wpmath0000007_2

Chemostat.html

  1. D = Medium flow rate Culture volume = F V D=\dfrac{\mbox{Medium flow rate}~{}}{\mbox{Culture volume}~{}}=\dfrac{\mbox{F}% ~{}}{\mbox{V}~{}}

Chess'n_Math_Association.html

  1. R n = R o + ( W - L ) × 16 + D * 0.04 R_{n}=R_{o}+(W-L)\times 16+D*0.04
  2. B o n u s = ( R n - R o - ( 18 + 2 × G ) ) ; m i n ( B o n u s ) = 0 Bonus=(R_{n}-R_{o}-(18+2\times G));\ \ \ min(Bonus)=0

Chetaev_instability_theorem.html

  1. 𝐱 ˙ = X ( 𝐱 ) \dot{\,\textbf{x}}=X(\,\textbf{x})
  2. V ˙ ( 𝐱 ) \dot{V}(\,\textbf{x})
  3. 𝐱 ˙ = X ( 𝐱 ) \dot{\,\textbf{x}}=X(\,\textbf{x})
  4. V ˙ \dot{V}

Chi-squared_target_models.html

  1. p ( σ ) = m Γ ( m ) σ a v ( m σ σ a v ) m - 1 e - m σ σ a v I [ 0 , ) ( σ ) p(\sigma)=\frac{m}{\Gamma(m)\sigma_{av}}\left(\frac{m\sigma}{\sigma_{av}}% \right)^{m-1}e^{-\frac{m\sigma}{\sigma_{av}}}I_{[0,\infty)}(\sigma)
  2. σ a v \sigma_{av}
  3. σ \sigma
  4. m m
  5. m m
  6. m m
  7. m m
  8. m m
  9. m = 1 m=1
  10. p ( σ ) = 1 σ a v e - σ σ a v p(\sigma)=\frac{1}{\sigma_{av}}e^{-\frac{\sigma}{\sigma_{av}}}
  11. m = 2 m=2
  12. p ( σ ) = 4 σ σ a v 2 e - 2 σ σ a v p(\sigma)=\frac{4\sigma}{\sigma_{av}^{2}}e^{-\frac{2\sigma}{\sigma_{av}}}
  13. m m\to\infty

Chien_search.html

  1. q q
  2. Λ ( x ) = λ 0 + λ 1 x + λ 2 x 2 + + λ t x t \ \Lambda(x)=\lambda_{0}+\lambda_{1}x+\lambda_{2}x^{2}+\cdots+\lambda_{t}x^{t}
  3. Λ ( β ) \Lambda(\beta)
  4. β \beta
  5. q q
  6. β \beta
  7. α i β \alpha^{i_{\beta}}
  8. i β i_{\beta}
  9. α \alpha
  10. GF ( q ) \mathrm{GF}(q)
  11. i β i_{\beta}
  12. α \alpha
  13. α i \alpha^{i}
  14. 0 i < ( q - 1 ) 0\leq i<(q-1)
  15. Λ ( α i ) = λ 0 + λ 1 ( α i ) + λ 2 ( α i ) 2 + + λ t ( α i ) t γ 0 , i + γ 1 , i + γ 2 , i + + γ t , i \begin{array}[]{lllllllllll}\Lambda(\alpha^{i})&=&\lambda_{0}&+&\lambda_{1}(% \alpha^{i})&+&\lambda_{2}(\alpha^{i})^{2}&+&\cdots&+&\lambda_{t}(\alpha^{i})^{% t}\\ &\triangleq&\gamma_{0,i}&+&\gamma_{1,i}&+&\gamma_{2,i}&+&\cdots&+&\gamma_{t,i}% \end{array}
  16. Λ ( α i + 1 ) = λ 0 + λ 1 ( α i + 1 ) + λ 2 ( α i + 1 ) 2 + + λ t ( α i + 1 ) t = λ 0 + λ 1 ( α i ) α + λ 2 ( α i ) 2 α 2 + + λ t ( α i ) t α t = γ 0 , i + γ 1 , i α + γ 2 , i α 2 + + γ t , i α t γ 0 , i + 1 + γ 1 , i + 1 + γ 2 , i + 1 + + γ t , i + 1 \begin{array}[]{lllllllllll}\Lambda(\alpha^{i+1})&=&\lambda_{0}&+&\lambda_{1}(% \alpha^{i+1})&+&\lambda_{2}(\alpha^{i+1})^{2}&+&\cdots&+&\lambda_{t}(\alpha^{i% +1})^{t}\\ &=&\lambda_{0}&+&\lambda_{1}(\alpha^{i})\,\alpha&+&\lambda_{2}(\alpha^{i})^{2}% \,\alpha^{2}&+&\cdots&+&\lambda_{t}(\alpha^{i})^{t}\,\alpha^{t}\\ &=&\gamma_{0,i}&+&\gamma_{1,i}\,\alpha&+&\gamma_{2,i}\,\alpha^{2}&+&\cdots&+&% \gamma_{t,i}\,\alpha^{t}\\ &\triangleq&\gamma_{0,i+1}&+&\gamma_{1,i+1}&+&\gamma_{2,i+1}&+&\cdots&+&\gamma% _{t,i+1}\end{array}
  17. Λ ( α i ) \Lambda(\alpha^{i})
  18. { γ j , i | 0 j t } \{\gamma_{j,i}|0\leq j\leq t\}
  19. γ j , i + 1 = γ j , i α j \ \gamma_{j,i+1}=\gamma_{j,i}\,\alpha^{j}
  20. i = 0 i=0
  21. γ j , 0 = λ j \gamma_{j,0}=\lambda_{j}
  22. i i
  23. ( q - 1 ) (q-1)
  24. j = 0 t γ j , i = 0 , \ \sum_{j=0}^{t}\gamma_{j,i}=0,
  25. Λ ( α i ) = 0 \Lambda(\alpha^{i})=0
  26. α i \alpha_{i}

Chinese_restaurant_process.html

  1. Pr ( B n = B ) = b B ( | b | - 1 ) ! n ! \Pr(B_{n}=B)=\dfrac{\prod_{b\in B}(|b|-1)!}{n!}
  2. θ + | B | α n + θ , \dfrac{\theta+|B|\alpha}{n+\theta},
  3. | b | - α n + θ . \dfrac{|b|-\alpha}{n+\theta}.
  4. Pr ( B n = B ) = ( θ + α ) | B | - 1 , α ( θ + 1 ) n - 1 , 1 b B ( 1 - α ) | b | - 1 , 1 \Pr(B_{n}=B)=\dfrac{(\theta+\alpha)_{|B|-1,\alpha}}{(\theta+1)_{n-1,1}}\prod_{% b\in B}(1-\alpha)_{|b|-1,1}
  5. ( a ) 0 , c = 1 (a)_{0,c}=1
  6. b > 0 b>0
  7. ( a ) b , c = i = 0 b - 1 ( a + i c ) = { a b if c = 0 , c b Γ ( a / c + b ) Γ ( a / c ) otherwise . (a)_{b,c}=\prod_{i=0}^{b-1}(a+ic)=\begin{cases}a^{b}&\,\text{if }c=0,\\ \\ \dfrac{c^{b}\,\Gamma(a/c+b)}{\Gamma(a/c)}&\,\text{otherwise}.\end{cases}
  8. θ > 0 \theta>0
  9. Pr ( B n = B ) = Γ ( θ ) Γ ( θ + n ) α | B | Γ ( θ / α + | B | ) Γ ( θ / α ) b B Γ ( | b | - α ) Γ ( 1 - α ) . \Pr(B_{n}=B)=\dfrac{\Gamma(\theta)}{\Gamma(\theta+n)}\dfrac{\alpha^{|B|}\,% \Gamma(\theta/\alpha+|B|)}{\Gamma(\theta/\alpha)}\prod_{b\in B}\dfrac{\Gamma(|% b|-\alpha)}{\Gamma(1-\alpha)}.
  10. α \alpha
  11. Pr ( B n = B ) = Γ ( θ ) θ | B | Γ ( θ + n ) b B Γ ( | b | ) . \Pr(B_{n}=B)=\dfrac{\Gamma(\theta)\,\theta^{|B|}}{\Gamma(\theta+n)}\prod_{b\in B% }\Gamma(|b|).
  12. θ \theta
  13. Pr ( B n = B ) = α | B | - 1 Γ ( | B | ) < m t p l > Γ ( n ) b B Γ ( | b | - α ) Γ ( 1 - α ) . \Pr(B_{n}=B)=\dfrac{\alpha^{|B|-1}\,\Gamma(|B|)}{<}mtpl>{{\Gamma(n)}}\prod_{b% \in B}\dfrac{\Gamma(|b|-\alpha)}{\Gamma(1-\alpha)}.
  14. Pr ( C i = c | C 1 , , C i - 1 ) = { θ + | B | α θ + i - 1 if c new block , | b | - α θ + i - 1 if c b ; \Pr(C_{i}=c|C_{1},\ldots,C_{i-1})=\begin{cases}\dfrac{\theta+|B|\alpha}{\theta% +i-1}&\,\text{if }c\in\,\text{new block},\\ \\ \dfrac{|b|-\alpha}{\theta+i-1}&\,\text{if }c\in b;\end{cases}
  15. k = 1 n θ θ + k - 1 = θ ( Ψ ( θ + n ) - Ψ ( θ ) ) \begin{aligned}\displaystyle\sum_{k=1}^{n}\frac{\theta}{\theta+k-1}=\theta% \cdot(\Psi(\theta+n)-\Psi(\theta))\end{aligned}
  16. Ψ ( θ ) \Psi(\theta)
  17. Γ ( θ + n + α ) Γ ( θ + 1 ) α Γ ( θ + n ) Γ ( θ + α ) - θ α . \begin{aligned}\displaystyle\frac{\Gamma(\theta+n+\alpha)\Gamma(\theta+1)}{% \alpha\Gamma(\theta+n)\Gamma(\theta+\alpha)}-\frac{\theta}{\alpha}.\end{aligned}

Chiral_Potts_curve.html

  1. x N + y N = k ( 1 + x N y N ) x^{N}+y^{N}=k(1+x^{N}y^{N})

Chiral_symmetry_breaking.html

  1. q ¯ R a q L b = v δ a b , \langle\bar{q}^{a}_{R}q^{b}_{L}\rangle=v\delta^{ab}~{},
  2. U ( 2 ) L × U ( 2 ) R U(2)_{L}\times U(2)_{R}
  3. S U ( 2 ) L × S U ( 2 ) R × U ( 1 ) V × U ( 1 ) A . SU(2)_{L}\times SU(2)_{R}\times U(1)_{V}\times U(1)_{A}~{}.
  4. S U ( 2 ) L × S U ( 2 ) R SU(2)_{L}\times SU(2)_{R}
  5. v m ¯ \overline{vm}
  6. S U ( 3 ) L × S U ( 3 ) R × U ( 1 ) V × U ( 1 ) A SU(3)_{L}\times SU(3)_{R}\times U(1)_{V}\times U(1)_{A}
  7. S U ( 3 ) L × S U ( 3 ) R SU(3)_{L}\times SU(3)_{R}

Choquet_theory.html

  1. c = e E w ( e ) e c=\sum_{e\in E}w(e)e
  2. e E w ( e ) = 1. \sum_{e\in E}w(e)=1.
  3. f ( c ) = f ( e ) d w ( e ) . f(c)=\int f(e)dw(e).
  4. f ( c ) = f ( e ) d w ( e ) . f(c)=\int f(e)dw(e).

Chow_ring.html

  1. 0 k dim X 0\leq k\leq\dim{X}
  2. A * ( X ) = k = 0 dim X A k ( X ) . A^{*}(X)=\bigoplus_{k=0}^{\dim{X}}A^{k}(X).
  3. [ Y ] , [ Z ] [Y],[Z]
  4. [ Y ] [ Z ] = [ Y Z ] . [Y]\cdot[Z]=[Y\cap Z].
  5. A * ( n ) = [ ω ] / ( ω n + 1 ) A^{*}(\mathbb{P}^{n})=\mathbb{Z}[\omega]/(\omega^{n+1})
  6. ω \omega
  7. d ω k d\omega^{k}
  8. [ Y ] [ Z ] = d e ω n [Y]\cdot[Z]=de\;\omega^{n}
  9. ω n \omega^{n}
  10. f * : A k ( X ) A k ( X ) f^{*}\colon A^{k}(X^{\prime})\to A^{k}(X)\,\!
  11. f * : A k ( X ) A k ( X ) . f_{*}\colon A_{k}(X)\to A_{k}(X^{\prime}).\,\!
  12. f * f^{*}
  13. f * f_{*}
  14. f ( Y Z ) = f ( Y ) f ( Z ) f(Y\cap Z)=f(Y)\cap f(Z)
  15. f * ( [ Y ] f * ( [ Y ] ) ) = f * ( [ Y ] ) [ Y ] . f_{*}([Y]\cdot f^{*}([Y^{\prime}]))=f_{*}([Y])\cdot[Y^{\prime}].
  16. f : A * ( X ) H 2 * ( X ) f\colon A^{*}(X)\to H^{2*}(X)\,\!
  17. [ Y ] [Y]
  18. ω \omega
  19. H J H\cap J
  20. ω 2 \omega^{2}
  21. ω \omega
  22. P = [ 0 : : 0 : 1 ] P=[0:\dots:0:1]
  23. t = [ t 0 : t 1 ] t=[t_{0}:t_{1}]
  24. [ 0 : 1 ] [0:1]
  25. f t : n { P } n { P } , f t ( [ a 0 : : a n - 1 : a n ] ) = [ t 0 a 0 : : t 0 a n - 1 : t 1 a n ] . f_{t}\colon\mathbb{P}^{n}\setminus\{P\}\to\mathbb{P}^{n}\setminus\{P\},f_{t}([% a_{0}:\dots:a_{n-1}:a_{n}])=[t_{0}a_{0}:\dots:t_{0}a_{n-1}:t_{1}a_{n}].
  26. d ω k d\omega^{k}
  27. I I
  28. I I
  29. I d I^{d}

Christofides_algorithm.html

  1. G = ( V , w ) G=(V,w)
  2. G G
  3. V V
  4. w w
  5. G G
  6. T T
  7. G G
  8. O O
  9. T T
  10. M M
  11. O O
  12. M M
  13. T T
  14. H H
  15. H H
  16. B B
  17. O O
  18. O O
  19. M M
  20. M M
  21. O O
  22. ( O , B ) (O,B)
  23. G = ( V , E ) G=\left(V,E\right)
  24. T T
  25. V V^{\prime}
  26. T T
  27. G G
  28. V V^{\prime}
  29. G | V G|_{V^{\prime}}
  30. M M
  31. G | V G|_{V^{\prime}}
  32. T M T\cup M
  33. T M T\cup M

Chua's_circuit.html

  1. d x d t = α [ y - x - f ( x ) ] \frac{dx}{dt}=\alpha[y-x-f(x)]
  2. d y d t = x - y + z \frac{dy}{dt}=x-y+z
  3. d z d t = - β y \frac{dz}{dt}=-\beta y

Circulant_graph.html

  1. n n
  2. n 1 n−1
  3. x x
  4. ( x + d ) m o d n (x+d)modn
  5. z z
  6. ( z + d ) m o d n (z+d)modn
  7. n n
  8. n n
  9. n 1 n−1
  10. n n
  11. n n
  12. m m
  13. n n
  14. m × n m×n
  15. m × n m×n
  16. m m
  17. n n
  18. C n s 1 , , s k C_{n}^{s_{1},\ldots,s_{k}}
  19. s 1 , , s k s_{1},\ldots,s_{k}
  20. n n
  21. 0 , 1 , , n - 1 0,1,\ldots,n-1
  22. i ± s 1 , , i ± s k mod n i\pm s_{1},\ldots,i\pm s_{k}\mod n
  23. C n s 1 , , s k C_{n}^{s_{1},\ldots,s_{k}}
  24. gcd ( n , s 1 , , s k ) = 1 \gcd(n,s_{1},\ldots,s_{k})=1
  25. 1 s 1 < < s k 1\leq s_{1}<\cdots<s_{k}
  26. t ( C n s 1 , , s k ) = n a n 2 t(C_{n}^{s_{1},\ldots,s_{k}})=na_{n}^{2}
  27. a n a_{n}
  28. 2 s k - 1 2^{s_{k}-1}
  29. t ( C n 1 , 2 ) = n F n 2 t(C_{n}^{1,2})=nF_{n}^{2}
  30. F n F_{n}
  31. n n
  32. n n
  33. n n
  34. n n
  35. n 1 n−1
  36. x x
  37. y y
  38. z z
  39. ( z x + y ) m o d n (z−x+y)modn
  40. a a
  41. n n
  42. b b
  43. x x
  44. a x + b ax+b
  45. G G
  46. H H
  47. G G
  48. H H
  49. G G
  50. H H
  51. x x
  52. G G
  53. x ± 1 x±1
  54. x ± 2 x±2
  55. x ± 7 x±7
  56. H H
  57. x ± 2 x±2
  58. x ± 3 x±3
  59. x ± 5 x±5

Circular_convolution.html

  1. x T ( t ) = def k = - x ( t - k T ) = k = - x ( t + k T ) . x_{T}(t)\ \stackrel{\mathrm{def}}{=}\ \sum_{k=-\infty}^{\infty}x(t-kT)=\sum_{k% =-\infty}^{\infty}x(t+kT).
  2. ( x T * h ) ( t ) = def - h ( τ ) x T ( t - τ ) d τ t o t o + T h T ( τ ) x T ( t - τ ) d τ , \begin{aligned}\displaystyle(x_{T}*h)(t)&\displaystyle\stackrel{\mathrm{def}}{% =}\ \int_{-\infty}^{\infty}h(\tau)\cdot x_{T}(t-\tau)\,d\tau\\ &\displaystyle\equiv\int_{t_{o}}^{t_{o}+T}h_{T}(\tau)\cdot x_{T}(t-\tau)\,d% \tau,\end{aligned}
  3. - h ( τ ) x T ( t - τ ) d τ \int_{-\infty}^{\infty}h(\tau)\cdot x_{T}(t-\tau)\,d\tau
  4. = k = - [ t o + k T t o + ( k + 1 ) T h ( τ ) x T ( t - τ ) d τ ] \displaystyle=\sum_{k=-\infty}^{\infty}\left[\int_{t_{o}+kT}^{t_{o}+(k+1)T}h(% \tau)\cdot x_{T}(t-\tau)\ d\tau\right]
  5. ( x N * h ) [ n ] = def m = - h [ m ] x N [ n - m ] = m = - ( h [ m ] k = - x [ n - m - k N ] ) . \begin{aligned}\displaystyle(x_{N}*h)[n]&\displaystyle\stackrel{\mathrm{def}}{% =}\ \sum_{m=-\infty}^{\infty}h[m]\cdot x_{N}[n-m]\\ &\displaystyle=\sum_{m=-\infty}^{\infty}\left(h[m]\cdot\sum_{k=-\infty}^{% \infty}x[n-m-kN]\right).\end{aligned}

Circular_mil.html

  1. A = d 2 A=d^{2}
  2. d = 0.46 d=0.46
  3. A = d 2 A=d^{2}
  4. A = 460 2 A=460^{2}
  5. d = 0.46 d=0.46
  6. r = d 2 r={d\over 2}
  7. A = π r 2 A=\pi r^{2}
  8. A = π × 230 2 = 52 , 900 π 166 , 190.25 A=\pi\times 230^{2}=52,900\pi\approx 166,190.25
  9. d = 0.46 d=0.46
  10. r = d 2 r={d\over 2}
  11. A = π r 2 A=\pi r^{2}
  12. A = π × ( 0.23 ) 2 = .0529 π 0.16619 A=\pi\times(0.23)^{2}=.0529\pi\approx 0.16619
  13. A = A A=A
  14. 211 , 600 211,600
  15. = 52 , 900 π =52,900\pi
  16. = 52 , 900 π 211 , 600 ={52,900\pi\over 211,600}
  17. = π 4 ={\pi\over 4}
  18. = 4 π ={4\over\pi}
  19. A = 250 A=250
  20. A = 250 , 000 A=250,000
  21. d = A d=\sqrt{A}
  22. d = 250 , 000 d=\sqrt{250,000}
  23. A n = ( 5 × 92 36 - n 39 ) 2 A_{n}=(5\times 92^{\frac{36-n}{39}})^{2}

Clause_(logic).html

  1. l i l_{i}
  2. l 1 l n l_{1}\vee\cdots\vee l_{n}
  3. $\empty$
  4. \bot
  5. \Box
  6. f a l s e false
  7. b 1 , , b m b_{1},\ldots,b_{m}
  8. h 1 , , h n h_{1},\ldots,h_{n}
  9. h 1 , , h n b 1 , , b m h_{1},\ldots,h_{n}\leftarrow b_{1},\ldots,b_{m}

Clausius_theorem.html

  1. δ Q T 0 , \oint\frac{\delta Q}{T}\leq 0,
  2. Q Q
  3. T T
  4. δ Q T = 0 \oint\frac{\delta Q}{T}=0
  5. δ Q T 0 \oint\frac{\delta Q}{T}\leq 0
  6. Δ S < m t p l δ Q T \Delta S<mtpl>{{=}}\oint\frac{\delta Q}{T}
  7. ( Δ S < 0 ) (\Delta S<0)
  8. ( Δ S = 0 ) (\Delta S=0)
  9. d S T o t a l = d S S y s + d S R e s 0 dS_{Total}=dS_{Sys}+dS_{Res}\geq 0
  10. δ Q 1 \delta Q_{1}
  11. 0 \geq 0
  12. d S T o t a l 1 dS_{Total_{1}}
  13. T H o t T_{Hot}
  14. T 1 T_{1}
  15. d S S y s 1 = δ Q 1 T 1 dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}
  16. T H o t T 1 T_{Hot}\geq T_{1}
  17. - d S R e s 1 = δ Q 1 T H o t δ Q 1 T 1 = d S S y s 1 -dS_{Res_{1}}=\frac{\delta Q_{1}}{T_{Hot}}\leq\frac{\delta Q_{1}}{T_{1}}=dS_{% Sys_{1}}
  18. | d S R e s 1 | = δ Q 1 T H o t |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}}
  19. d S S y s 1 dS_{Sys_{1}}
  20. 0 \geq 0
  21. T 2 T_{2}
  22. - δ Q 2 -\delta Q_{2}
  23. δ Q 2 0 \delta Q_{2}\leq 0
  24. T C o l d T 2 T_{Cold}\leq T_{2}
  25. - d S R e s 2 = δ Q 2 T C o l d δ Q 2 T 2 = d S S y s 2 -dS_{Res_{2}}=\frac{\delta Q_{2}}{T_{Cold}}\leq\frac{\delta Q_{2}}{T_{2}}=dS_{% Sys_{2}}
  26. δ Q 2 \delta Q_{2}
  27. 0 \leq 0
  28. d S S y s 2 0 dS_{Sys_{2}}\leq 0
  29. d S R e s 2 = | δ Q 2 | T c o l d dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}
  30. | d S S y s 2 | |dS_{Sys_{2}}|
  31. T T
  32. - d S R e s = δ Q T d S S y s = 0 -\oint dS_{Res}=\oint\frac{\delta Q}{T}\leq\oint dS_{Sys}=0
  33. δ Q T 0 \oint\frac{\delta Q}{T}\leq 0
  34. d S R e s 0 \oint dS_{Res}\geq 0
  35. d S S y s = 0 \oint dS_{Sys}=0
  36. d S T o t a l = d S R e s + d S S y s 0 \oint dS_{Total}=\oint dS_{Res}+\oint dS_{Sys}\geq 0
  37. δ Q r e v T = 0 \oint\frac{\delta Q_{rev}}{T}=0

Claw_(disambiguation).html

  1. K 1 , 3 K_{1,3}

Clifford_theory.html

  1. μ ( g ) ( n ) = μ ( g n g - 1 ) \mu^{(g)}(n)=\mu(gng^{-1})
  2. χ N , μ 0 , \langle\chi_{N},\mu\rangle\neq 0,
  3. χ N = e ( i = 1 t μ ( g i ) ) , \chi_{N}=e\left(\sum_{i=1}^{t}\mu^{(g_{i})}\right),
  4. { g G : μ ( g ) = μ } \{g\in G:\mu^{(g)}=\mu\}
  5. I G ( μ ) . I_{G}(\mu).
  6. [ I G ( μ ) : N ] , [I_{G}(\mu):N],
  7. g G U . g \sum_{g\in G}U.g
  8. χ N , μ ( g ) = χ N ( g ) , μ ( g ) = χ N , μ \langle\chi_{N},\mu^{(g)}\rangle=\langle\chi_{N}^{(g)},\mu^{(g)}\rangle=% \langle\chi_{N},\mu\rangle

Co-occurrence_matrix.html

  1. C Δ x , Δ y ( i , j ) = p = 1 n q = 1 m { 1 , if I ( p , q ) = i and I ( p + Δ x , q + Δ y ) = j 0 , otherwise C_{\Delta x,\Delta y}(i,j)=\sum_{p=1}^{n}\sum_{q=1}^{m}\begin{cases}1,&\,\text% {if }I(p,q)=i\,\text{ and }I(p+\Delta x,q+\Delta y)=j\\ 0,&\,\text{otherwise}\end{cases}
  2. θ \theta

Coadjoint_representation.html

  1. K K
  2. G G
  3. 𝔤 \mathfrak{g}
  4. G G
  5. G G
  6. 𝔤 * \mathfrak{g}^{*}
  7. 𝔤 \mathfrak{g}
  8. G G
  9. G G
  10. G G
  11. G G
  12. G G
  13. 𝔤 \mathfrak{g}
  14. Ad : G Aut ( 𝔤 ) \mathrm{Ad}:G\rightarrow\mathrm{Aut}(\mathfrak{g})
  15. G G
  16. K : G Aut ( 𝔤 * ) K:G\rightarrow\mathrm{Aut}(\mathfrak{g}^{*})
  17. Ad * ( g - 1 ) := Ad ( g - 1 ) * \mathrm{Ad}^{*}(g^{-1}):=\mathrm{Ad}(g^{-1})^{*}
  18. K ( g ) F , Y = F , Ad ( g - 1 ) Y \langle K(g)F,Y\rangle=\langle F,\mathrm{Ad}(g^{-1})Y\rangle
  19. g G , Y 𝔤 , F 𝔤 * , g\in G,Y\in\mathfrak{g},F\in\mathfrak{g}^{*},
  20. F , Y \langle F,Y\rangle
  21. F F
  22. Y Y
  23. K * K_{*}
  24. 𝔤 \mathfrak{g}
  25. 𝔤 * \mathfrak{g}^{*}
  26. G G
  27. X 𝔤 , K * ( X ) = - ad ( X ) * X\in\mathfrak{g},K_{*}(X)=-\mathrm{ad}(X)^{*}
  28. ad \mathrm{ad}
  29. 𝔤 \mathfrak{g}
  30. K K
  31. K * ( X ) F , Y = F , - ad ( X ) Y \langle K_{*}(X)F,Y\rangle=\langle F,-\mathrm{ad}(X)Y\rangle
  32. X , Y 𝔤 , F 𝔤 * X,Y\in\mathfrak{g},F\in\mathfrak{g}^{*}
  33. Ω := 𝒪 ( F ) \Omega:=\mathcal{O}(F)
  34. F F
  35. 𝔤 * \mathfrak{g}^{*}
  36. 𝔤 \mathfrak{g}
  37. K ( G ) ( F ) K(G)(F)
  38. 𝔤 * \mathfrak{g}^{*}
  39. G / Stab ( F ) G/\mathrm{Stab}(F)
  40. Stab ( F ) \mathrm{Stab}(F)
  41. F F
  42. 𝔤 * \mathfrak{g}^{*}
  43. Ω \Omega
  44. G G
  45. σ Ω \sigma_{\Omega}
  46. 𝔤 \mathfrak{g}
  47. B F B_{F}
  48. 𝔤 \mathfrak{g}
  49. B F ( X , Y ) := F , [ X , Y ] , X , Y 𝔤 B_{F}(X,Y):=\langle F,[X,Y]\rangle,X,Y\in\mathfrak{g}
  50. σ Ω Hom ( Λ 2 ( Ω ) , ) \sigma_{\Omega}\in\mathrm{Hom}(\Lambda^{2}(\Omega),\mathbb{R})
  51. σ Ω ( F ) ( K * ( X ) ( F ) , K * ( Y ) ( F ) ) := B F ( X , Y ) \sigma_{\Omega}(F)(K_{*}(X)(F),K_{*}(Y)(F)):=B_{F}(X,Y)
  52. G G
  53. σ Ω \sigma_{\Omega}
  54. T F ( Ω ) T_{F}(\Omega)
  55. 𝔤 / stab ( F ) \mathfrak{g}/\mathrm{stab}(F)
  56. stab ( F ) \mathrm{stab}(F)
  57. Stab ( F ) \mathrm{Stab}(F)
  58. B F B_{F}
  59. stab ( F ) \mathrm{stab}(F)
  60. B F B_{F}
  61. Stab ( F ) \mathrm{Stab}(F)
  62. σ Ω \sigma_{\Omega}
  63. σ Ω \sigma_{\Omega}
  64. ( Ω , σ Ω ) (\Omega,\sigma_{\Omega})
  65. G G
  66. Ω 𝔤 * \Omega\hookrightarrow\mathfrak{g}^{*}
  67. G G

Coalescent_theory.html

  1. P c ( t ) = ( 1 - 1 2 N e ) t - 1 ( 1 2 N e ) . P_{c}(t)=\left(1-\frac{1}{2N_{e}}\right)^{t-1}\left(\frac{1}{2N_{e}}\right).
  2. P c ( t ) = 1 2 N e e - t - 1 2 N e . P_{c}(t)=\frac{1}{2N_{e}}e^{-\frac{t-1}{2N_{e}}}.
  3. H ¯ \bar{H}
  4. 2 μ 2\mu
  5. H ¯ = 2 μ 2 μ + 1 2 N e = 4 N e μ 1 + 4 N e μ = θ 1 + θ \begin{aligned}\displaystyle\bar{H}&\displaystyle=\frac{2\mu}{2\mu+\frac{1}{2N% _{e}}}\\ &\displaystyle=\frac{4N_{e}\mu}{1+4N_{e}\mu}\\ &\displaystyle=\frac{\theta}{1+\theta}\end{aligned}
  6. 4 N e μ 1 4N_{e}\mu\gg 1

Coanalytic_set.html

  1. \scriptstylesymbol Π 1 1 \scriptstylesymbol{\Pi}^{1}_{1}

Cochrane–Orcutt_estimation.html

  1. y t = α + X t β + ε t , y_{t}=\alpha+X_{t}\beta+\varepsilon_{t},\,
  2. y t y_{t}
  3. β \beta
  4. X t X_{t}
  5. ε t \varepsilon_{t}
  6. ε t = ρ ε t - 1 + e t , | ρ | < 1 \varepsilon_{t}=\rho\varepsilon_{t-1}+e_{t},\ |\rho|<1
  7. e t e_{t}
  8. y t - ρ y t - 1 = α ( 1 - ρ ) + β ( X t - ρ X t - 1 ) + e t . y_{t}-\rho y_{t-1}=\alpha(1-\rho)+\beta(X_{t}-\rho X_{t-1})+e_{t}.\,
  9. e t 2 e_{t}^{2}
  10. ( α , β ) (\alpha,\beta)
  11. ρ \rho
  12. ρ \rho
  13. ε ^ t \hat{\varepsilon}_{t}
  14. ε ^ t \hat{\varepsilon}_{t}
  15. ε ^ t - 1 \hat{\varepsilon}_{t-1}
  16. ρ \rho
  17. ρ \rho
  18. ρ \rho

Cochran–Armitage_test_for_trend.html

  1. T i = 1 k t i ( N 1 i R 2 - N 2 i R 1 ) , T\equiv\sum_{i=1}^{k}t_{i}(N_{1i}R_{2}-N_{2i}R_{1}),
  2. Pr ( A = 1 | B = 1 ) = = Pr ( A = 1 | B = k ) . \Pr(A=1|B=1)=\cdots=\Pr(A=1|B=k).
  3. E ( T ) = E ( E ( T | R 1 , R 2 ) ) = E ( 0 ) = 0. \operatorname{E}(T)=\operatorname{E}\left(\operatorname{E}(T|R_{1},R_{2})% \right)=\operatorname{E}(0)=0.
  4. Var ( T ) = R 1 R 2 N ( i = 1 k t i 2 C i ( N - C i ) - 2 i = 1 k - 1 j = i + 1 k t i t j C i C j ) , {\rm Var}(T)=\frac{R_{1}R_{2}}{N}\left(\sum_{i=1}^{k}t_{i}^{2}C_{i}(N-C_{i})-2% \sum_{i=1}^{k-1}\sum_{j=i+1}^{k}t_{i}t_{j}C_{i}C_{j}\right),
  5. T Var ( T ) N ( 0 , 1 ) . \frac{T}{\sqrt{\mathrm{Var}(T)}}\sim\mathrm{N}(0,1).

Cocurvature.html

  1. R ¯ P \bar{R}_{P}
  2. R ¯ P ( X , Y ) = ( Id - P ) [ P X , P Y ] \bar{R}_{P}(X,Y)=(\operatorname{Id}-P)[PX,PY]

Coefficient_matrix.html

  1. a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=b_{1}\,
  2. a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=b_{2}\,
  3. \vdots\,
  4. a m 1 x 1 + a m 2 x 2 + + a m n x n = b m a_{m1}x_{1}+a_{m2}x_{2}+\cdots+a_{mn}x_{n}=b_{m}\,
  5. x 1 , x 2 , , x n x_{1},\ x_{2},...,x_{n}
  6. a 11 , a 12 , , a m n a_{11},\ a_{12},...,\ a_{mn}
  7. a i j a_{ij}
  8. [ a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ] \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{bmatrix}

Coefficient_of_restitution.html

  1. Coefficient of restitution ( e ) = Relative speed after collision Relative speed before collision \,\text{Coefficient of restitution }(e)=\frac{\,\text{Relative speed after % collision}}{\,\text{Relative speed before collision}}
  2. Speed of separation = e × Speed of approach \,\text{Speed of separation}=e\times\,\text{Speed of approach}
  3. C R = v b - v a u a - u b C_{R}=\frac{v_{b}-v_{a}}{u_{a}-u_{b}}
  4. v a v_{a}
  5. v b v_{b}
  6. u a u_{a}
  7. u b u_{b}
  8. C R C_{R}
  9. C R C_{R}
  10. C R = v u C_{R}=\frac{v}{u}
  11. v v
  12. u u
  13. C R = h H C_{R}=\sqrt{\frac{h}{H}}
  14. h h
  15. H H
  16. C R C_{R}
  17. C R = K E (after impact) K E (before impact) = 1 2 m v 2 1 2 m u 2 = v 2 u 2 = v u C_{R}=\sqrt{\frac{KE\text{(after impact)}}{KE\text{(before impact)}}}=\sqrt{% \frac{\frac{1}{2}mv^{2}}{\frac{1}{2}mu^{2}}}=\sqrt{\frac{v^{2}}{u^{2}}}=\frac{% v}{u}
  18. C R = P E (at bounce height) P E (at drop height) = m g h m g H = h H C_{R}=\sqrt{\frac{PE\text{(at bounce height)}}{PE\text{(at drop height)}}}=% \sqrt{\frac{mgh}{mgH}}=\sqrt{\frac{h}{H}}
  19. v a = m a u a + m b u b + m b C R ( u b - u a ) m a + m b v_{a}=\frac{m_{a}u_{a}+m_{b}u_{b}+m_{b}C_{R}(u_{b}-u_{a})}{m_{a}+m_{b}}
  20. v b = m a u a + m b u b + m a C R ( u a - u b ) m a + m b v_{b}=\frac{m_{a}u_{a}+m_{b}u_{b}+m_{a}C_{R}(u_{a}-u_{b})}{m_{a}+m_{b}}
  21. v a v_{a}
  22. v b v_{b}
  23. u a u_{a}
  24. u b u_{b}
  25. m a m_{a}
  26. m b m_{b}
  27. u u
  28. v v
  29. m a u a + m b u b = m a v a + m b v b \displaystyle m_{a}u_{a}+m_{b}u_{b}=m_{a}v_{a}+m_{b}v_{b}
  30. v a v_{a}
  31. v b v_{b}
  32. m a u a + m b u b - m b v b m a = v a \displaystyle\frac{m_{a}u_{a}+m_{b}u_{b}-m_{b}v_{b}}{m_{a}}=v_{a}
  33. v b v_{b}
  34. v a v_{a}
  35. m a u a + m b u b - m b C R ( u a - u b ) - m b v a m a = v a \displaystyle\frac{m_{a}u_{a}+m_{b}u_{b}-m_{b}C_{R}(u_{a}-u_{b})-m_{b}v_{a}}{m% _{a}}=v_{a}
  36. v b v_{b}

Cofunction.html

  1. sin ( π 2 - A ) = cos ( A ) \sin\left(\frac{\pi}{2}-A\right)=\cos(A)
  2. cos ( π 2 - A ) = sin ( A ) \cos\left(\frac{\pi}{2}-A\right)=\sin(A)
  3. sec ( π 2 - A ) = csc ( A ) \sec\left(\frac{\pi}{2}-A\right)=\csc(A)
  4. csc ( π 2 - A ) = sec ( A ) \csc\left(\frac{\pi}{2}-A\right)=\sec(A)
  5. tan ( π 2 - A ) = cot ( A ) \tan\left(\frac{\pi}{2}-A\right)=\cot(A)
  6. cot ( π 2 - A ) = tan ( A ) \cot\left(\frac{\pi}{2}-A\right)=\tan(A)

Cole–Cole_equation.html

  1. ε * ( ω ) - ε = ε s - ε 1 + ( i ω τ ) 1 - α \varepsilon^{*}(\omega)-\varepsilon_{\infty}=\frac{\varepsilon_{s}-\varepsilon% _{\infty}}{1+(i\omega\tau)^{1-\alpha}}
  2. ε * \varepsilon^{*}
  3. ε s \varepsilon_{s}
  4. ε \varepsilon_{\infty}
  5. ω \omega
  6. τ \tau
  7. α \alpha
  8. α = 0 \alpha=0
  9. α > 0 \alpha>0
  10. ω \omega

Collineation.html

  1. α ( v ) = β ( v ) \alpha(\langle v\rangle)=\langle\beta(v)\rangle
  2. n 3 , n\geq 3,
  3. P Γ L P\Gamma L
  4. P Γ L P G L Gal ( K / k ) , P\Gamma L\cong PGL\rtimes\operatorname{Gal}(K/k),
  5. 𝔽 p \mathbb{F}_{p}
  6. \mathbb{Q}
  7. P G L = P Γ L , PGL=P\Gamma L,
  8. 𝔽 p n \mathbb{F}_{p^{n}}
  9. n 2 n\geq 2
  10. \mathbb{C}
  11. P Γ L / P G L Gal ( K / k ) P\Gamma L/PGL\cong\operatorname{Gal}(K/k)
  12. P G L < P Γ L , PGL<P\Gamma L,
  13. Gal ( K / k ) . \operatorname{Gal}(K/k).
  14. f ( z ) = a z * + b c z * + d . f(z)=\frac{az^{*}+b}{cz^{*}+d}.
  15. f ( z ) = 1 / z * f(z)=1/z^{*}
  16. f ( p ) f(p)
  17. g ( l ) g(l)
  18. ( p , l ) I (p,l)\in I
  19. ( f ( p ) , g ( l ) ) I \big(f(p),g(l)\big)\in I^{\prime}

Color_quantization.html

  1. ( r 1 , g 1 , b 1 ) (r_{1},g_{1},b_{1})
  2. ( r 2 , g 2 , b 2 ) (r_{2},g_{2},b_{2})
  3. ( r 1 - r 2 ) 2 + ( g 1 - g 2 ) 2 + ( b 1 - b 2 ) 2 . \sqrt{(r_{1}-r_{2})^{2}+(g_{1}-g_{2})^{2}+(b_{1}-b_{2})^{2}}.

Commitment_ordering.html

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

Commotio_cordis.html

  1. E k = 1 2 m v 2 E\text{k}=\tfrac{1}{2}mv^{2}

Commutation_cell.html

  1. E = 1 2 C V 2 E=\frac{1}{2}C\cdot V^{2}
  2. C U C\cdot U
  3. 2 C 2C
  4. Q 2 C = V 2 \frac{Q}{2C}=\frac{V}{2}
  5. 1 2 ( 2 C ) ( V 2 ) 2 = E 2 \frac{1}{2}(2C)\left(\frac{V}{2}\right)^{2}=\frac{E}{2}
  6. Φ = L I \Phi=L\cdot I
  7. I 2 \frac{I}{2}
  8. 1 2 L I 2 \frac{1}{2}L\cdot I^{2}
  9. 1 2 L ( I 2 ) 2 \frac{1}{2}L\cdot\left(\frac{I}{2}\right)^{2}

Commutativity_of_conjunction.html

  1. ( P and Q ) ( Q and P ) (P\and Q)\vdash(Q\and P)
  2. ( Q and P ) ( P and Q ) (Q\and P)\vdash(P\and Q)
  3. \vdash
  4. ( Q and P ) (Q\and P)
  5. ( P and Q ) (P\and Q)
  6. ( P and Q ) (P\and Q)
  7. ( Q and P ) (Q\and P)
  8. P and Q Q and P \frac{P\and Q}{\therefore Q\and P}
  9. Q and P P and Q \frac{Q\and P}{\therefore P\and Q}
  10. ( P and Q ) (P\and Q)
  11. ( Q and P ) (Q\and P)
  12. ( Q and P ) (Q\and P)
  13. ( P and Q ) (P\and Q)
  14. ( P and Q ) ( Q and P ) (P\and Q)\to(Q\and P)
  15. ( Q and P ) ( P and Q ) (Q\and P)\to(P\and Q)
  16. P P
  17. Q Q
  18. \land
  19. \land
  20. \land
  21. \land
  22. \land

Compact_quantum_group.html

  1. G G
  2. Δ : C ( G ) C ( G ) C ( G ) \Delta:C(G)\to C(G)\otimes C(G)
  3. C ( G ) C ( G ) C(G)⊗C(G)
  4. C ( G ) C(G)
  5. C ( G ) C(G)
  6. Δ ( f ) ( x , y ) = f ( x y ) \Delta(f)(x,y)=f(xy)
  7. f C ( G ) f\in C(G)
  8. x , y G x,y\in G
  9. ( f g ) ( x , y ) = f ( x ) g ( y ) (f\otimes g)(x,y)=f(x)g(y)
  10. f , g C ( G ) f,g\in C(G)
  11. x , y G x,y\in G
  12. κ : C ( G ) C ( G ) \kappa:C(G)\to C(G)
  13. κ ( f ) ( x ) = f ( x - 1 ) \kappa(f)(x)=f(x^{-1})
  14. f C ( G ) f\in C(G)
  15. x G x\in G
  16. C ( G ) C(G)
  17. G G
  18. G G
  19. C ( G ) C(G)
  20. g ( u i j ( g ) ) i , j g\mapsto(u_{ij}(g))_{i,j}
  21. n n
  22. G G
  23. u i j C ( G ) u_{ij}\in C(G)
  24. i , j i,j
  25. Δ ( u i j ) = k u i k u k j \Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj}
  26. i , j i,j
  27. u i j u_{ij}
  28. i , j i,j
  29. κ ( u i j ) \kappa(u_{ij})
  30. i , j i,j
  31. ϵ ( u i j ) = δ i j \epsilon(u_{ij})=\delta_{ij}
  32. i , j i,j
  33. δ i j \delta_{ij}
  34. κ κ
  35. 1 = k u 1 k κ ( u k 1 ) = k κ ( u 1 k ) u k 1 . 1=\sum_{k}u_{1k}\kappa(u_{k1})=\sum_{k}\kappa(u_{1k})u_{k1}.
  36. ( C , u ) (C,u)
  37. C C
  38. u = ( u i j ) i , j = 1 , , n u=(u_{ij})_{i,j=1,\dots,n}
  39. C C
  40. C C
  41. u u
  42. C C
  43. Δ : C C C Δ:C→C⊗C
  44. C C C⊗C
  45. C C
  46. C C
  47. i , j : Δ ( u i j ) = k u i k u k j ; \forall i,j:\qquad\Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj};
  48. κ ( κ ( v * ) * ) = v \kappa(\kappa(v*)*)=v
  49. v C 0 v\in C_{0}
  50. k κ ( u i k ) u k j = k u i k κ ( u k j ) = δ i j I , \sum_{k}\kappa(u_{ik})u_{kj}=\sum_{k}u_{ik}\kappa(u_{kj})=\delta_{ij}I,
  51. I I
  52. C C
  53. κ κ
  54. κ ( v w ) = κ ( w ) κ ( v ) κ(vw)=κ(w)κ(v)
  55. v , w C 0 v,w\in C_{0}
  56. C C
  57. C C
  58. C C
  59. u u
  60. A A
  61. B B
  62. H H
  63. K K
  64. A B A⊗B
  65. B ( H K ) B(H⊗K)
  66. A B A⊗B
  67. ( C , Δ ) (C,Δ)
  68. C C
  69. Δ : C C C Δ:C→C⊗C
  70. ( Δ i d ) Δ = ( i d Δ ) Δ (Δ⊗id)Δ=(id⊗Δ)Δ
  71. C C C⊗C
  72. A A
  73. v = ( v i j ) i , j = 1 , , n v=(v_{ij})_{i,j=1,\dots,n}
  74. A A
  75. v M ( n , A ) v∈M(n,A)
  76. i , j : Δ ( v i j ) = k = 1 n v i k v k j \forall i,j:\qquad\Delta(v_{ij})=\sum_{k=1}^{n}v_{ik}\otimes v_{kj}
  77. i , j : ϵ ( v i j ) = δ i j . \forall i,j:\qquad\epsilon(v_{ij})=\delta_{ij}.
  78. i , j : κ ( v i j ) = v j i * . \forall i,j:\qquad\kappa(v_{ij})=v^{*}_{ji}.
  79. μ μ
  80. α α
  81. γ γ
  82. γ γ * = γ * γ , α γ = μ γ α , α γ * = μ γ * α , α α * + μ γ * γ = α * α + μ - 1 γ * γ = I , \gamma\gamma^{*}=\gamma^{*}\gamma,\ \alpha\gamma=\mu\gamma\alpha,\ \alpha% \gamma^{*}=\mu\gamma^{*}\alpha,\ \alpha\alpha^{*}+\mu\gamma^{*}\gamma=\alpha^{% *}\alpha+\mu^{-1}\gamma^{*}\gamma=I,
  83. u = ( α γ - γ * α * ) , u=\left(\begin{matrix}\alpha&\gamma\\ -\gamma^{*}&\alpha^{*}\end{matrix}\right),
  84. Δ ( α ) = α α - γ γ * , Δ ( γ ) = α γ + γ α * \Delta(\alpha)=\alpha\otimes\alpha-\gamma\otimes\gamma^{*},\Delta(\gamma)=% \alpha\otimes\gamma+\gamma\otimes\alpha^{*}
  85. κ ( α ) = α * , κ ( γ ) = - μ - 1 γ , κ ( γ * ) = - μ γ * , κ ( α * ) = α \kappa(\alpha)=\alpha^{*},\kappa(\gamma)=-\mu^{-1}\gamma,\kappa(\gamma^{*})=-% \mu\gamma^{*},\kappa(\alpha^{*})=\alpha
  86. u u
  87. u u
  88. v = ( α μ γ - 1 μ γ * α * ) . v=\left(\begin{matrix}\alpha&\sqrt{\mu}\gamma\\ -\frac{1}{\sqrt{\mu}}\gamma^{*}&\alpha^{*}\end{matrix}\right).
  89. α α
  90. β β
  91. β β * = β * β , α β = μ β α , α β * = μ β * α , α α * + μ 2 β * β = α * α + β * β = I , \beta\beta^{*}=\beta^{*}\beta,\ \alpha\beta=\mu\beta\alpha,\ \alpha\beta^{*}=% \mu\beta^{*}\alpha,\ \alpha\alpha^{*}+\mu^{2}\beta^{*}\beta=\alpha^{*}\alpha+% \beta^{*}\beta=I,
  92. w = ( α μ β - β * α * ) , w=\left(\begin{matrix}\alpha&\mu\beta\\ -\beta^{*}&\alpha^{*}\end{matrix}\right),
  93. Δ ( α ) = α α - μ β β * , Δ ( β ) = α β + β α * \Delta(\alpha)=\alpha\otimes\alpha-\mu\beta\otimes\beta^{*},\Delta(\beta)=% \alpha\otimes\beta+\beta\otimes\alpha^{*}
  94. κ ( α ) = α * , κ ( β ) = - μ - 1 β , κ ( β * ) = - μ β * \kappa(\alpha)=\alpha^{*},\kappa(\beta)=-\mu^{-1}\beta,\kappa(\beta^{*})=-\mu% \beta^{*}
  95. κ ( α * ) = α \kappa(\alpha^{*})=\alpha
  96. w w
  97. γ = μ β \gamma=\sqrt{\mu}\beta
  98. μ = 1 μ=1
  99. S U < s u b > μ ( 2 ) SU<sub>μ(2)

Compactness_measure_of_a_shape.html

  1. ( surfacearea ) 1.5 / ( volume ) (\mathrm{surfacearea})^{1.5}/(\mathrm{volume})
  2. ψ - 3 / 2 \psi^{-3/2}

Competition_(biology).html

  1. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\qquad\!

Complex_affine_space.html

  1. \C n = \C × \C × × \C n -times \C^{n}=\underbrace{\C\times\C\times\cdots\times\C}_{n\,\text{-times}}
  2. \C n = { ( z 1 , , z n ) : z i \C for all 1 i n } \C^{n}=\{(z_{1},\ldots,z_{n}):z_{i}\in\C\,\text{ for all }1\leq i\leq n\}
  3. \C n \C^{n}
  4. 2 × n 2\times n
  5. 2 × n 2\times n
  6. \C 1 \C^{1}
  7. \C 2 \C^{2}
  8. \C 1 \C^{1}
  9. \C n \C^{n}
  10. \C 2 \C^{2}

Complex_beam_parameter.html

  1. 1 q ( z ) = 1 R ( z ) - i λ 0 π n w ( z ) 2 \frac{1}{q(z)}=\frac{1}{R(z)}-\frac{i\lambda_{0}}{\pi nw(z)^{2}}
  2. q ( z ) = z + z 0 i , q(z)=z+z_{0}i\ ,
  3. ( A B C D ) \begin{pmatrix}A&B\\ C&D\end{pmatrix}
  4. q f = A q i + B C q i + D q_{f}=\frac{Aq_{i}+B}{Cq_{i}+D}
  5. 1 q f = C + D / q i A + B / q i {1\over q_{f}}=\frac{C+D/q_{i}}{A+B/q_{i}}
  6. C q f 2 + ( D - A ) q f - B = 0 C{q_{f}}^{2}+(D-A)q_{f}-B=0

Complex_logarithm.html

  1. , w - 4 π i , w - 2 π i , w , w + 2 π i , w + 4 π i , , \ldots,\;w-4\pi i,\;w-2\pi i,\;w,\;w+2\pi i,\;w+4\pi i,\;\ldots,
  2. Log z := ln r + i θ = ln | z | + i Arg z = ln x 2 + y 2 + i atan2 ( y , x ) . \operatorname{Log}z:=\,\text{ln }r+i\theta=\ln|z|+i\operatorname{Arg}z=% \operatorname{ln}\sqrt{x^{2}+y^{2}}+i\operatorname{atan2}(y,x).
  3. Log : × S \operatorname{Log}\colon\mathbb{C}^{\times}\to S
  4. Log ( ( - 1 ) i ) = Log ( - i ) = ln ( 1 ) - π i 2 = - π i 2 , \operatorname{Log}((-1)i)=\operatorname{Log}(-i)=\ln(1)-\frac{\pi i}{2}=-\frac% {\pi i}{2},
  5. Log ( - 1 ) + Log ( i ) = ( ln ( 1 ) + π i ) + ( ln ( 1 ) + π i 2 ) = 3 π i 2 - π i 2 . \operatorname{Log}(-1)+\operatorname{Log}(i)=\left(\ln(1)+\pi i\right)+\left(% \ln(1)+\frac{\pi i}{2}\right)=\frac{3\pi i}{2}\neq-\frac{\pi i}{2}.
  6. × \mathbb{C}^{\times}
  7. × \mathbb{C}^{\times}
  8. 2 π i 2\pi i\mathbb{Z}
  9. - 0 \mathbb{C}-\mathbb{R}_{\leq 0}
  10. log ( 1 + u ) = n = 1 ( - 1 ) n + 1 n u n = u - u 2 2 + u 3 3 - \log(1+u)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}u^{n}=u-\frac{u^{2}}{2}+\frac% {u^{3}}{3}-\cdots\,
  11. ln x = 1 x 1 u d u . \ln x=\int_{1}^{x}\frac{1}{u}\,du.
  12. ln x = ln a + a x 1 u d u \ln x=\ln a+\int_{a}^{x}\frac{1}{u}\,du
  13. \mathbb{C}
  14. L ( z ) := b + a z 1 w d w L(z):=b+\int_{a}^{z}\frac{1}{w}\,dw
  15. f : U f\colon U\to\mathbb{C}
  16. f ( z ) 0 f^{\prime}(z)\neq 0
  17. z U z\in U
  18. - 0 \mathbb{C}-\mathbb{R}_{\leq 0}
  19. log : × \log\colon\mathbb{C}^{\times}\to\mathbb{C}
  20. - 0 \mathbb{C}-\mathbb{R}_{\leq 0}
  21. - 0 \mathbb{C}-\mathbb{R}_{\geq 0}
  22. × 3 \mathbb{C}\times\mathbb{R}\approx\mathbb{R}^{3}
  23. log R : R \log_{R}\colon R\to\mathbb{C}
  24. × \mathbb{C}^{\times}
  25. z × z\in\mathbb{C}^{\times}
  26. L 1 : U 1 L_{1}\colon U_{1}\to\mathbb{C}
  27. L 2 : U 2 L_{2}\colon U_{2}\to\mathbb{C}
  28. U 1 U 2 U_{1}\cap U_{2}
  29. × \mathbb{C}^{\times}
  30. R × R\to\mathbb{C}^{\times}
  31. × \mathbb{C}^{\times}
  32. \mathbb{Z}
  33. \mathbb{C}
  34. × \mathbb{C}^{\times}
  35. log a b = log b log a , \log_{a}b=\frac{\log b}{\log a}\,,
  36. log i e = ln e ln i = 1 π i / 2 = - 2 i π . \log_{i}e=\frac{\ln e}{\ln i}=\frac{1}{\pi i/2}=-\frac{2i}{\pi}.
  37. \mathbb{C}
  38. \mathbb{C}
  39. g ( z ) := b + a z f ( w ) f ( w ) d w g(z):=b+\int_{a}^{z}\frac{f^{\prime}(w)}{f(w)}\,dw

Complex_modulus.html

  1. | x + i y | = x 2 + y 2 . |x+iy|=\sqrt{x^{2}+y^{2}}.

Complexity_of_constraint_satisfaction.html

  1. A = ( V , R 1 A , , R n A ) A=(V,R^{A}_{1},\ldots,R^{A}_{n})
  2. R i A R^{A}_{i}
  3. V V
  4. V V
  5. h h
  6. A = ( V , R 1 A , , R n A ) A=(V,R^{A}_{1},\ldots,R^{A}_{n})
  7. B = ( D , R 1 B , , R n B ) B=(D,R^{B}_{1},\ldots,R^{B}_{n})
  8. V V
  9. D D
  10. ( x 1 , , x k ) R i A (x_{1},\ldots,x_{k})\in R^{A}_{i}
  11. ( h ( x 1 ) , , h ( x k ) ) R i B (h(x_{1}),\ldots,h(x_{k}))\in R^{B}_{i}
  12. L ( a 1 , , a n ) L(a_{1},\ldots,a_{n})
  13. R 1 , , R n R_{1},\ldots,R_{n}
  14. R i ( x 1 , , x n ) R_{i}(x_{1},\ldots,x_{n})
  15. k k
  16. k k
  17. b b
  18. d d
  19. a a
  20. b b
  21. ( 1 , 1 ) (1,1)
  22. ( 1 , 1 ) (1,1)
  23. ( 1 , 0 ) (1,0)
  24. b b
  25. d d
  26. ( 0 , 0 ) (0,0)
  27. k k
  28. k k
  29. k k
  30. b b
  31. d d
  32. k k
  33. k k
  34. 1 , 0 , 1 1,0,1
  35. D 1 , , D n D_{1},\ldots,D_{n}
  36. D D
  37. s : D D s:D\rightarrow D
  38. s ( x ) = s ( y ) s(x)=s(y)
  39. x , y x,y
  40. x 1 , , x n R \langle x_{1},\ldots,x_{n}\rangle\in R
  41. s ( x 1 ) , , s ( x n ) R \langle s(x_{1}),\ldots,s(x_{n})\rangle\in R

Compliance_(medicine).html

  1. V M P R = All days supply Elapsed days (inclusive of last prescription) VMPR=\dfrac{\,\text{All days supply}}{\,\text{Elapsed days (inclusive of last % prescription)}}
  2. F M P R = All days supply 365 365 FMPR=\dfrac{\,\text{All days supply}\leq 365}{365}

Computation_tree.html

  1. x X x\in X

Computational_aeroacoustics.html

  1. ρ 0 \rho_{0}
  2. p 0 p_{0}
  3. u 0 u_{0}
  4. 𝐔 t + 𝐅 x + 𝐆 y = 𝐒 \frac{\partial\mathbf{U}}{\partial t}+\frac{\partial\mathbf{F}}{\partial x}+% \frac{\partial\mathbf{G}}{\partial y}=\mathbf{S}
  5. 𝐔 = [ ρ u v p ] , 𝐅 = [ ρ 0 u + ρ u 0 u 0 u + p / ρ 0 u 0 v u 0 p + γ p 0 u ] , 𝐆 = [ ρ 0 v 0 p / ρ 0 γ p 0 v ] , \mathbf{U}=\begin{bmatrix}\rho\\ u\\ v\\ p\\ \end{bmatrix}\ ,\ \mathbf{F}=\begin{bmatrix}\rho_{0}u+\rho u_{0}\\ u_{0}u+p/\rho_{0}\\ u_{0}v\\ u_{0}p+\gamma p_{0}u\\ \end{bmatrix}\ ,\ \mathbf{G}=\begin{bmatrix}\rho_{0}v\\ 0\\ p/\rho_{0}\\ \gamma p_{0}v\\ \end{bmatrix},
  6. ρ \rho
  7. u u
  8. v v
  9. p p
  10. γ \gamma
  11. c p / c v c_{p}/c_{v}
  12. c p / c v = 1.4 c_{p}/c_{v}=1.4
  13. 𝐒 \mathbf{S}

Computational_electromagnetics.html

  1. t u ¯ + A x u ¯ + B y u ¯ + C u ¯ = g ¯ \frac{\partial}{\partial t}\bar{u}+A\frac{\partial}{\partial x}\bar{u}+B\frac{% \partial}{\partial y}\bar{u}+C\bar{u}=\bar{g}
  2. u ¯ = ( E x E y H z ) , \bar{u}=\left(\begin{matrix}E_{x}\\ E_{y}\\ H_{z}\end{matrix}\right),
  3. A = ( 0 0 0 0 0 1 ϵ 0 1 μ 0 ) , A=\left(\begin{matrix}0&0&0\\ 0&0&\frac{1}{\epsilon}\\ 0&\frac{1}{\mu}&0\end{matrix}\right),
  4. B = ( 0 0 - 1 ϵ 0 0 0 - 1 μ 0 0 ) , B=\left(\begin{matrix}0&0&\frac{-1}{\epsilon}\\ 0&0&0\\ \frac{-1}{\mu}&0&0\end{matrix}\right),
  5. C = ( σ ϵ 0 0 0 σ ϵ 0 0 0 0 ) . C=\left(\begin{matrix}\frac{\sigma}{\epsilon}&0&0\\ 0&\frac{\sigma}{\epsilon}&0\\ 0&0&0\end{matrix}\right).
  6. g ¯ \bar{g}
  7. u ¯ \bar{u}
  8. g ¯ = ( E x , c o n s t r a i n t E y , c o n s t r a i n t H z , c o n s t r a i n t ) . \bar{g}=\left(\begin{matrix}E_{x,constraint}\\ E_{y,constraint}\\ H_{z,constraint}\end{matrix}\right).
  9. g ¯ \bar{g}
  10. RCSPlate = 4 π A 2 λ 2 , \,\text{RCS}\text{Plate}=\frac{4\pi A^{2}}{\lambda^{2}},
  11. λ \lambda

Concurrent_constraint_logic_programming.html

  1. A A
  2. H H

Conditional_random_field.html

  1. s y m b o l X symbol{X}
  2. s y m b o l Y symbol{Y}
  3. G = ( V , E ) G=(V,E)
  4. s y m b o l Y = ( s y m b o l Y v ) v V symbol{Y}=(symbol{Y}_{v})_{v\in V}
  5. s y m b o l Y symbol{Y}
  6. G G
  7. ( s y m b o l X , s y m b o l Y ) (symbol{X},symbol{Y})
  8. s y m b o l Y v symbol{Y}_{v}
  9. s y m b o l X symbol{X}
  10. p ( s y m b o l Y v | s y m b o l X , s y m b o l Y w , w v ) = p ( s y m b o l Y v | s y m b o l X , s y m b o l Y w , w v ) p(symbol{Y}_{v}|symbol{X},symbol{Y}_{w},w\neq v)=p(symbol{Y}_{v}|symbol{X},% symbol{Y}_{w},w\sim v)
  11. w v \mathit{w}\sim v
  12. w w
  13. v v
  14. G G
  15. s y m b o l X symbol{X}
  16. s y m b o l Y symbol{Y}
  17. p ( s y m b o l Y | s y m b o l X ) p(symbol{Y}|symbol{X})
  18. θ \theta
  19. p ( Y i | X i ; θ ) p(Y_{i}|X_{i};\theta)
  20. X X
  21. Y Y
  22. Y i Y_{i}
  23. Y i - 1 Y_{i-1}
  24. Y i Y_{i}
  25. Y i Y_{i}
  26. Y i Y_{i}
  27. Y Y
  28. X X
  29. Y Y
  30. X X
  31. Y i Y_{i}
  32. X X
  33. f ( i , Y i - 1 , Y i , X ) f(i,Y_{i-1},Y_{i},X)
  34. Y i Y_{i}
  35. Y i Y_{i}
  36. X X
  37. Y i Y_{i}
  38. o o
  39. Y i - o , , Y i - 1 Y_{i-o},...,Y_{i-1}
  40. o o
  41. o o
  42. Y Y
  43. Y i Y_{i}
  44. x x
  45. x xₙ
  46. y y
  47. y yₙ
  48. Y Y
  49. P P
  50. P ( 𝐲 | 𝐱 ) = 𝐡 P ( 𝐲 | 𝐡 , 𝐱 ) P ( 𝐡 | 𝐱 ) P(\mathbf{y}|\mathbf{x})=\sum_{\mathbf{h}}P(\mathbf{y}|\mathbf{h},\mathbf{x})P% (\mathbf{h}|\mathbf{x})

Conductance_quantum.html

  1. G 0 = 2 e 2 h G_{0}=\frac{2e^{2}}{h}
  2. d n / d ϵ = 2 / h v dn/d\epsilon=2/hv
  3. V = - ( u 1 - u 2 ) / e V=-(u_{1}-u_{2})/e
  4. j = - e v ( u 1 - u 2 ) d n / d ϵ j=-ev(u_{1}-u_{2})dn/d\epsilon
  5. G = I / V = j / V = 2 e 2 / h G=I/V=j/V=2e^{2}/h
  6. λ L \lambda\gg L

Configuration_state_function.html

  1. Ψ \Psi
  2. Ψ = k c k ψ k \Psi=\sum_{k}c_{k}\psi_{k}
  3. ψ k \psi_{k}
  4. c k c_{k}
  5. Ψ \Psi
  6. L ^ 2 \hat{L}^{2}
  7. L ^ z \hat{L}_{z}
  8. S ^ 2 \hat{S}^{2}
  9. S ^ z \hat{S}_{z}
  10. L ^ 2 \hat{L}^{2}
  11. L ^ 2 \hat{L}^{2}
  12. L ^ z , S ^ 2 \hat{L}_{z},\hat{S}^{2}
  13. S ^ z \hat{S}_{z}
  14. L ^ 2 \hat{L}^{2}
  15. L ^ z \hat{L}_{z}
  16. S ^ 2 \hat{S}^{2}
  17. S ^ z \hat{S}_{z}
  18. 1 s 2 1s^{2}
  19. 1 π 2 1\pi^{2}
  20. N N
  21. 1 s 1s
  22. 1 s α 1 s β 1s\alpha\;\;\;1s\beta
  23. α , β \alpha,\;\;\;\beta
  24. 1 π 1\pi
  25. C v C_{\infty v}
  26. 1 π ( + ) α , 1 π ( + ) β , 1 π ( - ) α , 1 π ( - ) β 1\pi(+)\alpha,\;1\pi(+)\beta,\;1\pi(-)\alpha,\;1\pi(-)\beta
  27. π \pi
  28. + 1 +1
  29. - 1 -1
  30. M M
  31. N N
  32. M M
  33. D i D_{i}
  34. N M N<<M
  35. M M
  36. N N
  37. c i c_{i}
  38. i c i D i \sum_{i}c_{i}\;D_{i}
  39. D i D_{i}
  40. M M
  41. N N
  42. N N
  43. M M
  44. S z S_{z}
  45. C v C_{\infty v}
  46. D h D_{\infty h}
  47. λ \lambda
  48. D 2 h D_{2h}
  49. N N
  50. N N
  51. S z S_{z}
  52. N N
  53. M M
  54. σ \sigma
  55. S z S_{z}
  56. σ 1 \sigma^{1}
  57. Σ + 2 {}^{2}\Sigma^{+}
  58. 1 2 \;\;\frac{1}{2}
  59. σ 1 \sigma^{1}
  60. Σ - 2 {}^{2}\Sigma^{-}
  61. - 1 2 -\frac{1}{2}
  62. σ 2 \sigma^{2}
  63. Σ + 1 {}^{1}\Sigma^{+}
  64. 0
  65. π \pi
  66. δ , ϕ , γ , \delta,\phi,\gamma,\ldots
  67. S z S_{z}
  68. π 1 \pi^{1}
  69. Π 2 {}^{2}\Pi
  70. + 1 +1
  71. 1 2 \frac{1}{2}
  72. π 1 \pi^{1}
  73. Π 2 {}^{2}\Pi
  74. + 1 +1
  75. - 1 2 -\frac{1}{2}
  76. π 1 \pi^{1}
  77. Π 2 {}^{2}\Pi
  78. - 1 -1
  79. 1 2 \frac{1}{2}
  80. π 1 \pi^{1}
  81. Π 2 {}^{2}\Pi
  82. - 1 -1
  83. - 1 2 -\frac{1}{2}
  84. π 2 \pi^{2}
  85. Σ - 3 {}^{3}\Sigma^{-}
  86. 0
  87. + 1 +1
  88. π 2 \pi^{2}
  89. Σ - 3 {}^{3}\Sigma^{-}
  90. 0
  91. 0
  92. π 2 \pi^{2}
  93. Σ - 3 {}^{3}\Sigma^{-}
  94. 0
  95. - 1 -1
  96. π 2 \pi^{2}
  97. Δ 1 {}^{1}\Delta
  98. + 2 +2
  99. 0
  100. π 2 \pi^{2}
  101. Δ 1 {}^{1}\Delta
  102. - 2 -2
  103. 0
  104. π 2 \pi^{2}
  105. Σ + 1 {}^{1}\Sigma^{+}
  106. 0
  107. 0
  108. π 3 \pi^{3}
  109. Π 2 {}^{2}\Pi
  110. + 1 +1
  111. 1 2 \frac{1}{2}
  112. π 3 \pi^{3}
  113. Π 2 {}^{2}\Pi
  114. + 1 +1
  115. - 1 2 -\frac{1}{2}
  116. π 3 \pi^{3}
  117. Π 2 {}^{2}\Pi
  118. - 1 -1
  119. 1 2 \frac{1}{2}
  120. π 3 \pi^{3}
  121. Π 2 {}^{2}\Pi
  122. - 1 -1
  123. - 1 2 -\frac{1}{2}
  124. π 4 \pi^{4}
  125. Σ + 1 {}^{1}\Sigma^{+}
  126. 0
  127. 0
  128. \ldots

Confluence_(abstract_rewriting).html

  1. \color M i d n i g h t B l u e eval left ( 11 + 9 ) × ( 2 + 4 ) \color M i d n i g h t B l u e eval right \color M i d n i g h t B l u e \color M i d n i g h t B l u e 20 × ( 2 + 4 ) ( 11 + 9 ) × 6 \color M i d n i g h t B l u e \color M i d n i g h t B l u e \color M i d n i g h t B l u e eval right 20 × 6 \color M i d n i g h t B l u e eval left \color M i d n i g h t B l u e 120 \begin{array}[]{|rcccl|}\hline\color{MidnightBlue}{\mbox{eval left}~{}}&&(11+9% )\times(2+4)&&\color{MidnightBlue}{\mbox{eval right}~{}}\\ &\color{MidnightBlue}{\swarrow}&&\color{MidnightBlue}{\searrow}&\\ 20\times(2+4)&&&&(11+9)\times 6\\ &\color{MidnightBlue}{\searrow}&&\color{MidnightBlue}{\swarrow}&\\ \color{MidnightBlue}{\mbox{eval right}~{}}&&20\times 6&&\color{MidnightBlue}{% \mbox{eval left}~{}}\\ &&\color{MidnightBlue}{\downarrow}&&\\ &&120&&\\ \hline\end{array}
  2. a a
  3. b b
  4. c c
  5. d d

Conformal_gravity.html

  1. g a b Ω 2 ( x ) g a b g_{ab}\rightarrow\Omega^{2}(x)g_{ab}
  2. g a b g_{ab}
  3. Ω ( x ) \Omega(x)
  4. 𝒮 = d 4 x - g C a b c d C a b c d , \mathcal{S}=\int\mathrm{d}^{4}x\sqrt{-g}C_{abcd}C^{abcd},
  5. C a b c d C_{abcd}
  6. 2 a d C a b c d + C a b c d R a d = 0 , 2\nabla_{a}\nabla_{d}{{C^{a}}_{bc}}^{d}+{{C^{a}}_{bc}}^{d}R_{ad}=0,
  7. R a b R_{ab}
  8. 2 Φ = 0 \Box^{2}\Phi=0
  9. Φ ( r ) = 1 - 2 m r + a r + b r 2 \Phi(r)=1-\frac{2m}{r}+ar+br^{2}
  10. ϕ ( r ) = g 00 = ( 1 - 6 b c ) 1 2 - 2 b r + c r + d 3 r 2 \phi(r)=g^{00}=(1-6bc)^{\frac{1}{2}}-\frac{2b}{r}+cr+\frac{d}{3}r^{2}
  11. ε \varepsilon
  12. Φ + ε 2 2 Φ = 0 \Box\Phi+\varepsilon^{2}\Box^{2}\Phi=0
  13. Φ = 1 - 2 m r ( 1 + α sin ( r / ε + β ) ) \Phi=1-\frac{2m}{r}(1+\alpha\sin(r/\varepsilon+\beta))

Congruum.html

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

Conical_coordinates.html

  1. r r
  2. ( r , μ , ν ) (r,\mu,\nu)
  3. x = r μ ν b c x=\frac{r\mu\nu}{bc}
  4. y = r b ( μ 2 - b 2 ) ( ν 2 - b 2 ) ( b 2 - c 2 ) y=\frac{r}{b}\sqrt{\frac{\left(\mu^{2}-b^{2}\right)\left(\nu^{2}-b^{2}\right)}% {\left(b^{2}-c^{2}\right)}}
  5. z = r c ( μ 2 - c 2 ) ( ν 2 - c 2 ) ( c 2 - b 2 ) z=\frac{r}{c}\sqrt{\frac{\left(\mu^{2}-c^{2}\right)\left(\nu^{2}-c^{2}\right)}% {\left(c^{2}-b^{2}\right)}}
  6. ν 2 < c 2 < μ 2 < b 2 \nu^{2}<c^{2}<\mu^{2}<b^{2}
  7. r r
  8. x 2 + y 2 + z 2 = r 2 x^{2}+y^{2}+z^{2}=r^{2}
  9. μ \mu
  10. ν \nu
  11. x 2 μ 2 + y 2 μ 2 + b 2 + z 2 μ 2 - c 2 = 0 \frac{x^{2}}{\mu^{2}}+\frac{y^{2}}{\mu^{2}+b^{2}}+\frac{z^{2}}{\mu^{2}-c^{2}}=0
  12. x 2 ν 2 + y 2 ν 2 - b 2 + z 2 ν 2 + c 2 = 0 \frac{x^{2}}{\nu^{2}}+\frac{y^{2}}{\nu^{2}-b^{2}}+\frac{z^{2}}{\nu^{2}+c^{2}}=0
  13. r r
  14. h r = 1 h_{r}=1
  15. h μ = r μ 2 - ν 2 ( b 2 - μ 2 ) ( μ 2 - c 2 ) h_{\mu}=r\sqrt{\frac{\mu^{2}-\nu^{2}}{\left(b^{2}-\mu^{2}\right)\left(\mu^{2}-% c^{2}\right)}}
  16. h ν = r μ 2 - ν 2 ( b 2 - ν 2 ) ( c 2 - ν 2 ) h_{\nu}=r\sqrt{\frac{\mu^{2}-\nu^{2}}{\left(b^{2}-\nu^{2}\right)\left(c^{2}-% \nu^{2}\right)}}

Conical_pendulum.html

  1. T sin θ = m v 2 r T\sin\theta=\frac{mv^{2}}{r}\,
  2. T cos θ = m g T\cos\theta=mg\,
  3. g cos θ = v 2 r sin θ \frac{g}{\cos\theta}=\frac{v^{2}}{r\sin\theta}
  4. v = 2 π r t v=\frac{2\pi r}{t}
  5. g cos θ = ( 2 π r t ) 2 r sin θ = ( 2 π ) 2 r t 2 sin θ \frac{g}{\cos\theta}=\frac{(\frac{2\pi r}{t})^{2}}{r\sin\theta}=\frac{(2\pi)^{% 2}r}{t^{2}\sin\theta}
  6. t = 2 π r g tan θ t=2\pi\sqrt{\frac{r}{g\tan\theta}}
  7. r = L sin θ r=L\sin\theta\,
  8. t = 2 π L cos θ g t=2\pi\sqrt{\frac{L\cos\theta}{g}}

Conjugacy_problem.html

  1. y = z x z - 1 . y=zxz^{-1}.\,\!
  2. x y - 1 xy^{-1}

Conjugate_index.html

  1. p , q > 1 p,q>1
  2. 1 p + 1 q = 1. \frac{1}{p}+\frac{1}{q}=1.
  3. q = q=\infty
  4. p = 1 p=1
  5. p , q > 1 p,q>1

Conjunctive_grammar.html

  1. A α 1 & & α m A\to\alpha_{1}\And\ldots\And\alpha_{m}
  2. A A
  3. α 1 \alpha_{1}
  4. α m \alpha_{m}
  5. Σ \Sigma
  6. N N
  7. w w
  8. Σ \Sigma
  9. α 1 \alpha_{1}
  10. α m \alpha_{m}
  11. A A

Connect6.html

  1. ( 300 × 300 2 ) \left(\frac{300\times 300}{2}\right)
  2. ( 300 × 300 2 ) 30 \left(\frac{300\times 300}{2}\right)^{30}
  3. ( 300 × 300 2 ) 15 \left(\frac{300\times 300}{2}\right)^{15}

Connectedness_locus.html

  1. f c ( z ) = z 2 + c f_{c}(z)=z^{2}+c\,
  2. z z d + c z\mapsto z^{d}+c\,
  3. d 3 d\geq 3\,

Consolidation_(soil).html

  1. δ c = C c 1 + e 0 H log ( σ z f σ z 0 ) \delta_{c}=\frac{C_{c}}{1+e_{0}}H\log\left(\frac{\sigma_{zf}^{\prime}}{\sigma_% {z0}^{\prime}}\right)
  2. δ c = C r 1 + e 0 H log ( σ z c σ z 0 ) + C c 1 + e 0 H log ( σ z f σ z c ) \delta_{c}=\frac{C_{r}}{1+e_{0}}H\log\left(\frac{\sigma_{zc}^{\prime}}{\sigma_% {z0}^{\prime}}\right)+\frac{C_{c}}{1+e_{0}}H\log\left(\frac{\sigma_{zf}^{% \prime}}{\sigma_{zc}^{\prime}}\right)
  3. S s = H 0 1 + e 0 C a log ( t t 90 ) S_{s}=\frac{H_{0}}{1+e_{0}}C_{a}\log\left(\frac{t}{t_{90}}\right)

Constant_curvature.html

  1. R = 0 \nabla\mathrm{R}=0
  2. 1 2 n ( n + 1 ) \frac{1}{2}n(n+1)
  3. 1 2 n ( n + 1 ) \frac{1}{2}n(n+1)

Constant_fraction_discriminator.html

  1. t r t_{r}
  2. t d t_{d}
  3. 0 t d t r 0\ll t_{d}\leq t_{r}

Constrained_optimization.html

  1. min f ( 𝐱 ) subject to g i ( 𝐱 ) = c i for i = 1 , , n Equality constraints h j ( 𝐱 ) d j for j = 1 , , m Inequality constraints \begin{array}[]{rcll}\min&&f(\mathbf{x})&\\ \mathrm{subject~{}to}&&g_{i}(\mathbf{x})=c_{i}&\,\text{for }i=1,\ldots,n\quad% \,\text{Equality constraints}\\ &&h_{j}(\mathbf{x})\geqq d_{j}&\,\text{for }j=1,\ldots,m\quad\,\text{% Inequality constraints}\end{array}
  2. g i ( 𝐱 ) = c i for i = 1 , , n g_{i}(\mathbf{x})=c_{i}~{}\mathrm{for~{}}i=1,\ldots,n
  3. h j ( 𝐱 ) d j for j = 1 , , m h_{j}(\mathbf{x})\geq d_{j}~{}\mathrm{for~{}}j=1,\ldots,m
  4. x = a x=a
  5. x = b x=b
  6. n n
  7. n n
  8. x 1 , , x i x_{1},\ldots,x_{i}
  9. x i + 1 , , x n x_{i+1},\ldots,x_{n}
  10. x i + 1 , , x n x_{i+1},\ldots,x_{n}
  11. x x
  12. C 1 , , C n C_{1},\ldots,C_{n}
  13. y 1 , , y m y_{1},\ldots,y_{m}
  14. x x
  15. C ( y 1 = a 1 , , y n = a n ) = max a i C i ( x = a , y 1 = a 1 , , y n = a n ) C(y_{1}=a_{1},\ldots,y_{n}=a_{n})=\max_{a}\sum_{i}C_{i}(x=a,y_{1}=a_{1},\ldots% ,y_{n}=a_{n})

Constraint_inference.html

  1. D D
  2. C C
  3. D D
  4. C C
  5. V V
  6. D D
  7. D D
  8. V V
  9. V V
  10. C C
  11. ( ( x , y ) , R ) ((x,y),R)
  12. ( ( y , z ) , S ) ((y,z),S)
  13. ( ( x , z ) , Q ) ((x,z),Q)
  14. y y
  15. ( ( x , y ) , R ) ((x,y),R)
  16. ( ( y , z ) , S ) ((y,z),S)
  17. ( t , R ) (t,R)
  18. t t^{\prime}
  19. ( t , R ) (t^{\prime},R^{\prime})
  20. ( t , R ) (t,R)
  21. C 1 , , C m C_{1},\ldots,C_{m}
  22. A A
  23. ( A , R ) (A,R)
  24. A A
  25. C 1 , , C m C_{1},\ldots,C_{m}

Constraint_learning.html

  1. x 1 = a 1 , , x k = a k x_{1}=a_{1},\ldots,x_{k}=a_{k}
  2. x i = a i x_{i}=a_{i}
  3. i [ 1 , k ] i\in[1,k]
  4. x 2 = a 2 , x 5 = a 5 , x k - 1 = a k - 1 x_{2}=a_{2},x_{5}=a_{5},x_{k-1}=a_{k-1}
  5. x 1 x_{1}
  6. x 4 x_{4}
  7. x 1 x_{1}
  8. x 4 x_{4}
  9. x k + 1 x_{k+1}
  10. x 1 = a 1 , , x k = a k x_{1}=a_{1},\ldots,x_{k}=a_{k}
  11. x k + 1 x_{k+1}
  12. x k + 1 x_{k+1}
  13. x 1 = a 1 , , x k = a k x_{1}=a_{1},\ldots,x_{k}=a_{k}
  14. x k + 1 x_{k+1}
  15. x 1 , , x k x_{1},\ldots,x_{k}
  16. x k + 1 x_{k+1}

Constraint_logic_programming.html

  1. G , S \langle G,S\rangle
  2. X = 2 X=2
  3. X > 2 X>2
  4. G , S G , S \langle G,S\rangle\rightarrow\langle G^{\prime},S^{\prime}\rangle
  5. G , S \langle G,S\rangle
  6. G , S \langle G^{\prime},S^{\prime}\rangle
  7. G G
  8. C C
  9. G = G \ { C } G^{\prime}=G\backslash\{C\}
  10. S = S { C } S^{\prime}=S\cup\{C\}
  11. G G
  12. L ( t 1 , , t n ) L(t_{1},\ldots,t_{n})
  13. L ( t 1 , , t n ) : - B L(t_{1}^{\prime},\ldots,t_{n}^{\prime}):-B
  14. G G^{\prime}
  15. G G
  16. L ( t 1 , , t n ) L(t_{1},\ldots,t_{n})
  17. t 1 = t 1 , , t n = t n , B t_{1}=t_{1}^{\prime},\ldots,t_{n}=t_{n}^{\prime},B
  18. S = S S^{\prime}=S
  19. S S
  20. S S^{\prime}
  21. G G
  22. G , \langle G,\emptyset\rangle
  23. , S \langle\emptyset,S\rangle
  24. S S
  25. L ( t 1 , , t n ) = L ( t 1 , , t n ) L(t_{1},\ldots,t_{n})=L(t_{1}^{\prime},\ldots,t_{n}^{\prime})
  26. t 1 = t 1 , , t n = t n t_{1}=t_{1}^{\prime},\ldots,t_{n}=t_{n}^{\prime}
  27. L ( t 1 , , t n ) = L ( t 1 , , t n ) L(t_{1},\ldots,t_{n})=L(t_{1}^{\prime},\ldots,t_{n}^{\prime})
  28. Y = 1 Y=1

Constraint_satisfaction_dual_problem.html

  1. x = 1 , y = 2 x=1,y=2
  2. y = 3 , z = 1 y=3,z=1
  3. y y
  4. C 1 C_{1}
  5. C 2 C_{2}
  6. y y
  7. C 1 C_{1}
  8. C 2 C_{2}
  9. ( 1 , 2 ) (1,2)
  10. ( 2 , 1 ) (2,1)
  11. y = 2 y=2
  12. x , y x,y
  13. C 1 C_{1}
  14. C 2 C_{2}
  15. C 1 C_{1}
  16. C 2 C_{2}
  17. x x
  18. y y
  19. C 2 C_{2}
  20. C 1 C_{1}
  21. y y
  22. C 1 C_{1}
  23. C 3 C_{3}
  24. y y
  25. C 2 C_{2}
  26. C 3 C_{3}
  27. y y
  28. y y
  29. C 2 C_{2}
  30. C 3 C_{3}
  31. n n
  32. n n

Constructive_set_theory.html

  1. A ( [ x A y ϕ ( x , y ) ] B x A y B ϕ ( x , y ) ) \forall A\;([\forall x\in A\;\exists y\;\phi(x,y)]\to\exists B\;\forall x\in A% \;\exists y\in B\;\phi(x,y))
  2. [ y ( [ x y ϕ ( x ) ] ϕ ( y ) ) ] y ϕ ( y ) [\forall y\;([\forall x\in y\;\phi(x)]\to\phi(y))]\to\forall y\;\phi(y)
  3. a ( ( x a y ϕ ( x , y ) ) b ( x a y b ϕ ( x , y ) ) ( y b x a ϕ ( x , y ) ) ) \forall a((\forall x\in a\;\exists y\;\phi(x,y))\to\exists b\;(\forall x\in a% \;\exists y\in b\;\phi(x,y))\wedge(\forall y\in b\;\exists x\in a\;\phi(x,y)))
  4. a , b u z ( ( x a y b ϕ ( x , y , z ) ) v u ( x a y v ϕ ( x , y , z ) ) ( y v x a ϕ ( x , y , z ) ) ) \forall a,b\;\exists u\;\forall z((\forall x\in a\;\exists y\in b\;\phi(x,y,z)% )\to\exists v\in u\;(\forall x\in a\;\exists y\in v\;\phi(x,y,z))\wedge(% \forall y\in v\;\exists x\in a\;\phi(x,y,z)))
  5. a , b c P ( a , b ) R P ( a , b ) S c S R \forall a,b\;\exists c\subseteq P(a,b)\;\forall R\in P(a,b)\;\exists S\in c\;S\subseteq R
  6. R P ( a , b ) ( u R x a y b x , y = u ) ( x a y b x , y R ) R\in P(a,b)\iff(\forall u\in R\;\exists x\in a\;\exists y\in b\;\langle x,y% \rangle=u)\wedge(\forall x\in a\;\exists y\in b\;\langle x,y\rangle\in R)
  7. c P ( a , b ) R c R P ( a , b ) c\subseteq P(a,b)\iff\forall R\in c\;R\in P(a,b)

Continuously_compounded_nominal_and_real_returns.html

  1. 1 + R S t = P t P t - 1 . 1+RS_{t}=\frac{P_{t}}{P_{t-1}}.
  2. R C t = ln ( P t P t - 1 ) . RC_{t}=\ln\left(\frac{P_{t}}{P_{t-1}}\right).
  3. P t = P t - 1 e R C t P_{t}=P_{t-1}\cdot e^{RC_{t}}
  4. π t \pi_{t}
  5. π t = 1 / ( P L t ) \pi_{t}=1/(PL_{t})
  6. P L t P L t - 1 - 1 \tfrac{PL_{t}}{PL_{t-1}}-1
  7. P t r e a l = P t P L t - 1 P L t . P_{t}^{real}=P_{t}\cdot\frac{PL_{t-1}}{PL_{t}}.
  8. R C r e a l RC^{real}
  9. R C t r e a l = ln ( P t r e a l P t - 1 ) . RC_{t}^{real}=\ln\left(\frac{P_{t}^{real}}{P_{t-1}}\right).

Contrast_resolution.html

  1. C = S A - S B S A + S B C=\frac{S_{A}-S_{B}}{S_{A}+S_{B}}
  2. C = | S A - S B | S ref C=\frac{|S_{A}-S_{B}|}{S_{\mathrm{ref}}}

Convenience_yield.html

  1. F t , T F_{t,T}
  2. S t S_{t}
  3. T T
  4. r r
  5. F t , T = S t e r ( T - t ) F_{t,T}=S_{t}\cdot e^{r(T-t)}
  6. S t S_{t}
  7. c c
  8. F t , T = S t e ( r - c ) ( T - t ) F_{t,T}=S_{t}\cdot e^{(r-c)(T-t)}
  9. T T
  10. r r
  11. F = S [ 1 + ( r - c ) T ] F=S\left[1+(r-c)T\right]
  12. c c
  13. c = r + 1 T ( 1 - F S ) c=r+\frac{1}{T}\left(1-\frac{F}{S}\right)
  14. c = 0.035 + 1 0.5 ( 1 - 1300 1371 ) = 0.13857 = 13.9 % c=0.035+\frac{1}{0.5}\left(1-\frac{1300}{1371}\right)=0.13857=13.9\%
  15. F = S e ( r - c ) T F=S\cdot e^{(r-c)T}
  16. c = r - 1 T ln ( F S ) c=r-\frac{1}{T}\ln\left(\frac{F}{S}\right)
  17. c = 0.035 - 1 0.5 × ln ( 1300 1371 ) = 0.14135 = 14.1 % c=0.035-\frac{1}{0.5}\times\ln\left(\frac{1300}{1371}\right)=0.14135=14.1\%
  18. F t , T F_{t,T}
  19. S t S_{t}
  20. r - c > 0 r-c>0
  21. F t , T < S t F_{t,T}<S_{t}
  22. r - c < 0 r-c<0

Convergence_tests.html

  1. lim n a n 0 \lim_{n\to\infty}a_{n}\neq 0
  2. r r
  3. lim n | a n + 1 a n | = r . \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|=r.
  4. r = lim sup n | a n | n , r=\limsup_{n\to\infty}\sqrt[n]{|a_{n}|},
  5. f : [ 1 , ) \R + f:[1,\infty)\to\R_{+}
  6. f ( n ) = a n f(n)=a_{n}
  7. 1 f ( x ) d x = lim t 1 t f ( x ) d x < , \int_{1}^{\infty}f(x)\,dx=\lim_{t\to\infty}\int_{1}^{t}f(x)\,dx<\infty,
  8. a n {a_{n}}
  9. n = 1 b n \sum_{n=1}^{\infty}b_{n}
  10. | a n | | b n | |a_{n}|\leq|b_{n}|
  11. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  12. { a n } , { b n } > 0 \left\{a_{n}\right\},\left\{b_{n}\right\}>0
  13. lim n a n b n \lim_{n\to\infty}\frac{a_{n}}{b_{n}}
  14. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  15. n = 1 b n \sum_{n=1}^{\infty}b_{n}
  16. { a n } \left\{a_{n}\right\}
  17. A = n = 1 a n A=\sum_{n=1}^{\infty}a_{n}
  18. A * = n = 0 2 n a 2 n A^{*}=\sum_{n=0}^{\infty}2^{n}a_{2^{n}}
  19. A A * 2 A A\leq A^{*}\leq 2A
  20. a n \sum a_{n}
  21. a n b n \sum a_{n}b_{n}
  22. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  23. a n a_{n}
  24. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  25. b n = n ( a n a n + 1 - 1 ) b_{n}=n\left(\frac{a_{n}}{a_{n+1}}-1\right)
  26. L = lim n b n L=\lim_{n\to\infty}b_{n}
  27. | a n + 1 a n | 1 - b n |\frac{a_{n+1}}{a_{n}}|\leq 1-\frac{b}{n}
  28. ( * ) n = 1 1 n α (*)\;\;\;\sum_{n=1}^{\infty}\frac{1}{n^{\alpha}}
  29. ( * * ) n = 1 2 n ( 1 2 n ) α (**)\;\;\;\sum_{n=1}^{\infty}2^{n}\left(\frac{1}{2^{n}}\right)^{\alpha}
  30. n = 1 2 n ( 1 2 n ) α = n = 1 2 n - n α = n = 1 2 ( 1 - α ) n \sum_{n=1}^{\infty}2^{n}\left(\frac{1}{2^{n}}\right)^{\alpha}=\sum_{n=1}^{% \infty}2^{n-n\alpha}=\sum_{n=1}^{\infty}2^{(1-\alpha)n}
  31. 2 ( 1 - α ) 2^{(1-\alpha)}
  32. α > 1 \alpha>1
  33. α > 1 \alpha>1
  34. { a n } n = 1 \left\{a_{n}\right\}_{n=1}^{\infty}
  35. n = 1 ( 1 + a n ) \prod_{n=1}^{\infty}(1+a_{n})
  36. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  37. 0 < a n < 1 0<a_{n}<1
  38. n = 1 ( 1 - a n ) \prod_{n=1}^{\infty}(1-a_{n})
  39. n = 1 a n \sum_{n=1}^{\infty}a_{n}

Convex_analysis.html

  1. λ x + ( 1 - λ ) y C \lambda x+(1-\lambda)y\in C
  2. f ( λ x + ( 1 - λ ) y ) λ f ( x ) + ( 1 - λ ) f ( y ) f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)
  3. { ( x , r ) X × 𝐑 : f ( x ) r } \left\{(x,r)\in X\times\mathbf{R}:f(x)\leq r\right\}
  4. f * ( x * ) = sup x X { x * , x - f ( x ) } . f^{*}(x^{*})=\sup_{x\in X}\left\{\langle x^{*},x\rangle-f(x)\right\}.
  5. inf x M f ( x ) \inf_{x\in M}f(x)
  6. inf x X f ( x ) . \inf_{x\in X}f(x).
  7. f = f + I constraints f=f+I_{\mathrm{constraints}}
  8. sup y * Y * - F * ( 0 , y * ) \sup_{y^{*}\in Y^{*}}-F^{*}(0,y^{*})
  9. sup y * Y * - F * ( 0 , y * ) inf x X F ( x , 0 ) . \sup_{y^{*}\in Y^{*}}-F^{*}(0,y^{*})\leq\inf_{x\in X}F(x,0).
  10. L ( x , u ) = f ( x ) + j = 1 m u j g j ( x ) L(x,u)=f(x)+\sum_{j=1}^{m}u_{j}g_{j}(x)

Convex_cone.html

  1. S = { x V : | x | = 1 } . S=\{x\in V\;:\;|x|=1\}.
  2. { v V : w C , w , v 0 } . \{v\in V\;:\;\forall w\in C,\langle w,v\rangle\geq 0\}.
  3. C * := { v V * : w C , v ( w ) 0 } . C^{*}:=\left\{v\in V^{*}\;:\;\forall w\in C,v(w)\geq 0\right\}.
  4. N K ( x ) = { p V : x * K , p , x - x * 0 } . N_{K}(x)=\left\{p\in V\;:\;\forall x^{*}\in K,\langle p,x-x^{*}\rangle\geq 0% \right\}.
  5. T K ( x ) = h > 0 1 h ( K - x ) ¯ . T_{K}(x)=\overline{\bigcup_{h>0}\tfrac{1}{h}(K-x)}.
  6. N K ( x ) = { p V : x * K , p , x - x * 0 } . N_{K}(x)=\left\{p\in V\;:\;\forall x^{*}\in K,\langle p,x-x^{*}\rangle% \leqslant 0\right\}.
  7. N K ( x ) N_{K}(x)
  8. T K ( x ) = N K * ( x ) = def { y V | ξ N K ( x ) : y , ξ 0 } T_{K}(x)=N_{K}^{*}(x)\overset{\underset{\mathrm{def}}{}}{=}\{y\in V|\forall\xi% \in N_{K}(x):\langle y,\xi\rangle\leqslant 0\}

Cook's_distance.html

  1. D i = j = 1 n ( Y ^ j - Y ^ j ( i ) ) 2 p MSE , D_{i}=\frac{\sum_{j=1}^{n}(\hat{Y}_{j}\ -\hat{Y}_{j(i)})^{2}}{p\ \mathrm{MSE}},
  2. Y ^ j \hat{Y}_{j}\,
  3. Y ^ j ( i ) \hat{Y}_{j(i)}\,
  4. p p
  5. MSE \mathrm{MSE}\,
  6. D i = e i 2 p MSE [ h i i ( 1 - h i i ) 2 ] , D_{i}=\frac{e_{i}^{2}}{p\ \mathrm{MSE}}\left[\frac{h_{ii}}{(1-h_{ii})^{2}}% \right],
  7. D i = ( β ^ - β ^ ( - i ) ) T ( X T X ) ( β ^ - β ^ ( - i ) ) ( 1 + p ) s 2 , D_{i}=\frac{(\hat{\beta}-\hat{\beta}^{(-i)})^{T}(X^{T}X)(\hat{\beta}-\hat{% \beta}^{(-i)})}{(1+p)s^{2}},
  8. h i i h_{ii}\,
  9. 𝐗 ( 𝐗 T 𝐗 ) - 1 𝐗 T \mathbf{X}\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}
  10. e i e_{i}\,
  11. D i > 1 D_{i}>1
  12. D i > 4 / n D_{i}>4/n
  13. n n
  14. H 0 : β i = β 0 H_{0}:\beta_{i}=\beta_{0}
  15. β ^ [ - i ] \hat{\beta}_{[-i]}
  16. F p , n - p F_{p,n-p}
  17. F p , n - p , 1 - α F_{p,n-p,1-\alpha}
  18. D i D_{i}

Coordinate_time.html

  1. U = i G M i r i U=\sum_{i}\frac{GM_{i}}{r_{i}}

Copeland–Erdős_constant.html

  1. n = 1 p n 10 - ( n + k = 1 n log 10 p k ) \displaystyle\sum_{n=1}^{\infty}p_{n}10^{-\left(n+\sum_{k=1}^{n}\lfloor\log_{1% 0}{p_{k}}\rfloor\right)}
  2. n = 1 b - p n , \displaystyle\sum_{n=1}^{\infty}b^{-p_{n}},\,

Copper_loss.html

  1. Copper Loss I 2 R \mbox{Copper Loss}~{}\propto I^{2}\cdot R
  2. Copper Loss = I 2 R t \mbox{Copper Loss}~{}=I^{2}\cdot R\cdot t

Coppock_curve.html

  1. C o p p o c k = W M A [ 10 ] o f ( R O C [ 14 ] + R O C [ 11 ] ) Coppock=WMA[10]\;of\;(ROC[14]+ROC[11])

Coriolis_frequency.html

  1. f = 2 Ω sin φ . f=2\Omega\sin\varphi.\,
  2. f f
  3. φ \varphi
  4. v v
  5. 2 s y m b o l Ω × v 2\,symbol{\Omega\times v}
  6. s y m b o l Ω symbol{\Omega}
  7. | s y m b o l Ω | = Ω |symbol{\Omega}|=\Omega
  8. v v
  9. φ \varphi
  10. Ω sin φ \Omega\sin\varphi
  11. v v
  12. s y m b o l Ω symbol{\Omega}
  13. v 2 / r = 2 ( Ω sin φ ) v v^{2}/r=2(\Omega\sin\varphi)v
  14. r r
  15. v v
  16. v = r ω v=r\omega
  17. f = ω = 2 Ω sin φ . f=\omega=2\Omega\sin\varphi.
  18. f f
  19. f f
  20. f f
  21. β = f y \beta=\frac{\partial f}{\partial y}
  22. y y

Corner_detection.html

  1. I I
  2. ( u , v ) (u,v)
  3. ( x , y ) (x,y)
  4. S S
  5. S ( x , y ) = u v w ( u , v ) ( I ( u + x , v + y ) - I ( u , v ) ) 2 S(x,y)=\sum_{u}\sum_{v}w(u,v)\,\left(I(u+x,v+y)-I(u,v)\right)^{2}
  6. I ( u + x , v + y ) I(u+x,v+y)
  7. I x I_{x}
  8. I y I_{y}
  9. I I
  10. I ( u + x , v + y ) I ( u , v ) + I x ( u , v ) x + I y ( u , v ) y I(u+x,v+y)\approx I(u,v)+I_{x}(u,v)x+I_{y}(u,v)y
  11. S ( x , y ) u v w ( u , v ) ( I x ( u , v ) x + I y ( u , v ) y ) 2 , S(x,y)\approx\sum_{u}\sum_{v}w(u,v)\,\left(I_{x}(u,v)x+I_{y}(u,v)y\right)^{2},
  12. S ( x , y ) ( x y ) A ( x y ) , S(x,y)\approx\begin{pmatrix}x&y\end{pmatrix}A\begin{pmatrix}x\\ y\end{pmatrix},
  13. A = u v w ( u , v ) [ I x 2 I x I y I x I y I y 2 ] = [ I x 2 I x I y I x I y I y 2 ] A=\sum_{u}\sum_{v}w(u,v)\begin{bmatrix}I_{x}^{2}&I_{x}I_{y}\\ I_{x}I_{y}&I_{y}^{2}\end{bmatrix}=\begin{bmatrix}\langle I_{x}^{2}\rangle&% \langle I_{x}I_{y}\rangle\\ \langle I_{x}I_{y}\rangle&\langle I_{y}^{2}\rangle\end{bmatrix}
  14. ( u , v ) (u,v)
  15. S S
  16. ( x y ) \begin{pmatrix}x&y\end{pmatrix}
  17. A A
  18. A A
  19. λ 1 0 \lambda_{1}\approx 0
  20. λ 2 0 \lambda_{2}\approx 0
  21. ( x , y ) (x,y)
  22. λ 1 0 \lambda_{1}\approx 0
  23. λ 2 \lambda_{2}
  24. λ 1 \lambda_{1}
  25. λ 2 \lambda_{2}
  26. M c M_{c}
  27. κ \kappa
  28. M c = λ 1 λ 2 - κ ( λ 1 + λ 2 ) 2 = det ( A ) - κ trace 2 ( A ) M_{c}=\lambda_{1}\lambda_{2}-\kappa\,(\lambda_{1}+\lambda_{2})^{2}=% \operatorname{det}(A)-\kappa\,\operatorname{trace}^{2}(A)
  29. A A
  30. A A
  31. m i n ( λ 1 , λ 2 ) min(\lambda_{1},\lambda_{2})
  32. κ \kappa
  33. κ \kappa
  34. M c M_{c}^{\prime}
  35. M c = 2 det ( A ) trace ( A ) + ϵ , M_{c}^{\prime}=2\frac{\operatorname{det}(A)}{\operatorname{trace}(A)+\epsilon},
  36. ϵ \epsilon
  37. A - 1 A^{-1}
  38. 1 I x 2 I y 2 - I x I y 2 [ I y 2 - I x I y - I x I y I x 2 ] . \frac{1}{\langle I_{x}^{2}\rangle\langle I_{y}^{2}\rangle-\langle I_{x}I_{y}% \rangle^{2}}\begin{bmatrix}\langle I_{y}^{2}\rangle&-\langle I_{x}I_{y}\rangle% \\ -\langle I_{x}I_{y}\rangle&\langle I_{x}^{2}\rangle\end{bmatrix}.
  39. T 𝐱 ( 𝐱 ) T_{\mathbf{x^{\prime}}}(\mathbf{x})
  40. 𝐱 \mathbf{x^{\prime}}
  41. T 𝐱 ( 𝐱 ) = I ( 𝐱 ) ( 𝐱 - 𝐱 ) = 0 T_{\mathbf{x^{\prime}}}(\mathbf{x})=\nabla I(\mathbf{x^{\prime}})^{\top}(% \mathbf{x}-\mathbf{x^{\prime}})=0
  42. I ( 𝐱 ) = [ I 𝐱 , I 𝐲 ] \nabla I(\mathbf{x^{\prime}})=[I_{\mathbf{x}},I_{\mathbf{y}}]^{\top}
  43. I I
  44. 𝐱 \mathbf{x^{\prime}}
  45. 𝐱 0 \mathbf{x}_{0}
  46. N N
  47. 𝐱 0 = argmin 𝐱 2 × 2 𝐱 N T 𝐱 ( 𝐱 ) 2 d 𝐱 \mathbf{x}_{0}=\underset{\mathbf{x}\in\mathbb{R}^{2\times 2}}{\operatorname{% argmin}}\int_{\mathbf{x^{\prime}}\in N}T_{\mathbf{x^{\prime}}}(\mathbf{x})^{2}% d\mathbf{x^{\prime}}
  48. 𝐱 0 \mathbf{x}_{0}
  49. T 𝐱 T_{\mathbf{x^{\prime}}}
  50. 𝐱 0 \mathbf{x}_{0}
  51. 𝐱 0 \displaystyle\mathbf{x}_{0}
  52. A 2 × 2 , 𝐛 2 × 1 , c A\in\mathbb{R}^{2\times 2},\,\textbf{b}\in\mathbb{R}^{2\times 1},c\in\mathbb{R}
  53. A \displaystyle A
  54. x x
  55. 2 A 𝐱 - 2 𝐛 = 0 A 𝐱 = 𝐛 2A\mathbf{x}-2\mathbf{b}=0\Rightarrow A\mathbf{x}=\mathbf{b}
  56. A 2 × 2 A\in\mathbb{R}^{2\times 2}
  57. A A
  58. A A
  59. x 0 = A - 1 𝐛 x_{0}=A^{-1}\mathbf{b}
  60. N N
  61. d ~ m i n = c - b T A - 1 b trace A \tilde{d}_{min}=\frac{c-b^{T}A^{-1}b}{\mbox{trace}~{}A}
  62. c c
  63. c = 0 c=0
  64. A A
  65. I x , I y I_{x},I_{y}
  66. I I
  67. L L
  68. I I
  69. g ( x , y , t ) = 1 2 π t e - ( x 2 + y 2 ) / 2 t g(x,y,t)=\frac{1}{2{\pi}t}e^{-(x^{2}+y^{2})/2t}
  70. t t
  71. L ( x , y , t ) = g ( x , y , t ) * I ( x , y ) L(x,y,t)\ =g(x,y,t)*I(x,y)
  72. L x = x L L_{x}=\partial_{x}L
  73. L y = y L L_{y}=\partial_{y}L
  74. L L
  75. g ( x , y , s ) g(x,y,s)
  76. s s
  77. μ ( x , y ; t , s ) = ξ = - η = - [ L x 2 ( x - ξ , y - η ; t ) L x ( x - ξ , y - η ; t ) L y ( x - ξ , y - η ; t ) L x ( x - ξ , y - η ; t ) L y ( x - ξ , y - η ; t ) L y 2 ( x - ξ , y - η ; t ) ] g ( ξ , η ; s ) d ξ d η . \mu(x,y;t,s)=\int_{\xi=-\infty}^{\infty}\int_{\eta=-\infty}^{\infty}\begin{% bmatrix}L_{x}^{2}(x-\xi,y-\eta;t)&L_{x}(x-\xi,y-\eta;t)\,L_{y}(x-\xi,y-\eta;t)% \\ L_{x}(x-\xi,y-\eta;t)\,L_{y}(x-\xi,y-\eta;t)&L_{y}^{2}(x-\xi,y-\eta;t)\end{% bmatrix}g(\xi,\eta;s)\,d\xi\,d\eta.
  78. μ \mu
  79. A A
  80. M c ( x , y ; t , s ) = det ( μ ( x , y ; t , s ) ) - κ trace 2 ( μ ( x , y ; t , s ) ) M_{c}(x,y;t,s)=\operatorname{det}(\mu(x,y;t,s))-\kappa\,\operatorname{trace}^{% 2}(\mu(x,y;t,s))
  81. t t
  82. s s
  83. γ \gamma
  84. s = γ 2 t s=\gamma^{2}t
  85. γ \gamma
  86. [ 1 , 2 ] [1,2]
  87. M c ( x , y ; t , γ 2 t ) M_{c}(x,y;t,\gamma^{2}t)
  88. t t
  89. n o r m 2 L ( x , y ; t ) = t 2 L ( x , y , t ) = t ( L x x ( x , y , t ) + L y y ( x , y , t ) ) \nabla^{2}_{norm}L(x,y;t)\ =t\nabla^{2}L(x,y,t)=t(L_{xx}(x,y,t)+L_{yy}(x,y,t))
  90. M c ( x , y ; t , γ 2 t ) M_{c}(x,y;t,\gamma^{2}t)
  91. ( x ^ , y ^ ; t ) = argmaxlocal ( x , y ) M c ( x , y ; t , γ 2 t ) (\hat{x},\hat{y};t)=\operatorname{argmaxlocal}_{(x,y)}M_{c}(x,y;t,\gamma^{2}t)
  92. n o r m 2 ( x , y , t ) \nabla^{2}_{norm}(x,y,t)
  93. t ^ = argmaxminlocal t n o r m 2 L ( x ^ , y ^ ; t ) \hat{t}=\operatorname{argmaxminlocal}_{t}\nabla^{2}_{norm}L(\hat{x},\hat{y};t)
  94. κ ~ ( x , y ; t ) = L x 2 L y y + L y 2 L x x - 2 L x L y L x y \tilde{\kappa}(x,y;t)=L_{x}^{2}L_{yy}+L_{y}^{2}L_{xx}-2L_{x}L_{y}L_{xy}
  95. t t
  96. L L
  97. γ \gamma
  98. κ n o r m ~ ( x , y ; t ) = t 2 γ ( L x 2 L y y + L y 2 L x x - 2 L x L y L x y ) \tilde{\kappa_{norm}}(x,y;t)=t^{2\gamma}(L_{x}^{2}L_{yy}+L_{y}^{2}L_{xx}-2L_{x% }L_{y}L_{xy})
  99. γ = 7 / 8 \gamma=7/8
  100. ( x ^ , y ^ ; t ^ ) = argminmaxlocal ( x , y ; t ) κ ~ n o r m ( x , y ; t ) (\hat{x},\hat{y};\hat{t})=\operatorname{argminmaxlocal}_{(x,y;t)}\tilde{\kappa% }_{norm}(x,y;t)
  101. C C
  102. C = 2 I - c | I | 2 , C=\nabla^{2}I-c|\nabla I|^{2},
  103. c c
  104. C C
  105. M M
  106. m M \vec{m}\in M
  107. m 0 \vec{m}_{0}
  108. c ( m ) = e - ( I ( m ) - I ( m 0 ) t ) 6 c(\vec{m})=e^{-\left(\frac{I(\vec{m})-I(\vec{m}_{0})}{t}\right)^{6}}
  109. t t
  110. I I
  111. n ( M ) = m M c ( m ) n(M)=\sum_{\vec{m}\in M}c(\vec{m})
  112. c c
  113. n n
  114. t t
  115. R ( M ) = { g - n ( M ) if n ( M ) < g 0 otherwise, R(M)=\begin{cases}g-n(M)&\mbox{if}~{}\ n(M)<g\\ 0&\mbox{otherwise,}\end{cases}
  116. g g
  117. t t
  118. g g
  119. g g
  120. c \vec{c}
  121. p P \vec{p}\in P
  122. P P
  123. c \vec{c}
  124. p \vec{p^{\prime}}
  125. p \vec{p}
  126. r ( c ) = min p P ( I ( p ) - I ( c ) ) 2 + ( I ( p ) - I ( c ) ) 2 r(\vec{c})=\min_{\vec{p}\in P}\quad(I(\vec{p})-I(\vec{c}))^{2}+(I(\vec{p^{% \prime}})-I(\vec{c}))^{2}
  127. P P
  128. min \min
  129. c c
  130. r r
  131. n n
  132. t t
  133. t t
  134. r r
  135. n n
  136. n n

Corona_theorem.html

  1. | f 1 | + + | f n | δ |f_{1}|+\cdots+|f_{n}|\geq\delta

Correlated_equilibrium.html

  1. N N
  2. ( N , A i , u i ) \displaystyle(N,A_{i},u_{i})
  3. A i A_{i}
  4. u i u_{i}
  5. i i
  6. i i
  7. a i A i a_{i}\in A_{i}
  8. N - 1 N-1
  9. a - i a_{-i}
  10. i i
  11. u i ( a i , a - i ) \displaystyle u_{i}(a_{i},a_{-i})
  12. i i
  13. ϕ : A i A i \phi\colon A_{i}\to A_{i}
  14. ϕ \phi
  15. i i
  16. ϕ ( a i ) \phi(a_{i})
  17. a i a_{i}
  18. ( Ω , π ) (\Omega,\pi)
  19. i i
  20. P i P_{i}
  21. q i q_{i}
  22. i i
  23. s i : Ω A i s_{i}\colon\Omega\rightarrow A_{i}
  24. i i
  25. ( ( Ω , π ) , P i ) ((\Omega,\pi),P_{i})
  26. ( N , A i , u i ) (N,A_{i},u_{i})
  27. i i
  28. ϕ \phi
  29. ω Ω q i ( ω ) u i ( s i , s - i ) ω Ω q i ( ω ) u i ( ϕ ( s i ) , s - i ) \sum_{\omega\in\Omega}q_{i}(\omega)u_{i}(s_{i},s_{-i})\geq\sum_{\omega\in% \Omega}q_{i}(\omega)u_{i}(\phi(s_{i}),s_{-i})
  30. ( ( Ω , π ) , P i ) ((\Omega,\pi),P_{i})

Correlation_immunity.html

  1. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  2. f ( x 1 , x 2 , , x n ) f(x_{1},x_{2},\ldots,x_{n})
  3. f : 𝔽 2 n 𝔽 2 f:\mathbb{F}_{2}^{n}\rightarrow\mathbb{F}_{2}
  4. k k
  5. n n
  6. X 0 X n - 1 X_{0}\ldots X_{n-1}
  7. Z = f ( X 0 , , X n - 1 ) Z=f(X_{0},\ldots,X_{n-1})
  8. ( X i 1 X i k ) (X_{i_{1}}\ldots X_{i_{k}})
  9. 0 i 1 < < i k < n 0\leq i_{1}<\ldots<i_{k}<n

Correlation_integral.html

  1. C ( ε ) = lim N 1 N 2 i j i , j = 1 N Θ ( ε - || x ( i ) - x ( j ) || ) , x ( i ) m , C(\varepsilon)=\lim_{N\rightarrow\infty}\frac{1}{N^{2}}\sum_{\stackrel{i,j=1}{% i\neq j}}^{N}\Theta(\varepsilon-||\vec{x}(i)-\vec{x}(j)||),\quad\vec{x}(i)\in% \mathbb{R}^{m},
  2. N N
  3. x ( i ) \vec{x}(i)
  4. ε \varepsilon
  5. | | | | ||\cdot||
  6. Θ ( ) \Theta(\cdot)
  7. x ( i ) = ( u ( i ) , u ( i + τ ) , , u ( i + τ ( m - 1 ) ) , \vec{x}(i)=(u(i),u(i+\tau),\ldots,u(i+\tau(m-1)),
  8. u ( i ) u(i)
  9. m m
  10. τ \tau
  11. C ( ε ) = 1 N 2 i j i , j = 1 N Θ ( ε - || x ( i ) - x ( j ) || ) , x ( i ) m . C(\varepsilon)=\frac{1}{N^{2}}\sum_{\stackrel{i,j=1}{i\neq j}}^{N}\Theta(% \varepsilon-||\vec{x}(i)-\vec{x}(j)||),\quad\vec{x}(i)\in\mathbb{R}^{m}.

Correlation_sum.html

  1. C ( ε ) = 1 N 2 i j i , j = 1 N Θ ( ε - || x ( i ) - x ( j ) || ) , x ( i ) m , C(\varepsilon)=\frac{1}{N^{2}}\sum_{\stackrel{i,j=1}{i\neq j}}^{N}\Theta(% \varepsilon-||\vec{x}(i)-\vec{x}(j)||),\quad\vec{x}(i)\in\mathbb{R}^{m},
  2. N N
  3. x ( i ) \vec{x}(i)
  4. ε \varepsilon
  5. | | | | ||\cdot||
  6. Θ ( ) \Theta(\cdot)
  7. x ( i ) = ( u ( i ) , u ( i + τ ) , , u ( i + τ ( m - 1 ) ) , \vec{x}(i)=(u(i),u(i+\tau),\ldots,u(i+\tau(m-1)),
  8. u ( i ) u(i)
  9. m m
  10. τ \tau

Correspondence_rule.html

  1. i t i\hbar\frac{\partial}{\partial t}
  2. - i -i\hbar\nabla

Cost_efficiency.html

  1. O ( n ) O(n)
  2. O ( n p ) O\left(\frac{n}{p}\right)
  3. p p

Coulomb_operator.html

  1. J ^ j ( 1 ) f ( 1 ) = f ( 1 ) | φ j ( 2 ) | 2 1 r 12 d v 2 \widehat{J}_{j}(1)f(1)=f(1)\int{\left|\varphi_{j}(2)\right|}^{2}\frac{1}{r_{12% }}\,dv_{2}
  2. J ^ j ( 1 ) \widehat{J}_{j}(1)
  3. φ j ( 2 ) \varphi_{j}(2)
  4. r 12 r_{12}

Course-of-values_recursion.html

  1. f ( 1 ) , f ( 2 ) , , f ( n ) \langle f(1),f(2),\ldots,f(n)\rangle
  2. g ( n + 1 ) = i = 0 n g ( i ) n - i g(n+1)=\sum_{i=0}^{n}g(i)^{n-i}
  3. f ( n ) = h ( n , f ( 0 ) , f ( 1 ) , , f ( n - 1 ) ) f(n)=h(n,\langle f(0),f(1),\ldots,f(n-1)\rangle)
  4. f ( 0 ) , f ( 1 ) , , f ( n - 1 ) \langle f(0),f(1),\ldots,f(n-1)\rangle
  5. f ( 0 ) = h ( 0 , ) , f(0)=h(0,\langle\rangle),
  6. h ( n , s ) = { n if n < 2 s [ n - 2 ] + s [ n - 1 ] if n 2 h(n,s)=\begin{cases}n&\,\text{if }n<2\\ s[n-2]+s[n-1]&\,\text{if }n\geq 2\end{cases}
  7. f ( n ) = h ( n , f ( 0 ) , f ( 1 ) , , f ( n - 1 ) ) f(n)=h(n,\langle f(0),f(1),\ldots,f(n-1)\rangle)
  8. f f
  9. f ¯ ( n ) = f ( 0 ) , f ( 1 ) , , f ( n - 1 ) . \bar{f}(n)=\langle f(0),f(1),\ldots,f(n-1)\rangle.
  10. f ¯ ( n ) \bar{f}(n)
  11. n n
  12. f f
  13. f ¯ \bar{f}
  14. f ¯ ( n + 1 ) \bar{f}(n+1)
  15. f ¯ ( n ) \bar{f}(n)
  16. h ( n , f ¯ ( n ) ) h(n,\bar{f}(n))
  17. f ¯ ( 0 ) = \bar{f}(0)=\langle\rangle
  18. f ¯ ( n + 1 ) = 𝑎𝑝𝑝𝑒𝑛𝑑 ( n , f ¯ ( n ) , h ( n , f ¯ ( n ) ) ) , \bar{f}(n+1)=\mathit{append}(n,\bar{f}(n),h(n,\bar{f}(n))),
  19. a p p e n d ( n , s , x ) append(n,s,x)
  20. s s
  21. n n
  22. t t
  23. n + 1 n+1
  24. t n n = x tnn=x
  25. t i i = s i i tii=sii
  26. i Align l t ; n i&lt;n
  27. f ¯ \bar{f}
  28. f f
  29. f ( n ) = f ¯ ( n + 1 ) [ n ] f(n)=\bar{f}(n+1)[n]
  30. n 1 , n 2 , , n k \langle n_{1},n_{2},\ldots,n_{k}\rangle
  31. i = 1 k p i n i \prod_{i=1}^{k}p_{i}^{n_{i}}
  32. f ( n ) = h ( f ( 1 ) , f ( 2 ) , , f ( n - 1 ) ) f(n)=h(\langle f(1),f(2),\ldots,f(n-1)\rangle)
  33. n 1 , n 2 , , n k \langle n_{1},n_{2},\ldots,n_{k}\rangle
  34. i = 1 k p i ( n i + 1 ) \prod_{i=1}^{k}p_{i}^{(n_{i}+1)}
  35. 0 \langle 0\rangle
  36. 0 , 0 \langle 0,0\rangle

Covariant_formulation_of_classical_electromagnetism.html

  1. x α = ( c t , x , y , z ) . x^{\alpha}=(ct,x,y,z)\,.
  2. u α = γ ( c , u ) u^{\alpha}=\gamma(c,{u})\,
  3. p α = ( E / c , - p ) = m u α p_{\alpha}=(E/c,-{p})=mu_{\alpha}\,
  4. ν = x ν = ( 1 c t , - ) , \partial^{\nu}=\frac{\partial}{\partial x_{\nu}}=\left(\frac{1}{c}\frac{% \partial}{\partial t},-{\nabla}\right)\,,
  5. \Box
  6. η μ ν = ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) \eta^{\mu\nu}=\begin{pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}\,
  7. F α β = ( 0 E x / c E y / c E z / c - E x / c 0 - B z B y - E y / c B z 0 - B x - E z / c - B y B x 0 ) F_{\alpha\beta}=\left(\begin{matrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\ -E_{x}/c&0&-B_{z}&B_{y}\\ -E_{y}/c&B_{z}&0&-B_{x}\\ -E_{z}/c&-B_{y}&B_{x}&0\end{matrix}\right)\,
  8. F μ ν = def η μ α F α β η β ν = ( 0 - E x / c - E y / c - E z / c E x / c 0 - B z B y E y / c B z 0 - B x E z / c - B y B x 0 ) . F^{\mu\nu}\,\stackrel{\mathrm{def}}{=}\,\eta^{\mu\alpha}\,F_{\alpha\beta}\,% \eta^{\beta\nu}=\left(\begin{matrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\ E_{x}/c&0&-B_{z}&B_{y}\\ E_{y}/c&B_{z}&0&-B_{x}\\ E_{z}/c&-B_{y}&B_{x}&0\end{matrix}\right)\,.
  9. J α = ( c ρ , J ) J^{\alpha}=\,(c\rho,{J})\,
  10. A α = ( ϕ / c , - A ) A_{\alpha}=\left(\phi/c,-{A}\right)\,
  11. F α β = α A β - β A α F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\,
  12. T α β = ( ϵ 0 E 2 / 2 + B 2 / 2 μ 0 S x / c S y / c S z / c S x / c - σ x x - σ x y - σ x z S y / c - σ y x - σ y y - σ y z S z / c - σ z x - σ z y - σ z z ) T^{\alpha\beta}=\begin{pmatrix}\epsilon_{0}E^{2}/2+B^{2}/2\mu_{0}&S_{x}/c&S_{y% }/c&S_{z}/c\\ S_{x}/c&-\sigma_{xx}&-\sigma_{xy}&-\sigma_{xz}\\ S_{y}/c&-\sigma_{yx}&-\sigma_{yy}&-\sigma_{yz}\\ S_{z}/c&-\sigma_{zx}&-\sigma_{zy}&-\sigma_{zz}\end{pmatrix}\,
  13. S = 1 μ 0 E × B {S}=\frac{1}{\mu_{0}}{E}\times{B}
  14. σ i j = ϵ 0 E i E j + 1 μ 0 B i B j - ( 1 2 ϵ 0 E 2 + 1 2 μ 0 B 2 ) δ i j . \sigma_{ij}=\epsilon_{0}E_{i}E_{j}+\frac{1}{\mu_{0}}B_{i}B_{j}-\left(\frac{1}{% 2}\epsilon_{0}E^{2}+\frac{1}{2\mu_{0}}B^{2}\right)\delta_{ij}\,.
  15. T α β = 1 μ 0 ( η γ ν F α γ F ν β - 1 4 η α β F γ ν F γ ν ) T^{\alpha\beta}=\frac{1}{\mu_{0}}\left(\eta_{\gamma\nu}F^{\alpha\gamma}F^{\nu% \beta}-\frac{1}{4}\eta^{\alpha\beta}F_{\gamma\nu}F^{\gamma\nu}\right)
  16. ϵ 0 μ 0 c 2 = 1 \epsilon_{0}\mu_{0}c^{2}=1\,
  17. F [ α β , γ ] = 0 F_{[\alpha\beta,\gamma]}=0
  18. ν ν F α β = def F α β = def 1 c 2 2 F α β t 2 - 2 F α β = 0 , \partial^{\nu}\partial_{\nu}F^{\alpha\beta}\,\ \stackrel{\mathrm{def}}{=}\ \,% \Box F^{\alpha\beta}\,\ \stackrel{\mathrm{def}}{=}\ {1\over c^{2}}{\partial^{2% }F^{\alpha\beta}\over{\partial t}^{2}}-\nabla^{2}F^{\alpha\beta}=0\,,
  19. α A α = α A α = 0 . \partial_{\alpha}A^{\alpha}=\partial^{\alpha}A_{\alpha}=0\,.
  20. A σ = μ 0 J σ \Box A^{\sigma}=\mu_{0}\,J^{\sigma}\,
  21. d p α d t = q F α β d x β d t {dp_{\alpha}\over{dt}}=q\,F_{\alpha\beta}\,\frac{dx^{\beta}}{dt}\,
  22. d p α d τ = q F α β u β \frac{dp_{\alpha}}{d\tau}\,=q\,F_{\alpha\beta}\,u^{\beta}
  23. - f = - ( ρ E + J × B ) -{f}=-(\rho{E}+{J}\times{B})\,
  24. f α = F α β J β . f_{\alpha}=F_{\alpha\beta}J^{\beta}.\!
  25. f α = - T α β , β - T α β x β . f^{\alpha}=-{T^{\alpha\beta}}_{,\beta}\equiv-\frac{\partial T^{\alpha\beta}}{% \partial x^{\beta}}.
  26. J α , α = def α J α = 0 . {J^{\alpha}}_{,\alpha}\,\stackrel{\mathrm{def}}{=}\,\partial_{\alpha}J^{\alpha% }\,=\,0\,.
  27. T α β , β + F α β J β = 0 {T^{\alpha\beta}}_{,\beta}+F^{\alpha\beta}J_{\beta}=0
  28. η α ν T ν β , β + F α β J β = 0 , \eta_{\alpha\nu}{T^{\nu\beta}}_{,\beta}+F_{\alpha\beta}J^{\beta}=0,
  29. J ν = J ν free + J ν bound , J^{\nu}={J^{\nu}}_{\,\text{free}}+{J^{\nu}}_{\,\text{bound}}\,,
  30. J ν free = ( c ρ free , 𝐉 free ) = ( c 𝐃 , - 𝐃 t + × 𝐇 ) , {J^{\nu}}_{\,\text{free}}=(c\rho_{\,\text{free}},\mathbf{J}_{\,\text{free}})=% \left(c\nabla\cdot\mathbf{D},-\ \frac{\partial\mathbf{D}}{\partial t}+\nabla% \times\mathbf{H}\right)\,,
  31. J ν bound = ( c ρ bound , 𝐉 bound ) = ( - c 𝐏 , 𝐏 t + × 𝐌 ) . {J^{\nu}}_{\,\text{bound}}=(c\rho_{\,\text{bound}},\mathbf{J}_{\,\text{bound}}% )=\left(-\ c\nabla\cdot\mathbf{P},\frac{\partial\mathbf{P}}{\partial t}+\nabla% \times\mathbf{M}\right)\,.
  32. D = ϵ 0 E + P {D}=\epsilon_{0}{E}+{P}\,
  33. H = 1 μ 0 B - M . {H}=\frac{1}{\mu_{0}}{B}-{M}\,.
  34. μ ν = ( 0 P x c P y c P z c - P x c 0 - M z M y - P y c M z 0 - M x - P z c - M y M x 0 ) , \mathcal{M}^{\mu\nu}=\begin{pmatrix}0&P_{x}c&P_{y}c&P_{z}c\\ -P_{x}c&0&-M_{z}&M_{y}\\ -P_{y}c&M_{z}&0&-M_{x}\\ -P_{z}c&-M_{y}&M_{x}&0\end{pmatrix},
  35. J ν bound = μ μ ν . {J^{\nu}}_{\,\text{bound}}=\partial_{\mu}\mathcal{M}^{\mu\nu}\,.
  36. 𝒟 μ ν = ( 0 - D x c - D y c - D z c D x c 0 - H z H y D y c H z 0 - H x D z c - H y H x 0 ) . \mathcal{D}^{\mu\nu}=\begin{pmatrix}0&-D_{x}c&-D_{y}c&-D_{z}c\\ D_{x}c&0&-H_{z}&H_{y}\\ D_{y}c&H_{z}&0&-H_{x}\\ D_{z}c&-H_{y}&H_{x}&0\end{pmatrix}.
  37. 𝒟 μ ν = 1 μ 0 F μ ν - μ ν \mathcal{D}^{\mu\nu}=\frac{1}{\mu_{0}}F^{\mu\nu}-\mathcal{M}^{\mu\nu}\,
  38. × H = J free + D t {\nabla}\times{H}={J}_{\,\text{free}}+\frac{\partial{D}}{\partial t}
  39. D = ρ free {\nabla}\cdot{D}=\rho_{\,\text{free}}
  40. ν J ν bound = 0 \partial_{\nu}{J^{\nu}}_{\,\text{bound}}=0\,
  41. ν J ν free = 0 . \partial_{\nu}{J^{\nu}}_{\,\text{free}}=0\,.
  42. μ 0 𝒟 μ ν = η μ α F α β η β ν . \mu_{0}\mathcal{D}^{\mu\nu}=\eta^{\mu\alpha}F_{\alpha\beta}\eta^{\beta\nu}\,.
  43. F μ ν = η μ α F α β η β ν F^{\mu\nu}=\eta^{\mu\alpha}F_{\alpha\beta}\eta^{\beta\nu}\,
  44. β F α β = μ 0 J α . \partial_{\beta}F^{\alpha\beta}=\mu_{0}J^{\alpha}.\!
  45. T α π = F α β 𝒟 π β - 1 4 δ α π F μ ν 𝒟 μ ν T_{\alpha}^{\pi}=F_{\alpha\beta}\mathcal{D}^{\pi\beta}-\frac{1}{4}\delta_{% \alpha}^{\pi}F_{\mu\nu}\mathcal{D}^{\mu\nu}
  46. μ ν \mathcal{M}^{\mu\nu}
  47. J free = σ E {J}_{\,\text{free}}=\sigma{E}\,
  48. P = ϵ 0 χ e E {P}=\epsilon_{0}\chi_{e}{E}\,
  49. M = χ m H {M}=\chi_{m}{H}\,
  50. 𝒟 \mathcal{D}
  51. 𝒟 μ ν u ν = c 2 ϵ F μ ν u ν \mathcal{D}^{\mu\nu}u_{\nu}=c^{2}\epsilon F^{\mu\nu}u_{\nu}
  52. 𝒟 μ ν u ν = 1 μ F μ ν u ν {\star\mathcal{D}^{\mu\nu}}u_{\nu}=\frac{1}{\mu}{\star F^{\mu\nu}}u_{\nu}
  53. \star
  54. = field + int = - 1 4 μ 0 F α β F α β - A α J α . \mathcal{L}\,=\,\mathcal{L}_{\mathrm{field}}+\mathcal{L}_{\mathrm{int}}=-\frac% {1}{4\mu_{0}}F^{\alpha\beta}F_{\alpha\beta}-A_{\alpha}J^{\alpha}\,.
  55. ( A α , β A α ) \mathcal{L}(A_{\alpha},\partial_{\beta}A_{\alpha})\,
  56. β [ ( β A α ) ] - A α = 0 . \partial_{\beta}\left[\frac{\partial\mathcal{L}}{\partial(\partial_{\beta}A_{% \alpha})}\right]-\frac{\partial\mathcal{L}}{\partial A_{\alpha}}=0\,.
  57. F μ ν = μ A ν - ν A μ F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,
  58. ( β A α ) = - 1 4 μ 0 ( F μ ν η μ λ η ν σ F λ σ ) ( β A α ) = - 1 4 μ 0 η μ λ η ν σ ( F λ σ ( δ μ β δ ν α - δ ν β δ μ α ) + F μ ν ( δ λ β δ σ α - δ σ β δ λ α ) ) = - F β α μ 0 . \begin{aligned}\displaystyle\frac{\partial\mathcal{L}}{\partial(\partial_{% \beta}A_{\alpha})}&\displaystyle=-\ \frac{1}{4\mu_{0}}\ \frac{\partial(F_{\mu% \nu}\eta^{\mu\lambda}\eta^{\nu\sigma}F_{\lambda\sigma})}{\partial(\partial_{% \beta}A_{\alpha})}\\ &\displaystyle=-\ \frac{1}{4\mu_{0}}\ \eta^{\mu\lambda}\eta^{\nu\sigma}\left(F% _{\lambda\sigma}(\delta^{\beta}_{\mu}\delta^{\alpha}_{\nu}-\delta^{\beta}_{\nu% }\delta^{\alpha}_{\mu})+F_{\mu\nu}(\delta^{\beta}_{\lambda}\delta^{\alpha}_{% \sigma}-\delta^{\beta}_{\sigma}\delta^{\alpha}_{\lambda})\right)\\ &\displaystyle=-\ \frac{F^{\beta\alpha}}{\mu_{0}}\,.\end{aligned}
  59. A α = - J α . \frac{\partial\mathcal{L}}{\partial A_{\alpha}}=-J^{\alpha}\,.
  60. F β α x β = μ 0 J α . \frac{\partial F^{\beta\alpha}}{\partial x^{\beta}}=\mu_{0}J^{\alpha}\,.
  61. = - 1 4 μ 0 F α β F α β - A α J free α + 1 2 F α β α β . \mathcal{L}\,=\,-\frac{1}{4\mu_{0}}F^{\alpha\beta}F_{\alpha\beta}-A_{\alpha}J^% {\alpha}_{\,\text{free}}+\frac{1}{2}F_{\alpha\beta}\mathcal{M}^{\alpha\beta}\,.
  62. 𝒟 μ ν \mathcal{D}^{\mu\nu}
  63. = 1 2 ( ϵ 0 E 2 - 1 μ 0 B 2 ) - ϕ ρ free + A J free + E P + B M . \mathcal{L}\,=\,\frac{1}{2}\left(\epsilon_{0}E^{2}-\frac{1}{\mu_{0}}B^{2}% \right)-\phi\,\rho_{\,\text{free}}+{A}\cdot{J}_{\,\text{free}}+{E}\cdot{P}+{B}% \cdot{M}\,.

Crash_simulation.html

  1. 𝐌𝐚 = 𝐅 e x t - 𝐅 i n t \mathbf{Ma}=\mathbf{F}_{ext}-\mathbf{F}_{int}
  2. 𝐌𝐚 \mathbf{Ma}
  3. 𝐅 e x t \mathbf{F}_{ext}
  4. 𝐅 i n t \mathbf{F}_{int}
  5. 𝐚 n = 𝐌 - 1 ( 𝐅 e x t - 𝐅 i n t ) n \mathbf{a}_{n}=\mathbf{M}^{-1}(\mathbf{F}_{ext}-\mathbf{F}_{int})_{n}
  6. 𝐯 n + 1 / 2 = 𝐯 n - 1 / 2 + 𝐚 n Δ t n \mathbf{v}_{n+1/2}=\mathbf{v}_{n-1/2}+\mathbf{a}_{n}\Delta t_{n}
  7. 𝐮 n + 1 = 𝐮 n + 𝐯 n + 1 / 2 Δ t n + 1 / 2 \mathbf{u}_{n+1}=\mathbf{u}_{n}+\mathbf{v}_{n+1/2}\Delta t_{n+1/2}
  8. Δ t n \Delta t_{n}
  9. Δ t n + 1 / 2 \Delta t_{n+1/2}
  10. 𝐚 n \mathbf{a}_{n}
  11. 𝐯 n + 1 / 2 \mathbf{v}_{n+1/2}
  12. 𝐮 n + 1 \mathbf{u}_{n+1}
  13. Δ t \Delta t
  14. c = E 0 / ρ c=\sqrt{E_{0}/\rho}
  15. E 0 E_{0}
  16. ρ \rho
  17. Δ t = d m i n ρ / E 0 \Delta t=d_{min}\sqrt{\rho/E_{0}}
  18. d m i n d_{min}

Credibility_theory.html

  1. X ¯ \overline{X}
  2. j j
  3. X j ¯ \overline{X_{j}}
  4. C = z j X j ¯ + ( 1 - z j ) X ¯ C=z_{j}\overline{X_{j}}+(1-z_{j})\overline{X}\,
  5. z j z_{j}
  6. j j
  7. z j z_{j}
  8. X j ¯ \overline{X_{j}}
  9. z j z_{j}
  10. z j = 0 z_{j}=0
  11. z j = 1 z_{j}=1

Credible_interval.html

  1. t t
  2. t t
  3. 35 t 45 35\leq t\leq 45
  4. Pr ( x | μ ) = f ( x - μ ) \mathrm{Pr}(x|\mu)=f(x-\mu)
  5. Pr ( x | s ) = f ( x / s ) \mathrm{Pr}(x|s)=f(x/s)
  6. Pr ( s | I ) 1 / s \mathrm{Pr}(s|I)\;\propto\;1/s

Criterion-referenced_test.html

  1. 2 + 3 = ? 2+3=?
  2. 9 + 5 = ? 9+5=?

Critical_heat_flux.html

  1. q = h ( T w - T f ) q=h(T_{w}-T_{f})\,
  2. q q
  3. h h
  4. T w T_{w}
  5. T f T_{f}
  6. h h
  7. T w T_{w}
  8. q q
  9. T f T_{f}
  10. q q
  11. Δ T \Delta T
  12. q A m a x = C h f g ρ v [ σ g ( ρ L - ρ v ) ρ v 2 ] 1 4 {{\frac{q}{A_{max}}}}=C{{h}_{fg}}{{\rho}_{v}}{{\left[\frac{\sigma g\left({{% \rho}_{L}}-{{\rho}_{v}}\right)}{{{\rho}_{v}}^{2}}\right]}^{{}^{1}\!\!\diagup\!% \!{}_{4}\;}}
  13. C = π 24 = 0.131 C=\frac{\pi}{24}=0.131
  14. C = 0.149 C=0.149

Critical_point_(set_theory).html

  1. { A | A κ κ j ( A ) } . \{A|A\subseteq\kappa\land\kappa\in j(A)\}\,.

Critical_resolved_shear_stress.html

  1. ( 1 ¯ 11 ) (\overline{1}11)
  2. [ 101 ] [101]
  3. [ 123 ] [123]
  4. M [ 1 0 1 ] ( 1 ¯ 11 ) = \displaystyle M_{[\,\text{1 0 1}](\overline{1}11)}=

Crofton_formula.html

  1. length ( γ ) = 1 4 n γ ( φ , p ) d φ d p . \,\text{length}(\gamma)=\frac{1}{4}\iint n_{\gamma}(\varphi,p)\;d\varphi\;dp.
  2. d φ d p d\varphi\wedge dp

Cross-multiplication.html

  1. a b = c d \frac{a}{b}=\frac{c}{d}
  2. b b
  3. d d
  4. a d = b c or a = b c d . ad=bc\qquad\mathrm{or}\qquad a=\frac{bc}{d}.
  5. a b c d a b c d . \frac{a}{b}\nwarrow\frac{c}{d}\quad\frac{a}{b}\nearrow\frac{c}{d}.
  6. a b = c d \frac{a}{b}=\frac{c}{d}
  7. b d bd
  8. a b × b d = c d × b d . \frac{a}{b}\times bd=\frac{c}{d}\times bd.
  9. b b
  10. d d
  11. a d = b c ad=bc
  12. d d
  13. a = b c d . a=\frac{bc}{d}.
  14. a b = c d \frac{a}{b}=\frac{c}{d}
  15. d d = 1 \frac{d}{d}=1
  16. b b = 1 \frac{b}{b}=1
  17. a b × d d = c d × b b \frac{a}{b}\times\frac{d}{d}=\frac{c}{d}\times\frac{b}{b}
  18. a d b d = c b d b . \frac{ad}{bd}=\frac{cb}{db}.
  19. b d = d b bd=db
  20. a d = c b . ad=cb.
  21. x x
  22. x b = c d \frac{x}{b}=\frac{c}{d}
  23. x = b c d . x=\frac{bc}{d}.
  24. x 7 hours = 90 miles 3 hours . \frac{x}{7\ \mathrm{hours}}=\frac{90\ \mathrm{miles}}{3\ \mathrm{hours}}.
  25. x = 7 hours × 90 miles 3 hours x=\frac{7\ \mathrm{hours}\times 90\ \mathrm{miles}}{3\ \mathrm{hours}}
  26. x = 210 miles . x=210\ \mathrm{miles}.
  27. a = x d a=\frac{x}{d}
  28. b b
  29. a 1 = x d . \frac{a}{1}=\frac{x}{d}.
  30. a b = c x \frac{a}{b}=\frac{c}{x}
  31. x = b c a . x=\frac{bc}{a}.
  32. a a
  33. b b
  34. c c
  35. x x
  36. 4 yards 12 shillings = 6 yards x \frac{4\ \mathrm{yards}}{12\ \mathrm{shillings}}=\frac{6\ \mathrm{yards}}{x}
  37. x x
  38. x = 12 shillings × 6 yards 4 yards = 18 shillings . x=\frac{12\ \mathrm{shillings}\times 6\ \mathrm{yards}}{4\ \mathrm{yards}}=18% \ \mathrm{shillings}.

Crosscap_number.html

  1. 1 - χ ( S ) , 1-\chi(S),\,
  2. χ \chi
  3. C ( k 1 ) + C ( k 2 ) - 1 C ( k 1 # k 2 ) C ( k 1 ) + C ( k 2 ) . C(k_{1})+C(k_{2})-1\leq C(k_{1}\#k_{2})\leq C(k_{1})+C(k_{2}).\,

Crowds.html

  1. C C
  2. N N
  3. N - C N-C
  4. p f p_{f}
  5. p f > 1 2 p_{f}>\frac{1}{2}
  6. n p f p f - 1 2 ( c + 1 ) n\geq\frac{p_{f}}{p_{f}-\frac{1}{2}}\left(c+1\right)
  7. n p f p f - 1 2 ( c + 1 ) n\geq\frac{p_{f}}{p_{f}-\frac{1}{2}}\left(c+1\right)
  8. ( p f n - c n ) i - 1 ( c n ) (p_{f}\frac{n-c}{n})^{i-1}(\frac{c}{n})
  9. n - c n \frac{n-c}{n}
  10. c n \frac{c}{n}
  11. c n k = 0 ( p f n - c n ) k = ( c n ) ( 1 1 - p f ( n - c ) n ) \frac{c}{n}\sum_{k=0}(p_{f}\frac{n-c}{n})^{k}=(\frac{c}{n})(\frac{1}{1-\frac{p% _{f}(n-c)}{n}})
  12. c n k = 1 ( p f n - c n ) k = ( c n ) ( p f ( n - c ) n 1 - p f ( n - c ) n ) \frac{c}{n}\sum_{k=1}(p_{f}\frac{n-c}{n})^{k}=(\frac{c}{n})(\frac{\frac{p_{f}(% n-c)}{n}}{1-\frac{p_{f}(n-c)}{n}})
  13. c n \frac{c}{n}
  14. 1 1
  15. 1 n - c \frac{1}{n-c}
  16. c ( n - n p f n + c p f + p f ) n 2 - p f n ( n - c ) \frac{c(n-np_{f}n+cp_{f}+pf)}{n^{2}-pfn(n-c)}
  17. P ( I and H 1 + ) P ( H 1 + ) \frac{P(I\and H1+)}{P(H1+)}
  18. P ( I ) P ( H 1 + ) \frac{P(I)}{P(H1+)}
  19. n - p f ( n - c - 1 ) n \frac{n-p_{f}(n-c-1)}{n}
  20. n p f p f - 1 2 ( c + 1 ) n\geq\frac{p_{f}}{p_{f}-\frac{1}{2}}\left(c+1\right)
  21. O ( 1 ( 1 - p f ) 2 ( 1 + 1 n ) ) O(\frac{1}{(1-p_{f})^{2}}(1+\frac{1}{n}))
  22. d 1 d\leftarrow 1
  23. N p f p f - 1 2 ( C + 1 ) N\geq\frac{p_{f}}{p_{f}-\frac{1}{2}}\left(C+1\right)
  24. d N - p f ( N - C - 1 ) N lg [ N N - p f ( N - C - 1 ) ] + p f N - C - 1 N lg [ N / p f ] lg ( N - C ) d\leftarrow\frac{\frac{N-p_{f}\cdot(N-C-1)}{N}\cdot\lg\left[\frac{N}{N-p_{f}% \cdot(N-C-1)}\right]+p_{f}\cdot\frac{N-C-1}{N}\cdot\lg\left[N/p_{f}\right]}{% \lg(N-C)}
  25. O ( N C lg ( N ) ) O\left(\frac{N}{C}\lg(N)\right)
  26. O ( N 2 ) O(N^{2})

Cubic_honeycomb.html

  1. 3 ¯ \overline{3}
  2. C ~ 3 {\tilde{C}}_{3}
  3. 3 ¯ \overline{3}
  4. 3 ¯ \overline{3}
  5. 3 ¯ \overline{3}
  6. 3 ¯ \overline{3}
  7. 3 ¯ \overline{3}
  8. A ~ 3 {\tilde{A}}_{3}
  9. 3 ¯ \overline{3}
  10. C ~ 3 {\tilde{C}}_{3}
  11. C ~ 3 {\tilde{C}}_{3}
  12. B ~ 3 {\tilde{B}}_{3}
  13. B ~ 3 {\tilde{B}}_{3}
  14. A ~ 3 {\tilde{A}}_{3}
  15. 3 ¯ \overline{3}
  16. 3 ¯ \overline{3}
  17. 3 ¯ \overline{3}
  18. 4 ¯ \overline{4}
  19. 3 ¯ \overline{3}
  20. C ~ 3 {\tilde{C}}_{3}
  21. C ~ 3 {\tilde{C}}_{3}
  22. B ~ 3 {\tilde{B}}_{3}
  23. 3 ¯ \overline{3}
  24. 3 ¯ \overline{3}

Cuckoo_hashing.html

  1. h ( k ) = k mod 11 h\left(k\right)=k\mod 11
  2. h ( k ) = k 11 mod 11 h^{\prime}\left(k\right)=\left\lfloor\frac{k}{11}\right\rfloor\mod 11
  3. h ( 6 ) = 6 mod 11 = 6 h\left(6\right)=6\mod 11=6
  4. h ( 6 ) = 6 11 mod 11 = 0 h^{\prime}\left(6\right)=\left\lfloor\frac{6}{11}\right\rfloor\mod 11=0

Cuntz_algebra.html

  1. 𝒪 n \mathcal{O}_{n}
  2. 𝒪 n \mathcal{O}_{n}
  3. 𝒜 \mathcal{A}
  4. { S i } i = 1 n \{S_{i}\}_{i=1}^{n}
  5. i = 1 n S i S i * = I . \sum_{i=1}^{n}S_{i}S_{i}^{*}=I.
  6. S i * S j = δ i j I . S_{i}^{*}S_{j}=\delta_{ij}I.
  7. 𝒜 \mathcal{A}
  8. \mathcal{L}
  9. \mathcal{F}
  10. \mathcal{F}
  11. s i 1 s i k s j 1 * s j k * , k 0. s_{i_{1}}\cdots s_{i_{k}}s_{j_{1}}^{*}\cdots s_{j_{k}}^{*},k\geq 0.
  12. \mathcal{F}
  13. \mathcal{F}
  14. \mathcal{L}
  15. 𝒜 \mathcal{A}
  16. 𝒪 n \mathcal{O}_{n}
  17. 𝒪 n \mathcal{O}_{n}
  18. 𝒪 n \mathcal{O}_{n}
  19. \mathcal{F}
  20. \mathcal{F}^{\prime}
  21. \mathcal{F}
  22. \mathcal{F}
  23. \mathcal{F}
  24. P P = P\mathcal{F}P=\mathcal{F^{\prime}}
  25. \mathcal{F}
  26. \mathcal{F}
  27. \mathcal{F}^{\prime}
  28. 𝒪 n \;\mathcal{O}_{n}
  29. \mathcal{F}
  30. 𝒪 n \mathcal{O}_{n}
  31. 𝒪 m \mathcal{O}_{m}
  32. 𝒪 n \mathcal{O}_{n}
  33. 𝒪 n \mathcal{O}_{n}
  34. 𝒪 m \mathcal{O}_{m}

Current_divider.html

  1. I X = R T ( R X ) + ( R T ) I T I_{X}=\frac{R_{T}}{(R_{X})+(R_{T})}I_{T}
  2. 1 R T = 1 R 1 + 1 R 2 + 1 R 3 + . \frac{1}{R_{T}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}+...\ .
  3. I X = Z T Z X + Z T I T , I_{X}=\frac{Z_{T}}{Z_{X}+Z_{T}}I_{T}\ ,
  4. I X = Y X Y T o t a l I T I_{X}=\frac{Y_{X}}{Y_{Total}}I_{T}
  5. I X = Y X Y T o t a l I T = 1 R X 1 R X + 1 R 1 + 1 R 2 + 1 R 3 I T I_{X}=\frac{Y_{X}}{Y_{Total}}I_{T}=\frac{\frac{1}{R_{X}}}{\frac{1}{R_{X}}+% \frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}}I_{T}
  6. I R = 1 j ω C R + 1 j ω C I T I_{R}=\frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}I_{T}
  7. = 1 1 + j ω C R I T , =\frac{1}{1+j\omega CR}I_{T}\ ,
  8. i i = R S R S + R i n i S , i_{i}=\frac{R_{S}}{R_{S}+R_{in}}i_{S}\ ,
  9. i L = R o u t R o u t + R L A i i i . i_{L}=\frac{R_{out}}{R_{out}+R_{L}}A_{i}i_{i}\ .
  10. A l o a d e d = i L i S = R S R S + R i n A_{loaded}=\frac{i_{L}}{i_{S}}=\frac{R_{S}}{R_{S}+R_{in}}
  11. R o u t R o u t + R L A i . \frac{R_{out}}{R_{out}+R_{L}}A_{i}\ .
  12. A f b = i L i S = A l o a d e d 1 + β ( R L / R S ) A l o a d e d . A_{fb}=\frac{i_{L}}{i_{S}}=\frac{A_{loaded}}{1+{\beta}(R_{L}/R_{S})A_{loaded}}\ .

Current_ratio.html

  1. Current ratio = Current Assets Current Liabilities \mbox{Current ratio}~{}=\frac{\mbox{Current Assets}~{}}{\mbox{Current % Liabilities}~{}}

Curtin–Hammett_principle.html

  1. C k 1 A K B k 2 D \mathrm{C\ \xleftarrow{k_{1}}\ A\ \overset{K}{\rightleftharpoons}\ B\ % \xrightarrow{k_{2}}\ D}
  2. C k 1 A K B k 2 D \mathrm{C\ \xleftarrow{k_{1}}\ A\ \overset{K}{\rightleftharpoons}\ B\ % \xrightarrow{k_{2}}\ D}
  3. d [ C ] d t = k 1 [ A ] \frac{d[C]}{dt}=k_{1}[A]
  4. d [ D ] d t = k 2 [ B ] = k 2 K [ A ] \frac{d[D]}{dt}=k_{2}[B]=k_{2}K[A]
  5. d [ D ] d t d [ C ] d t = k 2 K [ A ] k 1 [ A ] = k 2 K k 1 \frac{\frac{d[D]}{dt}}{\frac{d[C]}{dt}}=\frac{k_{2}K[A]}{k_{1}[A]}=\frac{k_{2}% K}{k_{1}}
  6. [ D ] [ C ] = k 2 K k 1 = e - Δ G 2 / R T e - Δ G / R T e - Δ G 1 / R T \frac{[D]}{[C]}=\frac{k_{2}K}{k_{1}}=\frac{e^{-\Delta G_{2}^{\ddagger}/RT}e^{-% \Delta G/RT}}{e^{-\Delta G_{1}^{\ddagger}/RT}}
  7. Δ \Delta
  8. [ D ] [ C ] = e - Δ Δ G R T \frac{[D]}{[C]}=e^{-\frac{\Delta\Delta G^{\ddagger}}{RT}}

Curve_orientation.html

  1. 𝐎 = [ 1 x A y A 1 x B y B 1 x C y C ] . \mathbf{O}=\begin{bmatrix}1&x_{A}&y_{A}\\ 1&x_{B}&y_{B}\\ 1&x_{C}&y_{C}\end{bmatrix}.
  2. det ( O ) \displaystyle\det(O)
  3. det ( O ) \displaystyle\det(O)
  4. 𝐎 = [ 1 x F y F 1 x G y G 1 x H y H ] . \mathbf{O}=\begin{bmatrix}1&x_{F}&y_{F}\\ 1&x_{G}&y_{G}\\ 1&x_{H}&y_{H}\end{bmatrix}.

Custodial_symmetry.html

  1. V S M = - λ ( H H ) + μ ( H H ) 2 V_{SM}=-\lambda(H^{\dagger}H)+\mu(H^{\dagger}H)^{2}
  2. ρ \rho
  3. ρ \rho
  4. ρ \rho
  5. | H D μ H | 2 / Λ 2 \left|H^{\dagger}D_{\mu}H\right|^{2}/\Lambda^{2}
  6. | H D μ H | 2 / Λ 2 \left|H^{\dagger}D_{\mu}H\right|^{2}/\Lambda^{2}
  7. D μ H D μ H D^{\mu}H^{\dagger}D_{\mu}H
  8. H H H^{\dagger}H
  9. ( H H ) 2 \left(H^{\dagger}H\right)^{2}
  10. | H D μ H | 2 / Λ 2 \left|H^{\dagger}D_{\mu}H\right|^{2}/\Lambda^{2}

Cyanimide.html

  1. \equiv

Cycle_index.html

  1. ( 1 2 3 4 5 2 3 4 5 1 ) \left(\begin{matrix}1&2&3&4&5\\ 2&3&4&5&1\end{matrix}\right)
  2. ( 1 2 3 4 5 6 2 1 3 5 6 4 ) = ( 12 ) ( 3 ) ( 456 ) . \left(\begin{matrix}1&2&3&4&5&6\\ 2&1&3&5&6&4\end{matrix}\right)=(12)(3)(456).
  3. k = 1 n a k j k ( g ) \prod_{k=1}^{n}a_{k}^{j_{k}(g)}
  4. 0 j k ( g ) n / k and k = 1 n k j k ( g ) = n . 0\leq j_{k}(g)\leq\lfloor n/k\rfloor\mbox{ and }~{}\sum_{k=1}^{n}k\,j_{k}(g)\;% =n.
  5. c g a | c | = k = 1 n a k j k ( g ) \prod_{c\in g}a_{|c|}=\prod_{k=1}^{n}a_{k}^{j_{k}(g)}
  6. Z ( G ) = 1 | G | g G k = 1 n a k j k ( g ) . Z(G)=\frac{1}{|G|}\sum_{g\in G}\prod_{k=1}^{n}a_{k}^{j_{k}(g)}.
  7. Z ( C 4 ) = 1 4 ( a 1 4 + a 2 2 + 2 a 4 ) . Z(C_{4})=\frac{1}{4}\left(a_{1}^{4}+a_{2}^{2}+2a_{4}\right).
  8. Z ( C 4 ) = 1 4 ( a 1 6 + a 1 2 a 2 2 + 2 a 2 a 4 ) . Z(C_{4})=\frac{1}{4}\left(a_{1}^{6}+a_{1}^{2}a_{2}^{2}+2a_{2}a_{4}\right).
  9. Z ( C 4 ) = 1 4 ( a 1 16 + a 2 8 + 2 a 4 4 ) . Z(C_{4})=\frac{1}{4}\left(a_{1}^{16}+a_{2}^{8}+2a_{4}^{4}\right).
  10. S 3 = { e , ( 23 ) , ( 12 ) , ( 123 ) , ( 132 ) , ( 13 ) } S_{3}=\{e,(23),(12),(123),(132),(13)\}
  11. Z ( S 3 ) = 1 6 ( a 1 3 + 3 a 1 a 2 + 2 a 3 ) . Z(S_{3})=\frac{1}{6}\left(a_{1}^{3}+3a_{1}a_{2}+2a_{3}\right).
  12. Z ( C 6 ) = 1 6 ( a 1 6 + a 2 3 + 2 a 3 2 + 2 a 6 ) . Z(C_{6})=\frac{1}{6}\left(a_{1}^{6}+a_{2}^{3}+2a_{3}^{2}+2a_{6}\right).
  13. a 1 3 . a_{1}^{3}.\,
  14. 3 a 1 a 2 . 3a_{1}a_{2}.\,
  15. 2 a 3 . 2a_{3}.\,
  16. Z ( G ) = 1 6 ( a 1 3 + 3 a 1 a 2 + 2 a 3 ) . Z(G)=\frac{1}{6}\left(a_{1}^{3}+3a_{1}a_{2}+2a_{3}\right).
  17. ( n 2 ) {\left({{n}\atop{2}}\right)}
  18. a 1 6 . a_{1}^{6}.\,
  19. 6 a 1 2 a 2 2 . 6a_{1}^{2}a_{2}^{2}.\,
  20. 8 a 3 2 . 8a_{3}^{2}.\,
  21. 3 a 1 2 a 2 2 . 3a_{1}^{2}a_{2}^{2}.\,
  22. 6 a 2 a 4 . 6a_{2}a_{4}.\,
  23. Z ( G ) = 1 24 ( a 1 6 + 9 a 1 2 a 2 2 + 8 a 3 2 + 6 a 2 a 4 ) . Z(G)=\frac{1}{24}\left(a_{1}^{6}+9a_{1}^{2}a_{2}^{2}+8a_{3}^{2}+6a_{2}a_{4}% \right).
  24. a 1 6 . a_{1}^{6}.
  25. 6 a 1 2 a 4 . 6a_{1}^{2}a_{4}.
  26. 3 a 1 2 a 2 2 . 3a_{1}^{2}a_{2}^{2}.
  27. 8 a 3 2 . 8a_{3}^{2}.
  28. 6 a 2 3 . 6a_{2}^{3}.
  29. Z ( C ) = 1 24 ( a 1 6 + 6 a 1 2 a 4 + 3 a 1 2 a 2 2 + 8 a 3 2 + 6 a 2 3 ) . Z(C)=\frac{1}{24}\left(a_{1}^{6}+6a_{1}^{2}a_{4}+3a_{1}^{2}a_{2}^{2}+8a_{3}^{2% }+6a_{2}^{3}\right).
  30. Z ( E n ) = a 1 n . Z(E_{n})=a_{1}^{n}.
  31. Z ( C n ) = 1 n d | n φ ( d ) a d n / d . Z(C_{n})=\frac{1}{n}\sum_{d|n}\varphi(d)a_{d}^{n/d}.
  32. Z ( D n ) = 1 2 Z ( C n ) + { 1 2 a 1 a 2 ( n - 1 ) / 2 , n odd, 1 4 ( a 1 2 a 2 ( n - 2 ) / 2 + a 2 n / 2 ) , n even. Z(D_{n})=\frac{1}{2}Z(C_{n})+\begin{cases}\frac{1}{2}a_{1}a_{2}^{(n-1)/2},&n% \mbox{ odd, }\\ \frac{1}{4}\left(a_{1}^{2}a_{2}^{(n-2)/2}+a_{2}^{n/2}\right),&n\mbox{ even.}% \end{cases}
  33. Z ( A n ) = j 1 + 2 j 2 + 3 j 3 + + n j n = n 1 + ( - 1 ) j 2 + j 4 + k = 1 n k j k j k ! k = 1 n a k j k . Z(A_{n})=\sum_{j_{1}+2j_{2}+3j_{3}+\cdots+nj_{n}=n}\frac{1+(-1)^{j_{2}+j_{4}+% \cdots}}{\prod_{k=1}^{n}k^{j_{k}}j_{k}!}\prod_{k=1}^{n}a_{k}^{j_{k}}.
  34. 1 | A n | = 2 n ! \frac{1}{|A_{n}|}=\frac{2}{n!}
  35. Z ( S n ) = j 1 + 2 j 2 + 3 j 3 + + n j n = n 1 k = 1 n k j k j k ! k = 1 n a k j k Z(S_{n})=\sum_{j_{1}+2j_{2}+3j_{3}+\cdots+nj_{n}=n}\frac{1}{\prod_{k=1}^{n}k^{% j_{k}}j_{k}!}\prod_{k=1}^{n}a_{k}^{j_{k}}
  36. Z ( S n ) = B n ( 0 ! a 1 , 1 ! a 2 , , ( n - 1 ) ! a n ) n ! . Z(S_{n})=\frac{B_{n}(0!\,a_{1},1!\,a_{2},\dots,(n-1)!\,a_{n})}{n!}.
  37. j k j_{k}
  38. k ! / k k!/k
  39. S j k S_{j_{k}}
  40. n ! k = 1 n ( k ! ) j k k = 1 n ( k ! k ) j k k = 1 n 1 j k ! = n ! k = 1 n k j k j k ! . \frac{n!}{\prod_{k=1}^{n}(k!)^{j_{k}}}\prod_{k=1}^{n}\left(\frac{k!}{k}\right)% ^{j_{k}}\prod_{k=1}^{n}\frac{1}{j_{k}!}=\frac{n!}{\prod_{k=1}^{n}k^{j_{k}}j_{k% }!}.
  41. Z ( S 0 ) = 1 Z(S_{0})=1
  42. 1 l n . \begin{matrix}1\leq l\leq n.\end{matrix}
  43. ( n - 1 l - 1 ) \begin{matrix}{n-1\choose l-1}\end{matrix}
  44. l - 1 l-1
  45. l ! l \begin{matrix}\frac{l!}{l}\end{matrix}
  46. Z ( S n ) = 1 n ! g S n k = 1 n a k j k ( g ) = 1 n ! l = 1 n ( n - 1 l - 1 ) l ! l a l ( n - l ) ! Z ( S n - l ) Z(S_{n})=\frac{1}{n!}\sum_{g\in S_{n}}\prod_{k=1}^{n}a_{k}^{j_{k}(g)}=\frac{1}% {n!}\sum_{l=1}^{n}{n-1\choose l-1}\;\frac{l!}{l}\;a_{l}\;(n-l)!\;Z(S_{n-l})
  47. Z ( S n ) = 1 n l = 1 n a l Z ( S n - l ) . Z(S_{n})=\frac{1}{n}\sum_{l=1}^{n}a_{l}\;Z(S_{n-l}).
  48. Z ( G ) = Z ( G ; a 1 , a 2 , , a n ) . Z(G)=Z(G;a_{1},a_{2},\ldots,a_{n}).
  49. k = 0 n f k t k = Z ( G ; 1 + t , 1 + t 2 , , 1 + t n ) , \sum_{k=0}^{n}f_{k}t^{k}=Z(G;1+t,1+t^{2},\ldots,1+t^{n}),
  50. k = 0 n F k t k / k ! = Z ( G ; 1 + t , 1 , 1 , , 1 ) . \sum_{k=0}^{n}F_{k}t^{k}/k!=Z(G;1+t,1,1,\ldots,1).

Cycle_per_second.html

  1. s - 1 \,\text{s}^{-1}

Cyclic_code.html

  1. 𝒞 \mathcal{C}
  2. G F ( q ) GF(q)
  3. 𝒞 \mathcal{C}
  4. G F ( q ) n GF(q)^{n}
  5. 𝒞 \mathcal{C}
  6. R = A [ x ] / ( x n - 1 ) R=A[x]/(x^{n}-1)
  7. A = G F ( q ) A=GF(q)
  8. ( c 0 , , c n - 1 ) (c_{0},\ldots,c_{n-1})
  9. c 0 + c 1 x + + c n - 1 x n - 1 c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}
  10. x n - 1 x^{n}-1
  11. n - d n-d
  12. 𝔽 2 \mathbb{F}_{2}
  13. ( ( 0 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 0 , 1 , 1 ) , ( 1 , 0 , 1 ) ) ((0,0,0),(1,1,0),(0,1,1),(1,0,1))\,
  14. 𝔽 2 [ x ] / ( x 3 - 1 ) \mathbb{F}_{2}[x]/(x^{3}-1)
  15. ( 1 + x ) (1+x)
  16. ( 1 + x ) (1+x)
  17. x + x 2 x+x^{2}
  18. x n - 1 x^{n}-1
  19. x n - 1 x^{n}-1
  20. x + 1 x+1
  21. x n - 1 x^{n}-1
  22. ( n , k ) (n,k)
  23. b b
  24. n n
  25. x b c ( x ) ( mod x n - 1 ) x^{b}c(x)\;\;(\mathop{{\rm mod}}x^{n}-1)
  26. c ( x ) c(x)
  27. c ( x ) = a ( x ) g ( x ) c(x)=a(x)g(x)
  28. ( c 0 , . . , c n - 1 ) (c_{0},..,c_{n-1})
  29. i = 0 n - 1 c i * x i \sum_{i=0}^{n-1}c_{i}*x^{i}
  30. [ n , k ] [n,k]
  31. b b
  32. ( n + b , k + b ) (n+b,k+b)
  33. n n
  34. k k
  35. k k
  36. n n
  37. ( n , k ) (n,k)
  38. ( n - b , k - b ) (n-b,k-b)
  39. b b
  40. b b
  41. b b
  42. n n
  43. g ( x ) = x 3 + x + 1 g(x)=x^{3}+x+1
  44. G F ( 8 ) GF(8)
  45. α \alpha
  46. 𝒞 ( α ) = 0 \mathcal{C}(\alpha)=0
  47. G F ( 2 ) GF(2)
  48. n n
  49. 2 m - 1 2^{m}-1
  50. α \alpha
  51. α 3 \alpha^{3}
  52. G F ( 2 m ) GF(2^{m})
  53. n - 1 n-1
  54. v ( x ) = a ( x ) g ( x ) + e ( x ) v(x)=a(x)g(x)+e(x)
  55. e ( x ) e(x)
  56. S ( x ) S(x)
  57. v ( x ) v(x)
  58. g ( x ) g(x)
  59. S ( x ) S(x)
  60. v ( x ) mod g ( x ) = ( a ( x ) g ( x ) + e ( x ) ) mod g ( x ) = e ( x ) mod g ( x ) v(x)\mod g(x)=(a(x)g(x)+e(x))\mod g(x)=e(x)\mod g(x)
  61. ( a ( x ) g ( x ) ) mod g ( x ) (a(x)g(x))\mod g(x)
  62. X 1 X_{1}
  63. X 2 X_{2}
  64. X 2 X_{2}
  65. S 1 = v ( α ) S_{1}={v}(\alpha)
  66. S 3 = v ( α 3 ) S_{3}={v}(\alpha^{3})
  67. g ( x ) g(x)
  68. α \alpha
  69. α 3 \alpha^{3}
  70. S 1 = e ( α ) S_{1}=e(\alpha)
  71. S 3 = e ( α 3 ) S_{3}=e(\alpha^{3})
  72. S 1 = α i + α i S_{1}=\alpha^{i}+\alpha^{i^{\prime}}
  73. S 3 = α 3 i + α 3 i S_{3}=\alpha^{3i}+\alpha^{3i^{\prime}}
  74. G F ( 2 m ) GF(2^{m})
  75. S 1 = X 1 + X 2 S_{1}=X_{1}+X_{2}
  76. S 3 = ( X 1 ) 3 + ( X 2 ) 3 S_{3}=(X_{1})^{3}+(X_{2})^{3}
  77. 1 + x + x 3 1+x+x^{3}
  78. r 3 r\geq 3
  79. [ 2 r - 1 , 2 r - r - 2 , 4 ] [2^{r}-1,2^{r}-r-2,4]
  80. m m
  81. m m
  82. 2 m - 1 2^{m}-1
  83. d m i n d_{min}
  84. m m
  85. 2 m - 1 2^{m}-1
  86. ( 2 m - 1 , 2 m - 1 - m ) (2^{m}-1,2^{m}-1-m)
  87. q q
  88. H H
  89. q q
  90. m m
  91. ( q m - 1 ) / ( q - 1 ) (q^{m}-1)/(q-1)
  92. [ ( q m - 1 ) / ( q - 1 ) , ( q m - 1 ) / ( q - 1 ) - m ] [(q^{m}-1)/(q-1),(q^{m}-1)/(q-1)-m]
  93. α \alpha
  94. G F ( q m ) GF(q^{m})
  95. β = α q - 1 \beta=\alpha^{q-1}
  96. β ( q m - 1 ) / ( q - 1 ) = 1 \beta^{(q^{m}-1)/(q-1)}=1
  97. β \beta
  98. x ( q m - 1 ) / ( q - 1 ) - 1 x^{(q^{m}-1)/(q-1)}-1
  99. n = ( q m - 1 ) / ( q - 1 ) n=(q^{m}-1)/(q-1)
  100. q = 2 q=2
  101. α = β \alpha=\beta
  102. n - 1 n-1
  103. v ( x ) = a ( x ) g ( x ) + e ( x ) v(x)=a(x)g(x)+e(x)
  104. e ( x ) = 0 e(x)=0
  105. x i x^{i}
  106. i i
  107. α i \alpha^{i}
  108. G F ( 2 m ) GF(2^{m})
  109. g ( α ) = 0 g(\alpha)=0
  110. v ( α ) = α i v(\alpha)=\alpha^{i}
  111. α \alpha
  112. 0
  113. 2 m - 2 2^{m}-2
  114. i i
  115. α i \alpha^{i}
  116. v ( α ) = 0 v(\alpha)=0
  117. G F ( 2 ) GF(2)
  118. n = 2 m - 1 n=2^{m}-1
  119. k = n - m k=n-m
  120. 2 t + 1 2t+1
  121. t t
  122. t t
  123. t t
  124. t t
  125. e ( x ) = x i b ( x ) mod ( x n - 1 ) e(x)=x^{i}b(x)\mod(x^{n}-1)
  126. b ( x ) b(x)
  127. t - 1 t-1
  128. b 0 b_{0}
  129. b ( x ) b(x)
  130. x i x^{i}
  131. b ( x ) + 1 b(x)+1
  132. s ( x ) = e ( x ) mod g ( x ) s(x)=e(x)\mod g(x)
  133. t t
  134. 2 t 2t
  135. t t
  136. 2 t 2t
  137. t t
  138. t t
  139. t t
  140. 2 t 2t
  141. 2 t 2t
  142. q 2 t q^{2t}
  143. q 2 t q^{2t}
  144. 2 t 2t
  145. x c + 1 x^{c}+1
  146. c c
  147. G F ( q ) GF(q)
  148. g ( x ) = ( x 2 t - 1 - 1 ) p ( x ) g(x)=(x^{2t-1}-1)p(x)
  149. p ( x ) p(x)
  150. m m
  151. t t
  152. p ( x ) p(x)
  153. x 2 t - 1 - 1 x^{2t-1}-1
  154. n n
  155. g ( x ) g(x)
  156. x n - 1 x^{n}-1
  157. b ( x ) b(x)
  158. x j b ( x ) x^{j}b^{\prime}(x)
  159. b ( x ) b(x)
  160. x j b ( x ) x^{j}b^{\prime}(x)
  161. t t
  162. g ( x ) g(x)
  163. x 2 t - 1 - 1 x^{2t-1}-1
  164. b ( x ) = x j b ( x ) mod ( x 2 t - 1 - 1 ) b(x)=x^{j}b^{\prime}(x)\mod(x^{2t-1}-1)
  165. j j
  166. 2 t - 1 2t-1
  167. b ( x ) = x l ( 2 t - 1 ) b ( x ) b(x)=x^{l(2t-1)}b^{\prime}(x)
  168. l l
  169. l ( 2 t - 1 ) l(2t-1)
  170. t t
  171. l l
  172. q m - 1 q^{m}-1
  173. ( x l ( 2 t - 1 ) - 1 ) b ( x ) (x^{l(2t-1)}-1)b(x)
  174. ( x l ( 2 t - 1 ) - 1 ) b ( x ) = a ( x ) ( x 2 t - 1 - 1 ) p ( x ) (x^{l(2t-1)}-1)b(x)=a(x)(x^{2t-1}-1)p(x)
  175. b ( x ) b(x)
  176. p ( x ) p(x)
  177. p ( x ) p(x)
  178. b ( x ) b(x)
  179. l l
  180. p ( x ) p(x)
  181. x l ( 2 t - 1 ) - 1 x^{l(2t-1)}-1
  182. l l
  183. q m - 1 q^{m}-1
  184. m m
  185. p ( x ) p(x)
  186. x l ( 2 t - 1 ) - 1 x^{l(2t-1)}-1
  187. l l
  188. q m - 1 q^{m}-1
  189. l l
  190. j j
  191. m m
  192. t t
  193. 3 t - 1 3t-1
  194. t - 1 t-1
  195. G F ( q ) GF(q)
  196. v = v 0 , v 1 , . , v n - 1 v=v_{0},v_{1},....,v_{n-1}
  197. V = V 0 , V 1 , . . , V n - 1 V=V_{0},V_{1},.....,V_{n-1}
  198. V k V_{k}
  199. Σ i = 0 n - 1 e - j 2 π n - 1 i k v i \Sigma_{i=0}^{n-1}e^{-j2\pi n^{-1}ik}v_{i}
  200. k = 0 , . . , n - 1 k=0,.....,n-1
  201. - j 2 π / n -j2\pi/n
  202. n n
  203. n n
  204. ω \omega
  205. n n
  206. v = ( v 0 , v 1 , . , v n - 1 ) v=(v_{0},v_{1},....,v_{n-1})
  207. G F ( q ) GF(q)
  208. ω \omega
  209. G F ( q ) GF(q)
  210. n n
  211. v v
  212. V = ( V 0 , V 1 , . . , V n - 1 ) V=(V_{0},V_{1},.....,V_{n-1})
  213. V j V_{j}
  214. Σ i = 0 n - 1 ω i j v i \Sigma_{i=0}^{n-1}\omega^{ij}v_{i}
  215. k = 0 , . . , n - 1 k=0,.....,n-1
  216. i i
  217. j j
  218. V V
  219. ω \omega
  220. n n
  221. ω \omega
  222. n n
  223. q - 1 q-1
  224. G F ( q m ) GF(q^{m})
  225. n n
  226. q m - 1 q^{m}-1
  227. m m
  228. v v
  229. G F ( q ) GF(q)
  230. V V
  231. G F ( q m ) GF(q^{m})
  232. n n
  233. c ( x ) c(x)
  234. n - 1 n-1
  235. c ( x ) = a ( x ) g ( x ) c(x)=a(x)g(x)
  236. C j = A j G j C_{j}=A_{j}G_{j}
  237. C j C_{j}
  238. G F ( q m ) GF(q^{m})
  239. G F ( q ) GF(q)
  240. A j A_{j}
  241. G j G_{j}
  242. j j
  243. C j C_{j}
  244. A = ( j 1 , . , j n - k ) A=(j_{1},....,j_{n-k})
  245. C C
  246. G F ( q ) GF(q)
  247. j 1 , , j n - k j_{1},...,j_{n-k}
  248. C C
  249. A j G j A_{j}G_{j}
  250. G F ( q ) GF(q)
  251. G F ( q m ) GF(q^{m})
  252. G F ( q m ) GF(q^{m})
  253. G F ( q ) GF(q)
  254. n n
  255. ( q m - 1 ) (q^{m}-1)
  256. m m
  257. G F ( q ) n GF(q)^{n}
  258. d - 1 d-1
  259. d - 1 d-1
  260. n n
  261. ( q m - 1 ) (q^{m}-1)
  262. m m
  263. b b
  264. n n
  265. v v
  266. G F ( q ) n GF(q)^{n}
  267. d - 1 d-1
  268. V j V_{j}
  269. j = 1 + 2 b mod n j=\ell_{1}+\ell_{2}b(\mod n)
  270. 1 = 0 , . , d - s - 1 \ell_{1}=0,....,d-s-1
  271. 2 = 0 , . , s - 1 \ell_{2}=0,....,s-1
  272. n n
  273. q m - 1 q^{m}-1
  274. m m
  275. G C D ( n , b ) = 1 GCD(n,b)=1
  276. G F ( q ) n GF(q)^{n}
  277. d - 1 d-1
  278. V j V_{j}
  279. j = l 1 + l 2 b mod n j=l_{1}+l_{2}b(\mod n)
  280. l 1 = 0 , , d - s - 2 l_{1}=0,...,d-s-2
  281. l 2 l_{2}
  282. s + 1 s+1
  283. 0 , . , d - 2 0,....,d-2
  284. l l
  285. p p
  286. p p
  287. ( p + 1 ) / 2 (p+1)/2
  288. p \sqrt{p}
  289. G F ( l ) GF(l)

Cyclic_prefix.html

  1. N N
  2. 𝐝 = [ d 0 , d 1 , d N - 1 ] T \mathbf{d}=[d_{0},d_{1},\ldots d_{N-1}]^{T}
  3. 𝐱 = [ x [ 0 ] , x [ 1 ] , x [ N - 1 ] ] T \mathbf{x}^{\prime}=[x[0],x[1],\ldots x[N-1]]^{T}
  4. L - 1 L-1
  5. 𝐱 = [ x [ N - L + 1 ] , x [ N - 2 ] , x [ N - 1 ] , x [ 0 ] , x [ 1 ] , x [ N - 1 ] ] T \mathbf{x}=[x[N-L+1],\ldots x[N-2],x[N-1],x[0],x[1],\ldots x[N-1]]^{T}
  6. 𝐡 = [ h 0 , h 1 , h L - 1 ] T \mathbf{h}=[h_{0},h_{1},\ldots h_{L-1}]^{T}
  7. y [ m ] = l = 0 L - 1 h [ l ] x [ m - l ] L - 1 m N - 1 y[m]=\sum_{l=0}^{L-1}h[l]x[m-l]\quad L-1\leq m\leq N-1
  8. x [ m - l ] x[m-l]
  9. x [ ( m - l ) mod N ] x^{\prime}[(m-l)\mod N]
  10. Y [ k ] = H [ k ] X [ k ] Y[k]=H[k]\cdot X[k]
  11. X [ k ] X[k]
  12. 𝐱 \mathbf{x}
  13. H [ k ] H[k]
  14. { H [ k ] } \{H[k]\}
  15. H [ k ] H[k]
  16. { X [ k ] } \{X[k]\}
  17. [ d 0 , d 1 , d N - 1 ] T [d_{0},d_{1},\ldots d_{N-1}]^{T}

Cycloidal_drive.html

  1. r = P - L L r=\frac{P-L}{L}

Cyclostationary_process.html

  1. x ( t ) x(t)
  2. R x ( t ; τ ) = E { x ( t - τ / 2 ) x * ( t + τ / 2 ) } . R_{x}(t;\tau)=E\{x(t-\tau/2)x^{*}(t+\tau/2)\}.\,
  3. x ( t ) x(t)
  4. T 0 T_{0}
  5. R x ( t ; τ ) R_{x}(t;\tau)
  6. t t
  7. T 0 , T_{0},
  8. R x ( t ; τ ) = R x ( t + T 0 ; τ ) for all t , τ . R_{x}(t;\tau)=R_{x}(t+T_{0};\tau)\,\text{ for all }t,\tau.
  9. x ( t ) x(t)
  10. R ^ x α ( τ ) = lim T 1 T - T / 2 T / 2 x ( u + τ / 2 ) x * ( u - τ / 2 ) e - i 2 π α u d u . \hat{R}_{x}^{\alpha}(\tau)=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/% 2}x(u+\tau/2)x^{*}(u-\tau/2)e^{-i2\pi\alpha u}\,du.
  11. x ( t ) x(t)
  12. T 0 T_{0}
  13. R ^ x α \hat{R}_{x}^{\alpha}
  14. α = n / T 0 \alpha=n/T_{0}
  15. n n
  16. α \alpha

Cylindrical_multipole_moments.html

  1. ln R \ln\ R
  2. ( ρ , θ ) (\rho^{\prime},\theta^{\prime})
  3. ( ρ , θ ) (\rho,\theta)
  4. 𝐫 \mathbf{r}
  5. ( ρ , θ , z ) (\rho,\theta,z)
  6. ρ \rho
  7. z z
  8. θ \theta
  9. z z
  10. z z
  11. z z^{\prime}
  12. λ \lambda
  13. ( ρ , θ ) (\rho^{\prime},\theta^{\prime})
  14. Φ ( ρ , θ ) = - λ 2 π ϵ ln R = - λ 4 π ϵ ln | ρ 2 + ( ρ ) 2 - 2 ρ ρ cos ( θ - θ ) | \Phi(\rho,\theta)=\frac{-\lambda}{2\pi\epsilon}\ln R=\frac{-\lambda}{4\pi% \epsilon}\ln\left|\rho^{2}+\left(\rho^{\prime}\right)^{2}-2\rho\rho^{\prime}% \cos(\theta-\theta^{\prime})\right|
  15. R R
  16. z z
  17. λ \lambda
  18. z z
  19. ρ \rho
  20. ρ \rho^{\prime}
  21. ρ 2 \rho^{2}
  22. Φ ( ρ , θ ) = - λ 4 π ϵ { 2 ln ρ + ln ( 1 - ρ ρ e i ( θ - θ ) ) ( 1 - ρ ρ e - i ( θ - θ ) ) } \Phi(\rho,\theta)=\frac{-\lambda}{4\pi\epsilon}\left\{2\ln\rho+\ln\left(1-% \frac{\rho^{\prime}}{\rho}e^{i\left(\theta-\theta^{\prime}\right)}\right)\left% (1-\frac{\rho^{\prime}}{\rho}e^{-i\left(\theta-\theta^{\prime}\right)}\right)\right\}
  23. ( ρ / ρ ) < 1 (\rho^{\prime}/\rho)<1
  24. Φ ( ρ , θ ) = - λ 2 π ϵ { ln ρ - k = 1 ( 1 k ) ( ρ ρ ) k [ cos k θ cos k θ + sin k θ sin k θ ] } \Phi(\rho,\theta)=\frac{-\lambda}{2\pi\epsilon}\left\{\ln\rho-\sum_{k=1}^{% \infty}\left(\frac{1}{k}\right)\left(\frac{\rho^{\prime}}{\rho}\right)^{k}% \left[\cos k\theta\cos k\theta^{\prime}+\sin k\theta\sin k\theta^{\prime}% \right]\right\}
  25. Φ ( ρ , θ ) = - Q 2 π ϵ ln ρ + ( 1 2 π ϵ ) k = 1 C k cos k θ + S k sin k θ ρ k \Phi(\rho,\theta)=\frac{-Q}{2\pi\epsilon}\ln\rho+\left(\frac{1}{2\pi\epsilon}% \right)\sum_{k=1}^{\infty}\frac{C_{k}\cos k\theta+S_{k}\sin k\theta}{\rho^{k}}
  26. Q = λ , Q=\lambda,
  27. C k = λ k ( ρ ) k cos k θ , C_{k}=\frac{\lambda}{k}\left(\rho^{\prime}\right)^{k}\cos k\theta^{\prime},
  28. S k = λ k ( ρ ) k sin k θ . S_{k}=\frac{\lambda}{k}\left(\rho^{\prime}\right)^{k}\sin k\theta^{\prime}.
  29. ρ \rho
  30. ρ \rho^{\prime}
  31. ( ρ ) 2 \left(\rho^{\prime}\right)^{2}
  32. ( ρ / ρ ) < 1 (\rho/\rho^{\prime})<1
  33. Φ ( ρ , θ ) = - λ 2 π ϵ { ln ρ - k = 1 ( 1 k ) ( ρ ρ ) k [ cos k θ cos k θ + sin k θ sin k θ ] } \Phi(\rho,\theta)=\frac{-\lambda}{2\pi\epsilon}\left\{\ln\rho^{\prime}-\sum_{k% =1}^{\infty}\left(\frac{1}{k}\right)\left(\frac{\rho}{\rho^{\prime}}\right)^{k% }\left[\cos k\theta\cos k\theta^{\prime}+\sin k\theta\sin k\theta^{\prime}% \right]\right\}
  34. Φ ( ρ , θ ) = - Q 2 π ϵ ln ρ + ( 1 2 π ϵ ) k = 1 ρ k [ I k cos k θ + J k sin k θ ] \Phi(\rho,\theta)=\frac{-Q}{2\pi\epsilon}\ln\rho^{\prime}+\left(\frac{1}{2\pi% \epsilon}\right)\sum_{k=1}^{\infty}\rho^{k}\left[I_{k}\cos k\theta+J_{k}\sin k% \theta\right]
  35. Q = λ , Q=\lambda,
  36. I k = λ k cos k θ ( ρ ) k , I_{k}=\frac{\lambda}{k}\frac{\cos k\theta^{\prime}}{\left(\rho^{\prime}\right)% ^{k}},
  37. J k = λ k sin k θ ( ρ ) k . J_{k}=\frac{\lambda}{k}\frac{\sin k\theta^{\prime}}{\left(\rho^{\prime}\right)% ^{k}}.
  38. λ ( ρ , θ ) \lambda(\rho^{\prime},\theta^{\prime})
  39. Φ ( 𝐫 ) = - Q 2 π ϵ ln ρ + ( 1 2 π ϵ ) k = 1 C k cos k θ + S k sin k θ ρ k \Phi(\mathbf{r})=\frac{-Q}{2\pi\epsilon}\ln\rho+\left(\frac{1}{2\pi\epsilon}% \right)\sum_{k=1}^{\infty}\frac{C_{k}\cos k\theta+S_{k}\sin k\theta}{\rho^{k}}
  40. Q = d θ ρ d ρ λ ( ρ , θ ) Q=\int d\theta^{\prime}\int\rho^{\prime}d\rho^{\prime}\lambda(\rho^{\prime},% \theta^{\prime})
  41. C k = ( 1 k ) d θ d ρ ( ρ ) k + 1 λ ( ρ , θ ) cos k θ C_{k}=\left(\frac{1}{k}\right)\int d\theta^{\prime}\int d\rho^{\prime}\left(% \rho^{\prime}\right)^{k+1}\lambda(\rho^{\prime},\theta^{\prime})\cos k\theta^{\prime}
  42. S k = ( 1 k ) d θ d ρ ( ρ ) k + 1 λ ( ρ , θ ) sin k θ S_{k}=\left(\frac{1}{k}\right)\int d\theta^{\prime}\int d\rho^{\prime}\left(% \rho^{\prime}\right)^{k+1}\lambda(\rho^{\prime},\theta^{\prime})\sin k\theta^{\prime}
  43. λ ( ρ , θ ) \lambda(\rho^{\prime},\theta^{\prime})
  44. ( ρ - θ ) (\rho-\theta)
  45. Φ ( ρ , θ ) = - Q 2 π ϵ ln ρ + ( 1 2 π ϵ ) k = 1 ρ k [ I k cos k θ + J k sin k θ ] \Phi(\rho,\theta)=\frac{-Q}{2\pi\epsilon}\ln\rho^{\prime}+\left(\frac{1}{2\pi% \epsilon}\right)\sum_{k=1}^{\infty}\rho^{k}\left[I_{k}\cos k\theta+J_{k}\sin k% \theta\right]
  46. Q = d θ ρ d ρ λ ( ρ , θ ) Q=\int d\theta^{\prime}\int\rho^{\prime}d\rho^{\prime}\lambda(\rho^{\prime},% \theta^{\prime})
  47. I k = ( 1 k ) d θ d ρ [ cos k θ ( ρ ) k - 1 ] λ ( ρ , θ ) I_{k}=\left(\frac{1}{k}\right)\int d\theta^{\prime}\int d\rho^{\prime}\left[% \frac{\cos k\theta^{\prime}}{\left(\rho^{\prime}\right)^{k-1}}\right]\lambda(% \rho^{\prime},\theta^{\prime})
  48. J k = ( 1 k ) d θ d ρ [ sin k θ ( ρ ) k - 1 ] λ ( ρ , θ ) J_{k}=\left(\frac{1}{k}\right)\int d\theta^{\prime}\int d\rho^{\prime}\left[% \frac{\sin k\theta^{\prime}}{\left(\rho^{\prime}\right)^{k-1}}\right]\lambda(% \rho^{\prime},\theta^{\prime})
  49. f ( 𝐫 ) f(\mathbf{r}^{\prime})
  50. λ ( ρ , θ ) \lambda(\rho,\theta)
  51. λ ( ρ , θ ) = d z f ( ρ , θ , z ) \lambda(\rho,\theta)=\int dz\ f(\rho,\theta,z)
  52. U = d θ ρ d ρ λ ( ρ , θ ) Φ ( ρ , θ ) U=\int d\theta\int\rho d\rho\ \lambda(\rho,\theta)\Phi(\rho,\theta)
  53. U = - Q 1 2 π ϵ ρ d ρ λ ( ρ , θ ) ln ρ U=\frac{-Q_{1}}{2\pi\epsilon}\int\rho d\rho\ \lambda(\rho,\theta)\ln\rho
  54. + ( 1 2 π ϵ ) k = 1 C 1 k d θ d ρ [ cos k θ ρ k - 1 ] λ ( ρ , θ ) \ \ \ \ \ \ \ \ \ \ +\ \left(\frac{1}{2\pi\epsilon}\right)\sum_{k=1}^{\infty}C% _{1k}\int d\theta\int d\rho\left[\frac{\cos k\theta}{\rho^{k-1}}\right]\lambda% (\rho,\theta)
  55. + ( 1 2 π ϵ ) k = 1 S 1 k d θ d ρ [ sin k θ ρ k - 1 ] λ ( ρ , θ ) \ \ \ \ \ \ \ \ +\ \left(\frac{1}{2\pi\epsilon}\right)\sum_{k=1}^{\infty}S_{1k% }\int d\theta\int d\rho\left[\frac{\sin k\theta}{\rho^{k-1}}\right]\lambda(% \rho,\theta)
  56. Q 1 Q_{1}
  57. C 1 k C_{1k}
  58. S 1 k S_{1k}
  59. U = - Q 1 2 π ϵ ρ d ρ λ ( ρ , θ ) ln ρ + ( 1 2 π ϵ ) k = 1 k ( C 1 k I 2 k + S 1 k J 2 k ) U=\frac{-Q_{1}}{2\pi\epsilon}\int\rho d\rho\ \lambda(\rho,\theta)\ln\rho+\left% (\frac{1}{2\pi\epsilon}\right)\sum_{k=1}^{\infty}k\left(C_{1k}I_{2k}+S_{1k}J_{% 2k}\right)
  60. I 2 k I_{2k}
  61. J 2 k J_{2k}
  62. U = - Q 1 ln ρ 2 π ϵ ρ d ρ λ ( ρ , θ ) + ( 1 2 π ϵ ) k = 1 k ( C 2 k I 1 k + S 2 k J 1 k ) U=\frac{-Q_{1}\ln\rho^{\prime}}{2\pi\epsilon}\int\rho d\rho\ \lambda(\rho,% \theta)+\left(\frac{1}{2\pi\epsilon}\right)\sum_{k=1}^{\infty}k\left(C_{2k}I_{% 1k}+S_{2k}J_{1k}\right)
  63. I 1 k I_{1k}
  64. J 1 k J_{1k}
  65. C 2 k C_{2k}
  66. S 2 k S_{2k}

D'Alembert's_formula.html

  1. u t t - c 2 u x x = 0 , u ( x , 0 ) = g ( x ) , u t ( x , 0 ) = h ( x ) , u_{tt}-c^{2}u_{xx}=0,\,u(x,0)=g(x),\,u_{t}(x,0)=h(x),
  2. - < x < , t > 0 -\infty<x<\infty,\,\,t>0
  3. x ± c t = const x\pm ct=\mathrm{const}\,
  4. μ = x + c t , η = x - c t \mu=x+ct,\eta=x-ct\,
  5. u μ η = 0 u_{\mu\eta}=0\,
  6. u ( μ , η ) = F ( μ ) + G ( η ) u(\mu,\eta)=F(\mu)+G(\eta)\,
  7. F F\,
  8. G G\,
  9. C 1 C^{1}\,
  10. x , t x,t\,
  11. u ( x , t ) = F ( x + c t ) + G ( x - c t ) u(x,t)=F(x+ct)+G(x-ct)\,
  12. u u\,
  13. C 2 C^{2}\,
  14. F F\,
  15. G G\,
  16. C 2 C^{2}\,
  17. u u\,
  18. c c\,
  19. u ( x , 0 ) = g ( x ) , u t ( x , 0 ) = h ( x ) u(x,0)=g(x),u_{t}(x,0)=h(x)\,
  20. u ( x , 0 ) = g ( x ) u(x,0)=g(x)\,
  21. F ( x ) + G ( x ) = g ( x ) F(x)+G(x)=g(x)\,
  22. u t ( x , 0 ) = h ( x ) u_{t}(x,0)=h(x)\,
  23. c F ( x ) - c G ( x ) = h ( x ) cF^{\prime}(x)-cG^{\prime}(x)=h(x)\,
  24. c F ( x ) - c G ( x ) = - x h ( ξ ) d ξ + c 1 . cF(x)-cG(x)=\int_{-\infty}^{x}h(\xi)\,d\xi+c_{1}.\,
  25. F ( x ) = - 1 2 c ( - c g ( x ) - ( - x h ( ξ ) d ξ + c 1 ) ) F(x)=\frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^{x}h(\xi)\,d\xi+c_{1}% \right)\right)\,
  26. G ( x ) = - 1 2 c ( - c g ( x ) + ( - x h ( ξ ) d ξ + c 1 ) ) . G(x)=\frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^{x}h(\xi)d\xi+c_{1}\right)% \right).\,
  27. u ( x , t ) = F ( x + c t ) + G ( x - c t ) u(x,t)=F(x+ct)+G(x-ct)\,
  28. u ( x , t ) = 1 2 [ g ( x - c t ) + g ( x + c t ) ] + 1 2 c x - c t x + c t h ( ξ ) d ξ . u(x,t)=\frac{1}{2}\left[g(x-ct)+g(x+ct)\right]+\frac{1}{2c}\int_{x-ct}^{x+ct}h% (\xi)\,d\xi.

D-module.html

  1. 𝒪 ( 𝐂 n ) \mathcal{O}(\mathbf{C}^{n})
  2. Hom ( M , 𝒪 ( 𝐂 ) ) \mathrm{Hom}(M,\mathcal{O}(\mathbf{C}))
  3. : D X E n d K ( M ) , v v \nabla:D_{X}\rightarrow End_{K}(M),v\mapsto\nabla_{v}
  4. f v ( m ) = f v ( m ) \nabla_{fv}(m)=f\nabla_{v}(m)
  5. v ( f m ) = v ( f ) m + f v ( m ) \nabla_{v}(fm)=v(f)m+f\nabla_{v}(m)
  6. [ v , w ] ( m ) = [ v , w ] ( m ) \nabla_{[v,w]}(m)=[\nabla_{v},\nabla_{w}](m)
  7. f i 1 , , i n 1 i 1 n i n \sum f_{i_{1},\dots,i_{n}}\partial_{1}^{i_{1}}\cdots\partial_{n}^{i_{n}}
  8. f i 1 , , i n f_{i_{1},\dots,i_{n}}
  9. D ( M ) := Hom ( M , D X ) Ω X - 1 [ dim X ] . \mathrm{D}(M):=\mathcal{R}\mathrm{Hom}(M,D_{X})\otimes\Omega^{-1}_{X}[% \operatorname{dim}X].

Dadda_multiplier.html

  1. n 2 n^{2}
  2. a 2 b 3 a_{2}b_{3}
  3. n = 4 n=4
  4. a 3 a 2 a 1 a 0 a_{3}a_{2}a_{1}a_{0}
  5. b 3 b 2 b 1 b 0 b_{3}b_{2}b_{1}b_{0}

Damerau–Levenshtein_distance.html

  1. a a
  2. b b
  3. d a , b ( | a | , | b | ) d_{a,b}(|a|,|b|)
  4. d a , b ( i , j ) = { max ( i , j ) if min ( i , j ) = 0 , min { d a , b ( i - 1 , j ) + 1 d a , b ( i , j - 1 ) + 1 d a , b ( i - 1 , j - 1 ) + 1 ( a i b j ) d a , b ( i - 2 , j - 2 ) + 1 if i , j > 1 and a i = b j - 1 and a i - 1 = b j min { d a , b ( i - 1 , j ) + 1 d a , b ( i , j - 1 ) + 1 d a , b ( i - 1 , j - 1 ) + 1 ( a i b j ) otherwise. \qquad d_{a,b}(i,j)=\begin{cases}\max(i,j)&\,\text{ if}\min(i,j)=0,\\ \min\begin{cases}d_{a,b}(i-1,j)+1\\ d_{a,b}(i,j-1)+1\\ d_{a,b}(i-1,j-1)+1_{(a_{i}\neq b_{j})}\\ d_{a,b}(i-2,j-2)+1\end{cases}&\,\text{ if }i,j>1\,\text{ and }a_{i}=b_{j-1}\,% \text{ and }a_{i-1}=b_{j}\\ \min\begin{cases}d_{a,b}(i-1,j)+1\\ d_{a,b}(i,j-1)+1\\ d_{a,b}(i-1,j-1)+1_{(a_{i}\neq b_{j})}\end{cases}&\,\text{ otherwise.}\end{cases}
  5. 1 ( a i b j ) 1_{(a_{i}\neq b_{j})}
  6. a i = b j a_{i}=b_{j}
  7. d a , b ( i - 1 , j ) + 1 d_{a,b}(i-1,j)+1
  8. d a , b ( i , j - 1 ) + 1 d_{a,b}(i,j-1)+1
  9. d a , b ( i - 1 , j - 1 ) + 1 ( a i b j ) d_{a,b}(i-1,j-1)+1_{(a_{i}\neq b_{j})}
  10. d a , b ( i - 2 , j - 2 ) + 1 d_{a,b}(i-2,j-2)+1
  11. W T W_{T}
  12. 2 W T W I + W D 2W_{T}\geq W_{I}+W_{D}
  13. O ( M N max ( M , N ) ) O\left(M\cdot N\cdot\max(M,N)\right)
  14. O ( M N ) O\left(M\cdot N\right)

Darboux's_theorem_(analysis).html

  1. I I
  2. f : I \R f\colon I\to\R
  3. f f^{\prime}
  4. a a
  5. b b
  6. I I
  7. a < b a<b
  8. y y
  9. f ( a ) f^{\prime}(a)
  10. f ( b ) f^{\prime}(b)
  11. x x
  12. [ a , b ] [a,b]
  13. f ( x ) = y f^{\prime}(x)=y
  14. y y
  15. f ( a ) f^{\prime}(a)
  16. f ( b ) f^{\prime}(b)
  17. x x
  18. a a
  19. b b
  20. y y
  21. f ( a ) f^{\prime}(a)
  22. f ( b ) f^{\prime}(b)
  23. f ( a ) > y > f ( b ) f^{\prime}(a)>y>f^{\prime}(b)
  24. ϕ : I \R \phi\colon I\to\R
  25. ϕ ( t ) = f ( t ) - y t . \phi(t)=f(t)-yt.
  26. ϕ \phi
  27. [ a , b ] [a,b]
  28. x x
  29. x [ a , b ] x\in[a,b]
  30. ϕ ( a ) = f ( a ) - y > y - y = 0 \phi^{\prime}(a)=f^{\prime}(a)-y>y-y=0
  31. ϕ ( b ) = f ( b ) - y < y - y = 0 \phi^{\prime}(b)=f^{\prime}(b)-y<y-y=0
  32. a a
  33. b b
  34. x x
  35. ϕ \phi
  36. x ( a , b ) x\in(a,b)
  37. ϕ ( x ) = 0 \phi^{\prime}(x)=0
  38. f ( x ) = y f^{\prime}(x)=y
  39. x sin ( 1 / x ) x\mapsto\sin(1/x)
  40. x x 2 sin ( 1 / x ) x\mapsto x^{2}\sin(1/x)

Dasymeter.html

  1. density of sphere density of gas = weight of sphere weight of sphere - weight of immersed sphere \frac{\mbox{density of sphere}~{}}{\mbox{density of gas}}=\frac{\mbox{weight % of sphere}~{}}{\mbox{weight of sphere}~{}-\mbox{weight of immersed sphere}~{}}\,

Data_integration.html

  1. G , S , M \left\langle G,S,M\right\rangle
  2. G G
  3. S S
  4. M M
  5. G G
  6. S S
  7. M M
  8. G G
  9. S S
  10. G G
  11. S S
  12. G G
  13. M M
  14. G G
  15. S S
  16. S S
  17. M M
  18. G G
  19. S S
  20. G G
  21. S S
  22. G G
  23. M M
  24. S S
  25. G G
  26. G G
  27. S S
  28. f ( A , B ) f(A,B)
  29. A < B A<B
  30. A A
  31. B B
  32. A B A\supset B
  33. B B
  34. A A

David_Harbater.html

  1. p ( t ) \mathbb{Q}_{p}(t)

Davidson_correction.html

  1. Δ E Q = ( 1 - a 0 2 ) ( E CISD - E HF ) , \Delta E_{Q}=(1-a_{0}^{2})(E_{\rm CISD}-E_{\rm HF}),
  2. E CISDTQ E CISD + Δ E Q , E_{\rm CISDTQ}\approx E_{\rm CISD}+\Delta E_{Q},
  3. a 0 2 a_{0}^{2}

De_Rham_curve.html

  1. ( M , d ) (M,d)
  2. \mathbb{R}
  3. d 0 : M M d_{0}:\ M\to M
  4. d 1 : M M . d_{1}:\ M\to M.
  5. p 0 p_{0}
  6. p 1 p_{1}
  7. [ 0 , 1 ] [0,1]
  8. x = k = 1 b k 2 k , x=\sum_{k=1}^{\infty}\frac{b_{k}}{2^{k}},
  9. b k b_{k}
  10. c x : M M c_{x}:\ M\to M
  11. c x = d b 1 d b 2 d b k , c_{x}=d_{b_{1}}\circ d_{b_{2}}\circ\cdots\circ d_{b_{k}}\circ\cdots,
  12. \circ
  13. c x c_{x}
  14. d 0 d_{0}
  15. d 1 d_{1}
  16. p x p_{x}
  17. M M
  18. p x p_{x}
  19. d 0 ( p 1 ) = d 1 ( p 0 ) d_{0}(p_{1})=d_{1}(p_{0})
  20. p x p_{x}
  21. p ( x ) = d 0 ( p ( 2 x ) ) p(x)=d_{0}(p(2x))
  22. x [ 0 , 0.5 ] x\in[0,0.5]
  23. p ( x ) = d 1 ( p ( 2 x - 1 ) ) p(x)=d_{1}(p(2x-1))
  24. x [ 0.5 , 1 ] . x\in[0.5,1].
  25. { p ( x ) , x [ 0 , 1 ] } \{p(x),x\in[0,1]\}
  26. { d 0 , d 1 } \{d_{0},\ d_{1}\}
  27. p 0 = 0 p_{0}=0
  28. p 1 = 1 p_{1}=1
  29. a a
  30. | a | < 1 |a|<1
  31. | 1 - a | < 1 |1-a|<1
  32. d 0 d_{0}
  33. d 1 d_{1}
  34. d 0 ( z ) = a z d_{0}(z)=az
  35. d 1 ( z ) = a + ( 1 - a ) z . d_{1}(z)=a+(1-a)z.
  36. a = ( 1 + i ) / 2 a=(1+i)/2
  37. p 0 = 0 p_{0}=0
  38. p 1 = 1 p_{1}=1
  39. z ¯ \overline{z}
  40. z z
  41. d 0 ( z ) = a z ¯ d_{0}(z)=a\overline{z}
  42. d 1 ( z ) = a + ( 1 - a ) z ¯ . d_{1}(z)=a+(1-a)\overline{z}.
  43. a Koch = 1 2 + i 3 6 , a\text{Koch}=\frac{1}{2}+i\frac{\sqrt{3}}{6},
  44. a Peano = ( 1 + i ) 2 . a\text{Peano}=\frac{(1+i)}{2}.
  45. d 0 = ( 1 0 0 0 α δ 0 β ϵ ) d_{0}=\begin{pmatrix}1&0&0\\ 0&\alpha&\delta\\ 0&\beta&\epsilon\end{pmatrix}
  46. d 1 = ( 1 0 0 α 1 - α ζ β - β η ) . d_{1}=\begin{pmatrix}1&0&0\\ \alpha&1-\alpha&\zeta\\ \beta&-\beta&\eta\end{pmatrix}.
  47. ( u , v ) (u,v)
  48. ( 1 u v ) . \begin{pmatrix}1\\ u\\ v\end{pmatrix}.
  49. ( u , v ) = ( α , β ) (u,v)=(\alpha,\beta)
  50. w w
  51. α = β = ϵ = 1 / 2 \alpha=\beta=\epsilon=1/2
  52. δ = ζ = 0 \delta=\zeta=0
  53. η = w \eta=w
  54. d 0 = ( 1 0 0 0 1 / 2 0 0 1 / 2 w ) d_{0}=\begin{pmatrix}1&0&0\\ 0&1/2&0\\ 0&1/2&w\end{pmatrix}
  55. d 1 = ( 1 0 0 1 / 2 1 / 2 0 1 / 2 - 1 / 2 w ) . d_{1}=\begin{pmatrix}1&0&0\\ 1/2&1/2&0\\ 1/2&-1/2&w\end{pmatrix}.
  56. w = 1 / 4 w=1/4
  57. f ( x ) = 4 x ( 1 - x ) f(x)=4x(1-x)
  58. d 0 ( z ) = z z + 1 d_{0}(z)=\frac{z}{z+1}
  59. d 1 ( z ) = 1 z + 1 . d_{1}(z)=\frac{1}{z+1}.
  60. d i ( p ( n - 1 ) ) = d ( i + 1 ) ( p 0 ) d_{i}(p_{(n-1)})=d_{(i+1)}(p_{0})
  61. i = 0 n - 2. i=0\ldots n-2.