 \frac{1-\sin(\theta)}{2}
  91. hacovercosin ( θ ) \operatorname{hacovercosin}(\theta)
  92. 1 + sin ( θ ) 2 \frac{1+\sin(\theta)}{2}
  93. exsec ( θ ) \operatorname{exsec}(\theta)
  94. sec ( θ ) - 1 \sec(\theta)-1
  95. excsc ( θ ) \operatorname{excsc}(\theta)
  96. csc ( θ ) - 1 \csc(\theta)-1
  97. crd ( θ ) \operatorname{crd}(\theta)
  98. 2 sin θ 2 2\sin\frac{\theta}{2}
  99. θ = 0 \theta=0
  100. θ = π / 4 \theta=\pi/4
  101. θ = π / 2 \theta=\pi/2
  102. sin ( - θ ) = - sin θ cos ( - θ ) = + cos θ tan ( - θ ) = - tan θ csc ( - θ ) = - csc θ sec ( - θ ) = + sec θ cot ( - θ ) = - cot θ \begin{aligned}\displaystyle\sin(-\theta)&\displaystyle=-\sin\theta\\ \displaystyle\cos(-\theta)&\displaystyle=+\cos\theta\\ \displaystyle\tan(-\theta)&\displaystyle=-\tan\theta\\ \displaystyle\csc(-\theta)&\displaystyle=-\csc\theta\\ \displaystyle\sec(-\theta)&\displaystyle=+\sec\theta\\ \displaystyle\cot(-\theta)&\displaystyle=-\cot\theta\\ \end{aligned}
  103. sin ( π 2 - θ ) = + cos θ cos ( π 2 - θ ) = + sin θ tan ( π 2 - θ ) = + cot θ csc ( π 2 - θ ) = + sec θ sec ( π 2 - θ ) = + csc θ cot ( π 2 - θ ) = + tan θ \begin{aligned}\displaystyle\sin(\tfrac{\pi}{2}-\theta)&\displaystyle=+\cos% \theta\\ \displaystyle\cos(\tfrac{\pi}{2}-\theta)&\displaystyle=+\sin\theta\\ \displaystyle\tan(\tfrac{\pi}{2}-\theta)&\displaystyle=+\cot\theta\\ \displaystyle\csc(\tfrac{\pi}{2}-\theta)&\displaystyle=+\sec\theta\\ \displaystyle\sec(\tfrac{\pi}{2}-\theta)&\displaystyle=+\csc\theta\\ \displaystyle\cot(\tfrac{\pi}{2}-\theta)&\displaystyle=+\tan\theta\\ \end{aligned}
  104. sin ( π - θ ) = + sin θ cos ( π - θ ) = - cos θ tan ( π - θ ) = - tan θ csc ( π - θ ) = + csc θ sec ( π - θ ) = - sec θ cot ( π - θ ) = - cot θ \begin{aligned}\displaystyle\sin(\pi-\theta)&\displaystyle=+\sin\theta\\ \displaystyle\cos(\pi-\theta)&\displaystyle=-\cos\theta\\ \displaystyle\tan(\pi-\theta)&\displaystyle=-\tan\theta\\ \displaystyle\csc(\pi-\theta)&\displaystyle=+\csc\theta\\ \displaystyle\sec(\pi-\theta)&\displaystyle=-\sec\theta\\ \displaystyle\cot(\pi-\theta)&\displaystyle=-\cot\theta\\ \end{aligned}
  105. + cos θ +\cos\theta
  106. cos θ \cos\theta
  107. θ = π \theta=\pi
  108. + cos θ = - 1 +\cos\theta=-1
  109. sin ( θ + π 2 ) = + cos θ cos ( θ + π 2 ) = - sin θ tan ( θ + π 2 ) = - cot θ csc ( θ + π 2 ) = + sec θ sec ( θ + π 2 ) = - csc θ cot ( θ + π 2 ) = - tan θ \begin{aligned}\displaystyle\sin(\theta+\tfrac{\pi}{2})&\displaystyle=+\cos% \theta\\ \displaystyle\cos(\theta+\tfrac{\pi}{2})&\displaystyle=-\sin\theta\\ \displaystyle\tan(\theta+\tfrac{\pi}{2})&\displaystyle=-\cot\theta\\ \displaystyle\csc(\theta+\tfrac{\pi}{2})&\displaystyle=+\sec\theta\\ \displaystyle\sec(\theta+\tfrac{\pi}{2})&\displaystyle=-\csc\theta\\ \displaystyle\cot(\theta+\tfrac{\pi}{2})&\displaystyle=-\tan\theta\end{aligned}
  110. sin ( θ + π ) = - sin θ cos ( θ + π ) = - cos θ tan ( θ + π ) = + tan θ csc ( θ + π ) = - csc θ sec ( θ + π ) = - sec θ cot ( θ + π ) = + cot θ \begin{aligned}\displaystyle\sin(\theta+\pi)&\displaystyle=-\sin\theta\\ \displaystyle\cos(\theta+\pi)&\displaystyle=-\cos\theta\\ \displaystyle\tan(\theta+\pi)&\displaystyle=+\tan\theta\\ \displaystyle\csc(\theta+\pi)&\displaystyle=-\csc\theta\\ \displaystyle\sec(\theta+\pi)&\displaystyle=-\sec\theta\\ \displaystyle\cot(\theta+\pi)&\displaystyle=+\cot\theta\\ \end{aligned}
  111. sin ( θ + 2 π ) = + sin θ cos ( θ + 2 π ) = + cos θ tan ( θ + 2 π ) = + tan θ csc ( θ + 2 π ) = + csc θ sec ( θ + 2 π ) = + sec θ cot ( θ + 2 π ) = + cot θ \begin{aligned}\displaystyle\sin(\theta+2\pi)&\displaystyle=+\sin\theta\\ \displaystyle\cos(\theta+2\pi)&\displaystyle=+\cos\theta\\ \displaystyle\tan(\theta+2\pi)&\displaystyle=+\tan\theta\\ \displaystyle\csc(\theta+2\pi)&\displaystyle=+\csc\theta\\ \displaystyle\sec(\theta+2\pi)&\displaystyle=+\sec\theta\\ \displaystyle\cot(\theta+2\pi)&\displaystyle=+\cot\theta\end{aligned}
  112. sin ( α + β ) = sin α cos β + cos α sin β \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta
  113. cos ( α + β ) = cos α cos β - sin α sin β \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
  114. sin ( α ± β ) = sin α cos β ± cos α sin β \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta\!
  115. cos ( α ± β ) = cos α cos β sin α sin β \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta\,
  116. tan ( α ± β ) = tan α ± tan β 1 tan α tan β \tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}
  117. arcsin α ± arcsin β = arcsin ( α 1 - β 2 ± β 1 - α 2 ) \arcsin\alpha\pm\arcsin\beta=\arcsin\left(\alpha\sqrt{1-\beta^{2}}\pm\beta% \sqrt{1-\alpha^{2}}\right)
  118. arccos α ± arccos β = arccos ( α β ( 1 - α 2 ) ( 1 - β 2 ) ) \arccos\alpha\pm\arccos\beta=\arccos\left(\alpha\beta\mp\sqrt{(1-\alpha^{2})(1% -\beta^{2})}\right)
  119. arctan α ± arctan β = arctan ( α ± β 1 α β ) \arctan\alpha\pm\arctan\beta=\arctan\left(\frac{\alpha\pm\beta}{1\mp\alpha% \beta}\right)
  120. ( cos α - sin α sin α cos α ) ( cos β - sin β sin β cos β ) \displaystyle{}\quad\left(\begin{array}[]{rr}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{rr}\cos\beta&-% \sin\beta\\ \sin\beta&\cos\beta\end{array}\right)
  121. sin ( i = 1 θ i ) = odd k 1 ( - 1 ) ( k - 1 ) / 2 A { 1 , 2 , 3 , } | A | = k ( i A sin θ i i A cos θ i ) \sin\left(\sum_{i=1}^{\infty}\theta_{i}\right)=\sum_{\,\text{odd}\ k\geq 1}(-1% )^{(k-1)/2}\sum_{\begin{smallmatrix}A\subseteq\{\,1,2,3,\dots\,\}\\ \left|A\right|=k\end{smallmatrix}}\left(\prod_{i\in A}\sin\theta_{i}\prod_{i% \not\in A}\cos\theta_{i}\right)
  122. cos ( i = 1 θ i ) = even k 0 ( - 1 ) k / 2 A { 1 , 2 , 3 , } | A | = k ( i A sin θ i i A cos θ i ) \cos\left(\sum_{i=1}^{\infty}\theta_{i}\right)=\sum_{\,\text{even}\ k\geq 0}~{% }(-1)^{k/2}~{}~{}\sum_{\begin{smallmatrix}A\subseteq\{\,1,2,3,\dots\,\}\\ \left|A\right|=k\end{smallmatrix}}\left(\prod_{i\in A}\sin\theta_{i}\prod_{i% \not\in A}\cos\theta_{i}\right)
  123. x i = tan θ i x_{i}=\tan\theta_{i}\,
  124. e 0 \displaystyle e_{0}
  125. tan ( i θ i ) = e 1 - e 3 + e 5 - e 0 - e 2 + e 4 - . \tan\left(\sum_{i}\theta_{i}\right)=\frac{e_{1}-e_{3}+e_{5}-\cdots}{e_{0}-e_{2% }+e_{4}-\cdots}.\!
  126. tan ( θ 1 + θ 2 ) \displaystyle\tan(\theta_{1}+\theta_{2})
  127. sec ( i θ i ) \displaystyle\sec\left(\sum_{i}\theta_{i}\right)
  128. e 0 - e 2 + e 4 - = i sec θ i sec ( i θ i ) e_{0}-e_{2}+e_{4}-\cdots=\frac{\prod_{i}\sec\theta_{i}}{\sec\left(\sum_{i}% \theta_{i}\right)}
  129. sec ( α + β + γ ) \displaystyle\sec(\alpha+\beta+\gamma)
  130. cos n θ = T n ( cos θ ) \cos n\theta=T_{n}(\cos\theta)\,
  131. sin 2 n θ = S n ( sin 2 θ ) \sin^{2}n\theta=S_{n}(\sin^{2}\theta)\,
  132. i i
  133. cos n θ + i sin n θ = ( cos ( θ ) + i sin ( θ ) ) n \cos n\theta+i\sin n\theta=(\cos(\theta)+i\sin(\theta))^{n}\,
  134. sin 2 θ = 2 sin θ cos θ = 2 tan θ 1 + tan 2 θ \sin 2\theta=2\sin\theta\cos\theta=\frac{2\tan\theta}{1+\tan^{2}\theta}
  135. cos 2 θ = cos 2 θ - sin 2 θ = 2 cos 2 θ - 1 = 1 - 2 sin 2 θ = 1 - tan 2 θ 1 + tan 2 θ \cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta% =\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}
  136. tan 2 θ = 2 tan θ 1 - tan 2 θ \tan 2\theta=\frac{2\tan\theta}{1-\tan^{2}\theta}
  137. cot 2 θ = cot 2 θ - 1 2 cot θ \cot 2\theta=\frac{\cot^{2}\theta-1}{2\cot\theta}
  138. sin 3 θ = - sin 3 θ + 3 cos 2 θ sin θ = - 4 sin 3 θ + 3 sin θ \sin 3\theta=-\sin^{3}\theta+3\cos^{2}\theta\sin\theta=-4\sin^{3}\theta+3\sin\theta
  139. cos 3 θ = cos 3 θ - 3 sin 2 θ cos θ = 4 cos 3 θ - 3 cos θ \cos 3\theta=\cos^{3}\theta-3\sin^{2}\theta\cos\theta=4\cos^{3}\theta-3\cos\theta
  140. tan 3 θ = 3 tan θ - tan 3 θ 1 - 3 tan 2 θ \tan 3\theta=\frac{3\tan\theta-\tan^{3}\theta}{1-3\tan^{2}\theta}
  141. cot 3 θ = 3 cot θ - cot 3 θ 1 - 3 cot 2 θ \cot 3\theta=\frac{3\cot\theta-\cot^{3}\theta}{1-3\cot^{2}\theta}
  142. sin θ 2 = sgn ( 2 π - θ + 4 π θ 4 π ) 1 - cos θ 2 \sin\frac{\theta}{2}=\operatorname{sgn}\left(2\pi-\theta+4\pi\left\lfloor\frac% {\theta}{4\pi}\right\rfloor\right)\sqrt{\frac{1\!-\!\cos\theta}{2}}
  143. sin 2 θ 2 = 1 - cos θ 2 \sin^{2}\frac{\theta}{2}=\frac{1-\cos\theta}{2}
  144. cos θ 2 = sgn ( π + θ + 4 π π - θ 4 π ) 1 + cos θ 2 \cos\frac{\theta}{2}=\operatorname{sgn}\left(\pi+\theta+4\pi\left\lfloor\frac{% \pi-\theta}{4\pi}\right\rfloor\right)\sqrt{\frac{1+\cos\theta}{2}}
  145. cos 2 θ 2 = 1 + cos θ 2 \cos^{2}\frac{\theta}{2}=\frac{1+\cos\theta}{2}
  146. tan θ 2 = csc θ - cot θ = ± 1 - cos θ 1 + cos θ = sin θ 1 + cos θ = 1 - cos θ sin θ \tan\frac{\theta}{2}=\csc\theta-\cot\theta=\pm\,\sqrt{1-\cos\theta\over 1+\cos% \theta}=\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}
  147. tan η + θ 2 = sin η + sin θ cos η + cos θ \tan\frac{\eta+\theta}{2}=\frac{\sin\eta+\sin\theta}{\cos\eta+\cos\theta}
  148. tan ( θ 2 + π 4 ) = sec θ + tan θ \tan\left(\frac{\theta}{2}+\frac{\pi}{4}\right)=\sec\theta+\tan\theta
  149. 1 - sin θ 1 + sin θ = 1 - tan ( θ / 2 ) 1 + tan ( θ / 2 ) \sqrt{\frac{1-\sin\theta}{1+\sin\theta}}=\frac{1-\tan(\theta/2)}{1+\tan(\theta% /2)}
  150. tan 1 2 θ = tan θ 1 + sec θ \tan\tfrac{1}{2}\theta=\frac{\tan\theta}{1+\sec{\theta}}
  151. cot θ 2 = csc θ + cot θ = ± 1 + cos θ 1 - cos θ = sin θ 1 - cos θ = 1 + cos θ sin θ \cot\frac{\theta}{2}=\csc\theta+\cot\theta=\pm\,\sqrt{1+\cos\theta\over 1-\cos% \theta}=\frac{\sin\theta}{1-\cos\theta}=\frac{1+\cos\theta}{\sin\theta}
  152. sin 2 θ = 2 sin θ cos θ = 2 tan θ 1 + tan 2 θ \begin{aligned}\displaystyle\sin 2\theta&\displaystyle=2\sin\theta\cos\theta\\ &\displaystyle=\frac{2\tan\theta}{1+\tan^{2}\theta}\end{aligned}
  153. cos 2 θ = cos 2 θ - sin 2 θ = 2 cos 2 θ - 1 = 1 - 2 sin 2 θ = 1 - tan 2 θ 1 + tan 2 θ \begin{aligned}\displaystyle\cos 2\theta&\displaystyle=\cos^{2}\theta-\sin^{2}% \theta\\ &\displaystyle=2\cos^{2}\theta-1\\ &\displaystyle=1-2\sin^{2}\theta\\ &\displaystyle=\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}\end{aligned}
  154. tan 2 θ = 2 tan θ 1 - tan 2 θ \tan 2\theta=\frac{2\tan\theta}{1-\tan^{2}\theta}
  155. cot 2 θ = cot 2 θ - 1 2 cot θ \cot 2\theta=\frac{\cot^{2}\theta-1}{2\cot\theta}
  156. sin 3 θ = - sin 3 θ + 3 cos 2 θ sin θ = - 4 sin 3 θ + 3 sin θ \begin{aligned}\displaystyle\sin 3\theta&\displaystyle=-\sin^{3}\theta+3\cos^{% 2}\theta\sin\theta\\ &\displaystyle=-4\sin^{3}\theta+3\sin\theta\end{aligned}
  157. cos 3 θ = cos 3 θ - 3 sin 2 θ cos θ = 4 cos 3 θ - 3 cos θ \begin{aligned}\displaystyle\cos 3\theta&\displaystyle=\cos^{3}\theta-3\sin^{2% }\theta\cos\theta\\ &\displaystyle=4\cos^{3}\theta-3\cos\theta\end{aligned}
  158. tan 3 θ = 3 tan θ - tan 3 θ 1 - 3 tan 2 θ \tan 3\theta=\frac{3\tan\theta-\tan^{3}\theta}{1-3\tan^{2}\theta}
  159. cot 3 θ = 3 cot θ - cot 3 θ 1 - 3 cot 2 θ \cot 3\theta=\frac{3\cot\theta-\cot^{3}\theta}{1-3\cot^{2}\theta}
  160. sin θ 2 = sgn ( 2 π - θ + 4 π θ 4 π ) 1 - cos θ 2 ( or sin 2 θ 2 = 1 - cos θ 2 ) \begin{aligned}&\displaystyle\sin\frac{\theta}{2}=\operatorname{sgn}\left(2\pi% -\theta+4\pi\left\lfloor\frac{\theta}{4\pi}\right\rfloor\right)\sqrt{\frac{1\!% -\!\cos\theta}{2}}\\ \\ &\displaystyle\left(\mathrm{or}\,\,\sin^{2}\frac{\theta}{2}=\frac{1-\cos\theta% }{2}\right)\end{aligned}
  161. cos θ 2 = sgn ( π + θ + 4 π π - θ 4 π ) 1 + cos θ 2 ( or cos 2 θ 2 = 1 + cos θ 2 ) \begin{aligned}&\displaystyle\cos\frac{\theta}{2}=\operatorname{sgn}\left(\pi+% \theta+4\pi\left\lfloor\frac{\pi-\theta}{4\pi}\right\rfloor\right)\sqrt{\frac{% 1+\cos\theta}{2}}\\ \\ &\displaystyle\left(\mathrm{or}\,\,\cos^{2}\frac{\theta}{2}=\frac{1+\cos\theta% }{2}\right)\end{aligned}
  162. tan θ 2 = csc θ - cot θ = ± 1 - cos θ 1 + cos θ = sin θ 1 + cos θ = 1 - cos θ sin θ tan η + θ 2 = sin η + sin θ cos η + cos θ tan ( θ 2 + π 4 ) = sec θ + tan θ 1 - sin θ 1 + sin θ = 1 - tan ( θ / 2 ) 1 + tan ( θ / 2 ) tan 1 2 θ = tan θ 1 + 1 + tan 2 θ for θ ( - π 2 , π 2 ) \begin{aligned}\displaystyle\tan\frac{\theta}{2}&\displaystyle=\csc\theta-\cot% \theta\\ &\displaystyle=\pm\,\sqrt{1-\cos\theta\over 1+\cos\theta}\\ &\displaystyle=\frac{\sin\theta}{1+\cos\theta}\\ &\displaystyle=\frac{1-\cos\theta}{\sin\theta}\\ \displaystyle\tan\frac{\eta+\theta}{2}&\displaystyle=\frac{\sin\eta+\sin\theta% }{\cos\eta+\cos\theta}\\ \displaystyle\tan\left(\frac{\theta}{2}+\frac{\pi}{4}\right)&\displaystyle=% \sec\theta+\tan\theta\\ \displaystyle\sqrt{\frac{1-\sin\theta}{1+\sin\theta}}&\displaystyle=\frac{1-% \tan(\theta/2)}{1+\tan(\theta/2)}\\ \displaystyle\tan\tfrac{1}{2}\theta&\displaystyle=\frac{\tan\theta}{1+\sqrt{1+% \tan^{2}\theta}}\\ &\displaystyle\mbox{for}~{}\quad\theta\in\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}% \right)\end{aligned}
  163. cot θ 2 = csc θ + cot θ = ± 1 + cos θ 1 - cos θ = sin θ 1 - cos θ = 1 + cos θ sin θ \begin{aligned}\displaystyle\cot\frac{\theta}{2}&\displaystyle=\csc\theta+\cot% \theta\\ &\displaystyle=\pm\,\sqrt{1+\cos\theta\over 1-\cos\theta}\\ &\displaystyle=\frac{\sin\theta}{1-\cos\theta}\\ &\displaystyle=\frac{1+\cos\theta}{\sin\theta}\end{aligned}
  164. x 3 - 3 x + d 4 = 0 x^{3}-\frac{3x+d}{4}=0
  165. sin n θ = k = 0 n ( n k ) cos k θ sin n - k θ sin ( 1 2 ( n - k ) π ) \sin n\theta=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\cos^{k}\theta\,\sin^{n% -k}\theta\,\sin\left(\frac{1}{2}(n-k)\pi\right)
  166. cos n θ = k = 0 n ( n k ) cos k θ sin n - k θ cos ( 1 2 ( n - k ) π ) \cos n\theta=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\cos^{k}\theta\,\sin^{n% -k}\theta\,\cos\left(\frac{1}{2}(n-k)\pi\right)
  167. tan ( n + 1 ) θ = tan n θ + tan θ 1 - tan n θ tan θ . \tan\,(n{+}1)\theta=\frac{\tan n\theta+\tan\theta}{1-\tan n\theta\,\tan\theta}.
  168. cot ( n + 1 ) θ = cot n θ cot θ - 1 cot n θ + cot θ . \cot\,(n{+}1)\theta=\frac{\cot n\theta\,\cot\theta-1}{\cot n\theta+\cot\theta}.
  169. cos n x = 2 cos x cos ( ( n - 1 ) x ) - cos ( ( n - 2 ) x ) \cos nx=2\cdot\cos x\cdot\cos((n-1)x)-\cos((n-2)x)\,
  170. sin n x = 2 cos x sin ( ( n - 1 ) x ) - sin ( ( n - 2 ) x ) \sin nx=2\cdot\cos x\cdot\sin((n-1)x)-\sin((n-2)x)\,
  171. tan n x = H + K tan x K - H tan x \tan nx=\frac{H+K\tan x}{K-H\tan x}\,
  172. tan ( α + β 2 ) = sin α + sin β cos α + cos β = - cos α - cos β sin α - sin β \tan\left(\frac{\alpha+\beta}{2}\right)=\frac{\sin\alpha+\sin\beta}{\cos\alpha% +\cos\beta}=-\,\frac{\cos\alpha-\cos\beta}{\sin\alpha-\sin\beta}
  173. cos ( θ 2 ) cos ( θ 4 ) cos ( θ 8 ) = n = 1 cos ( θ 2 n ) = sin θ θ = sinc θ . \cos\left({\theta\over 2}\right)\cdot\cos\left({\theta\over 4}\right)\cdot\cos% \left({\theta\over 8}\right)\cdots=\prod_{n=1}^{\infty}\cos\left({\theta\over 2% ^{n}}\right)={\sin\theta\over\theta}=\operatorname{sinc}\,\theta.
  174. sin 2 θ = 1 - cos 2 θ 2 \sin^{2}\theta=\frac{1-\cos 2\theta}{2}\!
  175. cos 2 θ = 1 + cos 2 θ 2 \cos^{2}\theta=\frac{1+\cos 2\theta}{2}\!
  176. sin 2 θ cos 2 θ = 1 - cos 4 θ 8 \sin^{2}\theta\cos^{2}\theta=\frac{1-\cos 4\theta}{8}\!
  177. sin 3 θ = 3 sin θ - sin 3 θ 4 \sin^{3}\theta=\frac{3\sin\theta-\sin 3\theta}{4}\!
  178. cos 3 θ = 3 cos θ + cos 3 θ 4 \cos^{3}\theta=\frac{3\cos\theta+\cos 3\theta}{4}\!
  179. sin 3 θ cos 3 θ = 3 sin 2 θ - sin 6 θ 32 \sin^{3}\theta\cos^{3}\theta=\frac{3\sin 2\theta-\sin 6\theta}{32}\!
  180. sin 4 θ = 3 - 4 cos 2 θ + cos 4 θ 8 \sin^{4}\theta=\frac{3-4\cos 2\theta+\cos 4\theta}{8}\!
  181. cos 4 θ = 3 + 4 cos 2 θ + cos 4 θ 8 \cos^{4}\theta=\frac{3+4\cos 2\theta+\cos 4\theta}{8}\!
  182. sin 4 θ cos 4 θ = 3 - 4 cos 4 θ + cos 8 θ 128 \sin^{4}\theta\cos^{4}\theta=\frac{3-4\cos 4\theta+\cos 8\theta}{128}\!
  183. sin 5 θ = 10 sin θ - 5 sin 3 θ + sin 5 θ 16 \sin^{5}\theta=\frac{10\sin\theta-5\sin 3\theta+\sin 5\theta}{16}\!
  184. cos 5 θ = 10 cos θ + 5 cos 3 θ + cos 5 θ 16 \cos^{5}\theta=\frac{10\cos\theta+5\cos 3\theta+\cos 5\theta}{16}\!
  185. sin 5 θ cos 5 θ = 10 sin 2 θ - 5 sin 6 θ + sin 10 θ 512 \sin^{5}\theta\cos^{5}\theta=\frac{10\sin 2\theta-5\sin 6\theta+\sin 10\theta}% {512}\!
  186. if n is odd \,\text{if }n\,\text{ is odd}
  187. cos n θ = 2 2 n k = 0 n - 1 2 ( n k ) cos ( ( n - 2 k ) θ ) \cos^{n}\theta=\frac{2}{2^{n}}\sum_{k=0}^{\frac{n-1}{2}}{\left({{n}\atop{k}}% \right)}\cos{((n-2k)\theta)}
  188. sin n θ = 2 2 n k = 0 n - 1 2 ( - 1 ) ( n - 1 2 - k ) ( n k ) sin ( ( n - 2 k ) θ ) \sin^{n}\theta=\frac{2}{2^{n}}\sum_{k=0}^{\frac{n-1}{2}}(-1)^{(\frac{n-1}{2}-k% )}{\left({{n}\atop{k}}\right)}\sin{((n-2k)\theta)}
  189. if n is even \,\text{if }n\,\text{ is even}
  190. cos n θ = 1 2 n ( n n 2 ) + 2 2 n k = 0 n 2 - 1 ( n k ) cos ( ( n - 2 k ) θ ) \cos^{n}\theta=\frac{1}{2^{n}}{\left({{n}\atop{\frac{n}{2}}}\right)}+\frac{2}{% 2^{n}}\sum_{k=0}^{\frac{n}{2}-1}{\left({{n}\atop{k}}\right)}\cos{((n-2k)\theta)}
  191. sin n θ = 1 2 n ( n n 2 ) + 2 2 n k = 0 n 2 - 1 ( - 1 ) ( n 2 - k ) ( n k ) cos ( ( n - 2 k ) θ ) \sin^{n}\theta=\frac{1}{2^{n}}{\left({{n}\atop{\frac{n}{2}}}\right)}+\frac{2}{% 2^{n}}\sum_{k=0}^{\frac{n}{2}-1}(-1)^{(\frac{n}{2}-k)}{\left({{n}\atop{k}}% \right)}\cos{((n-2k)\theta)}
  192. 2 cos θ cos φ = < m t p l > cos ( θ - φ ) + cos ( θ + φ ) 2\cos\theta\cos\varphi=<mtpl>{{\cos(\theta-\varphi)+\cos(\theta+\varphi)}}
  193. 2 sin θ sin φ = cos ( θ - φ ) - cos ( θ + φ ) 2\sin\theta\sin\varphi={{\cos(\theta-\varphi)-\cos(\theta+\varphi)}}
  194. 2 sin θ cos φ = sin ( θ + φ ) + sin ( θ - φ ) 2\sin\theta\cos\varphi={{\sin(\theta+\varphi)+\sin(\theta-\varphi)}}
  195. 2 cos θ sin φ = sin ( θ + φ ) - sin ( θ - φ ) 2\cos\theta\sin\varphi={{\sin(\theta+\varphi)-\sin(\theta-\varphi)}}
  196. tan θ tan φ = cos ( θ - φ ) - cos ( θ + φ ) cos ( θ - φ ) + cos ( θ + φ ) \tan\theta\tan\varphi=\frac{\cos(\theta-\varphi)-\cos(\theta+\varphi)}{\cos(% \theta-\varphi)+\cos(\theta+\varphi)}
  197. k = 1 n cos θ k = 1 2 n e S cos ( e 1 θ 1 + + e n θ n ) where S = { 1 , - 1 } n \begin{aligned}\displaystyle\prod_{k=1}^{n}\cos\theta_{k}&\displaystyle=\frac{% 1}{2^{n}}\sum_{e\in S}\cos(e_{1}\theta_{1}+\cdots+e_{n}\theta_{n})\\ &\displaystyle\,\text{where }S=\{1,-1\}^{n}\end{aligned}
  198. sin θ ± sin φ = 2 sin ( θ ± φ 2 ) cos ( θ φ 2 ) \sin\theta\pm\sin\varphi=2\sin\left(\frac{\theta\pm\varphi}{2}\right)\cos\left% (\frac{\theta\mp\varphi}{2}\right)
  199. cos θ + cos φ = 2 cos ( θ + φ 2 ) cos ( θ - φ 2 ) \cos\theta+\cos\varphi=2\cos\left(\frac{\theta+\varphi}{2}\right)\cos\left(% \frac{\theta-\varphi}{2}\right)
  200. cos θ - cos φ = - 2 sin ( θ + φ 2 ) sin ( θ - φ 2 ) \cos\theta-\cos\varphi=-2\sin\left({\theta+\varphi\over 2}\right)\sin\left({% \theta-\varphi\over 2}\right)
  201. x + y + z = π x+y+z=\pi
  202. sin ( 2 x ) + sin ( 2 y ) + sin ( 2 z ) = 4 sin ( x ) sin ( y ) sin ( z ) . \sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\sin(y)\sin(z).\,
  203. If x + y + z = π = half circle, \,\text{If }x+y+z=\pi=\,\text{half circle,}\,
  204. then tan ( x ) + tan ( y ) + tan ( z ) = tan ( x ) tan ( y ) tan ( z ) . \,\text{then }\tan(x)+\tan(y)+\tan(z)=\tan(x)\tan(y)\tan(z).\,
  205. If x + y + z = π 2 = right angle (quarter circle), \,\text{If }x+y+z=\tfrac{\pi}{2}=\,\text{right angle (quarter circle),}\,
  206. then cot ( x ) + cot ( y ) + cot ( z ) = cot ( x ) cot ( y ) cot ( z ) . \,\text{then }\cot(x)+\cot(y)+\cot(z)=\cot(x)\cot(y)\cot(z).\,
  207. A n , k = 1 j n j k cot ( a k - a j ) A_{n,k}=\prod_{\begin{smallmatrix}1\leq j\leq n\\ j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})
  208. cot ( z - a 1 ) cot ( z - a n ) = cos n π 2 + k = 1 n A n , k cot ( z - a k ) . \cot(z-a_{1})\cdots\cot(z-a_{n})=\cos\frac{n\pi}{2}+\sum_{k=1}^{n}A_{n,k}\cot(% z-a_{k}).
  209. cot ( z - a 1 ) cot ( z - a 2 ) = - 1 + cot ( a 1 - a 2 ) cot ( z - a 1 ) + cot ( a 2 - a 1 ) cot ( z - a 2 ) . \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})% \cot(z-a_{2}).
  210. If w + x + y + z = π = half circle, then: \,\text{If }w+x+y+z=\pi=\,\text{half circle, then:}
  211. sin ( w + x ) sin ( x + y ) \displaystyle\sin(w+x)\sin(x+y)
  212. a sin x + b cos x = c sin ( x + φ ) a\sin x+b\cos x=c\cdot\sin(x+\varphi)\,
  213. c = a 2 + b 2 , c=\sqrt{a^{2}+b^{2}},\,
  214. φ = atan2 ( b , a ) . \varphi=\operatorname{atan2}\left(b,a\right).
  215. a sin x + b sin ( x + θ ) = c sin ( x + φ ) a\sin x+b\sin(x+\theta)=c\sin(x+\varphi)\,
  216. c = a 2 + b 2 + 2 a b cos θ , c=\sqrt{a^{2}+b^{2}+2ab\cos\theta},\,
  217. φ = atan2 ( b sin θ , a + b cos θ ) . \varphi=\operatorname{atan2}\left(b\,\sin\theta,a+b\cos\theta\right).
  218. i a i sin ( x + θ i ) = a sin ( x + θ ) , \sum_{i}a_{i}\sin(x+\theta_{i})=a\sin(x+\theta),
  219. a 2 = i , j a i a j cos ( θ i - θ j ) a^{2}=\sum_{i,j}a_{i}a_{j}\cos(\theta_{i}-\theta_{j})
  220. tan θ = i a i sin θ i i a i cos θ i . \tan\theta=\frac{\sum_{i}a_{i}\sin\theta_{i}}{\sum_{i}a_{i}\cos\theta_{i}}.
  221. n = 1 N sin n θ = 1 2 cot θ 2 - cos ( N + 1 2 ) θ 2 sin 1 2 θ n = 1 N cos n θ = - 1 2 + sin ( N + 1 2 ) θ 2 sin 1 2 θ \begin{aligned}\displaystyle\sum_{n=1}^{N}\sin n\theta&\displaystyle=\frac{1}{% 2}\cot\frac{\theta}{2}-\frac{\cos(N+\frac{1}{2})\theta}{2\sin\frac{1}{2}\theta% }\\ \displaystyle\sum_{n=1}^{N}\cos n\theta&\displaystyle=-\frac{1}{2}+\frac{\sin(% N+\frac{1}{2})\theta}{2\sin\frac{1}{2}\theta}\end{aligned}
  222. 1 + 2 cos ( x ) + 2 cos ( 2 x ) + 2 cos ( 3 x ) + + 2 cos ( n x ) = sin ( ( n + 1 2 ) x ) sin ( x / 2 ) . 1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots+2\cos(nx)=\frac{\sin\left(\left(n+\frac{% 1}{2}\right)x\right)}{\sin(x/2)}.
  223. α 0 \alpha\neq 0
  224. sin φ + sin ( φ + α ) + sin ( φ + 2 α ) + + sin ( φ + n α ) = sin ( ( n + 1 ) α 2 ) sin ( φ + n α 2 ) sin α 2 and cos φ + cos ( φ + α ) + cos ( φ + 2 α ) + + cos ( φ + n α ) = sin ( ( n + 1 ) α 2 ) cos ( φ + n α 2 ) sin α 2 . \begin{aligned}&\displaystyle\sin{\varphi}+\sin{(\varphi+\alpha)}+\sin{(% \varphi+2\alpha)}+\cdots\\ &\displaystyle{}\qquad\qquad\cdots+\sin{(\varphi+n\alpha)}=\frac{\sin{\left(% \frac{(n+1)\alpha}{2}\right)}\cdot\sin{(\varphi+\frac{n\alpha}{2})}}{\sin{% \frac{\alpha}{2}}}\quad\hbox{and}\\ &\displaystyle\cos{\varphi}+\cos{(\varphi+\alpha)}+\cos{(\varphi+2\alpha)}+% \cdots\\ &\displaystyle{}\qquad\qquad\cdots+\cos{(\varphi+n\alpha)}=\frac{\sin{\left(% \frac{(n+1)\alpha}{2}\right)}\cdot\cos{(\varphi+\frac{n\alpha}{2})}}{\sin{% \frac{\alpha}{2}}}.\end{aligned}
  225. a cos ( x ) + b sin ( x ) = a 2 + b 2 cos ( x - atan2 ( b , a ) ) a\cos(x)+b\sin(x)=\sqrt{a^{2}+b^{2}}\cos(x-\operatorname{atan2}\,(b,a))\;
  226. tan ( x ) + sec ( x ) = tan ( x 2 + π 4 ) . \tan(x)+\sec(x)=\tan\left({x\over 2}+{\pi\over 4}\right).
  227. cot ( x ) cot ( y ) + cot ( y ) cot ( z ) + cot ( z ) cot ( x ) = 1. \cot(x)\cot(y)+\cot(y)\cot(z)+\cot(z)\cot(x)=1.\,
  228. f ( x ) = ( cos α ) x - sin α ( sin α ) x + cos α , f(x)=\frac{(\cos\alpha)x-\sin\alpha}{(\sin\alpha)x+\cos\alpha},
  229. g ( x ) = ( cos β ) x - sin β ( sin β ) x + cos β , g(x)=\frac{(\cos\beta)x-\sin\beta}{(\sin\beta)x+\cos\beta},
  230. f ( g ( x ) ) = g ( f ( x ) ) = ( cos ( α + β ) ) x - sin ( α + β ) ( sin ( α + β ) ) x + cos ( α + β ) . f(g(x))=g(f(x))=\frac{(\cos(\alpha+\beta))x-\sin(\alpha+\beta)}{(\sin(\alpha+% \beta))x+\cos(\alpha+\beta)}.
  231. f α f β = f α + β . f_{\alpha}\circ f_{\beta}=f_{\alpha+\beta}.\,
  232. arcsin ( x ) + arccos ( x ) = π / 2 \arcsin(x)+\arccos(x)=\pi/2\;
  233. arctan ( x ) + \arccot ( x ) = π / 2. \arctan(x)+\arccot(x)=\pi/2.\;
  234. arctan ( x ) + arctan ( 1 / x ) = { π / 2 , if x > 0 - π / 2 , if x < 0 \arctan(x)+\arctan(1/x)=\left\{\begin{matrix}\pi/2,&\mbox{if }~{}x>0\\ -\pi/2,&\mbox{if }~{}x<0\end{matrix}\right.
  235. sin [ arccos ( x ) ] = 1 - x 2 \sin[\arccos(x)]=\sqrt{1-x^{2}}\,
  236. tan [ arcsin ( x ) ] = x 1 - x 2 \tan[\arcsin(x)]=\frac{x}{\sqrt{1-x^{2}}}
  237. sin [ arctan ( x ) ] = x 1 + x 2 \sin[\arctan(x)]=\frac{x}{\sqrt{1+x^{2}}}
  238. tan [ arccos ( x ) ] = 1 - x 2 x \tan[\arccos(x)]=\frac{\sqrt{1-x^{2}}}{x}
  239. cos [ arctan ( x ) ] = 1 1 + x 2 \cos[\arctan(x)]=\frac{1}{\sqrt{1+x^{2}}}
  240. cot [ arcsin ( x ) ] = 1 - x 2 x \cot[\arcsin(x)]=\frac{\sqrt{1-x^{2}}}{x}
  241. cos [ arcsin ( x ) ] = 1 - x 2 \cos[\arcsin(x)]=\sqrt{1-x^{2}}\,
  242. cot [ arccos ( x ) ] = x 1 - x 2 \cot[\arccos(x)]=\frac{x}{\sqrt{1-x^{2}}}
  243. e i x = cos ( x ) + i sin ( x ) e^{ix}=\cos(x)+i\sin(x)\,
  244. e - i x = cos ( - x ) + i sin ( - x ) = cos ( x ) - i sin ( x ) e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin(x)
  245. e i π = - 1 e^{i\pi}=-1
  246. e 2 π i = 1 e^{2\pi i}=1
  247. cos ( x ) = e i x + e - i x 2 \cos(x)=\frac{e^{ix}+e^{-ix}}{2}
  248. sin ( x ) = e i x - e - i x 2 i \sin(x)=\frac{e^{ix}-e^{-ix}}{2i}
  249. tan ( x ) = sin ( x ) cos ( x ) = e i x - e - i x i ( e i x + e - i x ) \tan(x)=\frac{\sin(x)}{\cos(x)}=\frac{e^{ix}-e^{-ix}}{i({e^{ix}+e^{-ix}})}
  250. i 2 = - 1 i^{2}=-1
  251. sin x = x n = 1 ( 1 - x 2 π 2 n 2 ) \sin x=x\prod_{n=1}^{\infty}\left(1-\frac{x^{2}}{\pi^{2}n^{2}}\right)
  252. sinh x = x n = 1 ( 1 + x 2 π 2 n 2 ) \sinh x=x\prod_{n=1}^{\infty}\left(1+\frac{x^{2}}{\pi^{2}n^{2}}\right)
  253. sin x x = n = 1 cos ( x 2 n ) \frac{\sin x}{x}=\prod_{n=1}^{\infty}\cos\left(\frac{x}{2^{n}}\right)
  254. cos x = n = 1 ( 1 - x 2 π 2 ( n - 1 2 ) 2 ) \cos x=\prod_{n=1}^{\infty}\left(1-\frac{x^{2}}{\pi^{2}(n-\frac{1}{2})^{2}}\right)
  255. cosh x = n = 1 ( 1 + x 2 π 2 ( n - 1 2 ) 2 ) \cosh x=\prod_{n=1}^{\infty}\left(1+\frac{x^{2}}{\pi^{2}(n-\frac{1}{2})^{2}}\right)
  256. | sin x | = 1 2 n = 0 | tan ( 2 n x ) | 2 n + 1 |\sin x|=\frac{1}{2}\prod_{n=0}^{\infty}\sqrt[2^{n+1}]{\left|\tan\left(2^{n}x% \right)\right|}
  257. cos 20 cos 40 cos 80 = 1 8 \cos 20^{\circ}\cdot\cos 40^{\circ}\cdot\cos 80^{\circ}=\frac{1}{8}
  258. j = 0 k - 1 cos ( 2 j x ) = sin ( 2 k x ) 2 k sin ( x ) . \prod_{j=0}^{k-1}\cos(2^{j}x)=\frac{\sin(2^{k}x)}{2^{k}\sin(x)}.
  259. sin 20 sin 40 sin 80 = 3 8 . \sin 20^{\circ}\cdot\sin 40^{\circ}\cdot\sin 80^{\circ}=\frac{\sqrt{3}}{8}.
  260. cos π 9 cos 2 π 9 cos 4 π 9 = 1 8 , \cos\frac{\pi}{9}\cos\frac{2\pi}{9}\cos\frac{4\pi}{9}=\frac{1}{8},
  261. tan 50 tan 60 tan 70 = tan 80 . \tan 50^{\circ}\cdot\tan 60^{\circ}\cdot\tan 70^{\circ}=\tan 80^{\circ}.
  262. tan 40 tan 30 tan 20 = tan 10 . \tan 40^{\circ}\cdot\tan 30^{\circ}\cdot\tan 20^{\circ}=\tan 10^{\circ}.
  263. cos 24 + cos 48 + cos 96 + cos 168 = 1 2 . \cos 24^{\circ}+\cos 48^{\circ}+\cos 96^{\circ}+\cos 168^{\circ}=\frac{1}{2}.
  264. cos ( 2 π 21 ) + cos ( 2 2 π 21 ) + cos ( 4 2 π 21 ) \displaystyle\cos\left(\frac{2\pi}{21}\right)+\cos\left(2\cdot\frac{2\pi}{21}% \right)+\cos\left(4\cdot\frac{2\pi}{21}\right)
  265. 2 cos π 3 = 1 , 2\cos\frac{\pi}{3}=1,
  266. 2 cos π 5 × 2 cos 2 π 5 = 1 , 2\cos\frac{\pi}{5}\times 2\cos\frac{2\pi}{5}=1,
  267. 2 cos π 7 × 2 cos 2 π 7 × 2 cos 3 π 7 = 1 , 2\cos\frac{\pi}{7}\times 2\cos\frac{2\pi}{7}\times 2\cos\frac{3\pi}{7}=1,
  268. cos π 3 + cos π 5 × cos 2 π 5 + cos π 7 × cos 2 π 7 × cos 3 π 7 + = 1. \cos\frac{\pi}{3}+\cos\frac{\pi}{5}\times\cos\frac{2\pi}{5}+\cos\frac{\pi}{7}% \times\cos\frac{2\pi}{7}\times\cos\frac{3\pi}{7}+\dots=1.
  269. k = 1 n - 1 sin ( k π n ) = n 2 n - 1 \prod_{k=1}^{n-1}\sin\left(\frac{k\pi}{n}\right)=\frac{n}{2^{n-1}}
  270. k = 1 n - 1 cos ( k π n ) = sin ( π n / 2 ) 2 n - 1 \prod_{k=1}^{n-1}\cos\left(\frac{k\pi}{n}\right)=\frac{\sin(\pi n/2)}{2^{n-1}}
  271. k = 1 n - 1 tan ( k π n ) = n sin ( π n / 2 ) \prod_{k=1}^{n-1}\tan\left(\frac{k\pi}{n}\right)=\frac{n}{\sin(\pi n/2)}
  272. k = 1 m tan ( k π 2 m + 1 ) = 2 m + 1 \prod_{k=1}^{m}\tan\left(\frac{k\pi}{2m+1}\right)=\sqrt{2m+1}
  273. k = 1 n sin ( ( 2 k - 1 ) π 4 n ) = k = 1 n cos ( ( 2 k - 1 ) π 4 n ) = 2 2 n \prod_{k=1}^{n}\sin\left(\frac{\left(2k-1\right)\pi}{4n}\right)=\prod_{k=1}^{n% }\cos\left(\frac{\left(2k-1\right)\pi}{4n}\right)=\frac{\sqrt{2}}{2^{n}}
  274. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}
  275. π 4 = 5 arctan 1 7 + 2 arctan 3 79 \frac{\pi}{4}=5\arctan\frac{1}{7}+2\arctan\frac{3}{79}
  276. π = arccos 4 5 + arccos 5 13 + arccos 16 65 = arcsin 3 5 + arcsin 12 13 + arcsin 63 65 . \pi=\arccos\frac{4}{5}+\arccos\frac{5}{13}+\arccos\frac{16}{65}=\arcsin\frac{3% }{5}+\arcsin\frac{12}{13}+\arcsin\frac{63}{65}.
  277. n / 2 \scriptstyle\sqrt{n}/2
  278. sin 0 = sin 0 = 0 / 2 = cos 90 = cos ( π 2 ) sin ( π 6 ) = sin 30 = 1 / 2 = cos 60 = cos ( π 3 ) sin ( π 4 ) = sin 45 = 2 / 2 = cos 45 = cos ( π 4 ) sin ( π 3 ) = sin 60 = 3 / 2 = cos 30 = cos ( π 6 ) sin ( π 2 ) = sin 90 = 4 / 2 = cos 0 = cos 0 \begin{matrix}\sin 0&=&\sin 0^{\circ}&=&\sqrt{0}/2&=&\cos 90^{\circ}&=&\cos% \left(\frac{\pi}{2}\right)\\ \\ \sin\left(\frac{\pi}{6}\right)&=&\sin 30^{\circ}&=&\sqrt{1}/2&=&\cos 60^{\circ% }&=&\cos\left(\frac{\pi}{3}\right)\\ \\ \sin\left(\frac{\pi}{4}\right)&=&\sin 45^{\circ}&=&\sqrt{2}/2&=&\cos 45^{\circ% }&=&\cos\left(\frac{\pi}{4}\right)\\ \\ \sin\left(\frac{\pi}{3}\right)&=&\sin 60^{\circ}&=&\sqrt{3}/2&=&\cos 30^{\circ% }&=&\cos\left(\frac{\pi}{6}\right)\\ \\ \sin\left(\frac{\pi}{2}\right)&=&\sin 90^{\circ}&=&\sqrt{4}/2&=&\cos 0^{\circ}% &=&\cos 0\end{matrix}
  279. cos ( π 5 ) = cos 36 = 1 4 ( 5 + 1 ) = 1 2 φ \cos\left(\frac{\pi}{5}\right)=\cos 36^{\circ}=\tfrac{1}{4}(\sqrt{5}+1)=\tfrac% {1}{2}\varphi
  280. sin ( π 10 ) = sin 18 = 1 4 ( 5 - 1 ) = 1 2 φ - 1 \sin\left(\frac{\pi}{10}\right)=\sin 18^{\circ}=\tfrac{1}{4}(\sqrt{5}-1)=% \tfrac{1}{2}\varphi^{-1}
  281. sin 2 ( 18 ) + sin 2 ( 30 ) = sin 2 ( 36 ) . \sin^{2}(18^{\circ})+\sin^{2}(30^{\circ})=\sin^{2}(36^{\circ}).\,
  282. cos ( t sin ( x ) ) = J 0 ( t ) + 2 k = 1 J 2 k ( t ) cos ( 2 k x ) \cos(t\sin(x))=J_{0}(t)+2\sum_{k=1}^{\infty}J_{2k}(t)\cos(2kx)
  283. sin ( t sin ( x ) ) = 2 k = 0 J 2 k + 1 ( t ) sin ( ( 2 k + 1 ) x ) \sin(t\sin(x))=2\sum_{k=0}^{\infty}J_{2k+1}(t)\sin((2k+1)x)
  284. cos ( t cos ( x ) ) = J 0 ( t ) + 2 k = 1 ( - 1 ) k J 2 k ( t ) cos ( 2 k x ) \cos(t\cos(x))=J_{0}(t)+2\sum_{k=1}^{\infty}(-1)^{k}J_{2k}(t)\cos(2kx)
  285. sin ( t cos ( x ) ) = 2 k = 0 ( - 1 ) k J 2 k + 1 ( t ) cos ( ( 2 k + 1 ) x ) \sin(t\cos(x))=2\sum_{k=0}^{\infty}(-1)^{k}J_{2k+1}(t)\cos((2k+1)x)
  286. lim x 0 sin x x = 1 , \lim_{x\rightarrow 0}\frac{\sin x}{x}=1,
  287. lim x 0 1 - cos x x = 0 , \lim_{x\rightarrow 0}\frac{1-\cos x}{x}=0,
  288. d d x sin x = cos x {\mathrm{d}\over\mathrm{d}x}\sin x=\cos x
  289. d d x sin x = cos x , d d x arcsin x = 1 1 - x 2 d d x cos x = - sin x , d d x arccos x = - 1 1 - x 2 d d x tan x = sec 2 x , d d x arctan x = 1 1 + x 2 d d x cot x = - csc 2 x , d d x \arccot x = - 1 1 + x 2 d d x sec x = tan x sec x , d d x \arcsec x = 1 | x | x 2 - 1 d d x csc x = - csc x cot x , d d x \arccsc x = - 1 | x | x 2 - 1 \begin{aligned}\displaystyle{\mathrm{d}\over\mathrm{d}x}\sin x&\displaystyle=% \cos x,&\displaystyle{\mathrm{d}\over\mathrm{d}x}\arcsin x&\displaystyle={1% \over\sqrt{1-x^{2}}}\\ \\ \displaystyle{\mathrm{d}\over\mathrm{d}x}\cos x&\displaystyle=-\sin x,&% \displaystyle{\mathrm{d}\over\mathrm{d}x}\arccos x&\displaystyle={-1\over\sqrt% {1-x^{2}}}\\ \\ \displaystyle{\mathrm{d}\over\mathrm{d}x}\tan x&\displaystyle=\sec^{2}x,&% \displaystyle{\mathrm{d}\over\mathrm{d}x}\arctan x&\displaystyle={1\over 1+x^{% 2}}\\ \\ \displaystyle{\mathrm{d}\over\mathrm{d}x}\cot x&\displaystyle=-\csc^{2}x,&% \displaystyle{\mathrm{d}\over\mathrm{d}x}\arccot x&\displaystyle={-1\over 1+x^% {2}}\\ \\ \displaystyle{\mathrm{d}\over\mathrm{d}x}\sec x&\displaystyle=\tan x\sec x,&% \displaystyle{\mathrm{d}\over\mathrm{d}x}\arcsec x&\displaystyle={1\over|x|% \sqrt{x^{2}-1}}\\ \\ \displaystyle{\mathrm{d}\over\mathrm{d}x}\csc x&\displaystyle=-\csc x\cot x,&% \displaystyle{\mathrm{d}\over\mathrm{d}x}\arccsc x&\displaystyle={-1\over|x|% \sqrt{x^{2}-1}}\end{aligned}
  290. d u a 2 - u 2 = sin - 1 ( u a ) + C \int\frac{\mathrm{d}u}{\sqrt{a^{2}-u^{2}}}=\sin^{-1}\left(\frac{u}{a}\right)+C
  291. d u a 2 + u 2 = 1 a tan - 1 ( u a ) + C \int\frac{\mathrm{d}u}{a^{2}+u^{2}}=\frac{1}{a}\tan^{-1}\left(\frac{u}{a}% \right)+C
  292. d u u u 2 - a 2 = 1 a sec - 1 | u a | + C \int\frac{\mathrm{d}u}{u\sqrt{u^{2}-a^{2}}}=\frac{1}{a}\sec^{-1}\left|\frac{u}% {a}\right|+C
  293. \circ
  294. k = 0 n ( n k ) ( d d x - sin x ) ( d d x - sin x + i ) ( d d x - sin x + ( k - 1 ) i ) ( sin x ) n - k = 0. \sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\left(\frac{\mathrm{d}}{\mathrm{d}x}% -\sin x\right)\circ\left(\frac{\mathrm{d}}{\mathrm{d}x}-\sin x+i\right)\circ% \cdots\circ\left(\frac{\mathrm{d}}{\mathrm{d}x}-\sin x+(k-1)i\right)(\sin x)^{% n-k}=0.
  295. sin θ = e i θ - e - i θ 2 i \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}\,
  296. arcsin x = - i ln ( i x + 1 - x 2 ) \arcsin x=-i\ln\left(ix+\sqrt{1-x^{2}}\right)\,
  297. cos θ = e i θ + e - i θ 2 \cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}\,
  298. arccos x = i ln ( x - i 1 - x 2 ) \arccos x=i\,\ln\left(x-i\,\sqrt{1-x^{2}}\right)\,
  299. tan θ = e i θ - e - i θ i ( e i θ + e - i θ ) \tan\theta=\frac{e^{i\theta}-e^{-i\theta}}{i(e^{i\theta}+e^{-i\theta})}\,
  300. arctan x = i 2 ln ( i + x i - x ) \arctan x=\frac{i}{2}\ln\left(\frac{i+x}{i-x}\right)\,
  301. csc θ = 2 i e i θ - e - i θ \csc\theta=\frac{2i}{e^{i\theta}-e^{-i\theta}}\,
  302. \arccsc x = - i ln ( i x + 1 - 1 x 2 ) \arccsc x=-i\ln\left(\tfrac{i}{x}+\sqrt{1-\tfrac{1}{x^{2}}}\right)\,
  303. sec θ = 2 e i θ + e - i θ \sec\theta=\frac{2}{e^{i\theta}+e^{-i\theta}}\,
  304. \arcsec x = - i ln ( 1 x + 1 - i x 2 ) \arcsec x=-i\ln\left(\tfrac{1}{x}+\sqrt{1-\tfrac{i}{x^{2}}}\right)\,
  305. cot θ = i ( e i θ + e - i θ ) e i θ - e - i θ \cot\theta=\frac{i(e^{i\theta}+e^{-i\theta})}{e^{i\theta}-e^{-i\theta}}\,
  306. \arccot x = i 2 ln ( x - i x + i ) \arccot x=\frac{i}{2}\ln\left(\frac{x-i}{x+i}\right)\,
  307. cis θ = e i θ \operatorname{cis}\,\theta=e^{i\theta}\,
  308. arccis x = ln x i = - i ln x = arg x \operatorname{arccis}\,x=\frac{\ln x}{i}=-i\ln x=\operatorname{arg}\,x\,
  309. 1 + 2 cos ( x ) + 2 cos ( 2 x ) + 2 cos ( 3 x ) + + 2 cos ( n x ) = sin [ ( n + 1 2 ) x ] sin ( x 2 ) . 1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots+2\cos(nx)=\frac{\sin\left[\left(n+\frac{% 1}{2}\right)x\right]}{\sin\left(\frac{x}{2}\right)}.
  310. π \pi
  311. t = tan ( x 2 ) , t=\tan\left(\frac{x}{2}\right),
  312. sin ( x ) = 2 t 1 + t 2 and cos ( x ) = 1 - t 2 1 + t 2 and e i x = 1 + i t 1 - i t \sin(x)=\frac{2t}{1+t^{2}}\,\text{ and }\cos(x)=\frac{1-t^{2}}{1+t^{2}}\,\text% { and }e^{ix}=\frac{1+it}{1-it}
  313. 55 / 8 5{5}/{8}

Literal_movement_grammar.html

  1. S α S\to\alpha
  2. S S
  3. α \alpha
  4. X ( x 1 , , x n ) α X(x_{1},...,x_{n})\to\alpha
  5. α \alpha
  6. x i x_{i}
  7. f ( x y ) f(xy)
  8. f ( a b ) f(ab)
  9. x = ϵ , y = a b ; x = a , y = b ; x = a b , y = ϵ x=\epsilon,\ y=ab;\ x=a,\ y=b;\ x=ab,\ y=\epsilon
  10. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  11. x : f ( x 1 , , x n ) x\,\text{:}f(x_{1},\ldots,x_{n})
  12. f ( x 1 , , x n ) f(x_{1},...,x_{n})
  13. f ( x 1 , , x n ) / α f(x_{1},\ldots,x_{n})/\alpha
  14. f ( x 1 , , x n ) f(x_{1},...,x_{n})
  15. α \alpha
  16. f ( x 1 , , x m ) α y : g ( z 1 , z n ) β f(x_{1},...,x_{m})\to\alpha\ y\,\text{:}g(z_{1},...z_{n})\ \beta
  17. α \alpha
  18. β \beta
  19. α \alpha
  20. β \beta
  21. x / y x/y
  22. ϵ \epsilon
  23. g ( y 1 , , y n ) = z g(y_{1},...,y_{n})=z
  24. { a n b n c n : n 1 } \{a^{n}b^{n}c^{n}:n\geq 1\}
  25. S ( ) x : A ( ) B ( x ) S()\to x\,\text{:}A()\ B(x)
  26. A ( ) a A ( ) A()\to a\ A()
  27. A ( ) ϵ A()\to\epsilon
  28. B ( x y ) a / x b B ( y ) c B(xy)\to a/x\ b\ B(y)c
  29. B ( ϵ ) ϵ B(\epsilon)\to\epsilon
  30. S ( ) x : A ( ) B ( x ) x : ( a A ( ) ) B ( x ) x : ( a a A ( ) ) B ( x ) x : a a B ( x ) a a B ( a a ) S()\to x\,\text{:}A()\ B(x)\to x\,\text{:}(a\ A())\ B(x)\to x\,\text{:}(aa\ A(% ))\ B(x)\to x\,\text{:}aa\ B(x)\to aa\ B(aa)
  31. a a a / a b B ( a ) c a a b B ( a ) c a a b a / a b B ( ) c c a a b b B ( ) c c a a b b c c \to aa\ a/a\ b\ B(a)\ c\to aab\ B(a)\ c\to aab\ a/a\ b\ B()\ cc\to aabb\ B()\ % cc\ \to aabbcc

Littlewood's_rule.html

  1. R 1 R_{1}
  2. R 2 R_{2}
  3. R 1 > R 2 R_{1}>R_{2}
  4. C C
  5. j j
  6. D j D_{j}
  7. F j ( ) F_{j}()
  8. R 2 R 1 * Prob ( D 1 > x ) R_{2}\geq R_{1}*\,\text{Prob}(D_{1}>x)
  9. R 2 R_{2}
  10. R 1 R_{1}
  11. D 1 D_{1}
  12. x x
  13. y 1 y_{1}^{\star}
  14. F j ( x ) F_{j}(x)
  15. y 1 y_{1}^{\star}
  16. π ( x ) = R 1 * Prob ( D 1 > x ) \pi(x)=R_{1}*\,\text{Prob}(D_{1}>x)
  17. n n

Littlewood–Paley_theory.html

  1. f ^ ρ = χ ρ f ^ \hat{f}_{\rho}=\chi_{\rho}\hat{f}
  2. ρ = [ - 2 k + 1 , - 2 k ] [ 2 k , 2 k + 1 ] \rho=[-2^{k+1},-2^{k}]\cup[2^{k},2^{k+1}]
  3. u ( x , y ) = R n P y ( t ) f ( x - t ) d t u(x,y)=\int_{R^{n}}P_{y}(t)f(x-t)dt
  4. P y ( x ) = R n e - 2 π i t x - 2 π | t | y d t = Γ ( ( n + 1 ) / 2 ) π ( n + 1 ) / 2 y ( | x | 2 + y 2 ) ( n + 1 ) / 2 P_{y}(x)=\int_{R^{n}}e^{-2\pi itx-2\pi|t|y}dt=\frac{\Gamma((n+1)/2)}{\pi^{(n+1% )/2}}\frac{y}{(|x|^{2}+y^{2})^{(n+1)/2}}
  5. g ( f ) ( x ) = ( 0 | u ( x , y ) | 2 y d y ) 1 2 g(f)(x)=(\int_{0}^{\infty}|\nabla u(x,y)|^{2}ydy)^{\frac{1}{2}}

Living_anionic_polymerization.html

  1. M n = M o [ M ] o [ I ] M_{n}=M_{o}\frac{[\mbox{M}~{}]_{o}}{[\mbox{I}~{}]}
  2. PLi + X-Y P-X + LiY \displaystyle\mbox{PLi}~{}+\mbox{X-Y}~{}{\longrightarrow}\mbox{P-X}~{}+\mbox{LiY}

Ljubljana_graph.html

  1. ( x - 3 ) x 14 ( x + 3 ) ( x 2 - x - 4 ) 7 ( x 2 - 2 ) 6 ( x 2 + x - 4 ) 7 ( x 4 - 6 x 2 + 4 ) 14 . (x-3)x^{14}(x+3)(x^{2}-x-4)^{7}(x^{2}-2)^{6}(x^{2}+x-4)^{7}(x^{4}-6x^{2}+4)^{1% 4}.

Lobb_numbers.html

  1. L m , n = 2 m + 1 m + n + 1 ( 2 n m + n ) for n m 0. L_{m,n}=\frac{2m+1}{m+n+1}{\left({{2n}\atop{m+n}}\right)}\qquad\,\text{ for }n% \geq m\geq 0.

LOBPCG.html

  1. A x = λ B x , Ax=\lambda Bx,
  2. ( A , B ) (A,B)
  3. B B
  4. ρ ( x ) := ρ ( A , B ; x ) := x T A x x T B x , \rho(x):=\rho(A,B;x):=\frac{x^{T}Ax}{x^{T}Bx},
  5. A x = λ B x . Ax=\lambda Bx.
  6. r := A x - ρ ( x ) B x , r:=Ax-\rho(x)Bx,
  7. T T
  8. w := T r , w:=Tr,
  9. T := I T:=I
  10. w := r , w:=r,
  11. x i + 1 := x i + α i T ( A x i - ρ ( x i ) B x i ) , x^{i+1}:=x^{i}+\alpha^{i}T(Ax^{i}-\rho(x^{i})Bx^{i}),
  12. x i + 1 := x i + α i w i , x^{i+1}:=x^{i}+\alpha^{i}w^{i},\,
  13. w i := T r i , w^{i}:=Tr^{i},\,
  14. r i := A x i - ρ ( x i ) B x i , r^{i}:=Ax^{i}-\rho(x^{i})Bx^{i},
  15. α i \alpha^{i}
  16. x i + 1 := arg max y s p a n { x i , w i } ρ ( y ) x^{i+1}:=\arg\max_{y\in span\{x^{i},w^{i}\}}\rho(y)
  17. arg min \arg\min
  18. x i + 1 := arg max y s p a n { x i , w i , x i - 1 } ρ ( y ) x^{i+1}:=\arg\max_{y\in span\{x^{i},w^{i},x^{i-1}\}}\rho(y)
  19. arg min \arg\min
  20. x i x^{i}
  21. x i - 1 x^{i-1}
  22. x i - 1 x^{i-1}
  23. p i = x i - 1 - x i p^{i}=x^{i-1}-x^{i}
  24. T = I T=I
  25. B = I B=I
  26. i > 3 i>3

Local_asymptotic_normality.html

  1. ln d P n , θ + r n - 1 h n d P n , θ = h Δ n , θ - 1 2 h I θ h + o P n , θ ( 1 ) , \ln\frac{dP_{\!n,\theta+r_{n}^{-1}h_{n}}}{dP_{n,\theta}}=h^{\prime}\Delta_{n,% \theta}-\frac{1}{2}h^{\prime}I_{\theta}\,h+o_{P_{n,\theta}}(1),
  2. ln d P n , θ + r n - 1 h n d P n , θ 𝑑 𝒩 ( - 1 2 h I θ h , h I θ h ) . \ln\frac{dP_{\!n,\theta+r_{n}^{-1}h_{n}}}{dP_{n,\theta}}\ \ \xrightarrow{d}\ % \ \mathcal{N}\Big({-\tfrac{1}{2}}h^{\prime}I_{\theta}\,h,\ h^{\prime}I_{\theta% }\,h\Big).
  3. P n , θ + r n - 1 h n P_{\!n,\theta+r_{n}^{-1}h_{n}}
  4. P n , θ P_{n,\theta}
  5. p n , θ ( x 1 , , x n ; θ ) = i = 1 n f ( x i , θ ) . p_{n,\theta}(x_{1},\ldots,x_{n};\,\theta)=\prod_{i=1}^{n}f(x_{i},\theta).
  6. ln p n , θ + δ θ \displaystyle\ln p_{n,\theta+\delta\theta}
  7. ln p n , θ + h / n p n , θ = h ( 1 n i = 1 n ln f ( x i , θ ) θ ) - 1 2 h ( 1 n i = 1 n - 2 ln f ( x i , θ ) θ θ ) h + o p ( 1 ) . \ln\frac{p_{n,\theta+h/\sqrt{n}}}{p_{n,\theta}}=h^{\prime}\Bigg(\frac{1}{\sqrt% {n}}\sum_{i=1}^{n}\frac{\partial\ln f(x_{i},\theta)}{\partial\theta}\Bigg)\;-% \;\frac{1}{2}h^{\prime}\Bigg(\frac{1}{n}\sum_{i=1}^{n}-\frac{\partial^{2}\ln f% (x_{i},\theta)}{\partial\theta\,\partial\theta^{\prime}}\Bigg)h\;+\;o_{p}(1).
  8. I θ = E [ - 2 ln f ( X i , θ ) θ θ ] = E [ ( ln f ( X i , θ ) θ ) ( ln f ( X i , θ ) θ ) ] . I_{\theta}=\mathrm{E}\bigg[{-\frac{\partial^{2}\ln f(X_{i},\theta)}{\partial% \theta\,\partial\theta^{\prime}}}\bigg]=\mathrm{E}\bigg[\bigg(\frac{\partial% \ln f(X_{i},\theta)}{\partial\theta}\bigg)\bigg(\frac{\partial\ln f(X_{i},% \theta)}{\partial\theta}\bigg)^{\prime}\,\bigg].

Local_Euler_characteristic_formula.html

  1. χ ( G K , M ) = # H 0 ( K , M ) # H 2 ( K , M ) # H 1 ( K , M ) \chi(G_{K},M)=\frac{\#H^{0}(K,M)\cdot\#H^{2}(K,M)}{\#H^{1}(K,M)}
  2. χ ( G K , M ) = ( # R / m R ) - 1 , \chi(G_{K},M)=\left(\#R/mR\right)^{-1},
  3. χ ( G K , M ) = p - [ K : 𝐐 p ] v p ( m ) \chi(G_{K},M)=p^{-[K:\mathbf{Q}_{p}]v_{p}(m)}
  4. χ ( G K , M ) = # H 0 ( K , M ) # H 0 ( K , M ) # H 1 ( K , M ) \chi(G_{K},M)=\frac{\#H^{0}(K,M)\cdot\#H^{0}(K,M^{\prime})}{\#H^{1}(K,M)}

Local_outlier_factor.html

  1. k k
  2. k-distance ( A ) \mbox{k-distance}~{}(A)
  3. A A
  4. N k ( A ) N_{k}(A)
  5. reachability-distance ( A , B ) k = max { k-distance ( B ) , d ( A , B ) } \mbox{reachability-distance}~{}_{k}(A,B)=\max\{\mbox{k-distance}~{}(B),d(A,B)\}
  6. A A
  7. B B
  8. k-distance \mbox{k-distance}~{}
  9. B B
  10. B B
  11. B B
  12. k-distance \mbox{k-distance}~{}
  13. A A
  14. lrd ( A ) := 1 / ( B N k ( A ) reachability-distance ( A , B ) k | N k ( A ) | ) \mbox{lrd}~{}(A):=1/\left(\frac{\sum_{B\in N_{k}(A)}\mbox{reachability-% distance}~{}_{k}(A,B)}{|N_{k}(A)|}\right)
  15. A A
  16. A A
  17. k-distance ( A ) \mbox{k-distance}~{}(A)
  18. LOF ( A ) k := B N k ( A ) lrd ( B ) lrd ( A ) | N k ( A ) | = B N k ( A ) lrd ( B ) | N k ( A ) | / lrd ( A ) \mbox{LOF}~{}_{k}(A):=\frac{\sum_{B\in N_{k}(A)}\frac{\mbox{lrd}~{}(B)}{\mbox{% lrd}~{}(A)}}{|N_{k}(A)|}=\frac{\sum_{B\in N_{k}(A)}\mbox{lrd}~{}(B)}{|N_{k}(A)% |}/\mbox{lrd}~{}(A)
  19. 1 1
  20. 1 1
  21. 1 1
  22. [ 0 : 1 ] [0:1]
  23. [ 0 : 1 ] [0:1]

Local_oxidation_nanolithography.html

  1. M + n H 2 O M O n + 2 n H + + 2 n e - M+nH_{2}O\rightarrow MO_{n}+2nH^{+}+2ne^{-}
  2. M n + 2 n H 2 O + 2 n e - n H 2 + 2 n O H - + M M^{n}+2nH_{2}O+2ne^{-}\rightarrow nH_{2}+2nOH^{-}+M
  3. 2 H + + 2 e - H 2 2H^{+}+2e^{-}\rightarrow H_{2}

Local_Tate_duality.html

  1. A = Hom ( A , μ ) A^{\prime}=\mathrm{Hom}(A,\mu)
  2. H i ( K , A ) × H 2 - i ( K , A ) H 2 ( K , μ ) = 𝐐 / 𝐙 H^{i}(K,A)\times H^{2-i}(K,A^{\prime})\rightarrow H^{2}(K,\mu)=\mathbf{Q}/% \mathbf{Z}
  3. ρ : G K GL ( V ) \rho:G_{K}\rightarrow\mathrm{GL}(V)
  4. V = Hom ( V , 𝐐 p ( 1 ) ) V^{\prime}=\mathrm{Hom}(V,\mathbf{Q}_{p}(1))
  5. H i ( K , V ) × H 2 - i ( K , V ) H 2 ( K , 𝐐 p ( 1 ) ) = 𝐐 p H^{i}(K,V)\times H^{2-i}(K,V^{\prime})\rightarrow H^{2}(K,\mathbf{Q}_{p}(1))=% \mathbf{Q}_{p}

Localization_theorem.html

  1. D F ( x ) d x = 0 D Ω F ( x ) = 0 x Ω \int\limits_{D}F(x)\,dx=0~{}\forall D\subset\Omega~{}\Rightarrow~{}F(x)=0~{}% \forall x\in\Omega
  2. V f ρ ( x , t ) d Ω = 0 \int\limits_{V_{f}}\rho(\vec{x},t)\,d\Omega=0
  3. V c [ ρ t + ( ρ v ) ] d Ω = 0 \int\limits_{V_{c}}\left[{{\partial\rho}\over{\partial t}}+\nabla\cdot(\rho% \vec{v})\right]\,d\Omega=0
  4. ρ t + ( ρ v ) = 0 {\partial\rho\over\partial t}+\nabla\cdot(\rho\vec{v})=0

Locally_catenative_sequence.html

  1. w ( n ) = w ( n - i 1 ) w ( n - i 2 ) w ( n - i k ) for n max { i 1 , , i k } . w(n)=w(n-i_{1})w(n-i_{2})\ldots w(n-i_{k})\,\text{ for }n\geq\max\{i_{1},% \ldots,i_{k}\}\,.
  2. S ( n ) = S ( n - 1 ) S ( n - 2 ) for n 2 . S(n)=S(n-1)S(n-2)\,\text{ for }n\geq 2\,.
  3. T ( n ) = T ( n - 1 ) μ ( T ( n - 1 ) ) for n 1 , T(n)=T(n-1)\mu(T(n-1))\,\text{ for }n\geq 1\,,

Locally_decodable_code.html

  1. ( q , δ , ϵ ) (q,\delta,\epsilon)
  2. n n
  3. x x
  4. N N
  5. C ( x ) C(x)
  6. x i x_{i}
  7. 1 - ϵ 1-\epsilon
  8. q q
  9. C ( x ) C(x)
  10. δ N \delta N
  11. j [ q ] j\in[q]
  12. i [ n ] i\in[n]
  13. j t h j^{th}
  14. i t h i^{th}
  15. [ N ] [N]
  16. [ y ] [y]
  17. { 1 , , y } \{1,\ldots,y\}
  18. δ / 2 \delta/2
  19. δ \delta
  20. δ \delta
  21. ϵ \epsilon
  22. ϵ \epsilon
  23. δ \delta
  24. ϵ \epsilon
  25. | x | | C ( x ) | \frac{|x|}{|C(x)|}
  26. k ϵ k^{\epsilon}
  27. ϵ > 0 \epsilon>0
  28. L : { 0 , 1 } n { 0 , 1 } L:\{0,1\}^{n}\rightarrow\{0,1\}
  29. L : { 0 , 1 } N { 0 , 1 } L^{\prime}:\{0,1\}^{N}\rightarrow\{0,1\}
  30. L L
  31. t t
  32. L L
  33. 2 t 2^{t}
  34. L ( x ) L(x)
  35. x { 0 , 1 } t x\in\{0,1\}^{t}
  36. C C
  37. L L
  38. 2 O ( t ) = 2 t 2^{O(t)}=2^{t^{\prime}}
  39. L L^{\prime}
  40. t t^{\prime}
  41. L L^{\prime}
  42. L L^{\prime}
  43. 1 - ϵ 1-\epsilon
  44. L L^{\prime}
  45. L L
  46. L L^{\prime}
  47. L L
  48. L L
  49. k k
  50. k k
  51. k k
  52. C C
  53. n n
  54. N N
  55. k k
  56. S 1 , , S k S_{1},\ldots,S_{k}
  57. n n
  58. x x
  59. C C
  60. N N
  61. C ( x ) C(x)
  62. i t h i^{th}
  63. x x
  64. k k
  65. q 1 , q k q_{1},\ldots q_{k}
  66. x i x_{i}
  67. C ( x ) q 1 , C ( x ) q k C(x)_{q_{1}},\ldots C(x)_{q_{k}}
  68. A A
  69. C C
  70. A A
  71. x i x_{i}
  72. q j q_{j}
  73. k k
  74. 2 k 2^{k}
  75. x { 0 , 1 } k x\in\{0,1\}^{k}
  76. a j { 0 , 1 } k a_{j}\in\{0,1\}^{k}
  77. j t h j^{th}
  78. x a j x\odot a_{j}
  79. x y = i = 1 k x i y i x\odot y=\sum\limits_{i=1}^{k}x_{i}y_{i}
  80. n 2 \frac{n}{2}
  81. 1 4 \frac{1}{4}
  82. ρ < 1 4 \rho<\frac{1}{4}
  83. ρ \rho
  84. i t h i^{th}
  85. 1 - 2 ρ 1-2\rho
  86. H H
  87. i i
  88. i t h i^{th}
  89. x x
  90. e j e^{j}
  91. { 0 , 1 } k \{0,1\}^{k}
  92. j t h j^{th}
  93. y { 0 , 1 } k y\in\{0,1\}^{k}
  94. f ( y ) f(y)
  95. H H
  96. x y x\odot y
  97. y { 0 , 1 } k y\in\{0,1\}^{k}
  98. y = y e i y^{\prime}=y\otimes e^{i}
  99. \otimes
  100. f ( y ) f ( y ) f(y)\otimes f(y^{\prime})
  101. ( x y ) ( x y ) = ( x y ) ( x ( y e i ) ) = ( x y ) ( x y ) ( x e i ) = x e i (x\odot y)\otimes(x\odot y^{\prime})=(x\odot y)\otimes(x\odot(y\otimes e^{i}))% =(x\odot y)\otimes(x\odot y)\otimes(x\odot e^{i})=x\odot e^{i}
  102. ( x e i ) = x i (x\odot e^{i})=x_{i}
  103. f ( y ) = x y f(y)=x\odot y
  104. f ( y ) = x y f(y^{\prime})=x\odot y^{\prime}
  105. y y
  106. y y^{\prime}
  107. f ( y ) = x y f(y)=x\odot y
  108. f ( y ) = x y f(y^{\prime})=x\odot y^{\prime}
  109. 1 - 2 ρ 1-2\rho
  110. d d
  111. l l
  112. S S
  113. x x
  114. S S
  115. x x
  116. x x
  117. 𝔽 \mathbb{F}
  118. l , d l,d
  119. d < | 𝔽 | d<|\mathbb{F}|
  120. 𝔽 , l , d \mathbb{F},l,d
  121. 𝔽 ( l + d d ) 𝔽 | 𝔽 | l \mathbb{F}^{{\left({{l+d}\atop{d}}\right)}}\rightarrow\mathbb{F}^{|\mathbb{F}|% ^{l}}
  122. l l
  123. P P
  124. 𝔽 \mathbb{F}
  125. d d
  126. P P
  127. 𝔽 l \mathbb{F}^{l}
  128. P ( x 1 , , x l ) = i 1 + + i l d c i 1 , , i l x 1 i 1 x 2 i 2 x l i l P(x_{1},\ldots,x_{l})=\sum\limits_{i_{1}+\ldots+i_{l}\leq d}c_{i_{1},\ldots,i_% {l}}x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{l}^{i_{l}}
  129. ( l + d d ) {\left({{l+d}\atop{d}}\right)}
  130. { P ( x 1 , , x l ) } \{P(x_{1},\ldots,x_{l})\}
  131. x 1 , , x l 𝔽 x_{1},\ldots,x_{l}\in\mathbb{F}
  132. d d
  133. w 𝔽 n w\in\mathbb{F}^{n}
  134. w w
  135. d + 1 d+1
  136. w w
  137. v 𝔽 n v\in\mathbb{F}^{n}
  138. L = { w + λ v λ 𝔽 } L=\{w+\lambda v\mid\lambda\in\mathbb{F}\}
  139. w w
  140. S S
  141. 𝔽 \mathbb{F}
  142. | S | = d + 1 |S|=d+1
  143. w + λ v w+\lambda v
  144. λ S \lambda\in S
  145. { e λ } \{e_{\lambda}\}
  146. h h
  147. d d
  148. h ( λ ) = e λ h(\lambda)=e_{\lambda}
  149. λ S \lambda\in S
  150. w w
  151. h ( 0 ) h(0)
  152. w w
  153. δ \delta
  154. 1 - ( d + 1 ) δ 1-(d+1)\delta

Locally_testable_code.html

  1. Δ ( x , y ) = | { i : x i y i } | \Delta(x,y)=|\{i:x_{i}\neq y_{i}\}|
  2. w w
  3. C : { 0 , 1 } k { 0 , 1 } n C:\{0,1\}^{k}\to\{0,1\}^{n}
  4. Δ ( w , C ) = min x { w , C ( x ) } \Delta(w,C)=\min_{x}\{w,C(x)\}
  5. δ ( x , y ) = Δ ( x , y ) / n and δ ( w , C ) = Δ ( w , C ) / n \delta(x,y)=\Delta(x,y)/n\,\text{ and }\delta(w,C)=\Delta(w,C)/n
  6. C : { 0 , 1 } k 0 , 1 n C:\{0,1\}^{k}\to{0,1}^{n}
  7. q q
  8. δ \delta
  9. w w
  10. q q
  11. w w
  12. x { 0 , 1 } k x\in\{0,1\}^{k}
  13. w = C ( x ) w=C(x)
  14. P r [ M w ( 1 k ) = 1 ] = 1 Pr[M^{w}(1^{k})=1]=1
  15. w { 0 , 1 } n w\in\{0,1\}^{n}
  16. δ ( w , C ) > δ \delta(w,C)>\delta
  17. P r [ M w ( 1 k ) = 1 ] 1 / 2 Pr[M^{w}(1^{k})=1]\leq 1/2
  18. δ \delta
  19. ϵ > 0 \epsilon>0
  20. n = k 1 + ϵ n=k^{1+\epsilon}
  21. n = k 1 + ϵ ( k ) n=k^{1+\epsilon(k)}
  22. ϵ ( k ) \epsilon(k)
  23. 1 / log log k 1/\log\log k
  24. 1 / ( log k ) c 1/(\log k)^{c}
  25. c ( 0 , 1 ) c\in(0,1)
  26. exp ( ( log log log k ) c ) / log k \exp((\log\log\log k)^{c})/\log k
  27. c ( 0 , 1 ) c\in(0,1)
  28. n = k 1 + 1 / ( log k ) c n=k^{1+1/(\log k)^{c}}
  29. c ( 0 , 1 ) c\in(0,1)
  30. n = poly ( log k ) * k n=\,\text{poly}(\log k)*k
  31. q q
  32. x x
  33. w w
  34. M w ( x ) = 1 M^{w}(x)=1
  35. w w
  36. f ( y ) = i x i y i f(y)={\sum_{i}{x_{i}y_{i}}}
  37. w ( x ) w ( y ) = w ( x y ) w(x)\oplus w(y)=w(x\oplus y)
  38. \oplus
  39. w w
  40. δ \delta
  41. δ \delta
  42. w ( x ) w(x)
  43. w ( y ) w(y)
  44. w ( x y ) w(x\oplus y)
  45. P ( E ) P ( A B C ) - 3 * P ( A B ) 3 * P ( A ) - 3 * P ( A B ) - 3 * P ( A B ) 3 δ - 6 δ 2 \begin{aligned}\displaystyle P(E)&\displaystyle\geq P(A\cup B\cup C)-3*P(A\cup B% )\\ &\displaystyle\geq 3*P(A)-3*P(A\cup B)-3*P(A\cup B)\\ &\displaystyle\geq 3\delta-6\delta^{2}\end{aligned}
  46. 0 < δ 5 / 16 0<\delta\leq 5/16
  47. 3 δ - 6 δ 2 < δ 3\delta-6\delta^{2}<\delta
  48. f ( x ) = { 3 δ - 6 δ 2 : 0 δ 5 / 16 45 / 128 : 5 / 16 δ 45 / 128 δ : 45 / 128 δ 1 / 2 f(x)=\begin{cases}3\delta-6\delta^{2}&:0\leq\delta\leq 5/16\\ 45/128&:5/16\leq\delta\leq 45/128\\ \delta&:45/128\leq\delta\leq 1/2\end{cases}
  49. δ \delta
  50. 0 i 2 k 0\leq i\leq 2^{k}
  51. k k
  52. i t h i^{th}
  53. f i ( x ) = x i f_{i}(x)=x_{i}
  54. 2 k 2^{k}
  55. 0 x 2 2 k 0\leq x\leq 2^{2^{k}}
  56. n = 2 2 k n=2^{2^{k}}
  57. x x
  58. y = x y=x
  59. μ > 0 \mu>0
  60. f f
  61. f ( x ) = f ( y ) f(x)=f(y)
  62. f f
  63. f f
  64. x i x_{i}
  65. 1 - μ 1-\mu

Log_sum_inequality.html

  1. a 1 , , a n a_{1},\ldots,a_{n}
  2. b 1 , , b n b_{1},\ldots,b_{n}
  3. a i a_{i}\;
  4. a a
  5. b i b_{i}\;
  6. b b
  7. i = 1 n a i log a i b i a log a b , \sum_{i=1}^{n}a_{i}\log\frac{a_{i}}{b_{i}}\geq a\log\frac{a}{b},
  8. a i b i \frac{a_{i}}{b_{i}}
  9. i i
  10. f ( x ) = x log x f(x)=x\log x
  11. i = 1 n a i log a i b i = i = 1 n b i f ( a i b i ) = b i = 1 n b i b f ( a i b i ) b f ( i = 1 n b i b a i b i ) = b f ( 1 b i = 1 n a i ) = b f ( a b ) = a log a b , \begin{aligned}\displaystyle\sum_{i=1}^{n}a_{i}\log\frac{a_{i}}{b_{i}}&% \displaystyle{}=\sum_{i=1}^{n}b_{i}f\left(\frac{a_{i}}{b_{i}}\right)=b\sum_{i=% 1}^{n}\frac{b_{i}}{b}f\left(\frac{a_{i}}{b_{i}}\right)\\ &\displaystyle{}\geq bf\left(\sum_{i=1}^{n}\frac{b_{i}}{b}\frac{a_{i}}{b_{i}}% \right)=bf\left(\frac{1}{b}\sum_{i=1}^{n}a_{i}\right)=bf\left(\frac{a}{b}% \right)\\ &\displaystyle{}=a\log\frac{a}{b},\end{aligned}
  12. b i b 0 \frac{b_{i}}{b}\geq 0
  13. i b i b = 1 \sum_{i}\frac{b_{i}}{b}=1
  14. f f
  15. p i p_{i}\;
  16. a i a_{i}\;
  17. q i q_{i}\;
  18. b i b_{i}\;
  19. D KL ( P Q ) i = 1 n p i log 2 p i q i 1 log 1 1 = 0. D_{\mathrm{KL}}(P\|Q)\equiv\sum_{i=1}^{n}p_{i}\log_{2}\frac{p_{i}}{q_{i}}\geq 1% \log\frac{1}{1}=0.
  20. n = n=\infty
  21. a < a<\infty
  22. b < b<\infty
  23. g g
  24. f ( x ) = x g ( x ) f(x)=xg(x)

Logarithmic_differentiation.html

  1. [ ln ( f ) ] = f f f = f [ ln ( f ) ] . [\ln(f)]^{\prime}=\frac{f^{\prime}}{f}\quad\rightarrow\quad f^{\prime}=f\cdot[% \ln(f)]^{\prime}.
  2. y = f ( x ) y=f(x)\,\!
  3. ln | y | = ln | f ( x ) | \ln|y|=\ln|f(x)|\,\!
  4. 1 y d y d x = f ( x ) f ( x ) \frac{1}{y}\frac{dy}{dx}=\frac{f^{\prime}(x)}{f(x)}
  5. d y d x = y × f ( x ) f ( x ) = f ( x ) . \frac{dy}{dx}=y\times\frac{f^{\prime}(x)}{f(x)}=f^{\prime}(x).
  6. ln ( a b ) = ln ( a ) + ln ( b ) , ln ( a b ) = ln ( a ) - ln ( b ) , ln ( a n ) = n ln ( a ) \ln(ab)=\ln(a)+\ln(b),\qquad\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b),\qquad% \ln(a^{n})=n\ln(a)
  7. f ( x ) = i ( f i ( x ) ) α i ( x ) . f(x)=\prod_{i}(f_{i}(x))^{\alpha_{i}(x)}.
  8. ln ( f ( x ) ) = i α i ( x ) ln ( f i ( x ) ) , \ln(f(x))=\sum_{i}\alpha_{i}(x)\cdot\ln(f_{i}(x)),
  9. f ( x ) f ( x ) = i [ α i ( x ) ln ( f i ( x ) ) + α i ( x ) f i ( x ) f i ( x ) ] . \frac{f^{\prime}(x)}{f(x)}=\sum_{i}\left[\alpha_{i}^{\prime}(x)\cdot\ln(f_{i}(% x))+\alpha_{i}(x)\cdot\frac{f_{i}^{\prime}(x)}{f_{i}(x)}\right].
  10. f ( x ) = i ( f i ( x ) ) α i ( x ) f ( x ) × i { α i ( x ) ln ( f i ( x ) ) + α i ( x ) f i ( x ) f i ( x ) } [ ln ( f ( x ) ) ] f^{\prime}(x)=\overbrace{\prod_{i}(f_{i}(x))^{\alpha_{i}(x)}}^{f(x)}\times% \overbrace{\sum_{i}\left\{\alpha_{i}^{\prime}(x)\cdot\ln(f_{i}(x))+\alpha_{i}(% x)\cdot\frac{f_{i}^{\prime}(x)}{f_{i}(x)}\right\}}^{[\ln(f(x))]^{\prime}}
  11. f ( x ) = g ( x ) h ( x ) f(x)=g(x)h(x)\,\!
  12. ln ( f ( x ) ) = ln ( g ( x ) h ( x ) ) = ln ( g ( x ) ) + ln ( h ( x ) ) \ln(f(x))=\ln(g(x)h(x))=\ln(g(x))+\ln(h(x))\,\!
  13. f ( x ) f ( x ) = g ( x ) g ( x ) + h ( x ) h ( x ) \frac{f^{\prime}(x)}{f(x)}=\frac{g^{\prime}(x)}{g(x)}+\frac{h^{\prime}(x)}{h(x)}
  14. f ( x ) = f ( x ) × { g ( x ) g ( x ) + h ( x ) h ( x ) } = g ( x ) h ( x ) × { g ( x ) g ( x ) + h ( x ) h ( x ) } f^{\prime}(x)=f(x)\times\Bigg\{\frac{g^{\prime}(x)}{g(x)}+\frac{h^{\prime}(x)}% {h(x)}\Bigg\}=g(x)h(x)\times\Bigg\{\frac{g^{\prime}(x)}{g(x)}+\frac{h^{\prime}% (x)}{h(x)}\Bigg\}
  15. f ( x ) = g ( x ) h ( x ) f(x)=\frac{g(x)}{h(x)}\,\!
  16. ln ( f ( x ) ) = ln ( g ( x ) h ( x ) ) = ln ( g ( x ) ) - ln ( h ( x ) ) \ln(f(x))=\ln\Bigg(\frac{g(x)}{h(x)}\Bigg)=\ln(g(x))-\ln(h(x))\,\!
  17. f ( x ) f ( x ) = g ( x ) g ( x ) - h ( x ) h ( x ) \frac{f^{\prime}(x)}{f(x)}=\frac{g^{\prime}(x)}{g(x)}-\frac{h^{\prime}(x)}{h(x)}
  18. f ( x ) = f ( x ) × { g ( x ) g ( x ) - h ( x ) h ( x ) } = g ( x ) h ( x ) × { g ( x ) g ( x ) - h ( x ) h ( x ) } f^{\prime}(x)=f(x)\times\Bigg\{\frac{g^{\prime}(x)}{g(x)}-\frac{h^{\prime}(x)}% {h(x)}\Bigg\}=\frac{g(x)}{h(x)}\times\Bigg\{\frac{g^{\prime}(x)}{g(x)}-\frac{h% ^{\prime}(x)}{h(x)}\Bigg\}
  19. f ( x ) f(x)
  20. f ( x ) = g ( x ) h ( x ) f(x)=g(x)^{h(x)}\,\!
  21. ln ( f ( x ) ) = ln ( g ( x ) h ( x ) ) = h ( x ) ln ( g ( x ) ) \ln(f(x))=\ln\left(g(x)^{h(x)}\right)=h(x)\ln(g(x))\,\!
  22. f ( x ) f ( x ) = h ( x ) ln ( g ( x ) ) + h ( x ) g ( x ) g ( x ) \frac{f^{\prime}(x)}{f(x)}=h^{\prime}(x)\ln(g(x))+h(x)\frac{g^{\prime}(x)}{g(x)}
  23. f ( x ) = f ( x ) × { h ( x ) ln ( g ( x ) ) + h ( x ) g ( x ) g ( x ) } = g ( x ) h ( x ) × { h ( x ) ln ( g ( x ) ) + h ( x ) g ( x ) g ( x ) } . f^{\prime}(x)=f(x)\times\Bigg\{h^{\prime}(x)\ln(g(x))+h(x)\frac{g^{\prime}(x)}% {g(x)}\Bigg\}=g(x)^{h(x)}\times\Bigg\{h^{\prime}(x)\ln(g(x))+h(x)\frac{g^{% \prime}(x)}{g(x)}\Bigg\}.

Long_code_(mathematics).html

  1. f 1 , , f 2 n : { 0 , 1 } k { 0 , 1 } f_{1},\dots,f_{2^{n}}:\{0,1\}^{k}\to\{0,1\}
  2. k = log n k=\log n
  3. { 0 , 1 } k { 0 , 1 } \{0,1\}^{k}\to\{0,1\}
  4. x { 0 , 1 } k x\in\{0,1\}^{k}
  5. f 1 ( x ) f 2 ( x ) f 2 n ( x ) f_{1}(x)\circ f_{2}(x)\circ\dots\circ f_{2^{n}}(x)
  6. \circ
  7. 2 n = 2 2 k 2^{n}=2^{2^{k}}
  8. f i f_{i}
  9. 𝔽 2 k 𝔽 2 \mathbb{F}_{2}^{k}\to\mathbb{F}_{2}
  10. 2 k 2^{k}
  11. 2 k 2^{k}
  12. j [ n ] j\in[n]
  13. j j
  14. f : { 0 , 1 } n { 0 , 1 } f:\{0,1\}^{n}\to\{0,1\}
  15. f ( x 1 , , x n ) = x i f(x_{1},\dots,x_{n})=x_{i}
  16. ( log n ) (\log n)
  17. 2 n 2^{n}
  18. f i f_{i}
  19. i i
  20. f j f_{j}
  21. j j
  22. j i j\neq i

Lorentz_space.html

  1. p p
  2. q q
  3. ( X , μ ) (X,μ)
  4. f L p , q ( X , μ ) = p 1 q t μ { | f | t } 1 p L q ( 𝐑 + , d t t ) \|f\|_{L^{p,q}(X,\mu)}=p^{\frac{1}{q}}\left\|t\mu\{|f|\geq t\}^{\frac{1}{p}}% \right\|_{L^{q}\left(\mathbf{R}^{+},\frac{dt}{t}\right)}
  5. q = q=∞
  6. f L p , ( X , μ ) p = sup t > 0 ( t p μ { x : | f ( x ) | > t } ) . \|f\|_{L^{p,\infty}(X,\mu)}^{p}=\sup_{t>0}\left(t^{p}\mu\left\{x:|f(x)|>t% \right\}\right).
  7. ( X , μ ) (X,μ)
  8. f * : [ 0 , ) [ 0 , ] f^{*}:[0,\infty)\to[0,\infty]
  9. f * ( t ) = inf { α 𝐑 + : d f ( α ) t } f^{*}(t)=\inf\{\alpha\in\mathbf{R}^{+}:d_{f}(\alpha)\leq t\}
  10. d f ( α ) = μ ( { x X : | f ( x ) | > α } ) . d_{f}(\alpha)=\mu(\{x\in X:|f(x)|>\alpha\}).
  11. i n f inf∅
  12. μ ( { x X : | f ( x ) | > α } ) = λ ( { t > 0 : f * ( t ) > α } ) , α > 0 , \mu\bigl(\{x\in X:|f(x)|>\alpha\}\bigr)=\lambda\bigl(\{t>0:f^{*}(t)>\alpha\}% \bigr),\quad\alpha>0,
  13. t 𝐑 1 2 f * ( | t | ) . t\in\mathbf{R}\ \longrightarrow\ \tfrac{1}{2}f^{*}(|t|).
  14. p , q ( 0 , ) p,q∈(0,∞)
  15. f L p , q = { ( 0 ( t 1 p f * ( t ) ) q d t t ) 1 q q ( 0 , ) , sup t > 0 t 1 p f * ( t ) q = . \|f\|_{L^{p,q}}=\begin{cases}\left(\int_{0}^{\infty}\left(t^{\frac{1}{p}}f^{*}% (t)\right)^{q}\,\frac{dt}{t}\right)^{\frac{1}{q}}&q\in(0,\infty),\\ \sup\limits_{t>0}\,t^{\frac{1}{p}}f^{*}(t)&q=\infty.\end{cases}
  16. p p
  17. p ( 1 , ) , q 1 , p∈(1,∞),q∈1,∞∞
  18. p = 1 p=1
  19. f ( x ) = 1 x χ ( 0 , 1 ) ( x ) and g ( x ) = 1 1 - x χ ( 0 , 1 ) ( x ) , f(x)=\tfrac{1}{x}\chi_{(0,1)}(x)\quad\,\text{and}\quad g(x)=\tfrac{1}{1-x}\chi% _{(0,1)}(x),
  20. L < s u p > 1 , L<sup>1,∞

Lothar_Göttsche.html

  1. S S
  2. 0
  3. S S
  4. n = 0 [ S [ n ] ] t n = m = 1 Z ( S , 𝕃 m - 1 t m ) \sum_{n=0}^{\infty}[S^{[n]}]t^{n}=\prod_{m=1}^{\infty}Z(S,{\mathbb{L}}^{m-1}t^% {m})
  5. S [ n ] S^{[n]}
  6. n n
  7. S S

Low_Exercise_Price_Option.html

  1. L 0 , 1 = S 0 e r ( n / 365 ) - D e r ( n - y ) / 365 - X L_{0,1}=S_{0}e^{r(n/365)}-De^{r(n-y)/365}-X
  2. L 0 , 1 L_{0,1}
  3. S 0 S_{0}
  4. L 0 , 1 = [ S 0 N ( d 1 ) - X e - r n / 365 N ( d 2 ) ] e r n / 365 L_{0,1}=[S_{0}N(d_{1})-Xe^{-rn/365}N(d_{2})]e^{rn/365}
  5. d 1 = ln ( S 0 / X ) + ( r + σ 2 / 2 ) ( n / 365 ) σ n / 365 d_{1}=\dfrac{\ln(S_{0}/X)+(r+{\sigma^{2}}/2)(n/365)}{\sigma\sqrt{n/365}}
  6. d 2 = d 1 - σ n / 365 d_{2}=d_{1}-\sigma\sqrt{n/365}
  7. S 0 S_{0}
  8. N ( d 1 ) N(d_{1})
  9. L 0 , 1 = S 0 e r n / 365 - X L_{0,1}=S_{0}e^{rn/365}-X

Lukacs's_proportion-sum_independence_theorem.html

  1. W = Y 1 + Y 2 and P = Y 1 Y 1 + Y 2 W=Y_{1}+Y_{2}\,\text{ and }P=\frac{Y_{1}}{Y_{1}+Y_{2}}
  2. P i = Y i i = 1 k Y i P_{i}=\frac{Y_{i}}{\sum_{i=1}^{k}Y_{i}}
  3. W = i = 1 k Y i W=\sum_{i=1}^{k}Y_{i}

Lupanov_representation.html

  1. | A p | = s s |A_{p}|=s^{\prime}\leq s
  2. B i , w B_{i,w}
  3. A i A_{i}
  4. w w

Luttinger's_theorem.html

  1. G ( ω = 0 , p ) 0 G(\omega=0,\,p)\to 0
  2. , \infty,
  3. G G
  4. n = 2 G ( ω = 0 , p ) > 0 d D k ( 2 π ) D n=2\int_{G(\omega=0,p)>0}\frac{d^{D}k}{(2\pi)^{D}}
  5. G G
  6. d D k d^{D}k
  7. k k
  8. D D

Luttinger_parameter.html

  1. p 3 / 2 p_{3/2}
  2. p 1 / 2 p_{1/2}
  3. | 3 2 , ± 3 2 , | 3 2 , ± 1 2 , | 1 2 , ± 1 2 |{3\over 2},\pm{3\over 2}\rangle,|{3\over 2},\pm{1\over 2}\rangle,|{1\over 2},% \pm{1\over 2}\rangle
  4. H = 2 2 m 0 [ ( γ 1 + 5 2 γ 2 ) 𝐤 2 - 2 γ 2 ( 𝐤 𝐉 ) 2 ] H={{\hbar^{2}}\over{2m_{0}}}[(\gamma_{1}+{{5}\over{2}}\gamma_{2})\mathbf{k}^{2% }-2\gamma_{2}(\mathbf{k}\cdot\mathbf{J})^{2}]
  5. γ i \gamma_{i}
  6. α = γ 1 + 5 2 γ 2 \alpha=\gamma_{1}+{5\over 2}\gamma_{2}
  7. β = γ 2 \beta=\gamma_{2}
  8. 𝐤 \mathbf{k}
  9. 𝐤 = k e ^ z \mathbf{k}=k\hat{e}_{z}
  10. E = 2 k 2 2 m 0 ( γ 1 + 5 2 γ 2 - 2 γ 2 m j 2 ) E={{\hbar^{2}k^{2}}\over{2m_{0}}}(\gamma_{1}+{{5}\over{2}}\gamma_{2}-2\gamma_{% 2}m_{j}^{2})
  11. E h h = 2 k 2 2 m 0 ( γ 1 - 2 γ 2 ) E_{hh}={{\hbar^{2}k^{2}}\over{2m_{0}}}(\gamma_{1}-2\gamma_{2})
  12. m j = ± 3 2 m_{j}=\pm{3\over 2}
  13. E l h = 2 k 2 2 m 0 ( γ 1 + 2 γ 2 ) E_{lh}={{\hbar^{2}k^{2}}\over{2m_{0}}}(\gamma_{1}+2\gamma_{2})
  14. m j = ± 1 2 m_{j}=\pm{1\over 2}
  15. E h h E_{hh}
  16. E l h E_{lh}
  17. ϵ h , l = - 1 2 γ 1 k 2 ± [ γ 2 2 k 4 + 3 ( γ 3 2 - γ 2 2 ) × ( k x 2 k z 2 + k x 2 k y 2 + k y 2 k z 2 ) ] 1 / 2 \epsilon_{h,l}=-{{1}\over{2}}\gamma_{1}k^{2}\pm[{\gamma_{2}}^{2}k^{4}+3({% \gamma_{3}}^{2}-{\gamma_{2}}^{2})\times({k_{x}}^{2}{k_{z}}^{2}+{k_{x}}^{2}{k_{% y}}^{2}+{k_{y}}^{2}{k_{z}}^{2})]^{1/2}

Luttinger–Kohn_model.html

  1. ϕ \phi
  2. ϕ i ( 0 ) \phi^{(0)}_{i}
  3. ϕ = n A , B a n ϕ i ( 0 ) \phi=\sum^{A,B}_{n}a_{n}\phi^{(0)}_{i}
  4. ( E - H m m ) a m = n m A H m n a n + α m B H m α a α (E-H_{mm})a_{m}=\sum^{A}_{n\neq m}H_{mn}a_{n}+\sum^{B}_{\alpha\neq m}H_{m% \alpha}a_{\alpha}
  5. H m n = ϕ m ( 0 ) H ϕ n ( 0 ) d 3 𝐫 = E n ( 0 ) δ m n + H m n H_{mn}=\int\phi^{(0)\dagger}_{m}H\phi^{(0)}_{n}d^{3}\mathbf{r}=E^{(0)}_{n}% \delta_{mn}+H^{{}^{\prime}}_{mn}
  6. a m = n m A H m n E - H m m a n + α m B H m α E - H m m a α a_{m}=\sum^{A}_{n\neq m}\frac{H_{mn}}{E-H_{mm}}a_{n}+\sum^{B}_{\alpha\neq m}% \frac{H_{m\alpha}}{E-H_{mm}}a_{\alpha}
  7. a m a_{m}
  8. a m = n m A U m n A - H m n E - H m m a n a_{m}=\sum^{A}_{n\neq m}\frac{U^{A}_{mn}-H_{mn}}{E-H_{mm}}a_{n}
  9. U m n A = H m n + α m B H m α H α n E - H α α + α , β m , n ; α β H m α H α β H β n ( E - H α α ) ( E - H β β ) + U^{A}_{mn}=H_{mn}+\sum^{B}_{\alpha\neq m}\frac{H_{m\alpha}H_{\alpha n}}{E-H_{% \alpha\alpha}}+\sum_{\alpha,\beta\neq m,n;\alpha\neq\beta}\frac{H_{m\alpha}H_{% \alpha\beta}H_{\beta n}}{(E-H_{\alpha\alpha})(E-H_{\beta\beta})}+\ldots
  10. a n a_{n}
  11. n A n\in A
  12. a n = n A ( U m n A - E δ m n ) a n = 0 , m A a_{n}=\sum^{A}_{n}(U^{A}_{mn}-E\delta_{mn})a_{n}=0,m\in A
  13. a γ = n A U γ n A - H γ n δ γ n E - H γ γ a n = 0 , γ B a_{\gamma}=\sum^{A}_{n}\frac{U^{A}_{\gamma n}-H_{\gamma n}\delta_{\gamma n}}{E% -H_{\gamma\gamma}}a_{n}=0,\gamma\in B
  14. a n a_{n}
  15. a γ a_{\gamma}
  16. H = H 0 + 4 m 0 2 c 2 σ ¯ V × 𝐩 H=H_{0}+\frac{\hbar}{4m_{0}^{2}c^{2}}\bar{\sigma}\cdot\nabla V\times\mathbf{p}
  17. σ ¯ \bar{\sigma}
  18. H u n 𝐤 ( 𝐫 ) = ( H 0 + m 0 𝐤 𝚷 + 2 k 2 4 m 0 2 c 2 V × 𝐩 σ ¯ ) u n 𝐤 ( 𝐫 ) = E n ( 𝐤 ) u n 𝐤 ( 𝐫 ) Hu_{n\mathbf{k}}(\mathbf{r})=\left(H_{0}+\frac{\hbar}{m_{0}}\mathbf{k}\cdot% \mathbf{\Pi}+\frac{\hbar^{2}k^{2}}{4m_{0}^{2}c^{2}}\nabla V\times\mathbf{p}% \cdot\bar{\sigma}\right)u_{n\mathbf{k}}(\mathbf{r})=E_{n}(\mathbf{k})u_{n% \mathbf{k}}(\mathbf{r})
  19. 𝚷 = 𝐩 + 4 m 0 2 c 2 σ ¯ × V \mathbf{\Pi}=\mathbf{p}+\frac{\hbar}{4m_{0}^{2}c^{2}}\bar{\sigma}\times\nabla V
  20. H = m 0 𝐤 𝚷 . H^{\prime}=\frac{\hbar}{m_{0}}\mathbf{k}\cdot\mathbf{\Pi}.
  21. | S |S\rangle
  22. | X |X\rangle
  23. | Y |Y\rangle
  24. | Z |Z\rangle
  25. u n 𝐤 ( 𝐫 ) = j A a j ( 𝐤 ) u j 0 ( 𝐫 ) + γ B a γ ( 𝐤 ) u γ 0 ( 𝐫 ) u_{n\mathbf{k}}(\mathbf{r})=\sum^{A}_{j^{\prime}}a_{j^{\prime}}(\mathbf{k})u_{% j^{\prime}0}(\mathbf{r})+\sum^{B}_{\gamma}a_{\gamma}(\mathbf{k})u_{\gamma 0}(% \mathbf{r})
  26. γ \gamma
  27. u 10 ( 𝐫 ) = u e l ( 𝐫 ) = | S 1 2 , 1 2 = | S u_{10}(\mathbf{r})=u_{el}(\mathbf{r})=\left|S\frac{1}{2},\frac{1}{2}\right% \rangle=\left|S\uparrow\right\rangle
  28. u 20 ( 𝐫 ) = u S O ( 𝐫 ) = | 1 2 , 1 2 = 1 3 | ( X + i Y ) + 1 3 | Z u_{20}(\mathbf{r})=u_{SO}(\mathbf{r})=\left|\frac{1}{2},\frac{1}{2}\right% \rangle=\frac{1}{\sqrt{3}}|(X+iY)\downarrow\rangle+\frac{1}{\sqrt{3}}|Z\uparrow\rangle
  29. u 30 ( 𝐫 ) = u l h ( 𝐫 ) = | 1 2 , 1 2 = - 1 6 | ( X + i Y ) + 3 2 | Z u_{30}(\mathbf{r})=u_{lh}(\mathbf{r})=\left|\frac{1}{2},\frac{1}{2}\right% \rangle=-\frac{1}{\sqrt{6}}|(X+iY)\downarrow\rangle+\sqrt{\frac{3}{2}}|Z\uparrow\rangle
  30. u 40 ( 𝐫 ) = u h h ( 𝐫 ) = | S 3 2 , 3 2 = - 1 2 | ( X + i Y ) u_{40}(\mathbf{r})=u_{hh}(\mathbf{r})=\left|S\frac{3}{2},\frac{3}{2}\right% \rangle=-\frac{1}{\sqrt{2}}|(X+iY)\uparrow\rangle
  31. u 50 ( 𝐫 ) = u ¯ e l ( 𝐫 ) = | S 1 2 , - 1 2 = - | S u_{50}(\mathbf{r})=\bar{u}_{el}(\mathbf{r})=\left|S\frac{1}{2},-\frac{1}{2}% \right\rangle=-|S\downarrow\rangle
  32. u 60 ( 𝐫 ) = u ¯ S O ( 𝐫 ) = | 1 2 , - 1 2 = 1 3 | ( X - i Y ) - 1 3 | Z u_{60}(\mathbf{r})=\bar{u}_{SO}(\mathbf{r})=\left|\frac{1}{2},-\frac{1}{2}% \right\rangle=\frac{1}{\sqrt{3}}|(X-iY)\uparrow\rangle-\frac{1}{\sqrt{3}}|Z\downarrow\rangle
  33. u 70 ( 𝐫 ) = u ¯ l h ( 𝐫 ) = | 3 2 , - 1 2 = 1 6 | ( X - i Y ) + 3 2 | Z u_{70}(\mathbf{r})=\bar{u}_{lh}(\mathbf{r})=\left|\frac{3}{2},-\frac{1}{2}% \right\rangle=\frac{1}{\sqrt{6}}|(X-iY)\uparrow\rangle+\sqrt{\frac{3}{2}}|Z\downarrow\rangle
  34. u 80 ( 𝐫 ) = u ¯ h h ( 𝐫 ) = | 3 2 , - 3 2 = - 1 2 | ( X - i Y ) u_{80}(\mathbf{r})=\bar{u}_{hh}(\mathbf{r})=\left|\frac{3}{2},-\frac{3}{2}% \right\rangle=-\frac{1}{\sqrt{2}}|(X-iY)\downarrow\rangle
  35. j A ( U j j A - E δ j j ) a j ( 𝐤 ) = 0 , \sum^{A}_{j^{\prime}}(U^{A}_{jj^{\prime}}-E\delta_{jj^{\prime}})a_{j^{\prime}}% (\mathbf{k})=0,
  36. U j j A = H j j + γ j , j B H j γ H γ j E 0 - E γ = H j j + γ j , j B H j γ H γ j E 0 - E γ U^{A}_{jj^{\prime}}=H_{jj^{\prime}}+\sum^{B}_{\gamma\neq j,j^{\prime}}\frac{H_% {j\gamma}H_{\gamma j^{\prime}}}{E_{0}-E_{\gamma}}=H_{jj^{\prime}}+\sum^{B}_{% \gamma\neq j,j^{\prime}}\frac{H^{{}^{\prime}}_{j\gamma}H^{{}^{\prime}}_{\gamma j% ^{\prime}}}{E_{0}-E_{\gamma}}
  37. H j γ = u j 0 | m 0 𝐤 ( 𝐩 + 4 m 0 c 2 σ ¯ × V ) | u γ 0 α k α m 0 p j γ α . H^{{}^{\prime}}_{j\gamma}=\left\langle u_{j0}\right|\frac{\hbar}{m_{0}}\mathbf% {k}\cdot\left(\mathbf{p}+\frac{\hbar}{4m_{0}c^{2}}\bar{\sigma}\times\nabla V% \right)\left|u_{\gamma 0}\right\rangle\approx\sum_{\alpha}\frac{\hbar k_{% \alpha}}{m_{0}}p^{\alpha}_{j\gamma}.
  38. Π \Pi
  39. U j j A U^{A}_{jj^{\prime}}
  40. D j j U j j A = E j ( 0 ) δ j j + α β D j j α β k α k β , D_{jj^{\prime}}\equiv U^{A}_{jj^{\prime}}=E_{j}(0)\delta_{jj^{\prime}}+\sum_{% \alpha\beta}D^{\alpha\beta}_{jj^{\prime}}k_{\alpha}k_{\beta},
  41. D j j α β = 2 2 m 0 [ δ j j δ α β + γ B p j γ α p γ j β + p j γ β p γ j α m 0 ( E 0 - E γ ) ] . D^{\alpha\beta}_{jj^{\prime}}=\frac{\hbar^{2}}{2m_{0}}\left[\delta_{jj^{\prime% }}\delta_{\alpha\beta}+\sum^{B}_{\gamma}\frac{p^{\alpha}_{j\gamma}p^{\beta}_{% \gamma j^{\prime}}+p^{\beta}_{j\gamma}p^{\alpha}_{\gamma j^{\prime}}}{m_{0}(E_% {0}-E_{\gamma})}\right].
  42. A 0 = 2 2 m 0 + 2 m 0 2 γ B p x γ x p γ x x E 0 - E γ , A_{0}=\frac{\hbar^{2}}{2m_{0}}+\frac{\hbar^{2}}{m_{0}^{2}}\sum^{B}_{\gamma}% \frac{p^{x}_{x\gamma}p^{x}_{\gamma x}}{E_{0}-E_{\gamma}},
  43. B 0 = 2 2 m 0 + 2 m 0 2 γ B p x γ y p γ x y E 0 - E γ , B_{0}=\frac{\hbar^{2}}{2m_{0}}+\frac{\hbar^{2}}{m_{0}^{2}}\sum^{B}_{\gamma}% \frac{p^{y}_{x\gamma}p^{y}_{\gamma x}}{E_{0}-E_{\gamma}},
  44. C 0 = 2 m 0 2 γ B p x γ x p γ y y + p x γ y p γ y x E 0 - E γ , C_{0}=\frac{\hbar^{2}}{m_{0}^{2}}\sum^{B}_{\gamma}\frac{p^{x}_{x\gamma}p^{y}_{% \gamma y}+p^{y}_{x\gamma}p^{x}_{\gamma y}}{E_{0}-E_{\gamma}},
  45. γ 1 = - 1 3 2 m 0 2 ( A 0 + 2 B 0 ) , \gamma_{1}=-\frac{1}{3}\frac{2m_{0}}{\hbar^{2}}(A_{0}+2B_{0}),
  46. γ 2 = - 1 6 2 m 0 2 ( A 0 - B 0 ) , \gamma_{2}=-\frac{1}{6}\frac{2m_{0}}{\hbar^{2}}(A_{0}-B_{0}),
  47. γ 3 = - 1 6 2 m 0 2 C 0 , \gamma_{3}=-\frac{1}{6}\frac{2m_{0}}{\hbar^{2}}C_{0},
  48. γ 1 \gamma_{1}
  49. γ 2 \gamma_{2}
  50. | X |X\rangle
  51. | Y |Y\rangle
  52. | Z |Z\rangle
  53. γ 3 \gamma_{3}
  54. Γ \Gamma
  55. γ 2 γ 3 \gamma_{2}\neq\gamma_{3}
  56. 𝐃 𝐣𝐣 \mathbf{D_{jj^{\prime}}}
  57. 𝐇 = ( E e l P z 2 P z - 3 P + 0 2 P - P - 0 P z P + Δ 2 Q - S / 2 - 2 P + 0 - 3 / 2 S - 2 R E e l P z 2 P z - 3 P + 0 2 P - P - 0 E e l P z 2 P z - 3 P + 0 2 P - P - 0 E e l P z 2 P z - 3 P + 0 2 P - P - 0 E e l P z 2 P z - 3 P + 0 2 P - P - 0 E e l P z 2 P z - 3 P + 0 2 P - P - 0 E e l P z 2 P z - 3 P + 0 2 P - P - 0 ) \mathbf{H}=\left(\begin{array}[]{cccccccc}E_{el}&P_{z}&\sqrt{2}P_{z}&-\sqrt{3}% P_{+}&0&\sqrt{2}P_{-}&P_{-}&0\\ P_{z}^{\dagger}&P+\Delta&\sqrt{2}Q^{\dagger}&-S^{\dagger}/\sqrt{2}&-\sqrt{2}P_% {+}^{\dagger}&0&-\sqrt{3/2}S&-\sqrt{2}R\\ E_{el}&P_{z}&\sqrt{2}P_{z}&-\sqrt{3}P_{+}&0&\sqrt{2}P_{-}&P_{-}&0\\ E_{el}&P_{z}&\sqrt{2}P_{z}&-\sqrt{3}P_{+}&0&\sqrt{2}P_{-}&P_{-}&0\\ E_{el}&P_{z}&\sqrt{2}P_{z}&-\sqrt{3}P_{+}&0&\sqrt{2}P_{-}&P_{-}&0\\ E_{el}&P_{z}&\sqrt{2}P_{z}&-\sqrt{3}P_{+}&0&\sqrt{2}P_{-}&P_{-}&0\\ E_{el}&P_{z}&\sqrt{2}P_{z}&-\sqrt{3}P_{+}&0&\sqrt{2}P_{-}&P_{-}&0\\ E_{el}&P_{z}&\sqrt{2}P_{z}&-\sqrt{3}P_{+}&0&\sqrt{2}P_{-}&P_{-}&0\\ \end{array}\right)

Lyapunov–Malkin_theorem.html

  1. x ˙ = A x + X ( x , y ) , y ˙ = Y ( x , y ) \dot{x}=Ax+X(x,y),\quad\dot{y}=Y(x,y)
  2. x m x\in\mathbb{R}^{m}
  3. y n y\in\mathbb{R}^{n}
  4. A A
  5. A A
  6. lim t x ( t ) = 0 , lim t y ( t ) = c . \lim_{t\to\infty}x(t)=0,\quad\lim_{t\to\infty}y(t)=c.

Lydersen_method.html

  1. T c = T b 0.567 + G i - ( G i ) 2 T_{c}=\frac{T_{b}}{0.567+\sum G_{i}-\left(\sum G_{i}\right)^{2}}
  2. P c = M ( 0.34 + G i ) 2 P_{c}=\frac{M}{\left(0.34+\sum G_{i}\right)^{2}}
  3. V c = 40 + G i V_{c}\,=\,40+\sum G_{i}

M-spline.html

  1. M i ( x | 1 , t ) = 1 t i + 1 - t i M_{i}(x|1,t)=\frac{1}{t_{i+1}-t_{i}}
  2. M i ( x | k , t ) = k [ ( x - t i ) M i ( x | k - 1 , t ) + ( t i + k - x ) M i + 1 ( x | k - 1 , t ) ] ( k - 1 ) ( t i + k - t i ) . M_{i}(x|k,t)=\frac{k\left[(x-t_{i})M_{i}(x|k-1,t)+(t_{i+k}-x)M_{i+1}(x|k-1,t)% \right]}{(k-1)(t_{i+k}-t_{i})}.

M::M::c_queue.html

  1. Q = ( - λ λ μ - ( μ + λ ) λ 2 μ - ( 2 μ + λ ) λ 3 μ - ( 3 μ + λ ) λ c μ - ( c μ + λ ) λ c μ - ( c μ + λ ) λ c μ - ( c μ + λ ) λ ) Q=\begin{pmatrix}-\lambda&\lambda\\ \mu&-(\mu+\lambda)&\lambda\\ &2\mu&-(2\mu+\lambda)&\lambda\\ &&3\mu&-(3\mu+\lambda)&\lambda\\ &&&&\ddots\\ &&&&c\mu&-(c\mu+\lambda)&\lambda\\ &&&&&c\mu&-(c\mu+\lambda)&\lambda\\ &&&&&&c\mu&-(c\mu+\lambda)&\lambda\\ &&&&&&&\ddots\\ \end{pmatrix}
  2. ( ρ = λ c μ ) < 1 \left(\rho=\frac{\lambda}{c\mu}\right)<1
  3. π 0 = [ k = 0 c - 1 ( c ρ ) k k ! + ( c ρ ) c c ! 1 1 - ρ ] - 1 \pi_{0}=\left[\sum_{k=0}^{c-1}\frac{(c\rho)^{k}}{k!}+\frac{(c\rho)^{c}}{c!}% \frac{1}{1-\rho}\right]^{-1}
  4. π k = { π 0 ( c ρ ) k k ! , if 0 < k < c π 0 ρ k c c c ! , if c k \pi_{k}=\begin{cases}\pi_{0}\dfrac{(c\rho)^{k}}{k!},&\mbox{if }~{}0<k<c\\ \pi_{0}\dfrac{\rho^{k}c^{c}}{c!},&\mbox{if }~{}c\leq k\end{cases}
  5. C ( c , λ / μ ) = ( ( c ρ ) c c ! ) ( 1 1 - ρ ) k = 0 c - 1 ( c ρ ) k k ! + ( ( c ρ ) c c ! ) ( 1 1 - ρ ) \,\text{ C}(c,\lambda/\mu)=\frac{\left(\frac{(c\rho)^{c}}{c!}\right)\left(% \frac{1}{1-\rho}\right)}{\sum_{k=0}^{c-1}\frac{(c\rho)^{k}}{k!}+\left(\frac{(c% \rho)^{c}}{c!}\right)\left(\frac{1}{1-\rho}\right)}
  6. ρ 1 - ρ C ( c , λ / μ ) + c ρ . \frac{\rho}{1-\rho}\,\text{ C}(c,\lambda/\mu)+c\rho.
  7. η c ( s ) = c μ k μ + s + λ - λ η c ( s ) . \eta_{c}(s)=\frac{c\mu}{k\mu+s+\lambda-\lambda\eta_{c}(s)}.
  8. η k ( s ) = k μ k μ + s + λ - λ η k + 1 ( s ) . \eta_{k}(s)=\frac{k\mu}{k\mu+s+\lambda-\lambda\eta_{k+1}(s)}.
  9. C ( c , λ / μ ) c μ - λ + 1 μ . \frac{\,\text{ C}(c,\lambda/\mu)}{c\mu-\lambda}+\frac{1}{\mu}.
  10. c μ n \frac{c\mu}{n}
  11. Q = ( - λ λ μ - ( μ + λ ) λ 2 μ - ( 2 μ + λ ) λ 3 μ - ( 3 μ + λ ) λ c μ - ( c μ + λ ) λ c μ - ( c μ + λ ) λ c μ - ( c μ ) ) Q=\begin{pmatrix}-\lambda&\lambda\\ \mu&-(\mu+\lambda)&\lambda\\ &2\mu&-(2\mu+\lambda)&\lambda\\ &&3\mu&-(3\mu+\lambda)&\lambda\\ &&&&\ddots\\ &&&&c\mu&-(c\mu+\lambda)&\lambda\\ &&&&&c\mu&-(c\mu+\lambda)&\lambda\\ &&&&&&&\ddots\\ &&&&&&&c\mu&-(c\mu)\\ \end{pmatrix}
  12. π 0 = [ k = 0 c λ k μ k k ! + λ c μ c c ! k = c + 1 K λ k - c μ k - c c k - c ] - 1 \pi_{0}=\left[\sum_{k=0}^{c}\frac{\lambda^{k}}{\mu^{k}k!}+\frac{\lambda^{c}}{% \mu^{c}c!}\sum_{k=c+1}^{K}\frac{\lambda^{k-c}}{\mu^{k-c}c^{k-c}}\right]^{-1}
  13. π k = { ( λ / μ ) k k ! π 0 for k = 1 , 2 , , c ( λ / μ ) k c k - c c ! π 0 for k = c + 1 , , K . \pi_{k}=\begin{cases}\frac{(\lambda/\mu)^{k}}{k!}\pi_{0}&\,\text{for }k=1,2,% \ldots,c\\ \frac{(\lambda/\mu)^{k}}{c^{k-c}c!}\pi_{0}&\,\text{for }k=c+1,\ldots,K.\end{cases}
  14. λ μ + π 0 ρ ( c ρ ) c ( 1 - ρ ) 2 c ! \frac{\lambda}{\mu}+\pi_{0}\frac{\rho(c\rho)^{c}}{(1-\rho)^{2}c!}
  15. 1 μ + π 0 ρ ( c ρ ) c λ ( 1 - ρ ) 2 c ! . \frac{1}{\mu}+\pi_{0}\frac{\rho(c\rho)^{c}}{\lambda(1-\rho)^{2}c!}.
  16. X ^ n ( t ) = X ( n t ) - 𝔼 ( X ( n t ) ) n \hat{X}_{n}(t)=\frac{X(nt)-\mathbb{E}(X(nt))}{\sqrt{n}}
  17. β = ( 1 - ρ ) s \beta=(1-\rho)\sqrt{s}

MacMahon_Master_theorem.html

  1. A = ( a i j ) m × m A=(a_{ij})_{m\times m}
  2. x 1 , , x m x_{1},\ldots,x_{m}
  3. G ( k 1 , , k m ) = [ x 1 k 1 x m k m ] i = 1 m ( a i 1 x 1 + + a i m x m ) k i . G(k_{1},\dots,k_{m})\,=\,\bigl[x_{1}^{k_{1}}\cdots x_{m}^{k_{m}}\bigr]\,\prod_% {i=1}^{m}\bigl(a_{i1}x_{1}+\dots+a_{im}x_{m}\bigl)^{k_{i}}.
  4. [ f ] g [f]g
  5. f f
  6. g g
  7. t 1 , , t m t_{1},\ldots,t_{m}
  8. T = ( δ i j t i ) m × m T=(\delta_{ij}t_{i})_{m\times m}
  9. ( k 1 , , k m ) G ( k 1 , , k m ) t 1 k 1 t m k m = 1 det ( I m - T A ) , \sum_{(k_{1},\dots,k_{m})}G(k_{1},\dots,k_{m})\,t_{1}^{k_{1}}\cdots t_{m}^{k_{% m}}\,=\,\frac{1}{\det(I_{m}-TA)},
  10. ( k 1 , , k m ) (k_{1},\dots,k_{m})
  11. I m I_{m}
  12. m m
  13. A = ( 0 1 - 1 - 1 0 1 1 - 1 0 ) . A=\begin{pmatrix}0&1&-1\\ -1&0&1\\ 1&-1&0\end{pmatrix}.
  14. G ( 2 n , 2 n , 2 n ) = [ x 1 2 n x 2 2 n x 3 2 n ] ( x 2 - x 3 ) 2 n ( x 3 - x 1 ) 2 n ( x 1 - x 2 ) 2 n = k = 0 2 n ( - 1 ) k ( 2 n k ) 3 , G(2n,2n,2n)=\bigl[x_{1}^{2n}x_{2}^{2n}x_{3}^{2n}\bigl](x_{2}-x_{3})^{2n}(x_{3}% -x_{1})^{2n}(x_{1}-x_{2})^{2n}\,=\,\sum_{k=0}^{2n}(-1)^{k}{\left({{2n}\atop{k}% }\right)}^{3},
  15. [ x 2 k x 3 2 n - k ] ( x 2 - x 3 ) 2 n , [ x 3 k x 1 2 n - k ] ( x 3 - x 1 ) 2 n , [ x 1 k x 2 2 n - k ] ( x 1 - x 2 ) 2 n , [x_{2}^{k}x_{3}^{2n-k}](x_{2}-x_{3})^{2n},\ \ [x_{3}^{k}x_{1}^{2n-k}](x_{3}-x_% {1})^{2n},\ \ [x_{1}^{k}x_{2}^{2n-k}](x_{1}-x_{2})^{2n},
  16. det ( I - T A ) = det ( 1 - t 1 t 1 t 2 1 - t 2 - t 3 t 3 1 ) = 1 + ( t 1 t 2 + t 1 t 3 + t 2 t 3 ) . \det(I-TA)\,=\,\det\begin{pmatrix}1&-t_{1}&t_{1}\\ t_{2}&1&-t_{2}\\ -t_{3}&t_{3}&1\end{pmatrix}\,=\,1+\bigl(t_{1}t_{2}+t_{1}t_{3}+t_{2}t_{3}\bigr).
  17. G ( 2 n , 2 n , 2 n ) = [ t 1 2 n t 2 2 n t 3 2 n ] ( - 1 ) 3 n ( t 1 t 2 + t 1 t 3 + t 2 t 3 ) 3 n = ( - 1 ) n ( 3 n n , n , n ) , G(2n,2n,2n)\,=\,\bigl[t_{1}^{2n}t_{2}^{2n}t_{3}^{2n}\bigl](-1)^{3n}\bigl(t_{1}% t_{2}+t_{1}t_{3}+t_{2}t_{3}\bigr)^{3n}\,=\,(-1)^{n}{\left({{3n}\atop{n,n,n}}% \right)},
  18. k = 0 2 n ( - 1 ) k ( 2 n k ) 3 = ( - 1 ) n ( 3 n n , n , n ) . \sum_{k=0}^{2n}(-1)^{k}{\left({{2n}\atop{k}}\right)}^{3}\,=\,(-1)^{n}{\left({{% 3n}\atop{n,n,n}}\right)}.

MacRobert_E_function.html

  1. E ( 𝐚 𝐩 𝐛 𝐪 | x ) = j = 1 p Γ ( a j ) j = 1 q Γ ( b j ) p F q ( 𝐚 𝐩 𝐛 𝐪 | - x - 1 ) E\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,x\right)=\frac{\prod_{j=1}^{p}\Gamma(a_{j% })}{\prod_{j=1}^{q}\Gamma(b_{j})}\;_{p}F_{q}\!\left(\left.\begin{matrix}% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,-x^{-1}\right)
  2. E ( 𝐚 𝐩 𝐛 𝐪 | x ) = h = 1 p j = 1 p Γ ( a j - a h ) * j = 1 q Γ ( b j - a h ) Γ ( a h ) x q + 1 a h F p - 1 ( a h , 1 + a h - b 1 , , 1 + a h - b q 1 + a h - a 1 , , * , , 1 + a h - a p | ( - 1 ) p - q x ) . E\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,x\right)=\sum_{h=1}^{p}\frac{\prod_{j=1}^% {p}\Gamma(a_{j}-a_{h})^{*}}{\prod_{j=1}^{q}\Gamma(b_{j}-a_{h})}\Gamma(a_{h})\;% x^{a_{h}}\;_{q+1}F_{p-1}\!\left(\left.\begin{matrix}a_{h},1+a_{h}-b_{1},\dots,% 1+a_{h}-b_{q}\\ 1+a_{h}-a_{1},\dots,*,\dots,1+a_{h}-a_{p}\end{matrix}\;\right|\,(-1)^{p-q}\;x% \right).
  3. E ( 𝐚 𝐩 𝐛 𝐪 | x ) = G q + 1 , p p , 1 ( 1 , 𝐛 𝐪 𝐚 𝐩 | x ) E\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,x\right)=G_{q+1,\,p}^{\,p,\,1}\!\left(% \left.\begin{matrix}1,\mathbf{b_{q}}\\ \mathbf{a_{p}}\end{matrix}\;\right|\,x\right)

Madhava's_sine_table.html

  1. \angle
  2. P O S = m arcminutes, s arcseconds, t sixtieths of an arcsecond \angle POS=m\,\text{ arcminutes, }s\,\text{ arcseconds, }t\,\text{ sixtieths % of an arcsecond}
  3. sin ( A ) = R Q = length of arc P S = P O S in radians = π 180 × 60 ( m + s 60 + t 60 × 60 ) . \begin{aligned}\displaystyle\sin(A)&\displaystyle=RQ\\ &\displaystyle=\,\text{length of arc }PS\\ &\displaystyle=\angle POS\,\text{ in radians}\\ &\displaystyle=\frac{\pi}{180\times 60}\left(m+\frac{s}{60}+\frac{t}{60\times 6% 0}\right).\end{aligned}
  4. d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d_{1}\quad d_{2}\quad d_{3}\quad d_{4}\quad d_{5}\quad d_{6}\quad d_{7}\quad d% _{8}
  5. m = d 8 × 1000 + d 7 × 100 + d 6 × 10 + d 5 s = d 4 × 10 + d 3 t = d 2 × 10 + d 1 \begin{aligned}\displaystyle m&\displaystyle=d_{8}\times 1000+d_{7}\times 100+% d_{6}\times 10+d_{5}\\ \displaystyle s&\displaystyle=d_{4}\times 10+d_{3}\\ \displaystyle t&\displaystyle=d_{2}\times 10+d_{1}\end{aligned}
  6. π \pi
  7. 5 1 1 5 0 3 4 2 5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2
  8. m = 2 × 1000 + 4 × 100 + 3 × 10 + 0 arcminutes = 2430 arcminutes s = 5 × 10 + 1 arcseconds = 51 arcseconds t = 1 × 10 + 5 sixtieths of an arcsecond = 15 sixtieths of an arcsecond \begin{aligned}\displaystyle m&\displaystyle=2\times 1000+4\times 100+3\times 1% 0+0\,\text{ arcminutes}\\ &\displaystyle=2430\,\text{ arcminutes}\\ \displaystyle s&\displaystyle=5\times 10+1\,\text{ arcseconds}\\ &\displaystyle=51\,\text{ arcseconds}\\ \displaystyle t&\displaystyle=1\times 10+5\,\text{ sixtieths of an arcsecond}% \\ &\displaystyle=15\,\text{ sixtieths of an arcsecond}\end{aligned}
  9. sin 45 = π 180 × 60 ( 2430 + 51 60 + 15 60 × 60 ) \sin 45^{\circ}=\frac{\pi}{180\times 60}\left(2430+\frac{51}{60}+\frac{15}{60% \times 60}\right)
  10. 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

Madhava_series.html

  1. x < s u p > 3 3 ! x\frac{<sup>3}{3!}
  2. s s 2 , s s 2 s 2 , s s 2 s 2 s 2 , s\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2}% \cdot s^{2},\cdot
  3. s s 2 ( 2 2 + 2 ) r 2 , s s 2 ( 2 2 + 2 ) r 2 s 2 ( 4 2 + 4 ) r 2 , s s 2 ( 2 2 + 2 ) r 2 s 2 ( 4 2 + 4 ) r 2 s 2 ( 6 2 + 6 ) r 2 , s\cdot\frac{s^{2}}{(2^{2}+2)r^{2}},\qquad s\cdot\frac{s^{2}}{(2^{2}+2)r^{2}}% \cdot\frac{s^{2}}{(4^{2}+4)r^{2}},\qquad s\cdot\frac{s^{2}}{(2^{2}+2)r^{2}}% \cdot\frac{s^{2}}{(4^{2}+4)r^{2}}\cdot\frac{s^{2}}{(6^{2}+6)r^{2}},\cdots
  4. jiva = s - [ s s 2 ( 2 2 + 2 ) r 2 - [ s s 2 ( 2 2 + 2 ) r 2 s 2 ( 4 2 + 4 ) r 2 - [ s s 2 ( 2 2 + 2 ) r 2 s 2 ( 4 2 + 4 ) r 2 s 2 ( 6 2 + 6 ) r 2 - ] ] ] \,\text{jiva}=s-\Big[s\cdot\frac{s^{2}}{(2^{2}+2)r^{2}}-\Big[s\cdot\frac{s^{2}% }{(2^{2}+2)r^{2}}\cdot\frac{s^{2}}{(4^{2}+4)r^{2}}-\Big[s\cdot\frac{s^{2}}{(2^% {2}+2)r^{2}}\cdot\frac{s^{2}}{(4^{2}+4)r^{2}}\cdot\frac{s^{2}}{(6^{2}+6)r^{2}}% -\cdots\Big]\Big]\Big]
  5. sin θ = θ - θ 3 3 ! + θ 5 5 ! - θ 7 7 ! + \sin\theta=\theta-\frac{\theta^{3}}{3!}+\frac{\theta^{5}}{5!}-\frac{\theta^{7}% }{7!}+\quad\cdots
  6. jiva \displaystyle\,\text{jiva }
  7. r s 2 , r s 2 s 2 , r s 2 s 2 s 2 , r\cdot s^{2},\qquad r\cdot s^{2}\cdot s^{2},\qquad r\cdot s^{2}\cdot s^{2}% \cdot s^{2},\cdot
  8. r s 2 ( 2 2 - 2 ) r 2 , r s 2 ( 2 2 - 2 ) r 2 s 2 ( 4 2 - 4 ) r 2 , r s 2 ( 2 2 - 2 ) r 2 s 2 ( 4 2 - 4 ) r 2 s 2 ( 6 2 - 6 ) r 2 , r\cdot\frac{s^{2}}{(2^{2}-2)r^{2}},\qquad r\cdot\frac{s^{2}}{(2^{2}-2)r^{2}}% \cdot\frac{s^{2}}{(4^{2}-4)r^{2}},\qquad r\cdot\frac{s^{2}}{(2^{2}-2)r^{2}}% \cdot\frac{s^{2}}{(4^{2}-4)r^{2}}\cdot\frac{s^{2}}{(6^{2}-6)r^{2}},\cdots
  9. sara = r s 2 ( 2 2 - 2 ) r 2 - [ r s 2 ( 2 2 - 2 ) r 2 s 2 ( 4 2 - 4 ) r 2 - [ r s 2 ( 2 2 - 2 ) r 2 s 2 ( 4 2 - 4 ) r 2 s 2 ( 6 2 - 6 ) r 2 - ] ] \,\text{sara}=r\cdot\frac{s^{2}}{(2^{2}-2)r^{2}}-\Big[r\cdot\frac{s^{2}}{(2^{2% }-2)r^{2}}\cdot\frac{s^{2}}{(4^{2}-4)r^{2}}-\Big[r\cdot\frac{s^{2}}{(2^{2}-2)r% ^{2}}\cdot\frac{s^{2}}{(4^{2}-4)r^{2}}\cdot\frac{s^{2}}{(6^{2}-6)r^{2}}-\cdots% \Big]\Big]
  10. 1 - cos θ = θ 2 2 ! - θ 4 4 ! + θ 6 6 ! + 1-\cos\theta=\frac{\theta^{2}}{2!}-\frac{\theta^{4}}{4!}+\frac{\theta^{6}}{6!}% +\quad\cdots
  11. jiva \displaystyle\,\text{jiva }
  12. y r x \frac{y\cdot r}{x}
  13. y 2 x 2 \frac{y^{2}}{x^{2}}
  14. y r x y 2 x 2 , y r x y 2 x 2 y 2 x 2 , \frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}},\quad\frac{y\cdot r}{x}\cdot\frac{y% ^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}},\quad\cdots
  15. 1 1 y r x , 1 3 y r x y 2 x 2 , 1 5 y r x y 2 x 2 y 2 x 2 , \frac{1}{1}\frac{y\cdot r}{x},\quad\frac{1}{3}\frac{y\cdot r}{x}\cdot\frac{y^{% 2}}{x^{2}},\quad\frac{1}{5}\frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}}\cdot% \frac{y^{2}}{x^{2}},\quad\cdots
  16. 1 1 y r x + 1 5 y r x y 2 x 2 y 2 x 2 + \frac{1}{1}\frac{y\cdot r}{x}+\frac{1}{5}\frac{y\cdot r}{x}\cdot\frac{y^{2}}{x% ^{2}}\cdot\frac{y^{2}}{x^{2}}+\quad\cdots
  17. 1 3 y r x y 2 x 2 + 1 7 y r x y 2 x 2 y 2 x 2 y 2 x 2 + \frac{1}{3}\frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}}+\frac{1}{7}\frac{y\cdot r% }{x}\cdot\frac{y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}}+\quad\cdots
  18. s = ( 1 1 y r x + 1 5 y r x y 2 x 2 y 2 x 2 + ) - ( 1 3 y r x y 2 x 2 + 1 7 y r x y 2 x 2 y 2 x 2 y 2 x 2 + ) s=\left(\frac{1}{1}\frac{y\cdot r}{x}+\frac{1}{5}\frac{y\cdot r}{x}\cdot\frac{% y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}}+\quad\cdots\right)-\left(\frac{1}{3}% \frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}}+\frac{1}{7}\frac{y\cdot r}{x}\cdot% \frac{y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}}+\quad% \cdots\right)
  19. θ = tan θ - tan 3 θ 3 + tan 5 θ 5 - tan 7 θ 7 + \theta=\tan\theta-\frac{\tan^{3}\theta}{3}+\frac{\tan^{5}\theta}{5}-\frac{\tan% ^{7}\theta}{7}+\quad\cdots
  20. tan - 1 q = q - q 3 3 + q 5 5 - q 7 7 + \tan^{-1}q=q-\frac{q^{3}}{3}+\frac{q^{5}}{5}-\frac{q^{7}}{7}+\quad\cdots
  21. c = 12 d 2 - 12 d 2 3 3 + 12 d 2 3 2 5 - 12 d 2 3 3 7 + c=\sqrt{12d^{2}}-\frac{\sqrt{12d^{2}}}{3\cdot 3}+\frac{\sqrt{12d^{2}}}{3^{2}% \cdot 5}-\frac{\sqrt{12d^{2}}}{3^{3}\cdot 7}+\quad\cdots
  22. π = 12 ( 1 - 1 3 3 + 1 3 2 5 - 1 3 3 7 + ) \pi=\sqrt{12}\left(1-\frac{1}{3\cdot 3}+\frac{1}{3^{2}\cdot 5}-\frac{1}{3^{3}% \cdot 7}+\quad\cdots\right)
  23. 1 / 3 1/\sqrt{3}

Magic_tee.html

  1. S = 1 2 ( 0 0 1 - 1 0 0 1 1 1 1 0 0 - 1 1 0 0 ) S=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&1&-1\\ 0&0&1&1\\ 1&1&0&0\\ -1&1&0&0\end{pmatrix}

Magnetic_braking.html

  1. L L
  2. 𝐅 \mathbf{F}
  3. 𝐅 = q 𝐯 × 𝐁 \mathbf{F}=q\mathbf{v}\times\mathbf{B}
  4. q q
  5. 𝐯 \mathbf{v}
  6. 𝐁 \mathbf{B}
  7. P B P_{B}
  8. P B = B 2 2 μ 0 P_{B}=\frac{B^{2}}{2\mu_{0}}
  9. P g P_{g}
  10. P g = n m v 2 P_{g}=nmv^{2}
  11. B ( r ) = D r 3 B(r)=\frac{D_{\odot}}{r^{3}}
  12. P B = P g P_{B}=P_{g}\Rightarrow
  13. B ( r c ) 2 2 μ 0 = n m v 2 \frac{B(rc)^{2}}{2\mu_{0}}=nmv^{2}\Rightarrow
  14. D 2 2 μ 0 r c 6 = n m v 2 \frac{D_{\odot}^{2}}{2\mu_{0}r_{c}^{6}}=nmv^{2}
  15. n m = d M / d t 4 π r 2 v nm=\frac{dM/dt}{4\pi r^{2}v}
  16. r c = ( 2 π D 2 μ 0 v M ˙ ) 1 4 r_{c}=\left(\frac{2\pi D_{\odot}^{2}}{\mu_{0}v\dot{M}}\right)^{1\over 4}
  17. M ˙ = d M / d t = 2 10 9 k g / s \dot{M}=dM/dt=2\cdot 10^{9}kg/s
  18. v = 5 10 5 m / s v=5\cdot 10^{5}m/s
  19. D = 8 10 22 T m 3 D_{\odot}=8\cdot 10^{22}Tm^{3}
  20. r c = 3.4 R r_{c}=3.4R_{\odot}
  21. j j c = ( R r c ) 2 10 % \frac{j_{\odot}}{j_{c}}=\left(\frac{R_{\odot}}{r_{c}}\right)^{2}\approx 10\%

Magnetic_damping.html

  1. M a = M g - K v Ma=Mg-Kv\,

Magnetic_dipole_transition.html

  1. H = 1 2 m [ 𝐏 - q 𝐀 ( 𝐑 , t ) ] 2 + V ( R ) - q m 𝐒 𝐁 ( 𝐑 , t ) H=\frac{1}{2m}[\mathbf{P}-q\mathbf{A}(\mathbf{R},t)]^{2}+V(R)-\frac{q}{m}% \mathbf{S}\cdot\mathbf{B}(\mathbf{R},t)
  2. H = H 0 + W ( t ) H=H_{0}+W(t)
  3. H 0 = 1 2 m 𝐏 2 + V ( R ) H_{0}=\frac{1}{2m}\mathbf{P}^{2}+V(R)
  4. W ( t ) = - q m 𝐏 𝐀 ( 𝐑 , t ) - q m 𝐒 𝐁 ( 𝐑 , t ) + q 2 2 m 𝐀 2 ( 𝐑 , t ) W(t)=-\frac{q}{m}\mathbf{P}\cdot\mathbf{A}(\mathbf{R},t)-\frac{q}{m}\mathbf{S}% \cdot\mathbf{B}(\mathbf{R},t)+\frac{q^{2}}{2m}\mathbf{A}^{2}(\mathbf{R},t)
  5. W D M ( t ) = - q 2 m ( 𝐋 + 2 𝐒 ) 𝐁 ( 𝐑 , t ) W_{DM}(t)=-\frac{q}{2m}(\mathbf{L}+2\mathbf{S})\cdot\mathbf{B}(\mathbf{R},t)
  6. Δ J = 0 , ± 1 ( except J = 0 J = 0 ) \Delta J=0,\pm 1(\,\text{except }J=0\rightarrow J=0)
  7. Δ M J = 0 , ± 1 \Delta M_{J}=0,\pm 1
  8. M J M_{J}

Magnetic_energy.html

  1. E p , m = - m B E_{\rm p,m}=-m\cdot B
  2. E p , m = 1 2 L I 2 E_{\rm p,m}={1\over 2}LI^{2}
  3. μ \mu
  4. u = 1 2 B 2 μ u={1\over 2}{B^{2}\over\mu}

Magnetic_hyperthermia.html

  1. τ N = τ 0 e K V k B T \tau_{N}=\tau_{0}e^{\frac{KV}{k_{B}T}}
  2. τ 0 \tau_{0}
  3. M = χ H M=\chi H
  4. χ \chi
  5. τ \tau
  6. 1 τ = 1 τ B + 1 τ N \frac{1}{\tau}=\frac{1}{\tau_{B}}+\frac{1}{\tau_{N}}
  7. χ ′′ = μ 0 M S 2 V 3 k B T 2 π f τ 1 + ( 2 π f τ ) 2 \chi^{\prime\prime}=\frac{\mu_{0}M_{S}^{2}V}{3k_{B}T}\frac{2\pi f\tau}{1+(2\pi f% \tau)^{2}}
  8. A = π μ 0 H 2 χ ′′ A=\pi\mu_{0}H^{2}\chi^{\prime\prime}
  9. H C ( 0 ) = K M S H_{C}(0)=\frac{K}{M_{S}}
  10. A = 2 μ 0 M S H C ( 0 ) A=2\mu_{0}M_{S}H_{C}(0)
  11. A = 4 3 η H 3 A=\frac{4}{3}\eta H^{3}
  12. η \eta

Magnetic_translation.html

  1. ( X , Y ) (X,Y)
  2. ( R x , R y ) (R_{x},R_{y})
  3. [ X , Y ] = - i B 2 [X,Y]=-i\ell_{B}^{2}
  4. B = / e B \ell_{B}=\sqrt{\hbar/eB}
  5. Q = q Q=q
  6. P = - i d d q P=-i\hbar\frac{d}{dq}
  7. f ( q ) f(q)
  8. e i a P e^{iaP}
  9. e i b Q e^{ibQ}
  10. e i ( p x X + p y Y ) , e^{i(p_{x}X+p_{y}Y)},
  11. ( p x , p y ) (p_{x},p_{y})
  12. ( p x , p y ) (p_{x},p_{y})
  13. ( p x , p y ) (p^{\prime}_{x},p^{\prime}_{y})

Magnetization_dynamics.html

  1. m m
  2. H H
  3. τ \tau
  4. s y m b o l τ = μ 0 𝐦 × 𝐇 symbol{\tau}=\mu_{0}\mathbf{m}\times\mathbf{H}
  5. L L
  6. s y m b o l τ = d 𝐋 d t symbol{\tau}=\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}
  7. γ \gamma
  8. 𝐦 = - γ 𝐋 \mathbf{m}=-\gamma\mathbf{L}
  9. γ e = 1.760859770 ( 44 ) × 10 11 s - 1 T - 1 \gamma_{e}=1.760859770(44)\times 10^{11}\mathrm{s^{-1}T^{-1}}
  10. d 𝐦 d t = - γ d 𝐋 d t = - γ s y m b o l τ \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\frac{\mathrm{d}\mathbf{L}}{% \mathrm{d}t}=-\gamma symbol{\tau}
  11. d 𝐦 d t = - γ μ 0 ( 𝐦 × 𝐇 ) \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\mu_{0}\left(\mathbf{m}\times% \mathbf{H}\right)
  12. z z
  13. d m x d t = - γ μ 0 m y H z d m y d t = γ μ 0 m x H z d m z d t = 0 \frac{\mathrm{d}m_{x}}{\mathrm{d}t}=-\gamma\mu_{0}m_{y}H_{z}\qquad\frac{% \mathrm{d}m_{y}}{\mathrm{d}t}=\gamma\mu_{0}m_{x}H_{z}\qquad\frac{\mathrm{d}m_{% z}}{\mathrm{d}t}=0
  14. H e f f H_{eff}
  15. δ t \delta{t}
  16. d 𝐦 ( t ) d t = - γ μ 0 𝐦 ( t ) × 𝐇 𝐞𝐟𝐟 ( t - δ t ) \frac{\mathrm{d}\mathbf{m}\left(t\right)}{\mathrm{d}t}=-\gamma\mu_{0}\mathbf{m% }\left(t\right)\times\mathbf{H_{eff}}\left(t-\delta t\right)
  17. d 𝐇 𝐞𝐟𝐟 d t = d 𝐇 𝐞𝐟𝐟 d 𝐦 d 𝐦 d t \tfrac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}t}=\tfrac{\mathrm{d}\mathbf{H_{% eff}}}{\mathrm{d}\mathbf{m}}\tfrac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}
  18. 𝐇 𝐞𝐟𝐟 ( t - δ t ) = 𝐇 𝐞𝐟𝐟 ( t ) - δ t d 𝐇 𝐞𝐟𝐟 d 𝐦 d 𝐦 d t + \mathbf{H_{eff}}\left(t-\delta t\right)=\mathbf{H_{eff}}\left(t\right)-\delta t% \frac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}\mathbf{m}}\frac{\mathrm{d}\mathbf% {m}}{\mathrm{d}t}+\dots
  19. d 𝐦 d t = - γ μ 0 𝐦 × 𝐇 𝐞𝐟𝐟 + 𝐦 m × ( α ^ d 𝐦 d t ) \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\mu_{0}\mathbf{m}\times\mathbf% {H_{eff}}+\frac{\mathbf{m}}{m}\times\left(\hat{\alpha}\frac{\mathrm{d}\mathbf{% m}}{\mathrm{d}t}\right)
  20. α ^ = γ μ 0 m d 𝐇 𝐞𝐟𝐟 d 𝐦 δ t \hat{\alpha}=\gamma\mu_{0}m\frac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}\mathbf% {m}}\delta{t}
  21. α ^ = [ α 0 0 0 α 0 0 0 α ] \hat{\alpha}=\begin{bmatrix}\alpha&0&0\\ 0&\alpha&0\\ 0&0&\alpha\end{bmatrix}
  22. α ^ d 𝐦 d t = α d 𝐦 d t \hat{\alpha}\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=\alpha\frac{\mathrm{d}% \mathbf{m}}{\mathrm{d}t}
  23. d 𝐦 d t = - γ μ 0 𝐦 × 𝐇 𝐞𝐟𝐟 + α m ( 𝐦 × d 𝐦 d t ) \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\mu_{0}\mathbf{m}\times\mathbf% {H_{eff}}+\frac{\alpha}{m}\left(\mathbf{m}\times\frac{\mathrm{d}\mathbf{m}}{% \mathrm{d}t}\right)
  24. d 𝐦 d t \tfrac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}
  25. d 𝐦 d t = - γ ( s y m b o l τ + s y m b o l τ d ) \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\left(symbol{\tau}+symbol{\tau% _{d}}\right)
  26. s y m b o l τ d = - α γ m ( 𝐦 × d 𝐦 d t ) symbol{\tau_{d}}=-\frac{\alpha}{\gamma m}\left(\mathbf{m}\times\frac{\mathrm{d% }\mathbf{m}}{\mathrm{d}t}\right)
  27. M M
  28. d 𝐌 d t = - γ μ 0 𝐌 × 𝐇 𝐞𝐟𝐟 + α M ( 𝐌 × d 𝐌 d t ) \frac{\mathrm{d}\mathbf{M}}{\mathrm{d}t}=-\gamma\mu_{0}\mathbf{M}\times\mathbf% {H_{eff}}+\frac{\alpha}{M}\left(\mathbf{M}\times\frac{\mathrm{d}\mathbf{M}}{% \mathrm{d}t}\right)

Magnetohydrodynamic_turbulence.html

  1. 𝐮 t + 𝐮 𝐮 = - p + 𝐁 𝐁 + ν 2 𝐮 𝐁 t + 𝐮 𝐁 = 𝐁 𝐮 + η 2 𝐁 𝐮 = 0 𝐁 = 0. \begin{array}[]{lcl}\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot% \nabla\mathbf{u}&=&-\nabla p+\mathbf{B}\cdot\nabla\mathbf{B}+\nu\nabla^{2}% \mathbf{u}\\ \frac{\partial\mathbf{B}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{B}&=&% \mathbf{B}\cdot\nabla\mathbf{u}+\eta\nabla^{2}\mathbf{B}\\ \nabla\cdot\mathbf{u}&=&0\\ \nabla\cdot\mathbf{B}&=&0.\end{array}
  2. ν \nu
  3. η \eta
  4. 𝐁 = 𝐁 𝟎 + 𝐛 \mathbf{B}=\mathbf{B_{0}}+\mathbf{b}
  5. 𝐳 ± = 𝐮 ± 𝐛 \mathbf{z}^{\pm}=\mathbf{u}\pm\mathbf{b}
  6. 𝐳 ± t ( 𝐁 0 ) 𝐳 ± + ( 𝐳 ) 𝐳 ± = - p + ν + 2 𝐳 ± + ν - 2 𝐳 \frac{\partial{\mathbf{z}^{\pm}}}{\partial t}\mp\left(\mathbf{B}_{0}\cdot{% \mathbf{\nabla}}\right){\mathbf{z}^{\pm}}+\left({\mathbf{z}^{\mp}}\cdot{% \mathbf{\nabla}}\right){\mathbf{z}^{\pm}}=-{\mathbf{\nabla}}p+\nu_{+}\nabla^{2% }\mathbf{z}^{\pm}+\nu_{-}\nabla^{2}\mathbf{z}^{\mp}
  7. ν ± = ν ± η \nu_{\pm}=\nu\pm\eta
  8. z z^{\mp}
  9. Reynolds number R e = U L / ν Magnetic Reynolds number R e M = U L / η Magnetic Prandtl number P M = ν / η . \begin{array}[]{lcl}\,\text{Reynolds number }Re&=&UL/\nu\\ \,\text{Magnetic Reynolds number }Re_{M}&=&UL/\eta\\ \,\text{Magnetic Prandtl number }P_{M}&=&\nu/\eta.\end{array}
  10. P M P_{M}
  11. 10 - 5 10^{-5}
  12. P M P_{M}
  13. 𝐮 𝐮 \mathbf{u}\cdot\nabla\mathbf{u}
  14. R e Re
  15. R e M Re_{M}
  16. R e M Re_{M}
  17. z + z^{+}
  18. z - z^{-}
  19. B 0 B_{0}
  20. ( B 0 k ) - 1 (B_{0}k)^{-1}
  21. Π \Pi
  22. z ± z^{\pm}
  23. τ ± \tau^{\pm}
  24. z ± z^{\pm}
  25. τ ± 1 / ( k V A ) \tau^{\pm}\approx 1/(kV_{A})
  26. T N L ± ( k z k ) - 1 T_{NL}^{\pm}\approx(kz_{k}^{\mp})^{-1}
  27. Π + \Pi^{+}
  28. Π - \Pi^{-}
  29. z + z^{+}
  30. z - z^{-}
  31. K ± K^{\pm}
  32. k - 1 / 3 k^{-1/3}
  33. k - 4 / 3 k^{-4/3}
  34. k - 5 / 3 k^{-5/3}
  35. δ z ± B 0 \delta z^{\pm}\ll B_{0}
  36. E ( k ) ( Π B 0 ) 1 / 2 k | | 1 / 2 k - 2 E(k)\sim(\Pi B_{0})^{1/2}k_{||}^{1/2}k_{\perp}^{-2}
  37. k | | k_{||}
  38. k k_{\perp}
  39. δ z ± B 0 \delta z^{\pm}\sim B_{0}
  40. k z k k | | B 0 k_{\perp}z_{k_{\perp}}\sim k_{||}B_{0}
  41. E ( k ) k - 5 / 3 ; k | | k 2 / 3 \begin{array}[]{lcl}E(k)&\propto&k_{\perp}^{-5/3};\\ k_{||}&\propto&k_{\perp}^{2/3}\end{array}
  42. E ± E^{\pm}
  43. k - 5 / 3 k^{-5/3}
  44. k - 3 / 2 k^{-3/2}
  45. Π ± \Pi^{\pm}
  46. E + ( k ) E - ( k ) E^{+}(k)\gg E^{-}(k)
  47. Π ± \Pi^{\pm}
  48. k | | k 2 / 3 k_{||}\sim k_{\perp}^{2/3}
  49. k - 3 / 2 k^{-3/2}

Magnus_expansion.html

  1. n n
  2. n × n n×n
  3. A ( t ) A(t)
  4. Y ( t ) = A ( t ) Y ( t ) , Y ( t 0 ) = Y 0 Y^{\prime}(t)=A(t)Y(t),\qquad\qquad Y(t_{0})=Y_{0}
  5. n n
  6. Y ( t ) Y(t)
  7. Y ( t ) = exp ( t 0 t A ( s ) d s ) Y 0 . Y(t)=\exp\left(\int_{t_{0}}^{t}A(s)\,ds\right)Y_{0}.
  8. A ( t ) A(t)
  9. A A
  10. t t
  11. n × n n×n
  12. Y ( t ) = exp ( Ω ( t , t 0 ) ) Y 0 , Y(t)=\exp\left(\Omega(t,t_{0})\right)\,Y_{0}~{},
  13. Ω ( t ) = k = 1 Ω k ( t ) , \Omega(t)=\sum_{k=1}^{\infty}\Omega_{k}(t),
  14. Ω ( t ) Ω(t)
  15. Ω Ω
  16. Ω Ω
  17. Ω = ad Ω exp ( ad Ω ) - 1 A , \Omega^{\prime}=\frac{\operatorname{ad}_{\Omega}}{\exp(\operatorname{ad}_{% \Omega})-1}~{}A~{},
  18. Ω Ω
  19. A A
  20. Ω 1 ( t ) = 0 t A ( t 1 ) d t 1 , Ω 2 ( t ) = 1 2 0 t d t 1 0 t 1 d t 2 [ A ( t 1 ) , A ( t 2 ) ] , Ω 3 ( t ) = 1 6 0 t d t 1 0 t 1 d t 2 0 t 2 d t 3 ( [ A ( t 1 ) , [ A ( t 2 ) , A ( t 3 ) ] ] + [ A ( t 3 ) , [ A ( t 2 ) , A ( t 1 ) ] ] ) , Ω 4 ( t ) = 1 12 0 t d t 1 0 t 1 d t 2 0 t 2 d t 3 0 t 3 d t 4 ( [ [ [ A 1 , A 2 ] , A 3 ] , A 4 ] + [ A 1 , [ [ A 2 , A 3 ] , A 4 ] ] + [ A 1 , [ A 2 , [ A 3 , A 4 ] ] ] + [ A 2 , [ A 3 , [ A 4 , A 1 ] ] ] ) \begin{aligned}\displaystyle\Omega_{1}(t)&\displaystyle=\int_{0}^{t}A(t_{1})\,% dt_{1},\\ \displaystyle\Omega_{2}(t)&\displaystyle=\frac{1}{2}\int_{0}^{t}dt_{1}\int_{0}% ^{t_{1}}dt_{2}\ \left[A(t_{1}),A(t_{2})\right],\\ \displaystyle\Omega_{3}(t)&\displaystyle=\frac{1}{6}\int_{0}^{t}dt_{1}\int_{0}% ^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\Bigl(\left[A(t_{1}),\left[A(t_{2}),A(t_{3% })\right]\right]+\left[A(t_{3}),\left[A(t_{2}),A(t_{1})\right]\right]\Bigr),\\ \displaystyle\Omega_{4}(t)&\displaystyle=\frac{1}{12}\int_{0}^{t}dt_{1}\int_{0% }^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\int_{0}^{t_{3}}dt_{4}\Bigl(\left[\left[% \left[A_{1},A_{2}\right],A_{3}\right],A_{4}\right]\\ &\displaystyle\quad+\left[A_{1},\left[\left[A_{2},A_{3}\right],A_{4}\right]% \right]+\left[A_{1},\left[A_{2},\left[A_{3},A_{4}\right]\right]\right]+\left[A% _{2},\left[A_{3},\left[A_{4},A_{1}\right]\right]\right]\Bigr)\end{aligned}
  21. A A , B A B B A AA,B≡AB−BA
  22. n n
  23. Ω Ω
  24. A ( t ) A(t)
  25. Ω ( t ) Ω(t)
  26. t 0 , , T ) t∈0,,T)
  27. 0 T A ( s ) 2 d s < π \int_{0}^{T}\|A(s)\|_{2}ds<\pi
  28. 2 \|\cdot\|_{2}
  29. A ( t ) A(t)
  30. t > T t>T
  31. S n ( j ) = m = 1 n - j [ Ω m , S n - m ( j - 1 ) ] , 2 j n - 1 S_{n}^{(j)}=\sum_{m=1}^{n-j}\left[\Omega_{m},S_{n-m}^{(j-1)}\right],\qquad% \qquad 2\leq j\leq n-1
  32. S n ( 1 ) = [ Ω n - 1 , A ] , S n ( n - 1 ) = ad Ω 1 n - 1 ( A ) , S_{n}^{(1)}=\left[\Omega_{n-1},A\right],\qquad S_{n}^{(n-1)}=\mathrm{ad}_{% \Omega_{1}}^{n-1}(A)~{},
  33. Ω 1 = 0 t A ( τ ) d τ \Omega_{1}=\int_{0}^{t}A(\tau)d\tau
  34. Ω n = j = 1 n - 1 B j j ! 0 t S n ( j ) ( τ ) d τ , n 2. \Omega_{n}=\sum_{j=1}^{n-1}\frac{B_{j}}{j!}\int_{0}^{t}S_{n}^{(j)}(\tau)d\tau,% \qquad\qquad n\geq 2.
  35. ad Ω 0 A = A , ad Ω k + 1 A = [ Ω , ad Ω k A ] , \mathrm{ad}_{\Omega}^{0}A=A,\qquad\mathrm{ad}_{\Omega}^{k+1}A=[\Omega,\mathrm{% ad}_{\Omega}^{k}A],
  36. n n
  37. A A
  38. Ω n ( t ) = j = 1 n - 1 B j j ! k 1 + + k j = n - 1 k 1 1 , , k j 1 0 t ad Ω k 1 ( τ ) ad Ω k 2 ( τ ) ad Ω k j ( τ ) A ( τ ) d τ n 2 , \Omega_{n}(t)=\sum_{j=1}^{n-1}\frac{B_{j}}{j!}\,\sum_{k_{1}+\cdots+k_{j}=n-1% \atop k_{1}\geq 1,\ldots,k_{j}\geq 1}\,\int_{0}^{t}\,\mathrm{ad}_{\Omega_{k_{1% }}(\tau)}\,\mathrm{ad}_{\Omega_{k_{2}}(\tau)}\cdots\,\mathrm{ad}_{\Omega_{k_{j% }}(\tau)}A(\tau)\,d\tau\qquad n\geq 2,
  39. n n

Mahler_volume.html

  1. { x x y 1 for all y B } . \left\{x\mid x\cdot y\leq 1\,\text{ for all }y\in B\right\}.
  2. ( T B ) = ( T - 1 ) B (TB)^{\circ}=(T^{-1})^{\ast}B^{\circ}
  3. det T \det T
  4. det ( T - 1 ) \det(T^{-1})^{\ast}
  5. Γ ( 3 / 2 ) 2 n 4 n Γ ( n 2 + 1 ) 2 . \frac{\Gamma(3/2)^{2n}4^{n}}{\Gamma(\frac{n}{2}+1)^{2}}.
  6. 4 n Γ ( n + 1 ) . \frac{4^{n}}{\Gamma(n+1)}.
  7. ( π 2 ) n \left(\tfrac{\pi}{2}\right)^{n}

Maier's_theorem.html

  1. π ( x + ( log x ) λ ) - π ( x ) ( log x ) λ - 1 \frac{\pi(x+(\log x)^{\lambda})-\pi(x)}{(\log x)^{\lambda-1}}
  2. z = x 1 / u z=x^{1/u}
  3. u u
  4. 2 Y ( 2 < p x log p - 2 < n x 1 ) 2 d x \int_{2}^{Y}\left(\sum_{2<p\leq x}\log p-\sum_{2<n\leq x}1\right)^{2}\,dx

Majumdar–Ghosh_model.html

  1. H ^ = J j = 1 N S j S j + 1 + J 2 j = 1 N S j S j + 2 \hat{H}=J\sum_{j=1}^{N}\vec{S}_{j}\cdot\vec{S}_{j+1}+\frac{J}{2}\sum_{j=1}^{N}% \vec{S}_{j}\cdot\vec{S}_{j+2}

Make_Compatible.html

  1. 7 / 8 7/8

Maker-Breaker_game.html

  1. | V | = |V|=\infty
  2. | H | = |H|=\infty
  3. G = ( V , E ) G=(V,E)
  4. = { F E | G [ F ] has property 𝒫 } \mathcal{F}=\{F\subset E|G[F]\hbox{ has property }\mathcal{P}\}
  5. 𝒫 \mathcal{P}

Malcev_Lie_algebra.html

  1. L L
  2. {\mathbb{Q}}
  3. { F r L } r 1 \{F_{r}L\}_{r\geq 1}
  4. F 1 L = L F_{1}L=L
  5. [ F r L , F s L ] F r + s L [F_{r}L,F_{s}L]\subset F_{r+s}L
  6. r 1 F r L / F r + 1 L \oplus_{r\geq 1}F_{r}L/F_{r+1}L

Mamyshev_2R_regenerator.html

  1. L L
  2. γ \gamma
  3. α \alpha
  4. δ f \delta f
  5. Δ λ \Delta\lambda
  6. λ 0 \lambda_{0}

Mandelbulb.html

  1. 𝐯 = x , y , z {\mathbf{v}}=\langle x,y,z\rangle
  2. 𝐯 n := r n sin ( n θ ) cos ( n ϕ ) , sin ( n θ ) sin ( n ϕ ) , cos ( n θ ) {\mathbf{v}}^{n}:=r^{n}\langle\sin(n\theta)\cos(n\phi),\sin(n\theta)\sin(n\phi% ),\cos(n\theta)\rangle
  3. r = x 2 + y 2 + z 2 r=\sqrt{x^{2}+y^{2}+z^{2}}
  4. ϕ = arctan ( y / x ) = arg ( x + y i ) \phi=\arctan(y/x)=\arg(x+yi)
  5. θ = arctan ( x 2 + y 2 / z ) = arccos ( z / r ) \theta=\arctan(\sqrt{x^{2}+y^{2}}/z)=\arccos(z/r)
  6. 𝐜 {\mathbf{c}}
  7. 0 , 0 , 0 \langle 0,0,0\rangle
  8. 𝐯 𝐯 n + 𝐜 {\mathbf{v}}\mapsto{\mathbf{v}}^{n}+{\mathbf{c}}
  9. x , y , z 3 = ( 3 z 2 - x 2 - y 2 ) x ( x 2 - 3 y 2 ) x 2 + y 2 , ( 3 z 2 - x 2 - y 2 ) y ( 3 x 2 - y 2 ) x 2 + y 2 , z ( z 2 - 3 x 2 - 3 y 2 ) \langle x,y,z\rangle^{3}=\left\langle\ \frac{(3z^{2}-x^{2}-y^{2})x(x^{2}-3y^{2% })}{x^{2}+y^{2}},\frac{(3z^{2}-x^{2}-y^{2})y(3x^{2}-y^{2})}{x^{2}+y^{2}},z(z^{% 2}-3x^{2}-3y^{2})\right\rangle
  10. ( x 2 - y 2 - z 2 ) 2 + ( 2 x z ) 2 + ( 2 x y ) 2 = ( x 2 + y 2 + z 2 ) 2 (x^{2}-y^{2}-z^{2})^{2}+(2xz)^{2}+(2xy)^{2}=(x^{2}+y^{2}+z^{2})^{2}
  11. x x 2 - y 2 - z 2 + x 0 x\rightarrow x^{2}-y^{2}-z^{2}+x_{0}
  12. y 2 x z + y 0 y\rightarrow 2xz+y_{0}
  13. z 2 x y + z 0 z\rightarrow 2xy+z_{0}
  14. ( x 3 - 3 x y 2 - 3 x z 2 ) 2 + ( y 3 - 3 y x 2 + y z 2 ) 2 + ( z 3 - 3 z x 2 + z y 2 ) 2 = ( x 2 + y 2 + z 2 ) 3 (x^{3}-3xy^{2}-3xz^{2})^{2}+(y^{3}-3yx^{2}+yz^{2})^{2}+(z^{3}-3zx^{2}+zy^{2})^% {2}=(x^{2}+y^{2}+z^{2})^{3}
  15. x x 3 - 3 x ( y 2 + z 2 ) + x 0 x\rightarrow x^{3}-3x(y^{2}+z^{2})+x_{0}
  16. y - y 3 + 3 y x 2 - y z 2 + y 0 y\rightarrow-y^{3}+3yx^{2}-yz^{2}+y_{0}
  17. z z 3 - 3 z x 2 + z y 2 + z 0 z\rightarrow z^{3}-3zx^{2}+zy^{2}+z_{0}
  18. w w 3 + w 0 w\rightarrow w^{3}+w_{0}
  19. w w ¯ 3 + w 0 w\rightarrow\overline{w}^{3}+w_{0}
  20. z z 4 m + 1 + z 0 z\rightarrow z^{4m+1}+z_{0}
  21. i 4 = 1 i^{4}=1
  22. z z 5 + z 0 z\rightarrow z^{5}+z_{0}
  23. z = x + i y z=x+iy
  24. x x 5 - 10 x 3 y 2 + 5 x y 4 + x 0 x\rightarrow x^{5}-10x^{3}y^{2}+5xy^{4}+x_{0}
  25. y y 5 - 10 y 3 x 2 + 5 y x 4 + y 0 y\rightarrow y^{5}-10y^{3}x^{2}+5yx^{4}+y_{0}
  26. x x 5 - 10 x 3 ( y 2 + A y z + z 2 ) + 5 x ( y 4 + B y 3 z + C y 2 z 2 + B y z 3 + z 4 ) + D x 2 y z ( y + z ) + x 0 x\rightarrow x^{5}-10x^{3}(y^{2}+Ayz+z^{2})+5x(y^{4}+By^{3}z+Cy^{2}z^{2}+Byz^{% 3}+z^{4})+Dx^{2}yz(y+z)+x_{0}
  27. y y 5 - 10 y 3 ( z 2 + A x z + x 2 ) + 5 y ( z 4 + B z 3 x + C z 2 x 2 + B z x 3 + x 4 ) + D y 2 z x ( z + x ) + y 0 y\rightarrow y^{5}-10y^{3}(z^{2}+Axz+x^{2})+5y(z^{4}+Bz^{3}x+Cz^{2}x^{2}+Bzx^{% 3}+x^{4})+Dy^{2}zx(z+x)+y_{0}
  28. z z 5 - 10 z 3 ( x 2 + A x y + y 2 ) + 5 z ( x 4 + B x 3 y + C x 2 y 2 + B x y 3 + y 4 ) + D z 2 x y ( x + y ) + z 0 z\rightarrow z^{5}-10z^{3}(x^{2}+Axy+y^{2})+5z(x^{4}+Bx^{3}y+Cx^{2}y^{2}+Bxy^{% 3}+y^{4})+Dz^{2}xy(x+y)+z_{0}
  29. z z 9 z\rightarrow z^{9}
  30. z - z 5 + z 0 z\rightarrow-z^{5}+z_{0}
  31. x x 9 - 36 x 7 ( y 2 + z 2 ) + 126 x 5 ( y 2 + z 2 ) 2 - 84 x 3 ( y 2 + z 2 ) 3 + 9 x ( y 2 + z 2 ) 4 + x 0 x\rightarrow x^{9}-36x^{7}(y^{2}+z^{2})+126x^{5}(y^{2}+z^{2})^{2}-84x^{3}(y^{2% }+z^{2})^{3}+9x(y^{2}+z^{2})^{4}+x_{0}
  32. y y 9 - 36 y 7 ( z 2 + x 2 ) + 126 y 5 ( z 2 + x 2 ) 2 - 84 y 3 ( z 2 + x 2 ) 3 + 9 y ( z 2 + x 2 ) 4 + y 0 y\rightarrow y^{9}-36y^{7}(z^{2}+x^{2})+126y^{5}(z^{2}+x^{2})^{2}-84y^{3}(z^{2% }+x^{2})^{3}+9y(z^{2}+x^{2})^{4}+y_{0}
  33. z z 9 - 36 z 7 ( x 2 + y 2 ) + 126 z 5 ( x 2 + y 2 ) 2 - 84 z 3 ( x 2 + y 2 ) 3 + 9 z ( x 2 + y 2 ) 4 + z 0 z\rightarrow z^{9}-36z^{7}(x^{2}+y^{2})+126z^{5}(x^{2}+y^{2})^{2}-84z^{3}(x^{2% }+y^{2})^{3}+9z(x^{2}+y^{2})^{4}+z_{0}
  34. x 1 2 ( x + i y 2 + z 2 ) 9 + 1 2 ( x - i y 2 + z 2 ) 9 + x 0 x\rightarrow\frac{1}{2}(x+i\sqrt{y^{2}+z^{2}})^{9}+\frac{1}{2}(x-i\sqrt{y^{2}+% z^{2}})^{9}+x_{0}
  35. ( x , y , z ) ( f ( x , y , z ) + x 0 , g ( x , y , z ) + y 0 , h ( x , y , z ) + z 0 ) (x,y,z)\rightarrow(f(x,y,z)+x_{0},g(x,y,z)+y_{0},h(x,y,z)+z_{0})
  36. ( x 2 + y 2 + z 2 ) n = f ( x , y , z ) 2 + g ( x , y , z ) 2 + h ( x , y , z ) 2 (x^{2}+y^{2}+z^{2})^{n}=f(x,y,z)^{2}+g(x,y,z)^{2}+h(x,y,z)^{2}

Manuscripts_of_the_Austrian_National_Library.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}

Maps_of_manifolds.html

  1. f : M \scriptstyle f\colon M\to\mathbb{R}
  2. f : M , \scriptstyle f\colon M\to\mathbb{C},
  3. M \scriptstyle M\to\mathbb{R}
  4. M , \scriptstyle\mathbb{R}\to M,
  5. [ a , b ] , [a,b],
  6. [ 0 , 1 ] , [0,1],

Marchenko–Pastur_distribution.html

  1. X X
  2. M × N M\times N
  3. σ 2 < \sigma^{2}<\infty
  4. Y N = N - 1 X X T Y_{N}=N^{-1}XX^{T}\,
  5. λ 1 , λ 2 , , λ M \lambda_{1},\,\lambda_{2},\,\dots,\,\lambda_{M}
  6. Y N Y_{N}
  7. μ M ( A ) = 1 M # { λ j A } , A . \mu_{M}(A)=\frac{1}{M}\#\left\{\lambda_{j}\in A\right\},\quad A\subset\mathbb{% R}.
  8. M , N M,\,N\,\to\,\infty
  9. M / N λ ( 0 , + ) M/N\,\to\,\lambda\in(0,+\infty)
  10. μ M μ \mu_{M}\,\to\,\mu
  11. μ ( A ) = { ( 1 - 1 λ ) 𝟏 0 A + ν ( A ) , if λ > 1 ν ( A ) , if 0 λ 1 , \mu(A)=\begin{cases}(1-\frac{1}{\lambda})\mathbf{1}_{0\in A}+\nu(A),&\,\text{% if }\lambda>1\\ \nu(A),&\,\text{if }0\leq\lambda\leq 1,\end{cases}
  12. d ν ( x ) = 1 2 π σ 2 ( λ + - x ) ( x - λ - ) λ x 1 [ λ - , λ + ] d x d\nu(x)=\frac{1}{2\pi\sigma^{2}}\frac{\sqrt{(\lambda_{+}-x)(x-\lambda_{-})}}{% \lambda x}\,\mathbf{1}_{[\lambda_{-},\lambda_{+}]}\,dx
  13. λ ± = σ 2 ( 1 ± λ ) 2 . \lambda_{\pm}=\sigma^{2}(1\pm\sqrt{\lambda})^{2}.\,
  14. 1 / λ 1/\lambda
  15. σ 2 \sigma^{2}

Marcinkiewicz–Zygmund_inequality.html

  1. x i \textstyle x_{i}
  2. i = 1 , , n \textstyle i=1,\ldots,n
  3. E ( x i ) = 0 \textstyle E\left(x_{i}\right)=0
  4. E ( | x i | p ) < + \textstyle E\left(\left|x_{i}\right|^{p}\right)<+\infty
  5. 1 p < + \textstyle 1\leq p<+\infty
  6. A p E ( ( i = 1 n | x i | 2 ) < m t p l > ) p / 2 E ( | i = 1 n x i | p ) B p E ( ( i = 1 n | x i | 2 ) p / 2 ) A_{p}E\left(\left(\sum_{i=1}^{n}\left|x_{i}\right|^{2}\right)_{<}mtpl>{}^{p/2}% \right)\leq E\left(\left|\sum_{i=1}^{n}x_{i}\right|^{p}\right)\leq B_{p}E\left% (\left(\sum_{i=1}^{n}\left|x_{i}\right|^{2}\right)^{p/2}\right)
  7. A p \textstyle A_{p}
  8. B p \textstyle B_{p}
  9. p \textstyle p
  10. p = 2 \textstyle p=2
  11. A 2 = B 2 = 1 \textstyle A_{2}=B_{2}=1
  12. E ( x i ) = 0 \textstyle E\left(x_{i}\right)=0
  13. E ( | x i | 2 ) < + \textstyle E\left(\left|x_{i}\right|^{2}\right)<+\infty
  14. Var ( i = 1 n x i ) = E ( | i = 1 n x i | 2 ) = i = 1 n j = 1 n E ( x i x ¯ j ) = i = 1 n E ( | x i | 2 ) = i = 1 n Var ( x i ) . \mathrm{Var}\left(\sum_{i=1}^{n}x_{i}\right)=E\left(\left|\sum_{i=1}^{n}x_{i}% \right|^{2}\right)=\sum_{i=1}^{n}\sum_{j=1}^{n}E\left(x_{i}\overline{x}_{j}% \right)=\sum_{i=1}^{n}E\left(\left|x_{i}\right|^{2}\right)=\sum_{i=1}^{n}% \mathrm{Var}\left(x_{i}\right).

Margin_Infused_Relaxed_Algorithm.html

  1. T = { x i , y i } T=\{x_{i},y_{i}\}
  2. w w
  3. i i
  4. w ( 0 ) w^{(0)}
  5. n n
  6. N N
  7. t t
  8. | T | |T|
  9. w ( i + 1 ) w^{(i+1)}
  10. w ( i ) w^{(i)}
  11. { x t , y t } \{x_{t},y_{t}\}
  12. i i
  13. i + 1 i+1
  14. j = 1 N × | T | w ( j ) N × | T | \frac{\sum_{j=1}^{N\times|T|}w^{(j)}}{N\times|T|}
  15. m i n w ( i + 1 ) - w ( i ) min\|w^{(i+1)}-w^{(i)}\|
  16. s c o r e ( x t , y t ) - s c o r e ( x t , y ) L ( y t , y ) y score(x_{t},y_{t})-score(x_{t},y^{\prime})\geq L(y_{t},y^{\prime})\ \forall y^% {\prime}
  17. y y
  18. y y^{\prime}
  19. y y^{\prime}
  20. y y

Marginal_product_of_labor.html

  1. Δ Y Δ L . \frac{\Delta Y}{\Delta L}.
  2. Y L . \frac{\partial Y}{\partial L}.
  3. Q = 90 L - L 2 Q=90L-L^{2}
  4. M C L = 30 MC_{L}=30
  5. M P L = 90 - 2 L MP_{L}=90-2L
  6. M R P L = 40 ( 90 - 2 L ) MRP_{L}=40(90-2L)
  7. M R P L = 3600 - 80 L MRP_{L}=3600-80L
  8. M R P L = M C L MRP_{L}=MC_{L}
  9. 3600 - 80 L = 30 3600-80L=30
  10. 3570 = 80 L 3570=80L
  11. L = 44.625 L=44.625
  12. Q = 90 L - L 2 Q=90L-L^{2}
  13. Q = 90 ( 44.625 ) - ( 44.625 ) 2 Q=90(44.625)-(44.625)^{2}
  14. Q = 4016.25 - 1991.39 Q=4016.25-1991.39
  15. Q = 2024.86 Q=2024.86
  16. T R - T C = Π TR-TC=\Pi
  17. Π = 40 ( 2025 ) - 30 ( 2025 ) \Pi=40(2025)-30(2025)
  18. Π = 81 , 000 - 60 , 750 \Pi=81,000-60,750
  19. Π = 20 , 250 \Pi=20,250
  20. L = 44.625 L=44.625

Marginal_utility.html

  1. S 1 S_{1}
  2. S 2 S_{2}
  3. Δ U = U ( S 2 ) - U ( S 1 ) \Delta U=U(S_{2})-U(S_{1})\,
  4. S 1 S_{1}
  5. S 2 S_{2}
  6. g g\,
  7. g g\,
  8. Δ U Δ g | c . p . \left.\frac{\Delta U}{\Delta g}\right|_{c.p.}
  9. g g\,
  10. lim Δ g 0 Δ U Δ g | c . p . \lim_{\Delta g\to 0}\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}
  11. U g = lim Δ g 0 Δ U Δ g | c . p . \frac{\partial U}{\partial g}=\lim_{\Delta g\to 0}\left.\frac{\Delta U}{\Delta g% }\right|_{c.p.}
  12. 2 U g 2 < 0 \frac{\partial^{2}U}{\partial g^{2}}<0

Margules_activity_model.html

  1. γ i \gamma_{i}
  2. G e x R T = X 1 X 2 ( A 21 X 1 + A 12 X 2 ) + X 1 2 X 2 2 ( B 21 X 1 + B 12 X 2 ) + + X 1 m X 2 m ( M 21 X 1 + M 12 X 2 ) \frac{G^{ex}}{RT}=X_{1}X_{2}(A_{21}X_{1}+A_{12}X_{2})+X_{1}^{2}X_{2}^{2}(B_{21% }X_{1}+B_{12}X_{2})+...+X_{1}^{m}X_{2}^{m}(M_{21}X_{1}+M_{12}X_{2})
  3. X 1 X 2 X_{1}X_{2}
  4. { ln γ 1 = [ A 12 + 2 ( A 21 - A 12 ) x 1 ] x 2 2 ln γ 2 = [ A 21 + 2 ( A 12 - A 21 ) x 2 ] x 1 2 \left\{\begin{matrix}\ln\ \gamma_{1}=[A_{12}+2(A_{21}-A_{12})x_{1}]x^{2}_{2}\\ \ln\ \gamma_{2}=[A_{21}+2(A_{12}-A_{21})x_{2}]x^{2}_{1}\end{matrix}\right.
  5. ln ( γ 1 ) \ln\ (\gamma_{1}^{\infty})
  6. ln ( γ 2 ) \ln\ (\gamma_{2}^{\infty})
  7. A 12 = A 21 = A A_{12}=A_{21}=A
  8. { ln γ 1 = A x 2 2 ln γ 2 = A x 1 2 \left\{\begin{matrix}\ln\ \gamma_{1}=Ax^{2}_{2}\\ \ln\ \gamma_{2}=Ax^{2}_{1}\end{matrix}\right.
  9. d l n γ 1 / d x 1 dln\gamma_{1}/dx_{1}
  10. x 1 x_{1}
  11. A 12 < 0 A_{12}<0
  12. A 21 > 0 A_{21}>0
  13. 0.5 < A 12 / A 21 < 2 0.5<A_{12}/A_{21}<2
  14. A 12 < A 21 / 2 A_{12}<A_{21}/2
  15. A 12 < 0 A_{12}<0
  16. x 1 = 1 - 2 A 12 / A 21 3 ( 1 - A 12 / A 21 ) x_{1}=\frac{1-2A_{12}/A_{21}}{3(1-A_{12}/A_{21})}
  17. A 12 < A 21 / 2 A_{12}<A_{21}/2
  18. A 12 > 0 A_{12}>0
  19. A = ln γ 1 = ln γ 2 A=\ln\gamma_{1}^{\infty}=\ln\gamma_{2}^{\infty}
  20. γ 1 = γ 2 > exp ( 2 ) 7.38 \gamma_{1}^{\infty}=\gamma_{2}^{\infty}>\exp(2)\approx 7.38
  21. A 21 + A 12 > 4 A_{21}+A_{12}>4
  22. γ 1 γ 2 > exp ( 4 ) 54.6 \gamma_{1}^{\infty}\gamma_{2}^{\infty}>\exp(4)\approx 54.6

Markov_switching_multifractal.html

  1. P t P_{t}
  2. r t = ln ( P t / P t - 1 ) r_{t}=\ln(P_{t}/P_{t-1})
  3. r t = μ + σ ¯ ( M 1 , t M 2 , t M k ¯ , t ) 1 / 2 ϵ t , r_{t}=\mu+\bar{\sigma}(M_{1,t}M_{2,t}...M_{\bar{k},t})^{1/2}\epsilon_{t},
  4. μ \mu
  5. σ \sigma
  6. ϵ t \epsilon_{t}
  7. M t = ( M 1 , t M 2 , t M k ¯ , t ) R + k ¯ . M_{t}=(M_{1,t}M_{2,t}\dots M_{\bar{k},t})\in R_{+}^{\bar{k}}.
  8. M t M_{t}
  9. M k , t + 1 M_{k,t+1}
  10. M M
  11. γ k \gamma_{k}
  12. M k , t M_{k,t}
  13. M M
  14. γ k \gamma_{k}
  15. M k , t = M k , t - 1 M_{k,t}=M_{k,t-1}
  16. 1 - γ k 1-\gamma_{k}
  17. γ k = 1 - ( 1 - γ 1 ) ( b k - 1 ) \gamma_{k}=1-(1-\gamma_{1})^{(b^{k-1})}
  18. γ k \gamma_{k}
  19. γ k γ 1 b k - 1 \gamma_{k}\approx\gamma_{1}b^{k-1}
  20. M M
  21. k k
  22. M M
  23. m 0 m_{0}
  24. 2 - m 0 2-m_{0}
  25. r t r_{t}
  26. θ = ( m 0 , μ , σ ¯ , b , γ 1 ) \theta=(m_{0},\mu,\bar{\sigma},b,\gamma_{1})
  27. k ¯ > 1 \bar{k}>1
  28. d P t P t = μ d t + σ ( M t ) d W t , \frac{dP_{t}}{P_{t}}=\mu dt+\sigma(M_{t})\,dW_{t},
  29. σ ( M t ) = σ ¯ ( M 1 , t M k ¯ , t ) 1 / 2 \sigma(M_{t})=\bar{\sigma}(M_{1,t}\dots M_{\bar{k},t})^{1/2}
  30. W t W_{t}
  31. μ \mu
  32. σ ¯ \bar{\sigma}
  33. M k , t M_{k,t}
  34. M M
  35. γ k d t \gamma_{k}dt
  36. M k , t + d t = M k , t M_{k,t+dt}=M_{k,t}
  37. 1 - γ k d t 1-\gamma_{k}dt
  38. k k
  39. γ k = γ 1 b k - 1 . \gamma_{k}=\gamma_{1}b^{k-1}.
  40. k ¯ \bar{k}
  41. M M
  42. M t M_{t}
  43. m 1 , , m d R + k ¯ m^{1},...,m^{d}\in R_{+}^{\bar{k}}
  44. d = 2 k ¯ d=2^{\bar{k}}
  45. A = ( a i , j ) 1 i , j d A=(a_{i,j})_{1\leq i,j\leq d}
  46. a i , j = P ( M t + 1 = m j | M t = m i ) a_{i,j}=P\left(M_{t+1}=m^{j}|M_{t}=m^{i}\right)
  47. r t r_{t}
  48. f ( r t | M t = m i ) = 1 2 π σ 2 ( m i ) exp [ - ( r t - μ ) 2 2 σ 2 ( m i ) ] . f(r_{t}|M_{t}=m^{i})=\frac{1}{\sqrt{2\pi\sigma^{2}(m^{i})}}\exp\left[-\frac{(r% _{t}-\mu)^{2}}{2\sigma^{2}(m^{i})}\right].
  49. M t M_{t}
  50. Π t j P ( M t = m j | r 1 , , r t ) . \Pi_{t}^{j}\equiv P(M_{t}=m^{j}|r_{1},...,r_{t}).
  51. Π t = ( Π t 1 , , Π t d ) \Pi_{t}=(\Pi_{t}^{1},...,\Pi_{t}^{d})
  52. Π t = ω ( r t ) * ( Π t - 1 A ) [ ω ( r t ) * ( Π t - 1 A ) ] 𝟏 . \Pi_{t}=\frac{\omega(r_{t})*(\Pi_{t-1}A)}{[\omega(r_{t})*(\Pi_{t-1}A)]\,% \textbf{1}^{\prime}}.
  53. 𝟏 = ( 1 , , 1 ) R d \,\textbf{1}=(1,\dots,1)\in R^{d}
  54. x * y = ( x 1 y 1 , , x d y d ) x*y=\left(x_{1}y_{1},\dots,x_{d}y_{d}\right)
  55. x , y R d x,y\in R^{d}
  56. ω ( r t ) = [ f ( r t | M t = m 1 ) ; ; f ( r t | M t = m d ) ] . \omega(r_{t})=\left[f(r_{t}|M_{t}=m^{1});\dots;f(r_{t}|M_{t}=m^{d})\right].
  57. Π 0 \Pi_{0}
  58. M t M_{t}
  59. Π 0 j = 1 / d = 2 - k ¯ \Pi_{0}^{j}=1/d=2^{-\bar{k}}
  60. j j
  61. ln L ( r 1 , , r T ; θ ) = t = 1 T ln [ ω ( r t ) . ( Π t - 1 A ) ] . \ln L(r_{1},\dots,r_{T};\theta)=\sum_{t=1}^{T}\ln[\omega(r_{t}).(\Pi_{t-1}A)].
  62. M M
  63. r 1 , , r t r_{1},\dots,r_{t}
  64. t + n t+n
  65. Π ^ t , n = Π t A n . \hat{\Pi}_{t,n}=\Pi_{t}A^{n}.\,

Markup_rule.html

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

Mass_action_law_(electronics).html

  1. n n
  2. p p
  3. n i n_{i}
  4. n p = n i 2 np=n_{i}^{2}
  5. n = N c exp [ - ( E c - E F ) k T ] n=N_{c}\,\text{ exp}\left[-\frac{(E_{c}-E_{F})}{kT}\right]
  6. N c = 2 ( 2 π m e * k T h 2 ) 3 / 2 \textstyle N_{c}=2\left(\frac{2\pi m_{e}^{*}kT}{h^{2}}\right)^{3/2}
  7. p = N v exp [ - ( E F - E v ) k T ] p=N_{v}\,\text{ exp}\left[-\frac{(E_{F}-E_{v})}{kT}\right]
  8. N v = 2 ( 2 π m h * k T h 2 ) 3 / 2 \textstyle N_{v}=2\left(\frac{2\pi m_{h}^{*}kT}{h^{2}}\right)^{3/2}
  9. n p = N c N v exp ( - E g k T ) = n i 2 np=N_{c}N_{v}\,\text{ exp}\left(-\frac{E_{g}}{kT}\right)=n_{i}^{2}

Mass_concentration_(chemistry).html

  1. ρ i \rho_{i}
  2. γ i \gamma_{i}
  3. m i m_{i}
  4. V V
  5. ρ i = m i V . \rho_{i}=\frac{m_{i}}{V}.
  6. V V
  7. ρ i = ρ i , T 0 < m t p l > ( 1 + α Δ T ) \rho_{i}=\frac{{\rho_{i,T_{0}}}}{<}mtpl>{{(1+\alpha\cdot\Delta T)}}
  8. ρ i , T 0 \rho_{i,T_{0}}
  9. α \alpha
  10. ρ \rho
  11. ρ = i ρ i \rho=\sum_{i}\rho_{i}\,
  12. i ρ i v i ¯ = 1 \sum_{i}\rho_{i}\cdot\bar{v_{i}}=1
  13. ρ i = ρ i * V i V \rho_{i}=\rho_{i}^{*}\frac{V_{i}}{V}\,
  14. ρ i * \rho_{i}^{*}
  15. V i V_{i}
  16. ν = V m = ρ - 1 \ \nu=\frac{V}{m}\ ={\rho}^{-1}
  17. c i c_{i}
  18. c i = ρ i M i c_{i}=\frac{\rho_{i}}{M_{i}}
  19. M i M_{i}
  20. i i
  21. w i w_{i}
  22. w i = ρ i ρ w_{i}=\frac{\rho_{i}}{\rho}
  23. x i x_{i}
  24. x i = ρ i ρ M M i x_{i}=\frac{\rho_{i}}{\rho}\cdot\frac{M}{M_{i}}
  25. M M
  26. b i b_{i}
  27. b i = ρ i M i ( ρ - ρ i ) b_{i}=\frac{\rho_{i}}{M_{i}(\rho-\rho_{i})}

Mass_matrix.html

  1. q ˙ \dot{q}
  2. T = 1 2 q ˙ T M q ˙ T=\frac{1}{2}\dot{q}^{\mathrm{T}}M\dot{q}
  3. q ˙ T \dot{q}^{\mathrm{T}}
  4. q ˙ \dot{q}
  5. m m
  6. T = 1 2 m | v | 2 = 1 2 v m v T\;=\;\frac{1}{2}m|v|^{2}\;=\;\frac{1}{2}v\cdot mv
  7. q = [ x 1 x 2 ] T q=[x_{1}\,x_{2}]^{\mathrm{T}}
  8. T = i = 1 2 1 2 m i x ˙ i 2 T=\sum_{i=1}^{2}\frac{1}{2}m_{i}\dot{x}_{i}{}^{2}
  9. T = 1 2 q ˙ T M q ˙ T=\frac{1}{2}\dot{q}^{\mathrm{T}}M\dot{q}
  10. M = [ m 1 0 0 m 2 ] M=\begin{bmatrix}m_{1}&0\\ 0&m_{2}\end{bmatrix}
  11. M = diag [ m 1 I n 1 , m 2 I n 2 , , m N I n N ] M=\mathrm{diag}[m_{1}I_{n_{1}},m_{2}I_{n_{2}},\cdots,m_{N}I_{n_{N}}]
  12. M = [ m 1 0 0 0 0 0 0 m 1 0 0 0 0 0 0 m 2 0 0 0 0 0 0 m 2 0 0 0 0 0 0 m N 0 0 0 0 0 0 m N ] M=\begin{bmatrix}m_{1}&\cdots&0&0&\cdots&0&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\cdots&m_{1}&0&\cdots&0&\cdots&0&\cdots&0\\ 0&\cdots&0&m_{2}&\cdots&0&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\cdots&0&0&\cdots&m_{2}&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\cdots&0&0&\cdots&0&\cdots&m_{N}&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\cdots&0&0&\cdots&0&\cdots&0&\cdots&m_{N}\\ \end{bmatrix}
  13. q = [ x , y , α ] q=[x,y,\alpha]
  14. p 1 = ( x , y ) + R ( cos α , sin α ) v 1 = ( x ˙ , y ˙ ) + R α ˙ ( - sin α , cos α ) p 2 = ( x , y ) - R ( cos α , sin α ) v 2 = ( x ˙ , y ˙ ) - R α ˙ ( - sin α , cos α ) \begin{array}[]{ll}p_{1}=(x,y)+R(\cos\alpha,\sin\alpha)&v_{1}=(\dot{x},\dot{y}% )+R\dot{\alpha}(-\sin\alpha,\cos\alpha)\\ p_{2}=(x,y)-R(\cos\alpha,\sin\alpha)&v_{2}=(\dot{x},\dot{y})-R\dot{\alpha}(-% \sin\alpha,\cos\alpha)\end{array}
  15. T = m x ˙ 2 + m y ˙ 2 + m R 2 α ˙ 2 + 2 R d cos α x ˙ α ˙ + 2 R d sin α y ˙ α ˙ T=m\dot{x}^{2}+m\dot{y}^{2}+mR^{2}\dot{\alpha}^{2}+2Rd\cos\alpha\dot{x}\dot{% \alpha}+2Rd\sin\alpha\dot{y}\dot{\alpha}
  16. m = m 1 + m 2 m=m_{1}+m_{2}
  17. d = m 1 - m 2 d=m_{1}-m_{2}
  18. T = 1 2 q ˙ T M q ˙ T=\frac{1}{2}\dot{q}^{\mathrm{T}}M\dot{q}
  19. M = [ m 0 R d cos α 0 m R d sin α R d cos α R d sin α R 2 m ] M=\begin{bmatrix}m&0&Rd\cos\alpha\\ 0&m&Rd\sin\alpha\\ Rd\cos\alpha&Rd\sin\alpha&R^{2}m\end{bmatrix}

Mass_point_geometry.html

  1. ( m , P ) (m,P)
  2. m P mP
  3. m m
  4. P P
  5. m P mP
  6. n Q nQ
  7. m = n m=n
  8. P = Q P=Q
  9. m P mP
  10. n Q nQ
  11. m + n m+n
  12. R R
  13. R R
  14. P Q PQ
  15. P R : R Q = n : m PR:RQ=n:m
  16. R R
  17. P P
  18. Q Q
  19. m P mP
  20. k k
  21. k ( m , P ) = ( k m , P ) k(m,P)=(km,P)
  22. A B C ABC
  23. E E
  24. A C AC
  25. C E = 3 A E CE=3AE
  26. F F
  27. A B AB
  28. B F = 3 A F BF=3AF
  29. B E BE
  30. C F CF
  31. O O
  32. A O AO
  33. B C BC
  34. D D
  35. O B O E \tfrac{OB}{OE}
  36. O D O A \tfrac{OD}{OA}
  37. A A
  38. 3 3
  39. B B
  40. C C
  41. 1 1
  42. E E
  43. F F
  44. 4 4
  45. O O
  46. 4 + 1 = 5 4+1=5
  47. D D
  48. 5 - 3 = 2 5-3=2
  49. O B O E \tfrac{OB}{OE}
  50. = 4 =4
  51. O D O A = 3 2 \tfrac{OD}{OA}=\tfrac{3}{2}
  52. A B C ABC
  53. D D
  54. E E
  55. F F
  56. B C BC
  57. C A CA
  58. A B AB
  59. A E = A F = C D = 2 AE=AF=CD=2
  60. B D = C E = 3 BD=CE=3
  61. B F = 5 BF=5
  62. D E DE
  63. C F CF
  64. O O
  65. O D O E \tfrac{OD}{OE}
  66. O C O F \tfrac{OC}{OF}
  67. C C
  68. A A
  69. 15 15
  70. B B
  71. 6 6
  72. C C
  73. 10 10
  74. A A
  75. 9 9
  76. B B
  77. D D
  78. E E
  79. F F
  80. 15 15
  81. 25 25
  82. 21 21
  83. O D O E = 25 15 = 5 3 \tfrac{OD}{OE}=\tfrac{25}{15}=\tfrac{5}{3}
  84. O C O F = 21 10 + 9 = 21 19 \tfrac{OC}{OF}=\tfrac{21}{10+9}=\tfrac{21}{19}
  85. A B C ABC
  86. D D
  87. E E
  88. B C BC
  89. C A CA
  90. F F
  91. G G
  92. A B AB
  93. G G
  94. F F
  95. B B
  96. B E BE
  97. C F CF
  98. O 1 O_{1}
  99. B E BE
  100. D G DG
  101. O 2 O_{2}
  102. F G = 1 FG=1
  103. A E = A F = D B = D C = 2 AE=AF=DB=DC=2
  104. B G = C E = 3 BG=CE=3
  105. O 1 O 2 B E \tfrac{O_{1}O_{2}}{BE}
  106. O 1 O_{1}
  107. O 2 O_{2}
  108. O 1 O_{1}
  109. D G DG
  110. D D
  111. G G
  112. O 2 O_{2}
  113. A A
  114. 6 6
  115. B B
  116. C C
  117. 3 3
  118. 4 4
  119. E E
  120. F F
  121. O 1 O_{1}
  122. E O 1 B O 1 = 3 10 \tfrac{EO_{1}}{BO_{1}}=\tfrac{3}{10}
  123. E O 1 B E = 3 13 \tfrac{EO_{1}}{BE}=\tfrac{3}{13}
  124. O 2 O_{2}
  125. C F CF
  126. F F
  127. O 1 O_{1}
  128. B B
  129. A A
  130. 3 3
  131. C C
  132. 2 2
  133. B B
  134. 3 3
  135. A A
  136. C C
  137. D D
  138. G G
  139. O 2 O_{2}
  140. B O 2 E O 2 = 5 3 + 2 = 1 \tfrac{BO_{2}}{EO_{2}}=\tfrac{5}{3+2}=1
  141. B O 2 B E = 1 2 \tfrac{BO_{2}}{BE}=\tfrac{1}{2}
  142. O 1 O 2 B E = B E - B O 2 - E O 1 B E = 1 - B O 2 B E - E O 1 B E = 1 - 1 2 - 3 13 = 7 26 . \tfrac{O_{1}O_{2}}{BE}=\tfrac{BE-BO_{2}-EO_{1}}{BE}=1-\tfrac{BO_{2}}{BE}-% \tfrac{EO_{1}}{BE}=1-\tfrac{1}{2}-\tfrac{3}{13}=\tfrac{7}{26}.

Massieu_function.html

  1. Ψ \Psi
  2. Ψ = Ψ ( X 1 , , X i , Y i + 1 , Y r ) \Psi=\Psi\big(X_{1},\dots,X_{i},Y_{i+1},\dots Y_{r}\big)\,
  3. ( X 1 , , X i , Y i + 1 , Y r ) \big(X_{1},\dots,X_{i},Y_{i+1},\dots Y_{r}\big)\,
  4. Ψ \Psi

Mass–action_ratio.html

  1. Γ \Gamma
  2. Γ = P 1 P 2 S 1 S 2 \Gamma=\frac{P_{1}P_{2}\ldots}{S_{1}S_{2}\ldots}
  3. Γ = K e q \Gamma=K_{eq}
  4. ρ \rho
  5. ρ = Γ K e q \rho=\frac{\Gamma}{K_{eq}}
  6. ρ = 1 \rho=1
  7. ρ < 1 \rho<1

Mass–luminosity_relation.html

  1. L L = ( M M ) a \frac{L}{L_{\odot}}=\left(\frac{M}{M_{\odot}}\right)^{a}
  2. L L .23 ( M M ) 2.3 ( M < .43 M ) \frac{L}{L_{\odot}}\approx.23\left(\frac{M}{M_{\odot}}\right)^{2.3}\qquad(M<.4% 3M_{\odot})
  3. L L = ( M M ) 4 ( .43 M < M < 2 M ) \frac{L}{L_{\odot}}=\left(\frac{M}{M_{\odot}}\right)^{4}\qquad\qquad(.43M_{% \odot}<M<2M_{\odot})
  4. L L 1.5 ( M M ) 3.5 ( 2 M < M < 20 M ) \frac{L}{L_{\odot}}\approx 1.5\left(\frac{M}{M_{\odot}}\right)^{3.5}\qquad(2M_% {\odot}<M<20M_{\odot})
  5. L L 3200 M M ( M > 20 M ) \frac{L}{L_{\odot}}\approx 3200\frac{M}{M_{\odot}}\qquad(M>20M_{\odot})
  6. L = 4 π R 2 σ T E 4 , L=4\pi R^{2}\sigma T_{E}^{4},
  7. d P d r = - G m ( r ) ρ ( r ) r 2 \frac{dP}{dr}=-\frac{Gm(r)\rho(r)}{r^{2}}
  8. P = - 1 3 E G R V \langle P\rangle=-\frac{1}{3}\frac{E_{GR}}{V}
  9. E G R = - 3 5 G M 2 R . E_{GR}=-\frac{3}{5}\frac{GM^{2}}{R}.
  10. P G M 2 4 π R 4 \langle P\rangle\approx\frac{GM^{2}}{4\pi R^{4}}
  11. P = ρ m ¯ k T , \langle P\rangle=\frac{\langle\rho\rangle}{\bar{m}}kT,
  12. k T = G M m ¯ 3 R kT=\frac{GM\bar{m}}{3R}
  13. m ¯ \bar{m}
  14. R = ( 3 4 1 ρ π M ) 1 3 R=\left(\frac{3}{4}\frac{1}{\rho\pi}M\right)^{\frac{1}{3}}
  15. L M 3.33 L\varpropto M^{3.33}
  16. l l
  17. 𝐃 = 𝐥 𝟏 + 𝐥 𝟐 + + 𝐥 𝐧 \mathbf{D=l_{1}+l_{2}+\cdots+l_{n}}
  18. D 2 = l 1 2 + l 2 2 + + l n 2 + 2 ( 𝐥 𝟏 𝐥 𝟐 + 𝐥 𝟏 𝐥 𝟑 + ) D^{2}=l_{1}^{2}+l_{2}^{2}+\cdots+l_{n}^{2}+2(\mathbf{l_{1}\cdot l_{2}+l_{1}% \cdot l_{3}+\cdots)}
  19. N N
  20. D 2 = l 1 2 + l 2 2 + + l n 2 = N l 2 . D^{2}=l_{1}^{2}+l_{2}^{2}+\cdots+l_{n}^{2}=Nl^{2}.
  21. R 2 l 2 \frac{R^{2}}{l^{2}}
  22. t R 2 c l t\approx\frac{R^{2}}{cl}
  23. R c \frac{R}{c}
  24. l R \frac{l}{R}
  25. T E [ l R ] 1 4 T I T_{E}\approx\Big[\frac{l}{R}\Big]^{\frac{1}{4}}T_{I}
  26. L 4 π R 2 σ T I 4 l R ( 4 π ) 2 3 5 σ k 4 G 4 m ¯ 4 ρ l M 3 L\approx 4\pi R^{2}\sigma T_{I}^{4}\frac{l}{R}\approx\frac{(4\pi)^{2}}{3^{5}}% \frac{\sigma}{k^{4}}G^{4}\bar{m}^{4}\langle\rho\rangle lM^{3}
  27. l ρ - 1 l\varpropto\langle\rho\rangle^{-1}
  28. L M 3 . L\varpropto M^{3}.
  29. κ \kappa
  30. d P r a d d r = - κ ρ c L 4 π r 2 , \frac{dP_{rad}}{dr}=-\frac{\kappa\rho}{c}\frac{L}{4\pi r^{2}},
  31. 1 / κ ρ = l 1/\kappa\rho=l
  32. P r a d = 4 σ 3 c T I 4 P_{rad}=\frac{4\sigma}{3c}{T_{I}}^{4}
  33. T I 3 d T I d r = - 3 κ ρ 16 σ L 4 π r 2 , {T_{I}}^{3}\frac{dT_{I}}{dr}=-\frac{3\kappa\rho}{16\sigma}\frac{L}{4\pi r^{2}},
  34. L T I 4 R ρ T I 4 R 4 M L\varpropto{T_{I}}^{4}\frac{R}{\rho}\varpropto{T_{I}}^{4}\frac{R^{4}}{M}
  35. T I M R T_{I}\varpropto\frac{M}{R}
  36. T I - T E T_{I}-T_{E}
  37. L M 3 . L\varpropto M^{3}.
  38. T I 4 M 2 R 4 , {T_{I}}^{4}\varpropto\frac{M^{2}}{R^{4}},
  39. L M . L\varpropto M.

Matching_polynomial.html

  1. m G ( x ) := k 0 m k x k . m_{G}(x):=\sum_{k\geq 0}m_{k}x^{k}.
  2. M G ( x ) := k 0 ( - 1 ) k m k x n - 2 k . M_{G}(x):=\sum_{k\geq 0}(-1)^{k}m_{k}x^{n-2k}.
  3. μ G ( x , y ) := k 0 m k x k y n - 2 k . \mu_{G}(x,y):=\sum_{k\geq 0}m_{k}x^{k}y^{n-2k}.
  4. M G ( x ) = x n m G ( - x - 2 ) M_{G}(x)=x^{n}m_{G}(-x^{-2})
  5. μ G ( x , y ) = y n m G ( x / y 2 ) . \mu_{G}(x,y)=y^{n}m_{G}(x/y^{2}).
  6. M K m , n ( x ) = n ! L n ( m - n ) ( x 2 ) . M_{K_{m,n}}(x)=n!L_{n}^{(m-n)}(x^{2}).\,
  7. M K n ( x ) = H n ( x ) , M_{K_{n}}(x)=H_{n}(x),\,

Matérn_covariance_function.html

  1. C ( d ) = σ 2 1 Γ ( ν ) 2 ν - 1 ( 2 ν d ρ ) ν K ν ( 2 ν d ρ ) , C(d)=\sigma^{2}\frac{1}{\Gamma(\nu)2^{\nu-1}}\Bigg(\sqrt{2\nu}\frac{d}{\rho}% \Bigg)^{\nu}K_{\nu}\Bigg(\sqrt{2\nu}\frac{d}{\rho}\Bigg),
  2. ν - 1 \lfloor\nu-1\rfloor
  3. ν \nu\rightarrow\infty
  4. C ( d ) = σ 2 exp ( - d 2 / 2 ρ 2 ) . C(d)=\sigma^{2}\exp(-d^{2}/2\rho^{2}).
  5. ν = 1 / 2 \nu=1/2
  6. C ( d ) = σ 2 exp ( - d ρ ) ν = 1 2 , C(d)=\sigma^{2}\exp\Bigg(-\frac{d}{\rho}\Bigg)\quad\quad\nu=\tfrac{1}{2},
  7. C ( d ) = σ 2 ( 1 + 3 d ρ ) exp ( - 3 d ρ ) ν = 3 2 , C(d)=\sigma^{2}\Bigg(1+\frac{\sqrt{3}d}{\rho}\Bigg)\exp\Bigg(-\frac{\sqrt{3}d}% {\rho}\Bigg)\quad\quad\nu=\tfrac{3}{2},
  8. C ( d ) = σ 2 ( 1 + 5 d ρ + 5 d 2 3 ρ 2 ) exp ( - 5 d ρ ) ν = 5 2 . C(d)=\sigma^{2}\Bigg(1+\frac{\sqrt{5}d}{\rho}+\frac{5d^{2}}{3\rho^{2}}\Bigg)% \exp\Bigg(-\frac{\sqrt{5}d}{\rho}\Bigg)\quad\quad\nu=\tfrac{5}{2}.

Mathematical_exposure_modeling.html

  1. 0 T g ( t ) d t \int_{0}^{T}g(t)\,dt
  2. 0 T w x ( t ) d t \int_{0}^{T}wx(t)\,dt
  3. i = 1 m \sum_{i=1}^{m}