wpmath0000003_3

Crank_(mechanism).html

  1. x = r cos α + l x=r\cos\alpha+l
  2. x = r cos α + l 2 - r 2 sin 2 α x=r\cos\alpha+\sqrt{l^{2}-r^{2}\sin^{2}\alpha}
  3. τ = F r sin α \tau=Fr\sin\alpha\,
  4. τ \tau\,
  5. x = - r sin α - r 2 sin α cos α l 2 - r 2 sin 2 α x^{\prime}=-r\sin\alpha-\frac{r^{2}\sin\alpha\cos\alpha}{\sqrt{l^{2}-r^{2}\sin% ^{2}\alpha}}

Cristobalite.html

  1. 3 ¯ \overline{3}
  2. α - β \alpha-\beta

Critical_phenomena.html

  1. 2 D 2D
  2. T T
  3. H = - J [ i , j ] S i S j H=-J\sum_{[i,j]}S_{i}\cdot S_{j}
  4. J J
  5. T c T_{c}
  6. T c T_{c}
  7. ξ \xi
  8. T c T_{c}
  9. T T c T\to T_{c}
  10. ξ \xi\to\infty
  11. A ( T ) ( T - T c ) α A(T)\propto(T-T_{c})^{\alpha}
  12. α , \alpha\,,
  13. τ \tau
  14. ξ \xi
  15. τ = ξ z \tau=\xi^{\,z}
  16. T c T_{c}
  17. T < T c T<T_{c}
  18. T c T_{c}

Cron.html

  1. O ( n ) O\left(\sqrt{n}\right)

Cronbach's_alpha.html

  1. α \alpha
  2. α \alpha
  3. α \alpha
  4. α \alpha
  5. K K
  6. X = Y 1 + Y 2 + + Y K X=Y_{1}+Y_{2}+\cdots+Y_{K}
  7. α \alpha
  8. α = K K - 1 ( 1 - i = 1 K σ Y i 2 σ X 2 ) \alpha={K\over K-1}\left(1-{\sum_{i=1}^{K}\sigma^{2}_{Y_{i}}\over\sigma^{2}_{X% }}\right)
  9. σ X 2 \sigma^{2}_{X}
  10. σ Y i 2 \sigma^{2}_{Y_{i}}
  11. α = K K - 1 ( 1 - i = 1 K P i Q i σ X 2 ) \alpha={K\over K-1}\left(1-{\sum_{i=1}^{K}P_{i}Q_{i}\over\sigma^{2}_{X}}\right)
  12. P i P_{i}
  13. Q i = 1 - P i Q_{i}=1-P_{i}
  14. α \alpha
  15. α = K c ¯ ( v ¯ + ( K - 1 ) c ¯ ) \alpha={K\bar{c}\over(\bar{v}+(K-1)\bar{c})}
  16. K K
  17. v ¯ \bar{v}
  18. c ¯ \bar{c}
  19. α standardized = K r ¯ ( 1 + ( K - 1 ) r ¯ ) \alpha\text{standardized}={K\bar{r}\over(1+(K-1)\bar{r})}
  20. K K
  21. r ¯ \bar{r}
  22. K ( K - 1 ) / 2 K(K-1)/2
  23. α \alpha
  24. ρ X X = σ T 2 σ X 2 \rho_{XX}={\sigma^{2}_{T}\over\sigma_{X}^{2}}
  25. α \alpha
  26. τ \tau

Cross-ratio.html

  1. ( A , B ; C , D ) = A C B D B C A D (A,B;C,D)=\frac{AC\cdot BD}{BC\cdot AD}
  2. ( z 1 , z 2 ; z 3 , z 4 ) = ( z 1 - z 3 ) ( z 2 - z 4 ) ( z 2 - z 3 ) ( z 1 - z 4 ) . (z_{1},z_{2};z_{3},z_{4})=\frac{(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1% }-z_{4})}.
  3. ( z 1 , z 2 ; z 3 , z 4 ) = z 1 - z 3 z 2 - z 3 : z 1 - z 4 z 2 - z 4 . (z_{1},z_{2};z_{3},z_{4})=\frac{z_{1}-z_{3}}{z_{2}-z_{3}}:\frac{z_{1}-z_{4}}{z% _{2}-z_{4}}.
  4. z 1 , z 2 , z 3 , z 4 z_{1},\ z_{2},\ z_{3},\ z_{4}
  5. ( A , B ; C , D ) = A C B D B C A D (A,B;C,D)=\frac{AC\cdot BD}{BC\cdot AD}
  6. ( A , B ; C , D ) = A C C B : A D D B (A,B;C,D)=\frac{AC}{CB}:\frac{AD}{DB}
  7. A C C B \frac{AC}{CB}
  8. A D D B \frac{AD}{DB}
  9. d ( p , q ) = 1 2 log | q - a | | b - p | | p - a | | b - q | d(p,q)=\frac{1}{2}\log\frac{|q-a||b-p|}{|p-a||b-q|}
  10. ( z 1 , z 2 ; z 3 , z 4 ) = ( z 2 , z 1 ; z 4 , z 3 ) = ( z 3 , z 4 ; z 1 , z 2 ) = ( z 4 , z 3 ; z 2 , z 1 ) . (z_{1},z_{2};z_{3},z_{4})=(z_{2},z_{1};z_{4},z_{3})=(z_{3},z_{4};z_{1},z_{2})=% (z_{4},z_{3};z_{2},z_{1}).\,
  11. ( z 1 , z 2 ; z 3 , z 4 ) \displaystyle(z_{1},z_{2};z_{3},z_{4})
  12. 1 λ \frac{1}{\lambda}
  13. 1 - λ 1-\lambda\,
  14. λ λ - 1 \frac{\lambda}{\lambda-1}
  15. 1 1 - λ \frac{1}{1-\lambda}
  16. λ - 1 λ \frac{\lambda-1}{\lambda}
  17. e ± i π / 3 e^{\pm i\pi/3}
  18. S 3 PGL ( 2 , 2 ) \mathrm{S}_{3}\approx\mathrm{PGL}(2,2)
  19. - 1 = [ - 1 : 1 ] -1=[-1:1]
  20. 2 = 1 / 2 = - 1 2=1/2=-1
  21. S 4 PGL ( 2 , 3 ) \mathrm{S}_{4}\approx\mathrm{PGL}(2,3)
  22. S 3 S 4 \mathrm{S}_{3}\hookrightarrow\mathrm{S}_{4}
  23. - 1 -1
  24. ( a b ) ( c d ) (ab)(cd)
  25. S 4 / K \mathrm{S}_{4}/\mathrm{K}
  26. ( z , z 2 ; z , z 4 ) = ( z 1 , z ; z 3 , z ) = 0 (z,z_{2};z,z_{4})=(z_{1},z;z_{3},z)=0
  27. ( z , z ; z 3 , z 4 ) = ( z 1 , z 2 ; z , z ) = 1 (z,z;z_{3},z_{4})=(z_{1},z_{2};z,z)=1
  28. ( z , z 2 ; z 3 , z ) = ( z 1 , z ; z , z 4 ) = . (z,z_{2};z_{3},z)=(z_{1},z;z,z_{4})=\infty.
  29. λ = e ± i π / 3 \lambda=e^{\pm i\pi/3}
  30. f ( z ) = a z + b c z + d , where a , b , c , d and a d - b c 0. f(z)=\frac{az+b}{cz+d}\;,\quad\mbox{where }~{}a,b,c,d\in\mathbb{C}\mbox{ and }% ~{}ad-bc\neq 0.
  31. ( f ( z 1 ) , f ( z 2 ) ; f ( z 3 ) , f ( z 4 ) ) = ( z 1 , z 2 ; z 3 , z 4 ) . (f(z_{1}),f(z_{2});f(z_{3}),f(z_{4}))=(z_{1},z_{2};z_{3},z_{4}).
  32. f ( z ) = ( z , z 2 ; z 3 , z 4 ) . f(z)=(z,z_{2};z_{3},z_{4}).\,
  33. z z + a z\mapsto z+a
  34. z b z z\mapsto bz
  35. T : z z - 1 , T:z\mapsto z^{-1},
  36. 𝐏 n = 𝐏 ( K n + 1 ) \mathbf{P}^{n}=\mathbf{P}(K^{n+1})
  37. PGL ( n , K ) = 𝐏 ( GL ( n + 1 , K ) ) , \mathrm{PGL}(n,K)=\mathbf{P}(\mathrm{GL}(n+1,K)),

Cross-validation_(statistics).html

  1. 1 n i = 1 n ( y i - a - s y m b o l β T 𝐱 i ) 2 = 1 n i = 1 n ( y i - a - β 1 x i 1 - - β p x i p ) 2 \frac{1}{n}\sum_{i=1}^{n}(y_{i}-a-symbol\beta^{T}\mathbf{x}_{i})^{2}=\frac{1}{% n}\sum_{i=1}^{n}(y_{i}-a-\beta_{1}x_{i1}-\dots-\beta_{p}x_{ip})^{2}
  2. C p n C_{p}^{n}
  3. C 1 n = n C_{1}^{n}=n

Crossover_(genetic_algorithm).html

  1. i i
  2. i i

Cryptosystem.html

  1. ( 𝒫 , 𝒞 , 𝒦 , , 𝒟 ) (\mathcal{P},\mathcal{C},\mathcal{K},\mathcal{E},\mathcal{D})
  2. 𝒫 \mathcal{P}
  3. 𝒞 \mathcal{C}
  4. 𝒦 \mathcal{K}
  5. = { E k : k 𝒦 } \mathcal{E}=\{E_{k}:k\in\mathcal{K}\}
  6. E k : 𝒫 𝒞 E_{k}:\mathcal{P}\rightarrow\mathcal{C}
  7. 𝒟 = { D k : k 𝒦 } \mathcal{D}=\{D_{k}:k\in\mathcal{K}\}
  8. D k : 𝒞 𝒫 D_{k}:\mathcal{C}\rightarrow\mathcal{P}
  9. e 𝒦 e\in\mathcal{K}
  10. d 𝒦 d\in\mathcal{K}
  11. D d ( E e ( p ) ) = p D_{d}(E_{e}(p))=p
  12. p 𝒫 p\in\mathcal{P}

Crystal_system.html

  1. 1 \mathbb{Z}_{1}
  2. 1 ¯ \overline{1}
  3. 2 \mathbb{Z}_{2}
  4. 2 \mathbb{Z}_{2}
  5. 2 \mathbb{Z}_{2}
  6. 𝕍 = 2 × 2 \mathbb{V}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}
  7. 𝕍 = 2 × 2 \mathbb{V}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}
  8. 𝕍 = 2 × 2 \mathbb{V}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}
  9. 𝕍 × 2 \mathbb{V}\times\mathbb{Z}_{2}
  10. 4 \mathbb{Z}_{4}
  11. 4 ¯ \overline{4}
  12. 4 \mathbb{Z}_{4}
  13. 4 × 2 \mathbb{Z}_{4}\times\mathbb{Z}_{2}
  14. 𝔻 8 = 4 2 \mathbb{D}_{8}=\mathbb{Z}_{4}\rtimes\mathbb{Z}_{2}
  15. 𝔻 8 = 4 2 \mathbb{D}_{8}=\mathbb{Z}_{4}\rtimes\mathbb{Z}_{2}
  16. 4 ¯ \overline{4}
  17. 4 ¯ \overline{4}
  18. 𝔻 8 = 4 2 \mathbb{D}_{8}=\mathbb{Z}_{4}\rtimes\mathbb{Z}_{2}
  19. 𝔻 8 × 2 \mathbb{D}_{8}\times\mathbb{Z}_{2}
  20. 3 \mathbb{Z}_{3}
  21. 3 ¯ \overline{3}
  22. 6 = 3 × 2 \mathbb{Z}_{6}=\mathbb{Z}_{3}\times\mathbb{Z}_{2}
  23. 𝔻 6 = 3 2 \mathbb{D}_{6}=\mathbb{Z}_{3}\rtimes\mathbb{Z}_{2}
  24. 𝔻 6 = 3 2 \mathbb{D}_{6}=\mathbb{Z}_{3}\rtimes\mathbb{Z}_{2}
  25. 3 ¯ \overline{3}
  26. 3 ¯ \overline{3}
  27. 3 ¯ \overline{3}
  28. 𝔻 12 = 6 2 \mathbb{D}_{12}=\mathbb{Z}_{6}\rtimes\mathbb{Z}_{2}
  29. 6 = 3 × 2 \mathbb{Z}_{6}=\mathbb{Z}_{3}\times\mathbb{Z}_{2}
  30. 6 ¯ \overline{6}
  31. 6 = 3 × 2 \mathbb{Z}_{6}=\mathbb{Z}_{3}\times\mathbb{Z}_{2}
  32. 6 × 2 \mathbb{Z}_{6}\times\mathbb{Z}_{2}
  33. 𝔻 12 = 6 2 \mathbb{D}_{12}=\mathbb{Z}_{6}\rtimes\mathbb{Z}_{2}
  34. 𝔻 12 = 6 2 \mathbb{D}_{12}=\mathbb{Z}_{6}\rtimes\mathbb{Z}_{2}
  35. 6 ¯ \overline{6}
  36. 6 ¯ \overline{6}
  37. 𝔻 12 = 6 2 \mathbb{D}_{12}=\mathbb{Z}_{6}\rtimes\mathbb{Z}_{2}
  38. 𝔻 12 × 2 \mathbb{D}_{12}\times\mathbb{Z}_{2}
  39. 𝔸 4 \mathbb{A}_{4}
  40. 4 ¯ \overline{4}
  41. 𝕊 4 \mathbb{S}_{4}
  42. 3 ¯ \overline{3}
  43. 𝔸 4 × 2 \mathbb{A}_{4}\times\mathbb{Z}_{2}
  44. 𝕊 4 \mathbb{S}_{4}
  45. 3 ¯ \overline{3}
  46. 𝕊 4 × 2 \mathbb{S}_{4}\times\mathbb{Z}_{2}
  47. 𝐑 = n 1 𝐚 1 + n 2 𝐚 2 + n 3 𝐚 3 , \mathbf{R}=n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}+n_{3}\mathbf{a}_{3},
  48. a , b , c , d a,b,c,d
  49. α , β , γ , δ , ϵ , ζ \alpha,\beta,\gamma,\delta,\epsilon,\zeta
  50. a b c d , α β γ δ ϵ ζ 90 a\neq b\neq c\neq d,\alpha\neq\beta\neq\gamma\neq\delta\neq\epsilon\neq\zeta% \neq 90^{\circ}
  51. a b c d , α β γ 90 , δ = ϵ = ζ = 90 a\neq b\neq c\neq d,\alpha\neq\beta\neq\gamma\neq 90^{\circ},\delta=\epsilon=% \zeta=90^{\circ}
  52. a b c d , α 90 , β = γ = δ = ϵ = 90 , ζ 90 a\neq b\neq c\neq d,\alpha\neq 90^{\circ},\beta=\gamma=\delta=\epsilon=90^{% \circ},\zeta\neq 90^{\circ}
  53. a b c d , α 90 , β = γ = δ = ϵ = ζ = 90 a\neq b\neq c\neq d,\alpha\neq 90^{\circ},\beta=\gamma=\delta=\epsilon=\zeta=9% 0^{\circ}
  54. a b c d , α = β = γ = δ = ϵ = ζ = 90 a\neq b\neq c\neq d,\alpha=\beta=\gamma=\delta=\epsilon=\zeta=90^{\circ}
  55. a b = c d , α 90 , β = γ = δ = ϵ = ζ = 90 a\neq b=c\neq d,\alpha\neq 90^{\circ},\beta=\gamma=\delta=\epsilon=\zeta=90^{\circ}
  56. a b = c d , α 90 , β = γ = δ = ϵ = 90 , ζ = 120 a\neq b=c\neq d,\alpha\neq 90^{\circ},\beta=\gamma=\delta=\epsilon=90^{\circ},% \zeta=120^{\circ}
  57. a = d b = c , α = ζ = 90 , β = ϵ 90 , γ 90 , δ = 180 - γ a=d\neq b=c,\alpha=\zeta=90^{\circ},\beta=\epsilon\neq 90^{\circ},\gamma\neq 9% 0^{\circ},\delta=180^{\circ}-\gamma
  58. a = d b = c , α = ζ = 120 , β = ϵ 90 , γ δ 90 , c o s δ = c o s β - c o s γ a=d\neq b=c,\alpha=\zeta=120^{\circ},\beta=\epsilon\neq 90^{\circ},\gamma\neq% \delta\neq 90^{\circ},cos\delta=cos\beta-cos\gamma
  59. a b = c d , α = β = γ = δ = ϵ = ζ = 90 a\neq b=c\neq d,\alpha=\beta=\gamma=\delta=\epsilon=\zeta=90^{\circ}
  60. a b = c d , α = β = γ = δ = ϵ = 90 , ζ = 120 a\neq b=c\neq d,\alpha=\beta=\gamma=\delta=\epsilon=90^{\circ},\zeta=120^{\circ}
  61. a = d b = c , α = γ = δ = ζ = 90 , β = ϵ 90 a=d\neq b=c,\alpha=\gamma=\delta=\zeta=90^{\circ},\beta=\epsilon\neq 90^{\circ}
  62. a = d b = c , α = ζ = 120 , β = ϵ 90 , γ = δ 90 , c o s γ = - 1 2 c o s β a=d\neq b=c,\alpha=\zeta=120^{\circ},\beta=\epsilon\neq 90^{\circ},\gamma=% \delta\neq 90^{\circ},cos\gamma=-\tfrac{1}{2}cos\beta
  63. a = d b = c , α = β = γ = δ = ϵ = ζ = 90 a=d\neq b=c,\alpha=\beta=\gamma=\delta=\epsilon=\zeta=90^{\circ}
  64. a = d b = c , α = β = γ = δ = ϵ = 90 , ζ = 120 a=d\neq b=c,\alpha=\beta=\gamma=\delta=\epsilon=90^{\circ},\zeta=120^{\circ}
  65. a = d b = c , α = ζ = 120 , β = γ = δ = ϵ = 90 , a=d\neq b=c,\alpha=\zeta=120^{\circ},\beta=\gamma=\delta=\epsilon=90^{\circ},
  66. a = b = c d , α = β = γ = δ = ϵ = ζ = 90 a=b=c\neq d,\alpha=\beta=\gamma=\delta=\epsilon=\zeta=90^{\circ}
  67. a = b = c = d , α = γ = ζ 90 , β = ϵ = 90 , δ = 180 - α a=b=c=d,\alpha=\gamma=\zeta\neq 90^{\circ},\beta=\epsilon=90^{\circ},\delta=18% 0^{\circ}-\alpha
  68. a = b = c = d , α = γ = ζ β = δ = ϵ , c o s β = - 0.5 - c o s α a=b=c=d,\alpha=\gamma=\zeta\neq\beta=\delta=\epsilon,cos\beta=-0.5-cos\alpha
  69. a = b = c = d , α = ζ = 90 , β = ϵ = 120 , γ = δ 90 a=b=c=d,\alpha=\zeta=90^{\circ},\beta=\epsilon=120^{\circ},\gamma=\delta\neq 9% 0^{\circ}
  70. a = b = c = d , α = ζ = 120 , β = γ = δ = ϵ = 90 a=b=c=d,\alpha=\zeta=120^{\circ},\beta=\gamma=\delta=\epsilon=90^{\circ}
  71. a = b = c = d , α = β = γ = δ = ϵ = ζ , c o s α = - 1 4 a=b=c=d,\alpha=\beta=\gamma=\delta=\epsilon=\zeta,cos\alpha=-\tfrac{1}{4}
  72. a = b = c = d , α = β = γ = δ = ϵ = ζ = 90 a=b=c=d,\alpha=\beta=\gamma=\delta=\epsilon=\zeta=90^{\circ}

Cuban_prime.html

  1. p = x 3 - y 3 x - y , x = y + 1 , y > 0 p=\frac{x^{3}-y^{3}}{x-y},\ x=y+1,\ y>0
  2. ( y + 1 ) 3 - y 3 y + 1 - y \tfrac{(y+1)^{3}-y^{3}}{y+1-y}
  3. 3 y 2 + 3 y + 1 3y^{2}+3y+1
  4. y = 100000845 4096 y=100000845^{4096}
  5. p = x 3 - y 3 x - y , x = y + 2 , y > 0. p=\frac{x^{3}-y^{3}}{x-y},\ x=y+2,\ y>0.
  6. 3 y 2 + 6 y + 4 3y^{2}+6y+4
  7. y = n - 1 y=n-1
  8. 3 n 2 + 1 , n > 1 3n^{2}+1,\ n>1

Cube_(algebra).html

  1. n n
  2. n n
  3. n n
  4. n n
  5. n n
  6. m m
  7. m m
  8. x y z 0 xyz≠0
  9. n n
  10. n n
  11. 1 3 + 2 3 + + n 3 = ( 1 + 2 + + n ) 2 = ( n ( n + 1 ) 2 ) 2 . 1^{3}+2^{3}+\dots+n^{3}=(1+2+\dots+n)^{2}=\left(\frac{n(n+1)}{2}\right)^{2}.
  12. 1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 15 2 1^{3}+2^{3}+3^{3}+4^{3}+5^{3}=15^{2}\,
  13. y y
  14. 1 3 + 3 3 + + ( 2 y - 1 ) 3 = ( x y ) 2 1^{3}+3^{3}+\dots+(2y-1)^{3}=(xy)^{2}
  15. x x
  16. y y
  17. x 2 - 2 y 2 = - 1 x^{2}-2y^{2}=-1
  18. y = 5 y=5
  19. 29 29
  20. 1 3 + 3 3 + + 9 3 = ( 7 5 ) 2 1^{3}+3^{3}+\dots+9^{3}=(7\cdot 5)^{2}\,
  21. 1 3 + 3 3 + + 57 3 = ( 41 29 ) 2 1^{3}+3^{3}+\dots+57^{3}=(41\cdot 29)^{2}
  22. 28 = 2 2 ( 2 3 - 1 ) = 1 3 + 3 3 28=2^{2}(2^{3}-1)=1^{3}+3^{3}
  23. 496 = 2 4 ( 2 5 - 1 ) = 1 3 + 3 3 + 5 3 + 7 3 496=2^{4}(2^{5}-1)=1^{3}+3^{3}+5^{3}+7^{3}
  24. 8128 = 2 6 ( 2 7 - 1 ) = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + 11 3 + 13 3 + 15 3 8128=2^{6}(2^{7}-1)=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}
  25. 3 3 + 4 3 + 5 3 = 6 3 3^{3}+4^{3}+5^{3}=6^{3}
  26. 11 3 + 12 3 + 13 3 + 14 3 = 20 3 11^{3}+12^{3}+13^{3}+14^{3}=20^{3}
  27. 31 3 + 33 3 + 35 3 + 37 3 + 39 3 + 41 3 = 66 3 31^{3}+33^{3}+35^{3}+37^{3}+39^{3}+41^{3}=66^{3}
  28. F F
  29. n n
  30. d d
  31. F ( d , a , n ) = a 3 + ( a + d ) 3 + ( a + 2 d ) 3 + + ( a + d n - d ) 3 F(d,a,n)=a^{3}+(a+d)^{3}+(a+2d)^{3}+\cdots+(a+dn-d)^{3}
  32. F ( d , a , n ) = ( n / 4 ) ( 2 a - d + d n ) ( 2 a 2 - 2 a d + 2 a d n - d 2 n + d 2 n 2 ) F(d,a,n)=(n/4)(2a-d+dn)(2a^{2}-2ad+2adn-d^{2}n+d^{2}n^{2})
  33. F ( d , a , n ) = y 3 F(d,a,n)=y^{3}
  34. d = 1 d=1
  35. d > 1 d>1
  36. d d
  37. x x < s u p > 3 : 𝐑 𝐑 x↦x<sup>3:\mathbf{R}→\mathbf{R}

Cubic_surface.html

  1. 3 \mathbb{P}^{3}
  2. [ X : Y : Z : W ] [X:Y:Z:W]
  3. X 3 + Y 3 + Z 3 + W 3 = 0 X^{3}+Y^{3}+Z^{3}+W^{3}=0
  4. X 3 + Y 3 + Z 3 + W 3 = ( X + Y + Z + W ) 3 X^{3}+Y^{3}+Z^{3}+W^{3}=(X+Y+Z+W)^{3}
  5. W X Y + X Y Z + Y Z W + Z W X = 0 WXY+XYZ+YZW+ZWX=0
  6. 2 \mathbb{P}^{2}
  7. 2 \mathbb{P}^{2}
  8. W X Y + X Y Z + Y Z W + Z W X = 0 WXY+XYZ+YZW+ZWX=0
  9. [ 0 : 0 : 0 : 1 ] [0:0:0:1]

Cullen_number.html

  1. n 2 n + 1 n\cdot 2^{n}+1
  2. C n C_{n}
  3. n x n\leq x
  4. x x\to\infty

Cumulant.html

  1. g ( t ) = log 𝔼 [ e t X ] . g(t)=\log\mathbb{E}\!\left[e^{tX}\right].
  2. g ( t ) = n = 1 κ n t n n ! . g(t)=\sum_{n=1}^{\infty}\kappa_{n}\frac{t^{n}}{n!}.
  3. κ n = g ( n ) ( 0 ) \kappa_{n}=g^{(n)}(0)
  4. h ( t ) = log 𝔼 [ e i t X ] = n = 1 κ n ( i t ) n n ! = μ i t - σ 2 t 2 2 + h(t)=\log\mathbb{E}\!\left[e^{itX}\right]=\sum_{n=1}^{\infty}\kappa_{n}\frac{(% it)^{n}}{n!}=\mu it-\sigma^{2}\frac{t^{2}}{2}+\cdots
  5. g X + Y ( t ) = log 𝔼 [ e t ( X + Y ) ] = log ( 𝔼 [ e t X ] 𝔼 [ e t Y ] ) = log 𝔼 [ e t X ] + log 𝔼 [ e t Y ] = g X ( t ) + g Y ( t ) \begin{aligned}\displaystyle g_{X+Y}(t)&\displaystyle=\log\mathbb{E}\!\left[e^% {t(X+Y)}\right]\\ &\displaystyle=\log\left(\mathbb{E}\left[e^{tX}\right]\mathbb{E}\left[e^{tY}% \right]\right)\\ &\displaystyle=\log\mathbb{E}\left[e^{tX}\right]+\log\mathbb{E}\left[e^{tY}% \right]\\ &\displaystyle=g_{X}(t)+g_{Y}(t)\end{aligned}
  6. g X Y ( t ) = log 𝔼 [ e t i = 1 X Y i ] = g X ( g Y ( t ) ) \begin{aligned}\displaystyle g_{XY}(t)&\displaystyle=\log\mathbb{E}\left[e^{t% \sum_{i=1}^{X}Y_{i}}\right]\\ &\displaystyle=g_{X}(g_{Y}(t))\end{aligned}
  7. X = μ X=μ
  8. κ n + 1 = p ( 1 - p ) d κ n d p . \kappa_{n+1}=p(1-p)\frac{d\kappa_{n}}{dp}.
  9. n = 1 n=1
  10. n = 1 n=1
  11. ε = μ - 1 σ 2 = κ 1 - 1 κ 2 , \varepsilon=\mu^{-1}\sigma^{2}=\kappa_{1}^{-1}\kappa_{2},
  12. g ( t ) = μ ( 1 + ε ( e - t - 1 ) ) - 1 . g^{\prime}(t)=\mu\cdot(1+\varepsilon\cdot(e^{-t}-1))^{-1}.
  13. g ′′ ( t ) = g ( t ) ( 1 + e t ( ε - 1 - 1 ) ) - 1 g^{\prime\prime}(t)=g^{\prime}(t)\cdot(1+e^{t}\cdot(\varepsilon^{-1}-1))^{-1}
  14. X = μ X=μ
  15. ε = 0 ε=0
  16. ε = 1 p ε=1−p
  17. ε > 1 ε>1
  18. ε = 0 ε=0
  19. ε > 1 ε>1
  20. c > 0 , F ( x ) = O ( e c x ) , x - ; \exists c>0,F(x)=O(e^{cx}),x\to-\infty;
  21. d > 0 , 1 - F ( x ) = O ( e - d x ) , x + ; \exists d>0,1-F(x)=O(e^{-dx}),x\to+\infty;
  22. F F
  23. y = ( t + 1 ) inf supp X - μ ( X ) , y=(t+1)\inf\mathrm{supp}X-\mu(X),
  24. y = ( t - 1 ) sup supp X + μ ( X ) , y=(t-1)\sup\mathrm{supp}X+\mu(X),
  25. - 0 [ t inf supp X - g ( t ) ] d t , 0 [ t inf supp X - g ( t ) ] d t \int_{-\infty}^{0}\left[t\inf\mathrm{supp}X-g^{\prime}(t)\right]dt,\qquad\int_% {\infty}^{0}\left[t\inf\mathrm{supp}X-g^{\prime}(t)\right]dt
  26. g X + k ( t ) = g X ( t ) + k t . g_{X+k}(t)=g_{X}(t)+kt.
  27. g k ( t ) = k t g_{k}(t)=kt
  28. g X + Y = g X + g Y g_{X+Y}=g_{X}+g_{Y}
  29. g ( t ) = log M ( t ) , g(t)=\log M(t),
  30. f | θ f|\theta
  31. f ( x | θ ) = 1 M ( θ ) e θ x f ( x ) , f(x|\theta)=\frac{1}{M(\theta)}e^{\theta x}f(x),
  32. g ( t | θ ) = g ( t + θ ) - g ( θ ) . g(t|\theta)=g(t+\theta)-g(\theta).
  33. κ 1 ( X + c ) = κ 1 ( X ) + c and \kappa_{1}(X+c)=\kappa_{1}(X)+c~{}\,\text{ and}
  34. κ n ( X + c ) = κ n ( X ) for n 2. \kappa_{n}(X+c)=\kappa_{n}(X)~{}\,\text{ for }~{}n\geq 2.
  35. κ n ( c X ) = c n κ n ( X ) . \kappa_{n}(cX)=c^{n}\kappa_{n}(X).\,
  36. M ( t ) = 1 + n = 1 μ n t n n ! = exp ( n = 1 κ n t n n ! ) = exp ( g ( t ) ) . M(t)=1+\sum_{n=1}^{\infty}\frac{\mu^{\prime}_{n}t^{n}}{n!}=\exp\left(\sum_{n=1% }^{\infty}\frac{\kappa_{n}t^{n}}{n!}\right)=\exp(g(t)).
  37. g ( t ) = log M ( t ) . g(t)=\log M(t).
  38. exp ( g ( t ) ) \exp(g(t))
  39. μ n = M ( n ) ( 0 ) = d n exp ( g ( t ) ) d t n | t = 0 . \mu^{\prime}_{n}=M^{(n)}(0)=\frac{\mathrm{d}^{n}\exp(g(t))}{\mathrm{d}t^{n}}% \Big|_{t=0}.
  40. log M ( t ) \log M(t)
  41. κ n = g ( n ) ( 0 ) = d n log M ( t ) d t n | t = 0 . \kappa_{n}=g^{(n)}(0)=\frac{\mathrm{d}^{n}\log M(t)}{\mathrm{d}t^{n}}\Big|_{t=% 0}.
  42. μ n = k = 1 n B n , k ( κ 1 , , κ n - k + 1 ) \mu^{\prime}_{n}=\sum_{k=1}^{n}B_{n,k}(\kappa_{1},\ldots,\kappa_{n-k+1})
  43. κ n = k = 1 n ( - 1 ) k - 1 ( k - 1 ) ! B n , k ( μ 1 , , μ n - k + 1 ) , \kappa_{n}=\sum_{k=1}^{n}(-1)^{k-1}(k-1)!B_{n,k}(\mu^{\prime}_{1},\ldots,\mu^{% \prime}_{n-k+1}),
  44. B n , k B_{n,k}
  45. μ \mu
  46. C ( t ) = 𝔼 [ e t ( x - μ ) ] = e - μ t M ( t ) = exp ( g ( t ) - μ t ) , C(t)=\mathbb{E}[e^{t(x-\mu)}]=e^{-\mu t}M(t)=\exp(g(t)-\mu t),
  47. μ n = C ( n ) ( 0 ) = d n d t n exp ( g ( t ) - μ t ) | t = 0 = k = 1 n B n , k ( 0 , κ 2 , , κ n - k + 1 ) . \mu_{n}=C^{(n)}(0)=\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\exp(g(t)-\mu t)\Big|% _{t=0}=\sum_{k=1}^{n}B_{n,k}(0,\kappa_{2},\ldots,\kappa_{n-k+1}).
  48. κ n = g ( n ) ( 0 ) = d n d t n ( log C ( t ) + μ t ) | t = 0 = k = 1 n ( - 1 ) k - 1 ( k - 1 ) ! B n , k ( 0 , μ 2 , , μ n - k + 1 ) . \kappa_{n}=g^{(n)}(0)=\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}(\log C(t)+\mu t)% \Big|_{t=0}=\sum_{k=1}^{n}(-1)^{k-1}(k-1)!B_{n,k}(0,\mu_{2},\ldots,\mu_{n-k+1}).
  49. μ 1 = κ 1 \mu^{\prime}_{1}=\kappa_{1}\,
  50. μ 2 = κ 2 + κ 1 2 \mu^{\prime}_{2}=\kappa_{2}+\kappa_{1}^{2}\,
  51. μ 3 = κ 3 + 3 κ 2 κ 1 + κ 1 3 \mu^{\prime}_{3}=\kappa_{3}+3\kappa_{2}\kappa_{1}+\kappa_{1}^{3}\,
  52. μ 4 = κ 4 + 4 κ 3 κ 1 + 3 κ 2 2 + 6 κ 2 κ 1 2 + κ 1 4 \mu^{\prime}_{4}=\kappa_{4}+4\kappa_{3}\kappa_{1}+3\kappa_{2}^{2}+6\kappa_{2}% \kappa_{1}^{2}+\kappa_{1}^{4}\,
  53. μ 5 = κ 5 + 5 κ 4 κ 1 + 10 κ 3 κ 2 + 10 κ 3 κ 1 2 + 15 κ 2 2 κ 1 + 10 κ 2 κ 1 3 + κ 1 5 \mu^{\prime}_{5}=\kappa_{5}+5\kappa_{4}\kappa_{1}+10\kappa_{3}\kappa_{2}+10% \kappa_{3}\kappa_{1}^{2}+15\kappa_{2}^{2}\kappa_{1}+10\kappa_{2}\kappa_{1}^{3}% +\kappa_{1}^{5}\,
  54. μ 6 = κ 6 + 6 κ 5 κ 1 + 15 κ 4 κ 2 + 15 κ 4 κ 1 2 + 10 κ 3 2 + 60 κ 3 κ 2 κ 1 + 20 κ 3 κ 1 3 + 15 κ 2 3 + 45 κ 2 2 κ 1 2 + 15 κ 2 κ 1 4 + κ 1 6 . \mu^{\prime}_{6}=\kappa_{6}+6\kappa_{5}\kappa_{1}+15\kappa_{4}\kappa_{2}+15% \kappa_{4}\kappa_{1}^{2}+10\kappa_{3}^{2}+60\kappa_{3}\kappa_{2}\kappa_{1}+20% \kappa_{3}\kappa_{1}^{3}+15\kappa_{2}^{3}+45\kappa_{2}^{2}\kappa_{1}^{2}+15% \kappa_{2}\kappa_{1}^{4}+\kappa_{1}^{6}.\,
  55. μ 1 = 0 \mu_{1}=0\,
  56. μ 2 = κ 2 \mu_{2}=\kappa_{2}\,
  57. μ 3 = κ 3 \mu_{3}=\kappa_{3}\,
  58. μ 4 = κ 4 + 3 κ 2 2 \mu_{4}=\kappa_{4}+3\kappa_{2}^{2}\,
  59. μ 5 = κ 5 + 10 κ 3 κ 2 \mu_{5}=\kappa_{5}+10\kappa_{3}\kappa_{2}\,
  60. μ 6 = κ 6 + 15 κ 4 κ 2 + 10 κ 3 2 + 15 κ 2 3 . \mu_{6}=\kappa_{6}+15\kappa_{4}\kappa_{2}+10\kappa_{3}^{2}+15\kappa_{2}^{3}.\,
  61. κ 1 = μ 1 \kappa_{1}=\mu^{\prime}_{1}\,
  62. κ 2 = μ 2 - μ 1 2 \kappa_{2}=\mu^{\prime}_{2}-{\mu^{\prime}_{1}}^{2}\,
  63. κ 3 = μ 3 - 3 μ 2 μ 1 + 2 μ 1 3 \kappa_{3}=\mu^{\prime}_{3}-3\mu^{\prime}_{2}\mu^{\prime}_{1}+2{\mu^{\prime}_{% 1}}^{3}\,
  64. κ 4 = μ 4 - 4 μ 3 μ 1 - 3 μ 2 2 + 12 μ 2 μ 1 2 - 6 μ 1 4 \kappa_{4}=\mu^{\prime}_{4}-4\mu^{\prime}_{3}\mu^{\prime}_{1}-3{\mu^{\prime}_{% 2}}^{2}+12\mu^{\prime}_{2}{\mu^{\prime}_{1}}^{2}-6{\mu^{\prime}_{1}}^{4}\,
  65. κ 5 = μ 5 - 5 μ 4 μ 1 - 10 μ 3 μ 2 + 20 μ 3 μ 1 2 + 30 μ 2 2 μ 1 - 60 μ 2 μ 1 3 + 24 μ 1 5 \kappa_{5}=\mu^{\prime}_{5}-5\mu^{\prime}_{4}\mu^{\prime}_{1}-10\mu^{\prime}_{% 3}\mu^{\prime}_{2}+20\mu^{\prime}_{3}{\mu^{\prime}_{1}}^{2}+30{\mu^{\prime}_{2% }}^{2}\mu^{\prime}_{1}-60\mu^{\prime}_{2}{\mu^{\prime}_{1}}^{3}+24{\mu^{\prime% }_{1}}^{5}\,
  66. κ 6 = μ 6 - 6 μ 5 μ 1 - 15 μ 4 μ 2 + 30 μ 4 μ 1 2 - 10 μ 3 2 + 120 μ 3 μ 2 μ 1 - 120 μ 3 μ 1 3 + 30 μ 2 3 - 270 μ 2 2 μ 1 2 + 360 μ 2 μ 1 4 - 120 μ 1 6 . \kappa_{6}=\mu^{\prime}_{6}-6\mu^{\prime}_{5}\mu^{\prime}_{1}-15\mu^{\prime}_{% 4}\mu^{\prime}_{2}+30\mu^{\prime}_{4}{\mu^{\prime}_{1}}^{2}-10{\mu^{\prime}_{3% }}^{2}+120\mu^{\prime}_{3}\mu^{\prime}_{2}\mu^{\prime}_{1}-120\mu^{\prime}_{3}% {\mu^{\prime}_{1}}^{3}+30{\mu^{\prime}_{2}}^{3}-270{\mu^{\prime}_{2}}^{2}{\mu^% {\prime}_{1}}^{2}+360\mu^{\prime}_{2}{\mu^{\prime}_{1}}^{4}-120{\mu^{\prime}_{% 1}}^{6}\,.
  67. n > 1 n>1
  68. κ 2 = μ 2 \kappa_{2}=\mu_{2}\,
  69. κ 3 = μ 3 \kappa_{3}=\mu_{3}\,
  70. κ 4 = μ 4 - 3 μ 2 2 \kappa_{4}=\mu_{4}-3{\mu_{2}}^{2}\,
  71. κ 5 = μ 5 - 10 μ 3 μ 2 \kappa_{5}=\mu_{5}-10\mu_{3}\mu_{2}\,
  72. κ 6 = μ 6 - 15 μ 4 μ 2 - 10 μ 3 2 + 30 μ 2 3 . \kappa_{6}=\mu_{6}-15\mu_{4}\mu_{2}-10{\mu_{3}}^{2}+30{\mu_{2}}^{3}\,.
  73. n > 2 n>2
  74. κ 3 = μ 3 \kappa_{3}=\mu_{3}\,
  75. κ 4 = μ 4 - 3 \kappa_{4}=\mu_{4}-3\,
  76. κ 5 = μ 5 - 10 μ 3 \kappa_{5}=\mu_{5}-10\mu_{3}\,
  77. κ 6 = μ 6 - 15 μ 4 - 10 μ 3 2 + 30 . \kappa_{6}=\mu_{6}-15\mu_{4}-10{\mu_{3}}^{2}+30\,.
  78. κ n = μ n - m = 1 n - 1 ( n - 1 m - 1 ) κ m μ n - m . \kappa_{n}=\mu^{\prime}_{n}-\sum_{m=1}^{n-1}{n-1\choose m-1}\kappa_{m}\mu_{n-m% }^{\prime}.
  79. 𝔼 [ e t X ] = 1 + m = 1 μ m t m m ! = e g ( t ) . \mathbb{E}\!\left[e^{tX}\right]=1+\sum_{m=1}^{\infty}\mu^{\prime}_{m}\frac{t^{% m}}{m!}=e^{g(t)}.
  80. g ( t ) \displaystyle g(t)
  81. κ 1 \displaystyle\kappa_{1}
  82. μ n = 𝔼 [ X n ] = X n \mu^{\prime}_{n}=\mathbb{E}\!\left[X^{n}\right]=\langle X^{n}\rangle\,
  83. κ n = X n c . \kappa_{n}=\langle X^{n}\rangle_{c}.\,
  84. μ n = Π B π κ | B | \mu^{\prime}_{n}=\sum_{\Pi}\prod_{B\in\pi}\kappa_{\left|B\right|}
  85. g ( t 1 , t 2 , , t n ) = log E ( e j = 1 n t j X j ) . g(t_{1},t_{2},\dots,t_{n})=\log E(\mathrm{e}^{\sum_{j=1}^{n}t_{j}X_{j}}).
  86. κ ( X 1 , , X n ) = π ( | π | - 1 ) ! ( - 1 ) | π | - 1 B π E ( i B X i ) \kappa(X_{1},\dots,X_{n})=\sum_{\pi}(|\pi|-1)!(-1)^{|\pi|-1}\prod_{B\in\pi}E% \left(\prod_{i\in B}X_{i}\right)
  87. κ ( X , Y , Z ) = E ( X Y Z ) - E ( X Y ) E ( Z ) - E ( X Z ) E ( Y ) - E ( Y Z ) E ( X ) + 2 E ( X ) E ( Y ) E ( Z ) . \kappa(X,Y,Z)=E(XYZ)-E(XY)E(Z)-E(XZ)E(Y)-E(YZ)E(X)+2E(X)E(Y)E(Z).\,
  88. κ ( X , X , Z ) = E ( X 2 Z ) - 2 E ( X Z ) E ( X ) - E ( X 2 ) E ( Z ) + 2 E ( X ) 2 E ( Z ) , \kappa(X,X,Z)=E(X^{2}Z)-2E(XZ)E(X)-E(X^{2})E(Z)+2E(X)^{2}E(Z),\,
  89. κ ( X , Y , Z ) = E ( X Y Z ) . \kappa(X,Y,Z)=E(XYZ).\,
  90. κ ( X , Y , Z , W ) = E ( X Y Z W ) - E ( X Y ) E ( Z W ) - E ( X Z ) E ( Y W ) - E ( X W ) E ( Y Z ) . \kappa(X,Y,Z,W)=E(XYZW)-E(XY)E(ZW)-E(XZ)E(YW)-E(XW)E(YZ).\,
  91. E ( X 1 X n ) = π B π κ ( X i : i B ) . E(X_{1}\cdots X_{n})=\sum_{\pi}\prod_{B\in\pi}\kappa(X_{i}:i\in B).
  92. E ( X Y Z ) = κ ( X , Y , Z ) + κ ( X , Y ) κ ( Z ) + κ ( X , Z ) κ ( Y ) + κ ( Y , Z ) κ ( X ) + κ ( X ) κ ( Y ) κ ( Z ) . E(XYZ)=\kappa(X,Y,Z)+\kappa(X,Y)\kappa(Z)+\kappa(X,Z)\kappa(Y)+\kappa(Y,Z)% \kappa(X)+\kappa(X)\kappa(Y)\kappa(Z).\,
  93. κ ( X + Y , Z 1 , Z 2 , ) = κ ( X , Z 1 , Z 2 , ) + κ ( Y , Z 1 , Z 2 , ) . \kappa(X+Y,Z_{1},Z_{2},\dots)=\kappa(X,Z_{1},Z_{2},\dots)+\kappa(Y,Z_{1},Z_{2}% ,\dots).\,
  94. var ( X + Y ) = var ( X ) + 2 cov ( X , Y ) + var ( Y ) \operatorname{var}(X+Y)=\operatorname{var}(X)+2\operatorname{cov}(X,Y)+% \operatorname{var}(Y)\,
  95. κ n ( X + Y ) = j = 0 n ( n j ) κ ( X , , X j , Y , , Y n - j ) . \kappa_{n}(X+Y)=\sum_{j=0}^{n}{n\choose j}\kappa(\,\underbrace{X,\dots,X}_{j},% \underbrace{Y,\dots,Y}_{n-j}\,).\,
  96. μ 3 ( X ) = E ( μ 3 ( X Y ) ) + μ 3 ( E ( X Y ) ) + 3 cov ( E ( X Y ) , var ( X Y ) ) . \mu_{3}(X)=E(\mu_{3}(X\mid Y))+\mu_{3}(E(X\mid Y))+3\,\operatorname{cov}(E(X% \mid Y),\operatorname{var}(X\mid Y)).
  97. κ ( X 1 , , X n ) = π κ ( κ ( X π 1 Y ) , , κ ( X π b Y ) ) \kappa(X_{1},\dots,X_{n})=\sum_{\pi}\kappa(\kappa(X_{\pi_{1}}\mid Y),\dots,% \kappa(X_{\pi_{b}}\mid Y))
  98. Z ( β ) = exp ( - β E ) , Z(\beta)=\langle\exp(-\beta E)\rangle,\,
  99. A \langle A\rangle
  100. 𝔼 [ A ] \mathbb{E}\!\left[A\right]
  101. F ( β ) = - β - 1 log Z F(\beta)=-\beta^{-1}\log Z\,
  102. E = E c E=\langle E\rangle_{c}
  103. C = d E / d T = k β 2 E 2 c = k β 2 ( E 2 - E 2 ) C=dE/dT=k\beta^{2}\langle E^{2}\rangle_{c}=k\beta^{2}(\langle E^{2}\rangle-% \langle E\rangle^{2})
  104. E 2 c \langle E^{2}\rangle_{c}
  105. μ \mu
  106. Ω = - β - 1 log ( exp ( - β E - β μ N ) ) , \Omega=-\beta^{-1}\log(\langle\exp(-\beta E-\beta\mu N)\rangle),\,
  107. Ω \Omega
  108. 1 + n = 1 m n t n / n ! = exp ( n = 1 κ n t n / n ! ) , 1+\sum_{n=1}^{\infty}m_{n}t^{n}/n!=\exp\left(\sum_{n=1}^{\infty}\kappa_{n}t^{n% }/n!\right),
  109. μ 6 = κ 6 + 6 κ 5 κ 1 + 15 κ 4 κ 2 + 15 κ 4 κ 1 2 + 10 κ 3 2 + 60 κ 3 κ 2 κ 1 + 20 κ 3 κ 1 3 + 15 κ 2 3 + 45 κ 2 2 κ 1 2 + 15 κ 2 κ 1 4 + κ 1 6 \mu^{\prime}_{6}=\kappa_{6}+6\kappa_{5}\kappa_{1}+15\kappa_{4}\kappa_{2}+15% \kappa_{4}\kappa_{1}^{2}+10\kappa_{3}^{2}+60\kappa_{3}\kappa_{2}\kappa_{1}+20% \kappa_{3}\kappa_{1}^{3}+15\kappa_{2}^{3}+45\kappa_{2}^{2}\kappa_{1}^{2}+15% \kappa_{2}\kappa_{1}^{4}+\kappa_{1}^{6}
  110. p 6 ( x ) = κ 6 x + ( 6 κ 5 κ 1 + 15 κ 4 κ 2 + 10 κ 3 2 ) x 2 m + ( 15 κ 4 κ 1 2 + 60 κ 3 κ 2 κ 1 + 15 κ 2 3 ) x 3 + ( 45 κ 2 2 κ 1 2 ) x 4 + ( 15 κ 2 κ 1 4 ) x 5 + ( κ 1 6 ) x 6 , p_{6}(x)=\kappa_{6}\,x+(6\kappa_{5}\kappa_{1}+15\kappa_{4}\kappa_{2}+10\kappa_% {3}^{2})\,x^{2}m+(15\kappa_{4}\kappa_{1}^{2}+60\kappa_{3}\kappa_{2}\kappa_{1}+% 15\kappa_{2}^{3})\,x^{3}+(45\kappa_{2}^{2}\kappa_{1}^{2})\,x^{4}+(15\kappa_{2}% \kappa_{1}^{4})\,x^{5}+(\kappa_{1}^{6})\,x^{6},
  111. E ( X 1 X n ) = π B π κ ( X i : i B ) E(X_{1}\cdots X_{n})=\sum_{\pi}\prod_{B\in\pi}\kappa(X_{i}:i\in B)

Cunningham_chain.html

  1. p 2 \displaystyle p_{2}
  2. a = p 1 + 1 2 a=\frac{p_{1}+1}{2}
  3. a a
  4. p i = 2 i a - 1 p_{i}=2^{i}a-1
  5. p i = 2 i - 1 p 1 - ( 2 i - 1 - 1 ) p_{i}=2^{i-1}p_{1}-(2^{i-1}-1)\,
  6. a = p 1 - 1 2 a=\frac{p_{1}-1}{2}
  7. p i = 2 i a + 1 p_{i}=2^{i}a+1
  8. 2 a , - 1 2a,-1
  9. a a
  10. 2 a - 1 = p 2a-1=p
  11. p 1 p_{1}
  12. p 1 1 ( mod 2 ) p_{1}\equiv 1\;\;(\mathop{{\rm mod}}2)
  13. p i + 1 = 2 p i + 1 p_{i+1}=2p_{i}+1
  14. p i 2 i - 1 ( mod 2 i ) p_{i}\equiv 2^{i}-1\;\;(\mathop{{\rm mod}}2^{i})
  15. p 2 3 ( mod 4 ) p_{2}\equiv 3\;\;(\mathop{{\rm mod}}4)
  16. p 3 7 ( mod 8 ) p_{3}\equiv 7\;\;(\mathop{{\rm mod}}8)
  17. p i + 1 = 2 p i + 1 p_{i+1}=2p_{i}+1
  18. p i p_{i}
  19. p i p_{i}
  20. p i + 1 p_{i+1}
  21. p i p_{i}
  22. p i + 1 p_{i+1}
  23. p i + 1 p_{i+1}
  24. 2 p i 2p_{i}
  25. p 1 1 ( mod 2 ) p_{1}\equiv 1\;\;(\mathop{{\rm mod}}2)
  26. p i + 1 = 2 p i - 1 p_{i+1}=2p_{i}-1
  27. p i 1 ( mod 2 i ) p_{i}\equiv 1\;\;(\mathop{{\rm mod}}2^{i})
  28. i i
  29. p i + 1 p_{i+1}
  30. p i p_{i}
  31. p i = 2 i - 1 p 1 + ( 2 i - 1 - 1 ) p_{i}=2^{i-1}p_{1}+(2^{i-1}-1)\,
  32. p i 2 i - 1 - 1 ( mod p 1 ) p_{i}\equiv 2^{i-1}-1\;\;(\mathop{{\rm mod}}p_{1})
  33. 2 p 1 - 1 1 ( mod p 1 ) 2^{p_{1}-1}\equiv 1\;\;(\mathop{{\rm mod}}p_{1})
  34. p 1 p_{1}
  35. p p 1 p_{p_{1}}
  36. i = p 1 i=p_{1}

Current_account.html

  1. C A = ( X - M ) + N Y + N C T CA=(X-M)+NY+NCT

Curvature_form.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 \mathfrak{g}
  3. 𝔤 \mathfrak{g}
  4. Ω = d ω + 1 2 [ ω ω ] = D ω . \Omega=d\omega+{1\over 2}[\omega\wedge\omega]=D\omega.
  5. d d
  6. [ ] [\cdot\wedge\cdot]
  7. Ω ( X , Y ) = d ω ( X , Y ) + 1 2 [ ω ( X ) , ω ( Y ) ] \,\Omega(X,Y)=d\omega(X,Y)+{1\over 2}[\omega(X),\omega(Y)]
  8. 2 Ω ( X , Y ) = - [ h X , h Y ] + h [ X , Y ] 2\Omega(X,Y)=-[hX,hY]+h[X,Y]
  9. Ω = d ω + ω ω , \,\Omega=d\omega+\omega\wedge\omega,
  10. \wedge
  11. ω j i \omega^{i}_{\ j}
  12. Ω j i \Omega^{i}_{\ j}
  13. ω j i \omega^{i}_{\ j}
  14. Ω j i \Omega^{i}_{\ j}
  15. Ω j i = d ω j i + k ω k i ω j k . \Omega^{i}_{\ j}=d\omega^{i}_{\ j}+\sum_{k}\omega^{i}_{\ k}\wedge\omega^{k}_{% \ j}.
  16. R ( X , Y ) = Ω ( X , Y ) , \,R(X,Y)=\Omega(X,Y),
  17. θ \theta
  18. Θ \Theta
  19. ω \omega
  20. Θ = d θ + ω θ = D θ , \Theta=d\theta+\omega\wedge\theta=D\theta,
  21. D Θ = Ω θ . D\Theta=\Omega\wedge\theta.
  22. D Ω = 0 \,D\Omega=0

Curve_fitting.html

  1. y = a x + b . y=ax+b\;.
  2. y = a x 2 + b x + c . y=ax^{2}+bx+c\;.
  3. y = a x 3 + b x 2 + c x + d . y=ax^{3}+bx^{2}+cx+d\;.

Curve_of_constant_width.html

  1. 1 : 2 1:\sqrt{2}
  2. f ( x , y ) f(x,y)
  3. R 2 R^{2}
  4. f ( x , y ) = 0 f(x,y)=0
  5. f ( x , y ) = ( x 2 + y 2 ) 4 - 45 ( x 2 + y 2 ) 3 - 41283 ( x 2 + y 2 ) 2 + 7950960 ( x 2 + y 2 ) + 16 ( x 2 - 3 y 2 ) 3 f(x,y)=(x^{2}+y^{2})^{4}-45(x^{2}+y^{2})^{3}-41283(x^{2}+y^{2})^{2}+7950960(x^% {2}+y^{2})+16(x^{2}-3y^{2})^{3}
  6. + 48 ( x 2 + y 2 ) ( x 2 - 3 y 2 ) 2 + ( x 2 - 3 y 2 ) x [ 16 ( x 2 + y 2 ) 2 - 5544 ( x 2 + y 2 ) + 266382 ] - 720 3 . +48(x^{2}+y^{2})(x^{2}-3y^{2})^{2}+(x^{2}-3y^{2})x[16(x^{2}+y^{2})^{2}-5544(x^% {2}+y^{2})+266382]-720^{3}.

Curvilinear_perspective.html

  1. ( x , y , z ) \left(x,y,z\right)
  2. P 3 D = ( x , y , z ) P_{3D}=(x,y,z)
  3. d = x 2 + y 2 + z 2 d=\sqrt{x^{2}+y^{2}+z^{2}}
  4. R R
  5. P 2 D = ( x R / d , y R / d ) P_{2D}=(xR/d,yR/d)
  6. d = 0 d=0
  7. R R
  8. P s p h e r e = ( x , y , z ) * ( R / d ) P_{sphere}=(x,y,z)*(R/d)
  9. z = R z=R
  10. P i m a g e = ( x R / d , y R / d , R ) P_{image}=(xR/d,yR/d,R)
  11. z = R z=R
  12. P 2 D = ( x R / d , y R / d ) = R * ( x x 2 + y 2 + z 2 , y x 2 + y 2 + z 2 ) P_{2D}=(xR/d,yR/d)=R*(\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}},\frac{y}{\sqrt{x^{2}+% y^{2}+z^{2}}})
  13. R R
  14. P 2 D = ( x / d , y / d ) = ( x x 2 + y 2 + z 2 , y x 2 + y 2 + z 2 ) P_{2D}=(x/d,y/d)=(\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}},\frac{y}{\sqrt{x^{2}+y^{2% }+z^{2}}})

Cusp_form.html

  1. Σ a n q n \Sigma a_{n}q^{n}
  2. z z + 1. z\mapsto z+1.

Customer_lifetime_value.html

  1. CLV = GC i = 1 n r i ( 1 + d ) i - M i = 1 n r i - 1 ( 1 + d ) i - 0.5 \,\text{CLV}=\,\text{GC}\cdot\sum_{i=1}^{n}\frac{r^{i}}{(1+d)^{i}}-\,\text{M}% \cdot\sum_{i=1}^{n}\frac{r^{i-1}}{(1+d)^{i-0.5}}
  2. GC \,\text{GC}
  3. M \,\text{M}
  4. n n
  5. r r
  6. d d
  7. CLV = GC ( r 1 + d - r ) \,\text{CLV}=\,\text{GC}\cdot\left(\frac{r}{1+d-r}\right)

CW_complex.html

  1. S 0 S_{0}
  2. S 0 S_{0}
  3. S n - 1 S_{n-1}
  4. S n - 1 S_{n-1}
  5. \mathbb{Z}
  6. { [ n , n + 1 ] : n } \{[n,n+1]:n\in\mathbb{Z}\}
  7. n \mathbb{R}^{n}
  8. \mathbb{R}
  9. n \mathbb{R}^{n}
  10. f : { 0 , 1 } X f:\{0,1\}\to X
  11. f ( 0 ) f(0)
  12. f ( 1 ) f(1)
  13. S n - 1 S^{n-1}
  14. S n - 1 S n S^{n-1}\to S^{n}
  15. S n S^{n}
  16. 0 k n 0\leq k\leq n
  17. { r e 2 π i θ : 0 r 1 , θ } \{re^{2\pi i\theta}:0\leq r\leq 1,\theta\in\mathbb{Q}\}\subset\mathbb{C}
  18. S n S^{n}
  19. C * C_{*}
  20. C k = { k { 0 , n } 0 k { 0 , n } C_{k}=\left\{\begin{array}[]{lr}\mathbb{Z}&k\in\{0,n\}\\ 0&k\notin\{0,n\}\end{array}\right.
  21. H k = { k { 0 , n } 0 k { 0 , n } H_{k}=\left\{\begin{array}[]{lr}\mathbb{Z}&k\in\{0,n\}\\ 0&k\notin\{0,n\}\end{array}\right.
  22. C k = { 2 0 k n 0 otherwise C_{k}=\left\{\begin{array}[]{lr}\mathbb{Z}^{2}&0\leq k\leq n\\ 0&\,\text{otherwise}\end{array}\right.
  23. ( 1 - 1 1 - 1 ) \begin{pmatrix}1&-1\\ 1&-1\end{pmatrix}
  24. C 0 C_{0}
  25. C n C_{n}
  26. n \mathbb{P}^{n}\mathbb{C}
  27. H k ( n ) = { for 0 k 2 n , even 0 otherwise . H^{k}(\mathbb{P}^{n}\mathbb{C})=\begin{cases}\mathbb{Z}\quad\,\text{for }0\leq k% \leq 2n,\,\text{even}\\ 0\quad\,\text{otherwise}.\end{cases}
  28. X / X/\sim
  29. x y x\sim y
  30. X X / X\to X/\sim
  31. X / X/\sim
  32. X X
  33. X / X/\sim
  34. X 1 X^{1}
  35. X 1 X^{1}
  36. X 2 X^{2}
  37. X 1 X^{1}
  38. X 1 X^{1}
  39. X ~ \tilde{X}
  40. X ~ = X e 1 e 2 \tilde{X}=X\cup e^{1}\cup e^{2}
  41. X ~ X \tilde{X}\to X
  42. X ~ = X e 2 e 3 \tilde{X}=X\cup e^{2}\cup e^{3}
  43. X 2 X^{2}
  44. X ~ X \tilde{X}\to X
  45. X ~ \tilde{X}
  46. X n X^{n}
  47. n 2 n\geq 2
  48. n = 1 n=1
  49. H n ( X ; ) H_{n}(X;\mathbb{Z})

Cyclic_permutation.html

  1. ( 1 2 3 4 5 6 7 8 4 2 7 6 5 8 1 3 ) = ( 1 4 6 8 3 7 2 5 4 6 8 3 7 1 2 5 ) = ( 146837 ) ( 2 ) ( 5 ) \begin{pmatrix}1&2&3&4&5&6&7&8\\ 4&2&7&6&5&8&1&3\end{pmatrix}=\begin{pmatrix}1&4&6&8&3&7&2&5\\ 4&6&8&3&7&1&2&5\end{pmatrix}=(146837)(2)(5)
  2. ( 1 2 3 4 5 6 7 8 4 5 7 6 8 2 1 3 ) = ( 1 4 6 2 5 8 3 7 4 6 2 5 8 3 7 1 ) = ( 14625837 ) \begin{pmatrix}1&2&3&4&5&6&7&8\\ 4&5&7&6&8&2&1&3\end{pmatrix}=\begin{pmatrix}1&4&6&2&5&8&3&7\\ 4&6&2&5&8&3&7&1\end{pmatrix}=(14625837)
  3. σ : X X \sigma:X\to X
  4. σ \sigma
  5. s 0 s_{0}
  6. s i = σ i ( s 0 ) s_{i}=\sigma^{i}(s_{0})\,
  7. i 𝐙 i\in\mathbf{Z}
  8. k 1 k\geq 1
  9. s k = s 0 s_{k}=s_{0}
  10. S = { s 0 , s 1 , , s k - 1 } S=\{s_{0},s_{1},\ldots,s_{k-1}\}
  11. σ \sigma
  12. σ ( s i ) = s i + 1 for 0 i < k \sigma(s_{i})=s_{i+1}\quad\mbox{for }~{}0\leq i<k
  13. σ ( x ) = x \sigma(x)=x
  14. X S X\setminus S
  15. σ \sigma
  16. s 0 s 1 s 2 s k - 1 s k = s 0 s_{0}\mapsto s_{1}\mapsto s_{2}\mapsto\cdots\mapsto s_{k-1}\mapsto s_{k}=s_{0}
  17. σ = ( s 0 s 1 s k - 1 ) \sigma=(s_{0}~{}s_{1}~{}\dots~{}s_{k-1})
  18. s 0 s_{0}
  19. 1 k n 1\leq k\leq n
  20. ( n k ) ( k - 1 ) ! = n ( n - 1 ) ( n - k + 1 ) k = n ! ( n - k ) ! k {\left({{n}\atop{k}}\right)}(k-1)!=\frac{n(n-1)\cdots(n-k+1)}{k}=\frac{n!}{(n-% k)!k}
  21. a = 1 a=1
  22. b = 2 b=2
  23. e = 5 e=5
  24. ( k k + 1 ) , (k~{}~{}k+1),
  25. ( 1 2 ) (1~{}2)
  26. ( 2 3 ) (2~{}3)
  27. ( 3 4 ) (3~{}4)
  28. ( 4 5 ) . (4~{}5).
  29. ( k l ) (k~{}~{}l)
  30. k < l k<l
  31. ( k l ) = ( k k + 1 ) ( k + 1 k + 2 ) ( l - 1 l ) ( l - 2 l - 1 ) ( k k + 1 ) . (k~{}~{}l)=(k~{}~{}k+1)\cdot(k+1~{}~{}k+2)\cdots(l-1~{}~{}l)\cdot(l-2~{}~{}l-1% )\cdots(k~{}~{}k+1).
  32. ( a b c d y z ) = ( a b ) ( b c d y z ) (a~{}b~{}c~{}d~{}\ldots~{}y~{}z)=(a~{}b)\cdot(b~{}c~{}d~{}\ldots~{}y~{}z)
  33. a a
  34. b b
  35. b b
  36. c c
  37. y y
  38. z z
  39. z z
  40. a a
  41. a a
  42. z z
  43. b b
  44. a a
  45. z z
  46. ( a b ) (a~{}b)
  47. z z
  48. b b
  49. a a
  50. z z

Cyclotomic_identity.html

  1. 1 1 - α z = j = 1 ( 1 1 - z j ) M ( α , j ) {1\over 1-\alpha z}=\prod_{j=1}^{\infty}\left({1\over 1-z^{j}}\right)^{M(% \alpha,j)}
  2. M ( α , n ) = 1 n d | n μ ( n d ) α d , M(\alpha,n)={1\over n}\sum_{d\,|\,n}\mu\left({n\over d}\right)\alpha^{d},
  3. j = 1 ( 1 1 - α z j ) M ( β , j ) = j = 1 ( 1 1 - β z j ) M ( α , j ) \prod_{j=1}^{\infty}\left({1\over 1-\alpha z^{j}}\right)^{M(\beta,j)}=\prod_{j% =1}^{\infty}\left({1\over 1-\beta z^{j}}\right)^{M(\alpha,j)}

Ćuk_converter.html

  1. E = 1 2 L I 2 E=\frac{1}{2}LI^{2}
  2. V L = L d I d t V_{L}=L\frac{dI}{dt}
  3. V L 1 = V i - V C V_{L1}=V_{i}-V_{C}
  4. V L 2 = V o V_{L2}=V_{o}
  5. V L 1 = V i V_{L1}=V_{i}
  6. V L 2 = V o + V C V_{L2}=V_{o}+V_{C}
  7. V ¯ L 1 = D V i + ( 1 - D ) ( V i - V C ) = ( V i - ( 1 - D ) V C ) \bar{V}_{L1}=D\cdot V_{i}+\left(1-D\right)\cdot\left(V_{i}-V_{C}\right)=\left(% V_{i}-(1-D)\cdot V_{C}\right)
  8. V ¯ L 2 = D ( V o + V C ) + ( 1 - D ) V o = ( V o + D V C ) \bar{V}_{L2}=D\left(V_{o}+V_{C}\right)+\left(1-D\right)\cdot V_{o}=\left(V_{o}% +D\cdot V_{C}\right)
  9. V C = - V o D V_{C}=-\frac{V_{o}}{D}
  10. V ¯ L 1 = ( V i + ( 1 - D ) V o D ) = 0 \bar{V}_{L1}=\left(V_{i}+(1-D)\cdot\frac{V_{o}}{D}\right)=0
  11. V o V i = - D 1 - D \frac{V_{o}}{V_{i}}=\frac{-D}{1-D}
  12. L 1 m i n = ( 1 - D ) 2 R 2 D f s L_{1}min=\frac{(1-D)^{2}R}{2Df_{s}}
  13. f s f_{s}

D'Alembert's_paradox.html

  1. s y m b o l s y m b o l u = 0 (incompressibility) s y m b o l \timessymbol u = 0 (irrotational) t s y m b o l u + ( s y m b o l u s y m b o l ) s y m b o l u = - 1 ρ s y m b o l p (Euler equation) \begin{aligned}&\displaystyle symbol{\nabla}\cdot symbol{u}=0&&\displaystyle\,% \text{(incompressibility)}\\ &\displaystyle symbol{\nabla}\timessymbol{u}=0&&\displaystyle\,\text{(% irrotational)}\\ &\displaystyle\frac{\partial}{\partial t}symbol{u}+\left(symbol{u}\cdot symbol% {\nabla}\right)symbol{u}=-\frac{1}{\rho}symbol{\nabla}p&&\displaystyle\,\text{% (Euler equation)}\end{aligned}
  2. ( s y m b o l u s y m b o l ) s y m b o l u = 1 2 s y m b o l ( s y m b o l u s y m b o l u ) - s y m b o l u × s y m b o l × s y m b o l u = 1 2 s y m b o l ( s y m b o l u s y m b o l u ) ( 1 ) \left(symbol{u}\cdot symbol{\nabla}\right)symbol{u}=\tfrac{1}{2}symbol{\nabla}% \left(symbol{u}\cdot symbol{u}\right)-symbol{u}\times symbol{\nabla}\times symbol% {u}=\tfrac{1}{2}symbol{\nabla}\left(symbol{u}\cdot symbol{u}\right)\qquad(1)
  3. s y m b o l ( φ t + 1 2 s y m b o l u s y m b o l u + p ρ ) = s y m b o l 0. symbol{\nabla}\left(\frac{\partial\varphi}{\partial t}+\tfrac{1}{2}symbol{u}% \cdot symbol{u}+\frac{p}{\rho}\right)=symbol{0}.
  4. φ t + 1 2 s y m b o l u s y m b o l u + p ρ = 0 , ( 2 ) \frac{\partial\varphi}{\partial t}+\tfrac{1}{2}symbol{u}\cdot symbol{u}+\frac{% p}{\rho}=0,\qquad(2)
  5. s y m b o l u t + ( s y m b o l v s y m b o l ) s y m b o l u = s y m b o l 0. \frac{\partial symbol{u}}{\partial t}+\left(symbol{v}\cdot symbol{\nabla}% \right)symbol{u}=symbol{0}.
  6. φ t = - s y m b o l v s y m b o l φ + R ( t ) = - s y m b o l v s y m b o l u + R ( t ) . \frac{\partial\varphi}{\partial t}=-symbol{v}\cdot symbol{\nabla}\varphi+R(t)=% -symbol{v}\cdot symbol{u}+R(t).
  7. s y m b o l F = - A p s y m b o l n d S symbol{F}=-\int_{A}p\,symbol{n}\;\mathrm{d}S
  8. p = - ρ ( φ t + 1 2 s y m b o l u s y m b o l u ) = ρ ( s y m b o l v s y m b o l u - 1 2 s y m b o l u s y m b o l u - R ( t ) ) , p=-\rho\Bigl(\frac{\partial\varphi}{\partial t}+\tfrac{1}{2}symbol{u}\cdot symbol% {u}\Bigr)=\rho\Bigl(symbol{v}\cdot symbol{u}-\tfrac{1}{2}symbol{u}\cdot symbol% {u}-R(t)\Bigr),
  9. s y m b o l F = - A p s y m b o l n d S = ρ A ( 1 2 s y m b o l u s y m b o l u - s y m b o l v s y m b o l u ) s y m b o l n d S , symbol{F}=-\int_{A}p\,symbol{n}\;\mathrm{d}S=\rho\int_{A}\left(\tfrac{1}{2}% symbol{u}\cdot symbol{u}-symbol{v}\cdot symbol{u}\right)symbol{n}\;\mathrm{d}S,
  10. F k = ρ A i ( 1 2 u i 2 - u i v i ) n k d S . ( 3 ) F_{k}=\rho\int_{A}\sum_{i}(\tfrac{1}{2}u_{i}^{2}-u_{i}v_{i})n_{k}\,\mathrm{d}S% .\qquad(3)
  11. 1 2 A i u i 2 n k d S = - 1 2 V x k ( i u i 2 ) d V . \frac{1}{2}\int_{A}\sum_{i}u_{i}^{2}n_{k}\,\mathrm{d}S=-\frac{1}{2}\int_{V}% \frac{\partial}{\partial x_{k}}\left(\sum_{i}u_{i}^{2}\right)\,\mathrm{d}V.
  12. 1 2 x k ( i u i 2 ) = i u i u k x i = i ( u i u k ) x i \frac{1}{2}\frac{\partial}{\partial x_{k}}\left(\sum_{i}u_{i}^{2}\right)=\sum_% {i}u_{i}\frac{\partial u_{k}}{\partial x_{i}}=\sum_{i}\frac{\partial(u_{i}u_{k% })}{\partial x_{i}}
  13. - 1 2 V x k ( i u i 2 ) d V = - V i ( u i u k ) x i d V = A u k i u i n i d S . -\frac{1}{2}\int_{V}\frac{\partial}{\partial x_{k}}\left(\sum_{i}u_{i}^{2}% \right)\,\mathrm{d}V=-\int_{V}\sum_{i}\frac{\partial(u_{i}u_{k})}{\partial x_{% i}}\,\mathrm{d}V=\int_{A}u_{k}\sum_{i}u_{i}n_{i}\,\mathrm{d}S.
  14. F k = ρ A i ( u k u i n i - v i u i n k ) d S . F_{k}=\rho\int_{A}\sum_{i}(u_{k}u_{i}n_{i}-v_{i}u_{i}n_{k})\,\mathrm{d}S.
  15. F k = ρ A i ( u k v i n i - v i u i n k ) d S . F_{k}=\rho\int_{A}\sum_{i}(u_{k}v_{i}n_{i}-v_{i}u_{i}n_{k})\,\mathrm{d}S.
  16. s y m b o l v s y m b o l F = i v i F i = 0. symbol{v}\cdot symbol{F}=\sum_{i}v_{i}F_{i}=0.

D'Alembert's_principle.html

  1. i ( 𝐅 i - m i 𝐚 i ) δ 𝐫 i = 0 , \sum_{i}(\mathbf{F}_{i}-m_{i}\mathbf{a}_{i})\cdot\delta\mathbf{r}_{i}=0,
  2. i i
  3. 𝐅 i \mathbf{F}_{i}
  4. i i
  5. m i m_{i}\scriptstyle
  6. i i
  7. 𝐚 i \mathbf{a}_{i}
  8. i i
  9. m i 𝐚 i m_{i}\mathbf{a}_{i}
  10. i i
  11. δ 𝐫 i \delta\mathbf{r}_{i}
  12. i i
  13. 𝐐 j {\mathbf{Q}}_{j}
  14. 𝐩 i = m i 𝐯 i \mathbf{p}_{i}=m_{i}\mathbf{v}_{i}
  15. 𝐩 i ˙ = m ˙ i 𝐯 i + m i 𝐯 ˙ i \dot{\mathbf{p}_{i}}=\dot{m}_{i}\mathbf{v}_{i}+m_{i}\dot{\mathbf{v}}_{i}
  16. 𝐩 i ˙ = m i 𝐯 ˙ i = m i 𝐚 i \dot{\mathbf{p}_{i}}=m_{i}\dot{\mathbf{v}}_{i}=m_{i}\mathbf{a}_{i}
  17. m ˙ i 𝐯 i \dot{m}_{i}\mathbf{v}_{i}
  18. m i 𝐯 ˙ i m_{i}\dot{\mathbf{v}}_{i}
  19. i ( 𝐅 i - m i 𝐚 i - m ˙ i 𝐯 i ) δ 𝐫 i = 0. \sum_{i}(\mathbf{F}_{i}-m_{i}\mathbf{a}_{i}-\dot{m}_{i}\mathbf{v}_{i})\cdot% \delta\mathbf{r}_{i}=0.
  20. 𝐅 i ( T ) = m i 𝐚 i , \mathbf{F}_{i}^{(T)}=m_{i}\mathbf{a}_{i},
  21. 𝐅 i ( T ) \mathbf{F}_{i}^{(T)}
  22. m i 𝐚 i m_{i}\mathbf{a}_{i}
  23. 𝐅 i ( T ) - m i 𝐚 i = 0. \mathbf{F}_{i}^{(T)}-m_{i}\mathbf{a}_{i}=\mathbf{0}.
  24. δ W \delta W
  25. δ 𝐫 i \delta\mathbf{r}_{i}
  26. δ W = i 𝐅 i ( T ) δ 𝐫 i - i m i 𝐚 i δ 𝐫 i = 0 \delta W=\sum_{i}\mathbf{F}_{i}^{(T)}\cdot\delta\mathbf{r}_{i}-\sum_{i}m_{i}% \mathbf{a}_{i}\cdot\delta\mathbf{r}_{i}=0
  27. 𝐅 i \mathbf{F}_{i}
  28. 𝐂 i \mathbf{C}_{i}
  29. δ W = i 𝐅 i δ 𝐫 i + i 𝐂 i δ 𝐫 i - i m i 𝐚 i δ 𝐫 i = 0. \delta W=\sum_{i}\mathbf{F}_{i}\cdot\delta\mathbf{r}_{i}+\sum_{i}\mathbf{C}_{i% }\cdot\delta\mathbf{r}_{i}-\sum_{i}m_{i}\mathbf{a}_{i}\cdot\delta\mathbf{r}_{i% }=0.
  30. δ W = i ( 𝐅 i - m i 𝐚 i ) δ 𝐫 i = 0. \delta W=\sum_{i}(\mathbf{F}_{i}-m_{i}\mathbf{a}_{i})\cdot\delta\mathbf{r}_{i}% =0.
  31. W W
  32. W = m g W=mg
  33. T T
  34. a a
  35. T - W = m a T-W=ma
  36. T = W + m a T=W+ma
  37. T T
  38. W W
  39. m a ma
  40. ( W / g ) a (W/g)a
  41. 𝐅 i = - m 𝐫 ¨ c \mathbf{F}_{i}=-m\ddot{\mathbf{r}}_{c}
  42. 𝐫 c \mathbf{r}_{c}
  43. m m
  44. T i = - I θ ¨ T_{i}=-I\ddot{\theta}
  45. I I
  46. F x = 0 , F y = 0 , T = 0 \begin{aligned}\displaystyle\sum F_{x}&\displaystyle=0,\\ \displaystyle\sum F_{y}&\displaystyle=0,\\ \displaystyle\sum T&\displaystyle=0\end{aligned}
  47. T \sum T
  48. δ W = ( Q 1 + Q 1 * ) δ q 1 + + ( Q m + Q m * ) δ q m = 0 , \delta W=(Q_{1}+Q^{*}_{1})\delta q_{1}+\ldots+(Q_{m}+Q^{*}_{m})\delta q_{m}=0,
  49. Q j + Q j * = 0 , j = 1 , , m , Q_{j}+Q^{*}_{j}=0,\quad j=1,\ldots,m,
  50. d d t T q ˙ j - T q j = Q j , j = 1 , , m . \frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{j}}-\frac{\partial T}{\partial q% _{j}}=Q_{j},\quad j=1,\ldots,m.

D-brane.html

  1. N N
  2. N × N N\times N
  3. N N
  4. ϕ 2 χ 2 \phi^{2}\chi^{2}
  5. ϕ \phi
  6. χ \chi
  7. T H = c 3 8 π G M k B ( 1.227 × 10 23 k g M K ) T_{\rm H}=\frac{\hbar c^{3}}{8\pi GMk_{B}}\;\quad(\approx{1.227\times 10^{23}% \;kg\over M}\;K)
  8. S B = k B 4 π G c M 2 . S_{\rm B}=\frac{k_{B}4\pi G}{\hbar c}M^{2}.
  9. S B = A k B 4 l P 2 , S_{\rm B}=\frac{Ak_{B}}{4l_{\rm P}^{2}},
  10. l P l_{\rm P}
  11. e < s u p > [ u f r a c t i o n , u - 1 , u g ] e<sup>[u^{\prime}fraction^{\prime},u^{\prime}-1^{\prime},u^{\prime}g^{\prime}]

Damköhler_numbers.html

  1. Da = reaction rate convective mass transport rate \mathrm{Da}=\frac{\,\text{reaction rate}}{\,\text{convective mass transport % rate}}
  2. Da II = reaction rate diffusive mass transfer rate \mathrm{Da}_{\mathrm{II}}=\frac{\,\text{reaction rate}}{\,\text{diffusive mass% transfer rate}}
  3. Da = flow time scale chemical time scale \mathrm{Da}=\frac{\,\text{flow time scale}}{\,\text{chemical time scale}}
  4. Da = k C 0 n - 1 τ \mathrm{Da}=kC_{0}^{\ n-1}\tau
  5. τ \tau
  6. Da II = k C 0 n - 1 k g a \mathrm{Da}_{\mathrm{II}}=\frac{kC_{0}^{n-1}}{k_{g}a}

Damping.html

  1. F F
  2. v v
  3. F = - c v , F=-cv\,,
  4. v 2 v^{2}
  5. c l i n c_{lin}
  6. x x
  7. d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = 0 , \frac{d^{2}x}{dt^{2}}+2\zeta\omega_{0}\frac{dx}{dt}+\omega_{0}^{2}x=0,
  8. F s = - k x F_{\mathrm{s}}=-kx\,
  9. F d = - c v = - c d x d t = - c x ˙ . F_{\mathrm{d}}=-cv=-c\frac{dx}{dt}=-c\dot{x}.
  10. F tot = m a = m d 2 x d t 2 = m x ¨ . F_{\mathrm{tot}}=ma=m\frac{d^{2}x}{dt^{2}}=m\ddot{x}.
  11. m x ¨ = - k x + - c x ˙ . m\ddot{x}=-kx+-c\dot{x}.
  12. x ¨ + c m x ˙ + k m x = 0. \ddot{x}+{c\over m}\dot{x}+{k\over m}x=0.\,
  13. ω 0 = k m \omega_{0}=\sqrt{k\over m}
  14. ζ = c 2 m k . \zeta={c\over 2\sqrt{mk}}.
  15. x ¨ + 2 ζ ω 0 x ˙ + ω 0 2 x = 0. \ddot{x}+2\zeta\omega_{0}\dot{x}+\omega_{0}^{2}x=0.\,
  16. x = e γ t x=e^{\gamma t}\,
  17. γ \gamma
  18. γ 2 + 2 ζ ω 0 γ + ω 0 2 = 0 , \gamma^{2}+2\zeta\omega_{0}\gamma+\omega_{0}^{2}=0\,,
  19. γ + \gamma_{+}
  20. γ - \gamma_{-}
  21. x ( t ) = A e γ + t + B e γ - t , x(t)=Ae^{\gamma_{+}t}+Be^{\gamma_{-}t}\,,
  22. A = x ( 0 ) + γ + x ( 0 ) - x ˙ ( 0 ) γ - - γ + A=x(0)+\frac{\gamma_{+}x(0)-\dot{x}(0)}{\gamma_{-}-\gamma_{+}}
  23. B = - γ + x ( 0 ) - x ˙ ( 0 ) γ - - γ + . B=-\frac{\gamma_{+}x(0)-\dot{x}(0)}{\gamma_{-}-\gamma_{+}}.
  24. x ( t ) = ( A + B t ) e - ω 0 t x(t)=(A+Bt)\,e^{-\omega_{0}t}\,
  25. A A
  26. B B
  27. A = x ( 0 ) A=x(0)\,
  28. B = x ˙ ( 0 ) + ω 0 x ( 0 ) B=\dot{x}(0)+\omega_{0}x(0)\,
  29. x ( t ) = A e γ + t + B e γ - t x(t)=Ae^{\gamma_{+}t}+Be^{\gamma_{-}t}
  30. A A
  31. B B
  32. A = x ( 0 ) + γ + x ( 0 ) - x ˙ ( 0 ) γ - - γ + A=x(0)+\frac{\gamma_{+}x(0)-\dot{x}(0)}{\gamma_{-}-\gamma_{+}}
  33. B = - γ + x ( 0 ) - x ˙ ( 0 ) γ - - γ + . B=-\frac{\gamma_{+}x(0)-\dot{x}(0)}{\gamma_{-}-\gamma_{+}}.
  34. x ( t ) = e - ζ ω 0 t ( A cos ( ω d t ) + B sin ( ω d t ) ) x(t)=e^{-\zeta\omega_{0}t}(A\cos\,(\omega_{\mathrm{d}}\,t)+B\sin\,(\omega_{% \mathrm{d}}\,t))\,
  35. ω d = ω 0 1 - ζ 2 \omega_{\mathrm{d}}=\omega_{0}\sqrt{1-\zeta^{2}}\,
  36. A = x ( 0 ) A=x(0)\,
  37. B = 1 ω d ( ζ ω 0 x ( 0 ) + x ˙ ( 0 ) ) . B=\frac{1}{\omega_{\mathrm{d}}}(\zeta\omega_{0}x(0)+\dot{x}(0)).\,
  38. ω p e a k = ω 0 1 - 2 ζ 2 \omega_{peak}=\omega_{0}\sqrt{1-2\zeta^{2}}
  39. m x ¨ + h x i + k x = 0 m\ddot{x}+hxi+kx=0
  40. m x ¨ + k ( 1 + i η ) x = 0 m\ddot{x}+k(1+i\eta)x=0
  41. η = E d / ( π k X 2 ) \eta=Ed/(\pi kX^{2})
  42. m x ¨ + A d r x d t r i + k x = 0 m\ddot{x}+A\frac{d^{r}x}{dt^{r}}i+kx=0

Data_center.html

  1. PUE = Total Facility Power IT Equipment Power \mathrm{PUE}={\mbox{Total Facility Power}~{}\over\mbox{IT Equipment Power}~{}}

Daubechies_wavelet.html

  1. 2 \sqrt{2}
  2. a ( Z ) = 2 1 - A ( 1 + Z ) A p ( Z ) a(Z)=2^{1-A}(1+Z)^{A}\,p(Z)
  3. a ( Z ) a ( Z - 1 ) + a ( - Z ) a ( - Z - 1 ) = 4 a(Z)\,a(Z^{-1})+a(-Z)\,a(-Z^{-1})=4
  4. ( 2 - X ) A P ( X ) + X A P ( 2 - X ) = 2 A (2-X)^{A}P(X)+X^{A}\,P(2-X)=2^{A}
  5. X := 1 / 2 ( 2 - Z - Z - 1 ) X:=1/2\cdot(2-Z-Z^{-1})
  6. X ( - Z ) = 2 - X ( Z ) X(-Z)=2-X(Z)
  7. P ( X ( Z ) ) = p ( Z ) p ( Z - 1 ) P(X(Z))=p(Z)p(Z^{-1})
  8. X ( e i w ) = 1 - c o s ( w ) X(e^{iw})=1-cos(w)
  9. p ( e i w ) p ( e - i w ) = | p ( e i w ) | 2 p(e^{iw})p(e^{-iw})=|p(e^{iw})|^{2}
  10. P A ( X ) = k = 0 A - 1 ( A + k - 1 A - 1 ) 2 - k X k P_{A}(X)=\sum_{k=0}^{A-1}\left({{A+k-1}\atop{A-1}}\right)2^{-k}X^{k}
  11. X A ( X - 1 ) R ( ( X - 1 ) 2 ) X^{A}(X-1)R((X-1)^{2})
  12. P ( X ) = P A ( X ) + X A ( X - 1 ) R ( ( X - 1 ) 2 ) P(X)=P_{A}(X)+X^{A}(X-1)R((X-1)^{2})
  13. 4 A - r 4^{A-r}
  14. P ( X ( Z ) ) = p ( Z ) p ( Z - 1 ) P(X(Z))=p(Z)p(Z^{-1})
  15. P ( X ) = ( X - μ 1 ) ( X - μ N ) P(X)=(X-\mu_{1})\dots(X-\mu_{N})
  16. ( X ( Z ) - μ ) = - 1 2 Z + 1 - μ - 1 2 Z - 1 (X(Z)-\mu)=-\frac{1}{2}Z+1-\mu-\frac{1}{2}Z^{-1}
  17. c i c_{i}
  18. c 0 = 1 + 3 4 2 c_{0}=\frac{1+\sqrt{3}}{4\sqrt{2}}
  19. c 1 = 3 + 3 4 2 c_{1}=\frac{3+\sqrt{3}}{4\sqrt{2}}
  20. c 2 = 3 - 3 4 2 c_{2}=\frac{3-\sqrt{3}}{4\sqrt{2}}
  21. c 1 = 1 - 3 4 2 c_{1}=\frac{1-\sqrt{3}}{4\sqrt{2}}
  22. b k = ( - 1 ) k a N - 1 - k b_{k}=(-1)^{k}a_{N-1-k}

Dawson_function.html

  1. F ( x ) = D + ( x ) = e - x 2 0 x e t 2 d t F(x)=D_{+}(x)=e^{-x^{2}}\int_{0}^{x}e^{t^{2}}\,dt
  2. D - ( x ) = e x 2 0 x e - t 2 d t D_{-}(x)=e^{x^{2}}\int_{0}^{x}e^{-t^{2}}\,dt\!
  3. D + ( x ) = 1 2 0 e - t 2 / 4 sin ( x t ) d t . D_{+}(x)=\frac{1}{2}\int_{0}^{\infty}e^{-t^{2}/4}\,\sin{(xt)}\,dt.
  4. D + ( x ) = π 2 e - x 2 erfi ( x ) = - i π 2 e - x 2 erf ( i x ) D_{+}(x)={\sqrt{\pi}\over 2}e^{-x^{2}}\mathrm{erfi}(x)=-{i\sqrt{\pi}\over 2}e^% {-x^{2}}\mathrm{erf}(ix)
  5. D - ( x ) = π 2 e x 2 erf ( x ) D_{-}(x)=\frac{\sqrt{\pi}}{2}e^{x^{2}}\mathrm{erf}(x)
  6. F ( z ) = π 2 e - z 2 erfi ( z ) = i π 2 [ e - z 2 - w ( z ) ] F(z)={\sqrt{\pi}\over 2}e^{-z^{2}}\mathrm{erfi}(z)=\frac{i\sqrt{\pi}}{2}\left[% e^{-z^{2}}-w(z)\right]
  7. D + ( x ) = F ( x ) = π 2 Im [ w ( x ) ] D_{+}(x)=F(x)=\frac{\sqrt{\pi}}{2}\operatorname{Im}[w(x)]
  8. D - ( x ) = i F ( - i x ) = - π 2 [ e x 2 - w ( - i x ) ] D_{-}(x)=iF(-ix)=-\frac{\sqrt{\pi}}{2}\left[e^{x^{2}}-w(-ix)\right]
  9. F ( x ) = k = 0 ( - 1 ) k 2 k ( 2 k + 1 ) ! ! x 2 k + 1 = x - 2 3 x 3 + 4 15 x 5 - F(x)=\sum_{k=0}^{\infty}\frac{(-1)^{k}\,2^{k}}{(2k+1)!!}\,x^{2k+1}=x-\frac{2}{% 3}x^{3}+\frac{4}{15}x^{5}-\cdots
  10. F ( x ) = k = 0 ( 2 k - 1 ) ! ! 2 k + 1 x 2 k + 1 = 1 2 x + 1 4 x 3 + 3 8 x 5 + F(x)=\sum_{k=0}^{\infty}\frac{(2k-1)!!}{2^{k+1}x^{2k+1}}=\frac{1}{2x}+\frac{1}% {4x^{3}}+\frac{3}{8x^{5}}+\cdots
  11. d F d x + 2 x F = 1 \frac{dF}{dx}+2xF=1\,\!
  12. F ( x ) = 1 2 x F(x)=\frac{1}{2x}
  13. F ( x ) = x 2 x 2 - 1 F(x)=\frac{x}{2x^{2}-1}
  14. H ( y ) = π - 1 P . V . - e - x 2 y - x d x H(y)=\pi^{-1}P.V.\int_{-\infty}^{\infty}{e^{-x^{2}}\over y-x}dx
  15. y y
  16. H ( y ) H(y)
  17. 1 / u 1/u
  18. 1 u = 0 d k sin k u = 0 d k e i k u {1\over u}=\int_{0}^{\infty}dk\sin ku=\int_{0}^{\infty}dk\Im e^{iku}
  19. u = 1 / ( y - x ) u=1/(y-x)
  20. sin ( k u ) \sin(ku)
  21. x x
  22. π H ( y ) = 0 d k exp [ - k 2 / 4 + i k y ] - d x exp [ - ( x + i k / 2 ) 2 ] \pi H(y)=\Im\int_{0}^{\infty}dk\exp[-k^{2}/4+iky]\int_{-\infty}^{\infty}dx\exp% [-(x+ik/2)^{2}]
  23. x x
  24. π 1 / 2 \pi^{1/2}
  25. π 1 / 2 H ( y ) = 0 d k exp [ - k 2 / 4 + i k y ] \pi^{1/2}H(y)=\Im\int_{0}^{\infty}dk\exp[-k^{2}/4+iky]
  26. k k
  27. π 1 / 2 H ( y ) = e - y 2 0 d k exp [ - ( k / 2 - i y ) 2 ] \pi^{1/2}H(y)=e^{-y^{2}}\Im\int_{0}^{\infty}dk\exp[-(k/2-iy)^{2}]
  28. u = i k / 2 + y u=ik/2+y
  29. π 1 / 2 H ( y ) = - 2 e - y 2 y i + y d u e u 2 \pi^{1/2}H(y)=-2e^{-y^{2}}\Im\int_{y}^{i\infty+y}due^{u^{2}}
  30. H ( y ) = 2 π - 1 / 2 F ( y ) H(y)=2\pi^{-1/2}F(y)
  31. F ( y ) F(y)
  32. x 2 n e - x 2 x^{2n}e^{-x^{2}}
  33. H n = π - 1 P . V . - x 2 n e - x 2 y - x d x H_{n}=\pi^{-1}P.V.\int_{-\infty}^{\infty}{x^{2n}e^{-x^{2}}\over y-x}dx
  34. H a = π - 1 P . V . - e - a x 2 y - x d x H_{a}=\pi^{-1}P.V.\int_{-\infty}^{\infty}{e^{-ax^{2}}\over y-x}dx
  35. n H a a n = ( - 1 ) n π - 1 P . V . - x 2 n e - a x 2 y - x d x {\partial^{n}H_{a}\over\partial a^{n}}=(-1)^{n}\pi^{-1}P.V.\int_{-\infty}^{% \infty}{x^{2n}e^{-ax^{2}}\over y-x}dx
  36. H n = ( - 1 ) n n H a a n | a = 1 H_{n}=(-1)^{n}{\partial^{n}H_{a}\over\partial a^{n}}|_{a=1}
  37. a = 1 a=1
  38. H a = 2 π - 1 / 2 F ( y a ) H_{a}=2\pi^{-1/2}F(y\sqrt{a})
  39. F ( y ) = 1 - 2 y F ( y ) F^{\prime}(y)=1-2yF(y)
  40. H n = P 1 ( y ) + P 2 ( y ) F ( y ) H_{n}=P_{1}(y)+P_{2}(y)F(y)
  41. P 1 P_{1}
  42. P 2 P_{2}
  43. H 1 = - π - 1 / 2 y + 2 π - 1 / 2 y 2 F ( y ) H_{1}=-\pi^{-1/2}y+2\pi^{-1/2}y^{2}F(y)
  44. H n H_{n}
  45. n 0 n\geq 0
  46. H n + 1 ( y ) = y 2 H n ( y ) - ( 2 n - 1 ) ! ! π 2 n y H_{n+1}(y)=y^{2}H_{n}(y)-\frac{(2n-1)!!}{\sqrt{\pi}2^{n}}y

Dc_(computer_program).html

  1. ( 12 + ( - 3 ) 4 ) 11 - 22 \sqrt{(12+(-3)^{4})\over 11}-22
  2. n ! = i = 1 n i n!=\prod_{i=1}^{n}i

De_Boor's_algorithm.html

  1. 𝐝 0 , 𝐝 1 , , 𝐝 p - 1 \mathbf{d}_{0},\mathbf{d}_{1},\dots,\mathbf{d}_{p-1}
  2. 𝐬 ( x ) \mathbf{s}(x)
  3. 𝐬 ( u 0 ) = 𝐝 0 , , 𝐬 ( u p - 1 ) = 𝐝 p - 1 \mathbf{s}(u_{0})=\mathbf{d}_{0},\dots,\mathbf{s}(u_{p-1})=\mathbf{d}_{p-1}
  4. 𝐝 0 , 𝐝 1 , , 𝐝 p - 1 \mathbf{d}_{0},\mathbf{d}_{1},\dots,\mathbf{d}_{p-1}
  5. 𝐝 0 , 𝐝 1 , , 𝐝 p - 1 \mathbf{d}_{0},\mathbf{d}_{1},\dots,\mathbf{d}_{p-1}
  6. 𝐬 ( x ) \mathbf{s}(x)
  7. x [ u , u + 1 ] x\in[u_{\ell},u_{\ell+1}]
  8. 𝐬 ( x ) = i = 0 p - 1 𝐝 i N i n ( x ) , \mathbf{s}(x)=\sum_{i=0}^{p-1}\mathbf{d}_{i}N_{i}^{n}(x),
  9. N i n ( x ) = x - u i u i + n - u i N i n - 1 ( x ) + u i + n + 1 - x u i + n + 1 - u i + 1 N i + 1 n - 1 ( x ) , N_{i}^{n}(x)=\frac{x-u_{i}}{u_{i+n}-u_{i}}N_{i}^{n-1}(x)+\frac{u_{i+n+1}-x}{u_% {i+n+1}-u_{i+1}}N_{i+1}^{n-1}(x),
  10. N i 0 ( x ) = { 1 , if x [ u i , u i + 1 ) 0 , otherwise N_{i}^{0}(x)=\left\{\begin{matrix}1,&\mbox{if }~{}x\in[u_{i},u_{i+1})\\ 0,&\mbox{otherwise }\end{matrix}\right.
  11. 𝐬 ( x ) = i = - n 𝐝 i N i n ( x ) \mathbf{s}(x)=\sum_{i=\ell-n}^{\ell}\mathbf{d}_{i}N_{i}^{n}(x)
  12. 𝐬 ( x ) \mathbf{s}(x)
  13. 𝐝 - n , 𝐝 - n + 1 , , 𝐝 \mathbf{d}_{\ell-n},\mathbf{d}_{\ell-n+1},\dots,\mathbf{d}_{\ell}
  14. 𝐝 i \mathbf{d}_{i}
  15. 𝐬 ( x ) \mathbf{s}(x)
  16. x [ u , u + 1 ) x\in[u_{\ell},u_{\ell+1})
  17. 𝐝 i [ 0 ] = 𝐝 i \mathbf{d}_{i}^{[0]}=\mathbf{d}_{i}
  18. i = - n , , i=\ell-n,\dots,\ell
  19. 𝐝 i [ k ] = ( 1 - α k , i ) 𝐝 i - 1 [ k - 1 ] + α k , i 𝐝 i [ k - 1 ] ; k = 1 , , n ; i = - n + k , , \mathbf{d}_{i}^{[k]}=(1-\alpha_{k,i})\mathbf{d}_{i-1}^{[k-1]}+\alpha_{k,i}% \mathbf{d}_{i}^{[k-1]};\qquad k=1,\dots,n;\quad i=\ell-n+k,\dots,\ell
  20. α k , i = x - u i u i + n + 1 - k - u i . \alpha_{k,i}=\frac{x-u_{i}}{u_{i+n+1-k}-u_{i}}.
  21. 𝐬 ( x ) = 𝐝 [ n ] \mathbf{s}(x)=\mathbf{d}_{\ell}^{[n]}

De_dicto_and_de_re.html

  1. x A x \Box\exists{x}Ax
  2. x A x \exists{x}\Box Ax

De_Laval_nozzle.html

  1. v e = T R M 2 γ γ - 1 [ 1 - ( p e p ) γ - 1 γ ] v_{e}=\sqrt{\frac{TR}{M}\cdot\frac{2\gamma}{\gamma-1}\cdot\left[1-\left(\frac{% p_{e}}{p}\right)^{\frac{\gamma-1}{\gamma}}\right]}
  2. v e v_{e}
  3. T T
  4. R R
  5. M M
  6. γ \gamma
  7. c p c v \frac{c_{p}}{c_{v}}
  8. c p c_{p}
  9. c v c_{v}
  10. p e p_{e}
  11. p p

De_Sitter_universe.html

  1. t = 10 - 33 t=10^{-33}
  2. Λ \Lambda
  3. H H
  4. H Λ H\propto\sqrt{\Lambda}
  5. a ( t ) = e H t a(t)=e^{Ht}
  6. H H
  7. t t
  8. a ( t ) a(t)
  9. q = - 1 q=-1

Debye_function.html

  1. D n ( x ) = n x n 0 x t n e t - 1 d t . D_{n}(x)=\frac{n}{x^{n}}\int_{0}^{x}\frac{t^{n}}{e^{t}-1}\,dt.
  2. D n ( x ) = 1 - n 2 ( n + 1 ) x + n k = 1 B 2 k ( 2 k + n ) ( 2 k ) ! x 2 k , | x | < 2 π , n 1. D_{n}(x)=1-\frac{n}{2(n+1)}x+n\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k+n)(2k)!}x^{% 2k},\quad|x|<2\pi,\ n\geq 1.
  3. x 0 x\rightarrow 0
  4. D n ( 0 ) = 1. D_{n}(0)=1.
  5. x 1 x\ll 1
  6. D n D_{n}
  7. D n ( x ) 0 d t t n exp ( t ) - 1 = Γ ( n + 1 ) ζ ( n + 1 ) . [ n > 0 ] D_{n}(x)\propto\int_{0}^{\infty}{\rm d}t\frac{t^{n}}{\exp(t)-1}=\Gamma(n+1)% \zeta(n+1).\quad[\Re\,n>0]
  8. g D ( ω ) = 9 ω 2 ω D 3 g_{\rm D}(\omega)=\frac{9\omega^{2}}{\omega_{\rm D}^{3}}
  9. 0 ω ω D 0\leq\omega\leq\omega_{\rm D}
  10. U = 0 d ω g ( ω ) ω n ( ω ) U=\int_{0}^{\infty}{\rm d}\omega\,g(\omega)\,\hbar\omega\,n(\omega)
  11. n ( ω ) = 1 exp ( ω / k B T ) - 1 n(\omega)=\frac{1}{\exp(\hbar\omega/k_{\rm B}T)-1}
  12. U = 3 k B T D 3 ( ω D / k B T ) U=3k_{\rm B}T\,D_{3}(\hbar\omega_{\rm D}/k_{\rm B}T)
  13. exp ( - 2 W ( q ) ) = exp ( - q 2 u x 2 \exp(-2W(q))=\exp(-q^{2}\langle u_{x}^{2}\rangle
  14. 2 W ( q ) = 2 q 2 6 M k B T 0 d ω k B T ω g ( ω ) coth ω 2 k B T = 2 q 2 6 M k B T 0 d ω k B T ω g ( ω ) [ 2 exp ( ω / k B T ) - 1 + 1 ] . 2W(q)=\frac{\hbar^{2}q^{2}}{6Mk_{\rm B}T}\int_{0}^{\infty}{\rm d}\omega\frac{k% _{\rm B}T}{\hbar\omega}g(\omega)\coth\frac{\hbar\omega}{2k_{\rm B}T}=\frac{% \hbar^{2}q^{2}}{6Mk_{\rm B}T}\int_{0}^{\infty}{\rm d}\omega\frac{k_{\rm B}T}{% \hbar\omega}g(\omega)\left[\frac{2}{\exp(\hbar\omega/k_{\rm B}T)-1}+1\right].
  15. 2 W ( q ) = 3 2 2 q 2 M ω D [ 2 ( k B T ω D ) D 1 ( ω D k B T ) + 1 2 ] 2W(q)=\frac{3}{2}\frac{\hbar^{2}q^{2}}{M\hbar\omega_{\rm D}}\left[2\left(\frac% {k_{\rm B}T}{\hbar\omega_{\rm D}}\right)D_{1}\left(\frac{\hbar\omega_{\rm D}}{% k_{\rm B}T}\right)+\frac{1}{2}\right]

Debye_model.html

  1. T 3 T^{3}
  2. L L
  3. λ n = 2 L n , \lambda_{n}={2L\over n}\,,
  4. n n
  5. E n = h ν n , E_{n}\ =h\nu_{n}\,,
  6. h h
  7. ν n \nu_{n}
  8. E n = h ν n = h c s λ n = h c s n 2 L , E_{n}=h\nu_{n}={hc_{s}\over\lambda_{n}}={hc_{s}n\over 2L}\,,
  9. c s c_{s}
  10. E n 2 = p n 2 c s 2 = ( h c s 2 L ) 2 ( n x 2 + n y 2 + n z 2 ) , E_{n}^{2}={p_{n}^{2}c_{s}^{2}}=\left({hc_{s}\over 2L}\right)^{2}\left(n_{x}^{2% }+n_{y}^{2}+n_{z}^{2}\right)\,,
  11. p n p_{n}
  12. E = n E n N ¯ ( E n ) , E=\sum_{n}E_{n}\,\bar{N}(E_{n})\,,
  13. N ¯ ( E n ) \bar{N}(E_{n})
  14. E n E_{n}
  15. U = n x n y n z E n N ¯ ( E n ) . U=\sum_{n_{x}}\sum_{n_{y}}\sum_{n_{z}}E_{n}\,\bar{N}(E_{n})\,.
  16. N N
  17. N 3 \sqrt[3]{N}
  18. L / N 3 L/\sqrt[3]{N}
  19. λ min = 2 L N 3 , \lambda_{\rm min}={2L\over\sqrt[3]{N}}\,,
  20. n n
  21. n max = N 3 . n_{\rm max}=\sqrt[3]{N}\,.
  22. U = n x N 3 n y N 3 n z N 3 E n N ¯ ( E n ) . U=\sum_{n_{x}}^{\sqrt[3]{N}}\sum_{n_{y}}^{\sqrt[3]{N}}\sum_{n_{z}}^{\sqrt[3]{N% }}E_{n}\,\bar{N}(E_{n})\,.
  23. U 0 N 3 0 N 3 0 N 3 E ( n ) N ¯ ( E ( n ) ) d n x d n y d n z . U\approx\int_{0}^{\sqrt[3]{N}}\int_{0}^{\sqrt[3]{N}}\int_{0}^{\sqrt[3]{N}}E(n)% \,\bar{N}\left(E(n)\right)\,dn_{x}\,dn_{y}\,dn_{z}\,.
  24. N ¯ ( E ) \bar{N}(E)
  25. E . E\,.
  26. N B E = 1 e E / k T - 1 . \langle N\rangle_{BE}={1\over e^{E/kT}-1}\,.
  27. N ¯ ( E ) = 3 e E / k T - 1 . \bar{N}(E)={3\over e^{E/kT}-1}\,.
  28. c s := c < m t p l > eff c_{s}:=c_{<}mtpl>{{\rm eff}}
  29. T D T_{D}
  30. c < m t p l > eff c_{<}mtpl>{{\rm eff}}
  31. T D - 3 c < m t p l > eff - 3 := ( 1 / 3 ) c long - 3 + ( 2 / 3 ) c trans - 3 T_{D}^{-3}\propto c_{<}mtpl>{{\rm eff}}^{-3}:=(1/3)c_{{\rm long}}^{-3}+(2/3)c_% {{\rm trans}}^{-3}
  32. U = 0 N 3 0 N 3 0 N 3 E ( n ) 3 e E ( n ) / k T - 1 d n x d n y d n z . U=\int_{0}^{\sqrt[3]{N}}\int_{0}^{\sqrt[3]{N}}\int_{0}^{\sqrt[3]{N}}E(n)\,{3% \over e^{E(n)/kT}-1}\,dn_{x}\,dn_{y}\,dn_{z}\,.
  33. ( n x , n y , n z ) = ( n sin θ cos ϕ , n sin θ sin ϕ , n cos θ ) \ (n_{x},n_{y},n_{z})=(n\sin\theta\cos\phi,n\sin\theta\sin\phi,n\cos\theta)
  34. U 0 π / 2 0 π / 2 0 R E ( n ) 3 e E ( n ) / k T - 1 n 2 sin θ d n d θ d ϕ , U\approx\int_{0}^{\pi/2}\int_{0}^{\pi/2}\int_{0}^{R}E(n)\,{3\over e^{E(n)/kT}-% 1}n^{2}\sin\theta\,dn\,d\theta\,d\phi\,,
  35. R R
  36. N N
  37. N = 1 8 4 3 π R 3 , N={1\over 8}{4\over 3}\pi R^{3}\,,
  38. R = 6 N π 3 . R=\sqrt[3]{6N\over\pi}\,.
  39. U = 3 π 2 0 R h c s n 2 L n 2 e h c s n / 2 L k T - 1 d n U={3\pi\over 2}\int_{0}^{R}\,{hc_{s}n\over 2L}{n^{2}\over e^{hc_{s}n/2LkT}-1}% \,dn
  40. x = h c s n 2 L k T x={hc_{s}n\over 2LkT}
  41. U = 3 π 2 k T ( 2 L k T h c s ) 3 0 h c s R / 2 L k T x 3 e x - 1 d x U={3\pi\over 2}kT\left({2LkT\over hc_{s}}\right)^{3}\int_{0}^{hc_{s}R/2LkT}{x^% {3}\over e^{x}-1}\,dx
  42. T D T_{D}
  43. T D = def h c s R 2 L k = h c s 2 L k 6 N π 3 = h c s 2 k 6 π N V 3 T_{D}\ \stackrel{\mathrm{def}}{=}\ {hc_{s}R\over 2Lk}={hc_{s}\over 2Lk}\sqrt[3% ]{6N\over\pi}={hc_{s}\over 2k}\sqrt[3]{{6\over\pi}{N\over V}}
  44. k T D kT_{D}
  45. U N k = 9 T ( T T D ) 3 0 T D / T x 3 e x - 1 d x = 3 T D 3 ( T D T ) , \frac{U}{Nk}=9T\left({T\over T_{D}}\right)^{3}\int_{0}^{T_{D}/T}{x^{3}\over e^% {x}-1}\,dx=3TD_{3}\left({T_{D}\over T}\right)\,,
  46. D 3 ( x ) D_{3}(x)
  47. T T
  48. C V N k = 9 ( T T D ) 3 0 T D / T x 4 e x ( e x - 1 ) 2 d x . \frac{C_{V}}{Nk}=9\left({T\over T_{D}}\right)^{3}\int_{0}^{T_{D}/T}{x^{4}e^{x}% \over\left(e^{x}-1\right)^{2}}\,dx\,.
  49. E ( ν ) E(\nu)
  50. ( g ( ν ) d ν 3 N ) , (\int g(\nu)\,{\rm d\nu}\equiv 3N)\,,
  51. n 1 3 ν 3 V F , n\sim{1\over 3}\nu^{3}VF\,,
  52. V V
  53. F F
  54. U = 0 h ν 3 V F e h ν / k T - 1 d ν , U=\int_{0}^{\infty}\,{h\nu^{3}VF\over e^{h\nu/kT}-1}\,d\nu\,,
  55. T 3 T^{3}
  56. 3 N 3N
  57. ν m \nu_{m}
  58. 3 N 3N
  59. 3 N = 1 3 ν m 3 V F . 3N={1\over 3}\nu_{m}^{3}VF\,.
  60. U = 0 ν m h ν 3 V F e h ν / k T - 1 d ν , U=\int_{0}^{\nu_{m}}\,{h\nu^{3}VF\over e^{h\nu/kT}-1}\,d\nu\,,
  61. = V F k T ( k T / h ) 3 0 T D / T x 3 e x - 1 d x , =VFkT(kT/h)^{3}\int_{0}^{T_{D}/T}\,{x^{3}\over e^{x}-1}\,dx\,,
  62. T D T_{D}
  63. h ν m / k h\nu_{m}/k
  64. = 9 N k T ( T / T D ) 3 0 T D / T x 3 e x - 1 d x , =9NkT(T/T_{D})^{3}\int_{0}^{T_{D}/T}\,{x^{3}\over e^{x}-1}\,dx\,,
  65. = 3 N k T D 3 ( T D / T ) , =3NkTD_{3}(T_{D}/T)\,,
  66. D 3 D_{3}
  67. L x , L y , L z L_{x},L_{y},L_{z}
  68. 𝐤 = ( k x , k y , k z ) \mathbf{k}=(k_{x},k_{y},k_{z})
  69. l x = k x | 𝐤 | , l y = k y | 𝐤 | , l z = k z | 𝐤 | l_{x}=\frac{k_{x}}{|\mathbf{k}|},l_{y}=\frac{k_{y}}{|\mathbf{k}|},l_{z}=\frac{% k_{z}}{|\mathbf{k}|}
  70. u ( x , y , z , t ) = sin ( 2 π ν t ) sin ( 2 π l x x λ ) sin ( 2 π l y y λ ) sin ( 2 π l z z λ ) u(x,y,z,t)=\sin(2\pi\nu t)\sin\left(\frac{2\pi l_{x}x}{\lambda}\right)\sin% \left(\frac{2\pi l_{y}y}{\lambda}\right)\sin\left(\frac{2\pi l_{z}z}{\lambda}\right)
  71. u = 0 u=0
  72. x , y , z = 0 , x = L x , y = L y , z = L z x,y,z=0,x=L_{x},y=L_{y},z=L_{z}
  73. 2 l x L x λ = n x ; 2 l y L y λ = n y ; 2 l z L z λ = n z \frac{2l_{x}L_{x}}{\lambda}=n_{x};\frac{2l_{y}L_{y}}{\lambda}=n_{y};\frac{2l_{% z}L_{z}}{\lambda}=n_{z}
  74. n x , n y , n z n_{x},n_{y},n_{z}
  75. c s = λ ν c_{s}=\lambda\nu
  76. n x 2 ( 2 ν L x / c s ) 2 + n y 2 ( 2 ν L y / c s ) 2 + n z 2 ( 2 ν L z / c s ) 2 = 1. \frac{n_{x}^{2}}{(2\nu L_{x}/c_{s})^{2}}+\frac{n_{y}^{2}}{(2\nu L_{y}/c_{s})^{% 2}}+\frac{n_{z}^{2}}{(2\nu L_{z}/c_{s})^{2}}=1.
  77. ν \nu
  78. n x , n y , n z n_{x},n_{y},n_{z}
  79. ν \nu
  80. L x , L y , L z L_{x},L_{y},L_{z}\to\infty
  81. N ( ν ) N(\nu)
  82. [ 0 , ν ] [0,\nu]
  83. N ( ν ) = 1 8 4 π 3 ( 2 ν c s ) 3 L x L y L z = 4 π ν 3 V 3 c s 3 , N(\nu)=\frac{1}{8}\frac{4\pi}{3}\left(\frac{2\nu}{c_{s}}\right)^{3}L_{x}L_{y}L% _{z}=\frac{4\pi\nu^{3}V}{3c_{s}^{3}},
  84. V = L x L y L z V=L_{x}L_{y}L_{z}
  85. 3 c s 3 = 1 c long 3 + 2 c trans 3 \frac{3}{c_{s}^{3}}=\frac{1}{c\text{long}^{3}}+\frac{2}{c\text{trans}^{3}}
  86. ν D \nu_{D}
  87. [ 0 , ν D ] [0,\nu_{D}]
  88. ν D \nu_{D}
  89. 3 N = N ( ν D ) = 4 π ν D 3 V 3 c s 3 3N=N(\nu_{D})=\frac{4\pi\nu_{D}^{3}V}{3c_{s}^{3}}
  90. ν D = k T D h \nu_{D}=\frac{kT_{D}}{h}
  91. N ( ν ) = 3 N h 3 ν 3 k 3 T D 3 , N(\nu)=\frac{3Nh^{3}\nu^{3}}{k^{3}T_{D}^{3}},
  92. ν \nu
  93. E i = ( i + 1 / 2 ) h ν i = 0 , 1 , 2 , E_{i}=(i+1/2)h\nu i=0,1,2,\ldots
  94. E i E_{i}
  95. n i = 1 A e - E i / ( k T ) = 1 A e - ( i + 1 / 2 ) h ν / ( k T ) n_{i}=\frac{1}{A}e^{-E_{i}/(kT)}=\frac{1}{A}e^{-(i+1/2)h\nu/(kT)}
  96. ν \nu
  97. d U ( ν ) = i = 0 E i 1 A e - E i / ( k T ) dU(\nu)=\sum_{i=0}^{\infty}E_{i}\frac{1}{A}e^{-E_{i}/(kT)}
  98. i = 0 n i = d N ( ν ) \sum_{i=0}^{\infty}n_{i}=dN(\nu)
  99. d N ( ν ) dN(\nu)
  100. ν \nu
  101. 1 A e - 1 / 2 h ν / ( k T ) i = 0 e - i h ν / ( k T ) = 1 A e - 1 / 2 h ν / ( k T ) 1 1 - e - h ν / ( k T ) = d N ( ν ) \frac{1}{A}e^{-1/2h\nu/(kT)}\sum_{i=0}^{\infty}e^{-ih\nu/(kT)}=\frac{1}{A}e^{-% 1/2h\nu/(kT)}\frac{1}{1-e^{-h\nu/(kT)}}=dN(\nu)
  102. d U = d N ( ν ) e 1 / 2 h ν / ( k T ) ( 1 - e - h ν / ( k T ) ) i = 0 h ν ( i + 1 / 2 ) e - h ν ( i + 1 / 2 ) / ( k T ) = dU=dN(\nu)e^{1/2h\nu/(kT)}(1-e^{-h\nu/(kT)})\sum_{i=0}^{\infty}h\nu(i+1/2)e^{-% h\nu(i+1/2)/(kT)}=
  103. = d N ( ν ) ( 1 - e - h ν / ( k T ) ) i = 0 h ν ( i + 1 / 2 ) e - h ν i / ( k T ) = d N ( ν ) h ν ( 1 2 + ( 1 - e - h ν / ( k T ) ) i = 0 i e - h ν i / ( k T ) ) = =dN(\nu)(1-e^{-h\nu/(kT)})\sum_{i=0}^{\infty}h\nu(i+1/2)e^{-h\nu i/(kT)}=dN(% \nu)h\nu\left(\frac{1}{2}+(1-e^{-h\nu/(kT)})\sum_{i=0}^{\infty}ie^{-h\nu i/(kT% )}\right)=
  104. d N ( ν ) h ν ( 1 2 + 1 e h ν / ( k T ) - 1 ) dN(\nu)h\nu\left(\frac{1}{2}+\frac{1}{e^{h\nu/(kT)}-1}\right)
  105. U = 9 N h 4 k 3 T D 3 0 ν D ( 1 2 + 1 e h ν / ( k T ) - 1 ) ν 3 d ν U=\frac{9Nh^{4}}{k^{3}T_{D}^{3}}\int_{0}^{\nu_{D}}\left(\frac{1}{2}+\frac{1}{e% ^{h\nu/(kT)}-1}\right)\nu^{3}d\nu
  106. T T D T\ll T_{D}
  107. C V N k 9 ( T T D ) 3 0 x 4 e x ( e x - 1 ) 2 d x \frac{C_{V}}{Nk}\sim 9\left({T\over T_{D}}\right)^{3}\int_{0}^{\infty}{x^{4}e^% {x}\over\left(e^{x}-1\right)^{2}}\,dx
  108. C V N k 12 π 4 5 ( T T D ) 3 \frac{C_{V}}{Nk}\sim{12\pi^{4}\over 5}\left({T\over T_{D}}\right)^{3}
  109. T T D T\gg T_{D}
  110. e x - 1 x e^{x}-1\approx x
  111. | x | 1 |x|\ll 1
  112. C V N k 9 ( T T D ) 3 0 T D / T x 4 x 2 d x \frac{C_{V}}{Nk}\sim 9\left({T\over T_{D}}\right)^{3}\int_{0}^{T_{D}/T}{x^{4}% \over x^{2}}\,dx
  113. C V N k 3 . \frac{C_{V}}{Nk}\sim 3\,.
  114. C V = 3 N k ( ϵ k T ) 2 e ϵ / k T ( e ϵ / k T - 1 ) 2 C_{V}=3Nk\left({\epsilon\over kT}\right)^{2}{e^{\epsilon/kT}\over\left(e^{% \epsilon/kT}-1\right)^{2}}
  115. ϵ / k \epsilon/k
  116. T D T_{D}
  117. ϵ k T D , {\epsilon\over k}\neq T_{D}\,,
  118. T E = def ϵ k , T_{E}\ \stackrel{\mathrm{def}}{=}\ {\epsilon\over k}\,,
  119. T E T D , T_{E}\neq T_{D}\,,
  120. T E T D = ? \frac{T_{E}}{T_{D}}=?
  121. ϵ = ω = h ν \epsilon=\hbar\omega=h\nu
  122. λ m i n \lambda_{min}
  123. ν = c s λ = c s N 3 2 L = c s 2 N V 3 \nu={c_{s}\over\lambda}={c_{s}\sqrt[3]{N}\over 2L}={c_{s}\over 2}\sqrt[3]{N% \over V}
  124. T E = ϵ k = h ν k = h c s 2 k N V 3 , T_{E}={\epsilon\over k}={h\nu\over k}={hc_{s}\over 2k}\sqrt[3]{N\over V}\,,
  125. T E T D = π 6 3 = 0.805995977... {T_{E}\over T_{D}}=\sqrt[3]{\pi\over 6}\ =0.805995977...
  126. T T
  127. T 3 T^{3}
  128. E ( ν ) k 2 E(\nu)\propto k^{2}
  129. E ( ν ) k E(\nu)\propto k
  130. k = 2 π / λ k=2\pi/\lambda
  131. g ( ν ) d ν N \int g(\nu){\rm d}\nu\equiv N\,
  132. Δ C V | magnon T 3 / 2 \Delta C_{\,{\rm V|\,magnon}}\,\propto T^{3/2}
  133. Δ C V | phonon T 3 \,\Delta C_{\,{\rm V|\,phonon}}\propto T^{3}
  134. T \propto T

Decagon.html

  1. A = 5 2 t 2 cot π 10 = 5 t 2 2 5 + 2 5 7.694208843 t 2 . A=\frac{5}{2}t^{2}\cot\frac{\pi}{10}=\frac{5t^{2}}{2}\sqrt{5+2\sqrt{5}}\simeq 7% .694208843t^{2}.
  2. A = 2.5 d t A=2.5dt
  3. d = 2 t ( cos 3 π 10 + cos π 10 ) , d=2t\left(\cos\tfrac{3\pi}{10}+\cos\tfrac{\pi}{10}\right),
  4. d = t 5 + 2 5 . d=t\sqrt{5+2\sqrt{5}}.
  5. - 1 + 5 2 = 1 ϕ \tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\phi}
  6. 1 + 5 2 \tfrac{1+\sqrt{5}}{2}

Decision_tree_learning.html

  1. ( 𝐱 , Y ) = ( x 1 , x 2 , x 3 , , x k , Y ) (\,\textbf{x},Y)=(x_{1},x_{2},x_{3},...,x_{k},Y)
  2. i { 1 , 2 , , m } i\in\{1,2,...,m\}
  3. f i f_{i}
  4. i i
  5. I G ( f ) = i = 1 m f i ( 1 - f i ) = i = 1 m ( f i - f i 2 ) = i = 1 m f i - i = 1 m f i 2 = 1 - i = 1 m f i 2 = i k f i f k I_{G}(f)=\sum_{i=1}^{m}f_{i}(1-f_{i})=\sum_{i=1}^{m}(f_{i}-{f_{i}}^{2})=\sum_{% i=1}^{m}f_{i}-\sum_{i=1}^{m}{f_{i}}^{2}=1-\sum^{m}_{i=1}{f_{i}}^{2}=\sum_{i% \neq k}f_{i}f_{k}
  6. I E ( f ) = - i = 1 m f i log 2 f i I_{E}(f)=-\sum^{m}_{i=1}f_{i}\log_{2}f_{i}
  7. N N
  8. x x
  9. I V ( N ) = 1 | S | 2 i S j S 1 2 ( x i - x j ) 2 - ( 1 | S t | 2 i S t j S t 1 2 ( x i - x j ) 2 + 1 | S f | 2 i S f j S f 1 2 ( x i - x j ) 2 ) I_{V}(N)=\frac{1}{|S|^{2}}\sum_{i\in S}\sum_{j\in S}\frac{1}{2}(x_{i}-x_{j})^{% 2}-\left(\frac{1}{|S_{t}|^{2}}\sum_{i\in S_{t}}\sum_{j\in S_{t}}\frac{1}{2}(x_% {i}-x_{j})^{2}+\frac{1}{|S_{f}|^{2}}\sum_{i\in S_{f}}\sum_{j\in S_{f}}\frac{1}% {2}(x_{i}-x_{j})^{2}\right)
  10. S S
  11. S t S_{t}
  12. S f S_{f}

Dedekind_eta_function.html

  1. η ( τ ) = e π i τ 12 n = 1 ( 1 - q n ) . \eta(\tau)=e^{\frac{\pi\rm{i}\tau}{12}}\prod_{n=1}^{\infty}(1-q^{n}).
  2. q e 2 π i τ q\equiv e^{2\pi\rm{i}\tau}\,
  3. e π i τ e^{\pi\rm{i}\tau}\,
  4. Δ = ( 2 π ) 12 η 24 ( τ ) \Delta=(2\pi)^{12}\eta^{24}(\tau)
  5. η ( τ + 1 ) = e π i 12 η ( τ ) , \eta(\tau+1)=e^{\frac{\pi{\rm{i}}}{12}}\eta(\tau),\,
  6. η ( - 1 τ ) = - i τ η ( τ ) . \eta(-\tfrac{1}{\tau})=\sqrt{-{\rm{i}}\tau}\eta(\tau).\,
  7. τ a τ + b c τ + d \tau\mapsto\frac{a\tau+b}{c\tau+d}
  8. η ( a τ + b c τ + d ) = ϵ ( a , b , c , d ) ( c τ + d ) 1 2 η ( τ ) , \eta\left(\frac{a\tau+b}{c\tau+d}\right)=\epsilon(a,b,c,d)(c\tau+d)^{\frac{1}{% 2}}\eta(\tau),
  9. ϵ ( a , b , c , d ) = e b i π 12 ( c = 0 , d = 1 ) ; \epsilon(a,b,c,d)=e^{\frac{b{\rm{i}}\pi}{12}}\quad(c=0,d=1);
  10. ϵ ( a , b , c , d ) = e i π [ a + d 12 c - s ( d , c ) - 1 4 ] ( c > 0 ) . \epsilon(a,b,c,d)=e^{{\rm{i}}\pi[\frac{a+d}{12c}-s(d,c)-\frac{1}{4}]}\quad(c>0).
  11. s ( h , k ) s(h,k)\,
  12. s ( h , k ) = n = 1 k - 1 n k ( h n k - h n k - 1 2 ) . s(h,k)=\sum_{n=1}^{k-1}\frac{n}{k}\left(\frac{hn}{k}-\left\lfloor\frac{hn}{k}% \right\rfloor-\frac{1}{2}\right).
  13. Δ ( τ ) = ( 2 π ) 12 η ( τ ) 24 \Delta(\tau)=(2\pi)^{12}\eta(\tau)^{24}\,
  14. η ( τ ) = n = 1 χ ( n ) exp ( 1 12 π i n 2 τ ) , \eta(\tau)=\sum_{n=1}^{\infty}\chi(n)\exp(\tfrac{1}{12}\pi in^{2}\tau),
  15. χ ( n ) \chi(n)
  16. χ ( ± 1 ) = 1 \chi(\pm 1)=1
  17. χ ( ± 5 ) = - 1 \chi(\pm 5)=-1
  18. η ( τ ) = e π i τ 12 ϑ 3 ( π ( τ + 1 ) 2 , e 3 π i τ ) . \eta(\tau)=e^{\tfrac{\pi i\tau}{12}}\vartheta_{3}(\tfrac{\pi(\tau+1)}{2},e^{3% \pi i\tau}).
  19. ϕ ( q ) = n = 1 ( 1 - q n ) , \phi(q)=\prod_{n=1}^{\infty}\left(1-q^{n}\right),
  20. η \eta\,
  21. ϕ ( q ) = q - 1 / 24 η ( τ ) \phi(q)=q^{-1/24}\eta(\tau)\,
  22. ϕ ( q ) = n = - ( - 1 ) n q ( 3 n 2 - n ) / 2 . \phi(q)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(3n^{2}-n)/2}.
  23. q 1 / 24 q^{1/24}
  24. η ( i ) = Γ ( 1 4 ) 2 π 3 / 4 , \eta(i)=\frac{\Gamma\left(\frac{1}{4}\right)}{2\pi^{3/4}},
  25. η ( 1 2 i ) = Γ ( 1 4 ) 2 7 / 8 π 3 / 4 , \eta\left(\tfrac{1}{2}i\right)=\frac{\Gamma\left(\frac{1}{4}\right)}{2^{7/8}% \pi^{3/4}},
  26. η ( 2 i ) = Γ ( 1 4 ) 2 11 / 8 π 3 / 4 , \eta(2i)=\frac{\Gamma\left(\frac{1}{4}\right)}{2^{{11}/8}\pi^{3/4}},
  27. η ( 4 i ) = - 1 + 2 4 Γ ( 1 4 ) 2 29 / 16 π 3 / 4 . \eta(4i)=\frac{\sqrt[4]{-1+\sqrt{2}}\;\Gamma\left(\frac{1}{4}\right)}{2^{{29}/% 16}\pi^{3/4}}.
  28. j ( τ ) = ( ( η ( τ ) η ( 2 τ ) ) 8 + 2 8 ( η ( 2 τ ) η ( τ ) ) 16 ) 3 j(\tau)=\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{8}+2^{8}\big(\tfrac{% \eta(2\tau)}{\eta(\tau)}\big)^{16}\Big)^{3}
  29. j 2 A ( τ ) = ( ( η ( τ ) η ( 2 τ ) ) 12 + 2 6 ( η ( 2 τ ) η ( τ ) ) 12 ) 2 j_{2A}(\tau)=\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{12}+2^{6}\big(% \tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{12}\Big)^{2}
  30. j 3 A ( τ ) = ( ( η ( τ ) η ( 3 τ ) ) 6 + 3 3 ( η ( 3 τ ) η ( τ ) ) 6 ) 2 j_{3A}(\tau)=\Big(\big(\tfrac{\eta(\tau)}{\eta(3\tau)}\big)^{6}+3^{3}\big(% \tfrac{\eta(3\tau)}{\eta(\tau)}\big)^{6}\Big)^{2}
  31. j ( 1 + - 163 2 ) = - 640320 3 , e π 163 640320 3 + 743.99999999999925 j\Big(\tfrac{1+\sqrt{-163}}{2}\Big)=-640320^{3},\quad e^{\pi\sqrt{163}}\approx 6% 40320^{3}+743.99999999999925\dots
  32. j 2 A ( - 58 2 ) = 396 4 , e π 58 396 4 - 104.00000017 j_{2A}\Big(\tfrac{\sqrt{-58}}{2}\Big)=396^{4},\qquad\quad e^{\pi\sqrt{58}}% \approx 396^{4}-104.00000017\dots
  33. j 3 A ( 1 + - 89 / 3 2 ) = - 300 3 , e π 89 / 3 300 3 + 41.999971 j_{3A}\Big(\tfrac{1+\sqrt{-89/3}}{2}\Big)=-300^{3},\quad e^{\pi\sqrt{89/3}}% \approx 300^{3}+41.999971\dots

Deduction_theorem.html

  1. Δ { A } \Delta\cup\{A\}
  2. Δ . \Delta\,.
  3. Δ { A } B \Delta\cup\{A\}\vdash B
  4. Δ A B . \Delta\vdash A\to B\,.
  5. Δ \Delta\,
  6. { A } B \{A\}\vdash B
  7. A B . \vdash A\to B.
  8. E 1 , E 2 , , E n - 1 , E n S , E_{1},E_{2},...,E_{n-1},E_{n}\vdash S,
  9. E 1 , E 2 , , E n - 1 , E n , H H . E_{1},E_{2},...,E_{n-1},E_{n},H\vdash H.
  10. E F \vdash E\rightarrow F
  11. E F E\vdash F
  12. ¬ F ¬ E \neg F\vdash\neg E

Deficient_number.html

  1. [ n , n + ( log n ) 2 ] [n,n+(\log n)^{2}]

Definite_description.html

  1. ι x \scriptstyle\iota x
  2. ι \scriptstyle\iota
  3. ι x ( ϕ x ) \iota x(\phi x)
  4. x \scriptstyle x
  5. ϕ x \scriptstyle\phi x
  6. ψ ( ι x ( ϕ x ) ) \psi(\iota x(\phi x))
  7. ϕ \scriptstyle\phi
  8. ψ \scriptstyle\psi
  9. x y ( ϕ ( y ) y = x and ψ ( y ) ) \exists x\forall y(\phi(y)\iff y=x\and\psi(y))

Degenerate_conic.html

  1. x 2 - y 2 = 0 x^{2}-y^{2}=0
  2. ( x - y ) ( x + y ) = 0 (x-y)(x+y)=0
  3. x 2 + y 2 = 0 x^{2}+y^{2}=0
  4. ( 0 , 0 ) (0,0)
  5. x 2 + y 2 = - 1 x^{2}+y^{2}=-1
  6. x 2 + y 2 = 0 x^{2}+y^{2}=0
  7. ( x + i y ) ( x - i y ) = 0 (x+iy)(x-iy)=0
  8. x 2 - y 2 = 0 ( x + y ) ( x - y ) = 0 x^{2}-y^{2}=0\Leftrightarrow(x+y)(x-y)=0
  9. x 2 - 1 = 0 ( x + 1 ) ( x - 1 ) = 0 x^{2}-1=0\Leftrightarrow(x+1)(x-1)=0
  10. x 2 = 0 x^{2}=0
  11. x 2 + y 2 = 0. x^{2}+y^{2}=0.
  12. x 2 + y 2 = - 1 x^{2}+y^{2}=-1
  13. A x 2 + 2 B x y + C y 2 + 2 D x + 2 E y + F Ax^{2}+2Bxy+Cy^{2}+2Dx+2Ey+F
  14. [ A B B C ] , \begin{bmatrix}A&B\\ B&C\\ \end{bmatrix},
  15. ( x , y ) (x,y)
  16. ( x , y , z ) (x,y,z)
  17. A x 2 + 2 B x y + C y 2 + 2 D x z + 2 E y z + F z 2 ; Ax^{2}+2Bxy+Cy^{2}+2Dxz+2Eyz+Fz^{2};
  18. [ A B D B C E D E F ] . \begin{bmatrix}A&B&D\\ B&C&E\\ D&E&F\\ \end{bmatrix}.
  19. x 2 + a y 2 = 1 x^{2}+ay^{2}=1
  20. a 0 a\neq 0
  21. a = 0 ; a=0;
  22. a > 0 , a>0,
  23. a = 0 , a=0,
  24. a < 0 a<0
  25. 1 / | a | , 1/\sqrt{|a|},
  26. a = 0. a=0.
  27. ( 4 2 , 2 ) = 3 \textstyle{{\left({{4}\atop{2,2}}\right)}=3}
  28. ( ± 1 , ± 1 ) , (\pm 1,\pm 1),
  29. ( 1 + a ) x 2 + ( 1 - a ) y 2 = 2 , (1+a)x^{2}+(1-a)y^{2}=2,
  30. a > 1 : a>1:
  31. a = 1 : a=1:
  32. x = - 1 , x = 1 ; x=-1,x=1;
  33. 0 < a < 1 : 0<a<1:
  34. a = 0 : a=0:
  35. 2 \sqrt{2}
  36. - 1 < a < 0 : -1<a<0:
  37. a = - 1 : a=-1:
  38. y = - 1 , y = 1 ; y=-1,y=1;
  39. a < - 1 : a<-1:
  40. a = : a=\infty:
  41. y = x , y = - x ; y=x,y=-x;
  42. a a
  43. a a\to\infty
  44. x 2 - y 2 = 0 x^{2}-y^{2}=0
  45. a > 1 , a>1,
  46. x 2 - y 2 = a 2 , x^{2}-y^{2}=a^{2},
  47. x 2 - a 2 y 2 = 1 , x^{2}-a^{2}y^{2}=1,
  48. x 2 - a 2 y 2 = a 2 , x^{2}-a^{2}y^{2}=a^{2},
  49. x 2 - a y - 1 = 0 x^{2}-ay-1=0
  50. x 2 - a y = 0 , x^{2}-ay=0,
  51. x 2 + a 2 y 2 - 1 = 0 x^{2}+a^{2}y^{2}-1=0
  52. x 2 + a 2 y 2 - a 2 = 0 , x^{2}+a^{2}y^{2}-a^{2}=0,
  53. x 2 - a y 2 - 1 = 0 , x^{2}-ay^{2}-1=0,
  54. x 2 - a y 2 = 0. x^{2}-ay^{2}=0.
  55. x 2 - a 2 = 0 , x^{2}-a^{2}=0,
  56. x 2 a 2 + y 2 b 2 = 1 \tfrac{x^{2}}{a^{2}}+\tfrac{y^{2}}{b^{2}}=1
  57. a x 2 + ( 1 - a ) y 2 = 1 , ax^{2}+(1-a)y^{2}=1,
  58. x 2 = 1 x^{2}=1
  59. y 2 = 1 , y^{2}=1,
  60. 0 , 1 , . 0,1,\infty.

Degree_(music).html

  1. 1 ^ , 2 ^ , 3 ^ \hat{1},\hat{2},\hat{3}...

Density_estimation.html

  1. p ( diabetes = 1 | glu ) = p ( glu | db. = 1 ) p ( db. = 1 ) p ( glu | db. = 1 ) p ( db. = 1 ) + p ( glu | db. = 0 ) p ( db. = 0 ) p(\mbox{diabetes}~{}=1|\mbox{glu}~{})=\frac{p(\mbox{glu}~{}|\mbox{db.}~{}=1)\,% p(\mbox{db.}~{}=1)}{p(\mbox{glu}~{}|\mbox{db.}~{}=1)\,p(\mbox{db.}~{}=1)+p(% \mbox{glu}~{}|\mbox{db.}~{}=0)\,p(\mbox{db.}~{}=0)}

Density_matrix_renormalization_group.html

  1. s 1 s N Tr ( A s 1 A s N ) | s 1 s N \sum_{s_{1}\cdots s_{N}}\operatorname{Tr}(A^{s_{1}}\cdots A^{s_{N}})|s_{1}% \cdots s_{N}\rangle
  2. s 1 s N s_{1}\cdots s_{N}

Density_of_air.html

  1. ρ = p R specific T \rho=\frac{p}{R_{\rm specific}T}
  2. ρ = \rho=
  3. p = p=
  4. T = T=
  5. R specific = R_{\rm specific}=
  6. ρ humid air = p d R d T + p v R v T = p d M d + p v M v R T \rho_{\,\mathrm{humid~{}air}}=\frac{p_{d}}{R_{d}T}+\frac{p_{v}}{R_{v}T}=\frac{% p_{d}M_{d}+p_{v}M_{v}}{RT}\,
  7. ρ humid air = \rho_{\,\mathrm{humid~{}air}}=
  8. p d = p_{d}=
  9. R d = R_{d}=
  10. T = T=
  11. p v = p_{v}=
  12. R v = R_{v}=
  13. M d = M_{d}=
  14. M v = M_{v}=
  15. R = R=
  16. p v = ϕ p sat p_{v}=\phi p_{\mathrm{sat}}\,
  17. p v = p_{v}=
  18. ϕ = \phi=
  19. p sat = p_{\mathrm{sat}}=
  20. p sat = 6.1078 × 10 7.5 T T + 237.3 p_{\mathrm{sat}}=6.1078\times 10^{\frac{7.5T}{T+237.3}}
  21. T = T=
  22. p d p_{d}
  23. p d = p - p v p_{d}=p-p_{v}\,
  24. p p
  25. ρ {}_{\rho}
  26. p 0 = p_{0}=
  27. T 0 = T_{0}=
  28. g = g=
  29. L = L=
  30. R = R=
  31. M = M=
  32. h h
  33. T = T 0 - L h T=T_{0}-Lh\,
  34. h h
  35. p = p 0 ( 1 - L h T 0 ) g M R L p=p_{0}\left(1-\frac{Lh}{T_{0}}\right)^{\frac{gM}{RL}}
  36. ρ = p M R T \rho=\frac{pM}{RT}\,
  37. M = M=
  38. R = R=
  39. T = T=
  40. p = p=

Density_of_states.html

  1. E g E_{g}
  2. Ω n , k \Omega_{n,k}
  3. Ω n ( k ) = c n k n \Omega_{n}(k)=c_{n}k^{n}
  4. c 1 = 2 , c 2 = π , c 3 = 4 π 3 c_{1}=2,\,\text{ }c_{2}=\pi,\,\text{ }c_{3}=\frac{4\pi}{3}
  5. Ω n , k \Omega_{n,k}
  6. N n ( k ) = d Ω n ( k ) d k = n c n k ( n - 1 ) N_{n}(k)=\frac{d\Omega_{n}(k)}{dk}=n\,\text{ }c_{n}\,\text{ }k^{(n-1)}
  7. N 1 ( k ) = 2 N_{1}(k)=2
  8. N 2 ( k ) = 2 π k N_{2}(k)=2\pi k
  9. N 3 ( k ) = 4 π k 2 N_{3}(k)=4\pi k^{2}
  10. k = 2 π λ k=\frac{2\pi}{\lambda}
  11. s / V k s/V_{k}
  12. s = 1 s=1
  13. E E
  14. [ E , E + d E ] [E,E+dE]
  15. D n ( E ) = d Ω n ( E ) d E D_{n}\left(E\right)=\frac{d\Omega_{n}(E)}{dE}
  16. E ( k ) E(k)
  17. E ( k ) E(k)
  18. Ω n ( k ) \Omega_{n}(k)
  19. Ω n ( E ) \Omega_{n}(E)
  20. E = E 0 + ( k ) 2 2 m , E=E_{0}+\frac{(\hbar k)^{2}}{2m}\ ,
  21. E = 2 ω 0 | sin ( k a / 2 ) | E=2\hbar\omega_{0}|\sin(ka/2)|
  22. ω 0 = k F / m \omega_{0}=\sqrt{k_{F}/m}
  23. m m
  24. k F k_{F}
  25. a a
  26. k π / a k\ll\pi/a
  27. E = ω 0 k a E=\hbar\omega_{0}ka
  28. k π / a k\approx\pi/a
  29. E = 2 ω 0 | cos ( π / 2 - k a / 2 ) | E=2\hbar\omega_{0}|\cos(\pi/2-ka/2)|
  30. q = k - π / a q=k-\pi/a
  31. q q
  32. E = 2 ω 0 [ 1 - ( q a / 2 ) 2 ] E=2\hbar\omega_{0}[1-(qa/2)^{2}]
  33. E = E 0 + c k k p E=E_{0}+c_{k}k^{p}
  34. k = ( E - E 0 c k ) 1 / p , k=\left(\frac{E-E_{0}}{c_{k}}\right)^{1/p}\ ,
  35. Ω n ( k ) = c n k n \Omega_{n}(k)=c_{n}k^{n}
  36. Ω n ( E ) = c n c k n / p ( E - E 0 ) n / p , \Omega_{n}(E)=\frac{c_{n}}{c_{k}^{n/p}}\left(E-E_{0}\right)^{n/p}\ ,
  37. D n ( E ) = d d E Ω n ( E ) = n c n p c k n / p ( E - E 0 ) ( n / p - 1 ) D_{n}\left(E\right)=\frac{d}{dE}\Omega_{n}(E)=\frac{nc_{n}}{pc_{k}^{n/p}}\left% (E-E_{0}\right)^{(n/p-1)}
  38. D n ( E ) D_{n}\left(E\right)
  39. D 1 ( E ) = 1 c k ( E - E 0 ) . D_{1}\left(E\right)=\frac{1}{\sqrt{c_{k}(E-E_{0})}}\ .
  40. D 2 ( E ) = π c k D_{2}\left(E\right)=\frac{\pi}{c_{k}}
  41. D 3 ( E ) = 2 π E - E 0 c k 3 . D_{3}\left(E\right)=2\pi\sqrt{\frac{E-E_{0}}{c_{k}^{3}}}\ .
  42. E > E 0 E>E_{0}
  43. D ( E ) = 0 D(E)=0
  44. E < E 0 E<E_{0}
  45. E E
  46. E 0 E_{0}
  47. E E
  48. N ( E ) = V 2 π 2 ( 2 m 2 ) 3 / 2 E - E 0 N(E)=\frac{V}{2\pi^{2}}\left(\frac{2m}{\hbar^{2}}\right)^{3/2}\sqrt{E-E_{0}}
  49. V V
  50. N ( E - E 0 ) N(E-E_{0})
  51. D 1 ( E ) = 1 c k D_{1}\left(E\right)=\frac{1}{c_{k}}
  52. D 2 ( E ) = 2 π c k 2 ( E - E 0 ) D_{2}\left(E\right)=\frac{2\pi}{c_{k}^{2}}\left(E-E_{0}\right)
  53. D 3 ( E ) = 4 π c k 3 ( E - E 0 ) 2 D_{3}\left(E\right)=\frac{4\pi}{c_{k}^{3}}\left(E-E_{0}\right)^{2}
  54. f FD ( E ) = 1 exp ( E - μ k B T ) + 1 f_{\mathrm{FD}}(E)=\frac{1}{\exp\left(\frac{E-\mu}{k_{\mathrm{B}}T}\right)+1}
  55. μ \mu
  56. k B k_{\mathrm{B}}
  57. T T
  58. f BE ( E ) = 1 exp ( E - μ k B T ) - 1 f_{\mathrm{BE}}(E)=\frac{1}{\exp\left(\frac{E-\mu}{k_{B}T}\right)-1}
  59. U U
  60. n n
  61. C C
  62. k k
  63. g ( E ) g(E)
  64. D ( E ) D(E)
  65. U = E f ( E ) g ( E ) d E U=\int E\,f(E)\,g(E)\,dE
  66. n = f ( E ) g ( E ) d E n=\int f(E)\,g(E)\,dE
  67. C = T E f ( E ) g ( E ) d E C=\frac{\partial}{\partial T}\int E\,f(E)\,g(E)\,dE
  68. k = 1 d T E f ( E ) g ( E ) ν ( E ) Λ ( E ) d E k=\frac{1}{d}\frac{\partial}{\partial T}\int Ef(E)\,g(E)\,\nu(E)\,\Lambda(E)\,dE
  69. d d
  70. ν \nu
  71. Λ \Lambda
  72. g ( i ) g(i)
  73. g ( i ) g ( i ) + f g(i)\rightarrow g(i)+f
  74. f n + 1 ( 1 / 2 ) f n f_{n+1}\rightarrow(1/2)f_{n}
  75. f n < 10 - 8 f_{n}<10^{-8}

Dependency_ratio.html

  1. ( T o t a l ) D e p e n d e n c y r a t i o = ( n u m b e r o f p e o p l e a g e d 0 - 14 a n d t h o s e a g e d 65 a n d o v e r ) n u m b e r o f p e o p l e a g e d 15 - 64 × 100 (Total)\ Dependency\ ratio=\frac{(number\ of\ people\ aged\ 0-14\ and\ those\ % aged\ 65\ and\ over)}{number\ of\ people\ aged\ 15-64}\times 100
  2. C h i l d d e p e n d e n c y r a t i o = n u m b e r o f p e o p l e a g e d 0 - 14 n u m b e r o f p e o p l e a g e d 15 - 64 × 100 Child\ dependency\ ratio\ =\frac{number\ of\ people\ aged\ 0-14}{number\ of\ % people\ aged\ 15-64}\times 100
  3. A g e d d e p e n d e n c y r a t i o = n u m b e r o f p e o p l e a g e d 65 a n d o v e r n u m b e r o f p e o p l e a g e d 15 - 64 × 100 Aged\ dependency\ ratio\ =\frac{number\ of\ people\ aged\ 65\ and\ over}{% number\ of\ people\ aged\ 15-64}\times 100

Dependent_and_independent_variables.html

  1. y = f ( x ) y=f(x)
  2. z = f ( x , y ) z=f(x,y)
  3. X × Y X\times Y
  4. y i = a + b x i + e i y_{i}=a+bx_{i}+e_{i}
  5. y i y_{i}
  6. x i x_{i}
  7. e i e_{i}
  8. y i = a + b x 1 + b x 2 + + b x n + e i y_{i}=a+bx_{1}+bx_{2}+...+bx_{n}+e_{i}
  9. e i e_{i}

Deposition_(geology).html

  1. 4 3 π R 3 ρ s g = 4 3 π R 3 ρ g + 1 2 \C d ρ π R 2 w s 2 \frac{4}{3}\pi R^{3}\rho_{s}g=\frac{4}{3}\pi R^{3}\rho g+\frac{1}{2}\C_{d}\rho% \pi R^{2}w_{s}^{2}\,

Derived_functor.html

  1. 0 X I 0 I 1 I 2 0\to X\to I^{0}\to I^{1}\to I^{2}\to\cdots
  2. 0 F ( I 0 ) F ( I 1 ) F ( I 2 ) 0\to F(I^{0})\to F(I^{1})\to F(I^{2})\to\cdots
  3. P 2 P 1 P 0 X 0 \cdots\to P_{2}\to P_{1}\to P_{0}\to X\to 0
  4. 0 A B C 0 0\to A\to B\to C\to 0
  5. L 2 G ( C ) L 1 G ( A ) L 1 G ( B ) L 1 G ( C ) G ( A ) G ( B ) G ( C ) 0 \cdots\to L_{2}G(C)\to L_{1}G(A)\to L_{1}G(B)\to L_{1}G(C)\to G(A)\to G(B)\to G% (C)\to 0
  6. 0 A B C 0 0\to A\to B\to C\to 0
  7. 0 F ( C ) F ( B ) F ( A ) R 1 F ( C ) R 1 F ( B ) R 1 F ( A ) R 2 F ( C ) 0\to F(C)\to F(B)\to F(A)\to R^{1}F(C)\to R^{1}F(B)\to R^{1}F(A)\to R^{2}F(C)\to\cdots
  8. 0 A 1 f 1 B 1 g 1 C 1 0 α β γ 0 A 2 f 2 B 2 g 2 C 2 0 \begin{array}[]{ccccccccc}0&\xrightarrow{}&A_{1}&\xrightarrow{f_{1}}&B_{1}&% \xrightarrow{g_{1}}&C_{1}&\xrightarrow{}&0\\ &&\alpha\downarrow&&\beta\downarrow&&\gamma\downarrow&&\\ 0&\xrightarrow{}&A_{2}&\xrightarrow{f_{2}}&B_{2}&\xrightarrow{g_{2}}&C_{2}&% \xrightarrow{}&0\end{array}
  9. 0 A 𝑓 B 𝑔 C 0 0\xrightarrow{}A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{}0

DES-X.html

  1. DES-X ( M ) = K 2 DES ( M K 1 ) K \mbox{DES-X}~{}(M)=K_{2}\oplus\mbox{DES}~{}_{K}(M\oplus K_{1})

Descartes'_theorem.html

  1. k 4 = k 1 + k 2 + k 3 ± 2 k 1 k 2 + k 2 k 3 + k 3 k 1 . k_{4}=k_{1}+k_{2}+k_{3}\pm 2\sqrt{k_{1}k_{2}+k_{2}k_{3}+k_{3}k_{1}}.\,
  2. k 4 = k 1 + k 2 ± 2 k 1 k 2 . k_{4}=k_{1}+k_{2}\pm 2\sqrt{k_{1}k_{2}}.
  3. k 4 = k 1 . \displaystyle k_{4}=k_{1}.
  4. ( v 2 + x 2 + y 2 + z 2 ) 2 = 2 ( v 4 + x 4 + y 4 + z 4 ) (v^{2}+x^{2}+y^{2}+z^{2})^{2}=2\,(v^{4}+x^{4}+y^{4}+z^{4})
  5. ( 2 v x ) 2 + ( 2 y z ) 2 = ( v 2 + x 2 - y 2 - z 2 ) 2 (2vx)^{2}+(2yz)^{2}=\,(v^{2}+x^{2}-y^{2}-z^{2})^{2}
  6. ( 2 v y ) 2 + ( 2 x z ) 2 = ( v 2 - x 2 + y 2 - z 2 ) 2 (2vy)^{2}+(2xz)^{2}=\,(v^{2}-x^{2}+y^{2}-z^{2})^{2}
  7. ( 2 v z ) 2 + ( 2 x y ) 2 = ( v 2 - x 2 - y 2 + z 2 ) 2 (2vz)^{2}+(2xy)^{2}=\,(v^{2}-x^{2}-y^{2}+z^{2})^{2}
  8. ( - v 2 + x 2 + y 2 + z 2 ) 2 = 2 ( v 4 + x 4 + y 4 + z 4 ) (-v^{2}+x^{2}+y^{2}+z^{2})^{2}=2\,(v^{4}+x^{4}+y^{4}+z^{4})
  9. [ v , x , y , z ] = [ 2 ( a b - c d ) ( a b + c d ) , ( a 2 + b 2 + c 2 + d 2 ) ( a 2 - b 2 + c 2 - d 2 ) , 2 ( a c - b d ) ( a 2 + c 2 ) , 2 ( a c - b d ) ( b 2 + d 2 ) ] [v,x,y,z]=\,[2(ab-cd)(ab+cd),(a^{2}+b^{2}+c^{2}+d^{2})(a^{2}-b^{2}+c^{2}-d^{2}% ),2(ac-bd)(a^{2}+c^{2}),2(ac-bd)(b^{2}+d^{2})]
  10. a 4 + b 4 = c 4 + d 4 a^{4}+b^{4}=\,c^{4}+d^{4}
  11. z 4 = z 1 k 1 + z 2 k 2 + z 3 k 3 ± 2 k 1 k 2 z 1 z 2 + k 2 k 3 z 2 z 3 + k 1 k 3 z 1 z 3 k 4 . z_{4}=\frac{z_{1}k_{1}+z_{2}k_{2}+z_{3}k_{3}\pm 2\sqrt{k_{1}k_{2}z_{1}z_{2}+k_% {2}k_{3}z_{2}z_{3}+k_{1}k_{3}z_{1}z_{3}}}{k_{4}}.
  12. n n
  13. ( n 1 ) (n−1)
  14. n + 2 n+2
  15. ( i = 1 n + 2 k i ) 2 = n i = 1 n + 2 k i 2 \left(\sum_{i=1}^{n+2}k_{i}\right)^{2}=n\,\sum_{i=1}^{n+2}k_{i}^{2}
  16. k < s u b > i = 0 k<sub>i=0

Descent_(mathematics).html

  1. p : Y X . p:Y\rightarrow X.
  2. X i j , X_{ij},
  3. f i j : V i V j f_{ij}:V_{i}\rightarrow V_{j}
  4. f j k f i j = f i k f_{jk}\circ f_{ij}=f_{ik}
  5. f i j = f j i - 1 f_{ij}=f^{-1}_{ji}
  6. X i j X_{ij}
  7. Y × X Y , Y\times_{X}Y,
  8. p : X X p:X^{\prime}\to X
  9. ( F = p * F , α : p 0 * F p 1 * F ) , p i : X ′′ = X × X X X (F^{\prime}=p^{*}F,\alpha:p_{0}^{*}F^{\prime}\simeq p_{1}^{*}F^{\prime}),\,p_{% i}:X^{\prime\prime}=X^{\prime}\times_{X}X^{\prime}\to X
  10. α \alpha
  11. p 02 * α = p 12 * α p 01 * α , p i j : X ′′ × X X ′′ × X X ′′ X ′′ × X X ′′ p_{02}^{*}\alpha=p_{12}^{*}\alpha\circ p_{01}^{*}\alpha,\,p_{ij}:X^{\prime% \prime}\times_{X^{\prime}}X^{\prime\prime}\times_{X^{\prime}}X^{\prime\prime}% \to X^{\prime\prime}\times_{X^{\prime}}X^{\prime\prime}
  12. F ( F , α ) F\mapsto(F^{\prime},\alpha)

Descriptive_set_theory.html

  1. \mathbb{R}
  2. 𝒩 \mathcal{N}
  3. 𝒞 \mathcal{C}
  4. I I^{\mathbb{N}}
  5. 𝒩 \mathcal{N}
  6. 𝒩 ω \mathcal{N}^{\omega}
  7. X A X\setminus A
  8. A n \bigcup A_{n}
  9. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  10. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  11. 𝚫 α 0 \mathbf{\Delta}^{0}_{\alpha}
  12. 𝚺 1 0 \mathbf{\Sigma}^{0}_{1}
  13. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  14. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  15. 𝚺 δ 0 \mathbf{\Sigma}^{0}_{\delta}
  16. 𝚷 λ ( i ) 0 \mathbf{\Pi}^{0}_{\lambda(i)}
  17. 𝚫 α 0 \mathbf{\Delta}^{0}_{\alpha}
  18. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  19. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  20. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  21. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  22. 𝚫 α + 1 0 \mathbf{\Delta}^{0}_{\alpha+1}
  23. Δ β 0 \Delta^{0}_{\beta}
  24. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  25. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  26. 𝚺 1 0 𝚺 2 0 𝚫 1 0 𝚫 2 0 𝚷 1 0 𝚷 2 0 𝚺 α 0 𝚫 α 0 𝚫 α + 1 0 𝚷 α 0 \begin{matrix}&&\mathbf{\Sigma}^{0}_{1}&&&&\mathbf{\Sigma}^{0}_{2}&&\cdots\\ &\nearrow&&\searrow&&\nearrow\\ \mathbf{\Delta}^{0}_{1}&&&&\mathbf{\Delta}^{0}_{2}&&&&\cdots\\ &\searrow&&\nearrow&&\searrow\\ &&\mathbf{\Pi}^{0}_{1}&&&&\mathbf{\Pi}^{0}_{2}&&\cdots\end{matrix}\begin{% matrix}&&\mathbf{\Sigma}^{0}_{\alpha}&&&\cdots\\ &\nearrow&&\searrow\\ \quad\mathbf{\Delta}^{0}_{\alpha}&&&&\mathbf{\Delta}^{0}_{\alpha+1}&\cdots\\ &\searrow&&\nearrow\\ &&\mathbf{\Pi}^{0}_{\alpha}&&&\cdots\end{matrix}
  27. 𝚺 1 1 \mathbf{\Sigma}^{1}_{1}
  28. 𝚷 1 1 \mathbf{\Pi}^{1}_{1}
  29. 𝚺 n + 1 1 \mathbf{\Sigma}^{1}_{n+1}
  30. 𝚷 n 1 \mathbf{\Pi}^{1}_{n}
  31. X × X X\times X
  32. 𝚷 n + 1 1 \mathbf{\Pi}^{1}_{n+1}
  33. 𝚺 n 1 \mathbf{\Sigma}^{1}_{n}
  34. X × X X\times X
  35. 𝚫 n 1 \mathbf{\Delta}^{1}_{n}
  36. 𝚷 n 1 \mathbf{\Pi}^{1}_{n}
  37. 𝚺 n 1 \mathbf{\Sigma}^{1}_{n}
  38. 𝚫 n 1 \mathbf{\Delta}^{1}_{n}
  39. 𝚺 n + 1 1 \mathbf{\Sigma}^{1}_{n+1}
  40. 𝚷 n + 1 1 . \mathbf{\Pi}^{1}_{n+1}.
  41. X × X X\times X

Determination_of_the_day_of_the_week.html

  1. ( d + m + y + y 4 + c ) mod 7 (d+m+y+\left\lfloor\frac{y}{4}\right\rfloor+c)\bmod 7
  2. A A
  3. R ( 1 + 5 R ( A - 1 , 4 ) + 4 R ( A - 1 , 100 ) + 6 R ( A - 1 , 400 ) , 7 ) R(1+5R(A-1,4)+4R(A-1,100)+6R(A-1,400),7)
  4. R ( y , m ) R(y,m)
  5. y y
  6. m m
  7. y y
  8. m m
  9. w = d + [ 2.6 m - 0.2 ] + 5 R ( Y , 4 ) + 4 R ( Y , 100 ) + 6 R ( Y , 400 ) mod 7. w=d+[2.6m-0.2]+5R(Y,4)+4R(Y,100)+6R(Y,400)\bmod 7.
  10. w = d + [ 2.6 m - 0.2 ] + 5 R ( y , 4 ) + 3 R ( y , 7 ) + 5 R ( c , 4 ) mod 7. w=d+[2.6m-0.2]+5R(y,4)+3R(y,7)+5R(c,4)\bmod 7.
  11. w = d + [ 2.6 m - 2.2 ] + 5 R ( Y , 4 ) + 3 R ( Y , 7 ) mod 7 w=d+[2.6m-2.2]+5R(Y,4)+3R(Y,7)\bmod 7
  12. w = d + [ 2.6 m - 2.2 ] + 5 R ( y , 4 ) + 3 R ( y , 7 ) + 6 R ( c , 7 ) mod 7 w=d+[2.6m-2.2]+5R(y,4)+3R(y,7)+6R(c,7)\bmod 7
  13. w = ( d + 2.6 m - 0.2 + y + y 4 + c 4 - 2 c ) mod 7 , w=(d+\lfloor 2.6m-0.2\rfloor+y+\left\lfloor\frac{y}{4}\right\rfloor+\left% \lfloor\frac{c}{4}\right\rfloor-2c)\bmod 7,
  14. w = ( 1 + 2.6 11 - 0.2 + ( 0 - 1 ) + 0 - 1 4 + 20 4 - 2 20 ) mod 7 w=(1+\lfloor 2.6\cdot 11-0.2\rfloor+(0-1)+\left\lfloor\frac{0-1}{4}\right% \rfloor+\left\lfloor\frac{20}{4}\right\rfloor-2\cdot 20)\bmod 7
  15. = ( 1 + 28 - 1 - 1 + 5 - 40 ) mod 7 = 6 = Saturday =(1+28-1-1+5-40)\bmod 7=6=\,\text{Saturday}
  16. w = ( 1 + 2.6 11 - 0.2 + ( 100 - 1 ) + 100 - 1 4 + 20 - 1 4 - 2 ( 20 - 1 ) mod 7 w=(1+\lfloor 2.6\cdot 11-0.2\rfloor+(100-1)+\left\lfloor\frac{100-1}{4}\right% \rfloor+\left\lfloor\frac{20-1}{4}\right\rfloor-2\cdot(20-1)\bmod 7
  17. = ( 1 + 28 + 99 + 24 + 4 - 38 ) mod 7 = 6 = Saturday =(1+28+99+24+4-38)\bmod 7=6=\,\text{Saturday}
  18. w = d + m + y + c mod 7 w=d+m+y+c\bmod 7
  19. w = 1 + 0 + 5 + 0 mod 7 = 6 = Saturday w=1+0+5+0\bmod 7=6=\,\text{Saturday}
  20. w = 1 + 0 + 4 + 1 mod 7 = 6 = Saturday w=1+0+4+1\bmod 7=6=\,\text{Saturday}
  21. w = d + m + c + y mod 7 , w=d+m+c+y\bmod 7,
  22. w w
  23. d d
  24. m m
  25. c c
  26. y y
  27. m m
  28. c c
  29. c c
  30. y = ( s 4 + s ) mod 7 y=(\lfloor\frac{s}{4}\rfloor+s)\bmod 7
  31. y = ( 87 4 + 87 ) mod 7 y=(\lfloor\frac{87}{4}\rfloor+87)\bmod 7
  32. y = ( 21.75 + 87 ) mod 7 y=(21.75+87)\bmod 7
  33. y = 108.75 mod 7 y=108.75\bmod 7
  34. y = 3.75 y=3.75
  35. y = 3 y=3
  36. c = y 100 and g = y - 100 c , c=\left\lfloor\frac{y}{100}\right\rfloor\quad\,\text{and}\quad g=y-100c,
  37. c = y - 1 100 and g = y - 1 - 100 c . c=\left\lfloor\frac{y-1}{100}\right\rfloor\quad\,\text{and}\quad g=y-1-100c.
  38. w = d + e + f + g + g 4 mod 7 , w=d+e+f+g+\left\lfloor\frac{g}{4}\right\rfloor\bmod 7,
  39. e e
  40. m m
  41. e e
  42. f f
  43. c m o d 4 cmod4
  44. f f
  45. c m o d 7 cmod7
  46. f f
  47. w = ( d + 13 ( m + 1 ) 5 + y + y 4 + c 4 - 2 c ) mod 7 , w=\left(d+\left\lfloor\frac{13(m+1)}{5}\right\rfloor+y+\left\lfloor\frac{y}{4}% \right\rfloor+\left\lfloor\frac{c}{4}\right\rfloor-2c\right)\bmod 7,
  48. ( d + [ ( m + 1 ) 2.6 ] + y + [ y / 4 ] + [ c / 4 ] - 2 c ) mod 7 - ( d + [ 2.6 m - 0.2 ] + y + [ y / 4 ] + [ c / 4 ] - 2 c ) mod 7 (d+[(m+1)2.6]+y+[y/4]+[c/4]-2c)\bmod 7-(d+[2.6m-0.2]+y+[y/4]+[c/4]-2c)\bmod 7
  49. = ( [ ( m + 2 + 1 ) 2.6 - ( 2.6 m - 0.2 ) ] ) mod 7 =([(m+2+1)2.6-(2.6m-0.2)])\bmod 7
  50. = ( [ 2.6 m + 7.8 - 2.6 m + 0.2 ] ) mod 7 =([2.6m+7.8-2.6m+0.2])\bmod 7
  51. = 8 mod 7 = 1 =8\bmod 7=1

Deutsch–Jozsa_algorithm.html

  1. f : { 0 , 1 } n { 0 , 1 } f:\{0,1\}^{n}\rightarrow\{0,1\}
  2. f f
  3. 2 n - 1 + 1 2^{n-1}+1
  4. f f
  5. f f
  6. k k
  7. ϵ 1 / 2 k - 1 \epsilon\leq 1/2^{k-1}
  8. k = 2 n - 1 + 1 k=2^{n-1}+1
  9. f f
  10. f : { 0 , 1 } { 0 , 1 } f:\{0,1\}\rightarrow\{0,1\}
  11. n n
  12. f f
  13. | 0 n | 1 |0\rangle^{\otimes n}|1\rangle
  14. | 0 |0\rangle
  15. | 1 |1\rangle
  16. 1 2 n + 1 x = 0 2 n - 1 | x ( | 0 - | 1 ) \frac{1}{\sqrt{2^{n+1}}}\sum_{x=0}^{2^{n}-1}|x\rangle(|0\rangle-|1\rangle)
  17. f f
  18. | x | y |x\rangle|y\rangle
  19. | x | y f ( x ) |x\rangle|y\oplus f(x)\rangle
  20. \oplus
  21. 1 2 n + 1 x = 0 2 n - 1 | x ( | f ( x ) - | 1 f ( x ) ) \frac{1}{\sqrt{2^{n+1}}}\sum_{x=0}^{2^{n}-1}|x\rangle(|f(x)\rangle-|1\oplus f(% x)\rangle)
  22. x x
  23. f ( x ) f(x)
  24. 0
  25. 1 1
  26. 1 2 n + 1 x = 0 2 n - 1 ( - 1 ) f ( x ) | x ( | 0 - | 1 ) \frac{1}{\sqrt{2^{n+1}}}\sum_{x=0}^{2^{n}-1}(-1)^{f(x)}|x\rangle(|0\rangle-|1\rangle)
  27. 1 2 n x = 0 2 n - 1 ( - 1 ) f ( x ) [ y = 0 2 n - 1 ( - 1 ) x y | y ] = 1 2 n y = 0 2 n - 1 [ x = 0 2 n - 1 ( - 1 ) f ( x ) ( - 1 ) x y ] | y \frac{1}{2^{n}}\sum_{x=0}^{2^{n}-1}(-1)^{f(x)}\left[\sum_{y=0}^{2^{n}-1}(-1)^{% x\cdot y}|y\rangle\right]=\frac{1}{2^{n}}\sum_{y=0}^{2^{n}-1}\left[\sum_{x=0}^% {2^{n}-1}(-1)^{f(x)}(-1)^{x\cdot y}\right]|y\rangle
  28. x y = x 0 y 0 x 1 y 1 x n - 1 y n - 1 x\cdot y=x_{0}y_{0}\oplus x_{1}y_{1}\oplus\cdots\oplus x_{n-1}y_{n-1}
  29. | 0 n |0\rangle^{\otimes n}
  30. | 1 2 n x = 0 2 n - 1 ( - 1 ) f ( x ) | 2 \bigg|\frac{1}{2^{n}}\sum_{x=0}^{2^{n}-1}(-1)^{f(x)}\bigg|^{2}
  31. f ( x ) f(x)
  32. f ( x ) f(x)
  33. f ( 0 ) = f ( 1 ) f(0)=f(1)
  34. f ( 0 ) f ( 1 ) f(0)\oplus f(1)
  35. \oplus
  36. f f
  37. f f
  38. | 0 | 1 |0\rangle|1\rangle
  39. 1 2 ( | 0 + | 1 ) ( | 0 - | 1 ) . \frac{1}{2}(|0\rangle+|1\rangle)(|0\rangle-|1\rangle).
  40. f f
  41. | x | y |x\rangle|y\rangle
  42. | x | f ( x ) y |x\rangle|f(x)\oplus y\rangle
  43. 1 2 ( | 0 ( | f ( 0 ) 0 - | f ( 0 ) 1 ) + | 1 ( | f ( 1 ) 0 - | f ( 1 ) 1 ) ) \frac{1}{2}(|0\rangle(|f(0)\oplus 0\rangle-|f(0)\oplus 1\rangle)+|1\rangle(|f(% 1)\oplus 0\rangle-|f(1)\oplus 1\rangle))
  44. = 1 2 ( ( - 1 ) f ( 0 ) | 0 ( | 0 - | 1 ) + ( - 1 ) f ( 1 ) | 1 ( | 0 - | 1 ) ) =\frac{1}{2}((-1)^{f(0)}|0\rangle(|0\rangle-|1\rangle)+(-1)^{f(1)}|1\rangle(|0% \rangle-|1\rangle))
  45. = ( - 1 ) f ( 0 ) 1 2 ( | 0 + ( - 1 ) f ( 0 ) f ( 1 ) | 1 ) ( | 0 - | 1 ) . =(-1)^{f(0)}\frac{1}{2}\left(|0\rangle+(-1)^{f(0)\oplus f(1)}|1\rangle\right)(% |0\rangle-|1\rangle).
  46. 1 2 ( | 0 + ( - 1 ) f ( 0 ) f ( 1 ) | 1 ) . \frac{1}{\sqrt{2}}(|0\rangle+(-1)^{f(0)\oplus f(1)}|1\rangle).
  47. 1 2 ( | 0 + | 1 + ( - 1 ) f ( 0 ) f ( 1 ) | 0 - ( - 1 ) f ( 0 ) f ( 1 ) | 1 ) \frac{1}{2}(|0\rangle+|1\rangle+(-1)^{f(0)\oplus f(1)}|0\rangle-(-1)^{f(0)% \oplus f(1)}|1\rangle)
  48. = 1 2 ( ( 1 + ( - 1 ) f ( 0 ) f ( 1 ) ) | 0 + ( 1 - ( - 1 ) f ( 0 ) f ( 1 ) ) | 1 ) . =\frac{1}{2}((1+(-1)^{f(0)\oplus f(1)})|0\rangle+(1-(-1)^{f(0)\oplus f(1)})|1% \rangle).
  49. f ( 0 ) f ( 1 ) = 0 f(0)\oplus f(1)=0
  50. f ( 0 ) f ( 1 ) = 1 f(0)\oplus f(1)=1
  51. f ( x ) f(x)

Deviance_(statistics).html

  1. D ( y ) = - 2 ( log ( p ( y θ ^ 0 ) ) - log ( p ( y θ ^ s ) ) ) . D(y)=-2\Big(\log\big(p(y\mid\hat{\theta}_{0})\big)-\log\big(p(y\mid\hat{\theta% }_{s})\big)\Big).\,
  2. θ ^ 0 \hat{\theta}_{0}
  3. θ ^ s \hat{\theta}_{s}
  4. - 2 log ( p ( y θ ^ 0 ) ) -2\log\big(p(y\mid\hat{\theta}_{0})\big)
  5. - 2 log ( p ( y θ ^ 0 ) ) -2\log\big(p(y\mid\hat{\theta}_{0})\big)

Diagonal.html

  1. n 2 - 3 n 2 \frac{n^{2}-3n}{2}\,
  2. n ( n - 3 ) 2 \frac{n(n-3)}{2}\,
  3. A A
  4. i i
  5. j j
  6. A i j A_{ij}
  7. i = j i=j
  8. ( 1 0 0 0 1 0 0 0 1 ) \begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix}
  9. A i j A_{ij}
  10. j = i j=i
  11. j = i + 1 j=i+1
  12. ( 0 2 0 0 0 3 0 0 0 ) \begin{pmatrix}0&2&0\\ 0&0&3\\ 0&0&0\end{pmatrix}
  13. A i j A_{ij}
  14. j = i - 1 j=i-1
  15. k k
  16. k = 0 k=0
  17. k = 1 k=1
  18. k = - 1 k=-1
  19. k k
  20. A i j A_{ij}
  21. j = i + k j=i+k

Dialetheism.html

  1. p ¬ p p\wedge\neg p
  2. p p
  3. ¬ p \neg p
  4. q q
  5. p p
  6. p q p\vee q
  7. p q p\vee q
  8. ¬ p \neg p
  9. q q

Dickson's_lemma.html

  1. n n
  2. K K
  3. S = { ( x , y ) x y K } S=\{(x,y)\mid xy\geq K\}
  4. K K
  5. S S
  6. ( x , K / x ) (x,K/x)
  7. x x
  8. S S
  9. ( x , K / x ) (x,K/x)
  10. ( x , y ) (x,y)
  11. x K x\leq K
  12. y K y\leq K
  13. S S
  14. K 2 K^{2}
  15. \mathbb{N}
  16. n \mathbb{N}^{n}
  17. n n
  18. ( a 1 , a 2 , , a n ) ( b 1 , b 2 , b n ) (a_{1},a_{2},\dots,a_{n})\leq(b_{1},b_{2},\dots b_{n})
  19. i i
  20. a i b i a_{i}\leq b_{i}
  21. ( a 1 , a 2 , , a n ) (a_{1},a_{2},\dots,a_{n})
  22. S S
  23. n \mathbb{N}^{n}
  24. S S
  25. n n
  26. ( a 1 , a 2 , , a n ) (a_{1},a_{2},\dots,a_{n})
  27. ( b 1 , b 2 , b n ) (b_{1},b_{2},\dots b_{n})
  28. i i
  29. a i b i a_{i}\leq b_{i}
  30. ( n , ) (\mathbb{N}^{n},\leq)
  31. S S
  32. n \mathbb{N}^{n}
  33. S S
  34. n n
  35. n n
  36. ( | P | , ) (\mathbb{N}^{|P|},\leq)
  37. ( a 1 , a 2 , , a n ) (a_{1},a_{2},\dots,a_{n})
  38. n \mathbb{N}^{n}
  39. x 1 a 1 x 2 a 2 x n a n x_{1}^{a_{1}}x_{2}^{a_{2}}\dots x_{n}^{a_{n}}
  40. n n
  41. x 1 , x 2 , x n x_{1},x_{2},\dots x_{n}
  42. x x
  43. y y
  44. K \sqrt{K}
  45. 2 K 2\sqrt{K}

Dietary_Reference_Intake.html

  1. R D A = E A R + 2 S D ( E A R ) RDA=EAR+2SD(EAR)
  2. R D A = 1.2 E A R RDA=1.2EAR

Differentiable_function.html

  1. f f
  2. f f
  3. f ( x ) = { x 2 sin ( 1 / x ) if x 0 0 if x = 0 f(x)\;=\;\begin{cases}x^{2}\sin(1/x)&\,\text{if }x\neq 0\\ 0&\,\text{if }x=0\end{cases}
  4. f ( 0 ) = lim ϵ 0 ( ϵ 2 sin ( 1 / ϵ ) - 0 ϵ ) = 0 , f^{\prime}(0)=\lim_{\epsilon\to 0}\left(\frac{\epsilon^{2}\sin(1/\epsilon)-0}{% \epsilon}\right)=0,
  5. f ( x ) = 2 x sin ( 1 / x ) - cos ( 1 / x ) f^{\prime}(x)=2x\sin(1/x)-\cos(1/x)
  6. lim 𝐡 𝟎 𝐟 ( 𝐱 𝟎 + 𝐡 ) - 𝐟 ( 𝐱 𝟎 ) - 𝐉 ( 𝐡 ) 𝐑 n 𝐡 𝐑 m = 0. \lim_{\mathbf{h}\to\mathbf{0}}\frac{\|\mathbf{f}(\mathbf{x_{0}}+\mathbf{h})-% \mathbf{f}(\mathbf{x_{0}})-\mathbf{J}\mathbf{(h)}\|_{\mathbf{R}^{n}}}{\|% \mathbf{h}\|_{\mathbf{R}^{m}}}=\mathbf{0}.
  7. 𝐉 \mathbf{J}
  8. f ( x , y ) = { x if y x 2 0 if y = x 2 f(x,y)=\begin{cases}x&\,\text{if }y\neq x^{2}\\ 0&\,\text{if }y=x^{2}\end{cases}
  9. ( 0 , 0 ) (0,0)
  10. f ( x , y ) = { y 3 / ( x 2 + y 2 ) if ( x , y ) ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) f(x,y)=\begin{cases}y^{3}/(x^{2}+y^{2})&\,\text{if }(x,y)\neq(0,0)\\ 0&\,\text{if }(x,y)=(0,0)\end{cases}
  11. ( 0 , 0 ) (0,0)

Differential_amplifier.html

  1. V in - \scriptstyle V\text{in}^{-}
  2. V in + \scriptstyle V\text{in}^{+}
  3. V out \scriptstyle V\text{out}
  4. V out = A ( V in + - V in - ) V\text{out}=A(V\text{in}^{+}-V\text{in}^{-})
  5. A \scriptstyle A
  6. V out = A d ( V in + - V in - ) V\text{out}=A\text{d}(V\text{in}^{+}-V\text{in}^{-})
  7. V in + V\text{in}^{+}
  8. V in - V\text{in}^{-}
  9. A d A\text{d}
  10. V in + V\text{in}^{+}
  11. V in - V\text{in}^{-}
  12. V out = A d ( V in + - V in - ) + A c ( V in + + V in - 2 ) V\text{out}=A\text{d}(V\text{in}^{+}-V\text{in}^{-})+A\text{c}\left(\frac{V% \text{in}^{+}+V\text{in}^{-}}{2}\right)
  13. A c A\text{c}
  14. CMRR = 10 log 10 ( A d A c ) 2 = 20 log 10 ( A d | A c | ) \mathrm{CMRR}=10\log_{10}\left(\frac{A_{\mathrm{d}}}{A_{\mathrm{c}}}\right)^{2% }=20\log_{10}\left(\frac{A_{\mathrm{d}}}{|A_{\mathrm{c}}|}\right)
  15. A c A\text{c}
  16. R e R_{\,\text{e}}
  17. A c A_{\,\text{c}}
  18. V + = V in + * R / / + / R i + - I b + * R / / + ; R + = R / / + = R i + / / R f + V^{+\prime}=V^{+}_{\,\text{in}}*R^{+}_{/\!\!/}/R^{+}_{\,\text{i}}-I^{+}_{\,% \text{b}}*R^{+}_{/\!\!/};\quad R^{+\prime}=R^{+}_{/\!\!/}=R^{+}_{\,\text{i}}{/% \!\!/}R^{+}_{\,\text{f}}
  19. V - = V in - * R / / - / R i - + V out * R / / - / R f - - I b - * R / / - ; R - = R / / - = R i - / / R f - . V^{-\prime}=V^{-}_{\,\text{in}}*R^{-}_{/\!\!/}/R^{-}_{\,\text{i}}+V_{\,\text{% out}}*R^{-}_{/\!\!/}/R^{-}_{\,\text{f}}-I^{-}_{\,\text{b}}*R^{-}_{/\!\!/};% \quad R^{-\prime}=R^{-}_{/\!\!/}=R^{-}_{\,\text{i}}{/\!\!/}R^{-}_{\,\text{f}}.
  20. V out = A ol * 2 R d V + - V - 2 R / / + 2 R d = ( V + - V - ) * A ol R / / / ( R / / / / R d ) V_{\,\text{out}}=A_{\,\text{ol}}*2R_{\,\text{d}}\frac{V^{+\prime}-V^{-\prime}}% {2R_{/\!\!/}+2R_{\,\text{ d }}}=(V^{+\prime}-V^{-\prime})*A_{\,\text{ol}}R_{/% \!\!/}/(R_{/\!\!/}/\!\!/R_{\,\text{d}})
  21. R i + = R i - and R f + = R f - R^{+}_{\,\text{i}}=R^{-}_{\,\text{i}}\quad\,\text{and}\quad R^{+}_{\,\text{f}}% =R^{-}_{\,\text{f}}
  22. V in + - V in - - R i * I b Δ = V out [ R i R f + 1 A ol * R i R i / / R f / / R d ] V^{+}_{\,\text{in}}-V^{-}_{\,\text{in}}-R_{\,\text{i}}*I^{\Delta}_{\,\text{b}}% =V_{\,\text{out}}\left[\frac{R_{\,\text{i}}}{R_{\,\text{f}}}+\frac{1}{A_{\,% \text{ol}}*\frac{R_{\,\text{i}}}{R_{\,\text{i}}/\!\!/R_{\,\text{f}}/\!\!/R_{\,% \text{d}}}}\right]

Differential_centrifugation.html

  1. D = 18 η ln ( R f / R i ) ( ρ p - ρ f ) ω 2 t D=\sqrt{\frac{18\eta\,\ln(R_{f}/R_{i})}{(\rho_{p}-\rho_{f})\omega^{2}t}}

Differential_geometry_of_curves.html

  1. γ : I n \gamma\colon I\to{\mathbb{R}}^{n}
  2. C ω C^{\omega}
  3. γ : I n \gamma\colon I\rightarrow\mathbb{R}^{n}
  4. { γ ( t ) , γ ′′ ( t ) , , γ ( m ) ( t ) } , m k \{\gamma^{\prime}(t),\gamma^{\prime\prime}(t),...,\gamma^{(m)}(t)\}\mbox{, }m\leq k
  5. γ ( t ) 0 \gamma^{\prime}(t)\neq 0
  6. t I t\in I
  7. γ 𝟏 : I 1 R n \mathbf{\gamma_{1}}:I_{1}\to R^{n}
  8. γ 𝟐 : I 2 R n \mathbf{\gamma_{2}}:I_{2}\to R^{n}
  9. ϕ : I 1 I 2 \phi:I_{1}\to I_{2}
  10. ϕ ( t ) 0 ( t I 1 ) \phi^{\prime}(t)\neq 0\qquad(t\in I_{1})
  11. γ 𝟐 ( ϕ ( t ) ) = γ 𝟏 ( t ) ( t I 1 ) \mathbf{\gamma_{2}}(\phi(t))=\mathbf{\gamma_{1}}(t)\qquad(t\in I_{1})
  12. l = a b | γ ( t ) | d t . l=\int_{a}^{b}|\mathbf{\gamma}\,^{\prime}(t)|dt.
  13. s ( t ) = t 0 t | γ ( x ) | d x . s(t)=\int_{t_{0}}^{t}|\mathbf{\gamma}\,^{\prime}(x)|dx.
  14. γ ¯ ( s ) = γ ( t ( s ) ) \bar{\mathbf{\gamma}}(s)=\gamma(t(s))
  15. γ ¯ \bar{\gamma}
  16. | γ ¯ ( s ( t ) ) | = 1 ( t I ) . |\bar{\mathbf{\gamma}}\,^{\prime}(s(t))|=1\qquad(\forall t\in I).
  17. E ( γ ) = 1 2 a b | γ ( t ) | 2 d t E(\gamma)=\frac{1}{2}\int_{a}^{b}|\mathbf{\gamma}^{\prime}(t)|^{2}dt
  18. 𝐞 1 ( t ) , , 𝐞 n ( t ) \mathbf{e}_{1}(t),\ldots,\mathbf{e}_{n}(t)
  19. 𝐞 1 ( t ) = γ ( t ) γ ( t ) \mathbf{e}_{1}(t)=\frac{\mathbf{\gamma}^{\prime}(t)}{\|\mathbf{\gamma}^{\prime% }(t)\|}
  20. 𝐞 j ( t ) = 𝐞 j ¯ ( t ) 𝐞 j ¯ ( t ) , 𝐞 j ¯ ( t ) = γ ( j ) ( t ) - i = 1 j - 1 γ ( j ) ( t ) , 𝐞 i ( t ) 𝐞 i ( t ) \mathbf{e}_{j}(t)=\frac{\overline{\mathbf{e}_{j}}(t)}{\|\overline{\mathbf{e}_{% j}}(t)\|}\mbox{, }~{}\overline{\mathbf{e}_{j}}(t)=\mathbf{\gamma}^{(j)}(t)-% \sum_{i=1}^{j-1}\langle\mathbf{\gamma}^{(j)}(t),\mathbf{e}_{i}(t)\rangle\,% \mathbf{e}_{i}(t)
  21. χ i ( t ) = 𝐞 i ( t ) , 𝐞 i + 1 ( t ) γ ( t ) \chi_{i}(t)=\frac{\langle\mathbf{e}_{i}^{\prime}(t),\mathbf{e}_{i+1}(t)\rangle% }{\|\mathbf{\gamma}^{^{\prime}}(t)\|}
  22. γ ( t 0 ) = d d t γ ( t ) \gamma^{\prime}(t_{0})=\frac{d}{d\,t}\mathbf{\gamma}(t)
  23. t = t 0 {t=t_{0}}
  24. γ ( t 0 ) , \|\mathbf{\gamma}^{\prime}(t_{0})\|,
  25. 𝐞 1 ( t ) = γ ( t ) γ ( t ) . \mathbf{e}_{1}(t)=\frac{\mathbf{\gamma}^{\prime}(t)}{\|\mathbf{\gamma}^{\prime% }(t)\|}.
  26. 𝐞 1 ( s ) = γ ( s ) . \mathbf{e}_{1}(s)=\mathbf{\gamma}^{\prime}(s).
  27. 𝐞 2 ¯ ( t ) = γ ′′ ( t ) - γ ′′ ( t ) , 𝐞 1 ( t ) 𝐞 1 ( t ) . \overline{\mathbf{e}_{2}}(t)=\mathbf{\gamma}^{\prime\prime}(t)-\langle\mathbf{% \gamma}^{\prime\prime}(t),\mathbf{e}_{1}(t)\rangle\,\mathbf{e}_{1}(t).
  28. 𝐞 2 ( t ) = 𝐞 2 ¯ ( t ) 𝐞 2 ¯ ( t ) . \mathbf{e}_{2}(t)=\frac{\overline{\mathbf{e}_{2}}(t)}{\|\overline{\mathbf{e}_{% 2}}(t)\|}.
  29. κ ( t ) = χ 1 ( t ) = 𝐞 1 ( t ) , 𝐞 2 ( t ) γ ( t ) \kappa(t)=\chi_{1}(t)=\frac{\langle\mathbf{e}_{1}^{\prime}(t),\mathbf{e}_{2}(t% )\rangle}{\|\mathbf{\gamma}^{\prime}(t)\|}
  30. 1 κ ( t ) \frac{1}{\kappa(t)}
  31. κ ( t ) = 1 r \kappa(t)=\frac{1}{r}
  32. 𝐞 3 ( t ) = 𝐞 3 ¯ ( t ) 𝐞 3 ¯ ( t ) , 𝐞 3 ¯ ( t ) = γ ′′′ ( t ) - γ ′′′ ( t ) , 𝐞 1 ( t ) 𝐞 1 ( t ) - γ ′′′ ( t ) , 𝐞 2 ( t ) 𝐞 2 ( t ) \mathbf{e}_{3}(t)=\frac{\overline{\mathbf{e}_{3}}(t)}{\|\overline{\mathbf{e}_{% 3}}(t)\|}\mbox{, }~{}\overline{\mathbf{e}_{3}}(t)=\mathbf{\gamma}^{\prime% \prime\prime}(t)-\langle\mathbf{\gamma}^{\prime\prime\prime}(t),\mathbf{e}_{1}% (t)\rangle\,\mathbf{e}_{1}(t)-\langle\mathbf{\gamma}^{\prime\prime\prime}(t),% \mathbf{e}_{2}(t)\rangle\,\mathbf{e}_{2}(t)
  33. 𝐞 3 ( t ) = 𝐞 1 ( t ) × 𝐞 2 ( t ) \mathbf{e}_{3}(t)=\mathbf{e}_{1}(t)\times\mathbf{e}_{2}(t)
  34. 𝐞 3 ( t ) = - 𝐞 1 ( t ) × 𝐞 2 ( t ) \mathbf{e}_{3}(t)=-\mathbf{e}_{1}(t)\times\mathbf{e}_{2}(t)
  35. τ ( t ) = χ 2 ( t ) = 𝐞 2 ( t ) , 𝐞 3 ( t ) γ ( t ) \tau(t)=\chi_{2}(t)=\frac{\langle\mathbf{e}_{2}^{\prime}(t),\mathbf{e}_{3}(t)% \rangle}{\|\mathbf{\gamma}^{\prime}(t)\|}
  36. χ i C n - i ( [ a , b ] , n ) , 1 i n - 1 \chi_{i}\in C^{n-i}([a,b],\mathbb{R}^{n})\mbox{, }~{}1\leq i\leq n-1
  37. χ i ( t ) > 0 , 1 i n - 1 \chi_{i}(t)>0\mbox{, }~{}1\leq i\leq n-1
  38. γ ( t ) = 1 ( t [ a , b ] ) \|\gamma^{\prime}(t)\|=1\mbox{ }~{}(t\in[a,b])
  39. χ i ( t ) = 𝐞 i ( t ) , 𝐞 i + 1 ( t ) γ ( t ) \chi_{i}(t)=\frac{\langle\mathbf{e}_{i}^{\prime}(t),\mathbf{e}_{i+1}(t)\rangle% }{\|\mathbf{\gamma}^{\prime}(t)\|}
  40. 𝐞 1 ( t ) , , 𝐞 n ( t ) \mathbf{e}_{1}(t),\ldots,\mathbf{e}_{n}(t)
  41. γ ( t 0 ) = 𝐩 0 \mathbf{\gamma}(t_{0})=\mathbf{p}_{0}
  42. 𝐞 i ( t 0 ) = 𝐞 i , 1 i n - 1 \mathbf{e}_{i}(t_{0})=\mathbf{e}_{i}\mbox{, }~{}1\leq i\leq n-1
  43. [ 𝐞 1 ( t ) 𝐞 2 ( t ) ] = γ ( t ) [ 0 κ ( t ) - κ ( t ) 0 ] [ 𝐞 1 ( t ) 𝐞 2 ( t ) ] \begin{bmatrix}\mathbf{e}_{1}^{\prime}(t)\\ \mathbf{e}_{2}^{\prime}(t)\\ \end{bmatrix}=\left\|\gamma^{\prime}\left(t\right)\right\|\begin{bmatrix}0&% \kappa(t)\\ -\kappa(t)&0\\ \end{bmatrix}\begin{bmatrix}\mathbf{e}_{1}(t)\\ \mathbf{e}_{2}(t)\\ \end{bmatrix}
  44. [ 𝐞 1 ( t ) 𝐞 2 ( t ) 𝐞 3 ( t ) ] = γ ( t ) [ 0 κ ( t ) 0 - κ ( t ) 0 τ ( t ) 0 - τ ( t ) 0 ] [ 𝐞 1 ( t ) 𝐞 2 ( t ) 𝐞 3 ( t ) ] \begin{bmatrix}\mathbf{e}_{1}^{\prime}(t)\\ \mathbf{e}_{2}^{\prime}(t)\\ \mathbf{e}_{3}^{\prime}(t)\\ \end{bmatrix}=\left\|\gamma^{\prime}\left(t\right)\right\|\begin{bmatrix}0&% \kappa(t)&0\\ -\kappa(t)&0&\tau(t)\\ 0&-\tau(t)&0\\ \end{bmatrix}\begin{bmatrix}\mathbf{e}_{1}(t)\\ \mathbf{e}_{2}(t)\\ \mathbf{e}_{3}(t)\\ \end{bmatrix}
  45. [ 𝐞 1 ( t ) 𝐞 2 ( t ) 𝐞 n - 1 ( t ) 𝐞 n ( t ) ] = γ ( t ) [ 0 χ 1 ( t ) 0 0 - χ 1 ( t ) 0 0 0 0 0 0 χ n - 1 ( t ) 0 0 - χ n - 1 ( t ) 0 ] [ 𝐞 1 ( t ) 𝐞 2 ( t ) 𝐞 n - 1 ( t ) 𝐞 n ( t ) ] \begin{bmatrix}\mathbf{e}_{1}^{\prime}(t)\\ \mathbf{e}_{2}^{\prime}(t)\\ \vdots\\ \mathbf{e}_{n-1}^{\prime}(t)\\ \mathbf{e}_{n}^{\prime}(t)\\ \end{bmatrix}=\left\|\gamma^{\prime}\left(t\right)\right\|\begin{bmatrix}0&% \chi_{1}(t)&\cdots&0&0\\ -\chi_{1}(t)&0&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&0&\chi_{n-1}(t)\\ 0&0&\cdots&-\chi_{n-1}(t)&0\\ \end{bmatrix}\begin{bmatrix}\mathbf{e}_{1}(t)\\ \mathbf{e}_{2}(t)\\ \vdots\\ \mathbf{e}_{n-1}(t)\\ \mathbf{e}_{n}(t)\\ \end{bmatrix}

Diffraction-limited_system.html

  1. d = λ 2 n sin θ d=\frac{\lambda}{2n\sin{\theta}}
  2. θ \theta
  3. d = λ 2 n sin θ d=\frac{\lambda}{2n\sin\theta}
  4. n sin θ n\sin\theta
  5. d / 2 = 1.22 λ N , d/2=1.22\lambda N,\,

Digamma_function.html

  1. ψ ( z ) \psi(z)
  2. ψ ( x ) = d d x ln Γ ( x ) = Γ ( x ) Γ ( x ) . \psi(x)=\frac{d}{dx}\ln{\Gamma(x)}=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}.
  3. ϝ \digamma
  4. ψ ( n ) = H n - 1 - γ \psi(n)=H_{n-1}-\gamma
  5. n n
  6. ψ ( n + 1 2 ) = - γ - 2 ln 2 + k = 1 n 2 2 k - 1 \psi\left(n+{\frac{1}{2}}\right)=-\gamma-2\ln 2+\sum_{k=1}^{n}\frac{2}{2k-1}
  7. x x
  8. ψ ( x ) = 0 ( e - t t - e - x t 1 - e - t ) d t \psi(x)=\int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-xt}}{1-e^{-t}}\right% )\,dt
  9. ψ ( s + 1 ) = - γ + 0 1 1 - x s 1 - x d x \psi(s+1)=-\gamma+\int_{0}^{1}\frac{1-x^{s}}{1-x}dx
  10. ψ ( z + 1 ) = - γ + n = 1 z n ( n + z ) z - 1 , - 2 , - 3 , \psi(z+1)=-\gamma+\sum_{n=1}^{\infty}\frac{z}{n(n+z)}\qquad z\neq-1,-2,-3,\ldots
  11. ψ ( z ) = - γ + n = 0 z - 1 ( n + 1 ) ( n + z ) = - γ + n = 0 ( 1 n + 1 - 1 n + z ) z 0 , - 1 , - 2 , - 3 , \psi(z)=-\gamma+\sum_{n=0}^{\infty}\frac{z-1}{(n+1)(n+z)}=-\gamma+\sum_{n=0}^{% \infty}\left(\frac{1}{n+1}-\frac{1}{n+z}\right)\qquad z\neq 0,-1,-2,-3,\ldots
  12. n = 0 u n = n = 0 p ( n ) q ( n ) , \sum_{n=0}^{\infty}u_{n}=\sum_{n=0}^{\infty}\frac{p(n)}{q(n)},
  13. u n = p ( n ) q ( n ) = k = 1 m a k n + b k . u_{n}=\frac{p(n)}{q(n)}=\sum_{k=1}^{m}\frac{a_{k}}{n+b_{k}}.
  14. lim n n u n = 0 , \lim_{n\to\infty}nu_{n}=0,
  15. k = 1 m a k = 0 , \sum_{k=1}^{m}a_{k}=0,
  16. n = 0 u n = n = 0 k = 1 m a k n + b k = n = 0 k = 1 m a k ( 1 n + b k - 1 n + 1 ) = k = 1 m ( a k n = 0 ( 1 n + b k - 1 n + 1 ) ) = - k = 1 m a k ( ψ ( b k ) + γ ) = - k = 1 m a k ψ ( b k ) . \begin{aligned}\displaystyle\sum_{n=0}^{\infty}u_{n}&\displaystyle=\sum_{n=0}^% {\infty}\sum_{k=1}^{m}\frac{a_{k}}{n+b_{k}}\\ &\displaystyle=\sum_{n=0}^{\infty}\sum_{k=1}^{m}a_{k}\left(\frac{1}{n+b_{k}}-% \frac{1}{n+1}\right)\\ &\displaystyle=\sum_{k=1}^{m}\left(a_{k}\sum_{n=0}^{\infty}\left(\frac{1}{n+b_% {k}}-\frac{1}{n+1}\right)\right)\\ &\displaystyle=-\sum_{k=1}^{m}a_{k}\left(\psi(b_{k})+\gamma\right)\\ &\displaystyle=-\sum_{k=1}^{m}a_{k}\psi(b_{k}).\end{aligned}
  17. n = 0 u n = n = 0 k = 1 m a k ( n + b k ) r k = k = 1 m ( - 1 ) r k ( r k - 1 ) ! a k ψ ( r k - 1 ) ( b k ) , \sum_{n=0}^{\infty}u_{n}=\sum_{n=0}^{\infty}\sum_{k=1}^{m}\frac{a_{k}}{(n+b_{k% })^{r_{k}}}=\sum_{k=1}^{m}\frac{(-1)^{r_{k}}}{(r_{k}-1)!}a_{k}\psi^{(r_{k}-1)}% (b_{k}),
  18. z = 1 z=1
  19. ψ ( z + 1 ) = - γ - k = 1 ζ ( k + 1 ) ( - z ) k \psi(z+1)=-\gamma-\sum_{k=1}^{\infty}\zeta(k+1)\;(-z)^{k}
  20. ζ ( n ) ζ(n)
  21. ψ ( s + 1 ) = - γ - k = 1 ( - 1 ) k k ( s k ) \psi(s+1)=-\gamma-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}{s\choose k}
  22. ( s k ) \textstyle{s\choose k}
  23. ψ ( 1 - x ) - ψ ( x ) = π cot ( π x ) \psi(1-x)-\psi(x)=\pi\,\!\cot{\left(\pi x\right)}
  24. ψ ( x + 1 ) = ψ ( x ) + 1 x . \psi(x+1)=\psi(x)+\frac{1}{x}.
  25. Δ [ ψ ] ( x ) = 1 x \Delta[\psi](x)=\frac{1}{x}
  26. ψ ( n ) = H n - 1 - γ \psi(n)\ =\ H_{n-1}-\gamma
  27. γ γ
  28. ψ ( x + 1 ) = - γ + k = 1 ( 1 k - 1 x + k ) . \psi(x+1)=-\gamma+\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{x+k}\right).
  29. ψ ψ
  30. F ( x + 1 ) = F ( x ) + 1 x F(x+1)=F(x)+\frac{1}{x}
  31. F ( 1 ) = γ F(1)=−γ
  32. Γ Γ
  33. ψ ( x + N ) - ψ ( x ) = k = 0 N - 1 1 x + k \psi(x+N)-\psi(x)=\sum_{k=0}^{N-1}\frac{1}{x+k}
  34. r = 1 m ψ ( r m ) = - m ( γ + ln m ) , \sum_{r=1}^{m}\psi\left(\frac{r}{m}\right)=-m(\gamma+\ln m),
  35. r = 1 m ψ ( r m ) exp 2 π r k i m = m ln ( 1 - exp 2 π k i m ) , k , m , k m . \sum_{r=1}^{m}\psi\left(\frac{r}{m}\right)\cdot\exp\dfrac{2\pi rki}{m}=m\ln\!% \left(\!1-\exp\dfrac{2\pi ki}{m}\!\right)\,,\qquad\quad k\in\mathbb{Z}\,,% \qquad m\in\mathbb{N}\,,\qquad k\neq m.
  36. r = 1 m - 1 ψ ( r m ) cos 2 π r k m = m ln ( 2 sin k π m ) + γ , k = 1 , 2 , , m - 1 \sum_{r=1}^{m-1}\psi\left(\frac{r}{m}\right)\cdot\cos\dfrac{2\pi rk}{m}\,=\,m% \ln\!\left(\!2\sin\frac{\,k\pi\,}{m}\!\right)+\,\gamma\,,\qquad\qquad\qquad k=% 1,2,\ldots,m-1
  37. r = 1 m - 1 ψ ( r m ) sin 2 π r k m = π 2 ( 2 k - m ) , k = 1 , 2 , , m - 1 \sum_{r=1}^{m-1}\psi\left(\frac{r}{m}\right)\cdot\sin\dfrac{2\pi rk}{m}=\frac{% \pi}{2}(2k-m)\,,\qquad\qquad\qquad k=1,2,\ldots,m-1
  38. r = 0 m - 1 ψ ( 2 r + 1 2 m ) cos ( 2 r + 1 ) k π m = m ln ( tan π k 2 m ) , k = 1 , 2 , , m - 1 \sum_{r=0}^{m-1}\psi\left(\frac{2r+1}{2m}\right)\cdot\cos\dfrac{(2r+1)k\pi}{m}% =m\ln\left(\tan\frac{\,\pi k\,}{2m}\right)\,,\qquad\qquad\qquad k=1,2,\ldots,m-1
  39. r = 0 m - 1 ψ ( 2 r + 1 2 m ) sin ( 2 r + 1 ) k π m = - π m 2 , k = 1 , 2 , , m - 1 \sum_{r=0}^{m-1}\psi\left(\frac{2r+1}{2m}\right)\cdot\sin\dfrac{(2r+1)k\pi}{m}% =-\frac{\pi m}{2}\,,\qquad\qquad\qquad k=1,2,\ldots,m-1
  40. r = 1 m - 1 ψ ( r m ) cot π r m = - π ( m - 1 ) ( m - 2 ) 6 \sum_{r=1}^{m-1}\psi\left(\frac{r}{m}\right)\cdot\cot\frac{\pi r}{m}=-\frac{% \pi(m-1)(m-2)}{6}
  41. r = 1 m - 1 ψ ( r m ) r m = - γ 2 ( m - 1 ) - m 2 ln m - π 2 r = 1 m - 1 r m cot π r m \sum_{r=1}^{m-1}\psi\left(\frac{r}{m}\right)\cdot\frac{r}{m}=-\frac{\gamma}{2}% (m-1)-\frac{m}{2}\ln m-\frac{\pi}{2}\sum_{r=1}^{m-1}\dfrac{r}{m}\cdot\cot% \dfrac{\pi r}{m}
  42. r = 1 m - 1 ψ ( r m ) cos ( 2 l + 1 ) π r m = - π m r = 1 m - 1 r sin 2 π r m cos 2 π r m - cos ( 2 l + 1 ) π m , l \sum_{r=1}^{m-1}\psi\left(\frac{r}{m}\right)\cdot\cos\dfrac{(2l+1)\pi r}{m}=-% \frac{\pi}{m}\sum_{r=1}^{m-1}\frac{r\cdot\sin\dfrac{2\pi r}{m}}{\,\cos\dfrac{2% \pi r}{m}-\cos\dfrac{(2l+1)\pi}{m}\,},\qquad\quad l\in\mathbb{Z}
  43. r = 1 m - 1 ψ ( r m ) sin ( 2 l + 1 ) π r m = - ( γ + ln 2 m ) cot ( 2 l + 1 ) π 2 m + sin ( 2 l + 1 ) π m r = 1 m - 1 ln sin π r m cos 2 π r m - cos ( 2 l + 1 ) π m , l \sum_{r=1}^{m-1}\psi\left(\frac{r}{m}\right)\cdot\sin\dfrac{(2l+1)\pi r}{m}=-(% \gamma+\ln 2m)\cot\frac{(2l+1)\pi}{2m}+\sin\dfrac{(2l+1)\pi}{m}\sum_{r=1}^{m-1% }\frac{\ln\sin\dfrac{\pi r}{m}}{\,\cos\dfrac{2\pi r}{m}-\cos\dfrac{(2l+1)\pi}{% m}\,},\qquad\quad l\in\mathbb{Z}
  44. r = 1 m - 1 ψ 2 ( r m ) = ( m - 1 ) γ 2 + m ( 2 γ + ln 4 m ) ln m - m ( m - 1 ) ln 2 2 + π 2 ( m 2 - 3 m + 2 ) 12 + m l = 1 m - 1 ln 2 sin π l m \sum_{r=1}^{m-1}\psi^{2}\!\left(\frac{r}{m}\right)=(m-1)\gamma^{2}+m(2\gamma+% \ln 4m)\ln{m}-m(m-1)\ln^{2}2+\frac{\pi^{2}(m^{2}-3m+2)}{12}+m\sum_{l=1}^{m-1}% \ln^{2}\sin\frac{\pi l}{m}
  45. ψ ( r m ) = - γ - ln ( 2 m ) - π 2 cot ( r π m ) + 2 n = 1 m - 1 2 cos ( 2 π n r m ) ln sin ( π n m ) \psi\left(\frac{r}{m}\right)=-\gamma-\ln(2m)-\frac{\pi}{2}\cot\left(\frac{r\pi% }{m}\right)+2\sum_{n=1}^{\lfloor\frac{m-1}{2}\rfloor}\cos\left(\frac{2\pi nr}{% m}\right)\ln\sin\left(\frac{\pi n}{m}\right)
  46. n = 1 x 1 n \sum_{n=1}^{x}\frac{1}{n}
  47. x x
  48. ψ ( x ) = ln ( x ) - 1 2 x - 1 12 x 2 + 1 120 x 4 - 1 252 x 6 + 1 240 x 8 - 5 660 x 10 + 691 32760 x 12 - 1 12 x 14 + O ( 1 x 16 ) \psi(x)=\ln(x)-\frac{1}{2x}-\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac{1}{252x% ^{6}}+\frac{1}{240x^{8}}-\frac{5}{660x^{10}}+\frac{691}{32760x^{12}}-\frac{1}{% 12x^{14}}+O\left(\frac{1}{x^{16}}\right)
  49. ψ ( x ) ψ(x)
  50. ψ ( x ) = ln ( x ) - 1 2 x + n = 1 ζ ( 1 - 2 n ) x 2 n = ln ( x ) - 1 2 x - n = 1 B 2 n 2 n x 2 n \psi(x)=\ln(x)-\frac{1}{2x}+\sum_{n=1}^{\infty}\frac{\zeta(1-2n)}{x^{2n}}=\ln(% x)-\frac{1}{2x}-\sum_{n=1}^{\infty}\frac{B_{2n}}{2n\,x^{2n}}
  51. B k B_{k}
  52. k k
  53. ζ ζ
  54. x x
  55. x x
  56. ψ ( x ) ψ(x)
  57. x x
  58. ψ ( x + 1 ) = 1 x + ψ ( x ) \psi(x+1)=\frac{1}{x}+\psi(x)
  59. x x
  60. x x
  61. x 14 x^{14}
  62. ψ ( x ) [ ln ( x - 1 ) , ln x ] exp ( ψ ( x ) ) { x 2 2 x [ 0 , 1 ] x - 1 2 x > 1 \begin{aligned}\displaystyle\psi(x)&\displaystyle\in[\ln(x-1),\ln x]\\ \displaystyle\exp(\psi(x))&\displaystyle\approx\begin{cases}\frac{x^{2}}{2}&x% \in[0,1]\\ x-\frac{1}{2}&x>1\end{cases}\end{aligned}
  63. ψ ψ
  64. exp ψ \exp\circ\,\psi
  65. exp ψ \exp\circ\,\psi
  66. exp ( - ψ ( 1 / y ) ) \exp(-\psi(1/y))
  67. y = 0 y=0
  68. 1 exp ( ψ ( x ) ) = 1 x + 1 2 x 2 + 5 4 3 ! x 3 + 3 2 4 ! x 4 + 47 48 5 ! x 5 - 5 16 6 ! x 6 + \frac{1}{\exp(\psi(x))}=\frac{1}{x}+\frac{1}{2\cdot x^{2}}+\frac{5}{4\cdot 3!% \cdot x^{3}}+\frac{3}{2\cdot 4!\cdot x^{4}}+\frac{47}{48\cdot 5!\cdot x^{5}}-% \frac{5}{16\cdot 6!\cdot x^{6}}+\cdots
  69. exp ( ψ ( x + 1 2 ) ) = x + 1 4 ! x - 37 8 6 ! x 3 + 10313 72 8 ! x 5 - 5509121 384 10 ! x 7 + O ( 1 x 9 ) for x > 1 \exp(\psi(x+\tfrac{1}{2}))=x+\frac{1}{4!\cdot x}-\frac{37}{8\cdot 6!\cdot x^{3% }}+\frac{10313}{72\cdot 8!\cdot x^{5}}-\frac{5509121}{384\cdot 10!\cdot x^{7}}% +O\left(\frac{1}{x^{9}}\right)\qquad\mbox{for }~{}x>1
  70. ψ ( 1 ) \displaystyle\psi(1)
  71. ( ψ ( i ) ) = - γ - n = 0 n - 1 n 3 + n 2 + n + 1 , ( ψ ( i ) ) = n = 0 1 n 2 + 1 = 1 2 + π 2 coth ( π ) . \begin{aligned}\displaystyle\Re\left(\psi(i)\right)&\displaystyle=-\gamma-\sum% _{n=0}^{\infty}\frac{n-1}{n^{3}+n^{2}+n+1},\\ \displaystyle\Im\left(\psi(i)\right)&\displaystyle=\sum_{n=0}^{\infty}\frac{1}% {n^{2}+1}=\frac{1}{2}+\frac{\pi}{2}\coth(\pi).\end{aligned}
  72. x 0 = 1.461632144968 x_{0}=1.461632144968\ldots
  73. x 1 \displaystyle x_{1}
  74. x n = - n + 1 ln n + o ( 1 ln 2 n ) x_{n}=-n+\frac{1}{\ln n}+o\left(\frac{1}{\ln^{2}n}\right)
  75. x n - n + 1 π arctan ( π ln n ) n 2 x_{n}\approx-n+\frac{1}{\pi}\arctan\left(\frac{\pi}{\ln n}\right)\qquad n\geq 2
  76. x n - n + 1 π arctan ( π ln n + 1 8 n ) n 1 x_{n}\approx-n+\frac{1}{\pi}\arctan\left(\frac{\pi}{\ln n+\frac{1}{8n}}\right)% \qquad n\geq 1
  77. 0 = ψ ( 1 - x n ) = ψ ( x n ) + π tan ( π x n ) 0=\psi(1-x_{n})=\psi(x_{n})+\frac{\pi}{\tan(\pi x_{n})}
  78. ψ ( x n ) \psi(x_{n})
  79. 1 2 n \tfrac{1}{2n}
  80. 0 d x x + a , \int_{0}^{\infty}\frac{dx}{x+a},
  81. n = 0 1 n + a = - ψ ( a ) . \sum_{n=0}^{\infty}\frac{1}{n+a}=-\psi(a).

Digital_control.html

  1. s = 2 ( z - 1 ) T ( z + 1 ) s=\frac{2(z-1)}{T(z+1)}
  2. z = e s T \begin{aligned}\displaystyle z&\displaystyle=e^{sT}\end{aligned}
  3. z \displaystyle z
  4. s \displaystyle s

Digital_geometry.html

  1. Σ \Sigma
  2. { A 1 , , A m } \{A_{1},\dots,A_{m}\}
  3. A 1 < < A m A_{1}<\cdots<A_{m}
  4. A i A_{i}
  5. Σ \Sigma
  6. f ( x ) = A i f(x)=A_{i}
  7. f ( y ) = A i f(y)=A_{i}
  8. f ( x ) = A i + 1 f(x)=A_{i+1}
  9. A i - 1 A_{i-1}
  10. D Σ D\subset\Sigma
  11. f : D { A 1 , , A m } f:D\rightarrow\{A_{1},\dots,A_{m}\}
  12. F F
  13. f f
  14. x x
  15. y y
  16. D D
  17. f ( x ) = A i f(x)=A_{i}
  18. f ( y ) = A j f(y)=A_{j}
  19. | i - j | d ( x , y ) |i-j|\leq d(x,y)
  20. d ( x , y ) d(x,y)
  21. x x
  22. y y

Digital_waveguide_synthesis.html

  1. y ( m , n ) = y + ( m - n ) + y - ( m + n ) y(m,n)=y^{+}(m-n)+y^{-}(m+n)
  2. y + y^{+}
  3. y - y^{-}
  4. y y
  5. m m
  6. n n

Dihedral_(aeronautics).html

  1. β \beta
  2. β \beta

Dihedral_angle.html

  1. 𝐛 1 \mathbf{b}_{1}
  2. 𝐛 2 \mathbf{b}_{2}
  3. 𝐛 2 \mathbf{b}_{2}
  4. 𝐛 3 \mathbf{b}_{3}
  5. φ A B \varphi_{AB}
  6. 𝐧 A \mathbf{n}_{A}
  7. 𝐧 B \mathbf{n}_{B}
  8. cos ( φ A B ) = 𝐧 A 𝐧 B \cos\left(\varphi_{AB}\right)=\mathbf{n}_{A}\cdot\mathbf{n}_{B}
  9. φ A B \varphi_{AB}
  10. φ A B = - φ B A \varphi_{AB}\;=\;-\varphi_{BA}
  11. 𝐛 1 \mathbf{b}_{1}
  12. 𝐛 2 \mathbf{b}_{2}
  13. 𝐛 3 \mathbf{b}_{3}
  14. 𝐛 1 \mathbf{b}_{1}
  15. 𝐛 2 \mathbf{b}_{2}
  16. 𝐛 2 \mathbf{b}_{2}
  17. 𝐛 3 \mathbf{b}_{3}
  18. φ = atan2 ( ( [ 𝐛 1 × 𝐛 2 ] × [ 𝐛 2 × 𝐛 3 ] ) 𝐛 2 | 𝐛 2 | , [ 𝐛 1 × 𝐛 2 ] [ 𝐛 2 × 𝐛 3 ] ) , \varphi=\operatorname{atan2}\left(\left([\mathbf{b}_{1}\times\mathbf{b}_{2}]% \times[\mathbf{b}_{2}\times\mathbf{b}_{3}]\right)\cdot\frac{\mathbf{b}_{2}}{|% \mathbf{b}_{2}|},[\mathbf{b}_{1}\times\mathbf{b}_{2}]\cdot[\mathbf{b}_{2}% \times\mathbf{b}_{3}]\right),
  19. φ \varphi
  20. φ \varphi
  21. 𝐛 1 \mathbf{b}_{1}
  22. 𝐛 3 \mathbf{b}_{3}
  23. φ A B = arccos ( U A U B | U A | | U B | ) = arcsin ( | U A × U B | | U A | | U B | ) \varphi_{AB}=\arccos\left(\frac{U_{A}\cdot U_{B}}{|U_{A}||U_{B}|}\right)=% \arcsin\left(\frac{|U_{A}\times U_{B}|}{|U_{A}||U_{B}|}\right)
  24. 𝐀 \mathbf{A}
  25. 𝐁 \mathbf{B}
  26. u = u\;=
  27. ( 𝐁 T 𝐀 ) T ( 𝐁 T 𝐀 ) \left(\mathbf{B}^{T}\mathbf{A}\right)^{T}\left(\mathbf{B}^{T}\mathbf{A}\right)
  28. v = v\;=
  29. ( 𝐀 T 𝐁 ) T ( 𝐀 T 𝐁 ) \left(\mathbf{A}^{T}\mathbf{B}\right)^{T}\left(\mathbf{A}^{T}\mathbf{B}\right)
  30. u u
  31. v v

Dilated_cardiomyopathy.html

  1. F = F e F g F=F^{e}\cdot F^{g}\,
  2. F e F^{e}
  3. F g F^{g}
  4. F g = 𝕀 + [ λ g - 1 ] f 0 f 0 F^{g}=\mathbb{I}+[\lambda^{g}-1]f_{0}\otimes f_{0}\,
  5. f 0 f_{0}
  6. λ g \lambda^{g}
  7. λ = λ e F λ g \lambda=\lambda^{e}\cdot F\lambda^{g}\,
  8. μ \mu
  9. μ \mu

Dilaton.html

  1. g = exp ( ϕ ) g=\exp(\langle\phi\rangle)
  2. d D x - g [ 1 2 κ ( Φ R - ω [ Φ ] g μ ν μ Φ ν Φ Φ ) - V [ Φ ] ] \int d^{D}x\sqrt{-g}\left[\frac{1}{2\kappa}\left(\Phi R-\omega\left[\Phi\right% ]\frac{g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi}{\Phi}\right)-V[\Phi]\right]

Dilution_refrigerator.html

  1. Q ˙ m [ W ] = ( n ˙ 3 [ mol/s ] ) ( 95 ( T m [ K ] ) 2 - 11 ( T i [ K ] ) 2 ) \dot{Q}_{m}\;[\,\text{W}]=\left(\dot{n}_{3}\;[\,\text{mol/s}]\right)\left(95(T% _{m}\;[\,\text{K}])^{2}-11(T_{i}\;[\,\text{K}])^{2}\right)
  2. n ˙ 3 \dot{n}_{3}
  3. T m = T i 2.8 . T_{m}=\frac{T_{i}}{2.8}.
  4. m 3 {}^{−3}_{m}
  5. m 8 {}^{−8}_{m}

Dimension_theorem_for_vector_spaces.html

  1. b j = i E j λ i , j a i b_{j}=\sum_{i\in E_{j}}\lambda_{i,j}a_{i}
  2. E j E_{j}
  3. I I
  4. j J E j \bigcup_{j\in J}E_{j}
  5. i 0 I i_{0}\in I
  6. E j E_{j}
  7. a i 0 a_{i_{0}}
  8. b j b_{j}
  9. a i a_{i}
  10. a i 0 a_{i_{0}}
  11. a i 0 a_{i_{0}}
  12. a i a_{i}
  13. a i = j J μ i , j b j a_{i}=\sum_{j\in J}\mu_{i,j}b_{j}
  14. ( μ i , j : i I , j J ) (\mu_{i,j}:i\in I,j\in J)
  15. ( μ i , j : i I ) (\mu_{i,j}:i\in I)
  16. r i = ( μ i , j : j J ) r_{i}=(\mu_{i,j}:j\in J)
  17. i I ν i r i = 0 \sum_{i\in I}\nu_{i}r_{i}=0
  18. i I ν i a i = i I ν i j J μ i , j b j = j J ( i I ν i μ i , j ) b j = 0 , \sum_{i\in I}\nu_{i}a_{i}=\sum_{i\in I}\nu_{i}\sum_{j\in J}\mu_{i,j}b_{j}=\sum% _{j\in J}\biggl(\sum_{i\in I}\nu_{i}\mu_{i,j}\biggr)b_{j}=0,
  19. a i a_{i}
  20. A = ( a 1 , , a n ) V A=(a_{1},\dots,a_{n})\subseteq V
  21. V V
  22. B 0 = ( b 1 , , b r ) B_{0}=(b_{1},...,b_{r})
  23. V V
  24. n r n\leq r
  25. B 0 B_{0}
  26. V V
  27. ( a 1 , b 1 , , b r ) (a_{1},b_{1},\dots,b_{r})
  28. a 1 0 a_{1}\neq 0
  29. A A
  30. t { 1 , , r } t\in\{1,\ldots,r\}
  31. b t b_{t}
  32. B 1 = ( a 1 , b 1 , , b t - 1 , b t + 1 , b r ) B_{1}=(a_{1},b_{1},\dots,b_{t-1},b_{t+1},...b_{r})
  33. B 1 B_{1}
  34. B 0 B_{0}
  35. A A
  36. B i B_{i}
  37. a j a_{j}
  38. A A
  39. b i b_{i}
  40. n n
  41. B n = ( a 1 , , a n , b m 1 , , b m k ) B_{n}=(a_{1},\ldots,a_{n},b_{m_{1}},\ldots,b_{m_{k}})
  42. k = 0 k=0
  43. r r
  44. A B n A\subseteq B_{n}
  45. | A | | B n | |A|\leq|B_{n}|
  46. n r n\leq r
  47. V V
  48. S = { v 1 , , v n } S=\{v_{1},\ldots,v_{n}\}
  49. T = { w 1 , , w m } T=\{w_{1},\ldots,w_{m}\}
  50. V V
  51. S S
  52. T T
  53. m n m\geq n
  54. T T
  55. S S
  56. n m n\geq m
  57. m = n m=n

Dimensionless_physical_constant.html

  1. α = e 2 c 4 π ε 0 1 137.03599908 , \alpha=\frac{e^{2}}{\hbar c\ 4\pi\varepsilon_{0}}\approx\frac{1}{137.03599908},
  2. Ω Λ \Omega_{\Lambda}

Dipole_antenna.html

  1. l = 1 2 k λ = 1 2 k c f l=\frac{1}{2}k\lambda=\frac{1}{2}k\frac{c}{f}
  2. E θ = - i I 0 sin θ 4 ε 0 c r L λ e i ( ω t - k r ) . E_{\theta}={-iI_{0}\sin\theta\over 4\varepsilon_{0}cr}{L\over\lambda}e^{i\left% (\omega t-kr\right)}.
  3. I 0 e i ω t I_{0}e^{i\omega t}
  4. P total = π 12 I 0 2 Z 0 ( L λ ) 2 P\text{total}={\pi\over 12}I_{0}^{2}Z_{0}\left({L\over\lambda}\right)^{2}
  5. 1 2 I 0 2 R r a d i a t i o n \frac{1}{2}I_{0}^{2}R_{radiation}
  6. R radiation = π 6 Z 0 ( L λ ) 2 ( L λ ) 2 ( 197 Ω ) . R\text{radiation}={\pi\over 6}Z_{0}\left({L\over\lambda}\right)^{2}\approx% \left({L\over\lambda}\right)^{2}(197\Omega).
  7. I ( z ) = I 0 e i ω t cos k z , I(z)=I_{0}e^{i\omega t}\cos kz,
  8. E θ = - i Z 0 I 0 2 π r cos ( π 2 cos θ ) sin θ e i ( ω t - k r ) . E_{\theta}=\frac{-iZ_{0}I_{0}}{2\pi r}\frac{\cos\left(\frac{\pi}{2}\cos\theta% \right)}{\sin\theta}e^{i(\omega t-kr)}.
  9. R r a d i a t i o n 73.1 Ω . R_{radiation}\approx 73.1\ \Omega.
  10. R r a d i a t i o n = Z 0 4 π Cin ( 2 π ) = Z 0 4 π 0 2 π 1 - cos ( θ ) θ d θ 73.1 Ω . R_{radiation}=\frac{Z_{0}}{4\pi}\operatorname{Cin}(2\pi)=\frac{Z_{0}}{4\pi}% \int_{0}^{2\pi}\frac{1-\cos(\theta)}{\theta}d\theta\approx 73.1\ \Omega.
  11. R r a d i a t i o n = 73.1 Ω sin 2 ( k x ) R_{radiation}=\frac{73.1\ \Omega}{\sin^{2}(kx)}
  12. G = 4 Cin ( 2 π ) 1.64 G=\frac{4}{\operatorname{Cin}(2\pi)}\approx 1.64\;
  13. λ \scriptstyle{\lambda}
  14. \scriptstyle{\ll}
  15. < m t p l > λ 4 \scriptstyle<mtpl>{{\lambda\over 4}}
  16. 73 + i 43 2 = 36 + i 21 \scriptstyle{{73+i43\over 2}=36+i21}
  17. < m t p l > λ 2 \scriptstyle<mtpl>{{\lambda\over 2}}
  18. R f d R_{fd}
  19. I 0 I_{0}
  20. 1 2 R λ 2 I 0 2 = 1 2 R f d ( I 0 / 2 ) 2 . \frac{1}{2}R_{\frac{\lambda}{2}}I_{0}^{2}=\frac{1}{2}R_{fd}\left(I_{0}/2\right% )^{2}.
  21. R f d = 4 R λ 2 292.32 Ω . R_{fd}=4R_{\frac{\lambda}{2}}\approx 292.32\ \Omega.
  22. I e i ω t Ie^{i\omega t}
  23. e i ω t e^{i\omega t}
  24. e - i k r e^{-ikr}
  25. 𝐀 ( 𝐫 ) = 𝐳 ^ μ 0 4 π r e - i k r δ \mathbf{A}(\mathbf{r})=\hat{\mathbf{z}}\,\frac{\mu_{0}}{4\pi r}e^{-ikr}\delta\ell
  26. μ 𝐇 = 𝐁 = × 𝐀 \mu\mathbf{H}=\mathbf{B}=\nabla\times\mathbf{A}
  27. 𝐄 = × 𝐀 i ω ϵ \mathbf{E}=\frac{\nabla\times\mathbf{A}}{i\omega\epsilon}\,
  28. H ϕ = i I δ 4 π ( k r - i r 2 ) e - i k r sin ( θ ) H_{\phi}=i\frac{I\delta\ell}{4\pi}\left(\frac{k}{r}-\frac{i}{r^{2}}\right)e^{-% ikr}\,\sin(\theta)
  29. E θ = i Z 0 I δ 4 π ( k r - i r 2 - 1 k r 3 ) e - i k r sin ( θ ) E_{\theta}=i\frac{Z_{0}\,I\delta\ell}{4\pi}\left(\frac{k}{r}-\frac{i}{r^{2}}-% \frac{1}{kr^{3}}\right)e^{-ikr}\,\sin(\theta)
  30. E r = Z 0 I δ 2 π ( 1 r 2 - i k r 3 ) e - i k r cos ( θ ) E_{r}=\frac{Z_{0}\,I\delta\ell}{2\pi}\left(\frac{1}{r^{2}}-\frac{i}{kr^{3}}% \right)e^{-ikr}\,\cos(\theta)
  31. H ϕ = i I δ k 4 π r e - i k r sin ( θ ) H_{\phi}=i\frac{I\delta\ell\,k}{4\pi r}e^{-ikr}\,\sin(\theta)
  32. E θ = i Z 0 I δ k 4 π r e - i k r sin ( θ ) E_{\theta}=i\frac{Z_{0}\,I\delta\ell\,k}{4\pi r}e^{-ikr}\,\sin(\theta)\;
  33. 𝐒 = 1 2 Re ( 𝐄 × 𝐇 * ) . \langle\mathbf{S}\rangle=\frac{1}{2}\mathrm{Re}\left(\mathbf{E}\times\mathbf{H% }^{*}\right).
  34. 𝐫 ^ \hat{\mathbf{r}}
  35. 𝐒 𝗋 = Z 0 2 k 2 | I | 2 ( δ ) 2 ( 4 π r ) 2 sin 2 θ \langle\mathbf{S}_{\mathsf{r}}\rangle=\frac{Z_{0}}{2}\frac{k^{2}|I|^{2}\ (% \delta\ell)^{2}}{(4\pi r)^{2}}\sin^{2}\theta
  36. P n e t = 0 2 π 0 π 𝐒 𝗋 r 2 sin θ d ϕ d θ P_{net}=\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\langle\mathbf{S}_{\mathsf{r}}\rangle r% ^{2}\sin\theta\,d\phi\,d\theta
  37. = Z 0 12 π k 2 | I | 2 ( δ ) 2 = π Z 0 3 | I | 2 ( δ λ ) 2 \;\;=\frac{Z_{0}}{12\pi}k^{2}|I|^{2}\ (\delta\ell)^{2}=\frac{\pi Z_{0}}{3}|I|^% {2}\ \left(\frac{\delta\ell}{\lambda}\right)^{2}
  38. λ = 2 π / k \lambda=2\pi/k
  39. R rad = 2 π 3 Z 0 ( δ λ ) 2 . R_{\mathrm{rad}}=\frac{2\pi}{3}Z_{0}\left(\frac{\delta\ell}{\lambda}\right)^{2}.
  40. P a v g = P n e t 4 π r 2 = Z 0 48 π 2 r 2 k 2 | I | 2 ( δ ) 2 P_{avg}=\frac{P_{net}}{4\pi r^{2}}=\frac{Z_{0}}{48\pi^{2}r^{2}}k^{2}|I|^{2}\ (% \delta\ell)^{2}
  41. 𝖦 ( θ ) = 𝐒 𝗋 P a v g = 3 2 sin 2 θ \mathsf{G}(\theta)=\frac{\langle\mathbf{S}_{\mathsf{r}}\rangle}{P_{avg}}=\frac% {3}{2}\sin^{2}\theta
  42. 𝐒 𝗋 \langle\mathbf{S}_{\mathsf{r}}\rangle
  43. I ( z ) = A sin ( k ( L / 2 - | z | ) ) I(z)=A\sin(k(L/2-|z|))
  44. R dipole \displaystyle R_{\mathrm{dipole}}
  45. X dipole = Z m 4 π sin 2 ( k L / 2 ) { 2 Si ( k L ) + cos ( k L ) [ 2 Si ( k L ) - Si ( 2 k L ) ] - sin ( k L ) [ 2 Ci ( k L ) - Ci ( 2 k L ) - Ci ( 2 k a 2 / L ) ] } , \begin{aligned}\displaystyle X_{\mathrm{dipole}}&\displaystyle=\frac{Z_{m}}{4% \pi\sin^{2}(kL/2)}\Big\{2\operatorname{Si}(kL)+\cos(kL)\big[2\operatorname{Si}% (kL)-\operatorname{Si}(2kL)\big]\\ &\displaystyle\qquad\qquad\qquad\qquad-\sin(kL)\big[2\operatorname{Ci}(kL)-% \operatorname{Ci}(2kL)-\operatorname{Ci}(2ka^{2}/L)\big]\Big\},\end{aligned}

Directed-energy_weapon.html

  1. 1 / c 1/c

Directional_derivative.html

  1. f ( x , y ) = x 2 + y 2 f(x,y)=x^{2}+y^{2}
  2. u {u}
  3. u {u}
  4. f ( x ) = f ( x 1 , x 2 , , x n ) f({x})=f(x_{1},x_{2},\ldots,x_{n})
  5. v = ( v 1 , , v n ) {v}=(v_{1},\ldots,v_{n})
  6. v f ( x ) = lim h 0 f ( x + h v ) - f ( x ) h . \nabla_{{v}}{f}({x})=\lim_{h\rightarrow 0}{\frac{f({x}+h{v})-f({x})}{h}}.
  7. v f ( x ) = f ( x ) v \nabla_{{v}}{f}({x})=\nabla f({x})\cdot{v}
  8. \nabla
  9. \cdot
  10. v f ( x ) = lim h 0 f ( x + h v ) - f ( x ) h | v | , \nabla_{{v}}{f}({x})=\lim_{h\rightarrow 0}{\frac{f({x}+h{v})-f({x})}{h|{v}|}},
  11. v f ( x ) = f ( x ) v | v | . \nabla_{{v}}{f}({x})=\nabla f({x})\cdot\frac{{v}}{|{v}|}.
  12. v f ( x ) f ( x ) v f 𝐯 ( x ) D v f ( x ) 𝐯 f ( x ) v f ( x ) x \nabla_{{v}}{f}({x})\sim\frac{\partial{f({x})}}{\partial{v}}\sim f^{\prime}_{% \mathbf{v}}({x})\sim D_{v}f({x})\sim\mathbf{v}\cdot{\nabla f({x})}\sim{v}\cdot% \frac{\partial f({x})}{\partial{x}}
  13. v ( c f ) = c v f \nabla_{{v}}(cf)=c\nabla_{{v}}f
  14. v ( f g ) = g v f + f v g \nabla_{{v}}(fg)=g\nabla_{{v}}f+f\nabla_{{v}}g
  15. v ( h g ) ( p ) = h ( g ( p ) ) v g ( p ) \nabla_{{v}}(h\circ g)({p})=h^{\prime}(g({p}))\nabla_{{v}}g({p})
  16. v f ( p ) \nabla_{{v}}f({p})
  17. L v f ( p ) L_{{v}}f({p})
  18. v p ( f ) {{v}}_{{p}}(f)
  19. v f ( p ) = d d τ f γ ( τ ) | τ = 0 \nabla_{{v}}f({p})=\left.\frac{d}{d\tau}f\circ\gamma(\tau)\right|_{\tau=0}
  20. W μ ( x ) \scriptstyle W^{\mu}(x)
  21. V μ ( x ) \scriptstyle V^{\mu}(x)
  22. V W μ = ( V ) W μ - ( W ) V μ \mathcal{L}_{V}W^{\mu}=(V\cdot\nabla)W^{\mu}-(W\cdot\nabla)V^{\mu}
  23. ϕ ( x ) \scriptstyle\phi(x)
  24. V ϕ = ( V ) ϕ \mathcal{L}_{V}\phi=(V\cdot\nabla)\phi
  25. 1 + ν δ ν D ν = 1 + δ D 1+\sum_{\nu}\delta^{\nu}D_{\nu}=1+\delta\cdot D
  26. 1 + μ δ μ D μ = 1 + δ D 1+\sum_{\mu}\delta^{\prime\mu}D_{\mu}=1+\delta^{\prime}\cdot D
  27. ( 1 + δ D ) ( 1 + δ D ) S ρ - ( 1 + δ D ) ( 1 + δ D ) S ρ = μ , ν δ μ δ ν [ D μ , D ν ] S ρ (1+\delta^{\prime}\cdot D)(1+\delta\cdot D)S^{\rho}-(1+\delta\cdot D)(1+\delta% ^{\prime}\cdot D)S^{\rho}=\sum_{\mu,\nu}\delta^{\prime\mu}\delta^{\nu}[D_{\mu}% ,D_{\nu}]S_{\rho}
  28. [ D μ , D ν ] S ρ = ± σ R ρ μ ν σ S σ [D_{\mu},D_{\nu}]S_{\rho}=\pm\sum_{\sigma}R^{\sigma}_{\rho\mu\nu}S_{\sigma}
  29. 𝐏 = i \mathbf{P}=i\nabla
  30. U ( s y m b o l λ ) = exp ( - i s y m b o l λ 𝐏 ) U(symbol{\lambda})=\exp\left(-isymbol{\lambda}\cdot\mathbf{P}\right)
  31. U ( s y m b o l λ ) = exp ( s y m b o l λ ) U(symbol{\lambda})=\exp\left(symbol{\lambda}\cdot\nabla\right)
  32. U ( s y m b o l λ ) f ( 𝐱 ) = exp ( s y m b o l λ ) f ( 𝐱 ) = f ( 𝐱 + s y m b o l λ ) U(symbol{\lambda})f(\mathbf{x})=\exp\left(symbol{\lambda}\cdot\nabla\right)f(% \mathbf{x})=f(\mathbf{x}+symbol{\lambda})
  33. d f d x = f ( x + ϵ ) - f ( x ) ϵ \frac{df}{dx}=\frac{f(x+\epsilon)-f(x)}{\epsilon}
  34. f ( x + ϵ ) = f ( x ) + ϵ d f d x = ( 1 + ϵ d d x ) f ( x ) f(x+\epsilon)=f(x)+\epsilon\,\frac{df}{dx}=\left(1+\epsilon\,\frac{d}{dx}% \right)f(x)
  35. [ 1 + ϵ ( d / d x ) ] [1+\epsilon\,(d/dx)]
  36. f ( 𝐱 + s y m b o l ϵ ) = ( 1 + s y m b o l ϵ ) f ( 𝐱 ) f(\mathbf{x}+symbol{\epsilon})=\left(1+symbol{\epsilon}\cdot\nabla\right)f(% \mathbf{x})
  37. s y m b o l ϵ symbol{\epsilon}\cdot\nabla
  38. U ( s y m b o l ϵ ) = 1 + s y m b o l ϵ U(symbol{\epsilon})=1+symbol{\epsilon}\cdot\nabla
  39. U ( 𝐚 ) U ( 𝐛 ) = U ( 𝐚 + 𝐛 ) U(\mathbf{a})U(\mathbf{b})=U(\mathbf{a+b})
  40. s y m b o l λ = N s y m b o l ϵ symbol{\lambda}=Nsymbol{\epsilon}
  41. [ U ( s y m b o l ϵ ) ] N = U ( N s y m b o l ϵ ) = U ( s y m b o l λ ) [U(symbol{\epsilon})]^{N}=U(Nsymbol{\epsilon})=U(symbol{\lambda})
  42. [ U ( s y m b o l ϵ ) ] N = [ 1 + s y m b o l ϵ ] N = [ 1 + s y m b o l λ N ] N [U(symbol{\epsilon})]^{N}=\left[1+symbol{\epsilon}\cdot\nabla\right]^{N}=\left% [1+\frac{symbol{\lambda}\cdot\nabla}{N}\right]^{N}
  43. exp ( x ) = [ 1 + x N ] N \exp(x)=\left[1+\frac{x}{N}\right]^{N}
  44. U ( s y m b o l λ ) = exp ( s y m b o l λ ) U(symbol{\lambda})=\exp\left(symbol{\lambda}\cdot\nabla\right)
  45. [ U ( s y m b o l ϵ ) ] N f ( 𝐱 ) = f ( 𝐱 + N s y m b o l ϵ ) = f ( 𝐱 + s y m b o l λ ) = U ( s y m b o l λ ) f ( 𝐱 ) = exp ( s y m b o l λ ) f ( 𝐱 ) [U(symbol{\epsilon})]^{N}f(\mathbf{x})=f(\mathbf{x}+Nsymbol{\epsilon})=f(% \mathbf{x}+symbol{\lambda})=U(symbol{\lambda})f(\mathbf{x})=\exp\left(symbol{% \lambda}\cdot\nabla\right)f(\mathbf{x})
  46. ξ a \scriptstyle\xi^{a}
  47. T ( ξ ¯ ) T ( ξ ) = T ( f ( ξ ¯ , ξ ) ) T(\bar{\xi})T(\xi)=T(f(\bar{\xi},\xi))
  48. ξ a \scriptstyle\xi^{a}
  49. f a ( ξ , 0 ) = f a ( 0 , ξ ) = ξ a f^{a}(\xi,0)=f^{a}(0,\xi)=\xi^{a}
  50. U ( T ( ξ ) ) = 1 + i a ξ a t a + 1 2 b , c ξ b ξ c t b c + U(T(\xi))=1+i\sum_{a}\xi^{a}t_{a}+\frac{1}{2}\sum_{b,c}\xi^{b}\xi^{c}t_{bc}+\cdots
  51. U ( T ( ξ ¯ ) ) U ( T ( ξ ) ) = U ( T ( f ( ξ ¯ , ξ ) ) ) U(T(\bar{\xi}))U(T(\xi))=U(T(f(\bar{\xi},\xi)))
  52. f a ( ξ ¯ , ξ ) = ξ a + ξ ¯ a + b , c f a b c ξ ¯ b ξ c f^{a}(\bar{\xi},\xi)=\xi^{a}+\bar{\xi}^{a}+\sum_{b,c}f^{abc}\bar{\xi}^{b}\xi^{c}
  53. t b c = - t b t c - i a f a b c t a t_{bc}=-t_{b}t_{c}-i\sum_{a}f^{abc}t_{a}
  54. t a b \scriptstyle t_{ab}
  55. [ t b , t c ] = i a ( - f a b c + f a c b ) t a = i a C a b c t a [t_{b},t_{c}]=i\sum_{a}(-f^{abc}+f^{acb})t_{a}=i\sum_{a}C^{abc}t_{a}
  56. [ x b , x c ] = 0 \left[\frac{\partial}{\partial x^{b}},\frac{\partial}{\partial x^{c}}\right]=0
  57. f a abelian ( ξ ¯ , ξ ) = ξ a + ξ ¯ a f^{a}\text{abelian}(\bar{\xi},\xi)=\xi^{a}+\bar{\xi}^{a}
  58. U ( T ( ξ ¯ ) ) U ( T ( ξ ) ) = U ( T ( ξ ¯ + ξ ) ) U(T(\bar{\xi}))U(T(\xi))=U(T(\bar{\xi}+\xi))
  59. θ ^ \scriptstyle\hat{\theta}
  60. U ( R ( s y m b o l θ ) ) = exp ( - i s y m b o l θ 𝐋 ) U(R(symbol{\theta}))=\exp(-isymbol{\theta}\cdot\mathbf{L})
  61. 𝐋 = ( 0 0 0 0 0 1 0 - 1 0 ) 𝐢 + ( 0 0 - 1 0 0 0 1 0 0 ) 𝐣 + ( 0 1 0 - 1 0 0 0 0 0 ) 𝐤 \mathbf{L}=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&-1&0\end{pmatrix}\mathbf{i}+\begin{pmatrix}0&0&-1\\ 0&0&0\\ 1&0&0\end{pmatrix}\mathbf{j}+\begin{pmatrix}0&1&0\\ -1&0&0\\ 0&0&0\end{pmatrix}\mathbf{k}
  62. 𝐱 𝐱 - \deltasymbol θ × 𝐱 \mathbf{x}\rightarrow\mathbf{x}-\deltasymbol{\theta}\times\mathbf{x}
  63. U ( R ( \deltasymbol θ ) ) f ( 𝐱 ) = f ( 𝐱 - \deltasymbol θ × 𝐱 ) = f ( 𝐱 ) - ( \deltasymbol θ × 𝐱 ) f U(R(\deltasymbol{\theta}))f(\mathbf{x})=f(\mathbf{x}-\deltasymbol{\theta}% \times\mathbf{x})=f(\mathbf{x})-(\deltasymbol{\theta}\times\mathbf{x})\cdot\nabla f
  64. U ( R ( \deltasymbol θ ) ) = 1 - ( \deltasymbol θ × 𝐱 ) U(R(\deltasymbol{\theta}))=1-(\deltasymbol{\theta}\times\mathbf{x})\cdot\nabla
  65. U ( R ( s y m b o l θ ) ) = exp ( - ( s y m b o l θ × 𝐱 ) ) U(R(symbol{\theta}))=\exp(-(symbol{\theta}\times\mathbf{x})\cdot\nabla)
  66. n {n}
  67. f n \frac{\partial f}{\partial n}
  68. f n = f ( x ) n = n f ( x ) = f x n = D f ( x ) [ n ] \frac{\partial f}{\partial n}=\nabla f({x})\cdot{n}=\nabla_{{n}}{f}({x})=\frac% {\partial f}{\partial{x}}\cdot{n}=Df({x})[{n}]
  69. f ( 𝐯 ) f(\mathbf{v})
  70. 𝐯 \mathbf{v}
  71. f ( 𝐯 ) f(\mathbf{v})
  72. 𝐯 \mathbf{v}
  73. 𝐯 \mathbf{v}
  74. 𝐮 \mathbf{u}
  75. f 𝐯 𝐮 = D f ( 𝐯 ) [ 𝐮 ] = [ d d α f ( 𝐯 + α 𝐮 ) ] α = 0 \frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=Df(\mathbf{v})[\mathbf{u}% ]=\left[\frac{d}{d\alpha}~{}f(\mathbf{v}+\alpha~{}\mathbf{u})\right]_{\alpha=0}
  76. 𝐮 \mathbf{u}
  77. f ( 𝐯 ) = f 1 ( 𝐯 ) f 2 ( 𝐯 ) f(\mathbf{v})=f_{1}(\mathbf{v})~{}f_{2}(\mathbf{v})
  78. f 𝐯 𝐮 = ( f 1 𝐯 𝐮 ) f 2 ( 𝐯 ) + f 1 ( 𝐯 ) ( f 2 𝐯 𝐮 ) \frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=\left(\frac{\partial f_{1% }}{\partial\mathbf{v}}\cdot\mathbf{u}\right)~{}f_{2}(\mathbf{v})+f_{1}(\mathbf% {v})~{}\left(\frac{\partial f_{2}}{\partial\mathbf{v}}\cdot\mathbf{u}\right)
  79. f ( 𝐯 ) = f 1 ( f 2 ( 𝐯 ) ) f(\mathbf{v})=f_{1}(f_{2}(\mathbf{v}))
  80. f 𝐯 𝐮 = f 1 f 2 f 2 𝐯 𝐮 \frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=\frac{\partial f_{1}}{% \partial f_{2}}~{}\frac{\partial f_{2}}{\partial\mathbf{v}}\cdot\mathbf{u}
  81. 𝐟 ( 𝐯 ) \mathbf{f}(\mathbf{v})
  82. 𝐯 \mathbf{v}
  83. 𝐟 ( 𝐯 ) \mathbf{f}(\mathbf{v})
  84. 𝐯 \mathbf{v}
  85. 𝐯 \mathbf{v}
  86. 𝐮 \mathbf{u}
  87. 𝐟 𝐯 𝐮 = D 𝐟 ( 𝐯 ) [ 𝐮 ] = [ d d α 𝐟 ( 𝐯 + α 𝐮 ) ] α = 0 \frac{\partial\mathbf{f}}{\partial\mathbf{v}}\cdot\mathbf{u}=D\mathbf{f}(% \mathbf{v})[\mathbf{u}]=\left[\frac{d}{d\alpha}~{}\mathbf{f}(\mathbf{v}+\alpha% ~{}\mathbf{u})\right]_{\alpha=0}
  88. 𝐮 \mathbf{u}
  89. 𝐟 ( 𝐯 ) = 𝐟 1 ( 𝐯 ) × 𝐟 2 ( 𝐯 ) \mathbf{f}(\mathbf{v})=\mathbf{f}_{1}(\mathbf{v})\times\mathbf{f}_{2}(\mathbf{% v})
  90. 𝐟 𝐯 𝐮 = ( 𝐟 1 𝐯 𝐮 ) × 𝐟 2 ( 𝐯 ) + 𝐟 1 ( 𝐯 ) × ( 𝐟 2 𝐯 𝐮 ) \frac{\partial\mathbf{f}}{\partial\mathbf{v}}\cdot\mathbf{u}=\left(\frac{% \partial\mathbf{f}_{1}}{\partial\mathbf{v}}\cdot\mathbf{u}\right)\times\mathbf% {f}_{2}(\mathbf{v})+\mathbf{f}_{1}(\mathbf{v})\times\left(\frac{\partial% \mathbf{f}_{2}}{\partial\mathbf{v}}\cdot\mathbf{u}\right)
  91. 𝐟 ( 𝐯 ) = 𝐟 1 ( 𝐟 2 ( 𝐯 ) ) \mathbf{f}(\mathbf{v})=\mathbf{f}_{1}(\mathbf{f}_{2}(\mathbf{v}))
  92. 𝐟 𝐯 𝐮 = 𝐟 1 𝐟 2 ( 𝐟 2 𝐯 𝐮 ) \frac{\partial\mathbf{f}}{\partial\mathbf{v}}\cdot\mathbf{u}=\frac{\partial% \mathbf{f}_{1}}{\partial\mathbf{f}_{2}}\cdot\left(\frac{\partial\mathbf{f}_{2}% }{\partial\mathbf{v}}\cdot\mathbf{u}\right)
  93. f ( s y m b o l S ) f(symbol{S})
  94. s y m b o l S symbol{S}
  95. f ( s y m b o l S ) f(symbol{S})
  96. s y m b o l S symbol{S}
  97. s y m b o l S symbol{S}
  98. s y m b o l T symbol{T}
  99. f s y m b o l S : s y m b o l T = D f ( s y m b o l S ) [ s y m b o l T ] = [ d d α f ( s y m b o l S + \alphasymbol T ) ] α = 0 \frac{\partial f}{\partial symbol{S}}:symbol{T}=Df(symbol{S})[symbol{T}]=\left% [\frac{d}{d\alpha}~{}f(symbol{S}+\alphasymbol{T})\right]_{\alpha=0}
  100. s y m b o l T symbol{T}
  101. f ( s y m b o l S ) = f 1 ( s y m b o l S ) f 2 ( s y m b o l S ) f(symbol{S})=f_{1}(symbol{S})~{}f_{2}(symbol{S})
  102. f s y m b o l S : s y m b o l T = ( f 1 s y m b o l S : s y m b o l T ) f 2 ( s y m b o l S ) + f 1 ( s y m b o l S ) ( f 2 s y m b o l S : s y m b o l T ) \frac{\partial f}{\partial symbol{S}}:symbol{T}=\left(\frac{\partial f_{1}}{% \partial symbol{S}}:symbol{T}\right)~{}f_{2}(symbol{S})+f_{1}(symbol{S})~{}% \left(\frac{\partial f_{2}}{\partial symbol{S}}:symbol{T}\right)
  103. f ( s y m b o l S ) = f 1 ( f 2 ( s y m b o l S ) ) f(symbol{S})=f_{1}(f_{2}(symbol{S}))
  104. f s y m b o l S : s y m b o l T = f 1 f 2 ( f 2 s y m b o l S : s y m b o l T ) \frac{\partial f}{\partial symbol{S}}:symbol{T}=\frac{\partial f_{1}}{\partial f% _{2}}~{}\left(\frac{\partial f_{2}}{\partial symbol{S}}:symbol{T}\right)
  105. s y m b o l F ( s y m b o l S ) symbol{F}(symbol{S})
  106. s y m b o l S symbol{S}
  107. s y m b o l F ( s y m b o l S ) symbol{F}(symbol{S})
  108. s y m b o l S symbol{S}
  109. s y m b o l S symbol{S}
  110. s y m b o l T symbol{T}
  111. s y m b o l F s y m b o l S : s y m b o l T = D s y m b o l F ( s y m b o l S ) [ s y m b o l T ] = [ d d α s y m b o l F ( s y m b o l S + \alphasymbol T ) ] α = 0 \frac{\partial symbol{F}}{\partial symbol{S}}:symbol{T}=Dsymbol{F}(symbol{S})[% symbol{T}]=\left[\frac{d}{d\alpha}~{}symbol{F}(symbol{S}+\alphasymbol{T})% \right]_{\alpha=0}
  112. s y m b o l T symbol{T}
  113. s y m b o l F ( s y m b o l S ) = s y m b o l F 1 ( s y m b o l S ) \cdotsymbol F 2 ( s y m b o l S ) symbol{F}(symbol{S})=symbol{F}_{1}(symbol{S})\cdotsymbol{F}_{2}(symbol{S})
  114. s y m b o l F s y m b o l S : s y m b o l T = ( s y m b o l F 1 s y m b o l S : s y m b o l T ) \cdotsymbol F 2 ( s y m b o l S ) + s y m b o l F 1 ( s y m b o l S ) ( s y m b o l F 2 s y m b o l S : s y m b o l T ) \frac{\partial symbol{F}}{\partial symbol{S}}:symbol{T}=\left(\frac{\partial symbol% {F}_{1}}{\partial symbol{S}}:symbol{T}\right)\cdotsymbol{F}_{2}(symbol{S})+% symbol{F}_{1}(symbol{S})\cdot\left(\frac{\partial symbol{F}_{2}}{\partial symbol% {S}}:symbol{T}\right)
  115. s y m b o l F ( s y m b o l S ) = s y m b o l F 1 ( s y m b o l F 2 ( s y m b o l S ) ) symbol{F}(symbol{S})=symbol{F}_{1}(symbol{F}_{2}(symbol{S}))
  116. s y m b o l F s y m b o l S : s y m b o l T = s y m b o l F 1 s y m b o l F 2 : ( s y m b o l F 2 s y m b o l S : s y m b o l T ) \frac{\partial symbol{F}}{\partial symbol{S}}:symbol{T}=\frac{\partial symbol{% F}_{1}}{\partial symbol{F}_{2}}:\left(\frac{\partial symbol{F}_{2}}{\partial symbol% {S}}:symbol{T}\right)
  117. f ( s y m b o l S ) = f 1 ( s y m b o l F 2 ( s y m b o l S ) ) f(symbol{S})=f_{1}(symbol{F}_{2}(symbol{S}))
  118. f s y m b o l S : s y m b o l T = f 1 s y m b o l F 2 : ( s y m b o l F 2 s y m b o l S : s y m b o l T ) \frac{\partial f}{\partial symbol{S}}:symbol{T}=\frac{\partial f_{1}}{\partial symbol% {F}_{2}}:\left(\frac{\partial symbol{F}_{2}}{\partial symbol{S}}:symbol{T}\right)

Dirichlet's_unit_theorem.html

  1. O K O_{K}
  2. O O K O\subset O_{K}
  3. 𝐐 O K , S 𝐙 𝐐 \mathbf{Q}\oplus O_{K,S}\otimes_{\mathbf{Z}}\mathbf{Q}
  4. N j log | u i j | N_{j}\log|u_{i}^{j}|
  5. N j log | u j | N_{j}\log|u^{j}|
  6. [ 1 × log | 5 + 1 2 | , 1 × log | - 5 + 1 2 | ] . \left[1\times\log\left|{\sqrt{5}+1\over 2}\right|,\quad 1\times\log\left|{-% \sqrt{5}+1\over 2}\right|\ \right].
  7. U 1 = P | p U 1 , P . U_{1}=\prod_{P|p}U_{1,P}.
  8. E 1 E_{1}
  9. r 1 + r 2 - 1 r_{1}+r_{2}-1

Dirichlet_boundary_condition.html

  1. y ′′ + y = 0 y^{\prime\prime}+y=0~{}
  2. [ a , b ] [a,\,b]
  3. y ( a ) = α and y ( b ) = β y(a)=\alpha\ \,\text{and}\ y(b)=\beta
  4. α \alpha
  5. β \beta
  6. 2 y + y = 0 \nabla^{2}y+y=0
  7. 2 \nabla^{2}
  8. Ω n \Omega\subset\mathbb{R}^{n}
  9. y ( x ) = f ( x ) x Ω y(x)=f(x)\quad\forall x\in\partial\Omega
  10. Ω \partial\Omega

Dirichlet_eta_function.html

  1. η ( s ) = n = 1 ( - 1 ) n - 1 n s = 1 1 s - 1 2 s + 1 3 s - 1 4 s + \eta(s)=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^% {s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots
  2. η ( s ) = ( 1 - 2 1 - s ) ζ ( s ) \eta(s)=\left(1-2^{1-s}\right)\zeta(s)
  3. 1 - 2 1 - s 1-2^{1-s}
  4. η ( s ) = 1 Γ ( s ) 0 x s - 1 e x + 1 d x \eta(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}{dx}
  5. η ( - s ) = 2 1 - 2 - s - 1 1 - 2 - s π - s - 1 s sin ( π s 2 ) Γ ( s ) η ( s + 1 ) . \eta(-s)=2\frac{1-2^{-s-1}}{1-2^{-s}}\pi^{-s-1}s\sin\left({\pi s\over 2}\right% )\Gamma(s)\eta(s+1).
  6. 1 - 2 1 - s 1-2^{1-s}
  7. ( s ) = 1 \Re(s)=1
  8. s n = 1 + 2 n π i / log ( 2 ) s_{n}=1+2n\pi i/\log(2)
  9. ( s ) = 1 / 2 , ( s ) = 1 \Re(s)=1/2,\Re(s)=1
  10. ζ ( s ) = η ( s ) 1 - 2 1 - s , \zeta(s)=\frac{\eta(s)}{1-2^{1-s}},
  11. s n = 1 + n 2 π ln 2 i , n 0 , n Z s_{n}=1+n\frac{2\pi}{\ln{2}}i,n\neq 0,n\in Z
  12. s n 1 s_{n}\neq 1
  13. λ \lambda
  14. ( s ) > 0 \Re(s)>0
  15. ( s ) = 1 \Re(s)=1
  16. λ ( s ) = ( 1 - 3 3 s ) ζ ( s ) = ( 1 + 1 2 s ) - 2 3 s + ( 1 4 s + 1 5 s ) - 2 6 s + \begin{aligned}\displaystyle\lambda(s)=(1-\frac{3}{3^{s}})\zeta(s)=(1+\frac{1}% {2^{s}})-\frac{2}{3^{s}}+(\frac{1}{4^{s}}+\frac{1}{5^{s}})-\frac{2}{6^{s}}+% \ldots\end{aligned}
  17. s s
  18. λ ( s ) \lambda(s)
  19. ( s ) > 0 \Re(s)>0
  20. R e ( s ) = 1 Re(s)=1
  21. ζ ( s ) = η ( s ) 1 - 2 2 s \zeta(s)=\frac{\eta(s)}{1-\frac{2}{2^{s}}}
  22. ζ ( s ) = λ ( s ) 1 - 3 3 s \zeta(s)=\frac{\lambda(s)}{1-\frac{3}{3^{s}}}
  23. log 3 log 2 \frac{\log 3}{\log 2}
  24. s = 1 s=1
  25. ζ ( s ) \zeta(s)\,
  26. ( s ) > 0 \Re(s)>0
  27. s = 1 s=1
  28. η ( s n ) = 0 \eta(s_{n})=0
  29. s n 1 s_{n}\neq 1
  30. η ( s n ) = ( 1 - 2 2 s n ) ζ ( s n ) = 1 - 2 2 s n 1 - 3 3 s n λ ( s n ) = 0. \eta(s_{n})=(1-\frac{2}{2^{s_{n}}})\zeta(s_{n})=\frac{1-\frac{2}{2^{s_{n}}}}{1% -\frac{3}{3^{s_{n}}}}\lambda(s_{n})=0.
  31. ζ \zeta\,
  32. s n 1 s_{n}\neq 1
  33. ( s ) > 1 \Re(s)>1
  34. η 2 n ( s ) = k = 1 2 n ( - 1 ) k - 1 k s = 1 - 1 2 s + 1 3 s - 1 4 s + + ( - 1 ) 2 n - 1 ( 2 n ) s = 1 + 1 2 s + 1 3 s + 1 4 s + + 1 ( 2 n ) s - 2 ( 1 2 s + 1 4 s + + 1 ( 2 n ) s ) \eta_{2n}(s)=\sum_{k=1}^{2n}\frac{(-1)^{k-1}}{k^{s}}=1-\frac{1}{2^{s}}+\frac{1% }{3^{s}}-\frac{1}{4^{s}}+\ldots+\frac{(-1)^{2n-1}}{{(2n)}^{s}}=1+\frac{1}{2^{s% }}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\ldots+\frac{1}{{(2n)}^{s}}-2(\frac{1}{2^{s% }}+\frac{1}{4^{s}}+\ldots+\frac{1}{{(2n)}^{s}})
  35. = ( 1 - 2 2 s ) ζ 2 n ( s ) + 2 2 s ( 1 ( n + 1 ) s + + 1 ( 2 n ) s ) = ( 1 - 2 2 s ) ζ 2 n ( s ) + 2 n ( 2 n ) s 1 n ( 1 ( 1 + 1 / n ) s + + 1 ( 1 + n / n ) s ) . =(1-\frac{2}{2^{s}})\zeta_{2n}(s)+\frac{2}{2^{s}}(\frac{1}{{(n+1)}^{s}}+\ldots% +\frac{1}{{(2n)}^{s}})=(1-\frac{2}{2^{s}})\zeta_{2n}(s)+\frac{2n}{{(2n)}^{s}}% \,\frac{1}{n}\,(\frac{1}{{(1+1/n)}^{s}}+\ldots+\frac{1}{{(1+n/n)}^{s}}).
  36. s = 1 + i t s=1+it
  37. 2 s = 2 2^{s}=2
  38. ζ 2 n ( s ) \zeta_{2n}(s)\,
  39. η 2 n ( s ) = 1 n i t R n ( 1 ( 1 + x ) s , 0 , 1 ) , \eta_{2n}(s)=\frac{1}{n^{it}}R_{n}(\frac{1}{{(1+x)}^{s}},0,1),
  40. η ( 1 ) = lim n η 2 n ( 1 ) = lim n R n ( 1 1 + x , 0 , 1 ) = 0 1 d x 1 + x = log 2 0. \eta(1)=\lim_{n\to\infty}\eta_{2n}(1)=\lim_{n\to\infty}R_{n}(\frac{1}{1+x},0,1% )=\int_{0}^{1}\frac{dx}{1+x}=\log 2\neq 0.
  41. t 0 t\neq 0
  42. | n 1 - s | = | n - i t | = 1 |n^{1-s}|=|n^{-it}|=1
  43. | η ( s ) | = lim n | η 2 n ( s ) | = lim n | R n ( 1 ( 1 + x ) s , 0 , 1 ) | = | 0 1 d x ( 1 + x ) s | = | 2 1 - s - 1 1 - s | = | 1 - 1 - i t | = 0. |\eta(s)|=\lim_{n\to\infty}|\eta_{2n}(s)|=\lim_{n\to\infty}|R_{n}(\frac{1}{{(1% +x)}^{s}},0,1)|=|\int_{0}^{1}\frac{dx}{{(1+x)}^{s}}|=|\frac{2^{1-s}-1}{1-s}|=|% \frac{1-1}{-it}|=0.
  44. η ( s n ) = 0 \eta(s_{n})=0
  45. s n 1 s_{n}\neq 1
  46. 2 s n = 2 2^{s_{n}}=2
  47. ζ ( s n ) \zeta(s_{n})\,
  48. ζ ( s n ) = lim s s n η ( s ) 1 - 2 2 s = lim s s n η ( s ) - η ( s n ) 2 2 s n - 2 2 s = lim s s n η ( s ) - η ( s n ) s - s n s - s n 2 2 s n - 2 2 s = η ( s n ) log ( 2 ) . \zeta(s_{n})=\lim_{s\to s_{n}}\frac{\eta(s)}{1-\frac{2}{2^{s}}}=\lim_{s\to s_{% n}}\frac{\eta(s)-\eta(s_{n})}{\frac{2}{2^{s_{n}}}-\frac{2}{2^{s}}}=\lim_{s\to s% _{n}}\frac{\eta(s)-\eta(s_{n})}{s-s_{n}}\,\frac{s-s_{n}}{\frac{2}{2^{s_{n}}}-% \frac{2}{2^{s}}}=\frac{\eta^{\prime}(s_{n})}{\log(2)}.
  49. s n 1 s_{n}\neq 1
  50. s > 0 \Re{s}>0
  51. s = 1 s=1
  52. lim s 1 ( s - 1 ) ζ ( s ) = lim s 1 η ( s ) 1 - 2 1 - s s - 1 = η ( 1 ) log 2 = 1. \lim_{s\to 1}(s-1)\zeta(s)=\lim_{s\to 1}\frac{\eta(s)}{\frac{1-2^{1-s}}{s-1}}=% \frac{\eta(1)}{\log 2}=1.
  53. s > 0. \Re s>0.
  54. Γ ( s ) η ( s ) = 0 x s - 1 e x + 1 d x = 0 0 x x s - 2 e x + 1 d y d x = 0 0 ( t + r ) s - 2 e t + r + 1 d r d t = 0 1 0 1 ( - log ( x y ) ) s - 2 1 + x y d x d y . \begin{aligned}\displaystyle\Gamma(s)\eta(s)&\displaystyle=\int_{0}^{\infty}% \frac{x^{s-1}}{e^{x}+1}\,dx=\int_{0}^{\infty}\int_{0}^{x}\frac{x^{s-2}}{e^{x}+% 1}\,dy\,dx\\ &\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\frac{(t+r)^{s-2}}{e^{t+r}+1}% {dr}\,dt=\int_{0}^{1}\int_{0}^{1}\frac{(-\log(xy))^{s-2}}{1+xy}\,dx\,dy.\end{aligned}
  55. s > - 1 \Re s>-1
  56. 2 1 - s Γ ( s + 1 ) η ( s ) = 2 0 x 2 s + 1 cosh 2 ( x 2 ) d x = 0 t s cosh 2 ( t ) d t . 2^{1-s}\,\Gamma(s+1)\,\eta(s)=2\int_{0}^{\infty}\frac{x^{2s+1}}{\cosh^{2}(x^{2% })}\,dx=\int_{0}^{\infty}\frac{t^{s}}{\cosh^{2}(t)}\,dt.
  57. η ( s ) = - ( 1 / 2 + i t ) - s e π t + e - π t d t . \eta(s)=\int_{-\infty}^{\infty}\frac{(1/2+it)^{-s}}{e^{\pi t}+e^{-\pi t}}\,dt.
  58. ( s - 1 ) ζ ( s ) (s-1)\,\zeta(s)
  59. ( s - 1 ) ζ ( s ) = 2 π - ( 1 / 2 + i t ) 1 - s ( e π t + e - π t ) 2 d t . (s-1)\zeta(s)=2\pi\,\int_{-\infty}^{\infty}\frac{(1/2+it)^{1-s}}{(e^{\pi t}+e^% {-\pi t})^{2}}\,dt.
  60. 0 < c < 1 0<c<1
  61. s s
  62. η ( s ) = 1 2 - ( c + i t ) - s sin ( π ( c + i t ) ) d t . \eta(s)=\frac{1}{2}\int_{-\infty}^{\infty}\frac{(c+it)^{-s}}{\sin{(\pi(c+it))}% }\,dt.
  63. c 0 + c\to 0^{+}
  64. s < 0 \Re s<0
  65. η ( s ) = - sin ( s π 2 ) 0 t - s sinh ( π t ) d t . \eta(s)=-\sin\left(\frac{s\pi}{2}\right)\int_{0}^{\infty}\frac{t^{-s}}{\sinh{(% \pi t)}}\,dt.
  66. η ( s ) = n = 0 1 2 n + 1 k = 0 n ( - 1 ) k ( n k ) 1 ( k + 1 ) s . \eta(s)=\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\sum_{k=0}^{n}(-1)^{k}{n\choose k}% \frac{1}{(k+1)^{s}}.
  67. d k = n i = 0 k ( n + i - 1 ) ! 4 i ( n - i ) ! ( 2 i ) ! d_{k}=n\sum_{i=0}^{k}\frac{(n+i-1)!4^{i}}{(n-i)!(2i)!}
  68. η ( s ) = - 1 d n k = 0 n - 1 ( - 1 ) k ( d k - d n ) ( k + 1 ) s + γ n ( s ) , \eta(s)=-\frac{1}{d_{n}}\sum_{k=0}^{n-1}\frac{(-1)^{k}(d_{k}-d_{n})}{(k+1)^{s}% }+\gamma_{n}(s),
  69. ( s ) 1 2 \Re(s)\geq\frac{1}{2}
  70. | γ n ( s ) | 3 ( 3 + 8 ) n ( 1 + 2 | ( s ) | ) exp ( π 2 | ( s ) | ) . |\gamma_{n}(s)|\leq\frac{3}{(3+\sqrt{8})^{n}}(1+2|\Im(s)|)\exp(\frac{\pi}{2}|% \Im(s)|).
  71. 3 + 8 5.8 3+\sqrt{8}\approx 5.8
  72. η ( 1 - k ) = 2 k - 1 k B k . \eta(1-k)=\frac{2^{k}-1}{k}B_{k}.
  73. η ( 1 ) = ln 2 \!\ \eta(1)=\ln 2
  74. η ( 2 ) = π 2 12 \eta(2)={\pi^{2}\over 12}
  75. η ( 4 ) = 7 π 4 720 0.94703283 \eta(4)={{7\pi^{4}}\over 720}\approx 0.94703283
  76. η ( 6 ) = 31 π 6 30240 0.98555109 \eta(6)={{31\pi^{6}}\over 30240}\approx 0.98555109
  77. η ( 8 ) = 127 π 8 1209600 0.99623300 \eta(8)={{127\pi^{8}}\over 1209600}\approx 0.99623300
  78. η ( 10 ) = 73 π 10 6842880 0.99903951 \eta(10)={{73\pi^{10}}\over 6842880}\approx 0.99903951
  79. η ( 12 ) = 1414477 π 12 1307674368000 0.99975769 \eta(12)={{1414477\pi^{12}}\over{1307674368000}}\approx 0.99975769
  80. η ( 2 n ) = ( - 1 ) n + 1 B 2 n π 2 n ( 2 2 n - 1 - 1 ) ( 2 n ) ! . \eta(2n)=(-1)^{n+1}{{B_{2n}\pi^{2n}(2^{2n-1}-1)}\over{(2n)!}}.
  81. s s
  82. s 1 s\neq 1
  83. η ( s ) = n = 1 ( - 1 ) n ln n n s = 2 1 - s ln 2 ζ ( s ) + ( 1 - 2 1 - s ) ζ ( s ) \eta^{\prime}(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n}\ln n}{n^{s}}=2^{1-s}\ln 2% \zeta(s)+(1-2^{1-s})\zeta^{\prime}(s)
  84. η ( 1 ) = ln ( 2 ) γ - ln ( 2 ) 2 / 2 \eta^{\prime}(1)=\ln(2)\gamma-\ln(2)^{2}/2

Dirichlet_L-function.html

  1. L ( s , χ ) = n = 1 χ ( n ) n s . L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}.
  2. L ( s , χ ) = p ( 1 - χ ( p ) p - s ) - 1 for Re ( s ) > 1 , L(s,\chi)=\prod_{p}\left(1-\chi(p)p^{-s}\right)^{-1}\,\text{ for }\,\text{Re}(% s)>1,
  3. Λ ( s , χ ) = ( π k ) - ( s + a ) / 2 Γ ( s + a 2 ) L ( s , χ ) , \Lambda(s,\chi)=\left(\frac{\pi}{k}\right)^{-(s+a)/2}\Gamma\left(\frac{s+a}{2}% \right)L(s,\chi),
  4. a = { 0 ; if χ ( - 1 ) = 1 , 1 ; if χ ( - 1 ) = - 1 , a=\begin{cases}0;&\mbox{if }~{}\chi(-1)=1,\\ 1;&\mbox{if }~{}\chi(-1)=-1,\end{cases}
  5. Λ ( 1 - s , χ ¯ ) = i a k 1 / 2 τ ( χ ) Λ ( s , χ ) . \Lambda(1-s,\overline{\chi})=\frac{i^{a}k^{1/2}}{\tau(\chi)}\Lambda(s,\chi).
  6. n = 1 k χ ( n ) exp ( 2 π i n / k ) . \sum_{n=1}^{k}\chi(n)\exp(2\pi in/k).
  7. L ( s , χ ) = n = 1 χ ( n ) n s = 1 k s m = 1 k χ ( m ) ζ ( s , m k ) . L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=\frac{1}{k^{s}}\sum_{m=1}^{% k}\chi(m)\;\zeta\left(s,\frac{m}{k}\right).
  8. ζ ( s ) = 1 k s m = 1 k ζ ( s , m k ) . \zeta(s)=\frac{1}{k^{s}}\sum_{m=1}^{k}\zeta\left(s,\frac{m}{k}\right).

Dirichlet_problem.html

  1. D D
  2. D \partial D
  3. u ( x ) = D ν ( s ) G ( x , s ) n d s u(x)=\int_{\partial D}\nu(s)\frac{\partial G(x,s)}{\partial n}ds
  4. G ( x , y ) G(x,y)
  5. G ( x , s ) n = n ^ s G ( x , s ) = i n i G ( x , s ) s i \frac{\partial G(x,s)}{\partial n}=\widehat{n}\cdot\nabla_{s}G(x,s)=\sum_{i}n_% {i}\frac{\partial G(x,s)}{\partial s_{i}}
  6. n ^ \widehat{n}
  7. d s ds
  8. ν ( s ) \nu(s)
  9. f ( x ) = - ν ( x ) 2 + D ν ( s ) G ( x , s ) n d s . f(x)=-\frac{\nu(x)}{2}+\int_{\partial D}\nu(s)\frac{\partial G(x,s)}{\partial n% }ds.
  10. G ( x , s ) = 0 G(x,s)=0
  11. s D s\in\partial D
  12. x D x\in D
  13. f ( s ) f(s)
  14. D C 1 , α \partial D\in C^{1,\alpha}
  15. α ( 0 , 1 ) \alpha\in(0,1)
  16. C 1 , α C^{1,\alpha}
  17. f f
  18. D \partial D
  19. D D
  20. u ( z ) u(z)
  21. u ( z ) = { 1 2 π 0 2 π f ( e i ψ ) 1 - | z | 2 | 1 - z e - i ψ | 2 d ψ if z D f ( z ) if z D . u(z)=\begin{cases}\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{i\psi})\frac{1-|z|^{2}}{|1% -ze^{-i\psi}|^{2}}d\psi&\mbox{if }~{}z\in D\\ f(z)&\mbox{if }~{}z\in\partial D.\end{cases}
  22. u u
  23. D ¯ \bar{D}
  24. D . D.
  25. G ( z , x ) = - 1 2 π log | z - x | + γ ( z , x ) G(z,x)=-\frac{1}{2\pi}\log|z-x|+\gamma(z,x)
  26. γ ( z , x ) \gamma(z,x)
  27. Δ x γ ( z , x ) = 0 \Delta_{x}\gamma(z,x)=0
  28. G ( z , x ) = 0 G(z,x)=0
  29. x D x\in\partial D
  30. 2 t 2 u ( x , t ) - 2 x 2 u ( x , t ) = 0 \frac{\partial{}^{2}}{\partial t^{2}}u(x,t)-\frac{\partial{}^{2}}{\partial x^{% 2}}u(x,t)=0
  31. u ( 0 , t ) = 0 u(0,t)=0
  32. u ( λ t , t ) = 0 u(\lambda t,t)=0
  33. u ( x , t ) = f ( t - x ) - f ( x + t ) u(x,t)=f(t-x)-f(x+t)
  34. f ( t - λ t ) - f ( λ t + t ) = 0 f(t-\lambda t)-f(\lambda t+t)=0
  35. τ = ( λ + 1 ) t \tau=(\lambda+1)t
  36. f ( γ τ ) = f ( τ ) f(\gamma\tau)=f(\tau)
  37. γ = 1 - λ λ + 1 \gamma=\frac{1-\lambda}{\lambda+1}
  38. sin [ log ( e 2 π x ) ] = sin [ log ( x ) ] \sin[\log(e^{2\pi}x)]=\sin[\log(x)]
  39. λ = e 2 π = 1 - i \lambda=e^{2\pi}=1^{-i}
  40. f ( τ ) = g [ log ( γ τ ) ] f(\tau)=g[\log(\gamma\tau)]
  41. g g
  42. log ( γ ) \log(\gamma)
  43. g [ τ + log ( γ ) ] = g ( τ ) g[\tau+\log(\gamma)]=g(\tau)
  44. u ( x , t ) = g [ log ( t - x ) ] - g [ log ( x + t ) ] u(x,t)=g[\log(t-x)]-g[\log(x+t)]

Dirichlet_series.html

  1. n = 1 a n n s , \sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}},
  2. 𝔇 w A ( s ) = a A 1 w ( a ) s = n = 1 a n n s \mathfrak{D}^{A}_{w}(s)=\sum_{a\in A}\frac{1}{w(a)^{s}}=\sum_{n=1}^{\infty}% \frac{a_{n}}{n^{s}}
  3. 𝔇 w A B ( s ) = 𝔇 w A ( s ) + 𝔇 w B ( s ) . \mathfrak{D}^{A\uplus B}_{w}(s)=\mathfrak{D}^{A}_{w}(s)+\mathfrak{D}^{B}_{w}(s).
  4. w ( a , b ) = u ( a ) v ( b ) , w(a,b)=u(a)v(b),
  5. 𝔇 w A × B ( s ) = 𝔇 u A ( s ) 𝔇 v B ( s ) . \mathfrak{D}^{A\times B}_{w}(s)=\mathfrak{D}^{A}_{u}(s)\cdot\mathfrak{D}^{B}_{% v}(s).
  6. n - s m - s = ( n m ) - s . n^{-s}\cdot m^{-s}=(nm)^{-s}.
  7. ζ ( s ) = n = 1 1 n s , \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},
  8. ζ ( s ) = 𝔇 id ( s ) = p prime 𝔇 id { p n : n } ( s ) = p prime n 𝔇 id { p n } ( s ) = p prime n 1 ( p n ) s = p prime n ( 1 p s ) n = p prime 1 1 - p - s , \begin{aligned}\displaystyle\zeta(s)&\displaystyle=\mathfrak{D}^{\mathbb{N}}_{% \mathrm{id}}(s)=\prod_{p\,\mathrm{prime}}\mathfrak{D}^{\{p^{n}:n\in\mathbb{N}% \}}_{\mathrm{id}}(s)=\prod_{p\,\mathrm{prime}}\sum_{n\in\mathbb{N}}\mathfrak{D% }^{\{p^{n}\}}_{\mathrm{id}}(s)\\ &\displaystyle=\prod_{p\,\mathrm{prime}}\sum_{n\in\mathbb{N}}\frac{1}{(p^{n})^% {s}}=\prod_{p\,\mathrm{prime}}\sum_{n\in\mathbb{N}}\left(\frac{1}{p^{s}}\right% )^{n}=\prod_{p\,\mathrm{prime}}\frac{1}{1-p^{-s}}\end{aligned},
  9. 1 ζ ( s ) = n = 1 μ ( n ) n s \frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}
  10. 1 L ( χ , s ) = n = 1 μ ( n ) χ ( n ) n s \frac{1}{L(\chi,s)}=\sum_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^{s}}
  11. ζ ( s - 1 ) ζ ( s ) = n = 1 φ ( n ) n s \frac{\zeta(s-1)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\varphi(n)}{n^{s}}
  12. ζ ( s - k ) ζ ( s ) = n = 1 J k ( n ) n s \frac{\zeta(s-k)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{J_{k}(n)}{n^{s}}
  13. ζ ( s ) ζ ( s - a ) = n = 1 σ a ( n ) n s \zeta(s)\zeta(s-a)=\sum_{n=1}^{\infty}\frac{\sigma_{a}(n)}{n^{s}}
  14. ζ ( s ) ζ ( s - a ) ζ ( s - 2 a ) ζ ( 2 s - 2 a ) = n = 1 σ a ( n 2 ) n s \frac{\zeta(s)\zeta(s-a)\zeta(s-2a)}{\zeta(2s-2a)}=\sum_{n=1}^{\infty}\frac{% \sigma_{a}(n^{2})}{n^{s}}
  15. ζ ( s ) ζ ( s - a ) ζ ( s - b ) ζ ( s - a - b ) ζ ( 2 s - a - b ) = n = 1 σ a ( n ) σ b ( n ) n s \frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}=\sum_{n=1}^{% \infty}\frac{\sigma_{a}(n)\sigma_{b}(n)}{n^{s}}
  16. ζ 2 ( s ) = n = 1 d ( n ) n s \zeta^{2}(s)=\sum_{n=1}^{\infty}\frac{d(n)}{n^{s}}
  17. ζ 3 ( s ) ζ ( 2 s ) = n = 1 d ( n 2 ) n s \frac{\zeta^{3}(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{d(n^{2})}{n^{s}}
  18. ζ 4 ( s ) ζ ( 2 s ) = n = 1 d ( n ) 2 n s . \frac{\zeta^{4}(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{d(n)^{2}}{n^{s}}.
  19. log ζ ( s ) = n = 2 Λ ( n ) log ( n ) 1 n s \log\zeta(s)=\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^{s}}
  20. ζ ( s ) ζ ( s ) = - n = 1 Λ ( n ) n s . \frac{\zeta^{\prime}(s)}{\zeta(s)}=-\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{s}}.
  21. ζ ( 2 s ) ζ ( s ) = n = 1 λ ( n ) n s . \frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}.
  22. σ 1 - s ( m ) ζ ( s ) = n = 1 c n ( m ) n s . \frac{\sigma_{1-s}(m)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{c_{n}(m)}{n^{s}}.
  23. ζ ( s ) ζ ( 2 s ) = n = 1 | μ ( n ) | n s n = 1 μ 2 ( n ) n s . \frac{\zeta(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{s}}\equiv\sum% _{n=1}^{\infty}\frac{\mu^{2}(n)}{n^{s}}.
  24. D ( a , s ) = n = 1 a ( n ) n - s D(a,s)=\sum_{n=1}^{\infty}a(n)n^{-s}
  25. D ( a , s ) + D ( b , s ) = n = 1 ( a + b ) ( n ) n - s D(a,s)+D(b,s)=\sum_{n=1}^{\infty}(a+b)(n)n^{-s}
  26. D ( a , s ) D ( b , s ) = n = 1 ( a * b ) ( n ) n - s D(a,s)\cdot D(b,s)=\sum_{n=1}^{\infty}(a*b)(n)n^{-s}
  27. ( a + b ) ( n ) = a ( n ) + b ( n ) (a+b)(n)=a(n)+b(n)
  28. ( a * b ) ( n ) = k | n a ( k ) b ( n / k ) (a*b)(n)=\sum_{k|n}a(k)b(n/k)
  29. f ( s ) = n = 1 a n n s f(s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}
  30. F ( s ) = n = 1 f ( n ) n s F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}
  31. F ( s ) = - n = 1 f ( n ) log ( n ) n s F^{\prime}(s)=-\sum_{n=1}^{\infty}\frac{f(n)\log(n)}{n^{s}}
  32. F ( s ) F ( s ) = - n = 1 f ( n ) Λ ( n ) n s \frac{F^{\prime}(s)}{F(s)}=-\sum_{n=1}^{\infty}\frac{f(n)\Lambda(n)}{n^{s}}
  33. F ( s ) = n = 1 f ( n ) n - s F(s)=\sum_{n=1}^{\infty}f(n)n^{-s}
  34. G ( s ) = n = 1 g ( n ) n - s . G(s)=\sum_{n=1}^{\infty}g(n)n^{-s}.
  35. 1 2 T - T T F ( a + i t ) G ( b - i t ) d t = n = 1 f ( n ) g ( n ) n - a - b as T . \frac{1}{2T}\int_{-T}^{T}\,F(a+it)G(b-it)\,dt=\sum_{n=1}^{\infty}f(n)g(n)n^{-a% -b}\,\text{ as }T\sim\infty.
  36. 1 2 T - T T | F ( a + i t ) | 2 d t = n = 1 [ f ( n ) ] 2 n - 2 a as T . \frac{1}{2T}\int_{-T}^{T}|F(a+it)|^{2}dt=\sum_{n=1}^{\infty}[f(n)]^{2}n^{-2a}% \,\text{ as }T\sim\infty.
  37. ζ ( s ) m = n = 1 a n n s \zeta(s)^{m}=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}
  38. n = 1 a n x n = x + ( m 1 ) a = 2 x a + ( m 2 ) a = 2 b = 2 x a b + ( m 3 ) a = 2 b = 2 c = 2 x a b c + ( m 4 ) a = 2 b = 2 c = 2 d = 2 x a b c d + \sum\limits_{n=1}^{\infty}a_{n}x^{n}=x+{m\choose 1}\sum\limits_{a=2}^{\infty}x% ^{a}+{m\choose 2}\sum\limits_{a=2}^{\infty}\sum\limits_{b=2}^{\infty}x^{ab}+{m% \choose 3}\sum\limits_{a=2}^{\infty}\sum\limits_{b=2}^{\infty}\sum\limits_{c=2% }^{\infty}x^{abc}+{m\choose 4}\sum\limits_{a=2}^{\infty}\sum\limits_{b=2}^{% \infty}\sum\limits_{c=2}^{\infty}\sum\limits_{d=2}^{\infty}x^{abcd}+...

Disc_integration.html

  1. x x
  2. π a b [ R ( x ) ] 2 d x \pi\int_{a}^{b}[R(x)]^{2}\ \mathrm{d}x
  3. R ( x ) R(x)
  4. y = 3 y=3
  5. y y
  6. π c d [ R ( y ) ] 2 d y \pi\int_{c}^{d}[R(y)]^{2}\ \mathrm{d}y
  7. R ( y ) R(y)
  8. x = 4 x=4
  9. π a b ( [ R O ( x ) ] 2 - [ R I ( x ) ] 2 ) d x \pi\int_{a}^{b}(\left[R_{O}(x)\right]^{2}-\left[R_{I}(x)\right]^{2})\mathrm{d}x
  10. R O ( x ) R_{O}(x)
  11. R I ( x ) R_{I}(x)
  12. [ R O ( x ) ] 2 - [ R I ( x ) ] 2 [ R O ( x ) - R I ( x ) ] 2 [R_{O}(x)]^{2}-[R_{I}(x)]^{2}\ \not\equiv\;[R_{O}(x)-R_{I}(x)]^{2}
  13. h h
  14. π a b ( [ h - R O ( x ) ] 2 - [ h - R I ( x ) ] 2 ) d x . \pi\int_{a}^{b}([h-R_{O}(x)]^{2}-[h-R_{I}(x)]^{2})\,\mathrm{d}x.
  15. y = - 2 x + x 2 y=-2x+x^{2}
  16. y = x y=x
  17. y = 4 y=4
  18. π 0 3 ( [ 4 - ( - 2 x + x 2 ) ] 2 - [ 4 - x ] 2 ) d x . \pi\int_{0}^{3}([4-(-2x+x^{2})]^{2}-[4-x]^{2})\,\mathrm{d}x.
  19. x x
  20. y = x y=x
  21. y = - 2 x + x 2 y=-2x+x^{2}
  22. y = 4 y=4
  23. x x

Discrete_geometry.html

  1. C = ( P , L , I ) . C=(P,L,I).\,
  2. I P × L I\subseteq P\times L
  3. I I
  4. ( p , l ) I , (p,l)\in I,
  5. l l

Discretization.html

  1. 𝐱 ˙ ( t ) = 𝐀𝐱 ( t ) + 𝐁𝐮 ( t ) + 𝐰 ( t ) \dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)+\mathbf{w}% (t)
  2. 𝐲 ( t ) = 𝐂𝐱 ( t ) + 𝐃𝐮 ( t ) + 𝐯 ( t ) \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{u}(t)+\mathbf{v}(t)
  3. 𝐰 ( t ) N ( 0 , 𝐐 ) \mathbf{w}(t)\sim N(0,\mathbf{Q})
  4. 𝐯 ( t ) N ( 0 , 𝐑 ) \mathbf{v}(t)\sim N(0,\mathbf{R})
  5. 𝐱 [ k + 1 ] = 𝐀 d 𝐱 [ k ] + 𝐁 d 𝐮 [ k ] + 𝐰 [ k ] \mathbf{x}[k+1]=\mathbf{A}_{d}\mathbf{x}[k]+\mathbf{B}_{d}\mathbf{u}[k]+% \mathbf{w}[k]
  6. 𝐲 [ k ] = 𝐂 d 𝐱 [ k ] + 𝐃 d 𝐮 [ k ] + 𝐯 [ k ] \mathbf{y}[k]=\mathbf{C}_{d}\mathbf{x}[k]+\mathbf{D}_{d}\mathbf{u}[k]+\mathbf{% v}[k]
  7. 𝐰 [ k ] N ( 0 , 𝐐 d ) \mathbf{w}[k]\sim N(0,\mathbf{Q}_{d})
  8. 𝐯 [ k ] N ( 0 , 𝐑 d ) \mathbf{v}[k]\sim N(0,\mathbf{R}_{d})
  9. 𝐀 d = e 𝐀 T = - 1 { ( s 𝐈 - 𝐀 ) - 1 } t = T \mathbf{A}_{d}=e^{\mathbf{A}T}=\mathcal{L}^{-1}\{(s\mathbf{I}-\mathbf{A})^{-1}% \}_{t=T}
  10. 𝐁 d = ( τ = 0 T e 𝐀 τ d τ ) 𝐁 = 𝐀 - 1 ( 𝐀 d - I ) 𝐁 \mathbf{B}_{d}=\left(\int_{\tau=0}^{T}e^{\mathbf{A}\tau}d\tau\right)\mathbf{B}% =\mathbf{A}^{-1}(\mathbf{A}_{d}-I)\mathbf{B}
  11. 𝐀 \mathbf{A}
  12. 𝐂 d = 𝐂 \mathbf{C}_{d}=\mathbf{C}
  13. 𝐃 d = 𝐃 \mathbf{D}_{d}=\mathbf{D}
  14. 𝐐 d = τ = 0 T e 𝐀 τ 𝐐 e 𝐀 T τ d τ \mathbf{Q}_{d}=\int_{\tau=0}^{T}e^{\mathbf{A}\tau}\mathbf{Q}e^{\mathbf{A}^{T}% \tau}d\tau
  15. 𝐑 d = 1 T 𝐑 \mathbf{R}_{d}=\frac{1}{T}\mathbf{R}
  16. T T
  17. 𝐀 T \mathbf{A}^{T}
  18. 𝐀 \mathbf{A}
  19. e [ 𝐀 𝐁 𝟎 𝟎 ] T = [ 𝐌 𝟏𝟏 𝐌 𝟏𝟐 𝟎 𝐈 ] e^{\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}T}=\begin{bmatrix}\mathbf{M_{11}}&\mathbf{M_% {12}}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}
  20. 𝐀 d = 𝐌 11 \mathbf{A}_{d}=\mathbf{M}_{11}
  21. 𝐁 d = 𝐌 12 \mathbf{B}_{d}=\mathbf{M}_{12}
  22. 𝐐 d \mathbf{Q}_{d}
  23. 𝐅 = [ - 𝐀 𝐐 𝟎 𝐀 T ] T \mathbf{F}=\begin{bmatrix}-\mathbf{A}&\mathbf{Q}\\ \mathbf{0}&\mathbf{A}^{T}\end{bmatrix}T
  24. 𝐆 = e 𝐅 = [ 𝐀 d - 1 𝐐 d 𝟎 𝐀 d T ] . \mathbf{G}=e^{\mathbf{F}}=\begin{bmatrix}\dots&\mathbf{A}_{d}^{-1}\mathbf{Q}_{% d}\\ \mathbf{0}&\mathbf{A}_{d}^{T}\end{bmatrix}.
  25. 𝐐 d = ( 𝐀 d T ) T ( 𝐀 d - 1 𝐐 d ) . \mathbf{Q}_{d}=(\mathbf{A}_{d}^{T})^{T}(\mathbf{A}_{d}^{-1}\mathbf{Q}_{d}).
  26. 𝐱 ˙ ( t ) = 𝐀𝐱 ( t ) + 𝐁𝐮 ( t ) \mathbf{\dot{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)
  27. d d t e 𝐀 t = 𝐀 e 𝐀 t = e 𝐀 t 𝐀 \frac{d}{dt}e^{\mathbf{A}t}=\mathbf{A}e^{\mathbf{A}t}=e^{\mathbf{A}t}\mathbf{A}
  28. e - 𝐀 t 𝐱 ˙ ( t ) = e - 𝐀 t 𝐀𝐱 ( t ) + e - 𝐀 t 𝐁𝐮 ( t ) e^{-\mathbf{A}t}\mathbf{\dot{x}}(t)=e^{-\mathbf{A}t}\mathbf{A}\mathbf{x}(t)+e^% {-\mathbf{A}t}\mathbf{B}\mathbf{u}(t)
  29. d d t ( e - 𝐀 t 𝐱 ( t ) ) = e - 𝐀 t 𝐁𝐮 ( t ) \frac{d}{dt}(e^{-\mathbf{A}t}\mathbf{x}(t))=e^{-\mathbf{A}t}\mathbf{B}\mathbf{% u}(t)
  30. e - 𝐀 t 𝐱 ( t ) - e 0 𝐱 ( 0 ) = 0 t e - 𝐀 τ 𝐁𝐮 ( τ ) d τ e^{-\mathbf{A}t}\mathbf{x}(t)-e^{0}\mathbf{x}(0)=\int_{0}^{t}e^{-\mathbf{A}% \tau}\mathbf{B}\mathbf{u}(\tau)d\tau
  31. 𝐱 ( t ) = e 𝐀 t 𝐱 ( 0 ) + 0 t e 𝐀 ( t - τ ) 𝐁𝐮 ( τ ) d τ \mathbf{x}(t)=e^{\mathbf{A}t}\mathbf{x}(0)+\int_{0}^{t}e^{\mathbf{A}(t-\tau)}% \mathbf{B}\mathbf{u}(\tau)d\tau
  32. 𝐱 [ k ] = def 𝐱 ( k T ) \mathbf{x}[k]\ \stackrel{\mathrm{def}}{=}\ \mathbf{x}(kT)
  33. 𝐱 [ k ] = e 𝐀 k T 𝐱 ( 0 ) + 0 k T e 𝐀 ( k T - τ ) 𝐁𝐮 ( τ ) d τ \mathbf{x}[k]=e^{\mathbf{A}kT}\mathbf{x}(0)+\int_{0}^{kT}e^{\mathbf{A}(kT-\tau% )}\mathbf{B}\mathbf{u}(\tau)d\tau
  34. 𝐱 [ k + 1 ] = e 𝐀 ( k + 1 ) T 𝐱 ( 0 ) + 0 ( k + 1 ) T e 𝐀 ( ( k + 1 ) T - τ ) 𝐁𝐮 ( τ ) d τ \mathbf{x}[k+1]=e^{\mathbf{A}(k+1)T}\mathbf{x}(0)+\int_{0}^{(k+1)T}e^{\mathbf{% A}((k+1)T-\tau)}\mathbf{B}\mathbf{u}(\tau)d\tau
  35. 𝐱 [ k + 1 ] = e 𝐀 T [ e 𝐀 k T 𝐱 ( 0 ) + 0 k T e 𝐀 ( k T - τ ) 𝐁𝐮 ( τ ) d τ ] + k T ( k + 1 ) T e 𝐀 ( k T + T - τ ) 𝐁𝐮 ( τ ) d τ \mathbf{x}[k+1]=e^{\mathbf{A}T}\left[e^{\mathbf{A}kT}\mathbf{x}(0)+\int_{0}^{% kT}e^{\mathbf{A}(kT-\tau)}\mathbf{B}\mathbf{u}(\tau)d\tau\right]+\int_{kT}^{(k% +1)T}e^{\mathbf{A}(kT+T-\tau)}\mathbf{B}\mathbf{u}(\tau)d\tau
  36. 𝐱 [ k ] \mathbf{x}[k]
  37. v = k T + T - τ v=kT+T-\tau
  38. 𝐮 \mathbf{u}
  39. 𝐱 [ k + 1 ] = e 𝐀 T 𝐱 [ k ] + ( 0 T e 𝐀 v d v ) 𝐁𝐮 [ k ] = e 𝐀 T 𝐱 [ k ] + 𝐀 - 1 ( e 𝐀 T - 𝐈 ) 𝐁𝐮 [ k ] \begin{matrix}\mathbf{x}[k+1]&=&e^{\mathbf{A}T}\mathbf{x}[k]+\left(\int_{0}^{T% }e^{\mathbf{A}v}dv\right)\mathbf{B}\mathbf{u}[k]\\ &=&e^{\mathbf{A}T}\mathbf{x}[k]+\mathbf{A}^{-1}\left(e^{\mathbf{A}T}-\mathbf{I% }\right)\mathbf{B}\mathbf{u}[k]\end{matrix}
  40. e 𝐀 T 𝐈 + 𝐀 T e^{\mathbf{A}T}\approx\mathbf{I}+\mathbf{A}T
  41. 𝐱 [ k + 1 ] ( 𝐈 + 𝐀 T ) 𝐱 [ k ] + T 𝐁𝐮 [ k ] \mathbf{x}[k+1]\approx(\mathbf{I}+\mathbf{A}T)\mathbf{x}[k]+T\mathbf{B}\mathbf% {u}[k]
  42. e 𝐀 T ( 𝐈 - 𝐀 T ) - 1 e^{\mathbf{A}T}\approx\left(\mathbf{I}-\mathbf{A}T\right)^{-1}
  43. e 𝐀 T ( 𝐈 + 1 2 𝐀 T ) ( 𝐈 - 1 2 𝐀 T ) - 1 e^{\mathbf{A}T}\approx\left(\mathbf{I}+\frac{1}{2}\mathbf{A}T\right)\left(% \mathbf{I}-\frac{1}{2}\mathbf{A}T\right)^{-1}

Discretization_error.html

  1. f ( x ) \,\!f(x)
  2. f ( x ) = lim h 0 f ( x + h ) - f ( x ) h f^{\prime}(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-f(x)}{h}}
  3. f ( x ) f ( x + h ) - f ( x ) h f^{\prime}(x)\approx\frac{f(x+h)-f(x)}{h}
  4. h \,\!h

Diseconomies_of_scale.html

  1. n ( n - 1 ) 2 \frac{n(n-1)}{2}

Disk_(mathematics).html

  1. ( a , b ) (a,b)
  2. D = { ( x , y ) 2 : ( x - a ) 2 + ( y - b ) 2 < R 2 } D=\{(x,y)\in{\mathbb{R}^{2}}:(x-a)^{2}+(y-b)^{2}<R^{2}\}
  3. D ¯ = { ( x , y ) 2 : ( x - a ) 2 + ( y - b ) 2 R 2 } . \overline{D}=\{(x,y)\in{\mathbb{R}^{2}}:(x-a)^{2}+(y-b)^{2}\leq R^{2}\}.
  4. f ( x , y ) = ( x + 1 - y 2 2 , y ) f(x,y)=\left(\frac{x+\sqrt{1-y^{2}}}{2},y\right)

Displacement_current.html

  1. s y m b o l D = ε 0 s y m b o l E + s y m b o l P . symbol{D}=\varepsilon_{0}symbol{E}+symbol{P}\ .
  2. s y m b o l J s y m b o l D = ε 0 s y m b o l E t + s y m b o l P t . symbol{J}_{s}ymbol{D}=\varepsilon_{0}\frac{\partial symbol{E}}{\partial t}+% \frac{\partial symbol{P}}{\partial t}\ .
  3. s y m b o l D = ε s y m b o l E , symbol{D}=\varepsilon symbol{E}\ ,
  4. I D = ε Φ E t . I_{\mathrm{D}}=\varepsilon\frac{\partial\Phi_{E}}{\partial t}.
  5. s y m b o l P = ε 0 χ e s y m b o l E = ε 0 ( ε r - 1 ) s y m b o l E symbol{P}=\varepsilon_{0}\chi_{e}symbol{E}=\varepsilon_{0}(\varepsilon_{r}-1)% symbol{E}
  6. ε = ε r ε 0 = ( 1 + χ e ) ε 0 . \varepsilon=\varepsilon_{r}\varepsilon_{0}=(1+\chi_{e})\varepsilon_{0}.
  7. C 𝐁 s y m b o l d s y m b o l = μ 0 I D . \oint_{C}\mathbf{B}\ symbol{\cdot}\ \mathrm{d}symbol{\ell}=\mu_{0}I_{D}\ .
  8. C \oint_{C}
  9. 𝐁 \mathbf{B}
  10. s y m b o l symbol{\cdot}
  11. d s y m b o l \mathrm{d}symbol{\ell}
  12. μ 0 \mu_{0}\!
  13. I D I_{D}\!
  14. I D = I , I_{D}=I\ ,
  15. Q ( t ) = ε 0 𝒮 d 𝒮 s y m b o l s y m b o l E ( t ) , Q(t)=\varepsilon_{0}\oint_{\mathcal{S}}d\mathbf{\mathcal{S}}\ symbol{\cdot}\ % symbol{E}(t)\ ,
  16. d Q d t = I = ε 0 𝒮 d 𝒮 s y m b o l s y m b o l E t - S ε 0 E t , \frac{dQ}{dt}=\mathit{I}=\varepsilon_{0}\oint_{\mathcal{S}}d\mathbf{\mathcal{S% }}\ symbol{\cdot}\ \frac{\partial symbol{E}}{\partial t}\approx-{S}\ % \varepsilon_{0}\frac{\partial E}{\partial t}\ ,
  17. J D = I D S = - I S = ε 0 E t = D t , J_{D}=\frac{I_{D}}{S}=-\frac{I}{S}=\varepsilon_{0}\frac{\partial E}{\partial t% }=\frac{\partial D}{\partial t}\ ,
  18. S s y m b o l B d s y m b o l = μ 0 S ( s y m b o l J + ϵ 0 s y m b o l E t ) d s y m b o l S \oint_{\partial S}symbol{B}\cdot dsymbol{\ell}=\mu_{0}\int_{S}(symbol{J}+% \epsilon_{0}\frac{\partial symbol{E}}{\partial t})\cdot dsymbol{S}\,
  19. B = μ 0 I 2 π r B=\frac{\mu_{0}I}{2\pi r}\,
  20. S \partial S
  21. B = μ 0 I D 2 π r B=\frac{\mu_{0}I_{D}}{2\pi r}\,
  22. s y m b o l J f = - ρ f t , \nabla symbol{\cdot J_{f}}=-\frac{\partial\rho_{f}}{\partial t}\ ,
  23. s y m b o l × B = μ 0 s y m b o l J f , symbol{\nabla\times B}=\mu_{0}symbolJ_{f}\ ,
  24. s y m b o l × B = μ 0 ( s y m b o l J + ε 0 s y m b o l E t ) = μ 0 ( s y m b o l J f + s y m b o l D t ) , symbol{\nabla\times B}=\mu_{0}\left(symbolJ+\varepsilon_{0}\frac{\partial symbolE% }{\partial t}\right)=\mu_{0}\left(symbolJ_{f}+\frac{\partial symbolD}{\partial t% }\right)\ ,
  25. s y m b o l ( s y m b o l × B ) = 0 = μ 0 ( s y m b o l J f + t s y m b o l D ) , symbol{\nabla\cdot}\left(symbol{\nabla\times B}\right)=0=\mu_{0}\left(\nabla% \cdot symbolJ_{f}+\frac{\partial}{\partial t}symbol{\nabla\cdot D}\right)\ ,
  26. s y m b o l D = ρ f . symbol{\nabla\cdot D}=\rho_{f}\ .
  27. s y m b o l J D = ϵ 0 s y m b o l E t symbol{J_{D}}=\epsilon_{0}\frac{\partial symbol{E}}{\partial t}
  28. s y m b o l × B = μ 0 s y m b o l J D , symbol{\nabla\times B}=\mu_{0}symbol{J_{D}}\ ,
  29. s y m b o l × ( s y m b o l × B ) = μ 0 ϵ 0 t s y m b o l × E . symbol{\nabla\times}\left(symbol{\nabla\times B}\right)=\mu_{0}\epsilon_{0}% \frac{\partial}{\partial t}symbol{\nabla\times E}\ .
  30. s y m b o l × E = - t s y m b o l B , symbol{\nabla\times E}=-\frac{\partial}{\partial t}symbolB\ ,
  31. - s y m b o l × ( s y m b o l × B ) = 2 s y m b o l B = μ 0 ϵ 0 2 t 2 s y m b o l B = 1 c 2 2 t 2 s y m b o l B , -symbol{\nabla\times}\left(symbol{\nabla\times B}\right)=\nabla^{2}symbolB=\mu% _{0}\epsilon_{0}\frac{\partial^{2}}{\partial t^{2}}symbol{B}=\frac{1}{c^{2}}% \frac{\partial^{2}}{\partial t^{2}}symbol{B}\ ,
  32. s y m b o l × ( s y m b o l × V ) = s y m b o l ( s y m b o l V ) - 2 s y m b o l V , symbol{\nabla\times}\left(symbol{\nabla\times V}\right)=symbol{\nabla}\left(% symbol{\nabla\cdot V}\right)-\nabla^{2}symbolV\ ,
  33. s y m b o l × ( s y m b o l × E ) = - t s y m b o l × s y m b o l B = - μ 0 t ( s y m b o l J + ϵ 0 t s y m b o l E ) . symbol{\nabla\times}\left(symbol{\nabla\times E}\right)=-\frac{\partial}{% \partial t}symbol{\nabla\times}symbol{B}=-\mu_{0}\frac{\partial}{\partial t}% \left(symbolJ+\epsilon_{0}\frac{\partial}{\partial t}symbolE\right)\ .
  34. 2 s y m b o l E = μ 0 ϵ 0 2 t 2 s y m b o l E = 1 c 2 2 t 2 s y m b o l E . \nabla^{2}symbolE=\mu_{0}\epsilon_{0}\frac{\partial^{2}}{\partial t^{2}}symbol% {E}=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}symbol{E}\ .
  35. s y m b o l E = - s y m b o l φ - s y m b o l A t , symbol{E}=-symbol{\nabla}\varphi-\frac{\partial symbol{A}}{\partial t}\ ,
  36. J = d d t 1 4 π E 2 E = d d t ε r ε 0 E = d d t D . J=\frac{d}{dt}\frac{1}{4\pi\mathrm{E}^{2}}\mathit{E}=\frac{d}{dt}\varepsilon_{% r}\varepsilon_{0}\mathit{E}=\frac{d}{dt}\mathit{D}\ .

Display_size.html

  1. h e i g h t = H × D W 2 + H 2 height=H\times{\frac{D}{\sqrt{W^{2}+H^{2}}}}
  2. w i d t h = W H × h e i g h t width=\frac{W}{H}\times height

Distributed_coordination_function.html

  1. BackoffTime = random ( ) × aSlotTime \mathrm{BackoffTime}=\mathrm{random}()\times\mathrm{aSlotTime}

Distribution_function.html

  1. f ( x , y , z , t ; v x , v y , v z ) f(x,y,z,t;v_{x},v_{y},v_{z})
  2. ( v x , v y , v z ) (v_{x},v_{y},v_{z})
  3. ( x , y , z ) (x,y,z)
  4. ( t ) (t)
  5. n ( x , y , z , t ) = f d v x d v y d v z n(x,y,z,t)=\int f\,dv_{x}\,dv_{y}\,dv_{z}
  6. N ( t ) = n d x d y d z . N(t)=\int n\,dx\,dy\,dz.
  7. f ( x , y , z ; p x , p y , p z ) f(x,y,z;p_{x},p_{y},p_{z})
  8. ( p x , p y , p z ) (p_{x},p_{y},p_{z})
  9. k k
  10. T T
  11. f = n ( m 2 π k T ) 3 / 2 exp ( - m ( v x 2 + v y 2 + v z 2 ) 2 k T ) . f=n\left(\frac{m}{2\pi kT}\right)^{3/2}\exp\left({-\frac{m(v_{x}^{2}+v_{y}^{2}% +v_{z}^{2})}{2kT}}\right).
  12. m ( ( v x - u x ) 2 + ( v y - u y ) 2 + ( v z - u z ) 2 ) m((v_{x}-u_{x})^{2}+(v_{y}-u_{y})^{2}+(v_{z}-u_{z})^{2})
  13. ( u x , u y , u z ) (u_{x},u_{y},u_{z})

Distributivity_(order_theory).html

  1. \vee
  2. \wedge
  3. x ( y z ) = ( x y ) ( x z ) x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)
  4. x ( y z ) = ( x y ) ( x z ) x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z)
  5. x S = { x s s S } x\wedge\bigvee S=\bigvee\{x\wedge s\mid s\in S\}
  6. x S = { x s s S } x\vee\bigwedge S=\bigwedge\{x\vee s\mid s\in S\}
  7. j J k K ( j ) x j , k = f F j J x j , f ( j ) \bigwedge_{j\in J}\bigvee_{k\in K(j)}x_{j,k}=\bigvee_{f\in F}\bigwedge_{j\in J% }x_{j,f(j)}

Dividend_yield.html

  1. Current Dividend Yield = Most Recent Full-Year Dividend Current Share Price \mbox{Current Dividend Yield}~{}=\frac{\mbox{Most Recent Full-Year Dividend}~{% }}{\mbox{Current Share Price}~{}}
  2. Current Dividend Yield = Most Recent Full-Year Dividend Current Share Price = $ 1 $ 20 = 0.05 = 5 % \begin{array}[]{lcl}\mbox{Current Dividend Yield}&=&\frac{\mbox{Most Recent % Full-Year Dividend}~{}}{\mbox{Current Share Price}~{}}\\ &=&\frac{\$1}{\$20}\\ &=&0.05\\ &=&5\%\\ \end{array}

Dividing_a_circle_into_areas.html

  1. f ( n ) = f ( n - 1 ) + i = 1 n - 1 ( 1 + ( n - i - 1 ) ( i - 1 ) ) f(n)=f(n-1)+\sum^{n-1}_{i=1}\left(1+\left(n-i-1\right)\left(i-1\right)\right)
  2. f ( n ) = f ( n - 1 ) + i = 1 n - 1 ( 2 - n + n i - i 2 ) f(n)=f(n-1)+\sum^{n-1}_{i=1}\left(2-n+ni-i^{2}\right)
  3. ( n - 1 ) (n-1)
  4. ( n - 1 ) (n-1)
  5. f ( n ) = f ( n - 1 ) + 1 6 n 3 - n 2 + 17 6 n - 2 f(n)=f(n-1)+\frac{1}{6}n^{3}-n^{2}+\frac{17}{6}n-2
  6. f ( n ) = k = 1 n ( 1 6 k 3 - k 2 + 17 6 k - 2 ) + 1 f(n)=\sum^{n}_{k=1}\left(\frac{1}{6}k^{3}-k^{2}+\frac{17}{6}k-2\right)+1
  7. f ( 0 ) = 1 f(0)=1
  8. f ( n ) = n 24 ( n 3 - 6 n 2 + 23 n - 18 ) + 1 f(n)=\frac{n}{24}(n^{3}-6n^{2}+23n-18)+1
  9. V - E + F = 2 \,V-E+F=2
  10. V interior = ( n 4 ) , V_{\,\text{interior}}={n\choose 4},
  11. V = V exterior + V interior = n + ( n 4 ) . V=V_{\,\text{exterior}}+V_{\,\text{interior}}=n+{n\choose 4}.
  12. 4 ( n 4 ) 4{n\choose 4}
  13. 2 ( ( n 2 ) - n ) 2\left({n\choose 2}-n\right)
  14. E = 4 ( n 4 ) + 2 ( ( n 2 ) - n ) 2 + n + n = 2 ( n 4 ) + ( n 2 ) + n . E=\frac{4{n\choose 4}+2\left({n\choose 2}-n\right)}{2}+n+n=2{n\choose 4}+{n% \choose 2}+n.
  15. F = E - V + 2 , \,F=E-V+2,
  16. F = ( n 4 ) + ( n 2 ) + 2. F={n\choose 4}+{n\choose 2}+2.
  17. r G = ( n 4 ) + ( n 2 ) + 1 , r_{G}={n\choose 4}+{n\choose 2}+1,
  18. r G = n ! ( n - 4 ) ! 4 ! + n ! ( n - 2 ) ! 2 ! + 1 r_{G}=\frac{n!}{(n-4)!4!}+\frac{n!}{(n-2)!2!}+1
  19. r G = 1 24 n ( n 3 - 6 n 2 + 23 n - 18 ) + 1 r_{G}=\frac{1}{24}n(n^{3}-6n^{2}+23n-18)+1

Divisible_group.html

  1. \mathbb{Q}
  2. \mathbb{Q}
  3. / \mathbb{Q}/\mathbb{Z}
  4. [ 1 / p ] / \mathbb{Z}[1/p]/\mathbb{Z}
  5. / \mathbb{Q}/\mathbb{Z}
  6. [ p ] \mathbb{Z}[p^{\infty}]
  7. * \mathbb{C}^{*}
  8. A A
  9. T T
  10. Hom 𝐙 ( A , T ) \mathrm{Hom}_{\mathbf{Z}}(A,T)
  11. A A
  12. G = Tor ( G ) G / Tor ( G ) . G=\mathrm{Tor}(G)\oplus G/\mathrm{Tor}(G).
  13. G / Tor ( G ) = i I = ( I ) . G/\mathrm{Tor}(G)=\oplus_{i\in I}\mathbb{Q}=\mathbb{Q}^{(I)}.
  14. I p I_{p}
  15. ( Tor ( G ) ) p = i I p [ p ] = [ p ] ( I p ) , (\mathrm{Tor}(G))_{p}=\oplus_{i\in I_{p}}\mathbb{Z}[p^{\infty}]=\mathbb{Z}[p^{% \infty}]^{(I_{p})},
  16. ( Tor ( G ) ) p (\mathrm{Tor}(G))_{p}
  17. G = ( p 𝐏 [ p ] ( I p ) ) ( I ) . G=\left(\bigoplus_{p\in\mathbf{P}}\mathbb{Z}[p^{\infty}]^{(I_{p})}\right)% \oplus\mathbb{Q}^{(I)}.

Divisor_function.html

  1. σ x ( n ) = d | n d x , \sigma_{x}(n)=\sum_{d|n}d^{x}\,\!,
  2. d | n {d|n}
  3. σ 0 ( 12 ) \displaystyle\sigma_{0}(12)
  4. σ 1 ( 12 ) \displaystyle\sigma_{1}(12)
  5. s ( 12 ) \displaystyle s(12)
  6. x = 2 x=2
  7. x = 3 x=3
  8. σ 0 ( n ) \sigma_{0}(n)
  9. n \sqrt{n}
  10. σ 0 ( n ) \sigma_{0}(n)
  11. σ 0 ( p ) = 2 σ 0 ( p n ) = n + 1 σ 1 ( p ) = p + 1 \begin{aligned}\displaystyle\sigma_{0}(p)&\displaystyle=2\\ \displaystyle\sigma_{0}(p^{n})&\displaystyle=n+1\\ \displaystyle\sigma_{1}(p)&\displaystyle=p+1\end{aligned}
  12. σ 0 ( p n # ) = 2 n \sigma_{0}(p_{n}\#)=2^{n}\,
  13. p i p_{i}
  14. 1 < σ 0 ( n ) < n 1<\sigma_{0}(n)<n
  15. n = i = 1 r p i a i n=\prod_{i=1}^{r}p_{i}^{a_{i}}
  16. σ x ( n ) = i = 1 r p i ( a i + 1 ) x - 1 p i x - 1 \sigma_{x}(n)=\prod_{i=1}^{r}\frac{p_{i}^{(a_{i}+1)x}-1}{p_{i}^{x}-1}
  17. σ x ( n ) = i = 1 r j = 0 a i p i j x = i = 1 r ( 1 + p i x + p i 2 x + + p i a i x ) . \sigma_{x}(n)=\prod_{i=1}^{r}\sum_{j=0}^{a_{i}}p_{i}^{jx}=\prod_{i=1}^{r}(1+p_% {i}^{x}+p_{i}^{2x}+\cdots+p_{i}^{a_{i}x}).
  18. σ 0 ( n ) = i = 1 r ( a i + 1 ) . \sigma_{0}(n)=\prod_{i=1}^{r}(a_{i}+1).
  19. σ 0 ( 24 ) \sigma_{0}(24)
  20. σ 0 ( 24 ) \displaystyle\sigma_{0}(24)
  21. σ ( n ) = 2 × 2 k - 1 = 2 n - 1 , \sigma(n)=2\times 2^{k}-1=2n-1,
  22. σ ( n ) = ( p + 1 ) ( q + 1 ) = n + 1 + ( p + q ) , \sigma(n)=(p+1)(q+1)=n+1+(p+q),\,
  23. φ ( n ) = ( p - 1 ) ( q - 1 ) = n + 1 - ( p + q ) , \varphi(n)=(p-1)(q-1)=n+1-(p+q),\,
  24. n + 1 = ( σ ( n ) + φ ( n ) ) / 2 , n+1=(\sigma(n)+\varphi(n))/2,\,
  25. p + q = ( σ ( n ) - φ ( n ) ) / 2 , p+q=(\sigma(n)-\varphi(n))/2,\,
  26. ( x - p ) ( x - q ) = x 2 - ( p + q ) x + n = x 2 - [ ( σ ( n ) - φ ( n ) ) / 2 ] x + [ ( σ ( n ) + φ ( n ) ) / 2 - 1 ] = 0 (x-p)(x-q)=x^{2}-(p+q)x+n=x^{2}-[(\sigma(n)-\varphi(n))/2]x+[(\sigma(n)+% \varphi(n))/2-1]=0\,
  27. p = ( σ ( n ) - φ ( n ) ) / 4 - [ ( σ ( n ) - φ ( n ) ) / 4 ] 2 - [ ( σ ( n ) + φ ( n ) ) / 2 - 1 ] , p=(\sigma(n)-\varphi(n))/4-\sqrt{[(\sigma(n)-\varphi(n))/4]^{2}-[(\sigma(n)+% \varphi(n))/2-1]},\,
  28. q = ( σ ( n ) - φ ( n ) ) / 4 + [ ( σ ( n ) - φ ( n ) ) / 4 ] 2 - [ ( σ ( n ) + φ ( n ) ) / 2 - 1 ] . q=(\sigma(n)-\varphi(n))/4+\sqrt{[(\sigma(n)-\varphi(n))/4]^{2}-[(\sigma(n)+% \varphi(n))/2-1]}.\,
  29. σ 0 ( n ) = σ 0 ( n + 1 ) \sigma_{0}(n)=\sigma_{0}(n+1)
  30. n = 1 σ a ( n ) n s = ζ ( s ) ζ ( s - a ) , \sum_{n=1}^{\infty}\frac{\sigma_{a}(n)}{n^{s}}=\zeta(s)\zeta(s-a),
  31. n = 1 d ( n ) n s = ζ 2 ( s ) , \sum_{n=1}^{\infty}\frac{d(n)}{n^{s}}=\zeta^{2}(s),
  32. n = 1 σ a ( n ) σ b ( n ) n s = ζ ( s ) ζ ( s - a ) ζ ( s - b ) ζ ( s - a - b ) ζ ( 2 s - a - b ) . \sum_{n=1}^{\infty}\frac{\sigma_{a}(n)\sigma_{b}(n)}{n^{s}}=\frac{\zeta(s)% \zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}.
  33. n = 1 q n σ a ( n ) = n = 1 n a q n 1 - q n \sum_{n=1}^{\infty}q^{n}\sigma_{a}(n)=\sum_{n=1}^{\infty}\frac{n^{a}q^{n}}{1-q% ^{n}}
  34. for all ϵ > 0 , d ( n ) = o ( n ϵ ) . \mbox{for all }~{}\epsilon>0,\quad d(n)=o(n^{\epsilon}).
  35. lim sup n log d ( n ) log n / log log n = log 2. \limsup_{n\to\infty}\frac{\log d(n)}{\log n/\log\log n}=\log 2.
  36. lim inf n d ( n ) = 2. \liminf_{n\to\infty}d(n)=2.
  37. for all x 1 , n x d ( n ) = x log x + ( 2 γ - 1 ) x + O ( x ) , \mbox{for all }~{}x\geq 1,\sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}),
  38. γ \gamma
  39. O ( x ) O(\sqrt{x})
  40. lim sup n σ ( n ) n log log n = e γ , \limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log\log n}=e^{\gamma},
  41. lim n 1 log n p n p p - 1 = e γ , \lim_{n\to\infty}\frac{1}{\log n}\prod_{p\leq n}\frac{p}{p-1}=e^{\gamma},
  42. σ ( n ) < e γ n log log n \ \sigma(n)<e^{\gamma}n\log\log n
  43. σ ( n ) < H n + ln ( H n ) e H n \sigma(n)<H_{n}+\ln(H_{n})e^{H_{n}}
  44. H n H_{n}
  45. σ ( n ) < e γ n log log n + 0.6483 n log log n \ \sigma(n)<e^{\gamma}n\log\log n+\frac{0.6483\ n}{\log\log n}

Dixon's_Q_test.html

  1. Q = gap range Q=\frac{\,\text{gap}}{\,\text{range}}
  2. 0.189 , 0.167 , 0.187 , 0.183 , 0.186 , 0.182 , 0.181 , 0.184 , 0.181 , 0.177 0.189,\ 0.167,\ 0.187,\ 0.183,\ 0.186,\ 0.182,\ 0.181,\ 0.184,\ 0.181,\ 0.177\,
  3. 0.167 , 0.177 , 0.181 , 0.181 , 0.182 , 0.183 , 0.184 , 0.186 , 0.187 , 0.189 0.167,\ 0.177,\ 0.181,\ 0.181,\ 0.182,\ 0.183,\ 0.184,\ 0.186,\ 0.187,\ 0.189\,
  4. Q = gap range = 0.177 - 0.167 0.189 - 0.167 = 0.455. Q=\frac{\,\text{gap}}{\,\text{range}}=\frac{0.177-0.167}{0.189-0.167}=0.455.

Dodecagon.html

  1. A \displaystyle A
  2. A = 6 sin ( π 6 ) R 2 = 3 R 2 . A=6\sin\left(\frac{\pi}{6}\right)R^{2}=3R^{2}.
  3. A = 12 tan ( π 12 ) r 2 = 12 ( 2 - 3 ) r 2 3.2153903 r 2 . \begin{aligned}\displaystyle A&\displaystyle=12\tan\left(\frac{\pi}{12}\right)% r^{2}=12\left(2-\sqrt{3}\right)r^{2}\\ &\displaystyle\simeq 3.2153903\,r^{2}.\end{aligned}
  4. A = 3 a d \scriptstyle A\,=\,3ad
  5. d = a ( 1 + 2 c o s 30 + 2 c o s 60 ) \scriptstyle d\,=\,a(1\,+\,2cos{30^{\circ}}\,+\,2cos{60^{\circ}})

Dolby_Pro_Logic.html

  1. π / 2 {π}/{2}
  2. 1 1
  3. 0
  4. - j 1 2 -j\sqrt{\frac{1}{2}}
  5. 0
  6. 1 1
  7. j 1 2 j\sqrt{\frac{1}{2}}
  8. 1 1
  9. 0
  10. 1 2 \sqrt{\frac{1}{2}}
  11. - j 1 2 -j\sqrt{\frac{1}{2}}
  12. 0
  13. 1 1
  14. 1 2 \sqrt{\frac{1}{2}}
  15. j 1 2 j\sqrt{\frac{1}{2}}
  16. 1 1
  17. 0
  18. 1 2 \sqrt{\frac{1}{2}}
  19. - j 19 25 -j\sqrt{\frac{19}{25}}
  20. - j 6 25 -j\sqrt{\frac{6}{25}}
  21. 0
  22. 1 1
  23. 1 2 \sqrt{\frac{1}{2}}
  24. j 6 25 j\sqrt{\frac{6}{25}}
  25. j 19 25 j\sqrt{\frac{19}{25}}

Domain_(ring_theory).html

  1. 𝕂 x 1 , , x n , \mathbb{K}\langle x_{1},\ldots,x_{n}\rangle,
  2. ( 1 - g ) ( 1 + g + + g n - 1 ) = 1 - g n , (1-g)(1+g+\ldots+g^{n-1})=1-g^{n},

Domain_theory.html

  1. x d x\sqsubseteq d
  2. x y x\ll y
  3. x y x\sqsubseteq y
  4. { 0 } , { 0 , 1 } , { 0 , 1 , 2 } , \{0\},\{0,1\},\{0,1,2\},\ldots
  5. x x x\ll x
  6. fix ( f ) = n f n ( ) \operatorname{fix}(f)=\bigsqcup_{n\in\mathbb{N}}f^{n}(\bot)

Dominated_convergence_theorem.html

  1. | f n ( x ) | g ( x ) |f_{n}(x)|\leq g(x)
  2. lim n S | f n - f | d μ = 0 \lim_{n\to\infty}\int_{S}|f_{n}-f|\,d\mu=0
  3. lim n S f n d μ = S f d μ \lim_{n\to\infty}\int_{S}f_{n}\,d\mu=\int_{S}f\,d\mu
  4. S | g | d μ < . \int_{S}|g|\,d\mu<\infty.
  5. | f - f n | | f | + | f n | 2 g |f-f_{n}|\leq|f|+|f_{n}|\leq 2g
  6. lim sup n | f - f n | = 0. \limsup_{n\to\infty}|f-f_{n}|=0.
  7. | S f d μ - S f n d μ | = | S ( f - f n ) d μ | S | f - f n | d μ . \left|\int_{S}{f\,d\mu}-\int_{S}{f_{n}\,d\mu}\right|=\left|\int_{S}{(f-f_{n})% \,d\mu}\right|\leq\int_{S}{|f-f_{n}|\,d\mu}.
  8. lim sup n S | f - f n | d μ S lim sup n | f - f n | d μ = 0 , \limsup_{n\to\infty}\int_{S}|f-f_{n}|\,d\mu\leq\int_{S}\limsup_{n\to\infty}|f-% f_{n}|\,d\mu=0,
  9. lim n S | f - f n | d μ = 0. \lim_{n\to\infty}\int_{S}|f-f_{n}|\,d\mu=0.
  10. 0 1 h ( x ) d x 1 m 1 h ( x ) d x = n = 1 m - 1 ( 1 n + 1 , 1 n ] h ( x ) d x n = 1 m - 1 ( 1 n + 1 , 1 n ] n d x = n = 1 m - 1 1 n + 1 as m \int_{0}^{1}h(x)\,dx\geq\int_{\frac{1}{m}}^{1}{h(x)\,dx}=\sum_{n=1}^{m-1}\int_% {\left(\frac{1}{n+1},\frac{1}{n}\right]}{h(x)\,dx}\geq\sum_{n=1}^{m-1}\int_{% \left(\frac{1}{n+1},\frac{1}{n}\right]}{n\,dx}=\sum_{n=1}^{m-1}\frac{1}{n+1}% \to\infty\qquad\,\text{as }m\to\infty
  11. 0 1 lim n f n ( x ) d x = 0 1 = lim n 0 1 f n ( x ) d x , \int_{0}^{1}\lim_{n\to\infty}f_{n}(x)\,dx=0\neq 1=\lim_{n\to\infty}\int_{0}^{1% }f_{n}(x)\,dx,
  12. lim n S f n d μ = S f d μ . \lim_{n\to\infty}\int_{S}{f_{n}\,d\mu}=\int_{S}{f\,d\mu}.
  13. ( Ω , 𝒜 , μ ) (\Omega,\mathcal{A},\mu)
  14. 𝒜 \mathcal{A}
  15. f n : Ω \R { } f_{n}:\Omega\to\R\cup\{\infty\}
  16. 𝒜 \mathcal{A}
  17. g L p g\in L^{p}
  18. L p L^{p}
  19. L p L^{p}
  20. lim n f n - f p = lim n ( Ω | f n - f | p d μ ) 1 p = 0. \lim_{n\to\infty}\|f_{n}-f\|_{p}=\lim_{n\to\infty}\left(\int_{\Omega}|f_{n}-f|% ^{p}\,d\mu\right)^{\frac{1}{p}}=0.
  21. h n = | f n - f | p h_{n}=|f_{n}-f|^{p}
  22. ( 2 g ) p (2g)^{p}

Dominator_(graph_theory).html

  1. \gg
  2. Dom ( n o ) = { n o } \operatorname{Dom}(n_{o})=\left\{n_{o}\right\}
  3. Dom ( n ) = ( p preds ( n ) Dom ( p ) ) { n } \operatorname{Dom}(n)=\left(\bigcap_{p\in\,\text{preds}(n)}\operatorname{Dom}(% p)\right)\bigcup\left\{n\right\}
  4. n o n_{o}

Doomsday_argument.html

  1. P ( N ) = k N P(N)=\frac{k}{N}
  2. P ( N n ) = P ( n N ) P ( N ) P ( n ) . P(N\mid n)=\frac{P(n\mid N)P(N)}{P(n)}.
  3. P ( n N ) = 1 N P(n\mid N)=\frac{1}{N}
  4. P ( n ) = N = n N = P ( n N ) P ( N ) d N = n k N 2 d N P(n)=\int_{N=n}^{N=\infty}P(n\mid N)P(N)\,dN=\int_{n}^{\infty}\frac{k}{N^{2}}% \,dN
  5. = k n . =\frac{k}{n}.
  6. P ( n ) = k n P(n)=\frac{k}{n}
  7. P ( N n ) = n N 2 P(N\mid n)=\frac{n}{N^{2}}
  8. P ( N Z ) = N = n N = Z P ( N | n ) d N P(N\leq Z)=\int_{N=n}^{N=Z}P(N|n)\,dN
  9. = Z - n Z =\frac{Z-n}{Z}
  10. P ( N 20 n ) = 19 20 P(N\leq 20n)=\frac{19}{20}
  11. P ( N 40 [ 200000 ] ) = 39 40 P(N\leq 40[200000])=\frac{39}{40}
  12. Pr ( N ) = k N α , 0 < α < 1. \Pr(N)=\frac{k}{N^{\alpha}},0<\alpha<1.
  13. α \alpha
  14. Pr ( n ) = N = n N = Pr ( n N ) Pr ( N ) d N = n k N ( α + 1 ) d N = k α n α \Pr(n)=\int_{N=n}^{N=\infty}\Pr(n\mid N)\Pr(N)\,dN=\int_{n}^{\infty}\frac{k}{N% ^{(\alpha+1)}}\,dN=\frac{k}{{\alpha}n^{\alpha}}
  15. Pr ( N n ) = α n α N ( 1 + α ) . \Pr(N\mid n)=\frac{{\alpha}n^{\alpha}}{N^{(1+\alpha)}}.
  16. Pr ( N > x n ) = N = x n N = Pr ( N n ) d N = 1 x α . \Pr(N>xn)=\int_{N=xn}^{N=\infty}\Pr(N\mid n)\,dN=\frac{1}{x^{\alpha}}.
  17. α \alpha
  18. Pr ( N > 20 n ) = 1 20 22.3 % . \Pr(N>20n)=\frac{1}{\sqrt{20}}\simeq 22.3\%.
  19. α \alpha
  20. α \alpha
  21. α \alpha
  22. α 0 \alpha\to 0
  23. α 1 \alpha\leq 1
  24. α \alpha
  25. E ( N ) = 0 1 n f d f = n ln ( 1 ) - n ln ( 0 ) = + . E(N)=\int_{0}^{1}{n\over f}\,df=n\ln(1)-n\ln(0)=+\infty.
  26. P ( H T S | D p X ) / P ( H T L | D p X ) = [ P ( H F S | X ) / P ( H F L | X ) ] [ P ( D p | H T S X ) / P ( D p | H T L X ) ] P(H_{TS}|D_{p}X)/P(H_{TL}|D_{p}X)=[P(H_{FS}|X)/P(H_{FL}|X)]\cdot[P(D_{p}|H_{TS% }X)/P(D_{p}|H_{TL}X)]
  27. ( A E ) = ( E A ) ( A ) ( E ) \mathbb{P}(A\mid E)=\frac{\mathbb{P}(E\mid A)\cdot\mathbb{P}(A)}{\mathbb{P}(E)}
  28. ( E A ) = 10 % , ( E B ) = 0.10 % \mathbb{P}(E\mid A)=10\%\ ,\ \mathbb{P}(E\mid B)=0.10\%
  29. ( A ) = 1 100 , ( B ) = 99 100 \mathbb{P}(A)=\frac{1}{100}\ ,\ \mathbb{P}(B)=\frac{99}{100}
  30. ( E ) = ( E A ) + ( E B ) = ( E A ) ( A ) + ( E B ) ( B ) = 19.9 10000 \mathbb{P}(E)=\mathbb{P}(E\cap A)+\mathbb{P}(E\cap B)=\mathbb{P}(E\mid A)\cdot% \mathbb{P}(A)+\mathbb{P}(E\mid B)\cdot\mathbb{P}(B)=\frac{19.9}{10000}
  31. ( A E ) = 10 19.2 = 50.25 % \mathbb{P}(A\mid E)=\frac{10}{19.2}=50.25\%
  32. ( B E ) = 9.9 19.2 = 49.75 % \mathbb{P}(B\mid E)=\frac{9.9}{19.2}=49.75\%
  33. ( B E ) = 0.1 % × 5 10 12 × 99 % 0.1 % × 5 10 12 × 99 % + 10 % × 50 10 9 × 1 % = 99 % 99 % + 1 % = 99 % = ( B ) \mathbb{P}(B\mid E)=\frac{0.1\%\times 5\cdot 10^{12}\times 99\%}{0.1\%\times 5% \cdot 10^{12}\times 99\%+10\%\times 50\cdot 10^{9}\times 1\%}=\frac{99\%}{99\%% +1\%}=99\%=\mathbb{P}(B)
  34. ( A E ) = 1 % 99 % + 1 % = 1 % = ( A ) \mathbb{P}(A\mid E)=\frac{1\%}{99\%+1\%}=1\%=\mathbb{P}(A)

Doomsday_rule.html

  1. ( y 12 + y mod 12 + y mod 12 4 ) mod 7 + anchor = Doomsday \begin{matrix}\left({\left\lfloor{\frac{y}{12}}\right\rfloor+y\bmod 12+\left% \lfloor{\frac{y\bmod 12}{4}}\right\rfloor}\right)\bmod 7+\rm{anchor}=\rm{% Doomsday}\end{matrix}
  2. ( 66 12 + 66 mod 12 + 66 mod 12 4 ) mod 7 + Wednesday = ( 5 + 6 + 1 ) mod 7 + Wednesday = Monday \begin{matrix}\left({\left\lfloor{\frac{66}{12}}\right\rfloor+66\bmod 12+\left% \lfloor{\frac{66\bmod 12}{4}}\right\rfloor}\right)\bmod 7+\rm{Wednesday}&=&% \left(5+6+1\right)\bmod 7+\rm{Wednesday}\\ &=&\rm{Monday}\end{matrix}
  3. ( 66 + 66 4 ) mod 7 + Wednesday = ( 66 + 16 ) mod 7 + Wednesday = Monday \begin{matrix}\left({66+\left\lfloor{\frac{66}{4}}\right\rfloor}\right)\bmod 7% +\rm{Wednesday}&=&\left(66+16\right)\bmod 7+\rm{Wednesday}\\ &=&\rm{Monday}\end{matrix}
  4. ( 5 12 + 5 mod 12 + 5 mod 12 4 ) mod 7 + Tuesday = Monday \left({\left\lfloor{\frac{5}{12}}\right\rfloor+5\bmod 12+\left\lfloor{\frac{5% \bmod 12}{4}}\right\rfloor}\right)\bmod 7+\rm{Tuesday}=\rm{Monday}
  5. ( y + y 4 ) mod 7. (y+\lfloor\frac{y}{4}\rfloor)\bmod 7.
  6. ( 66 + 66 4 ) mod 7 = ( 66 + 16 ) mod 7 = 82 mod 7 = 5 (66+\lfloor\frac{66}{4}\rfloor)\bmod 7=(66+16)\bmod 7=82\bmod 7=5
  7. ( y + y 4 ) mod 7 (y+\lfloor\frac{y}{4}\rfloor)\bmod 7
  8. y 12 \lfloor\frac{y}{12}\rfloor
  9. ( y + y 4 ) mod 7. (y+\lfloor\frac{y}{4}\rfloor)\,\operatorname{mod}\,7.
  10. - [ y + 11 ( y mod 2 ) 2 + 11 ( y + 11 ( y mod 2 ) 2 mod 2 ) ] mod 7. -\left[\frac{y+11(y\,\bmod 2)}{2}+11\left(\frac{y+11(y\,\bmod 2)}{2}\bmod 2% \right)\right]\bmod 7.
  11. y + 11 ( y mod 2 ) 2 \frac{y+11(y\,\bmod 2)}{2}
  12. DD = ( 3 - DL ) mod 7 \mbox{DD}~{}=(3-\mbox{DL}~{})\bmod 7
  13. 5 × ( c mod 4 ) mod 7 + Tuesday = anchor . 5\times(c\bmod 4)\bmod 7+\rm{Tuesday=anchor.}
  14. 6 × ( c mod 7 ) mod 7 + Sunday = anchor . 6\times(c\bmod 7)\bmod 7+\rm{Sunday=anchor}.
  15. c = y e a r / 100 . c=\lfloor year/100\rfloor.
  16. Doomsday = Tuesday + y + y 4 - y 100 + y 400 = Tuesday + 5 × ( y mod 4 ) + 4 × ( y mod 100 ) + 6 × ( y mod 400 ) \mbox{Doomsday}~{}=\mbox{Tuesday}~{}+y+\left\lfloor\frac{y}{4}\right\rfloor-% \left\lfloor\frac{y}{100}\right\rfloor+\left\lfloor\frac{y}{400}\right\rfloor=% \mbox{Tuesday}~{}+5\times(y\mod 4)+4\times(y\mod 100)+6\times(y\mod 400)
  17. Saturday (6) mod 7 = Tuesday (2) + 2009 + 2009 4 - 2009 100 + 2009 400 \mbox{Saturday (6)}~{}\mod 7=\mbox{Tuesday (2)}~{}+2009+\left\lfloor\frac{2009% }{4}\right\rfloor-\left\lfloor\frac{2009}{100}\right\rfloor+\left\lfloor\frac{% 2009}{400}\right\rfloor
  18. Thursday (4) mod 7 = Tuesday (2) + 1946 + 1946 4 - 1946 100 + 1946 400 \mbox{Thursday (4)}~{}\mod 7=\mbox{Tuesday (2)}~{}+1946+\left\lfloor\frac{1946% }{4}\right\rfloor-\left\lfloor\frac{1946}{100}\right\rfloor+\left\lfloor\frac{% 1946}{400}\right\rfloor
  19. Doomsday = Sunday + y + y 4 = Sunday + 5 × ( y mod 4 ) + 3 × ( y mod 7 ) \mbox{Doomsday}~{}=\mbox{Sunday}~{}+y+\left\lfloor\frac{y}{4}\right\rfloor=% \mbox{Sunday}~{}+5\times(y\mod 4)+3\times(y\mod 7)

Dot_(diacritic).html

  1. v = x ˙ v=\dot{x}
  2. 0. 3 ˙ 0.\dot{3}
  3. 1 / 3 {1}/{3}
  4. 0. 1 ˙ 4 ˙ 2 ˙ 8 ˙ 5 ˙ 7 ˙ 0.\dot{1}\dot{4}\dot{2}\dot{8}\dot{5}\dot{7}
  5. 1 / 7 {1}/{7}

Dotted_note.html

  1. a n = a ( 1 + 1 2 + 1 4 + + 1 2 n ) = a ( 2 - 1 2 n ) a_{n}=a\left(1+\tfrac{1}{2}+\tfrac{1}{4}+\cdots+\tfrac{1}{2^{n}}\right)=a(2-% \frac{1}{2^{n}})
  2. 7 / 8 {7}/{8}

Double.html

  1. x + y j x+yj
  2. j 2 = + 1 j^{2}=+1
  3. ( a , b ) (a,b)

Double_factorial.html

  1. n ! ! = k = 0 m ( n - 2 k ) = n ( n - 2 ) ( n - 4 ) n!!=\prod_{k=0}^{m}(n-2k)=n(n-2)(n-4)\cdots
  2. m = n / 2 - 1. m=\lceil n/2\rceil-1.
  3. 0 ! ! = 1. 0!!=1.
  4. n ! ! = k = 1 n / 2 ( 2 k ) = n ( n - 2 ) 2. n!!=\prod_{k=1}^{n/2}(2k)=n(n-2)\cdots 2.
  5. n ! ! = k = 1 ( n + 1 ) / 2 ( 2 k - 1 ) = n ( n - 2 ) 1. n!!=\prod_{k=1}^{(n+1)/2}(2k-1)=n(n-2)\cdots 1.
  6. n ! ! = 2 k k ! . n!!=2^{k}k!.
  7. n ! ! = ( 2 k ) ! 2 k k ! = n ! ( n - 1 ) ! ! . n!!=\frac{(2k)!}{2^{k}k!}=\frac{n!}{(n-1)!!}.
  8. ( 2 k - 1 ) ! ! = P k 2 k 2 k = ( 2 k ) k ¯ 2 k . (2k-1)!!=\frac{{}_{2k}P_{k}}{2^{k}}=\frac{{(2k)}^{\underline{k}}}{2^{k}}.
  9. n ! ! = n × ( n - 2 ) ! ! n!!=n\times(n-2)!!
  10. n ! ! = ( n + 2 ) ! ! n + 2 . n!!=\frac{(n+2)!!}{n+2}.
  11. ( - n ) ! ! × n ! ! = ( - 1 ) ( n - 1 ) / 2 × n . (-n)!!\times n!!=(-1)^{(n-1)/2}\times n.
  12. z ! ! = z ( z - 2 ) ( 3 ) = 2 ( z - 1 ) / 2 ( z 2 ) ( z - 2 2 ) ( 3 2 ) z!!=z(z-2)\cdots(3)=2^{(z-1)/2}\left(\frac{z}{2}\right)\left(\frac{z-2}{2}% \right)\cdots\left(\frac{3}{2}\right)
  13. = 2 ( z - 1 ) / 2 Γ ( z 2 + 1 ) Γ ( 1 2 + 1 ) = 2 z + 1 π Γ ( z 2 + 1 ) = ( z 2 ) ! 2 z + 1 π . =2^{(z-1)/2}\frac{\Gamma\left(\frac{z}{2}+1\right)}{\Gamma\left(\frac{1}{2}+1% \right)}=\sqrt{\frac{2^{z+1}}{\pi}}\Gamma\left(\frac{z}{2}+1\right)=\left(% \frac{z}{2}\right)!\sqrt{\frac{2^{z+1}}{\pi}}\,.
  14. ( 2 k ) ! ! = 2 π i = 1 k ( 2 i ) = 2 k k ! 2 π , (2k)!!=\sqrt{\frac{2}{\pi}}\prod_{i=1}^{k}(2i)=2^{k}k!\sqrt{\frac{2}{\pi}}\,,
  15. 0 ! ! = 2 π 0.7978845608... . 0!!=\sqrt{\frac{2}{\pi}}\approx 0.7978845608...\,.
  16. V n = 2 ( 2 π ) ( n - 1 ) / 2 n ! ! R n . V_{n}=\frac{2(2\pi)^{(n-1)/2}}{n!!}R^{n}.
  17. 0 π / 2 sin n x d x = 0 π / 2 cos n x d x = ( n - 1 ) ! ! n ! ! { 1 n odd π 2 n even . \int_{0}^{\pi/2}\sin^{n}x\,dx=\int_{0}^{\pi/2}\cos^{n}x\,dx=\frac{(n-1)!!}{n!!% }\cdot{\begin{cases}1&n\,\text{ odd}\\ \frac{\pi}{2}&n\,\text{ even}\end{cases}}.
  18. 0 π / 2 sin n x d x = 0 π / 2 cos n x d x = ( n - 1 ) ! ! n ! ! π 2 . \int_{0}^{\pi/2}\sin^{n}x\,dx=\int_{0}^{\pi/2}\cos^{n}x\,dx=\frac{(n-1)!!}{n!!% }\cdot\sqrt{\frac{\pi}{2}}.
  19. ( 2 n - 1 ) ! ! = 2 n Γ ( 1 2 + n ) π = ( - 2 ) n π Γ ( 1 2 - n ) (2n-1)!!=2^{n}\cdot\frac{\Gamma(\frac{1}{2}+n)}{\sqrt{\pi}}=(-2)^{n}\cdot\frac% {\sqrt{\pi}}{\Gamma(\frac{1}{2}-n)}
  20. ( 2 n - 1 ) ! ! = k = 1 n - 1 ( n k + 1 ) ( 2 k - 1 ) ! ! ( 2 n - 2 k - 3 ) ! ! . (2n-1)!!=\sum_{k=1}^{n-1}{\left({{n}\atop{k+1}}\right)}(2k-1)!!(2n-2k-3)!!.
  21. ( 2 n - 1 ) ! ! = k = 0 n ( 2 n - k - 1 k - 1 ) ( 2 k - 1 ) ( 2 n - k + 1 ) k + 1 ( 2 n - 2 k - 3 ) ! ! . (2n-1)!!=\sum_{k=0}^{n}{\left({{2n-k-1}\atop{k-1}}\right)}\frac{(2k-1)(2n-k+1)% }{k+1}(2n-2k-3)!!.
  22. ( 2 n - 1 ) ! ! = k = 1 n ( n - 1 ) ! ( k - 1 ) ! k ( 2 k - 3 ) ! ! . (2n-1)!!=\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}k(2k-3)!!.