wpmath0000014_1

Bernoulli_stochastics.html

  1. โ„ฌ X , D \mathcal{B}_{X,D}
  2. ๐’Ÿ \mathcal{D}
  3. ๐’ณ \mathcal{X}
  4. ๐’ซ \mathcal{P}
  5. โ„ฌ X , D = ( ๐’Ÿ , ๐’ณ , ๐’ซ ) \mathcal{B}_{X,D}=(\mathcal{D},\mathcal{X},\mathcal{P})
  6. ( X , D ) (X,D)
  7. ๐’Ÿ \mathcal{D}
  8. ๐’ณ \mathcal{X}
  9. ๐’ซ \mathcal{P}
  10. ๐’Ÿ \mathcal{D}
  11. ๐’ณ \mathcal{X}
  12. ๐’ซ \mathcal{P}
  13. ๐”‡ = { d } \mathfrak{D}=\{d\}
  14. { x ( d ) } \{x(d)\}
  15. A X A_{X}
  16. A X A_{X}
  17. ฮฒ \beta
  18. ฮฒ \beta
  19. ฮฒ \beta
  20. ฮฒ \beta
  21. A X ( ฮฒ ) A_{X}^{(\beta)}
  22. โ„ฌ X , D \mathcal{B}_{X,D}
  23. ๐’Ÿ \mathcal{D}
  24. ฮฒ \beta
  25. A X ( ฮฒ ) ( { d } ) A^{(\beta)}_{X}(\{d\})
  26. C D C_{D}
  27. ฮฒ \beta
  28. ฮฒ \beta
  29. C D ( ฮฒ ) C^{(\beta)}_{D}
  30. ฮฒ \beta
  31. ฮฒ \beta
  32. C D ( ฮฒ ) C^{(\beta)}_{D}
  33. ฮฒ \beta
  34. A X ( ฮฒ ) A_{X}^{(\beta)}
  35. C D ( ฮฒ ) ( { x } ) = { d | x ฯต A X ( ฮฒ ) ( { d } ) } C^{(\beta)}_{D}(\{x\})=\{d|x\epsilon A_{X}^{(\beta)}(\{d\})\}
  36. { x } \{x\}
  37. C D ( ฮฒ ) ( { x } ) C^{(\beta)}_{D}(\{x\})
  38. { x } \{x\}
  39. ฮฒ \beta

Bernstein's_problem.html

  1. โˆ‘ i = 1 n - 1 โˆ‚ โˆ‚ x i โˆ‚ f โˆ‚ x i 1 + โˆ‘ j = 1 n - 1 ( โˆ‚ f โˆ‚ x j ) 2 = 0 \sum_{i=1}^{n-1}\frac{\partial}{\partial x_{i}}\frac{\frac{\partial f}{% \partial x_{i}}}{\sqrt{1+\sum_{j=1}^{n-1}(\frac{\partial f}{\partial x_{j}})^{% 2}}}=0

Bernstein's_theorem_(approximation_theory).html

  1. deg P n = n , sup 0 โ‰ค x โ‰ค 2 ฯ€ | f ( x ) - P n ( x ) | โ‰ค C ( f ) n r + ฮฑ , \deg\,P_{n}=n~{},\quad\sup_{0\leq x\leq 2\pi}|f(x)-P_{n}(x)|\leq\frac{C(f)}{n^% {r+\alpha}}~{},

Berry_connection_and_curvature.html

  1. H ( ๐‘ ) H(\mathbf{R})
  2. ๐‘ \mathbf{R}
  3. t t
  4. n n
  5. ฮต n ( ๐‘ ) \varepsilon_{n}(\mathbf{R})
  6. | n ( ๐‘ ( 0 ) ) โŸฉ \,|n(\mathbf{R}(0))\rangle
  7. | n ( ๐‘ ( t ) ) โŸฉ \,|n(\mathbf{R}(t))\rangle
  8. H ( ๐‘ ( t ) ) \,H(\mathbf{R}(t))
  9. | ฮจ n ( t ) โŸฉ = e i ฮณ n ( t ) e - i โ„ โˆซ 0 t d t โ€ฒ ฮต n ( ๐‘ ( t โ€ฒ ) ) | n ( ๐‘ ( t ) ) โŸฉ , |\Psi_{n}(t)\rangle=e^{i\gamma_{n}(t)}\,e^{-{i\over\hbar}\int_{0}^{t}dt^{% \prime}\varepsilon_{n}(\mathbf{R}(t^{\prime}))}\,|n(\mathbf{R}(t))\rangle,
  10. ฮณ n \gamma_{n}
  11. ฮณ n ( t ) = i โˆซ 0 t d t โ€ฒ โŸจ n ( ๐‘ ( t โ€ฒ ) ) | d d t โ€ฒ | n ( ๐‘ ( t โ€ฒ ) ) โŸฉ = i โˆซ ๐‘ ( 0 ) ๐‘ ( t ) d ๐‘ โŸจ n ( ๐‘ ) | โˆ‡ ๐‘ | n ( ๐‘ ) โŸฉ , \gamma_{n}(t)=i\int_{0}^{t}dt^{\prime}\,\langle n(\mathbf{R}(t^{\prime}))|{d% \over dt^{\prime}}|n(\mathbf{R}(t^{\prime}))\rangle=i\int_{\mathbf{R}(0)}^{% \mathbf{R}(t)}d\mathbf{R}\,\langle n(\mathbf{R})|\nabla_{\mathbf{R}}|n(\mathbf% {R})\rangle,
  12. ๐’ž \mathcal{C}
  13. ๐‘ ( T ) = ๐‘ ( 0 ) \mathbf{R}(T)=\mathbf{R}(0)
  14. ฮณ n = i โˆฎ ๐’ž d ๐‘ โŸจ n ( ๐‘ ) | โˆ‡ ๐‘ | n ( ๐‘ ) โŸฉ . \gamma_{n}=i\oint_{\mathcal{C}}d\mathbf{R}\,\langle n(\mathbf{R})|\nabla_{% \mathbf{R}}|n(\mathbf{R})\rangle.
  15. | n ~ ( ๐‘ ) โŸฉ = e - i ฮฒ ( ๐‘ ) | n ( ๐‘ ) โŸฉ |\tilde{n}(\mathbf{R})\rangle=e^{-i\beta(\mathbf{R})}|n(\mathbf{R})\rangle
  16. ๐‘ \mathbf{R}
  17. ฮณ ~ n ( t ) = ฮณ n ( t ) + ฮฒ ( t ) - ฮฒ ( 0 ) \tilde{\gamma}_{n}(t)=\gamma_{n}(t)+\beta(t)-\beta(0)
  18. ฮฒ ( T ) - ฮฒ ( 0 ) = 2 ฯ€ m \beta(T)-\beta(0)=2\pi m
  19. m m
  20. ฮณ n \gamma_{n}
  21. 2 ฯ€ 2\pi
  22. ฮณ n = โˆซ ๐’ž d ๐‘ โ‹… ๐’œ n ( ๐‘ ) \gamma_{n}=\int_{\mathcal{C}}d\mathbf{R}\cdot\mathcal{A}_{n}(\mathbf{R})
  23. ๐’œ n ( ๐‘ ) = i โŸจ n ( ๐‘ ) | โˆ‡ ๐‘ | n ( ๐‘ ) โŸฉ \mathcal{A}_{n}(\mathbf{R})=i\langle n(\mathbf{R})|\nabla_{\mathbf{R}}|n(% \mathbf{R})\rangle
  24. ๐’œ ~ n ( ๐‘ ) = ๐’œ n ( ๐‘ ) + โˆ‡ ๐‘ ฮฒ ( ๐‘ ) \tilde{\mathcal{A}}_{n}(\mathbf{R})=\mathcal{A}_{n}(\mathbf{R})+\nabla_{% \mathbf{R}\,}\beta(\mathbf{R})
  25. ๐’œ n ( ๐‘ ) \mathcal{A}_{n}(\mathbf{R})
  26. ฮณ n \gamma_{n}
  27. 2 ฯ€ 2\pi
  28. e i ฮณ n e^{i\gamma_{n}}
  29. ฮฉ n , ฮผ ฮฝ ( ๐‘ ) = โˆ‚ โˆ‚ R ฮผ ๐’œ n , ฮฝ ( ๐‘ ) - โˆ‚ โˆ‚ R ฮฝ ๐’œ n , ฮผ ( ๐‘ ) . \Omega_{n,\mu\nu}(\mathbf{R})={\partial\over\partial R^{\mu}}\mathcal{A}_{n,% \nu}(\mathbf{R})-{\partial\over\partial R^{\nu}}\mathcal{A}_{n,\mu}(\mathbf{R}).
  30. ๐›€ n ( ๐‘ ) = โˆ‡ ๐‘ ร— ๐’œ n ( ๐‘ ) . \mathbf{\Omega}_{n}(\mathbf{R})=\nabla_{\mathbf{R}}\times\mathcal{A}_{n}(% \mathbf{R}).
  31. ฮฉ n , ฮผ ฮฝ = ฯต ฮผ ฮฝ ฮพ ๐›€ n , ฮพ \Omega_{n,\mu\nu}=\epsilon_{\mu\nu\xi}\,\mathbf{\Omega}_{n,\xi}
  32. ๐’ž \mathcal{C}
  33. ๐’ฎ \mathcal{S}
  34. ฮณ n = โˆซ ๐’ฎ d ๐’ โ‹… ๐›€ n ( ๐‘ ) . \gamma_{n}=\int_{\mathcal{S}}d\mathbf{S}\cdot\mathbf{\Omega}_{n}(\mathbf{R}).
  35. 2 ฯ€ 2\pi
  36. 2 ฯ€ 2\pi
  37. ฮฉ n , ฮผ ฮฝ ( ๐‘ ) = i โˆ‘ n โ€ฒ โ‰  n โŸจ n | ( โˆ‚ H / โˆ‚ R ฮผ ) | n โ€ฒ โŸฉ โŸจ n โ€ฒ | ( โˆ‚ H / โˆ‚ R ฮฝ ) | n โŸฉ - ( ฮฝ โ†” ฮผ ) ( ฮต n - ฮต n โ€ฒ ) 2 . \Omega_{n,\mu\nu}(\mathbf{R})=i\sum_{n^{\prime}\neq n}{\langle n|(\partial H/% \partial R_{\mu})|n^{\prime}\rangle\langle n^{\prime}|(\partial H/\partial R_{% \nu})|n\rangle-(\nu\leftrightarrow\mu)\over(\varepsilon_{n}-\varepsilon_{n^{% \prime}})^{2}}.
  38. H = ฮผ ฯƒ โ‹… ๐ , H=\mu\mathbf{\sigma}\cdot\mathbf{B},
  39. ฯƒ \mathbf{\sigma}
  40. ฮผ \mu
  41. ยฑ ฮผ B \pm\mu B
  42. | u - โŸฉ = ( sin ฮธ 2 e - i ฯ• - cos ฮธ 2 ) , | u + โŸฉ = ( cos ฮธ 2 e - i ฯ• sin ฮธ 2 ) . |u_{-}\rangle=\begin{pmatrix}\sin{\theta\over 2}e^{-i\phi}\\ -\cos{\theta\over 2}\end{pmatrix},|u_{+}\rangle=\begin{pmatrix}\cos{\theta% \over 2}e^{-i\phi}\\ \sin{\theta\over 2}\end{pmatrix}.
  43. | u - โŸฉ |u_{-}\rangle
  44. ๐’œ ฮธ = โŸจ u | i โˆ‚ ฮธ u โŸฉ = 0 , \mathcal{A}_{\theta}=\langle u|i\partial_{\theta}u\rangle=0,
  45. ๐’œ ฯ• = โŸจ u | i โˆ‚ ฯ• u โŸฉ = sin 2 ฮธ 2 \mathcal{A}_{\phi}=\langle u|i\partial_{\phi}u\rangle=\sin^{2}{\theta\over 2}
  46. ฮฉ ฮธ ฯ• = โˆ‚ ฮธ ๐’œ ฯ• - โˆ‚ ฯ• ๐’œ ฮธ = 1 2 sin ฮธ . \Omega_{\theta\phi}=\partial_{\theta}\mathcal{A}_{\phi}-\partial_{\phi}% \mathcal{A}_{\theta}={1\over 2}\sin\theta.
  47. | u - โŸฉ |u_{-}\rangle
  48. e i ฯ• e^{i\phi}
  49. ๐’œ ฮธ = 0 \mathcal{A}_{\theta}=0
  50. ๐’œ ฯ• = - cos 2 ฮธ 2 \mathcal{A}_{\phi}=-\cos^{2}{\theta\over 2}
  51. ฮฉ ยฏ ฮธ ฯ• = ฮฉ ฮธ ฯ• / sin ฮธ = 1 / 2 \overline{\Omega}_{\theta\phi}=\Omega_{\theta\phi}/\sin\theta=1/2
  52. ๐’ฎ 2 \mathcal{S}^{2}
  53. 2 ฯ€ 2\pi
  54. ฯˆ n ๐ค ( ๐ซ ) = e i ๐ค โ‹… ๐ซ u n ๐ค ( ๐ซ ) , \psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(% \mathbf{r}),
  55. n n
  56. ๐ค \mathbf{k}
  57. u n ๐ค ( ๐ซ ) u_{n\mathbf{k}}(\mathbf{r})
  58. ๐ซ \mathbf{r}
  59. ๐ค \mathbf{k}
  60. ๐‘ \mathbf{R}
  61. ๐’œ n ( ๐ค ) = i โŸจ n ( ๐ค ) | โˆ‡ ๐ค | n ( ๐ค ) โŸฉ . \mathcal{A}_{n}(\mathbf{k})=i\langle n(\mathbf{k})|\nabla_{\mathbf{k}}|n(% \mathbf{k})\rangle.

Besselโ€“Maitland_function.html

  1. J ฮผ , ฮฝ ( z ) = โˆ‘ k โ‰ฅ 0 ( - z ) k ฮ“ ( k ฮผ + ฮฝ + 1 ) k ! . J^{\mu,\nu}(z)=\sum_{k\geq 0}\frac{(-z)^{k}}{\Gamma(k\mu+\nu+1)k!}.

Beta_negative_binomial_distribution.html

  1. B ( ฮฑ , ฮฒ + r ) B ( ฮฑ , ฮฒ ) F 1 2 ( r , ฮฑ ; ฮฑ + ฮฒ + r ; e i t ) \frac{\mathrm{B}(\alpha,\beta+r)}{\mathrm{B}(\alpha,\beta)}{}_{2}F_{1}(r,% \alpha;\alpha+\beta+r;e^{it})\!
  2. X โˆฃ p โˆผ NB ( r , p ) , X\mid p\sim\mathrm{NB}(r,p),
  3. p โˆผ B ( ฮฑ , ฮฒ ) , p\sim\textrm{B}(\alpha,\beta),
  4. X โˆผ BNB ( r , ฮฑ , ฮฒ ) . X\sim\mathrm{BNB}(r,\alpha,\beta).
  5. { ( k + 1 ) p ( k + 1 ) ( ฮฑ + ฮฒ + k + r ) + ( ฮฒ + k ) ( - k - r ) p ( k ) = 0 , p ( 0 ) = ( ฮฑ ) r ( ฮฑ + ฮฒ ) r } \left\{(k+1)p(k+1)(\alpha+\beta+k+r)+(\beta+k)(-k-r)p(k)=0,p(0)=\frac{(\alpha)% _{r}}{(\alpha+\beta)_{r}}\right\}
  6. r r
  7. f ( k | ฮฑ , ฮฒ , r ) = ( r + k - 1 k ) B ( ฮฑ + r , ฮฒ + k ) B ( ฮฑ , ฮฒ ) f(k|\alpha,\beta,r)={\left({{r+k-1}\atop{k}}\right)}\frac{B(\alpha+r,\beta+k)}% {B(\alpha,\beta)}
  8. f ( k | ฮฑ , ฮฒ , r ) = ฮ“ ( r + k ) k ! ฮ“ ( r ) B ( ฮฑ + r , ฮฒ + k ) B ( ฮฑ , ฮฒ ) f(k|\alpha,\beta,r)=\frac{\Gamma(r+k)}{k!\;\Gamma(r)}\frac{B(\alpha+r,\beta+k)% }{B(\alpha,\beta)}
  9. r r
  10. f ( k | ฮฑ , ฮฒ , r ) = ( r + k - 1 k ) ฮ“ ( ฮฑ + r ) ฮ“ ( ฮฒ + k ) ฮ“ ( ฮฑ + ฮฒ ) ฮ“ ( ฮฑ + r + ฮฒ + k ) ฮ“ ( ฮฑ ) ฮ“ ( ฮฒ ) f(k|\alpha,\beta,r)={\left({{r+k-1}\atop{k}}\right)}\frac{\Gamma(\alpha+r)% \Gamma(\beta+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha+r+\beta+k)\Gamma(\alpha)% \Gamma(\beta)}
  11. f ( k | ฮฑ , ฮฒ , r ) = ฮ“ ( r + k ) k ! ฮ“ ( r ) ฮ“ ( ฮฑ + r ) ฮ“ ( ฮฒ + k ) ฮ“ ( ฮฑ + ฮฒ ) ฮ“ ( ฮฑ + r + ฮฒ + k ) ฮ“ ( ฮฑ ) ฮ“ ( ฮฒ ) f(k|\alpha,\beta,r)=\frac{\Gamma(r+k)}{k!\;\Gamma(r)}\frac{\Gamma(\alpha+r)% \Gamma(\beta+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha+r+\beta+k)\Gamma(\alpha)% \Gamma(\beta)}
  12. r r
  13. f ( k | ฮฑ , ฮฒ , r ) = r ( k ) ฮฑ ( r ) ฮฒ ( k ) k ! ( ฮฑ + ฮฒ ) ( r ) ( r + ฮฑ + ฮฒ ) ( k ) f(k|\alpha,\beta,r)=\frac{r^{(k)}\alpha^{(r)}\beta^{(k)}}{k!(\alpha+\beta)^{(r% )}(r+\alpha+\beta)^{(k)}}
  14. r = 1 r=1
  15. ฮฑ \alpha
  16. ฮฒ \beta
  17. ฮฑ \alpha
  18. ฮฒ \beta
  19. r r
  20. f ( k | ฮฑ , ฮฒ , r ) โˆผ ฮ“ ( ฮฑ + r ) ฮ“ ( r ) B ( ฮฑ , ฮฒ ) k r - 1 ( ฮฒ + k ) ฮฑ + r f(k|\alpha,\beta,r)\sim\frac{\Gamma(\alpha+r)}{\Gamma(r)B(\alpha,\beta)}\frac{% k^{r-1}}{(\beta+k)^{\alpha+r}}

Beta_rectangular_distribution.html

  1. { 0 for x โ‰ค a ฮธ I z ( ฮฑ , ฮฒ ) + ( 1 - ฮธ ) ( x - a ) b - a for x โˆˆ [ a , b ] 1 for x โ‰ฅ b \begin{cases}0&\,\text{for }x\leq a\\ \theta I_{z}(\alpha,\beta)+\frac{(1-\theta)(x-a)}{b-a}&\,\text{for }x\in[a,b]% \\ 1&\,\text{for }x\geq b\end{cases}
  2. z = ( x - a ) / ( b - a ) z=(x-a)/(b-a)
  3. a + ( b - a ) ( ฮธ ฮฑ ฮฑ + ฮฒ + 1 - ฮธ 2 ) a+(b-a)\left(\frac{\theta\alpha}{\alpha+\beta}+\frac{1-\theta}{2}\right)
  4. ( b - a ) 2 ( ฮธ ฮฑ ( ฮฑ + 1 ) k ( k + 1 ) + 1 - ฮธ 3 - ( k + ฮธ ( ฮฑ - ฮฒ ) ) 2 4 k 2 ) (b-a)^{2}\left(\frac{\theta\,\alpha(\alpha+1)}{k(k+1)}+\frac{1-\theta}{3}-% \frac{\bigl(k+\theta(\alpha-\beta)\bigr)^{2}}{4k^{2}}\right)
  5. k = ฮฑ + ฮฒ k=\alpha+\beta
  6. p ( x | ฮฑ , ฮฒ , ฮธ ) = { ฮธ ฮ“ ( ฮฑ + ฮฒ ) ฮ“ ( ฮฑ ) ฮ“ ( ฮฒ ) ( x - a ) ฮฑ - 1 ( b - x ) ฮฒ - 1 ( b - a ) ฮฑ + ฮฒ + 1 + 1 - ฮธ b - a for a โ‰ค x โ‰ค b , 0 for x < a or x > b p(x|\alpha,\beta,\theta)=\begin{cases}\frac{\theta\Gamma(\alpha+\beta)}{\Gamma% (\alpha)\Gamma(\beta)}\frac{(x-a)^{\alpha-1}(b-x)^{\beta-1}}{(b-a)^{\alpha+% \beta+1}}+\frac{1-\theta}{b-a}&\mathrm{for}\ a\leq x\leq b,\\ 0&\mathrm{for}\ x<a\ \mathrm{or}\ x>b\end{cases}
  7. ฮ“ ( โ‹… ) \Gamma(\cdot)
  8. F ( x | ฮฑ , ฮฒ , ฮธ ) = ฮธ I z ( ฮฑ , ฮฒ ) + ( 1 - ฮธ ) ( x - a ) b - a for a โ‰ค x โ‰ค b , F(x|\alpha,\beta,\theta)=\theta I_{z}(\alpha,\beta)+\frac{(1-\theta)(x-a)}{b-a% }\quad\quad\mathrm{for}\ a\leq x\leq b,
  9. z = x - a b - a z=\dfrac{x-a}{b-a}
  10. I z ( ฮฑ , ฮฒ ) I_{z}(\alpha,\beta)
  11. E ( x ) \displaystyle E(x)
  12. E ( x ) = ฮธ ( a + 4 m + b ) + 3 ( 1 - ฮธ ) ( a + b ) 6 . E(x)=\frac{\theta(a+4m+b)+3(1-\theta)(a+b)}{6}.
  13. Var ( x ) = ( b - a ) 2 ( 3 - 2 ฮธ ) 36 , \operatorname{Var}(x)=\frac{(b-a)^{2}(3-2\theta)}{36},
  14. Var ( x ) = ( b - a ) 2 ( 3 - 2 ฮธ 2 ) 36 . \operatorname{Var}(x)=\frac{(b-a)^{2}(3-2\theta^{2})}{36}.

Betheโ€“Feynman_formula.html

  1. a โ‰ˆ ( b c ) 2 f a\approx(bc)^{2}f

Betweenness_centrality.html

  1. v v
  2. g ( v ) = โˆ‘ s โ‰  v โ‰  t ฯƒ s t ( v ) ฯƒ s t g(v)=\sum_{s\neq v\neq t}\frac{\sigma_{st}(v)}{\sigma_{st}}
  3. ฯƒ s t \sigma_{st}
  4. s s
  5. t t
  6. ฯƒ s t ( v ) \sigma_{st}(v)
  7. v v
  8. v v
  9. g โˆˆ [ 0 , 1 ] g\in[0,1]
  10. ( N - 1 ) ( N - 2 ) (N-1)(N-2)
  11. ( N - 1 ) ( N - 2 ) / 2 (N-1)(N-2)/2
  12. N N
  13. normal ( g ( v ) ) = g ( v ) - min ( g ) max ( g ) - min ( g ) \mbox{normal}~{}(g(v))=\frac{g(v)-\min(g)}{\max(g)-\min(g)}
  14. max ( n o r m a l ) = 1 \max(normal)=1
  15. min ( n o r m a l ) = 0 \min(normal)=0
  16. ฮด \delta
  17. P ( g ) โ‰ˆ g - ฮด P(g)\approx g^{-\delta}
  18. g ( k ) โ‰ˆ k ฮท g(k)\approx k^{\eta}
  19. g ( k ) g(k)
  20. k k
  21. ฮด \delta
  22. ฮท \eta
  23. P ( g ) = โˆซ d k P ( k ) ฮด ( g - k ฮท ) P(g)=\int dkP(k)\delta(g-k^{\eta})
  24. P ( g โ‰ซ 1 ) = โˆซ d k k - ฮณ ฮด ( g - k ฮท ) P(g\gg 1)=\int dkk^{-\gamma}\delta(g-k^{\eta})
  25. โˆผ g - 1 - ฮณ - 1 ฮท \sim g^{-1-\frac{\gamma-1}{\eta}}
  26. ฮท = ฮณ - 1 ฮด - 1 \eta=\frac{\gamma-1}{\delta-1}
  27. ฮท \eta
  28. ฮท = 2 \eta=2
  29. ฮท = 2 โ†’ ฮด = ฮณ + 1 2 \eta=2\rightarrow\delta=\frac{\gamma+1}{2}
  30. ฮท \eta
  31. ฮด \delta
  32. ฮท โ‰ค 2 โ†’ ฮด โ‰ฅ ฮณ + 1 2 \eta\leq 2\rightarrow\delta\geq\frac{\gamma+1}{2}
  33. ฮด , ฮท \delta,\eta
  34. s i = โˆ‘ j = 1 N a i j w i j s_{i}=\sum_{j=1}^{N}a_{ij}w_{ij}
  35. a i j a_{ij}
  36. w i j w_{ij}
  37. i i
  38. j j
  39. s ( k ) โ‰ˆ k ฮฒ s(k)\approx k^{\beta}\,
  40. s ( b ) s(b)
  41. b b
  42. s ( b ) โ‰ˆ b ฮฑ s(b)\approx b^{\alpha}
  43. ฮ˜ ( | V | 3 ) \Theta(|V|^{3})
  44. O ( | V | 2 log | V | + | V | | E | ) O(|V|^{2}\log|V|+|V||E|)
  45. O ( | V | | E | ) O(|V||E|)

Beฬla_Karlovitz.html

  1. ๐พ๐‘Ž = k t c \mathit{Ka}=kt_{c}
  2. t c t_{c}
  3. k k
  4. k = ( d A / d t ) / A k=(dA/dt)/A
  5. A A

Bidimensionality.html

  1. ฮ  \Pi
  2. ฮ“ * ร— โ„• \Gamma^{*}\times\mathbb{N}
  3. ฮ“ \Gamma
  4. ฮ  \Pi
  5. H , G H,G
  6. H H
  7. G G
  8. k k
  9. ( G , k ) โˆˆ ฮ  (G,k)\in\Pi
  10. ( H , k ) โˆˆ ฮ  (H,k)\in\Pi
  11. G G
  12. ฮด > 0 \delta>0
  13. ( r ร— r ) (r\times r)
  14. R R
  15. ( R , k ) โˆ‰ ฮ  (R,k)\not\in\Pi
  16. k โ‰ค ฮด r 2 k\leq\delta r^{2}
  17. R R
  18. ฮด r 2 \delta r^{2}
  19. ฮ“ 6 \Gamma_{6}
  20. ฮ“ r \Gamma_{r}
  21. ( r ร— r ) (r\times r)
  22. ฮ  \Pi
  23. H , G H,G
  24. H H
  25. G G
  26. k k
  27. ( G , k ) โˆˆ ฮ  (G,k)\in\Pi
  28. ( H , k ) โˆˆ ฮ  (H,k)\in\Pi
  29. G G
  30. ฮด > 0 \delta>0
  31. ( ฮ“ r , k ) โˆ‰ ฮ  (\Gamma_{r},k)\not\in\Pi
  32. k โ‰ค ฮด r 2 k\leq\delta r^{2}
  33. ฮ“ r \Gamma_{r}
  34. ฮ  \Pi
  35. ( G , k ) โˆˆ ฮ  (G,k)\in\Pi
  36. 2 O ( t ) โ‹… | G | O ( 1 ) 2^{O(t)}\cdot|G|^{O(1)}
  37. ( G , k ) โˆˆ ฮ  (G,k)\in\Pi
  38. 2 O ( k ) โ‹… | G | O ( 1 ) 2^{O(\sqrt{k})}\cdot|G|^{O(1)}
  39. ( G , k ) โˆˆ ฮ  (G,k)\in\Pi
  40. 2 O ( k ) โ‹… | G | O ( 1 ) 2^{O(\sqrt{k})}\cdot|G|^{O(1)}
  41. 2 O ( k ) โ‹… | G | O ( 1 ) 2^{O(\sqrt{k})}\cdot|G|^{O(1)}
  42. ฮ  \Pi
  43. ฮ  \Pi

Bidโ€“ask_matrix.html

  1. ( i , j ) (i,j)
  2. i i
  3. j j
  4. d ร— d d\times d
  5. ฮ  = [ ฯ€ i j ] 1 โ‰ค i , j โ‰ค d \Pi=\left[\pi_{ij}\right]_{1\leq i,j\leq d}
  6. ฯ€ i j > 0 \pi_{ij}>0
  7. 1 โ‰ค i , j โ‰ค d 1\leq i,j\leq d
  8. ฯ€ i i = 1 \pi_{ii}=1
  9. 1 โ‰ค i โ‰ค d 1\leq i\leq d
  10. ฯ€ i j โ‰ค ฯ€ i k ฯ€ k j \pi_{ij}\leq\pi_{ik}\pi_{kj}
  11. 1 โ‰ค i , j , k โ‰ค d 1\leq i,j,k\leq d
  12. x x
  13. y y
  14. ฮ  \Pi
  15. ฮ  = [ 1 x y 1 ] \Pi=\begin{bmatrix}1&x\\ y&1\end{bmatrix}
  16. ฮ  \Pi
  17. d d
  18. ฮ  = ( ฯ€ i j ) 1 โ‰ค i , j โ‰ค d \Pi=\left(\pi^{ij}\right)_{1\leq i,j\leq d}
  19. m โ‰ค d m\leq d
  20. m = d m=d
  21. K ( ฮ  ) โŠ‚ โ„ d K(\Pi)\subset\mathbb{R}^{d}
  22. e i , 1 โ‰ค i โ‰ค m e^{i},1\leq i\leq m
  23. ฯ€ i j e i - e j , 1 โ‰ค i , j โ‰ค d \pi^{ij}e^{i}-e^{j},1\leq i,j\leq d
  24. ( i , j ) (i,j)
  25. { 1 ฯ€ j i , ฯ€ i j } \left\{\frac{1}{\pi_{ji}},\pi_{ij}\right\}
  26. ฯ€ i j = 1 ฯ€ j i \pi_{ij}=\frac{1}{\pi_{ji}}

Big_M_method.html

  1. x - y โ‰ค M ( 1 - z ) x-y\leq M(1-z)

Big_q-Jacobi_polynomials.html

  1. P n ( x ; a , b , c ; q ) = ฯ• 2 3 ( q - n , a b q n + 1 , x ; a q , c q ; q , q ) \displaystyle P_{n}(x;a,b,c;q)={}_{3}\phi_{2}(q^{-n},abq^{n+1},x;aq,cq;q,q)

Big_q-Laguerre_polynomials.html

  1. P n ( x ; a , b ; q ) = 1 ( b - 1 * q - n ; q , n ) * 2 ฮฆ 1 ( q - n , a q x - 1 ; a q | q ; x b ) P_{n}(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_{2}\Phi_{1}(q^{-n},aqx^{-1};aq|q% ;\frac{x}{b})

Bihamโ€“Middletonโ€“Levine_traffic_model.html

  1. y โˆˆ { 0 , โ€ฆ , N - 1 } y\in\{0,...,N-1\}
  2. ( N - 1 , y ) (N-1,y)
  3. ( 0 , N - y - 1 ) (0,N-y-1)
  4. x โˆˆ { 0 , โ€ฆ , N - 1 } x\in\{0,...,N-1\}
  5. ( x , N - 1 ) (x,N-1)
  6. ( N - x - 1 , 0 ) (N-x-1,0)
  7. L 2 L^{2}
  8. L L
  9. ฮฑ \alpha
  10. ฮฒ \beta

Binary_expression_tree.html

  1. โˆง \land
  2. โˆจ \lor
  3. ยฌ \neg

Binomial_ring.html

  1. ( x n ) = x ( x - 1 ) โ‹ฏ ( x - n + 1 ) n ! {\left({{x}\atop{n}}\right)}=\frac{x(x-1)\cdots(x-n+1)}{n!}

Biological_applications_of_bifurcation_theory.html

  1. x 1 ห™ = d x 1 d t = f 1 ( x 1 , โ€ฆ , x n ) \dot{x_{1}}=\frac{dx_{1}}{dt}=f_{1}(x_{1},\ldots,x_{n})
  2. โ‹ฎ \vdots
  3. x i ห™ = d x i d t = f i ( x 1 , โ€ฆ , x n ) \dot{x_{i}}=\frac{dx_{i}}{dt}=f_{i}(x_{1},\ldots,x_{n})
  4. โ‹ฎ \vdots
  5. x n ห™ = d x n d t = f n ( x 1 , โ€ฆ , x n ) \dot{x_{n}}=\frac{dx_{n}}{dt}=f_{n}(x_{1},\ldots,x_{n})
  6. f i = a i 1 x 1 + a i 2 x 2 + โ‹ฏ + a i n x n f_{i}=a_{i1}x_{1}+a_{i2}x_{2}+\cdots+a_{in}x_{n}\,
  7. x ห™ = d x d t = f ( x ) \dot{x}=\frac{dx}{dt}=f(x)\,
  8. x ห™ = - x 2 + ฮต \dot{x}=-x^{2}+\varepsilon

Biology_Monte_Carlo_method.html

  1. v โ†’ ( t + d t 2 ) = v โ†’ ( t - d t 2 ) + F โ†’ ( t ) d t \vec{v}(t+\frac{dt}{2})=\vec{v}(t-\frac{dt}{2})+\vec{F}(t)\,dt
  2. r โ†’ ( t + d t ) = r โ†’ ( t - d t ) + v โ†’ ( t + d t 2 ) d t \vec{r}(t+dt)=\vec{r}(t-dt)+\vec{v}(t+\frac{dt}{2})\,dt
  3. โˆ‡ ( ฮต ( r ) โˆ‡ ฯ• ( r , t ) ) = - ( ฯ ions ( r , t ) + ฯ perm ( r ) ) \nabla(\varepsilon(r)\nabla\phi(r,t))=-(\rho\text{ions}(r,t)+\rho\text{perm}(r))
  4. ฯ ions ( r , t ) \rho\text{ions}(r,t)
  5. ฯ perm ( r ) \rho\text{perm}(r)
  6. ฯต ( r ) \epsilon(r)
  7. ฯ• ( r , t ) \phi(r,t)
  8. โˆ‡ ( ฮต โˆ‡ ฯ† ) = ฯ \nabla(\varepsilon\nabla\varphi)=\rho
  9. ฮต \varepsilon
  10. ฮฉ \Omega
  11. โˆฎ โˆ‚ ฮฉ n ^ ( ฮต โˆ‡ ฯ† ) = - โˆซ ฮฉ ฯ \oint_{\partial\Omega}\hat{n}(\varepsilon\nabla\varphi)=-\int_{\Omega}\rho
  12. ฯต \epsilon
  13. ฯ† \varphi
  14. ฮฉ \Omega
  15. โˆ‡ ฯ† \nabla\varphi
  16. ฮฉ \Omega
  17. ฯต \epsilon
  18. โˆ‚ ฮฉ \partial\Omega
  19. โˆฎ โˆ‚ ฮฉ n ^ ( ฮต โˆ‡ ฯ† ) = ฯ† i + 1 , j - ฯ† i , j h i x ( h j y 2 ฯต i , j x + h j - 1 y 2 ฮต i , j - 1 x ) \oint_{\partial\Omega}\hat{n}(\varepsilon\nabla\varphi)=\frac{\varphi_{i+1,j}-% \varphi_{i,j}}{h_{i}^{x}}\left(\frac{h^{y}_{j}}{2}\epsilon^{x}_{i,j}+\frac{h^{% y}_{j-1}}{2}\varepsilon^{x}_{i,j-1}\right)
  20. - ฯ† i , j - ฯ† i - 1 , j h i - 1 x ( h j y 2 ฯต i - 1 , j x + h j - 1 y 2 ฮต i - 1 , j - 1 x ) {}-\frac{\varphi_{i,j}-\varphi_{i-1,j}}{h_{i-1}^{x}}\left(\frac{h^{y}_{j}}{2}% \epsilon^{x}_{i-1,j}+\frac{h^{y}_{j-1}}{2}\varepsilon^{x}_{i-1,j-1}\right)
  21. + ฯ† i , j + 1 - ฯ† i , j h j y ( h i x 2 ฮต i , j y + h i - 1 x 2 ฮต i - 1 , j y ) {}+\frac{\varphi_{i,j+1}-\varphi_{i,j}}{h_{j}^{y}}\left(\frac{h^{x}_{i}}{2}% \varepsilon^{y}_{i,j}+\frac{h^{x}_{i-1}}{2}\varepsilon^{y}_{i-1,j}\right)
  22. - ฯ† i , j - ฯ† i , j - 1 h j - 1 y ( h i x 2 ฮต i , j - 1 y + h i - 1 x 2 ฮต i - 1 , j - 1 y ) {}-\frac{\varphi_{i,j}-\varphi_{i,j-1}}{h_{j-1}^{y}}\left(\frac{h^{x}_{i}}{2}% \varepsilon^{y}_{i,j-1}+\frac{h^{x}_{i-1}}{2}\varepsilon^{y}_{i-1,j-1}\right)
  23. ฮต x \varepsilon^{x}
  24. ฮต y \varepsilon^{y}
  25. ฯต \epsilon
  26. ฯ \rho
  27. ฯ† \varphi
  28. โˆซ ฮฉ i ฯ = Volume ( ฮฉ i ) ฯ i \int_{\Omega_{i}}\rho=\,\text{Volume}(\Omega_{i})\rho_{i}
  29. U L J ( r i j ) = { 4 ฯต L J ( ( ฯƒ i j r i j ) 12 - ( ฯƒ i j r i j ) 6 ) + ฯต L J r i j < 2 1 / 6 ฯƒ i j 0 r i j > 2 1 / 6 ฯƒ i j U_{LJ}(r_{ij})=\begin{cases}4\epsilon_{LJ}\left(\left(\frac{\sigma_{ij}}{r_{ij% }}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right)+\epsilon_{LJ% }&r_{ij}<2^{1/6}\sigma_{ij}\\ 0&r_{ij}>2^{1/6}\sigma_{ij}\end{cases}
  30. ฯต L J \epsilon_{LJ}
  31. ฯƒ i j = ( ฯƒ i + ฯƒ j ) / 2 \sigma_{ij}=(\sigma_{i}+\sigma_{j})/2
  32. - ln ( r ) = โˆซ 0 T f ฮป ( p โ†’ ( t ) ) d t -\ln(r)=\int^{T_{f}}_{0}\lambda(\vec{p}(t))\,dt
  33. ฮป \lambda
  34. ฮป \lambda
  35. ฮป = k T m D \lambda=\frac{kT}{mD}
  36. U h y = c 0 exp ( c 1 - r c 2 ) c o s ( c 3 ( c 1 - r ) ฯ€ ) + c 4 ( c 1 r ) 6 U_{hy}=c_{0}\exp\left(\frac{c_{1}-r}{c_{2}}\right)cos(c_{3}(c_{1}-r)\pi)+c_{4}% \left(\frac{c_{1}}{r}\right)^{6}
  37. P D B F = โˆซ - d z ^ . E โ†’ P_{DBF}=\int-d\hat{z}.\vec{E}

Biomechanics_of_sprint_running.html

  1. V = f step F avg / W b L c , V=f\text{step}F\text{avg}/W\text{b}L\text{c},
  2. V V
  3. f step f\text{step}
  4. F avg F\text{avg}
  5. W b W\text{b}
  6. L c L\text{c}
  7. g โ€ฒ g^{\prime}
  8. g โ€ฒ = ( ( a f 2 + g 2 ) ) 0.5 g^{\prime}=((a\text{f}^{2}+g^{2}))^{0.5}
  9. ES = tan ( 90 - arctan ( g + a f ) ) \,\text{ES}=\tan\left(90-\arctan\left(g+a\text{f}\right)\right)
  10. EM = g โ€ฒ g = [ ( a f 2 g 2 + 1 ) ] 0.5 \,\text{EM}=\frac{g^{\prime}}{g}=\left[\left(\frac{a\text{f}^{2}}{g^{2}}+1% \right)\right]^{0.5}
  11. C sr C\text{sr}
  12. C sr = ( ( 155.4 ES ) 5 - ( 30.4 ES ) 4 - ( 43.3 ES ) 3 + ( 46.3 ES ) 2 + 19.5 ES + 3.6 ) EM C\text{sr}=((155.4\,\,\text{ES})^{5}-(30.4\,\,\text{ES})^{4}-(43.3\,\,\text{ES% })^{3}+(46.3\,\,\text{ES})^{2}+19.5\,\,\text{ES}+3.6)\,\,\text{EM}
  13. C aer C\text{aer}
  14. C aer = k โ€ฒ v 2 C\text{aer}=k^{\prime}v^{2}
  15. C sr = ( ( 155.4 ES ) 5 - ( 30.4 ES ) 4 - ( 43.3 ES ) 3 + ( 46.3 ES ) 2 + 19.5 ES + 3.6 ) EM + k โ€ฒ v 2 C\text{sr}=((155.4\,\,\text{ES})^{5}-(30.4\,\,\text{ES})^{4}-(43.3\,\,\text{ES% })^{3}+(46.3\,\,\text{ES})^{2}+19.5\,\,\text{ES}+3.6)\,\,\text{EM}+k^{\prime}v% ^{2}
  16. g โ€ฒ g^{\prime}
  17. a f a\text{f}
  18. g g
  19. k โ€ฒ k^{\prime}
  20. v v

Biometric_points.html

  1. d d
  2. P 1 P1
  3. P 2 P2
  4. d = || P i - P j || d=||Pi-Pj||

Biomimetic_antifouling_coating.html

  1. Total biocide release = L a ร— a ร— W a ร— 100 S V R ร— S P G ร— D F T \,\text{Total biocide release}={L\text{a}\times a\times W\text{a}\times 100% \over\ SVR\times SPG\times DFT}
  2. E R I = r ร— n 1 - ฯ• ERI={r\times n\over 1-\phi}

Biorthogonal_polynomial.html

  1. โˆซ p ( x ) d ฮผ i ( x ) = 0 \int p(x)\,d\mu_{i}(x)=0
  2. โˆซ ฯ• m ( x ) ฯˆ n ( x ) d ฮผ ( x ) = 0 \int\phi_{m}(x)\psi_{n}(x)\,d\mu(x)=0

Biotโ€“Tolstoyโ€“Medwin_diffraction_model.html

  1. p ( t ) = โˆซ 0 โˆž h ( ฯ„ ) q ( t - ฯ„ ) d ฯ„ p(t)=\int_{0}^{\infty}h(\tau)q(t-\tau)\,d\tau
  2. q ( t ) q(t)
  3. h ( t ) h(t)
  4. ( r S , ฮธ S , z S ) (r_{S},\theta_{S},z_{S})
  5. z z
  6. ฮธ \theta
  7. ( r R , ฮธ R , z R ) (r_{R},\theta_{R},z_{R})
  8. ฮธ W \theta_{W}
  9. ฮฝ = ฯ€ / ฮธ W \nu=\pi/\theta_{W}
  10. c c
  11. z z
  12. h ( ฯ„ ) = - ฮฝ 4 ฯ€ โˆ‘ ฯ• i = ฯ€ ยฑ ฯ• S ยฑ ฯ• R โˆซ z 1 z 2 ฮด ( ฯ„ - m + l c ) ฮฒ i m l d z h(\tau)=-\frac{\nu}{4\pi}\sum_{\phi_{i}=\pi\pm\phi_{S}\pm\phi_{R}}\int_{z_{1}}% ^{z_{2}}\delta\left(\tau-\frac{m+l}{c}\right)\frac{\beta_{i}}{ml}\,dz
  13. m m
  14. l l
  15. z z
  16. ฮด \delta
  17. ฮฒ i = sin ( ฮฝ ฯ• i ) cosh ( ฮฝ ฮท ) - cos ( ฮฝ ฯ• i ) \beta_{i}=\frac{\sin(\nu\phi_{i})}{\cosh(\nu\eta)-\cos(\nu\phi_{i})}
  18. ฮท = cosh - 1 m l + ( z - z S ) ( z - z R ) r S r R \eta=\cosh^{-1}\frac{ml+(z-z_{S})(z-z_{R})}{r_{S}r_{R}}

Bipolar_theorem.html

  1. C โŠ‚ X C\subset X
  2. X X
  3. C o o = ( C o ) o C^{oo}=(C^{o})^{o}
  4. C o o = cl ( co { ฮป c : ฮป โ‰ฅ 0 , c โˆˆ C } ) C^{oo}=\operatorname{cl}(\operatorname{co}\{\lambda c:\lambda\geq 0,c\in C\})
  5. co \operatorname{co}
  6. C โŠ‚ X C\subset X
  7. C + + = C o o = C C^{++}=C^{oo}=C
  8. C + + = ( C + ) + C^{++}=(C^{+})^{+}
  9. ( โ‹… ) + (\cdot)^{+}
  10. C C
  11. C o o = cl C . C^{oo}=\operatorname{cl}C.
  12. f ( x ) = ฮด ( x | C ) = { 0 if x โˆˆ C + โˆž else f(x)=\delta(x|C)=\begin{cases}0&\,\text{if }x\in C\\ +\infty&\,\text{else}\end{cases}
  13. C C
  14. f * ( x * ) = ฮด ( x * | C o ) = ฮด * ( x * | C ) = sup x โˆˆ C โŸจ x * , x โŸฉ f^{*}(x^{*})=\delta(x^{*}|C^{o})=\delta^{*}(x^{*}|C)=\sup_{x\in C}\langle x^{*% },x\rangle
  15. C C
  16. f * * ( x ) = ฮด ( x | C o o ) f^{**}(x)=\delta(x|C^{oo})
  17. C = C o o C=C^{oo}
  18. f = f * * f=f^{**}

Bistochastic_quantum_channel.html

  1. ฯ• ( ฯ ) \phi(\rho)
  2. ฯ• ( I ) = I \phi(I)=I

Black-oil_equations.html

  1. โˆ‚ โˆ‚ t [ ฯ• ( S o B o + R V S g B g ) ] + โˆ‡ โ‹… ( 1 B o u โ†’ o + R V B g u โ†’ g ) = 0 \frac{\partial}{\partial t}\left[\phi\left(\frac{S_{o}}{B_{o}}+\frac{R_{V}S_{g% }}{B_{g}}\right)\right]+\nabla\cdot\left(\frac{1}{B_{o}}\vec{u}_{o}+\frac{R_{V% }}{B_{g}}\vec{u}_{g}\right)=0
  2. โˆ‚ โˆ‚ t [ ฯ• ( S w B w ) ] + โˆ‡ โ‹… ( 1 B w u โ†’ w ) = 0 \frac{\partial}{\partial t}\left[\phi\left(\frac{S_{w}}{B_{w}}\right)\right]+% \nabla\cdot\left(\frac{1}{B_{w}}\vec{u}_{w}\right)=0
  3. โˆ‚ โˆ‚ t [ ฯ• ( R S S o B o + S g B g ) ] + โˆ‡ โ‹… ( R S B o u โ†’ o + 1 B g u โ†’ g ) = 0 \frac{\partial}{\partial t}\left[\phi\left(\frac{R_{S}S_{o}}{B_{o}}+\frac{S_{g% }}{B_{g}}\right)\right]+\nabla\cdot\left(\frac{R_{S}}{B_{o}}\vec{u}_{o}+\frac{% 1}{B_{g}}\vec{u}_{g}\right)=0
  4. ฯ• \phi
  5. S w S_{w}
  6. S o , S g S_{o},S_{g}
  7. u โ†’ o , u โ†’ w , u โ†’ g \vec{u}_{o},\vec{u}_{w},\vec{u}_{g}
  8. B o B_{o}
  9. B w B_{w}
  10. B g B_{g}
  11. R S R_{S}
  12. R V R_{V}

Black_Widow_Pulsar.html

  1. 1.66 M โŠ™ 1.66M_{\odot}
  2. 2.4 M โŠ™ 2.4M_{\odot}

Blake_number.html

  1. B = u ฯ D h ฮผ ( 1 - ฯต ) B=\frac{u\rho D_{h}}{\mu(1-\epsilon)}
  2. < v a r > ฮต <var>ฮต

Blakersโ€“Massey_theorem.html

  1. A โ† f C โ†’ g B A\stackrel{f}{\leftarrow}C\stackrel{g}{\rightarrow}B
  2. ( A , C ) โ†’ ( X , B ) (A,C)\rightarrow(X,B)\,

Blasius_function.html

  1. 2 f x x x + f f x x = 0 2f_{xxx}+f\,f_{xx}=0

Bloch's_principle.html

  1. { f n } \{f_{n}\}
  2. z n z_{n}
  3. ฯ n \rho_{n}
  4. lim n โ†’ โˆž ฯ n = 0 \lim_{n\rightarrow\infty}\rho_{n}=0
  5. f n ( z n + ฯ n z ) โ†’ f , f_{n}(z_{n}+\rho_{n}z)\to f,\,

Bloch_group.html

  1. Li 2 ( z ) = โˆ‘ k = 1 โˆž z k k 2 . \operatorname{Li}_{2}(z)=\sum_{k=1}^{\infty}{z^{k}\over k^{2}}.
  2. Li 2 ( z ) = - โˆซ 0 z log ( 1 - t ) t d t . \operatorname{Li}_{2}(z)=-\int_{0}^{z}{\log(1-t)\over t}\,\mathrm{d}t.
  3. D 2 ( z ) = Im ( Li 2 ( z ) ) + arg ( 1 - z ) log | z | \operatorname{D}_{2}(z)=\operatorname{Im}(\operatorname{Li}_{2}(z))+\arg(1-z)% \log|z|
  4. z โˆˆ โ„‚ โˆ– { 0 , 1 } . z\in\mathbb{C}\setminus\{0,1\}.
  5. D 2 ( z ) \operatorname{D}_{2}(z)
  6. โ„‚ โˆ– { 0 , 1 } . \mathbb{C}\setminus\{0,1\}.
  7. D 2 ( z ) = D 2 ( 1 - 1 z ) = D 2 ( 1 1 - z ) = - D 2 ( 1 z ) = - D 2 ( 1 - z ) = - D 2 ( - z 1 - z ) . \operatorname{D}_{2}(z)=\operatorname{D}_{2}\left(1-\frac{1}{z}\right)=% \operatorname{D}_{2}\left(\frac{1}{1-z}\right)=-\operatorname{D}_{2}\left(% \frac{1}{z}\right)=-\operatorname{D}_{2}(1-z)=-\operatorname{D}_{2}\left(\frac% {-z}{1-z}\right).
  8. D 2 ( x ) + D 2 ( y ) + D 2 ( 1 - x 1 - x y ) + D 2 ( 1 - x y ) + D 2 ( 1 - y 1 - x y ) = 0. \operatorname{D}_{2}(x)+\operatorname{D}_{2}(y)+\operatorname{D}_{2}\left(% \frac{1-x}{1-xy}\right)+\operatorname{D}_{2}(1-xy)+\operatorname{D}_{2}\left(% \frac{1-y}{1-xy}\right)=0.
  9. โ„ค ( K ) = โ„ค [ K โˆ– { 0 , 1 } ] \mathbb{Z}(K)=\mathbb{Z}[K\setminus\{0,1\}]
  10. [ x ] + [ y ] + [ 1 - x 1 - x y ] + [ 1 - x y ] + [ 1 - y 1 - x y ] [x]+[y]+\left[\frac{1-x}{1-xy}\right]+[1-xy]+\left[\frac{1-y}{1-xy}\right]
  11. B โˆ™ : A ( K ) โŸถ d โˆง 2 K * \operatorname{B}^{\bullet}:A(K)\stackrel{d}{\longrightarrow}\wedge^{2}K^{*}
  12. d [ x ] = x โˆง ( 1 - x ) d[x]=x\wedge(1-x)
  13. B 2 ( K ) = H 1 ( Spec ( K ) , B โˆ™ ) \operatorname{B}_{2}(K)=\operatorname{H}^{1}(\operatorname{Spec}(K),% \operatorname{B}^{\bullet})
  14. 0 โŸถ B 2 ( K ) โŸถ A ( K ) โŸถ d โˆง 2 K * โŸถ K 2 ( K ) โŸถ 0 0\longrightarrow\operatorname{B}_{2}(K)\longrightarrow A(K)\stackrel{d}{% \longrightarrow}\wedge^{2}K^{*}\longrightarrow\operatorname{K}_{2}(K)\longrightarrow 0
  15. [ x ] + [ 1 - x ] โˆˆ B 2 ( K ) [x]+[1-x]\in\operatorname{B}_{2}(K)
  16. coker ( ฯ€ 3 ( BGM ( K ) + ) โ†’ K 3 ( K ) ) = B 2 ( K ) / 2 c \operatorname{coker}(\pi_{3}(\operatorname{BGM}(K)^{+})\rightarrow% \operatorname{K}_{3}(K))=\operatorname{B}_{2}(K)/2c
  17. 0 โ†’ Tor ( K * , K * ) โˆผ โ†’ K 3 ( K ) i n d โ†’ B 2 ( K ) โ†’ 0 0\rightarrow\operatorname{Tor}(K^{*},K^{*})^{\sim}\rightarrow\operatorname{K}_% {3}(K)_{ind}\rightarrow\operatorname{B}_{2}(K)\rightarrow 0
  18. D 2 ( z ) D_{2}(z)
  19. โ„‚ โˆ– { 0 , 1 } = โ„‚ P 1 โˆ– { 0 , 1 , โˆž } \mathbb{C}\setminus\{0,1\}=\mathbb{C}P^{1}\setminus\{0,1,\infty\}
  20. โ„ 3 \mathbb{H}^{3}
  21. โ„ 3 = โ„‚ ร— โ„ > 0 \mathbb{H}^{3}=\mathbb{C}\times\mathbb{R}_{>0}
  22. โ„‚ โˆช { โˆž } = โ„‚ P 1 \mathbb{C}\cup\{\infty\}=\mathbb{C}P^{1}
  23. โ„ 3 \mathbb{H}^{3}
  24. ( p 0 , p 1 , p 2 , p 3 ) (p_{0},p_{1},p_{2},p_{3})
  25. โŸจ p 0 , p 1 , p 2 , p 3 โŸฉ \left\langle p_{0},p_{1},p_{2},p_{3}\right\rangle
  26. p 1 , โ€ฆ , p 3 โˆˆ โ„‚ P 1 p_{1},\ldots,p_{3}\in\mathbb{C}P^{1}
  27. โŸจ p 0 , p 1 , p 2 , p 3 โŸฉ = D 2 ( ( p 0 - p 2 ) ( p 1 - p 3 ) ( p 0 - p 1 ) ( p 2 - p 3 ) ) . \left\langle p_{0},p_{1},p_{2},p_{3}\right\rangle=D_{2}\left(\frac{(p_{0}-p_{2% })(p_{1}-p_{3})}{(p_{0}-p_{1})(p_{2}-p_{3})}\right)\ .
  28. D 2 ( z ) = โŸจ 0 , 1 , z , โˆž โŸฉ D_{2}(z)=\left\langle 0,1,z,\infty\right\rangle
  29. D 2 ( z ) D_{2}(z)
  30. ( p 0 , p 1 , p 2 , p 3 , p 4 ) (p_{0},p_{1},p_{2},p_{3},p_{4})
  31. โŸจ โˆ‚ ( p 0 , p 1 , p 2 , p 3 , p 4 ) โŸฉ = โˆ‘ i = 0 4 ( - 1 ) i โŸจ p 0 , . . , p ^ i , . . , p 4 โŸฉ = 0โ€…. \left\langle\partial(p_{0},p_{1},p_{2},p_{3},p_{4})\right\rangle=\sum_{i=0}^{4% }(-1)^{i}\left\langle p_{0},..,\hat{p}_{i},..,p_{4}\right\rangle=0\ .
  32. X = โ„ 3 / ฮ“ X=\mathbb{H}^{3}/\Gamma
  33. X = โ‹ƒ j = 1 n ฮ” ( z j ) X=\bigcup^{n}_{j=1}\Delta(z_{j})
  34. ฮ” ( z j ) \Delta(z_{j})
  35. โˆ‚ โ„ 3 \partial\mathbb{H}^{3}
  36. z j z_{j}
  37. Im z > 0 \,\text{Im}\ z>0
  38. 0 , 1 , z , โˆž 0,1,z,\infty
  39. z z
  40. Im z > 0 \,\text{Im}\ z>0
  41. z z
  42. z z
  43. ฮ” \Delta
  44. v o l ( ฮ” ( z ) ) = D 2 ( z ) vol(\Delta(z))=D_{2}(z)
  45. D 2 ( z ) D_{2}(z)
  46. v o l ( X ) = โˆ‘ j = 1 n vol(X)=\sum^{n}_{j=1}
  47. Im z j > 0 \,\text{Im}\ z_{j}>0
  48. j j

Bloch_wave_โ€“_MoM_method.html

  1. E = - j k ฮท [ A + 1 k 2 โˆ‡ ( โˆ‡ โˆ™ A ) ] ( 1.1 ) E~{}=~{}-jk\eta\left[A+\frac{1}{k^{2}}\nabla(\nabla\bullet A)\right]~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1.1)
  2. โˆ‡ 2 A + k 2 A = - J ( 1.2 ) \nabla^{2}A+k^{2}A~{}=~{}-J~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1.2)
  3. k 2 = ฯ‰ 2 ( ฮผ ฯต ) ( 1.3 ) k^{2}~{}=~{}\omega^{2}(\mu\epsilon)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(% 1.3)
  4. J ( x , y , z ) = โˆ‘ m n p J ( ฮฑ m , ฮฒ n , ฮณ p ) e j ( ฮฑ m x + ฮฒ n y + ฮณ p z ) ( 2.1 a ) J(x,y,z)~{}=~{}\sum_{mnp}~{}J(\alpha_{m},\beta_{n},\gamma_{p})~{}e^{j(\alpha_{% m}x+\beta_{n}y+\gamma_{p}z)}~{}~{}~{}~{}~{}(2.1a)
  5. E ( x , y , z ) = โˆ‘ m n p E ( ฮฑ m , ฮฒ n , ฮณ p ) e j ( ฮฑ m x + ฮฒ n y + ฮณ p z ) ( 2.1 b ) E(x,y,z)~{}=~{}\sum_{mnp}~{}E(\alpha_{m},\beta_{n},\gamma_{p})~{}e^{j(\alpha_{% m}x+\beta_{n}y+\gamma_{p}z)}~{}~{}~{}~{}(2.1b)
  6. A ( x , y , z ) = โˆ‘ m n p A ( ฮฑ m , ฮฒ n , ฮณ p ) e j ( ฮฑ m x + ฮฒ n y + ฮณ p z ) ( 2.1 c ) A(x,y,z)~{}=~{}\sum_{mnp}~{}A(\alpha_{m},\beta_{n},\gamma_{p})~{}e^{j(\alpha_{% m}x+\beta_{n}y+\gamma_{p}z)}~{}~{}~{}(2.1c)
  7. ฮฑ m = k 0 sin ฮธ 0 cos ฯ• 0 + 2 m ฯ€ l x ( 2.2 a ) \alpha_{m}~{}=~{}k_{0}~{}\sin\theta_{0}~{}\cos\phi_{0}~{}+~{}\frac{2m\pi}{l_{x% }}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.2a)
  8. ฮฒ n = k 0 sin ฮธ 0 sin ฯ• 0 + 2 n ฯ€ l y ( 2.2 b ) \beta_{n}~{}=~{}k_{0}~{}\sin\theta_{0}~{}\sin\phi_{0}~{}+~{}\frac{2n\pi}{l_{y}% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.2b)
  9. ฮณ p = k 0 cos ฮธ 0 + 2 p ฯ€ l z ( 2.2 c ) \gamma_{p}~{}=~{}k_{0}~{}\cos\theta_{0}~{}+~{}\frac{2p\pi}{l_{z}}~{}~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.2c)
  10. k 0 = 2 ฯ€ / ฮป ( 2.3 ) k_{0}~{}=~{}2\pi/\lambda~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.3)
  11. E ( ฮฑ m , ฮฒ n , ฮณ p ) = j k ฮท k 2 - ฮฑ m 2 - ฮฒ n 2 - ฮณ p 2 G m n p J ( ฮฑ m , ฮฒ n , ฮณ p ) ( 3.1 ) E(\alpha_{m},\beta_{n},\gamma_{p})~{}=~{}\frac{jk\eta}{k^{2}-\alpha_{m}^{2}-% \beta_{n}^{2}-\gamma_{p}^{2}}~{}G_{mnp}~{}J(\alpha_{m},\beta_{n},\gamma_{p})~{% }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.1)
  12. G m n p = ( 1 - ฮฑ m 2 k 2 - ฮฑ m ฮฒ n k 2 - ฮฑ m ฮณ p k 2 - ฮฑ m ฮฒ n k 2 1 - ฮฒ n 2 k 2 - ฮฒ n ฮณ p k 2 - ฮฑ m ฮณ p k 2 - ฮฒ n ฮณ p k 2 1 - ฮณ p 2 k 2 ) ( 3.2 ) G_{mnp}~{}=~{}\left(\begin{matrix}1-\frac{\alpha_{m}^{2}}{k^{2}}&-\frac{\alpha% _{m}\beta_{n}}{k^{2}}&-\frac{\alpha_{m}\gamma_{p}}{k^{2}}\\ -\frac{\alpha_{m}\beta_{n}}{k^{2}}&1-\frac{\beta_{n}^{2}}{k^{2}}&-\frac{\beta_% {n}\gamma_{p}}{k^{2}}\\ -\frac{\alpha_{m}\gamma_{p}}{k^{2}}&-\frac{\beta_{n}\gamma_{p}}{k^{2}}&1-\frac% {\gamma_{p}^{2}}{k^{2}}\end{matrix}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.2)
  13. โˆ‘ m n p 1 k 2 - ฮฑ m 2 - ฮฒ n 2 - ฮณ p 2 G m n p J ( ฮฑ m , ฮฒ n , ฮณ p ) e j ( ฮฑ m x + ฮฒ n y + ฮณ p z ) = 0 ( 3.3 ) ~{}\sum_{mnp}~{}\frac{1}{k^{2}-\alpha_{m}^{2}-\beta_{n}^{2}-\gamma_{p}^{2}}~{}% G_{mnp}~{}J(\alpha_{m},\beta_{n},\gamma_{p})~{}e^{j(\alpha_{m}x+\beta_{n}y+% \gamma_{p}z)}~{}=~{}0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.3)
  14. J ( x , y , z ) = โˆ‘ j J j J j ( x , y , z ) ( 3.4 ) ~{}J(x,y,z)~{}=~{}\sum_{j}~{}J_{j}~{}J_{j}(x,y,z)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}(3.4)
  15. โˆ‘ j J j [ โˆ‘ m n p J i ( - ฮฑ m , - ฮฒ n , - ฮณ p ) G m n p J j ( ฮฑ m , ฮฒ n , ฮณ p ) k 2 - ฮฑ m 2 - ฮฒ n 2 - ฮณ p 2 ] = 0 ( 3.5 ) ~{}\sum_{j}~{}J_{j}~{}\left[~{}\sum_{mnp}~{}\frac{J_{i}(-\alpha_{m},-\beta_{n}% ,-\gamma_{p})~{}G_{mnp}~{}J_{j}(\alpha_{m},\beta_{n},\gamma_{p})}{k^{2}-\alpha% _{m}^{2}-\beta_{n}^{2}-\gamma_{p}^{2}}\right]~{}=~{}0~{}~{}~{}(3.5)

Blood_gas_tension.html

  1. C a O 2 = 1.36 * H g b * S a O 2 100 + 0.0031 * P a O 2 C_{a}O_{2}=1.36*Hgb*\frac{S_{a}O_{2}}{100}+0.0031*P_{a}O_{2}
  2. S O 2 = ( 23 , 400 p O 2 3 + 150 p O 2 + 1 ) - 1 SO_{2}=(\frac{23,400}{pO_{2}^{3}+150pO_{2}}+1)^{-1}

Boasโ€“Buck_polynomials.html

  1. C ( z t r B ( t ) ) = โˆ‘ n โ‰ฅ 0 ฮฆ n ( r ) ( z ) t n \displaystyle C(zt^{r}B(t))=\sum_{n\geq 0}\Phi_{n}^{(r)}(z)t^{n}

Bochner_measurable_function.html

  1. f ( t ) = lim n โ†’ โˆž f n ( t ) for almost every t , f(t)=\lim_{n\rightarrow\infty}f_{n}(t)\,\text{ for almost every }t,\,
  2. f n f_{n}
  3. f - 1 { x } f^{-1}\{x\}
  4. ฮผ \mu

Bochnerโ€“Martinelli_formula.html

  1. ฮถ ฮถ
  2. z z
  3. ฯ‰ ( ฮถ , z ) ฯ‰(ฮถ,z)
  4. ฮถ ฮถ
  5. ( n , n โˆ’ 1 ) (n,nโˆ’1)
  6. ฯ‰ ( ฮถ , z ) = ( n - 1 ) ! ( 2 ฯ€ i ) n 1 | z - ฮถ | 2 n โˆ‘ 1 โ‰ค j โ‰ค n ( ฮถ ยฏ j - z ยฏ j ) d ฮถ ยฏ 1 and d ฮถ 1 and โ‹ฏ and d ฮถ j and โ‹ฏ and d ฮถ ยฏ n and d ฮถ n \omega(\zeta,z)=\frac{(n-1)!}{(2\pi i)^{n}}\frac{1}{|z-\zeta|^{2n}}\sum_{1\leq j% \leq n}(\overline{\zeta}_{j}-\overline{z}_{j})\,d\overline{\zeta}_{1}\and d% \zeta_{1}\and\cdots\and d\zeta_{j}\and\cdots\and d\overline{\zeta}_{n}\and d% \zeta_{n}
  7. f f
  8. D D
  9. โˆ‚ D โˆ‚D
  10. z z
  11. D D
  12. f ( z ) = โˆซ โˆ‚ D f ( ฮถ ) ฯ‰ ( ฮถ , z ) - โˆซ D โˆ‚ ยฏ f ( ฮถ ) and ฯ‰ ( ฮถ , z ) . \displaystyle f(z)=\int_{\partial D}f(\zeta)\omega(\zeta,z)-\int_{D}\overline{% \partial}f(\zeta)\and\omega(\zeta,z).
  13. f f
  14. f ( z ) = โˆซ โˆ‚ D f ( ฮถ ) ฯ‰ ( ฮถ , z ) . \displaystyle f(z)=\int_{\partial D}f(\zeta)\omega(\zeta,z).

Bockstein_spectral_sequence.html

  1. 0 โ†’ C โ†’ ๐‘ C โ†’ mod p C โŠ— โ„ค / p โ†’ 0 0\to C\overset{p}{\to}C\overset{\,\text{mod }p}{\to}C\otimes\mathbb{Z}/p\to 0
  2. H * ( C ) โ†’ i = p H * ( C ) โ†’ ๐‘— H * ( C โŠ— โ„ค / p ) โ†’ ๐‘˜ H_{*}(C)\overset{i=p}{\to}H_{*}(C)\overset{j}{\to}H_{*}(C\otimes\mathbb{Z}/p)% \overset{k}{\to}
  3. H * ( C ) s , t = H s + t ( C ) H_{*}(C)_{s,t}=H_{s+t}(C)
  4. H * ( C โŠ— โ„ค / p ) H_{*}(C\otimes\mathbb{Z}/p)
  5. deg i = ( 1 , - 1 ) \operatorname{deg}i=(1,-1)
  6. deg j = ( 0 , 0 ) \operatorname{deg}j=(0,0)
  7. deg k = ( - 1 , 0 ) \operatorname{deg}k=(-1,0)
  8. E s , t 1 = H s + t ( C โŠ— โ„ค / p ) E_{s,t}^{1}=H_{s+t}(C\otimes\mathbb{Z}/p)
  9. d 1 = j โˆ˜ k {}^{1}d=j\circ k
  10. D r = p r - 1 H * ( C ) D^{r}=p^{r-1}H_{*}(C)
  11. D r โ†’ i = p D r โ†’ j r E r โ†’ ๐‘˜ D^{r}\overset{i=p}{\to}D^{r}\overset{{}^{r}j}{\to}E^{r}\overset{k}{\to}
  12. j r {}^{r}j
  13. ( mod p ) โˆ˜ p - r + 1 (\,\text{mod }p)\circ p^{-{r+1}}
  14. deg j r = ( - ( r - 1 ) , r - 1 ) \operatorname{deg}{}^{r}j=(-(r-1),r-1)
  15. D n r โŠ— - D_{n}^{r}\otimes-
  16. 0 โ†’ โ„ค โ†’ ๐‘ โ„ค โ†’ โ„ค / p โ†’ 0 0\to\mathbb{Z}\overset{p}{\to}\mathbb{Z}\to\mathbb{Z}/p\to 0
  17. 0 โ†’ Tor 1 โ„ค ( D n r , โ„ค / p ) โ†’ D n r โ†’ ๐‘ D n r โ†’ D n r โŠ— โ„ค / p โ†’ 0 0\to\operatorname{Tor}_{1}^{\mathbb{Z}}(D_{n}^{r},\mathbb{Z}/p)\to D_{n}^{r}% \overset{p}{\to}D_{n}^{r}\to D_{n}^{r}\otimes\mathbb{Z}/p\to 0
  18. D n r โ†’ ๐‘ D n r D^{r}_{n}\overset{p}{\to}D^{r}_{n}
  19. 0 โ†’ ( p r - 1 H n ( C ) ) โŠ— โ„ค / p โ†’ E n , 0 r โ†’ Tor ( p r - 1 H n - 1 ( C ) , โ„ค / p ) โ†’ 0 0\to(p^{r-1}H_{n}(C))\otimes\mathbb{Z}/p\to E^{r}_{n,0}\to\operatorname{Tor}(p% ^{r-1}H_{n-1}(C),\mathbb{Z}/p)\to 0
  20. r = 1 r=1
  21. H * ( C ) H_{*}(C)
  22. โ„ค / p s \mathbb{Z}/p^{s}
  23. H * ( C ) H_{*}(C)
  24. r โ†’ โˆž r\to\infty
  25. E โˆž E^{\infty}
  26. ( free part of H * ( C ) ) โŠ— โ„ค / p (\,\text{free part of }H_{*}(C))\otimes\mathbb{Z}/p

Body_adiposity_index.html

  1. 100 ร— hip circumference in m height in m ร— height - 18 \frac{100\times\,\text{hip circumference in m}}{\,\text{height in m}\times% \sqrt{\,\text{height}}}-18

Boehmians.html

  1. [ 0 , โˆž ) [0,\infty)
  2. [ 0 , โˆž ) [0,\infty)
  3. โ„ N \mathbb{R}^{N}
  4. X X
  5. G G
  6. X X
  7. ฮ” \Delta
  8. G G
  9. ( ฯ• n ) , ( ฯˆ n ) โˆˆ ฮ” (\phi_{n}),(\psi_{n})\in\Delta
  10. ( ฯ• n ฯˆ n ) โˆˆ ฮ” (\phi_{n}\psi_{n})\in\Delta
  11. x , y โˆˆ X x,y\in X
  12. ฯ• n x = ฯ• n y \phi_{n}x=\phi_{n}y
  13. ( ฯ• n ) โˆˆ ฮ” (\phi_{n})\in\Delta
  14. n โˆˆ โ„• n\in\mathbb{N}
  15. x = y x=y
  16. ๐’œ = { ( ( x n ) , ( ฯ• n ) ) : x n โˆˆ X , ( ฯ• n ) โˆˆ ฮ” , ฯ• m x n = ฯ• n x m for all m , n โˆˆ โ„• } \mathcal{A}=\{((x_{n}),(\phi_{n})):x_{n}\in X,(\phi_{n})\in\Delta,\phi_{m}x_{n% }=\phi_{n}x_{m}\,\text{ for all }m,n\in\mathbb{N}\}
  17. ๐’œ \mathcal{A}
  18. ( ( x n ) , ( ฯ• n ) ) ((x_{n}),(\phi_{n}))
  19. ( ( y n ) , ( ฯˆ n ) ) ((y_{n}),(\psi_{n}))
  20. ฯ• m y n = ฯˆ n x m for all m , n โˆˆ โ„• \phi_{m}y_{n}=\psi_{n}x_{m}\,\text{ for all }m,n\in\mathbb{N}
  21. โ„ฌ ( X , ฮ” ) \mathcal{B}(X,\Delta)
  22. ๐’œ \mathcal{A}
  23. โ„ฌ ( X , ฮ” ) = ๐’œ / \mathcal{B}(X,\Delta)=\mathcal{A}/

Bollobaฬsโ€“Riordan_polynomial.html

  1. R G ( x , y , z ) = โˆ‘ F x r ( G ) - r ( F ) y n ( F ) z k ( F ) - b c ( F ) + n ( F ) RG(x,y,z)=\sum_{F}x^{r(G)-r(F)}y^{n(F)}z^{k(F)-bc(F)+n(F)}

Bond_characterization.html

  1. L L
  2. ฮณ \gamma
  3. ฮณ = 3 t b 2 E 1 t w 1 3 E 2 t w 2 3 16 L 4 ( E 1 t w 1 3 + E 2 t w 2 3 ) \gamma=\frac{3t_{b}^{2}E_{1}t_{w1}^{3}E_{2}t_{w2}^{3}}{16L^{4}(E_{1}t_{w1}^{3}% +E_{2}t_{w2}^{3})}
  4. E E
  5. t w t_{w}
  6. t b t_{b}
  7. L L
  8. C = 8 L 3 E * w t 3 C=\frac{8L^{3}}{E^{*}wt^{3}}
  9. L L
  10. w w
  11. t t
  12. E * E^{*}
  13. G I C G_{IC}
  14. G I C = 3 E * ฮด 2 t 3 16 L 4 G_{IC}=\frac{3E^{*}\delta^{2}t^{3}}{16L^{4}}
  15. ฮด \delta
  16. C = 8 L 3 E * w t 3 ( 1 + 3 c ( t L ) 2 - n + ฮฑ s E * 4 ฮผ ( t L ) 2 ) C=\frac{8L^{3}}{E^{*}wt^{3}}\left(1+3c\left(\frac{t}{L}\right)^{2-n}+\frac{% \alpha_{s}E^{*}}{4\mu}\left(\frac{t}{L}\right)^{2}\right)
  17. 8 L 3 E * w t 3 \frac{8L^{3}}{E^{*}wt^{3}}
  18. n n
  19. c c
  20. ฮฑ S \alpha_{S}
  21. K I C K_{IC}
  22. K I C K_{IC}
  23. G I G_{I}
  24. K I K_{I}
  25. K I K_{I}
  26. G I G_{I}
  27. K I C K_{IC}
  28. G I C G_{IC}
  29. G I C G_{IC}
  30. K I C K_{IC}
  31. a a
  32. a c a_{c}
  33. a c a_{c}
  34. F m a x F_{max}
  35. Y M I N Y_{MIN}
  36. K I C K_{IC}
  37. w w
  38. t t
  39. K I C = F M A X t โ‹… w โ‹… Y M I N K_{IC}=\frac{F_{MAX}}{t\cdot\sqrt{w}}\cdot Y_{MIN}
  40. F M A X F_{MAX}
  41. Y M I N Y_{MIN}
  42. G I C G_{IC}
  43. E E
  44. v v
  45. G I C = K I C 2 E โ‹… ( 1 - v 2 ) G_{IC}=\frac{K^{2}_{IC}}{E}\cdot(1-v^{2})
  46. d d
  47. d d
  48. d = ฮป 1 ฮป 2 2 n ( ฮป 2 - ฮป 1 ) d=\frac{\lambda_{1}\lambda_{2}}{2n(\lambda_{2}-\lambda_{1})}
  49. n n
  50. ฮป 1 \lambda_{1}
  51. ฮป 2 \lambda_{2}

Bone_remodeling_period.html

  1. ฯƒ f = M W T M f \sigma_{f}=\frac{MWT}{M_{f}}
  2. ฯƒ r = M W T M r \sigma_{r}=\frac{MWT}{M_{r}}
  3. ฯƒ ( r + f ) = M W T M r + M W T M f \sigma_{(r+f)}=\frac{MWT}{M_{r}}+\frac{MWT}{M_{f}}

Boole_polynomials.html

  1. โˆ‘ s n ( x ) t n / n ! = ( 1 + t ) x 1 + ( 1 + t ) ฮป \displaystyle\sum s_{n}(x)t^{n}/n!=\frac{(1+t)^{x}}{1+(1+t)^{\lambda}}

Born_series.html

  1. V V
  2. G 0 V , G_{0}V,
  3. G 0 G_{0}
  4. V โ†’ ฮป V V\to\lambda V
  5. G 0 V G_{0}V
  6. V V
  7. | ฯˆ โŸฉ = | ฯ• โŸฉ + G 0 ( E ) V | ฯ• โŸฉ + [ G 0 ( E ) V ] 2 | ฯ• โŸฉ + [ G 0 ( E ) V ] 3 | ฯ• โŸฉ + โ€ฆ |\psi\rangle=|\phi\rangle+G_{0}(E)V|\phi\rangle+[G_{0}(E)V]^{2}|\phi\rangle+[G% _{0}(E)V]^{3}|\phi\rangle+\dots
  8. | ฯˆ โŸฉ = | ฯ• โŸฉ + G 0 ( E ) V | ฯˆ โŸฉ . |\psi\rangle=|\phi\rangle+G_{0}(E)V|\psi\rangle.
  9. G 0 G_{0}
  10. | ฯˆ ( + ) โŸฉ |\psi^{(+)}\rangle
  11. | ฯˆ ( - ) โŸฉ |\psi^{(-)}\rangle
  12. | ฯˆ ( P ) โŸฉ |\psi^{(P)}\rangle
  13. | ฯˆ โŸฉ |\psi\rangle
  14. | ฯ• โŸฉ |\phi\rangle
  15. G 0 V G_{0}V
  16. | ฯˆ โŸฉ = [ I - G 0 ( E ) V ] - 1 | ฯ• โŸฉ = [ V - V G 0 ( E ) V ] - 1 V | ฯ• โŸฉ . |\psi\rangle=[I-G_{0}(E)V]^{-1}|\phi\rangle=[V-VG_{0}(E)V]^{-1}V|\phi\rangle.
  17. T ( E ) = V + V G 0 ( E ) V + V [ G 0 ( E ) V ] 2 + V [ G 0 ( E ) V ] 3 + โ€ฆ T(E)=V+VG_{0}(E)V+V[G_{0}(E)V]^{2}+V[G_{0}(E)V]^{3}+\dots
  18. G 0 G_{0}
  19. G 0 ( + ) ( E ) G_{0}^{(+)}(E)
  20. G ( E ) = G 0 ( E ) + G 0 ( E ) V G ( E ) . G(E)=G_{0}(E)+G_{0}(E)VG(E).
  21. G ( E ) = ( E - H + i ฯต ) - 1 G(E)=(E-H+i\epsilon)^{-1}
  22. G ( E ) = G 0 ( E ) + G 0 ( E ) V G 0 ( E ) + [ G 0 ( E ) V ] 2 G 0 ( E ) + [ G 0 ( E ) V ] 3 G 0 ( E ) + โ€ฆ G(E)=G_{0}(E)+G_{0}(E)VG_{0}(E)+[G_{0}(E)V]^{2}G_{0}(E)+[G_{0}(E)V]^{3}G_{0}(E% )+\dots

Borwein_integral.html

  1. โˆซ 0 โˆž sin ( x ) x d x = ฯ€ / 2 \displaystyle\int_{0}^{\infty}\frac{\sin(x)}{x}\,dx=\pi/2
  2. โˆซ 0 โˆž sin ( x ) x sin ( x / 3 ) x / 3 โ‹ฏ sin ( x / 13 ) x / 13 d x = ฯ€ / 2 \int_{0}^{\infty}\frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3}\cdots\frac{\sin(x/13)}% {x/13}\,dx=\pi/2
  3. โˆซ 0 โˆž sin ( x ) x sin ( x / 3 ) x / 3 โ‹ฏ sin ( x / 15 ) x / 15 d x \displaystyle\int_{0}^{\infty}\frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3}\cdots% \frac{\sin(x/15)}{x/15}\,dx
  4. โˆซ 0 โˆž 2 cos ( x ) sin ( x ) x sin ( x / 3 ) x / 3 โ‹ฏ sin ( x / 111 ) x / 111 d x = ฯ€ / 2 , \int_{0}^{\infty}2\cos(x)\frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3}\cdots\frac{% \sin(x/111)}{x/111}\,dx=\pi/2,
  5. โˆซ 0 โˆž 2 cos ( x ) sin ( x ) x sin ( x / 3 ) x / 3 โ‹ฏ sin ( x / 111 ) x / 111 sin ( x / 113 ) x / 113 d x < ฯ€ / 2 , \int_{0}^{\infty}2\cos(x)\frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3}\cdots\frac{% \sin(x/111)}{x/111}\frac{\sin(x/113)}{x/113}\,dx<\pi/2,

Bott_residue_formula.html

  1. โˆ‘ v ( p ) = 0 P ( A p ) det A p = โˆซ M P ( i ฮ˜ / 2 ฯ€ ) \sum_{v(p)=0}\frac{P(A_{p})}{\det A_{p}}=\int_{M}P(i\Theta/2\pi)

Boucโ€“Wen_model_of_hysteresis.html

  1. m u ยจ ( t ) + c u ห™ ( t ) + F ( t ) = f ( t ) m\ddot{u}(t)+c\dot{u}(t)+F(t)=f(t)
  2. m \textstyle m
  3. u ( t ) \textstyle u(t)
  4. c \textstyle c
  5. F ( t ) \textstyle F(t)
  6. f ( t ) \textstyle f(t)
  7. a := k f k i \textstyle a:=\frac{k_{f}}{k_{i}}
  8. k f \textstyle k_{f}
  9. k i := F y u y \textstyle k_{i}:=\frac{F_{y}}{u_{y}}
  10. F y \textstyle F_{y}
  11. u y \textstyle u_{y}
  12. z ( t ) \textstyle z(t)
  13. z ( 0 ) = 0 \textstyle z(0)=0
  14. z ห™ ( t ) = A u ห™ ( t ) - ฮฒ | u ห™ ( t ) | | z ( t ) | n - 1 z ( t ) - ฮณ u ห™ ( t ) | z ( t ) | n \dot{z}(t)=A\dot{u}(t)-\beta|\dot{u}(t)||z(t)|^{n-1}z(t)-\gamma\dot{u}(t)|z(t)% |^{n}
  15. sign \textstyle\operatorname{sign}
  16. A \textstyle A
  17. ฮฒ > 0 \textstyle\beta>0
  18. ฮณ \textstyle\gamma
  19. n \textstyle n
  20. n = โˆž \textstyle n=\infty
  21. ฮฒ \textstyle\beta
  22. ฮฑ \textstyle\alpha
  23. ฮณ \textstyle\gamma
  24. ฮฒ \textstyle\beta
  25. ฮฒ \textstyle\beta
  26. ฮณ \textstyle\gamma
  27. F ( t ) \textstyle F(t)
  28. n \textstyle n
  29. A \textstyle A
  30. ฮฒ \textstyle\beta
  31. ฮณ \textstyle\gamma
  32. A = 1 \textstyle A=1
  33. n \textstyle n
  34. n \textstyle n
  35. ฮฒ \textstyle\beta
  36. ฮณ \textstyle\gamma
  37. ฮณ := [ - ฮฒ , ฮฒ ] \textstyle\gamma:=[-\beta,\beta]
  38. ฯ‰ 2 := k i m \textstyle\omega^{2}:=\frac{k_{i}}{m}
  39. J / k g \textstyle J/kg
  40. ฮฝ ( ฮต ) \textstyle\nu(\varepsilon)
  41. ฮท ( ฮต ) \textstyle\eta(\varepsilon)
  42. h ( z ) \textstyle h(z)
  43. ฮฝ ( ฮต ) \textstyle\nu(\varepsilon)
  44. A ( ฮต ) \textstyle A(\varepsilon)
  45. ฮท ( ฮต ) \textstyle\eta(\varepsilon)
  46. ฮต \textstyle\varepsilon
  47. h ( z ) \textstyle h(z)
  48. z u \textstyle z_{u}
  49. z \textstyle z
  50. ฮด ฮฝ > 0 \textstyle\delta_{\nu}>0
  51. ฮด A > 0 \textstyle\delta_{A}>0
  52. ฮด ฮท > 0 \textstyle\delta_{\eta}>0
  53. ฮฝ 0 \textstyle\nu_{0}
  54. A 0 \textstyle A_{0}
  55. ฮท 0 \textstyle\eta_{0}
  56. ฯˆ 0 \textstyle\psi_{0}
  57. ฮด ฯˆ \textstyle\delta_{\psi}
  58. ฮป \textstyle\lambda
  59. p \textstyle p
  60. ฯ‚ \textstyle\varsigma
  61. ฮด ฮฝ = 0 \textstyle\delta_{\nu}=0
  62. ฮด ฮท = 0 \textstyle\delta_{\eta}=0
  63. h ( z ) = 1 \textstyle h(z)=1
  64. m \textstyle m
  65. ฯ• \textstyle\phi
  66. C 1 ( u ห™ ( t ) , u ( t ) , z ( t ) , ฮฒ 1 , ฮฒ 2 , ฮฒ 3 ) = ฮฒ 1 sign ( u ห™ ( t ) z ( t ) ) + ฮฒ 2 sign ( u ( t ) u ห™ ( t ) ) + ฮฒ 3 sign ( u ( t ) z ( t ) ) C_{1}(\dot{u}(t),u(t),z(t),\beta_{1},\beta_{2},\beta_{3})=\beta_{1}% \operatorname{sign}(\dot{u}(t)z(t))+\beta_{2}\operatorname{sign}(u(t)\dot{u}(t% ))+\beta_{3}\operatorname{sign}(u(t)z(t))
  67. C 2 ( u ห™ ( t ) , u ( t ) , z ( t ) , ฮฒ 4 , ฮฒ 5 , ฮฒ 6 ) = ฮฒ 4 sign ( u ห™ ( t ) ) + ฮฒ 5 sign ( z ( t ) ) + ฮฒ 6 sign ( u ( t ) ) C_{2}(\dot{u}(t),u(t),z(t),\beta_{4},\beta_{5},\beta_{6})=\beta_{4}% \operatorname{sign}(\dot{u}(t))+\beta_{5}\operatorname{sign}(z(t))+\beta_{6}% \operatorname{sign}(u(t))
  68. ฮฒ i \textstyle\beta_{i}
  69. i = 1 , 2 , โ€ฆ , 6 \textstyle i=1,2,\ldots,6
  70. ฮฒ 2 \textstyle\beta_{2}
  71. ฮฒ 3 \textstyle\beta_{3}
  72. ฮฒ 6 \textstyle\beta_{6}
  73. u ( t ) \textstyle u(t)
  74. u ห™ ( t ) \textstyle\dot{u}(t)
  75. x 1 ( t ) = u ( t ) \textstyle x_{1}(t)=u(t)
  76. x ห™ 1 ( t ) = u ห™ ( t ) = x 2 ( t ) \textstyle\dot{x}_{1}(t)=\dot{u}(t)=x_{2}(t)
  77. x ห™ 2 ( t ) = u ยจ ( t ) \textstyle\dot{x}_{2}(t)=\ddot{u}(t)
  78. x 3 ( t ) = z ( t ) \textstyle x_{3}(t)=z(t)
  79. u ห™ ( t ) \dot{u}(t)
  80. sign \operatorname{sign}
  81. F 1 2 ( a , b , c ; w ) {}_{2}F_{1}(a,b,c;w)
  82. q = ฮฒ sign ( z ( t ) u ห™ ( t ) ) + ฮณ q=\beta\operatorname{sign}(z(t)\dot{u}(t))+\gamma
  83. A = 1 A=1
  84. u 0 u_{0}
  85. z 0 z_{0}
  86. z z
  87. n n
  88. n = 1 n=1
  89. n = 2 n=2
  90. n n
  91. a โˆˆ ( 0 , 1 ) \textstyle a\in(0,1)
  92. k i > 0 \textstyle k_{i}>0
  93. k f > 0 \textstyle k_{f}>0
  94. c > 0 \textstyle c>0
  95. A > 0 \textstyle A>0
  96. n > 1 \textstyle n>1
  97. ฮฒ > 0 \textstyle\beta>0
  98. ฮณ โˆˆ [ - ฮฒ , ฮฒ ] \textstyle\gamma\in[-\beta,\beta]
  99. A = 1 \textstyle A=1
  100. A ฮฒ + ฮณ = 1 \textstyle\frac{A}{\beta+\gamma}=1
  101. ฮณ \textstyle\gamma
  102. n \textstyle n
  103. a \textstyle a
  104. k i \textstyle k_{i}
  105. c \textstyle c

Bounded_type_(mathematics).html

  1. ฮฉ \Omega
  2. f f
  3. ฮฉ \Omega
  4. log + | f ( z ) | \log^{+}|f(z)|
  5. ฮฉ , \Omega,
  6. log + ( x ) = max { 0 , log ( x ) } \log^{+}(x)=\max\{0,\log(x)\}
  7. ฮฉ \Omega
  8. f f
  9. ฮฉ \Omega
  10. N ( ฮฉ ) N(\Omega)
  11. ฮฉ \Omega
  12. f ( z ) f(z)
  13. f ( z ) = P ( z ) / Q ( z ) f(z)=P(z)/Q(z)
  14. P ( z ) = f ( z ) / ( z + i ) n P(z)=f(z)/(z+i)^{n}
  15. Q ( z ) = 1 / ( z + i ) n . Q(z)=1/(z+i)^{n}.
  16. e x p ( a i z ) exp(aiz)
  17. Q ( z ) = 1 Q(z)=1
  18. Q ( z ) = e x p ( | a | i z ) Q(z)=exp(|a|iz)
  19. sin ( z ) = P ( z ) / Q ( z ) \sin(z)=P(z)/Q(z)
  20. P ( z ) = sin ( z ) e x p ( i z ) P(z)=\sin(z)exp(iz)
  21. Q ( z ) = e x p ( i z ) Q(z)=exp(iz)
  22. P ( z ) P(z)
  23. Q ( z ) , Q(z),
  24. f ( z ) = P ( z ) / Q ( z ) f(z)=P(z)/Q(z)
  25. P ( z ) = f ( z ) f ( z ) + i P(z)=\frac{f(z)}{f(z)+i}
  26. Q ( z ) = 1 f ( z ) + i Q(z)=\frac{1}{f(z)+i}
  27. f ( z ) / i f(z)/i
  28. P ( z ) / Q ( z ) P(z)/Q(z)
  29. P ( z ) P(z)
  30. Q ( z ) Q(z)
  31. P ( z ) / Q ( z ) = e x p ( - U ( z ) ) / e x p ( - V ( z ) ) P(z)/Q(z)=exp(-U(z))/exp(-V(z))
  32. U ( z ) U(z)
  33. V ( z ) V(z)
  34. U ( z ) = c - i p z - i โˆซ โ„ ( 1 ฮป - z - ฮป 1 + ฮป 2 ) d ฮผ ( ฮป ) U(z)=c-ipz-i\int_{\mathbb{R}}\left(\frac{1}{\lambda-z}-\frac{\lambda}{1+% \lambda^{2}}\right)d\mu(\lambda)
  35. V ( z ) = d - i q z - i โˆซ โ„ ( 1 ฮป - z - ฮป 1 + ฮป 2 ) d ฮฝ ( ฮป ) V(z)=d-iqz-i\int_{\mathbb{R}}\left(\frac{1}{\lambda-z}-\frac{\lambda}{1+% \lambda^{2}}\right)d\nu(\lambda)
  36. f ( z ) = exp ( i z ) exp ( i z ) exp ( - i / z ) f(z)=\exp(iz)\frac{\exp(i\sqrt{z})}{\exp(-i/\sqrt{z})}
  37. exp ( a i z ) \exp(aiz)
  38. sin ( z ) \sin(z)
  39. 1 2 ฯ€ i โˆซ - โˆž โˆž f ( t ) d t t - z \frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{f(t)dt}{t-z}
  40. f ( z ) f(z)

Box_counting.html

  1. \Epsilon \Epsilon
  2. ฯต \epsilon
  3. ฯต \epsilon
  4. ฯต \epsilon
  5. \Epsilon \Epsilon
  6. ฯต \epsilon
  7. ฯต \epsilon
  8. ฯต \epsilon
  9. ฯต \epsilon
  10. ฯต \epsilon
  11. ฯต \epsilon
  12. \Epsilon \Epsilon

Boฬˆhmer_integral.html

  1. C ( x , ฮฑ ) = โˆซ x โˆž t ฮฑ - 1 cos ( t ) d t \displaystyle C(x,\alpha)=\int_{x}^{\infty}t^{\alpha-1}\cos(t)\,dt
  2. S ( x , ฮฑ ) = โˆซ x โˆž t ฮฑ - 1 sin ( t ) d t \displaystyle S(x,\alpha)=\int_{x}^{\infty}t^{\alpha-1}\sin(t)\,dt

Boฬˆttcher's_equation.html

  1. F ( h ( z ) ) = ( F ( z ) ) n , F(h(z))=(F(z))^{n}~{},
  2. h h
  3. n n
  4. a a
  5. h ( z ) = a + c ( z - a ) n + O ( ( z - a ) n + 1 ) , h(z)=a+c(z-a)^{n}+O((z-a)^{n+1})~{},
  6. a a
  7. F F
  8. h ( z ) h(z)
  9. z < s u p > n z<sup>n

Braided_vector_space.html

  1. V \;V
  2. ฯ„ \tau\;
  3. ฯ„ : V โŠ— V โŸถ V โŠ— V \tau:\;V\otimes V\longrightarrow V\otimes V\,
  4. ฯ„ \tau\;
  5. V \;V
  6. x i x_{i}\;
  7. ฯ„ ( x i โŠ— x j ) = q i j ( x j โŠ— x i ) \tau(x_{i}\otimes x_{j})=q_{ij}(x_{j}\otimes x_{i})\,
  8. ฯ„ V , W \tau_{V,W}\;
  9. โ„ค 2 \mathbb{Z}_{2}
  10. V V\;
  11. [ x , y ] ฯ„ := ฮผ ( ( x โŠ— y ) - ฯ„ ( x โŠ— y ) ) ฮผ ( x โŠ— y ) := x y \;[x,y]_{\tau}:=\mu((x\otimes y)-\tau(x\otimes y))\qquad\mu(x\otimes y):=xy

Brake_specific_fuel_consumption.html

  1. B S F C = r P BSFC=\frac{r}{P}
  2. P = ฯ„ ฯ‰ P=\tau\omega
  3. ฯ‰ \omega
  4. ฯ„ \tau
  5. ฯ‰ \omega
  6. ฯ„ \tau

Bramsโ€“Taylor_procedure.html

  1. X X
  2. Y Y
  3. Y Y
  4. Y Y
  5. Y Y
  6. A , B , C A,B,C
  7. A A
  8. Y Y
  9. A โˆ– Y A\setminus Y
  10. ( A โˆ– Y ) , B , C (A\setminus Y),B,C
  11. ( A โˆ– Y ) (A\setminus Y)
  12. Y Y
  13. ( A โˆ– Y ) (A\setminus Y)
  14. B B
  15. C C
  16. A A
  17. ( A โˆ– Y ) (A\setminus Y)
  18. Y Y

Brandt_semigroup.html

  1. I , J I,J
  2. P P
  3. | I | ร— | J | |I|\times|J|
  4. G 0 = G โˆช { 0 } . G^{0}=G\cup\{0\}.
  5. S = ( I ร— G 0 ร— J ) S=(I\times G^{0}\times J)
  6. ( i , a , j ) * ( k , b , n ) = ( i , a p j k b , n ) (i,a,j)*(k,b,n)=(i,ap_{jk}b,n)
  7. S = ( I ร— G 0 ร— I ) S=(I\times G^{0}\times I)
  8. ( i , a , j ) * ( k , b , n ) = ( i , a p j k b , n ) (i,a,j)*(k,b,n)=(i,ap_{jk}b,n)
  9. P P

Breath_gas_analysis.html

  1. C A = C v ยฏ ฮป b:air + V ห™ A / Q ห™ c , C_{A}=\frac{C_{\bar{v}}}{\lambda\text{b:air}+\dot{V}_{A}/\dot{Q}_{c}},
  2. C A C_{A}
  3. C v ยฏ C_{\bar{v}}
  4. ฮป b:air \lambda\text{b:air}
  5. V ห™ A / Q ห™ c \dot{V}_{A}/\dot{Q}_{c}
  6. ฮป b:air = 340 \lambda\text{b:air}=340
  7. ฮป b:air = 0.75 \lambda\text{b:air}=0.75
  8. 1 ฮผ g / l 1\mu g/l
  9. ฮป b:air = 340 \lambda\text{b:air}=340
  10. 1 m g / l 1mg/l

Brenkeโ€“Chihara_polynomials.html

  1. A ( w ) B ( x w ) = โˆ‘ n = 0 โˆž P n ( x ) w n . A(w)B(xw)=\sum_{n=0}^{\infty}P_{n}(x)w^{n}.

Brian_Launder.html

  1. k k
  2. ฯต \epsilon
  3. k k
  4. ฯต \epsilon
  5. k k
  6. ฯต \epsilon

Brieskorn_manifold.html

  1. x 1 k 1 + โ‹ฏ + x n k n = 0 x_{1}^{k_{1}}+\cdots+x_{n}^{k_{n}}=0

Bril_(unit).html

  1. 1 10 7 ฯ€ \frac{1}{10^{7}\pi}
  2. 1 10 11 ฯ€ \frac{1}{10^{11}\pi}

Brjuno_number.html

  1. โˆ‘ n = 0 โˆž log q n + 1 q n < โˆž \sum_{n=0}^{\infty}\frac{\log q_{n+1}}{q_{n}}<\infty
  2. B ( x ) = B ( x + 1 ) B(x)=B(x+1)
  3. B ( x ) = - log x + x B ( 1 / x ) B(x)=-\log x+xB(1/x)

Brodal_queue.html

  1. O ( 1 ) O(1)
  2. O ( log ( n ) ) O(\mathrm{log}(n))

Brus_equation.html

  1. ฮ” E ( r ) = E gap + h 2 8 r 2 ( 1 / m e * + 1 / m h * ) . \Delta E(r)=E_{\mathrm{gap}}+\frac{h^{2}}{8r^{2}}\left(1/m_{\mathrm{e}}^{*}+1/% m_{\mathrm{h}}^{*}\right).

Budan's_theorem.html

  1. c 0 , c 1 , c 2 , โ€ฆ c_{0},c_{1},c_{2},\ldots
  2. l < r l<r
  3. r = l + 1 r=l+1
  4. c l c_{l}
  5. c r c_{r}
  6. r โ‰ฅ l + 2 r\geq l+2
  7. c l + 1 , โ€ฆ , c r - 1 c_{l+1},\ldots,c_{r-1}
  8. c l c_{l}
  9. c r c_{r}
  10. c l c_{l}
  11. c r c_{r}
  12. p ( x ) p(x)
  13. p ( x ) p(x)
  14. x x
  15. p ( x ) = 0 p(x)=0
  16. n > 0 n>0
  17. x โ† x + l x\leftarrow x+l
  18. x โ† x + r x\leftarrow x+r
  19. l l
  20. r r
  21. l < r l<r
  22. v l v_{l}
  23. v r v_{r}
  24. p ( x + l ) p(x+l)
  25. p ( x + r ) p(x+r)
  26. p ( r ) โ‰  0 p(r)\neq 0
  27. p ( x + l ) p(x+l)
  28. p ( x + r ) p(x+r)
  29. v l โ‰ฅ v r v_{l}\geq v_{r}
  30. ฯ \rho
  31. p ( x ) = 0 p(x)=0
  32. ( l , r ) (l,r)
  33. p ( x + l ) p(x+l)
  34. p ( x + r ) p(x+r)
  35. ฯ โ‰ค v l - v r \rho\leq v_{l}-v_{r}
  36. ฯ \rho
  37. p ( x ) = 0 p(x)=0
  38. ( l , r ) (l,r)
  39. p ( x + l ) p(x+l)
  40. p ( x + r ) p(x+r)
  41. ฯ = v l - v r - 2 ฮป \rho=v_{l}-v_{r}-2\lambda
  42. ฮป \lambda
  43. โ„ค + \mathbb{Z}_{+}
  44. x โ† x + l x\leftarrow x+l
  45. x โ† x + r x\leftarrow x+r
  46. ( l , r ) (l,r)
  47. ( l , r ) (l,r)
  48. ( l , r ) (l,r)
  49. p ( x ) = x 3 - 7 x + 7 p(x)=x^{3}-7x+7
  50. ( 0 , 2 ) (0,2)
  51. x โ† x + 0 x\leftarrow x+0
  52. x โ† x + 2 x\leftarrow x+2
  53. p ( x + 0 ) = ( x + 0 ) 3 - 7 ( x + 0 ) + 7 โ‡’ p ( x + 0 ) = x 3 - 7 x + 7 , v 0 = 2 p(x+0)=(x+0)^{3}-7(x+0)+7\Rightarrow p(x+0)=x^{3}-7x+7,v_{0}=2
  54. p ( x + 2 ) = ( x + 2 ) 3 - 7 ( x + 2 ) + 7 โ‡’ p ( x + 2 ) = x 3 + 6 x 2 + 5 x + 1 , v 2 = 0 p(x+2)=(x+2)^{3}-7(x+2)+7\Rightarrow p(x+2)=x^{3}+6x^{2}+5x+1,v_{2}=0
  55. ฯ โ‰ค v 0 - v 2 = 2 \rho\leq v_{0}-v_{2}=2
  56. p ( x ) p(x)
  57. ( 0 , 2 ) (0,2)
  58. p ( x ) = x 3 - 7 x + 7 p(x)=x^{3}-7x+7
  59. ( 0 , 1 ) (0,1)
  60. x โ† x + 0 x\leftarrow x+0
  61. x โ† x + 1 x\leftarrow x+1
  62. p ( x + 0 ) = ( x + 0 ) 3 - 7 ( x + 0 ) + 7 โ‡’ p ( x + 0 ) = x 3 - 7 x + 7 , v 0 = 2 p(x+0)=(x+0)^{3}-7(x+0)+7\Rightarrow p(x+0)=x^{3}-7x+7,v_{0}=2
  63. p ( x + 1 ) = ( x + 1 ) 3 - 7 ( x + 1 ) + 7 โ‡’ p ( x + 1 ) = x 3 + 3 x 2 - 4 x + 1 , v 2 = 2 p(x+1)=(x+1)^{3}-7(x+1)+7\Rightarrow p(x+1)=x^{3}+3x^{2}-4x+1,v_{2}=2
  64. ฯ = v 0 - v 2 = 0 \rho=v_{0}-v_{2}=0
  65. ( 0 , 1 ) (0,1)
  66. x x
  67. p ( x ) = 0 p(x)=0
  68. n > 0 n>0
  69. p ( x ) p(x)
  70. F seq ( x ) F\text{seq}(x)
  71. n + 1 n+1
  72. p ( x ) , p ( 1 ) ( x ) , โ€ฆ , p ( n ) ( x ) p(x),p^{(1)}(x),\ldots,p^{(n)}(x)
  73. p ( i ) p^{(i)}
  74. p ( x ) p(x)
  75. F seq ( x ) = { p ( x ) , p ( 1 ) ( x ) , โ€ฆ , p ( n ) ( x ) } F\text{seq}(x)=\big\{p(x),p^{(1)}(x),\ldots,p^{(n)}(x)\big\}
  76. p ( x ) = x 3 - 7 x + 7 p(x)=x^{3}-7x+7
  77. F seq ( x ) = { x 3 - 7 x + 7 , 3 x 2 - 7 , 6 x , 6 } F\text{seq}(x)=\big\{x^{3}-7x+7,3x^{2}-7,6x,6\big\}
  78. x x
  79. p ( x ) = 0 p(x)=0
  80. n > 0 n>0
  81. F seq ( x ) = { p ( x ) , p ( 1 ) ( x ) , โ€ฆ , p ( n ) ( x ) } F\text{seq}(x)=\big\{p(x),p^{(1)}(x),\ldots,p^{(n)}(x)\big\}
  82. x x
  83. l , r l,r
  84. ( l < r ) (l<r)
  85. F seq ( l ) F\text{seq}(l)
  86. F seq ( r ) F\text{seq}(r)
  87. v l , v r v_{l},v_{r}
  88. p ( r ) โ‰  0 p(r)\neq 0
  89. F seq ( l ) F\text{seq}(l)
  90. F seq ( r ) F\text{seq}(r)
  91. v l โ‰ฅ v r v_{l}\geq v_{r}
  92. ฯ \rho
  93. p ( x ) = 0 p(x)=0
  94. ( l , r ) (l,r)
  95. F seq ( l ) F\text{seq}(l)
  96. F seq ( r ) F\text{seq}(r)
  97. ฯ โ‰ค v l - v r \rho\leq v_{l}-v_{r}
  98. ฯ \rho
  99. p ( x ) = 0 p(x)=0
  100. ( l , r ) (l,r)
  101. F seq ( l ) F\text{seq}(l)
  102. F seq ( r ) F\text{seq}(r)
  103. ฯ = v l - v r - 2 ฮป \rho=v_{l}-v_{r}-2\lambda
  104. ฮป โˆˆ โ„ค + \lambda\in\mathbb{Z}_{+}
  105. p ( x ) = x 3 - 7 x + 7 p(x)=x^{3}-7x+7
  106. ( 0 , 2 ) (0,2)
  107. F seq ( 0 ) = { 7 , - 7 , 0 , 6 } , v 0 = 2 F\text{seq}(0)=\big\{7,-7,0,6\big\},v_{0}=2
  108. F seq ( 2 ) = { 1 , 5 , 12 , 6 } , v 2 = 0 F\text{seq}(2)=\big\{1,5,12,6\big\},v_{2}=0
  109. ฯ โ‰ค v 0 - v 2 = 2 \rho\leq v_{0}-v_{2}=2
  110. p ( x ) p(x)
  111. ( 0 , 2 ) (0,2)
  112. p ( x + a ) p(x+a)
  113. p ( x + a + 1 ) p(x+a+1)
  114. p ( x ) p(x)
  115. ( a , a + 1 ) (a,a+1)
  116. p ( x ) p(x)
  117. x โŸต 1 x + 1 x\longleftarrow\frac{1}{x+1}
  118. p ( x ) p(x)
  119. ( 0 , 1 ) (0,1)
  120. ฯ 01 ( p ) \rho_{01}(p)
  121. ( 0 , 1 ) (0,1)
  122. p ( x ) โˆˆ โ„ [ x ] p(x)\in\mathbb{R}[x]
  123. d e g ( p ) deg(p)
  124. v a r 01 ( p ) var_{01}(p)
  125. v a r 01 ( p ) = v a r ( ( x + 1 ) d e g ( p ) p ( 1 x + 1 ) ) var_{01}(p)=var((x+1)^{deg(p)}p(\frac{1}{x+1}))
  126. v a r 01 ( p ) โ‰ฅ ฯ 01 ( p ) var_{01}(p)\geq\rho_{01}(p)
  127. v a r 01 ( p ) = 0 var_{01}(p)=0
  128. ฯ 01 ( p ) = 0 \rho_{01}(p)=0
  129. v a r 01 ( p ) = 1 var_{01}(p)=1
  130. ฯ 01 ( p ) = 1 \rho_{01}(p)=1
  131. p ( x ) p(x)
  132. n > 0 n>0
  133. n + 1 n+1
  134. F seq ( a ) F\text{seq}(a)
  135. x โ† a x\leftarrow a
  136. F seq ( x ) F\text{seq}(x)
  137. p ( x + a ) = โˆ‘ i = 0 n p ( i ) ( a ) i ! x i p(x+a)=\sum_{i=0}^{n}\frac{p^{(i)}(a)}{i!}\ x^{i}
  138. x โ† x + p x\leftarrow x+p
  139. p p
  140. p ( x ) p(x)
  141. ( a , a + 1 ) (a,a+1)
  142. p ( x + a ) p(x+a)
  143. p ( x + a + 1 ) p(x+a+1)
  144. x = a + 1 x โ€ฒ , x โ€ฒ = b + 1 x โ€ฒโ€ฒ , x โ€ฒโ€ฒ = c + 1 x โ€ฒโ€ฒโ€ฒ , โ€ฆ x=a+\frac{1}{x^{\prime}},\quad x^{\prime}=b+\frac{1}{x^{\prime\prime}},\quad x% ^{\prime\prime}=c+\frac{1}{x^{\prime\prime\prime}},\ldots
  145. a + 1 b + 1 c + 1 โ‹ฑ a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{\ddots}}}
  146. a , b , c , โ€ฆ a,b,c,\ldots
  147. x โ† x + 1 x\leftarrow x+1
  148. p ( x ) p(x)
  149. p ( x + 1 ) p(x+1)
  150. p ( x + 1 ) p(x+1)
  151. ( 5 , 6 ) (5,6)
  152. p ( x + 6 ) p(x+6)
  153. p ( x + 5 ) p(x+5)
  154. x = 5 + 1 x โ€ฒ x=5+\frac{1}{x^{\prime}}
  155. p ( x + 1 ) p(x+1)
  156. p ( x ) p(x)
  157. p ( x ) p(x)
  158. ( 0 , 1 ) (0,1)
  159. x โ† 1 1 + x x\leftarrow\frac{1}{1+x}
  160. p ( x ) p(x)
  161. x โ† x + 1 x\leftarrow x+1
  162. x โ† 1 1 + x x\leftarrow\frac{1}{1+x}
  163. p ( x ) p(x)
  164. ( 0 , 1 ) (0,1)

Building_block_model.html

  1. R e v t = W A C C t ร— R A B t - 1 + D e p t + O p e x t Rev_{t}=WACC_{t}\times RAB_{t-1}+Dep_{t}+Opex_{t}
  2. R e v t Rev_{t}
  3. W A C C t WACC_{t}
  4. R A B t - 1 RAB_{t-1}
  5. D e p t Dep_{t}
  6. O p e x t Opex_{t}
  7. R A B t = R A B t - 1 + C a p e x t - D e p t RAB_{t}=RAB_{t-1}+Capex_{t}-Dep_{t}\,\!
  8. R A B t RAB_{t}
  9. R A B t - 1 RAB_{t-1}
  10. C a p e x t Capex_{t}
  11. D e p t Dep_{t}
  12. R A B t = ( 1 + i n f l a t i o n t ) ร— R A B t - 1 + C a p e x t - D e p t RAB_{t}=(1+inflation_{t})\times RAB_{t-1}+Capex_{t}-Dep_{t}
  13. i n f l a t i o n t inflation_{t}

Bumblebee_models.html

  1. โ„’ B \displaystyle{\mathcal{L}}_{B}
  2. D ฮผ . D_{\mu}\Big.
  3. B ฮผ ฮฝ = D ฮผ B ฮฝ - D ฮฝ B ฮผ . B_{\mu\nu}=D_{\mu}B_{\nu}-D_{\nu}B_{\mu}\Big.
  4. ฯƒ 1 . \sigma_{1}\Big.
  5. ฯƒ 2 . \sigma_{2}\Big.
  6. ฯ„ 1 . \tau_{1}\Big.
  7. ฯ„ 2 . \tau_{2}\Big.
  8. ฯ„ 3 . \tau_{3}\Big.
  9. โ„’ M {\mathcal{L}}_{\rm M}
  10. V ( B ฮผ B ฮผ โˆ“ b 2 ) V(B_{\mu}B^{\mu}\mp b^{2})
  11. B ฮผ B ฮผ โˆ“ b 2 = 0. B_{\mu}B^{\mu}\mp b^{2}=0.
  12. V = 1 2 ฮบ ( B ฮผ B ฮผ โˆ“ b 2 ) 2 , V=\frac{1}{2}\kappa(B_{\mu}B^{\mu}\mp b^{2})^{2},
  13. ฮบ . \kappa\Big.
  14. V = ฮป ( B ฮผ B ฮผ โˆ“ b 2 ) . V=\lambda(B_{\mu}B^{\mu}\mp b^{2}).
  15. โ„’ B {\mathcal{L}}_{B}
  16. ฯ„ 1 = 1. \tau_{1}=1\Big.
  17. ฯƒ 1 = ฯƒ 2 = ฯ„ 2 = ฯ„ 3 = 0. \sigma_{1}=\sigma_{2}=\tau_{2}=\tau_{3}=0\Big.
  18. โ„’ KS = 1 16 ฯ€ G ( R - 2 ฮ› ) - 1 4 B ฮผ ฮฝ B ฮผ ฮฝ - V ( B ฮผ B ฮผ ยฑ b 2 ) + B ฮผ J ฮผ + โ„’ M . {\mathcal{L}}_{\rm KS}=\frac{1}{16\pi G}(R-2\Lambda)-\frac{1}{4}B_{\mu\nu}B^{% \mu\nu}-V(B_{\mu}B^{\mu}\pm b^{2})+B_{\mu}J^{\mu}+{\mathcal{L}}_{\rm M}.
  19. J ฮผ . J^{\mu}\Big.
  20. โ„’ B {\mathcal{L}}_{B}
  21. ฯ„ 1 = 1. \tau_{1}=1\Big.
  22. ฯƒ 1 = ฮพ / 16 ฯ€ G . \sigma_{1}=\xi/16\pi G\Big.
  23. ฯƒ 2 = ฯ„ 2 = ฯ„ 3 = 0. \sigma_{2}=\tau_{2}=\tau_{3}=0\Big.
  24. ฮพ . \xi\Big.
  25. โ„’ = 1 16 ฯ€ G ( R - 2 ฮ› + ฮพ B ฮผ B ฮฝ R ฮผ ฮฝ ) - 1 4 B ฮผ ฮฝ B ฮผ ฮฝ - V ( B ฮผ B ฮผ ยฑ b 2 ) + B ฮผ J ฮผ + โ„’ M . {\mathcal{L}}=\frac{1}{16\pi G}(R-2\Lambda+\xi B^{\mu}B^{\nu}R_{\mu\nu})-\frac% {1}{4}B_{\mu\nu}B^{\mu\nu}-V(B_{\mu}B^{\mu}\pm b^{2})+B_{\mu}J^{\mu}+{\mathcal% {L}}_{\rm M}.
  26. โ„’ B {\mathcal{L}}_{B}
  27. ฯƒ 1 . \sigma_{1}\Big.
  28. ฯƒ 2 . \sigma_{2}\Big.
  29. ฯ„ 1 . \tau_{1}\Big.
  30. ฯ„ 2 . \tau_{2}\Big.
  31. ฯ„ 3 . \tau_{3}\Big.
  32. ฮ“ ฮผ ฮฝ ฮป \Gamma^{\lambda}_{\,\,\mu\nu}

Busemannโ€“Petty_problem.html

  1. Vol n - 1 ( K โˆฉ A ) โ‰ค Vol n - 1 ( T โˆฉ A ) \mathrm{Vol}_{n-1}\,(K\cap A)\leq\mathrm{Vol}_{n-1}\,(T\cap A)

Bussgang_theorem.html

  1. { X ( t ) } \left\{X(t)\right\}
  2. { Y ( t ) } = g ( X ( t ) ) \left\{Y(t)\right\}=g(X(t))
  3. g ( โ‹… ) g(\cdot)
  4. R X ( ฯ„ ) R_{X}(\tau)
  5. { X ( t ) } \left\{X(t)\right\}
  6. { X ( t ) } \left\{X(t)\right\}
  7. { Y ( t ) } \left\{Y(t)\right\}
  8. R X Y ( ฯ„ ) = C R X ( ฯ„ ) , R_{XY}(\tau)=CR_{X}(\tau),
  9. C C
  10. g ( โ‹… ) g(\cdot)
  11. C = 1 ฯƒ 3 2 ฯ€ โˆซ - โˆž โˆž u g ( u ) e - u 2 2 ฯƒ 2 d u . C=\frac{1}{\sigma^{3}\sqrt{2\pi}}\int_{-\infty}^{\infty}ug(u)e^{-\frac{u^{2}}{% 2\sigma^{2}}}\,du.

C::2010_X1.html

  1. ( 100 5 ) 19.5 - 6.5 โ‰ˆ 158489 (\sqrt[5]{100})^{19.5-6.5}\approx 158489

Calculus_of_moving_surfaces.html

  1. ฮด / ฮด t \delta/\delta t
  2. โˆ‡ ฮฑ \nabla_{\alpha}
  3. S t S_{t}
  4. S S
  5. t t
  6. C C
  7. ฮด / ฮด t \delta/\delta t
  8. S S
  9. C C
  10. P P
  11. C = lim h โ†’ 0 Distance ( P , P * ) h C=\lim_{h\to 0}\frac{\,\text{Distance}(P,P^{*})}{h}
  12. P * P^{*}
  13. S t + h S_{t+h}
  14. S t S_{t}
  15. C C
  16. P P * ยฏ \overline{PP^{*}}
  17. S t S_{t}
  18. C C
  19. ฮด / ฮด t \delta/\delta t
  20. S t S_{t}
  21. F F
  22. ฮด F ฮด t = lim h โ†’ 0 F ( P * ) - F ( P ) h \frac{\delta F}{\delta t}=\lim_{h\to 0}\frac{F(P^{*})-F(P)}{h}
  23. ฮด / ฮด t \delta/\delta t
  24. C C
  25. ฮด / ฮด t \delta/\delta t
  26. S S
  27. Z i = Z i ( t , S ) Z^{i}=Z^{i}\left(t,S\right)\,
  28. Z i Z^{i}
  29. S ฮฑ S^{\alpha}
  30. S S
  31. S ฮฑ S^{\alpha}
  32. v i v^{i}
  33. v i = โˆ‚ Z i ( t , S ) โˆ‚ t v^{i}=\frac{\partial Z^{i}\left(t,S\right)}{\partial t}
  34. C C
  35. C = v i N i C=v^{i}N_{i}\,
  36. N i N_{i}
  37. N โ†’ \vec{N}
  38. ฮด / ฮด t \delta/\delta t
  39. ฮด F ฮด t = โˆ‚ F ( t , S ) โˆ‚ t - v i Z i ฮฑ โˆ‡ ฮฑ F \frac{\delta F}{\delta t}=\frac{\partial F\left(t,S\right)}{\partial t}-v^{i}Z% ^{\alpha}_{i}\nabla_{\alpha}F
  40. Z i ฮฑ Z^{\alpha}_{i}
  41. โˆ‡ ฮฑ \nabla_{\alpha}
  42. T j ฮฒ i ฮฑ T^{i\alpha}_{j\beta}
  43. ฮด T j ฮฒ i ฮฑ ฮด t = โˆ‚ T j ฮฒ i ฮฑ โˆ‚ t - v ฮท โˆ‡ ฮท T j ฮฒ i ฮฑ + v m ฮ“ m k i T j ฮฒ k ฮฑ - v m ฮ“ m j k T k ฮฒ i ฮฑ + โˆ‡ ฮท v ฮฑ T j ฮฒ i ฮท - โˆ‡ ฮฒ v ฮท T j ฮท i ฮฑ \frac{\delta T^{i\alpha}_{j\beta}}{\delta t}=\frac{\partial T^{i\alpha}_{j% \beta}}{\partial t}-v^{\eta}\nabla_{\eta}T^{i\alpha}_{j\beta}+v^{m}\Gamma^{i}_% {mk}T^{k\alpha}_{j\beta}-v^{m}\Gamma^{k}_{mj}T^{i\alpha}_{k\beta}+\nabla_{\eta% }v^{\alpha}T^{i\eta}_{j\beta}-\nabla_{\beta}v^{\eta}T^{i\alpha}_{j\eta}
  44. ฮ“ m j k \Gamma^{k}_{mj}
  45. ฮด / ฮด t \delta/\delta t
  46. ฮด ฮด t ( S ฮฑ i T j ฮฒ ) = ฮด S ฮฑ i ฮด t T j ฮฒ + S ฮฑ i ฮด T j ฮฒ ฮด t \frac{\delta}{\delta t}\left(S^{i}_{\alpha}T^{\beta}_{j}\right)=\frac{\delta S% ^{i}_{\alpha}}{\delta t}T^{\beta}_{j}+S^{i}_{\alpha}\frac{\delta T^{\beta}_{j}% }{\delta t}
  47. ฮด F k j ฮด t = โˆ‚ F k j โˆ‚ t + C N i โˆ‡ i F k j \frac{\delta F^{j}_{k}}{\delta t}=\frac{\partial F^{j}_{k}}{\partial t}+CN^{i}% \nabla_{i}F^{j}_{k}
  48. ฮด / ฮด t \delta/\delta t
  49. ฮด ฮด j i ฮด t , ฮด Z i j ฮด t , ฮด Z i j ฮด t , ฮด ฮต i j k ฮด t , ฮด ฮต i j k ฮด t = 0 \frac{\delta\delta^{i}_{j}}{\delta t},\frac{\delta Z_{ij}}{\delta t},\frac{% \delta Z^{ij}}{\delta t},\frac{\delta\varepsilon_{ijk}}{\delta t},\frac{\delta% \varepsilon^{ijk}}{\delta t}=0
  50. Z i j Z_{ij}
  51. Z i j Z^{ij}
  52. ฮด j i \delta^{i}_{j}
  53. ฮต i j k \varepsilon_{ijk}
  54. ฮต i j k \varepsilon^{ijk}
  55. Z i j Z_{ij}
  56. ฮด / ฮด t \delta/\delta t
  57. S ฮฑ ฮฒ S_{\alpha\beta}
  58. S ฮฑ ฮฒ S^{\alpha\beta}
  59. ฮด S ฮฑ ฮฒ ฮด t \displaystyle\frac{\delta S_{\alpha\beta}}{\delta t}
  60. B ฮฑ ฮฒ B_{\alpha\beta}
  61. B ฮฑ ฮฒ B^{\alpha\beta}
  62. B ฮฒ ฮฑ B^{\alpha}_{\beta}
  63. ฮด B ฮฑ ฮฒ ฮด t \displaystyle\frac{\delta B_{\alpha\beta}}{\delta t}
  64. Z ฮฑ i Z^{i}_{\alpha}
  65. N i N^{i}
  66. ฮด Z ฮฑ i ฮด t \displaystyle\frac{\delta Z^{i}_{\alpha}}{\delta t}
  67. ฮต ฮฑ ฮฒ \varepsilon_{\alpha\beta}
  68. ฮต ฮฑ ฮฒ \varepsilon^{\alpha\beta}
  69. ฮด ฮต ฮฑ ฮฒ ฮด t \displaystyle\frac{\delta\varepsilon_{\alpha\beta}}{\delta t}

Cameron_Leigh_Stewart.html

  1. f ( x , y ) = h f(x,y)=h
  2. 2800 ( 1 + 1 / 4 ฯต deg f ) ( deg f ) 1 + ฯ‰ ( g ) 2800(1+1/4\epsilon\deg f)(\deg f)^{1+\omega(g)}
  3. ฯต \epsilon
  4. ฯ‰ ( g ) \omega(g)
  5. g g
  6. h h
  7. P ( n ) P(n)
  8. 2 n - 1 2^{n}-1
  9. lim n โ†’ โˆž P ( n ) / n = โˆž \lim_{n\rightarrow\infty}P(n)/n=\infty

Campbell's_theorem_(probability).html

  1. N {N}
  2. ๐‘ d \,\textbf{R}^{d}
  3. f f
  4. ๐‘ d \,\textbf{R}^{d}
  5. N {N}
  6. E [ โˆ‘ x โˆˆ N f ( x ) ] E[\sum_{x\in{N}}f(x)]
  7. E E
  8. N {N}
  9. N {N}
  10. N {N}
  11. ฮ› ( B ) = E [ N ( B ) ] \Lambda(B)=E[{N}(B)]
  12. N {N}
  13. N {N}
  14. N {N}
  15. f : ๐‘ d โ†’ ๐‘ f:\,\textbf{R}^{d}\rightarrow\,\textbf{R}
  16. S = โˆ‘ x โˆˆ N f ( x ) S=\sum_{x\in{N}}f(x)
  17. โˆซ ๐‘ d min ( | f ( x ) | , 1 ) ฮ› ( d x ) < โˆž . \int_{\,\textbf{R}^{d}}\min(|f(x)|,1)\Lambda(dx)<\infty.
  18. ฮธ \theta
  19. E ( e ฮธ S ) = exp ( โˆซ ๐‘ d [ e ฮธ f ( x ) - 1 ] ฮ› ( d x ) ) , E(e^{\theta S})=\textrm{exp}\left(\int_{\,\textbf{R}^{d}}[e^{\theta f(x)}-1]% \Lambda(dx)\right),
  20. ฮธ \theta
  21. E ( S ) = โˆซ ๐‘ d f ( x ) ฮ› ( d x ) , E(S)=\int_{\,\textbf{R}^{d}}f(x)\Lambda(dx),
  22. Var ( S ) = โˆซ ๐‘ d f ( x ) 2 ฮ› ( d x ) , \,\text{Var}(S)=\int_{\,\textbf{R}^{d}}f(x)^{2}\Lambda(dx),
  23. Var ( S ) \,\text{Var}(S)
  24. S S
  25. N {N}
  26. ฮ› ( B ) = E [ N ( B ) ] , \Lambda(B)=E[{N}(B)],
  27. f f
  28. N {N}
  29. f : ๐‘ d โ†’ ๐‘ f:\,\textbf{R}^{d}\rightarrow\,\textbf{R}
  30. f f
  31. E [ โˆ‘ x โˆˆ N f ( x ) ] = โˆซ ๐‘ d f ( x ) ฮ› ( d x ) , E\left[\sum_{x\in{N}}f(x)\right]=\int_{\,\textbf{R}^{d}}f(x)\Lambda(dx),
  32. E [ โˆ‘ x โˆˆ N f ( x ) ] = โˆซ ๐‘ d f d ฮ› , E\left[\sum_{x\in{N}}f(x)\right]=\int_{\,\textbf{R}^{d}}fd\Lambda,
  33. ฮ› \Lambda
  34. N {N}
  35. ฮป ( x ) \lambda(x)
  36. E [ โˆ‘ x โˆˆ N f ( x ) ] = โˆซ ๐‘ d f ( x ) ฮป ( x ) d x E\left[\sum_{x\in{N}}f(x)\right]=\int_{\,\textbf{R}^{d}}f(x)\lambda(x)dx
  37. N {N}
  38. ฮป > 0 \lambda>0
  39. E [ โˆ‘ x โˆˆ N f ( x ) ] = ฮป โˆซ ๐‘ d f ( x ) d x E\left[\sum_{x\in{N}}f(x)\right]=\lambda\int_{\,\textbf{R}^{d}}f(x)dx
  40. N {N}
  41. ฮ› \Lambda
  42. โ„’ < m t p l > N \mathcal{L}_{<}mtpl>{{N}}
  43. โ„’ < m t p l > N \mathcal{L}_{<}mtpl>{{N}}
  44. S = โˆ‘ x โˆˆ N a n f ( x ) S=\sum_{x\in{N}}a_{n}f(x)
  45. ฮบ i \kappa_{i}
  46. S S
  47. ฮบ i = ฮป a i ยฏ โˆซ f ( x ) d x \kappa_{i}=\lambda\overline{a^{i}}\int f(x)dx
  48. a i ยฏ \overline{a^{i}}
  49. a a

CAMPUS_(database).html

  1. ฯƒ Y \sigma_{\mathrm{Y}}
  2. ฯต Y \epsilon_{\mathrm{Y}}
  3. ฯต t B \epsilon_{\mathrm{t}B}
  4. ฯƒ 50 \sigma_{\mathrm{5}0}
  5. ฯƒ B \sigma_{\mathrm{B}}
  6. ฯต B \epsilon_{\mathrm{B}}
  7. E t c E_{\mathrm{t}c}
  8. E t c E_{\mathrm{t}c}
  9. a c U a_{\mathrm{c}U}
  10. a c A a_{\mathrm{c}A}
  11. a t1 a_{\mathrm{t}1}
  12. F M F_{\mathrm{M}}
  13. W P W_{\mathrm{P}}
  14. E f E_{\mathrm{f}}
  15. ฯƒ f \sigma_{\mathrm{f}}
  16. T m T_{\mathrm{m}}
  17. T g T_{\mathrm{g}}
  18. T f T_{\mathrm{f}}
  19. T f T_{\mathrm{f}}
  20. T f T_{\mathrm{f}}
  21. T V T_{\mathrm{V}}
  22. โ‰ฅ \geq
  23. ฮฑ p \alpha_{\mathrm{p}}
  24. ฮฑ n \alpha_{\mathrm{n}}
  25. โ‰ฅ \geq
  26. โ‰ฅ \geq
  27. โ‰ฅ \geq
  28. โ‰ฅ \geq
  29. ฯต r \epsilon_{\mathrm{r}}
  30. โ‰ฅ \geq
  31. โ‰ฅ \geq
  32. ฯต r \epsilon_{\mathrm{r}}
  33. ฮด \delta
  34. ฮด \delta
  35. ฯ e \rho_{\mathrm{e}}
  36. ฮฉ \Omega
  37. ฯƒ e \sigma_{\mathrm{e}}
  38. ฮฉ \Omega
  39. E B E_{\mathrm{B}}
  40. โ‰ฅ \geq
  41. โ‰ฅ \geq
  42. โ‰ฅ \geq
  43. โ‰ฅ \geq
  44. โ‰ฅ \geq
  45. ฯ \rho
  46. tan ฮด ( T ) \tan\delta(T)
  47. E t ( T ) E_{\mathrm{t}}(T)
  48. ฯƒ ( ฮต ) \sigma(\varepsilon)
  49. E t S ( ฮต , T ) E_{\mathrm{t}S}(\varepsilon,T)
  50. ฯƒ c ( ฮต , t , T ) \sigma_{\mathrm{c}}(\varepsilon,t,T)
  51. E t c S ( ฮต , t , T ) E_{\mathrm{t}cS}(\varepsilon,t,T)
  52. ฮ” H ( T ) / m \Delta H(T)/m
  53. ฮท ( ฮณ , T ) \eta(\gamma,T)
  54. ฯ„ ( ฮณ , T ) \tau(\gamma,T)
  55. v ( T , p ) v(T,p)

Canberra_distance.html

  1. d ( ๐ฉ , ๐ช ) = โˆ‘ i = 1 n | p i - q i | | p i | + | q i | , d(\mathbf{p},\mathbf{q})=\sum_{i=1}^{n}\frac{|p_{i}-q_{i}|}{|p_{i}|+|q_{i}|},
  2. ๐ฉ = ( p 1 , p 2 , โ€ฆ , p n ) and ๐ช = ( q 1 , q 2 , โ€ฆ , q n ) \mathbf{p}=(p_{1},p_{2},\dots,p_{n})\,\text{ and }\mathbf{q}=(q_{1},q_{2},% \dots,q_{n})\,

Cancer_slope_factor.html

  1. Risk i = C โ‹… I R i โ‹… E F i โ‹… E D i B W i โ‹… A T โ‹… S F โ‹… A D A F i \,\text{Risk}_{i}=C\cdot\frac{IR_{i}\cdot EF_{i}\cdot ED_{i}}{BW_{i}\cdot AT}% \cdot SF\cdot ADAF_{i}

Cantellated_5-cubes.html

  1. ( ยฑ 1 , ยฑ 1 , ยฑ ( 1 + 2 ) , ยฑ ( 1 + 2 ) , ยฑ ( 1 + 2 ) ) \left(\pm 1,\ \pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)
  2. ( 1 , 1 + 2 , 1 + 2 2 , 1 + 2 2 , 1 + 2 2 ) \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2}\right)

Cantor's_intersection_theorem.html

  1. S S
  2. S S
  3. C 0 โЇ C 1 โЇ โ‹ฏ C k โЇ C k + 1 โ‹ฏ , C_{0}\supseteq C_{1}\supseteq\cdots C_{k}\supseteq C_{k+1}\cdots,\,
  4. ( โ‹‚ k C k ) โ‰  โˆ… . \left(\bigcap_{k}C_{k}\right)\neq\emptyset.\,
  5. C k = [ 2 , 2 + 1 / k ] = ( 2 , 2 + 1 / k ) C_{k}=[\sqrt{2},\sqrt{2}+1/k]=(\sqrt{2},\sqrt{2}+1/k)
  6. โ‹‚ C n = โˆ… \bigcap C_{n}=\emptyset
  7. U n = X โˆ– C n U_{n}=X\setminus C_{n}
  8. โ‹ƒ U n = X โˆ– โ‹‚ C n \bigcup U_{n}=X\setminus\bigcap C_{n}
  9. โ‹‚ C n = โˆ… \bigcap C_{n}=\emptyset
  10. โ‹ƒ U n = X \bigcup U_{n}=X
  11. X X
  12. ( U n ) (U_{n})
  13. U k U_{k}
  14. X = U k X=U_{k}
  15. C k = X โˆ– U k = โˆ… C_{k}=X\setminus U_{k}=\emptyset
  16. lim n โ†’ โˆž diam ( C n ) = 0 \lim_{n\to\infty}\operatorname{diam}(C_{n})=0
  17. diam ( C n ) = sup { d ( x , y ) | x , y โˆˆ C n } . \operatorname{diam}(C_{n})=\sup\{d(x,y)|x,y\in C_{n}\}.
  18. โˆฉ n = 1 โˆž C n = { x } \cap_{n=1}^{\infty}C_{n}=\{x\}

Capital_market_line.html

  1. CML : E ( r ) = r f + ฯƒ E ( r M ) - r f ฯƒ M . \mathrm{CML}:E(r)=r_{f}+\sigma\frac{E(r_{M})-r_{f}}{\sigma_{M}}.
  2. E ( r ) - r f ฯƒ = E ( r M ) - r f ฯƒ M . \frac{E(r)-r_{f}}{\sigma}=\frac{E(r_{M})-r_{f}}{\sigma_{M}}.
  3. ฯƒ \sigma

Capital_structure_substitution_theory.html

  1. [ โˆ‚ D โˆ‚ n ] x,t = - P x,t \left[\frac{\partial D}{\partial n}\right]_{\,\text{x,t}}=-P_{\,\text{x,t}}
  2. E / n E/n
  3. P โ‹… R โ‹… [ 1 - T ] / n P\cdot R\cdot[1-T]/n
  4. [ โˆ‚ E โˆ‚ n ] x,t = - E x,t n + P x,t R x,t [ 1 - T ] n \left[\frac{\partial E}{\partial n}\right]_{\,\text{x,t}}=-\frac{E_{\,\text{x,% t}}}{n}+\frac{P_{\,\text{x,t}}\ R_{\,\text{x,t}}\ [1-T]}{n}
  5. E x,t P x,t = R x,t [ 1 - T ] \frac{E_{\,\text{x,t}}}{P_{\,\text{x,t}}}=R_{\,\text{x,t}}\ [1-T]
  6. R x,t R_{\,\text{x,t}}
  7. P x,t = E x,t R x,t [ 1 - T ] P_{\,\text{x,t}}=\frac{E_{\,\text{x,t}}}{R_{\,\text{x,t}}[1-T]}
  8. B e t a x,t = R t ยฏ โ‹… [ P E ] x,t โ‹… [ 1 - T ] Beta_{\,\text{x,t}}=\overline{R_{\,\text{t}}}\cdot\left[\frac{P}{E}\right]_{\,% \text{x,t}}\cdot[1-T]
  9. R t ยฏ \overline{R_{\,\text{t}}}

Caratheฬodory_kernel_theorem.html

  1. { 0 } \{0\}
  2. { 0 } \{0\}
  3. { 0 } \{0\}
  4. { 0 } \{0\}
  5. | f n ( z ) | โ‰ค f n โ€ฒ ( 0 ) | z | ( 1 - | z | ) 2 . |f_{n}(z)|\leq f_{n}^{\prime}(0){|z|\over(1-|z|)^{2}}.

Carbon_nanotube_field-effect_transistor.html

  1. d t = 3 a C - C m 2 + m n + n 2 ฯ€ d_{t}=\frac{\sqrt{3}a_{C-C}\sqrt{m^{2}+mn+n^{2}}}{\pi}
  2. ฮธ = tan - 1 ( 3 n 2 m + n ) \theta=\tan^{-1}\left(\frac{\sqrt{3}\,n}{2m+n}\right)
  3. E g = ฮณ ( 2 a d t ) E_{g}=\gamma\left(\frac{2a}{d_{t}}\right)
  4. N S = 1 2 โˆซ - โˆž + โˆž D ( E ) f ( E - U S F ) d E N_{S}=\frac{1}{2}\int_{-\infty}^{+\infty}D(E)f(E-U_{SF})\,dE
  5. N D = 1 2 โˆซ - โˆž + โˆž D ( E ) f ( E - U D F ) d E N_{D}=\frac{1}{2}\int_{-\infty}^{+\infty}D(E)f(E-U_{DF})\,dE
  6. N 0 = 1 2 โˆซ - โˆž + โˆž D ( E ) f ( E - E F ) d E N_{0}=\frac{1}{2}\int_{-\infty}^{+\infty}D(E)f(E-E_{F})\,dE
  7. D ( E ) = D 0 E E 2 - ( E g / 2 ) 2 ฮ˜ ( E - E g / 2 ) D(E)=D_{0}\frac{E}{\sqrt{E^{2}-(E_{g}/2)^{2}}}\Theta(E-E_{g}/2)
  8. U S F = E F - q V S C U_{SF}=E_{F}-qV_{SC}
  9. U D F = E F - q V S C - q V D S . U_{DF}=E_{F}-qV_{SC}-qV_{DS}.
  10. ฮ˜ ( E - E g / 2 ) \Theta(E-E_{g}/2)
  11. V S C = - Q t + q N S ( V S C ) + q N D ( V S C ) - q N 0 C ฮฃ V_{SC}=\frac{-Q_{t}+qN_{S}(V_{SC})+qN_{D}(V_{SC})-qN_{0}}{C_{\Sigma}}
  12. I D S = 2 q k T ฯ€ โ„ [ F 0 ( U S F k T ) - F 0 ( U D F k T ) ] I_{DS}=\frac{2qkT}{\pi{\hbar}}\left[F_{0}\left(\frac{U_{SF}}{kT}\right)-F_{0}% \left(\frac{U_{DF}}{kT}\right)\right]

Carbon_nanotube_springs.html

  1. r i = r - n h 2 r_{i}=r-\frac{nh}{2}
  2. r o = r + n h 2 r_{o}=r+\frac{nh}{2}
  3. A = ฯ€ ( r o 2 - r i 2 ) A=\pi(r_{o}^{2}-r_{i}^{2})
  4. A t = ฯ€ r o 2 A_{t}=\pi r_{o}^{2}
  5. ฯต \epsilon\,
  6. U = 1 2 โˆญ ( ฯƒ x ฯต x U=\frac{1}{2}\iiint(\sigma_{x}\,\epsilon_{x}\,
  7. d x d y d z = 1 2 E ฯต 2 A L \,dx\,dy\,dz=\frac{1}{2}E\epsilon^{2}\ AL
  8. 1 2 E ฯต 2 ฯ€ ( r o 2 - r i 2 ) L \frac{1}{2}E\epsilon^{2}\pi(r_{o}^{2}-r_{i}^{2})L
  9. A / A t A/A_{t}
  10. A / A t A/A_{t}
  11. A h A_{h}
  12. r i = r - n h 2 r_{i}=r-\frac{nh}{2}
  13. r o = r + n h 2 r_{o}=r+\frac{nh}{2}
  14. A = ฯ€ ( r o 2 - r i 2 ) A=\pi(r_{o}^{2}-r_{i}^{2})
  15. A t = ฯ€ r o 2 A_{t}=\pi r_{o}^{2}
  16. ฯต \epsilon\,
  17. U = 1 2 โˆญ ( ฯƒ x ฯต x U=\frac{1}{2}\iiint(\sigma_{x}\,\epsilon_{x}\,
  18. d x d y d z = 1 2 E ฯต 2 A L \,dx\,dy\,dz=\frac{1}{2}E\epsilon^{2}\ AL
  19. 1 2 E ฯต 2 ฯ€ ( r o 2 - r i 2 ) L \frac{1}{2}E\epsilon^{2}\pi(r_{o}^{2}-r_{i}^{2})L
  20. A / A t A/A_{t}
  21. ร— 10 6 \times 10^{6}
  22. ร— 10 6 \times 10^{6}

Cardinal_characteristic_of_the_continuum.html

  1. โ„ต 0 \aleph_{0}
  2. โ„ \mathbb{R}
  3. 2 โ„ต 0 2^{\aleph_{0}}
  4. ๐”  \mathfrak{c}
  5. ๐”  \mathfrak{c}
  6. โ„ต 0 \aleph_{0}
  7. โ„ต 0 \aleph_{0}
  8. โ„ต 1 \aleph_{1}
  9. ๐”  = โ„ต 1 \mathfrak{c}=\aleph_{1}
  10. ๐”  \mathfrak{c}
  11. โ„ต 2 \aleph_{2}
  12. โ„ต 0 \aleph_{0}
  13. ๐”  \mathfrak{c}
  14. ๐”  \mathfrak{c}
  15. โ„ต 0 \aleph_{0}
  16. ๐”  \mathfrak{c}
  17. โ„ต 1 \aleph_{1}
  18. ฯ‰ \omega
  19. โ„ต 0 \aleph_{0}
  20. ๐’ฉ \mathcal{N}
  21. ๐”Ÿ \mathfrak{b}
  22. ๐”ก \mathfrak{d}
  23. ฯ‰ ฯ‰ \omega^{\omega}
  24. ฯ‰ \omega
  25. ฯ‰ \omega
  26. f : ฯ‰ โ†’ ฯ‰ f:\omega\to\omega
  27. g : ฯ‰ โ†’ ฯ‰ g:\omega\to\omega
  28. f โ‰ค * g f\leq^{*}g
  29. n โˆˆ ฯ‰ , f ( n ) โ‰ค g ( n ) n\in\omega,f(n)\leq g(n)
  30. ๐”Ÿ \mathfrak{b}
  31. ๐”Ÿ = min ( { | F | : F โІ ฯ‰ ฯ‰ โˆง โˆ€ f : ฯ‰ โ†’ ฯ‰ โˆƒ g โˆˆ F ( g โ‰ฐ * f ) } ) . \mathfrak{b}=\min(\{|F|:F\subseteq\omega^{\omega}\land\forall f:\omega\to% \omega\exists g\in F(g\nleq^{*}f)\}).
  32. ๐”ก \mathfrak{d}
  33. ฯ‰ \omega
  34. ฯ‰ \omega
  35. โ‰ค * \leq^{*}
  36. ๐”ก = min ( { | F | : F โІ ฯ‰ ฯ‰ โˆง โˆ€ f : ฯ‰ โ†’ ฯ‰ โˆƒ g โˆˆ F ( f โ‰ค * g ) } ) . \mathfrak{d}=\min(\{|F|:F\subseteq\omega^{\omega}\land\forall f:\omega\to% \omega\exists g\in F(f\leq^{*}g)\}).
  37. F F
  38. ๐”Ÿ \mathfrak{b}
  39. ๐”ก \mathfrak{d}
  40. ๐”Ÿ > โ„ต 0 \mathfrak{b}>\aleph_{0}
  41. ๐”  = โ„ต 1 \mathfrak{c}=\aleph_{1}
  42. ๐”Ÿ = ๐”ก = โ„ต 1 \mathfrak{b}=\mathfrak{d}=\aleph_{1}
  43. ๐”Ÿ \mathfrak{b}
  44. ๐”ก \mathfrak{d}
  45. ๐”ฐ \mathfrak{s}
  46. ๐”ฏ \mathfrak{r}
  47. [ ฯ‰ ] ฯ‰ [\omega]^{\omega}
  48. ฯ‰ \omega
  49. a , b โˆˆ [ ฯ‰ ] ฯ‰ a,b\in[\omega]^{\omega}
  50. a a
  51. b b
  52. b โˆฉ a b\cap a
  53. b โˆ– a b\setminus a
  54. ๐”ฐ \mathfrak{s}
  55. S S
  56. [ ฯ‰ ] ฯ‰ [\omega]^{\omega}
  57. b โˆˆ [ ฯ‰ ] ฯ‰ b\in[\omega]^{\omega}
  58. a โˆˆ S a\in S
  59. a a
  60. b b
  61. ๐”ฐ = min ( { | S | : S โІ [ ฯ‰ ] ฯ‰ โˆง โˆ€ b โˆˆ [ ฯ‰ ] ฯ‰ โˆƒ a โˆˆ S ( | b โˆฉ a | = โ„ต 0 โˆง | b โˆ– a | = โ„ต 0 ) } ) . \mathfrak{s}=\min(\{|S|:S\subseteq[\omega]^{\omega}\land\forall b\in[\omega]^{% \omega}\exists a\in S(|b\cap a|=\aleph_{0}\land|b\setminus a|=\aleph_{0})\}).
  62. ๐”ฏ \mathfrak{r}
  63. R R
  64. [ ฯ‰ ] ฯ‰ [\omega]^{\omega}
  65. a a
  66. [ ฯ‰ ] ฯ‰ [\omega]^{\omega}
  67. R R
  68. ๐”ฏ = min ( { | R | : R โІ [ ฯ‰ ] ฯ‰ โˆง โˆ€ a โˆˆ [ ฯ‰ ] ฯ‰ โˆƒ b โˆˆ R ( | b โˆฉ a | < โ„ต 0 โˆจ | b โˆ– a | < โ„ต 0 ) } ) . \mathfrak{r}=\min(\{|R|:R\subseteq[\omega]^{\omega}\land\forall a\in[\omega]^{% \omega}\exists b\in R(|b\cap a|<\aleph_{0}\lor|b\setminus a|<\aleph_{0})\}).
  69. ๐”ฒ \mathfrak{u}
  70. ๐”ฒ \mathfrak{u}
  71. ฯ‰ \omega
  72. ๐”ฒ = โ„ต 1 \mathfrak{u}=\aleph_{1}
  73. ๐”  = โ„ต โ„ต 1 \mathfrak{c}=\aleph_{\aleph_{1}}
  74. ๐”ฒ = โ„ต 1 \mathfrak{u}=\aleph_{1}
  75. ๐”  = โ„ต 2 \mathfrak{c}=\aleph_{2}
  76. ๐”ž \mathfrak{a}
  77. A A
  78. B B
  79. ฯ‰ \omega
  80. | A โˆฉ B | |A\cap B|
  81. ฯ‰ \omega
  82. ฯ‰ \omega
  83. ๐’œ \mathcal{A}
  84. X X
  85. ฯ‰ \omega
  86. A โˆˆ ๐’œ A\in\mathcal{A}
  87. A A
  88. X X
  89. ๐”ž \mathfrak{a}
  90. ๐”Ÿ โ‰ค ๐”ž \mathfrak{b}\leq\mathfrak{a}
  91. ๐”Ÿ < ๐”ž \mathfrak{b}<\mathfrak{a}
  92. ฯ‰ ฯ‰ {}^{\omega}\omega

Carl_Severin_Wigert.html

  1. lim sup n โ†’ โˆž log d ( n ) log n / log log n = log 2. \limsup_{n\to\infty}\frac{\log d(n)}{\log n/\log\log n}=\log 2.

Cartan's_lemma.html

  1. v 1 โˆง w 1 + โ‹ฏ + v p โˆง w p = 0 v_{1}\wedge w_{1}+\cdots+v_{p}\wedge w_{p}=0
  2. w i = โˆ‘ j = 1 p h i j v j . w_{i}=\sum_{j=1}^{p}h_{ij}v_{j}.
  3. K 1 = { z 1 = x 1 + i y 1 | a 2 < x 1 < a 3 , b 1 < y 1 < b 2 } K 1 โ€ฒ = { z 1 = x 1 + i y 1 | a 1 < x 1 < a 3 , b 1 < y 1 < b 2 } K 1 โ€ฒโ€ฒ = { z 1 = x 1 + i y 1 | a 2 < x 1 < a 4 , b 1 < y 1 < b 2 } \begin{aligned}\displaystyle K_{1}&\displaystyle=\{z_{1}=x_{1}+iy_{1}|a_{2}<x_% {1}<a_{3},b_{1}<y_{1}<b_{2}\}\\ \displaystyle K_{1}^{\prime}&\displaystyle=\{z_{1}=x_{1}+iy_{1}|a_{1}<x_{1}<a_% {3},b_{1}<y_{1}<b_{2}\}\\ \displaystyle K_{1}^{\prime\prime}&\displaystyle=\{z_{1}=x_{1}+iy_{1}|a_{2}<x_% {1}<a_{4},b_{1}<y_{1}<b_{2}\}\end{aligned}
  4. K 1 = K 1 โ€ฒ โˆฉ K 1 โ€ฒโ€ฒ K_{1}=K_{1}^{\prime}\cap K_{1}^{\prime\prime}
  5. K = K 1 ร— K 2 ร— โ‹ฏ ร— K n K โ€ฒ = K 1 โ€ฒ ร— K 2 ร— โ‹ฏ ร— K n K โ€ฒโ€ฒ = K 1 โ€ฒโ€ฒ ร— K 2 ร— โ‹ฏ ร— K n \begin{aligned}\displaystyle K&\displaystyle=K_{1}\times K_{2}\times\cdots% \times K_{n}\\ \displaystyle K^{\prime}&\displaystyle=K_{1}^{\prime}\times K_{2}\times\cdots% \times K_{n}\\ \displaystyle K^{\prime\prime}&\displaystyle=K_{1}^{\prime\prime}\times K_{2}% \times\cdots\times K_{n}\end{aligned}
  6. K = K โ€ฒ โˆฉ K โ€ฒโ€ฒ K=K^{\prime}\cap K^{\prime\prime}
  7. F โ€ฒ F^{\prime}\,
  8. K โ€ฒ K^{\prime}\,
  9. F โ€ฒโ€ฒ F^{\prime\prime}\,
  10. K โ€ฒโ€ฒ K^{\prime\prime}\,
  11. F ( z ) = F โ€ฒ ( z ) / F โ€ฒโ€ฒ ( z ) F(z)=F^{\prime}(z)/F^{\prime\prime}(z)\,

Cartesian_monoid.html

  1. โŸจ * , e , ( - , - ) , L , R โŸฉ \langle*,e,(-,-),L,R\rangle
  2. * *
  3. ( - , - ) (-,-)
  4. L , R L,R
  5. e e
  6. x , y , z x,y,z
  7. * *
  8. e e
  9. L * ( x , y ) = x L*(x,\,y)=x
  10. R * ( x , y ) = y R*(x,\,y)=y
  11. ( L * x , R * x ) = x (L*x,\,R*x)=x
  12. ( x * z , y * z ) = ( x , y ) * z (x*z,\,y*z)=(x,\,y)*z
  13. L L
  14. R R
  15. ( - , - ) (-,-)

Cartesian_oval.html

  1. P P
  2. Q Q
  3. d ( P , S ) d(P,S)
  4. d ( Q , S ) d(Q,S)
  5. S S
  6. m m
  7. a a
  8. d ( P , S ) + m d ( Q , S ) = a d(P,S)+md(Q,S)=a
  9. d ( P , S ) + m d ( Q , S ) = ยฑ a d(P,S)+md(Q,S)=ยฑa
  10. d ( P , S ) โˆ’ m d ( Q , S ) = ยฑ a d(P,S)โˆ’md(Q,S)=ยฑa
  11. d ( P , S ) + m d ( Q , S ) = a d(P,S)+md(Q,S)=a
  12. m = 1 m=1
  13. a > d ( P , Q ) a>d(P,Q)
  14. m = a / d ( P , Q ) m=a/\,\text{d}(P,Q)
  15. m = - 1 m=-1
  16. 0 < a < d ( P , Q ) 0<a<\,\text{d}(P,Q)
  17. ( x , y ) (x,y)
  18. c c
  19. d ( P , Q ) \,\text{d}(P,Q)
  20. P = ( 0 , 0 ) P=(0,0)
  21. Q = ( c , 0 ) Q=(c,0)
  22. d ( P , S ) ยฑ m d ( Q , S ) = a d(P,S)ยฑmd(Q,S)=a
  23. d ( P , S ) ยฑ m d ( Q , S ) = โˆ’ a d(P,S)ยฑmd(Q,S)=โˆ’a
  24. P P
  25. Q Q
  26. P Q PQ
  27. P P
  28. Q Q
  29. P P
  30. Q Q
  31. P P
  32. Q Q
  33. P P
  34. Q Q
  35. R R

CASA_ratio.html

  1. C A S A R a t i o = ( C A S A D e p o s i t s T o t a l D e p o s i t s ) {CASARatio}=\left(\frac{CASADeposits}{TotalDeposits}\right)

Castelnuovo_curve.html

  1. g โ‰ค ( n - 1 ) m ( m - 1 ) / 2 + m ฯต g\leq(n-1)m(m-1)/2+m\epsilon

Catalecticant.html

  1. [ a b c d e b c d e f c d e f g d e f g h e f g h i ] . \begin{bmatrix}a&b&c&d&e\\ b&c&d&e&f\\ c&d&e&f&g\\ d&e&f&g&h\\ e&f&g&h&i\end{bmatrix}.

Category:Superparticular_intervals.html

  1. n + 1 n = 1 + 1 n . {n+1\over n}=1+{1\over n}.

Category_O.html

  1. ๐’ช \mathcal{O}
  2. ๐”ค \mathfrak{g}
  3. ๐”ฅ \mathfrak{h}
  4. ฮฆ \Phi
  5. ฮฆ + \Phi^{+}
  6. ๐”ค ฮฑ \mathfrak{g}_{\alpha}
  7. ฮฑ โˆˆ ฮฆ \alpha\in\Phi
  8. ๐”ซ := โŠ• ฮฑ โˆˆ ฮฆ + ๐”ค ฮฑ \mathfrak{n}:=\oplus_{\alpha\in\Phi^{+}}\mathfrak{g}_{\alpha}
  9. M M
  10. ๐”ค \mathfrak{g}
  11. ฮป โˆˆ ๐”ฅ * \lambda\in\mathfrak{h}^{*}
  12. M ฮป M_{\lambda}
  13. M ฮป = { v โˆˆ M ; โˆ€ h โˆˆ ๐”ฅ h โ‹… v = ฮป ( h ) v } . M_{\lambda}=\{v\in M;\,\,\forall\,h\in\mathfrak{h}\,\,h\cdot v=\lambda(h)v\}.
  14. ๐”ค \mathfrak{g}
  15. M M
  16. M M
  17. M = โŠ• ฮป โˆˆ ๐”ฅ * M ฮป M=\oplus_{\lambda\in\mathfrak{h}^{*}}M_{\lambda}
  18. M M
  19. ๐”ซ \mathfrak{n}
  20. v โˆˆ M v\in M
  21. ๐”ซ \mathfrak{n}
  22. v v
  23. ๐”ค \mathfrak{g}
  24. Z ( ๐”ค ) Z(\mathfrak{g})
  25. M M
  26. v โˆˆ M v\in M
  27. Z ( ๐”ค ) v โІ M Z(\mathfrak{g})v\subseteq M
  28. v v
  29. ๐”ค \mathfrak{g}
  30. ๐”ค \mathfrak{g}
  31. ๐”ค \mathfrak{g}

Cationic_polymerization.html

  1. E \mathit{E}
  2. E i \mathit{E_{i}}
  3. E p \mathit{E_{p}}
  4. E t \mathit{E_{t}}
  5. E = E i + E p - E t \textstyle E=E_{i}+E_{p}-E_{t}
  6. E t \mathit{E_{t}}
  7. E i \mathit{E_{i}}
  8. E p \mathit{E_{p}}
  9. I + + M โ†’ k i M + \,\text{I}^{+}~{}+~{}\,\text{M}~{}~{}\xrightarrow{k_{i}}~{}~{}\,\text{M}^{+}
  10. M + + M โ†’ k p M + \,\text{M}^{+}~{}+~{}\,\text{M}~{}~{}\xrightarrow{k_{p}}~{}~{}\,\text{M}^{+}
  11. M + โ†’ k t M \,\text{M}^{+}~{}~{}\xrightarrow{k_{t}}~{}~{}\,\text{M}
  12. M + + M โ†’ k t r M + M + \,\text{M}^{+}~{}+~{}\,\text{M}~{}~{}\xrightarrow{k_{tr}}~{}~{}\,\text{M}~{}+~% {}\,\text{M}^{+}
  13. k i \mathit{k_{i}}
  14. k p \mathit{k_{p}}
  15. k t \mathit{k_{t}}
  16. k ๐‘ก๐‘Ÿ \mathit{k_{tr}}
  17. rate(initiation) = k i [ I + ] [ M ] \textstyle\,\text{rate(initiation)}=k_{i}[\,\text{I}^{+}][\,\text{M}]
  18. rate(propagation) = k p [ M + ] [ M ] \textstyle\,\text{rate(propagation)}=k_{p}[\,\text{M}^{+}][\,\text{M}]
  19. rate(termination) = k t [ M + ] \textstyle\,\text{rate(termination)}=k_{t}[\,\text{M}^{+}]
  20. rate(chain transfer) = k t r [ M + ] [ M ] \textstyle\,\text{rate(chain transfer)}=k_{tr}[\,\text{M}^{+}][\,\text{M}]
  21. [ M + ] = k i [ I + ] [ M ] k t [\,\text{M}^{+}]={k_{i}[\,\text{I}^{+}][\,\text{M}]\over k_{t}}
  22. rate(propagation) = k p k i [ M ] 2 [ I + ] k t \,\text{rate(propagation)}={k_{p}k_{i}[\,\text{M}]^{2}[\,\text{I}^{+}]\over k_% {t}}
  23. X n \mathit{X_{n}}
  24. X n = rate(propagation) rate(termination) = k p [ M ] k t Xn={\,\text{rate(propagation)}\over\,\text{rate(termination)}}={k_{p}[\,\text{% M}]\over k_{t}}
  25. X n \mathit{X_{n}}
  26. X n = rate(propagation) rate(chain transfer) = k p k t r Xn={\,\text{rate(propagation)}\over\,\text{rate(chain transfer)}}={k_{p}\over k% _{tr}}

Cavalieri's_quadrature_formula.html

  1. โˆซ 0 a x n d x = 1 n + 1 a n + 1 n โ‰ฅ 0 , \int_{0}^{a}x^{n}\,dx=\tfrac{1}{n+1}\,a^{n+1}\qquad n\geq 0,
  2. โˆซ x n d x = 1 n + 1 x n + 1 + C n โ‰  - 1. \int x^{n}\,dx=\tfrac{1}{n+1}\,x^{n+1}+C\qquad n\neq-1.
  3. โˆซ 1 a x n d x = 1 n + 1 ( a n + 1 - 1 ) n โ‰  - 1. \int_{1}^{a}x^{n}\,dx=\tfrac{1}{n+1}(a^{n+1}-1)\qquad n\neq-1.
  4. โˆซ 1 a 1 x d x = ln a , \int_{1}^{a}\frac{1}{x}\,dx=\ln a,
  5. โˆซ 1 x d x = ln x + C , x > 0 \int\frac{1}{x}\,dx=\ln x+C,\qquad x>0
  6. โˆซ 1 x d x = ln | x | + C , x โ‰  0. \int\frac{1}{x}\,dx=\ln|x|+C,\qquad x\neq 0.
  7. โˆซ 1 x d x = { ln | x | + C - x < 0 ln | x | + C + x > - 0 \int\frac{1}{x}\,dx=\begin{cases}\ln|x|+C^{-}&x<0\\ \ln|x|+C^{+}&x>-0\end{cases}
  8. โˆซ - c c 1 x d x = 0 , \int_{-c}^{c}\frac{1}{x}\,dx=0,
  9. โˆซ - 1 1 1 x d x = 0 , \int_{-1}^{1}\frac{1}{x}\,dx=0,
  10. โˆซ 0 a x n - 1 d x = 1 n a n n โ‰ฅ 1. \int_{0}^{a}x^{n-1}\,dx=\tfrac{1}{n}a^{n}\qquad n\geq 1.
  11. โˆซ x n - 1 d x = 1 n x n + C n โ‰  0. \int x^{n-1}\,dx=\tfrac{1}{n}x^{n}+C\qquad n\neq 0.
  12. โˆซ ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C (for n โ‰  - 1 ) \int(ax+b)^{n}dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C\qquad\mbox{(for }~{}n\neq-1% \mbox{)}~{}\,\!
  13. โˆซ 1 a x + b d x = 1 a ln | a x + b | + C \int\frac{1}{ax+b}dx=\frac{1}{a}\ln\left|ax+b\right|+C
  14. โˆซ 1 a x + b d x = { 1 a ln | a x + b | + C - x < - b / a 1 a ln | a x + b | + C + x > - b / a \int\frac{1}{ax+b}\,dx=\begin{cases}\frac{1}{a}\ln\left|ax+b\right|+C^{-}&x<-b% /a\\ \frac{1}{a}\ln\left|ax+b\right|+C^{+}&x>-b/a\end{cases}
  15. โˆซ 1 x d x , \int\frac{1}{x}\,dx,
  16. 1 0 โ‹… x 0 \frac{1}{0}\cdot x^{0}
  17. y = x p / q , y=x^{p/q},
  18. x p = k y q x^{p}=ky^{q}
  19. x p y q = k x^{p}y^{q}=k

Cavity_optomechanics.html

  1. k k
  2. p = โ„ k p=\hbar k
  3. โ„ \hbar
  4. ฮ” p = 2 โ„ k \Delta p=2\hbar k
  5. โ„ ฯ‰ m \hbar\omega_{m}
  6. H = โ„ ฯ‰ c a โ€  a + 1 2 โ„ ฯ‰ m ( p m 2 + x m 2 ) - โ„ g 0 a โ€  a x m + i โ„ E ( a โ€  e - i ฯ‰ l t - a e i ฯ‰ l t ) H=\hbar\omega_{c}a^{\dagger}a+\frac{1}{2}\hbar\omega_{m}(p_{m}^{2}+x_{m}^{2})-% \hbar g_{0}a^{\dagger}ax_{m}+i\hbar E(a^{\dagger}e^{-i\omega_{l}t}-ae^{i\omega% _{l}t})
  7. ฯ‰ c \omega_{c}
  8. a a
  9. a โ€  a^{\dagger}
  10. p m p_{m}
  11. x m x_{m}
  12. g 0 g_{0}
  13. E E
  14. ฯ‰ l \omega_{l}
  15. 10 5 10^{5}
  16. 10 8 10^{8}

Cavity_perturbation_theory.html

  1. ฯ‰ - ฯ‰ 0 ฯ‰ 0 โ‰ˆ - โˆญ V ( ฮ” ฮผ | H 0 | 2 + ฮ” ฯต | E 0 | 2 ) d v โˆญ V ( ฮผ | H 0 | 2 + ฯต | E 0 | 2 ) d v \frac{\omega-\omega_{0}}{\omega_{0}}\thickapprox-\frac{\iiint_{V}(\Delta\mu|H_% {0}|^{2}+\Delta\epsilon|E_{0}|^{2})dv}{\iiint_{V}(\mu|H_{0}|^{2}+\epsilon|E_{0% }|^{2})dv}\,
  2. ฯ‰ \omega
  3. ฯ‰ 0 \omega_{0}
  4. E 0 E_{0}
  5. H 0 H_{0}
  6. ฮผ \mu
  7. ฯต \epsilon
  8. ฮ” ฮผ \Delta\mu
  9. ฮ” ฯต \Delta\epsilon
  10. ฯ‰ - ฯ‰ 0 ฯ‰ 0 โ‰ˆ - 1 W โˆญ V ( ฮ” ฯต ฯต โ‹… w e ยฏ + ฮ” ฮผ ฮผ โ‹… w m ยฏ ) d v \frac{\omega-\omega_{0}}{\omega_{0}}\thickapprox-\frac{1}{W}\iiint_{V}(\frac{% \Delta\epsilon}{\epsilon}\cdot\bar{w_{e}}+\frac{\Delta\mu}{\mu}\cdot\bar{w_{m}% })dv\,
  11. w e ยฏ \bar{w_{e}}
  12. w m ยฏ \bar{w_{m}}
  13. ฯ‰ - ฯ‰ 0 ฯ‰ 0 โ‰ˆ โˆญ ฮ” V ( ฮผ | H 0 | 2 - ฯต | E 0 | 2 ) d v โˆญ V ( ฮผ | H 0 | 2 + ฯต | E 0 | 2 ) d v \frac{\omega-\omega_{0}}{\omega_{0}}\thickapprox\frac{\iiint_{\Delta V}(\mu|H_% {0}|^{2}-\epsilon|E_{0}|^{2})dv}{\iiint_{V}(\mu|H_{0}|^{2}+\epsilon|E_{0}|^{2}% )dv}\,
  14. ฯ‰ - ฯ‰ 0 ฯ‰ 0 โ‰ˆ ฮ” W m - ฮ” W e W m + W e \frac{\omega-\omega_{0}}{\omega_{0}}\thickapprox\frac{\Delta W_{m}-\Delta W_{e% }}{W_{m}+W_{e}}\,
  15. ฮ” W m \Delta W_{m}
  16. ฮ” W e \Delta W_{e}
  17. ฮ” V \Delta V
  18. ฯ‰ - ฯ‰ 0 ฯ‰ 0 โ‰ˆ ( w m ยฏ - w e ยฏ ) โ‹… ฮ” V W \frac{\omega-\omega_{0}}{\omega_{0}}\thickapprox\frac{(\bar{w_{m}}-\bar{w_{e}}% )\cdot\Delta V}{W}\,
  19. T E 10 n TE_{10n}
  20. T E 10 n TE_{10n}
  21. ฯต r = ฯต r โ€ฒ + j ฯต r โ€ฒโ€ฒ \epsilon_{r}=\epsilon_{r}^{\prime}+j\epsilon_{r}^{\prime\prime}
  22. ฯต r โ€ฒ - 1 = f c - f s 2 f s V c V s \epsilon_{r}^{\prime}-1=\frac{f_{c}-f_{s}}{2f_{s}}\frac{V_{c}}{V_{s}}\,
  23. ฯต r โ€ฒโ€ฒ = V c 4 V s Q c - Q s Q c Q s \epsilon_{r}^{\prime\prime}=\frac{V_{c}}{4V_{s}}\frac{Q_{c}-Q_{s}}{Q_{c}Q_{s}}\,
  24. f c f_{c}
  25. f s f_{s}
  26. V c V_{c}
  27. V s V_{s}
  28. Q c Q_{c}
  29. Q s Q_{s}
  30. ฯƒ e \sigma_{e}
  31. tan ฮด \tan\delta
  32. t a n ฮด = ฯต r โ€ฒโ€ฒ ฯต r โ€ฒ tan\delta=\frac{\epsilon_{r}^{\prime\prime}}{\epsilon_{r}^{\prime}}\,
  33. ฯต 0 \epsilon_{0}
  34. ฮผ r = ฮผ r โ€ฒ + j ฮผ r โ€ฒโ€ฒ \mu_{r}=\mu_{r}^{\prime}+j\mu_{r}^{\prime\prime}
  35. ฮผ r โ€ฒ - 1 = ( ฮป g 2 + 4 a 2 8 a 2 ) ( f c - f s f s ) ( V c V s ) \mu_{r}^{\prime}-1=(\frac{\lambda_{g}^{2}+4a^{2}}{8a^{2}})(\frac{f_{c}-f_{s}}{% f_{s}})(\frac{V_{c}}{V_{s}})\,
  36. ฮผ r โ€ฒโ€ฒ = ( ฮป g 2 + 4 a 2 16 a 2 ) ( V c V s ) ( Q c - Q s Q c Q s ) \mu_{r}^{\prime\prime}=(\frac{\lambda_{g}^{2}+4a^{2}}{16a^{2}})(\frac{V_{c}}{V% _{s}})(\frac{Q_{c}-Q_{s}}{Q_{c}Q_{s}})\,
  37. ฮป g \lambda_{g}
  38. ฮป g = 2 l n \lambda_{g}=\frac{2l}{n}

Cayleyโ€“Klein_metric.html

  1. d ( p , q ) = C log | q a | | b p | | p a | | b q | d(p,q)=C\log\frac{|qa||bp|}{|pa||bq|}
  2. { ( ฮพ , ฮท , ฮถ , ฯ„ ) : ฯ„ = 0 , ฮพ 2 + ฮท 2 + ฮถ 2 = 0 } โŠ‚ P 3 ( R ) . \{(\xi,\eta,\zeta,\tau):\quad\tau=0,\quad\xi^{2}+\eta^{2}+\zeta^{2}=0\}\subset P% ^{3}(R).
  3. x 2 + y 2 + z 2 - t 2 = 0 x^{2}+y^{2}+z^{2}-t^{2}=0
  4. d x 2 + d y 2 + d z 2 - d t 2 = 0 dx^{2}+dy^{2}+dz^{2}-dt^{2}=0
  5. ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 2 = 1 \left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}+\left(\frac{dz}{% dt}\right)^{2}=1

Caฬ€dlaฬ€g.html

  1. r r
  2. r r
  3. โ„™ [ x โ‰ค r ] \mathbb{P}[x\leq r]
  4. ( - โˆž , r ] (-\infty,r]
  5. w f ( F ) := sup s , t โˆˆ F | f ( s ) - f ( t ) | w_{f}(F):=\sup_{s,t\in F}|f(s)-f(t)|
  6. ฯ– f โ€ฒ ( ฮด ) := inf ฮ  max 1 โ‰ค i โ‰ค k w f ( [ t i - 1 , t i ) ) , \varpi^{\prime}_{f}(\delta):=\inf_{\Pi}\max_{1\leq i\leq k}w_{f}([t_{i-1},t_{i% })),
  7. โˆฅ f โˆฅ := sup t โˆˆ E | f ( t ) | \|f\|:=\sup_{t\in E}|f(t)|
  8. ฯƒ ( f , g ) := inf ฮป โˆˆ ฮ› max { โˆฅ ฮป - I โˆฅ , โˆฅ f - g โˆ˜ ฮป โˆฅ } , \sigma(f,g):=\inf_{\lambda\in\Lambda}\max\{\|\lambda-I\|,\|f-g\circ\lambda\|\},
  9. lim a โ†’ โˆž lim sup n โ†’ โˆž ฮผ n ( { f โˆˆ D | โˆฅ f โˆฅ โ‰ฅ a } ) = 0 , \lim_{a\to\infty}\limsup_{n\to\infty}\mu_{n}\big(\{f\in D\;|\;\|f\|\geq a\}% \big)=0,
  10. lim ฮด โ†’ 0 lim sup n โ†’ โˆž ฮผ n ( { f โˆˆ D | ฯ– f โ€ฒ ( ฮด ) โ‰ฅ ฮต } ) = 0 for all ฮต > 0. \lim_{\delta\to 0}\limsup_{n\to\infty}\mu_{n}\big(\{f\in D\;|\;\varpi^{\prime}% _{f}(\delta)\geq\varepsilon\}\big)=0\,\text{ for all }\varepsilon>0.
  11. E = [ 0 , 2 ) E=[0,2)
  12. f n = ฯ‡ [ 1 - 1 / n , 2 ) โˆˆ D f_{n}=\chi_{[1-1/n,2)}\in D
  13. f n โ†’ ฯ‡ [ 1 , 2 ) f_{n}\rightarrow\chi_{[1,2)}
  14. f n - ฯ‡ [ 1 , 2 ) f_{n}-\chi_{[1,2)}

Ceiling_temperature.html

  1. T c T_{c}
  2. ฮ” G p = ฮ” H p - T ฮ” S p \Delta G_{p}=\Delta H_{p}-T\Delta S_{p}
  3. ฮ” S p \Delta S_{p}
  4. ฮ” H p \Delta H_{p}
  5. ฮ” H p = E p - E d p \Delta H_{p}=E_{p}-E_{dp}
  6. E p E_{p}
  7. E d p E_{dp}
  8. T ฮ” S p T\Delta S_{p}
  9. T c = ฮ” H p ฮ” S p T_{c}=\frac{\Delta H_{p}}{\Delta S_{p}}

Cell-free_marginal_layer_model.html

  1. R R
  2. - ฮ” P L = 1 r d d r ( ฮผ c r d u c d r ) ; \frac{-\Delta P}{L}=\frac{1}{r}\frac{d}{dr}(\mu_{c}r\frac{du_{c}}{dr});
  3. 0 โ‰ค r โ‰ค R - ฮด 0\leq r\ \leq R-\delta\,
  4. - ฮ” P L = 1 r d d r ( ฮผ p r d u p d r ) ; \frac{-\Delta P}{L}=\frac{1}{r}\frac{d}{dr}(\mu_{p}r\frac{du_{p}}{dr});
  5. R - ฮด โ‰ค r โ‰ค R R-\delta\leq r\ \leq R\ \,
  6. ฮ” P \Delta P
  7. L L
  8. u c u_{c}
  9. u p u_{p}
  10. ฮผ c \mu_{c}
  11. ฮผ p \mu_{p}
  12. ฮด \delta
  13. d u c d r | r = 0 = 0 \left.\frac{du_{c}}{dr}\right|_{r=0}=0
  14. u p | r = R = 0 \left.u_{p}\right|_{r=R}=0
  15. u p | r = R - ฮด = u c | r = R - ฮด \left.u_{p}\right|_{r=R-\delta}=\left.u_{c}\right|_{r=R-\delta}
  16. ฯ„ p | r = R - ฮด = ฯ„ c | r = R - ฮด \left.\tau_{p}\right|_{r=R-\delta}=\left.\tau_{c}\right|_{r=R-\delta}
  17. u c = ฮ” P R 2 4 ฮผ p L [ 1 - ( R - ฮด R ) 2 - ฮผ p ฮผ c ( r R ) 2 + ฮผ p ฮผ c ( R - ฮด R ) 2 ] u_{c}=\frac{\Delta PR^{2}}{4\mu_{p}L}[1-(\frac{R-\delta}{R})^{2}-\frac{\mu_{p}% }{\mu_{c}}(\frac{r}{R})^{2}+\frac{\mu_{p}}{\mu_{c}}(\frac{R-\delta}{R})^{2}]
  18. u p = ฮ” P R 2 4 ฮผ p L [ 1 - ( r R ) 2 ] u_{p}=\frac{\Delta PR^{2}}{4\mu_{p}L}[1-(\frac{r}{R})^{2}]
  19. Q = ฯ€ ฮ” P R 4 8 ฮผ p L [ 1 - ( 1 - ฮด R ) 4 ( 1 - ฮผ p ฮผ c ) ] Q=\frac{\pi\Delta PR^{4}}{8\mu_{p}L}[1-(1-\frac{\delta}{R})^{4}(1-\frac{\mu_{p% }}{\mu_{c}})]
  20. ฮผ e \mu_{e}
  21. ฮผ e = ฮผ p [ 1 - ( 1 - ฮด R ) 4 ( 1 - ฮผ p ฮผ c ) ] \mu_{e}=\frac{\mu_{p}}{[1-(1-\frac{\delta}{R})^{4}(1-\frac{\mu_{p}}{\mu_{c}})]}
  22. ฮผ c \mu_{c}

Cellular_algebra.html

  1. R R
  2. A A
  3. R R
  4. A A
  5. ( ฮ› , i , M , C ) (\Lambda,i,M,C)
  6. ฮ› \Lambda
  7. R R
  8. i : A โ†’ A i:A\to A
  9. i 2 = i d A i^{2}=id_{A}
  10. ฮป โˆˆ ฮ› \lambda\in\Lambda
  11. M ( ฮป ) M(\lambda)
  12. C : โ‹ƒ ห™ ฮป โˆˆ ฮ› M ( ฮป ) ร— M ( ฮป ) โ†’ A C:\dot{\bigcup}_{\lambda\in\Lambda}M(\lambda)\times M(\lambda)\to A
  13. ฮป โˆˆ ฮ› \lambda\in\Lambda
  14. ๐”ฐ , ๐”ฑ โˆˆ M ( ฮป ) \mathfrak{s},\mathfrak{t}\in M(\lambda)
  15. C ๐”ฐ ๐”ฑ ฮป C_{\mathfrak{st}}^{\lambda}
  16. C C
  17. R R
  18. A A
  19. i ( C ๐”ฐ ๐”ฑ ฮป ) = C ๐”ฑ ๐”ฐ ฮป i(C_{\mathfrak{st}}^{\lambda})=C_{\mathfrak{ts}}^{\lambda}
  20. ฮป โˆˆ ฮ› \lambda\in\Lambda
  21. ๐”ฐ , ๐”ฑ โˆˆ M ( ฮป ) \mathfrak{s},\mathfrak{t}\in M(\lambda)
  22. a โˆˆ A a\in A
  23. a C ๐”ฐ ๐”ฑ ฮป โ‰ก โˆ‘ ๐”ฒ โˆˆ M ( ฮป ) r a ( ๐”ฒ , ๐”ฐ ) C ๐”ฒ ๐”ฑ ฮป mod A ( < ฮป ) aC_{\mathfrak{st}}^{\lambda}\equiv\sum_{\mathfrak{u}\in M(\lambda)}r_{a}(% \mathfrak{u},\mathfrak{s})C_{\mathfrak{ut}}^{\lambda}\mod A(<\lambda)
  24. r a ( ๐”ฒ , ๐”ฐ ) โˆˆ R r_{a}(\mathfrak{u},\mathfrak{s})\in R
  25. a a
  26. ๐”ฒ \mathfrak{u}
  27. ๐”ฐ \mathfrak{s}
  28. ๐”ฑ \mathfrak{t}
  29. A ( < ฮป ) A(<\lambda)
  30. R R
  31. ฮป \lambda
  32. i : A โ†’ A i:A\to A
  33. R R
  34. i 2 = i d i^{2}=id
  35. A A
  36. i i
  37. J โІ A J\subseteq A
  38. i ( J ) = J i(J)=J
  39. ฮ” โІ J \Delta\subseteq J
  40. R R
  41. ฮฑ : ฮ” โŠ— R i ( ฮ” ) โ†’ J \alpha:\Delta\otimes_{R}i(\Delta)\to J
  42. A A
  43. A A
  44. ฮฑ \alpha
  45. i i
  46. โˆ€ x , y โˆˆ ฮ” : i ( ฮฑ ( x โŠ— i ( y ) ) ) = ฮฑ ( y โŠ— i ( x ) ) \forall x,y\in\Delta:i(\alpha(x\otimes i(y)))=\alpha(y\otimes i(x))
  47. A A
  48. i i
  49. A = โŠ• k = 1 m U k A=\bigoplus_{k=1}^{m}U_{k}
  50. R R
  51. i ( U k ) = U k i(U_{k})=U_{k}
  52. J k := โŠ• j = 1 k U j J_{k}:=\bigoplus_{j=1}^{k}U_{j}
  53. A A
  54. J k / J k - 1 J_{k}/J_{k-1}
  55. A / J k - 1 A/J_{k-1}
  56. ( A , i ) (A,i)
  57. ฮ› \Lambda
  58. ฮ” / J k - 1 โІ J k / J k - 1 \Delta/J_{k-1}\subseteq J_{k}/J_{k-1}
  59. A A
  60. R [ x ] / ( x n ) R[x]/(x^{n})
  61. i = i d i=id
  62. ฮ› := { 0 , โ€ฆ , n - 1 } \Lambda:=\{0,\ldots,n-1\}
  63. M ( ฮป ) := { 1 } M(\lambda):=\{1\}
  64. C 11 ฮป := x ฮป C_{11}^{\lambda}:=x^{\lambda}
  65. 0 โІ ( x n - 1 ) โІ ( x n - 2 ) โІ โ€ฆ โІ ( x 1 ) โІ ( x 0 ) = R 0\subseteq(x^{n-1})\subseteq(x^{n-2})\subseteq\ldots\subseteq(x^{1})\subseteq(% x^{0})=R
  66. R d ร— d R^{d\times d}
  67. i ( A ) = A T i(A)=A^{T}
  68. ฮ› := { 1 } \Lambda:=\{1\}
  69. M ( 1 ) := { 1 , โ€ฆ , d } M(1):=\{1,\dots,d\}
  70. C s t 1 := E s t C_{st}^{1}:=E_{st}
  71. C s t 1 C_{st}^{1}
  72. 0 โІ R d ร— d 0\subseteq R^{d\times d}
  73. ฮ› \Lambda
  74. T w โ†ฆ T w - 1 T_{w}\mapsto T_{w^{-1}}
  75. ๐’ช \mathcal{O}
  76. A A
  77. ( ฮ› , i , M , C ) (\Lambda,i,M,C)
  78. A A
  79. W ( ฮป ) W(\lambda)
  80. R R
  81. { C ๐”ฐ | ๐”ฐ โˆˆ M ( ฮป ) } \{C_{\mathfrak{s}}|\mathfrak{s}\in M(\lambda)\}
  82. a C ๐”ฐ := โˆ‘ ๐”ฒ r a ( ๐”ฒ , ๐”ฐ ) C ๐”ฒ aC_{\mathfrak{s}}:=\sum_{\mathfrak{u}}r_{a}(\mathfrak{u},\mathfrak{s})C_{% \mathfrak{u}}
  83. r a ( ๐”ฒ , ๐”ฐ ) r_{a}(\mathfrak{u},\mathfrak{s})
  84. W ( ฮป ) W(\lambda)
  85. A A
  86. ฯ• ฮป : W ( ฮป ) ร— W ( ฮป ) โ†’ R \phi_{\lambda}:W(\lambda)\times W(\lambda)\to R
  87. C ๐”ฐ ๐”ฑ ฮป C ๐”ฒ ๐”ณ ฮป โ‰ก ฯ• ฮป ( C ๐”ฑ , C ๐”ฒ ) C ๐”ฐ ๐”ณ ฮป mod A ( < ฮป ) C_{\mathfrak{st}}^{\lambda}C_{\mathfrak{uv}}^{\lambda}\equiv\phi_{\lambda}(C_{% \mathfrak{t}},C_{\mathfrak{u}})C_{\mathfrak{sv}}^{\lambda}\mod A(<\lambda)
  88. s , t , u , v โˆˆ M ( ฮป ) s,t,u,v\in M(\lambda)
  89. ฯ• ฮป \phi_{\lambda}
  90. ฯ• ฮป ( x , y ) = ฯ• ฮป ( y , x ) \phi_{\lambda}(x,y)=\phi_{\lambda}(y,x)
  91. x , y โˆˆ W ( ฮป ) x,y\in W(\lambda)
  92. A A
  93. ฯ• ฮป ( i ( a ) x , y ) = ฯ• ฮป ( x , a y ) \phi_{\lambda}(i(a)x,y)=\phi_{\lambda}(x,ay)
  94. a โˆˆ A a\in A
  95. x , y โˆˆ W ( ฮป ) x,y\in W(\lambda)
  96. R R
  97. A A
  98. ฮ› 0 := { ฮป โˆˆ ฮ› | ฯ• ฮป โ‰  0 } \Lambda_{0}:=\{\lambda\in\Lambda|\phi_{\lambda}\neq 0\}
  99. L ( ฮป ) := W ( ฮป ) / rad ( ฯ• ฮป ) L(\lambda):=W(\lambda)/\operatorname{rad}(\phi_{\lambda})
  100. ฮป โˆˆ ฮ› 0 \lambda\in\Lambda_{0}
  101. L ( ฮป ) L(\lambda)
  102. A A
  103. A A
  104. R R
  105. R R
  106. A A
  107. A o p A^{op}
  108. A A
  109. ( ฮ› , i , M , C ) (\Lambda,i,M,C)
  110. ฮฆ โІ ฮ› \Phi\subseteq\Lambda
  111. ฮ› \Lambda
  112. A ( ฮฆ ) := โˆ‘ R C ๐”ฐ ๐”ฑ ฮป A(\Phi):=\sum RC_{\mathfrak{st}}^{\lambda}
  113. ฮป โˆˆ ฮ› \lambda\in\Lambda
  114. s , t โˆˆ M ( ฮป ) s,t\in M(\lambda)
  115. i i
  116. A A
  117. A / A ( ฮฆ ) A/A(\Phi)
  118. ( ฮ› โˆ– ฮฆ , i , M , C ) (\Lambda\setminus\Phi,i,M,C)
  119. A A
  120. R R
  121. R โ†’ S R\to S
  122. S โŠ— R A S\otimes_{R}A
  123. S S
  124. R R
  125. R R
  126. ( A , i ) (A,i)
  127. R R
  128. A = A 1 โŠ• A 2 A=A_{1}\oplus A_{2}
  129. i i
  130. ( A , i ) (A,i)
  131. ( A 1 , i ) (A_{1},i)
  132. ( A 2 , i ) (A_{2},i)
  133. A A
  134. i i
  135. ( A , i ) (A,i)
  136. R R
  137. i i
  138. i i
  139. i i
  140. A A
  141. R R
  142. R โ†’ k R\to k
  143. k k
  144. K := Q u o t ( R ) K:=Quot(R)
  145. R R
  146. k A kA
  147. K A KA
  148. R R
  149. A A
  150. i i
  151. e โˆˆ A e\in A
  152. i ( e ) = e i(e)=e
  153. e A e eAe
  154. R R
  155. A A
  156. i i
  157. A A
  158. A A
  159. A A
  160. โˆ€ ฮป โˆˆ ฮ› : W ( ฮป ) \forall\lambda\in\Lambda:W(\lambda)
  161. โˆ€ ฮป โˆˆ ฮ› : ฯ• ฮป \forall\lambda\in\Lambda:\phi_{\lambda}
  162. C A C_{A}
  163. A A
  164. A A
  165. ฮ› = ฮ› 0 \Lambda=\Lambda_{0}
  166. ( A , i ) (A,i)
  167. ( A , j ) (A,j)
  168. j : A โ†’ A j:A\to A
  169. A A
  170. det ( C A ) = 1 \det(C_{A})=1
  171. A A
  172. B B
  173. R R
  174. B B
  175. A A
  176. e โˆˆ A e\in A
  177. i ( e ) i(e)
  178. A e โ‰… A i ( e ) Ae\cong Ai(e)
  179. c h a r ( R ) โ‰  2 char(R)\neq 2
  180. i i

Cellular_noise.html

  1. ฮท X = ฯƒ X ฮผ X , \eta_{X}=\frac{\sigma_{X}}{\mu_{X}},
  2. ฮท X \eta_{X}
  3. X X
  4. ฮผ X \mu_{X}
  5. X X
  6. ฯƒ X \sigma_{X}
  7. X X
  8. F X = ฯƒ X 2 ฮผ X . F_{X}=\frac{\sigma_{X}^{2}}{\mu_{X}}.
  9. P ( ๐ฑ , t ) P(\mathbf{x},t)
  10. ๐ฑ \mathbf{x}
  11. t t

Center_vortex.html

  1. z n = e 2 ฯ€ i n N I , z_{n}=e^{\frac{2\pi in}{N}}I\;,
  2. ฯˆ โ†’ e 2 ฯ€ i n N ฯˆ , \psi\to e^{\frac{2\pi in}{N}}\psi\;,
  3. โŸจ W ( C ) โŸฉ โˆ e - ฯƒ A , \langle W(C)\rangle\propto e^{-\sigma A}\;,
  4. โŸจ W ( C ) โŸฉ โˆ e - ฮฑ L , \langle W(C)\rangle\propto e^{-\alpha L}\;,

Central_limit_theorem_for_directional_statistics.html

  1. ฮธ i \theta_{i}
  2. 2 ฯ€ 2\pi
  3. z i = e i ฮธ i = cos ( ฮธ i ) + i sin ( ฮธ i ) z_{i}=e^{i\theta_{i}}=\cos(\theta_{i})+i\sin(\theta_{i})
  4. m n = E ( z n ) = C n + i S n = R n e i ฮธ n m_{n}=E(z^{n})=C_{n}+iS_{n}=R_{n}e^{i\theta_{n}}\,
  5. C n = E ( cos ( n ฮธ ) ) C_{n}=E(\cos(n\theta))\,
  6. S n = E ( sin ( n ฮธ ) ) S_{n}=E(\sin(n\theta))\,
  7. R n = | E ( z n ) | = C n 2 + S n 2 R_{n}=|E(z^{n})|=\sqrt{C_{n}^{2}+S_{n}^{2}}\,
  8. ฮธ n = arg ( E ( z n ) ) \theta_{n}=\arg(E(z^{n}))\,
  9. m n ยฏ = 1 N โˆ‘ i = 1 N z i n = C n ยฏ + i S n ยฏ = R n ยฏ e i ฮธ n ยฏ \overline{m_{n}}=\frac{1}{N}\sum_{i=1}^{N}z_{i}^{n}=\overline{C_{n}}+i% \overline{S_{n}}=\overline{R_{n}}e^{i\overline{\theta_{n}}}
  10. C n ยฏ = 1 N โˆ‘ i = 1 N cos ( n ฮธ i ) \overline{C_{n}}=\frac{1}{N}\sum_{i=1}^{N}\cos(n\theta_{i})
  11. S n ยฏ = 1 N โˆ‘ i = 1 N sin ( n ฮธ i ) \overline{S_{n}}=\frac{1}{N}\sum_{i=1}^{N}\sin(n\theta_{i})
  12. R n ยฏ = 1 N โˆ‘ i = 1 N | z i n | \overline{R_{n}}=\frac{1}{N}\sum_{i=1}^{N}|z_{i}^{n}|
  13. ฮธ n ยฏ = 1 N โˆ‘ i = 1 N arg ( z i n ) \overline{\theta_{n}}=\frac{1}{N}\sum_{i=1}^{N}\arg(z_{i}^{n})
  14. C 1 ยฏ , S 1 ยฏ \overline{C_{1}},\overline{S_{1}}
  15. ( m 1 ยฏ ) (\overline{m_{1}})
  16. C 1 ยฏ \overline{C_{1}}
  17. S 1 ยฏ \overline{S_{1}}
  18. [ C 1 ยฏ , S 1 ยฏ ] โ†’ ๐‘‘ ๐’ฉ ( [ C 1 , S 1 ] , ฮฃ / N ) [\overline{C_{1}},\overline{S_{1}}]\xrightarrow{d}\mathcal{N}([C_{1},S_{1}],% \Sigma/N)
  19. ๐’ฉ ( ) \mathcal{N}()
  20. ฮฃ \Sigma
  21. ฮฃ = [ ฯƒ C C ฯƒ C S ฯƒ S C ฯƒ S S ] \Sigma=\begin{bmatrix}\sigma_{CC}&\sigma_{CS}\\ \sigma_{SC}&\sigma_{SS}\end{bmatrix}\quad
  22. ฯƒ C C = E ( cos 2 ฮธ ) - E ( cos ฮธ ) 2 \sigma_{CC}=E(\cos^{2}\theta)-E(\cos\theta)^{2}\,
  23. ฯƒ C S = ฯƒ S C = E ( cos ฮธ sin ฮธ ) - E ( cos ฮธ ) E ( sin ฮธ ) \sigma_{CS}=\sigma_{SC}=E(\cos\theta\sin\theta)-E(\cos\theta)E(\sin\theta)\,
  24. ฯƒ S S = E ( sin 2 ฮธ ) - E ( sin ฮธ ) 2 \sigma_{SS}=E(\sin^{2}\theta)-E(\sin\theta)^{2}\,
  25. C 2 = E ( cos ( 2 ฮธ ) ) = E ( cos 2 ฮธ - 1 ) = E ( 1 - sin 2 ฮธ ) C_{2}=E(\cos(2\theta))=E(\cos^{2}\theta-1)=E(1-\sin^{2}\theta)\,
  26. S 2 = E ( sin ( 2 ฮธ ) ) = E ( 2 cos ฮธ sin ฮธ ) S_{2}=E(\sin(2\theta))=E(2\cos\theta\sin\theta)\,
  27. ฯƒ C C = E ( cos 2 ฮธ ) - E ( cos ฮธ ) 2 = 1 2 ( 1 + C 2 - 2 C 1 2 ) \sigma_{CC}=E(\cos^{2}\theta)-E(\cos\theta)^{2}=\frac{1}{2}\left(1+C_{2}-2C_{1% }^{2}\right)
  28. ฯƒ C S = E ( cos ฮธ sin ฮธ ) - E ( cos ฮธ ) E ( sin ฮธ ) = 1 2 ( S 2 - 2 C 1 S 1 ) \sigma_{CS}=E(\cos\theta\sin\theta)-E(\cos\theta)E(\sin\theta)=\frac{1}{2}% \left(S_{2}-2C_{1}S_{1}\right)
  29. ฯƒ S S = E ( sin 2 ฮธ ) - E ( sin ฮธ ) 2 = 1 2 ( 1 - C 2 - 2 S 1 2 ) \sigma_{SS}=E(\sin^{2}\theta)-E(\sin\theta)^{2}=\frac{1}{2}\left(1-C_{2}-2S_{1% }^{2}\right)
  30. P ( C 1 ยฏ , S 1 ยฏ ) d C 1 ยฏ d S 1 ยฏ P(\overline{C_{1}},\overline{S_{1}})d\overline{C_{1}}d\overline{S_{1}}
  31. d C 1 ยฏ d S 1 ยฏ d\overline{C_{1}}d\overline{S_{1}}
  32. P ( R 1 ยฏ cos ( ฮธ 1 ยฏ ) , R 1 ยฏ sin ( ฮธ 1 ยฏ ) ) R 1 ยฏ d R 1 ยฏ d ฮธ 1 ยฏ P(\overline{R_{1}}\cos(\overline{\theta_{1}}),\overline{R_{1}}\sin(\overline{% \theta_{1}}))\overline{R_{1}}d\overline{R_{1}}d\overline{\theta_{1}}

Centrifugal_micro-fluidic_biochip.html

  1. f < m t p l > ฯ‰ = N ฯ‰ 2 r . f_{\mathrm{<}}mtpl>{{\omega}}=N{\omega}^{2}r.
  2. f C = 2 N ฯ‰ u . f_{\mathrm{C}}=2N{\omega}u.
  3. f E = N r d ฯ‰ d t . f_{\mathrm{E}}=Nr\frac{d{\omega}}{dt}.
  4. f v = v โˆ‚ 2 u โˆ‚ x 2 . f_{\mathrm{v}}=v\frac{\partial^{2}u}{\partial x^{2}}.
  5. r = t 79.7 ( 58 2 - 25 2 ) + 25 2 r=\sqrt{{t\over 79.7}(58^{2}-25^{2})+25^{2}}

CGHS_model.html

  1. S = 1 2 ฯ€ โˆซ d 2 x - g { e - 2 ฯ• [ R + 4 ( โˆ‡ ฯ• ) 2 + 4 ฮป 2 ] - โˆ‘ i = 1 N 1 2 ( โˆ‡ f i ) 2 } S=\frac{1}{2\pi}\int d^{2}x\,\sqrt{-g}\left\{e^{-2\phi}\left[R+4\left(\nabla% \phi\right)^{2}+4\lambda^{2}\right]-\sum^{N}_{i=1}\frac{1}{2}\left(\nabla f_{i% }\right)^{2}\right\}
  2. f i ( u , v ) = A i ( u ) + B i ( v ) f_{i}\left(u,v\right)=A_{i}\left(u\right)+B_{i}\left(v\right)
  3. e - 2 ฯ• ( - 2 ฯ• , v v + 4 ฯ , v ฯ• , v ) + f i , v f i , v / 2 = 0 e^{-2\phi}\left(-2\phi_{,vv}+4\rho_{,v}\phi_{,v}\right)+f_{i,v}f_{i,v}/2=0
  4. e - 2 ฯ• ( - 2 ฯ• , u u + 4 ฯ , u ฯ• , u ) + f i , u f i , u / 2 = 0 e^{-2\phi}\left(-2\phi_{,uu}+4\rho_{,u}\phi_{,u}\right)+f_{i,u}f_{i,u}/2=0
  5. ( e - 2 ฯ• ) , u v = - ฮป 2 e - 2 ฯ• e 2 ฯ \left(e^{-2\phi}\right)_{,uv}=-\lambda^{2}e^{-2\phi}e^{2\rho}
  6. 2 ฯ , u v - 4 ฯ• , u v + 4 ฯ• , u ฯ• , v + ฮป 2 e 2 ฯ = 0 2\rho_{,uv}-4\phi_{,uv}+4\phi_{,u}\phi_{,v}+\lambda^{2}e^{2\rho}=0
  7. d s 2 = - ( M ฮป - ฮป 2 u v ) - 1 d u d v ds^{2}=-\left(\frac{M}{\lambda}-\lambda^{2}uv\right)^{-1}du\,dv
  8. e - 2 ฯ• = M ฮป - ฮป 2 u v e^{-2\phi}=\frac{M}{\lambda}-\lambda^{2}uv

Chaotic_scattering.html

  1. N ( T ) โˆผ e - ฮณ T N(T)\sim e^{-\gamma T}
  2. ฮณ \gamma
  3. ฮณ = lim n โ†’ โˆž - ln N ( T ) T \gamma=\lim_{n\rightarrow\infty}-\frac{\ln N(T)}{T}
  4. ฮณ = 0.739 \gamma=0.739
  5. ฮธ โˆˆ [ - ฯ€ , ฯ€ ] \theta\in[-\pi,\pi]
  6. ฯ• โˆˆ [ - ฯ€ / 2 , ฯ€ / 2 ] \phi\in[-\pi/2,\pi/2]
  7. ฯต \epsilon
  8. ฮธ \theta
  9. ฯ• \phi
  10. ฮธ \theta
  11. ฯ• \phi
  12. ฯต \epsilon
  13. ฮณ = 0.380 \gamma=0.380
  14. D 0 = N - ฮณ = 2 - 0.380 = 1.62 D_{0}=N-\gamma=2-0.380=1.62
  15. D = D s + D u - N = 2 D s - N = N - 2 ฮณ D=D_{s}+D_{u}-N=2D_{s}-N=N-2\gamma
  16. D 1 = ( h 1 - 1 ฮณ ) ( 1 h 1 - 1 h 2 ) D_{1}=\left(h_{1}-\frac{1}{\gamma}\right)\left(\frac{1}{h_{1}}-\frac{1}{h_{2}}\right)
  17. ฮณ = โˆž \gamma=\infty

Characteristic_2_type.html

  1. C M ( O 2 ( M ) ) โ‰ค O 2 ( M ) C_{M}(O_{2}(M))\leq O_{2}(M)

Characteristic_equation_(calculus).html

  1. n n\,
  2. n n\,
  3. y y\,
  4. a n , a n - 1 , โ€ฆ , a 1 , a 0 a_{n},a_{n-1},\ldots,a_{1},a_{0}
  5. a n y ( n ) + a n - 1 y ( n - 1 ) + โ‹ฏ + a 1 y โ€ฒ + a 0 y = 0 a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_{1}y^{\prime}+a_{0}y=0
  6. a n r n + a n - 1 r n - 1 + โ‹ฏ + a 1 r + a 0 = 0 a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots+a_{1}r+a_{0}=0
  7. r n , r n - 1 , โ€ฆ , r r^{n},r^{n-1},\ldots,r
  8. a n , a n - 1 , โ€ฆ , a 1 , a 0 a_{n},a_{n-1},\ldots,a_{1},a_{0}
  9. a n y ( n ) + a n - 1 y ( n - 1 ) + โ‹ฏ + a 1 y โ€ฒ + a 0 y = 0 a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_{1}y^{{}^{\prime}}+a_{0}y=0
  10. y ( x ) = e r x y(x)=e^{rx}\,
  11. e r x e^{rx}\,
  12. e r x e^{rx}\,
  13. y โ€ฒ = r e r x y^{\prime}=re^{rx}\,
  14. y โ€ฒโ€ฒ = r 2 e r x y^{\prime\prime}=r^{2}e^{rx}\,
  15. y ( n ) = r n e r x y^{(n)}=r^{n}e^{rx}\,
  16. r r\,
  17. e r x e^{rx}\,
  18. r r\,
  19. y = e r x y=e^{rx}\,
  20. a n r n e r x + a n - 1 r n - 1 e r x + โ‹ฏ + a 1 r e r x + a 0 e r x = 0 a_{n}r^{n}e^{rx}+a_{n-1}r^{n-1}e^{rx}+\cdots+a_{1}re^{rx}+a_{0}e^{rx}=0
  21. e r x e^{rx}\,
  22. a n r n + a n - 1 r n - 1 + โ‹ฏ + a 1 r + a 0 = 0 a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots+a_{1}r+a_{0}=0
  23. r r\,
  24. r r\,
  25. y ( x ) = c e 3 x y(x)=ce^{3x}\,
  26. c c\,
  27. r 1 , โ€ฆ , r n r_{1},\ldots,r_{n}
  28. h h\,
  29. k k\,
  30. y D ( x ) y_{D}(x)\,
  31. y R 1 ( x ) , โ€ฆ , y R h ( x ) y_{R_{1}}(x),\ldots,y_{R_{h}}(x)
  32. y C 1 ( x ) , โ€ฆ , y C k ( x ) y_{C_{1}}(x),\ldots,y_{C_{k}}(x)
  33. y ( x ) = y D ( x ) + y R 1 ( x ) + โ‹ฏ + y R h ( x ) + y C 1 ( x ) + โ‹ฏ + y C k ( x ) y(x)=y_{D}(x)+y_{R_{1}}(x)+\cdots+y_{R_{h}}(x)+y_{C_{1}}(x)+\cdots+y_{C_{k}}(x)
  34. u 1 , โ€ฆ , u n u_{1},\ldots,u_{n}
  35. n n\,
  36. c 1 u 1 + โ‹ฏ + c n u n c_{1}u_{1}+\cdots+c_{n}u_{n}
  37. c 1 , โ€ฆ , c n c_{1},\ldots,c_{n}
  38. r 1 , โ€ฆ , r n r_{1},\ldots,r_{n}
  39. y D ( x ) = c 1 e r 1 x + c 2 e r 2 x + โ‹ฏ + c n e r n x y_{D}(x)=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+\cdots+c_{n}e^{r_{n}x}
  40. r 1 r_{1}\,
  41. k k\,
  42. y p ( x ) = c 1 e r 1 x y_{p}(x)=c_{1}e^{r_{1}x}
  43. k - 1 k-1\,
  44. r 1 r_{1}\,
  45. k k\,
  46. ( d d x - r 1 ) k y = 0 \left(\frac{d}{dx}-r_{1}\right)^{k}y=0
  47. y p ( x ) = c 1 e r 1 x y_{p}(x)=c_{1}e^{r_{1}x}
  48. y ( x ) = u ( x ) e r 1 x y(x)=u(x)e^{r_{1}x}\,
  49. u ( x ) u(x)\,
  50. u e r 1 x ue^{r_{1}x}\,
  51. ( d d x - r 1 ) u e r 1 x = d d x ( u e r 1 x ) - r 1 u e r 1 x = d d x ( u ) e r 1 x + r 1 u e r 1 x - r 1 u e r 1 x = d d x ( u ) e r 1 x \left(\frac{d}{dx}-r_{1}\right)ue^{r_{1}x}=\frac{d}{dx}(ue^{r_{1}x})-r_{1}ue^{% r_{1}x}=\frac{d}{dx}(u)e^{r_{1}x}+r_{1}ue^{r_{1}x}-r_{1}ue^{r_{1}x}=\frac{d}{% dx}(u)e^{r_{1}x}
  52. k = 1 k=1\,
  53. k k\,
  54. ( d d x - r 1 ) k u e r 1 x = d k d x k ( u ) e r 1 x = 0 \left(\frac{d}{dx}-r_{1}\right)^{k}ue^{r_{1}x}=\frac{d^{k}}{dx^{k}}(u)e^{r_{1}% x}=0
  55. e r 1 x e^{r_{1}x}\,
  56. d k d x k ( u ) = u ( k ) = 0 \frac{d^{k}}{dx^{k}}(u)=u^{(k)}=0
  57. u ( x ) u(x)\,
  58. k - 1 k-1\,
  59. u ( x ) = c 1 + c 2 x + c 3 x 2 + โ‹ฏ + c k x k - 1 u(x)=c_{1}+c_{2}x+c_{3}x^{2}+\cdots+c_{k}x^{k-1}
  60. y ( x ) = u e r 1 x y(x)=ue^{r_{1}x}\,
  61. r 1 r_{1}
  62. y R ( x ) = e r 1 x ( c 1 + c 2 x + โ‹ฏ + c k x k - 1 ) y_{R}(x)=e^{r_{1}x}(c_{1}+c_{2}x+\cdots+c_{k}x^{k-1})
  63. r 1 = a + b i r_{1}=a+bi
  64. r 2 = a - b i r_{2}=a-bi
  65. y ( x ) = c 1 e ( a + b i ) x + c 2 e ( a - b i ) x y(x)=c_{1}e^{(a+bi)x}+c_{2}e^{(a-bi)x}\,
  66. e i ฮธ = cos ฮธ + i sin ฮธ e^{i\theta}=\cos\theta+i\sin\theta\,
  67. y ( x ) = c 1 e ( a + b i ) x + c 2 e ( a - b i ) x = c 1 e a x ( cos b x + i sin b x ) + c 2 e a x ( cos b x - i sin b x ) = ( c 1 + c 2 ) e a x cos b x + i ( c 1 - c 2 ) e a x sin b x \begin{array}[]{rcl}y(x)&=&c_{1}e^{(a+bi)x}+c_{2}e^{(a-bi)x}\\ &=&c_{1}e^{ax}(\cos bx+i\sin bx)+c_{2}e^{ax}(\cos bx-i\sin bx)\\ &=&(c_{1}+c_{2})e^{ax}\cos bx+i(c_{1}-c_{2})e^{ax}\sin bx\end{array}
  68. c 1 c_{1}\,
  69. c 2 c_{2}\,
  70. c 1 = c 2 = 1 2 c_{1}=c_{2}=\tfrac{1}{2}
  71. y 1 ( x ) = e a x cos b x y_{1}(x)=e^{ax}\cos bx\,
  72. c 1 = 1 2 i c_{1}=\tfrac{1}{2i}
  73. c 2 = - 1 2 i c_{2}=-\tfrac{1}{2i}
  74. y 2 ( x ) = e a x sin b x y_{2}(x)=e^{ax}\sin bx\,
  75. r = a ยฑ b i r=a\pm bi\,
  76. y C ( x ) = e a x ( c 1 cos b x + c 2 sin b x ) y_{C}(x)=e^{ax}(c_{1}\cos bx+c_{2}\sin bx)\,

Charge_radius.html

  1. R d = r d 2 + 3 4 ( m e m d ) 2 ( ฮป C 2 ฯ€ ) 2 R_{\rm d}=\sqrt{r_{\rm d}^{2}+\frac{3}{4}\left(\frac{m_{\rm e}}{m_{\rm d}}% \right)^{2}\left(\frac{\lambda_{\rm C}}{2\pi}\right)^{2}}

Chebyshev's_bias.html

  1. ฯ€ ( x ; 4 , 1 ) โˆผ ฯ€ ( x ; 4 , 3 ) โˆผ 1 2 x log x , \pi(x;4,1)\sim\pi(x;4,3)\sim\frac{1}{2}\frac{x}{\log x},

CheiRank.html

  1. G * G^{*}
  2. G G
  3. x = log 10 P i x=\log_{10}P_{i}
  4. y = log 10 P i * y=\log_{10}{P}^{*}_{i}
  5. N = 285509 N=285509
  6. ฮฑ = 0.85 \alpha=0.85
  7. ฮป = 1 \lambda=1
  8. G * G^{*}
  9. G G
  10. G G
  11. G * G^{*}
  12. P i * P^{*}_{i}
  13. P i P_{i}
  14. N N
  15. i i
  16. K i * , K i K^{*}_{i},K_{i}
  17. P i * , P i P^{*}_{i},P_{i}
  18. P i P_{i}
  19. P i โˆ 1 / K i ฮฒ P_{i}\propto 1/{K_{i}}^{\beta}
  20. ฮฒ = 1 / ( ฮฝ - 1 ) โ‰ˆ 0.9 \beta=1/(\nu-1)\approx 0.9
  21. ฮฝ โ‰ˆ 2.1 \nu\approx 2.1
  22. P i * โˆ 1 / K i * ฮฒ * P^{*}_{i}\propto 1/{K^{*}_{i}}^{\beta^{*}}
  23. ฮฒ * = 1 / ( ฮฝ * - 1 ) โ‰ˆ 0.6 \beta^{*}=1/(\nu^{*}-1)\approx 0.6
  24. ฮฝ * โ‰ˆ 2.7 \nu^{*}\approx 2.7
  25. P i P_{i}
  26. P i * P^{*}_{i}
  27. P P
  28. P * P^{*}
  29. K K
  30. K * K^{*}
  31. ฮฒ = 0.92 ; 0.57 \beta=0.92;0.57
  32. ฮฒ = 1 / ( ฮฝ - 1 ) \beta=1/(\nu-1)
  33. 0 < ln K , ln K * < ln N 0<\ln K,\ln K^{*}<\ln N
  34. N = 3282257 N=3282257
  35. P , P * P,P^{*}
  36. K , K * K,K^{*}
  37. G G
  38. G * G^{*}
  39. S S
  40. G = ฮฑ S + ( 1 - ฮฑ ) e e T / N G=\alpha S+(1-\alpha)ee^{T}/N
  41. G G
  42. G P = P GP=P
  43. S * S^{*}
  44. G * = ฮฑ S * + ( 1 - ฮฑ ) e e T / N G^{*}=\alpha S^{*}+(1-\alpha)ee^{T}/N
  45. G * G^{*}
  46. G * P * = P * G^{*}P^{*}=P^{*}
  47. ฮฑ โ‰ˆ 0.85 \alpha\approx 0.85

Chemistry_of_biofilm_prevention.html

  1. ฮจ \Psi\,\!
  2. ฮฑ \alpha\,\!
  3. \Tau \Tau\,\!
  4. ฮจ = < m t p l > ฮฑ \Tau \Psi=\frac{<}{m}tpl>{{\alpha}}{\Tau}
  5. ฯ‡ = ( ฮด 1 - ฮด 2 ) 2 ร— V ยฏ R T . \chi=\frac{{(\delta_{1}-\delta_{2}})^{2}\times\bar{V}}{RT}.
  6. ฮด 1 \delta_{1}
  7. ฮด 2 \delta_{2}
  8. V ยฏ \bar{V}
  9. V 1 ยฏ = V 2 ยฏ \bar{V_{1}}=\bar{V_{2}}
  10. ฮณ SL + ฮณ LV cos ฮธ c = ฮณ SV \gamma_{\mathrm{SL}}+\gamma_{\mathrm{LV}}\cos{\theta_{\mathrm{c}}}=\gamma_{% \mathrm{SV}}\,
  11. ฮณ SL \gamma_{\mathrm{SL}}
  12. ฮณ LV \gamma_{\mathrm{LV}}
  13. ฮณ SV \gamma_{\mathrm{SV}}
  14. ฮธ c \theta_{\mathrm{c}}
  15. cos ฮธ o b s = R cos ฮธ \cos\,{\theta_{obs}}=R\cos\,{\theta}
  16. ฮธ o b s \theta_{obs}
  17. ฮธ o b s \theta_{obs}

Cherry_blossom_front.html

  1. D T S = exp ( ( 9.5 ร— 10 3 ) . ( T - 288.2 288.2 T ) ) DTS=\exp((9.5\times 10^{3}).(\frac{T-288.2}{288.2T}))

Chlorophyll_fluorescence.html

  1. F 0 \,F_{0}
  2. F m \,F_{m}
  3. F m \,F_{m}
  4. F t \,F_{t}
  5. F 0 \,F_{0}
  6. F m \,F_{m}
  7. F m 0 F_{m}^{0}
  8. F 0 \,F_{0}
  9. F m \,F_{m}
  10. F 0 โ€ฒ \,{F_{0}}^{\prime}
  11. F 0 \,F_{0}
  12. F m โ€ฒ \,{F_{m}}^{\prime}
  13. F t r \,F_{tr}
  14. T 1 / 2 \,T_{1/2}
  15. F 0 \,F_{0}
  16. F m \,F_{m}
  17. F v \,F_{v}
  18. F v \,F_{v}
  19. F m \,F_{m}
  20. F 0 \,F_{0}
  21. F v F m \tfrac{F_{v}}{F_{m}}
  22. F m - F 0 F m \frac{F_{m}-F_{0}}{F_{m}}
  23. F v F m \tfrac{F_{v}}{F_{m}}
  24. ฮฆ P S I I \,\Phi_{PSII}
  25. Y ( I I ) \,Y_{(II)}
  26. , F m โ€ฒ - F F m โ€ฒ ,\frac{{F_{m}}^{\prime}-F}{{F_{m}}^{\prime}}
  27. q P \,qP
  28. F m โ€ฒ - F F m โ€ฒ - F 0 โ€ฒ \,\frac{{F_{m}}^{\prime}-F}{{F_{m}}^{\prime}-{F_{0}}^{\prime}}
  29. ฮฆ P S I I \,\Phi_{PSII}
  30. q P \,qP
  31. F v F m \tfrac{F_{v}}{F_{m}}
  32. q P \,qP
  33. F v F m \tfrac{F_{v}}{F_{m}}
  34. F m \,F_{m}
  35. F m \,F_{m}
  36. C i C_{i}
  37. F 0 \,F_{0}
  38. F m \,F_{m}
  39. F v \,F_{v}
  40. F v F m \tfrac{F_{v}}{F_{m}}
  41. F 0 \,F_{0}
  42. F m \,F_{m}
  43. F v \,F_{v}

Chord_names_and_symbols_(popular_music).html

  1. 4 4 \textstyle\frac{4}{4}
  2. 4 4 \textstyle\frac{4}{4}
  3. u p p e r c h o r d l o w e r c h o r d \textstyle\frac{upperchord}{lowerchord}
  4. B C \textstyle\frac{B}{C}
  5. E b m / G b C d r o p 2 \textstyle\frac{Ebm/Gb}{Cdrop2}

Chordal_problem.html

  1. O O
  2. [ X , Y ] [X,Y]
  3. O O
  4. [ X , O ] [X,O]
  5. [ O , Y ] [O,Y]
  6. ฮฑ \alpha
  7. | X - O | ฮฑ + | O - Y | ฮฑ = c |X-O|^{\alpha}+|O-Y|^{\alpha}=c
  8. c c
  9. O O
  10. ฮฑ \alpha
  11. ฮฑ \alpha
  12. ฮฑ \alpha
  13. ฮฑ = 1 \alpha=1
  14. r = f ( ฮธ ) r=f(\theta)
  15. ฮฑ = 1 \alpha=1
  16. r = r 0 ( ฮธ ) r=r_{0}(\theta)
  17. 0 โ‰ค ฮธ โ‰ค ฯ€ 0\leq\theta\leq\pi
  18. r = r ( ฮธ ) r=r(\theta)
  19. ฮธ โˆˆ [ 0 , 2 ฯ€ ] \theta\in[0,2\pi]
  20. r 0 ( ฮธ ) r_{0}(\theta)
  21. r 0 ( ฮธ ) = x + ( 1 2 - x ) cos 2 ฮธ r_{0}(\theta)=x+\left(\frac{1}{2}-x\right)\cos 2\theta
  22. x x
  23. x โˆˆ ( 1 / 4 , 1 / 2 ] x\in(1/4,1/2]
  24. ฮธ \theta
  25. 0 โ‰ค ฮธ โ‰ค ฯ€ 0\leq\theta\leq\pi
  26. r ( 0 ) = r ( ฯ€ ) r(0)=r(\pi)
  27. r ( ฮธ ) r(\theta)
  28. r ( ฮธ ) = { r 0 ( ฮธ ) if 0 โ‰ค ฮธ โ‰ค ฯ€ 1 - r 0 ( ฮธ - ฯ€ ) if ฯ€ โ‰ค ฮธ โ‰ค 2 ฯ€ . r(\theta)=\begin{cases}r_{0}(\theta)&\,\text{ if }0\leq\theta\leq\pi\\ 1-r_{0}(\theta-\pi)&\,\text{ if }\pi\leq\theta\leq 2\pi.\end{cases}
  29. r ( ฮธ ) > 0 r(\theta)>0
  30. r ( ฮธ ) r(\theta)
  31. [ 0 , 2 ฯ€ ] [0,2\pi]
  32. r ( 0 ) = r ( 2 ฯ€ ) r(0)=r(2\pi)
  33. r ( ฮธ ) r(\theta)
  34. 2 ฯ€ 2\pi
  35. ( - โˆž , โˆž ) (-\infty,\infty)
  36. r r
  37. r ( ฮธ ) + r ( ฮธ + ฯ€ ) = 1 r(\theta)+r(\theta+\pi)=1
  38. ฮธ โˆˆ ( - โˆž , โˆž ) \theta\in(-\infty,\infty)
  39. r = r ( ฮธ ) r=r(\theta)
  40. C 1 C^{1}
  41. C 2 C^{2}
  42. x = 1 x=1
  43. r ( ฮธ ) r(\theta)
  44. r = 0.5 + 0.2 sin ฮธ + 0.2 cos 3 ฮธ r=0.5+0.2\,\sin\theta+0.2\,\cos 3\theta
  45. r ( ฮธ ) r(\theta)
  46. ฮฑ = 1 \alpha=1
  47. ฮฑ = - 1 \alpha=-1
  48. | X - O | |X-O|
  49. | O - Y | |O-Y|
  50. | X - O | | O - Y | = c . |X-O||O-Y|=c.\,
  51. log | X - O | + log | O - Y | = c โ€ฒ . \log|X-O|+\log|O-Y|=c^{\prime}\,.
  52. f : โ„ + โ†’ โ„ f:\mathbb{R}^{+}\to\mathbb{R}
  53. f ( | X - O | ) + f ( | O - Y ) ) = c . f(|X-O|)+f(|O-Y))=c.
  54. f ( x , y ) f(x,y)
  55. f f
  56. f ( x , y ) = f ( y , x ) f(x,y)=f(y,x)
  57. f ( | X - O | , | O - Y | ) = c . f(|X-O|,|O-Y|)=c.
  58. f ( x , y ) f(x,y)
  59. x x
  60. y y
  61. O O
  62. ฮฑ \alpha
  63. ฮฑ \alpha

Christensen_failure_criterion.html

  1. ( ฯƒ T ) \left(\sigma_{T}\right)
  2. ( ฯƒ C ) \left(\sigma_{C}\right)
  3. 0 โ‰ค T C โ‰ค 1 0\leq\frac{T}{C}\leq 1
  4. ( 1 T - 1 C ) ( ฯƒ 1 + ฯƒ 2 + ฯƒ 3 ) + 1 2 T C [ ( ฯƒ 1 - ฯƒ 2 ) 2 + ( ฯƒ 2 - ฯƒ 3 ) 2 + ( ฯƒ 3 - ฯƒ 1 ) 2 ] โ‰ค 1 \left(\frac{1}{T}-\frac{1}{C}\right)\left(\sigma_{1}+\sigma_{2}+\sigma_{3}% \right)+\frac{1}{2TC}\left[\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma% _{2}-\sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2}\right]\leq 1
  5. 0 โ‰ค T C โ‰ค 1 2 0\leq\frac{T}{C}\leq\frac{1}{2}
  6. ฯƒ 1 โ‰ค T ฯƒ 2 โ‰ค T ฯƒ 3 โ‰ค T \begin{array}[]{lcl}\sigma_{1}&\leq&T\\ \sigma_{2}&\leq&T\\ \sigma_{3}&\leq&T\end{array}

Chuโ€“Harrington_limit.html

  1. ฮป 2 ฯ€ \frac{\lambda}{2\pi}
  2. Q = 1 ( k 3 a 3 ) + 1 ( k a ) Q=\frac{1}{(k^{3}a^{3})}+\frac{1}{(ka)}
  3. k = 2 ฯ€ ฮป k=\frac{2\pi}{\lambda}

Circle_packing_in_a_circle.html

  1. 1 + 2 3 3 1+\frac{2}{3}\sqrt{3}
  2. 1 + 2 1+\sqrt{2}
  3. 1 + 2 ( 1 + 1 5 ) 1+\sqrt{2(1+\frac{1}{\sqrt{5}})}
  4. 1 + 1 sin ( ฯ€ 7 ) 1+\frac{1}{\sin(\frac{\pi}{7})}
  5. 1 + 2 ( 2 + 2 ) 1+\sqrt{2(2+\sqrt{2})}
  6. 1 + 1 sin ( ฯ€ 9 ) 1+\frac{1}{\sin(\frac{\pi}{9})}
  7. 2 + 5 2+\sqrt{5}
  8. 1 + 6 + 2 5 + 4 1 + 2 5 1+\sqrt{6+\frac{2}{\sqrt{5}}+4\sqrt{1+\frac{2}{\sqrt{5}}}}
  9. 1 + 2 + 6 1+\sqrt{2}+\sqrt{6}
  10. 1 + 2 + 6 1+\sqrt{2}+\sqrt{6}

Circle_packing_in_a_square.html

  1. L = 2 + 2 d n L=2+\frac{2}{d_{n}}
  2. 2 + 2 2+\sqrt{2}
  3. 2 \sqrt{2}
  4. 2 + 2 2 + 6 2 2+\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2}
  5. 6 - 2 \sqrt{6}-\sqrt{2}
  6. 2 + 2 2 2+2\sqrt{2}
  7. 1 2 2 \frac{1}{2}\sqrt{2}
  8. 2 + 12 13 2+\frac{12}{\sqrt{13}}
  9. 1 6 13 \frac{1}{6}\sqrt{13}
  10. 4 + 3 4+\sqrt{3}
  11. 4 - 2 3 4-2\sqrt{3}
  12. 2 + 2 + 6 2+\sqrt{2}+\sqrt{6}
  13. 1 2 ( 6 - 2 ) \frac{1}{2}(\sqrt{6}-\sqrt{2})
  14. 2 + 15 2 17 2+15\sqrt{\frac{2}{17}}
  15. 6 + 3 6+\sqrt{3}
  16. 4 + 2 + 6 4+\sqrt{2}+\sqrt{6}
  17. 2 + 24 13 2+\frac{24}{\sqrt{13}}
  18. 130 17 + 16 17 2 \frac{130}{17}+\frac{16}{17}\sqrt{2}

Circuit_(computer_science).html

  1. ( M , L , G ) (M,L,G)
  2. M M
  3. L L
  4. M i M^{i}
  5. M M
  6. i i
  7. i i
  8. G G
  9. L L
  10. g g
  11. i i
  12. g g
  13. โ„“ \ell
  14. L L
  15. โ„“ \ell
  16. M i M^{i}
  17. g g
  18. h h
  19. G G
  20. h h
  21. g g
  22. k k
  23. k k
  24. g g
  25. G G
  26. g g
  27. i i
  28. i i
  29. i i
  30. i + 1 i+1
  31. V ( g ) V(g)
  32. g g
  33. i i
  34. l l
  35. g g
  36. V ( g ) = { l if g is an input l ( V ( g 1 ) , โ€ฆ , V ( g i ) ) otherwise, V(g)=\begin{cases}l&\,\text{if }g\,\text{ is an input}\\ l(V(g_{1}),\ldots,V(g_{i}))&\,\text{otherwise,}\end{cases}
  37. g j g_{j}
  38. g g
  39. M M
  40. n n
  41. M n M^{n}
  42. M M
  43. ( C n ) n โˆˆ โ„• (C_{n})_{n\in\mathbb{N}}
  44. C n C_{n}
  45. n n
  46. M * M^{*}
  47. M M
  48. โ„• \mathbb{N}
  49. โ„• \mathbb{N}
  50. s i z e ( n ) size(n)
  51. n n

Circuit_quantum_electrodynamics.html

  1. ฮฉ \Omega
  2. V m V_{m}
  3. E 0 E_{0}
  4. E 0 = โ„ ฯ‰ r 2 ฮต 0 V m E_{0}=\sqrt{\frac{\hbar\omega_{r}}{2\varepsilon_{0}V_{m}}}
  5. ฮป / 2 \lambda/2
  6. ฮป / 4 \lambda/4
  7. โ„“ \ell
  8. E E
  9. ฮป / 2 : ฮฝ n = c ฮต eff n 2 โ„“ ( n = 1 , 2 , 3 , โ€ฆ ) ฮป / 4 : ฮฝ n = c ฮต eff 2 n + 1 4 โ„“ ( n = 0 , 1 , 2 , โ€ฆ ) \lambda/2:\quad\nu_{n}=\frac{c}{\sqrt{\varepsilon_{\,\text{eff}}}}\frac{n}{2% \ell}\quad(n=1,2,3,\ldots)\qquad\lambda/4:\quad\nu_{n}=\frac{c}{\sqrt{% \varepsilon_{\,\text{eff}}}}\frac{2n+1}{4\ell}\quad(n=0,1,2,\ldots)
  10. ฮต eff \varepsilon_{\,\text{eff}}
  11. N N
  12. โˆฃ g โŸฉ \mid g\rangle
  13. N + 1 N+1
  14. โˆฃ e โŸฉ \mid e\rangle
  15. ฯ‰ a \omega_{a}
  16. d d
  17. n n
  18. โ„‹ JC = โ„ ฯ‰ r ( a โ€  a + 1 2 ) โŸ cavity term + 1 2 โ„ ฯ‰ a ฯƒ z โŸ atomic term + โ„ g ( ฯƒ + a + a โ€  ฯƒ - ) โŸ interaction term \mathcal{H}_{\,\text{JC}}=\underbrace{\hbar\omega_{r}\left(a^{\dagger}a+\frac{% 1}{2}\right)}_{\,\text{cavity term}}+\underbrace{\frac{1}{2}\hbar\omega_{a}% \sigma_{z}}_{\,\text{atomic term}}+\underbrace{\hbar g\left(\sigma_{+}a+a^{% \dagger}\sigma_{-}\right)}_{\,\text{interaction term}}
  19. ฯ‰ r \omega_{r}
  20. a โ€  a^{\dagger}
  21. a a
  22. ฯ‰ a \omega_{a}
  23. ฯƒ z \sigma_{z}
  24. ฯƒ ยฑ \sigma_{\pm}
  25. ฯ‰ r = ฯ‰ a \omega_{r}=\omega_{a}
  26. โˆฃ n โŸฉ \mid n\rangle
  27. โˆฃ g โŸฉ \mid g\rangle
  28. โˆฃ e โŸฉ \mid e\rangle
  29. โˆฃ n , ยฑ โŸฉ = 1 2 ( โˆฃ g โŸฉ โˆฃ n โŸฉ ยฑ โˆฃ e โŸฉ โˆฃ n - 1 โŸฉ ) \mid n,\pm\rangle=\frac{1}{\sqrt{2}}\left(\mid g\rangle\mid n\rangle\pm\mid e% \rangle\mid n-1\rangle\right)
  30. 2 g n 2g\sqrt{n}
  31. ยฑ g 2 / ฮ” \pm g^{2}/\Delta
  32. ฮ” = ฯ‰ a - ฯ‰ r \Delta=\omega_{a}-\omega_{r}
  33. g = E โ‹… d g=E\cdot d
  34. ฮบ = ฯ‰ r Q \kappa=\frac{\omega_{r}}{Q}
  35. Q Q
  36. Q Q
  37. ฮณ \gamma

Circuit_satisfiability_problem.html

  1. O ( 2 0.4058 m ) O(2^{0.4058m})

Circuits_over_sets_of_natural_numbers.html

  1. A ยฏ = { x โˆˆ โ„• | x โˆ‰ A } \overline{A}=\{x\in\mathbb{N}|x\not\in A\}
  2. A + B = { a + b | a โˆˆ A , b โˆˆ B } A+B=\{a+b|a\in A,b\in B\}
  3. A ร— B = { a ร— b | a โˆˆ A , b โˆˆ B } A\times B=\{a\times b|a\in A,b\in B\}
  4. โ„• \mathbb{N}
  5. A โˆฉ B ยฏ A\cap\overline{B}
  6. E 0 = { 2 } , E i + 1 = E i ร— E i E_{0}=\{2\},E_{i+1}=E_{i}\times E_{i}
  7. i i
  8. E i = { 2 2 i } E_{i}=\{2^{2^{i}}\}
  9. E 0 = { 2 } = { 2 1 } = { 2 2 0 } E_{0}=\{2\}=\{2^{1}\}=\{2^{2^{0}}\}
  10. E i + 1 = E i ร— E i = { 2 2 i } ร— { 2 2 i } = { ( 2 2 i ) 2 } = { 2 2 i ร— 2 } = { 2 2 i + 1 } E_{i+1}=E_{i}\times E_{i}=\{2^{2^{i}}\}\times\{2^{2^{i}}\}=\{(2^{2^{i}})^{2}\}% =\{2^{2^{i}\times 2}\}=\{2^{2^{i+1}}\}
  11. S 0 = { 0 , 1 , 2 } , S i + 1 = ( S i ร— S i ) + S i S_{0}=\{0,1,2\},S_{i+1}=(S_{i}\times S_{i})+S_{i}
  12. { x | 0 < x < 2 2 i } โŠ‚ S i \{x|0<x<2^{2^{i}}\}\subset S_{i}
  13. S i S_{i}
  14. 2 2 i 2^{2^{i}}
  15. i i
  16. S 0 S_{0}
  17. x โˆˆ { x | 0 < x < 2 2 i + 1 } x\in\{x|0<x<2^{2^{i+1}}\}
  18. x x
  19. 2 2 i 2^{2^{i}}
  20. x = 2 2 i ร— d + r x=2^{2^{i}}\times d+r
  21. d , r < 2 2 i d,r<2^{2^{i}}
  22. 2 2 i , d 2^{2^{i}},d
  23. r r
  24. S i S_{i}
  25. x โˆˆ ( S i ร— S i ) + S i x\in(S_{i}\times S_{i})+S_{i}

Circular_law.html

  1. n ร— n nร—n
  2. n โ†’ โˆž nโ†’โˆž
  3. n ร— n nร—n
  4. 1 / n 1/n
  5. ( X n ) n = 1 โˆž (X_{n})_{n=1}^{\infty}
  6. n ร— n nร—n
  7. x x
  8. ฮป 1 , โ€ฆ , ฮป n , 1 โ‰ค j โ‰ค n \lambda_{1},\ldots,\lambda_{n},1\leq j\leq n
  9. 1 n X n \displaystyle\frac{1}{\sqrt{n}}X_{n}
  10. 1 n X n \displaystyle\frac{1}{\sqrt{n}}X_{n}
  11. ฮผ 1 n X n ( A ) = n - 1 # { j โ‰ค n : ฮป j โˆˆ A } , A โˆˆ โ„ฌ ( โ„‚ ) \displaystyle\mu_{\frac{1}{\sqrt{n}}X_{n}}(A)=n^{-1}\#\{j\leq n:\lambda_{j}\in A% \}~{},\quad A\in\mathcal{B}(\mathbb{C})
  12. ฮผ 1 n X n ( x , y ) \displaystyle\mu_{\frac{1}{\sqrt{n}}X_{n}}(x,y)

Cladding_(fiber_optics).html

  1. NA = n core 2 - n clad 2 \mathrm{NA}=\sqrt{n_{\mathrm{core}}^{2}-n_{\mathrm{clad}}^{2}}

Class_kappa-ell_function.html

  1. ๐’ฆ \mathcal{K}
  2. ฮฒ : [ 0 , a ) ร— [ 0 , โˆž ) โ†’ [ 0 , โˆž ) \beta:[0,a)\times[0,\infty)\rightarrow[0,\infty)
  3. ๐’ฆ โ„’ \mathcal{KL}
  4. s s
  5. ฮฒ ( r , s ) \beta(r,s)
  6. r r
  7. ฮฒ ( r , s ) \beta(r,s)
  8. s s
  9. ฮฒ ( r , s ) โ†’ 0 \beta(r,s)\rightarrow 0
  10. s โ†’ โˆž s\rightarrow\infty

Class_kappa_function.html

  1. ๐’ฆ \mathcal{K}
  2. ฮฑ : [ 0 , a ) โ†’ [ 0 , โˆž ) \alpha:[0,a)\rightarrow[0,\infty)
  3. ๐’ฆ \mathcal{K}
  4. ฮฑ ( 0 ) = 0 \alpha(0)=0
  5. ฮฑ : [ 0 , a ) โ†’ [ 0 , โˆž ) \alpha:[0,a)\rightarrow[0,\infty)
  6. ๐’ฆ โˆž \mathcal{K}_{\infty}
  7. ๐’ฆ \mathcal{K}
  8. a = โˆž a=\infty
  9. lim r โ†’ โˆž ฮฑ ( r ) = โˆž \lim_{r\rightarrow\infty}\alpha(r)=\infty

Class_of_accuracy_in_electrical_measurements.html

  1. I = E R I=\frac{E}{R}
  2. I 2 = E R + r I_{2}=\frac{E}{R+r}
  3. ฮ” I = I - I 2 โ‰ˆ E โ‹… r R 2 \Delta I=I-I_{2}\approx\frac{E\cdot r}{R^{2}}
  4. ฮ” I I โ‰ˆ r R \frac{\Delta I}{I}\approx\frac{r}{R}

Classical_central-force_problem.html

  1. ๐… = F ( r ) ๐ซ ^ \mathbf{F}=F(r)\hat{\mathbf{r}}
  2. r ห™ = d r d t \dot{r}=\frac{dr}{dt}
  3. r ยจ = d 2 r d t 2 \ddot{r}=\frac{d^{2}r}{dt^{2}}
  4. ๐… = F ( r ) ๐ซ ^ = m ๐š = m ๐ซ ยจ \mathbf{F}=F(r)\hat{\mathbf{r}}=m\mathbf{a}=m\ddot{\mathbf{r}}
  5. F ( r ) = - d U d r F(r)=-\frac{dU}{dr}
  6. W = โˆซ ๐ซ 1 ๐ซ 2 ๐… โ‹… d ๐ซ = โˆซ ๐ซ 1 ๐ซ 2 F ( r ) ๐ซ ^ โ‹… d ๐ซ = โˆซ r 1 r 2 F d r = U ( r 1 ) - U ( r 2 ) W=\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{F}\cdot d\mathbf{r}=\int_{% \mathbf{r}_{1}}^{\mathbf{r}_{2}}F(r)\hat{\mathbf{r}}\cdot d\mathbf{r}=\int_{r_% {1}}^{r_{2}}Fdr=U(r_{1})-U(r_{2})
  7. โˆ‡ ร— ๐… = 1 r sin ฮธ ( โˆ‚ F โˆ‚ ฯ† ) s y m b o l ฮธ ^ - 1 r ( โˆ‚ F โˆ‚ ฮธ ) s y m b o l ฯ† ^ = 0 \nabla\times\mathbf{F}=\frac{1}{r\sin\theta}\left(\frac{\partial F}{\partial% \varphi}\right)\hat{symbol\theta}-\frac{1}{r}\left(\frac{\partial F}{\partial% \theta}\right)\hat{symbol\varphi}=0
  8. m r ยจ = F ( r ) m\ddot{r}=F(r)
  9. | r ห™ | = | d r d t | = 2 m E tot - U ( r ) |\dot{r}|=\Big|\frac{dr}{dt}\Big|=\sqrt{\frac{2}{m}}\sqrt{E_{\mathrm{tot}}-U(r)}
  10. | t - t 0 | = m 2 โˆซ | d r | E tot - U ( r ) |t-t_{0}|={\sqrt{\frac{m}{2}}}\int\frac{|dr|}{\sqrt{E_{\mathrm{tot}}-U(r)}}
  11. m v 2 r = F ( r ) \frac{mv^{2}}{r}=F(r)
  12. ๐ซ ยจ = ๐ฑ ยจ 1 - ๐ฑ ยจ 2 = ( ๐… 21 m 1 - ๐… 12 m 2 ) = ( 1 m 1 + 1 m 2 ) ๐… 21 \ddot{\mathbf{r}}=\ddot{\mathbf{x}}_{1}-\ddot{\mathbf{x}}_{2}=\left(\frac{% \mathbf{F}_{21}}{m_{1}}-\frac{\mathbf{F}_{12}}{m_{2}}\right)=\left(\frac{1}{m_% {1}}+\frac{1}{m_{2}}\right)\mathbf{F}_{21}
  13. ฮผ ๐ซ ยจ = ๐… \mu\ddot{\mathbf{r}}=\mathbf{F}
  14. ฮผ \mu
  15. ฮผ = 1 1 m 1 + 1 m 2 = m 1 m 2 m 1 + m 2 \mu=\frac{1}{\frac{1}{m_{1}}+\frac{1}{m_{2}}}=\frac{m_{1}m_{2}}{m_{1}+m_{2}}
  16. ๐‹ = ๐ซ ร— ๐ฉ = ๐ซ ร— m ๐ฏ \mathbf{L}=\mathbf{r}\times\mathbf{p}=\mathbf{r}\times m\mathbf{v}
  17. d ๐‹ d t = ๐ซ ห™ ร— m ๐ฏ + ๐ซ ร— m ๐ฏ ห™ = ๐ฏ ร— m ๐ฏ + ๐ซ ร— ๐… = ๐ซ ร— ๐… , \frac{d\mathbf{L}}{dt}=\dot{\mathbf{r}}\times m\mathbf{v}+\mathbf{r}\times m% \dot{\mathbf{v}}=\mathbf{v}\times m\mathbf{v}+\mathbf{r}\times\mathbf{F}=% \mathbf{r}\times\mathbf{F}\ ,
  18. ๐ซ = ( x , y ) = r ( cos ฯ† , sin ฯ† ) \mathbf{r}=(x,\ y)=r(\cos\varphi,\ \sin\varphi)
  19. ๐ฏ = d ๐ซ d t = r ห™ ( cos ฯ† , sin ฯ† ) + r ฯ† ห™ ( - sin ฯ† , cos ฯ† ) \mathbf{v}=\frac{d\mathbf{r}}{dt}=\dot{r}(\cos\varphi,\ \sin\varphi)+r\dot{% \varphi}(-\sin\varphi,\cos\varphi)
  20. ๐š = r ยจ ( cos ฯ† , sin ฯ† ) + 2 r ห™ ฯ† ห™ ( - sin ฯ† , cos ฯ† ) + r ฯ† ยจ ( - sin ฯ† , cos ฯ† ) - r ฯ† ห™ 2 ( cos ฯ† , sin ฯ† ) \mathbf{a}=\ddot{r}(\cos\varphi,\ \sin\varphi)+2\dot{r}\dot{\varphi}(-\sin% \varphi,\ \cos\varphi)+r\ddot{\varphi}(-\sin\varphi,\cos\varphi)-r\dot{\varphi% }^{2}(\cos\varphi,\sin\varphi)
  21. ๐ซ ^ = ( cos ฯ† , sin ฯ† ) \mathbf{\hat{r}}=(\cos\varphi,\ \sin\varphi)
  22. s y m b o l ฯ† ^ โ‹… s y m b o l ฯ† ^ = ( - sin ฯ† ) 2 + ( cos ฯ† ) 2 = 1 \hat{symbol\varphi}\cdot\hat{symbol\varphi}=(-\sin\varphi)^{2}+(\cos\varphi)^{% 2}=1
  23. s y m b o l ฯ† ^ โ‹… ๐ซ ^ = - sin ฯ† cos ฯ† + cos ฯ† sin ฯ† = 0 \hat{symbol\varphi}\cdot\mathbf{\hat{r}}=-\sin\varphi\cos\varphi+\cos\varphi% \sin\varphi=0
  24. s y m b o l ฯ† ^ = ( - sin ฯ† , cos ฯ† ) \hat{symbol\varphi}=(-\sin\varphi,\ \cos\varphi)
  25. ๐ฏ = v r ๐ซ ^ + v ฯ† s y m b o l ฯ† ^ = r ห™ ๐ซ ^ + r ฯ† ห™ s y m b o l ฯ† ^ \mathbf{v}=v_{r}\mathbf{\hat{r}}+v_{\varphi}\hat{symbol\varphi}=\dot{r}\mathbf% {\hat{r}}+r\dot{\varphi}\hat{symbol\varphi}
  26. ๐š = a r ๐ซ ^ + a ฯ† s y m b o l ฯ† ^ = ( r ยจ - r ฯ† ห™ 2 ) ๐ซ ^ + ( 2 r ห™ ฯ† ห™ + r ฯ† ยจ ) s y m b o l ฯ† ^ \mathbf{a}=a_{r}\mathbf{\hat{r}}+a_{\varphi}\hat{symbol\varphi}=(\ddot{r}-r% \dot{\varphi}^{2})\mathbf{\hat{r}}+(2\dot{r}\dot{\varphi}+r\ddot{\varphi})\hat% {symbol\varphi}
  27. a ฯ† = 2 r ห™ ฯ† ห™ + r ฯ† ยจ = 0 a_{\varphi}=2\dot{r}\dot{\varphi}+r\ddot{\varphi}=0
  28. d d t ( r 2 ฯ† ห™ ) = r ( 2 r ห™ ฯ† ห™ + r ฯ† ยจ ) = r a ฯ† = 0 \frac{d}{dt}\left(r^{2}\dot{\varphi}\right)=r(2\dot{r}\dot{\varphi}+r\ddot{% \varphi})=ra_{\varphi}=0
  29. h = r 2 ฯ† ห™ = r v ฯ† = | ๐ซ ร— ๐ฏ | = v r โŸ‚ = L m h=r^{2}\dot{\varphi}=rv_{\varphi}=\left|\mathbf{r}\times\mathbf{v}\right|=vr_{% \perp}=\frac{L}{m}
  30. ฯ‰ = ฯ† ห™ = d ฯ† d t \omega=\dot{\varphi}=\frac{d\varphi}{dt}
  31. ฯ‰ = h r 2 \omega=\frac{h}{r^{2}}
  32. ฯ„ = โˆซ d t y 2 \tau=\int\frac{dt}{y^{2}}
  33. d ฮพ d ฯ„ = d d t ( x y ) d t d ฯ„ = ( x ห™ y - y ห™ x y 2 ) y 2 = - h \frac{d\xi}{d\tau}=\frac{d}{dt}\left(\frac{x}{y}\right)\frac{dt}{d\tau}=\left(% \frac{\dot{x}y-\dot{y}x}{y^{2}}\right)y^{2}=-h
  34. d ฮท d ฯ„ = d d t ( 1 y ) d t d ฯ„ = - y ห™ y 2 y 2 = - y ห™ \frac{d\eta}{d\tau}=\frac{d}{dt}\left(\frac{1}{y}\right)\frac{dt}{d\tau}=-% \frac{\dot{y}}{y^{2}}y^{2}=-\dot{y}
  35. d 2 ฮพ d ฯ„ 2 = 0 \frac{d^{2}\xi}{d\tau^{2}}=0
  36. d 2 ฮท d ฯ„ 2 = d t d ฯ„ d d t ( d ฮท d ฯ„ ) = - y 2 y ยจ = - y 3 m r F ( r ) \frac{d^{2}\eta}{d\tau^{2}}=\frac{dt}{d\tau}\frac{d}{dt}\left(\frac{d\eta}{d% \tau}\right)=-y^{2}\ddot{y}=-\frac{y^{3}}{mr}F(r)
  37. y ยจ = 1 m F y = 1 m F ( r ) y r \ddot{y}=\frac{1}{m}F_{y}=\frac{1}{m}F(r)\,\frac{y}{r}
  38. F ( r ) = m r ยจ - m r ฯ‰ 2 = m d 2 r d t 2 - m h 2 r 3 F(r)=m\ddot{r}-mr\omega^{2}=m\frac{d^{2}r}{dt^{2}}-\frac{mh^{2}}{r^{3}}
  39. d r d t \frac{dr}{dt}
  40. F ( r ) d r = F ( r ) d r d t d t = m ( d r d t d 2 r d t 2 - h 2 r 3 d r d t ) d t = m 2 d [ ( d r d t ) 2 + ( h r ) 2 ] \begin{aligned}\displaystyle F(r)\,dr&\displaystyle=F(r)\frac{dr}{dt}\,dt\\ &\displaystyle=m\left(\frac{dr}{dt}\frac{d^{2}r}{dt^{2}}-\frac{h^{2}}{r^{3}}% \frac{dr}{dt}\right)\,dt\\ &\displaystyle=\frac{m}{2}\,d\left[\left(\frac{dr}{dt}\right)^{2}+\left(\frac{% h}{r}\right)^{2}\right]\end{aligned}
  41. โˆซ r F ( r ) d r = m 2 [ ( d r d t ) 2 + ( h r ) 2 ] \int^{r}F(r)\,dr=\frac{m}{2}\left[\left(\frac{dr}{dt}\right)^{2}+\left(\frac{h% }{r}\right)^{2}\right]
  42. d d t = ฯ‰ d d ฯ† = h r 2 d d ฯ† \frac{d}{dt}=\omega\frac{d}{d\varphi}=\frac{h}{r^{2}}\frac{d}{d\varphi}
  43. โˆซ r F ( r ) d r = m h 2 2 [ ( - 1 r 2 d r d ฯ† ) 2 + ( 1 r ) 2 ] \int^{r}F(r)\,dr=\frac{mh^{2}}{2}\left[\left(-\frac{1}{r^{2}}\frac{dr}{d% \varphi}\right)^{2}+\left(\frac{1}{r}\right)^{2}\right]
  44. ( d u d ฯ† ) 2 = C - u 2 - G ( u ) \left(\frac{du}{d\varphi}\right)^{2}=C-u^{2}-G\left(u\right)
  45. G ( u ) = - 2 m h 2 โˆซ 1 u F ( r ) d r G(u)=-\frac{2}{mh^{2}}\int^{\frac{1}{u}}F(r)\,dr
  46. d 2 u d ฯ† 2 + u = - 1 m h 2 u 2 F ( 1 / u ) \frac{d^{2}u}{d\varphi^{2}}+u=-\frac{1}{mh^{2}u^{2}}F(1/u)
  47. ฯ† = ฯ† 0 + โˆซ 1 r d u C - u 2 - G ( u ) \varphi=\varphi_{0}+\int^{\frac{1}{r}}\frac{du}{\sqrt{C-u^{2}-G(u)}}
  48. E tot = 1 2 m r ห™ 2 + 1 2 m r 2 ฯ† ห™ 2 + U ( r ) = 1 2 m r ห™ 2 + m h 2 2 r 2 + U ( r ) E_{\mathrm{tot}}=\frac{1}{2}m\dot{r}^{2}+\frac{1}{2}mr^{2}\dot{\varphi}^{2}+U(% r)=\frac{1}{2}m\dot{r}^{2}+\frac{mh^{2}}{2r^{2}}+U(r)
  49. r ห™ = d r d t = 2 m E tot - U ( r ) - m h 2 2 r 2 \dot{r}=\frac{dr}{dt}=\sqrt{\frac{2}{m}}\sqrt{E_{\mathrm{tot}}-U(r)-\frac{mh^{% 2}}{2r^{2}}}
  50. d r d ฯ† = r 2 h d r d t \frac{dr}{d\varphi}=\frac{r^{2}}{h}\frac{dr}{dt}
  51. ฯ† = ฯ† 0 + L 2 m โˆซ r d r r 2 E tot - U ( r ) - L 2 2 m r 2 \varphi=\varphi_{0}+\frac{L}{\sqrt{2m}}\int^{r}\frac{dr}{r^{2}\sqrt{E_{\mathrm% {tot}}-U(r)-\frac{L^{2}}{2mr^{2}}}}
  52. U eff = U ( r ) + L 2 2 m r 2 U_{\mathrm{eff}}=U(r)+\frac{L^{2}}{2mr^{2}}
  53. ฯ† = ฯ† 0 + โˆซ u d u 2 m L 2 E tot - 2 m L 2 U ( 1 / u ) - u 2 \varphi=\varphi_{0}+\int^{u}\frac{du}{\sqrt{\frac{2m}{L^{2}}E_{\mathrm{tot}}-% \frac{2m}{L^{2}}U(1/u)-u^{2}}}
  54. E tot = U ( r ) + L 2 2 m r 2 E_{\mathrm{tot}}=U(r)+\frac{L^{2}}{2mr^{2}}
  55. ฮ” ฯ† = L 2 m โˆซ r min r max d r r 2 E - U ( r ) - L 2 2 m r 2 \Delta\varphi=\frac{L}{\sqrt{2m}}\int_{r_{\mathrm{min}}}^{r_{\mathrm{max}}}% \frac{dr}{r^{2}\sqrt{E-U(r)-\frac{L^{2}}{2mr^{2}}}}
  56. ฮ” ฯ† = 2 ฯ€ m n \Delta\varphi=2\pi\frac{m}{n}
  57. F = ฮฑ r 2 = ฮฑ u 2 F=\frac{\alpha}{r^{2}}=\alpha u^{2}
  58. d 2 u d ฯ† 2 + u = - ฮฑ m h 2 . \frac{d^{2}u}{d\varphi^{2}}+u=-\frac{\alpha}{mh^{2}}.
  59. u ( ฯ† ) = - ฮฑ m h 2 [ 1 - e cos ( ฯ† - ฯ† 0 ) ] u(\varphi)=-\frac{\alpha}{mh^{2}}\left[1-e\cos\left(\varphi-\varphi_{0}\right)\right]
  60. u ( ฯ† ) = u 1 + ( u 2 - u 1 ) sin 2 ( ฯ† - ฯ† 0 2 ) u(\varphi)=u_{1}+(u_{2}-u_{1})\sin^{2}\left(\frac{\varphi-\varphi_{0}}{2}\right)
  61. u 1 + u 2 = - 2 ฮฑ m h 2 u_{1}+u_{2}=\frac{-2\alpha}{mh^{2}}
  62. e = u 2 - u 1 u 2 + u 1 e=\frac{u_{2}-u_{1}}{u_{2}+u_{1}}
  63. ฯ† = ฯ† 0 + L 2 m โˆซ u d u E tot - U ( 1 / u ) - L 2 u 2 2 m \varphi=\varphi_{0}+\frac{L}{\sqrt{2m}}\int^{u}\frac{du}{\sqrt{E_{\mathrm{tot}% }-U(1/u)-\frac{L^{2}u^{2}}{2m}}}
  64. F ( r ) = A r - 3 + B r + C r 3 + D r 5 F(r)=Ar^{-3}+Br+Cr^{3}+Dr^{5}
  65. F ( r ) = A r - 3 + B r + C r - 5 + D r - 7 F(r)=Ar^{-3}+Br+Cr^{-5}+Dr^{-7}
  66. F ( r ) = A r - 3 + B r - 2 + C r + D F(r)=Ar^{-3}+Br^{-2}+Cr+D
  67. F ( r ) = A r - 3 + B r - 2 + C r - 4 + D r - 5 F(r)=Ar^{-3}+Br^{-2}+Cr^{-4}+Dr^{-5}
  68. F ( r ) = A r - 3 + B r - 2 + C r - 3 / 2 + D r - 5 / 2 F(r)=Ar^{-3}+Br^{-2}+Cr^{-3/2}+Dr^{-5/2}
  69. F ( r ) = A r - 3 + B r - 1 / 3 + C r - 5 / 3 + D r - 7 / 3 F(r)=Ar^{-3}+Br^{-1/3}+Cr^{-5/3}+Dr^{-7/3}
  70. F ( r ) = A r - 3 + B r F(r)=Ar^{-3}+Br
  71. F ( r ) = A r - 3 + B r - 2 F(r)=Ar^{-3}+Br^{-2}
  72. F ( r ) = A r - 5 F(r)=Ar^{-5}
  73. F 2 ( r ) = F 1 ( r ) + L 1 2 m r 3 ( 1 - k 2 ) F_{2}(r)=F_{1}(r)+\frac{L_{1}^{2}}{mr^{3}}\left(1-k^{2}\right)
  74. L = 1 2 m r ห™ 2 + 1 2 m r 2 ฯ† ห™ 2 - U ( r ) L=\frac{1}{2}m\dot{r}^{2}+\frac{1}{2}mr^{2}\dot{\varphi}^{2}-U(r)
  75. d d t ( โˆ‚ L โˆ‚ r ห™ ) = โˆ‚ L โˆ‚ r \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{r}}\right)=\frac{\partial L}{% \partial r}
  76. m r ยจ = m r ฯ† ห™ 2 - d U d r = m h 2 r 3 + F ( r ) m\ddot{r}=mr\dot{\varphi}^{2}-\frac{dU}{dr}=\frac{mh^{2}}{r^{3}}+F(r)
  77. H = 1 2 m ( p r 2 + p ฯ• 2 r 2 ) + U ( r ) H=\frac{1}{2m}\left(p_{r}^{2}+\frac{p_{\phi}^{2}}{r^{2}}\right)+U(r)
  78. d ฯ† d t = โˆ‚ H โˆ‚ p ฯ† = p ฯ† m r 2 = L m r 2 \frac{d\varphi}{dt}=\frac{\partial H}{\partial p_{\varphi}}=\frac{p_{\varphi}}% {mr^{2}}=\frac{L}{mr^{2}}
  79. d r d t = โˆ‚ H โˆ‚ p r = p r m \frac{dr}{dt}=\frac{\partial H}{\partial p_{r}}=\frac{p_{r}}{m}
  80. d 2 r d t 2 = 1 m d p r d t = - 1 m ( โˆ‚ H โˆ‚ r ) = p ฯ† 2 m 2 r 3 - 1 m d U d r = L 2 m 2 r 3 + 1 m F ( r ) \frac{d^{2}r}{dt^{2}}=\frac{1}{m}\frac{dp_{r}}{dt}=-\frac{1}{m}\left(\frac{% \partial H}{\partial r}\right)=\frac{p_{\varphi}^{2}}{m^{2}r^{3}}-\frac{1}{m}% \frac{dU}{dr}=\frac{L^{2}}{m^{2}r^{3}}+\frac{1}{m}F(r)
  81. 1 2 m ( d S r d r ) 2 + 1 2 m r 2 ( d S ฯ† d ฯ† ) 2 + U ( r ) = E tot \frac{1}{2m}\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{1}{2mr^{2}}\left(\frac{dS% _{\varphi}}{d\varphi}\right)^{2}+U(r)=E_{\mathrm{tot}}
  82. d S ฯ† d ฯ† = p ฯ† = L \frac{dS_{\varphi}}{d\varphi}=p_{\varphi}=L
  83. 1 2 m ( d S r d r ) 2 + L 2 2 m r 2 + U ( r ) = E tot \frac{1}{2m}\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{L^{2}}{2mr^{2}}+U(r)=E_{% \mathrm{tot}}
  84. S r ( r ) = 2 m โˆซ d r E tot - U ( r ) - L 2 2 m r 2 S_{r}(r)=\sqrt{2m}\int dr\sqrt{E_{\mathrm{tot}}-U(r)-\frac{L^{2}}{2mr^{2}}}
  85. ฯ† 0 = โˆ‚ S โˆ‚ L = โˆ‚ S ฯ† โˆ‚ L + โˆ‚ S r โˆ‚ L = ฯ† - L 2 m โˆซ r d r r 2 E tot - U ( r ) - L 2 2 m r 2 \varphi_{0}=\frac{\partial S}{\partial L}=\frac{\partial S_{\varphi}}{\partial L% }+\frac{\partial S_{r}}{\partial L}=\varphi-\frac{L}{\sqrt{2m}}\int^{r}\frac{% dr}{r^{2}\sqrt{E_{\mathrm{tot}}-U(r)-\frac{L^{2}}{2mr^{2}}}}

Classical_Cepheid_variable.html

  1. P P
  2. M v M_{v}
  3. M v = ( - 2.43 ยฑ 0.12 ) ( log 10 ( P ) - 1 ) - ( 4.05 ยฑ 0.02 ) M_{v}=(-2.43\pm 0.12)(\log_{10}(P)-1)-(4.05\pm 0.02)\,
  4. P P
  5. d d
  6. 5 log 10 d = V + ( 3.34 ) log 10 P - ( 2.45 ) ( V - I ) + 7.52โ€‰. 5\log_{10}{d}=V+(3.34)\log_{10}{P}-(2.45)(V-I)+7.52\,.
  7. 5 log 10 d = V + ( 3.37 ) log 10 P - ( 2.55 ) ( V - I ) + 7.48โ€‰. 5\log_{10}{d}=V+(3.37)\log_{10}{P}-(2.55)(V-I)+7.48\,.
  8. I I
  9. V V

Classifier_chains.html

  1. L \mathit{L}\,
  2. ( x , Y ) \mathit{(x,Y)}\,
  3. x \mathit{x}\,
  4. Y โІ L Y\subseteq L
  5. | L | \left|L\right|
  6. | L | \left|L\right|
  7. H : X โ†’ { l , ยฌ l } H:X\rightarrow\{l,\neg l\}
  8. l โˆˆ L l\in L
  9. H : X โ†’ ๐’ซ ( L ) H:X\rightarrow\mathcal{P}(L)
  10. ๐’ซ ( L ) \mathcal{P}(L)
  11. L \mathit{L}\,
  12. 2 10 = 1024 2^{10}=1024
  13. L \mathit{L}\,
  14. | L | \left|L\right|
  15. i i
  16. ( x i , Y i ) \mathit{(x_{i},Y_{i})}\,
  17. Y i \mathit{Y_{i}}\,
  18. x i \mathit{x_{i}}\,
  19. | L | \left|L\right|
  20. j j
  21. ( ( x i , l 1 , โ€ฆ , l j - 1 ) , l j ) , l j โˆˆ { 0 , 1 } ((x_{i},l_{1},...,l_{j-1}),l_{j}),l_{j}\in\{0,1\}
  22. j j
  23. l j \mathit{l_{j}}\,
  24. 1 1
  25. 0
  26. C 1 \mathit{C_{1}}\,
  27. C | L | \mathit{C_{|L|}}\,