wpmath0000004_5

Electronic_band_structure.html

  1. ψ n 𝐤 ( 𝐫 ) = e i 𝐤 𝐫 u n 𝐤 ( 𝐫 ) \psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(% \mathbf{r})
  2. f ( E ) = 1 1 + e ( E - μ ) / k B T f(E)=\frac{1}{1+e^{{(E-\mu)}/{k_{\rm B}T}}}
  3. N / V = - g ( E ) f ( E ) d E N/V=\int_{-\infty}^{\infty}g(E)f(E)\,dE
  4. V ( 𝐫 ) = 𝐊 V 𝐊 e i 𝐊 𝐫 V(\mathbf{r})=\sum_{\mathbf{K}}{V_{\mathbf{K}}e^{i\mathbf{K}\cdot\mathbf{r}}}
  5. Ψ n , 𝐤 ( 𝐫 ) = e i 𝐤 𝐫 u n ( 𝐫 ) {\Psi}_{n,\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n}(\mathbf{% r})
  6. u n ( 𝐫 ) u_{n}(\mathbf{r})
  7. u n ( 𝐫 ) = u n ( 𝐫 - 𝐑 ) u_{n}(\mathbf{r})=u_{n}(\mathbf{r-R})
  8. Ψ \Psi
  9. ψ n ( 𝐫 ) \psi_{n}(\mathbf{r})
  10. Ψ ( 𝐫 ) = n , 𝐑 b n , 𝐑 ψ n ( 𝐫 - 𝐑 ) \Psi(\mathbf{r})=\sum_{n,\mathbf{R}}b_{n,\mathbf{R}}\psi_{n}(\mathbf{r-R})
  11. b n , 𝐑 b_{n,\mathbf{R}}
  12. a n ( 𝐫 - 𝐑 ) = V C ( 2 π ) 3 B Z d 𝐤 e - i 𝐤 ( 𝐑 - 𝐫 ) u n 𝐤 a_{n}(\mathbf{r-R})=\frac{V_{C}}{(2\pi)^{3}}\int_{BZ}d\mathbf{k}e^{-i\mathbf{k% }\cdot(\mathbf{R-r})}u_{n\mathbf{k}}
  13. u n 𝐤 u_{n\mathbf{k}}
  14. Ψ n , 𝐤 ( 𝐫 ) = 𝐑 e - i 𝐤 ( 𝐑 - 𝐫 ) a n ( 𝐫 - 𝐑 ) \Psi_{n,\mathbf{k}}(\mathbf{r})=\sum_{\mathbf{R}}e^{-i\mathbf{k}\cdot(\mathbf{% R-r})}a_{n}(\mathbf{r-R})

Electroweak_scale.html

  1. v = ( G F 2 ) - 1 / 2 v=(G_{F}\sqrt{2})^{-1/2}
  2. G F G_{F}

Element_(mathematics).html

  1. x A x\in A
  2. A x , A\ni x,
  3. x A x\notin A

ELEMENTARY.html

  1. ELEMENTARY \displaystyle\mathrm{ELEMENTARY}
  2. \subsetneq
  3. \subsetneq
  4. \subsetneq
  5. \subsetneq
  6. O ( 2 2 n ) O(2^{2^{n}})
  7. f ( m , x 1 , , x n ) = i = 0 m g ( i , x 1 , , x n ) f(m,x_{1},\ldots,x_{n})=\sum\limits_{i=0}^{m}g(i,x_{1},\ldots,x_{n})
  8. f ( m , x 1 , , x n ) = i = 0 m g ( i , x 1 , , x n ) f(m,x_{1},\ldots,x_{n})=\prod\limits_{i=0}^{m}g(i,x_{1},\ldots,x_{n})
  9. { n + 1 , n - . m , n / m , n m } \{n+1,n\,\stackrel{.}{-}\,m,\lfloor n/m\rfloor,n^{m}\}
  10. { n + m , n - . m , n / m , 2 n } \{n+m,n\,\stackrel{.}{-}\,m,\lfloor n/m\rfloor,2^{n}\}
  11. { n + m , n 2 , n mod m , 2 n } \{n+m,n^{2},n\,\bmod\,m,2^{n}\}
  12. n - . m = max { n - m , 0 } n\,\stackrel{.}{-}\,m=\max\{n-m,0\}
  13. NTIME ( 2 2 2 O ( n ) ) = H O i \mathrm{NTIME}(2^{2^{\cdots{2^{O(n)}}}})=\exists{}HO^{i}
  14. \cdots
  15. i i
  16. H O i \exists{}HO^{i}
  17. i i

Elementary_equivalence.html

  1. \models
  2. \models
  3. \preceq
  4. \succeq
  5. \models
  6. \models
  7. \models
  8. \models

Eliyahu_Rips.html

  1. \mathbb{R}
  2. \mathbb{R}

Ellipsometry.html

  1. ρ \rho
  2. Ψ \Psi
  3. Δ \Delta
  4. r s r_{s}
  5. r p r_{p}
  6. r p r_{p}
  7. r s r_{s}
  8. ρ \rho
  9. r p r_{p}
  10. r s r_{s}
  11. ρ = r p r s = tan ( Ψ ) e i Δ \rho=\frac{r_{p}}{r_{s}}=\tan(\Psi)e^{i\Delta}
  12. tan ( Ψ ) \tan(\Psi)
  13. Δ \Delta
  14. Ψ \Psi
  15. Δ \Delta
  16. Ψ \Psi
  17. Δ \Delta
  18. Ψ \Psi
  19. Δ \Delta
  20. Ψ \Psi
  21. Δ \Delta
  22. Ψ \Psi
  23. Δ \Delta
  24. E x E_{x}
  25. E y E_{y}
  26. Ψ = A \Psi=A
  27. Δ = 2 P + π / 2 \Delta=2P+\pi/2

Elliptic_orbit.html

  1. v v\,
  2. v = μ ( 2 r - 1 a ) v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}
  3. μ \mu\,
  4. r r\,
  5. a a\,\!
  6. 1 a {1\over{a}}
  7. T T\,\!
  8. T = 2 π a 3 μ T=2\pi\sqrt{a^{3}\over{\mu}}
  9. μ \mu\,
  10. a a\,\!
  11. a a\,\!
  12. ϵ \epsilon\,
  13. v 2 2 - μ r = - μ 2 a = ϵ < 0 {v^{2}\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0
  14. v v\,
  15. r r\,
  16. a a\,
  17. μ \mu\,
  18. ϕ \phi
  19. h = r v cos ϕ h\,=r\,v\,\cos\phi
  20. h h\,
  21. v v\,
  22. r r\,
  23. ϕ \phi\,

Elliptic_partial_differential_equation.html

  1. A u x x + 2 B u x y + C u y y + D u x + E u y + F = 0 Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0\,
  2. B 2 - A C < 0. B^{2}-AC<0.
  3. u x y = u y x u_{xy}=u_{yx}
  4. B 2 - 4 A C B^{2}-4AC
  5. B 2 - A C , B^{2}-AC,
  6. A x 2 + 2 B x y + C y 2 + = 0 Ax^{2}+2Bxy+Cy^{2}+\cdots=0
  7. u x y = u y x = 0 u_{xy}=u_{yx}=0
  8. A u x x + C u y y + D u x + E u y + F = 0 Au_{xx}+Cu_{yy}+Du_{x}+Eu_{y}+F=0
  9. A x 2 + C y 2 + = 0 Ax^{2}+Cy^{2}+\cdots=0
  10. x 2 a 2 + y 2 b 2 - 1 = 0. {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-1=0.
  11. L u = i = 1 n j = 1 n a i , j 2 u x i x j + (lower-order terms) = 0 Lu=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}\frac{\partial^{2}u}{\partial x_{i}% \partial x_{j}}\quad\,\text{ + (lower-order terms)}=0\,
  12. a 2 u x 2 + b 2 u x y + c 2 u y 2 + d 2 u y z + e 2 u z 2 + (lower-order terms) = 0 , a\frac{\partial^{2}u}{\partial x^{2}}+b\frac{\partial^{2}u}{\partial x\partial y% }+c\frac{\partial^{2}u}{\partial y^{2}}+d\frac{\partial^{2}u}{\partial y% \partial z}+e\frac{\partial^{2}u}{\partial z^{2}}\,\text{ + (lower-order terms% )}=0,
  13. a 2 u x 2 + c 2 u y 2 + e 2 u z 2 + (lower-order terms) = 0. a\frac{\partial^{2}u}{\partial x^{2}}+c\frac{\partial^{2}u}{\partial y^{2}}+e% \frac{\partial^{2}u}{\partial z^{2}}\,\text{ + (lower-order terms)}=0.
  14. x 2 a 2 + y 2 b 2 + z 2 c 2 = 1. {x^{2}\over a^{2}}+{y^{2}\over b^{2}}+{z^{2}\over c^{2}}=1.

Elongated_pentagonal_cupola.html

  1. V = ( 1 6 ( 5 + 4 5 + 15 5 + 2 5 ) ) a 3 10.0183... a 3 V=(\frac{1}{6}(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}}))a^{3}\approx 10.0183...a^{3}
  2. A = ( 1 4 ( 60 + 10 ( 80 + 31 5 + 2175 + 930 5 ) ) ) a 2 26.5797... a 2 A=(\frac{1}{4}(60+\sqrt{10(80+31\sqrt{5}+\sqrt{2175+930\sqrt{5}})}))a^{2}% \approx 26.5797...a^{2}

Elongated_pentagonal_gyrobicupola.html

  1. V = 1 6 ( 10 + 8 5 + 15 5 + 2 5 ) a 3 12.3423... a 3 V=\frac{1}{6}\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^{3}\approx 12.342% 3...a^{3}
  2. A = ( 20 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 27.7711... a 2 A=\left(20+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}% \right)a^{2}\approx 27.7711...a^{2}

Elongated_pentagonal_gyrobirotunda.html

  1. V = 1 6 ( 45 + 17 5 + 15 5 + 2 5 ) a 3 21.5297... a 3 V=\frac{1}{6}(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}})a^{3}\approx 21.5297...a^{3}
  2. A = 10 + 30 ( 10 + 3 5 + 75 + 30 5 ) a 2 39.306... a 2 A=10+\sqrt{30(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}})a^{2}\approx 39.306...a^{2}

Elongated_pentagonal_orthobicupola.html

  1. V = 1 6 ( 10 + 8 5 + 15 5 + 2 5 ) a 3 12.3423... a 3 V=\frac{1}{6}(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}})a^{3}\approx 12.3423...a^{3}
  2. A = ( 20 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 27.7711... a 2 A=(20+\sqrt{\frac{5}{2}(10+\sqrt{5}+\sqrt{75+30\sqrt{5}})})a^{2}\approx 27.771% 1...a^{2}

Elongated_pentagonal_orthobirotunda.html

  1. V = 1 6 ( 45 + 17 5 + 15 5 + 2 5 ) a 3 21.5297... a 3 V=\frac{1}{6}(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}})a^{3}\approx 21.5297...a^{3}
  2. A = 10 + 30 ( 10 + 3 5 + 75 + 30 5 ) a 2 39.306... a 2 A=10+\sqrt{30(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}})a^{2}\approx 39.306...a^{2}

Elongated_pentagonal_rotunda.html

  1. V = 1 12 ( 45 + 17 5 + 30 5 + 2 5 ) a 3 14.612... a 3 V=\frac{1}{12}(45+17\sqrt{5}+30\sqrt{5+2\sqrt{5}})a^{3}\approx 14.612...a^{3}
  2. A = 1 2 ( 20 + 5 ( 145 + 58 5 + 2 30 ( 65 + 29 5 ) ) ) a 2 32.3472... a 2 A=\frac{1}{2}(20+\sqrt{5(145+58\sqrt{5}+2\sqrt{30(65+29\sqrt{5})})})a^{2}% \approx 32.3472...a^{2}

Elongated_square_cupola.html

  1. V = ( 3 + 8 2 3 ) a 3 6.77124... a 3 V=(3+\frac{8\sqrt{2}}{3})a^{3}\approx 6.77124...a^{3}
  2. A = ( 15 + 2 2 + 3 ) a 2 19.5605... a 2 A=(15+2\sqrt{2}+\sqrt{3})a^{2}\approx 19.5605...a^{2}
  3. C = ( 1 2 5 + 2 2 ) a 1.39897... a C=(\frac{1}{2}\sqrt{5+2\sqrt{2}})a\approx 1.39897...a

Elongated_triangular_bipyramid.html

  1. V = ( 1 12 ( 2 2 + 3 3 ) ) a 3 0.668715... a 3 V=(\frac{1}{12}(2\sqrt{2}+3\sqrt{3}))a^{3}\approx 0.668715...a^{3}
  2. A = ( 3 2 ( 2 + 3 ) ) a 2 5.59808... a 2 A=(\frac{3}{2}(2+\sqrt{3}))a^{2}\approx 5.59808...a^{2}

Elongated_triangular_cupola.html

  1. V = ( 1 6 ( 5 2 + 9 3 ) ) a 3 3.77659... a 3 V=(\frac{1}{6}(5\sqrt{2}+9\sqrt{3}))a^{3}\approx 3.77659...a^{3}
  2. A = ( 9 + 5 3 2 ) a 2 13.3301... a 2 A=(9+\frac{5\sqrt{3}}{2})a^{2}\approx 13.3301...a^{2}

Elongated_triangular_gyrobicupola.html

  1. V = ( 5 2 3 + 3 3 2 ) a 3 4.9551... a 3 V=(\frac{5\sqrt{2}}{3}+\frac{3\sqrt{3}}{2})a^{3}\approx 4.9551...a^{3}
  2. A = 2 ( 6 + 3 ) a 2 15.4641... a 2 A=2(6+\sqrt{3})a^{2}\approx 15.4641...a^{2}

Elongated_triangular_orthobicupola.html

  1. 1 2 \frac{1}{\sqrt{2}}
  2. V t e t r a h e d r o n = 1 3 V c u b e = 1 3 1 2 3 = 2 12 V_{tetrahedron}=\frac{1}{3}V_{cube}=\frac{1}{3}\frac{1}{{\sqrt{2}}^{3}}=\frac{% \sqrt{2}}{12}
  3. 6 3 4 6\frac{\sqrt{3}}{4}
  4. V p r i s m = 3 3 2 V_{prism}=\frac{3\sqrt{3}}{2}
  5. V J 35 = 20 V t e t r a h e d r o n + V p r i s m = 5 2 3 + 3 3 2 V_{J_{35}}=20V_{tetrahedron}+V_{prism}=\frac{5\sqrt{2}}{3}+\frac{3\sqrt{3}}{2}
  6. V J 35 = 4.9550988153084743549606507192748 V_{J_{35}}=4.9550988153084743549606507192748

Elongated_triangular_pyramid.html

  1. V = ( 1 12 ( 2 + 3 3 ) ) a 3 0.550864... a 3 V=(\frac{1}{12}(\sqrt{2}+3\sqrt{3}))a^{3}\approx 0.550864...a^{3}
  2. A = ( 3 + 3 ) a 2 4.73205... a 2 A=(3+\sqrt{3})a^{2}\approx 4.73205...a^{2}

Emission_theory.html

  1. c = c ± k v c^{\prime}=c\pm kv\,
  2. k < 2 × 10 - 3 k<2\times 10^{-3}
  3. k < 2 × 10 - 9 k<2\times 10^{-9}
  4. k = ( - 3 ± 13 ) × 10 - 5 k=(-3\pm 13)\times 10^{-5}
  5. k 10 - 6 k\leq 10^{-6}

Emissivity.html

  1. ε = M e M e , \varepsilon=\frac{M_{\mathrm{e}}}{M_{\mathrm{e}}^{\circ}},
  2. ε ν = M e , ν M e , ν , \varepsilon_{\nu}=\frac{M_{\mathrm{e},\nu}}{M_{\mathrm{e},\nu}^{\circ}},
  3. ε λ = M e , λ M e , λ , \varepsilon_{\lambda}=\frac{M_{\mathrm{e},\lambda}}{M_{\mathrm{e},\lambda}^{% \circ}},
  4. ε Ω = L e , Ω L e , Ω , \varepsilon_{\Omega}=\frac{L_{\mathrm{e},\Omega}}{L_{\mathrm{e},\Omega}^{\circ% }},
  5. ε ν , Ω = L e , Ω , ν L e , Ω , ν , \varepsilon_{\nu,\Omega}=\frac{L_{\mathrm{e},\Omega,\nu}}{L_{\mathrm{e},\Omega% ,\nu}^{\circ}},
  6. ε λ , Ω = L e , Ω , λ L e , Ω , λ , \varepsilon_{\lambda,\Omega}=\frac{L_{\mathrm{e},\Omega,\lambda}}{L_{\mathrm{e% },\Omega,\lambda}^{\circ}},

Empirical_Bayes_method.html

  1. y = { y 1 , y 2 , , y N } y=\{y_{1},y_{2},\dots,y_{N}\}
  2. θ = { θ 1 , θ 2 , , θ n } \theta=\{\theta_{1},\theta_{2},\dots,\theta_{n}\}
  3. p ( y | θ ) p(y|\theta)\,
  4. θ \theta
  5. η \eta\,
  6. p ( θ | η ) p(\theta|\eta)\,
  7. η \eta\,
  8. p ( η ) p(\eta)\,
  9. θ i \theta_{i}\;
  10. θ \theta\;
  11. η \eta\;
  12. p ( θ | y ) = p ( y | θ ) p ( θ ) p ( y ) = p ( y | θ ) p ( y ) p ( θ | η ) p ( η ) d η . p(\theta|y)=\frac{p(y|\theta)p(\theta)}{p(y)}=\frac{p(y|\theta)}{p(y)}\int p(% \theta|\eta)p(\eta)\,d\eta\,.
  13. p ( θ | y ) = p ( θ | η , y ) p ( η | y ) d η = p ( y | θ ) p ( θ | η ) p ( y | η ) p ( η | y ) d η , p(\theta|y)=\int p(\theta|\eta,y)p(\eta|y)\;d\eta=\int\frac{p(y|\theta)p(% \theta|\eta)}{p(y|\eta)}p(\eta|y)\;d\eta\,,
  14. p ( η | y ) = p ( η | θ ) p ( θ | y ) d θ . p(\eta|y)=\int p(\eta|\theta)p(\theta|y)\;d\theta.
  15. p ( θ | y ) p(\theta|y)\;
  16. p ( η | y ) p(\eta|y)\;
  17. p ( θ | y ) p(\theta|y)\;
  18. η \eta
  19. p ( η | y ) p(\eta|y)\;
  20. p ( θ | y ) p(\theta|y)\;
  21. p ( η | y ) p(\eta|y)\;
  22. p ( θ | y ) p(\theta|y)\;
  23. p ( η | y ) p(\eta|y)\;
  24. p ( η | y ) p(\eta|y)\;
  25. p ( θ | y ) p(\theta|y)\;
  26. η \eta\;
  27. η * \eta^{*}\;
  28. p ( θ | y ) p ( y | θ ) p ( θ | η * ) p ( y | η * ) . p(\theta|y)\simeq\frac{p(y|\theta)\;p(\theta|\eta^{*})}{p(y|\eta^{*})}\,.
  29. η \eta\;
  30. η * \eta^{*}\;
  31. p ( θ | y ) p(\theta|y)\;
  32. η * \eta^{*}\;
  33. η \eta
  34. y i y_{i}
  35. θ i \theta_{i}
  36. p ( y i | θ i ) = θ i y i e - θ i y i ! p(y_{i}|\theta_{i})={{\theta_{i}}^{y_{i}}e^{-\theta_{i}}\over{y_{i}}!}
  37. G ( θ ) G(\theta)
  38. θ i \theta_{i}
  39. E ( θ i | y i ) = ( θ y + 1 e - θ / y i ! ) d G ( θ ) ( θ y e - θ / y i ! ) d G ( θ ) . \operatorname{E}(\theta_{i}|y_{i})={\int(\theta^{y+1}e^{-\theta}/{y_{i}}!)\,dG% (\theta)\over{\int(\theta^{y}e^{-\theta}/{y_{i}}!)\,dG(\theta})}.
  40. ( y i + 1 ) / ( y i + 1 ) ({y_{i}}+1)/({y_{i}}+1)
  41. E ( θ i | y i ) = ( y i + 1 ) p G ( y i + 1 ) p G ( y i ) , \operatorname{E}(\theta_{i}|y_{i})={{(y_{i}+1)p_{G}(y_{i}+1)}\over{p_{G}(y_{i}% )}},
  42. E ( θ i | y i ) ( y i + 1 ) # { Y j = y i + 1 } # { Y j = y i } , \operatorname{E}(\theta_{i}|y_{i})\approx(y_{i}+1){{\#\{Y_{j}=y_{i}+1\}}\over{% \#\{Y_{j}=y_{i}\}}},
  43. # \#
  44. m ( y | η ) m(y|\eta)
  45. η \eta
  46. η \eta
  47. θ \theta
  48. θ \theta
  49. G ( α , β ) G(\alpha,\beta)
  50. η = ( α , β ) \eta=(\alpha,\beta)
  51. ρ ( θ | α , β ) = θ α - 1 e - θ / β β α Γ ( α ) for θ > 0 , α > 0 , β > 0 . \rho(\theta|\alpha,\beta)=\frac{\theta^{\alpha-1}\,e^{-\theta/\beta}}{\beta^{% \alpha}\Gamma(\alpha)}\ \mathrm{for}\ \theta>0,\alpha>0,\beta>0\,\!.
  52. ρ ( θ | y ) ρ ( y | θ ) ρ ( θ | α , β ) , \rho(\theta|y)\propto\rho(y|\theta)\rho(\theta|\alpha,\beta),
  53. θ \theta
  54. θ \theta
  55. ρ ( θ | y ) ( θ y e - θ ) ( θ α - 1 e - θ / β ) = θ y + α - 1 e - θ ( 1 + 1 / β ) . \rho(\theta|y)\propto(\theta^{y}\,e^{-\theta})(\theta^{\alpha-1}\,e^{-\theta/% \beta})=\theta^{y+\alpha-1}\,e^{-\theta(1+1/\beta)}.
  56. G ( α , β ) G(\alpha^{\prime},\beta^{\prime})
  57. α = y + α \alpha^{\prime}=y+\alpha
  58. β = ( 1 + 1 / β ) - 1 \beta^{\prime}=(1+1/\beta)^{-1}
  59. Θ \Theta
  60. E ( θ | y ) \operatorname{E}(\theta|y)
  61. μ \mu
  62. G ( α , β ) G(\alpha^{\prime},\beta^{\prime})
  63. α β \alpha^{\prime}\beta^{\prime}
  64. E ( θ | y ) = α β = y ¯ + α 1 + 1 / β = β 1 + β y ¯ + 1 1 + β ( α β ) . \operatorname{E}(\theta|y)=\alpha^{\prime}\beta^{\prime}=\frac{\bar{y}+\alpha}% {1+1/\beta}=\frac{\beta}{1+\beta}\bar{y}+\frac{1}{1+\beta}(\alpha\beta).
  65. α \alpha
  66. β \beta
  67. α β \alpha\beta
  68. α β 2 \alpha\beta^{2}
  69. E ( θ | y ) \operatorname{E}(\theta|y)
  70. y ¯ \bar{y}
  71. μ = α β \mu=\alpha\beta

Encephalization.html

  1. E = C S r E=CS^{r}

Encephalization_quotient.html

  1. E w ( b r a i n ) 1 g = 0.12 ( w ( b o d y ) 1 g ) 2 3 {{Ew(brain)}\over{1g}}=0.12{\left({{w(body)}\over{1g}}\right)^{\frac{2}{3}}}

Endemic_(epidemiology).html

  1. R 0 × S = 1 \ {R_{0}}\times{S}={1}

Endergonic.html

  1. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\cdot\Delta S
  2. Δ R G < 0 \mathrm{\Delta}_{\mathrm{R}}G<0
  3. Δ R G > 0 \mathrm{\Delta}_{\mathrm{R}}G>0
  4. Δ R G \mathrm{\Delta}_{\mathrm{R}}G

Energy–maneuverability_theory.html

  1. P S = V ( T - D W ) V = Velocity T = Thrust D = Drag W = Weight \begin{array}[]{rcl}P_{S}&=&V\left(\frac{T-D}{W}\right)\\ \\ V&=&\,\text{Velocity}\\ T&=&\,\text{Thrust}\\ D&=&\,\text{Drag}\\ W&=&\,\text{Weight}\end{array}
  2. P S P_{S}

Enterprise_application_integration.html

  1. n n
  2. ( n 2 ) = n ( n - 1 ) 2 {\textstyle\left({{n}\atop{2}}\right)}=\frac{n(n-1)}{2}
  3. 10 × 9 2 \frac{10\times 9}{2}

Enthalpy_change_of_solution.html

  1. H d i l = i ν i R T l n γ i ( 1 + T ϵ ϵ T ) H_{dil}=\sum_{i}\nu_{i}RTln\gamma_{i}(1+\frac{T}{\epsilon}\frac{\partial% \epsilon}{\partial T})

Entropic_uncertainty.html

  1. g ( y ) - exp ( - 2 π i x y ) f ( x ) d x , f ( x ) - exp ( 2 π i x y ) g ( y ) d y , g(y)\approx\int_{-\infty}^{\infty}\exp(-2\pi ixy)f(x)\,dx,\qquad f(x)\approx% \int_{-\infty}^{\infty}\exp(2\pi ixy)g(y)\,dy~{},
  2. L L
  3. - | f ( x ) | 2 d x = - | g ( y ) | 2 d y = 1 . \int_{-\infty}^{\infty}|f(x)|^{2}\,dx=\int_{-\infty}^{\infty}|g(y)|^{2}\,dy=1~% {}.
  4. H ( | f | 2 ) + H ( | g | 2 ) - - | f ( x ) | 2 log | f ( x ) | 2 d x - - | g ( y ) | 2 log | g ( y ) | 2 d y 0. H(|f|^{2})+H(|g|^{2})\equiv-\int_{-\infty}^{\infty}|f(x)|^{2}\log|f(x)|^{2}\,% dx-\int_{-\infty}^{\infty}|g(y)|^{2}\log|g(y)|^{2}\,dy\geq 0.
  5. 1 / α + 1 / β = 2 1/α+1/β=2
  6. q , p = sup f L p ( ) f q f p , \|\mathcal{F}\|_{q,p}=\sup_{f\in L^{p}(\mathbb{R})}\frac{\|\mathcal{F}f\|_{q}}% {\|f\|_{p}},
  7. 1 < p 2 , 1<p\leq 2~{},
  8. 1 p + 1 q = 1. \frac{1}{p}+\frac{1}{q}=1.
  9. q , p = p 1 / p / q 1 / q . \|\mathcal{F}\|_{q,p}=\sqrt{p^{1/p}/q^{1/q}}.
  10. f q ( p 1 / p / q 1 / q ) 1 / 2 f p . \|\mathcal{F}f\|_{q}\leq\left(p^{1/p}/q^{1/q}\right)^{1/2}\|f\|_{p}.
  11. g = f g=\mathcal{F}f
  12. 1 / α + 1 / β = 2 1/α+1/β=2
  13. 1 β log ( | g ( y ) | 2 β d y ) 1 2 log ( 2 α ) 1 / α ( 2 β ) 1 / β + 1 α log ( | f ( x ) | 2 α d x ) . \frac{1}{\beta}\log\left(\int_{\mathbb{R}}|g(y)|^{2\beta}\,dy\right)\leq\frac{% 1}{2}\log\frac{(2\alpha)^{1/\alpha}}{(2\beta)^{1/\beta}}+\frac{1}{\alpha}\log% \left(\int_{\mathbb{R}}|f(x)|^{2\alpha}\,dx\right).
  14. β 1 - β = - α 1 - α \frac{\beta}{1-\beta}=-\frac{\alpha}{1-\alpha}
  15. 1 1 - β log ( | g ( y ) | 2 β d y ) α 2 ( α - 1 ) log ( 2 α ) 1 / α ( 2 β ) 1 / β - 1 1 - α log ( | f ( x ) | 2 α d x ) . \frac{1}{1-\beta}\log\left(\int_{\mathbb{R}}|g(y)|^{2\beta}\,dy\right)\geq% \frac{\alpha}{2(\alpha-1)}\log\frac{(2\alpha)^{1/\alpha}}{(2\beta)^{1/\beta}}-% \frac{1}{1-\alpha}\log\left(\int_{\mathbb{R}}|f(x)|^{2\alpha}\,dx\right)~{}.
  16. 1 1 - α log ( | f ( x ) | 2 α d x ) + 1 1 - β log ( | g ( y ) | 2 β d y ) α 2 ( α - 1 ) log ( 2 α ) 1 / α ( 2 β ) 1 / β ; \frac{1}{1-\alpha}\log\left(\int_{\mathbb{R}}|f(x)|^{2\alpha}\,dx\right)+\frac% {1}{1-\beta}\log\left(\int_{\mathbb{R}}|g(y)|^{2\beta}\,dy\right)\geq\frac{% \alpha}{2(\alpha-1)}\log\frac{(2\alpha)^{1/\alpha}}{(2\beta)^{1/\beta}};
  17. H α ( | f | 2 ) + H β ( | g | 2 ) 1 2 ( log α α - 1 + log β β - 1 ) - log 2 . H_{\alpha}(|f|^{2})+H_{\beta}(|g|^{2})\geq\frac{1}{2}\left(\frac{\log\alpha}{% \alpha-1}+\frac{\log\beta}{\beta-1}\right)-\log 2~{}.
  18. α α
  19. β β
  20. H ( | f | 2 ) + H ( | g | 2 ) log e 2 , where g ( y ) e - 2 π i x y f ( x ) d x , H(|f|^{2})+H(|g|^{2})\geq\log\frac{e}{2},\quad\textrm{where}\quad g(y)\approx% \int_{\mathbb{R}}e^{-2\pi ixy}f(x)\,dx~{},
  21. H ( | f | 2 ) + H ( | g | 2 ) log ( π e ) for g ( y ) 1 2 π e - i x y f ( x ) d x . H(|f|^{2})+H(|g|^{2})\geq\log(\pi e)\quad\textrm{for}\quad g(y)\approx\frac{1}% {\sqrt{2\pi}}\int_{\mathbb{R}}e^{-ixy}f(x)\,dx~{}.
  22. π π
  23. π π
  24. H ( ϕ ) log 2 π e V ( ϕ ) , H(\phi)\leq\log\sqrt{2\pi eV(\phi)},
  25. 1 / 2 exp ( H ( | f | 2 ) + H ( | g | 2 ) ) / ( 2 e π ) V ( | f | 2 ) V ( | g | 2 ) . 1/2\leq\exp(H(|f|^{2})+H(|g|^{2}))/(2e\pi)\leq\sqrt{V(|f|^{2})V(|g|^{2})}~{}.
  26. δ > 0 , μ { x | ϕ 1 ( x ) δ } = μ { x | ϕ 2 ( x ) δ } , \forall\delta>0,\,\mu\{x\in\mathbb{R}|\phi_{1}(x)\geq\delta\}=\mu\{x\in\mathbb% {R}|\phi_{2}(x)\geq\delta\},
  27. μ μ

Entropy_of_fusion.html

  1. Δ G f u s = Δ H f u s - T × Δ S f u s < 0 \Delta G_{fus}=\Delta H_{fus}-T\times\Delta S_{fus}<0
  2. Δ H f u s \Delta H_{fus}
  3. T = T f T=T_{f}
  4. Δ G f u s = Δ H f u s - T f × Δ S f u s = 0 \Delta G_{fus}=\Delta H_{fus}-T_{f}\times\Delta S_{fus}=0
  5. Δ S f u s = Δ H f u s T f \Delta S_{fus}=\frac{\Delta H_{fus}}{T_{f}}

Envelope_(mathematics).html

  1. F ( t , x , y ) = F t ( t , x , y ) = 0 F(t,x,y)={\partial F\over\partial t}(t,x,y)=0
  2. F / t \partial F/\partial t
  3. F ( t , x , y ) = F ( u , x , y ) = 0 F(t,x,y)=F(u,x,y)=0\,
  4. F ( t , x , y ) = F ( u , x , y ) - F ( t , x , y ) u - t = 0. F(t,x,y)=\frac{F(u,x,y)-F(t,x,y)}{u-t}=0.
  5. x t + y 1 - t = 1 \frac{x}{t}+\frac{y}{1-t}=1
  6. x ( 1 - t ) + y t - t ( 1 - t ) = t 2 + ( - x + y - 1 ) t + x = 0. x(1-t)+yt-t(1-t)=t^{2}+(-x+y-1)t+x=0.\,
  7. ( - x + y - 1 ) 2 - 4 x = ( x - y ) 2 - 2 ( x + y ) + 1 = 0. (-x+y-1)^{2}-4x=(x-y)^{2}-2(x+y)+1=0.\,
  8. u 1 2 ( t + 1 t ) + v 1 2 i ( t - 1 t ) = w u{1\over 2}(t+{1\over t})+v{1\over 2i}(t-{1\over t})=w
  9. ( u - i v ) t 2 - 2 w t + ( u + i v ) = 0. (u-iv)t^{2}-2wt+(u+iv)=0.\,
  10. ( u - i v ) ( u + i v ) - w 2 = 0 (u-iv)(u+iv)-w^{2}=0\,
  11. u 2 + v 2 = w 2 . u^{2}+v^{2}=w^{2}.\,
  12. E 1 𝒟 E_{1}\subseteq\mathcal{D}
  13. E 2 𝒟 E_{2}\subseteq\mathcal{D}
  14. E 3 𝒟 E_{3}\subseteq\mathcal{D}
  15. 𝒟 \mathcal{D}
  16. 𝒟 \mathcal{D}
  17. F ( t , ( x , y ) ) = 3 t 2 x - y - 2 t 3 . F(t,(x,y))=3t^{2}x-y-2t^{3}.
  18. F ( t , ( t , y ) ) = t 3 - y F(t,(t,y))=t^{3}-y\,
  19. 𝒟 = { ( x , y ) \R 2 : y = x 3 } { ( x , y ) \R 2 : y = 0 } . \mathcal{D}=\{(x,y)\in\R^{2}:y=x^{3}\}\cup\{(x,y)\in\R^{2}:y=0\}\ .
  20. L := F ( t , ( x , y ) ) - F ( t + ε , ( x , y ) ) = 2 ε 3 + 6 ε t 2 + 6 ε 2 t - ( 3 ε 2 + 6 ε t ) x = 0. L:=F(t,(x,y))-F(t+\varepsilon,(x,y))=2\varepsilon^{3}+6\varepsilon t^{2}+6% \varepsilon^{2}t-(3\varepsilon^{2}+6\varepsilon t)x=0.
  21. lim ε 0 1 ε L = 6 t ( t - x ) . \lim_{\varepsilon\to 0}\frac{1}{\varepsilon}L=6t(t-x)\ .
  22. lim ε 0 1 ε 2 L = 3 x . \lim_{\varepsilon\to 0}\frac{1}{\varepsilon^{2}}L=3x\ .
  23. E 1 = { ( x , y ) \R 2 : y = x 3 } . E_{1}=\{(x,y)\in\R^{2}:y=x^{3}\}\ .
  24. E 2 = { ( x , y ) \R 2 : y = x 3 } . E_{2}=\{(x,y)\in\R^{2}:y=x^{3}\}\ .
  25. F ( t , ( x 0 , y 0 ) ) = 3 t 2 x 0 - y 0 - 2 t 3 = 0 . F(t,(x_{0},y_{0}))=3t^{2}x_{0}-y_{0}-2t^{3}=0\ .
  26. E 3 = . E_{3}=\varnothing.
  27. F ( x , y , t ) = t 2 + t ( y - x - k ) + k x = 0 F(x,y,t)=t^{2}+t(y-x-k)+kx=0\,
  28. F ( x , y , t ) t = 2 t + y - x - k = 0 \frac{\partial F(x,y,t)}{\partial t}=2t+y-x-k=0\,
  29. x 2 - 2 x y + y 2 - 2 k x - 2 k y + k 2 = 0 x^{2}-2xy+y^{2}-2kx-2ky+k^{2}=0\,
  30. F ( t , x ) = ( x - γ ( t ) ) T ( t ) . F(t,{x})=({x}-\gamma(t))\cdot{T}(t)\ .
  31. L t 0 := { x \R 2 : F ( t 0 , x ) = 0 } L_{t_{0}}:=\{{x}\in\R^{2}:F(t_{0},{x})=0\}
  32. F t ( t , x ) = κ ( t ) ( x - γ ( t ) ) N ( t ) - 1 , \frac{\partial F}{\partial t}(t,{x})=\kappa(t)({x}-\gamma(t))\cdot{N}(t)-1\ ,
  33. F t = λ κ ( t ) - 1 . \frac{\partial F}{\partial t}=\lambda\kappa(t)-1\ .
  34. 𝒟 = γ ( t ) + 1 κ ( t ) N ( t ) . \mathcal{D}=\gamma(t)+\frac{1}{\kappa(t)}{N}(t)\ .
  35. s > 0 s>0
  36. t > 0 t>0
  37. ( 0 , 0 ) (0,0)
  38. ( s , 0 ) (s,0)
  39. ( 0 , t ) (0,t)
  40. T s , t := { ( x , y ) \R + 2 : x s + y t < 1 } . T_{s,t}:=\left\{(x,y)\in\R_{+}^{2}:\ \frac{x}{s}+\frac{y}{t}<1\right\}.
  41. α > 0 \alpha>0
  42. T s , t T_{s,t}
  43. s α + t α = 1 \textstyle s^{\alpha}+t^{\alpha}=1
  44. Δ α := s α + t α = 1 T s , t . \Delta_{\alpha}:=\bigcup_{s^{\alpha}+t^{\alpha}=1}T_{s,t}.
  45. Δ α \textstyle\Delta_{\alpha}
  46. s > 0 \textstyle s>0
  47. t > 0 \textstyle t>0
  48. s α + t α = 1 \textstyle s^{\alpha}+t^{\alpha}=1
  49. ( x , y ) \R + 2 \textstyle(x,y)\in\R_{+}^{2}
  50. \R 2 \textstyle\R^{2}
  51. p := 1 + 1 α p:=1+\frac{1}{\alpha}
  52. q := 1 + α \textstyle q:={1+\alpha}
  53. x α α + 1 + y α α + 1 ( x s + y t ) α α + 1 ( s α + t α ) 1 α + 1 = ( x s + y t ) α α + 1 x^{\frac{\alpha}{\alpha+1}}+y^{\frac{\alpha}{\alpha+1}}\leq\left(\frac{x}{s}+% \frac{y}{t}\right)^{\frac{\alpha}{\alpha+1}}\Big(s^{\alpha}+t^{\alpha}\Big)^{% \frac{1}{\alpha+1}}=\left(\frac{x}{s}+\frac{y}{t}\right)^{\frac{\alpha}{\alpha% +1}}
  54. s : t = x 1 1 + α : y 1 1 + α \textstyle s:\,t=x^{\frac{1}{1+\alpha}}:\,y^{\frac{1}{1+\alpha}}
  55. ( x , y ) \R + 2 (x,y)\in\R_{+}^{2}
  56. Δ α \textstyle\Delta_{\alpha}
  57. T s , t \textstyle T_{s,t}
  58. s α + t α = 1 \textstyle s^{\alpha}+t^{\alpha}=1
  59. x α α + 1 + y α α + 1 < 1. x^{\frac{\alpha}{\alpha+1}}+y^{\frac{\alpha}{\alpha+1}}<1.
  60. \R + 2 \R_{+}^{2}
  61. Δ α \textstyle\Delta_{\alpha}
  62. { ( x , y ) \R + 2 : x s + y t = 1 } , s α + t α = 1 \left\{(x,y)\in\R_{+}^{2}:\ \frac{x}{s}+\frac{y}{t}=1\right\}\ ,\qquad s^{% \alpha}+t^{\alpha}=1
  63. x α α + 1 + y α α + 1 = 1. x^{\frac{\alpha}{\alpha+1}}+y^{\frac{\alpha}{\alpha+1}}=1.
  64. α = 1 \alpha=1
  65. α = 2 \alpha=2
  66. d 2 y d t 2 = - g , d 2 x d t 2 = 0 , \frac{d^{2}y}{dt^{2}}=-g,\;\frac{d^{2}x}{dt^{2}}=0,
  67. d x d t | t = 0 = v cos θ , d y d t | t = 0 = v sin θ , x | t = 0 = y | t = 0 = 0. \frac{dx}{dt}\bigg|_{t=0}=v\cos\theta,\;\frac{dy}{dt}\bigg|_{t=0}=v\sin\theta,% \;x\bigg|_{t=0}=y\bigg|_{t=0}=0.
  68. F ( x , y , θ ) = x tan θ - g x 2 2 v 2 cos 2 θ - y = 0. F(x,y,\theta)=x\tan\theta-\frac{gx^{2}}{2v^{2}\cos^{2}\theta}-y=0.
  69. F θ = x cos 2 θ - g x 2 tan θ v 2 cos 2 θ = 0. \frac{\partial F}{\partial\theta}=\frac{x}{\cos^{2}\theta}-\frac{gx^{2}\tan% \theta}{v^{2}\cos^{2}\theta}=0.
  70. y = v 2 2 g - g 2 v 2 x 2 . y=\frac{v^{2}}{2g}-\frac{g}{2v^{2}}x^{2}.
  71. F ( x , y , z , a ) = 0 F(x,y,z,a)=0
  72. F ( x , y , z , a ) = 0 , F ( x , y , z , a ) - F ( x , y , z , a ) a - a = 0. F(x,y,z,a)=0,\,\,{F(x,y,z,a^{\prime})-F(x,y,z,a)\over a^{\prime}-a}=0.
  73. F ( x , y , z , a ) = 0 , F a ( x , y , z , a ) = 0. F(x,y,z,a)=0,\,\,{\partial F\over\partial a}(x,y,z,a)=0.
  74. t 2 - 2 t x + y ( x ) = 0. t^{2}-2tx+y(x)=0.
  75. t = ( d y d x ) / 2 t=\left(\frac{dy}{dx}\right)/2
  76. ( d y d x ) 2 - 4 x d y d x + 4 y = 0. \left(\frac{dy}{dx}\right)^{2}\!\!-4x\frac{dy}{dx}+4y=0.
  77. D a u ( x ; a ) = 0 D_{a}u(x;a)=0\,
  78. v ( x ) = u ( x ; φ ( x ) ) , x Ω , v(x)=u(x;\varphi(x)),\quad x\in\Omega,
  79. L [ γ ] = a b | γ ( t ) | d t L[\gamma]=\int_{a}^{b}|\gamma^{\prime}(t)|\,dt

EP.html

  1. E p E_{p}
  2. E p E_{p}

Epicycloid.html

  1. x ( θ ) = ( R + r ) cos θ - r cos ( R + r r θ ) x(\theta)=(R+r)\cos\theta-r\cos\left(\frac{R+r}{r}\theta\right)
  2. y ( θ ) = ( R + r ) sin θ - r sin ( R + r r θ ) , y(\theta)=(R+r)\sin\theta-r\sin\left(\frac{R+r}{r}\theta\right),
  3. x ( θ ) = r ( k + 1 ) cos θ - r cos ( ( k + 1 ) θ ) x(\theta)=r(k+1)\cos\theta-r\cos\left((k+1)\theta\right)\,
  4. y ( θ ) = r ( k + 1 ) sin θ - r sin ( ( k + 1 ) θ ) . y(\theta)=r(k+1)\sin\theta-r\sin\left((k+1)\theta\right).\,
  5. p p
  6. α \alpha
  7. p p
  8. θ \theta
  9. R = r \ell_{R}=\ell_{r}
  10. R = θ R , r = α r \ell_{R}=\theta R,\ell_{r}=\alpha r
  11. θ R = α r \theta R=\alpha r
  12. α \alpha
  13. θ \theta
  14. α = R r θ \alpha=\frac{R}{r}\theta
  15. p p
  16. x = ( R + r ) cos θ - r cos ( θ + α ) = ( R + r ) cos θ - r cos ( R + r r θ ) x=\left(R+r\right)\cos\theta-r\cos\left(\theta+\alpha\right)=\left(R+r\right)% \cos\theta-r\cos\left(\frac{R+r}{r}\theta\right)
  17. y = ( R + r ) sin θ - r sin ( θ + α ) = ( R + r ) sin θ - r sin ( R + r r θ ) y=\left(R+r\right)\sin\theta-r\sin\left(\theta+\alpha\right)=\left(R+r\right)% \sin\theta-r\sin\left(\frac{R+r}{r}\theta\right)

Eqn.html

  1. a 2 a^{2}
  2. k = 1 N k 2 \sum_{k=1}^{N}k^{2}
  3. x = - b ± b 2 - 4 a c 2 a x={-b\pm\sqrt{b^{2}-4ac}\over 2a}
  4. f ( p i r 2 ) \scriptstyle{f(pir^{2)}}
  5. f ( π r 2 ) \scriptstyle{f(\pi r^{2})}

Equally_spaced_polynomial.html

  1. E S P ( x ) = i = 0 m x s i ESP(x)=\sum_{i=0}^{m}x^{si}
  2. i = 0 , 1 , , m i=0,1,\ldots,m
  3. E S P ( x ) = x s m + x s ( m - 1 ) + + x s + 1. ESP(x)=x^{sm}+x^{s(m-1)}+\cdots+x^{s}+1.

Equant.html

  1. α ( t ) = Ω t - arcsin ( E R sin ( Ω t ) ) \alpha(t)=\Omega t-\arcsin\left(\frac{E}{R}\sin(\Omega t)\right)

Equidistributed_sequence.html

  1. lim n | { s 1 , , s n } [ c , d ] | n = d - c b - a . \lim_{n\to\infty}{\left|\{\,s_{1},\dots,s_{n}\,\}\cap[c,d]\right|\over n}={d-c% \over b-a}.\,
  2. D N = sup a c d b | | { s 1 , , s N } [ c , d ] | N - d - c b - a | . D_{N}=\sup_{a\leq c\leq d\leq b}\left|\frac{\left|\{\,s_{1},\dots,s_{N}\,\}% \cap[c,d]\right|}{N}-\frac{d-c}{b-a}\right|.\,
  3. lim N 1 N n = 1 N f ( s n ) = 1 b - a a b f ( x ) d x \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f\left(s_{n}\right)=\frac{1}{b-a}% \int_{a}^{b}f(x)\,dx
  4. d - c b - a . \textstyle\frac{d-c}{b-a}.
  5. a b ( f 2 ( x ) - f 1 ( x ) ) d x ε . \textstyle\int_{a}^{b}(f_{2}(x)-f_{1}(x))\,dx\leq\varepsilon.
  6. a b f 1 ( x ) d x = lim N 1 N n = 1 N f 1 ( s n ) lim inf N 1 N n = 1 N f ( s n ) \int_{a}^{b}f_{1}(x)\,dx=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1}(s_{n}% )\leq\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(s_{n})
  7. a b f 2 ( x ) d x = lim N 1 N n = 1 N f 2 ( s n ) lim sup N 1 N n = 1 N f ( s n ) \int_{a}^{b}f_{2}(x)\,dx=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{2}(s_{n}% )\geq\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(s_{n})
  8. 1 N n = 1 N f ( s n ) \textstyle\frac{1}{N}\sum_{n=1}^{N}f(s_{n})
  9. lim n 1 n j = 1 n e 2 π i a j = 0. \lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}e^{2\pi i\ell a_{j}}=0.
  10. f ( x ) = e 2 π i x , \textstyle f(x)=e^{2\pi i\ell x},
  11. lim n 1 n j = 0 n - 1 e 2 π i v j = 0. \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}e^{2\pi i\ell\cdot v_{j}}=0.
  12. e 2 π i α \textstyle e^{2\pi i\ell\alpha}
  13. | j = 0 n - 1 e 2 π i j α | = | j = 0 n - 1 ( e 2 π i α ) j | = | 1 - e 2 π i n α 1 - e 2 π i α | 2 | 1 - e 2 π i α | , \left|\sum_{j=0}^{n-1}e^{2\pi i\ell j\alpha}\right|=\left|\sum_{j=0}^{n-1}% \left(e^{2\pi i\ell\alpha}\right)^{j}\right|=\left|\frac{1-e^{2\pi i\ell n% \alpha}}{1-e^{2\pi i\ell\alpha}}\right|\leq\frac{2}{\left|1-e^{2\pi i\ell% \alpha}\right|},
  14. π \pi
  15. lim n | { s k + 1 , , s k + n } [ c , d ] | n = d - c b - a \lim_{n\to\infty}{\left|\{\,s_{k+1},\dots,s_{k+n}\,\}\cap[c,d]\right|\over n}=% {d-c\over b-a}\,
  16. ( X , μ ) (X,\mu)
  17. x n x_{n}
  18. μ \mu
  19. μ \mu
  20. k = 1 n δ x k n μ . \frac{\sum_{k=1}^{n}\delta_{x_{k}}}{n}\Rightarrow\mu\ .

Equilibrium_constant.html

  1. α A + β B ρ R + σ S \alpha A+\beta B...\rightleftharpoons\rho R+\sigma S...
  2. K = { R } ρ { S } σ { A } α { B } β K^{\ominus}=\frac{{\{R\}}^{\rho}{\{S\}}^{\sigma}...}{{\{A\}}^{\alpha}{\{B\}}^{% \beta}...}
  3. Δ G = - R T ln K \Delta G^{\ominus}=-RT\ln K^{\ominus}
  4. K c = [ R ] ρ [ S ] σ [ A ] α [ B ] β K\text{c}=\frac{{[R]}^{\rho}{[S]}^{\sigma}...}{{[A]}^{\alpha}{[B]}^{\beta}...}
  5. M + 2 L M L 2 ; [ M L 2 ] = β 12 [ M ] [ L ] 2 M+2L\rightleftharpoons ML_{2};\quad[ML_{2}]=\beta_{12}[M][L]^{2}
  6. M L + L M L 2 ; [ M L 2 ] = K [ M L ] [ L ] = K β 11 [ M ] [ L ] [ L ] = K β 11 [ M ] [ L ] 2 ML+L\rightleftharpoons ML_{2};\quad[ML_{2}]=K[ML][L]=K\beta_{11}[M][L][L]=K% \beta_{11}[M][L]^{2}
  7. β 12 = K β 11 \beta_{12}=K\beta_{11}\,
  8. M L + L M L + L ML+L^{\prime}\rightleftharpoons ML^{\prime}+L
  9. [ M L ] = K [ M L ] [ L ] [ L ] = K β M L [ M ] [ L ] [ L ] [ L ] = K β M L [ M ] [ L ] ; β M L = K β M L [ML^{\prime}]=K\frac{[ML][L^{\prime}]}{[L]}=K\frac{\beta_{ML}[M][L][L^{\prime}% ]}{[L]}=K\beta_{ML}[M][L^{\prime}];\quad\beta_{ML^{\prime}}=K\beta_{ML}
  10. p K a = - log K diss = log ( 1 / K diss ) pK\text{a}=-\log K_{\,\text{diss}}=\log(1/K_{\,\text{diss}})\,
  11. M + H L M L + H M+HL\rightleftharpoons ML+H
  12. H 3 L H 2 L + H ; p K 1 = - log ( [ H 2 L ] [ H ] [ H 3 L ] ) H_{3}L\rightleftharpoons H_{2}L+H;\quad pK_{1}=-\log\left(\frac{[H_{2}L][H]}{[% H_{3}L]}\right)
  13. H 2 L H L + H ; p K 2 = - log ( [ H L ] [ H ] [ H 2 L ] ) H_{2}L\rightleftharpoons HL+H;\quad pK_{2}=-\log\left(\frac{[HL][H]}{[H_{2}L]}\right)
  14. H L L + H ; p K 3 = - log ( [ L ] [ H ] [ H L ] ) HL\rightleftharpoons L+H;\quad pK_{3}=-\log\left(\frac{[L][H]}{[HL]}\right)
  15. L + H H L ; log β 011 = log ( [ H L ] [ L ] [ H ] ) = p K 3 L+H\rightleftharpoons HL;\quad\log\beta_{011}=\log\left(\frac{[HL]}{[L][H]}% \right)=pK_{3}
  16. L + 2 H H 2 L ; log β 012 = log ( [ H 2 L ] [ L ] [ H ] 2 ) = p K 3 + p K 2 L+2H\rightleftharpoons H_{2}L;\quad\log\beta_{012}=\log\left(\frac{[H_{2}L]}{[% L][H]^{2}}\right)=pK_{3}+pK_{2}
  17. L + 3 H H 3 L ; log β 013 = log ( [ H 3 L ] [ L ] [ H ] 3 ) = p K 3 + p K 2 + p K 1 L+3H\rightleftharpoons H_{3}L;\quad\log\beta_{013}=\log\left(\frac{[H_{3}L]}{[% L][H]^{3}}\right)=pK_{3}+pK_{2}+pK_{1}
  18. M + L M L ; log β 110 = log ( [ M L ] [ M ] [ L ] ) M+L\rightleftharpoons ML;\quad\log\beta_{110}=\log\left(\frac{[ML]}{[M][L]}\right)
  19. M + L + H M L H ; log β 111 = log ( [ M L H ] [ M ] [ L ] [ H ] ) M+L+H\rightleftharpoons MLH;\quad\log\beta_{111}=\log\left(\frac{[MLH]}{[M][L]% [H]}\right)
  20. p H = - log { H + } pH=-\log\{H^{+}\}\,
  21. H L L + H ; p K = - log ( [ L ] { H } [ H L ] ) HL\rightleftharpoons L+H;\quad pK=-\log\left(\frac{[L]\{H\}}{[HL]}\right)
  22. K W = [ H ] [ OH ] K\text{W}=[H][\,\text{OH}]
  23. [ OH ] = K W [ H ] - 1 [\,\text{OH}]=K\text{W}[\,\text{H}]^{-1}\,
  24. M ( H 2 O ) M ( OH ) + H ; [ M ( OH ) ] = β * [ M ] [ H ] - 1 M(\,\text{H}_{2}\,\text{O})\rightleftharpoons M(\,\text{OH})+H;\quad[M(\,\text% {OH})]=\beta^{*}[M][\,\text{H}]^{-1}
  25. M + OH M ( OH ) ; [ M ( OH ) ] = K [ M ] [ OH ] = K K W [ M ] [ H ] - 1 M+\,\text{OH}\rightleftharpoons M(\,\text{OH});\quad[M(\,\text{OH})]=K[M][\,% \text{OH}]=KK\text{W}[M][\,\text{H}]^{-1}
  26. K cond = [ Total Fe bound to EDTA ] [ Total Fe not bound to EDTA ] × [ Total EDTA not bound to Fe ] K_{\,\text{cond}}=\frac{[\mbox{Total Fe bound to EDTA}~{}]}{[\mbox{Total Fe % not bound to EDTA}~{}]\times[\mbox{Total EDTA not bound to Fe}~{}]}
  27. f N 2 O 4 / p = K ( f NO 2 / p ) 2 f_{\,\text{N}_{2}\,\text{O}_{4}}/p^{\ominus}=K\left(f_{\,\text{NO}_{2}}/{p^{% \ominus}}\right)^{2}
  28. K = ϕ N 2 O 4 p N 2 O 4 / p ( ϕ NO 2 p NO 2 / p ) 2 K=\frac{\phi_{\,\text{N}_{2}\,\text{O}_{4}}p_{\,\text{N}_{2}\,\text{O}_{4}}/{p% ^{\ominus}}}{\left(\phi_{\,\text{NO}_{2}}p_{\,\text{NO}_{2}}/{p^{\ominus}}% \right)^{2}}
  29. K = ϕ N 2 O 4 p N 2 O 4 ( ϕ NO 2 p NO 2 ) 2 K=\frac{\phi_{\,\text{N}_{2}\,\text{O}_{4}}p_{\,\text{N}_{2}\,\text{O}_{4}}}{% \left(\phi_{\,\text{NO}_{2}}p_{\,\text{NO}_{2}}\right)^{2}}
  30. α A + β B ρ R + σ S \alpha A+\beta B...\rightleftharpoons\rho R+\sigma S...
  31. K = { R } ρ { S } σ { A } α { B } β K^{\ominus}=\frac{{\{R\}}^{\rho}{\{S\}}^{\sigma}...}{{\{A\}}^{\alpha}{\{B\}}^{% \beta}...}
  32. Δ G = - R T ln K \Delta G^{\ominus}=-RT\ln K^{\ominus}
  33. K = [ R ] ρ [ S ] σ [ A ] α [ B ] β × γ R ρ γ S σ γ A α γ B β × ( c A ) α ( c B ) β ( c R ) ρ ( c S ) σ = Q E Γ C 0 K^{\ominus}=\frac{[R]^{\rho}[S]^{\sigma}...}{[A]^{\alpha}[B]^{\beta}...}\times% \frac{{\gamma_{R}}^{\rho}{\gamma_{S}}^{\sigma}...}{{\gamma_{A}}^{\alpha}{% \gamma_{B}}^{\beta}...}\times\frac{\left({c^{\ominus}_{A}}\right)^{\alpha}% \left({c^{\ominus}_{B}}\right)^{\beta}...}{\left({c^{\ominus}_{R}}\right)^{% \rho}\left({c^{\ominus}_{S}}\right)^{\sigma}...}=Q^{E}\Gamma C^{0}
  34. K = Q E × Γ × 1 = Q E Γ K^{\ominus}=Q^{E}\times\Gamma\times 1=Q^{E}\Gamma
  35. K c = [ R ] ρ [ S ] σ [ A ] α [ B ] β × C 0 = K Γ K\text{c}=\frac{[R]^{\rho}[S]^{\sigma}...}{[A]^{\alpha}[B]^{\beta}...}\times C% ^{0}=\frac{K^{\ominus}}{\Gamma}
  36. A + B A B ; K c = [ A B ] [ A ] [ B ] / M - 1 A+B\rightleftharpoons AB;\quad K\text{c}=\frac{[AB]}{[A][B]}/\,\text{M}^{-1}
  37. A + 2 B A B 2 ; K c = [ A B 2 ] [ A ] [ B ] 2 / M - 2 A+2B\rightleftharpoons AB_{2};\quad K\text{c}=\frac{[AB_{2}]}{[A][B]^{2}}/\,% \text{M}^{-2}
  38. T A = [ A ] + K [ A ] [ B ] T B = [ B ] + K [ A ] [ B ] \begin{aligned}\displaystyle T_{A}&\displaystyle=[A]+K[A][B]\\ \displaystyle T_{B}&\displaystyle=[B]+K[A][B]\end{aligned}
  39. δ G r = ( G ξ ) T , P ; δ G r ( Eq ) = 0 \delta G\text{r}=\left(\frac{\partial G}{\partial\xi}\right)_{T,P};\quad\delta G% _{r}(\,\text{Eq})=0
  40. μ i = ( G N i ) T , P \mu_{i}=\left(\frac{\partial G}{\partial N_{i}}\right)_{T,P}
  41. j n j Reactant j k m k Product k α A + β B ρ R + σ S \sum_{j}n_{j}\,\text{Reactant}_{j}\rightleftharpoons\sum_{k}m_{k}\,\text{% Product}_{k}\quad\equiv\quad\alpha A+\beta B...\rightleftharpoons\rho R+\sigma S% ...
  42. δ G r = k m k μ k - j n j μ j \delta G_{r}=\sum_{k}m_{k}\mu_{k}\,-\sum_{j}n_{j}\mu_{j}
  43. μ i = μ i + R T ln a i \mu_{i}=\mu_{i}^{\ominus}+RT\ln a_{i}
  44. j n j ( μ j + R T ln a j ) = k m k ( μ k + R T ln a k ) \sum_{j}n_{j}(\mu_{j}^{\ominus}+RT\ln a_{j})=\sum_{k}m_{k}(\mu_{k}^{\ominus}+% RT\ln a_{k})
  45. k m k μ k - j n j μ j = - R T ( k ln a k m k - j ln a j n j ) \sum_{k}m_{k}\mu_{k}^{\ominus}-\sum_{j}n_{j}\mu_{j}^{\ominus}=-RT\left(\sum_{k% }\ln{a_{k}}^{m_{k}}-\sum_{j}\ln{a_{j}}^{n_{j}}\right)
  46. Δ G = - R T ln K . \Delta G^{\ominus}=-RT\ln K.
  47. Δ G = k m k μ k - j n j μ j \Delta G^{\ominus}=\sum_{k}m_{k}\mu_{k}^{\ominus}-\sum_{j}n_{j}\mu_{j}^{\ominus}
  48. ln K = k ln a k m k - j ln a j n j ; K = k a k m k j a j n j { R } ρ { S } σ { A } α { B } β \ln K=\sum_{k}\ln{a_{k}}^{m_{k}}-\sum_{j}\ln{a_{j}}^{n_{j}};K=\frac{\prod_{k}{% a_{k}}^{m_{k}}}{\prod_{j}{a_{j}}^{n_{j}}}\equiv\frac{{\{R\}}^{\rho}{\{S\}}^{% \sigma}...}{{\{A\}}^{\alpha}{\{B\}}^{\beta}...}
  49. d ln K d T = Δ H R T 2 or d ln K d ( 1 / T ) = - Δ H R \frac{d\ln K}{dT}\ =\frac{\Delta H^{\ominus}}{RT^{2}}\mbox{ or }~{}\frac{d\ln K% }{d(1/T)}\ =-\frac{\Delta H^{\ominus}}{R}
  50. ( H T ) p = C p \left(\frac{\partial H}{\partial T}\right)_{p}=C_{p}
  51. Δ G = - R T ln K \ \Delta G^{\ominus}=-RT\ln K
  52. ln K = - Δ r G R T \ln K={{-\Delta_{\mathrm{r}}G^{\ominus}}\over{RT}}
  53. G T 2 = G T 1 - S T 1 ( T 2 - T 1 ) - T 2 T 1 T 2 C P T d T + T 1 T 2 C p d T G_{T_{2}}^{\ominus}=G_{T_{1}}^{\ominus}-S_{T_{1}}^{\ominus}(T_{2}-T_{1})-T_{2}% \int^{T_{2}}_{T_{1}}{{C_{P}^{\ominus}}\over{T}}\,dT+\int^{T_{2}}_{T_{1}}C_{p}^% {\ominus}\,dT
  54. C p = A + B T + C T - 2 C_{p}^{\ominus}=A+BT+CT^{-2}
  55. C p = ( 4.186 a + b S ˘ T 1 ) ( T 2 - T 1 ) ln ( T 2 / T 1 ) C_{p}^{\ominus}=(4.186a+b\breve{S}^{\ominus}_{T_{1}}){{(T_{2}-T_{1})}\over{\ln% (T_{2}/T_{1})}}
  56. G T 2 = G T 1 + ( C p - S T 1 ) ( T 2 - T 1 ) - T 2 ln ( T 2 / T 1 ) C p G_{T_{2}}^{\ominus}=G_{T_{1}}^{\ominus}+(C_{p}^{\ominus}-S_{T_{1}}^{\ominus})(% T_{2}-T_{1})-T_{2}\ln{({T_{2}}/{T_{1}})}C_{p}^{\ominus}
  57. S ˘ \breve{S}
  58. α A + β B σ S + τ T \alpha A+\beta B\rightleftharpoons\sigma S+\tau T\,
  59. K p = p S σ p T τ p A α p B β = X S σ X T τ X A α X B β P σ + τ - α - β = K X P σ + τ - α - β K_{p}=\frac{{p_{S}}^{\sigma}{p_{T}}^{\tau}}{{p_{A}}^{\alpha}{p_{B}}^{\beta}}=% \frac{{X_{S}}^{\sigma}{X_{T}}^{\tau}}{{X_{A}}^{\alpha}{X_{B}}^{\beta}}P^{% \sigma+\tau-\alpha-\beta}=K_{X}P^{\sigma+\tau-\alpha-\beta}
  60. α A + β B σ S + τ T \alpha A+\beta B\rightleftharpoons\sigma S+\tau T
  61. Δ V ¯ = σ V ¯ S + τ V ¯ T - α V ¯ A - β V ¯ B \Delta\bar{V}=\sigma\bar{V}_{S}+\tau\bar{V}_{T}-\alpha\bar{V}_{A}-\beta\bar{V}% _{B}
  62. V ¯ \overline{V}
  63. ( ln K X P ) T = - Δ V ¯ R T . \left(\frac{\partial\ln K_{X}}{\partial P}\right)_{T}=\frac{-\Delta\bar{V}}{RT}.

Equity_premium_puzzle.html

  1. E 0 [ t = 0 β t U ( c t ) ] E_{0}\left[\sum_{t=0}^{\infty}\beta^{t}U(c_{t})\right]
  2. U ( c , α ) = c ( 1 - α ) 1 - α U(c,\alpha)=\frac{c^{(1-\alpha)}}{1-\alpha}
  3. U t = [ c t 1 - ρ + β ( E t U t + 1 1 - α ) ( 1 - ρ ) / ( 1 - α ) ] 1 / ( 1 - ρ ) U_{t}=\left[c_{t}^{1-\rho}+\beta(E_{t}U_{t+1}^{1-\alpha})^{(1-\rho)/(1-\alpha)% }\right]^{1/(1-\rho)}
  4. p t U ( c t ) = β E t [ ( p t + 1 + y t + 1 ) U ( c t + 1 ) ] p_{t}U^{\prime}(c_{t})=\beta E_{t}[(p_{t+1}+y_{t+1})U^{\prime}(c_{t+1})]
  5. 1 = β E t [ U ( c t + 1 ) U ( c t ) R e , t + 1 ] 1=\beta E_{t}\left[\frac{U^{\prime}(c_{t+1})}{U^{\prime}(c_{t})}R_{e,t+1}\right]
  6. R e , t + 1 = ( p t + 1 + y t + 1 ) / p t R_{e,t+1}=(p_{t+1}+y_{t+1})/p_{t}

Equivalence_principle.html

  1. M 1 M_{1}
  2. M 0 M_{0}
  3. F 1 = M 0 act M 1 pass r 2 F_{1}=\frac{M_{0}^{\mathrm{act}}M_{1}^{\mathrm{pass}}}{r^{2}}
  4. F 2 = M 0 act M 2 pass r 2 F_{2}=\frac{M_{0}^{\mathrm{act}}M_{2}^{\mathrm{pass}}}{r^{2}}
  5. F = m inert a F=m^{\mathrm{inert}}a
  6. m 1 m_{1}
  7. m 2 m_{2}
  8. r r
  9. m 0 m_{0}
  10. a 1 = F 1 m 1 inert = a 2 = F 2 m 2 inert a_{1}=\frac{F_{1}}{m_{1}^{\mathrm{inert}}}=a_{2}=\frac{F_{2}}{m_{2}^{\mathrm{% inert}}}
  11. M 0 act M 1 pass r 2 m 1 inert = M 0 act M 2 pass r 2 m 2 inert \frac{M_{0}^{\mathrm{act}}M_{1}^{\mathrm{pass}}}{r^{2}m_{1}^{\mathrm{inert}}}=% \frac{M_{0}^{\mathrm{act}}M_{2}^{\mathrm{pass}}}{r^{2}m_{2}^{\mathrm{inert}}}
  12. M 1 pass m 1 inert = M 2 pass m 2 inert \frac{M_{1}^{\mathrm{pass}}}{m_{1}^{\mathrm{inert}}}=\frac{M_{2}^{\mathrm{pass% }}}{m_{2}^{\mathrm{inert}}}
  13. F 1 = M 0 act M 1 pass r 2 F_{1}=\frac{M_{0}^{\mathrm{act}}M_{1}^{\mathrm{pass}}}{r^{2}}
  14. F 0 = M 1 act M 0 pass r 2 F_{0}=\frac{M_{1}^{\mathrm{act}}M_{0}^{\mathrm{pass}}}{r^{2}}
  15. M 0 act M 0 pass = M 1 act M 1 pass \frac{M_{0}^{\mathrm{act}}}{M_{0}^{\mathrm{pass}}}=\frac{M_{1}^{\mathrm{act}}}% {M_{1}^{\mathrm{pass}}}
  16. η ( A , B ) \eta(A,B)
  17. η ( A , B ) = 2 ( m g m i ) A - ( m g m i ) B ( m g m i ) A + ( m g m i ) B \eta(A,B)=2\frac{\left(\frac{m_{g}}{m_{i}}\right)_{A}-\left(\frac{m_{g}}{m_{i}% }\right)_{B}}{\left(\frac{m_{g}}{m_{i}}\right)_{A}+\left(\frac{m_{g}}{m_{i}}% \right)_{B}}
  18. | η ( Al , Au ) | = ( 1.3 ± 1.0 ) × 10 - 11 |\eta(\mathrm{Al},\mathrm{Au})|=(1.3\pm 1.0)\times 10^{-11}
  19. η ( Earth , Be-Ti ) = ( 0.3 ± 1.8 ) × 10 - 13 \eta(\,\text{Earth},\,\text{Be-Ti})=(0.3\pm 1.8)\times 10^{-13}

Equivalent_air_depth.html

  1. p p N 2 ( n i t r o x , d e p t h ) = p p N 2 ( a i r , E A D ) ppN_{2}(nitrox,depth)=ppN_{2}(air,EAD)
  2. F N 2 ( n i t r o x ) P d e p t h = F N 2 ( a i r ) P E A D FN_{2}(nitrox)\cdot P_{depth}=FN_{2}(air)\cdot P_{EAD}
  3. F N 2 FN_{2}
  4. P d e p t h P_{depth}
  5. P E A D P_{EAD}
  6. P E A D = F N 2 ( n i t r o x ) F N 2 ( a i r ) P d e p t h P_{EAD}={FN_{2}(nitrox)\over FN_{2}(air)}\cdot P_{depth}
  7. P E A D P_{EAD}\,
  8. P d e p t h P_{depth}\,
  9. P d e p t h = P a t m o s p h e r e + ρ s e a w a t e r g h d e p t h P_{depth}=P_{atmosphere}+\rho_{seawater}\cdot g\cdot h_{depth}\,
  10. P d e p t h ( P a ) = P a t m o s p h e r e ( P a ) + ρ s e a w a t e r g h d e p t h ( m ) P_{depth}(Pa)=P_{atmosphere}(Pa)+\rho_{seawater}\cdot g\cdot h_{depth}(m)\,
  11. P d e p t h ( a t m ) = 1 + ρ s e a w a t e r g h d e p t h P a t m o s p h e r e ( P a ) = 1 + 1027 9.8 h d e p t h 101325 1 + h d e p t h ( m ) 10 P_{depth}(atm)=1+\frac{\rho_{seawater}\cdot g\cdot h_{depth}}{P_{atmosphere}(% Pa)}=1+\frac{1027\cdot 9.8\cdot h_{depth}}{101325}\ \approx 1+\frac{h_{depth}(% m)}{10}
  12. F N 2 ( n i t r o x ) F N 2 ( a i r ) = R \frac{FN_{2}(nitrox)}{FN_{2}(air)}=R
  13. 1 + h E A D 10 = R ( 1 + h d e p t h 10 ) 1+\frac{h_{EAD}}{10}=R\cdot(1+\frac{h_{depth}}{10})
  14. h E A D = 10 ( R + R h d e p t h 10 - 1 ) = R ( h d e p t h + 10 ) - 10 h_{EAD}=10\cdot(R+R\cdot\frac{h_{depth}}{10}-1)=R\cdot(h_{depth}+10)-10
  15. h ( f t ) 3.3 h ( m ) h(ft)\approx 3.3\cdot h(m)\,
  16. h E A D ( f t ) = 3.3 ( R ( h d e p t h ( f t ) 3.3 + 10 ) - 10 ) = R ( h d e p t h ( f t ) + 33 ) - 33 h_{EAD}(ft)=3.3\cdot\Bigl(R\cdot(\frac{h_{depth}(ft)}{3.3}+10)-10\Bigr)=R\cdot% (h_{depth}(ft)+33)-33
  17. F N 2 ( a i r ) = 0.79 FN_{2}(air)=0.79
  18. h E A D ( m ) = F N 2 ( n i t r o x ) 0.79 ( h d e p t h ( m ) + 10 ) - 10 h_{EAD}(m)=\frac{FN_{2}(nitrox)}{0.79}\cdot(h_{depth}(m)+10)-10
  19. h E A D ( f t ) = F N 2 ( n i t r o x ) 0.79 ( h d e p t h ( f t ) + 33 ) - 33 h_{EAD}(ft)=\frac{FN_{2}(nitrox)}{0.79}\cdot(h_{depth}(ft)+33)-33

Equivalent_airspeed.html

  1. E A S = T A S × ρ ρ 0 EAS=TAS\times\sqrt{\frac{\rho}{\rho_{0}}}
  2. ρ \rho\,
  3. ρ 0 \rho_{0}\,
  4. E A S = 2 q ρ 0 EAS=\sqrt{\frac{2q}{\rho_{0}}}
  5. q {q}\,
  6. q = 1 2 ρ v 2 , q=\tfrac{1}{2}\,\rho\,v^{2},
  7. E A S = a 0 M P P 0 EAS={a_{0}}M\sqrt{P\over P_{0}}
  8. a 0 {a_{0}}\,
  9. M M\,
  10. P P\,
  11. P 0 P_{0}\,
  12. E A S = a 0 5 P P 0 [ ( q c P + 1 ) 2 7 - 1 ] EAS={a_{0}}\sqrt{{5P\over P_{0}}[(\frac{q_{c}}{P}+1)^{\frac{2}{7}}-1]}
  13. q c {q_{c}}\,
  14. C A S = E A S × [ 1 + 1 8 ( 1 - δ ) M 2 + 3 640 ( 1 - 10 δ + 9 δ 2 ) M 4 ] CAS={EAS\times[1+\frac{1}{8}(1-\delta)M^{2}+\frac{3}{640}(1-10\delta+9\delta^{% 2})M^{4}]}
  15. δ = P P 0 \delta=\frac{P}{P_{0}}
  16. C A S CAS\,
  17. E A S EAS\,

Erasure_code.html

  1. O ( n log n ) O(n\log n)
  2. { v i } 1 i k \{v_{i}\}_{1\leq i\leq k}
  3. v k + 1 = - i = 1 k v i . v_{k+1}=-\sum_{i=1}^{k}v_{i}.
  4. { v i } 1 i k + 1 \{v_{i}\}_{1\leq i\leq k+1}
  5. v e v_{e}
  6. v e = - i = 1 , i e k + 1 v i . v_{e}=-\sum_{i=1,i\neq e}^{k+1}v_{i}.
  7. f ( i ) = a + ( b - a ) ( i - 1 ) f(i)=a+(b-a)(i-1)
  8. f ( i ) = 555 + 74 ( i - 1 ) f(i)=555+74(i-1)
  9. f ( 1 ) = 555 f(1)=555
  10. f ( 2 ) = 629 f(2)=629
  11. f ( i ) = a + ( b - a ) ( i - 1 ) f(i)=a+(b-a)(i-1)

Ernst_Ising.html

  1. S i S_{i}
  2. E = - i j J i j S i S j E=-\sum_{ij}J_{ij}S_{i}S_{j}\,
  3. J i j > 0 J_{ij}>0
  4. J i j < 0 J_{ij}<0
  5. J i j = 0 J_{ij}=0
  6. E = i S i S i + 1 E=\sum_{i}S_{i}S_{i+1}\,

Erosion_(morphology).html

  1. d \mathbb{R}^{d}
  2. d \mathbb{Z}^{d}
  3. A B = { z E | B z A } A\ominus B=\{z\in E|B_{z}\subseteq A\}
  4. B z = { b + z | b B } B_{z}=\{b+z|b\in B\}
  5. z E \forall z\in E
  6. A B = b B A - b A\ominus B=\bigcap_{b\in B}A_{-b}
  7. A C A\subseteq C
  8. A B C B A\ominus B\subseteq C\ominus B
  9. A B A A\ominus B\subseteq A
  10. ( A B ) C = A ( B C ) (A\ominus B)\ominus C=A\ominus(B\oplus C)
  11. \oplus
  12. { , - } \mathbb{R}\cup\{\infty,-\infty\}
  13. \mathbb{R}
  14. \infty
  15. - -\infty
  16. ( f b ) ( x ) = inf y B [ f ( x + y ) - b ( y ) ] (f\ominus b)(x)=\inf_{y\in B}[f(x+y)-b(y)]
  17. ( L , ) (L,\leq)
  18. \wedge
  19. \vee
  20. \emptyset
  21. { X i } \{X_{i}\}
  22. ( L , ) (L,\leq)
  23. ε : L L \varepsilon:L\rightarrow L
  24. i ε ( X i ) = ε ( i X i ) \bigwedge_{i}\varepsilon(X_{i})=\varepsilon\left(\bigwedge_{i}X_{i}\right)
  25. ε ( U ) = U \varepsilon(U)=U

Estimation_of_covariance_matrices.html

  1. cov ( X ) = E [ ( X - E [ X ] ) ( X - E [ X ] ) T ] \operatorname{cov}(X)=\operatorname{E}\left[\left(X-\operatorname{E}[X])(X-% \operatorname{E}[X]\right)^{\mathrm{T}}\right]
  2. 𝐐 = 1 n - 1 i = 1 n ( x i - x ¯ ) ( x i - x ¯ ) T , \mathbf{Q}={1\over{n-1}}\sum_{i=1}^{n}(x_{i}-\overline{x})(x_{i}-\overline{x})% ^{\mathrm{T}},
  3. x i \textstyle x_{i}
  4. x ¯ = [ x ¯ 1 x ¯ p ] = 1 n i = 1 n x i \overline{x}=\left[\begin{array}[c]{c}\bar{x}_{1}\\ \vdots\\ \bar{x}_{p}\end{array}\right]={1\over{n}}\sum_{i=1}^{n}x_{i}
  5. 𝐐 𝐧 = 1 n i = 1 n ( x i - x ¯ ) ( x i - x ¯ ) T . \mathbf{Q_{n}}={1\over n}\sum_{i=1}^{n}(x_{i}-\overline{x})(x_{i}-\overline{x}% )^{\mathrm{T}}.
  6. f ( x ) = ( 2 π ) - p / 2 det ( Σ ) - 1 / 2 exp ( - 1 2 ( x - μ ) T Σ - 1 ( x - μ ) ) f(x)=(2\pi)^{-p/2}\,\det(\Sigma)^{-1/2}\exp\left(-{1\over 2}(x-\mu)^{\mathrm{T% }}\Sigma^{-1}(x-\mu)\right)
  7. ( 2 π ) - p / 2 det ( Σ ) - 1 / 2 (2\pi)^{-p/2}\det(\Sigma)^{-1/2}
  8. f ( x ) f(x)
  9. ( μ , Σ ) = ( 2 π ) - n p / 2 i = 1 n det ( Σ ) - 1 / 2 exp ( - 1 2 ( x i - μ ) T Σ - 1 ( x i - μ ) ) \mathcal{L}(\mu,\Sigma)=(2\pi)^{-np/2}\,\prod_{i=1}^{n}\det(\Sigma)^{-1/2}\exp% \left(-{1\over 2}(x_{i}-\mu)^{\mathrm{T}}\Sigma^{-1}(x_{i}-\mu)\right)
  10. x ¯ = ( x 1 + + x n ) / n . \overline{x}=(x_{1}+\cdots+x_{n})/n.
  11. x ¯ \bar{x}
  12. ( x ¯ , Σ ) det ( Σ ) - n / 2 exp ( - 1 2 i = 1 n ( x i - x ¯ ) T Σ - 1 ( x i - x ¯ ) ) , \mathcal{L}(\overline{x},\Sigma)\propto\det(\Sigma)^{-n/2}\exp\left(-{1\over 2% }\sum_{i=1}^{n}(x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1}(x_{i}-\overline{x}% )\right),
  13. \mathcal{L}
  14. ( x i - x ¯ ) T Σ - 1 ( x i - x ¯ ) (x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1}(x_{i}-\overline{x})
  15. ( x ¯ , Σ ) det ( Σ ) - n / 2 exp ( - 1 2 i = 1 n tr ( ( x i - x ¯ ) T Σ - 1 ( x i - x ¯ ) ) ) \mathcal{L}(\overline{x},\Sigma)\propto\det(\Sigma)^{-n/2}\exp\left(-{1\over 2% }\sum_{i=1}^{n}\operatorname{tr}((x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1}(% x_{i}-\overline{x}))\right)
  16. = det ( Σ ) - n / 2 exp ( - 1 2 i = 1 n tr ( ( x i - x ¯ ) ( x i - x ¯ ) T Σ - 1 ) ) =\det(\Sigma)^{-n/2}\exp\left(-{1\over 2}\sum_{i=1}^{n}\operatorname{tr}((x_{i% }-\overline{x})(x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1})\right)
  17. = det ( Σ ) - n / 2 exp ( - 1 2 tr ( i = 1 n ( x i - x ¯ ) ( x i - x ¯ ) T Σ - 1 ) ) =\det(\Sigma)^{-n/2}\exp\left(-{1\over 2}\operatorname{tr}\left(\sum_{i=1}^{n}% (x_{i}-\overline{x})(x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1}\right)\right)
  18. = det ( Σ ) - n / 2 exp ( - 1 2 tr ( S Σ - 1 ) ) =\det(\Sigma)^{-n/2}\exp\left(-{1\over 2}\operatorname{tr}\left(S\Sigma^{-1}% \right)\right)
  19. S = i = 1 n ( x i - x ¯ ) ( x i - x ¯ ) T 𝐑 p × p . S=\sum_{i=1}^{n}(x_{i}-\overline{x})(x_{i}-\overline{x})^{\mathrm{T}}\in% \mathbf{R}^{p\times p}.
  20. S S
  21. p p
  22. det ( Σ ) - n / 2 exp ( - 1 2 tr ( S 1 / 2 Σ - 1 S 1 / 2 ) ) . \det(\Sigma)^{-n/2}\exp\left(-{1\over 2}\operatorname{tr}\left(S^{1/2}\Sigma^{% -1}S^{1/2}\right)\right).
  23. det ( S ) - n / 2 det ( B ) n / 2 exp ( - 1 2 tr ( B ) ) . \det(S)^{-n/2}\det(B)^{n/2}\exp\left(-{1\over 2}\operatorname{tr}(B)\right).
  24. det ( B ) n / 2 exp ( - 1 2 tr ( B ) ) \det(B)^{n/2}\exp\left(-{1\over 2}\operatorname{tr}(B)\right)
  25. λ i n / 2 exp ( - λ i / 2 ) . \lambda_{i}^{n/2}\exp(-\lambda_{i}/2).
  26. B = Q ( n I p ) Q - 1 = n I p B=Q(nI_{p})Q^{-1}=nI_{p}
  27. Σ = S 1 / 2 B - 1 S 1 / 2 = S 1 / 2 ( 1 n I p ) S 1 / 2 = S n , \Sigma=S^{1/2}B^{-1}S^{1/2}=S^{1/2}\left(\frac{1}{n}I_{p}\right)S^{1/2}=\frac{% S}{n},
  28. S n = 1 n i = 1 n ( X i - X ¯ ) ( X i - X ¯ ) T {S\over n}={1\over n}\sum_{i=1}^{n}(X_{i}-\overline{X})(X_{i}-\overline{X})^{% \mathrm{T}}
  29. i = 1 n ( X i - X ¯ ) ( X i - X ¯ ) T W p ( Σ , n - 1 ) . \sum_{i=1}^{n}(X_{i}-\overline{X})(X_{i}-\overline{X})^{\mathrm{T}}\sim W_{p}(% \Sigma,n-1).
  30. ln ( μ , Σ ) = const - n 2 ln det ( Σ ) - 1 2 tr [ Σ - 1 i = 1 n ( x i - μ ) ( x i - μ ) T ] . \ln\mathcal{L}(\mu,\Sigma)=\operatorname{const}-{n\over 2}\ln\det(\Sigma)-{1% \over 2}\operatorname{tr}\left[\Sigma^{-1}\sum_{i=1}^{n}(x_{i}-\mu)(x_{i}-\mu)% ^{\mathrm{T}}\right].
  31. d ln ( μ , Σ ) = - n 2 tr [ Σ - 1 { d Σ } ] d\ln\mathcal{L}(\mu,\Sigma)=-{n\over 2}\operatorname{tr}\left[\Sigma^{-1}\left% \{d\Sigma\right\}\right]
  32. - 1 2 tr [ - Σ - 1 { d Σ } Σ - 1 i = 1 n ( x i - μ ) ( x i - μ ) T - 2 Σ - 1 i = 1 n ( x i - μ ) { d μ } T ] . -{1\over 2}\operatorname{tr}\left[-\Sigma^{-1}\{d\Sigma\}\Sigma^{-1}\sum_{i=1}% ^{n}(x_{i}-\mu)(x_{i}-\mu)^{\mathrm{T}}-2\Sigma^{-1}\sum_{i=1}^{n}(x_{i}-\mu)% \{d\mu\}^{\mathrm{T}}\right].
  33. d ln ( μ , Σ ) = 0 d\ln\mathcal{L}(\mu,\Sigma)=0
  34. d μ d\mu
  35. d Σ d\Sigma
  36. Σ \Sigma
  37. i = 1 n ( x i - μ ) = 0 , \sum_{i=1}^{n}(x_{i}-\mu)=0,
  38. μ ^ = X ¯ = 1 n i = 1 n X i . \widehat{\mu}=\bar{X}={1\over n}\sum_{i=1}^{n}X_{i}.
  39. i = 1 n ( x i - μ ) ( x i - μ ) T = i = 1 n ( x i - x ¯ ) ( x i - x ¯ ) T = S \sum_{i=1}^{n}(x_{i}-\mu)(x_{i}-\mu)^{\mathrm{T}}=\sum_{i=1}^{n}(x_{i}-\bar{x}% )(x_{i}-\bar{x})^{\mathrm{T}}=S
  40. d Σ d\Sigma
  41. d ln L d\ln L
  42. - 1 2 tr ( Σ - 1 { d Σ } [ n I p - Σ - 1 S ] ) . -{1\over 2}\operatorname{tr}\left(\Sigma^{-1}\left\{d\Sigma\right\}\left[nI_{p% }-\Sigma^{-1}S\right]\right).
  43. d ln ( μ , Σ ) = 0 d\ln\mathcal{L}(\mu,\Sigma)=0
  44. Σ \Sigma
  45. n n
  46. Σ ^ = 1 n S , \widehat{\Sigma}={1\over n}S,
  47. 𝐑 ^ = 1 n i = 1 n x i x i T . \hat{\mathbf{R}}={1\over n}\sum_{i=1}^{n}x_{i}x_{i}^{\mathrm{T}}.
  48. E 𝐑 [ 𝐑 ^ ] = def exp 𝐑 E [ exp 𝐑 - 1 𝐑 ^ ] \mathrm{E}_{\mathbf{R}}[\hat{\mathbf{R}}]\ \stackrel{\mathrm{def}}{=}\ \exp_{% \mathbf{R}}\mathrm{E}\left[\exp_{\mathbf{R}}^{-1}\hat{\mathbf{R}}\right]
  49. exp 𝐑 ( 𝐑 ^ ) = 𝐑 1 2 exp ( 𝐑 - 1 2 𝐑 ^ 𝐑 - 1 2 ) 𝐑 1 2 \exp_{\mathbf{R}}(\hat{\mathbf{R}})=\mathbf{R}^{\frac{1}{2}}\exp\left(\mathbf{% R}^{-\frac{1}{2}}\hat{\mathbf{R}}\mathbf{R}^{-\frac{1}{2}}\right)\mathbf{R}^{% \frac{1}{2}}
  50. exp 𝐑 - 1 ( 𝐑 ^ ) = 𝐑 1 2 ( log 𝐑 - 1 2 𝐑 ^ 𝐑 - 1 2 ) 𝐑 1 2 \exp_{\mathbf{R}}^{-1}(\hat{\mathbf{R}})=\mathbf{R}^{\frac{1}{2}}\left(\log% \mathbf{R}^{-\frac{1}{2}}\hat{\mathbf{R}}\mathbf{R}^{-\frac{1}{2}}\right)% \mathbf{R}^{\frac{1}{2}}
  51. 𝐁 ( 𝐑 ^ ) = exp 𝐑 - 1 E 𝐑 [ 𝐑 ^ ] = E [ exp 𝐑 - 1 𝐑 ^ ] \mathbf{B}(\hat{\mathbf{R}})=\exp_{\mathbf{R}}^{-1}\mathrm{E}_{\mathbf{R}}% \left[\hat{\mathbf{R}}\right]=\mathrm{E}\left[\exp_{\mathbf{R}}^{-1}\hat{% \mathbf{R}}\right]
  52. exp 𝐑 𝐁 ( 𝐑 ^ ) \exp_{\mathbf{R}}\mathbf{B}(\hat{\mathbf{R}})
  53. 𝐁 ( 𝐑 ^ ) = - β ( p , n ) 𝐑 \mathbf{B}(\hat{\mathbf{R}})=-\beta(p,n)\mathbf{R}
  54. β ( p , n ) = 1 p ( p log n + p - ψ ( n - p + 1 ) + ( n - p + 1 ) ψ ( n - p + 2 ) + ψ ( n + 1 ) - ( n + 1 ) ψ ( n + 2 ) ) \beta(p,n)=\frac{1}{p}\left(p\log n+p-\psi(n-p+1)+(n-p+1)\psi(n-p+2)+\psi(n+1)% -(n+1)\psi(n+2)\right)
  55. exp 𝐑 𝐁 ( 𝐑 ^ ) = e - β ( p , n ) 𝐑 \exp_{\mathbf{R}}\mathbf{B}(\hat{\mathbf{R}})=e^{-\beta(p,n)}\mathbf{R}
  56. B B
  57. δ \delta
  58. δ A + ( 1 - δ ) B \delta A+(1-\delta)B

Euclid's_lemma.html

  1. r x + s y = 1. rx+sy=1.
  2. r n + s a = 1. rn+sa=1.
  3. r n b + s a b = b . rnb+sab=b.

Euclidean_division.html

  1. b b
  2. b 0 b≠0
  3. q q
  4. r r
  5. a = b q + r a=bq+r
  6. | b | |b|
  7. b b
  8. a a
  9. b b
  10. q q
  11. r r
  12. q q
  13. r r
  14. b = 0 b=0
  15. m m
  16. a a
  17. d d
  18. m > 0 m>0
  19. q q
  20. r r
  21. d r < m + d d\leq r<m+d
  22. a = m q + r a=mq+r
  23. d = - m 2 d=-\left\lfloor\frac{m}{2}\right\rfloor
  24. - m 2 r < m - m 2 -\left\lfloor\frac{m}{2}\right\rfloor\leq r<m-\left\lfloor\frac{m}{2}\right\rfloor
  25. r r
  26. a a
  27. m m
  28. R R
  29. m > 0 m>0
  30. R - 1 m * R^{-1}\in\mathbb{Z}_{m}^{*}
  31. R m * R\in\mathbb{Z}_{m}^{*}
  32. q q
  33. r r
  34. 0 r < m 0\leq r<m
  35. a = m q + R - 1 r a=mq+R^{-1}\cdot r
  36. r r

Euclidean_group.html

  1. x A ( x + b ) x\mapsto A(x+b)
  2. x A x + b x\mapsto Ax+b

Euclidean_minimum_spanning_tree.html

  1. f f
  2. n n
  3. d 1 d\neq 1
  4. c ( d ) n d - 1 d d f ( x ) d - 1 d d x c(d)n^{\frac{d-1}{d}}\int_{\mathbb{R}^{d}}f(x)^{\frac{d-1}{d}}dx
  5. c ( d ) c(d)
  6. d d

Euclidean_quantum_gravity.html

  1. i i
  2. π / 2 \pi/2
  3. 𝒟 g 𝒟 ϕ exp ( d 4 x | g | ( R + matter ) ) \int\mathcal{D}{g}\,\mathcal{D}\phi\,\exp\left(\int d^{4}x\sqrt{|{g}|}(R+% \mathcal{L}_{\mathrm{matter}})\right)
  4. ϕ \phi
  5. Z = 𝒟 g 𝒟 ϕ exp ( d 4 x | g | ( R + matter ) ) Z=\int\mathcal{D}{g}\,\mathcal{D}\phi\,\exp\left(\int d^{4}x\sqrt{|{g}|}(R+% \mathcal{L}_{\mathrm{matter}})\right)
  6. 𝒟 g \mathcal{D}{g}
  7. N N
  8. N a N^{a}
  9. Z Z
  10. δ Z δ N = 0 = 𝒟 g 𝒟 ϕ δ S δ N | Σ exp ( d 4 x | g | ( R + matter ) ) \frac{\delta Z}{\delta N}=0=\int\mathcal{D}{g}\,\mathcal{D}\phi\,\left.\frac{% \delta S}{\delta N}\right|_{\Sigma}\exp\left(\int d^{4}x\sqrt{|{g}|}(R+% \mathcal{L}_{\mathrm{matter}})\right)
  11. Σ \Sigma

Eugène_Charles_Catalan.html

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

Euler's_Disk.html

  1. ω \omega
  2. ω \omega
  3. ω \omega
  4. t t
  5. t 0 t_{0}
  6. τ \tau
  7. t 0 t_{0}
  8. τ [ ( 2 a 9 g ) 3 2 π μ a M ] 1 5 \tau\simeq\left[\left(\frac{2a}{9g}\right)^{3}\frac{2\pi\mu a}{M}\right]^{% \frac{1}{5}}
  9. a a
  10. g g
  11. μ \mu
  12. M M
  13. τ \tau
  14. 10 - 2 10^{-2}
  15. α \alpha
  16. Ω \Omega
  17. t 0 = α 0 3 M 2 π μ a t_{0}=\frac{\alpha_{0}^{3}M}{2\pi\mu a}
  18. α 0 \alpha_{0}
  19. t 0 - t > τ t_{0}-t>\tau
  20. Ω \Omega
  21. Ω ( t 0 - t ) - 1 6 \Omega\sim(t_{0}-t)^{-\frac{1}{6}}
  22. α \alpha

Euler's_equations_(rigid_body_dynamics).html

  1. 𝐈 s y m b o l ω ˙ + s y m b o l ω × ( 𝐈 s y m b o l ω ) = 𝐌 . \mathbf{I}\cdot\dot{symbol\omega}+symbol\omega\times\left(\mathbf{I}\cdot symbol% \omega\right)=\mathbf{M}.
  2. I 1 ω ˙ 1 + ( I 3 - I 2 ) ω 2 ω 3 \displaystyle I_{1}\dot{\omega}_{1}+(I_{3}-I_{2})\omega_{2}\omega_{3}
  3. d 𝐋 in d t = def d d t ( 𝐈 in s y m b o l ω ) = 𝐌 in \frac{d\mathbf{L}_{\,\text{in}}}{dt}\ \stackrel{\mathrm{def}}{=}\ \frac{d}{dt}% \left(\mathbf{I}_{\,\text{in}}\cdot symbol\omega\right)=\mathbf{M}_{\,\text{in}}
  4. 𝐋 = def L 1 𝐞 1 + L 2 𝐞 2 + L 3 𝐞 3 = I 1 ω 1 𝐞 1 + I 2 ω 2 𝐞 2 + I 3 ω 3 𝐞 3 \mathbf{L}\ \stackrel{\mathrm{def}}{=}\ L_{1}\mathbf{e}_{1}+L_{2}\mathbf{e}_{2% }+L_{3}\mathbf{e}_{3}=I_{1}\omega_{1}\mathbf{e}_{1}+I_{2}\omega_{2}\mathbf{e}_% {2}+I_{3}\omega_{3}\mathbf{e}_{3}
  5. ( d 𝐋 d t ) rot + s y m b o l ω × 𝐋 = 𝐌 \left(\frac{d\mathbf{L}}{dt}\right)_{\mathrm{rot}}+symbol\omega\times\mathbf{L% }=\mathbf{M}
  6. 𝐌 in = 𝐐𝐌 , \mathbf{M}_{\,\text{in}}=\mathbf{Q}\mathbf{M},
  7. s y m b o l ω × s y m b o l v = 𝐐 ˙ 𝐐 - 1 s y m b o l v symbol\omega\times symbol{v}=\dot{\mathbf{Q}}\mathbf{Q}^{-1}symbol{v}
  8. 𝐈 s y m b o l ω ˙ + s y m b o l ω × ( 𝐈 s y m b o l ω ) = 𝐌 . \mathbf{I}\cdot\dot{symbol\omega}+symbol\omega\times\left(\mathbf{I}\cdot symbol% \omega\right)=\mathbf{M}.
  9. L k = def I k ω k L_{k}\ \stackrel{\mathrm{def}}{=}\ I_{k}\omega_{k}
  10. 𝐧 ^ \mathbf{\hat{n}}
  11. 𝐌 = def I d ω d t 𝐧 ^ = I α 𝐧 ^ \mathbf{M}\ \stackrel{\mathrm{def}}{=}\ I\frac{d\omega}{dt}\mathbf{\hat{n}}=I% \alpha\mathbf{\hat{n}}
  12. 𝐧 ^ \mathbf{\hat{n}}
  13. ( d 𝐋 d t ) relative \left(\frac{d\mathbf{L}}{dt}\right)_{\mathrm{relative}}

Euler's_rotation_theorem.html

  1. 𝐞 ^ \mathbf{\hat{e}}
  2. 𝐑 T 𝐑 = 𝐑𝐑 T = 𝐈 , \mathbf{R}^{\mathrm{T}}\mathbf{R}=\mathbf{R}\mathbf{R}^{\mathrm{T}}=\mathbf{I},
  3. 1 = det ( 𝐈 ) = det ( 𝐑 T 𝐑 ) = det ( 𝐑 T ) det ( 𝐑 ) = det ( 𝐑 ) 2 det ( 𝐑 ) = ± 1. 1=\det(\mathbf{I})=\det(\mathbf{R}^{\mathrm{T}}\mathbf{R})=\det(\mathbf{R}^{% \mathrm{T}})\det(\mathbf{R})=\det(\mathbf{R})^{2}\quad\Longrightarrow\quad\det% (\mathbf{R})=\pm 1.
  4. det ( - 𝐑 ) = ( - 1 ) 3 det ( 𝐑 ) = - det ( 𝐑 ) and det ( 𝐑 - 1 ) = 1 , \det(-\mathbf{R})=(-1)^{3}\det(\mathbf{R})=-\det(\mathbf{R})\quad\hbox{and}% \quad\det(\mathbf{R}^{-1})=1,
  5. det ( 𝐑 - 𝐈 ) = det ( ( 𝐑 - 𝐈 ) T ) = det ( ( 𝐑 T - 𝐈 ) ) = det ( ( 𝐑 - 1 - 𝐈 ) ) = det ( - 𝐑 - 1 ( 𝐑 - 𝐈 ) ) = - det ( 𝐑 - 1 ) det ( 𝐑 - 𝐈 ) = - det ( 𝐑 - 𝐈 ) det ( 𝐑 - 𝐈 ) = 0. \begin{aligned}\displaystyle\det(\mathbf{R}-\mathbf{I})=&\displaystyle\det\big% ((\mathbf{R}-\mathbf{I})^{\mathrm{T}}\big)=\det\big((\mathbf{R}^{\mathrm{T}}-% \mathbf{I})\big)=\det\big((\mathbf{R}^{-1}-\mathbf{I})\big)=\det\big(-\mathbf{% R}^{-1}(\mathbf{R}-\mathbf{I})\big)\\ \displaystyle=&\displaystyle-\det(\mathbf{R}^{-1})\;\det(\mathbf{R}-\mathbf{I}% )=-\det(\mathbf{R}-\mathbf{I})\quad\Longrightarrow\quad\det(\mathbf{R}-\mathbf% {I})=0.\end{aligned}
  6. det ( 𝐑 - λ 𝐈 ) = 0 for λ = 1. \det(\mathbf{R}-\lambda\mathbf{I})=0\quad\hbox{for}\quad\lambda=1.
  7. ( 𝐑 - 𝐈 ) 𝐧 = 𝟎 𝐑𝐧 = 𝐧 . (\mathbf{R}-\mathbf{I})\mathbf{n}=\mathbf{0}\quad\Longleftrightarrow\quad% \mathbf{R}\mathbf{n}=\mathbf{n}.
  8. 𝐑 ( cos ϕ - sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) , 0 ϕ 2 π . \mathbf{R}\sim\begin{pmatrix}\cos\phi&-\sin\phi&0\\ \sin\phi&\cos\phi&0\\ 0&0&1\\ \end{pmatrix},\qquad 0\leq\phi\leq 2\pi.
  9. ϕ = 0 \phi=0
  10. 𝐀𝐔 = 𝐔 diag ( α 1 , , α m ) 𝐔 𝐀𝐔 = diag ( α 1 , , α m ) , \mathbf{A}\mathbf{U}=\mathbf{U}\;\mathrm{diag}(\alpha_{1},\ldots,\alpha_{m})% \quad\Longleftrightarrow\quad\mathbf{U}^{\dagger}\mathbf{A}\mathbf{U}=% \operatorname{diag}(\alpha_{1},\ldots,\alpha_{m}),
  11. 𝐔 = 𝐔 - 1 . \mathbf{U}^{\dagger}=\mathbf{U}^{-1}.
  12. ( 𝐔 𝐀𝐔 ) = diag ( α 1 * , , α m * ) = 𝐔 𝐀 - 1 𝐔 = diag ( 1 / α 1 , , 1 / α m ) \left(\mathbf{U}^{\dagger}\mathbf{A}\mathbf{U}\right)^{\dagger}=\mathrm{diag}(% \alpha^{*}_{1},\ldots,\alpha^{*}_{m})=\mathbf{U}^{\dagger}\mathbf{A}^{-1}% \mathbf{U}=\mathrm{diag}(1/\alpha_{1},\ldots,1/\alpha_{m})
  13. α k * = 1 / α k α k * α k = | α k | 2 = 1 , k = 1 , , m . \alpha^{*}_{k}=1/\alpha_{k}\quad\Longleftrightarrow\alpha^{*}_{k}\alpha_{k}=|% \alpha_{k}|^{2}=1,\qquad k=1,\ldots,m.
  14. 𝐑𝐔 = 𝐔 ( e i ϕ 0 0 0 e - i ϕ 0 0 0 ± 1 ) \mathbf{R}\mathbf{U}=\mathbf{U}\begin{pmatrix}e^{i\phi}&0&0\\ 0&e^{-i\phi}&0\\ 0&0&\pm 1\\ \end{pmatrix}
  15. 𝐑𝐮 1 = e i ϕ 𝐮 1 and 𝐑𝐮 2 = e - i ϕ 𝐮 2 . \mathbf{R}\mathbf{u}_{1}=e^{i\phi}\,\mathbf{u}_{1}\quad\hbox{and}\quad\mathbf{% R}\mathbf{u}_{2}=e^{-i\phi}\,\mathbf{u}_{2}.
  16. 𝐑𝐔 ( 1 2 i 2 0 1 2 - i 2 0 0 0 1 ) = 𝐔 ( 1 2 i 2 0 1 2 - i 2 0 0 0 1 ) ( 1 2 1 2 0 - i 2 i 2 0 0 0 1 ) = 𝐈 ( e i ϕ 0 0 0 e - i ϕ 0 0 0 1 ) ( 1 2 i 2 0 1 2 - i 2 0 0 0 1 ) \mathbf{R}\mathbf{U}\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\ \frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\ 0&0&1\\ \end{pmatrix}=\mathbf{U}\underbrace{\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{i}% {\sqrt{2}}&0\\ \frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\ 0&0&1\\ \end{pmatrix}\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\ \frac{-i}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\ 0&0&1\\ \end{pmatrix}}_{=\;\mathbf{I}}\begin{pmatrix}e^{i\phi}&0&0\\ 0&e^{-i\phi}&0\\ 0&0&1\\ \end{pmatrix}\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\ \frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\ 0&0&1\\ \end{pmatrix}
  17. 𝐔 𝐑𝐔 = ( cos ϕ - sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) with 𝐔 = 𝐔 ( 1 2 i 2 0 1 2 - i 2 0 0 0 1 ) . \mathbf{U^{\prime}}^{\dagger}\mathbf{R}\mathbf{U^{\prime}}=\begin{pmatrix}\cos% \phi&-\sin\phi&0\\ \sin\phi&\cos\phi&0\\ 0&0&1\\ \end{pmatrix}\quad\,\text{ with }\quad\mathbf{U^{\prime}}=\mathbf{U}\begin{% pmatrix}\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\ \frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\ 0&0&1\\ \end{pmatrix}.
  18. 1 + 2 c o s φ 1+2cosφ
  19. Tr [ 𝐀𝐑𝐀 T ] = Tr [ 𝐑𝐀 T 𝐀 ] = Tr [ 𝐑 ] with 𝐀 T = 𝐀 - 1 , \mathrm{Tr}[\mathbf{A}\mathbf{R}\mathbf{A}^{\mathrm{T}}]=\mathrm{Tr}[\mathbf{R% }\mathbf{A}^{\mathrm{T}}\mathbf{A}]=\mathrm{Tr}[\mathbf{R}]\quad\,\text{ with % }\quad\mathbf{A}^{\mathrm{T}}=\mathbf{A}^{-1},
  20. Δ R = [ 1 0 0 0 1 0 0 0 1 ] + [ 0 z - y - z 0 x y - x 0 ] Δ θ = 𝐈 + 𝐀 Δ θ . \Delta R=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}+\begin{bmatrix}0&z&-y\\ -z&0&x\\ y&-x&0\end{bmatrix}\,\Delta\theta=\mathbf{I}+\mathbf{A}\,\Delta\theta.
  21. R = ( 𝟏 + 𝐀 θ N ) N e 𝐀 θ . R=\left(\mathbf{1}+\frac{\mathbf{A}\theta}{N}\right)^{N}\approx e^{\mathbf{A}% \theta}.
  22. 𝐀 θ \mathbf{A}\theta
  23. x = a cos ( α ) + b sin ( α ) y = - a sin ( α ) + b cos ( α ) cos ( α ) = a T x sin ( α ) = b T x y = - a b T x + b a T x = ( b a T - a b T ) x x = x cos ( β ) + y sin ( β ) = [ I cos ( β ) + ( b a T - a b T ) sin ( β ) ] x R = I cos ( β ) + ( b a T - a b T ) sin ( β ) = I cos ( β ) + G sin ( β ) G = b a T - a b T \begin{aligned}&\displaystyle x=a\cos\left(\alpha\right)+b\sin\left(\alpha% \right)\\ &\displaystyle y=-a\sin\left(\alpha\right)+b\cos\left(\alpha\right)\\ &\displaystyle\cos\left(\alpha\right)={{a}^{T}}x\quad\sin\left(\alpha\right)={% {b}^{T}}x\\ &\displaystyle y=-a{{b}^{T}}x+b{{a}^{T}}x=\left(b{{a}^{T}}-a{{b}^{T}}\right)x% \\ &\\ &\displaystyle{x}^{\prime}=x\cos\left(\beta\right)+y\sin\left(\beta\right)\\ &\displaystyle\ \ \ =\left[I\cos\left(\beta\right)+\left(b{{a}^{T}}-a{{b}^{T}}% \right)\sin\left(\beta\right)\right]x\\ &\\ &\displaystyle R=I\cos\left(\beta\right)+\left(b{{a}^{T}}-a{{b}^{T}}\right)% \sin\left(\beta\right)\\ &\displaystyle\quad=I\cos\left(\beta\right)+G\sin\left(\beta\right)\\ &\\ &\displaystyle G=b{{a}^{T}}-a{{b}^{T}}\\ \end{aligned}
  24. P a b = - G 2 R = I - P a b + [ I cos ( β ) + G sin ( β ) ] P a b = e G β \begin{aligned}&\displaystyle{{P}_{ab}}=-{{G}^{2}}\\ &\displaystyle R=I-{{P}_{ab}}+\left[I\cos\left(\beta\right)+G\sin\left(\beta% \right)\right]{{P}_{ab}}={{e}^{G\beta}}\\ \end{aligned}

Euler_system.html

  1. cor G / F ( c G ) = q Σ ( G / F ) P ( Fr q - 1 | Hom O ( T , O ( 1 ) ) ; Fr q - 1 ) c F {\rm cor}_{G/F}(c_{G})=\prod_{q\in\Sigma(G/F)}P(\mathrm{Fr}_{q}^{-1}|{\rm Hom}% _{O}(T,O(1));\mathrm{Fr}_{q}^{-1})c_{F}
  2. N Q ( ζ n l ) / Q ( ζ l ) ( α n l ) = α n F l - 1 N_{Q(\zeta_{nl})/Q(\zeta_{l})}(\alpha_{nl})=\alpha_{n}^{F_{l}-1}
  3. α n l α n \alpha_{nl}\equiv\alpha_{n}

Euler–Tricomi_equation.html

  1. u x x = x u y y . u_{xx}=xu_{yy}.\,
  2. y ± 2 3 x 3 / 2 = C , y\pm\frac{2}{3}x^{3/2}=C,
  3. u = A x y + B x + C y + D , u=Axy+Bx+Cy+D,\,
  4. u = A ( 3 y 2 + x 3 ) + B ( y 3 + x 3 y ) + C ( 6 x y 2 + x 4 ) , u=A(3y^{2}+x^{3})+B(y^{3}+x^{3}y)+C(6xy^{2}+x^{4}),\,

European_Union_energy_label.html

  1. P 0.240 Φ + 0.0103 Φ . P\leq 0.240\cdot\sqrt{\Phi}+0.0103\cdot\Phi.
  2. P 0.150 Φ + 0.0097 Φ . P\leq 0.150\cdot\sqrt{\Phi}+0.0097\cdot\Phi.
  3. P R = { 0.88 Φ + 0.049 Φ ( Φ > 34 lm ) 0.2 Φ ( Φ 34 lm ) P_{\mathrm{R}}=\begin{cases}0.88\cdot\sqrt{\Phi}+0.049\cdot\Phi&(\Phi>34\mbox{% lm}~{})\\ 0.2\cdot\Phi&(\Phi\leq 34\mbox{ lm}~{})\end{cases}
  4. P ref = 20 [ W ] + 4.3224 [ W / dm 2 ] A . P_{\mathrm{ref}}=20\,\mathrm{[W]}+4.3224\,\mathrm{[W/dm^{2}]}\cdot A.
  5. P basic P_{\mathrm{basic}}
  6. P basic = 24 [ W ] + 4.3224 [ W / dm 2 ] A . P_{\mathrm{basic}}=24\,\mathrm{[W]}+4.3224\,\mathrm{[W/dm^{2}]}\cdot A.
  7. P basic = 28 [ W ] + 4.3224 [ W / dm 2 ] A . P_{\mathrm{basic}}=28\,\mathrm{[W]}+4.3224\,\mathrm{[W/dm^{2}]}\cdot A.

Evaporative_cooler.html

  1. ϵ \epsilon
  2. ϵ = T e , d b - T l , d b T e , d b - T e , w b \epsilon=\frac{T_{e,db}-T_{l,db}}{T_{e,db}-T_{e,wb}}
  3. ϵ \epsilon
  4. T e , d b T_{e,db}
  5. T l , d b T_{l,db}
  6. T e , w b T_{e,wb}
  7. T l , d b T_{l,db}

Evolute.html

  1. 𝐓 ( s ) = γ ( s ) \mathbf{T}(s)=\gamma^{\prime}(s)
  2. 𝐓 ( s ) = k ( s ) 𝐍 ( s ) \mathbf{T}^{\prime}(s)=k(s)\mathbf{N}(s)
  3. R ( s ) = 1 k ( s ) . R(s)=\frac{1}{k(s)}.
  4. E ( s ) = γ ( s ) + R ( s ) 𝐍 ( s ) = γ ( s ) + 1 k ( s ) 𝐍 ( s ) . E(s)=\gamma(s)+R(s)\mathbf{N}(s)=\gamma(s)+\frac{1}{k(s)}\mathbf{N}(s).
  5. ( X , Y ) = ( x , y ) + R 𝐍 = ( x - R sin φ , y + R cos φ ) (X,Y)=(x,y)+R\mathbf{N}=(x-R\sin\varphi,y+R\cos\varphi)
  6. ( cos φ , sin φ ) = ( x , y ) ( x 2 + y 2 ) 1 / 2 (\cos\varphi,\sin\varphi)=\frac{(x^{\prime},y^{\prime})}{(x^{\prime 2}+y^{% \prime 2})^{1/2}}
  7. R = 1 / k = ( x 2 + y 2 ) 3 / 2 x y ′′ - x ′′ y , R=1/k=\frac{(x^{\prime 2}+y^{\prime 2})^{3/2}}{x^{\prime}y^{\prime\prime}-x^{% \prime\prime}y^{\prime}},
  8. X [ x , y ] = x - y x 2 + y 2 x y ′′ - x ′′ y X[x,y]=x-y^{\prime}\frac{x^{\prime 2}+y^{\prime 2}}{x^{\prime}y^{\prime\prime}% -x^{\prime\prime}y^{\prime}}
  9. Y [ x , y ] = y + x x 2 + y 2 x y ′′ - x ′′ y Y[x,y]=y+x^{\prime}\frac{x^{\prime 2}+y^{\prime 2}}{x^{\prime}y^{\prime\prime}% -x^{\prime\prime}y^{\prime}}
  10. s 1 s 2 | d R d s | d s . \int_{s_{1}}^{s_{2}}\left|\frac{dR}{ds}\right|ds.
  11. s 1 s 2 | d R d s | d s = | R ( s 2 ) - R ( s 1 ) | . \int_{s_{1}}^{s_{2}}\left|\frac{dR}{ds}\right|ds=|R(s_{2})-R(s_{1})|.
  12. d σ d s = | d R d s | . \frac{d\sigma}{ds}=\left|\frac{dR}{ds}\right|.
  13. E ( s ) = γ ( s ) + R ( s ) 𝐍 ( s ) E(s)=\gamma(s)+R(s)\mathbf{N}(s)
  14. E ( s ) = γ ( s ) + R ( s ) 𝐍 ( s ) - 𝐓 ( s ) = R ( s ) 𝐍 ( s ) E^{\prime}(s)=\gamma^{\prime}(s)+R^{\prime}(s)\mathbf{N}(s)-\mathbf{T}(s)=R^{% \prime}(s)\mathbf{N}(s)
  15. d E d s = d R d s 𝐍 ( s ) \frac{dE}{ds}=\frac{dR}{ds}\mathbf{N}\left(s\right)
  16. d E d σ = d E d s / d σ d s = ± 𝐍 \frac{dE}{d\sigma}=\left.\frac{dE}{ds}\right/\frac{d\sigma}{ds}=\pm\mathbf{N}
  17. d 2 E d σ 2 = ± d 𝐍 d s / d σ d s = - 1 R R d E d σ . \frac{d^{2}E}{d\sigma^{2}}=\pm\left.\frac{d\mathbf{N}}{ds}\right/\frac{d\sigma% }{ds}=-\frac{1}{RR^{\prime}}\frac{dE}{d\sigma}.
  18. k E = - 1 R R k_{E}=-\frac{1}{RR^{\prime}}
  19. ( X , Y ) = ( - y x 2 + y 2 x y ′′ - x ′′ y , x x 2 + y 2 x y ′′ - x ′′ y ) . (X,Y)=\left(-y^{\prime}\frac{x^{\prime 2}+y^{\prime 2}}{x^{\prime}y^{\prime% \prime}-x^{\prime\prime}y^{\prime}},x^{\prime}\frac{x^{\prime 2}+y^{\prime 2}}% {x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}}\right).

Evolution_of_sexual_reproduction.html

  1. G \surd G
  2. G G

Evolutionary_game_theory.html

  1. p ( x ) = e - x / V V . p(x)=\frac{e^{-x/V}}{V}.
  2. p ( m ) = 1 - e m / V , p(m)=1-e^{m/V},
  3. w i = a i + j r j b j . w_{i}=a_{i}+\sum_{j}r_{j}b_{j}.
  4. < v a r > W H a w k = < v a r > V · ( 1 - p ) + ( V / 2 - C / 2 ) · p < / v a r > <var>WHawk=<var>V·(1-p)+(V/2-C/2)·p</var>

Ewald's_sphere.html

  1. λ \lambda
  2. K i K_{i}
  3. 2 π / λ 2\pi/\lambda
  4. K f K_{f}
  5. K f K_{f}
  6. K i K_{i}
  7. Δ K = K f - K i \Delta{K}=K_{f}-K_{i}
  8. K i K_{i}
  9. K f K_{f}
  10. 2 π / λ 2\pi/\lambda
  11. K i K_{i}

Exact_coloring.html

  1. ( k 2 ) {\textstyle\left({{k}\atop{2}}\right)}
  2. m ( k 2 ) m\leq{\textstyle\left({{k}\atop{2}}\right)}
  3. ( k 2 ) - m {\textstyle\left({{k}\atop{2}}\right)}-m
  4. m ( k 2 ) m\geq{\textstyle\left({{k}\atop{2}}\right)}

Exact_trigonometric_constants.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π \pi
  5. sin 0 = 0 \sin 0=0\,
  6. cos 0 = 1 \cos 0=1\,
  7. tan 0 = 0 \tan 0=0\,
  8. cot 0 is undefined \cot 0\,\text{ is undefined}\,
  9. sin π 80 = sin 2.25 = 10 - 2 5 ( ( 2 + 2 ) ( 2 + 2 + 2 ) - 2 + 2 + 2 ) - ( 5 + 1 ) ( ( 2 + 2 ) ( 2 - 2 + 2 ) + 2 - 2 + 2 ) 8 \sin\frac{\pi}{80}=\sin 2.25^{\circ}=\frac{\sqrt{10-2\sqrt{5}}\left(\sqrt{(2+% \sqrt{2})(2+\sqrt{2+\sqrt{2}})}-\sqrt{2+\sqrt{2+\sqrt{2}}}\right)-(\sqrt{5}+1)% \left(\sqrt{(2+\sqrt{2})(2-\sqrt{2+\sqrt{2}})}+\sqrt{2-\sqrt{2+\sqrt{2}}}% \right)}{8}
  10. cos π 80 = cos 2.25 = 1 2 + 4 + 8 + 40 + 320 32 \cos\frac{\pi}{80}=\cos 2.25^{\circ}=\sqrt{\frac{1}{2}+\sqrt{\frac{4+\sqrt{8+% \sqrt{40+\sqrt{320}}}}{32}}}
  11. sin π 64 = sin 2.8125 = 2 - 2 + 2 + 2 + 2 2 \sin\frac{\pi}{64}=\sin 2.8125^{\circ}=\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+% \sqrt{2}}}}}}{2}
  12. cos π 64 = cos 2.8125 = 2 + 2 + 2 + 2 + 2 2 \cos\frac{\pi}{64}=\cos 2.8125^{\circ}=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+% \sqrt{2}}}}}}{2}
  13. sin π 60 = sin 3 = ( 2 - 12 ) 5 + 5 + ( 10 - 2 ) ( 3 + 1 ) 16 \sin\frac{\pi}{60}=\sin 3^{\circ}=\frac{(2-\sqrt{12})\sqrt{5+\sqrt{5}}+(\sqrt{% 10}-\sqrt{2})(\sqrt{3}+1)}{16}\,
  14. cos π 60 = cos 3 = ( 2 + 12 ) 5 + 5 + ( 10 - 2 ) ( 3 - 1 ) 16 \cos\frac{\pi}{60}=\cos 3^{\circ}=\frac{(2+\sqrt{12})\sqrt{5+\sqrt{5}}+(\sqrt{% 10}-\sqrt{2})(\sqrt{3}-1)}{16}\,
  15. tan π 60 = tan 3 = [ ( 2 - 3 ) ( 3 + 5 ) - 2 ] [ 2 - 10 - 20 ] 4 \tan\frac{\pi}{60}=\tan 3^{\circ}=\frac{\left[(2-\sqrt{3})(3+\sqrt{5})-2\right% ]\left[2-\sqrt{10-\sqrt{20}}\right]}{4}\,
  16. cot π 60 = cot 3 = [ ( 2 + 3 ) ( 3 + 5 ) - 2 ] [ 2 + 10 - 20 ] 4 \cot\frac{\pi}{60}=\cot 3^{\circ}=\frac{\left[(2+\sqrt{3})(3+\sqrt{5})-2\right% ]\left[2+\sqrt{10-\sqrt{20}}\right]}{4}\,
  17. sin π 40 = sin 4.5 = ( 2 - 1 ) ( 4 + 8 ) ( 5 + 5 ) - ( 2 + 1 ) ( 5 - 1 ) 2 - 2 8 \sin\frac{\pi}{40}=\sin 4.5^{\circ}=\frac{(\sqrt{2}-1)\sqrt{(4+\sqrt{8})(5+% \sqrt{5})}-(\sqrt{2}+1)(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}
  18. cos π 40 = cos 4.5 = ( 2 + 1 ) ( 4 - 8 ) ( 5 + 5 ) + ( 2 - 1 ) ( 5 - 1 ) 2 + 2 8 \cos\frac{\pi}{40}=\cos 4.5^{\circ}=\frac{(\sqrt{2}+1)\sqrt{(4-\sqrt{8})(5+% \sqrt{5})}+(\sqrt{2}-1)(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}
  19. sin π 32 = sin 5.625 = 2 - 2 + 2 + 2 2 \sin\frac{\pi}{32}=\sin 5.625^{\circ}=\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}% }}{2}
  20. cos π 32 = cos 5.625 = 2 + 2 + 2 + 2 2 \cos\frac{\pi}{32}=\cos 5.625^{\circ}=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}% }}{2}
  21. sin π 30 = sin 6 = 30 - 180 - 5 - 1 8 \sin\frac{\pi}{30}=\sin 6^{\circ}=\frac{\sqrt{30-\sqrt{180}}-\sqrt{5}-1}{8}\,
  22. cos π 30 = cos 6 = 10 - 20 + 3 + 15 8 \cos\frac{\pi}{30}=\cos 6^{\circ}=\frac{\sqrt{10-\sqrt{20}}+\sqrt{3}+\sqrt{15}% }{8}\,
  23. tan π 30 = tan 6 = 10 - 20 + 3 - 15 2 \tan\frac{\pi}{30}=\tan 6^{\circ}=\frac{\sqrt{10-\sqrt{20}}+\sqrt{3}-\sqrt{15}% }{2}\,
  24. cot π 30 = cot 6 = 27 + 15 + 50 + 2420 2 \cot\frac{\pi}{30}=\cot 6^{\circ}=\frac{\sqrt{27}+\sqrt{15}+\sqrt{50+\sqrt{242% 0}}}{2}\,
  25. sin π 24 = sin 7.5 = 1 2 2 - 2 + 3 = 1 2 2 - 6 + 2 2 = 1 4 2 ( 4 - 6 - 2 ) \sin\frac{\pi}{24}=\sin 7.5^{\circ}=\tfrac{1}{2}\sqrt{2-\sqrt{2+\sqrt{3}}}=% \tfrac{1}{2}\sqrt{2-\tfrac{\sqrt{6}+\sqrt{2}}{2}}=\tfrac{1}{4}\sqrt{2(4-\sqrt{% 6}-\sqrt{2})}\,
  26. cos π 24 = cos 7.5 = 1 2 2 + 2 + 3 = 1 2 2 + 6 + 2 2 = 1 4 2 ( 4 + 6 + 2 ) \cos\frac{\pi}{24}=\cos 7.5^{\circ}=\tfrac{1}{2}\sqrt{2+\sqrt{2+\sqrt{3}}}=% \tfrac{1}{2}\sqrt{2+\tfrac{\sqrt{6}+\sqrt{2}}{2}}=\tfrac{1}{4}\sqrt{2(4+\sqrt{% 6}+\sqrt{2})}\,
  27. tan π 24 = tan 7.5 = 6 - 3 + 2 - 2 = ( 2 - 1 ) ( 3 - 2 ) \tan\frac{\pi}{24}=\tan 7.5^{\circ}=\sqrt{6}-\sqrt{3}+\sqrt{2}-2\ =(\sqrt{2}-1% )(\sqrt{3}-\sqrt{2})
  28. cot π 24 = cot 7.5 = 6 + 3 + 2 + 2 = ( 2 + 1 ) ( 3 + 2 ) \cot\frac{\pi}{24}=\cot 7.5^{\circ}=\sqrt{6}+\sqrt{3}+\sqrt{2}+2\ =(\sqrt{2}+1% )(\sqrt{3}+\sqrt{2})
  29. sin π 20 = sin 9 = 1 8 [ 10 + 2 - 2 5 - 5 ] \sin\frac{\pi}{20}=\sin 9^{\circ}=\tfrac{1}{8}\left[\sqrt{10}+\sqrt{2}-2\sqrt{% 5-\sqrt{5}}\right]\,
  30. cos π 20 = cos 9 = 1 8 [ 10 + 2 + 2 5 - 5 ] \cos\frac{\pi}{20}=\cos 9^{\circ}=\tfrac{1}{8}\left[\sqrt{10}+\sqrt{2}+2\sqrt{% 5-\sqrt{5}}\right]\,
  31. tan π 20 = tan 9 = 5 + 1 - 5 + 2 5 \tan\frac{\pi}{20}=\tan 9^{\circ}=\sqrt{5}+1-\sqrt{5+2\sqrt{5}}\,
  32. cot π 20 = cot 9 = 5 + 1 + 5 + 2 5 \cot\frac{\pi}{20}=\cot 9^{\circ}=\sqrt{5}+1+\sqrt{5+2\sqrt{5}}\,
  33. sin π 16 = sin 11.25 = 1 2 2 - 2 + 2 \sin\frac{\pi}{16}=\sin 11.25^{\circ}=\tfrac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2}}}
  34. cos π 16 = cos 11.25 = 1 2 2 + 2 + 2 \cos\frac{\pi}{16}=\cos 11.25^{\circ}=\tfrac{1}{2}\sqrt{2+\sqrt{2+\sqrt{2}}}
  35. tan π 16 = tan 11.25 = 4 + 2 2 - 2 - 1 \tan\frac{\pi}{16}=\tan 11.25^{\circ}=\sqrt{4+2\sqrt{2}}-\sqrt{2}-1
  36. cot π 16 = cot 11.25 = 4 + 2 2 + 2 + 1 \cot\frac{\pi}{16}=\cot 11.25^{\circ}=\sqrt{4+2\sqrt{2}}+\sqrt{2}+1
  37. sin π 15 = sin 12 = 1 8 [ 2 ( 5 + 5 ) + 3 - 15 ] \sin\frac{\pi}{15}=\sin 12^{\circ}=\tfrac{1}{8}\left[\sqrt{2(5+\sqrt{5})}+% \sqrt{3}-\sqrt{15}\right]\,
  38. cos π 15 = cos 12 = 1 8 [ 6 ( 5 + 5 ) + 5 - 1 ] \cos\frac{\pi}{15}=\cos 12^{\circ}=\tfrac{1}{8}\left[\sqrt{6(5+\sqrt{5})}+% \sqrt{5}-1\right]\,
  39. tan π 15 = tan 12 = 1 2 [ 3 3 - 15 - 2 ( 25 - 11 5 ) ] \tan\frac{\pi}{15}=\tan 12^{\circ}=\tfrac{1}{2}\left[3\sqrt{3}-\sqrt{15}-\sqrt% {2(25-11\sqrt{5})}\right]\,
  40. cot π 15 = cot 12 = 1 2 [ 15 + 3 + 2 ( 5 + 5 ) ] \cot\frac{\pi}{15}=\cot 12^{\circ}=\tfrac{1}{2}\left[\sqrt{15}+\sqrt{3}+\sqrt{% 2(5+\sqrt{5})}\right]\,
  41. sin π 12 = sin 15 = 1 4 ( 6 - 2 ) \sin\frac{\pi}{12}=\sin 15^{\circ}=\tfrac{1}{4}(\sqrt{6}-\sqrt{2})\,
  42. cos π 12 = cos 15 = 1 4 ( 6 + 2 ) \cos\frac{\pi}{12}=\cos 15^{\circ}=\tfrac{1}{4}(\sqrt{6}+\sqrt{2})\,
  43. tan π 12 = tan 15 = 2 - 3 \tan\frac{\pi}{12}=\tan 15^{\circ}=2-\sqrt{3}\,
  44. cot π 12 = cot 15 = 2 + 3 \cot\frac{\pi}{12}=\cot 15^{\circ}=2+\sqrt{3}\,
  45. sin π 10 = sin 18 = 1 4 ( 5 - 1 ) \sin\frac{\pi}{10}=\sin 18^{\circ}=\tfrac{1}{4}\left(\sqrt{5}-1\right)\,
  46. cos π 10 = cos 18 = 1 4 2 ( 5 + 5 ) \cos\frac{\pi}{10}=\cos 18^{\circ}=\tfrac{1}{4}\sqrt{2(5+\sqrt{5})}\,
  47. tan π 10 = tan 18 = 1 5 5 ( 5 - 2 5 ) \tan\frac{\pi}{10}=\tan 18^{\circ}=\tfrac{1}{5}\sqrt{5(5-2\sqrt{5})}\,
  48. cot π 10 = cot 18 = 5 + 2 5 \cot\frac{\pi}{10}=\cot 18^{\circ}=\sqrt{5+2\sqrt{5}}\,
  49. sin 7 π 60 = sin 21 = 1 16 [ 2 ( 3 + 1 ) 5 - 5 - ( 6 - 2 ) ( 1 + 5 ) ] \sin\frac{7\pi}{60}=\sin 21^{\circ}=\tfrac{1}{16}\left[2(\sqrt{3}+1)\sqrt{5-% \sqrt{5}}-(\sqrt{6}-\sqrt{2})(1+\sqrt{5})\right]\,
  50. cos 7 π 60 = cos 21 = 1 16 [ 2 ( 3 - 1 ) 5 - 5 + ( 6 + 2 ) ( 1 + 5 ) ] \cos\frac{7\pi}{60}=\cos 21^{\circ}=\tfrac{1}{16}\left[2(\sqrt{3}-1)\sqrt{5-% \sqrt{5}}+(\sqrt{6}+\sqrt{2})(1+\sqrt{5})\right]\,
  51. tan 7 π 60 = tan 21 = 1 4 [ 2 - ( 2 + 3 ) ( 3 - 5 ) ] [ 2 - 2 ( 5 + 5 ) ] \tan\frac{7\pi}{60}=\tan 21^{\circ}=\tfrac{1}{4}\left[2-(2+\sqrt{3})(3-\sqrt{5% })\right]\left[2-\sqrt{2(5+\sqrt{5})}\right]\,
  52. cot 7 π 60 = cot 21 = 1 4 [ 2 - ( 2 - 3 ) ( 3 - 5 ) ] [ 2 + 2 ( 5 + 5 ) ] \cot\frac{7\pi}{60}=\cot 21^{\circ}=\tfrac{1}{4}\left[2-(2-\sqrt{3})(3-\sqrt{5% })\right]\left[2+\sqrt{2(5+\sqrt{5})}\right]\,
  53. sin π 8 = sin 22.5 = 1 2 2 - 2 , \sin\frac{\pi}{8}=\sin 22.5^{\circ}=\tfrac{1}{2}\sqrt{2-\sqrt{2}},
  54. cos π 8 = cos 22.5 = 1 2 2 + 2 \cos\frac{\pi}{8}=\cos 22.5^{\circ}=\tfrac{1}{2}\sqrt{2+\sqrt{2}}\,
  55. tan π 8 = tan 22.5 = 2 - 1 \tan\frac{\pi}{8}=\tan 22.5^{\circ}=\sqrt{2}-1\,
  56. cot π 8 = cot 22.5 = 2 + 1 \cot\frac{\pi}{8}=\cot 22.5^{\circ}=\sqrt{2}+1\,
  57. sin 2 π 15 = sin 24 = 1 8 [ 15 + 3 - 2 ( 5 - 5 ) ] \sin\frac{2\pi}{15}=\sin 24^{\circ}=\tfrac{1}{8}\left[\sqrt{15}+\sqrt{3}-\sqrt% {2(5-\sqrt{5})}\right]\,
  58. cos 2 π 15 = cos 24 = 1 8 ( 6 ( 5 - 5 ) + 5 + 1 ) \cos\frac{2\pi}{15}=\cos 24^{\circ}=\tfrac{1}{8}\left(\sqrt{6(5-\sqrt{5})}+% \sqrt{5}+1\right)\,
  59. tan 2 π 15 = tan 24 = 1 2 [ 50 + 22 5 - 3 3 - 15 ] \tan\frac{2\pi}{15}=\tan 24^{\circ}=\tfrac{1}{2}\left[\sqrt{50+22\sqrt{5}}-3% \sqrt{3}-\sqrt{15}\right]\,
  60. cot 2 π 15 = cot 24 = 1 2 [ 15 - 3 + 2 ( 5 - 5 ) ] \cot\frac{2\pi}{15}=\cot 24^{\circ}=\tfrac{1}{2}\left[\sqrt{15}-\sqrt{3}+\sqrt% {2(5-\sqrt{5})}\right]\,
  61. sin 3 π 20 = sin 27 = 1 8 [ 2 5 + 5 - 2 ( 5 - 1 ) ] \sin\frac{3\pi}{20}=\sin 27^{\circ}=\tfrac{1}{8}\left[2\sqrt{5+\sqrt{5}}-\sqrt% {2}\;(\sqrt{5}-1)\right]\,
  62. cos 3 π 20 = cos 27 = 1 8 [ 2 5 + 5 + 2 ( 5 - 1 ) ] \cos\frac{3\pi}{20}=\cos 27^{\circ}=\tfrac{1}{8}\left[2\sqrt{5+\sqrt{5}}+\sqrt% {2}\;(\sqrt{5}-1)\right]\,
  63. tan 3 π 20 = tan 27 = 5 - 1 - 5 - 2 5 \tan\frac{3\pi}{20}=\tan 27^{\circ}=\sqrt{5}-1-\sqrt{5-2\sqrt{5}}\,
  64. cot 3 π 20 = cot 27 = 5 - 1 + 5 - 2 5 \cot\frac{3\pi}{20}=\cot 27^{\circ}=\sqrt{5}-1+\sqrt{5-2\sqrt{5}}\,
  65. sin π 6 = sin 30 = 1 2 \sin\frac{\pi}{6}=\sin 30^{\circ}=\tfrac{1}{2}\,
  66. cos π 6 = cos 30 = 1 2 3 \cos\frac{\pi}{6}=\cos 30^{\circ}=\tfrac{1}{2}\sqrt{3}\,
  67. tan π 6 = tan 30 = 1 3 3 = 1 3 \tan\frac{\pi}{6}=\tan 30^{\circ}=\tfrac{1}{3}\sqrt{3}=\frac{1}{\sqrt{3}}\,
  68. cot π 6 = cot 30 = 3 \cot\frac{\pi}{6}=\cot 30^{\circ}=\sqrt{3}\,
  69. sin 11 π 60 = sin 33 = 1 16 [ 2 ( 3 - 1 ) 5 + 5 + 2 ( 1 + 3 ) ( 5 - 1 ) ] \sin\frac{11\pi}{60}=\sin 33^{\circ}=\tfrac{1}{16}\left[2(\sqrt{3}-1)\sqrt{5+% \sqrt{5}}+\sqrt{2}(1+\sqrt{3})(\sqrt{5}-1)\right]\,
  70. cos 11 π 60 = cos 33 = 1 16 [ 2 ( 3 + 1 ) 5 + 5 + 2 ( 1 - 3 ) ( 5 - 1 ) ] \cos\frac{11\pi}{60}=\cos 33^{\circ}=\tfrac{1}{16}\left[2(\sqrt{3}+1)\sqrt{5+% \sqrt{5}}+\sqrt{2}(1-\sqrt{3})(\sqrt{5}-1)\right]\,
  71. tan 11 π 60 = tan 33 = 1 4 [ 2 - ( 2 - 3 ) ( 3 + 5 ) ] [ 2 + 2 ( 5 - 5 ) ] \tan\frac{11\pi}{60}=\tan 33^{\circ}=\tfrac{1}{4}\left[2-(2-\sqrt{3})(3+\sqrt{% 5})\right]\left[2+\sqrt{2(5-\sqrt{5})}\right]\,
  72. cot 11 π 60 = cot 33 = 1 4 [ 2 - ( 2 + 3 ) ( 3 + 5 ) ] [ 2 - 2 ( 5 - 5 ) ] \cot\frac{11\pi}{60}=\cot 33^{\circ}=\tfrac{1}{4}\left[2-(2+\sqrt{3})(3+\sqrt{% 5})\right]\left[2-\sqrt{2(5-\sqrt{5})}\right]\,
  73. sin π 5 = sin 36 = 10 - 20 4 \sin\frac{\pi}{5}=\sin 36^{\circ}=\frac{\sqrt{10-\sqrt{20}}}{4}\,
  74. cos π 5 = cos 36 = 5 + 1 4 = 1 2 φ \cos\frac{\pi}{5}=\cos 36^{\circ}=\frac{\sqrt{5}+1}{4}=\tfrac{1}{2}\varphi\,
  75. φ \varphi
  76. tan π 5 = tan 36 = 5 - 20 \tan\frac{\pi}{5}=\tan 36^{\circ}=\sqrt{5-\sqrt{20}}\,
  77. cot π 5 = cot 36 = 25 + 500 5 \cot\frac{\pi}{5}=\cot 36^{\circ}=\frac{\sqrt{25+\sqrt{500}}}{5}\,
  78. sin 13 π 60 = sin 39 = 1 16 [ 2 ( 1 - 3 ) 5 - 5 + 2 ( 3 + 1 ) ( 5 + 1 ) ] \sin\frac{13\pi}{60}=\sin 39^{\circ}=\tfrac{1}{16}[2(1-\sqrt{3})\sqrt{5-\sqrt{% 5}}+\sqrt{2}(\sqrt{3}+1)(\sqrt{5}+1)]\,
  79. cos 13 π 60 = cos 39 = 1 16 [ 2 ( 1 + 3 ) 5 - 5 + 2 ( 3 - 1 ) ( 5 + 1 ) ] \cos\frac{13\pi}{60}=\cos 39^{\circ}=\tfrac{1}{16}[2(1+\sqrt{3})\sqrt{5-\sqrt{% 5}}+\sqrt{2}(\sqrt{3}-1)(\sqrt{5}+1)]\,
  80. tan 13 π 60 = tan 39 = 1 4 [ ( 2 - 3 ) ( 3 - 5 ) - 2 ] [ 2 - 2 ( 5 + 5 ) ] \tan\frac{13\pi}{60}=\tan 39^{\circ}=\tfrac{1}{4}\left[(2-\sqrt{3})(3-\sqrt{5}% )-2\right]\left[2-\sqrt{2(5+\sqrt{5})}\right]\,
  81. cot 13 π 60 = cot 39 = 1 4 [ ( 2 + 3 ) ( 3 - 5 ) - 2 ] [ 2 + 2 ( 5 + 5 ) ] \cot\frac{13\pi}{60}=\cot 39^{\circ}=\tfrac{1}{4}\left[(2+\sqrt{3})(3-\sqrt{5}% )-2\right]\left[2+\sqrt{2(5+\sqrt{5})}\right]\,
  82. sin 7 π 30 = sin 42 = 30 + 180 - 5 + 1 8 \sin\frac{7\pi}{30}=\sin 42^{\circ}=\frac{\sqrt{30+\sqrt{180}}-\sqrt{5}+1}{8}\,
  83. cos 7 π 30 = cos 42 = 15 - 3 + 10 + 20 8 \cos\frac{7\pi}{30}=\cos 42^{\circ}=\frac{\sqrt{15}-\sqrt{3}+\sqrt{10+\sqrt{20% }}}{8}\,
  84. tan 7 π 30 = tan 42 = 15 + 3 - 10 + 20 2 \tan\frac{7\pi}{30}=\tan 42^{\circ}=\frac{\sqrt{15}+\sqrt{3}-\sqrt{10+\sqrt{20% }}}{2}\,
  85. cot 7 π 30 = cot 42 = 50 - 2420 + 27 - 15 2 \cot\frac{7\pi}{30}=\cot 42^{\circ}=\frac{\sqrt{50-\sqrt{2420}}+\sqrt{27}-% \sqrt{15}}{2}\,
  86. sin π 4 = sin 45 = 1 2 2 = 1 2 \sin\frac{\pi}{4}=\sin 45^{\circ}=\tfrac{1}{2}\sqrt{2}=\frac{1}{\sqrt{2}}\,
  87. cos π 4 = cos 45 = 1 2 2 = 1 2 \cos\frac{\pi}{4}=\cos 45^{\circ}=\tfrac{1}{2}\sqrt{2}=\frac{1}{\sqrt{2}}\,
  88. tan π 4 = tan 45 = 1 \tan\frac{\pi}{4}=\tan 45^{\circ}=1\,
  89. cot π 4 = cot 45 = 1 \cot\frac{\pi}{4}=\cot 45^{\circ}=1\,
  90. sin π 3 = sin 60 = 1 2 3 \sin\frac{\pi}{3}=\sin 60^{\circ}=\tfrac{1}{2}\sqrt{3}\,
  91. cos π 3 = cos 60 = 1 2 \cos\frac{\pi}{3}=\cos 60^{\circ}=\tfrac{1}{2}\,
  92. tan π 3 = tan 60 = 3 \tan\frac{\pi}{3}=\tan 60^{\circ}=\sqrt{3}\,
  93. cot π 3 = cot 60 = 1 3 3 = 1 3 \cot\frac{\pi}{3}=\cot 60^{\circ}=\tfrac{1}{3}\sqrt{3}=\frac{1}{\sqrt{3}}\,
  94. sin π 2 = sin 90 = 1 \sin\frac{\pi}{2}=\sin 90^{\circ}=1\,
  95. cos π 2 = cos 90 = 0 \cos\frac{\pi}{2}=\cos 90^{\circ}=0\,
  96. tan π 2 = tan 90 is undefined \tan\frac{\pi}{2}=\tan 90^{\circ}\,\text{ is undefined}\,
  97. cot π 2 = cot 90 = 0 \cot\frac{\pi}{2}=\cot 90^{\circ}=0\,
  98. V = 5 a 3 cos 36 tan 2 36 V=\frac{5a^{3}\cos 36^{\circ}}{\tan^{2}{36^{\circ}}}
  99. cos 36 = 5 + 1 4 \cos 36^{\circ}=\frac{\sqrt{5}+1}{4}\,
  100. tan 36 = 5 - 2 5 \tan 36^{\circ}=\sqrt{5-2\sqrt{5}}\,
  101. V = a 3 ( 15 + 7 5 ) 4 V=\frac{a^{3}(15+7\sqrt{5})}{4}\,
  102. π \pi
  103. ϕ \phi
  104. crd 36 = crd ( ADB ) = a b = 2 1 + 5 = 5 - 1 2 \operatorname{crd}36^{\circ}=\operatorname{crd}(\angle\mathrm{ADB})=\frac{a}{b% }=\frac{2}{1+\sqrt{5}}=\frac{\sqrt{5}-1}{2}
  105. crd θ = 2 sin θ 2 . \operatorname{crd}\ {\theta}=2\sin\frac{\theta}{2}.\,
  106. sin 18 = 1 1 + 5 = 5 - 1 4 . \sin 18^{\circ}=\frac{1}{1+\sqrt{5}}=\frac{\sqrt{5}-1}{4}.
  107. crd 108 = crd ( ABC ) = b a = 1 + 5 2 , \operatorname{crd}\ 108^{\circ}=\operatorname{crd}(\angle\mathrm{ABC})=\frac{b% }{a}=\frac{1+\sqrt{5}}{2},
  108. sin 54 = cos 36 = 1 + 5 4 . \sin 54^{\circ}=\cos 36^{\circ}=\frac{1+\sqrt{5}}{4}.
  109. 5 x 5x\,
  110. x { 18 , 36 , 54 , 72 , 90 } x\in\{18,36,54,72,90\}\,
  111. 5 x { 90 , 180 , 270 , 360 , 450 } 5x\in\{90,180,270,360,450\}\,
  112. x x
  113. 5 x 5x\,
  114. sin 5 x = 16 sin 5 x - 20 sin 3 x + 5 sin x , \sin 5x=16\sin^{5}x-20\sin^{3}x+5\sin x,\,
  115. cos 5 x = 16 cos 5 x - 20 cos 3 x + 5 cos x . \cos 5x=16\cos^{5}x-20\cos^{3}x+5\cos x.\,
  116. sin 5 x = 0 \sin 5x=0\,
  117. cos 5 x = 0 \cos 5x=0\,
  118. y = sin x y=\sin x\,
  119. y = cos x y=\cos x\,
  120. y y\,
  121. 16 y 5 - 20 y 3 + 5 y = 0. 16y^{5}-20y^{3}+5y=0.\,
  122. y 2 y^{2}\,
  123. sin 5 x = 1 \sin 5x=1\,
  124. cos 5 x = 1 \cos 5x=1\,
  125. y = sin x y=\sin x\,
  126. y = cos x y=\cos x\,
  127. y y\,
  128. 16 y 5 - 20 y 3 + 5 y - 1 = 0 , 16y^{5}-20y^{3}+5y-1=0,\,
  129. ( y - 1 ) ( 4 y 2 + 2 y - 1 ) 2 = 0. (y-1)(4y^{2}+2y-1)^{2}=0.\,
  130. π \pi
  131. π \pi
  132. π \pi
  133. a ± b c \sqrt{a\pm b\sqrt{c}}\,
  134. R = a 2 - b 2 c R=\sqrt{a^{2}-b^{2}c}\,
  135. d = a + R 2 and e = a - R 2 d=\frac{a+R}{2}\,\text{ and }e=\frac{a-R}{2}\,
  136. a ± b c = d ± e . \sqrt{a\pm b\sqrt{c}}=\sqrt{d}\pm\sqrt{e}.\,
  137. 4 sin 18 = 6 - 2 5 = 5 - 1. 4\sin 18^{\circ}=\sqrt{6-2\sqrt{5}}=\sqrt{5}-1.\,
  138. 4 sin 15 = 2 2 - 3 = 2 ( 3 - 1 ) . 4\sin 15^{\circ}=2\sqrt{2-\sqrt{3}}=\sqrt{2}(\sqrt{3}-1).
  139. π \pi
  140. π \pi
  141. π \pi
  142. π \pi
  143. π \pi
  144. π \pi
  145. π \pi
  146. π \pi
  147. π \pi
  148. π \pi
  149. π \pi
  150. π \pi
  151. π \pi
  152. π \pi
  153. π \pi
  154. π \pi
  155. π \pi
  156. π \pi
  157. π \pi
  158. π \pi
  159. π \pi
  160. π \pi

Exergonic_process.html

  1. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\cdot\Delta S
  2. Δ R G < 0 \mathrm{\Delta}_{\mathrm{R}}G<0
  3. Δ R G > 0 \mathrm{\Delta}_{\mathrm{R}}G>0
  4. Δ R G \mathrm{\Delta}_{\mathrm{R}}G

Exergy.html

  1. d B d t 0 is equivalent to d S t o t a l d t 0 (1) \frac{\mathrm{d}B}{\mathrm{d}t}\leq 0\mbox{ is equivalent to }~{}\frac{\mathrm% {d}S_{total}}{\mathrm{d}t}\geq 0\qquad\mbox{(1)}~{}
  2. B = U + P R V - T R S - i μ i , R N i (2) B=U+P_{R}V-T_{R}S-\sum_{i}\mu_{i,R}N_{i}\qquad\mbox{(2)}~{}
  3. P R V P_{R}V
  4. T R S T_{R}S
  5. B = U [ μ 1 , μ 2 , μ n ] + P R V - T R S = U [ s y m b o l μ ] + P R V - T R S (3) B=U[\mu_{1},\mu_{2},...\mu_{n}]+P_{R}V-T_{R}S=U[symbol{\mu}]+P_{R}V-T_{R}S% \qquad\mbox{(3)}~{}
  6. d B d T R = - S (4) \frac{\mathrm{d}B}{\mathrm{d}T_{R}}=-S\qquad\mbox{(4)}~{}
  7. U U
  8. U [ s y m b o l μ ] U[symbol{\mu}]
  9. B = H - T R S (5) B=H-T_{R}S\qquad\mbox{(5)}~{}
  10. B = A + P R V (6) B=A+P_{R}V\qquad\mbox{(6)}~{}
  11. B = G (7) B=G\qquad\mbox{(7)}~{}
  12. e ¯ c h = [ g ¯ F + ( a + b 4 - c 2 ) g ¯ O 2 - a g ¯ CO 2 - b 2 g ¯ H 2 O ( g ) ] ( T 0 , p 0 ) + R ¯ T 0 l n [ ( y O 2 e ) a + b 4 - c 2 ( y CO 2 e ) a ( y H 2 O e ) b 2 ] (8) \bar{e}^{ch}=\left[\bar{g}_{\mathrm{F}}+\left(a+\frac{b}{4}-\frac{c}{2}\right)% \bar{g}_{\mathrm{O_{2}}}-a\bar{g}_{\mathrm{CO_{2}}}-\,\frac{b}{2}\bar{g}_{% \mathrm{H_{2}O}(g)}\right]\,\left(T_{0,}p_{0}\right)+\bar{R}T_{0}\,ln\left[% \frac{{{(y}_{\mathrm{O_{2}}}^{e})}^{a+\frac{b}{4}-\,\frac{c}{2}}}{\left(y_{% \mathrm{CO_{2}}}^{e}\right)^{a}\left(y_{\mathrm{H_{2}O}}^{e}\right)^{\frac{b}{% 2}}}\right]\mbox{(8)}~{}
  13. g ¯ x = \bar{g}_{x}=
  14. ( T 0 , p 0 ) \left(T_{0,}p_{0}\right)
  15. g ¯ F \bar{g}_{F}
  16. R ¯ = \bar{R}=
  17. T 0 = T_{0}=
  18. y x e = y_{x}^{e}=\,
  19. e ¯ c h = [ g ¯ F + ( a + b 4 - c 2 ) g ¯ O 2 - a g ¯ CO 2 - b 2 g ¯ H 2 O ( g ) ] ( T 0 , p 0 ) + a e ¯ CO 2 c h + ( b 2 ) e ¯ H 2 O ( l ) c h - ( a + b 4 ) e ¯ O 2 c h (9) \bar{e}^{ch}=\left[\bar{g}_{\mathrm{F}}+\left(a+\frac{b}{4}-\frac{c}{2}\right)% \bar{g}_{\mathrm{O_{2}}}-a\bar{g}_{\mathrm{CO_{2}}}-\,\frac{b}{2}\bar{g}_{% \mathrm{H_{2}O}(g)}\right]\,\left(T_{0,}p_{0}\right)+a\bar{e}_{\mathrm{CO_{2}}% }^{ch}+\,\left(\frac{b}{2}\right)\bar{e}_{\mathrm{H_{2}O}(l)}^{ch}-\,\left(a+% \,\frac{b}{4}\right)\bar{e}_{\mathrm{O_{2}}}^{ch}\mbox{(9)}~{}
  20. e ¯ x c h = \bar{e}_{x}^{ch}=\,
  21. I = T 0 s g e n (10) I=T_{0}s_{gen}\qquad\mbox{(10)}~{}
  22. s g e n s_{gen}
  23. I > 0 I>0
  24. I = 0 I=0
  25. w a c t = w m a x + I (11) w_{act}=w_{max}+I\qquad\mbox{(11)}~{}
  26. w m a x = Δ B = B i n - B o u t (12) w_{max}=\Delta B=B_{in}-B_{out}\qquad\mbox{(12)}~{}
  27. Δ B = - I (13) \Delta B=-I\qquad\mbox{(13)}~{}
  28. If d B d t { > 0 , d B d t = maximum power generated < 0 , d B d t = minimum power required (14) \mbox{If }~{}\frac{\mathrm{d}B}{\mathrm{d}t}\begin{cases}>0,&\frac{\mathrm{d}B% }{\mathrm{d}t}=\mbox{ maximum power generated}\\ <0,&\frac{\mathrm{d}B}{\mathrm{d}t}=\mbox{ minimum power required}\end{cases}% \qquad\mbox{(14)}~{}
  29. η = 1 - T C T H (15) \eta=1-\frac{T_{C}}{T_{H}}\qquad\mbox{(15)}~{}
  30. B = Q ( 1 - T o T s o u r c e ) (16) \ B=Q(1-\frac{T_{o}}{T_{source}})\qquad\mbox{(16)}~{}

Exotic_baryon.html

  1. S 0 \mbox{S}~{}^{0}

Exotic_sphere.html

  1. b P n + 1 bP_{n+1}
  2. Θ n / b P n + 1 \Theta_{n}/bP_{n+1}
  3. n = 4 k + 3 , n=4k+3,
  4. 2 2 n - 2 ( 2 2 n - 1 - 1 ) B 2^{2n-2}(2^{2n-1}-1)B\,\!
  5. Θ n / b P n + 1 π n S / J \Theta_{n}/bP_{n+1}\to\pi_{n}^{S}/J\,
  6. n = 4 k + 3 , n=4k+3,
  7. π n S / J \pi_{n}^{S}/J\,
  8. Θ n / b P n + 1 \Theta_{n}/bP_{n+1}
  9. 254 = 2 8 - 2 254=2^{8}-2
  10. a 2 + b 2 + c 2 + d 3 + e 6 k - 1 = 0 a^{2}+b^{2}+c^{2}+d^{3}+e^{6k-1}=0
  11. Γ n \Gamma_{n}
  12. π 0 Diff + ( D n ) π 0 Diff + ( S n - 1 ) Γ n 0. \pi_{0}\,\,\text{Diff}^{+}(D^{n})\to\pi_{0}\,\,\text{Diff}^{+}(S^{n-1})\to% \Gamma_{n}\to 0.\,\!
  13. π 0 Diff + ( D n ) \pi_{0}\,\,\text{Diff}^{+}(D^{n})
  14. n 6 n\geq 6
  15. Γ n π 0 Diff + ( S n - 1 ) \Gamma_{n}\simeq\pi_{0}\,\,\text{Diff}^{+}(S^{n-1})
  16. n 6 n\geq 6

Expected_utility_hypothesis.html

  1. A B A\succeq B
  2. A B A\preceq B
  3. A B A\succeq B
  4. B C B\succeq C
  5. A C A\succeq C
  6. A B A\succeq B
  7. t ( 0 , 1 ] t\in(0,1]
  8. t A + ( 1 - t ) C t B + ( 1 - t ) C tA+(1-t)C\succeq tB+(1-t)C
  9. A B C A\succeq B\succeq C
  10. p A + ( 1 - p ) C pA+(1-p)C
  11. \succeq
  12. 𝐴𝑅𝐴 ( w ) = - u ′′ ( w ) u ( w ) \mathit{ARA}(w)=-\frac{u^{\prime\prime}(w)}{u^{\prime}(w)}
  13. 𝑅𝑅𝐴 ( w ) = - w u ′′ ( w ) u ( w ) \mathit{RRA}(w)=-\frac{wu^{\prime\prime}(w)}{u^{\prime}(w)}
  14. u ( w ) = log ( w ) u(w)=\log(w)
  15. u ( w ) = - e - a w u(w)=-e^{-aw}
  16. K - e - a w K-e^{-aw}
  17. - e - a w -e^{-aw}
  18. - e - a w -e^{-aw}
  19. u ( w ) = log ( w ) u(w)=\log(w)
  20. u ( w ) = w α u(w)=w^{\alpha}
  21. α ( 0 , 1 ) \alpha\in(0,1)
  22. 1 - α 1-\alpha
  23. u ( w ) = - w α u(w)=-w^{\alpha}
  24. α < 0 \alpha<0
  25. 1 - α 1-\alpha
  26. u ( w ) = w - b e - a w u(w)=w-be^{-aw}
  27. E [ u ( w ) ] = E [ w ] - b E [ e - a w ] = E [ w ] - b E [ e - a E [ w ] - a ( w - E [ w ] ) ] = E [ w ] - b e - a E [ w ] E [ e - a ( w - E [ w ] ) ] = Expected wealth - b e - a Expected wealth Risk . \begin{aligned}\displaystyle\operatorname{E}[u(w)]&\displaystyle=\operatorname% {E}[w]-b\operatorname{E}[e^{-aw}]\\ &\displaystyle=\operatorname{E}[w]-b\operatorname{E}[e^{-a\operatorname{E}[w]-% a(w-\operatorname{E}[w])}]\\ &\displaystyle=\operatorname{E}[w]-be^{-a\operatorname{E}[w]}\operatorname{E}[% e^{-a(w-\operatorname{E}[w])}]\\ &\displaystyle=\,\text{Expected wealth}-b\cdot e^{-a\cdot\,\text{Expected % wealth}}\cdot\,\text{Risk}.\end{aligned}
  28. E ( e - a ( w - E w ) ) \operatorname{E}(e^{-a(w-\operatorname{E}w)})
  29. a a

Expenditure_function.html

  1. u u
  2. e ( p , u * ) : 𝐑 + n × 𝐑 𝐑 e(p,u^{*}):\textbf{R}^{n}_{+}\times\textbf{R}\rightarrow\textbf{R}
  3. u * u^{*}
  4. p p
  5. e ( p , u * ) = min x ( u * ) p x e(p,u^{*})=\min_{x\in\geq(u^{*})}p\cdot x
  6. ( u * ) = { x 𝐑 + n : u ( x ) u * } \geq(u^{*})=\{x\in\textbf{R}^{n}_{+}:u(x)\geq u^{*}\}
  7. u * u^{*}
  8. x 1 p 1 + + x n p n x_{1}p_{1}+\dots+x_{n}p_{n}
  9. u ( x 1 , , x n ) u * , u(x_{1},\dots,x_{n})\geq u^{*},
  10. x 1 * , x n * x_{1}^{*},\dots x_{n}^{*}
  11. u * u^{*}
  12. e ( p 1 , , p n ; u * ) = p 1 x 1 * + + p n x n * . e(p_{1},\dots,p_{n};u^{*})=p_{1}x_{1}^{*}+\dots+p_{n}x_{n}^{*}.

Expenditure_minimization_problem.html

  1. u u
  2. L L
  3. p p
  4. u * u^{*}
  5. e ( p , u * ) = min x u * p x e(p,u^{*})=\min_{x\in\geq{u^{*}}}p\cdot x
  6. u * = { x + L : u ( x ) u * } \geq{u^{*}}=\{x\in\mathbb{R}^{L}_{+}:u(x)\geq u^{*}\}
  7. u * u^{*}
  8. h ( p , u * ) h(p,u^{*})
  9. h ( p , u * ) = x ( p , e ( p , u * ) ) . h(p,u^{*})=x(p,e(p,u^{*})).\,

Exponential_backoff.html

  1. 1 N + 1 i = 0 N i \frac{1}{N+1}\sum_{i=0}^{N}i
  2. N = 2 c - 1 N=2^{c}-1
  3. N = 2 3 - 1 = 8 - 1 N=2^{3}-1=8-1
  4. N = 7 N=7
  5. E ( c ) = 1 N + 1 i = 0 N i \operatorname{E}(c)=\frac{1}{N+1}\sum_{i=0}^{N}i
  6. E ( 3 ) = 1 7 + 1 i = 0 7 i = 1 8 ( 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 ) = 28 8 \operatorname{E}(3)=\frac{1}{7+1}\sum_{i=0}^{7}i=\frac{1}{8}(0+1+2+3+4+5+6+7)=% \frac{28}{8}
  7. E ( 3 ) = 3.5 \operatorname{E}(3)=3.5
  8. E ( c ) = 2 c - 1 2 \operatorname{E}(c)=\frac{2^{c}-1}{2}
  9. ( 2 c - 1 ) 2 c 2 = N ( N + 1 ) 2 \frac{(2^{c}-1)2^{c}}{2}=\frac{N(N+1)}{2}
  10. N 2 = 2 c - 1 2 \frac{N}{2}=\frac{2^{c}-1}{2}

Exponential_formula.html

  1. f ( x ) = a 1 x + a 2 2 x 2 + a 3 6 x 3 + + a n n ! x n + f(x)=a_{1}x+{a_{2}\over 2}x^{2}+{a_{3}\over 6}x^{3}+\cdots+{a_{n}\over n!}x^{n% }+\cdots\,
  2. exp f ( x ) = e f ( x ) = n = 0 b n n ! x n , \exp f(x)=e^{f(x)}=\sum_{n=0}^{\infty}{b_{n}\over n!}x^{n},\,
  3. b n = π = { S 1 , , S k } a | S 1 | a | S k | , b_{n}=\sum_{\pi=\left\{\,S_{1},\,\dots,\,S_{k}\,\right\}}a_{\left|S_{1}\right|% }\cdots a_{\left|S_{k}\right|},
  4. b n = B n ( a 1 , a 2 , , a n ) , b_{n}=B_{n}(a_{1},a_{2},\dots,a_{n}),
  5. exp ( n = 1 a n n ! x n ) = n = 0 B n ( a 1 , , a n ) n ! x n , \exp\left(\sum_{n=1}^{\infty}{a_{n}\over n!}x^{n}\right)=\sum_{n=0}^{\infty}{B% _{n}(a_{1},\dots,a_{n})\over n!}x^{n},
  6. b 3 = B 3 ( a 1 , a 2 , a 3 ) = a 3 + 3 a 2 a 1 + a 1 3 , b_{3}=B_{3}(a_{1},a_{2},a_{3})=a_{3}+3a_{2}a_{1}+a_{1}^{3},

Exponential_hierarchy.html

  1. 2 c n 2^{cn}
  2. 2 n c 2^{n^{c}}
  3. Σ k E \Sigma^{E}_{k}
  4. Σ k E = NE Σ k - 1 P \Sigma^{E}_{k}=\mathrm{NE}^{\Sigma^{P}_{k-1}}
  5. 2 c n 2^{cn}
  6. Σ k - 1 P \Sigma^{P}_{k-1}
  7. Π k E = coNE Σ k - 1 P \Pi^{E}_{k}=\mathrm{coNE}^{\Sigma^{P}_{k-1}}
  8. Δ k E = E Σ k - 1 P \Delta^{E}_{k}=\mathrm{E}^{\Sigma^{P}_{k-1}}
  9. Σ k E \Sigma^{E}_{k}
  10. x L y 1 y 2 Q y k R ( x , y 1 , , y k ) , x\in L\iff\exists y_{1}\,\forall y_{2}\dots Qy_{k}\,R(x,y_{1},\dots,y_{k}),
  11. R ( x , y 1 , , y n ) R(x,y_{1},\dots,y_{n})
  12. 2 c | x | 2^{c|x|}
  13. 2 c n 2^{cn}
  14. Σ k E X P \Sigma^{EXP}_{k}
  15. Σ k E X P = NEXP Σ k - 1 P \Sigma^{EXP}_{k}=\mathrm{NEXP}^{\Sigma^{P}_{k-1}}
  16. 2 n c 2^{n^{c}}
  17. Σ k - 1 P \Sigma^{P}_{k-1}
  18. Π k E X P = coNEXP Σ k - 1 P \Pi^{EXP}_{k}=\mathrm{coNEXP}^{\Sigma^{P}_{k-1}}
  19. Δ k E X P = EXP Σ k - 1 P \Delta^{EXP}_{k}=\mathrm{EXP}^{\Sigma^{P}_{k-1}}
  20. Σ k E X P \Sigma^{EXP}_{k}
  21. x L y 1 y 2 Q y k R ( x , y 1 , , y k ) , x\in L\iff\exists y_{1}\,\forall y_{2}\dots Qy_{k}\,R(x,y_{1},\dots,y_{k}),
  22. R ( x , y 1 , , y k ) R(x,y_{1},\dots,y_{k})
  23. 2 | x | c 2^{|x|^{c}}
  24. 2 n c 2^{n^{c}}

Exponential_integral.html

  1. Ei ( x ) = - - x e - t t d t . \operatorname{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}{t}\,dt.\,
  2. \infty
  3. E 1 ( z ) = z e - t t d t , | Arg ( z ) | < π \mathrm{E}_{1}(z)=\int_{z}^{\infty}\frac{e^{-t}}{t}\,dt,\qquad|{\rm Arg}(z)|<\pi
  4. z z
  5. E 1 ( z ) = 1 e - t z t d t = 0 1 e - z / u u d u , ( z ) 0. \mathrm{E}_{1}(z)=\int_{1}^{\infty}\frac{e^{-tz}}{t}\,dt=\int_{0}^{1}\frac{e^{% -z/u}}{u}\,du,\qquad\Re(z)\geq 0.
  6. lim δ 0 + E 1 ( - x ± i δ ) = - Ei ( x ) i π , x > 0 , \lim_{\delta\to 0+}\mathrm{E_{1}}(-x\pm i\delta)=-\mathrm{Ei}(x)\mp i\pi,% \qquad x>0,
  7. e - t / t e^{-t}/t
  8. E 1 ( x ) \mathrm{E_{1}}(x)
  9. x x
  10. Ei ( x ) = γ + ln | x | + k = 1 x k k k ! x 0 \mathrm{Ei}(x)=\gamma+\ln|x|+\sum_{k=1}^{\infty}\frac{x^{k}}{k\;k!}\qquad x\neq 0
  11. E 1 ( z ) = - γ - ln z - k = 1 ( - z ) k k k ! ( | Arg ( z ) | < π ) \mathrm{E_{1}}(z)=-\gamma-\ln z-\sum_{k=1}^{\infty}\frac{(-z)^{k}}{k\;k!}% \qquad(|\mathrm{Arg}(z)|<\pi)
  12. γ \gamma
  13. z z
  14. E 1 ( x ) \mathrm{E_{1}}(x)
  15. x x
  16. x > 2.5 x>2.5
  17. Ei ( x ) = γ + ln x + exp ( x / 2 ) n = 1 ( - 1 ) n - 1 x n n ! 2 n - 1 k = 0 ( n - 1 ) / 2 1 2 k + 1 {\rm Ei}(x)=\gamma+\ln x+\exp{(x/2)}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}x^{n}}{% n!\,2^{n-1}}\sum_{k=0}^{\lfloor(n-1)/2\rfloor}\frac{1}{2k+1}
  18. N ~{}N~{}
  19. z e z E 1 ( z ) ze^{z}\mathrm{E_{1}}(z)
  20. E 1 ( z ) = exp ( - z ) z n = 0 N - 1 n ! ( - z ) n \mathrm{E_{1}}(z)=\frac{\exp(-z)}{z}\sum_{n=0}^{N-1}\frac{n!}{(-z)^{n}}
  21. O ( N ! z - N ) O(N!z^{-N})
  22. Re ( z ) \mathrm{Re}(z)
  23. N N
  24. N = 1 N=1
  25. N = 5 N=5
  26. E 1 \mathrm{E_{1}}
  27. E 1 \mathrm{E_{1}}
  28. E 1 \mathrm{E_{1}}
  29. 1 2 e - x ln ( 1 + 2 x ) < E 1 ( x ) < e - x ln ( 1 + 1 x ) x > 0 \frac{1}{2}e^{-x}\,\ln\!\left(1+\frac{2}{x}\right)<\mathrm{E_{1}}(x)<e^{-x}\,% \ln\!\left(1+\frac{1}{x}\right)\qquad x>0
  30. E 1 ( x ) \mathrm{E_{1}}(x)
  31. Ei \mathrm{Ei}
  32. E 1 \mathrm{E_{1}}
  33. Ein \mathrm{Ein}
  34. Ein ( z ) = 0 z ( 1 - e - t ) d t t = k = 1 ( - 1 ) k + 1 z k k k ! \mathrm{Ein}(z)=\int_{0}^{z}(1-e^{-t})\frac{dt}{t}=\sum_{k=1}^{\infty}\frac{(-% 1)^{k+1}z^{k}}{k\;k!}
  35. E 1 \mathrm{E_{1}}
  36. E 1 ( z ) = - γ - ln z + Ein ( z ) | Arg ( z ) | < π \mathrm{E_{1}}(z)\,=\,-\gamma-\ln z+{\rm Ein}(z)\qquad|\mathrm{Arg}(z)|<\pi
  37. Ei ( x ) = γ + ln x - Ein ( - x ) x > 0 \mathrm{Ei}(x)\,=\,\gamma+\ln x-\mathrm{Ein}(-x)\qquad x>0
  38. li ( x ) = Ei ( ln x ) \mathrm{li}(x)=\mathrm{Ei}(\ln x)\,
  39. x x
  40. E n ( x ) = 1 e - x t t n d t , {\rm E}_{n}(x)=\int_{1}^{\infty}\frac{e^{-xt}}{t^{n}}\,dt,
  41. E n ( x ) = x n - 1 Γ ( 1 - n , x ) . {\rm E}_{n}(x)=x^{n-1}\Gamma(1-n,x).\,
  42. φ m ( x ) \varphi_{m}(x)
  43. φ m ( x ) = E - m ( x ) . \varphi_{m}(x)={\rm E}_{-m}(x).\,
  44. E s j ( z ) = 1 Γ ( j + 1 ) 1 ( log t ) j e - z t t s d t E_{s}^{j}(z)=\frac{1}{\Gamma(j+1)}\int_{1}^{\infty}(\log t)^{j}\frac{e^{-zt}}{% t^{s}}\,dt
  45. Ei ( a b ) = e a b d a d b \mathrm{Ei}(a\cdot b)=\iint e^{ab}\,da\,db
  46. d ( n ) d(n)
  47. n n
  48. n = 1 d ( n ) x n = a = 1 b = 1 x a b \sum\limits_{n=1}^{\infty}d(n)x^{n}=\sum\limits_{a=1}^{\infty}\sum\limits_{b=1% }^{\infty}x^{ab}
  49. E n \mathrm{E_{n}}
  50. E n ( z ) = - E n - 1 ( z ) ( n = 1 , 2 , 3 , ) \mathrm{E_{n}}^{\prime}(z)=-\mathrm{E_{n-1}}(z)\qquad(n=1,2,3,\ldots)
  51. E 0 \mathrm{E_{0}}
  52. e - z / z e^{-z}/z
  53. E 1 ( i x ) \mathrm{E_{1}}(ix)
  54. x x
  55. z z
  56. E 1 ( z ) = 1 e - t z t d t \mathrm{E_{1}}(z)=\int_{1}^{\infty}\frac{e^{-tz}}{t}dt
  57. Si \mathrm{Si}
  58. Ci \mathrm{Ci}
  59. E 1 ( i x ) = i ( - 1 2 π + Si ( x ) ) - Ci ( x ) ( x > 0 ) \mathrm{E_{1}}(ix)=i\left(-\tfrac{1}{2}\pi+\mathrm{Si}(x)\right)-\mathrm{Ci}(x% )\qquad(x>0)
  60. E 1 ( x ) \mathrm{E_{1}}(x)

Ext_functor.html

  1. Ext R n ( A , B ) = ( R n T ) ( B ) . \operatorname{Ext}_{R}^{n}(A,B)=(R^{n}T)(B).
  2. 0 B I 0 I 1 , 0\rightarrow B\rightarrow I^{0}\rightarrow I^{1}\rightarrow\dots,
  3. 0 Hom R ( A , I 0 ) Hom R ( A , I 1 ) . 0\rightarrow\operatorname{Hom}_{R}(A,I^{0})\rightarrow\operatorname{Hom}_{R}(A% ,I^{1})\rightarrow\dots.
  4. Ext R n ( A , B ) = ( R n G ) ( A ) . \operatorname{Ext}_{R}^{n}(A,B)=(R^{n}G)(A).
  5. P 1 P 0 A 0 , \dots\rightarrow P^{1}\rightarrow P^{0}\rightarrow A\rightarrow 0,
  6. 0 Hom R ( P 0 , B ) Hom R ( P 1 , B ) . 0\rightarrow\operatorname{Hom}_{R}(P^{0},B)\rightarrow\operatorname{Hom}_{R}(P% ^{1},B)\rightarrow\dots.
  7. 0 B E A 0. 0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0.
  8. 0 B E A 0 0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0
  9. 0 B E A 0 0\rightarrow B\rightarrow E^{\prime}\rightarrow A\rightarrow 0
  10. 0 B A B A 0. 0\rightarrow B\rightarrow A\oplus B\rightarrow A\rightarrow 0.
  11. 0 B E A 0 0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0
  12. Ext R 1 ( A , B ) . \operatorname{Ext}_{R}^{1}(A,B).
  13. 0 B E A 0 0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0
  14. 0 B E A 0 0\rightarrow B\rightarrow E^{\prime}\rightarrow A\rightarrow 0
  15. A A
  16. Γ = { ( e , e ) E E | g ( e ) = g ( e ) } . \Gamma=\left\{(e,e^{\prime})\in E\oplus E^{\prime}\;|\;g(e)=g^{\prime}(e^{% \prime})\right\}.
  17. Y = Γ / { ( f ( b ) , 0 ) - ( 0 , f ( b ) ) | b B } Y=\Gamma/\{(f(b),0)-(0,f^{\prime}(b))\;|\;b\in B\}
  18. ( f ( b ) + e , e ) ( e , f ( b ) + e ) (f(b)+e,e^{\prime})\sim(e,f^{\prime}(b)+e^{\prime})
  19. 0 B Y A 0 0\rightarrow B\rightarrow Y\rightarrow A\rightarrow 0
  20. b [ ( f ( b ) , 0 ) ] = [ ( 0 , f ( b ) ) ] b\mapsto[(f(b),0)]=[(0,f^{\prime}(b))]
  21. ( e , e ) g ( e ) = g ( e ) (e,e^{\prime})\mapsto g(e)=g^{\prime}(e^{\prime})
  22. 0 B X n X 1 A 0 0\rightarrow B\rightarrow X_{n}\rightarrow\cdots\rightarrow X_{1}\rightarrow A\rightarrow 0
  23. ξ : 0 B X n X 1 A 0 \xi:0\rightarrow B\rightarrow X_{n}\rightarrow\cdots\rightarrow X_{1}% \rightarrow A\rightarrow 0
  24. ξ : 0 B X n X 1 A 0 \xi^{\prime}:0\rightarrow B\rightarrow X^{\prime}_{n}\rightarrow\cdots% \rightarrow X^{\prime}_{1}\rightarrow A\rightarrow 0
  25. ξ \xi
  26. ξ \xi
  27. 0 B X n ′′ X n - 1 X n - 1 X 2 X 2 X 1 ′′ A 0. 0\rightarrow B\rightarrow X^{\prime\prime}_{n}\rightarrow X_{n-1}\oplus X^{% \prime}_{n-1}\rightarrow\cdots\rightarrow X_{2}\oplus X^{\prime}_{2}% \rightarrow X^{\prime\prime}_{1}\rightarrow A\rightarrow 0.
  28. Ext R n ( α A α , B ) α Ext R n ( A α , B ) \operatorname{Ext}^{n}_{R}\left(\bigoplus_{\alpha}A_{\alpha},B\right)\cong% \prod_{\alpha}\operatorname{Ext}^{n}_{R}(A_{\alpha},B)
  29. Ext R n ( A , β B β ) β Ext R n ( A , B β ) \operatorname{Ext}^{n}_{R}\left(A,\prod_{\beta}B_{\beta}\right)\cong\prod_{% \beta}\operatorname{Ext}^{n}_{R}(A,B_{\beta})
  30. X 1 Y n \cdots\rightarrow X_{1}\rightarrow Y_{n}\rightarrow\cdots
  31. 𝔤 \mathfrak{g}
  32. H * ( 𝔤 , M ) \operatorname{H}^{*}(\mathfrak{g},M)

Exterior_covariant_derivative.html

  1. T u P = H u V u T_{u}P=H_{u}\oplus V_{u}
  2. h : T u P H u h:T_{u}P\to H_{u}
  3. D ϕ ( v 0 , v 1 , , v k ) = d ϕ ( h v 0 , h v 1 , , h v k ) D\phi(v_{0},v_{1},\dots,v_{k})=d\phi(hv_{0},hv_{1},\dots,hv_{k})
  4. R g * ϕ = ρ ( g ) - 1 ϕ R_{g}^{*}\phi=\rho(g)^{-1}\phi
  5. R g ( u ) = u g R_{g}(u)=ug
  6. ρ : 𝔤 𝔤 𝔩 ( V ) . \rho:\mathfrak{g}\to\mathfrak{gl}(V).
  7. D ϕ = d ϕ + ρ ( ω ) ϕ , D\phi=d\phi+\rho(\omega)\cdot\phi,
  8. X # X^{\#}
  9. 𝔤 \mathfrak{g}
  10. d ϕ ( X u # ) = d d t | 0 ϕ ( u exp ( t X ) ) = - ρ ( X ) ϕ ( u ) = - ρ ( ω ( X u # ) ) ϕ ( u ) d\phi(X^{\#}_{u})={d\over dt}|_{0}\phi(u\operatorname{exp}(tX))=-\rho(X)\phi(u% )=-\rho(\omega(X^{\#}_{u}))\phi(u)
  11. D ϕ ( X ) = d ϕ ( X ) D\phi(X)=d\phi(X)
  12. ω ( X ) = 0 \omega(X)=0
  13. D ϕ ( X 0 , , X k ) - d ϕ ( X 0 , , X k ) = 1 k + 1 0 k ( - 1 ) i ρ ( ω ( X i ) ) ϕ ( X 0 , , X i ^ , , X k ) = 1 ( k + 1 ) ! 0 k ( - 1 ) i ρ ( ω ( X i ) ) σ : { 0 , , i ^ , , k } sgn ( σ ) ϕ ( X σ ( 0 ) , , X σ ( i ) ^ , , X σ ( k ) ) \begin{aligned}&\displaystyle D\phi(X_{0},\dots,X_{k})-d\phi(X_{0},\dots,X_{k}% )={1\over k+1}\sum_{0}^{k}(-1)^{i}\rho(\omega(X_{i}))\phi(X_{0},\dots,\widehat% {X_{i}},\dots,X_{k})\\ &\displaystyle={1\over(k+1)!}\sum_{0}^{k}(-1)^{i}\rho(\omega(X_{i}))\sum_{% \sigma:\{0,\dots,\widehat{i},\dots,k\}}\operatorname{sgn}(\sigma)\phi(X_{% \sigma(0)},\dots,\widehat{X_{\sigma(i)}},\dots,X_{\sigma(k)})\end{aligned}
  14. ( ρ ( ω ) ϕ ) ( X 0 , , X k ) (\rho(\omega)\cdot\phi)(X_{0},\cdots,X_{k})
  15. ρ ( ω ) \rho(\omega)
  16. 𝔤 𝔩 ( V ) \mathfrak{gl}(V)
  17. ( ρ ( ω ) ϕ ) ( v 1 , , v k + 1 ) = 1 / ( k + 1 ) ! σ sgn ( σ ) ρ ( ω ( v σ ( 1 ) ) ) ϕ ( v σ ( 2 ) , , v σ ( k + 1 ) ) . (\rho(\omega)\cdot\phi)(v_{1},\dots,v_{k+1})=1/{(k+1)}!\sum_{\sigma}% \operatorname{sgn}(\sigma)\rho(\omega(v_{\sigma(1)}))\phi(v_{\sigma(2)},\dots,% v_{\sigma(k+1)}).
  18. d Ω + ad ( ω ) Ω = 0 d\Omega+\operatorname{ad}(\omega)\cdot\Omega=0
  19. D 2 ϕ = F ϕ , D^{2}\phi=F\cdot\phi,
  20. ρ ( Ω ) = F = F i j e j i \rho(\Omega)=F=\sum{F^{i}}_{j}{e^{j}}_{i}
  21. e i j {e^{i}}_{j}
  22. F i j {F^{i}}_{j}
  23. : Γ ( M , E ) Γ ( M , T M E ) \nabla:\Gamma(M,E)\to\Gamma(M,TM\otimes E)
  24. X s = i X s \nabla_{X}s=i_{X}\nabla s
  25. i X i_{X}
  26. X s = ( h X ) s \nabla_{X}s=(hX)s
  27. X ϕ ¯ = D ϕ ( X ) ¯ = d ϕ ( h X ) ¯ = ( h X ) s \nabla_{X}\overline{\phi}=\overline{D\phi(X)}=\overline{d\phi(hX)}=(hX)s
  28. s = ϕ ¯ s=\overline{\phi}
  29. 2 F ( X , Y ) s = ( - [ X , Y ] + [ X , Y ] ) s 2F(X,Y)s=(-[\nabla_{X},\nabla_{Y}]+\nabla_{[X,Y]})s
  30. D 2 ϕ = ρ ( d ω ) ϕ + ρ ( ω ) ( ρ ( ω ) ϕ ) = ρ ( d ω ) ϕ + 1 2 ρ ( [ ω ω ] ) ϕ , D^{2}\phi=\rho(d\omega)\cdot\phi+\rho(\omega)\cdot(\rho(\omega)\cdot\phi)=\rho% (d\omega)\cdot\phi+{1\over 2}\rho([\omega\wedge\omega])\cdot\phi,
  31. ρ ( Ω ) ϕ \rho(\Omega)\cdot\phi

Extreme_physical_information.html

  1. \mathcal{I}
  2. 𝒥 \mathcal{J}
  3. - 𝒥 = Extremum \mathcal{I}-\mathcal{J}=\mathrm{Extremum}

Eyepiece.html

  1. MA = f O f E \mathrm{MA}=\frac{f_{O}}{f_{E}}
  2. f O f_{O}
  3. f E f_{E}
  4. MA = D D EO f O f E = D f E × D EO f O \mathrm{MA}=\frac{DD_{\mathrm{EO}}}{f_{O}f_{E}}=\frac{D}{f_{E}}\times\frac{D_{% \mathrm{EO}}}{f_{O}}
  5. D D
  6. D EO D_{\mathrm{EO}}
  7. f O f_{O}
  8. f E f_{E}
  9. P E P_{\mathrm{E}}
  10. P O P_{\mathrm{O}}
  11. P E = D f E , P O = D EO f O P_{\mathrm{E}}=\frac{D}{f_{E}},\qquad P_{\mathrm{O}}=\frac{D_{\mathrm{EO}}}{f_% {O}}
  12. MA = P E × P O \mathrm{MA}=P_{\mathrm{E}}\times P_{\mathrm{O}}
  13. D D
  14. F O V C = F O V P m a g FOV_{C}=\frac{FOV_{P}}{mag}
  15. F O V C = F O V P ( f T f E ) FOV_{C}=\frac{FOV_{P}}{(\frac{f_{T}}{f_{E}})}
  16. F O V C FOV_{C}
  17. F O V P FOV_{P}
  18. F O V P FOV_{P}
  19. m a g mag
  20. f T f_{T}
  21. f E f_{E}
  22. f T f_{T}
  23. F O V C = 57.3 d f T FOV_{C}=\frac{57.3d}{f_{T}}
  24. F O V C FOV_{C}
  25. d d
  26. f T f_{T}
  27. t a n A A O V 2 = m a g × t a n A O V 2 tan\frac{AAOV}{2}=mag\times tan\frac{AOV}{2}
  28. A A O V = 2 × a r c t a n 0.5 d f E {AAOV}=2\times arctan\frac{0.5d}{f_{E}}
  29. d = 1 2 ( f A + f B ) d=\frac{1}{2}(f_{A}+f_{B})
  30. f A f_{A}
  31. f B f_{B}

Émilie_du_Châtelet.html

  1. E k = 1 2 m v 2 E_{k}=\frac{1}{2}mv^{2}
  2. E k E_{k}
  3. m m
  4. v v
  5. E α 1 2 m v 2 E\alpha\,\!\frac{1}{2}mv^{2}

F-statistics.html

  1. p 2 ( 1 - F ) + p F for 𝐀𝐀 ; 2 p q ( 1 - F ) for 𝐀𝐚 ; and q 2 ( 1 - F ) + q F for 𝐚𝐚 . p^{2}(1-F)+pF\,\text{ for }\mathbf{AA};\ 2pq(1-F)\,\text{ for }\mathbf{Aa};\,% \text{ and }q^{2}(1-F)+qF\,\text{ for }\mathbf{aa}.
  2. F = 1 - O ( f ( 𝐀𝐚 ) ) E ( f ( 𝐀𝐚 ) ) = 1 - ObservedNumber ( 𝐀𝐚 ) n E ( f ( 𝐀𝐚 ) ) , F=1-\frac{\operatorname{O}(f(\mathbf{Aa}))}{\operatorname{E}(f(\mathbf{Aa}))}=% 1-\frac{\operatorname{ObservedNumber}(\mathbf{Aa})}{n\operatorname{E}(f(% \mathbf{Aa}))},\!
  3. E ( f ( 𝐀𝐚 ) ) = 2 p q , \operatorname{E}(f(\mathbf{Aa}))=2pq,\!
  4. p = 2 × obs ( A A ) + obs ( A a ) 2 × ( obs ( A A ) + obs ( A a ) + obs ( a a ) ) = 0.954 p={2\times\mathrm{obs}(AA)+\mathrm{obs}(Aa)\over 2\times(\mathrm{obs}(AA)+% \mathrm{obs}(Aa)+\mathrm{obs}(aa))}=0.954
  5. q = 1 - p = 0.046 q=1-p=0.046\,
  6. F = 1 - obs ( A a ) n 2 p q = 1 - 138 1612 * 2 ( 0.954 ) ( 0.046 ) = 0.023 F=1-\frac{\mathrm{obs}(Aa)}{n2pq}=1-{138\over 1612*2(0.954)(0.046)}=0.023
  7. ( 1 - F I S ) ( 1 - F S T ) = 1 - F I T , (1-F_{IS})(1-F_{ST})=1-F_{IT},\,
  8. 1 - F I T = ( 1 - F I S ) ( 1 - F S T ) . 1-F_{IT}=(1-F_{IS})\,(1-F_{ST}).\!
  9. 1 - F = i = 0 i = I ( 1 - F i , i + 1 ) 1-F=\prod_{i=0}^{i=I}(1-F_{i,i+1})\!
  10. F S T = var ( 𝐩 ) p ( 1 - p ) F_{ST}=\frac{\operatorname{var}(\mathbf{p})}{p\,(1-p)}\!

F-theory.html

  1. 10 500 10^{500}

Factorial_moment.html

  1. r r
  2. r r
  3. X X
  4. E [ ( X ) r ] = E [ X ( X - 1 ) ( X - 2 ) ( X - r + 1 ) ] , \operatorname{E}\bigl[(X)_{r}\bigr]=\operatorname{E}\bigl[X(X-1)(X-2)\cdots(X-% r+1)\bigr],
  5. E E
  6. ( x ) r = x ( x - 1 ) ( x - 2 ) ( x - r + 1 ) (x)_{r}=x(x-1)(x-2)\cdots(x-r+1)
  7. X X
  8. λ 0 λ≥0
  9. X X
  10. E [ ( X ) r ] = λ r , r 0 . \operatorname{E}\bigl[(X)_{r}\bigr]=\lambda^{r},\qquad r\in\mathbb{N}_{0}.
  11. X X
  12. p p∈
  13. [ 0 , 1 ] [0,1]
  14. n n
  15. X X
  16. E [ ( X ) r ] = n ! ( n - r ) ! p r , r { 0 , 1 , , n } , \operatorname{E}\bigl[(X)_{r}\bigr]=\frac{n!}{(n-r)!}p^{r},\qquad r\in\{0,1,% \ldots,n\},
  17. ! !
  18. r > n r>n
  19. X X
  20. N N
  21. X X
  22. E [ ( X ) r ] = K ! ( K - r ) ! n ! ( n - r ) ! ( N - r ) ! N ! , r { 0 , 1 , , min { n , K } } . \operatorname{E}\bigl[(X)_{r}\bigr]=\frac{K!}{(K-r)!}\frac{n!}{(n-r)!}\frac{(N% -r)!}{N!},\qquad r\in\{0,1,\ldots,\min\{n,K\}\}.
  23. r r
  24. X X
  25. α > 0 α>0
  26. β > 0 β>0
  27. n n
  28. X X
  29. E [ ( X ) r ] = n ! ( n - r ) ! B ( α + r , β ) B ( α , β ) , r { 0 , 1 , , n } , \operatorname{E}\bigl[(X)_{r}\bigr]=\frac{n!}{(n-r)!}\frac{B(\alpha+r,\beta)}{% B(\alpha,\beta)},\qquad r\in\{0,1,\ldots,n\},
  30. B B
  31. r > n r>n
  32. n n
  33. X X
  34. E [ X n ] = r = 0 n { n r } E [ ( X ) r ] , n 0 , \operatorname{E}\bigl[X^{n}\bigr]=\sum_{r=0}^{n}\biggl\{{n\atop r}\biggr\}% \operatorname{E}\bigl[(X)_{r}\bigr],\qquad n\in\mathbb{N}_{0},

Factorial_number_system.html

  1. \leq
  2. i = 0 n i i ! = ( n + 1 ) ! - 1. \sum_{i=0}^{n}{i\cdot i!}={(n+1)!}-1.
  3. 1 / 2 = 0.0 1 ! 1/2=0.0\ 1_{!}
  4. 1 / 3 = 0.0 0 2 ! 1/3=0.0\ 0\ 2_{!}
  5. 1 / 4 = 0.0 0 1 2 ! 1/4=0.0\ 0\ 1\ 2_{!}
  6. 1 / 5 = 0.0 0 1 0 4 ! 1/5=0.0\ 0\ 1\ 0\ 4_{!}
  7. 1 / 6 = 0.0 0 1 ! 1/6=0.0\ 0\ 1_{!}
  8. 1 / 11 = 0.0 0 0 2 0 5 3 1 4 0 A ! 1/11=0.0\ 0\ 0\ 2\ 0\ 5\ 3\ 1\ 4\ 0\ A_{!}
  9. 2 / 11 = 0.0 0 1 0 1 4 6 2 8 1 9 ! 2/11=0.0\ 0\ 1\ 0\ 1\ 4\ 6\ 2\ 8\ 1\ 9_{!}
  10. 9 / 11 = 0.0 1 1 3 3 1 0 5 0 8 2 ! 9/11=0.0\ 1\ 1\ 3\ 3\ 1\ 0\ 5\ 0\ 8\ 2_{!}
  11. 1 / 15 = 0.0 0 0 1 3 ! 1/15=0.0\ 0\ 0\ 1\ 3_{!}
  12. 237 / 360 = 0.0 1 0 3 4 ! 237/360=0.0\ 1\ 0\ 3\ 4_{!}
  13. e = 1 0.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1... ! e=1\ 0.0\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1..._{!}
  14. e - 1 = 0.0 0 2 0 4 0 6 0 8 0 A 0 C 0 E ! e^{-1}=0.0\ 0\ 2\ 0\ 4\ 0\ 6\ 0\ 8\ 0\ A\ 0\ C\ 0\ E..._{!}
  15. sin ( 1 ) = 0.0 1 2 0 0 5 6 0 0 9 A 0 0 D E ! \sin(1)=0.0\ 1\ 2\ 0\ 0\ 5\ 6\ 0\ 0\ 9\ A\ 0\ 0\ D\ E..._{!}
  16. cos ( 1 ) = 0.0 1 0 0 4 5 0 0 8 9 0 0 C D 0... ! \cos(1)=0.0\ 1\ 0\ 0\ 4\ 5\ 0\ 0\ 8\ 9\ 0\ 0\ C\ D\ 0..._{!}
  17. sinh ( 1 ) = 1.0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0... ! \sinh(1)=1.0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0..._{!}
  18. cosh ( 1 ) = 1.0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1... ! \cosh(1)=1.0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1..._{!}

Faddeev–Popov_ghost.html

  1. c a ( x ) c^{a}(x)\,
  2. a a
  3. ghost = μ c ¯ a μ c a + g f a b c ( μ c ¯ a ) A μ b c c . \mathcal{L}_{\,\text{ghost}}=\partial_{\mu}\bar{c}^{a}\partial^{\mu}c^{a}+gf^{% abc}\left(\partial^{\mu}\bar{c}^{a}\right)A_{\mu}^{b}c^{c}\;.
  4. f a b c = 0 f^{abc}=0

Fair_division.html

  1. X X
  2. n n
  3. X X
  4. n n
  5. X = X 1 X 2 X n X=X_{1}\sqcup X_{2}\sqcup\cdots\sqcup X_{n}
  6. X X
  7. X = { p i a n o , c a r , a p a r t m e n t } X=\{piano,car,apartment\}
  8. n n
  9. V i V_{i}
  10. X X
  11. V i ( ) = 0 V_{i}()=0
  12. V i ( X ) = 1 V_{i}(X)=1
  13. X X
  14. X X
  15. V i ( X i ) 1 / n V_{i}(X_{i})\geq 1/n
  16. V i ( X i ) V i ( X j ) V_{i}(X_{i})\geq V_{i}(X_{j})
  17. V i ( X i ) = 1 / n V_{i}(X_{i})=1/n
  18. V i ( X i ) = V j ( X j ) V_{i}(X_{i})=V_{j}(X_{j})

Fall_factor.html

  1. f = h L f=\frac{h}{L}
  2. F m a x = m g + ( m g ) 2 + 2 m g h k = m g + ( m g ) 2 + 2 m g E q f F_{max}=mg+\sqrt{(mg)^{2}+2mghk}=mg+\sqrt{(mg)^{2}+2mgEqf}
  3. m g h = 1 2 k x m a x 2 - m g x m a x ; F m a x = k x m a x mgh=\frac{1}{2}kx_{max}^{2}-mgx_{max}\ ;\ F_{max}=kx_{max}
  4. F m a x = m g + ( m g ) 2 + F 0 ( F 0 - 2 m 0 g ) m m 0 f f 0 F_{max}=mg+\sqrt{(mg)^{2}+F_{0}(F_{0}-2m_{0}g)\frac{m}{m_{0}}\frac{f}{f_{0}}}

False_position_method.html

  1. a x = b , ax=b,
  2. f ( x ) = b , f(x)=b,
  3. f ( x 1 ) = b 1 , f ( x 2 ) = b 2 . f(x_{1})=b_{1},\qquad f(x_{2})=b_{2}.
  4. f ( x ) = a x + c , f(x)=ax+c,
  5. c k = b k - f ( b k ) ( b k - a k ) f ( b k ) - f ( a k ) c_{k}=b_{k}-f(b_{k})\frac{(b_{k}-a_{k})}{f(b_{k})-f(a_{k})}
  6. y - f ( b k ) = f ( b k ) - f ( a k ) b k - a k ( x - b k ) . y-f(b_{k})=\frac{f(b_{k})-f(a_{k})}{b_{k}-a_{k}}(x-b_{k}).
  7. y = 0 y=0
  8. f ( b k ) + f ( b k ) - f ( a k ) b k - a k ( c k - b k ) = 0. f(b_{k})+\frac{f(b_{k})-f(a_{k})}{b_{k}-a_{k}}(c_{k}-b_{k})=0.
  9. f ( b k ) + f ( b k ) - f ( a k ) b k - a k ( c k - b k ) = 0 f ( b k ) - f ( a k ) b k - a k ( c k - b k ) = - f ( b k ) ( c k - b k ) = - f ( b k ) b k - a k f ( b k ) - f ( a k ) c k = b k - f ( b k ) b k - a k f ( b k ) - f ( a k ) \begin{aligned}\displaystyle f(b_{k})+\frac{f(b_{k})-f(a_{k})}{b_{k}-a_{k}}(c_% {k}-b_{k})&\displaystyle=0\\ \displaystyle\frac{f(b_{k})-f(a_{k})}{b_{k}-a_{k}}(c_{k}-b_{k})&\displaystyle=% -f(b_{k})\\ \displaystyle(c_{k}-b_{k})&\displaystyle=-f(b_{k})\frac{b_{k}-a_{k}}{f(b_{k})-% f(a_{k})}\\ \displaystyle c_{k}&\displaystyle=b_{k}-f(b_{k})\frac{b_{k}-a_{k}}{f(b_{k})-f(% a_{k})}\\ \end{aligned}
  10. f ( x ) = 2 x 3 - 4 x 2 + 3 x f(x)=2x^{3}-4x^{2}+3x
  11. c k = 1 2 f ( b k ) a k - f ( a k ) b k 1 2 f ( b k ) - f ( a k ) c_{k}=\frac{\frac{1}{2}f(b_{k})a_{k}-f(a_{k})b_{k}}{\frac{1}{2}f(b_{k})-f(a_{k% })}
  12. c k = f ( b k ) a k - 1 2 f ( a k ) b k f ( b k ) - 1 2 f ( a k ) c_{k}=\frac{f(b_{k})a_{k}-\frac{1}{2}f(a_{k})b_{k}}{f(b_{k})-\frac{1}{2}f(a_{k% })}

False_vacuum.html

  1. 4 π r 2 4\pi r^{2}
  2. 4 3 π r 3 \textstyle\frac{4}{3}\pi r^{3}

Faraday's_law_of_induction.html

  1. Φ B = Σ ( t ) 𝐁 ( 𝐫 , t ) d 𝐀 , \Phi_{B}=\iint\limits_{\Sigma(t)}\mathbf{B}(\mathbf{r},t)\cdot d\mathbf{A}\ ,
  2. \mathcal{E}
  3. = - d Φ B d t \mathcal{E}=-{{d\Phi_{B}}\over dt}
  4. \mathcal{E}
  5. = - N d Φ B d t \mathcal{E}=-N{{d\Phi_{B}}\over dt}
  6. × \nabla\times
  7. Σ 𝐄 d s y m b o l = - d d t Σ 𝐁 d 𝐀 . \oint_{\partial\Sigma}\mathbf{E}\cdot dsymbol{\ell}=-\frac{d}{dt}\int_{\Sigma}% \mathbf{B}\cdot d\mathbf{A}.
  8. Σ ( t ) \Sigma(t)
  9. d Φ B d t = d d t Σ ( t ) 𝐁 ( t ) d 𝐀 \frac{d\Phi_{B}}{dt}=\frac{d}{dt}\int_{\Sigma(t)}\mathbf{B}(t)\cdot d\mathbf{A}
  10. d Φ B d t | t = t 0 = ( Σ ( t 0 ) 𝐁 t | t = t 0 d 𝐀 ) + ( d d t Σ ( t ) 𝐁 ( t 0 ) d 𝐀 ) \left.\frac{d\Phi_{B}}{dt}\right|_{t=t_{0}}=\left(\int_{\Sigma(t_{0})}\left.% \frac{\partial\mathbf{B}}{\partial t}\right|_{t=t_{0}}\cdot d\mathbf{A}\right)% +\left(\frac{d}{dt}\int_{\Sigma(t)}\mathbf{B}(t_{0})\cdot d\mathbf{A}\right)
  11. Σ ( t 0 ) 𝐁 t | t = t 0 d 𝐀 = - Σ ( t 0 ) 𝐄 ( t 0 ) d s y m b o l \int_{\Sigma(t_{0})}\left.\frac{\partial\mathbf{B}}{\partial t}\right|_{t=t_{0% }}\cdot d\mathbf{A}=-\oint_{\partial\Sigma(t_{0})}\mathbf{E}(t_{0})\cdot dsymbol% {\ell}
  12. d d t Σ ( t ) 𝐁 ( t 0 ) d 𝐀 \frac{d}{dt}\int_{\Sigma(t)}\mathbf{B}(t_{0})\cdot d\mathbf{A}
  13. d s y m b o l dsymbol{\ell}
  14. d t dt
  15. d 𝐀 = 𝐯 d t × d s y m b o l d\mathbf{A}=\mathbf{v}\,dt\times dsymbol{\ell}
  16. 𝐁 ( 𝐯 d t × d s y m b o l ) = - d t d s y m b o l ( 𝐯 × 𝐁 ) \mathbf{B}\cdot(\mathbf{v}\,dt\times dsymbol{\ell})=-dt\,dsymbol{\ell}\cdot(% \mathbf{v}\times\mathbf{B})
  17. d d t Σ ( t ) 𝐁 ( t 0 ) d 𝐀 = - Σ ( t 0 ) ( 𝐯 ( t 0 ) × 𝐁 ( t 0 ) ) d s y m b o l \frac{d}{dt}\int_{\Sigma(t)}\mathbf{B}(t_{0})\cdot d\mathbf{A}=-\oint_{% \partial\Sigma(t_{0})}(\mathbf{v}(t_{0})\times\mathbf{B}(t_{0}))\cdot dsymbol{\ell}
  18. Σ \partial\Sigma
  19. d Φ B d t | t = t 0 = ( - Σ ( t 0 ) 𝐄 ( t 0 ) d s y m b o l ) + ( - Σ ( t 0 ) ( 𝐯 ( t 0 ) × 𝐁 ( t 0 ) ) d s y m b o l ) \left.\frac{d\Phi_{B}}{dt}\right|_{t=t_{0}}=\left(-\oint_{\partial\Sigma(t_{0}% )}\mathbf{E}(t_{0})\cdot dsymbol{\ell}\right)+\left(-\oint_{\partial\Sigma(t_{% 0})}(\mathbf{v}(t_{0})\times\mathbf{B}(t_{0}))\cdot dsymbol{\ell}\right)
  20. E M F = ( 𝐄 + 𝐯 × 𝐁 ) d s y m b o l EMF=\oint\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\cdot\,\text{d}% symbol{\ell}
  21. d Φ B d t = - E M F \frac{d\Phi_{B}}{dt}=-EMF
  22. × 𝐄 = - 𝐭 𝐁 \stackrel{\mathbf{\nabla\times E=-\partial_{t}B}}{}

Feature_selection.html

  1. 2 \sqrt{2}
  2. log n \sqrt{\log{n}}
  3. log n \sqrt{\log{n}}
  4. 2 log p \sqrt{2\log{p}}
  5. 2 log p q \sqrt{2\log{\frac{p}{q}}}
  6. D ( S , c ) = 1 | S | f i S I ( f i ; c ) D(S,c)=\frac{1}{|S|}\sum_{f_{i}\in S}I(f_{i};c)
  7. R ( S ) = 1 | S | 2 f i , f j S I ( f i ; f j ) R(S)=\frac{1}{|S|^{2}}\sum_{f_{i},f_{j}\in S}I(f_{i};f_{j})
  8. mRMR = max S [ 1 | S | f i S I ( f i ; c ) - 1 | S | 2 f i , f j S I ( f i ; f j ) ] . \mathrm{mRMR}=\max_{S}\left[\frac{1}{|S|}\sum_{f_{i}\in S}I(f_{i};c)-\frac{1}{% |S|^{2}}\sum_{f_{i},f_{j}\in S}I(f_{i};f_{j})\right].
  9. mRMR = max x { 0 , 1 } n [ i = 1 n c i x i i = 1 n x i - i , j = 1 n a i j x i x j ( i = 1 n x i ) 2 ] . \mathrm{mRMR}=\max_{x\in\{0,1\}^{n}}\left[\frac{\sum^{n}_{i=1}c_{i}x_{i}}{\sum% ^{n}_{i=1}x_{i}}-\frac{\sum^{n}_{i,j=1}a_{ij}x_{i}x_{j}}{(\sum^{n}_{i=1}x_{i})% ^{2}}\right].
  10. QPFS : min 𝐱 { α 𝐱 T H 𝐱 - 𝐱 T F } s.t. i = 1 n x i = 1 , x i 0 \mathrm{QPFS}:\min_{\mathbf{x}}\left\{\alpha\mathbf{x}^{T}H\mathbf{x}-\mathbf{% x}^{T}F\right\}\quad\mbox{s.t.}~{}\ \sum_{i=1}^{n}x_{i}=1,x_{i}\geq 0
  11. F n × 1 = [ I ( f 1 ; c ) , , I ( f n ; c ) ] T F_{n\times 1}=[I(f_{1};c),\ldots,I(f_{n};c)]^{T}
  12. n n
  13. H n × n = [ I ( f i ; f j ) ] i , j = 1 n H_{n\times n}=[I(f_{i};f_{j})]_{i,j=1\ldots n}
  14. 𝐱 n × 1 \mathbf{x}_{n\times 1}
  15. I ( f i ; f i ) I(f_{i};f_{i})
  16. H H
  17. SPEC CMI : max 𝐱 { 𝐱 T Q 𝐱 } s.t. 𝐱 = 1 , x i 0 \mathrm{SPEC_{CMI}}:\max_{\mathbf{x}}\left\{\mathbf{x}^{T}Q\mathbf{x}\right\}% \quad\mbox{s.t.}~{}\ \|\mathbf{x}\|=1,x_{i}\geq 0
  18. Q i i = I ( f i ; c ) Q_{ii}=I(f_{i};c)
  19. Q i j = I ( f i ; c | f j ) , i j Q_{ij}=I(f_{i};c|f_{j}),i\neq j
  20. SPEC CMI \mathrm{SPEC_{CMI}}
  21. Q Q
  22. SPEC CMI \mathrm{SPEC_{CMI}}
  23. HSICLasso : min 𝐱 1 2 k , l = 1 n x k x l HSIC ( f k , f l ) - k = 1 n x k HSIC ( f k , c ) + λ 𝐱 1 , s.t. x 1 , , x n 0 , \mathrm{HSICLasso}:\min_{\mathbf{x}}\frac{1}{2}\sum_{k,l=1}^{n}x_{k}x_{l}{% \mbox{HSIC}~{}}(f_{k},f_{l})-\sum_{k=1}^{n}x_{k}{\mbox{HSIC}~{}}(f_{k},c)+% \lambda\|\mathbf{x}\|_{1},\quad\mbox{s.t.}~{}\ x_{1},\ldots,x_{n}\geq 0,
  24. HSIC ( f k , c ) = tr ( 𝐊 ¯ ( k ) 𝐋 ¯ ) {\mbox{HSIC}~{}}(f_{k},c)=\mbox{tr}~{}(\bar{\mathbf{K}}^{(k)}\bar{\mathbf{L}})
  25. tr ( ) \mbox{tr}~{}(\cdot)
  26. λ \lambda
  27. 𝐊 ¯ ( k ) = 𝚪 𝐊 ( k ) 𝚪 \bar{\mathbf{K}}^{(k)}=\mathbf{\Gamma}\mathbf{K}^{(k)}\mathbf{\Gamma}
  28. 𝐋 ¯ = 𝚪 𝐋 𝚪 \bar{\mathbf{L}}=\mathbf{\Gamma}\mathbf{L}\mathbf{\Gamma}
  29. K i , j ( k ) = K ( u k , i , u k , j ) K^{(k)}_{i,j}=K(u_{k,i},u_{k,j})
  30. L i , j = L ( c i , c j ) L_{i,j}=L(c_{i},c_{j})
  31. K ( u , u ) K(u,u^{\prime})
  32. L ( c , c ) L(c,c^{\prime})
  33. 𝚪 = 𝐈 m - 1 m 𝟏 m 𝟏 m T \mathbf{\Gamma}=\mathbf{I}_{m}-\frac{1}{m}\mathbf{1}_{m}\mathbf{1}_{m}^{T}
  34. 𝐈 m \mathbf{I}_{m}
  35. m m
  36. 𝟏 m \mathbf{1}_{m}
  37. m m
  38. 1 \|\cdot\|_{1}
  39. 1 \ell_{1}
  40. HSICLasso : min 𝐱 1 2 𝐋 ¯ - k = 1 n x k 𝐊 ¯ ( k ) F 2 + λ 𝐱 1 , s.t. x 1 , , x n 0 , \mathrm{HSICLasso}:\min_{\mathbf{x}}\frac{1}{2}\|\bar{\mathbf{L}}-\sum_{k=1}^{% n}x_{k}\bar{\mathbf{K}}^{(k)}\|^{2}_{F}+\lambda\|\mathbf{x}\|_{1},\quad\mbox{s% .t.}~{}\ x_{1},\ldots,x_{n}\geq 0,
  41. F \|\cdot\|_{F}
  42. M e r i t S k = k r c f ¯ k + k ( k - 1 ) r f f ¯ . Merit_{S_{k}}=\frac{k\overline{r_{cf}}}{\sqrt{k+k(k-1)\overline{r_{ff}}}}.
  43. r c f ¯ \overline{r_{cf}}
  44. r f f ¯ \overline{r_{ff}}
  45. CFS = max S k [ r c f 1 + r c f 2 + + r c f k k + 2 ( r f 1 f 2 + + r f i f j + + r f k f 1 ) ] . \mathrm{CFS}=\max_{S_{k}}\left[\frac{r_{cf_{1}}+r_{cf_{2}}+\cdots+r_{cf_{k}}}{% \sqrt{k+2(r_{f_{1}f_{2}}+\cdots+r_{f_{i}f_{j}}+\cdots+r_{f_{k}f_{1}})}}\right].
  46. r c f i r_{cf_{i}}
  47. r f i f j r_{f_{i}f_{j}}
  48. CFS = max x { 0 , 1 } n [ ( i = 1 n a i x i ) 2 i = 1 n x i + i j 2 b i j x i x j ] . \mathrm{CFS}=\max_{x\in\{0,1\}^{n}}\left[\frac{(\sum^{n}_{i=1}a_{i}x_{i})^{2}}% {\sum^{n}_{i=1}x_{i}+\sum_{i\neq j}2b_{ij}x_{i}x_{j}}\right].

Feature_structure.html

  1. [ category n o u n p h r a s e agreement [ number s i n g u l a r person t h i r d ] ] \begin{bmatrix}\mbox{category}&noun\ phrase\\ \mbox{agreement}&\begin{bmatrix}\mbox{number}&singular\\ \mbox{person}&third\end{bmatrix}\end{bmatrix}

Fejér_kernel.html

  1. F n ( x ) = 1 n k = 0 n - 1 D k ( x ) , F_{n}(x)=\frac{1}{n}\sum_{k=0}^{n-1}D_{k}(x),
  2. D k ( x ) = s = - k k e i s x D_{k}(x)=\sum_{s=-k}^{k}{\rm e}^{isx}
  3. F n ( x ) = 1 n ( sin n x 2 sin x 2 ) 2 = 1 n ( 1 - cos ( n x ) 1 - cos x ) F_{n}(x)=\frac{1}{n}\left(\frac{\sin\frac{nx}{2}}{\sin\frac{x}{2}}\right)^{2}=% \frac{1}{n}\left(\frac{1-\cos(nx)}{1-\cos x}\right)
  4. F n ( x ) = | j | n ( 1 - | j | n ) e i j x F_{n}(x)=\sum_{|j|\leq n}\left(1-\frac{|j|}{n}\right)e^{ijx}
  5. F n ( x ) 0 F_{n}(x)\geq 0
  6. 1 1
  7. f 0 f\geq 0
  8. 2 π 2\pi
  9. 0 ( f * F n ) ( x ) = 1 2 π - π π f ( y ) F n ( x - y ) d y . 0\leq(f*F_{n})(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(y)F_{n}(x-y)\,dy.
  10. f * D n = S n ( f ) = | j | n f ^ j e i j x f*D_{n}=S_{n}(f)=\sum_{|j|\leq n}\widehat{f}_{j}e^{ijx}
  11. f * F n = 1 n k = 0 n - 1 S k ( f ) f*F_{n}=\frac{1}{n}\sum_{k=0}^{n-1}S_{k}(f)
  12. F n * f L p ( [ - π , π ] ) f L p ( [ - π , π ] ) \|F_{n}*f\|_{L^{p}([-\pi,\pi])}\leq\|f\|_{L^{p}([-\pi,\pi])}
  13. 1 p 1\leq p\leq\infty
  14. f L p f\in L^{p}
  15. f L 1 ( [ - π , π ] ) f\in L^{1}([-\pi,\pi])
  16. f * F n f f*F_{n}\rightarrow f
  17. [ - π , π ] [-\pi,\pi]
  18. L 1 ( [ - π , π ] ) L 2 ( [ - π , π ] ) L ( [ - π , π ] ) L^{1}([-\pi,\pi])\supset L^{2}([-\pi,\pi])\supset\cdots\supset L^{\infty}([-% \pi,\pi])
  19. L p L^{p}
  20. p 1 p\geq 1
  21. f f
  22. f , g L 1 f,g\in L^{1}
  23. f ^ = g ^ \hat{f}=\hat{g}
  24. f = g f=g
  25. f * F n = | j | n ( 1 - | j | n ) e i j t f*F_{n}=\sum_{|j|\leq n}\left(1-\frac{|j|}{n}\right)e^{ijt}
  26. lim n S n ( f ) \lim_{n\to\infty}S_{n}(f)
  27. lim n F n ( f ) = f \lim_{n\to\infty}F_{n}(f)=f
  28. F n * f F_{n}*f

Fermat_curve.html

  1. X n + Y n = Z n . X^{n}+Y^{n}=Z^{n}.
  2. x n + y n = 1. x^{n}+y^{n}=1.
  3. ( n - 1 ) ( n - 2 ) / 2. (n-1)(n-2)/2.
  4. n - 1. n-1.

Fermi's_golden_rule.html

  1. | i |i\rangle
  2. H H
  3. H H
  4. H H
  5. H H
  6. ω ω
  7. ħ ω ħω
  8. | i |i\rangle
  9. | f |f\rangle
  10. Γ i f = 2 π | f | H | i | 2 ρ \Gamma_{i\rightarrow f}=\frac{2\pi}{\hbar}\left|\langle f|H^{\prime}|i\rangle% \right|^{2}\rho
  11. ρ ρ
  12. f | H | i \langle f|H^{\prime}|i\rangle
  13. H H
  14. H H
  15. ( H 0 + H - i t ) n a n ( t ) ψ n e - i t E n / = 0 , \left(H_{0}+H^{\prime}-i\hbar\frac{\partial}{\partial t}\right)\sum_{n}a_{n}(t% )~{}\psi_{n}e^{-itE_{n}/\hbar}=0~{},
  16. H H
  17. H H
  18. t t
  19. i d a k ( t ) d t = n k | H | n a n ( t ) e i t ( E k - E n ) / , i\hbar\frac{da_{k}(t)}{dt}=\sum_{n}\langle k|H^{\prime}|n\rangle~{}a_{n}(t)~{}% e^{it(E_{k}-E_{n})/\hbar},
  20. i a k ( t ) = 2 k | H | i e i ω t / 2 sin ω t / 2 ω i\hbar a_{k}(t)=2\langle k|H^{\prime}|i\rangle~{}e^{i\omega t/2}~{}\frac{\sin% \omega t/2}{\omega}
  21. ω ( E k - E i ) / \omega\equiv(E_{k}-E_{i})/\hbar
  22. Γ i k = d d t | a k ( t ) | 2 = 2 | k | H | i | 2 2 sin ω t ω , \Gamma_{i\rightarrow k}=\frac{d}{dt}~{}|a_{k}(t)|^{2}=\frac{2|\langle k|H^{% \prime}|i\rangle|^{2}}{\hbar^{2}}\frac{\sin\omega t}{\omega}~{},
  23. ω ω
  24. t t
  25. | k |k\rangle
  26. E E
  27. ρ ( E ) ρ(E)
  28. ω ω
  29. Γ i f = 2 - d ω ρ ( ω ) | f | H | i | 2 sin ω t ω . \Gamma_{i\rightarrow f}=\frac{2}{\hbar}\int_{-\infty}^{\infty}d\omega\rho(% \omega)|\langle f|H^{\prime}|i\rangle|^{2}\frac{\sin\omega t}{\omega}~{}.
  30. t t
  31. ω ω
  32. 2 π / t , 2 π / t −−2π/t,2π/t
  33. Γ i f = 2 ρ | f | H | i | 2 - d ω sin ω t ω \Gamma_{i\rightarrow f}=\frac{2\rho|\langle f|H^{\prime}|i\rangle|^{2}}{\hbar}% \int_{-\infty}^{\infty}d\omega\frac{\sin\omega t}{\omega}
  34. π π
  35. f | H | i \langle f|H^{\prime}|i\rangle

Fermi's_interaction.html

  1. σ G F 2 E 2 \sigma\approx G_{\rm F}^{2}E^{2}
  2. G < s u b > F G<sub>F
  3. V A V−A
  4. G F ( c ) 3 = 2 8 g 2 m W 2 = 1.16637 ( 1 ) × 10 - 5 GeV - 2 . \frac{G_{\rm F}}{(\hbar c)^{3}}=\frac{\sqrt{2}}{8}\frac{g^{2}}{m_{\rm W}^{2}}=% 1.16637(1)\times 10^{-5}\;\textrm{GeV}^{-2}\ .
  5. g g
  6. v = ( 2 G F ) - 1 / 2 246.22 GeV v=(\sqrt{2}G_{\rm F})^{-1/2}\simeq 246.22\;\textrm{GeV}

Fermi_surface.html

  1. N N
  2. ϵ i \epsilon_{i}
  3. n i = 1 e ( ϵ i - μ ) / k B T + 1 , \langle n_{i}\rangle=\frac{1}{e^{(\epsilon_{i}-\mu)/k_{B}T}+1},
  4. n i \left\langle n_{i}\right\rangle
  5. i t h i^{th}
  6. ϵ i \epsilon_{i}
  7. i t h i^{th}
  8. μ \mu
  9. ϵ F \epsilon_{F}
  10. T T
  11. k B k_{B}
  12. T 0 T\to 0
  13. n i { 1 ( ϵ i < μ ) 0 ( ϵ i > μ ) . \left\langle n_{i}\right\rangle\approx\begin{cases}1&(\epsilon_{i}<\mu)\\ 0&(\epsilon_{i}>\mu)\end{cases}.
  14. ϵ F \epsilon_{F}
  15. ϵ F \epsilon_{F}
  16. N N
  17. p F p_{F}
  18. k F = 2 m E F k_{F}=\frac{\sqrt{2mE_{F}}}{\hbar}
  19. \hbar
  20. k z \vec{k}_{z}
  21. k F k_{F}
  22. k \vec{k}
  23. 2 π a \frac{2\pi}{a}
  24. k \vec{k}
  25. K \vec{K}
  26. k \vec{k}
  27. k \vec{k}
  28. K \vec{K}
  29. H H
  30. 1 / H 1/H
  31. ω c \hbar\omega_{c}
  32. ω c = e H / m * c \omega_{c}=eH/m^{*}c
  33. e e
  34. m * m^{*}
  35. c c
  36. Δ H \Delta H
  37. Å - 2 \AA^{-2}
  38. A A_{\perp}
  39. A = 2 π e Δ H c A_{\perp}=\frac{2\pi e\Delta H}{\hbar c}

Fick_principle.html

  1. V O 2 = ( C O × C a ) - ( C O × C v ) VO_{2}=(CO\times\ C_{a})-(CO\times\ C_{v})
  2. C O = V O 2 C a - C v CO=\frac{VO_{2}}{C_{a}-C_{v}}
  3. Cardiac Output = oxygen consumption arteriovenous oxygen difference \,\text{Cardiac Output}=\frac{\,\text{oxygen consumption}}{\,\text{% arteriovenous oxygen difference}}
  4. Oxygen Content of blood = [ Hb ] ( g/dl ) × 1.36 ( ml O 2 / g of Hb ) × O 2 saturation fraction + 0.0032 × P O 2 ( torr ) \,\text{Oxygen Content of blood}=\left[\,\text{Hb}\right]\left(\,\text{g/dl}% \right)\ \times\ 1.36\left(\,\text{ml}\ O_{2}/\,\text{g of Hb}\right)\times\ O% _{2}^{\,\text{saturation fraction}}+\ 0.0032\ \times\ P_{O_{2}}(\,\text{torr})

Fictitious_force.html

  1. 𝐱 B = j = 1 3 x j 𝐮 j . \mathbf{x}_{\mathrm{B}}=\sum_{j=1}^{3}x_{j}\mathbf{u}_{j}\ .
  2. 𝐱 A = 𝐗 AB + j = 1 3 x j 𝐮 j . \mathbf{x}_{\mathrm{A}}=\mathbf{X}_{\mathrm{AB}}+\sum_{j=1}^{3}x_{j}\mathbf{u}% _{j}\ .
  3. d 𝐱 A d t = d 𝐗 AB d t + j = 1 3 d x j d t 𝐮 j + j = 1 3 x j d 𝐮 j d t . \frac{d\mathbf{x}_{\mathrm{A}}}{dt}=\frac{d\mathbf{X}_{\mathrm{AB}}}{dt}+\sum_% {j=1}^{3}\frac{dx_{j}}{dt}\mathbf{u}_{j}+\sum_{j=1}^{3}x_{j}\frac{d\mathbf{u}_% {j}}{dt}\ .
  4. d 𝐱 A d t = 𝐯 AB + 𝐯 B + j = 1 3 x j d 𝐮 j d t . \frac{d\mathbf{x}_{\mathrm{A}}}{dt}=\mathbf{v}_{\mathrm{AB}}+\mathbf{v}_{% \mathrm{B}}+\sum_{j=1}^{3}x_{j}\frac{d\mathbf{u}_{j}}{dt}.
  5. d 2 𝐱 A d t 2 = 𝐚 AB + d 𝐯 B d t + j = 1 3 d x j d t d 𝐮 j d t + j = 1 3 x j d 2 𝐮 j d t 2 . \frac{d^{2}\mathbf{x}_{\mathrm{A}}}{dt^{2}}=\mathbf{a}_{\mathrm{AB}}+\frac{d% \mathbf{v}_{\mathrm{B}}}{dt}+\sum_{j=1}^{3}\frac{dx_{j}}{dt}\frac{d\mathbf{u}_% {j}}{dt}+\sum_{j=1}^{3}x_{j}\frac{d^{2}\mathbf{u}_{j}}{dt^{2}}.
  6. d 𝐯 B d t = j = 1 3 d v j d t 𝐮 j + j = 1 3 v j d 𝐮 j d t = 𝐚 B + j = 1 3 v j d 𝐮 j d t . \frac{d\mathbf{v}_{\mathrm{B}}}{dt}=\sum_{j=1}^{3}\frac{dv_{j}}{dt}\mathbf{u}_% {j}+\sum_{j=1}^{3}v_{j}\frac{d\mathbf{u}_{j}}{dt}=\mathbf{a}_{\mathrm{B}}+\sum% _{j=1}^{3}v_{j}\frac{d\mathbf{u}_{j}}{dt}.
  7. d 2 𝐱 A d t 2 = 𝐚 AB + 𝐚 B + 2 j = 1 3 v j d 𝐮 j d t + j = 1 3 x j d 2 𝐮 j d t 2 . \frac{d^{2}\mathbf{x}_{\mathrm{A}}}{dt^{2}}=\mathbf{a}_{\mathrm{AB}}+\mathbf{a% }_{\mathrm{B}}+2\ \sum_{j=1}^{3}v_{j}\frac{d\mathbf{u}_{j}}{dt}+\sum_{j=1}^{3}% x_{j}\frac{d^{2}\mathbf{u}_{j}}{dt^{2}}.
  8. x j d 𝐮 j / d t \sum x_{j}\,d\mathbf{u}_{j}/dt
  9. 𝐅 A = 𝐅 B + m 𝐚 AB + 2 m j = 1 3 v j d 𝐮 j d t + m j = 1 3 x j d 2 𝐮 j d t 2 . \mathbf{F}_{\mathrm{A}}=\mathbf{F}_{\mathrm{B}}+m\mathbf{a}_{\mathrm{AB}}+2m% \sum_{j=1}^{3}v_{j}\frac{d\mathbf{u}_{j}}{dt}+m\sum_{j=1}^{3}x_{j}\frac{d^{2}% \mathbf{u}_{j}}{dt^{2}}\ .
  10. 𝐅 B = 𝐅 A + 𝐅 fictitious , \mathbf{F}_{\mathrm{B}}=\mathbf{F}_{\mathrm{A}}+\mathbf{F}_{\mathrm{fictitious% }},
  11. 𝐅 fictitious = - m 𝐚 AB - 2 m j = 1 3 v j d 𝐮 j d t - m j = 1 3 x j d 2 𝐮 j d t 2 . \mathbf{F}_{\mathrm{fictitious}}=-m\mathbf{a}_{\mathrm{AB}}-2m\sum_{j=1}^{3}v_% {j}\frac{d\mathbf{u}_{j}}{dt}-m\sum_{j=1}^{3}x_{j}\frac{d^{2}\mathbf{u}_{j}}{% dt^{2}}\ .
  12. | s y m b o l Ω | = d θ d t = ω ( t ) , |symbol{\Omega}|=\frac{d\theta}{dt}=\omega(t),
  13. d 𝐮 j ( t ) d t = s y m b o l Ω × 𝐮 j ( t ) , \frac{d\mathbf{u}_{j}(t)}{dt}=symbol{\Omega}\times\mathbf{u}_{j}(t),
  14. d 2 𝐮 j ( t ) d t 2 = d s y m b o l Ω d t × 𝐮 j + s y m b o l Ω × d 𝐮 j ( t ) d t = d s y m b o l Ω d t × 𝐮 j + s y m b o l Ω × [ s y m b o l Ω × 𝐮 j ( t ) ] , \frac{d^{2}\mathbf{u}_{j}(t)}{dt^{2}}=\frac{dsymbol{\Omega}}{dt}\times\mathbf{% u}_{j}+symbol{\Omega}\times\frac{d\mathbf{u}_{j}(t)}{dt}=\frac{dsymbol{\Omega}% }{dt}\times\mathbf{u}_{j}+symbol{\Omega}\times\left[symbol{\Omega}\times% \mathbf{u}_{j}(t)\right],
  15. d 2 𝐱 A d t 2 = 𝐚 B + 2 j = 1 3 v j d 𝐮 j d t + j = 1 3 x j d 2 𝐮 j d t 2 , \frac{d^{2}\mathbf{x}_{\mathrm{A}}}{dt^{2}}=\mathbf{a}_{\mathrm{B}}+2\sum_{j=1% }^{3}v_{j}\ \frac{d\mathbf{u}_{j}}{dt}+\sum_{j=1}^{3}x_{j}\frac{d^{2}\mathbf{u% }_{j}}{dt^{2}},
  16. 𝐚 A = 𝐚 B + 2 j = 1 3 v j s y m b o l Ω × 𝐮 j ( t ) + j = 1 3 x j d s y m b o l Ω d t × 𝐮 j + j = 1 3 x j s y m b o l Ω × [ s y m b o l Ω × 𝐮 j ( t ) ] \mathbf{a}_{\mathrm{A}}=\mathbf{a}_{\mathrm{B}}+\ 2\sum_{j=1}^{3}v_{j}symbol{% \Omega}\times\mathbf{u}_{j}(t)+\sum_{j=1}^{3}x_{j}\frac{dsymbol{\Omega}}{dt}% \times\mathbf{u}_{j}\ +\sum_{j=1}^{3}x_{j}symbol{\Omega}\times\left[symbol{% \Omega}\times\mathbf{u}_{j}(t)\right]
  17. = 𝐚 B + 2 s y m b o l Ω × j = 1 3 v j 𝐮 j ( t ) + d s y m b o l Ω d t × j = 1 3 x j 𝐮 j + s y m b o l Ω × [ s y m b o l Ω × j = 1 3 x j 𝐮 j ( t ) ] . =\mathbf{a}_{\mathrm{B}}+2symbol{\Omega}\times\sum_{j=1}^{3}v_{j}\mathbf{u}_{j% }(t)+\frac{dsymbol{\Omega}}{dt}\times\sum_{j=1}^{3}x_{j}\mathbf{u}_{j}+symbol{% \Omega}\times\left[symbol{\Omega}\times\sum_{j=1}^{3}x_{j}\mathbf{u}_{j}(t)% \right].
  18. 𝐚 A = 𝐚 B + 2 s y m b o l Ω × 𝐯 B + d s y m b o l Ω d t × 𝐱 B + s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) . \mathbf{a}_{A}=\mathbf{a}_{B}+2symbol{\Omega}\times\mathbf{v}_{\mathrm{B}}+% \frac{dsymbol{\Omega}}{dt}\times\mathbf{x}_{\mathrm{B}}+symbol{\Omega}\times% \left(symbol{\Omega}\times\mathbf{x}_{B}\right)\ .
  19. 𝐚 B = 𝐚 A - 2 s y m b o l Ω × 𝐯 B - s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) - d s y m b o l Ω d t × 𝐱 B . \mathbf{a}_{\mathrm{B}}=\mathbf{a}_{\mathrm{A}}-2symbol{\Omega}\times\mathbf{v% }_{\mathrm{B}}-symbol{\Omega}\times(symbol\Omega\times\mathbf{x}_{\mathrm{B}})% -\frac{dsymbol\Omega}{dt}\times\mathbf{x}_{\mathrm{B}}.
  20. 𝐅 fict = - 2 m s y m b o l Ω × 𝐯 B - m s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) - m d s y m b o l Ω d t × 𝐱 B . \mathbf{F}_{\mathrm{fict}}=-2msymbol\Omega\times\mathbf{v}_{\mathrm{B}}-% msymbol\Omega\times(symbol\Omega\times\mathbf{x}_{\mathrm{B}})-m\frac{dsymbol% \Omega}{dt}\times\mathbf{x}_{\mathrm{B}}.
  21. d 2 𝐱 A d t 2 = 𝐚 A B + 𝐚 B + 2 j = 1 3 v j d 𝐮 j d t \frac{d^{2}\mathbf{x}_{A}}{dt^{2}}=\mathbf{a}_{AB}+\mathbf{a}_{B}+2\ \sum_{j=1% }^{3}v_{j}\ \frac{d\mathbf{u}_{j}}{dt}
  22. + j = 1 3 x j d 2 𝐮 j d t 2 . +\sum_{j=1}^{3}x_{j}\ \frac{d^{2}\mathbf{u}_{j}}{dt^{2}}\ .
  23. = 𝐚 A B + 𝐚 B , =\mathbf{a}_{AB}\ +\mathbf{a}_{B}\ ,
  24. 𝐗 A B = R ( cos ( ω t ) , sin ( ω t ) ) , \mathbf{X}_{AB}=R\left(\cos(\omega t),\ \sin(\omega t)\right)\ ,
  25. 𝐯 A B = d d t 𝐗 A B = 𝛀 × 𝐗 A B , \mathbf{v}_{AB}=\frac{d}{dt}\mathbf{X}_{AB}=\mathbf{\Omega\times X}_{AB}\ ,
  26. 𝐚 A B = d 2 d t 2 𝐗 A B \mathbf{a}_{AB}=\frac{d^{2}}{dt^{2}}\mathbf{X}_{AB}
  27. = 𝛀 × ( 𝛀 × 𝐗 A B ) =\mathbf{\Omega\ \times}\left(\mathbf{\Omega\times X}_{AB}\right)
  28. = - ω 2 𝐗 A B . =-\omega^{2}\mathbf{X}_{AB}\ .
  29. 𝛀 × ( 𝛀 × 𝐗 A B ) , \mathbf{\Omega\ \times}\left(\mathbf{\Omega\times X}_{AB}\right)\ ,
  30. s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) , symbol{\Omega}\times\left(symbol{\Omega}\times\mathbf{x}_{B}\right)\ ,
  31. 𝐅 fict = m ω 2 𝐗 A B , \mathbf{F}_{\mathrm{fict}}=m\omega^{2}\mathbf{X}_{AB}\ ,
  32. | 𝐅 fict | = m ω 2 R . |\mathbf{F}_{\mathrm{fict}}|=m\omega^{2}R\ .
  33. 𝐮 1 = ( - cos ω t , - sin ω t ) ; \mathbf{u}_{1}=(-\cos\omega t,\ -\sin\omega t)\ ;
  34. 𝐮 2 = ( sin ω t , - cos ω t ) . \mathbf{u}_{2}=(\sin\omega t,\ -\cos\omega t)\ .
  35. d d t 𝐮 1 = 𝛀 × 𝐮 𝟏 = ω 𝐮 2 ; \frac{d}{dt}\mathbf{u}_{1}=\mathbf{\Omega\times u_{1}}=\omega\mathbf{u}_{2}\ ;
  36. d d t 𝐮 2 = 𝛀 × 𝐮 𝟐 = - ω 𝐮 1 . \ \frac{d}{dt}\mathbf{u}_{2}=\mathbf{\Omega\times u_{2}}=-\omega\mathbf{u}_{1}% \ \ .
  37. d 2 𝐱 A d t 2 = 𝐚 A B + 𝐚 B + 2 j = 1 3 v j d 𝐮 j d t \frac{d^{2}\mathbf{x}_{A}}{dt^{2}}=\mathbf{a}_{AB}+\mathbf{a}_{B}+2\ \sum_{j=1% }^{3}v_{j}\ \frac{d\mathbf{u}_{j}}{dt}
  38. + j = 1 3 x j d 2 𝐮 j d t 2 +\ \sum_{j=1}^{3}x_{j}\ \frac{d^{2}\mathbf{u}_{j}}{dt^{2}}
  39. = 𝛀 × ( 𝛀 × 𝐗 A B ) + 𝐚 B + 2 j = 1 3 v j 𝛀 × 𝐮 𝐣 =\mathbf{\Omega\ \times}\left(\mathbf{\Omega\times X}_{AB}\right)+\mathbf{a}_{% B}+2\ \sum_{j=1}^{3}v_{j}\ \mathbf{\Omega\times u_{j}}
  40. + j = 1 3 x j s y m b o l Ω × ( s y m b o l Ω × 𝐮 j ) \ +\ \sum_{j=1}^{3}x_{j}\ symbol{\Omega}\times\left(symbol{\Omega}\times% \mathbf{u}_{j}\right)
  41. = 𝛀 × ( 𝛀 × 𝐗 A B ) + 𝐚 B + 2 s y m b o l Ω × 𝐯 B =\mathbf{\Omega\ \times}\left(\mathbf{\Omega\times X}_{AB}\right)+\mathbf{a}_{% B}+2\ symbol{\Omega}\times\mathbf{v}_{B}
  42. + s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) \ +\ symbol{\Omega}\times\left(symbol{\Omega}\times\mathbf{x}_{B}\right)
  43. = 𝛀 × ( 𝛀 × ( 𝐗 A B + 𝐱 B ) ) + 𝐚 B + 2 s y m b o l Ω × 𝐯 B , =\mathbf{\Omega\ \times}\left(\mathbf{\Omega\times}(\mathbf{X}_{AB}+\mathbf{x}% _{B})\right)+\mathbf{a}_{B}+2\ symbol{\Omega}\times\mathbf{v}_{B}\ \ ,
  44. 𝛀 × ( 𝛀 × ( 𝐗 A B + 𝐱 B ) ) , \mathbf{\Omega\ \times}\left(\mathbf{\Omega\times}(\mathbf{X}_{AB}+\mathbf{x}_% {B})\right)\ ,
  45. s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) , symbol{\Omega}\times\left(symbol{\Omega}\times\mathbf{x}_{B}\right)\ ,
  46. 𝐫 ( t ) = R ( t ) 𝐮 R = [ x ( t ) y ( t ) ] = [ R ( t ) cos ( ω t + π / 4 ) R ( t ) sin ( ω t + π / 4 ) ] , \mathbf{r}(t)=R(t)\mathbf{u}_{R}=\begin{bmatrix}x(t)\\ y(t)\end{bmatrix}=\begin{bmatrix}R(t)\cos(\omega t+\pi/4)\\ R(t)\sin(\omega t+\pi/4)\end{bmatrix},
  47. R ( t ) = s t , R(t)=st,
  48. 𝐯 ( t ) = d R d t [ cos ( ω t + π / 4 ) sin ( ω t + π / 4 ) ] + ω R ( t ) [ - sin ( ω t + π / 4 ) cos ( ω t + π / 4 ) ] \mathbf{v}(t)=\frac{dR}{dt}\begin{bmatrix}\cos(\omega t+\pi/4)\\ \sin(\omega t+\pi/4)\end{bmatrix}+\omega R(t)\begin{bmatrix}-\sin(\omega t+\pi% /4)\\ \cos(\omega t+\pi/4)\end{bmatrix}
  49. = d R d t 𝐮 R + ω R ( t ) 𝐮 θ , =\frac{dR}{dt}\mathbf{u}_{R}+\omega R(t)\mathbf{u}_{\theta},
  50. 𝐚 ( t ) = d 2 R d t 2 [ cos ( ω t + π / 4 ) sin ( ω t + π / 4 ) ] + 2 d R d t ω [ - sin ( ω t + π / 4 ) cos ( ω t + π / 4 ) ] - ω 2 R ( t ) [ cos ( ω t + π / 4 ) sin ( ω t + π / 4 ) ] \mathbf{a}(t)=\frac{d^{2}R}{dt^{2}}\begin{bmatrix}\cos(\omega t+\pi/4)\\ \sin(\omega t+\pi/4)\end{bmatrix}+2\frac{dR}{dt}\omega\begin{bmatrix}-\sin(% \omega t+\pi/4)\\ \cos(\omega t+\pi/4)\end{bmatrix}-\omega^{2}R(t)\begin{bmatrix}\cos(\omega t+% \pi/4)\\ \sin(\omega t+\pi/4)\end{bmatrix}
  51. = 2 s ω [ - sin ( ω t + π / 4 ) cos ( ω t + π / 4 ) ] - ω 2 R ( t ) [ cos ( ω t + π / 4 ) sin ( ω t + π / 4 ) ] =2s\omega\begin{bmatrix}-\sin(\omega t+\pi/4)\\ \cos(\omega t+\pi/4)\end{bmatrix}-\omega^{2}R(t)\begin{bmatrix}\cos(\omega t+% \pi/4)\\ \sin(\omega t+\pi/4)\end{bmatrix}
  52. = 2 s ω 𝐮 θ - ω 2 R ( t ) 𝐮 R . =2s\ \omega\ \mathbf{u}_{\theta}-\omega^{2}R(t)\ \mathbf{u}_{R}\ .
  53. 𝐅 fict = - 2 m s y m b o l Ω × 𝐯 B - m s y m b o l Ω × ( s y m b o l Ω × 𝐱 B ) - m d s y m b o l Ω d t × 𝐱 B . \mathbf{F}_{\mathrm{fict}}=-2msymbol\Omega\times\mathbf{v}_{\mathrm{B}}-% msymbol\Omega\times(symbol\Omega\times\mathbf{x}_{\mathrm{B}})-m\frac{dsymbol% \Omega}{dt}\times\mathbf{x}_{\mathrm{B}}.
  54. 𝐯 B = s 𝐮 R , \mathbf{v}_{\mathrm{B}}=s\mathbf{u}_{R},
  55. 𝐱 B = R ( t ) 𝐮 R , \mathbf{x}_{\mathrm{B}}=R(t)\mathbf{u}_{R},
  56. s y m b o l Ω × 𝐮 R = ω 𝐮 θ symbol\Omega\times\mathbf{u}_{R}=\omega\mathbf{u}_{\theta}
  57. s y m b o l Ω × 𝐮 θ = - ω 𝐮 R , symbol\Omega\times\mathbf{u}_{\theta}=-\omega\mathbf{u}_{R}\ ,
  58. 𝐅 fict = - 2 m ω s 𝐮 θ + m ω 2 R ( t ) 𝐮 R . \mathbf{F}_{\mathrm{fict}}=-2m\omega s\mathbf{u}_{\theta}+m\omega^{2}R(t)% \mathbf{u}_{R}.

Field_of_sets.html

  1. X , \langle X,\mathcal{F}\rangle
  2. X X
  3. \mathcal{F}
  4. X X
  5. X X
  6. \mathcal{F}
  7. X X
  8. \mathcal{F}
  9. X X
  10. \mathcal{F}
  11. X X
  12. X X
  13. 𝐗 = X , \mathbf{X}=\langle X,\mathcal{F}\rangle
  14. T ( 𝐗 ) T(\mathbf{X})
  15. T ( 𝐗 ) T(\mathbf{X})
  16. T ( 𝐗 ) T(\mathbf{X})
  17. 𝐗 \mathbf{X}
  18. T ( 𝐗 ) T(\mathbf{X})
  19. \mathcal{F}
  20. 𝐗 \mathbf{X}
  21. T ( 𝐗 ) T(\mathbf{X})
  22. \mathcal{F}
  23. 𝐗 \mathbf{X}
  24. X , , μ \langle X,\mathcal{F},\mu\rangle
  25. X , \langle X,\mathcal{F}\rangle
  26. μ \mu
  27. μ \mu
  28. X , 𝒯 , \langle X,\mathcal{T},\mathcal{F}\rangle
  29. X , 𝒯 \langle X,\mathcal{T}\rangle
  30. X , \langle X,\mathcal{F}\rangle
  31. 𝒯 \mathcal{T}
  32. \mathcal{F}
  33. X , 𝒯 \langle X,\mathcal{T}\rangle
  34. X , , \langle X,\leq,\mathcal{F}\rangle
  35. X , \langle X,\leq\rangle
  36. X , \langle X,\mathcal{F}\rangle
  37. Int ( S ) = { x X : \mbox{Int}~{}(S)=\{x\in X:
  38. y S y\in S
  39. y x } y\leq x\}
  40. Cl ( S ) = { x X : \mbox{Cl}~{}(S)=\{x\in X:
  41. y S y\in S
  42. x y } x\leq y\}
  43. S S\in\mathcal{F}
  44. 𝒜 \mathcal{A}
  45. x y x\leq y
  46. S 𝒜 S\in\mathcal{A}
  47. x S x\in S
  48. y S y\in S
  49. X , ( R i ) I , \langle X,(R_{i})_{I},\mathcal{F}\rangle
  50. X , ( R i ) I \langle X,(R_{i})_{I}\rangle
  51. X , \langle X,\mathcal{F}\rangle
  52. 𝐗 = X , ( R i ) I , \mathbf{X}=\langle X,(R_{i})_{I},\mathcal{F}\rangle
  53. 𝒞 ( 𝐗 ) = , , , , , X , ( f i ) I \mathcal{C}(\mathbf{X})=\langle\mathcal{F},\cap,\cup,\prime,,X,(f_{i})_{I}\rangle
  54. i I i\in I
  55. R i R_{i}
  56. n + 1 n+1
  57. f i f_{i}
  58. n n
  59. S 1 , , S n S_{1},...,S_{n}\in\mathcal{F}
  60. f i ( S 1 , , S n ) = { x X : f_{i}(S_{1},...,S_{n})=\{x\in X:
  61. x 1 S 1 , , x n S n x_{1}\in S_{1},...,x_{n}\in S_{n}
  62. R i ( x 1 , , x n , x ) } R_{i}(x_{1},...,x_{n},x)\}
  63. \mathcal{F}
  64. X X
  65. 𝒞 ( 𝐗 ) \mathcal{C}(\mathbf{X})

FIFA_Women's_World_Rankings.html

  1. R a f t = R b e f + K ( S a c t - S e x p ) R_{aft}=R_{bef}+K(S_{act}-S_{exp})
  2. S e x p = 1 1 + 10 - x / 2 S_{exp}=\frac{1}{1+10^{-x/2}}
  3. x = R b e f - O b e f ± H c x=\frac{R_{bef}-O_{bef}\pm H}{c}
  4. R a f t R_{aft}
  5. R b e f R_{bef}
  6. K K
  7. 15 M 15M
  8. S a c t S_{act}
  9. S e x p S_{exp}
  10. x x
  11. O b e f O_{bef}
  12. H H
  13. c c
  14. M M
  15. R b e f R_{bef}
  16. c c

FIFA_World_Rankings.html

  1. Opposition strength multiplier = 200 - ranking position \,\text{Opposition strength multiplier}={200-\,\text{ranking position}}
  2. Regional strength multiplier = Team 1 regional weighting + Team 2 regional weighting 2 \,\text{Regional strength multiplier}=\frac{\,\text{Team 1 regional weighting}% +\,\text{Team 2 regional weighting}}{2}
  3. Ranking points = Result points × Match status × Opposition strength × Regional strength \,\text{Ranking points}=\,\text{Result points}\times\,\text{Match status}% \times\,\text{Opposition strength}\times\,\text{Regional strength}

Files-11.html

  1. i i
  2. i + k i+k
  3. k k

Filtration_(mathematics).html

  1. \mathcal{F}
  2. \mathcal{F}
  3. S i S_{i}
  4. S i S j S i + j S_{i}\cdot S_{j}\subset S_{i+j}
  5. S i S_{i}
  6. S S
  7. S i S_{i}
  8. S S
  9. S i S j S_{i}\supseteq S_{j}
  10. S i S j S_{i}\subseteq S_{j}
  11. i I S i = 0 \bigcap_{i\in I}S_{i}=0
  12. i I S i = S \bigcup_{i\in I}S_{i}=S
  13. { 0 } { 0 , 1 } { 0 , 1 , 2 } \{0\}\subset\{0,1\}\subset\{0,1,2\}
  14. ( 0 , 1 , 2 ) (0,1,2)
  15. ( Ω , ) (\Omega,\mathcal{F})
  16. { t } t 0 \{\mathcal{F}_{t}\}_{t\geq 0}
  17. t \mathcal{F}_{t}\subseteq\mathcal{F}
  18. t 1 t 2 t 1 t 2 . t_{1}\leq t_{2}\implies\mathcal{F}_{t_{1}}\subseteq\mathcal{F}_{t_{2}}.
  19. t { 0 , 1 , , N } , 0 , [ 0 , T ] or [ 0 , + ) . t\in\{0,1,\dots,N\},\mathbb{N}_{0},[0,T]\mbox{ or }~{}[0,+\infty).
  20. ( Ω , , { t } t 0 , ) \left(\Omega,\mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t\geq 0},\mathbb{P}\right)
  21. { t } t 0 \left\{\mathcal{F}_{t}\right\}_{t\geq 0}
  22. \mathcal{F}
  23. 0 \mathcal{F}_{0}
  24. \mathbb{P}
  25. t = t + := s > t s \mathcal{F}_{t}=\mathcal{F}_{t+}:=\bigcap_{s>t}\mathcal{F}_{s}
  26. t t
  27. \mathcal{F}_{\infty}
  28. t \mathcal{F}_{t}
  29. \mathcal{F}
  30. = σ ( t 0 t ) . \mathcal{F}_{\infty}=\sigma\left(\bigcup_{t\geq 0}\mathcal{F}_{t}\right)% \subseteq\mathcal{F}.
  31. ( Ω , , { t } t 0 , ) \left(\Omega,\mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t\geq 0},\mathbb{P}\right)
  32. τ : Ω [ 0 , ] \tau:\Omega\rightarrow[0,\infty]
  33. { t } t 0 \left\{\mathcal{F}_{t}\right\}_{t\geq 0}
  34. { τ t } t \{\tau\leq t\}\in\mathcal{F}_{t}
  35. t 0 t\geq 0
  36. σ \sigma
  37. τ := { A : A { τ t } t , t 0 } \mathcal{F}_{\tau}:=\left\{A\in\mathcal{F}:A\cap\{\tau\leq t\}\in\mathcal{F}_{% t},\ \forall t\geq 0\right\}
  38. τ \mathcal{F}_{\tau}
  39. σ \sigma
  40. τ \mathcal{F}_{\tau}
  41. τ \tau
  42. τ \tau
  43. τ \mathcal{F}_{\tau}
  44. \mathcal{F}
  45. τ \mathcal{F}_{\tau}
  46. t 0 t\geq 0
  47. t \mathcal{F}_{t}
  48. { τ = t } \{\tau=t\}
  49. τ \tau
  50. τ \mathcal{F}_{\tau}
  51. σ ( τ ) τ \sigma(\tau)\neq\mathcal{F}_{\tau}
  52. τ 1 \tau_{1}
  53. τ 2 \tau_{2}
  54. ( Ω , , { t } t 0 , ) \left(\Omega,\mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t\geq 0},\mathbb{P}\right)
  55. τ 1 τ 2 \tau_{1}\leq\tau_{2}
  56. τ 1 τ 2 \mathcal{F}_{\tau_{1}}\subseteq\mathcal{F}_{\tau_{2}}

Finite_state_transducer.html

  1. δ Q × ( Σ { ϵ } ) × ( Γ { ϵ } ) × Q \delta\subseteq Q\times(\Sigma\cup\{\epsilon\})\times(\Gamma\cup\{\epsilon\})\times Q
  2. ( q , a , b , r ) δ (q,a,b,r)\in\delta
  3. δ * \delta^{*}
  4. δ δ * \delta\subseteq\delta^{*}
  5. ( q , ϵ , ϵ , q ) δ * (q,\epsilon,\epsilon,q)\in\delta^{*}
  6. q Q q\in Q
  7. ( q , x , y , r ) δ * (q,x,y,r)\in\delta^{*}
  8. ( r , a , b , s ) δ (r,a,b,s)\in\delta
  9. ( q , x a , y b , s ) δ * (q,xa,yb,s)\in\delta^{*}
  10. δ * \delta^{*}
  11. x [ T ] y x[T]y
  12. i I i\in I
  13. f F f\in F
  14. ( i , x , y , f ) δ * (i,x,y,f)\in\delta^{*}
  15. x Σ * x\in\Sigma^{*}
  16. y Γ * y\in\Gamma^{*}
  17. E Q × ( Σ { ϵ } ) × ( Γ { ϵ } ) × Q × K E\subseteq Q\times(\Sigma\cup\{\epsilon\})\times(\Gamma\cup\{\epsilon\})\times Q\times K
  18. λ : I K \lambda:I\rightarrow K
  19. ρ : F K \rho:F\rightarrow K
  20. T S T\cup S
  21. x [ T S ] y x[T\cup S]y
  22. x [ T ] y x[T]y
  23. x [ S ] y x[S]y
  24. T S T\cdot S
  25. w x [ T S ] y z wx[T\cdot S]yz
  26. w [ T ] y w[T]y
  27. x [ S ] z x[S]z
  28. T * T^{*}
  29. ϵ [ T * ] ϵ \epsilon[T^{*}]\epsilon
  30. w [ T * ] y w[T^{*}]y
  31. x [ T ] z x[T]z
  32. w x [ T * ] y z wx[T^{*}]yz
  33. x [ T * ] y x[T^{*}]y
  34. T S T\cap S
  35. x [ T S ] y x[T\cap S]y
  36. x [ T ] y x[T]y
  37. x [ S ] y x[S]y
  38. T S T\circ S
  39. x [ T S ] z x[T\circ S]z
  40. y Γ * y\in\Gamma^{*}
  41. x [ T ] y x[T]y
  42. y [ S ] z y[S]z
  43. T T
  44. S S
  45. ( x , z ) T S (x,z)\in T\circ S
  46. y y
  47. ( x , y ) S (x,y)\in S
  48. ( y , z ) T (y,z)\in T
  49. π 1 \pi_{1}
  50. π 2 \pi_{2}
  51. π 1 \pi_{1}
  52. π 1 T \pi_{1}T
  53. π 1 T \pi_{1}T
  54. x [ T ] y x[T]y
  55. π 2 \pi_{2}
  56. L = ( Σ { ϵ } ) × ( Γ { ϵ } ) L=(\Sigma\cup\{\epsilon\})\times(\Gamma\cup\{\epsilon\})
  57. L L
  58. L = [ ( Σ { ϵ } ) × Γ ] [ Σ × ( Γ { ϵ } ) ] L=[(\Sigma\cup\{\epsilon\})\times\Gamma]\cup[\Sigma\times(\Gamma\cup\{\epsilon% \})]

Firefighting.html

  1. V v = 3571 V w V_{v}=3571\cdot V_{w}
  2. V v = 1723 V w V_{v}=1723\cdot V_{w}
  3. V v = V r V_{v}=V_{r}
  4. V v = 0.286 V r V_{v}=0.286\cdot V_{r}

First-countable_space.html

  1. / \mathbb{R}/\mathbb{N}

First_class_constraint.html

  1. f i ( x ) = 0 , f_{i}(x)=0,
  2. { f i } i = 1 n \{f_{i}\}_{i=1}^{n}
  3. { f i , f j } \{f_{i},f_{j}\}
  4. { f i , H } \{f_{i},H\}
  5. { f i , f j } = k c i j k f k \{f_{i},f_{j}\}=\sum_{k}c_{ij}^{k}f_{k}
  6. c i j k c_{ij}^{k}
  7. { f i , H } = j v i j f j \{f_{i},H\}=\sum_{j}v_{i}^{j}f_{j}
  8. v i j v_{i}^{j}
  9. C ( M ) C^{\infty}(M)
  10. V ¯ \bar{V}
  11. V ¯ \bar{V}
  12. S [ A , σ ] = d d x 1 4 g 2 η ( ( g - 1 g - 1 ) ( F , F ) ) + 1 2 α ( g - 1 ( D σ , D σ ) ) S[{A},\sigma]=\int d^{d}x\frac{1}{4g^{2}}\eta(({g}^{-1}\otimes{g}^{-1})({F},{F% }))+\frac{1}{2}\alpha({g}^{-1}(D\sigma,D\sigma))
  13. d A + A A d{A}+{A}\wedge{A}
  14. A \vec{A}
  15. ρ ¯ \bar{\rho}
  16. A \vec{A}
  17. π A \vec{\pi}_{A}
  18. D π A - ρ ( π σ , σ ) = 0 \vec{D}\cdot\vec{\pi}_{A}-\rho^{\prime}(\pi_{\sigma},\sigma)=0
  19. ρ : L V V \rho:L\otimes V\rightarrow V
  20. ρ : V ¯ V L \rho^{\prime}:\bar{V}\otimes V\rightarrow L
  21. H f = d d - 1 x 1 2 α - 1 ( π σ , π σ ) + 1 2 α ( D σ D σ ) - g 2 2 η ( π A , π A ) - 1 2 g 2 η ( B B ) - η ( π ϕ , f ) - < π σ , ϕ [ σ ] > - η ( ϕ , D π A ) . H_{f}=\int d^{d-1}x\frac{1}{2}\alpha^{-1}(\pi_{\sigma},\pi_{\sigma})+\frac{1}{% 2}\alpha(\vec{D}\sigma\cdot\vec{D}\sigma)-\frac{g^{2}}{2}\eta(\vec{\pi}_{A},% \vec{\pi}_{A})-\frac{1}{2g^{2}}\eta({B}\cdot{B})-\eta(\pi_{\phi},f)-<\pi_{% \sigma},\phi[\sigma]>-\eta(\phi,\vec{D}\cdot\vec{\pi}_{A}).
  22. S = d t L = d t [ m 2 ( x ˙ 2 + y ˙ 2 + z ˙ 2 ) - m g z + λ 2 ( x 2 + y 2 + z 2 - R 2 ) ] S=\int dtL=\int dt\left[\frac{m}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-mgz+% \frac{\lambda}{2}(x^{2}+y^{2}+z^{2}-R^{2})\right]
  23. S = d t [ m R 2 2 ( θ ˙ 2 + sin 2 ( θ ) ϕ ˙ 2 ) + m g R cos ( θ ) ] S=\int dt\left[\frac{mR^{2}}{2}(\dot{\theta}^{2}+\sin^{2}(\theta)\dot{\phi}^{2% })+mgR\cos(\theta)\right]
  24. p x = m x ˙ p_{x}=m\dot{x}
  25. p y = m y ˙ p_{y}=m\dot{y}
  26. p z = m z ˙ p_{z}=m\dot{z}
  27. p λ = 0 p_{\lambda}=0
  28. λ ˙ \dot{\lambda}
  29. H = p r ˙ + p λ λ ˙ - L = p 2 2 m + p λ λ ˙ + m g z - λ 2 ( r 2 - R 2 ) H=\vec{p}\cdot\dot{\vec{r}}+p_{\lambda}\dot{\lambda}-L=\frac{p^{2}}{2m}+p_{% \lambda}\dot{\lambda}+mgz-\frac{\lambda}{2}(r^{2}-R^{2})
  30. λ ˙ \dot{\lambda}
  31. λ ˙ \dot{\lambda}
  32. u 1 = λ ˙ u_{1}=\dot{\lambda}
  33. λ ˙ = u 1 \dot{\lambda}=u_{1}
  34. u u
  35. H = p 2 2 m + m g z - λ 2 ( r 2 - R 2 ) + u 1 p λ + u 2 ( r 2 - R 2 ) H=\frac{p^{2}}{2m}+mgz-\frac{\lambda}{2}(r^{2}-R^{2})+u_{1}p_{\lambda}+u_{2}(r% ^{2}-R^{2})
  36. p r = 0 \vec{p}\cdot\vec{r}=0
  37. { r 2 - R 2 , H } P B = 0 \{r^{2}-R^{2},\,H\}_{PB}=0
  38. H = p 2 2 m + m g z - λ 2 ( r 2 - R 2 ) + u 1 p λ + u 2 ( r 2 - R 2 ) + u 3 p r , H=\frac{p^{2}}{2m}+mgz-\frac{\lambda}{2}(r^{2}-R^{2})+u_{1}p_{\lambda}+u_{2}(r% ^{2}-R^{2})+u_{3}\vec{p}\cdot\vec{r},
  39. u 1 u_{1}
  40. u 2 u_{2}
  41. u 3 u_{3}
  42. { p r , H } P B = p 2 m - m g z + λ r 2 - 2 u 2 r 2 = 0. \{\vec{p}\cdot\vec{r},\,H\}_{PB}=\frac{p^{2}}{m}-mgz+\lambda r^{2}-2u_{2}r^{2}% =0.
  43. u 2 = λ 2 + 1 r 2 ( p 2 2 m - 1 2 m g z ) . u_{2}=\frac{\lambda}{2}+\frac{1}{r^{2}}\left(\frac{p^{2}}{2m}-\frac{1}{2}mgz% \right).
  44. 2 m r p + 2 u 3 r 2 = 0. \frac{2}{m}\vec{r}\cdot\vec{p}+2u_{3}r^{2}=0.
  45. u 3 = - r p m r 2 . u_{3}=-\frac{\vec{r}\cdot\vec{p}}{mr^{2}}.
  46. H = ( 2 - R 2 r 2 ) p 2 2 m + 1 2 ( 1 + R 2 r 2 ) m g z - ( r p ) 2 m r 2 + u 1 p λ H=\left(2-\frac{R^{2}}{r^{2}}\right)\frac{p^{2}}{2m}+\frac{1}{2}\left(1+\frac{% R^{2}}{r^{2}}\right)mgz-\frac{(\vec{r}\cdot\vec{p})^{2}}{mr^{2}}+u_{1}p_{\lambda}
  47. r ˙ = { r , H } P B , p ˙ = { p , H } P B , λ ˙ = { λ , H } P B , p ˙ λ = { p λ , H } P B . \dot{\vec{r}}=\{\vec{r},\,H\}_{PB},\quad\dot{\vec{p}}=\{\vec{p},\,H\}_{PB},% \quad\dot{\lambda}=\{\lambda,\,H\}_{PB},\quad\dot{p}_{\lambda}=\{p_{\lambda},H% \}_{PB}.
  48. ϕ 1 = p λ , ϕ 2 = r 2 - R 2 , ϕ 3 = p r . \phi_{1}=p_{\lambda},\quad\phi_{2}=r^{2}-R^{2},\quad\phi_{3}=\vec{p}\cdot\vec{% r}.
  49. { ϕ 2 , ϕ 3 } = 2 r 2 0. \{\phi_{2},\phi_{3}\}=2r^{2}\neq 0.
  50. ϕ 2 \phi_{2}
  51. ϕ 3 \phi_{3}
  52. ϕ 1 \phi_{1}
  53. λ \lambda
  54. ϕ 1 \phi_{1}
  55. λ \lambda
  56. λ \lambda
  57. u 1 u_{1}
  58. ϕ 1 = p λ \phi_{1}=p_{\lambda}
  59. r 2 - R 2 r^{2}-R^{2}
  60. λ λ
  61. A μ = ( A , ϕ ) A^{\mu}=(\vec{A},\phi)
  62. S = d d x d t [ 1 2 E 2 - 1 4 B i j B i j - m 2 2 A 2 + m 2 2 ϕ 2 ] S=\int d^{d}xdt\left[\frac{1}{2}E^{2}-\frac{1}{4}B_{ij}B_{ij}-\frac{m^{2}}{2}A% ^{2}+\frac{m^{2}}{2}\phi^{2}\right]
  63. E - ϕ - A ˙ \vec{E}\equiv-\nabla\phi-\dot{\vec{A}}
  64. B i j A j x i - A i x j B_{ij}\equiv\frac{\partial A_{j}}{\partial x_{i}}-\frac{\partial A_{i}}{% \partial x_{j}}
  65. ( A , - E ) (\vec{A},-\vec{E})
  66. ( ϕ , π ) (\phi,\pi)
  67. π 0 \pi\approx 0
  68. E + m 2 ϕ 0 \nabla\cdot\vec{E}+m^{2}\phi\approx 0
  69. H = d d x [ 1 2 E 2 + 1 4 B i j B i j - π A + E ϕ + m 2 2 A 2 - m 2 2 ϕ 2 ] H=\int d^{d}x\left[\frac{1}{2}E^{2}+\frac{1}{4}B_{ij}B_{ij}-\pi\nabla\cdot\vec% {A}+\vec{E}\cdot\nabla\phi+\frac{m^{2}}{2}A^{2}-\frac{m^{2}}{2}\phi^{2}\right]

Fisher's_exact_test.html

  1. p = ( a + b a ) ( c + d c ) ( n a + c ) = ( a + b ) ! ( c + d ) ! ( a + c ) ! ( b + d ) ! a ! b ! c ! d ! n ! p=\frac{\displaystyle{{a+b}\choose{a}}\displaystyle{{c+d}\choose{c}}}{% \displaystyle{{n}\choose{a+c}}}=\frac{(a+b)!~{}(c+d)!~{}(a+c)!~{}(b+d)!}{a!~{}% ~{}b!~{}~{}c!~{}~{}d!~{}~{}n!}
  2. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  3. p = ( 10 1 ) ( 14 11 ) / ( 24 12 ) = 10 ! 14 ! 12 ! 12 ! 1 ! 9 ! 11 ! 3 ! 24 ! 0.001346076 p={{{\textstyle\left({{10}\atop{1}}\right)}}{{\textstyle\left({{14}\atop{11}}% \right)}}}/{{{\textstyle\left({{24}\atop{12}}\right)}}}=\tfrac{10!~{}14!~{}12!% ~{}12!}{1!~{}9!~{}11!~{}3!~{}24!}\approx 0.001346076
  4. p = ( 10 0 ) ( 14 12 ) / ( 24 12 ) 0.000033652 {p={{\textstyle\left({{10}\atop{0}}\right)}}{{\textstyle\left({{14}\atop{12}}% \right)}}}/{{{\textstyle\left({{24}\atop{12}}\right)}}}\approx 0.000033652
  5. α e \alpha_{e}
  6. α e \alpha_{e}
  7. α e \alpha_{e}

Fisher_transformation.html

  1. r = i = 1 N ( X i - X ¯ ) ( Y i - Y ¯ ) i = 1 N ( X i - X ¯ ) 2 i = 1 N ( Y i - Y ¯ ) 2 r=\frac{\sum^{N}_{i=1}(X_{i}-\bar{X})(Y_{i}-\bar{Y})}{\sqrt{\sum^{N}_{i=1}(X_{% i}-\bar{X})^{2}}\sqrt{\sum^{N}_{i=1}(Y_{i}-\bar{Y})^{2}}}
  2. z := 1 2 ln ( 1 + r 1 - r ) = arctanh ( r ) , z:={1\over 2}\ln\left({1+r\over 1-r}\right)=\operatorname{arctanh}(r),
  3. 1 2 ln ( 1 + ρ 1 - ρ ) , {1\over 2}\ln\left({{1+\rho}\over{1-\rho}}\right),
  4. 1 N - 3 , {1\over\sqrt{N-3}},
  5. r = exp ( 2 z ) - 1 exp ( 2 z ) + 1 = tanh ( z ) , r={{\exp(2z)-1}\over{\exp(2z)+1}}=\operatorname{tanh}(z),

Fisheye_lens.html

  1. r = f tan ( θ ) r=f\tan(\theta)
  2. θ \theta
  3. r = 2 f tan ( θ / 2 ) r=2f\tan(\theta/2)
  4. r = f θ r=f\cdot\theta
  5. r = 2 f sin ( θ / 2 ) r=2f\sin(\theta/2)
  6. r = f sin ( θ ) r=f\sin(\theta)
  7. FOV = 4 arcsin ( frame size 4 focal length ) \,\text{FOV}=4\cdot\arcsin\left(\frac{\,\text{frame size}}{4\cdot\,\text{focal% length}}\right)

Fitting_lemma.html

  1. ker ( f k ) im ( f k ) = 0 \mathrm{ker}\left(f^{k}\right)\cap\mathrm{im}\left(f^{k}\right)=0
  2. x ker ( f k ) im ( f k ) x\in\mathrm{ker}\left(f^{k}\right)\cap\mathrm{im}\left(f^{k}\right)
  3. x = f k ( y ) x=f^{k}\left(y\right)
  4. y M y\in M
  5. f k ( x ) = 0 f^{k}\left(x\right)=0
  6. 0 = f k ( x ) = f k ( f k ( y ) ) = f 2 k ( y ) 0=f^{k}\left(x\right)=f^{k}\left(f^{k}\left(y\right)\right)=f^{2k}\left(y\right)
  7. y ker ( f 2 k ) = ker ( f k ) y\in\mathrm{ker}\left(f^{2k}\right)=\mathrm{ker}\left(f^{k}\right)
  8. 0 = f k ( y ) = x 0=f^{k}\left(y\right)=x
  9. ker ( f k ) + im ( f k ) = M \mathrm{ker}\left(f^{k}\right)+\mathrm{im}\left(f^{k}\right)=M
  10. x M x\in M
  11. y M y\in M
  12. f k ( x ) = f 2 k ( y ) f^{k}\left(x\right)=f^{2k}\left(y\right)
  13. f k ( x ) im ( f k ) = im ( f 2 k ) f^{k}\left(x\right)\in\mathrm{im}\left(f^{k}\right)=\mathrm{im}\left(f^{2k}\right)
  14. f k ( x - f k ( y ) ) = f k ( x ) - f k ( f k ( y ) ) = f k ( x ) - f 2 k ( y ) = 0 f^{k}\left(x-f^{k}\left(y\right)\right)=f^{k}\left(x\right)-f^{k}\left(f^{k}% \left(y\right)\right)=f^{k}\left(x\right)-f^{2k}\left(y\right)=0
  15. x - f k ( y ) ker ( f k ) x-f^{k}\left(y\right)\in\mathrm{ker}\left(f^{k}\right)
  16. x ker ( f k ) + f k ( y ) ker ( f k ) + im ( f k ) x\in\mathrm{ker}\left(f^{k}\right)+f^{k}\left(y\right)\subseteq\mathrm{ker}% \left(f^{k}\right)+\mathrm{im}\left(f^{k}\right)

Flap_(aeronautics).html

  1. L = 1 2 ρ V 2 S C L L=\tfrac{1}{2}\rho V^{2}SC_{L}
  2. ρ \rho
  3. C L C_{L}
  4. C L C_{L}

Flatness_problem.html

  1. H 2 = 8 π G 3 ρ - k c 2 a 2 H^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{a^{2}}
  2. H H
  3. ρ \rho
  4. a a
  5. k k
  6. k k
  7. G G
  8. c c
  9. ρ c \rho_{c}
  10. H H
  11. ρ c = 3 H 2 8 π G \rho_{c}=\frac{3H^{2}}{8\pi G}
  12. G G
  13. H H
  14. ρ c \rho_{c}
  15. 3 a 2 8 π G H 2 = ρ a 2 - 3 k c 2 8 π G \frac{3a^{2}}{8\pi G}H^{2}=\rho a^{2}-\frac{3kc^{2}}{8\pi G}
  16. ρ c a 2 - ρ a 2 = - 3 k c 2 8 π G \rho_{c}a^{2}-\rho a^{2}=-\frac{3kc^{2}}{8\pi G}
  17. ( Ω - 1 - 1 ) ρ a 2 = - 3 k c 2 8 π G . (\Omega^{-1}-1)\rho a^{2}=\frac{-3kc^{2}}{8\pi G}.
  18. a a
  19. ρ \rho
  20. ρ \rho
  21. a 2 a^{2}
  22. 10 60 , 10^{60},
  23. ρ = ρ c \rho=\rho_{c}
  24. ρ c \rho_{c}
  25. ρ c r i t \rho_{crit}
  26. a a
  27. e λ t e^{\lambda t}
  28. t t
  29. λ \lambda
  30. ρ a 2 \rho a^{2}
  31. a a
  32. ( Ω - 1 - 1 ) ρ a 2 = - 3 k c 2 8 π G (\Omega^{-1}-1)\rho a^{2}=\frac{-3kc^{2}}{8\pi G}
  33. | Ω - 1 - 1 | |\Omega^{-1}-1|
  34. | Ω - 1 - 1 | |\Omega^{-1}-1|
  35. 10 - 62 10^{-62}

Flesch–Kincaid_readability_tests.html

  1. 206.835 - 1.015 ( total words total sentences ) - 84.6 ( total syllables total words ) . 206.835-1.015\left(\frac{\,\text{total words}}{\,\text{total sentences}}\right% )-84.6\left(\frac{\,\text{total syllables}}{\,\text{total words}}\right).
  2. 0.39 ( total words total sentences ) + 11.8 ( total syllables total words ) - 15.59 0.39\left(\frac{\mbox{total words}~{}}{\mbox{total sentences}~{}}\right)+11.8% \left(\frac{\mbox{total syllables}~{}}{\mbox{total words}~{}}\right)-15.59

Floating_rate_note.html

  1. 100 - Clean price Maturity in years + Spread . \frac{100-\,\text{Clean price}}{\,\text{Maturity in years}}+\,\text{Spread}.
  2. 100 Clean price × ( 100 - Clean price Maturity in years + Spread ) . \frac{100}{\,\text{Clean price}}\times\left(\frac{100-\,\text{Clean price}}{\,% \text{Maturity in years}}+\,\text{Spread}\right).

Flow_network.html

  1. G = ( V , E ) G=(V,E)
  2. ( u , v ) E \ (u,v)\in E
  3. c ( u , v ) \ c(u,v)
  4. ( u , v ) E \ (u,v)\not\in E
  5. c ( u , v ) = 0 \ c(u,v)=0
  6. s \ s
  7. t \ t
  8. f : V × V \ f:V\times V\rightarrow\mathbb{R}
  9. u \ u
  10. v \ v
  11. f ( u , v ) c ( u , v ) \ f(u,v)\leq c(u,v)
  12. f ( u , v ) = - f ( v , u ) \ f(u,v)=-f(v,u)
  13. u \ u
  14. v \ v
  15. v \ v
  16. u \ u
  17. w V f ( u , w ) = 0 \ \sum_{w\in V}f(u,w)=0
  18. u = s \ u=s
  19. u = t \ u=t
  20. ( u , v ) E f ( u , v ) = ( v , z ) E f ( v , z ) \ \sum_{(u,v)\in E}f(u,v)=\sum_{(v,z)\in E}f(v,z)
  21. v V { s , t } \ {v\in V\setminus\{s,t\}}
  22. f ( u , v ) \ f(u,v)
  23. u \ u
  24. v \ v
  25. u \ u
  26. v \ v
  27. v \ v
  28. u \ u
  29. f ( u , v ) = 1 \ f(u,v)=1
  30. f ( v , u ) = - 1 \ f(v,u)=-1
  31. ( s , v ) E f ( s , v ) \ \sum_{(s,v)\in E}f(s,v)
  32. c f ( u , v ) = c ( u , v ) - f ( u , v ) \ c_{f}(u,v)=c(u,v)-f(u,v)
  33. G f ( V , E f ) \ G_{f}(V,E_{f})
  34. u \ u
  35. v \ v
  36. u \ u
  37. v \ v
  38. v \ v
  39. u \ u
  40. u \ u
  41. v \ v
  42. ( u 1 , u 2 , , u k ) \ (u_{1},u_{2},\dots,u_{k})
  43. u 1 = s \ u_{1}=s
  44. u k = t \ u_{k}=t
  45. c f ( u i , u i + 1 ) > 0 \ c_{f}(u_{i},u_{i+1})>0
  46. G f \ G_{f}
  47. G f \ G_{f}
  48. G f \ G_{f}
  49. V \ V
  50. G f \ G_{f}
  51. E f \ E_{f}
  52. ( x , y ) E \ (x,y)\in E
  53. f ( x , y ) < c ( x , y ) , \ f(x,y)<c(x,y),
  54. ( x , y ) E f \ (x,y)\in E_{f}
  55. c f = c ( x , y ) - f ( x , y ) \ c_{f}=c(x,y)-f(x,y)
  56. f ( x , y ) > 0 , \ f(x,y)>0,
  57. ( y , x ) E f \ (y,x)\in E_{f}
  58. c f = f ( x , y ) \ c_{f}=f(x,y)
  59. s s
  60. t t
  61. f / c f/c
  62. s s
  63. t t
  64. s s
  65. t t
  66. ( d , c ) (d,c)
  67. ( s , a , c , t ) (s,a,c,t)
  68. ( s , a , b , d , t ) (s,a,b,d,t)
  69. ( s , a , b , d , c , t ) (s,a,b,d,c,t)
  70. min ( c ( s , a ) - f ( s , a ) , c ( a , c ) - f ( a , c ) , c ( c , t ) - f ( c , t ) ) \min(c(s,a)-f(s,a),c(a,c)-f(a,c),c(c,t)-f(c,t))
  71. = min ( 5 - 3 , 3 - 2 , 2 - 1 ) = min ( 2 , 1 , 1 ) = 1 =\min(5-3,3-2,2-1)=\min(2,1,1)=1
  72. u , v u,v
  73. k ( u , v ) k(u,v)
  74. f ( u , v ) f(u,v)
  75. f ( u , v ) k ( u , v ) f(u,v)\cdot k(u,v)
  76. l ( u , v ) l(u,v)
  77. c ( u , v ) c(u,v)
  78. l ( t , s ) l(t,s)
  79. c ( t , s ) c(t,s)

Flux_quantization.html

  1. h / 2 e h/2e

Focus_(geometry).html

  1. x 2 1 + c + y 2 c = 1. \frac{x^{2}}{1+c}+\frac{y^{2}}{c}=1.

Force-directed_graph_drawing.html

  1. ( i , j ) (i,j)
  2. δ i j \delta_{ij}

Force_of_infection.html

  1. λ = number of new infections number of susceptible persons exposed × average duration of exposure \lambda=\frac{\mbox{number of new infections}~{}}{\mbox{number of susceptible % persons exposed}~{}\times\mbox{average duration of exposure}~{}}
  2. λ = 1 A \lambda=\frac{1}{A}
  3. A A

Forgetful_functor.html

  1. 𝒞 \mathcal{C}
  2. Ob ( 𝒞 ) \operatorname{Ob}(\mathcal{C})
  3. 𝒞 \mathcal{C}
  4. Fl ( 𝒞 ) \operatorname{Fl}(\mathcal{C})
  5. A A
  6. Ob ( 𝒞 ) , A | A | = \operatorname{Ob}(\mathcal{C}),\quad A\mapsto|A|=
  7. A , A,
  8. u u
  9. Fl ( 𝒞 ) , u | u | = \operatorname{Fl}(\mathcal{C}),\quad u\mapsto|u|=
  10. u u
  11. | | |\cdot|
  12. 𝒞 \mathcal{C}
  13. 𝐒𝐞𝐭 \mathbf{Set}
  14. 𝐌𝐨𝐝 ( R ) \mathbf{Mod}(R)
  15. R R
  16. 𝐒𝐞𝐭 \mathbf{Set}
  17. Free R \operatorname{Free}_{R}
  18. X Free R ( X ) X\mapsto\operatorname{Free}_{R}(X)
  19. R R
  20. X X
  21. X M X\to M
  22. Free R ( X ) M \operatorname{Free}_{R}(X)\to M
  23. Hom 𝐌𝐨𝐝 R ( Free R ( X ) , M ) = Hom 𝐒𝐞𝐭 ( X , Forget ( M ) ) . \operatorname{Hom}_{\mathbf{Mod}_{R}}(\operatorname{Free}_{R}(X),M)=% \operatorname{Hom}_{\mathbf{Set}}(X,\operatorname{Forget}(M)).
  24. X Free R ( X ) X\to\operatorname{Free}_{R}(X)

Forward_algorithm.html

  1. P ( x t | y 1 : t ) P(x_{t}|y_{1:t})
  2. x ( t ) x(t)
  3. x t x_{t}
  4. y 1 : t y_{1:t}
  5. 1 1
  6. t t
  7. p ( x t , y 1 : t ) p(x_{t},y_{1:t})
  8. x ( t ) x(t)
  9. x t x_{t}
  10. ( y ( 1 ) , y ( 2 ) , , y ( t ) ) (y(1),y(2),...,y(t))
  11. y 1 : t y_{1:t}
  12. p ( x t , y 1 : t ) p(x_{t},y_{1:t})
  13. { x 1 : t - 1 } \{x_{1:t-1}\}
  14. t t
  15. α t ( x t ) = p ( x t , y 1 : t ) = x t - 1 p ( x t , x t - 1 , y 1 : t ) \alpha_{t}(x_{t})=p(x_{t},y_{1:t})=\sum_{x_{t-1}}p(x_{t},x_{t-1},y_{1:t})
  16. p ( x t , x t - 1 , y 1 : t ) p(x_{t},x_{t-1},y_{1:t})
  17. α t ( x t ) = x t - 1 p ( y t | x t , x t - 1 , y 1 : t - 1 ) p ( x t | x t - 1 , y 1 : t - 1 ) p ( x t - 1 , y 1 : t - 1 ) \alpha_{t}(x_{t})=\sum_{x_{t-1}}p(y_{t}|x_{t},x_{t-1},y_{1:t-1})p(x_{t}|x_{t-1% },y_{1:t-1})p(x_{t-1},y_{1:t-1})
  18. y t y_{t}
  19. x t x_{t}
  20. x t x_{t}
  21. x t - 1 x_{t-1}
  22. α t ( x t ) = p ( y t | x t ) x t - 1 p ( x t | x t - 1 ) α t - 1 ( x t - 1 ) \alpha_{t}(x_{t})=p(y_{t}|x_{t})\sum_{x_{t-1}}p(x_{t}|x_{t-1})\alpha_{t-1}(x_{% t-1})
  23. p ( y t | x t ) p(y_{t}|x_{t})
  24. p ( x t | x t - 1 ) p(x_{t}|x_{t-1})
  25. α t ( x t ) \alpha_{t}(x_{t})
  26. α t - 1 ( x t - 1 ) \alpha_{t-1}(x_{t-1})
  27. P ( x k | y 1 : t ) P(x_{k}|y_{1:t})
  28. 1 < k < t 1<k<t
  29. P ( x 0 : t | y 0 : t ) P(x_{0:t}|y_{0:t})

Forward_osmosis.html

  1. J w = A ( Δ π - Δ P ) J_{w}=A\left(\Delta\pi-\Delta P\right)
  2. J w J_{w}
  3. J w J_{w}
  4. J s J_{s}
  5. J s = B Δ c J_{s}=B\Delta c
  6. B B
  7. Δ c \Delta c

Förster_resonance_energy_transfer.html

  1. E E
  2. E = k E T k f + k E T + k i E=\frac{k_{ET}}{k_{f}+k_{ET}+\sum{k_{i}}}
  3. k E T k_{ET}
  4. k f k_{f}
  5. k i k_{i}
  6. E E
  7. r r
  8. E = 1 1 + ( r / R 0 ) 6 E=\frac{1}{1+(r/R_{0})^{6}}\!
  9. R 0 R_{0}
  10. R 0 6 = 9000 Q 0 ( ln 10 ) κ 2 J 128 π 5 n 4 N A {R_{0}}^{6}=\frac{9000\,Q_{0}\,(\ln 10)\kappa^{2}\,J}{128\,\pi^{5}\,n^{4}\,N_{% A}}
  11. Q 0 Q_{0}
  12. n n
  13. N A N_{A}
  14. J J
  15. J = f D ( λ ) ϵ A ( λ ) λ 4 d λ J=\int f_{\rm D}(\lambda)\,\epsilon_{\rm A}(\lambda)\,\lambda^{4}\,d\lambda
  16. f D f_{\rm D}
  17. ϵ A \epsilon_{\rm A}
  18. κ = μ ^ A μ ^ D - 3 ( μ ^ D R ^ ) ( μ ^ A R ^ D A ) \kappa=\hat{\mu}_{A}\cdot\hat{\mu}_{D}-3(\hat{\mu}_{D}\cdot\hat{R})(\hat{\mu}_% {A}\cdot\hat{R}_{DA})
  19. μ ^ i \hat{\mu}_{i}
  20. R ^ \hat{R}
  21. E = 1 - τ D / τ D E=1-{\tau^{\prime}_{\rm D}}/{\tau_{\rm D}}\!
  22. τ D \tau^{\prime}_{\rm D}
  23. τ D \tau_{\rm D}
  24. E = 1 - F D / F D E=1-{F\,^{\prime}_{\rm D}}/{F_{\rm D}}\!
  25. F D F\,^{\prime}_{\rm D}
  26. F D F_{\rm D}
  27. ( background ) + ( constant ) * e - ( time ) / τ pb (\mbox{background}~{})+(\mbox{constant}~{})*e^{-(\mbox{time}~{})/{\tau_{\rm pb% }}}
  28. τ pb {\tau_{\rm pb}}
  29. E = 1 - τ pb / τ pb E=1-{\tau_{\rm pb}}/{\tau^{\prime}_{\rm pb}}\!
  30. τ pb {\tau^{\prime}_{\rm pb}}
  31. τ pb {\tau_{\rm pb}}

Fractional_coloring.html

  1. χ f ( G ) = lim b χ b ( G ) b = inf b χ b ( G ) b \chi_{f}(G)=\lim_{b\to\infty}\frac{\chi_{b}(G)}{b}=\inf_{b}\frac{\chi_{b}(G)}{b}
  2. Pr ( v S ) 1 k \Pr(v\in S)\geq\frac{1}{k}
  3. χ f ( G ) n ( G ) / α ( G ) \chi_{f}(G)\geq n(G)/\alpha(G)
  4. ω ( G ) χ f ( G ) χ ( G ) \omega(G)\leq\chi_{f}(G)\leq\chi(G)
  5. \mathcal{I}
  6. \mathcal{I}
  7. I ( G ) x I \sum_{I\in\mathcal{I}(G)}x_{I}\,
  8. I ( G , x ) x I 1 \sum_{I\in\mathcal{I}(G,x)}x_{I}\geq 1
  9. x x
  10. x I x_{I}

Fractional_Fourier_transform.html

  1. \mathcal{F}
  2. f ^ ( ξ ) = - f ( x ) e - 2 π i x ξ d x \hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)\ e^{-2\pi ix\xi}\,\mathrm{d}x
  3. ξ \xi
  4. - 1 \mathcal{F}^{-1}
  5. f ( x ) = - f ^ ( ξ ) e 2 π i ξ x d ξ , f(x)=\int_{-\infty}^{\infty}\hat{f}(\xi)\ e^{2\pi i\xi x}\,\mathrm{d}\xi,
  6. n \mathcal{F}^{n}
  7. n [ f ] = [ n - 1 [ f ] ] \mathcal{F}^{n}[f]=\mathcal{F}[\mathcal{F}^{n-1}[f]]
  8. - n = ( - 1 ) n \mathcal{F}^{-n}=(\mathcal{F}^{-1})^{n}
  9. 0 [ f ] = f \mathcal{F}^{0}[f]=f
  10. \mathcal{F}
  11. 4 [ f ] = f \mathcal{F}^{4}[f]=f
  12. 𝒫 \mathcal{P}
  13. 𝒫 [ f ] : t f ( - t ) \mathcal{P}[f]\colon t\mapsto f(-t)
  14. 0 = Id , 1 = , 2 = 𝒫 , 4 = Id \mathcal{F}^{0}=\mathrm{Id},\qquad\mathcal{F}^{1}=\mathcal{F},\qquad\mathcal{F% }^{2}=\mathcal{P},\qquad\mathcal{F}^{4}=\mathrm{Id}
  15. 3 = - 1 = 𝒫 = 𝒫 . \mathcal{F}^{3}=\mathcal{F}^{-1}=\mathcal{P}\circ\mathcal{F}=\mathcal{F}\circ% \mathcal{P}.
  16. n = 2 α / π n=2α/π
  17. α α
  18. α α
  19. α ( u ) \mathcal{F}_{\alpha}(u)
  20. [ - π / 2 , π / 2 ] [-\pi/2,\pi/2]
  21. α α
  22. 2 ( f ) = f ( - t ) \mathcal{F}^{2}(f)=f(-t)
  23. α ( f ) \mathcal{F}_{\alpha}(f)
  24. f ( t ) f(t)
  25. f ( t ) f(−t)
  26. α α
  27. π π
  28. α = π / 2 α=π/2
  29. α = π / 2 α=−π/2
  30. u u
  31. x x
  32. ξ ξ
  33. ( x , ξ ) (x,ξ)
  34. α α
  35. x a x_{a}
  36. α \mathcal{F}_{\alpha}
  37. α ( f ) ( ω ) = 1 - i cot ( α ) 2 π e i cot ( α ) ω 2 / 2 - e - i csc ( α ) ω t + i cot ( α ) t 2 / 2 f ( t ) d t . \mathcal{F}_{\alpha}(f)(\omega)=\sqrt{\frac{1-i\cot(\alpha)}{2\pi}}e^{i\cot(% \alpha)\omega^{2}/2}\int_{-\infty}^{\infty}e^{-i\csc(\alpha)\omega t+i\cot(% \alpha)t^{2}/2}f(t)\,dt~{}.
  38. α \mathcal{F}_{\alpha}
  39. α k π / 2 [ 2 π ] \alpha\equiv k\pi/2\,[2\pi]
  40. k k
  41. α = k \mathcal{F}_{\alpha}=\mathcal{F}^{k}
  42. α , β α,β
  43. α + β = α β = β α . \mathcal{F}_{\alpha+\beta}=\mathcal{F}_{\alpha}\circ\mathcal{F}_{\beta}=% \mathcal{F}_{\beta}\circ\mathcal{F}_{\alpha}.
  44. α \mathcal{F}_{\alpha}
  45. α α
  46. α [ k b k f k ( u ) ] = k b k α [ f k ( u ) ] \mathcal{F}_{\alpha}[\sum_{k}{b_{k}f_{k}(u)}]=\sum_{k}{b_{k}\mathcal{F}_{% \alpha}[f_{k}(u)]}
  47. α α
  48. π / 2 \pi/2
  49. α α
  50. k k
  51. α = k π / 2 = k = ( ) k \mathcal{F}_{\alpha}=\mathcal{F}_{k\pi/2}=\mathcal{F}^{k}=(\mathcal{F})^{k}
  52. 2 = 𝒫 \mathcal{F}^{2}=\mathcal{P}
  53. 3 = - 1 = ( ) - 1 \mathcal{F}^{3}=\mathcal{F}^{-1}=(\mathcal{F})^{-1}
  54. 4 = 0 = \mathcal{F}^{4}=\mathcal{F}^{0}=\mathcal{I}
  55. j = j mod 4 \mathcal{F}^{j}=\mathcal{F}^{j\mod 4}
  56. ( α ) - 1 = - α (\mathcal{F}_{\alpha})^{-1}=\mathcal{F}_{-\alpha}
  57. α 1 α 2 = α 2 α 1 \mathcal{F}_{\alpha_{1}}\mathcal{F}_{\alpha_{2}}=\mathcal{F}_{\alpha_{2}}% \mathcal{F}_{\alpha_{1}}
  58. ( α 1 α 2 ) α 3 = α 1 ( α 2 α 3 ) (\mathcal{F}_{\alpha_{1}}\mathcal{F}_{\alpha_{2}})\mathcal{F}_{\alpha_{3}}=% \mathcal{F}_{\alpha_{1}}(\mathcal{F}_{\alpha_{2}}\mathcal{F}_{\alpha_{3}})
  59. f * ( u ) g ( u ) d u = f α * ( u ) g α ( u ) d u \int f^{*}(u)g(u)du=\int f_{\alpha}^{*}(u)g_{\alpha}(u)du
  60. 𝒫 \mathcal{P}
  61. 𝒫 [ f ( u ) ] = f ( - u ) \mathcal{P}[f(u)]=f(-u)
  62. α 𝒫 = 𝒫 α \mathcal{F}_{\alpha}\mathcal{P}=\mathcal{P}\mathcal{F}_{\alpha}
  63. α [ f ( - u ) ] = f α ( - u ) \mathcal{F}_{\alpha}[f(-u)]=f_{\alpha}(-u)
  64. 𝒮 ( u 0 ) \mathcal{SH}(u_{0})
  65. 𝒫 ( v 0 ) \mathcal{PH}(v_{0})
  66. 𝒮 ( u 0 ) \mathcal{SH}(u_{0})
  67. 𝒫 ( v 0 ) \mathcal{PH}(v_{0})
  68. 𝒮 ( u 0 ) [ f ( u ) ] = f ( u + u 0 ) \mathcal{SH}(u_{0})[f(u)]=f(u+u_{0})
  69. 𝒫 ( v 0 ) [ f ( u ) ] = e j 2 π v 0 u f ( u ) \mathcal{PH}(v_{0})[f(u)]=e^{j2\pi v_{0}u}f(u)
  70. α 𝒮 ( u 0 ) = e j π u 0 2 sin α cos α 𝒫 ( u 0 sin α ) 𝒮 ( u 0 cos α ) F α \mathcal{F}_{\alpha}\mathcal{SH}(u_{0})=e^{j\pi u_{0}^{2}\sin\alpha\cos\alpha}% \mathcal{PH}(u_{0}\sin\alpha)\mathcal{SH}(u_{0}\cos\alpha)F_{\alpha}
  71. α [ f ( u + u 0 ) ] = e j π u 0 2 sin α cos α e j 2 π u u 0 sin α f α ( u + u 0 cos α ) \mathcal{F}_{\alpha}[f(u+u_{0})]=e^{j\pi u_{0}^{2}\sin\alpha\cos\alpha}e^{j2% \pi uu_{0}\sin\alpha}f_{\alpha}(u+u_{0}\cos\alpha)
  72. M ( M ) M(M)
  73. Q ( q ) Q(q)
  74. M ( M ) M(M)
  75. Q ( q ) Q(q)
  76. M ( M ) [ f ( u ) ] = | M | - 1 / 2 f ( u / M ) M(M)[f(u)]=|M|^{-1/2}f(u/M)
  77. Q ( q ) [ f ( u ) ] = e - j π q u 2 f ( u ) Q(q)[f(u)]=e^{-j\pi qu^{2}}f(u)
  78. α M ( M ) = Q ( - cot ( 1 - cos 2 α cos 2 α α ) ) × M ( sin α M sin α ) α \mathcal{F}_{\alpha}M(M)=Q(-\cot(\frac{1-\cos^{2}\alpha^{\prime}}{\cos^{2}% \alpha}\alpha))\times M(\frac{\sin\alpha}{M\sin\alpha^{\prime}})\mathcal{F}_{% \alpha^{\prime}}
  79. α [ | M | - 1 / 2 f ( u / M ) ] = 1 - j cot α 1 - j M 2 cot α e j π u 2 cot ( 1 - cos 2 α cos 2 α α ) × f a ( M u sin α sin α ) \mathcal{F}_{\alpha}[|M|^{-1/2}f(u/M)]=\sqrt{\frac{1-j\cot\alpha}{1-jM^{2}\cot% \alpha}}e^{j\pi u^{2}\cot(\frac{1-\cos^{2}\alpha^{\prime}}{\cos^{2}\alpha}% \alpha)}\times f_{a}(\frac{Mu\sin\alpha^{\prime}}{\sin\alpha})
  80. f ( u / M ) f(u/M)
  81. f α ( u ) f_{\alpha}(u)
  82. f ( u / M ) f(u/M)
  83. f α ( u ) f_{\alpha}^{\prime}(u)
  84. α α \alpha\neq\alpha^{\prime}
  85. α f ( u ) = K α ( u , x ) f ( x ) d x \mathcal{F}_{\alpha}f(u)=\int K_{\alpha}(u,x)f(x)\,\mathrm{d}x
  86. K α ( u , x ) = { 1 - i cot ( α ) exp ( i π ( cot ( α ) ( x 2 + u 2 ) - 2 csc ( α ) u x ) ) if α is not a multiple of π , δ ( u - x ) if α is a multiple of 2 π , δ ( u + x ) if α + π is a multiple of 2 π , K_{\alpha}(u,x)=\begin{cases}\sqrt{1-i\cot(\alpha)}\exp\left(i\pi(\cot(\alpha)% (x^{2}+u^{2})-2\csc(\alpha)ux)\right)&\mbox{if }~{}\alpha\mbox{ is not a % multiple of }~{}\pi,\\ \delta(u-x)&\mbox{if }~{}\alpha\mbox{ is a multiple of }~{}2\pi,\\ \delta(u+x)&\mbox{if }~{}\alpha+\pi\mbox{ is a multiple of }~{}2\pi,\\ \end{cases}
  87. [ - π / 2 , π / 2 ] [-\pi/2,\pi/2]
  88. α α
  89. π π
  90. K α ( u , u ) = K α ( u , u ) K_{\alpha}(u,u^{\prime})=K_{\alpha}(u^{\prime},u)
  91. K α - 1 ( u , u ) = K α * ( u , u ) = K - α ( u , u ) K_{\alpha}^{-1}(u,u^{\prime})=K_{\alpha}^{*}(u,u^{\prime})=K_{-\alpha}(u^{% \prime},u)
  92. K α + β ( u , u ) = K α ( u , u ′′ ) K β ( u ′′ , u ) d u ′′ . K_{\alpha+\beta}(u,u^{\prime})=\int K_{\alpha}(u,u^{\prime\prime})K_{\beta}(u^% {\prime\prime},u^{\prime})\,\mathrm{d}u^{\prime\prime}.

Fracture_mechanics.html

  1. a a
  2. σ f a C \sigma_{f}\sqrt{a}\approx C
  3. C = 2 E γ π C=\sqrt{\cfrac{2E\gamma}{\pi}}
  4. σ y a = C \sigma_{y}\sqrt{a}=C
  5. G = 2 γ + G p G=2\gamma+G_{p}
  6. σ f a = E G π . \sigma_{f}\sqrt{a}=\sqrt{\cfrac{E~{}G}{\pi}}.
  7. G 2 γ = 2 J / m 2 G\approx 2\gamma=2\,\,J/m^{2}
  8. G G p = 1000 J / m 2 G\approx G_{p}=1000\,\,J/m^{2}
  9. G 2 - 1000 J / m 2 G\approx 2-1000\,\,J/m^{2}
  10. σ i j ( K 2 π r ) f i j ( θ ) \sigma_{ij}\approx\left(\cfrac{K}{\sqrt{2\pi r}}\right)~{}f_{ij}(\theta)
  11. MPa- m \,\text{MPa-}\sqrt{\,\text{m}}
  12. G := [ U a ] P = - [ U a ] u G:=\left[\cfrac{\partial U}{\partial a}\right]_{P}=-\left[\cfrac{\partial U}{% \partial a}\right]_{u}
  13. G = G I = { K I 2 E plane stress ( 1 - ν 2 ) K I 2 E plane strain G=G_{I}=\begin{cases}\cfrac{K_{I}^{2}}{E}&\,\text{plane stress}\\ \cfrac{(1-\nu^{2})K_{I}^{2}}{E}&\,\text{plane strain}\end{cases}
  14. r p = K C 2 σ Y 2 . r_{p}=\frac{K_{C}^{2}}{\sigma_{Y}^{2}}.
  15. J = Γ ( w d y - T i u i x d s ) with w = 0 ε i j σ i j d ε i j J=\int_{\Gamma}(w\,dy-T_{i}\frac{\partial u_{i}}{\partial x}\,ds)\quad\,\text{% with}\quad w=\int^{\varepsilon_{ij}}_{0}\sigma_{ij}\,d\varepsilon_{ij}
  16. Γ \Gamma
  17. w w
  18. T i T_{i}
  19. u i u_{i}
  20. d s ds
  21. Γ \Gamma
  22. σ i j \sigma_{ij}
  23. ε i j \varepsilon_{ij}
  24. σ Y \sigma_{Y}
  25. K I C K_{IC}
  26. σ f a i l = K I C / π a \sigma_{fail}=K_{IC}/\sqrt{\pi a}
  27. σ f a i l = σ Y \sigma_{fail}=\sigma_{Y}
  28. a = K I C 2 / π σ Y 2 a=K_{IC}^{2}/\pi\sigma_{Y}^{2}
  29. a a
  30. a t a_{t}
  31. a < a t a<a_{t}
  32. a > a t a>a_{t}
  33. a t a_{t}
  34. σ i j = ( σ i j ) + Q * δ i j * σ y i e l d \sigma_{ij}=(\sigma_{ij})+Q*\delta_{ij}*\sigma_{yield}
  35. δ i j = 1 \delta_{ij}=1
  36. i = j i=j
  37. G = π σ 2 a E G=\frac{\pi\sigma^{2}a}{E}\,
  38. G G
  39. σ \sigma
  40. a a
  41. E E
  42. G c = π σ f 2 a E G_{c}=\frac{\pi\sigma_{f}^{2}a}{E}\,
  43. G G
  44. G c G_{c}
  45. K I = σ π a K_{I}=\sigma\sqrt{\pi a}\,
  46. K c = E G c K_{c}=\sqrt{EG_{c}}\,
  47. K c = E G c 1 - ν 2 K_{c}=\sqrt{\frac{EG_{c}}{1-\nu^{2}}}\,
  48. ν \nu
  49. K I K c K_{I}\geq K_{c}
  50. K c K_{c}
  51. K I c K_{Ic}
  52. K I K_{I}
  53. K I = Y σ π a K_{I}=Y\sigma\sqrt{\pi a}\,
  54. Y ( a W ) = sec ( π a W ) Y\left(\frac{a}{W}\right)=\sqrt{\sec\left(\frac{\pi a}{W}\right)}\,
  55. Y ( a W ) = 1.12 - 0.41 π a W + 18.7 π ( a W ) 2 - Y\left(\frac{a}{W}\right)=1.12-\frac{0.41}{\sqrt{\pi}}\frac{a}{W}+\frac{18.7}{% \sqrt{\pi}}\left(\frac{a}{W}\right)^{2}-\cdots\,
  56. K I c = E * J I c K_{Ic}=\sqrt{E^{*}J_{Ic}}\,
  57. E * = E E^{*}=E
  58. E * = E 1 - ν 2 E^{*}=\frac{E}{1-\nu^{2}}
  59. K = c σ a K=c\sigma\sqrt{a}

Francis_turbine.html

  1. 2 V f 1 2 ( cot α 1 ( cot α 1 + cot β 1 ) ) V f 2 2 + 2 V f 1 2 ( cot α 1 ( cot α 1 + cot β 1 ) ) \frac{2V_{f1}^{2}(\cot\alpha_{1}(\cot\alpha_{1}+\cot\beta_{1}))}{V_{f2}^{2}+2V% _{f1}^{2}(\cot\alpha_{1}(\cot\alpha_{1}+\cot\beta_{1}))}\,

Fraunhofer_diffraction.html

  1. W 2 L λ 1 \frac{W^{2}}{L\lambda}\ll 1
  2. W W
  3. λ \lambda
  4. L L
  5. λ λ
  6. W W
  7. W W
  8. θ θ
  9. θ 0 θ0
  10. α α
  11. α 2 λ W \alpha\approx{\frac{2\lambda}{W}}
  12. α α
  13. z z
  14. d f = 2 λ z W d_{f}=\frac{2\lambda z}{W}
  15. y y
  16. θ θ
  17. C D CD
  18. θ θ
  19. B B
  20. θ θ
  21. A A
  22. B B
  23. θ θ
  24. θ m i n C D A C = λ W \theta_{min}\approx\frac{CD}{AC}=\frac{\lambda}{W}
  25. α = 2 θ m i n = 2 λ W \alpha=2\theta_{min}=\frac{2\lambda}{W}
  26. α 1.22 λ W \alpha\approx\frac{1.22\lambda}{W}
  27. W W
  28. θ f = λ / d ~{}\theta_{f}=\lambda/d
  29. z z
  30. w f = z θ f = z λ / d ~{}w_{f}=z\theta_{f}=z\lambda/d
  31. d d
  32. θ θ
  33. d sin θ d θ d\sin\theta\approx d\theta
  34. d θ n = n λ , n = 0 , 1 , 2..... ~{}d\theta_{n}=n\lambda,~{}n=0,1,2.....
  35. θ f λ / d \theta_{f}\approx\lambda/d
  36. z z
  37. z θ
  38. w = z λ / d ~{}w=z\lambda/d
  39. S S
  40. sin θ n = n λ / S , n = 0 , ± 1 , ± 2...... ~{}\sin\theta_{n}=n\lambda/S,n=0,\pm 1,\pm 2......
  41. sin θ n = n λ S + sin θ 0 , n = 0 , ± 1 , ± 2.... \sin\theta_{n}=\frac{n\lambda}{S}+\sin\theta_{0},n=0,\pm 1,\pm 2....
  42. n n
  43. θ θ
  44. W sin θ = n λ , n = 0 , ± 1 , ± 2 , . . W\sin\theta=n\lambda,n=0,\pm 1,\pm 2,.....

Free-electron_laser.html

  1. λ r \lambda_{r}
  2. λ r = λ u 2 γ 2 ( 1 + K 2 ) \lambda_{r}=\frac{\lambda_{u}}{2\gamma^{2}}(1+K^{2})
  3. λ r λ u 2 γ 2 \lambda_{r}\propto\frac{\lambda_{u}}{2\gamma^{2}}
  4. λ u \lambda_{u}
  5. γ \gamma
  6. γ \gamma
  7. λ u / γ \lambda_{u}/\gamma
  8. γ \gamma
  9. γ \gamma
  10. K = γ λ u 2 π ρ = e B 0 λ u 8 π m e c K=\frac{\gamma\lambda_{u}}{2\pi\rho}=\frac{eB_{0}\lambda_{u}}{\sqrt{8}\pi m_{e% }c}
  11. ρ \rho
  12. B 0 B_{0}
  13. m e m_{e}

Free_cash_flow.html

  1. F C F t = O C B t - I t FCF_{t}=OCB_{t}-I_{t}\,
  2. I t = K t - K t - 1 I_{t}=K_{t}-K_{t-1}\,

Free_electron_model.html

  1. V ( r ) = 0 V({r})=0
  2. - 2 2 m 2 Ψ ( r , t ) = i t Ψ ( r , t ) -\frac{\hbar^{2}}{2m}\nabla^{2}\Psi({r},t)=i\hbar\frac{\partial}{\partial t}% \Psi({r},t)
  3. Ψ ( r , t ) \Psi({r},t)
  4. Ψ ( r , t ) = ψ ( r ) e - i ω t \Psi({r},t)=\psi({r})e^{-i\omega t}
  5. E = ω E=\hbar\omega
  6. ψ k ( r ) = 1 Ω r e i k r \psi_{{k}}({r})=\frac{1}{\sqrt{\Omega_{r}}}e^{i{k}\cdot{r}}
  7. k {k}
  8. Ω r \Omega_{r}
  9. E = 2 k 2 2 m E=\frac{\hbar^{2}k^{2}}{2m}
  10. Ψ ( r , t ) = 1 Ω r e i k r - i ω t \Psi({r},t)=\frac{1}{\sqrt{\Omega_{r}}}e^{i{k}\cdot{r}-i\omega t}
  11. ψ k ( r ) \psi_{{k}}({r})
  12. ρ s = - n e x \rho_{s}=-nex
  13. E = n e x ϵ 0 E=\frac{nex}{\epsilon_{0}}
  14. ϵ ( ω ) = D ( ω ) ϵ 0 E ( ω ) = 1 + P ( ω ) ϵ 0 E ( ω ) \epsilon(\omega)=\frac{D(\omega)}{\epsilon_{0}E(\omega)}=1+\frac{P(\omega)}{% \epsilon_{0}E(\omega)}
  15. D ( ω ) D(\omega)
  16. P ( ω ) P(\omega)
  17. E ( ω ) = E 0 e - i ω t , P ( ω ) = P 0 e - i ω t E(\omega)=E_{0}e^{-i\omega t},\quad P(\omega)=P_{0}e^{-i\omega t}
  18. P = - n e x P=-nex
  19. F = - e E = m a = m d 2 x d t 2 F=-eE=ma=m\frac{d^{2}x}{dt^{2}}
  20. P ( ω ) = - n e 2 m ω 2 E ( ω ) P(\omega)=-\frac{ne^{2}}{m\omega^{2}}E(\omega)
  21. ϵ ( ω ) = 1 - n e 2 ϵ 0 m ω 2 \epsilon(\omega)=1-\frac{ne^{2}}{\epsilon_{0}m\omega^{2}}
  22. ω p \omega_{p}
  23. ω p = n e 2 ϵ 0 m \omega_{p}=\sqrt{\frac{ne^{2}}{\epsilon_{0}m}}
  24. V ( r ) = 0 V({r})=0
  25. - 2 2 m 2 Ψ ( r , t ) = i t Ψ ( r , t ) -\frac{\hbar^{2}}{2m}\nabla^{2}\Psi({r},t)=i\hbar\frac{\partial}{\partial t}% \Psi({r},t)
  26. i t Ψ ( r , t ) = E Ψ ( r , t ) i\hbar\frac{\partial}{\partial t}\Psi({r},t)=E\Psi({r},t)
  27. - 2 2 m 2 Ψ ( r , t ) = E Ψ ( r , t ) -\frac{\hbar^{2}}{2m}\nabla^{2}\Psi({r},t)=E\Psi({r},t)
  28. Ψ ( r , t ) = ψ ( r ) e α t \Psi({r},t)=\psi({r})e^{\alpha t}
  29. i t e α t = E e α t i\hbar\frac{\partial}{\partial t}e^{\alpha t}=Ee^{\alpha t}
  30. α = - i E \alpha=-\frac{iE}{\hbar}
  31. E = i α = ω E=i\hbar\alpha=\hbar\omega
  32. Ψ ( r , t ) = ψ ( r ) e - i ω t \Psi({r},t)=\psi({r})e^{-i\omega t}
  33. - 2 2 m 2 ψ ( r ) = E ψ ( r ) -\frac{\hbar^{2}}{2m}\nabla^{2}\psi({r})=E\psi({r})
  34. 2 = 2 x 2 + 2 y 2 + 2 z 2 \nabla^{2}=\frac{\partial^{2}}{{\partial x}^{2}}+\frac{\partial^{2}}{{\partial y% }^{2}}+\frac{\partial^{2}}{{\partial z}^{2}}
  35. ψ ( r ) = ϕ x ( x ) ϕ y ( y ) ϕ z ( z ) \psi({r})=\phi_{x}(x)\phi_{y}(y)\phi_{z}(z)
  36. - 2 2 m 2 x 2 ϕ x ( x ) = E x ϕ x ( x ) -\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{{\partial x}^{2}}\phi_{x}(x)=E_{x}% \phi_{x}(x)
  37. ϕ x ( x ) = N x e κ x \phi_{x}(x)=N_{x}e^{\kappa x}
  38. 2 x 2 ϕ x ( x ) = κ 2 N x e κ x = - 2 m 2 E x N x e κ x \frac{\partial^{2}}{{\partial x}^{2}}\phi_{x}(x)=\kappa^{2}N_{x}e^{\kappa x}=-% \frac{2m}{\hbar^{2}}E_{x}N_{x}e^{\kappa x}
  39. κ = i k x = i 2 m E x 2 \kappa=ik_{x}=i\sqrt{\frac{2mE_{x}}{\hbar^{2}}}
  40. ψ ( r ) = N x N y N z e i ( k x x + k y y + k z z ) \psi({r})=N_{x}N_{y}N_{z}e^{i(k_{x}x+k_{y}y+k_{z}z)}
  41. E = 2 2 m ( k x 2 + k y 2 + k z 2 ) E=\frac{\hbar^{2}}{2m}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})
  42. Ω r ψ k * ( r ) ψ k ( r ) d r = δ k , k \int_{\Omega_{r}}\psi_{{k}}^{*}({r})\psi_{{k}^{\prime}}({r})d{r}=\delta_{{k},{% k}^{\prime}}
  43. k = k x 2 + k y 2 + k z 2 k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}
  44. ψ k ( r ) = 1 Ω r e i k r \psi_{{k}}({r})=\frac{1}{\sqrt{\Omega_{r}}}e^{i{k}\cdot{r}}
  45. k {k}
  46. E = 2 k 2 2 m E=\frac{\hbar^{2}k^{2}}{2m}
  47. Ω r \Omega_{r}
  48. Ψ ( r , t ) = ψ ( r ) e - i ω t \Psi({r},t)=\psi({r})e^{-i\omega t}
  49. Ψ ( r , t ) = 1 Ω r e i k r - i ω t \Psi({r},t)=\frac{1}{\sqrt{\Omega_{r}}}e^{i{k}\cdot{r}-i\omega t}
  50. E F E_{F}
  51. E ( k ) = 2 k 2 2 m E({k})=\frac{\hbar^{2}k^{2}}{2m}
  52. | k | < k F |{k}|<k_{F}
  53. k F k_{F}
  54. k F = ( 3 π 2 N e / V ) 1 / 3 k_{F}=(3\pi^{2}N_{e}/V)^{1/3}
  55. N e N_{e}
  56. E F = 2 2 m ( 3 π 2 N e V ) 2 / 3 E_{F}=\frac{\hbar^{2}}{2m}\left(\frac{3\pi^{2}N_{e}}{V}\right)^{2/3}\,
  57. Z Z
  58. N e N_{e}
  59. N Z NZ
  60. N N
  61. E ( k ) = 2 k 2 2 m E({k})=\frac{\hbar^{2}k^{2}}{2m}
  62. Δ V Δ k = ( 2 π ) 3 \Delta V\Delta{k}=(2\pi)^{3}
  63. N ( E ) = V 2 π 2 ( 2 m 2 ) 3 / 2 E N(E)=\frac{V}{2\pi^{2}}\left(\frac{2m}{\hbar^{2}}\right)^{3/2}\sqrt{E}
  64. V V
  65. N ( E F ) = 3 Z N 2 1 E F , N(E_{F})=\frac{3ZN}{2}\frac{1}{E_{F}},

Frenet–Serret_formulas.html

  1. d 𝐓 d s \displaystyle\dfrac{d\mathbf{T}}{ds}
  2. d 𝐓 d s \tfrac{d\mathbf{T}}{ds}
  3. s ( t ) = 0 t 𝐫 ( σ ) d σ . s(t)=\int_{0}^{t}\|\mathbf{r}^{\prime}(\sigma)\|d\sigma.
  4. 𝐓 = d 𝐫 d s . ( 1 ) \mathbf{T}={d\mathbf{r}\over ds}.\qquad\qquad(1)
  5. 𝐍 = d 𝐓 d s d 𝐓 d s . ( 2 ) \mathbf{N}={\frac{d\mathbf{T}}{ds}\over\left\|\frac{d\mathbf{T}}{ds}\right\|}.% \qquad\qquad(2)
  6. 𝐁 = 𝐓 × 𝐍 . ( 3 ) \mathbf{B}=\mathbf{T}\times\mathbf{N}.\qquad\qquad(3)
  7. d 𝐓 d s = κ 𝐍 d 𝐍 d s = - κ 𝐓 + τ 𝐁 d 𝐁 d s = - τ 𝐍 \begin{matrix}\frac{d\mathbf{T}}{ds}&=&&\kappa\mathbf{N}&\\ &&&&\\ \frac{d\mathbf{N}}{ds}&=&-\kappa\mathbf{T}&&+\,\tau\mathbf{B}\\ &&&&\\ \frac{d\mathbf{B}}{ds}&=&&-\tau\mathbf{N}&\end{matrix}
  8. κ \kappa
  9. τ \tau
  10. [ 𝐓 𝐍 𝐁 ] = [ 0 κ 0 - κ 0 τ 0 - τ 0 ] [ 𝐓 𝐍 𝐁 ] . \begin{bmatrix}\mathbf{T^{\prime}}\\ \mathbf{N^{\prime}}\\ \mathbf{B^{\prime}}\end{bmatrix}=\begin{bmatrix}0&\kappa&0\\ -\kappa&0&\tau\\ 0&-\tau&0\end{bmatrix}\begin{bmatrix}\mathbf{T}\\ \mathbf{N}\\ \mathbf{B}\end{bmatrix}.
  11. 𝐞 1 ( s ) = 𝐫 ( s ) \mathbf{e}_{1}(s)=\mathbf{r}^{\prime}(s)
  12. 𝐞 2 ¯ ( s ) = 𝐫 ′′ ( s ) - 𝐫 ′′ ( s ) , 𝐞 1 ( s ) 𝐞 1 ( s ) \overline{\mathbf{e}_{2}}(s)=\mathbf{r}^{\prime\prime}(s)-\langle\mathbf{r}^{% \prime\prime}(s),\mathbf{e}_{1}(s)\rangle\,\mathbf{e}_{1}(s)
  13. 𝐞 2 ( s ) = 𝐞 2 ¯ ( s ) 𝐞 2 ¯ ( s ) \mathbf{e}_{2}(s)=\frac{\overline{\mathbf{e}_{2}}(s)}{\|\overline{\mathbf{e}_{% 2}}(s)\|}
  14. 𝐞 j ( s ) = 𝐞 j ¯ ( s ) 𝐞 j ¯ ( s ) , \begin{aligned}\displaystyle\mathbf{e}_{j}(s)=\frac{\overline{\mathbf{e}_{j}}(% s)}{\|\overline{\mathbf{e}_{j}}(s)\|}\mbox{, }\end{aligned}
  15. 𝐞 j ¯ ( s ) = 𝐫 ( j ) ( s ) - i = 1 j - 1 𝐫 ( j ) ( s ) , 𝐞 i ( s ) 𝐞 i ( s ) . \begin{aligned}\displaystyle\overline{\mathbf{e}_{j}}(s)=\mathbf{r}^{(j)}(s)-% \sum_{i=1}^{j-1}\langle\mathbf{r}^{(j)}(s),\mathbf{e}_{i}(s)\rangle\,\mathbf{e% }_{i}(s).\end{aligned}
  16. χ i ( s ) = 𝐞 i ( s ) , 𝐞 i + 1 ( s ) 𝐫 ( s ) \chi_{i}(s)=\frac{\langle\mathbf{e}_{i}^{\prime}(s),\mathbf{e}_{i+1}(s)\rangle% }{\|\mathbf{r}^{\prime}(s)\|}
  17. [ 𝐞 1 ( s ) 𝐞 n ( s ) ] = \begin{aligned}\displaystyle\begin{bmatrix}\mathbf{e}_{1}^{\prime}(s)\\ \vdots\\ \mathbf{e}_{n}^{\prime}(s)\\ \end{bmatrix}=\\ \end{aligned}
  18. [ 0 χ 1 ( s ) 0 - χ 1 ( s ) 0 χ n - 1 ( s ) 0 - χ n - 1 ( s ) 0 ] [ 𝐞 1 ( s ) 𝐞 n ( s ) ] \begin{aligned}\displaystyle\begin{bmatrix}0&\chi_{1}(s)&&0\\ -\chi_{1}(s)&\ddots&\ddots&\\ &\ddots&0&\chi_{n-1}(s)\\ 0&&-\chi_{n-1}(s)&0\\ \end{bmatrix}\begin{bmatrix}\mathbf{e}_{1}(s)\\ \vdots\\ \mathbf{e}_{n}(s)\\ \end{bmatrix}\end{aligned}
  19. Q = [ 𝐓 𝐍 𝐁 ] Q=\left[\begin{matrix}\mathbf{T}\\ \mathbf{N}\\ \mathbf{B}\end{matrix}\right]
  20. ( d Q d s ) Q T = [ 0 κ 0 - κ 0 τ 0 - τ 0 ] \left(\frac{dQ}{ds}\right)Q^{T}=\left[\begin{matrix}0&\kappa&0\\ -\kappa&0&\tau\\ 0&-\tau&0\end{matrix}\right]
  21. 0 = d I d s = ( d Q d s ) Q T + Q ( d Q d s ) T ( d Q d s ) Q T = - ( ( d Q d s ) Q T ) T \begin{aligned}\displaystyle 0=\frac{dI}{ds}=\left(\frac{dQ}{ds}\right)Q^{T}+Q% \left(\frac{dQ}{ds}\right)^{T}\implies\\ \displaystyle\left(\frac{dQ}{ds}\right)Q^{T}=-\left(\left(\frac{dQ}{ds}\right)% Q^{T}\right)^{T}\\ \end{aligned}
  22. κ = r r 2 + h 2 \kappa=\frac{r}{r^{2}+h^{2}}
  23. τ = ± h r 2 + h 2 . \tau=\pm\frac{h}{r^{2}+h^{2}}.
  24. h 2 + r 2 \sqrt{h^{2}+r^{2}}
  25. A 2 = h 2 + r 2 A^{2}=h^{2}+r^{2}
  26. 𝐫 ( s ) = 𝐫 ( 0 ) + ( s - s 3 κ 2 ( 0 ) 6 ) 𝐓 ( 0 ) + ( s 2 κ ( 0 ) 2 + s 3 κ ( 0 ) 6 ) 𝐍 ( 0 ) + ( s 3 κ ( 0 ) τ ( 0 ) 6 ) 𝐁 ( 0 ) + o ( s 3 ) . \mathbf{r}(s)=\mathbf{r}(0)+\left(s-\frac{s^{3}\kappa^{2}(0)}{6}\right)\mathbf% {T}(0)+\left(\frac{s^{2}\kappa(0)}{2}+\frac{s^{3}\kappa^{\prime}(0)}{6}\right)% \mathbf{N}(0)+\left(\frac{s^{3}\kappa(0)\tau(0)}{6}\right)\mathbf{B}(0)+o(s^{3% }).
  27. 𝐫 ( 0 ) + s 𝐓 ( 0 ) + s 2 κ ( 0 ) 2 𝐍 ( 0 ) + o ( s 2 ) . \mathbf{r}(0)+s\mathbf{T}(0)+\frac{s^{2}\kappa(0)}{2}\mathbf{N}(0)+o(s^{2}).
  28. 𝐫 ( 0 ) + ( s 2 κ ( 0 ) 2 + s 3 κ ( 0 ) 6 ) 𝐍 ( 0 ) + ( s 3 κ ( 0 ) τ ( 0 ) 6 ) 𝐁 ( 0 ) + o ( s 3 ) \mathbf{r}(0)+\left(\frac{s^{2}\kappa(0)}{2}+\frac{s^{3}\kappa^{\prime}(0)}{6}% \right)\mathbf{N}(0)+\left(\frac{s^{3}\kappa(0)\tau(0)}{6}\right)\mathbf{B}(0)% +o(s^{3})
  29. 𝐫 ( 0 ) + ( s - s 3 κ 2 ( 0 ) 6 ) 𝐓 ( 0 ) + ( s 3 κ ( 0 ) τ ( 0 ) 6 ) 𝐁 ( 0 ) + o ( s 3 ) \mathbf{r}(0)+\left(s-\frac{s^{3}\kappa^{2}(0)}{6}\right)\mathbf{T}(0)+\left(% \frac{s^{3}\kappa(0)\tau(0)}{6}\right)\mathbf{B}(0)+o(s^{3})
  30. R ( s , t ) = C ( s ) + t 𝐍 , - 1 t 1. R(s,t)=C(s)+t\mathbf{N},\quad-1\leq t\leq 1.
  31. κ P ( s ) = ± κ ( s ) 2 + τ ( s ) 2 . \kappa_{P}(s)=\pm\sqrt{\kappa(s)^{2}+\tau(s)^{2}}.
  32. Q Q M . Q\rightarrow QM.
  33. ( d ( Q M ) d s ) ( Q M ) T = ( d Q d s ) M M T Q T = ( d Q d s ) Q T \left(\frac{d(QM)}{ds}\right)(QM)^{T}=\left(\frac{dQ}{ds}\right)MM^{T}Q^{T}=% \left(\frac{dQ}{ds}\right)Q^{T}
  34. 𝐓 ( t ) = 𝐫 ( t ) 𝐫 ( t ) . \mathbf{T}(t)=\frac{\mathbf{r}^{\prime}(t)}{\|\mathbf{r}^{\prime}(t)\|}.
  35. 𝐍 ( t ) = 𝐓 ( t ) 𝐓 ( t ) = 𝐫 ( t ) × ( 𝐫 ′′ ( t ) × 𝐫 ( t ) ) 𝐫 ( t ) 𝐫 ′′ ( t ) × 𝐫 ( t ) . \mathbf{N}(t)=\frac{\mathbf{T}^{\prime}(t)}{\|\mathbf{T}^{\prime}(t)\|}=\frac{% \mathbf{r}^{\prime}(t)\times\left(\mathbf{r}^{\prime\prime}(t)\times\mathbf{r}% ^{\prime}(t)\right)}{\left\|\mathbf{r}^{\prime}(t)\right\|\,\left\|\mathbf{r}^% {\prime\prime}(t)\times\mathbf{r}^{\prime}(t)\right\|}.
  36. 𝐁 ( t ) = 𝐓 ( t ) × 𝐍 ( t ) = 𝐫 ( t ) × 𝐫 ′′ ( t ) 𝐫 ( t ) × 𝐫 ′′ ( t ) . \mathbf{B}(t)=\mathbf{T}(t)\times\mathbf{N}(t)=\frac{\mathbf{r}^{\prime}(t)% \times\mathbf{r}^{\prime\prime}(t)}{\|\mathbf{r}^{\prime}(t)\times\mathbf{r}^{% \prime\prime}(t)\|}.
  37. d d t [ 𝐓 𝐍 𝐁 ] = 𝐫 ( t ) [ 0 κ 0 - κ 0 τ 0 - τ 0 ] [ 𝐓 𝐍 𝐁 ] . \frac{d}{dt}\begin{bmatrix}\mathbf{T}\\ \mathbf{N}\\ \mathbf{B}\end{bmatrix}=\|\mathbf{r}^{\prime}(t)\|\begin{bmatrix}0&\kappa&0\\ -\kappa&0&\tau\\ 0&-\tau&0\end{bmatrix}\begin{bmatrix}\mathbf{T}\\ \mathbf{N}\\ \mathbf{B}\end{bmatrix}.

Friedmann_equations.html

  1. ρ \!\rho
  2. p \!p
  3. d s 2 = a ( t ) 2 d s 3 2 - c 2 d t 2 ds^{2}=a(t)^{2}\,ds_{3}^{2}-c^{2}\,dt^{2}
  4. d s 3 2 \!ds_{3}^{2}
  5. k \!k
  6. a ( t ) \!a(t)
  7. a ˙ 2 + k c 2 a 2 = 8 π G ρ + Λ c 2 3 \frac{\dot{a}^{2}+kc^{2}}{a^{2}}=\frac{8\pi G\rho+\Lambda c^{2}}{3}
  8. a ¨ a = - 4 π G 3 ( ρ + 3 p c 2 ) + Λ c 2 3 \frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^{2}}\right)+\frac{% \Lambda c^{2}}{3}
  9. H a ˙ a H\equiv\frac{\dot{a}}{a}
  10. k a 2 k\over a^{2}
  11. R = 6 c 2 a 2 ( a ¨ a + a ˙ 2 + k c 2 ) R=\frac{6}{c^{2}a^{2}}(\ddot{a}a+\dot{a}^{2}+kc^{2})
  12. a a
  13. a a
  14. i i
  15. a a
  16. k k
  17. a = 1 a=1
  18. R t R_{t}
  19. R 0 R_{0}
  20. a = R t / R 0 a=R_{t}/R_{0}
  21. k k
  22. k k
  23. k k
  24. ρ ˙ = - 3 H ( ρ + p c 2 ) , \dot{\rho}=-3H\left(\rho+\frac{p}{c^{2}}\right),
  25. Λ \Lambda\!
  26. T α β = ; β 0 T^{\alpha\beta}{}_{;\beta}\,=0
  27. ρ ρ - Λ c 2 8 π G \rho\rightarrow\rho-\frac{\Lambda c^{2}}{8\pi G}
  28. p p + Λ c 4 8 π G p\rightarrow p+\frac{\Lambda c^{4}}{8\pi G}
  29. H 2 = ( a ˙ a ) 2 = 8 π G 3 ρ - k c 2 a 2 H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{a^% {2}}
  30. H ˙ + H 2 = a ¨ a = - 4 π G 3 ( ρ + 3 p c 2 ) . \dot{H}+H^{2}=\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^{2}}% \right).
  31. Ω \Omega
  32. ρ \rho
  33. ρ c \rho_{c}
  34. ρ c = 3 H 2 8 π G . \rho_{c}=\frac{3H^{2}}{8\pi G}.
  35. Ω ρ ρ c = 8 π G ρ 3 H 2 . \Omega\equiv\frac{\rho}{\rho_{c}}=\frac{8\pi G\rho}{3H^{2}}.
  36. ρ c \rho_{c}
  37. Ω \Omega
  38. Ω \Omega
  39. Ω \Omega
  40. Ω \Omega
  41. k k
  42. H 2 H 0 2 = Ω R a - 4 + Ω M a - 3 + Ω k a - 2 + Ω Λ . \frac{H^{2}}{H_{0}^{2}}=\Omega_{R}a^{-4}+\Omega_{M}a^{-3}+\Omega_{k}a^{-2}+% \Omega_{\Lambda}.
  43. Ω R \Omega_{R}
  44. a = 1 a=1
  45. Ω M \Omega_{M}
  46. Ω k = 1 - Ω \Omega_{k}=1-\Omega
  47. Ω Λ \Omega_{\Lambda}
  48. p = w ρ c 2 , p=w\rho c^{2},
  49. p p
  50. ρ \rho
  51. w w
  52. a ( t ) = a 0 t 2 3 ( w + 1 ) a(t)=a_{0}\,t^{\frac{2}{3(w+1)}}
  53. a 0 a_{0}
  54. w w
  55. w = 0 w=0
  56. a ( t ) t 2 / 3 a(t)\propto t^{2/3}
  57. w = 1 / 3 w=1/3
  58. a ( t ) t 1 / 2 a(t)\propto t^{1/2}
  59. w = - 1 w=-1
  60. ρ ˙ f = - 3 H ( ρ f + p f c 2 ) \dot{\rho}_{f}=-3H\left(\rho_{f}+\frac{p_{f}}{c^{2}}\right)\,
  61. ρ ˙ f = - 3 H ( ρ f + w f ρ f ) \dot{\rho}_{f}=-3H\left(\rho_{f}+w_{f}\rho_{f}\right)\,
  62. ρ f a - 3 ( 1 + w f ) . {\rho}_{f}\propto a^{-3(1+w_{f})}\,.
  63. ρ = A a - 3 + B a - 4 + C a 0 \rho=Aa^{-3}+Ba^{-4}+Ca^{0}\,
  64. ( a ˙ a ) 2 = 8 π G 3 ρ - k c 2 a 2 \left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{a^{2}}\,
  65. a ~ = a a 0 , ρ c = 3 H 0 2 8 π G , Ω = ρ ρ c , t = t ~ H 0 , Ω c = - k c 2 H 0 2 a 0 2 \tilde{a}=\frac{a}{a_{0}},\;\rho_{c}=\frac{3H_{0}^{2}}{8\pi G},\;\Omega=\frac{% \rho}{\rho_{c}},\;t=\frac{\tilde{t}}{H_{0}},\;\Omega_{c}=-\frac{kc^{2}}{H_{0}^% {2}a_{0}^{2}}\;
  66. a 0 a_{0}
  67. H 0 H_{0}
  68. 1 2 ( d a ~ d t ~ ) 2 + U eff ( a ~ ) = 1 2 Ω c \frac{1}{2}\left(\frac{d\tilde{a}}{d\tilde{t}}\right)^{2}+U\text{eff}(\tilde{a% })=\frac{1}{2}\Omega_{c}
  69. U eff ( a ~ ) = Ω a ~ 2 2 U\text{eff}(\tilde{a})=\frac{\Omega\tilde{a}^{2}}{2}\;
  70. U eff ( a ~ ) U\text{eff}(\tilde{a})\;
  71. p = p ( ρ ) p=p(\rho)