wpmath0000016_1

Artin–Schreier_curve.html

  1. p p
  2. y p - y = f ( x ) y^{p}-y=f(x)
  3. f f
  4. y 2 + h ( x ) y = f ( x ) y^{2}+h(x)y=f(x)
  5. f f
  6. h h
  7. p p
  8. C 1 C\to\mathbb{P}^{1}
  9. p p
  10. / p \mathbb{Z}/p\mathbb{Z}
  11. k ( C ) / k ( x ) k(C)/k(x)
  12. k k
  13. y p - y = f ( x ) , y^{p}-y=f(x),
  14. f k ( x ) f\in k(x)
  15. z p - z z^{p}-z
  16. z z
  17. g ( z ) = z p - z g(z)=z^{p}-z
  18. f k ( x ) \ g ( k ( x ) ) f\in k(x)\backslash g(k(x))
  19. C : y p - y = f ( x ) C:y^{p}-y=f(x)
  20. f f
  21. k k
  22. f ( x ) = f ( x ) + α B f α ( 1 x - α ) f(x)=f_{\infty}(x)+\sum_{\alpha\in B^{\prime}}f_{\alpha}\left(\frac{1}{x-% \alpha}\right)
  23. B B^{\prime}
  24. k k
  25. f α f_{\alpha}
  26. k k
  27. f f_{\infty}
  28. f f
  29. p p
  30. f f_{\infty}
  31. B = { B if f = 0 , B { } otherwise. B=\begin{cases}B^{\prime}&\,\text{ if }f_{\infty}=0,\\ B^{\prime}\cup\{\infty\}&\,\text{ otherwise.}\end{cases}
  32. B 1 ( k ) B\subset\mathbb{P}^{1}(k)
  33. C 1 C\to\mathbb{P}^{1}
  34. y p - y = f ( x ) y^{p}-y=f(x)
  35. f f
  36. α B \alpha\in B
  37. P α P_{\alpha}
  38. e ( P α ) = ( p - 1 ) ( deg ( f α ) + 1 ) + 1. e(P_{\alpha})=(p-1)\big(\deg(f_{\alpha})+1\big)+1.
  39. p p
  40. deg ( f α ) \deg(f_{\alpha})
  41. e ( P α ) e(P_{\alpha})
  42. p p
  43. g = p - 1 2 ( α B ( deg ( f α ) + 1 ) - 2 ) . g=\frac{p-1}{2}\left(\sum_{\alpha\in B}\big(\deg(f_{\alpha})+1\big)-2\right).
  44. p = 2 p=2
  45. y 2 - y = f ( x ) y^{2}-y=f(x)
  46. f f
  47. g = α B deg ( f α ) + 1 2 - 1. g=\sum_{\alpha\in B}\frac{\deg(f_{\alpha})+1}{2}-1.
  48. k k
  49. p p
  50. g ( y p ) = f ( x ) g(y^{p})=f(x)
  51. g k [ x ] g\in k[x]
  52. f k ( x ) \ g ( k ( x ) ) f\in k(x)\backslash g(k(x))
  53. ( x , y ) x (x,y)\mapsto x
  54. C C
  55. 1 \mathbb{P}^{1}
  56. g g
  57. k ( C ) / k ( x ) k(C)/k(x)
  58. g ( y p ) = a m y p m + a m - 1 y p m - 1 + + a 1 y p + a 0 g(y^{p})=a_{m}y^{p^{m}}+a_{m-1}y^{p^{m-1}}+\cdots+a_{1}y^{p}+a_{0}
  59. a m = a 1 = 1 a_{m}=a_{1}=1
  60. a 0 = 0 a_{0}=0
  61. C C m - 1 C 0 = 1 , C\to C_{m-1}\to\cdots\to C_{0}=\mathbb{P}^{1},
  62. p p

Artin–Tate_lemma.html

  1. B \sub C B\sub C
  2. x 1 , , x m x_{1},...,x_{m}
  3. C C
  4. A A
  5. y 1 , , y n y_{1},...,y_{n}
  6. C C
  7. B B
  8. x i = j b i j y j x_{i}=\sum_{j}b_{ij}y_{j}
  9. y i y j = k b i j k y k y_{i}y_{j}=\sum_{k}b_{ijk}y_{k}
  10. b i j , b i j k B b_{ij},b_{ijk}\in B
  11. C C
  12. A A
  13. B 0 B_{0}
  14. b i j , b i j k b_{ij},b_{ijk}
  15. A A
  16. B 0 B_{0}
  17. B B
  18. B 0 B_{0}
  19. B 0 B_{0}
  20. A A
  21. B B
  22. A A

Asano_contraction.html

  1. Φ ( z 1 , z 2 , , z n ) \Phi(z_{1},z_{2},\ldots,z_{n})
  2. a + b z 1 + c z 2 + d z 1 z 2 a+bz_{1}+cz_{2}+dz_{1}z_{2}
  3. Φ ( z i , z j ) = a + b z i + c z j + d z i z j \Phi(z_{i},z_{j})=a+bz_{i}+cz_{j}+dz_{i}z_{j}
  4. ( z i , z j ) z (z_{i},z_{j})\mapsto z
  5. Φ \Phi
  6. Φ ~ = a + d z \tilde{\Phi}=a+dz
  7. M 1 , M 2 , , M n M_{1},M_{2},\ldots,M_{n}
  8. Φ \Phi
  9. z i M i z_{i}\in M_{i}
  10. i i
  11. Φ ~ = ( ( z j , z k ) z ) ( Φ ) \tilde{\Phi}=((z_{j},z_{k})\mapsto z)(\Phi)
  12. z i M i z_{i}\in M_{i}
  13. i k , j i\neq k,j
  14. z - M j M k z\in-M_{j}M_{k}
  15. - M j M k = { - a b ; a M j , b M k } -M_{j}M_{k}=\{-ab;a\in M_{j},b\in M_{k}\}
  16. Λ \Lambda
  17. z x z_{x}
  18. P ( z Λ ) = X Λ c X z X P(z_{\Lambda})=\sum_{X\subseteq\Lambda}c_{X}z^{X}
  19. z X = x X z x z^{X}=\prod_{x\in X}z_{x}
  20. c X = e - β U ( X ) c_{X}=e^{-\beta U(X)}
  21. U ( X ) U(X)
  22. X X
  23. Λ = Λ 1 Λ 2 \Lambda=\Lambda_{1}\cap\Lambda_{2}
  24. P ( z Λ ) P(z_{\Lambda})
  25. P ( z Λ 1 ) P ( z Λ 2 ) P(z_{\Lambda_{1}})P(z_{\Lambda_{2}})
  26. P ( z Λ 1 ) P(z_{\Lambda_{1}})
  27. P ( z Λ 2 ) P(z_{\Lambda_{2}})

Asparagine_synthase_(glutamine-hydrolysing).html

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

Assemble-to-order_system.html

  1. minimize G ( 𝐲 , 𝐝 ) = 𝐡 𝐱 + 𝐩 𝐰 subject to A 𝐳 + 𝐱 = 𝐲 𝐳 + 𝐰 = 𝐝 𝐰 , 𝐱 , 𝐳 0 , \begin{aligned}\displaystyle\,\text{minimize }G(\mathbf{y},\mathbf{d})&% \displaystyle=\mathbf{h}^{\prime}\mathbf{x}+\mathbf{p}^{\prime}\mathbf{w}\\ \displaystyle\,\text{subject to }A\mathbf{z}+\mathbf{x}&\displaystyle=\mathbf{% y}\\ \displaystyle\mathbf{z}+\mathbf{w}&\displaystyle=\mathbf{d}\\ \displaystyle\mathbf{w},\mathbf{x},\mathbf{z}&\displaystyle\geq 0,\end{aligned}
  2. minimize c ( 𝐲 - 𝐱 0 ) + 𝔼 𝐝 [ G ( 𝐲 , 𝐝 ) ] subject to 𝐲 𝐱 0 , \begin{aligned}\displaystyle\,\text{minimize }&\displaystyle c(\mathbf{y}-% \mathbf{x}_{0})+\mathbb{E}_{\mathbf{d}}[G(\mathbf{y},\mathbf{d})]\\ \displaystyle\,\text{subject to }&\displaystyle\mathbf{y}\geq\mathbf{x}_{0},% \end{aligned}

Assouad_dimension.html

  1. 0 < r < R ρ , 0<r<R\leq\rho,
  2. sup x E N r ( B R ( x ) E ) C ( R r ) α . \sup_{x\in E}N_{r}(B_{R}(x)\cap E)\leq C\left(\frac{R}{r}\right)^{\alpha}.

Assured_Clear_Distance_Ahead.html

  1. A C D A s i = V ( 2 d i a i + t p c ) ACDA_{si}=V(\sqrt{\frac{2d_{i}}{a_{i}}}+t_{pc})
  2. A C D A s i = V d h s d V i ACDA_{si}=\frac{Vd_{hsd}}{V_{i}}
  3. V B S L = { ( μ + e ) 2 g 2 t p r t 2 + 2 ( μ + e ) g d A C D A s - ( μ + e ) g t p r t , if V A C D A s V A C D A s i 1 or V A C D A s i 2 or V A C D A d or V c s or V c l 2 g ( μ + e ) ( d h s d v i - t p r t ) , if V A C D A s i 1 < V A C D A s or V A C D A s i 2 or V A C D A d or V c s or V c l 2 g ( μ + e ) ( 2 d s l a i + t p c - t p r t ) , if V A C D A s i 2 < V A C D A s or V A C D A s i 1 or V A C D A d or V c s or V c l d A C D A d t g , if V A C D A d < V A C D A s or V A C D A s i 1 or V A C D A s i 2 or V c s or V c l ( μ + e ) g r 1 - μ e , if V c s < V A C D A s or V A C D A s i 1 or V A C D A s i 2 or V A C D A d or V c l V c l , if V c l < V A C D A s or V A C D A s i 1 or V A C D A s i 2 or V A C D A d or V c s V_{BSL}=\begin{cases}\sqrt{(\mu+e)^{2}g^{2}t_{prt}^{2}+2(\mu+e)gd_{ACDA_{s}}}-% (\mu+e)gt_{prt},&\mbox{if }~{}V_{ACDA_{s}}\leq V_{ACDA_{si1}}\mbox{ or }~{}V_{% ACDA_{si2}}\mbox{ or }~{}V_{ACDA_{d}}\mbox{ or }~{}V_{cs}\mbox{ or }~{}V_{cl}% \\ \\ 2g(\mu+e)(\frac{d_{hsd}}{v_{i}}-t_{prt}),&\mbox{if }~{}V_{ACDA_{si1}}<V_{ACDA_% {s}}\mbox{ or }~{}V_{ACDA_{si2}}\mbox{ or }~{}V_{ACDA_{d}}\mbox{ or }~{}V_{cs}% \mbox{ or }~{}V_{cl}\\ \\ 2g(\mu+e)(\sqrt{\frac{2d_{sl}}{a_{i}}}+t_{pc}-t_{prt}),&\mbox{if }~{}V_{ACDA_{% si2}}<V_{ACDA_{s}}\mbox{ or }~{}V_{ACDA_{si1}}\mbox{ or }~{}V_{ACDA_{d}}\mbox{% or }~{}V_{cs}\mbox{ or }~{}V_{cl}\\ \\ \frac{d_{ACDA_{d}}}{t_{g}},&\mbox{if }~{}V_{ACDA_{d}}<V_{ACDA_{s}}\mbox{ or }~% {}V_{ACDA_{si1}}\mbox{ or }~{}V_{ACDA_{si2}}\mbox{ or }~{}V_{cs}\mbox{ or }~{}% V_{cl}\\ \\ \sqrt{\frac{(\mu+e)gr}{1-\mu e}},&\mbox{if }~{}V_{cs}<V_{ACDA_{s}}\mbox{ or }~% {}V_{ACDA_{si1}}\mbox{ or }~{}V_{ACDA_{si2}}\mbox{ or }~{}V_{ACDA_{d}}\mbox{ % or }~{}V_{cl}\\ \\ V_{cl},&\mbox{if }~{}V_{cl}<V_{ACDA_{s}}\mbox{ or }~{}V_{ACDA_{si1}}\mbox{ or % }~{}V_{ACDA_{si2}}\mbox{ or }~{}V_{ACDA_{d}}\mbox{ or }~{}V_{cs}\end{cases}
  4. e = s i n ( θ ) θ t a n ( θ ) = % g r a d e 100 e=sin(\theta)\approx\theta\approx tan(\theta)=\frac{\%grade}{100}
  5. V A C D A s V_{ACDA_{s}}
  6. μ \mu
  7. d A C D A s d_{ACDA_{s}}
  8. g g
  9. t p r t t_{prt}
  10. F t o t a l = F f r i c t i o n + F g r a v i t y sin θ F_{total}=F_{friction}+F_{gravity}\sin{\theta}
  11. F t o t a l = μ F n o r m a l + m g sin θ F_{total}=\mu F_{normal}+mg\sin{\theta}
  12. F t o t a l = μ m g cos θ + m g sin θ F_{total}=\mu mg\cos{\theta}+mg\sin{\theta}
  13. F t o t a l = m a F_{total}=ma
  14. μ m g cos θ + m g sin θ = m a \mu mg\cos{\theta}+mg\sin{\theta}=ma
  15. a = g ( μ cos θ + sin θ ) a=g(\mu\cos{\theta}+\sin{\theta})
  16. d = v 2 2 a d=\frac{v^{2}}{2a}
  17. d = v 2 2 g ( μ cos θ + sin θ ) d=\frac{v^{2}}{2g(\mu\cos{\theta}+\sin{\theta})}
  18. sin θ θ \sin{\theta}\approx\theta
  19. cos θ 1 - θ 2 2 \cos{\theta}\approx 1-\frac{\theta^{2}}{2}
  20. d v 2 2 g [ μ ( 1 - θ 2 2 ) + θ ] v 2 2 g ( μ + θ ) d\approx\frac{v^{2}}{2g[\mu(1-\frac{\theta^{2}}{2})+\theta]}\approx\frac{v^{2}% }{2g(\mu+\theta)}
  21. d t o t a l = d b r e a k i n g + d p e r c e p t i o n - r e a c t i o n d_{total}=d_{breaking}+d_{perception-reaction}
  22. d t o t a l v 2 2 g ( μ + θ ) + v t p r d_{total}\approx\frac{v^{2}}{2g(\mu+\theta)}+vt_{pr}
  23. 1 2 g ( μ + θ ) v 2 + v t p r t - d t o t a l 0 \frac{1}{2g(\mu+\theta)}v^{2}+vt_{prt}-d_{total}\approx 0
  24. v ( μ + θ ) 2 g 2 t p r t 2 + 2 ( μ + θ ) g d t o t a l - ( μ + θ ) g t p r t v\approx\sqrt{(\mu+\theta)^{2}g^{2}t_{prt}^{2}+2(\mu+\theta)gd_{total}}-(\mu+% \theta)gt_{prt}
  25. θ t a n ( θ ) = % g r a d e 100 \theta\approx tan(\theta)=\frac{\%grade}{100}
  26. V B S L 1 ( μ + e ) 2 g 2 t p r t 2 + 2 ( μ + e ) g d A C D A - ( μ + e ) g t p r t V_{BSL1}\approx\sqrt{(\mu+e)^{2}g^{2}t_{prt}^{2}+2(\mu+e)gd_{ACDA}}-(\mu+e)gt_% {prt}
  27. V B S L 1 = μ 2 g 2 t p r t 2 + 2 μ g d A C D A - μ g t p r t V_{BSL1}=\sqrt{\mu^{2}g^{2}t_{prt}^{2}+2\mu gd_{ACDA}}-\mu gt_{prt}
  28. t = d i v i t=\frac{d_{i}}{v_{i}}
  29. t = d v t=\frac{d}{v}
  30. d v = d i v i \frac{d}{v}=\frac{d_{i}}{v_{i}}
  31. d = v d i v i d=\frac{vd_{i}}{v_{i}}
  32. v d i v i = v 2 2 g ( μ + e ) + v t p r t \frac{vd_{i}}{v_{i}}=\frac{v^{2}}{2g(\mu+e)}+vt_{prt}
  33. v [ v 2 g ( μ + e ) + ( t p r t - d i v i ) ] = 0 v[\frac{v}{2g(\mu+e)}+(t_{prt}-\frac{d_{i}}{v_{i}})]=0
  34. v 2 g ( μ + e ) + ( t p r t - d i v i ) = 0 \frac{v}{2g(\mu+e)}+(t_{prt}-\frac{d_{i}}{v_{i}})=0
  35. v = 2 g ( μ + e ) ( d i v i - t p r t ) v=2g(\mu+e)(\frac{d_{i}}{v_{i}}-t_{prt})
  36. t = t p + t c + t a = t p c + t a t=t_{p}+t_{c}+t_{a}=t_{pc}+t_{a}
  37. t a = 2 d i a i t_{a}=\sqrt{\frac{2d_{i}}{a_{i}}}
  38. t = d v t=\frac{d}{v}
  39. d v = 2 d i a i + t p c \frac{d}{v}=\sqrt{\frac{2d_{i}}{a_{i}}}+t_{pc}
  40. d = v ( 2 d i a i + t p c ) d=v(\sqrt{\frac{2d_{i}}{a_{i}}}+t_{pc})
  41. v ( 2 d i a i + t p c ) = v 2 2 g ( μ + e ) + v t p r t v(\sqrt{\frac{2d_{i}}{a_{i}}}+t_{pc})=\frac{v^{2}}{2g(\mu+e)}+vt_{prt}
  42. v [ v 2 g ( μ + e ) + ( t p r t - 2 d i a i - t p c ) ] = 0 v[\frac{v}{2g(\mu+e)}+(t_{prt}-\sqrt{\frac{2d_{i}}{a_{i}}}-t_{pc})]=0
  43. v 2 g ( μ + e ) + ( t p r t - 2 d i a i - t p c ) = 0 \frac{v}{2g(\mu+e)}+(t_{prt}-\sqrt{\frac{2d_{i}}{a_{i}}}-t_{pc})=0
  44. v = 2 g ( μ + e ) ( 2 d i a i + t p c - t p r t ) v=2g(\mu+e)(\sqrt{\frac{2d_{i}}{a_{i}}}+t_{pc}-t_{prt})
  45. t g = d v t_{g}=\frac{d}{v}
  46. v = d t g v=\frac{d}{t_{g}}
  47. F c e n t r i p e t a l cos θ = F f r i c t i o n + F g r a v i t y sin θ F_{centripetal}\cos{\theta}=F_{friction}+F_{gravity}\sin{\theta}
  48. m v 2 r cos θ = μ F n o r m a l + m g sin θ m\frac{v^{2}}{r}\cos{\theta}=\mu F_{normal}+mg\sin{\theta}
  49. m v 2 r cos θ = μ ( m g cos θ + m v 2 r sin θ ) + m g sin θ m\frac{v^{2}}{r}\cos{\theta}=\mu(mg\cos{\theta}+m\frac{v^{2}}{r}\sin{\theta})+% mg\sin{\theta}
  50. v 2 r cos θ - μ v 2 r sin θ = g ( μ cos θ + sin θ ) \frac{v^{2}}{r}\cos{\theta}-\mu\frac{v^{2}}{r}\sin{\theta}=g(\mu\cos{\theta}+% \sin{\theta})
  51. v 2 ( cos θ - μ sin θ ) = g r ( μ cos θ + sin θ ) v^{2}(\cos{\theta}-\mu\sin{\theta})=gr(\mu\cos{\theta}+\sin{\theta})
  52. v = g r ( μ cos θ + sin θ ) cos θ - μ sin θ v=\sqrt{\frac{gr(\mu\cos{\theta}+\sin{\theta})}{\cos{\theta}-\mu\sin{\theta}}}
  53. v g r [ μ ( 1 - θ 2 2 ) + θ ] 1 - θ 2 2 - μ θ v\approx\sqrt{\frac{gr[\mu(1-\frac{\theta^{2}}{2})+\theta]}{1-\frac{\theta^{2}% }{2}-\mu\theta}}
  54. v g r ( μ + θ ) 1 - μ θ g r ( μ + e ) 1 - μ e v\approx\sqrt{\frac{gr(\mu+\theta)}{1-\mu\theta}}\approx\sqrt{\frac{gr(\mu+e)}% {1-\mu e}}

ASTM_smoke_pump.html

  1. S d = ( 154 × ( O b 2.8 ) ) + 3.6 Sd=(154\times(Ob^{2.8}))+3.6

Atiyah–Bott_formula.html

  1. H * ( Bun G ( X ) , l ) \operatorname{H}^{*}(\operatorname{Bun}_{G}(X),\mathbb{Q}_{l})

Atmospheric_lidar.html

  1. δ = I I \delta=\frac{I_{\perp}}{I_{\parallel}}
  2. β aer / mol ( r ) \beta_{\mathrm{aer/mol}}(r)
  3. α aer / mol ( r ) \alpha_{\mathrm{aer/mol}}(r)
  4. β aer / mol ( r ) \beta_{\mathrm{aer/mol}}(r)
  5. α aer / mol ( r ) \alpha_{\mathrm{aer/mol}}(r)
  6. Å \AA
  7. α \alpha
  8. β \beta
  9. λ \lambda
  10. [ r min , r max ] \left[r_{\min},r_{\max}\right]
  11. λ i = { 355 , 532 , 1064 } \lambda_{i}=\{355,532,1064\}
  12. i = { 1 , 2 , 3 } i=\{1,2,3\}

Attack_tolerance.html

  1. d e g G ( i ; t 1 , t n ) = 1 ( N - 1 ) j = 1 n d e g G ( t j ) ( i ) deg_{G}(i;t_{1},t_{n})=\textstyle\frac{1}{(N-1)}\sum_{j=1}^{n}{deg_{G(t_{j})}(% i)}
  2. N p G ( i ; t 1 , t n ) = 1 n j = 1 n δ t j ( i ) Np_{G}(i;t_{1},t_{n})=\textstyle\frac{1}{n}\sum_{j=1}^{n}{\delta_{t_{j}}(i)}
  3. δ t j ( i ) = { 1 , if i V t j at t j t h time step. 0 , otherwise. \delta_{t_{j}}(i)=\begin{cases}1,&\,\text{if }i\in V_{t_{j}}\,\text{at }t_{j}^% {th}\,\text{ time step.}\\ 0,&\,\text{otherwise.}\end{cases}
  4. C G ( i ; t 1 , t n ) = 1 ( N - 1 ) j ; j i d j i ( t 1 , t n ) C_{G}(i;t_{1},t_{n})=\frac{1}{(N-1)}\sum_{j;j\neq i}{d_{ji}(t_{1},t_{n})}

Augmentation_(algebra).html

  1. A k A\to k
  2. A = k [ G ] A=k[G]
  3. A k , a i x i a i A\to k,\,\sum a_{i}x_{i}\mapsto\sum a_{i}

Automotive_Safety_Integrity_Level.html

  1. Risk = ( expected loss in case of the accident ) × ( probability of the accident occurring ) \,\text{Risk}=(\,\text{expected loss in case of the accident})\times(\,\text{% probability of the accident occurring})
  2. Risk = Severity × ( Exposure × Likelihood ) \,\text{Risk}=\,\text{Severity}\times(\,\text{Exposure}\times\,\text{% Likelihood})
  3. ASIL = Severity × ( Exposure × Controllability ) \,\text{ASIL}=\,\text{Severity}\times(\,\text{Exposure}\times\,\text{% Controllability})
  4. probability of failure < Tolerable Risk Risk \,\text{probability of failure}<{\,\text{Tolerable Risk}\over\,\text{Risk}}

Average_memory_access_time.html

  1. A M A T = H + M R A M P AMAT=H+MR\cdot AMP
  2. A M A T = H 1 + M R 1 A M P 1 AMAT=H_{1}+MR_{1}\cdot AMP_{1}
  3. A M P 1 = H 2 + M R 2 A M P 2 AMP_{1}=H_{2}+MR_{2}\cdot AMP_{2}

Averaged_Lagrangian.html

  1. \mathcal{L}
  2. θ ( s y m b o l x , t ) \theta(symbol{x},t)
  3. 𝒜 - ( t θ ) = + ω \mathcal{A}\equiv-\frac{\partial\mathcal{L}}{\partial(\partial_{t}\theta)}=+% \frac{\partial\mathcal{L}}{\partial\omega}
  4. s y m b o l - ( s y m b o l θ ) = - \partialsymbol k symbol{\mathcal{B}}\equiv-\frac{\partial\mathcal{L}}{\partial(symbol{\nabla}% \theta)}=-\frac{\partial\mathcal{L}}{\partialsymbol{k}}
  5. 𝒜 \mathcal{A}
  6. s y m b o l symbol{\mathcal{B}}
  7. s y m b o l x symbol{x}
  8. t t
  9. s y m b o l symbol{\nabla}
  10. ω ( s y m b o l x , t ) \omega(symbol{x},t)
  11. s y m b o l k ( s y m b o l x , t ) symbol{k}(symbol{x},t)
  12. ω ( s y m b o l x , t ) \omega(symbol{x},t)
  13. s y m b o l k ( s y m b o l x , t ) symbol{k}(symbol{x},t)
  14. s y m b o l k ( s y m b o l x , t ) symbol{k}(symbol{x},t)
  15. φ ( s y m b o l x , t ) \varphi(symbol{x},t)
  16. L ( t φ , s y m b o l φ , φ ) , L\left(\partial_{t}\varphi,symbol{\nabla}\varphi,\varphi\right),
  17. δ ( L ( t φ ( s y m b o l x , t ) , s y m b o l φ ( s y m b o l x , t ) , φ ( s y m b o l x , t ) ) d s y m b o l x d t ) = 0 , \delta\left(\int\,\iiint\,L\left(\partial_{t}\varphi(symbol{x},t),symbol{% \nabla}\varphi(symbol{x},t),\varphi(symbol{x},t)\right)\,\,\text{d}symbol{x}\,% \,\text{d}t\right)=0,
  18. s y m b o l symbol{\nabla}
  19. t \partial_{t}
  20. t ( L ( t φ ) ) + s y m b o l ( L ( s y m b o l φ ) ) - L φ = 0 , \partial_{t}\left(\frac{\partial L}{\partial\left(\partial_{t}\varphi\right)}% \right)+symbol{\nabla}\cdot\left(\frac{\partial L}{\partial\left(symbol{\nabla% }\varphi\right)}\right)-\frac{\partial L}{\partial\varphi}=0,
  21. φ . \varphi.
  22. x x
  23. σ = - 1 24 . \sigma=-\tfrac{1}{24}.
  24. σ = 0 \sigma=0
  25. φ { A e i θ } = a cos ( θ + α ) , \varphi\sim\Re\left\{A\,\,\text{e}^{i\theta}\right\}=a\,\cos\left(\theta+% \alpha\right),
  26. a = | A | a=\left|A\right|
  27. α = arg { A } , \alpha=\arg\left\{A\right\},
  28. θ \theta
  29. | A | |A|
  30. A , A,
  31. arg { A } \arg\{A\}
  32. { A } \Re\{A\}
  33. a a
  34. α \alpha
  35. ω \omega
  36. s y m b o l k symbol{k}
  37. θ ( s y m b o l x , t ) \theta(symbol{x},t)
  38. ω - t θ \omega\equiv-\partial_{t}\theta\,
  39. s y m b o l k + s y m b o l θ . symbol{k}\equiv+symbol{\nabla}\theta.\,
  40. ω ( s y m b o l x , t ) \omega(symbol{x},t)
  41. s y m b o l k ( s y m b o l x , t ) symbol{k}(symbol{x},t)
  42. t s y m b o l k + s y m b o l ω = s y m b o l 0 \partial_{t}symbol{k}+symbol{\nabla}\omega=symbol{0}
  43. s y m b o l × s y m b o l k = s y m b o l 0. symbol{\nabla}\times symbol{k}=symbol{0}.
  44. A , A,
  45. a , a,
  46. ω , \omega,
  47. s y m b o l k symbol{k}
  48. α \alpha
  49. s y m b o l x symbol{x}
  50. t t
  51. θ \theta
  52. a , a,
  53. ω , \omega,
  54. s y m b o l k symbol{k}
  55. α \alpha
  56. φ ( s y m b o l x , t ) \varphi(symbol{x},t)
  57. t φ + ω a sin ( θ + α ) \partial_{t}\varphi\approx+\omega\,a\,\sin(\theta+\alpha)
  58. s y m b o l φ - s y m b o l k a sin ( θ + α ) . symbol{\nabla}\varphi\approx-symbol{k}\,a\,\sin(\theta+\alpha).
  59. φ ( s y m b o l x , t ) \varphi(symbol{x},t)
  60. L ( t φ , s y m b o l φ , φ ) . L\left(\partial_{t}\varphi,symbol{\nabla}\varphi,\varphi\right).
  61. φ ( s y m b o l x , t ) \varphi(symbol{x},t)
  62. φ = a cos ( θ + α ) + a 2 cos ( 2 θ + α 2 ) + a 3 cos ( 3 θ + α 3 ) + , \varphi=a\,\cos\left(\theta+\alpha\right)+a_{2}\,\cos\left(2\theta+\alpha_{2}% \right)+a_{3}\,\cos\left(3\theta+\alpha_{3}\right)+\cdots,
  63. a , a,
  64. a 2 , a_{2},
  65. α , \alpha,
  66. α 2 , \alpha_{2},
  67. ω \omega
  68. s y m b o l k symbol{k}
  69. θ : \theta:
  70. t φ + ω a sin ( θ + α ) + 2 ω a 2 sin ( 2 θ + α 2 ) + 3 ω a 3 sin ( 3 θ + α 3 ) + , \partial_{t}\varphi\approx+\omega a\,\sin\left(\theta+\alpha\right)+2\omega a_% {2}\,\sin\left(2\theta+\alpha_{2}\right)+3\omega a_{3}\,\sin\left(3\theta+% \alpha_{3}\right)+\cdots,
  71. s y m b o l φ - s y m b o l k a sin ( θ + α ) - 2 s y m b o l k a 2 sin ( 2 θ + α 2 ) - 3 s y m b o l k a 3 sin ( 3 θ + α 3 ) + . symbol{\nabla}\varphi\approx-symbol{k}a\,\sin\left(\theta+\alpha\right)-2% symbol{k}a_{2}\,\sin\left(2\theta+\alpha_{2}\right)-3symbol{k}a_{3}\,\sin\left% (3\theta+\alpha_{3}\right)+\cdots.
  72. L L
  73. L ¯ . \overline{L}.
  74. L ( t φ , s y m b o l φ , φ ) L\left(\partial_{t}\varphi,symbol{\nabla}\varphi,\varphi\right)
  75. φ ( s y m b o l x , t ) \varphi(symbol{x},t)
  76. φ ( s y m b o l x , t ) \varphi(symbol{x},t)
  77. θ : \theta:
  78. L ¯ = 1 2 π 0 2 π L ( t φ , s y m b o l φ , φ ) d θ . \overline{L}=\frac{1}{2\pi}\int_{0}^{2\pi}L\left(\partial_{t}\varphi,symbol{% \nabla}\varphi,\varphi\right)\;\,\text{d}\theta.
  79. L ¯ \overline{L}
  80. ( ω , s y m b o l k , a ) \mathcal{L}(\omega,symbol{k},a)
  81. ω , \omega,
  82. s y m b o l k symbol{k}
  83. a a
  84. θ \theta
  85. \mathcal{L}
  86. \mathcal{L}
  87. φ , \varphi,
  88. t φ \partial_{t}\varphi
  89. x φ \partial_{x}\varphi
  90. θ \theta
  91. L ¯ = 1 4 ( ω 2 - k 2 - 1 ) a 2 - 3 32 σ a 4 + ( ω 2 - k 2 - 1 4 ) a 2 2 + 𝒪 ( a 6 ) , \overline{L}=\tfrac{1}{4}(\omega^{2}-k^{2}-1)a^{2}-\tfrac{3}{32}\sigma a^{4}+(% \omega^{2}-k^{2}-\tfrac{1}{4})a_{2}^{2}+\mathcal{O}(a^{6}),
  92. a 2 = 𝒪 ( a 2 ) a_{2}=\mathcal{O}(a^{2})
  93. a 3 = 𝒪 ( a 3 ) a_{3}=\mathcal{O}(a^{3})
  94. L ¯ \overline{L}
  95. a 2 a_{2}
  96. a 2 = 0. a_{2}=0.
  97. = 1 4 ( ω 2 - k 2 - 1 ) a 2 - 3 32 σ a 4 + 𝒪 ( a 6 ) . \mathcal{L}=\tfrac{1}{4}(\omega^{2}-k^{2}-1)a^{2}-\tfrac{3}{32}\sigma a^{4}+% \mathcal{O}(a^{6}).
  98. σ \sigma
  99. θ \theta
  100. t ( + ω ) + s y m b o l ( - s y m b o l k ) = 0 , \partial_{t}\left(+\frac{\partial\mathcal{L}}{\partial\omega}\right)+symbol{% \nabla}\cdot\left(-\frac{\partial\mathcal{L}}{\partial symbol{k}}\right)=0,
  101. ω = - t θ \omega=-\partial_{t}\theta
  102. s y m b o l k = s y m b o l θ symbol{k}=symbol{\nabla}\theta
  103. θ \theta
  104. \mathcal{L}
  105. 𝒜 + / ω \mathcal{A}\equiv+\partial\mathcal{L}/\partial\omega
  106. s y m b o l - / \partialsymbol k symbol{\mathcal{B}}\equiv-\partial\mathcal{L}/\partialsymbol{k}
  107. t 𝒜 + s y m b o l s y m b o l = 0. \partial_{t}\mathcal{A}+symbol{\nabla}\cdot symbol{\mathcal{B}}=0.
  108. ω \omega
  109. s y m b o l k symbol{k}
  110. t s y m b o l k + s y m b o l ω = s y m b o l 0 \partial_{t}symbol{k}+symbol{\nabla}\omega=symbol{0}
  111. s y m b o l × s y m b o l k = s y m b o l 0. symbol{\nabla}\times symbol{k}=symbol{0}.
  112. a a
  113. / a = 0. \partial\mathcal{L}/\partial a=0.
  114. θ : \theta:
  115. t ( 1 2 ω a 2 ) + x ( 1 2 k a 2 ) = 0 , \partial_{t}\left(\tfrac{1}{2}\omega a^{2}\right)+\partial_{x}\left(\tfrac{1}{% 2}ka^{2}\right)=0,
  116. a : a:
  117. ω 2 = k 2 + 1 + 3 4 σ a 2 . \omega^{2}=k^{2}+1+\tfrac{3}{4}\sigma a^{2}.
  118. 𝒜 = 1 2 ω a 2 \mathcal{A}=\tfrac{1}{2}\omega a^{2}
  119. = 1 2 k a 2 . \mathcal{B}=\tfrac{1}{2}ka^{2}.
  120. v g v_{g}
  121. v g / 𝒜 = k / ω . v_{g}\equiv\mathcal{B}/\mathcal{A}=k/\omega.
  122. θ \theta
  123. ¯ \overline{\mathcal{L}}
  124. 𝒜 δ ¯ δ ω = - δ ¯ δ ( t θ ) \displaystyle\mathcal{A}\equiv\frac{\delta\overline{\mathcal{L}}}{\delta\omega% }=-\frac{\delta\overline{\mathcal{L}}}{\delta\left(\partial_{t}\theta\right)}
  125. s y m b o l - δ ¯ δ s y m b o l k = - δ ¯ δ ( s y m b o l θ ) , \displaystyle symbol{\mathcal{B}}\equiv-\frac{\delta\overline{\mathcal{L}}}{% \delta symbol{k}}=-\frac{\delta\overline{\mathcal{L}}}{\delta\left(symbol{% \nabla}\theta\right)},
  126. 𝒜 \mathcal{A}
  127. s y m b o l symbol{\mathcal{B}}
  128. t \partial_{t}
  129. s y m b o l symbol{\nabla}
  130. s y m b o l v g symbol{v}_{g}
  131. s y m b o l s y m b o l v g 𝒜 . symbol{\mathcal{B}}\equiv symbol{v}_{g}\mathcal{A}.\,
  132. = G ( ω , s y m b o l k ) a 2 . \mathcal{L}=G(\omega,symbol{k})a^{2}.
  133. / a = 0 \partial\mathcal{L}/\partial a=0
  134. G ( ω , s y m b o l k ) = 0. G(\omega,symbol{k})=0.
  135. G ( ω , s y m b o l k ) G(\omega,symbol{k})
  136. t a \partial_{t}a
  137. x a \partial_{x}a
  138. a ( μ x , μ t ) a(\mu x,\mu t)
  139. μ 1 \mu\ll 1
  140. = G ( ω , k ) a 2 + G 2 ( ω , k ) a 4 + 1 2 μ 2 ( G ω ω ( T a ) 2 + 2 G ω k ( T a ) ( X a ) + G k k ( X a ) 2 ) , \mathcal{L}=G(\omega,k)a^{2}+G_{2}(\omega,k)a^{4}+\tfrac{1}{2}\mu^{2}\left(G_{% \omega\omega}(\partial_{T}a)^{2}+2G_{\omega k}(\partial_{T}a)(\partial_{X}a)+G% _{kk}(\partial_{X}a)^{2}\right),
  141. X = μ x X=\mu x
  142. T = μ t . T=\mu t.

Avermitilol_synthase.html

  1. \rightleftharpoons

Axial_Turbine_Stages.html

  1. E = U ( c y 2 + c y 3 ) 1 2 c 2 E=U\frac{(c_{y2}+c_{y3})}{\frac{1}{2}c_{2}}

Axiality_(geometry).html

  1. 2 ( 2 - 1 ) 0.828 2(\sqrt{2}-1)\approx 0.828
  2. x x
  3. 0
  4. 2 \sqrt{2}
  5. 1 1
  6. 2 ( 2 - 1 ) 2(\sqrt{2}-1)
  7. y y
  8. O ( n 4 ) O(n^{4})
  9. O ( n 3 ) O(n^{3})
  10. O ( n ) O(n)
  11. O ( n 4 ) O(n^{4})
  12. O ( n 5 ) O(n^{5})
  13. w w
  14. 0 w ( p ) 1 0\leq w(p)\leq 1
  15. σ \sigma
  16. w ( p ) w ( σ ( p ) ) w ( p ) 2 d x d y . \int\int\frac{w(p)w(\sigma(p))}{w(p)^{2}}dx\,dy.
  17. w w

Axiom_of_adjunction.html

  1. x y w z [ z w ( z x z = y ) ] . \forall x\,\forall y\,\exists w\,\forall z\,[z\in w\leftrightarrow(z\in xz=y)].

Baccharis_oxide_synthase.html

  1. \rightleftharpoons

Balances_Mechanics.html

  1. I = > S I=>S

Balassa_index.html

  1. RCA i j = x i j X i x a j X a \,\text{RCA}_{ij}=\dfrac{\dfrac{x_{ij}}{X_{i}}}{\dfrac{x_{aj}}{X_{a}}}

Ball-pen_probe.html

  1. Φ \Phi
  2. V f l V_{fl}
  3. T e T_{e}
  4. V f l = Φ - α * T e V_{fl}=\Phi-\alpha*T_{e}
  5. α \alpha
  6. j e s a t j^{sat}_{e}
  7. j i s a t j^{sat}_{i}
  8. A e A_{e}
  9. A i A_{i}
  10. α = l n ( A e j e s a t A i j i s a t ) = l n ( R ) \alpha=ln(\frac{A_{e}j^{sat}_{e}}{A_{i}j^{sat}_{i}})=ln(R)
  11. R R
  12. α \alpha
  13. V f l = Φ V_{fl}=\Phi
  14. R = 1 R=1
  15. R R
  16. R R
  17. T e = Φ - V f l α T_{e}=\frac{\Phi-V_{fl}}{\alpha}
  18. α \alpha
  19. α \alpha
  20. R R
  21. α \alpha
  22. R R
  23. R = 0 R=0
  24. T e = Φ B P P - V f l α ¯ T_{e}=\frac{\Phi_{BPP}-V_{fl}}{\bar{\alpha}}
  25. α ¯ \bar{\alpha}
  26. α ¯ \bar{\alpha}

Balls_into_bins.html

  1. log n log log n ( 1 + o ( 1 ) ) \frac{\log n}{\log\log n}\cdot(1+o(1))
  2. log log n log d ( 1 + o ( 1 ) ) + Θ ( m n ) \frac{\log\log n}{\log d}\cdot(1+o(1))+\Theta(\frac{m}{n})
  3. log log n log d ( 1 + o ( 1 ) ) + Θ ( 1 ) \frac{\log\log n}{\log d}\cdot(1+o(1))+\Theta(1)
  4. log n log log n \frac{\log n}{\log\log n}
  5. log log n log 2 \frac{\log\log n}{\log 2}
  6. O ( log n log log n ) O\left(\frac{\log n}{\log\log n}\right)
  7. O ( log log n ) O(\log\log n)

Barban–Davenport–Halberstam_theorem.html

  1. ϑ ( x ; q , a ) = p x ; p a mod q log p \vartheta(x;q,a)=\sum_{p\leq x\,;\,p\equiv a\bmod q}\log p
  2. ϑ ( x ; q , a ) = x φ ( q ) + E ( x ; q , a ) \vartheta(x;q,a)=\frac{x}{\varphi(q)}+E(x;q,a)
  3. V ( x , Q ) = q Q a mod q | E ( x ; q , a ) | 2 . V(x,Q)=\sum_{q\leq Q}\sum_{a\bmod{q}}|E(x;q,a)|^{2}\ .
  4. V ( x , Q ) = O ( Q x log x ) + O ( x 2 ( log x ) - A ) V(x,Q)=O(Qx\log x)+O(x^{2}(\log x)^{-A})
  5. 1 Q x 1\leq Q\leq x
  6. Q x ( log x ) - B Q\leq x(\log x)^{-B}

Baruol_synthase.html

  1. \rightleftharpoons

Basic_element.html

  1. ( C * , d ) (C^{*},d)
  2. d x = 0 dx=0

Bateman_Equation.html

  1. N i ( t ) N_{i}(t)
  2. i i
  3. i + 1 i+1
  4. λ i \lambda_{i}
  5. d N 1 ( t ) d t = - λ 1 N 1 ( t ) \frac{dN_{1}(t)}{dt}=-\lambda_{1}N_{1}(t)
  6. d N i ( t ) d t = - λ i N i ( t ) + λ i - 1 N i - 1 ( t ) \frac{dN_{i}(t)}{dt}=-\lambda_{i}N_{i}(t)+\lambda_{i-1}N_{i-1}(t)
  7. d N k ( t ) d t = λ k - 1 N k - 1 ( t ) \frac{dN_{k}(t)}{dt}=\lambda_{k-1}N_{k-1}(t)
  8. N n ( t ) = j = 1 n - 1 λ j i = 1 n j = i n ( N i ( 0 ) e - λ j t p = i , p j n ( λ p - λ j ) ) N_{n}(t)=\prod_{j=1}^{n-1}\lambda_{j}\sum_{i=1}^{n}\sum_{j=i}^{n}\left(\frac{N% _{i}(0)e^{-\lambda_{j}t}}{\prod_{p=i,p\neq j}^{n}(\lambda_{p}-\lambda_{j})}\right)
  9. λ p λ j \lambda_{p}\approx\lambda_{j}

Bayesian_hierarchical_modeling.html

  1. θ j \theta_{j}
  2. θ j \theta_{j}
  3. θ j \theta_{j}
  4. P ( θ ) P(\theta)
  5. P ( y θ ) P(y\mid\theta)
  6. P ( θ , y ) = P ( θ ) P ( y θ ) P(\theta,y)=P(\theta)P(y\mid\theta)
  7. P ( θ y ) = P ( θ , y ) P ( y ) = P ( y θ ) P ( θ ) P ( y ) P(\theta\mid y)=\frac{P(\theta,y)}{P(y)}=\frac{P(y\mid\theta)P(\theta)}{P(y)}
  8. P ( θ y ) P(\theta\mid y)
  9. y n y_{n}
  10. θ j \theta_{j}
  11. θ \theta
  12. P ( θ ) P(\theta)
  13. y 1 , y 2 , , y n y_{1},y_{2},\ldots,y_{n}
  14. P ( y 1 , y 2 , , y n ) P(y_{1},y_{2},\ldots,y_{n})
  15. π \pi
  16. ( π 1 , π 2 , , π n ) (\pi_{1},\pi_{2},\ldots,\pi_{n})
  17. P ( y 1 , y 2 , , y n ) = P ( y π 1 , y π 2 , , y π n ) . P(y_{1},y_{2},\ldots,y_{n})=P(y_{\pi_{1}},y_{\pi_{2}},\ldots,y_{\pi_{n}}).
  18. 1 2 \frac{1}{2}
  19. Let Y i = { 1 , if the i th ball is red , 0 , otherwise . \,\text{Let }Y_{i}=\begin{cases}1,&\,\text{if the }i\,\text{th ball is red},\\ 0,&\,\text{otherwise}.\end{cases}
  20. [ P ( y 1 = 1 , y 2 = 0 ) = P ( y 1 = 0 , y 2 = 1 ) = 1 2 ] [P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)=\frac{1}{2}]
  21. y 1 y_{1}
  22. y 2 y_{2}
  23. [ P ( y 2 = 1 y 2 = 1 ) = 0 P ( y 2 = 1 ) = 1 2 ] [P(y_{2}=1\mid y_{2}=1)=0\neq P(y_{2}=1)=\frac{1}{2}]
  24. y 1 y_{1}
  25. y 2 y_{2}
  26. x 1 , , x n x_{1},\ldots,x_{n}
  27. y 1 y_{1}
  28. y 2 , y_{2},\ldots
  29. y 1 , y 2 , , y n y_{1},y_{2},\ldots,y_{n}
  30. Y θ N ( θ , 1 ) Y\mid\theta\sim N(\theta,1)
  31. θ \theta
  32. μ \mu
  33. θ μ N ( μ , 1 ) \theta\mid\mu\sim N(\mu,1)
  34. μ \mu
  35. N ( 0 , 1 ) \,\text{N}(0,1)
  36. μ \mu
  37. Y θ , μ N ( θ , 1 ) Y\mid\theta,\mu\sim N(\theta,1)
  38. μ \mu
  39. β \beta
  40. ϵ \epsilon
  41. μ N ( β , ϵ ) \mu\sim N(\beta,\epsilon)
  42. $\mbox { }$
  43. β \beta
  44. ϵ \epsilon
  45. y j y_{j}
  46. θ j \theta_{j}
  47. y j y_{j}
  48. θ 1 , θ 2 , , θ j \theta_{1},\theta_{2},\ldots,\theta_{j}
  49. ϕ \phi
  50. θ \theta
  51. ϕ \phi
  52. θ \theta
  53. ϕ \phi
  54. Stage I: y j θ j , ϕ P ( y j θ j , ϕ ) \,\text{Stage I: }y_{j}\mid\theta_{j},\phi\sim P(y_{j}\mid\theta_{j},\phi)
  55. Stage II: θ j ϕ P ( θ j ϕ ) \,\text{Stage II: }\theta_{j}\mid\phi\sim P(\theta_{j}\mid\phi)
  56. Stage III: ϕ P ( ϕ ) \,\text{Stage III: }\phi\sim P(\phi)
  57. P ( y j θ j , ϕ ) P(y_{j}\mid\theta_{j},\phi)
  58. P ( θ j , ϕ ) P(\theta_{j},\phi)
  59. ϕ \phi
  60. θ j \theta_{j}
  61. P ( θ j , ϕ ) = P ( θ j ϕ ) P ( ϕ ) P(\theta_{j},\phi)=P(\theta_{j}\mid\phi)P(\phi)
  62. ϕ \phi
  63. P ( ϕ ) P(\phi)
  64. P ( ϕ , θ j y ) P ( y j θ j , ϕ ) P ( θ j ϕ ) P(\phi,\theta_{j}\mid y)\propto P(y_{j}\mid\theta_{j},\phi)P(\theta_{j}\mid\phi)
  65. P ( ϕ , θ j y ) P ( y j θ j ) P ( θ j , ϕ ) P(\phi,\theta_{j}\mid y)\propto P(y_{j}\mid\theta_{j})P(\theta_{j},\phi)
  66. θ \theta
  67. Y θ P ( Y θ ) Y\mid\theta\sim P(Y\mid\theta)
  68. θ \theta
  69. ϕ \phi
  70. θ ϕ P ( θ ϕ ) \theta\mid\phi\sim P(\theta\mid\phi)
  71. ϕ \phi
  72. P ( ϕ ) P(\phi)
  73. P ( θ , ϕ Y ) P ( Y θ , ϕ ) P ( θ , ϕ ) P(\theta,\phi\mid Y)\propto P(Y\mid\theta,\phi)P(\theta,\phi)
  74. P ( θ , ϕ Y ) P ( Y θ ) P ( θ ϕ ) P ( ϕ ) P(\theta,\phi\mid Y)\propto P(Y\mid\theta)P(\theta\mid\phi)P(\phi)
  75. P ( θ , ϕ Y ) = P ( Y θ , ϕ ) P ( θ , ϕ ) P ( Y ) = P ( Y θ ) P ( θ ϕ ) P ( ϕ ) P ( Y ) P(\theta,\phi\mid Y)={P(Y\mid\theta,\phi)P(\theta,\phi)\over P(Y)}={P(Y\mid% \theta)P(\theta\mid\phi)P(\phi)\over P(Y)}
  76. P ( θ , ϕ Y ) P ( Y θ ) P ( θ ϕ ) P ( ϕ ) P(\theta,\phi\mid Y)\propto P(Y\mid\theta)P(\theta\mid\phi)P(\phi)
  77. P ( θ , ϕ , X Y ) = P ( Y θ ) P ( θ ϕ ) P ( ϕ X ) P ( X ) P ( Y ) P(\theta,\phi,X\mid Y)={P(Y\mid\theta)P(\theta\mid\phi)P(\phi\mid X)P(X)\over P% (Y)}
  78. P ( θ , ϕ , X Y ) P ( Y θ ) P ( θ ϕ ) P ( ϕ X ) P ( X ) P(\theta,\phi,X\mid Y)\propto P(Y\mid\theta)P(\theta\mid\phi)P(\phi\mid X)P(X)

Bayesian_programming.html

  1. Program { Description { Specification ( π ) { Variables Decomposition Forms Identification (based on δ ) Question \,\text{Program}\begin{cases}\,\text{Description}\begin{cases}\,\text{% Specification}(\pi)\begin{cases}\,\text{Variables}\\ \,\text{Decomposition}\\ \,\text{Forms}\\ \end{cases}\\ \,\text{Identification (based on }\delta)\end{cases}\\ \,\text{Question}\end{cases}
  2. π \pi
  3. δ \delta
  4. { X 1 , X 2 , , X N } \left\{X_{1},X_{2},\cdots,X_{N}\right\}
  5. δ \delta
  6. π \pi
  7. P ( X 1 X 2 X N δ π ) P\left(X_{1}\wedge X_{2}\wedge\cdots\wedge X_{N}\mid\delta\wedge\pi\right)
  8. π \pi
  9. { X 1 , X 2 , , X N } \left\{X_{1},X_{2},\cdots,X_{N}\right\}
  10. { X 1 , X 2 , , X N } \left\{X_{1},X_{2},\ldots,X_{N}\right\}
  11. K K
  12. K K
  13. L 1 , , L K L_{1},\cdots,L_{K}
  14. L k L_{k}
  15. { X k 1 , X k 2 , } \left\{X_{k_{1}},X_{k_{2}},\cdots\right\}
  16. k t h k^{th}
  17. P ( X 1 X 2 X N δ π ) \displaystyle P\left(X_{1}\wedge X_{2}\wedge\cdots\wedge X_{N}\mid\delta\wedge% \pi\right)
  18. L k L_{k}
  19. X n X_{n}
  20. L k - 1 L 2 L 1 L_{k-1}\wedge\cdots\wedge L_{2}\wedge L_{1}
  21. R k R_{k}
  22. P ( L k L k - 1 L 1 δ π ) = P ( L k R k δ π ) P\left(L_{k}\mid L_{k-1}\wedge\cdots\wedge L_{1}\wedge\delta\wedge\pi\right)=P% \left(L_{k}\mid R_{k}\wedge\delta\wedge\pi\right)
  23. P ( X 1 X 2 X N δ π ) \displaystyle P\left(X_{1}\wedge X_{2}\wedge\cdots\wedge X_{N}\mid\delta\wedge% \pi\right)
  24. P ( L k R k δ π ) P\left(L_{k}\mid R_{k}\wedge\delta\wedge\pi\right)
  25. f μ ( L k ) f_{\mu}\left(L_{k}\right)
  26. P ( L k R k δ π ) = P ( L R δ ^ π ^ ) P\left(L_{k}\mid R_{k}\wedge\delta\wedge\pi\right)=P\left(L\mid R\wedge% \widehat{\delta}\wedge\widehat{\pi}\right)
  27. f μ ( L k ) f_{\mu}\left(L_{k}\right)
  28. μ \mu
  29. R k R_{k}
  30. δ \delta
  31. δ \delta
  32. P ( L k R k δ π ) P\left(L_{k}\mid R_{k}\wedge\delta\wedge\pi\right)
  33. π ^ \widehat{\pi}
  34. δ ^ \widehat{\delta}
  35. P ( X 1 X 2 X N δ π ) P\left(X_{1}\wedge X_{2}\wedge\cdots\wedge X_{N}\mid\delta\wedge\pi\right)
  36. { X 1 , X 2 , , X N } \left\{X_{1},X_{2},\cdots,X_{N}\right\}
  37. S e a r c h e d Searched
  38. K n o w n Known
  39. F r e e Free
  40. P ( S e a r c h e d Known δ π ) P\left(Searched\mid\,\text{Known}\wedge\delta\wedge\pi\right)
  41. K n o w n Known
  42. P ( Searched Known δ π ) P\left(\,\text{Searched}\mid\,\text{Known}\wedge\delta\wedge\pi\right)
  43. P ( X 1 X 2 X N δ π ) P\left(X_{1}\wedge X_{2}\wedge\cdots\wedge X_{N}\mid\delta\wedge\pi\right)
  44. P ( Searched Known δ π ) \displaystyle P\left(\,\text{Searched}\mid\,\text{Known}\wedge\delta\wedge\pi\right)
  45. Z Z
  46. P ( Searched Known δ π ) P\left(\,\text{Searched}\mid\,\text{Known}\wedge\delta\wedge\pi\right)
  47. P ( Searched Known δ π ) \displaystyle P\left(\,\text{Searched}\mid\,\text{Known}\wedge\delta\wedge\pi\right)
  48. S p a m Spam
  49. W 0 , W 1 , , W N - 1 W_{0},W_{1},\ldots,W_{N-1}
  50. N N
  51. W n W_{n}
  52. n t h n^{th}
  53. N + 1 N+1
  54. P ( Spam W 0 W N - 1 ) \displaystyle P(\,\text{Spam}\wedge W_{0}\wedge\cdots\wedge W_{N-1})
  55. P ( W 1 Spam W 0 ) = P ( W 1 Spam ) P(W_{1}\mid\,\text{Spam}\land W_{0})=P(W_{1}\mid\,\text{Spam})
  56. P ( Spam W 0 W N - 1 ) = P ( Spam ) n = 0 N - 1 [ P ( W n Spam ) ] P(\,\text{Spam}\land W_{0}\land\ldots\land W_{N-1})=P(\,\text{Spam})\prod_{n=0% }^{N-1}[P(W_{n}\mid\,\text{Spam})]
  57. N + 1 N+1
  58. P ( Spam ) P(\,\text{Spam})
  59. P ( [ Spam = 1 ] ) = 0.75 P([\,\text{Spam}=1])=0.75
  60. N N
  61. P ( W n Spam ) P(W_{n}\mid\,\text{Spam})
  62. P ( W n [ Spam = false ] ) = 1 + a f n 2 + a f P(W_{n}\mid[\,\text{Spam}=\,\text{false}])=\frac{1+a^{n}_{f}}{2+a_{f}}
  63. P ( W n [ Spam = true ] ) = 1 + a t n 2 + a t P(W_{n}\mid[\,\text{Spam}=\,\text{true}])=\frac{1+a^{n}_{t}}{2+a_{t}}
  64. a f n a^{n}_{f}
  65. n t h n^{th}
  66. a f a_{f}
  67. a t n a_{t}^{n}
  68. n t h n^{th}
  69. a t a_{t}
  70. N N
  71. P ( W n Spam ) P(W_{n}\mid\,\text{Spam})
  72. 2 N + 2 2N+2
  73. a f n = 0 , , N - 1 a_{f}^{n=0,\ldots,N-1}
  74. a t n = 0 , , N - 1 a_{t}^{n=0,\ldots,N-1}
  75. a f a_{f}
  76. a t a_{t}
  77. P ( Spam w 0 w N - 1 ) P(\,\text{Spam}\mid w_{0}\wedge\cdots\wedge w_{N-1})
  78. P ( Spam w 0 w N - 1 ) \displaystyle P(\,\text{Spam}\mid w_{0}\wedge\cdots\wedge w_{N-1})
  79. P ( [ Spam = true ] w 0 w N - 1 ) P ( [ Spam = false ] w 0 w N - 1 ) \displaystyle\frac{P([\,\text{Spam}=\,\text{true}]\mid w_{0}\wedge\cdots\wedge w% _{N-1})}{P([\,\text{Spam}=\,\text{false}]\mid w_{0}\wedge\cdots\wedge w_{N-1})}
  80. 2 N 2N
  81. Pr { D s { S p ( π ) { V a : Spam , W 0 , W 1 W N - 1 D c : { P ( Spam W 0 W n W N - 1 ) = P ( Spam ) n = 0 N - 1 P ( W n Spam ) F o : { P ( Spam ) : { P ( [ Spam = false ] ) = 0.25 P ( [ Spam = true ] ) = 0.75 P ( W n Spam ) : { P ( W n [ Spam = false ] ) = 1 + a f n 2 + a f P ( W n [ Spam = true ] ) = 1 + a t n 2 + a t Identification (based on δ ) Q u : P ( Spam w 0 w n w N - 1 ) \Pr\begin{cases}Ds\begin{cases}Sp(\pi)\begin{cases}Va:\,\text{Spam},W_{0},W_{1% }\ldots W_{N-1}\\ Dc:\begin{cases}P(\,\text{Spam}\land W_{0}\land\ldots\land W_{n}\land\ldots% \land W_{N-1})\\ =P(\,\text{Spam})\prod_{n=0}^{N-1}P(W_{n}\mid\,\text{Spam})\end{cases}\\ Fo:\begin{cases}P(\,\text{Spam}):\begin{cases}P([\,\text{Spam}=\,\text{false}]% )=0.25\\ P([\,\text{Spam}=\,\text{true}])=0.75\end{cases}\\ P(W_{n}\mid\,\text{Spam}):\begin{cases}P(W_{n}\mid[\,\text{Spam}=\,\text{false% }])\\ =\frac{1+a^{n}_{f}}{2+a_{f}}\\ P(W_{n}\mid[\,\text{Spam}=\,\text{true}])\\ =\frac{1+a^{n}_{t}}{2+a_{t}}\end{cases}\\ \end{cases}\\ \end{cases}\\ \,\text{Identification (based on }\delta)\end{cases}\\ Qu:P(\,\text{Spam}\mid w_{0}\land\ldots\land w_{n}\land\ldots\land w_{N-1})% \end{cases}
  82. S 0 , , S T S^{0},\ldots,S^{T}
  83. 0
  84. T T
  85. O 0 , , O T O^{0},\ldots,O^{T}
  86. P ( S t S t - 1 ) P(S^{t}\mid S^{t-1})
  87. t - 1 t-1
  88. t t
  89. P ( O t S t ) P(O^{t}\mid S^{t})
  90. t t
  91. S t S^{t}
  92. 0
  93. P ( S 0 O 0 ) P(S^{0}\wedge O^{0})
  94. P ( S t + k O 0 O t ) P\left(S^{t+k}\mid O^{0}\wedge\cdots\wedge O^{t}\right)
  95. t + k t+k
  96. 0
  97. t t
  98. k = 0 k=0
  99. ( k > 0 ) (k>0)
  100. ( k < 0 ) (k<0)
  101. ( k = 0 ) (k=0)
  102. P ( S t | O 0 O t ) P\left(S^{t}|O^{0}\wedge\cdots\wedge O^{t}\right)
  103. P ( S t 1 O 0 O t - 1 ) P\left(S^{t1}\mid O^{0}\wedge\cdots\wedge O^{t-1}\right)
  104. P ( S t | O 0 O t ) = P ( O t | S t ) × S t - 1 [ P ( S t | S t - 1 ) × P ( S t - 1 | O 0 O t - 1 ) ] \begin{array}[]{ll}&P\left(S^{t}|O^{0}\wedge\cdots\wedge O^{t}\right)\\ =&P\left(O^{t}|S^{t}\right)\times\sum_{S^{t-1}}\left[P\left(S^{t}|S^{t-1}% \right)\times P\left(S^{t-1}|O^{0}\wedge\cdots\wedge O^{t-1}\right)\right]\end% {array}
  105. P ( S t | O 0 O t - 1 ) = S t - 1 [ P ( S t | S t - 1 ) × P ( S t - 1 | O 0 O t - 1 ) ] \begin{array}[]{ll}&P\left(S^{t}|O^{0}\wedge\cdots\wedge O^{t-1}\right)\\ =&\sum_{S^{t-1}}\left[P\left(S^{t}|S^{t-1}\right)\times P\left(S^{t-1}|O^{0}% \wedge\cdots\wedge O^{t-1}\right)\right]\end{array}
  106. P ( S t O 0 O t ) \displaystyle P\left(S^{t}\mid O^{0}\wedge\cdots\wedge O^{t}\right)
  107. P r { D s { S p ( π ) { V a : S 0 , , S T , O 0 , , O T D c : { P ( S 0 S T O 0 O T | π ) = P ( S 0 O 0 ) × t = 1 T [ P ( S t | S t - 1 ) × P ( O t | S t ) ] F o : { P ( S 0 O 0 ) P ( S t | S t - 1 ) P ( O t | S t ) I d Q u : { P ( S t + k | O 0 O t ) ( k = 0 ) Filtering ( k > 0 ) Prediction ( k < 0 ) Smoothing Pr\begin{cases}Ds\begin{cases}Sp(\pi)\begin{cases}Va:\\ S^{0},\cdots,S^{T},O^{0},\cdots,O^{T}\\ Dc:\\ \begin{cases}&P\left(S^{0}\wedge\cdots\wedge S^{T}\wedge O^{0}\wedge\cdots% \wedge O^{T}|\pi\right)\\ =&P\left(S^{0}\wedge O^{0}\right)\times\prod_{t=1}^{T}\left[P\left(S^{t}|S^{t-% 1}\right)\times P\left(O^{t}|S^{t}\right)\right]\end{cases}\\ Fo:\\ \begin{cases}P\left(S^{0}\wedge O^{0}\right)\\ P\left(S^{t}|S^{t-1}\right)\\ P\left(O^{t}|S^{t}\right)\end{cases}\end{cases}\\ Id\end{cases}\\ Qu:\\ \begin{cases}\begin{array}[]{l}P\left(S^{t+k}|O^{0}\wedge\cdots\wedge O^{t}% \right)\\ \left(k=0\right)\equiv\,\text{Filtering}\\ \left(k>0\right)\equiv\,\text{Prediction}\\ \left(k<0\right)\equiv\,\text{Smoothing}\end{array}\end{cases}\end{cases}
  108. P r { D s { S p ( π ) { V a : S 0 , , S T , O 0 , , O T D c : { P ( S 0 O T | π ) = [ P ( S 0 O 0 | π ) t = 1 T [ P ( S t | S t - 1 π ) × P ( O t | S t π ) ] ] F o : { P ( S t S t - 1 π ) G ( S t , A S t - 1 , Q ) P ( O t S t π ) G ( O t , H S t , R ) I d Q u : P ( S T O 0 O T π ) Pr\begin{cases}Ds\begin{cases}Sp(\pi)\begin{cases}Va:\\ S^{0},\cdots,S^{T},O^{0},\cdots,O^{T}\\ Dc:\\ \begin{cases}&P\left(S^{0}\wedge\cdots\wedge O^{T}|\pi\right)\\ =&\left[\begin{array}[]{c}P\left(S^{0}\wedge O^{0}|\pi\right)\\ \prod_{t=1}^{T}\left[P\left(S^{t}|S^{t-1}\wedge\pi\right)\times P\left(O^{t}|S% ^{t}\wedge\pi\right)\right]\end{array}\right]\end{cases}\\ Fo:\\ \begin{cases}P\left(S^{t}\mid S^{t-1}\wedge\pi\right)\equiv G\left(S^{t},A% \bullet S^{t-1},Q\right)\\ P\left(O^{t}\mid S^{t}\wedge\pi\right)\equiv G\left(O^{t},H\bullet S^{t},R% \right)\end{cases}\end{cases}\\ Id\end{cases}\\ Qu:\\ P\left(S^{T}\mid O^{0}\wedge\cdots\wedge O^{T}\wedge\pi\right)\end{cases}
  109. P ( S t S t - 1 π ) P(S^{t}\mid S^{t-1}\wedge\pi)
  110. P ( O t S t π ) P(O^{t}\mid S^{t}\wedge\pi)
  111. P ( S T O 0 O T π ) P(S^{T}\mid O^{0}\wedge\cdots\wedge O^{T}\wedge\pi)
  112. Pr { D s { S p ( π ) { V a : S 0 , , S T , O 0 , , O T D c : { P ( S 0 O T π ) = [ P ( S 0 O 0 π ) t = 1 T [ P ( S t S t - 1 π ) × P ( O t S t π ) ] ] F o : { P ( S 0 O 0 π ) Matrix P ( S t S t - 1 π ) Matrix P ( O t S t π ) Matrix I d Q u : max S 1 S T - 1 [ P ( S 1 S T - 1 S T O 0 O T π ) ] \Pr\begin{cases}Ds\begin{cases}Sp(\pi)\begin{cases}Va:\\ S^{0},\ldots,S^{T},O^{0},\ldots,O^{T}\\ Dc:\\ \begin{cases}&P\left(S^{0}\wedge\cdots\wedge O^{T}\mid\pi\right)\\ =&\left[\begin{array}[]{c}P\left(S^{0}\wedge O^{0}\mid\pi\right)\\ \prod_{t=1}^{T}\left[P\left(S^{t}\mid S^{t-1}\wedge\pi\right)\times P\left(O^{% t}\mid S^{t}\wedge\pi\right)\right]\end{array}\right]\end{cases}\\ Fo:\\ \begin{cases}P\left(S^{0}\wedge O^{0}\mid\pi\right)\equiv\,\text{Matrix}\\ P\left(S^{t}\mid S^{t-1}\wedge\pi\right)\equiv\,\text{Matrix}\\ P\left(O^{t}\mid S^{t}\wedge\pi\right)\equiv\,\text{Matrix}\end{cases}\end{% cases}\\ Id\end{cases}\\ Qu:\\ \max_{S^{1}\wedge\cdots\wedge S^{T-1}}\left[P\left(S^{1}\wedge\cdots\wedge S^{% T-1}\mid S^{T}\wedge O^{0}\wedge\cdots\wedge O^{T}\wedge\pi\right)\right]\end{cases}
  113. P ( S t S t - 1 π ) P\left(S^{t}\mid S^{t-1}\wedge\pi\right)
  114. P ( O t S t π ) P\left(O^{t}\mid S^{t}\wedge\pi\right)
  115. max S 1 S T - 1 [ P ( S 1 S T - 1 S T O 0 O T π ) ] \max_{S^{1}\wedge\cdots\wedge S^{T-1}}\left[P\left(S^{1}\wedge\cdots\wedge S^{% T-1}\mid S^{T}\wedge O^{0}\wedge\cdots\wedge O^{T}\wedge\pi\right)\right]

Beam_and_Warming_scheme.html

  1. u t = - u u x with x R \frac{\partial u}{\partial t}=-u\frac{\partial u}{\partial x}\quad\,\text{with% }x\in R
  2. u t = - E x \frac{\partial u}{\partial t}=-\frac{\partial E}{\partial x}
  3. E = u 2 2 E=\frac{u^{2}}{2}
  4. u i n + 1 u^{n+1}_{i}
  5. u i n + 1 = u i n + 1 2 [ u t | i n + u t | i n + 1 ] Δ t + O ( Δ t 3 ) u^{n+1}_{i}=u^{n}_{i}+\frac{1}{2}\left[\left.\frac{\partial u}{\partial t}% \right|^{n}_{i}+\left.\frac{\partial u}{\partial t}\right|^{n+1}_{i}\right]% \Delta t+O(\Delta t^{3})
  6. u i n + 1 - u i n Δ t = - 1 2 ( E x | i n + E x | i n + x [ A ( u i n + 1 - u i n ) ] ) \therefore\frac{u^{n+1}_{i}-u^{n}_{i}}{\Delta t}=-\frac{1}{2}\left(\left.\frac% {\partial E}{\partial x}\right|^{n}_{i}+\left.\frac{\partial E}{\partial x}% \right|^{n}_{i}+\frac{\partial}{\partial x}\left[A(u^{n+1}_{i}-u^{n}_{i})% \right]\right)
  7. u t = - E x \because\frac{\partial u}{\partial t}=-\frac{\partial E}{\partial x}
  8. - Δ t 4 Δ x ( A i - 1 n u i - 1 n + 1 ) + u i n + 1 + Δ t 4 Δ x ( A i + 1 n u i + 1 n + 1 ) = u i n - 1 2 Δ t Δ x ( E i + 1 n - E i - 1 n ) + Δ t 4 Δ x ( A i + 1 n u i + 1 n - A i - 1 n u i - 1 n ) -\frac{\Delta t}{4\Delta x}\left(A^{n}_{i-1}u^{n+1}_{i-1}\right)+u^{n+1}_{i}+% \frac{\Delta t}{4\Delta x}\left(A^{n}_{i+1}u^{n+1}_{i+1}\right)=u^{n}_{i}-% \frac{1}{2}\frac{\Delta t}{\Delta x}\left(E^{n}_{i+1}-E^{n}_{i-1}\right)+\frac% {\Delta t}{4\Delta x}\left(A^{n}_{i+1}u^{n}_{i+1}-A^{n}_{i-1}u^{n}_{i-1}\right)
  9. D = - ϵ e ( u i + 2 n - 4 u i + 1 n + 6 u i n - 4 i - 1 n + u i - 2 n ) D=-\epsilon_{e}(u^{n}_{i+2}-4u^{n}_{i+1}+6u^{n}_{i}-4^{n}_{i-1}+u^{n}_{i-2})
  10. n n
  11. n ( U ) = 0 \nabla^{n}(U)=0
  12. L 2 L^{2}
  13. | a | Δ t 2 Δ x |a|\Delta t\leq 2\Delta x
  14. O ( ( Δ t ) 2 + ( Δ x ) 2 ) O((\Delta t)^{2}+(\Delta x)^{2})

Behavior_Trees_(artificial_intelligence,_robotics_and_control).html

  1. T i = { f i , r i , Δ t } , T_{i}=\{f_{i},r_{i},\Delta t\},
  2. i i\in\mathbb{N}
  3. f i : n n f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}
  4. Δ t \Delta t
  5. r i : n { R i , S i , F i } r_{i}:\mathbb{R}^{n}\rightarrow\{R_{i},S_{i},F_{i}\}
  6. R i R_{i}
  7. S i S_{i}
  8. F i F_{i}
  9. x k + t ( t k + 1 ) = f i ( x k ( t k ) ) x_{k+t}(t_{k+1})=f_{i}(x_{k}(t_{k}))
  10. t k + 1 = t k + Δ t t_{k+1}=t_{k}+\Delta t
  11. k k\in\mathbb{N}
  12. x n x\in\mathbb{R}^{n}
  13. T i T_{i}
  14. T j T_{j}
  15. T 0 T_{0}
  16. T 0 = fallback ( T i , T j ) . T_{0}=\mbox{fallback}~{}(T_{i},T_{j}).
  17. r 0 r_{0}
  18. f 0 f_{0}
  19. T 0 T_{0}
  20. r 0 ( x k ) = { r j ( x k ) if x k 1 r i ( x k ) otherwise . r_{0}(x_{k})=\begin{cases}r_{j}(x_{k})&\,\text{ if }x_{k}\in\mathcal{F}_{1}\\ r_{i}(x_{k})&\,\text{ otherwise }.\end{cases}
  21. f 0 ( x k ) = { f j ( x k ) if x k 1 f i ( x k ) otherwise . f_{0}(x_{k})=\begin{cases}f_{j}(x_{k})&\,\text{ if }x_{k}\in\mathcal{F}_{1}\\ f_{i}(x_{k})&\,\text{ otherwise }.\end{cases}
  22. T i T_{i}
  23. T j T_{j}
  24. T 0 T_{0}
  25. T 0 = sequence ( T i , T j ) . T_{0}=\mbox{sequence}~{}(T_{i},T_{j}).
  26. r 0 r_{0}
  27. f 0 f_{0}
  28. T 0 T_{0}
  29. r 0 ( x k ) = { r j ( x k ) if x k 𝒮 1 r i ( x k ) otherwise . r_{0}(x_{k})=\begin{cases}r_{j}(x_{k})&\,\text{ if }x_{k}\in\mathcal{S}_{1}\\ r_{i}(x_{k})&\,\text{ otherwise }.\end{cases}
  30. f 0 ( x k ) = { f j ( x k ) if x k 𝒮 1 f i ( x k ) otherwise . f_{0}(x_{k})=\begin{cases}f_{j}(x_{k})&\,\text{ if }x_{k}\in\mathcal{S}_{1}\\ f_{i}(x_{k})&\,\text{ otherwise }.\end{cases}

Behrend's_trace_formula.html

  1. C C
  2. # C = p 1 # Aut ( p ) , \#C=\sum_{p}{1\over\#\operatorname{Aut}(p)},
  3. 𝐅 q \mathbf{F}_{q}
  4. ϕ - 1 : X X \phi^{-1}:X\to X
  5. ϕ \phi
  6. # X ( 𝐅 q ) = q dim X i = 0 ( - 1 ) i tr ( ϕ - 1 ; H i ( X , l ) ) . \#X(\mathbf{F}_{q})=q^{\operatorname{dim}X}\sum_{i=0}^{\infty}(-1)^{i}% \operatorname{tr}(\phi^{-1};H^{i}(X,\mathbb{Q}_{l})).
  7. B 𝔾 m = [ Spec 𝐅 q / 𝔾 m ] B\mathbb{G}_{m}=[\operatorname{Spec}\mathbf{F}_{q}/\mathbb{G}_{m}]
  8. 𝔾 m ( R ) = R × \mathbb{G}_{m}(R)=R^{\times}
  9. B 𝔾 m ( 𝐅 q ) B\mathbb{G}_{m}(\mathbf{F}_{q})
  10. 𝔾 m \mathbb{G}_{m}
  11. Spec 𝐅 q \operatorname{Spec}\mathbf{F}_{q}
  12. 𝔾 m \mathbb{G}_{m}
  13. 𝐅 q \mathbf{F}_{q}
  14. # 𝔾 m ( 𝐅 q ) = 𝐅 q × = q - 1 \#\mathbb{G}_{m}(\mathbf{F}_{q})=\mathbf{F}_{q}^{\times}=q-1
  15. B 𝔾 m B\mathbb{G}_{m}
  16. B × B\mathbb{C}^{\times}\cong\mathbb{CP}^{\infty}
  17. B × B\mathbb{C}^{\times}
  18. B 𝔾 m B\mathbb{G}_{m}
  19. B 𝔾 m N B\mathbb{G}_{m}\to\mathbb{P}^{N}
  20. 𝔾 m \mathbb{G}_{m}
  21. 𝒪 ( 1 ) \mathcal{O}(1)
  22. B 𝔾 m B\mathbb{G}_{m}
  23. N \mathbb{P}^{N}
  24. i 0 ( - 1 ) i tr ( ϕ - 1 ; H i ( B 𝔾 m , l ) ) = 1 + 1 / q + 1 / q 2 + = q q - 1 . \sum_{i\geq 0}(-1)^{i}\operatorname{tr}(\phi^{-1};H^{i}(B\mathbb{G}_{m},% \mathbb{Q}_{l}))=1+1/q+1/q^{2}+\cdots={q\over{q-1}}.
  25. dim B 𝔾 m = dim Spec 𝐅 q - dim 𝔾 m = - 1 \dim B\mathbb{G}_{m}=\dim\operatorname{Spec}\mathbf{F}_{q}-\dim\mathbb{G}_{m}=-1
  26. q - 1 q^{-1}
  27. ϕ \phi
  28. ϕ : 𝐅 q ¯ 𝐅 q ¯ , x x q \phi:\overline{\mathbf{F}_{q}}\to\overline{\mathbf{F}_{q}},x\mapsto x^{q}
  29. i d × ϕ : X × 𝐅 q 𝐅 q ¯ X × 𝐅 q 𝐅 q ¯ id\times\phi:X\times_{\mathbf{F}_{q}}\overline{\mathbf{F}_{q}}\to X\times_{% \mathbf{F}_{q}}\overline{\mathbf{F}_{q}}
  30. ϕ \phi

Beilinson_regulator.html

  1. K n ( X ) p 0 H D 2 p - n ( X , 𝐐 ( p ) ) . K_{n}(X)\rightarrow\oplus_{p\geq 0}H_{D}^{2p-n}(X,\mathbf{Q}(p)).
  2. 𝒪 F \mathcal{O}_{F}
  3. 𝒪 F × 𝐑 r 1 + r 2 , x ( log | σ ( x ) | ) σ \mathcal{O}_{F}^{\times}\rightarrow\mathbf{R}^{r_{1}+r_{2}},\ \ x\mapsto(\log|% \sigma(x)|)_{\sigma}
  4. σ : F 𝐂 \sigma:F\subset\mathbf{C}

Bernstein–Kushnirenko_theorem.html

  1. A A
  2. n \mathbb{Z}^{n}
  3. L A L_{A}
  4. [ x 1 ± 1 , , x n ± 1 ] \mathbb{C}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]
  5. A A
  6. L A = { f f ( x ) = α A c α x α } , L_{A}=\{f\mid f(x)=\sum_{\alpha\in A}c_{\alpha}x^{\alpha}\},
  7. c α c_{\alpha}\in\mathbb{C}
  8. α = ( a 1 , , a n ) n \alpha=(a_{1},\ldots,a_{n})\in\mathbb{Z}^{n}
  9. x α x^{\alpha}
  10. x 1 a 1 x n a n x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}
  11. n n
  12. A 1 , , A n A_{1},\ldots,A_{n}
  13. L A 1 , , L A n L_{A_{1}},\ldots,L_{A_{n}}
  14. f 1 ( x ) = = f n ( x ) = 0 , f_{1}(x)=\ldots=f_{n}(x)=0,
  15. f i f_{i}
  16. L A i L_{A_{i}}
  17. x ( 0 ) n x\in(\mathbb{C}\setminus 0)^{n}
  18. n ! V ( Δ 1 , , V n ) n!V(\Delta_{1},\ldots,V_{n})
  19. V V
  20. i i
  21. Δ i \Delta_{i}
  22. A i A_{i}
  23. A i A_{i}
  24. L A i L_{A_{i}}
  25. A i A_{i}
  26. A = A 1 = = A n A=A_{1}=\cdots=A_{n}
  27. L A L_{A}
  28. n ! v o l ( Δ ) n!vol(\Delta)
  29. Δ \Delta
  30. A A
  31. n n
  32. n ! n!

Beta-amyrin_synthase.html

  1. \rightleftharpoons

Beta-carotene_isomerase.html

  1. \rightleftharpoons

Beta-chamigrene_synthase.html

  1. \rightleftharpoons

Beta-copaene_synthase.html

  1. \rightleftharpoons

Beta-cubebene_synthase.html

  1. \rightleftharpoons

Beta-eudesmol_synthase.html

  1. \rightleftharpoons

Beta-farnesene_synthase.html

  1. \rightleftharpoons

Beta-phellandrene_synthase_(neryl-diphosphate-cyclizing).html

  1. \rightleftharpoons

Beta-santalene_synthase.html

  1. \rightleftharpoons

Beta-seco-amyrin_synthase.html

  1. \rightleftharpoons

Beta-selinene_cyclase.html

  1. \rightleftharpoons

Beta-sesquiphellandrene_synthase.html

  1. \rightleftharpoons

Bhargava_cube.html

  1. Q ( x , y ) = a x 2 + b x y + c y 2 Q(x,y)=ax^{2}+bxy+cy^{2}
  2. D = b 2 - 4 a c . D=b^{2}-4ac.
  3. Q ( x , y ) = a x 2 + b x y + c y 2 , Q ( x , y ) = a x 2 + b x y + c y 2 Q(x,y)=ax^{2}+bxy+cy^{2},\quad Q^{\prime}(x,y)=a^{\prime}x^{2}+b^{\prime}xy+c^% {\prime}y^{2}
  4. x α x + β y , y γ x + δ y x\mapsto\alpha x+\beta y,\quad y\mapsto\gamma x+\delta y
  5. α δ - β γ = 1 \alpha\delta-\beta\gamma=1
  6. Q ( x , y ) Q(x,y)
  7. Q ( x , y ) Q^{\prime}(x,y)
  8. Q ( x , y ) Q(x,y)
  9. Q ( x , y ) Q^{\prime}(x,y)
  10. [ Q ( x , y ) ] [Q(x,y)]
  11. [ Q ( x , y ) ] [Q^{\prime}(x,y)]
  12. p , q , r , s , p , q , r , s , a ′′ , b ′′ , c ′′ p,q,r,s,p^{\prime},q^{\prime},r^{\prime},s^{\prime},a^{\prime\prime},b^{\prime% \prime},c^{\prime\prime}
  13. X = p x 1 x 2 + q x 1 y 2 + r y 1 x 2 + s y 1 y 2 X=px_{1}x_{2}+qx_{1}y_{2}+ry_{1}x_{2}+sy_{1}y_{2}
  14. Y = p x 1 x 2 + q x 1 y 2 + r y 1 x 2 + s y 1 y 2 Y=p^{\prime}x_{1}x_{2}+q^{\prime}x_{1}y_{2}+r^{\prime}y_{1}x_{2}+s^{\prime}y_{% 1}y_{2}
  15. Q ′′ ( x , y ) = a ′′ x 2 + b ′′ x y + c ′′ y 2 Q^{\prime\prime}(x,y)=a^{\prime\prime}x^{2}+b^{\prime\prime}xy+c^{\prime\prime% }y^{2}
  16. Q ′′ ( X , Y ) = Q ( x 1 , y 1 ) Q ( x 2 , y 2 ) Q^{\prime\prime}(X,Y)=Q(x_{1},y_{1})Q^{\prime}(x_{2},y_{2})
  17. [ Q ′′ ( x , y ) ] [Q^{\prime\prime}(x,y)]
  18. [ Q ( x , y ) ] [Q(x,y)]
  19. [ Q ( x , y ) ] [Q^{\prime}(x,y)]
  20. [ Q ′′ ( x , y ) ] = [ Q ( x , y ) ] Q ( x , y ) ] [Q^{\prime\prime}(x,y)]=[Q(x,y)]\ast Q^{\prime}(x,y)]
  21. Q I d ( D ) ( x , y ) = { x 2 - D 4 y 2 D 0 ( mod 4 ) x 2 + x y + 1 - D 4 y 2 D 1 ( mod 4 ) Q_{Id}^{(D)}(x,y)=\begin{cases}x^{2}-\frac{D}{4}y^{2}&D\equiv 0\;\;(\mathop{{% \rm mod}}4)\\ x^{2}+xy+\frac{1-D}{4}y^{2}&D\equiv 1\;\;(\mathop{{\rm mod}}4)\end{cases}
  22. [ a x 2 + h x y + b y 2 ] [ax^{2}+hxy+by^{2}]
  23. [ a x 2 - h x y + b y 2 ] [ax^{2}-hxy+by^{2}]
  24. Q = - det ( M x + N y ) Q=-\det(Mx+Ny)
  25. Q = - det ( M x - N y ) Q=-\det(Mx-Ny)
  26. M 1 = [ a b c d ] , N 1 = [ e f g h ] M_{1}=\begin{bmatrix}a&b\\ c&d\end{bmatrix},N_{1}=\begin{bmatrix}e&f\\ g&h\end{bmatrix}
  27. Q 1 = - det ( M 1 x + N 1 y ) = - ( det ( M 1 ) x 2 + ( a h + e d - b g - f c ) x y + det ( N 1 ) y 2 ) Q_{1}=-\det(M_{1}x+N_{1}y)=-(\det(M_{1})x^{2}+(ah+ed-bg-fc)xy+\det(N_{1})y^{2})
  28. D 1 = ( a h - b g + c f - d e ) 2 - 4 det ( M 1 ) det ( N 1 ) . D_{1}=(ah-bg+cf-de)^{2}-4\det(M_{1})\det(N_{1}).
  29. M 2 = [ a c e g ] , N 2 = [ b d f h ] M_{2}=\begin{bmatrix}a&c\\ e&g\end{bmatrix},N_{2}=\begin{bmatrix}b&d\\ f&h\end{bmatrix}
  30. Q 2 = - det ( M 2 x + N 2 y ) = - ( det ( M 2 ) x 2 + ( a h + b g - f c - e d ) x y + det ( N 2 ) y 2 ) Q_{2}=-\det(M_{2}x+N_{2}y)=-(\det(M_{2})x^{2}+(ah+bg-fc-ed)xy+\det(N_{2})y^{2})
  31. D 2 = ( a h + b g - f c - e d ) 2 - 4 det ( M 2 ) det ( N 2 ) . D_{2}=(ah+bg-fc-ed)^{2}-4\det(M_{2})\det(N_{2}).
  32. M 3 = [ a e b f ] , N 3 = [ c g d h ] M_{3}=\begin{bmatrix}a&e\\ b&f\end{bmatrix},N_{3}=\begin{bmatrix}c&g\\ d&h\end{bmatrix}
  33. Q 3 = - det ( M 3 x + N 3 y ) = - ( det ( M 3 ) x 2 + ( a h + f c - e d - b g ) x y + det ( N 3 ) y 2 ) Q_{3}=-\det(M_{3}x+N_{3}y)=-(\det(M_{3})x^{2}+(ah+fc-ed-bg)xy+\det(N_{3})y^{2})
  34. D 3 = ( a h + f c - e d - b g ) 2 - 4 det ( M 3 ) det ( N 3 ) . D_{3}=(ah+fc-ed-bg)^{2}-4\det(M_{3})\det(N_{3}).
  35. Q 1 ( x , y ) = - det ( M 1 x + N 1 y ) \displaystyle Q_{1}(x,y)=-\det(M_{1}x+N_{1}y)
  36. [ Q 1 ( x , y ) ] [ Q 2 ( x , y ) ] [Q_{1}(x,y)]\ast[Q_{2}(x,y)]
  37. [ Q ( x , y ) ] [Q(x,y)]
  38. Q ( x , y ) = - 3 x 2 + 5 x y - 8 y 2 Q(x,y)=-3x^{2}+5xy-8y^{2}
  39. X = - 2 x 1 x 2 + 4 y 1 x 2 + y 1 y 2 X=-2x_{1}x_{2}+4y_{1}x_{2}+y_{1}y_{2}
  40. Y = x 1 x 2 + 3 y 1 y 2 Y=x_{1}x_{2}+3y_{1}y_{2}
  41. Q ( X , Y ) = Q 1 ( x 1 , y 1 ) Q 2 ( x 2 , y 2 ) Q(X,Y)=Q_{1}(x_{1},y_{1})Q_{2}(x_{2},y_{2})
  42. [ Q 3 ( x , y ) ] - 1 = Q ( x , y ) [Q_{3}(x,y)]^{-1}=Q(x,y)
  43. [ Q 1 ( x , y ) ] [ Q 2 ( x , y ) ] [ Q 3 ( x , y ) ] [Q_{1}(x,y)]\ast[Q_{2}(x,y)]\ast[Q_{3}(x,y)]
  44. p x 3 + 3 q x 2 y + 3 r x y 2 + s y 3 px^{3}+3qx^{2}y+3rxy^{2}+sy^{3}
  45. ( a x 2 + 2 b x y + c y 2 , d x 2 + 2 e x y + f y 2 ) (ax^{2}+2bxy+cy^{2},dx^{2}+2exy+fy^{2})

Bhattacharyya_angle.html

  1. Δ ( p , q ) = arccos BC ( p , q ) \Delta(p,q)=\arccos\operatorname{BC}(p,q)
  2. BC ( p , q ) = i = 1 n p i q i \operatorname{BC}(p,q)=\sum_{i=1}^{n}\sqrt{p_{i}q_{i}}
  3. S n - 1 S^{n-1}
  4. p i p i , i = 1 , , n p_{i}\mapsto\sqrt{p_{i}},\ i=1,\ldots,n
  5. Δ ( ρ , σ ) = arccos F ( ρ , σ ) . \Delta(\rho,\sigma)=\arccos\sqrt{F(\rho,\sigma)}.

BHT_algorithm.html

  1. f : { 1 , , n } { 1 , , n } f:\,\{1,\ldots,n\}\rightarrow\{1,\ldots,n\}
  2. O ( n 1 / 3 ) O(n^{1/3})
  3. Ω ( n 1 / 3 ) \Omega(n^{1/3})
  4. O ( n 2 / 3 ) = O ( n 1 / 3 ) O(\sqrt{n^{2/3}})=O(n^{1/3})

Bi-twin_chain.html

  1. n - 1 , n + 1 , 2 n - 1 , 2 n + 1 , , 2 k n - 1 , 2 k n + 1 n-1,n+1,2n-1,2n+1,\dots,2^{k}n-1,2^{k}n+1\,
  2. n - 1 , 2 n - 1 , , 2 k n - 1 n-1,2n-1,\dots,2^{k}n-1
  3. k + 1 k+1
  4. n + 1 , 2 n + 1 , , 2 k n + 1 n+1,2n+1,\dots,2^{k}n+1
  5. 2 i n - 1 , 2 i n + 1 2^{i}n-1,2^{i}n+1
  6. 2 i n - 1 2^{i}n-1
  7. 0 i k - 1 0\leq i\leq k-1
  8. 2 i n - 1 2^{i}n-1
  9. 1 i k 1\leq i\leq k
  10. p p
  11. 2 p + 1 2p+1
  12. p p
  13. ( p - 1 ) / 2 (p-1)/2

Bianconi–Barabási_model.html

  1. η j \eta_{j}
  2. η j \eta_{j}
  3. k i k_{i}
  4. η i \eta_{i}
  5. Π i = η i k i j η j k j . \Pi_{i}=\frac{\eta_{i}k_{i}}{\sum_{j}\eta_{j}k_{j}}.
  6. k i t = m η i k i j η j k j \frac{\partial k_{i}}{\partial t}=m\frac{\eta_{i}k_{i}}{\sum_{j}\eta_{j}k_{j}}
  7. k i k_{i}
  8. β ( η i ) \beta(\eta_{i})
  9. k ( t , t i , η i ) = m ( t t i ) β ( η i ) k(t,t_{i},\eta_{i})=m\left(\frac{t}{t_{i}}\right)^{\beta(\eta_{i})}
  10. β ( η ) = η C and C = ρ ( η ) η 1 - β ( η ) d η . \beta(\eta)=\frac{\eta}{C}\,\text{ and }C=\int\rho(\eta)\frac{\eta}{1-\beta(% \eta)}\,d\eta.
  11. P i i Pi_{i}
  12. i i
  13. k i k_{i}
  14. i i
  15. Π i = k i j k j . \Pi_{i}=\frac{k_{i}}{\sum_{j}k_{j}}.

Biased_random_walk_on_a_graph.html

  1. j j
  2. i i
  3. α i \alpha_{i}
  4. j j
  5. i i
  6. T i j α = α i A i j k α k A k j T_{ij}^{\alpha}=\tfrac{\alpha_{i}A_{ij}}{\sum_{k}\alpha_{k}A_{kj}}
  7. A i j A_{ij}
  8. j j
  9. i i
  10. α \alpha
  11. α \alpha
  12. α \alpha
  13. α i \alpha_{i}
  14. i i
  15. C ( i ) = Total number of shortest paths through i Total number of shortest paths C(i)=\tfrac{\,\text{Total number of shortest paths through i}}{\,\text{Total % number of shortest paths}}
  16. r i = 1 C ( i ) r_{i}=\tfrac{1}{C(i)}

Bias–variance_tradeoff.html

  1. x 1 , , x n x_{1},\dots,x_{n}
  2. y i y_{i}
  3. x i x_{i}
  4. y i = f ( x i ) + ϵ y_{i}=f(x_{i})+\epsilon
  5. ϵ \epsilon
  6. σ 2 \sigma^{2}
  7. f ^ ( x ) \hat{f}(x)
  8. y = f ( x ) y=f(x)
  9. y y
  10. f ^ ( x ) \hat{f}(x)
  11. ( y - f ^ ( x ) ) 2 (y-\hat{f}(x))^{2}
  12. x 1 , , x n x_{1},\dots,x_{n}
  13. y i y_{i}
  14. ϵ \epsilon
  15. f ^ \hat{f}
  16. f ^ \hat{f}
  17. x x
  18. E [ ( y - f ^ ( x ) ) 2 ] = Bias [ f ^ ( x ) ] 2 + Var [ f ^ ( x ) ] + σ 2 \begin{aligned}\displaystyle\mathrm{E}\Big[\big(y-\hat{f}(x)\big)^{2}\Big]&% \displaystyle=\mathrm{Bias}\big[\hat{f}(x)\big]^{2}+\mathrm{Var}\big[\hat{f}(x% )\big]+\sigma^{2}\\ \end{aligned}
  19. Bias [ f ^ ( x ) ] = E [ f ^ ( x ) ] - f ( x ) \displaystyle\mathrm{Bias}\big[\hat{f}(x)\big]=\mathrm{E}\big[\hat{f}(x)\big]-% f(x)
  20. Var [ f ^ ( x ) ] = E [ ( f ^ ( x ) - E [ f ^ ( x ) ] ) 2 ] \begin{aligned}\displaystyle\mathrm{Var}\big[\hat{f}(x)\big]=\mathrm{E}\Big[% \big(\hat{f}(x)-\mathrm{E}[\hat{f}(x)]\big)^{2}\Big]\end{aligned}
  21. x 1 , , x n , y 1 , , y n x_{1},\dots,x_{n},y_{1},\dots,y_{n}
  22. f ( x ) f(x)
  23. f ^ ( x ) \hat{f}(x)
  24. f ^ ( x ) \hat{f}(x)
  25. σ 2 \sigma^{2}
  26. f ^ ( x ) \hat{f}(x)
  27. f = f ( x ) f=f(x)
  28. f ^ = f ^ ( x ) \hat{f}=\hat{f}(x)
  29. X X
  30. E [ X 2 ] = E [ X 2 ] - E [ 2 X E [ X ] ] + E [ E [ X ] 2 ] + E [ 2 X E [ X ] ] - E [ E [ X ] 2 ] = E [ X 2 - 2 X E [ X ] + E [ X ] 2 ] + 2 E [ X ] 2 - E [ X ] 2 = E [ ( X - E [ X ] ) 2 ] + E [ X ] 2 = Var [ X ] + E [ X ] 2 \begin{aligned}\displaystyle\mathrm{E}[X^{2}]&\displaystyle=\mathrm{E}[X^{2}]-% \mathrm{E}[2X\mathrm{E}[X]]+\mathrm{E}[\mathrm{E}[X]^{2}]+\mathrm{E}[2X\mathrm% {E}[X]]-\mathrm{E}[\mathrm{E}[X]^{2}]\\ &\displaystyle=\mathrm{E}[X^{2}-2X\mathrm{E}[X]+\mathrm{E}[X]^{2}]+2\mathrm{E}% [X]^{2}-\mathrm{E}[X]^{2}\\ &\displaystyle=\mathrm{E}[(X-\mathrm{E}[X])^{2}]+\mathrm{E}[X]^{2}\\ &\displaystyle=\mathrm{Var}[X]+\mathrm{E}[X]^{2}\end{aligned}
  31. f f
  32. 0 = Var [ f ] = E [ ( f - E [ f ] ) 2 ] f - E [ f ] = 0 E [ f ] = f \begin{aligned}\displaystyle 0=\mathrm{Var}[f]=\mathrm{E}[(f-\mathrm{E}[f])^{2% }]\Rightarrow f-\mathrm{E}[f]=0\Rightarrow\mathrm{E}[f]=f\end{aligned}
  33. y = f + ϵ y=f+\epsilon
  34. E [ ϵ ] = 0 \mathrm{E}[\epsilon]=0
  35. E [ y ] = E [ f + ϵ ] = E [ f ] = f \mathrm{E}[y]=\mathrm{E}[f+\epsilon]=\mathrm{E}[f]=f
  36. Var [ ϵ ] = σ 2 \mathrm{Var}[\epsilon]=\sigma^{2}
  37. Var [ y ] = E [ ( y - E [ y ] ) 2 ] = E [ ( y - f ) 2 ] = E [ ( f + ϵ - f ) 2 ] = E [ ϵ 2 ] = Var [ ϵ ] + E [ ϵ ] 2 = σ 2 \begin{aligned}\displaystyle\mathrm{Var}[y]=\mathrm{E}[(y-\mathrm{E}[y])^{2}]=% \mathrm{E}[(y-f)^{2}]=\mathrm{E}[(f+\epsilon-f)^{2}]=\mathrm{E}[\epsilon^{2}]=% \mathrm{Var}[\epsilon]+\mathrm{E}[\epsilon]^{2}=\sigma^{2}\end{aligned}
  38. ϵ \epsilon
  39. f ^ \hat{f}
  40. E [ ( y - f ^ ) 2 ] = E [ y 2 + f ^ 2 - 2 y f ^ ] = E [ y 2 ] + E [ f ^ 2 ] - E [ 2 y f ^ ] = Var [ y ] + E [ y ] 2 + Var [ f ^ ] + E [ f ^ ] 2 - 2 f E [ f ^ ] = Var [ y ] + Var [ f ^ ] + ( f - E [ f ^ ] ) 2 = Var [ y ] + Var [ f ^ ] + E [ f - f ^ ] 2 = σ 2 + Var [ f ^ ] + Bias [ f ^ ] 2 \begin{aligned}\displaystyle\mathrm{E}\big[(y-\hat{f})^{2}\big]&\displaystyle=% \mathrm{E}[y^{2}+\hat{f}^{2}-2y\hat{f}]\\ &\displaystyle=\mathrm{E}[y^{2}]+\mathrm{E}[\hat{f}^{2}]-\mathrm{E}[2y\hat{f}]% \\ &\displaystyle=\mathrm{Var}[y]+\mathrm{E}[y]^{2}+\mathrm{Var}[\hat{f}]+\mathrm% {E}[\hat{f}]^{2}-2f\mathrm{E}[\hat{f}]\\ &\displaystyle=\mathrm{Var}[y]+\mathrm{Var}[\hat{f}]+(f-\mathrm{E}[\hat{f}])^{% 2}\\ &\displaystyle=\mathrm{Var}[y]+\mathrm{Var}[\hat{f}]+\mathrm{E}[f-\hat{f}]^{2}% \\ &\displaystyle=\sigma^{2}+\mathrm{Var}[\hat{f}]+\mathrm{Bias}[\hat{f}]^{2}\end% {aligned}
  41. k k
  42. k k
  43. k k
  44. E [ ( y - f ^ ( x ) ) 2 ] = ( f ( x ) - 1 k i = 1 k f ( N i ( x ) ) ) 2 + σ 2 k + σ 2 \mathrm{E}[(y-\hat{f}(x))^{2}]=\left(f(x)-\frac{1}{k}\sum_{i=1}^{k}f(N_{i}(x))% \right)^{2}+\frac{\sigma^{2}}{k}+\sigma^{2}
  45. N 1 ( x ) , , N k ( x ) N_{1}(x),\dots,N_{k}(x)
  46. k k
  47. x x
  48. k k
  49. k k

Biclique_attack.html

  1. 2 126.1 2^{126.1}
  2. 2 189.7 2^{189.7}
  3. 2 254.4 2^{254.4}
  4. 2 126.1 2^{126.1}
  5. 2 56 * 2 2^{56*2}
  6. 2 * 2 56 2*2^{56}
  7. K 1 K_{1}
  8. K 2 K_{2}
  9. K 1 K_{1}
  10. K 2 K_{2}
  11. 2 d 2^{d}
  12. 2 d 2^{d}
  13. 2 2 d 2^{2d}
  14. S S
  15. C C
  16. K [ i , j ] K[i,j]
  17. i , j : S j K [ i , j ] 𝑓 C i \forall i,j:S_{j}\xrightarrow[f]{K[i,j]}C_{i}
  18. S 0 S_{0}
  19. C 0 C_{0}
  20. K [ 0 , 0 ] K[0,0]
  21. S 0 K [ 0 , 0 ] 𝑓 C o S_{0}\xrightarrow[f]{K[0,0]}C_{o}
  22. f f
  23. 2 d 2^{d}
  24. f f
  25. 0 Δ i K 𝑓 Δ i 0\xrightarrow[f]{\Delta^{K}_{i}}\Delta_{i}
  26. f f
  27. j j K 𝑓 0 \nabla_{j}\xrightarrow[f]{\nabla^{K}_{j}}0
  28. Δ i \Delta_{i}
  29. j \nabla_{j}
  30. Δ i \Delta_{i}
  31. Δ i K \Delta^{K}_{i}
  32. j \nabla_{j}
  33. J K \nabla^{K}_{J}
  34. 0 Δ i K 𝑓 Δ i j j K 𝑓 0 = j Δ i K j K 𝑓 Δ i 0\xrightarrow[f]{\Delta^{K}_{i}}\Delta_{i}\oplus\nabla_{j}\xrightarrow[f]{% \nabla^{K}_{j}}0=\nabla_{j}\xrightarrow[f]{\Delta^{K}_{i}\oplus\nabla^{K}_{j}}% \Delta_{i}
  35. ( S 0 , C 0 , K [ 0 , 0 ] ) (S_{0},C_{0},K[0,0])
  36. S 0 , C 0 S_{0},C_{0}
  37. K [ 0 , 0 ] K[0,0]
  38. 0 𝑓 0 0 0\xrightarrow[f]{0}0
  39. Δ 0 = 0 , 0 = 0 \Delta_{0}=0,\nabla_{0}=0
  40. Δ 0 K = 0 \Delta^{K}_{0}=0
  41. S 0 j K [ 0 , 0 ] Δ i K j K 𝑓 C 0 Δ i S_{0}\oplus\nabla_{j}\xrightarrow[f]{K[0,0]\oplus\Delta^{K}_{i}\oplus\nabla^{K% }_{j}}C_{0}\oplus\Delta_{i}
  42. S j = S 0 j S_{j}=S_{0}\oplus\nabla_{j}
  43. K [ i , j ] = K [ 0 , 0 ] Δ i K j K K[i,j]=K[0,0]\oplus\Delta^{K}_{i}\oplus\nabla^{K}_{j}
  44. C i = C 0 Δ i C_{i}=C_{0}\oplus\Delta_{i}
  45. S j K [ i , j ] 𝑓 C i S_{j}\xrightarrow[f]{K[i,j]}C_{i}
  46. i , j : S j K [ i , j ] 𝑓 C i \forall i,j:S_{j}\xrightarrow[f]{K[i,j]}C_{i}
  47. 2 2 d 2^{2d}
  48. 2 2 d 2^{2d}
  49. 2 d 2^{d}
  50. 2 d 2^{d}
  51. 2 2 d 2^{2d}
  52. 2 * 2 d 2*2^{d}
  53. Δ i \Delta_{i}
  54. j \nabla_{j}
  55. f f
  56. Δ i j \Delta_{i}\neq\nabla_{j}
  57. i + j > 0 i+j>0
  58. K [ i , j ] K[i,j]
  59. 2 2 d 2^{2d}
  60. d d
  61. K [ i , j ] K[i,j]
  62. 2 d × 2 d 2^{d}\times 2^{d}
  63. f f
  64. g g
  65. E = f g E=f\circ g
  66. 2 d 2^{d}
  67. K [ i , 0 ] K[i,0]
  68. K [ 0 , j ] K[0,j]
  69. K [ i , j ] K[i,j]
  70. 2 2 d 2^{2d}
  71. 2 d 2^{d}
  72. S j S_{j}
  73. 2 d 2^{d}
  74. C i C_{i}
  75. 2 2 d 2^{2d}
  76. K [ i , 0 ] K[i,0]
  77. K [ 0 , j ] K[0,j]
  78. 2 d 2^{d}
  79. C i C_{i}
  80. P i P_{i}
  81. S j S_{j}
  82. P i P_{i}
  83. f f
  84. g g
  85. S j S_{j}
  86. P i P_{i}
  87. 2 112 2^{112}
  88. 2 16 2^{16}
  89. 2 112 2^{112}
  90. K [ 0 , 0 ] K[0,0]
  91. [ - - - 0 0 - - - - - - - - - - - ] \begin{bmatrix}-&-&-&0\\ 0&-&-&-\\ -&-&-&-\\ -&-&-&-\end{bmatrix}
  92. 2 112 2^{112}
  93. 2 16 2^{16}
  94. i i
  95. j j
  96. [ - - i i j - j - - - - - - - - - ] \begin{bmatrix}-&-&i&i\\ j&-&j&-\\ -&-&-&-\\ -&-&-&-\end{bmatrix}
  97. 2 8 K [ i , 0 ] 2^{8}K[i,0]
  98. 2 8 K [ 0 , j ] 2^{8}K[0,j]
  99. 2 16 2^{16}
  100. K [ i , j ] K[i,j]
  101. 2 16 2^{16}
  102. 2 112 2^{112}
  103. Δ i \Delta_{i}
  104. K [ i , 0 ] K[i,0]
  105. j \nabla_{j}
  106. K [ 0 , j ] K[0,j]
  107. 2 d 2^{d}
  108. P i K [ i , 0 ] v i P_{i}\xrightarrow[]{K[i,0]}\xrightarrow[v_{i}]{}
  109. 2 d 2^{d}
  110. v j K [ 0 , j ] S j \xleftarrow[v_{j}]{}\xleftarrow[]{K[0,j]}S_{j}
  111. K [ i , 0 ] K[i,0]
  112. K [ 0 , j ] K[0,j]
  113. K [ i , j ] K[i,j]
  114. P i K [ i , 0 ] v i P_{i}\xrightarrow[]{K[i,0]}\xrightarrow[v_{i}]{}
  115. P i K [ i , j ] v i P_{i}\xrightarrow[]{K[i,j]}\xrightarrow[v_{i}]{}
  116. S j S_{j}
  117. v j \xleftarrow[v_{j}]{}
  118. P i P_{i}
  119. v i \xrightarrow[v_{i}]{}
  120. K [ i , j ] K[i,j]
  121. P i P_{i}
  122. S j S_{j}
  123. 2 126.18 2^{126.18}
  124. 2 88 2^{88}
  125. 2 8 2^{8}

Biconvex_optimization.html

  1. B X × Y B\subset X\times Y
  2. X × Y X\times Y
  3. y Y y\in Y
  4. B y = { x X : ( x , y ) B } B_{y}=\{x\in X:(x,y)\in B\}
  5. X X
  6. x X x\in X
  7. B x = { y Y : ( x , y ) B } B_{x}=\{y\in Y:(x,y)\in B\}
  8. Y Y
  9. f ( x , y ) : B f(x,y):B\to\mathbb{R}
  10. x x
  11. f x ( y ) = f ( x , y ) f_{x}(y)=f(x,y)
  12. Y Y
  13. y y
  14. f y ( x ) = f ( x , y ) f_{y}(x)=f(x,y)
  15. X X
  16. x , y x,y

Bicyclogermacrene_synthase.html

  1. \rightleftharpoons

Bifilar_sundial.html

  1. h 2 = h 1 sin φ h_{2}=h_{1}\sin\varphi\quad
  2. f 1 f_{1}\,
  3. h 1 h_{1}\,
  4. Π \Pi\,
  5. f 2 f_{2}\,
  6. h 2 h_{2}\,
  7. Π \Pi\,
  8. f 2 f_{2}\,
  9. f 1 f_{1}\,
  10. φ \varphi
  11. ( 𝒟 1 ) (\mathcal{D}_{1})
  12. ( 𝒟 2 ) (\mathcal{D}_{2})
  13. f 1 f_{1}\,
  14. f 2 f_{2}\,
  15. Π \Pi\,
  16. O O\,
  17. t t_{\odot}
  18. δ \delta\,
  19. x I x_{I}\,
  20. y I y_{I}\,
  21. I I\,
  22. Π \Pi\,
  23. x I = h 1 sin t sin φ tan δ + cos φ cos t y I = h 2 - cos φ tan δ + sin φ cos t sin φ tan δ + cos φ cos t \begin{matrix}x_{I}&=&h_{1}\frac{\sin t_{\odot}}{\sin\varphi\ \operatorname{% tan}\delta\ +\ \cos\varphi\cos t_{\odot}}\\ &&\\ y_{I}&=&h_{2}\frac{-\cos\varphi\ \operatorname{tan}\delta\ +\ \sin\varphi\cos t% _{\odot}}{\sin\varphi\ \operatorname{tan}\delta\ +\ \cos\varphi\cos t_{\odot}}% \end{matrix}
  24. δ \delta\,
  25. x I x_{I}\,
  26. y I y_{I}\,
  27. φ \varphi
  28. t t_{\odot}
  29. x I y I + h 2 / tan φ = h 1 sin φ h 2 tan t \frac{x_{I}}{y_{I}+h_{2}/\operatorname{tan}\varphi}=\frac{h_{1}\sin\varphi}{h_% {2}}\ \operatorname{tan}t_{\odot}
  30. C C\,
  31. φ \varphi
  32. h 2 h_{2}\,
  33. h 1 h_{1}\,
  34. x I - x C y I - y C = tan t \frac{x_{I}-x_{C}}{y_{I}-y_{C}}=\operatorname{tan}t_{\odot}
  35. I I\,
  36. Π \Pi\,
  37. O C I ^ \widehat{OCI}
  38. t t_{\odot}
  39. h 2 = h 1 sin φ h_{2}=h_{1}\sin\varphi\quad
  40. C C\,
  41. h 1 h_{1}
  42. h 2 h_{2}
  43. h 2 h_{2}

Big_q-Legendre_polynomials.html

  1. P n ( x ; c ; q ) = ϕ 2 3 ( q - n , q n + 1 , x ; q , c q ; q , q ) \displaystyle P_{n}(x;c;q)={}_{3}\phi_{2}(q^{-n},q^{n+1},x;q,cq;q,q)
  2. c q q P m ( x ; c ; q ) P n ( x ; c ; q ) d x = q ( 1 - c ) 1 - q 1 - q 2 n + 1 ( c - 1 q ; q ) n ( c q ; q ) n ( - c q 2 ) n q ( n 2 ) δ m n \int_{cq}^{q}P_{m}(x;c;q)P_{n}(x;c;q)\,dx=q(1-c)\frac{1-q}{1-q^{2n+1}}\frac{(c% ^{-1}q;q)_{n}}{(cq;q)_{n}}(-cq^{2})^{n}q^{n\choose 2}\delta_{mn}
  3. lim q 1 P n ( x ; 0 ; q ) = P n ( 2 x - 1 ) \displaystyle\lim_{q\to 1}P_{n}(x;0;q)=P_{n}(2x-1)
  4. P n P_{n}
  5. n n

Bimetric_gravity.html

  1. γ i j \gamma_{ij}
  2. g i j g_{ij}
  3. 1. d s 2 = g i j d x i d x j 1.~{}~{}~{}~{}ds^{2}=g_{ij}dx^{i}dx^{j}
  4. 2. d σ 2 = γ i j d x i d x j 2.~{}~{}~{}~{}d\sigma^{2}=\gamma_{ij}dx^{i}dx^{j}
  5. g i j g_{ij}
  6. γ i j \gamma_{ij}
  7. g i j g_{ij}
  8. γ i j \gamma_{ij}
  9. { j k i } \{^{i}_{jk}\}
  10. Γ j k i \Gamma^{i}_{jk}
  11. Δ \Delta
  12. Δ j k i = { j k i } - Γ j k i ( 1 ) \Delta^{i}_{jk}=\{^{i}_{jk}\}-\Gamma^{i}_{jk}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}(1)
  13. g g
  14. g i j g_{ij}
  15. γ i j \gamma_{ij}
  16. R i j σ λ R^{\lambda}_{ij\sigma}
  17. P i j σ λ P^{\lambda}_{ij\sigma}
  18. g i j g_{ij}
  19. γ i j \gamma_{ij}
  20. P i j σ λ P^{\lambda}_{ij\sigma}
  21. γ i j \gamma_{ij}
  22. Γ \Gamma
  23. Δ \Delta
  24. R i j k h = - Δ i j / k h + Δ i k / j h + Δ m j h Δ i k m - Δ m k h Δ i j m R^{h}_{ijk}=-\Delta^{h}_{ij/k}+\Delta^{h}_{ik/j}+\Delta^{h}_{mj}\Delta^{m}_{ik% }-\Delta^{h}_{mk}\Delta^{m}_{ij}
  25. Δ \Delta
  26. - g \sqrt{-g}
  27. g γ \sqrt{\frac{g}{\gamma}}
  28. d 4 x d^{4}x
  29. - γ d 4 x \sqrt{-\gamma}d^{4}x
  30. g = d e t ( g i j ) g=det(g_{ij})
  31. 3 = d e t ( γ i j ) 3=det(\gamma_{ij})
  32. d 4 x = d x 1 d x 2 d x 3 d x 4 d^{4}x=dx^{1}dx^{2}dx^{3}dx^{4}
  33. γ i j \gamma_{ij}
  34. d 2 x d s 2 + Γ j k i d x j d s d x k d s + Δ j k i d x j d s d x k d s = 0 ( 2 ) \frac{d^{2}x}{ds^{2}}+\Gamma^{i}_{jk}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds}+\Delta% ^{i}_{jk}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}% ~{}~{}~{}(2)
  35. Γ \Gamma
  36. Δ \Delta
  37. K j i = N j i - 1 2 δ j i N = - 8 π κ T j i ( 3 ) K^{i}_{j}=N^{i}_{j}-\frac{1}{2}\delta^{i}_{j}N=-8\pi\kappa T^{i}_{j}~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3)
  38. N j i = 1 2 γ α β ( g h i g h j / α ) / β N^{i}_{j}=\frac{1}{2}\gamma^{\alpha\beta}(g^{hi}g_{hj/\alpha})_{/\beta}
  39. N j i = 1 2 γ α β { ( g h i g h j , α ) , β - ( g h i g m j Γ h α m ) , β - γ α β ( Γ j α i ) , β + Γ λ β i [ g h λ g h j , α - g h λ g m j Γ h α m - Γ j α λ ] - Γ j β λ [ g h i g h λ , α - g h i g m λ Γ h α m - Γ λ α i ] + Γ α β λ [ g h i g h j , λ - g h i g m j Γ h λ m - Γ j λ i ] } \begin{aligned}\displaystyle N^{i}_{j}=\frac{1}{2}\gamma^{\alpha\beta}\Big\{(g% ^{hi}g_{hj,\alpha})_{,\beta}-(g^{hi}g_{mj}\Gamma^{m}_{h\alpha})_{,\beta}-% \gamma^{\alpha\beta}(\Gamma^{i}_{j\alpha})_{,\beta}+\Gamma^{i}_{\lambda\beta}[% g^{h\lambda}g_{hj,\alpha}-g^{h\lambda}g_{mj}\Gamma^{m}_{h\alpha}-\Gamma^{% \lambda}_{j\alpha}]\\ \displaystyle\qquad-\Gamma^{\lambda}_{j\beta}[g^{hi}g_{h\lambda,\alpha}-g^{hi}% g_{m\lambda}\Gamma^{m}_{h\alpha}-\Gamma^{i}_{\lambda\alpha}]+\Gamma^{\lambda}_% {\alpha\beta}[g^{hi}g_{hj,\lambda}-g^{hi}g_{mj}\Gamma^{m}_{h\lambda}-\Gamma^{i% }_{j\lambda}]\Big\}\end{aligned}
  40. N = g i j N i j , κ = g γ , N=g^{ij}N_{ij},\qquad\kappa=\sqrt{\frac{g}{\gamma}},
  41. T j i T^{i}_{j}
  42. T j ; i i = 0. T^{i}_{j;i}=0.
  43. K j ; i i = 0 , K^{i}_{j;i}=0,
  44. g i j . g_{ij}.
  45. g μ ν g_{\mu\nu}
  46. f μ ν f_{\mu\nu}
  47. g - 1 f g^{-1}f
  48. f μ ν f_{\mu\nu}
  49. g μ ν g_{\mu\nu}
  50. S = - M g 2 2 d 4 x - g R ( g ) - M f 2 2 d 4 x - f R ( f ) + m 2 M g 2 d 4 x - g n = 0 4 β n e n ( 𝕏 ) + d 4 x - g m ( g , Φ i ) . S=-\frac{M_{g}^{2}}{2}\int d^{4}x\sqrt{-g}R(g)-\frac{M_{f}^{2}}{2}\int d^{4}x% \sqrt{-f}R(f)+m^{2}M_{g}^{2}\int d^{4}x\sqrt{-g}\displaystyle\sum_{n=0}^{4}% \beta_{n}e_{n}(\mathbb{X})+\int d^{4}x\sqrt{-g}\mathcal{L}_{\mathrm{m}}(g,\Phi% _{i}).
  51. g μ ν g_{\mu\nu}
  52. R ( g ) R(g)
  53. m \mathcal{L}_{\mathrm{m}}
  54. Φ i \Phi_{i}
  55. f μ ν f_{\mu\nu}
  56. M g M_{g}
  57. M f M_{f}
  58. β i \beta_{i}
  59. m m
  60. β i 1 / 2 m \beta_{i}^{1/2}m
  61. e n e_{n}
  62. 𝕂 = 𝕀 - g - 1 f \mathbb{K}=\mathbb{I}-\sqrt{g^{-1}f}
  63. 𝕏 = g - 1 f \mathbb{X}=\sqrt{g^{-1}f}
  64. α i \alpha_{i}
  65. β i \beta_{i}
  66. g - 1 f \sqrt{g^{-1}f}
  67. g - 1 f g^{-1}f
  68. 𝕏 \mathbb{X}
  69. X μ X α α = ν g μ α f ν α . X^{\mu}{}_{\alpha}X^{\alpha}{}_{\nu}=g^{\mu\alpha}f_{\nu\alpha}.
  70. e n e_{n}
  71. 𝕏 \mathbb{X}
  72. e 0 ( 𝕏 ) = 1 , e 1 ( 𝕏 ) = [ 𝕏 ] , e 2 ( 𝕏 ) = 1 2 ( [ 𝕏 ] 2 - [ 𝕏 2 ] ) , e 3 ( 𝕏 ) = 1 6 ( [ 𝕏 ] 3 - 3 [ 𝕏 ] [ 𝕏 2 ] + 2 [ 𝕏 3 ] ) , e 4 ( 𝕏 ) = det 𝕏 , \begin{aligned}\displaystyle e_{0}(\mathbb{X})&\displaystyle=1,\\ \displaystyle e_{1}(\mathbb{X})&\displaystyle=[\mathbb{X}],\\ \displaystyle e_{2}(\mathbb{X})&\displaystyle=\frac{1}{2}\left([\mathbb{X}]^{2% }-[\mathbb{X}^{2}]\right),\\ \displaystyle e_{3}(\mathbb{X})&\displaystyle=\frac{1}{6}\left([\mathbb{X}]^{3% }-3[\mathbb{X}][\mathbb{X}^{2}]+2[\mathbb{X}^{3}]\right),\\ \displaystyle e_{4}(\mathbb{X})&\displaystyle=\operatorname{det}\mathbb{X},% \end{aligned}
  73. [ 𝕏 ] X μ μ [\mathbb{X}]\equiv X^{\mu}{}_{\mu}
  74. e n e_{n}

Binding_neuron.html

  1. n n
  2. t 1 , t 2 , , t n t_{1},t_{2},\dots,t_{n}
  3. t c tc
  4. t c = 1 t n - t 1 . tc=\frac{1}{t_{n}-t_{1}}\,.
  5. n n
  6. t 1 , t 2 , , t n t_{1},t_{2},\dots,t_{n}
  7. U ( t ) U(t)
  8. W W
  9. U ( t ) = k = 1 N P V ( t - t k ) , t k [ 0 ; W ] . U(t)=\sum_{k=1}^{NP}V(t-t_{k}),\qquad t_{k}\in[0;W].
  10. V ( t ) V(t)
  11. t t
  12. t k t_{k}
  13. k k
  14. N P NP
  15. t k t_{k}
  16. [ 0 ; W ] [0;W]
  17. I ( t ) = - C M d U ( t ) d t , I(t)=-C_{M}\ \frac{dU(t)}{dt},
  18. C M C_{M}
  19. W W
  20. T C = 1 W TC=\frac{1}{W}
  21. f p fp
  22. N P NP
  23. τ \tau
  24. N t h N_{th}
  25. N t h N_{th}
  26. τ \tau
  27. τ \tau
  28. τ \tau

Binomial_sum_variance_inequality.html

  1. Var ( Z ) = n p ¯ ( 1 - p ¯ ) - n s 2 , \operatorname{Var}(Z)=n\bar{p}(1-\bar{p})-ns^{2},
  2. s 2 = 1 n i = 1 n ( p i - p ¯ ) 2 s^{2}=\frac{1}{n}\sum_{i=1}^{n}(p_{i}-\bar{p})^{2}
  3. p = p ¯ p=\bar{p}

Biogeography-based_optimization.html

  1. λ \lambda
  2. μ \mu
  3. I I
  4. S max S_{\max}
  5. S max S_{\max}
  6. E E
  7. λ \lambda
  8. μ \mu
  9. S 0 S_{0}
  10. S max S_{\max}
  11. I I
  12. E E
  13. λ k \lambda_{k}
  14. k k
  15. λ k \lambda_{k}
  16. x k x_{k}
  17. μ k \mu_{k}
  18. Prob ( x j ) is selected for emigration = μ j i = 1 N μ i \,\text{Prob}(x_{j})\,\text{ is selected for emigration}=\frac{\mu_{j}}{\sum_{% i=1}^{N}\mu_{i}}
  19. j = 1 , , N j=1,\cdots,N
  20. N N
  21. N N
  22. n n
  23. N N
  24. N N
  25. N N
  26. { x k } k = 1 N \{x_{k}\}_{k=1}^{N}
  27. { z k } \{z_{k}\}
  28. { x k } \{x_{k}\}
  29. 𝔼 \mathbb{E}
  30. 𝔼 \mathbb{E}
  31. 𝔼 \mathbb{E}
  32. 𝔼 \mathbb{E}
  33. O ( N 2 ) O(N^{2})
  34. z k ( s ) z_{k}(s)
  35. x j ( s ) x_{j}(s)
  36. z k ( s ) z_{k}(s)
  37. x j ( s ) x_{j}(s)
  38. z k ( s ) α z k ( s ) + ( 1 - α ) x j ( s ) z_{k}(s)\leftarrow\alpha z_{k}(s)+(1-\alpha)x_{j}(s)
  39. α [ 0 , 1 ] \alpha\in[0,1]
  40. α = 0 \alpha=0

Biological_motion_perception.html

  1. F t c ( t ) = i = 1 n e ( ( μ t c - p i ( t ) ) 2 2 X σ ) F_{tc}(t)=\sum_{i=1}^{n}e^{\left(}\frac{(\mu_{tc}-p_{i}(t))^{2}}{2X\sigma}\right)
  2. p i p_{i}
  3. μ t c \mu_{tc}
  4. σ \sigma
  5. τ δ u 1 , 2 ( t ) δ t = - u 1 , 2 + i 1 , 2 + w + f ( u 1 , 2 ( t ) ) - w - f ( u 2 , 1 ( t ) ) \tau\frac{\delta u_{1,2}(t)}{\delta t}=-u_{1,2}+i_{1,2}+w_{+}f(u_{1,2}(t))-w_{% -}f(u_{2,1}(t))
  6. w + w_{+}
  7. w - w_{-}
  8. u 1 , 2 u_{1,2}
  9. τ δ v 1 , 2 ( t ) δ t = - v 1 , 2 ( t ) + w m , n u ( t ) \tau\frac{\delta v_{1,2}(t)}{\delta t}=-v_{1,2}(t)+w_{m,n}u(t)
  10. ( u ) (u)
  11. ( v 1 , 2 ) (v_{1,2})
  12. n n
  13. m m
  14. R ψ ( t ) = i = 1 N e x p ( - | ( x i ( t ) , y i ( t ) ) - ( \Chi i , ψ , \Rho i , ψ ) | 2 2 \sdot σ ) R_{\psi}(t)=\sum_{i=1}^{N}exp\left(-\frac{\left|(x_{i}(t),y_{i}(t))-(\Chi_{i},% _{\psi},\Rho_{i},_{\psi})\right|^{2}}{2\sdot\sigma}\right)
  15. ( x i , y i ) (x_{i},y_{i})
  16. ( c i , r i ) (c_{i},r_{i})
  17. t t
  18. ψ {}_{\psi}
  19. R R
  20. N N
  21. σ \sigma
  22. ν ψ ( t ) = R ψ ( t ) - R ¯ R ¯ \nu_{\psi}(t)=\frac{R_{\psi}(t)-\bar{R}}{\bar{R}}
  23. R y ( t ) R_{y}(t)
  24. ψ {}_{\psi}
  25. t t
  26. R ¯ \bar{R}
  27. t t
  28. n y ( t ) n_{y}(t)
  29. g f , g b g^{f},g^{b}
  30. r ψ ( τ ) = t = 0 m s τ p = 1 100 g τ , ψ ( t , p ) \sdot ν ψ ( t ) r_{\psi}(\tau)=\sum_{t=0ms}^{\tau}\sum_{p=1}^{100}g_{\tau,\psi}(t,p)\sdot\nu_{% \psi}(t)
  31. r r
  32. τ \tau
  33. p p
  34. N ψ ( τ ) = max [ ( r ψ ( τ ) t p g τ , ψ ( t , p ) 2 ) , 0 ] N_{\psi}(\tau)=\max\left[\left(\frac{r_{\psi}(\tau)}{\sum_{t}\sum_{p}g_{\tau,% \psi}(t,p)^{2}}\right),0\right]
  35. N N
  36. ε ψ ( τ ) = N ψ F ( τ ) 2 - N ψ B ( τ ) 2 \varepsilon_{\psi}(\tau)=N_{\psi}^{F}(\tau)^{2}-N_{\psi}^{B}(\tau)^{2}
  37. ε \varepsilon
  38. G p ( x ) = H ( v ( x ) , v 1 , v 2 ) b ( θ , θ p ) G_{p}(x)=H(v(x),v_{1},v_{2})\cdot b(\theta,\theta_{p})
  39. x x
  40. θ p , \theta_{p},
  41. v v
  42. θ \theta
  43. H H
  44. H ( v , v 1 , v 2 ) = 1 H(v,v_{1},v_{2})=1
  45. v 1 < v < v 2 v_{1}<v<v_{2}
  46. H ( v , v 1 , v 2 ) = 0 H(v,v_{1},v_{2})=0
  47. b ( θ , θ p ) = { ( 1 2 ) [ 1 + c o s ( θ , θ p ) ] } q b(\theta,\theta_{p})=\left\{\left(\frac{1}{2}\right)\left[\ 1+cos(\theta,% \theta_{p})\right]\ \right\}^{q}
  48. q q
  49. o l ( x ) = m a x ( g p ( x i ) ) m a x ( g r ( x j ) ) o_{l}(x)=\sqrt{max(g_{p}(x_{i}))max(g_{r}(x_{j}))}
  50. x x
  51. p p
  52. r r
  53. i , j i,j
  54. o l ( x ) = m a x ( o l ( x k ) ) o_{l}(x)=max(o_{l}(x_{k}))
  55. l l
  56. x k x_{k}
  57. G ( u ) = e ( u - u 0 ) T C ( u - u 0 ) G(u)=e^{(u-u_{0})^{T}C(u-u_{0})}
  58. u 0 u_{0}
  59. C C
  60. τ H k l ( t ) = - H k l ( t ) + m w ( k - m ) f ( H k l ( t ) + G k l ( t ) ) \tau H_{k}^{l}(t)=-H_{k}^{l}(t)+\sum_{m}w(k-m)f(H_{k}^{l}(t)+G_{k}^{l}(t))
  61. k k
  62. l l
  63. τ \tau
  64. f ( H ) f(H)
  65. w ( m ) w(m)
  66. G k l ( t ) G_{k}^{l}(t)
  67. H l l ( t ) H_{l}^{l}(t)
  68. τ s P l ( t ) = - P l ( t ) + k H l l ( t ) \tau_{s}P^{l}(t)=-P^{l}(t)+\sum_{k}H_{l}^{l}(t)
  69. P l ( t ) P^{l}(t)
  70. l l
  71. τ s \tau_{s}
  72. H k l ( t ) H_{k}^{l}(t)

Biomass_partitioning.html

  1. ( d W / d t ) i = P c i * d W / d t (dW/dt)_{i}=P_{ci}*dW/dt

Bipartite_realization_problem.html

  1. ( a 1 , , a n ) (a_{1},\dots,a_{n})
  2. ( b 1 , , b n ) (b_{1},\dots,b_{n})
  3. ( a 1 , , a n ) , ( b 1 , , b n ) (a_{1},\dots,a_{n}),(b_{1},\dots,b_{n})
  4. n n
  5. ( a 1 , , a n ) (a_{1},\ldots,a_{n})
  6. ( b 1 , , b n ) (b_{1},\ldots,b_{n})

Biquaternion_algebra.html

  1. q = - a 1 , - a 2 , a 1 a 2 , b 1 , b 2 , - b 1 b 2 . q=\left\langle{-a_{1},-a_{2},a_{1}a_{2},b_{1},b_{2},-b_{1}b_{2}}\right\rangle\ .

Birectified_16-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}
  3. D ~ 4 {\tilde{D}}_{4}
  4. B ~ 4 {\tilde{B}}_{4}
  5. D ~ 4 {\tilde{D}}_{4}
  6. F ~ 4 {\tilde{F}}_{4}
  7. F ~ 4 {\tilde{F}}_{4}
  8. B ~ 4 {\tilde{B}}_{4}
  9. D ~ 4 {\tilde{D}}_{4}

Bishop–Phelps_theorem.html

  1. { e * E * e * attains its supremum on B } \{e^{*}\in E^{*}\mid e^{*}\,\text{ attains its supremum on }B\}
  2. E * E^{*}

Bistritz_stability_criterion.html

  1. D n ( z ) = d 0 + d 1 z + d 2 z 2 + + d n - 1 z n - 1 + d n z n D_{n}(z)=d_{0}+d_{1}z+d_{2}z^{2}+\cdots+d_{n-1}z^{n-1}+d_{n}z^{n}
  2. D n ( z ) D_{n}(z)
  3. | z k | < 1 , k = 1 , , n |z_{k}|<1,k=1,\ldots,n
  4. D n ( z ) = d n k = 1 n ( z - z k ) D_{n}(z)=d_{n}\prod_{k=1}^{n}(z-z_{k})
  5. D n ( z ) D_{n}(z)
  6. ( | z k | < 1 ) (~{}|z_{k}|<1~{})
  7. ( | z k | = 1 ) (~{}|z_{k}|=1~{})
  8. ( | z k | > 1 ) (~{}|z_{k}|>1~{})
  9. D n ( z ) D_{n}(z)
  10. D n ( 1 ) 0 D_{n}(1)\neq 0
  11. D n ( 1 ) = 0 D_{n}(1)=0
  12. D n ( z ) = z n D n ( 1 / z ) = d n + d n - 1 z + d n - 2 z 2 + + d n - 1 z n - 1 + d 0 z n D^{\sharp}_{n}(z)=z^{n}D_{n}(1/z)=d_{n}+d_{n-1}z+d_{n-2}z^{2}+\cdots+d_{n-1}z^% {n-1}+d_{0}z^{n}
  13. D n ( z ) D_{n}(z)
  14. T m ( z ) = T m ( z ) , m = n , n - 1 , , 0 T_{m}(z)=T^{\sharp}_{m}(z),m=n,n-1,\ldots,0
  15. T m ( z ) = k = 1 m t m , k z k T_{m}(z)=\sum_{k=1}^{m}t_{m,k}z^{k}
  16. T m ( z ) = t m , 0 + t m , 1 z + + t m , 1 z m - 1 + t m , 0 z m T_{m}(z)=t_{m,0}+t_{m,1}z+\cdots+t_{m,1}z^{m-1}+t_{m,0}z^{m}
  17. T n ( z ) = D n ( z ) + D n ( z ) , T n - 1 ( z ) = D n ( z ) - D n ( z ) z - 1 T_{n}(z)=D_{n}(z)+D^{\sharp}_{n}(z)\quad,\quad T_{n-1}(z)=\frac{D_{n}(z)-D^{% \sharp}_{n}(z)}{z-1}
  18. m = n - 1 , , 1 m=n-1,\ldots,1
  19. δ m + 1 = T m + 1 ( 0 ) T m ( 0 ) \delta_{m+1}=\frac{T_{m+1}(0)}{T_{m}(0)}
  20. T m - 1 ( z ) = δ m + 1 ( 1 + z ) T m ( z ) - T m + 1 ( z ) z T_{m-1}(z)=\frac{\delta_{m+1}(1+z)T_{m}(z)-T_{m+1}(z)}{z}
  21. T m ( 0 ) 0 , m = n - 1 , , 1 T_{m}(0)\neq 0,\quad m=n-1,\ldots,1
  22. T m ( 0 ) 0 , m = n , , 0 T_{m}(0)\neq 0,\quad m=n,\ldots,0
  23. T m ( 0 ) = t m , 0 = t m , m = 0 T_{m}(0)=t_{m,0}=t_{m,m}=0
  24. D n ( z ) D_{n}(z)
  25. ν = V a r { T n ( 1 ) , T n - 1 ( 1 ) , , T 1 ( 1 ) , t 0 , 0 } \nu=Var\{T_{n}(1),T_{n-1}(1),\ldots,T_{1}(1),t_{0,0}\}
  26. D n ( z ) D_{n}(z)
  27. ν = 0 \nu=0
  28. D n ( z ) D_{n}(z)
  29. ν \nu
  30. n - ν n-\nu
  31. T m ( 0 ) = 0 T_{m}(0)=0
  32. δ m < 0 \delta_{m}<0
  33. T m ( 1 ) T_{m}(1)
  34. D 3 ( z ) = 2 + K z - 22 z 2 + 24 z 3 D_{3}(z)=2+Kz-22z^{2}+24z^{3}
  35. K K
  36. K K
  37. T 3 ( z ) = 26 + ( K - 22 ) z + ( K - 22 ) z 2 + 26 z 3 T_{3}(z)=26+(K-22)z+(K-22)z^{2}+26z^{3}
  38. T 2 ( z ) = 22 - K z + 22 z 2 T_{2}(z)=22-Kz+22z^{2}
  39. T 1 ( z ) = 24 ( 22 - K ) 11 ( 1 + z ) T_{1}(z)=\frac{24(22-K)}{11}(1+z)
  40. T 0 ( z ) = 44 + k T_{0}(z)=44+k
  41. Var ( 8 + 2 K , 44 - K , 48 ( 22 - K ) / 11 , 44 + k ) \operatorname{Var}(8+2K,44-K,48(22-K)/11,44+k)\,

Bitruncated_16-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}
  3. D ~ 4 {\tilde{D}}_{4}
  4. B ~ 4 {\tilde{B}}_{4}
  5. D ~ 4 {\tilde{D}}_{4}
  6. F ~ 4 {\tilde{F}}_{4}
  7. F ~ 4 {\tilde{F}}_{4}
  8. B ~ 4 {\tilde{B}}_{4}
  9. D ~ 4 {\tilde{D}}_{4}

Bitruncated_24-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}

Bitruncated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}
  3. D ~ 4 {\tilde{D}}_{4}

Blackman's_theorem.html

  1. Z = Z D 1 + | T S C | 1 + | T O C | , Z=Z_{D}\frac{1+|T_{SC}|}{1+|T_{OC}|}\ ,
  2. Z i n = Z i n [ 1 + Z e 0 / Z 1 + Z e / Z ] Z_{in}=Z^{\infty}_{in}\left[\frac{1+Z^{0}_{e}/Z}{1+Z^{\infty}_{e}/Z}\right]
  3. Z Z
  4. Z i n Z^{\infty}_{in}
  5. Z Z
  6. Z e 0 Z^{0}_{e}
  7. Z Z
  8. Z e Z^{\infty}_{e}
  9. Z Z
  10. Z S S = Z S 0 [ 1 + T I 1 + T Z ] , Z_{SS}=Z_{S0}\left[\frac{1+T_{I}}{1+T_{Z}}\right]\ ,
  11. Z S 0 Z_{S0}
  12. Z S S Z_{SS}
  13. T Z T_{Z}
  14. T I T_{I}
  15. T Z T_{Z}

Blade_solidity.html

  1. s = 2 π r m / Z b s=2\pi r_{m}/Z_{b}
  2. r m r_{m}
  3. Z b Z_{b}
  4. r h r_{h}
  5. r t r_{t}
  6. r m = [ ( r t 2 + r h 2 ) / 2 ] 0.5 r_{m}=[(r_{t}^{2}+r_{h}^{2})/2]^{0.5}
  7. C L = 2 ( s / c ) ( tan β 1 - tan β 2 ) c o s β m C_{L}=2(s/c)(\tan\beta_{1}-\tan\beta_{2})cos\beta_{m}
  8. C d = ( s c ) ( Δ p 0 ρ W 1 2 / 2 ) C_{d}=\left(\frac{s}{c}\right)\left(\frac{\Delta p_{0}}{\rho W_{1}^{2}/2}\right)
  9. C L C_{L}
  10. C d C_{d}
  11. β 1 \beta_{1}
  12. β 2 \beta_{2}
  13. β m \beta_{m}
  14. W 1 W_{1}
  15. W m W_{m}
  16. Δ p 0 \Delta p_{0}
  17. tan β m = 1 2 ( tan β 1 + tan β 2 ) \tan\beta_{m}=\frac{1}{2}(\tan\beta_{1}+\tan\beta_{2})
  18. F L = C L b c ( 1 2 ρ W m 2 ) F_{L}=C_{L}bc\left(\frac{1}{2}\rho W_{m}^{2}\right)
  19. F d = C d b c ( 1 2 ρ W m 2 ) F_{d}=C_{d}bc\left(\frac{1}{2}\rho W_{m}^{2}\right)
  20. c s = 10 ( D h / D t ) ( N s / 1000 ) 1.5 \frac{c}{s}=\frac{10}{(D_{h}/D_{t})(N_{s}/1000)^{1.5}}
  21. D h D t \frac{D_{h}}{D_{t}}
  22. N s N_{s}
  23. D t D_{t}

Bloch's_formula.html

  1. K 2 K_{2}
  2. 𝒪 X \mathcal{O}_{X}
  3. CH q ( X ) = H q ( X , K q ( 𝒪 X ) ) \operatorname{CH}^{q}(X)=\operatorname{H}^{q}(X,K_{q}(\mathcal{O}_{X}))
  4. K q ( 𝒪 X ) K_{q}(\mathcal{O}_{X})
  5. U K q ( U ) U\mapsto K_{q}(U)
  6. Pic ( X ) = H 1 ( X , 𝒪 X * ) \operatorname{Pic}(X)=H^{1}(X,\mathcal{O}_{X}^{*})

Block_sort.html

  1. A \sqrt{A}
  2. A \sqrt{A}
  3. A \sqrt{A}
  4. A \sqrt{A}
  5. ( c o u n t + 1 ) / 2 \sqrt{(count+1)/2}
  6. A \sqrt{A}
  7. A \sqrt{A}
  8. A \sqrt{A}
  9. n \sqrt{n}
  10. A \sqrt{A}
  11. A \sqrt{A}
  12. A \sqrt{A}
  13. n \sqrt{n}
  14. n \sqrt{n}
  15. A \sqrt{A}
  16. A \sqrt{A}
  17. A \sqrt{A}
  18. A \sqrt{A}
  19. A \sqrt{A}
  20. A \sqrt{A}
  21. A \sqrt{A}
  22. A \sqrt{A}
  23. A \sqrt{A}

Body_Shape_Index.html

  1. A B S I = W C B M I 2 3 × H e i g h t 1 2 ABSI={WC\over{BMI^{2\over 3}\times Height^{1\over 2}}}

Bogdanov_map.html

  1. { x n + 1 = x n + y n + 1 y n + 1 = y n + ϵ y n + k x n ( x n - 1 ) + μ x n y n \begin{cases}x_{n+1}=x_{n}+y_{n+1}\\ y_{n+1}=y_{n}+\epsilon y_{n}+kx_{n}(x_{n}-1)+\mu x_{n}y_{n}\end{cases}

Boggs_eumorphic_projection.html

  1. x = R 2 k λ - λ 0 sec φ + π 2 4 sec θ x=R2k\frac{\lambda-\lambda_{0}}{\sec\varphi+\frac{\pi\sqrt{2}}{4}\sec\theta}
  2. y = R φ + 2 sin θ 2 k y=R\frac{\varphi+\sqrt{2}\sin\theta}{2k}
  3. 2 θ + sin 2 θ = π sin φ 2\theta+\sin 2\theta=\pi\sin\varphi

Borel_distribution.html

  1. 1 1 - μ \frac{1}{1-\mu}
  2. μ ( 1 - μ ) 3 \frac{\mu}{(1-\mu)^{3}}
  3. P μ ( n ) = Pr ( X = n ) = e - μ n ( μ n ) n - 1 n ! P_{\mu}(n)=\Pr(X=n)=\frac{e^{-\mu n}(\mu n)^{n-1}}{n!}
  4. Pr ( X = n ) = 1 n Pr ( S n = n - 1 ) \Pr(X=n)=\frac{1}{n}\Pr(S_{n}=n-1)
  5. 1 + μ + μ 2 + = 1 1 - μ . 1+\mu+\mu^{2}+\cdots=\frac{1}{1-\mu}.
  6. P μ * ( n ) = ( 1 - μ ) e - μ n ( μ n ) n - 1 ( n - 1 ) ! . P_{\mu}^{*}(n)=(1-\mu)\frac{e^{-\mu n}(\mu n)^{n-1}}{(n-1)!}.
  7. P μ ( n ) = 1 μ 0 μ P λ * ( n ) d λ . P_{\mu}(n)=\frac{1}{\mu}\int_{0}^{\mu}P_{\lambda}^{*}(n)\,d\lambda.
  8. E ( 1 / X ) = 1 - μ / 2. E(1/X)=1-\mu/2.
  9. Pr ( W = n ) = k n Pr ( S n = n - k ) \Pr(W=n)=\frac{k}{n}\Pr(S_{n}=n-k)
  10. Pr ( W = n ) = k n e - μ n ( μ n ) n - k ( n - k ) ! \Pr(W=n)=\frac{k}{n}\frac{e^{-\mu n}(\mu n)^{n-k}}{(n-k)!}

Born–Mayer_equation.html

  1. E = - N A M z + z - e 2 4 π ϵ 0 r 0 ( 1 - ρ r 0 ) E=-\frac{N_{A}Mz^{+}z^{-}e^{2}}{4\pi\epsilon_{0}r_{0}}\left(1-\frac{\rho}{r_{0% }}\right)
  2. × 10 19 \times 10^{−}19
  3. × 10 10 \times 10^{−}10

Bose-Einstein_condensation_of_quasiparticles.html

  1. k B T = 2 n 2 / 3 / M k_{B}T=~{}\hbar^{2}n^{2/3}/M
  2. N ( T / 2 π ) 3 u 1 / 2 P / v 3 N\propto(T/2\pi)^{3}u^{1/2}P/v\hbar^{3}
  3. T c < 32 π 3 6 V 2 u 0 P 2 T_{c}<32\pi^{3}\hbar^{6}V^{2}u_{0}P^{2}
  4. V ( r ) = M ω 2 / 2 V(r)=M\omega^{2}/2
  5. f ( 0 ) = N 0 ( t ) / N = 1 - ( T / T c ) 3 f(0)=N_{0}(t)/N=1-(T/T_{c})^{3}
  6. λ \lambda
  7. λ d B \lambda_{dB}
  8. n n
  9. m 𝑒𝑓𝑓 m_{\mathit{eff}}
  10. \hbar
  11. k k
  12. g g
  13. n = g τ n=g\tau
  14. d n / d t = - a n 2 dn/dt=-an^{2}
  15. N c = ζ ( 3 ) ( k T / ω ) 3 N_{c}=\zeta(3)(kT/\hbar\omega)^{3}
  16. τ \tau
  17. λ \lambda

Bott–Samelson_resolution.html

  1. w W = N G ( T ) / T . w\in W=N_{G}(T)/T.
  2. w ¯ = ( s i 1 , s i 2 , , s i l ) \underline{w}=(s_{i_{1}},s_{i_{2}},\ldots,s_{i_{l}})
  3. w = s i 1 s i 2 s i l w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{l}}
  4. P i j G P_{i_{j}}\subset G
  5. s i j s_{i_{j}}
  6. Z w ¯ Z_{\underline{w}}
  7. Z w ¯ = P i 1 × × P i l / B l Z_{\underline{w}}=P_{i_{1}}\times\cdots\times P_{i_{l}}/B^{l}
  8. B l B^{l}
  9. ( b 1 , , b l ) ( p 1 , , p l ) = ( p 1 b 1 - 1 , b 1 p 2 b 2 - 1 , , b l - 1 p l b l - 1 ) (b_{1},\ldots,b_{l})\cdot(p_{1},\ldots,p_{l})=(p_{1}b_{1}^{-1},b_{1}p_{2}b_{2}% ^{-1},\ldots,b_{l-1}p_{l}b_{l}^{-1})
  10. X w = B w B ¯ / B = ( P i 1 P i l ) / B X_{w}=\overline{BwB}/B=(P_{i_{1}}\cdots P_{i_{l}})/B
  11. π : Z w ¯ X w \pi:Z_{\underline{w}}\to X_{w}
  12. π \pi
  13. π * 𝒪 Z w ¯ = 𝒪 X w \pi_{*}\mathcal{O}_{Z_{\underline{w}}}=\mathcal{O}_{X_{w}}
  14. R i π * 𝒪 Z w ¯ = 0 , i 1. R^{i}\pi_{*}\mathcal{O}_{Z_{\underline{w}}}=0,\,i\geq 1.
  15. X w X_{w}

Bousfield_class.html

  1. X Y = 0 X\otimes Y=0

Bousfield_localization.html

  1. f : X Y f:X\to Y
  2. f * : m a p ( Y , W ) m a p ( X , W ) f^{*}:map(Y,W)\to map(X,W)
  3. s * : m a p ( B , W ) m a p ( A , W ) s^{*}:map(B,W)\to map(A,W)
  4. f : A B f:A\to B
  5. m a p ( - , - ) map(-,-)
  6. π 0 ( m a p ( X , Y ) ) = H o m H o ( M ) ( X , Y ) . \pi_{0}(map(X,Y))=Hom_{Ho(M)}(X,Y).
  7. C [ W - 1 ] C[W^{-1}]
  8. C C [ W - 1 ] C\to C[W^{-1}]
  9. C D C\to D
  10. L C M L_{C}M
  11. M L C M M\to L_{C}M
  12. M N M\to N
  13. M L C M M\to L_{C}M
  14. S ( p ) S_{(p)}

Boyfriend_Maker_(Smartphone_App).html

  1. x = - b ± b 2 - 4 a c 2 a x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a}

Bradley–Terry_model.html

  1. i i
  2. j j
  3. i > j i>j
  4. P ( i > j ) = p i p i + p j P(i>j)=\frac{p_{i}}{p_{i}+p_{j}}
  5. i i
  6. i > j i>j
  7. i i
  8. j j
  9. i i
  10. j j
  11. i i
  12. j j
  13. i i
  14. P ( i > j ) P(i>j)
  15. i i
  16. j j
  17. P ( i > j ) P(i>j)
  18. i i
  19. j j
  20. n n
  21. p i = e β i p_{i}=e^{\beta_{i}}
  22. P ( i > j ) = e β i e β i + e β j P(i>j)=\frac{e^{\beta_{i}}}{e^{\beta_{i}}+e^{\beta_{j}}}
  23. P ( i > j ) P ( j > i ) = β i - β j \frac{P(i>j)}{P(j>i)}=\beta_{i}-\beta_{j}
  24. ( i , j ) (i,j)
  25. i i
  26. j j
  27. i i
  28. j j
  29. L ( 𝐩 ) = i n j n w i j ln p i - w i j ln ( p i + p j ) . L(\mathbf{p})=\sum_{i}^{n}\sum_{j}^{n}w_{ij}\ln p_{i}-w_{ij}\ln(p_{i}+p_{j}).
  30. i i
  31. i i
  32. j j
  33. 𝐩 \mathbf{p}
  34. p i = W i ( j i N i j p i + p j ) - 1 p^{\prime}_{i}=W_{i}\left(\sum_{j\neq i}\frac{N_{ij}}{p_{i}+p_{j}}\right)^{-1}
  35. i i
  36. p i p i j n p j . p_{i}\leftarrow\frac{p^{\prime}_{i}}{\sum_{j}^{n}p^{\prime}_{j}}.

Brams–Taylor–Zwicker_procedure.html

  1. k k
  2. 1 / k 1/k
  3. k = 4 k=4

Brauner_space.html

  1. X X
  2. K n K_{n}
  3. T X T\subseteq X
  4. K n K_{n}
  5. X X
  6. X X^{\star}
  7. X X
  8. X X^{\star}
  9. M M
  10. σ \sigma
  11. 𝒞 ( M ) {\mathcal{C}}(M)
  12. M M
  13. {\mathbb{R}}
  14. {\mathbb{C}}
  15. M M
  16. 𝒞 ( M ) {\mathcal{C}}^{\star}(M)
  17. M M
  18. 𝒞 ( M ) {\mathcal{C}}(M)
  19. M M
  20. ( M ) {\mathcal{E}}(M)
  21. M M
  22. {\mathbb{R}}
  23. {\mathbb{C}}
  24. M M
  25. ( M ) {\mathcal{E}}^{\star}(M)
  26. M M
  27. ( M ) {\mathcal{E}}(M)
  28. M M
  29. 𝒪 ( M ) {\mathcal{O}}(M)
  30. M M
  31. M M
  32. 𝒪 ( M ) {\mathcal{O}}^{\star}(M)
  33. M M
  34. 𝒪 ( M ) {\mathcal{O}}(M)
  35. G G
  36. 𝒪 exp ( G ) {\mathcal{O}}_{\exp}(G)
  37. G G
  38. X X
  39. X X^{\star}
  40. f : X f:X\to\mathbb{C}
  41. X X

Brazilian_Swap.html

  1. N o t i o n a l fixed leg ( t ) = N × i = 1 n - 1 ( 1 + κ ) ( 1 252 ) Notional\text{fixed leg}(t)=N\times\prod_{i=1}^{n-1}\left(1+\kappa\right)^{(% \frac{1}{252})}
  2. N o t i o n a l floating leg ( t ) = N × i = 1 n - 1 ( 1 + r i ) ( 1 252 ) Notional\text{floating leg}(t)=N\times\prod_{i=1}^{n-1}\left(1+r_{i}\right)^{(% \frac{1}{252})}
  3. [ 1 + R ( t ) ] ( 1 252 ) \left[1+R(t)\right]^{(\frac{1}{252})}

Breakover_angle.html

  1. breakover angle a p p r o x i m a t e = 2 arctan ( 2 ground clearance wheelbase ) \,\text{breakover angle}_{approximate}=2\cdot\arctan\left(\frac{2\cdot\,\text{% ground clearance}}{\,\text{wheelbase}}\right)

Breed_method.html

  1. M F = 100 m m f ´ i e l d A r e a i n m m MF={{100\ mm}\over\acute{f}ield\ \ Area\ in\ mm}
  2. N u m b e r o f s o m a t i c s C e l l s = T o t a l s o m a t i c c e l l s N u m b e r o f c o u n t e d f i e l d s Number\ of\ somatics\ Cells={Total\ somatic\ cells\ \over Number\ of\ counted% \ fields}

Bretherton_equation.html

  1. u t t + u x x + u x x x x + u = u p , u_{tt}+u_{xx}+u_{xxxx}+u=u^{p},
  2. p p
  3. p 2. p\geq 2.
  4. u t , u x u_{t},u_{x}
  5. u x x u_{xx}
  6. u ( x , t ) . u(x,t).
  7. p = 2. p=2.
  8. p = 3 p=3
  9. = 1 2 ( u t ) 2 + 1 2 ( u x ) 2 - 1 2 ( u x x ) 2 - 1 2 u 2 + 1 p + 1 u p + 1 \mathcal{L}=\tfrac{1}{2}\left(u_{t}\right)^{2}+\tfrac{1}{2}\left(u_{x}\right)^% {2}-\tfrac{1}{2}\left(u_{xx}\right)^{2}-\tfrac{1}{2}u^{2}+\tfrac{1}{p+1}u^{p+1}
  10. t ( u t ) + x ( u x ) - 2 x 2 ( u x x ) - u = 0. \frac{\partial}{\partial t}\left(\frac{\partial\mathcal{L}}{\partial u_{t}}% \right)+\frac{\partial}{\partial x}\left(\frac{\partial\mathcal{L}}{\partial u% _{x}}\right)-\frac{\partial^{2}}{\partial x^{2}}\left(\frac{\partial\mathcal{L% }}{\partial u_{xx}}\right)-\frac{\partial\mathcal{L}}{\partial u}=0.
  11. u t - δ H δ v = 0 , v t + δ H δ u = 0 , \begin{aligned}\displaystyle u_{t}&\displaystyle-\frac{\delta{H}}{\delta v}=0,% \\ \displaystyle v_{t}&\displaystyle+\frac{\delta{H}}{\delta u}=0,\end{aligned}
  12. H : H:
  13. H ( u , v ) = ( u , v ; x , t ) d x H(u,v)=\int\mathcal{H}(u,v;x,t)\;\mathrm{d}x
  14. ( u , v ; x , t ) = 1 2 v 2 - 1 2 ( u x ) 2 + 1 2 ( u x x ) 2 + 1 2 u 2 - 1 p + 1 u p + 1 \mathcal{H}(u,v;x,t)=\tfrac{1}{2}v^{2}-\tfrac{1}{2}\left(u_{x}\right)^{2}+% \tfrac{1}{2}\left(u_{xx}\right)^{2}+\tfrac{1}{2}u^{2}-\tfrac{1}{p+1}u^{p+1}
  15. \mathcal{H}
  16. v = u t . v=u_{t}.
  17. H H

Brezis–Gallouet_inequality.html

  1. u H 2 ( Ω ) u\in H^{2}(\Omega)
  2. Ω 2 \Omega\subset\mathbb{R}^{2}
  3. C C
  4. u L ( Ω ) C u H 1 ( Ω ) ( 1 + log Δ u λ 1 u H 1 ( Ω ) ) 1 / 2 , \displaystyle\|u\|_{L^{\infty}(\Omega)}\leq C\|u\|_{H^{1}(\Omega)}\left(1+\log% \frac{\|\Delta u\|}{\lambda_{1}\|u\|_{H^{1}(\Omega)}}\right)^{1/2},
  5. Δ \Delta
  6. λ 1 \lambda_{1}

Brian_Alspach.html

  1. K n K_{n}
  2. K n - I K_{n}-I

Briggs–Bers_criterion.html

  1. L y = 0 Ly=0
  2. y = y ( x , t ) y=y(x,t)
  3. x x
  4. t t
  5. L = L ( x , t ) L=L(\partial_{x},\partial_{t})
  6. x x
  7. t t
  8. y y
  9. y = y ^ exp ( i k x - i ω t ) . y=\hat{y}\exp(ikx-i\omega t).
  10. L ( i k , - i ω ) y ^ exp ( i k x - i ω t ) = 0 ; L(ik,-i\omega)\hat{y}\exp(ikx-i\omega t)=0;
  11. L ( i k , - i ω ) = 0. L(ik,-i\omega)=0.
  12. k k
  13. ω \omega
  14. ω \omega
  15. k k
  16. y ( x , 0 ) y(x,0)
  17. exp ( i k x ) \exp(ikx)
  18. y ( x , t ) y(x,t)
  19. L L
  20. ω ( k ) \omega(k)
  21. k k
  22. y ( x , t ) = 1 2 π y ^ ( k ) exp ( i k x - i ω ( k ) t ) d k y(x,t)=\frac{1}{2\pi}\int\hat{y}(k)\exp(ikx-i\omega(k)t)\,dk
  23. y ^ ( k ) = y ( x , 0 ) exp ( - i k x ) d x \hat{y}(k)=\int y(x,0)\exp(-ikx)\,dx
  24. k k
  25. ω \omega
  26. y ( x , t ) y(x,t)
  27. t t\rightarrow\infty
  28. y ( x , t ) y(x,t)
  29. ω \Im\omega
  30. ω < 0 \Im\omega<0
  31. t t\rightarrow\infty
  32. y = 0 y=0
  33. ω > 0 \Im\omega>0
  34. x = 0 x=0
  35. x = 0 x=0
  36. y ( 0 , t ) 0 y(0,t)\rightarrow 0
  37. y ( 0 , t ) y(0,t)\rightarrow\infty
  38. y t = A y y_{t}=Ay
  39. A A
  40. x x
  41. A A
  42. y y
  43. A A
  44. A A
  45. y y
  46. y = 0 y=0
  47. y y
  48. y y
  49. L L
  50. x x

Broadband_acoustic_resonance_dissolution_spectroscopy.html

  1. υ = 1 K ρ \upsilon=\frac{1}{\sqrt{K\rho}}
  2. K = ( d V d p ) V K=\frac{\left({dV\over dp}\right)}{V}
  3. υ w υ = f w f = ( 1 + α V a ) 1 2 {\upsilon_{w}\over\upsilon}={f_{w}\over f}={(1+\alpha V_{a})}^{1\over 2}

Broer-Kaup_equation.html

  1. u y , t + ( 2 * u * u x ) x + 2 * v x x - u x x y = 0 u_{y,t}+(2*u*u_{x})_{x}+2*v_{xx}-u_{xxy}=0
  2. v t + 2 * ( v u ) x + v x x = 0 v_{t}+2*(vu)_{x}+v_{xx}=0

Brown_clustering.html

  1. n n
  2. c cᵢ
  3. w wᵢ
  4. w wᵢ
  5. w wᵢ₋₁
  6. P ( w i | w i - 1 ) = P ( w i | c i ) P ( c i | c i - 1 ) P(w_{i}|w_{i-1})=P(w_{i}|c_{i})P(c_{i}|c_{i-1})

BSTAR.html

  1. F D = 1 2 ρ C d A v 2 F_{D}=\frac{1}{2}\rho C_{d}Av^{2}
  2. ρ \rho
  3. C d C_{d}
  4. A A
  5. v v
  6. a D = F D m = ρ C d A v 2 2 m a_{D}=\frac{F_{D}}{m}=\frac{\rho C_{d}Av^{2}}{2m}
  7. B = C d A m B=\frac{C_{d}A}{m}
  8. B * = ρ B 2 = ρ C d A 2 m B^{*}=\frac{\rho B}{2}=\frac{\rho C_{d}A}{2m}
  9. a D = B * v 2 a_{D}=B^{*}v^{2}
  10. B * B^{*}

Buchstab_function.html

  1. ω : \R 1 \R > 0 \omega:\R_{\geq 1}\rightarrow\R_{>0}
  2. ω ( u ) = 1 u , 1 u 2 , \omega(u)=\frac{1}{u},\qquad\qquad\qquad 1\leq u\leq 2,
  3. d d u ( u ω ( u ) ) = ω ( u - 1 ) , u 2. {\frac{d}{du}}(u\omega(u))=\omega(u-1),\qquad u\geq 2.
  4. e - γ e^{-\gamma}
  5. u , u\to\infty,
  6. γ \gamma
  7. | ω ( u ) - e - γ | ρ ( u - 1 ) u , u 1 , |\omega(u)-e^{-\gamma}|\leq\frac{\rho(u-1)}{u},\qquad u\geq 1,
  8. ω ( u ) - e - γ \omega(u)-e^{-\gamma}
  9. Φ ( x , x 1 / u ) ω ( u ) x log x 1 / u , x . \Phi(x,x^{1/u})\sim\omega(u)\frac{x}{\log x^{1/u}},\qquad x\to\infty.

Bulk_dispatch_lapse.html

  1. f ( x ) = { m x + y 1 - m x 1 if 0 x < x 1 0 if x x 1 f(x)=\begin{cases}mx+y_{1}-mx_{1}&\,\text{if }0\leq x<x_{1}\\ 0&\,\text{if }x\geq x_{1}\end{cases}

Bunch–Nielsen–Sorensen_formula.html

  1. A A
  2. v v T vv^{T}
  3. v v
  4. λ i \lambda_{i}
  5. A A
  6. λ ~ i \tilde{\lambda}_{i}
  7. A ~ = A + v v T \tilde{A}=A+vv^{T}
  8. A A
  9. q ~ i \tilde{q}_{i}
  10. A ~ \tilde{A}
  11. ( q ~ i ) k = N i v k λ k - λ ~ i (\tilde{q}_{i})_{k}=\frac{N_{i}v_{k}}{\lambda_{k}-\tilde{\lambda}_{i}}
  12. N i N_{i}
  13. q ~ i \tilde{q}_{i}
  14. ( A - λ ~ + v v T ) - 1 (A-\tilde{\lambda}+vv^{T})^{-1}
  15. A ~ \tilde{A}

Bundle_theorem.html

  1. x 4 + y 4 + z 4 = 1 x^{4}+y^{4}+z^{4}=1
  2. \infty
  3. 𝔐 \mathfrak{M}
  4. A 1 , A 2 , A 3 , A 4 , B 1 , B 2 , B 3 , B 4 A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},B_{3},B_{4}
  5. Q i j := { A i , B i , A j , B j } , i < j , Q_{ij}:=\{A_{i},B_{i},A_{j},B_{j}\},\ i<j,
  6. c i j c_{ij}
  7. c 23 , c 34 , c 24 c_{23},c_{34},c_{24}
  8. P P
  9. P P
  10. A 2 B 2 , A 4 B 4 A_{2}B_{2},\ A_{4}B_{4}
  11. c 12 , c 14 , c 24 c_{12},c_{14},c_{24}
  12. P P^{\prime}
  13. P P^{\prime}
  14. A 2 B 2 , A 4 B 4 A_{2}B_{2},\ A_{4}B_{4}
  15. P = P P=P^{\prime}
  16. A 1 B 1 , A 3 B 3 A_{1}B_{1},\ A_{3}B_{3}
  17. P P
  18. A 1 , B 1 , A 3 , B 3 A_{1},B_{1},A_{3},B_{3}
  19. P P

Burgers_vortex.html

  1. ( r , z , ϕ ) (r,z,\phi)
  2. ϕ \phi
  3. v r = - 1 2 α r , v_{r}=-\frac{1}{2}\alpha r,
  4. v z = α z , v_{z}=\alpha z,
  5. v ϕ = v ϕ ( r ) , v_{\phi}=v_{\phi}(r),
  6. α > 0 \alpha>0
  7. z z
  8. D ζ D t = ζ v z z + ν 2 ζ , \frac{D\zeta}{Dt}=\zeta\frac{\partial v_{z}}{\partial z}+\nu\nabla^{2}\zeta,
  9. D / D t D/Dt
  10. ν \nu
  11. ζ = ζ 0 exp ( - α r 2 4 ν ) , \zeta=\zeta_{0}\exp(-\frac{\alpha r^{2}}{4\nu}),
  12. ζ 0 \zeta_{0}
  13. R = 2 ν α . R=2\sqrt{\frac{\nu}{\alpha}}.

Burnside_category.html

  1. X U Y X\leftarrow U\rightarrow Y
  2. X U Y X\leftarrow U\rightarrow Y
  3. X W Y X\leftarrow W\rightarrow Y
  4. A ( G ) ( X , Y ) A(G)(X,Y)
  5. A ( G ) ( X , Y ) × A ( G ) ( Y , Z ) A ( G ) ( X , Z ) A(G)(X,Y)\times A(G)(Y,Z)\rightarrow A(G)(X,Z)
  6. A ( G ) ( X , Y ) A(G)(X,Y)

Burst_suppression.html

  1. p i = e x i 1 + e x i , p_{i}=\frac{e^{x_{i}}}{1+e^{x_{i}}},

Byers-Yang_theorem.html

  1. Φ \Phi
  2. Φ 0 = h / e \Phi_{0}=h/e
  3. Φ \Phi
  4. A ( r ) A(r)
  5. C A d l = Φ \oint_{C}A\cdot dl=\Phi
  6. C C
  7. ψ ( { r n } ) = exp ( i e j χ ( r j ) ) ψ ( { r n } ) \psi^{\prime}(\{r_{n}\})=\exp\left(\frac{ie}{\hbar}\sum_{j}\chi(r_{j})\right)% \psi(\{r_{n}\})
  8. ψ ( { r n } ) \psi(\{r_{n}\})
  9. r 1 , r 2 , r_{1},r_{2},\ldots
  10. A ( r ) = A ( r ) + χ ( r ) A^{\prime}(r)=A(r)+\nabla\chi(r)
  11. B ( r ) = × A ( r ) = 0 B(r)=\nabla\times A(r)=0
  12. r r
  13. χ ( r ) \chi(r)
  14. A ( r ) = 0 A^{\prime}(r)=0
  15. Φ \Phi
  16. Φ \Phi
  17. ψ \psi^{\prime}
  18. δ ϕ = ( e / ) C χ ( r ) d l = 2 π Φ / Φ 0 \delta\phi=(e/\hbar)\oint_{C}\nabla\chi(r)\cdot dl=2\pi\Phi/\Phi_{0}
  19. r n r_{n}
  20. Φ \Phi
  21. Φ 0 \Phi_{0}
  22. h / e h/e

BzK_galaxy.html

  1. B z K ( z - K ) A B - ( B - z ) A B - 0.2 BzK\equiv(z-K)_{AB}-(B-z)_{AB}\geq-0.2
  2. B z K < - 0.2 BzK<-0.2
  3. ( z - K ) A B > 2.5 (z-K)_{AB}>2.5

Cache-oblivious_distribution_sort.html

  1. N N
  2. Z Z
  3. L L
  4. O ( N L log Z N ) O(\frac{N}{L}\log_{Z}N)
  5. Z = Ω ( L 2 ) Z=\Omega(L^{2})
  6. Θ ( N log N ) \Theta(N\log N)
  7. N N
  8. N \sqrt{N}
  9. N \sqrt{N}
  10. q N q\leq\sqrt{N}
  11. B 1 , B 2 , , B q B_{1},B_{2},\ldots,B_{q}
  12. 2 N 2\sqrt{N}
  13. B i B_{i}
  14. B i + 1 . B_{i+1}.
  15. B 1 , B 2 , , B q . B_{1},B_{2},\ldots,B_{q}.
  16. 2 N 2\sqrt{N}
  17. B i B_{i}
  18. B i + 1 . B_{i+1}.
  19. \infty
  20. ( 2 N + 1 ) (2\sqrt{N}+1)
  21. b n u m = bnum=\infty
  22. B j B_{j}
  23. i , , i + m - 1 i,\ldots,i+m-1
  24. b n u m [ r ] j bnum[r]\geq j
  25. b n u m [ r ] j + m bnum[r]\geq j+m
  26. ( 1 , 1 , N ) (1,1,\sqrt{N})

Cache-oblivious_matrix_multiplication.html

  1. A A
  2. B B
  3. m × n m\times n
  4. n × p n\times p
  5. Z Z
  6. L L
  7. Θ ( m + n + p + m n + n p + m p L + m n p L Z ) \Theta(m+n+p+\frac{mn+np+mp}{L}+\frac{mnp}{L\sqrt{Z}})
  8. Z = Ω ( L 2 ) Z=\Omega(L^{2})
  9. Z Z
  10. L L
  11. m n , p m\geq n,p
  12. A B = ( A 1 A 2 ) B = ( A 1 B A 2 B ) AB=\begin{pmatrix}A_{1}\\ A_{2}\end{pmatrix}B=\begin{pmatrix}A_{1}B\\ A_{2}B\end{pmatrix}
  13. n m , p n\geq m,p
  14. A B = ( A 1 A 2 ) ( B 1 B 2 ) = A 1 B 1 + A 2 B 2 AB=\begin{pmatrix}A_{1}&A_{2}\end{pmatrix}\begin{pmatrix}B_{1}\\ B_{2}\end{pmatrix}=A_{1}B_{1}+A_{2}B_{2}
  15. p m , n p\geq m,n
  16. A B = A ( B 1 B 2 ) = ( A B 1 A B 2 ) AB=A\begin{pmatrix}B_{1}&B_{2}\end{pmatrix}=\begin{pmatrix}AB_{1}&AB_{2}\end{pmatrix}
  17. m = n = p = 1 m=n=p=1
  18. A A
  19. B B
  20. Θ ( m + n + p + m n + n p + m p L + m n p L Z ) \Theta(m+n+p+\frac{mn+np+mp}{L}+\frac{mnp}{L\sqrt{Z}})
  21. Θ ( m n p ) \Theta(mnp)

Calcium_looping.html

  1. C a C O 3 C a O + C O 2 Δ H = + 178 k J / m o l CaCO_{3}\leftrightharpoons CaO+CO_{2}\qquad\qquad\Delta H=+178kJ/mol
  2. C a O + S O 2 + 1 / 2 O 2 C a S O 4 CaO+SO_{2}+1/2O_{2}\Rightarrow CaSO_{4}
  3. C a C O 2 + S O 2 + 1 / 2 O 2 C a S O 4 + C O 2 CaCO_{2}+SO_{2}+1/2O_{2}\Rightarrow CaSO_{4}+CO_{2}

Calculation_of_radiocarbon_dates.html

  1. A s = A s t d ( M s - M b M s t d - M b ) A_{s}=A_{std}\left(\frac{M_{s}-M_{b}}{M_{std}-M_{b}}\right)
  2. F r a c 13 / 12 ( s a m p l e ) = ( 13 C / 12 C ) w o o d ( 13 C / 12 C ) s a m p l e Frac_{13/12(sample)}=\frac{(^{13}C/^{12}C)_{wood}}{{(^{13}C/^{12}C)_{sample}}}
  3. F r a c 14 / 12 ( s a m p l e ) = ( F r a c 13 / 12 ( s a m p l e ) ) 2 Frac_{14/12(sample)}=(Frac_{13/12(sample)})^{2}
  4. A s n = A s F r a c 14 / 12 ( s ) A_{sn}=A_{s}Frac_{14/12(s)}
  5. ( C 13 C 12 ) s a m p l e = ( 1 + δ 13 C 1000 ) ( C 13 C 12 ) P D B \left(\frac{{}^{13}C}{{}^{12}C}\right)_{sample}=\left(1+\frac{\delta^{13}C}{10% 00}\right)\left(\frac{{}^{13}C}{{}^{12}C}\right)_{PDB}
  6. A s n = A s ( ( 1 - 25 1000 ) ( C 13 C 12 ) P D B ( 1 + δ 13 C 1000 ) ( C 13 C 12 ) P D B ) 2 A_{sn}=A_{s}\left(\frac{\left(1-\frac{25}{1000}\right)\left(\frac{{}^{13}C}{{}% ^{12}C}\right)_{PDB}}{\left(1+\frac{\delta^{13}C}{1000}\right)\left(\frac{{}^{% 13}C}{{}^{12}C}\right)_{PDB}}\right)^{2}
  7. A s n = A s ( ( 1 - 25 1000 ) ( 1 + δ 13 C 1000 ) ) 2 A_{sn}=A_{s}\left(\frac{\left(1-\frac{25}{1000}\right)}{\left(1+\frac{\delta^{% 13}C}{1000}\right)}\right)^{2}
  8. F m = R n o r m R m o d e r n F_{m}=\frac{R_{norm}}{R_{modern}}
  9. R s = R s - R m b R^{\prime}_{s}=R_{s}-R_{mb}
  10. R s t d = R s t d - R m b R^{\prime}_{std}=R_{std}-R_{mb}
  11. R p b = R p b - R m b R^{\prime}_{pb}=R_{pb}-R_{mb}
  12. R H O x I , - 19 = R H o x I ( 1 + - 19 1000 1 + δ 13 C H o X I 1000 ) 2 R_{HOxI,-19}=R^{\prime}_{HoxI}\left(\frac{1+\frac{-19}{1000}}{1+\frac{\delta^{% 13}C_{HoXI}}{1000}}\right)^{2}
  13. R H O x I I , - 25 = R H o x I I ( 1 + - 25 1000 1 + δ 13 C H o X I I 1000 ) 2 R_{HOxII,-25}=R^{\prime}_{HoxII}\left(\frac{1+\frac{-25}{1000}}{1+\frac{\delta% ^{13}C_{HoXII}}{1000}}\right)^{2}
  14. R H O x I , - 19 = R H o x I ( 1 + - 19 1000 1 + δ 13 C H o X I 1000 ) R_{HOxI,-19}=R^{\prime}_{HoxI}\left(\frac{1+\frac{-19}{1000}}{1+\frac{\delta^{% 13}C_{HoXI}}{1000}}\right)
  15. R H O x I I , - 25 = R H o x I I ( 1 + - 25 1000 1 + δ 13 C H o X I I 1000 ) R_{HOxII,-25}=R^{\prime}_{HoxII}\left(\frac{1+\frac{-25}{1000}}{1+\frac{\delta% ^{13}C_{HoXII}}{1000}}\right)
  16. R m o d e r n = 0.95 R H O x I , - 19 = .7459 R H O x 2 , - 25 R_{modern}=0.95R_{HOxI,-19}=.7459R_{HOx2,-25}
  17. F m u c = R s R m o d e r n Fm_{uc}=\frac{R^{\prime}_{s}}{R_{modern}}
  18. F m m s = F m u c ( 1 + - 25 1000 1 + δ 13 C s 1000 ) 2 Fm_{ms}=Fm_{uc}\left(\frac{1+\frac{-25}{1000}}{1+\frac{\delta 13C_{s}}{1000}}% \right)^{2}
  19. F m m s = F m u c ( 1 + - 25 1000 1 + δ 13 C s 1000 ) Fm_{ms}=Fm_{uc}\left(\frac{1+\frac{-25}{1000}}{1+\frac{\delta 13C_{s}}{1000}}\right)
  20. F m s = F m m s C m s - F m p b C p b C s Fm_{s}=\frac{Fm_{ms}C_{ms}-Fm_{pb}C_{pb}}{C_{s}}
  21. A g e = - 8033 l n ( F m ) Age=-8033ln(Fm)

Calibration_of_radiocarbon_dates.html

  1. σ t o t a l = ( σ s a m p l e 2 + σ c a l i b 2 ) 1 2 \sigma_{total}={\bigl(\sigma_{sample}^{2}+\sigma_{calib}^{2}\bigr)}^{\frac{1}{% 2}}

Calkin_correspondence.html

  1. diag ( a ) = n = 0 a n | e n e n | , {\rm diag}(a)=\sum_{n=0}^{\infty}a_{n}|e_{n}\rangle\langle e_{n}|,
  2. = =
  3. j = { a l : diag ( μ ( a ) ) J } . j=\{a\in l_{\infty}:{\rm diag}(\mu(a))\in J\}.
  4. J = { A B ( H ) : μ ( A ) j } . J=\{A\in B(H):\mu(A)\in j\}.

Callier_effect.html

  1. Q = D d i r < m t p l > D d i f Q=\frac{{D_{dir}}}{<}mtpl>{{D_{{dif}}}}

Calvo_(staggered)_contracts.html

  1. P r [ i ] = ( 1 - h ) i - 1 h Pr[i]=(1-h)^{i-1}h
  2. E [ P r [ i ] ] = h - 1 E[Pr[i]]=h^{-1}
  3. i = 1... i=1...\infty
  4. ( 1 - h ) 2 (1-h)^{2}
  5. ( 1 - h ) i (1-h)^{i}
  6. π t = β E t [ π t + 1 ] + κ y t \pi_{t}=\beta E_{t}[\pi_{t+1}]+\kappa y_{t}
  7. κ = h [ 1 - ( 1 - h ) β ] 1 - h γ \kappa=\frac{h[1-(1-h)\beta]}{1-h}\gamma
  8. β E t [ π t + 1 ] \beta E_{t}[\pi_{t+1}]
  9. κ \kappa
  10. α i \alpha^{i}
  11. α i = ( 1 - h ) i - 1 . h \alpha^{i}=(1-h)^{i-1}.h
  12. A * A^{*}
  13. A * = i = 0 i . h i ( 1 - h ) i - 1 = 1 h A^{*}=\sum_{i=0}^{\infty}i.h^{i}(1-h)^{i-1}=\frac{1}{h}
  14. T = 2 h - 1 = 2. A * - 1 T=\frac{2}{h}-1=2.A^{*}-1
  15. π t = ( 1 - ψ ) β E t [ π t + 1 ] + ψ . π t - 1 + κ y t \pi_{t}=(1-\psi)\beta E_{t}[\pi_{t+1}]+\psi.\pi_{t-1}+\kappa y_{t}
  16. h ( i ) h(i)

Calvo_contract.html

  1. P r [ i ] = ( 1 - h ) i - 1 h Pr[i]=(1-h)^{i-1}h
  2. E [ P r [ i ] ] = h - 1 E[Pr[i]]=h^{-1}
  3. i = 1... i=1...\infty
  4. ( 1 - h ) 2 (1-h)^{2}
  5. ( 1 - h ) i (1-h)^{i}
  6. π t = β E t [ π t + 1 ] + κ y t \pi_{t}=\beta E_{t}[\pi_{t+1}]+\kappa y_{t}
  7. κ = h [ 1 - ( 1 - h ) β ] 1 - h γ \kappa=\frac{h[1-(1-h)\beta]}{1-h}\gamma
  8. β E t [ π t + 1 ] \beta E_{t}[\pi_{t+1}]
  9. κ \kappa
  10. α i \alpha^{i}
  11. α i = ( 1 - h ) i - 1 . h \alpha^{i}=(1-h)^{i-1}.h
  12. A * A^{*}
  13. A * = i = 0 i . h i ( 1 - h ) i - 1 = 1 h A^{*}=\sum_{i=0}^{\infty}i.h^{i}(1-h)^{i-1}=\frac{1}{h}
  14. T = 2 h - 1 = 2. A * - 1 T=\frac{2}{h}-1=2.A^{*}-1
  15. π t = ( 1 - ψ ) β E t [ π t + 1 ] + ψ . π t - 1 + κ y t \pi_{t}=(1-\psi)\beta E_{t}[\pi_{t+1}]+\psi.\pi_{t-1}+\kappa y_{t}
  16. h ( i ) h(i)

Camelliol_C_synthase.html

  1. \rightleftharpoons

Cameron_graph.html

  1. ( 231 , 30 , 9 , 3 ) (231,30,9,3)

Canopy_conductance.html

  1. g c g_{c}
  2. g s g_{s}
  3. g c = l a y e r 1 l a y e r N ( g s , s u n i l s u n i ) + ( g s , s h a d e l s h a d e ) g_{c}=\sum_{layer_{1}}^{layer_{N}}(g^{i}_{s,sun}\cdot l_{sun}^{i})+(g_{s,shade% }\cdot l_{shade})

Cantellated_24-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}

Cantellated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}

Cantitruncated_24-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}

Cantitruncated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}

Capillary_bridges.html

  1. sin ϕ = ( r 2 - r 1 r 2 ) r ( r 1 - r 2 ) \sin\phi=\frac{\left(r^{2}-r_{1}r_{2}\right)}{r\left(r_{1}-r_{2}\right)}
  2. z = ± [ r 1 F ( r , ϕ ) - E ( r , ϕ ) ] z=\pm\left[r_{1}F\left(r,\phi\right)-E\left(r,\phi\right)\right]
  3. k 2 = r 2 2 - r 1 2 r 2 2 k^{2}=\frac{r_{2}^{2}-r_{1}^{2}}{r_{2}^{2}}
  4. sin 2 ϕ = r 2 2 - r 2 r 2 2 - r 1 2 \sin^{2}\phi=\frac{r_{2}^{2}-r^{2}}{r_{2}^{2}-r_{1}^{2}}
  5. d ( r sin r ) r d r = 0 \frac{d\left(r\sin r\right)}{rdr}=0
  6. C = X sin θ - 1 X 2 - 1 C=\frac{X\sin{\theta}-1}{X^{2}-1}
  7. C = P y r m 2 γ C=P_{y}\frac{r_{m}}{2\gamma}
  8. X = R r m X=\frac{R}{r_{m}}
  9. d y d x = ± C ( x 2 - 1 ) + 1 x 2 - [ C ( x 2 - 1 ) + 1 ] 2 \frac{dy}{dx}=\pm\frac{C\left(x^{2}-1\right)+1}{\sqrt{x^{2}-\left[C\left(x^{2}% -1\right)+1\right]^{2}}}
  10. sin ϕ = d y d x cos ϕ \sin\phi=\frac{dy}{dx}\cos\phi
  11. x = r r m x=\frac{r}{r_{m}}
  12. C ( X = 1 - Δ ) - 1 - sin θ 2 Δ + 1 + sin θ 4 C\left(X=1-\Delta\right)\approx-\frac{1-\sin\theta}{2\Delta}+\frac{1+\sin% \theta}{4}
  13. d y d x = ± 1 + 2 C ( x - 1 ) 1 - [ 2 C ( x - 1 ) + 1 ] 2 \frac{dy}{dx}=\pm\frac{1+2C\left(x-1\right)}{\sqrt{1-\left[2C\left(x-1\right)+% 1\right]^{2}}}
  14. ( H V 3 ) = 1 X ( R V 3 ) { π 4 C - A - 1 X ζ - C ( X 2 - 1 ) + 1 ζ + C ( X 2 - 1 ) + 1 d ζ } \left(\frac{H}{\sqrt[3]{V}}\right)=\frac{1}{X}\left(\frac{R}{\sqrt[3]{V}}% \right)\left\{\frac{\pi}{4C}-A-\int\limits_{1}^{X}\sqrt{\frac{\zeta-C\left(X^{% 2}-1\right)+1}{\zeta+C\left(X^{2}-1\right)+1}}d\zeta\right\}
  15. ( R V 3 ) = X 2 π 3 { [ 1 - ( 1 - 2 C ) 2 C 2 ] π 4 C - ( 1 - 2 C ) ( X 2 - 1 ) 2 C 2 - [ 1 - ( 1 - 2 C ) 2 C 2 ] A - 1 X ζ 2 ζ - C ( X 2 - 1 ) + 1 ζ + C ( X 2 - 1 ) + 1 d ζ } - 1 3 \left(\frac{R}{\sqrt[3]{V}}\right)=\frac{X}{\sqrt[3]{2\pi}}\left\{\left[1-% \frac{\left(1-2C\right)}{2C^{2}}\right]\frac{\pi}{4C}-\frac{\sqrt{\left(1-2C% \right)\left(X^{2}-1\right)}}{2C^{2}}-\left[1-\frac{\left(1-2C\right)}{2C^{2}}% \right]A-\int\limits_{1}^{X}\zeta^{2}\sqrt{\frac{\zeta-C\left(X^{2}-1\right)+1% }{\zeta+C\left(X^{2}-1\right)+1}}d\zeta\right\}^{-\frac{1}{3}}
  16. A = 1 2 C arcsin [ 1 - 2 C - 2 C 2 ( X 2 - 1 ) 1 - 2 C ] A=\frac{1}{2C}\arcsin\left[\frac{1-2C-2C^{2}\left(X^{2}-1\right)}{1-2C}\right]
  17. B o = ρ g R 2 γ Bo=\frac{\rho gR^{2}}{\gamma}
  18. H 2 R \frac{H}{2R}
  19. V π \R 2 H \frac{V}{\pi\R^{2}H}

Capsanthin::capsorubin_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Caratheodory-π_solution.html

  1. π \pi
  2. π \pi
  3. x ˙ = g ( x , t ) \dot{x}=g(x,t)
  4. x ˙ = f ( x , u ) \dot{x}=f(x,u)
  5. u = k ( x , t ) u=k(x,t)
  6. x ˙ = g ( x , t ) \dot{x}=g(x,t)
  7. x ˙ = u \dot{x}=u
  8. u = g ( x , t ) u=g(x,t)
  9. [ 0 , ) [0,\infty)
  10. π = { t i } i 0 \pi=\{t_{i}\}_{i\geq 0}
  11. t i as i t_{i}\to\infty\,\text{ as }i\to\infty
  12. t 0 t_{0}
  13. t 1 t_{1}
  14. u ( t ) = g ( x 0 , t ) , x ( t 0 ) = x 0 , t 0 t t 1 u(t)=g(x_{0},t),\quad x(t_{0})=x_{0},\quad t_{0}\leq t\leq t_{1}
  15. x ˙ = u ( t ) , x ( t 0 ) = x 0 \dot{x}=u(t),\quad x(t_{0})=x_{0}
  16. t u t\mapsto u
  17. t = t 1 t=t_{1}
  18. x 1 = x ( t 1 ) x_{1}=x(t_{1})
  19. u ( t ) = g ( x 1 , t ) u(t)=g(x_{1},t)
  20. x ˙ ( t ) = u ( t ) , x ( t 1 ) = x 1 , t 1 t t 2 \dot{x}(t)=u(t),\quad x(t_{1})=x^{1},\quad t_{1}\leq t\leq t_{2}
  21. π \pi
  22. π \pi

Carbon_nanotubes_for_water_transport.html

  1. Q < m t p l > s l i p = π ( d 2 ) 4 + 4 ( d 2 ) 3 L ( s ) 8 μ Δ P L Q_{<}mtpl>{{slip}}=\frac{\pi\left(\tfrac{d}{2}\right)^{4}+4\left(\tfrac{d}{2}% \right)^{3}\cdot L(s)}{8\mu}\cdot\frac{\Delta P}{L}
  2. Q < m t p l > s l i p Q_{<}mtpl>{{slip}}
  3. d d
  4. Δ P \Delta P
  5. μ \mu
  6. L L
  7. L s ( d ) = L s + C d 3 L_{s}(d)=L_{s\infty}+\frac{C}{d^{3}}
  8. L s L_{s\infty}
  9. C C

Carbonic_anhydrase.html

  1. CO 2 + H 2 O Carbonic anhydrase H 2 CO 3 \rm CO_{2}+H_{2}O\xrightarrow{Carbonic\ anhydrase}H_{2}CO_{3}
  2. HCO 3 - + H + H 2 CO 3 CO 2 + H 2 O \rm HCO_{3}^{-}+H^{+}\rightarrow H_{2}CO_{3}\rightarrow CO_{2}+H_{2}O
  3. \rightleftharpoons
  4. \rightleftharpoons

Carboxybiotin_decarboxylase.html

  1. \rightleftharpoons

Cardy_formula.html

  1. S = 2 π c 6 ( L 0 - c 24 ) , S=2\pi\sqrt{\tfrac{c}{6}\bigl(L_{0}-\tfrac{c}{24}\bigr)},
  2. d s 2 = - d t 2 + R 2 Ω n 2 ds^{2}=-dt^{2}+R^{2}\Omega^{2}_{n}
  3. S = 2 π R n E c ( 2 E - E c ) , S=\frac{2\pi R}{n}\sqrt{E_{c}(2E-E_{c})},
  4. S S m a x = 2 π R E n , S\leq S_{max}=\frac{2\pi RE}{n},

Carleman's_equation.html

  1. a b ln | x - t | y ( t ) d t = f ( x ) \int_{a}^{b}\ln|x-t|\,y(t)\,dt=f(x)
  2. y ( x ) = 1 π 2 ( x - a ) ( b - x ) [ a b ( t - a ) ( b - t ) f t ( t ) d t t - x + 1 ln [ 1 4 ( b - a ) ] a b f ( t ) d t ( t - a ) ( b - t ) ] y(x)=\frac{1}{\pi^{2}\sqrt{(x-a)(b-x)}}\left[\int_{a}^{b}\frac{\sqrt{(t-a)(b-t% )}f^{\prime}_{t}(t)\,dt}{t-x}+\frac{1}{\ln\left[\frac{1}{4}(b-a)\right]}\int_{% a}^{b}\frac{f(t)\,dt}{\sqrt{(t-a)(b-t)}}\right]
  3. a b f ( t ) d t ( t - a ) ( b - t ) = 0 \int_{a}^{b}\frac{f(t)\,dt}{\sqrt{(t-a)(b-t)}}=0
  4. y ( x ) = 1 π 2 ( x - a ) ( b - x ) [ a b ( t - a ) ( b - t ) f t ( t ) d t t - x + C ] y(x)=\frac{1}{\pi^{2}\sqrt{(x-a)(b-x)}}\left[\int_{a}^{b}\frac{\sqrt{(t-a)(b-t% )}f^{\prime}_{t}(t)\,dt}{t-x}+C\right]
  5. y ( x ) = 1 π ln [ 1 4 ( b - a ) ] 1 ( x - a ) ( b - x ) y(x)=\frac{1}{\pi\ln\left[\frac{1}{4}(b-a)\right]}\frac{1}{\sqrt{(x-a)(b-x)}}

Carotenoid_1,2-hydratase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Carotid–Kundalini_function.html

  1. C K ( n , x ) = cos ( n x arccos ( x ) ) CK(n,x)=\cos(nx\arccos(x))

Carrier-envelope_phase.html

  1. ϕ 0 \phi_{0}
  2. 2 π 2\pi
  3. T CEO T_{\mathrm{CEO}}
  4. T rep T_{\mathrm{rep}}
  5. f CEO = 1 T CEO = d ϕ 0 d t = f_{\mathrm{CEO}}=\frac{1}{T_{\mathrm{CEO}}}=\frac{\mathrm{d}\phi_{0}}{\mathrm{% d}t}=

Carrier_frequency_offset.html

  1. f f
  2. z i , n = z ( t ) e j 2 π Δ f t | t = i ( N + N g ) T s + N g T s + n T s z_{i,n}=z(t)e^{j2\pi\Delta ft}|_{t=i(N+N_{g})T_{s}+N_{g}T_{s}+nT_{s}}
  3. f S = 1 / ( N T s ) ) f_{S}=1/(NT_{s}))
  4. ( ϵ I ) (\epsilon_{I})
  5. ( ϵ f ) (\epsilon_{f})
  6. Δ f = ( ϵ I + ϵ f ) f S \Delta f=(\epsilon_{I}+\epsilon_{f})f_{S}
  7. - 0.5 ϵ f < 0.5 -0.5\leq\epsilon_{f}<0.5
  8. Z i , k = X i , k - ϵ I H k - ϵ I sin ( π ϵ f ) N sin ( π ϵ f / N ) exp ( j 2 π i ( N + N g ) + N g N ( ϵ f + ϵ I ) ) exp ( j π N - 1 N ϵ f ) + l = - N / 2 , l k - ϵ I N / 2 - 1 X i , l H l sin ( π ( ϵ I + ϵ f + l - k ) ) N sin ( π ( ϵ I + ϵ f + l - k ) / N ) × exp ( j 2 π i ( N + N g ) + N g N ( ϵ I + ϵ f ) exp ( j π N - 1 N ( ϵ I + ϵ f + l - k ) ) + V i , k Z_{i,k}=X_{i,k-\epsilon_{I}}H_{k-\epsilon_{I}}\frac{\sin(\pi\epsilon_{f})}{N% \sin(\pi\epsilon_{f}/N)}\exp(j2\pi\frac{i(N+N_{g})+N_{g}}{N}(\epsilon_{f}+% \epsilon_{I}))\exp(j\pi\frac{N-1}{N}\epsilon_{f})+\sum_{l=-N/2,l\neq k-% \epsilon_{I}}^{N/2-1}{X_{i,l}H_{l}\frac{\sin(\pi(\epsilon_{I}+\epsilon_{f}+l-k% ))}{N\sin(\pi(\epsilon_{I}+\epsilon_{f}+l-k)/N)}}\times\exp(j2\pi\frac{i(N+N_{% g})+N_{g}}{N}{(\epsilon_{I}+\epsilon_{f}})\exp(j\pi\frac{N-1}{N}(\epsilon_{I}+% \epsilon_{f}+l-k))+V_{i,k}
  9. V i , k V_{i,k}
  10. ϵ f \epsilon_{f}
  11. ϵ I \epsilon_{I}
  12. i i
  13. Δ f ^ = 1 2 π L T s ( r = 0 R - 1 z m - r z m - r - L * ) \widehat{\Delta f}=\frac{1}{2\pi LT_{s}}\angle\ (\sum_{r=0}^{R-1}{z_{m-r}z^{*}% _{m-r-L}})
  14. [ - π , π ] \left[-\pi,\pi\right]
  15. [ - 1 / ( 2 L T s ) , 1 / ( 2 L T s ) ] \left[-1/(2LT_{s}),1/(2LT_{s})\right]
  16. H z Hz
  17. L = N L=N
  18. Δ f ^ = ϵ ^ f f S \widehat{\Delta f}=\widehat{\epsilon}_{f}f_{S}
  19. L > 1 / ( Δ f T s ) L>1/(\Delta fT_{s})
  20. U > 2 U>2
  21. U R = N UR=N
  22. u R uR
  23. Φ B L U E ( u ) = 1 N - u R m = u R N - 1 z m z m - u R * , 0 u K . \Phi_{BLUE(u)}=\frac{1}{N-uR}\sum_{m=uR}^{N-1}{z_{m}z^{*}_{m-uR}},\ \ \ \ \ \ % \ 0\leq u\leq K.
  24. R R
  25. ϕ ( u ) = [ \ang { Φ B L U E ( u ) } ] - \ang { Φ B L U E ( u - 1 ) } ] 2 π 1 u K , \phi(u)=[\ang\left\{\Phi_{BLUE}(u)\right\}]-\ang\left\{\Phi_{BLUE}(u-1)\right% \}]_{2\pi}\ \ \ \ \ 1\leq u\leq K,
  26. [ ] 2 π [\ \cdot\ ]_{2\pi}
  27. 2 π 2\pi
  28. K K
  29. U U
  30. ϕ ( u ) \phi(u)
  31. u u
  32. ϕ ( u ) \phi(u)
  33. Δ f ^ / f S = U 2 π ( u = 1 K w u ϕ ( u ) ) , \widehat{\Delta f}/f_{S}=\frac{U}{2\pi}(\sum_{u=1}^{K}{w_{u}\phi(u)}),
  34. w u = 3 ( U - u ) ( U - u + 1 ) - K ( U - K ) K ( 4 K 2 - 6 U K + 3 U 2 - 1 ) w_{u}=3\frac{(U-u)(U-u+1)-K(U-K)}{K(4K^{2}-6UK+3U^{2}-1)}
  35. K K
  36. Δ f ^ \widehat{\Delta f}
  37. U / 2 U/2
  38. - U f S / 2 Δ f ^ U f S / 2 -Uf_{S}/2\leq\widehat{\Delta f}\leq Uf_{S}/2
  39. U U
  40. ( f S ) (f_{S})
  41. ( f S ) (f_{S})
  42. [ - 2 f S , 2 f S ] [-2f_{S},2f_{S}]
  43. ± 12 f S \pm 12f_{S}
  44. ± 8 f S \pm 8f_{S}
  45. ± 4 f S \pm 4f_{S}
  46. ± 11 f S \pm 11f_{S}

Carr–Madan_formula.html

  1. log S t \log S_{t}
  2. S t S_{t}
  3. t t

Cartan_pair.html

  1. 𝔤 \mathfrak{g}
  2. 𝔨 \mathfrak{k}
  3. 𝔤 \mathfrak{g}
  4. ( 𝔤 , 𝔨 ) (\mathfrak{g},\mathfrak{k})
  5. H * ( 𝔤 , 𝔨 ) H^{*}(\mathfrak{g},\mathfrak{k})
  6. im ( S ( 𝔨 * ) H * ( 𝔤 , 𝔨 ) ) \mathrm{im}\big(S(\mathfrak{k}^{*})\to H^{*}(\mathfrak{g},\mathfrak{k})\big)
  7. P ^ \bigwedge\hat{P}
  8. H * ( 𝔤 ) H^{*}(\mathfrak{g})
  9. P ^ \hat{P}
  10. P 𝜏 S ( 𝔤 * ) S ( 𝔨 * ) P\overset{\tau}{\to}S(\mathfrak{g}^{*})\to S(\mathfrak{k}^{*})
  11. P P
  12. H * ( 𝔤 ) H^{*}(\mathfrak{g})
  13. τ \tau
  14. S ( 𝔤 * ) S ( 𝔨 * ) S(\mathfrak{g}^{*})\to S(\mathfrak{k}^{*})
  15. 𝔤 * 𝔨 * \mathfrak{g}^{*}\to\mathfrak{k}^{*}
  16. G G K B K G\to G_{K}\to BK
  17. G K := ( E K × G ) / K G / K G_{K}:=(EK\times G)/K\simeq G/K
  18. G / K 𝜒 B K 𝑟 B G G/K\overset{\chi}{\to}BK\overset{r}{\to}BG
  19. χ * : H * ( B K ) H * ( G / K ) \chi^{*}\colon H^{*}(BK)\to H^{*}(G/K)
  20. τ : P H * ( B G ) \tau\colon P\to H^{*}(BG)
  21. H * ( G ) H^{*}(G)
  22. G E G B G G\to EG\to BG
  23. P ^ \hat{P}
  24. H * ( G / K ) H^{*}(G/K)
  25. r * τ r^{*}\circ\tau

Cartesian_monoidal_category.html

  1. X = j 1 , , n X j . X=\bigoplus_{j\in{1,\ldots,n}}X_{j}.
  2. X = j 1 , , n X j X=\coprod_{j\in{1,\ldots,n}}X_{j}
  3. X = j 1 , , n X j X=\prod_{j\in{1,\ldots,n}}X_{j}

Cascade_Learning_Based_on_Adaboost.html

  1. g 1 [ f 1 , f 2 ] ( x ) = f 1 ( x ) u [ - f 1 ( x ) ] + u [ f 1 ( x ) ] f 2 ( x ) = { f 1 ( x ) if f 1 ( x ) < 0 f 2 ( x ) if f 1 ( x ) > 0 g_{1}[f_{1},f_{2}](x)=f_{1}(x)u[-f_{1}(x)]+u[f_{1}(x)]f_{2}(x)=\begin{cases}f_% {1}(x)&\mbox{if }~{}f_{1}(x)<0\\ f_{2}(x)&\mbox{if }~{}f_{1}(x)>0\end{cases}
  2. F k ( x ) = { f m ( x ) , k = m f k ( x ) u [ - f k ( x ) ] + u [ f k ( x ) ] F k + 1 ( x ) , 1 k < m F_{k}(x)=\begin{cases}f_{m}(x),&k=m\\ f_{k}(x)u[-f_{k}(x)]+u[f_{k}(x)]F_{k}+1(x),&1\leq k<m\end{cases}
  3. g 2 [ f 1 , f 2 ] ( x ) = f 1 ( x ) u [ - f 1 ( x ) ] + u [ f 1 ( x ) ] f 1 ( x ) f 2 ( x ) = { f 1 ( x ) if f 1 ( x ) < 0 f 2 ( x ) f 1 ( x ) if f 1 ( x ) > 0 g_{2}[f_{1},f_{2}](x)=f_{1}(x)u[-f_{1}(x)]+u[f_{1}(x)]f_{1}(x)f_{2}(x)=\begin{% cases}f_{1}(x)&\mbox{if }~{}f_{1}(x)<0\\ f_{2}(x)f_{1}(x)&\mbox{if }~{}f_{1}(x)>0\end{cases}
  4. F k ( x ) = { f m ( x ) , k = m f k ( x ) u [ - f k ( x ) ] + u [ f k ( x ) ] f k ( x ) F k + 1 ( x ) , 1 k < m F_{k}(x)=\begin{cases}f_{m}(x),&k=m\\ f_{k}(x)u[-f_{k}(x)]+u[f_{k}(x)]f_{k}(x)F_{k}+1(x),&1\leq k<m\end{cases}
  5. [ f ] = E [ F ] + η C [ F ] \mathcal{L}[f]=\mathcal{R}_{E}[F]+\eta\mathcal{R}_{C}[F]
  6. F ( x ) F(x)
  7. R E [ F ] R_{E}[F]
  8. g k * ( x ) g_{k}^{*}(x)
  9. g k * ( x ) g_{k}^{*}(x)
  10. f k + 1 ( x ) = f k ( x ) = w ( x ) f_{k+1}(x)=f_{k}(x)=w(x)

Categorical_quotient.html

  1. π : X Y \pi:X\to Y
  2. π σ = π p 2 \pi\circ\sigma=\pi\circ p_{2}
  3. σ : G × X X \sigma:G\times X\to X
  4. X Z X\to Z
  5. π \pi
  6. π \pi
  7. π \pi
  8. Y Y Y^{\prime}\to Y
  9. π : X = X × Y Y Y \pi^{\prime}:X^{\prime}=X\times_{Y}Y^{\prime}\to Y^{\prime}
  10. G / H G/H
  11. X / / G X/\!/G

Cauchy–Euler_operator.html

  1. p ( x ) d d x p(x)\cdot{d\over dx}

Cauchy–Kowalevski_theorem.html

  1. x n f = A 1 ( x , f ) x 1 f + + A n - 1 ( x , f ) x n - 1 f + b ( x , f ) \partial_{x_{n}}f=A_{1}(x,f)\partial_{x_{1}}f+\cdots+A_{n-1}(x,f)\partial_{x_{% n-1}}f+b(x,f)\,
  2. f ( x ) = 0 f(x)=0\,
  3. x n = 0 x_{n}=0\,
  4. t k h = F ( x , t , t j x α h ) , where j < k and | α | + j k , \partial_{t}^{k}h=F\left(x,t,\partial_{t}^{j}\,\partial_{x}^{\alpha}h\right),% \,\text{ where }j<k\,\text{ and }|\alpha|+j\leq k,\,
  5. t j h ( x , 0 ) = f j ( x ) , 0 j < k , \partial_{t}^{j}h(x,0)=f_{j}(x),\qquad 0\leq j<k,
  6. t h = x 2 h \partial_{t}h=\partial_{x}^{2}h\,
  7. h ( 0 , x ) = 1 1 + x 2 for t = 0 h(0,x)={1\over 1+x^{2}}\,\text{ for }t=0\,
  8. E x t 1 Ext^{1}
  9. n m n\leq m
  10. Y = { x 1 = = x n } Y=\{x_{1}=\cdots=x_{n}\}
  11. x i f = g i , i = 1 , , n , \partial_{x_{i}}f=g_{i},i=1,\ldots,n,
  12. f { x 1 , , x m } f\in\mathbb{C}\{x_{1},\ldots,x_{m}\}
  13. x i g j = x j g i \partial_{x_{i}}g_{j}=\partial_{x_{j}}g_{i}
  14. f | Y = h f|_{Y}=h
  15. h { x n + 1 , , x m } h\in\mathbb{C}\{x_{n+1},\ldots,x_{m}\}

Causal_fermion_system.html

  1. M ^ \hat{M}
  2. F ( x ) , x M ^ , F(x),x\in\hat{M},
  3. ( ψ i ) (\psi_{i})
  4. ( F ( x ) ) j i = - ψ i ( x ) ¯ ψ j ( x ) \big(F(x)\big)^{i}_{j}=-\overline{\psi_{i}(x)}\psi_{j}(x)
  5. ψ ¯ \overline{\psi}
  6. { F ( x ) | x M ^ } \{F(x)\,|\,x\in\hat{M}\}
  7. d 4 x d^{4}x
  8. n n\in\mathbb{N}
  9. ( , , ρ ) (\mathcal{H},\mathcal{F},\rho)
  10. ( , . | . ) (\mathcal{H},\langle.|.\rangle_{\mathcal{H}})
  11. \mathcal{F}
  12. \mathcal{H}
  13. n n
  14. n n
  15. ρ \rho
  16. \mathcal{F}
  17. ρ \rho
  18. ( , . | . ) (\mathcal{H},\langle.|.\rangle_{\mathcal{H}})
  19. n n
  20. \mathcal{F}
  21. x , y x,y\in{\mathcal{F}}
  22. x y xy
  23. 2 n 2n
  24. ( x y ) * = y x x y (xy)^{*}=yx\neq xy
  25. x y xy
  26. λ 1 x y , , λ 2 n x y . \lambda^{xy}_{1},\ldots,\lambda^{xy}_{2n}\in{\mathbb{C}}.
  27. | . | |.|
  28. | x y | = i = 1 2 n | λ i x y | and | ( x y ) 2 | = i = 1 2 n | λ i x y | 2 . |xy|=\sum_{i=1}^{2n}|\lambda^{xy}_{i}|\quad\,\text{and}\quad\big|(xy)^{2}\big|% =\sum_{i=1}^{2n}|\lambda^{xy}_{i}|^{2}{\,}.
  29. ( x , y ) = | ( x y ) 2 | - 1 2 n | x y | 2 = 1 4 n i , j = 1 2 n ( | λ i x y | - | λ j x y | ) 2 0 . {\mathcal{L}}(x,y)=\big|(xy)^{2}\big|-\frac{1}{2n}{\,}|xy|^{2}=\frac{1}{4n}% \sum_{i,j=1}^{2n}\big(|\lambda^{xy}_{i}|-|\lambda^{xy}_{j}|\big)^{2}\geq 0{\,}.
  30. 𝒮 = × ( x , y ) d ρ ( x ) d ρ ( y ) . {\mathcal{S}}=\iint_{{\mathcal{F}}\times{\mathcal{F}}}{\mathcal{L}}(x,y){\,}d% \rho(x){\,}d\rho(y){\,}.
  31. 𝒮 {\mathcal{S}}
  32. ρ \rho
  33. × | x y | 2 d ρ ( x ) d ρ ( y ) C \iint_{{\mathcal{F}}\times{\mathcal{F}}}|xy|^{2}{\,}d\rho(x){\,}d\rho(y)\leq C
  34. C C
  35. tr ( x ) d ρ ( x ) \;\;\;\int_{\mathcal{F}}\,\text{tr}(x){\,}d\rho(x)
  36. ρ ( ) \rho({\mathcal{F}})
  37. L ( ) {\mathcal{F}}\subset{\mathrm{L}}({\mathcal{H}})
  38. sup \sup
  39. {\mathcal{H}}
  40. {\mathcal{H}}
  41. ρ ( ) \rho({\mathcal{F}})
  42. δ ρ \delta\rho
  43. ( δ ρ ) ( ) = 0 (\delta\rho)({\mathcal{F}})=0
  44. ( M , g ) (M,g)
  45. M M
  46. \mathcal{F}
  47. ( , , ρ ) (\mathcal{H},\mathcal{F},\rho)
  48. M M
  49. M := supp ρ . M:=\,\text{supp}\,\rho\subset\mathcal{F}.
  50. \mathcal{F}
  51. M M
  52. x , y M x,y\in M
  53. x y xy
  54. λ 1 x y , , λ 2 n x y \lambda^{xy}_{1},\ldots,\lambda^{xy}_{2n}\in{\mathbb{C}}
  55. x x
  56. y y
  57. λ j x y \lambda^{xy}_{j}
  58. λ j x y \lambda^{xy}_{j}
  59. x x
  60. y y
  61. x , y M x,y\in M
  62. ( x , y ) {\mathcal{L}}(x,y)
  63. π x \pi_{x}
  64. S x := x ( ) S_{x}:=x({\mathcal{H}})\subset{\mathcal{H}}
  65. i Tr ( x y π x π y - y x π y π x ) i\,\text{Tr}\big(x\,y\,\pi_{x}\,\pi_{y}-y\,x\,\pi_{y}\,\pi_{x})
  66. x M x\in M
  67. S x = x ( ) S_{x}=x({\mathcal{H}})
  68. {\mathcal{H}}
  69. 2 n 2n
  70. . | . x {\prec}.|.{\succ}_{x}
  71. u | v x = - u | x u for all u , v S x {\prec}u|v{\succ}_{x}=-{\langle}u|xu{\rangle}_{\mathcal{H}}\qquad\,\text{for % all }u,v\in S_{x}
  72. S x S_{x}
  73. ( p , q ) (p,q)
  74. p , q n p,q\leq n
  75. ψ \psi
  76. ψ : M with ψ ( x ) S x for all x M . \psi{\,}:{\,}M\rightarrow{\mathcal{H}}\qquad\,\text{with}\qquad\psi(x)\in S_{x% }\quad\,\text{for all }x\in M{\,}.
  77. | | | . | | | {|\!|\!|}.{|\!|\!|}
  78. | | | ψ | | | 2 = M ψ ( x ) | | x | ψ ( x ) d ρ ( x ) {|\!|\!|}\psi{|\!|\!|}^{2}=\int_{M}{\langle}\psi(x)|\,|x|\,\psi(x){\rangle}_{% \mathcal{H}}{\,}d\rho(x)
  79. | x | = x 2 |x|=\sqrt{x^{2}}
  80. x x
  81. < ψ | ϕ > = M ψ ( x ) | ϕ ( x ) x d ρ ( x ) . {\mathopen{<}}\psi|\phi{\mathclose{>}}=\int_{M}{\prec}\psi(x)|\phi(x){\succ}_{% x}{\,}d\rho(x){\,}.
  82. | | | . | | | {|\!|\!|}.{|\!|\!|}
  83. ( 𝒦 , < . | . > ) ({{\mathcal{K}}},{\mathopen{<}}.|.{\mathclose{>}})
  84. u u\in\mathcal{H}
  85. ψ u ( x ) := π x u \psi^{u}(x):=\pi_{x}u
  86. π x : S x \pi_{x}:\mathcal{H}\rightarrow S_{x}
  87. P ( x , y ) P(x,y)
  88. P ( x , y ) = π x y | S y : S y S x P(x,y)=\pi_{x}\,y|_{S_{y}}{\,}:{\,}S_{y}\rightarrow S_{x}
  89. π x : S x \pi_{x}:\mathcal{H}\rightarrow S_{x}
  90. | S y |_{S_{y}}
  91. S y S_{y}
  92. P P
  93. P : 𝒦 𝒦 , ( P ψ ) ( x ) = M P ( x , y ) ψ ( y ) d ρ ( y ) , P{\,}:{\,}{{\mathcal{K}}}\rightarrow{{\mathcal{K}}}{\,},\qquad(P\psi)(x)=\int_% {M}P(x,y)\,\psi(y)\,d\rho(y){\,},
  94. ψ 𝒦 \psi\in{{\mathcal{K}}}
  95. ϕ := M x ψ ( x ) d ρ ( x ) and || | ϕ | || < . \phi:=\int_{M}x\,\psi(x)\,d\rho(x){\,}\in{\,}{\mathcal{H}}\quad\,\text{and}% \quad{|\!|\!|}\phi{|\!|\!|}<\infty{\,}.
  96. D x , y : S y S x unitary . D_{x,y}\,:\,S_{y}\rightarrow S_{x}\quad\,\text{unitary}\,.
  97. P ( x , y ) P(x,y)
  98. x , y : T y T x isometric , \nabla_{x,y}\,:\,T_{y}\rightarrow T_{x}\quad\,\text{isometric}\,,
  99. T x T_{x}
  100. S x S_{x}
  101. ( x , y , z ) = D x , y D y , z D z , x : S x S x . \mathfrak{R}(x,y,z)=D_{x,y}\,D_{y,z}\,D_{z,x}\,:\,S_{x}\rightarrow S_{x}\,.
  102. {\mathcal{H}}
  103. f f
  104. u 1 , , u f u_{1},\ldots,u_{f}
  105. {\mathcal{H}}
  106. ( ψ u 1 ψ u f ) ( x 1 , , x f ) \big(\psi^{u_{1}}\wedge\cdots\wedge\psi^{u_{f}}\big)(x_{1},\ldots,x_{f})
  107. f f
  108. {\mathcal{H}}
  109. x x
  110. ( 𝔭 x , 𝔮 x ) ({\mathfrak{p}}_{x},{\mathfrak{q}}_{x})
  111. ( 𝔢 α ( x ) ) α = 1 , , 𝔭 x + 𝔮 x (\mathfrak{e}_{\alpha}(x))_{\alpha=1,\ldots,{\mathfrak{p}}_{x}+{\mathfrak{q}}_% {x}}
  112. S x S_{x}
  113. 𝔢 α | 𝔢 β = s α δ α β with s 1 , , s 𝔭 x = 1 , s 𝔭 x + 1 , , s 𝔭 x + 𝔮 x = - 1 . {\prec}\mathfrak{e}_{\alpha}|\mathfrak{e}_{\beta}{\succ}=s_{\alpha}{\,}\delta_% {\alpha\beta}\quad\,\text{with}\quad s_{1},\ldots,s_{{\mathfrak{p}}_{x}}=1,\;% \;s_{{\mathfrak{p}}_{x}+1},\ldots,s_{{\mathfrak{p}}_{x}+{\mathfrak{q}}_{x}}=-1% {\,}.
  114. ψ \psi
  115. ψ ( x ) = α = 1 𝔭 x + 𝔮 x ψ α ( x ) 𝔢 α ( x ) . \psi(x)=\sum_{\alpha=1}^{{\mathfrak{p}}_{x}+{\mathfrak{q}}_{x}}\psi^{\alpha}(x% ){\,}\mathfrak{e}_{\alpha}(x){\,}.
  116. ( 𝔢 α ( x ) ) (\mathfrak{e}_{\alpha}(x))
  117. ψ α ( x ) β = 1 𝔭 x + 𝔮 x U ( x ) β α ψ β ( x ) with U ( x ) U ( 𝔭 x , 𝔮 x ) . \psi^{\alpha}(x)\rightarrow\sum_{\beta=1}^{{\mathfrak{p}}_{x}+{\mathfrak{q}}_{% x}}U(x)^{\alpha}_{\beta}\,\,\psi^{\beta}(x)\quad\,\text{with}\quad U(x)\in\,% \text{U}({\mathfrak{p}}_{x},{\mathfrak{q}}_{x}){\,}.
  118. ( M ^ , g ) (\hat{M},g)
  119. S M ^ S\hat{M}
  120. ( , . | . ) ({\mathcal{H}},{\langle}.|.{\rangle}_{\mathcal{H}})
  121. F ( p ) F(p)
  122. p M ^ p\in\hat{M}
  123. ψ | F ( p ) ϕ = - ψ | ϕ p {\langle}\psi|F(p)\phi{\rangle}_{\mathcal{H}}=-{\prec}\psi|\phi{\succ}_{p}
  124. ψ | ϕ p {\prec}\psi|\phi{\succ}_{p}
  125. S p M ^ S_{p}\hat{M}
  126. M ^ \hat{M}
  127. ρ = F * d μ , \rho=F_{*}d\mu{\,},
  128. {\mathcal{H}}
  129. ε \varepsilon
  130. ε 0 \varepsilon\searrow 0
  131. M := supp ρ M:=\,\text{supp}\,\rho
  132. {\mathcal{H}}
  133. A A
  134. ( i / + γ 5 A / - m ) ψ = 0 C 0 ( j k A j - A k ) - C 2 A k = 12 π 2 ψ ¯ γ 5 γ k ψ . \begin{aligned}\displaystyle(i\partial\!\!\!/\ +\gamma^{5}A\!\!\!/\ -m)\psi&% \displaystyle=0\\ \displaystyle C_{0}(\partial^{k}_{j}A^{j}-\Box A^{k})-C_{2}A^{k}&\displaystyle% =12\pi^{2}\bar{\psi}\gamma^{5}\gamma^{k}\psi\,.\end{aligned}
  135. C 0 C_{0}
  136. C 2 C_{2}
  137. S U ( 2 ) SU(2)
  138. R j k - 1 2 R g j k + Λ g j k = κ T j k [ Ψ , A ] , R_{jk}-\frac{1}{2}\,R\,g_{jk}+\Lambda\,g_{jk}=\kappa\,T_{jk}[\Psi,A]\,,
  139. Λ \Lambda
  140. T j k T_{jk}
  141. S U ( 2 ) SU(2)
  142. κ \kappa

Causal_graph.html

  1. Q 1 \displaystyle Q_{1}
  2. Q 1 Q_{1}
  3. Q 2 Q_{2}
  4. C C
  5. S S
  6. Q 1 Q_{1}
  7. Q 2 Q_{2}
  8. C C
  9. S S
  10. C \displaystyle C
  11. S S
  12. U S U_{S}
  13. U C U_{C}
  14. U S U_{S}
  15. U C U_{C}
  16. C C
  17. C C
  18. β \beta
  19. A A
  20. Q 1 \displaystyle Q_{1}
  21. A \displaystyle A
  22. U A U_{A}
  23. U S U_{S}
  24. β \beta
  25. S S
  26. C C
  27. A A
  28. β \beta

Cavitation_modelling.html

  1. R R ¨ + 3 2 R R ˙ = p ( R ) - p ρ L R\ddot{R}+\frac{3}{2}R\dot{R}=\frac{p(R)-p_{\infty}}{\rho_{L}}
  2. R R ¨ + 3 2 R R ˙ = p i - p - 2 σ R - 4 ν R R ˙ ρ L R\ddot{R}+\frac{3}{2}R\dot{R}=\frac{p_{i}-p_{\infty}-\frac{2\sigma}{R}-\frac{4% \nu}{R}\dot{R}}{\rho_{L}}
  3. ( 1 - R ˙ ( t ) c ( R ) ) R ( t ) R ¨ ( t ) + 3 2 ( 1 - R ˙ ( t ) 3 c ( R ) ) R ( t ) R ˙ ( t ) = ( 1 + R ˙ ( t ) c ( R ) ) H ( R ) + ( 1 - R ˙ c ( R ) ) R c ( R ) H ˙ ( R ) (1-\frac{\dot{R}(t)}{c(R)})R(t)\ddot{R}(t)+\frac{3}{2}(1-\frac{\dot{R}(t)}{3c(% R)})R(t)\dot{R}(t)=(1+\frac{\dot{R}(t)}{c(R)})H(R)+(1-\frac{\dot{R}}{c(R)})% \frac{R}{c(R)}\dot{H}(R)
  4. H = n n - 1 p ( t ) + B ρ L [ ( P + B p ( t ) + B ) n - 1 n - 1 ] H=\frac{n}{n-1}\frac{p_{\infty}(t)+B}{\rho_{L}}\left[(\frac{P+B}{p_{\infty}(t)% +B})^{\frac{n-1}{n}}-1\right]
  5. c = c 0 ( p g ( t ) - 2 σ / R + B p ( t ) + B ) n - 1 2 n c=c_{0}\left(\frac{p_{g}(t)-2\sigma/R+B}{p_{\infty}(t)+B}\right)^{\frac{n-1}{2% n}}
  6. H ˙ = D p ( t ) + B H - D ρ ( P + B p ( t ) + B ) n - 1 n + R ˙ ρ L R [ p ( t ) + B P + B ] 1 n [ 2 σ R - 3 k p g ( t ) ] \dot{H}=\frac{D}{p_{\infty}(t)+B}H-\frac{D}{\rho}(\frac{P+B}{p_{\infty}(t)+B})% ^{\frac{n-1}{n}}+\frac{\dot{R}}{\rho_{L}R}\left[\frac{p_{\infty}(t)+B}{P+B}% \right]^{\frac{1}{n}}\left[\frac{2\sigma}{R}-3kp_{g}(t)\right]

Center_of_mass_(relativistic).html

  1. ( t , x ) (t,x)
  2. m i m_{i}
  3. x i ( t ) {\vec{x}}_{i}(t)
  4. x ( n r ) ( t ) = i = 1 N m i x i ( t ) i = 1 N m i {\vec{x}}_{(nr)}(t)={\frac{{\sum_{i=1}^{N}\,m_{i}\,{\vec{x}}_{i}(t)}}{{\sum_{i% =1}^{N}\,m_{i}}}}
  5. x μ = ( x 0 , x ) x^{\mu}=(x^{0},x)
  6. x ~ \vec{\tilde{x}}
  7. { x ~ i , x ~ j } = 0 \{{\tilde{x}}^{i},{\tilde{x}}^{j}\}=0
  8. x ~ μ = ( x ~ o , x ~ ) {\tilde{x}}^{\mu}=({\tilde{x}}^{o},{\vec{\tilde{x}}})
  9. Y \vec{Y}
  10. Y μ = ( Y 0 , Y ) Y^{\mu}=(Y^{0},\vec{Y})
  11. { Y i , Y j } 0 \{Y^{i},Y^{j}\}\not=0
  12. R \vec{R}
  13. m i m_{i}
  14. { R i , R j } 0 \{R^{i},R^{j}\}\not=0
  15. x i ( t ) {\vec{x}_{i}(t)}
  16. p i ( t ) {\vec{p}_{i}(t)}
  17. m i ( i = 1.. N ) m_{i}(i=1..N)
  18. ( t , x ) (t,x)
  19. ( V ( t ) = V ( x i ( t ) - x j ( t ) ) (V(t)=V({\vec{x}}_{i}(t)-{\vec{x}}_{j}(t))
  20. E G = i = 1 N p i 2 ( t ) 2 m i + V ( t ) , P G = i = 1 N p i ( t ) , E_{G}=\sum_{i=1}^{N}\,{\frac{{{\vec{p}}_{i}^{2}(t)}}{{2m_{i}}}}+V(t),\qquad{% \vec{P}}_{G}=\sum_{i=1}^{N}\,{\vec{p}}_{i}(t),
  21. J G = i = 1 N x i ( t ) × p i ( t ) , K G = P t - i = 1 N m i x i ( t ) . {\vec{J}}_{G}=\sum_{i=1}^{N}\,{\vec{x}}_{i}(t)\times{\vec{p}}_{i}(t),\qquad{% \vec{K}}_{G}=\vec{P}\,t-\sum_{i=1}^{N}\,m_{i}\,{\vec{x}}_{i}(t).
  22. t = 0 t=0
  23. x ( n r ) = - K G M , M = i = 1 N m i {\vec{x}}_{(nr)}=-\frac{{\vec{K}}_{G}}{M},M=\sum_{i=1}^{N}m_{i}
  24. P μ , J μ ν P^{\mu},J^{\mu\nu}
  25. P μ , J μ ν P^{\mu},J^{\mu\nu}
  26. P μ P^{\mu}
  27. M M
  28. S \vec{S}
  29. M 2 c 2 = ( P 0 ) 2 - P 2 , M^{2}c^{2}=(P^{0})^{2}-\vec{P}^{2},
  30. P 0 P^{0}
  31. W μ = 1 2 ε μ ν κ λ P ν J κ λ W^{\mu}=\frac{1}{2}\varepsilon^{\mu\nu\kappa\lambda}P_{\nu}J_{\kappa\lambda}
  32. W | P = 0 = M c S , \vec{W}|_{\vec{P}=0}=Mc\vec{S},
  33. W 2 = M 2 c 2 S 2 W^{2}=M^{2}c^{2}S^{2}
  34. x μ = ( x 0 , x ) x^{\mu}=(x^{0},\vec{x})
  35. P μ , J μ ν , M P^{\mu},J^{\mu\nu},M
  36. S \vec{S}
  37. J i = 1 2 j k ϵ i j k J j k , K i = J 0 i J^{i}={\frac{1}{2}}\,\sum_{jk}\,\epsilon^{ijk}\,J^{jk},K^{i}=J^{0i}
  38. x 0 = 0 x^{0}=0
  39. R = - < m t p l > K M c \vec{R}=-{\frac{<}{m}tpl>{{\vec{K}}}{{Mc}}}
  40. x ~ = - < m t p l > K M 2 c 2 - P 2 + J × P M 2 c 2 - P 2 ( M c + M 2 c 2 - P 2 ) + K P P M c M 2 c 2 - P 2 ( M c + M 2 c 2 - P 2 ) {\vec{\tilde{x}}}=-{\frac{<}{m}tpl>{{\vec{K}}}{\sqrt{M^{2}c^{2}-{\vec{P}}^{2}}% }}+{\frac{{\vec{J}\times\vec{P}}}{{\sqrt{M^{2}c^{2}-{\vec{P}}^{2}}(Mc+\sqrt{M^% {2}c^{2}-{\vec{P}}^{2}})}}}+{\frac{{\vec{K}\cdot\vec{P}\,\vec{P}}}{{Mc\,\sqrt{% M^{2}c^{2}-{\vec{P}}^{2}}(Mc+\sqrt{M^{2}c^{2}-{\vec{P}}^{2}})}}}
  41. Y = ( M c + M 2 c 2 - P 2 ) x ~ - M c R M 2 c 2 - P 2 \vec{Y}={\frac{{(Mc+\sqrt{M^{2}c^{2}-{\vec{P}}^{2}})\,{\vec{\tilde{x}}}-Mc\,% \vec{R}}}{\sqrt{M^{2}c^{2}-{\vec{P}}^{2}}}}
  42. x μ x^{\mu}
  43. Y μ Y^{\mu}
  44. τ \tau
  45. σ \vec{\sigma}
  46. Σ τ \vec{\Sigma}_{\tau}
  47. z W μ ( τ , σ ) = Y μ ( τ ) + r = 1 3 ϵ r μ ( h ) σ r , z^{\mu}_{W}(\tau,\vec{\sigma})=Y^{\mu}(\tau)+\sum_{r=1}^{3}\epsilon^{\mu}_{r}(% \vec{h})\sigma^{r},
  48. h = P / M c \vec{h}=\vec{P}/Mc
  49. h μ = P μ / M c h^{\mu}=P^{\mu}/Mc
  50. ϵ r μ ( h ) \epsilon^{\mu}_{r}(\vec{h})
  51. σ \vec{\sigma}
  52. Σ ( W ) τ \Sigma_{(W)\tau}
  53. x ~ μ ( τ ) , Y μ ( τ ) , R μ ( τ ) \tilde{x}^{\mu}(\tau),Y^{\mu}(\tau),R^{\mu}(\tau)
  54. τ = h μ x ~ μ ( τ ) = h μ Y μ ( τ ) = h μ R μ ( τ ) \tau=h_{\mu}\tilde{x}^{\mu}(\tau)=h_{\mu}Y^{\mu}(\tau)=h_{\mu}R^{\mu}(\tau)
  55. τ , z = M c x ~ ( 0 ) \tau,\vec{z}=Mc{\vec{\tilde{x}}}(0)
  56. τ = 0 \tau=0
  57. h , M \vec{h},M
  58. S \vec{S}
  59. x ~ μ ( τ ) = ( x ~ 0 ( τ ) ; x ~ ( τ ) ) = ( 1 + h 2 ( τ + < m t p l > h z M c ) ; z M c + ( τ + h z M c ) h ) = z W μ ( τ , σ ~ ) = Y μ ( τ ) + ( 0 , - S × h M c ( 1 + 1 + h 2 ) ) \begin{aligned}\displaystyle{\tilde{x}}^{\mu}(\tau)&\displaystyle=\left({% \tilde{x}}^{0}(\tau);{\tilde{\vec{x}}}(\tau)\right)=\left(\sqrt{1+{\vec{h}}^{2% }}(\tau+{\frac{<}{m}tpl>{{\vec{h}\cdot\vec{z}}}{{Mc}}});{\frac{{\vec{z}}}{{Mc}% }}+(\tau+{\frac{{\vec{h}\cdot\vec{z}}}{{Mc}}})\vec{h}\right)\\ &\displaystyle=z^{\mu}_{W}(\tau,{\tilde{\vec{\sigma}}})=Y^{\mu}(\tau)+\left(0,% {\frac{{-\vec{S}\times\vec{h}}}{{Mc(1+\sqrt{1+{\vec{h}}^{2}})}}}\right)\\ \end{aligned}
  60. Y μ ( τ ) = ( x ~ 0 ( τ ) ; Y ( τ ) ) = ( 1 + h 2 ( τ + h z M c ) ; z M c + ( τ + h z M c ) h + S × h M c ( 1 + 1 + h 2 ) ) = z W μ ( τ , 0 ) , \begin{aligned}\displaystyle Y^{\mu}(\tau)&\displaystyle=\left({\tilde{x}}^{0}% (\tau);\vec{Y}(\tau)\right)=\left(\sqrt{1+{\vec{h}}^{2}}(\tau+\frac{\vec{h}% \cdot\vec{z}}{Mc});\frac{\vec{z}}{Mc}+(\tau+\frac{\vec{h}\cdot\vec{z}}{Mc})% \vec{h}+\frac{\vec{S}\times\vec{h}}{Mc(1+\sqrt{1+{\vec{h}}^{2}})}\right)\\ &\displaystyle=z_{W}^{\mu}(\tau,\vec{0})\end{aligned},
  61. R μ ( τ ) = ( x ~ 0 ( τ ) ; R ( τ ) ) = ( 1 + h 2 ( τ + h z M c ) ; z M c + ( τ + h z M c ) h - S × h M c 1 + h 2 ( 1 + 1 + h 2 ) ) = z W μ ( τ , σ R ) = Y μ ( τ ) + ( 0 ; < m t p l > - S × h M c 1 + h 2 ) \begin{aligned}\displaystyle R^{\mu}(\tau)&\displaystyle=\left({\tilde{x}}^{0}% (\tau);\vec{R}(\tau)\right)=\left(\sqrt{1+{\vec{h}}^{2}}(\tau+\frac{\vec{h}% \cdot\vec{z}}{Mc});\frac{\vec{z}}{Mc}+(\tau+\frac{\vec{h}\cdot\vec{z}}{Mc})% \vec{h}-\frac{\ \vec{S}\times\vec{h}}{Mc\sqrt{1+{\vec{h}}^{2}}(1+\sqrt{1+{\vec% {h}}^{2}})}\right)\\ &\displaystyle=z^{\mu}_{W}(\tau,{\vec{\sigma}}_{R})=Y^{\mu}(\tau)+\left(0;{% \frac{<}{m}tpl>{{-\vec{S}\times\vec{h}}}{{Mc\sqrt{1+{\vec{h}}^{2}}}}}\right)% \end{aligned}
  62. σ ~ = - S × h M c ( 1 + 1 + h 2 ) \tilde{\vec{\sigma}}=\frac{-\vec{S}\times\vec{h}}{Mc(1+\sqrt{1+{\vec{h}}^{2}})}
  63. σ R = - S × h M c 1 + h 2 \vec{\sigma}_{R}=\frac{-\,\vec{S}\times\vec{h}}{Mc\sqrt{1+{\vec{h}}^{2}}}
  64. x ~ μ ( τ ) \tilde{x}^{\mu}(\tau)
  65. R μ ( τ ) R^{\mu}(\tau)
  66. Y μ ( τ ) Y^{\mu}(\tau)
  67. ρ = | S | / M c \rho=|\vec{S}|/Mc
  68. Y μ ( τ ) Y^{\mu}(\tau)
  69. R μ ( τ ) R^{\mu}(\tau)
  70. x ~ μ ( τ ) \tilde{x}^{\mu}(\tau)
  71. S \vec{S}
  72. h \vec{h}
  73. Y μ ( τ ) Y^{\mu}(\tau)

Centered_dodecahedral_number.html

  1. ( 2 n + 1 ) × ( 5 n 2 + 5 n + 1 ) (2n+1)\times(5n^{2}+5n+1)

Centered_icosahedral_number.html

  1. ( 2 n + 1 ) × ( 5 n 2 + 5 n + 3 ) 3 (2n+1)\times{(5n^{2}+5n+3)\over 3}

Centered_octahedral_number.html

  1. ( 2 n + 1 ) × ( 2 n 2 + 2 n + 3 ) 3 \frac{(2n+1)\times(2n^{2}+2n+3)}{3}
  2. ( 1 + x ) 3 ( 1 - x ) 4 . \frac{(1+x)^{3}}{(1-x)^{4}}.
  3. C ( n ) = C ( n - 1 ) + 4 n 2 + 2. C(n)=C(n-1)+4n^{2}+2.

Centered_tetrahedral_number.html

  1. ( 2 n + 1 ) × ( n 2 + n + 3 ) 3 (2n+1)\times{(n^{2}+n+3)\over 3}

Central_differencing_scheme.html

  1. Φ e = ( Φ P + Φ E ) / 2 \Phi_{e}=(\Phi_{P}+\Phi_{E})/2
  2. Φ w = ( Φ W + Φ P ) / 2 \Phi_{w}=(\Phi_{W}+\Phi_{P})/2
  3. div ( ρ u φ ) = div ( Γ φ ) + S φ ; \operatorname{div}(\rho u\varphi)=\operatorname{div}(\Gamma\nabla\varphi)+S_{% \varphi};\,
  4. A n ( ρ u φ ) d A = A n ( Γ φ ) d A + C V S φ d V \int\limits_{A}\,n\cdot(\rho u\varphi)\,dA=\int\limits_{A}\,n\cdot(\Gamma% \nabla\varphi)\,dA+\int\limits_{CV}\,S_{\varphi}\,dV
  5. d d x ( ρ u φ ) = d d x ( d φ d x ) {d\over dx}(\rho u\varphi)={d\over dx}\left({d\varphi\over dx}\right)
  6. d d x ( ρ u ) = 0 {d\over dx}(\rho u)=0
  7. ( ρ u φ A ) e - ( ρ u φ A ) w = ( Γ A d φ / d x ) e - ( Γ A d φ / d x ) w (\rho u\varphi A)_{e}-(\rho u\varphi A)_{w}=(\Gamma Ad\varphi/dx)_{e}-(\Gamma Ad% \varphi/dx)_{w}
  8. ( ρ u A ) e - ( ρ u A ) w = 0 (\rho uA)_{e}-(\rho uA)_{w}=0
  9. F = ρ u F=\rho u
  10. D = Γ / δ x D=\Gamma/\delta x
  11. A e = A w A_{e}=A_{w}
  12. F e φ e - F w φ w = D e ( φ E - φ P ) - D w ( φ P - φ W ) F_{e}\varphi_{e}-F_{w}\varphi_{w}=D_{e}(\varphi_{E}-\varphi_{P})-D_{w}(\varphi% _{P}-\varphi_{W})
  13. F e - F w = 0 F_{e}-F_{w}=0
  14. φ e = ( φ E + φ P ) / 2 , φ w = ( φ P + φ W ) / 2 \varphi_{e}=(\varphi_{E}+\varphi_{P})/2,\quad\varphi_{w}=(\varphi_{P}+\varphi_% {W})/2
  15. F e φ E + φ P 2 - F w φ W + φ P 2 = D e ( φ E - φ P ) - D w ( φ P - φ W ) F_{e}\frac{\varphi_{E}+\varphi_{P}}{2}-F_{w}\frac{\varphi_{W}+\varphi_{P}}{2}=% D_{e}(\varphi_{E}-\varphi_{P})-D_{w}(\varphi_{P}-\varphi_{W})
  16. [ ( D w + F w 2 ) + ( D e - F e 2 ) + ( F e - F w ) ] φ P = ( D w + F w 2 ) φ W + ( D e - F e 2 ) φ E \left[\left(D_{w}+\frac{F_{w}}{2}\right)+\left(D_{e}-\frac{F_{e}}{2}\right)+(F% _{e}-F_{w})\right]\varphi_{P}=\left(D_{w}+\frac{F_{w}}{2}\right)\varphi_{W}+% \left(D_{e}-\frac{F_{e}}{2}\right)\varphi_{E}
  17. a P φ P = a W φ W + a E φ E a_{P}\varphi_{P}=a_{W}\varphi_{W}+a_{E}\varphi_{E}
  18. [ Γ e 1 ( φ 2 - φ 1 ) δ x - q A ] + [ Γ e 2 ( φ 3 - φ 2 ) δ x - Γ w 2 ( φ 2 - φ 1 ) δ x ] \displaystyle\left[\frac{\Gamma_{e_{1}}(\varphi_{2}-\varphi_{1})}{\delta x}-q_% {A}\right]+\left[\frac{\Gamma_{e_{2}}(\varphi_{3}-\varphi_{2})}{\delta x}-% \frac{\Gamma_{w_{2}}(\varphi_{2}-\varphi_{1})}{\delta x}\right]
  19. Γ e 1 = Γ w 2 , Γ e 2 = Γ w 3 , Γ e 3 = Γ w 4 \Gamma_{e_{1}}=\Gamma_{w_{2}},\Gamma_{e_{2}}=\Gamma_{w_{3}},\Gamma_{e_{3}}=% \Gamma_{w_{4}}
  20. F e - F w = 0 F_{e}-F_{w}=0
  21. a P φ P = a W φ W + a E φ E a_{P}\varphi_{P}=a_{W}\varphi_{W}+a_{E}\varphi_{E}
  22. F e / D e < 2 F_{e}/D_{e}<2
  23. F e > 0 , F w > 0 F_{e}>0,F_{w}>0
  24. a E = ( D e - F e / 2 ) a_{E}=(D_{e}-F_{e}/2)
  25. D e > F e / 2 D_{e}>F_{e}/2
  26. φ \varphi
  27. φ \varphi

Central_polynomial.html

  1. ( x y - y x ) 2 (xy-yx)^{2}
  2. ( x y - y x ) 2 = - det ( x y - y x ) I (xy-yx)^{2}=-\det(xy-yx)I

Centrifugal_mechanism_of_acceleration.html

  1. γ = γ 0 1 - Ω 2 r 2 / c 2 \gamma=\frac{\gamma_{0}}{1-\Omega^{2}r^{2}/c^{2}}
  2. γ 0 \gamma_{0}
  3. r r
  4. c c
  5. γ I C S m a x 10 8 \gamma_{ICS}^{max}\sim 10^{8}
  6. L < 8 × 10 40 e r g / s L<8\times 10^{40}erg/s
  7. γ B B W m a x 10 7 \gamma_{BBW}^{max}\sim 10^{7}
  8. γ K N m a x 10 7 \gamma_{KN}^{max}\sim 10^{7}
  9. ϵ n p F r e a c δ r n G J \epsilon\approx\frac{n_{p}F_{reac}\delta r}{n_{{}_{GJ}}}
  10. δ r c / Γ \delta r\sim c/\Gamma
  11. Γ \Gamma
  12. F r e a c 2 m c Ω ξ ( r ) - 3 F_{reac}\approx 2mc\Omega\xi(r)^{-3}
  13. ξ ( r ) = ( 1 - Ω 2 r 2 / c 2 ) 1 / 2 \xi(r)=\left(1-\Omega^{2}r^{2}/c^{2}\right)^{1/2}
  14. n p n_{p}
  15. m m
  16. n G J n_{{}_{GJ}}
  17. 100 s 100s
  18. T e V s TeVs
  19. P e V s PeVs
  20. 10 21 e V 10^{21}eV
  21. ϵ p ( e V ) 6.4 × 10 17 × M 8 - 5 / 2 × L 42 5 / 2 , \epsilon_{p}\left(eV\right)\approx 6.4\times 10^{17}\times M_{8}^{-5/2}\times L% _{42}^{5/2},
  22. L 42 L / 10 42 e r g / s L_{42}\equiv L/10^{42}erg/s
  23. M 8 M / ( 10 8 M ) M_{8}\equiv M/(10^{8}M_{\odot})
  24. M M_{\odot}
  25. 10 21 e V 10^{21}eV

Centripetal_Catmull–Rom_spline.html

  1. 𝐏 0 , 𝐏 1 , 𝐏 2 , 𝐏 3 \mathbf{P}_{0},\mathbf{P}_{1},\mathbf{P}_{2},\mathbf{P}_{3}
  2. 𝐏 1 \mathbf{P}_{1}
  3. 𝐏 2 \mathbf{P}_{2}
  4. 𝐏 i = [ x i y i ] T \mathbf{P}_{i}=[x_{i}\quad y_{i}]^{T}
  5. 𝐂 \mathbf{C}
  6. 𝐏 0 , 𝐏 1 , 𝐏 2 , 𝐏 3 \mathbf{P}_{0},\mathbf{P}_{1},\mathbf{P}_{2},\mathbf{P}_{3}
  7. t 0 , t 1 , t 2 , t 3 t_{0},t_{1},t_{2},t_{3}
  8. 𝐂 = t 2 - t t 2 - t 1 𝐁 1 + t - t 1 t 2 - t 1 𝐁 2 \mathbf{C}=\frac{t_{2}-t}{t_{2}-t_{1}}\mathbf{B}_{1}+\frac{t-t_{1}}{t_{2}-t_{1% }}\mathbf{B}_{2}
  9. 𝐁 1 = t 2 - t t 2 - t 0 𝐀 1 + t - t 0 t 2 - t 0 𝐀 2 \mathbf{B}_{1}=\frac{t_{2}-t}{t_{2}-t_{0}}\mathbf{A}_{1}+\frac{t-t_{0}}{t_{2}-% t_{0}}\mathbf{A}_{2}
  10. 𝐁 2 = t 3 - t t 3 - t 1 𝐀 2 + t - t 1 t 3 - t 1 𝐀 3 \mathbf{B}_{2}=\frac{t_{3}-t}{t_{3}-t_{1}}\mathbf{A}_{2}+\frac{t-t_{1}}{t_{3}-% t_{1}}\mathbf{A}_{3}
  11. 𝐀 1 = t 1 - t t 1 - t 0 𝐏 0 + t - t 0 t 1 - t 0 𝐏 1 \mathbf{A}_{1}=\frac{t_{1}-t}{t_{1}-t_{0}}\mathbf{P}_{0}+\frac{t-t_{0}}{t_{1}-% t_{0}}\mathbf{P}_{1}
  12. 𝐀 2 = t 2 - t t 2 - t 1 𝐏 1 + t - t 1 t 2 - t 1 𝐏 2 \mathbf{A}_{2}=\frac{t_{2}-t}{t_{2}-t_{1}}\mathbf{P}_{1}+\frac{t-t_{1}}{t_{2}-% t_{1}}\mathbf{P}_{2}
  13. 𝐀 3 = t 3 - t t 3 - t 2 𝐏 2 + t - t 2 t 3 - t 2 𝐏 3 \mathbf{A}_{3}=\frac{t_{3}-t}{t_{3}-t_{2}}\mathbf{P}_{2}+\frac{t-t_{2}}{t_{3}-% t_{2}}\mathbf{P}_{3}
  14. t i + 1 = [ ( x i + 1 - x i ) 2 + ( y i + 1 - y i ) 2 ] α + t i t_{i+1}=\left[\sqrt{(x_{i+1}-x_{i})^{2}+(y_{i+1}-y_{i})^{2}}\right]^{\alpha}+t% _{i}
  15. α \alpha
  16. i = 0 , 1 , 2 , 3 i=0,1,2,3
  17. t 0 = 0 t_{0}=0
  18. α \alpha
  19. 0.5 0.5
  20. α = 0 \alpha=0
  21. α = 1 \alpha=1
  22. t = t 1 t=t_{1}
  23. 𝐀 1 , 𝐀 2 , 𝐀 3 , 𝐁 1 , 𝐁 2 , \mathbf{A}_{1},\mathbf{A}_{2},\mathbf{A}_{3},\mathbf{B}_{1},\mathbf{B}_{2},
  24. 𝐂 \mathbf{C}
  25. t 1 t_{1}
  26. 𝐂 = 𝐏 1 \mathbf{C}=\mathbf{P}_{1}
  27. t = t 2 t=t_{2}
  28. 𝐂 = 𝐏 2 \mathbf{C}=\mathbf{P}_{2}
  29. t 2 t_{2}
  30. α \alpha
  31. t i + 1 t_{i+1}
  32. 𝐂 \mathbf{C}
  33. t 1 t_{1}
  34. t 2 t_{2}

Ceramide_phosphoethanolamine_synthase.html

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

Chaffee-Infante_equation.html

  1. u t - u x x + λ * ( u 3 - u ) = 0 u_{t}-u_{xx}+\lambda*(u^{3}-u)=0

Chamfer_(geometry).html

  1. cos - 1 ( - 1 3 ) \cos^{-1}(-\frac{1}{3})
  2. ( ± 1 , ± 1 , ± 1 ) (\pm 1,\pm 1,\pm 1)
  3. ( ± 2 , 0 , 0 ) (\pm 2,0,0)

Change_of_rings.html

  1. f : R S f:R\to S
  2. f ! M = M R S f_{!}M=M\otimes_{R}S
  3. f * M = Hom R ( S , M ) f_{*}M=\operatorname{Hom}_{R}(S,M)
  4. f * N = N R f^{*}N=N_{R}
  5. f ! : Mod R Mod S : f * f_{!}:\,\text{Mod}_{R}\leftrightarrows\,\text{Mod}_{S}:f^{*}
  6. f * : Mod S Mod R : f * . f^{*}:\,\text{Mod}_{S}\leftrightarrows\,\text{Mod}_{R}:f_{*}.

Chaplygin_sleigh.html

  1. v ˙ = ω 2 \dot{v}=\omega^{2}
  2. ω ˙ = - b v ω \dot{\omega}=-bv\omega

Charlie_Hughes.html

  1. S = A x 2 + B r 2 + x 2 + C S=Ax^{2}+B\sqrt{r^{2}+x^{2}}\ +C

Chazy_equation.html

  1. d 3 y d x 3 = 2 y d 2 y d x 2 - 3 ( d y d x ) 2 . \frac{d^{3}y}{dx^{3}}=2y\frac{d^{2}y}{dx^{2}}-3\left(\frac{dy}{dx}\right)^{2}.
  2. E 2 ( τ ) = 1 - 24 σ 1 ( n ) q n = 1 - 24 q - 72 q 2 - . E_{2}(\tau)=1-24\sum\sigma_{1}(n)q^{n}=1-24q-72q^{2}-\cdots.

Chebyshev_integral.html

  1. x p ( 1 - x ) q d x = B ( x ; 1 + p , 1 + q ) , \int x^{p}(1-x)^{q}\,dx=B(x;1+p,1+q),
  2. B ( x ; a , b ) B(x;a,b)

Checkerboard_score.html

  1. C i j = ( r i - S i j ) ( r j - S i j ) C_{ij}=(r_{i}-S_{ij})(r_{j}-S_{ij})
  2. C = j = 0 M i < j C i j / P C=\sum_{j=0}^{M}\sum_{i<j}C_{ij}/P
  3. C i j = ( r i - S i j ) ( r j - S i j ) / ( r i + r j - S i j ) C_{ij}=(r_{i}-S_{ij})(r_{j}-S_{ij})/(r_{i}+r_{j}-S_{ij})

Chemical_reaction_model.html

  1. ( ρ Y i ) t + ( ρ v Y i ) = - J i + R i + S i \frac{\partial(\rho Y_{i})}{\partial t}+\nabla\cdot(\rho\vec{v}Y_{i})=-\nabla% \cdot\vec{J}_{i}+R_{i}+S_{i}

Chemistry_of_pressure-sensitive_adhesives.html

  1. T g {T_{g}}
  2. γ \gamma
  3. m J m 2 {mJ\over m^{2}}
  4. T g {T_{g}}
  5. γ \gamma
  6. T g T_{g}
  7. ϕ 1 {\phi_{1}}
  8. ϕ 2 {\phi_{2}}
  9. T g , 1 {T_{g,1}}
  10. T g , 2 {T_{g,2}}
  11. T g = ϕ 1 T g , 1 + ϕ 2 T g , 2 {T_{g}}={\phi_{1}T_{g,1}}+{\phi_{2}T_{g,2}}
  12. H H
  13. k w k_{w}
  14. ν \nu
  15. F L F_{L}
  16. Ψ \Psi
  17. Ψ = k w F L ν H \Psi=k_{w}{F_{L}\nu\over H}

Chessboard_detection.html

  1. 𝐗 \mathbf{X}
  2. 𝐱 \mathbf{x}
  3. 𝐱 = K [ R t ] 𝐗 , 𝐱 2 , 𝐗 3 , \mathbf{x}=K\begin{bmatrix}R&t\end{bmatrix}\mathbf{X}\quad,\quad\mathbf{x}\in% \mathbb{P}^{2}\quad,\quad\mathbf{X}\in\mathbb{P}^{3},
  4. n \mathbb{P}^{n}
  5. n n
  6. 3 × 4 3\times 4
  7. M = K [ R t ] M=K\begin{bmatrix}R&t\end{bmatrix}
  8. ( ρ , θ ) (\rho,\theta)
  9. ( ρ , θ ) (\rho,\theta)
  10. ( i , j ) (i,j)
  11. ( ρ i , θ j ) (\rho_{i},\theta_{j})

Chiral_homology.html

  1. 𝒟 X \mathcal{D}_{X}

Chiral_Lie_algebra.html

  1. 2 \mathcal{E}_{2}

Chiral_Potts_model.html

  1. σ n = ( 1 - k 2 ) β , β = n ( N - n ) / 2 N 2 . \langle\sigma^{n}\rangle=(1-k^{\prime 2})^{\beta},\quad\beta=n(N-n)/2N^{2}.
  2. L i 1 α j 1 β ( x ) L i 2 β j 2 γ ( y ) R j 1 j 2 k 1 k 2 ( y / x ) = R i 1 i 2 j 1 j 2 ( y / x ) L j 2 α k 2 β ( y ) L j 1 β k 1 γ ( x ) , 0 < i , j , k 1 , 0 α , β , γ N - 1. L_{i_{1}\alpha}^{j_{1}\beta}(x)L_{i_{2}\beta}^{j_{2}\gamma}(y)R_{j_{1}j_{2}}^{% k_{1}k_{2}}(y/x)=R_{i_{1}i_{2}}^{j_{1}j_{2}}(y/x)L_{j_{2}\alpha}^{k_{2}\beta}(% y)L_{j_{1}\beta}^{k_{1}\gamma}(x),\quad 0<i,j,k\leq 1,\quad 0\leq\alpha,\beta,% \gamma\leq N-1.
  3. L i 1 α 1 i 2 α 2 L ^ i 2 β 1 i 3 β 2 S α 2 β 2 α 3 β 3 = S α 1 β 1 α 2 β 2 L ^ i 1 β 2 i 2 β 3 L i 2 α 2 i 3 α 3 , 0 < i i 1 , 0 α i , β i N - 1. L_{i_{1}\alpha_{1}}^{i_{2}\alpha_{2}}{\hat{L}}_{i_{2}\beta_{1}}^{i_{3}\beta_{2% }}S_{\alpha_{2}\beta_{2}}^{\alpha_{3}\beta_{3}}=S_{\alpha_{1}\beta_{1}}^{% \alpha_{2}\beta_{2}}{\hat{L}}_{i_{1}\beta_{2}}^{i_{2}\beta_{3}}L_{i_{2}\alpha_% {2}}^{i_{3}\alpha_{3}},\quad 0<i_{i}\leq 1,\quad 0\leq\alpha_{i},\beta_{i}\leq N% -1.

Chirp_compression.html

  1. Y ( f ) = e x p [ j π ( f - f 0 ) 2 / k ] Y(f)=exp[j\pi(f-f_{0})^{2}/k]
  2. ψ ( f ) = π . ( f - f 0 ) 2 / k \psi(f)=\pi.(f-f_{0})^{2}/k
  3. t d = - 1 2 π . d ψ d f = 1 k . ( f 0 - f ) t_{d}=-\frac{1}{2\pi}.\frac{d\psi}{df}=\frac{1}{k}.(f_{0}-f)
  4. y ( t ) = - h ( τ ) . h * ( t - τ ) . d τ y(t)=\int_{-\infty}^{\infty}h(\tau).h^{*}(t-\tau).d\tau
  5. y ( t ) = 1 2 π . - | H ( ω ) | 2 . e x p ( j ω t ) . d ω y(t)=\frac{1}{2\pi}.\int_{-\infty}^{\infty}|H(\omega)|^{2}.exp(j\omega t).d\omega
  6. y ( t ) = - h ( τ ) . g * ( t - τ ) . d τ y(t)=\int_{-\infty}^{\infty}h(\tau).g^{*}(t-\tau).d\tau
  7. y ( t ) = 1 2 π . - H ( ω ) . G * ( ω ) . e x p ( j ω t ) . d ω y(t)=\frac{1}{2\pi}.\int_{-\infty}^{\infty}H(\omega).G*(\omega).exp(j\omega t)% .d\omega
  8. c 1 ( n ) = I F F T [ F F T { a ( n ) } * F F T { b ( n ) } ] c_{1}(n)=IFFT[FFT\left\{a(n)\right\}^{*}FFT\left\{b(n)\right\}]
  9. s 1 ( t ) = r e c t ( t T ) . e 2 π j ( f 0 t + k t 2 / 2 ) s_{1}(t)=rect(\frac{t}{T}).e^{2\pi j(f_{0}t+kt^{2}/2)}
  10. ϕ ( t ) = 2 π ( f 0 t + k t 2 / 2 ) \phi(t)=\frac{2}{\pi}(f_{0}t+kt^{2}/2)
  11. f i = 2 π . d ϕ d ω = f 0 + k . t f_{i}={2\pi}.\frac{d\phi}{d\omega}=f_{0}+k.t
  12. S 1 ( f ) = - s 1 ( t ) . e - j 2 π . f t . d t = - e - j 2 π . [ ( f 0 - f ) t + k t 2 ] . d t S_{1}(f)=\int_{-\infty}^{\infty}s_{1}(t).e^{-j2\pi.ft}.dt=\int_{-\infty}^{% \infty}e^{-j2\pi.[(f_{0}-f)t+kt^{2}]}.dt
  13. S out ( f ) = Y ( f ) . S 1 ( f ) S_{\,\text{out}}(f)=Y(f).S_{1}(f)
  14. s out ( t ) s_{\,\text{out}}(t)
  15. S out ( f ) S_{\,\text{out}}(f)
  16. s out ( t ) s_{\,\text{out}}(t)
  17. s out ( t ) = s 1 ( t ) * y ( t ) s_{\,\text{out}}(t)=s_{1}(t)^{*}y(t)
  18. f ( t ) * g ( t ) = - f ( τ ) . g ( t - τ ) . d τ f(t)^{*}g(t)=\int_{-\infty}^{\infty}f(\tau).g(t-\tau).d\tau
  19. y ( t ) = - Y ( f ) . e 2 π . f t . d f = - e j ( f - f 0 ) 2 / k . e j 2 π . f t . d f y(t)=\int_{-\infty}^{\infty}Y(f).e^{2\pi.ft}.df=\int_{-\infty}^{\infty}e^{j(f-% f_{0})^{2}/k}.e^{j2\pi.ft}.df
  20. P u t f - f 0 = u s o f = u - f 0 a n d d f = d u . A l s o w h e n f = ± t h e n u = ± Put\quad f-f_{0}=u\quad so\quad f=u-f_{0}\quad and\quad df=du.\qquad Also\quad when% \quad f=\pm\infty\quad then\quad u=\pm\infty
  21. y ( t ) = - e j π . u 2 / k . e j π . ( u - f 0 ) t . d u = e j 2 π . f 0 t - e j π . u 2 / k . e j 2 π . u t . d u y(t)=\int_{-\infty}^{\infty}e^{j\pi.u^{2}/k}.e^{j\pi.(u-f_{0})t}.du=e^{j2\pi.f% _{0}t}\int_{-\infty}^{\infty}e^{j\pi.u^{2}/k}.e^{j2\pi.ut}.du
  22. - e x p ( - π β . u 2 ) . e x p ( j 2 π . u t ) . d u = 1 β . e x p ( - π . t 2 β ) \int_{-\infty}^{\infty}exp(-\pi\beta.u^{2}).exp(j2\pi.ut).du=\frac{1}{\sqrt{% \beta}}.exp(-\frac{\pi.t^{2}}{\beta})
  23. y ( t ) = j k . e 2 π . f 0 t . e - j π . t 2 k = j B T . e j 2 π . f 0 t . e - j π . t 2 . k = j B T . e j 2 π ( f 0 t - t 2 k / 2 ) y(t)=\sqrt{jk}.e^{2\pi.f_{0}t}.e^{-j\pi.t^{2}k}=\sqrt{\frac{jB}{T}}.e^{j2\pi.f% _{0}t}.e^{-j\pi.t^{2}.k}=\sqrt{\frac{jB}{T}}.e^{j2\pi(f_{0}t-t^{2}k/2)}
  24. s out = j B T . - T / 2 T / 2 e j 2 π . ( f 0 τ + k τ 2 / 2 . e j 2 π ( f 0 ( t - τ ) - k ( t - τ ) 2 / 2 ) . d τ s_{\,\text{out}}=\sqrt{\frac{jB}{T}}.\int_{-T/2}^{T/2}e^{j2\pi.(f_{0}\tau+k% \tau^{2}/2}.e^{j2\pi(f_{0}(t-\tau)-k(t-\tau)^{2}/2)}.d\tau
  25. s out = j B T . e j 2 π ( f 0 t - k t 2 / 2 ) . - T / 2 T / 2 e j 2 π k t τ . d τ s_{\,\text{out}}=\sqrt{\frac{jB}{T}}.e^{j2\pi(f_{0}t-kt^{2}/2)}.\int_{-T/2}^{T% /2}e^{j2\pi kt\tau}.d\tau
  26. e a x . d x = e a x a \int e^{ax}.dx=\frac{e^{ax}}{a}
  27. - T / 2 T / 2 e j 2 π . k t τ . d τ = 1 j 2 π k t . [ e j 2 π . k t τ ] -T/2 T/2 = 1 j 2 π k t [ e j 2 π k T / 2 - e - j 2 π k T / 2 ] \int_{-T/2}^{T/2}e^{j2\pi.kt\tau}.d\tau=\frac{1}{j2\pi kt}.\Bigl[e^{j2\pi.kt% \tau}\Bigr]^{\,\text{T/2}}_{\,\text{-T/2}}=\frac{1}{j2\pi kt}[e^{j2\pi kT/2}-e% ^{-j2\pi kT/2}]
  28. = 1 π k t . e j π B t - e - j π B t 2 j = 1 π k t . s i n ( π B t ) =\frac{1}{\pi kt}.\frac{e^{j\pi Bt}-e^{-j\pi Bt}}{2j}=\frac{1}{\pi kt}.sin(\pi Bt)
  29. s out = 1 π k t . j B T . s i n ( π B T ) . e x p [ j 2 π ( f 0 t - k t 2 / 2 ) s_{\,\text{out}}=\frac{1}{\pi kt}.\sqrt{\frac{jB}{T}}.sin(\pi BT).exp[j2\pi(f_% {0}t-kt^{2}/2)
  30. = T . B . j . s i n ( π B T ) π B t . e x p [ j 2 π ( f 0 t - k t 2 / 2 ) =\sqrt{T.B}.\sqrt{j}.\frac{sin(\pi BT)}{\pi Bt}.exp[j2\pi(f_{0}t-kt^{2}/2)
  31. | Compressed Output | T . B × s i n ( π B t ) ( π B t ) \left|{\,\text{Compressed Output}}\right|\approx\sqrt{T.B}\times\frac{sin(\pi Bt% )}{(\pi Bt)}
  32. T τ T × B \frac{T}{\tau}\approx T\times B
  33. T × B \sqrt{T\times B}
  34. F a r S L ( d B ) - 20 × l o g 10 ( T . B ) FarSL(dB)\approx-20\times log_{\,\text{10}}(T.B)
  35. K ( ω ) = C ( ω ) H ( ω ) K(\omega)=\frac{C^{\prime}(\omega)}{H(\omega)}
  36. t shift = - f d B . T w h e r e f d = 2. V r λ = 2. f m . V r c t_{\,\text{shift}}=-\frac{f_{d}}{B}.T\quad where\quad f_{d}=2.\frac{V_{r}}{% \lambda}=2.\frac{f_{m}.V_{r}}{c}
  37. 2 × T × f d 1 2\times T\times f_{d}\approx 1
  38. f d 0.06 T f_{d}\leq\frac{0.06}{T}
  39. K ( ω ) = C ( ω ) Φ ( ω ) . A ( ω ) . H ( ω ) K(\omega)=\frac{C^{\prime}(\omega)}{\Phi(\omega).A(\omega).H(\omega)}
  40. c o s ( ω . t 0 ) ± j . s i n ( ω . t 0 ) = e x p ( ± j ω . t 0 ) cos(\omega.t_{0})\pm j.sin(\omega.t_{0})\quad=\quad exp(\pm j\omega.t_{0})