wpmath0000008_1

Beta-binomial_distribution.html

  1. F 1 2 ( - n , α ; α + β ; 1 - e t ) {}_{2}F_{1}(-n,\alpha;\alpha+\beta;1-e^{t})\!
  2. for t < log e ( 2 ) \,\text{for }t<\log_{e}(2)
  3. F 1 2 ( - n , α ; α + β ; 1 - e i t ) {}_{2}F_{1}(-n,\alpha;\alpha+\beta;1-e^{it})\!
  4. for | t | < log e ( 2 ) \,\text{for }|t|<\log_{e}(2)
  5. p p
  6. X Bin ( n , p ) then P ( X = k | p , n ) = L ( k | p ) = ( n k ) p k ( 1 - p ) n - k \begin{aligned}\displaystyle X&\displaystyle\sim\operatorname{Bin}(n,p)\\ \displaystyle\,\text{then }P(X=k|p,n)&\displaystyle=L(k|p)={n\choose k}p^{k}(1% -p)^{n-k}\end{aligned}
  7. π ( p | α , β ) = Beta ( α , β ) = p α - 1 ( 1 - p ) β - 1 B ( α , β ) \begin{aligned}\displaystyle\pi(p|\alpha,\beta)&\displaystyle=\mathrm{Beta}(% \alpha,\beta)\\ &\displaystyle=\frac{p^{\alpha-1}(1-p)^{\beta-1}}{\mathrm{B}(\alpha,\beta)}% \end{aligned}
  8. f ( k | n , α , β ) = 0 1 L ( k | p ) π ( p | α , β ) d p = ( n k ) 1 B ( α , β ) 0 1 p k + α - 1 ( 1 - p ) n - k + β - 1 d p = ( n k ) B ( k + α , n - k + β ) B ( α , β ) . \begin{aligned}\displaystyle f(k|n,\alpha,\beta)&\displaystyle=\int_{0}^{1}L(k% |p)\pi(p|\alpha,\beta)\,dp\\ &\displaystyle={n\choose k}\frac{1}{\mathrm{B}(\alpha,\beta)}\int_{0}^{1}p^{k+% \alpha-1}(1-p)^{n-k+\beta-1}\,dp\\ &\displaystyle={n\choose k}\frac{\mathrm{B}(k+\alpha,n-k+\beta)}{\mathrm{B}(% \alpha,\beta)}.\end{aligned}
  9. f ( k | n , α , β ) = Γ ( n + 1 ) Γ ( k + 1 ) Γ ( n - k + 1 ) Γ ( k + α ) Γ ( n - k + β ) Γ ( n + α + β ) Γ ( α + β ) Γ ( α ) Γ ( β ) . f(k|n,\alpha,\beta)=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)}\frac{\Gamma(k% +\alpha)\Gamma(n-k+\beta)}{\Gamma(n+\alpha+\beta)}\frac{\Gamma(\alpha+\beta)}{% \Gamma(\alpha)\Gamma(\beta)}.
  10. μ 1 \displaystyle\mu_{1}
  11. β 2 = ( α + β ) 2 ( 1 + α + β ) n α β ( α + β + 2 ) ( α + β + 3 ) ( α + β + n ) [ ( α + β ) ( α + β - 1 + 6 n ) + 3 α β ( n - 2 ) + 6 n 2 - 3 α β n ( 6 - n ) α + β - 18 α β n 2 ( α + β ) 2 ] . \beta_{2}=\frac{(\alpha+\beta)^{2}(1+\alpha+\beta)}{n\alpha\beta(\alpha+\beta+% 2)(\alpha+\beta+3)(\alpha+\beta+n)}\left[(\alpha+\beta)(\alpha+\beta-1+6n)+3% \alpha\beta(n-2)+6n^{2}-\frac{3\alpha\beta n(6-n)}{\alpha+\beta}-\frac{18% \alpha\beta n^{2}}{(\alpha+\beta)^{2}}\right].
  12. π = α α + β \pi=\frac{\alpha}{\alpha+\beta}\!
  13. μ = n α α + β = n π \mu=\frac{n\alpha}{\alpha+\beta}=n\pi\!
  14. σ 2 = n α β ( α + β + n ) ( α + β ) 2 ( α + β + 1 ) = n π ( 1 - π ) α + β + n α + β + 1 = n π ( 1 - π ) [ 1 + ( n - 1 ) ρ ] \sigma^{2}=\frac{n\alpha\beta(\alpha+\beta+n)}{(\alpha+\beta)^{2}(\alpha+\beta% +1)}=n\pi(1-\pi)\frac{\alpha+\beta+n}{\alpha+\beta+1}=n\pi(1-\pi)[1+(n-1)\rho]\!
  15. ρ = 1 α + β + 1 \rho=\tfrac{1}{\alpha+\beta+1}\!
  16. { ( α + k ) ( n - k ) p ( k ) - ( k + 1 ) p ( k + 1 ) ( β - k + n - 1 ) = 0 , p ( 0 ) = ( β ) n ( α + β ) n } \left\{(\alpha+k)(n-k)p(k)-(k+1)p(k+1)(\beta-k+n-1)=0,p(0)=\frac{(\beta)_{n}}{% (\alpha+\beta)_{n}}\right\}
  17. μ 1 \displaystyle\mu_{1}
  18. μ ^ 1 \displaystyle\hat{\mu}_{1}
  19. α ^ \displaystyle\hat{\alpha}
  20. m 1 \displaystyle m_{1}
  21. α ^ \displaystyle\hat{\alpha}
  22. α ^ mle \displaystyle\hat{\alpha}_{\mathrm{mle}}
  23. log = - 12492.9 \log\mathcal{L}=-12492.9
  24. 𝐴𝐼𝐶 = 24989.74. \mathit{AIC}=24989.74.
  25. π ( θ | μ , M ) = Beta ( M μ , M ( 1 - μ ) ) = Γ ( M ) Γ ( M μ ) Γ ( M ( 1 - μ ) ) θ M μ - 1 ( 1 - θ ) M ( 1 - μ ) - 1 \begin{aligned}\displaystyle\pi(\theta|\mu,M)&\displaystyle=\operatorname{Beta% }(M\mu,M(1-\mu))\\ &\displaystyle=\frac{\Gamma(M)}{\Gamma(M\mu)\Gamma(M(1-\mu))}\theta^{M\mu-1}(1% -\theta)^{M(1-\mu)-1}\end{aligned}
  26. μ \displaystyle\mu
  27. E ( θ | μ , M ) \displaystyle\operatorname{E}(\theta|\mu,M)
  28. ρ ( θ | k ) \displaystyle\rho(\theta|k)
  29. E ( θ | k ) = k + M μ n + M . \operatorname{E}(\theta|k)=\frac{k+M\mu}{n+M}.
  30. m ( k | μ , M ) = 0 1 l ( k | θ ) π ( θ | μ , M ) d θ = Γ ( M ) Γ ( M μ ) Γ ( M ( 1 - μ ) ) ( n k ) 0 1 θ k + M μ - 1 ( 1 - θ ) n - k + M ( 1 - μ ) - 1 d θ = Γ ( M ) Γ ( M μ ) Γ ( M ( 1 - μ ) ) ( n k ) Γ ( k + M μ ) Γ ( n - k + M ( 1 - μ ) ) Γ ( n + M ) . \begin{aligned}\displaystyle m(k|\mu,M)&\displaystyle=\int_{0}^{1}l(k|\theta)% \pi(\theta|\mu,M)\,d\theta\\ &\displaystyle=\frac{\Gamma(M)}{\Gamma(M\mu)\Gamma(M(1-\mu))}{n\choose k}\int_% {0}^{1}\theta^{k+M\mu-1}(1-\theta)^{n-k+M(1-\mu)-1}d\theta\\ &\displaystyle=\frac{\Gamma(M)}{\Gamma(M\mu)\Gamma(M(1-\mu))}{n\choose k}\frac% {\Gamma(k+M\mu)\Gamma(n-k+M(1-\mu))}{\Gamma(n+M)}.\end{aligned}
  31. k i \displaystyle k_{i}
  32. E ( k n ) = E [ E ( k n | θ ) ] = E ( θ ) = μ \operatorname{E}\left(\frac{k}{n}\right)=\operatorname{E}\left[\operatorname{E% }\left(\left.\frac{k}{n}\right|\theta\right)\right]=\operatorname{E}(\theta)=\mu
  33. var ( k n ) \displaystyle\operatorname{var}\left(\frac{k}{n}\right)
  34. μ \mu
  35. M M
  36. μ ^ \hat{\mu}
  37. μ ^ = i = 1 N k i i = 1 N n i . \hat{\mu}=\frac{\sum_{i=1}^{N}k_{i}}{\sum_{i=1}^{N}n_{i}}.
  38. s 2 = 1 N i = 1 N var ( k i n i ) = 1 N i = 1 N μ ^ ( 1 - μ ^ ) n i [ 1 + n i - 1 M ^ + 1 ] s^{2}=\frac{1}{N}\sum_{i=1}^{N}\operatorname{var}\left(\frac{k_{i}}{n_{i}}% \right)=\frac{1}{N}\sum_{i=1}^{N}\frac{\hat{\mu}(1-\hat{\mu})}{n_{i}}\left[1+% \frac{n_{i}-1}{\widehat{M}+1}\right]
  39. M ^ = μ ^ ( 1 - μ ^ ) - s 2 s 2 - μ ^ ( 1 - μ ^ ) N i = 1 N 1 / n i , \widehat{M}=\frac{\hat{\mu}(1-\hat{\mu})-s^{2}}{s^{2}-\frac{\hat{\mu}(1-\hat{% \mu})}{N}\sum_{i=1}^{N}1/n_{i}},
  40. s 2 = N i = 1 N n i ( θ i ^ - μ ^ ) 2 ( N - 1 ) i = 1 N n i . s^{2}=\frac{N\sum_{i=1}^{N}n_{i}(\hat{\theta_{i}}-\hat{\mu})^{2}}{(N-1)\sum_{i% =1}^{N}n_{i}}.
  41. μ ^ \hat{\mu}
  42. M ^ \widehat{M}
  43. θ ~ i \tilde{\theta}_{i}
  44. θ i ^ = k i / n i \hat{\theta_{i}}=k_{i}/n_{i}
  45. μ ^ \hat{\mu}
  46. θ i ~ = E ( θ | k i ) = k i + M ^ μ ^ n i + M ^ = M ^ n i + M ^ μ ^ + n i n i + M ^ k i n i . \tilde{\theta_{i}}=E(\theta|k_{i})=\frac{k_{i}+\widehat{M}\hat{\mu}}{n_{i}+% \widehat{M}}=\frac{\widehat{M}}{n_{i}+\widehat{M}}\hat{\mu}+\frac{n_{i}}{n_{i}% +\widehat{M}}\frac{k_{i}}{n_{i}}.
  47. θ ~ i = B ^ i μ ^ + ( 1 - B ^ i ) θ ^ i \tilde{\theta}_{i}=\hat{B}_{i}\,\hat{\mu}+(1-\hat{B}_{i})\hat{\theta}_{i}
  48. B ^ i \hat{B}_{i}
  49. B i ^ = M ^ M ^ + n i \hat{B_{i}}=\frac{\hat{M}}{\hat{M}+n_{i}}
  50. B B ( 1 , 1 , n ) U ( 0 , n ) BB(1,1,n)\sim U(0,n)\,
  51. U ( a , b ) U(a,b)\,

Beta_(plasma_physics).html

  1. β = p p m a g = n k B T B 2 / ( 2 μ 0 ) \beta=\frac{p}{p_{mag}}=\frac{nk_{B}T}{B^{2}/(2\mu_{0})}
  2. β \beta
  3. β \beta
  4. β m a x = β N I a B 0 \beta_{max}=\frac{\beta_{N}I}{aB_{0}}
  5. B 0 B_{0}
  6. β N \beta_{N}
  7. β m a x \beta_{max}
  8. β m a x \beta_{max}
  9. β m a x \beta_{max}
  10. β N \beta_{N}

Beta_diversity.html

  1. β A = ( S 1 - c ) + ( S 2 - c ) \beta_{A}=(S_{1}-c)+(S_{2}-c)

Beta_helix.html

  1. [ LIV ] - [ GAED ] - X 2 - [ STAV ] - X \mathrm{[LIV]-[GAED]-X_{2}-[STAV]-X}

Beta_wavelet.html

  1. α \alpha
  2. β \beta
  3. 0 t 1 0\leq t\leq 1
  4. α \alpha
  5. β \beta
  6. P ( t ) = 1 B ( α , β ) t α - 1 ( 1 - t ) β - 1 , 1 α , β + P(t)=\frac{1}{B(\alpha,\beta)}t^{\alpha-1}\cdot(1-t)^{\beta-1},\quad 1\leq% \alpha,\beta\leq+\infty
  7. B ( α , β ) = Γ ( α ) Γ ( β ) Γ ( α + β ) B(\alpha,\beta)=\frac{\Gamma(\alpha)\cdot\Gamma(\beta)}{\Gamma(\alpha+\beta)}
  8. Γ ( ) \Gamma(\cdot)
  9. B ( , ) B(\cdot,\cdot)
  10. p i ( t ) p_{i}(t)
  11. t i t_{i}
  12. i = 1 , 2 , 3.. N i=1,2,3..N
  13. p i ( t ) 0 p_{i}(t)\geq 0
  14. ( t ) (\forall t)
  15. - + p i ( t ) d t = 1 \int_{-\infty}^{+\infty}p_{i}(t)dt=1
  16. t i t_{i}
  17. m i = - + τ p i ( τ ) d τ , m_{i}=\int_{-\infty}^{+\infty}\tau\cdot p_{i}(\tau)d\tau,
  18. σ i 2 = - + ( τ - m i ) 2 p i ( τ ) d τ \sigma_{i}^{2}=\int_{-\infty}^{+\infty}(\tau-m_{i})^{2}\cdot p_{i}(\tau)d\tau
  19. t t
  20. m = i = 1 N m i m=\sum_{i=1}^{N}m_{i}
  21. σ 2 = i = 1 N σ i 2 \sigma^{2}=\sum_{i=1}^{N}\sigma_{i}^{2}
  22. p ( t ) p(t)
  23. t = i = 1 N t i t=\sum_{i=1}^{N}t_{i}
  24. { p i ( t ) } \{p_{i}(t)\}
  25. S u p p { ( p i ( t ) ) } = ( a i , b i ) ( i ) Supp\{(p_{i}(t))\}=(a_{i},b_{i})(\forall i)
  26. a = i = 1 N a i < + a=\sum_{i=1}^{N}a_{i}<+\infty
  27. b = i = 1 N b i < + b=\sum_{i=1}^{N}b_{i}<+\infty
  28. a = 0 a=0
  29. b = 1 b=1
  30. t t
  31. N N\rightarrow\infty
  32. p ( t ) p(t)\approx
  33. { k t α ( 1 - t ) β , o t h e r w i s e \begin{cases}{k\cdot t^{\alpha}(1-t)^{\beta}},\\ otherwise\end{cases}
  34. α = m ( m - m 2 - σ 2 ) σ 2 , \alpha=\frac{m(m-m^{2}-\sigma^{2})}{\sigma^{2}},
  35. β = ( 1 - m ) ( α + 1 ) m . \beta=\frac{(1-m)(\alpha+1)}{m}.
  36. P ( | α , β ) P(\cdot|\alpha,\beta)
  37. ψ b e t a ( t | α , β ) = ( - 1 ) d P ( t | α , β ) d t \psi_{beta}(t|\alpha,\beta)=(-1)\frac{dP(t|\alpha,\beta)}{dt}
  38. α \alpha
  39. β \beta
  40. S u p p ( ψ ) = [ - α β α + β + 1 , β α α + β + 1 ] = [ a , b ] . Supp(\psi)=[-\sqrt{\frac{\alpha}{\beta}}\sqrt{\alpha+\beta+1},\sqrt{\frac{% \beta}{\alpha}}\sqrt{\alpha+\beta+1}]=[a,b].
  41. l e n g t h S u p p ( ψ ) = T ( α , β ) = ( α + β ) α + β + 1 α β . lengthSupp(\psi)=T(\alpha,\beta)=(\alpha+\beta)\sqrt{\frac{\alpha+\beta+1}{% \alpha\beta}}.
  42. R = b / | a | = β / α R=b/|a|=\beta/\alpha
  43. t z e r o c r o s s t_{zerocross}
  44. t z e r o c r o s s = ( α - β ) ( α + β - 2 ) α + β + 1 α β . t_{zerocross}=\frac{(\alpha-\beta)}{(\alpha+\beta-2)}\sqrt{\frac{\alpha+\beta+% 1}{\alpha\beta}}.
  45. ϕ b e t a ( t | α , β ) = 1 B ( α , β ) T α + β - 1 ( t - a ) α - 1 ( b - t ) β - 1 , \phi_{beta}(t|\alpha,\beta)=\frac{1}{B(\alpha,\beta)T^{\alpha+\beta-1}}\cdot(t% -a)^{\alpha-1}\cdot(b-t)^{\beta-1},
  46. a t b a\leq t\leq b
  47. ψ b e t a ( t | α , β ) = - 1 B ( α , β ) T α + β - 1 [ α - 1 t - a - β - 1 b - t ] ( t - a ) α - 1 ( b - t ) β - 1 \psi_{beta}(t|\alpha,\beta)=\frac{-1}{B(\alpha,\beta)T^{\alpha+\beta-1}}\cdot[% \frac{\alpha-1}{t-a}-\frac{\beta-1}{b-t}]\cdot(t-a)^{\alpha-1}\cdot(b-t)^{% \beta-1}
  48. α = 4 \alpha=4
  49. β = 3 \beta=3
  50. α = 3 \alpha=3
  51. β = 7 \beta=7
  52. α = 5 \alpha=5
  53. β = 17 \beta=17
  54. ψ b e t a ( t | α , β ) Ψ B E T A ( ω | α , β ) \psi_{beta}(t|\alpha,\beta)\leftrightarrow\Psi_{BETA}(\omega|\alpha,\beta)
  55. Ψ B E T A ( ω ) \Psi_{BETA}(\omega)
  56. Ψ B E T A ( ω ) = - j ω M ( α , α + β , - j ω ( α + β ) α + β + 1 α β ) e x p { ( j ω α ( α + β + 1 ) β ) } \Psi_{BETA}(\omega)=-j\omega\cdot M(\alpha,\alpha+\beta,-j\omega(\alpha+\beta)% \sqrt{\frac{\alpha+\beta+1}{\alpha\beta}})\cdot exp\{(j\omega\sqrt{\frac{% \alpha(\alpha+\beta+1)}{\beta}})\}
  57. M ( α , α + β , j ν ) = Γ ( α + β ) Γ ( α ) Γ ( β ) 0 1 e j ν t t α - 1 ( 1 - t ) β - 1 d t M(\alpha,\alpha+\beta,j\nu)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\cdot% \Gamma(\beta)}\cdot\int_{0}^{1}e^{j\nu t}t^{\alpha-1}(1-t)^{\beta-1}dt
  58. ( α = β ) (\alpha=\beta)
  59. ( α β ) (\alpha\neq\beta)
  60. | Ψ B E T A ( ω | α , β ) | = | Ψ B E T A ( ω | β , α ) | . |\Psi_{BETA}(\omega|\alpha,\beta)|=|\Psi_{BETA}(\omega|\beta,\alpha)|.
  61. ψ b e t a ( t | α , β ) = ( - 1 ) N d N P ( t | α , β ) d t N . \psi_{beta}(t|\alpha,\beta)=(-1)^{N}\frac{d^{N}P(t|\alpha,\beta)}{dt^{N}}.
  62. N N
  63. N M i n ( α , β ) - 1 N\leq Min(\alpha,\beta)-1
  64. Ψ b e t a ( t | α , β ) = ( - 1 ) N B ( α , β ) T α + β - 1 n = 0 N s g n ( 2 n - N ) Γ ( α ) Γ ( α - ( N - n ) ) ( t - a ) α - 1 - ( N - n ) Γ ( β ) Γ ( β - n ) ( b - t ) β - 1 - n . \Psi_{beta}(t|\alpha,\beta)=\frac{(-1)^{N}}{B(\alpha,\beta)\cdot T^{\alpha+% \beta-1}}\sum_{n=0}^{N}sgn(2n-N)\cdot\frac{\Gamma(\alpha)}{\Gamma(\alpha-(N-n)% )}(t-a)^{\alpha-1-(N-n)}\cdot\frac{\Gamma(\beta)}{\Gamma(\beta-n)}(b-t)^{\beta% -1-n}.
  65. Ψ B E T A ( ω ) \Psi_{BETA}(\omega)
  66. | Ψ B E T A ( ω α , β ) | |\Psi_{BETA}(\omega\alpha,\beta)|
  67. × ω \times\omega
  68. α = β = 3 \alpha=\beta=3
  69. α = β = 4 \alpha=\beta=4
  70. α = β = 5 \alpha=\beta=5
  71. Ψ B E T A ( ω ) \Psi_{BETA}(\omega)
  72. | Ψ B E T A ( ω α , β ) | |\Psi_{BETA}(\omega\alpha,\beta)|
  73. × ω \times\omega
  74. α = 3 \alpha=3
  75. β = 4 \beta=4
  76. α = 3 \alpha=3
  77. β = 5 \beta=5

Betti's_theorem.html

  1. F i P F^{P}_{i}
  2. F i Q F^{Q}_{i}
  3. d i P d^{P}_{i}
  4. d i Q d^{Q}_{i}
  5. F i P F^{P}_{i}
  6. 1 2 i = 1 n F i P d i P = 1 2 Ω σ i j P ϵ i j P d Ω \frac{1}{2}\sum^{n}_{i=1}F^{P}_{i}d^{P}_{i}=\frac{1}{2}\int_{\Omega}\sigma^{P}% _{ij}\epsilon^{P}_{ij}\,d\Omega
  7. F i Q F^{Q}_{i}
  8. 1 2 i = 1 n F i Q d i Q = 1 2 Ω σ i j Q ϵ i j Q d Ω \frac{1}{2}\sum^{n}_{i=1}F^{Q}_{i}d^{Q}_{i}=\frac{1}{2}\int_{\Omega}\sigma^{Q}% _{ij}\epsilon^{Q}_{ij}\,d\Omega
  9. F i P F^{P}_{i}
  10. F i Q F^{Q}_{i}
  11. F i P F^{P}_{i}
  12. 1 2 i = 1 n F i Q d i Q + i = 1 n F i P d i Q = 1 2 Ω σ i j Q ϵ i j Q d Ω + Ω σ i j P ϵ i j Q d Ω \frac{1}{2}\sum^{n}_{i=1}F^{Q}_{i}d^{Q}_{i}+\sum^{n}_{i=1}F^{P}_{i}d^{Q}_{i}=% \frac{1}{2}\int_{\Omega}\sigma^{Q}_{ij}\epsilon^{Q}_{ij}\,d\Omega+\int_{\Omega% }\sigma^{P}_{ij}\epsilon^{Q}_{ij}\,d\Omega
  13. F i Q F^{Q}_{i}
  14. F i P F^{P}_{i}
  15. 1 2 i = 1 n F i P d i P + i = 1 n F i Q d i P = 1 2 Ω σ i j P ϵ i j P d Ω + Ω σ i j Q ϵ i j P d Ω \frac{1}{2}\sum^{n}_{i=1}F^{P}_{i}d^{P}_{i}+\sum^{n}_{i=1}F^{Q}_{i}d^{P}_{i}=% \frac{1}{2}\int_{\Omega}\sigma^{P}_{ij}\epsilon^{P}_{ij}\,d\Omega+\int_{\Omega% }\sigma^{Q}_{ij}\epsilon^{P}_{ij}\,d\Omega
  16. i = 1 n F i P d i Q = Ω σ i j P ϵ i j Q d Ω \sum^{n}_{i=1}F^{P}_{i}d^{Q}_{i}=\int_{\Omega}\sigma^{P}_{ij}\epsilon^{Q}_{ij}% \,d\Omega
  17. i = 1 n F i Q d i P = Ω σ i j Q ϵ i j P d Ω \sum^{n}_{i=1}F^{Q}_{i}d^{P}_{i}=\int_{\Omega}\sigma^{Q}_{ij}\epsilon^{P}_{ij}% \,d\Omega
  18. σ i j = D i j k l ϵ k l \sigma_{ij}=D_{ijkl}\epsilon_{kl}
  19. i = 1 n F i P d i Q = Ω D i j k l ϵ i j P ϵ k l Q d Ω \sum^{n}_{i=1}F^{P}_{i}d^{Q}_{i}=\int_{\Omega}D_{ijkl}\epsilon^{P}_{ij}% \epsilon^{Q}_{kl}\,d\Omega
  20. i = 1 n F i Q d i P = Ω D i j k l ϵ i j Q ϵ k l P d Ω \sum^{n}_{i=1}F^{Q}_{i}d^{P}_{i}=\int_{\Omega}D_{ijkl}\epsilon^{Q}_{ij}% \epsilon^{P}_{kl}\,d\Omega
  21. i = 1 n F i P d i Q = i = 1 n F i Q d i P \sum^{n}_{i=1}F^{P}_{i}d^{Q}_{i}=\sum^{n}_{i=1}F^{Q}_{i}d^{P}_{i}
  22. Δ P 2 \Delta_{P2}
  23. Δ Q 1 \Delta_{Q1}
  24. P Δ Q 1 = Q Δ P 2 . P\,\Delta_{Q1}=Q\,\Delta_{P2}.

Betz's_law.html

  1. m ˙ = ρ A 1 v 1 = ρ S v = ρ A 2 v 2 \dot{m}=\rho A_{1}v_{1}=\rho Sv=\rho A_{2}v_{2}
  2. A 1 A_{1}
  3. A 2 A_{2}
  4. F \displaystyle F
  5. d E = F d x dE=F\cdot dx
  6. P = d E d t = F d x d t = F v P=\begin{matrix}\frac{dE}{dt}\end{matrix}=F\cdot\begin{matrix}\frac{dx}{dt}% \end{matrix}=F\cdot v
  7. P = ρ S v 2 ( v 1 - v 2 ) P=\rho\cdot S\cdot v^{2}\cdot(v_{1}-v_{2})
  8. P = Δ E Δ t P=\begin{matrix}\frac{\Delta E}{\Delta t}\end{matrix}
  9. = 1 2 m ˙ ( v 1 2 - v 2 2 ) =\begin{matrix}\frac{1}{2}\end{matrix}\cdot\dot{m}\cdot(v_{1}^{2}-v_{2}^{2})
  10. P = 1 2 ρ S v ( v 1 2 - v 2 2 ) P=\begin{matrix}\frac{1}{2}\end{matrix}\cdot\rho\cdot S\cdot v\cdot(v_{1}^{2}-% v_{2}^{2})
  11. P = 1 2 ρ S v ( v 1 2 - v 2 2 ) = ρ S v 2 ( v 1 - v 2 ) P=\begin{matrix}\frac{1}{2}\end{matrix}\cdot\rho\cdot S\cdot v\cdot(v_{1}^{2}-% v_{2}^{2})=\rho\cdot S\cdot v^{2}\cdot(v_{1}-v_{2})
  12. 1 2 ( v 1 2 - v 2 2 ) = 1 2 ( v 1 - v 2 ) ( v 1 + v 2 ) = v ( v 1 - v 2 ) \begin{matrix}\frac{1}{2}\end{matrix}\cdot(v_{1}^{2}-v_{2}^{2})=\begin{matrix}% \frac{1}{2}\end{matrix}\cdot(v_{1}-v_{2})\cdot(v_{1}+v_{2})=v\cdot(v_{1}-v_{2})
  13. v = 1 2 ( v 1 + v 2 ) v=\begin{matrix}\frac{1}{2}\end{matrix}\cdot(v_{1}+v_{2})
  14. E ˙ = 1 2 m ˙ ( v 1 2 - v 2 2 ) \dot{E}=\begin{matrix}\frac{1}{2}\end{matrix}\cdot\dot{m}\cdot\left(v_{1}^{2}-% v_{2}^{2}\right)
  15. = 1 2 ρ S v ( v 1 2 - v 2 2 ) =\begin{matrix}\frac{1}{2}\end{matrix}\cdot\rho\cdot S\cdot v\cdot\left(v_{1}^% {2}-v_{2}^{2}\right)
  16. = 1 4 ρ S ( v 1 + v 2 ) ( v 1 2 - v 2 2 ) =\begin{matrix}\frac{1}{4}\end{matrix}\cdot\rho\cdot S\cdot\left(v_{1}+v_{2}% \right)\cdot\left(v_{1}^{2}-v_{2}^{2}\right)
  17. = 1 4 ρ S v 1 3 ( 1 - ( v 2 v 1 ) 2 + ( v 2 v 1 ) - ( v 2 v 1 ) 3 ) =\begin{matrix}\frac{1}{4}\end{matrix}\cdot\rho\cdot S\cdot v_{1}^{3}\cdot% \left(1-\left(\frac{v_{2}}{v_{1}}\right)^{2}+\left(\frac{v_{2}}{v_{1}}\right)-% \left(\frac{v_{2}}{v_{1}}\right)^{3}\right)
  18. E ˙ \dot{E}
  19. v 2 v 1 \frac{v_{2}}{v_{1}}
  20. E ˙ \dot{E}
  21. E ˙ \dot{E}
  22. v 2 v 1 = 1 3 \begin{matrix}\frac{v_{2}}{v_{1}}=\frac{1}{3}\end{matrix}
  23. P max = 16 27 1 2 ρ S v 1 3 . P_{\max}=\begin{matrix}\frac{16}{27}\cdot\frac{1}{2}\end{matrix}\cdot\rho\cdot S% \cdot v_{1}^{3}.
  24. P = C p 1 2 ρ S v 1 3 . P=C_{\mathrm{p}}\cdot\begin{matrix}\frac{1}{2}\end{matrix}\cdot\rho\cdot S% \cdot v_{1}^{3}.
  25. P wind = 1 2 ρ S v 1 3 . P_{\rm wind}=\begin{matrix}\frac{1}{2}\end{matrix}\cdot\rho\cdot S\cdot v_{1}^% {3}.

Bhabha_scattering.html

  1. e + e - e + e - e^{+}e^{-}\rightarrow e^{+}e^{-}
  2. d σ d ( cos θ ) = π α 2 s ( u 2 ( 1 s + 1 t ) 2 + ( t s ) 2 + ( s t ) 2 ) \frac{\mathrm{d}\sigma}{\mathrm{d}(\cos\theta)}=\frac{\pi\alpha^{2}}{s}\left(u% ^{2}\left(\frac{1}{s}+\frac{1}{t}\right)^{2}+\left(\frac{t}{s}\right)^{2}+% \left(\frac{s}{t}\right)^{2}\right)\,
  3. α \alpha
  4. θ \theta
  5. s = s=\,
  6. ( k + p ) 2 = (k+p)^{2}=\,
  7. ( k + p ) 2 (k^{\prime}+p^{\prime})^{2}\approx\,
  8. 2 k p 2k\cdot p\approx\,
  9. 2 k p 2k^{\prime}\cdot p^{\prime}\,
  10. t = t=\,
  11. ( k - k ) 2 = (k-k^{\prime})^{2}=\,
  12. ( p - p ) 2 (p-p^{\prime})^{2}\approx\,
  13. - 2 k k -2k\cdot k^{\prime}\approx\,
  14. - 2 p p -2p\cdot p^{\prime}\,
  15. u = u=\,
  16. ( k - p ) 2 = (k-p^{\prime})^{2}=\,
  17. ( p - k ) 2 (p-k^{\prime})^{2}\approx\,
  18. - 2 k p -2k\cdot p^{\prime}\approx\,
  19. - 2 k p -2k^{\prime}\cdot p\,
  20. γ μ \gamma^{\mu}\,
  21. u , and u ¯ u,\ \mathrm{and}\ \bar{u}\,
  22. v , and v ¯ v,\ \mathrm{and}\ \bar{v}\,
  23. = \mathcal{M}=\,
  24. - e 2 ( v ¯ k γ μ v k ) 1 ( k - k ) 2 ( u ¯ p γ μ u p ) -e^{2}\left(\bar{v}_{k}\gamma^{\mu}v_{k^{\prime}}\right)\frac{1}{(k-k^{\prime}% )^{2}}\left(\bar{u}_{p^{\prime}}\gamma_{\mu}u_{p}\right)
  25. + e 2 ( v ¯ k γ ν u p ) 1 ( k + p ) 2 ( u ¯ p γ ν v k ) +e^{2}\left(\bar{v}_{k}\gamma^{\nu}u_{p}\right)\frac{1}{(k+p)^{2}}\left(\bar{u% }_{p^{\prime}}\gamma_{\nu}v_{k^{\prime}}\right)
  26. | | 2 ¯ \overline{|\mathcal{M}|^{2}}\,
  27. = 1 ( 2 s e - + 1 ) ( 2 s e + + 1 ) spins | | 2 =\frac{1}{(2s_{e-}+1)(2s_{e+}+1)}\sum_{\mathrm{spins}}|\mathcal{M}|^{2}\,
  28. = 1 4 s = 1 2 s = 1 2 r = 1 2 r = 1 2 | | 2 =\frac{1}{4}\sum_{s=1}^{2}\sum_{s^{\prime}=1}^{2}\sum_{r=1}^{2}\sum_{r^{\prime% }=1}^{2}|\mathcal{M}|^{2}\,
  29. | | 2 |\mathcal{M}|^{2}\,
  30. | | 2 |\mathcal{M}|^{2}\,
  31. e 4 | ( v ¯ k γ μ v k ) ( u ¯ p γ μ u p ) ( k - k ) 2 | 2 e^{4}\left|\frac{(\bar{v}_{k}\gamma^{\mu}v_{k^{\prime}})(\bar{u}_{p^{\prime}}% \gamma_{\mu}u_{p})}{(k-k^{\prime})^{2}}\right|^{2}\,
  32. - e 4 ( ( v ¯ k γ μ v k ) ( u ¯ p γ μ u p ) ( k - k ) 2 ) * ( ( v ¯ k γ ν u p ) ( u ¯ p γ ν v k ) ( k + p ) 2 ) {}-e^{4}\left(\frac{(\bar{v}_{k}\gamma^{\mu}v_{k^{\prime}})(\bar{u}_{p^{\prime% }}\gamma_{\mu}u_{p})}{(k-k^{\prime})^{2}}\right)^{*}\left(\frac{(\bar{v}_{k}% \gamma^{\nu}u_{p})(\bar{u}_{p^{\prime}}\gamma_{\nu}v_{k^{\prime}})}{(k+p)^{2}}% \right)\,
  33. - e 4 ( ( v ¯ k γ μ v k ) ( u ¯ p γ μ u p ) ( k - k ) 2 ) ( ( v ¯ k γ ν u p ) ( u ¯ p γ ν v k ) ( k + p ) 2 ) * {}-e^{4}\left(\frac{(\bar{v}_{k}\gamma^{\mu}v_{k^{\prime}})(\bar{u}_{p^{\prime% }}\gamma_{\mu}u_{p})}{(k-k^{\prime})^{2}}\right)\left(\frac{(\bar{v}_{k}\gamma% ^{\nu}u_{p})(\bar{u}_{p^{\prime}}\gamma_{\nu}v_{k^{\prime}})}{(k+p)^{2}}\right% )^{*}\,
  34. + e 4 | ( v ¯ k γ ν u p ) ( u ¯ p γ ν v k ) ( k + p ) 2 | 2 {}+e^{4}\left|\frac{(\bar{v}_{k}\gamma^{\nu}u_{p})(\bar{u}_{p^{\prime}}\gamma_% {\nu}v_{k^{\prime}})}{(k+p)^{2}}\right|^{2}\,
  35. | | 2 |\mathcal{M}|^{2}\,
  36. = e 4 ( k - k ) 4 ( ( v ¯ k γ μ v k ) ( u ¯ p γ μ u p ) ) * ( ( v ¯ k γ ν v k ) ( u ¯ p γ ν u p ) ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\Big((\bar{v}_{k}\gamma^{\mu}v_{k^{\prime}})(% \bar{u}_{p^{\prime}}\gamma_{\mu}u_{p})\Big)^{*}\Big((\bar{v}_{k}\gamma^{\nu}v_% {k^{\prime}})(\bar{u}_{p^{\prime}}\gamma_{\nu}u_{p})\Big)\,
  37. ( 1 ) (1)\,
  38. = e 4 ( k - k ) 4 ( ( v ¯ k γ μ v k ) * ( u ¯ p γ μ u p ) * ) ( ( v ¯ k γ ν v k ) ( u ¯ p γ ν u p ) ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\Big((\bar{v}_{k}\gamma^{\mu}v_{k^{\prime}})^% {*}(\bar{u}_{p^{\prime}}\gamma_{\mu}u_{p})^{*}\Big)\Big((\bar{v}_{k}\gamma^{% \nu}v_{k^{\prime}})(\bar{u}_{p^{\prime}}\gamma_{\nu}u_{p})\Big)\,
  39. ( 2 ) (2)\,
  40. = e 4 ( k - k ) 4 ( ( v ¯ k γ μ v k ) ( u ¯ p γ μ u p ) ) ( ( v ¯ k γ ν v k ) ( u ¯ p γ ν u p ) ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\Big(\left(\bar{v}_{k^{\prime}}\gamma^{\mu}v_% {k}\right)\left(\bar{u}_{p}\gamma_{\mu}u_{p^{\prime}}\right)\Big)\Big(\left(% \bar{v}_{k}\gamma^{\nu}v_{k^{\prime}}\right)\left(\bar{u}_{p^{\prime}}\gamma_{% \nu}u_{p}\right)\Big)\,
  41. ( 3 ) (3)\,
  42. = e 4 ( k - k ) 4 ( v ¯ k γ μ v k ) ( v ¯ k γ ν v k ) ( u ¯ p γ μ u p ) ( u ¯ p γ ν u p ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\left(\bar{v}_{k^{\prime}}\gamma^{\mu}v_{k}% \right)\left(\bar{v}_{k}\gamma^{\nu}v_{k^{\prime}}\right)\left(\bar{u}_{p}% \gamma_{\mu}u_{p^{\prime}}\right)\left(\bar{u}_{p^{\prime}}\gamma_{\nu}u_{p}% \right)\,
  43. ( 4 ) (4)\,
  44. spins | | 2 \sum_{\mathrm{spins}}|\mathcal{M}|^{2}\,
  45. = e 4 ( k - k ) 4 ( r v ¯ k γ μ ( r v k v ¯ k ) γ ν v k ) ( s u ¯ p γ μ ( s u p u ¯ p ) γ ν u p ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\left(\sum_{r^{\prime}}\bar{v}_{k^{\prime}}% \gamma^{\mu}(\sum_{r}v_{k}\bar{v}_{k})\gamma^{\nu}v_{k^{\prime}}\right)\left(% \sum_{s}\bar{u}_{p}\gamma_{\mu}(\sum_{s^{\prime}}{u_{p^{\prime}}\bar{u}_{p^{% \prime}}})\gamma_{\nu}u_{p}\right)\,
  46. ( 5 ) (5)\,
  47. = e 4 ( k - k ) 4 Tr ( ( r v k v ¯ k ) γ μ ( r v k v ¯ k ) γ ν ) Tr ( ( s u p u ¯ p ) γ μ ( s u p u ¯ p ) γ ν ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\operatorname{Tr}\left(\Big(\sum_{r^{\prime}}% v_{k^{\prime}}\bar{v}_{k^{\prime}}\Big)\gamma^{\mu}\Big(\sum_{r}v_{k}\bar{v}_{% k}\Big)\gamma^{\nu}\right)\operatorname{Tr}\left(\Big(\sum_{s}u_{p}\bar{u}_{p}% \Big)\gamma_{\mu}\Big(\sum_{s^{\prime}}{u_{p^{\prime}}\bar{u}_{p^{\prime}}}% \Big)\gamma_{\nu}\right)\,
  48. ( 6 ) (6)\,
  49. = e 4 ( k - k ) 4 Tr ( ( k / - m ) γ μ ( k / - m ) γ ν ) Tr ( ( p / + m ) γ μ ( p / + m ) γ ν ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\operatorname{Tr}\left((k\!\!\!/^{\prime}-m)% \gamma^{\mu}(k\!\!\!/-m)\gamma^{\nu}\right)\cdot\operatorname{Tr}\left((p\!\!% \!/^{\prime}+m)\gamma_{\mu}(p\!\!\!/+m)\gamma_{\nu}\right)\,
  50. ( 7 ) (7)\,
  51. = e 4 ( k - k ) 4 ( 4 ( k μ k ν - ( k k ) η μ ν + k ν k μ ) + 4 m 2 η μ ν ) ( 4 ( p μ p ν - ( p p ) η μ ν + p ν p μ ) + 4 m 2 η μ ν ) =\frac{e^{4}}{(k-k^{\prime})^{4}}\left(4\left({k^{\prime}}^{\mu}k^{\nu}-(k^{% \prime}\cdot k)\eta^{\mu\nu}+k^{\prime\nu}k^{\mu}\right)+4m^{2}\eta^{\mu\nu}% \right)\left(4\left({p^{\prime}}_{\mu}p_{\nu}-(p^{\prime}\cdot p)\eta_{\mu\nu}% +p^{\prime}_{\nu}p_{\mu}\right)+4m^{2}\eta_{\mu\nu}\right)\,
  52. ( 8 ) (8)\,
  53. = 32 e 4 ( k - k ) 4 ( ( k p ) ( k p ) + ( k p ) ( k p ) - m 2 p p - m 2 k k + 2 m 4 ) =\frac{32{e^{4}}}{(k-k^{\prime})^{4}}\left((k^{\prime}\cdot p^{\prime})(k\cdot p% )+(k^{\prime}\cdot p)(k\cdot p^{\prime})-m^{2}p^{\prime}\cdot p-m^{2}k^{\prime% }\cdot k+2m^{4}\right)\,
  54. ( 9 ) (9)\,
  55. 1 4 spins | | 2 \frac{1}{4}\sum_{\mathrm{spins}}|\mathcal{M}|^{2}\,
  56. = 32 e 4 4 ( k - k ) 4 ( ( k p ) ( k p ) + ( k p ) ( k p ) ) =\frac{32e^{4}}{4(k-k^{\prime})^{4}}\left((k^{\prime}\cdot p^{\prime})(k\cdot p% )+(k^{\prime}\cdot p)(k\cdot p^{\prime})\right)\,
  57. = 8 e 4 t 2 ( 1 2 s 1 2 s + 1 2 u 1 2 u ) =\frac{8e^{4}}{t^{2}}\left(\tfrac{1}{2}s\tfrac{1}{2}s+\tfrac{1}{2}u\tfrac{1}{2% }u\right)\,
  58. = 2 e 4 s 2 + u 2 t 2 =2e^{4}\frac{s^{2}+u^{2}}{t^{2}}\,
  59. 1 4 spins | | 2 \frac{1}{4}\sum_{\mathrm{spins}}|\mathcal{M}|^{2}\,
  60. = 32 e 4 4 ( k + p ) 4 ( ( k k ) ( p p ) + ( k p ) ( k p ) ) =\frac{32e^{4}}{4(k+p)^{4}}\left((k\cdot k^{\prime})(p\cdot p^{\prime})+(k^{% \prime}\cdot p)(k\cdot p^{\prime})\right)\,
  61. = 8 e 4 s 2 ( 1 2 t 1 2 t + 1 2 u 1 2 u ) =\frac{8e^{4}}{s^{2}}\left(\tfrac{1}{2}t\tfrac{1}{2}t+\tfrac{1}{2}u\tfrac{1}{2% }u\right)\,
  62. = 2 e 4 t 2 + u 2 s 2 =2e^{4}\frac{t^{2}+u^{2}}{s^{2}}\,
  63. ( 1 + cos 2 θ ) (1+\cos^{2}\theta)
  64. θ \theta
  65. | | 2 ¯ 2 e 4 = u 2 + s 2 t 2 + 2 u 2 s t + u 2 + t 2 s 2 \frac{\overline{|\mathcal{M}|^{2}}}{2e^{4}}=\frac{u^{2}+s^{2}}{t^{2}}+\frac{2u% ^{2}}{st}+\frac{u^{2}+t^{2}}{s^{2}}\,
  66. s = 1 , 2 u p ( s ) u ¯ p ( s ) = p / + m \sum_{s=1,2}{u^{(s)}_{p}\bar{u}^{(s)}_{p}}=p\!\!\!/+m\,
  67. s = 1 , 2 v p ( s ) v ¯ p ( s ) = p / - m \sum_{s=1,2}{v^{(s)}_{p}\bar{v}^{(s)}_{p}}=p\!\!\!/-m\,
  68. p / = γ μ p μ p\!\!\!/=\gamma^{\mu}p_{\mu}\,
  69. u ¯ = u γ 0 \bar{u}=u^{\dagger}\gamma^{0}\,
  70. γ μ \gamma_{\mu}\,
  71. Tr ( γ μ γ ν ) = 4 η μ ν \operatorname{Tr}(\gamma^{\mu}\gamma^{\nu})=4\eta^{\mu\nu}
  72. Tr ( γ ρ γ μ γ σ γ ν ) = 4 ( η ρ μ η σ ν - η ρ σ η μ ν + η ρ ν η μ σ ) \operatorname{Tr}\left(\gamma_{\rho}\gamma_{\mu}\gamma_{\sigma}\gamma_{\nu}% \right)=4\left(\eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+% \eta_{\rho\nu}\eta_{\mu\sigma}\right)\,
  73. Tr ( ( p / + m ) γ μ ( p / + m ) γ ν ) \operatorname{Tr}\left((p\!\!\!/^{\prime}+m)\gamma_{\mu}(p\!\!\!/+m)\gamma_{% \nu}\right)\,
  74. = Tr ( p / γ μ p / γ ν ) + Tr ( m γ μ p / γ ν ) =\operatorname{Tr}\left(p\!\!\!/^{\prime}\gamma_{\mu}p\!\!\!/\gamma_{\nu}% \right)+\operatorname{Tr}\left(m\gamma_{\mu}p\!\!\!/\gamma_{\nu}\right)\,
  75. + Tr ( p / γ μ m γ ν ) + Tr ( m 2 γ μ γ ν ) +\operatorname{Tr}\left(p\!\!\!/^{\prime}\gamma_{\mu}m\gamma_{\nu}\right)+% \operatorname{Tr}\left(m^{2}\gamma_{\mu}\gamma_{\nu}\right)\,
  76. = Tr ( p / γ μ p / γ ν ) + m 2 Tr ( γ μ γ ν ) =\operatorname{Tr}\left(p\!\!\!/^{\prime}\gamma_{\mu}p\!\!\!/\gamma_{\nu}% \right)+m^{2}\operatorname{Tr}\left(\gamma_{\mu}\gamma_{\nu}\right)\,
  77. = p ρ p σ Tr ( γ ρ γ μ γ σ γ ν ) + m 2 4 η μ ν ={p^{\prime}}^{\rho}p^{\sigma}\operatorname{Tr}\left(\gamma_{\rho}\gamma_{\mu}% \gamma_{\sigma}\gamma_{\nu}\right)+m^{2}\cdot 4\eta_{\mu\nu}\,
  78. = p ρ p σ 4 ( η ρ μ η σ ν - η ρ σ η μ ν + η ρ ν η μ σ ) + 4 m 2 η μ ν ={p^{\prime}}^{\rho}p^{\sigma}4\left(\eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho% \sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma}\right)+4m^{2}\eta_{\mu\nu}\,
  79. = 4 ( p μ p ν - 𝐩 𝐩 η μ ν + p ν p μ ) + 4 m 2 η μ ν =4\left({p^{\prime}}_{\mu}p_{\nu}-\mathbf{p^{\prime}\cdot p}\eta_{\mu\nu}+p^{% \prime}_{\nu}p_{\mu}\right)+4m^{2}\eta_{\mu\nu}\,

Bhatnagar–Gross–Krook_operator.html

  1. Ω i = - τ - 1 ( n i - n i E Q ) \Omega_{i}=-\tau^{-1}(n_{i}-n_{i}^{EQ})
  2. n i E Q n_{i}^{EQ}
  3. 𝐞 i \mathbf{e}_{i}
  4. τ \tau

Bianchi_classification.html

  1. ξ i ( a ) \xi^{(a)}_{i}
  2. ( ξ i ( c ) x k - ξ k ( c ) x i ) ξ ( a ) i ξ ( b ) k = C a b c \left(\frac{\partial\xi^{(c)}_{i}}{\partial x^{k}}-\frac{\partial\xi^{(c)}_{k}% }{\partial x^{i}}\right)\xi^{i}_{(a)}\xi^{k}_{(b)}=C^{c}_{\ ab}
  3. C a b c C^{c}_{\ ab}
  4. C a b c C^{c}_{\ ab}
  5. C a b c = ε a b d n c d - δ a c a b + δ b c a a C^{c}_{\ ab}=\varepsilon_{abd}n^{cd}-\delta^{c}_{a}a_{b}+\delta^{c}_{b}a_{a}
  6. ε a b d \varepsilon_{abd}
  7. δ a c \delta^{c}_{a}
  8. a a = ( a , 0 , 0 ) a_{a}=(a,0,0)
  9. n c d n^{cd}
  10. n ( i ) n^{(i)}
  11. n c d n^{cd}
  12. a a
  13. n ( 1 ) n^{(1)}
  14. n ( 2 ) n^{(2)}
  15. n ( 3 ) n^{(3)}
  16. a = 1 a=1
  17. a a
  18. a = 1 a=1
  19. a a
  20. VII h \scriptstyle\,\text{VII}_{h}
  21. d s 2 = γ a b ξ i ( a ) ξ k ( b ) d x i d x k ds^{2}=\gamma_{ab}\xi^{(a)}_{i}\xi^{(b)}_{k}dx^{i}dx^{k}
  22. ξ i ( a ) d x i \xi^{(a)}_{i}dx^{i}
  23. R i k R_{ik}
  24. R i k = R ( a ) ( b ) ξ i ( a ) ξ k ( b ) R_{ik}=R_{(a)(b)}\xi^{(a)}_{i}\xi^{(b)}_{k}
  25. R ( a ) ( b ) = 1 2 [ C b c d ( C c d a + C d c a ) + C c d c ( C a b d + C b a d ) - 1 2 C b c d C a c d ] R_{(a)(b)}=\frac{1}{2}\left[C^{cd}_{\ \ b}\left(C_{cda}+C_{dca}\right)+C^{c}_{% \ cd}\left(C^{\ \ d}_{ab}+C^{\ \ d}_{ba}\right)-\frac{1}{2}C^{\ cd}_{b}C_{acd}\right]
  26. γ a b \gamma_{ab}
  27. x i x^{i}

Bias_tee.html

  1. X C = 1 j ω C = 1 j 2 π f C ; X L = j ω L = j 2 π f L X_{C}={1\over j\omega C}={1\over j2\pi fC}\ ;\ X_{L}=j\omega L=j2\pi fL

Biasing.html

  1. β \beta

Bicycle_and_motorcycle_dynamics.html

  1. Trail = ( R w cos ( A h ) - O f ) sin ( A h ) \,\text{Trail}=\frac{(R_{w}\cos(A_{h})-O_{f})}{\sin(A_{h})}
  2. R w R_{w}
  3. A h A_{h}
  4. O f O_{f}
  5. r = w δ cos ( ϕ ) r=\frac{w}{\delta\cos\left(\phi\right)}
  6. r r\,\!
  7. w w\,\!
  8. δ \delta\,\!
  9. ϕ \phi\,\!
  10. θ = arctan ( v 2 g r ) \theta=\arctan\left(\frac{v^{2}}{gr}\right)
  11. r = w cos ( θ ) δ cos ( ϕ ) r=\frac{w\cos\left(\theta\right)}{\delta\cos\left(\phi\right)}
  12. arcsin ( t sin ( ϕ ) h - t ) \arcsin\left(t\frac{\sin(\phi)}{h-t}\right)
  13. Δ = δ cos ( ϕ ) \Delta=\delta\cos\left(\phi\right)
  14. Δ \Delta\,\!
  15. δ \delta\,\!
  16. ϕ \phi\,\!
  17. r = w cos ( θ ) δ cos ( ϕ ) r=\frac{w\cos\left(\theta\right)}{\delta\cos\left(\phi\right)}
  18. r r\,\!
  19. w w\,\!
  20. θ \theta\,\!
  21. δ \delta\,\!
  22. ϕ \phi\,\!
  23. M θ θ θ r ¨ + K θ θ θ r + M θ ψ ψ ¨ + C θ ψ ψ ˙ + K θ ψ ψ = M θ M_{\theta\theta}\ddot{\theta_{r}}+K_{\theta\theta}\theta_{r}+M_{\theta\psi}% \ddot{\psi}+C_{\theta\psi}\dot{\psi}+K_{\theta\psi}\psi=M_{\theta}
  24. M ψ ψ ψ ¨ + C ψ ψ ψ ˙ + K ψ ψ ψ + M ψ θ θ r ¨ + C ψ θ θ r ˙ + K ψ θ θ r = M ψ , M_{\psi\psi}\ddot{\psi}+C_{\psi\psi}\dot{\psi}+K_{\psi\psi}\psi+M_{\psi\theta}% \ddot{\theta_{r}}+C_{\psi\theta}\dot{\theta_{r}}+K_{\psi\theta}\theta_{r}=M_{% \psi}\mbox{,}~{}
  25. θ r \theta_{r}
  26. ψ \psi
  27. M θ M_{\theta}
  28. M ψ M_{\psi}
  29. M 𝐪 ¨ + C 𝐪 ˙ + K 𝐪 = 𝐟 M\mathbf{\ddot{q}}+C\mathbf{\dot{q}}+K\mathbf{q}=\mathbf{f}
  30. M M
  31. C C
  32. v v
  33. K K
  34. g g
  35. v 2 v^{2}
  36. g g
  37. v 2 v^{2}
  38. 𝐪 \mathbf{q}
  39. 𝐟 \mathbf{f}
  40. L L
  41. h h
  42. b b
  43. N r = m g ( L - b L - μ h L ) N_{r}=mg\left(\frac{L-b}{L}-\mu\frac{h}{L}\right)
  44. N f = m g ( b L + μ h L ) N_{f}=mg\left(\frac{b}{L}+\mu\frac{h}{L}\right)
  45. F r = μ N r F_{r}=\mu N_{r}\,
  46. F f = μ N f F_{f}=\mu N_{f}\,
  47. μ \mu
  48. m m
  49. g g
  50. μ L - b h , \mu\geq\frac{L-b}{h},
  51. θ = tan - 1 ( 1 μ ) \theta=\tan^{-1}\left(\frac{1}{\mu}\right)\,

Bicycle_performance.html

  1. P D P_{D}
  2. P D = v r 1 2 ρ v a 2 C D A P_{D}\,=v_{r}\,\tfrac{1}{2}\,\rho\,v_{a}^{2}\,C_{D}\,A
  3. ρ \rho
  4. v r v_{r}
  5. v a v_{a}
  6. C D A C_{D}\,A
  7. v a v_{a}
  8. A A
  9. C D C_{D}
  10. P R P_{R}
  11. P R = v r m g cos ( arctan s ) C r r v r m g C r r P_{R}=v_{r}\,mg\cos(\arctan s)C_{rr}\approx v_{r}mgC_{rr}
  12. C r r C_{rr}
  13. v r v_{r}
  14. P S P_{S}
  15. s s
  16. P S = v r m g sin ( arctan s ) v r m g s P_{S}=v_{r}mg\sin(\arctan s)\approx v_{r}mgs
  17. P A P_{A}
  18. m w m_{w}
  19. P A v r ( m + m w ) a P_{A}\approx v_{r}(m+m_{w})a
  20. m w m_{w}
  21. P = ( P D + P R + P S + P A ) / η P\,=(P_{D}\,+P_{R}\,+P_{S}\,+P_{A}\,)/\eta\,
  22. η \eta\,

Bifluoride.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Biholomorphism.html

  1. ϕ \phi
  2. n n
  3. V V
  4. ϕ - 1 : V U \phi^{-1}:V\to U
  5. ϕ : U V \phi\colon U\to V
  6. n = 1 , n=1,
  7. n > 1. n>1.

Binary_entropy_function.html

  1. H ( p ) H(p)\,
  2. H b ( p ) H_{\mathrm{b}}(p)\,
  3. X = 1 X=1
  4. X = 0 X=0
  5. Pr ( X = 1 ) = p , \mathrm{Pr}(X=1)=p,
  6. Pr ( X = 0 ) = 1 - p \mathrm{Pr}(X=0)=1-p
  7. H ( X ) = H b ( p ) = - p log 2 p - ( 1 - p ) log 2 ( 1 - p ) . H(X)=H_{\mathrm{b}}(p)=-p\log_{2}p-(1-p)\log_{2}(1-p).\,
  8. 0 log 2 0 0\log_{2}0
  9. p = 1 2 , p=\frac{1}{2},
  10. H ( p ) H(p)
  11. H ( X ) H(X)
  12. H 2 ( p ) H_{2}(p)
  13. H 2 ( X ) H_{2}(X)
  14. p = 0 p=0
  15. p = 1 p=1
  16. p = 1 / 2 p=1/2
  17. p = 1 / 4 p=1/4
  18. d d p H b ( p ) = - logit 2 ( p ) = - log 2 ( p 1 - p ) . {d\over dp}H_{\mathrm{b}}(p)=-\operatorname{logit}_{2}(p)=-\log_{2}\left(\frac% {p}{1-p}\right).\,
  19. H b ( p ) = 1 - 1 2 ln 2 n = 1 ( 1 - 2 p ) 2 n n ( 2 n - 1 ) H_{\mathrm{b}}(p)=1-\frac{1}{2\ln 2}\sum^{\infty}_{n=1}\frac{(1-2p)^{2n}}{n(2n% -1)}
  20. 0 p 1 0\leq p\leq 1

Binary_erasure_channel.html

  1. p e p_{e}
  2. p p
  3. 1 - p 1-p

Binary_scaling.html

  1. π 2 \frac{\pi}{2}
  2. 3 π 2 \frac{3\pi}{2}

Binding_constant.html

  1. K K
  2. R + L RL {\rm R}+{\rm L}\rightleftharpoons{\rm RL}
  3. k on k_{\rm on}
  4. k off k_{\rm off}
  5. R + L RL {\rm R}+{\rm L}\to{\rm RL}
  6. RL R + L {\rm RL}\to{\rm R}+{\rm L}
  7. k on [ R ] [ L ] = k off [ RL ] k_{\rm on}\,[{\rm R}]\,[{\rm L}]=k_{\rm off}\,[{\rm RL}]
  8. [ R ] [{\rm R}]
  9. [ L ] [{\rm L}]
  10. [ RL ] [{\rm RL}]
  11. K a K_{\rm a}
  12. K a = k on k off = [ RL ] [ R ] [ L ] K_{\rm a}={k_{\rm on}\over k_{\rm off}}={[{\rm RL}]\over{[{\rm R}]\,[{\rm L}]}}
  13. K d 1 / K a K_{\rm d}\equiv 1/K_{\rm a}
  14. Δ G \Delta G
  15. K d K_{\rm d}
  16. Δ G = R T ln K d c \Delta G=RT\ln{{K_{\rm d}\over c^{\ominus}}}
  17. R R
  18. T T
  19. c c^{\ominus}

Binding_potential.html

  1. B P = R L L | L 0 BP=\frac{RL}{L}\bigg|_{L\approx 0}
  2. B P = R K i BP=\frac{R}{K_{i}}
  3. B P 1 BP_{1}
  4. B P 2 BP_{2}
  5. B P 2 BP_{2}
  6. V 3 ′′ V_{3}^{\prime\prime}
  7. B P 2 = k 3 / k 4 BP_{2}=k_{3}/k_{4}
  8. B P 2 = f 2 B P BP_{2}=f_{2}BP
  9. f 2 f_{2}
  10. B P 1 BP_{1}
  11. B P BP^{\prime}
  12. B P 1 BP_{1}
  13. B P 1 = f 1 B P BP_{1}=f_{1}BP
  14. f 1 f_{1}
  15. f 1 f_{1}
  16. B P 1 BP_{1}
  17. B P 2 BP_{2}
  18. B P 2 BP_{2}
  19. B P 2 BP_{2}
  20. B m a x B_{max}
  21. R + R L R+RL
  22. k 3 k_{3}
  23. k 4 k_{4}

Biquandle.html

  1. X X
  2. a b a^{b}
  3. a b a_{b}
  4. a b c b = a c b c a^{bc_{b}}={a^{c}}^{b^{c}}
  5. a b c b = a c b c {a_{b}}_{c_{b}}={a_{c}}_{b^{c}}
  6. a b c b = a c b c {a_{b}}^{c_{b}}={a^{c}}_{b^{c}}
  7. a * b a*b
  8. a b a_{b}
  9. a * * b a**b
  10. a b a^{b}
  11. ( a * * b ) * * ( c * b ) = ( a * * c ) * * ( b * * c ) (a**b)**(c*b)=(a**c)**(b**c)
  12. ( a * b ) * ( c * b ) = ( a * c ) * ( b * * c ) (a*b)*(c*b)=(a*c)*(b**c)
  13. ( a * b ) * * ( c * b ) = ( a * * c ) * ( b * * c ) (a*b)**(c*b)=(a**c)*(b**c)
  14. a , b a,b
  15. X X
  16. x , y x,y
  17. X X
  18. x b = a x^{b}=a
  19. y b = a y_{b}=a
  20. X X
  21. X X
  22. a b a^{b}
  23. a b = a a_{b}=a
  24. S : X 2 X 2 S:X^{2}\rightarrow X^{2}
  25. S ( a , b a ) = ( b , a b ) . S(a,b_{a})=(b,a^{b}).\,
  26. S S
  27. S 1 S 2 S 1 = S 2 S 1 S 2 S_{1}S_{2}S_{1}=S_{2}S_{1}S_{2}\,
  28. S 1 S_{1}
  29. S 2 S_{2}
  30. S 1 ( a , b , c ) = ( S ( a , b ) , c ) S_{1}(a,b,c)=(S(a,b),c)
  31. S 2 ( a , b , c ) = ( a , S ( b , c ) ) S_{2}(a,b,c)=(a,S(b,c))
  32. S S^{\prime}
  33. S ( b , a b ) = ( a , b a ) S^{\prime}(b,a^{b})=(a,b_{a})\,
  34. S S\,
  35. ( c , b c , a b c b ) (c,b_{c},a_{bc^{b}})
  36. S 1 S 2 S 1 S_{1}S_{2}S_{1}
  37. ( c , b c , a b c b ) ( b , c b , a b c b ) ( b , a b , c b a b ) ( a , b a , c b a b ) . (c,b_{c},a_{bc^{b}})\to(b,c^{b},a_{bc^{b}})\to(b,a_{b},c^{ba_{b}})\to(a,b^{a},% c^{ba_{b}}).
  38. ( c , b c , a b c b ) = ( c , b c , a c b c ) (c,b_{c},a_{bc^{b}})=(c,b_{c},a_{cb_{c}})
  39. S 2 S 1 S 2 S_{2}S_{1}S_{2}
  40. ( c , b c , a c b c ) ( c , a c , b c a c ) ( a , c a , b c a c ) = ( a , c a , b a c a ) ( a , b a , c a b a ) = ( a , b a , c b a b ) . (c,b_{c},a_{cb_{c}})\to(c,a_{c},{b_{c}}^{a_{c}})\to(a,c^{a},{b_{c}}^{a_{c}})=(% a,c^{a},{b^{a}}_{c^{a}})\to(a,b_{a},c_{ab_{a}})=(a,b^{a},c^{ba_{b}}).
  41. S S
  42. T ( a , b ) = ( b , a ) T(a,b)=(b,a)
  43. S ( a , b ) = ( b , a b ) S(a,b)=(b,a^{b})
  44. a b a^{b}

Birch's_theorem.html

  1. n ψ ( r 1 , , r k , l , K ) n\geq\psi(r_{1},\ldots,r_{k},l,K)
  2. f 1 ( x ) = = f k ( x ) = 0 , x V . f_{1}(x)=\cdots=f_{k}(x)=0,\quad\forall x\in V.
  3. c 1 x 1 r + + c n x n r = 0 , c i , i = 1 , , n c_{1}x_{1}^{r}+\cdots+c_{n}x_{n}^{r}=0,\quad c_{i}\in\mathbb{Z},i=1,\ldots,n

Birch–Tate_conjecture.html

  1. ζ F \zeta_{F}
  2. # K 2 = | N ζ F ( - 1 ) | . \#K_{2}=|N\zeta_{F}(-1)|.

Birkhoff_polytope.html

  1. K n , n K_{n,n}
  2. ( σ , ω ) (\sigma,\omega)
  3. σ - 1 ω \sigma^{-1}\omega
  4. vol ( B n ) = exp ( - ( n - 1 ) 2 ln n + n 2 - ( n - 1 2 ) ln ( 2 π ) + 1 3 + o ( 1 ) ) . \mathop{\mathrm{vol}}(B_{n})\,=\,\exp\left(-(n-1)^{2}\ln n+n^{2}-(n-\frac{1}{2% })\ln(2\pi)+\frac{1}{3}+o(1)\right).

Bistatic_Doppler_shift.html

  1. f = 1 λ d d t ( R t x + R r x ) f=\frac{1}{\lambda}\frac{d}{dt}(R_{tx}+R_{rx})

Bitangents_of_a_quartic.html

  1. 144 ( x 4 + y 4 ) - 225 ( x 2 + y 2 ) + 350 x 2 y 2 + 81 = 0. \displaystyle 144(x^{4}+y^{4})-225(x^{2}+y^{2})+350x^{2}y^{2}+81=0.
  2. [ a b c d e f ] \left[\begin{array}[]{ccc}a&b&c\\ d&e&f\\ \end{array}\right]

Bitopological_space.html

  1. X X
  2. σ \sigma
  3. τ \tau
  4. ( X , σ , τ ) (X,\sigma,\tau)
  5. f : X X \scriptstyle f:X\to X^{\prime}
  6. ( X , τ 1 , τ 2 ) \scriptstyle(X,\tau_{1},\tau_{2})
  7. ( X , τ 1 , τ 2 ) \scriptstyle(X^{\prime},\tau_{1}^{\prime},\tau_{2}^{\prime})
  8. f \scriptstyle f
  9. ( X , τ 1 ) \scriptstyle(X,\tau_{1})
  10. ( X , τ 1 ) \scriptstyle(X^{\prime},\tau_{1}^{\prime})
  11. ( X , τ 2 ) \scriptstyle(X,\tau_{2})
  12. ( X , τ 2 ) \scriptstyle(X^{\prime},\tau_{2}^{\prime})
  13. ( X , τ 1 , τ 2 ) \scriptstyle(X,\tau_{1},\tau_{2})
  14. { U i i I } \scriptstyle\{U_{i}\mid i\in I\}
  15. X \scriptstyle X
  16. U i τ 1 τ 2 \scriptstyle U_{i}\in\tau_{1}\cup\tau_{2}
  17. { U i i I } \scriptstyle\{U_{i}\mid i\in I\}
  18. τ 1 \tau_{1}
  19. τ 2 \tau_{2}
  20. ( X , τ 1 , τ 2 ) \scriptstyle(X,\tau_{1},\tau_{2})
  21. x , y X \scriptstyle x,y\in X
  22. U 1 τ 1 \scriptstyle U_{1}\in\tau_{1}
  23. U 2 τ 2 \scriptstyle U_{2}\in\tau_{2}
  24. x U 1 \scriptstyle x\in U_{1}
  25. y U 2 \scriptstyle y\in U_{2}
  26. ( X , τ 1 , τ 2 ) \scriptstyle(X,\tau_{1},\tau_{2})
  27. ( X , τ 1 ) \scriptstyle(X,\tau_{1})
  28. ( X , τ 2 ) \scriptstyle(X,\tau_{2})
  29. ( X , τ 1 ) \scriptstyle(X,\tau_{1})
  30. ( X , τ 2 ) \scriptstyle(X,\tau_{2})
  31. ( X , τ 1 ) \scriptstyle(X,\tau_{1})
  32. ( X , τ 2 ) \scriptstyle(X,\tau_{2})
  33. ( X , σ , τ ) \scriptstyle(X,\sigma,\tau)
  34. F σ \scriptstyle F_{\sigma}
  35. σ \scriptstyle\sigma
  36. F τ \scriptstyle F_{\tau}
  37. τ \scriptstyle\tau
  38. G σ \scriptstyle G_{\sigma}
  39. σ \scriptstyle\sigma
  40. G τ \scriptstyle G_{\tau}
  41. τ \scriptstyle\tau
  42. F σ G τ \scriptstyle F_{\sigma}\subseteq G_{\tau}
  43. F τ G σ \scriptstyle F_{\tau}\subseteq G_{\sigma}
  44. G σ G τ = . \scriptstyle G_{\sigma}\cap G_{\tau}=.

BKL_singularity.html

  1. R 0 0 = T 0 0 - 1 2 T R_{0}^{0}=T_{0}^{0}-\tfrac{1}{2}T
  2. d l 2 = t 2 p 1 d x 2 + t 2 p 2 d y 2 + t 2 p 3 d z 2 dl^{2}=t^{2p_{1}}dx^{2}+t^{2p_{2}}dy^{2}+t^{2p_{3}}dz^{2}
  3. ( - 1 3 , 2 3 , 2 3 ) \scriptstyle{(-\frac{1}{3},\frac{2}{3},\frac{2}{3})}
  4. - 1 3 p 1 0 , 0 p 2 2 3 , 2 3 p 3 1. \begin{matrix}-\tfrac{1}{3}\leq p_{1}\leq 0,\\ \ 0\leq p_{2}\leq\tfrac{2}{3},\\ \frac{2}{3}\leq p_{3}\leq 1.\end{matrix}
  5. p 1 ( u ) = - u 1 + u + u 2 , p 2 ( u ) = 1 + u 1 + u + u 2 , p 3 ( u ) = u ( 1 + u ) 1 + u + u 2 p_{1}(u)=\frac{-u}{1+u+u^{2}},\ p_{2}(u)=\frac{1+u}{1+u+u^{2}},\ p_{3}(u)=% \frac{u(1+u)}{1+u+u^{2}}
  6. d l 2 = ( a 2 l α l β + b 2 m α m β + c 2 n α n β ) d x α d x β dl^{2}=\left(a^{2}l_{\alpha}l_{\beta}+b^{2}m_{\alpha}m_{\beta}+c^{2}n_{\alpha}% n_{\beta}\right)dx^{\alpha}dx^{\beta}
  7. a = t p l , b = t p m , c = t p n a=t^{p_{l}},\ b=t^{p_{m}},\ c=t^{p_{n}}
  8. λ = 𝐥 rot 𝐥 v , μ = 𝐦 rot 𝐦 v , ν = 𝐧 rot 𝐧 v . \lambda=\frac{\mathbf{l}\ \mathrm{rot}\ \mathbf{l}}{v},\ \mu=\frac{\mathbf{m}% \ \mathrm{rot}\ \mathbf{m}}{v},\ \nu=\frac{\mathbf{n}\ \mathrm{rot}\ \mathbf{n% }}{v}.
  9. R 0 0 = - 1 2 ϰ α α t - 1 4 ϰ α β ϰ β α = 0 , R_{0}^{0}=-\frac{1}{2}\frac{\partial\varkappa_{\alpha}^{\alpha}}{\partial t}-% \frac{1}{4}\varkappa_{\alpha}^{\beta}\varkappa_{\beta}^{\alpha}=0,
  10. R α β = - ( 1 2 - g ) t ( - g ϰ α β ) - P α β = 0 , R_{\alpha}^{\beta}=-\left(\frac{1}{2}\sqrt{-g}\right)\frac{\partial}{\partial t% }\left(\sqrt{-g}\varkappa_{\alpha}^{\beta}\right)-P_{\alpha}^{\beta}=0,
  11. R α 0 = 1 2 ( ϰ α ; β β - ϰ β ; α β ) = 0 , R_{\alpha}^{0}=\frac{1}{2}\left(\varkappa_{\alpha;\beta}^{\beta}-\varkappa_{% \beta;\alpha}^{\beta}\right)=0,
  12. ϰ α β \scriptstyle{\varkappa_{\alpha}^{\beta}}
  13. ϰ α β = γ α β t \scriptstyle{\varkappa_{\alpha}^{\beta}=\frac{\partial\gamma_{\alpha}^{\beta}}% {\partial t}}
  14. ϰ α β = ( 2 a ˙ a ) l α l β + ( 2 b ˙ b ) m α m β + ( 2 c ˙ c ) n α n β \varkappa_{\alpha}^{\beta}=\left(\frac{2\dot{a}}{a}\right)l_{\alpha}l^{\beta}+% \left(\frac{2\dot{b}}{b}\right)m_{\alpha}m^{\beta}+\left(\frac{2\dot{c}}{c}% \right)n_{\alpha}n^{\beta}
  15. - R 0 0 = a ¨ a + b ¨ b + c ¨ c = 0. -R_{0}^{0}=\frac{\ddot{a}}{a}+\frac{\ddot{b}}{b}+\frac{\ddot{c}}{c}=0.
  16. - R l l = ( a ˙ b c ) ˙ a b c = 0 , - R m m = ( a b ˙ c ) ˙ a b c = 0 , - R n n = ( a b c ˙ ) ˙ a b c = 0 -R_{l}^{l}=\frac{(\dot{a}bc)\dot{}}{abc}=0,\ -R_{m}^{m}=\frac{(a\dot{b}c)\dot{% }}{abc}=0,\ -R_{n}^{n}=\frac{(ab\dot{c})\dot{}}{abc}=0
  17. - R l l = ( a ˙ b c ) ˙ a b c + λ 2 a 2 2 b 2 c 2 = 0 , - R m m = ( a b ˙ c ) ˙ a b c - λ 2 a 2 2 b 2 c 2 = 0 , - R n n = ( a b c ˙ ) ˙ a b c - λ 2 a 2 2 b 2 c 2 = 0. \begin{aligned}\displaystyle-R_{l}^{l}&\displaystyle=\frac{(\dot{a}bc)\dot{}}{% abc}+\frac{\lambda^{2}a^{2}}{2b^{2}c^{2}}=0,\\ \displaystyle-R_{m}^{m}&\displaystyle=\frac{(a\dot{b}c)\dot{}}{abc}-\frac{% \lambda^{2}a^{2}}{2b^{2}c^{2}}=0,\\ \displaystyle-R_{n}^{n}&\displaystyle=\frac{(ab\dot{c})\dot{}}{abc}-\frac{% \lambda^{2}a^{2}}{2b^{2}c^{2}}=0.\\ \end{aligned}
  18. 1 - g x i ( - g σ u i ) = 0 , \frac{1}{\sqrt{-g}}\frac{\partial}{\partial x^{i}}\left(\sqrt{-g}\sigma u^{i}% \right)=0,
  19. ( p + ε ) u k { u i x k - 1 2 u l g k l x i } = - p x i - u i u k p x k , (p+\varepsilon)u^{k}\left\{\frac{\partial u_{i}}{\partial x^{k}}-\frac{1}{2}u^% {l}\frac{\partial g_{kl}}{\partial x^{i}}\right\}=-\frac{\partial p}{\partial x% ^{i}}-u_{i}u^{k}\frac{\partial p}{\partial x^{k}},
  20. t ( - g u 0 ε 3 4 ) = 0 , 4 ε u α t + u α ε t = 0 , \frac{\partial}{\partial t}\left(\sqrt{-g}u_{0}\varepsilon^{\frac{3}{4}}\right% )=0,\ 4\varepsilon\cdot\frac{\partial u_{\alpha}}{\partial t}+u_{\alpha}\cdot% \frac{\partial\varepsilon}{\partial t}=0,
  21. a b c u 0 ε 3 4 = const , u α ε 1 4 = const , abcu_{0}\varepsilon^{\frac{3}{4}}=\mathrm{const},\ u_{\alpha}\varepsilon^{% \frac{1}{4}}=\mathrm{const},
  22. u 0 2 u n u n = u n 2 c 2 , u_{0}^{2}\approx u_{n}u^{n}=\frac{u_{n}^{2}}{c^{2}},
  23. ε 1 a 2 b 2 , u α a b \varepsilon\sim\frac{1}{a^{2}b^{2}},\ u_{\alpha}\sim\sqrt{ab}
  24. ε t - 2 ( p 1 + p 2 ) = t - 2 ( 1 - p 3 ) , u α t ( 1 - p 3 ) 2 . \varepsilon\sim t^{-2(p_{1}+p_{2})}=t^{-2(1-p_{3})},\ u_{\alpha}\sim t^{\frac{% (1-p_{3})}{2}}.
  25. R 0 0 = T 0 0 - 1 2 T , R α β = T α β - 1 2 δ α β T , R_{0}^{0}=T_{0}^{0}-\frac{1}{2}T,\ R_{\alpha}^{\beta}=T_{\alpha}^{\beta}-\frac% {1}{2}\delta_{\alpha}^{\beta}T,
  26. R α 0 = T α 0 \scriptstyle{R_{\alpha}^{0}=T_{\alpha}^{0}}
  27. R α 0 \scriptstyle{R_{\alpha}^{0}}
  28. - R l l = ( a ˙ b c ) ˙ a b c + 1 2 ( a 2 b 2 c 2 ) [ λ 2 a 4 - ( μ b 2 - ν c 2 ) 2 ] = 0 , - R m m = ( a b ˙ c ) ˙ a b c + 1 2 ( a 2 b 2 c 2 ) [ μ 2 b 4 - ( λ a 2 - ν c 2 ) 2 ] = 0 , - R n n = ( a b c ˙ ) ˙ a b c + 1 2 ( a 2 b 2 c 2 ) [ ν 2 c 4 - ( λ a 2 - μ b 2 ) 2 ] = 0 , \begin{aligned}\displaystyle-R_{l}^{l}&\displaystyle=\frac{\left(\dot{a}bc% \right)\dot{}}{abc}+\frac{1}{2}\left(a^{2}b^{2}c^{2}\right)\left[\lambda^{2}a^% {4}-\left(\mu b^{2}-\nu c^{2}\right)^{2}\right]=0,\\ \displaystyle-R_{m}^{m}&\displaystyle=\frac{(a\dot{b}c)\dot{}}{abc}+\frac{1}{2% }\left(a^{2}b^{2}c^{2}\right)\left[\mu^{2}b^{4}-\left(\lambda a^{2}-\nu c^{2}% \right)^{2}\right]=0,\\ \displaystyle-R_{n}^{n}&\displaystyle=\frac{\left(ab\dot{c}\right)\dot{}}{abc}% +\frac{1}{2}\left(a^{2}b^{2}c^{2}\right)\left[\nu^{2}c^{4}-\left(\lambda a^{2}% -\mu b^{2}\right)^{2}\right]=0,\\ \end{aligned}
  29. - R 0 0 = a ¨ a + b ¨ b + c ¨ c = 0 -R_{0}^{0}=\frac{\ddot{a}}{a}+\frac{\ddot{b}}{b}+\frac{\ddot{c}}{c}=0
  30. R l 0 \scriptstyle{R_{l}^{0}}
  31. R m 0 \scriptstyle{R_{m}^{0}}
  32. R n 0 \scriptstyle{R_{n}^{0}}
  33. R l m \scriptstyle{R_{l}^{m}}
  34. R l n \scriptstyle{R_{l}^{n}}
  35. R m n \scriptstyle{R_{m}^{n}}
  36. 2 α τ τ \displaystyle 2\alpha_{\tau\tau}
  37. 1 2 ( α + β + γ ) τ τ = α τ β τ + α τ γ τ + β τ γ τ . \frac{1}{2}\left(\alpha+\beta+\gamma\right)_{\tau\tau}=\alpha_{\tau}\beta_{% \tau}+\alpha_{\tau}\gamma_{\tau}+\beta_{\tau}\gamma_{\tau}.
  38. α τ β τ + α τ γ τ + β τ γ τ = 1 4 ( λ 2 a 4 + μ 2 b 4 + ν 2 c 4 - 2 λ μ a 2 b 2 - 2 λ ν a 2 c 2 - 2 μ ν b 2 c 2 ) . \alpha_{\tau}\beta_{\tau}+\alpha_{\tau}\gamma_{\tau}+\beta_{\tau}\gamma_{\tau}% =\frac{1}{4}\left(\lambda^{2}a^{4}+\mu^{2}b^{4}+\nu^{2}c^{4}-2\lambda\mu a^{2}% b^{2}-2\lambda\nu a^{2}c^{2}-2\mu\nu b^{2}c^{2}\right).
  39. α τ τ = - 1 2 λ 2 e 4 α , β τ τ = γ τ τ = 1 2 λ 2 e 4 α \alpha_{\tau\tau}=-\frac{1}{2}\lambda^{2}e^{4\alpha},\ \beta_{\tau\tau}=\gamma% _{\tau\tau}=\frac{1}{2}\lambda^{2}e^{4\alpha}
  40. a t p 1 , b t p 2 , c t p 3 . a\sim t^{p_{1}},\ b\sim t^{p_{2}},\ c\sim t^{p_{3}}.
  41. a b c = Λ t , τ = Λ - 1 ln t + const abc=\Lambda t,\ \tau=\Lambda^{-1}\ln t+\mathrm{const}
  42. α τ = Λ p 1 , β τ = Λ p 2 , γ τ = Λ p 3 at τ \alpha_{\tau}=\Lambda p_{1},\ \beta_{\tau}=\Lambda p_{2},\ \gamma_{\tau}=% \Lambda p_{3}\ \mathrm{at}\ \tau\to\infty
  43. { a 2 = 2 | p 1 | Λ ch ( 2 | p 1 | Λ τ ) , b 2 = b 0 2 e 2 Λ ( p 2 - | p 1 | ) τ ch ( 2 | p 1 | Λ τ ) , c 2 = c 0 2 e 2 Λ ( p 2 - | p 1 | ) τ ch ( 2 | p 1 | Λ τ ) , \begin{cases}a^{2}=\frac{2|p_{1}|\Lambda}{\operatorname{ch}(2|p_{1}|\Lambda% \tau)},\\ b^{2}=b_{0}^{2}e^{2\Lambda(p_{2}-|p_{1}|)\tau}\operatorname{ch}(2|p_{1}|% \Lambda\tau),\\ c^{2}=c_{0}^{2}e^{2\Lambda(p_{2}-|p_{1}|)\tau}\operatorname{ch}(2|p_{1}|% \Lambda\tau),\end{cases}
  44. a e - Λ p 1 τ , b e Λ ( p 2 + 2 p 1 ) τ , c e Λ ( p 3 + 2 p 1 ) τ , t e Λ ( 1 + 2 p 1 ) τ . a\sim e^{-\Lambda p_{1}\tau},\ b\sim e^{\Lambda(p_{2}+2p_{1})\tau},\ c\sim e^{% \Lambda(p_{3}+2p_{1})\tau},\ t\sim e^{\Lambda(1+2p_{1})\tau}.
  45. a t p l , b t p m , c t p n a\sim t^{p^{\prime}_{l}},b\sim t^{p^{\prime}_{m}},c\sim t^{p^{\prime}_{n}}
  46. p l = | p 1 | 1 - 2 | p 1 | , p m = - 2 | p 1 | - p 2 1 - 2 | p 1 | , p n = p 3 - 2 | p 1 | 1 - 2 | p 1 | . p^{\prime}_{l}=\frac{|p_{1}|}{1-2|p_{1}|},p^{\prime}_{m}=-\frac{2|p_{1}|-p_{2}% }{1-2|p_{1}|},p^{\prime}_{n}=\frac{p_{3}-2|p_{1}|}{1-2|p_{1}|}.
  47. if p l = p 1 ( u ) p m = p 2 ( u ) p n = p 3 ( u ) then p l = p 2 ( u - 1 ) p m = p 1 ( u - 1 ) p n = p 3 ( u - 1 ) \begin{matrix}\mathrm{if}&p_{l}=p_{1}(u)&p_{m}=p_{2}(u)&p_{n}=p_{3}(u)\\ \mathrm{then}&p^{\prime}_{l}=p_{2}(u-1)&p^{\prime}_{m}=p_{1}(u-1)&p^{\prime}_{% n}=p_{3}(u-1)\end{matrix}
  48. a max = 2 Λ | p 1 ( u ) | a_{\max}=\sqrt{2\Lambda|p_{1}(u)|}
  49. a max a max = [ p 1 ( u - 1 ) p 1 ( u ) ( 1 - 2 | p 1 ( u ) | ) ] 1 2 ; \frac{a^{\prime}_{\max}}{a_{\max}}=\left[\frac{p_{1}(u-1)}{p_{1}(u)}\left(1-2|% p_{1}(u)|\right)\right]^{\frac{1}{2}};
  50. a max a max = u - 1 u u u . \frac{a^{\prime}_{\max}}{a_{\max}}=\sqrt{\frac{u-1}{u}}\equiv\sqrt{\frac{u^{% \prime}}{u}}.
  51. R α 0 \scriptstyle{R_{\alpha}^{0}}
  52. T α 0 \scriptstyle{T_{\alpha}^{0}}
  53. ε = t - 2 ( 1 - p 3 ) , \varepsilon=t^{-2(1-p_{3})},
  54. u max ( s ) \scriptstyle{u_{\max}^{(s)}}
  55. u max ( s ) \scriptstyle{u_{\max}^{(s)}}
  56. u max ( s ) \scriptstyle{u_{\max}^{(s)}}
  57. u min ( s ) \scriptstyle{u_{\min}^{(s)}}
  58. u max ( s ) \scriptstyle{u_{\max}^{(s)}}
  59. u max ( s + 1 ) = 1 x ( s ) , k ( s + 1 ) = [ 1 x ( s ) ] . u_{\max}^{(s+1)}=\frac{1}{x^{(s)}},\ k^{(s+1)}=\left[\frac{1}{x^{(s)}}\right].
  60. p 1 - 1 u , p 2 1 u , p 2 1 - 1 u 2 , p_{1}\approx-\frac{1}{u},\ p_{2}\approx\frac{1}{u},\ p_{2}\approx 1-\frac{1}{u% ^{2}},
  61. α τ τ + β τ τ = 0 , \alpha_{\tau\tau}+\beta_{\tau\tau}=0,\,
  62. α τ τ - β τ τ = e 4 β - e 4 α , \alpha_{\tau\tau}-\beta_{\tau\tau}=e^{4\beta}-e^{4\alpha},\,
  63. γ τ τ ( α τ τ + β τ τ ) = - α τ β τ + 1 4 ( e 2 α - e 2 β ) 2 . \gamma_{\tau\tau}\left(\alpha_{\tau\tau}+\beta_{\tau\tau}\right)=-\alpha_{\tau% }\beta_{\tau}+\frac{1}{4}\left(e^{2\alpha}-e^{2\beta}\right)^{2}.
  64. α + β = ( 2 a 0 2 ξ 0 ) ( τ - τ 0 ) + 2 ln a 0 , \alpha+\beta=\left(\frac{2a_{0}^{2}}{\xi_{0}}\right)\left(\tau-\tau_{0}\right)% +2\ln a_{0},
  65. ξ = ξ 0 exp [ 2 a 0 2 ξ 0 ( τ - τ 0 ) ] . \xi=\xi_{0}\exp\left[\frac{2a_{0}^{2}}{\xi_{0}}\left(\tau-\tau_{0}\right)% \right].
  66. α + β = ln ( ξ ξ 0 ) + 2 ln a 0 . \alpha+\beta=\ln\left(\frac{\xi}{\xi_{0}}\right)+2\ln a_{0}.
  67. χ ξ ξ = χ ξ ξ + 1 2 sh 2 χ = 0 , \chi_{\xi\xi}=\frac{\chi_{\xi}}{\xi}+\frac{1}{2}\operatorname{sh}2\chi=0,
  68. γ ξ = - 1 4 ξ + 1 8 ξ ( 2 χ ξ 2 + ch 2 χ - 1 ) . \gamma_{\xi}=-\frac{1}{4}\xi+\frac{1}{8}\xi\left(2\chi_{\xi}^{2}+\operatorname% {ch}2\chi-1\right).
  69. χ = α - β = ( 2 A ξ ) sin ( ξ - ξ 0 ) , \chi=\alpha-\beta=\left(\frac{2A}{\sqrt{\xi}}\right)\sin\left(\xi-\xi_{0}% \right),
  70. 1 ξ \tfrac{1}{\sqrt{\xi}}
  71. γ ξ = 1 4 ξ ( 2 χ ξ 2 + χ 2 ) = A 2 , γ = A 2 ( ξ - ξ 0 ) + const . \gamma_{\xi}=\frac{1}{4}\xi\left(2\chi_{\xi}^{2}+\chi^{2}\right)=A^{2},\ % \gamma=A^{2}\left(\xi-\xi_{0}\right)+\mathrm{const}.
  72. { a b = a 0 ξ ξ 0 [ 1 ± A ξ sin ( ξ - ξ 0 ) ] , \begin{cases}a\\ b\end{cases}=a_{0}\sqrt{\frac{\xi}{\xi_{0}}}\left[1\pm\frac{A}{\sqrt{\xi}}\sin% \left(\xi-\xi_{0}\right)\right],
  73. c = c 0 e - A 2 ( ξ 0 - ξ ) . c=c_{0}e^{-A^{2}\left(\xi_{0}-\xi\right)}.
  74. t t 0 = e - A 2 ( ξ 0 - ξ ) . \frac{t}{t_{0}}=e^{-A^{2}\left(\xi_{0}-\xi\right)}.
  75. \scriptstyle{\ll}
  76. \scriptstyle{\ll}
  77. χ = α - β = k ln ξ + const , \chi=\alpha-\beta=k\ln\xi+\mathrm{const},\,
  78. a ξ 1 + k 2 , b ξ 1 - k 2 , c ξ - 1 - k 2 4 , t ξ 3 + k 2 4 . a\sim\xi^{\frac{1+k}{2}},\ b\sim\xi^{\frac{1-k}{2}},\ c\sim\xi^{-\frac{1-k^{2}% }{4}},\ t\sim\xi^{\frac{3+k^{2}}{4}}.
  79. a - b a 1 ξ \tfrac{a-b}{a}\sim\tfrac{1}{\sqrt{\xi}}
  80. ξ \scriptstyle{\sim\sqrt{\xi}}
  81. A 2 ξ 0 = ln t 0 t 1 . A^{2}\xi_{0}=\ln\frac{t_{0}}{t_{1}}.
  82. ξ ξ 0 = ln t t 1 ln t 0 t 1 . \frac{\xi}{\xi_{0}}=\frac{\ln\tfrac{t}{t_{1}}}{\ln\tfrac{t_{0}}{t_{1}}}.
  83. ε ( ξ 0 ξ ) 2 . \varepsilon\sim\left(\frac{\xi_{0}}{\xi}\right)^{2}.
  84. ξ 0 2 \scriptstyle{\xi^{2}_{0}}
  85. a 2 = b 2 = p 2 ch ( 2 p τ + δ 1 ) ch 2 ( p τ + δ 2 ) , c 2 = 2 p ch ( 2 p τ + δ 1 ) , a^{2}=b^{2}=\frac{p}{2}\frac{\mathrm{ch}(2p\tau+\delta_{1})}{\mathrm{ch}^{2}(p% \tau+\delta_{2})},\;c^{2}=\frac{2p}{\mathrm{ch}(2p\tau+\delta_{1})},
  86. γ ξ = - 1 4 ξ + 1 8 ξ ( 2 χ ξ 2 + ch2 χ + 1 ) . \gamma_{\xi}=-\frac{1}{4}\xi+\frac{1}{8}\xi\left(2\chi_{\xi}^{2}+\mathrm{ch}2% \chi+1\right).
  87. γ ξ 1 8 ξ 2 , γ 1 8 ( ξ 2 - ξ 0 2 ) , \gamma_{\xi}\approx\frac{1}{8}\xi\cdot 2,\quad\gamma\approx\frac{1}{8}\left(% \xi^{2}-\xi_{0}^{2}\right),
  88. c c 0 = t t 0 = e - 1 8 ( ξ 0 2 - ξ 2 ) . \frac{c}{c_{0}}=\frac{t}{t_{0}}=e^{-\frac{1}{8}\left(\xi_{0}^{2}-\xi^{2}\right% )}.
  89. ξ 0 = 8 ln t t 0 . \xi_{0}=\sqrt{8\ln\frac{t}{t_{0}}}.
  90. 1 / 2 {1}/{2}
  91. const + | p 1 ( u n ) | Ω \mathrm{const}+|p_{1}(u_{n})|\Omega\,
  92. const - p 2 ( u n ) Ω \mathrm{const}-p_{2}(u_{n})\Omega\,
  93. δ n + 1 Ω n + 1 = 1 + u n u n δ n Ω n = 1 + u 0 u n δ 0 Ω 0 \delta_{n+1}\Omega_{n+1}=\frac{1+u_{n}}{u_{n}}\delta_{n}\Omega_{n}=\frac{1+u_{% 0}}{u_{n}}\delta_{0}\Omega_{0}
  94. Δ n + 1 Ω n + 1 - Ω n = f ( u n ) u n δ n Ω n = f ( u n ) ( 1 + u n - 1 ) f ( u n - 1 ) u n Δ n , \Delta_{n+1}\equiv\Omega_{n+1}-\Omega_{n}=\frac{f(u_{n})}{u_{n}}\delta_{n}% \Omega_{n}=\frac{f(u_{n})(1+u_{n-1})}{f(u_{n-1})u_{n}}\Delta_{n},
  95. Ω n - Ω 0 = [ n ( n - 1 ) + n f ( u n - 1 ) u n - 1 ] δ 0 Ω 0 . \Omega_{n}-\Omega_{0}=\left[n(n-1)+\frac{nf(u_{n-1})}{u_{n-1}}\right]\delta_{0% }\Omega_{0}.
  96. A 0 k / ( 1 + x ) \scriptstyle{A_{0}^{k/(1+x)}}
  97. Ω 0 - Ω 0 Ω k - Ω 0 = k ( k + x + 1 x ) δ 0 Ω 0 \Omega^{\prime}_{0}-\Omega_{0}\equiv\Omega_{k}-\Omega_{0}=k\left(k+x+\frac{1}{% x}\right)\delta_{0}\Omega_{0}
  98. δ 0 Ω 0 = ( δ 0 - 1 + k 2 + k x - 1 ) δ 0 Ω 0 . \delta^{\prime}_{0}\Omega^{\prime}_{0}=\left(\delta_{0}^{-1}+k^{2}+kx-1\right)% \delta_{0}\Omega_{0}.
  99. A 0 A 0 k 2 \scriptstyle{A_{0}^{\prime}\sim A_{0}^{k^{2}}}
  100. ln ( ε n + 1 ε n ) = 2 [ 1 - p 3 ( u n ) ] Δ n + 1 . \ln\left(\frac{\varepsilon_{n+1}}{\varepsilon_{n}}\right)=2\left[1-p_{3}(u_{n}% )\right]\Delta_{n+1}.
  101. ln ( ε k ε 0 ) ln ( ε 0 ε 0 ) = 2 ( k - 1 + x ) δ 0 Ω 0 . \ln\left(\frac{\varepsilon_{k}}{\varepsilon_{0}}\right)\equiv\ln\left(\frac{% \varepsilon_{0}^{\prime}}{\varepsilon_{0}}\right)=2(k-1+x)\delta_{0}\Omega_{0}.
  102. ε 0 / ε 0 A 0 2 k \scriptstyle{\varepsilon_{0}^{\prime}/\varepsilon_{0}\sim A_{0}^{2k}}
  103. ε 0 ′′ / ε 0 A 0 2 k ′′ A 0 2 k 2 k \scriptstyle{\varepsilon_{0}^{\prime\prime}/\varepsilon_{0}^{\prime}\sim A_{0}% ^{\prime 2k^{\prime\prime}}\sim A_{0}^{2k^{2}k^{\prime}}}
  104. w ( x ) = 1 ( 1 + x ) ln 2 . w(x)=\frac{1}{(1+x)\ln 2}.
  105. W ( k ) = ( ln 2 ) - 1 ln [ ( k + 1 ) 2 k ( k + 2 ) ] . W(k)=\left(\ln 2\right)^{-1}\ln\left[\frac{(k+1)^{2}}{k(k+2)}\right].
  106. W ( k ) 1 k 2 ln 2 . W(k)\approx\frac{1}{k^{2}\ln 2}.
  107. k ¯ \scriptstyle{\bar{k}}
  108. k ¯ ln N \scriptstyle{\bar{k}\sim\ln N}
  109. 1 K N \scriptstyle{1\ll K\ll N}
  110. Ω ( s + 1 ) / Ω ( s ) = 1 + δ ( s ) k ( s ) ( k ( s ) + x ( s ) + 1 / x ( s ) ) ε ξ s , \Omega^{(s+1)}/\Omega^{(s)}=1+\delta^{(s)}k^{(s)}\left(k^{(s)}+x^{(s)}+1/x^{(s% )}\right)\equiv\varepsilon^{\xi_{s}},
  111. δ ( s + 1 ) = 1 - ( k ( s ) / x ( s ) + 1 ) δ ( s ) 1 + δ ( s ) k ( s ) ( 1 + x ( s ) + 1 / x ( s ) ) , \delta^{(s+1)}=1-\frac{\left(k^{(s)}/x^{(s)}+1\right)\delta^{(s)}}{1+\delta^{(% s)}k^{(s)}\left(1+x^{(s)}+1/x^{(s)}\right)},
  112. ln ( ε ( s + 1 ) ε ( s ) ) = 2 ( k ( s ) + x ( s ) - 1 ) δ ( s ) Ω ( s ) \ln\left(\frac{\varepsilon^{(s+1)}}{\varepsilon^{(s)}}\right)=2\left(k^{(s)}+x% ^{(s)}-1\right)\delta^{(s)}\Omega^{(s)}
  113. δ ¯ = 0.52. \scriptstyle{\bar{\delta}=0.52.}
  114. Ω ( s ) Ω ( 0 ) = exp ( p = 0 s - 1 ξ p ) . \frac{\Omega^{(s)}}{\Omega^{(0)}}=\exp\left(\sum_{p=0}^{s-1}\xi_{p}\right).
  115. τ s ln ( Ω ( s ) Ω ( 0 ) ) = p = 0 s - 1 ξ p \tau_{s}\equiv\ln\left(\frac{\Omega^{(s)}}{\Omega^{(0)}}\right)=\sum_{p=0}^{s-% 1}\xi_{p}
  116. ξ ¯ = 2.1 , ξ ¯ 2 = 6.8. \scriptstyle{\bar{\xi}=2.1,\quad\bar{\xi}^{2}=6.8.}
  117. τ ¯ s = 2.1 s , \bar{\tau}_{s}=2.1s,
  118. ( τ s - τ ¯ s ) 2 ¯ = p , q = 0 s - 1 ( ξ p ξ ¯ q - ξ ¯ p ξ ¯ q ) = s p = 0 s - 1 ( ξ 0 ξ ¯ p - ξ ¯ 2 ) . \overline{\left(\tau_{s}-\bar{\tau}_{s}\right)^{2}}=\sum_{p,q=0}^{s-1}\left(% \overline{\xi_{p}\xi}_{q}-\bar{\xi}_{p}\bar{\xi}_{q}\right)=s\sum_{p=0}^{s-1}% \left(\overline{\xi_{0}\xi}_{p}-\bar{\xi}^{2}\right).
  119. ξ s + 1 ξ ¯ s - ξ ¯ 2 \scriptstyle{\overline{\xi_{s+1}\xi}_{s}-\bar{\xi}^{2}}
  120. [ ( τ s - τ ¯ s ) 2 ] ¯ 1 2 = 1.4 s , \overline{\left[\left(\tau_{s}-\bar{\tau}_{s}\right)^{2}\right]}^{\frac{1}{2}}% =1.4\sqrt{s},
  121. ρ ( τ s ) exp { - ( τ s - 2.1 s ) 2 4 s } . \rho(\tau_{s})\propto\exp\left\{-\frac{\left(\tau_{s}-2.1s\right)^{2}}{4s}% \right\}.
  122. s ¯ τ \scriptstyle{\bar{s}_{\tau}}
  123. s ¯ τ = 0.47 τ . \bar{s}_{\tau}=0.47\tau.
  124. ρ ( s τ ) exp { - ( s τ - 0.47 τ ) 2 / 0.43 τ } . \rho(s_{\tau})\propto\exp\left\{-\left(s_{\tau}-0.47\tau\right)^{2}/0.43\tau% \right\}.
  125. ln ln ε ( s + 1 ) ε ( s ) = η s + p = 0 s - 1 ξ p , η s = ln [ 2 δ ( s ) ( k ( s ) + x ( s ) - 1 ) Ω ( 0 ) ] \ln\ln\frac{\varepsilon^{(s+1)}}{\varepsilon^{(s)}}=\eta_{s}+\sum_{p=0}^{s-1}% \xi_{p},\quad\eta_{s}=\ln\left[2\delta^{(s)}\left(k^{(s)}+x^{(s)}-1\right)% \Omega^{(0)}\right]
  126. ln ln ε ( s ) ε ( 0 ) = ln p = 0 s - 1 exp { q = 0 p ξ q + η p } . \ln\ln\frac{\varepsilon^{(s)}}{\varepsilon^{(0)}}=\ln\sum_{p=0}^{s-1}\exp\left% \{\sum_{q=0}^{p}\xi_{q}+\eta_{p}\right\}.
  127. s ξ ¯ \scriptstyle{s\bar{\xi}}
  128. ln ln ( ε ( s ) ε ( 0 ) ) ¯ = ln ( Ω ( s ) Ω ( 0 ) ) ¯ . \overline{\ln\ln\left(\frac{\varepsilon^{(s)}}{\varepsilon^{(0)}}\right)}=% \overline{\ln\left(\frac{\Omega^{(s)}}{\Omega^{(0)}}\right)}.
  129. ln ln ( ε τ / ε ( 0 ) ) ¯ = τ or ln ln ( ε ( s ) / ε ( 0 ) ) ¯ = 2.1 s , \overline{\ln\ln\left(\varepsilon_{\tau}/\varepsilon^{(0)}\right)}=\tau\quad\,% \text{or}\quad\overline{\ln\ln\left(\varepsilon^{(s)}/\varepsilon^{(0)}\right)% }=2.1s,
  130. x ( s ) exp | α ( s ) | < 1 , x^{(s)}\exp\left|\alpha^{(s)}\right|<1,
  131. λ = exp ( | - α ( s ) | ) + s = 1 k exp ( | - α ( s ) | ) k ( 1 ) 2 k ( 2 ) 2 k ( s ) 2 \lambda=\exp\left(\left|-\alpha^{(s)}\right|\right)+\sum_{s=1}^{\infty}\sum_{k% }\frac{\exp\left(\left|-\alpha^{(s)}\right|\right)}{k^{(1)^{2}}k^{(2)^{2}}% \cdot\cdot\cdot k^{(s)^{2}}}
  132. \scriptstyle{\ll}
  133. k 1 / k ( 1 ) 2 k ( 2 ) 2 k ( s ) 2 = ( π 2 / 6 ) s \sum_{k}1/k^{(1)^{2}}k^{(2)^{2}}\cdot\cdot\cdot k^{(s)^{2}}=\left(\pi^{2}/6% \right)^{s}
  134. λ = exp ( | - α ( 0 ) | ) s = 0 [ ( π 2 / 6 ) exp ( | - α ( 0 ) | ) ] s exp ( | - α ( 0 ) | ) . \lambda=\exp\left(\left|-\alpha^{(0)}\right|\right)\sum_{s=0}^{\infty}\left[% \left(\pi^{2}/6\right)\exp\left(\left|-\alpha^{(0)}\right|\right)\right]^{s}% \approx\exp\left(\left|-\alpha^{(0)}\right|\right).
  135. g 00 = γ 33 , g 0 α = 0. g_{00}=\gamma_{33},\quad g_{0\alpha}=0.
  136. d s 2 = γ 33 ( d ξ 2 - d z 2 ) - γ a b d x a d x b - 2 γ a 3 d x a d z . ds^{2}=\gamma_{33}\left(d\xi^{2}-dz^{2}\right)-\gamma_{ab}dx^{a}dx^{b}-2\gamma% _{a3}dx^{a}dz.
  137. γ 33 γ a b , \gamma_{33}\ll\gamma_{ab},
  138. γ a 3 2 γ a a γ 33 \gamma_{a3}^{2}\ll\gamma_{aa}\gamma_{33}
  139. d s 2 = γ 33 ( d ξ 2 - d z 2 ) - γ a b d x a d x b . ds^{2}=\gamma_{33}\left(d\xi^{2}-dz^{2}\right)-\gamma_{ab}dx^{a}dx^{b}.
  140. R 0 0 \scriptstyle{R_{0}^{0}}
  141. R 3 0 \scriptstyle{R_{3}^{0}}
  142. R 3 3 \scriptstyle{R_{3}^{3}}
  143. R a b \scriptstyle{R_{a}^{b}}
  144. γ 33 = e ψ , γ ˙ a b = ϰ a b , γ a b = λ a b , | γ a b | = G 2 , \gamma_{33}=e^{\psi},\quad\dot{\gamma}_{ab}=\varkappa_{ab},\quad\gamma_{ab}^{% \prime}=\lambda_{ab},\quad|\gamma_{ab}|=G^{2},
  145. 2 e ψ R a b = G - 1 ( G λ a b ) - G - 1 ( G ϰ a b ) ˙ = 0 , 2e^{\psi}R_{a}^{b}=G^{-1}\left(G\lambda_{a}^{b}\right)^{\prime}-G^{-1}\left(G% \varkappa_{a}^{b}\right)\dot{}=0,
  146. 2 e ψ R 3 0 = 1 2 ϰ ψ + 1 2 λ ψ ˙ - ϰ - 1 2 ϰ a b λ b a = 0 , 2e^{\psi}R_{3}^{0}=\frac{1}{2}\varkappa\psi^{\prime}+\frac{1}{2}\lambda\dot{% \psi}-\varkappa^{\prime}-\frac{1}{2}\varkappa_{a}^{b}\lambda_{b}^{a}=0,
  147. 2 e ψ ( R 0 0 - R 3 3 ) = λ ψ + ϰ ψ ˙ - ϰ ˙ - λ - 1 2 ϰ a b ϰ b a - 1 2 λ a b λ b a = 0. 2e^{\psi}(R_{0}^{0}-R_{3}^{3})=\lambda\psi^{\prime}+\varkappa\dot{\psi}-\dot{% \varkappa}-\lambda^{\prime}-\frac{1}{2}\varkappa_{a}^{b}\varkappa_{b}^{a}-% \frac{1}{2}\lambda_{a}^{b}\lambda_{b}^{a}=0.
  148. ϰ \scriptstyle{\varkappa}
  149. ϰ a a \scriptstyle{\varkappa_{a}^{a}}
  150. λ a a \scriptstyle{\lambda_{a}^{a}}
  151. R a 0 \scriptstyle{R_{a}^{0}}
  152. R a 3 \scriptstyle{R_{a}^{3}}
  153. G ′′ + G ¨ = 0 \scriptstyle{G^{\prime\prime}+\ddot{G}=0}
  154. \scriptstyle{\gg}
  155. N g 00 ( G ˙ ) 2 - γ 33 ( G ) 2 = 4 γ 33 f ˙ 1 f ˙ 2 \scriptstyle{N\approx g^{00}\left(\dot{G}\right)^{2}-\gamma^{33}\left(G^{% \prime}\right)^{2}=4\gamma^{33}\dot{f}_{1}\dot{f}_{2}}
  156. ϰ ˙ a b + ξ - 1 ϰ a b - λ a b = 0 , \dot{\varkappa}_{a}^{b}+\xi^{-1}\varkappa_{a}^{b}-{\lambda_{a}^{b}}^{\prime}=0,
  157. ψ ˙ = - ξ - 1 + 1 4 ξ ( ϰ a b ϰ b a + λ a b λ b a ) . \dot{\psi}=-\xi^{-1}+\frac{1}{4}\xi\left(\varkappa_{a}^{b}\varkappa_{b}^{a}+% \lambda_{a}^{b}\lambda_{b}^{a}\right).
  158. ψ = 1 2 ξ a b λ b a . \psi^{\prime}=\frac{1}{2}\xi_{a}^{b}\lambda_{b}^{a}.
  159. \scriptstyle{\gg}
  160. \scriptstyle{\ll}
  161. γ a b = ξ [ a a b ( x , y , z ) + O ( 1 / ξ ) ] , \gamma_{ab}=\xi\left[a_{ab}(x,y,z)+O(1/\sqrt{\xi})\right],
  162. | a a b | = sin 2 y |a_{ab}|=\sin^{2}y\,
  163. ( a a c a b c ) = 0 , {\left({a^{ac}}^{\prime}a_{bc}\right)}^{\prime}=0,
  164. a a b = l a l b e - 2 ρ z + m a m b e 2 ρ z , a_{ab}=l_{a}l_{b}e^{-2\rho z}+m_{a}m_{b}e^{2\rho z},
  165. γ a b = ξ ( L ~ e H L ) a b , \gamma_{ab}=\xi\left(\tilde{L}e^{H}L\right)_{ab},
  166. L = [ l 1 e - ρ z l 2 e - ρ z m 1 e ρ z m 2 e ρ z ] , L=\begin{bmatrix}l_{1}e^{-\rho z}&l_{2}e^{-\rho z}\\ m_{1}e^{\rho z}&m_{2}e^{\rho z}\end{bmatrix},
  167. H = [ σ φ φ - σ ] . H=\begin{bmatrix}\sigma&\varphi\\ \varphi&-\sigma\end{bmatrix}.
  168. σ ¨ + ξ - 1 σ ˙ - σ ′′ = 0 , \ddot{\sigma}+\xi^{-1}\dot{\sigma}-\sigma^{\prime\prime}=0,
  169. φ ¨ + ξ - 1 φ ˙ - φ ′′ + 4 ρ 2 φ = 0. \ddot{\varphi}+\xi^{-1}\dot{\varphi}-\varphi^{\prime\prime}+4\rho^{2}\varphi=0.
  170. σ = 1 ξ n = - ( A 1 n e i n ω ξ + B 1 n e - i n ω ξ ) e i n ω z , \sigma=\frac{1}{\sqrt{\xi}}\sum_{n=-\infty}^{\infty}\left(A_{1n}e^{in\omega\xi% }+B_{1n}e^{-in\omega\xi}\right)e^{in\omega z},
  171. φ = 1 ξ n = - ( A 2 n e i n ω ξ + B 2 n e - i n ω ξ ) e i n ω z , \varphi=\frac{1}{\sqrt{\xi}}\sum_{n=-\infty}^{\infty}\left(A_{2n}e^{in\omega% \xi}+B_{2n}e^{-in\omega\xi}\right)e^{in\omega z},
  172. ω n 2 = n 2 ω 2 + 4 ρ 2 . \omega_{n}^{2}=n^{2}\omega^{2}+4\rho^{2}.
  173. \scriptstyle{\ll}
  174. ϰ ˙ a b + ξ - 1 ϰ a b = 0. \dot{\varkappa}_{a}^{b}+\xi^{-1}\varkappa_{a}^{b}=0.
  175. γ a b = λ a λ b ξ 2 s 1 + μ a μ b ξ 2 s 2 , \gamma_{ab}=\lambda_{a}\lambda_{b}\xi^{2s_{1}}+\mu_{a}\mu_{b}\xi^{2s_{2}},\,
  176. γ 33 = e ψ ξ - ( 1 - s 1 2 - s 2 2 ) . \gamma_{33}=e^{\psi}\sim\xi^{-(1-s_{1}^{2}-s_{2}^{2})}.\,
  177. λ a b \scriptstyle{{\lambda_{a}^{b}}^{\prime}}
  178. s 1 2 - s 2 2 \scriptstyle{s_{1}^{2}-s_{2}^{2}}
  179. R α 0 \scriptstyle{R_{\alpha}^{0}}
  180. R α 3 \scriptstyle{R_{\alpha}^{3}}
  181. \scriptstyle{\ll}
  182. \scriptstyle{\gg}
  183. \scriptstyle{\ll}
  184. \scriptstyle{\ll}
  185. d s I X 2 = 1 4 c 2 ( d ξ 2 - d z 2 ) - a b { sin 2 y ( 1 - χ cos z ) d x 2 + ( 1 + χ cos z ) + 2 χ sin z sin y d x d y } . ds_{IX}^{2}=\tfrac{1}{4}c^{2}\left(d\xi^{2}-dz^{2}\right)-ab\left\{\sin^{2}{y}% \left(1-\chi\cos{z}\right)dx^{2}+\left(1+\chi\cos{z}\right)+2\chi\sin{z}\sin{y% }\ dx\ dy\right\}.
  186. d s V I I I 2 = 1 4 c 2 ( d ξ 2 - d z 2 ) - a b { sin 2 y ( ch z - χ ) d x 2 + ( ch z + χ ) + 2 sh z sin y d x d y } . ds_{VIII}^{2}=\tfrac{1}{4}c^{2}\left(d\xi^{2}-dz^{2}\right)-ab\left\{\sin^{2}{% y}\left(\operatorname{ch}z-\chi\right)dx^{2}+\left(\operatorname{ch}z+\chi% \right)+2\operatorname{sh}z\sin{y}\ dx\ dy\right\}.
  187. a 0 2 \scriptstyle{a_{0}^{2}}
  188. l 1 = m 1 = 1 2 sin y , l 2 = m 2 = 1 2 l_{1}=m_{1}=\frac{1}{\sqrt{2}}\sin{y},\qquad l_{2}=m_{2}=\frac{1}{\sqrt{2}}
  189. ρ = 1 2 , A 20 * = B 20 = i A e i ξ 0 , A 1 n = A 2 n = B 1 n = B 2 n = 0 ( n 0 ) . \rho=\tfrac{1}{2},\quad A_{20}^{*}=B_{20}=iAe^{i\xi_{0}},\quad A_{1n}=A_{2n}=B% _{1n}=B_{2n}=0\quad(n\neq 0).
  190. d l ¯ 2 \scriptstyle{d\bar{l}^{2}}
  191. d l ¯ 2 \scriptstyle{d\bar{l}^{2}}
  192. d s 2 = η 2 ( d η 2 - d l ¯ 2 ) . ds^{2}=\eta^{2}\left(d\eta^{2}-d\bar{l}^{2}\right).
  193. l ¯ \scriptstyle{\bar{l}}
  194. l ¯ \scriptstyle{\bar{l}}
  195. l ¯ 2 \scriptstyle{\bar{l}^{2}}
  196. l ¯ 2 \scriptstyle{\bar{l}^{2}}
  197. d s 2 = e 2 η ( d η 2 - d l ¯ 2 ) , ds^{2}=e^{2\eta}\left(d\eta^{2}-d\bar{l}^{2}\right),
  198. \scriptstyle{\gg}
  199. \scriptstyle{\gg}
  200. R 0 0 + R 3 3 = 0 \scriptstyle{R_{0}^{0}+R_{3}^{3}=0}
  201. G ˙ 0 \scriptstyle{\dot{G}\neq 0}
  202. G 0 \scriptstyle{G^{\prime}\neq 0}
  203. G ˙ = G = 0 \scriptstyle{\dot{G}=G^{\prime}=0}

Black's_equation.html

  1. M T T F = A j - n e ( Q k T ) MTTF=Aj^{-n}e^{\left(}\frac{Q}{kT}\right)
  2. A A
  3. j j
  4. n n
  5. Q Q
  6. k k
  7. T T

Blast_wave.html

  1. P ( t ) = P s e - t t * ( 1 - t t * ) . P(t)=P_{s}e^{-\frac{t}{t^{*}}}\left(1-\frac{t}{t^{*}}\right).
  2. r 5 ρ o t 2 E \frac{r^{5}\rho_{o}}{t^{2}E}
  3. ρ o \rho_{o}
  4. E E
  5. E = ( ρ o t 2 ) ( r C ) 5 E=\left(\frac{\rho_{o}}{t^{2}}\right)\left(\frac{r}{C}\right)^{5}
  6. C C
  7. R = 14 ( E 0 n ) 1 / 5 t 2 / 5 p c R=14\left(\frac{E_{0}}{n}\right)^{1/5}t^{2/5}\,pc
  8. T = 1.0 × 10 10 ( E 0 n ) R - 3 K . T=1.0\times 10^{10}\left(\frac{E_{0}}{n}\right)R^{-3}\;\,\,\text{K}\,.\;

Blob_detection.html

  1. f ( x , y ) f(x,y)
  2. g ( x , y , t ) = 1 2 π t 2 e - x 2 + y 2 2 t 2 g(x,y,t)=\frac{1}{2\pi t^{2}}e^{-\frac{x^{2}+y^{2}}{2t^{2}}}
  3. t t
  4. L ( x , y ; t ) = g ( x , y , t ) * f ( x , y ) L(x,y;t)\ =g(x,y,t)*f(x,y)
  5. 2 L = L x x + L y y \nabla^{2}L=L_{xx}+L_{yy}
  6. 2 t \sqrt{2t}
  7. n o r m 2 L ( x , y ; t ) = t ( L x x + L y y ) \nabla^{2}_{norm}L(x,y;t)=t(L_{xx}+L_{yy})
  8. n o r m 2 L \nabla^{2}_{norm}L
  9. f ( x , y ) f(x,y)
  10. L ( x , y , t ) L(x,y,t)
  11. ( x ^ , y ^ ) (\hat{x},\hat{y})
  12. t ^ \hat{t}
  13. ( x ^ , y ^ ; t ^ ) = argmaxminlocal ( x , y ; t ) ( n o r m 2 L ( x , y ; t ) ) (\hat{x},\hat{y};\hat{t})=\operatorname{argmaxminlocal}_{(x,y;t)}(\nabla^{2}_{% norm}L(x,y;t))
  14. ( x 0 , y 0 ; t 0 ) (x_{0},y_{0};t_{0})
  15. s s
  16. ( s x 0 , s y 0 ; s 2 t 0 ) (sx_{0},sy_{0};s^{2}t_{0})
  17. L ( x , y , t ) L(x,y,t)
  18. t L = 1 2 2 L \partial_{t}L=\frac{1}{2}\nabla^{2}L
  19. 2 L ( x , y , t ) \nabla^{2}L(x,y,t)
  20. n o r m 2 L ( x , y ; t ) t Δ t ( L ( x , y ; t + Δ t ) - L ( x , y ; t - Δ t ) ) \begin{aligned}\displaystyle\nabla^{2}_{norm}L(x,y;t)&\displaystyle\approx% \frac{t}{\Delta t}\left(L(x,y;t+\Delta t)-L(x,y;t-\Delta t)\right)\end{aligned}
  21. det H L ( x , y ; t ) = t 2 ( L x x L y y - L x y 2 ) \operatorname{det}HL(x,y;t)=t^{2}(L_{xx}L_{yy}-L_{xy}^{2})
  22. H L HL
  23. L L
  24. ( x ^ , y ^ ; t ^ ) = argmaxlocal ( x , y ; t ) ( det H L ( x , y ; t ) ) (\hat{x},\hat{y};\hat{t})=\operatorname{argmaxlocal}_{(x,y;t)}(\operatorname{% det}HL(x,y;t))
  25. ( x ^ , y ^ ) (\hat{x},\hat{y})
  26. t ^ \hat{t}
  27. ( x ^ , y ^ ) = argmaxlocal ( x , y ) ( det H L ( x , y ; t ) ) (\hat{x},\hat{y})=\operatorname{argmaxlocal}_{(x,y)}(\operatorname{det}HL(x,y;% t))
  28. t ^ = argmaxminlocal t ( n o r m 2 L ( x ^ , y ^ ; t ) ) \hat{t}=\operatorname{argmaxminlocal}_{t}(\nabla^{2}_{norm}L(\hat{x},\hat{y};t))

Block_nested_loop.html

  1. R R
  2. S S
  3. | R | < | S | |R|<|S|
  4. S S
  5. R R
  6. R R
  7. S S
  8. S S
  9. R R
  10. R R
  11. S S
  12. S S
  13. R R
  14. S S
  15. R R
  16. S S
  17. S S
  18. O ( P r P s / M ) O(P_{r}P_{s}/M)
  19. M M
  20. P r P_{r}
  21. P s P_{s}
  22. R R
  23. S S
  24. O ( P r + P s ) O(P_{r}+P_{s})
  25. R R

BLOSUM.html

  1. L o g O d d R a t i o = 2 log 2 ( P ( O ) P ( E ) ) LogOddRatio=2\log_{2}{\left(\frac{P\left(O\right)}{P\left(E\right)}\right)}
  2. P ( O ) P\left(O\right)
  3. P ( E ) P\left(E\right)
  4. S i j = ( 1 λ ) log ( p i j q i * q j ) S_{ij}=\left(\frac{1}{\lambda}\right)\log{\left(\frac{p_{ij}}{q_{i}*q_{j}}% \right)}
  5. p i j p_{ij}
  6. i i
  7. j j
  8. q i q_{i}
  9. q j q_{j}
  10. i i
  11. j j
  12. λ \lambda

Bochner_space.html

  1. 1 p 1\leq p\leq\infty
  2. g ( t , x ) g(t,x)
  3. ( f ( t ) ) ( x ) := g ( t , x ) (f(t))(x):=g(t,x)
  4. u L p ( T ; X ) := ( T u ( t ) X p d μ ( t ) ) 1 / p < + for 1 p < , \|u\|_{L^{p}(T;X)}:=\left(\int_{T}\|u(t)\|_{X}^{p}\,\mathrm{d}\mu(t)\right)^{1% /p}<+\infty\mbox{ for }~{}1\leq p<\infty,
  5. u L ( T ; X ) := ess sup t T u ( t ) X < + . \|u\|_{L^{\infty}(T;X)}:=\mathrm{ess\,sup}_{t\in T}\|u(t)\|_{X}<+\infty.
  6. u L 2 ( [ 0 , T ] ; H 0 1 ( Ω ) ) u\in L^{2}\left([0,T];H_{0}^{1}(\Omega)\right)
  7. u t L 2 ( [ 0 , T ] ; H - 1 ( Ω ) ) . \frac{\partial u}{\partial t}\in L^{2}\left([0,T];H^{-1}(\Omega)\right).
  8. H 0 1 ( Ω ) H_{0}^{1}(\Omega)
  9. H - 1 ( Ω ) H^{-1}(\Omega)
  10. H 0 1 ( Ω ) H_{0}^{1}(\Omega)

Bond_albedo.html

  1. A = p q A=pq\frac{}{}
  2. q = 2 0 π I ( α ) I ( 0 ) sin α d α . q=2\int_{0}^{\pi}\frac{I(\alpha)}{I(0)}\sin\alpha\,d\alpha.

Bond_valence_method.html

  1. V = ( v i ) V=\sum(v_{i})
  2. v i = exp ( R 0 - R i b ) v_{i}=\exp\left(\frac{R_{0}-R_{i}}{b}\right)
  3. v i = ( R i R 0 ) - 6 v_{i}=\left(\frac{R_{i}}{R_{0}}\right)^{-6}
  4. V = s u m ( S j ) V=sum(S_{j})
  5. S = e x p ( ( R o - R ) / b ) S=exp((Ro-R)/b)
  6. S a = V / N S_{a}=V/N
  7. 0.5 < ( S 1 / S 2 ) < 2.0 0.5<(S_{1}/S_{2})<2.0
  8. S E = V E / N E S_{E}=V_{E}/N_{E}

Bondi_k-calculus.html

  1. k k
  2. k k
  3. k k

Bonferroni_correction.html

  1. m m
  2. 1 / m 1/m
  3. α \alpha
  4. α / m \alpha/m
  5. α = 0.05 \alpha=0.05
  6. α = 0.05 / 8 = 0.00625 \alpha=0.05/8=0.00625
  7. H 1 , , H m H_{1},...,H_{m}
  8. p 1 , , p m p_{1},...,p_{m}
  9. I 0 I_{0}
  10. m 0 m_{0}
  11. I 0 I_{0}
  12. p i α m p_{i}\leq\frac{\alpha}{m}
  13. 𝐹𝑊𝐸𝑅 α \mathit{FWER}\leq\alpha
  14. 𝐹𝑊𝐸𝑅 = 𝑃𝑟 { I o ( p i α m ) } I o { 𝑃𝑟 ( p i α m ) } m 0 α m m α m = α \mathit{FWER}=\mathit{Pr}\left\{\bigcup_{I_{o}}\left(p_{i}\leq\frac{\alpha}{m}% \right)\right\}\leq\sum_{I_{o}}\left\{\mathit{Pr}\left(p_{i}\leq\frac{\alpha}{% m}\right)\right\}\leq m_{0}\frac{\alpha}{m}\leq m\frac{\alpha}{m}=\alpha
  15. i = 1 n α n = α \sum_{i=1}^{n}\frac{\alpha}{n}=\alpha
  16. i = 1 n a i = α \sum_{i=1}^{n}a_{i}=\alpha
  17. m m
  18. 1 - α 1-\alpha
  19. 1 - α m 1-\frac{\alpha}{m}

Boolean_algebras_canonically_defined.html

  1. f 0 0 {}^{0}\!f_{0}
  2. f 1 0 {}^{0}\!f_{1}
  3. x 0 x_{0}
  4. f 0 1 {}^{1}\!f_{0}
  5. f 1 1 {}^{1}\!f_{1}
  6. f 2 1 {}^{1}\!f_{2}
  7. f 3 1 {}^{1}\!f_{3}
  8. x 0 x_{0}
  9. x 1 x_{1}
  10. f 0 2 {}^{2}\!f_{0}
  11. f 1 2 {}^{2}\!f_{1}
  12. f 2 2 {}^{2}\!f_{2}
  13. f 3 2 {}^{2}\!f_{3}
  14. f 4 2 {}^{2}\!f_{4}
  15. f 5 2 {}^{2}\!f_{5}
  16. f 6 2 {}^{2}\!f_{6}
  17. f 7 2 {}^{2}\!f_{7}
  18. f 8 2 {}^{2}\!f_{8}
  19. f 9 2 {}^{2}\!f_{9}
  20. f 10 2 {}^{2}\!f_{10}
  21. f 11 2 {}^{2}\!f_{11}
  22. f 12 2 {}^{2}\!f_{12}
  23. f 13 2 {}^{2}\!f_{13}
  24. f 14 2 {}^{2}\!f_{14}
  25. f 15 2 {}^{2}\!f_{15}

Bootstrapping_(statistics).html

  1. ( 2 n - 1 n ) {\left({{2n-1}\atop{n}}\right)}
  2. x ¯ = 1 10 ( x 1 + x 2 + + x 10 ) \bar{x}=\frac{1}{10}(x_{1}+x_{2}+\ldots+x_{10})
  3. x ¯ \bar{x}
  4. N N
  5. i i
  6. 𝒟 J \mathcal{D}^{J}
  7. w i J = x i J - x i - 1 J w^{J}_{i}=x^{J}_{i}-x^{J}_{i-1}
  8. 𝐱 J \mathbf{x}^{J}
  9. N - 1 N-1
  10. [ 0 , 1 ] [0,1]
  11. 𝒟 J \mathcal{D}^{J}
  12. y ^ i \hat{y}_{i}
  13. ϵ ^ i = y i - y ^ i , ( i = 1 , , n ) \hat{\epsilon}_{i}=y_{i}-\hat{y}_{i},(i=1,\dots,n)
  14. ϵ ^ j \hat{\epsilon}_{j}
  15. y i * = y ^ i + ϵ ^ j y^{*}_{i}=\hat{y}_{i}+\hat{\epsilon}_{j}
  16. y i * y^{*}_{i}
  17. μ ^ i * \hat{\mu}^{*}_{i}
  18. y i * y^{*}_{i}
  19. y y
  20. y i * = y ^ i + ϵ ^ i v i y^{*}_{i}=\hat{y}_{i}+\hat{\epsilon}_{i}v_{i}
  21. v i v_{i}
  22. v i v_{i}
  23. v i = { - ( 5 - 1 ) / 2 with prob. ( 5 + 1 ) / ( 2 5 ) ( 5 + 1 ) / 2 with prob. ( 5 - 1 ) / ( 2 5 ) v_{i}=\left\{\begin{matrix}-(\sqrt{5}-1)/2&\mbox{with prob. }(\sqrt{5}+1)/(2% \sqrt{5})\\ (\sqrt{5}+1)/2&\mbox{with prob. }(\sqrt{5}-1)/(2\sqrt{5})\end{matrix}\right.
  24. v i = { - 1 with prob. 1 / 2 1 with prob. 1 / 2 v_{i}=\left\{\begin{matrix}-1&\mbox{with prob. }1/2\\ 1&\mbox{with prob. }1/2\end{matrix}\right.
  25. ( 2 θ - θ ( 1 - α / 2 ) * ; 2 θ - θ ( α / 2 ) * ) (2\theta-\theta^{*}_{(1-\alpha/2)};2\theta-\theta^{*}_{(\alpha/2)})
  26. θ ( 1 - α / 2 ) * \theta^{*}_{(1-\alpha/2)}
  27. 1 - α / 2 1-\alpha/2
  28. θ * \theta^{*}
  29. ( θ ( α / 2 ) * ; θ ( 1 - α / 2 ) * ) (\theta^{*}_{(\alpha/2)};\theta^{*}_{(1-\alpha/2)})
  30. θ ( 1 - α / 2 ) * \theta^{*}_{(1-\alpha/2)}
  31. 1 - α / 2 1-\alpha/2
  32. θ * \theta^{*}
  33. ( θ - t ( 1 - α / 2 ) * s e ^ θ ; θ - t ( α / 2 ) * s e ^ θ ) (\theta-t^{*}_{(1-\alpha/2)}\cdot\hat{se}_{\theta};\theta-t^{*}_{(\alpha/2)}% \cdot\hat{se}_{\theta})
  34. t ( 1 - α / 2 ) * t^{*}_{(1-\alpha/2)}
  35. 1 - α / 2 1-\alpha/2
  36. t * = ( θ ^ * - θ ^ ) / s e ^ θ ^ * t^{*}=(\hat{\theta}^{*}-\hat{\theta})/\hat{se}_{\hat{\theta}^{*}}
  37. s e ^ θ \hat{se}_{\theta}
  38. σ = 1 / n \sigma=1/\sqrt{n}

Borel_conjecture.html

  1. M M
  2. N N
  3. f : M N f:M\to N
  4. f f
  5. f : M B G f:M\to BG
  6. S 3 S^{3}
  7. S 3 S^{3}
  8. S 3 S^{3}
  9. T 3 = S 1 × S 1 × S 1 T^{3}=S^{1}\times S^{1}\times S^{1}
  10. S 3 S^{3}

Borel_hierarchy.html

  1. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  2. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  3. 𝚫 α 0 \mathbf{\Delta}^{0}_{\alpha}
  4. α \alpha
  5. 𝚺 1 0 \mathbf{\Sigma}^{0}_{1}
  6. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  7. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  8. A A
  9. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  10. α > 1 \alpha>1
  11. A 1 , A 2 , A_{1},A_{2},\ldots
  12. A i A_{i}
  13. 𝚷 α i 0 \mathbf{\Pi}^{0}_{\alpha_{i}}
  14. α i < α \alpha_{i}<\alpha
  15. A = A i A=\bigcup A_{i}
  16. 𝚫 α 0 \mathbf{\Delta}^{0}_{\alpha}
  17. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  18. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  19. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  20. α < ω 1 𝚺 α 0 = α < ω 1 𝚷 α 0 = α < ω 1 𝚫 α 0 \bigcup_{\alpha<\omega_{1}}\mathbf{\Sigma}^{0}_{\alpha}=\bigcup_{\alpha<\omega% _{1}}\mathbf{\Pi}^{0}_{\alpha}=\bigcup_{\alpha<\omega_{1}}\mathbf{\Delta}^{0}_% {\alpha}
  21. 𝚺 α 0 𝚷 α 0 𝚫 α + 1 0 \mathbf{\Sigma}^{0}_{\alpha}\cup\mathbf{\Pi}^{0}_{\alpha}\subseteq\mathbf{% \Delta}^{0}_{\alpha+1}
  22. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  23. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  24. X X
  25. 𝚺 α 0 \mathbf{\Sigma}^{0}_{\alpha}
  26. 𝚷 α 0 \mathbf{\Pi}^{0}_{\alpha}
  27. α < ω 1 \alpha<\omega_{1}
  28. 𝚺 1 0 \mathbf{\Sigma}^{0}_{1}
  29. 𝚷 1 0 \mathbf{\Pi}^{0}_{1}
  30. 𝚺 2 0 \mathbf{\Sigma}^{0}_{2}
  31. 𝚷 2 0 \mathbf{\Pi}^{0}_{2}
  32. Σ α 0 \Sigma^{0}_{\alpha}
  33. Π α 0 \Pi^{0}_{\alpha}
  34. Δ α 0 \Delta^{0}_{\alpha}
  35. α \alpha
  36. ω 1 CK \omega^{\mathrm{CK}}_{1}
  37. Σ 1 0 \Sigma^{0}_{1}
  38. Π α 0 \Pi^{0}_{\alpha}
  39. Σ α 0 \Sigma^{0}_{\alpha}
  40. Σ α 0 \Sigma^{0}_{\alpha}
  41. A 1 , A 2 , A_{1},A_{2},\ldots
  42. A i A_{i}
  43. Π α i 0 \Pi^{0}_{\alpha_{i}}
  44. α i < α \alpha_{i}<\alpha
  45. A = A i A=\bigcup A_{i}
  46. Σ α 0 \Sigma^{0}_{\alpha}
  47. A i A_{i}
  48. α < ω 1 CK \alpha<\omega^{\mathrm{CK}}_{1}
  49. Σ α 0 Π α 0 \Sigma^{0}_{\alpha}\setminus\Pi^{0}_{\alpha}
  50. ω 1 CK \omega^{\mathrm{CK}}_{1}
  51. Δ 1 1 \Delta^{1}_{1}
  52. ω < ω \omega^{<\omega}\,
  53. ω 1 CK \omega^{\mathrm{CK}}_{1}

Borel–Moore_homology.html

  1. 0 Ext 𝐙 1 ( H c i + 1 ( X , 𝐙 ) , 𝐙 ) H i B M ( X , 𝐙 ) Hom ( H c i ( X , 𝐙 ) , 𝐙 ) 0. 0\to\,\text{Ext}^{1}_{\mathbf{Z}}(H^{i+1}_{c}(X,\mathbf{Z}),\mathbf{Z})\to H_{% i}^{BM}(X,\mathbf{Z})\to\,\text{Hom}(H^{i}_{c}(X,\mathbf{Z}),\mathbf{Z})\to 0.
  2. u = σ a σ σ , u=\sum_{\sigma}a_{\sigma}\sigma,
  3. C 2 B M ( X ) C 1 B M ( X ) C 0 B M ( X ) 0. \cdots\to C_{2}^{BM}(X)\to C_{1}^{BM}(X)\to C_{0}^{BM}(X)\to 0.
  4. H i B M ( X ) = ker ( : C i B M ( X ) C i - 1 B M ( X ) ) / im ( : C i + 1 B M ( X ) C i B M ( X ) ) . H^{BM}_{i}(X)=\ker\left(\partial:C_{i}^{BM}(X)\to C_{i-1}^{BM}(X)\right)/\,% \text{im}\left(\partial:C_{i+1}^{BM}(X)\to C_{i}^{BM}(X)\right).
  5. H i B M ( X ) = H m - i ( M , M X ) , H^{BM}_{i}(X)=H^{m-i}(M,M\setminus X),
  6. X X
  7. H i B M ( X ) = H - i ( X , D X ) , H^{BM}_{i}(X)=H^{-i}(X,D_{X}),
  8. H i B M ( F ) H i B M ( X ) H i B M ( U ) H i - 1 B M ( F ) \cdots\to H^{BM}_{i}(F)\to H^{BM}_{i}(X)\to H^{BM}_{i}(U)\to H^{BM}_{i-1}(F)\to\cdots
  9. H i ( X ) H n - i B M ( X ) H^{i}(X)\to H_{n-i}^{BM}(X)
  10. H c i ( X ) H n - i ( X ) . H^{i}_{c}(X)\to H_{n-i}(X).
  11. M < s u p > r e g M M<sup>reg⊂M

Born_coordinates.html

  1. d s 2 = - d T 2 + d Z 2 + d R 2 + R 2 d Φ 2 , - < T , Z < , 0 < R < , - π < Φ < π ds^{2}=-dT^{2}+dZ^{2}+dR^{2}+R^{2}\,d\Phi^{2},\;\;-\infty<T,\,Z<\infty,\;0<R<% \infty,\;-\pi<\Phi<\pi
  2. e 0 = T , e 1 = Z , e 2 = R , e 3 = 1 R Φ \vec{e}_{0}=\partial_{T},\;\;\vec{e}_{1}=\partial_{Z},\;\;\vec{e}_{2}=\partial% _{R},\;\;\vec{e}_{3}=\frac{1}{R}\,\partial_{\Phi}
  3. e 0 \vec{e}_{0}
  4. e 3 \vec{e}_{3}
  5. p 0 = 1 1 - ω 2 R 2 T + ω R 1 - ω 2 R 2 1 R Φ \vec{p}_{0}=\frac{1}{\sqrt{1-\omega^{2}\,R^{2}}}\,\partial_{T}+\frac{\omega\,R% }{\sqrt{1-\omega^{2}\,R^{2}}}\;\frac{1}{R}\partial_{\Phi}
  6. p 1 = Z , p 2 = R \vec{p}_{1}=\partial_{Z},\;\;\vec{p}_{2}=\partial_{R}
  7. p 3 = 1 1 - ω 2 R 2 1 R Φ + ω R 1 - ω 2 R 2 T \vec{p}_{3}=\frac{1}{\sqrt{1-\omega^{2}\,R^{2}}}\;\frac{1}{R}\,\partial_{\Phi}% +\frac{\omega\,R}{\sqrt{1-\omega^{2}\,R^{2}}}\,\partial_{T}
  8. p 0 \vec{p}_{0}
  9. p 3 \vec{p}_{3}
  10. p 0 p 0 = - ω 2 R 1 - ω 2 R 2 p 2 \nabla_{\vec{p}_{0}}\vec{p}_{0}=\frac{-\omega^{2}\,R}{1-\omega^{2}\,R^{2}}\;% \vec{p}_{2}
  11. Ω = ω 1 - ω 2 R 2 p 1 \vec{\Omega}=\frac{\omega}{1-\omega^{2}\,R^{2}}\;\vec{p}_{1}
  12. T = T 0 T=T_{0}
  13. Φ \partial_{\Phi}
  14. p 2 , p 3 \vec{p}_{2},\;\vec{p}_{3}
  15. p 1 \vec{p}_{1}
  16. p 2 , p 3 \vec{p}_{2},\;\vec{p}_{3}
  17. R \partial_{R}
  18. p 1 \vec{p}_{1}
  19. t = T , z = Z , r = R , ϕ = Φ - ω T t=T,\;\;z=Z,\;\;r=R,\;\;\phi=\Phi-\omega\,T
  20. d s 2 = - ( 1 - ω 2 r 2 ) d t 2 + 2 ω r 2 d t d ϕ + d z 2 + d r 2 + r 2 d ϕ 2 ds^{2}=-\left(1-\omega^{2}\,r^{2}\right)\,dt^{2}+2\,\omega\,r^{2}\,dt\,d\phi+% dz^{2}+dr^{2}+r^{2}\,d\phi^{2}
  21. - < t , z < , 0 < r < 1 ω , - π < ϕ < π -\infty<t,\,z<\infty,0<r<\frac{1}{\omega},\;-\pi<\phi<\pi
  22. d t d ϕ dt\,d\phi
  23. p 0 = 1 1 - ω 2 r 2 t \vec{p}_{0}=\frac{1}{\sqrt{1-\omega^{2}\,r^{2}}}\,\partial_{t}
  24. p 1 = z , p 2 = r \vec{p}_{1}=\partial_{z},\;\;\vec{p}_{2}=\partial_{r}
  25. p 3 = 1 - ω 2 r 2 r ϕ + ω r 1 - ω 2 r 2 t \vec{p}_{3}=\frac{\sqrt{1-\omega^{2}\,r^{2}}}{r}\,\partial_{\phi}+\frac{\omega% \,r}{\sqrt{1-\omega^{2}\,r^{2}}}\,\partial_{t}
  26. p 0 p 0 = - ω 2 r 1 - ω 2 r 2 p 2 \nabla_{\vec{p}_{0}}\,\vec{p}_{0}=\frac{-\omega^{2}\,r}{1-\omega^{2}\,r^{2}}\,% \vec{p}_{2}
  27. Ω = ω 1 - ω 2 r 2 p 1 \vec{\Omega}=\frac{\omega}{1-\omega^{2}\,r^{2}}\;\vec{p}_{1}
  28. t = t 0 t=t_{0}
  29. p 3 = 1 - ω 2 r 2 1 r ϕ + ω r 1 - ω 2 r 2 t \vec{p}_{3}=\sqrt{1-\omega^{2}\,r^{2}}\,\frac{1}{r}\,\partial_{\phi}+\frac{% \omega\,r}{\sqrt{1-\omega^{2}\,r^{2}}}\,\partial_{t}
  30. ϕ = 0 , t = 0 \phi=0,\,t=0
  31. ϕ + ω t - t ω r 2 = 0 , - π < ϕ < π \phi+\omega\,t-\frac{t}{\omega\,r^{2}}=0,\;\;-\pi<\phi<\pi
  32. p 2 , p 3 \vec{p}_{2},\,\vec{p}_{3}
  33. p 2 \vec{p}_{2}
  34. p 2 , p 3 \vec{p}_{2},\,\vec{p}_{3}
  35. d s = d z = d r = 0 ds=dz=dr=0
  36. ( 1 - ω 2 r 0 2 ) d t 2 = 2 ω r 0 2 d t d ϕ + r 0 2 d ϕ 2 (1-\omega^{2}\,r_{0}^{2})\,dt^{2}=2\omega\,r_{0}^{2}\,dt\,d\phi+r_{0}^{2}\,d% \phi^{2}
  37. d t = r 0 d ϕ 1 ± ω r 0 dt=\frac{r_{0}\,d\phi}{1\pm\omega\,r_{0}}
  38. Δ t + = 2 π r 0 1 + ω r 0 , Δ t - = 2 π r 0 1 - ω r 0 \Delta t_{+}=\frac{2\pi r_{0}}{1+\omega\,r_{0}},\;\;\Delta t_{-}=\frac{2\pi r_% {0}}{1-\omega\,r_{0}}
  39. δ = Δ t + - Δ t - 2 π r \delta=\frac{\Delta t_{+}-\Delta t_{-}}{2\,\pi\,r}
  40. ω = - 1 + 1 + δ 2 δ r \omega=\frac{-1+\sqrt{1+\delta^{2}}}{\delta\,r}
  41. T ¨ = 0 , Z ¨ = 0 , R ¨ - R Φ ˙ 2 = 0 , Φ ¨ + 2 R Φ ˙ R ˙ = 0 \ddot{T}=0,\;\;\ddot{Z}=0,\;\;\ddot{R}-R\,\dot{\Phi}^{2}=0,\;\;\ddot{\Phi}+% \frac{2}{R}\,\dot{\Phi}\,\dot{R}=0
  42. T ˙ = E , Z ˙ = P , Φ ˙ = L R 2 \dot{T}=E,\;\;\dot{Z}=P,\;\;\dot{\Phi}=\frac{L}{R^{2}}
  43. d s 2 = 0 ds^{2}=0
  44. R ˙ 2 = E 2 - P 2 - L 2 R 2 \dot{R}^{2}=E^{2}-P^{2}-\frac{L^{2}}{R^{2}}
  45. R 0 = L E 2 - P 2 R_{0}=\frac{L}{\sqrt{E^{2}-P^{2}}}
  46. R = ( E 2 - P 2 ) s 2 + 2 s ( E 2 - P 2 ) R 0 2 - L 2 + R 0 2 R=\sqrt{(E^{2}-P^{2})\,s^{2}+2\,s\,\sqrt{(E^{2}-P^{2})\;R_{0}^{2}-L^{2}+R_{0}^% {2}}}
  47. T = T 0 + E s , Z = Z 0 + P T=T_{0}+E\,s,\;\;Z=Z_{0}+P
  48. Φ = Φ 0 + arctan ( 2 ( E 2 - P 2 ) R 0 2 - L 2 + 2 ( E 2 - P 2 ) s 2 L ) \Phi=\Phi_{0}+\operatorname{arctan}\left(\frac{2\,\sqrt{(E^{2}-P^{2})\,R_{0}^{% 2}-L^{2}}+2\,(E^{2}-P^{2})\,s}{2\,L}\right)
  49. - arctan ( ( E 2 - P 2 ) R 0 2 - L 2 L ) \;\;\;-\operatorname{arctan}\left(\frac{\sqrt{(E^{2}-P^{2})\,R_{0}^{2}-L^{2}}}% {L}\right)
  50. T = T 0 T=T_{0}
  51. R = R 0 sec ( Φ - Φ 0 ) R=R_{0}\,\sec(\Phi-\Phi_{0})
  52. R 1 = R 0 sec ( Φ 1 - Φ 0 ) , R 2 = R 0 sec ( Φ 2 - Φ 0 ) R_{1}=R_{0}\,\sec(\Phi_{1}-\Phi_{0}),\;\;R_{2}=R_{0}\,\sec(\Phi_{2}-\Phi_{0})
  53. R 0 = R 1 R 2 | sin ( Φ 2 - Φ 1 ) | R 1 2 - 2 R 1 R 2 cos ( Φ 2 - Φ 1 ) + R 2 2 R_{0}=\frac{R_{1}\,R_{2}\,|\sin(\Phi_{2}-\Phi_{1})|}{\sqrt{R_{1}^{2}-2\,R_{1}% \,R_{2}\,\cos(\Phi_{2}-\Phi_{1})+R_{2}^{2}}}
  54. T = s , Z = Z 0 , R = s T=s,\;Z=Z_{0},\;R=s
  55. Φ = ω R 0 \Phi=\omega\,R_{0}
  56. r = ϕ ω - r 0 r=\frac{\phi}{\omega}-r_{0}
  57. T = R 0 tan ( Φ ) T=R_{0}\,\tan(\Phi)
  58. R = R 0 sec ( Φ ) R=R_{0}\,\sec(\Phi)
  59. t = r 0 tan ( ϕ + ω t ) t=r_{0}\,\tan(\phi+\omega\,t)
  60. r = r 0 sec ( ϕ + ω t ) r=r_{0}\,\sec(\phi+\omega\,t)
  61. r = r 0 2 + t 2 r=\sqrt{r_{0}^{2}+t^{2}}
  62. ϕ = - ω r 2 - r 0 2 + arctan ( t / r 0 ) \phi=-\omega\,\sqrt{r^{2}-r_{0}^{2}}+\operatorname{arctan}(t/r_{0})
  63. R 0 1 - ω 2 R 0 2 R_{0}\sqrt{1-\omega^{2}\,R_{0}^{2}}
  64. r 0 1 - ω 2 r 0 2 r_{0}\sqrt{1-\omega^{2}\,r_{0}^{2}}
  65. 1 - ω 2 r 0 2 \sqrt{1-\omega^{2}\,r_{0}^{2}}
  66. T = 0 , Z = 0 , X = R 0 , Y = 0 T=0,\;Z=0,\;X=R_{0},Y=0
  67. T = 2 R 0 sin ( Φ 2 ) , Z = 0 , T=2\,R_{0}\,\sin\left(\frac{\Phi}{2}\right),\;Z=0,
  68. X = R 0 cos ( Φ ) , Y = R 0 sin ( Φ ) X=R_{0}\cos(\Phi),\;Y=R_{0}\sin(\Phi)
  69. Δ s \Delta s
  70. T = Δ s 1 - ω 2 R 0 2 , Z = 0 T=\frac{\Delta s}{\sqrt{1-\omega^{2}\,R_{0}^{2}}},\;Z=0
  71. X = R 0 cos ( ω Δ s 1 - ω 2 R 0 2 ) , Y = R 0 sin ( ω Δ s 1 - ω 2 R 0 2 ) X=R_{0}\cos\left(\frac{\omega\,\Delta s}{\sqrt{1-\omega^{2}\,R_{0}^{2}}}\right% ),\;Y=R_{0}\sin\left(\frac{\omega\,\Delta s}{\sqrt{1-\omega^{2}\,R_{0}^{2}}}\right)
  72. 2 1.414 \sqrt{2}\approx 1.414
  73. - < t , z < , 0 < r < 1 ω , - π < ϕ < π -\infty<t,z<\infty,\;\;0<r<\frac{1}{\omega},\;\;-\pi<\phi<\pi
  74. d t + = - ω r 2 + ( 1 - ω 2 r 2 ) ( d z 2 + d r 2 ) + r 2 d ϕ 2 1 - ω 2 r 2 dt_{+}=\frac{-\omega\,r^{2}+\sqrt{(1-\omega^{2}\,r^{2})\;(dz^{2}+dr^{2})+r^{2}% \,d\phi^{2}}}{1-\omega^{2}\,r^{2}}
  75. d t - = - ω r 2 - ( 1 - ω 2 r 2 ) ( d z 2 + d r 2 ) + r 2 d ϕ 2 1 - ω 2 r 2 dt_{-}=\frac{-\omega\,r^{2}-\sqrt{(1-\omega^{2}\,r^{2})\;(dz^{2}+dr^{2})+r^{2}% \,d\phi^{2}}}{1-\omega^{2}\,r^{2}}
  76. 1 - ω 2 r 2 d t + - d t - 2 = d z 2 + d r 2 + r 2 d ϕ 2 1 - ω 2 r 2 \sqrt{1-\omega^{2}\,r^{2}}\;\frac{dt_{+}-dt_{-}}{2}=\sqrt{dz^{2}+dr^{2}+\frac{% r^{2}\,d\phi^{2}}{1-\omega^{2}\,r^{2}}}
  77. d σ 2 = d z 2 + d r 2 + r 2 d ϕ 2 1 - ω 2 r 2 d\sigma^{2}=dz^{2}+dr^{2}+\frac{r^{2}\,d\phi^{2}}{1-\omega^{2}\,r^{2}}
  78. - < t < , 0 < r < 1 ω , - π < ϕ < π -\infty<t<\infty,\;\;0<r<\frac{1}{\omega},\;\;-\pi<\phi<\pi
  79. θ 1 ^ = d z , θ 2 ^ = d r , θ 3 ^ = r d ϕ 1 - ω 2 r 2 \theta^{\hat{1}}=dz,\;\;\theta^{\hat{2}}=dr,\;\;\theta^{\hat{3}}=\frac{r\,d% \phi}{\sqrt{1-\omega^{2}\,r^{2}}}
  80. R 2 ^ 3 ^ 2 ^ 3 ^ = - 3 ω 2 ( 1 - ω 2 r 2 ) 2 = - 3 ω 2 ( 1 + 2 ω 2 r 2 ) + O ( ω 6 r 6 ) {R^{\hat{2}}}_{\hat{3}\hat{2}\hat{3}}=\frac{-3\,\omega^{2}}{(1-\omega^{2}r^{2}% )^{2}}=-3\,\omega^{2}\;\left(1+2\,\omega^{2}\,r^{2}\right)+O(\omega^{6}\,r^{6})
  81. d r = d ϕ ω dr=\frac{d\phi}{\omega}
  82. Δ = r 0 0 d r 1 - ω 2 r 2 = arcsin ( ω r 0 ) ω = r 0 + ω 2 r 0 3 6 + O ( r 0 4 ) \Delta=\int_{r_{0}}^{0}\frac{dr}{\sqrt{1-\omega^{2}\,r^{2}}}=\frac{\arcsin(% \omega r_{0})}{\omega}=r_{0}+\frac{\omega^{2}\,r_{0}^{3}}{6}+O(r_{0}^{4})

Borsuk's_conjecture.html

  1. n \mathbb{R}^{n}
  2. n \mathbb{R}^{n}
  3. α ( d ) > d + 1 \alpha(d)>d+1
  4. α ( d ) \alpha(d)
  5. α ( d ) ( 1.2 ) d \alpha(d)\geq(1.2)^{\sqrt{d}}
  6. α ( d ) ( 3 / 2 + ε ) d \alpha(d)\leq\left(\sqrt{3/2}+\varepsilon\right)^{d}
  7. α ( d ) > c d \alpha(d)>c^{d}

Bounce_rate.html

  1. R b = T v T e R_{b}=\frac{T_{v}}{T_{e}}
  2. R b = T v T e P s R_{b}=\frac{T_{v}}{T_{e}}\leq P_{s}

Bounded_quantifier.html

  1. n < t \forall n<t
  2. n < t \exists n<t
  3. ϕ \phi
  4. n < t ϕ n ( n < t ϕ ) \exists n<t\,\phi\Leftrightarrow\exists n(n<t\land\phi)
  5. n < t ϕ n ( n < t ϕ ) \forall n<t\,\phi\Leftrightarrow\forall n(n<t\rightarrow\phi)
  6. ϕ \phi
  7. n < t ϕ \exists n<t\,\phi
  8. n < t ϕ \forall n<t\,\phi
  9. 0 , 1 , + , × , < , = \langle 0,1,+,\times,<,=\rangle
  10. Σ 0 0 \Sigma^{0}_{0}
  11. Δ 0 0 \Delta^{0}_{0}
  12. Π 0 0 \Pi^{0}_{0}
  13. , , = \langle\in,\ldots,=\rangle
  14. x t \forall x\in t
  15. x t \exists x\in t
  16. x t ( ϕ ) x ( x t ϕ ) \exists x\in t\ (\phi)\Leftrightarrow\exists x(x\in t\land\phi)
  17. x t ( ϕ ) x ( x t ϕ ) \forall x\in t\ (\phi)\Leftrightarrow\forall x(x\in t\rightarrow\phi)
  18. Σ 0 \Sigma_{0}
  19. Δ 0 \Delta_{0}
  20. Π 0 \Pi_{0}

Bovine_pancreatic_ribonuclease.html

  1. α + β \alpha+\beta

Box_spread.html

  1. S = F e - r T S=Fe^{-rT}
  2. K e - r T Ke^{-rT}
  3. c - p = S - K e - r T c-p=S-Ke^{-rT}
  4. K 1 K_{1}
  5. K 2 K_{2}
  6. K 1 K_{1}
  7. K 1 K_{1}
  8. K 1 K_{1}
  9. K 2 K_{2}
  10. K 2 K_{2}
  11. K 1 K_{1}
  12. K 2 K_{2}
  13. K 1 K_{1}
  14. K 2 K_{2}
  15. K 1 = 90 K_{1}=90
  16. K 2 = 110 K_{2}=110
  17. S T S_{T}
  18. S T < K 1 S_{T}<K_{1}
  19. K 1 < S T < K 2 K_{1}<S_{T}<K_{2}
  20. K 2 < S T K_{2}<S_{T}
  21. 0
  22. S T - 90 S_{T}-90
  23. S T - 90 S_{T}-90
  24. 0
  25. S T - 90 S_{T}-90
  26. 0
  27. 0
  28. 110 - S T 110-S_{T}
  29. 0
  30. 110 - S T 110-S_{T}
  31. 110 - S T 110-S_{T}
  32. 0

Bra_size.html

  1. V = 2 π r 3 3 V={2\pi r^{3}\over 3}
  2. V = π D 3 12 V={\pi D^{3}\over 12}
  3. V = 2 π a b c 3 V={2\pi abc\over 3}
  4. V π × c w × c d × w l 12 V\approx{\pi\times cw\times cd\times wl\over 12}

Bracketing_paradox.html

  1. [ un- ] [ [ easi ] [ -er ] ] \Big[\mbox{un-}~{}\Big]\Big[\big[\mbox{easi}~{}\big]\big[\mbox{-er}~{}\big]\Big]
  2. [ [ un- ] [ easi ] ] [ -er ] \Big[\big[\mbox{un-}~{}\big]\big[\mbox{easi}~{}\big]\Big]\Big[\mbox{-er}~{}\Big]
  3. [ nuclear ] [ [ physic(s) ] [ -ist ] ] \Big[\mbox{nuclear}~{}\Big]\Big[\big[\mbox{physic(s)}~{}\big]\big[\mbox{-ist}~% {}\big]\Big]
  4. [ [ nuclear ] [ physic(s) ] ] [ -ist ] \Big[\big[\mbox{nuclear}~{}\big]\big[\mbox{physic(s)}~{}\big]\Big]\Big[\mbox{-% ist}~{}\Big]

Bradford_Factor.html

  1. B = S 2 × D B=S^{2}\times D

Brahmagupta's_identity.html

  1. a 2 + n b 2 a^{2}+nb^{2}
  2. ( a 2 + n b 2 ) ( c 2 + n d 2 ) \displaystyle\left(a^{2}+nb^{2}\right)\left(c^{2}+nd^{2}\right)
  3. ( x 1 2 - N y 1 2 ) ( x 2 2 - N y 2 2 ) = ( x 1 x 2 + N y 1 y 2 ) 2 - N ( x 1 y 2 + x 2 y 1 ) 2 , (x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_% {1}y_{2}+x_{2}y_{1})^{2},\,
  4. ( x 1 x 2 + N y 1 y 2 , x 1 y 2 + x 2 y 1 , k 1 k 2 ) . (x_{1}x_{2}+Ny_{1}y_{2}\,,\,x_{1}y_{2}+x_{2}y_{1}\,,\,k_{1}k_{2}).

Bram_van_Leer.html

  1. F up = 1 2 ( F j + F j + 1 ) - | A | j + 1 2 ( u j + 1 - u j ) . F^{\hbox{up}}=\frac{1}{2}(F_{j}+F_{j+1})-|A|_{j+\frac{1}{2}}(u_{j+1}-u_{j}).
  2. | A | |A|
  3. A A
  4. A A
  5. j + 1 2 j+\frac{1}{2}
  6. ( x j , x j + 1 ) (x_{j},x_{j+1})

Branching_theorem.html

  1. X X
  2. Y Y
  3. f : X Y f:X\to Y
  4. a X a\in X
  5. b := f ( a ) Y b:=f(a)\in Y
  6. k 𝒩 k\in\mathcal{N}
  7. ψ 1 : U 1 V 1 \psi_{1}:U_{1}\to V_{1}
  8. X X
  9. ψ 2 : U 2 V 2 \psi_{2}:U_{2}\to V_{2}
  10. Y Y
  11. ψ 1 ( a ) = ψ 2 ( b ) = 0 \psi_{1}(a)=\psi_{2}(b)=0
  12. ψ 2 f ψ 1 - 1 : V 1 V 2 \psi_{2}\circ f\circ\psi_{1}^{-1}:V_{1}\to V_{2}
  13. z z k . z\mapsto z^{k}.
  14. k k
  15. f f
  16. a a
  17. ν ( f , a ) \nu(f,a)
  18. k > 1 k>1
  19. a a
  20. f f
  21. f f

Brauer–Siegel_theorem.html

  1. K 1 , K 2 , . K_{1},K_{2},\ldots.
  2. [ K i : Q ] log | D i | 0 as i . \frac{[K_{i}:Q]}{\log|D_{i}|}\to 0\,\text{ as }i\to\infty.
  3. log ( h i R i ) log | D i | 1 as i \frac{\log(h_{i}R_{i})}{\log\sqrt{|D_{i}|}}\to 1\,\text{ as }i\to\infty

Bretschneider's_formula.html

  1. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) - a b c d cos 2 ( α + γ 2 ) K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cdot\cos^{2}\left(\frac{\alpha+\gamma}{2}% \right)}
  2. = ( s - a ) ( s - b ) ( s - c ) ( s - d ) - 1 2 a b c d [ 1 + cos ( α + γ ) ] . =\sqrt{(s-a)(s-b)(s-c)(s-d)-\tfrac{1}{2}abcd[1+\cos(\alpha+\gamma)]}.
  3. α \alpha\,
  4. γ \gamma\,
  5. K = area of A D B + area of B D C = a d sin α 2 + b c sin γ 2 . \begin{aligned}\displaystyle K&\displaystyle=\,\text{area of }\triangle ADB+\,% \text{area of }\triangle BDC\\ &\displaystyle=\frac{ad\sin\alpha}{2}+\frac{bc\sin\gamma}{2}.\end{aligned}
  6. 4 K 2 = ( a d ) 2 sin 2 α + ( b c ) 2 sin 2 γ + 2 a b c d sin α sin γ . 4K^{2}=(ad)^{2}\sin^{2}\alpha+(bc)^{2}\sin^{2}\gamma+2abcd\sin\alpha\sin\gamma.\,
  7. a 2 + d 2 - 2 a d cos α = b 2 + c 2 - 2 b c cos γ , a^{2}+d^{2}-2ad\cos\alpha=b^{2}+c^{2}-2bc\cos\gamma,\,
  8. ( a 2 + d 2 - b 2 - c 2 ) 2 4 = ( a d ) 2 cos 2 α + ( b c ) 2 cos 2 γ - 2 a b c d cos α cos γ . \frac{(a^{2}+d^{2}-b^{2}-c^{2})^{2}}{4}=(ad)^{2}\cos^{2}\alpha+(bc)^{2}\cos^{2% }\gamma-2abcd\cos\alpha\cos\gamma.\,
  9. 4 K 2 4K^{2}
  10. 4 K 2 + ( a 2 + d 2 - b 2 - c 2 ) 2 4 = ( a d ) 2 + ( b c ) 2 - 2 a b c d cos ( α + γ ) = ( a d + b c ) 2 - 4 a b c d cos 2 ( α + γ 2 ) . \begin{aligned}\displaystyle 4K^{2}+\frac{(a^{2}+d^{2}-b^{2}-c^{2})^{2}}{4}&% \displaystyle=(ad)^{2}+(bc)^{2}-2abcd\cos(\alpha+\gamma)\\ &\displaystyle=(ad+bc)^{2}-4abcd\cos^{2}\left(\frac{\alpha+\gamma}{2}\right).% \end{aligned}
  11. 16 K 2 = ( a + b + c - d ) ( a + b - c + d ) ( a - b + c + d ) ( - a + b + c + d ) - 16 a b c d cos 2 ( α + γ 2 ) . 16K^{2}=(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)-16abcd\cos^{2}\left(\frac{\alpha% +\gamma}{2}\right).
  12. s = a + b + c + d 2 , s=\frac{a+b+c+d}{2},
  13. 16 K 2 = 16 ( s - a ) ( s - b ) ( s - c ) ( s - d ) - 16 a b c d cos 2 ( α + γ 2 ) 16K^{2}=16(s-a)(s-b)(s-c)(s-d)-16abcd\cos^{2}\left(\frac{\alpha+\gamma}{2}\right)
  14. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) - 1 4 ( a c + b d + p q ) ( a c + b d - p q ) . K=\sqrt{(s-a)(s-b)(s-c)(s-d)-\tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)}.

Bridge_probabilities.html

  1. 52 ! / ( 13 ! ) 4 52!/(13!)^{4}

Brinkman_number.html

  1. Br = μ u 2 κ ( T w - T 0 ) = Pr Ec \mathrm{Br}=\frac{\mu u^{2}}{\kappa(T_{w}-T_{0})}=\mathrm{Pr}\,\mathrm{Ec}

Brocard_points.html

  1. P A B = P B C = P C A . \angle PAB=\angle PBC=\angle PCA.\,
  2. cot ω = cot α + cot β + cot γ . \cot\omega=\cot\alpha+\cot\beta+\cot\gamma.\,
  3. Q C B = Q B A = Q A C \angle QCB=\angle QBA=\angle QAC
  4. P B C = P C A = P A B \angle PBC=\angle PCA=\angle PAB
  5. Q C B = Q B A = Q A C . \angle QCB=\angle QBA=\angle QAC.

Brownian_dynamics.html

  1. M X ¨ = - U ( X ) - γ X ˙ + 2 γ k B T R ( t ) M\ddot{X}=-\nabla U(X)-\gamma\dot{X}+\sqrt{2\gamma k_{B}T}R(t)
  2. γ \gamma
  3. U ( X ) U(X)
  4. \nabla
  5. - U ( X ) -\nabla U(X)
  6. X ˙ \dot{X}
  7. X ¨ \ddot{X}
  8. R ( t ) R(t)
  9. R ( t ) = 0 \left\langle R(t)\right\rangle=0
  10. R ( t ) R ( t ) = δ ( t - t ) . \left\langle R(t)R(t^{\prime})\right\rangle=\delta(t-t^{\prime}).
  11. M M
  12. M X ¨ ( t ) M\ddot{X}(t)
  13. 0 = - U ( X ) - γ X ˙ + 2 γ k B T R ( t ) 0=-\nabla U(X)-\gamma\dot{X}+\sqrt{2\gamma k_{B}T}R(t)
  14. D = k B T / γ D=k_{B}T/\gamma
  15. X ˙ ( t ) = - U ( X ) / γ + 2 D R ( t ) . \dot{X}(t)=-\nabla U(X)/\gamma+\sqrt{2D}R(t).

BRST_quantization.html

  1. i i\hbar
  2. 0 \mathcal{H}_{0}
  3. \mathcal{H}
  4. 0 \mathcal{H}_{0}
  5. 0 \mathcal{H}_{0}
  6. \mathcal{H}
  7. μ A μ = 0 \partial^{\mu}A_{\mu}=0
  8. 𝔤 \mathfrak{g}
  9. 0 𝔤 * 0\in\mathfrak{g}^{*}
  10. Φ : M 𝔤 * \Phi:M\to\mathfrak{g}^{*}
  11. M 0 = Φ - 1 ( 0 ) M_{0}=\Phi^{-1}(0)
  12. M ~ = M 0 / G \widetilde{M}=M_{0}/G
  13. M ~ = M / / G \widetilde{M}=M//G
  14. Λ 𝔤 C ( M ) . \Lambda^{\cdot}{\mathfrak{g}}\otimes C^{\infty}(M).
  15. Λ 𝔤 C ( M ) \Lambda^{\cdot}{\mathfrak{g}}\otimes C^{\infty}(M)
  16. 𝔤 C ( M ) {\mathfrak{g}}\to C^{\infty}(M)
  17. S ( 𝔤 ) S({\mathfrak{g}})
  18. S ( 𝔤 ) S(\mathfrak{g})
  19. 𝔤 \mathfrak{g}
  20. S ( 𝔤 ) C ( M ) S({\mathfrak{g}})\to C^{\infty}(M)
  21. 𝔤 C ( M ) \mathfrak{g}\to C^{\infty}(M)
  22. S ( 𝔤 ) S({\mathfrak{g}})
  23. C ( M 0 ) C^{\infty}(M_{0})
  24. H j ( Λ 𝔤 C ( M ) , δ ) = { C ( M 0 ) j = 0 0 j 0 H^{j}(\Lambda^{\cdot}{\mathfrak{g}}\otimes C^{\infty}(M),\delta)=\begin{cases}% C^{\infty}(M_{0})&j=0\\ 0&j\neq 0\end{cases}
  25. Λ 𝔤 C ( M ) \Lambda^{\cdot}{\mathfrak{g}}\otimes C^{\infty}(M)
  26. 𝔤 \mathfrak{g}
  27. K , = C ( 𝔤 , Λ 𝔤 C ( M ) ) = Λ 𝔤 * Λ 𝔤 C ( M ) . K^{\cdot,\cdot}=C^{\cdot}\left(\mathfrak{g},\Lambda^{\cdot}{\mathfrak{g}}% \otimes C^{\infty}(M)\right)=\Lambda^{\cdot}{\mathfrak{g}}^{*}\otimes\Lambda^{% \cdot}{\mathfrak{g}}\otimes C^{\infty}(M).
  28. d : K i , K i + 1 , d:K^{i,\cdot}\to K^{i+1,\cdot}
  29. Λ 𝔤 C ( M ) \Lambda^{\cdot}{\mathfrak{g}}\otimes C^{\infty}(M)
  30. 𝔤 \mathfrak{g}
  31. Λ 𝔤 * \Lambda^{\cdot}{\mathfrak{g}}^{*}
  32. 𝔤 \mathfrak{g}
  33. Tot ( K ) n = i - j = n K i , j \operatorname{Tot}(K)^{n}=\bigoplus\nolimits_{i-j=n}K^{i,j}
  34. ( K , , d , δ ) (K^{\cdot,\cdot},d,\delta)
  35. E 1 i , j = H j ( K i , , δ ) = Λ i 𝔤 * C ( M 0 ) E_{1}^{i,j}=H^{j}(K^{i,\cdot},\delta)=\Lambda^{i}{\mathfrak{g}}^{*}\otimes C^{% \infty}(M_{0})
  36. ( Ω vert ( M 0 ) , d vert ) (\Omega^{\cdot}_{\operatorname{vert}}(M_{0}),d_{\operatorname{vert}})
  37. M 0 M ~ M_{0}\to\widetilde{M}
  38. E 1 , E_{1}^{\cdot,\cdot}
  39. E 2 i , j H i ( E 1 , j , d ) = C ( M 0 ) g = C ( M ~ ) E_{2}^{i,j}\cong H^{i}(E_{1}^{\cdot,j},d)=C^{\infty}(M_{0})^{g}=C^{\infty}(% \widetilde{M})
  40. i = j = 0 i=j=0
  41. E i , j = E 2 i , j E_{\infty}^{i,j}=E_{2}^{i,j}
  42. H p ( Tot ( K ) , D ) = C ( M 0 ) g = C ( M ~ ) H^{p}(\operatorname{Tot}(K),D)=C^{\infty}(M_{0})^{g}=C^{\infty}(\widetilde{M})
  43. 𝔤 \mathfrak{g}
  44. [ i δ λ , s B ] s B X = i δ λ ( s B s B X ) + s B ( i δ λ ( s B X ) ) = s B ( i δ λ ( s B X ) ) , \left[i_{\delta\lambda},s_{B}\right]s_{B}X=i_{\delta\lambda}(s_{B}s_{B}X)+s_{B% }\left(i_{\delta\lambda}(s_{B}X)\right)=s_{B}\left(i_{\delta\lambda}(s_{B}X)% \right),
  45. [ Q B , ] = 0 [Q_{B},\mathcal{H}]=0
  46. Q B | Ψ i = 0 Q_{B}|\Psi_{i}\rangle=0
  47. Q B | Ψ f 0 Q_{B}|\Psi_{f}\rangle\neq 0
  48. 0 \mathcal{H}_{0}
  49. = matter ( ψ , A μ a ) - 1 4 F μ ν a F a , μ ν - ( i ( μ c ¯ a ) D μ a b c b + ( μ B a ) A μ a ) + 1 2 α 0 B a B a \mathcal{L}=\mathcal{L}_{\textrm{matter}}(\psi,\,A_{\mu}^{a})-\tfrac{1}{4}F^{a% }_{\mu\nu}F^{a,\,\mu\nu}-(i(\partial^{\mu}\bar{c}^{a})D_{\mu}^{ab}c^{b}+(% \partial^{\mu}B^{a})A_{\mu}^{a})+\tfrac{1}{2}\alpha_{0}B^{a}B^{a}
  50. V 𝔈 V\mathfrak{E}
  51. δ λ V 𝔈 \delta\lambda\in V\mathfrak{E}
  52. 𝔤 \mathfrak{g}
  53. δ ψ i \displaystyle\delta\psi_{i}
  54. μ A μ = 0 \partial^{\mu}A_{\mu}=0
  55. c ¯ \bar{c}
  56. δ c ¯ = i δ λ B \delta\bar{c}=i\delta\lambda B
  57. 𝔤 \mathfrak{g}
  58. V 𝔈 V\mathfrak{E}
  59. - i ( μ c ¯ ) D μ c -i(\partial^{\mu}\bar{c})D_{\mu}c
  60. c ¯ \bar{c}
  61. V 𝔈 V\mathfrak{E}
  62. 𝔤 \mathfrak{g}
  63. c ¯ \bar{c}
  64. 𝔤 \mathfrak{g}
  65. V 𝔈 V\mathfrak{E}
  66. 𝔤 \mathfrak{g}
  67. c ¯ \bar{c}
  68. V 𝔈 V\mathfrak{E}
  69. 𝔤 \mathfrak{g}
  70. V 𝔈 V\mathfrak{E}
  71. c ¯ \bar{c}
  72. c ¯ \bar{c}
  73. s B ( c ¯ ( i μ A μ - 1 2 α 0 s B c ¯ ) ) . s_{B}\left(\bar{c}\left(i\partial^{\mu}A_{\mu}-\tfrac{1}{2}\alpha_{0}s_{B}\bar% {c}\right)\right).
  74. - i ( μ c ¯ ) D μ c -i(\partial^{\mu}\bar{c})D_{\mu}c
  75. 𝔈 \mathfrak{E}
  76. V 𝔈 V\mathfrak{E}
  77. 𝔈 \mathfrak{E}
  78. 𝔤 \mathfrak{g}
  79. ϵ V 𝔈 \epsilon\in V\mathfrak{E}
  80. 𝔤 \mathfrak{g}
  81. G = ξ μ A μ G=\xi\partial^{\mu}A_{\mu}
  82. 𝔤 \mathfrak{g}
  83. 𝔤 \mathfrak{g}
  84. 𝔤 \mathfrak{g}
  85. Q A = D c QA=Dc
  86. Q c = i 2 [ c , c ] L Qc=\tfrac{i}{2}[c,c]_{L}
  87. Q B = 0 QB=0
  88. Q b = B Qb=B
  89. = - 1 4 g 2 Tr [ F μ ν F μ ν ] + 1 2 g 2 Tr [ B B ] - 1 g 2 Tr [ B G ] - ξ g 2 Tr [ μ b D μ c ] \mathcal{L}=-\frac{1}{4g^{2}}\operatorname{Tr}[F^{\mu\nu}F_{\mu\nu}]+{1\over 2% g^{2}}\operatorname{Tr}[BB]-{1\over g^{2}}\operatorname{Tr}[BG]-{\xi\over g^{2% }}\operatorname{Tr}[\partial^{\mu}bD_{\mu}c]
  90. Q = c i ( L i - 1 2 f i j k b j c k ) Q=c^{i}\left(L_{i}-\frac{1}{2}{{f_{i}}^{j}}_{k}b_{j}c^{k}\right)
  91. c i , b i c^{i},b_{i}
  92. f i j k f_{ij}{}^{k}

Brunn–Minkowski_theorem.html

  1. [ μ ( A + B ) ] 1 / n [ μ ( A ) ] 1 / n + [ μ ( B ) ] 1 / n , [\mu(A+B)]^{1/n}\geq[\mu(A)]^{1/n}+[\mu(B)]^{1/n},
  2. A + B := { a + b n a A , b B } . A+B:=\{\,a+b\in\mathbb{R}^{n}\mid a\in A,\ b\in B\,\}.
  3. A [ μ ( A ) ] 1 / n A\mapsto[\mu(A)]^{1/n}
  4. [ μ ( t A + ( 1 - t ) B ) ] 1 / n t [ μ ( A ) ] 1 / n + ( 1 - t ) [ μ ( B ) ] 1 / n . \left[\mu(tA+(1-t)B)\right]^{1/n}\geq t[\mu(A)]^{1/n}+(1-t)[\mu(B)]^{1/n}.

Bubble_rafts.html

  1. U ( ρ ) = - π R 4 ρ s o l u t i o n g ( B α ) 2 A K 0 ( α ρ ) + { 0 ρ 2 π R 4 ρ s o l u t i o n g ( ( 2 - ρ ) 2 α 2 ) ρ 2 U(\rho)=-\pi R^{4}\rho_{solution}g\left(\frac{B}{\alpha}\right)^{2}\mathit{A}K% _{0}(\alpha\rho)+\begin{cases}0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho\geq\ 2\\ \pi R^{4}\rho_{solution}g\left(\frac{(2-\rho)^{2}}{\alpha^{2}}\right)~{}~{}~{}% \rho\leq\ 2\end{cases}
  2. U ( ρ ) U(\rho)
  3. R R
  4. ρ s o l u t i o n \rho_{solution}
  5. g g
  6. ρ \rho
  7. B B
  8. α \alpha
  9. a 2 = T ρ s o l u t i o n g a^{2}=\frac{T}{\rho_{solution}g}
  10. T T
  11. A A
  12. K 0 K_{0}

Bulgarian_conjugation.html

  1. verb form = stem + thematic vowel + inflectional suffix \mathrm{verb\ form}=\mbox{stem}~{}+\mbox{thematic vowel}~{}+\mbox{inflectional% suffix}~{}

Bulk_synchronous_parallel.html

  1. L L
  2. h h
  3. g g
  4. h g hg
  5. h h
  6. m m
  7. m m
  8. m m
  9. m g mg
  10. g g
  11. g g
  12. g g
  13. l l
  14. l < L l<L
  15. l l
  16. p p
  17. m a x i = 1 p ( w i ) + m a x i = 1 p ( h i g ) + l max_{i=1}^{p}(w_{i})+max_{i=1}^{p}(h_{i}g)+l
  18. w i w_{i}
  19. i i
  20. h i h_{i}
  21. i i
  22. w + h g + l w+hg+l
  23. w w
  24. h h
  25. W + H g + S l = s = 1 S w s + g s = 1 S h s + S l W+Hg+Sl=\sum_{s=1}^{S}w_{s}+g\sum_{s=1}^{S}h_{s}+Sl
  26. S S
  27. W W
  28. H H
  29. S S
  30. H O ( n / p ) H\in O(n/p)

Bundle_map.html

  1. π F φ = π E \pi_{F}\circ\varphi=\pi_{E}
  2. π F φ = f π E \pi_{F}\circ\varphi=f\circ\pi_{E}

Butson-type_Hadamard_matrix.html

  1. ( H j k ) q = 1 for j , k = 1 , 2 , , N . (H_{jk})^{q}=1{\quad\rm for\quad}j,k=1,2,\dots,N.
  2. H ( p , N ) H(p,N)
  3. N = m p N=mp
  4. p 3 p\geq 3
  5. { q , N } \{q,N\}
  6. H ( q , N ) H(q,N)
  7. H ( 2 , N ) H(2,N)
  8. H ( 4 , N ) H(4,N)
  9. ± 1 , ± i \pm 1,\pm i
  10. q q\to\infty
  11. [ F N ] j k := exp [ ( 2 π i ( j - 1 ) ( k - 1 ) / N ] for j , k = 1 , 2 , , N [F_{N}]_{jk}:=\exp[(2\pi i(j-1)(k-1)/N]{\quad\rm for\quad}j,k=1,2,\dots,N
  12. F N H ( N , N ) , F_{N}\in H(N,N),
  13. F N F N H ( N , N 2 ) , F_{N}\otimes F_{N}\in H(N,N^{2}),
  14. F N F N F N H ( N , N 3 ) . F_{N}\otimes F_{N}\otimes F_{N}\in H(N,N^{3}).
  15. D 6 := [ 1 1 1 1 1 1 1 - 1 i - i - i i 1 i - 1 i - i - i 1 - i i - 1 i - i 1 - i - i i - 1 i 1 i - i - i i - 1 ] H ( 4 , 6 ) D_{6}:=\begin{bmatrix}1&1&1&1&1&1\\ 1&-1&i&-i&-i&i\\ 1&i&-1&i&-i&-i\\ 1&-i&i&-1&i&-i\\ 1&-i&-i&i&-1&i\\ 1&i&-i&-i&i&-1\\ \end{bmatrix}\in H(4,6)
  16. S 6 := [ 1 1 1 1 1 1 1 1 z z z 2 z 2 1 z 1 z 2 z 2 z 1 z z 2 1 z z 2 1 z 2 z 2 z 1 z 1 z 2 z z 2 z 1 ] H ( 3 , 6 ) S_{6}:=\begin{bmatrix}1&1&1&1&1&1\\ 1&1&z&z&z^{2}&z^{2}\\ 1&z&1&z^{2}&z^{2}&z\\ 1&z&z^{2}&1&z&z^{2}\\ 1&z^{2}&z^{2}&z&1&z\\ 1&z^{2}&z&z^{2}&z&1\\ \end{bmatrix}\in H(3,6)
  17. z = exp ( 2 π i / 3 ) . z=\exp(2\pi i/3).

Butterfly_curve_(algebraic).html

  1. x 6 + y 6 = x 2 . x^{6}+y^{6}=x^{2}.\,\!
  2. Γ \Gamma
  3. 4 0 1 ( x 2 - x 6 ) 1 6 d x = Γ ( 1 6 ) Γ ( 1 3 ) 3 π 2.804 , 4\cdot\int_{0}^{1}(x^{2}-x^{6})^{\frac{1}{6}}dx=\frac{\Gamma(\frac{1}{6})\cdot% \Gamma(\frac{1}{3})}{3\sqrt{\pi}}\approx 2.804,
  4. s 9.017. s\approx 9.017.

C-mole.html

  1. C H 2 O CH_{2}O

C-theorem.html

  1. C ( g i , μ ) C(g_{i},\mu)
  2. g i g_{i}
  3. μ \mu
  4. C ( g i , μ ) C(g_{i},\mu)
  5. g i * g^{*}_{i}
  6. C ( g i * , μ ) = C * C(g^{*}_{i},\mu)=C_{*}

Cable_theory.html

  1. c m c_{m}
  2. r m r_{m}
  3. r l r_{l}
  4. r m = R m 2 π a r_{m}=\frac{R_{m}}{2\pi a\ }
  5. r l = R l π a 2 r_{l}=\frac{R_{l}}{\pi a^{2}\ }
  6. V x = - i l r l \frac{\partial V}{\partial x}=-i_{l}r_{l}
  7. 1 r l V x = - i l \frac{1}{r_{l}}\frac{\partial V}{\partial x}=-i_{l}
  8. i l x = - i m \frac{\partial i_{l}}{\partial x}=-i_{m}
  9. i m i_{m}
  10. i c i_{c}
  11. i c i_{c}
  12. i c = c m V t i_{c}=c_{m}\frac{\partial V}{\partial t}
  13. c m c_{m}
  14. V / t {\partial V}/{\partial t}
  15. i r i_{r}
  16. i r = V r m i_{r}=\frac{V}{r_{m}}
  17. i m = i r + i c i_{m}=i_{r}+i_{c}
  18. i m i_{m}
  19. i l x = - i m = - ( V r m + c m V t ) \frac{\partial i_{l}}{\partial x}=-i_{m}=-\left(\frac{V}{r_{m}}+c_{m}\frac{% \partial V}{\partial t}\right)
  20. i l / x {\partial i_{l}}/{\partial x}
  21. 1 r l 2 V x 2 = c m V t + V r m \frac{1}{r_{l}}\frac{\partial^{2}V}{\partial x^{2}}=c_{m}\frac{\partial V}{% \partial t}+\frac{V}{r_{m}}
  22. λ \lambda
  23. τ \tau
  24. λ \lambda
  25. λ \lambda
  26. λ = r m r l \lambda=\sqrt{\frac{r_{m}}{r_{l}}}
  27. λ \lambda
  28. r l r_{l}
  29. λ \lambda
  30. V x = V 0 e - x λ V_{x}=V_{0}e^{-\frac{x}{\lambda}}
  31. V 0 V_{0}
  32. x = 0 x=0
  33. V x V_{x}
  34. x = λ x=\lambda
  35. x λ = 1 \frac{x}{\lambda}=1
  36. V x = V 0 e - 1 V_{x}=V_{0}e^{-1}
  37. V V
  38. λ \lambda
  39. x = 0 x=0
  40. V λ = V 0 e = 0.368 V 0 V_{\lambda}=\frac{V_{0}}{e}=0.368V_{0}
  41. V λ V_{\lambda}
  42. V 0 V_{0}
  43. V m V_{m}
  44. τ \tau
  45. τ \tau
  46. c m c_{m}
  47. r m r_{m}
  48. r m r l 2 V x 2 = c m r m V t + V \frac{r_{m}}{r_{l}}\frac{\partial^{2}V}{\partial x^{2}}=c_{m}r_{m}\frac{% \partial V}{\partial t}+V
  49. λ 2 = r m / r l \lambda^{2}={r_{m}}/{r_{l}}
  50. τ = c m r m \tau=c_{m}r_{m}
  51. λ 2 2 V x 2 = τ V t + V \lambda^{2}\frac{\partial^{2}V}{\partial x^{2}}=\tau\frac{\partial V}{\partial t% }+V

Cage_(graph_theory).html

  1. 1 + r i = 0 ( g - 3 ) / 2 ( r - 1 ) i 1+r\sum_{i=0}^{(g-3)/2}(r-1)^{i}
  2. 2 i = 0 ( g - 2 ) / 2 ( r - 1 ) i 2\sum_{i=0}^{(g-2)/2}(r-1)^{i}
  3. g 2 log r - 1 n + O ( 1 ) . g\leq 2\log_{r-1}n+O(1).
  4. g 4 3 log r - 1 n + O ( 1 ) . g\geq\frac{4}{3}\log_{r-1}n+O(1).

Cahen's_constant.html

  1. C = ( - 1 ) i s i - 1 = 1 1 - 1 2 + 1 6 - 1 42 + 1 1806 - 0.64341054629. C=\sum\frac{(-1)^{i}}{s_{i}-1}=\frac{1}{1}-\frac{1}{2}+\frac{1}{6}-\frac{1}{42% }+\frac{1}{1806}-\cdots\approx 0.64341054629.
  2. C = 1 s 2 i = 1 2 + 1 7 + 1 1807 + 1 10650056950807 + C=\sum\frac{1}{s_{2i}}=\frac{1}{2}+\frac{1}{7}+\frac{1}{1807}+\frac{1}{1065005% 6950807}+\cdots
  3. q n + 2 = q n 2 q n + 1 + q n q_{n+2}=q_{n}^{2}q_{n+1}+q_{n}
  4. [ 0 , 1 , q 0 2 , q 1 2 , q 2 2 , ] [0,1,q_{0}^{2},q_{1}^{2},q_{2}^{2},\ldots]

Calabi_conjecture.html

  1. g g\;
  2. ω \omega\;
  3. g ~ \tilde{g}
  4. ω ~ \tilde{\omega}
  5. ω \omega\;
  6. ω ~ \tilde{\omega}
  7. ω ~ \tilde{\omega}
  8. ω + d d ϕ \omega+dd^{\prime}\phi
  9. ( ω + d d ϕ ) m = e f ω m (\omega+dd^{\prime}\phi)^{m}=e^{f}\omega^{m}
  10. F = ( ω + d d ϕ ) m / ω m F=(\omega+dd^{\prime}\phi)^{m}/\omega^{m}
  11. ( ω + d d φ 1 ) m = ( ω + d d φ 2 ) m (\omega+dd^{\prime}\varphi_{1})^{m}=(\omega+dd^{\prime}\varphi_{2})^{m}
  12. | d ( φ 1 - φ 2 ) | 2 |d(\varphi_{1}-\varphi_{2})|^{2}
  13. d ( φ 1 - φ 2 ) = 0 d(\varphi_{1}-\varphi_{2})=0

Calculated_Carbon_Aromaticity_Index.html

  1. C C A I = D - 140.7 log ( log ( V + 0.85 ) ) - 80.6 - 210 ln ( t + 273 323 ) CCAI=D-140.7\log(\log(V+0.85))-80.6-210\ln\left(\frac{t+273}{323}\right)
  2. C C A I = D - 140.7 log ( log ( V + 0.85 ) ) - 80.6 - 483.5 log ( t + 273 323 ) CCAI=D-140.7\log(\log(V+0.85))-80.6-483.5\log\left(\frac{t+273}{323}\right)

Calculated_Ignition_Index.html

  1. C I I = ( 270.795 + 0.1038 T ) - 0.254565 D + 23.708 log log ( V + 0.7 ) CII=(270.795+0.1038T)-0.254565D+23.708\log\log(V+0.7)\,

Camera_resectioning.html

  1. [ u v 1 ] T [u\,v\,1]^{T}
  2. [ x w y w z w 1 ] T [x_{w}\,y_{w}\,z_{w}\,1]^{T}
  3. z c [ u v 1 ] = A [ R T ] [ x w y w z w 1 ] z_{c}\begin{bmatrix}u\\ v\\ 1\end{bmatrix}=A\begin{bmatrix}R&T\end{bmatrix}\begin{bmatrix}x_{w}\\ y_{w}\\ z_{w}\\ 1\end{bmatrix}
  4. A = [ α x γ u 0 0 α y v 0 0 0 1 ] A=\begin{bmatrix}\alpha_{x}&\gamma&u_{0}\\ 0&\alpha_{y}&v_{0}\\ 0&0&1\end{bmatrix}
  5. α x = f m x \alpha_{x}=f\cdot m_{x}
  6. α y = f m y \alpha_{y}=f\cdot m_{y}
  7. m x m_{x}
  8. m y m_{y}
  9. f f
  10. γ \gamma
  11. u 0 u_{0}
  12. v 0 v_{0}
  13. R , T R,T
  14. T T
  15. T T
  16. C C
  17. C = - R - 1 T = - R T T C=-R^{-1}T=-R^{T}T
  18. R R
  19. 𝐇 \,\textbf{H}
  20. x π x_{\pi}
  21. π \pi
  22. x x
  23. I , J = [ 1 ± j 0 ] T I,J=[1\,\pm j\,0]^{T}
  24. π \pi
  25. Ω \Omega_{\infty}
  26. Ω \Omega_{\infty}
  27. ω \omega
  28. x 1 T ω x 1 = 0 x_{1}^{T}\omega x_{1}=0
  29. x 2 T ω x 2 = 0 x_{2}^{T}\omega x_{2}=0
  30. x 1 = 𝐇 I = [ h 1 h 2 h 3 ] [ 1 j 0 ] = h 1 + j h 2 x 2 = 𝐇 J = [ h 1 h 2 h 3 ] [ 1 - j 0 ] = h 1 - j h 2 \begin{aligned}\displaystyle x_{1}&\displaystyle=\,\textbf{H}I=\begin{bmatrix}% h_{1}&h_{2}&h_{3}\end{bmatrix}\begin{bmatrix}1\\ j\\ 0\end{bmatrix}=h_{1}+jh_{2}\\ \displaystyle x_{2}&\displaystyle=\,\textbf{H}J=\begin{bmatrix}h_{1}&h_{2}&h_{% 3}\end{bmatrix}\begin{bmatrix}1\\ -j\\ 0\end{bmatrix}=h_{1}-jh_{2}\end{aligned}
  31. x 2 x_{2}
  32. x 1 x_{1}
  33. x 1 T ω x 1 \displaystyle x_{1}^{T}\omega x_{1}

Cameron–Martin_theorem.html

  1. γ n ( A ) = 1 ( 2 π ) n / 2 A exp ( - 1 2 x , x 𝐑 n ) d x . \gamma_{n}(A)=\frac{1}{(2\pi)^{n/2}}\int_{A}\exp\left(-\tfrac{1}{2}\langle x,x% \rangle_{\mathbf{R}^{n}}\right)\,dx.
  2. x , x 𝐑 n \langle x,x\rangle_{\mathbf{R}^{n}}
  3. γ n ( A - h ) = 1 ( 2 π ) n / 2 A exp ( - 1 2 x - h , x - h 𝐑 n ) d x = 1 ( 2 π ) n / 2 A exp ( 2 x , h 𝐑 n - h , h 𝐑 n 2 ) exp ( - 1 2 x , x 𝐑 n ) d x . \begin{aligned}\displaystyle\gamma_{n}(A-h)&\displaystyle=\frac{1}{(2\pi)^{n/2% }}\int_{A}\exp\left(-\tfrac{1}{2}\langle x-h,x-h\rangle_{\mathbf{R}^{n}}\right% )\,dx\\ &\displaystyle=\frac{1}{(2\pi)^{n/2}}\int_{A}\exp\left(\frac{2\langle x,h% \rangle_{\mathbf{R}^{n}}-\langle h,h\rangle_{\mathbf{R}^{n}}}{2}\right)\exp% \left(-\tfrac{1}{2}\langle x,x\rangle_{\mathbf{R}^{n}}\right)\,dx.\end{aligned}
  4. exp ( 2 x , h 𝐑 n - h , h 𝐑 n 2 ) = exp ( x , h 𝐑 n - 1 2 h 𝐑 n 2 ) . \exp\left(\frac{2\langle x,h\rangle_{\mathbf{R}^{n}}-\langle h,h\rangle_{% \mathbf{R}^{n}}}{2}\right)=\exp\left(\langle x,h\rangle_{\mathbf{R}^{n}}-% \tfrac{1}{2}\|h\|_{\mathbf{R}^{n}}^{2}\right).
  5. d ( T h ) * ( γ n ) d γ n ( x ) = exp ( h , x 𝐑 n - 1 2 h 𝐑 n 2 ) . \frac{\mathrm{d}(T_{h})_{*}(\gamma^{n})}{\mathrm{d}\gamma^{n}}(x)=\exp\left(% \left\langle h,x\right\rangle_{\mathbf{R}^{n}}-\tfrac{1}{2}\|h\|_{\mathbf{R}^{% n}}^{2}\right).
  6. d ( T h ) * ( γ ) d γ ( x ) = exp ( h , x - 1 2 h H 2 ) , \frac{\mathrm{d}(T_{h})_{*}(\gamma)}{\mathrm{d}\gamma}(x)=\exp\left(\langle h,% x\rangle^{\sim}-\tfrac{1}{2}\|h\|_{H}^{2}\right),
  7. h , x = I ( h ) ( x ) \langle h,x\rangle^{\sim}=I(h)(x)
  8. E F ( x + t i ( h ) ) d γ ( x ) = E F ( x ) exp ( t h , x - 1 2 t 2 h H 2 ) d γ ( x ) \int_{E}F(x+ti(h))\,\mathrm{d}\gamma(x)=\int_{E}F(x)\exp\left(t\langle h,x% \rangle^{\sim}-\tfrac{1}{2}t^{2}\|h\|_{H}^{2}\right)\,\mathrm{d}\gamma(x)
  9. E D F ( x ) ( i ( h ) ) d γ ( x ) = E F ( x ) h , x d γ ( x ) . \int_{E}\mathrm{D}F(x)(i(h))\,\mathrm{d}\gamma(x)=\int_{E}F(x)\langle h,x% \rangle^{\sim}\,\mathrm{d}\gamma(x).
  10. div [ V h ] ( x ) = - h , x , \mathop{\mathrm{div}}[V_{h}](x)=-\langle h,x\rangle^{\sim},
  11. 0 1 j , k = 1 q | H j , k ( t ) | d t < , \int_{0}^{1}\sum_{j,k=1}^{q}|H_{j,k}(t)|\,dt<\infty,
  12. E [ exp ( - 0 1 w ( t ) H ( t ) w ( t ) d t ) ] = exp [ 1 2 0 1 tr ( G ( t ) ) d t ] , E\left[\exp\left(-\int_{0}^{1}w^{\prime}(t)H(t)w(t)\,dt\right)\right]=\exp% \left[\tfrac{1}{2}\int_{0}^{1}\operatorname{tr}(G(t))\,dt\right],
  13. d G ( t ) d t = 2 H ( t ) - G 2 ( t ) . \frac{dG(t)}{dt}=2H(t)-G^{2}(t).

Canadian_weather_radar_network.html

  1. μ \mu

Canonical_Huffman_code.html

  1. j = 1 i - 1 2 - l i . \sum_{j=1}^{i-1}2^{-l_{i}}.

Cantellated_5-cell.html

  1. ( 2 2 5 , 2 2 3 , 1 3 , ± 1 ) \left(2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  2. ( 2 2 5 , 2 2 3 , - 2 3 , 0 ) \left(2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  3. ( 2 2 5 , 0 , ± 3 , ± 1 ) \left(2\sqrt{\frac{2}{5}},\ 0,\ \pm\sqrt{3},\ \pm 1\right)
  4. ( 2 2 5 , 0 , 0 , ± 2 ) \left(2\sqrt{\frac{2}{5}},\ 0,\ 0,\ \pm 2\right)
  5. ( 2 2 5 , - 2 2 3 , 2 3 , 0 ) \left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\ 0\right)
  6. ( 2 2 5 , - 2 2 3 , - 1 3 , ± 1 ) \left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm 1\right)
  7. ( - 1 10 , 3 2 , ± 3 , ± 1 ) \left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm 1\right)
  8. ( - 1 10 , 3 2 , 0 , ± 2 ) \left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ 0,\ \pm 2\right)
  9. ( - 1 10 , - 1 6 , 2 3 , ± 2 ) \left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm 2\right)
  10. ( - 1 10 , - 1 6 , - 4 3 , 0 ) \left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right)
  11. ( - 1 10 , - 5 6 , 1 3 , ± 1 ) \left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  12. ( - 1 10 , - 5 6 , - 2 3 , 0 ) \left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  13. ( - 3 2 5 , 0 , 0 , 0 ) ± ( 0 , 2 3 , 2 3 , 0 ) \left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right)\pm\left(0,\ \sqrt{\frac{2}{3}},% \ \frac{2}{\sqrt{3}},\ 0\right)
  14. ( - 3 2 5 , 0 , 0 , 0 ) ± ( 0 , 2 3 , - 1 3 , ± 1 ) \left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right)\pm\left(0,\ \sqrt{\frac{2}{3}},% \ \frac{-1}{\sqrt{3}},\ \pm 1\right)
  15. ( 3 2 5 , ± 6 , ± 3 , ± 1 ) \left(3\sqrt{\frac{2}{5}},\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm 1\right)
  16. ( 3 2 5 , ± 6 , 0 , ± 2 ) \left(3\sqrt{\frac{2}{5}},\ \pm\sqrt{6},\ 0,\ \pm 2\right)
  17. ( 3 2 5 , 0 , 0 , 0 ) ± ( 0 , 2 3 , 5 3 , ± 1 ) \left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right)\pm\left(0,\ \sqrt{\frac{2}{3}},\ % \frac{5}{\sqrt{3}},\ \pm 1\right)
  18. ( 3 2 5 , 0 , 0 , 0 ) ± ( 0 , 2 3 , - 1 3 , ± 3 ) \left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right)\pm\left(0,\ \sqrt{\frac{2}{3}},\ % \frac{-1}{\sqrt{3}},\ \pm 3\right)
  19. ( 3 2 5 , 0 , 0 , 0 ) ± ( 0 , 2 3 , - 4 3 , ± 2 ) \left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right)\pm\left(0,\ \sqrt{\frac{2}{3}},\ % \frac{-4}{\sqrt{3}},\ \pm 2\right)
  20. ( 1 10 , 5 6 , 5 3 , ± 1 ) \left(\frac{1}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{5}{\sqrt{3}},\ \pm 1\right)
  21. ( 1 10 , 5 6 , - 1 3 , ± 3 ) \left(\frac{1}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm 3\right)
  22. ( 1 10 , 5 6 , - 4 3 , ± 2 ) \left(\frac{1}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ \pm 2\right)
  23. ( 1 10 , - 3 2 , 3 , ± 3 ) \left(\frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ \sqrt{3},\ \pm 3\right)
  24. ( 1 10 , - 3 2 , - 2 3 , 0 ) \left(\frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ -2\sqrt{3},\ 0\right)
  25. ( 1 10 , - 7 6 , 2 3 , ± 2 ) \left(\frac{1}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm 2\right)
  26. ( 1 10 , - 7 6 , - 4 3 , 0 ) \left(\frac{1}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right)
  27. ( - 2 2 5 , 2 2 3 , 4 3 , ± 2 ) \left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{4}{\sqrt{3}},\ \pm 2\right)
  28. ( - 2 2 5 , 2 2 3 , 1 3 , ± 3 ) \left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm 3\right)
  29. ( - 2 2 5 , 2 2 3 , - 5 3 , ± 1 ) \left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{-5}{\sqrt{3}},\ \pm 1\right)
  30. ( - 2 2 5 , 0 , 3 , ± 3 ) \left(-2\sqrt{\frac{2}{5}},\ 0,\ \sqrt{3},\ \pm 3\right)
  31. ( - 2 2 5 , 0 , - 2 3 , 0 ) \left(-2\sqrt{\frac{2}{5}},\ 0,\ -2\sqrt{3},\ 0\right)
  32. ( - 2 2 5 , - 4 2 3 , 1 3 , ± 1 ) \left(-2\sqrt{\frac{2}{5}},\ -4\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  33. ( - 2 2 5 , - 4 2 3 , - 2 3 , 0 ) \left(-2\sqrt{\frac{2}{5}},\ -4\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  34. ( - 9 10 , 3 2 , ± 3 , ± 1 ) \left(\frac{-9}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm 1\right)
  35. ( - 9 10 , 3 2 , 0 , ± 2 ) \left(\frac{-9}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ 0,\ \pm 2\right)
  36. ( - 9 10 , - 1 6 , 2 3 , ± 2 ) \left(\frac{-9}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm 2\right)
  37. ( - 9 10 , - 1 6 , - 4 3 , 0 ) \left(\frac{-9}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right)
  38. ( - 9 10 , - 5 6 , 1 3 , ± 1 ) \left(\frac{-9}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  39. ( - 9 10 , - 5 6 , - 2 3 , 0 ) \left(\frac{-9}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)

Cantellation_(geometry).html

  1. r { p q } r\begin{Bmatrix}p\\ q\end{Bmatrix}

Cantor–Zassenhaus_algorithm.html

  1. f ( x ) f(x)
  2. 𝔽 q \mathbb{F}_{q}
  3. g ( x ) g(x)
  4. g ( x ) g(x)
  5. f ( x ) f(x)
  6. f ( x ) f(x)
  7. f ( x ) f(x)
  8. R = 𝔽 q [ x ] f ( x ) R=\frac{\mathbb{F}_{q}[x]}{\langle f(x)\rangle}
  9. f ( x ) f(x)
  10. p 1 ( x ) , p 2 ( x ) , , p s ( x ) p_{1}(x),p_{2}(x),\ldots,p_{s}(x)
  11. S = i = 1 s 𝔽 q [ x ] p i ( x ) S=\prod_{i=1}^{s}\frac{\mathbb{F}_{q}[x]}{\langle p_{i}(x)\rangle}
  12. ϕ \phi
  13. g ( x ) R g(x)\in R
  14. p i ( x ) p_{i}(x)
  15. g ( x ) \displaystyle g(x)
  16. ϕ ( g ( x ) + f ( x ) ) = ( g 1 ( x ) + p 1 ( x ) , , g s ( x ) + p s ( x ) ) \phi(g(x)+\langle f(x)\rangle)=(g_{1}(x)+\langle p_{1}(x)\rangle,\ldots,g_{s}(% x)+\langle p_{s}(x)\rangle)
  17. p i ( x ) p_{i}(x)
  18. q d q^{d}
  19. a ( x ) R a(x)\in R
  20. a ( x ) 0 , ± 1 a(x)\neq 0,\pm 1
  21. a i ( x ) { 0 , - 1 , 1 } for i = 1 , 2 , , s , a_{i}(x)\in\{0,-1,1\}\,\text{ for }i=1,2,\ldots,s,
  22. a i ( x ) a_{i}(x)
  23. a ( x ) a(x)
  24. p i ( x ) p_{i}(x)
  25. A = { i | a i ( x ) = 0 } , A=\{i|a_{i}(x)=0\},
  26. B = { i | a i ( x ) = - 1 } , B=\{i|a_{i}(x)=-1\},
  27. C = { i | a i ( x ) = 1 } , C=\{i|a_{i}(x)=1\},
  28. f ( x ) f(x)
  29. gcd ( f ( x ) , a ( x ) ) = i A p i ( x ) , \gcd(f(x),a(x))=\prod_{i\in A}p_{i}(x),
  30. gcd ( f ( x ) , a ( x ) + 1 ) = i B p i ( x ) , \gcd(f(x),a(x)+1)=\prod_{i\in B}p_{i}(x),
  31. gcd ( f ( x ) , a ( x ) - 1 ) = i C p i ( x ) . \gcd(f(x),a(x)-1)=\prod_{i\in C}p_{i}(x).
  32. a ( x ) a(x)
  33. 𝔽 q \mathbb{F}_{q}
  34. b ( x ) R b(x)\in R
  35. b ( x ) 0 , ± 1 b(x)\neq 0,\pm 1
  36. m = ( q d - 1 ) / 2 m=(q^{d}-1)/2
  37. b ( x ) m b(x)^{m}
  38. ϕ \phi
  39. ϕ ( b ( x ) m ) = ( b 1 m ( x ) + p 1 ( x ) , , b s m ( x ) + p s ( x ) ) . \phi(b(x)^{m})=(b_{1}^{m}(x)+\langle p_{1}(x)\rangle,\ldots,b^{m}_{s}(x)+% \langle p_{s}(x)\rangle).
  40. b i ( x ) + p i ( x ) b_{i}(x)+\langle p_{i}(x)\rangle
  41. q d q^{d}
  42. q d - 1 q^{d}-1
  43. b i ( x ) = 0 b_{i}(x)=0
  44. b i ( x ) q d - 1 = 1 b_{i}(x)^{q^{d}-1}=1
  45. b i ( x ) m = ± 1 b_{i}(x)^{m}=\pm 1
  46. b i ( x ) = 0 b_{i}(x)=0
  47. b i ( x ) m = 0 b_{i}(x)^{m}=0
  48. b ( x ) m b(x)^{m}
  49. a ( x ) a(x)
  50. b ( x ) 0 , ± 1 b(x)\neq 0,\pm 1
  51. A , B A,B

Cap_product.html

  1. : H p ( X ; R ) × H q ( X ; R ) H p - q ( X ; R ) . \frown\;:H_{p}(X;R)\times H^{q}(X;R)\rightarrow H_{p-q}(X;R).
  2. σ : Δ p X \sigma:\Delta\ ^{p}\rightarrow\ X
  3. ψ C q ( X ; R ) , \psi\in C^{q}(X;R),
  4. σ ψ = ψ ( σ | [ v 0 , , v q ] ) σ | [ v q , , v p ] . \sigma\frown\psi=\psi(\sigma|_{[v_{0},\ldots,v_{q}]})\sigma|_{[v_{q},\ldots,v_% {p}]}.
  5. σ | [ v 0 , , v q ] \sigma|_{[v_{0},\ldots,v_{q}]}
  6. σ \sigma
  7. C ( X ) C ( X ) Δ * Id C ( X ) C ( X ) C ( X ) Id ε C ( X ) C_{\bullet}(X)\otimes C^{\bullet}(X)\overset{\Delta_{*}\otimes\mathrm{Id}}{% \longrightarrow}C_{\bullet}(X)\otimes C_{\bullet}(X)\otimes C^{\bullet}(X)% \overset{\mathrm{Id}\otimes\varepsilon}{\longrightarrow}C_{\bullet}(X)
  8. X X
  9. Δ : X X × X \Delta\colon X\to X\times X
  10. Δ * \Delta_{*}
  11. ε : C p ( X ) C q ( X ) \varepsilon\colon C_{p}(X)\otimes C^{q}(X)\to\mathbb{Z}
  12. p = q p=q
  13. : H ( X ) × H ( X ) H ( X ) \frown\colon H_{\bullet}(X)\times H^{\bullet}(X)\to H_{\bullet}(X)
  14. : H p ( X ) × H q ( X ) H p - q ( X ) \frown\colon H_{p}(X)\times H^{q}(X)\to H_{p-q}(X)
  15. p < q p<q
  16. X × Y X\times Y
  17. \ : H p ( X ; R ) H q ( X × Y ; R ) H q - p ( Y ; R ) . \backslash\;:H_{p}(X;R)\otimes H^{q}(X\times Y;R)\rightarrow H^{q-p}(Y;R).
  18. ( σ ψ ) = ( - 1 ) q ( σ ψ - σ δ ψ ) . \partial(\sigma\frown\psi)=(-1)^{q}(\partial\sigma\frown\psi-\sigma\frown% \delta\psi).
  19. f * ( σ ) ψ = f * ( σ f * ( ψ ) ) . f_{*}(\sigma)\frown\psi=f_{*}(\sigma\frown f^{*}(\psi)).
  20. ψ ( σ φ ) = ( φ ψ ) ( σ ) \psi(\sigma\frown\varphi)=(\varphi\smile\psi)(\sigma)
  21. σ : Δ p + q X \sigma:\Delta^{p+q}\rightarrow X
  22. ψ C q ( X ; R ) \psi\in C^{q}(X;R)
  23. φ C p ( X ; R ) . \varphi\in C^{p}(X;R).
  24. H ( X ; R ) H_{\ast}(X;R)
  25. H ( X ; R ) - H^{\ast}(X;R)-

Capital_adequacy_ratio.html

  1. CAR = Tier 1 capital + Tier 2 capital Risk weighted assets \mbox{CAR}~{}=\cfrac{\mbox{Tier 1 capital + Tier 2 capital}~{}}{\mbox{Risk % weighted assets}~{}}
  2. a \,a
  3. CAR = T 1 + T 2 a \mbox{CAR}~{}=\cfrac{T_{1}+T_{2}}{a}
  4. T 1 T_{1}
  5. T 2 T_{2}
  6. 10 * 0 % = 0 10*0\%=0
  7. 15 * 0 % = 0 15*0\%=0
  8. 20 * 50 % = 10 20*50\%=10
  9. 50 * 100 % = 50 50*100\%=50
  10. 5 * 100 % = 5 5*100\%=5

Capital_recovery_factor.html

  1. C R F = i ( 1 + i ) n ( 1 + i ) n - 1 CRF=\frac{i(1+i)^{n}}{(1+i)^{n}-1}
  2. n n
  3. n = 1 n=1
  4. C R F CRF
  5. 1 + i 1+i
  6. n n\to\infty
  7. C R F i CRF\to i

Caramel_color.html

  1. Color Intensity = A * 100 T S \,\text{Color Intensity}=\frac{A*100}{TS}

Carleson_measure.html

  1. μ ( Ω 𝔹 r ( p ) ) C σ ( Ω 𝔹 r ( p ) ) , \mu\left(\Omega\cap\mathbb{B}_{r}(p)\right)\leq C\sigma\left(\partial\Omega% \cap\mathbb{B}_{r}(p)\right),
  2. 𝔹 r ( p ) := { x n | x - p n < r } \mathbb{B}_{r}(p):=\left\{x\in\mathbb{R}^{n}\left|\|x-p\|_{\mathbb{R}^{n}}<r% \right.\right\}
  3. P : H p ( D ) L p ( D , μ ) P:H^{p}(\partial D)\to L^{p}(D,\mu)
  4. P ( f ) ( z ) = 1 2 π 0 2 π Re e i t + z e i t - z f ( e i t ) d t . P(f)(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{Re}\frac{e^{it}+z}{e^{it}-z}f(e^{% it})\,\mathrm{d}t.
  5. r > 0 , p Ω , μ ( Ω 𝔹 r ( p ) ) C σ ( Ω 𝔹 r ( p ) ) \forall r>0,\forall p\in\partial\Omega,\mu\left(\Omega\cap\mathbb{B}_{r}(p)% \right)\leq C\sigma\left(\partial\Omega\cap\mathbb{B}_{r}(p)\right)
  6. r ( 0 , R ) , p Ω , μ ( Ω 𝔹 r ( p ) ) C σ ( Ω 𝔹 r ( p ) ) \forall r\in(0,R),\forall p\in\partial\Omega,\mu\left(\Omega\cap\mathbb{B}_{r}% (p)\right)\leq C\sigma\left(\partial\Omega\cap\mathbb{B}_{r}(p)\right)

Carnot_cycle.html

  1. Δ S = Q 1 / T 1 \Delta S=Q_{1}/T_{1}
  2. Δ S = Q 2 / T 2 \Delta S=Q_{2}/T_{2}
  3. Q = A B T d S ( 1 ) Q=\int_{A}^{B}T\,dS\quad\quad(1)
  4. W = P d V = ( d Q - d U ) = ( T d S - d U ) ( 2 ) W=\oint PdV=\oint(dQ-dU)=\oint(TdS-dU)\quad\quad\quad\quad(2)
  5. W = P d V = T d s = ( T H - T C ) ( S B - S A ) W=\oint PdV=\oint Tds=(T_{H}-T_{C})(S_{B}-S_{A})
  6. Q H = T H ( S B - S A ) Q_{H}=T_{H}(S_{B}-S_{A})\,
  7. Q C = T C ( S B - S A ) Q_{C}=T_{C}(S_{B}-S_{A})\,
  8. η \eta
  9. η = W Q H = 1 - T C T H ( 3 ) \eta=\frac{W}{Q_{H}}=1-\frac{T_{C}}{T_{H}}\quad\quad\quad\quad\quad\quad\quad% \quad\quad(3)
  10. W W
  11. Q C Q_{C}
  12. Q H Q_{H}
  13. T C T_{C}
  14. T H T_{H}
  15. S B S_{B}
  16. S A S_{A}
  17. T H T_{H}
  18. T C T_{C}
  19. T H = 1 Δ S Q i n T d S \langle T_{H}\rangle=\frac{1}{\Delta S}\int_{Q_{in}}TdS
  20. T C = 1 Δ S Q o u t T d S \langle T_{C}\rangle=\frac{1}{\Delta S}\int_{Q_{out}}TdS

Caroline_era.html

  1. × \times

Carrier_recovery.html

  1. ω R F \omega_{RF}
  2. V B P S K ( t ) \displaystyle V_{BPSK}(t)
  3. 2 π 2\pi
  4. V Q P S K ( t ) \displaystyle V_{QPSK}(t)
  5. 4 ω R F 4\omega_{RF}
  6. π / 2 \pi/2
  7. π / 2 \pi/2

Cascaded_integrator–comb_filter.html

  1. H ( z ) \displaystyle H(z)
  2. x [ n ] x[n]
  3. y [ n - 1 ] y[n-1]
  4. R M RM
  5. y [ n ] \displaystyle y[n]
  6. c [ n ] = x [ n ] - x [ n - R M ] c[n]=x[n]-x[n-RM]
  7. y [ n ] = y [ n - 1 ] + c [ n ] y[n]=y[n-1]+c[n]
  8. N N
  9. R R
  10. N log 2 ( R M ) N\log_{2}(RM)

Cash_accumulation_equation.html

  1. y y\,\!
  2. P P\,\!
  3. i i\,\!
  4. t t\,\!
  5. y y
  6. y y
  7. P P
  8. 1 = t = 0 1=t=0
  9. F F\,\!
  10. d y = i y d t + F d t dy=iy\,dt+F\,dt
  11. t = d y i y + F t=\int\frac{dy}{iy+F}
  12. t = 1 i ln ( i y + F ) + k t=\frac{1}{i}\ln(iy+F)+k
  13. k k\,\!
  14. P P\,\!
  15. ( t , y ) = ( 0 , P ) (t,y)=(0,P)\,\!
  16. k = - 1 i ln ( i P + F ) k=-\frac{1}{i}\ln(iP+F)
  17. k k\,\!
  18. ln ( a ) - ln ( b ) = ln ( a b ) \ln(a)-\ln(b)=\ln\left(\frac{a}{b}\right)
  19. i t = ln ( i y + F i P + F ) it=\ln\left(\frac{iy+F}{iP+F}\right)
  20. F = 0 F=0\,
  21. i = 0 i=0\,\!
  22. e i t - 1 i , i = 0 \frac{e^{it}-1}{i}\mbox{ , }~{}i=0\,
  23. e i t = 1 + i t + ( i t ) 2 2 ! + e^{it}=1+it+\frac{(it)^{2}}{2!}+\cdots
  24. 1 1\,\!
  25. i i\,\!
  26. e i t - 1 i = t , i = 0 \frac{e^{it}-1}{i}=t\mbox{ , }~{}i=0\,
  27. y = P + F t , i = 0 y=P+Ft\mbox{ , }~{}i=0\,
  28. F = - i P F=-iP\,\!
  29. y = P y=P\,
  30. F F\,\!
  31. P P\,\!

Cash_flow_return_on_investment.html

  1. CFROI = Cash Flow Market Recapitalization \,\text{CFROI}=\frac{\,\text{Cash Flow}}{\,\text{Market Recapitalization}}
  2. CFROI = Gross Cash Flow Gross Investment \,\text{CFROI}=\frac{\,\text{Gross Cash Flow}}{\,\text{Gross Investment}}
  3. CFROI = Gross Cash Flow - Economic Depreciation Gross Investment \,\text{CFROI}=\frac{\,\text{Gross Cash Flow}-\,\text{Economic Depreciation}}{% \,\text{Gross Investment}}
  4. Economic Depreciation = K c ( 1 + K c ) n - 1 \,\text{Economic Depreciation}=\frac{K_{c}}{\left(1+K_{c}\right)^{n}-1}

Casimir_pressure.html

  1. F F
  2. A A
  3. F = P A . F=PA.
  4. P P

Cassini_projection.html

  1. x = arcsin ( cos ( ϕ ) sin ( λ ) ) x=\arcsin(\cos(\phi)\sin(\lambda))\,
  2. y = arctan ( tan ( ϕ ) cos ( λ ) ) y=\arctan\left(\frac{\tan(\phi)}{\cos(\lambda)}\right)
  3. λ \lambda
  4. ϕ \phi
  5. sin ( ϕ ) \sin(\phi)
  6. cos ( ϕ ) cos ( λ ) \cos(\phi)\cos(\lambda)

Casson_invariant.html

  1. λ ( Σ + 1 n + 1 K ) - λ ( Σ + 1 n K ) \lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n% }\cdot K\right)
  2. Σ + 1 m K \Sigma+\frac{1}{m}\cdot K
  3. 1 m \frac{1}{m}
  4. λ ( Σ + 1 m + 1 K + 1 n + 1 L ) - λ ( Σ + 1 m K + 1 n + 1 L ) - λ ( Σ + 1 m + 1 K + 1 n L ) + λ ( Σ + 1 m K + 1 n L ) \lambda\left(\Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n+1}\cdot L\right)-\lambda% \left(\Sigma+\frac{1}{m}\cdot K+\frac{1}{n+1}\cdot L\right)-\lambda\left(% \Sigma+\frac{1}{m+1}\cdot K+\frac{1}{n}\cdot L\right)+\lambda\left(\Sigma+% \frac{1}{m}\cdot K+\frac{1}{n}\cdot L\right)
  5. λ ( Σ + 1 n + 1 K ) - λ ( Σ + 1 n K ) = ± 1 \lambda\left(\Sigma+\frac{1}{n+1}\cdot K\right)-\lambda\left(\Sigma+\frac{1}{n% }\cdot K\right)=\pm 1
  6. λ ( M + 1 n + 1 K ) - λ ( M + 1 n K ) = ϕ 1 ( K ) , \lambda\left(M+\frac{1}{n+1}\cdot K\right)-\lambda\left(M+\frac{1}{n}\cdot K% \right)=\phi_{1}(K),
  7. ϕ 1 ( K ) \phi_{1}(K)
  8. z 2 z^{2}
  9. K ( z ) \nabla_{K}(z)
  10. Σ ( p , q , r ) \Sigma(p,q,r)
  11. λ ( Σ ( p , q , r ) ) = - 1 8 [ 1 - 1 3 p q r ( 1 - p 2 q 2 r 2 + p 2 q 2 + q 2 r 2 + p 2 r 2 ) - d ( p , q r ) - d ( q , p r ) - d ( r , p q ) ] \lambda(\Sigma(p,q,r))=-\frac{1}{8}\left[1-\frac{1}{3pqr}\left(1-p^{2}q^{2}r^{% 2}+p^{2}q^{2}+q^{2}r^{2}+p^{2}r^{2}\right)-d(p,qr)-d(q,pr)-d(r,pq)\right]
  12. d ( a , b ) = - 1 a k = 1 a - 1 cot ( π k a ) cot ( π b k a ) d(a,b)=-\frac{1}{a}\sum_{k=1}^{a-1}\cot\left(\frac{\pi k}{a}\right)\cot\left(% \frac{\pi bk}{a}\right)
  13. ( M ) = R irr ( M ) / S O ( 3 ) \mathcal{R}(M)=R^{\mathrm{irr}}(M)/SO(3)
  14. R irr ( M ) R^{\mathrm{irr}}(M)
  15. π 1 ( M ) \pi_{1}(M)
  16. Σ = M 1 F M 2 \Sigma=M_{1}\cup_{F}M_{2}
  17. M M
  18. ( - 1 ) g 2 \frac{(-1)^{g}}{2}
  19. ( M 1 ) \mathcal{R}(M_{1})
  20. ( M 2 ) \mathcal{R}(M_{2})
  21. λ C W ( M ) = λ C W ( M ) + m , μ m , ν μ , ν Δ W ′′ ( M - K ) ( 1 ) + τ W ( m , μ ; ν ) \lambda_{CW}(M^{\prime})=\lambda_{CW}(M)+\frac{\langle m,\mu\rangle}{\langle m% ,\nu\rangle\langle\mu,\nu\rangle}\Delta_{W}^{\prime\prime}(M-K)(1)+\tau_{W}(m,% \mu;\nu)
  22. , \langle\cdot,\cdot\rangle
  23. H 1 ( M - K ) / Torsion H_{1}(M-K)/\,\text{Torsion}
  24. τ W ( m , μ ; ν ) = - sgn y , m s ( x , m , y , m ) + sgn y , μ s ( x , μ , y , μ ) + ( δ 2 - 1 ) m , μ 12 m , ν μ , ν \tau_{W}(m,\mu;\nu)=-\mathrm{sgn}\langle y,m\rangle s(\langle x,m\rangle,% \langle y,m\rangle)+\mathrm{sgn}\langle y,\mu\rangle s(\langle x,\mu\rangle,% \langle y,\mu\rangle)+\frac{(\delta^{2}-1)\langle m,\mu\rangle}{12\langle m,% \nu\rangle\langle\mu,\nu\rangle}
  25. x , y = 1 \langle x,y\rangle=1
  26. λ C W ( M ) = 2 λ ( M ) \lambda_{CW}(M)=2\lambda(M)
  27. λ C W L ( M ) = 1 2 | H 1 ( M ) | λ C W ( M ) \lambda_{CWL}(M)=\tfrac{1}{2}\left|H_{1}(M)\right|\lambda_{CW}(M)
  28. λ C W L ( M ) = Δ M ′′ ( 1 ) 2 - torsion ( H 1 ( M , ) ) 12 \lambda_{CWL}(M)=\frac{\Delta^{\prime\prime}_{M}(1)}{2}-\frac{\mathrm{torsion}% (H_{1}(M,\mathbb{Z}))}{12}
  29. λ C W L ( M ) = | torsion ( H 1 ( M ) ) | Link M ( γ , γ ) \lambda_{CWL}(M)=\left|\mathrm{torsion}(H_{1}(M))\right|\mathrm{Link}_{M}(% \gamma,\gamma^{\prime})
  30. S 1 , S 2 S_{1},S_{2}
  31. H 2 ( M ; ) H_{2}(M;\mathbb{Z})
  32. γ \gamma^{\prime}
  33. S 1 , S 2 S_{1},S_{2}
  34. H 1 ( M ; ) H_{1}(M;\mathbb{Z})
  35. λ C W L ( M ) = | torsion ( H 1 ( M ; ) ) | ( ( a b c ) ( [ M ] ) ) 2 \lambda_{CWL}(M)=\left|\mathrm{torsion}(H_{1}(M;\mathbb{Z}))\right|\left((a% \cup b\cup c)([M])\right)^{2}
  36. λ C W L ( M ) = 0 \lambda_{CWL}(M)=0
  37. λ C W L ( M 1 # M 2 ) = | H 1 ( M 2 ) | λ C W L ( M 1 ) + | H 1 ( M 1 ) | λ C W L ( M 2 ) \lambda_{CWL}(M_{1}\#M_{2})=\left|H_{1}(M_{2})\right|\lambda_{CWL}(M_{1})+% \left|H_{1}(M_{1})\right|\lambda_{CWL}(M_{2})
  38. 𝒜 / 𝒢 \mathcal{A}/\mathcal{G}
  39. 𝒜 \mathcal{A}
  40. 𝒢 \mathcal{G}
  41. S 1 S^{1}
  42. 𝒜 / 𝒢 \mathcal{A}/\mathcal{G}

Catalyst_poisoning.html

  1. η = tanh ( h p ) h p \eta=\frac{\tanh(h_{p})}{h_{p}}
  2. F = tanh ( h t 1 - α ) 1 - α tanh ( h t ) F=\frac{\tanh(h_{t}\cdot\sqrt{1-\alpha})\cdot\sqrt{1-\alpha}}{\tanh(h_{t})}
  3. F = 1 - α F=1-\alpha
  4. F = 1 - α F=\sqrt{1-\alpha}
  5. D i f f u s i o n R a t e = - π r a v g 2 D c d C d x DiffusionRate=-\pi\cdot r_{avg}^{2}\cdot D_{c}\cdot\frac{dC}{dx}
  6. R e a c t i o n R a t e = η π r a v g ( 1 - α ) L a v g k 1 ′′ C c ReactionRate=\eta\cdot\pi\cdot r_{avg}\cdot(1-\alpha)\cdot L_{avg}\cdot k_{1}^% {\prime\prime}\cdot C_{c}
  7. F = r p o i s o n e d r u n p o i s o n e d F=\frac{r_{poisoned}}{r_{unpoisoned}}
  8. F = tanh [ ( 1 - α ) h t ] tanh ( h t ) 1 1 + α h t tanh [ ( 1 - α ) h t ] F=\frac{\tanh[(1-\alpha)\cdot h_{t}]}{\tanh(h_{t})}\cdot\frac{1}{1+\alpha\cdot h% _{t}\cdot\tanh[(1-\alpha)\cdot h_{t}]}

Cataphora.html

  1. x = y 3 + 2 z - 1 x=y^{3}+2z-1

Catechol_2,3-dioxygenase.html

  1. \rightleftharpoons

Category:Dagger_categories.html

  1. \dagger

Category:Graph_invariants.html

  1. f f\,
  2. f ( G 1 ) = f ( G 2 ) f(G_{1})=f(G_{2})\,
  3. G 1 G_{1}\,
  4. G 2 G_{2}\,

Category:Properties_of_groups.html

  1. G 1 G_{1}
  2. G 2 G_{2}

Category:Spiric_sections.html

  1. ( r 2 - a 2 + c 2 + x 2 + y 2 ) 2 = 4 r 2 ( x 2 + c 2 ) \left(r^{2}-a^{2}+c^{2}+x^{2}+y^{2}\right)^{2}=4r^{2}\left(x^{2}+c^{2}\right)
  2. r r
  3. a a
  4. c c

Category_of_relations.html

  1. A B A\otimes B
  2. A B A\Rightarrow B

Cation–pi_interaction.html

  1. r ion r_{\mathrm{ion}}
  2. Δ \Delta
  3. r ion r_{\mathrm{ion}}

Cauchy's_functional_equation.html

  1. f ( x + y ) = f ( x ) + f ( y ) . f(x+y)=f(x)+f(y).
  2. f ( x ) = c x f(x)=cx
  3. c c
  4. f f
  5. f f
  6. f f
  7. f f
  8. c c
  9. f ( c x ) c f ( x ) f(cx)\neq cf(x)
  10. f ( q ) = q f ( 1 ) , q f\left(q\right)=qf\left(1\right),q\in\mathbb{Q}
  11. f ( x + y ) = f ( x ) + f ( y ) f(x+y)=f(x)+f(y)
  12. x = 1 , y = 0 x=1,y=0
  13. f ( 0 ) = 0 \Rightarrow f(0)=0
  14. f ( x + x + + x ) = f ( α x ) f\left(x+x+...+x\right)=f\left(\alpha x\right)
  15. α f ( x ) = f ( α x ) , α + \alpha f\left(x\right)=f\left(\alpha x\right),\quad\alpha\in\mathbb{N^{+}}
  16. x x
  17. x α \frac{x}{\alpha}
  18. β α \frac{\beta}{\alpha}
  19. β f ( x α ) = β α f ( x ) , α + \beta f\left(\frac{x}{\alpha}\right)=\frac{\beta}{\alpha}f\left(x\right),\quad% \alpha\in\mathbb{N^{+}}
  20. f ( β α x ) = β α f ( x ) , α , β + f\left(\frac{\beta}{\alpha}x\right)=\frac{\beta}{\alpha}f\left(x\right),\quad% \alpha,\beta\in\mathbb{N^{+}}
  21. f ( q x ) = q f ( x ) , q , q > 0 \Rightarrow f\left(qx\right)=qf\left(x\right),\quad q\in\mathbb{Q},q>0
  22. f ( q ) = q f ( 1 ) , q , q > 0 \Rightarrow f\left(q\right)=qf\left(1\right),\quad q\in\mathbb{Q},q>0
  23. f ( - x ) = - f ( x ) \Rightarrow f(-x)=-f(x)
  24. - f ( q ) = - q f ( 1 ) , q , q > 0 -f\left(q\right)=-qf\left(1\right),\quad q\in\mathbb{Q},q>0
  25. f ( - q ) = - q f ( 1 ) , q , q > 0 \Rightarrow f\left(-q\right)=-qf\left(1\right),\quad q\in\mathbb{Q},q>0
  26. f ( q ) = q f ( 1 ) , q , q < 0. f\left(q\right)=qf\left(1\right),\quad q\in\mathbb{Q},q<0.\;\blacksquare
  27. y = f ( x ) y=f(x)
  28. 2 \mathbb{R}^{2}
  29. f ( q ) = q q f(q)=q\ \forall q\in\mathbb{Q}
  30. f ( α ) α f(\alpha)\neq\alpha
  31. α \alpha\in\mathbb{R}
  32. f ( α ) = α + δ , δ 0 f(\alpha)=\alpha+\delta,\delta\neq 0
  33. ( x , y ) (x,y)
  34. r r
  35. x , y , r , r > 0 , x y x,y,r\in\mathbb{Q},r>0,x\neq y
  36. β = y - x δ \beta=\frac{y-x}{\delta}
  37. b 0 b\neq 0
  38. β \beta
  39. | β - b | < r 2 | δ | \left|\beta-b\right|<\frac{r}{2\left|\delta\right|}
  40. a a
  41. α \alpha
  42. | α - a | < r 2 | b | \left|\alpha-a\right|<\frac{r}{2\left|b\right|}
  43. X = x + b ( α - a ) X=x+b(\alpha-a)
  44. Y = f ( X ) Y=f(X)
  45. Y = f ( x + b ( α - a ) ) Y=f(x+b(\alpha-a))
  46. = x + b f ( α ) - b f ( a ) =x+bf(\alpha)-bf(a)
  47. = y - δ β + b f ( α ) - b f ( a ) =y-\delta\beta+bf(\alpha)-bf(a)
  48. = y - δ β + b ( α + δ ) - b a =y-\delta\beta+b(\alpha+\delta)-ba
  49. = y + b ( α - a ) - δ ( β - b ) =y+b(\alpha-a)-\delta(\beta-b)
  50. ( X , Y ) (X,Y)
  51. α \alpha\mathbb{Q}
  52. \mathbb{Q}
  53. A \sub A\sub\mathbb{R}
  54. z z
  55. X = { x 1 , x n } \sub A X=\left\{x_{1},\dots x_{n}\right\}\sub A
  56. ( λ i ) \left(\lambda_{i}\right)
  57. \mathbb{Q}
  58. z = i = 1 n λ i x i z=\sum_{i=1}^{n}{\lambda_{i}x_{i}}
  59. x , x A x\mathbb{Q},x\in A
  60. f f
  61. f ( z ) = i = 1 n g ( x i ) λ i x i f(z)=\sum_{i=1}^{n}{g(x_{i})\lambda_{i}x_{i}}
  62. g : A g:A\rightarrow\mathbb{R}

Cauchy_matrix.html

  1. a i j = 1 x i - y j ; x i - y j 0 , 1 i m , 1 j n a_{ij}={\frac{1}{x_{i}-y_{j}}};\quad x_{i}-y_{j}\neq 0,\quad 1\leq i\leq m,% \quad 1\leq j\leq n
  2. x i x_{i}
  3. y j y_{j}
  4. \mathcal{F}
  5. ( x i ) (x_{i})
  6. ( y j ) (y_{j})
  7. x i - y j = i + j - 1. x_{i}-y_{j}=i+j-1.\;
  8. ( x i ) (x_{i})
  9. ( y j ) (y_{j})
  10. x i x_{i}
  11. y j y_{j}
  12. det 𝐀 = i = 2 n j = 1 i - 1 ( x i - x j ) ( y j - y i ) i = 1 n j = 1 n ( x i - y j ) \det\mathbf{A}={{\prod_{i=2}^{n}\prod_{j=1}^{i-1}(x_{i}-x_{j})(y_{j}-y_{i})}% \over{\prod_{i=1}^{n}\prod_{j=1}^{n}(x_{i}-y_{j})}}
  13. b i j = ( x j - y i ) A j ( y i ) B i ( x j ) b_{ij}=(x_{j}-y_{i})A_{j}(y_{i})B_{i}(x_{j})\,
  14. ( x i ) (x_{i})
  15. ( y j ) (y_{j})
  16. A i ( x ) = A ( x ) A ( x i ) ( x - x i ) and B i ( x ) = B ( x ) B ( y i ) ( x - y i ) , A_{i}(x)=\frac{A(x)}{A^{\prime}(x_{i})(x-x_{i})}\quad\,\text{and}\quad B_{i}(x% )=\frac{B(x)}{B^{\prime}(y_{i})(x-y_{i})},
  17. A ( x ) = i = 1 n ( x - x i ) and B ( x ) = i = 1 n ( x - y i ) . A(x)=\prod_{i=1}^{n}(x-x_{i})\quad\,\text{and}\quad B(x)=\prod_{i=1}^{n}(x-y_{% i}).
  18. C i j = r i s j x i - y j . C_{ij}=\frac{r_{i}s_{j}}{x_{i}-y_{j}}.
  19. 𝐗𝐂 - 𝐂𝐘 = r s T \mathbf{XC}-\mathbf{CY}=rs^{\mathrm{T}}
  20. r = s = ( 1 , 1 , , 1 ) r=s=(1,1,\ldots,1)
  21. O ( n log n ) O(n\log n)
  22. O ( n 2 ) O(n^{2})
  23. O ( n log 2 n ) O(n\log^{2}n)
  24. n n

Causal_model.html

  1. U , V , E \langle U,V,E\rangle

Cayley's_nodal_cubic_surface.html

  1. w x y + x y z + y z w + z w x = 0 wxy+xyz+yzw+zwx=0

Cellular_Potts_model.html

  1. H = \displaystyle H=

Center-of-momentum_frame.html

  1. v = v - V c , v^{\prime}=v-V_{c},
  2. V c = i m i v i i m i V_{c}=\frac{\sum_{i}m_{i}v_{i}}{\sum_{i}m_{i}}
  3. i p i = i m i v i = i m i ( v i - V c ) = i m i v i - i m i j m j v j j m j = i m i v i - j m j v j = 0. \sum_{i}p^{\prime}_{i}=\sum_{i}m_{i}v^{\prime}_{i}=\sum_{i}m_{i}(v_{i}-V_{c})=% \sum_{i}m_{i}v_{i}-\sum_{i}m_{i}\frac{\sum_{j}m_{j}v_{j}}{\sum_{j}m_{j}}=\sum_% {i}m_{i}v_{i}-\sum_{j}m_{j}v_{j}=0.
  4. m 0 = E 0 c 2 . m_{0}=\frac{E_{0}}{c^{2}}.
  5. m 0 = 2 ( E c 2 ) 2 - ( p c ) 2 m_{0}{}^{2}=\left(\frac{E}{c^{2}}\right)^{2}-\left(\frac{p}{c}\right)^{2}\,\!
  6. p = E c . p=\frac{E}{c}.\,\!
  7. u 1 = u 1 - V , u 2 = u 2 - V {u}_{1}^{\prime}={u}_{1}-{V},\quad{u}_{2}^{\prime}={u}_{2}-{V}
  8. d R d t = < m t p l > dd t ( m 1 r 1 + m 2 r 2 m 1 + m 2 ) = m 1 u 1 + m 2 u 2 m 1 + m 2 = V \begin{aligned}\displaystyle\frac{{\rm d}{R}}{{\rm d}t}&\displaystyle=\frac{<}% {m}tpl>{{\rm d}}{{\rm d}t}\left(\frac{m_{1}{r}_{1}+m_{2}{r}_{2}}{m_{1}+m_{2}}% \right)\\ &\displaystyle=\frac{m_{1}{u}_{1}+m_{2}{u}_{2}}{m_{1}+m_{2}}\\ &\displaystyle={V}\\ \end{aligned}\,\!
  9. m 1 u 1 + m 2 u 2 = s y m b o l 0 m_{1}{u}_{1}^{\prime}+m_{2}{u}_{2}^{\prime}=symbol{0}
  10. V = p 1 + p 2 m 1 + m 2 = m 1 u 1 + m 2 u 2 m 1 + m 2 {V}=\frac{{p}_{1}+{p}_{2}}{m_{1}+m_{2}}=\frac{m_{1}{u}_{1}+m_{2}{u}_{2}}{m_{1}% +m_{2}}\,\!
  11. p 1 + p 2 = m 1 u 1 + m 2 u 2 = s y m b o l 0 {p}_{1}^{\prime}+{p}_{2}^{\prime}=m_{1}{u}_{1}^{\prime}+m_{2}{u}_{2}^{\prime}=% symbol{0}
  12. p 1 = m 1 u 1 = m 1 ( u 1 - V ) = m 1 m 2 m 1 + m 2 ( u 1 - u 2 ) = - m 2 u 2 = - p 2 \begin{aligned}\displaystyle{p}_{1}^{\prime}&\displaystyle=m_{1}{u}_{1}^{% \prime}\\ &\displaystyle=m_{1}\left({u}_{1}-{V}\right)=\frac{m_{1}m_{2}}{m_{1}+m_{2}}% \left({u}_{1}-{u}_{2}\right)\\ &\displaystyle=-m_{2}{u}_{2}^{\prime}=-{p}_{2}^{\prime}\\ \end{aligned}\,\!
  13. Δ u = u 1 - u 2 \Delta{u}={u}_{1}-{u}_{2}
  14. μ = m 1 m 2 m 1 + m 2 \mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}\,\!
  15. p 1 = - p 2 = μ Δ u {p}_{1}^{\prime}=-{p}_{2}^{\prime}=\mu\Delta{u}\,\!
  16. d R d t = < m t p l > dd t ( m 1 r 1 + m 2 r 2 m 1 + m 2 ) = m 1 v 1 + m 2 v 2 m 1 + m 2 = V \begin{aligned}\displaystyle\frac{{\rm d}{R}}{{\rm d}t}&\displaystyle=\frac{<}% {m}tpl>{{\rm d}}{{\rm d}t}\left(\frac{m_{1}{r}_{1}+m_{2}{r}_{2}}{m_{1}+m_{2}}% \right)\\ &\displaystyle=\frac{m_{1}{v}_{1}+m_{2}{v}_{2}}{m_{1}+m_{2}}\\ &\displaystyle={V}\\ \end{aligned}\,\!
  17. m 1 v 1 + m 2 v 2 = s y m b o l 0 m_{1}{v}_{1}^{\prime}+m_{2}{v}_{2}^{\prime}=symbol{0}
  18. m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 = ( m 1 + m 2 ) V m_{1}{u}_{1}+m_{2}{u}_{2}=m_{1}{v}_{1}+m_{2}{v}_{2}=(m_{1}+m_{2}){V}
  19. m 1 u 1 = m 1 v 1 = m 1 V , m 2 u 2 = m 2 v 2 = m 2 V m_{1}{u}_{1}=m_{1}{v}_{1}=m_{1}{V},\quad m_{2}{u}_{2}=m_{2}{v}_{2}=m_{2}{V}
  20. P = p 1 + p 2 = ( m 1 + m 2 ) V = M V \begin{aligned}\displaystyle{P}&\displaystyle={p}_{1}+{p}_{2}\\ &\displaystyle=(m_{1}+m_{2}){V}\\ &\displaystyle=M{V}\end{aligned}\,\!
  21. p 1 = - p 2 = μ Δ v = μ Δ u {p}_{1}^{\prime}=-{p}_{2}^{\prime}=\mu\Delta{v}=\mu\Delta{u}\,\!
  22. Δ v = v 1 - v 2 = Δ u . \Delta{v}={v}_{1}-{v}_{2}=\Delta{u}.

Central_composite_design.html

  1. E = [ α 0 0 0 - α 0 0 0 0 α 0 0 0 - α 0 0 0 0 0 0 α 0 0 0 0 - α ] . E=\begin{bmatrix}\alpha&0&0&\cdots&\cdots&\cdots&0\\ {-\alpha}&0&0&\cdots&\cdots&\cdots&0\\ 0&\alpha&0&\cdots&\cdots&\cdots&0\\ 0&{-\alpha}&0&\cdots&\cdots&\cdots&0\\ \vdots&&&&&&\vdots\\ 0&0&0&0&\cdots&\cdots&\alpha\\ 0&0&0&0&\cdots&\cdots&{-\alpha}\\ \end{bmatrix}.
  2. d = [ F C E ] . d=\begin{bmatrix}F\\ C\\ E\end{bmatrix}.
  3. X = [ 1 d d ( 1 ) × d ( 2 ) d ( 1 ) × d ( 3 ) d ( k - 1 ) × d ( k ) d ( 1 ) 2 d ( 2 ) 2 d ( k ) 2 ] , X=\begin{bmatrix}1&d&d(1)\times d(2)&d(1)\times d(3)&\cdots&d(k-1)\times d(k)&% d(1)^{2}&d(2)^{2}&\cdots&d(k)^{2}\end{bmatrix},
  4. α = ( Q × F / 4 ) 1 / 4 \alpha=(Q\times F/4)^{1/4}\,\!
  5. Q = ( F + T - F ) 2 Q=(\sqrt{F+T}-\sqrt{F})^{2}

Centrosymmetric_matrix.html

  1. [ a b b a ] . \begin{bmatrix}a&b\\ b&a\end{bmatrix}.
  2. [ a b c d e d c b a ] . \begin{bmatrix}a&b&c\\ d&e&d\\ c&b&a\end{bmatrix}.

Cerebellar_model_articulation_controller.html

  1. f ( x 1 x n ) f(x_{1}...x_{n})
  2. n n

Cerebroside-sulfatase.html

  1. \rightleftharpoons

Cevian.html

  1. d d
  2. b 2 m + c 2 n = a ( d 2 + m n ) . \,b^{2}m+c^{2}n=a(d^{2}+mn).
  3. m ( b 2 + c 2 ) = a ( d 2 + m 2 ) \,m(b^{2}+c^{2})=a(d^{2}+m^{2})
  4. 2 ( b 2 + c 2 ) = 4 d 2 + a 2 \,2(b^{2}+c^{2})=4d^{2}+a^{2}
  5. a = 2 m . \,a=2m.
  6. d = 2 b 2 + 2 c 2 - a 2 4 . d=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}.
  7. ( b + c ) 2 = a 2 ( d 2 m n + 1 ) , \,(b+c)^{2}=a^{2}\left(\frac{d^{2}}{mn}+1\right),
  8. d 2 + m n = b c d^{2}+mn=bc
  9. d = 2 b c s ( s - a ) b + c d=\frac{2\sqrt{bcs(s-a)}}{b+c}
  10. d 2 = b 2 - n 2 = c 2 - m 2 \,d^{2}=b^{2}-n^{2}=c^{2}-m^{2}
  11. d = 2 s ( s - a ) ( s - b ) ( s - c ) a , d=\frac{2\sqrt{s(s-a)(s-b)(s-c)}}{a},
  12. A F F B B D D C C E E A = 1 ; \frac{AF}{FB}\cdot\frac{BD}{DC}\cdot\frac{CE}{EA}=1;
  13. A O O D = A E E C + A F F B ; \frac{AO}{OD}=\frac{AE}{EC}+\frac{AF}{FB};
  14. O D A D + O E B E + O F C F = 1 ; \frac{OD}{AD}+\frac{OE}{BE}+\frac{OF}{CF}=1;
  15. A O A D + B O B E + C O C F = 2. \frac{AO}{AD}+\frac{BO}{BE}+\frac{CO}{CF}=2.

Characteristic_function_(convex_analysis).html

  1. X X
  2. A A
  3. X X
  4. A A
  5. χ A : X { + } \chi_{A}:X\to\mathbb{R}\cup\{+\infty\}
  6. χ A ( x ) := { 0 , x A ; + , x A . \chi_{A}(x):=\begin{cases}0,&x\in A;\\ +\infty,&x\not\in A.\end{cases}
  7. 𝟏 A : X \mathbf{1}_{A}:X\to\mathbb{R}
  8. 𝟏 A ( x ) := { 1 , x A ; 0 , x A . \mathbf{1}_{A}(x):=\begin{cases}1,&x\in A;\\ 0,&x\not\in A.\end{cases}
  9. a { + } a\in\mathbb{R}\cup\{+\infty\}
  10. a + ( + ) = + a+(+\infty)=+\infty
  11. a ( + ) = + a(+\infty)=+\infty
  12. 1 0 = + \frac{1}{0}=+\infty
  13. 1 + = 0 \frac{1}{+\infty}=0
  14. 𝟏 A ( x ) = 1 1 + χ A ( x ) \mathbf{1}_{A}(x)=\frac{1}{1+\chi_{A}(x)}
  15. χ A ( x ) = ( + ) ( 1 - 𝟏 A ( x ) ) . \chi_{A}(x)=(+\infty)\left(1-\mathbf{1}_{A}(x)\right).

Characteristic_mode_analysis.html

  1. Z Z
  2. X ( J n ) = λ n R ( J n ) X(J_{n})=\lambda_{n}R(J_{n})
  3. R R
  4. X X
  5. Z Z
  6. λ n \lambda_{n}
  7. J n J_{n}
  8. R R
  9. X X
  10. λ n \lambda_{n}
  11. J n J_{n}
  12. m n m\neq n
  13. J m , R J n = 0 \left\langle J_{m},RJ_{n}\right\rangle=0
  14. J m , X J n = 0 \left\langle J_{m},XJ_{n}\right\rangle=0
  15. J m , Z J n = 0 \left\langle J_{m},ZJ_{n}\right\rangle=0

Characteristic_state_function.html

  1. P = exp ( - β Q ) P=\exp(-\beta Q)
  2. P = exp ( + β Q ) P=\exp(+\beta Q)
  3. Ω ( U , V , N ) = e β T S \Omega(U,V,N)=e^{\beta TS}\;\,
  4. T S TS
  5. Z ( T , V , N ) = e - β A Z(T,V,N)=e^{-\beta A}\,\;
  6. A A
  7. 𝒵 ( T , V , μ ) = e - β Φ \mathcal{Z}(T,V,\mu)=e^{-\beta\Phi}\,\;
  8. Φ \Phi
  9. Δ ( N , T , P ) = e - β G \Delta(N,T,P)=e^{-\beta G}\;\,
  10. G G

Charge-carrier_density.html

  1. V V
  2. N N
  3. N = V n ( 𝐫 ) d V N=\int_{V}n(\mathbf{r})\,\mathrm{d}V
  4. n ( 𝐫 ) n(\mathbf{r})
  5. n 0 n_{0}
  6. N = V n 0 N=V\cdot n_{0}

Charge_(physics).html

  1. 2 2 ¯ = 3 1. 2\otimes\overline{2}=3\oplus 1.

Charged_current.html

  1. 𝔐 CC J μ ( CC ) ( e - ν e ) J ( CC ) μ ( ν e e - ) \mathfrak{M}^{\mathrm{CC}}\propto J_{\mu}^{\mathrm{(CC)}}(\mathrm{e^{-}}\to\nu% _{\mathrm{e}})\;J^{\mathrm{(CC)}\mu}(\nu_{\mathrm{e}}\to\mathrm{e^{-}})
  2. J ( CC ) μ ( f f ) = u ¯ f γ μ 1 2 ( 1 - γ 5 ) u f . J^{\mathrm{(CC)\mu}}(f\to f^{\prime})=\bar{u}_{f^{\prime}}\gamma^{\mu}\frac{1}% {2}\left(1-\gamma^{5}\right)u_{f}.

Charlieplexing.html

  1. n + log 2 n n+\lceil\log_{2}n\rceil
  2. 1 + 1 + 4 L 2 \left\lceil\frac{1+\sqrt{1+4\cdot L}}{2}\right\rceil

Chebyshev_function.html

  1. ψ ( x ) = p k x log p = n x Λ ( n ) = p x log p x log p , \psi(x)=\sum_{p^{k}\leq x}\log p=\sum_{n\leq x}\Lambda(n)=\sum_{p\leq x}% \lfloor\log_{p}x\rfloor\log p,
  2. Λ \Lambda
  3. ψ ( x ) = p x k log p \psi(x)=\sum_{p\leq x}k\log p
  4. ψ ( x ) = n = 1 ϑ ( x 1 / n ) . \psi(x)=\sum_{n=1}^{\infty}\vartheta\left(x^{1/n}\right).
  5. ϑ ( x 1 / n ) = 0 for n > log 2 x = log x log 2 , . \vartheta\left(x^{1/n}\right)=0\,\text{ for }n>\log_{2}x\ =\frac{\log x}{\log 2% },.
  6. lcm ( 1 , 2 , , n ) = e ψ ( n ) . \operatorname{lcm}(1,2,\dots,n)=e^{\psi(n)}.
  7. lcm ( 1 , 2 , , n ) \operatorname{lcm}(1,2,\dots,n)
  8. ϑ ( p k ) k ( ln k + ln ln k - 1 + ln ln k - 2.050735 ln k ) \vartheta(p_{k})\geq k\left(\ln k+\ln\ln k-1+\frac{\ln\ln k-2.050735}{\ln k}\right)
  9. k 10 11 , k\geq 10^{11},
  10. ϑ ( p k ) k ( ln k + ln ln k - 1 + ln ln k - 2 ln k ) \vartheta(p_{k})\leq k\left(\ln k+\ln\ln k-1+\frac{\ln\ln k-2}{\ln k}\right)
  11. | ϑ ( x ) - x | 0.006788 x ln x |\vartheta(x)-x|\leq 0.006788\frac{x}{\ln x}
  12. | ψ ( x ) - x | 0.006409 x ln x |\psi(x)-x|\leq 0.006409\frac{x}{\ln x}
  13. 0.9999 x < ψ ( x ) - ϑ ( x ) < 1.00007 x + 1.78 x 3 0.9999\sqrt{x}<\psi(x)-\vartheta(x)<1.00007\sqrt{x}+1.78\sqrt[3]{x}
  14. x 121. x\geq 121.
  15. | ϑ ( x ) - x | = O ( x 1 / 2 + ε ) |\vartheta(x)-x|=O(x^{1/2+\varepsilon})
  16. | ψ ( x ) - x | = O ( x 1 / 2 + ε ) |\psi(x)-x|=O(x^{1/2+\varepsilon})
  17. ε > 0. \varepsilon>0.
  18. ϑ ( x ) \vartheta(x)
  19. ψ ( x ) \psi(x)
  20. ϑ ( x ) < 1.01624 x \vartheta(x)<1.01624x
  21. ψ ( x ) < 1.03883 x \psi(x)<1.03883x
  22. x > 0. x>0.
  23. ψ ( x ) \psi(x)
  24. ψ 0 ( x ) = x - ρ x ρ ρ - ζ ( 0 ) ζ ( 0 ) - 1 2 log ( 1 - x - 2 ) . \psi_{0}(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\frac{\zeta^{\prime}(0)}{\zeta(% 0)}-\frac{1}{2}\log(1-x^{-2}).
  25. ρ \rho
  26. ψ 0 ( x ) = 1 2 ( n x Λ ( n ) + n < x Λ ( n ) ) = { ψ ( x ) - 1 2 Λ ( x ) x = 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , ψ ( x ) otherwise. \psi_{0}(x)=\frac{1}{2}\left(\sum_{n\leq x}\Lambda(n)+\sum_{n<x}\Lambda(n)% \right)=\begin{cases}\psi(x)-\frac{1}{2}\Lambda(x)&x=2,3,4,5,7,8,9,11,13,16,% \dots\\ \psi(x)&\mbox{otherwise.}\end{cases}
  27. x ω / ω x^{\omega}/{\omega}
  28. ω = - 2 , - 4 , - 6 , \omega=-2,-4,-6,\ldots
  29. k = 1 x - 2 k - 2 k = 1 2 log ( 1 - x - 2 ) . \sum_{k=1}^{\infty}\frac{x^{-2k}}{-2k}=\frac{1}{2}\log(1-x^{-2}).
  30. ψ ( x ) - x < - K x \psi(x)-x<-K\sqrt{x}
  31. ψ ( x ) - x > K x . \psi(x)-x>K\sqrt{x}.
  32. ψ ( x ) - x o ( x ) . \psi(x)-x\neq o\left(\sqrt{x}\right).
  33. ψ ( x ) - x o ( x log log log x ) . \psi(x)-x\neq o\left(\sqrt{x}\log\log\log x\right).
  34. ϑ ( x ) = p x log p = log p x p = log ( x # ) . \vartheta(x)=\sum_{p\leq x}\log p=\log\prod_{p\leq x}p=\log(x\#).
  35. Π ( x ) = n x Λ ( n ) log n . \Pi(x)=\sum_{n\leq x}\frac{\Lambda(n)}{\log n}.
  36. Π ( x ) = n x Λ ( n ) n x d t t log 2 t + 1 log x n x Λ ( n ) = 2 x ψ ( t ) d t t log 2 t + ψ ( x ) log x . \Pi(x)=\sum_{n\leq x}\Lambda(n)\int_{n}^{x}\frac{dt}{t\log^{2}t}+\frac{1}{\log x% }\sum_{n\leq x}\Lambda(n)=\int_{2}^{x}\frac{\psi(t)\,dt}{t\log^{2}t}+\frac{% \psi(x)}{\log x}.
  37. Π \Pi
  38. π \pi
  39. Π ( x ) = π ( x ) + 1 2 π ( x 1 / 2 ) + 1 3 π ( x 1 / 3 ) + . \Pi(x)=\pi(x)+\frac{1}{2}\pi(x^{1/2})+\frac{1}{3}\pi(x^{1/3})+\cdots.
  40. π ( x ) x \pi(x)\leq x
  41. π ( x ) = Π ( x ) + O ( x ) . \pi(x)=\Pi(x)+O(\sqrt{x}).
  42. | x ρ | = x |x^{\rho}|=\sqrt{x}
  43. ρ x ρ ρ = O ( x log 2 x ) . \sum_{\rho}\frac{x^{\rho}}{\rho}=O(\sqrt{x}\log^{2}x).
  44. π ( x ) = li ( x ) + O ( x log x ) . \pi(x)=\operatorname{li}(x)+O(\sqrt{x}\log x).
  45. ζ ( 1 / 2 + i H ^ ) | n ζ ( 1 / 2 + i E n ) = 0 , \zeta(1/2+i\hat{H})|n\geq\zeta(1/2+iE_{n})=0,\,
  46. n e i u E n = Z ( u ) = e u / 2 - e - u / 2 d ψ 0 d u - e u / 2 e 3 u - e u = Tr ( e i u H ^ ) , \sum_{n}e^{iuE_{n}}=Z(u)=e^{u/2}-e^{-u/2}\frac{d\psi_{0}}{du}-\frac{e^{u/2}}{e% ^{3u}-e^{u}}=\operatorname{Tr}(e^{iu\hat{H}}),
  47. e i u H ^ e^{iu\hat{H}}
  48. ρ = 1 / 2 + i E ( n ) . \rho=1/2+iE(n).
  49. Z ( u ) u 1 / 2 π - e i ( u V ( x ) + π / 4 ) d x \frac{Z(u)u^{1/2}}{\sqrt{\pi}}\sim\int_{-\infty}^{\infty}e^{i(uV(x)+\pi/4)}\,dx
  50. V - 1 ( x ) ( 4 π ) d 1 / 2 N ( x ) d x 1 / 2 V^{-1}(x)\approx\sqrt{(}4\pi)\frac{d^{1/2}N(x)}{dx^{1/2}}
  51. π N ( E ) = A r g ξ ( 1 / 2 + i E ) \pi N(E)=Arg\xi(1/2+iE)
  52. ψ 1 ( x ) = 0 x ψ ( t ) d t . \psi_{1}(x)=\int_{0}^{x}\psi(t)\,dt.
  53. ψ 1 ( x ) x 2 2 . \psi_{1}(x)\sim\frac{x^{2}}{2}.
  54. J [ f ] = 0 f ( s ) ζ ( s + c ) ζ ( s + c ) ( s + c ) d s - 0 0 e - s t f ( s ) f ( t ) d s d t , J[f]=\int_{0}^{\infty}\frac{f(s)\zeta^{\prime}(s+c)}{\zeta(s+c)(s+c)}\,ds-\int% _{0}^{\infty}\!\!\!\int_{0}^{\infty}e^{-st}f(s)f(t)\,ds\,dt,
  55. f ( t ) = ψ ( e t ) e - c t , f(t)=\psi(e^{t})e^{-ct},\,

Chebyshev_rational_functions.html

  1. R n ( x ) = def T n ( x - 1 x + 1 ) R_{n}(x)\ \stackrel{\mathrm{def}}{=}\ T_{n}\left(\frac{x-1}{x+1}\right)
  2. T n ( x ) T_{n}(x)
  3. R n + 1 ( x ) = 2 x - 1 x + 1 R n ( x ) - R n - 1 ( x ) for n 1 R_{n+1}(x)=2\,\frac{x-1}{x+1}R_{n}(x)-R_{n-1}(x)\quad\mathrm{for\,n\geq 1}
  4. ( x + 1 ) 2 R n ( x ) = 1 n + 1 d d x R n + 1 ( x ) - 1 n - 1 d d x R n - 1 ( x ) for n 2 (x+1)^{2}R_{n}(x)=\frac{1}{n+1}\frac{d}{dx}\,R_{n+1}(x)-\frac{1}{n-1}\frac{d}{% dx}\,R_{n-1}(x)\quad\mathrm{for\,n\geq 2}
  5. ( x + 1 ) 2 x d 2 d x 2 R n ( x ) + ( 3 x + 1 ) ( x + 1 ) 2 d d x R n ( x ) + n 2 R n ( x ) = 0 (x+1)^{2}x\frac{d^{2}}{dx^{2}}\,R_{n}(x)+\frac{(3x+1)(x+1)}{2}\frac{d}{dx}\,R_% {n}(x)+n^{2}R_{n}(x)=0
  6. ω ( x ) = def 1 ( x + 1 ) x \omega(x)\ \stackrel{\mathrm{def}}{=}\ \frac{1}{(x+1)\sqrt{x}}
  7. 0 R m ( x ) R n ( x ) ω ( x ) d x = π c n 2 δ n m \int_{0}^{\infty}R_{m}(x)\,R_{n}(x)\,\omega(x)\,dx=\frac{\pi c_{n}}{2}\delta_{nm}
  8. c n c_{n}
  9. c n c_{n}
  10. n 1 n\geq 1
  11. δ n m \delta_{nm}
  12. f ( x ) L ω 2 f(x)\in L_{\omega}^{2}
  13. f ( x ) f(x)
  14. f ( x ) = n = 0 F n R n ( x ) f(x)=\sum_{n=0}^{\infty}F_{n}R_{n}(x)
  15. F n = 2 c n π 0 f ( x ) R n ( x ) ω ( x ) d x . F_{n}=\frac{2}{c_{n}\pi}\int_{0}^{\infty}f(x)R_{n}(x)\omega(x)\,dx.
  16. R 0 ( x ) = 1 R_{0}(x)=1\,
  17. R 1 ( x ) = x - 1 x + 1 R_{1}(x)=\frac{x-1}{x+1}\,
  18. R 2 ( x ) = x 2 - 6 x + 1 ( x + 1 ) 2 R_{2}(x)=\frac{x^{2}-6x+1}{(x+1)^{2}}\,
  19. R 3 ( x ) = x 3 - 15 x 2 + 15 x - 1 ( x + 1 ) 3 R_{3}(x)=\frac{x^{3}-15x^{2}+15x-1}{(x+1)^{3}}\,
  20. R 4 ( x ) = x 4 - 28 x 3 + 70 x 2 - 28 x + 1 ( x + 1 ) 4 R_{4}(x)=\frac{x^{4}-28x^{3}+70x^{2}-28x+1}{(x+1)^{4}}\,
  21. R n ( x ) = 1 ( x + 1 ) n m = 0 n ( - 1 ) m ( 2 n 2 m ) x n - m R_{n}(x)=\frac{1}{(x+1)^{n}}\sum_{m=0}^{n}(-1)^{m}{2n\choose 2m}x^{n-m}\,
  22. R n ( x ) = m = 0 n ( m ! ) 2 ( 2 m ) ! ( n + m - 1 m ) ( n m ) ( - 4 ) m ( x + 1 ) m R_{n}(x)=\sum_{m=0}^{n}\frac{(m!)^{2}}{(2m)!}{n+m-1\choose m}{n\choose m}\frac% {(-4)^{m}}{(x+1)^{m}}

Cheeger_constant_(graph_theory).html

  1. G G
  2. V ( G ) V(G)
  3. E ( G ) E(G)
  4. A V ( G ) A⊆V(G)
  5. A ∂A
  6. A A
  7. A A
  8. A := { ( x , y ) E ( G ) : x A , y V ( G ) A } . \partial A:=\{(x,y)\in E(G)\ :\ x\in A,y\in V(G)\setminus A\}.
  9. ( x , y ) (x,y)
  10. ( y , x ) (y,x)
  11. G G
  12. h ( G ) h(G)
  13. h ( G ) := min { | A | | A | : A V ( G ) , 0 < | A | 1 2 | V ( G ) | } . h(G):=\min\left\{\frac{|\partial A|}{|A|}\ :\ A\subseteq V(G),0<|A|\leq\tfrac{% 1}{2}|V(G)|\right\}.
  14. G G
  15. | V ( G ) | |V(G)|
  16. N 3 N≥3
  17. 1 , 2 , , N 1,2,...,N
  18. V ( G N ) \displaystyle V(G_{N})
  19. A A
  20. N 2 \left\lfloor\tfrac{N}{2}\right\rfloor
  21. A = { 1 , 2 , , N 2 } . A=\left\{1,2,\cdots,\left\lfloor\tfrac{N}{2}\right\rfloor\right\}.
  22. A = { ( N 2 , N 2 + 1 ) , ( N , 1 ) } , \partial A=\left\{\left(\left\lfloor\tfrac{N}{2}\right\rfloor,\left\lfloor% \tfrac{N}{2}\right\rfloor+1\right),(N,1)\right\},
  23. | A | | A | = 2 N 2 0 as N . \frac{|\partial A|}{|A|}=\frac{2}{\left\lfloor\tfrac{N}{2}\right\rfloor}\to 0% \mbox{ as }~{}N\to\infty.
  24. h ( G < s u b > N ) h(G<sub>N)

Cheerios_effect.html

  1. ρ s \rho_{s}
  2. R R
  3. \ell
  4. ρ \rho
  5. 2 π γ R B 5 / 2 Σ 2 K 1 ( L c ) 2\pi\gamma RB^{5/2}\Sigma^{2}K_{1}\left(\frac{\ell}{L_{c}}\right)
  6. γ \gamma
  7. K 1 K_{1}
  8. B = ρ g R 2 / γ B=\rho gR^{2}/\gamma
  9. Σ = 2 ρ s / ρ - 1 3 - cos θ 2 + cos 3 θ 6 \Sigma=\frac{2\rho_{s}/\rho-1}{3}-\frac{\cos\theta}{2}+\frac{\cos^{3}\theta}{6}
  10. θ \theta
  11. L C = R / B L_{C}=R/\sqrt{B}

Chen_model.html

  1. d r t = ( θ t - α t ) d t + r t σ t d W t , dr_{t}=(\theta_{t}-\alpha_{t})\,dt+\sqrt{r_{t}}\,\sigma_{t}\,dW_{t},
  2. d α t = ( ζ t - α t ) d t + α t σ t d W t , d\alpha_{t}=(\zeta_{t}-\alpha_{t})\,dt+\sqrt{\alpha_{t}}\,\sigma_{t}\,dW_{t},
  3. d σ t = ( β t - σ t ) d t + σ t η t d W t . d\sigma_{t}=(\beta_{t}-\sigma_{t})\,dt+\sqrt{\sigma_{t}}\,\eta_{t}\,dW_{t}.

Chevalley_basis.html

  1. ± α i \pm\alpha_{i}
  2. [ H α i , H α j ] = 0 [H_{\alpha_{i}},H_{\alpha_{j}}]=0
  3. [ H α i , E α j ] = α j ( H α i ) E α j [H_{\alpha_{i}},E_{\alpha_{j}}]=\alpha_{j}(H_{\alpha_{i}})E_{\alpha_{j}}
  4. [ E - α i , E α i ] = H α i [E_{-\alpha_{i}},E_{\alpha_{i}}]=H_{\alpha_{i}}
  5. [ E β , E γ ] = ± ( p + 1 ) E β + γ [E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}
  6. p p
  7. γ - p β \gamma-p\beta
  8. E β + γ = 0 E_{\beta+\gamma}=0
  9. β + γ \beta+\gamma
  10. β γ \beta\prec\gamma
  11. β + α γ + α \beta+\alpha\prec\gamma+\alpha
  12. ( β , γ ) (\beta,\gamma)
  13. β \beta
  14. β 0 \beta_{0}
  15. ( β 0 , γ 0 ) (\beta_{0},\gamma_{0})
  16. β 0 + γ 0 = β + γ \beta_{0}+\gamma_{0}=\beta+\gamma
  17. ( β , γ ) (\beta,\gamma)

Choi's_theorem_on_completely_positive_maps.html

  1. Φ : 𝐂 < s u p > n × n 𝐂 m × m Φ:\mathbf{C}<sup>n×n→\mathbf{C}^{m×m}
  2. Φ Φ
  3. n n
  4. C Φ = ( I n Φ ) ( i j E i j E i j ) = i j E i j Φ ( E i j ) n m × n m C_{\Phi}=\left(I_{n}\otimes\Phi\right)\left(\sum_{ij}E_{ij}\otimes E_{ij}% \right)=\sum_{ij}E_{ij}\otimes\Phi(E_{ij})\in\mathbb{C}^{nm\times nm}
  5. E < s u b > i j 𝐂 n × n E<sub>ij∈\mathbf{C}^{n×n}
  6. Φ Φ
  7. E = i j E i j E i j , E=\sum_{ij}E_{ij}\otimes E_{ij},
  8. n m × n m n m ( n m ) * n m ( n m ) * n ( n ) * m ( m ) * n × n m × m . \mathbb{C}^{nm\times nm}\cong\mathbb{C}^{nm}\otimes(\mathbb{C}^{nm})^{*}\cong% \mathbb{C}^{n}\otimes\mathbb{C}^{m}\otimes(\mathbb{C}^{n}\otimes\mathbb{C}^{m}% )^{*}\cong\mathbb{C}^{n}\otimes(\mathbb{C}^{n})^{*}\otimes\mathbb{C}^{m}% \otimes(\mathbb{C}^{m})^{*}\cong\mathbb{C}^{n\times n}\otimes\mathbb{C}^{m% \times m}.
  9. C Φ = i = 1 n m λ i v i v i * , C_{\Phi}=\sum_{i=1}^{nm}\lambda_{i}v_{i}v_{i}^{*},
  10. v i v_{i}
  11. λ i \lambda_{i}
  12. v i v_{i}
  13. C Φ = i = 1 n m v i v i * . \;C_{\Phi}=\sum_{i=1}^{nm}v_{i}v_{i}^{*}.
  14. i = 1 n m \textstyle\oplus_{i=1}^{n}\mathbb{C}^{m}
  15. n m n m \textstyle\mathbb{C}^{nm}\cong\mathbb{C}^{n}\otimes\mathbb{C}^{m}
  16. Φ ( E k l ) = P k C Φ P l * = i = 1 n m P k v i ( P l v i ) * . \;\Phi(E_{kl})=P_{k}\cdot C_{\Phi}\cdot P_{l}^{*}=\sum_{i=1}^{nm}P_{k}v_{i}(P_% {l}v_{i})^{*}.
  17. V i e k = P k v i , \;V_{i}e_{k}=P_{k}v_{i},
  18. Φ ( E k l ) = i = 1 n m P k v i ( P l v i ) * = i = 1 n m V i e k e l * V i * = i = 1 n m V i E k l V i * . \Phi(E_{kl})=\sum_{i=1}^{nm}P_{k}v_{i}(P_{l}v_{i})^{*}=\sum_{i=1}^{nm}V_{i}e_{% k}e_{l}^{*}V_{i}^{*}=\sum_{i=1}^{nm}V_{i}E_{kl}V_{i}^{*}.
  19. Φ ( A ) = i = 1 n m V i A V i * \Phi(A)=\sum_{i=1}^{nm}V_{i}AV_{i}^{*}
  20. B * = [ b 1 , , b n m ] , B^{*}=[b_{1},\ldots,b_{nm}],
  21. C Φ = i = 1 n m b i b i * . C_{\Phi}=\sum_{i=1}^{nm}b_{i}b_{i}^{*}.
  22. { U i j } i j n m 2 × n m 2 such that A i = i = 1 U i j B j . \{U_{ij}\}_{ij}\in\mathbb{C}^{nm^{2}\times nm^{2}}\quad\,\text{such that}\quad A% _{i}=\sum_{i=1}U_{ij}B_{j}.
  23. A i = i = 1 u i j B j . A_{i}=\sum_{i=1}u_{ij}B_{j}.
  24. Φ ( A ) = i V i A T V i * . \Phi(A)=\sum_{i}V_{i}A^{T}V_{i}^{*}.
  25. Φ ( A ) = i = 1 n m λ i V i A V i * \Phi(A)=\sum_{i=1}^{nm}\lambda_{i}V_{i}AV_{i}^{*}

Chronaxie.html

  1. Q ( d ) = I d = b ( d + c ) Q(d)=Id=b(d+c)
  2. I ( d ) = b / ( 1 - e - d / τ ) I(d)=b/(1-e^{-d/\tau})
  3. τ = R C \tau=RC
  4. C d v d t + v R = I , C\frac{dv}{dt}+\frac{v}{R}=I,
  5. v V - V r e s t . v\equiv V-V_{rest}.
  6. d τ d\ll\tau
  7. I ( d ) b τ / d I(d)\approx b\tau/d

Chvorinov's_rule.html

  1. t = B ( V A ) n t=B\left(\frac{V}{A}\right)^{n}
  2. s / m 2 s/m^{2}
  3. B = [ ρ m L ( T m - T o ) ] 2 [ π 4 k ρ c ] [ 1 + ( c m Δ T s L ) 2 ] B=\left[\frac{\rho_{m}L}{\left(T_{m}-T_{o}\right)}\right]^{2}\left[\frac{\pi}{% 4k\rho c}\right]\left[1+\left(\frac{c_{m}\Delta T_{s}}{L}\right)^{2}\right]

Cichoń's_diagram.html

  1. 1 \aleph_{1}
  2. 2 0 2^{\aleph_{0}}
  3. add ( I ) = min { | 𝒜 | : 𝒜 I 𝒜 I } {\rm add}(I)=\min\{|{\mathcal{A}}|:{\mathcal{A}}\subseteq I\wedge\bigcup{% \mathcal{A}}\notin I\big\}
  4. 0 \aleph_{0}
  5. 1 \aleph_{1}
  6. cov ( I ) = min { | 𝒜 | : 𝒜 I 𝒜 = X } {\rm cov}(I)=\min\{|{\mathcal{A}}|:{\mathcal{A}}\subseteq I\wedge\bigcup{% \mathcal{A}}=X\big\}
  7. non ( I ) = min { | A | : A X A I } {\rm non}(I)=\min\{|A|:A\subseteq X\ \wedge\ A\notin I\big\}
  8. unif ( I ) {\rm unif}(I)
  9. cof ( I ) = min { | | : I ( A I ) ( B ) ( A B ) } . {\rm cof}(I)=\min\{|{\mathcal{B}}|:{\mathcal{B}}\subseteq I\wedge(\forall A\in I% )(\exists B\in{\mathcal{B}})(A\subseteq B)\big\}.
  10. 𝔟 {\mathfrak{b}}
  11. 𝔡 {\mathfrak{d}}
  12. 𝔟 = min { | F | : F ( g ) ( f F ) ( n ) ( g ( n ) < f ( n ) ) } {\mathfrak{b}}=\min\big\{|F|:F\subseteq{\mathbb{N}}^{\mathbb{N}}\ \wedge\ (% \forall g\in{\mathbb{N}}^{\mathbb{N}})(\exists f\in F)(\exists^{\infty}n\in{% \mathbb{N}})(g(n)<f(n))\big\}
  13. 𝔡 = min { | F | : F ( g ) ( f F ) ( n ) ( g ( n ) < f ( n ) ) } {\mathfrak{d}}=\min\big\{|F|:F\subseteq{\mathbb{N}}^{\mathbb{N}}\ \wedge\ (% \forall g\in{\mathbb{N}}^{\mathbb{N}})(\exists f\in F)(\forall^{\infty}n\in{% \mathbb{N}})(g(n)<f(n))\big\}
  14. n \exists^{\infty}n\in{\mathbb{N}}
  15. n \forall^{\infty}n\in{\mathbb{N}}
  16. 𝒦 {\mathcal{K}}
  17. {\mathcal{L}}
  18. a a
  19. b b
  20. a b a≤b
  21. cov ( ) {\rm cov}({\mathcal{L}})
  22. \longrightarrow
  23. non ( 𝒦 ) {\rm non}({\mathcal{K}})
  24. \longrightarrow
  25. cof ( 𝒦 ) {\rm cof}({\mathcal{K}})
  26. \longrightarrow
  27. cof ( ) {\rm cof}({\mathcal{L}})
  28. \longrightarrow
  29. 2 0 2^{\aleph_{0}}
  30. \Bigg\uparrow
  31. \uparrow
  32. \uparrow
  33. \Bigg\uparrow
  34. 𝔟 {\mathfrak{b}}
  35. \longrightarrow
  36. 𝔡 {\mathfrak{d}}
  37. \uparrow
  38. \uparrow
  39. 1 \aleph_{1}
  40. \longrightarrow
  41. add ( ) {\rm add}({\mathcal{L}})
  42. \longrightarrow
  43. add ( 𝒦 ) {\rm add}({\mathcal{K}})
  44. \longrightarrow
  45. cov ( 𝒦 ) {\rm cov}({\mathcal{K}})
  46. \longrightarrow
  47. non ( ) {\rm non}({\mathcal{L}})
  48. add ( 𝒦 ) = min { cov ( 𝒦 ) , 𝔟 } {\rm add}({\mathcal{K}})=\min\{{\rm cov}({\mathcal{K}}),{\mathfrak{b}}\}
  49. cof ( 𝒦 ) = max { non ( 𝒦 ) , 𝔡 } {\rm cof}({\mathcal{K}})=\max\{{\rm non}({\mathcal{K}}),{\mathfrak{d}}\}
  50. 1 \aleph_{1}
  51. 2 \aleph_{2}
  52. 2 \aleph_{2}
  53. 1 \aleph_{1}
  54. cov ( 𝒦 ) non ( ) {\rm cov}({\mathcal{K}})\leq{\rm non}({\mathcal{L}})
  55. cov ( ) non ( 𝒦 ) {\rm cov}({\mathcal{L}})\leq{\rm non}({\mathcal{K}})
  56. 2 0 2^{\aleph_{0}}
  57. 1 \aleph_{1}
  58. 1 \aleph_{1}
  59. 2 0 2^{\aleph_{0}}

Circuit_complexity.html

  1. C 1 , C 2 , C_{1},C_{2},\ldots
  2. n n
  3. n n
  4. n n
  5. f f
  6. f f
  7. f f
  8. f f
  9. C 1 , C 2 , C_{1},C_{2},\ldots
  10. C n C_{n}
  11. n n
  12. 1 1
  13. 0
  14. n n
  15. C n C_{n}
  16. A A
  17. t : t:\mathbb{N}\to\mathbb{N}
  18. n n
  19. C n C_{n}
  20. A A
  21. C 1 , C 2 , C_{1},C_{2},\dots
  22. C n C_{n}
  23. C n C_{n}
  24. { C n : n } \{C_{n}:n\in\mathbb{N}\}
  25. n n\in\mathbb{N}
  26. C n C_{n}
  27. 1 n 1^{n}
  28. { C n : n } \{C_{n}:n\in\mathbb{N}\}
  29. n n\in\mathbb{N}
  30. C n C_{n}
  31. 1 n 1^{n}
  32. ( n 2 ) {n\choose 2}
  33. f k : { 0 , 1 } < m t p l > ( n 2 ) { 0 , 1 } f_{k}:\{0,1\}^{<}mtpl>{{n\choose 2}}\to\{0,1\}
  34. f k f_{k}
  35. Ω ( n k / 4 ) \Omega(n^{k/4})
  36. n k / 4 + O ( 1 ) n^{k/4+O(1)}
  37. f k f_{k}
  38. 𝖭𝖤𝖷𝖯 𝖠𝖢𝖢 0 \mathsf{NEXP}\not\subseteq\mathsf{ACC}^{0}
  39. 𝖭𝖤𝖷𝖯 𝖯 / 𝗉𝗈𝗅𝗒 \mathsf{NEXP}\not\subseteq\mathsf{P/poly}
  40. O ( log i ( n ) ) O(\log^{i}(n))
  41. A A
  42. TIME ( t ( n ) ) \,\text{TIME}(t(n))
  43. t : t:\mathbb{N}\to\mathbb{N}
  44. A A
  45. 𝒪 ( t 2 ( n ) ) \mathcal{O}(t^{2}(n))

Circular_points_at_infinity.html

  1. ( x - x 0 ) 2 + ( y - y 0 ) 2 = r 2 . (x-x_{0})^{2}+(y-y_{0})^{2}=r^{2}.
  2. A x 2 + A y 2 + 2 B 1 x z + 2 B 2 y z - C z 2 = 0. Ax^{2}+Ay^{2}+2B_{1}xz+2B_{2}yz-Cz^{2}=0.
  3. u : y = x tan θ , u : y = x tan θ . u:y=x\tan\theta,\quad u^{\prime}:y=x\tan\theta^{\prime}.
  4. ( u u , ω ω ) = tan θ - i tan θ + i ÷ tan θ - i tan θ + i , (uu^{\prime},\omega\omega^{\prime})=\frac{\tan\theta-i}{\tan\theta+i}\div\frac% {\tan\theta^{\prime}-i}{\tan\theta^{\prime}+i},
  5. ϕ = θ - θ = i 2 log ( u u , ω ω ) . \phi=\theta^{\prime}-\theta=\tfrac{i}{2}\log(uu^{\prime},\omega\omega^{\prime}).

Circulation_problem.html

  1. G ( V , E ) G(V,E)
  2. l ( v , w ) l(v,w)
  3. v v
  4. w w
  5. u ( v , w ) u(v,w)
  6. v v
  7. w w
  8. c ( v , w ) c(v,w)
  9. ( v , w ) (v,w)
  10. l ( v , w ) f ( v , w ) u ( v , w ) l(v,w)\leq f(v,w)\leq u(v,w)
  11. w V f ( u , w ) = 0 \sum_{w\in V}f(u,w)=0
  12. ( v , w ) E c ( v , w ) f ( v , w ) . \sum_{(v,w)\in E}c(v,w)\cdot f(v,w).
  13. f i ( v , w ) \,f_{i}(v,w)
  14. i i
  15. v v
  16. w w
  17. f ( v , w ) = i f i ( v , w ) \,f(v,w)=\sum_{i}f_{i}(v,w)
  18. l i ( v , w ) f i ( v , w ) \,l_{i}(v,w)\leq f_{i}(v,w)
  19. w V f i ( u , w ) = 0. \ \sum_{w\in V}f_{i}(u,w)=0.
  20. K i ( s i , t i , d i ) K_{i}(s_{i},t_{i},d_{i})
  21. d i d_{i}
  22. i i
  23. s i s_{i}
  24. t i t_{i}
  25. ( t i , s i ) (t_{i},s_{i})
  26. l i ( t i , s i ) = c ( t i , s i ) = d i l_{i}(t_{i},s_{i})=c(t_{i},s_{i})=d_{i}
  27. i i
  28. l i ( u , v ) = 0 l_{i}(u,v)=0
  29. t t
  30. s s
  31. l ( t , s ) = 0 l(t,s)=0
  32. u ( t , s ) = u(t,s)=
  33. c ( t , s ) = - 1 c(t,s)=-1
  34. m m
  35. l ( t , s ) = u ( t , s ) = m l(t,s)=u(t,s)=m
  36. c ( t , s ) = 0 c(t,s)=0
  37. l ( u , v ) = 0 l(u,v)=0
  38. c ( u , v ) = 1 c(u,v)=1
  39. ( t , s ) (t,s)
  40. l ( t , s ) = c ( t , s ) = 1 l(t,s)=c(t,s)=1
  41. a ( t , s ) = 0 a(t,s)=0
  42. v ( v - 1 ) / 2 v(v-1)/2

Citrabhanu.html

  1. x + y = a , x - y = b , x y = c , x 2 + y 2 = d , x 2 - y 2 = e , x 3 + y 3 = f , x 3 - y 3 = g \ x+y=a,x-y=b,xy=c,x^{2}+y^{2}=d,x^{2}-y^{2}=e,x^{3}+y^{3}=f,x^{3}-y^{3}=g

Clamper_(electronics).html

  1. τ = R C \tau=RC

Clark–Ocone_theorem.html

  1. F ( σ ) = C 0 F ( p ) d γ ( p ) + 0 T 𝐄 [ t H F t ( - ) | Σ t ] ( σ ) d σ t . F(\sigma)=\int_{C_{0}}F(p)\,\mathrm{d}\gamma(p)+\int_{0}^{T}\mathbf{E}\left[% \left.\frac{\partial}{\partial t}\nabla_{H}F_{t}(-)\right|\Sigma_{t}\right](% \sigma)\,\mathrm{d}\sigma_{t}.
  2. C 0 F ( p ) d γ ( p ) = 𝐄 [ F ] \int_{C_{0}}F(p)\,\mathrm{d}\gamma(p)=\mathbf{E}[F]
  3. 0 T d σ ( t ) \int_{0}^{T}\cdots\,\mathrm{d}\sigma(t)
  4. V ˙ = V t : [ 0 , T ] × C 0 \dot{V}=\frac{\partial V}{\partial t}:[0,T]\times C_{0}\to\mathbb{R}
  5. C 0 D F ( σ ) ( V ( σ ) ) d γ ( σ ) = C 0 F ( σ ) ( 0 T V ˙ t ( σ ) d σ t ) d γ ( σ ) , \int_{C_{0}}\mathrm{D}F(\sigma)(V(\sigma))\,\mathrm{d}\gamma(\sigma)=\int_{C_{% 0}}F(\sigma)\left(\int_{0}^{T}\dot{V}_{t}(\sigma)\,\mathrm{d}\sigma_{t}\right)% \,\mathrm{d}\gamma(\sigma),
  6. C 0 H F ( σ ) , V ( σ ) L 0 2 , 1 d γ ( σ ) = - C 0 F ( σ ) div ( V ) ( σ ) d γ ( σ ) \int_{C_{0}}\left\langle\nabla_{H}F(\sigma),V(\sigma)\right\rangle_{L_{0}^{2,1% }}\,\mathrm{d}\gamma(\sigma)=-\int_{C_{0}}F(\sigma)\operatorname{div}(V)(% \sigma)\,\mathrm{d}\gamma(\sigma)
  7. 𝔼 [ H F , V ] = - 𝔼 [ F div V ] , \mathbb{E}\big[\langle\nabla_{H}F,V\rangle\big]=-\mathbb{E}\big[F\operatorname% {div}V\big],
  8. div ( V ) ( σ ) := - 0 T V ˙ t ( σ ) d σ t . \operatorname{div}(V)(\sigma):=-\int_{0}^{T}\dot{V}_{t}(\sigma)\,\mathrm{d}% \sigma_{t}.

Class_formation.html

  1. 0 H 2 ( E / F ) H 2 ( E / E G ) H 2 ( E / E G ) H 2 ( F / F G ) 0\rightarrow H^{2}(E/F)\cap H^{2}(E/E\cap G_{\infty})\rightarrow H^{2}(E/E\cap G% _{\infty})\rightarrow H^{2}(F/F\cap G_{\infty})

Classical_modular_curve.html

  1. ( x , y ) = ( j ( n τ ) , j ( τ ) ) (x,y)=(j(nτ),j(τ))
  2. j ( τ ) j(τ)
  3. j j
  4. 𝐇 \mathbf{H}
  5. 2 n 2n
  6. n > 1 n>1
  7. n n
  8. x x
  9. 𝐙 y y \mathbf{Z}yy
  10. ψ ( n ) ψ(n)
  11. ψ ψ
  12. y = x y=x
  13. n > 2 n>2
  14. x = 0 , y = x=0,y=∞
  15. x = , y = 0 x=∞,y=0
  16. n = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 13 , 16 , 18 n=1,2,3,4,5,6,7,8,9,10,12,13,16,18
  17. 25 25
  18. j 2 ( q ) = q - 1 - 24 + 276 q - 2048 q 2 + 11202 q 3 + = ( η ( q ) η ( q 2 ) ) 24 j_{2}(q)=q^{-1}-24+276q-2048q^{2}+11202q^{3}+\cdots=\left(\frac{\eta(q)}{\eta(% q^{2})}\right)^{24}
  19. η η
  20. x = ( j 2 + 256 ) 3 j 2 2 , x=\frac{(j_{2}+256)^{3}}{j_{2}^{2}},
  21. y = ( j 2 + 16 ) 3 j 2 y=\frac{(j_{2}+16)^{3}}{j_{2}}
  22. C C
  23. 𝐐 \mathbf{Q}
  24. n n
  25. 𝐐 \mathbf{Q}
  26. n n
  27. n n
  28. j j
  29. j j
  30. j j
  31. y y
  32. x x
  33. y y
  34. x x 5 - 2 x 4 + 3 x 3 - 2 x + 1 x 2 ( x - 1 ) 2 x\mapsto\frac{x^{5}-2x^{4}+3x^{3}-2x+1}{x^{2}(x-1)^{2}}
  35. y y - ( 2 y + 1 ) ( x 4 + x 3 - 3 x 2 + 3 x - 1 ) x 3 ( x - 1 ) 3 y\mapsto y-\frac{(2y+1)(x^{4}+x^{3}-3x^{2}+3x-1)}{x^{3}(x-1)^{3}}
  36. x x
  37. j j
  38. 11 11
  39. E E
  40. n n
  41. 𝐐 \mathbf{Q}
  42. n n
  43. 𝐙 y y \mathbf{Z}yy
  44. ψ ( n ) ψ(n)
  45. x x
  46. 𝐐 ( y ) \mathbf{Q}(y)
  47. p p
  48. p p
  49. 𝐐 ( x , y ) / 𝐐 ( y ) \mathbf{Q}(x,y)/\mathbf{Q}(y)
  50. P G L ( 2 , p ) PGL(2,p)
  51. p p
  52. p + 1 p+1
  53. F / 𝐐 F/\mathbf{Q}
  54. p * = ( - 1 ) ( p - 1 ) / 2 p p^{*}=(-1)^{(p-1)/2}p
  55. F = 𝐐 ( p * ) . F=\mathbf{Q}\left(\sqrt{p^{*}}\right).
  56. F F
  57. P S L ( 2 , p ) PSL(2,p)
  58. p p
  59. y y
  60. P S L ( 2 , p ) PSL(2,p)
  61. F F
  62. P G L ( 2 , p ) PGL(2,p)
  63. 𝐐 \mathbf{Q}
  64. n n
  65. n n
  66. X < s u b > 0 ( n ) X<sub>0(n)

Classical_Wiener_space.html

  1. d ( f ( s ) , f ( t ) ) 0 d(f(s),f(t))\to 0
  2. | s - t | 0. |s-t|\to 0.
  3. f := sup t [ 0 , T ] | f ( t ) | \|f\|:=\sup_{t\in[0,T]}|f(t)|
  4. d ( f , g ) := f - g d(f,g):=\|f-g\|
  5. ω f ( δ ) := sup { | f ( s ) - f ( t ) | | s , t [ 0 , T ] , | s - t | δ } . \omega_{f}(\delta):=\sup\left\{|f(s)-f(t)|\left|s,t\in[0,T],|s-t|\leq\delta% \right.\right\}.
  6. f C ω f ( δ ) 0 f\in C\iff\omega_{f}(\delta)\to 0
  7. ( μ n ) n = 1 (\mu_{n})_{n=1}^{\infty}
  8. lim a lim sup n μ n { f C | | f ( 0 ) | a } = 0 , \lim_{a\to\infty}\limsup_{n\to\infty}\mu_{n}\{f\in C||f(0)|\geq a\}=0,
  9. lim δ 0 lim sup n μ n { f C | ω f ( δ ) ε } = 0 \lim_{\delta\to 0}\limsup_{n\to\infty}\mu_{n}\{f\in C|\omega_{f}(\delta)\geq% \varepsilon\}=0
  10. B t - B s Normal ( 0 , | t - s | ) , B_{t}-B_{s}\sim\mathrm{Normal}\left(0,|t-s|\right),

Clearing_the_neighbourhood.html

  1. Λ = M 2 a 3 2 k \Lambda=\frac{M^{2}}{a^{\frac{3}{2}}}\,k

Clenshaw–Curtis_quadrature.html

  1. x = cos θ x=\cos\theta
  2. f ( x ) f(x)
  3. N N
  4. O ( N log N ) O(N\log N)
  5. - 1 1 f ( x ) d x = 0 π f ( cos θ ) sin ( θ ) d θ . \int_{-1}^{1}f(x)\,dx=\int_{0}^{\pi}f(\cos\theta)\sin(\theta)\,d\theta.
  6. f ( x ) f(x)
  7. f ( cos θ ) sin θ f(\cos\theta)\sin\theta
  8. f ( cos θ ) f(\cos\theta)
  9. f ( cos θ ) = a 0 2 + k = 1 a k cos ( k θ ) f(\cos\theta)=\frac{a_{0}}{2}+\sum_{k=1}^{\infty}a_{k}\cos(k\theta)
  10. 0 π f ( cos θ ) sin ( θ ) d θ = a 0 + k = 1 2 a 2 k 1 - ( 2 k ) 2 . \int_{0}^{\pi}f(\cos\theta)\sin(\theta)\,d\theta=a_{0}+\sum_{k=1}^{\infty}% \frac{2a_{2k}}{1-(2k)^{2}}.
  11. a k = 2 π 0 π f ( cos θ ) cos ( k θ ) d θ a_{k}=\frac{2}{\pi}\int_{0}^{\pi}f(\cos\theta)\cos(k\theta)\,d\theta
  12. f ( cos θ ) f(\cos\theta)
  13. k = N k=N
  14. N + 1 N+1
  15. θ n = n π / N \theta_{n}=n\pi/N
  16. n = 0 , , N n=0,\ldots,N
  17. a k 2 N [ f ( 1 ) 2 + f ( - 1 ) 2 ( - 1 ) k + n = 1 N - 1 f ( cos [ n π / N ] ) cos ( n k π / N ) ] a_{k}\approx\frac{2}{N}\left[\frac{f(1)}{2}+\frac{f(-1)}{2}(-1)^{k}+\sum_{n=1}% ^{N-1}f(\cos[n\pi/N])\cos(nk\pi/N)\right]
  18. k = 0 , , N k=0,\ldots,N
  19. a k a_{k}
  20. a 2 k a_{2k}
  21. a 2 k 2 N [ f ( 1 ) + f ( - 1 ) 2 + f ( 0 ) ( - 1 ) k + n = 1 N / 2 - 1 { f ( cos [ n π / N ] ) + f ( - cos [ n π / N ] ) } cos ( n k π N / 2 ) ] a_{2k}\approx\frac{2}{N}\left[\frac{f(1)+f(-1)}{2}+f(0)(-1)^{k}+\sum_{n=1}^{N/% 2-1}\left\{f(\cos[n\pi/N])+f(-\cos[n\pi/N])\right\}\cos\left(\frac{nk\pi}{N/2}% \right)\right]
  22. a 2 k a_{2k}
  23. a 2 k a_{2k}
  24. a N a_{N}
  25. 0 π f ( cos θ ) sin ( θ ) d θ a 0 + k = 1 N / 2 - 1 2 a 2 k 1 - ( 2 k ) 2 + a N 1 - N 2 . \int_{0}^{\pi}f(\cos\theta)\sin(\theta)\,d\theta\approx a_{0}+\sum_{k=1}^{N/2-% 1}\frac{2a_{2k}}{1-(2k)^{2}}+\frac{a_{N}}{1-N^{2}}.
  26. T k ( x ) T_{k}(x)
  27. T k ( cos θ ) = cos ( k θ ) T_{k}(\cos\theta)=\cos(k\theta)
  28. f ( x ) f(x)
  29. f ( x ) = a 0 2 T 0 ( x ) + k = 1 a k T k ( x ) , f(x)=\frac{a_{0}}{2}T_{0}(x)+\sum_{k=1}^{\infty}a_{k}T_{k}(x),
  30. f ( x ) f(x)
  31. x n = cos ( n π / N ) x_{n}=\cos(n\pi/N)
  32. T N ( x ) T_{N}(x)
  33. T k ( x ) T_{k}(x)
  34. f ( cos θ ) f(\cos\theta)
  35. θ \theta
  36. f ( x ) f(x)
  37. [ - 1 , 1 ] [-1,1]
  38. f ( x ) f(x)
  39. f ( cos θ ) f(\cos\theta)
  40. f ( - 1 ) f(-1)
  41. f ( 1 ) f(1)
  42. a k a_{k}
  43. f ( cos θ ) f(\cos\theta)
  44. θ n = ( n + 0.5 ) π / N \theta_{n}=(n+0.5)\pi/N
  45. 0 n < N 0\leq n<N
  46. T N ( cos θ ) T_{N}(\cos\theta)
  47. a k 2 N n = 0 N - 1 f ( cos [ ( n + 0.5 ) π / N ] ) cos [ ( n + 0.5 ) k π / N ] a_{k}\approx\frac{2}{N}\sum_{n=0}^{N-1}f(\cos[(n+0.5)\pi/N])\cos[(n+0.5)k\pi/N]
  48. N + 1 N+1
  49. 2 N + 1 2N+1
  50. N + 1 N+1
  51. N N
  52. O ( [ 2 N ] - k / k ) O([2N]^{-k}/k)
  53. O ( N log N ) O(N\log N)
  54. O ( N 2 ) O(N^{2})
  55. O ( N ) O(N)
  56. N N
  57. f ( x ) f(x)
  58. w ( x ) w(x)
  59. - 1 1 f ( x ) w ( x ) d x = 0 π f ( cos θ ) w ( cos θ ) sin ( θ ) d θ . \int_{-1}^{1}f(x)w(x)\,dx=\int_{0}^{\pi}f(\cos\theta)w(\cos\theta)\sin(\theta)% \,d\theta.
  60. w ( x ) = 1 w(x)=1
  61. w ( x ) w(x)
  62. f ( x ) f(x)
  63. f ( cos θ ) f(\cos\theta)
  64. W k = 0 π w ( cos θ ) cos ( k θ ) sin ( θ ) d θ . W_{k}=\int_{0}^{\pi}w(\cos\theta)\cos(k\theta)\sin(\theta)\,d\theta.
  65. w ( x ) w(x)
  66. f ( x ) f(x)
  67. W k W_{k}
  68. w ( x ) w(x)
  69. W k W_{k}
  70. f ( x ) w ( x ) f(x)w(x)
  71. w ( x ) w(x)
  72. w ( x ) = 1 w(x)=1
  73. W k W_{k}
  74. f ( x ) f(x)
  75. w ( x ) w(x)
  76. W k W_{k}
  77. w ( x ) w(x)
  78. 0 f ( x ) d x \int_{0}^{\infty}f(x)dx
  79. - f ( x ) d x \int_{-\infty}^{\infty}f(x)dx
  80. f ( x ) f(x)
  81. - + f ( x ) d x = - 1 + 1 f ( t 1 - t 2 ) 1 + t 2 ( 1 - t 2 ) 2 d t , \int_{-\infty}^{+\infty}f(x)dx=\int_{-1}^{+1}f\left(\frac{t}{1-t^{2}}\right)% \frac{1+t^{2}}{(1-t^{2})^{2}}dt\;,
  82. x = L cot 2 ( θ / 2 ) x=L\cot^{2}(\theta/2)
  83. 0 f ( x ) d x = 2 L 0 π f [ L cot 2 ( θ / 2 ) ] [ 1 - cos ( θ ) ] 2 sin ( θ ) d θ . \int_{0}^{\infty}f(x)dx=2L\int_{0}^{\pi}\frac{f[L\cot^{2}(\theta/2)]}{[1-\cos(% \theta)]^{2}}\sin(\theta)d\theta.
  84. x = L cot ( θ ) x=L\cot(\theta)
  85. - f ( x ) d x = L 0 π f [ L cot ( θ ) ] sin 2 ( θ ) d θ L π N n = 1 N - 1 f [ L cot ( n π / N ) ] sin 2 ( n π / N ) . \int_{-\infty}^{\infty}f(x)dx=L\int_{0}^{\pi}\frac{f[L\cot(\theta)]}{\sin^{2}(% \theta)}d\theta\approx\frac{L\pi}{N}\sum_{n=1}^{N-1}\frac{f[L\cot(n\pi/N)]}{% \sin^{2}(n\pi/N)}.
  86. 0 e - x g ( x ) d x \int_{0}^{\infty}e^{-x}g(x)dx
  87. x = - ln [ ( 1 + cos θ ) / 2 ] x=-\ln[(1+\cos\theta)/2]
  88. 0 π f ( cos θ ) sin θ d θ \int_{0}^{\pi}f(\cos\theta)\sin\theta\,d\theta
  89. f ( u ) = g ( - ln [ ( 1 + u ) / 2 ] ) / 2 f(u)=g(-\ln[(1+u)/2])/2
  90. w n w_{n}
  91. - 1 1 f ( x ) d x n = 0 N / 2 w n { f ( cos [ n π / N ] ) + f ( - cos [ n π / N ] ) } . \int_{-1}^{1}f(x)\,dx\approx\sum_{n=0}^{N/2}w_{n}\left\{f(\cos[n\pi/N])+f(-% \cos[n\pi/N])\right\}.
  92. w n w_{n}
  93. a 2 k a_{2k}
  94. c = ( a 0 a 2 a 4 a N ) = D ( y 0 y 1 y 2 y N / 2 ) = D y , c=\begin{pmatrix}a_{0}\\ a_{2}\\ a_{4}\\ \vdots\\ a_{N}\end{pmatrix}=D\begin{pmatrix}y_{0}\\ y_{1}\\ y_{2}\\ \vdots\\ y_{N/2}\end{pmatrix}=Dy,
  95. D k n = 2 N cos ( n k π N / 2 ) × { 1 / 2 n = 0 , N / 2 1 otherwise D_{kn}=\frac{2}{N}\cos\left(\frac{nk\pi}{N/2}\right)\times\begin{cases}1/2&n=0% ,N/2\\ 1&\mathrm{otherwise}\end{cases}
  96. y n y_{n}
  97. y n = f ( cos [ n π / N ] ) + f ( - cos [ n π / N ] ) . y_{n}=f(\cos[n\pi/N])+f(-\cos[n\pi/N]).\!
  98. a N a_{N}
  99. ( N / 2 + 1 ) × ( N / 2 + 1 ) (N/2+1)\times(N/2+1)
  100. - 1 1 f ( x ) d x a 0 + k = 1 N / 2 - 1 2 a 2 k 1 - ( 2 k ) 2 + a N 1 - N 2 = d T c , \int_{-1}^{1}f(x)\,dx\approx a_{0}+\sum_{k=1}^{N/2-1}\frac{2a_{2k}}{1-(2k)^{2}% }+\frac{a_{N}}{1-N^{2}}=d^{T}c,
  101. a 2 k a_{2k}
  102. d = ( 1 2 / ( 1 - 4 ) 2 / ( 1 - 16 ) 2 / ( 1 - [ N - 2 ] 2 ) 1 / ( 1 - N 2 ) ) . d=\begin{pmatrix}1\\ 2/(1-4)\\ 2/(1-16)\\ \vdots\\ 2/(1-[N-2]^{2})\\ 1/(1-N^{2})\end{pmatrix}.
  103. d T c d^{T}c
  104. - 1 1 f ( x ) d x d T c = d T D y = ( D T d ) T y = w T y \int_{-1}^{1}f(x)\,dx\approx d^{T}c=d^{T}Dy=(D^{T}d)^{T}y=w^{T}y
  105. w n w_{n}
  106. w = D T d . w=D^{T}d.\!
  107. D T D^{T}
  108. w n w_{n}