wpmath0000009_1

Borel_right_process.html

  1. E E
  2. \mathcal{E}
  3. E E
  4. Ω \Omega
  5. [ 0 , ) [0,\infty)
  6. E E
  7. E E
  8. t [ 0 , ) t\in[0,\infty)
  9. X t X_{t}
  10. t t
  11. ω Ω \omega\in\Omega
  12. X t ( ω ) E X_{t}(\omega)\in E
  13. ω \omega
  14. t t
  15. \mathcal{E}
  16. * \mathcal{E}^{*}
  17. t [ 0 , ) t\in[0,\infty)
  18. t = σ { X s - 1 ( B ) : s [ 0 , t ] , B } , \mathcal{F}_{t}=\sigma\left\{X_{s}^{-1}(B):s\in[0,t],B\in\mathcal{E}\right\},
  19. t * = σ { X s - 1 ( B ) : s [ 0 , t ] , B * } , \mathcal{F}_{t}^{*}=\sigma\left\{X_{s}^{-1}(B):s\in[0,t],B\in\mathcal{E}^{*}% \right\},
  20. = σ { X s - 1 ( B ) : s [ 0 , ) , B } , \mathcal{F}_{\infty}=\sigma\left\{X_{s}^{-1}(B):s\in[0,\infty),B\in\mathcal{E}% \right\},
  21. * = σ { X s - 1 ( B ) : s [ 0 , ) , B * } . \mathcal{F}_{\infty}^{*}=\sigma\left\{X_{s}^{-1}(B):s\in[0,\infty),B\in% \mathcal{E}^{*}\right\}.
  22. f f
  23. E E
  24. x E x\in E
  25. U α f ( x ) = 𝐄 x [ 0 e - α t f ( X t ) d t ] . U^{\alpha}f(x)=\mathbf{E}^{x}\left[\int_{0}^{\infty}e^{-\alpha t}f(X_{t})\,dt% \right].
  26. P t f ( x ) = 𝐄 x [ f ( X t ) ] P_{t}f(x)=\mathbf{E}^{x}\left[f(X_{t})\right]
  27. t X t t\rightarrow X_{t}
  28. f f
  29. t P t f ( x ) t\rightarrow P_{t}f(x)
  30. f f
  31. ( t , x ) P t f ( x ) (t,x)\rightarrow P_{t}f(x)
  32. ( [ 0 , ) ) * \mathcal{B}([0,\infty))\otimes\mathcal{E}^{*}
  33. ( ( [ 0 , ) ) * ) λ μ \left(\mathcal{B}([0,\infty))\otimes\mathcal{E}^{*}\right)^{\lambda\otimes\mu}
  34. λ \lambda
  35. ( [ 0 , ) ) \mathcal{B}([0,\infty))
  36. μ \mu
  37. * \mathcal{E}^{*}
  38. ( ( [ 0 , ) ) * ) λ μ \left(\mathcal{B}([0,\infty))\otimes\mathcal{E}^{*}\right)^{\lambda\otimes\mu}
  39. ( [ 0 , ) ) * \mathcal{B}([0,\infty))\otimes\mathcal{E}^{*}
  40. λ μ \lambda\otimes\mu
  41. f f
  42. E E
  43. t P t f ( x ) t\rightarrow P_{t}f(x)
  44. α [ 0 , ) \alpha\in[0,\infty)
  45. U α f ( x ) = 0 e - α t P t f ( x ) d t . U^{\alpha}f(x)=\int_{0}^{\infty}e^{-\alpha t}P_{t}f(x)dt.
  46. { U α : α ( 0 , ) } \{U^{\alpha}:\alpha\in(0,\infty)\}
  47. ( E , * ) (E,\mathcal{E}^{*})
  48. { P t : t [ 0 , ) } \{P_{t}:t\in[0,\infty)\}
  49. U α f ( x ) = 𝐄 x [ 0 e - α t f ( X t ) d t ] . U^{\alpha}f(x)=\mathbf{E}^{x}\left[\int_{0}^{\infty}e^{-\alpha t}f(X_{t})dt% \right].
  50. μ \mu
  51. ( E , ) (E,\mathcal{E})
  52. 𝐏 μ \mathbf{P}^{\mu}
  53. ( Ω , * ) (\Omega,\mathcal{F}^{*})
  54. ( X t , t * , P μ ) (X_{t},\mathcal{F}_{t}^{*},P^{\mu})
  55. μ \mu
  56. { P t : t [ 0 , ) } \{P_{t}:t\in[0,\infty)\}
  57. f f
  58. α \alpha
  59. ( E , * ) (E,\mathcal{E}^{*})
  60. μ \mu
  61. ( E , ) (E,\mathcal{E})
  62. t f ( X t ) t\rightarrow f(X_{t})
  63. P μ P^{\mu}
  64. [ 0 , ) [0,\infty)

Borell–Brascamp–Lieb_inequality.html

  1. h ( ( 1 - λ ) x + λ y ) M p ( f ( x ) , g ( y ) , λ ) , h\left((1-\lambda)x+\lambda y\right)\geq M_{p}\left(f(x),g(y),\lambda\right),
  2. M p ( a , b , λ ) = ( ( 1 - λ ) a p + λ b p ) 1 / p , M 0 ( a , b , λ ) = a 1 - λ b λ . \begin{aligned}\displaystyle M_{p}(a,b,\lambda)&\displaystyle=\left((1-\lambda% )a^{p}+\lambda b^{p}\right)^{1/p},\\ \displaystyle M_{0}(a,b,\lambda)&\displaystyle=a^{1-\lambda}b^{\lambda}.\end{aligned}
  3. n h ( x ) d x M p / ( n p + 1 ) ( n f ( x ) d x , n g ( x ) d x , λ ) . \int_{\mathbb{R}^{n}}h(x)\,\mathrm{d}x\geq M_{p/(np+1)}\left(\int_{\mathbb{R}^% {n}}f(x)\,\mathrm{d}x,\int_{\mathbb{R}^{n}}g(x)\,\mathrm{d}x,\lambda\right).

Boroxine.html

  1. 3 CO + 1.5 B 2 H 6 LiBH 4 ( CH 3 BO ) 3 \rm\ 3CO+1.5B_{2}H_{6}\xrightarrow{LiBH_{4}}(CH_{3}BO)_{3}
  2. C 6 H 5 X + ( CH 3 BO ) 3 K 2 CO 3 , Pd ( PPh 3 ) 4 dioxane C 6 H 5 CH 3 ( X = Br , I ) \rm\ C_{6}H_{5}X+(CH_{3}BO)_{3}\xrightarrow[dioxane]{K_{2}CO_{3},Pd(PPh_{3})_{% 4}}C_{6}H_{5}CH_{3}(X=Br,I)

Bose–Einstein_condensation_(network_theory).html

  1. ε ε
  2. T = 1 β T=\frac{1}{β}
  3. n ( ε ) = 1 e β ( ε - μ ) - 1 n(\varepsilon)=\frac{1}{e^{\beta(\varepsilon-\mu)}-1}
  4. μ μ
  5. N = d ε g ( ε ) n ( ε ) N=\int d\varepsilon\,g(\varepsilon)\,n(\varepsilon)
  6. g ( ε ) g(ε)
  7. g ( ε ) 0 g(ε)→0
  8. ε 0 ε→0
  9. g ( ε ) g(ε)
  10. g ( ε ) ε d - 2 2 g(\varepsilon)\sim\varepsilon^{\frac{d-2}{2}}
  11. g ( ε ) 0 g(ε)→0
  12. ε 0 ε→0
  13. d > 2 d>2
  14. d > 2 d>2
  15. T c = ( n ζ ( 3 2 ) ) 2 3 h 2 2 π m k B T_{c}=\left(\frac{n}{\zeta\left(\tfrac{3}{2}\right)}\right)^{\tfrac{2}{3}}% \frac{h^{2}}{2\pi mk_{B}}
  16. n n
  17. m m
  18. h h
  19. ζ ζ
  20. ζ ( 3 2 ) 2.6124 ζ(\frac{3}{2})≈2.6124
  21. ε i = - 1 β ln η i \varepsilon_{i}=-\frac{1}{\beta}\ln{\eta_{i}}
  22. β = 1 β=1
  23. β = 0 β=0
  24. β 1 β≫1
  25. i i
  26. p ( ε ) p(ε)
  27. j j
  28. Π j = e - β ε j k j r e - β ε r k r . \Pi_{j}=\frac{e^{-\beta\varepsilon_{j}}k_{j}}{\sum_{r}e^{-\beta\varepsilon_{r}% }k_{r}}.
  29. j j
  30. i i
  31. k i ( ε i , t , t i ) t = m e - β ε i k i ( ε i , t , t i ) Z t \frac{\partial k_{i}(\varepsilon_{i},t,t_{i})}{\partial t}=m\frac{e^{-\beta% \varepsilon_{i}}k_{i}(\varepsilon_{i},t,t_{i})}{Z_{t}}
  32. k i ( ε i , t , t i ) k_{i}(\varepsilon_{i},t,t_{i})
  33. i i
  34. t i t_{i}
  35. Z t Z_{t}
  36. Z t = i e - β ε i k i ( ε i , t , t i ) . Z_{t}=\sum_{i}e^{-\beta\varepsilon_{i}}k_{i}(\varepsilon_{i},t,t_{i}).
  37. k i ( ε i , t , t i ) = m ( t t i ) f ( ε i ) k_{i}(\varepsilon_{i},t,t_{i})=m\left(\frac{t}{t_{i}}\right)^{f(\varepsilon_{i% })}
  38. f ( ε ) f(\varepsilon)
  39. f ( ε ) = e - β ( ε - μ ) f(\varepsilon)=e^{-\beta(\varepsilon-\mu)}
  40. μ μ
  41. d ε p ( ε ) 1 e β ( ε - μ ) - 1 = 1 \int d\varepsilon\,p(\varepsilon)\frac{1}{e^{\beta(\varepsilon-\mu)}-1}=1
  42. p ( ε ) p(ε)
  43. ε ε
  44. t t→∞
  45. ε ε
  46. n ( ε ) = 1 e β ( ε - μ ) - 1 . n(\varepsilon)=\frac{1}{e^{\beta(\varepsilon-\mu)}-1}.
  47. μ μ
  48. p ( ε ) p(ε)
  49. p ( ε ) 0 p(ε)→0
  50. ε 0 ε→0
  51. β β
  52. ρ ( η ) ρ(η)
  53. ρ ( η ) = ( 1 - η ) λ \rho(\eta)=(1-\eta)^{\lambda}
  54. λ = 1 λ=1
  55. β β
  56. ρ ( ν ) ρ(ν)
  57. β β

Bounded_deformation.html

  1. ε ( u ) = u + u 2 \varepsilon(u)=\frac{\nabla u+\nabla u^{\top}}{2}
  2. ε ( u ) = e ( u ) d x + ( u + ( x ) - u - ( x ) ) ν u ( x ) H n - 1 | J u , \varepsilon(u)=e(u)\,\mathrm{d}x+\big(u_{+}(x)-u_{-}(x)\big)\odot\nu_{u}(x)H^{% n-1}|J_{u},
  3. \odot
  4. a b = a b + b a 2 . a\odot b=\frac{a\otimes b+b\otimes a}{2}.

Bounded_inverse_theorem.html

  1. T x = ( x 1 , x 2 2 , x 3 3 , ) Tx=\left(x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},\dots\right)
  2. x ( n ) = ( 1 , 1 2 , , 1 n , 0 , 0 , ) x^{(n)}=\left(1,\frac{1}{2},\dots,\frac{1}{n},0,0,\dots\right)
  3. x ( ) = ( 1 , 1 2 , , 1 n , ) , x^{(\infty)}=\left(1,\frac{1}{2},\dots,\frac{1}{n},\dots\right),
  4. c 0 c_{0}
  5. x = ( 1 , 1 2 , 1 3 , ) , x=\left(1,\frac{1}{2},\frac{1}{3},\dots\right),
  6. c 0 c_{0}
  7. T : c 0 c 0 T:c_{0}\to c_{0}

Boyle_temperature.html

  1. Z Z
  2. T b = a R b T_{b}=\frac{a}{Rb}
  3. p = R T ( 1 V m + B 2 ( T ) V m 2 + B 3 ( T ) V m 3 + ) p=RT\left(\frac{1}{V_{m}}+\frac{B_{2}(T)}{V_{m}^{2}}+\frac{B_{3}(T)}{V_{m}^{3}% }+\dots\right)
  4. B 2 ( T ) B_{2}(T)
  5. c = 1 V m c=\frac{1}{V_{m}}
  6. d Z d p = 0 if p = 0 \frac{\mathrm{d}Z}{\mathrm{d}p}=0\qquad\mbox{if }~{}p=0
  7. Z Z

Bracket_(mathematics).html

  1. \langle\,\,\rangle
  2. \langle\,\,\rangle\,
  3. ( x + y ) × ( x - y ) (x+y)\times(x-y)
  4. f ( x ) f(x)
  5. sin x \sin x
  6. a , b \langle a,b\rangle
  7. [ a , c ) [a,c)
  8. a a
  9. c c
  10. [ 5 , 12 ) [5,12)
  11. [ 5 , 12 [ [5,12[
  12. ( 1 - 1 2 3 ) [ c d ] \begin{pmatrix}1&-1\\ 2&3\end{pmatrix}\quad\quad\begin{bmatrix}c&d\end{bmatrix}
  13. f ( n ) ( x ) f^{(n)}(x)\,
  14. f ( x ) = exp ( λ x ) f(x)=\exp(\lambda x)
  15. f ( n ) ( x ) = λ n exp ( λ x ) f^{(n)}(x)=\lambda^{n}\exp(\lambda x)
  16. f n ( x ) = f ( f ( ( f ( x ) ) ) ) f^{n}(x)=f(f(\ldots(f(x))\ldots))
  17. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) = x ! ( x - n ) ! . (x)_{n}=x(x-1)(x-2)\cdots(x-n+1)=\frac{x!}{(x-n)!}.
  18. x ( n ) = x ( x + 1 ) ( x + 2 ) ( x + n - 1 ) = ( x + n - 1 ) ! ( x - 1 ) ! . x^{(n)}=x(x+1)(x+2)\cdots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}.
  19. A | \left\langle A\right|
  20. | B \left|B\right\rangle
  21. [ x ] \mathbb{R}[x]
  22. x x
  23. [ , ] : 𝔤 × 𝔤 𝔤 [\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}
  24. π = 3 \lfloor\pi\rfloor=3
  25. π = 4 \lceil\pi\rceil=4

Bracketing_(linguistics).html

  1. [ [ un- ] [ [ event ] [ -ful ] ] ] \left[[\mbox{un-}~{}]\left[[\mbox{event}~{}][\mbox{-ful}~{}]\right]\right]
  2. [ [ [ un- ] [ event ] ] [ -ful ] ] \left[\left[[\mbox{un-}~{}][\mbox{event}~{}]\right][\mbox{-ful}~{}]\right]
  3. [ [ min ] [ ed ] ] \left[[\mbox{min}~{}][\mbox{ed}~{}]\right]
  4. [ [ mi ] [ ned ] ] \left[[\mbox{mi}~{}][\mbox{ned}~{}]\right]

Braided_Hopf_algebra.html

  1. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}
  2. ( R , , η ) (R,\cdot,\eta)
  3. : R × R R \cdot:R\times R\to R
  4. η : k R \eta:k\to R
  5. ( R , Δ , ε ) (R,\Delta,\varepsilon)
  6. ε \varepsilon
  7. Δ \Delta
  8. ε \varepsilon
  9. Δ : R R R \Delta:R\to R\otimes R
  10. ε : R k \varepsilon:R\to k
  11. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}
  12. R R R\otimes R
  13. η η ( 1 ) : k R R \eta\otimes\eta(1):k\to R\otimes R
  14. ( R R ) × ( R R ) R R , ( r s , t u ) i r t i s i u , and c ( s t ) = i t i s i . (R\otimes R)\times(R\otimes R)\to R\otimes R,\quad(r\otimes s,t\otimes u)% \mapsto\sum_{i}rt_{i}\otimes s_{i}u,\quad\,\text{and}\quad c(s\otimes t)=\sum_% {i}t_{i}\otimes s_{i}.
  15. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}
  16. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}
  17. S : R R S:R\to R
  18. S ( r ( 1 ) ) r ( 2 ) = r ( 1 ) S ( r ( 2 ) ) = η ( ε ( r ) ) S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\eta(\varepsilon(r))
  19. r R , r\in R,
  20. Δ R ( r ) = r ( 1 ) r ( 2 ) \Delta_{R}(r)=r^{(1)}\otimes r^{(2)}
  21. H = k H=k
  22. H = k [ / 2 ] H=k[\mathbb{Z}/2\mathbb{Z}]
  23. T V TV
  24. V 𝒴 H H 𝒟 V\in{}^{H}_{H}\mathcal{YD}
  25. Δ \Delta
  26. T V TV
  27. Δ ( v ) = 1 v + v 1 for all v V . \Delta(v)=1\otimes v+v\otimes 1\quad\,\text{for all}\quad v\in V.
  28. ε : T V k \varepsilon:TV\to k
  29. ε ( v ) = 0 \varepsilon(v)=0
  30. v V . v\in V.
  31. T V TV
  32. V V
  33. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}
  34. R # H R\#H
  35. R # H R\#H
  36. R H R\otimes H
  37. R # H R\#H
  38. ( r # h ) ( r # h ) = r ( h ( 1 ) s y m b o l . r ) # h ( 2 ) h , (r\#h)(r^{\prime}\#h^{\prime})=r(h_{(1)}symbol{.}r^{\prime})\#h_{(2)}h^{\prime},
  39. r , r R , h , h H r,r^{\prime}\in R,\quad h,h^{\prime}\in H
  40. Δ ( h ) = h ( 1 ) h ( 2 ) \Delta(h)=h_{(1)}\otimes h_{(2)}
  41. h H h\in H
  42. s y m b o l . : H R R symbol{.}:H\otimes R\to R
  43. R # H R\#H
  44. Δ ( r # h ) = ( r ( 1 ) # r ( 2 ) h ( 1 ) ( - 1 ) ) ( r ( 2 ) # ( 0 ) h ( 2 ) ) , r R , h H . \Delta(r\#h)=(r^{(1)}\#r^{(2)}{}_{(-1)}h_{(1)})\otimes(r^{(2)}{}_{(0)}\#h_{(2)% }),\quad r\in R,h\in H.
  45. Δ R ( r ) = r ( 1 ) r ( 2 ) \Delta_{R}(r)=r^{(1)}\otimes r^{(2)}
  46. δ ( r ( 2 ) ) = r ( 2 ) ( - 1 ) r ( 2 ) ( 0 ) \delta(r^{(2)})=r^{(2)}{}_{(-1)}\otimes r^{(2)}{}_{(0)}
  47. r ( 2 ) R . r^{(2)}\in R.

Braking_distance.html

  1. μ = 2.25 T W 0.15 \mu=\frac{2.25}{TW^{0.15}}
  2. E E
  3. E = 1 2 m v 2 E=\frac{1}{2}mv^{2}
  4. m m
  5. v v
  6. W W
  7. W = μ m g d W=\mu mgd
  8. μ μ
  9. g g
  10. d d
  11. v v
  12. W = E W=E
  13. d = v 2 2 μ g d=\frac{v^{2}}{2\mu g}
  14. d d
  15. v = 2 μ g d v=\sqrt{2\mu gd}
  16. F = m a F=ma
  17. μ \mu
  18. F f r i c t = - μ m g F_{frict}=-\mu mg
  19. a = - μ g a=-\mu g
  20. d f ( d i , v i , v f ) d_{f}(d_{i},v_{i},v_{f})
  21. d f = d i + v f 2 - v i 2 2 a d_{f}=d_{i}+\frac{v_{f}^{2}-v_{i}^{2}}{2a}
  22. d i , v f = 0 d_{i},v_{f}=0
  23. a a
  24. d f = - v i 2 2 a = v i 2 2 μ g d_{f}=\frac{-v_{i}^{2}}{2a}=\frac{v_{i}^{2}}{2\mu g}
  25. D t o t a l = D p - r + D b r a k i n g = v t p - r + v 2 2 μ g D_{total}=D_{p-r}+D_{braking}=vt_{p-r}+\frac{v^{2}}{2\mu g}
  26. t p - r = 1.5 [ s ] , μ = 0.7 t_{p-r}=1.5[s],\mu=0.7
  27. D p - r = D b r a k i n g D_{p-r}=D_{braking}
  28. v t p - r = v 2 2 μ g vt_{p-r}=\frac{v^{2}}{2\mu g}
  29. v = 2 μ g t p - r v=2\mu gt_{p-r}

Brascamp–Lieb_inequality.html

  1. i = 1 m c i n i = n . \sum_{i=1}^{m}c_{i}n_{i}=n.
  2. f i L 1 ( n i ; [ 0 , + ] ) f_{i}\in L^{1}\left(\mathbb{R}^{n_{i}};[0,+\infty]\right)
  3. B i : n n i . B_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n_{i}}.
  4. n i = 1 m f i ( B i x ) c i d x D - 1 / 2 i = 1 m ( n i f i ( y ) d y ) c i , \int_{\mathbb{R}^{n}}\prod_{i=1}^{m}f_{i}\left(B_{i}x\right)^{c_{i}}\,\mathrm{% d}x\leq D^{-1/2}\prod_{i=1}^{m}\left(\int_{\mathbb{R}^{n_{i}}}f_{i}(y)\,% \mathrm{d}y\right)^{c_{i}},
  5. D = inf { det ( i = 1 m c i B i * A i B i ) i = 1 m ( det A i ) c i | A i is a positive-definite n i × n i matrix } . D=\inf\left\{\left.\frac{\det\left(\sum_{i=1}^{m}c_{i}B_{i}^{*}A_{i}B_{i}% \right)}{\prod_{i=1}^{m}(\det A_{i})^{c_{i}}}\right|A_{i}\mbox{ is a positive-% definite }~{}n_{i}\times n_{i}\mbox{ matrix}~{}\right\}.
  6. f i f_{i}
  7. f i ( y ) = exp { - ( y , A i y ) } . f_{i}(y)=\exp\{-(y,\,A_{i}\,y)\}.
  8. x = i = 1 m c i ( x u i ) u i x=\sum_{i=1}^{m}c_{i}(x\cdot u_{i})u_{i}
  9. n i = 1 m f i ( x u i ) c i d x i = 1 m ( f i ( y ) d y ) c i . \int_{\mathbb{R}^{n}}\prod_{i=1}^{m}f_{i}(x\cdot u_{i})^{c_{i}}\,\mathrm{d}x% \leq\prod_{i=1}^{m}\left(\int_{\mathbb{R}}f_{i}(y)\,\mathrm{d}y\right)^{c_{i}}.
  10. B i * ( z i ) = z i u i . B_{i}^{*}(z_{i})=z_{i}u_{i}.
  11. i = 1 m 1 p i = 1 \sum_{i=1}^{m}\frac{1}{p_{i}}=1
  12. n i = 1 m f i ( x ) d x i = 1 m f i p i . \int_{\mathbb{R}^{n}}\prod_{i=1}^{m}f_{i}(x)\,\mathrm{d}x\leq\prod_{i=1}^{m}\|% f_{i}\|_{p_{i}}.

Brauer's_three_main_theorems.html

  1. Q C G ( Q ) QC_{G}(Q)
  2. N G ( Q ) N_{G}(Q)
  3. C G ( Q ) C_{G}(Q)
  4. Z ( F G ) Z(FG)
  5. F G FG
  6. F C G ( Q ) FC_{G}(Q)
  7. C G ( Q ) C_{G}(Q)
  8. Z ( F H ) Z(FH)
  9. G G
  10. D D
  11. p p
  12. G G
  13. G G
  14. D D
  15. N G ( D ) N_{G}(D)
  16. H = N G ( D ) H=N_{G}(D)
  17. C G ( t ) C_{G}(t)
  18. G G
  19. G G
  20. C G ( t ) C_{G}(t)
  21. C G ( t ) C_{G}(t)
  22. C G ( t ) C_{G}(t)
  23. Q C G ( Q ) QC_{G}(Q)
  24. N G ( Q ) N_{G}(Q)

British_flag_theorem.html

  1. A P 2 + C P 2 = B P 2 + D P 2 . AP^{2}+CP^{2}=BP^{2}+DP^{2}.\,
  2. A P 2 = A w 2 + w P 2 = A w 2 + A z 2 AP^{2}=Aw^{2}+wP^{2}=Aw^{2}+Az^{2}
  3. P C 2 = w B 2 + z D 2 , PC^{2}=wB^{2}+zD^{2},
  4. B P 2 = w B 2 + A z 2 , BP^{2}=wB^{2}+Az^{2},
  5. P D 2 = z D 2 + A w 2 . PD^{2}=zD^{2}+Aw^{2}.
  6. A P 2 + P C 2 = ( A w 2 + A z 2 ) + ( w B 2 + z D 2 ) = ( w B 2 + A z 2 ) + ( z D 2 + A w 2 ) = B P 2 + P D 2 . AP^{2}+PC^{2}=(Aw^{2}+Az^{2})+(wB^{2}+zD^{2})=(wB^{2}+Az^{2})+(zD^{2}+Aw^{2})=% BP^{2}+PD^{2}.\,

Brocard's_problem.html

  1. n ! + 1 = m 2 , n!+1=m^{2},
  2. n ! + A = k 2 n!+A=k^{2}
  3. n ! = P ( x ) n!=P(x)

BrownBoost.html

  1. c c
  2. t t
  3. α \alpha
  4. T T
  5. c c
  6. c c
  7. c c
  8. α \alpha
  9. t t
  10. α \alpha
  11. t t
  12. r i ( x j ) r_{i}(x_{j})
  13. s s
  14. 1 m j = 1 m 1 - erf ( c ) = 1 - erf ( c ) \frac{1}{m}\sum_{j=1}^{m}1-\mbox{erf}~{}(\sqrt{c})=1-\mbox{erf}~{}(\sqrt{c})
  15. 1 m j = 1 m 1 - erf ( r i ( x j ) / c ) = 1 - erf ( c ) \frac{1}{m}\sum_{j=1}^{m}1-\mbox{erf}~{}(r_{i}(x_{j})/\sqrt{c})=1-\mbox{erf}~{% }(\sqrt{c})
  16. 1 - erf ( c ) 1-\mbox{erf}~{}(\sqrt{c})
  17. 1 - erf ( c ) 1-\mbox{erf}~{}(\sqrt{c})
  18. m m
  19. ( x 1 , y 1 ) , , ( x m , y m ) (x_{1},y_{1}),\ldots,(x_{m},y_{m})
  20. x j X , y j Y = { - 1 , + 1 } x_{j}\in X,\,y_{j}\in Y=\{-1,+1\}
  21. c c
  22. s = c s=c
  23. s s
  24. r i ( x j ) = 0 r_{i}(x_{j})=0
  25. j \forall j
  26. r i ( x j ) r_{i}(x_{j})
  27. i i
  28. x j x_{j}
  29. s > 0 s>0
  30. W i ( x j ) = e - ( r i ( x j ) + s ) 2 c W_{i}(x_{j})=e^{-\frac{(r_{i}(x_{j})+s)^{2}}{c}}
  31. r i ( x j ) r_{i}(x_{j})
  32. x j x_{j}
  33. h i : X { - 1 , + 1 } h_{i}:X\to\{-1,+1\}
  34. j W i ( x j ) h i ( x j ) y j > 0 \sum_{j}W_{i}(x_{j})h_{i}(x_{j})y_{j}>0
  35. α , t \alpha,t
  36. j h i ( x j ) y j e - ( r i ( x j ) + α h i ( x j ) y j + s - t ) 2 c = 0 \sum_{j}h_{i}(x_{j})y_{j}e^{-\frac{(r_{i}(x_{j})+\alpha h_{i}(x_{j})y_{j}+s-t)% ^{2}}{c}}=0
  37. E W i + 1 [ h i ( x j ) y j ] = 0 E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0
  38. W i + 1 = exp ( ) W_{i+1}=\exp(\frac{\ldots}{\ldots})
  39. E W i + 1 [ h i ( x j ) y j ] = 0 E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0
  40. ( Φ ( r i ( x j ) + α h ( x j ) y j + s - t ) - Φ ( r i ( x j ) + s ) ) = 0 \sum\left(\Phi\left(r_{i}(x_{j})+\alpha h(x_{j})y_{j}+s-t\right)-\Phi\left(r_{% i}(x_{j})+s\right)\right)=0
  41. Φ ( z ) = 1 - erf ( z / c ) \Phi(z)=1-\mbox{erf}~{}(z/\sqrt{c})
  42. r i ( x j ) r_{i}(x_{j})
  43. r i + 1 ( x j ) = r i ( x j ) + α h ( x j ) y j r_{i+1}(x_{j})=r_{i}(x_{j})+\alpha h(x_{j})y_{j}
  44. s = s - t s=s-t
  45. H ( x ) = sign ( i α i h i ( x ) ) H(x)=\textrm{sign}\left(\sum_{i}\alpha_{i}h_{i}(x)\right)

Broyden's_method.html

  1. k k
  2. 𝐟 ( 𝐱 ) = 𝟎 \mathbf{f}(\mathbf{x})=\mathbf{0}
  3. 𝐉 \mathbf{J}
  4. n × n n×n
  5. 2 n 2n
  6. f f′
  7. f ( x n ) f ( x n ) - f ( x n - 1 ) x n - x n - 1 , f^{\prime}(x_{n})\simeq\frac{f(x_{n})-f(x_{n-1})}{x_{n}-x_{n-1}},
  8. x n + 1 = x n - 1 f ( x n ) f ( x n ) x_{n+1}=x_{n}-\frac{1}{f^{\prime}(x_{n})}f(x_{n})
  9. n n
  10. k k
  11. 𝐟 ( 𝐱 ) = 𝟎 , \mathbf{f}(\mathbf{x})=\mathbf{0},
  12. 𝐟 \mathbf{f}
  13. 𝐱 \mathbf{x}
  14. 𝐱 = ( x 1 , x 2 , x 3 , , x k ) \mathbf{x}=(x_{1},x_{2},x_{3},\ldots,x_{k})
  15. 𝐟 ( 𝐱 ) = ( f 1 ( x 1 , x 2 , , x k ) , f 2 ( x 1 , x 2 , , x k ) , , f k ( x 1 , x 2 , , x k ) ) \mathbf{f}(\mathbf{x})=(f_{1}(x_{1},x_{2},\ldots,x_{k}),f_{2}(x_{1},x_{2},% \ldots,x_{k}),\ldots,f_{k}(x_{1},x_{2},\ldots,x_{k}))
  16. 𝐉 \mathbf{J}
  17. 𝐉 n ( 𝐱 n - 𝐱 n - 1 ) 𝐟 ( 𝐱 n ) - 𝐟 ( 𝐱 n - 1 ) , \mathbf{J}_{n}(\mathbf{x}_{n}-\mathbf{x}_{n-1})\simeq\mathbf{f}(\mathbf{x}_{n}% )-\mathbf{f}(\mathbf{x}_{n-1}),
  18. n n
  19. 𝐟 n = 𝐟 ( 𝐱 n ) , \mathbf{f}_{n}=\mathbf{f}(\mathbf{x}_{n}),
  20. Δ 𝐱 n = 𝐱 n - 𝐱 n - 1 , \Delta\mathbf{x}_{n}=\mathbf{x}_{n}-\mathbf{x}_{n-1},
  21. Δ 𝐟 n = 𝐟 n - 𝐟 n - 1 , \Delta\mathbf{f}_{n}=\mathbf{f}_{n}-\mathbf{f}_{n-1},
  22. 𝐉 n Δ 𝐱 n Δ 𝐟 n . \mathbf{J}_{n}\Delta\mathbf{x}_{n}\simeq\Delta\mathbf{f}_{n}.
  23. k k
  24. 𝐉 n = 𝐉 n - 1 + Δ 𝐟 n - 𝐉 n - 1 Δ 𝐱 n Δ 𝐱 n 2 Δ 𝐱 n T \mathbf{J}_{n}=\mathbf{J}_{n-1}+\frac{\Delta\mathbf{f}_{n}-\mathbf{J}_{n-1}% \Delta\mathbf{x}_{n}}{\|\Delta\mathbf{x}_{n}\|^{2}}\Delta\mathbf{x}_{n}^{% \mathrm{T}}
  25. 𝐉 n - 𝐉 n - 1 f . \|\mathbf{J}_{n}-\mathbf{J}_{n-1}\|_{\mathrm{f}}.
  26. 𝐱 n + 1 = 𝐱 n - 𝐉 n - 1 𝐟 ( 𝐱 n ) . \mathbf{x}_{n+1}=\mathbf{x}_{n}-\mathbf{J}_{n}^{-1}\mathbf{f}(\mathbf{x}_{n}).
  27. 𝐉 n - 1 = 𝐉 n - 1 - 1 + Δ 𝐱 n - 𝐉 n - 1 - 1 Δ 𝐟 n Δ 𝐱 n T 𝐉 n - 1 - 1 Δ 𝐟 n Δ 𝐱 n T 𝐉 n - 1 - 1 \mathbf{J}_{n}^{-1}=\mathbf{J}_{n-1}^{-1}+\frac{\Delta\mathbf{x}_{n}-\mathbf{J% }^{-1}_{n-1}\Delta\mathbf{f}_{n}}{\Delta\mathbf{x}_{n}^{\mathrm{T}}\mathbf{J}^% {-1}_{n-1}\Delta\mathbf{f}_{n}}\Delta\mathbf{x}_{n}^{\mathrm{T}}\mathbf{J}^{-1% }_{n-1}
  28. 𝐉 n - 1 = 𝐉 n - 1 - 1 + Δ 𝐱 n - 𝐉 n - 1 - 1 Δ 𝐟 n Δ 𝐟 n 2 Δ 𝐟 n T \mathbf{J}_{n}^{-1}=\mathbf{J}_{n-1}^{-1}+\frac{\Delta\mathbf{x}_{n}-\mathbf{J% }^{-1}_{n-1}\Delta\mathbf{f}_{n}}{\|\Delta\mathbf{f}_{n}\|^{2}}\Delta\mathbf{f% }_{n}^{\mathrm{T}}
  29. 𝐉 n - 1 - 𝐉 n - 1 - 1 f . \|\mathbf{J}_{n}^{-1}-\mathbf{J}_{n-1}^{-1}\|_{\mathrm{f}}.

Brushed_DC_electric_motor.html

  1. I = V a p p l i e d - V c e m f R a r m a t u r e I=\frac{V_{applied}-V_{cemf}}{R_{armature}}
  2. P = I V c e m f P=I\cdot V_{cemf}
  3. π \pi
  4. π \pi

Buckley–Leverett_equation.html

  1. S ( x , t ) S(x,t)
  2. S t = U ( S ) S x \frac{\partial S}{\partial t}=U(S)\frac{\partial S}{\partial x}
  3. U ( S ) = Q ϕ A d f d S . U(S)=\frac{Q}{\phi A}\frac{\mathrm{d}f}{\mathrm{d}S}.
  4. f f
  5. Q Q
  6. ϕ \phi
  7. A A
  8. p c ( S ) p_{c}(S)
  9. S S
  10. d p c / d S = 0 \mathrm{d}p_{c}/\mathrm{d}S=0
  11. S ( x , t ) = S ( x + U ( S ) t ) S(x,t)=S(x+U(S)t)
  12. U ( S ) U(S)
  13. S S

Builder's_Old_Measurement.html

  1. < m t p l > T o n n a g e = ( L e n g t h - ( B e a m × 3 5 ) ) × B e a m × B e a m 2 94 <mtpl>{{Tonnage}}=\frac{({Length}-({{Beam}\times\frac{3}{5}}))\times{Beam}% \times\frac{Beam}{2}}{94}
  2. < m t p l > T o n n a g e = L e n g t h × B e a m × D e p t h 100 <mtpl>{{Tonnage}}=\frac{{Length}\times{Beam}\times{Depth}}{100}
  3. < m t p l > T o n n a g e = L e n g t h × B e a m × B e a m 2 × 3 5 × 0.62 35 <mtpl>{{Tonnage}}=\frac{{Length}\times\ {Beam}\times\frac{Beam}{2}\times\frac{% 3}{5}\times{0.62}}{35}
  4. < m t p l > T o n n a g e = L e n g t h × B e a m × B e a m 2 94 <mtpl>{{Tonnage}}=\frac{{Length}\times\ {Beam}\times\frac{Beam}{2}}{94}
  5. < m t p l > T o n n a g e = L e n g t h × B e a m × D e p t h 94 <mtpl>{{Tonnage}}=\frac{{Length}\times\ {Beam}\times{Depth}}{94}
  6. 7 / 94 {7}/{94}

Bunyakovsky_conjecture.html

  1. f ( x ) f(x)
  2. f ( x ) f(x)
  3. n n
  4. f ( n ) f(n)
  5. f ( x ) f(x)
  6. f ( x ) f(x)
  7. f ( n ) f(n)
  8. n n
  9. x x
  10. Φ n ( x ) \Phi_{n}(x)
  11. Φ n ( x ) \Phi_{n}(x)
  12. Φ n ( x ) \Phi_{n}(x)
  13. Φ n ( x ) \Phi_{n}(x)
  14. Φ n ( x ) \Phi_{n}(x)
  15. Φ n ( x ) \Phi_{n}(x)
  16. f ( x ) < 0 f(x)<0
  17. x x
  18. f ( n ) f(n)
  19. n n
  20. f ( n ) f(n)
  21. f ( x ) = g ( x ) h ( x ) f(x)=g(x)h(x)
  22. g ( x ) g(x)
  23. h ( x ) h(x)
  24. ± 1 \pm 1
  25. f ( n ) = g ( n ) h ( n ) f(n)=g(n)h(n)
  26. n n
  27. f ( n ) f(n)
  28. n n
  29. g ( x ) g(x)
  30. h ( x ) h(x)
  31. ± 1 \pm 1
  32. f ( n ) f(n)
  33. f ( x ) = x 2 + x + 2 f(x)=x^{2}+x+2
  34. f ( n ) f(n)
  35. n n
  36. n = 1 n=1
  37. f ( x ) f(x)
  38. m m
  39. n n
  40. f ( m ) f(m)
  41. f ( n ) f(n)
  42. f ( x ) f(x)
  43. f ( m ) f(m)
  44. f ( n ) f(n)
  45. n 2 + 1 n^{2}+1
  46. f ( 1 ) , f ( 2 ) , f ( 3 ) , f(1),f(2),f(3),\dots
  47. f ( x ) f(x)
  48. x 2 - x + 2 x^{2}-x+2
  49. f ( 1 ) , f ( 2 ) , f ( 3 ) , f(1),f(2),f(3),\dots
  50. f ( n ) f(n)
  51. n 1 n\geq 1
  52. f ( x ) = c 0 + c 1 x + + c d x d f(x)=c_{0}+c_{1}x+\cdots+c_{d}x^{d}
  53. ( x k ) {\left({{x}\atop{k}}\right)}
  54. f ( x ) = a 0 + a 1 ( x 1 ) + + a d ( x d ) f(x)=a_{0}+a_{1}{\left({{x}\atop{1}}\right)}+\cdots+a_{d}{\left({{x}\atop{d}}% \right)}
  55. c i c_{i}
  56. a i a_{i}
  57. gcd { f ( n ) : n 1 } = gcd ( a 0 , a 1 , , a d ) . \gcd\{f(n):n\geq 1\}=\gcd(a_{0},a_{1},\dots,a_{d}).
  58. x 2 - x + 2 = 2 ( x 2 ) + 2 x^{2}-x+2=2{\left({{x}\atop{2}}\right)}+2
  59. x 2 - x + 2 x^{2}-x+2
  60. gcd { f ( n ) : n 1 } \gcd\{f(n):n\geq 1\}
  61. m m
  62. n n
  63. f ( m ) f(m)
  64. f ( n ) f(n)
  65. a a
  66. m m
  67. p a ( mod m ) p\equiv a\!\!\;\;(\mathop{{\rm mod}}m)
  68. f ( x ) = a + m x f(x)=a+mx
  69. a - m x a-mx
  70. m < 0 m<0
  71. m x + a mx+a
  72. a a
  73. m m

Burgers_vector.html

  1. 𝐛 = ( a / 2 ) h 2 + k 2 + l 2 \|\mathbf{b}\|\ =(a/2)\sqrt{h^{2}+k^{2}+l^{2}}

Burr_distribution.html

  1. 1 - ( 1 + x c ) - k 1-\left(1+x^{c}\right)^{-k}
  2. k B ( k - 1 / c , 1 + 1 / c ) k\operatorname{B}(k-1/c,\,1+1/c)
  3. ( 2 1 k - 1 ) 1 c \left(2^{\frac{1}{k}}-1\right)^{\frac{1}{c}}
  4. ( c - 1 k c + 1 ) 1 c \left(\frac{c-1}{kc+1}\right)^{\frac{1}{c}}
  5. f ( x ; c , k ) = c k x c - 1 ( 1 + x c ) k + 1 f(x;c,k)=ck\frac{x^{c-1}}{(1+x^{c})^{k+1}}\!
  6. F ( x ; c , k ) = 1 - ( 1 + x c ) - k . F(x;c,k)=1-\left(1+x^{c}\right)^{-k}.

Butanone_(data_page).html

  1. log e P m m H g = log e ( 760 101.325 ) - 7.783651 log e ( T + 273.15 ) - 6160.169 T + 273.15 + 66.97868 + 6.139268 × 10 - 6 ( T + 273.15 ) 2 \scriptstyle\log_{e}P_{mmHg}=\log_{e}(\frac{760}{101.325})-7.783651\log_{e}(T+% 273.15)-\frac{6160.169}{T+273.15}+66.97868+6.139268\times 10^{-6}(T+273.15)^{2}

BWF_Super_Series.html

  1. T o t a l p r i z e m o n e y × P e r c e n t a g e 100 Total\ prize\ money\ \times\frac{Percentage}{100}

Caccioppoli_set.html

  1. n n
  2. n n
  3. n n
  4. Ω \Omega
  5. n \scriptstyle\mathbb{R}^{n}
  6. E E
  7. E E
  8. Ω \Omega
  9. P ( E , Ω ) = V ( χ E , Ω ) := sup { Ω χ E ( x ) div s y m b o l ϕ ( x ) d x : s y m b o l ϕ C c 1 ( Ω , n ) , \Vertsymbol ϕ | L ( Ω ) 1 } P(E,\Omega)=V\left(\chi_{E},\Omega\right):=\sup\left\{\int_{\Omega}\chi_{E}(x)% \mathrm{div}symbol{\phi}(x)\,\mathrm{d}x\colon symbol{\phi}\in C_{c}^{1}(% \Omega,\mathbb{R}^{n}),\ \Vertsymbol{\phi}\|_{L^{\infty}(\Omega)}\leq 1\right\}
  10. χ E \chi_{E}
  11. E E
  12. E E
  13. Ω \Omega
  14. Ω = n \Omega=\mathbb{R}^{n}
  15. P ( E ) = P ( E , n ) P(E)=P(E,\mathbb{R}^{n})
  16. E E
  17. Ω \Omega
  18. n \mathbb{R}^{n}
  19. P ( E , Ω ) < + P(E,\Omega)<+\infty
  20. Ω n \Omega\subset\mathbb{R}^{n}
  21. D χ E D\chi_{E}
  22. Ω χ E ( x ) div s y m b o l ϕ ( x ) d x = E div s y m b o l ϕ ( x ) d x = - Ω \langlesymbol ϕ , D χ E ( x ) \forallsymbol ϕ C c 1 ( Ω , n ) \int_{\Omega}\chi_{E}(x)\mathrm{div}symbol{\phi}(x)\mathrm{d}x=\int_{E}\mathrm% {div}symbol{\phi}(x)\,\mathrm{d}x=-\int_{\Omega}\langlesymbol{\phi},D\chi_{E}(% x)\rangle\qquad\forallsymbol{\phi}\in C_{c}^{1}(\Omega,\mathbb{R}^{n})
  23. D χ E D\chi_{E}
  24. χ E \chi_{E}
  25. D χ E D\chi_{E}
  26. | D χ E | |D\chi_{E}|
  27. Ω n \Omega\subset\mathbb{R}^{n}
  28. | D χ E | ( Ω ) |D\chi_{E}|(\Omega)
  29. P ( E , Ω ) = V ( χ E , Ω ) P(E,\Omega)=V(\chi_{E},\Omega)
  30. W λ χ E ( x ) = n g λ ( x - y ) χ E ( y ) d y = ( π λ ) - n 2 E e - ( x - y ) 2 λ d y W_{\lambda}\chi_{E}(x)=\int_{\mathbb{R}^{n}}g_{\lambda}(x-y)\chi_{E}(y)\mathrm% {d}y=(\pi\lambda)^{-\frac{n}{2}}\int_{E}e^{-\frac{(x-y)^{2}}{\lambda}}\mathrm{% d}y
  31. W λ χ ( x ) W_{\lambda}\chi(x)
  32. x n \scriptstyle x\in\mathbb{R}^{n}
  33. lim λ 0 W λ χ E ( x ) = χ E ( x ) \lim_{\lambda\to 0}W_{\lambda}\chi_{E}(x)=\chi_{E}(x)
  34. W λ χ E ( x ) = grad W λ χ E ( x ) = D W λ χ E ( x ) = ( W λ χ E ( x ) x 1 W λ χ E ( x ) x n ) | D W λ χ E ( x ) | = k = 1 n | W λ χ E ( x ) x k | 2 \nabla W_{\lambda}\chi_{E}(x)=\mathrm{grad}W_{\lambda}\chi_{E}(x)=DW_{\lambda}% \chi_{E}(x)=\begin{pmatrix}\frac{\partial W_{\lambda}\chi_{E}(x)}{\partial x_{% 1}}\\ \vdots\\ \frac{\partial W_{\lambda}\chi_{E}(x)}{\partial x_{n}}\\ \end{pmatrix}\Longleftrightarrow\left|DW_{\lambda}\chi_{E}(x)\right|=\sqrt{% \sum_{k=1}^{n}\left|\frac{\partial W_{\lambda}\chi_{E}(x)}{\partial x_{k}}% \right|^{2}}
  35. Ω \Omega
  36. n \scriptstyle\mathbb{R}^{n}
  37. E E
  38. E E
  39. Ω \Omega
  40. P ( E , Ω ) = lim λ 0 Ω | D W λ χ E ( x ) | d x P(E,\Omega)=\lim_{\lambda\to 0}\int_{\Omega}|DW_{\lambda}\chi_{E}(x)|\mathrm{d}x
  41. Ω = n \scriptstyle\Omega=\mathbb{R}^{n}
  42. Ω Ω 1 \Omega\subseteq\Omega_{1}
  43. P ( E , Ω ) P ( E , Ω 1 ) P(E,\Omega)\leq P(E,\Omega_{1})
  44. E E
  45. Ω \Omega
  46. E 1 E_{1}
  47. E 2 E_{2}
  48. P ( E 1 E 2 , Ω ) P ( E 1 , Ω ) + P ( E 2 , Ω 1 ) P(E_{1}\cup E_{2},\Omega)\leq P(E_{1},\Omega)+P(E_{2},\Omega_{1})
  49. d ( E 1 , E 2 ) > 0 d(E_{1},E_{2})>0
  50. d d
  51. E E
  52. 0
  53. P ( E ) = 0 P(E)=0
  54. E 1 E 2 E_{1}\triangle E_{2}
  55. P ( E 1 ) = P ( E 2 ) P(E_{1})=P(E_{2})
  56. E n E\subset\mathbb{R}^{n}
  57. D χ E D\chi_{E}
  58. | D χ E | |D\chi_{E}|
  59. P ( E , Ω ) = Ω | D χ E | P(E,\Omega)=\int_{\Omega}|D\chi_{E}|
  60. Ω \Omega
  61. D χ E D\chi_{E}
  62. E E
  63. D χ E D\chi_{E}
  64. | D χ E | |D\chi_{E}|
  65. E \partial E
  66. D χ E D\chi_{E}
  67. | D χ E | |D\chi_{E}|
  68. E \partial E
  69. D χ E D\chi_{E}
  70. E \partial E
  71. E E
  72. x 0 E x_{0}\notin\partial E
  73. x 0 x_{0}
  74. n E \mathbb{R}^{n}\setminus\partial E
  75. A A
  76. E E
  77. n E \mathbb{R}^{n}\setminus E
  78. ϕ C c 1 ( A ; n ) \phi\in C^{1}_{c}(A;\mathbb{R}^{n})
  79. A ( n E ) = n E - A\subseteq(\mathbb{R}^{n}\setminus E)^{\circ}=\mathbb{R}^{n}\setminus E^{-}
  80. E - E^{-}
  81. E E
  82. χ E ( x ) = 0 \chi_{E}(x)=0
  83. x A x\in A
  84. Ω \langlesymbol ϕ , D χ E ( x ) = - A χ E ( x ) div s y m b o l ϕ ( x ) d x = 0 \int_{\Omega}\langlesymbol{\phi},D\chi_{E}(x)\rangle=-\int_{A}\chi_{E}(x)\,% \operatorname{div}symbol{\phi}(x)\,\mathrm{d}x=0
  85. A E A\subseteq E^{\circ}
  86. χ E ( x ) = 1 \chi_{E}(x)=1
  87. x A x\in A
  88. Ω \langlesymbol ϕ , D χ E ( x ) = - A div s y m b o l ϕ ( x ) d x = 0 \int_{\Omega}\langlesymbol{\phi},D\chi_{E}(x)\rangle=-\int_{A}\operatorname{% div}symbol{\phi}(x)\,\mathrm{d}x=0
  89. ϕ C c 1 ( A , n ) \phi\in C^{1}_{c}(A,\mathbb{R}^{n})
  90. x 0 x_{0}
  91. D χ E D\chi_{E}
  92. E \partial E
  93. P ( E ) P(E)
  94. E = { ( x , y ) : 0 x , y 1 } { ( x , 0 ) : - 1 x 1 } 2 E=\{(x,y):0\leq x,y\leq 1\}\cup\{(x,0):-1\leq x\leq 1\}\subset\mathbb{R}^{2}
  95. P ( E ) = 4 P(E)=4
  96. E = { ( x , 0 ) : - 1 x 1 } { ( x , 1 ) : 0 x 1 } { ( x , y ) : x { 0 , 1 } , 0 y 1 } \partial E=\{(x,0):-1\leq x\leq 1\}\;\cup\;\{(x,1):0\leq x\leq 1\}\;\cup\;\{(x% ,y):x\in\{0,1\},\;0\leq y\leq 1\}
  97. 1 ( E ) = 5 \mathcal{H}^{1}(\partial E)=5
  98. E \partial E
  99. E n E\subset\mathbb{R}^{n}
  100. * E \partial^{*}E
  101. x x
  102. ν E ( x ) := lim ρ 0 D χ E ( B ρ ( x ) ) | D χ E | ( B ρ ( x ) ) n \nu_{E}(x):=\lim_{\rho\downarrow 0}\frac{D\chi_{E}(B_{\rho}(x))}{|D\chi_{E}|(B% _{\rho}(x))}\in\mathbb{R}^{n}
  103. | ν E ( x ) | = 1 |\nu_{E}(x)|=1
  104. * E \partial^{*}E
  105. D χ E D\chi_{E}
  106. E \partial E
  107. * E support D χ E E \partial^{*}E\subseteq\operatorname{support}D\chi_{E}\subseteq\partial E
  108. Ω = n \Omega=\mathbb{R}^{n}
  109. E E
  110. P ( E ) ( = | D χ E | ) = n - 1 ( * E ) P(E)\left(=\int|D\chi_{E}|\right)=\mathcal{H}^{n-1}(\partial^{*}E)
  111. E n E\subset\mathbb{R}^{n}
  112. x x
  113. * E \partial^{*}E
  114. T x T_{x}
  115. | D χ E | |D\chi_{E}|
  116. T x T_{x}
  117. n \mathbb{R}^{n}
  118. lim λ 0 n f ( λ - 1 ( z - x ) ) | D χ E | ( z ) = T x f ( y ) d n - 1 ( y ) \lim_{\lambda\downarrow 0}\int_{\mathbb{R}^{n}}f(\lambda^{-1}(z-x))|D\chi_{E}|% (z)=\int_{T_{x}}f(y)\,d\mathcal{H}^{n-1}(y)
  119. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  120. T x T_{x}
  121. ν E ( x ) = lim ρ 0 D χ E ( B ρ ( x ) ) | D χ E | ( B ρ ( x ) ) n \nu_{E}(x)=\lim_{\rho\downarrow 0}\frac{D\chi_{E}(B_{\rho}(x))}{|D\chi_{E}|(B_% {\rho}(x))}\in\mathbb{R}^{n}
  122. lim λ 0 { λ - 1 ( z - x ) : z E } { y n : y ν E ( x ) > 0 } \lim_{\lambda\downarrow 0}\{\lambda^{-1}(z-x):z\in E\}\to\{y\in\mathbb{R}^{n}:% y\cdot\nu_{E}(x)>0\}
  123. L 1 L^{1}
  124. * E \partial^{*}E
  125. * E \partial^{*}E
  126. n - 1 \mathcal{H}^{n-1}
  127. * E \partial^{*}E
  128. | D χ E | |D\chi_{E}|
  129. | D χ E | ( A ) = n - 1 ( A * E ) |D\chi_{E}|(A)=\mathcal{H}^{n-1}(A\cap\partial^{*}E)
  130. A n A\subset\mathbb{R}^{n}
  131. n - 1 \mathcal{H}^{n-1}
  132. * E \partial^{*}E
  133. D χ E D\chi_{E}
  134. D χ E D\chi_{E}
  135. E div s y m b o l ϕ ( x ) d x = - E \langlesymbol ϕ , D χ E ( x ) s y m b o l ϕ C c 1 ( Ω , n ) \int_{E}\operatorname{div}symbol{\phi}(x)\,\mathrm{d}x=-\int_{\partial E}% \langlesymbol{\phi},D\chi_{E}(x)\rangle\qquad symbol{\phi}\in C_{c}^{1}(\Omega% ,\mathbb{R}^{n})
  136. * E \partial^{*}E
  137. ν E \nu_{E}
  138. E div s y m b o l ϕ ( x ) d x = - * E s y m b o l ϕ ( x ) ν E ( x ) d n - 1 ( x ) s y m b o l ϕ C c 1 ( Ω , n ) \int_{E}\operatorname{div}symbol{\phi}(x)\,\mathrm{d}x=-\int_{\partial^{*}E}% symbol{\phi}(x)\cdot\nu_{E}(x)\,\mathrm{d}\mathcal{H}^{n-1}(x)\qquad symbol{% \phi}\in C^{1}_{c}(\Omega,\mathbb{R}^{n})
  139. B V ( Ω ) BV(\Omega)

Cahn–Hilliard_equation.html

  1. c c
  2. c = ± 1 c=\pm 1
  3. c t = D 2 ( c 3 - c - γ 2 c ) , \frac{\partial c}{\partial t}=D\nabla^{2}\left(c^{3}-c-\gamma\nabla^{2}c\right),
  4. D D
  5. Length 2 / Time \,\text{Length}^{2}/\,\text{Time}
  6. γ \sqrt{\gamma}
  7. / t \partial/{\partial t}
  8. 2 \nabla^{2}
  9. n n
  10. μ = c 3 - c - γ 2 c \mu=c^{3}-c-\gamma\nabla^{2}c
  11. F [ c ] = d n x [ 1 4 ( c 2 - 1 ) 2 + γ 2 | c | 2 ] , F[c]=\int d^{n}x\left[\frac{1}{4}\left(c^{2}-1\right)^{2}+\frac{\gamma}{2}% \left|\nabla c\right|^{2}\right],
  12. d F d t = - d n x | μ | 2 , \frac{dF}{dt}=-\int d^{n}x\left|\nabla\mu\right|^{2},
  13. γ = 0.5 \gamma=0.5
  14. C = 0 C=0
  15. c ( x ) = tanh ( x 2 γ ) , c(x)=\tanh\left(\frac{x}{\sqrt{2\gamma}}\right),
  16. γ \sqrt{\gamma}
  17. L ( t ) L(t)
  18. L ( t ) t 1 / 3 L(t)\propto t^{1/3}
  19. c t = j ( x ) , \frac{\partial c}{\partial t}=\nabla\cdot{j}(x),
  20. j ( x ) = D μ {j}(x)=D\nabla\mu
  21. C = d n x c ( x , t ) C=\int d^{n}xc\left(x,t\right)
  22. d C d t = 0 \frac{dC}{dt}=0

Calcium_hexaboride.html

  1. μ B \mu_{\mathrm{B}}

Calo_tester.html

  1. t = x y d t=\frac{xy}{d}

Camera_matrix.html

  1. 3 × 4 3\times 4
  2. 𝐱 \mathbf{x}
  3. 𝐲 \mathbf{y}
  4. 𝐲 𝐂 𝐱 \mathbf{y}\sim\mathbf{C}\,\mathbf{x}
  5. 𝐂 \mathbf{C}
  6. \,\sim
  7. 𝐂 \mathbf{C}
  8. ( y 1 y 2 ) = f x 3 ( x 1 x 2 ) \begin{pmatrix}y_{1}\\ y_{2}\end{pmatrix}=\frac{f}{x_{3}}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}
  9. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  10. ( y 1 , y 2 ) (y_{1},y_{2})
  11. ( y 1 , y 2 ) (y_{1},y_{2})
  12. 𝐲 = ( y 1 , y 2 , 1 ) \mathbf{y}=(y_{1},y_{2},1)
  13. \,\sim
  14. ( y 1 y 2 1 ) = f x 3 ( x 1 x 2 x 3 f ) ( x 1 x 2 x 3 f ) \begin{pmatrix}y_{1}\\ y_{2}\\ 1\end{pmatrix}=\frac{f}{x_{3}}\begin{pmatrix}x_{1}\\ x_{2}\\ \frac{x_{3}}{f}\end{pmatrix}\sim\begin{pmatrix}x_{1}\\ x_{2}\\ \frac{x_{3}}{f}\end{pmatrix}
  15. 𝐱 \mathbf{x}
  16. ( y 1 y 2 1 ) ( 1 0 0 0 0 1 0 0 0 0 1 f 0 ) ( x 1 x 2 x 3 1 ) \begin{pmatrix}y_{1}\\ y_{2}\\ 1\end{pmatrix}\sim\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\frac{1}{f}&0\end{pmatrix}\,\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\\ 1\end{pmatrix}
  17. 𝐲 𝐂 𝐱 \mathbf{y}\sim\mathbf{C}\,\mathbf{x}
  18. 𝐂 \mathbf{C}
  19. 𝐂 = ( 1 0 0 0 0 1 0 0 0 0 1 f 0 ) \mathbf{C}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\frac{1}{f}&0\end{pmatrix}
  20. 𝐂 = ( 1 0 0 0 0 1 0 0 0 0 1 f 0 ) ( f 0 0 0 0 f 0 0 0 0 1 0 ) \mathbf{C}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\frac{1}{f}&0\end{pmatrix}\sim\begin{pmatrix}f&0&0&0\\ 0&f&0&0\\ 0&0&1&0\end{pmatrix}
  21. 𝐂 \mathbf{C}
  22. 𝐂 \mathbf{C}
  23. 𝐧 = ( 0 0 0 1 ) \mathbf{n}=\begin{pmatrix}0\\ 0\\ 0\\ 1\end{pmatrix}
  24. x 3 = 0 x_{3}=0
  25. 𝐲 𝐂 𝐱 \mathbf{y}\sim\mathbf{C}\,\mathbf{x}
  26. 𝐲 = ( y 1 y 2 0 ) \mathbf{y}=(y_{1}\,y_{2}\,0)^{\top}
  27. 𝐂 0 = ( 1 0 0 0 0 1 0 0 0 0 1 0 ) = ( 𝐈 𝟎 ) \mathbf{C}_{0}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{pmatrix}=\left(\begin{array}[]{c|c}\mathbf{I}&\mathbf{0}\end{array% }\right)
  28. 𝐈 \mathbf{I}
  29. 3 × 3 3\times 3
  30. 3 × 4 3\times 4
  31. 𝐂 \mathbf{C}
  32. 3 × 3 3\times 3
  33. 𝐂 0 \mathbf{C}_{0}
  34. 4 × 4 4\times 4
  35. ( 𝐑 𝟎 𝟎 1 ) \left(\begin{array}[]{c|c}\mathbf{R}&\mathbf{0}\\ \hline\mathbf{0}&1\end{array}\right)
  36. ( 𝐈 𝐭 𝟎 1 ) \left(\begin{array}[]{c|c}\mathbf{I}&\mathbf{t}\\ \hline\mathbf{0}&1\end{array}\right)
  37. 𝐑 \mathbf{R}
  38. 3 × 3 3\times 3
  39. 𝐭 \mathbf{t}
  40. ( 𝐑 𝐭 𝟎 1 ) \left(\begin{array}[]{c|c}\mathbf{R}&\mathbf{t}\\ \hline\mathbf{0}&1\end{array}\right)
  41. 𝐑 \mathbf{R}
  42. 𝐭 \mathbf{t}
  43. 𝐱 = ( 𝐑 𝐭 𝟎 1 ) 𝐱 \mathbf{x}=\left(\begin{array}[]{c|c}\mathbf{R}&\mathbf{t}\\ \hline\mathbf{0}&1\end{array}\right)\mathbf{x}^{\prime}
  44. 𝐱 \mathbf{x}^{\prime}
  45. 𝐂 0 \mathbf{C}_{0}
  46. 𝐲 𝐂 0 𝐱 = ( 𝐈 𝟎 ) ( 𝐑 𝐭 𝟎 1 ) 𝐱 = ( 𝐑 𝐭 ) 𝐱 \mathbf{y}\sim\mathbf{C}_{0}\,\mathbf{x}=\left(\begin{array}[]{c|c}\mathbf{I}&% \mathbf{0}\end{array}\right)\,\left(\begin{array}[]{c|c}\mathbf{R}&\mathbf{t}% \\ \hline\mathbf{0}&1\end{array}\right)\mathbf{x}^{\prime}=\left(\begin{array}[]{% c|c}\mathbf{R}&\mathbf{t}\end{array}\right)\,\mathbf{x}^{\prime}
  47. 𝐂 N = ( 𝐑 𝐭 ) \mathbf{C}_{N}=\left(\begin{array}[]{c|c}\mathbf{R}&\mathbf{t}\end{array}\right)
  48. 𝐂 N \mathbf{C}_{N}
  49. 𝐧 = ( - 𝐑 - 1 𝐭 1 ) = ( 𝐧 ~ 1 ) \mathbf{n}=\begin{pmatrix}-\mathbf{R}^{-1}\,\mathbf{t}\\ 1\end{pmatrix}=\begin{pmatrix}\tilde{\mathbf{n}}\\ 1\end{pmatrix}
  50. 𝐧 ~ \tilde{\mathbf{n}}
  51. 𝐂 N \mathbf{C}_{N}
  52. 𝐂 N = 𝐑 ( 𝐈 𝐑 - 1 𝐭 ) = 𝐑 ( 𝐈 - 𝐧 ~ ) \mathbf{C}_{N}=\mathbf{R}\,\left(\begin{array}[]{c|c}\mathbf{I}&\mathbf{R}^{-1% }\,\mathbf{t}\end{array}\right)=\mathbf{R}\,\left(\begin{array}[]{c|c}\mathbf{% I}&-\tilde{\mathbf{n}}\end{array}\right)
  53. 𝐧 ~ \tilde{\mathbf{n}}
  54. 3 × 3 3\times 3
  55. 𝐇 \mathbf{H}
  56. 𝐲 \mathbf{y}
  57. 𝐲 \mathbf{y}^{\prime}
  58. 𝐲 = 𝐇 𝐲 \mathbf{y}^{\prime}=\mathbf{H}\,\mathbf{y}
  59. 𝐲 = 𝐇 𝐂 N 𝐱 \mathbf{y}^{\prime}=\mathbf{H}\,\mathbf{C}_{N}\,\mathbf{x}^{\prime}
  60. 𝐂 = 𝐇 𝐂 N = 𝐇 ( 𝐑 𝐭 ) \mathbf{C}=\mathbf{H}\,\mathbf{C}_{N}=\mathbf{H}\,\left(\begin{array}[]{c|c}% \mathbf{R}&\mathbf{t}\end{array}\right)

Canonical_units.html

  1. D U DU
  2. T U TU
  3. μ \mu
  4. μ = G * M \mu=G*M\,\!
  5. G G
  6. M M
  7. μ = 1 * D U 3 T U 2 \mu=1*\frac{DU^{3}}{TU^{2}}
  8. μ \mu
  9. μ = G * M = 1 * D U 3 T U 2 \mu=G*M=1*\frac{DU^{3}}{TU^{2}}
  10. T U = D U 3 G * M TU=\sqrt{\frac{DU^{3}}{G*M}}

Cant_deficiency.html

  1. v 2 R cos α = g sin α {v^{2}\over R}\cos\alpha=g\sin\alpha
  2. V b a l = ( R g tan α ) 1 2 V_{bal}=\left({Rg\tan\alpha}\right)^{\tfrac{1}{2}}
  3. C D = g a u g e s e ( 1 + R 2 g 2 V a c t 4 ) 1 2 - s u p e r _ e l CD={gauge_{se}\over{\left(1+{R^{2}g^{2}\over{V_{act}^{4}}}\right)^{\tfrac{1}{2% }}}}-super\_el
  4. V b a l = 1746.4 9.80665 tan ( arcsin ( 152.4 / 1511.3 ) ) V_{bal}=\sqrt{1746.4\cdot 9.80665\cdot\tan(\arcsin(152.4/1511.3))}
  5. = 41.6638 m / s = 149.99 km / h = 93.20 miles / h =41.6638\,\mathrm{m/s}=149.99\,\mathrm{km/h}=93.20\,\mathrm{miles/h}
  6. C D = 1511.3 1 + ( 1746.4 2 9.8066 2 / 55.8 4 ) - 152.4 CD=\frac{1511.3}{\sqrt{1+(1746.4^{2}\cdot 9.8066^{2}/55.8^{4})}}-152.4
  7. = 118.7 mm ( = 4.67 inches ) =118.7\,\mathrm{mm}(=4.67\,\mathrm{inches})

Capillary_length.html

  1. λ c = γ ρ g \lambda_{c}=\sqrt{\frac{\gamma}{\rho g}}
  2. g g
  3. ρ \rho
  4. γ \gamma

Capital,_Volume_I.html

  1. [ 1 coat 10 lb. of tea 40 lb. of coffee 1 quarter of corn 2 ounces of gold 1 / 2 ton of iron x commodity A, etc. ] = 20 yards of linen \begin{bmatrix}1&\mbox{coat}\\ 10&\mbox{lb. of tea}\\ 40&\mbox{lb. of coffee}\\ 1&\mbox{quarter of corn}\\ 2&\mbox{ounces of gold}\\ 1/2&\mbox{ton of iron}\\ x&\mbox{commodity A, etc.}\\ \end{bmatrix}=20\mbox{ yards of linen}~{}
  2. [ 1 coat 10 lb. of tea 40 lb. of coffee 1 quarter of corn 20 yards of linen 1 / 2 ton of iron x commodity A, etc. ] = 2 ounces of gold \begin{bmatrix}1&\mbox{coat}\\ 10&\mbox{lb. of tea}\\ 40&\mbox{lb. of coffee}\\ 1&\mbox{quarter of corn}\\ 20&\mbox{yards of linen}\\ 1/2&\mbox{ton of iron}\\ x&\mbox{commodity A, etc.}\\ \end{bmatrix}=2\mbox{ ounces of gold}~{}
  3. C M C C\to M\to C
  4. C M C\to M
  5. M C M\to C
  6. M C M\to C
  7. = =
  8. C M C\to M
  9. C = c + v C=c+v
  10. C C
  11. = c + v + s =c+v+s

Carathéodory_metric.html

  1. ρ ( a , b ) = tanh - 1 | a - b | | 1 - a ¯ b | \rho(a,b)=\tanh^{-1}\frac{|a-b|}{|1-\bar{a}b|}
  2. d ( x , y ) = sup { ρ ( f ( x ) , f ( y ) ) | f : B Δ is holomorphic } . d(x,y)=\sup\{\rho(f(x),f(y))|f:B\to\Delta\mbox{ is holomorphic}~{}\}.
  3. d ( 0 , x ) = ρ ( 0 , x ) . d(0,x)=\rho(0,\|x\|).
  4. d ( x , y ) = sup { 2 tanh - 1 f ( x ) - f ( y ) 2 | f : B Δ is holomorphic } d(x,y)=\sup\left\{\left.2\tanh^{-1}\left\|\frac{f(x)-f(y)}{2}\right\|\right|f:% B\to\Delta\mbox{ is holomorphic}~{}\right\}
  5. a - b 2 tanh d ( a , b ) 2 , ( 1 ) \|a-b\|\leq 2\tanh\frac{d(a,b)}{2},\qquad\qquad(1)
  6. ρ ( ( a ) , ( b ) ) = d ( a , b ) . \rho(\ell(a),\ell(b))=d(a,b).
  7. α ( x , v ) = sup { | D f ( x ) v | | f : B Δ is holomorphic } . \alpha(x,v)=\sup\big\{|\mathrm{D}f(x)v|\big|f:B\to\Delta\mbox{ is holomorphic}% ~{}\big\}.

Carbamoyl_phosphate_synthase_II.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Carbon_dioxide_equivalent.html

  1. R F = α l n ( C / C 0 ) RF=\alpha ln(C/C_{0})
  2. α \alpha
  3. C 0 C_{0}
  4. C 0 e x p ( R F / α ) C_{0}exp(RF/\alpha)

Cardiac_index.html

  1. CI = CO BSA = SV × HR BSA \,\text{CI}=\frac{\,\text{CO}}{\,\text{BSA}}=\frac{\,\text{SV}\times\,\text{HR% }}{\,\text{BSA}}

Caristi_fixed-point_theorem.html

  1. d ( x , T ( x ) ) f ( x ) - f ( T ( x ) ) . d\big(x,T(x)\big)\leq f(x)-f\big(T(x)\big).

Carl_Johan_Malmsten.html

  1. 0 1 ln ln 1 x 1 + x 2 d x = 1 ln ln x 1 + x 2 d x = π 2 ln { Γ ( 3 / 4 ) Γ ( 1 / 4 ) 2 π } \int\limits_{0}^{1}\!\frac{\,\ln\ln\frac{1}{x}\,}{1+x^{2}}\,dx\,=\,\int\limits% _{1}^{\infty}\!\frac{\,\ln\ln{x}\,}{1+x^{2}}\,dx\,=\,\frac{\pi}{\,2\,}\ln\left% \{\frac{\Gamma{(3/4)}}{\Gamma{(1/4)}}\sqrt{2\pi\,}\right\}
  2. 0 1 ln ln 1 x ( 1 + x ) 2 d x = 1 ln ln x ( 1 + x ) 2 d x = 1 2 ( ln π - ln 2 - γ ) , \int\limits_{0}^{1}\frac{\ln\ln\frac{1}{x}}{(1+x)^{2}}\,dx=\int\limits_{1}^{% \infty}\!\frac{\ln\ln{x}}{(1+x)^{2}}\,dx=\frac{1}{2}\bigl(\ln\pi-\ln 2-\gamma% \bigr),
  3. 0 1 ln ln 1 x 1 - x + x 2 d x = 1 ln ln x 1 - x + x 2 d x = 2 π 3 ln { 32 π 5 6 Γ ( 1 / 6 ) } \int\limits_{0}^{1}\!\frac{\ln\ln\frac{1}{x}}{1-x+x^{2}}\,dx=\int\limits_{1}^{% \infty}\!\frac{\ln\ln{x}}{1-x+x^{2}}\,dx=\frac{2\pi}{\sqrt{3}}\ln\biggl\{\frac% {\sqrt[6]{32\pi^{5}}}{\Gamma{(1/6)}}\biggr\}
  4. 0 1 ln ln 1 x 1 + x + x 2 d x = 1 ln ln x 1 + x + x 2 d x = π 3 ln { Γ ( 2 / 3 ) Γ ( 1 / 3 ) 2 π 3 } \int\limits_{0}^{1}\!\frac{\ln\ln\frac{1}{x}}{1+x+x^{2}}\,dx=\int\limits_{1}^{% \infty}\!\frac{\ln\ln{x}}{1+x+x^{2}}\,dx=\frac{\pi}{\sqrt{3}}\ln\biggl\{\frac{% \Gamma{(2/3)}}{\Gamma{(1/3)}}\sqrt[3]{2\pi}\biggr\}
  5. 0 1 ln ln 1 x 1 + 2 x cos φ + x 2 d x = 1 ln ln x 1 + 2 x cos φ + x 2 d x = π 2 sin φ ln { ( 2 π ) φ π Γ ( 1 2 + φ 2 π ) Γ ( 1 2 - φ 2 π ) } , - π < φ < π \int\limits_{0}^{1}\!\frac{\ln\ln\frac{1}{x}}{1+2x\cos\varphi+x^{2}}\,dx\,=% \int\limits_{1}^{\infty}\!\frac{\ln\ln{x}}{1+2x\cos\varphi+x^{2}}\,dx=\frac{% \pi}{2\sin\varphi}\ln\left\{\frac{(2\pi)^{\frac{\scriptstyle\varphi}{% \scriptstyle\pi}}\,\Gamma\!\left(\!\displaystyle\frac{1}{\,2\,}+\frac{\varphi}% {\,2\pi\,}\!\right)}{\Gamma\!\left(\!\displaystyle\frac{1}{\,2\,}-\frac{% \varphi}{\,2\pi\,}\!\right)}\right\},\qquad-\pi<\varphi<\pi
  6. 0 1 x n - 2 ln ln 1 x 1 - x 2 + x 4 - + x 2 n - 2 d x = 1 x n - 2 ln ln x 1 - x 2 + x 4 - + x 2 n - 2 d x = \int\limits_{0}^{1}\!\frac{x^{n-2}\ln\ln\frac{1}{x}}{1-x^{2}+x^{4}-\cdots+x^{2% n-2}}\,dx\,=\int\limits_{1}^{\infty}\!\frac{x^{n-2}\ln\ln{x}}{1-x^{2}+x^{4}-% \cdots+x^{2n-2}}\,dx=
  7. = π 2 n sec π 2 n ln π + π n l = 1 1 2 ( n - 1 ) ( - 1 ) l - 1 cos ( 2 l - 1 ) π 2 n ln { Γ ( 1 - 2 l - 1 2 n ) Γ ( 2 l - 1 2 n ) } , n = 3 , 5 , 7 , \quad=\,\frac{\pi}{\,2n\,}\sec\frac{\,\pi\,}{2n}\!\cdot\ln\pi+\frac{\pi}{\,n\,% }\cdot\!\!\!\!\!\!\sum_{l=1}^{\;\;\frac{1}{2}(n-1)}\!\!\!\!(-1)^{l-1}\cos\frac% {\,(2l-1)\pi\,}{2n}\cdot\ln\left\{\!\frac{\Gamma\!\left(1-\displaystyle\frac{2% l-1}{2n}\right)}{\Gamma\!\left(\displaystyle\frac{2l-1}{2n}\right)}\right\},% \qquad n=3,5,7,\ldots
  8. 0 1 x n - 2 ln ln 1 x 1 + x 2 + x 4 + + x 2 n - 2 d x = 1 x n - 2 ln ln x 1 + x 2 + x 4 + + x 2 n - 2 d x = \int\limits_{0}^{1}\!\frac{x^{n-2}\ln\ln\frac{1}{x}}{1+x^{2}+x^{4}+\cdots+x^{2% n-2}}\,dx\,=\int\limits_{1}^{\infty}\!\frac{x^{n-2}\ln\ln{x}}{1+x^{2}+x^{4}+% \cdots+x^{2n-2}}\,dx=
  9. = { π 2 n tan π 2 n ln 2 π + π n l = 1 n - 1 ( - 1 ) l - 1 sin π l n ln { Γ ( 1 2 + l 2 n ) Γ ( l 2 n ) } , n = 2 , 4 , 6 , π 2 n tan π 2 n ln π + π n l = 1 1 2 ( n - 1 ) ( - 1 ) l - 1 sin π l n ln { Γ ( 1 - l n ) Γ ( l n ) } , n = 3 , 5 , 7 , \qquad=\begin{cases}\displaystyle\frac{\,\pi\,}{2n}\tan\frac{\,\pi\,}{2n}\ln 2% \pi+\frac{\pi}{n}\sum_{l=1}^{n-1}(-1)^{l-1}\sin\frac{\,\pi l\,}{n}\cdot\ln% \left\{\!\frac{\Gamma\!\left(\!\displaystyle\frac{1}{\,2\,}+\displaystyle\frac% {l}{\,2n}\!\right)}{\Gamma\!\left(\!\displaystyle\frac{l}{\,2n}\!\right)}% \right\},\quad n=2,4,6,\ldots\\ \displaystyle\frac{\,\pi\,}{2n}\tan\frac{\,\pi\,}{2n}\ln\pi+\frac{\pi}{n}\!\!% \!\!\!\sum_{l=1}^{\;\;\;\frac{1}{2}(n-1)}\!\!\!\!(-1)^{l-1}\sin\frac{\,\pi l\,% }{n}\cdot\ln\left\{\!\frac{\Gamma\!\left(1-\displaystyle\frac{\,l}{n}\!\right)% }{\Gamma\!\left(\!\displaystyle\frac{\,l}{n}\!\right)}\right\},\qquad n=3,5,7,% \ldots\end{cases}
  10. 0 1 ln ln 1 x 1 + x 3 d x = 1 x ln ln x 1 + x 3 d x = ln 2 6 ln 3 2 - π 6 3 { ln 54 - 8 ln 2 π + 12 ln Γ ( 1 3 ) } \int\limits_{0}^{1}\frac{\ln\ln\frac{1}{x}}{1+x^{3}}\,dx=\int\limits_{1}^{% \infty}\frac{x\ln\ln x}{1+x^{3}}\,dx=\frac{\ln 2}{6}\ln\frac{3}{2}-\frac{\pi}{% 6\sqrt{3}}\left\{\ln 54-8\ln 2\pi+12\ln\Gamma\left(\frac{1}{3}\right)\right\}
  11. 0 1 x ln ln 1 x ( 1 - x + x 2 ) 2 d x = 1 x ln ln x ( 1 - x + x 2 ) 2 d x = - γ 3 - 1 3 ln 6 3 π + π 3 27 { 5 ln 2 π - 6 ln Γ ( 1 6 ) } \int\limits_{0}^{1}\!\frac{x\ln\ln\frac{1}{x}}{(1-x+x^{2})^{2}}\,dx=\int% \limits_{1}^{\infty}\!\frac{x\ln\ln x}{(1-x+x^{2})^{2}}\,dx=-\frac{\gamma}{3}-% \frac{1}{3}\ln\frac{6\sqrt{3}}{\pi}+\frac{\pi\sqrt{3}}{27}\left\{5\ln 2\pi-6% \ln\Gamma\left(\frac{1}{6}\right)\right\}
  12. 0 1 ( x 4 - 6 x 2 + 1 ) ln ln 1 x ( 1 + x 2 ) 3 d x = 1 ( x 4 - 6 x 2 + 1 ) ln ln x ( 1 + x 2 ) 3 d x = 2 G π \int\limits_{0}^{1}\frac{\left(x^{4}-6x^{2}+1\right)\ln\ln\frac{1}{x}}{\,(1+x^% {2})^{3}\,}\,dx=\int\limits_{1}^{\infty}\frac{\left(x^{4}-6x^{2}+1\right)\ln% \ln{x}}{\,(1+x^{2})^{3}\,}\,dx=\frac{2\,\mathrm{G}}{\pi}
  13. 0 1 x ( x 4 - 4 x 2 + 1 ) ln ln 1 x ( 1 + x 2 ) 4 d x = 1 x ( x 4 - 4 x 2 + 1 ) ln ln x ( 1 + x 2 ) 4 d x = 7 ζ ( 3 ) 8 π 2 \int\limits_{0}^{1}\frac{x\left(x^{4}-4x^{2}+1\right)\ln\ln\frac{1}{x}}{\,(1+x% ^{2})^{4}\,}\,dx=\int\limits_{1}^{\infty}\frac{x\left(x^{4}-4x^{2}+1\right)\ln% \ln{x}}{\,(1+x^{2})^{4}\,}\,dx=\frac{7\zeta(3)}{8\pi^{2}}
  14. 0 1 x ( x m n - x - m n ) 2 ln ln 1 x ( 1 - x 2 ) 2 d x = 1 x ( x m n - x - m n ) 2 ln ln x ( 1 - x 2 ) 2 d x = m π n l = 1 n - 1 sin 2 π m l n ln Γ ( l n ) - π m 2 n cot π m n ln π n - 1 2 ln ( 2 π sin m π n ) - γ 2 \begin{array}[]{ll}\displaystyle\int\limits_{0}^{1}\frac{x\!\left(x^{\frac{m}{% n}}-x^{-\frac{m}{n}}\right)^{\!2}\ln\ln\frac{1}{x}}{\,(1-x^{2})^{2}\,}\,dx=% \int\limits_{1}^{\infty}\frac{x\!\left(x^{\frac{m}{n}}-x^{-\frac{m}{n}}\right)% ^{\!2}\ln\ln{x}}{\,(1-x^{2})^{2}\,}\,dx=&\displaystyle\frac{\,m\pi\,}{\,n\,}% \sum_{l=1}^{n-1}\sin\dfrac{2\pi ml}{n}\cdot\ln\Gamma\!\left(\!\frac{l}{n}\!% \right)-\,\frac{\pi m}{\,2n\,}\cot\frac{\pi m}{n}\cdot\ln\pi n\\ &\displaystyle-\,\frac{\,1\,}{2}\ln\!\left(\!\frac{\,2\,}{\pi}\sin\frac{\,m\pi% \,}{n}\!\right)-\,\frac{\gamma}{2}\end{array}
  15. 0 1 x 2 ( x m n + x - m n ) ln ln 1 x ( 1 + x 2 ) 3 d x = 1 x 2 ( x m n + x - m n ) ln ln x ( 1 + x 2 ) 3 d x = - π ( n 2 - m 2 ) 8 n 2 l = 0 2 n - 1 ( - 1 ) l cos ( 2 l + 1 ) m π 2 n ln Γ ( 2 l + 1 4 n ) + m 8 n 2 l = 0 2 n - 1 ( - 1 ) l sin ( 2 l + 1 ) m π 2 n Ψ ( 2 l + 1 4 n ) - 1 32 π n 2 l = 0 2 n - 1 ( - 1 ) l cos ( 2 l + 1 ) m π 2 n Ψ 1 ( 2 l + 1 4 n ) + π ( n 2 - m 2 ) 16 n 2 sec m π 2 n ln 2 π n \begin{array}[]{l}\displaystyle\int\limits_{0}^{1}\frac{x^{2}\!\left(x^{\frac{% m}{n}}+x^{-\frac{m}{n}}\right)\ln\ln\frac{1}{x}}{\,(1+x^{2})^{3}\,}\,dx=\int% \limits_{1}^{\infty}\frac{x^{2}\!\left(x^{\frac{m}{n}}+x^{-\frac{m}{n}}\right)% \ln\ln{x}}{\,(1+x^{2})^{3}\,}\,dx=-\frac{\,\pi\left(n^{2}-m^{2}\right)\,}{8n^{% 2}}\!\sum_{l=0}^{2n-1}\!(-1)^{l}\cos\dfrac{(2l+1)m\pi}{2n}\cdot\ln\Gamma\!% \left(\!\frac{2l+1}{4n}\right)\\ \displaystyle\,\,+\frac{\,m\,}{\,8n^{2}\,}\!\sum_{l=0}^{2n-1}\!(-1)^{l}\sin% \dfrac{(2l+1)m\pi}{2n}\cdot\Psi\!\left(\!\frac{2l+1}{4n}\right)-\frac{\,1\,}{% \,32\pi n^{2}\,}\!\sum_{l=0}^{2n-1}(-1)^{l}\cos\dfrac{(2l+1)m\pi}{2n}\cdot\Psi% _{1}\!\left(\!\frac{2l+1}{4n}\right)+\,\frac{\,\pi(n^{2}-m^{2})\,}{16n^{2}}% \sec\dfrac{m\pi}{2n}\cdot\ln 2\pi n\end{array}
  16. 0 1 x ln ln 1 x 1 + 4 x 2 + x 4 d x = 1 x ln ln x 1 + 4 x 2 + x 4 d x = π 2 3 Im [ ln Γ ( 1 2 - ln ( 2 + 3 ) 2 π i ) ] + ln ( 2 + 3 ) 4 3 ln π \int\limits_{0}^{1}\!\frac{x\ln\ln\frac{1}{x}}{1+4x^{2}+x^{4}}\,dx=\int\limits% _{1}^{\infty}\!\frac{x\ln\ln{x}}{1+4x^{2}+x^{4}}\,dx=\frac{\,\pi\,}{\,2\sqrt{3% \,}\,}\mathrm{Im}\!\left[\ln\Gamma\!\left(\!\frac{1}{2}-\frac{\ln(2+\sqrt{3\,}% )}{2\pi i}\right)\!\right]+\,\frac{\ln(2+\sqrt{3\,})}{\,4\sqrt{3\,}\,}\ln\pi
  17. 0 1 x ln ln 1 x x 4 - 2 x 2 cosh 2 + 1 d x = 1 x ln ln x x 4 - 2 x 2 cosh 2 + 1 d x = - π 2 sinh 2 Im [ ln Γ ( i 2 π ) - ln Γ ( 1 2 - i 2 π ) ] - π 2 8 sinh 2 - ln 2 π 2 sinh 2 \int\limits_{0}^{1}\!\frac{\,x\ln\ln\frac{1}{x}\,}{\,x^{4}-2x^{2}\cosh{2}+1\,}% \,dx=\int\limits_{1}^{\infty}\!\frac{\,x\ln\ln{x}\,}{\,x^{4}-2x^{2}\cosh{2}+1% \,}\,dx=-\frac{\,\pi\,}{2\,\sinh{2}\,}\mathrm{Im}\!\left[\ln\Gamma\!\left(\!% \frac{i}{2\pi}\right)-\ln\Gamma\!\left(\!\frac{1}{2}-\frac{i}{2\pi}\right)\!% \right]-\frac{\,\pi^{2}}{8\,\sinh{2}\,}-\frac{\,\ln 2\pi\,}{2\,\sinh{2}\,}
  18. n = 0 ( - 1 ) n ln ( 2 n + 1 ) 2 n + 1 = π 4 ( ln π - γ ) - π ln Γ ( 3 4 ) \sum_{n=0}^{\infty}(-1)^{n}\frac{\ln(2n+1)}{2n+1}\,=\,\frac{\pi}{4}\big(\ln\pi% -\gamma)-\pi\ln\Gamma\left(\frac{3}{4}\right)
  19. n = 1 ( - 1 ) n - 1 sin a n ln n n = π ln { π 1 2 - a 2 π Γ ( 1 2 + a 2 π ) } - a 2 ( γ + ln 2 ) - π 2 ln cos a 2 , - π < a < π . \sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin an\cdot\ln{n}}{n}\,=\,\pi\ln\left\{% \frac{\pi^{\frac{1}{2}-\frac{a}{2\pi}}}{\Gamma\left(\displaystyle\frac{1}{2}+% \frac{a}{2\pi}\right)}\right\}-\frac{a}{2}\big(\gamma+\ln 2\big)-\frac{\pi}{2}% \ln\cos\frac{a}{2}\,,\qquad-\pi<a<\pi.
  20. 1 π n = 1 sin 2 π n x ln n n = ln Γ ( x ) - 1 2 ln ( 2 π ) + 1 2 ln ( 2 sin π x ) - 1 2 ( γ + ln 2 π ) ( 1 - 2 x ) , 0 < x < 1 , \frac{1}{\pi}\sum_{n=1}^{\infty}\frac{\sin 2\pi nx\cdot\ln{n}}{n}=\ln\Gamma(x)% -\frac{1}{2}\ln(2\pi)+\frac{1}{2}\ln(2\sin\pi x)-\frac{1}{2}(\gamma+\ln 2\pi)(% 1-2x)\,,\qquad 0<x<1,
  21. L ( s ) n = 0 ( - 1 ) n ( 2 n + 1 ) s L ( 1 - s ) = L ( s ) Γ ( s ) 2 s π - s sin π s 2 , L(s)\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}\qquad\qquad L(1-s)=L(% s)\Gamma(s)2^{s}\pi^{-s}\sin\frac{\pi s}{2},
  22. M ( s ) 2 3 n = 1 ( - 1 ) n + 1 n s sin π n 3 M ( 1 - s ) = 2 3 M ( s ) Γ ( s ) 3 s ( 2 π ) - s sin π s 2 , M(s)\equiv\frac{2}{\sqrt{3}}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{s}}\sin% \frac{\pi n}{3}\qquad\qquad M(1-s)=\displaystyle\frac{2}{\sqrt{3}}\,M(s)\Gamma% (s)3^{s}(2\pi)^{-s}\sin\frac{\pi s}{2},
  23. γ 1 ( m n ) - γ 1 ( 1 - m n ) = 2 π l = 1 n - 1 sin 2 π m l n ln Γ ( l n ) - π ( γ + ln 2 π n ) cot m π n \gamma_{1}\biggl(\frac{m}{n}\biggr)-\gamma_{1}\biggl(1-\frac{m}{n}\biggr)=2\pi% \sum_{l=1}^{n-1}\sin\frac{2\pi ml}{n}\cdot\ln\Gamma\biggl(\frac{l}{n}\biggr)-% \pi(\gamma+\ln 2\pi n)\cot\frac{m\pi}{n}

Carleman's_condition.html

  1. m n = - + x n d μ ( x ) , n = 0 , 1 , 2 , m_{n}=\int_{-\infty}^{+\infty}x^{n}\,d\mu(x)~{},\quad n=0,1,2,\cdots
  2. n = 1 m 2 n - 1 2 n = + , \sum_{n=1}^{\infty}m_{2n}^{-\frac{1}{2n}}=+\infty,
  3. n = 1 m n - 1 2 n = + . \sum_{n=1}^{\infty}m_{n}^{-\frac{1}{2n}}=+\infty.\,

Carleman's_inequality.html

  1. n = 1 ( a 1 a 2 a n ) 1 / n e n = 1 a n . \sum_{n=1}^{\infty}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}\leq e\sum_{n=1}^{% \infty}a_{n}.
  2. 0 exp { 1 x 0 x ln f ( t ) d t } d x e 0 f ( x ) d x \int_{0}^{\infty}\exp\left\{\frac{1}{x}\int_{0}^{x}\ln f(t)dt\right\}dx\leq e% \int_{0}^{\infty}f(x)dx
  3. 1 a 1 , 2 a 2 , , n a n 1\cdot a_{1},2\cdot a_{2},\dots,n\cdot a_{n}
  4. MG ( a 1 , , a n ) = MG ( 1 a 1 , 2 a 2 , , n a n ) ( n ! ) - 1 / n MA ( 1 a 1 , 2 a 2 , , n a n ) ( n ! ) - 1 / n \mathrm{MG}(a_{1},\dots,a_{n})=\mathrm{MG}(1a_{1},2a_{2},\dots,na_{n})(n!)^{-1% /n}\leq\mathrm{MA}(1a_{1},2a_{2},\dots,na_{n})(n!)^{-1/n}\,
  5. n ! 2 π n n n e - n n!\geq\sqrt{2\pi n}\,n^{n}e^{-n}
  6. n + 1 n+1
  7. ( n ! ) - 1 / n e n + 1 (n!)^{-1/n}\leq\frac{e}{n+1}
  8. n 1. n\geq 1.
  9. M G ( a 1 , , a n ) e n ( n + 1 ) 1 k n k a k , MG(a_{1},\dots,a_{n})\leq\frac{e}{n(n+1)}\,\sum_{1\leq k\leq n}ka_{k}\,,
  10. n 1 M G ( a 1 , , a n ) e k 1 ( n k 1 n ( n + 1 ) ) k a k = e k 1 a k , \sum_{n\geq 1}MG(a_{1},\dots,a_{n})\leq\,e\,\sum_{k\geq 1}\bigg(\sum_{n\geq k}% \frac{1}{n(n+1)}\bigg)\,ka_{k}=\,e\,\sum_{k\geq 1}\,a_{k}\,,
  11. n n
  12. a k = C / k a_{k}=C/k
  13. k = 1 , , n k=1,\dots,n
  14. a n a_{n}
  15. n = 1 ( a 1 + a 2 + + a n n ) p ( p p - 1 ) p n = 1 a n p \sum_{n=1}^{\infty}\left(\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)^{p}\leq% \left(\frac{p}{p-1}\right)^{p}\sum_{n=1}^{\infty}a_{n}^{p}

Carrier_lifetime.html

  1. 1 τ n ( N ) = A + B N + C N 2 \frac{1}{\tau_{n}(N)}=A+BN+CN^{2}
  2. τ n ( N ) \tau_{n}(N)

Carry_operator.html

  1. ( G 1 , P 1 ) (G_{1},P_{1})
  2. ( G 2 , P 2 ) = ( G 1 G 2 P 1 , P 2 P 1 ) (G_{2},P_{2})=(G_{1}G_{2}P_{1},P_{2}P_{1})

Cartan–Hadamard_theorem.html

  1. d ( z , γ ( 1 / 2 ) ) 2 1 2 d ( z , γ ( 0 ) ) 2 + 1 2 d ( z , γ ( 1 ) ) 2 - 1 4 d ( γ ( 0 ) , γ ( 1 ) ) 2 . d(z,\gamma(1/2))^{2}\leq\frac{1}{2}d(z,\gamma(0))^{2}+\frac{1}{2}d(z,\gamma(1)% )^{2}-\frac{1}{4}d(\gamma(0),\gamma(1))^{2}.
  2. t d ( a ( t ) , b ( t ) ) t\mapsto d(a(t),b(t))

Cascade_algorithm.html

  1. φ ( k + 1 ) ( t ) = n = 0 N - 1 h [ n ] 2 φ ( k ) ( 2 t - n ) \varphi^{(k+1)}(t)=\sum_{n=0}^{N-1}h[n]\sqrt{2}\varphi^{(k)}(2t-n)
  2. Φ ( k + 1 ) ( ω ) = 1 2 H ( ω 2 ) Φ ( k ) ( ω 2 ) \Phi^{(k+1)}(\omega)=\frac{1}{\sqrt{2}}H\left(\frac{\omega}{2}\right)\Phi^{(k)% }\left(\frac{\omega}{2}\right)
  3. Φ ( ) ( ω ) = k = 1 1 2 H ( ω 2 k ) Φ ( ) ( 0 ) . \Phi^{(\infty)}(\omega)=\prod_{k=1}^{\infty}\frac{1}{\sqrt{2}}H\left(\frac{% \omega}{2^{k}}\right)\Phi^{(\infty)}(0).
  4. Φ ( ω ) = k = 1 1 2 H ( ω 2 k ) Φ ( ) ( 0 ) \Phi(\omega)=\prod_{k=1}^{\infty}\frac{1}{\sqrt{2}}H\left(\frac{\omega}{2^{k}}% \right)\Phi^{(\infty)}(0)
  5. ψ ( t ) = n = - g [ n ] 2 φ ( k ) ( 2 t - n ) . \psi(t)=\sum_{n=-\infty}^{\infty}g[n]{\sqrt{2}}\varphi^{(k)}(2t-n).

Casus_irreducibilis.html

  1. a x 3 + b x 2 + c x + d = 0. ax^{3}+bx^{2}+cx+d=0.\,
  2. D = 18 a b c d - 4 b 3 d + b 2 c 2 - 4 a c 3 - 27 a 2 d 2 . D=18abcd-4b^{3}d+b^{2}c^{2}-4ac^{3}-27a^{2}d^{2}.\,
  3. F \sub F ( Δ ) \sub F ( Δ , α 1 p 1 ) \sub \sub K \sub K ( α 3 ) F\sub F(\sqrt{\Delta})\sub F(\sqrt{\Delta},\sqrt[p_{1}]{\alpha_{1}})\sub\cdots% \sub K\sub K(\sqrt[3]{\alpha})
  4. a x 3 + b x 2 + c x + d = 0 ax^{3}+bx^{2}+cx+d=0\,
  5. a a
  6. x = t - b 3 a x=t-\frac{b}{3a}
  7. t 3 + p t + q = 0 t^{3}+pt+q=0
  8. p = 3 a c - b 2 3 a 2 p=\frac{3ac-b^{2}}{3a^{2}}
  9. q = 2 b 3 - 9 a b c + 27 a 2 d 27 a 3 . q=\frac{2b^{3}-9abc+27a^{2}d}{27a^{3}}.
  10. t k = ω k - q 2 + q 2 4 + p 3 27 3 + ω k 2 - q 2 - q 2 4 + p 3 27 3 t_{k}=\omega_{k}\sqrt[3]{-{q\over 2}+\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}+% \omega_{k}^{2}\sqrt[3]{-{q\over 2}-\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}
  11. ω k \omega_{k}
  12. ω 1 = 1 \omega_{1}=1
  13. ω 2 = - 1 2 + 3 2 i \omega_{2}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i
  14. ω 3 = - 1 2 - 3 2 i \omega_{3}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i
  15. q 2 4 + p 3 27 < 0 {q^{2}\over 4}+{p^{3}\over 27}<0
  16. α + β i \alpha+\beta i
  17. α \alpha
  18. β \beta
  19. t 3 + p t + q = 0 t^{3}+pt+q=0
  20. t k = 2 - p 3 cos ( 1 3 arccos ( 3 q 2 p - 3 p ) - k 2 π 3 ) for k = 0 , 1 , 2 . t_{k}=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt% {\frac{-3}{p}}\right)-k\frac{2\pi}{3}\right)\quad\,\text{for}\quad k=0,1,2\,.
  21. q 2 4 + p 3 27 < 0 {q^{2}\over 4}+{p^{3}\over 27}<0
  22. θ \theta
  23. 4 x 3 - 3 x - cos ( θ ) = 0. 4x^{3}-3x-\cos(\theta)=0.
  24. θ \theta
  25. - 4 y 3 + 3 y - sin ( θ ) = 0. -4y^{3}+3y-\sin(\theta)=0.
  26. cos ( θ / 3 ) \cos(\theta/3)
  27. sin ( θ / 3 ) \sin(\theta/3)
  28. θ / 3 \theta/3
  29. cos ( θ / 3 ) \cos(\theta/3)
  30. sin ( θ / 3 ) \sin(\theta/3)
  31. θ / 3 \theta/3
  32. θ \theta
  33. K R K\subseteq R

Catalycity.html

  1. X 2 2 X X_{2}\rightarrow 2X
  2. 2 X X 2 2X\rightarrow X_{2}
  3. X ( g ) + ( s ) X ( s ) X(g)+(s)\rightarrow X(s)
  4. 2 X ( s ) X 2 ( s ) + ( s ) 2X(s)\rightarrow X_{2}(s)+(s)
  5. X 2 ( s ) X ( g ) + ( s ) X_{2}(s)\rightarrow X(g)+(s)
  6. γ = r e c \gamma=\frac{\mathcal{M}_{rec}}{\mathcal{M}^{\downarrow}}

Categorical_distribution.html

  1. Cov ( [ x = i ] , [ x = j ] ) = - p i p j ( i j ) \textstyle{\mathrm{Cov}}([x=i],[x=j])=-p_{i}p_{j}~{}~{}(i\neq j)
  2. i = 1 k p i e t i \sum_{i=1}^{k}p_{i}e^{t_{i}}
  3. j = 1 k p j e i t j \sum_{j=1}^{k}p_{j}e^{it_{j}}
  4. i 2 = - 1 i^{2}=-1
  5. i = 1 k p i z i for ( z 1 , , z k ) k \sum_{i=1}^{k}p_{i}z_{i}\,\text{ for }(z_{1},\ldots,z_{k})\in\mathbb{C}^{k}
  6. Dir ( s y m b o l α + ( [ x = i ] , , [ x = k ] ) ) \mathrm{Dir}\left(symbol\alpha+([x=i],\dots,[x=k])\right)
  7. f ( x = i | s y m b o l p ) = p i , f(x=i|symbol{p})=p_{i},
  8. s y m b o l p = ( p 1 , , p k ) symbol{p}=(p_{1},...,p_{k})
  9. p i p_{i}
  10. i = 1 k p i = 1 \textstyle{\sum_{i=1}^{k}p_{i}=1}
  11. f ( x | s y m b o l p ) = i = 1 k p i [ x = i ] , f(x|symbol{p})=\prod_{i=1}^{k}p_{i}^{[x=i]},
  12. [ x = i ] [x=i]
  13. x = i x=i
  14. f ( 𝐱 | s y m b o l p ) = i = 1 k p i x i , f(\mathbf{x}|symbol{p})=\prod_{i=1}^{k}p_{i}^{x_{i}},
  15. p i p_{i}
  16. i p i = 1 \textstyle{\sum_{i}p_{i}=1}
  17. k = 3 k=3
  18. p 1 + p 2 + p 3 = 1 p_{1}+p_{2}+p_{3}=1
  19. p i = P ( X = i ) p_{i}=P(X=i)
  20. i p i = 1 \textstyle{\sum_{i}p_{i}=1}
  21. ( k - 1 ) (k-1)
  22. p 1 + p 2 = 1 , 0 p 1 , p 2 1. p_{1}+p_{2}=1,0\leq p_{1},p_{2}\leq 1.
  23. 𝔼 [ 𝐱 ] = s y m b o l p \mathbb{E}\left[\mathbf{x}\right]=symbol{p}
  24. s y m b o l X symbol{X}
  25. Y i = I ( s y m b o l X = i ) , Y_{i}=I(symbol{X}=i),
  26. n = 1 n=1
  27. n n
  28. s y m b o l p symbol{p}
  29. n n
  30. s y m b o l p . symbol{p}.
  31. [ x = i ] [x=i]
  32. δ x i , \delta_{xi},
  33. p i . p_{i}.
  34. s y m b o l α = ( α 1 , , α K ) = concentration hyperparameter 𝐩 \midsymbol α = ( p 1 , , p K ) Dir ( K , s y m b o l α ) 𝕏 𝐩 = ( x 1 , , x K ) Cat ( K , 𝐩 ) \begin{array}[]{lclcl}symbol\alpha&=&(\alpha_{1},\ldots,\alpha_{K})&=&\,\text{% concentration hyperparameter}\\ \mathbf{p}\midsymbol\alpha&=&(p_{1},\ldots,p_{K})&\sim&\operatorname{Dir}(K,% symbol\alpha)\\ \mathbb{X}\mid\mathbf{p}&=&(x_{1},\ldots,x_{K})&\sim&\operatorname{Cat}(K,% \mathbf{p})\end{array}
  35. 𝐜 = ( c 1 , , c K ) = number of occurrences of category i = j = 1 N [ x j = i ] 𝐩 𝕏 , s y m b o l α Dir ( K , 𝐜 + s y m b o l α ) = Dir ( K , c 1 + α 1 , , c K + α K ) \begin{array}[]{lclcl}\mathbf{c}&=&(c_{1},\ldots,c_{K})&=&\,\text{number of % occurrences of category }i=\sum_{j=1}^{N}[x_{j}=i]\\ \mathbf{p}\mid\mathbb{X},symbol\alpha&\sim&\operatorname{Dir}(K,\mathbf{c}+% symbol\alpha)&=&\operatorname{Dir}(K,c_{1}+\alpha_{1},\ldots,c_{K}+\alpha_{K})% \end{array}
  36. 𝔼 [ p i 𝕏 , s y m b o l α ] = c i + α i N + k α k \mathbb{E}[p_{i}\mid\mathbb{X},symbol\alpha]=\frac{c_{i}+\alpha_{i}}{N+\sum_{k% }\alpha_{k}}
  37. α i \alpha_{i}
  38. α i - 1 \alpha_{i}-1
  39. i i
  40. c i + α i c_{i}+\alpha_{i}
  41. c i + α i - 1 c_{i}+\alpha_{i}-1
  42. s y m b o l α = ( 1 , 1 , ) symbol\alpha=(1,1,\ldots)
  43. - 1 \dots-1
  44. α i \alpha_{i}
  45. arg max 𝐩 p ( 𝐩 | 𝕏 ) = α i + c i - 1 i ( α i + c i - 1 ) , i α i + c i > 1 \arg\max_{\mathbf{p}}p(\mathbf{p}|\mathbb{X})=\frac{\alpha_{i}+c_{i}-1}{\sum_{% i}(\alpha_{i}+c_{i}-1)},\qquad\forall i\;\alpha_{i}+c_{i}>1
  46. i α i + c i > 1 \forall i\;\alpha_{i}+c_{i}>1
  47. α i > 1 \alpha_{i}>1
  48. p ( 𝕏 \midsymbol α ) = 𝐩 p ( 𝕏 𝐩 ) p ( 𝐩 \midsymbol α ) d 𝐩 = Γ ( k α k ) Γ ( N + k α k ) k = 1 K Γ ( c k + α k ) Γ ( α k ) \begin{aligned}\displaystyle p(\mathbb{X}\midsymbol{\alpha})&\displaystyle=% \int_{\mathbf{p}}p(\mathbb{X}\mid\mathbf{p})p(\mathbf{p}\midsymbol{\alpha})% \textrm{d}\mathbf{p}\\ &\displaystyle=\frac{\Gamma\left(\sum_{k}\alpha_{k}\right)}{\Gamma\left(N+\sum% _{k}\alpha_{k}\right)}\prod_{k=1}^{K}\frac{\Gamma(c_{k}+\alpha_{k})}{\Gamma(% \alpha_{k})}\end{aligned}
  49. x ~ \tilde{x}
  50. 𝕏 \mathbb{X}
  51. p ( x ~ = i 𝕏 , s y m b o l α ) = 𝐩 p ( x ~ = i 𝐩 ) p ( 𝐩 𝕏 , s y m b o l α ) d 𝐩 = c i + α i N + k α k = 𝔼 [ p i 𝕏 , s y m b o l α ] c i + α i . \begin{aligned}\displaystyle p(\tilde{x}=i\mid\mathbb{X},symbol{\alpha})&% \displaystyle=\int_{\mathbf{p}}p(\tilde{x}=i\mid\mathbf{p})\,p(\mathbf{p}\mid% \mathbb{X},symbol{\alpha})\,\textrm{d}\mathbf{p}\\ &\displaystyle=\,\frac{c_{i}+\alpha_{i}}{N+\sum_{k}\alpha_{k}}\\ &\displaystyle=\,\mathbb{E}[p_{i}\mid\mathbb{X},symbol\alpha]\\ &\displaystyle\propto\,c_{i}+\alpha_{i}.\\ \end{aligned}
  52. p ( x ~ = i 𝕏 , s y m b o l α ) \displaystyle p(\tilde{x}=i\mid\mathbb{X},symbol{\alpha})
  53. 𝕏 \mathbb{X}
  54. x n x_{n}
  55. 𝕏 ( - n ) \mathbb{X}^{(-n)}
  56. p ( x n = i 𝕏 ( - n ) , s y m b o l α ) = c i ( - n ) + α i N - 1 + i α i c i ( - n ) + α i \begin{aligned}\displaystyle p(x_{n}=i\mid\mathbb{X}^{(-n)},symbol{\alpha})&% \displaystyle=\,\frac{c_{i}^{(-n)}+\alpha_{i}}{N-1+\sum_{i}\alpha_{i}}&% \displaystyle\propto\,c_{i}^{(-n)}+\alpha_{i}\\ \end{aligned}
  57. c i ( - n ) c_{i}^{(-n)}
  58. p 1 , , p k p_{1},\ldots,p_{k}
  59. k \mathbb{R}^{k}
  60. γ i = log p i + α \gamma_{i}=\log p_{i}+\alpha
  61. α \alpha
  62. p 1 , , p k p_{1},\ldots,p_{k}
  63. g 1 , , g k g_{1},\ldots,g_{k}
  64. c = arg max i γ i + g i c=\arg\max_{i}\gamma_{i}+g_{i}
  65. u i u_{i}
  66. g i = - log ( - log u i ) g_{i}=-\log(-\log u_{i})

Categories_of_New_Testament_manuscripts.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 \mathfrak{P}
  25. 𝔓 \mathfrak{P}
  26. 𝔓 \mathfrak{P}
  27. 𝔓 \mathfrak{P}
  28. 𝔓 \mathfrak{P}
  29. 𝔓 \mathfrak{P}
  30. 𝔓 \mathfrak{P}
  31. 𝔓 \mathfrak{P}
  32. 𝔓 \mathfrak{P}
  33. 𝔓 \mathfrak{P}
  34. 𝔓 \mathfrak{P}
  35. 𝔓 \mathfrak{P}
  36. 𝔓 \mathfrak{P}
  37. 𝔓 \mathfrak{P}
  38. 𝔓 \mathfrak{P}
  39. 𝔓 \mathfrak{P}
  40. 𝔓 \mathfrak{P}
  41. 𝔓 \mathfrak{P}
  42. 𝔓 \mathfrak{P}
  43. 𝔓 \mathfrak{P}
  44. 𝔓 \mathfrak{P}
  45. 𝔓 \mathfrak{P}
  46. 𝔓 \mathfrak{P}
  47. 𝔓 \mathfrak{P}
  48. 𝔓 \mathfrak{P}
  49. 𝔓 \mathfrak{P}
  50. 𝔓 \mathfrak{P}
  51. 𝔓 \mathfrak{P}
  52. 𝔓 \mathfrak{P}
  53. 𝔓 \mathfrak{P}
  54. 𝔓 \mathfrak{P}
  55. 𝔓 \mathfrak{P}
  56. 𝔓 \mathfrak{P}
  57. 𝔓 \mathfrak{P}
  58. 𝔓 \mathfrak{P}
  59. 𝔓 \mathfrak{P}
  60. 𝔓 \mathfrak{P}
  61. 𝔓 \mathfrak{P}
  62. 𝔓 \mathfrak{P}
  63. 𝔓 \mathfrak{P}
  64. 𝔓 \mathfrak{P}
  65. 𝔓 \mathfrak{P}
  66. 𝔓 \mathfrak{P}
  67. 𝔓 \mathfrak{P}
  68. 𝔓 \mathfrak{P}
  69. 𝔓 \mathfrak{P}
  70. 𝔓 \mathfrak{P}
  71. 𝔓 \mathfrak{P}
  72. 𝔓 \mathfrak{P}
  73. 𝔓 \mathfrak{P}
  74. 𝔓 \mathfrak{P}
  75. 𝔓 \mathfrak{P}
  76. 𝔓 \mathfrak{P}
  77. 𝔓 \mathfrak{P}
  78. 𝔓 \mathfrak{P}
  79. 𝔓 \mathfrak{P}
  80. 𝔓 \mathfrak{P}
  81. 𝔓 \mathfrak{P}
  82. 𝔓 \mathfrak{P}
  83. 𝔓 \mathfrak{P}
  84. 𝔓 \mathfrak{P}
  85. 𝔓 \mathfrak{P}
  86. 𝔓 \mathfrak{P}
  87. 𝔓 \mathfrak{P}
  88. 𝔓 \mathfrak{P}
  89. 𝔓 \mathfrak{P}
  90. 𝔓 \mathfrak{P}
  91. 𝔓 \mathfrak{P}
  92. 𝔓 \mathfrak{P}

Category_utility.html

  1. C U ( C , F ) = 1 p c j C p ( c j ) [ f i F k = 1 m p ( f i k | c j ) 2 - f i F k = 1 m p ( f i k ) 2 ] CU(C,F)=\tfrac{1}{p}\sum_{c_{j}\in C}p(c_{j})\left[\sum_{f_{i}\in F}\sum_{k=1}% ^{m}p(f_{ik}|c_{j})^{2}-\sum_{f_{i}\in F}\sum_{k=1}^{m}p(f_{ik})^{2}\right]
  2. F = { f i } , i = 1 n F=\{f_{i}\},\ i=1\ldots n
  3. n n
  4. m m
  5. C = { c j } j = 1 p C=\{c_{j}\}\ j=1\ldots p
  6. p p
  7. p ( f i k ) p(f_{ik})
  8. f i f_{i}
  9. k k
  10. p ( f i k | c j ) p(f_{ik}|c_{j})
  11. f i f_{i}
  12. k k
  13. c j c_{j}
  14. 1 p \textstyle\tfrac{1}{p}
  15. p ( c j ) f i F k = 1 m p ( f i k | c j ) 2 \textstyle p(c_{j})\sum_{f_{i}\in F}\sum_{k=1}^{m}p(f_{ik}|c_{j})^{2}
  16. p ( c j ) f i F k = 1 m p ( f i k ) 2 \textstyle p(c_{j})\sum_{f_{i}\in F}\sum_{k=1}^{m}p(f_{ik})^{2}
  17. n n
  18. F = { f i } , i = 1 n F=\{f_{i}\},\ i=1\ldots n
  19. C = { c , c ¯ } C=\{c,\bar{c}\}
  20. C U ( C , F ) = [ p ( c ) i = 1 n p ( f i | c ) log p ( f i | c ) + p ( c ¯ ) i = 1 n p ( f i | c ¯ ) log p ( f i | c ¯ ) ] - i = 1 n p ( f i ) log p ( f i ) CU(C,F)=\left[p(c)\sum_{i=1}^{n}p(f_{i}|c)\log p(f_{i}|c)+p(\bar{c})\sum_{i=1}% ^{n}p(f_{i}|\bar{c})\log p(f_{i}|\bar{c})\right]-\sum_{i=1}^{n}p(f_{i})\log p(% f_{i})
  21. p ( c ) p(c)
  22. c c
  23. p ( f i | c ) p(f_{i}|c)
  24. f i f_{i}
  25. c c
  26. p ( f i | c ¯ ) p(f_{i}|\bar{c})
  27. f i f_{i}
  28. c ¯ \bar{c}
  29. p ( f i ) p(f_{i})
  30. f i f_{i}
  31. p ( c ) i = 1 n p ( f i | c ) log p ( f i | c ) p(c)\textstyle\sum_{i=1}^{n}p(f_{i}|c)\log p(f_{i}|c)
  32. c c
  33. p ( c ¯ ) i = 1 n p ( f i | c ¯ ) log p ( f i | c ¯ ) p(\bar{c})\textstyle\sum_{i=1}^{n}p(f_{i}|\bar{c})\log p(f_{i}|\bar{c})
  34. c ¯ \bar{c}
  35. i = 1 n p ( f i ) log p ( f i ) \textstyle\sum_{i=1}^{n}p(f_{i})\log p(f_{i})
  36. n n
  37. F = { f i } , i = 1 n F=\{f_{i}\},\ i=1\ldots n
  38. m m
  39. m m
  40. m = 2 m=2
  41. m m
  42. F F
  43. F a F_{a}
  44. m n m^{n}
  45. v i , i = 1 m n v_{i},\ i=1\ldots m^{n}
  46. F \otimes F
  47. p ( F a = v i ) p(F_{a}=v_{i})
  48. p ( v i ) p(v_{i})
  49. F a F_{a}
  50. v i v_{i}
  51. F a F_{a}
  52. C C
  53. p p
  54. p p
  55. p = 2 p=2
  56. I ( F a ; C ) I(F_{a};C)
  57. F a F_{a}
  58. C C
  59. I ( F a ; C ) = v i F a c j C p ( v i , c j ) log p ( v i , c j ) p ( v i ) p ( c j ) I(F_{a};C)=\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i},c_{j})\log\frac{p(v_{% i},c_{j})}{p(v_{i})\,p(c_{j})}
  60. p ( v i ) p(v_{i})
  61. F a F_{a}
  62. v i v_{i}
  63. p ( c j ) p(c_{j})
  64. C C
  65. c j c_{j}
  66. p ( v i , c j ) p(v_{i},c_{j})
  67. F a F_{a}
  68. C C
  69. I ( F a ; C ) = v i F a c j C p ( v i , c j ) log p ( v i | c j ) p ( v i ) = v i F a c j C p ( v i | c j ) p ( c j ) [ log p ( v i | c j ) - log p ( v i ) ] = v i F a c j C p ( v i | c j ) p ( c j ) log p ( v i | c j ) - v i F a c j C p ( v i | c j ) p ( c j ) log p ( v i ) = v i F a c j C p ( v i | c j ) p ( c j ) log p ( v i | c j ) - v i F a c j C p ( v i , c j ) log p ( v i ) = v i F a c j C p ( v i | c j ) p ( c j ) log p ( v i | c j ) - v i F a log p ( v i ) c j C p ( v i , c j ) = \color B l u e v i F a c j C p ( v i | c j ) p ( c j ) log p ( v i | c j ) - v i F a p ( v i ) log p ( v i ) \begin{aligned}\displaystyle I(F_{a};C)&\displaystyle=\sum_{v_{i}\in F_{a}}% \sum_{c_{j}\in C}p(v_{i},c_{j})\log\frac{p(v_{i}|c_{j})}{p(v_{i})}\\ &\displaystyle=\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})% \left[\log p(v_{i}|c_{j})-\log p(v_{i})\right]\\ &\displaystyle=\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})% \log p(v_{i}|c_{j})-\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i}|c_{j})p(c_{j% })\log p(v_{i})\\ &\displaystyle=\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})% \log p(v_{i}|c_{j})-\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i},c_{j})\log p% (v_{i})\\ &\displaystyle=\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})% \log p(v_{i}|c_{j})-\sum_{v_{i}\in F_{a}}\log p(v_{i})\sum_{c_{j}\in C}p(v_{i}% ,c_{j})\\ &\displaystyle={\color{Blue}\sum_{v_{i}\in F_{a}}\sum_{c_{j}\in C}p(v_{i}|c_{j% })p(c_{j})\log p(v_{i}|c_{j})-\sum_{v_{i}\in F_{a}}p(v_{i})\log p(v_{i})}\\ \end{aligned}
  70. C = { c , c ¯ } C=\{c,\bar{c}\}
  71. C U ( C , F ) = f i F c j C p ( f i | c j ) p ( c j ) log p ( f i | c j ) - f i F p ( f i ) log p ( f i ) CU(C,F)=\sum_{f_{i}\in F}\sum_{c_{j}\in C}p(f_{i}|c_{j})p(c_{j})\log p(f_{i}|c% _{j})-\sum_{f_{i}\in F}p(f_{i})\log p(f_{i})
  72. f i F \textstyle\sum_{f_{i}\in F}
  73. F = { f i } , i = 1 n F=\{f_{i}\},\ i=1\ldots n
  74. v i F a \textstyle\sum_{v_{i}\in F_{a}}
  75. m n m^{n}
  76. F a F_{a}
  77. { f i } \{f_{i}\}
  78. p ( f i ¯ ) p(\bar{f_{i}})
  79. f i f_{i}
  80. c j c_{j}
  81. p ( c j | f i ) p(c_{j}|f_{i})
  82. p ( c j | f i ) - p ( c j ) p(c_{j}|f_{i})-p(c_{j})
  83. p ( f i | c j ) p(f_{i}|c_{j})
  84. p ( c j | f i ) p ( f i | c j ) p(c_{j}|f_{i})p(f_{i}|c_{j})

Catenary_ring.html

  1. height ( P ) height ( p ) + tr.deg. A ( B ) - tr.deg. κ ( p ) ( κ ( P ) ) . \,\text{height}(P)\leq\,\text{height}(p)+\,\text{tr.deg.}_{A}(B)-\,\text{tr.% deg.}_{\kappa(p)}(\kappa(P)).
  2. B = A [ x 1 , , x n ] B=A[x_{1},\dots,x_{n}]

Cauchy's_theorem_(geometry).html

  1. 3 \mathbb{R}^{3}

Causal_sets.html

  1. C C
  2. \preceq
  3. x C x\in C
  4. x x x\preceq x
  5. x , y C x,y\in C
  6. x y x x = y x\preceq y\preceq x\implies x=y
  7. x , y , z C x,y,z\in C
  8. x y z x\preceq y\preceq z
  9. x z x\preceq z
  10. x , z C x,z\in C
  11. ( { y C | x y z } ) < (\{y\in C|x\preceq y\preceq z\})<\infty
  12. A A
  13. A A
  14. x y x\prec y
  15. x y x\preceq y
  16. x y x\neq y
  17. C C
  18. \preceq
  19. n n
  20. V V
  21. P ( n ) = ( ρ V ) n e - ρ V n ! P(n)=\frac{(\rho V)^{n}e^{-\rho V}}{n!}
  22. ρ \rho
  23. x , y C x,y\in C\,\!
  24. x y x\prec y
  25. z C z\in C\,\!
  26. x z y x\prec z\prec y
  27. x 0 , x 1 , , x n x_{0},x_{1},\ldots,x_{n}
  28. x i x i + 1 x_{i}\prec x_{i+1}
  29. i = 0 , , n - 1 i=0,\ldots,n-1
  30. n n
  31. x i , x i + 1 x_{i},x_{i+1}
  32. x y C x\preceq y\in C
  33. x 0 = x x_{0}=x\,\!
  34. x n = y x_{n}=y\,\!
  35. n n
  36. x x\,
  37. y y\,
  38. k k
  39. d d
  40. k k
  41. d d
  42. x x\,
  43. y y\,
  44. z z\,
  45. x z y x\prec z\prec y
  46. d d

Cayley's_mousetrap.html

  1. 1 1
  2. n n
  3. 1 , 2 , 3 , 1,2,3,...
  4. 1 1
  5. n + 1 n+1

Cebeci–Smith_model.html

  1. μ t \mu_{t}
  2. μ t = { μ t inner if y y crossover μ t outer if y > y crossover \mu_{t}=\begin{cases}{\mu_{t}}\text{inner}&\mbox{if }~{}y\leq y\text{crossover% }\\ {\mu_{t}}\text{outer}&\mbox{if }~{}y>y\text{crossover}\end{cases}
  3. y crossover y\text{crossover}
  4. μ t inner {\mu_{t}}\text{inner}
  5. μ t outer {\mu_{t}}\text{outer}
  6. μ t inner = ρ 2 [ ( U y ) 2 + ( V x ) 2 ] 1 / 2 {\mu_{t}}\text{inner}=\rho\ell^{2}\left[\left(\frac{\partial U}{\partial y}% \right)^{2}+\left(\frac{\partial V}{\partial x}\right)^{2}\right]^{1/2}
  7. = κ y ( 1 - e - y + / A + ) \ell=\kappa y\left(1-e^{-y^{+}/A^{+}}\right)
  8. κ \kappa
  9. A + = 26 [ 1 + y d P / d x ρ u τ 2 ] - 1 / 2 A^{+}=26\left[1+y\frac{dP/dx}{\rho u_{\tau}^{2}}\right]^{-1/2}
  10. μ t outer = α ρ U e δ v * F K {\mu_{t}}\text{outer}=\alpha\rho U_{e}\delta_{v}^{*}F_{K}
  11. α = 0.0168 \alpha=0.0168
  12. δ v * \delta_{v}^{*}
  13. δ v * = 0 δ ( 1 - U U e ) d y \delta_{v}^{*}=\int_{0}^{\delta}\left(1-\frac{U}{U_{e}}\right)\,dy
  14. F K = [ 1 + 5.5 ( y δ ) 6 ] - 1 F_{K}=\left[1+5.5\left(\frac{y}{\delta}\right)^{6}\right]^{-1}

Cell_lists.html

  1. r c r_{c}
  2. ( C α , C β ) (C_{\alpha},C_{\beta})
  3. p α C α p_{\alpha}\in C_{\alpha}
  4. p β C β p_{\beta}\in C_{\beta}
  5. r 2 = 𝐱 [ p α ] - 𝐱 [ p β ] 2 2 r^{2}=\|\mathbf{x}[p_{\alpha}]-\mathbf{x}[p_{\beta}]\|_{2}^{2}
  6. r 2 r c 2 r^{2}\leq r_{c}^{2}
  7. p α p_{\alpha}
  8. p β p_{\beta}
  9. r c r_{c}
  10. r c r_{c}
  11. N N
  12. m m
  13. N N
  14. N N
  15. c ¯ = N / m \overline{c}=N/m
  16. 𝒪 ( c ¯ 2 ) \mathcal{O}(\overline{c}^{2})
  17. N N
  18. 𝒪 ( N c ) 𝒪 ( N ) \mathcal{O}(Nc)\in\mathcal{O}(N)
  19. 𝒪 ( N 2 ) \mathcal{O}(N^{2})
  20. 𝐪 α β \mathbf{q}_{\alpha\beta}
  21. ( C α , C β ) (C_{\alpha},C_{\beta})
  22. p α C α p_{\alpha}\in C_{\alpha}
  23. p β C β p_{\beta}\in C_{\beta}
  24. r 2 = 𝐱 [ p α ] - 𝐱 [ p β ] - 𝐪 α β 2 2 r^{2}=\|\mathbf{x}[p_{\alpha}]-\mathbf{x}[p_{\beta}]-\mathbf{q}_{\alpha\beta}% \|^{2}_{2}
  25. 𝐪 α β \mathbf{q}_{\alpha\beta}
  26. 𝒪 ( N 2 ) \mathcal{O}(N^{2})
  27. 𝒪 ( N ) \mathcal{O}(N)
  28. r c r_{c}
  29. r c r_{c}
  30. r c r_{c}
  31. r c r_{c}
  32. C β C_{\beta}
  33. C α C_{\alpha}
  34. r c / 2 r_{c}/2

Cellular_waste_product.html

  1. \to

Centering_matrix.html

  1. C n = I n - 1 n 𝕆 C_{n}=I_{n}-\tfrac{1}{n}\mathbb{O}
  2. I n I_{n}\,
  3. 𝕆 \mathbb{O}
  4. C n = I n - 1 n 𝟏𝟏 C_{n}=I_{n}-\tfrac{1}{n}\mathbf{1}\mathbf{1}^{\top}
  5. 𝟏 \mathbf{1}
  6. \top
  7. C 1 = [ 0 ] C_{1}=\begin{bmatrix}0\end{bmatrix}
  8. C 2 = [ 1 0 0 1 ] - 1 2 [ 1 1 1 1 ] = [ 1 2 - 1 2 - 1 2 1 2 ] C_{2}=\left[\begin{array}[]{rrr}1&0\\ \\ 0&1\end{array}\right]-\frac{1}{2}\left[\begin{array}[]{rrr}1&1\\ \\ 1&1\end{array}\right]=\left[\begin{array}[]{rrr}\frac{1}{2}&-\frac{1}{2}\\ \\ -\frac{1}{2}&\frac{1}{2}\end{array}\right]
  9. C 3 = [ 1 0 0 0 1 0 0 0 1 ] - 1 3 [ 1 1 1 1 1 1 1 1 1 ] = [ 2 3 - 1 3 - 1 3 - 1 3 2 3 - 1 3 - 1 3 - 1 3 2 3 ] C_{3}=\left[\begin{array}[]{rrr}1&0&0\\ \\ 0&1&0\\ \\ 0&0&1\end{array}\right]-\frac{1}{3}\left[\begin{array}[]{rrr}1&1&1\\ \\ 1&1&1\\ \\ 1&1&1\end{array}\right]=\left[\begin{array}[]{rrr}\frac{2}{3}&-\frac{1}{3}&-% \frac{1}{3}\\ \\ -\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}\\ \\ -\frac{1}{3}&-\frac{1}{3}&\frac{2}{3}\end{array}\right]
  10. 𝐯 \mathbf{v}\,
  11. C n C_{n}\,
  12. C n 𝐯 = 𝐯 - ( 1 n 𝟏 𝐯 ) 𝟏 C_{n}\,\mathbf{v}=\mathbf{v}-(\tfrac{1}{n}\mathbf{1}^{\prime}\mathbf{v})% \mathbf{1}
  13. 1 n 𝟏 𝐯 \tfrac{1}{n}\mathbf{1}^{\prime}\mathbf{v}
  14. 𝐯 \mathbf{v}\,
  15. C n C_{n}\,
  16. C n C_{n}\,
  17. C n k = C n C_{n}^{k}=C_{n}
  18. k = 1 , 2 , k=1,2,\ldots
  19. C n C_{n}\,
  20. C n 𝐯 C_{n}\,\mathbf{v}
  21. C n C_{n}\,
  22. C n C_{n}\,
  23. 𝟏 \mathbf{1}
  24. C n C_{n}\,
  25. C n 𝐯 C_{n}\mathbf{v}
  26. 𝐯 \mathbf{v}\,
  27. 𝟏 \mathbf{1}
  28. X X\,
  29. C m X C_{m}\,X
  30. X C n X\,C_{n}
  31. S = ( X - μ 𝟏 ) ( X - μ 𝟏 ) S=(X-\mu\mathbf{1}^{\prime})(X-\mu\mathbf{1}^{\prime})^{\prime}
  32. X X\,
  33. μ = 1 n X 𝟏 \mu=\tfrac{1}{n}X\mathbf{1}
  34. S = X C n ( X C n ) = X C n C n X = X C n X . S=X\,C_{n}(X\,C_{n})^{\prime}=X\,C_{n}\,C_{n}\,X\,^{\prime}=X\,C_{n}\,X\,^{% \prime}.
  35. C n C_{n}
  36. k = n k=n
  37. p 1 = p 2 = = p n = 1 n p_{1}=p_{2}=\cdots=p_{n}=\frac{1}{n}

Centimorgan.html

  1. Pr [ recombination | linkage of d cM ] = k = 0 Pr [ 2 k + 1 crossovers | linkage of d cM ] \Pr[\,\text{recombination}|\,\text{linkage of }d\,\text{ cM}]=\sum_{k=0}^{% \infty}\Pr[2k+1\,\text{ crossovers}|\,\text{linkage of }d\,\text{ cM}]
  2. = k = 0 e - d / 100 ( d / 100 ) 2 k + 1 ( 2 k + 1 ) ! = e - d / 100 sinh ( d / 100 ) = 1 - e - 2 d / 100 2 , {}=\sum_{k=0}^{\infty}e^{-d/100}\frac{(d/100)^{2\,k+1}}{(2\,k+1)!}=e^{-d/100}% \sinh(d/100)=\frac{1-e^{-2d/100}}{2}\,,
  3. d = 50 ln ( 1 1 - 2 Pr [ recombination ] ) . d=50\ln\left({\frac{1}{1-2\Pr[\,\text{recombination}]}}\right)\,.

Centre_(category).html

  1. 𝒞 = ( 𝒞 , , I ) \mathcal{C}=(\mathcal{C},\otimes,I)
  2. 𝒞 \mathcal{C}
  3. 𝒵 ( 𝒞 ) \mathcal{Z(C)}
  4. 𝒞 \mathcal{C}
  5. u X : A X X A u_{X}:A\otimes X\rightarrow X\otimes A
  6. u X Y = ( 1 u Y ) ( u X 1 ) u_{X\otimes Y}=(1\otimes u_{Y})(u_{X}\otimes 1)
  7. u I = 1 A u_{I}=1_{A}
  8. 𝒵 ( 𝒞 ) \mathcal{Z(C)}
  9. f : A B f:A\rightarrow B
  10. 𝒞 \mathcal{C}
  11. v X ( f 1 X ) = ( 1 X f ) u X v_{X}(f\otimes 1_{X})=(1_{X}\otimes f)u_{X}
  12. 𝒵 ( 𝒞 ) \mathcal{Z(C)}
  13. ( A , u ) ( B , v ) = ( A B , w ) (A,u)\otimes(B,v)=(A\otimes B,w)
  14. w X = ( u X 1 ) ( 1 v X ) w_{X}=(u_{X}\otimes 1)(1\otimes v_{X})

Centrifugal_fan.html

  1. r p m f a n = r p m m o t o r ( D m o t o r D f a n ) rpm_{fan}=rpm_{motor}\,\bigg(\frac{\,D_{motor}}{D_{fan}}\bigg)
  2. r p m f a n rpm_{fan}
  3. r p m m o t o r rpm_{motor}
  4. D m o t o r D_{motor}
  5. D f a n D_{fan}
  6. V = U + V r V=U+V_{r}

Ceramic_capacitor.html

  1. C = ε n A d C=\varepsilon\cdot{{n\cdot A}\over{d}}
  2. Z Z
  3. Z = u ^ ı ^ = U AC I AC . Z=\frac{\hat{u}}{\hat{\imath}}=\frac{U_{\mathrm{AC}}}{I_{\mathrm{AC}}}.
  4. C C
  5. L L
  6. R R
  7. Z Z
  8. Z = E S R 2 + ( X C + ( - X L ) ) 2 Z=\sqrt{{ESR}^{2}+(X_{\mathrm{C}}+(-X_{\mathrm{L}}))^{2}}
  9. X C = - 1 ω C X_{C}=-\frac{1}{\omega C}
  10. X L = ω L ESL X_{L}=\omega L_{\mathrm{ESL}}
  11. X C = X L X_{C}=X_{L}
  12. E S R {ESR}
  13. Z Z
  14. X C X_{C}
  15. X L X_{L}
  16. E S L ESL
  17. tan δ = E S R ω C \tan\delta=ESR\cdot\omega C
  18. Q = 1 t a n δ = f 0 B Q=\frac{1}{tan\delta}=\frac{f_{0}}{B}
  19. B B
  20. f 0 f_{0}
  21. ω = 1 L C \omega=\frac{1}{\sqrt{LC}}
  22. u ( t ) = U 0 e - t / τ s , u(t)=U_{0}\cdot\mathrm{e}^{-t/\tau_{\mathrm{s}}},
  23. U 0 U_{0}
  24. τ s = R ins C \tau_{\mathrm{s}}=R_{\mathrm{ins}}\cdot C
  25. τ s \tau_{\mathrm{s}}\,
  26. U 0 U_{0}

Cerf_theory.html

  1. f : M f:M\to\mathbb{R}
  2. M M
  3. f : M f:M\to\mathbb{R}
  4. M M
  5. f t ( x ) = ( 1 / 3 ) x 3 - t x , f_{t}(x)=(1/3)x^{3}-tx,\,
  6. M = M=\mathbb{R}
  7. t = - 1 t=-1\,
  8. t = 1 t=1\,
  9. x = ± 1. x=\pm 1.\,
  10. t = 0 t=0
  11. t t
  12. M M
  13. Morse ( M ) \operatorname{Morse}(M)
  14. f : M f:M\to\mathbb{R}\,
  15. Func ( M ) \operatorname{Func}(M)
  16. f : M . f:M\to\mathbb{R}.\,
  17. Morse ( M ) Func ( M ) \operatorname{Morse}(M)\subset\operatorname{Func}(M)\,
  18. C C^{\infty}
  19. Func ( M ) \operatorname{Func}(M)
  20. Func ( M ) \operatorname{Func}(M)
  21. M M
  22. Func ( M ) \operatorname{Func}(M)
  23. Morse ( M ) \operatorname{Morse}(M)
  24. Func ( M ) 0 = Morse ( M ) \operatorname{Func}(M)^{0}=\operatorname{Morse}(M)
  25. X X
  26. X 0 X^{0}
  27. X 1 X^{1}
  28. X X
  29. X 0 X^{0}
  30. X 1 X^{1}
  31. X i X^{i}
  32. i > 1 i>1
  33. Func ( M ) \operatorname{Func}(M)
  34. Func ( M ) i \operatorname{Func}(M)^{i}
  35. i > 0 i>0
  36. f t ( x ) = x 3 - t x , f_{t}(x)=x^{3}-tx,\,
  37. t = 0 t=0
  38. f 0 ( x ) = x 3 f_{0}(x)=x^{3}\,
  39. f : M f:M\to\mathbb{R}
  40. p p
  41. g : n g:\mathbb{R}^{n}\to\mathbb{R}
  42. g ( x 1 , x 2 , , x n ) = f ( p ) + ϵ 1 x 1 2 + ϵ 2 x 2 2 + + ϵ n x n 2 g(x_{1},x_{2},\cdots,x_{n})=f(p)+\epsilon_{1}x_{1}^{2}+\epsilon_{2}x_{2}^{2}+% \cdots+\epsilon_{n}x_{n}^{2}
  43. ϵ i { ± 1 } \epsilon_{i}\in\{\pm 1\}
  44. f t : M f_{t}:M\to\mathbb{R}
  45. M M
  46. t [ 0 , 1 ] t\in[0,1]
  47. f 0 , f 1 f_{0},f_{1}
  48. F t : M F_{t}:M\to\mathbb{R}
  49. F 0 = f 0 , F 1 = f 1 F_{0}=f_{0},F_{1}=f_{1}
  50. F F
  51. f f
  52. C k C^{k}
  53. M × [ 0 , 1 ] M\times[0,1]\to\mathbb{R}
  54. F t F_{t}
  55. p p
  56. F t F_{t}
  57. g t ( x 1 , x 2 , , x n ) = f ( p ) + x 1 3 + ϵ 1 t x 1 + ϵ 2 x 2 2 + + ϵ n x n 2 g_{t}(x_{1},x_{2},\cdots,x_{n})=f(p)+x_{1}^{3}+\epsilon_{1}tx_{1}+\epsilon_{2}% x_{2}^{2}+\cdots+\epsilon_{n}x_{n}^{2}
  58. ϵ i { ± 1 } , t [ - 1 , 1 ] \epsilon_{i}\in\{\pm 1\},t\in[-1,1]
  59. ϵ 1 = - 1 \epsilon_{1}=-1
  60. t t
  61. ϵ 1 = 1 \epsilon_{1}=1
  62. S 2 3 S^{2}\subset\mathbb{R}^{3}
  63. S 3 S^{3}
  64. S 2 3 S^{2}\subset\mathbb{R}^{3}
  65. Γ 4 = 0 \Gamma_{4}=0
  66. { f : M } \{f:M\to\mathbb{R}\}
  67. K i K_{i}
  68. π 1 M \pi_{1}M
  69. i = 2 i=2
  70. π 2 M \pi_{2}M
  71. i = 1 i=1
  72. π 1 M \pi_{1}M
  73. i = 3 i=3

Cesàro_equation.html

  1. κ \kappa
  2. s s
  3. R R
  4. R = 1 / κ R=1/\kappa
  5. κ = 0 \kappa=0
  6. κ = 1 / α \kappa=1/\alpha
  7. α \alpha
  8. κ = C / s \kappa=C/s
  9. C C
  10. κ = C / s \kappa=C/\sqrt{s}
  11. C C
  12. κ = C s \kappa=Cs
  13. C C
  14. κ = a s 2 + a 2 \kappa=\frac{a}{s^{2}+a^{2}}
  15. φ = f ( s ) \varphi=f(s)\!
  16. κ = f ( s ) \kappa=f^{\prime}(s)\!

Céa's_lemma.html

  1. V V
  2. . \|\cdot\|.
  3. a : V × V a:V\times V\to\mathbb{R}
  4. | a ( v , w ) | γ v w |a(v,w)|\leq\gamma\|v\|\,\|w\|
  5. γ > 0 \gamma>0
  6. v , w v,w
  7. V V
  8. a ( v , v ) α v 2 a(v,v)\geq\alpha\|v\|^{2}
  9. α > 0 \alpha>0
  10. v v
  11. V V
  12. V V
  13. L : V L:V\to\mathbb{R}
  14. u u
  15. V V
  16. a ( u , v ) = L ( v ) a(u,v)=L(v)\,
  17. v v
  18. V . V.\,
  19. V h V_{h}
  20. V , V,
  21. u h u_{h}
  22. V h V_{h}
  23. a ( u h , v ) = L ( v ) a(u_{h},v)=L(v)\,
  24. v v
  25. V h . V_{h}.\,
  26. u - u h γ α u - v \|u-u_{h}\|\leq\frac{\gamma}{\alpha}\|u-v\|
  27. v v
  28. V h . V_{h}.
  29. u h u_{h}
  30. u u
  31. V h , V_{h},
  32. γ / α . \gamma/\alpha.
  33. α u - u h 2 a ( u - u h , u - u h ) = a ( u - u h , u - v ) + a ( u - u h , v - u h ) = a ( u - u h , u - v ) γ u - u h u - v \alpha\|u-u_{h}\|^{2}\leq a(u-u_{h},u-u_{h})=a(u-u_{h},u-v)+a(u-u_{h},v-u_{h})% =a(u-u_{h},u-v)\leq\gamma\|u-u_{h}\|\|u-v\|
  34. v v
  35. V h . V_{h}.
  36. a a
  37. u - u h u-u_{h}
  38. V h V_{h}
  39. a ( u - u h , v ) = 0 , v a(u-u_{h},v)=0,\ \forall\ v
  40. V h V_{h}
  41. V h V V_{h}\subset V
  42. a ( u , v ) = L ( v ) = a ( u h , v ) a(u,v)=L(v)=a(u_{h},v)
  43. v v
  44. V h V_{h}
  45. a ( , ) a(\cdot,\cdot)
  46. | a ( v , v ) | α v 2 |a(v,v)|\geq\alpha\|v\|^{2}
  47. v v
  48. V V
  49. a ( v , v ) a(v,v)
  50. a : V × V a:V\times V\to\mathbb{R}
  51. a ( v , w ) = a ( w , v ) a(v,w)=a(w,v)\,
  52. v , w v,w
  53. V . V.
  54. a ( , ) a(\cdot,\cdot)
  55. V . V.
  56. v a = a ( v , v ) \|v\|_{a}=\sqrt{a(v,v)}
  57. . \|\cdot\|.
  58. a a
  59. u - u h u-u_{h}
  60. V h V_{h}
  61. u - u h a 2 = a ( u - u h , u - u h ) = a ( u - u h , u - v ) u - u h a u - v a \|u-u_{h}\|_{a}^{2}=a(u-u_{h},u-u_{h})=a(u-u_{h},u-v)\leq\|u-u_{h}\|_{a}\cdot% \|u-v\|_{a}
  62. v v
  63. V h V_{h}
  64. u - u h a u - v a \|u-u_{h}\|_{a}\leq\|u-v\|_{a}
  65. v v
  66. V h V_{h}
  67. γ / α \gamma/\alpha
  68. u h u_{h}
  69. u u
  70. u h u_{h}
  71. u u
  72. V h V_{h}
  73. a ( , ) a(\cdot,\cdot)
  74. \|\cdot\|
  75. α u - u h 2 a ( u - u h , u - u h ) = u - u h a 2 u - v a 2 γ u - v 2 \alpha\|u-u_{h}\|^{2}\leq a(u-u_{h},u-u_{h})=\|u-u_{h}\|_{a}^{2}\leq\|u-v\|_{a% }^{2}\leq\gamma\|u-v\|^{2}
  76. v v
  77. V h V_{h}
  78. u - u h γ α u - v \|u-u_{h}\|\leq\sqrt{\frac{\gamma}{\alpha}}\|u-v\|
  79. v v
  80. V h V_{h}
  81. u : [ a , b ] u:[a,b]\to\mathbb{R}
  82. { - u ′′ = f in [ a , b ] u ( a ) = u ( b ) = 0 \begin{cases}-u^{\prime\prime}=f\mbox{ in }[a,b]\\ u(a)=u(b)=0\end{cases}
  83. f : [ a , b ] f:[a,b]\to\mathbb{R}
  84. u u
  85. x x
  86. a a
  87. b b
  88. f ( x ) 𝐞 f(x)\mathbf{e}
  89. 𝐞 \mathbf{e}
  90. f f
  91. V V
  92. H 0 1 ( a , b ) , H^{1}_{0}(a,b),
  93. v v
  94. [ a , b ] [a,b]
  95. [ a , b ] [a,b]
  96. v v^{\prime}
  97. v v
  98. v ( a ) = v ( b ) = 0. v(a)=v(b)=0.
  99. ( v , w ) = a b v ( x ) w ( x ) d x (v,w)=\int_{a}^{b}\!v^{\prime}(x)w^{\prime}(x)\,dx
  100. v v
  101. w w
  102. V . V.
  103. v v
  104. a ( u , v ) = L ( v ) a(u,v)=L(v)\,
  105. v v
  106. V , V,
  107. a ( u , v ) = a b u ( x ) v ( x ) d x a(u,v)=\int_{a}^{b}\!u^{\prime}(x)v^{\prime}(x)\,dx
  108. L ( v ) = a b f ( x ) v ( x ) d x . L(v)=\int_{a}^{b}\!f(x)v(x)\,dx.
  109. a ( , ) a(\cdot,\cdot)
  110. L L
  111. V h V_{h}
  112. V , V,
  113. a = x 0 < x 1 < < x n - 1 < x n = b a=x_{0}<x_{1}<\cdots<x_{n-1}<x_{n}=b
  114. [ a , b ] , [a,b],
  115. V h V_{h}
  116. V h V_{h}
  117. [ a , b ] . [a,b].
  118. V h V_{h}
  119. V V
  120. n - 1 n-1
  121. u h u_{h}
  122. a ( u h , v ) = L ( v ) a(u_{h},v)=L(v)\,
  123. v v
  124. V h , V_{h},
  125. u h u_{h}
  126. u . u.
  127. C > 0 C>0
  128. a ( , ) , a(\cdot,\cdot),
  129. u - u h C u - v \|u-u_{h}\|\leq C\|u-v\|
  130. v v
  131. V h . V_{h}.
  132. u u
  133. u h , u_{h},
  134. π u \pi u
  135. V h V_{h}
  136. u u
  137. π u \pi u
  138. [ x i , x i + 1 ] [x_{i},x_{i+1}]
  139. u u
  140. K K
  141. a a
  142. b , b,
  143. | u ( x ) - ( π u ) ( x ) | K h u ′′ L 2 ( a , b ) |u^{\prime}(x)-(\pi u)^{\prime}(x)|\leq Kh\|u^{\prime\prime}\|_{L^{2}(a,b)}
  144. x x
  145. [ a , b ] , [a,b],
  146. h h
  147. [ x i , x i + 1 ] [x_{i},x_{i+1}]
  148. u - π u . \|u-\pi u\|.\,
  149. v = π u v=\pi u
  150. u - u h C h u ′′ L 2 ( a , b ) , \|u-u_{h}\|\leq Ch\|u^{\prime\prime}\|_{L^{2}(a,b)},
  151. C C
  152. [ a , b ] [a,b]
  153. h . h.
  154. u u
  155. V h . V_{h}.

Chain_rule_for_Kolmogorov_complexity.html

  1. H ( X , Y ) = H ( X ) + H ( Y | X ) H(X,Y)=H(X)+H(Y|X)
  2. P ( X , Y ) = P ( X ) P ( Y | X ) P(X,Y)=P(X)P(Y|X)\,
  3. K ( x , y ) = K ( x ) + K ( y | x ) + O ( log ( K ( x , y ) ) ) K(x,y)=K(x)+K(y|x)+O(\log(K(x,y)))

Chain_sequence.html

  1. a 1 = ( 1 - g 0 ) g 1 a 2 = ( 1 - g 1 ) g 2 a n = ( 1 - g n - 1 ) g n a_{1}=(1-g_{0})g_{1}\quad a_{2}=(1-g_{1})g_{2}\quad a_{n}=(1-g_{n-1})g_{n}
  2. f ( z ) = a 1 z 1 + a 2 z 1 + a 3 z 1 + a 4 z f(z)=\cfrac{a_{1}z}{1+\cfrac{a_{2}z}{1+\cfrac{a_{3}z}{1+\cfrac{a_{4}z}{\ddots}% }}}\,
  3. g 0 = 0 g 1 = 1 4 g 2 = 1 3 g 3 = 3 8 g_{0}=0\quad g_{1}={\textstyle\frac{1}{4}}\quad g_{2}={\textstyle\frac{1}{3}}% \quad g_{3}={\textstyle\frac{3}{8}}\;\dots

Chan's_algorithm.html

  1. m = min ( n , 2 2 t ) m=\min(n,2^{2^{t}})
  2. t = 0 log log h O ( n log ( 2 2 t ) ) = O ( n ) t = 0 log log h O ( 2 t ) = O ( n 2 1 + log log h ) = O ( n log h ) . \sum_{t=0}^{\lceil\log\log h\rceil}O\left(n\log(2^{2^{t}})\right)=O(n)\sum_{t=% 0}^{\lceil\log\log h\rceil}O(2^{t})=O\left(n\cdot 2^{1+\lceil\log\log h\rceil}% \right)=O(n\log h).

Chandrasekhar_number.html

  1. Q \ Q
  2. 1 σ ( 𝐮 t + ( 𝐮 ) 𝐮 ) = - p + 2 𝐮 + σ ζ Q ( 𝐁 ) 𝐁 , \frac{1}{\sigma}\left(\frac{\partial\mathbf{u}}{\partial t}\ +\ (\mathbf{u}% \cdot\nabla)\mathbf{u}\right)\ =\ -{\mathbf{\nabla}}p\ +\ \nabla^{2}\mathbf{u}% \ +\frac{\sigma}{\zeta}{Q}\ ({\mathbf{\nabla}}\wedge\mathbf{B})\wedge\mathbf{B},
  3. σ \ \sigma
  4. ζ \ \zeta
  5. Q = B 0 2 d 2 μ 0 ρ ν λ {Q}\ =\ \frac{{B_{0}}^{2}d^{2}}{\mu_{0}\rho\nu\lambda}
  6. μ 0 \ \mu_{0}
  7. ρ \ \rho
  8. ν \ \nu
  9. λ \ \lambda
  10. B 0 \ B_{0}
  11. d \ d
  12. H \ H
  13. Q = H 2 Q\ {=}\ H^{2}

Channel-state_duality.html

  1. n × n A . \mathbb{C}^{n\times n}\otimes A.
  2. Φ : L ( H 1 ) L ( H 2 ) \Phi:L(H_{1})\rightarrow L(H_{2})
  3. ρ Φ = ( Φ ( E i j ) ) i j L ( H 1 ) L ( H 2 ) \rho_{\Phi}=(\Phi(E_{ij}))_{ij}\in L(H_{1})\otimes L(H_{2})
  4. Φ ρ Φ , \Phi\rightarrow\rho_{\Phi},

Chaperonin_ATPase.html

  1. \rightleftharpoons

Characterisation_of_pore_space_in_soil.html

  1. ρ = M s t V \rho=\frac{M_{s}}{t_{V}}
  2. f = V f V t f=\frac{V_{f}}{V_{t}}
  3. f = V a + V w V s + V a + V w f=\frac{V_{a}+V_{w}}{V_{s}+V_{a}+V_{w}}

Charles_Loewner.html

  1. sys 2 2 3 area ( 𝕋 2 ) , \operatorname{sys}^{2}\leq\frac{2}{\sqrt{3}}\operatorname{area}(\mathbb{T}^{2}),
  2. \mathbb{C}

Chebyshev–Markov–Stieltjes_inequalities.html

  1. x k d μ ( x ) = m k \int x^{k}d\mu(x)=m_{k}
  2. ρ m - 1 ( z ) = 1 / k = 0 m - 1 | P k ( z ) | 2 \rho_{m-1}(z)=1\Big/\sum_{k=0}^{m-1}|P_{k}(z)|^{2}
  3. μ ( - , ξ j ] ρ m - 1 ( ξ 1 ) + + ρ m - 1 ( ξ j ) μ ( - , ξ j + 1 ) . \mu(-\infty,\xi_{j}]\leq\rho_{m-1}(\xi_{1})+\cdots+\rho_{m-1}(\xi_{j})\leq\mu(% -\infty,\xi_{j+1}).

Chemical_transport_reaction.html

  1. \overrightarrow{\leftarrow}

Chief_series.html

  1. 1 = N 0 N 1 N 2 N n = G , 1=N_{0}\subseteq N_{1}\subseteq N_{2}\subseteq\cdots\subseteq N_{n}=G,

Chilton_and_Colburn_J-factor_analogy.html

  1. J M = f 2 = J H = h c p G P r 2 3 = J D = k c v ¯ S c 2 3 J_{M}=\frac{f}{2}=J_{H}=\frac{h}{c_{p}\,G}\,{Pr}^{\frac{2}{3}}=J_{D}=\frac{k^{% \prime}_{c}}{\overline{v}}\cdot{Sc}^{\frac{2}{3}}
  2. J d = f 2 = S h R e S c 1 3 = J h = f 2 = N u R e P r 1 3 J_{d}=\frac{f}{2}=\frac{Sh}{Re\,Sc^{\frac{1}{3}}}=J_{h}=\frac{f}{2}=\frac{Nu}{% Re\,Pr^{\frac{1}{3}}}

Chinese_hypothesis.html

  1. 2 n 2 ( mod n ) 2^{n}\equiv 2\;\;(\mathop{{\rm mod}}n)\,
  2. 2 n 2 ( mod n ) 2^{n}\equiv 2\;\;(\mathop{{\rm mod}}n)\,
  3. 2 n 2 ( mod n ) \,2^{n}\equiv 2\;\;(\mathop{{\rm mod}}n)

Chip_(CDMA).html

  1. SF = chip rate symbol rate \ \mbox{SF}~{}=\frac{\mbox{chip rate}~{}}{\mbox{symbol rate}~{}}

Choquet_integral.html

  1. S S
  2. \mathcal{F}
  3. S S
  4. f : S f:S\to\mathbb{R}
  5. ν : + \nu:\mathcal{F}\to\mathbb{R}^{+}
  6. f f
  7. ν \nu
  8. x : { s | f ( s ) x } \forall x\in\mathbb{R}\colon\{s|f(s)\geq x\}\in\mathcal{F}
  9. f f
  10. ν \nu
  11. ( C ) f d ν := - 0 ( ν ( { s | f ( s ) x } ) - ν ( S ) ) d x + 0 ν ( { s | f ( s ) x } ) d x (C)\int fd\nu:=\int_{-\infty}^{0}(\nu(\{s|f(s)\geq x\})-\nu(S))\,dx+\int^{% \infty}_{0}\nu(\{s|f(s)\geq x\})\,dx
  12. x x
  13. ν \nu
  14. f d ν + g d ν ( f + g ) d ν . \int f\,d\nu+\int g\,d\nu\neq\int(f+g)\,d\nu.
  15. f f
  16. g g
  17. f g f\leq g
  18. ( C ) f d ν ( C ) g d ν (C)\int f\,d\nu\leq(C)\int g\,d\nu
  19. λ 0 \lambda\geq 0
  20. ( C ) λ f d ν = λ ( C ) f d ν , (C)\int\lambda f\,d\nu=\lambda(C)\int f\,d\nu,
  21. f , g : S f,g:S\rightarrow\mathbb{R}
  22. s , s S s,s^{\prime}\in S
  23. ( f ( s ) - f ( s ) ) ( g ( s ) - g ( s ) ) 0 (f(s)-f(s^{\prime}))(g(s)-g(s^{\prime}))\geq 0
  24. ( C ) f d ν + ( C ) g d ν = ( C ) ( f + g ) d ν . (C)\int\,fd\nu+(C)\int g\,d\nu=(C)\int(f+g)\,d\nu.
  25. ν \nu
  26. ( C ) f d ν + ( C ) g d ν ( C ) ( f + g ) d ν . (C)\int\,fd\nu+(C)\int g\,d\nu\geq(C)\int(f+g)\,d\nu.
  27. ν \nu
  28. ( C ) f d ν + ( C ) g d ν ( C ) ( f + g ) d ν . (C)\int\,fd\nu+(C)\int g\,d\nu\leq(C)\int(f+g)\,d\nu.
  29. G G
  30. G - 1 G^{-1}
  31. d H dH
  32. - G - 1 ( α ) d H ( α ) = - - a H ( G ( x ) ) d x + a H ^ ( 1 - G ( x ) ) d x , \int_{-\infty}^{\infty}G^{-1}(\alpha)dH(\alpha)=-\int_{-\infty}^{a}H(G(x))dx+% \int_{a}^{\infty}\hat{H}(1-G(x))dx,
  33. H ^ ( x ) = H ( 1 ) - H ( 1 - x ) \hat{H}(x)=H(1)-H(1-x)
  34. H ( x ) := x H(x):=x
  35. 0 1 G - 1 ( x ) d x = E [ X ] \int_{0}^{1}G^{-1}(x)dx=E[X]
  36. H ( x ) := 1 [ α , x ] H(x):=1_{[\alpha,x]}
  37. 0 1 G - 1 ( x ) d H ( x ) = G - 1 ( α ) \int_{0}^{1}G^{-1}(x)dH(x)=G^{-1}(\alpha)

Chou–Fasman_method.html

  1. p ( t ) = p t ( j ) × p t ( j + 1 ) × p t ( j + 2 ) × p t ( j + 3 ) p(t)=p_{t}(j)\times p_{t}(j+1)\times p_{t}(j+2)\times p_{t}(j+3)

Chow's_lemma.html

  1. X X
  2. S S
  3. S S
  4. X X^{\prime}
  5. S S
  6. f : X X f\colon X^{\prime}\to X
  7. f - 1 ( U ) U f^{-1}(U)\simeq U
  8. U X U\subseteq X
  9. X X
  10. X X
  11. X i X_{i}
  12. S S
  13. X X
  14. S S
  15. U i X i U_{i}\subset X_{i}
  16. U := ( U i i j X j ) U:=\bigsqcup(U_{i}\setminus\bigcup\limits_{i\neq j}X_{j})
  17. X i X_{i}
  18. U U
  19. X X
  20. g g
  21. X X
  22. X = i = 1 n U i X=\bigcup_{i=1}^{n}U_{i}
  23. U i U_{i}
  24. S S
  25. S S
  26. ϕ i : U i P i \phi_{i}\colon U_{i}\to P_{i}
  27. S S
  28. P i P_{i}
  29. U = U i U=\cap U_{i}
  30. U U
  31. X X
  32. ϕ : U P = P 1 × S × S P n . \phi\colon U\to P=P_{1}\times_{S}\cdots\times_{S}P_{n}.
  33. ϕ i \phi_{i}
  34. U U
  35. S S
  36. ψ : U X × S P . \psi\colon U\to X\times_{S}P.
  37. U X U\hookrightarrow X
  38. ϕ \phi
  39. S S
  40. ψ \psi
  41. X X × S P X^{\prime}\to X\times_{S}P
  42. f : X X f\colon X^{\prime}\to X
  43. f f
  44. f - 1 ( U ) U f^{-1}(U)\simeq U
  45. f - 1 ( U ) = ψ ( U ) f^{-1}(U)=\psi(U)
  46. ψ ( U ) \psi(U)
  47. U × S P U\times_{S}P
  48. ψ \psi
  49. U Γ ϕ U × S P X × S P U\overset{\Gamma_{\phi}}{\to}U\times_{S}P\to X\times_{S}P
  50. P P
  51. S S
  52. Γ ϕ \Gamma_{\phi}
  53. X X^{\prime}
  54. S S
  55. g : X P g\colon X^{\prime}\to P
  56. g g
  57. X X^{\prime}
  58. S S
  59. U i U_{i}
  60. P i P_{i}
  61. ϕ i \phi_{i}
  62. V i = p i - 1 ( U i ) V_{i}=p_{i}^{-1}(U_{i})
  63. p i : P P i p_{i}\colon P\to P_{i}
  64. g - 1 ( V i ) g^{-1}(V_{i})
  65. X X^{\prime}
  66. f - 1 ( U i ) g - 1 ( V i ) f^{-1}(U_{i})\subset g^{-1}(V_{i})
  67. f = p i g f=p_{i}\circ g
  68. U i U_{i}
  69. X X
  70. S S
  71. U i U_{i}
  72. X × S P P X\times_{S}P\to P
  73. X S X\to S
  74. g ( X ) g(X^{\prime})
  75. V i V_{i}
  76. i i
  77. g : g - 1 ( V i ) V i g\colon g^{-1}(V_{i})\to V_{i}
  78. h h
  79. i i
  80. Z Z
  81. u : V i p i U i X u\colon V_{i}\overset{p_{i}}{\to}U_{i}\hookrightarrow X
  82. X × S V i X\times_{S}V_{i}
  83. X X
  84. S S
  85. q 1 : X × S P X , q 2 : X × S P P q_{1}\colon X\times_{S}P\to X,\ q_{2}\colon X\times_{S}P\to P
  86. h h
  87. Z Z
  88. h h
  89. w : U V i w\colon U^{\prime}\to V_{i}
  90. v = Γ u w q 1 v = u q 2 v q 1 ψ = u q 2 ψ q 1 ψ = u ϕ . v=\Gamma_{u}\circ w\quad\Leftrightarrow\quad q_{1}\circ v=u\circ q_{2}\circ v% \quad\Leftrightarrow\quad q_{1}\circ\psi=u\circ q_{2}\circ\psi\quad% \Leftrightarrow\quad q_{1}\circ\psi=u\circ\phi.
  91. w w
  92. \square
  93. X X
  94. X X^{\prime}
  95. X X
  96. X X^{\prime}
  97. f : X X f\colon X^{\prime}\to X

Chromatographic_response_function.html

  1. Δ R F \Delta R_{F}
  2. Δ R F \Delta R_{F}
  3. M R F = ( U - h R F n ) ( h R F 1 - L ) i = 1 n - 1 ( h R F i + 1 - h R F i ) [ ( U - L ) / ( n + 1 ) ] n + 1 MRF=\frac{(U-hR_{Fn})(hR_{F1}-L)\prod^{n-1}_{i=1}(hR_{Fi+1}-hR_{Fi})}{[(U-L)/(% n+1)]^{n+1}}
  4. R D = [ ( n + 1 ) ( n + 1 ) i = 0 n ( R F ( i + 1 ) - R F i ) ] 1 n R_{D}=\Bigg[(n+1)^{(n+1)}\prod^{n}_{i=0}{(R_{F(i+1)}-R_{Fi})\Bigg]^{\frac{1}{n% }}}
  5. D = i = 1 n ( R F i - i - 1 n - 1 ) D=\sqrt{\sum^{n}_{i=1}\left(R_{Fi}-\frac{i-1}{n-1}\right)}
  6. I p = ( Δ h R F i - Δ h R F t ) 2 n ( n + 1 ) I_{p}=\sqrt{\frac{\sum(\Delta hR_{Fi}-\Delta hR_{Ft})^{2}}{n(n+1)}}
  7. s m = ( Δ h R F i - Δ h R F t ) 2 n + 1 s_{m}=\sqrt{\frac{\sum(\Delta hR_{Fi}-\Delta hR_{Ft})^{2}}{n+1}}
  8. R U = 1 - 6 ( n + 1 ) n ( 2 n + 1 ) i = 1 n ( R F i - i n + 1 ) 2 R_{U}=1-\sqrt{\frac{6(n+1)}{n(2n+1)}\sum_{i=1}^{n}{\left(R_{Fi}-\frac{i}{n+1}% \right)^{2}}}

CIECAM02.html

  1. L A = E w π Y b Y w = L W Y b Y w L_{A}=\frac{E_{w}}{\pi}\frac{Y_{b}}{Y_{w}}=\frac{L_{W}Y_{b}}{Y_{w}}
  2. [ L M S ] = 𝐌 C A T 02 [ X Y Z ] , 𝐌 C A T 02 = [ 0.7328 0.4296 - 0.1624 - 0.7036 1.6975 0.0061 0.0030 0.0136 0.9834 ] \begin{bmatrix}L\\ M\\ S\end{bmatrix}=\mathbf{M}_{CAT02}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix},\quad\mathbf{M}_{CAT02}=\begin{bmatrix}\;\;\,0.7328&0.4296&-0.1% 624\\ -0.7036&1.6975&\;\;\,0.0061\\ \;\;\,0.0030&0.0136&\;\;\,0.9834\end{bmatrix}
  3. L c \displaystyle L_{c}
  4. D = F ( 1 - 1 3.6 e - ( L A + 42 ) / 92 ) D=F\left(1-\textstyle{\frac{1}{3.6}}e^{-(L_{A}+42)/92}\right)
  5. L c \displaystyle L_{c}
  6. L c \displaystyle L_{c}
  7. [ L M S ] = 𝐌 H [ X c Y c Z c ] = 𝐌 H 𝐌 C A T 02 - 1 [ L c M c S c ] \begin{bmatrix}L^{\prime}\\ M^{\prime}\\ S^{\prime}\end{bmatrix}=\mathbf{M}_{H}\begin{bmatrix}X_{c}\\ Y_{c}\\ Z_{c}\end{bmatrix}=\mathbf{M}_{H}\mathbf{M}_{CAT02}^{-1}\begin{bmatrix}L_{c}\\ M_{c}\\ S_{c}\end{bmatrix}
  8. 𝐌 H = [ 0.38971 0.68898 - 0.07868 - 0.22981 1.18340 0.04641 0.00000 0.00000 1.00000 ] \mathbf{M}_{H}=\begin{bmatrix}\;\;\,0.38971&0.68898&-0.07868\\ -0.22981&1.18340&\;\;\,0.04641\\ \;\;\,0.00000&0.00000&\;\;\,1.00000\end{bmatrix}
  9. k = 1 5 L A + 1 k=\frac{1}{5L_{A}+1}
  10. F L = 1 5 k 4 ( 5 L A ) + 1 10 ( 1 - k 4 ) 2 ( 5 L A ) 1 / 3 F_{L}=\textstyle{\frac{1}{5}}k^{4}\left(5L_{A}\right)+\textstyle{\frac{1}{10}}% {(1-k^{4})}^{2}{\left(5L_{A}\right)}^{1/3}
  11. L a = 400 ( F L L / 100 ) 0.42 27.13 + ( F L L / 100 ) 0.42 + 0.1 M a = 400 ( F L M / 100 ) 0.42 27.13 + ( F L M / 100 ) 0.42 + 0.1 S a = 400 ( F L S / 100 ) 0.42 27.13 + ( F L S / 100 ) 0.42 + 0.1 \begin{aligned}\displaystyle L^{\prime}_{a}&\displaystyle=\frac{400{\left(F_{L% }L^{\prime}/100\right)}^{0.42}}{27.13+{\left(F_{L}L^{\prime}/100\right)}^{0.42% }}+0.1\\ \displaystyle M^{\prime}_{a}&\displaystyle=\frac{400{\left(F_{L}M^{\prime}/100% \right)}^{0.42}}{27.13+{\left(F_{L}M^{\prime}/100\right)}^{0.42}}+0.1\\ \displaystyle S^{\prime}_{a}&\displaystyle=\frac{400{\left(F_{L}S^{\prime}/100% \right)}^{0.42}}{27.13+{\left(F_{L}S^{\prime}/100\right)}^{0.42}}+0.1\end{aligned}
  12. C 1 = L a - M a C 2 = M a - S a C 3 = S a - L a \begin{aligned}\displaystyle C_{1}&\displaystyle=L^{\prime}_{a}-M^{\prime}_{a}% \\ \displaystyle C_{2}&\displaystyle=M^{\prime}_{a}-S^{\prime}_{a}\\ \displaystyle C_{3}&\displaystyle=S^{\prime}_{a}-L^{\prime}_{a}\end{aligned}
  13. a = C 1 - 1 11 C 2 = L a - 12 11 M a + 1 11 S a b = 1 2 ( C 2 - C 1 + C 1 - C 3 ) / 4.5 = 1 9 ( L a + M a - 2 S a ) \begin{aligned}\displaystyle a&\displaystyle=C_{1}-\textstyle{\frac{1}{11}}C_{% 2}&\displaystyle=L^{\prime}_{a}-\textstyle{\frac{12}{11}}M^{\prime}_{a}+% \textstyle{\frac{1}{11}}S^{\prime}_{a}\\ \displaystyle b&\displaystyle=\textstyle{\frac{1}{2}}\left(C_{2}-C_{1}+C_{1}-C% _{3}\right)/4.5&\displaystyle=\textstyle{\frac{1}{9}}\left(L^{\prime}_{a}+M^{% \prime}_{a}-2S^{\prime}_{a}\right)\end{aligned}
  14. h = ( a , b ) , ( 0 < h < 360 ) h=\angle(a,b),\ (0<h<360^{\circ})
  15. H = H i + 100 ( h - h i ) / e i ( h - h i ) / e i + ( h i + 1 - h ) / e i + 1 e t = 1 4 [ cos ( π 180 h + 2 ) + 3.8 ] \begin{aligned}\displaystyle H&\displaystyle=H_{i}+\frac{100(h^{\prime}-h_{i})% /e_{i}}{(h^{\prime}-h_{i})/e_{i}+(h_{i+1}-h^{\prime})/e_{i+1}}\\ \displaystyle e_{t}&\displaystyle=\textstyle{\frac{1}{4}}\left[\cos\left(% \textstyle{\frac{\pi}{180}}h+2\right)+3.8\right]\end{aligned}
  16. A = ( 2 L a + M a + 1 20 S a - 0.305 ) N b b A=(2L^{\prime}_{a}+M^{\prime}_{a}+\textstyle{\frac{1}{20}}S^{\prime}_{a}-0.305% )N_{bb}
  17. N b b = N c b = 0.725 n - 0.2 n = Y b / Y w \begin{aligned}&\displaystyle N_{bb}=N_{cb}=0.725n^{-0.2}\\ &\displaystyle n=Y_{b}/Y_{w}\end{aligned}
  18. J = 100 ( A / A w ) c z J=100\left(A/A_{w}\right)^{cz}
  19. z = 1.48 + n z=1.48+\sqrt{n}
  20. Q = ( 4 / c ) 1 100 J ( A w + 4 ) F L 1 / 4 Q=\left(4/c\right)\sqrt{\textstyle{\frac{1}{100}}J}\left(A_{w}+4\right)F_{L}^{% 1/4}
  21. t = 50 000 13 N c N c b e t a 2 + b 2 L a + M a + 21 20 S a t=\frac{\textstyle{\frac{50\,000}{13}}N_{c}N_{cb}e_{t}\sqrt{a^{2}+b^{2}}}{L_{a% }^{\prime}+M_{a}^{\prime}+\textstyle{\frac{21}{20}}S_{a}^{\prime}}
  22. C = t 0.9 1 100 J ( 1.64 - 0.29 n ) 0.73 C=t^{0.9}\sqrt{\textstyle{\frac{1}{100}}J}(1.64-0.29^{n})^{0.73}
  23. M = C F L 1 / 4 M=C\cdot F_{L}^{1/4}
  24. s = 100 M / Q s=100\sqrt{M/Q}

CIELUV.html

  1. L * = { ( 29 3 ) 3 Y / Y n , Y / Y n ( 6 29 ) 3 116 ( Y / Y n ) 1 / 3 - 16 , Y / Y n > ( 6 29 ) 3 u * = 13 L * ( u - u n ) v * = 13 L * ( v - v n ) \begin{aligned}\displaystyle L^{*}&\displaystyle=\begin{cases}\left(\frac{29}{% 3}\right)^{3}Y/Y_{n},&Y/Y_{n}\leq\left(\frac{6}{29}\right)^{3}\\ 116\left(Y/Y_{n}\right)^{1/3}-16,&Y/Y_{n}>\left(\frac{6}{29}\right)^{3}\end{% cases}\\ \displaystyle u^{*}&\displaystyle=13L^{*}\cdot(u^{\prime}-u_{n}^{\prime})\\ \displaystyle v^{*}&\displaystyle=13L^{*}\cdot(v^{\prime}-v_{n}^{\prime})\end{aligned}
  2. u = 4 X X + 15 Y + 3 Z = 4 x - 2 x + 12 y + 3 v = 9 Y X + 15 Y + 3 Z = 9 y - 2 x + 12 y + 3 \begin{aligned}\displaystyle u^{\prime}&\displaystyle=\frac{4X}{X+15Y+3Z}&% \displaystyle=\frac{4x}{-2x+12y+3}\\ \displaystyle v^{\prime}&\displaystyle=\frac{9Y}{X+15Y+3Z}&\displaystyle=\frac% {9y}{-2x+12y+3}\end{aligned}
  3. x = 9 u 6 u - 16 v + 12 y = 4 v 6 u - 16 v + 12 \begin{aligned}\displaystyle x&\displaystyle=\frac{9u^{\prime}}{6u^{\prime}-16% v^{\prime}+12}\\ \displaystyle y&\displaystyle=\frac{4v^{\prime}}{6u^{\prime}-16v^{\prime}+12}% \end{aligned}
  4. u = u * 13 L * + u n v = v * 13 L * + v n Y = { Y n L * ( 3 29 ) 3 , L * 8 Y n ( L * + 16 116 ) 3 , L * > 8 X = Y 9 u 4 v Z = Y 12 - 3 u - 20 v 4 v \begin{aligned}\displaystyle u^{\prime}&\displaystyle=\frac{u^{*}}{13L^{*}}+u^% {\prime}_{n}\\ \displaystyle v^{\prime}&\displaystyle=\frac{v^{*}}{13L^{*}}+v^{\prime}_{n}\\ \displaystyle Y&\displaystyle=\begin{cases}Y_{n}\cdot L^{*}\cdot\left(\frac{3}% {29}\right)^{3},&L^{*}\leq 8\\ Y_{n}\cdot\left(\frac{L^{*}+16}{116}\right)^{3},&L^{*}>8\end{cases}\\ \displaystyle X&\displaystyle=Y\cdot\frac{9u^{\prime}}{4v^{\prime}}\\ \displaystyle Z&\displaystyle=Y\cdot\frac{12-3u^{\prime}-20v^{\prime}}{4v^{% \prime}}\\ \end{aligned}
  5. C u v * = ( u * ) 2 + ( v * ) 2 C_{uv}^{*}=\sqrt{(u^{*})^{2}+(v^{*})^{2}}
  6. h u v = atan2 ( v * , u * ) , h_{uv}=\operatorname{atan2}(v^{*},u^{*}),
  7. s u v = C * L * = 13 ( u - u n ) 2 + ( v - v n ) 2 s_{uv}=\frac{C^{*}}{L^{*}}=13\sqrt{(u^{\prime}-u^{\prime}_{n})^{2}+(v^{\prime}% -v^{\prime}_{n})^{2}}
  8. ( Δ u ) 2 + ( Δ v ) 2 = 1 / 13 \sqrt{(\Delta u^{\prime})^{2}+(\Delta v^{\prime})^{2}}=1/13

Circular_buffer.html

  1. ( size - 1 ) (\,\text{size}-1)

Circumconic_and_inconic.html

  1. ( α , β , γ ) (\alpha,\beta,\gamma)
  2. Area of inellipse Area of triangle = π ( 1 - 2 α ) ( 1 - 2 β ) ( 1 - 2 γ ) , \frac{\,\text{Area of inellipse}}{\,\text{Area of triangle}}=\pi\sqrt{(1-2% \alpha)(1-2\beta)(1-2\gamma)},
  3. α = β = γ = 1 / 3. \alpha=\beta=\gamma=1/3.

Classical_and_quantum_conductivity.html

  1. v = 8 k b T π m \left\langle v\right\rangle=\sqrt{\frac{8k_{b}T}{\pi m}}
  2. F = e ϵ F=e\epsilon
  3. e e
  4. ϵ \epsilon
  5. I = V / R I=V/R
  6. I = Δ Q / Δ t = n e A v d I=\Delta Q/\Delta t=neAv_{d}
  7. A A
  8. v d v_{d}
  9. e e
  10. n n
  11. I = ( 1 / ρ ) ϵ A I=(1/\rho)\epsilon A
  12. j = σ ϵ j=\sigma\epsilon
  13. v d = j / n e v_{d}=j/ne
  14. v \langle v\rangle
  15. ρ \rho
  16. σ \sigma
  17. ϵ \epsilon
  18. I I
  19. ϵ \epsilon
  20. ϵ \epsilon

Classical_group.html

  1. 𝐑 \mathbf{R}
  2. 𝐂 \mathbf{C}
  3. 𝐇 \mathbf{H}
  4. S O ( 3 ) SO(3)
  5. O ( 3 , 1 ) O(3,1)
  6. S U ( 3 ) SU(3)
  7. S p ( m ) Sp(m)
  8. 𝐑 , 𝐂 \mathbf{R},\mathbf{C}
  9. 𝐇 \mathbf{H}
  10. B < s u b > m B<sub>m
  11. S L ( n , 𝐂 ) SL(n,\mathbf{C})
  12. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  13. S p ( m , 𝐂 ) Sp(m,\mathbf{C})
  14. S U ( n ) SU(n)
  15. S O ( n ) SO(n)
  16. S p ( m ) Sp(m)
  17. 𝐠 \mathbf{g}
  18. 𝐠 = 𝐮 + i 𝐮 \mathbf{g}=\mathbf{u}+i\mathbf{u}
  19. 𝐮 \mathbf{u}
  20. K K
  21. e x p ( X ) : X 𝐮 exp(X):X∈\mathbf{u}
  22. K K
  23. S L ( n , 𝐂 ) , S O ( n , 𝐂 ) SL(n,\mathbf{C}),SO(n,\mathbf{C})
  24. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  25. S O ( 2 n , 𝐂 ) SO(2n,\mathbf{C})
  26. S U ( p , q ) SU(p,q)
  27. S l ( n , 𝐂 ) Sl(n,\mathbf{C})
  28. S l ( n , 𝐇 ) Sl(n,\mathbf{H})
  29. S O ( 2 n , 𝐂 ) SO(2n,\mathbf{C})
  30. 𝐑 \mathbf{R}
  31. 𝐂 \mathbf{C}
  32. 𝐇 \mathbf{H}
  33. V V
  34. 𝐑 \mathbf{R}
  35. 𝐂 \mathbf{C}
  36. 𝐇 \mathbf{H}
  37. 𝐇 \mathbf{H}
  38. V V
  39. 𝐑 \mathbf{R}
  40. 𝐂 \mathbf{C}
  41. φ : V × V F φ:V×V→F
  42. F = 𝐑 , 𝐂 F=\mathbf{R},\mathbf{C}
  43. 𝐇 \mathbf{H}
  44. φ ( x α , y β ) = α φ ( x , y ) β , x , y V , α , β F . \varphi(x\alpha,y\beta)=\alpha\varphi(x,y)\beta,\quad\forall x,y\in V,\forall% \alpha,\beta\in F.
  45. φ ( x α , y β ) = α ¯ φ ( x , y ) β , x , y V , α , β F . \varphi(x\alpha,y\beta)=\bar{\alpha}\varphi(x,y)\beta,\quad\forall x,y\in V,% \forall\alpha,\beta\in F.
  46. φ φ
  47. Α Α
  48. V V
  49. φ φ
  50. φ φ
  51. A u t ( φ ) Aut(φ)
  52. 𝐑 \mathbf{R}
  53. 𝐂 \mathbf{C}
  54. 𝐇 \mathbf{H}
  55. F = 𝐑 F=\mathbf{R}
  56. F = 𝐇 F=\mathbf{H}
  57. φ ( x , y ) = φ ( y , x ) . \varphi(x,y)=\varphi(y,x).
  58. φ ( x , y ) = - φ ( y , x ) . \varphi(x,y)=-\varphi(y,x).
  59. φ ( x , y ) = φ ( y , x ) ¯ \varphi(x,y)=\overline{\varphi(y,x)}
  60. φ ( x , y ) = - φ ( y , x ) ¯ . \varphi(x,y)=-\overline{\varphi(y,x)}.
  61. φ φ
  62. φ φ
  63. Bilinear symmetric form in (pseudo-)orthonormal basis: φ ( x , y ) \displaystyle\,\text{Bilinear symmetric form in (pseudo-)orthonormal basis:}% \qquad\varphi(x,y)
  64. 𝐣 \mathbf{j}
  65. ( 𝟏 , 𝐢 , 𝐣 , 𝐤 ) (\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k})
  66. 𝐇 \mathbf{H}
  67. p p
  68. q q
  69. ( p , q ) (p,q)
  70. p q p−q
  71. 𝐑 , 𝐂 , 𝐇 \mathbf{R},\mathbf{C},\mathbf{H}
  72. H H
  73. 𝐑 \mathbf{R}
  74. ( p , q ) (p,q)
  75. + +
  76. p q p−q
  77. i i
  78. 𝐇 \mathbf{H}
  79. 𝐑 \mathbf{R}
  80. 𝐂 \mathbf{C}
  81. 𝐇 \mathbf{H}
  82. φ φ
  83. V V
  84. 𝐑 , 𝐂 \mathbf{R},\mathbf{C}
  85. 𝐇 \mathbf{H}
  86. Aut ( φ ) = { A GL ( V ) : φ ( A u , A v ) = φ ( x , y ) , x , y V } . \mathrm{Aut}(\varphi)=\{A\in\mathrm{GL}(V):\varphi(Au,Av)=\varphi(x,y),\quad% \forall x,y\in V\}.
  87. φ φ
  88. V V
  89. φ ( x , y ) = ξ i φ i j η j \varphi(x,y)=\sum\xi_{i}\varphi_{ij}\eta_{j}
  90. x , y x,y
  91. φ ( x , y ) = x T Φ y \varphi(x,y)=x^{\mathrm{T}}\Phi y
  92. Φ Φ
  93. Φ Φ
  94. A u t ( φ ) Aut(φ)
  95. Aut ( φ ) = { A GL ( V ) : Φ - 1 A T Φ A = 1 } . \operatorname{Aut}(\varphi)=\{A\in\operatorname{GL}(V):\Phi^{-1}A^{\mathrm{T}}% \Phi A=1\}.
  96. 𝐚𝐮𝐭 ( φ ) \mathbf{aut}(φ)
  97. X 𝐚𝐮𝐭 ( φ ) X∈\mathbf{aut}(φ)
  98. ( e t X ) φ e t X = 1 (e^{tX})^{\varphi}e^{tX}=1
  99. t t
  100. 𝔞 𝔲 𝔱 ( φ ) = { X M n ( V ) : X φ = - X } , \mathfrak{aut}(\varphi)=\{X\in M_{n}(V):X^{\varphi}=-X\},
  101. 𝔞 𝔲 𝔱 ( φ ) = { X M n ( V ) : Φ - 1 X T Φ = - X } \mathfrak{aut}(\varphi)=\{X\in M_{n}(V):\Phi^{-1}X^{\mathrm{T}}\Phi=-X\}
  102. X 𝐚𝐮𝐭 ( φ ) X∈\mathbf{aut}(φ)
  103. 𝔞 𝔲 𝔱 ( φ ) = { X M n ( V ) : φ ( X x , y ) = - φ ( x , X y ) , x , y V } . \mathfrak{aut}(\varphi)=\{X\in M_{n}(V):\varphi(Xx,y)=-\varphi(x,Xy),\quad% \forall x,y\in V\}.
  104. φ φ
  105. Φ Φ
  106. A u t ( φ ) Aut(φ)
  107. O ( φ ) O(φ)
  108. A u t ( φ ) Aut(φ)
  109. S p ( φ ) Sp(φ)
  110. φ φ
  111. φ ( x , y ) = ± ξ 1 η 1 ± ξ 1 η 1 ± ξ n η n . \varphi(x,y)=\pm\xi_{1}\eta_{1}\pm\xi_{1}\eta_{1}\cdots\pm\xi_{n}\eta_{n}.
  112. O ( φ ) = O ( p , q ) O(φ)=O(p,q)
  113. p p
  114. q q
  115. p + q = n p+q=n
  116. q = 0 q=0
  117. O ( n ) O(n)
  118. Φ Φ
  119. Φ = ( I p 0 0 - I q ) I p , q \Phi=\left(\begin{matrix}I_{p}&0\\ 0&-I_{q}\end{matrix}\right)\equiv I_{p,q}
  120. A φ = ( I p 0 0 - I q ) ( A 11 A n n ) T ( I p 0 0 - I q ) , A^{\varphi}=\left(\begin{matrix}I_{p}&0\\ 0&-I_{q}\end{matrix}\right)\left(\begin{matrix}A_{11}&\cdots\\ \cdots&A_{nn}\end{matrix}\right)^{\mathrm{T}}\left(\begin{matrix}I_{p}&0\\ 0&-I_{q}\end{matrix}\right),
  121. p p
  122. q q
  123. S p ( m , 𝐑 Sp(m,\mathbf{R}
  124. 𝔬 ( p , q ) = { ( X p × p Y p × q Y T W q × q ) | X T = - X , W T = - W } , \mathfrak{o}(p,q)=\left\{\left.\left(\begin{matrix}X_{p\times p}&Y_{p\times q}% \\ Y^{\mathrm{T}}&W_{q\times q}\end{matrix}\right)\right|X^{\mathrm{T}}=-X,\quad W% ^{\mathrm{T}}=-W\right\},
  125. O ( p , q ) = { g GL ( n , ) | I p , q - 1 g T I p , q g = I } . \mathrm{O}(p,q)=\{g\in\mathrm{GL}(n,\mathbb{R})|I_{p,q}^{-1}g^{\mathrm{T}}I_{p% ,q}g=I\}.
  126. O ( p , q ) O(p,q)
  127. O ( q , p ) O(q,p)
  128. O ( p , q ) O ( q , p ) , g σ g σ - 1 , σ = [ 0 0 1 0 1 0 1 0 0 ] . \mathrm{O}(p,q)\rightarrow\mathrm{O}(q,p),\quad g\rightarrow\sigma g\sigma^{-1% },\quad\sigma=\left[\begin{smallmatrix}0&0&\cdots&1\\ \vdots&\vdots&\ddots&\vdots\\ 0&1&\cdots&0\\ 1&0&\cdots&0\end{smallmatrix}\right].
  129. 𝔬 ( 3 , 1 ) = span { ( 0 1 0 0 - 1 0 0 0 0 0 0 0 0 0 0 0 ) , ( 0 0 - 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) , ( 0 0 0 0 0 0 1 0 0 - 1 0 0 0 0 0 0 ) , ( 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ) , ( 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 ) , ( 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 ) } . \mathfrak{o}(3,1)=\mathrm{span}\left\{\left(\begin{smallmatrix}0&1&0&0\\ -1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&-1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&0\\ 0&0&1&0\\ 0&-1&0&0\\ 0&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&1&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&1&0\end{smallmatrix}\right)\right\}.
  130. q q
  131. φ φ
  132. φ ( x , y ) = ξ 1 η m + 1 + ξ 2 η m + 2 + ξ m η 2 m = n - ξ m + 1 η 1 - ξ m + 2 η 2 - ξ 2 m = n η m , \varphi(x,y)=\xi_{1}\eta_{m+1}+\xi_{2}\eta_{m+2}\cdots+\xi_{m}\eta_{2m=n}-\xi_% {m+1}\eta_{1}-\xi_{m+2}\eta_{2}\cdots-\xi_{2m=n}\eta_{m},
  133. n = 2 m n=2m
  134. A u t ( φ ) Aut(φ)
  135. S p ( φ ) = S p ( V ) Sp(φ)=Sp(V)
  136. S p ( m , 𝐑 ) Sp(m,\mathbf{R})
  137. S p ( 2 m , 𝐑 ) Sp(2m,\mathbf{R})
  138. Φ = ( 0 m I m - I m 0 m ) = J m . \Phi=\left(\begin{matrix}0_{m}&I_{m}\\ -I_{m}&0_{m}\end{matrix}\right)=J_{m}.
  139. V = ( X Y Z W ) , V=\left(\begin{matrix}X&Y\\ Z&W\end{matrix}\right),
  140. X , Y , Z , W X,Y,Z,W
  141. m m
  142. ( 0 m - I m I m 0 m ) ( X Y Z W ) T ( 0 m I m - I m 0 m ) = - ( X Y Z W ) \left(\begin{matrix}0_{m}&-I_{m}\\ I_{m}&0_{m}\end{matrix}\right)\left(\begin{matrix}X&Y\\ Z&W\end{matrix}\right)^{\mathrm{T}}\left(\begin{matrix}0_{m}&I_{m}\\ -I_{m}&0_{m}\end{matrix}\right)=-\left(\begin{matrix}X&Y\\ Z&W\end{matrix}\right)
  143. S p ( m , 𝐑 ) Sp(m,\mathbf{R})
  144. 𝔰 𝔭 ( m , ) = { X M n ( ) : J m X + X T J m = 0 } = { ( X Y Z - X T ) | Y T = Y , Z T = Z } , \mathfrak{sp}(m,\mathbb{R})=\{X\in M_{n}(\mathbb{R}):J_{m}X+X^{\mathrm{T}}J_{m% }=0\}=\left\{\left.\left(\begin{matrix}X&Y\\ Z&-X^{\mathrm{T}}\end{matrix}\right)\right|Y^{\mathrm{T}}=Y,Z^{\mathrm{T}}=Z% \right\},
  145. Sp ( m , ) = { g M n ( ) | g T J m g = J m } . \mathrm{Sp}(m,\mathbb{R})=\{g\in M_{n}(\mathbb{R})|g^{\mathrm{T}}J_{m}g=J_{m}\}.
  146. φ φ
  147. φ ( x , y ) = ξ 1 η 1 + ξ 1 η 1 + ξ n η n \varphi(x,y)=\xi_{1}\eta_{1}+\xi_{1}\eta_{1}\cdots+\xi_{n}\eta_{n}
  148. O ( n , 𝐂 ) O(n,\mathbf{C})
  149. 𝐨 ( p , q ) \mathbf{o}(p,q)
  150. 𝔬 ( n , ) = 𝔰 𝔬 ( n , ) = { X | X T = - X } , \mathfrak{o}(n,\mathbb{C})=\mathfrak{so}(n,\mathbb{C})=\{X|X^{\mathrm{T}}=-X\},
  151. O ( n , ) = { g | g T g = I n } . \mathrm{O}(n,\mathbb{C})=\{g|g^{\mathrm{T}}g=I_{n}\}.
  152. 𝐬𝐨 ( n ) \mathbf{so}(n)
  153. n n
  154. n n
  155. φ φ
  156. φ ( x , y ) = ξ 1 η m + 1 + ξ 2 η m + 2 + ξ m η 2 m = n - ξ m + 1 η 1 - ξ m + 2 η 2 - ξ 2 m = n η m , \varphi(x,y)=\xi_{1}\eta_{m+1}+\xi_{2}\eta_{m+2}\cdots+\xi_{m}\eta_{2m=n}-\xi_% {m+1}\eta_{1}-\xi_{m+2}\eta_{2}\cdots-\xi_{2m=n}\eta_{m},
  157. A u t ( φ ) Aut(φ)
  158. S p ( φ ) = S p ( V ) Sp(φ)=Sp(V)
  159. S p ( m , ) Sp(m,ℂ)
  160. S p ( 2 m , ) Sp(2m,ℂ)
  161. 𝐬𝐩 ( m , ) \mathbf{sp}(m,ℝ)
  162. 𝔰 𝔭 ( m , ) = { X M n ( ) : J m X + X T J m = 0 } = { ( X Y Z - X T ) | Y T = Y , Z T = Z } , \mathfrak{sp}(m,\mathbb{C})=\{X\in M_{n}(\mathbb{C}):J_{m}X+X^{\mathrm{T}}J_{m% }=0\}=\left\{\left.\left(\begin{matrix}X&Y\\ Z&-X^{\mathrm{T}}\end{matrix}\right)\right|Y^{\mathrm{T}}=Y,Z^{\mathrm{T}}=Z% \right\},
  163. Sp ( m , ) = { g M n ( ) | g T J m g = J m } . \mathrm{Sp}(m,\mathbb{C})=\{g\in M_{n}(\mathbb{C})|g^{\mathrm{T}}J_{m}g=J_{m}\}.
  164. φ ( x , y ) = ξ ¯ i φ i j η j . \varphi(x,y)=\sum\bar{\xi}_{i}\varphi_{ij}\eta_{j}.
  165. φ ( x , y ) = x * Φ y , A φ = Φ - 1 A * Φ , \varphi(x,y)=x^{*}\Phi y,\qquad A^{\varphi}=\Phi^{-1}A^{*}\Phi,
  166. Aut ( φ ) = { A GL ( V ) : Φ - 1 A * Φ A = 1 } , \operatorname{Aut}(\varphi)=\{A\in\operatorname{GL}(V):\Phi^{-1}A^{*}\Phi A=1\},
  167. 𝔞 𝔲 𝔱 ( φ ) = { X M n ( V ) : Φ - 1 X * Φ = - X } . \mathfrak{aut}(\varphi)=\{X\in M_{n}(V):\Phi^{-1}X^{*}\Phi=-X\}.
  168. i i
  169. φ ( x , y ) = ± ξ 1 ¯ η 1 ± ξ 2 ¯ η 2 ± ξ n ¯ η n . \varphi(x,y)=\pm\bar{\xi_{1}}\eta_{1}\pm\bar{\xi_{2}}\eta_{2}\cdots\pm\bar{\xi% _{n}}\eta_{n}.
  170. U ( V ) U(V)
  171. U ( p , q ) U(p,q)
  172. q = 0 q=0
  173. U ( n ) U(n)
  174. Φ Φ
  175. Φ = ( 1 p 0 0 - 1 q ) = I p , q , \Phi=\left(\begin{matrix}1_{p}&0\\ 0&-1_{q}\end{matrix}\right)=I_{p,q},
  176. 𝔲 ( p , q ) = { ( X p × p Z p × q Z ¯ T Y q × q ) | X ¯ T = - X , Y ¯ T = - Y } . \mathfrak{u}(p,q)=\left\{\left.\left(\begin{matrix}X_{p\times p}&Z_{p\times q}% \\ {\overline{Z}}^{\mathrm{T}}&Y_{q\times q}\end{matrix}\right)\right|{\overline{% X}}^{\mathrm{T}}=-X,\quad{\overline{Y}}^{\mathrm{T}}=-Y\right\}.
  177. U ( p , q ) = { g | I p , q - 1 g * I p , q g = I } . \mathrm{U}(p,q)=\{g|I_{p,q}^{-1}g^{*}I_{p,q}g=I\}.
  178. 𝐇 \mathbf{H}
  179. A ( v h ) = ( A v ) h A(vh)=(Av)h
  180. h h
  181. v v
  182. A A
  183. 𝐇 \mathbf{H}
  184. V V
  185. 𝐇 \mathbf{H}
  186. 𝐇 \mathbf{H}
  187. q = x + 𝐣 y q=x+\mathbf{j}y
  188. α α ¯ + β β ¯ = 1 = d e t Q α\overline{α}+β\overline{β}=1=detQ
  189. S U ( 2 ) SU(2)
  190. n × n n×n
  191. 2 n × 2 n 2n×2n
  192. n n
  193. n n
  194. n × n n×n
  195. 2 n × 2 n 2n×2n
  196. n × n n×n
  197. ( Q ) n × n = ( X ) n × n + j ( Y ) n × n ( X - Y ¯ Y X ¯ ) 2 n × 2 n . \left(Q\right)_{n\times n}=\left(X\right)_{n\times n}+\mathrm{j}\left(Y\right)% _{n\times n}\leftrightarrow\left(\begin{matrix}X&-\bar{Y}\\ Y&\bar{X}\end{matrix}\right)_{2n\times 2n}.
  198. T G L ( 2 n , 𝐂 ) T∈GL(2n,\mathbf{C})
  199. n 2 n , M n ( ) { T M 2 n ( ) | J n T = T ¯ J n , J n = ( 0 I n - I n 0 ) } . \mathbb{H}^{n}\approx\mathbb{C}^{2n},M_{n}(\mathbb{H})\approx\left\{\left.T\in M% _{2n}(\mathbb{C})\right|J_{n}T=\overline{T}J_{n},\quad J_{n}=\left(\begin{% matrix}0&I_{n}\\ -I_{n}&0\end{matrix}\right)\right\}.
  200. 𝐢 \mathbf{i}
  201. i i
  202. 𝐢 ( X + 𝐣 Y ) = ( 𝐢 X ) + 𝐣 ( - 𝐢 Y ) \mathbf{i}(X+\mathbf{j}Y)=(\mathbf{i}X)+\mathbf{j}(-\mathbf{i}Y)
  203. X X
  204. Y Y
  205. n n
  206. A O ( 2 n , 𝐂 ) A∈O(2n,\mathbf{C})
  207. S L ( n , 𝐇 ) SL(n,\mathbf{H})
  208. 𝐂 \mathbf{C}
  209. 𝐇 \mathbf{H}
  210. GL ( n , ) = { g GL ( 2 n , ) | J g = g ¯ J , det g 0 } U * ( 2 n ) . \mathrm{GL}(n,\mathbb{H})=\{g\in\mathrm{GL}(2n,\mathbb{C})|Jg=\overline{g}J,% \mathrm{det}\quad g\neq 0\}\equiv\mathrm{U}^{*}(2n).
  211. 𝐠𝐥 ( n , 𝐇 ) \mathbf{gl}(n,\mathbf{H})
  212. 𝔤 𝔩 ( n , ) = { ( X - Y ¯ Y X ¯ ) | X , Y 𝔤 𝔩 ( n , ) } 𝔲 * ( 2 n ) . \mathfrak{gl}(n,\mathbb{H})=\left\{\left.\left(\begin{matrix}X&-\overline{Y}\\ Y&\overline{X}\end{matrix}\right)\right|X,Y\in\mathfrak{gl}(n,\mathbb{C})% \right\}\equiv\mathfrak{u}^{*}(2n).
  213. SL ( n , ) = { g GL ( n , ) | det g = 1 } SU * ( 2 n ) , \mathrm{SL}(n,\mathbb{H})=\{g\in\mathrm{GL}(n,\mathbb{H})|\mathrm{det}\ g=1\}% \equiv\mathrm{SU}^{*}(2n),
  214. 𝔰 𝔩 ( n , ) = { ( X - Y ¯ Y X ¯ ) | Tr X = 0 } 𝔰 𝔲 * ( 2 n ) . \mathfrak{sl}(n,\mathbb{H})=\left\{\left.\left(\begin{matrix}X&-\overline{Y}\\ Y&\overline{X}\end{matrix}\right)\right|\operatorname{Tr}X=0\right\}\equiv% \mathfrak{su}^{*}(2n).
  215. φ ( x , y ) = ± ξ 1 ¯ η 1 ± ξ 2 ¯ η 2 ± ξ n ¯ η n \varphi(x,y)=\pm\bar{\xi_{1}}\eta_{1}\pm\bar{\xi_{2}}\eta_{2}\cdots\pm\bar{\xi% _{n}}\eta_{n}
  216. S p ( φ ) = S p ( p , q ) Sp(φ)=Sp(p,q)
  217. S p ( n , 𝐂 ) Sp(n,\mathbf{C})
  218. ( 2 p , 2 q ) (2p,2q)
  219. p p
  220. q = 0 q=0
  221. U ( n , 𝐇 ) U(n,\mathbf{H})
  222. Φ = ( I p 0 0 - I q ) = I p , q \Phi=\left(\begin{matrix}I_{p}&0\\ 0&-I_{q}\end{matrix}\right)=I_{p,q}
  223. 𝒬 = ( 𝒳 p × p 𝒵 p × q 𝒵 * 𝒴 q × q ) , 𝒳 * = - 𝒳 , 𝒴 * = - 𝒴 \mathcal{Q}=\left(\begin{matrix}\mathcal{X}_{p\times p}&\mathcal{Z}_{p\times q% }\\ \mathcal{Z}^{*}&\mathcal{Y}_{q\times q}\end{matrix}\right),\qquad\mathcal{X}^{% *}=-\mathcal{X},\mathcal{Y}^{*}=-\mathcal{Y}
  224. Φ - 1 𝒬 * Φ = - 𝒬 , \Phi^{-1}\mathcal{Q}^{*}\Phi=-\mathcal{Q},
  225. 𝐮 ( p , q ) \mathbf{u}(p,q)
  226. I I
  227. - I -I
  228. 𝒳 = ( X 1 ( p × p ) - X ¯ 2 X 2 X ¯ 1 ) , 𝒴 = ( Y 1 ( q × q ) - Y ¯ 2 Y 2 Y ¯ 1 ) , 𝒵 = ( Z 1 ( p × q ) - Z ¯ 2 Z 2 Z ¯ 1 ) , \mathcal{X}=\left(\begin{matrix}X_{1(p\times p)}&-\overline{X}_{2}\\ X_{2}&\overline{X}_{1}\end{matrix}\right),\mathcal{Y}=\left(\begin{matrix}Y_{1% (q\times q)}&-\overline{Y}_{2}\\ Y_{2}&\overline{Y}_{1}\end{matrix}\right),\mathcal{Z}=\left(\begin{matrix}Z_{1% (p\times q)}&-\overline{Z}_{2}\\ Z_{2}&\overline{Z}_{1}\end{matrix}\right),
  229. X 1 * = - X , Y 1 * = - Y . X_{1}^{*}=-X,Y_{1}^{*}=-Y.
  230. 𝔰 𝔭 ( p , q ) = { ( [ X 1 ( p × p ) - X ¯ 2 X 2 X ¯ 1 ] [ Z 1 ( p × q ) - Z ¯ 2 Z 2 Z ¯ 1 ] [ Z 1 ( p × q ) - Z ¯ 2 Z 2 Z ¯ 1 ] * [ Y 1 ( q × q ) - Y ¯ 2 Y 2 Y ¯ 1 ] ) | X 1 * = - X , Y 1 * = - Y } . \mathfrak{sp}(p,q)=\left\{\left(\left.\begin{matrix}\begin{bmatrix}X_{1(p% \times p)}&-\overline{X}_{2}\\ X_{2}&\overline{X}_{1}\end{bmatrix}&\begin{bmatrix}Z_{1(p\times q)}&-\overline% {Z}_{2}\\ Z_{2}&\overline{Z}_{1}\end{bmatrix}\\ \begin{bmatrix}Z_{1(p\times q)}&-\overline{Z}_{2}\\ Z_{2}&\overline{Z}_{1}\end{bmatrix}^{*}&\begin{bmatrix}Y_{1(q\times q)}&-% \overline{Y}_{2}\\ Y_{2}&\overline{Y}_{1}\end{bmatrix}\end{matrix}\right)\right|X_{1}^{*}=-X,Y_{1% }^{*}=-Y\right\}.
  231. Sp ( p , q ) = { g GL ( n , ) | I p , q - 1 g * I p , q g = I p + q } = { g GL ( 2 n , ) | K p , q - 1 g * K p , q g = I 2 ( p + q ) , K = diag ( I p , q , I p , q ) } . \mathrm{Sp}(p,q)=\{g\in\mathrm{GL}(n,\mathbb{H})|I_{p,q}^{-1}g^{*}I_{p,q}g=I_{% p+q}\}=\{g\in\mathrm{GL}(2n,\mathbb{C})|K_{p,q}^{-1}g^{*}K_{p,q}g=I_{2(p+q)},% \qquad K=\mathrm{diag}(I_{p,q},I_{p,q})\}.
  232. φ ( w , z ) φ(w,z)
  233. S p ( p , q ) Sp(p,q)
  234. w u + j v w→u+jv
  235. z x + j y z→x+jy
  236. φ ( w , z ) = [ u * v * ] K p , q [ x y ] + j [ u - v ] K p , q [ y x ] = φ 1 ( w , z ) + 𝐣 φ 2 ( w , z ) , K p , q = diag ( I p , q , I p , q ) \varphi(w,z)=\begin{bmatrix}u^{*}&v^{*}\end{bmatrix}K_{p,q}\begin{bmatrix}x\\ y\end{bmatrix}+j\begin{bmatrix}u&-v\end{bmatrix}K_{p,q}\begin{bmatrix}y\\ x\end{bmatrix}=\varphi_{1}(w,z)+\mathbf{j}\varphi_{2}(w,z),\qquad K_{p,q}=% \mathrm{diag}(I_{p,q},I_{p,q})
  237. 𝐇 \mathbf{H}
  238. S p ( p , q ) Sp(p,q)
  239. ( 2 p , 2 q ) (2p,2q)
  240. 𝐣 \mathbf{j}
  241. Sp ( p , q ) = U ( 2 n , φ 1 ) Sp ( 2 n , φ 2 ) \mathrm{Sp}(p,q)=\mathrm{U}(\mathbb{C}^{2n},\varphi_{1})\cap\mathrm{Sp}(% \mathbb{C}^{2n},\varphi_{2})
  242. φ ( x , y ) = ξ 1 ¯ 𝐣 η 1 + ξ 2 ¯ 𝐣 η 2 + ξ n ¯ 𝐣 η n , \varphi(x,y)=\bar{\xi_{1}}\mathbf{j}\eta_{1}+\bar{\xi_{2}}\mathbf{j}\eta_{2}% \cdots+\bar{\xi_{n}}\mathbf{j}\eta_{n},
  243. 𝐣 \mathbf{j}
  244. ( 𝟏 , 𝐢 , 𝐣 , 𝐤 ) (\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k})
  245. O ( 2 n , 𝐂 ) O(2n,\mathbf{C})
  246. ( n , n ) (n,n)
  247. Φ = ( 𝐣 0 0 0 𝐣 0 0 𝐣 ) j n \Phi=\left(\begin{smallmatrix}\mathbf{j}&0&\cdots&0\\ 0&\mathbf{j}&\cdots&\vdots\\ \vdots&&\ddots&&\\ 0&\cdots&0&\mathbf{j}\end{smallmatrix}\right)\equiv\mathrm{j}_{n}
  248. - Φ V * Φ = - V V * = j n V j n . -\Phi V^{*}\Phi=-V\Leftrightarrow V^{*}=\mathrm{j}_{n}V\mathrm{j}_{n}.
  249. V 𝐨 ( 2 n ) V∈\mathbf{o}(2n)
  250. V = X + 𝐣 Y ( X - Y ¯ Y X ¯ ) V=X+\mathbf{j}Y\leftrightarrow\left(\begin{matrix}X&-\overline{Y}\\ Y&\overline{X}\end{matrix}\right)
  251. Φ Φ
  252. Φ ( 0 - I n I n 0 ) J n . \Phi\leftrightarrow\left(\begin{matrix}0&-I_{n}\\ I_{n}&0\end{matrix}\right)\equiv J_{n}.
  253. ( X - Y ¯ Y X ¯ ) * = ( 0 - I n I n 0 ) ( X - Y ¯ Y X ¯ ) ( 0 - I n I n 0 ) X T = - X , Y ¯ T = Y . \left(\begin{matrix}X&-\overline{Y}\\ Y&\overline{X}\end{matrix}\right)^{*}=\left(\begin{matrix}0&-I_{n}\\ I_{n}&0\end{matrix}\right)\left(\begin{matrix}X&-\overline{Y}\\ Y&\overline{X}\end{matrix}\right)\left(\begin{matrix}0&-I_{n}\\ I_{n}&0\end{matrix}\right)\Leftrightarrow X^{\mathrm{T}}=-X,\quad\overline{Y}^% {\mathrm{T}}=Y.
  254. 𝔬 * ( 2 n ) = { ( X - Y ¯ Y X ¯ ) | X T = - X , Y ¯ T = Y } , \mathfrak{o}^{*}(2n)=\left\{\left.\left(\begin{matrix}X&-\overline{Y}\\ Y&\overline{X}\end{matrix}\right)\right|X^{\mathrm{T}}=-X,\quad\overline{Y}^{% \mathrm{T}}=Y\right\},
  255. O * ( 2 n ) = { g GL ( n , ) | j n - 1 g * j n g = I n } = { g GL ( 2 n , ) | J n - 1 g * J n g = I 2 n } . \mathrm{O}^{*}(2n)=\{g\in\mathrm{GL}(n,\mathbb{H})|\mathrm{j}_{n}^{-1}g^{*}% \mathrm{j}_{n}g=I_{n}\}=\{g\in\mathrm{GL}(2n,\mathbb{C})|J_{n}^{-1}g^{*}J_{n}g% =I_{2n}\}.
  256. O * ( 2 n ) = { g O ( 2 n , ) | θ ( g ¯ ) = g } , \mathrm{O}^{*}(2n)=\{g\in\mathrm{O}(2n,\mathbb{C})|\theta(\overline{g})=g\},
  257. θ : G L ( 2 n , 𝐂 ) G L ( 2 n , 𝐂 ) θ:GL(2n,\mathbf{C})→GL(2n,\mathbf{C})
  258. 𝐇 \mathbf{H}
  259. φ ( x , y ) = w ¯ 2 I n z 1 - w ¯ 1 I n z 2 + 𝐣 ( w 1 I n z 1 + w 2 I n z 2 ) = φ 1 ( w , z ) ¯ + 𝐣 φ 2 ( w , z ) . \varphi(x,y)=\overline{w}_{2}I_{n}z_{1}-\overline{w}_{1}I_{n}z_{2}+\mathbf{j}(% w_{1}I_{n}z_{1}+w_{2}I_{n}z_{2})=\overline{\varphi_{1}(w,z)}+\mathbf{j}\varphi% _{2}(w,z).
  260. ( n , n ) (n,n)
  261. ( 𝐞 , 𝐟 ) (\mathbf{e},\mathbf{f})
  262. ( ( 𝐞 + i 𝐟 ) / 2 , ( 𝐞 i 𝐟 ) / 2 ) ((\mathbf{e}+i\mathbf{f})/√2,(\mathbf{e}−i\mathbf{f})/√2)
  263. 𝐞 , 𝐟 \mathbf{e},\mathbf{f}
  264. n n
  265. 𝐣 \mathbf{j}
  266. O * ( 2 n ) = O ( 2 n , ) U ( 2 n , φ 1 ) , \mathrm{O}^{*}(2n)=\mathrm{O}(2n,\mathbb{C})\cap\mathrm{U}(\mathbb{C}^{2n},% \varphi_{1}),

Classification_of_Fatou_components.html

  1. f = P ( z ) Q ( z ) f=\frac{P(z)}{Q(z)}
  2. max ( deg ( P ) , deg ( Q ) ) 2 , \max(\deg(P),\,\deg(Q))\geq 2,
  3. U U
  4. U U
  5. U U
  6. U U
  7. U U
  8. f : z z - g g ( z ) f:z\mapsto z-\frac{g}{g^{\prime}}(z)
  9. g : z z 3 - 1 g:z\mapsto z^{3}-1
  10. f ( z ) = z - ( z 3 - 1 ) / 3 z 2 f(z)=z-(z^{3}-1)/3z^{2}
  11. z 3 = 1 z^{3}=1
  12. z 3 = 1 z^{3}=1
  13. f ( z ) = e 2 π i t z 2 ( z - 4 ) / ( 1 - 4 z ) f(z)=e^{2\pi it}z^{2}(z-4)/(1-4z)
  14. f ( z ) = z - 1 + ( 1 - 2 z ) e z f(z)=z-1+(1-2z)e^{z}

Clifford_bundle.html

  1. v 2 = - v , v v^{2}=-\langle v,v\rangle
  2. C ( E ) = x M C ( E x , g x ) C\ell(E)=\coprod_{x\in M}C\ell(E_{x},g_{x})
  3. ρ : O ( n ) Aut ( C n ) \rho:\mathrm{O}(n)\to\mathrm{Aut}(C\ell_{n}\mathbb{R})
  4. ρ ( A ) ( v 1 v 2 v k ) = ( A v 1 ) ( A v 2 ) ( A v k ) \rho(A)(v_{1}v_{2}\cdots v_{k})=(Av_{1})(Av_{2})\cdots(Av_{k})
  5. C ( E ) = F ( E ) × ρ C n C\ell(E)=F(E)\times_{\rho}C\ell_{n}\mathbb{R}
  6. C ( E ) = C 0 ( E ) C 1 ( E ) . C\ell(E)=C\ell^{0}(E)\oplus C\ell^{1}(E).
  7. C ( T * M ) Λ ( T * M ) . C\ell(T^{*}M)\cong\Lambda(T^{*}M).
  8. C 0 ( T * M ) = Λ even ( T * M ) C 1 ( T * M ) = Λ odd ( T * M ) . \begin{aligned}\displaystyle C\ell^{0}(T^{*}M)&\displaystyle=\Lambda^{\mathrm{% even}}(T^{*}M)\\ \displaystyle C\ell^{1}(T^{*}M)&\displaystyle=\Lambda^{\mathrm{odd}}(T^{*}M).% \end{aligned}

Clip_coordinates.html

  1. w w

Clock_angle_problem.html

  1. θ hr = 1 2 M Σ = 1 2 ( 60 H + M ) \theta_{\,\text{hr}}=\frac{1}{2}M_{\Sigma}=\frac{1}{2}(60H+M)
  2. θ \scriptstyle\theta
  3. H \scriptstyle H
  4. M \scriptstyle M
  5. M Σ \scriptstyle M_{\Sigma}
  6. θ min. = 6 M \theta_{\,\text{min.}}=6M
  7. θ \scriptstyle\theta
  8. M \scriptstyle M
  9. θ hr = 1 2 ( 60 × 5 + 24 ) = 162 \theta_{\,\text{hr}}=\frac{1}{2}(60\times 5+24)=162
  10. θ min. = 6 × 24 = 144 \theta_{\,\text{min.}}=6\times 24=144
  11. Δ θ \displaystyle\Delta\theta
  12. H \scriptstyle H
  13. M \scriptstyle M
  14. Δ θ = | 1 2 ( 60 × 2 - 11 × 20 ) | = | 1 2 ( 120 - 220 ) | = 50 \begin{aligned}\displaystyle\Delta\theta&\displaystyle=\left|\frac{1}{2}(60% \times 2-11\times 20)\right|\\ &\displaystyle=\left|\frac{1}{2}(120-220)\right|\\ &\displaystyle=50\end{aligned}
  15. θ hr = θ min. 1 2 ( 60 H + M ) = 6 M 11 M = 60 H M = 60 11 H M = 5. 45 ¯ H \begin{aligned}\displaystyle\theta_{\,\text{hr}}&\displaystyle=\theta_{\,\text% {min.}}\\ \displaystyle\Rightarrow\frac{1}{2}(60H+M)&\displaystyle=6M\\ \displaystyle\Rightarrow 11M&\displaystyle=60H\\ \displaystyle\Rightarrow M&\displaystyle=\frac{60}{11}H\\ \displaystyle\Rightarrow M&\displaystyle=5.\overline{45}H\end{aligned}
  16. H \scriptstyle H
  17. 45 ¯ \overline{45}
  18. 90 ¯ \overline{90}
  19. 36 ¯ \overline{36}
  20. 45 ¯ \overline{45}
  21. 27 ¯ \overline{27}

Cloth_modeling.html

  1. F = m a \vec{F}=m\vec{a}
  2. E ( P a r t i c l e i , j ) = k s E s , i , j + k b E b , i , j + k g E g , i , j E(Particle_{i,j})=k_{s}E_{s,i,j}+k_{b}E_{b,i,j}+k_{g}E_{g,i,j}
  3. U T o t a l = U R e p e l + U S t r e t c h + U B e n d + U T r e l l i s + U G r a v i t y U_{Total}=U_{Repel}+U_{Stretch}+U_{Bend}+U_{Trellis}+U_{Gravity}

Clutching_construction.html

  1. S n S^{n}
  2. D + n D^{n}_{+}
  3. D - n D^{n}_{-}
  4. S n - 1 S^{n-1}
  5. F F
  6. G G
  7. f : S n - 1 G f\colon S^{n-1}\to G
  8. S n - 1 × F D + n × F D - n × F S^{n-1}\times F\to D^{n}_{+}\times F\coprod D^{n}_{-}\times F
  9. ( x , v ) ( x , v ) D + n × F (x,v)\mapsto(x,v)\in D^{n}_{+}\times F
  10. ( x , v ) ( x , f ( x ) ( v ) ) D - n × F (x,v)\mapsto(x,f(x)(v))\in D^{n}_{-}\times F
  11. π n - 1 G Fib F ( S n ) \pi_{n-1}G\to\,\text{Fib}_{F}(S^{n})
  12. π n - 1 O ( k ) Vect k ( S n ) \pi_{n-1}O(k)\to\,\text{Vect}_{k}(S^{n})
  13. ( X ; A , B ) (X;A,B)
  14. A B A\cap B
  15. p : M N p:M\to N
  16. F F
  17. 𝒰 \mathcal{U}
  18. ( U i , q i ) (U_{i},q_{i})
  19. q i : p - 1 ( U i ) N × F q_{i}:p^{-1}(U_{i})\to N\times F
  20. p p
  21. U i N U_{i}\subset N
  22. U i U_{i}
  23. N N
  24. i U i = N \coprod_{i}U_{i}=N
  25. i U i × F \coprod_{i}U_{i}\times F
  26. ( u i , f i ) U i × F (u_{i},f_{i})\in U_{i}\times F
  27. ( u j , f j ) U j × F (u_{j},f_{j})\in U_{j}\times F
  28. U i U j ϕ U_{i}\cap U_{j}\neq\phi
  29. q i q j - 1 ( u j , f j ) = ( u i , f i ) q_{i}\circ q_{j}^{-1}(u_{j},f_{j})=(u_{i},f_{i})
  30. q i q_{i}
  31. p p
  32. i U i × H o m e o ( F ) \coprod_{i}U_{i}\times Homeo(F)
  33. ( u i , h i ) U i × H o m e o ( F ) (u_{i},h_{i})\in U_{i}\times Homeo(F)
  34. ( u j , h j ) U j × H o m e o ( F ) (u_{j},h_{j})\in U_{j}\times Homeo(F)
  35. U i U j ϕ U_{i}\cap U_{j}\neq\phi
  36. q i q j - 1 q_{i}\circ q_{j}^{-1}
  37. q i q j - 1 : U i U j H o m e o ( F ) q_{i}\circ q_{j}^{-1}:U_{i}\cap U_{j}\to Homeo(F)
  38. q i q j - 1 ( u j ) ( h j ) = h i q_{i}\circ q_{j}^{-1}(u_{j})(h_{j})=h_{i}
  39. p p
  40. F F
  41. H o m e o ( F ) Homeo(F)
  42. H o m e o ( F ) Homeo(F)
  43. B B
  44. H o m e o ( F ) Homeo(F)
  45. p : M p N p:M_{p}\to N
  46. ( M p × F ) / H o m e o ( F ) = M (M_{p}\times F)/Homeo(F)=M
  47. H o m e o ( F ) M p N Homeo(F)\to M_{p}\to N
  48. M p N B ( H o m e o ( F ) ) M_{p}\to N\to B(Homeo(F))
  49. B ( H o m e o ( F ) ) B(Homeo(F))
  50. H o m e o ( F ) Homeo(F)
  51. G G
  52. G M p N G\to M_{p}\to N
  53. M p × G E G M_{p}\times_{G}EG
  54. M p × G E G M p / G = N M_{p}\times_{G}EG\to M_{p}/G=N
  55. E G EG
  56. M p × G E G E G / G = B G M_{p}\times_{G}EG\to EG/G=BG
  57. M p M_{p}
  58. M p N M p × G E G B G M_{p}\to N\simeq M_{p}\times_{G}EG\to BG
  59. p : M N p:M\to N
  60. G M p N G\to M_{p}\to N
  61. G E G B G G\to EG\to BG
  62. p p
  63. S n - 1 S n - 1 S^{n-1}\to S^{n-1}
  64. S n - 1 G S^{n-1}\to G

Clutter_(radar).html

  1. R R
  2. τ \tau
  3. τ \tau
  4. R R
  5. θ / 2 \theta/2
  6. ϕ / 2 \phi/2
  7. V m = π R tan ( θ / 2 ) R tan ( ϕ / 2 ) ( c τ / 2 ) \ V_{m}=\pi R\tan(\theta/2)R\tan(\phi/2)(c\tau/2)
  8. V m π 4 ( R θ ) ( R ϕ ) ( c τ / 2 ) \ V_{m}\approx\frac{\pi}{4}(R\theta)(R\phi)(c\tau/2)
  9. η \eta
  10. C = P t G t A r ( 4 π ) 2 R 4 π 4 ( R θ ) ( R ϕ ) ( c τ / 2 ) η \ C=\frac{P_{t}G_{t}A_{r}}{(4\pi)^{2}R^{4}}\frac{\pi}{4}(R\theta)(R\phi)(c\tau% /2)\eta
  11. P t P_{t}
  12. G t G_{t}
  13. A r A_{r}
  14. R R
  15. C = P t G t A r 2 log 2 ( 4 π ) 2 R 4 π 4 ( R θ ) ( R ϕ ) ( c τ / 2 ) η \ C=\frac{P_{t}G_{t}A_{r}}{2\log 2(4\pi)^{2}R^{4}}\frac{\pi}{4}(R\theta)(R\phi% )(c\tau/2)\eta
  16. A r = G λ 2 4 π \ A_{r}=\frac{G\lambda^{2}}{4\pi}
  17. G = π 2 θ ϕ \ G=\frac{\pi^{2}}{\theta\phi}
  18. C = P t G 2 λ 2 1024 ( log 2 ) π 2 R 2 c τ η \ C=\frac{P_{t}G^{2}\lambda^{2}}{1024(\log 2)\pi^{2}R^{2}}c\tau\eta
  19. S = P t G 2 λ 2 ( 4 π ) 3 R 4 σ \ S=\frac{P_{t}G^{2}\lambda^{2}}{(4\pi)^{3}R^{4}}\sigma
  20. S C = 1024 ( log 2 ) G σ ( 4 π ) 3 R 2 c τ η \ \frac{S}{C}=\frac{1024(\log 2)G\sigma}{(4\pi)^{3}R^{2}c\tau\eta}
  21. G = π 2 θ ϕ \ G=\frac{\pi^{2}}{\theta\phi}
  22. S C = 16 ( log 2 ) G σ ( 4 π ) 3 R 2 θ ϕ c τ η \ \frac{S}{C}=\frac{16(\log 2)G\sigma}{(4\pi)^{3}R^{2}\theta\phi c\tau\eta}
  23. 2 R tan θ / 2 \ 2R\tan\theta/2
  24. ( c τ / 2 ) sec ψ \ (c\tau/2)\sec\psi
  25. A = 2 R ( c τ / 2 ) ( tan θ / 2 ) sec ψ \ A=2R(c\tau/2)(\tan\theta/2)\sec\psi
  26. A = R ( c τ / 2 ) θ sec ψ \ A=R(c\tau/2)\theta\sec\psi
  27. C = P t G 2 λ 2 ( 4 π ) 3 R 4 A σ o \ C=\frac{P_{t}G^{2}\lambda^{2}}{(4\pi)^{3}R^{4}}A\sigma^{o}
  28. A A
  29. C = c 2 7 π 3 P t G 2 λ 2 R 3 τ θ sec ψ σ o \ C=\frac{c}{2^{7}\pi^{3}}\frac{P_{t}G^{2}\lambda^{2}}{R^{3}}\tau\theta\sec% \psi\sigma^{o}
  30. σ o \sigma^{o}
  31. θ \theta
  32. C = 1300 P t G 2 λ 2 R 3 τ θ o sec ψ σ o \ C=1300\frac{P_{t}G^{2}\lambda^{2}}{R^{3}}\tau\theta^{o}\sec\psi\sigma^{o}
  33. S C = 1 1300 R 3 P t G 2 λ 2 1 τ θ sec ψ σ o P t G 2 λ 2 ( 4 π ) 3 R 4 σ \ \frac{S}{C}=\frac{1}{1300}\frac{R^{3}}{P_{t}G^{2}\lambda^{2}}\frac{1}{\tau% \theta\sec\psi\sigma^{o}}\frac{P_{t}G^{2}\lambda^{2}}{(4\pi)^{3}R^{4}}\sigma
  34. S C = 4 × 10 - 7 cos ψ R τ θ σ σ o \ \frac{S}{C}=4\times 10^{-7}\frac{\cos\psi}{R\tau\theta}\frac{\sigma}{\sigma^% {o}}
  35. A = π R 2 tan 2 θ / 2 \ A=\pi R^{2}\tan^{2}\theta/2
  36. A π R 2 θ 2 / 4 \ A\approx\pi R^{2}\theta^{2}/4
  37. C = P t G 2 λ 2 ( 4 π ) 3 R 4 A σ o \ C=\frac{P_{t}G^{2}\lambda^{2}}{(4\pi)^{3}R^{4}}A\sigma^{o}
  38. A A
  39. C = P t G 2 λ 2 ( 4 π ) 3 R 4 π R 2 ( θ / 2 ) 2 σ o \ C=\frac{P_{t}G^{2}\lambda^{2}}{(4\pi)^{3}R^{4}}\pi R^{2}(\theta/2)^{2}\sigma% ^{o}
  40. C = P t G 2 λ 2 4 4 π 2 R 2 θ 2 σ o \ C=\frac{P_{t}G^{2}\lambda^{2}}{4^{4}\pi^{2}R^{2}}\theta^{2}\sigma^{o}
  41. θ \theta
  42. C = P t G 2 λ 2 4 4 R 2 ( θ o / 180 ) 2 σ o \ C=\frac{P_{t}G^{2}\lambda^{2}}{4^{4}R^{2}}(\theta^{o}/180)^{2}\sigma^{o}
  43. S C = 4 4 R 2 P t G 2 λ 2 ( 180 / θ o ) 2 1 σ o P t G 2 λ 2 ( 4 π ) 3 R 4 σ \ \frac{S}{C}=\frac{4^{4}R^{2}}{P_{t}G^{2}\lambda^{2}}(180/\theta^{o})^{2}% \frac{1}{\sigma^{o}}\frac{P_{t}G^{2}\lambda^{2}}{(4\pi)^{3}R^{4}}\sigma
  44. S C = 5.25 × 10 4 1 θ o 2 R 2 σ σ o \ \frac{S}{C}=5.25\times 10^{4}\frac{1}{\theta^{o2}R^{2}}\frac{\sigma}{\sigma^% {o}}

CM-field.html

  1. \mathbb{C}
  2. \mathbb{R}
  3. \mathbb{R}
  4. α \sqrt{\alpha}
  5. \mathbb{Q}
  6. K K^{\prime}
  7. \mathbb{C}
  8. \mathbb{Z}
  9. ( ζ n ) \mathbb{Q}(\zeta_{n})
  10. ( ζ n + ζ n - 1 ) . \mathbb{Q}(\zeta_{n}+\zeta_{n}^{-1}).
  11. ( ζ n ) \mathbb{Q}(\zeta_{n})
  12. ζ n 2 + ζ n - 2 - 2 = ( ζ n - ζ n - 1 ) 2 . \zeta_{n}^{2}+\zeta_{n}^{-2}-2=(\zeta_{n}-\zeta_{n}^{-1})^{2}.
  13. 𝐐 ¯ \overline{\mathbf{Q}}
  14. 𝐐 ¯ \overline{\mathbf{Q}}
  15. 𝐐 ¯ \overline{\mathbf{Q}}

Co-orbital_configuration.html

  1. ( λ = ϖ + M ) ({\lambda}=\varpi+M)
  2. ( ϖ = Ω + ω ) (\varpi=\Omega+\omega)
  3. ( Δ λ , Δ ϖ ) ({\Delta}{\lambda},{\Delta}\varpi)

Cobalt(II)_nitrate.html

  1. 𝖢𝗈 + 𝟦 𝖧 𝖭 𝖮 𝟥 𝖢𝗈 ( 𝖭𝖮 𝟥 ) 𝟤 + 𝟤 𝖭 𝖮 𝟤 + 𝟤 𝖧 𝟤 𝖮 \mathsf{Co+4HNO_{3}\ \xrightarrow{}\ Co(NO_{3})_{2}+2NO_{2}+2H_{2}O}
  2. 𝖢𝗈𝖮 + 𝟤 𝖧 𝖭 𝖮 𝟥 𝖢𝗈 ( 𝖭𝖮 𝟥 ) 𝟤 + 𝖧 𝟤 𝖮 \mathsf{CoO+2HNO_{3}\ \xrightarrow{}\ Co(NO_{3})_{2}+H_{2}O}
  3. 𝖢𝗈𝖢𝖮 𝟥 + 𝟤 𝖧 𝖭 𝖮 𝟥 𝖢𝗈 ( 𝖭𝖮 𝟥 ) 𝟤 + 𝖢𝖮 𝟤 + 𝖧 𝟤 𝖮 \mathsf{CoCO_{3}+2HNO_{3}\ \xrightarrow{}\ Co(NO_{3})_{2}+CO_{2}\uparrow+H_{2}O}

Coble_creep.html

  1. d ϵ d t = σ d 3 D g b e - Q C o b l e / R T \frac{d\epsilon}{dt}=\frac{\sigma}{d^{3}}D_{gb}e^{-Q_{Coble}/RT}
  2. σ \sigma
  3. d d
  4. D g b D_{gb}
  5. - Q C o b l e -Q_{Coble}
  6. R R
  7. T T
  8. d ϵ d t \frac{d\epsilon}{dt}
  9. σ \sigma
  10. d d
  11. d - 3 d^{-3}
  12. d - 2 d^{-2}
  13. n n
  14. d ϵ d t α d n \frac{d\epsilon}{dt}~{}\alpha~{}d^{n}

Coding_gain.html

  1. ρ 2 \rho\leq 2
  2. γ eff ( A ) \gamma_{\mathrm{eff}}(A)
  3. A A
  4. P b ( E ) P_{b}(E)
  5. E b / N 0 E_{b}/N_{0}
  6. P b ( E ) P_{b}(E)
  7. A A
  8. E b / N 0 E_{b}/N_{0}
  9. P b ( E ) P_{b}(E)
  10. γ c ( A ) \gamma_{c}(A)
  11. γ c ( A ) = d min 2 ( A ) 4 E b . \gamma_{c}(A)=\frac{d^{2}_{\min}(A)}{4E_{b}}.
  12. γ c ( A ) = 1 \gamma_{c}(A)=1
  13. K b ( A ) K_{b}(A)
  14. γ eff ( A ) \gamma_{\mathrm{eff}}(A)
  15. γ c ( A ) \gamma_{c}(A)
  16. K b ( A ) > 1 K_{b}(A)>1
  17. γ eff ( A ) \gamma_{\mathrm{eff}}(A)
  18. γ c ( A ) \gamma_{c}(A)
  19. P b ( E ) P_{b}(E)
  20. E b / N 0 E_{b}/N_{0}
  21. P b ( E ) P_{b}(E)
  22. P b ( E ) K b ( A ) Q 2 γ c ( A ) E b N 0 , P_{b}(E)\approx K_{b}(A)Q\sqrt{\frac{2\gamma_{c}(A)E_{b}}{N_{0}}},
  23. C C
  24. ( n , k , d ) (n,k,d)
  25. ρ = 2 k / n \rho=2k/n
  26. P b ( E ) 10 - 5 P_{b}(E)\approx 10^{-5}
  27. n 64 n\leq 64
  28. ρ > 2 b / 2 D \rho>2b/2D
  29. γ eff ( A ) \gamma_{\mathrm{eff}}(A)
  30. A A
  31. P s ( E ) P_{s}(E)
  32. S N R norm SNR_{\mathrm{norm}}
  33. P s ( E ) P_{s}(E)
  34. A A
  35. S N R norm SNR_{\mathrm{norm}}
  36. P s ( E ) P_{s}(E)
  37. γ c ( A ) \gamma_{c}(A)
  38. γ c ( A ) = ( 2 ρ - 1 ) d min 2 ( A ) 6 E s . \gamma_{c}(A)={(2^{\rho}-1)d^{2}_{\min}(A)\over 6E_{s}}.
  39. γ c ( A ) = 1 \gamma_{c}(A)=1
  40. P s ( E ) K s ( A ) Q 3 γ c ( A ) S N R norm , P_{s}(E)\approx K_{s}(A)Q\sqrt{3\gamma_{c}(A)SNR_{\mathrm{norm}}},
  41. K s ( A ) K_{s}(A)

Coercive_function.html

  1. f ( x ) x x + as x + , \frac{f(x)\cdot x}{\|x\|}\to+\infty\mbox{ as }~{}\|x\|\to+\infty,
  2. \cdot
  3. x \|x\|
  4. f ( x ) ( f ( x ) x ) / x \|f(x)\|\geq(f(x)\cdot x)/\|x\|
  5. x n { 0 } x\in\mathbb{R}^{n}\setminus\{0\}
  6. f ( x ) x = 0 f(x)\cdot x=0
  7. x 2 x\in\mathbb{R}^{2}
  8. A : H H , A:H\to H,
  9. H H
  10. c > 0 c>0
  11. A x , x c x 2 \langle Ax,x\rangle\geq c\|x\|^{2}
  12. x x
  13. H . H.
  14. a : H × H a:H\times H\to\mathbb{R}
  15. c > 0 c>0
  16. a ( x , x ) c x 2 a(x,x)\geq c\|x\|^{2}
  17. x x
  18. H . H.
  19. a ( x , y ) = a ( y , x ) a(x,y)=a(y,x)
  20. x , y x,y
  21. H H
  22. | a ( x , y ) | k x y |a(x,y)|\leq k\|x\|\,\|y\|
  23. x , y x,y
  24. H H
  25. k > 0 k>0
  26. a a
  27. a ( x , y ) = A x , y a(x,y)=\langle Ax,y\rangle
  28. A : H H , A:H\to H,
  29. A , A,
  30. a a
  31. A : H H A:H\to H
  32. f : X X f:X\to X^{\prime}
  33. ( X , ) (X,\|\cdot\|)
  34. ( X , ) (X^{\prime},\|\cdot\|^{\prime})
  35. f ( x ) + as x + \|f(x)\|^{\prime}\to+\infty\mbox{ as }~{}\|x\|\to+\infty
  36. f : X X f:X\to X^{\prime}
  37. X X
  38. X X^{\prime}
  39. K K^{\prime}
  40. X X^{\prime}
  41. K K
  42. X X
  43. f ( X K ) X K . f(X\setminus K)\subseteq X^{\prime}\setminus K^{\prime}.
  44. f : n { - , + } f:\mathbb{R}^{n}\to\mathbb{R}\cup\{-\infty,+\infty\}
  45. f ( x ) + as x + . f(x)\to+\infty\mbox{ as }~{}\|x\|\to+\infty.
  46. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  47. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  48. \mathbb{R}

Coherence_time_(communications_systems).html

  1. x ( t ) x(t)
  2. t 1 t_{1}
  3. y t 1 ( t ) = x ( t - t 1 ) * h t 1 ( t ) , y_{t_{1}}(t)=x(t-t_{1})*h_{t_{1}}(t),
  4. h t 1 ( t ) h_{t_{1}}(t)
  5. t 1 t_{1}
  6. t 2 t_{2}
  7. y t 2 ( t ) = x ( t - t 2 ) * h t 2 ( t ) . y_{t_{2}}(t)=x(t-t_{2})*h_{t_{2}}(t).
  8. h t 1 ( t ) - h t 2 ( t ) h_{t_{1}}(t)-h_{t_{2}}(t)
  9. t 1 t_{1}
  10. t 2 t_{2}
  11. T c T_{c}
  12. T c = t 2 - t 1 . T_{c}=t_{2}-t_{1}.
  13. f d f_{d}
  14. T c = 9 16 π f d 2 T_{c}=\sqrt{\frac{9}{16\pi f_{d}^{2}}}
  15. T c = 9 16 π 1 f d 0.423 f d T_{c}=\sqrt{\frac{9}{16\pi}}\frac{1}{f_{d}}\simeq\frac{0.423}{f_{d}}

Coherent_sampling.html

  1. f i n f_{in}
  2. f s f_{s}
  3. M c y c l e s M_{cycles}
  4. N s a m p l e s N_{samples}
  5. f i n f s = M c y c l e s N s a m p l e s . \frac{f_{in}}{f_{s}}=\frac{M_{cycles}}{N_{samples}}.
  6. N s a m p l e s N_{samples}
  7. M c y c l e s M_{cycles}
  8. N s a m p l e s = 2 11 = 2048 N_{samples}=2^{11}=2048
  9. f s = 100 e 6 f_{s}=100e6
  10. f s / 2 f_{s}/2
  11. f i n = 44 M H z f_{in}=44MHz
  12. M c y c l e s = 901.12 M_{cycles}=901.12
  13. M c y c l e s = 901 M_{cycles}=901
  14. f i n = 43994140.625 H z f_{in}=43994140.625Hz
  15. M M

Cohomological_dimension.html

  1. \to
  2. \to
  3. 𝐙 ^ \mathbf{\hat{Z}}
  4. 𝐙 ^ \mathbf{\hat{Z}}

COIN-OR.html

  1. c 1 x 1 + c 2 x 2 c_{1}x_{1}+c_{2}x_{2}\,
  2. a 11 x 1 + a 12 x 2 b 1 a_{11}x_{1}+a_{12}x_{2}\leq b_{1}
  3. a 21 x 1 + a 22 x 2 b 2 a_{21}x_{1}+a_{22}x_{2}\leq b_{2}
  4. a 31 x 1 + a 32 x 2 b 3 a_{31}x_{1}+a_{32}x_{2}\leq b_{3}
  5. x 1 0 x_{1}\geq 0
  6. x 2 0 x_{2}\geq 0

Cold-air_damming.html

  1. 2 x = x 3 - x 2 d 2 - 3 - x 2 - x 1 d 1 - 2 1 2 ( d 2 - 3 + d 1 - 2 ) \nabla^{2}x=\frac{\frac{x_{3}-x_{2}}{d_{2-3}}-\frac{x_{2}-x_{1}}{d_{1-2}}}{% \frac{1}{2}(d_{2-3}+d_{1-2})}
  2. 2 x \nabla^{2}x
  3. R i = g Δ θ v / θ v [ ( Δ U ) 2 + ( Δ V ) 2 ] / Δ Z Ri=\frac{g\Delta\theta_{v}/\theta_{v}}{[(\Delta U)^{2}+(\Delta V)^{2}]/\Delta Z}

Colinear_map.html

  1. ρ M : M M C , ρ N : N N C \rho_{M}:M\rightarrow M\otimes C,\rho_{N}:N\rightarrow N\otimes C
  2. f : M N f:M\rightarrow N
  3. ρ N f = ( f 1 ) ρ M \rho_{N}\circ f=(f\otimes 1)\circ\rho_{M}

Collectionwise_normal_space.html

  1. X X
  2. X X
  3. \mathcal{F}
  4. X X
  5. X X
  6. \mathcal{F}
  7. X X

Collision_problem.html

  1. n n
  2. f : { 1 , , n } { 1 , , n } f:\,\{1,\ldots,n\}\rightarrow\{1,\ldots,n\}
  3. f ( i ) f(i)
  4. i { 1 , , n } i\in\{1,\ldots,n\}
  5. n / 2 + 1 n/2+1
  6. n / r + 1 n/r+1
  7. n / r + 1 n/r+1
  8. n / r + 1 n/r+1
  9. n / r n/r
  10. n / r + 1 n/r+1
  11. Θ ( n ) \Theta(\sqrt{n})
  12. O ( n 1 / 3 ) O(n^{1/3})

Color–color_diagram.html

  1. C - D = ν c - ν d ν a - ν b ( A - B ) + k , C-D=\frac{\nu_{c}-\nu_{d}}{\nu_{a}-\nu_{b}}(A-B)+k,
  2. ν a \nu_{a}
  3. ν b \nu_{b}
  4. ν c \nu_{c}
  5. ν d \nu_{d}
  6. k = - 2.5 log 10 [ ( < m t p l > ν c ν d ) 2 ( Δ c Δ d ) ( ν b ν a ) 2 ν c - ν d ν a - ν b ( Δ b Δ a ) ν c - ν d ν a - ν b ] k=-2.5\log_{10}\left[{\left({\frac{<}{m}tpl>{{\nu_{c}}}{{\nu_{d}}}}\right)^{2}% \left({\frac{{\Delta_{c}}}{{\Delta_{d}}}}\right)\left({\frac{{\nu_{b}}}{{\nu_{% a}}}}\right)^{2\frac{{\nu_{c}-\nu_{d}}}{{\nu_{a}-\nu_{b}}}}\left({\frac{{% \Delta_{b}}}{{\Delta_{a}}}}\right)^{\frac{{\nu_{c}-\nu_{d}}}{{\nu_{a}-\nu_{b}}% }}}\right]

Commutant_lifting_theorem.html

  1. R T n = P H S U n | H n 0 , RT^{n}=P_{H}SU^{n}|_{H}\;\forall n\geq 0,
  2. S = R . \|S\|=\|R\|.

Compact_complement_topology.html

  1. \scriptstyle\mathbb{R}
  2. X \scriptstyle X\subseteq\mathbb{R}
  3. X \scriptstyle\mathbb{R}\setminus X
  4. \scriptstyle\mathbb{R}

Competitive_equilibrium.html

  1. U ( x ) - P x U ( y ) - P y U(x)-Px\geq U(y)-Py
  2. ϵ \epsilon
  3. P P
  4. x x
  5. P ϵ x P^{x}_{\epsilon}
  6. ϵ \epsilon
  7. ϵ \epsilon
  8. P ϵ x P^{x}_{\epsilon}
  9. ϵ \epsilon
  10. P ϵ x P^{x}_{\epsilon}
  11. i = 1 , , n i=1,...,n
  12. i i
  13. u ( i ) u(i)
  14. u u
  15. i i
  16. k n k\leq n
  17. p p
  18. u ( n - k ) p u ( n - k + 1 ) u(n-k)\leq p\leq u(n-k+1)
  19. P b P_{b}
  20. P a P_{a}
  21. J 1 J_{1}
  22. K 1 K_{1}
  23. P b P_{b}
  24. P a P_{a}
  25. J 2 J_{2}
  26. K 2 K_{2}
  27. P b / P a P_{b}/P_{a}
  28. M R S J a n e = P b / P a MRS_{Jane}=P_{b}/P_{a}
  29. M R S K e l v i n = P b / P a MRS_{Kelvin}=P_{b}/P_{a}
  30. Δ demand ( X ) Δ price ( Y ) 0 \frac{\Delta\,\text{demand}(X)}{\Delta\,\text{price}(Y)}\geq 0

Complement_(group_theory).html

  1. G = H K = { h k : h H , k K } and H K = { e } . G=HK=\{hk:h\in H,k\in K\}\,\text{ and }H\cap K=\{e\}.

Complementary_sequences.html

  1. R x ( k ) = j = 0 N - k - 1 x j x j + k . R_{x}(k)=\sum_{j=0}^{N-k-1}x_{j}x_{j+k}.\,
  2. R a ( k ) + R b ( k ) = 0 , R_{a}(k)+R_{b}(k)=0,\,
  3. R a ( k ) + R b ( k ) = C δ ( k ) , R_{a}(k)+R_{b}(k)=C\delta(k),\,
  4. S a + S b = C S , S_{a}+S_{b}=C_{S},
  5. S a = C S - S b < C S , S_{a}=C_{S}-S_{b}<C_{S},
  6. S b < C S . S_{b}<C_{S}.
  7. | A ( z ) | 2 + | B ( z ) | 2 = 2 N |A(z)|^{2}+|B(z)|^{2}=2N\,

Complete_quotient.html

  1. x = [ a 0 ; a 1 , a 2 , a 3 , ] = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 , x=[a_{0};a_{1},a_{2},a_{3},\dots]=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac% {1}{a_{3}+\cfrac{1}{\ddots}}}},
  2. ζ 0 = [ a 0 ; a 1 , a 2 , a 3 , ] ζ 1 = [ a 1 ; a 2 , a 3 , a 4 , ] ζ 2 = [ a 2 ; a 3 , a 4 , a 5 , ] ζ k = [ a k ; a k + 1 , a k + 2 , a k + 3 , ] . \begin{aligned}\displaystyle\zeta_{0}&\displaystyle=[a_{0};a_{1},a_{2},a_{3},% \dots]\\ \displaystyle\zeta_{1}&\displaystyle=[a_{1};a_{2},a_{3},a_{4},\dots]\\ \displaystyle\zeta_{2}&\displaystyle=[a_{2};a_{3},a_{4},a_{5},\dots]\\ \displaystyle\zeta_{k}&\displaystyle=[a_{k};a_{k+1},a_{k+2},a_{k+3},\dots].% \end{aligned}
  3. ζ k = a k + 1 ζ k + 1 = [ a k ; ζ k + 1 ] , \zeta_{k}=a_{k}+\frac{1}{\zeta_{k+1}}=[a_{k};\zeta_{k+1}],\,
  4. ζ k + 1 = 1 ζ k - a k . \zeta_{k+1}=\frac{1}{\zeta_{k}-a_{k}}.\,
  5. x = A k ζ k + 1 + A k - 1 B k ζ k + 1 + B k - 1 x=\frac{A_{k}\zeta_{k+1}+A_{k-1}}{B_{k}\zeta_{k+1}+B_{k-1}}\,
  6. A 0 < A 2 B 2 < A 4 B 4 < < A 2 n B 2 n < x < A 2 n + 1 B 2 n + 1 < < A 5 B 5 < A 3 B 3 < A 1 B 1 . A_{0}<\frac{A_{2}}{B_{2}}<\frac{A_{4}}{B_{4}}<\cdots<\frac{A_{2n}}{B_{2n}}<x<% \frac{A_{2n+1}}{B_{2n+1}}<\cdots<\frac{A_{5}}{B_{5}}<\frac{A_{3}}{B_{3}}<\frac% {A_{1}}{B_{1}}.\,
  7. f ( x ) = a + b x c + d x f(x)=\frac{a+bx}{c+dx}\,
  8. y = f ( x ) = a + b x c + d x y=f(x)=\frac{a+bx}{c+dx}\,
  9. x = a + b ϕ c + d ϕ . x=\frac{a+b\phi}{c+d\phi}.\,
  10. y = a + b ϕ c + d ϕ y=\frac{a+b\phi}{c+d\phi}\,

Complete_set_of_invariants.html

  1. f i : X Y i f_{i}:X\to Y_{i}\,
  2. Y i Y_{i}
  3. x x x\sim x^{\prime}
  4. f i ( x ) = f i ( x ) f_{i}(x)=f_{i}(x^{\prime})
  5. f i : ( X / ) Y i \prod f_{i}:(X/\sim)\to\prod Y_{i}
  6. f i : X Y i . \prod f_{i}:X\to\prod Y_{i}.

Complete_theory.html

  1. S S\!
  2. A B S A\land B\in S
  3. A S A\in S
  4. B S B\in S
  5. S S\!
  6. A B S A\lor B\in S
  7. A S A\in S
  8. B S B\in S

Complete_topological_space.html

  1. X X
  2. X X
  3. X X

Completion_(algebra).html

  1. E = F 0 E F 1 E F 2 E E=F^{0}{E}\supset F^{1}{E}\supset F^{2}{E}\supset\cdots\,
  2. E ^ = lim ( E / F n E ) . \hat{E}=\underleftarrow{\lim}(E/F^{n}{E}).\,
  3. I = 𝔪 I=\mathfrak{m}
  4. F 0 R = R I I 2 , F n R = I n . F^{0}{R}=R\supset I\supset I^{2}\supset\cdots,\quad F^{n}{R}=I^{n}.
  5. R ^ I = lim ( R / I n ) \hat{R}_{I}=\underleftarrow{\lim}(R/I^{n})
  6. x + I n M for x M . x+I^{n}M\quad\,\text{for }x\in M.
  7. M ^ I = lim ( M / I n M ) . \hat{M}_{I}=\underleftarrow{\lim}(M/I^{n}{M}).
  8. R ^ I \hat{R}_{I}
  9. 𝔪 = ( x 1 , , x n ) \mathfrak{m}=(x_{1},\ldots,x_{n})
  10. R ^ 𝔪 \hat{R}_{\mathfrak{m}}
  11. f ^ : R ^ S ^ . \hat{f}:\hat{R}\to\hat{S}.
  12. f ^ : M ^ N ^ , \hat{f}:\hat{M}\to\hat{N},\quad
  13. M ^ , N ^ \hat{M},\hat{N}
  14. R ^ . \hat{R}.
  15. M ^ = M R R ^ . \hat{M}=M\otimes_{R}\hat{R}.
  16. 𝔪 \mathfrak{m}
  17. R K [ [ x 1 , , x n ] ] / I R\simeq K[[x_{1},\ldots,x_{n}]]/I

Complex_conjugate_root_theorem.html

  1. x 3 - 7 x 2 + 41 x - 87 x^{3}-7x^{2}+41x-87\,
  2. 3 , 2 + 5 i , 2 - 5 i , 3,\,2+5i,\,2-5i,
  3. ( x - 3 ) ( x - 2 - 5 i ) ( x - 2 + 5 i ) . (x-3)(x-2-5i)(x-2+5i).\,
  4. ( x - 3 ) ( x 2 - 4 x + 29 ) . (x-3)(x^{2}-4x+29).\,
  5. P ( z ) = a 0 + a 1 z + a 2 z 2 + + a n z n P(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots+a_{n}z^{n}
  6. P ( ζ ¯ ) = 0 P(\overline{\zeta})=0
  7. a 0 + a 1 ζ + a 2 ζ 2 + + a n ζ n = 0 a_{0}+a_{1}\zeta+a_{2}\zeta^{2}+\cdots+a_{n}\zeta^{n}=0\,
  8. r = 0 n a r ζ r = 0. \sum_{r=0}^{n}a_{r}\zeta^{r}=0.
  9. P ( ζ ¯ ) = r = 0 n a r ( ζ ¯ ) r P(\overline{\zeta})=\sum_{r=0}^{n}a_{r}\left(\overline{\zeta}\right)^{r}
  10. r = 0 n a r ( ζ ¯ ) r = r = 0 n a r ζ r ¯ = r = 0 n a r ζ r ¯ = r = 0 n a r ζ r ¯ . \sum_{r=0}^{n}a_{r}\left(\overline{\zeta}\right)^{r}=\sum_{r=0}^{n}a_{r}% \overline{\zeta^{r}}=\sum_{r=0}^{n}\overline{a_{r}\zeta^{r}}=\overline{\sum_{r% =0}^{n}a_{r}\zeta^{r}}.
  11. r = 0 n a r ζ r ¯ = 0 ¯ \overline{\sum_{r=0}^{n}a_{r}\zeta^{r}}=\overline{0}
  12. r = 0 n a r ( ζ ¯ ) r = 0 ¯ = 0. \sum_{r=0}^{n}a_{r}\left(\overline{\zeta}\right)^{r}=\overline{0}=0.
  13. P ( ζ ¯ ) = a 0 + a 1 ζ ¯ + a 2 ( ζ ¯ ) 2 + + a n ( ζ ¯ ) n = 0. P(\overline{\zeta})=a_{0}+a_{1}\overline{\zeta}+a_{2}\left(\overline{\zeta}% \right)^{2}+\cdots+a_{n}\left(\overline{\zeta}\right)^{n}=0.

Complex_gain.html

  1. V i ( t ) = 1 V sin ( ω t ) V_{i}(t)=1V\cdot\sin(\omega\cdot t)
  2. V o ( t ) = 2 V cos ( ω t ) V_{o}(t)=2V\cdot\cos(\omega\cdot t)
  3. G = 2 V j 1 V = - 2 j . G=\frac{2V}{j\cdot 1V}=-2j.

Complex_geodesic.html

  1. ρ ( a , b ) = tanh - 1 | a - b | | 1 - a ¯ b | \rho(a,b)=\tanh^{-1}\frac{|a-b|}{|1-\bar{a}b|}
  2. d ( f ( w ) , f ( z ) ) = ρ ( w , z ) d(f(w),f(z))=\rho(w,z)\,
  3. d ( f ( 0 ) , f ( z ) ) = ρ ( 0 , z ) d(f(0),f(z))=\rho(0,z)
  4. α ( f ( 0 ) , f ( 0 ) ) = 1 , \alpha(f(0),f^{\prime}(0))=1,

Complex_multiplier.html

  1. k = 1 / [ M P S + M R T + M P M ] = 1 / M P W k=1/[MPS+MRT+MPM]=1/MPW\,\!

Complex_polytope.html

  1. x 2 - 1 = 0 x^{2}-1=0
  2. x p - 1 = 0 x^{p}-1=0

Complex_quadratic_polynomial.html

  1. f ( x ) = a 2 x 2 + a 1 x + a 0 f(x)=a_{2}x^{2}+a_{1}x+a_{0}\qquad\,
  2. a 2 0 \qquad a_{2}\neq 0
  3. f r ( x ) = r x ( 1 - x ) f_{r}(x)=rx(1-x)\,
  4. f θ ( x ) = x 2 + e 2 π θ i x f_{\theta}(x)=x^{2}+e^{2\pi\theta i}x\,
  5. λ = e 2 π θ i \lambda=e^{2\pi\theta i}\,
  6. f c ( x ) = x 2 + c f_{c}(x)=x^{2}+c\,
  7. f c ( x ) f_{c}(x)\,
  8. θ \theta\,
  9. c c\,
  10. c = c ( θ ) = e 2 π θ i 2 ( 1 - e 2 π θ i 2 ) c=c(\theta)=\frac{e^{2\pi\theta i}}{2}\left(1-\frac{e^{2\pi\theta i}}{2}\right)
  11. r r\,
  12. c c\,
  13. c = c ( r ) = 1 - ( r - 1 ) 2 4 c=c(r)\,=\,\frac{1-(r-1)^{2}}{4}
  14. f c : z z 2 + c f_{c}:z\to z^{2}+c\,
  15. c c\in\mathbb{C}\,
  16. z z\,
  17. c c\,
  18. f c ( z ) = z 2 + c . f_{c}(z)=z^{2}+c.\,
  19. z n + 1 = f c ( z n ) z_{n+1}=f_{c}(z_{n})\,
  20. f c : z z 2 + c . f_{c}:z\to z^{2}+c.\,
  21. f n f^{n}\,
  22. f f\,
  23. f c n ( z ) = f c 1 ( f c n - 1 ( z ) ) f_{c}^{n}(z)=f_{c}^{1}(f_{c}^{n-1}(z))\,
  24. z n = f c n ( z 0 ) . z_{n}=f_{c}^{n}(z_{0}).\,
  25. f n f^{\circ n}\,
  26. f . f.\,
  27. f c f_{c}\,
  28. z c r z_{cr}\,
  29. f c ( z c r ) = 0. f_{c}^{\prime}(z_{cr})=0.\,
  30. f c ( z ) = d d z f c ( z ) = 2 z f_{c}^{\prime}(z)=\frac{d}{dz}f_{c}(z)=2z
  31. z c r = 0 z_{cr}=0\,
  32. f c f_{c}\,
  33. z c r = 0 z_{cr}=0\,
  34. z 0 z_{0}
  35. z c v z_{cv}
  36. f c f_{c}\,
  37. z c v = f c ( z c r ) z_{cv}=f_{c}(z_{cr})\,
  38. z c r = 0 z_{cr}=0\,
  39. z c v = c . z_{cv}=c.\,
  40. c c\,
  41. f c ( z ) . f_{c}(z).\,
  42. z 0 = z c r = 0 z_{0}=z_{cr}=0\,
  43. z 1 = f c ( z 0 ) = c z_{1}=f_{c}(z_{0})=c\,
  44. z 2 = f c ( z 1 ) = c 2 + c z_{2}=f_{c}(z_{1})=c^{2}+c\,
  45. z 3 = f c ( z 2 ) = ( c 2 + c ) 2 + c z_{3}=f_{c}(z_{2})=(c^{2}+c)^{2}+c\,
  46. ...\,
  47. P n ( c ) = f c n ( z c r ) = f c n ( 0 ) P_{n}(c)=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)\,
  48. P 0 ( c ) = 0 P_{0}(c)=0\,
  49. P 1 ( c ) = c P_{1}(c)=c\,
  50. P 2 ( c ) = c 2 + c P_{2}(c)=c^{2}+c\,
  51. P 3 ( c ) = ( c 2 + c ) 2 + c P_{3}(c)=(c^{2}+c)^{2}+c\,
  52. c e n t e r s = { c : P n ( c ) = 0 } centers=\{c:P_{n}(c)=0\}\,
  53. P n ( c ) P_{n}(c)\,
  54. M n , k = { c : P k ( c ) = P k + n ( c ) } M_{n,k}=\{c:P_{k}(c)=P_{k+n}(c)\}\,
  55. f c f_{c}\,
  56. z n + 1 = γ z n ( 1 - z n ) , z_{n+1}=\gamma z_{n}\left(1-z_{n}\right),
  57. z 0 = z c r z0=z_{cr}\,
  58. c c\,
  59. c c\,
  60. z z\,
  61. f 0 f_{0}
  62. c = 0 c=0
  63. f c f_{c}
  64. c 0 c\neq 0
  65. c c
  66. z 0 = 0 z_{0}=0
  67. f c n ( z 0 ) f_{c}^{n}(z_{0})
  68. z n = d d c f c n ( z 0 ) . z_{n}^{\prime}=\frac{d}{dc}f_{c}^{n}(z_{0}).
  69. z 0 = d d c f c 0 ( z 0 ) = 1 z_{0}^{\prime}=\frac{d}{dc}f_{c}^{0}(z_{0})=1
  70. z n + 1 = d d c f c n + 1 ( z 0 ) = 2 f c n ( z ) d d c f c n ( z 0 ) + 1 = 2 z n z n + 1. z_{n+1}^{\prime}=\frac{d}{dc}f_{c}^{n+1}(z_{0})=2\cdot{}f_{c}^{n}(z)\cdot\frac% {d}{dc}f_{c}^{n}(z_{0})+1=2\cdot z_{n}\cdot z_{n}^{\prime}+1.
  71. z z
  72. c c
  73. z 0 z_{0}\,
  74. f c ( z 0 ) = d d z f c ( z 0 ) = 2 z 0 f_{c}^{\prime}(z_{0})=\frac{d}{dz}f_{c}(z_{0})=2z_{0}
  75. ( f c p ) ( z 0 ) = d d z f c p ( z 0 ) = i = 0 p - 1 f c ( z i ) = 2 p i = 0 p - 1 z i . (f_{c}^{p})^{\prime}(z_{0})=\frac{d}{dz}f_{c}^{p}(z_{0})=\prod_{i=0}^{p-1}f_{c% }^{\prime}(z_{i})=2^{p}\prod_{i=0}^{p-1}z_{i}.
  76. z n z^{\prime}_{n}\,
  77. z 0 = 1 z^{\prime}_{0}=1\,
  78. z n = 2 * z n - 1 * z n - 1 z^{\prime}_{n}=2*z_{n-1}*z^{\prime}_{n-1}\,
  79. ( S f ) ( z ) = f ′′′ ( z ) f ( z ) - 3 2 ( f ′′ ( z ) f ( z ) ) 2 (Sf)(z)=\frac{f^{\prime\prime\prime}(z)}{f^{\prime}(z)}-\frac{3}{2}\left(\frac% {f^{\prime\prime}(z)}{f^{\prime}(z)}\right)^{2}