wpmath0000006_1

British_Association_screw_threads.html

  1. p = 0.9 K \scriptstyle p=0.9^{K}
  2. 6 p 1.2 \scriptstyle 6p^{1.2}

Britney_Gallivan.html

  1. W = π t 2 ( 3 / 2 ) ( n - 1 ) . W=\pi t2^{(3/2)\left(n-1\right)}.
  2. L = π t 6 ( 2 n + 4 ) ( 2 n - 1 ) , L=\frac{\pi t}{6}\left(2^{n}+4\right)\left(2^{n}-1\right),
  3. π \pi

Brocard's_conjecture.html

  1. p n p_{n}
  2. p n 2 p_{n}^{2}
  3. Δ \Delta
  4. Δ \Delta
  5. π ( p n + 1 2 ) - π ( p n 2 ) \pi(p_{n+1}^{2})-\pi(p_{n}^{2})

Brun's_theorem.html

  1. π 2 ( x ) \pi_{2}(x)
  2. π 2 ( x ) \pi_{2}(x)
  3. π 2 ( x ) = O ( x ( log log x ) 2 ( log x ) 2 ) . \pi_{2}(x)=O\left(\frac{x(\log\log x)^{2}}{(\log x)^{2}}\right).
  4. p : p + 2 ( 1 p + 1 < m t p l > p + 2 ) = ( 1 3 + 1 5 ) + ( 1 5 + 1 7 ) + ( 1 11 + 1 13 ) + \sum\limits_{p\,:\,p+2\in\mathbb{P}}{\left({\frac{1}{p}+\frac{1}{<}mtpl>{{p+2}% }}\right)}=\left({\frac{1}{3}+\frac{1}{5}}\right)+\left({\frac{1}{5}+\frac{1}{% 7}}\right)+\left({\frac{1}{{11}}+\frac{1}{{13}}}\right)+\cdots
  5. × 10 1 5 \times 10^{1}5
  6. B 4 = ( 1 5 + 1 7 + 1 11 + 1 13 ) + ( 1 11 + 1 13 + 1 17 + 1 19 ) + ( 1 101 + 1 103 + 1 107 + 1 109 ) + B_{4}=\left(\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}\right)+\left(% \frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}\right)+\left(\frac{1}{101}% +\frac{1}{103}+\frac{1}{107}+\frac{1}{109}\right)+\cdots
  7. C 2 = 0.6601 C_{2}=0.6601\ldots
  8. π 2 ( x ) 2 C 2 x ( log x ) 2 . \pi_{2}(x)\sim 2C_{2}\frac{x}{(\log x)^{2}}.
  9. π 2 ( x ) < ( 2 C 2 + ε ) x ( log x ) 2 \pi_{2}(x)<(2C_{2}+\varepsilon)\frac{x}{(\log x)^{2}}
  10. ε > 0 \varepsilon>0
  11. π 2 ( x ) < 4.5 x ( log x ) 2 \pi_{2}(x)<4.5\frac{x}{(\log x)^{2}}
  12. ε 3.18 \varepsilon\approx 3.18

Brunt–Väisälä_frequency.html

  1. ρ 0 \rho_{0}
  2. ρ = ρ ( z ) \rho=\rho(z)
  3. z z^{\prime}
  4. ρ 0 2 z t 2 = - g ( ρ ( z ) - ρ ( z + z ) ) \rho_{0}\frac{\partial^{2}z^{\prime}}{\partial t^{2}}=-g(\rho(z)-\rho(z+z^{% \prime}))
  5. ρ ( z + z ) - ρ ( z ) = ρ ( z ) z z \rho(z+z^{\prime})-\rho(z)=\frac{\partial\rho(z)}{\partial z}z^{\prime}
  6. ρ 0 \rho_{0}
  7. 2 z t 2 = g ρ 0 ρ ( z ) z z \frac{\partial^{2}z^{\prime}}{\partial t^{2}}=\frac{g}{\rho_{0}}\frac{\partial% \rho(z)}{\partial z}z^{\prime}
  8. z = z 0 e - N 2 t z^{\prime}=z^{\prime}_{0}e^{\sqrt{-N^{2}}t}\!
  9. N = - g ρ 0 ρ ( z ) z N=\sqrt{-\frac{g}{\rho_{0}}\frac{\partial\rho(z)}{\partial z}}
  10. ρ ( z ) z \frac{\partial\rho(z)}{\partial z}
  11. N g θ d θ d z N\equiv\sqrt{\frac{g}{\theta}\frac{d\theta}{dz}}
  12. θ \theta
  13. g g
  14. z z
  15. N - g ρ d ρ d z N\equiv\sqrt{-\frac{g}{\rho}\frac{d\rho}{dz}}
  16. ρ \rho
  17. < v a r > N 2 > 0 <var>N^{2}>0

Brønsted_catalysis_equation.html

  1. log k = α * log ( K a ) + C \ \log k=\alpha*\log(K_{a})+C

Bulgarian_solitaire.html

  1. N N
  2. N N
  3. N = 1 + 2 + + k N=1+2+\cdots+k
  4. k k
  5. 1 , 2 , , k 1,2,\ldots,k
  6. k 2 - k k^{2}-k
  7. N N
  8. N N
  9. p p

Bulk_density.html

  1. V t V_{t}
  2. M t M_{t}
  3. M s M_{s}
  4. M t = M s + M l M_{t}=M_{s}+M_{l}
  5. M l M_{l}
  6. ρ b = M s V t \rho_{b}=\frac{M_{s}}{V_{t}}
  7. ρ t = M t V t \rho_{t}=\frac{M_{t}}{V_{t}}

Bullet-nose_curve.html

  1. a 2 y 2 - b 2 x 2 = x 2 y 2 a^{2}y^{2}-b^{2}x^{2}=x^{2}y^{2}\,
  2. f ( z ) = n = 0 ( 2 n n ) z 2 n + 1 = z + 2 z 3 + 6 z 5 + 20 z 7 + f(z)=\sum_{n=0}^{\infty}{2n\choose n}z^{2n+1}=z+2z^{3}+6z^{5}+20z^{7}+\cdots
  3. y = f ( x 2 a ) ± 2 b y=f\left(\frac{x}{2a}\right)\pm 2b

Bunching_parameter.html

  1. η q = q q - 1 P q P q - 2 P q - 1 2 , \eta_{q}=\frac{q}{q-1}\frac{P_{q}P_{q-2}}{P_{q-1}^{2}},
  2. P n P_{n}
  3. n n
  4. P n P_{n}
  5. η q = 1 \eta_{q}=1
  6. F q = n - q n = q n ! ( n - q ) ! P n . F_{q}=\langle n\rangle^{-q}\sum^{\infty}_{n=q}\frac{n!}{(n-q)!}P_{n}.

Burgers_material.html

  1. a 1 a_{1}
  2. a 2 a_{2}
  3. a 3 a_{3}
  4. a 4 a_{4}
  5. σ + a 1 σ ˙ + a 2 σ ¨ = a 3 γ + a 4 γ ˙ \sigma+a_{1}\dot{\sigma}+a_{2}\ddot{\sigma}=a_{3}\gamma+a_{4}\dot{\gamma}
  6. σ \sigma
  7. γ \gamma

Busemann_function.html

  1. ( X , d ) (X,d)
  2. γ : [ 0 , ) X \gamma:[0,\infty)\to X
  3. t , t [ 0 , ) t,t^{\prime}\in[0,\infty)
  4. d ( γ ( t ) , γ ( t ) ) = | t - t | d\big(\gamma(t),\gamma(t^{\prime})\big)=\big|t-t^{\prime}\big|
  5. [ 0 , ) [0,\infty)
  6. B γ : X B_{\gamma}:X\to\mathbb{R}
  7. B γ ( x ) = lim t ( d ( γ ( t ) , x ) - t ) B_{\gamma}(x)=\lim_{t\to\infty}\big(d\big(\gamma(t),x\big)-t\big)
  8. d ( γ ( t ) , x ) d\big(\gamma(t),x\big)
  9. B γ ( x ) + t B_{\gamma}(x)+t

Butterfly_curve_(transcendental).html

  1. x = sin ( t ) ( e cos ( t ) - 2 cos ( 4 t ) - sin 5 ( t 12 ) ) x=\sin(t)\left(e^{\cos(t)}-2\cos(4t)-\sin^{5}\left({t\over 12}\right)\right)
  2. y = cos ( t ) ( e cos ( t ) - 2 cos ( 4 t ) - sin 5 ( t 12 ) ) y=\cos(t)\left(e^{\cos(t)}-2\cos(4t)-\sin^{5}\left({t\over 12}\right)\right)
  3. r = e sin θ - 2 cos ( 4 θ ) + sin 5 ( 2 θ - π 24 ) r=e^{\sin\theta}-2\cos(4\theta)+\sin^{5}\left(\frac{2\theta-\pi}{24}\right)

Butterfly_diagram.html

  1. y 0 = x 0 + x 1 y_{0}=x_{0}+x_{1}\,
  2. y 1 = x 0 - x 1 . y_{1}=x_{0}-x_{1}.\,
  3. ω n k = e - 2 π i k n \omega^{k}_{n}=e^{-\frac{2\pi ik}{n}}
  4. y 0 = x 0 + x 1 ω n k y_{0}=x_{0}+x_{1}\omega^{k}_{n}\,
  5. y 1 = x 0 - x 1 ω n k , y_{1}=x_{0}-x_{1}\omega^{k}_{n},\,
  6. x 0 = 1 2 ( y 0 + y 1 ) x_{0}=\frac{1}{2}(y_{0}+y_{1})\,
  7. x 1 = ω n - k 2 ( y 0 - y 1 ) , x_{1}=\frac{\omega^{-k}_{n}}{2}(y_{0}-y_{1}),\,

C_(disambiguation).html

  1. \mathbb{C}
  2. 𝔠 \mathfrak{c}
  3. C 0 , C 1 , C 2 , , C C^{0},C^{1},C^{2},\dots,C^{\infty}
  4. C ( n , k ) C(n,k)
  5. C k n C_{k}^{n}
  6. C k n {}^{n}C_{k}
  7. C k n {}_{n}C_{k}
  8. ( n k ) {n\choose k}

Caesium_chloride.html

  1. 3 ¯ \overline{3}
  2. CH 2 = CHCOOCH 3 + ArCH = N - CH ( CH 3 ) - COOC ( CH 3 ) 3 TBAB , CsCl , K 2 CO 3 CPME , 0 o C ArCH = N - C ( C 2 H 4 COOCH 3 ) ( CH 3 ) - COOC ( CH 3 ) 3 \mathrm{CH_{2}\!\!=\!\!CHCOOCH_{3}+ArCH\!\!=\!\!N\!\!-\!\!CH(CH_{3})\!\!-\!\!% COOC(CH_{3})_{3}\ \xrightarrow[CPME,\ 0^{o}C]{TBAB,\ CsCl,\ K_{2}CO_{3}}\ ArCH% \!\!=\!\!N\!\!-\!\!C(C_{2}H_{4}COOCH_{3})(CH_{3})\!\!-\!\!COOC(CH_{3})_{3}}
  3. C ( NO 2 ) 4 + CsCl DMF C ( NO 2 ) 3 Cl + CsNO 2 \mathrm{C(NO_{2})_{4}+CsCl\ \xrightarrow[]{DMF}\ C(NO_{2})_{3}Cl+CsNO_{2}}

Cairo_pentagonal_tiling.html

  1. 3 \sqrt{3}
  2. 3 \sqrt{3}

Calabi_flow.html

  1. g i j t = ( Δ R ) g i j \frac{\partial g_{ij}}{\partial t}=(\Delta R)g_{ij}
  2. g i j g_{ij}
  3. Δ \Delta

California_bearing_ratio.html

  1. C B R = p p s 100 CBR=\frac{p}{p_{s}}\cdot 100\quad
  2. C B R CBR\quad
  3. p p\quad
  4. p s p_{s}\quad

Calorimeter_constant.html

  1. C cal = Δ H Δ T C_{\mathrm{cal}}=\frac{\Delta{H}}{\Delta{T}}
  2. Δ H neutralization = C cal Δ T \Delta{H_{\mathrm{neutralization}}}=C_{\mathrm{cal}}\cdot\Delta{T}

Caml.html

  1. x 3 - x - 1 x^{3}-x-1
  2. x = 3 x=3
  3. f ( x ) = 3 x 2 - 1 f ( 3 ) = 27 - 1 = 26 f^{\prime}(x)=3x^{2}-1\rightarrow f^{\prime}(3)=27-1=26

Capacitor-input_filter.html

  1. f 0 = 1 2 π L C . f_{0}={1\over{2\pi\sqrt{LC}}}.

Capitalization_rate.html

  1. Capitalization Rate = annual net operating income cost (or value) \mbox{Capitalization Rate}~{}=\frac{\mbox{annual net operating income}~{}}{% \mbox{cost (or value)}~{}}

Carathéodory's_theorem_(conformal_mapping).html

  1. \to

Carbon_dioxide_(data_page).html

  1. [ CO 2 ] [ CO ] [ O 2 ] 1 2 \scriptstyle\frac{[\mathrm{CO_{2}}]}{[\mathrm{CO}][\mathrm{O_{2}}]^{\frac{1}{2% }}}

Carbonate_compensation_depth.html

  1. CaCO 3 + CO 2 + H 2 O Ca 2 + ( aq ) + 2 HCO 3 - ( aq ) \mathrm{CaCO_{3}+CO_{2}+H_{2}O\ \rightleftharpoons\ Ca^{2+}(aq)+2\ HCO_{3}^{-}% (aq)}

Carminati–McLenaghan_invariants.html

  1. C a b c d C_{abcd}
  2. C a c d b {{}^{\star}C}_{acdb}
  3. R a b R_{ab}
  4. S a b = R a b - 1 4 R g a b S_{ab}=R_{ab}-\frac{1}{4}\,R\,g_{ab}
  5. S a b {S^{a}}_{b}
  6. S a m S m b {S^{a}}_{m}\,{S^{m}}_{b}
  7. S a b S b a {S^{a}}_{b}\,{S^{b}}_{a}
  8. R = R m m R={R^{m}}_{m}
  9. R 1 = 1 4 S a b S b a R_{1}=\frac{1}{4}\,{S^{a}}_{b}\,{S^{b}}_{a}
  10. R 2 = - 1 8 S a b S b c S c a R_{2}=-\frac{1}{8}\,{S^{a}}_{b}\,{S^{b}}_{c}\,{S^{c}}_{a}
  11. R 3 = 1 16 S a b S b c S c d S d a R_{3}=\frac{1}{16}\,{S^{a}}_{b}\,{S^{b}}_{c}\,{S^{c}}_{d}\,{S^{d}}_{a}
  12. M 3 = 1 16 S b c S e f ( C a b c d C a e f d + C a b c d C a e f d ) M_{3}=\frac{1}{16}\,S^{bc}\,S_{ef}\left(C_{abcd}\,C^{aefd}+{{}^{\star}C}_{abcd% }\,{{}^{\star}C}^{aefd}\right)
  13. M 4 = - 1 32 S a g S e f S c d ( C a c d b C b e f g + C a c d b C b e f g ) M_{4}=-\frac{1}{32}\,S^{ag}\,S^{ef}\,{S^{c}}_{d}\,\left({C_{ac}}^{db}\,C_{befg% }+{{{}^{\star}C}_{ac}}^{db}\,{{}^{\star}C}_{befg}\right)
  14. W 1 = 1 8 ( C a b c d + i C a b c d ) C a b c d W_{1}=\frac{1}{8}\,\left(C_{abcd}+i\,{{}^{\star}C}_{abcd}\right)\,C^{abcd}
  15. W 2 = - 1 16 ( C a b c d + i C a b c d ) C c d e f C e f a b W_{2}=-\frac{1}{16}\,\left({C_{ab}}^{cd}+i\,{{{}^{\star}C}_{ab}}^{cd}\right)\,% {C_{cd}}^{ef}\,{C_{ef}}^{ab}
  16. M 1 = 1 8 S a b S c d ( C a c d b + i C a c d b ) M_{1}=\frac{1}{8}\,S^{ab}\,S^{cd}\,\left(C_{acdb}+i\,{{}^{\star}C}_{acdb}\right)
  17. M 2 = 1 16 S b c S e f ( C a b c d C a e f d - C a c d b C a e f d ) + 1 8 i S b c S e f C a b c d C a e f d M_{2}=\frac{1}{16}\,S^{bc}\,S_{ef}\,\left(C_{abcd}\,C^{aefd}-{{}^{\star}C}_{% acdb}\,{{}^{\star}C}^{aefd}\right)+\frac{1}{8}\,i\,S^{bc}\,S_{ef}\,{{}^{\star}% C}_{abcd}\,C^{aefd}
  18. M 5 = 1 32 S c d S e f ( C a g h b + i C a g h b ) ( C a c d b C g e f h + C a c d b C g e f h ) M_{5}=\frac{1}{32}\,S^{cd}\,S^{ef}\,\left(C^{aghb}+i\,{{}^{\star}C}^{aghb}% \right)\,\left(C_{acdb}\,C_{gefh}+{{}^{\star}C}_{acdb}\,{{}^{\star}C}_{gefh}\right)
  19. R R
  20. R 1 , W 1 R_{1},\,W_{1}
  21. R 2 , W 2 , M 1 R_{2},\,W_{2},\,M_{1}
  22. R 3 , M 2 , M 3 R_{3},\,M_{2},\,M_{3}
  23. M 4 , M 5 M_{4},\,M_{5}
  24. R , R 1 , R 2 , R 3 , ( W 1 ) , ( M 1 ) , ( M 2 ) R,\,R_{1},\,R_{2},\,R_{3},\,\Re(W_{1}),\,\Re(M_{1}),\,\Re(M_{2})
  25. 1 32 S c d S e f C a g h b C a c d b C g e f h \frac{1}{32}\,S^{cd}\,S^{ef}\,C^{aghb}\,C_{acdb}\,C_{gefh}

Carothers_equation.html

  1. X ¯ n = 1 1 - p \bar{X}_{n}=\frac{1}{1-p}
  2. X ¯ n \bar{X}_{n}
  3. X ¯ n = 2 n \bar{X}_{n}=2n
  4. p = N 0 - N N 0 p=\frac{N_{0}-N}{N_{0}}
  5. N 0 N_{0}
  6. N N
  7. X ¯ n = 50 \bar{X}_{n}=50
  8. X ¯ n = 100 \bar{X}_{n}=100
  9. X ¯ n = 1 + r 1 + r - 2 r p \bar{X}_{n}=\frac{1+r}{1+r-2rp}
  10. f a v = N i \sdot f i N i f_{av}=\frac{\sum N_{i}\sdot f_{i}}{\sum N_{i}}
  11. x n = 2 2 - p f a v x_{n}=\frac{2}{2-pf_{av}}
  12. 2 ( N 0 - N ) N 0 \sdot f a v \frac{2(N_{0}-N)}{N_{0}\sdot f_{av}}
  13. X ¯ w = 1 + p 1 - p M ¯ n = M o 1 1 - p M ¯ w = M o 1 + p 1 - p P D I = M ¯ w M ¯ n = 1 + p \begin{matrix}\bar{X}_{w}&=&\frac{1+p}{1-p}\\ \bar{M}_{n}&=&M_{o}\frac{1}{1-p}\\ \bar{M}_{w}&=&M_{o}\frac{1+p}{1-p}\\ PDI&=&\frac{\bar{M}_{w}}{\bar{M}_{n}}=1+p\\ \end{matrix}

Carreau_fluid.html

  1. μ eff \mu_{\operatorname{eff}}
  2. γ ˙ \dot{\gamma}
  3. μ eff ( γ ˙ ) = μ inf + ( μ 0 - μ inf ) ( 1 + ( λ γ ˙ ) 2 ) n - 1 2 \mu_{\operatorname{eff}}(\dot{\gamma})=\mu_{\operatorname{\inf}}+(\mu_{0}-\mu_% {\operatorname{\inf}})\left(1+\left(\lambda\dot{\gamma}\right)^{2}\right)^{% \frac{n-1}{2}}
  4. μ 0 \mu_{0}
  5. μ inf \mu_{\operatorname{\inf}}
  6. λ \lambda
  7. n n
  8. μ 0 \mu_{0}
  9. μ inf \mu_{\operatorname{\inf}}
  10. λ \lambda
  11. n n
  12. γ ˙ 1 / λ \dot{\gamma}\ll 1/\lambda
  13. γ ˙ 1 / λ \dot{\gamma}\gg 1/\lambda

Carrier_generation_and_recombination.html

  1. n n
  2. p p
  3. ( n o p o = n i 2 ) (n_{o}p_{o}=n_{i}^{2})
  4. n p > n i 2 np>n_{i}^{2}
  5. n p < n i 2 np<n_{i}^{2}
  6. U A u g = Γ n n ( n p - n i 2 ) + Γ p p ( n p - n i 2 ) U_{Aug}=\Gamma_{n}\,n(np-n_{i}^{2})+\Gamma_{p}\,p(np-n_{i}^{2})

Carry-lookahead_adder.html

  1. A + B A+B
  2. G ( A , B ) G(A,B)
  3. A + B A+B
  4. G ( A , B ) = A B G(A,B)=A\cdot B
  5. A + B A+B
  6. P ( A , B ) P(A,B)
  7. A + B A+B
  8. P ( A , B ) = A + B \ P(A,B)=A+B
  9. P ( A , B ) = A B P^{\prime}(A,B)=A\oplus B
  10. P ( A , B ) P^{\prime}(A,B)
  11. C i C_{i}
  12. P i P_{i}
  13. G i G_{i}
  14. C i + 1 = G i + ( P i C i ) C_{i+1}=G_{i}+\left(P_{i}\cdot C_{i}\right)
  15. C 1 = G 0 + P 0 C 0 C_{1}=G_{0}+P_{0}\cdot C_{0}
  16. C 2 = G 1 + P 1 C 1 C_{2}=G_{1}+P_{1}\cdot C_{1}
  17. C 3 = G 2 + P 2 C 2 C_{3}=G_{2}+P_{2}\cdot C_{2}
  18. C 4 = G 3 + P 3 C 3 C_{4}=G_{3}+P_{3}\cdot C_{3}
  19. C 1 C_{1}
  20. C 2 C_{2}
  21. C 2 C_{2}
  22. C 3 C_{3}
  23. C 3 C_{3}
  24. C 4 C_{4}
  25. C 1 = G 0 + P 0 C 0 C_{1}=G_{0}+P_{0}\cdot C_{0}
  26. C 2 = G 1 + G 0 P 1 + C 0 P 0 P 1 C_{2}=G_{1}+G_{0}\cdot P_{1}+C_{0}\cdot P_{0}\cdot P_{1}
  27. C 3 = G 2 + G 1 P 2 + G 0 P 1 P 2 + C 0 P 0 P 1 P 2 C_{3}=G_{2}+G_{1}\cdot P_{2}+G_{0}\cdot P_{1}\cdot P_{2}+C_{0}\cdot P_{0}\cdot P% _{1}\cdot P_{2}
  28. C 4 = G 3 + G 2 P 3 + G 1 P 2 P 3 + G 0 P 1 P 2 P 3 + C 0 P 0 P 1 P 2 P 3 C_{4}=G_{3}+G_{2}\cdot P_{3}+G_{1}\cdot P_{2}\cdot P_{3}+G_{0}\cdot P_{1}\cdot P% _{2}\cdot P_{3}+C_{0}\cdot P_{0}\cdot P_{1}\cdot P_{2}\cdot P_{3}
  29. G i = A i B i G_{i}=A_{i}\cdot B_{i}
  30. P i = A i B i P_{i}=A_{i}\oplus B_{i}
  31. P i = A i + B i \ P_{i}=A_{i}+B_{i}
  32. C 1 = G 0 + P 0 C 0 C_{1}=G_{0}+P_{0}\cdot C_{0}
  33. A B A\oplus B
  34. A + B A+B
  35. A A
  36. B B
  37. A A
  38. B B
  39. G 0 G_{0}
  40. A B A\cdot B
  41. P 0 C 0 P_{0}\cdot C_{0}
  42. P G PG
  43. G G GG
  44. P G = P 0 P 1 P 2 P 3 PG=P_{0}\cdot P_{1}\cdot P_{2}\cdot P_{3}
  45. G G = G 3 + G 2 P 3 + G 1 P 3 P 2 + G 0 P 3 P 2 P 1 GG=G_{3}+G_{2}\cdot P_{3}+G_{1}\cdot P_{3}\cdot P_{2}+G_{0}\cdot P_{3}\cdot P_% {2}\cdot P_{1}
  46. C i C_{i}
  47. C 16 C_{16}
  48. P i P_{i}
  49. G i G_{i}
  50. C i C_{i}
  51. P G PG
  52. G G GG
  53. S i S_{i}
  54. C 16 C_{16}
  55. S [ 8 - 15 ] S_{[8-15]}

Carry-select_adder.html

  1. ( n + 1 ) (n+1)
  2. n n
  3. O ( n ) O(\sqrt{n})
  4. n \lfloor\sqrt{n}\rfloor
  5. O ( n ) O(\sqrt{n})
  6. c n / 2 - 1 c_{n/2-1}
  7. s n / 2 s_{n/2}
  8. s n - 1 s_{n-1}

Carry-skip_adder.html

  1. ( a i , b i ) (a_{i},b_{i})
  2. n n
  3. τ C R A ( n ) n τ V A \tau_{CRA}(n)\approx n\cdot\tau_{VA}
  4. ( a i , b i ) (a_{i},b_{i})
  5. p i = a i b i p_{i}=a_{i}\oplus b_{i}
  6. c 0 c_{0}
  7. p i p_{i}
  8. c n c_{n}
  9. c 0 c_{0}
  10. c o u t c_{out}
  11. s = p n - 1 p n - 2 p 1 p 0 = p [ 0 : n - 1 ] s=p_{n-1}\wedge p_{n-2}\wedge\dots\wedge p_{1}\wedge p_{0}=p_{[0:n-1]}
  12. s n - 1 s_{n-1}
  13. n n
  14. n n
  15. τ C S A ( n ) = τ C R A ( n ) \tau_{CSA}(n)=\tau_{CRA}(n)
  16. m m
  17. T S K = T A N D ( m ) + T M U X T_{SK}=T_{AND}(m)+T_{MUX}
  18. T C S K = T M U X = 2 D T_{CSK}=T_{MUX}=2D
  19. A = ( a n - 1 , a n - 2 , , a 1 , a 0 ) A=(a_{n-1},a_{n-2},\dots,a_{1},a_{0})
  20. B = ( b n - 1 , b n - 2 , , b 1 , b 0 ) B=(b_{n-1},b_{n-2},\dots,b_{1},b_{0})
  21. k k
  22. ( m k , m k - 1 , , m 2 , m 1 ) (m_{k},m_{k-1},\dots,m_{2},m_{1})
  23. n + 1 n+1
  24. m + n m+n
  25. k = n m k=\frac{n}{m}
  26. T F C S A ( n ) = T C R A [ 0 : c o u t ] ( m ) + T C S K + ( k - 2 ) T C S K + T C R A ( m ) = 3 D + m 2 D + ( k - 1 ) 2 D + ( m + 2 ) 2 D = ( 2 m + k ) 2 D + 5 D T_{FCSA}(n)=T_{CRA_{[0:c_{out}]}}(m)+T_{CSK}+(k-2)\cdot T_{CSK}+T_{CRA}(m)=3D+% m\cdot 2D+(k-1)\cdot 2D+(m+2)2D=(2m+k)\cdot 2D+5D
  27. d T F C S A ( n ) d m = 0 \frac{dT_{FCSA}(n)}{dm}=0
  28. 2 D ( 2 - n 1 m 2 ) = 0 2D\cdot\left(2-n\cdot\frac{1}{m^{2}}\right)=0
  29. m 1 , 2 = ± n 2 \Rightarrow m_{1,2}=\pm\sqrt{\frac{n}{2}}
  30. m = n 2 \Rightarrow m=\sqrt{\frac{n}{2}}
  31. p [ i : i + 3 ] p_{[i:i+3]}
  32. p [ i : i + 15 ] = p [ i : i + 3 ] p [ i + 4 : i + 7 ] p [ i + 8 : i + 11 ] p [ i + 12 : i + 15 ] p_{[i:i+15]}=p_{[i:i+3]}\wedge p_{[i+4:i+7]}\wedge p_{[i+8:i+11]}\wedge p_{[i+% 12:i+15]}
  33. C 1 C_{1}
  34. C 1 C_{1}
  35. C 1 C_{1}
  36. X 2 X_{2}
  37. X 3 X_{3}

Cartan's_criterion.html

  1. 𝔤 \mathfrak{g}
  2. K ( u , v ) = tr ( ad ( u ) ad ( v ) ) , K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),
  3. 𝔤 \mathfrak{g}
  4. T r ( a b ) = 0 Tr(ab)=0
  5. a 𝔤 , b [ 𝔤 , 𝔤 ] . a\in\mathfrak{g},b\in[\mathfrak{g},\mathfrak{g}].
  6. T r ( a b ) = 0 Tr(ab)=0
  7. 𝔤 \mathfrak{g}
  8. K ( 𝔤 , [ 𝔤 , 𝔤 ] ) = 0 K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0
  9. 𝔤 \mathfrak{g}

Cartesian_product_of_graphs.html

  1. \square
  2. \square
  3. \square
  4. \square
  5. \square
  6. \square
  7. \square
  8. \square
  9. \square
  10. \square
  11. ( K 2 ) n = Q n . (K_{2})^{\square n}=Q_{n}.
  12. \square
  13. \square
  14. \square
  15. \square
  16. \square
  17. \square
  18. \square
  19. G 1 G_{1}
  20. n 1 n_{1}
  21. n 1 × n 1 n_{1}\times n_{1}
  22. 𝐀 1 \mathbf{A}_{1}
  23. G 2 G_{2}
  24. n 2 n_{2}
  25. n 2 × n 2 n_{2}\times n_{2}
  26. 𝐀 2 \mathbf{A}_{2}
  27. 𝐀 1 2 = 𝐀 1 𝐈 n 2 + 𝐈 n 1 𝐀 2 \mathbf{A}_{1\square 2}=\mathbf{A}_{1}\otimes\mathbf{I}_{n_{2}}+\mathbf{I}_{n_% {1}}\otimes\mathbf{A}_{2}
  28. \otimes
  29. 𝐈 n \mathbf{I}_{n}
  30. n × n n\times n

Car–Parrinello_method.html

  1. = 1 2 ( I nuclei M I 𝐑 ˙ I 2 + μ i orbitals d 𝐫 | ψ ˙ i ( 𝐫 , t ) | 2 ) - E [ { ψ i } , { 𝐑 I } ] , \mathcal{L}=\frac{1}{2}\left(\sum_{I}^{\mathrm{nuclei}}\ M_{I}\dot{\mathbf{R}}% _{I}^{2}+\mu\sum_{i}^{\mathrm{orbitals}}\int d\mathbf{r}\ |\dot{\psi}_{i}(% \mathbf{r},t)|^{2}\right)-E\left[\{\psi_{i}\},\{\mathbf{R}_{I}\}\right],
  2. d 𝐫 ψ i * ( 𝐫 , t ) ψ j ( 𝐫 , t ) = δ i j , \int d\mathbf{r}\ \psi_{i}^{*}(\mathbf{r},t)\psi_{j}(\mathbf{r},t)=\delta_{ij},
  3. M I 𝐑 ¨ I = - I E [ { ψ i } , { 𝐑 J } ] M_{I}\ddot{\mathbf{R}}_{I}=-\nabla_{I}\,E\left[\{\psi_{i}\},\{\mathbf{R}_{J}\}\right]
  4. μ ψ ¨ i ( 𝐫 , t ) = - δ E δ ψ i * ( 𝐫 , t ) + j Λ i j ψ j ( 𝐫 , t ) , \mu\ddot{\psi}_{i}(\mathbf{r},t)=-\frac{\delta E}{\delta\psi_{i}^{*}(\mathbf{r% },t)}+\sum_{j}\Lambda_{ij}\psi_{j}(\mathbf{r},t),

Cascode.html

  1. A v = g 21 = v out v in | i o u t = 0 {A_{\mathrm{v}}}=g_{21}=\begin{matrix}{v_{\mathrm{out}}\over v_{\mathrm{in}}}% \end{matrix}\Big|_{i_{out}=0}
  2. - g m 2 ( r π 1 / / r O2 ) ( g m 1 r O1 + 1 ) {-g_{m2}(r_{{\pi}1}//r_{\mathrm{O}2})\left(g_{m1}r_{\mathrm{O}1}+1\right)}
  3. R in = 1 g 11 = v i n i i n | i o u t = 0 R_{\mathrm{in}}=\begin{matrix}\frac{1}{g_{11}}\end{matrix}=\begin{matrix}\frac% {v_{in}}{i_{in}}\end{matrix}\Big|_{i_{out}=0}
  4. r π 2 r_{\pi 2}
  5. R out = g 22 = v o u t i o u t | v i n = 0 R_{\mathrm{out}}=g_{22}=\begin{matrix}\frac{v_{out}}{i_{out}}\end{matrix}\Big|% _{v_{in}=0}
  6. r O1 + ( g m 1 r O1 + 1 ) ( r π 1 / / r O2 ) r_{\mathrm{O}1}+\left(g_{m1}r_{\mathrm{O}1}+1\right)(r_{{\pi}1}//r_{\mathrm{O}% 2})
  7. I B 0 I_{B}\to 0
  8. \rArr \rArr
  9. r π = V T I B r_{\pi}=\begin{matrix}\frac{V_{T}}{I_{B}}\end{matrix}\to\infty
  10. A v = g 21 = v out v in | i o u t = 0 {A_{\mathrm{v}}}=g_{21}=\begin{matrix}{v_{\mathrm{out}}\over v_{\mathrm{in}}}% \end{matrix}\Big|_{i_{out}=0}
  11. - ( g m1 r O1 + 1 ) g m2 r O2 {-(g_{\mathrm{m1}}r_{\mathrm{O1}}+1)g_{\mathrm{m2}}r_{\mathrm{O2}}}
  12. R in = 1 g 11 = v i n i i n | i o u t = 0 R_{\mathrm{in}}=\begin{matrix}\frac{1}{g_{11}}\end{matrix}=\begin{matrix}\frac% {v_{in}}{i_{in}}\end{matrix}\Big|_{i_{out}=0}
  13. \infty
  14. R out = g 22 = v o u t i o u t | v i n = 0 R_{\mathrm{out}}=g_{22}=\begin{matrix}\frac{v_{out}}{i_{out}}\end{matrix}\Big|% _{v_{in}=0}
  15. ( r O1 + r O2 ) ( 1 + g m1 ( r O1 / / r O2 ) ) \left(r_{\mathrm{O1}}+r_{\mathrm{O2}}\right)\left(1+g_{\mathrm{m1}}(r_{\mathrm% {O1}}//r_{\mathrm{O2}})\right)
  16. g m r O = I C V T V A + V C E I C = V A + V C E V T g_{m}\ r_{O}=\begin{matrix}\frac{I_{C}}{V_{T}}\frac{V_{A}+V_{CE}}{I_{C}}\end{% matrix}=\begin{matrix}\frac{V_{A}+V_{CE}}{V_{T}}\end{matrix}
  17. g m r O = 2 I D V G S - V t h 1 / λ + V D S I D = 2 ( 1 / λ + V D S ) V G S - V t h g_{m}\ r_{O}=\begin{matrix}\frac{2I_{D}}{V_{GS}-V_{th}}\frac{1/\lambda+V_{DS}}% {I_{D}}\end{matrix}=\begin{matrix}\frac{2(1/\lambda+V_{DS})}{V_{GS}-V_{th}}% \end{matrix}
  18. υ i n = υ s R i n R S + R i n {\upsilon}_{in}={\upsilon}_{s}\begin{matrix}\frac{R_{in}}{R_{S}+R_{in}}\end{matrix}
  19. υ o u t = A v υ i n R L R L + R o u t {\upsilon}_{out}=A_{v}\ {\upsilon}_{in}\begin{matrix}\frac{R_{L}}{R_{L}+R_{out% }}\end{matrix}
  20. υ o u t = A v υ i n R L R L + R o u t A v υ i n R L R o u t = A v R o u t υ i n R L - g m 2 R L υ i n {\upsilon}_{out}=A_{v}\ {\upsilon}_{in}\begin{matrix}\frac{R_{L}}{R_{L}+R_{out% }}\approx A_{v}\ {\upsilon}_{in}\frac{R_{L}}{R_{out}}=\frac{A_{v}}{R_{out}}\ {% \upsilon}_{in}R_{L}\approx-g_{m2}R_{L}{\upsilon}_{in}\end{matrix}

Cash_on_cash_return.html

  1. cash-on-cash return = annual before-tax cash flow total cash invested \mbox{cash-on-cash return}~{}=\frac{\mbox{annual before-tax cash flow}~{}}{% \mbox{total cash invested}~{}}
  2. $ 60,000 $ 300,000 = 0.20 = 20 % \frac{\$\ \mbox{60,000}~{}}{\$\ \mbox{300,000}~{}}=0.20=20\%
  3. $ 36,000 $ 300,000 = 0.12 = 12 % \frac{\$\ \mbox{36,000}~{}}{\$\ \mbox{300,000}~{}}=0.12=12\%

Cassegrain_reflector.html

  1. R 1 = - 2 D F F - B R_{1}=-\frac{2DF}{F-B}
  2. R 2 = - 2 D B F - B - D R_{2}=-\frac{2DB}{F-B-D}
  3. F F
  4. B B
  5. D D
  6. B B
  7. D D
  8. f 1 f_{1}
  9. b b
  10. D = f 1 ( F - b ) / ( F + f 1 ) D=f_{1}(F-b)/(F+f_{1})
  11. B = D + b B=D+b
  12. K 1 = - 1 K_{1}=-1
  13. K 2 K_{2}
  14. K 2 = - 1 - α - α ( α + 2 ) K_{2}=-1-\alpha-\sqrt{\alpha(\alpha+2)}
  15. α = 1 2 [ 4 D B M ( F + B M - D M ) ( F - B - D ) ] 2 \alpha=\frac{1}{2}\left[\frac{4DBM}{(F+BM-DM)(F-B-D)}\right]^{2}
  16. M = ( F - B ) / D M=(F-B)/D

Cassie's_law.html

  1. cos θ c = f 1 cos θ 1 + f 2 cos θ 2 , \cos\theta_{c}=f_{1}\cos\theta_{1}+f_{2}\cos\theta_{2},
  2. cos θ c = f 1 ( cos θ 1 + 1 ) - 1 , \cos\theta_{c}=\mathit{f}_{1}(\cos\theta_{1}+1)-1,

Cassini_oval.html

  1. dist ( q 1 , p ) dist ( q 2 , p ) = b 2 . \operatorname{dist}(q_{1},p)\cdot\operatorname{dist}(q_{2},p)=b^{2}.\,
  2. ( ( x - a ) 2 + y 2 ) ( ( x + a ) 2 + y 2 ) = b 4 . ((x-a)^{2}+y^{2})((x+a)^{2}+y^{2})=b^{4}.\,
  3. ( x 2 + y 2 ) 2 - 2 a 2 ( x 2 - y 2 ) + a 4 = b 4 . (x^{2}+y^{2})^{2}-2a^{2}(x^{2}-y^{2})+a^{4}=b^{4}.\,
  4. r 4 - 2 a 2 r 2 cos 2 θ = b 4 - a 4 . r^{4}-2a^{2}r^{2}\cos 2\theta=b^{4}-a^{4}.\,
  5. 1 < e < 2 1<e<\sqrt{2}
  6. e 2 e\geq\sqrt{2}
  7. \infty
  8. ( ± a 1 - e 4 , 0 ) ( e < 1 ) (\pm a\sqrt{1-e^{4}},0)\quad(e<1)
  9. ( 0 , ± a e 4 - 1 ) ( e > 1 ) . (0,\pm a\sqrt{e^{4}-1})\quad(e>1).
  10. L 2 = { c : abs ( c 2 + c ) = E R } L_{2}=\{c:\operatorname{abs}(c^{2}+c)=ER\}\,

Category:Subgroup_properties.html

  1. G G
  2. G G^{\prime}
  3. H H
  4. G G
  5. H H^{\prime}
  6. H H
  7. G G
  8. H H^{\prime}
  9. G G^{\prime}

Cathetus.html

  1. c 1 / c 2 c_{1}/c_{2}
  2. c 2 c_{2}
  3. c 1 c_{1}

Cauchy's_equation.html

  1. n ( λ ) = B + C λ 2 + D λ 4 + , n(\lambda)=B+\frac{C}{\lambda^{2}}+\frac{D}{\lambda^{4}}+\cdots,
  2. n ( λ ) = B + C λ 2 , n(\lambda)=B+\frac{C}{\lambda^{2}},

Cauchy's_theorem_(group_theory).html

  1. X = { ( x 1 , , x p ) G p : x 1 x 2 x p = e } X=\{\,(x_{1},\cdots,x_{p})\in G^{p}:x_{1}x_{2}...x_{p}=e\,\}
  2. ( x 1 , x 2 , , x p ) ( x 2 , , x p , x 1 ) (x_{1},x_{2},\ldots,x_{p})\mapsto(x_{2},\ldots,x_{p},x_{1})

Cauchy_formula_for_repeated_integration.html

  1. f ( - n ) ( x ) = a x a σ 1 a σ n - 1 f ( σ n ) d σ n d σ 2 d σ 1 f^{(-n)}(x)=\int_{a}^{x}\int_{a}^{\sigma_{1}}\cdots\int_{a}^{\sigma_{n-1}}f(% \sigma_{n})\,\mathrm{d}\sigma_{n}\cdots\,\mathrm{d}\sigma_{2}\,\mathrm{d}% \sigma_{1}
  2. f ( - n ) ( x ) = 1 ( n - 1 ) ! a x ( x - t ) n - 1 f ( t ) d t f^{(-n)}(x)=\frac{1}{(n-1)!}\int_{a}^{x}\left(x-t\right)^{n-1}f(t)\,\mathrm{d}t
  3. d d x f ( - 1 ) ( x ) = d d x a x f ( t ) d t = f ( x ) \frac{\mathrm{d}}{\mathrm{d}x}f^{(-1)}(x)=\frac{\mathrm{d}}{\mathrm{d}x}\int_{% a}^{x}f(t)\,\mathrm{d}t=f(x)
  4. f ( - 1 ) ( a ) = a a f ( t ) d t = 0 f^{(-1)}(a)=\int_{a}^{a}f(t)\,\mathrm{d}t=0
  5. f - ( n + 1 ) ( x ) = a x a σ 1 a σ n f ( σ n + 1 ) d σ n + 1 d σ 2 d σ 1 = 1 ( n - 1 ) ! a x a σ 1 ( σ 1 - t ) n - 1 f ( t ) d t d σ 1 = 1 ( n - 1 ) ! a x t x ( σ 1 - t ) n - 1 f ( t ) d σ 1 d t = 1 n ! a x ( x - t ) n f ( t ) d t \begin{aligned}\displaystyle f^{-(n+1)}(x)&\displaystyle=\int_{a}^{x}\int_{a}^% {\sigma_{1}}\cdots\int_{a}^{\sigma_{n}}f(\sigma_{n+1})\,\mathrm{d}\sigma_{n+1}% \cdots\,\mathrm{d}\sigma_{2}\,\mathrm{d}\sigma_{1}\\ &\displaystyle=\frac{1}{(n-1)!}\int_{a}^{x}\int_{a}^{\sigma_{1}}\left(\sigma_{% 1}-t\right)^{n-1}f(t)\,\mathrm{d}t\,\mathrm{d}\sigma_{1}\\ &\displaystyle=\frac{1}{(n-1)!}\int_{a}^{x}\int_{t}^{x}\left(\sigma_{1}-t% \right)^{n-1}f(t)\,\mathrm{d}\sigma_{1}\,\mathrm{d}t\\ &\displaystyle=\frac{1}{n!}\int_{a}^{x}\left(x-t\right)^{n}f(t)\,\mathrm{d}t% \end{aligned}

Cauchy_index.html

  1. I s r = { + 1 , if lim x s r ( x ) = - lim x s r ( x ) = + , - 1 , if lim x s r ( x ) = + lim x s r ( x ) = - , 0 , otherwise. I_{s}r=\begin{cases}+1,&\,\text{if }\displaystyle\lim_{x\uparrow s}r(x)=-% \infty\;\land\;\lim_{x\downarrow s}r(x)=+\infty,\\ -1,&\,\text{if }\displaystyle\lim_{x\uparrow s}r(x)=+\infty\;\land\;\lim_{x% \downarrow s}r(x)=-\infty,\\ 0,&\,\text{otherwise.}\end{cases}
  2. I s I_{s}
  3. I a b r I_{a}^{b}r
  4. [ - , + ] [-\infty,+\infty]
  5. r ( x ) = 4 x 3 - 3 x 16 x 5 - 20 x 3 + 5 x = p ( x ) q ( x ) . r(x)=\frac{4x^{3}-3x}{16x^{5}-20x^{3}+5x}=\frac{p(x)}{q(x)}.
  6. x 1 = 0.9511 x_{1}=0.9511
  7. x 2 = 0.5878 x_{2}=0.5878
  8. x 3 = 0 x_{3}=0
  9. x 4 = - 0.5878 x_{4}=-0.5878
  10. x 5 = - 0.9511 x_{5}=-0.9511
  11. x j = cos ( ( 2 i - 1 ) π / 2 n ) x_{j}=\cos((2i-1)\pi/2n)
  12. j = 1 , , 5 j=1,...,5
  13. I x 1 r = I x 2 r = 1 I_{x_{1}}r=I_{x_{2}}r=1
  14. I x 4 r = I x 5 r = - 1 I_{x_{4}}r=I_{x_{5}}r=-1
  15. I x 3 r = 0 I_{x_{3}}r=0
  16. I - 1 1 r = 0 = I - + r I_{-1}^{1}r=0=I_{-\infty}^{+\infty}r

Cauchy_surface.html

  1. 𝒮 \mathcal{S}
  2. D + ( 𝒮 ) D^{+}(\mathcal{S})
  3. 𝒮 \mathcal{S}
  4. D + ( 𝒮 ) := { p such that every inextensible, past-directed, non-spacelike curve through p intersects 𝒮 ) D^{+}(\mathcal{S}):=\{p\in\mathcal{M}\ \ \,\text{such that every inextensible,% past-directed, non-spacelike curve through }p\,\text{ intersects }\mathcal{S})
  5. D - ( 𝒮 ) D^{-}(\mathcal{S})
  6. 𝒮 \mathcal{S}
  7. D + D^{+}
  8. D - D^{-}
  9. 𝒮 \mathcal{S}
  10. 𝒮 \mathcal{S}
  11. 𝒮 \mathcal{S}
  12. D + ( 𝒮 ) 𝒮 D - ( 𝒮 ) = D^{+}(\mathcal{S})\cup\mathcal{S}\cup D^{-}(\mathcal{S})=\mathcal{M}
  13. 𝒮 \mathcal{S}
  14. t t
  15. D + ( 𝒮 ) 𝒮 D - ( 𝒮 ) D^{+}(\mathcal{S})\cup\mathcal{S}\cup D^{-}(\mathcal{S})\not=\mathcal{M}
  16. D ± ( 𝒮 ) D^{\pm}(\mathcal{S})
  17. 𝒮 \mathcal{S}

Cayley–Bacharach_theorem.html

  1. C C
  2. C C
  3. 3 × 3 = 9 3×3=9
  4. d d
  5. d d
  6. d d
  7. d d
  8. ( d + 1 ) ( d + 2 ) 2 - 1 = d 2 + 3 d 2 . \frac{(d+1)(d+2)}{2}-1=\frac{d^{2}+3d}{2}.
  9. d = 3 d=3
  10. d < s u p > 2 d<sup> 2

CBC-MAC.html

  1. m m
  2. m m
  3. m 1 m 2 m x m_{1}\|m_{2}\|\cdots\|m_{x}
  4. k k
  5. E E
  6. ( m , (m,
  7. t ) t)
  8. ( m , (m^{\prime},
  9. t ) t^{\prime})
  10. m ′′ m^{\prime\prime}
  11. t t^{\prime}
  12. m m^{\prime}
  13. t t
  14. m m
  15. m m^{\prime}
  16. m ′′ = m [ ( m 1 t ) m 2 m x ] m^{\prime\prime}=m\|[(m_{1}^{\prime}\oplus t)\|m_{2}^{\prime}\|\dots\|m_{x}^{% \prime}]
  17. m ′′ m^{\prime\prime}
  18. m m
  19. t t
  20. E K MAC ( m 1 t ) E_{K\text{MAC}}(m_{1}^{\prime}\oplus t)
  21. m m
  22. E K MAC ( m 1 t t ) = E K MAC ( m 1 ) E_{K\text{MAC}}(m_{1}^{\prime}\oplus t\oplus t)=E_{K\text{MAC}}(m_{1}^{\prime})
  23. m ′′ m^{\prime\prime}
  24. t t^{\prime}
  25. CBC-MAC-ELB ( m , ( k 1 , k 2 ) ) = E ( k 2 , CBC-MAC ( k 1 , m ) ) \,\text{CBC-MAC-ELB}(m,(k_{1},k_{2}))=E(k_{2},\,\text{CBC-MAC}(k_{1},m))
  26. k k
  27. C = C 1 | | C 2 | | | | C n C=C_{1}\ ||\ C_{2}\ ||\ \dots\ ||\ C_{n}
  28. C 1 , , C n - 1 C_{1},\dots,C_{n-1}
  29. C n C_{n}
  30. C = C 1 | | C 2 | | | | C n C=C_{1}\ ||\ C_{2}\ ||\ \cdots\ ||\ C_{n}
  31. C 1 , , C n - 1 C_{1},\dots,C_{n-1}
  32. C n C_{n}
  33. C = C 1 | | | | C n - 1 | | C n C^{\prime}=C_{1}^{\prime}\ ||\ \dots\ ||\ C_{n-1}^{\prime}\ ||\ C_{n}
  34. K K
  35. P n P_{n}^{\prime}
  36. P n P_{n}
  37. C n C_{n}
  38. C n - 1 C n - 1 C_{n-1}^{\prime}\not=C_{n-1}
  39. P n P_{n}^{\prime}
  40. C n C_{n}
  41. P n = C n - 1 E K - 1 ( C n ) P_{n}^{\prime}=C_{n-1}^{\prime}\oplus E_{K}^{-1}(C_{n})
  42. t t^{\prime}
  43. t = E K ( P n E K ( P n - 1 E K ( E K ( P 1 ) ) ) ) t^{\prime}=E_{K}(P_{n}^{\prime}\oplus E_{K}(P_{n-1}^{\prime}\oplus E_{K}(\dots% \oplus E_{K}(P_{1}^{\prime}))))
  44. t = E K ( P n C n - 1 ) t^{\prime}=E_{K}(P_{n}^{\prime}\oplus C_{n-1}^{\prime})
  45. C n C_{n}
  46. t = E K ( C n - 1 E K - 1 ( C n ) C n - 1 ) = E K ( E K - 1 ( C n ) ) = C n t^{\prime}=E_{K}(C_{n-1}^{\prime}\oplus E_{K}^{-1}(C_{n})\oplus C_{n-1}^{% \prime})=E_{K}(E_{K}^{-1}(C_{n}))=C_{n}
  47. t = C n = t t^{\prime}=C_{n}=t
  48. P P^{\prime}
  49. P P^{\prime}
  50. P P
  51. P P P\not=P^{\prime}
  52. K 1 K_{1}
  53. K 2 K_{2}
  54. C i C_{i}^{\prime}
  55. P i P_{i}^{\prime}
  56. K 2 K_{2}
  57. P i P_{i}^{\prime}
  58. K 2 K_{2}
  59. M A C i C i MAC_{i}\not=C_{i}^{\prime}
  60. P 1 I V P_{1}\oplus IV
  61. M 1 = P 1 | P 2 | M_{1}=P_{1}|P_{2}|\dots
  62. I V 1 IV_{1}
  63. E K ( I V 1 P 1 ) E_{K}(IV_{1}\oplus P_{1})
  64. ( M 1 , T 1 ) (M_{1},T_{1})
  65. M 2 = P 1 | P 2 | M_{2}=P_{1}^{\prime}|P_{2}|\dots
  66. P 1 P_{1}^{\prime}
  67. I V 1 IV_{1}^{\prime}
  68. E K ( P 1 I V 1 ) E_{K}(P_{1}^{\prime}\oplus IV_{1}^{\prime})
  69. M 1 M_{1}

Cellular_homology.html

  1. X X
  2. X n X_{n}
  3. H n + 1 ( X n + 1 , X n ) H n ( X n , X n - 1 ) H n - 1 ( X n - 1 , X n - 2 ) , \cdots\to{H_{n+1}}(X_{n+1},X_{n})\to{H_{n}}(X_{n},X_{n-1})\to{H_{n-1}}(X_{n-1}% ,X_{n-2})\to\cdots,
  4. X - 1 X_{-1}
  5. H n ( X n , X n - 1 ) {H_{n}}(X_{n},X_{n-1})
  6. n n
  7. X X
  8. e n α e_{n}^{\alpha}
  9. n n
  10. X X
  11. χ n α : e n α 𝕊 n - 1 X n - 1 \chi_{n}^{\alpha}:\partial e_{n}^{\alpha}\cong\mathbb{S}^{n-1}\to X_{n-1}
  12. χ n α β : 𝕊 n - 1 e n α χ n α X n - 1 q X n - 1 / ( X n - 1 e n - 1 β ) 𝕊 n - 1 , \chi_{n}^{\alpha\beta}:\mathbb{S}^{n-1}\,\stackrel{\cong}{\longrightarrow}\,% \partial e_{n}^{\alpha}\,\stackrel{\chi_{n}^{\alpha}}{\longrightarrow}\,X_{n-1% }\,\stackrel{q}{\longrightarrow}\,X_{n-1}/\left(X_{n-1}\setminus e_{n-1}^{% \beta}\right)\,\stackrel{\cong}{\longrightarrow}\,\mathbb{S}^{n-1},
  13. 𝕊 n - 1 \mathbb{S}^{n-1}
  14. e n α \partial e_{n}^{\alpha}
  15. Φ n α \Phi_{n}^{\alpha}
  16. e n α e_{n}^{\alpha}
  17. e n - 1 β e_{n-1}^{\beta}
  18. ( n - 1 ) (n-1)
  19. q q
  20. X n - 1 e n - 1 β X_{n-1}\setminus e_{n-1}^{\beta}
  21. e n - 1 β e_{n-1}^{\beta}
  22. 𝕊 n - 1 \mathbb{S}^{n-1}
  23. X n - 1 / ( X n - 1 e n - 1 β ) X_{n-1}/\left(X_{n-1}\setminus e_{n-1}^{\beta}\right)
  24. 𝕊 n - 1 \mathbb{S}^{n-1}
  25. Φ n - 1 β \Phi_{n-1}^{\beta}
  26. e n - 1 β e_{n-1}^{\beta}
  27. d n : H n ( X n , X n - 1 ) H n - 1 ( X n - 1 , X n - 2 ) d_{n}:{H_{n}}(X_{n},X_{n-1})\to{H_{n-1}}(X_{n-1},X_{n-2})
  28. d n ( e n α ) = β deg ( χ n α β ) e n - 1 β , {d_{n}}(e_{n}^{\alpha})=\sum_{\beta}\deg\left(\chi_{n}^{\alpha\beta}\right)e_{% n-1}^{\beta},
  29. deg ( χ n α β ) \deg\left(\chi_{n}^{\alpha\beta}\right)
  30. χ n α β \chi_{n}^{\alpha\beta}
  31. ( n - 1 ) (n-1)
  32. X X
  33. H n - 1 ( X n - 1 , X n - 2 ) {H_{n-1}}(X_{n-1},X_{n-2})
  34. n n
  35. H k ( X ) H k ( X n ) {H_{k}}(X)\cong{H_{k}}(X_{n})
  36. k < n k<n
  37. n \mathbb{CP}^{n}
  38. 0 k n 0\leq k\leq n
  39. H 2 k ( n ; ) {H_{2k}}(\mathbb{CP}^{n};\mathbb{Z})\cong\mathbb{Z}
  40. H 2 k + 1 ( n ; ) = 0. {H_{2k+1}}(\mathbb{CP}^{n};\mathbb{Z})=0.
  41. X X
  42. X j X_{j}
  43. j j
  44. c j c_{j}
  45. j j
  46. H j ( X j , X j - 1 ) {H_{j}}(X_{j},X_{j-1})
  47. X X
  48. χ ( X ) = j = 0 n ( - 1 ) j c j . \chi(X)=\sum_{j=0}^{n}(-1)^{j}c_{j}.
  49. X X
  50. χ ( X ) = j = 0 n ( - 1 ) j Rank ( H j ( X ) ) . \chi(X)=\sum_{j=0}^{n}(-1)^{j}\operatorname{Rank}({H_{j}}(X)).
  51. ( X n , X n - 1 , ) (X_{n},X_{n-1},\varnothing)
  52. H i ( X n - 1 , ) H i ( X n , ) H i ( X n , X n - 1 ) . \cdots\to{H_{i}}(X_{n-1},\varnothing)\to{H_{i}}(X_{n},\varnothing)\to{H_{i}}(X% _{n},X_{n-1})\to\cdots.
  53. i = 0 n ( - 1 ) i Rank ( H i ( X n , ) ) = i = 0 n ( - 1 ) i Rank ( H i ( X n , X n - 1 ) ) + i = 0 n ( - 1 ) i Rank ( H i ( X n - 1 , ) ) . \sum_{i=0}^{n}(-1)^{i}\operatorname{Rank}({H_{i}}(X_{n},\varnothing))=\sum_{i=% 0}^{n}(-1)^{i}\operatorname{Rank}({H_{i}}(X_{n},X_{n-1}))+\sum_{i=0}^{n}(-1)^{% i}\operatorname{Rank}({H_{i}}(X_{n-1},\varnothing)).
  54. ( X n - 1 , X n - 2 , ) (X_{n-1},X_{n-2},\varnothing)
  55. ( X n - 2 , X n - 3 , ) (X_{n-2},X_{n-3},\varnothing)
  56. i = 0 n ( - 1 ) i Rank ( H i ( X n , ) ) = j = 0 n i = 0 j ( - 1 ) i Rank ( H i ( X j , X j - 1 ) ) = j = 0 n ( - 1 ) j c j . \sum_{i=0}^{n}(-1)^{i}\;\operatorname{Rank}({H_{i}}(X_{n},\varnothing))=\sum_{% j=0}^{n}\sum_{i=0}^{j}(-1)^{i}\operatorname{Rank}({H_{i}}(X_{j},X_{j-1}))=\sum% _{j=0}^{n}(-1)^{j}c_{j}.

Cem_Yıldırım.html

  1. lim inf n p n + 1 - p n log p n = 0 \liminf_{n\to\infty}\frac{p_{n+1}-p_{n}}{\log p_{n}}=0

Center_of_percussion.html

  1. F = M d V d t , F=M\frac{dV}{dt},
  2. F b = I d ω d t , Fb=I\frac{d\omega}{dt},
  3. ω \omega
  4. v = V - A ω , v=V-A\omega\,,
  5. d v d t = ( 1 M - A b I ) F . \frac{dv}{dt}=\left(\frac{1}{M}-\frac{Ab}{I}\right)F.
  6. v = ( 1 M - A b I ) F d t . v=\left(\frac{1}{M}-\frac{Ab}{I}\right)\int Fdt.
  7. v = 0 v=0
  8. b = I A M . b=\frac{I}{AM}.
  9. I = M L 2 12 I=\frac{ML^{2}}{12}
  10. b = L 2 12 A b=\frac{L^{2}}{12A}

Central_binomial_coefficient.html

  1. ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 for all n 0. {2n\choose n}=\frac{(2n)!}{(n!)^{2}}\,\text{ for all }n\geq 0.
  2. 1 1 - 4 x = 1 + 2 x + 6 x 2 + 20 x 3 + 70 x 4 + 252 x 5 + . \frac{1}{\sqrt{1-4x}}=1+2x+6x^{2}+20x^{3}+70x^{4}+252x^{5}+\cdots.
  3. ( 2 n n ) 4 n π n as n . {2n\choose n}\sim\frac{4^{n}}{\sqrt{\pi n}}\,\text{ as }n\rightarrow\infty.
  4. 2 π \sqrt{2\pi}
  5. 4 n 2 n + 1 ( 2 n n ) 4 n for all n 1 \frac{4^{n}}{2n+1}\leq{2n\choose n}\leq 4^{n}\,\text{ for all }n\geq 1
  6. 4 n 4 n ( 2 n n ) 4 n 3 n + 1 for all n 1 \frac{4^{n}}{\sqrt{4n}}\leq{2n\choose n}\leq\frac{4^{n}}{\sqrt{3n+1}}\,\text{ % for all }n\geq 1
  7. ( 2 n n ) = 4 n π n ( 1 - c n n ) where 1 9 < c n < 1 8 {2n\choose n}=\frac{4^{n}}{\sqrt{\pi n}}\left(1-\frac{c_{n}}{n}\right)\,\text{% where }\frac{1}{9}<c_{n}<\frac{1}{8}
  8. n 1. n\geq 1.
  9. C n = 1 n + 1 ( 2 n n ) = ( 2 n n ) - ( 2 n n + 1 ) for all n 0. C_{n}=\frac{1}{n+1}{2n\choose n}={2n\choose n}-{2n\choose n+1}\,\text{ for all% }n\geq 0.
  10. Γ ( 2 n + 1 ) Γ ( n + 1 ) 2 = 1 n B ( n + 1 , n ) \frac{\Gamma(2n+1)}{\Gamma(n+1)^{2}}=\frac{1}{nB(n+1,n)}
  11. Γ ( x ) \Gamma(x)
  12. B ( x , y ) B(x,y)

Central_field_approximation.html

  1. U ( r ) U(r)

Centrifugal_pump.html

  1. H d = U 2 2 2 g H_{d}=\frac{U_{2}^{2}}{2g}
  2. ω = d ϕ d t \omega=\frac{d\phi}{dt}
  3. F c = m ω 2 R F_{c}=m\omega^{2}R
  4. H s = g h H_{s}=gh
  5. H s = a c 1 + a c 2 2 R 2 - R 1 H_{s}=\frac{a_{c}1+a_{c}2}{2}R_{2}-R_{1}
  6. H s = ω 2 R 1 + ω 2 R 2 2 R 2 - R 1 = U 2 2 - U 1 2 2 H_{s}=\frac{\omega^{2}R_{1}+\omega^{2}R_{2}}{2}R_{2}-R_{1}=\frac{U_{2}^{2}-U_{% 1}^{2}}{2}
  7. a 1 = 1971 m / s 2 = 201 g a_{1}=1971m/s^{2}=201g
  8. a 2 = 804 g a_{2}=804g
  9. H = U 2 2 2 g + U 2 2 - U 1 2 2 H=\frac{U_{2}^{2}}{2g}+\frac{U_{2}^{2}-U_{1}^{2}}{2}
  10. ρ Q ( c 2 u . r 2 - c 1 u . r 1 ) = M + M τ \rho Q(c_{2}u.r_{2}-c_{1}u.r_{1})=M+M_{\tau}
  11. Y t h . g = H t = c 2 u . u 2 - c 1 u . u 1 Yth.g=H_{t}=c_{2}u.u_{2}-c_{1}u.u_{1}
  12. Y t h = 1 / 2 ( u 2 2 - u 1 2 + w 1 2 - w 2 2 + c 2 2 - c 1 2 ) Yth=1/2(u_{2}^{2}-u_{1}^{2}+w_{1}^{2}-w_{2}^{2}+c_{2}^{2}-c_{1}^{2})
  13. η = ρ . g Q H P m \eta=\frac{\rho.gQH}{P_{m}}
  14. P m P_{m}
  15. ρ \rho
  16. g g
  17. H H
  18. Q Q
  19. η \eta
  20. H H
  21. η p u m p \eta_{pump}
  22. P i P_{i}
  23. P i = ρ g H Q η P_{i}=\cfrac{\rho\ g\ H\ Q}{\eta}
  24. P i P_{i}
  25. ρ \rho
  26. g g
  27. H H
  28. Q Q
  29. η \eta
  30. H H
  31. η p u m p \eta_{pump}

Cerebral_perfusion_pressure.html

  1. C B F = C P P / C V R CBF=CPP/CVR
  2. C P P = M A P - I C P CPP=MAP-ICP
  3. C P P = M A P - I C P CPP=MAP-ICP
  4. C P P = M A P - J V P CPP=MAP-JVP

Change_of_variables.html

  1. x 6 - 9 x 3 + 8 = 0. x^{6}-9x^{3}+8=0.\,
  2. ( x 3 ) 2 - 9 ( x 3 ) + 8 = 0 (x^{3})^{2}-9(x^{3})+8=0
  3. u 3 \sqrt[3]{u}
  4. u 2 - 9 u + 8 = 0 , u^{2}-9u+8=0,
  5. u = 1 and u = 8. u=1\quad\,\text{and}\quad u=8.
  6. x 3 = 1 and x 3 = 8. x^{3}=1\quad\,\text{and}\quad x^{3}=8.
  7. x = ( 1 ) 1 / 3 = 1 and x = ( 8 ) 1 / 3 = 2. x=(1)^{1/3}=1\quad\,\text{and}\quad x=(8)^{1/3}=2.
  8. x y + x + y = 71 xy+x+y=71
  9. x 2 y + x y 2 = 880 x^{2}y+xy^{2}=880
  10. x x
  11. y y
  12. x > y x>y
  13. x y ( x + y ) = 880 xy(x+y)=880
  14. s = x + y , t = x y s=x+y,t=xy
  15. s + t = 71 , s t = 880. s+t=71,st=880.
  16. ( s , t ) = ( 16 , 55 ) (s,t)=(16,55)
  17. ( s , t ) = ( 55 , 16 ) . (s,t)=(55,16).
  18. x + y = 16 , x y = 55 x+y=16,xy=55
  19. ( x , y ) = ( 11 , 5 ) . (x,y)=(11,5).
  20. x + y = 55 , x y = 16 x+y=55,xy=16
  21. ( x , y ) = ( 11 , 5 ) (x,y)=(11,5)
  22. A A
  23. B B
  24. Φ : A B \Phi:A\rightarrow B
  25. C r C^{r}
  26. Φ \Phi
  27. r r
  28. A A
  29. B B
  30. r r
  31. B B
  32. A A
  33. r r
  34. \infty
  35. ω \omega
  36. Φ \Phi
  37. C r C^{r}
  38. Φ \Phi
  39. x = Φ ( y ) x=\Phi(y)
  40. x x
  41. y y
  42. Φ \Phi
  43. y y
  44. x x
  45. U ( x , y , z ) := ( x 2 + y 2 ) 1 - x 2 x 2 + y 2 = 0. U(x,y,z):=(x^{2}+y^{2})\sqrt{1-\frac{x^{2}}{x^{2}+y^{2}}}=0.
  46. ( x , y , z ) = Φ ( r , θ , z ) \displaystyle(x,y,z)=\Phi(r,\theta,z)
  47. Φ ( r , θ , z ) = ( r cos ( θ ) , r sin ( θ ) , z ) \displaystyle\Phi(r,\theta,z)=(r\cos(\theta),r\sin(\theta),z)
  48. θ \theta
  49. 2 π 2\pi
  50. [ 0 , 2 π ] [0,2\pi]
  51. Φ \Phi
  52. Φ \Phi
  53. ( 0 , ] × [ 0 , 2 π ) × [ - , ] (0,\infty]\times[0,2\pi)\times[-\infty,\infty]
  54. r = 0 r=0
  55. Φ \Phi
  56. θ \theta
  57. Φ \Phi
  58. sin 2 x + cos 2 x = 1 \sin^{2}x+\cos^{2}x=1
  59. V ( r , θ , z ) = r 2 1 - r 2 cos 2 θ r 2 = r 2 1 - cos 2 θ = r 2 | sin θ | V(r,\theta,z)=r^{2}\sqrt{1-\frac{r^{2}\cos^{2}\theta}{r^{2}}}=r^{2}\sqrt{1-% \cos^{2}\theta}=r^{2}\left|\sin\theta\right|
  60. sin ( θ ) = 0 \sin(\theta)=0
  61. θ = 0 \theta=0
  62. θ = π \theta=\pi
  63. Φ \Phi
  64. y = 0 y=0
  65. x 0 x\not=0
  66. y = 0 y=0
  67. r = 0 r=0
  68. Φ \Phi
  69. x , y , z \reals x,y,z\in\reals
  70. d d x ( sin ( x 2 ) ) \frac{d}{dx}\left(\sin(x^{2})\right)\,
  71. d d x = d d u d u d x = d d x ( u ) d d u = d d x ( x 2 ) d d u = 2 x d d u \frac{d}{dx}=\frac{d}{du}\frac{du}{dx}=\frac{d}{dx}\left(u\right)\frac{d}{du}=% \frac{d}{dx}\left(x^{2}\right)\frac{d}{du}=2x\frac{d}{du}\,
  72. d d x ( sin ( x 2 ) ) = 2 x d d u ( sin ( u ) ) = 2 x cos ( x 2 ) \frac{d}{dx}\left(\sin(x^{2})\right)=2x\frac{d}{du}\left(\sin(u)\right)=2x\cos% (x^{2})\,
  73. d n y d x n = y scale x scale n d n y ^ d x ^ n \frac{d^{n}y}{dx^{n}}=\frac{y\text{scale}}{x\text{scale}^{n}}\frac{d^{n}\hat{y% }}{d\hat{x}^{n}}
  74. x = x ^ x scale + x shift x=\hat{x}x\text{scale}+x\text{shift}
  75. y = y ^ y scale + y shift . y=\hat{y}y\text{scale}+y\text{shift}.
  76. μ d 2 u d y 2 = d p d x ; u ( 0 ) = u ( L ) = 0 \mu\frac{d^{2}u}{dy^{2}}=\frac{dp}{dx}\quad;\quad u(0)=u(L)=0
  77. d p / d x dp/dx
  78. d 2 u ^ d y ^ 2 = 1 ; u ^ ( 0 ) = u ^ ( 1 ) = 0 \frac{d^{2}\hat{u}}{d\hat{y}^{2}}=1\quad;\quad\hat{u}(0)=\hat{u}(1)=0
  79. y = y ^ L and u = u ^ L 2 μ d p d x . y=\hat{y}L\qquad\,\text{and}\qquad u=\hat{u}\frac{L^{2}}{\mu}\frac{dp}{dx}.
  80. m v ˙ = - H x m\dot{v}=-\frac{\partial H}{\partial x}
  81. m x ˙ = H v m\dot{x}=\frac{\partial H}{\partial v}
  82. H ( x , v ) H(x,v)
  83. Φ ( p ) = 1 / m v \Phi(p)=1/m\cdot v
  84. \mathbb{R}
  85. \mathbb{R}
  86. v = Φ ( p ) v=\Phi(p)
  87. p ˙ = - H x \dot{p}=-\frac{\partial H}{\partial x}
  88. x ˙ = H p \dot{x}=\frac{\partial H}{\partial p}
  89. ϕ ( t , x , v ) \phi(t,x,v)
  90. m x ¨ = ϕ ( t , x , v ) m\ddot{x}=\phi(t,x,v)
  91. x = Ψ ( t , y ) x=\Psi(t,y)
  92. v = Ψ ( t , y ) t + Ψ ( t , y ) y w v=\frac{\partial\Psi(t,y)}{\partial t}+\frac{\partial\Psi(t,y)}{\partial y}\cdot w
  93. L y = d d t L w \frac{\partial{L}}{\partial y}=\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial{L}% }{\partial{w}}
  94. L = T - V L=T-V

Channel_length_modulation.html

  1. I D I_{D}
  2. K n K^{\prime}_{n}
  3. V G S V_{GS}
  4. V t h V_{th}
  5. V D S V_{DS}
  6. V D S , s a t = V G S - V t h V_{DS,sat}=V_{GS}-V_{th}
  7. V t h V_{th}
  8. r O = 1 + λ V D S λ I D = 1 / λ + V D S I D = V E L + V D S I D r_{O}=\begin{matrix}\frac{1+\lambda V_{DS}}{\lambda I_{D}}\end{matrix}=\begin{% matrix}\frac{1/\lambda+V_{DS}}{I_{D}}\end{matrix}=\begin{matrix}\frac{V_{E}L+V% _{DS}}{I_{D}}\end{matrix}
  9. V D S V_{DS}
  10. I D I_{D}
  11. λ \lambda
  12. λ \lambda
  13. Δ L V E L \begin{matrix}\frac{\Delta L}{V_{E}L}\end{matrix}

Chapman_function.html

  1. s e c ( z ) , sec(z),

Character_sum.html

  1. Σ χ ( n ) \Sigma\chi(n)\,
  2. Σ \Sigma
  3. Σ = O ( N ) \Sigma=O(N)
  4. Σ = O ( N log N ) . \Sigma=O(\sqrt{N}\log N).
  5. Σ = O ( N log log N ) . \Sigma=O(\sqrt{N}\log\log N).
  6. Σ χ ( F ( n ) ) \Sigma\chi(F(n))\,
  7. F ( n ) = n ( n + 1 ) F(n)=n(n+1)\,
  8. O ( p ) . O(\sqrt{p}).
  9. Σ p 1 / 2 log p , Σ 2 R 1 / 2 p 3 / 16 log p , Σ r R 1 - 1 / r p ( r + 1 ) / 4 r 2 ( log p ) 1 / 2 r \begin{aligned}\displaystyle\Sigma&\displaystyle\ll p^{1/2}\log p,\\ \displaystyle\Sigma&\displaystyle\ll 2R^{1/2}p^{3/16}\log p,\\ \displaystyle\Sigma&\displaystyle\ll rR^{1-1/r}p^{(r+1)/4r^{2}}(\log p)^{1/2r}% \end{aligned}

Characteristic_energy_length_scale.html

  1. χ \chi

Characterizations_of_the_category_of_topological_spaces.html

  1. A cl ( A ) A\subseteq\operatorname{cl}(A)\!
  2. cl ( cl ( A ) ) = cl ( A ) \operatorname{cl}(\operatorname{cl}(A))=\operatorname{cl}(A)\!
  3. cl ( A B ) = cl ( A ) cl ( B ) \operatorname{cl}(A\cup B)=\operatorname{cl}(A)\cup\operatorname{cl}(B)\!
  4. cl ( ) = \operatorname{cl}(\varnothing)=\varnothing\!
  5. f : ( X , cl ) ( X , cl ) f:(X,\operatorname{cl})\to(X^{\prime},\operatorname{cl}^{\prime})
  6. A A
  7. X X
  8. f ( cl ( A ) ) cl ( f ( A ) ) f(\operatorname{cl}(A))\subset\operatorname{cl}^{\prime}(f(A))
  9. A int ( A ) A\supseteq\operatorname{int}(A)\!
  10. int ( int ( A ) ) = int ( A ) \operatorname{int}(\operatorname{int}(A))=\operatorname{int}(A)\!
  11. int ( A B ) = int ( A ) int ( B ) \operatorname{int}(A\cap B)=\operatorname{int}(A)\cap\operatorname{int}(B)\!
  12. int ( X ) = X \operatorname{int}(X)=X\!
  13. f : ( X , int ) ( X , int ) f:(X,\operatorname{int})\to(X^{\prime},\operatorname{int}^{\prime})
  14. A A
  15. X X^{\prime}
  16. f - 1 ( int ( A ) ) int ( f - 1 ( A ) ) f^{-1}(\operatorname{int}^{\prime}(A))\subset\operatorname{int}(f^{-1}(A))
  17. x A : x close A x\|A:\iff x\,\text{ close }A
  18. x A f ( x ) f ( A ) x\|A\Rightarrow f(x)\|f(A)
  19. x A x cl ( A ) x\|A\iff x\in\operatorname{cl}(A)

Charge_conservation.html

  1. Q ( t 2 ) = Q ( t 1 ) + Q IN - Q OUT . Q(t_{2})\ =\ Q(t_{1})+Q_{\rm{IN}}-Q_{\rm{OUT}}.
  2. Q ( t ) Q(t)
  3. t t
  4. ρ ρ
  5. 𝐉 \mathbf{J}
  6. ρ t + 𝐉 = 0. \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0.
  7. ρ ρ
  8. 𝐉 \mathbf{J}
  9. I = - S 𝐉 d 𝐒 I=-\iint\limits_{S}\mathbf{J}\cdot d\mathbf{S}
  10. S = V S=∂V
  11. V V
  12. d 𝐒 d\mathbf{S}
  13. 𝐍 d S \mathbf{N}dS
  14. V ∂V
  15. 𝐉 \mathbf{J}
  16. I = - V ( 𝐉 ) d V . I=-\iiint\limits_{V}\left(\nabla\cdot\mathbf{J}\right)dV.
  17. d q d t = - V ( 𝐉 ) d V . \frac{dq}{dt}=-\iiint\limits_{V}\left(\nabla\cdot\mathbf{J}\right)dV.
  18. q = V ρ d V . q=\iiint\limits_{V}\rho dV.
  19. 0 = V ( ρ t + 𝐉 ) d V . 0=\iiint\limits_{V}\left(\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}% \right)dV.
  20. ρ t + 𝐉 = 0. \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0.
  21. ϕ \phi
  22. 𝐀 \mathbf{A}
  23. χ \chi
  24. ϕ = ϕ - χ t 𝐀 = 𝐀 + χ . \phi^{\prime}=\phi-\frac{\partial\chi}{\partial t}\qquad\qquad\mathbf{A}^{% \prime}=\mathbf{A}+\nabla\chi.
  25. ψ = e i χ ψ \psi^{\prime}=e^{i\chi}\psi
  26. | ψ | 2 |\psi|^{2}

Charge_density.html

  1. λ q = d Q d , \lambda_{q}=\frac{dQ}{d\ell}\,,\quad
  2. σ q = d Q d S , \sigma_{q}=\frac{dQ}{dS}\,,\quad
  3. ρ q = d Q d V , \rho_{q}=\frac{dQ}{dV}\,,\quad
  4. Q = L λ q ( r ) d Q=\int\limits_{L}\lambda_{q}({r})\,d\ell
  5. Q = S σ q ( r ) d S Q=\int\limits_{S}\sigma_{q}({r})\,dS
  6. Q = V ρ q ( r ) d V Q=\int\limits_{V}\rho_{q}({r})\,dV
  7. λ q = Q , σ q = Q S , ρ q = Q V . \langle\lambda_{q}\rangle=\frac{Q}{\ell}\,,\quad\langle\sigma_{q}\rangle=\frac% {Q}{S}\,,\quad\langle\rho_{q}\rangle=\frac{Q}{V}\,.
  8. ρ = ρ f + ρ b . \rho=\rho_{f}+\rho_{b}\,.
  9. σ = σ f + σ b . \sigma=\sigma_{f}+\sigma_{b}\,.
  10. q b = d 𝐧 ^ | s | q_{b}=\frac{{d}\cdot\mathbf{\hat{n}}}{|{s}|}
  11. d q b = d d | s | 𝐧 ^ dq_{b}=\frac{d{d}}{|{s}|}\cdot\mathbf{\hat{n}}
  12. σ b = d q b d S = d d | s | d S 𝐧 ^ = d d d V 𝐧 ^ = P 𝐧 ^ . \sigma_{b}=\frac{dq_{b}}{dS}=\frac{d{d}}{|{s}|dS}\cdot\mathbf{\hat{n}}=\frac{d% {d}}{dV}\cdot\mathbf{\hat{n}}={P}\cdot\mathbf{\hat{n}}\,.
  13. ρ b = - 𝐏 . \rho_{b}=-\nabla\cdot\mathbf{P}\,.
  14. φ = 1 4 π ϵ 0 ( r - r ) d | r - r | 3 \varphi=\frac{1}{4\pi\epsilon_{0}}\frac{({r}-{r}^{\prime})\cdot{d}}{|{r}-{r}^{% \prime}|^{3}}
  15. d d = P d V = P d 3 r d{d}={P}dV={P}d^{3}{r}
  16. φ = 1 4 π ϵ 0 ( r - r ) P | r - r | 3 d 3 r \varphi=\frac{1}{4\pi\epsilon_{0}}\iiint\frac{({r}-{r}^{\prime})\cdot{P}}{|{r}% -{r}^{\prime}|^{3}}d^{3}{r^{\prime}}
  17. ( 1 | r - r | ) ( e x x + e y y + e z z ) ( 1 | r - r | ) = r - r | r - r | 3 \nabla^{\prime}\left(\frac{1}{|{r}-{r}^{\prime}|}\right)\equiv\left({e}_{x}% \frac{\partial}{\partial x^{\prime}}+{e}_{y}\frac{\partial}{\partial y^{\prime% }}+{e}_{z}\frac{\partial}{\partial z^{\prime}}\right)\left(\frac{1}{|{r}-{r}^{% \prime}|}\right)=\frac{{r}-{r}^{\prime}}{|{r}-{r}^{\prime}|^{3}}
  18. φ = 1 4 π ϵ 0 P ( 1 | r - r | ) d 3 r \varphi=\frac{1}{4\pi\epsilon_{0}}\iiint{P}\cdot\nabla^{\prime}\left(\frac{1}{% |{r}-{r}^{\prime}|}\right)d^{3}{r^{\prime}}
  19. φ = 1 4 π ϵ 0 [ ( P | r - r | ) - 1 r - r ( P ) ] d 3 r \varphi=\frac{1}{4\pi\epsilon_{0}}\iiint\left[\nabla^{\prime}\cdot\left(\frac{% {P}}{|{r}-{r}^{\prime}|}\right)-\frac{1}{{r}-{r}^{\prime}}(\nabla^{\prime}% \cdot{P})\right]d^{3}{r^{\prime}}
  20. σ b = P n ^ , ρ b = - P \sigma_{b}={P}\cdot{\hat{n}}\,,\quad\rho_{b}=-\nabla\cdot{P}
  21. Q = V ρ 0 . Q=V\cdot\rho_{0}.
  22. Q = V ρ q ( r ) d V . Q=\int\limits_{V}\rho_{q}({r})\,dV.
  23. Q = ρ q , 0 V d V = ρ 0 V Q=\rho_{q,0}\int\limits_{V}\,dV=\rho_{0}V
  24. Q = V ρ q , 0 . Q=V\cdot\rho_{q,0}.
  25. ρ q ( r ) = q δ ( 𝐫 - 𝐫 0 ) \rho_{q}({r})=q\delta(\mathbf{r}-\mathbf{r}_{0})
  26. R d 3 𝐫 f ( 𝐫 ) δ ( 𝐫 - 𝐫 0 ) = f ( 𝐫 0 ) \int_{R}d^{3}\mathbf{r}f(\mathbf{r})\delta(\mathbf{r}-\mathbf{r}_{0})=f(% \mathbf{r}_{0})
  27. Q = R d 3 𝐫 ρ q = R d 3 𝐫 q δ ( 𝐫 - 𝐫 0 ) = q R d 3 𝐫 δ ( 𝐫 - 𝐫 0 ) = q Q=\int_{R}d^{3}\mathbf{r}\,\rho_{q}=\int_{R}d^{3}\mathbf{r}\,q\delta(\mathbf{r% }-\mathbf{r}_{0})=q\int_{R}d^{3}\mathbf{r}\,\delta(\mathbf{r}-\mathbf{r}_{0})=q
  28. ρ q ( r ) = i = 1 N q i δ ( 𝐫 - 𝐫 i ) \rho_{q}({r})=\sum_{i=1}^{N}\ q_{i}\delta(\mathbf{r}-\mathbf{r}_{i})\,\!
  29. Q = R d 3 𝐫 i = 1 N q i δ ( 𝐫 - 𝐫 i ) = i = 1 N q i R d 3 𝐫 δ ( 𝐫 - 𝐫 i ) = i = 1 N q i Q=\int_{R}d^{3}\mathbf{r}\sum_{i=1}^{N}\ q_{i}\delta(\mathbf{r}-\mathbf{r}_{i}% )=\sum_{i=1}^{N}\ q_{i}\int_{R}d^{3}\mathbf{r}\delta(\mathbf{r}-\mathbf{r}_{i}% )=\sum_{i=1}^{N}\ q_{i}
  30. ρ q ( r ) = q n ( 𝐫 ) . \rho_{q}({r})=qn(\mathbf{r})\,.
  31. ρ q ( r ) = q | ψ ( 𝐫 ) | 2 \rho_{q}({r})=q|\psi(\mathbf{r})|^{2}
  32. Q = R q | ψ ( 𝐫 ) | 2 d 3 r Q=\int_{R}q|\psi(\mathbf{r})|^{2}\,{\rm d}^{3}{r}

Cheater_bar.html

  1. torque = radius × force \,\text{torque}=\,\text{radius}\times\,\text{force}

Cheletropic_reaction.html

  1. d [ 3 ] d t = k 2 [ 1 ] [ S O 2 ] 2 \frac{d[3]}{dt}=k_{2}[1][SO_{2}]^{2}

Chevalley_scheme.html

  1. X X^{\prime}
  2. 𝒪 x \mathcal{O}_{x}
  3. X = Spec ( A ) X=\mathrm{Spec}(A)
  4. X X^{\prime}
  5. L ( A ) L(A)
  6. X X^{\prime}
  7. M X M\in X^{\prime}
  8. A i A_{i}
  9. X = i L ( A i ) X^{\prime}=\cup_{i}L(A_{i})
  10. A i j A_{ij}
  11. A i A j A_{i}\cup A_{j}
  12. A i A_{i}
  13. M N M\subseteq N
  14. X X^{\prime}
  15. A i A_{i}

Chevalley–Warning_theorem.html

  1. 𝔽 \mathbb{F}
  2. { f j } j = 1 r 𝔽 [ X 1 , , X n ] \{f_{j}\}_{j=1}^{r}\subseteq\mathbb{F}[X_{1},\ldots,X_{n}]
  3. n > j = 1 r d j n>\sum_{j=1}^{r}d_{j}
  4. d j d_{j}
  5. f j f_{j}
  6. f j ( x 1 , , x n ) = 0 for j = 1 , , r . f_{j}(x_{1},\dots,x_{n})=0\quad\,\text{for}\,j=1,\ldots,r.
  7. ( a 1 , , a n ) 𝔽 n (a_{1},\dots,a_{n})\in\mathbb{F}^{n}
  8. p p
  9. 𝔽 \mathbb{F}
  10. { f j } j = 1 r \{f_{j}\}_{j=1}^{r}
  11. 0
  12. p p
  13. ( 0 , , 0 ) 𝔽 n (0,\dots,0)\in\mathbb{F}^{n}
  14. ( a 1 , , a n ) 𝔽 n \ { ( 0 , , 0 ) } (a_{1},\dots,a_{n})\in\mathbb{F}^{n}\backslash\{(0,\dots,0)\}
  15. p p
  16. n n
  17. f j = x j , j = 1 , , n f_{j}=x_{j},j=1,\dots,n
  18. n n
  19. i < p - 1 i<p-1
  20. x 𝔽 x i = 0 \sum_{x\in\mathbb{F}}x^{i}=0
  21. 𝔽 n \mathbb{F}^{n}
  22. x 1 , , x n x_{1},\ldots,x_{n}
  23. n ( p - 1 ) n(p-1)
  24. p p
  25. f 1 , , f r = 0 f_{1},\ldots,f_{r}=0
  26. x 𝔽 n ( 1 - f 1 p - 1 ( x ) ) ( 1 - f r p - 1 ( x ) ) \sum_{x\in\mathbb{F}^{n}}(1-f_{1}^{p-1}(x))\cdot\ldots\cdot(1-f_{r}^{p-1}(x))
  27. f i f_{i}
  28. q b q^{b}
  29. q q
  30. 𝔽 \mathbb{F}
  31. d d
  32. d j d_{j}
  33. b b
  34. n - j d j d . \frac{n-\sum_{j}d_{j}}{d}.
  35. q q

Chiral_perturbation_theory.html

  1. p p
  2. ϵ \epsilon
  3. δ , \delta,
  4. ϵ \epsilon^{\prime}
  5. p p
  6. p p
  7. ( π ) 2 + m π 2 π 2 (\partial\pi)^{2}+m_{\pi}^{2}\pi^{2}
  8. m π 4 π 2 + ( π ) 6 m_{\pi}^{4}\pi^{2}+(\partial\pi)^{6}
  9. U = exp { i F ( π 0 2 π + 2 π - - π 0 ) } U=\exp\left\{\frac{i}{F}\begin{pmatrix}\pi^{0}&\sqrt{2}\pi^{+}\\ \sqrt{2}\pi^{-}&-\pi^{0}\end{pmatrix}\right\}
  10. F = 93 F=93
  11. F F
  12. 2 = F 2 4 tr ( μ U μ U ) + λ F 3 4 tr ( m q U + m q U ) \mathcal{L}_{2}=\frac{F^{2}}{4}{\rm tr}(\partial_{\mu}U\partial^{\mu}U^{% \dagger})+\frac{\lambda F^{3}}{4}{\rm tr}(m_{q}U+m_{q}^{\dagger}U^{\dagger})
  13. F = 93 F=93
  14. m q m_{q}
  15. p p
  16. p Λ χ , m π Λ χ . \frac{p}{\Lambda_{\chi}},\frac{m_{\pi}}{\Lambda_{\chi}}.
  17. Λ χ \Lambda_{\chi}
  18. Λ χ = 4 π F \Lambda_{\chi}=4\pi F
  19. m q m_{q}
  20. 𝒪 ( p 2 ) \mathcal{O}(p^{2})
  21. m π 2 = λ m q F m_{\pi}^{2}=\lambda m_{q}F
  22. 𝒪 ( p 4 ) \mathcal{O}(p^{4})
  23. 𝒪 ( p 4 ) \mathcal{O}(p^{4})
  24. 𝒪 ( p 2 ) \mathcal{O}(p^{2})
  25. 𝒪 ( p 2 ) \mathcal{O}(p^{2})
  26. 𝒪 ( p 4 ) \mathcal{O}(p^{4})
  27. p 4 p^{4}
  28. p - 2 p^{-2}
  29. p 2 p^{2}
  30. 𝒪 ( p 4 ) \mathcal{O}(p^{4})
  31. 𝒪 ( p 4 ) \mathcal{O}(p^{4})
  32. 𝒪 ( p n ) \mathcal{O}(p^{n})
  33. 𝒪 ( p n ) \mathcal{O}(p^{n})

Chiral_superfield.html

  1. x μ x^{\mu}
  2. μ = 0 , , 3 \mu=0,\ldots,3
  3. θ 1 , θ 2 , θ ¯ 1 , θ ¯ 2 \theta^{1},\theta^{2},\bar{\theta}^{1},\bar{\theta}^{2}
  4. D ¯ f = 0 \overline{D}f=0
  5. Q ¯ \overline{Q}
  6. θ \theta
  7. D ¯ α ˙ f = 0 \overline{D}_{\dot{\alpha}}f=0

Chirped_pulse_amplification.html

  1. τ ( ω ) \tau(\omega)
  2. τ \tau
  3. ω \omega
  4. ϕ ( ω ) = 2 π τ ( ω ) c / λ ( ω ) \phi(\omega)=2\pi\tau(\omega)c/\lambda(\omega)
  5. λ \lambda
  6. L < f L<f
  7. 2 f 2f
  8. L L
  9. L < f L<f
  10. L > f L>f
  11. L = f L=f

Choked_flow.html

  1. p * p^{*}
  2. p * p 0 = ( 2 k + 1 ) k k - 1 \frac{p^{*}}{p_{0}}=\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}
  3. k k
  4. c p / c v c_{p}/c_{v}
  5. γ \gamma
  6. p 0 p_{0}
  7. k = 1.4 k=1.4
  8. p * = 0.528 p 0 p^{*}=0.528p_{0}
  9. k k
  10. 0.487 < p * / p 0 < 0.587 0.487<p^{*}/p_{0}<0.587
  11. m ˙ = C d A k ρ 0 P 0 ( 2 k + 1 ) k + 1 k - 1 \dot{m}=C_{d}A\sqrt{k\rho_{0}P_{0}\left(\frac{2}{k+1}\right)^{\frac{k+1}{k-1}}}
  12. m ˙ {\dot{m}}
  13. C d C_{d}
  14. A A
  15. k k
  16. c p / c v c_{p}/c_{v}
  17. c p c_{p}
  18. c v c_{v}
  19. ρ 0 \rho_{0}
  20. P 0 P_{0}
  21. T 0 T_{0}
  22. P 0 P_{0}
  23. k g / ( m . s 2 ) kg/(m.s^{2})
  24. T 0 T_{0}
  25. A A
  26. P P
  27. T T
  28. C d C_{d}
  29. C d A = m ˙ 2 ρ Δ P C_{d}A=\dfrac{\dot{m}}{\sqrt{{2}\rho{\Delta}{P}}}
  30. C d C_{d}
  31. A A
  32. m ˙ \dot{m}
  33. ρ \rho
  34. Δ P \Delta P

Chow_test.html

  1. x = 1.7 x=1.7
  2. [ 0 , 1.7 ] [0,1.7]
  3. [ 1.7 , 4 ] [1.7,4]
  4. y t = a + b x 1 t + c x 2 t + ε . y_{t}=a+bx_{1t}+cx_{2t}+\varepsilon.\,
  5. y t = a 1 + b 1 x 1 t + c 1 x 2 t + ε . y_{t}=a_{1}+b_{1}x_{1t}+c_{1}x_{2t}+\varepsilon.\,
  6. y t = a 2 + b 2 x 1 t + c 2 x 2 t + ε . y_{t}=a_{2}+b_{2}x_{1t}+c_{2}x_{2t}+\varepsilon.\,
  7. a 1 = a 2 a_{1}=a_{2}
  8. b 1 = b 2 b_{1}=b_{2}
  9. c 1 = c 2 c_{1}=c_{2}
  10. ε \varepsilon
  11. S C S_{C}
  12. S 1 S_{1}
  13. S 2 S_{2}
  14. N 1 N_{1}
  15. N 2 N_{2}
  16. k k
  17. ( S C - ( S 1 + S 2 ) ) / ( k ) ( S 1 + S 2 ) / ( N 1 + N 2 - 2 k ) . \frac{(S_{C}-(S_{1}+S_{2}))/(k)}{(S_{1}+S_{2})/(N_{1}+N_{2}-2k)}.
  18. k k
  19. N 1 + N 2 - 2 k N_{1}+N_{2}-2k

Church_encoding.html

  1. n n
  2. f f
  3. f n = f f f n times . f^{n}=\underbrace{f\circ f\circ\cdots\circ f}_{n\,\text{ times}}.\,
  4. 0 f x = x 0\ f\ x=x
  5. 0 = λ f . λ x . x 0=\lambda f.\lambda x.x
  6. 1 f x = f x 1\ f\ x=f\ x
  7. 1 = λ f . λ x . f x 1=\lambda f.\lambda x.f\ x
  8. 2 f x = f ( f x ) 2\ f\ x=f\ (f\ x)
  9. 2 = λ f . λ x . f ( f x ) 2=\lambda f.\lambda x.f\ (f\ x)
  10. 3 f x = f ( f ( f x ) ) 3\ f\ x=f\ (f\ (f\ x))
  11. 3 = λ f . λ x . f ( f ( f x ) ) 3=\lambda f.\lambda x.f\ (f\ (f\ x))
  12. n f x = f n x n\ f\ x=f^{n}\ x
  13. n = λ f . λ x . f n x n=\lambda f.\lambda x.f^{n}\ x
  14. plus ( m , n ) = m + n \operatorname{plus}(m,n)=m+n
  15. f ( m + n ) ( x ) = f m ( f n ( x ) ) f^{(m+n)}(x)=f^{m}(f^{n}(x))
  16. plus λ m . λ n . λ f . λ x . m f ( n f x ) \operatorname{plus}\equiv\lambda m.\lambda n.\lambda f.\lambda x.m\ f\ (n\ f\ x)
  17. succ ( n ) = n + 1 \operatorname{succ}(n)=n+1
  18. ( plus 1 ) (\operatorname{plus}\ 1)
  19. succ λ n . λ f . λ x . f ( n f x ) \operatorname{succ}\equiv\lambda n.\lambda f.\lambda x.f\ (n\ f\ x)
  20. mult ( m , n ) = m * n \operatorname{mult}(m,n)=m*n
  21. f ( m * n ) ( x ) = ( f n ) m ( x ) f^{(m*n)}(x)=(f^{n})^{m}(x)
  22. mult λ m . λ n . λ f . m ( n f ) \operatorname{mult}\equiv\lambda m.\lambda n.\lambda f.m\ (n\ f)
  23. exp ( m , n ) = m n \operatorname{exp}(m,n)=m^{n}
  24. n f x = f n x n\ f\ x=f^{n}\ x
  25. f m , x f f\to m,x\to f
  26. n m f = m n f n\ m\ f=m^{n}\ f
  27. exp m n = m n = n m \operatorname{exp}\ m\ n=m^{n}=n\ m
  28. exp λ m . λ n . n m \operatorname{exp}\equiv\lambda m.\lambda n.n\ m
  29. pred ( n ) \operatorname{pred}(n)
  30. pred λ n . λ f . λ x . n ( λ g . λ h . h ( g f ) ) ( λ u . x ) ( λ u . u ) \operatorname{pred}\equiv\lambda n.\lambda f.\lambda x.n\ (\lambda g.\lambda h% .h\ (g\ f))\ (\lambda u.x)\ (\lambda u.u)
  31. minus λ m . λ n . ( n pred ) m \operatorname{minus}\equiv\lambda m.\lambda n.(n\operatorname{pred})\ m
  32. n + 1 n+1
  33. f n + 1 x = f ( f n x ) f^{n+1}\ x=f(f^{n}x)
  34. succ n f x = f ( n f x ) \operatorname{succ}\ n\ f\ x=f\ (n\ f\ x)
  35. λ n . λ f . λ x . f ( n f x ) \lambda n.\lambda f.\lambda x.f\ (n\ f\ x)
  36. m + n m+n
  37. f m + n x = f m ( f n x ) f^{m+n}\ x=f^{m}(f^{n}x)
  38. plus m n f x = m f ( n f x ) \operatorname{plus}\ m\ n\ f\ x=m\ f\ (n\ f\ x)
  39. λ m . λ n . λ f . λ x . m f ( n f x ) \lambda m.\lambda n.\lambda f.\lambda x.m\ f\ (n\ f\ x)
  40. λ m . λ n . n succ m \lambda m.\lambda n.n\operatorname{succ}m
  41. m * n m*n
  42. f m * n x = ( f m ) n x f^{m*n}\ x=(f^{m})^{n}\ x
  43. multiply m n f x = m ( n f ) x \operatorname{multiply}\ m\ n\ f\ x=m\ (n\ f)\ x
  44. λ m . λ n . λ f . λ x . m ( n f ) x \lambda m.\lambda n.\lambda f.\lambda x.m\ (n\ f)\ x
  45. λ m . λ n . λ f . m ( n f ) \lambda m.\lambda n.\lambda f.m\ (n\ f)
  46. m n m^{n}
  47. n m f = m n f n\ m\ f=m^{n}\ f
  48. exp m n f x = ( n m ) f x \operatorname{exp}\ m\ n\ f\ x=(n\ m)\ f\ x
  49. λ m . λ n . λ f . λ x . ( n m ) f x \lambda m.\lambda n.\lambda f.\lambda x.(n\ m)\ f\ x
  50. λ m . λ n . n m \lambda m.\lambda n.n\ m
  51. n - 1 n-1
  52. inc n con = val ( f n - 1 x ) \operatorname{inc}^{n}\operatorname{con}=\operatorname{val}(f^{n-1}x)
  53. pred \operatorname{pred}
  54. λ n . λ f . λ x . n ( λ g . λ h . h ( g f ) ) ( λ u . x ) ( λ u . u ) \lambda n.\lambda f.\lambda x.n\ (\lambda g.\lambda h.h\ (g\ f))\ (\lambda u.x% )\ (\lambda u.u)
  55. m - n m-n
  56. f m - n x = ( f - 1 ) n ( f m x ) f^{m-n}\ x=(f^{-1})^{n}(f^{m}x)
  57. minus m n = ( n pred ) m \operatorname{minus}\ m\ n=(n\operatorname{pred})\ m
  58. λ m . λ n . n pred m \lambda m.\lambda n.n\operatorname{pred}m
  59. pred ( 0 ) = 0 \operatorname{pred}(0)=0
  60. m < n m - n = 0 m<n\to m-n=0
  61. true λ a . λ b . a \operatorname{true}\equiv\lambda a.\lambda b.a
  62. false λ a . λ b . b \operatorname{false}\equiv\lambda a.\lambda b.b
  63. predicate x then - clause else - clause \operatorname{predicate}\ x\ \operatorname{then-clause}\ \operatorname{else-clause}
  64. and = λ p . λ q . p q p \operatorname{and}=\lambda p.\lambda q.p\ q\ p
  65. or = λ p . λ q . p p q \operatorname{or}=\lambda p.\lambda q.p\ p\ q
  66. not 1 = λ p . λ a . λ b . p b a \operatorname{not}_{1}=\lambda p.\lambda a.\lambda b.p\ b\ a
  67. not 2 = λ p . p ( λ a . λ b . b ) ( λ a . λ b . a ) = λ p . p false true \operatorname{not}_{2}=\lambda p.p\ (\lambda a.\lambda b.b)\ (\lambda a.% \lambda b.a)=\lambda p.p\operatorname{false}\operatorname{true}
  68. xor = λ a . λ b . a ( not b ) b \operatorname{xor}=\lambda a.\lambda b.a\ (\operatorname{not}\ b)\ b
  69. if = λ p . λ a . λ b . p a b \operatorname{if}=\lambda p.\lambda a.\lambda b.p\ a\ b
  70. and true false = ( λ p . λ q . p q p ) true false = true false true = ( λ a . λ b . a ) false true = false \operatorname{and}\operatorname{true}\operatorname{false}=(\lambda p.\lambda q% .p\ q\ p)\ \operatorname{true}\ \operatorname{false}=\operatorname{true}% \operatorname{false}\operatorname{true}=(\lambda a.\lambda b.a)\operatorname{% false}\operatorname{true}=\operatorname{false}
  71. or true false = ( λ p . λ q . p p q ) ( λ a . λ b . a ) ( λ a . λ b . b ) = ( λ a . λ b . a ) ( λ a . λ b . a ) ( λ a . λ b . b ) = ( λ a . λ b . a ) = true \operatorname{or}\operatorname{true}\operatorname{false}=(\lambda p.\lambda q.% p\ p\ q)\ (\lambda a.\lambda b.a)\ (\lambda a.\lambda b.b)=(\lambda a.\lambda b% .a)\ (\lambda a.\lambda b.a)\ (\lambda a.\lambda b.b)=(\lambda a.\lambda b.a)=% \operatorname{true}
  72. not 1 true = ( λ p . λ a . λ b . p b a ) ( λ a . λ b . a ) = λ a . λ b . ( λ a . λ b . a ) b a = λ a . λ b . ( λ x . b ) a = λ a . λ b . b = false \operatorname{not}_{1}\ \operatorname{true}=(\lambda p.\lambda a.\lambda b.p\ % b\ a)(\lambda a.\lambda b.a)=\lambda a.\lambda b.(\lambda a.\lambda b.a)\ b\ a% =\lambda a.\lambda b.(\lambda x.b)\ a=\lambda a.\lambda b.b=\operatorname{false}
  73. not 2 true = ( λ p . p ( λ a . λ b . b ) ( λ a . λ b . a ) ) ( λ a . λ b . a ) = ( λ a . λ b . a ) ( λ a . λ b . b ) ( λ a . λ b . a ) = ( λ b . ( λ a . λ b . b ) ) ( λ a . λ b . a ) = λ a . λ b . b = false \operatorname{not}_{2}\ \operatorname{true}=(\lambda p.p\ (\lambda a.\lambda b% .b)(\lambda a.\lambda b.a))(\lambda a.\lambda b.a)=(\lambda a.\lambda b.a)(% \lambda a.\lambda b.b)(\lambda a.\lambda b.a)=(\lambda b.(\lambda a.\lambda b.% b))\ (\lambda a.\lambda b.a)=\lambda a.\lambda b.b=\operatorname{false}
  74. IsZero \operatorname{IsZero}
  75. true \operatorname{true}
  76. 0
  77. false \operatorname{false}
  78. IsZero = λ n . n ( λ x . false ) true \operatorname{IsZero}=\lambda n.n\ (\lambda x.\operatorname{false})\ % \operatorname{true}
  79. LEQ = λ m . λ n . IsZero ( minus m n ) \operatorname{LEQ}=\lambda m.\lambda n.\operatorname{IsZero}\ (\operatorname{% minus}\ m\ n)
  80. x = y ( x y and y x ) x=y\equiv(x<=y\and y<=x)
  81. EQ = λ m . λ n . and ( LEQ m n ) ( LEQ n m ) \operatorname{EQ}=\lambda m.\lambda n.\operatorname{and}\ (\operatorname{LEQ}% \ m\ n)\ (\operatorname{LEQ}\ n\ m)
  82. pair λ x . λ y . λ z . z x y \operatorname{pair}\equiv\lambda x.\lambda y.\lambda z.z\ x\ y
  83. first λ p . p ( λ x . λ y . x ) \operatorname{first}\equiv\lambda p.p\ (\lambda x.\lambda y.x)
  84. second λ p . p ( λ x . λ y . y ) \operatorname{second}\equiv\lambda p.p\ (\lambda x.\lambda y.y)
  85. first ( pair a b ) \operatorname{first}\ (\operatorname{pair}\ a\ b)
  86. = ( λ p . p ( λ x . λ y . x ) ) ( ( λ x . λ y . λ z . z x y ) a b ) =(\lambda p.p\ (\lambda x.\lambda y.x))\ ((\lambda x.\lambda y.\lambda z.z\ x% \ y)\ a\ b)
  87. = ( λ p . p ( λ x . λ y . x ) ) ( λ z . z a b ) =(\lambda p.p\ (\lambda x.\lambda y.x))\ (\lambda z.z\ a\ b)
  88. = ( λ z . z a b ) ( λ x . λ y . x ) =(\lambda z.z\ a\ b)\ (\lambda x.\lambda y.x)
  89. = ( λ x . λ y . x ) a b = a =(\lambda x.\lambda y.x)\ a\ b=a
  90. nil pair true true \operatorname{nil}\equiv\operatorname{pair}\ \operatorname{true}\ % \operatorname{true}
  91. isnil first \operatorname{isnil}\equiv\operatorname{first}
  92. cons λ h . λ t . pair false ( pair h t ) \operatorname{cons}\equiv\lambda h.\lambda t.\operatorname{pair}\operatorname{% false}\ (\operatorname{pair}h\ t)
  93. head λ z . first ( second z ) \operatorname{head}\equiv\lambda z.\operatorname{first}\ (\operatorname{second% }z)
  94. tail λ z . second ( second z ) \operatorname{tail}\equiv\lambda z.\operatorname{second}\ (\operatorname{% second}z)
  95. cons pair \operatorname{cons}\equiv\operatorname{pair}
  96. head first \operatorname{head}\equiv\operatorname{first}
  97. tail second \operatorname{tail}\equiv\operatorname{second}
  98. nil false \operatorname{nil}\equiv\operatorname{false}
  99. isnil λ l . l ( λ h . λ t . λ d . false ) true \operatorname{isnil}\equiv\lambda l.l(\lambda h.\lambda t.\lambda d.% \operatorname{false})\operatorname{true}
  100. process - list λ l . l ( λ h . λ t . λ d . head - and - tail - clause ) nil - clause \operatorname{process-list}\equiv\lambda l.l(\lambda h.\lambda t.\lambda d.% \operatorname{head-and-tail-clause})\operatorname{nil-clause}
  101. nil λ c . λ n . n \operatorname{nil}\equiv\lambda c.\lambda n.n
  102. isnil λ l . l ( λ h . λ t . false ) true \operatorname{isnil}\equiv\lambda l.l\ (\lambda h.\lambda t.\operatorname{% false})\ \operatorname{true}
  103. cons λ h . λ t . λ c . λ n . c h ( t c n ) \operatorname{cons}\equiv\lambda h.\lambda t.\lambda c.\lambda n.c\ h\ (t\ c\ n)
  104. head λ l . l ( λ h . λ t . h ) false \operatorname{head}\equiv\lambda l.l\ (\lambda h.\lambda t.h)\ \operatorname{false}
  105. tail λ l . λ c . λ n . l ( λ h . λ t . λ g . g h ( t c ) ) ( λ t . n ) ( λ h . λ t . t ) \operatorname{tail}\equiv\lambda l.\lambda c.\lambda n.l\ (\lambda h.\lambda t% .\lambda g.g\ h\ (t\ c))\ (\lambda t.n)\ (\lambda h.\lambda t.t)
  106. pred ( n ) = { 0 if n = 0 , n - 1 otherwise \operatorname{pred}(n)=\begin{cases}0&\mbox{if }~{}n=0,\\ n-1&\mbox{otherwise}\end{cases}
  107. init = value x \operatorname{init}=\operatorname{value}\ x
  108. inc init = value ( f x ) \operatorname{inc}\ \operatorname{init}=\operatorname{value}\ (f\ x)
  109. inc const = value x \operatorname{inc}\ \operatorname{const}=\operatorname{value}\ x
  110. inc ( inc init ) = value ( f ( f x ) ) \operatorname{inc}\ (\operatorname{inc}\ \operatorname{init})=\operatorname{% value}\ (f\ (f\ x))
  111. inc ( inc const ) = value ( f x ) \operatorname{inc}\ (\operatorname{inc}\ \operatorname{const})=\operatorname{% value}\ (f\ x)
  112. inc ( inc ( inc init ) ) = value ( f ( f ( f x ) ) ) \operatorname{inc}\ (\operatorname{inc}\ (\operatorname{inc}\ \operatorname{% init}))=\operatorname{value}\ (f\ (f\ (f\ x)))
  113. inc ( inc ( inc const ) ) = value ( f ( f x ) ) \operatorname{inc}\ (\operatorname{inc}\ (\operatorname{inc}\ \operatorname{% const}))=\operatorname{value}\ (f\ (f\ x))
  114. n inc init = value ( f n x ) = value ( n f x ) n\operatorname{inc}\ \operatorname{init}=\operatorname{value}\ (f^{n}\ x)=% \operatorname{value}\ (n\ f\ x)
  115. n inc const = value ( f n - 1 x ) = value ( ( n - 1 ) f x ) n\operatorname{inc}\ \operatorname{const}=\operatorname{value}\ (f^{n-1}\ x)=% \operatorname{value}\ ((n-1)\ f\ x)
  116. inc ( value v ) = value ( f v ) \operatorname{inc}\ (\operatorname{value}\ v)=\operatorname{value}\ (f\ v)
  117. extract ( value v ) = v \operatorname{extract}\ (\operatorname{value}\ v)=v
  118. samenum = λ n . λ f . λ x . extract ( n inc init ) = λ n . λ f . λ x . extract ( value ( n f x ) ) = λ n . λ f . λ x . n f x = λ n . n \operatorname{samenum}=\lambda n.\lambda f.\lambda x.\operatorname{extract}\ (% n\operatorname{inc}\operatorname{init})=\lambda n.\lambda f.\lambda x.% \operatorname{extract}\ (\operatorname{value}\ (n\ f\ x))=\lambda n.\lambda f.% \lambda x.n\ f\ x=\lambda n.n
  119. pred = λ n . λ f . λ x . extract ( n inc const ) = λ n . λ f . λ x . extract ( value ( ( n - 1 ) f x ) ) = λ n . λ f . λ x . ( n - 1 ) f x = λ n . ( n - 1 ) \operatorname{pred}=\lambda n.\lambda f.\lambda x.\operatorname{extract}\ (n% \operatorname{inc}\operatorname{const})=\lambda n.\lambda f.\lambda x.% \operatorname{extract}\ (\operatorname{value}\ ((n-1)\ f\ x))=\lambda n.% \lambda f.\lambda x.(n-1)\ f\ x=\lambda n.(n-1)
  120. value \operatorname{value}
  121. extract k \operatorname{extract}\ k
  122. inc \operatorname{inc}
  123. init \operatorname{init}
  124. const \operatorname{const}
  125. λ v . ( λ h . h v ) \lambda v.(\lambda h.h\ v)
  126. k λ u . u k\ \lambda u.u
  127. λ g . λ h . h ( g f ) \lambda g.\lambda h.h\ (g\ f)
  128. λ h . h x \lambda h.h\ x
  129. λ u . x \lambda u.x
  130. pred = λ n . λ f . λ x . n ( λ g . λ h . h ( g f ) ) ( λ u . x ) ( λ u . u ) \operatorname{pred}=\lambda n.\lambda f.\lambda x.n\ (\lambda g.\lambda h.h\ (% g\ f))\ (\lambda u.x)\ (\lambda u.u)
  131. value v h = h v \operatorname{value}\ v\ h=h\ v
  132. value = λ v . ( λ h . h v ) \operatorname{value}=\lambda v.(\lambda h.h\ v)
  133. inc ( value v ) = value ( f v ) \operatorname{inc}\ (\operatorname{value}\ v)=\operatorname{value}\ (f\ v)
  134. g = value v g=\operatorname{value}\ v
  135. g f = value v f = f v g\ f=\operatorname{value}\ v\ f=f\ v
  136. inc g = value ( g f ) \operatorname{inc}\ g=\operatorname{value}\ (g\ f)
  137. inc = λ g . λ h . h ( g f ) \operatorname{inc}=\lambda g.\lambda h.h\ (g\ f)
  138. I = λ u . u I=\lambda u.u
  139. value v I = v \operatorname{value}\ v\ I=v
  140. extract k = k I \operatorname{extract}\ k=k\ I
  141. inc const = value x \operatorname{inc}\ \operatorname{const}=\operatorname{value}\ x
  142. λ h . h ( const f ) = λ h . h x \lambda h.h\ (\operatorname{const}\ f)=\lambda h.h\ x
  143. const f = x \operatorname{const}\ f=x
  144. const = λ u . x \operatorname{const}=\lambda u.x
  145. n / m = if n m then 1 + ( n - m ) / m else 0 n/m=\operatorname{if}\ n>=m\ \operatorname{then}\ 1+(n-m)/m\ \operatorname{% else}\ 0
  146. n - m n-m
  147. IsZero ( minus n m ) \operatorname{IsZero}\ (\operatorname{minus}\ n\ m)
  148. n > m n>m
  149. n m n>=m
  150. divide1 n m f x = ( λ d . IsZero d ( 0 f x ) ( f ( divide1 d m f x ) ) ) ( minus n m ) \operatorname{divide1}\ n\ m\ f\ x=(\lambda d.\operatorname{IsZero}\ d\ (0\ f% \ x)\ (f\ (\operatorname{divide1}\ d\ m\ f\ x)))\ (\operatorname{minus}\ n\ m)
  151. minus n m \operatorname{minus}\ n\ m
  152. ( n - 1 ) / m (n-1)/m
  153. divide n = divide1 ( succ n ) \operatorname{divide}\ n=\operatorname{divide1}\ (\operatorname{succ}\ n)
  154. divide1 div c \operatorname{divide1}\rightarrow\operatorname{div}\ c
  155. divide1 c \operatorname{divide1}\rightarrow c
  156. div = λ c . λ n . λ m . λ f . λ x . ( λ d . IsZero d ( 0 f x ) ( f ( c d m f x ) ) ) ( minus n m ) \operatorname{div}=\lambda c.\lambda n.\lambda m.\lambda f.\lambda x.(\lambda d% .\operatorname{IsZero}\ d\ (0\ f\ x)\ (f\ (c\ d\ m\ f\ x)))\ (\operatorname{% minus}\ n\ m)
  157. divide = λ n . divide1 ( succ n ) \operatorname{divide}=\lambda n.\operatorname{divide1}\ (\operatorname{succ}\ n)
  158. divide1 = Y div \operatorname{divide1}=Y\ \operatorname{div}
  159. succ = λ n . λ f . λ x . f ( n f x ) \operatorname{succ}=\lambda n.\lambda f.\lambda x.f\ (n\ f\ x)
  160. Y = λ f . ( λ x . x x ) ( λ x . f ( x x ) ) Y=\lambda f.(\lambda x.x\ x)\ (\lambda x.f\ (x\ x))
  161. 0 = λ f . λ x . x 0=\lambda f.\lambda x.x
  162. IsZero = λ n . n ( λ x . false ) true \operatorname{IsZero}=\lambda n.n\ (\lambda x.\operatorname{false})\ % \operatorname{true}
  163. true λ a . λ b . a \operatorname{true}\equiv\lambda a.\lambda b.a
  164. false λ a . λ b . b \operatorname{false}\equiv\lambda a.\lambda b.b
  165. minus = λ m . λ n . n pred m \operatorname{minus}=\lambda m.\lambda n.n\operatorname{pred}m
  166. pred = λ n . λ f . λ x . n ( λ g . λ h . h ( g f ) ) ( λ u . x ) ( λ u . u ) \operatorname{pred}=\lambda n.\lambda f.\lambda x.n\ (\lambda g.\lambda h.h\ (% g\ f))\ (\lambda u.x)\ (\lambda u.u)
  167. divide = λ n . ( ( λ f . ( λ x . x x ) ( λ x . f ( x x ) ) ) ( λ c . λ n . λ m . λ f . λ x . ( λ d . ( λ n . n ( λ x . ( λ a . λ b . b ) ) ( λ a . λ b . a ) ) d ( ( λ f . λ x . x ) f x ) ( f ( c d m f x ) ) ) ( ( λ m . λ n . n ( λ n . λ f . λ x . n ( λ g . λ h . h ( g f ) ) ( λ u . x ) ( λ u . u ) ) m ) n m ) ) ) ( ( λ n . λ f . λ x . f ( n f x ) ) n ) \operatorname{divide}=\lambda n.((\lambda f.(\lambda x.x\ x)\ (\lambda x.f\ (x% \ x)))\ (\lambda c.\lambda n.\lambda m.\lambda f.\lambda x.(\lambda d.(\lambda n% .n\ (\lambda x.(\lambda a.\lambda b.b))\ (\lambda a.\lambda b.a))\ d\ ((% \lambda f.\lambda x.x)\ f\ x)\ (f\ (c\ d\ m\ f\ x)))\ ((\lambda m.\lambda n.n(% \lambda n.\lambda f.\lambda x.n\ (\lambda g.\lambda h.h\ (g\ f))\ (\lambda u.x% )\ (\lambda u.u))m)\ n\ m)))\ ((\lambda n.\lambda f.\lambda x.f\ (n\ f\ x))\ n)
  168. λ \lambda
  169. convert s = λ x . pair x 0 \operatorname{convert}_{s}=\lambda x.\operatorname{pair}\ x\ 0
  170. neg s = λ x . pair ( second x ) ( first x ) \operatorname{neg}_{s}=\lambda x.\operatorname{pair}\ (\operatorname{second}\ % x)\ (\operatorname{first}\ x)
  171. OneZero = λ x . IsZero ( first x ) x ( IsZero ( second x ) x ( OneZero pair ( pred ( first x ) ) ( pred ( second x ) ) ) ) \operatorname{OneZero}=\lambda x.\operatorname{IsZero}\ (\operatorname{first}% \ x)\ x\ (\operatorname{IsZero}\ (\operatorname{second}\ x)\ x\ (\operatorname% {OneZero}\ \operatorname{pair}\ (\operatorname{pred}\ (\operatorname{first}\ x% ))\ (\operatorname{pred}\ (\operatorname{second}\ x))))
  172. OneZ = λ c . λ x . IsZero ( first x ) x ( IsZero ( second x ) x ( c pair ( pred ( first x ) ) ( pred ( second x ) ) ) ) \operatorname{OneZ}=\lambda c.\lambda x.\operatorname{IsZero}\ (\operatorname{% first}\ x)\ x\ (\operatorname{IsZero}\ (\operatorname{second}\ x)\ x\ (c\ % \operatorname{pair}\ (\operatorname{pred}\ (\operatorname{first}\ x))\ (% \operatorname{pred}\ (\operatorname{second}\ x))))
  173. OneZero = Y OneZ \operatorname{OneZero}=Y\operatorname{OneZ}
  174. x + y = [ x p , x n ] + [ y p , y n ] = x p - x n + y p - y n = ( x p + y p ) - ( x n + y n ) = [ x p + y p , x n + y n ] x+y=[x_{p},x_{n}]+[y_{p},y_{n}]=x_{p}-x_{n}+y_{p}-y_{n}=(x_{p}+y_{p})-(x_{n}+y% _{n})=[x_{p}+y_{p},x_{n}+y_{n}]
  175. plus s = λ x . λ y . OneZero ( pair ( plus ( first x ) ( first y ) ) ( plus ( second x ) ( second y ) ) ) \operatorname{plus}_{s}=\lambda x.\lambda y.\operatorname{OneZero}\ (% \operatorname{pair}\ (\operatorname{plus}\ (\operatorname{first}\ x)\ (% \operatorname{first}\ y))\ (\operatorname{plus}\ (\operatorname{second}\ x)\ (% \operatorname{second}\ y)))
  176. x - y = [ x p , x n ] - [ y p , y n ] = x p - x n - y p + y n = ( x p + y n ) - ( x n + y p ) = [ x p + y n , x n + y p ] x-y=[x_{p},x_{n}]-[y_{p},y_{n}]=x_{p}-x_{n}-y_{p}+y_{n}=(x_{p}+y_{n})-(x_{n}+y% _{p})=[x_{p}+y_{n},x_{n}+y_{p}]
  177. minus s = λ x . λ y . OneZero ( pair ( plus ( first x ) ( second y ) ) ( plus ( second x ) ( first y ) ) ) \operatorname{minus}_{s}=\lambda x.\lambda y.\operatorname{OneZero}\ (% \operatorname{pair}\ (\operatorname{plus}\ (\operatorname{first}\ x)\ (% \operatorname{second}\ y))\ (\operatorname{plus}\ (\operatorname{second}\ x)\ % (\operatorname{first}\ y)))
  178. x * y = [ x p , x n ] * [ y p , y n ] = ( x p - x n ) * ( y p - y n ) = ( x p * y p + x n * y n ) - ( x p * y n + x n * y p ) = [ x p * y p + x n * y n , x p * y n + x n * y p ] x*y=[x_{p},x_{n}]*[y_{p},y_{n}]=(x_{p}-x_{n})*(y_{p}-y_{n})=(x_{p}*y_{p}+x_{n}% *y_{n})-(x_{p}*y_{n}+x_{n}*y_{p})=[x_{p}*y_{p}+x_{n}*y_{n},x_{p}*y_{n}+x_{n}*y% _{p}]
  179. mult s = λ x . λ y . pair ( plus ( mult ( first x ) ( first y ) ) ( mult ( second x ) ( second y ) ) ) ( plus ( mult ( first x ) ( second y ) ) ( mult ( second x ) ( first y ) ) ) \operatorname{mult}_{s}=\lambda x.\lambda y.\operatorname{pair}\ (% \operatorname{plus}\ (\operatorname{mult}\ (\operatorname{first}\ x)\ (% \operatorname{first}\ y))\ (\operatorname{mult}\ (\operatorname{second}\ x)\ (% \operatorname{second}\ y)))\ (\operatorname{plus}\ (\operatorname{mult}\ (% \operatorname{first}\ x)\ (\operatorname{second}\ y))\ (\operatorname{mult}\ (% \operatorname{second}\ x)\ (\operatorname{first}\ y)))
  180. divZ = λ x . λ y . IsZero y 0 ( divide x y ) \operatorname{divZ}=\lambda x.\lambda y.\operatorname{IsZero}\ y\ 0\ (% \operatorname{divide}\ x\ y)
  181. divide s = λ x . λ y . pair ( plus ( divZ ( first x ) ( first y ) ) ( divZ ( second x ) ( second y ) ) ) ( plus ( divZ ( first x ) ( second y ) ) ( divZ ( second x ) ( first y ) ) ) \operatorname{divide}_{s}=\lambda x.\lambda y.\operatorname{pair}\ (% \operatorname{plus}\ (\operatorname{divZ}\ (\operatorname{first}\ x)\ (% \operatorname{first}\ y))\ (\operatorname{divZ}\ (\operatorname{second}\ x)\ (% \operatorname{second}\ y)))\ (\operatorname{plus}\ (\operatorname{divZ}\ (% \operatorname{first}\ x)\ (\operatorname{second}\ y))\ (\operatorname{divZ}\ (% \operatorname{second}\ x)\ (\operatorname{first}\ y)))

CIE_1931_color_space.html

  1. x ¯ ( λ ) \overline{x}(\lambda)
  2. y ¯ ( λ ) \overline{y}(\lambda)
  3. z ¯ ( λ ) \overline{z}(\lambda)
  4. I ( λ ) I(\lambda)\,
  5. X = 380 780 I ( λ ) x ¯ ( λ ) d λ X=\int_{380}^{780}I(\lambda)\,\overline{x}(\lambda)\,d\lambda
  6. Y = 380 780 I ( λ ) y ¯ ( λ ) d λ Y=\int_{380}^{780}I(\lambda)\,\overline{y}(\lambda)\,d\lambda
  7. Z = 380 780 I ( λ ) z ¯ ( λ ) d λ Z=\int_{380}^{780}I(\lambda)\,\overline{z}(\lambda)\,d\lambda
  8. λ \lambda\,
  9. I ( λ ) I(\lambda)\,
  10. x = X X + Y + Z x=\frac{X}{X+Y+Z}
  11. y = Y X + Y + Z y=\frac{Y}{X+Y+Z}
  12. z = Z X + Y + Z = 1 - x - y z=\frac{Z}{X+Y+Z}=1-x-y
  13. X = Y y x X=\frac{Y}{y}x
  14. Z = Y y ( 1 - x - y ) Z=\frac{Y}{y}(1-x-y)
  15. r ¯ ( λ ) \overline{r}(\lambda)
  16. g ¯ ( λ ) \overline{g}(\lambda)
  17. b ¯ ( λ ) \overline{b}(\lambda)
  18. r ¯ ( λ ) \overline{r}(\lambda)
  19. g ¯ ( λ ) \overline{g}(\lambda)
  20. r ¯ ( λ ) \overline{r}(\lambda)
  21. b ¯ ( λ ) \overline{b}(\lambda)
  22. g ¯ ( λ ) \overline{g}(\lambda)
  23. b ¯ ( λ ) \overline{b}(\lambda)
  24. 0 r ¯ ( λ ) d λ = 0 g ¯ ( λ ) d λ = 0 b ¯ ( λ ) d λ \int_{0}^{\infty}\overline{r}(\lambda)\,d\lambda=\int_{0}^{\infty}\overline{g}% (\lambda)\,d\lambda=\int_{0}^{\infty}\overline{b}(\lambda)\,d\lambda
  25. I ( λ ) I(\lambda)
  26. R = 0 I ( λ ) r ¯ ( λ ) d λ R=\int_{0}^{\infty}I(\lambda)\,\overline{r}(\lambda)\,d\lambda
  27. G = 0 I ( λ ) g ¯ ( λ ) d λ G=\int_{0}^{\infty}I(\lambda)\,\overline{g}(\lambda)\,d\lambda
  28. B = 0 I ( λ ) b ¯ ( λ ) d λ B=\int_{0}^{\infty}I(\lambda)\,\overline{b}(\lambda)\,d\lambda
  29. r = R R + G + B , r=\frac{R}{R+G+B},
  30. g = G R + G + B . g=\frac{G}{R+G+B}.
  31. x ¯ ( λ ) \overline{x}(\lambda)
  32. y ¯ ( λ ) \overline{y}(\lambda)
  33. z ¯ ( λ ) \overline{z}(\lambda)
  34. y ¯ ( λ ) \overline{y}(\lambda)
  35. z ¯ ( λ ) \overline{z}(\lambda)
  36. y ¯ ( λ ) \overline{y}(\lambda)
  37. z ¯ ( λ ) \overline{z}(\lambda)
  38. [ X Y Z ] = 1 b 21 [ b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ] [ R G B ] = 1 0.17697 [ 0.49 0.31 0.20 0.17697 0.81240 0.01063 0.00 0.01 0.99 ] [ R G B ] \begin{bmatrix}X\\ Y\\ Z\end{bmatrix}=\frac{1}{b_{21}}\begin{bmatrix}b_{11}&b_{12}&b_{13}\\ b_{21}&b_{22}&b_{23}\\ b_{31}&b_{32}&b_{33}\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}=\frac{1}{0.17697}\begin{bmatrix}0.49&0.31&0.20\\ 0.17697&0.81240&0.01063\\ 0.00&0.01&0.99\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}
  39. [ R G B ] = [ 0.41847 - 0.15866 - 0.082835 - 0.091169 0.25243 0.015708 0.00092090 - 0.0025498 0.17860 ] [ X Y Z ] , \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}0.41847&-0.15866&-0.082835\\ -0.091169&0.25243&0.015708\\ 0.00092090&-0.0025498&0.17860\end{bmatrix}\cdot\begin{bmatrix}X\\ Y\\ Z\end{bmatrix},

Cifrão.html

  1. S \mathrm{S}\!\!\!\|

Ciphertext_indistinguishability.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. M 0 , M 1 \scriptstyle M_{0},M_{1}
  4. \scriptstyle\in
  5. M b \scriptstyle M_{b}
  6. ( 1 2 ) + ϵ ( k ) \scriptstyle\left(\frac{1}{2}\right)\,+\,\epsilon(k)
  7. ϵ ( k ) \scriptstyle\epsilon(k)
  8. p o l y ( ) \scriptstyle poly()
  9. k 0 \scriptstyle k_{0}
  10. | ϵ ( k ) | < | 1 p o l y ( k ) | \scriptstyle|\epsilon(k)|\;<\;\left|\frac{1}{poly(k)}\right|
  11. k > k 0 \scriptstyle k\;>\;k_{0}
  12. M 0 \scriptstyle M_{0}
  13. M 1 \scriptstyle M_{1}
  14. M b \scriptstyle M_{b}
  15. M 0 \scriptstyle M_{0}
  16. M 1 \scriptstyle M_{1}
  17. M 0 , M 1 \scriptstyle M_{0},\,M_{1}
  18. M b \scriptstyle M_{b}
  19. A B \scriptstyle A\;\Rightarrow\;B
  20. A B \scriptstyle A\;\Leftrightarrow\;B
  21. A ⇏ B \scriptstyle A\;\not\Rightarrow\;B
  22. \scriptstyle\Leftrightarrow
  23. \scriptstyle\Rightarrow
  24. ⇏ \scriptstyle\not\Rightarrow
  25. \scriptstyle\Leftrightarrow

Circle_bundle.html

  1. 𝐒 1 \scriptstyle\mathbf{S}^{1}
  2. π * F \pi^{\!*}F
  3. π * F = d A . \pi^{\!*}F=dA.
  4. π : P M \pi:P\to M
  5. π * : H 2 ( M , ) H 2 ( P , ) \pi^{*}:H^{2}(M,\mathbb{Z})\to H^{2}(P,\mathbb{Z})
  6. π * \pi^{\!*}
  7. U ( 1 ) U(1)
  8. M B O 2 M\to BO_{2}
  9. S O 2 O 2 2 SO_{2}\to O_{2}\to\mathbb{Z}_{2}
  10. S O 2 U ( 1 ) SO_{2}\equiv U(1)
  11. B U ( 1 ) BU(1)
  12. U ( 1 ) U(1)
  13. H 2 ( M , ) \scriptstyle H^{2}(M,\mathbb{Z})
  14. [ M , B U ( 1 ) ] [ M , P ] H 1 ( M ) [M,BU(1)]\equiv[M,\mathbb{C}P^{\infty}]\equiv H^{1}(M)
  15. U ( 1 ) U(1)
  16. M B 2 M\to B\mathbb{Z}_{2}

Circles_of_Apollonius.html

  1. 𝒞 1 , 𝒞 2 , 𝒞 3 \mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}
  2. A 1 A 2 A 3 \mathrm{A_{1}A_{2}A_{3}}
  3. 𝒞 1 \mathcal{C}_{1}
  4. A 1 \mathrm{A_{1}}
  5. A 2 \mathrm{A_{2}}
  6. A 3 \mathrm{A_{3}}
  7. 𝒞 2 \mathcal{C}_{2}
  8. A 2 \mathrm{A_{2}}
  9. A 1 \mathrm{A_{1}}
  10. A 3 \mathrm{A_{3}}
  11. 𝒞 3 \mathcal{C}_{3}
  12. S S
  13. S S^{\prime}
  14. S S
  15. S S^{\prime}

Circumscription_(logic).html

  1. T T
  2. T T
  3. a a
  4. b b
  5. c c
  6. { a , c } \{a,c\}
  7. a a
  8. c c
  9. M M
  10. N N
  11. N M N\subseteq M
  12. M M
  13. N N
  14. \subseteq
  15. N M N\subset M
  16. N M N\subseteq M
  17. M M
  18. T T
  19. N N
  20. T T
  21. N M N\subset M
  22. C I R C ( T ) = { M | M is a minimal model of T } CIRC(T)=\{M~{}|~{}M\mbox{ is a minimal model of }~{}T\}
  23. C I R C ( T ) CIRC(T)
  24. C I R C CIRC
  25. T M Q T\models_{M}Q
  26. T T
  27. Q Q
  28. T = a ( b c ) T=a\land(b\lor c)
  29. a a
  30. b b
  31. c c
  32. { a , b , c } \{a,b,c\}
  33. a a
  34. b b
  35. c c
  36. { a , b } \{a,b\}
  37. a a
  38. c c
  39. b b
  40. { a , c } \{a,c\}
  41. c c
  42. b b
  43. c c
  44. T T
  45. a ¬ ( b c ) a\land\neg(b\leftrightarrow c)
  46. b b
  47. c c
  48. T T
  49. a a
  50. T T
  51. P P
  52. Z Z
  53. P Z P\cup Z
  54. CIRC ( T ; P , Z ) = { M | M T and N such that N T , N P M P and N Z = M Z } \,\text{CIRC}(T;P,Z)=\{M~{}|~{}M\models T\,\text{ and }\not\exists N\,\text{ % such that }N\models T,~{}N\cap P\subset M\cap P\,\text{ and }N\cap Z=M\cap Z\}
  55. P P
  56. Z Z
  57. ¬ open 0 \neg\,\text{open}_{0}
  58. true open 2 \,\text{true}\rightarrow\,\text{open}_{2}
  59. ¬ o p e n 1 \neg open_{1}
  60. c h a n g e _ o p e n t change\_open_{t}
  61. change open 0 ( open 0 open 1 ) \,\text{change open}_{0}\equiv(\,\text{open}_{0}\not\equiv\,\text{open}_{1})
  62. change open 1 ( open 1 open 2 ) \,\text{change open}_{1}\equiv(\,\text{open}_{1}\not\equiv\,\text{open}_{2})
  63. ¬ open 1 \neg\,\text{open}_{1}
  64. change open 0 \,\text{change open}_{0}
  65. change open 1 \,\text{change open}_{1}
  66. T T
  67. P P
  68. T T
  69. P P
  70. P ( v 1 , , v n ) P(v_{1},\ldots,v_{n})
  71. v 1 , , v n \langle v_{1},\ldots,v_{n}\rangle
  72. P P
  73. P ( v 1 , , v n ) P(v_{1},\ldots,v_{n})
  74. P P
  75. T T
  76. T T
  77. P P
  78. P ( v 1 , , v n ) P(v_{1},\ldots,v_{n})
  79. v 1 , , v n \langle v_{1},\ldots,v_{n}\rangle
  80. P P
  81. T T
  82. P P
  83. P P
  84. T T
  85. T ( P ) p ¬ ( T ( p ) p < P ) T(P)\wedge\forall p\neg(T(p)\wedge p<P)
  86. p p
  87. P P
  88. p < P p<P
  89. x ( p ( x ) P ( x ) ) ¬ x ( P ( x ) p ( x ) ) \forall x(p(x)\rightarrow P(x))\wedge\neg\forall x(P(x)\rightarrow p(x))
  90. x x
  91. P P
  92. P P
  93. p p
  94. P P
  95. P P
  96. ¬ A b n o r m a l ( ) \neg Abnormal(...)
  97. P ( a ) P ( b ) P(a)\equiv P(b)
  98. { a , b } \{a,b\}
  99. P ( a ) = P ( b ) = f a l s e P(a)=P(b)=false
  100. P ( a ) = P ( b ) = t r u e P(a)=P(b)=true
  101. P P
  102. \emptyset
  103. { a , b } \{a,b\}
  104. P ( a ) P ( b ) P(a)\equiv P(b)
  105. P ( a ) P(a)
  106. P ( b ) P(b)
  107. P ( a ) = P ( b ) = t r u e P(a)=P(b)=true
  108. P ( a ) P(a)
  109. P ( b ) P(b)
  110. O n On
  111. O n ( coin , moon ) On(\,\text{coin},\,\text{moon})
  112. O n On
  113. O n On
  114. ( coin , white area ) (\,\text{coin},\,\text{white area})
  115. ( coin , black area ) (\,\text{coin},\,\text{black area})
  116. O n On
  117. O n On
  118. O n ( coin , white area ) On(\,\text{coin},\,\text{white area})
  119. O n ( coin , black area ) On(\,\text{coin},\,\text{black area})
  120. O n On

Class_number_formula.html

  1. K K
  2. K K
  3. K K
  4. K K
  5. K K
  6. K K
  7. K K
  8. K / 𝐐 K/\mathbf{Q}
  9. R e ( s ) > 1 Re(s)>1
  10. s s
  11. s = 1 s=1
  12. lim s 1 ( s - 1 ) ζ K ( s ) = 2 r 1 ( 2 π ) r 2 h K Reg K w K | D K | \lim_{s\to 1}(s-1)\zeta_{K}(s)=\frac{2^{r_{1}}\cdot(2\pi)^{r_{2}}\cdot h_{K}% \cdot\operatorname{Reg}_{K}}{w_{K}\cdot\sqrt{|D_{K}|}}
  13. K K
  14. 𝐐 \mathbf{Q}
  15. χ = ( d m ) \chi=\left(\!\frac{d}{m}\!\right)
  16. χ \chi
  17. L ( s , χ ) L(s,\chi)
  18. χ \chi
  19. t 2 - d u 2 = 4 t^{2}-du^{2}=4
  20. ϵ = 1 2 ( t + u d ) . \epsilon=\frac{1}{2}(t+u\sqrt{d}).
  21. ( d ) \mathbb{Q}(\sqrt{d})
  22. h ( d ) = { w | d | 2 π L ( 1 , χ ) , d < 0 ; d ln ϵ L ( 1 , χ ) , d > 0. h(d)=\begin{cases}\dfrac{w\sqrt{|d|}}{2\pi}L(1,\chi),&d<0;\\ \dfrac{\sqrt{d}}{\ln\epsilon}L(1,\chi),&d>0.\end{cases}
  23. ζ K ( s ) = ζ ( s ) L ( s , χ ) \zeta_{K}(s)=\zeta(s)L(s,\chi)
  24. L ( 1 , χ ) L(1,\chi)
  25. χ \chi
  26. q q
  27. L ( 1 , χ ) = { - π q 3 / 2 m = 1 q - 1 m ( m q ) , q 3 mod 4 ; - 1 q 1 / 2 m = 1 q - 1 ( m q ) ln 2 sin m π q , q 1 mod 4. L(1,\chi)=\begin{cases}-\dfrac{\pi}{q^{3/2}}\sum_{m=1}^{q-1}m\left(\dfrac{m}{q% }\right),&q\equiv 3\mod 4;\\ -\dfrac{1}{q^{1/2}}\sum_{m=1}^{q-1}\left(\dfrac{m}{q}\right)\ln 2\sin\dfrac{m% \pi}{q},&q\equiv 1\mod 4.\end{cases}
  28. ζ K ( s ) \zeta_{K}(s)

Classe_préparatoire_aux_grandes_écoles.html

  1. 1 2 x d x = 2 2 2 - 1 2 2 = 3 2 \int_{1}^{2}\!x\,dx\ =\frac{2^{2}}{2}-\frac{1^{2}}{2}=\frac{3}{2}
  2. 2 3 x d x = 3 2 2 - 2 2 2 = 5 2 \int_{2}^{3}\!x\,dx\ =\frac{3^{2}}{2}-\frac{2^{2}}{2}=\frac{5}{2}

Classical_dichotomy.html

  1. 𝐉 d y = d x \mathbf{J}dy=dx
  2. d x dx
  3. d y dy
  4. 𝐉 = [ A 0 B C ] \mathbf{J}=\begin{bmatrix}A&0\\ B&C\\ \end{bmatrix}
  5. A A

Classification_of_discontinuities.html

  1. f f
  2. x x
  3. f f
  4. f ( x ) = { x 2 for x < 1 0 for x = 1 2 - x for x > 1 f(x)=\begin{cases}x^{2}&\mbox{ for }~{}x<1\\ 0&\mbox{ for }~{}x=1\\ 2-x&\mbox{ for }~{}x>1\end{cases}
  5. L - = lim x x 0 - f ( x ) L^{-}=\lim_{x\to x_{0}^{-}}f(x)
  6. L + = lim x x 0 + f ( x ) L^{+}=\lim_{x\to x_{0}^{+}}f(x)
  7. L L
  8. L L
  9. f ( x ) f(x)
  10. x x
  11. L L
  12. f f
  13. g ( x ) = { f ( x ) x x 0 L x = x 0 g(x)=\begin{cases}f(x)&x\neq x_{0}\\ L&x=x_{0}\end{cases}
  14. x x
  15. f ( x ) = { x 2 for x < 1 0 for x = 1 2 - ( x - 1 ) 2 for x > 1 f(x)=\begin{cases}x^{2}&\mbox{ for }~{}x<1\\ 0&\mbox{ for }~{}x=1\\ 2-(x-1)^{2}&\mbox{ for }~{}x>1\end{cases}
  16. L L
  17. f f
  18. f ( x ) = { sin 5 x - 1 for x < 1 0 for x = 1 1 x - 1 for x > 1 f(x)=\begin{cases}\sin\frac{5}{x-1}&\mbox{ for }~{}x<1\\ 0&\mbox{ for }~{}x=1\\ \frac{1}{x-1}&\mbox{ for }~{}x>1\end{cases}
  19. x 0 = 1 \scriptstyle x_{0}\;=\;1
  20. L - \scriptstyle L^{-}
  21. L + \scriptstyle L^{+}

Classification_of_electromagnetic_fields.html

  1. F a r b b = λ r a F^{a}{}_{b}r^{b}\,=\lambda\,r^{a}
  2. F a b = ( 0 B z - B y E x / c - B z 0 B x E y / c B y - B x 0 E z / c - E x / c - E y / c - E z / c 0 ) F_{ab}=\left(\begin{matrix}0&B_{z}&-B_{y}&E_{x}/c\\ -B_{z}&0&B_{x}&E_{y}/c\\ B_{y}&-B_{x}&0&E_{z}/c\\ -E_{x}/c&-E_{y}/c&-E_{z}/c&0\end{matrix}\right)
  3. E x , E y , E z E_{x},E_{y},E_{z}
  4. B x , B y , B z B_{x},B_{y},B_{z}
  5. c = 1 c=1
  6. η \eta
  7. P 1 2 F a b F a b = B 2 - E 2 c 2 = - 1 2 F a b * F a b * P\equiv\frac{1}{2}F_{ab}\,F^{ab}=\|\vec{B}\|^{2}-\frac{\|\vec{E}\|^{2}}{c^{2}}% =-\frac{1}{2}{}^{*}F_{ab}\,{}^{*}F^{ab}
  8. Q 1 4 F a b F a b * = 1 8 ϵ a b c d F a b F c d = E B c Q\equiv\frac{1}{4}F_{ab}\,{}^{*}F^{ab}=\frac{1}{8}\epsilon^{abcd}F_{ab}F_{cd}=% \frac{\vec{E}\cdot\vec{B}}{c}
  9. P = Q = 0 P=Q=0
  10. P 2 + Q 2 0 P^{2}+Q^{2}\neq\,0
  11. P 0 = Q P\neq 0=Q
  12. Q 0 Q\neq 0

Classification_rule.html

  1. { ( x 1 , y 1 ) , , ( x n , y n ) } \{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  2. y ^ = h ( x ) , \hat{y}=h(x),
  3. y i ^ = h ( x i ) \hat{y_{i}}=h(x_{i})
  4. y j ^ = h ( x j ) y j \hat{y_{j}}=h(x_{j})\approx y_{j}
  5. P ( A | B ) = P ( B | A ) P ( A ) P ( B | A ) P ( A ) + P ( B | not A ) P ( not A ) < c o d e > = 0.99 × 0.001 0.99 × 0.001 + 0.05 × 0.999 0.019. < / c o d e > \begin{aligned}\displaystyle P(A|B)&\displaystyle=\frac{P(B|A)P(A)}{P(B|A)P(A)% +P(B|\,\text{not }A)P(\,\text{not }A)}\\ \\ \displaystyle\par <code>&\displaystyle= \frac{0.99\times 0.001}{0.99 \times 0.% 001 + 0.05\times 0.999}\\ \\ &\displaystyle\approx 0.019.\end{aligned}</code>
  6. P ( A | B ) = 0.99 × 0.001 0.99 × 0.001 + 0.001 × 0.999 0.5 , P(A|B)=\frac{0.99\times 0.001}{0.99\times 0.001+0.001\times 0.999}\approx 0.5,
  7. P ( A | not B ) = P ( not B | A ) P ( A ) P ( not B | A ) P ( A ) + P ( not B | not A ) P ( not A ) < c o d e > = 0.01 × 0.001 0.01 × 0.001 + 0.95 × 0.999 0.0000105. < / c o d e > \begin{aligned}\displaystyle P(A|\,\text{not }B)&\displaystyle=\frac{P(\,\text% {not }B|A)P(A)}{P(\,\text{not }B|A)P(A)+P(\,\text{not }B|\,\text{not }A)P(\,% \text{not }A)}\\ \\ \displaystyle\par <code>&\displaystyle= \frac{0.01\times 0.001}{0.01 \times 0.% 001 + 0.95\times 0.999}\\ \\ &\displaystyle\approx 0.0000105.\end{aligned}</code>
  8. P ( A | not B ) = P ( not B | A ) P ( A ) P ( not B | A ) P ( A ) + P ( not B | not A ) P ( not A ) < c o d e > = 0.01 × 0.6 0.01 × 0.6 + 0.95 × 0.4 0.0155. < / c o d e > \begin{aligned}\displaystyle P(A|\,\text{not }B)&\displaystyle=\frac{P(\,\text% {not }B|A)P(A)}{P(\,\text{not }B|A)P(A)+P(\,\text{not }B|\,\text{not }A)P(\,% \text{not }A)}\\ \\ \displaystyle\par <code>&\displaystyle= \frac{0.01\times 0.6}{0.01 \times 0.6 % + 0.95\times 0.4}\\ \\ &\displaystyle\approx 0.0155.\end{aligned}</code>
  9. p p
  10. q = 1 - p q=1-p
  11. p p
  12. q q
  13. p 2 p^{2}
  14. p q pq
  15. p 2 / ( p 2 + p q ) = p p^{2}/(p^{2}+pq)=p
  16. q 2 / ( q 2 + p q ) = q q^{2}/(q^{2}+pq)=q

Classifying_space_for_U(n).html

  1. E U ( n ) = { e 1 , , e n : ( e i , e j ) = δ i j , e i } . EU(n)=\left\{e_{1},\ldots,e_{n}\ :\ (e_{i},e_{j})=\delta_{ij},e_{i}\in\mathcal% {H}\right\}.
  2. δ i j \delta_{ij}
  3. ( , ) (\cdot,\cdot)
  4. B U ( n ) = E U ( n ) / U ( n ) BU(n)=EU(n)/U(n)
  5. B U ( n ) = { V : dim V = n } BU(n)=\{V\subset\mathcal{H}\ :\ \dim V=n\}
  6. B U ( 1 ) = P U ( ) , BU(1)=PU(\mathcal{H}),
  7. F n ( 𝐂 k ) \displaystyle F_{n}(\mathbf{C}^{k})
  8. π p ( 𝐒 2 k - 1 ) \pi_{p}(\mathbf{S}^{2k-1})
  9. π p ( F n ( 𝐂 k ) ) = π p ( F n - 1 ( 𝐂 k - 1 ) ) \pi_{p}(F_{n}(\mathbf{C}^{k}))=\pi_{p}(F_{n-1}(\mathbf{C}^{k-1}))
  10. p 2 k - 2 p\leq 2k-2
  11. k > 1 2 p + n - 1 k>\tfrac{1}{2}p+n-1
  12. π p ( F n ( 𝐂 k ) ) = π p ( F n - 1 ( 𝐂 k - 1 ) ) = = π p ( F 1 ( 𝐂 k + 1 - n ) ) = π p ( 𝐒 k - n ) . \pi_{p}(F_{n}(\mathbf{C}^{k}))=\pi_{p}(F_{n-1}(\mathbf{C}^{k-1}))=\cdots=\pi_{% p}(F_{1}(\mathbf{C}^{k+1-n}))=\pi_{p}(\mathbf{S}^{k-n}).
  13. E U ( n ) = lim F n ( 𝐂 k ) EU(n)={\lim_{\to}}\;_{k\to\infty}F_{n}(\mathbf{C}^{k})
  14. G n ( 𝐂 ) = lim G n ( 𝐂 k ) G_{n}(\mathbf{C}^{\infty})={\lim_{\to}}\;_{k\to\infty}G_{n}(\mathbf{C}^{k})
  15. π p ( E U ( n ) ) \pi_{p}(EU(n))
  16. \Box
  17. 𝐑 [ c 1 ] / c 1 k + 1 \mathbf{R}[c_{1}]/c_{1}^{k+1}
  18. x i x_{i}
  19. H * ( B U ( n ) ) = 𝐑 [ c 1 , , c n ] , H^{*}(BU(n))=\mathbf{R}[c_{1},\ldots,c_{n}],
  20. c i c_{i}
  21. x i x_{i}
  22. \Box
  23. H * ( B U ( n ) ) H^{*}(BU(n))
  24. H * ( B U ( n ) ) = [ [ c 1 , c 2 , , c n ] ] H^{*}(BU(n))=\mathbb{Z}[[c_{1},c_{2},...,c_{n}]]
  25. K U KU
  26. K U * ( B U ( n ) ) [ t , t - 1 ] [ [ c 1 , , c n ] ] KU^{*}(BU(n))\cong\mathbb{Z}[t,t^{-1}][[c_{1},...,c_{n}]]
  27. K U * ( B U ( n ) ) KU_{*}(BU(n))
  28. [ t , t - 1 ] \mathbb{Z}[t,t^{-1}]
  29. β 0 \beta_{0}
  30. β i 1 β i r \beta_{i_{1}}\ldots\beta_{i_{r}}
  31. n i j > 0 n\geq i_{j}>0
  32. r n r\leq n
  33. K U * ( B U ( n ) ) KU_{*}(BU(n))
  34. B U BU
  35. K * ( X ) = π * ( K ) K 0 ( X ) K_{*}(X)=\pi_{*}(K)\otimes K_{0}(X)
  36. π * ( K ) = 𝐙 [ t , t - 1 ] \pi_{*}(K)=\mathbf{Z}[t,t^{-1}]
  37. f ( w 1 , , w n ) 1 n ! σ S n f ( x σ ( 1 ) , , x σ ( n ) ) f(w_{1},\dots,w_{n})\mapsto\frac{1}{n!}\sum_{\sigma\in S_{n}}f(x_{\sigma(1)},% \dots,x_{\sigma(n)})
  38. ( n n 1 , n 2 , , n r ) f ( k 1 , , k n ) 𝐙 {n\choose n_{1},n_{2},\ldots,n_{r}}f(k_{1},\dots,k_{n})\in\mathbf{Z}
  39. ( n k 1 , k 2 , , k m ) = n ! k 1 ! k 2 ! k m ! {n\choose k_{1},k_{2},\ldots,k_{m}}=\frac{n!}{k_{1}!\,k_{2}!\cdots k_{m}!}
  40. k 1 , , k n k_{1},\dots,k_{n}
  41. n 1 , , n r n_{1},\dots,n_{r}
  42. K U * ( B U ( n ) ) KU^{*}(BU(n))
  43. K U * ( B U ( n ) ) KU_{*}(BU(n))
  44. K 0 ( B G ) K_{0}(BG)
  45. K 0 ( K ) K_{0}(K)
  46. K 0 ( B U ( n ) ) K_{0}(BU(n))

Clearance_(medicine).html

  1. V d C d t = - K C + m ˙ ( 1 ) V\frac{dC}{dt}=-K\cdot C+\dot{m}\qquad(1)
  2. m ˙ \dot{m}
  3. d C d t \frac{dC}{dt}
  4. Δ m b o d y = ( - m ˙ o u t + m ˙ i n + m ˙ g e n . ) Δ t ( 2 ) \Delta m_{body}=(-\dot{m}_{out}+\dot{m}_{in}+\dot{m}_{gen.})\Delta t\qquad(2)
  5. Δ t \Delta t
  6. Δ m b o d y \Delta m_{body}
  7. Δ t \Delta t
  8. m ˙ i n \dot{m}_{in}
  9. m ˙ o u t \dot{m}_{out}
  10. m ˙ g e n . \dot{m}_{gen.}
  11. Δ m \Delta m
  12. Δ t \Delta t
  13. m b o d y = C V ( 3 ) m_{body}=C\cdot V\qquad(3)
  14. m ˙ o u t = K C ( 4 ) \dot{m}_{out}=K\cdot C\qquad(4)
  15. Δ ( C V ) = ( - K C + m ˙ i n + m ˙ g e n . ) Δ t ( 5 ) \Delta(C\cdot V)=(-K\cdot C+\dot{m}_{in}+\dot{m}_{gen.})\Delta t\qquad(5)
  16. m ˙ = m ˙ i n + m ˙ g e n . \dot{m}=\dot{m}_{in}+\dot{m}_{gen.}
  17. Δ t \Delta t
  18. Δ ( C V ) Δ t = - K C + m ˙ ( 6 ) \frac{\Delta(C\cdot V)}{\Delta t}=-K\cdot C+\dot{m}\qquad(6)
  19. Δ t 0 \Delta t\rightarrow 0
  20. d ( C V ) d t = - K C + m ˙ ( 7 ) \frac{d(C\cdot V)}{dt}=-K\cdot C+\dot{m}\qquad(7)
  21. C d V d t + V d C d t = - K C + m ˙ ( 8 ) C\frac{dV}{dt}+V\frac{dC}{dt}=-K\cdot C+\dot{m}\qquad(8)
  22. C d V d t = 0 C\frac{dV}{dt}=0
  23. V d C d t = - K C + m ˙ ( 1 ) V\frac{dC}{dt}=-K\cdot C+\dot{m}\qquad(1)
  24. C = m ˙ K + ( C o - m ˙ K ) e - K t V ( 9 ) C=\frac{\dot{m}}{K}+\left(C_{o}-\frac{\dot{m}}{K}\right)e^{-\frac{K\cdot t}{V}% }\qquad(9)
  25. C = m ˙ K ( 10 a ) C_{\infty}=\frac{\dot{m}}{K}\qquad(10a)
  26. K = m ˙ C ( 10 b ) K=\frac{\dot{m}}{C_{\infty}}\qquad(10b)
  27. K = C U Q C B ( 11 ) K=\frac{C_{U}\cdot Q}{C_{B}}\qquad(11)

Clélie.html

  1. x = a sin ( m θ ) cos ( θ ) x=a\sin(m\theta)\cos(\theta)
  2. y = a sin ( m θ ) sin ( θ ) y=a\sin(m\theta)\sin(\theta)
  3. z = a cos ( m θ ) z=a\cos(m\theta)
  4. a a
  5. m = 1 m=1

Click-through_rate.html

  1. CTR = Clicks Impressions × 100 % \,\text{CTR}={\,\text{Clicks}\over\,\text{Impressions}}\times 100\,\%

Clifford's_theorem_on_special_divisors.html

  1. g - 1 2 . \frac{g-1}{2}.

Clock_hypothesis.html

  1. P P
  2. d τ = P d t 2 - d x 2 / c 2 - d y 2 / c 2 - d z 2 / c 2 d\tau=\int_{P}\sqrt{dt^{2}-dx^{2}/c^{2}-dy^{2}/c^{2}-dz^{2}/c^{2}}

Cloister_vault.html

  1. 1 3 s 3 \frac{1}{3}s^{3}

Closed-world_assumption.html

  1. { E n g l i s h ( F r e d ) I r i s h ( F r e d ) } \{English(Fred)\vee Irish(Fred)\}
  2. E n g l i s h ( F r e d ) English(Fred)
  3. I r i s h ( F r e d ) Irish(Fred)
  4. { E n g l i s h ( F r e d ) I r i s h ( F r e d ) , ¬ E n g l i s h ( F r e d ) , ¬ I r i s h ( F r e d ) } \{English(Fred)\vee Irish(Fred),\neg English(Fred),\neg Irish(Fred)\}
  5. K K
  6. K K
  7. K K
  8. K K
  9. K K
  10. K K
  11. K K
  12. K K
  13. K { ¬ f | f F } K\wedge\{\neg f~{}|~{}f\in F\}
  14. F F
  15. K K
  16. f f
  17. f f
  18. K K
  19. f f
  20. c c
  21. K ⊬ c K\not\vdash c
  22. K ⊬ c f K\not\vdash c\vee f
  23. f f
  24. f f

Closure_with_a_twist.html

  1. Y Y
  2. X X
  3. y 1 , y 2 Y y_{1},y_{2}\in Y
  4. ϕ \phi
  5. X X
  6. y 3 Y y_{3}\in Y
  7. y 1 y 2 = ϕ ( y 3 ) y_{1}\cdot y_{2}=\phi(y_{3})
  8. \cdot
  9. X X
  10. ϕ \phi
  11. c C : p c Sym ( n ) : e C : p c ( e + c ) C . \ \forall c\in C:\exists p_{c}\in\,\text{Sym}(n):\forall e\in C:p_{c}(e+c)\in C.
  12. c C : p c Sym ( n ) : p c ( C + c ) = C . \ \forall c\in C:\exists p_{c}\in\,\text{Sym}(n):p_{c}(C+c)=C.
  13. F = [ 0 0 0 1 1 0 1 0 1 ] . F=\begin{bmatrix}0&0&0\\ 1&1&0\\ 1&0&1\end{bmatrix}.
  14. F + 000 = [ 0 0 0 1 1 0 1 0 1 ] = F i d = F ( 2 , 3 ) R ( 2 , 3 ) C . F+000=\begin{bmatrix}0&0&0\\ 1&1&0\\ 1&0&1\end{bmatrix}=F^{id}=F^{(2,3)_{R}(2,3)_{C}}.
  15. F + 110 = [ 1 1 0 0 0 0 0 1 1 ] = F ( 1 , 2 ) R ( 1 , 2 ) C = F ( 1 , 2 , 3 ) R ( 1 , 2 , 3 ) C . F+110=\begin{bmatrix}1&1&0\\ 0&0&0\\ 0&1&1\end{bmatrix}=F^{(1,2)_{R}(1,2)_{C}}=F^{(1,2,3)_{R}(1,2,3)_{C}}.
  16. F + 101 = [ 1 0 1 0 1 1 0 0 0 ] = F ( 1 , 3 ) R ( 1 , 3 ) C = F ( 1 , 3 , 2 ) R ( 1 , 3 , 2 ) C . F+101=\begin{bmatrix}1&0&1\\ 0&1&1\\ 0&0&0\end{bmatrix}=F^{(1,3)_{R}(1,3)_{C}}=F^{(1,3,2)_{R}(1,3,2)_{C}}.
  17. π R \pi_{R}
  18. σ C \sigma_{C}
  19. n 3 n\geq 3
  20. K n K_{n}
  21. n n
  22. n n
  23. K n = [ 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 ] . K_{n}=\begin{bmatrix}0&0&0&\cdots&0&0\\ 1&1&0&\cdots&0&0\\ 1&0&1&\cdots&0&0\\ &&&\vdots&&\\ 1&0&0&\cdots&1&0\\ 1&0&0&\cdots&0&1\end{bmatrix}.
  24. n = 3 n=3
  25. K 3 = F K_{3}=F
  26. W = [ 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 0 1 ] . W=\begin{bmatrix}0&0&0\\ 1&0&0\\ 1&1&0\\ 1&1&1\\ 0&1&1\\ 0&0&1\end{bmatrix}.
  27. \subset
  28. ϕ h \phi_{h}
  29. ϕ h \phi_{h}
  30. \cdot
  31. ϕ h \phi_{h}
  32. ϕ \phi
  33. h 1 , h 2 , h_{1},h_{2},...
  34. h 1 h_{1}
  35. h n h_{n}
  36. h 1 h_{1}
  37. \cdot
  38. ϕ \phi
  39. h n - 1 h_{n-1}
  40. π \pi
  41. π - 1 \pi^{-1}
  42. π - 1 \pi^{-1}
  43. Z 10 Z_{10}
  44. Z 2 n Z_{2}^{n}
  45. p q p^{q}
  46. p 2 p^{2}
  47. p q p^{q}

Cluster_decay.html

  1. B = T a / T c B=T_{a}/T_{c}
  2. E k = Q A d / A E_{k}=QA_{d}/A
  3. λ \lambda
  4. λ = ν S P s \lambda=\nu SP_{s}
  5. ν \nu
  6. log T = - log P s - 22.169 + 0.598 ( A e - 1 ) \log T=-\log P_{s}-22.169+0.598(A_{e}-1)
  7. Q = [ M - ( M d + M e ) ] c 2 Q=[M-(M_{d}+M_{e})]c^{2}

Cochleoid.html

  1. r = a sin θ θ r=\frac{a\sin\theta}{\theta}
  2. ( x 2 + y 2 ) arctan y x = a y (x^{2}+y^{2})\arctan\frac{y}{x}=ay
  3. x = a sin t cos t t x=\frac{a\sin t\cos t}{t}
  4. y = a sin 2 t t y=\frac{a\sin^{2}t}{t}

Cockcroft–Walton_generator.html

  1. V o = 2 N V p = N V pp V_{o}=2NV_{p}=NV\text{pp}\,

Coframe.html

  1. M M
  2. M M
  3. v k : k T * M k T * M v_{k}:\bigoplus^{k}T^{*}M\to\bigwedge^{k}T^{*}M
  4. v k : ( ρ 1 , , ρ k ) ρ 1 ρ k v_{k}:(\rho_{1},\ldots,\rho_{k})\mapsto\rho_{1}\wedge\ldots\wedge\rho_{k}
  5. M M
  6. n n
  7. σ \sigma
  8. n T * M \bigoplus^{n}T^{*}M
  9. v n σ 0 v_{n}\circ\sigma\neq 0
  10. v n v_{n}
  11. n T * M \bigwedge^{n}T^{*}M
  12. G L ( n ) GL(n)
  13. M M

Cohen-Daubechies-Feauveau_wavelet.html

  1. q prim ( X ) = 1 q_{\mathrm{prim}}(X)=1
  2. Q A ( X ) Q_{A}(X)
  3. ( 1 - X 2 ) A Q A ( X ) + ( X 2 ) A Q A ( 2 - X ) = 1 \left(1-\frac{X}{2}\right)^{A}\,Q_{A}(X)+\left(\frac{X}{2}\right)^{A}\,Q_{A}(2% -X)=1
  4. Q A ( X ) = q prim ( X ) q dual ( X ) Q_{A}(X)=q_{\mathrm{prim}}(X)\,q_{\mathrm{dual}}(X)
  5. a prim ( Z ) = 2 Z d ( 1 + Z 2 ) A q prim ( 1 - Z + Z - 1 2 ) a_{\mathrm{prim}}(Z)=2Z^{d}\,\left(\frac{1+Z}{2}\right)^{A}\,q_{\mathrm{prim}}% \left(1-\frac{Z+Z^{-1}}{2}\right)
  6. a dual ( Z ) = 2 Z d ( 1 + Z 2 ) A q dual ( 1 - Z + Z - 1 2 ) a_{\mathrm{dual}}(Z)=2Z^{d}\,\left(\frac{1+Z}{2}\right)^{A}\,q_{\mathrm{dual}}% \left(1-\frac{Z+Z^{-1}}{2}\right)
  7. Q A ( X ) Q_{A}(X)
  8. 2 A - 1 2^{A-1}
  9. q prim ( X ) = 1 q_{\mathrm{prim}}(X)=1
  10. q dual ( X ) = Q A ( X ) q_{\mathrm{dual}}(X)=Q_{A}(X)
  11. primary \mathrm{primary}
  12. 1 + X 1+X
  13. 1 + X 1+X
  14. 1 2 ( 1 + Z ) 2 Z \frac{1}{2}(1+Z)^{2}\,Z
  15. 1 2 ( 1 + Z ) 2 ( - 1 2 + 2 Z - 1 2 Z 2 ) \frac{1}{2}(1+Z)^{2}\,\left(-\frac{1}{2}+2\,Z-\frac{1}{2}\,Z^{2}\right)
  16. Q 4 ( X ) = 1 + 2 X + 5 / 2 X 2 + 5 / 2 X 3 Q_{4}(X)=1+2\,X+5/2\,X^{2}+5/2\,X^{3}
  17. 1 - c X 1-c\,X
  18. 1 + 2 X + 5 / 2 X 2 + 5 / 2 X 3 1+2\,X+5/2\,X^{2}+5/2\,X^{3}
  19. 1 - c X 1-c\,X
  20. 1 + ( c + 2 ) * X + ( c 2 + 2 * c + 5 / 2 ) X 2 1+(c+2)*\,X+(c^{2}+2*c+5/2)\,X^{2}
  21. n n
  22. a 0 \displaystyle a_{0}
  23. s m ( z ) = a m ( 2 m + 1 ) ( 1 + z ( - 1 ) m ) s_{m}(z)=a_{m}\cdot(2\cdot m+1)\cdot(1+z^{(-1)^{m}})
  24. x - 1 ( z ) \displaystyle x_{-1}(z)
  25. x n / 2 ( z ) = 2 - n / 2 ( 1 + z ) n z n / 2 mod 2 - n / 2 x_{n/2}(z)=2^{-n/2}\cdot(1+z)^{n}\cdot z^{n/2\bmod 2-n/2}
  26. x n / 2 x_{n/2}
  27. x n / 2 - 1 x_{n/2-1}
  28. n n
  29. a 0 \displaystyle a_{0}
  30. s m ( z ) = a m ( ( 2 m + 1 ) + ( 2 m - 1 ) z ) / z m mod 2 s_{m}(z)=a_{m}\cdot((2\cdot m+1)+(2\cdot m-1)\cdot z)/z^{m\bmod 2}
  31. x - 1 ( z ) \displaystyle x_{-1}(z)
  32. x ( n + 1 ) / 2 ( z ) ( 1 + z ) n x_{(n+1)/2}(z)\sim(1+z)^{n}
  33. x ( n + 1 ) / 2 x_{(n+1)/2}
  34. x ( n - 1 ) / 2 x_{(n-1)/2}

Coherence_bandwidth.html

  1. W c W_{c}
  2. W c 2 π D W_{c}\approx{2\pi\over\ D}
  3. B c B_{c}
  4. B c 1 D B_{c}\approx{1\over\ D}

Coherent_control.html

  1. | ϕ f |\phi_{f}\rangle
  2. J = | < ψ ( T ) | ϕ f > | 2 J=|<\psi(T)|\phi_{f}>|^{2}
  3. | ϕ i > |\phi_{i}>
  4. H ( t ) = H 0 + μ ϵ ( t ) H(t)=H_{0}+\mu\cdot\epsilon(t)
  5. ϵ ( t ) \epsilon(t)
  6. ϵ ( t ) \epsilon(t)
  7. J = J + 0 T χ ( t ) | ( i t - H ( ϵ ( t ) ) ) | ψ ( t ) d t + λ o T | ϵ ( t ) | 2 d t J^{\prime}=J+\int_{0}^{T}\langle\chi(t)|(i\frac{\partial}{\partial t}-H(% \epsilon(t)))|\psi(t)\rangle dt+\lambda\int_{o}^{T}|\epsilon(t)|^{2}dt
  8. | χ > |\chi>
  9. λ \lambda
  10. J J^{\prime}
  11. δ ϵ \delta\epsilon
  12. δ ψ \delta\psi
  13. | ψ > |\psi>
  14. | ψ ( 0 ) | ϕ i > |\psi(0)>=|\phi_{i}>
  15. | χ > |\chi>
  16. | χ ( T ) | ϕ f > |\chi(T)>=|\phi_{f}>
  17. N a 2 Na_{2}
  18. I 2 I_{2}

Coin_problem.html

  1. g ( a 1 , a 2 ) = a 1 a 2 - a 1 - a 2 g(a_{1},a_{2})=a_{1}a_{2}-a_{1}-a_{2}
  2. N ( a 1 , a 2 ) = ( a 1 - 1 ) ( a 2 - 1 ) / 2 N(a_{1},a_{2})=(a_{1}-1)(a_{2}-1)/2
  3. g ( a 1 , a 2 ) g(a_{1},a_{2})
  4. a 1 , a 2 a_{1},a_{2}\in\mathbb{N}
  5. gcd ( a 1 , a 2 ) = 1 \gcd(a_{1},a_{2})=1
  6. n ( a 1 - 1 ) ( a 2 - 1 ) n\geq(a_{1}-1)(a_{2}-1)
  7. ρ \rho
  8. σ \sigma
  9. σ < a 1 \sigma<a_{1}
  10. n = ρ a 1 + σ a 2 n=\rho a_{1}+\sigma a_{2}
  11. j = 0 , 1 , , a 1 - 1 j=0,1,\ldots,a_{1}-1
  12. gcd ( a 1 , a 2 ) = 1 \gcd(a_{1},a_{2})=1
  13. n - j a 2 n-ja_{2}
  14. a 1 a_{1}
  15. j j
  16. j = σ j=\sigma
  17. n = ρ a 1 + σ a 2 n=\rho a_{1}+\sigma a_{2}
  18. ρ 0 \rho\geq 0
  19. ρ a 1 = n - σ a 2 ( a 1 - 1 ) ( a 2 - 1 ) - ( a 1 - 1 ) a 2 = - a 1 + 1 \rho a_{1}=n-\sigma a_{2}\geq(a_{1}-1)(a_{2}-1)-(a_{1}-1)a_{2}=-a_{1}+1
  20. g ( a 1 , a 2 , a 3 ) 3 a 1 a 2 a 3 - a 1 - a 2 - a 3 g(a_{1},a_{2},a_{3})\geq\sqrt{3a_{1}a_{2}a_{3}}-a_{1}-a_{2}-a_{3}
  21. g ( a , a + d , a + 2 d , , a + s d ) = ( a - 2 s + 1 ) a + ( d - 1 ) ( a - 1 ) - 1. g(a,a+d,a+2d,\dots,a+sd)=\left(\left\lfloor\frac{a-2}{s}\right\rfloor+1\right)% a+(d-1)(a-1)-1.
  22. g ( m k , m k - 1 n , m k - 2 n 2 , , n k ) = n k - 1 ( m n - m - n ) + m 2 ( n - 1 ) ( m k - 1 - n k - 1 ) m - n . g(m^{k},m^{k-1}n,m^{k-2}n^{2},\dots,n^{k})=n^{k-1}(mn-m-n)+\frac{m^{2}(n-1)(m^% {k-1}-n^{k-1})}{m-n}.
  23. 44 = 6 + 9 + 9 + 20 44=6+9+9+20
  24. 45 = 9 + 9 + 9 + 9 + 9 45=9+9+9+9+9
  25. 46 = 6 + 20 + 20 46=6+20+20
  26. 47 = 9 + 9 + 9 + 20 47=9+9+9+20
  27. 48 = 6 + 6 + 9 + 9 + 9 + 9 48=6+6+9+9+9+9
  28. 49 = 9 + 20 + 20 49=9+20+20

Coincidence_point.html

  1. f , g : X Y f,g\colon X\rightarrow Y

Coleman–Liau_index.html

  1. C L I = 0.0588 L - 0.296 S - 15.8 CLI=0.0588{L}-0.296{S}-15.8\,\!
  2. C L I = 0.0588 × 537 - 0.296 × 4.20 - 15.8 = 14.5 CLI=0.0588\times 537-0.296\times 4.20-15.8=14.5\,\!

Colin_de_Verdière_graph_invariant.html

  1. μ ( G ) \mu(G)
  2. G = ( V , E ) G=(V,E)
  3. V = { 1 , , n } V=\{1,\dots,n\}
  4. μ ( G ) \mu(G)
  5. M = ( M i , j ) ( n ) M=(M_{i,j})\in\mathbb{R}^{(n)}
  6. i , j i,j
  7. i j i\neq j
  8. M i , j < 0 M_{i,j}<0
  9. M i , j = 0 M_{i,j}=0
  10. X = ( X i , j ) ( n ) X=(X_{i,j})\in\mathbb{R}^{(n)}
  11. M X = 0 MX=0
  12. X i , j = 0 X_{i,j}=0
  13. i = j i=j
  14. M i , j 0 M_{i,j}\neq 0
  15. μ ( H ) μ ( G ) \mu(H)\leq\mu(G)

Collinearity.html

  1. P 1 , P 2 , P 3 P_{1},P_{2},P_{3}
  2. A 1 , A 2 , A 3 A_{1},A_{2},A_{3}
  3. P 1 A 2 P 2 A 3 P 3 A 1 = P 1 A 3 P 2 A 1 P 3 A 2 . P_{1}A_{2}\cdot P_{2}A_{3}\cdot P_{3}A_{1}=P_{1}A_{3}\cdot P_{2}A_{1}\cdot P_{% 3}A_{2}.
  4. [ x 1 x 2 x n y 1 y 2 y n z 1 z 2 z n ] \begin{bmatrix}x_{1}&x_{2}&\dots&x_{n}\\ y_{1}&y_{2}&\dots&y_{n}\\ z_{1}&z_{2}&\dots&z_{n}\end{bmatrix}
  5. [ 1 x 1 x 2 x n 1 y 1 y 2 y n 1 z 1 z 2 z n ] \begin{bmatrix}1&x_{1}&x_{2}&\dots&x_{n}\\ 1&y_{1}&y_{2}&\dots&y_{n}\\ 1&z_{1}&z_{2}&\dots&z_{n}\end{bmatrix}
  6. det [ 0 d ( A B ) 2 d ( A C ) 2 1 d ( A B ) 2 0 d ( B C ) 2 1 d ( A C ) 2 d ( B C ) 2 0 1 1 1 1 0 ] = 0. \det\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&1\\ d(AB)^{2}&0&d(BC)^{2}&1\\ d(AC)^{2}&d(BC)^{2}&0&1\\ 1&1&1&0\end{bmatrix}=0.
  7. X 1 X_{1}
  8. X 2 X_{2}
  9. λ 0 \lambda_{0}
  10. λ 1 \lambda_{1}
  11. X 2 i = λ 0 + λ 1 X 1 i . X_{2i}=\lambda_{0}+\lambda_{1}X_{1i}.
  12. X k i = λ 0 + λ 1 X 1 i + λ 2 X 2 i + + λ k - 1 X ( k - 1 ) , i X_{ki}=\lambda_{0}+\lambda_{1}X_{1i}+\lambda_{2}X_{2i}+\dots+\lambda_{k-1}X_{(% k-1),i}
  13. X k i = λ 0 + λ 1 X 1 i + λ 2 X 2 i + + λ k - 1 X ( k - 1 ) , i + ε i X_{ki}=\lambda_{0}+\lambda_{1}X_{1i}+\lambda_{2}X_{2i}+\dots+\lambda_{k-1}X_{(% k-1),i}+\varepsilon_{i}
  14. ε i \varepsilon_{i}

Collision_resistance.html

  1. 2 N \scriptstyle\sqrt{2^{N}}

Comb_drive.html

  1. E = 1 2 C V 2 E=\frac{1}{2}CV^{2}
  2. F = 1 2 C d x d r i v e V 2 F=\frac{1}{2}\frac{\partial C}{\partial dx_{drive}}V^{2}
  3. F = 1 2 n t o r V 2 d F=\frac{1}{2}\frac{nt\in_{o}\in_{r}V^{2}}{d}
  4. V V
  5. r \in_{r}
  6. o \in_{o}
  7. n n
  8. t t
  9. d d

Combinatorial_design.html

  1. / 7 \mathbb{Z}/7\mathbb{Z}
  2. b = 1 B m v b m w b = Λ \sum_{b=1}^{B}m_{vb}m_{wb}=\Lambda
  3. f ( x 1 , , x d ) f(x_{1},\ldots,x_{d})

Combined_gas_law.html

  1. P V T = k \qquad\frac{PV}{T}=k
  2. P 1 V 1 T 1 = P 2 V 2 T 2 \qquad\frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}
  3. P V = k 1 ( 1 ) PV=k_{1}\qquad(1)
  4. V T = k 2 ( 2 ) \frac{V}{T}=k_{2}\qquad(2)
  5. P = k 3 T ( 3 ) P=k_{3}T\qquad(3)
  6. P V T = k 1 ( T ) T \frac{PV}{T}=\frac{k_{1}(T)}{T}
  7. P V T = k 2 ( P ) P \frac{PV}{T}=k_{2}(P)P
  8. k 1 ( T ) T = k 2 ( P ) P \frac{k_{1}(T)}{T}=k_{2}(P)P
  9. P V T = c o n s t \frac{PV}{T}=const
  10. P = k V T P=k_{V}\,T\,\!
  11. V = k P T V=k_{P}T\,\!
  12. P V = k T PV=k_{T}\,\!
  13. P V P V = k V T k P T k T PVPV=k_{V}Tk_{P}Tk_{T}\,\!
  14. P V T = k P k V k T \frac{PV}{T}=\sqrt{k_{P}k_{V}k_{T}}\,\!
  15. k T k V k P = T 2 \frac{k_{T}}{k_{V}k_{P}}=T^{2}\,\!
  16. P = k V ( V ) T P=k_{V}(V)\,T\,\!
  17. V = k P ( P ) T V=k_{P}(P)\,T\,\!
  18. T 0 = P 0 k V ( V 0 ) a n d T 0 = V 0 k P ( P 0 ) T_{0}=\frac{P_{0}}{k_{V}(V_{0})}\quad and\quad T_{0}=\frac{V_{0}}{k_{P}(P_{0})}
  19. k V ( V 0 ) k P ( P 0 ) = P 0 V 0 \frac{k_{V}(V_{0})}{k_{P}(P_{0})}=\frac{P_{0}}{V_{0}}\,\!
  20. k V ( V ) k P ( P ) = P V P , V \frac{k_{V}(V)}{k_{P}(P)}=\frac{P}{V}\quad\forall P,\forall V
  21. V k V ( V ) = P k P ( P ) V\,k_{V}(V)=P\,k_{P}(P)
  22. V k V ( V ) = k arb V\,k_{V}(V)=k_{\,\text{arb}}\,\!
  23. k V ( V ) = k arb V k_{V}(V)=\frac{k_{\,\text{arb}}}{V}
  24. P V T = k arb \frac{PV}{T}=k_{\,\text{arb}}\,\!

Commodity_channel_index.html

  1. C C I = 1 0.015 p t - S M A ( p t ) σ ( p t ) CCI=\frac{1}{0.015}\frac{p_{t}-SMA(p_{t})}{\sigma(p_{t})}
  2. Typical Price = H + L + C 3 \,\text{Typical Price}=\frac{H+L+C}{3}

Common_value_auction.html

  1. θ i = θ + ν i \theta_{i}=\theta+\nu_{i}
  2. θ \theta
  3. ν i \nu_{i}

Comparison_sort.html

  1. Ω ( n l o g n ) Ω(nlogn)
  2. n n
  3. Ω ( n ² l o g n ) Ω(n²logn)
  4. O ( n ² l o g n ) O(n²logn)
  5. O ( n ² ) O(n²)
  6. log 2 ( n ! ) \lceil\log_{2}(n!)\rceil
  7. log 2 ( n ! ) \lceil\log_{2}(n!)\rceil
  8. n log 2 n - n ln 2 n\log_{2}n-\frac{n}{\ln 2}
  9. log 2 ( n ! ) \lceil\log_{2}(n!)\rceil
  10. n log 2 n - n ln 2 n\log_{2}n-\frac{n}{\ln 2}
  11. n log ( n ) n\log(n)
  12. n n
  13. 2 f ( n ) n ! 2^{f(n)}\geq n!
  14. f ( n ) log 2 ( n ! ) . f(n)\geq\log_{2}(n!).
  15. log 2 ( n ! ) \log_{2}(n!)
  16. Ω ( n log 2 n ) \Omega(n\log_{2}n)
  17. log 2 ( n ! ) \lceil\log_{2}(n!)\rceil
  18. log 2 ( 13 ! ) = 33 \lceil\log_{2}(13!)\rceil=33

Complementary_code_keying.html

  1. 𝐜 = ( c 0 , , c 7 ) = ( e j ( ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 ) , e j ( ϕ 1 + ϕ 3 + ϕ 4 ) , e j ( ϕ 1 + ϕ 2 + ϕ 4 ) , - e j ( ϕ 1 + ϕ 4 ) , e j ( ϕ 1 + ϕ 2 + ϕ 3 ) , e j ( ϕ 1 + ϕ 3 ) , - e j ( ϕ 1 + ϕ 2 ) , e j ϕ 1 ) \mathbf{c}=(c_{0},\ldots,c_{7})=(e^{j(\phi_{1}+\phi_{2}+\phi_{3}+\phi_{4})},e^% {j(\phi_{1}+\phi_{3}+\phi_{4})},e^{j(\phi_{1}+\phi_{2}+\phi_{4})},-e^{j(\phi_{% 1}+\phi_{4})},e^{j(\phi_{1}+\phi_{2}+\phi_{3})},e^{j(\phi_{1}+\phi_{3})},-e^{j% (\phi_{1}+\phi_{2})},e^{j\phi_{1}})
  2. ϕ 1 , , ϕ 4 \phi_{1},\ldots,\phi_{4}
  3. ϕ 1 \phi_{1}
  4. ϕ 2 \phi_{2}
  5. c 0 c_{0}
  6. ϕ 3 \phi_{3}
  7. ϕ 4 \phi_{4}

Complete_intersection.html

  1. 3 \mathbb{P}^{3}
  2. 2 × 2 = 4 , 2\times 2=4,
  3. x z - y 2 = 0 xz-y^{2}=0
  4. z ( y w - z 2 ) - w ( x w - y z ) = 0 z(yw-z^{2})-w(xw-yz)=0
  5. 3 \mathbb{P}^{3}

Complete_market.html

  1. s s
  2. s 0 s_{0}
  3. t t
  4. s s
  5. s 0 s_{0}

Complete_numbering.html

  1. ν \nu
  2. A A
  3. a A a\in A
  4. f f
  5. h h
  6. ν h ( i ) = { ν f ( i ) if i dom ( f ) , a otherwise . \nu\circ h(i)=\left\{\begin{matrix}\nu\circ f(i)&\mbox{if}~{}\ i\in\mathrm{dom% }(f),\\ a&\mbox{otherwise}~{}.\end{matrix}\right.
  7. ν \nu
  8. ν f ( i ) = ν h ( i ) i dom ( f ) . \nu\circ f(i)=\nu\circ h(i)\qquad i\in\mathrm{dom}(f).\,

Complex_differential_form.html

  1. d z j = d x j + i d y j , d z ¯ j = d x j - i d y j , dz^{j}=dx^{j}+idy^{j},\quad d\bar{z}^{j}=dx^{j}-idy^{j},
  2. j = 1 n f j d z j + g j d z ¯ j . \sum_{j=1}^{n}f_{j}dz^{j}+g_{j}d\bar{z}^{j}.
  3. d z dz
  4. d z ¯ d\bar{z}
  5. Ω p , q = Ω 1 , 0 Ω 1 , 0 Ω 0 , 1 Ω 0 , 1 \Omega^{p,q}=\Omega^{1,0}\wedge\cdots\wedge\Omega^{1,0}\wedge\Omega^{0,1}% \wedge\cdots\wedge\Omega^{0,1}
  6. E k = Ω k , 0 Ω k - 1 , 1 Ω 1 , k - 1 Ω 0 , k = p + q = k Ω p , q . E^{k}=\Omega^{k,0}\oplus\Omega^{k-1,1}\oplus\cdots\oplus\Omega^{1,k-1}\oplus% \Omega^{0,k}=\bigoplus_{p+q=k}\Omega^{p,q}.
  7. π p , q : E k Ω p , q . \pi^{p,q}:E^{k}\rightarrow\Omega^{p,q}.
  8. d : E r E p + q + 1 d:E^{r}\to E^{p+q+1}
  9. d ( E p , q ) = r + s = p + q + 1 E r , s d(E^{p,q})=\sum_{r+s=p+q+1}E^{r,s}
  10. = π p + 1 , q d : Ω p , q Ω p + 1 , q , ¯ = π p , q + 1 d : Ω p , q Ω p , q + 1 \partial=\pi^{p+1,q}\circ d:\Omega^{p,q}\rightarrow\Omega^{p+1,q},\quad\bar{% \partial}=\pi^{p,q+1}\circ d:\Omega^{p,q}\rightarrow\Omega^{p,q+1}
  11. α = | I | = p , | J | = q f I J d z I d z ¯ J Ω p , q \alpha=\sum_{|I|=p,|J|=q}\ f_{IJ}\,dz^{I}\wedge d\bar{z}^{J}\in\Omega^{p,q}
  12. α = | I | , | J | f I J z d z d z I d z ¯ J \partial\alpha=\sum_{|I|,|J|}\sum_{\ell}\frac{\partial f_{IJ}}{\partial z^{% \ell}}\,dz^{\ell}\wedge dz^{I}\wedge d\bar{z}^{J}
  13. ¯ α = | I | , | J | f I J z ¯ d z ¯ d z I d z ¯ J . \bar{\partial}\alpha=\sum_{|I|,|J|}\sum_{\ell}\frac{\partial f_{IJ}}{\partial% \bar{z}^{\ell}}d\bar{z}^{\ell}\wedge dz^{I}\wedge d\bar{z}^{J}.
  14. d = + ¯ d=\partial+\bar{\partial}
  15. 2 = ¯ 2 = ¯ + ¯ = 0. \partial^{2}=\bar{\partial}^{2}=\partial\bar{\partial}+\bar{\partial}\partial=0.
  16. α = | I | = p f I d z I \alpha=\sum_{|I|=p}f_{I}\,dz^{I}
  17. ¯ α = 0. \bar{\partial}\alpha=0.

Complex_dimension.html

  1. x 2 + y 2 + z 2 = 0 x^{2}+y^{2}+z^{2}=0

Complex_Lie_group.html

  1. G × G G , ( x , y ) x y - 1 G\times G\to G,(x,y)\mapsto xy^{-1}
  2. GL n ( ) \operatorname{GL}_{n}(\mathbb{C})
  3. * \mathbb{C}^{*}
  4. g / L \mathbb{C}^{g}/L
  5. 𝔞 \mathfrak{a}
  6. exp : 𝔞 A \operatorname{exp}:\mathfrak{a}\to A
  7. * , z e z \mathbb{C}\to\mathbb{C}^{*},z\mapsto e^{z}
  8. * = GL 1 ( ) \mathbb{C}^{*}=\operatorname{GL}_{1}(\mathbb{C})
  9. Aut ( X ) \operatorname{Aut}(X)
  10. Γ ( X , T X ) \Gamma(X,TX)
  11. Lie ( G ) = Lie ( K ) \operatorname{Lie}(G)=\operatorname{Lie}(K)\otimes_{\mathbb{R}}\mathbb{C}
  12. GL n ( ) \operatorname{GL}_{n}(\mathbb{C})

Complex_measure.html

  1. μ \mu
  2. ( X , Σ ) (X,\Sigma)
  3. μ : Σ \mu:\Sigma\to\mathbb{C}
  4. ( A n ) n (A_{n})_{n\in\mathbb{N}}
  5. Σ \Sigma
  6. n = 1 μ ( A n ) = μ ( n = 1 A n ) . \sum_{n=1}^{\infty}\mu(A_{n})=\mu\left(\bigcup_{n=1}^{\infty}A_{n}\right)\in% \mathbb{C}.
  7. n = 1 A n = n = 1 A σ ( n ) \displaystyle\bigcup_{n=1}^{\infty}A_{n}=\bigcup_{n=1}^{\infty}A_{\sigma(n)}
  8. σ : \sigma:\mathbb{N}\to\mathbb{N}
  9. n = 1 μ ( A n ) \displaystyle\sum_{n=1}^{\infty}\mu(A_{n})
  10. μ 1 = μ 1 + - μ 1 - \mu_{1}=\mu_{1}^{+}-\mu_{1}^{-}
  11. μ 2 = μ 2 + - μ 2 - \mu_{2}=\mu_{2}^{+}-\mu_{2}^{-}
  12. X f d μ = ( X f d μ 1 + - X f d μ 1 - ) + i ( X f d μ 2 + - X f d μ 2 - ) \int_{X}\!f\,d\mu=\left(\int_{X}\!f\,d\mu_{1}^{+}-\int_{X}\!f\,d\mu_{1}^{-}% \right)+i\left(\int_{X}\!f\,d\mu_{2}^{+}-\int_{X}\!f\,d\mu_{2}^{-}\right)
  13. X f d μ = X ( f ) d μ + i X ( f ) d μ . \int_{X}\!f\,d\mu=\int_{X}\!\Re(f)\,d\mu+i\int_{X}\!\Im(f)\,d\mu.
  14. | μ | ( A ) = sup n = 1 | μ ( A n ) | |\mu|(A)=\sup\sum_{n=1}^{\infty}|\mu(A_{n})|
  15. d μ = e i θ d | μ | d\mu=e^{i\theta}d|\mu|\,
  16. X f d μ = X f e i θ d | μ | \int_{X}f\,d\mu=\int_{X}fe^{i\theta}\,d|\mu|
  17. X | f | d | μ | < . \int_{X}|f|\,d|\mu|<\infty.
  18. u \|u\|
  19. μ = | μ | ( X ) \|\mu\|=|\mu|(X)\,

Complex_wavelet_transform.html

  1. 2 d 2^{d}
  2. d d
  3. M M
  4. M M
  5. M M
  6. M M

Composition_operator.html

  1. C ϕ C_{\phi}
  2. ϕ \phi
  3. C ϕ ( f ) = f ϕ C_{\phi}(f)=f\circ\phi
  4. f ϕ f\circ\phi
  5. C ϕ C_{\phi}

Compressibility_factor.html

  1. Z Z
  2. Z = p n T Z=\frac{p}{nT}
  3. T T
  4. R R
  5. Z = p ρ R T , Z=\frac{p}{\rho RT},
  6. ρ \rho
  7. R = R 0 m R=\frac{R_{0}}{m}
  8. m m
  9. Z = 1 Z=1
  10. Z Z
  11. V m V_{\mathrm{m}}
  12. ( V m ) ideal gas = R T / p (V_{\mathrm{m}})_{\,\text{ideal gas}}=RT/p
  13. Z Z
  14. Z < 1 Z<1
  15. Z Z
  16. T r T_{r}
  17. P r P_{r}
  18. T r T_{r}
  19. P r P_{r}
  20. T r = T T c T_{r}=\frac{T}{T_{c}}
  21. P r = P P c . P_{r}=\frac{P}{P_{c}}.
  22. T c T_{c}
  23. P c P_{c}
  24. T c T_{c}
  25. P c P_{c}
  26. P r P_{r}
  27. T r T_{r}
  28. Z Z
  29. Z Z
  30. Z Z
  31. T r = T T c + 8 T_{r}=\frac{T}{T_{c}+8}
  32. P r = P P c + 8 P_{r}=\frac{P}{P_{c}+8}
  33. Z = 1 + B V m + C V m 2 + D V m 3 + Z=1+\frac{B}{V_{\mathrm{m}}}+\frac{C}{V_{\mathrm{m}}^{2}}+\frac{D}{V_{\mathrm{% m}}^{3}}+\dots
  34. B B
  35. C C
  36. Z = 1 + 2 π N A V M 0 ( 1 - e x p ( φ k T ) ) r 2 d r Z=1+2\pi\frac{N_{A}}{V_{M}}\int_{0}^{\infty}(1-exp({\frac{\varphi}{kT}))r^{2}dr}
  37. Z = 0.9152 Z=0.9152
  38. Z = 1.0025 Z=1.0025
  39. Z Z

Compression_theorem.html

  1. φ \varphi
  2. Φ \Phi
  3. f f
  4. C ( f ) := { φ i 𝐑 ( 1 ) | ( x ) Φ i ( x ) f ( x ) } . \mathrm{C}(f):=\{\varphi_{i}\in\mathbf{R}^{(1)}|(\forall^{\infty}x)\,\Phi_{i}(% x)\leq f(x)\}.
  5. f f
  6. i i
  7. Dom ( φ i ) = Dom ( φ f ( i ) ) \mathrm{Dom}(\varphi_{i})=\mathrm{Dom}(\varphi_{f(i)})
  8. C ( φ i ) C ( φ f ( i ) ) . \mathrm{C}(\varphi_{i})\subsetneq\mathrm{C}(\varphi_{f(i)}).

Compressor_map.html

  1. S M = 100 % m w ˙ - m s ˙ m w ˙ SM=100\%\cdot\frac{\dot{m_{w}}-\dot{m_{s}}}{\dot{m_{w}}}
  2. m w ˙ \dot{m_{w}}
  3. m s ˙ \dot{m_{s}}
  4. m w ˙ \dot{m_{w}}

Computable_isomorphism.html

  1. A ; B 𝒩 A;B\subseteq\mathcal{N}
  2. f : 𝒩 𝒩 f\colon\mathcal{N}\to\mathcal{N}
  3. f ( A ) = B f(A)=B
  4. ν \nu
  5. μ \mu
  6. f f
  7. ν = μ f \nu=\mu\circ f

Computation_in_the_limit.html

  1. r ( x ) r(x)
  2. r ^ ( x , s ) \hat{r}(x,s)
  3. r ^ = lim s r ^ ( x , s ) \displaystyle\hat{r}=\lim_{s\to\infty}\hat{r}(x,s)
  4. r ( x ) r(x)
  5. r ^ ( x , s ) \hat{r}(x,s)
  6. r ^ = lim s r ^ ( x , s ) \displaystyle\hat{r}=\lim_{s\to\infty}\hat{r}(x,s)
  7. ϕ ( t , i ) \phi(t,i)
  8. ϕ ( t , i ) \phi(t,i)
  9. 0 0^{\prime}
  10. D D
  11. D D^{\prime}
  12. r ^ ( x , s ) \hat{r}(x,s)
  13. r ^ ( x ) \hat{r}(x)
  14. 0 0^{\prime}
  15. r ^ ( x ) \hat{r}(x)
  16. 0 0^{\prime}
  17. r ^ ( x , s ) \hat{r}(x,s)
  18. r ^ ( x ) \hat{r}(x)
  19. 0 0^{\prime}
  20. r ^ ( x , s ) = { 1 if by stage s , x has been enumerated into 0 0 if not \displaystyle\hat{r}(x,s)=\begin{cases}1&\,\text{if by stage }s,x\,\text{ has % been enumerated into }0^{\prime}\\ 0&\,\text{if not}\end{cases}
  21. r ( x ) r(x)
  22. s s
  23. 0 0^{\prime}
  24. 0 0^{\prime}
  25. X , Y X,Y
  26. X s X_{s}
  27. X X
  28. Y ( z ) = ϕ X ( z ) Y(z)=\phi^{X}(z)
  29. ϕ \phi
  30. Y s Y_{s}
  31. Y s ( z ) = { ϕ X s ( z ) if ϕ X s converges in at most s steps. 0 otherwise \displaystyle Y_{s}(z)=\begin{cases}\phi^{X_{s}}(z)&\,\text{if }\phi^{X_{s}}\,% \text{ converges in at most }s\,\text{ steps.}\\ 0&\,\text{otherwise }\end{cases}
  32. ϕ X ( z ) \phi^{X}(z)
  33. s s
  34. s s
  35. X X
  36. s > s s^{\prime}>s
  37. z < s + 1 z<s+1
  38. X s ( z ) = X ( z ) X_{s^{\prime}}(z)=X(z)
  39. t > s t>s^{\prime}
  40. ϕ X t ( z ) \phi^{X_{t}}(z)
  41. s < t s^{\prime}<t
  42. ϕ X ( z ) \phi^{X}(z)
  43. Y s ( z ) Y_{s}(z)
  44. ϕ X ( z ) = Y ( z ) \phi^{X}(z)=Y(z)
  45. Y Y
  46. Δ 2 0 \Delta^{0}_{2}
  47. 0 0^{\prime}
  48. Δ 2 0 \Delta^{0}_{2}
  49. r i r_{i}
  50. ω \omega
  51. ϕ ( t , i ) \phi(t,i)
  52. { 0 , 1 } \{0,1\}
  53. lim t ϕ ( t , i ) \lim_{t\rightarrow\infty}\phi(t,i)
  54. ω ( i ) \omega(i)
  55. ϕ ( t , i ) \phi(t,i)
  56. ω ( i ) \omega(i)

Conchospiral.html

  1. r = μ t a r=\mu^{t}a
  2. θ = t \theta=t
  3. z = μ t c . z=\mu^{t}c.

Condensed_detachment.html

  1. p p
  2. q q
  3. p p
  4. q q
  5. p p
  6. q q
  7. r r
  8. p p
  9. r r
  10. p p
  11. r r
  12. p p
  13. t t
  14. r r
  15. t t
  16. p p
  17. r r

Conditional_factor_demands.html

  1. Minimize w L + r K with respect to L and K , \,\text{Minimize}\,wL+rK\,\,\,\text{with respect to}\,\,L\,\,\,\text{and}\,\,K,
  2. f ( L , K ) = q , f(L,K)=q,
  3. L ( w , r ; q ) L(w,r\,;q)
  4. K ( w , r ; q ) . K(w,r\,;q).

Condorcet's_jury_theorem.html

  1. P ( n , p ) = 1 / 2 + c 1 ( p - 1 / 2 ) + c 3 ( p - 1 / 2 ) 3 + O ( ( p - 1 / 2 ) 5 ) P(n,p)=1/2+c_{1}(p-1/2)+c_{3}(p-1/2)^{3}+O((p-1/2)^{5})
  2. c 1 = ( n n / 2 ) n / 2 + 1 4 n / 2 = 2 n + 1 π ( 1 + 1 16 n 2 + O ( n - 3 ) ) c_{1}={n\choose{\lfloor n/2\rfloor}}\frac{\lfloor n/2\rfloor+1}{4^{\lfloor n/2% \rfloor}}=\sqrt{\frac{2n+1}{\pi}}(1+\frac{1}{16n^{2}}+O(n^{-3}))
  3. c 3 < 0 c_{3}<0
  4. n \sqrt{n}

Conformal_Killing_equation.html

  1. g g
  2. X X
  3. g g
  4. X g = λ g \mathcal{L}_{X}g=\lambda g
  5. λ \lambda
  6. X \mathcal{L}_{X}
  7. λ = 2 n div X \lambda=\frac{2}{n}\mathrm{div}X
  8. ( X - 2 div X n ) g = 0. \left(\mathcal{L}_{X}-\frac{2\,\mathrm{div}\,X}{n}\right)g=0.
  9. ( a X b ) - 1 n g a b c X c = 0 , \nabla_{(a}X_{b)}-\frac{1}{n}g_{ab}\nabla_{c}X^{c}=0,
  10. n n

Confounding.html

  1. P ( y | d o ( x ) ) P(y|do(x))
  2. P ( y | x ) P(y|x)
  3. P ( y | d o ( x ) ) P(y|do(x))
  4. P ( y | d o ( x ) ) = z P ( y | x , z ) P ( z ) P(y|do(x))=\sum_{z}P(y|x,z)P(z)
  5. \leftarrow
  6. \rightarrow
  7. P ( Y = recovered | d o ( x = give drug ) ) \displaystyle P(Y=\,\text{recovered}\ |\ do(x=\,\text{give drug}))

Congruence_(general_relativity).html

  1. X \vec{X}
  2. X X = 0 \nabla_{\vec{X}}\vec{X}=0
  3. f f
  4. X \vec{X}
  5. Y = f X \vec{Y}=\,f\,\vec{X}
  6. X f = f , a X a \vec{X}f=f_{,a}\,X^{a}
  7. X X \nabla_{\vec{X}}\vec{X}
  8. X ˙ a = X a ; b X b \dot{X}^{a}={X^{a}}_{;b}X^{b}
  9. ( X ˙ a X b + X a ; b ) X b = X a ; b X b - X ˙ a = 0 \left(\dot{X}^{a}\,X_{b}+{X^{a}}_{;b}\right)\,X^{b}={X^{a}}_{;b}\,X^{b}-\dot{X% }^{a}=0
  10. X a ; b {X^{a}}_{;b}
  11. h a b = g a b + X a X b h_{ab}=g_{ab}+X_{a}\,X_{b}
  12. X \vec{X}
  13. X ˙ a X b + X a ; b = h m a h n b X m ; n \dot{X}_{a}\,X_{b}+X_{a;b}={h^{m}}_{a}\,{h^{n}}_{b}X_{m;n}
  14. X ˙ a X b + X a ; b = θ a b + ω a b \dot{X}_{a}\,X_{b}+X_{a;b}=\theta_{ab}+\omega_{ab}
  15. θ a b = h m a h n b X ( m ; n ) \theta_{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{(m;n)}
  16. ω a b = h m a h n b X [ m ; n ] \omega_{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{[m;n]}
  17. X \vec{X}
  18. θ \theta
  19. θ a b = σ a b + 1 3 θ h a b \theta_{ab}=\sigma_{ab}+\frac{1}{3}\,\theta\,h_{ab}
  20. X a ; b = σ a b + ω a b + 1 3 θ h a b - X ˙ a X b X_{a;b}=\sigma_{ab}+\omega_{ab}+\frac{1}{3}\,\theta\,h_{ab}-\dot{X}_{a}\,X_{b}
  21. σ a b \sigma_{ab}
  22. X a ; b n - X a ; n b = R a m b n X m X_{a;bn}-X_{a;nb}=R_{ambn}\,X^{m}
  23. E [ X ] a b = R a m b n X m X n E[\vec{X}]_{ab}=R_{ambn}\,X^{m}\,X^{n}
  24. ( X a : b n - X a : n b ) X n = E [ X ] a b \left(X_{a:bn}-X_{a:nb}\right)\,X^{n}=E[\vec{X}]_{ab}
  25. E [ X ] a b = 2 3 θ σ a b - σ a m σ m b - ω a m ω m b E[\vec{X}]_{ab}=\frac{2}{3}\,\theta\,\sigma_{ab}-\sigma_{am}\,{\sigma^{m}}_{b}% -\omega_{am}\,{\omega^{m}}_{b}
  26. - 1 3 ( θ ˙ + θ 2 3 ) h a b - h m a h n b ( σ ˙ m n - X ˙ ( m ; n ) ) - X ˙ a X ˙ b \cdots-\frac{1}{3}\left(\dot{\theta}+\frac{\theta^{2}}{3}\right)\,h_{ab}-{h^{m% }}_{a}\,{h^{n}}_{b}\,\left(\dot{\sigma}_{mn}-\dot{X}_{(m;n)}\right)-\dot{X}_{a% }\,\dot{X}_{b}
  27. X ˙ a = W a \dot{X}^{a}=W^{a}
  28. J a b = X a : b = θ 3 h a b + σ a b + ω a b - X ˙ a X b J_{ab}=X_{a:b}=\frac{\theta}{3}\,h_{ab}+\sigma_{ab}+\omega_{ab}-\dot{X}_{a}\,X% _{b}
  29. J ˙ a b = J a n ; b X n - E [ X ] a b \dot{J}_{ab}=J_{an;b}\,X^{n}-E[\vec{X}]_{ab}
  30. ( J a n X n ) ; b = J a n ; b X n + J a n X n ; b = J a n ; b X n + J a m J m b \left(J_{an}\,X^{n}\right)_{;b}=J_{an;b}\,X^{n}+J_{an}\,{X^{n}}_{;b}=J_{an;b}% \,X^{n}+J_{am}\,{J^{m}}_{b}
  31. J ˙ a b = - J a m J m b - E [ X ] a b + W a ; b \dot{J}_{ab}=-J_{am}\,{J^{m}}_{b}-{E[\vec{X}]}_{ab}+W_{a;b}
  32. J a b J_{ab}
  33. J a m J m b = θ 2 9 h a b + 2 θ 3 ( σ a b + ω a b ) + ( σ a m σ m b + ω a m ω m b ) + ( σ a m ω m b + ω a m σ m b ) J_{am}\,{J^{m}}_{b}=\frac{\theta^{2}}{9}\,h_{ab}+\frac{2\theta}{3}\,\left(% \sigma_{ab}+\omega_{ab}\right)+\left(\sigma_{am}\,{\sigma^{m}}_{b}+\omega_{am}% \,{\omega^{m}}_{b}\right)+\left(\sigma_{am}\,{\omega^{m}}_{b}+\omega_{am}\,{% \sigma^{m}}_{b}\right)
  34. J ˙ a b = - θ 2 9 h a b - 2 θ 3 ( σ a b + ω a b ) - ( σ a m σ m b + ω a m ω m b ) - ( σ a m ω m b + ω a m σ m b ) - E [ X ] a b \dot{J}_{ab}=-\frac{\theta^{2}}{9}\,h_{ab}-\frac{2\theta}{3}\,\left(\sigma_{ab% }+\omega_{ab}\right)-\left(\sigma_{am}\,{\sigma^{m}}_{b}+\omega_{am}\,{\omega^% {m}}_{b}\right)-\left(\sigma_{am}\,{\omega^{m}}_{b}+\omega_{am}\,{\sigma^{m}}_% {b}\right)-{E[\vec{X}]}_{ab}
  35. Σ , Ω \Sigma,\Omega
  36. Σ 2 + Ω 2 \Sigma^{2}+\Omega^{2}
  37. Σ Ω + Ω Σ \Sigma\,\Omega+\Omega\,\Sigma
  38. θ ˙ = ω 2 - σ 2 - θ 2 3 - E [ X ] m m \dot{\theta}=\omega^{2}-\sigma^{2}-\frac{\theta^{2}}{3}-{E[\vec{X}]^{m}}_{m}
  39. σ ˙ a b = - 2 θ 3 σ a b - ( σ a m σ m b + ω a m ω m b ) - E [ X ] a b + σ 2 - ω 2 + E [ X ] m m 3 h a b \dot{\sigma}_{ab}=-\frac{2\theta}{3}\,\sigma_{ab}-\left(\sigma_{am}\,{\sigma^{% m}}_{b}+\omega_{am}\,{\omega^{m}}_{b}\right)-{E[\vec{X}]}_{ab}+\frac{\sigma^{2% }-\omega^{2}+{E[\vec{X}]^{m}}_{m}}{3}\,h_{ab}
  40. ω ˙ a b = - 2 θ 3 ω a b - ( σ a m ω m b + ω a m σ m b ) \dot{\omega}_{ab}=-\frac{2\theta}{3}\,\omega_{ab}-\left(\sigma_{am}\,{\omega^{% m}}_{b}+\omega_{am}\,{\sigma^{m}}_{b}\right)
  41. σ 2 = σ m n σ m n , ω 2 = ω m n ω m n \sigma^{2}=\sigma_{mn}\,\sigma^{mn},\;\omega^{2}=\omega_{mn}\,\omega^{mn}
  42. σ , ω \sigma,\omega
  43. E [ X ] a a = R m n X m X n {E[\vec{X}]^{a}}_{a}=R_{mn}\,X^{m}\,X^{n}

Congruence_(manifolds).html

  1. X = ( x 2 - y 2 ) x + 2 x y y \vec{X}=(x^{2}-y^{2})\,\partial_{x}+2\,xy\,\partial_{y}
  2. x ˙ = x 2 - y 2 , y ˙ = 2 x y \dot{x}=x^{2}-y^{2},\;\dot{y}=2\,xy
  3. x ( λ ) = x 0 - ( x 0 2 + y 0 2 ) λ 1 - 2 x 0 λ + ( x 0 2 + y 0 2 ) λ 2 x(\lambda)=\frac{x_{0}-(x_{0}^{2}+y_{0}^{2})\,\lambda}{1-2\,x_{0}\,\lambda+(x_% {0}^{2}+y_{0}^{2})\,\lambda^{2}}
  4. y ( λ ) = y 0 1 - 2 x 0 λ + ( x 0 2 + y 0 2 ) λ 2 y(\lambda)=\frac{y_{0}}{1-2\,x_{0}\,\lambda+(x_{0}^{2}+y_{0}^{2})\,\lambda^{2}}
  5. d s 2 = ( 2 1 + x 2 + y 2 ) 2 ( d x 2 + d y 2 ) ds^{2}=\left(\frac{2}{1+x^{2}+y^{2}}\right)^{2}\,\left(dx^{2}+dy^{2}\right)
  6. d s 2 = d x 2 + d y 2 ds^{2}=dx^{2}+dy^{2}
  7. X \vec{X}
  8. X X = 0 \nabla_{\vec{X}}\vec{X}=0

Congruent_number.html

  1. * / * 2 \mathbb{Q}^{*}/\mathbb{Q}^{*2}
  2. a 2 + b 2 = c 2 1 2 a b = n . \begin{matrix}a^{2}+b^{2}&=&c^{2}\\ \tfrac{1}{2}ab&=&n.\end{matrix}
  3. y 2 = x 3 - n 2 x y^{2}=x^{3}-n^{2}x\,\!

Conjugate_points.html

  1. γ \gamma
  2. γ \gamma
  3. γ \gamma
  4. γ \gamma
  5. γ s ( t ) \gamma_{s}(t)
  6. s = 0 s=0
  7. γ s ( 1 ) \gamma_{s}(1)
  8. S 2 S^{2}
  9. n \mathbb{R}^{n}

Conjugate_variables_(thermodynamics).html

  1. σ i j \sigma_{ij}\,
  2. V × ε i j V\times\varepsilon_{ij}
  3. i i
  4. d U = T d S - P d V + i μ i d N i , \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V+\sum_{i}\mu_{i}\mathrm{d}N_{i}\,,
  5. μ i \mu_{i}
  6. N i N_{i}
  7. d U \mathrm{d}U
  8. σ i j \sigma_{ij}
  9. ε i j \varepsilon_{ij}
  10. d ε i j d\varepsilon_{ij}
  11. δ w = - V i j σ i j d ε i j \delta w=-V\sum_{ij}\sigma_{ij}d\varepsilon_{ij}
  12. δ w = - V σ i j d ε i j \delta w=-V\sigma_{ij}d\varepsilon_{ij}
  13. δ w = - V ( - P δ i j ) d ε i j = P V d ε k k \delta w=-V\,(-P\delta_{ij})\,d\varepsilon_{ij}=PVd\varepsilon_{kk}
  14. ε k k \varepsilon_{kk}
  15. δ w = P d V \delta w=PdV

Conjugation_of_isometries_in_Euclidean_space.html

  1. \cong
  2. x A x + b x\mapsto Ax+b
  3. \cong
  4. \cong
  5. \cong
  6. \cong
  7. \cong
  8. \cong
  9. \cong
  10. \cong
  11. \cong
  12. \cong

Connected_dominating_set.html

  1. n = d + l . \displaystyle n=d+l.

Conservativity_theorem.html

  1. x 1 x m φ ( x 1 , , x m ) \exists x_{1}\ldots\exists x_{m}\,\varphi(x_{1},\ldots,x_{m})
  2. T T
  3. T 1 T_{1}
  4. T T
  5. a 1 , , a m a_{1},\ldots,a_{m}
  6. φ ( a 1 , , a m ) \varphi(a_{1},\ldots,a_{m})
  7. T 1 T_{1}
  8. T T
  9. T 1 T_{1}
  10. a i a_{i}\,\!
  11. T T
  12. y x φ ( x , y ) \forall\vec{y}\,\exists x\,\!\,\varphi(x,\vec{y})
  13. T T
  14. y := ( y 1 , , y n ) \vec{y}:=(y_{1},\ldots,y_{n})
  15. T 1 T_{1}
  16. T T
  17. f f\,\!
  18. n n
  19. y φ ( f ( y ) , y ) \forall\vec{y}\,\varphi(f(\vec{y}),\vec{y})
  20. T 1 T_{1}
  21. T T
  22. T T
  23. T 1 T_{1}
  24. f f\,\!

Conserved_quantity.html

  1. d 𝐫 d t = 𝐟 ( 𝐫 , t ) \frac{d\mathbf{r}}{dt}=\mathbf{f}(\mathbf{r},t)
  2. d H d t = 0 \frac{dH}{dt}=0
  3. d H d t = H d 𝐫 d t = H 𝐟 ( 𝐫 , t ) \frac{dH}{dt}=\nabla H\cdot\frac{d\mathbf{r}}{dt}=\nabla H\cdot\mathbf{f}(% \mathbf{r},t)
  4. H 𝐟 ( 𝐫 , t ) = 0 \nabla H\cdot\mathbf{f}(\mathbf{r},t)=0
  5. d f d t = { f , } + f t \frac{\mathrm{d}f}{\mathrm{d}t}=\{f,\mathcal{H}\}+\frac{\partial f}{\partial t}
  6. { f , } + f t = 0 \{f,\mathcal{H}\}+\frac{\partial f}{\partial t}=0
  7. { f , } \{f,\mathcal{H}\}
  8. L t = 0 \frac{\partial L}{\partial t}=0
  9. E = i [ q ˙ i L q ˙ i ] - L E=\sum_{i}\left[\dot{q}_{i}\frac{\partial L}{\partial\dot{q}_{i}}\right]-L
  10. L q = 0 \frac{\partial L}{\partial q}=0
  11. p = L q ˙ p=\frac{\partial L}{\partial\dot{q}}

Consistent_hashing.html

  1. K / n K/n
  2. K K
  3. n n
  4. K / n K/n
  5. n n
  6. o o
  7. hash ( o ) mod n \mbox{hash}~{}(o)\mod n
  8. n n

Constant_elasticity_of_substitution.html

  1. Q = F ( a K r + ( 1 - a ) L r ) 1 r Q=F\cdot\left(a\cdot K^{r}+(1-a)\cdot L^{r}\right)^{\frac{1}{r}}
  2. Q Q
  3. F F
  4. a a
  5. K K
  6. L L
  7. r r
  8. ( s - 1 ) s {\frac{(s-1)}{s}}
  9. s s
  10. 1 ( 1 - r ) {\frac{1}{(1-r)}}
  11. r = 1 r=1
  12. r r
  13. r r
  14. Q = F [ i = 1 n a i X i r ] 1 r Q=F\cdot\left[\sum_{i=1}^{n}a_{i}X_{i}^{r}\ \right]^{\frac{1}{r}}
  15. Q Q
  16. F F
  17. a i a_{i}
  18. i = 1 n a i = 1 \sum_{i=1}^{n}a_{i}=1
  19. X i X_{i}
  20. s = 1 1 - r s=\frac{1}{1-r}
  21. n n
  22. c i c_{i}
  23. C C
  24. C = [ i = 1 n a i 1 s c i ( s - 1 ) s ] s ( s - 1 ) C=\left[\sum_{i=1}^{n}a_{i}^{\frac{1}{s}}c_{i}^{\frac{(s-1)}{s}}\ \right]^{% \frac{s}{(s-1)}}
  25. a i a_{i}
  26. s s
  27. c i c_{i}
  28. s s
  29. s s

Constraint_counting.html

  1. u t t = u x x + u y y . u_{tt}=u_{xx}+u_{yy}.
  2. u ( t , x , y ) u(t,x,y)
  3. u u
  4. u t y t = u t t y = u x x y + u y y y u_{tyt}=u_{tty}=u_{xxy}+u_{yyy}
  5. s ( ξ ) = k = 0 s k ξ k s(\xi)=\sum_{k=0}^{\infty}s_{k}\xi^{k}
  6. s k s_{k}
  7. f ( ξ ) = 1 ( 1 - ξ ) 3 = 1 + 3 ξ + 6 ξ 2 + 10 ξ 3 + f(\xi)=\frac{1}{(1-\xi)^{3}}=1+3\xi+6\xi^{2}+10\xi^{3}+\dots
  8. g ( ξ ) = 1 - ξ 2 ( 1 - ξ ) 3 = 1 + 2 ξ + 5 ξ 2 + 7 ξ 3 + g(\xi)=\frac{1-\xi^{2}}{(1-\xi)^{3}}=1+2\xi+5\xi^{2}+7\xi^{3}+\dots
  9. u t t u_{tt}
  10. u t t t , u t t x , u t t y u_{ttt},\,u_{ttx},\,u_{tty}
  11. s [ n ] ( ξ ) = 1 / ( 1 - ξ ) n = 1 + n ξ + ( n 2 ) ξ 2 + ( n + 1 3 ) ξ 3 + s[n](\xi)=1/(1-\xi)^{n}=1+n\,\xi+\left(\begin{matrix}n\\ 2\end{matrix}\right)\,\xi^{2}+\left(\begin{matrix}n+1\\ 3\end{matrix}\right)\,\xi^{3}+\dots
  12. g ( ξ ) = 1 - ξ m ( 1 - ξ ) n g(\xi)=\frac{1-\xi^{m}}{(1-\xi)^{n}}
  13. 1 - ξ 2 ( 1 - ξ ) 3 = 1 + ξ ( 1 - ξ ) 2 \frac{1-\xi^{2}}{(1-\xi)^{3}}=\frac{1+\xi}{(1-\xi)^{2}}
  14. u t t = u x x + u y y , u ( 0 , x , y ) = p ( x , y ) , u t ( 0 , x , y ) = q ( x , y ) u_{tt}=u_{xx}+u_{yy},\;u(0,x,y)=p(x,y),\;u_{t}(0,x,y)=q(x,y)
  15. u ( t , x , y ) [ L u ] ( ω , x , y ) u(t,x,y)\mapsto[Lu](\omega,x,y)
  16. - ω 2 [ L u ] + ω p ( x , y ) + q ( x , y ) + [ L u ] x + [ L u ] y -\omega^{2}\,[Lu]+\omega\,p(x,y)+q(x,y)+[Lu]_{x}+[Lu]_{y}
  17. [ L u ] ( ω , x , y ) [ F L U ] ( ω , m , n ) [Lu](\omega,x,y)\mapsto[FLU](\omega,m,n)
  18. - ω 2 [ F L u ] + ω [ F p ] + [ F q ] - ( m 2 + n 2 ) [ F L u ] -\omega^{2}\,[FLu]+\omega\,[Fp]+[Fq]-(m^{2}+n^{2})\,[FLu]
  19. [ F L u ] ( ω , m , n ) = ω [ F p ] ( m , n ) + [ F q ] ( m , n ) ω 2 + m 2 + n 2 [FLu](\omega,m,n)=\frac{\omega\,[Fp](m,n)+[Fq](m,n)}{\omega^{2}+m^{2}+n^{2}}
  20. [ F u ] ( t , m , n ) = [ F p ] ( m , n ) cos ( m 2 + n 2 t ) + [ F q ] ( m , n ) sin ( m 2 + n 2 t ) m 2 + n 2 [Fu](t,m,n)=[Fp](m,n)\,\cos(\sqrt{m^{2}+n^{2}}\,t)+\frac{[Fq](m,n)\,\sin(\sqrt% {m^{2}+n^{2}}\,t)}{\sqrt{m^{2}+n^{2}}}
  21. u ( t , x , y ) = Q ( t , x , y ) + P t ( t , x , y ) u(t,x,y)=Q(t,x,y)+P_{t}(t,x,y)
  22. P ( t , x , y ) = 1 2 π ( x - x ) 2 + ( y - y ) 2 < t 2 p ( x , y ) d x d y [ t 2 - ( x - x ) 2 - ( y - y ) 2 ] 1 / 2 P(t,x,y)=\frac{1}{2\pi}\,\int_{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}<t^{2}}% \frac{p(x^{\prime},y^{\prime})\,dx^{\prime}dy^{\prime}}{\left[t^{2}-(x-x^{% \prime})^{2}-(y-y^{\prime})^{2}\right]^{1/2}}
  23. Q ( t , x , y ) = 1 2 π ( x - x ) 2 + ( y - y ) 2 < t 2 q ( x , y ) d x d y [ t 2 - ( x - x ) 2 - ( y - y ) 2 ] 1 / 2 Q(t,x,y)=\frac{1}{2\pi}\,\int_{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}<t^{2}}% \frac{q(x^{\prime},y^{\prime})\,dx^{\prime}dy^{\prime}}{\left[t^{2}-(x-x^{% \prime})^{2}-(y-y^{\prime})^{2}\right]^{1/2}}

Content_validity.html

  1. C V R = ( n e - N / 2 ) / ( N / 2 ) CVR=(n_{e}-N/2)/(N/2)
  2. C V R = CVR=
  3. n e = n_{e}=
  4. N = N=

Continuous_stirred-tank_reactor.html

  1. [ a c c u m u l a t i o n ] = [ i n ] - [ o u t ] + [ g e n e r a t i o n ] [accumulation]=[in]-[out]+[generation]
  2. d N i d t = F i o - F i + V ν i r i \frac{dN_{i}}{dt}=F_{io}-F_{i}+V\nu_{i}r_{i}
  3. ν i \nu_{i}
  4. τ \tau
  5. C A = C A o 1 + k τ C_{A}=\frac{C_{Ao}}{1+k\tau}

Control_bus.html

  1. R ¯ \overline{R}
  2. W ¯ \overline{W}
  3. E ¯ \overline{E}

Control_variates.html

  1. μ \mu
  2. m m
  3. 𝔼 [ m ] = μ \mathbb{E}\left[m\right]=\mu
  4. t t
  5. 𝔼 [ t ] = τ \mathbb{E}\left[t\right]=\tau
  6. m = m + c ( t - τ ) m^{\star}=m+c\left(t-\tau\right)\,
  7. μ \mu
  8. c c
  9. m m^{\star}
  10. Var ( m ) = Var ( m ) + c 2 Var ( t ) + 2 c Cov ( m , t ) ; \textrm{Var}\left(m^{\star}\right)=\textrm{Var}\left(m\right)+c^{2}\,\textrm{% Var}\left(t\right)+2c\,\textrm{Cov}\left(m,t\right);
  11. c = - Cov ( m , t ) Var ( t ) ; c^{\star}=-\frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)};
  12. m m^{\star}
  13. Var ( m ) = Var ( m ) - [ Cov ( m , t ) ] 2 Var ( t ) = ( 1 - ρ m , t 2 ) Var ( m ) ; \begin{aligned}\displaystyle\textrm{Var}\left(m^{\star}\right)&\displaystyle=% \textrm{Var}\left(m\right)-\frac{\left[\textrm{Cov}\left(m,t\right)\right]^{2}% }{\textrm{Var}\left(t\right)}\\ &\displaystyle=\left(1-\rho_{m,t}^{2}\right)\textrm{Var}\left(m\right);\end{aligned}
  14. ρ m , t = Corr ( m , t ) ; \rho_{m,t}=\textrm{Corr}\left(m,t\right);\,
  15. | ρ m , t | |\rho_{m,t}|
  16. Cov ( m , t ) \textrm{Cov}\left(m,t\right)
  17. Var ( t ) \textrm{Var}\left(t\right)
  18. ρ m , t \rho_{m,t}\;
  19. I = 0 1 1 1 + x d x I=\int_{0}^{1}\frac{1}{1+x}\,\mathrm{d}x
  20. f ( U ) f(U)
  21. f ( x ) = 1 1 + x f(x)=\frac{1}{1+x}
  22. u 1 , , u n u_{1},\cdots,u_{n}
  23. I 1 n i f ( u i ) ; I\approx\frac{1}{n}\sum_{i}f(u_{i});
  24. g ( x ) = 1 + x g(x)=1+x
  25. 𝔼 [ g ( U ) ] = 0 1 ( 1 + x ) d x = 3 2 \mathbb{E}\left[g\left(U\right)\right]=\int_{0}^{1}(1+x)\,\mathrm{d}x=\frac{3}% {2}
  26. I 1 n i f ( u i ) + c ( 1 n i g ( u i ) - 3 / 2 ) . I\approx\frac{1}{n}\sum_{i}f(u_{i})+c\left(\frac{1}{n}\sum_{i}g(u_{i})-3/2% \right).
  27. n = 1500 n=1500
  28. c 0.4773 c^{\star}\approx 0.4773
  29. I = ln 2 0.69314718 I=\ln 2\approx 0.69314718

Controlled_NOT_gate.html

  1. | 1 |1\rangle
  2. | 0 |0\rangle
  3. | 0 |0\rangle
  4. | 0 |0\rangle
  5. | 0 |0\rangle
  6. | 0 |0\rangle
  7. | 1 |1\rangle
  8. | 0 |0\rangle
  9. | 1 |1\rangle
  10. | 1 |1\rangle
  11. | 0 |0\rangle
  12. | 1 |1\rangle
  13. | 1 |1\rangle
  14. | 1 |1\rangle
  15. | 1 |1\rangle
  16. | 1 |1\rangle
  17. | 0 |0\rangle
  18. { | 0 , | 1 } \{|0\rangle,|1\rangle\}
  19. | 1 |1\rangle
  20. { | 0 , | 1 } \{|0\rangle,|1\rangle\}
  21. a | 00 + b | 01 + c | 10 + d | 11 a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle
  22. a | 00 + b | 01 + c | 11 + d | 10 a|00\rangle+b|01\rangle+c|11\rangle+d|10\rangle
  23. CNOT = [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] . \operatorname{CNOT}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0\end{bmatrix}.
  24. { | 0 , | 1 } \{|0\rangle,|1\rangle\}
  25. { | + , | - } \{|+\rangle,|-\rangle\}
  26. | + = 1 2 ( | 0 + | 1 ) , | - = 1 2 ( | 0 - | 1 ) , |+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\qquad|-\rangle=\frac{1}{% \sqrt{2}}(|0\rangle-|1\rangle),
  27. | + + = | + | + = 1 2 ( | 0 + | 1 ) ( | 0 + | 1 ) = 1 2 ( | 00 + | 01 + | 10 + | 11 ) |++\rangle=|+\rangle|+\rangle=\frac{1}{2}(|0\rangle+|1\rangle)(|0\rangle+|1% \rangle)=\frac{1}{2}(|00\rangle+|01\rangle+|10\rangle+|11\rangle)
  28. { | 0 , | 1 } \{|0\rangle,|1\rangle\}
  29. { | + , | - } \{|+\rangle,|-\rangle\}
  30. | + |+\rangle
  31. | - |-\rangle
  32. | - |-\rangle
  33. | + + |++\rangle
  34. 1 2 ( | 00 + | 01 + | 10 + | 11 ) \frac{1}{2}(|00\rangle+|01\rangle+|10\rangle+|11\rangle)
  35. 1 2 ( | 00 + | 01 + | 11 + | 10 ) \frac{1}{2}(|00\rangle+|01\rangle+|11\rangle+|10\rangle)
  36. | + + |++\rangle
  37. | + - |+-\rangle
  38. 1 2 ( | 00 - | 01 + | 10 - | 11 ) \frac{1}{2}(|00\rangle-|01\rangle+|10\rangle-|11\rangle)
  39. 1 2 ( | 00 - | 01 + | 11 - | 10 ) \frac{1}{2}(|00\rangle-|01\rangle+|11\rangle-|10\rangle)
  40. | - - |--\rangle
  41. | - + |-+\rangle
  42. 1 2 ( | 00 + | 01 - | 10 - | 11 ) \frac{1}{2}(|00\rangle+|01\rangle-|10\rangle-|11\rangle)
  43. 1 2 ( | 00 + | 01 - | 11 - | 10 ) \frac{1}{2}(|00\rangle+|01\rangle-|11\rangle-|10\rangle)
  44. | - + |-+\rangle
  45. | - - |--\rangle
  46. 1 2 ( | 00 - | 01 - | 10 + | 11 ) \frac{1}{2}(|00\rangle-|01\rangle-|10\rangle+|11\rangle)
  47. 1 2 ( | 00 - | 01 - | 11 + | 10 ) \frac{1}{2}(|00\rangle-|01\rangle-|11\rangle+|10\rangle)
  48. | + - |+-\rangle
  49. | 01 |01\rangle
  50. | 11 |11\rangle
  51. | 00 |00\rangle
  52. | 10 |10\rangle
  53. 1 4 \displaystyle\frac{1}{4}
  54. | Φ + |\Phi^{+}\rangle
  55. | Φ + |\Phi^{+}\rangle
  56. | Φ + |\Phi^{+}\rangle
  57. 1 2 ( | 0 + | 1 ) A \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)_{A}
  58. | 0 B |0\rangle_{B}
  59. 1 2 ( | 00 + | 11 ) \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)
  60. | + A |+\rangle_{A}
  61. 1 2 ( | + + | - ) B \frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)_{B}
  62. | - B |-\rangle_{B}
  63. 1 2 ( | + + + | - - ) \frac{1}{\sqrt{2}}(|++\rangle+|--\rangle)
  64. = 1 2 ( | + A | + B + | - A | - B ) =\frac{1}{\sqrt{2}}(|+\rangle_{A}|+\rangle_{B}+|-\rangle_{A}|-\rangle_{B})
  65. = 1 2 2 ( ( | 0 A + | 1 A ) ( | 0 B + | 1 B ) + ( | 0 A - | 1 A ) ( | 0 B - | 1 B ) ) =\frac{1}{2\sqrt{2}}((|0\rangle_{A}+|1\rangle_{A})(|0\rangle_{B}+|1\rangle_{B}% )+(|0\rangle_{A}-|1\rangle_{A})(|0\rangle_{B}-|1\rangle_{B}))
  66. = 1 2 2 ( ( | 00 + | 01 + | 10 + | 11 ) + ( | 00 - | 01 - | 10 + | 11 ) ) =\frac{1}{2\sqrt{2}}((|00\rangle+|01\rangle+|10\rangle+|11\rangle)+(|00\rangle% -|01\rangle-|10\rangle+|11\rangle))
  67. = 1 2 ( | 00 + | 11 ) =\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)
  68. 1 2 ( | 00 + | 11 ) \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)

Convective_inhibition.html

  1. T v , p a r c e l T_{v,parcel}
  2. T v , e n v T_{v,env}
  3. CIN = z bottom z top g ( T v,parcel - T v,env T v,env ) d z \,\text{CIN}=\int_{z\text{bottom}}^{z\text{top}}g\left(\frac{T\text{v,parcel}-% T\text{v,env}}{T\text{v,env}}\right)dz

Coppersmith–Winograd_algorithm.html

  1. n × n n\times n
  2. O ( n 2.375477 ) O(n^{2.375477})
  3. O ( n 3 ) O(n^{3})
  4. O ( n 2.807355 ) O(n^{2.807355})
  5. n × n n\times n
  6. n 2 n^{2}
  7. O ( n 2.374 ) . O(n^{2.374}).
  8. O ( n 2.3728642 ) . O(n^{2.3728642}).
  9. O ( n 2.3728639 ) . O(n^{2.3728639}).

Coriolis_field.html

  1. ω \vec{\omega}
  2. v \vec{v}
  3. 𝐚 r = 𝐚 i - 2 s y m b o l ω × 𝐯 - s y m b o l ω × ( s y m b o l ω × 𝐫 ) - d s y m b o l ω d t × 𝐫 \mathbf{a}_{\mathrm{r}}=\mathbf{a}_{\mathrm{i}}-2symbol\omega\times\mathbf{v}-% symbol\omega\times(symbol\omega\times\mathbf{r})-\frac{dsymbol\omega}{dt}% \times\mathbf{r}
  4. 𝐅 Coriolis = - 2 m ( s y m b o l ω × 𝐯 ) = - 2 ( s y m b o l ω × 𝐩 ) \mathbf{F}_{\mathrm{Coriolis}}=-2m(symbol\omega\times\mathbf{v})=-2(symbol% \omega\times\mathbf{p})
  5. p \vec{p}
  6. ω \omega
  7. 𝐅 Coriolis = - 2 ( ω × 𝐩 ) = - 2 ( ω × ) 𝐩 = [ 0 - 2 ω 3 2 ω 2 2 ω 3 0 - 2 ω 1 - 2 ω 2 2 ω 1 0 ] [ p 1 p 2 p 3 ] \mathbf{F}_{\mathrm{Coriolis}}=-2(\mathbf{\omega}\times\mathbf{p})=-2(\mathbf{% \omega}\times)\mathbf{p}=\begin{bmatrix}\,0&\!-2\omega_{3}&\,\,2\omega_{2}\\ \,\,2\omega_{3}&0&\!-2\omega_{1}\\ -2\omega_{2}&\,\,2\omega_{1}&\,0\end{bmatrix}\begin{bmatrix}p_{1}\\ p_{2}\\ p_{3}\end{bmatrix}

Corrected_flow.html

  1. w θ / δ w\sqrt{\theta}/{\delta}
  2. w θ / δ = w T / 518.7 / ( P / 14.696 ) w\sqrt{\theta}/{\delta}=w\sqrt{T/518.7}/(P/14.696)
  3. w c wc
  4. w r wr
  5. w T / P w\sqrt{T}/{P}
  6. w T / P = w θ / δ * 518.7 / 14.696 w\sqrt{T}/{P}=w\sqrt{\theta}/{\delta}*\sqrt{518.7}/{14.696}
  7. w θ / δ = w T / 288.15 / ( P / 101.325 ) w\sqrt{\theta}/{\delta}=w\sqrt{T/288.15}/(P/101.325)
  8. w T / P = w θ / δ * 288.15 / 101.325 w\sqrt{T}/{P}=w\sqrt{\theta}/{\delta}*\sqrt{288.15}/{101.325}
  9. P P
  10. T T
  11. w w
  12. δ {\delta}
  13. θ {\theta}
  14. w / ( δ * θ ) w/(\delta*\sqrt{\theta})

Corrected_fuel_flow.html

  1. w f e / ( δ . θ ) = w f e / [ ( P / 14.696 ) . ( T / 288.15 ) ] wfe/({\delta}.\sqrt{\theta})=wfe/[(P/14.696).(\sqrt{T}/\sqrt{288.15})]
  2. ( w / ( P . T (w/({P}.\sqrt{T}
  3. w f e / ( P . T ) = [ w f e / ( δ . θ ) ] * 288.15 / 14.696 ) wfe/({P}.\sqrt{T})=[wfe/({\delta}.\sqrt{\theta})]*\sqrt{288.15}/{14.696})
  4. w f e / ( δ . θ ) = w f e / [ ( P / 101.325 ) . ( T / 288.15 ) ] wfe/({\delta}.\sqrt{\theta})=wfe/[(P/101.325).(\sqrt{T}/\sqrt{288.15})]
  5. w f e / ( P . T ) = [ w f e / ( δ . θ ) ] * ( 288.15 / 101.325 ) wfe/({P}.\sqrt{T})=[wfe/({\delta}.\sqrt{\theta})]*(\sqrt{288.15}/{101.325})
  6. P P
  7. T T

Corrected_speed.html

  1. N / θ N/{\sqrt{\theta}}
  2. N / θ = N / T / 288.15 N/\sqrt{\theta}=N/\sqrt{T/288.15}
  3. N c Nc
  4. N r Nr
  5. ( N / T ) (N/\sqrt{T})
  6. N / T = ( N / θ ) / 288.15 N/\sqrt{T}=(N/\sqrt{\theta})/\sqrt{288.15}
  7. T T
  8. N N
  9. θ {\theta}

Correlation_dimension.html

  1. x ( i ) = [ x 1 ( i ) , x 2 ( i ) , , x m ( i ) ] , i = 1 , 2 , N \vec{x}(i)=[x_{1}(i),x_{2}(i),\ldots,x_{m}(i)],\qquad i=1,2,\ldots N
  2. C ( ε ) = lim N g N 2 C(\varepsilon)=\lim_{N\rightarrow\infty}\frac{g}{N^{2}}
  3. C ( ε ) ε ν C(\varepsilon)\sim\varepsilon^{\nu}\,

Cosmic_neutrino_background.html

  1. σ g T 3 \sigma\propto gT^{3}
  2. ( g 0 g 1 ) 1 / 3 = T 1 T 0 \left(\frac{g_{0}}{g_{1}}\right)^{1/3}=\frac{T_{1}}{T_{0}}
  3. T ν T γ = ( 4 11 ) 1 / 3 \frac{T_{\nu}}{T_{\gamma}}=\left(\frac{4}{11}\right)^{1/3}
  4. ρ R = π 2 15 T γ 4 ( 1 + z ) 4 [ 1 + 7 8 N ν ( 4 11 ) 4 / 3 ] , \rho_{\rm R}=\frac{\pi^{2}}{15}\,T_{\gamma}^{4}(1+z)^{4}\left[1+\frac{7}{8}N_{% \rm\nu}\left(\frac{4}{11}\right)^{4/3}\right],

Cosmological_horizon.html

  1. t = t 0 t=t_{0}
  2. χ = c t \chi=ct
  3. t t
  4. t = 0 a d a H 0 Ω R a - 2 + Ω m a - 1 + Ω k + Ω Λ a 2 t=\int^{a}_{0}{\frac{da}{H_{0}\sqrt{\Omega_{R}a^{-2}+\Omega_{m}a^{-1}+\Omega_{% k}+\Omega_{\Lambda}a^{2}}}}
  5. H 0 H_{0}
  6. Ω \Omega
  7. χ 0 = c H 0 \chi_{0}=\frac{c}{H_{0}}
  8. t t
  9. d e ( t ) = a ( t ) t t m a x c d t a ( t ) d_{e}(t)=a(t)\int_{t}^{t_{max}}\frac{cdt^{\prime}}{a(t^{\prime})}
  10. t m a x t_{max}
  11. d e ( t 0 ) < d_{e}(t_{0})<\infty
  12. t t\rightarrow\infty

Costas_array.html

  1. A i , j = 1 A_{i,j}=1
  2. j g i mod p j\equiv g^{i}\bmod p
  3. A i , j = 1 A_{i,j}=1
  4. α i + β j = 1 \alpha^{i}+\beta^{j}=1

COSYSMO.html

  1. P M N S = A × S i z e E × i = 1 n E M i PM_{NS}=A\times Size^{E}\times\prod^{n}_{i=1}EM_{i}

Couette_flow.html

  1. d 2 u d y 2 = 0 , \frac{d^{2}u}{dy^{2}}=0,
  2. ( u , v , w ) (u,v,w)
  3. u ( y ) = u 0 y h u(y)=u_{0}\frac{y}{h}
  4. d 2 u d y 2 = 1 μ d p d x , \frac{d^{2}u}{dy^{2}}=\frac{1}{\mu}\frac{dp}{dx},
  5. d p / d x dp\!/\!dx
  6. μ \mu
  7. u ( y ) = u 0 y h + 1 2 μ ( d p d x ) ( y 2 - h y ) . u(y)=u_{0}\frac{y}{h}+\frac{1}{2\mu}\left(\frac{dp}{dx}\right)\left(y^{2}-hy% \right).
  8. P = - h 2 2 μ u 0 ( d p d x ) . P=-\frac{h^{2}}{2\mu u_{0}}\left(\frac{dp}{dx}\right).
  9. u ( r ) = C 1 r + C 2 r , u(r)=C_{1}r+\frac{C_{2}}{r},
  10. R 1 R_{1}
  11. R 2 R_{2}
  12. R 2 R_{2}
  13. R 1 R_{1}

Coulomb's_constant.html

  1. k e s = 0 {k}_{e}^{s=0}
  2. k e s = 0 {k}_{e}^{s=0}
  3. × 10 9 \times 10^{9}
  4. 𝐅 = k e Q q r 2 𝐞 ^ r \mathbf{F}=k\text{e}\frac{Qq}{r^{2}}\mathbf{\hat{e}}_{r}
  5. r {}_{r}
  6. 𝐅 = 1 4 π ε 0 Q q r 2 𝐞 ^ r = k e Q q r 2 𝐞 ^ r \mathbf{F}=\frac{1}{4\pi\varepsilon_{0}}\frac{Qq}{r^{2}}\mathbf{\hat{e}}_{r}=k% \text{e}\frac{Qq}{r^{2}}\mathbf{\hat{e}}_{r}
  7. k e = 1 4 π ε 0 \therefore k\text{e}=\frac{1}{4\pi\varepsilon_{0}}
  8. k e s = 0 {k}_{e}^{s=0}
  9. c 0 s = 0 {c}_{0}^{s=0}
  10. μ 0 s = 0 {μ}_{0}^{s=0}
  11. ε 0 s = 0 {ε}_{0}^{s=0}
  12. 1 μ 0 ε 0 = c 0 2 . \frac{1}{\mu_{0}\varepsilon_{0}}=c_{0}^{2}.
  13. c 0 s = 0 {c}_{0}^{s=0}
  14. μ 0 s = 0 {μ}_{0}^{s=0}
  15. ε 0 s = 0 {ε}_{0}^{s=0}
  16. k e = 1 4 π ε 0 = c 0 2 μ 0 4 π = c 0 2 × 10 - 7 H m - 1 = 8.987 551 787 368 176 4 × 10 9 N m 2 C - 2 . \begin{aligned}\displaystyle k\text{e}=\frac{1}{4\pi\varepsilon_{0}}=\frac{c_{% 0}^{2}\mu_{0}}{4\pi}&\displaystyle=c_{0}^{2}\times 10^{-7}\ \mathrm{H\ m}^{-1}% \\ &\displaystyle=8.987\ 551\ 787\ 368\ 176\ 4\times 10^{9}\ \mathrm{N\ m^{2}\ C}% ^{-2}.\end{aligned}
  17. k e = 1 4 π ε 0 k\text{e}=\frac{1}{4\pi\varepsilon_{0}}
  18. 𝐅 = k e Q q r 2 𝐞 ^ r \mathbf{F}=k\text{e}{Qq\over r^{2}}\mathbf{\hat{e}}_{r}
  19. U E ( r ) = k e Q q r U\text{E}(r)=k\text{e}\frac{Qq}{r}
  20. 𝐄 = k e i = 1 N Q i r i 2 𝐫 ^ i \mathbf{E}=k\text{e}\sum_{i=1}^{N}\frac{Q_{i}}{r_{i}^{2}}\mathbf{\hat{r}}_{i}

Coulomb_blockade.html

  1. U = e / C U=e/C
  2. e e
  3. C C
  4. Δ E . \Delta E.
  5. C C
  6. C = e 2 Δ E . C=\frac{e^{2}}{\Delta E}.
  7. V bias < e C V\text{bias}<\frac{e}{C}
  8. k B T , k_{B}T,
  9. k B T < e 2 2 C , k_{B}T<\frac{e^{2}}{2C},
  10. R t , R_{t},
  11. h e 2 , \frac{h}{e^{2}},

Covariance_and_correlation.html

  1. σ X Y = E [ ( X - E [ X ] ) ( Y - E [ Y ] ) ] \sigma_{XY}=E[(X-E[X])\,(Y-E[Y])]
  2. ρ X Y = E [ ( X - E [ X ] ) ( Y - E [ Y ] ) ] / ( σ X σ Y ) \rho_{XY}=E[(X-E[X])\,(Y-E[Y])]/(\sigma_{X}\sigma_{Y})
  3. c o v X Y = σ X Y = ρ X Y σ X σ Y cov_{XY}=\sigma_{XY}=\rho_{XY}\sigma_{X}\sigma_{Y}
  4. σ X X \sigma_{XX}
  5. σ X 2 , \sigma_{X}^{2},
  6. σ X Y ( m ) = E [ ( X n - μ X ) ( Y n + m - μ Y ) ] . \sigma_{XY}(m)=E[(X_{n}-\mu_{X})\,(Y_{n+m}-\mu_{Y})].
  7. ρ X Y ( m ) = E [ ( X n - μ X ) ( Y n + m - μ Y ) ] / ( σ X σ Y ) , \rho_{XY}(m)=E[(X_{n}-\mu_{X})\,(Y_{n+m}-\mu_{Y})]/(\sigma_{X}\sigma_{Y}),

Covering_system.html

  1. { a 1 ( mod n 1 ) , , a k ( mod n k ) } \{a_{1}(\mathrm{mod}\ {n_{1}}),\ \ldots,\ a_{k}(\mathrm{mod}\ {n_{k}})\}
  2. a i ( mod n i ) = { a i + n i x : x \Z } a_{i}(\mathrm{mod}\ {n_{i}})=\{a_{i}+n_{i}x:\ x\in\Z\}
  3. { 0 ( mod 3 ) , 1 ( mod 3 ) , 2 ( mod 3 ) } , \{0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {3}),\ 2(\mathrm{mod}\ {3})\},
  4. { 1 ( mod 2 ) , 2 ( mod 4 ) , 4 ( mod 8 ) , 0 ( mod 8 ) } , \{1(\mathrm{mod}\ {2}),\ 2(\mathrm{mod}\ {4}),\ 4(\mathrm{mod}\ {8}),\ 0(% \mathrm{mod}\ {8})\},
  5. { 0 ( mod 2 ) , 0 ( mod 3 ) , 1 ( mod 4 ) , 5 ( mod 6 ) , 7 ( mod 12 ) } . \{0(\mathrm{mod}\ {2}),\ 0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {4}),\ 5(% \mathrm{mod}\ {6}),\ 7(\mathrm{mod}\ {12})\}.
  6. n i n_{i}
  7. { a 1 ( mod n 1 ) , , a k ( mod n k ) } \{a_{1}(\mathrm{mod}\ {n_{1}}),\ \ldots,\ a_{k}(\mathrm{mod}\ {n_{k}})\}
  8. m m
  9. m m
  10. m m
  11. m m
  12. m = 2 , 3 , m=2,3,\ldots
  13. m m
  14. { 1 ( mod 2 ) ; 0 ( mod 3 ) ; 2 ( mod 6 ) ; 0 , 4 , 6 , 8 ( mod 10 ) ; \{1(\mathrm{mod}\ {2});\ 0(\mathrm{mod}\ {3});\ 2(\mathrm{mod}\ {6});\ 0,4,6,8% (\mathrm{mod}\ {10});
  15. 1 , 2 , 4 , 7 , 10 , 13 ( mod 15 ) ; 5 , 11 , 12 , 22 , 23 , 29 ( mod 30 ) } 1,2,4,7,10,13(\mathrm{mod}\ {15});\ 5,11,12,22,23,29(\mathrm{mod}\ {30})\}
  16. 10 50 10^{50}

Coxeter_element.html

  1. A n - 1 S n A_{n-1}\cong S_{n}
  2. ( 1 , 2 ) , ( 2 , 3 ) , (1,2),(2,3),\dots
  3. ( 1 , 2 , 3 , , n ) (1,2,3,\dots,n)
  4. 2 π / 2 m 2\pi/2m
  5. 2 π / m 2\pi/m

CPO-STV.html

  1. votes seats + 1 \rm votes\over\rm{seats+1}

CPU_core_voltage.html

  1. I D = k ( ( V G S - V t n ) V D S - ( V D S / 2 ) 2 ) \,I_{D}=k((V_{GS}-V_{tn})V_{DS}-(V_{DS}/2)^{2})
  2. I D I_{D}
  3. V t n V_{tn}
  4. V t n V_{tn}

CR_manifold.html

  1. [ L , L ] L [L,L]\subseteq L
  2. L L ¯ = { 0 } L\cap\bar{L}=\{0\}
  3. F ( z , w ) := | z | 2 + | w | 2 = 1 , F(z,w):=|z|^{2}+|w|^{2}=1,
  4. z , w . \frac{\partial}{\partial z},\quad\frac{\partial}{\partial w}.
  5. w ¯ z - z ¯ w . \bar{w}\frac{\partial}{\partial z}-\bar{z}\frac{\partial}{\partial w}.
  6. L L ¯ = { 0 } L\cap\bar{L}=\{0\}
  7. z 1 , , z n \frac{\partial}{\partial z_{1}},...,\frac{\partial}{\partial z_{n}}
  8. L L ¯ = { 0 } L\cap\bar{L}=\{0\}
  9. T ( 1 , 0 ) n = span ( z 1 , , z n ) . T^{(1,0)}\mathbb{C}^{n}=\mathrm{span}\left(\frac{\partial}{\partial z_{1}},% \dots,\frac{\partial}{\partial z_{n}}\right).
  10. T ( 0 , 1 ) n = span ( z ¯ 1 , , z ¯ n ) . T^{(0,1)}\mathbb{C}^{n}=\mathrm{span}\left(\frac{\partial}{\partial\bar{z}_{1}% },\dots,\frac{\partial}{\partial\bar{z}_{n}}\right).
  11. Ω ( 1 , 0 ) n = span ( d z 1 , , d z n ) . \Omega^{(1,0)}\mathbb{C}^{n}=\mathrm{span}(dz_{1},\dots,dz_{n}).
  12. Ω ( 0 , 1 ) n = span ( d z ¯ 1 , , d z ¯ n ) . \Omega^{(0,1)}\mathbb{C}^{n}=\mathrm{span}(d\bar{z}_{1},\dots,d\bar{z}_{n}).
  13. : Ω ( p , q ) Ω ( p + 1 , q ) \partial:\Omega^{(p,q)}\rightarrow\Omega^{(p+1,q)}
  14. ¯ : Ω ( p , q ) Ω ( p , q + 1 ) \bar{\partial}:\Omega^{(p,q)}\rightarrow\Omega^{(p,q+1)}
  15. d = + ¯ d=\partial+\bar{\partial}
  16. F 1 ¯ F 1 F k ¯ F k 0. \partial F_{1}\wedge\bar{\partial}F_{1}\wedge\dots\wedge\partial F_{k}\wedge% \bar{\partial}F_{k}\not=0.
  17. h = i ¯ F | L . h=i\partial\bar{\partial}F|_{L}.
  18. V = T M L L ¯ V=\frac{TM\otimes{\mathbb{C}}}{L\oplus\bar{L}}
  19. h ( v , w ) = 1 2 i [ v , w ¯ ] mod L L ¯ , v , w L . h(v,w)=\frac{1}{2i}[v,\bar{w}]\mod L\oplus\bar{L},\quad v,w\in L.
  20. L L ¯ L\oplus\bar{L}
  21. H 0 M = V * = ( L L ¯ ) \sub T * M . H_{0}M=V^{*}=(L\oplus\bar{L})^{\perp}\sub T^{*}M\otimes{\mathbb{C}}.
  22. h α ( v , w ) = d α ( v , w ¯ ) = - α ( [ v , w ¯ ] ) , v , w L L ¯ . h_{\alpha}(v,w)=d\alpha(v,\bar{w})=-\alpha([v,\bar{w}]),\quad v,w\in L\oplus% \bar{L}.
  23. n \mathbb{C}^{n}
  24. n \mathbb{C}^{n}
  25. n \mathbb{C}^{n}
  26. 𝕊 3 , \mathbb{S}^{3},
  27. n \mathbb{C}^{n}
  28. L ¯ \overline{L}
  29. L ¯ = z ¯ - ı z t , ( z , t ) × , ı = - 1 . \overline{L}=\frac{\partial}{\partial\bar{z}}-\imath z\frac{\partial}{\partial t% },(z,t)\in\mathbb{C}\times\mathbb{R},\imath=\sqrt{-1}.
  30. L ¯ Z i = 0 , i = 1 , 2 , Z 1 = z , Z 2 = t + ı | z | 2 , d Z 1 d Z 2 0. \overline{L}Z_{i}=0,i=1,2,Z_{1}=z,Z_{2}=t+\imath|z|^{2},dZ_{1}\wedge dZ_{2}% \not=0.
  31. [ L ¯ , L ¯ ] = 0 [\overline{L},\overline{L}]=0
  32. [ L ¯ , L ] = 2 i t , [\overline{L},L]=2i\frac{\partial}{\partial t},
  33. L = z + ı z ¯ t . L=\frac{\partial}{\partial z}+\imath\overline{z}\frac{\partial}{\partial t}.
  34. 2 \mathbb{C}^{2}
  35. ( z , t ) ( z , t + ı | z | 2 ) (z,t)\to(z,t+\imath|z|^{2})
  36. 2 \mathbb{C}^{2}
  37. × \mathbb{C}\times\mathbb{R}
  38. P u = 0 Pu=0
  39. ϕ \phi
  40. P = L ¯ + ϕ ( z , z ¯ , t ) t P=\overline{L}+\phi(z,\overline{z},t)\frac{\partial}{\partial t}
  41. n \mathbb{C}^{n}
  42. b ¯ \overline{\partial_{b}}
  43. ¯ \overline{\partial}
  44. b ¯ \overline{\partial_{b}}
  45. b ¯ b ¯ = 0 \overline{\partial_{b}}\circ\overline{\partial_{b}}=0
  46. b = b ¯ b ¯ + b ¯ b ¯ \Box_{b}=\overline{\partial_{b}}\overline{\partial_{b}}^{\star}+\overline{% \partial_{b}}^{\star}\overline{\partial_{b}}
  47. b ¯ \overline{\partial_{b}}^{\star}
  48. b ¯ \overline{\partial_{b}}
  49. L 2 ( M ) L^{2}(M)
  50. b \Box_{b}
  51. b \Box_{b}
  52. b ¯ \overline{\partial_{b}}
  53. b \Box_{b}
  54. b ¯ \overline{\partial_{b}}
  55. n × \mathbb{C}^{n}\times\mathbb{R}
  56. L ¯ j = z j ¯ - ı z j t , j = 1 , 2 , , n , ( z 1 , z 2 , , z n ) n , t . \overline{L}_{j}=\frac{\partial}{\partial\overline{z_{j}}}-\imath z_{j}\frac{% \partial}{\partial t},j=1,2,\cdots,n,(z_{1},z_{2},\cdots,z_{n})\in\mathbb{C}^{% n},t\in\mathbb{R}.
  57. ω \omega
  58. ω = b ¯ u = j = 1 n L j ¯ u d z j ¯ . \omega=\overline{\partial_{b}}u=\sum_{j=1}^{n}\overline{L_{j}}u\ d\overline{z_% {j}}.
  59. b ¯ \overline{\partial_{b}}^{\star}
  60. b = - j = 1 n L j L j ¯ \Box_{b}=-\sum_{j=1}^{n}L_{j}\overline{L_{j}}
  61. L j = z j + ı z j ¯ t , L_{j}=\frac{\partial}{\partial z_{j}}+\imath\overline{z_{j}}\frac{\partial}{% \partial t},
  62. [ L j , L j ¯ ] = - 2 ı T , T = t , j = 1 , 2 , , n [L_{j},\overline{L_{j}}]=-2\imath T,T=\frac{\partial}{\partial t},j=1,2,\cdots,n
  63. b = - 1 2 j = 1 n ( L j L j ¯ + L j ¯ L j ) + ı n T \Box_{b}=-\frac{1}{2}\sum_{j=1}^{n}(L_{j}\overline{L_{j}}+\overline{L_{j}}L_{j% })+\imath nT
  64. Δ b = - 1 2 j = 1 n ( L j L j ¯ + L j ¯ L j ) \Delta_{b}=-\frac{1}{2}\sum_{j=1}^{n}(L_{j}\overline{L_{j}}+\overline{L_{j}}L_% {j})
  65. Δ b \Delta_{b}
  66. b = Δ b + ı n T \Box_{b}=\Delta_{b}+\imath nT
  67. 2 n + 1 2n+1
  68. n + 1 \mathbb{C}^{n+1}
  69. L L
  70. L = T S 2 n + 1 T 1 , 0 n + 1 L=\mathbb{C}TS^{2n+1}\cap T^{1,0}\mathbb{C}^{n+1}
  71. T 1 , 0 n + 1 T^{1,0}\mathbb{C}^{n+1}
  72. P = ( L L ¯ ) P=\Re(L\oplus\bar{L})
  73. p S 2 n + 1 p\in S^{2n+1}
  74. I I
  75. n + 1 \mathbb{C}^{n+1}
  76. P p = { X T p S 2 n + 1 : I X T p S 2 n + 1 T p n + 1 } , P_{p}=\{X\in T_{p}S^{2n+1}:IX\in T_{p}S^{2n+1}\subset T_{p}\mathbb{C}^{n+1}\},
  77. P P
  78. I I