wpmath0000008_7

Linear-quadratic_regulator.html

  1. t [ t 0 , t 1 ] t\in[t_{0},t_{1}]
  2. x ˙ = A x + B u \dot{x}=Ax+Bu
  3. J = 1 2 x T ( t 1 ) F ( t 1 ) x ( t 1 ) + t 0 t 1 ( x T Q x + u T R u + 2 x T N u ) d t J=\frac{1}{2}x^{T}(t_{1})F(t_{1})x(t_{1})+\int\limits_{t_{0}}^{t_{1}}\left(x^{% T}Qx+u^{T}Ru+2x^{T}Nu\right)dt
  4. u = - K x u=-Kx\,
  5. K K
  6. K = R - 1 ( B T P ( t ) + N T ) K=R^{-1}(B^{T}P(t)+N^{T})\,
  7. P P
  8. A T P ( t ) + P ( t ) A - ( P ( t ) B + N ) R - 1 ( B T P ( t ) + N T ) + Q = - P ˙ ( t ) A^{T}P(t)+P(t)A-(P(t)B+N)R^{-1}(B^{T}P(t)+N^{T})+Q=-\dot{P}(t)\,
  9. P ( t 1 ) = F ( t 1 ) . P(t_{1})=F(t_{1}).
  10. x ˙ = A x + B u \dot{x}=Ax+Bu
  11. - λ ˙ = Q x + N u + A T λ -\dot{\lambda}=Qx+Nu+A^{T}\lambda
  12. 0 = R u + N T x + B T λ 0=Ru+N^{T}x+B^{T}\lambda
  13. x ( t 0 ) = x 0 x(t_{0})=x_{0}
  14. λ ( t 1 ) = F ( t 1 ) x ( t 1 ) \lambda(t_{1})=F(t_{1})x(t_{1})
  15. x ˙ = A x + B u \dot{x}=Ax+Bu
  16. J = 0 ( x T Q x + u T R u + 2 x T N u ) d t J=\int_{0}^{\infty}\left(x^{T}Qx+u^{T}Ru+2x^{T}Nu\right)dt
  17. u = - K x u=-Kx\,
  18. K K
  19. K = R - 1 ( B T P + N T ) K=R^{-1}(B^{T}P+N^{T})\,
  20. P P
  21. A T P + P A - ( P B + N ) R - 1 ( B T P + N T ) + Q = 0 A^{T}P+PA-(PB+N)R^{-1}(B^{T}P+N^{T})+Q=0\,
  22. 𝒜 T P + P 𝒜 - P B R - 1 B T P + 𝒬 = 0 \mathcal{A}^{T}P+P\mathcal{A}-PBR^{-1}B^{T}P+\mathcal{Q}=0\,
  23. 𝒜 = A - B R - 1 N T 𝒬 = Q - N R - 1 N T \mathcal{A}=A-BR^{-1}N^{T}\qquad\mathcal{Q}=Q-NR^{-1}N^{T}\,
  24. x k + 1 = A x k + B u k x_{k+1}=Ax_{k}+Bu_{k}\,
  25. J = k = 0 N ( x k T Q x k + u k T R u k + 2 x k T N u k ) J=\sum\limits_{k=0}^{N}\left(x_{k}^{T}Qx_{k}+u_{k}^{T}Ru_{k}+2x_{k}^{T}Nu_{k}\right)
  26. u k = - F k x k u_{k}=-F_{k}x_{k}\,
  27. F k = ( R + B T P k B ) - 1 ( B T P k A + N T ) F_{k}=(R+B^{T}P_{k}B)^{-1}(B^{T}P_{k}A+N^{T})\,
  28. P k P_{k}
  29. P k - 1 = A T P k A - ( A T P k B + N ) ( R + B T P k B ) - 1 ( B T P k A + N T ) + Q P_{k-1}=A^{T}P_{k}A-(A^{T}P_{k}B+N)\left(R+B^{T}P_{k}B\right)^{-1}(B^{T}P_{k}A% +N^{T})+Q
  30. P N = Q P_{N}=Q
  31. u N u_{N}
  32. x x
  33. x N x_{N}
  34. A x N - 1 + B u N - 1 Ax_{N-1}+Bu_{N-1}
  35. x k + 1 = A x k + B u k x_{k+1}=Ax_{k}+Bu_{k}\,
  36. J = k = 0 ( x k T Q x k + u k T R u k + 2 x k T N u k ) J=\sum\limits_{k=0}^{\infty}\left(x_{k}^{T}Qx_{k}+u_{k}^{T}Ru_{k}+2x_{k}^{T}Nu% _{k}\right)
  37. u k = - F x k u_{k}=-Fx_{k}\,
  38. F = ( R + B T P B ) - 1 ( B T P A + N T ) F=(R+B^{T}PB)^{-1}(B^{T}PA+N^{T})\,
  39. P P
  40. P = A T P A - ( A T P B + N ) ( R + B T P B ) - 1 ( B T P A + N T ) + Q P=A^{T}PA-(A^{T}PB+N)\left(R+B^{T}PB\right)^{-1}(B^{T}PA+N^{T})+Q
  41. P = 𝒜 T P 𝒜 - 𝒜 T P B ( R + B T P B ) - 1 B T P 𝒜 + 𝒬 P=\mathcal{A}^{T}P\mathcal{A}-\mathcal{A}^{T}PB\left(R+B^{T}PB\right)^{-1}B^{T% }P\mathcal{A}+\mathcal{Q}
  42. 𝒜 = A - B R - 1 N T 𝒬 = Q - N R - 1 N T \mathcal{A}=A-BR^{-1}N^{T}\qquad\mathcal{Q}=Q-NR^{-1}N^{T}

Linear_circuit.html

  1. F ( a x 1 + b x 2 ) = a F ( x 1 ) + b F ( x 2 ) F(ax_{1}+bx_{2})=aF(x_{1})+bF(x_{2})\,

Linear_matrix_inequality.html

  1. LMI ( y ) := A 0 + y 1 A 1 + y 2 A 2 + + y m A m 0 \operatorname{LMI}(y):=A_{0}+y_{1}A_{1}+y_{2}A_{2}+\cdots+y_{m}A_{m}\geq 0\,
  2. y = [ y i , i = 1 , , m ] y=[y_{i}\,,~{}i\!=\!1,\dots,m]
  3. A 0 , A 1 , A 2 , , A m A_{0},A_{1},A_{2},\dots,A_{m}
  4. n × n n\times n
  5. 𝕊 n \mathbb{S}^{n}
  6. B 0 B\geq 0
  7. B B
  8. 𝕊 + \mathbb{S}_{+}
  9. 𝕊 \mathbb{S}

Linear_programming_relaxation.html

  1. x i { 0 , 1 } x_{i}\in\{0,1\}
  2. 0 x i 1. 0\leq x_{i}\leq 1.
  3. x i { 0 , 1 } \textstyle x_{i}\in\{0,1\}
  4. { i e j S i } x i 1 \textstyle\sum_{\{i\mid e_{j}\in S_{i}\}}x_{i}\geq 1
  5. min i x i . \textstyle\min\sum_{i}x_{i}.
  6. M i n t M_{int}
  7. M f r a c M_{frac}
  8. I G = M i n t M f r a c IG=\frac{M_{int}}{M_{frac}}
  9. M f r a c M_{frac}
  10. R R M f r a c RR\cdot M_{frac}
  11. M i n t = I G M f r a c M_{int}=IG\cdot M_{frac}
  12. R R I G RR\geq IG

Line–line_intersection.html

  1. L 1 L_{1}\,
  2. L 2 L_{2}\,
  3. L 1 L_{1}\,
  4. ( x 1 , y 1 ) (x_{1},y_{1})\,
  5. ( x 2 , y 2 ) (x_{2},y_{2})\,
  6. L 2 L_{2}\,
  7. ( x 3 , y 3 ) (x_{3},y_{3})\,
  8. ( x 4 , y 4 ) (x_{4},y_{4})\,
  9. P P\,
  10. L 1 L_{1}\,
  11. L 2 L_{2}\,
  12. P x = | | x 1 y 1 x 2 y 2 | | x 1 1 x 2 1 | | x 3 y 3 x 4 y 4 | | x 3 1 x 4 1 | | | | x 1 1 x 2 1 | | y 1 1 y 2 1 | | x 3 1 x 4 1 | | y 3 1 y 4 1 | | P y = | | x 1 y 1 x 2 y 2 | | y 1 1 y 2 1 | | x 3 y 3 x 4 y 4 | | y 3 1 y 4 1 | | | | x 1 1 x 2 1 | | y 1 1 y 2 1 | | x 3 1 x 4 1 | | y 3 1 y 4 1 | | P_{x}=\frac{\begin{vmatrix}\begin{vmatrix}x_{1}&y_{1}\\ x_{2}&y_{2}\end{vmatrix}&\begin{vmatrix}x_{1}&1\\ x_{2}&1\end{vmatrix}\\ \\ \begin{vmatrix}x_{3}&y_{3}\\ x_{4}&y_{4}\end{vmatrix}&\begin{vmatrix}x_{3}&1\\ x_{4}&1\end{vmatrix}\end{vmatrix}}{\begin{vmatrix}\begin{vmatrix}x_{1}&1\\ x_{2}&1\end{vmatrix}&\begin{vmatrix}y_{1}&1\\ y_{2}&1\end{vmatrix}\\ \\ \begin{vmatrix}x_{3}&1\\ x_{4}&1\end{vmatrix}&\begin{vmatrix}y_{3}&1\\ y_{4}&1\end{vmatrix}\end{vmatrix}}\,\!\qquad P_{y}=\frac{\begin{vmatrix}\begin% {vmatrix}x_{1}&y_{1}\\ x_{2}&y_{2}\end{vmatrix}&\begin{vmatrix}y_{1}&1\\ y_{2}&1\end{vmatrix}\\ \\ \begin{vmatrix}x_{3}&y_{3}\\ x_{4}&y_{4}\end{vmatrix}&\begin{vmatrix}y_{3}&1\\ y_{4}&1\end{vmatrix}\end{vmatrix}}{\begin{vmatrix}\begin{vmatrix}x_{1}&1\\ x_{2}&1\end{vmatrix}&\begin{vmatrix}y_{1}&1\\ y_{2}&1\end{vmatrix}\\ \\ \begin{vmatrix}x_{3}&1\\ x_{4}&1\end{vmatrix}&\begin{vmatrix}y_{3}&1\\ y_{4}&1\end{vmatrix}\end{vmatrix}}\,\!
  13. ( P x , P y ) = ( \displaystyle(P_{x},P_{y})=\bigg(
  14. ( x 1 - x 2 ) ( y 3 - y 4 ) - ( y 1 - y 2 ) ( x 3 - x 4 ) = 0 if the lines are parallel (x_{1}-x_{2})(y_{3}-y_{4})-(y_{1}-y_{2})(x_{3}-x_{4})=0\,\text{ if the lines % are parallel}
  15. x x
  16. y y
  17. y = a x + c y=ax+c
  18. y = b x + d y=bx+d
  19. a a
  20. b b
  21. c c
  22. d d
  23. y y
  24. a x + c = b x + d ax+c=bx+d
  25. x x
  26. a x - b x = d - c ax-bx=d-c
  27. x = d - c a - b x=\frac{d-c}{a-b}
  28. y = a d - c a - b + c y=a\frac{d-c}{a-b}+c
  29. P ( d - c a - b , a d - c a - b + c ) = P ( d - c a - b , a d - b c a - b ) P\left(\frac{d-c}{a-b},a\frac{d-c}{a-b}+c\right)=P\left(\frac{d-c}{a-b},\frac{% ad-bc}{a-b}\right)
  30. a X + b Y + c W = 0 L ( a , b , c ) P ( X , Y , W ) = 0 aX+bY+cW=0\Rightarrow L(a,b,c)\cdot P(X,Y,W)=0
  31. L ( a 1 , b 1 , c 1 ) × L ( a 2 , b 2 , c 2 ) = P ( X , Y , W ) L(a_{1},b_{1},c_{1})\times L(a_{2},b_{2},c_{2})=P(X,Y,W)
  32. P ( x 1 , y 1 , w 1 ) × P ( x 2 , y 2 , w 2 ) = L ( a , b , c ) P(x_{1},y_{1},w_{1})\times P(x_{2},y_{2},w_{2})=L(a,b,c)
  33. ( a i 1 a i 2 ) ( x y ) T = b i , (a_{i1}\quad a_{i2})(x\quad y)^{T}=b_{i},
  34. A w = b , Aw=b,
  35. ( a i 1 , a i 2 ) (a_{i1},a_{i2})
  36. w = A g b = ( A T A ) - 1 A T b , w=A^{g}b=(A^{T}A)^{-1}A^{T}b,
  37. A g A^{g}
  38. A A
  39. ( a i 1 a i 2 a i 3 ) ( x y z ) T = b i . (a_{i1}\quad a_{i2}\quad a_{i3})(x\quad y\quad z)^{T}=b_{i}.
  40. A w = b Aw=b
  41. w = ( A T A ) - 1 A T b . w=(A^{T}A)^{-1}A^{T}b.
  42. p i p_{i}
  43. n ^ i \hat{n}_{i}
  44. x 1 x_{1}
  45. x 2 x_{2}
  46. p 1 = x 1 p_{1}=x_{1}
  47. n ^ 1 := [ 0 - 1 1 0 ] ( x 2 - x 1 ) / x 2 - x 1 \hat{n}_{1}:=\begin{bmatrix}0&-1\\ 1&0\end{bmatrix}(x_{2}-x_{1})/\|x_{2}-x_{1}\|
  48. ( p , n ^ ) (p,\hat{n})
  49. d ( x , ( p , n ) ) = ( x - p ) n ^ = ( x - p ) n ^ = ( x - p ) n ^ n ^ ( x - p ) . d(x,(p,n))=\|(x-p)\cdot\hat{n}\|=\|(x-p)^{\top}\hat{n}\|=\sqrt{(x-p)^{\top}% \hat{n}\hat{n}^{\top}(x-p)}.
  50. d ( x , ( p , n ) ) 2 = ( x - p ) ( n ^ n ^ ) ( x - p ) . d(x,(p,n))^{2}=(x-p)^{\top}(\hat{n}\hat{n}^{\top})(x-p).
  51. E ( x ) = i ( x - p i ) ( n ^ i n ^ i ) ( x - p i ) . E(x)=\sum_{i}(x-p_{i})^{\top}(\hat{n}_{i}\hat{n}_{i}^{\top})(x-p_{i}).
  52. E ( x ) \displaystyle E(x)
  53. E ( x ) x = 0 = 2 ( i n ^ i n ^ i ) x - 2 ( i n ^ i n ^ i p i ) \frac{\partial E(x)}{\partial x}=0=2\left(\sum_{i}\hat{n}_{i}\hat{n}_{i}^{\top% }\right)x-2\left(\sum_{i}\hat{n}_{i}\hat{n}_{i}^{\top}p_{i}\right)
  54. ( i n ^ i n ^ i ) x = i n ^ i n ^ i p i \left(\sum_{i}\hat{n}_{i}\hat{n}_{i}^{\top}\right)x=\sum_{i}\hat{n}_{i}\hat{n}% _{i}^{\top}p_{i}
  55. x = ( i n ^ i n ^ i ) - 1 ( i n ^ i n ^ i p i ) . x=\left(\sum_{i}\hat{n}_{i}\hat{n}_{i}^{\top}\right)^{-1}\left(\sum_{i}\hat{n}% _{i}\hat{n}_{i}^{\top}p_{i}\right).
  56. n ^ i \hat{n}_{i}
  57. n ^ i n ^ i \hat{n}_{i}\hat{n}_{i}^{\top}
  58. p i p_{i}
  59. v ^ i \hat{v}_{i}
  60. n ^ i n ^ i \hat{n}_{i}\hat{n}_{i}^{\top}
  61. I - v ^ i v ^ i I-\hat{v}_{i}\hat{v}_{i}^{\top}
  62. x = ( i I - v ^ i v ^ i ) - 1 ( i ( I - v ^ i v ^ i ) p i ) . x=\left(\sum_{i}I-\hat{v}_{i}\hat{v}_{i}^{\top}\right)^{-1}\left(\sum_{i}(I-% \hat{v}_{i}\hat{v}_{i}^{\top})p_{i}\right).

Line–sphere_intersection.html

  1. 𝐱 - 𝐜 2 = r 2 \left\|\mathbf{x}-\mathbf{c}\right\|^{2}=r^{2}
  2. 𝐜 \mathbf{c}
  3. r r
  4. 𝐱 \mathbf{x}
  5. 𝐨 \mathbf{o}
  6. 𝐱 = 𝐨 + d 𝐥 \mathbf{x}=\mathbf{o}+d\mathbf{l}
  7. d d
  8. 𝐥 \mathbf{l}
  9. 𝐨 \mathbf{o}
  10. 𝐱 \mathbf{x}
  11. d d
  12. 𝐨 + d 𝐥 - 𝐜 2 = r 2 ( 𝐨 + d 𝐥 - 𝐜 ) ( 𝐨 + d 𝐥 - 𝐜 ) = r 2 \left\|\mathbf{o}+d\mathbf{l}-\mathbf{c}\right\|^{2}=r^{2}\Leftrightarrow(% \mathbf{o}+d\mathbf{l}-\mathbf{c})\cdot(\mathbf{o}+d\mathbf{l}-\mathbf{c})=r^{2}
  13. d 2 ( 𝐥 𝐥 ) + 2 d ( 𝐥 ( 𝐨 - 𝐜 ) ) + ( 𝐨 - 𝐜 ) ( 𝐨 - 𝐜 ) = r 2 d^{2}(\mathbf{l}\cdot\mathbf{l})+2d(\mathbf{l}\cdot(\mathbf{o}-\mathbf{c}))+(% \mathbf{o}-\mathbf{c})\cdot(\mathbf{o}-\mathbf{c})=r^{2}
  14. d 2 ( 𝐥 𝐥 ) + 2 d ( 𝐥 ( 𝐨 - 𝐜 ) ) + ( 𝐨 - 𝐜 ) ( 𝐨 - 𝐜 ) - r 2 = 0 d^{2}(\mathbf{l}\cdot\mathbf{l})+2d(\mathbf{l}\cdot(\mathbf{o}-\mathbf{c}))+(% \mathbf{o}-\mathbf{c})\cdot(\mathbf{o}-\mathbf{c})-r^{2}=0
  15. a d 2 + b d + c = 0 ad^{2}+bd+c=0
  16. a = 𝐥 𝐥 = 𝐥 2 a=\mathbf{l}\cdot\mathbf{l}=\left\|\mathbf{l}\right\|^{2}
  17. b = 2 ( 𝐥 ( 𝐨 - 𝐜 ) ) b=2(\mathbf{l}\cdot(\mathbf{o}-\mathbf{c}))
  18. c = ( 𝐨 - 𝐜 ) ( 𝐨 - 𝐜 ) - r 2 = 𝐨 - 𝐜 2 - r 2 c=(\mathbf{o}-\mathbf{c})\cdot(\mathbf{o}-\mathbf{c})-r^{2}=\left\|\mathbf{o}-% \mathbf{c}\right\|^{2}-r^{2}
  19. d = - ( 𝐥 ( 𝐨 - 𝐜 ) ) ± ( 𝐥 ( 𝐨 - 𝐜 ) ) 2 - 𝐥 2 ( 𝐨 - 𝐜 2 - r 2 ) 𝐥 2 d=\frac{-(\mathbf{l}\cdot(\mathbf{o}-\mathbf{c}))\pm\sqrt{(\mathbf{l}\cdot(% \mathbf{o}-\mathbf{c}))^{2}-\left\|\mathbf{l}\right\|^{2}(\left\|\mathbf{o}-% \mathbf{c}\right\|^{2}-r^{2})}}{\left\|\mathbf{l}\right\|^{2}}
  20. 𝐥 \mathbf{l}
  21. 𝐥 2 = 1 \left\|\mathbf{l}\right\|^{2}=1
  22. d = - ( 𝐥 ( 𝐨 - 𝐜 ) ) ± ( 𝐥 ( 𝐨 - 𝐜 ) ) 2 - 𝐨 - 𝐜 2 + r 2 d=-(\mathbf{l}\cdot(\mathbf{o}-\mathbf{c}))\pm\sqrt{(\mathbf{l}\cdot(\mathbf{o% }-\mathbf{c}))^{2}-\left\|\mathbf{o}-\mathbf{c}\right\|^{2}+r^{2}}
  23. ( 𝐥 ( 𝐨 - 𝐜 ) ) 2 - 𝐨 - 𝐜 2 + r 2 (\mathbf{l}\cdot(\mathbf{o}-\mathbf{c}))^{2}-\left\|\mathbf{o}-\mathbf{c}% \right\|^{2}+r^{2}

Lipschitz_domain.html

  1. B r ( p ) := { x n | x - p < r } B_{r}(p):=\{x\in\mathbb{R}^{n}|\|x-p\|<r\}
  2. Q 0 := { ( x 1 , , x n ) Q | x n = 0 } ; Q_{0}:=\{(x_{1},\dots,x_{n})\in Q|x_{n}=0\};
  3. Q + := { ( x 1 , , x n ) Q | x n > 0 } . Q_{+}:=\{(x_{1},\dots,x_{n})\in Q|x_{n}>0\}.

Liquid_bubble.html

  1. f 0 = 1 2 π R 0 3 γ p 0 ρ f_{0}={1\over 2\pi R_{0}}\sqrt{3\gamma p_{0}\over\rho}
  2. γ \gamma
  3. R 0 R_{0}
  4. p 0 p_{0}
  5. ρ \rho
  6. f 0 = 1 2 π R 0 3 p 0 ρ + 4 σ ρ R 0 f_{0}={1\over 2\pi R_{0}}\sqrt{{3p_{0}\over\rho}+{4\sigma\over\rho R_{0}}}

List_of_centroids.html

  1. X X
  2. n n
  3. X X
  4. X X
  5. x ¯ \bar{x}
  6. y ¯ \bar{y}
  7. b 3 \frac{b}{3}
  8. h 3 \frac{h}{3}
  9. b h 2 \frac{bh}{2}
  10. 4 r 3 π \frac{4r}{3\pi}
  11. 4 r 3 π \frac{4r}{3\pi}
  12. π r 2 4 \frac{\pi r^{2}}{4}
  13. 0 \,\!0
  14. 4 r 3 π \frac{4r}{3\pi}
  15. π r 2 2 \frac{\pi r^{2}}{2}
  16. 4 a 3 π \frac{4a}{3\pi}
  17. 4 b 3 π \frac{4b}{3\pi}
  18. π a b 4 \frac{\pi ab}{4}
  19. 0 \,\!0
  20. 4 b 3 π \frac{4b}{3\pi}
  21. π a b 2 \frac{\pi ab}{2}
  22. y = h b 2 x 2 y=\frac{h}{b^{2}}x^{2}
  23. y \,\!y
  24. x = 0 \,\!x=0
  25. x = b \,\!x=b
  26. 3 b 8 \frac{3b}{8}
  27. 3 h 5 \frac{3h}{5}
  28. 2 b h 3 \frac{2bh}{3}
  29. y = h b 2 x 2 \,\!y=\frac{h}{b^{2}}x^{2}
  30. y = h \,\!y=h
  31. 0 \,\!0
  32. 3 h 5 \frac{3h}{5}
  33. 4 b h 3 \frac{4bh}{3}
  34. y = h b 2 x 2 \,\!y=\frac{h}{b^{2}}x^{2}
  35. x \,\!x
  36. x = 0 \,\!x=0
  37. x = b \,\!x=b
  38. 3 b 4 \frac{3b}{4}
  39. 3 h 10 \frac{3h}{10}
  40. b h 3 \frac{bh}{3}
  41. y = h b n x n y=\frac{h}{b^{n}}x^{n}
  42. x \,\!x
  43. x = 0 \,\!x=0
  44. x = b \,\!x=b
  45. n + 1 n + 2 b \frac{n+1}{n+2}b
  46. n + 1 4 n + 2 h \frac{n+1}{4n+2}h
  47. b h n + 1 \frac{bh}{n+1}
  48. r = ρ \,\!r=\rho
  49. θ = - α \,\!\theta=-\alpha
  50. θ = α \,\!\theta=\alpha
  51. 2 ρ sin ( α ) 3 α \frac{2\rho\sin(\alpha)}{3\alpha}
  52. 0 \,\!0
  53. α ρ 2 \,\!\alpha\rho^{2}
  54. 0 \,\!0
  55. 4 R sin 3 θ 2 3 ( θ - sin θ ) \frac{4R\sin^{3}{\frac{\theta}{2}}}{3(\theta-\sin{\theta})}
  56. R 2 2 ( θ - s i n θ ) \frac{R^{2}}{2}(\theta-sin{\theta})
  57. x 2 + y 2 = r 2 \,\!x^{2}+y^{2}=r^{2}
  58. 2 r π \frac{2r}{\pi}
  59. 2 r π \frac{2r}{\pi}
  60. L = π r 2 L=\frac{\pi r}{2}
  61. x 2 + y 2 = r 2 \,\!x^{2}+y^{2}=r^{2}
  62. x \,\!x
  63. 0 \,\!0
  64. 2 r π \frac{2r}{\pi}
  65. L = π r L=\,\!\pi r
  66. r = ρ \,\!r=\rho
  67. θ = - α \,\!\theta=-\alpha
  68. θ = α \,\!\theta=\alpha
  69. ρ sin ( α ) α \frac{\rho\sin(\alpha)}{\alpha}
  70. 0 \,\!0
  71. L = 2 α ρ L=\,\!2\alpha\rho

List_of_common_physics_notations.html

  1. A A
  2. 𝐚 \mathbf{a}
  3. 𝐁 \mathbf{B}
  4. C C
  5. c c
  6. 𝐃 \mathbf{D}
  7. D D
  8. d d
  9. d x dx
  10. d 𝐀 d\mathbf{A}
  11. d V dV
  12. 𝐄 \mathbf{E}
  13. E E
  14. e e
  15. 𝐅 \mathbf{F}
  16. f f
  17. G G
  18. g g
  19. 𝐇 \mathbf{H}
  20. H H
  21. h h
  22. \hbar
  23. ( h 2 π ) \textstyle\left({\frac{h}{2\pi}}\right)
  24. I I
  25. i i
  26. 𝐢 ^ \mathbf{\hat{i}}
  27. 𝐉 \mathbf{J}
  28. 𝐣 ^ \mathbf{\hat{j}}
  29. K K
  30. k k
  31. 𝐤 ^ \mathbf{\hat{k}}
  32. L L
  33. l l
  34. M M
  35. m m
  36. N N
  37. n n
  38. P P
  39. 𝐩 \mathbf{p}
  40. Q Q
  41. q q
  42. R R
  43. 𝐫 \mathbf{r}
  44. r r
  45. S S
  46. s s
  47. T T
  48. t t
  49. 𝐔 \mathbf{U}
  50. U U
  51. u u
  52. V V
  53. 𝐯 \mathbf{v}
  54. W W
  55. w w
  56. x x
  57. Z Z
  58. α \alpha
  59. β \beta
  60. γ \gamma
  61. Δ \Delta
  62. Δ x \Delta x
  63. δ \delta
  64. ϵ \epsilon
  65. ζ \zeta
  66. η \eta
  67. θ \theta
  68. K K
  69. Λ \Lambda
  70. λ \lambda
  71. μ \mathbf{\mu}
  72. ν \nu
  73. π \pi
  74. ρ \rho
  75. Ω \Omega
  76. Σ \Sigma
  77. σ \sigma
  78. τ \tau
  79. Φ \Phi
  80. ϕ \phi
  81. Ψ \Psi
  82. ω \omega
  83. \nabla\cdot
  84. × \nabla\times
  85. \nabla
  86. \partial
  87. y / x \partial y/\partial x
  88. \Box
  89. 2 - t 2 \nabla^{2}-\partial_{t}^{2}

List_of_earthquakes_in_the_British_Isles.html

  1. M L M_{L}
  2. M L M_{L}
  3. M L M_{L}
  4. M L M_{L}
  5. M L M_{L}

List_of_fallacies.html

  1. P ¬ P P\lor\neg P
  2. P ¬ P P\lor\neg P
  3. P P

List_of_formulas_in_Riemannian_geometry.html

  1. Γ k i j = 1 2 ( x j g k i + x i g k j - x k g i j ) = 1 2 ( g k i , j + g k j , i - g i j , k ) , \Gamma_{kij}=\frac{1}{2}\left(\frac{\partial}{\partial x^{j}}g_{ki}+\frac{% \partial}{\partial x^{i}}g_{kj}-\frac{\partial}{\partial x^{k}}g_{ij}\right)=% \frac{1}{2}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,,
  2. Γ m i j = g m k Γ k i j = 1 2 g m k ( x j g k i + x i g k j - x k g i j ) = 1 2 g m k ( g k i , j + g k j , i - g i j , k ) . \begin{aligned}\displaystyle\Gamma^{m}{}_{ij}&\displaystyle=g^{mk}\Gamma_{kij}% \\ &\displaystyle=\frac{1}{2}\,g^{mk}\left(\frac{\partial}{\partial x^{j}}g_{ki}+% \frac{\partial}{\partial x^{i}}g_{kj}-\frac{\partial}{\partial x^{k}}g_{ij}% \right)=\frac{1}{2}\,g^{mk}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,.\end{aligned}
  3. g i j g^{ij}
  4. g i j g_{ij}
  5. δ i = j g i k g k j \delta^{i}{}_{j}=g^{ik}g_{kj}
  6. n = δ i = i g i = i g i j g i j n=\delta^{i}{}_{i}=g^{i}{}_{i}=g^{ij}g_{ij}
  7. Γ k i j = Γ k j i \Gamma_{kij}=\Gamma_{kji}
  8. Γ i = j k Γ i k j \Gamma^{i}{}_{jk}=\Gamma^{i}{}_{kj}
  9. Γ i = k i 1 2 g i m g i m x k = 1 2 g g x k = log | g | x k \Gamma^{i}{}_{ki}=\frac{1}{2}g^{im}\frac{\partial g_{im}}{\partial x^{k}}=% \frac{1}{2g}\frac{\partial g}{\partial x^{k}}=\frac{\partial\log\sqrt{|g|}}{% \partial x^{k}}
  10. g k Γ i = k - 1 | g | ( | g | g i k ) x k g^{k\ell}\Gamma^{i}{}_{k\ell}=\frac{-1}{\sqrt{|g|}}\;\frac{\partial\left(\sqrt% {|g|}\,g^{ik}\right)}{\partial x^{k}}
  11. g i k g_{ik}
  12. v i v^{i}
  13. v i = ; j j v i = v i x j + Γ i v k j k v^{i}{}_{;j}=\nabla_{j}v^{i}=\frac{\partial v^{i}}{\partial x^{j}}+\Gamma^{i}{% }_{jk}v^{k}
  14. ( 0 , 1 ) (0,1)
  15. v i v_{i}
  16. v i ; j = j v i = v i x j - Γ k v k i j v_{i;j}=\nabla_{j}v_{i}=\frac{\partial v_{i}}{\partial x^{j}}-\Gamma^{k}{}_{ij% }v_{k}
  17. ( 2 , 0 ) (2,0)
  18. v i j v^{ij}
  19. v i j = ; k k v i j = v i j x k + Γ i v j k + Γ j v i k v^{ij}{}_{;k}=\nabla_{k}v^{ij}=\frac{\partial v^{ij}}{\partial x^{k}}+\Gamma^{% i}{}_{k\ell}v^{\ell j}+\Gamma^{j}{}_{k\ell}v^{i\ell}
  20. ϕ \phi
  21. i ϕ = ϕ ; i = ϕ , i = ϕ x i \nabla_{i}\phi=\phi_{;i}=\phi_{,i}=\frac{\partial\phi}{\partial x^{i}}
  22. k g i j = k g i j = 0 \nabla_{k}g_{ij}=\nabla_{k}g^{ij}=0
  23. k | g | = 0 \nabla_{k}\sqrt{|g|}=0
  24. X ( t ) X(t)
  25. v i v^{i}
  26. X ( t ) i = t v i - t 2 2 Γ i v j j k v k + O ( t 3 ) X(t)^{i}=tv^{i}-\frac{t^{2}}{2}\Gamma^{i}{}_{jk}v^{j}v^{k}+O(t^{3})
  27. R ( U , V ) W = U V W - V U W - [ U , V ] W R(U,V)W=\nabla_{U}\nabla_{V}W-\nabla_{V}\nabla_{U}W-\nabla_{[U,V]}W
  28. ( 1 , 3 ) (1,3)
  29. ( R ( U , V ) W ) = R W i i j k U j V k (R(U,V)W)^{\ell}=R^{\ell}{}_{ijk}W^{i}U^{j}V^{k}
  30. R = i j k x j Γ - i k x k Γ + i j Γ Γ i k s j s - Γ Γ s k s i j R^{\ell}{}_{ijk}=\frac{\partial}{\partial x^{j}}\Gamma^{\ell}{}_{ik}-\frac{% \partial}{\partial x^{k}}\Gamma^{\ell}{}_{ij}+\Gamma^{\ell}{}_{js}\Gamma_{ik}^% {s}-\Gamma^{\ell}{}_{ks}\Gamma^{s}{}_{ij}
  31. R i j k = g s R s i j k R_{\ell ijk}=g_{\ell s}R^{s}{}_{ijk}
  32. R i k m = 1 2 ( 2 g i m x k x + 2 g k x i x m - 2 g i x k x m - 2 g k m x i x ) + g n p ( Γ n Γ p k - i m Γ n Γ p k m ) i . R_{ik\ell m}=\frac{1}{2}\left(\frac{\partial^{2}g_{im}}{\partial x^{k}\partial x% ^{\ell}}+\frac{\partial^{2}g_{k\ell}}{\partial x^{i}\partial x^{m}}-\frac{% \partial^{2}g_{i\ell}}{\partial x^{k}\partial x^{m}}-\frac{\partial^{2}g_{km}}% {\partial x^{i}\partial x^{\ell}}\right)+g_{np}\left(\Gamma^{n}{}_{k\ell}% \Gamma^{p}{}_{im}-\Gamma^{n}{}_{km}\Gamma^{p}{}_{i\ell}\right).
  33. R i k m = R m i k R_{ik\ell m}=R_{\ell mik}
  34. R i k m = - R k i m = - R i k m . R_{ik\ell m}=-R_{ki\ell m}=-R_{ikm\ell}.
  35. R i k m + R i m k + R i m k = 0. R_{ik\ell m}+R_{imk\ell}+R_{i\ell mk}=0.
  36. m R n + i k R n + i m k k R n = i m 0 , \nabla_{m}R^{n}{}_{ik\ell}+\nabla_{\ell}R^{n}{}_{imk}+\nabla_{k}R^{n}{}_{i\ell m% }=0,
  37. R n + i k ; m R n + i m k ; R n = i m ; k 0 R^{n}{}_{ik\ell;m}+R^{n}{}_{imk;\ell}+R^{n}{}_{i\ell m;k}=0
  38. R i j = R = i j g m R i j m = g m R i m j = Γ i j x - Γ i x j + Γ Γ m i j - m Γ m Γ i . j m R_{ij}=R^{\ell}{}_{i\ell j}=g^{\ell m}R_{i\ell jm}=g^{\ell m}R_{\ell imj}=% \frac{\partial\Gamma^{\ell}{}_{ij}}{\partial x^{\ell}}-\frac{\partial\Gamma^{% \ell}{}_{i\ell}}{\partial x^{j}}+\Gamma^{\ell}{}_{ij}\Gamma^{m}{}_{\ell m}-% \Gamma^{m}{}_{i\ell}\Gamma^{\ell}{}_{jm}.
  39. R i j R_{ij}
  40. R i k = Γ i k x - Γ m Γ i - k m k ( x i ( log | g | ) ) . R_{ik}=\frac{\partial\Gamma^{\ell}{}_{ik}}{\partial x^{\ell}}-\Gamma^{m}{}_{i% \ell}\Gamma^{\ell}{}_{km}-\nabla_{k}\left(\frac{\partial}{\partial x^{i}}\left% (\log\sqrt{|g|}\right)\right).
  41. R = g i j R i j = g i j g m R i j m R=g^{ij}R_{ij}=g^{ij}g^{\ell m}R_{i\ell jm}
  42. R = m 1 2 m R , \nabla_{\ell}R^{\ell}{}_{m}={1\over 2}\nabla_{m}R,
  43. R = m ; 1 2 R ; m . R^{\ell}{}_{m;\ell}={1\over 2}R_{;m}.
  44. G a b = R a b - 1 2 g a b R G^{ab}=R^{ab}-{1\over 2}g^{ab}R
  45. a G a b = G a b = ; a 0. \nabla_{a}G^{ab}=G^{ab}{}_{;a}=0.
  46. C i k m = R i k m + 1 n - 2 ( - R i g k m + R i m g k + R k g i m - R k m g i ) + 1 ( n - 1 ) ( n - 2 ) R ( g i g k m - g i m g k ) , C_{ik\ell m}=R_{ik\ell m}+\frac{1}{n-2}\left(-R_{i\ell}g_{km}+R_{im}g_{k\ell}+% R_{k\ell}g_{im}-R_{km}g_{i\ell}\right)+\frac{1}{(n-1)(n-2)}R\left(g_{i\ell}g_{% km}-g_{im}g_{k\ell}\right),
  47. n n
  48. C i j k l + C k i j l + C j k i l = 0. C_{ijkl}+C_{kijl}+C_{jkil}=0.
  49. C i j k l = - C j i k l C i j k l = C k l i j , C_{ijkl}=-C_{jikl}\qquad C_{ijkl}=C_{klij},
  50. C i = j k i 0 C^{i}{}_{jki}=0
  51. C = 0 C=0
  52. M M
  53. n 4 n\geq 4
  54. M M
  55. d s 2 ds^{2}
  56. d s 2 = f 2 ( d x 1 2 + d x 2 2 + d x n 2 ) ds^{2}=f^{2}\left(dx_{1}^{2}+dx_{2}^{2}+\ldots dx_{n}^{2}\right)
  57. C i j k l C^{i}{}_{jkl}
  58. ϕ \phi
  59. i ϕ d x i \partial_{i}\phi dx^{i}
  60. i ϕ = ϕ ; i = g i k ϕ ; k = g i k ϕ , k = g i k k ϕ = g i k ϕ x k \nabla^{i}\phi=\phi^{;i}=g^{ik}\phi_{;k}=g^{ik}\phi_{,k}=g^{ik}\partial_{k}% \phi=g^{ik}\frac{\partial\phi}{\partial x^{k}}
  61. V m V^{m}
  62. m V m = V m x m + V k log | g | x k = 1 | g | ( V m | g | ) x m . \nabla_{m}V^{m}=\frac{\partial V^{m}}{\partial x^{m}}+V^{k}\frac{\partial\log% \sqrt{|g|}}{\partial x^{k}}=\frac{1}{\sqrt{|g|}}\frac{\partial(V^{m}\sqrt{|g|}% )}{\partial x^{m}}.
  63. f f
  64. Δ f \displaystyle\Delta f
  65. ( 2 , 0 ) (2,0)
  66. k A i k = 1 | g | ( A i k | g | ) x k . \nabla_{k}A^{ik}=\frac{1}{\sqrt{|g|}}\frac{\partial(A^{ik}\sqrt{|g|})}{% \partial x^{k}}.
  67. ϕ : M N \phi:M\rightarrow N
  68. ( ( d ϕ ) ) i j γ = 2 ϕ γ x i x j - M Γ k ϕ γ x k i j + N Γ γ ϕ α x i α β ϕ β x j . \left(\nabla\left(d\phi\right)\right)_{ij}^{\gamma}=\frac{\partial^{2}\phi^{% \gamma}}{\partial x^{i}\partial x^{j}}-^{M}\Gamma^{k}{}_{ij}\frac{\partial\phi% ^{\gamma}}{\partial x^{k}}+^{N}\Gamma^{\gamma}{}_{\alpha\beta}\frac{\partial% \phi^{\alpha}}{\partial x^{i}}\frac{\partial\phi^{\beta}}{\partial x^{j}}.
  69. h h
  70. k k
  71. h i j = h j i k i j = k j i h_{ij}=h_{ji}\qquad\qquad k_{ij}=k_{ji}
  72. h k h{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}k
  73. ( h k ) i j k l = h i k k j l + h j l k i k - h i l k j k - h j k k i l \left(h{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}k\right)_{ijkl}=h_{ik}k_{jl}+h_{jl}k_% {ik}-h_{il}k_{jk}-h_{jk}k_{il}
  74. h k = k h h{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}k=k{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}h
  75. g i j = δ i j g_{ij}=\delta_{ij}
  76. Γ i = j k 0 \Gamma^{i}{}_{jk}=0
  77. R i k m = 1 2 ( 2 g i m x k x + 2 g k x i x m - 2 g i x k x m - 2 g k m x i x ) R_{ik\ell m}=\frac{1}{2}\left(\frac{\partial^{2}g_{im}}{\partial x^{k}\partial x% ^{\ell}}+\frac{\partial^{2}g_{k\ell}}{\partial x^{i}\partial x^{m}}-\frac{% \partial^{2}g_{i\ell}}{\partial x^{k}\partial x^{m}}-\frac{\partial^{2}g_{km}}% {\partial x^{i}\partial x^{\ell}}\right)
  78. g g
  79. M M
  80. φ \varphi
  81. M M
  82. g ~ = e 2 φ g \tilde{g}=e^{2\varphi}g
  83. M M
  84. g ~ \tilde{g}
  85. g g
  86. g ~ \tilde{g}
  87. g g
  88. g ~ i j = e 2 φ g i j \tilde{g}_{ij}=e^{2\varphi}g_{ij}
  89. Γ ~ k = i j Γ k + i j δ i k j φ + δ j k i φ - g i j k φ \tilde{\Gamma}^{k}{}_{ij}=\Gamma^{k}{}_{ij}+\delta^{k}_{i}\partial_{j}\varphi+% \delta^{k}_{j}\partial_{i}\varphi-g_{ij}\nabla^{k}\varphi
  90. ~ F * X F * Y = F * ( X Y + X ( φ ) Y + Y ( φ ) X - g ( X , Y ) grad φ ) \tilde{\nabla}_{F_{*}X}F_{*}Y=F_{*}\Bigl(\nabla_{X}Y+X(\varphi)Y+Y(\varphi)X-g% (X,Y)\operatorname{grad}\varphi\Bigr)
  91. F : M N F:M\to N
  92. F * g ~ = e 2 φ g F^{*}\tilde{g}=e^{2\varphi}g
  93. X , Y X,Y
  94. d V ~ = e n φ d V d\tilde{V}=e^{n\varphi}dV
  95. d V dV
  96. R ~ i j k l = e 2 φ ( R i j k l - [ g ( φ - φ φ + 1 2 φ 2 g ) ] i j k l ) \tilde{R}_{ijkl}=e^{2\varphi}\left(R_{ijkl}-\left[g{~{}\wedge\!\!\!\!\!\!% \bigcirc~{}}\left(\nabla\partial\varphi-\partial\varphi\partial\varphi+\frac{1% }{2}\|\nabla\varphi\|^{2}g\right)\right]_{ijkl}\right)
  97. {~{}\wedge\!\!\!\!\!\!\bigcirc~{}}
  98. k \partial_{k}
  99. k \nabla_{k}
  100. R ~ i j = R i j - ( n - 2 ) [ i j φ - ( i φ ) ( j φ ) ] + ( φ - ( n - 2 ) φ 2 ) g i j \tilde{R}_{ij}=R_{ij}-(n-2)\left[\nabla_{i}\partial_{j}\varphi-(\partial_{i}% \varphi)(\partial_{j}\varphi)\right]+\left(\triangle\varphi-(n-2)\|\nabla% \varphi\|^{2}\right)g_{ij}
  101. \triangle
  102. f = - i i f \triangle f=-\nabla^{i}\partial_{i}f
  103. - -\triangle
  104. g g
  105. ~ f = e - 2 φ ( f - ( n - 2 ) k φ k f ) \tilde{\triangle}f=e^{-2\varphi}\left(\triangle f-(n-2)\nabla^{k}\varphi\nabla% _{k}f\right)
  106. R ~ = e - 2 φ ( R + 2 ( n - 1 ) φ - ( n - 2 ) ( n - 1 ) φ 2 ) \tilde{R}=e^{-2\varphi}\left(R+2(n-1)\triangle\varphi-(n-2)(n-1)\|\nabla% \varphi\|^{2}\right)
  107. n > 2 n>2
  108. R ~ = e - 2 φ [ R + 4 ( n - 1 ) ( n - 2 ) e - ( n - 2 ) φ / 2 ( e ( n - 2 ) φ / 2 ) ] \tilde{R}=e^{-2\varphi}\left[R+\frac{4(n-1)}{(n-2)}e^{-(n-2)\varphi/2}% \triangle\left(e^{(n-2)\varphi/2}\right)\right]
  109. C ~ i = j k l C i j k l \tilde{C}^{i}{}_{jkl}=C^{i}{}_{jkl}
  110. ω \omega
  111. p p
  112. * *
  113. δ \delta
  114. * ~ = e ( n - 2 p ) φ * \tilde{*}=e^{(n-2p)\varphi}*
  115. [ δ ~ ω ] ( v 1 , v 2 , , v p - 1 ) = e - 2 φ [ δ ω - ( n - 2 p ) ω ( φ , v 1 , v 2 , , v p - 1 ) ] \left[\tilde{\delta}\omega\right](v_{1},v_{2},\ldots,v_{p-1})=e^{-2\varphi}% \left[\delta\omega-(n-2p)\omega\left(\nabla\varphi,v_{1},v_{2},\ldots,v_{p-1}% \right)\right]

List_of_Foucault_pendulums.html

  1. l l\,
  2. g g\,
  3. 2 π l g 2\pi\sqrt{\frac{l}{g}}
  4. ϕ \phi\,
  5. 23 h 56 sin ϕ \frac{23h56^{\prime}}{\sin\phi}

List_of_gravitationally_rounded_objects_of_the_Solar_System.html

  1. × 10 1 7 \times 10^{1}7
  2. × 10 1 2 \times 10^{1}2
  3. × 10 1 8 \times 10^{1}8
  4. × 10 3 0 \times 10^{3}0
  5. × 10 6 \times 10^{6}
  6. × 10 1 0 \times 10^{1}0
  7. × 10 1 1 \times 10^{1}1
  8. × 10 1 2 \times 10^{1}2
  9. × 10 1 1 \times 10^{1}1
  10. × 10 1 5 \times 10^{1}5
  11. × 10 1 4 \times 10^{1}4
  12. × 10 1 3 \times 10^{1}3
  13. × 10 1 3 \times 10^{1}3
  14. × 10 2 3 \times 10^{2}3
  15. × 10 2 4 \times 10^{2}4
  16. × 10 2 4 \times 10^{2}4
  17. × 10 2 3 \times 10^{2}3
  18. × 10 2 7 \times 10^{2}7
  19. × 10 2 6 \times 10^{2}6
  20. × 10 2 5 \times 10^{2}5
  21. × 10 2 6 \times 10^{2}6
  22. × 10 4 \times 10^{4}
  23. × 10 6 \times 10^{6}
  24. × 10 6 \times 10^{6}
  25. × 10 5 \times 10^{5}
  26. × 10 5 \times 10^{5}
  27. × 10 5 \times 10^{5}
  28. × 10 4 \times 10^{4}
  29. × 10 4 \times 10^{4}
  30. × 10 2 0 \times 10^{2}0
  31. × 10 1 0 \times 10^{1}0
  32. × 10 1 0 \times 10^{1}0
  33. × 10 1 0 \times 10^{1}0
  34. × 10 1 0 \times 10^{1}0
  35. × 10 1 0 \times 10^{1}0
  36. × 10 7 \times 10^{7}
  37. × 10 7 \times 10^{7}
  38. × 10 8 \times 10^{8}
  39. × 10 8 \times 10^{8}
  40. × 10 9 \times 10^{9}
  41. × 10 2 2 \times 10^{2}2
  42. × 10 2 2 \times 10^{2}2
  43. × 10 2 2 \times 10^{2}2
  44. × 10 2 3 \times 10^{2}3
  45. × 10 2 3 \times 10^{2}3
  46. × 10 1 9 \times 10^{1}9
  47. × 10 2 0 \times 10^{2}0
  48. × 10 2 0 \times 10^{2}0
  49. × 10 2 1 \times 10^{2}1
  50. × 10 2 1 \times 10^{2}1
  51. × 10 1 0 \times 10^{1}0
  52. × 10 9 \times 10^{9}
  53. × 10 7 \times 10^{7}
  54. × 10 8 \times 10^{8}
  55. × 10 8 \times 10^{8}
  56. × 10 9 \times 10^{9}
  57. × 10 9 \times 10^{9}
  58. × 10 1 0 \times 10^{1}0
  59. × 10 8 \times 10^{8}
  60. × 10 2 3 \times 10^{2}3
  61. × 10 2 1 \times 10^{2}1
  62. × 10 1 9 \times 10^{1}9
  63. × 10 2 1 \times 10^{2}1
  64. × 10 2 1 \times 10^{2}1
  65. × 10 2 1 \times 10^{2}1
  66. × 10 2 1 \times 10^{2}1
  67. × 10 2 2 \times 10^{2}2
  68. × 10 2 1 \times 10^{2}1
  69. p ν p_{\nu}
  70. r r
  71. π \pi
  72. D = 1329 p 10 - 0.2 H \begin{smallmatrix}D=\frac{1329}{\sqrt{p}}10^{-0.2H}\end{smallmatrix}
  73. α = arccos ( c a ) \begin{smallmatrix}\alpha=\arccos\left(\frac{c}{a}\right)\end{smallmatrix}
  74. m = b 2 - c 2 b 2 sin ( α ) 2 \begin{smallmatrix}m=\frac{b^{2}-c^{2}}{b^{2}\sin(\alpha)^{2}}\end{smallmatrix}
  75. F ( α , m ) \begin{smallmatrix}F(\alpha,m)\end{smallmatrix}
  76. E ( α , m ) \begin{smallmatrix}E(\alpha,m)\end{smallmatrix}

List_of_limits.html

  1. If lim x c f ( x ) = L 1 and lim x c g ( x ) = L 2 then: \,\text{If }\lim_{x\to c}f(x)=L_{1}\,\text{ and }\lim_{x\to c}g(x)=L_{2}\,% \text{ then:}
  2. lim x c [ f ( x ) ± g ( x ) ] = L 1 ± L 2 \lim_{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}
  3. lim x c [ f ( x ) g ( x ) ] = L 1 × L 2 \lim_{x\to c}\,[f(x)g(x)]=L_{1}\times L_{2}
  4. lim x c f ( x ) g ( x ) = L 1 L 2 if L 2 0 \lim_{x\to c}\frac{f(x)}{g(x)}=\frac{L_{1}}{L_{2}}\qquad\,\text{ if }L_{2}\neq 0
  5. lim x c f ( x ) n = L 1 n if n is a positive integer \lim_{x\to c}\,f(x)^{n}=L_{1}^{n}\qquad\,\text{ if }n\,\text{ is a positive integer}
  6. lim x c f ( x ) 1 n = L 1 1 n if n is a positive integer, and if n is even, then L 1 > 0 \lim_{x\to c}\,f(x)^{1\over n}=L_{1}^{1\over n}\qquad\,\text{ if }n\,\text{ is% a positive integer, and if }n\,\text{ is even, then }L_{1}>0
  7. lim x c f ( x ) g ( x ) = lim x c f ( x ) g ( x ) if lim x c f ( x ) = lim x c g ( x ) = 0 or ± \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f^{\prime}(x)}{g^{\prime}(x)% }\qquad\,\text{ if }\lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0\,\text{ or }\pm\infty
  8. lim h 0 f ( x + h ) - f ( x ) h = f ( x ) \lim_{h\to 0}{f(x+h)-f(x)\over h}=f^{\prime}(x)
  9. lim h 0 ( f ( x + h ) f ( x ) ) 1 h = exp ( f ( x ) f ( x ) ) \lim_{h\to 0}\left(\frac{f(x+h)}{f(x)}\right)^{\frac{1}{h}}=\exp\left(\frac{f^% {\prime}(x)}{f(x)}\right)
  10. lim h 0 ( f ( x ( 1 + h ) ) f ( x ) ) 1 h = exp ( x f ( x ) f ( x ) ) \lim_{h\to 0}{\left({f(x(1+h))\over{f(x)}}\right)^{1\over{h}}}=\exp\left(\frac% {xf^{\prime}(x)}{f(x)}\right)
  11. lim x + ( 1 + k x ) m x = e m k \lim_{x\to+\infty}\left(1+\frac{k}{x}\right)^{mx}=e^{mk}
  12. lim x + ( 1 + 1 x ) x = e \lim_{x\to+\infty}\left(1+\frac{1}{x}\right)^{x}=e
  13. lim x + ( 1 - 1 x ) x = 1 e \lim_{x\to+\infty}\left(1-\frac{1}{x}\right)^{x}=\frac{1}{e}
  14. lim x + ( 1 + k x ) x = e k \lim_{x\to+\infty}\left(1+\frac{k}{x}\right)^{x}=e^{k}
  15. lim n n n ! n = e \lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}=e
  16. lim n 2 n 2 - 2 + 2 + + 2 n = π \lim_{n\to\infty}\,2^{n}\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\,\text{...}+\sqrt% {2}}}}}_{n}=\pi
  17. lim x 0 ( a x - 1 x ) = ln a , a > 0 \lim_{x\to 0}\left(\frac{a^{x}-1}{x}\right)=\ln{a},\qquad\forall~{}a>0
  18. lim x c a = a \lim_{x\to c}a=a
  19. lim x c x = c \lim_{x\to c}x=c
  20. lim x c a x + b = a c + b \lim_{x\to c}ax+b=ac+b
  21. lim x c x r = c r if r is a positive integer \lim_{x\to c}x^{r}=c^{r}\qquad\mbox{ if }~{}r\mbox{ is a positive integer}~{}
  22. lim x 0 + 1 x r = + \lim_{x\to 0^{+}}\frac{1}{x^{r}}=+\infty
  23. lim x 0 - 1 x r = { - , if r is odd + , if r is even \lim_{x\to 0^{-}}\frac{1}{x^{r}}=\begin{cases}-\infty,&\,\text{if }r\,\text{ % is odd}\\ +\infty,&\,\text{if }r\,\text{ is even}\end{cases}
  24. lim x 1 ln ( x ) x - 1 = 1 \lim_{x\to 1}\frac{\ln(x)}{x-1}=1
  25. lim y 0 ln ( y + 1 ) y = 1 \lim_{y\to 0}\frac{\ln(y+1)}{y}=1
  26. For a > 1 : \mbox{For }~{}a>1:\,
  27. lim x 0 + log a x = - \lim_{x\to 0^{+}}\log_{a}x=-\infty
  28. lim x log a x = \lim_{x\to\infty}\log_{a}x=\infty
  29. lim x - a x = 0 \lim_{x\to-\infty}a^{x}=0
  30. If a < 1 : \mbox{If }~{}a<1:\,
  31. lim x - a x = \lim_{x\to-\infty}a^{x}=\infty
  32. lim x a sin x = sin a \lim_{x\to a}\sin x=\sin a
  33. lim x a cos x = cos a \lim_{x\to a}\cos x=\cos a
  34. x x
  35. lim x 0 sin x x = 1 \lim_{x\to 0}\frac{\sin x}{x}=1
  36. lim x 0 1 - cos x x = 0 \lim_{x\to 0}\frac{1-\cos x}{x}=0
  37. lim x 0 1 - cos x x 2 = 1 2 \lim_{x\to 0}\frac{1-\cos x}{x^{2}}=\frac{1}{2}
  38. lim x n ± tan ( π x + π 2 ) = for any integer n \lim_{x\to n^{\pm}}\tan\left(\pi x+\frac{\pi}{2}\right)=\mp\infty\qquad\,\text% {for any integer }n
  39. lim x 0 sin a x x = a \lim_{x\to 0}\frac{\sin ax}{x}=a
  40. lim x 0 sin a x b x = a b \lim_{x\to 0}\frac{\sin ax}{bx}=\frac{a}{b}
  41. lim x N / x = 0 for any real N \lim_{x\to\infty}N/x=0\,\text{ for any real }N
  42. lim x x / N = { , N > 0 does not exist , N = 0 - , N < 0 \lim_{x\to\infty}x/N=\begin{cases}\infty,&N>0\\ \,\text{does not exist},&N=0\\ -\infty,&N<0\end{cases}
  43. lim x x N = { , N > 0 1 , N = 0 0 , N < 0 \lim_{x\to\infty}x^{N}=\begin{cases}\infty,&N>0\\ 1,&N=0\\ 0,&N<0\end{cases}
  44. lim x N x = { , N > 1 1 , N = 1 0 , 0 < N < 1 \lim_{x\to\infty}N^{x}=\begin{cases}\infty,&N>1\\ 1,&N=1\\ 0,&0<N<1\end{cases}
  45. lim x N - x = lim x 1 / N x = 0 for any N > 1 \lim_{x\to\infty}N^{-x}=\lim_{x\to\infty}1/N^{x}=0\,\text{ for any }N>1
  46. lim x N x = { 1 , N > 0 0 , N = 0 does not exist , N < 0 \lim_{x\to\infty}\sqrt[x]{N}=\begin{cases}1,&N>0\\ 0,&N=0\\ \,\text{does not exist},&N<0\end{cases}
  47. lim x x N = for any N > 0 \lim_{x\to\infty}\sqrt[N]{x}=\infty\,\text{ for any }N>0
  48. lim x log x = \lim_{x\to\infty}\log x=\infty
  49. lim x 0 + log x = - \lim_{x\to 0^{+}}\log x=-\infty

List_of_psychoactive_plants.html

  1. \leq
  2. \leq

Literal_(mathematical_logic).html

  1. l l
  2. l l
  3. l ¯ \bar{l}
  4. l l
  5. l x l\equiv x
  6. l ¯ \bar{l}
  7. ¬ x \lnot x
  8. l ¬ x l\equiv\lnot x
  9. l ¯ \bar{l}
  10. x x
  11. P ( t 1 , , t n ) P(t_{1},\ldots,t_{n})
  12. ¬ Q ( f ( g ( x ) , y , 2 ) , x ) \neg Q(f(g(x),y,2),x)

LiveMath.html

  1. x x
  2. x x

Ljung–Box_test.html

  1. Q = n ( n + 2 ) k = 1 h ρ ^ k 2 n - k Q=n\left(n+2\right)\sum_{k=1}^{h}\frac{\hat{\rho}^{2}_{k}}{n-k}
  2. ρ ^ k \hat{\rho}_{k}
  3. H 0 H_{0}
  4. χ ( h ) 2 \chi^{2}_{(h)}
  5. Q > χ 1 - α , h 2 Q>\chi_{1-\alpha,h}^{2}
  6. χ 1 - α , h 2 \chi_{1-\alpha,h}^{2}
  7. h - p - q h-p-q
  8. Q BP = n k = 1 h ρ ^ k 2 , Q\text{BP}=n\sum_{k=1}^{h}\hat{\rho}^{2}_{k},

Lloyd_N._Trefethen.html

  1. B M I = 1.3 w e i g h t / h e i g h t 2.5 BMI=1.3weight/height^{2.5}

Local_convex_hull.html

  1. Δ t i j \Delta t_{ij}
  2. Ψ i j = Δ x i j 2 + Δ y i j 2 + ( s v m a x Δ t i j ) 2 \Psi_{ij}=\sqrt{\Delta x_{ij}^{2}+\Delta y_{ij}^{2}+(sv_{max}\Delta t_{ij})^{2}}

Local_Langlands_conjectures.html

  1. ( 𝔤 , K ) (\mathfrak{g},K)

Local_martingale.html

  1. X t τ k := X min { t , τ k } X_{t}^{\tau_{k}}:=X_{\min\{t,\tau_{k}\}}
  2. X t = { W min ( t 1 - t , T ) for 0 t < 1 , - 1 for 1 t < . \displaystyle X_{t}=\begin{cases}W_{\min(\tfrac{t}{1-t},T)}&\,\text{for }0\leq t% <1,\\ -1&\,\text{for }1\leq t<\infty.\end{cases}
  3. X t X_{t}
  4. 𝔼 X t = { 0 for 0 t < 1 , - 1 for 1 t < . \displaystyle\mathbb{E}X_{t}=\begin{cases}0&\,\text{for }0\leq t<1,\\ -1&\,\text{for }1\leq t<\infty.\end{cases}
  5. τ k = min { t : X t = k } \tau_{k}=\min\{t:X_{t}=k\}
  6. 𝔼 | f ( W 1 ) | < . \mathbb{E}|f(W_{1})|<\infty.
  7. X t = 𝔼 ( f ( W 1 ) | F t ) = { f 1 - t ( W t ) for 0 t < 1 , f ( W 1 ) for 1 t < ; \displaystyle X_{t}=\mathbb{E}(f(W_{1})|F_{t})=\begin{cases}f_{1-t}(W_{t})&\,% \text{for }0\leq t<1,\\ f(W_{1})&\,\text{for }1\leq t<\infty;\end{cases}
  8. f s ( x ) = 𝔼 f ( x + W s ) = f ( x + y ) 1 2 π s e - y 2 / ( 2 s ) . \displaystyle f_{s}(x)=\mathbb{E}f(x+W_{s})=\int f(x+y)\frac{1}{\sqrt{2\pi s}}% \mathrm{e}^{-y^{2}/(2s)}.
  9. δ \delta
  10. f , f,
  11. Y t = 𝔼 ( δ ( W 1 ) | F t ) Y_{t}=\mathbb{E}(\delta(W_{1})|F_{t})
  12. Y t = { δ 1 - t ( W t ) for 0 t < 1 , 0 for 1 t < , \displaystyle Y_{t}=\begin{cases}\delta_{1-t}(W_{t})&\,\text{for }0\leq t<1,\\ 0&\,\text{for }1\leq t<\infty,\end{cases}
  13. δ s ( x ) = 1 2 π s e - x 2 / ( 2 s ) . \displaystyle\delta_{s}(x)=\frac{1}{\sqrt{2\pi s}}\mathrm{e}^{-x^{2}/(2s)}.
  14. Y t Y_{t}
  15. W 1 0 W_{1}\neq 0
  16. 𝔼 Y t = { 1 / 2 π for 0 t < 1 , 0 for 1 t < . \displaystyle\mathbb{E}Y_{t}=\begin{cases}1/\sqrt{2\pi}&\,\text{for }0\leq t<1% ,\\ 0&\,\text{for }1\leq t<\infty.\end{cases}
  17. τ k = min { t : Y t = k } . \tau_{k}=\min\{t:Y_{t}=k\}.
  18. Z t Z_{t}
  19. X t = ln | Z t - 1 | . \displaystyle X_{t}=\ln|Z_{t}-1|\,.
  20. X t X_{t}
  21. Z t Z_{t}
  22. u ln | u - 1 | u\mapsto\ln|u-1|
  23. τ k = min { t : X t = - k } . \tau_{k}=\min\{t:X_{t}=-k\}.
  24. 𝔼 X t \displaystyle\mathbb{E}X_{t}\to\infty
  25. t , t\to\infty,
  26. ln | u - 1 | \ln|u-1|
  27. | u | = r |u|=r
  28. r r\to\infty
  29. ln r \ln r
  30. M t M_{t}
  31. M t τ k M t M_{t}^{\tau_{k}}\to M_{t}
  32. k k\to\infty
  33. 𝔼 | M t τ k - M t | 0 ; \mathbb{E}|M_{t}^{\tau_{k}}-M_{t}|\to 0;
  34. M t τ k = M t τ k M_{t}^{\tau_{k}}=M_{t\wedge\tau_{k}}
  35. τ k \tau_{k}\to\infty
  36. M t τ k M t M_{t}^{\tau_{k}}\to M_{t}
  37. ( * ) 𝔼 sup k | M t τ k | < \textstyle(*)\quad\mathbb{E}\sup_{k}|M_{t}^{\tau_{k}}|<\infty
  38. M t M_{t}
  39. ( * * ) 𝔼 sup s [ 0 , t ] | M s | < \textstyle(**)\quad\mathbb{E}\sup_{s\in[0,t]}|M_{s}|<\infty
  40. sup s [ 0 , t ] 𝔼 | M s | < \textstyle\sup_{s\in[0,t]}\mathbb{E}|M_{s}|<\infty
  41. sup t [ 0 , ) 𝔼 e | M t | < \textstyle\sup_{t\in[0,\infty)}\mathbb{E}\mathrm{e}^{|M_{t}|}<\infty
  42. M t = f ( t , W t ) , \textstyle M_{t}=f(t,W_{t}),
  43. W t W_{t}
  44. f : [ 0 , ) × f:[0,\infty)\times\mathbb{R}\to\mathbb{R}
  45. M t M_{t}
  46. ( t + 1 2 2 x 2 ) f ( t , x ) = 0. \Big(\frac{\partial}{\partial t}+\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}% }\Big)f(t,x)=0.
  47. M t M_{t}
  48. ε > 0 \varepsilon>0
  49. C = C ( ε , t ) C=C(\varepsilon,t)
  50. | f ( s , x ) | C e ε x 2 \textstyle|f(s,x)|\leq C\mathrm{e}^{\varepsilon x^{2}}
  51. s [ 0 , t ] s\in[0,t]
  52. x . x\in\mathbb{R}.

Local_time_(mathematics).html

  1. ( b s ) s 0 (b_{s})_{s\geq 0}
  2. b b
  3. x x
  4. L x ( t ) = 0 t δ ( x - b ( s ) ) d s , L^{x}(t)=\int_{0}^{t}\delta(x-b(s))\,ds,
  5. δ \delta
  6. L x ( t ) L^{x}(t)
  7. b ( s ) b(s)
  8. x x
  9. t t
  10. L x ( t ) = lim ε 0 1 2 ε 0 t 1 { x - ε < b ( s ) < x + ε } d s , L^{x}(t)=\lim_{\varepsilon\downarrow 0}\frac{1}{2\varepsilon}\int_{0}^{t}1_{\{% x-\varepsilon<b(s)<x+\varepsilon\}}\,ds,
  11. b b
  12. x x
  13. ( X s ) s 0 (X_{s})_{s\geq 0}
  14. L x ( t ) = 0 t 1 { x } ( X s ) d s . L^{x}(t)=\int_{0}^{t}1_{\{x\}}(X_{s})\,ds.
  15. ( X s ) s 0 (X_{s})_{s\geq 0}
  16. : \mathbb{R}:
  17. L x ( t ) = | X t - x | - | X 0 - x | - 0 t ( 1 ( 0 , ) ( X s - x ) - 1 ( - , 0 ] ( X s - x ) ) d X s , t 0. L^{x}(t)=|X_{t}-x|-|X_{0}-x|-\int_{0}^{t}\left(1_{(0,\infty)}(X_{s}-x)-1_{(-% \infty,0]}(X_{s}-x)\right)dX_{s},\qquad t\geq 0.
  18. F : F:\mathbb{R}\rightarrow\mathbb{R}
  19. F , F^{\prime},
  20. F ( X t ) = F ( X 0 ) + 0 t F - ( X s ) d X s + 1 2 - L x ( t ) d F ( x ) , F(X_{t})=F(X_{0})+\int_{0}^{t}F^{\prime}_{-}(X_{s})dX_{s}+\frac{1}{2}\int_{-% \infty}^{\infty}L^{x}(t)dF^{\prime}(x),
  21. F - F^{\prime}_{-}
  22. L = ( L x ( t ) ) x , t 0 L=(L^{x}(t))_{x\in\mathbb{R},t\geq 0}
  23. x x
  24. t t
  25. ( | B s | ) s 0 (|B_{s}|)_{s\geq 0}
  26. L t = ( L t x ) x E L_{t}=(L^{x}_{t})_{x\in E}
  27. E E
  28. T = inf { t 0 : B t = 0 } T=\inf\{t\geq 0\colon B_{t}=0\}
  29. { L x ( T ) : x [ 0 , a ] } = 𝒟 { | W x | 2 : x [ 0 , a ] } \left\{L^{x}(T)\colon x\in[0,a]\right\}\stackrel{\mathcal{D}}{=}\left\{|W_{x}|% ^{2}\colon x\in[0,a]\right\}\,
  30. T a = inf { t 0 : L t 0 > a } . T_{a}=\inf\{t\geq 0\colon L^{0}_{t}>a\}.
  31. { L T a x + W x 2 : x 0 } = 𝒟 { ( W x + a ) 2 : x 0 } . \left\{L^{x}_{T_{a}}+W_{x}^{2}\colon x\geq 0\right\}\stackrel{\mathcal{D}}{=}% \left\{(W_{x}+\sqrt{a})^{2}\colon x\geq 0\right\}.\,
  32. ( L T a x ) x 0 (L^{x}_{T_{a}})_{x\geq 0}
  33. x x

Locally_connected_space.html

  1. n \mathbb{R}^{n}
  2. x U V x\in U\subset V
  3. x U V x\in U\subset V
  4. n \mathbb{R}^{n}
  5. [ 0 , 1 ] [ 2 , 3 ] [0,1]\cup[2,3]
  6. 1 \mathbb{R}^{1}
  7. \mathbb{Q}
  8. i X i \coprod_{i}X_{i}
  9. { X i } \{X_{i}\}
  10. X i X_{i}
  11. { Y i } \{Y_{i}\}
  12. i Y i \bigcap_{i}Y_{i}
  13. Y i Y_{i}
  14. i Y i \bigcup_{i}Y_{i}
  15. x , y X x,y\in X
  16. x c y x\equiv_{c}y
  17. x p c y x\equiv_{pc}y
  18. A B A\cup B
  19. C x C_{x}
  20. y c x y\equiv_{c}x
  21. C x C_{x}
  22. C x C_{x}
  23. C x C_{x}
  24. C x = { x } C_{x}=\{x\}
  25. C x \coprod C_{x}
  26. P C x PC_{x}
  27. y p c x y\equiv_{pc}x
  28. P C x PC_{x}
  29. P C x C x PC_{x}\subset C_{x}
  30. C U C\setminus U
  31. C x C_{x}
  32. C x = P C x C_{x}=PC_{x}
  33. x q c y x\equiv_{qc}y
  34. Q C x QC_{x}
  35. Q C x QC_{x}
  36. Q C x QC_{x}
  37. C x Q C x C_{x}\subseteq QC_{x}
  38. P C x C x Q C x . PC_{x}\subseteq C_{x}\subseteq QC_{x}.
  39. C x C_{x}
  40. Q C x C x QC_{x}\subseteq C_{x}
  41. Q C x = C x QC_{x}=C_{x}
  42. P C x = C x = Q C x . PC_{x}=C_{x}=QC_{x}.
  43. Q C x = C x = { x } QC_{x}=C_{x}=\{x\}

Locally_finite_measure.html

  1. p X , N p T s.t. p N p and | μ ( N p ) | < + . \forall p\in X,\exists N_{p}\in T\mbox{ s.t. }~{}p\in N_{p}\mbox{ and }~{}% \left|\mu(N_{p})\right|<+\infty.

Log-distance_path_loss_model.html

  1. P L = P T x d B m - P R x d B m = P L 0 + 10 γ log 10 d d 0 + X g , PL\;=P_{Tx_{dBm}}-P_{Rx_{dBm}}\;=\;PL_{0}\;+\;10\gamma\;\log_{10}\frac{d}{d_{0% }}\;+\;X_{g},
  2. P L {PL}
  3. P T x d B m = 10 log 10 P T x 1 m W P_{Tx_{dBm}}\;=10\log_{10}\frac{P_{Tx}}{1mW}
  4. P T x P_{Tx}
  5. P R x d B m = 10 log 10 P R x 1 m W P_{Rx_{dBm}}\;=10\log_{10}\frac{P_{Rx}}{1mW}
  6. P R x {P_{Rx}}
  7. P L 0 PL_{0}
  8. d {d}
  9. d 0 {d_{0}}
  10. γ \gamma
  11. X g X_{g}
  12. σ \sigma\;
  13. F g = 10 - X g 10 F_{g}\;=\;10^{\frac{-X_{g}}{10}}
  14. P R x P T x = c 0 F g d γ \frac{P_{Rx}}{P_{Tx}}\;=\;\frac{c_{0}F_{g}}{d^{\gamma}}
  15. c 0 = d 0 γ 10 - L 0 10 c_{0}\;=\;{d_{0}^{\gamma}}10^{\frac{-L_{0}}{10}}
  16. d 0 d_{0}
  17. F g = 10 - X g 10 F_{g}\;=\;10^{\frac{-X_{g}}{10}}
  18. σ \sigma\;
  19. γ \gamma
  20. σ \sigma
  21. γ \gamma
  22. σ \sigma

Log_area_ratio.html

  1. r k r_{k}
  2. A k = log 1 + r k 1 - r k A_{k}=\log{1+r_{k}\over 1-r_{k}}

Logarithmic_convolution.html

  1. s ( t ) s(t)
  2. r ( t ) r(t)
  3. s * l r ( t ) = r * l s ( t ) = 0 s ( t a ) r ( a ) d a a s*_{l}r(t)=r*_{l}s(t)=\int_{0}^{\infty}s\left(\frac{t}{a}\right)r(a)\,\frac{da% }{a}
  4. t t
  5. v = log t v=\log t
  6. s * l r ( t ) = 0 s ( t a ) r ( a ) d a a = - s ( t e u ) r ( e u ) d u s*_{l}r(t)=\int_{0}^{\infty}s\left(\frac{t}{a}\right)r(a)\,\frac{da}{a}=\int_{% -\infty}^{\infty}s\left(\frac{t}{e^{u}}\right)r(e^{u})\,du
  7. = - s ( e log t - u ) r ( e u ) d u . =\int_{-\infty}^{\infty}s\left(e^{\log t-u}\right)r(e^{u})\,du.
  8. f ( v ) = s ( e v ) f(v)=s(e^{v})
  9. g ( v ) = r ( e v ) g(v)=r(e^{v})
  10. v = log t v=\log t
  11. s * l r ( v ) = f * g ( v ) = g * f ( v ) = r * l s ( v ) . s*_{l}r(v)=f*g(v)=g*f(v)=r*_{l}s(v).\,

Logic_optimization.html

  1. F 1 = A B + A C + A D , F_{1}=AB+AC+AD,\,
  2. F 2 = A B + A C + A E . F_{2}=A^{\prime}B+A^{\prime}C+A^{\prime}E.\,

London_equations.html

  1. 𝐣 s t = n s e 2 m 𝐄 , × 𝐣 s = - n s e 2 m c 𝐁 . \frac{\partial\mathbf{j}_{s}}{\partial t}=\frac{n_{s}e^{2}}{m}\mathbf{E},% \qquad\mathbf{\nabla}\times\mathbf{j}_{s}=-\frac{n_{s}e^{2}}{mc}\mathbf{B}.
  2. 𝐣 s {\mathbf{j}}_{s}
  3. e e\,
  4. m m\,
  5. n s n_{s}\,
  6. 𝐣 s = - n s e 2 m c 𝐀 . \mathbf{j}_{s}=-\frac{n_{s}e^{2}}{mc}\mathbf{A}.
  7. × 𝐁 = 4 π 𝐣 c \nabla\times\mathbf{B}=\frac{4\pi\mathbf{j}}{c}
  8. 2 𝐁 = 1 λ 2 𝐁 , λ m c 2 4 π n s e 2 . \nabla^{2}\mathbf{B}=\frac{1}{\lambda^{2}}\mathbf{B},\qquad\lambda\equiv\sqrt{% \frac{mc^{2}}{4\pi n_{s}e^{2}}}.
  9. λ \lambda
  10. B z ( x ) = B 0 e - x / λ . B_{z}(x)=B_{0}e^{-x/\lambda}.\,
  11. 𝐅 = e 𝐄 + e c 𝐯 × 𝐁 \mathbf{F}=e\mathbf{E}+\frac{e}{c}\mathbf{v}\times\mathbf{B}
  12. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}
  13. t ( × 𝐣 s + n s e 2 m c 𝐁 ) = 0. \frac{\partial}{\partial t}\left(\nabla\times\mathbf{j}_{s}+\frac{n_{s}e^{2}}{% mc}\mathbf{B}\right)=0.
  14. 𝐣 s = n s e 𝐯 . \mathbf{j}_{s}=n_{s}e\mathbf{v}.
  15. 𝐯 = 1 m ( 𝐩 - e c 𝐀 ) \mathbf{v}=\frac{1}{m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)
  16. 𝐣 s = - n s e s 2 m c 𝐀 , \mathbf{j}_{s}=-\frac{n_{s}e_{s}^{2}}{mc}\mathbf{A},

Long_wavelength_limit.html

  1. λ a \lambda>>a
  2. a a
  3. λ a \lambda>>a
  4. k a 1 ka<<1
  5. k k
  6. ω c a 1 \frac{\omega}{c}a<<1
  7. ω \omega
  8. c c

Longest_element_of_a_Coxeter_group.html

  1. w 0 - 1 = w 0 w_{0}^{-1}=w_{0}
  2. w W , w\in W,
  3. ( w 0 w ) = ( w 0 ) - ( w ) . \ell(w_{0}w)=\ell(w_{0})-\ell(w).
  4. A n A_{n}
  5. n 2 n\geq 2
  6. D n D_{n}
  7. E 6 , E_{6},
  8. I 2 ( p ) I_{2}(p)

Lookahead_carry_unit.html

  1. P G P_{G}
  2. G G G_{G}
  3. P G P_{G}
  4. G G G_{G}
  5. P G = P 3 P 2 P 1 P 0 P_{G}=P_{3}\cdot P_{2}\cdot P_{1}\cdot P_{0}
  6. G G = G 3 + P 3 G 2 + P 3 P 2 G 1 + P 3 P 2 P 1 G 0 G_{G}=G_{3}+P_{3}\cdot G_{2}+P_{3}\cdot P_{2}\cdot G_{1}+P_{3}\cdot P_{2}\cdot P% _{1}\cdot G_{0}
  7. P i P_{i}
  8. P G P_{G}
  9. G i G_{i}
  10. G G G_{G}
  11. C 4 = G 0 + P 0 C 0 C_{4}=G_{0}+P_{0}\cdot C_{0}
  12. C 8 = G 4 + P 4 C 4 C_{8}=G_{4}+P_{4}\cdot C_{4}
  13. C 12 = G 8 + P 8 C 8 C_{12}=G_{8}+P_{8}\cdot C_{8}
  14. C 16 = G 12 + P 12 C 12 C_{16}=G_{12}+P_{12}\cdot C_{12}
  15. C 4 C_{4}
  16. C 8 C_{8}
  17. C 8 C_{8}
  18. C 12 C_{12}
  19. C 12 C_{12}
  20. C 16 C_{16}
  21. C 4 = G 0 + P 0 C 0 C_{4}=G_{0}+P_{0}\cdot C_{0}
  22. C 8 = G 4 + G 0 P 4 + C 0 P 0 P 4 C_{8}=G_{4}+G_{0}\cdot P_{4}+C_{0}\cdot P_{0}\cdot P_{4}
  23. C 12 = G 8 + G 4 P 8 + G 0 P 4 P 8 + C 0 P 0 P 4 P 8 C_{12}=G_{8}+G_{4}\cdot P_{8}+G_{0}\cdot P_{4}\cdot P_{8}+C_{0}\cdot P_{0}% \cdot P_{4}\cdot P_{8}
  24. C 16 = G 12 + G 8 P 12 + G 4 P 8 P 12 + G 0 P 4 P 8 P 12 + C 0 P 0 P 4 P 8 P 12 C_{16}=G_{12}+G_{8}\cdot P_{12}+G_{4}\cdot P_{8}\cdot P_{12}+G_{0}\cdot P_{4}% \cdot P_{8}\cdot P_{12}+C_{0}\cdot P_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}
  25. C 4 C_{4}
  26. C 8 C_{8}
  27. C 12 C_{12}
  28. C 16 C_{16}
  29. P L C U = P 0 P 4 P 8 P 12 P_{LCU}=P_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}
  30. G L C U = G 12 + G 8 P 12 + G 4 P 8 P 12 + G 0 P 4 P 8 P 12 + C 0 P 0 P 4 P 8 P 12 = C 16 G_{LCU}=G_{12}+G_{8}\cdot P_{12}+G_{4}\cdot P_{8}\cdot P_{12}+G_{0}\cdot P_{4}% \cdot P_{8}\cdot P_{12}+C_{0}\cdot P_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}=C_{% 16}
  31. P L C U P_{LCU}
  32. G L C U G_{LCU}

Loomis–Whitney_inequality.html

  1. π j : d d - 1 , \pi_{j}:\mathbb{R}^{d}\to\mathbb{R}^{d-1},
  2. π j : x = ( x 1 , , x d ) x ^ j = ( x 1 , , x j - 1 , x j + 1 , , x d ) . \pi_{j}:x=(x_{1},\dots,x_{d})\mapsto\hat{x}_{j}=(x_{1},\dots,x_{j-1},x_{j+1},% \dots,x_{d}).
  3. g j : d - 1 [ 0 , + ) , g_{j}:\mathbb{R}^{d-1}\to[0,+\infty),
  4. g j L d - 1 ( d - 1 ) . g_{j}\in L^{d-1}(\mathbb{R}^{d-1}).
  5. d j = 1 d g j ( π j ( x ) ) d x j = 1 d g j L d - 1 ( d - 1 ) . \int_{\mathbb{R}^{d}}\prod_{j=1}^{d}g_{j}(\pi_{j}(x))\,\mathrm{d}x\leq\prod_{j% =1}^{d}\|g_{j}\|_{L^{d-1}(\mathbb{R}^{d-1})}.
  6. f j ( x ) = g j ( x ) d - 1 , f_{j}(x)=g_{j}(x)^{d-1},
  7. d j = 1 d f j ( π j ( x ) ) 1 / ( d - 1 ) d x j = 1 d ( d - 1 f j ( x ^ j ) d x ^ j ) 1 / ( d - 1 ) . \int_{\mathbb{R}^{d}}\prod_{j=1}^{d}f_{j}(\pi_{j}(x))^{1/(d-1)}\,\mathrm{d}x% \leq\prod_{j=1}^{d}\left(\int_{\mathbb{R}^{d-1}}f_{j}(\hat{x}_{j})\,\mathrm{d}% \hat{x}_{j}\right)^{1/(d-1)}.
  8. d \mathbb{R}^{d}
  9. d \mathbb{R}^{d}
  10. f j = 𝟏 π j ( E ) f_{j}=\mathbf{1}_{\pi_{j}(E)}
  11. j = 1 d f j ( π j ( x ) ) 1 / ( d - 1 ) = 1. \prod_{j=1}^{d}f_{j}(\pi_{j}(x))^{1/(d-1)}=1.
  12. | E | j = 1 d | π j ( E ) | 1 / ( d - 1 ) , |E|\leq\prod_{j=1}^{d}|\pi_{j}(E)|^{1/(d-1)},
  13. | E | j = 1 d | E | | π j ( E ) | . |E|\geq\prod_{j=1}^{d}\frac{|E|}{|\pi_{j}(E)|}.
  14. | E | | π j ( E ) | \frac{|E|}{|\pi_{j}(E)|}

Loop_entropy.html

  1. N N
  2. Δ S = α k B ln N \Delta S=\alpha k_{B}\ln N\,
  3. k B k_{B}
  4. α \alpha
  5. P N - α P\sim N^{-\alpha}
  6. α \alpha
  7. M M
  8. M M
  9. 𝐖 \mathbf{W}
  10. M × M M\times M
  11. W i j W_{ij}
  12. i i
  13. j j
  14. Δ S = α k B ln det 𝐖 \Delta S=\alpha k_{B}\ln\det\mathbf{W}
  15. 𝐖 = def [ 58 26 26 52 ] \mathbf{W}\ \overset{\underset{\mathrm{def}}{}}{=}\begin{bmatrix}58&&26\\ 26&&52\end{bmatrix}
  16. α k B \alpha k_{B}

Lorenz_system.html

  1. d x d t \displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}
  2. x x
  3. y y
  4. z z
  5. t t
  6. σ \sigma
  7. ρ \rho
  8. β \beta
  9. σ \sigma
  10. ρ \rho
  11. β \beta
  12. σ = 10 \sigma=10
  13. β = 8 / 3 \beta=8/3
  14. ρ = 28 \rho=28
  15. ρ < 1 \rho<1
  16. ρ < 1 \rho<1
  17. ρ = 1 \rho=1
  18. ρ > 1 \rho>1
  19. ( ± β ( ρ - 1 ) , ± β ( ρ - 1 ) , ρ - 1 ) . \left(\pm\sqrt{\beta(\rho-1)},\pm\sqrt{\beta(\rho-1)},\rho-1\right).
  20. ρ < σ σ + β + 3 σ - β - 1 , \rho<\sigma\frac{\sigma+\beta+3}{\sigma-\beta-1},
  21. ρ \rho
  22. σ > β + 1 \sigma>\beta+1
  23. ρ = 28 \rho=28
  24. σ = 10 \sigma=10
  25. β = 8 / 3 \beta=8/3
  26. ρ \rho
  27. ρ = 99.96 \rho=99.96

Lost_time.html

  1. l 2 = y + a r - e l_{2}=y+ar-e
  2. l 2 \mathit{l}_{2}
  3. y y
  4. a r ar
  5. e e

Louis_Kauffman.html

  1. F ( K ) ( a , z ) = a - w ( K ) L ( K ) F(K)(a,z)=a^{-w(K)}L(K)\,
  2. w ( K ) w(K)
  3. L ( K ) L(K)

LU_decomposition.html

  1. A = L U , A=LU,\,
  2. [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] = [ l 11 0 0 l 21 l 22 0 l 31 l 32 l 33 ] [ u 11 u 12 u 13 0 u 22 u 23 0 0 u 33 ] . \begin{bmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{bmatrix}=\begin{bmatrix}l_{11}&0&0\\ l_{21}&l_{22}&0\\ l_{31}&l_{32}&l_{33}\end{bmatrix}\begin{bmatrix}u_{11}&u_{12}&u_{13}\\ 0&u_{22}&u_{23}\\ 0&0&u_{33}\end{bmatrix}.
  3. a 11 = l 11 u 11 a_{11}=l_{11}u_{11}
  4. a 11 = 0 a_{11}=0
  5. l 11 l_{11}
  6. u 11 u_{11}
  7. P A = L U , PA=LU,\,
  8. P A Q = L U , PAQ=LU,\,
  9. A = L D U , A=LDU,\,
  10. [ 4 3 6 3 ] = [ l 11 0 l 21 l 22 ] [ u 11 u 12 0 u 22 ] . \begin{bmatrix}4&3\\ 6&3\end{bmatrix}=\begin{bmatrix}l_{11}&0\\ l_{21}&l_{22}\end{bmatrix}\begin{bmatrix}u_{11}&u_{12}\\ 0&u_{22}\end{bmatrix}.
  11. l 11 u 11 + 0 0 = 4 l_{11}\cdot u_{11}+0\cdot 0=4
  12. l 11 u 12 + 0 u 22 = 3 l_{11}\cdot u_{12}+0\cdot u_{22}=3
  13. l 21 u 11 + l 22 0 = 6 l_{21}\cdot u_{11}+l_{22}\cdot 0=6
  14. l 21 u 12 + l 22 u 22 = 3. l_{21}\cdot u_{12}+l_{22}\cdot u_{22}=3.
  15. l 21 = 1.5 l_{21}=1.5
  16. u 11 = 4 u_{11}=4
  17. u 12 = 3 u_{12}=3
  18. u 22 = - 1.5. u_{22}=-1.5.
  19. [ 4 3 6 3 ] = [ 1 0 1.5 1 ] [ 4 3 0 - 1.5 ] . \begin{bmatrix}4&3\\ 6&3\end{bmatrix}=\begin{bmatrix}1&0\\ 1.5&1\end{bmatrix}\begin{bmatrix}4&3\\ 0&-1.5\end{bmatrix}.
  20. A A
  21. A A
  22. A A
  23. k k
  24. k k
  25. L L
  26. U U
  27. A = L L * . A=LL^{*}.\,
  28. D 1 = A 1 , 1 D_{1}=A_{1,1}
  29. i = 2 , , n i=2,\ldots,n
  30. D i D_{i}
  31. i t h i^{th}
  32. ( i - 1 ) t h (i-1)^{th}
  33. A = ( a n , n ) A=(a_{n,n})
  34. A ( 0 ) := A A^{(0)}:=A
  35. l i , n := - a i , n ( n - 1 ) a n , n ( n - 1 ) l_{i,n}:=-\frac{a_{i,n}^{(n-1)}}{a_{n,n}^{(n-1)}}
  36. i = n + 1 , , N i=n+1,\ldots,N
  37. L n = ( 1 0 1 l n + 1 , n 0 l N , n 1 ) . L_{n}=\begin{pmatrix}1&&&&&0\\ &\ddots&&&&\\ &&1&&&\\ &&l_{n+1,n}&\ddots&&\\ &&\vdots&&\ddots&\\ 0&&l_{N,n}&&&1\end{pmatrix}.
  38. A ( n ) := L n A ( n - 1 ) . A^{(n)}:=L_{n}A^{(n-1)}.
  39. A = L 1 - 1 L 1 A ( 0 ) = L 1 - 1 A ( 1 ) = L 1 - 1 L 2 - 1 L 2 A ( 1 ) = L 1 - 1 L 2 - 1 A ( 2 ) = = L 1 - 1 L N - 1 - 1 A ( N - 1 ) . A=L_{1}^{-1}L_{1}A^{(0)}=L_{1}^{-1}A^{(1)}=L_{1}^{-1}L_{2}^{-1}L_{2}A^{(1)}=L_% {1}^{-1}L_{2}^{-1}A^{(2)}=\cdots=L_{1}^{-1}\ldots L_{N-1}^{-1}A^{(N-1)}.
  40. L = L 1 - 1 L N - 1 - 1 L=L_{1}^{-1}\ldots L_{N-1}^{-1}
  41. L = ( 1 0 - l 2 , 1 1 - l n + 1 , n 1 - l N , 1 - l N , n - l N , N - 1 1 ) . L=\begin{pmatrix}1&&&&&0\\ -l_{2,1}&\ddots&&&&\\ &&1&&&\\ \vdots&&-l_{n+1,n}&\ddots&&\\ &&\vdots&&1&\\ -l_{N,1}&&-l_{N,n}&&-l_{N,N-1}&1\end{pmatrix}.
  42. A = L U A=LU
  43. a n , n ( n - 1 ) 0 a_{n,n}^{(n-1)}\not=0
  44. l i , n l_{i,n}
  45. P - 1 A = L U P^{-1}A=LU
  46. A A
  47. P 1 P_{1}
  48. A P 1 AP_{1}
  49. P 1 P_{1}
  50. A 1 = A P 1 A_{1}=AP_{1}
  51. A 2 A_{2}
  52. A 1 A_{1}
  53. A 2 = L 2 U 2 P 2 A_{2}=L_{2}U_{2}P_{2}
  54. L L
  55. L 2 L_{2}
  56. A 1 A_{1}
  57. U 3 U_{3}
  58. U 2 U_{2}
  59. P 3 P_{3}
  60. P 2 P_{2}
  61. A 3 = A 1 / P 3 = A P 1 / P 3 A_{3}=A_{1}/P_{3}=AP_{1}/P_{3}
  62. P P
  63. P 1 / P 3 P_{1}/P_{3}
  64. A 3 A_{3}
  65. L U 3 LU_{3}
  66. A A
  67. A 3 = L U 3 A_{3}=LU_{3}
  68. A = L U 3 P A=LU_{3}P
  69. A 3 A_{3}
  70. L U 3 LU_{3}
  71. A 3 = L U 3 U 1 A_{3}=LU_{3}U_{1}
  72. U 1 U_{1}
  73. U 1 U_{1}
  74. L U 3 LU_{3}
  75. A 3 A_{3}
  76. A = L U 3 U 1 P A=LU_{3}U_{1}P
  77. A x = b , Ax=b,\,
  78. P A = L U L U x = P b . PA=LULUx=Pb.\,
  79. L y = P b Ly=Pb
  80. U x = y Ux=y
  81. 2 3 n 3 \frac{2}{3}n^{3}
  82. A A
  83. n n
  84. 4 3 n 3 \frac{4}{3}n^{3}
  85. A X = L U X = B . AX=LUX=B.
  86. A = P - 1 L U A=P^{-1}LU
  87. det ( A ) = det ( P - 1 ) det ( L ) det ( U ) = ( - 1 ) S ( i = 1 n l i i ) ( i = 1 n u i i ) . \det(A)=\det(P^{-1})\det(L)\det(U)=(-1)^{S}\left(\prod_{i=1}^{n}l_{ii}\right)% \left(\prod_{i=1}^{n}u_{ii}\right).
  88. det ( A ) \det(A)

Lucas'_theorem.html

  1. ( m n ) {\textstyle\left({{m}\atop{n}}\right)}
  2. ( m n ) i = 0 k ( m i n i ) ( mod p ) , {\left({{m}\atop{n}}\right)}\equiv\prod_{i=0}^{k}{\left({{m_{i}}\atop{n_{i}}}% \right)}\;\;(\mathop{{\rm mod}}p),
  3. m = m k p k + m k - 1 p k - 1 + + m 1 p + m 0 , m=m_{k}p^{k}+m_{k-1}p^{k-1}+\cdots+m_{1}p+m_{0},
  4. n = n k p k + n k - 1 p k - 1 + + n 1 p + n 0 n=n_{k}p^{k}+n_{k-1}p^{k-1}+\cdots+n_{1}p+n_{0}
  5. ( m n ) {\textstyle\left({{m}\atop{n}}\right)}
  6. ( m n ) {\textstyle\left({{m}\atop{n}}\right)}
  7. i = 0 k ( m i n i ) ( mod p ) \prod_{i=0}^{k}{\left({{m_{i}}\atop{n_{i}}}\right)}\;\;(\mathop{{\rm mod}}p)
  8. ( p n ) = p ( p - 1 ) ( p - n + 1 ) n ( n - 1 ) 1 {\left({{p}\atop{n}}\right)}=\frac{p\cdot(p-1)\cdots(p-n+1)}{n\cdot(n-1)\cdots 1}
  9. ( p n ) {\textstyle\left({{p}\atop{n}}\right)}
  10. ( 1 + X ) p 1 + X p mod p . (1+X)^{p}\equiv 1+X^{p}\,\text{ mod }p.
  11. ( 1 + X ) p i 1 + X p i mod p . (1+X)^{p^{i}}\equiv 1+X^{p^{i}}\,\text{ mod }p.
  12. m = i = 0 k m i p i m=\sum_{i=0}^{k}m_{i}p^{i}
  13. n = 0 m ( m n ) X n = ( 1 + X ) m = i = 0 k ( ( 1 + X ) p i ) m i i = 0 k ( 1 + X p i ) m i = i = 0 k ( n i = 0 m i ( m i n i ) X n i p i ) = i = 0 k ( n i = 0 p - 1 ( m i n i ) X n i p i ) = n = 0 m ( i = 0 k ( m i n i ) ) X n mod p , \begin{aligned}\displaystyle\sum_{n=0}^{m}{\left({{m}\atop{n}}\right)}X^{n}&% \displaystyle=(1+X)^{m}=\prod_{i=0}^{k}\left((1+X)^{p^{i}}\right)^{m_{i}}\\ &\displaystyle\equiv\prod_{i=0}^{k}\left(1+X^{p^{i}}\right)^{m_{i}}=\prod_{i=0% }^{k}\left(\sum_{n_{i}=0}^{m_{i}}{\left({{m_{i}}\atop{n_{i}}}\right)}X^{n_{i}p% ^{i}}\right)\\ &\displaystyle=\prod_{i=0}^{k}\left(\sum_{n_{i}=0}^{p-1}{\left({{m_{i}}\atop{n% _{i}}}\right)}X^{n_{i}p^{i}}\right)=\sum_{n=0}^{m}\left(\prod_{i=0}^{k}{\left(% {{m_{i}}\atop{n_{i}}}\right)}\right)X^{n}\,\text{ mod }p,\end{aligned}
  14. ( m n ) {\textstyle\left({{m}\atop{n}}\right)}

Lumazine_synthase.html

  1. \rightleftharpoons

Luzin_N_property.html

  1. N [ a , b ] N\subset[a,b]
  2. λ ( N ) = 0 \lambda(N)=0
  3. λ ( f ( N ) ) = 0 \lambda(f(N))=0
  4. λ \lambda

M-ratio.html

  1. M = stack small blind + big blind + total antes M=\frac{\mbox{stack}~{}}{\mbox{small blind}~{}+\mbox{big blind}~{}+\mbox{total% antes}~{}}
  2. M = 2300 50 + 100 + ( 10 × 8 ) = 2300 230 = 10 M=\frac{2300}{50+100+(10\times 8)}=\frac{2300}{230}=10
  3. M Effective = M × ( Players 10 ) M_{\mbox{Effective}~{}}=M\times\left(\frac{\mbox{Players}~{}}{10}\right)
  4. M Effective = 9 × ( 5 10 ) = 4.5 M_{\mbox{Effective}~{}}=9\times\left(\frac{5}{10}\right)=4.5

Macrodiversity.html

  1. N \scriptstyle N
  2. n R \scriptstyle n_{R}
  3. 𝐲 = 𝐇𝐱 + 𝐧 \mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}
  4. 𝐲 \scriptstyle\mathbf{y}
  5. 𝐱 \scriptstyle\mathbf{x}
  6. 𝐇 \scriptstyle\mathbf{H}
  7. 𝐧 \scriptstyle\mathbf{n}
  8. N 0 \scriptstyle N_{0}
  9. i , j \scriptstyle i,j
  10. 𝐇 \scriptstyle\mathbf{H}
  11. h i j h_{ij}
  12. i , j \scriptstyle i,j
  13. j \scriptstyle j
  14. i \scriptstyle i
  15. E { | h i j | 2 } = g i j i , j E\left\{\left|h_{ij}\right|^{2}\right\}=g_{ij}\quad\forall i,j
  16. g i , j \scriptstyle g_{i,j}
  17. g i j N 0 \scriptstyle\frac{g_{ij}}{N_{0}}
  18. 𝐆 = ( g 11 g 1 N g 21 g 2 N g n R 1 g n R N ) . \mathbf{G}=\begin{pmatrix}g_{11}&\dots&g_{1N}\\ g_{21}&\dots&g_{2N}\\ \dots&\dots&\dots\\ g_{n_{R}1}&\dots&g_{n_{R}N}\\ \end{pmatrix}.
  19. 𝐇 w \mathbf{H}_{w}
  20. 𝐲 = ( ( 𝐆 1 2 ) 𝐇 w ) 𝐱 + 𝐧 \mathbf{y}=\left(\left(\mathbf{G}^{\circ\frac{1}{2}}\right)\circ\mathbf{H}_{w}% \right)\mathbf{x}+\mathbf{n}
  21. 𝐆 1 2 \mathbf{G}^{\circ\frac{1}{2}}
  22. 𝐆 \mathbf{G}
  23. \circ
  24. i , j \scriptstyle i,j
  25. 𝐇 w \mathbf{H}_{w}
  26. h w , i j h_{w,ij}
  27. E { | h w , i j | 2 } = 1 i , j E\left\{\left|h_{w,ij}\right|^{2}\right\}=1\quad\forall i,j
  28. 𝐆 \mathbf{G}
  29. 𝐱 \mathbf{x}

Magnesium::Teflon::Viton.html

  1. Δ R H \Delta_{\mathrm{R}}H
  2. Δ f H o \Delta_{\mathrm{f}}H^{o}
  3. Δ R H \Delta_{\mathrm{R}}H

Magnetic_dipole–dipole_interaction.html

  1. H H
  2. H = - μ 0 4 π | r | 3 ( 3 ( m 1 r ^ ) ( m 2 r ^ ) - m 1 m 2 ) H=-\frac{\mu_{0}}{4\pi|r|^{3}}\left(3(m_{1}\cdot\hat{r})(m_{2}\cdot\hat{r})-m_% {1}\cdot m_{2}\right)
  3. \mathbf{r̂}
  4. 𝐫 \mathbf{r}
  5. H = - μ 0 γ 1 γ 2 2 4 π | r | 3 ( 3 ( I 1 r ^ ) ( I 2 r ^ ) - I 1 I 2 ) H=-\frac{\mu_{0}\gamma_{1}\gamma_{2}\hbar^{2}}{4\pi|r|^{3}}\left(3(I_{1}\cdot% \hat{r})(I_{2}\cdot\hat{r})-I_{1}\cdot I_{2}\right)
  6. \mathbf{r̂}
  7. 𝐫 \mathbf{r}
  8. 𝐅 \mathbf{F}
  9. F = 3 μ 0 4 π | r | 4 ( ( r ^ × m 1 ) × m 2 + ( r ^ × m 2 ) × m 1 - 2 r ^ ( m 1 m 2 ) + 5 r ^ ( ( r ^ × m 1 ) ( r ^ × m 2 ) ) ) F=\frac{3\mu_{0}}{4\pi|r|^{4}}((\hat{r}\times m_{1})\times m_{2}+(\hat{r}% \times m_{2})\times m_{1}-2\hat{r}(m_{1}\cdot m_{2})+5\hat{r}((\hat{r}\times m% _{1})\cdot(\hat{r}\times m_{2})))

Magnetic_domain.html

  1. H e = α M H_{e}=\alpha\ M
  2. α \alpha
  3. H e = α M s H_{e}=\alpha\ M_{s}
  4. M s M_{s}
  5. E = E e x + E D + E λ + E k + E H E=E_{ex}+E_{D}+E_{\lambda}+E_{k}+E_{H}\,

Magneto-optic_Kerr_effect.html

  1. ε \varepsilon
  2. v p = 1 ε μ v_{p}=\frac{1}{\sqrt{\varepsilon\mu}}
  3. v p v_{p}
  4. ε \varepsilon
  5. μ \mu
  6. E E
  7. r r
  8. k k
  9. r r
  10. | r + k | 2 |r+k|^{2}
  11. | r - k | 2 |r-k|^{2}

Magnitude_condition.html

  1. 1 + 𝐆 ( s ) = 0 1+\,\textbf{G}(s)=0
  2. 𝐆 ( s ) = 𝐏 ( s ) 𝐐 ( s ) \,\textbf{G}(s)=\frac{\,\textbf{P}(s)}{\,\textbf{Q}(s)}
  3. e j 2 π + 𝐆 ( s ) = 0 e^{j2\pi}+\,\textbf{G}(s)=0
  4. 𝐆 ( s ) = - 1 = e j ( π + 2 k π ) \,\textbf{G}(s)=-1=e^{j(\pi+2k\pi)}
  5. ( k = 0 , 1 , 2 , ) (k=0,1,2,...)
  6. 𝐆 ( s ) \,\textbf{G}(s)
  7. 𝐆 ( s ) = 𝐏 ( s ) 𝐐 ( s ) = K ( s - a 1 ) ( s - a 2 ) ( s - a n ) ( s - b 1 ) ( s - b 2 ) ( s - b m ) \,\textbf{G}(s)=\frac{\,\textbf{P}(s)}{\,\textbf{Q}(s)}=K\frac{(s-a_{1})(s-a_{% 2})\cdots(s-a_{n})}{(s-b_{1})(s-b_{2})\cdots(s-b_{m})}
  8. ( s - a p ) (s-a_{p})
  9. ( s - b q ) (s-b_{q})
  10. A p e j θ p A_{p}e^{j\theta_{p}}
  11. B q e j ϕ q B_{q}e^{j\phi_{q}}
  12. 𝐆 ( s ) \,\textbf{G}(s)
  13. 𝐆 ( s ) = K A 1 A 2 A n e j ( θ 1 + θ 2 + + θ n ) B 1 B 2 B m e j ( ϕ 1 + ϕ 2 + + ϕ m ) \,\textbf{G}(s)=K\frac{A_{1}A_{2}\cdots A_{n}e^{j(\theta_{1}+\theta_{2}+\cdots% +\theta_{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\phi_{1}+\phi_{2}+\cdots+\phi_{m})}}
  14. e j ( π + 2 k π ) = K A 1 A 2 A n e j ( θ 1 + θ 2 + + θ n ) B 1 B 2 B m e j ( ϕ 1 + ϕ 2 + + ϕ m ) = K A 1 A 2 A n B 1 B 2 B m e j ( θ 1 + θ 2 + + θ n - ( ϕ 1 + ϕ 2 + + ϕ m ) ) e^{j(\pi+2k\pi)}=K\frac{A_{1}A_{2}\cdots A_{n}e^{j(\theta_{1}+\theta_{2}+% \cdots+\theta_{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\phi_{1}+\phi_{2}+\cdots+\phi_% {m})}}=K\frac{A_{1}A_{2}\cdots A_{n}}{B_{1}B_{2}\cdots B_{m}}e^{j(\theta_{1}+% \theta_{2}+\cdots+\theta_{n}-(\phi_{1}+\phi_{2}+\cdots+\phi_{m}))}
  15. 1 = K A 1 A 2 A n B 1 B 2 B m 1=K\frac{A_{1}A_{2}\cdots A_{n}}{B_{1}B_{2}\cdots B_{m}}

Malament–Hogarth_spacetime.html

  1. λ \lambda
  2. p p
  3. λ \lambda
  4. p p
  5. λ \lambda
  6. p p
  7. p p
  8. λ \lambda
  9. λ \lambda
  10. p p
  11. p p
  12. p p
  13. p p

Malcev_algebra.html

  1. x y = - y x xy=-yx
  2. ( x y ) ( x z ) = ( ( x y ) z ) x + ( ( y z ) x ) x + ( ( z x ) x ) y . (xy)(xz)=((xy)z)x+((yz)x)x+((zx)x)y.

Malliavin_derivative.html

  1. H H
  2. C 0 C_{0}
  3. H := { f W 1 , 2 ( [ 0 , T ] ; n ) | f ( 0 ) = 0 } := { paths starting at 0 with first derivative in L 2 } H:=\{f\in W^{1,2}([0,T];\mathbb{R}^{n})\;|\;f(0)=0\}:=\{\,\text{paths starting% at 0 with first derivative in }L^{2}\}
  4. C 0 := C 0 ( [ 0 , T ] ; n ) := { continuous paths starting at 0 } ; C_{0}:=C_{0}([0,T];\mathbb{R}^{n}):=\{\,\text{continuous paths starting at 0}\};
  5. H C 0 H\subset C_{0}
  6. i : H C 0 i:H\to C_{0}
  7. F : C 0 F:C_{0}\to\mathbb{R}
  8. D F : C 0 Lin ( C 0 ; ) ; \mathrm{D}F:C_{0}\to\mathrm{Lin}(C_{0};\mathbb{R});
  9. σ C 0 \sigma\in C_{0}
  10. D F ( σ ) \mathrm{D}F(\sigma)\;
  11. C 0 * C_{0}^{*}
  12. C 0 C_{0}\;
  13. D H F ( σ ) \mathrm{D}_{H}F(\sigma)\;
  14. H H\to\mathbb{R}
  15. D H F ( σ ) := D F ( σ ) i : H , \mathrm{D}_{H}F(\sigma):=\mathrm{D}F(\sigma)\circ i:H\to\mathbb{R},
  16. H F : C 0 H \nabla_{H}F:C_{0}\to H
  17. D H F \mathrm{D}_{H}F\;
  18. 0 T ( t H F ( σ ) ) t h := H F ( σ ) , h H = ( D H F ) ( σ ) ( h ) = lim t 0 F ( σ + t i ( h ) ) - F ( σ ) t . \int_{0}^{T}\left(\partial_{t}\nabla_{H}F(\sigma)\right)\cdot\partial_{t}h:=% \langle\nabla_{H}F(\sigma),h\rangle_{H}=\left(\mathrm{D}_{H}F\right)(\sigma)(h% )=\lim_{t\to 0}\frac{F(\sigma+ti(h))-F(\sigma)}{t}.
  19. D t \mathrm{D}_{t}
  20. ( D t F ) ( σ ) := t ( ( H F ) ( σ ) ) . \left(\mathrm{D}_{t}F\right)(\sigma):=\frac{\partial}{\partial t}\left(\left(% \nabla_{H}F\right)(\sigma)\right).
  21. D t \mathrm{D}_{t}
  22. 𝐅 \mathbf{F}
  23. C 0 C_{0}\;
  24. L 2 ( [ 0 , T ] ; n ) L^{2}([0,T];\mathbb{R}^{n})
  25. δ \delta\;
  26. δ := ( D t ) * : image ( D t ) L 2 ( [ 0 , T ] ; n ) 𝐅 * = Lin ( 𝐅 ; ) . \delta:=\left(\mathrm{D}_{t}\right)^{*}:\operatorname{image}\left(\mathrm{D}_{% t}\right)\subseteq L^{2}([0,T];\mathbb{R}^{n})\to\mathbf{F}^{*}=\mathrm{Lin}(% \mathbf{F};\mathbb{R}).

Malmquist_index.html

  1. S a S_{a}
  2. Q Q
  3. Q = f a ( S a ) Q=f_{a}(S_{a})
  4. M I = ( Q 1 Q 2 ) / ( Q 3 Q 4 ) MI=\sqrt{(Q_{1}Q_{2})/(Q_{3}Q_{4})}
  5. Q 1 = f a ( S a ) Q_{1}=f_{a}(S_{a})
  6. Q 2 = f a ( S b ) Q_{2}=f_{a}(S_{b})
  7. Q 3 = f b ( S a ) Q_{3}=f_{b}(S_{a})
  8. Q 4 = f b ( S b ) Q_{4}=f_{b}(S_{b})

Many-sorted_logic.html

  1. 𝑝𝑙𝑎𝑛𝑡 \,\textit{plant}
  2. 𝑎𝑛𝑖𝑚𝑎𝑙 \,\textit{animal}
  3. 𝑚𝑜𝑡ℎ𝑒𝑟 : 𝑎𝑛𝑖𝑚𝑎𝑙 𝑎𝑛𝑖𝑚𝑎𝑙 \,\textit{mother}:\,\textit{animal}\longrightarrow\,\textit{animal}
  4. 𝑚𝑜𝑡ℎ𝑒𝑟 : 𝑝𝑙𝑎𝑛𝑡 𝑝𝑙𝑎𝑛𝑡 \,\textit{mother}:\,\textit{plant}\longrightarrow\,\textit{plant}
  5. 𝑚𝑜𝑡ℎ𝑒𝑟 ( 𝑙𝑎𝑠𝑠𝑖𝑒 ) \,\textit{mother}(\,\textit{lassie})
  6. 𝑚𝑜𝑡ℎ𝑒𝑟 ( my_favorite_oak ) \,\textit{mother}(\,\textit{my\_favorite\_oak})
  7. s 1 s_{1}
  8. s 2 s_{2}
  9. s 1 s 2 s_{1}\subseteq s_{2}
  10. 𝑑𝑜𝑔 𝑐𝑎𝑟𝑛𝑖𝑣𝑜𝑟𝑒 \,\textit{dog}\subseteq\,\textit{carnivore}
  11. 𝑑𝑜𝑔 𝑚𝑎𝑚𝑚𝑎𝑙 \,\textit{dog}\subseteq\,\textit{mammal}
  12. 𝑐𝑎𝑟𝑛𝑖𝑣𝑜𝑟𝑒 𝑎𝑛𝑖𝑚𝑎𝑙 \,\textit{carnivore}\subseteq\,\textit{animal}
  13. 𝑚𝑎𝑚𝑚𝑎𝑙 𝑎𝑛𝑖𝑚𝑎𝑙 \,\textit{mammal}\subseteq\,\textit{animal}
  14. 𝑎𝑛𝑖𝑚𝑎𝑙 𝑐𝑟𝑒𝑎𝑡𝑢𝑟𝑒 \,\textit{animal}\subseteq\,\textit{creature}
  15. 𝑝𝑙𝑎𝑛𝑡 𝑐𝑟𝑒𝑎𝑡𝑢𝑟𝑒 \,\textit{plant}\subseteq\,\textit{creature}
  16. s s
  17. s s
  18. 𝑚𝑜𝑡ℎ𝑒𝑟 : 𝑎𝑛𝑖𝑚𝑎𝑙 𝑎𝑛𝑖𝑚𝑎𝑙 \,\textit{mother}:\,\textit{animal}\longrightarrow\,\textit{animal}
  19. 𝑙𝑎𝑠𝑠𝑖𝑒 : 𝑑𝑜𝑔 \,\textit{lassie}:\,\textit{dog}
  20. 𝑚𝑜𝑡ℎ𝑒𝑟 ( 𝑙𝑎𝑠𝑠𝑖𝑒 ) \,\textit{mother}(\,\textit{lassie})
  21. 𝑎𝑛𝑖𝑚𝑎𝑙 \,\textit{animal}
  22. 𝑚𝑜𝑡ℎ𝑒𝑟 : 𝑑𝑜𝑔 𝑑𝑜𝑔 \,\textit{mother}:\,\textit{dog}\longrightarrow\,\textit{dog}
  23. p i ( x ) p_{i}(x)
  24. s i s_{i}
  25. x : p i ( x ) p j ( x ) \forall x:p_{i}(x)\rightarrow p_{j}(x)
  26. s i s j s_{i}\subseteq s_{j}
  27. s 1 , s 2 s_{1},s_{2}
  28. s 1 s 2 s_{1}\cap s_{2}
  29. x 1 x_{1}
  30. x 2 x_{2}
  31. s 1 s_{1}
  32. s 2 s_{2}
  33. x 1 = ? x 2 x_{1}\stackrel{?}{=}x_{2}
  34. { x 1 = x , x 2 = x } \{x_{1}=x,\;x_{2}=x\}
  35. x : s 1 s 2 x:s_{1}\cap s_{2}
  36. 𝑙𝑖𝑠𝑡 ( X ) \,\textit{list}(X)
  37. X X
  38. 𝑖𝑛𝑡 𝑓𝑙𝑜𝑎𝑡 \,\textit{int}\subseteq\,\textit{float}
  39. 𝑙𝑖𝑠𝑡 ( 𝑖𝑛𝑡 ) 𝑙𝑖𝑠𝑡 ( 𝑓𝑙𝑜𝑎𝑡 ) \,\textit{list}(\,\textit{int})\subseteq\,\textit{list}(\,\textit{float})
  40. 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡 \,\textit{even}\subseteq\,\textit{int}
  41. 𝑜𝑑𝑑 𝑖𝑛𝑡 \,\textit{odd}\subseteq\,\textit{int}
  42. i : 𝑖𝑛𝑡 . ( i + i ) : 𝑒𝑣𝑒𝑛 \forall i:\,\textit{int}.\;(i+i):\,\textit{even}

Map_(higher-order_function).html

  1. F F
  2. G G
  3. h : T . F T G T h:\,\forall T.\,F\,T\rightarrow G\,T
  4. h Y ( fmap k ) = ( fmap k ) h X h_{Y}\circ(\mathrm{fmap}\,k)=(\mathrm{fmap}\,k)\circ h_{X}
  5. k : X Y k:X\rightarrow Y
  6. ( map f ) ( map g ) = map ( f g ) \left(\,\text{map}\,f\right)\circ\left(\,\text{map}\,g\right)=\,\text{map}\,% \left(f\circ g\right)

Marchenko_equation.html

  1. K ( r , r ) + g ( r , r ) + r K ( r , r ′′ ) g ( r ′′ , r ) d r ′′ = 0 K(r,r^{\prime})+g(r,r^{\prime})+\int_{r}^{\infty}K(r,r^{\prime\prime})g(r^{% \prime\prime},r^{\prime})\mathrm{d}r^{\prime\prime}=0
  2. g ( r , r ) g(r,r^{\prime})\,
  3. g ( r , r ) = g ( r , r ) , g(r,r^{\prime})=g(r^{\prime},r),\,
  4. K ( r , r ) K(r,r^{\prime})

Margulis_lemma.html

  1. ϵ = ϵ ( S ) > 0 \epsilon=\epsilon(S)>0
  2. d ( f x , x ) < ϵ d(f\cdot x,x)<\epsilon
  3. Γ \Gamma
  4. Γ \Gamma
  5. Γ 0 \Gamma_{0}
  6. Γ \Gamma
  7. Γ \Gamma
  8. ϵ ( S ) \epsilon(S)

Marked_graph.html

  1. p P : | p | = | p | = 1 \forall p\in P:|p\bullet|=|\bullet p|=1

Markov_chain_mixing_time.html

  1. | Pr ( X t A ) - π ( A ) | 1 / 4 |\Pr(X_{t}\in A)-\pi(A)|\leq 1/4

Markov_renewal_process.html

  1. S . \mathrm{S}.
  2. ( X n , T n ) (X_{n},T_{n})
  3. T n T_{n}
  4. X n X_{n}
  5. τ n = T n - T n - 1 \tau_{n}=T_{n}-T_{n-1}
  6. Pr ( τ n + 1 t , X n + 1 = j | ( X 0 , T 0 ) , ( X 1 , T 1 ) , , ( X n = i , T n ) ) \Pr(\tau_{n+1}\leq t,X_{n+1}=j|(X_{0},T_{0}),(X_{1},T_{1}),\ldots,(X_{n}=i,T_{% n}))
  7. = Pr ( τ n + 1 t , X n + 1 = j | X n = i ) n 1 , t 0 , i , j S =\Pr(\tau_{n+1}\leq t,X_{n+1}=j|X_{n}=i)\,\forall n\geq 1,t\geq 0,i,j\in% \mathrm{S}
  8. Y t := X n Y_{t}:=X_{n}
  9. t [ T n , T n + 1 ) t\in[T_{n},T_{n+1})
  10. Y t Y_{t}
  11. Pr ( τ n + 1 t , X n + 1 = j | ( X 0 , T 0 ) , ( X 1 , T 1 ) , , ( X n = i , T n ) ) = Pr ( τ n + 1 t , X n + 1 = j | X n = i ) \Pr(\tau_{n+1}\leq t,X_{n+1}=j|(X_{0},T_{0}),(X_{1},T_{1}),\ldots,(X_{n}=i,T_{% n}))=\Pr(\tau_{n+1}\leq t,X_{n+1}=j|X_{n}=i)
  12. = Pr ( X n + 1 = j | X n = i ) ( 1 - e - λ i t ) , for all n 1 , t 0 , i , j S =\Pr(X_{n+1}=j|X_{n}=i)(1-e^{-\lambda_{i}t}),\,\text{ for all }n\geq 1,t\geq 0% ,i,j\in\mathrm{S}
  13. X n X_{n}
  14. Pr ( X n + 1 = j | X 0 , X 1 , , X n = i ) = Pr ( X n + 1 = j | X n = i ) n 1 , i , j S \Pr(X_{n+1}=j|X_{0},X_{1},\ldots,X_{n}=i)=\Pr(X_{n+1}=j|X_{n}=i)\,\forall n% \geq 1,i,j\in\mathrm{S}
  15. τ \tau
  16. X n X_{n}
  17. Pr ( τ n + 1 t | T 0 , T 1 , , T n ) = Pr ( τ n + 1 t ) n 1 , t 0 \Pr(\tau_{n+1}\leq t|T_{0},T_{1},\ldots,T_{n})=\Pr(\tau_{n+1}\leq t)\,\forall n% \geq 1,\forall t\geq 0

Markus–Yamabe_conjecture.html

  1. n n
  2. f : n n f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}
  3. C 1 C^{1}
  4. f ( 0 ) = 0 f(0)=0
  5. D f ( x ) Df(x)
  6. x n x\in\mathbb{R}^{n}
  7. 0
  8. x ˙ = f ( x ) \dot{x}=f(x)
  9. n = 2 n=2
  10. n > 2 n>2

Martin_Huxley.html

  1. p n + 1 - p n < p n θ , p_{n+1}-p_{n}<p_{n}^{\theta},

Martingale_pricing.html

  1. ( Ω , ( t ) t [ 0 , T ] , ~ ) (\Omega,(\mathcal{F}_{t})_{t\in[0,T]},\tilde{\mathbb{P}})
  2. { S ( t ) } t [ 0 , T ] \{S(t)\}_{t\in[0,T]}
  3. V ( t , S ( t ) ) V(t,S(t))
  4. D ( t ) V ( t , S ( t ) ) = 𝔼 ~ [ D ( T ) V ( T , S ( T ) ) | t ] , d D ( t ) = - r ( t ) D ( t ) d t D(t)V(t,S(t))=\tilde{\mathbb{E}}[D(T)V(T,S(T))|\mathcal{F}_{t}],\qquad dD(t)=-% r(t)D(t)\ dt
  5. ~ \tilde{\mathbb{P}}
  6. ( r ( t ) ) t [ 0 , T ] (r(t))_{t\in[0,T]}
  7. t \mathcal{F}_{t}
  8. T T
  9. { X ( t ) } t [ 0 , T ] \{X(t)\}_{t\in[0,T]}
  10. Δ ( t ) \Delta(t)
  11. t t
  12. X ( t ) - Δ ( t ) S ( t ) X(t)-\Delta(t)S(t)
  13. r ( t ) r(t)
  14. d X ( t ) = Δ ( t ) d S ( t ) + r ( t ) ( X ( t ) - Δ ( t ) S ( t ) ) d t dX(t)=\Delta(t)\ dS(t)+r(t)(X(t)-\Delta(t)S(t))\ dt
  15. d ~ d \frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}
  16. Δ ( t ) \Delta(t)
  17. V ( 0 , S ( 0 ) ) = X ( 0 ) V(0,S(0))=X(0)
  18. ~ [ X ( T ) = V ( T ) ] = 1 \tilde{\mathbb{P}}[X(T)=V(T)]=1
  19. \mathbb{P}

MASH-1.html

  1. e = 2 e=2
  2. e = 2 8 + 1 e=2^{8}+1

Mason_equation.html

  1. d M d t = 4 π r p D v ( ρ 0 - ρ w ) \frac{dM}{dt}=4\pi r_{p}D_{v}(\rho_{0}-\rho_{w})\,
  2. d Q d t = 4 π r p K ( T 0 - T w ) \frac{dQ}{dt}=4\pi r_{p}K(T_{0}-T_{w})\,
  3. r d r d t = ( S - 1 ) [ ( L / R T - 1 ) L ρ l / K T 0 + ( ρ l R T 0 ) / ( D ρ v ) ] r\frac{dr}{dt}=\frac{(S-1)}{[(L/RT-1)\cdot L\rho_{l}/KT_{0}+(\rho_{l}RT_{0})/(% D\rho_{v})]}

Mass_distribution.html

  1. g / f g/\mid\nabla f\mid

Mass_in_general_relativity.html

  1. m = E 2 - ( p c ) 2 c 2 m=\frac{\sqrt{E^{2}-\left(pc\right)^{2}}}{c^{2}}
  2. c = 1 c=1
  3. m c mc
  4. g μ ν g_{\mu\nu}\,
  5. E = v T 00 d V P i = V T 0 i d V E=\int_{v}T_{00}dV\qquad P^{i}=\int_{V}T_{0i}dV
  6. G ( E + 3 P V ) / r 2 c 2 G\left(E+3PV\right)/r^{2}c^{2}

Matching_law.html

  1. 1 {}_{1}
  2. 2 {}_{2}
  3. 1 {}_{1}
  4. 2 {}_{2}
  5. 1 {}_{1}
  6. 1 {}_{1}
  7. 2 {}_{2}
  8. 1 {}_{1}
  9. 1 {}_{1}
  10. 2 {}_{2}
  11. R 1 R 1 + R 2 = R f 1 R f 1 + R f 2 \frac{R_{1}}{R_{1}+R_{2}}=\frac{Rf_{1}}{Rf_{1}+Rf_{2}}
  12. R 1 R 2 = R f 1 R f 2 \frac{R_{1}}{R_{2}}=\frac{Rf_{1}}{Rf_{2}}
  13. R 1 R 2 = b ( R f 1 R f 2 ) s \frac{R_{1}}{R_{2}}=b\left(\frac{Rf_{1}}{Rf_{2}}\right)^{s}
  14. log ( R 1 R 2 ) = log ( b ) + s log ( R f 1 R f 2 ) \log\left(\frac{R_{1}}{R_{2}}\right)=\log\left(b\right)+s\cdot\log\left(\frac{% Rf_{1}}{Rf_{2}}\right)

Matching_pursuit.html

  1. D D
  2. f f
  3. H H
  4. g γ n g_{\gamma_{n}}
  5. D D
  6. f ( t ) = n = 0 + a n g γ n ( t ) f(t)=\sum_{n=0}^{+\infty}a_{n}g_{\gamma_{n}}(t)
  7. n n
  8. a n a_{n}
  9. f f
  10. f f
  11. f f
  12. D D
  13. γ n \gamma_{n}
  14. a n a_{n}
  15. R n + 1 R_{n+1}
  16. f ( t ) f(t)
  17. D D
  18. ( a n , g γ n ) \left(a_{n},g_{\gamma_{n}}\right)
  19. R 1 f ( t ) R_{1}\,\leftarrow\,f(t)
  20. n 1 n\,\leftarrow\,1
  21. g γ n D g_{\gamma_{n}}\in D
  22. | R n , g γ n | |\langle R_{n},g_{\gamma_{n}}\rangle|
  23. a n R n , g γ n a_{n}\,\leftarrow\,\langle R_{n},g_{\gamma_{n}}\rangle
  24. R n + 1 R n - a n g γ n R_{n+1}\,\leftarrow\,R_{n}-a_{n}g_{\gamma_{n}}
  25. n n + 1 n\,\leftarrow\,n+1
  26. R n < threshold \|R_{n}\|<\mathrm{threshold}
  27. f f
  28. m m
  29. f 2 = n = 0 m - 1 | a n | 2 + R m 2 \|f\|^{2}=\sum_{n=0}^{m-1}{|a_{n}|^{2}}+\|R_{m}\|^{2}
  30. R n \|R_{n}\|
  31. min x f - D x 2 2 subject to x 0 N , \min_{x}\|f-Dx\|_{2}^{2}\ \,\text{ subject to }\ \|x\|_{0}\leq N,
  32. x 0 \|x\|_{0}
  33. L 0 L_{0}
  34. x x

Material_selection.html

  1. E / ρ E/\rho
  2. E 3 / ρ \sqrt[3]{E}/\rho
  3. E 2 / ρ \sqrt[2]{E}/\rho
  4. E / ρ E/\rho
  5. E 3 / ρ \sqrt[3]{E}/\rho

Materiality_(auditing).html

  1. M = β ( max ( i = 1 N A s s e t s i , i = 1 N R e v e n u e s i ) ) α . M=\beta\cdot(\max{(\sum_{i=1}^{N}{Assets_{i}},\sum_{i=1}^{N}{Revenues_{i}})})^% {\alpha}.\,
  2. β = \beta=\,
  3. β > 1. \beta>1.\,
  4. α = \alpha=\,
  5. 0 < α < 1. 0<\alpha<1.\,
  6. i = 1 , 2 , , N i=1,2,...,N\,

Mathematical_discussion_of_rangekeeping.html

  1. R T P = R T + d R T d t t T O F = R T + d R T d t ( t T O F + t D e l a y ) \begin{array}[]{lcr}R_{TP}&=&R_{T}+\frac{dR_{T}}{dt}\cdot t_{TOF^{\prime}}\\ &=&R_{T}+\frac{dR_{T}}{dt}\cdot\left(t_{TOF}+t_{Delay}\right)\end{array}
  2. R T P R_{TP}\,
  3. R T R_{T}\,
  4. t T O F t_{TOF^{\prime}}\,
  5. ( t T O F ) \left(t_{TOF}\right)\,
  6. ( t D e l a y ) \left(t_{Delay}\right)\,
  7. t T O F = t T O F + t D e l a y t_{TOF^{\prime}}=t_{TOF}+t_{Delay}\,
  8. R T P = R T + d R T d t ( k T O F R T + t D e l a y ) R_{TP}=R_{T}+\frac{dR_{T}}{dt}\cdot\left(k_{TOF}\cdot R_{T}+t_{Delay}\right)\,\!
  9. k T O F k_{TOF}\,
  10. d R T d t = - s O y - s T y \frac{dR_{T}}{dt}=-s_{Oy}-s_{Ty}\,\!
  11. s O y s_{Oy}\,
  12. v O y = v O cos ( θ T ) v_{Oy}=\lVert v_{O}\rVert\cdot\cos(\theta_{T})
  13. s T y s_{Ty}\,
  14. v T y = v T cos ( θ A O B ) v_{Ty}=\lVert v_{T}\rVert\cdot\cos(\theta_{AOB})
  15. R T P = R T + ( - s O y - s T y ) ( k T O F R T + t D e l a y ) R_{TP}=R_{T}+\left(-s_{Oy}-s_{Ty}\right)\cdot\left(k_{TOF}\cdot R_{T}+t_{Delay% }\right)\,\!
  16. θ T P = θ T + d θ T d t t T O F = θ T + d θ T d t ( t T O F + t D e l a y ) \begin{array}[]{lcr}\theta_{TP}&=&\theta_{T}+\frac{d\theta_{T}}{dt}\cdot t_{% TOF^{\prime}}\\ &=&\theta_{T}+\frac{d\theta_{T}}{dt}\cdot\left(t_{TOF}+t_{Delay}\right)\\ \end{array}
  17. θ T \theta_{T}\,
  18. θ T P \theta_{TP}\,
  19. d θ T d t = s T x + s O x R T \frac{d\theta_{T}}{dt}=\frac{s_{Tx}+s_{Ox}}{R_{T}}\,\!
  20. s O x s_{Ox}\,
  21. s O x = v O sin ( θ T ) s_{Ox}=\lVert v_{O}\rVert\cdot\sin(\theta_{T})\,
  22. s T x s_{Tx}\,
  23. s T x = v T sin ( θ A O B ) s_{Tx}=\lVert v_{T}\rVert\cdot\sin(\theta_{AOB})\,
  24. t T O F = ˙ k T O F R T t_{TOF}\dot{=}k_{TOF}\cdot R_{T}
  25. θ T P = θ T + ( s T x + s O x R T ) ( k T O F R T + t D e l a y ) \theta_{TP}=\theta_{T}+\left(\frac{s_{Tx}+s_{Ox}}{R_{T}}\right)\cdot\left(k_{% TOF}\cdot R_{T}+t_{Delay}\right)

Mathieu_transformation.html

  1. i p i δ q i = i P i δ Q i \sum_{i}p_{i}\delta q_{i}=\sum_{i}P_{i}\delta Q_{i}\,
  2. q i q_{i}
  3. Q i Q_{i}
  4. p i , P i p_{i},P_{i}
  5. Ω 1 ( q 1 , q 2 , , q n , Q 1 , Q 2 , Q n ) = 0 Ω m ( q 1 , q 2 , , q n , Q 1 , Q 2 , Q n ) = 0 \begin{aligned}\displaystyle\Omega_{1}(q_{1},q_{2},\ldots,q_{n},Q_{1},Q_{2},% \ldots Q_{n})&\displaystyle=0\\ &\displaystyle{}\ \ \vdots\\ \displaystyle\Omega_{m}(q_{1},q_{2},\ldots,q_{n},Q_{1},Q_{2},\ldots Q_{n})&% \displaystyle=0\end{aligned}
  6. 1 < m n 1<m\leq n
  7. m = n m=n

Matrix_equivalence.html

  1. B = Q - 1 A P \!B=Q^{-1}AP

Matrix_pencil.html

  1. A 0 , A 1 , , A l A_{0},A_{1},\dots,A_{l}
  2. n × n n\times n
  3. l l
  4. A l 0 A_{l}\neq 0
  5. l l
  6. L ( λ ) = i = 0 l λ i A i . L(\lambda)=\sum_{i=0}^{l}\lambda^{i}A_{i}.
  7. A - λ B A-\lambda B\,
  8. λ (or ), \lambda\in\mathbb{C}\,\text{ (or }\mathbb{R}\,\text{),}
  9. A A
  10. B B
  11. n × n n\times n
  12. ( A , B ) (A,B)
  13. λ \lambda
  14. det ( A - λ B ) 0 \det(A-\lambda B)\neq 0
  15. ( A , B ) (A,B)
  16. λ \lambda
  17. det ( A - λ B ) = 0 \det(A-\lambda B)=0
  18. σ ( A , B ) \sigma(A,B)
  19. B B
  20. B - 1 A x = λ x B^{-1}Ax=\lambda x
  21. B - 1 A B^{-1}A
  22. B B
  23. A B = B A AB=BA
  24. A A
  25. B B

Maximal_arc.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. ( q + 1 ) - q d (q+1)-\frac{q}{d}
  5. S ( K ) = ( P , B , I ) S(K)=(P,B,I)
  6. p g ( q - d , q - q d , q - q d - d + 1 ) pg(q-d,q-\frac{q}{d},q-\frac{q}{d}-d+1)
  7. P G ( 3 , 2 h ) ( h 1 ) PG(3,2^{h})(h\geq 1)
  8. d = 2 s ( 1 s m ) d=2^{s}(1\leq s\leq m)
  9. π \pi
  10. T 2 * ( K ) = ( P , B , I ) T_{2}^{*}(K)=(P,B,I)
  11. π \pi
  12. π \pi
  13. π \pi
  14. T 2 * ( K ) T_{2}^{*}(K)
  15. p g ( 2 h - 1 , ( 2 h + 1 ) ( 2 m - 1 ) , 2 m - 1 ) pg(2^{h}-1,(2^{h}+1)(2^{m}-1),2^{m}-1)\,

Maximal_pair.html

  1. ( p 1 , p 2 , l ) (p_{1},p_{2},l)
  2. S S
  3. n n
  4. S [ p 1 . . p 1 + l - 1 ] = S [ p 2 . . p 2 + l - 1 ] S[p_{1}..p_{1}+l-1]=S[p_{2}..p_{2}+l-1]
  5. S [ p 1 - 1 ] S [ p 2 - 1 ] S[p_{1}-1]\neq S[p_{2}-1]
  6. S [ p 1 + l ] S [ p 2 + l ] S[p_{1}+l]\neq S[p_{2}+l]
  7. Θ ( n + z ) \Theta(n+z)
  8. z z
  9. ( 2 , 6 , 3 ) (2,6,3)
  10. ( 6 , 10 , 3 ) (6,10,3)
  11. ( 2 , 10 , 3 ) (2,10,3)

McCarthy_Formalism.html

  1. π i n ( x 1 , , x n ) = x i \pi_{i}^{n}(x_{1},\ldots,x_{n})=x_{i}

Mean_absolute_difference.html

  1. MD := E [ | X - Y | ] . \mathrm{MD}:=E[|X-Y|].
  2. MD = i = 1 n j = 1 n f ( y i ) f ( y j ) | y i - y j | . \mathrm{MD}=\sum_{i=1}^{n}\sum_{j=1}^{n}f(y_{i})f(y_{j})|y_{i}-y_{j}|.
  3. MD = - - f ( x ) f ( y ) | x - y | d x d y . \mathrm{MD}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x)\,f(y)\,|x-y|\,% dx\,dy.
  4. MD = 0 1 0 1 | F ( x ) - F ( y ) | d x d y . \mathrm{MD}=\int_{0}^{1}\int_{0}^{1}|F(x)-F(y)|\,dx\,dy.
  5. MD = 1 n 2 i = 1 n j = 1 n | y i - y j | . \mathrm{MD}=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}|y_{i}-y_{j}|.
  6. R M D = M D arithmetic mean . RMD=\frac{MD}{\mbox{arithmetic mean}~{}}.
  7. M D ( S ) = i = 1 n j = 1 n | y i - y j | n ( n - 1 ) MD(S)=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}|y_{i}-y_{j}|}{n(n-1)}
  8. R M D ( S ) = i = 1 n j = 1 n | y i - y j | ( n - 1 ) i = 1 n y i RMD(S)=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}|y_{i}-y_{j}|}{(n-1)\sum_{i=1}^{n}y_{% i}}
  9. E ( R ( S ) ) = i = 0 n p i ( 1 - p ) n - i r i , \operatorname{E}(R(S))=\sum_{i=0}^{n}p^{i}(1-p)^{n-i}r_{i},
  10. 1 12 0.2887 \frac{1}{\sqrt{12}}\approx 0.2887
  11. 2 π 1.1284 \frac{2}{\sqrt{\pi}}\approx 1.1284
  12. 2 π 1.1284 \frac{2}{\sqrt{\pi}}\approx 1.1284
  13. k k - 1 \frac{k}{k-1}
  14. 1 k - 1 k k - 2 \frac{1}{k-1}\,\sqrt{\frac{k}{k-2}}
  15. 2 k ( k - 1 ) ( 2 k - 1 ) \frac{2k}{(k-1)(2k-1)}\,
  16. 2 2 k - 1 \frac{2}{2k-1}\,
  17. k θ \sqrt{k}\,\theta
  18. k θ ( 2 - I 0.5 ( k + 1 , k ) ) k\theta(2-I_{0.5}(k+1,k))
  19. 2 - 4 I 0.5 ( k + 1 , k ) 2-4I_{0.5}(k+1,k)
  20. 2 1.4142 \sqrt{2}\approx 1.4142
  21. 3 1.7321 \sqrt{3}\approx 1.7321
  22. p ( 1 - p ) \sqrt{p(1-p)}
  23. \infty
  24. I z ( x , y ) I_{z}(x,y)

Medallion_knitting.html

  1. c c
  2. r r
  3. c = 2 π r c=2\pi r
  4. n n
  5. r r
  6. Δ n \Delta n
  7. Δ n = 2 π stitch gauge row gauge \Delta n=2\pi\frac{\mathrm{stitch\ gauge}}{\mathrm{row\ gauge}}
  8. Δ n \Delta n
  9. π \pi
  10. Δ n \Delta n

Mediation_(statistics).html

  1. \to
  2. Y = β 10 + β 11 X + ε 1 Y=\beta_{10}+\beta_{11}X+\varepsilon_{1}
  3. \to
  4. M e = β 20 + β 21 X + ε 2 Me=\beta_{20}+\beta_{21}X+\varepsilon_{2}
  5. Y = β 30 + β 31 X + β 32 M e + ε 3 Y=\beta_{30}+\beta_{31}X+\beta_{32}Me+\varepsilon_{3}
  6. \to
  7. \to
  8. \to
  9. \to
  10. \to
  11. Y = β 40 + β 41 X + β 42 M o + β 43 X M o + ε 4 Y=\beta_{40}+\beta_{41}X+\beta_{42}Mo+\beta_{43}XMo+\varepsilon_{4}
  12. M e = β 50 + β 51 X + β 52 M o + β 53 X M o + ε 5 Me=\beta_{50}+\beta_{51}X+\beta_{52}Mo+\beta_{53}XMo+\varepsilon_{5}
  13. Y = β 60 + β 61 X + β 62 M o + β 63 X M o + β 64 M e + β 65 M e M o + ε 6 Y=\beta_{60}+\beta_{61}X+\beta_{62}Mo+\beta_{63}XMo+\beta_{64}Me+\beta_{65}% MeMo+\varepsilon_{6}
  14. X = f ( ϵ 1 ) , M = g ( X , ϵ 2 ) , Y = h ( X , M , ϵ 3 ) , {X=f(\epsilon_{1}),~{}~{}M=g(X,\epsilon_{2}),~{}~{}Y=h(X,M,\epsilon_{3})},
  15. X = f ( ϵ 1 ) , M = m , Y = h ( X , m , ϵ 3 ) {X=f(\epsilon_{1}),~{}~{}M=m,~{}~{}Y=h(X,m,\epsilon_{3})}
  16. T E = E [ Y ( 1 ) - Y ( 0 ) ] TE=E[Y(1)-Y(0)]
  17. C D E ( m ) = E [ Y ( 1 , m ) - Y ( 0 , m ) ] CDE(m)=E[Y(1,m)-Y(0,m)]
  18. N D E = E [ Y ( 1 , M ( 0 ) ) - Y ( 0 , M ( 0 ) ) ] NDE=E[Y(1,M(0))-Y(0,M(0))]
  19. N I E = E [ Y ( 0 , M ( 1 ) - Y ( 0 , M ( 0 ) ] NIE=E[Y(0,M(1)-Y(0,M(0)]
  20. T E = N D E - N I E r TE=NDE-NIE_{r}
  21. T E \displaystyle TE
  22. T E \displaystyle TE
  23. X \displaystyle X
  24. c 3 c_{3}
  25. T E TE
  26. T E TE
  27. b 1 c 2 / T E b_{1}c_{2}/TE
  28. ( T E - c 1 ) / T E (TE-c_{1})/TE
  29. N D E \displaystyle NDE
  30. N I E / T E = b 1 c 2 / ( c 1 + b 0 c 3 + b 1 ( c 2 + c 3 ) ) , NIE/TE=b_{1}c_{2}/(c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})),
  31. 1 - N D E / T E = b 1 ( c 2 + c 3 ) / ( c 1 + b 0 c 3 + b 1 ( c 2 + c 3 ) ) . 1-NDE/TE=b_{1}(c_{2}+c_{3})/(c_{1}+b_{0}c_{3}+b_{1}(c_{2}+c_{3})).
  32. c 1 c_{1}

Meijer_G-function.html

  1. G p , q m , n ( a 1 , , a p b 1 , , b q | z ) = 1 2 π i L j = 1 m Γ ( b j - s ) j = 1 n Γ ( 1 - a j + s ) j = m + 1 q Γ ( 1 - b j + s ) j = n + 1 p Γ ( a j - s ) z s d s , G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}a_{1},\dots,a_{p}\\ b_{1},\dots,b_{q}\end{matrix}\;\right|\,z\right)=\frac{1}{2\pi i}\int_{L}\frac% {\prod_{j=1}^{m}\Gamma(b_{j}-s)\prod_{j=1}^{n}\Gamma(1-a_{j}+s)}{\prod_{j=m+1}% ^{q}\Gamma(1-b_{j}+s)\prod_{j=n+1}^{p}\Gamma(a_{j}-s)}\,z^{s}\,ds,
  2. G p , q m , n ( a 1 , , a p b 1 , , b q | z ) = G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) . G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}a_{1},\dots,a_{p}\\ b_{1},\dots,b_{q}\end{matrix}\;\right|\,z\right)=G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right).
  3. ν = j = 1 q b j - j = 1 p a j . \nu=\sum_{j=1}^{q}b_{j}-\sum_{j=1}^{p}a_{j}.
  4. [ ( - 1 ) p - m - n z j = 1 p ( z d d z - a j + 1 ) - j = 1 q ( z d d z - b j ) ] G ( z ) = 0. \left[(-1)^{p-m-n}\;z\prod_{j=1}^{p}\left(z\frac{d}{dz}-a_{j}+1\right)-\prod_{% j=1}^{q}\left(z\frac{d}{dz}-b_{j}\right)\right]G(z)=0.
  5. G p , q 1 , p ( a 1 , , a p b h , b 1 , , b h - 1 , b h + 1 , , b q | ( - 1 ) p - m - n + 1 z ) , h = 1 , 2 , , q , G_{p,q}^{\,1,p}\!\left(\left.\begin{matrix}a_{1},\dots,a_{p}\\ b_{h},b_{1},\dots,b_{h-1},b_{h+1},\dots,b_{q}\end{matrix}\;\right|\,(-1)^{p-m-% n+1}\;z\right),\quad h=1,2,\dots,q,
  6. G p , q q , 1 ( a h , a 1 , , a h - 1 , a h + 1 , , a p b 1 , , b q | ( - 1 ) q - m - n + 1 z ) , h = 1 , 2 , , p . G_{p,q}^{\,q,1}\!\left(\left.\begin{matrix}a_{h},a_{1},\dots,a_{h-1},a_{h+1},% \dots,a_{p}\\ b_{1},\dots,b_{q}\end{matrix}\;\right|\,(-1)^{q-m-n+1}\;z\right),\quad h=1,2,% \dots,p.
  7. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = h = 1 m j = 1 m Γ ( b j - b h ) * j = 1 n Γ ( 1 + b h - a j ) z b h j = m + 1 q Γ ( 1 + b h - b j ) j = n + 1 p Γ ( a j - b h ) × G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=\sum_{h=1}^{m}\frac{\prod_{j=1}^% {m}\Gamma(b_{j}-b_{h})^{*}\prod_{j=1}^{n}\Gamma(1+b_{h}-a_{j})\;z^{b_{h}}}{% \prod_{j=m+1}^{q}\Gamma(1+b_{h}-b_{j})\prod_{j=n+1}^{p}\Gamma(a_{j}-b_{h})}\times
  8. × p F q - 1 ( 1 + b h - 𝐚 𝐩 ( 1 + b h - 𝐛 𝐪 ) * | ( - 1 ) p - m - n z ) . \times\;_{p}F_{q-1}\!\left(\left.\begin{matrix}1+b_{h}-\mathbf{a_{p}}\\ (1+b_{h}-\mathbf{b_{q}})^{*}\end{matrix}\;\right|\,(-1)^{p-m-n}\;z\right).
  9. 1 + b h - 𝐛 𝐪 = ( 1 + b h - b 1 ) , , ( 1 + b h - b j ) , , ( 1 + b h - b q ) , 1+b_{h}-\mathbf{b_{q}}=(1+b_{h}-b_{1}),\,\dots,\,(1+b_{h}-b_{j}),\,\dots,\,(1+% b_{h}-b_{q}),
  10. G p , q 0 , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = 0 , G_{p,q}^{\,0,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=0,
  11. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = h = 1 n j = 1 n Γ ( a h - a j ) * j = 1 m Γ ( 1 - a h + b j ) z a h - 1 j = n + 1 p Γ ( 1 - a h + a j ) j = m + 1 q Γ ( a h - b j ) × G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=\sum_{h=1}^{n}\frac{\prod_{j=1}^% {n}\Gamma(a_{h}-a_{j})^{*}\prod_{j=1}^{m}\Gamma(1-a_{h}+b_{j})\;z^{a_{h}-1}}{% \prod_{j=n+1}^{p}\Gamma(1-a_{h}+a_{j})\prod_{j=m+1}^{q}\Gamma(a_{h}-b_{j})}\times
  12. × q F p - 1 ( 1 - a h + 𝐛 𝐪 ( 1 - a h + 𝐚 𝐩 ) * | ( - 1 ) q - m - n z - 1 ) . \times\;_{q}F_{p-1}\!\left(\left.\begin{matrix}1-a_{h}+\mathbf{b_{q}}\\ (1-a_{h}+\mathbf{a_{p}})^{*}\end{matrix}\;\right|\,(-1)^{q-m-n}z^{-1}\right).
  13. G p , q m , 0 ( 𝐚 𝐩 𝐛 𝐪 | z ) = 0 , G_{p,q}^{\,m,0}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=0,
  14. F q p ( 𝐚 𝐩 𝐛 𝐪 | z ) = Γ ( 𝐛 𝐪 ) Γ ( 𝐚 𝐩 ) G p , q + 1 1 , p ( 1 - 𝐚 𝐩 0 , 1 - 𝐛 𝐪 | - z ) = Γ ( 𝐛 𝐪 ) Γ ( 𝐚 𝐩 ) G q + 1 , p p , 1 ( 1 , 𝐛 𝐪 𝐚 𝐩 | - z - 1 ) , \;{}_{p}F_{q}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=\frac{\Gamma(\mathbf{b_{q}})}{% \Gamma(\mathbf{a_{p}})}\;G_{p,\,q+1}^{\,1,\,p}\!\left(\left.\begin{matrix}1-% \mathbf{a_{p}}\\ 0,1-\mathbf{b_{q}}\end{matrix}\;\right|\,-z\right)=\frac{\Gamma(\mathbf{b_{q}}% )}{\Gamma(\mathbf{a_{p}})}\;G_{q+1,\,p}^{\,p,\,1}\!\left(\left.\begin{matrix}1% ,\mathbf{b_{q}}\\ \mathbf{a_{p}}\end{matrix}\;\right|\,-z^{-1}\right),
  15. Γ ( 𝐚 𝐩 ) = j = 1 p Γ ( a j ) . \Gamma(\mathbf{a_{p}})=\prod_{j=1}^{p}\Gamma(a_{j}).
  16. F q p + 1 ( - h , 𝐚 𝐩 𝐛 𝐪 | z ) = h ! j = n + 1 p Γ ( 1 - a j ) j = m + 1 q Γ ( b j ) j = 1 n Γ ( a j ) j = 1 m Γ ( 1 - b j ) × \;{}_{p+1}F_{q}\!\left(\left.\begin{matrix}-h,\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=h!\;\frac{\prod_{j=n+1}^{p}% \Gamma(1-a_{j})\prod_{j=m+1}^{q}\Gamma(b_{j})}{\prod_{j=1}^{n}\Gamma(a_{j})% \prod_{j=1}^{m}\Gamma(1-b_{j})}\times
  17. × [ G p + 1 , q + 1 m + 1 , n ( 1 - 𝐚 𝐩 , h + 1 0 , 1 - 𝐛 𝐪 | ( - 1 ) p - m - n z ) + ( - 1 ) h G p + 1 , q + 1 m , n + 1 ( h + 1 , 1 - 𝐚 𝐩 1 - 𝐛 𝐪 , 0 | ( - 1 ) p - m - n z ) ] , \times\left[G_{p+1,\,q+1}^{\,m+1,\,n}\!\left(\left.\begin{matrix}1-\mathbf{a_{% p}},h+1\\ 0,1-\mathbf{b_{q}}\end{matrix}\;\right|\,(-1)^{p-m-n}\;z\right)+(-1)^{h}\;G_{p% +1,\,q+1}^{\,m,\,n+1}\!\left(\left.\begin{matrix}h+1,1-\mathbf{a_{p}}\\ 1-\mathbf{b_{q}},0\end{matrix}\;\right|\,(-1)^{p-m-n}\;z\right)\right],
  18. G p , q m , n ( a 1 , a 2 , , a p b 1 , , b q - 1 , a 1 | z ) = G p - 1 , q - 1 m , n - 1 ( a 2 , , a p b 1 , , b q - 1 | z ) , n , p , q 1. G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}a_{1},a_{2},\dots,a_{p}\\ b_{1},\dots,b_{q-1},a_{1}\end{matrix}\;\right|\,z\right)=G_{p-1,\,q-1}^{\,m,\,% n-1}\!\left(\left.\begin{matrix}a_{2},\dots,a_{p}\\ b_{1},\dots,b_{q-1}\end{matrix}\;\right|\,z\right),\quad n,p,q\geq 1.
  19. G p , q m , n ( a 1 , , a p - 1 , b 1 b 1 , b 2 , , b q | z ) = G p - 1 , q - 1 m - 1 , n ( a 1 , , a p - 1 b 2 , , b q | z ) , m , p , q 1. G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}a_{1},\dots,a_{p-1},b_{1}\\ b_{1},b_{2},\dots,b_{q}\end{matrix}\;\right|\,z\right)=G_{p-1,\,q-1}^{\,m-1,\,% n}\!\left(\left.\begin{matrix}a_{1},\dots,a_{p-1}\\ b_{2},\dots,b_{q}\end{matrix}\;\right|\,z\right),\quad m,p,q\geq 1.
  20. z ρ G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = G p , q m , n ( 𝐚 𝐩 + ρ 𝐛 𝐪 + ρ | z ) , z^{\rho}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}\mathbf{a_{p}}+\rho\\ \mathbf{b_{q}}+\rho\end{matrix}\;\right|\,z\right),
  21. G p + 2 , q m , n + 1 ( α , 𝐚 𝐩 , α 𝐛 𝐪 | z ) = ( - 1 ) α - α G p + 2 , q m , n + 1 ( α , 𝐚 𝐩 , α 𝐛 𝐪 | z ) , n p , α - α , G_{p+2,\,q}^{\,m,\,n+1}\!\left(\left.\begin{matrix}\alpha,\mathbf{a_{p}},% \alpha^{\prime}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=(-1)^{\alpha^{\prime}-\alpha}\;G% _{p+2,\,q}^{\,m,\,n+1}\!\left(\left.\begin{matrix}\alpha^{\prime},\mathbf{a_{p% }},\alpha\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right),\quad n\leq p,\;\alpha^{\prime}-% \alpha\in\mathbb{Z},
  22. G p , q + 2 m + 1 , n ( 𝐚 𝐩 β , 𝐛 𝐪 , β | z ) = ( - 1 ) β - β G p , q + 2 m + 1 , n ( 𝐚 𝐩 β , 𝐛 𝐪 , β | z ) , m q , β - β , G_{p,\,q+2}^{\,m+1,\,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \beta,\mathbf{b_{q}},\beta^{\prime}\end{matrix}\;\right|\,z\right)=(-1)^{\beta% ^{\prime}-\beta}\;G_{p,\,q+2}^{\,m+1,\,n}\!\left(\left.\begin{matrix}\mathbf{a% _{p}}\\ \beta^{\prime},\mathbf{b_{q}},\beta\end{matrix}\;\right|\,z\right),\quad m\leq q% ,\;\beta^{\prime}-\beta\in\mathbb{Z},
  23. G p + 1 , q + 1 m , n + 1 ( α , 𝐚 𝐩 𝐛 𝐪 , β | z ) = ( - 1 ) β - α G p + 1 , q + 1 m + 1 , n ( 𝐚 𝐩 , α β , 𝐛 𝐪 | z ) , m q , β - α = 0 , 1 , 2 , , G_{p+1,\,q+1}^{\,m,\,n+1}\!\left(\left.\begin{matrix}\alpha,\mathbf{a_{p}}\\ \mathbf{b_{q}},\beta\end{matrix}\;\right|\,z\right)=(-1)^{\beta-\alpha}\;G_{p+% 1,\,q+1}^{\,m+1,\,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}},\alpha\\ \beta,\mathbf{b_{q}}\end{matrix}\;\right|\,z\right),\quad m\leq q,\;\beta-% \alpha=0,1,2,\dots,
  24. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = G q , p n , m ( 1 - 𝐛 𝐪 1 - 𝐚 𝐩 | z - 1 ) , G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=G_{q,p}^{\,n,m}\!\left(\left.% \begin{matrix}1-\mathbf{b_{q}}\\ 1-\mathbf{a_{p}}\end{matrix}\;\right|\,z^{-1}\right),
  25. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = h 1 + ν + ( p - q ) / 2 ( 2 π ) ( h - 1 ) δ G h p , h q h m , h n ( a 1 / h , , ( a 1 + h - 1 ) / h , , a p / h , , ( a p + h - 1 ) / h b 1 / h , , ( b 1 + h - 1 ) / h , , b q / h , , ( b q + h - 1 ) / h | z h h h ( q - p ) ) , h . G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=\frac{h^{1+\nu+(p-q)/2}}{(2\pi)^% {(h-1)\delta}}\;G_{hp,\,hq}^{\,hm,\,hn}\!\left(\left.\begin{matrix}a_{1}/h,% \dots,(a_{1}+h-1)/h,\dots,a_{p}/h,\dots,(a_{p}+h-1)/h\\ b_{1}/h,\dots,(b_{1}+h-1)/h,\dots,b_{q}/h,\dots,(b_{q}+h-1)/h\end{matrix}\;% \right|\,\frac{z^{h}}{h^{h(q-p)}}\right),\quad h\in\mathbb{N}.
  26. d d z [ z 1 - a 1 G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) ] = z - a 1 G p , q m , n ( a 1 - 1 , a 2 , , a p 𝐛 𝐪 | z ) , n 1 , \frac{d}{dz}\left[z^{1-a_{1}}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)\right]=z^{-a_{1}}\;G_{p,q}^{\,m,% n}\!\left(\left.\begin{matrix}a_{1}-1,a_{2},\dots,a_{p}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right),\quad n\geq 1,
  27. d d z [ z 1 - a p G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) ] = - z - a p G p , q m , n ( a 1 , , a p - 1 , a p - 1 𝐛 𝐪 | z ) , n < p . \frac{d}{dz}\left[z^{1-a_{p}}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)\right]=-z^{-a_{p}}\;G_{p,q}^{\,m% ,n}\!\left(\left.\begin{matrix}a_{1},\dots,a_{p-1},a_{p}-1\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right),\quad n<p.
  28. d d z [ z - b 1 G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) ] = - z - 1 - b 1 G p , q m , n ( 𝐚 𝐩 b 1 + 1 , b 2 , , b q | z ) , m 1 , \frac{d}{dz}\left[z^{-b_{1}}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)\right]=-z^{-1-b_{1}}\;G_{p,q}^{% \,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ b_{1}+1,b_{2},\dots,b_{q}\end{matrix}\;\right|\,z\right),\quad m\geq 1,
  29. d d z [ z - b q G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) ] = z - 1 - b q G p , q m , n ( 𝐚 𝐩 b 1 , , b q - 1 , b q + 1 | z ) , m < q , \frac{d}{dz}\left[z^{-b_{q}}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)\right]=z^{-1-b_{q}}\;G_{p,q}^{\,% m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ b_{1},\dots,b_{q-1},b_{q}+1\end{matrix}\;\right|\,z\right),\quad m<q,
  30. z d d z G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = G p , q m , n ( a 1 - 1 , a 2 , , a p 𝐛 𝐪 | z ) + ( a 1 - 1 ) G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) , n 1. z\frac{d}{dz}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}a_{1}-1,a_{2},\dots,a_{p}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)+(a_{1}-1)\;G_{p,q}^{\,m,n}\!% \left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right),\quad n\geq 1.
  31. z h d h d z h G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = G p + 1 , q + 1 m , n + 1 ( 0 , 𝐚 𝐩 𝐛 𝐪 , h | z ) = ( - 1 ) h G p + 1 , q + 1 m + 1 , n ( 𝐚 𝐩 , 0 h , 𝐛 𝐪 | z ) , z^{h}\frac{d^{h}}{dz^{h}}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{% a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=G_{p+1,\,q+1}^{\,m,\,n+1}\!\left% (\left.\begin{matrix}0,\mathbf{a_{p}}\\ \mathbf{b_{q}},h\end{matrix}\;\right|\,z\right)=(-1)^{h}\;G_{p+1,\,q+1}^{\,m+1% ,\,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}},0\\ h,\mathbf{b_{q}}\end{matrix}\;\right|\,z\right),
  32. z h d h d z h G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z - 1 ) = G p + 1 , q + 1 m + 1 , n ( 𝐚 𝐩 , 1 - h 1 , 𝐛 𝐪 | z - 1 ) = ( - 1 ) h G p + 1 , q + 1 m , n + 1 ( 1 - h , 𝐚 𝐩 𝐛 𝐪 , 1 | z - 1 ) , z^{h}\frac{d^{h}}{dz^{h}}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{% a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z^{-1}\right)=G_{p+1,\,q+1}^{\,m+1,\,n}\!% \left(\left.\begin{matrix}\mathbf{a_{p}},1-h\\ 1,\mathbf{b_{q}}\end{matrix}\;\right|\,z^{-1}\right)=(-1)^{h}\;G_{p+1,\,q+1}^{% \,m,\,n+1}\!\left(\left.\begin{matrix}1-h,\mathbf{a_{p}}\\ \mathbf{b_{q}},1\end{matrix}\;\right|\,z^{-1}\right),
  33. ( b 1 - b q ) G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = G p , q m , n ( a 1 , , a p b 1 + 1 , b 2 , , b q | z ) + G p , q m , n ( a 1 , , a p b 1 , , b q - 1 , b q + 1 | z ) , 1 m < q , (b_{1}-b_{q})\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}a_{1},\dots,a_{p}\\ b_{1}+1,b_{2},\dots,b_{q}\end{matrix}\;\right|\,z\right)+G_{p,q}^{\,m,n}\!% \left(\left.\begin{matrix}a_{1},\dots,a_{p}\\ b_{1},\dots,b_{q-1},b_{q}+1\end{matrix}\;\right|\,z\right),\quad 1\leq m<q,
  34. ( b 1 - a 1 + 1 ) G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = G p , q m , n ( a 1 - 1 , a 2 , , a p b 1 , , b q | z ) + G p , q m , n ( a 1 , , a p b 1 + 1 , b 2 , , b q | z ) , n 1 , m 1 , (b_{1}-a_{1}+1)\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}a_{1}-1,a_{2},\dots,a_{p}\\ b_{1},\dots,b_{q}\end{matrix}\;\right|\,z\right)+G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}a_{1},\dots,a_{p}\\ b_{1}+1,b_{2},\dots,b_{q}\end{matrix}\;\right|\,z\right),\quad n\geq 1,\;m\geq 1,
  35. ( a p - b q - 1 ) G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z ) = G p , q m , n ( a 1 , , a p - 1 , a p - 1 b 1 , , b q | z ) + G p , q m , n ( a 1 , , a p b 1 , , b q - 1 , b q + 1 | z ) , n < p , m < q . (a_{p}-b_{q}-1)\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right)=G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}a_{1},\dots,a_{p-1},a_{p}-1\\ b_{1},\dots,b_{q}\end{matrix}\;\right|\,z\right)+G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}a_{1},\dots,a_{p}\\ b_{1},\dots,b_{q-1},b_{q}+1\end{matrix}\;\right|\,z\right),\quad n<p,\;m<q.
  36. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | w z ) = w b 1 h = 0 ( 1 - w ) h h ! G p , q m , n ( 𝐚 𝐩 b 1 + h , b 2 , , b q | z ) , m 1 , G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,wz\right)=w^{b_{1}}\sum_{h=0}^{\infty}% \frac{(1-w)^{h}}{h!}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}% }\\ b_{1}+h,b_{2},\dots,b_{q}\end{matrix}\;\right|\,z\right),\quad m\geq 1,
  37. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | w z ) = w b q h = 0 ( w - 1 ) h h ! G p , q m , n ( 𝐚 𝐩 b 1 , , b q - 1 , b q + h | z ) , m < q , G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,wz\right)=w^{b_{q}}\sum_{h=0}^{\infty}% \frac{(w-1)^{h}}{h!}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}% }\\ b_{1},\dots,b_{q-1},b_{q}+h\end{matrix}\;\right|\,z\right),\quad m<q,
  38. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z w ) = w 1 - a 1 h = 0 ( 1 - w ) h h ! G p , q m , n ( a 1 - h , a 2 , , a p 𝐛 𝐪 | z ) , n 1 , G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\frac{z}{w}\right)=w^{1-a_{1}}\sum_{h=0}^% {\infty}\frac{(1-w)^{h}}{h!}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}a_{1}% -h,a_{2},\dots,a_{p}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right),\quad n\geq 1,
  39. G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z w ) = w 1 - a p h = 0 ( w - 1 ) h h ! G p , q m , n ( a 1 , , a p - 1 , a p - h 𝐛 𝐪 | z ) , n < p . G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\frac{z}{w}\right)=w^{1-a_{p}}\sum_{h=0}^% {\infty}\frac{(w-1)^{h}}{h!}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}a_{1}% ,\dots,a_{p-1},a_{p}-h\\ \mathbf{b_{q}}\end{matrix}\;\right|\,z\right),\quad n<p.
  40. 0 x s - 1 G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | η x ) d x = η - s j = 1 m Γ ( b j + s ) j = 1 n Γ ( 1 - a j - s ) j = m + 1 q Γ ( 1 - b j - s ) j = n + 1 p Γ ( a j + s ) . \int_{0}^{\infty}x^{s-1}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a% _{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\eta x\right)dx=\frac{\eta^{-s}\prod_{j=1% }^{m}\Gamma(b_{j}+s)\prod_{j=1}^{n}\Gamma(1-a_{j}-s)}{\prod_{j=m+1}^{q}\Gamma(% 1-b_{j}-s)\prod_{j=n+1}^{p}\Gamma(a_{j}+s)}.
  41. 0 1 x - α ( 1 - x ) α - β - 1 G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z x ) d x = Γ ( α - β ) G p + 1 , q + 1 m , n + 1 ( α , 𝐚 𝐩 𝐛 𝐪 , β | z ) , \int_{0}^{1}x^{-\alpha}\;(1-x)^{\alpha-\beta-1}\;G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,zx\right)dx=\Gamma(\alpha-\beta)\;G_{p+1,% \,q+1}^{\,m,\,n+1}\!\left(\left.\begin{matrix}\alpha,\mathbf{a_{p}}\\ \mathbf{b_{q}},\beta\end{matrix}\;\right|\,z\right),
  42. 1 x - α ( x - 1 ) α - β - 1 G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | z x ) d x = Γ ( α - β ) G p + 1 , q + 1 m + 1 , n ( 𝐚 𝐩 , α β , 𝐛 𝐪 | z ) . \int_{1}^{\infty}x^{-\alpha}\;(x-1)^{\alpha-\beta-1}\;G_{p,q}^{\,m,n}\!\left(% \left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,zx\right)dx=\Gamma(\alpha-\beta)\;G_{p+1,% \,q+1}^{\,m+1,\,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}},\alpha\\ \beta,\mathbf{b_{q}}\end{matrix}\;\right|\,z\right).
  43. 0 G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | η x ) G σ , τ μ , ν ( 𝐜 σ 𝐝 τ | ω x ) d x = \int_{0}^{\infty}G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\eta x\right)G_{\sigma,\tau}^{\,\mu,\nu}% \!\left(\left.\begin{matrix}\mathbf{c_{\sigma}}\\ \mathbf{d_{\tau}}\end{matrix}\;\right|\,\omega x\right)dx=
  44. = 1 η G q + σ , p + τ n + μ , m + ν ( - b 1 , , - b m , 𝐜 σ , - b m + 1 , , - b q - a 1 , , - a n , 𝐝 τ , - a n + 1 , , - a p | ω η ) = =\frac{1}{\eta}\;G_{q+\sigma,\,p+\tau}^{\,n+\mu,\,m+\nu}\!\left(\left.\begin{% matrix}-b_{1},\dots,-b_{m},\mathbf{c_{\sigma}},-b_{m+1},\dots,-b_{q}\\ -a_{1},\dots,-a_{n},\mathbf{d_{\tau}},-a_{n+1},\dots,-a_{p}\end{matrix}\;% \right|\,\frac{\omega}{\eta}\right)=
  45. = 1 ω G p + τ , q + σ m + ν , n + μ ( a 1 , , a n , - 𝐝 τ , a n + 1 , , a p b 1 , , b m , - 𝐜 σ , b m + 1 , , b q | η ω ) . =\frac{1}{\omega}\;G_{p+\tau,\,q+\sigma}^{\,m+\nu,\,n+\mu}\!\left(\left.\begin% {matrix}a_{1},\dots,a_{n},-\mathbf{d_{\tau}},a_{n+1},\dots,a_{p}\\ b_{1},\dots,b_{m},-\mathbf{c_{\sigma}},b_{m+1},\dots,b_{q}\end{matrix}\;\right% |\,\frac{\eta}{\omega}\right).
  46. 0 e - ω x x - α G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | η x ) d x = ω α - 1 G p + 1 , q m , n + 1 ( α , 𝐚 𝐩 𝐛 𝐪 | η ω ) , \int_{0}^{\infty}e^{-\omega x}\;x^{-\alpha}\;G_{p,q}^{\,m,n}\!\left(\left.% \begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\eta x\right)dx=\omega^{\alpha-1}\;G_{p+1% ,\,q}^{\,m,\,n+1}\!\left(\left.\begin{matrix}\alpha,\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\frac{\eta}{\omega}\right),
  47. x - α G p , q + 1 m , n ( 𝐚 𝐩 𝐛 𝐪 , α | η x ) = 1 2 π i c - i c + i e ω x ω α - 1 G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | η ω ) d ω , x^{-\alpha}\;G_{p,\,q+1}^{\,m,\,n}\!\left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}},\alpha\end{matrix}\;\right|\,\eta x\right)=\frac{1}{2\pi i}\int% _{c-i\infty}^{c+i\infty}e^{\omega x}\;\omega^{\alpha-1}\;G_{p,q}^{\,m,n}\!% \left(\left.\begin{matrix}\mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\frac{\eta}{\omega}\right)d\omega,
  48. 0 e - ω x G p , q m , n ( 𝐚 𝐩 𝐛 𝐪 | η x 2 ) d x = 1 π ω G p + 2 , q m , n + 2 ( 0 , 1 2 , 𝐚 𝐩 𝐛 𝐪 | 4 η ω 2 ) , \int_{0}^{\infty}e^{-\omega x}\;G_{p,q}^{\,m,n}\!\left(\left.\begin{matrix}% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\eta x^{2}\right)dx=\frac{1}{\sqrt{\pi}% \omega}\;G_{p+2,\,q}^{\,m,\,n+2}\!\left(\left.\begin{matrix}0,\frac{1}{2},% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\,\frac{4\eta}{\omega^{2}}\right),
  49. g ( z ) = 0 k ( z , y ) f ( y ) d y , f ( z ) = 0 h ( z , y ) g ( y ) d y . g(z)=\int_{0}^{\infty}k(z,y)\,f(y)\;dy,\quad f(z)=\int_{0}^{\infty}h(z,y)\,g(y% )\;dy.
  50. k ( z , y ) = 2 γ ( z y ) γ - 1 / 2 G p + q , m + n m , p ( 𝐚 𝐩 , 𝐛 𝐪 𝐜 𝐦 , 𝐝 𝐧 | ( z y ) 2 γ ) , k(z,y)=2\gamma\;(zy)^{\gamma-1/2}\;G_{p+q,\,m+n}^{\,m,\,p}\!\left(\left.\begin% {matrix}\mathbf{a_{p}},\mathbf{b_{q}}\\ \mathbf{c_{m}},\mathbf{d_{n}}\end{matrix}\;\right|\,(zy)^{2\gamma}\right),
  51. h ( z , y ) = 2 γ ( z y ) γ - 1 / 2 G p + q , m + n n , q ( - 𝐛 𝐪 , - 𝐚 𝐩 - 𝐝 𝐧 , - 𝐜 𝐦 | ( z y ) 2 γ ) h(z,y)=2\gamma\;(zy)^{\gamma-1/2}\;G_{p+q,\,m+n}^{\,n,\,q}\!\left(\left.\begin% {matrix}-\mathbf{b_{q}},-\mathbf{a_{p}}\\ -\mathbf{d_{n}},-\mathbf{c_{m}}\end{matrix}\;\right|\,(zy)^{2\gamma}\right)
  52. j = 1 p a j + j = 1 q b j = j = 1 m c j + j = 1 n d j , \sum_{j=1}^{p}a_{j}+\sum_{j=1}^{q}b_{j}=\sum_{j=1}^{m}c_{j}+\sum_{j=1}^{n}d_{j},
  53. k ( z , y ) = G p + 2 , q m , n + 2 ( 1 - ν + i z , 1 - ν - i z , 𝐚 𝐩 𝐛 𝐪 | y ) , k(z,y)=G_{p+2,\,q}^{\,m,\,n+2}\!\left(\left.\begin{matrix}1-\nu+iz,1-\nu-iz,% \mathbf{a_{p}}\\ \mathbf{b_{q}}\end{matrix}\;\right|\;y\right),
  54. h ( z , y ) = i π y e - ν π i [ e π y A ( ν + i y , ν - i y | z e i π ) - e - π y A ( ν - i y , ν + i y | z e i π ) ] , h(z,y)=\frac{i}{\pi}ye^{-\nu\pi i}\left[e^{\pi y}A(\nu+iy,\nu-iy\,|\,ze^{i\pi}% )-e^{-\pi y}A(\nu-iy,\nu+iy\,|\,ze^{i\pi})\right],
  55. A ( α , β | z ) = G p + 2 , q q - m , p - n + 1 ( - a n + 1 , - a n + 2 , , - a p , α , - a 1 , - a 2 , , - a n , β - b m + 1 , - b m + 2 , , - b q , - b 1 , - b 2 , , - b m | z ) . A(\alpha,\beta\,|\,z)=G_{p+2,\,q}^{\,q-m,\,p-n+1}\!\left(\left.\begin{matrix}-% a_{n+1},-a_{n+2},\dots,-a_{p},\alpha,-a_{1},-a_{2},\dots,-a_{n},\beta\\ -b_{m+1},-b_{m+2},\dots,-b_{q},-b_{1},-b_{2},\dots,-b_{m}\end{matrix}\;\right|% \,z\right).
  56. g ( s ) = 2 γ 0 ( s t ) γ + ρ - 1 / 2 G p , q + 1 q + 1 , 0 ( 𝐚 𝐩 0 , 𝐛 𝐪 | ( s t ) 2 γ ) f ( t ) d t , g(s)=2\gamma\int_{0}^{\infty}(st)^{\gamma+\rho-1/2}\;G_{p,\,q+1}^{\,q+1,\,0}\!% \left(\left.\begin{matrix}\mathbf{a_{p}}\\ 0,\mathbf{b_{q}}\end{matrix}\;\right|\,(st)^{2\gamma}\right)f(t)\;dt,
  57. f ( t ) = γ π i c - i c + i ( t s ) γ - ρ - 1 / 2 G p , q + 1 1 , p ( - 𝐚 𝐩 0 , - 𝐛 𝐪 | - ( t s ) 2 γ ) g ( s ) d s , f(t)=\frac{\gamma}{\pi i}\int_{c-i\infty}^{c+i\infty}(ts)^{\gamma-\rho-1/2}\;G% _{p,\,q+1}^{\,1,\,p}\!\left(\left.\begin{matrix}-\mathbf{a_{p}}\\ 0,-\mathbf{b_{q}}\end{matrix}\;\right|\,-(ts)^{2\gamma}\right)g(s)\;ds,
  58. ( q + 1 - p ) ρ 2 γ = j = 1 p a j - j = 1 q b j , (q+1-p)\,{\rho\over 2\gamma}=\sum_{j=1}^{p}a_{j}-\sum_{j=1}^{q}b_{j},
  59. g ( s ) = 2 / π 0 ( s t ) 1 / 2 K ν ( s t ) f ( t ) d t , g(s)=\sqrt{2/\pi}\int_{0}^{\infty}(st)^{1/2}\,K_{\nu}(st)\,f(t)\;dt,
  60. f ( t ) = 1 2 π i c - i c + i ( t s ) 1 / 2 I ν ( t s ) g ( s ) d s , f(t)=\frac{1}{\sqrt{2\pi}\,i}\int_{c-i\infty}^{c+i\infty}(ts)^{1/2}\,I_{\nu}(% ts)\,g(s)\;ds,
  61. g ( s ) = 0 ( s t ) - k - 1 / 2 e - s t / 2 W k + 1 / 2 , m ( s t ) f ( t ) d t , g(s)=\int_{0}^{\infty}(st)^{-k-1/2}\,e^{-st/2}\,W_{k+1/2,\,m}(st)\,f(t)\;dt,
  62. f ( t ) = Γ ( 1 - k + m ) 2 π i Γ ( 1 + 2 m ) c - i c + i ( t s ) k - 1 / 2 e t s / 2 M k - 1 / 2 , m ( t s ) g ( s ) d s . f(t)=\frac{\Gamma(1-k+m)}{2\pi i\,\Gamma(1+2m)}\int_{c-i\infty}^{c+i\infty}(ts% )^{k-1/2}\,e^{ts/2}\,M_{k-1/2,\,m}(ts)\,g(s)\;ds.
  63. H ( 1 - | x | ) = G 1 , 1 1 , 0 ( 1 0 | x ) , x H(1-|x|)=G_{1,1}^{\,1,0}\!\left(\left.\begin{matrix}1\\ 0\end{matrix}\;\right|\,x\right),\qquad\forall x
  64. H ( | x | - 1 ) = G 1 , 1 0 , 1 ( 1 0 | x ) , x H(|x|-1)=G_{1,1}^{\,0,1}\!\left(\left.\begin{matrix}1\\ 0\end{matrix}\;\right|\,x\right),\qquad\forall x
  65. e x = G 0 , 1 1 , 0 ( - 0 | - x ) , x e^{x}=G_{0,1}^{\,1,0}\!\left(\left.\begin{matrix}-\\ 0\end{matrix}\;\right|\,-x\right),\qquad\forall x
  66. cos x = π G 0 , 2 1 , 0 ( - 0 , 1 2 | x 2 4 ) , x \cos x=\sqrt{\pi}\;G_{0,2}^{\,1,0}\!\left(\left.\begin{matrix}-\\ 0,\frac{1}{2}\end{matrix}\;\right|\,\frac{x^{2}}{4}\right),\qquad\forall x
  67. sin x = π G 0 , 2 1 , 0 ( - 1 2 , 0 | x 2 4 ) , - π 2 < arg x π 2 \sin x=\sqrt{\pi}\;G_{0,2}^{\,1,0}\!\left(\left.\begin{matrix}-\\ \frac{1}{2},0\end{matrix}\;\right|\,\frac{x^{2}}{4}\right),\qquad\frac{-\pi}{2% }<\arg x\leq\frac{\pi}{2}
  68. cosh x = π G 0 , 2 1 , 0 ( - 0 , 1 2 | - x 2 4 ) , x \cosh x=\sqrt{\pi}\;G_{0,2}^{\,1,0}\!\left(\left.\begin{matrix}-\\ 0,\frac{1}{2}\end{matrix}\;\right|\,-\frac{x^{2}}{4}\right),\qquad\forall x
  69. sinh x = - π i G 0 , 2 1 , 0 ( - 1 2 , 0 | - x 2 4 ) , - π < arg x 0 \sinh x=-\sqrt{\pi}i\;G_{0,2}^{\,1,0}\!\left(\left.\begin{matrix}-\\ \frac{1}{2},0\end{matrix}\;\right|\,-\frac{x^{2}}{4}\right),\qquad-\pi<\arg x\leq 0
  70. arcsin x = - i 2 π G 2 , 2 1 , 2 ( 1 , 1 1 2 , 0 | - x 2 ) , - π < arg x 0 \arcsin x=\frac{-i}{2\sqrt{\pi}}\;G_{2,2}^{\,1,2}\!\left(\left.\begin{matrix}1% ,1\\ \frac{1}{2},0\end{matrix}\;\right|\,-x^{2}\right),\qquad-\pi<\arg x\leq 0
  71. arctan x = 1 2 G 2 , 2 1 , 2 ( 1 2 , 1 1 2 , 0 | x 2 ) , - π 2 < arg x π 2 \arctan x=\frac{1}{2}\;G_{2,2}^{\,1,2}\!\left(\left.\begin{matrix}\frac{1}{2},% 1\\ \frac{1}{2},0\end{matrix}\;\right|\,x^{2}\right),\qquad\frac{-\pi}{2}<\arg x% \leq\frac{\pi}{2}
  72. \arccot x = 1 2 G 2 , 2 2 , 1 ( 1 2 , 1 1 2 , 0 | x 2 ) , - π 2 < arg x π 2 \arccot x=\frac{1}{2}\;G_{2,2}^{\,2,1}\!\left(\left.\begin{matrix}\frac{1}{2},% 1\\ \frac{1}{2},0\end{matrix}\;\right|\,x^{2}\right),\qquad\frac{-\pi}{2}<\arg x% \leq\frac{\pi}{2}
  73. ln ( 1 + x ) = G 2 , 2 1 , 2 ( 1 , 1 1 , 0 | x ) , x \ln(1+x)=G_{2,2}^{\,1,2}\!\left(\left.\begin{matrix}1,1\\ 1,0\end{matrix}\;\right|\,x\right),\qquad\forall x
  74. γ ( α , x ) = G 1 , 2 1 , 1 ( 1 α , 0 | x ) , x \gamma(\alpha,x)=G_{1,2}^{\,1,1}\!\left(\left.\begin{matrix}1\\ \alpha,0\end{matrix}\;\right|\,x\right),\qquad\forall x
  75. Γ ( α , x ) = G 1 , 2 2 , 0 ( 1 α , 0 | x ) , x \Gamma(\alpha,x)=G_{1,2}^{\,2,0}\!\left(\left.\begin{matrix}1\\ \alpha,0\end{matrix}\;\right|\,x\right),\qquad\forall x
  76. J ν ( x ) = G 0 , 2 1 , 0 ( - ν 2 , - ν 2 | x 2 4 ) , - π 2 < arg x π 2 J_{\nu}(x)=G_{0,2}^{\,1,0}\!\left(\left.\begin{matrix}-\\ \frac{\nu}{2},\frac{-\nu}{2}\end{matrix}\;\right|\,\frac{x^{2}}{4}\right),% \qquad\frac{-\pi}{2}<\arg x\leq\frac{\pi}{2}
  77. Y ν ( x ) = G 1 , 3 2 , 0 ( - ν - 1 2 ν 2 , - ν 2 , - ν - 1 2 | x 2 4 ) , - π 2 < arg x π 2 Y_{\nu}(x)=G_{1,3}^{\,2,0}\!\left(\left.\begin{matrix}\frac{-\nu-1}{2}\\ \frac{\nu}{2},\frac{-\nu}{2},\frac{-\nu-1}{2}\end{matrix}\;\right|\,\frac{x^{2% }}{4}\right),\qquad\frac{-\pi}{2}<\arg x\leq\frac{\pi}{2}
  78. I ν ( x ) = i - ν G 0 , 2 1 , 0 ( - ν 2 , - ν 2 | - x 2 4 ) , - π < arg x 0 I_{\nu}(x)=i^{-\nu}\;G_{0,2}^{\,1,0}\!\left(\left.\begin{matrix}-\\ \frac{\nu}{2},\frac{-\nu}{2}\end{matrix}\;\right|\,-\frac{x^{2}}{4}\right),% \qquad-\pi<\arg x\leq 0
  79. K ν ( x ) = 1 2 G 0 , 2 2 , 0 ( - ν 2 , - ν 2 | x 2 4 ) , - π 2 < arg x π 2 K_{\nu}(x)=\frac{1}{2}\;G_{0,2}^{\,2,0}\!\left(\left.\begin{matrix}-\\ \frac{\nu}{2},\frac{-\nu}{2}\end{matrix}\;\right|\,\frac{x^{2}}{4}\right),% \qquad\frac{-\pi}{2}<\arg x\leq\frac{\pi}{2}
  80. Φ ( x , n , a ) = G n + 1 , n + 1 1 , n + 1 ( 0 , 1 - a , , 1 - a 0 , - a , , - a | - x ) , x , n = 0 , 1 , 2 , \Phi(x,n,a)=G_{n+1,\,n+1}^{\,1,\,n+1}\!\left(\left.\begin{matrix}0,1-a,\dots,1% -a\\ 0,-a,\dots,-a\end{matrix}\;\right|\,-x\right),\qquad\forall x,\;n=0,1,2,\dots
  81. Φ ( x , - n , a ) = G n + 1 , n + 1 1 , n + 1 ( 0 , - a , , - a 0 , 1 - a , , 1 - a | - x ) , x , n = 0 , 1 , 2 , \Phi(x,-n,a)=G_{n+1,\,n+1}^{\,1,\,n+1}\!\left(\left.\begin{matrix}0,-a,\dots,-% a\\ 0,1-a,\dots,1-a\end{matrix}\;\right|\,-x\right),\qquad\forall x,\;n=0,1,2,\dots

Merton's_portfolio_problem.html

  1. max E [ 0 T e - ρ s u ( c s ) d s + e - ρ T u ( W T ) ] \max E\left[\int_{0}^{T}e^{-\rho s}u(c_{s})\,ds+e^{-\rho T}u(W_{T})\right]
  2. d W t = [ ( r + π t ( μ - r ) ) W t - c t ] d t + W t π t σ d B t dW_{t}=[(r+\pi_{t}(\mu-r))W_{t}-c_{t}]\,dt+W_{t}\pi_{t}\sigma\,dB_{t}
  3. u ( x ) = x 1 - γ 1 - γ , u(x)=\frac{x^{1-\gamma}}{1-\gamma},
  4. γ \gamma
  5. π ( W , t ) = μ - r σ 2 γ . \pi(W,t)=\frac{\mu-r}{\sigma^{2}\gamma}.
  6. c ( W , t ) = { ν ( 1 + ( ν ϵ - 1 ) e - ν ( T - t ) ) - 1 W if T < and ν 0 ( T - t + ϵ ) - 1 W if T < and ν = 0 ν W if T = c(W,t)=\begin{cases}\nu\left(1+(\nu\epsilon-1)e^{-\nu(T-t)}\right)^{-1}W&% \textrm{if}\;T<\infty\;\textrm{and}\;\nu\neq 0\\ (T-t+\epsilon)^{-1}W&\textrm{if}\;T<\infty\;\textrm{and}\;\nu=0\\ \nu W&\textrm{if}\;T=\infty\end{cases}
  7. 0 ϵ 1 0\leq\epsilon\ll 1
  8. ν = ( ρ - ( 1 - γ ) ( ( μ - r ) 2 2 σ 2 γ + r ) ) / γ = ρ / γ - ( 1 - γ ) ( ( μ - r ) 2 2 σ 2 γ 2 + r γ ) = ρ / γ - ( 1 - γ ) ( π ( W , t ) 2 / 2 σ 2 + r / γ ) = ρ / γ - ( 1 - γ ) ( ( μ - r ) π ( W , t ) / 2 γ + r / γ ) . \begin{aligned}\displaystyle\nu&\displaystyle=\left(\rho-(1-\gamma)\left(\frac% {(\mu-r)^{2}}{2\sigma^{2}\gamma}+r\right)\right)/\gamma\\ &\displaystyle=\rho/\gamma-(1-\gamma)\left(\frac{(\mu-r)^{2}}{2\sigma^{2}% \gamma^{2}}+\frac{r}{\gamma}\right)\\ &\displaystyle=\rho/\gamma-(1-\gamma)(\pi(W,t)^{2}/2\sigma^{2}+r/\gamma)\\ &\displaystyle=\rho/\gamma-(1-\gamma)((\mu-r)\pi(W,t)/2\gamma+r/\gamma).\end{aligned}
  9. ρ \rho
  10. r , μ , σ r,\mu,\sigma

Meshfree_methods.html

  1. u ( x , t ) u(x,t)
  2. u i n u_{i}^{n}
  3. x i x_{i}
  4. i = 0 , 1 , 2... i=0,1,2...
  5. n = 0 , 1 , 2... n=0,1,2...
  6. x i + 1 - x i = h i x_{i+1}-x_{i}=h\ \forall i
  7. t n + 1 - t n = k n t_{n+1}-t_{n}=k\ \forall n
  8. u x = u i + 1 n - u i - 1 n 2 h {\partial u\over\partial x}={u_{i+1}^{n}-u_{i-1}^{n}\over 2h}
  9. u t = u i n + 1 - u i n k {\partial u\over\partial t}={u_{i}^{n+1}-u_{i}^{n}\over k}
  10. u ( x , t ) u(x,t)
  11. h h
  12. k k
  13. x i x_{i}
  14. x i - 1 x_{i-1}
  15. x i + 1 x_{i+1}
  16. u i u_{i}
  17. u ( x , t ) u(x,t)
  18. u ( x , t n ) = i m i u i n ρ i W ( | x - x i | ) u(x,t_{n})=\sum_{i}m_{i}\frac{u_{i}^{n}}{\rho_{i}}W(|x-x_{i}|)
  19. m i m_{i}
  20. i i
  21. ρ i \rho_{i}
  22. i i
  23. W W
  24. u x = i m i u i n ρ i W ( | x - x i | ) x {\partial u\over\partial x}=\sum_{i}m_{i}\frac{u_{i}^{n}}{\rho_{i}}{\partial W% (|x-x_{i}|)\over\partial x}
  25. u ( x , t ) u(x,t)
  26. u ( x , t ) u(x,t)
  27. W W

Metabolic_control_analysis.html

  1. v i v_{i}
  2. C v i J = ( d J d p p J ) / ( v i p p v i ) = d ln J d ln v i C^{J}_{v_{i}}=\left(\frac{dJ}{dp}\frac{p}{J}\right)\bigg/\left(\frac{\partial v% _{i}}{\partial p}\frac{p}{v_{i}}\right)=\frac{d\ln J}{d\ln v_{i}}
  3. C v i S = ( d S d p p S ) / ( v i p p v i ) = d ln S d ln v i C^{S}_{v_{i}}=\left(\frac{dS}{dp}\frac{p}{S}\right)\bigg/\left(\frac{\partial v% _{i}}{\partial p}\frac{p}{v_{i}}\right)=\frac{d\ln S}{d\ln v_{i}}
  4. i C v i J = 1 \sum_{i}C^{J}_{v_{i}}=1
  5. i C v i S = 0 \sum_{i}C^{S}_{v_{i}}=0
  6. S n S_{n}
  7. S m S_{m}
  8. i C i J ε S i = 0 \sum_{i}C^{J}_{i}\varepsilon^{i}_{S}=0
  9. i C i S n ε S m i = 0 n m \sum_{i}C^{S_{n}}_{i}\varepsilon^{i}_{S_{m}}=0\quad n\neq m
  10. i C i S n ε S m i = - 1 n = m \sum_{i}C^{S_{n}}_{i}\varepsilon^{i}_{S_{m}}=-1\quad n=m
  11. X o S X 1 X_{o}\rightarrow S\rightarrow X_{1}
  12. X o X_{o}
  13. X 1 X_{1}
  14. v 1 v_{1}
  15. v 2 v_{2}
  16. C v 1 J + C v 2 J = 1 C^{J}_{v_{1}}+C^{J}_{v_{2}}=1
  17. C v 1 J ε S v 1 + C v 2 J ε S v 2 = 0 C^{J}_{v_{1}}\varepsilon^{v_{1}}_{S}+C^{J}_{v_{2}}\varepsilon^{v_{2}}_{S}=0
  18. C v 1 J = ε S 2 ε S 2 - ε S 1 C^{J}_{v_{1}}=\frac{\varepsilon^{2}_{S}}{\varepsilon^{2}_{S}-\varepsilon^{1}_{% S}}
  19. C v 2 J = - ε S 1 ε S 2 - ε S 1 C^{J}_{v_{2}}=\frac{-\varepsilon^{1}_{S}}{\varepsilon^{2}_{S}-\varepsilon^{1}_% {S}}
  20. ε S v 1 = 0 \varepsilon^{v_{1}}_{S}=0
  21. C v 1 J = 1 C^{J}_{v_{1}}=1
  22. C v 2 J = 0 C^{J}_{v_{2}}=0
  23. C v 1 S = 1 ε S 2 - ε S 1 C^{S}_{v_{1}}=\frac{1}{\varepsilon^{2}_{S}-\varepsilon^{1}_{S}}
  24. C v 2 S = - 1 ε S 2 - ε S 1 C^{S}_{v_{2}}=\frac{-1}{\varepsilon^{2}_{S}-\varepsilon^{1}_{S}}
  25. E 1 E_{1}
  26. v 1 v_{1}
  27. E 1 E_{1}
  28. v 1 v_{1}
  29. v 2 v_{2}
  30. δ v 1 v 1 = ε E 1 1 δ E 1 E 1 + ε S 1 δ S S \frac{\delta v_{1}}{v_{1}}=\varepsilon^{1}_{E_{1}}\frac{\delta E_{1}}{E_{1}}+% \varepsilon^{1}_{S}\frac{\delta S}{S}
  31. δ v 2 v 2 = ε S 2 δ S S \frac{\delta v_{2}}{v_{2}}=\varepsilon^{2}_{S}\frac{\delta S}{S}
  32. v 1 v_{1}
  33. E 1 E_{1}
  34. δ J J = δ E 1 E 1 + ε S 1 δ S S \frac{\delta J}{J}=\frac{\delta E_{1}}{E_{1}}+\varepsilon^{1}_{S}\frac{\delta S% }{S}
  35. δ J J = ε S 2 δ S S \frac{\delta J}{J}=\varepsilon^{2}_{S}\frac{\delta S}{S}
  36. E 1 E_{1}
  37. δ E 1 0 \delta E_{1}\rightarrow 0
  38. C E 1 J = 1 + ε S 1 C E 1 S C^{J}_{E_{1}}=1+\varepsilon^{1}_{S}C^{S}_{E_{1}}
  39. C E 1 J = ε S 2 C E 1 S C^{J}_{E_{1}}=\varepsilon^{2}_{S}C^{S}_{E_{1}}
  40. C E 1 J C^{J}_{E_{1}}
  41. C E 1 S C^{S}_{E_{1}}
  42. E 2 E_{2}
  43. E 1 E_{1}
  44. E 2 E_{2}
  45. δ v i / v i = δ E i / E i \delta v_{i}/v_{i}=\delta E_{i}/E_{i}
  46. v i v_{i}
  47. X o S 1 S 2 X 1 X_{o}\rightarrow S_{1}\rightarrow S_{2}\rightarrow X_{1}
  48. X o X_{o}
  49. X 1 X_{1}
  50. C E 1 J = ε 1 2 ε 2 3 / D C^{J}_{E_{1}}=\varepsilon^{2}_{1}\varepsilon^{3}_{2}/D
  51. C E 2 J = - ε 1 1 ε 2 3 / D C^{J}_{E_{2}}=-\varepsilon^{1}_{1}\varepsilon^{3}_{2}/D
  52. C E 3 J = ε 1 1 ε 2 2 / D C^{J}_{E_{3}}=\varepsilon^{1}_{1}\varepsilon^{2}_{2}/D
  53. D = ε 1 2 ε 2 3 - ε 1 1 ε 2 3 + ε 1 1 ε 2 2 D=\varepsilon^{2}_{1}\varepsilon^{3}_{2}-\varepsilon^{1}_{1}\varepsilon^{3}_{2% }+\varepsilon^{1}_{1}\varepsilon^{2}_{2}
  54. S 1 S_{1}
  55. C E 1 S 1 = ( ε 2 3 - ε 2 2 ) / D C^{S_{1}}_{E_{1}}=(\varepsilon^{3}_{2}-\varepsilon^{2}_{2})/D
  56. C E 2 S 1 = - ε 2 3 / D C^{S_{1}}_{E_{2}}=-\varepsilon^{3}_{2}/D
  57. C E 3 S 1 = ε 2 2 / D C^{S_{1}}_{E_{3}}=\varepsilon^{2}_{2}/D
  58. S 2 S_{2}
  59. C E 1 S 2 = ε 1 2 / D C^{S_{2}}_{E_{1}}=\varepsilon^{2}_{1}/D
  60. C E 2 S 2 = - ε 1 1 / D C^{S_{2}}_{E_{2}}=-\varepsilon^{1}_{1}/D
  61. C E 3 S 2 = ( ε 1 1 - ε 1 2 ) / D C^{S_{2}}_{E_{3}}=(\varepsilon^{1}_{1}-\varepsilon^{2}_{1})/D

Metacyclic_group.html

  1. 1 K G H 1 , 1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1,\,

Metamath.html

  1. φ & ( φ ψ ) ψ \vdash\varphi\quad\&\quad\vdash(\varphi\rightarrow\psi)\quad\Rightarrow\quad\vdash\psi
  2. ( 2 + 2 ) = ( 2 + ( 1 + 1 ) ) (2+2)=(2+(1+1))
  3. A = B A=B
  4. ( C F A ) = ( C F B ) (CFA)=(CFB)
  5. ( C F A ) = ( C F B ) (CFA)=(CFB)
  6. ( 2 + 2 ) = ( 2 + ( 1 + 1 ) ) (2+2)=(2+(1+1))
  7. C C
  8. 2 2
  9. F F
  10. + +
  11. A A
  12. 2 2
  13. B B
  14. ( 1 + 1 ) (1+1)
  15. A = B A=B
  16. A A
  17. 2 2
  18. B B
  19. ( 1 + 1 ) (1+1)
  20. A = B A=B
  21. 2 = ( 1 + 1 ) 2=(1+1)
  22. ( 2 + 2 ) (2+2)
  23. B B
  24. B B
  25. ( 2 + 2 ) (2+2)

METEOR.html

  1. P P
  2. P = m w t P=\frac{m}{w_{t}}
  3. m m
  4. w t w_{t}
  5. R R
  6. R = m w r R=\frac{m}{w_{r}}
  7. m m
  8. w r w_{r}
  9. F m e a n = 10 P R R + 9 P F_{mean}=\frac{10PR}{R+9P}
  10. p p
  11. p p
  12. p = 0.5 ( c u m ) 3 p=0.5\left(\frac{c}{u_{m}}\right)^{3}
  13. u m u_{m}
  14. M M
  15. F m e a n F_{mean}
  16. M = F m e a n ( 1 - p ) M=F_{mean}(1-p)
  17. P P
  18. R R
  19. p p

Methanol_reformer.html

  1. CH 3 OH ( g ) + H 2 O ( g ) CO 2 + 3 H 2 Δ H R 298 0 = 49.2 kJ / mol \mathrm{CH_{3}OH_{(g)}+H_{2}O_{(g)}\;\longrightarrow\;CO_{2}+3\ H_{2}\qquad}% \Delta H_{R\ 298}^{0}=49.2\ \mathrm{kJ/mol}

Method_of_moments_(probability_theory).html

  1. E ( X k ) \operatorname{E}(X^{k})\,
  2. lim n E ( X n k ) = E ( X k ) \lim_{n\to\infty}\operatorname{E}(X_{n}^{k})=\operatorname{E}(X^{k})\,

Methods_of_detecting_exoplanets.html

  1. M true * sin i M\text{true}*{\sin i}\,

Metric_derivative.html

  1. ( M , d ) (M,d)
  2. E E\subseteq\mathbb{R}
  3. t t\in\mathbb{R}
  4. γ : E M \gamma:E\to M
  5. γ \gamma
  6. t t
  7. | γ | ( t ) |\gamma^{\prime}|(t)
  8. | γ | ( t ) := lim s 0 d ( γ ( t + s ) , γ ( t ) ) | s | , |\gamma^{\prime}|(t):=\lim_{s\to 0}\frac{d(\gamma(t+s),\gamma(t))}{|s|},
  9. d ( γ ( s ) , γ ( t ) ) s t m ( τ ) d τ for all [ s , t ] I d\left(\gamma(s),\gamma(t)\right)\leq\int_{s}^{t}m(\tau)\,\mathrm{d}\tau\mbox{% for all }~{}[s,t]\subseteq I
  10. n \mathbb{R}^{n}
  11. - \|-\|
  12. γ ˙ : E V * \dot{\gamma}:E\to V^{*}
  13. | γ | ( t ) = γ ˙ ( t ) , |\gamma^{\prime}|(t)=\|\dot{\gamma}(t)\|,
  14. d ( x , y ) := x - y d(x,y):=\|x-y\|

Metric_expansion_of_space.html

  1. Ω m \Omega_{m}

Metzler_matrix.html

  1. i j x i j 0. \qquad\forall_{i\neq j}\,x_{ij}\geq 0.
  2. A = ( a i j ) ; a i j 0 , i j . A=(a_{ij});\quad a_{ij}\geq 0,\quad i\neq j.
  3. Z ( - ) Z^{(-)}

Meyer–Schuster_rearrangement.html

  1. 3 {}_{3}

MICEX_10.html

  1. t t
  2. MICEX10 = k 10 i = 1 10 ( P i P i 0 ) \,\text{MICEX10}=\frac{k}{10}\cdot\sum\limits_{i=1}^{10}\left(\frac{P_{i}}{P^{% 0}_{i}}\right)
  3. P i P_{i}
  4. i i
  5. t t
  6. P i 0 P^{0}_{i}
  7. i i
  8. k k

Micro_heat_exchanger.html

  1. h = N u c k d h=Nu_{c}\frac{k}{d}
  2. h h
  3. N u c Nu_{c}
  4. k k
  5. d d
  6. N u c = 3.657 Nu_{c}=3.657
  7. N u c = 4.364 Nu_{c}=4.364

Microparticles.html

  1. μ \mu
  2. μ \mu
  3. 10 - 6 10^{-6}
  4. 10 - 7.5 10^{-7.5}
  5. 10 - 4.5 10^{-4.5}
  6. μ \mu
  7. μ \mu

Midnight_Zoo.html

  1. 12 c 3 12c3

Mikhail_Yakovlevich_Suslin.html

  1. \R 2 \R^{2}

Military_equipment_of_Israel.html

  1. } \Bigg\}
  2. } \Big\}
  3. } \Big\}
  4. } \Big\}
  5. } \Big\}
  6. } \Big\}
  7. } \Big\}
  8. } \Big\}
  9. } \Big\}
  10. } \Big\}
  11. } \Big\}
  12. } \Big\}

Milne_model.html

  1. d s 2 = d t 2 - t 2 ( d χ 2 + sinh 2 χ d Ω 2 ) ds^{2}=dt^{2}-t^{2}(d\chi^{2}+\sinh^{2}{\chi}d\Omega^{2})
  2. d Ω 2 = d θ 2 + sin 2 θ d ϕ 2 d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}
  3. χ = sinh - 1 r \chi=\sinh^{-1}{r}
  4. + +\infty
  5. ρ = 0 \rho=0

Milstein_method.html

  1. d X t = a ( X t ) d t + b ( X t ) d W t , \mathrm{d}X_{t}=a(X_{t})\,\mathrm{d}t+b(X_{t})\,\mathrm{d}W_{t},
  2. Δ t > 0 \Delta t>0
  3. 0 = τ 0 < τ 1 < < τ N = T with τ n := n Δ t and Δ t = T N ; 0=\tau_{0}<\tau_{1}<\dots<\tau_{N}=T\,\text{ with }\tau_{n}:=n\Delta t\,\text{% and }\Delta t=\frac{T}{N};
  4. Y 0 = x 0 ; Y_{0}=x_{0};
  5. Y n Y_{n}
  6. 1 n N 1\leq n\leq N
  7. Y n + 1 = Y n + a ( Y n ) Δ t + b ( Y n ) Δ W n + 1 2 b ( Y n ) b ( Y n ) ( ( Δ W n ) 2 - Δ t ) , Y_{n+1}=Y_{n}+a(Y_{n})\Delta t+b(Y_{n})\Delta W_{n}+\frac{1}{2}b(Y_{n})b^{% \prime}(Y_{n})\left((\Delta W_{n})^{2}-\Delta t\right),
  8. b b^{\prime}
  9. b ( x ) b(x)
  10. x x
  11. Δ W n = W τ n + 1 - W τ n \Delta W_{n}=W_{\tau_{n+1}}-W_{\tau_{n}}
  12. Δ t \Delta t
  13. Y n Y_{n}
  14. X τ n X_{\tau_{n}}
  15. 0 n N 0\leq n\leq N
  16. N N
  17. Δ t \Delta t
  18. ( Δ t ) 1 / 2 (\Delta t)^{1/2}
  19. d X t = μ X d t + σ X d W t \mathrm{d}X_{t}=\mu X\mathrm{d}t+\sigma XdW_{t}
  20. μ \mu
  21. σ \sigma
  22. d ln X t = ( μ - 1 2 σ 2 ) d t + σ d W t , \mathrm{d}\ln X_{t}=\left(\mu-\frac{1}{2}\sigma^{2}\right)\mathrm{d}t+\sigma% \mathrm{d}W_{t},
  23. X t + Δ t = X t exp { t t + Δ t ( μ - 1 2 σ 2 ) d t + t t + Δ t σ d W u } X t ( 1 + μ Δ t - 1 2 σ 2 Δ t + σ Δ W t + 1 2 σ 2 ( Δ W t ) 2 ) = X t + a ( X t ) Δ t + b ( X t ) Δ W t + 1 2 b ( X t ) b ( X t ) ( ( Δ W t ) 2 - Δ t ) \begin{aligned}\displaystyle X_{t+\Delta t}&\displaystyle=X_{t}\exp\left\{\int% _{t}^{t+\Delta t}\left(\mu-\frac{1}{2}\sigma^{2}\right)\mathrm{d}t+\int_{t}^{t% +\Delta t}\sigma\mathrm{d}W_{u}\right\}\\ &\displaystyle\approx X_{t}\left(1+\mu\Delta t-\frac{1}{2}\sigma^{2}\Delta t+% \sigma\Delta W_{t}+\frac{1}{2}\sigma^{2}(\Delta W_{t})^{2}\right)\\ &\displaystyle=X_{t}+a(X_{t})\Delta t+b(X_{t})\Delta W_{t}+\frac{1}{2}b(X_{t})% b^{\prime}(X_{t})((\Delta W_{t})^{2}-\Delta t)\end{aligned}
  24. a ( x ) = μ x , b ( x ) = σ x a(x)=\mu x,~{}b(x)=\sigma x

Milü.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π \pi
  5. 22 7 \frac{22}{7}
  6. 355 113 \frac{355}{113}
  7. 355 113 \frac{355}{113}
  8. π \pi
  9. π \pi
  10. π \pi
  11. 1 3748629 \frac{1}{3748629}
  12. π \pi
  13. 52163 16604 \frac{52163}{16604}
  14. π \pi
  15. 355 113 \frac{355}{113}
  16. 86953 27678 \frac{86953}{27678}
  17. 102928 32763 \frac{102928}{32763}
  18. π \displaystyle\pi
  19. π \pi
  20. 355 113 \frac{355}{113}
  21. π \pi
  22. π \pi

Minimum-cost_flow_problem.html

  1. G = ( V , E ) G=(V,E)
  2. s V s\in V
  3. t V t\in V
  4. ( u , v ) E (u,v)\in E
  5. c ( u , v ) > 0 c(u,v)>0
  6. f ( u , v ) 0 f(u,v)\geq 0
  7. a ( u , v ) a(u,v)
  8. f ( u , v ) a ( u , v ) f(u,v)\cdot a(u,v)
  9. d d
  10. s s
  11. t t
  12. ( u , v ) E a ( u , v ) f ( u , v ) \sum_{(u,v)\in E}a(u,v)\cdot f(u,v)
  13. f ( u , v ) c ( u , v ) \,f(u,v)\leq c(u,v)
  14. f ( u , v ) = - f ( v , u ) \,f(u,v)=-f(v,u)
  15. w V f ( u , w ) = 0 for all u s , t \,\sum_{w\in V}f(u,w)=0\,\text{ for all }u\neq s,t
  16. w V f ( s , w ) = d and w V f ( w , t ) = d \,\sum_{w\in V}f(s,w)=d\,\text{ and }\sum_{w\in V}f(w,t)=d
  17. d d
  18. t t
  19. s s
  20. c ( t , s ) = d c(t,s)=d
  21. l ( t , s ) = d l(t,s)=d
  22. s s
  23. t t
  24. d d

Minkowski–Steiner_formula.html

  1. n 2 n\geq 2
  2. A n A\subsetneq\mathbb{R}^{n}
  3. μ ( A ) \mu(A)
  4. A A
  5. λ ( A ) \lambda(\partial A)
  6. λ ( A ) := lim inf δ 0 μ ( A + B δ ¯ ) - μ ( A ) δ , \lambda(\partial A):=\liminf_{\delta\to 0}\frac{\mu\left(A+\overline{B_{\delta% }}\right)-\mu(A)}{\delta},
  7. B δ ¯ := { x = ( x 1 , , x n ) n | | x | := x 1 2 + + x n 2 δ } \overline{B_{\delta}}:=\left\{x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\left||x|:% =\sqrt{x_{1}^{2}+\dots+x_{n}^{2}}\leq\delta\right.\right\}
  8. δ > 0 \delta>0
  9. A + B δ ¯ := { a + b n | a A , b B δ ¯ } A+\overline{B_{\delta}}:=\left\{a+b\in\mathbb{R}^{n}\left|a\in A,b\in\overline% {B_{\delta}}\right.\right\}
  10. A A
  11. B δ ¯ \overline{B_{\delta}}
  12. A + B δ ¯ = { x n | | x - a | δ for some a A } . A+\overline{B_{\delta}}=\left\{x\in\mathbb{R}^{n}\mathrel{|}\ \mathopen{|}x-a% \mathclose{|}\leq\delta\mbox{ for some }~{}a\in A\right\}.
  13. A A
  14. λ ( A ) \lambda(\partial A)
  15. ( n - 1 ) (n-1)
  16. A \partial A
  17. A A
  18. A A
  19. μ ( A + B δ ¯ ) = μ ( A ) + λ ( A ) δ + i = 2 n - 1 λ i ( A ) δ i + ω n δ n , \mu\left(A+\overline{B_{\delta}}\right)=\mu(A)+\lambda(\partial A)\delta+\sum_% {i=2}^{n-1}\lambda_{i}(A)\delta^{i}+\omega_{n}\delta^{n},
  20. λ i \lambda_{i}
  21. A A
  22. ω n \omega_{n}
  23. n \mathbb{R}^{n}
  24. ω n = 2 π n / 2 n Γ ( n / 2 ) , \omega_{n}=\frac{2\pi^{n/2}}{n\Gamma(n/2)},
  25. Γ \Gamma
  26. A = B R ¯ A=\overline{B_{R}}
  27. R R
  28. S R := B R S_{R}:=\partial B_{R}
  29. λ ( S R ) = lim δ 0 μ ( B R ¯ + B δ ¯ ) - μ ( B R ¯ ) δ \lambda(S_{R})=\lim_{\delta\to 0}\frac{\mu\left(\overline{B_{R}}+\overline{B_{% \delta}}\right)-\mu\left(\overline{B_{R}}\right)}{\delta}
  30. = lim δ 0 [ ( R + δ ) n - R n ] ω n δ =\lim_{\delta\to 0}\frac{[(R+\delta)^{n}-R^{n}]\omega_{n}}{\delta}
  31. = n R n - 1 ω n , =nR^{n-1}\omega_{n},
  32. ω n \omega_{n}

Missing_data.html

  1. X R y | ( R x , Z ) X\perp\!\!\!\perp R_{y}|(R_{x},Z)
  2. P ( X , Y ) \displaystyle P(X,Y)
  3. R x = 0 R_{x}=0
  4. R y = 0 R_{y}=0
  5. P ( X | Y ) P(X|Y)
  6. P ( Y ) P(Y)
  7. P ( X | Y ) P(X|Y)
  8. P ( Y | X ) P(Y|X)
  9. R y R_{y}
  10. R x R_{x}
  11. X R y | R x = 0 X\perp\!\!\!\perp R_{y}|R_{x}=0

Mix_network.html

  1. K b K_{b}
  2. K m K_{m}
  3. K b ( m e s s a g e , R ) K_{b}(message,R)
  4. K m ( R 1 , K b ( R 0 , m e s s a g e ) , B ) ( K b ( R 0 , m e s s a g e ) , B ) K_{m}(R1,K_{b}(R0,message),B)\longrightarrow(K_{b}(R0,message),B)
  5. K m K_{m}
  6. R 1 R1
  7. K b K_{b}
  8. R 1 R1
  9. R 0 R0
  10. ( K b ( m e s s a g e ) ) (K_{b}(message))
  11. B B
  12. m e s s a g e message^{\prime}
  13. K b ( m e s s a g e ) = K b ( m e s s a g e ) K_{b}(message^{\prime})=K_{b}(message)
  14. R 0 R0
  15. m e s s a g e = m e s s a g e message^{\prime}=message
  16. R 0 R0
  17. R 0 R0
  18. K m ( S 1 , A ) , K x K_{m}(S1,A),K_{x}
  19. A A
  20. K x K_{x}
  21. S 1 S1
  22. K m ( S 1 , A ) , K x ( S 0 , r e s p o n s e ) K_{m}(S1,A),K_{x}(S0,response)
  23. A , S 1 ( K x ( S 0 , r e s p o n s e ) A,S1(K_{x}(S0,response)
  24. S 1 S1
  25. K m ( S 1 , A ) K_{m}(S1,A)
  26. K x ( S 0 , r e s p o n s e ) K_{x}(S0,response)
  27. S 1 S1
  28. K x K_{x}
  29. K x K_{x}
  30. \longrightarrow
  31. K m ( R 1 , K b ( R 0 , m e s s a g e , K m ( S 1 , A ) , K x ) , B ) K b ( R 0 , m e s s a g e , K m ( S 1 , A ) , K x ) K_{m}(R1,K_{b}(R0,message,K_{m}(S1,A),K_{x}),B)\longrightarrow K_{b}(R0,% message,K_{m}(S1,A),K_{x})
  32. \longrightarrow
  33. K m ( S 1 , A ) , K x ( S 0 , r e s p o n s e ) A , S 1 ( K x ( S 0 , r e s p o n s e ) ) K_{m}(S1,A),K_{x}(S0,response)\longrightarrow A,S1(K_{x}(S0,response))
  34. K b K_{b}
  35. K m K_{m}

Mixed_complementarity_problem.html

  1. F ( x ) : n n F(x):\mathbb{R}^{n}\to\mathbb{R}^{n}
  2. i { - } \ell_{i}\in\mathbb{R}\cup\{-\infty\}
  3. u i { } u_{i}\in\mathbb{R}\cup\{\infty\}
  4. x n x\in\mathbb{R}^{n}
  5. i { 1 , , n } i\in\{1,\ldots,n\}
  6. x i = i , F i ( x ) 0 x_{i}=\ell_{i},\;F_{i}(x)\geq 0
  7. i < x i < u i , F i ( x ) = 0 \ell_{i}<x_{i}<u_{i},\;F_{i}(x)=0
  8. x i = u i , F i ( x ) 0 x_{i}=u_{i},\;F_{i}(x)\leq 0
  9. [ , u ] [\ell,u]

Mixed_model.html

  1. s y m b o l y = X s y m b o l β + Z s y m b o l u + s y m b o l ϵ symbol{y}=Xsymbol{\beta}+Zsymbol{u}+symbol{\epsilon}
  2. s y m b o l y symbol{y}
  3. E ( s y m b o l y ) = X s y m b o l β E(symbol{y})=Xsymbol{\beta}
  4. s y m b o l β symbol{\beta}
  5. s y m b o l u symbol{u}
  6. E ( s y m b o l u ) = s y m b o l 0 E(symbol{u})=symbol{0}
  7. var ( s y m b o l u ) = G \operatorname{var}(symbol{u})=G
  8. s y m b o l ϵ symbol{\epsilon}
  9. E ( s y m b o l ϵ ) = s y m b o l 0 E(symbol{\epsilon})=symbol{0}
  10. var ( s y m b o l ϵ ) = R \operatorname{var}(symbol{\epsilon})=R
  11. X X
  12. Z Z
  13. s y m b o l y symbol{y}
  14. s y m b o l β symbol{\beta}
  15. s y m b o l u symbol{u}
  16. s y m b o l y symbol{y}
  17. s y m b o l u symbol{u}
  18. f ( s y m b o l y , s y m b o l u ) = f ( s y m b o l y | s y m b o l u ) f ( s y m b o l u ) f(symbol{y},symbol{u})=f(symbol{y}|symbol{u})\,f(symbol{u})
  19. s y m b o l u 𝒩 ( s y m b o l 0 , G ) symbol{u}\sim\mathcal{N}(symbol{0},G)
  20. s y m b o l ϵ 𝒩 ( s y m b o l 0 , R ) symbol{\epsilon}\sim\mathcal{N}(symbol{0},R)
  21. C o v ( s y m b o l u , s y m b o l ϵ ) = s y m b o l 0 Cov(symbol{u},symbol{\epsilon})=symbol{0}
  22. s y m b o l β symbol{\beta}
  23. s y m b o l u symbol{u}
  24. ( X R - 1 X X R - 1 Z Z R - 1 X Z R - 1 Z + G - 1 ) ( s y m b o l β ^ s y m b o l u ^ ) = ( X R - 1 s y m b o l y Z R - 1 s y m b o l y ) \begin{pmatrix}X^{\prime}R^{-1}X&X^{\prime}R^{-1}Z\\ Z^{\prime}R^{-1}X&Z^{\prime}R^{-1}Z+G^{-1}\end{pmatrix}\begin{pmatrix}\hat{% symbol{\beta}}\\ \hat{symbol{u}}\end{pmatrix}=\begin{pmatrix}X^{\prime}R^{-1}symbol{y}\\ Z^{\prime}R^{-1}symbol{y}\end{pmatrix}
  25. s y m b o l β ^ \textstyle\hat{symbol{\beta}}
  26. s y m b o l u ^ \textstyle\hat{symbol{u}}
  27. s y m b o l β symbol{\beta}
  28. s y m b o l u symbol{u}

Mixmaster_universe.html

  1. S 3 S^{3}
  2. S 3 S^{3}
  3. S 3 S^{3}
  4. a ( t ) a(t)
  5. S 3 S^{3}
  6. a ( t ) a(t)
  7. β ± ( t ) \beta_{\pm}(t)
  8. S 3 S^{3}
  9. a , β ± a,\beta_{\pm}
  10. β ± ( t ) \beta_{\pm}(t)
  11. d s 2 = - d t 2 + k = 1 3 L k 2 ( t ) σ k σ k \,\text{d}s^{2}=-\,\text{d}t^{2}+\sum_{k=1}^{3}{L_{k}^{2}(t)}\sigma_{k}\otimes% \sigma_{k}
  12. L k = R ( t ) e β k L_{k}=R(t)e^{\beta_{k}}
  13. σ k \sigma_{k}
  14. σ 1 = sin ψ d θ - cos ψ sin θ d ϕ \sigma_{1}=\sin\psi\,\text{d}\theta-\cos\psi\sin\theta\,\text{d}\phi
  15. σ 2 = cos ψ d θ + sin ψ sin θ d ϕ \sigma_{2}=\cos\psi\,\text{d}\theta+\sin\psi\sin\theta\,\text{d}\phi
  16. σ 3 = - d ψ - cos θ d ϕ \sigma_{3}=-\,\text{d}\psi-\cos\theta\,\text{d}\phi
  17. ( θ , ψ , ϕ ) (\theta,\psi,\phi)
  18. d σ i = 1 2 ϵ i j k σ j σ k \,\text{d}\sigma_{i}=\frac{1}{2}\epsilon_{ijk}\sigma_{j}\wedge\sigma_{k}
  19. d \,\text{d}
  20. \wedge
  21. S 3 S^{3}
  22. σ k \sigma_{k}
  23. S 3 S^{3}
  24. L k ( t ) L_{k}(t)
  25. Ω ( t ) \Omega(t)
  26. R ( t ) R(t)
  27. β k \beta_{k}
  28. R ( t ) = e - Ω ( t ) = ( L 1 ( t ) L 2 ( t ) L 3 ( t ) ) 1 / 3 , k = 1 3 β k ( t ) = 0 R(t)=e^{-\Omega(t)}=(L_{1}(t)L_{2}(t)L_{3}(t))^{1/3},\quad\sum_{k=1}^{3}\beta_% {k}(t)=0
  29. β k \beta_{k}
  30. β ± \beta_{\pm}
  31. β + = β 1 + β 2 = - β 3 , β - = β 1 - β 2 3 \beta_{+}=\beta_{1}+\beta_{2}=-\beta_{3},\quad\beta_{-}=\frac{\beta_{1}-\beta_% {2}}{\sqrt{3}}
  32. β ± \beta_{\pm}
  33. Ω \Omega
  34. H - 1 H^{-1}

Mock_modular_form.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. f ( a τ + b c τ + d ) = ρ ( a b c d ) ( c τ + d ) k f ( τ ) f\left(\frac{a\tau+b}{c\tau+d}\right)=\rho{\begin{pmatrix}a&b\\ c&d\end{pmatrix}}(c\tau+d)^{k}f(\tau)
  5. τ y k τ ¯ \frac{\partial}{\partial\tau}y^{k}\frac{\partial}{\partial\overline{\tau}}
  6. g = y k F ¯ τ = n b n q n g=y^{k}\frac{\partial\overline{F}}{\partial\tau}=\sum_{n}b_{n}q^{n}
  7. g * ( τ ) = ( i 2 ) k - 1 - τ ¯ i ( z + τ ) - k g ( - z ¯ ) ¯ d z = n n k - 1 b n ¯ β k ( 4 n y ) q - n + 1 g^{*}(\tau)=\left(\frac{i}{2}\right)^{k-1}\int_{-\overline{\tau}}^{i\infty}(z+% \tau)^{-k}\overline{g(-\overline{z})}\,dz=\sum_{n}n^{k-1}\overline{b_{n}}\beta% _{k}(4ny)q^{-n+1}
  8. β k ( t ) = t u - k e - π u d u \displaystyle\beta_{k}(t)=\int_{t}^{\infty}u^{-k}e^{-\pi u}\,du
  9. n Z ε ( n ) n q κ n 2 \sum_{n\in Z}\varepsilon(n)nq^{\kappa n^{2}}
  10. μ ( u , v ; τ ) = a 1 2 θ ( v ; τ ) n Z ( - b ) n q 1 2 n ( n + 1 ) 1 - a q n \mu(u,v;\tau)=\frac{a^{\frac{1}{2}}}{\theta(v;\tau)}\sum_{n\in Z}\frac{(-b)^{n% }q^{\frac{1}{2}n(n+1)}}{1-aq^{n}}
  11. q = e 2 π i τ , a = e 2 π i u , b = e 2 π i v \displaystyle q=e^{2\pi i\tau},\quad a=e^{2\pi iu},\quad b=e^{2\pi iv}
  12. θ ( v , τ ) = n Z ( - 1 ) n b n + 1 2 q 1 2 ( n + 1 2 ) 2 . \theta(v,\tau)=\sum_{n\in Z}(-1)^{n}b^{n+\frac{1}{2}}q^{\frac{1}{2}\left(n+% \frac{1}{2}\right)^{2}}.
  13. μ ^ ( u , v ; τ ) = μ ( u , v ; τ ) - 1 2 R ( u - v ; τ ) \hat{\mu}(u,v;\tau)=\mu(u,v;\tau)-\frac{1}{2}R(u-v;\tau)
  14. R ( z ; τ ) = ν Z + 1 2 ( - 1 ) ν - 1 2 ( sign ( ν ) - E [ ( ν + ( z ) y ) 2 y ] ) e - 2 π i ν z q - 1 2 ν 2 R(z;\tau)=\sum_{\nu\in Z+\frac{1}{2}}(-1)^{\nu-\frac{1}{2}}\left({\rm sign}(% \nu)-E\left[\left(\nu+\frac{\Im(z)}{y}\right)\sqrt{2y}\right]\right)e^{-2\pi i% \nu z}q^{-\frac{1}{2}\nu^{2}}
  15. E ( z ) = 2 0 z e - π u 2 d u E(z)=2\int_{0}^{z}e^{-\pi u^{2}}\,du
  16. μ ^ ( u + 1 , v ; τ ) = a - 1 b q - 1 2 μ ^ ( u + τ , v ; τ ) = - μ ^ ( u , v ; τ ) e 2 8 π i μ ^ ( u , v ; τ + 1 ) = μ ^ ( u , v ; τ ) = - ( τ i ) - 1 2 e π i τ ( u - v ) 2 μ ^ ( u τ , v τ ; - 1 τ ) . \begin{aligned}\displaystyle\hat{\mu}(u+1,v;\tau)&\displaystyle=a^{-1}bq^{-% \frac{1}{2}}\hat{\mu}(u+\tau,v;\tau)\\ &\displaystyle{}=-\hat{\mu}(u,v;\tau)\\ \displaystyle e^{\frac{2}{8}\pi i}\hat{\mu}(u,v;\tau+1)&\displaystyle=\hat{\mu% }(u,v;\tau)\\ &\displaystyle{}=-\left(\frac{\tau}{i}\right)^{-\frac{1}{2}}e^{\frac{\pi i}{% \tau}(u-v)^{2}}\hat{\mu}\left(\frac{u}{\tau},\frac{v}{\tau};-\frac{1}{\tau}% \right).\end{aligned}
  17. E 2 ( τ ) = 1 - 24 n > 0 σ 1 ( n ) q n \displaystyle E_{2}(\tau)=1-24\sum_{n>0}\sigma_{1}(n)q^{n}
  18. E 2 ( τ ) - 3 / π y \displaystyle E_{2}(\tau)-3/\pi y
  19. F ( τ ) = N H ( N ) q n + y - 1 / 2 n Z β ( 4 π n 2 y ) q - n 2 F(\tau)=\sum_{N}H(N)q^{n}+y^{-1/2}\sum_{n\in Z}\beta(4\pi n^{2}y)q^{-n^{2}}
  20. β ( x ) = 1 16 π 1 u - 3 / 2 e - x u d u \beta(x)=\frac{1}{16\pi}\int_{1}^{\infty}u^{-3/2}e^{-xu}du
  21. ( a ; q ) n (a;q)_{n}
  22. ( a ; q ) n = 0 j < n ( 1 - a q j ) = ( 1 - a ) ( 1 - a q ) ( 1 - a q n - 1 ) . (a;q)_{n}=\prod_{0\leq j<n}(1-aq^{j})=(1-a)(1-aq)\cdots(1-aq^{n-1}).
  23. A ( q ) = n 0 q ( n + 1 ) 2 ( - q ; q 2 ) n ( q ; q 2 ) n + 1 2 = n 0 q n + 1 ( - q 2 ; q 2 ) n ( q ; q 2 ) n + 1 A(q)=\sum_{n\geq 0}\frac{q^{(n+1)^{2}}(-q;q^{2})_{n}}{(q;q^{2})^{2}_{n+1}}=% \sum_{n\geq 0}\frac{q^{n+1}(-q^{2};q^{2})_{n}}{(q;q^{2})_{n+1}}
  24. B ( q ) = n 0 q n ( n + 1 ) ( - q 2 ; q 2 ) n ( q ; q 2 ) n + 1 2 = n 0 q n ( - q ; q 2 ) n ( q ; q 2 ) n + 1 B(q)=\sum_{n\geq 0}\frac{q^{n(n+1)}(-q^{2};q^{2})_{n}}{(q;q^{2})^{2}_{n+1}}=% \sum_{n\geq 0}\frac{q^{n}(-q;q^{2})_{n}}{(q;q^{2})_{n+1}}
  25. μ ( q ) = n 0 ( - 1 ) n q n 2 ( q ; q 2 ) n ( - q 2 ; q 2 ) n 2 \mu(q)=\sum_{n\geq 0}\frac{(-1)^{n}q^{n^{2}}(q;q^{2})_{n}}{(-q^{2};q^{2})^{2}_% {n}}
  26. U 0 ( q ) - 2 U 1 ( q ) = μ ( q ) U_{0}(q)-2U_{1}(q)=\mu(q)
  27. V 0 ( q ) - V 0 ( - q ) = 4 q B ( q 2 ) V_{0}(q)-V_{0}(-q)=4qB(q^{2})
  28. V 1 ( q ) + V 1 ( - q ) = 2 A ( q 2 ) V_{1}(q)+V_{1}(-q)=2A(q^{2})
  29. f ( q ) = n 0 q n 2 ( - q ; q ) n 2 = 2 n > 0 ( 1 - q n ) n Z ( - 1 ) n q n ( 3 n + 1 ) / 2 1 + q n f(q)=\sum_{n\geq 0}{q^{n^{2}}\over(-q;q)_{n}^{2}}={2\over\prod_{n>0}(1-q^{n})}% \sum_{n\in Z}{(-1)^{n}q^{n(3n+1)/2}\over 1+q^{n}}
  30. ϕ ( q ) = n 0 q n 2 ( - q 2 ; q 2 ) n = 1 n > 0 ( 1 - q n ) n Z ( - 1 ) n ( 1 + q n ) q n ( 3 n + 1 ) / 2 1 + q 2 n \phi(q)=\sum_{n\geq 0}{q^{n^{2}}\over(-q^{2};q^{2})_{n}}={1\over\prod_{n>0}(1-% q^{n})}\sum_{n\in Z}{(-1)^{n}(1+q^{n})q^{n(3n+1)/2}\over 1+q^{2n}}
  31. ψ ( q ) = n > 0 q n 2 ( q ; q 2 ) n = 1 n > 0 ( 1 - q 4 n ) n Z ( - 1 ) n q 6 n ( n + 1 ) 1 - q 4 n + 1 \psi(q)=\sum_{n>0}{q^{n^{2}}\over(q;q^{2})_{n}}={1\over\prod_{n>0}(1-q^{4n})}% \sum_{n\in Z}{(-1)^{n}q^{6n(n+1)}\over 1-q^{4n+1}}
  32. χ ( q ) = n 0 q n 2 1 i n ( 1 - q i + q 2 i ) = 1 2 n > 0 ( 1 - q n ) n Z ( - 1 ) n ( 1 + q n ) q n ( 3 n + 1 ) / 2 1 - q n + q 2 n \chi(q)=\sum_{n\geq 0}{q^{n^{2}}\over\prod_{1\leq i\leq n}(1-q^{i}+q^{2i})}={1% \over 2\prod_{n>0}(1-q^{n})}\sum_{n\in Z}{(-1)^{n}(1+q^{n})q^{n(3n+1)/2}\over 1% -q^{n}+q^{2n}}
  33. ω ( q ) = n 0 q 2 n ( n + 1 ) ( q ; q 2 ) n + 1 2 = 1 n > 0 ( 1 - q 2 n ) n 0 ( - 1 ) n q 3 n ( n + 1 ) 1 + q 2 n + 1 1 - q 2 n + 1 \omega(q)=\sum_{n\geq 0}{q^{2n(n+1)}\over(q;q^{2})^{2}_{n+1}}={1\over\prod_{n>% 0}(1-q^{2n})}\sum_{n\geq 0}{(-1)^{n}q^{3n(n+1)}{1+q^{2n+1}\over 1-q^{2n+1}}}
  34. ν ( q ) = n 0 q n ( n + 1 ) ( - q ; q 2 ) n + 1 = 1 n > 0 ( 1 - q n ) n 0 ( - 1 ) n q 3 n ( n + 1 ) / 2 1 - q 2 n + 1 1 + q 2 n + 1 \nu(q)=\sum_{n\geq 0}{q^{n(n+1)}\over(-q;q^{2})_{n+1}}={1\over\prod_{n>0}(1-q^% {n})}\sum_{n\geq 0}{(-1)^{n}q^{3n(n+1)/2}{1-q^{2n+1}\over 1+q^{2n+1}}}
  35. ρ ( q ) = n 0 q 2 n ( n + 1 ) 0 i n ( 1 + q 2 i + 1 + q 4 i + 2 ) = 1 n > 0 ( 1 - q 2 n ) n 0 ( - 1 ) n q 3 n ( n + 1 ) 1 - q 4 n + 2 1 + q 2 n + 1 + q 4 n + 2 \rho(q)=\sum_{n\geq 0}{q^{2n(n+1)}\over\prod_{0\leq i\leq n}(1+q^{2i+1}+q^{4i+% 2})}={1\over\prod_{n>0}(1-q^{2n})}\sum_{n\geq 0}{(-1)^{n}q^{3n(n+1)}{1-q^{4n+2% }\over 1+q^{2n+1}+q^{4n+2}}}
  36. 2 ϕ ( - q ) - f ( q ) \displaystyle 2\phi(-q)-f(q)
  37. f 0 ( q ) = n 0 q n 2 ( - q ; q ) n f_{0}(q)=\sum_{n\geq 0}{q^{n^{2}}\over(-q;q)_{n}}
  38. f 1 ( q ) = n 0 q n 2 + n ( - q ; q ) n f_{1}(q)=\sum_{n\geq 0}{q^{n^{2}+n}\over(-q;q)_{n}}
  39. ϕ 0 ( q ) = n 0 q n 2 ( - q ; q 2 ) n \phi_{0}(q)=\sum_{n\geq 0}{q^{n^{2}}(-q;q^{2})_{n}}
  40. ϕ 1 ( q ) = n 0 q ( n + 1 ) 2 ( - q ; q 2 ) n \phi_{1}(q)=\sum_{n\geq 0}{q^{(n+1)^{2}}(-q;q^{2})_{n}}
  41. ψ 0 ( q ) = n 0 q ( n + 1 ) ( n + 2 ) / 2 ( - q ; q ) n \psi_{0}(q)=\sum_{n\geq 0}{q^{(n+1)(n+2)/2}(-q;q)_{n}}
  42. ψ 1 ( q ) = n 0 q n ( n + 1 ) / 2 ( - q ; q ) n \psi_{1}(q)=\sum_{n\geq 0}{q^{n(n+1)/2}(-q;q)_{n}}
  43. χ 0 ( q ) = n 0 q n ( q n + 1 ; q ) n = 2 F 0 ( q ) - ϕ 0 ( - q ) \chi_{0}(q)=\sum_{n\geq 0}{q^{n}\over(q^{n+1};q)_{n}}=2F_{0}(q)-\phi_{0}(-q)
  44. χ 1 ( q ) = n 0 q n ( q n + 1 ; q ) n + 1 = 2 F 1 ( q ) + q - 1 ϕ 1 ( - q ) \chi_{1}(q)=\sum_{n\geq 0}{q^{n}\over(q^{n+1};q)_{n+1}}=2F_{1}(q)+q^{-1}\phi_{% 1}(-q)
  45. F 0 ( q ) = n 0 q 2 n 2 ( q ; q 2 ) n F_{0}(q)=\sum_{n\geq 0}{q^{2n^{2}}\over(q;q^{2})_{n}}
  46. F 1 ( q ) = n 0 q 2 n 2 + 2 n ( q ; q 2 ) n + 1 F_{1}(q)=\sum_{n\geq 0}{q^{2n^{2}+2n}\over(q;q^{2})_{n+1}}
  47. Ψ 0 ( q ) = - 1 + n 0 q 5 n 2 ( 1 - q ) ( 1 - q 4 ) ( 1 - q 6 ) ( 1 - q 9 ) ( 1 - q 5 n + 1 ) \Psi_{0}(q)=-1+\sum_{n\geq 0}{q^{5n^{2}}\over(1-q)(1-q^{4})(1-q^{6})(1-q^{9}).% ..(1-q^{5n+1})}
  48. Ψ 1 ( q ) = - 1 + n 0 q 5 n 2 ( 1 - q 2 ) ( 1 - q 3 ) ( 1 - q 7 ) ( 1 - q 8 ) ( 1 - q 5 n + 2 ) \Psi_{1}(q)=-1+\sum_{n\geq 0}{q^{5n^{2}}\over(1-q^{2})(1-q^{3})(1-q^{7})(1-q^{% 8})...(1-q^{5n+2})}
  49. ϕ ( q ) = n 0 ( - 1 ) n q n 2 ( q ; q 2 ) n ( - q ; q ) 2 n \phi(q)=\sum_{n\geq 0}{(-1)^{n}q^{n^{2}}(q;q^{2})_{n}\over(-q;q)_{2n}}
  50. ψ ( q ) = n 0 ( - 1 ) n q ( n + 1 ) 2 ( q ; q 2 ) n ( - q ; q ) 2 n + 1 \psi(q)=\sum_{n\geq 0}{(-1)^{n}q^{(n+1)^{2}}(q;q^{2})_{n}\over(-q;q)_{2n+1}}
  51. ρ ( q ) = n 0 q n ( n + 1 ) / 2 ( - q ; q ) n ( q ; q 2 ) n + 1 \rho(q)=\sum_{n\geq 0}{q^{n(n+1)/2}(-q;q)_{n}\over(q;q^{2})_{n+1}}
  52. σ ( q ) = n 0 q ( n + 1 ) ( n + 2 ) / 2 ( - q ; q ) n ( q ; q 2 ) n + 1 \sigma(q)=\sum_{n\geq 0}{q^{(n+1)(n+2)/2}(-q;q)_{n}\over(q;q^{2})_{n+1}}
  53. λ ( q ) = n 0 ( - 1 ) n q n ( q ; q 2 ) n ( - q ; q ) n \lambda(q)=\sum_{n\geq 0}{(-1)^{n}q^{n}(q;q^{2})_{n}\over(-q;q)_{n}}
  54. 2 μ ( q ) = n 0 ( - 1 ) n q n + 1 ( 1 + q n ) ( q ; q 2 ) n ( - q ; q ) n + 1 2\mu(q)=\sum_{n\geq 0}{(-1)^{n}q^{n+1}(1+q^{n})(q;q^{2})_{n}\over(-q;q)_{n+1}}
  55. γ ( q ) = n 0 q n 2 ( q ; q ) n ( q 3 ; q 3 ) n \gamma(q)=\sum_{n\geq 0}{q^{n^{2}}(q;q)_{n}\over(q^{3};q^{3})_{n}}
  56. ϕ - ( q ) = n 1 q n ( - q ; q ) 2 n - 1 ( q ; q 2 ) n \phi_{-}(q)=\sum_{n\geq 1}{q^{n}(-q;q)_{2n-1}\over(q;q^{2})_{n}}
  57. ψ - ( q ) = n 1 q n ( - q ; q ) 2 n - 2 ( q ; q 2 ) n \psi_{-}(q)=\sum_{n\geq 1}{q^{n}(-q;q)_{2n-2}\over(q;q^{2})_{n}}
  58. F 0 ( q ) = n 0 q n 2 ( q n + 1 ; q ) n \displaystyle F_{0}(q)=\sum_{n\geq 0}{q^{n^{2}}\over(q^{n+1};q)_{n}}
  59. F 1 ( q ) = n 0 q n 2 ( q n ; q ) n \displaystyle F_{1}(q)=\sum_{n\geq 0}{q^{n^{2}}\over(q^{n};q)_{n}}
  60. F 2 ( q ) = n 0 q n ( n + 1 ) ( q n + 1 ; q ) n + 1 \displaystyle F_{2}(q)=\sum_{n\geq 0}{q^{n(n+1)}\over(q^{n+1};q)_{n+1}}
  61. M 1 ( τ ) = q - 1 / 168 F 1 ( q ) + R 7 , 1 ( τ ) \displaystyle M_{1}(\tau)=q^{-1/168}F_{1}(q)+R_{7,1}(\tau)
  62. M 2 ( τ ) = - q - 25 / 168 F 2 ( q ) + R 7 , 2 ( τ ) \displaystyle M_{2}(\tau)=-q^{-25/168}F_{2}(q)+R_{7,2}(\tau)
  63. M 3 ( τ ) = q 47 / 168 F 3 ( q ) + R 7 , 3 ( τ ) \displaystyle M_{3}(\tau)=q^{47/168}F_{3}(q)+R_{7,3}(\tau)
  64. R p , j ( τ ) = n j mod p ( 12 n ) sgn ( n ) β ( n 2 y / 6 p ) q - n 2 / 24 p R_{p,j}(\tau)=\sum_{n\equiv j\bmod p}{12\choose n}\operatorname{sgn}(n)\beta(n% ^{2}y/6p)q^{-n^{2}/24p}
  65. β ( x ) = x u - 1 / 2 e - π u d u \beta(x)=\int_{x}^{\infty}u^{-1/2}e^{-\pi u}du
  66. M j ( - 1 / τ ) = τ / 7 i k = 1 3 2 sin ( 6 π j k / 7 ) M k ( τ ) M_{j}(-1/\tau)=\sqrt{\tau/7i}\sum_{k=1}^{3}2\sin(6\pi jk/7)M_{k}(\tau)
  67. M 1 ( τ + 1 ) = e - 2 π i / 168 M 1 ( τ ) M_{1}(\tau+1)=e^{-2\pi i/168}M_{1}(\tau)
  68. M 2 ( τ + 1 ) = e - 2 × 25 π i / 168 M 2 ( τ ) M_{2}(\tau+1)=e^{-2\times 25\pi i/168}M_{2}(\tau)
  69. M 3 ( τ + 1 ) = e - 2 × 121 π i / 168 M 3 ( τ ) . M_{3}(\tau+1)=e^{-2\times 121\pi i/168}M_{3}(\tau).
  70. S 0 ( q ) = n 0 q n 2 ( - q ; q 2 ) n ( - q 2 ; q 2 ) n S_{0}(q)=\sum_{n\geq 0}{q^{n^{2}}(-q;q^{2})_{n}\over(-q^{2};q^{2})_{n}}
  71. S 1 ( q ) = n 0 q n ( n + 2 ) ( - q ; q 2 ) n ( - q 2 ; q 2 ) n S_{1}(q)=\sum_{n\geq 0}{q^{n(n+2)}(-q;q^{2})_{n}\over(-q^{2};q^{2})_{n}}
  72. T 0 ( q ) = n 0 q ( n + 1 ) ( n + 2 ) ( - q 2 ; q 2 ) n ( - q ; q 2 ) n + 1 T_{0}(q)=\sum_{n\geq 0}{q^{(n+1)(n+2)}(-q^{2};q^{2})_{n}\over(-q;q^{2})_{n+1}}
  73. T 1 ( q ) = n 0 q n ( n + 1 ) ( - q 2 ; q 2 ) n ( - q ; q 2 ) n + 1 T_{1}(q)=\sum_{n\geq 0}{q^{n(n+1)}(-q^{2};q^{2})_{n}\over(-q;q^{2})_{n+1}}
  74. U 0 ( q ) = n 0 q n 2 ( - q ; q 2 ) n ( - q 4 ; q 4 ) n U_{0}(q)=\sum_{n\geq 0}{q^{n^{2}}(-q;q^{2})_{n}\over(-q^{4};q^{4})_{n}}
  75. U 1 ( q ) = n 0 q ( n + 1 ) 2 ( - q ; q 2 ) n ( - q 2 ; q 4 ) n + 1 U_{1}(q)=\sum_{n\geq 0}{q^{(n+1)^{2}}(-q;q^{2})_{n}\over(-q^{2};q^{4})_{n+1}}
  76. V 0 ( q ) = - 1 + 2 n 0 q n 2 ( - q ; q 2 ) n ( q ; q 2 ) n = - 1 + 2 n 0 q 2 n 2 ( - q 2 ; q 4 ) n ( q ; q 2 ) 2 n + 1 V_{0}(q)=-1+2\sum_{n\geq 0}{q^{n^{2}}(-q;q^{2})_{n}\over(q;q^{2})_{n}}=-1+2% \sum_{n\geq 0}{q^{2n^{2}}(-q^{2};q^{4})_{n}\over(q;q^{2})_{2n+1}}
  77. V 1 ( q ) = n 0 q ( n + 1 ) 2 ( - q ; q 2 ) n ( q ; q 2 ) n + 1 = n 0 q 2 n 2 + 2 n + 1 ( - q 4 ; q 4 ) n ( q ; q 2 ) 2 n + 2 V_{1}(q)=\sum_{n\geq 0}{q^{(n+1)^{2}}(-q;q^{2})_{n}\over(q;q^{2})_{n+1}}=\sum_% {n\geq 0}{q^{2n^{2}+2n+1}(-q^{4};q^{4})_{n}\over(q;q^{2})_{2n+2}}
  78. ϕ ( q ) = n 0 q n ( n + 1 ) / 2 ( q ; q 2 ) n + 1 \phi(q)=\sum_{n\geq 0}{q^{n(n+1)/2}\over(q;q^{2})_{n+1}}
  79. ψ ( q ) = n 0 q ( n + 1 ) ( n + 2 ) / 2 ( q ; q 2 ) n + 1 \psi(q)=\sum_{n\geq 0}{q^{(n+1)(n+2)/2}\over(q;q^{2})_{n+1}}
  80. \Chi ( q ) = n 0 ( - 1 ) n q n 2 ( - q ; q ) 2 n \Chi(q)=\sum_{n\geq 0}{(-1)^{n}q^{n^{2}}\over(-q;q)_{2n}}
  81. χ ( q ) = n 0 ( - 1 ) n q ( n + 1 ) 2 ( - q ; q ) 2 n + 1 \chi(q)=\sum_{n\geq 0}{(-1)^{n}q^{(n+1)^{2}}\over(-q;q)_{2n+1}}

Modal_analysis_using_FEM.html

  1. [ M ] [ U ¨ ] + [ C ] [ U ˙ ] + [ K ] [ U ] = [ F ] [M][\ddot{U}]+[C][\dot{U}]+[K][U]=[F]
  2. [ M ] [M]
  3. [ U ¨ ] [\ddot{U}]
  4. [ U ] [U]
  5. [ U ˙ ] [\dot{U}]
  6. [ C ] [C]
  7. [ K ] [K]
  8. [ F ] [F]
  9. [ M ] [ U ¨ ] + [ K ] [ U ] = [ 0 ] [M][\ddot{U}]+[K][U]=[0]
  10. [ U ¨ ] [\ddot{U}]
  11. λ [ U ] \lambda[U]
  12. λ \lambda
  13. s - 2 \mathrm{s}^{-2}
  14. [ M ] [ U ] λ + [ K ] [ U ] = [ 0 ] [M][U]\lambda+[K][U]=[0]
  15. [ K ] [ U ] = [ F ] [K][U]=[F]
  16. [ A ] [ x ] = [ x ] λ [A][x]=[x]\lambda
  17. [ M ] - 1 [M]^{-1}
  18. [ K ] - 1 [K]^{-1}
  19. μ \mu
  20. μ = 1 λ \mu=\frac{1}{\lambda}

Modal_companion.html

  1. T ( p ) = p T(p)=\Box p
  2. p p
  3. T ( ) = , T(\bot)=\bot,
  4. T ( A B ) = T ( A ) T ( B ) , T(A\land B)=T(A)\land T(B),
  5. T ( A B ) = T ( A ) T ( B ) , T(A\lor B)=T(A)\lor T(B),
  6. T ( A B ) = ( T ( A ) T ( B ) ) . T(A\to B)=\Box(T(A)\to T(B)).
  7. A A\to\bot
  8. T ( ¬ A ) = ¬ T ( A ) . T(\neg A)=\Box\neg T(A).
  9. \Box
  10. ρ M = { A M T ( A ) } . \rho M=\{A\mid M\vdash T(A)\}.
  11. L = ρ M L=\rho M
  12. τ L = 𝐒𝟒 { T ( A ) L A } , \tau L=\mathbf{S4}\oplus\{T(A)\mid L\vdash A\},
  13. \oplus
  14. τ L M σ L \tau L\subseteq M\subseteq\sigma L
  15. ( ( A A ) A ) A \Box(\Box(A\to\Box A)\to A)\to A
  16. 𝐓𝐫𝐢𝐯 = 𝐊 ( A A ) . \mathbf{Triv}=\mathbf{K}\oplus(A\leftrightarrow\Box A).
  17. ρ : NExt 𝐒𝟒 Ext 𝐈𝐏𝐂 , \rho\colon\mathrm{NExt}\,\mathbf{S4}\to\mathrm{Ext}\,\mathbf{IPC},
  18. τ , σ : Ext 𝐈𝐏𝐂 NExt 𝐒𝟒 . \tau,\sigma\colon\mathrm{Ext}\,\mathbf{IPC}\to\mathrm{NExt}\,\mathbf{S4}.
  19. ρ τ = ρ σ \rho\circ\tau=\rho\circ\sigma
  20. σ L = τ L + 𝐆𝐫𝐳 . \sigma L=\tau L+\mathbf{Grz}.
  21. 𝐅 = F , R , V \mathbf{F}=\langle F,R,V\rangle
  22. x y x R y y R x x\sim y\iff x\,R\,y\land y\,R\,x
  23. ρ F , = F , R / \langle\rho F,\leq\rangle=\langle F,R\rangle/{\sim}
  24. \sim
  25. ρ V = { A / A V , A = A } . \rho V=\{A/{\sim}\mid A\in V,A=\Box A\}.
  26. ρ 𝐅 = ρ F , , ρ V \rho\mathbf{F}=\langle\rho F,\leq,\rho V\rangle
  27. { ρ 𝐅 ; 𝐅 C } \{\rho\mathbf{F};\,\mathbf{F}\in C\}
  28. 𝐅 = F , , V \mathbf{F}=\langle F,\leq,V\rangle
  29. \Box
  30. σ 𝐅 = F , , σ V \sigma\mathbf{F}=\langle F,\leq,\sigma V\rangle
  31. { σ 𝐅 ; 𝐅 C } \{\sigma\mathbf{F};\,\mathbf{F}\in C\}
  32. ρ - 1 ( L ) \rho^{-1}(L)
  33. ρ - 1 ( 𝐂𝐏𝐂 ) \rho^{-1}(\mathbf{CPC})
  34. L ( C n ) L(C_{n})
  35. C n C_{n}
  36. ρ - 1 ( L ) \rho^{-1}(L)
  37. R = A 1 , , A n B R=\frac{A_{1},\dots,A_{n}}{B}
  38. T ( R ) = T ( A 1 ) , , T ( A n ) T ( B ) . T(R)=\frac{T(A_{1}),\dots,T(A_{n})}{T(B)}.

Modal_μ-calculus.html

  1. ν \nu
  2. ϕ \phi
  3. ψ \psi
  4. ϕ ψ \phi\wedge\psi
  5. ϕ \phi
  6. ¬ ϕ \neg\phi
  7. ϕ \phi
  8. a a
  9. [ a ] ϕ [a]\phi
  10. a a
  11. ϕ \phi
  12. a a
  13. ϕ \phi
  14. ϕ \phi
  15. Z Z
  16. ν Z . ϕ \nu Z.\phi
  17. Z Z
  18. ϕ \phi
  19. ν \nu
  20. ϕ ψ \phi\lor\psi
  21. ¬ ( ¬ ϕ ¬ ψ ) \neg(\neg\phi\land\neg\psi)
  22. a ϕ \langle a\rangle\phi
  23. a a
  24. ϕ \phi
  25. a a
  26. ϕ \phi
  27. ¬ [ a ] ¬ ϕ \neg[a]\neg\phi
  28. μ Z . ϕ \mu Z.\phi
  29. ¬ ν Z . ¬ ϕ [ Z := ¬ Z ] \neg\nu Z.\neg\phi[Z:=\neg Z]
  30. ϕ [ Z := ¬ Z ] \phi[Z:=\neg Z]
  31. ¬ Z \neg Z
  32. ϕ \phi
  33. μ Z . ϕ \mu Z.\phi
  34. ϕ \phi
  35. λ Z . ϕ \lambda Z.\phi
  36. ϕ \phi
  37. ( S , R , V ) (S,R,V)
  38. S S
  39. R R
  40. a a
  41. S S
  42. V : Var 2 S V:\mbox{Var}~{}\rightarrow 2^{S}
  43. p Prop p\in\mbox{Prop}~{}
  44. ( S , R , V ) (S,R,V)
  45. i i
  46. ϕ \phi
  47. μ \mu
  48. [ [ ¯ ] ] i : ϕ 2 S [\![\underline{~{}\,}]\!]_{i}:\phi\rightarrow 2^{S}
  49. [ [ p ] ] i = V ( p ) [\![p]\!]_{i}=V(p)
  50. [ [ ϕ ψ ] ] i = [ [ ϕ ] ] i [ [ ψ ] ] i [\![\phi\wedge\psi]\!]_{i}=[\![\phi]\!]_{i}\cap[\![\psi]\!]_{i}
  51. [ [ ¬ ϕ ] ] i = S [ [ ϕ ] ] i [\![\neg\phi]\!]_{i}=S\smallsetminus[\![\phi]\!]_{i}
  52. [ [ [ a ] ϕ ] ] i = { s S t S , ( s , t ) R a t [ [ ϕ ] ] i } [\![[a]\phi]\!]_{i}=\{s\in S\mid\forall t\in S,(s,t)\in R_{a}\rightarrow t\in[% \![\phi]\!]_{i}\}
  53. [ [ ν Z . ϕ ] ] i = { T S T [ [ ϕ ] ] i [ Z := T ] } [\![\nu Z.\phi]\!]_{i}=\bigcup\{T\subseteq S\mid T\subseteq[\![\phi]\!]_{i[Z:=% T]}\}
  54. i [ Z := T ] i[Z:=T]
  55. i i
  56. [ [ ϕ ψ ] ] i = [ [ ϕ ] ] i [ [ ψ ] ] i [\![\phi\vee\psi]\!]_{i}=[\![\phi]\!]_{i}\cup[\![\psi]\!]_{i}
  57. [ [ a ϕ ] ] i = { s S t S , ( s , t ) R a t [ [ ϕ ] ] i } [\![\langle a\rangle\phi]\!]_{i}=\{s\in S\mid\exists t\in S,(s,t)\in R_{a}% \wedge t\in[\![\phi]\!]_{i}\}
  58. [ [ μ Z . ϕ ] ] i = { T S [ [ ϕ ] ] i [ Z := T ] T } [\![\mu Z.\phi]\!]_{i}=\bigcap\{T\subseteq S\mid[\![\phi]\!]_{i[Z:=T]}% \subseteq T\}
  59. ( S , R , V ) (S,R,V)
  60. p p
  61. V ( p ) V(p)
  62. ϕ ψ \phi\wedge\psi
  63. ϕ \phi
  64. ψ \psi
  65. ¬ ϕ \neg\phi
  66. ϕ \phi
  67. [ a ] ϕ [a]\phi
  68. s s
  69. a a
  70. s s
  71. ϕ \phi
  72. a ϕ \langle a\rangle\phi
  73. s s
  74. a a
  75. s s
  76. ϕ \phi
  77. ν Z . ϕ \nu Z.\phi
  78. T T
  79. Z Z
  80. T T
  81. ϕ \phi
  82. T T
  83. [ [ ν Z . ϕ ] ] i [\![\nu Z.\phi]\!]_{i}
  84. [ [ ϕ ] ] i [ Z := T ] [\![\phi]\!]_{i[Z:=T]}
  85. [ [ μ Z . ϕ ] ] i [\![\mu Z.\phi]\!]_{i}
  86. [ a ] ϕ [a]\phi
  87. a ϕ \langle a\rangle\phi
  88. ν Z . ϕ [ a ] Z \nu Z.\phi\wedge[a]Z
  89. ϕ \phi
  90. μ Z . ϕ a Z \mu Z.\phi\vee\langle a\rangle Z
  91. ϕ \phi
  92. ν Z . ( a A a a A [ a ] Z ) \nu Z.(\bigvee_{a\in A}\langle a\rangle\top\wedge\bigwedge_{a\in A}[a]Z)

Models_of_DNA_evolution.html

  1. t t
  2. E 1 , , E 4 E_{1},\ldots,E_{4}
  3. P ( t ) = ( P i j ( t ) ) P(t)=\big(P_{ij}(t)\big)
  4. P i j ( t ) P_{ij}(t)
  5. E i E_{i}
  6. E j E_{j}
  7. t t
  8. P ( t ) = ( p A A ( t ) p G A ( t ) p C A ( t ) p T A ( t ) p A G ( t ) p G G ( t ) p C G ( t ) p T G ( t ) p A C ( t ) p G C ( t ) p C C ( t ) p T C ( t ) p A T ( t ) p G T ( t ) p C T ( t ) p T T ( t ) ) P(t)=\begin{pmatrix}p_{AA}(t)&p_{GA}(t)&p_{CA}(t)&p_{TA}(t)\\ p_{AG}(t)&p_{GG}(t)&p_{CG}(t)&p_{TG}(t)\\ p_{AC}(t)&p_{GC}(t)&p_{CC}(t)&p_{TC}(t)\\ p_{AT}(t)&p_{GT}(t)&p_{CT}(t)&p_{TT}(t)\end{pmatrix}
  9. t 0 t_{0}
  10. E i E_{i}
  11. t 0 + t t_{0}+t
  12. E j E_{j}
  13. i i
  14. j j
  15. t t
  16. p i j ( t ) p_{ij}(t)
  17. P ( t + τ ) = P ( t ) P ( τ ) P(t+\tau)=P(t)P(\tau)
  18. 𝐏 ( t ) = ( p A ( t ) , p G ( t ) , p C ( t ) , p T ( t ) ) T \mathbf{P}(t)=(p_{A}(t),\ p_{G}(t),\ p_{C}(t),\ p_{T}(t))^{T}
  19. A , A,
  20. G , \ G,
  21. C , \ C,
  22. T \ T
  23. t t
  24. = { A , G , C , T } \mathcal{E}=\{A,\ G,\ C,\ T\}
  25. x , y x,y\in\mathcal{E}
  26. μ x y \mu_{xy}
  27. x x
  28. y y
  29. x x
  30. μ x = y x μ x y \mu_{x}=\sum_{y\neq x}\mu_{xy}
  31. p A ( t ) p_{A}(t)
  32. Δ t \Delta t
  33. p A ( t + Δ t ) = p A ( t ) - p A ( t ) μ A Δ t + x A p x ( t ) μ x A Δ t p_{A}(t+\Delta t)=p_{A}(t)-p_{A}(t)\mu_{A}\Delta t+\sum_{x\neq A}p_{x}(t)\mu_{% xA}\Delta t
  34. A A
  35. t + Δ t t+\Delta t
  36. t t
  37. A A
  38. A A
  39. p G ( t ) , p C ( t ) , and p T ( t ) p_{G}(t),\ p_{C}(t),\ \mathrm{and}\ p_{T}(t)
  40. 𝐏 ( t + Δ t ) = 𝐏 ( t ) + Q 𝐏 ( t ) Δ t \mathbf{P}(t+\Delta t)=\mathbf{P}(t)+Q\mathbf{P}(t)\Delta t
  41. Q = ( - μ A μ G A μ C A μ T A μ A G - μ G μ C G μ T G μ A C μ G C - μ C μ T C μ A T μ G T μ C T - μ T ) Q=\begin{pmatrix}-\mu_{A}&\mu_{GA}&\mu_{CA}&\mu_{TA}\\ \mu_{AG}&-\mu_{G}&\mu_{CG}&\mu_{TG}\\ \mu_{AC}&\mu_{GC}&-\mu_{C}&\mu_{TC}\\ \mu_{AT}&\mu_{GT}&\mu_{CT}&-\mu_{T}\end{pmatrix}
  42. 𝐏 ( t ) = Q 𝐏 ( t ) \mathbf{P}^{\prime}(t)=Q\mathbf{P}(t)
  43. Q Q
  44. Q Q
  45. Q Q
  46. P ( t ) = exp ( Q t ) P(t)=\exp(Qt)
  47. 𝐏 ( t ) = P ( t ) 𝐏 ( 0 ) = exp ( Q t ) 𝐏 ( 0 ) . \mathbf{P}(t)=P(t)\mathbf{P}(0)=\exp(Qt)\mathbf{P}(0)\,.
  48. μ x y \mu_{xy}
  49. x , y x,y\in\mathcal{E}
  50. 𝚷 = { π x , x } \mathbf{\Pi}=\{\pi_{x},\ x\in\mathcal{E}\}
  51. π x \pi_{x}
  52. x x
  53. π A , π G , π C , π T \pi_{A},\pi_{G},\pi_{C},\pi_{T}
  54. 𝐏 ( t ) \mathbf{P}(t)
  55. 𝚷 \mathbf{\Pi}
  56. Q 𝚷 = 0 Q\mathbf{\Pi}=0
  57. Q 𝚷 = Q 𝐏 ( t ) = d 𝐏 ( t ) d t = 0 . Q\mathbf{\Pi}=Q\mathbf{P}(t)=\frac{d\mathbf{P}(t)}{dt}=0\,.
  58. x x
  59. y y
  60. y y
  61. x x
  62. π x μ x y = π y μ y x \pi_{x}\mu_{xy}=\pi_{y}\mu_{yx}
  63. s x y = μ x y / π y s_{xy}=\mu_{xy}/\pi_{y}
  64. s x y = s y x s_{xy}=s_{yx}
  65. s x y s_{xy}
  66. x x
  67. y y
  68. s x y s_{xy}
  69. x x
  70. y y
  71. x x
  72. Q Q
  73. Q Q
  74. π x \pi_{x}
  75. β = 1 / ( - i π i μ i i ) \beta=1/\left(-\sum_{i}\pi_{i}\mu_{ii}\right)
  76. ( π A = π G = π C = π T = 1 4 ) \left(\pi_{A}=\pi_{G}=\pi_{C}=\pi_{T}={1\over 4}\right)
  77. μ \mu
  78. Q = ( * μ 4 μ 4 μ 4 μ 4 * μ 4 μ 4 μ 4 μ 4 * μ 4 μ 4 μ 4 μ 4 * ) Q=\begin{pmatrix}{*}&{\mu\over 4}&{\mu\over 4}&{\mu\over 4}\\ {\mu\over 4}&{*}&{\mu\over 4}&{\mu\over 4}\\ {\mu\over 4}&{\mu\over 4}&{*}&{\mu\over 4}\\ {\mu\over 4}&{\mu\over 4}&{\mu\over 4}&{*}\end{pmatrix}
  79. P = ( 1 4 + 3 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 + 3 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 + 3 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 - 1 4 e - t μ 1 4 + 3 4 e - t μ ) P=\begin{pmatrix}{{1\over 4}+{3\over 4}e^{-t\mu}}&{{1\over 4}-{1\over 4}e^{-t% \mu}}&{{1\over 4}-{1\over 4}e^{-t\mu}}&{{1\over 4}-{1\over 4}e^{-t\mu}}\\ \\ {{1\over 4}-{1\over 4}e^{-t\mu}}&{{1\over 4}+{3\over 4}e^{-t\mu}}&{{1\over 4}-% {1\over 4}e^{-t\mu}}&{{1\over 4}-{1\over 4}e^{-t\mu}}\\ \\ {{1\over 4}-{1\over 4}e^{-t\mu}}&{{1\over 4}-{1\over 4}e^{-t\mu}}&{{1\over 4}+% {3\over 4}e^{-t\mu}}&{{1\over 4}-{1\over 4}e^{-t\mu}}\\ \\ {{1\over 4}-{1\over 4}e^{-t\mu}}&{{1\over 4}-{1\over 4}e^{-t\mu}}&{{1\over 4}-% {1\over 4}e^{-t\mu}}&{{1\over 4}+{3\over 4}e^{-t\mu}}\end{pmatrix}
  80. ν \nu
  81. P i j ( ν ) = { 1 4 + 3 4 e - 4 ν / 3 if i = j 1 4 - 1 4 e - 4 ν / 3 if i j P_{ij}(\nu)=\left\{\begin{array}[]{cc}{1\over 4}+{3\over 4}e^{-4\nu/3}&\mbox{ % if }~{}i=j\\ {1\over 4}-{1\over 4}e^{-4\nu/3}&\mbox{ if }~{}i\neq j\end{array}\right.
  82. ν = 3 4 t μ = ( μ 4 + μ 4 + μ 4 ) t \nu={3\over 4}t\mu=({\mu\over 4}+{\mu\over 4}+{\mu\over 4})t
  83. Q Q
  84. t t
  85. μ \mu
  86. p p
  87. d ^ = - 3 4 ln ( 1 - 4 3 p ) = ν ^ \hat{d}=-{3\over 4}\ln({1-{4\over 3}p})=\hat{\nu}
  88. p p
  89. p p
  90. p p
  91. p p
  92. Q = ( * κ 1 1 κ * 1 1 1 1 * κ 1 1 κ * ) Q=\begin{pmatrix}{*}&{\kappa}&{1}&{1}\\ {\kappa}&{*}&{1}&{1}\\ {1}&{1}&{*}&{\kappa}\\ {1}&{1}&{\kappa}&{*}\end{pmatrix}
  93. d ^ = - 1 2 ln ( 1 - 2 p - q ) - 1 4 ln ( 1 - 2 q ) \hat{d}=-{1\over 2}\ln(1-2p-q)-{1\over 4}\ln(1-2q)
  94. π T π C π A π G \pi_{T}\neq\pi_{C}\neq\pi_{A}\neq\pi_{G}
  95. Q = ( * π C π A π G π T * π A π G π T π C * π G π T π C π A * ) Q=\begin{pmatrix}{*}&{\pi_{C}}&{\pi_{A}}&{\pi_{G}}\\ {\pi_{T}}&{*}&{\pi_{A}}&{\pi_{G}}\\ {\pi_{T}}&{\pi_{C}}&{*}&{\pi_{G}}\\ {\pi_{T}}&{\pi_{C}}&{\pi_{A}}&{*}\end{pmatrix}
  96. β = 1 / ( 1 - π A 2 - π C 2 - π G 2 - π T 2 ) \beta=1/(1-\pi_{A}^{2}-\pi_{C}^{2}-\pi_{G}^{2}-\pi_{T}^{2})
  97. P i j ( ν ) = { e - β ν + π j ( 1 - e - β ν ) if i = j π j ( 1 - e - β ν ) if i j P_{ij}(\nu)=\left\{\begin{array}[]{cc}e^{-\beta\nu}+\pi_{j}\left(1-e^{-\beta% \nu}\right)&\mbox{ if }~{}i=j\\ \pi_{j}\left(1-e^{-\beta\nu}\right)&\mbox{ if }~{}i\neq j\end{array}\right.
  98. π T π C π A π G \pi_{T}\neq\pi_{C}\neq\pi_{A}\neq\pi_{G}
  99. Q = ( * κ π C π A π G κ π T * π A π G π T π C * κ π G π T π C κ π A * ) Q=\begin{pmatrix}{*}&{\kappa\pi_{C}}&{\pi_{A}}&{\pi_{G}}\\ {\kappa\pi_{T}}&{*}&{\pi_{A}}&{\pi_{G}}\\ {\pi_{T}}&{\pi_{C}}&{*}&{\kappa\pi_{G}}\\ {\pi_{T}}&{\pi_{C}}&{\kappa\pi_{A}}&{*}\end{pmatrix}
  100. β = 1 2 ( π A + π G ) ( π C + π T ) + 2 κ [ ( π A π G ) + ( π C π T ) ] \beta=\frac{1}{2(\pi_{A}+\pi_{G})(\pi_{C}+\pi_{T})+2\kappa[(\pi_{A}\pi_{G})+(% \pi_{C}\pi_{T})]}
  101. P A A ( ν , κ , π ) = [ π A ( π A + π G + ( π C + π T ) e - β ν ) + π G e - ( 1 + ( π A + π G ) ( κ - 1.0 ) ) β ν ] / ( π A + π G ) P_{AA}(\nu,\kappa,\pi)=\left[\pi_{A}\left(\pi_{A}+\pi_{G}+(\pi_{C}+\pi_{T})e^{% -\beta\nu}\right)+\pi_{G}e^{-(1+(\pi_{A}+\pi_{G})(\kappa-1.0))\beta\nu}\right]% /(\pi_{A}+\pi_{G})
  102. P A C ( ν , κ , π ) = π C ( 1.0 - e - β ν ) P_{AC}(\nu,\kappa,\pi)=\pi_{C}\left(1.0-e^{-\beta\nu}\right)
  103. P A G ( ν , κ , π ) = [ π G ( π A + π G + ( π C + π T ) e - β ν ) - π G e - ( 1 + ( π A + π G ) ( κ - 1.0 ) ) β ν ] / ( π A + π G ) P_{AG}(\nu,\kappa,\pi)=\left[\pi_{G}\left(\pi_{A}+\pi_{G}+(\pi_{C}+\pi_{T})e^{% -\beta\nu}\right)-\pi_{G}e^{-(1+(\pi_{A}+\pi_{G})(\kappa-1.0))\beta\nu}\right]% /\left(\pi_{A}+\pi_{G}\right)
  104. P A T ( ν , κ , π ) = π T ( 1.0 - e - β ν ) P_{AT}(\nu,\kappa,\pi)=\pi_{T}\left(1.0-e^{-\beta\nu}\right)
  105. π G C \pi_{GC}
  106. π G = π C = π G C 2 \pi_{G}=\pi_{C}={\pi_{GC}\over 2}
  107. π A = π T = ( 1 - π G C ) 2 \pi_{A}=\pi_{T}={(1-\pi_{GC})\over 2}
  108. Q = ( * κ ( 1 - π G C ) / 2 ( 1 - π G C ) / 2 ( 1 - π G C ) / 2 κ π G C / 2 * π G C / 2 π G C / 2 ( 1 - π G C ) / 2 ( 1 - π G C ) / 2 * κ ( 1 - π G C ) / 2 π G C / 2 π G C / 2 κ π G C / 2 * ) Q=\begin{pmatrix}{*}&{\kappa(1-\pi_{GC})/2}&{(1-\pi_{GC})/2}&{(1-\pi_{GC})/2}% \\ {\kappa\pi_{GC}/2}&{*}&{\pi_{GC}/2}&{\pi_{GC}/2}\\ {(1-\pi_{GC})/2}&{(1-\pi_{GC})/2}&{*}&{\kappa(1-\pi_{GC})/2}\\ {\pi_{GC}/2}&{\pi_{GC}/2}&{\kappa\pi_{GC}/2}&{*}\end{pmatrix}
  109. d = - h ln ( 1 - p h - q ) - 1 2 ( 1 - h ) ln ( 1 - 2 q ) d=-h\ln(1-{p\over h}-q)-{1\over 2}(1-h)\ln(1-2q)
  110. h = 2 θ ( 1 - θ ) h=2\theta(1-\theta)
  111. θ ( 0 , 1 ) \theta\in(0,1)
  112. π T π C π A π G \pi_{T}\neq\pi_{C}\neq\pi_{A}\neq\pi_{G}
  113. Q = ( * κ 1 π C π A π G κ 1 π T * π A π G π T π C * κ 2 π G π T π C κ 2 π A * ) Q=\begin{pmatrix}{*}&{\kappa_{1}\pi_{C}}&{\pi_{A}}&{\pi_{G}}\\ {\kappa_{1}\pi_{T}}&{*}&{\pi_{A}}&{\pi_{G}}\\ {\pi_{T}}&{\pi_{C}}&{*}&{\kappa_{2}\pi_{G}}\\ {\pi_{T}}&{\pi_{C}}&{\kappa_{2}\pi_{A}}&{*}\end{pmatrix}
  114. Π = ( π 1 , π 2 , π 3 , π 4 ) \Pi=(\pi_{1},\pi_{2},\pi_{3},\pi_{4})
  115. Q = ( - ( α π C + β π A + γ π G ) α π C β π A γ π G α π T - ( α π T + δ π A + ϵ π G ) δ π A ϵ π G β π T δ π C - ( β π T + δ π C + η π G ) η π G γ π T ϵ π C η π A - ( γ π T + ϵ π C + η π A ) ) Q=\begin{pmatrix}{-(\alpha\pi_{C}+\beta\pi_{A}+\gamma\pi_{G})}&{\alpha\pi_{C}}% &{\beta\pi_{A}}&{\gamma\pi_{G}}\\ {\alpha\pi_{T}}&{-(\alpha\pi_{T}+\delta\pi_{A}+\epsilon\pi_{G})}&{\delta\pi_{A% }}&{\epsilon\pi_{G}}\\ {\beta\pi_{T}}&{\delta\pi_{C}}&{-(\beta\pi_{T}+\delta\pi_{C}+\eta\pi_{G})}&{% \eta\pi_{G}}\\ {\gamma\pi_{T}}&{\epsilon\pi_{C}}&{\eta\pi_{A}}&{-(\gamma\pi_{T}+\epsilon\pi_{% C}+\eta\pi_{A})}\end{pmatrix}
  116. α = r ( T C ) = r ( C T ) β = r ( T A ) = r ( A T ) γ = r ( T G ) = r ( G T ) δ = r ( C A ) = r ( A C ) ϵ = r ( C G ) = r ( G C ) η = r ( A G ) = r ( G A ) \begin{aligned}\displaystyle\alpha=r(T\rightarrow C)=r(C\rightarrow T)\\ \displaystyle\beta=r(T\rightarrow A)=r(A\rightarrow T)\\ \displaystyle\gamma=r(T\rightarrow G)=r(G\rightarrow T)\\ \displaystyle\delta=r(C\rightarrow A)=r(A\rightarrow C)\\ \displaystyle\epsilon=r(C\rightarrow G)=r(G\rightarrow C)\\ \displaystyle\eta=r(A\rightarrow G)=r(G\rightarrow A)\end{aligned}
  117. μ \mu
  118. μ \mu
  119. n 2 - n 2 {{n^{2}-n}\over 2}
  120. μ \mu
  121. n 2 - n 2 + n - 1 = 1 2 n 2 + 1 2 n - 1. {{n^{2}-n}\over 2}+n-1={1\over 2}n^{2}+{1\over 2}n-1.
  122. 4 3 = 64 4^{3}=64
  123. 20 × 19 × 3 2 + 64 - 1 = 633 {{20\times 19\times 3}\over 2}+64-1=633

Modulo-N_code.html

  1. M o = D o M_{o}=D_{o}
  2. M e = ( D e ) m o d ( N ) M_{e}=(D_{e})mod(N)
  3. l o g 2 ( K ) log_{2}(K)
  4. l o g 2 ( M o ) + l o g 2 ( M e ) log_{2}(M_{o})+log_{2}(M_{e})
  5. l o g 2 ( M e ) l o g 2 ( M o ) log_{2}(M_{e})\leq log_{2}(M_{o})
  6. M e N M_{e}\leq N
  7. C . R = l o g 2 ( M o ) + l o g 2 ( M e ) 2 l o g 2 ( M o ) C.R=\frac{log_{2}(M_{o})+log_{2}(M_{e})}{2log_{2}(M_{o})}
  8. C L O S E S T ( M o , N . k + M e ) CLOSEST(M_{o},N.k+M_{e})
  9. M o N . k + M e M_{o}\simeq N.k+M_{e}
  10. N . k + M e N.k+M_{e}
  11. 43 8.5 + 7 43\simeq 8.5+7
  12. π \pi

Molecular_Hamiltonian.html

  1. - i \hbarsymbol -i\hbarsymbol{\nabla}
  2. s y m b o l symbol{\nabla}
  3. r i j | 𝐫 i - 𝐫 j | = ( 𝐫 i - 𝐫 j ) ( 𝐫 i - 𝐫 j ) = ( x i - x j ) 2 + ( y i - y j ) 2 + ( z i - z j ) 2 . r_{ij}\equiv|\mathbf{r}_{i}-\mathbf{r}_{j}|=\sqrt{(\mathbf{r}_{i}-\mathbf{r}_{% j})\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}=\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})% ^{2}+(z_{i}-z_{j})^{2}}.
  4. T ^ n = - i 2 2 M i 𝐑 i 2 \hat{T}_{n}=-\sum_{i}\frac{\hbar^{2}}{2M_{i}}\nabla^{2}_{\mathbf{R}_{i}}
  5. T ^ e = - i 2 2 m e 𝐫 i 2 \hat{T}_{e}=-\sum_{i}\frac{\hbar^{2}}{2m_{e}}\nabla^{2}_{\mathbf{r}_{i}}
  6. U ^ e n = - i j Z i e 2 4 π ϵ 0 | 𝐑 i - 𝐫 j | \hat{U}_{en}=-\sum_{i}\sum_{j}\frac{Z_{i}e^{2}}{4\pi\epsilon_{0}\left|\mathbf{% R}_{i}-\mathbf{r}_{j}\right|}
  7. U ^ e e = 1 2 i j i e 2 4 π ϵ 0 | 𝐫 i - 𝐫 j | = i j > i e 2 4 π ϵ 0 | 𝐫 i - 𝐫 j | \hat{U}_{ee}={1\over 2}\sum_{i}\sum_{j\neq i}\frac{e^{2}}{4\pi\epsilon_{0}% \left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|}=\sum_{i}\sum_{j>i}\frac{e^{2}}{4% \pi\epsilon_{0}\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|}
  8. U ^ n n = 1 2 i j i Z i Z j e 2 4 π ϵ 0 | 𝐑 i - 𝐑 j | = i j > i Z i Z j e 2 4 π ϵ 0 | 𝐑 i - 𝐑 j | . \hat{U}_{nn}={1\over 2}\sum_{i}\sum_{j\neq i}\frac{Z_{i}Z_{j}e^{2}}{4\pi% \epsilon_{0}\left|\mathbf{R}_{i}-\mathbf{R}_{j}\right|}=\sum_{i}\sum_{j>i}% \frac{Z_{i}Z_{j}e^{2}}{4\pi\epsilon_{0}\left|\mathbf{R}_{i}-\mathbf{R}_{j}% \right|}.
  9. 𝐫 i 2 s y m b o l 𝐫 i s y m b o l 𝐫 i = 2 x i 2 + 2 y i 2 + 2 z i 2 \nabla^{2}_{\mathbf{r}_{i}}\equiv symbol{\nabla}_{\mathbf{r}_{i}}\cdot symbol{% \nabla}_{\mathbf{r}_{i}}=\frac{\partial^{2}}{\partial x_{i}^{2}}+\frac{% \partial^{2}}{\partial y_{i}^{2}}+\frac{\partial^{2}}{\partial z_{i}^{2}}
  10. H = - 2 2 M tot 𝐗 2 + H with H = - 2 2 i = 1 N tot - 1 1 m i i 2 + 2 2 M tot i , j = 1 N tot - 1 i j + V ( 𝐭 ) . H=-\frac{\hbar^{2}}{2M_{\textrm{tot}}}\nabla^{2}_{\mathbf{X}}+H^{\prime}\quad% \,\text{with }\quad H^{\prime}=-\frac{\hbar^{2}}{2}\sum_{i=1}^{N_{\textrm{tot}% }-1}\frac{1}{m_{i}}\nabla^{2}_{i}+\frac{\hbar^{2}}{2M_{\textrm{tot}}}\sum_{i,j% =1}^{N_{\textrm{tot}}-1}\nabla_{i}\cdot\nabla_{j}+V(\mathbf{t}).
  11. H H
  12. H H^{\prime}
  13. H H^{\prime}
  14. H H^{\prime}
  15. H H^{\prime}
  16. H H
  17. H H^{\prime}
  18. H ^ el = T ^ e + U ^ e n + U ^ e e + U ^ n n . \hat{H}_{\mathrm{el}}=\hat{T}_{e}+\hat{U}_{en}+\hat{U}_{ee}+\hat{U}_{nn}.
  19. H el H_{\textrm{el}}
  20. V ( 𝐑 1 , 𝐑 2 , , 𝐑 N ) V(\mathbf{R}_{1},\mathbf{R}_{2},\ldots,\mathbf{R}_{N})
  21. V ( 𝐑 1 , 𝐑 2 , , 𝐑 N ) = V ( 𝐑 1 , 𝐑 2 , , 𝐑 N ) V(\mathbf{R}_{1},\mathbf{R}_{2},\ldots,\mathbf{R}_{N})=V(\mathbf{R}^{\prime}_{% 1},\mathbf{R}^{\prime}_{2},\ldots,\mathbf{R}^{\prime}_{N})
  22. 𝐑 i = 𝐑 i + 𝐭 (translation)   and 𝐑 i = 𝐑 i + Δ ϕ | 𝐬 | ( 𝐬 × 𝐑 i ) (infinitesimal   rotation) , \mathbf{R}^{\prime}_{i}=\mathbf{R}_{i}+\mathbf{t}\;\;\textrm{(translation)\;\;% and}\;\;\mathbf{R}^{\prime}_{i}=\mathbf{R}_{i}+\frac{\Delta\phi}{|\mathbf{s}|% }\;(\mathbf{s}\times\mathbf{R}_{i})\;\;\textrm{(infinitesimal\;\; rotation)},
  23. H ^ nuc = - 2 2 i = 1 N α = 1 3 1 M i 2 R i α 2 + V ( 𝐑 1 , , 𝐑 N ) \hat{H}_{\mathrm{nuc}}=-\frac{\hbar^{2}}{2}\sum_{i=1}^{N}\sum_{\alpha=1}^{3}% \frac{1}{M_{i}}\frac{\partial^{2}}{\partial R_{i\alpha}^{2}}+V(\mathbf{R}_{1},% \ldots,\mathbf{R}_{N})
  24. s y m b o l ρ i M i ( 𝐑 i - 𝐑 i 0 ) symbol{\rho}_{i}\equiv\sqrt{M_{i}}(\mathbf{R}_{i}-\mathbf{R}_{i}^{0})
  25. ρ i α = M i ( R i α - R i α 0 ) = 1 M i R i α , \frac{\partial}{\partial\rho_{i\alpha}}=\frac{\partial}{\sqrt{M_{i}}(\partial R% _{i\alpha}-\partial R^{0}_{i\alpha})}=\frac{1}{\sqrt{M_{i}}}\frac{\partial}{% \partial R_{i\alpha}},
  26. T = - 2 2 i = 1 N α = 1 3 2 ρ i α 2 . T=-\frac{\hbar^{2}}{2}\sum_{i=1}^{N}\sum_{\alpha=1}^{3}\frac{\partial^{2}}{% \partial\rho_{i\alpha}^{2}}.
  27. V = V 0 + i = 1 N α = 1 3 ( V ρ i α ) 0 ρ i α + 1 2 i , j = 1 N α , β = 1 3 ( 2 V ρ i α ρ j β ) 0 ρ i α ρ j β + , V=V_{0}+\sum_{i=1}^{N}\sum_{\alpha=1}^{3}\Big(\frac{\partial V}{\partial\rho_{% i\alpha}}\Big)_{0}\;\rho_{i\alpha}+\frac{1}{2}\sum_{i,j=1}^{N}\sum_{\alpha,% \beta=1}^{3}\Big(\frac{\partial^{2}V}{\partial\rho_{i\alpha}\partial\rho_{j% \beta}}\Big)_{0}\;\rho_{i\alpha}\rho_{j\beta}+\cdots,
  28. 𝐐𝐅𝐐 T = s y m b o l Φ with s y m b o l Φ = diag ( f 1 , , f 3 N - 6 , 0 , , 0 ) . \mathbf{Q}\mathbf{F}\mathbf{Q}^{\mathrm{T}}=symbol{\Phi}\quad\mathrm{with}% \quad symbol{\Phi}=\operatorname{diag}(f_{1},\dots,f_{3N-6},0,\ldots,0).
  29. q t i = 1 N α = 1 3 Q t , i α ρ i α , q_{t}\equiv\sum_{i=1}^{N}\sum_{\alpha=1}^{3}\;Q_{t,i\alpha}\rho_{i\alpha},
  30. H ^ nuc 1 2 t = 1 3 N - 6 [ - 2 2 q t 2 + f t q t 2 ] . \hat{H}_{\mathrm{nuc}}\approx\frac{1}{2}\sum_{t=1}^{3N-6}\left[-\hbar^{2}\frac% {\partial^{2}}{\partial q_{t}^{2}}+f_{t}q_{t}^{2}\right].
  31. 2 T = i j g i j s ˙ i s ˙ j . 2T=\sum_{ij}g_{ij}\dot{s}_{i}\dot{s}_{j}.
  32. - 2 -\hbar^{2}
  33. H ^ = - 2 2 M tot α = 1 3 2 X α 2 + 1 2 α , β = 1 3 μ α β ( 𝒫 α - Π α ) ( 𝒫 β - Π β ) + U - 2 2 s = 1 3 N - 6 2 q s 2 + V . \hat{H}=-\frac{\hbar^{2}}{2M_{\mathrm{tot}}}\sum_{\alpha=1}^{3}\frac{\partial^% {2}}{\partial X_{\alpha}^{2}}+\frac{1}{2}\sum_{\alpha,\beta=1}^{3}\mu_{\alpha% \beta}(\mathcal{P}_{\alpha}-\Pi_{\alpha})(\mathcal{P}_{\beta}-\Pi_{\beta})+U-% \frac{\hbar^{2}}{2}\sum_{s=1}^{3N-6}\frac{\partial^{2}}{\partial q_{s}^{2}}+V.
  34. 𝐗 1 M tot i = 1 N M i 𝐑 i with M tot i = 1 N M i . \mathbf{X}\equiv\frac{1}{M_{\mathrm{tot}}}\sum_{i=1}^{N}M_{i}\mathbf{R}_{i}% \quad\mathrm{with}\quad M_{\mathrm{tot}}\equiv\sum_{i=1}^{N}M_{i}.
  35. 𝒫 α \mathcal{P}_{\alpha}
  36. Π α \Pi_{\alpha}\,
  37. Π α = - i s , t = 1 3 N - 6 ζ s t α q s q t \Pi_{\alpha}=-i\hbar\sum_{s,t=1}^{3N-6}\zeta^{\alpha}_{st}\;q_{s}\frac{% \partial}{\partial q_{t}}
  38. ζ s t α = i = 1 N β , γ = 1 3 ϵ α β γ Q s , i β Q t , i γ and α = 1 , 2 , 3. \zeta^{\alpha}_{st}=\sum_{i=1}^{N}\sum_{\beta,\gamma=1}^{3}\epsilon_{\alpha% \beta\gamma}Q_{s,i\beta}\,Q_{t,i\gamma}\;\;\mathrm{and}\quad\alpha=1,2,3.
  39. 𝒫 α \mathcal{P}_{\alpha}
  40. 𝒫 α \mathcal{P}_{\alpha}
  41. Π β \Pi_{\beta}\,
  42. s y m b o l μ symbol{\mu}
  43. s y m b o l μ symbol{\mu}
  44. U = - 1 8 α = 1 3 μ α α U=-\frac{1}{8}\sum_{\alpha=1}^{3}\mu_{\alpha\alpha}
  45. q s = i = 1 N α = 1 3 Q s , i α ρ i α for s = 1 , , 3 N - 6. q_{s}=\sum_{i=1}^{N}\sum_{\alpha=1}^{3}Q_{s,i\alpha}\rho_{i\alpha}\quad\mathrm% {for}\quad s=1,\ldots,3N-6.
  46. V 1 2 s = 1 3 N - 6 f s q s 2 . V\approx\frac{1}{2}\sum_{s=1}^{3N-6}f_{s}q_{s}^{2}.

Molecular_orbital_diagram.html

  1. Bond Order = ( No. of electrons in bonding MOs ) - ( No. of electrons in anti-bonding MOs ) 2 \ \mbox{Bond Order}~{}=\frac{(\mbox{No. of electrons in bonding MOs}~{})-(% \mbox{No. of electrons in anti-bonding MOs}~{})}{2}
  2. n n
  3. n 1 n−1

Monadic_predicate_calculus.html

  1. P ( x ) P(x)
  2. P P
  3. x x
  4. ( x D ( x ) M ( x ) ) ¬ ( y M ( y ) B ( y ) ) ¬ ( z D ( z ) B ( z ) ) (\forall x\,D(x)\Rightarrow M(x))\land\neg(\exists y\,M(y)\land B(y))% \Rightarrow\neg(\exists z\,D(z)\land B(z))
  5. D D
  6. M M
  7. B B
  8. x P 1 ( x ) P n ( x ) ¬ P 1 ( x ) ¬ P m ( x ) \forall x\,P_{1}(x)\lor\cdots\lor P_{n}(x)\lor\neg P^{\prime}_{1}(x)\lor\cdots% \lor\neg P^{\prime}_{m}(x)
  9. x ¬ P 1 ( x ) ¬ P n ( x ) P 1 ( x ) P m ( x ) , \exists x\,\neg P_{1}(x)\land\cdots\land\neg P_{n}(x)\land P^{\prime}_{1}(x)% \land\cdots\land P^{\prime}_{m}(x),
  10. ( x ¬ M ( x ) H ( x ) C ( x ) ) (\forall x\,\neg M(x)\lor H(x)\lor C(x))

Monetary_conditions_index.html

  1. y = a 0 + a 1 r + a 2 q + ν , \ y=a_{0}+a_{1}r+a_{2}q+\nu,
  2. M C I t = M C I 0 e x p [ ( r t - r 0 ) + ( a 1 / a 2 ) q t ] . \ MCI_{t}=MCI_{0}exp[(r_{t}-r_{0})+(a_{1}/a_{2})q_{t}].

Monin–Obukhov_length.html

  1. L = - u * 3 θ ¯ v k g ( w θ v ¯ ) s L=-\frac{u^{3}_{*}\bar{\theta}_{v}}{kg(\overline{w^{^{\prime}}\theta^{^{\prime% }}_{v}})_{s}}
  2. u * u_{*}
  3. θ ¯ v \bar{\theta}_{v}
  4. ( w θ v ¯ ) s (\overline{w^{^{\prime}}\theta^{^{\prime}}_{v}})_{s}
  5. w θ v ¯ = w θ ¯ + 0.61 T ¯ w q ¯ \overline{w^{^{\prime}}\theta^{^{\prime}}_{v}}=\overline{w^{^{\prime}}\theta^{% ^{\prime}}}+0.61\overline{T}\;\overline{w^{^{\prime}}q^{^{\prime}}}
  6. θ \theta
  7. T ¯ \overline{T}
  8. q q
  9. L L
  10. w θ v ¯ \overline{w^{^{\prime}}\theta^{^{\prime}}_{v}}
  11. w θ v ¯ \overline{w^{^{\prime}}\theta^{^{\prime}}_{v}}
  12. w θ v ¯ \overline{w^{^{\prime}}\theta^{^{\prime}}_{v}}
  13. L L
  14. - L -L

Monotone_likelihood_ratio.html

  1. f ( x ) f(x)
  2. g ( x ) g(x)
  3. x x
  4. f ( x ) / g ( x ) f(x)/g(x)
  5. for every x 1 > x 0 , f ( x 1 ) g ( x 1 ) f ( x 0 ) g ( x 0 ) \,\text{for every }x_{1}>x_{0},\quad\frac{f(x_{1})}{g(x_{1})}\geq\frac{f(x_{0}% )}{g(x_{0})}
  6. x x
  7. x ( f ( x ) g ( x ) ) 0 \frac{\partial}{\partial x}\left(\frac{f(x)}{g(x)}\right)\geq 0
  8. f ( x ) f(x)
  9. g ( x ) g(x)
  10. x x
  11. f f
  12. g g
  13. e e
  14. q q
  15. e e
  16. e { H , L } e\in\{H,L\}
  17. q q
  18. f ( q e ) f(q\mid e)
  19. P r [ e = H q ] = f ( q H ) f ( q H ) + f ( q L ) Pr[e=H\mid q]=\frac{f(q\mid H)}{f(q\mid H)+f(q\mid L)}
  20. f ( q e ) f(q\mid e)
  21. 1 1 + f ( q L ) / f ( q H ) \frac{1}{1+f(q\mid L)/f(q\mid H)}
  22. q q
  23. { f θ ( x ) } θ Θ \{f_{\theta}(x)\}_{\theta\in\Theta}
  24. θ \theta
  25. Θ \Theta
  26. T ( X ) T(X)
  27. θ 1 < θ 2 \theta_{1}<\theta_{2}
  28. f θ 2 ( X = x 1 , x 2 , x 3 , ) f θ 1 ( X = x 1 , x 2 , x 3 , ) \frac{f_{\theta_{2}}(X=x_{1},x_{2},x_{3},\dots)}{f_{\theta_{1}}(X=x_{1},x_{2},% x_{3},\dots)}
  29. T ( X ) T(X)
  30. T ( X ) T(X)
  31. T ( X ) T(X)
  32. f θ ( X ) f_{\theta}(X)
  33. [ λ ] [\lambda]
  34. x i \sum x_{i}
  35. [ n , p ] [n,p]
  36. x i \sum x_{i}
  37. [ λ ] [\lambda]
  38. x i \sum x_{i}
  39. [ μ , σ ] [\mu,\sigma]
  40. σ \sigma
  41. x i \sum x_{i}
  42. T ( X ) T(X)
  43. H 0 : θ θ 0 H_{0}:\theta\leq\theta_{0}
  44. H 1 : θ > θ 0 H_{1}:\theta>\theta_{0}
  45. e e
  46. y y
  47. f ( y ; e ) . f(y;e).
  48. f f
  49. e 1 , e 2 e_{1},e_{2}
  50. e 2 > e 1 e_{2}>e_{1}
  51. f ( y ; e 2 ) / f ( y ; e 1 ) f(y;e_{2})/f(y;e_{1})
  52. y y
  53. f θ ( x ) f_{\theta}(x)
  54. T ( X ) T(X)
  55. θ \theta
  56. T ( X ) T(X)
  57. x x
  58. θ \theta
  59. T ( X ) T(X)
  60. f θ f_{\theta}
  61. θ 1 > θ 0 \theta_{1}>\theta_{0}
  62. x 1 > x 0 x_{1}>x_{0}
  63. f θ 1 ( x 1 ) f θ 0 ( x 1 ) f θ 1 ( x 0 ) f θ 0 ( x 0 ) , \frac{f_{\theta_{1}}(x_{1})}{f_{\theta_{0}}(x_{1})}\geq\frac{f_{\theta_{1}}(x_% {0})}{f_{\theta_{0}}(x_{0})},
  64. f θ 1 ( x 1 ) f θ 0 ( x 0 ) f θ 1 ( x 0 ) f θ 0 ( x 1 ) . f_{\theta_{1}}(x_{1})f_{\theta_{0}}(x_{0})\geq f_{\theta_{1}}(x_{0})f_{\theta_% {0}}(x_{1}).\,
  65. x 1 x_{1}
  66. x 0 x_{0}
  67. min x X x 1 f θ 1 ( x 1 ) f θ 0 ( x 0 ) d x 0 \int_{\min_{x}\in X}^{x_{1}}f_{\theta_{1}}(x_{1})f_{\theta_{0}}(x_{0})\,dx_{0}
  68. min x X x 1 f θ 1 ( x 0 ) f θ 0 ( x 1 ) d x 0 \geq\int_{\min_{x}\in X}^{x_{1}}f_{\theta_{1}}(x_{0})f_{\theta_{0}}(x_{1})\,dx% _{0}
  69. f θ 1 f θ 0 ( x ) F θ 1 F θ 0 ( x ) \frac{f_{\theta_{1}}}{f_{\theta_{0}}}(x)\geq\frac{F_{\theta_{1}}}{F_{\theta_{0% }}}(x)
  70. x 0 x_{0}
  71. x 1 x_{1}
  72. x 0 max x X f θ 1 ( x 1 ) f θ 0 ( x 0 ) d x 1 \int_{x_{0}}^{\max_{x}\in X}f_{\theta_{1}}(x_{1})f_{\theta_{0}}(x_{0})\,dx_{1}
  73. x 0 max x X f θ 1 ( x 0 ) f θ 0 ( x 1 ) d x 1 \geq\int_{x_{0}}^{\max_{x}\in X}f_{\theta_{1}}(x_{0})f_{\theta_{0}}(x_{1})\,dx% _{1}
  74. 1 - F θ 1 ( x ) 1 - F θ 0 ( x ) f θ 1 f θ 0 ( x ) \frac{1-F_{\theta_{1}}(x)}{1-F_{\theta_{0}}(x)}\geq\frac{f_{\theta_{1}}}{f_{% \theta_{0}}}(x)
  75. F θ 1 ( x ) F θ 0 ( x ) x F_{\theta_{1}}(x)\leq F_{\theta_{0}}(x)\ \forall x
  76. f θ 1 ( x ) 1 - F θ 1 ( x ) f θ 0 ( x ) 1 - F θ 0 ( x ) x \frac{f_{\theta_{1}}(x)}{1-F_{\theta_{1}}(x)}\leq\frac{f_{\theta_{0}}(x)}{1-F_% {\theta_{0}}(x)}\ \forall x

Monty_Hall_problem.html

  1. P ( H 3 | C 1 , X 1 ) = 1 / 2 P(H3|C1,X1)=1/2
  2. P ( H 3 | C 2 , X 1 ) = 1 P(H3|C2,X1)=1
  3. P ( H 3 | C 3 , X 1 ) = 0 P(H3|C3,X1)=0
  4. P ( C 2 | H 3 , X 1 ) = P ( H 3 | C 2 , X 1 ) P ( C 2 | X 1 ) P ( H 3 | X 1 ) P(C2|H3,X1)=\frac{P(H3|C2,X1)P(C2|X1)}{P(H3|X1)}
  5. = P ( H 3 | C 2 , X 1 ) P ( C 2 | X 1 ) P ( H 3 | C 1 , X 1 ) P ( C 1 | X 1 ) + P ( H 3 | C 2 , X 1 ) P ( C 2 | X 1 ) + P ( H 3 | C 3 , X 1 ) P ( C 3 | X 1 ) =\frac{P(H3|C2,X1)P(C2|X1)}{P(H3|C1,X1)P(C1|X1)+P(H3|C2,X1)P(C2|X1)+P(H3|C3,X1% )P(C3|X1)}
  6. = P ( H 3 | C 2 , X 1 ) P ( H 3 | C 1 , X 1 ) + P ( H 3 | C 2 , X 1 ) + P ( H 3 | C 3 , X 1 ) = 1 1 / 2 + 1 + 0 = 2 3 =\frac{P(H3|C2,X1)}{P(H3|C1,X1)+P(H3|C2,X1)+P(H3|C3,X1)}=\frac{1}{1/2+1+0}=% \frac{2}{3}

Moore_plane.html

  1. Γ \Gamma
  2. Γ = { ( x , y ) \R 2 | y 0 } \Gamma=\{(x,y)\in\R^{2}|y\geq 0\}
  3. Γ \Gamma
  4. ( p , q ) \mathcal{B}(p,q)
  5. ( x , y ) (x,y)
  6. y > 0 y>0
  7. Γ \Gamma
  8. Γ \ { ( x , 0 ) | x \R } \Gamma\backslash\{(x,0)|x\in\R\}
  9. p = ( x , 0 ) p=(x,0)
  10. { p } A \{p\}\cup A
  11. ( p , q ) = { { U ϵ ( p , q ) := { ( x , y ) : ( x - p ) 2 + ( y - q ) 2 < ϵ 2 } ϵ > 0 } , if q > 0 ; { V ϵ ( p ) := { ( p , 0 ) } { ( x , y ) : ( x - p ) 2 + ( y - ϵ ) 2 < ϵ 2 } ϵ > 0 } , if q = 0. \mathcal{B}(p,q)=\begin{cases}\{U_{\epsilon}(p,q):=\{(x,y):(x-p)^{2}+(y-q)^{2}% <\epsilon^{2}\}\mid\epsilon>0\},&\mbox{if }~{}q>0;\\ \{V_{\epsilon}(p):=\{(p,0)\}\cup\{(x,y):(x-p)^{2}+(y-\epsilon)^{2}<\epsilon^{2% }\}\mid\epsilon>0\},&\mbox{if }~{}q=0.\end{cases}
  12. Γ \Gamma
  13. { ( x , 0 ) Γ | x R } \{(x,0)\in\Gamma|x\in R\}
  14. Γ \Gamma
  15. S := { ( p , q ) × : q > 0 } S:=\{(p,q)\in\mathbb{Q}\times\mathbb{Q}:q>0\}
  16. f : M f:M\to\mathbb{R}
  17. S S
  18. | | | S | = 2 0 |\mathbb{R}|^{|S|}=2^{\aleph_{0}}
  19. L := { ( p , 0 ) : p } L:=\{(p,0):p\in\mathbb{R}\}
  20. 2 0 2^{\aleph_{0}}
  21. 2 2 0 > 2 0 2^{2^{\aleph_{0}}}>2^{\aleph_{0}}
  22. \mathbb{R}