wpmath0000005_4

Elementary_mathematics.html

  1. 1 2 \tfrac{1}{2}
  2. b i g = 1 V = 4 3 π r < s u p > 3 big=1V=\frac{4}{3}πr<sup>3
  3. y x \tfrac{y}{x}
  4. x y xy
  5. b n = b × × b n b^{n}=\underbrace{b\times\cdots\times b}_{n}
  6. x n \sqrt[n]{x}
  7. x n = r r n = x , \sqrt[n]{x}=r\iff r^{n}=x,
  8. \mathbb{Q}

Elementary_symmetric_polynomial.html

  1. n n
  2. e 0 ( X 1 , X 2 , , X n ) = 1 , e 1 ( X 1 , X 2 , , X n ) = 1 j n X j , e 2 ( X 1 , X 2 , , X n ) = 1 j < k n X j X k , e 3 ( X 1 , X 2 , , X n ) = 1 j < k < l n X j X k X l , \begin{aligned}\displaystyle e_{0}(X_{1},X_{2},\dots,X_{n})&\displaystyle=1,\\ \displaystyle e_{1}(X_{1},X_{2},\dots,X_{n})&\displaystyle=\textstyle\sum_{1% \leq j\leq n}X_{j},\\ \displaystyle e_{2}(X_{1},X_{2},\dots,X_{n})&\displaystyle=\textstyle\sum_{1% \leq j<k\leq n}X_{j}X_{k},\\ \displaystyle e_{3}(X_{1},X_{2},\dots,X_{n})&\displaystyle=\textstyle\sum_{1% \leq j<k<l\leq n}X_{j}X_{k}X_{l},\\ \end{aligned}
  3. e n ( X 1 , X 2 , , X n ) = X 1 X 2 X n e_{n}(X_{1},X_{2},\dots,X_{n})=X_{1}X_{2}\ldots X_{n}
  4. e k ( X 1 , , X n ) = 1 j 1 < j 2 < < j k n X j 1 X j k , e_{k}(X_{1},\ldots,X_{n})=\sum_{1\leq j_{1}<j_{2}<\ldots<j_{k}\leq n}X_{j_{1}}% \cdots X_{j_{k}},
  5. k > n k>n
  6. k k
  7. n n
  8. k k
  9. n n
  10. k k
  11. k k
  12. n n
  13. e λ ( X 1 , , X n ) e_{\lambda}(X_{1},\dots,X_{n})
  14. e λ ( X 1 , , X n ) = e λ 1 ( X 1 , , X n ) e λ 2 ( X 1 , , X n ) e λ m ( X 1 , , X n ) e_{\lambda}(X_{1},\dots,X_{n})=e_{\lambda_{1}}(X_{1},\dots,X_{n})\cdot e_{% \lambda_{2}}(X_{1},\dots,X_{n})\cdots e_{\lambda_{m}}(X_{1},\dots,X_{n})
  15. e 1 ( X 1 ) = X 1 . e_{1}(X_{1})=X_{1}.\,
  16. e 1 ( X 1 , X 2 ) \displaystyle e_{1}(X_{1},X_{2})
  17. e 1 ( X 1 , X 2 , X 3 ) \displaystyle e_{1}(X_{1},X_{2},X_{3})
  18. e 1 ( X 1 , X 2 , X 3 , X 4 ) \displaystyle e_{1}(X_{1},X_{2},X_{3},X_{4})
  19. j = 1 n ( λ - X j ) = λ n - e 1 ( X 1 , , X n ) λ n - 1 + e 2 ( X 1 , , X n ) λ n - 2 + + ( - 1 ) n e n ( X 1 , , X n ) . \prod_{j=1}^{n}(\lambda-X_{j})=\lambda^{n}-e_{1}(X_{1},\ldots,X_{n})\lambda^{n% -1}+e_{2}(X_{1},\ldots,X_{n})\lambda^{n-2}+\cdots+(-1)^{n}e_{n}(X_{1},\ldots,X% _{n}).
  20. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  21. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  22. n n
  23. n n
  24. [ e 1 ( X 1 , , X n ) , , e n ( X 1 , , X n ) ] . \mathbb{Z}[e_{1}(X_{1},\ldots,X_{n}),\ldots,e_{n}(X_{1},\ldots,X_{n})].
  25. X 1 , , X n X_{1},\ldots,X_{n}
  26. A [ X 1 , , X n ] S n A[X_{1},\ldots,X_{n}]^{S_{n}}
  27. A [ X 1 , , X n ] S n A[X_{1},\ldots,X_{n}]^{S_{n}}
  28. e k ( X 1 , , X n ) e_{k}(X_{1},\ldots,X_{n})
  29. e 0 e_{0}
  30. e 0 = 1 e_{0}=1
  31. P ( X 1 , , X n ) A [ X 1 , , X n ] S n P(X_{1},\ldots,X_{n})\in A[X_{1},\ldots,X_{n}]^{S_{n}}
  32. P ( X 1 , , X n ) = Q ( e 1 ( X 1 , , X n ) , , e n ( X 1 , , X n ) ) P(X_{1},\ldots,X_{n})=Q(e_{1}(X_{1},\ldots,X_{n}),\ldots,e_{n}(X_{1},\ldots,X_% {n}))
  33. Q A [ Y 1 , , Y n ] Q\in A[Y_{1},\ldots,Y_{n}]
  34. A [ X 1 , , X n ] S n A[X_{1},\ldots,X_{n}]^{S_{n}}
  35. A [ Y 1 , , Y n ] A[Y_{1},\ldots,Y_{n}]
  36. Y k Y_{k}
  37. e k ( X 1 , , X n ) e_{k}(X_{1},\ldots,X_{n})
  38. k = 1 , , n k=1,\ldots,n
  39. m < n m<n
  40. A [ X 1 , , X n ] S n A[X_{1},\ldots,X_{n}]^{S_{n}}
  41. P ( X 1 , , X n ) = P lacunary ( X 1 , , X n ) + X 1 X n Q ( X 1 , , X n ) . P(X_{1},\ldots,X_{n})=P_{\mbox{lacunary}~{}}(X_{1},\ldots,X_{n})+X_{1}\cdots X% _{n}\cdot Q(X_{1},\ldots,X_{n}).
  42. P lacunary P_{\mbox{lacunary}~{}}
  43. P ( X 1 , , X n - 1 , 0 ) P(X_{1},\ldots,X_{n-1},0)
  44. P ~ ( X 1 , , X n - 1 ) \tilde{P}(X_{1},\ldots,X_{n-1})
  45. P ~ ( X 1 , , X n - 1 ) = Q ~ ( σ 1 , n - 1 , , σ n - 1 , n - 1 ) \tilde{P}(X_{1},\ldots,X_{n-1})=\tilde{Q}(\sigma_{1,n-1},\ldots,\sigma_{n-1,n-% 1})
  46. Q ~ \tilde{Q}
  47. σ j , n - 1 \sigma_{j,n-1}
  48. R ( X 1 , , X n ) := Q ~ ( σ 1 , n , , σ n - 1 , n ) . R(X_{1},\ldots,X_{n}):=\tilde{Q}(\sigma_{1,n},\ldots,\sigma_{n-1,n})\ .
  49. R ( X 1 , , X n ) R(X_{1},\ldots,X_{n})
  50. P lacunary P_{\mbox{lacunary}~{}}
  51. R ( X 1 , , X n - 1 , 0 ) = Q ~ ( σ 1 , n - 1 , , σ n - 1 , n - 1 ) = P ( X 1 , , X n - 1 , 0 ) R(X_{1},\ldots,X_{n-1},0)=\tilde{Q}(\sigma_{1,n-1},\ldots,\sigma_{n-1,n-1})=P(% X_{1},\ldots,X_{n-1},0)
  52. σ j , n \sigma_{j,n}
  53. σ j , n - 1 \sigma_{j,n-1}
  54. j < n j<n
  55. X 1 X n X_{1}\cdots X_{n}
  56. σ n , n \sigma_{n,n}
  57. P - R = σ n , n Q P-R=\sigma_{n,n}\,Q
  58. e 1 , , e n e_{1},\ldots,e_{n}
  59. A [ X 1 , , X n ] S n A[X_{1},\ldots,X_{n}]^{S_{n}}
  60. A [ Y 1 , , Y n ] A[Y_{1},\ldots,Y_{n}]
  61. d d
  62. X X

ElGamal_signature_scheme.html

  1. Z p * Z_{p}^{*}
  2. r g k ( mod p ) r\,\equiv\,g^{k}\;\;(\mathop{{\rm mod}}p)
  3. s ( H ( m ) - x r ) k - 1 ( mod p - 1 ) s\,\equiv\,(H(m)-xr)k^{-1}\;\;(\mathop{{\rm mod}}p-1)
  4. s = 0 s=0
  5. 0 < r < p 0<r<p
  6. 0 < s < p - 1 0<s<p-1
  7. g H ( m ) y r r s ( mod p ) . g^{H(m)}\equiv y^{r}r^{s}\;\;(\mathop{{\rm mod}}p).
  8. H ( m ) x r + s k ( mod p - 1 ) . H(m)\,\equiv\,xr+sk\;\;(\mathop{{\rm mod}}p-1).
  9. g H ( m ) g x r g k s ( g x ) r ( g k ) s ( y ) r ( r ) s ( mod p ) . \begin{aligned}\displaystyle g^{H(m)}&\displaystyle\equiv g^{xr}g^{ks}\\ &\displaystyle\equiv(g^{x})^{r}(g^{k})^{s}\\ &\displaystyle\equiv(y)^{r}(r)^{s}\;\;(\mathop{{\rm mod}}p).\\ \end{aligned}
  10. H ( m ) H ( M ) ( mod p - 1 ) H(m)\equiv H(M)\;\;(\mathop{{\rm mod}}p-1)
  11. 1 < e < p - 1 1<e<p-1
  12. r = g e y ( mod p ) r=g^{e}y\;\;(\mathop{{\rm mod}}p)
  13. s = - r ( mod p - 1 ) s=-r\;\;(\mathop{{\rm mod}}p-1)
  14. ( r , s ) (r,s)
  15. m = e s ( mod p - 1 ) m=es\;\;(\mathop{{\rm mod}}p-1)
  16. 1 < e , v < p - 1 1<e,v<p-1
  17. gcd ( v , p - 1 ) = 1 \gcd(v,p-1)=1
  18. r = g e y v ( mod p ) r=g^{e}y^{v}\;\;(\mathop{{\rm mod}}p)
  19. s = - r v - 1 ( mod p - 1 ) s=-rv^{-1}\;\;(\mathop{{\rm mod}}p-1)
  20. ( r , s ) (r,s)
  21. m = e s ( mod p - 1 ) m=es\;\;(\mathop{{\rm mod}}p-1)

Elliott–Halberstam_conjecture.html

  1. π ( x ) \pi(x)
  2. π ( x ; q , a ) \pi(x;q,a)
  3. π ( x ; q , a ) π ( x ) φ ( q ) \pi(x;q,a)\approx\frac{\pi(x)}{\varphi(q)}
  4. φ \varphi
  5. E ( x ; q ) = max ( a , q ) = 1 | π ( x ; q , a ) - π ( x ) φ ( q ) | E(x;q)=\max_{(a,q)=1}\left|\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\right|
  6. 1 q x θ E ( x ; q ) C x log A x \sum_{1\leq q\leq x^{\theta}}E(x;q)\leq\frac{Cx}{\log^{A}x}

Elliptic_filter.html

  1. G n ( ω ) = 1 1 + ϵ 2 R n 2 ( ξ , ω / ω 0 ) G_{n}(\omega)={1\over\sqrt{1+\epsilon^{2}R_{n}^{2}(\xi,\omega/\omega_{0})}}
  2. ω 0 \omega_{0}
  3. ϵ \epsilon
  4. ξ \xi
  5. 1 / 1 + ϵ 2 1/\sqrt{1+\epsilon^{2}}
  6. L n L_{n}
  7. L n = R n ( ξ , ξ ) L_{n}=R_{n}(\xi,\xi)\,
  8. 1 / 1 + ϵ 2 L n 2 1/\sqrt{1+\epsilon^{2}L_{n}^{2}}
  9. ξ \xi\rightarrow\infty
  10. ξ \xi\rightarrow\infty
  11. ω 0 0 \omega_{0}\rightarrow 0
  12. ϵ 0 \epsilon\rightarrow 0
  13. ϵ R n ( ξ , 1 / ω 0 ) = 1 \epsilon\,R_{n}(\xi,1/\omega_{0})=1
  14. ξ \xi\rightarrow\infty
  15. ϵ 0 \epsilon\rightarrow 0
  16. ω 0 0 \omega_{0}\rightarrow 0
  17. ξ ω 0 = 1 \xi\omega_{0}=1
  18. ϵ L n = α \epsilon L_{n}=\alpha
  19. G ( ω ) = 1 1 + 1 α 2 T n 2 ( 1 / ω ) G(\omega)=\frac{1}{\sqrt{1+\frac{1}{\alpha^{2}T^{2}_{n}(1/\omega)}}}
  20. ( ω p m ) (\omega_{pm})
  21. s = σ + j ω s=\sigma+j\omega
  22. 1 + ϵ 2 R n 2 ( - j s , ξ ) = 0 1+\epsilon^{2}R_{n}^{2}(-js,\xi)=0\,
  23. - j s = cd ( w , 1 / ξ ) -js=\mathrm{cd}(w,1/\xi)
  24. 1 + ϵ 2 cd 2 ( n w K n K , 1 L n ) = 0 1+\epsilon^{2}\mathrm{cd}^{2}\left(\frac{nwK_{n}}{K},\frac{1}{L_{n}}\right)=0\,
  25. K = K ( 1 / ξ ) K=K(1/\xi)
  26. K n = K ( 1 / L n ) K_{n}=K(1/L_{n})
  27. w = K n K n cd - 1 ( ± j ϵ , 1 L n ) + m K n w=\frac{K}{nK_{n}}\mathrm{cd}^{-1}\left(\frac{\pm j}{\epsilon},\frac{1}{L_{n}}% \right)+\frac{mK}{n}
  28. s p m = i cd ( w , 1 / ξ ) s_{pm}=i\,\mathrm{cd}(w,1/\xi)\,
  29. s p m = a + j b c s_{pm}=\frac{a+jb}{c}
  30. a = - ζ n 1 - ζ n 2 1 - x m 2 1 - x m 2 / ξ 2 a=-\zeta_{n}\sqrt{1-\zeta_{n}^{2}}\sqrt{1-x_{m}^{2}}\sqrt{1-x_{m}^{2}/\xi^{2}}
  31. b = x m 1 - ζ n 2 ( 1 - 1 / ξ 2 ) b=x_{m}\sqrt{1-\zeta_{n}^{2}(1-1/\xi^{2})}
  32. c = 1 - ζ n 2 + x i 2 ζ n 2 / ξ 2 c=1-\zeta_{n}^{2}+x_{i}^{2}\zeta_{n}^{2}/\xi^{2}
  33. ζ n \zeta_{n}
  34. n , ϵ n,\,\epsilon
  35. ξ \xi
  36. x m x_{m}
  37. ζ n \zeta_{n}
  38. ζ 1 = 1 1 + ϵ 2 \zeta_{1}=\frac{1}{\sqrt{1+\epsilon^{2}}}
  39. ζ 2 = 2 ( 1 + t ) 1 + ϵ 2 + ( 1 - t ) 2 + ϵ 2 ( 1 + t ) 2 \zeta_{2}=\frac{2}{(1+t)\sqrt{1+\epsilon^{2}}+\sqrt{(1-t)^{2}+\epsilon^{2}(1+t% )^{2}}}
  40. t = 1 - 1 / ξ 2 t=\sqrt{1-1/\xi^{2}}
  41. ζ 3 \zeta_{3}
  42. ζ n \zeta_{n}
  43. ζ m n ( ξ , ϵ ) = ζ m ( ξ , 1 ζ n 2 ( L m , ϵ ) - 1 ) \zeta_{m\cdot n}(\xi,\epsilon)=\zeta_{m}\left(\xi,\sqrt{\frac{1}{\zeta_{n}^{2}% (L_{m},\epsilon)}-1}\right)
  44. L m = R m ( ξ , ξ ) L_{m}=R_{m}(\xi,\xi)
  45. Q = - | s p m | 2 R e ( s p m ) = - 1 2 cos ( arg ( s p m ) ) Q=-\frac{|s_{pm}|}{2\mathrm{Re}(s_{pm})}=-\frac{1}{2\cos(\arg(s_{pm}))}
  46. ϵ Q m i n = 1 L n ( ξ ) \epsilon_{Qmin}=\frac{1}{\sqrt{L_{n}(\xi)}}

Ellsberg_paradox.html

  1. R U ( $ 100 ) + ( 1 - R ) U ( $ 0 ) > B U ( $ 100 ) + ( 1 - B ) U ( $ 0 ) R\cdot U(\$100)+(1-R)\cdot U(\$0)>B\cdot U(\$100)+(1-B)\cdot U(\$0)
  2. U ( ) U(\cdot)
  3. U ( $ 100 ) > U ( $ 0 ) U(\$100)>U(\$0)
  4. R [ U ( $ 100 ) - U ( $ 0 ) ] > B [ U ( $ 100 ) - U ( $ 0 ) ] R[U(\$100)-U(\$0)]>B[U(\$100)-U(\$0)]
  5. R > B \Longleftrightarrow R>B\;
  6. B U ( $ 100 ) + Y U ( $ 100 ) + R U ( $ 0 ) > R U ( $ 100 ) + Y U ( $ 100 ) + B U ( $ 0 ) B\cdot U(\$100)+Y\cdot U(\$100)+R\cdot U(\$0)>R\cdot U(\$100)+Y\cdot U(\$100)+% B\cdot U(\$0)
  7. B [ U ( $ 100 ) - U ( $ 0 ) ] > R [ U ( $ 100 ) - U ( $ 0 ) ] B[U(\$100)-U(\$0)]>R[U(\$100)-U(\$0)]
  8. B > R \Longleftrightarrow B>R\;

Elongated_pentagonal_gyrocupolarotunda.html

  1. V = 5 12 ( 11 + 5 5 + 6 5 + 2 5 ) a 3 16.936... a 3 V=\frac{5}{12}(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}})a^{3}\approx 16.936...a^{3}
  2. A = 1 4 ( 60 + 10 ( 190 + 49 5 + 21 75 + 30 5 ) ) a 2 33.5385... a 2 A=\frac{1}{4}(60+\sqrt{10(190+49\sqrt{5}+21\sqrt{75+30\sqrt{5}})})a^{2}\approx 3% 3.5385...a^{2}

Elongated_pentagonal_orthocupolarotunda.html

  1. V = 5 12 ( 11 + 5 5 + 6 5 + 2 5 ) a 3 16.936... a 3 V=\frac{5}{12}(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}})a^{3}\approx 16.936...a^{3}
  2. A = 1 4 ( 60 + 10 ( 190 + 49 5 + 21 75 + 30 5 ) ) a 2 33.5385... a 2 A=\frac{1}{4}(60+\sqrt{10(190+49\sqrt{5}+21\sqrt{75+30\sqrt{5}}}))a^{2}\approx 3% 3.5385...a^{2}

Emergy.html

  1. n 1 E m i \sum_{n}^{1}Em_{i}

Empirical_risk_minimization.html

  1. X X
  2. Y Y
  3. h : X Y \!h:X\to Y
  4. y Y y\in Y
  5. x X x\in X
  6. ( x 1 , y 1 ) , , ( x m , y m ) \!(x_{1},y_{1}),\ldots,(x_{m},y_{m})
  7. x i X x_{i}\in X
  8. y i Y y_{i}\in Y
  9. h ( x i ) \!h(x_{i})
  10. P ( x , y ) P(x,y)
  11. X X
  12. Y Y
  13. m m
  14. ( x 1 , y 1 ) , , ( x m , y m ) \!(x_{1},y_{1}),\ldots,(x_{m},y_{m})
  15. P ( x , y ) P(x,y)
  16. y y
  17. x x
  18. P ( y | x ) P(y|x)
  19. x x
  20. L ( y ^ , y ) L(\hat{y},y)
  21. y ^ \hat{y}
  22. y y
  23. h ( x ) h(x)
  24. R ( h ) = 𝐄 [ L ( h ( x ) , y ) ] = L ( h ( x ) , y ) d P ( x , y ) . R(h)=\mathbf{E}[L(h(x),y)]=\int L(h(x),y)\,dP(x,y).
  25. L ( y ^ , y ) = I ( y ^ y ) L(\hat{y},y)=I(\hat{y}\neq y)
  26. I ( ) I(...)
  27. h * h^{*}
  28. \mathcal{H}
  29. R ( h ) R(h)
  30. h * = arg min h R ( h ) . h^{*}=\arg\min_{h\in\mathcal{H}}R(h).
  31. R ( h ) R(h)
  32. P ( x , y ) P(x,y)
  33. R emp ( h ) = 1 m i = 1 m L ( h ( x i ) , y i ) . \!R_{\mbox{emp}}~{}(h)=\frac{1}{m}\sum_{i=1}^{m}L(h(x_{i}),y_{i}).
  34. h ^ \hat{h}
  35. h ^ = arg min h R emp ( h ) . \hat{h}=\arg\min_{h\in\mathcal{H}}R_{\mbox{emp}~{}}(h).
  36. P ( x , y ) P(x,y)

Enantiomeric_excess.html

  1. e e = | F + - F - | \ ee=|F_{+}-F_{-}|
  2. F + + F - = 1 \ F_{+}+F_{-}=1
  3. e e = ( [ α ] o b s / [ α ] m a x ) × 100 \ ee=([\alpha]_{obs}/[\alpha]_{max})\times 100
  4. e e = ( ( R - S ) / ( R + S ) ) × 100 \ ee=((R-S)/(R+S))\times 100
  5. R \ R
  6. S \ S
  7. R + S = 1 \ R+S=1
  8. e e \ ee
  9. R \ R
  10. R = + e e 2 + 50 % \ R=+\frac{ee}{2}+50\%
  11. S = - e e 2 + 50 % \ S=-\frac{ee}{2}+50\%

Endogeneity_(econometrics).html

  1. α \alpha
  2. α \alpha
  3. β \beta
  4. y i = α + β x i + γ z i + u i y_{i}=\alpha+\beta x_{i}+\gamma z_{i}+u_{i}
  5. z i z_{i}
  6. z i z_{i}
  7. y i = α + β x i + ε i y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}
  8. ε i = γ z i + u i \varepsilon_{i}=\gamma z_{i}+u_{i}
  9. x x
  10. z z
  11. z z
  12. y y
  13. γ 0 \gamma\neq 0
  14. x x
  15. ε \varepsilon
  16. x i * x^{*}_{i}
  17. x i = x i * + ν i x_{i}=x^{*}_{i}+\nu_{i}
  18. ν i \nu_{i}
  19. y i = α + β x i * + ε i y_{i}=\alpha+\beta x^{*}_{i}+\varepsilon_{i}
  20. y i = α + β ( x i - ν i ) + ε i y_{i}=\alpha+\beta(x_{i}-\nu_{i})+\varepsilon_{i}
  21. y i = α + β x i + ( ε i - β ν i ) y_{i}=\alpha+\beta x_{i}+(\varepsilon_{i}-\beta\nu_{i})
  22. y i = α + β x i + u i y_{i}=\alpha+\beta x_{i}+u_{i}
  23. u i = ε i - β ν i u_{i}=\varepsilon_{i}-\beta\nu_{i}
  24. x i x_{i}
  25. u i u_{i}
  26. ν i \nu_{i}
  27. y i = β 1 x i + γ 1 z i + u i y_{i}=\beta_{1}x_{i}+\gamma_{1}z_{i}+u_{i}
  28. z i = β 2 x i + γ 2 y i + v i z_{i}=\beta_{2}x_{i}+\gamma_{2}y_{i}+v_{i}
  29. E ( z i u i ) 0 E(z_{i}u_{i})\neq 0
  30. z i z_{i}
  31. 1 - γ 1 γ 2 0 1-\gamma_{1}\gamma_{2}\neq 0
  32. z i = β 2 + γ 2 β 1 1 - γ 1 γ 2 x i + 1 1 - γ 1 γ 2 v i + γ 2 1 - γ 1 γ 2 u i z_{i}=\frac{\beta_{2}+\gamma_{2}\beta_{1}}{1-\gamma_{1}\gamma_{2}}x_{i}+\frac{% 1}{1-\gamma_{1}\gamma_{2}}v_{i}+\frac{\gamma_{2}}{1-\gamma_{1}\gamma_{2}}u_{i}
  33. x i x_{i}
  34. v i v_{i}
  35. u i u_{i}
  36. E ( z i u i ) = γ 2 1 - γ 1 γ 2 E ( u i u i ) E(z_{i}u_{i})=\frac{\gamma_{2}}{1-\gamma_{1}\gamma_{2}}E(u_{i}u_{i})
  37. E ( z i u i ) 0 E(z_{i}u_{i})\neq 0
  38. α \alpha
  39. α \alpha

Energy_charge.html

  1. Energy charge = [ ATP ] + 1 2 [ ADP ] [ ATP ] + [ ADP ] + [ AMP ] \mbox{Energy charge}~{}=\frac{[\mbox{ATP}~{}]+\frac{1}{2}[\mbox{ADP}~{}]}{[% \mbox{ATP}~{}]+[\mbox{ADP}~{}]+[\mbox{AMP}~{}]}

Energy_density.html

  1. U = ε 0 2 𝐄 2 + 1 2 μ 0 𝐁 2 U=\frac{\varepsilon_{0}}{2}\mathbf{E}^{2}+\frac{1}{2\mu_{0}}\mathbf{B}^{2}
  2. U = 1 2 ( 𝐄 𝐃 + 𝐇 𝐁 ) U=\frac{1}{2}(\mathbf{E}\cdot\mathbf{D}+\mathbf{H}\cdot\mathbf{B})

Energy_harvesting.html

  1. P b ( Z r , T i ) O 3 Pb(Zr,Ti)O3
  2. d 31 d_{31}
  3. d 33 d_{33}
  4. d 31 d_{31}
  5. ( 3 5 m a s k s ) (3~{}5masks)
  6. 6 m m 6mm
  7. 4 m m 4mm
  8. m m 2 mm2
  9. 6 6
  10. 4 4
  11. 3 m 3m
  12. 3 3
  13. 2 2
  14. m m
  15. X 3 X3
  16. L 6 L6
  17. L 4 L4
  18. L 3 L3
  19. L 2 L2
  20. P ( c l a s s m ) P(class“m”)
  21. P s = P 3 Ps=P3
  22. Δ P s \Delta P_{s}
  23. Δ P s = ( Δ P 1 , Δ P 2 , Δ P 3 ) \Delta P_{s}=(\Delta P_{1},\Delta P_{2},\Delta P_{3})
  24. Δ P s = ( Δ P 1 , Δ P 2 , Δ P 3 ) \Delta P_{s}=(\Delta P_{1},\Delta P_{2},\Delta P_{3})
  25. Δ P i = d i k l T k l \Delta P_{i}=diklTkl
  26. T k l Tkl
  27. d i k l dikl

Energy_level_splitting.html

  1. I I
  2. 2 × 2 2 × 2
  3. H ^ ε = H ^ 0 + ε σ 3 = ( E 0 + ε 0 0 E 0 - ε ) \hat{H}_{\varepsilon}=\hat{H}_{0}+\varepsilon\sigma_{3}=\begin{pmatrix}E_{0}+% \varepsilon&0\\ 0&E_{0}-\varepsilon\end{pmatrix}
  4. 0 0\rangle
  5. E < s u b > 0 + ε E<sub>0+ε

Energy_returned_on_energy_invested.html

  1. E R O E I = Usable Acquired Energy Energy Expended EROEI=\frac{\hbox{Usable Acquired Energy}}{\hbox{Energy Expended}}
  2. GrossEnergyYield ÷ EnergyExpended = E R O E I \hbox{GrossEnergyYield}\div\hbox{EnergyExpended}=EROEI
  3. ( NetEnergy ÷ EnergyExpended ) + 1 = E R O E I (\hbox{NetEnergy}\div\hbox{EnergyExpended})+1=EROEI

Energy_Tax_Act.html

  1. UnadjustedMPG ( combined ) = 1 .495 CityMPG + .351 HighwayMPG + .15 \mathrm{UnadjustedMPG(combined)}=\frac{\mathrm{1}}{\mathrm{\frac{\mathrm{.495}% }{\mathrm{CityMPG}}+\frac{\mathrm{.351}}{\mathrm{HighwayMPG}}}}+.15
  2. p t \mathit{p_{t}}
  3. p i \mathit{p_{i}}
  4. q i \mathit{q_{i}}
  5. p t = i p i q i p_{t}=\sum_{i}{p_{i}q_{i}}\,

Engel's_theorem.html

  1. [ 0 a 12 a 13 a 1 n 0 0 a 23 a 2 n 0 0 a n - 1 n 0 0 0 ] , \begin{bmatrix}0&a_{12}&a_{13}&\cdots&a_{1n}\\ 0&0&a_{23}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&&\ddots&a_{n-1n}\\ 0&0&\cdots&\cdots&0\end{bmatrix},
  2. ad x ( y ) = [ x , y ] \operatorname{ad}_{x}(y)=[x,y]
  3. ad I V = 0 : L ( V ) L ( V ) \operatorname{ad}_{I_{V}}=0:L(V)\rightarrow L(V)
  4. 𝐋 0 = 𝐋 , 𝐋 i + 1 = [ 𝐋 , 𝐋 i ] \mathbf{L}^{0}=\mathbf{L},\quad\mathbf{L}^{i+1}=[\mathbf{L},\mathbf{L}^{i}]
  5. V 0 V 1 V n V_{0}\subsetneq V_{1}\subsetneq\cdots\subsetneq V_{n}
  6. V 0 = 0 V_{0}=0
  7. V n = V V_{n}=V
  8. 𝐋 V i + 1 V i , i n - 1. \mathbf{L}\,V_{i+1}\subseteq V_{i},\quad\forall i\leq n-1.

Entropic_force.html

  1. 𝐅 \mathbf{F}
  2. { 𝐗 } \{\mathbf{X}\}
  3. 𝐅 ( 𝐗 𝟎 ) = T 𝐗 S ( 𝐗 ) | 𝐗 0 \mathbf{F}(\mathbf{X_{0}})=T\nabla_{\mathbf{X}}S(\mathbf{X})|_{\mathbf{X}_{0}}
  4. T T
  5. S ( 𝐗 ) S(\mathbf{X})
  6. 𝐗 \mathbf{X}
  7. 𝐗 𝟎 \mathbf{X_{0}}

Enumerator_polynomial.html

  1. C 𝔽 2 n C\subset\mathbb{F}_{2}^{n}
  2. n n
  3. A t = # { c C w ( c ) = t } A_{t}=\#\{c\in C\mid w(c)=t\}
  4. W ( C ; x , y ) = w = 0 n A w x w y n - w . W(C;x,y)=\sum_{w=0}^{n}A_{w}x^{w}y^{n-w}.
  5. W ( C ; 0 , 1 ) = A 0 = 1 W(C;0,1)=A_{0}=1
  6. W ( C ; 1 , 1 ) = w = 0 n A w = | C | W(C;1,1)=\sum_{w=0}^{n}A_{w}=|C|
  7. W ( C ; 1 , 0 ) = A n = 1 iff ( 1 , , 1 ) C and 0 otherwise. W(C;1,0)=A_{n}=1\mbox{ iff }~{}(1,\ldots,1)\in C\ \mbox{ and }~{}0\mbox{ % otherwise.}~{}
  8. W ( C ; 1 , - 1 ) = w = 0 n A w ( - 1 ) n - w = A n + ( - 1 ) 1 A n - 1 + + ( - 1 ) n - 1 A 1 + ( - 1 ) n A 0 W(C;1,-1)=\sum_{w=0}^{n}A_{w}(-1)^{n-w}=A_{n}+(-1)^{1}A_{n-1}+\ldots+(-1)^{n-1% }A_{1}+(-1)^{n}A_{0}
  9. C 𝔽 2 n C\subset\mathbb{F}_{2}^{n}
  10. C = { x 𝔽 2 n x , c = 0 c C } C^{\perp}=\{x\in\mathbb{F}_{2}^{n}\,\mid\,\langle x,c\rangle=0\mbox{ }~{}% \forall c\in C\}
  11. < , > <,>
  12. 𝔽 2 \mathbb{F}_{2}
  13. W ( C ; x , y ) = 1 C W ( C ; y - x , y + x ) . W(C^{\perp};x,y)=\frac{1}{\mid C\mid}W(C;y-x,y+x).
  14. A i = 1 M # { ( c 1 , c 2 ) C × C d ( c 1 , c 2 ) = i } A_{i}=\frac{1}{M}\#\left\{(c_{1},c_{2})\in C\times C\mid d(c_{1},c_{2})=i\right\}
  15. A ( C ; x , y ) = i = 0 n A i x i y n - i A(C;x,y)=\sum_{i=0}^{n}A_{i}x^{i}y^{n-i}
  16. B x , i = # { c C d ( c , x ) = i } . B_{x,i}=\#\left\{c\in C\mid d(c,x)=i\right\}.

Epigram_(programming_language).html

  1. data ¯ ( 𝖭𝖺𝗍 : ) where ¯ ( 𝗓𝖾𝗋𝗈 : 𝖭𝖺𝗍 ) ; ( n : 𝖭𝖺𝗍 𝗌𝗎𝖼 n : 𝖭𝖺𝗍 ) \underline{\mathrm{data}}\;\left(\frac{}{\mathsf{Nat}:\star}\right)\;% \underline{\mathrm{where}}\;\left(\frac{}{\mathsf{zero}:\mathsf{Nat}}\right)\;% ;\;\left(\frac{n:\mathsf{Nat}}{\mathsf{suc}\ n:\mathsf{Nat}}\right)
  2. 𝖭𝖺𝗍𝖨𝗇𝖽 : P : 𝖭𝖺𝗍 P 𝗓𝖾𝗋𝗈 ( n : 𝖭𝖺𝗍 P n P ( 𝗌𝗎𝖼 n ) ) n : 𝖭𝖺𝗍 P n \mathsf{NatInd}:\begin{matrix}\forall P:\mathsf{Nat}\rightarrow\star% \Rightarrow P\ \mathsf{zero}\rightarrow\\ (\forall n:\mathsf{Nat}\Rightarrow P\ n\rightarrow P\ (\mathsf{suc}\ n))% \rightarrow\\ \forall n:\mathsf{Nat}\Rightarrow P\ n\end{matrix}
  3. 𝖭𝖺𝗍𝖨𝗇𝖽 P m z m s 𝗓𝖾𝗋𝗈 m z \mathsf{NatInd}\ P\ mz\ ms\ \mathsf{zero}\equiv mz
  4. 𝖭𝖺𝗍𝖨𝗇𝖽 P m z m s ( 𝗌𝗎𝖼 n ) m s n ( N a t I n d P m z m s n ) \mathsf{NatInd}\ P\ mz\ ms\ (\mathsf{suc}\ n)\equiv ms\ n\ (NatInd\ P\ mz\ ms% \ n)
  5. 𝗉𝗅𝗎𝗌 x y rec ¯ x { \mathsf{plus}\ x\ y\Leftarrow\underline{\mathrm{rec}}\ x\ \{
  6. 𝗉𝗅𝗎𝗌 x y case ¯ x { \mathsf{plus}\ x\ y\Leftarrow\underline{\mathrm{case}}\ x\ \{
  7. 𝗉𝗅𝗎𝗌 𝗓𝖾𝗋𝗈 y y \mathsf{plus\ zero}\ y\Rightarrow y
  8. 𝗉𝗅𝗎𝗌 ( 𝗌𝗎𝖼 x ) y s u c ( 𝗉𝗅𝗎𝗌 x y ) } } \quad\quad\mathsf{plus}\ (\mathsf{suc}\ x)\ y\Rightarrow suc\ (\mathsf{plus}\ % x\ y)\ \}\ \}
  9. \star
  10. \star
  11. P Q P\rightarrow Q
  12. x : P Q \forall x:P\Rightarrow Q
  13. x x
  14. Q Q
  15. P P

Epitrochoid.html

  1. x ( θ ) = ( R + r ) cos θ - d cos ( R + r r θ ) , x(\theta)=(R+r)\cos\theta-d\cos\left({R+r\over r}\theta\right),\,
  2. y ( θ ) = ( R + r ) sin θ - d sin ( R + r r θ ) . y(\theta)=(R+r)\sin\theta-d\sin\left({R+r\over r}\theta\right).\,
  3. θ \theta

Equation_of_state_(cosmology).html

  1. w \!w
  2. p \!p
  3. ρ \!\rho
  4. w = p / ρ \!w=p/\rho
  5. p = ρ m R T = ρ m C 2 \!p=\rho_{m}RT=\rho_{m}C^{2}
  6. ρ m \!\rho_{m}
  7. R \!R
  8. T \!T
  9. C = R T \!C=\sqrt{RT}
  10. w = p ρ = ρ m C 2 ρ m c 2 = C 2 c 2 0 w=\frac{p}{\rho}=\frac{\rho_{m}C^{2}}{\rho_{m}c^{2}}=\frac{C^{2}}{c^{2}}\approx 0
  11. ρ = ρ m c 2 \!\rho=\rho_{m}c^{2}
  12. C c \!C<<c
  13. c \!c
  14. a \!a
  15. ρ a - 3 ( 1 + w ) . \rho\propto a^{-3(1+w)}.
  16. a t 2 3 ( 1 + w ) , a\propto t^{\frac{2}{3(1+w)}},
  17. t \!t
  18. 3 a ¨ a = Λ - 4 π G ( ρ + 3 p ) 3\frac{\ddot{a}}{a}=\Lambda-4\pi G(\rho+3p)
  19. Λ \!\Lambda
  20. G \!G
  21. a ¨ \ddot{a}
  22. ρ ρ + Λ 8 π G \rho^{\prime}\equiv\rho+\frac{\Lambda}{8\pi G}
  23. p p - Λ 8 π G p^{\prime}\equiv p-\frac{\Lambda}{8\pi G}
  24. p = w ρ p^{\prime}=w^{\prime}\rho^{\prime}
  25. a ¨ a = - 4 3 π G ( ρ + 3 p ) = - 4 3 π G ( 1 + 3 w ) ρ \frac{\ddot{a}}{a}=-\frac{4}{3}\pi G\left(\rho^{\prime}+3p^{\prime}\right)=-% \frac{4}{3}\pi G(1+3w^{\prime})\rho^{\prime}
  26. w = 0 \!w=0
  27. ρ a - 3 = V - 1 \rho\propto a^{-3}=V^{-1}
  28. V \!V
  29. w = 1 / 3 \!w=1/3
  30. ρ a - 4 \rho\propto a^{-4}
  31. w = - 1 \!w=-1
  32. a e H t a\propto e^{Ht}
  33. w < - 1 / 3 \!w<-1/3
  34. w < - 1 \!w<-1
  35. w < - 1 \!w<-1
  36. w - 1 \!w\geq-1
  37. w = - 1 / 3 \!w=-1/3
  38. w = 0 \!w=0
  39. w - 1 \!w\approx-1
  40. w \!w
  41. w - 1 \!w\neq-1
  42. ϕ \!\phi
  43. w = 1 2 ϕ ˙ 2 - V ( ϕ ) 1 2 ϕ ˙ 2 + V ( ϕ ) , {w=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)},}
  44. ϕ ˙ \!\dot{\phi}
  45. ϕ \!\phi
  46. V ( ϕ ) \!V(\phi)
  47. ( V = 0 ) \!(V=0)
  48. w = 1 \!w=1
  49. w = - 1 \!w=-1
  50. w = - 1 \!w=-1

Equianharmonic.html

  1. Γ 3 ( 1 / 3 ) 4 π \frac{\Gamma^{3}(1/3)}{4\pi}
  2. Γ \Gamma
  3. ω 1 = 1 2 ( - 1 + 3 i ) ω 2 . \omega_{1}=\tfrac{1}{2}(-1+\sqrt{3}i)\omega_{2}.
  4. e 1 = 4 - 1 / 3 e ( 2 / 3 ) π i , e 2 = 4 - 1 / 3 , e 3 = 4 - 1 / 3 e - ( 2 / 3 ) π i . e_{1}=4^{-1/3}e^{(2/3)\pi i},\qquad e_{2}=4^{-1/3},\qquad e_{3}=4^{-1/3}e^{-(2% /3)\pi i}.

Equilibrium_moisture_content.html

  1. M = m - m o d m o d M=\frac{m-m_{od}}{m_{od}}
  2. m o d m_{od}
  3. M eq = 1800 W [ k h 1 - k h + k 1 k h + 2 k 1 k 2 k 2 h 2 1 + k 1 k h + k 1 k 2 k 2 h 2 ] M_{\mathrm{eq}}=\frac{1800}{W}\left[\frac{kh}{1-kh}\,+\,\frac{k_{1}kh+2k_{1}k_% {2}k^{2}h^{2}}{1+k_{1}kh+k_{1}k_{2}k^{2}h^{2}}\right]
  4. W = 330 + 0.452 T + 0.00415 T 2 W=330+0.452\,T+0.00415\,T^{2}
  5. k = 0.791 + 4.63 × 10 - 4 T - 8.44 × 10 - 7 T 2 k=0.791+4.63\times 10^{-4}\,T-8.44\times 10^{-7}\,T^{2}
  6. k 1 = 6.34 + 7.75 × 10 - 4 T - 9.35 × 10 - 5 T 2 k_{1}=6.34+7.75\times 10^{-4}\,T-9.35\times 10^{-5}\,T^{2}
  7. k 2 = 1.09 + 2.84 × 10 - 2 T - 9.04 × 10 - 5 T 2 k_{2}=1.09+2.84\times 10^{-2}\,T-9.04\times 10^{-5}\,T^{2}

Equilibrium_point.html

  1. 𝐱 ~ n \tilde{\mathbf{x}}\in\mathbb{R}^{n}
  2. d 𝐱 d t = 𝐟 ( t , 𝐱 ) \frac{d\mathbf{x}}{dt}=\mathbf{f}(t,\mathbf{x})
  3. 𝐟 ( t , 𝐱 ~ ) = 0 \mathbf{f}(t,\tilde{\mathbf{x}})=0
  4. t t\,\!
  5. 𝐱 ~ n \tilde{\mathbf{x}}\in\mathbb{R}^{n}
  6. 𝐱 k + 1 = 𝐟 ( k , 𝐱 k ) \mathbf{x}_{k+1}=\mathbf{f}(k,\mathbf{x}_{k})
  7. 𝐟 ( k , 𝐱 ~ ) = 𝐱 ~ \mathbf{f}(k,\tilde{\mathbf{x}})=\tilde{\mathbf{x}}
  8. k = 0 , 1 , 2 , k=0,1,2,\ldots

Equirectangular_projection.html

  1. x = λ cos φ 1 y = φ \begin{aligned}\displaystyle x&\displaystyle=\lambda\cos\varphi_{1}\\ \displaystyle y&\displaystyle=\varphi\end{aligned}
  2. λ \scriptstyle\lambda
  3. φ \scriptstyle\varphi
  4. φ 1 \scriptstyle\varphi_{1}
  5. x x
  6. y y
  7. φ 1 \scriptstyle\varphi_{1}
  8. x x
  9. y y

Equivalence_partitioning.html

  1. C b a {}_{a}C_{b}
  2. a , b a,b
  3. a , b a,b
  4. C C
  5. N N
  6. N N
  7. [ a , b ] [a,b]
  8. z m i n x + y z m a x z_{min}\leq x+y\leq z_{max}
  9. I N T _ M I N x + y I N T _ M A X INT\_MIN\leq x+y\leq INT\_MAX
  10. x { I N T _ M I N , , I N T _ M A X } x\in\{INT\_MIN,...,INT\_MAX\}
  11. y { I N T _ M I N , , I N T _ M A X } y\in\{INT\_MIN,...,INT\_MAX\}
  12. I N T _ M I N = x + y INT\_MIN=x+y
  13. I N T _ M A X = x + y INT\_MAX=x+y

Equivalent_annual_cost.html

  1. E A C = N P V A t , r EAC=\frac{NPV}{A_{t,r}}
  2. A t , r = 1 - 1 ( 1 + r ) t r {A_{t,r}}=\frac{1-\frac{1}{(1+r)^{t}}}{r}
  3. $ 50 , 000 A 3 , 5 + $ 13 , 000 = $ 31 , 360 \frac{\$50,000}{A_{3,5}}+\$13,000=\$31,360
  4. $ 150 , 000 A 8 , 5 + $ 7 , 500 = $ 30 , 708 \frac{\$150,000}{A_{8,5}}+\$7,500=\$30,708
  5. E A C = I [ 1 - ( t d i + d ) ( 1 + 1 2 i 1 + i ) ] A n , i + 1 A n , i n = 0 N R n ( 1 - t ) ( 1 + i ) n - S [ 1 - ( t d i + d ) ( 1 + 1 2 i 1 + i ) ] F n , i EAC=\frac{I\left[1-\left(\frac{td}{i+d}\right)\left(\frac{1+\frac{1}{2}i}{1+i}% \right)\right]}{A_{n,i}}+\frac{1}{A_{n,i}}\sum_{n=0}^{N}\frac{{R_{n}}\left(1-t% \right)}{(1+i)^{n}}-\frac{S\left[1-\left(\frac{td}{i+d}\right)\left(\frac{1+% \frac{1}{2}i}{1+i}\right)\right]}{F_{n,i}}
  6. R n {R_{n}}

Equivalent_carbon_content.html

  1. C E = % C + ( % M n + % S i 6 ) + ( % C r + % M o + % V 5 ) + ( % C u + % N i 15 ) CE=\%C+\left(\frac{\%Mn+\%Si}{6}\right)+\left(\frac{\%Cr+\%Mo+\%V}{5}\right)+% \left(\frac{\%Cu+\%Ni}{15}\right)
  2. C E = % C + % M n 6 + ( % C r + % M o + % V 5 ) + ( % C u + % N i 15 ) CE=\%C+\frac{\%Mn}{6}+\left(\frac{\%Cr+\%Mo+\%V}{5}\right)+\left(\frac{\%Cu+\%% Ni}{15}\right)
  3. P c m = % C + % S i 30 + % M n + % C u + % C r 20 + % N i 60 + % M o 15 + % V 10 + 5 B Pcm=\%C+\frac{\%Si}{30}+\frac{\%Mn+\%Cu+\%Cr}{20}+\frac{\%Ni}{60}+\frac{\%Mo}{% 15}+\frac{\%V}{10}+5B
  4. C E = % C + % M n 6 + 0.05 CE=\%C+\frac{\%Mn}{6}+0.05
  5. C E = % C + % M n 6 + ( % C r + % M o + % Z r 10 ) + % T i 2 + % C b 3 + % V 7 + U T S 900 + h 20 CE=\%C+\frac{\%Mn}{6}+\left(\frac{\%Cr+\%Mo+\%Zr}{10}\right)+\frac{\%Ti}{2}+% \frac{\%Cb}{3}+\frac{\%V}{7}+\frac{UTS}{900}+\frac{h}{20}
  6. C E * = % C * + % M n 3.6 + % C u 20 + % N i 9 + % C r 5 + % M o 4 CE*=\%C*+\frac{\%Mn}{3.6}+\frac{\%Cu}{20}+\frac{\%Ni}{9}+\frac{\%Cr}{5}+\frac{% \%Mo}{4}
  7. % C * = 5 % C for % C 0.30 % \%C*=5\%C\mbox{ for }\%C\leq 0.30\%
  8. % C * = % C / 6 for % C 0.30 % \%C*=\%C/6\mbox{ for }\%C\geq 0.30\%
  9. log 10 Δ t 8 - 5 = 2.69 C E * \log_{10}\Delta t_{8-5}=2.69CE*
  10. C E = % C + 0.33 ( % S i ) + 0.33 ( % P ) - 0.027 ( % M n ) + 0.4 ( % S ) CE=\%C+0.33\left(\%Si\right)+0.33\left(\%P\right)-0.027\left(\%Mn\right)+0.4% \left(\%S\right)
  11. C E = % C + 0.33 ( % S i ) CE=\%C+0.33\left(\%Si\right)
  12. C E = % C + 0.33 ( % S i + % P ) CE=\%C+0.33\left(\%Si+\%P\right)

Equivalent_isotropically_radiated_power.html

  1. E I R P | l o g = P T - L c + G a EIRP|_{log}=P_{T}-L_{c}+G_{a}
  2. E I R P \scriptstyle EIRP
  3. P T \scriptstyle P_{T}
  4. L c \scriptstyle L_{c}
  5. G a \scriptstyle G_{a}
  6. dBm = 10 log ( power out 1 mW ) \,\text{dBm}=10\log\left(\frac{\,\text{power out}}{1\,\mathrm{mW}}\right)
  7. dBW = 10 log ( power out 1 W ) \,\text{dBW}=10\log\left(\frac{\,\text{power out}}{1\,\mathrm{W}}\right)
  8. 16.9897 dBW = 10 log ( 50 W 1 W ) 16.9897\,\mathrm{dBW}=10\log\left(\frac{50\,\mathrm{W}}{1\,\mathrm{W}}\right)

Equivalent_variation.html

  1. e ( , ) e(\cdot,\cdot)
  2. E V = e ( p 0 , u 1 ) - e ( p 0 , u 0 ) EV=e(p_{0},u_{1})-e(p_{0},u_{0})
  3. = e ( p 0 , u 1 ) - w =e(p_{0},u_{1})-w
  4. = e ( p 0 , u 1 ) - e ( p 1 , u 1 ) =e(p_{0},u_{1})-e(p_{1},u_{1})
  5. w w
  6. p 0 p_{0}
  7. p 1 p_{1}
  8. u 0 u_{0}
  9. u 1 u_{1}
  10. v ( , ) v(\cdot,\cdot)
  11. v ( p 0 , w + E V ) = u 1 v(p_{0},w+EV)=u_{1}
  12. p 0 p_{0}
  13. e ( p 0 , v ( p 0 , w + E V ) ) = e ( p 0 , u 1 ) e(p_{0},v(p_{0},w+EV))=e(p_{0},u_{1})
  14. w + E V = e ( p 0 , u 1 ) w+EV=e(p_{0},u_{1})
  15. E V = e ( p 0 , u 1 ) - w EV=e(p_{0},u_{1})-w

Equivalent_weight.html

  1. u r 2 = ( u ( V ) V ) 2 + ( u ( m ) m ) 2 = ( 0.03 22.45 ) 2 + ( 0.1 781.4 ) 2 = ( 0.001336 ) 2 + ( 0.000128 ) 2 u_{\rm r}^{2}=\left(\frac{u(V)}{V}\right)^{2}+\left(\frac{u(m)}{m}\right)^{2}=% \left(\frac{0.03}{22.45}\right)^{2}+\left(\frac{0.1}{781.4}\right)^{2}=(0.0013% 36)^{2}+(0.000128)^{2}
  2. u r = 0.00134 u ( c ) = u r c = 0.1 meq / l \Longrightarrow u_{\rm r}=0.00134\Longrightarrow u(c)=u_{\rm r}c=0.1\ {\rm meq% /l}
  3. equivalent weight = m acid c ( NaOH ) V eq = 52.0 ± 0.1 g {\rm equivalent\ weight}=\frac{m_{\rm acid}}{c({\rm NaOH})V_{\rm eq}}=52.0\pm 0% .1\ {\rm g}

Erasing_rule.html

  1. A ϵ A\to\epsilon

Ernst_Mally.html

  1. I. ( ( A f B ) & ( B C ) ) ( A f C ) II. ( ( A f B ) & ( A f C ) ) ( A f ( B & C ) ) III. ( A f B ) ! ( A B ) IV. U ! U V. ¬ ( U f ) \begin{array}[]{rl}\mbox{I.}&((A\;\operatorname{f}\;B)\And(B\to C))\to(A\;% \operatorname{f}\;C)\\ \mbox{II.}&((A\;\operatorname{f}\;B)\And(A\;\operatorname{f}\;C))\to(A\;% \operatorname{f}\;(B\And C))\\ \mbox{III.}&(A\;\operatorname{f}\;B)\leftrightarrow\;!(A\to B)\\ \mbox{IV.}&\exists U\;!U\\ \mbox{V.}&\neg(U\;\operatorname{f}\;\cap)\end{array}

Error_catastrophe.html

  1. x ˙ j = i a i Q i j x i \dot{x}_{j}=\sum_{i}a_{i}Q_{ij}x_{i}
  2. { x ˙ = a ( 1 - Q ) x + b R y y ˙ = a Q x + b ( 1 - R ) y \begin{cases}\dot{x}=&a(1-Q)x+bRy\\ \dot{y}=&aQx+b(1-R)y\\ \end{cases}
  3. { x ˙ = a ( 1 - Q ) x y ˙ = a Q x + b y \begin{cases}\dot{x}=&a(1-Q)x\\ \dot{y}=&aQx+by\\ \end{cases}
  4. z t = x ˙ y - x y ˙ y 2 = a ( 1 - Q ) x y - x ( a Q x + b y ) y 2 = a ( 1 - Q ) z - ( a Q z 2 + b z ) = z ( a ( 1 - Q ) - a Q z - b ) \begin{matrix}\frac{\partial z}{\partial t}&=&\frac{\dot{x}y-x\dot{y}}{y^{2}}% \\ &&\\ &=&\frac{a(1-Q)xy-x(aQx+by)}{y^{2}}\\ &&\\ &=&a(1-Q)z-(aQz^{2}+bz)\\ &&\\ &=&z(a(1-Q)-aQz-b)\\ \end{matrix}
  5. z ( ) = a ( 1 - Q ) - b a Q z(\infty)=\frac{a(1-Q)-b}{aQ}
  6. z ( ) > 0 a ( 1 - Q ) - b > 0 ( 1 - Q ) > b / a . z(\infty)>0\iff a(1-Q)-b>0\iff(1-Q)>b/a.
  7. z ( ) > 0 ( 1 - Q ) = ( 1 - q ) L > 1 - s z(\infty)>0\iff(1-Q)=(1-q)^{L}>1-s
  8. L ln ( 1 - q ) - L q > ln ( 1 - s ) - s L\ln{(1-q)}\approx-Lq>\ln{(1-s)}\approx-s
  9. L q < s Lq<s
  10. L q < - ln S Lq<-\ln{S}

Essential_spectrum.html

  1. λ I - T \lambda\,I-T
  2. lim k T ψ k - λ ψ k = 0. \lim_{k\to\infty}\left\|T\psi_{k}-\lambda\psi_{k}\right\|=0.
  3. { ψ k } \{\psi_{k}\}
  4. σ discr ( T ) = σ ( T ) σ ess ( T ) . \sigma_{\mathrm{discr}}(T)=\sigma(T)\setminus\sigma_{\mathrm{ess}}(T).
  5. { ψ X : T ψ = λ ψ } \{\psi\in X:T\psi=\lambda\psi\}
  6. σ ess , 1 ( T ) σ ess , 2 ( T ) σ ess , 3 ( T ) σ ess , 4 ( T ) σ ess , 5 ( T ) σ ( T ) 𝐂 , \sigma_{\mathrm{ess},1}(T)\subset\sigma_{\mathrm{ess},2}(T)\subset\sigma_{% \mathrm{ess},3}(T)\subset\sigma_{\mathrm{ess},4}(T)\subset\sigma_{\mathrm{ess}% ,5}(T)\subset\sigma(T)\subset\mathbf{C},
  7. r ess , k ( T ) = max { | λ | : λ σ ess , k ( T ) } . r_{\mathrm{ess},k}(T)=\max\{|\lambda|:\lambda\in\sigma_{\mathrm{ess},k}(T)\}.
  8. σ ess , 4 ( T ) = K K ( X ) σ ( T + K ) , \sigma_{\mathrm{ess},4}(T)=\bigcap_{K\in K(X)}\sigma(T+K),

Estimation_theory.html

  1. 𝐱 = [ x [ 0 ] x [ 1 ] x [ N - 1 ] ] . \mathbf{x}=\begin{bmatrix}x[0]\\ x[1]\\ \vdots\\ x[N-1]\end{bmatrix}.
  2. θ = [ θ 1 θ 2 θ M ] , \mathbf{\theta}=\begin{bmatrix}\theta_{1}\\ \theta_{2}\\ \vdots\\ \theta_{M}\end{bmatrix},
  3. p ( 𝐱 | θ ) . p(\mathbf{x}|\mathbf{\theta}).\,
  4. π ( θ ) . \pi(\mathbf{\theta}).\,
  5. θ ^ \hat{\mathbf{\theta}}
  6. 𝐞 = θ ^ - θ \mathbf{e}=\hat{\mathbf{\theta}}-\mathbf{\theta}
  7. x [ n ] x[n]
  8. N N
  9. A A
  10. w [ n ] w[n]
  11. σ 2 \sigma^{2}
  12. 𝒩 ( 0 , σ 2 ) \mathcal{N}(0,\sigma^{2})
  13. A A
  14. x [ n ] = A + w [ n ] n = 0 , 1 , , N - 1 x[n]=A+w[n]\quad n=0,1,\dots,N-1
  15. A ^ 1 = x [ 0 ] \hat{A}_{1}=x[0]
  16. A ^ 2 = 1 N n = 0 N - 1 x [ n ] \hat{A}_{2}=\frac{1}{N}\sum_{n=0}^{N-1}x[n]
  17. A A
  18. E [ A ^ 1 ] = E [ x [ 0 ] ] = A \mathrm{E}\left[\hat{A}_{1}\right]=\mathrm{E}\left[x[0]\right]=A
  19. E [ A ^ 2 ] = E [ 1 N n = 0 N - 1 x [ n ] ] = 1 N [ n = 0 N - 1 E [ x [ n ] ] ] = 1 N [ N A ] = A \mathrm{E}\left[\hat{A}_{2}\right]=\mathrm{E}\left[\frac{1}{N}\sum_{n=0}^{N-1}% x[n]\right]=\frac{1}{N}\left[\sum_{n=0}^{N-1}\mathrm{E}\left[x[n]\right]\right% ]=\frac{1}{N}\left[NA\right]=A
  20. var ( A ^ 1 ) = var ( x [ 0 ] ) = σ 2 \mathrm{var}\left(\hat{A}_{1}\right)=\mathrm{var}\left(x[0]\right)=\sigma^{2}
  21. var ( A ^ 2 ) = var ( 1 N n = 0 N - 1 x [ n ] ) = independence 1 N 2 [ n = 0 N - 1 var ( x [ n ] ) ] = 1 N 2 [ N σ 2 ] = σ 2 N \mathrm{var}\left(\hat{A}_{2}\right)=\mathrm{var}\left(\frac{1}{N}\sum_{n=0}^{% N-1}x[n]\right)\overset{\,\text{independence}}{=}\frac{1}{N^{2}}\left[\sum_{n=% 0}^{N-1}\mathrm{var}(x[n])\right]=\frac{1}{N^{2}}\left[N\sigma^{2}\right]=% \frac{\sigma^{2}}{N}
  22. w [ n ] w[n]
  23. p ( w [ n ] ) = 1 σ 2 π exp ( - 1 2 σ 2 w [ n ] 2 ) p(w[n])=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2\sigma^{2}}w[n]^{2}\right)
  24. x [ n ] x[n]
  25. x [ n ] x[n]
  26. 𝒩 ( A , σ 2 ) \mathcal{N}(A,\sigma^{2})
  27. p ( x [ n ] ; A ) = 1 σ 2 π exp ( - 1 2 σ 2 ( x [ n ] - A ) 2 ) p(x[n];A)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2\sigma^{2}}(x[n]-A)^% {2}\right)
  28. 𝐱 \mathbf{x}
  29. p ( 𝐱 ; A ) = n = 0 N - 1 p ( x [ n ] ; A ) = 1 ( σ 2 π ) N exp ( - 1 2 σ 2 n = 0 N - 1 ( x [ n ] - A ) 2 ) p(\mathbf{x};A)=\prod_{n=0}^{N-1}p(x[n];A)=\frac{1}{\left(\sigma\sqrt{2\pi}% \right)^{N}}\exp\left(-\frac{1}{2\sigma^{2}}\sum_{n=0}^{N-1}(x[n]-A)^{2}\right)
  30. ln p ( 𝐱 ; A ) = - N ln ( σ 2 π ) - 1 2 σ 2 n = 0 N - 1 ( x [ n ] - A ) 2 \ln p(\mathbf{x};A)=-N\ln\left(\sigma\sqrt{2\pi}\right)-\frac{1}{2\sigma^{2}}% \sum_{n=0}^{N-1}(x[n]-A)^{2}
  31. A ^ = arg max ln p ( 𝐱 ; A ) \hat{A}=\arg\max\ln p(\mathbf{x};A)
  32. A ln p ( 𝐱 ; A ) = 1 σ 2 [ n = 0 N - 1 ( x [ n ] - A ) ] = 1 σ 2 [ n = 0 N - 1 x [ n ] - N A ] \frac{\partial}{\partial A}\ln p(\mathbf{x};A)=\frac{1}{\sigma^{2}}\left[\sum_% {n=0}^{N-1}(x[n]-A)\right]=\frac{1}{\sigma^{2}}\left[\sum_{n=0}^{N-1}x[n]-NA\right]
  33. 0 = 1 σ 2 [ n = 0 N - 1 x [ n ] - N A ] = n = 0 N - 1 x [ n ] - N A 0=\frac{1}{\sigma^{2}}\left[\sum_{n=0}^{N-1}x[n]-NA\right]=\sum_{n=0}^{N-1}x[n% ]-NA
  34. A ^ = 1 N n = 0 N - 1 x [ n ] \hat{A}=\frac{1}{N}\sum_{n=0}^{N-1}x[n]
  35. N N
  36. ( A ) = E ( [ A ln p ( 𝐱 ; A ) ] 2 ) = - E [ 2 A 2 ln p ( 𝐱 ; A ) ] \mathcal{I}(A)=\mathrm{E}\left(\left[\frac{\partial}{\partial A}\ln p(\mathbf{% x};A)\right]^{2}\right)=-\mathrm{E}\left[\frac{\partial^{2}}{\partial A^{2}}% \ln p(\mathbf{x};A)\right]
  37. A ln p ( 𝐱 ; A ) = 1 σ 2 [ n = 0 N - 1 x [ n ] - N A ] \frac{\partial}{\partial A}\ln p(\mathbf{x};A)=\frac{1}{\sigma^{2}}\left[\sum_% {n=0}^{N-1}x[n]-NA\right]
  38. 2 A 2 ln p ( 𝐱 ; A ) = 1 σ 2 ( - N ) = - N σ 2 \frac{\partial^{2}}{\partial A^{2}}\ln p(\mathbf{x};A)=\frac{1}{\sigma^{2}}(-N% )=\frac{-N}{\sigma^{2}}
  39. - E [ 2 A 2 ln p ( 𝐱 ; A ) ] = N σ 2 -\mathrm{E}\left[\frac{\partial^{2}}{\partial A^{2}}\ln p(\mathbf{x};A)\right]% =\frac{N}{\sigma^{2}}
  40. var ( A ^ ) 1 \mathrm{var}\left(\hat{A}\right)\geq\frac{1}{\mathcal{I}}
  41. var ( A ^ ) σ 2 N \mathrm{var}\left(\hat{A}\right)\geq\frac{\sigma^{2}}{N}
  42. N N
  43. A A
  44. 1 , 2 , , N 1,2,\dots,N
  45. k + 1 k m - 1 = m + m k - 1 \frac{k+1}{k}m-1=m+\frac{m}{k}-1
  46. 1 k ( N - k ) ( N + 1 ) ( k + 2 ) N 2 k 2 for small samples k N \frac{1}{k}\frac{(N-k)(N+1)}{(k+2)}\approx\frac{N^{2}}{k^{2}}\,\text{ for % small samples }k\ll N
  47. N / k N/k
  48. m k \frac{m}{k}

Etendue.html

  1. d G = n 2 d S cos θ d Ω . \mathrm{d}G=n^{2}\,\mathrm{d}S\cos\theta\,\mathrm{d}\Omega.
  2. d G Σ = n 2 d Σ cos θ Σ d Ω Σ = n 2 d Σ cos θ Σ d S cos θ S d 2 \mathrm{d}G_{\Sigma}=n^{2}\,\mathrm{d}\Sigma\cos\theta_{\Sigma}\,\mathrm{d}% \Omega_{\Sigma}=n^{2}\,\mathrm{d}\Sigma\cos\theta_{\Sigma}\frac{\mathrm{d}S% \cos\theta_{S}}{d^{2}}
  3. d G S = n 2 d S cos θ S d Ω S = n 2 d S cos θ S d Σ cos θ Σ d 2 , \mathrm{d}G_{S}=n^{2}\,\mathrm{d}S\cos\theta_{S}\,\mathrm{d}\Omega_{S}=n^{2}\,% \mathrm{d}S\cos\theta_{S}\frac{\mathrm{d}\Sigma\cos\theta_{\Sigma}}{d^{2}},
  4. d G Σ = d G S , \mathrm{d}G_{\Sigma}=\mathrm{d}G_{S},
  5. G = Σ S d G . G=\int_{\Sigma}\!\int_{S}\mathrm{d}G.
  6. d G = d Σ cos θ Σ d S cos θ S d 2 = π d Σ ( cos θ Σ cos θ S π d 2 d S ) = π d Σ F d Σ d S , \mathrm{d}G=\mathrm{d}\Sigma\cos\theta_{\Sigma}\frac{\mathrm{d}S\cos\theta_{S}% }{d^{2}}=\pi\mathrm{d}\Sigma\!\left(\frac{\cos\theta_{\Sigma}\cos\theta_{S}}{% \pi d^{2}}\mathrm{d}S\right)=\pi\mathrm{d}\Sigma F_{\mathrm{d}\Sigma% \rightarrow\mathrm{d}S},
  7. n Σ sin θ Σ = n S sin θ S , n_{\Sigma}\sin\theta_{\Sigma}=n_{S}\sin\theta_{S},
  8. n Σ cos θ Σ d θ Σ = n S cos θ S d θ S , n_{\Sigma}\cos\theta_{\Sigma}\,\mathrm{d}\theta_{\Sigma}=n_{S}\cos\theta_{S}% \mathrm{d}\theta_{S},
  9. n Σ 2 cos θ Σ ( sin θ Σ d θ Σ d φ ) = n S 2 cos θ S ( sin θ S d θ S d φ ) , n_{\Sigma}^{2}\cos\theta_{\Sigma}\!\left(\sin\theta_{\Sigma}\,\mathrm{d}\theta% _{\Sigma}\,\mathrm{d}\varphi\right)=n_{S}^{2}\cos\theta_{S}\!\left(\sin\theta_% {S}\,\mathrm{d}\theta_{S}\,\mathrm{d}\varphi\right),
  10. n Σ 2 cos θ Σ d Ω Σ = n S 2 cos θ S d Ω S , n_{\Sigma}^{2}\cos\theta_{\Sigma}\,\mathrm{d}\Omega_{\Sigma}=n_{S}^{2}\cos% \theta_{S}\,\mathrm{d}\Omega_{S},
  11. n Σ 2 d S cos θ Σ d Ω Σ = n S 2 d S cos θ S d Ω S n_{\Sigma}^{2}\,\mathrm{d}S\cos\theta_{\Sigma}\,\mathrm{d}\Omega_{\Sigma}=n_{S% }^{2}\,\mathrm{d}S\cos\theta_{S}\,\mathrm{d}\Omega_{S}
  12. d G Σ = d G S , \mathrm{d}G_{\Sigma}=\mathrm{d}G_{S},
  13. L e , Ω = n 2 Φ e G , L_{\mathrm{e},\Omega}=n^{2}\frac{\partial\Phi_{\mathrm{e}}}{\partial G},
  14. L e , Ω * = L e , Ω n 2 L_{\mathrm{e},\Omega}^{*}=\frac{L_{\mathrm{e},\Omega}}{n^{2}}
  15. 𝐩 = n ( cos α X , cos α Y , cos α Z ) = ( p , q , r ) , \mathbf{p}=n(\cos\alpha_{X},\cos\alpha_{Y},\cos\alpha_{Z})=(p,q,r),
  16. 𝐩 = n ( sin θ cos φ , sin θ sin φ , cos θ ) , \mathbf{p}=n\!\left(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta% \right),
  17. d p d q = ( p , q ) ( θ , φ ) d θ d φ = ( p θ q φ - p φ q θ ) d θ d φ = n 2 cos θ sin θ d θ d φ = n 2 cos θ d Ω , \mathrm{d}p\,\mathrm{d}q=\frac{\partial(p,q)}{\partial(\theta,\varphi)}\mathrm% {d}\theta\,\mathrm{d}\varphi=\left(\frac{\partial p}{\partial\theta}\frac{% \partial q}{\partial\varphi}-\frac{\partial p}{\partial\varphi}\frac{\partial q% }{\partial\theta}\right)\mathrm{d}\theta\,\mathrm{d}\varphi=n^{2}\cos\theta% \sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi=n^{2}\cos\theta\,\mathrm{d}\Omega,
  18. d G = n 2 d S cos θ d Ω = d x d y d p d q , \mathrm{d}G=n^{2}\,\mathrm{d}S\cos\theta\,\mathrm{d}\Omega=\mathrm{d}x\,% \mathrm{d}y\,\mathrm{d}p\,\mathrm{d}q,
  19. d G = n 2 d S cos θ d Ω = n 2 d S 0 2 π 0 α cos θ sin θ d θ d φ = π n 2 d S sin 2 α . \mathrm{d}G=n^{2}\,\mathrm{d}S\int\cos\theta\,\mathrm{d}\Omega=n^{2}dS\int_{0}% ^{2\pi}\!\int_{0}^{\alpha}\cos\theta\sin\theta\,\mathrm{d}\theta\,\mathrm{d}% \varphi=\pi n^{2}\mathrm{d}S\sin^{2}\alpha.
  20. d G = π d S NA 2 . \mathrm{d}G=\pi\,\mathrm{d}S\mathrm{NA}^{2}.
  21. G = π n 2 sin 2 α d S = π n 2 S sin 2 α = π S NA 2 . G=\pi n^{2}\sin^{2}\alpha\int\mathrm{d}S=\pi n^{2}S\sin^{2}\alpha=\pi S\mathrm% {NA}^{2}.
  22. G i = π S sin 2 α G_{\mathrm{i}}=\pi S\sin^{2}\alpha
  23. G r = π n 2 Σ sin 2 β . G_{\mathrm{r}}=\pi n^{2}\Sigma\sin^{2}\beta.
  24. C = S Σ = n 2 sin 2 β sin 2 α , C=\frac{S}{\Sigma}=n^{2}\frac{\sin^{2}\beta}{\sin^{2}\alpha},
  25. C max = n 2 sin 2 α . C_{\mathrm{max}}=\frac{n^{2}}{\sin^{2}\alpha}.
  26. G i = π n i S sin 2 α = G r = π n r Σ sin 2 β , G_{\mathrm{i}}=\pi n_{\mathrm{i}}S\sin^{2}\alpha=G_{\mathrm{r}}=\pi n_{\mathrm% {r}}\Sigma\sin^{2}\beta,
  27. C = ( NA r NA i ) 2 , C=\left(\frac{\mathrm{NA}_{\mathrm{r}}}{\mathrm{NA}_{\mathrm{i}}}\right)^{2},
  28. C max = n r 2 NA i 2 . C_{\mathrm{max}}=\frac{n_{\mathrm{r}}^{2}}{\mathrm{NA}_{\mathrm{i}}^{2}}.

Euclidean_plane_isometry.html

  1. M : 𝐑 2 𝐑 2 M:\,\textbf{R}^{2}\to\,\textbf{R}^{2}
  2. d ( p , q ) = d ( M ( p ) , M ( q ) ) , d(p,q)=d(M(p),M(q)),\,\!
  3. T v ( p ) = p + v , T_{v}(p)=p+v,\,\!
  4. T v ( p ) = [ p x + v x p y + v y ] . T_{v}(p)=\begin{bmatrix}p_{x}+v_{x}\\ p_{y}+v_{y}\end{bmatrix}.
  5. R 0 , θ ( p ) = ( cos θ - sin θ sin θ cos θ ) [ p x p y ] . R_{0,\theta}(p)=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\begin{bmatrix}p_{x}\\ p_{y}\end{bmatrix}.
  6. G G T = G T G = I 2 . GG^{T}=G^{T}G=I_{2}.
  7. R c , θ = T c R 0 , θ T - c , R_{c,\theta}=T_{c}\circ R_{0,\theta}\circ T_{-c},
  8. R c , θ ( p ) = c + R 0 , θ ( p - c ) . R_{c,\theta}(p)=c+R_{0,\theta}(p-c).\,\!
  9. R c , θ ( p ) = R 0 , θ p + v . R_{c,\theta}(p)=R_{0,\theta}p+v.\,\!
  10. t = ( p - c ) v = ( p x - c x ) v x + ( p y - c y ) v y , t=(p-c)\cdot v=(p_{x}-c_{x})v_{x}+(p_{y}-c_{y})v_{y},
  11. F c , v ( p ) = p - 2 t v . F_{c,v}(p)=p-2tv.\,
  12. R 0 , θ ( p ) = ( cos θ sin θ sin θ - cos θ ) [ p x p y ] . R_{0,\theta}(p)=\begin{pmatrix}\cos\theta&\sin\theta\\ \sin\theta&-\cos\theta\end{pmatrix}\begin{bmatrix}p_{x}\\ p_{y}\end{bmatrix}.
  13. G c , v , w = T w F c , v , G_{c,v,w}=T_{w}\circ F_{c,v},
  14. G c , v , w ( p ) = w + F c , v ( p ) . G_{c,v,w}(p)=w+F_{c,v}(p).\,
  15. G c , v , w ( p ) = F c , v ( p + w ) ; G_{c,v,w}(p)=F_{c,v}(p+w);\,
  16. z a + ω z \begin{array}[]{ccc}\mathbb{C}&\longrightarrow&\mathbb{C}\\ z&\mapsto&a+\omega z\end{array}
  17. z a + ω z ¯ , \begin{array}[]{ccc}\mathbb{C}&\longrightarrow&\mathbb{C}\\ z&\mapsto&a+\omega\overline{z}\mbox{,}\end{array}
  18. g : z f ( z ) - a ω , \begin{array}[]{rccc}g\colon&\mathbb{C}&\longrightarrow&\mathbb{C}\\ &z&\mapsto&\frac{f(z)-a}{\omega}\mbox{,}\end{array}
  19. z ω ( z - p ) + p = ω z + p ( 1 - ω ) z\mapsto\omega(z-p)+p=\omega z+p(1-\omega)
  20. z ω z + a , z\mapsto\omega z+a,
  21. p ( 1 - ω ) = a p(1-\omega)=a
  22. p = a / ( 1 - ω ) p=a/(1-\omega)
  23. ω 1 \omega\neq 1

Euclidean_tilings_of_convex_regular_polygons.html

  1. n n
  2. k k
  3. k k
  4. t t
  5. t t
  6. e e
  7. e e
  8. n n\,\!
  9. ( 1 - 2 n ) 180 \left(1-\frac{2}{n}\right)180

Euclidean_topology.html

  1. S x , y = { r R : x < r < y } . S_{x,y}=\{r\in{R}:x<r<y\}.

Euler's_three-body_problem.html

  1. V ( 𝐫 ) = - μ 1 r 1 - μ 2 r 2 V(\mathbf{r})=\frac{-\mu_{1}}{r_{1}}-\frac{\mu_{2}}{r_{2}}
  2. E = 1 2 m | 𝐩 | 2 + V ( 𝐫 ) E=\frac{1}{2m}\left|\mathbf{p}\right|^{2}+V(\mathbf{r})
  3. r 1 2 r 2 2 ( d θ 1 d t ) ( d θ 2 d t ) - 2 a [ μ 1 cos θ 1 + μ 2 cos θ 2 ] , r_{1}^{2}r_{2}^{2}\left(\frac{d\theta_{1}}{dt}\right)\left(\frac{d\theta_{2}}{% dt}\right)-2a\left[\mu_{1}\cos\theta_{1}+\mu_{2}\cos\theta_{2}\right],
  4. B = | 𝐋 | 2 + a 2 | 𝐩 | 2 - 2 a [ μ 1 cos θ 1 + μ 2 cos θ 2 ] B=\left|\mathbf{L}\right|^{2}+a^{2}\left|\mathbf{p}\right|^{2}-2a\left[\mu_{1}% \cos\theta_{1}+\mu_{2}\cos\theta_{2}\right]
  5. H 2 + \,\text{H}_{2}^{+}
  6. V ( x , y ) = - μ 1 ( x - a ) 2 + y 2 - μ 2 ( x + a ) 2 + y 2 . V(x,y)=\frac{-\mu_{1}}{\sqrt{\left(x-a\right)^{2}+y^{2}}}-\frac{\mu_{2}}{\sqrt% {\left(x+a\right)^{2}+y^{2}}}.
  7. x = a cosh ξ cos η , \,x=\,a\cosh\xi\cos\eta,
  8. y = a sinh ξ sin η , \,y=\,a\sinh\xi\sin\eta,
  9. V ( ξ , η ) = - μ 1 a ( cosh ξ - cos η ) - μ 2 a ( cosh ξ + cos η ) = - μ 1 ( cosh ξ + cos η ) - μ 2 ( cosh ξ - cos η ) a ( cosh 2 ξ - cos 2 η ) , \begin{aligned}\displaystyle V(\xi,\eta)&\displaystyle=\frac{-\mu_{1}}{a\left(% \cosh\xi-\cos\eta\right)}-\frac{\mu_{2}}{a\left(\cosh\xi+\cos\eta\right)}\\ &\displaystyle=\frac{-\mu_{1}\left(\cosh\xi+\cos\eta\right)-\mu_{2}\left(\cosh% \xi-\cos\eta\right)}{a\left(\cosh^{2}\xi-\cos^{2}\eta\right)},\end{aligned}
  10. T = m a 2 2 ( cosh 2 ξ - cos 2 η ) ( ξ ˙ 2 + η ˙ 2 ) . T=\frac{ma^{2}}{2}\left(\cosh^{2}\xi-\cos^{2}\eta\right)\left(\dot{\xi}^{2}+% \dot{\eta}^{2}\right).
  11. Y = cosh 2 ξ - cos 2 η \,Y=\cosh^{2}\xi-\cos^{2}\eta
  12. W = - μ 1 ( cosh ξ + cos η ) - μ 2 ( cosh ξ - cos η ) . W=-\mu_{1}\left(\cosh\xi+\cos\eta\right)-\mu_{2}\left(\cosh\xi-\cos\eta\right).
  13. m a 2 2 ( cosh 2 ξ - cos 2 η ) 2 ξ ˙ 2 = E cosh 2 ξ + ( μ 1 + μ 2 a ) cosh ξ - γ \frac{ma^{2}}{2}\left(\cosh^{2}\xi-\cos^{2}\eta\right)^{2}\dot{\xi}^{2}=E\cosh% ^{2}\xi+\left(\frac{\mu_{1}+\mu_{2}}{a}\right)\cosh\xi-\gamma
  14. m a 2 2 ( cosh 2 ξ - cos 2 η ) 2 η ˙ 2 = - E cos 2 η + ( μ 1 - μ 2 a ) cos η + γ \frac{ma^{2}}{2}\left(\cosh^{2}\xi-\cos^{2}\eta\right)^{2}\dot{\eta}^{2}=-E% \cos^{2}\eta+\left(\frac{\mu_{1}-\mu_{2}}{a}\right)\cos\eta+\gamma
  15. d u = d ξ E cosh 2 ξ + ( μ 1 + μ 2 a ) cosh ξ - γ = d η - E cos 2 η + ( μ 1 - μ 2 a ) cos η + γ , du=\frac{d\xi}{\sqrt{E\cosh^{2}\xi+\left(\frac{\mu_{1}+\mu_{2}}{a}\right)\cosh% \xi-\gamma}}=\frac{d\eta}{\sqrt{-E\cos^{2}\eta+\left(\frac{\mu_{1}-\mu_{2}}{a}% \right)\cos\eta+\gamma}},
  16. u = d ξ E cosh 2 ξ + ( μ 1 + μ 2 a ) cosh ξ - γ = d η - E cos 2 η + ( μ 1 - μ 2 a ) cos η + γ . u=\int\frac{d\xi}{\sqrt{E\cosh^{2}\xi+\left(\frac{\mu_{1}+\mu_{2}}{a}\right)% \cosh\xi-\gamma}}=\int\frac{d\eta}{\sqrt{-E\cos^{2}\eta+\left(\frac{\mu_{1}-% \mu_{2}}{a}\right)\cos\eta+\gamma}}.

Euler_brick.html

  1. { a 2 + b 2 = d 2 a 2 + c 2 = e 2 b 2 + c 2 = f 2 \begin{cases}a^{2}+b^{2}=d^{2}\\ a^{2}+c^{2}=e^{2}\\ b^{2}+c^{2}=f^{2}\end{cases}
  2. u 2 + v 2 = w 2 . u^{2}+v^{2}=w^{2}.
  3. a = u | 4 v 2 - w 2 | , b = v | 4 u 2 - w 2 | , c = 4 u v w a=u|4v^{2}-w^{2}|,\quad b=v|4u^{2}-w^{2}|,\quad c=4uvw
  4. d = w 3 , e = u ( 4 v 2 + w 2 ) , f = v ( 4 u 2 + w 2 ) . d=w^{3},\quad e=u(4v^{2}+w^{2}),\quad f=v(4u^{2}+w^{2}).
  5. ( a , b , c ) = ( 44 , 117 , 240 ) (a,b,c)=(44,117,240)
  6. a 2 + b 2 + c 2 = g 2 , a^{2}+b^{2}+c^{2}=g^{2},\,
  7. ( a , b , c ) = ( 672 , 153 , 104 ) . (a,b,c)=(672,153,104).\,
  8. ( a , b , c ) = ( 18720 , 211773121 , 7800 ) (a,b,c)=(18720,\sqrt{211773121},7800)
  9. ( a , b , c ) = ( 520 , 576 , 618849 ) . (a,b,c)=(520,576,\sqrt{618849}).

Ewens's_sampling_formula.html

  1. Pr ( a 1 , , a n ; θ ) = n ! θ ( θ + 1 ) ( θ + n - 1 ) j = 1 n θ a j j a j a j ! , \operatorname{Pr}(a_{1},\dots,a_{n};\theta)={n!\over\theta(\theta+1)\cdots(% \theta+n-1)}\prod_{j=1}^{n}{\theta^{a_{j}}\over j^{a_{j}}a_{j}!},
  2. a 1 + 2 a 2 + 3 a 3 + + n a n = n . a_{1}+2a_{2}+3a_{3}+\cdots+na_{n}=n.\,

Exact_differential_equation.html

  1. I ( x , y ) d x + J ( x , y ) d y = 0 , I(x,y)\,\mathrm{d}x+J(x,y)\,\mathrm{d}y=0,\,\!
  2. F x ( x , y ) = I \frac{\partial F}{\partial x}(x,y)=I
  3. F y ( x , y ) = J . \frac{\partial F}{\partial y}(x,y)=J.
  4. F ( x 0 , x 1 , , x n - 1 , x n ) F(x_{0},x_{1},...,x_{n-1},x_{n})
  5. x 0 x_{0}
  6. d F d x 0 = F x 0 + i = 1 n F x i d x i d x 0 . \frac{\mathrm{d}F}{\mathrm{d}x_{0}}=\frac{\partial F}{\partial x_{0}}+\sum_{i=% 1}^{n}\frac{\partial F}{\partial x_{i}}\frac{\mathrm{d}x_{i}}{\mathrm{d}x_{0}}.
  7. F ( x , y ) := 1 2 ( x 2 + y 2 ) F(x,y):=\frac{1}{2}(x^{2}+y^{2})
  8. x d x + y d y = 0. xdx+ydy=0.\,
  9. I ( x , y ) d x + J ( x , y ) d y = 0 , I(x,y)\,\mathrm{d}x+J(x,y)\,\mathrm{d}y=0,\,\!
  10. I y ( x , y ) = J x ( x , y ) . \frac{\partial I}{\partial y}(x,y)=\frac{\partial J}{\partial x}(x,y).
  11. F ( x , f ( x ) ) = c . F(x,f(x))=c.\,
  12. y ( x 0 ) = y 0 y(x_{0})=y_{0}\,
  13. F ( x , y ) = x 0 x I ( t , y 0 ) d t + y 0 y [ J ( x , t ) - x 0 x I t ( u , t ) d u ] d t . F(x,y)=\int_{x_{0}}^{x}I(t,y_{0})\mathrm{d}t+\int_{y_{0}}^{y}\left[J(x,t)-\int% _{x_{0}}^{x}\frac{\partial I}{\partial t}(u,t)\,\mathrm{d}u\,\right]\mathrm{d}t.
  14. F ( x , y ) = c F(x,y)=c\,

Exact_solutions_in_general_relativity.html

  1. T α β T^{\alpha\beta}
  2. G α β = 8 π T α β . G^{\alpha\beta}=8\pi\,T^{\alpha\beta}.
  3. G α β G^{\alpha\beta}
  4. G α β G^{\alpha\beta}
  5. 8 π 8\pi
  6. T α β T^{\alpha\beta}
  7. T α β = 0 T^{\alpha\beta}=0
  8. T α β T^{\alpha\beta}
  9. T α β T^{\alpha\beta}
  10. T α β T^{\alpha\beta}
  11. T α β T^{\alpha\beta}
  12. T α β T^{\alpha\beta}
  13. { ( 1 , 1 ) ( 11 ) } \{\,(1,1)(11)\}
  14. { ( 2 , 11 ) } \{\,(2,11)\}
  15. { 1 , ( 111 ) } \{\,1,(111)\}
  16. { ( 1 , 111 ) } \{\,(1,111)\}

Examples_of_vector_spaces.html

  1. x = ( x 1 , x 2 , , x n ) x=(x_{1},x_{2},\ldots,x_{n})\,
  2. x + y = ( x 1 + y 1 , x 2 + y 2 , , x n + y n ) x+y=(x_{1}+y_{1},x_{2}+y_{2},\ldots,x_{n}+y_{n})\,
  3. α x = ( α x 1 , α x 2 , , α x n ) \alpha x=(\alpha x_{1},\alpha x_{2},\ldots,\alpha x_{n})\,
  4. 0 = ( 0 , 0 , , 0 ) 0=(0,0,\ldots,0)\,
  5. - x = ( - x 1 , - x 2 , , - x n ) -x=(-x_{1},-x_{2},\ldots,-x_{n})\,
  6. e 1 = ( 1 , 0 , , 0 ) e_{1}=(1,0,\ldots,0)\,
  7. e 2 = ( 0 , 1 , , 0 ) e_{2}=(0,1,\ldots,0)\,
  8. \vdots\,
  9. e n = ( 0 , 0 , , 1 ) e_{n}=(0,0,\ldots,1)\,
  10. x = ( x 1 , x 2 , x 3 , ) x=(x_{1},x_{2},x_{3},\ldots)\,
  11. ( f + g ) ( x ) = f ( x ) + g ( x ) (f+g)(x)=f(x)+g(x)\,
  12. ( α f ) ( x ) = α f ( x ) (\alpha f)(x)=\alpha f(x)\,
  13. δ x ( y ) = { 1 x = y 0 x y \delta_{x}(y)=\begin{cases}1\quad x=y\\ 0\quad x\neq y\end{cases}
  14. ( 𝐅 X ) 0 = x X 𝐅 . (\mathbf{F}^{X})_{0}=\bigoplus_{x\in X}\mathbf{F}.

Expected_return.html

  1. E [ R ] = i = 1 n R i P i E[R]=\sum_{i=1}^{n}R_{i}P_{i}
  2. R i R_{i}
  3. i i
  4. P i P_{i}
  5. R i R_{i}
  6. i i
  7. i i
  8. E [ R ] = R 1 P 1 + R 2 P 2 + R 3 P 3 = 10 * 0.5 + 20 * 0.25 + ( - 10 ) * 0.25 = 7.5. E[R]=R_{1}P_{1}+R_{2}P_{2}+R_{3}P_{3}=10*0.5+20*0.25+(-10)*0.25=7.5.
  9. E [ R ] = 1 3 1 - 2 3 0.5 = 0. E[R]=\frac{1}{3}\cdot 1-\frac{2}{3}\cdot 0.5=0.

Exponential_dichotomy.html

  1. 𝐱 ˙ = A ( t ) 𝐱 \dot{\mathbf{x}}=A(t)\mathbf{x}
  2. || Φ ( t ) P Φ - 1 ( s ) || K e - α ( t - s ) for s t < ||\Phi(t)P\Phi^{-1}(s)||\leq Ke^{-\alpha(t-s)}\mbox{ for }~{}s\leq t<\infty
  3. || Φ ( t ) ( I - P ) Φ - 1 ( s ) || L e - β ( s - t ) for s t > - . ||\Phi(t)(I-P)\Phi^{-1}(s)||\leq Le^{-\beta(s-t)}\mbox{ for }~{}s\geq t>-\infty.
  4. P ( I - P ) = n \scriptstyle P\oplus(I-P)=\mathbb{R}^{n}

Exponential_sheaf_sequence.html

  1. exp : 𝒪 M 𝒪 M * , \exp:\mathcal{O}_{M}\to\mathcal{O}_{M}^{*},
  2. 0 2 π i 𝒪 M 𝒪 M * 0. 0\to 2\pi i\,\mathbb{Z}\to\mathcal{O}_{M}\to\mathcal{O}_{M}^{*}\to 0.
  3. H 0 ( 𝒪 U ) H 0 ( 𝒪 U * ) H 1 ( 2 π i | U ) \cdots\to H^{0}(\mathcal{O}_{U})\to H^{0}(\mathcal{O}_{U}^{*})\to H^{1}(2\pi i% \,\mathbb{Z}|_{U})\to\cdots
  4. H 1 ( 𝒪 M ) H 1 ( 𝒪 M * ) H 2 ( 2 π i ) . \cdots\to H^{1}(\mathcal{O}_{M})\to H^{1}(\mathcal{O}_{M}^{*})\to H^{2}(2\pi i% \,\mathbb{Z})\to\cdots.

Exponential_smoothing.html

  1. { x t } \{x_{t}\}
  2. t = 0 t=0
  3. { s t } \{s_{t}\}
  4. x x
  5. t = 0 t=0
  6. s 0 = x 0 s t = α x t + ( 1 - α ) s t - 1 , t > 0 \begin{aligned}\displaystyle s_{0}&\displaystyle=x_{0}\\ \displaystyle s_{t}&\displaystyle=\alpha x_{t}+(1-\alpha)s_{t-1},\ t>0\end{aligned}
  7. α \alpha
  8. 0 < α < 1 0<\alpha<1
  9. s t = 1 k n = 0 k - 1 x t - n = x t + x t - 1 + x t - 2 + + x t - k + 1 k = s t - 1 + x t - x t - k k , s_{t}=\frac{1}{k}\,\sum_{n=0}^{k-1}x_{t-n}=\frac{x_{t}+x_{t-1}+x_{t-2}+\cdots+% x_{t-k+1}}{k}=s_{t-1}+\frac{x_{t}-x_{t-k}}{k},
  10. { w 1 , w 2 , , w k } \{w_{1},w_{2},\dots,w_{k}\}
  11. n = 1 k w n = 1 \sum_{n=1}^{k}w_{n}=1
  12. s t = n = 1 k w n x t + 1 - n = w 1 x t + w 2 x t - 1 + + w k x t - k + 1 . s_{t}=\sum_{n=1}^{k}w_{n}x_{t+1-n}=w_{1}x_{t}+w_{2}x_{t-1}+\cdots+w_{k}x_{t-k+% 1}.
  13. s t = α x t + ( 1 - α ) s t - 1 s_{t}=\alpha\cdot x_{t}+(1-\alpha)\cdot s_{t-1}
  14. 1 - 1 / e 63.2 % 1-1/e\approx 63.2\,\%
  15. α = 1 - e - Δ T T C \alpha=1-e^{-\Delta T\over TC}
  16. Δ T \Delta T
  17. α Δ T T C \alpha\approx{\Delta T\over TC}
  18. e t = y t - y ^ t | t - 1 e_{t}=y_{t}-\hat{y}_{t|t-1}
  19. S S E = t = 1 T ( y t - y ^ t | t - 1 ) 2 = t = 1 T e t 2 SSE=\sum_{t=1}^{T}(y_{t}-\hat{y}_{t|t-1})^{2}=\sum_{t=1}^{T}e_{t}^{2}
  20. s t = α x t + ( 1 - α ) s t - 1 = α x t + α ( 1 - α ) x t - 1 + ( 1 - α ) 2 s t - 2 = α [ x t + ( 1 - α ) x t - 1 + ( 1 - α ) 2 x t - 2 + ( 1 - α ) 3 x t - 3 + + ( 1 - α ) t - 1 x 1 ] + ( 1 - α ) t x 0 . \begin{aligned}\displaystyle s_{t}&\displaystyle=\alpha x_{t}+(1-\alpha)s_{t-1% }\\ &\displaystyle=\alpha x_{t}+\alpha(1-\alpha)x_{t-1}+(1-\alpha)^{2}s_{t-2}\\ &\displaystyle=\alpha\left[x_{t}+(1-\alpha)x_{t-1}+(1-\alpha)^{2}x_{t-2}+(1-% \alpha)^{3}x_{t-3}+\cdots+(1-\alpha)^{t-1}x_{1}\right]+(1-\alpha)^{t}x_{0}.% \end{aligned}
  21. s 1 = x 1 b 1 = x 1 - x 0 \begin{aligned}\displaystyle s_{1}&\displaystyle=x_{1}\\ \displaystyle b_{1}&\displaystyle=x_{1}-x_{0}\\ \end{aligned}
  22. s t = α x t + ( 1 - α ) ( s t - 1 + b t - 1 ) b t = β ( s t - s t - 1 ) + ( 1 - β ) b t - 1 \begin{aligned}\displaystyle s_{t}&\displaystyle=\alpha x_{t}+(1-\alpha)(s_{t-% 1}+b_{t-1})\\ \displaystyle b_{t}&\displaystyle=\beta(s_{t}-s_{t-1})+(1-\beta)b_{t-1}\\ \end{aligned}
  23. F t + m = s t + m b t \begin{aligned}\displaystyle F_{t+m}&\displaystyle=s_{t}+mb_{t}\end{aligned}
  24. s 0 = x 0 s 0 ′′ = x 0 s t = α x t + ( 1 - α ) s t - 1 s t ′′ = α s t + ( 1 - α ) s t - 1 ′′ F t + m = a t + m b t , \begin{aligned}\displaystyle s^{\prime}_{0}&\displaystyle=x_{0}\\ \displaystyle s^{\prime\prime}_{0}&\displaystyle=x_{0}\\ \displaystyle s^{\prime}_{t}&\displaystyle=\alpha x_{t}+(1-\alpha)s^{\prime}_{% t-1}\\ \displaystyle s^{\prime\prime}_{t}&\displaystyle=\alpha s^{\prime}_{t}+(1-% \alpha)s^{\prime\prime}_{t-1}\\ \displaystyle F_{t+m}&\displaystyle=a_{t}+mb_{t},\end{aligned}
  25. a t \displaystyle a_{t}
  26. s 0 = x 0 s t = α x t c t - L + ( 1 - α ) ( s t - 1 + b t - 1 ) b t = β ( s t - s t - 1 ) + ( 1 - β ) b t - 1 c t = γ x t s t + ( 1 - γ ) c t - L F t + m = ( s t + m b t ) c t - L + 1 + ( m - 1 ) mod L , \begin{aligned}\displaystyle s_{0}&\displaystyle=x_{0}\\ \displaystyle s_{t}&\displaystyle=\alpha\frac{x_{t}}{c_{t-L}}+(1-\alpha)(s_{t-% 1}+b_{t-1})\\ \displaystyle b_{t}&\displaystyle=\beta(s_{t}-s_{t-1})+(1-\beta)b_{t-1}\\ \displaystyle c_{t}&\displaystyle=\gamma\frac{x_{t}}{s_{t}}+(1-\gamma)c_{t-L}% \\ \displaystyle F_{t+m}&\displaystyle=(s_{t}+mb_{t})c_{t-L+1+(m-1)\mod L},\end{aligned}
  27. b 0 \displaystyle b_{0}
  28. c i \displaystyle c_{i}
  29. A j = i = 1 L x L ( j - 1 ) + i L j = 1 , 2 , , N \begin{aligned}\displaystyle A_{j}&\displaystyle=\frac{\sum_{i=1}^{L}x_{L(j-1)% +i}}{L}\quad\forall j&\displaystyle=1,2,\ldots,N\end{aligned}

Exponential_stability.html

  1. y ( t ) = e - t 5 y(t)=e^{-\frac{t}{5}}
  2. y ( t ) = e - t 5 sin ( t ) y(t)=e^{-\frac{t}{5}}\sin(t)

Exsecant.html

  1. exsec ( θ ) = sec ( θ ) - 1. \operatorname{exsec}(\theta)=\sec(\theta)-1.\,
  2. excsc ( θ ) = exsec ( π / 2 - θ ) = csc ( θ ) - 1. \operatorname{excsc}(\theta)=\operatorname{exsec}(\pi/2-\theta)=\csc(\theta)-1.\!
  3. exsec ( θ ) = 1 - cos ( θ ) cos ( θ ) = versin ( θ ) cos ( θ ) = 2 sin 2 ( θ / 2 ) sec ( θ ) . \operatorname{exsec}(\theta)=\frac{1-\cos(\theta)}{\cos(\theta)}=\frac{% \operatorname{versin}(\theta)}{\cos(\theta)}=2\sin^{2}(\theta/2)\sec(\theta).
  4. O E ¯ \overline{OE}
  5. D E ¯ \overline{DE}

Extended_Hückel_method.html

  1. H i j = K S i j H i i + H j j 2 H_{ij}=KS_{ij}\dfrac{H_{ii}+H_{jj}}{2}
  2. H i j = K S i j H_{ij}=KS_{ij}

Extended_X-ray_absorption_fine_structure.html

  1. μ ( E ) \mu(E)
  2. μ \mu
  3. x x
  4. I t = I 0 e - μ x I_{t}=I_{0}e^{-\mu x}
  5. μ = - l n ( I t / I 0 ) x \mu=\frac{-ln({I}_{t}/{I}_{0})}{x}

Extensive-form_game.html

  1. Γ = 𝒦 , 𝐇 , [ ( 𝐇 i ) i ] , { A ( H ) } H 𝐇 , a , ρ , u \Gamma=\langle\mathcal{K},\mathbf{H},[(\mathbf{H}_{i})_{i\in\mathcal{I}}],\{A(% H)\}_{H\in\mathbf{H}},a,\rho,u\rangle
  2. 𝒦 = V , v 0 , T , p \mathcal{K}=\langle V,v^{0},T,p\rangle
  3. V V
  4. v 0 V v^{0}\in V
  5. T V T\subset V
  6. D = V T D=V\setminus T
  7. p : V D p:V\rightarrow D
  8. 𝐇 \mathbf{H}
  9. D D
  10. A ( H ) A(H)
  11. H 𝐇 H\in\mathbf{H}
  12. 𝒜 \mathcal{A}
  13. a : V { v 0 } 𝒜 a:V\setminus\{v^{0}\}\rightarrow\mathcal{A}
  14. v v
  15. a ( v ) a(v)
  16. H 𝐇 , v H \forall H\in\mathbf{H},\forall v\in H
  17. a v : s ( v ) A ( H ) a_{v}:s(v)\rightarrow A(H)
  18. a a
  19. s ( v ) s(v)
  20. s ( v ) s(v)
  21. = { 1 , , I } \mathcal{I}=\{1,...,I\}
  22. 0
  23. ( 𝐇 i ) i I { 0 } (\mathbf{H}_{i})_{i\in I\cup\{0\}}
  24. 𝐇 \mathbf{H}
  25. ι ( v ) = ι ( H ) \iota(v)=\iota(H)
  26. v H v\in H
  27. ρ = { ρ H : A ( H ) [ 0 , 1 ] | H 𝐇 0 } \rho=\{\rho_{H}:A(H)\rightarrow[0,1]|H\in\mathbf{H}_{0}\}
  28. u = ( u i ) i : T u=(u_{i})_{i\in\mathcal{I}}:T\rightarrow\Re^{\mathcal{I}}
  29. q 2 ( q 1 ) = ( 5000 - q 1 - c 2 ) / 2 q2(q1)=(5000-q1-c2)/2
  30. q 1 * = ( 5000 + c 2 - 2 c 1 ) / 2 q1*=(5000+c2-2c1)/2
  31. q 2 * = ( 5000 + 2 c 1 - 3 c 2 ) / 4 q2*=(5000+2c1-3c2)/4

External_sorting.html

  1. log n \log n

Extinction_(astronomy).html

  1. N H A ( V ) 1.8 × 10 21 atoms cm mag - 2 - 1 \frac{N_{H}}{A(V)}\approx 1.8\times 10^{21}~{}\mbox{atoms}~{}~{}\mbox{cm}~{}^{% -2}~{}\mbox{mag}~{}^{-1}
  2. E B - V E_{B-V}
  3. E B - V = ( B - V ) observed - ( B - V ) intrinsic E_{B-V}=(B-V)_{\textrm{observed}}-(B-V)_{\textrm{intrinsic}}\,

Édouard_Goursat.html

  1. S ω = T d ω \int_{S}\omega=\int_{T}d\omega
  2. ω \omega
  3. ω \omega
  4. d ω = 0 d\omega=0
  5. η \eta
  6. d η = ω d\eta=\omega
  7. ω \omega

F-coalgebra.html

  1. F F
  2. F F
  3. F F
  4. F F
  5. F F
  6. F F
  7. F F
  8. 𝒞 \mathcal{C}
  9. F : 𝒞 𝒞 F:\mathcal{C}\longrightarrow\mathcal{C}
  10. A A
  11. 𝒞 \mathcal{C}
  12. α : A F A \alpha:A\longrightarrow FA
  13. ( A , α ) (A,\alpha)
  14. F F
  15. ( A , α ) (A,\alpha)
  16. F F
  17. ( B , β ) (B,\beta)
  18. f : A B f:A\longrightarrow B
  19. 𝒞 \mathcal{C}
  20. F f α = β f Ff\circ\alpha=\beta\circ f
  21. F F
  22. F : 𝐒𝐞𝐭 𝐒𝐞𝐭 F:\mathbf{Set}\longrightarrow\mathbf{Set}
  23. X X
  24. X × A { 1 } X\times A\cup\{1\}
  25. F F
  26. α : X X × A { 1 } = F X \alpha:X\longrightarrow X\times A\cup\{1\}=FX
  27. A A
  28. X X
  29. α \alpha
  30. A A
  31. { 1 } \{1\}
  32. X f 1 × f 2 × × f n X\rightarrow f_{1}\times f_{2}\times\ldots\times f_{n}
  33. X f 1 , X f 2 X f n X\rightarrow f_{1},\,X\rightarrow f_{2}\,\ldots\,X\rightarrow f_{n}
  34. X X A 1 × × A n X\rightarrow X^{A_{1}\times\ldots\times A_{n}}
  35. F F

F2.html

  1. 𝔽 2 \mathbb{F}_{2}

Factor_graph.html

  1. g ( X 1 , X 2 , , X n ) g(X_{1},X_{2},\dots,X_{n})
  2. g ( X 1 , X 2 , , X n ) = j = 1 m f j ( S j ) , g(X_{1},X_{2},\dots,X_{n})=\prod_{j=1}^{m}f_{j}(S_{j}),
  3. S j { X 1 , X 2 , , X n } S_{j}\subseteq\{X_{1},X_{2},\dots,X_{n}\}
  4. G = ( X , F , E ) G=(X,F,E)
  5. X = { X 1 , X 2 , , X n } X=\{X_{1},X_{2},\dots,X_{n}\}
  6. F = { f 1 , f 2 , , f m } F=\{f_{1},f_{2},\dots,f_{m}\}
  7. E E
  8. f j f_{j}
  9. X k X_{k}
  10. X k S j X_{k}\in S_{j}
  11. g ( X 1 , X 2 , , X n ) g(X_{1},X_{2},\dots,X_{n})\in\mathbb{R}
  12. g ( X 1 , X 2 , , X n ) g(X_{1},X_{2},\dots,X_{n})
  13. g ( X 1 , X 2 , X 3 ) = f 1 ( X 1 ) f 2 ( X 1 , X 2 ) f 3 ( X 1 , X 2 ) f 4 ( X 2 , X 3 ) g(X_{1},X_{2},X_{3})=f_{1}(X_{1})f_{2}(X_{1},X_{2})f_{3}(X_{1},X_{2})f_{4}(X_{% 2},X_{3})
  14. f 2 ( X 1 , X 2 ) f 3 ( X 1 , X 2 ) f_{2}(X_{1},X_{2})f_{3}(X_{1},X_{2})
  15. X k X_{k}
  16. g k ( X k ) = X k ¯ g ( X 1 , X 2 , , X n ) g_{k}(X_{k})=\sum_{X_{\bar{k}}}g(X_{1},X_{2},\dots,X_{n})
  17. X k ¯ X_{\bar{k}}
  18. X k X_{k}
  19. g ( X 1 , X 2 , , X n ) g(X_{1},X_{2},\dots,X_{n})

Factor_theorem.html

  1. f ( x ) f(x)
  2. ( x - k ) (x-k)
  3. f ( k ) = 0 f(k)=0
  4. k k
  5. a a
  6. f f
  7. ( x - a ) (x-a)
  8. f ( x ) f(x)
  9. g ( x ) = f ( x ) / ( x - a ) g(x)=f(x)\big/(x-a)
  10. x a x\neq a
  11. f ( x ) = 0 f(x)=0
  12. g ( x ) = 0 g(x)=0
  13. g g
  14. f f
  15. g g
  16. x 3 + 7 x 2 + 8 x + 2. x^{3}+7x^{2}+8x+2.
  17. ( x - 1 ) (x-1)
  18. x = 1 x=1
  19. x 3 + 7 x 2 + 8 x + 2 = ( 1 ) 3 + 7 ( 1 ) 2 + 8 ( 1 ) + 2 x^{3}+7x^{2}+8x+2=(1)^{3}+7(1)^{2}+8(1)+2
  20. = 1 + 7 + 8 + 2 =1+7+8+2
  21. = 18. =18.
  22. ( x - 1 ) (x-1)
  23. x 3 + 7 x 2 + 8 x + 2 x^{3}+7x^{2}+8x+2
  24. ( x + 1 ) (x+1)
  25. x = - 1 x=-1
  26. ( - 1 ) 3 + 7 ( - 1 ) 2 + 8 ( - 1 ) + 2. (-1)^{3}+7(-1)^{2}+8(-1)+2.
  27. 0
  28. x - ( - 1 ) x-(-1)
  29. x + 1 x+1
  30. - 1 -1
  31. x 3 + 7 x 2 + 8 x + 2. x^{3}+7x^{2}+8x+2.
  32. x 3 + 7 x 2 + 8 x + 2 x^{3}+7x^{2}+8x+2
  33. ( x + 1 ) (x+1)
  34. x 3 + 7 x 2 + 8 x + 2 x + 1 = x 2 + 6 x + 2 {x^{3}+7x^{2}+8x+2\over x+1}=x^{2}+6x+2
  35. ( x + 1 ) (x+1)
  36. x 2 + 6 x + 2 x^{2}+6x+2
  37. x 3 + 7 x 2 + 8 x + 2. x^{3}+7x^{2}+8x+2.

Factorial_experiment.html

  1. - -
  2. + +
  3. - - --
  4. + - +-
  5. - + -+
  6. + + ++

Failure_rate.html

  1. λ ( t ) \lambda(t)
  2. t t
  3. R ( t ) R(t)
  4. t t
  5. λ ( t ) = f ( t ) R ( t ) \lambda(t)=\frac{f(t)}{R(t)}
  6. f ( t ) f(t)
  7. R ( t ) = 1 - F ( t ) R(t)=1-F(t)
  8. λ ( t ) = R ( t 1 ) - R ( t 2 ) ( t 2 - t 1 ) R ( t 1 ) = R ( t ) - R ( t + t ) t R ( t ) \lambda(t)=\frac{R(t_{1})-R(t_{2})}{(t_{2}-t_{1})\cdot R(t_{1})}=\frac{R(t)-R(% t+\triangle t)}{\triangle t\cdot R(t)}\!
  9. ( t 2 - t 1 ) (t_{2}-t_{1})
  10. t 1 t_{1}
  11. t t
  12. t 2 t_{2}
  13. Δ t \Delta t
  14. ( t 2 - t 1 ) (t_{2}-t_{1})
  15. R ( t ) R(t)
  16. λ ( t ) \lambda(t)
  17. t t
  18. h ( t ) h(t)
  19. Δ t \scriptstyle\Delta t
  20. h ( t ) = lim Δ t 0 R ( t ) - R ( t + Δ t ) Δ t R ( t ) . h(t)=\lim_{\Delta t\to 0}\frac{R(t)-R(t+\Delta t)}{\Delta t\cdot R(t)}.
  21. F ( t ) \scriptstyle F(t)
  22. Pr ( T t ) = F ( t ) = 1 - R ( t ) , t 0. \operatorname{Pr}(T\leq t)=F(t)=1-R(t),\quad t\geq 0.\!
  23. T {T}
  24. F ( t ) = 0 t f ( τ ) d τ . F(t)=\int_{0}^{t}f(\tau)\,d\tau.\!
  25. h ( t ) = f ( t ) 1 - F ( t ) = f ( t ) R ( t ) . h(t)=\frac{f(t)}{1-F(t)}=\frac{f(t)}{R(t)}.
  26. F ( t ) = 0 t λ e - λ τ d τ = 1 - e - λ t , F(t)=\int_{0}^{t}\lambda e^{-\lambda\tau}\,d\tau=1-e^{-\lambda t},\!
  27. h ( t ) = f ( t ) R ( t ) = λ e - λ t e - λ t = λ . h(t)=\frac{f(t)}{R(t)}=\frac{\lambda e^{-\lambda t}}{e^{-\lambda t}}=\lambda.
  28. 6 failures 7502 hours = 0.0007998 failures hour = 799.8 × 10 - 6 failures hour , \frac{6\,\text{ failures}}{7502\,\text{ hours}}=0.0007998\frac{\,\text{% failures}}{\,\text{hour}}=799.8\times 10^{-6}\frac{\,\text{failures}}{\,\text{% hour}},

Faraday_cup.html

  1. N t = I e \frac{N}{t}=\frac{I}{e}
  2. S F = π D F 2 / 4 S_{F}=\pi D^{2}_{F}/4
  3. B e s B_{es}
  4. U e s U_{es}
  5. R F R_{F}
  6. U g ( t ) U_{g}(t)
  7. C F C_{F}
  8. R F R_{F}
  9. h D F h\geq D_{F}
  10. h λ i h\ll\lambda_{i}
  11. λ i \lambda_{i}
  12. R F R_{F}
  13. i Σ ( U g ) = i i ( U g ) - C F d U g d t i_{\Sigma}(U_{g})=i_{i}(U_{g})-C_{F}\frac{dU_{g}}{dt}
  14. C F C_{F}
  15. R F R_{F}
  16. i Σ i_{\Sigma}
  17. R F R_{F}
  18. i i i_{i}
  19. i c ( U g ) = - C F ( d U g / d t ) i_{c}(U_{g})=-C_{F}(dU_{g}/dt)
  20. C F C_{F}
  21. U g U_{g}
  22. i c ( U g ) i_{c}(U_{g})
  23. i Σ ( U g ) i_{\Sigma}(U_{g})
  24. i i ( U g ) i_{i}(U_{g})
  25. d i i di_{i}
  26. d n ( v ) dn(v)
  27. v v
  28. v + d v v+dv
  29. S F S_{F}
  30. d i i = e Z i S F v d n ( v ) di_{i}=eZ_{i}S_{F}vdn(v)
  31. d n ( v ) = n f ( v ) d v dn(v)=nf(v)dv
  32. e e
  33. Z i Z_{i}
  34. f ( v ) f(v)
  35. v v
  36. U g U_{g}
  37. i i ( U g ) = e Z i n i S F 2 e Z i U g / M i f ( v ) v d v i_{i}(U_{g})=eZ_{i}n_{i}S_{F}\int\limits_{\sqrt{2eZ_{i}U_{g}/M_{i}}}^{\infty}f% (v)vdv
  38. M i v i , s 2 / 2 = e Z i U g M_{i}v^{2}_{i,s}/2=eZ_{i}U_{g}
  39. v i , s v_{i,s}
  40. U g U_{g}
  41. M i M_{i}
  42. U g U_{g}
  43. d i i ( U g ) d U g = - e n i S F e Z i M i f ( 2 e Z i U g / M i ) \frac{di_{i}(U_{g})}{dU_{g}}=-en_{i}S_{F}\frac{eZ_{i}}{M_{i}}f\left(\sqrt{2eZ_% {i}U_{g}/M_{i}}\right)
  44. - n i S F ( e Z i / M i ) = C i -n_{i}S_{F}(eZ_{i}/M_{i})=C_{i}
  45. v i \langle v_{i}\rangle
  46. i \langle\mathcal{E}_{i}\rangle
  47. v i = 1.389 × 10 6 Z i M A 0 i i ( U g ) d U g ( 0 i i U g d U g ) - 1 \langle v_{i}\rangle=1.389\times 10^{6}\sqrt{\frac{Z_{i}}{M_{A}}}\int\limits_{% 0}^{\infty}i^{\prime}_{i}(U_{g})dU_{g}\left(\int\limits_{0}^{\infty}\frac{i^{% \prime}_{i}}{\sqrt{U_{g}}}dU_{g}\right)^{-1}
  48. i = 0 i i ( U g ) U g d U g ( 0 i i U g d U g ) - 1 \langle\mathcal{E}_{i}\rangle=\int\limits_{0}^{\infty}i^{\prime}_{i}(U_{g})% \sqrt{U_{g}}dU_{g}\left(\int\limits_{0}^{\infty}\frac{i^{\prime}_{i}}{\sqrt{U_% {g}}}dU_{g}\right)^{-1}
  49. M A M_{A}
  50. n i n_{i}
  51. n i = i i ( 0 ) e Z i v i S F n_{i}=\frac{i_{i}(0)}{eZ_{i}\langle v_{i}\rangle S_{F}}
  52. U g = 0 U_{g}=0
  53. 0 f ( v ) v d v = v \int\limits_{0}^{\infty}f(v)vdv=\langle v\rangle
  54. 0 f ( v ) d v = 1 \int\limits_{0}^{\infty}f(v)dv=1
  55. i i ( V ) i_{i}(V)
  56. i i ( V ) i^{\prime}_{i}(V)
  57. S F = 0.5 c m 2 S_{F}=0.5cm^{2}
  58. U e s = - 170 V U_{es}=-170V

Fast_Kalman_filter.html

  1. [ A B C D ] - 1 = [ A - 1 + A - 1 B ( D - C A - 1 B ) - 1 C A - 1 - A - 1 B ( D - C A - 1 B ) - 1 - ( D - C A - 1 B ) - 1 C A - 1 ( D - C A - 1 B ) - 1 ] \begin{bmatrix}A&B\\ C&D\end{bmatrix}^{-1}=\begin{bmatrix}A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1}&-A% ^{-1}B(D-CA^{-1}B)^{-1}\\ -(D-CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{bmatrix}
  2. A = A=
  3. ( D - C A - 1 B ) = (D-CA^{-1}B)=
  4. A A

Fast_wavelet_transform.html

  1. V J V_{J}
  2. s n ( J ) := 2 J f ( t ) , ϕ ( 2 J t - n ) , s^{(J)}_{n}:=2^{J}\langle f(t),\phi(2^{J}t-n)\rangle,
  3. ϕ \phi
  4. P J [ f ] ( x ) := n \Z s n ( J ) ϕ ( 2 J x - n ) P_{J}[f](x):=\sum_{n\in\Z}s^{(J)}_{n}\,\phi(2^{J}x-n)
  5. V J V_{J}
  6. a = ( a - N , , a 0 , , a N ) a=(a_{-N},\dots,a_{0},\dots,a_{N})
  7. a ( z ) = n = - N N a n z - n a(z)=\sum_{n=-N}^{N}a_{n}z^{-n}
  8. b = ( b - N , , b 0 , , b N ) b=(b_{-N},\dots,b_{0},\dots,b_{N})
  9. b ( z ) = n = - N N b n z - n b(z)=\sum_{n=-N}^{N}b_{n}z^{-n}
  10. d n ( k ) d^{(k)}_{n}
  11. s ( J ) s^{(J)}
  12. s ( J ) s^{(J)}
  13. s n ( k ) := 1 2 m = - N N a m s 2 n + m ( k + 1 ) s^{(k)}_{n}:=\frac{1}{2}\sum_{m=-N}^{N}a_{m}s^{(k+1)}_{2n+m}
  14. s ( k ) ( z ) := ( 2 ) ( a * ( z ) s ( k + 1 ) ( z ) ) s^{(k)}(z):=(\downarrow 2)(a^{*}(z)\cdot s^{(k+1)}(z))
  15. d n ( k ) := 1 2 m = - N N b m s 2 n + m ( k + 1 ) d^{(k)}_{n}:=\frac{1}{2}\sum_{m=-N}^{N}b_{m}s^{(k+1)}_{2n+m}
  16. d ( k ) ( z ) := ( 2 ) ( b * ( z ) s ( k + 1 ) ( z ) ) d^{(k)}(z):=(\downarrow 2)(b^{*}(z)\cdot s^{(k+1)}(z))
  17. n \Z n\in\Z
  18. ( 2 ) (\downarrow 2)
  19. ( 2 ) ( c ( z ) ) = k \Z c 2 k z - k (\downarrow 2)(c(z))=\sum_{k\in\Z}c_{2k}z^{-k}
  20. a * ( z ) a^{*}(z)
  21. a * ( z ) = n = - N N a - n * z - n a^{*}(z)=\sum_{n=-N}^{N}a_{-n}^{*}z^{-n}
  22. P k [ f ] ( x ) := n \Z s n ( k ) ϕ ( 2 k x - n ) P_{k}[f](x):=\sum_{n\in\Z}s^{(k)}_{n}\,\phi(2^{k}x-n)
  23. P J [ f ] ( x ) P_{J}[f](x)
  24. V k V_{k}
  25. P J [ f ] ( x ) = P k [ f ] ( x ) + D k [ f ] ( x ) + + D J - 1 [ f ] ( x ) P_{J}[f](x)=P_{k}[f](x)+D_{k}[f](x)+\dots+D_{J-1}[f](x)
  26. D k [ f ] ( x ) := n \Z d n ( k ) ψ ( 2 k x - n ) D_{k}[f](x):=\sum_{n\in\Z}d^{(k)}_{n}\,\psi(2^{k}x-n)
  27. ψ \psi
  28. s ( M ) s^{(M)}
  29. d ( k ) d^{(k)}
  30. s n ( k + 1 ) := k = - N N a k s 2 n - k ( k ) + k = - N N b k d 2 n - k ( k ) s^{(k+1)}_{n}:=\sum_{k=-N}^{N}a_{k}s^{(k)}_{2n-k}+\sum_{k=-N}^{N}b_{k}d^{(k)}_% {2n-k}
  31. s ( k + 1 ) ( z ) = a ( z ) ( 2 ) ( s ( k ) ( z ) ) + b ( z ) ( 2 ) ( d ( k ) ( z ) ) s^{(k+1)}(z)=a(z)\cdot(\uparrow 2)(s^{(k)}(z))+b(z)\cdot(\uparrow 2)(d^{(k)}(z))
  32. n \Z n\in\Z
  33. ( 2 ) (\uparrow 2)
  34. ( 2 ) ( c ( z ) ) := n \Z c n z - 2 n (\uparrow 2)(c(z)):=\sum_{n\in\Z}c_{n}z^{-2n}
  35. 2 ( \Z , \R ) \ell^{2}(\Z,\R)
  36. ( 2 ) (\downarrow 2)

FastICA.html

  1. 𝐱 \mathbf{x}
  2. 𝐱 \mathbf{x}
  3. 𝐱 \mathbf{x}
  4. 𝐱 𝐱 - E { 𝐱 } \mathbf{x}\leftarrow\mathbf{x}-E\left\{\mathbf{x}\right\}
  5. 𝐱 ~ \widetilde{\mathbf{x}}
  6. E { 𝐱 ~ 𝐱 ~ T } = 𝐈 E\left\{\widetilde{\mathbf{x}}\widetilde{\mathbf{x}}^{T}\right\}=\mathbf{I}
  7. E { 𝐱𝐱 T } = 𝐄𝐃𝐄 T E\left\{\mathbf{x}\mathbf{x}^{T}\right\}=\mathbf{E}\mathbf{D}\mathbf{E}^{T}
  8. 𝐄 \mathbf{E}
  9. 𝐃 \mathbf{D}
  10. 𝐱 𝐄𝐃 - 1 / 2 𝐄 T 𝐱 \mathbf{x}\leftarrow\mathbf{E}\mathbf{D}^{-1/2}\mathbf{E}^{T}\mathbf{x}
  11. 𝐰 \mathbf{w}
  12. 𝐰 T 𝐱 \mathbf{w}^{T}\mathbf{x}
  13. 𝐱 \mathbf{x}
  14. g ( u ) g(u)
  15. f ( u ) f(u)
  16. f f
  17. g g
  18. g {g}^{\prime}
  19. f ( u ) = log cosh ( u ) ; g ( u ) = tanh ( u ) ; g ( u ) = 1 - tanh 2 ( u ) f(u)=\log\cosh(u);\quad g(u)=\tanh(u);\quad{g}^{\prime}(u)=1-\tanh^{2}(u)
  20. f ( u ) = - e - u 2 / 2 ; g ( u ) = u e - u 2 / 2 ; g ( u ) = ( 1 - u 2 ) e - u 2 / 2 f(u)=-e^{-u^{2}/2};\quad g(u)=ue^{-u^{2}/2};\quad{g}^{\prime}(u)=(1-u^{2})e^{-% u^{2}/2}
  21. 𝐰 \mathbf{w}
  22. 𝐰 + E { 𝐱 g ( 𝐰 T 𝐱 ) T } - E { g ( 𝐰 T 𝐱 ) } 𝐰 \mathbf{w}^{+}\leftarrow E\left\{\mathbf{x}g(\mathbf{w}^{T}\mathbf{x})^{T}% \right\}-E\left\{g^{\prime}(\mathbf{w}^{T}\mathbf{x})\right\}\mathbf{w}
  23. E { } E\left\{...\right\}
  24. 𝐗 \mathbf{X}
  25. 𝐰 𝐰 + / 𝐰 + \mathbf{w}\leftarrow\mathbf{w}^{+}/\|\mathbf{w}^{+}\|
  26. 𝟏 \mathbf{1}
  27. C C
  28. 𝐗 N × M \mathbf{X}\in\mathbb{R}^{N\times M}
  29. C N C<=N
  30. 𝐖 C × N \mathbf{W}\in\mathbb{R}^{C\times N}
  31. 𝐒 C × M \mathbf{S}\in\mathbb{R}^{C\times M}
  32. 𝐰 𝐩 \mathbf{w_{p}}\leftarrow
  33. 𝐰 𝐩 \mathbf{w_{p}}
  34. 𝐰 𝐩 1 M 𝐗 g ( 𝐰 𝐩 T 𝐗 ) T - 1 M g ( 𝐰 𝐩 T 𝐗 ) 𝟏 𝐰 𝐩 \mathbf{w_{p}}\leftarrow\frac{1}{M}\mathbf{X}g(\mathbf{w_{p}}^{T}\mathbf{X})^{% T}-\frac{1}{M}g^{\prime}(\mathbf{w_{p}}^{T}\mathbf{X})\mathbf{1}\mathbf{w_{p}}
  35. 𝐰 𝐩 𝐰 𝐩 - j = 1 p - 1 𝐰 𝐣 𝐰 𝐩 T 𝐰 𝐣 \mathbf{w_{p}}\leftarrow\mathbf{w_{p}}-\sum_{j=1}^{p-1}\mathbf{w_{j}}\mathbf{w% _{p}}^{T}\mathbf{w_{j}}
  36. 𝐰 𝐩 𝐰 𝐩 𝐰 𝐩 \mathbf{w_{p}}\leftarrow\frac{\mathbf{w_{p}}}{\|\mathbf{w_{p}}\|}
  37. 𝐖 = [ 𝐰 𝟏 𝐰 𝐂 ] T \mathbf{W}=\begin{bmatrix}\mathbf{w_{1}}\\ \vdots\\ \mathbf{w_{C}}\end{bmatrix}^{T}
  38. 𝐒 = 𝐖𝐗 \mathbf{S}=\mathbf{W}\mathbf{X}

Federated_database_system.html

  1. n ( n - 1 ) 2 n(n-1)\over 2

Feedback_vertex_set.html

  1. G = ( V , E ) G=(V,E)
  2. k k
  3. X V X\subseteq V
  4. | X | k |X|\leq k
  5. G G
  6. X X
  7. G [ V X ] G[V\setminus X]
  8. X X
  9. G G

Feedforward_neural_network.html

  1. y = 1 1 + e - x y=\frac{1}{1+e^{-x}}
  2. y = y ( 1 - y ) y^{\prime}=y(1-y)
  3. d f d X \frac{df}{dX}

Fermat's_theorem_on_sums_of_two_squares.html

  1. p = x 2 + y 2 , p=x^{2}+y^{2},\,
  2. p 1 ( mod 4 ) . p\equiv 1\;\;(\mathop{{\rm mod}}4).
  3. 5 = 1 2 + 2 2 , 13 = 2 2 + 3 2 , 17 = 1 2 + 4 2 , 29 = 2 2 + 5 2 , 37 = 1 2 + 6 2 , 41 = 4 2 + 5 2 . 5=1^{2}+2^{2},\quad 13=2^{2}+3^{2},\quad 17=1^{2}+4^{2},\quad 29=2^{2}+5^{2},% \quad 37=1^{2}+6^{2},\quad 41=4^{2}+5^{2}.
  4. p p
  5. p = x 2 + 2 y 2 p 1 or p 3 ( mod 8 ) , p=x^{2}+2y^{2}\Leftrightarrow p\equiv 1\mbox{ or }~{}p\equiv 3\;\;(\mathop{{% \rm mod}}8),
  6. p = x 2 + 3 y 2 p 1 ( mod 3 ) . p=x^{2}+3y^{2}\Leftrightarrow p\equiv 1\;\;(\mathop{{\rm mod}}3).
  7. p = x 2 + 5 y 2 p 1 or p 9 ( mod 20 ) , p=x^{2}+5y^{2}\Leftrightarrow p\equiv 1\mbox{ or }~{}p\equiv 9\;\;(\mathop{{% \rm mod}}20),
  8. 2 p = x 2 + 5 y 2 p 3 or p 7 ( mod 20 ) . 2p=x^{2}+5y^{2}\Leftrightarrow p\equiv 3\mbox{ or }~{}p\equiv 7\;\;(\mathop{{% \rm mod}}20).

Fermion_doubling.html

  1. S = d d x ψ ¯ ( / + m ) ψ . S=\int d^{d}x\;\bar{\psi}(\partial\!\!\!/+m)\psi\;.
  2. γ μ a μ = a / \gamma_{\mu}a^{\mu}=a\!\!\!/
  3. S = a d x , μ 1 2 a ( ψ ¯ x γ μ ψ x + μ ^ - ψ ¯ x + μ ^ γ μ ψ x ) + a d x m ψ ¯ x ψ x , S=a^{d}\sum_{x,\mu}\frac{1}{2a}(\bar{\psi}_{x}\gamma_{\mu}\psi_{x+\hat{\mu}}-% \bar{\psi}_{x+\hat{\mu}}\gamma_{\mu}\psi_{x})+a^{d}\sum_{x}m\bar{\psi}_{x}\psi% _{x}\;,
  4. μ ^ \hat{\mu}
  5. S - 1 ( p ) = m + i a μ γ μ sin ( p μ a ) . S^{-1}(p)=m+\frac{i}{a}\sum_{\mu}\gamma_{\mu}\sin(p_{\mu}a)\;.

Feynman_slash_notation.html

  1. A / = def γ μ A μ A\!\!\!/\ \stackrel{\mathrm{def}}{=}\ \gamma^{\mu}A_{\mu}
  2. a μ a_{\mu}
  3. b μ b_{\mu}
  4. a / a / = a μ a μ = a 2 a\!\!\!/a\!\!\!/=a^{\mu}a_{\mu}=a^{2}
  5. a / b / + b / a / = 2 a b a\!\!\!/b\!\!\!/+b\!\!\!/a\!\!\!/=2a\cdot b\,
  6. / 2 = 2 . \partial\!\!\!/^{2}=\partial^{2}.
  7. tr ( a / b / ) = 4 a b \operatorname{tr}(a\!\!\!/b\!\!\!/)=4a\cdot b
  8. tr ( a / b / c / d / ) = 4 [ ( a b ) ( c d ) - ( a c ) ( b d ) + ( a d ) ( b c ) ] \operatorname{tr}(a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/)=4\left[(a\cdot b)(c\cdot d% )-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)\right]
  9. tr ( γ 5 a / b / c / d / ) = 4 i ϵ μ ν λ σ a μ b ν c λ d σ \operatorname{tr}(\gamma_{5}a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/)=4i\epsilon_{\mu% \nu\lambda\sigma}a^{\mu}b^{\nu}c^{\lambda}d^{\sigma}
  10. γ μ a / γ μ = - 2 a / \gamma_{\mu}a\!\!\!/\gamma^{\mu}=-2a\!\!\!/
  11. γ μ a / b / γ μ = 4 a b \gamma_{\mu}a\!\!\!/b\!\!\!/\gamma^{\mu}=4a\cdot b\,
  12. γ μ a / b / c / γ μ = - 2 c / b / a / \gamma_{\mu}a\!\!\!/b\!\!\!/c\!\!\!/\gamma^{\mu}=-2c\!\!\!/b\!\!\!/a\!\!\!/\,
  13. ϵ μ ν λ σ \epsilon_{\mu\nu\lambda\sigma}\,
  14. γ \gamma\,
  15. γ 0 = ( I 0 0 - I ) , γ i = ( 0 σ i - σ i 0 ) \gamma^{0}=\begin{pmatrix}I&0\\ 0&-I\end{pmatrix},\quad\gamma^{i}=\begin{pmatrix}0&\sigma^{i}\\ -\sigma^{i}&0\end{pmatrix}\,
  16. p μ = ( E , - p x , - p y , - p z ) p_{\mu}=\left(E,-p_{x},-p_{y},-p_{z}\right)\,
  17. p / = γ μ p μ = γ 0 p 0 - γ i p i = [ p 0 0 0 - p 0 ] - [ 0 σ i p i - σ i p i 0 ] = [ E - σ p σ p - E ] \begin{aligned}\displaystyle p\!\!/&\displaystyle=\gamma^{\mu}p_{\mu}=\gamma^{% 0}p_{0}-\gamma^{i}p_{i}\\ &\displaystyle=\begin{bmatrix}p_{0}&0\\ 0&-p_{0}\end{bmatrix}-\begin{bmatrix}0&\sigma^{i}p_{i}\\ -\sigma^{i}p_{i}&0\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}E&-\sigma\cdot\vec{p}\\ \sigma\cdot\vec{p}&-E\end{bmatrix}\end{aligned}

Fibered_knot.html

  1. K K
  2. S 3 S^{3}
  3. F t F_{t}
  4. K K
  5. t t
  6. S 1 S^{1}
  7. s s
  8. t t
  9. F s F_{s}
  10. F t F_{t}
  11. K K
  12. z 2 + w 3 z^{2}+w^{3}
  13. z 2 + w 2 z^{2}+w^{2}
  14. S 3 S^{3}

Fibonacci_word.html

  1. S 0 S_{0}
  2. S 1 S_{1}
  3. S n = S n - 1 S n - 2 S_{n}=S_{n-1}S_{n-2}
  4. S S_{\infty}
  5. S 0 S_{0}
  6. S 1 S_{1}
  7. S 2 S_{2}
  8. S 3 S_{3}
  9. S 4 S_{4}
  10. S 5 S_{5}
  11. 2 + n φ - ( n + 1 ) φ 2+\left\lfloor{{n}\,\varphi}\right\rfloor-\left\lfloor{\left({n+1}\right)\,% \varphi}\right\rfloor
  12. φ \varphi
  13. x \left\lfloor x\right\rfloor
  14. n φ - ( n - 1 ) φ - 1 \left\lfloor{{n}\,\varphi}\right\rfloor-\left\lfloor{\left({n-1}\right)\,% \varphi}\right\rfloor-1
  15. φ \varphi
  16. x \left\lfloor x\right\rfloor
  17. S 4 S_{4}
  18. 1 / ϕ 2 1/\phi^{2}
  19. u u
  20. u R u^{R}
  21. u u
  22. u u
  23. S n + 1 = S n S n - 1 S_{n+1}=S_{n}S_{n-1}
  24. S n - 1 S n S_{n-1}S_{n}
  25. ϕ \phi
  26. ϕ - 1 \phi-1
  27. n ϕ 2 \lfloor n\phi^{2}\rfloor
  28. n ϕ \lfloor n\phi\rfloor
  29. 2 + ϕ = 3.618 2+\phi=3.618

Fibrant_object.html

  1. c c
  2. c \varnothing\to c
  3. c c

Field_equation.html

  1. 2 Φ = - 4 π G ρ g , 2 Φ = - ρ e ϵ 0 {\nabla}^{2}\Phi=-4\pi G\rho_{g}\,,\quad{\nabla}^{2}\Phi=-{\rho_{e}\over% \epsilon_{0}}
  2. g d S = - 4 π G m g = - 4 π G ρ m \iint{g}\cdot{\rm d}{S}=-4\pi Gm\Rightarrow{\nabla}\cdot{g}=-4\pi G\rho_{m}
  3. E d S = q e ϵ 0 E = ρ e ϵ 0 \iint{E}\cdot{\rm d}{S}=\frac{q_{e}}{\epsilon_{0}}\Rightarrow{\nabla}\cdot{E}=% \frac{\rho_{e}}{\epsilon_{0}}
  4. g = - ϕ g E = - ϕ e {g}=-{\nabla}\phi_{g}\,\quad{E}=-{\nabla}\phi_{e}\quad
  5. 2 ϕ = 0 \nabla^{2}\phi=0
  6. g = - G i M i ( r - r i ) | r - r i | 3 , {g}=-G\sum_{i}\frac{M_{i}({r}-{r_{i}})}{|{r}-{r}_{i}|^{3}},
  7. G a b = κ T a b . G_{ab}=\kappa T_{ab}.
  8. S [ g ] = k R - g d 4 x S[g]=k\int R\sqrt{-g}\,d^{4}x
  9. F a b = , a k J b F^{ab}{}_{,a}\,=kJ^{b}
  10. = - 1 4 μ 0 F a b F a b + j a A a . \mathcal{L}=\frac{-1}{4\mu_{0}}F^{ab}F_{ab}+j^{a}A_{a}.

File:Congruent_triangles.png.html

  1. A B C D E F \triangle\ ABC\cong\ \triangle\ DEF

Filter_bank.html

  1. M M
  2. M t h M^{th}
  3. M M
  4. x ( n ) M = x ( M . n ) {x(n)}_{\downarrow{}M}=x(M.n)
  5. X ( z ) M = 1 M m = 0 M - 1 X ( z 1 M ) X(z)_{\downarrow M}=\frac{1}{M}\sum_{m=0}^{M-1}X(z^{\frac{1}{M}})
  6. x ( n ) M = { x ( n M ) 0 n M o t h e r w i s e x(n)_{\uparrow M}=\begin{cases}\begin{array}[]{c}x(\frac{n}{M})\\ 0\end{array}&\begin{array}[]{c}\frac{n}{M}\\ otherwise\end{array}\end{cases}
  7. X ( z ) M = X ( z M ) {X(z)}_{\uparrow{}M}=X(z^{M})
  8. x ( n ) x\left(n\right)
  9. x 1 ( n ) , x 2 ( n ) , x 3 ( n ) , x_{1}(n),x_{2}(n),x_{3}(n),...
  10. x ( n ) x\left(n\right)
  11. x 1 ( n ) , x 2 ( n ) , x 3 ( n ) , x_{1}(n),x_{2}(n),x_{3}(n),...
  12. B W 1 , B W 2 , B W 3 , BW_{1},BW_{2},BW_{3},...
  13. f c 1 , f c 2 , f c 3 , f_{c1},f_{c2},f_{c3},...
  14. H k ( z ) H_{k}(z)
  15. F k ( z ) F_{k}(z)
  16. { h k [ n ] } k = 1 K \left\{h_{k}[n]\right\}_{k=1}^{K}
  17. { g k [ n ] } k = 1 K \left\{g_{k}[n]\right\}_{k=1}^{K}
  18. { M k [ n ] } k = 1 K \left\{M_{k}[n]\right\}_{k=1}^{K}
  19. l 2 ( Z d ) l^{2}(Z^{d})
  20. φ k , m [ n ] = d e f h k * [ M k m - n ] \varphi_{k,m}[n]\stackrel{def}{=}h_{k}^{*}[M_{k}m-n]
  21. 1 k K 1\leq k\leq K
  22. m Z 2 m\in Z^{2}
  23. g k [ n ] g_{k}[n]
  24. ψ k , m [ n ] = d e f h k * [ M k m - n ] \psi_{k,m}[n]\stackrel{def}{=}h_{k}^{*}[M_{k}m-n]
  25. c k [ m ] = < x [ n ] , φ k , m [ n ] Align g t ; c_{k}[m]=<x[n],\varphi_{k,m}[n]&gt;
  26. x ^ [ n ] = 1 k K , m Z 2 c k [ m ] ψ k , m [ n ] \hat{x}[n]=\sum_{1\leq k\leq K,m\in Z^{2}}c_{k}[m]\psi_{k,m}[n]
  27. x ^ [ n ] = 1 k K , m Z 2 < x [ n ] , φ k , m [ n ] > ψ k , m [ n ] \hat{x}[n]=\sum_{1\leq k\leq K,m\in Z^{2}}<x[n],\varphi_{k,m}[n]>\psi_{k,m}[n]
  28. x [ n ] = x [ n ] ^ x[n]=\hat{x[n]}
  29. x [ n ] x[n]
  30. y j [ n ] , y_{j}[n],
  31. j = 0 , 1 , , N - 1 j=0,1,...,N-1
  32. y j [ n ] y_{j}[n]
  33. x [ n ] x[n]
  34. x ( z ) = d e f ( X 0 ( z ) , , X | M | - 1 ( z ) ) T x(z)\stackrel{def}{=}(X_{0}(z),...,X_{|M|-1}(z))^{T}
  35. y ( z ) = d e f ( Y 0 ( z ) , , Y | N | - 1 ( z ) ) T . y(z)\stackrel{def}{=}(Y_{0}(z),...,Y_{|N|-1}(z))^{T}.
  36. y ( z ) = H ( z ) x ( z ) y(z)=H(z)x(z)
  37. H i , j ( z ) H_{i,j}(z)
  38. H i ( z ) H_{i}(z)
  39. x ^ ( z ) = G ( z ) y ( z ) \hat{x}(z)=G(z)y(z)
  40. x ^ ( z ) = d e f ( X ^ 0 ( z ) , , X ^ | M | - 1 ( z ) ) T \hat{x}(z)\stackrel{def}{=}(\hat{X}_{0}(z),...,\hat{X}_{|M|-1}(z))^{T}
  41. G i , j ( z ) G_{i,j}(z)
  42. x ( z ) = x ^ ( z ) x(z)=\hat{x}(z)
  43. I | M | = G ( z ) H ( z ) I_{|M|}=G(z)H(z)
  44. { H 1 , , H N } \{H_{1},...,H_{N}\}
  45. { G 1 , , G N } \{G_{1},...,G_{N}\}
  46. H ( z ) H(z)
  47. G ( z ) G(z)
  48. N × M N\times M
  49. M × N M\times N
  50. M = d e f | M | M\stackrel{def}{=}|M|
  51. H ( z ) H(z)
  52. G ( z ) G(z)
  53. F ( z ) = k Z d f [ k ] z k = k Z d f [ k 1 , , k d ] z 1 k 1 z d k d F(z)=\sum_{k\in Z^{d}}f[k]z^{k}=\sum_{k\in Z^{d}}f[k_{1},...,k_{d}]z_{1}^{k_{1% }}...z_{d}^{k_{d}}
  54. G ( z ) H ( z ) = I | M | G(z)H(z)=I_{|M|}
  55. G ( z ) H ( z ) = I | M | G(z)H(z)=I_{|M|}
  56. M o d u l e { h 1 ( z ) , , h N ( z ) } = d e f { c 1 ( z ) h 1 ( z ) + + c N ( z ) h N ( z ) } Module\left\{h_{1}(z),...,h_{N}(z)\right\}\stackrel{def}{=}\{c_{1}(z)h_{1}(z)+% ...+c_{N}(z)h_{N}(z)\}
  57. : c 1 ( z ) , , c N ( z ) :c_{1}(z),...,c_{N}(z)
  58. { b 1 ( z ) , , b N ( z ) } \left\{b_{1}(z),...,b_{N}(z)\right\}
  59. { h 1 ( z ) , , h N ( z ) } \left\{h_{1}(z),...,h_{N}(z)\right\}
  60. b i ( z ) b_{i}(z)
  61. h j ( z ) h_{j}(z)
  62. K × N K\times N
  63. W i j ( z ) W_{ij}(z)
  64. b i ( z ) = j = 1 N W i j ( z ) h j ( z ) , i = 1 , , K b_{i}(z)=\sum_{j=1}^{N}W_{ij}(z)h_{j}(z),i=1,...,K
  65. D 1 = [ 2 0 0 1 ] D_{1}=\left[\begin{array}[]{cc}2&0\\ 0&1\end{array}\right]
  66. H 0 ( ξ ) H_{0}(\xi)
  67. G 0 ( ξ ) G_{0}(\xi)
  68. H 1 ( ξ ) H_{1}(\xi)
  69. G 1 ( ξ ) G_{1}(\xi)
  70. D 1 D_{1}
  71. H 0 ( z 1 , z 2 ) G 0 ( z 1 , z 2 ) + H 0 ( - z 1 , z 2 ) G 0 ( - z 1 , z 2 ) = 2 H_{0}(z_{1},z_{2})G_{0}(z_{1},z_{2})+H_{0}(-z_{1},z_{2})G_{0}(-z_{1},z_{2})=2
  72. H H T = I n HH^{T}=I_{n}
  73. 2 l 2^{l}

Final_topology.html

  1. X X
  2. X X
  3. X X
  4. Y i Y_{i}
  5. f i : Y i X f_{i}:Y_{i}\to X
  6. τ \tau
  7. X X
  8. f i : Y i ( X , τ ) f_{i}:Y_{i}\to(X,\tau)
  9. f i - 1 ( U ) f_{i}^{-1}(U)
  10. X X
  11. Y i Y_{i}
  12. g g
  13. X X
  14. Z Z
  15. g f i g\circ f_{i}
  16. f : i Y i X f\colon\coprod_{i}Y_{i}\to X

Fineness.html

  1. K = 24 M g M m K=24\,\frac{M\text{g}}{M\text{m}}
  2. K K
  3. M g M\text{g}
  4. M m M\text{m}
  5. 381 384 \tfrac{381}{384}
  6. V Au = M a × k t 24 19.32 V\text{Au}=\frac{M\text{a}\times\tfrac{kt}{24}}{19.32}
  7. V Au V\text{Au}
  8. M a M\text{a}
  9. k t kt
  10. 1 / 72 {1}/{72}
  11. 1 / 1728 {1}/{1728}

Finite_model_theory.html

  1. x y G ( x , y ) . \forall_{x}\exists_{y}G(x,y).
  2. x , y ( G ( x , y ) x y ) . \forall_{x,y}(G(x,y)\Rightarrow x\neq y).
  3. x y ( x y G ( x , y ) ) . \exists_{x}\forall_{y}(x\neq y\Rightarrow G(x,y)).
  4. R R
  5. n n
  6. φ 1 = i j ¬ ( x i = x j ) \varphi_{1}=\bigwedge_{i\neq j}\neg(x_{i}=x_{j})
  7. n n
  8. φ 2 = y i ( x i = y ) \varphi_{2}=\forall_{y}\bigvee_{i}(x_{i}=y)
  9. R R
  10. φ 3 = ( a i , a j ) R R ( x i , x j ) \varphi_{3}=\bigwedge_{(a_{i},a_{j})\in R}R(x_{i},x_{j})
  11. R R
  12. φ 4 = ( a i , a j ) R ¬ R ( x i , x j ) \varphi_{4}=\bigwedge_{(a_{i},a_{j})\notin R}\neg R(x_{i},x_{j})
  13. x 1 . . x n x_{1}..x_{n}
  14. x 1 x n ( φ 1 φ 2 φ 3 φ 4 ) \exists_{x_{1}}\dots\exists_{x_{n}}(\varphi_{1}\land\varphi_{2}\land\varphi_{3% }\land\varphi_{4})
  15. x 1 x n ( φ 1 φ 2 φ 3 φ 4 ) x 1 x n ( ϱ 1 ϱ 2 ϱ 3 ϱ 4 ) . \exists_{x_{1}}...\exists_{x_{n}}(\varphi_{1}\land\varphi_{2}\land\varphi_{3}% \land\varphi_{4})\lor\exists_{x_{1}}...\exists_{x_{n}}(\varrho_{1}\land\varrho% _{2}\land\varrho_{3}\land\varrho_{4}).
  16. A P , B P A\in P,B\not\in P
  17. A B A\equiv B
  18. A B A\equiv B
  19. A α B α A\models\alpha\Leftrightarrow B\models\alpha
  20. A P , B P A\in P,B\not\in P
  21. A m B A\equiv_{m}B
  22. A , B A,B
  23. m m
  24. A α B α A\models\alpha\Leftrightarrow B\models\alpha
  25. A m B A\equiv_{m}B
  26. \models
  27. \models

Fisher's_equation.html

  1. u t = r u ( 1 - u ) + D 2 u x 2 . \frac{\partial u}{\partial t}=ru(1-u)+D\frac{\partial^{2}u}{\partial x^{2}}.\,
  2. c 2 r D c\geq 2\sqrt{rD}
  3. c 2 c\geq 2
  4. u ( x , t ) = v ( x ± c t ) v ( z ) , u(x,t)=v(x\pm ct)\equiv v(z),\,
  5. v \textstyle v
  6. lim z - v ( z ) = 0 , lim z v ( z ) = 1. \lim_{z\rightarrow-\infty}v\left(z\right)=0,\quad\lim_{z\rightarrow\infty}v% \left(z\right)=1.
  7. c = ± 5 / 6 c=\pm 5/\sqrt{6}
  8. v ( z ) = ( 1 + C exp ( ± z / 6 ) ) - 2 v(z)=\left(1+C\mathrm{exp}\left(\pm{z}/{\sqrt{6}}\right)\right)^{-2}
  9. C C
  10. C > 0 C>0
  11. u t = Δ u + F ( u ) , \frac{\partial u}{\partial t}=\Delta u+F\left(u\right),
  12. f ( u ) = 0 f(u)=0

Fisher_hypothesis.html

  1. r = R - π e . r=R-\pi^{e}.
  2. r r
  3. R R
  4. π e \pi^{e}
  5. 1 + i = ( 1 + ρ ) × ( 1 + π e ) 1+i=(1+\rho)\times(1+\pi^{e})
  6. i i
  7. ρ \rho
  8. r r
  9. R R
  10. π e \pi^{e}
  11. π e \pi^{e}
  12. i i
  13. r r

Fitting_subgroup.html

  1. Fit ( G ) = { C G ( H / K ) : H / K a chief factor of G } . \operatorname{Fit}(G)=\bigcap\{C_{G}(H/K):H/K\,\text{ a chief factor of }G\}.
  2. Fit * ( G ) = { H C G ( H / K ) : H / K a chief factor of G } . \operatorname{Fit}^{*}(G)=\bigcap\{HC_{G}(H/K):H/K\,\text{ a chief factor of }% G\}.

Five_color_theorem.html

  1. G G
  2. G G
  3. v v
  4. v v
  5. G G
  6. G G^{\prime}
  7. G G
  8. v v
  9. G G
  10. v 1 v_{1}
  11. v 2 v_{2}
  12. v 3 v_{3}
  13. v 4 v_{4}
  14. v 5 v_{5}
  15. v v
  16. v v
  17. v 1 v_{1}
  18. v 2 v_{2}
  19. v 3 v_{3}
  20. v 4 v_{4}
  21. v 5 v_{5}
  22. G 13 G_{13}
  23. G G^{\prime}
  24. v 1 v_{1}
  25. v 3 v_{3}
  26. G 13 G_{13}
  27. v 1 v_{1}
  28. v v
  29. v 1 v_{1}
  30. v 3 v_{3}
  31. G 13 G_{13}
  32. G 13 G_{13}
  33. G 24 G_{24}
  34. G G^{\prime}
  35. G 24 G_{24}
  36. v v
  37. v 2 v_{2}
  38. v 4 v_{4}
  39. G 13 G_{13}
  40. G G

Fixed-point_space.html

  1. f : X X f:X\rightarrow X
  2. \mathbb{R}
  3. f ( x ) = a + 1 b - a ( x - a ) 2 f(x)=a+\frac{1}{b-a}\cdot(x-a)^{2}

FK-AK_space.html

  1. c 0 c_{0}
  2. l p ( 1 p < ) l^{p}(1\leq p<\infty)
  3. p \|\cdot\|_{p}
  4. l l^{\infty}
  5. E E β E^{\prime}\simeq E^{\beta}

FK-space.html

  1. X X
  2. X X
  3. ( x n ) n (x_{n})_{n\in\mathbb{N}}
  4. x n x_{n}\in\mathbb{C}
  5. ( a n ) n ( k ) (a_{n})_{n\in\mathbb{N}}^{(k)}
  6. X X
  7. ( x n ) n (x_{n})_{n\in\mathbb{N}}
  8. n n
  9. lim k ( a n ) n ( k ) = ( x n ) n \lim_{k\to\infty}(a_{n})_{n\in\mathbb{N}}^{(k)}=(x_{n})_{n\in\mathbb{N}}
  10. n : lim k a n ( k ) = x n \forall n\in\mathbb{N}:\lim_{k\to\infty}a_{n}^{(k)}=x_{n}
  11. ω \omega
  12. X X
  13. ω \omega
  14. ι : X ω \iota:X\to\omega
  15. ( X n , P n ) (X_{n},P_{n})
  16. P n P_{n}
  17. X := n = 1 X n X:=\bigcap_{n=1}^{\infty}X_{n}
  18. P := { p | X p P n } P:=\{p_{|X}\mid p\in P_{n}\}
  19. ( X , P ) (X,P)

Flag_field.html

  1. 2 5 = 32 2^{5}=32

Flat-field_correction.html

  1. C = ( R - D ) * m ( F - D ) = ( R - D ) * G C={{(R-D)*m}\over{(F-D)}}={(R-D)*G}
  2. m ( F - D ) m\over(F-D)

Flavour_(particle_physics).html

  1. M ( u d ) M\left({u\atop d}\right)
  2. u u
  3. d d
  4. M M
  5. L = 1 L=1
  6. T < s u b > 3 T<sub>3

Flexural_rigidity.html

  1. E I d y d x = 0 x M ( x ) d x + C 1 \ EI{dy\over dx}\ =\int_{0}^{x}M(x)dx+C_{1}
  2. E E
  3. I I
  4. y y
  5. M ( x ) M(x)
  6. D = E h e 3 12 ( 1 - ν 2 ) D=\dfrac{Eh_{e}^{3}}{12(1-\nu^{2})}
  7. E E
  8. h e h_{e}
  9. ν \nu

Flight_dynamics_(spacecraft).html

  1. 𝐅 = m 𝐚 , \mathbf{F}=m\mathbf{a},
  2. 𝐋 = C L q A r e f \mathbf{L}=C_{L}qA_{ref}
  3. 𝐃 = C D q A r e f \mathbf{D}=C_{D}qA_{ref}
  4. v ˙ = ( F cos α ) / m - D / m - g cos θ \dot{v}=(F\cos\alpha)/m-D/m-g\cos\theta\,
  5. θ ˙ = ( F sin α ) / m v + L / m v + ( g / v - v / r ) sin θ , \dot{\theta}=(F\sin\alpha)/mv+L/mv+(g/v-v/r)\sin\theta,\,
  6. g = g 0 ( r 0 / r ) 2 g=g_{0}(r_{0}/r)^{2}\,
  7. v = t 0 t v ˙ d t v=\int_{t_{0}}^{t}\dot{v}\,dt
  8. θ = t 0 t θ ˙ d t \theta=\int_{t_{0}}^{t}\dot{\theta}\,dt
  9. h = t 0 t v cos θ d t h=\int_{t_{0}}^{t}v\cos\theta\,dt
  10. r = r 0 + h r=r_{0}+h\,
  11. s = r 0 t 0 t v / r sin θ d t s=r_{0}\int_{t_{0}}^{t}v/r\sin\theta\,dt
  12. v s = v 2 + 2 ω r v cos ϕ sin θ sin A z + ( ω r cos θ ) 2 , v_{s}=\sqrt{v^{2}+2\omega rv\cos\phi\sin\theta\sin A_{z}+(\omega r\cos\theta)^% {2}},
  13. θ s = arccos ( v cos θ / v s ) \theta_{s}=\arccos(v\cos\theta/v_{s})\,
  14. τ = I x α , \mathbf{\tau}=I_{x}\mathbf{\alpha},
  15. α = ( 180 / π ) 𝐓 / I x , \mathbf{\alpha}=(180/\pi)\mathbf{T}/I_{x},
  16. τ = i = 1 N ( r i × 𝐅 𝐢 ) , \mathbf{\tau}=\sum_{i=1}^{N}(r_{i}\times\mathbf{F_{i}}),
  17. a = ( r a + r p ) / 2 a=(r_{a}+r_{p})/2\,
  18. e = r a / a - 1 e=r_{a}/a-1\,
  19. T P = 2 π a 3 / μ TP=2\pi\sqrt{a^{3}/\mu}\,
  20. p = a ( 1 - e 2 ) p=a(1-e^{2})\,
  21. r = p ( 1 + e cos ν ) r=\frac{p}{(1+e\cos\nu)}\,
  22. v = μ ( 2 / r - 1 / a ) v=\sqrt{\mu(2/r-1/a)}\,
  23. v c = μ / r v_{c}=\sqrt{\mu/r}\,
  24. v p = μ ( 1 + e ) a ( 1 - e ) v_{p}=\sqrt{\frac{\mu(1+e)}{a(1-e)}}\,
  25. v a = μ ( 1 - e ) a ( 1 + e ) v_{a}=\sqrt{\frac{\mu(1-e)}{a(1+e)}}\,
  26. v e = 2 μ / r p v_{e}=\sqrt{2\mu/r_{p}}\,
  27. h = r v cos ϕ , h=rv\cos{\phi},\,
  28. ϕ = arccos ( r p v p r v ) \phi=\arccos{(\frac{r_{p}v_{p}}{rv})}\,
  29. ν = r p v p t p t 1 r 2 d t \nu=r_{p}v_{p}\int_{t_{p}}^{t}\frac{1}{r^{2}}dt
  30. t = 1 r p v p 0 ν r 2 d ν t=\frac{1}{r_{p}v_{p}}\int_{0}^{\nu}r^{2}d\nu
  31. r S O I = D ( m s m c ) 2 / 5 , r_{SOI}=D\left(\frac{m_{s}}{m_{c}}\right)^{2/5},
  32. v ˙ = ( F cos α ) / m - g cos θ \dot{v}=(F\cos\alpha)/m-g\cos\theta\,
  33. θ ˙ = ( F sin α ) / m v + ( g / v - v / r ) sin θ , \dot{\theta}=(F\sin\alpha)/mv+(g/v-v/r)\sin\theta,\,
  34. v p = 2 μ / r p + v 2 v_{p}=\sqrt{2\mu/r_{p}+v_{\infty}^{2}}\,
  35. δ = arcsin ( 1 / e ) \delta=\arcsin(1/e)\,
  36. e = 1 + 2 ϵ h 2 μ 2 , e=\sqrt{1+\frac{2\epsilon h^{2}}{\mu^{2}}},
  37. h = r p v p , h=r_{p}v_{p},\,
  38. ϵ = v 2 / 2 - μ / r \epsilon=v^{2}/2-\mu/r\,
  39. p = r p ( 1 + e ) , p=r_{p}(1+e),\,
  40. r = r p ( 1 + e ) ( 1 + e cos ν ) r=\frac{r_{p}(1+e)}{(1+e\cos\nu)}\,
  41. v = μ ( 2 r - ( 1 - e 2 ) r p ( 1 + e ) ) v=\sqrt{\mu\left(\frac{2}{r}-\frac{(1-e^{2})}{r_{p}(1+e)}\right)}\,

Flight_plan.html

  1. C = D * O * sec θ / 2 A , C=D*O*\sec\theta/2A,

Flownet.html

  1. 2 ϕ = 0 \nabla^{2}\phi=0

Fluorescence-lifetime_imaging_microscopy.html

  1. I ( t ) = I 0 e - t / τ I(t)=I_{0}e^{-t/\tau}
  2. 1 τ = k i \frac{1}{\tau}=\sum k_{i}
  3. t t
  4. τ \tau
  5. I 0 I_{0}
  6. t = 0 t=0
  7. k i k_{i}
  8. k f k_{f}
  9. τ \tau
  10. τ \tau
  11. d ( t ) = I R F ( t ) F ( t ) {d}(t)={IRF}(t)\otimes{F}(t)
  12. d ( t i ) d({{t}_{i}})
  13. χ 2 = i [ d i ( t i ) - d 0 i ( t i , a , τ ) ] 2 {{\chi}^{2}}=\sum\limits_{i}{{{\left[{{d}_{i}}({{t}_{i}})-{{d}_{0i}}({{t}_{i}}% ,a,\tau)\right]}^{2}}}
  14. δ \delta
  15. δ \delta
  16. D 0 = i = 1 K / 2 I i δ t D 1 = i = K / 2 K I i δ t \begin{matrix}{{D}_{0}}=\sum\limits_{i=1}^{K/2}{{{I}_{i}}\delta t}&{{D}_{1}}=% \sum\limits_{i=K/2}^{K}{{{I}_{i}}\delta t}\\ \end{matrix}
  17. τ = δ t / ln ( D 0 / D 1 ) \tau=\delta t/\ln({{D}_{0}}/{{D}_{1}})
  18. τ = 1 n ω A n B n \tau=\frac{1}{n\omega}\frac{{{A}_{n}}}{{{B}_{n}}}
  19. A n = t d ( t ) sin ( n ω t ) t I R F ( t ) sin ( n ω t ) = ω τ 1 + ω 2 τ 2 , B n = t d ( t ) cos ( n ω t ) t I R F cos ( n ω t ) = 1 1 + n ω 2 τ 2 , ω = 2 π T \begin{matrix}{{A}_{n}}=\frac{\sum\limits_{t}{d(t)\sin(n\omega t)}}{\sum% \limits_{t}{IRF(t)\sin(n\omega t)}}=\frac{\omega\tau}{1+{{\omega}^{2}}{{\tau}^% {2}}},&{{B}_{n}}=\frac{\sum\limits_{t}{d(t)\cos(n\omega t)}}{\sum\limits_{t}{% IRF\cos(n\omega t)}}=\frac{1}{1+n{{\omega}^{2}}{{\tau}^{2}}},&\omega=\frac{2% \pi}{T}\\ \end{matrix}

FM_broadcasting.html

  1. [ 0.9 [ A + B 2 + A - B 2 sin 4 π f p t ] + 0.1 sin 2 π f p t ] × 75 kHz \left[0.9\left[\frac{A+B}{2}+\frac{A-B}{2}\sin 4\pi f_{p}t\right]+0.1\sin 2\pi f% _{p}t\right]\times 75~{}\mathrm{kHz}
  2. f p f_{p}

Fock_matrix.html

  1. F ^ ( i ) = h ^ ( i ) + j = 1 n / 2 [ 2 J ^ j ( i ) - K ^ j ( i ) ] \hat{F}(i)=\hat{h}(i)+\sum_{j=1}^{n/2}[2\hat{J}_{j}(i)-\hat{K}_{j}(i)]
  2. F ^ ( i ) \hat{F}(i)
  3. h ^ ( i ) {\hat{h}}(i)
  4. n n
  5. n 2 \frac{n}{2}
  6. J ^ j ( i ) \hat{J}_{j}(i)
  7. K ^ j ( i ) \hat{K}_{j}(i)

Focus_(linguistics).html

  1. l i k e M a r y , S u e like\langle Mary,Sue\rangle
  2. { l i k e M a r y , y | y E } \{like\langle Mary,y\rangle|y\in E\}
  3. { l i k e M a r y , S u e , l i k e M a r y , B i l l , l i k e M a r y , L i s a } \{like\langle Mary,Sue\rangle,like\langle Mary,Bill\rangle,like\langle Mary,% Lisa\rangle\}
  4. B , F \langle B,F\rangle
  5. λ x . x , A \langle\lambda x.x,A\rangle
  6. i n t r o d ( j , b , x ) , s \langle introd(j,b,x),s\rangle
  7. \exists
  8. \exists
  9. \exists
  10. \exists
  11. \exists
  12. \exists
  13. \exists
  14. \exists
  15. \exists
  16. \exists
  17. \exists

Fodor's_lemma.html

  1. κ \kappa
  2. S S
  3. κ \kappa
  4. f : S κ f:S\rightarrow\kappa
  5. f ( α ) < α f(\alpha)<\alpha
  6. α S \alpha\in S
  7. α 0 \alpha\neq 0
  8. γ \gamma
  9. S 0 S S_{0}\subseteq S
  10. f ( α ) = γ f(\alpha)=\gamma
  11. α S 0 \alpha\in S_{0}
  12. 0 S 0\notin S
  13. α < κ \alpha<\kappa
  14. C α C_{\alpha}
  15. C α f - 1 ( α ) = C_{\alpha}\cap f^{-1}(\alpha)=\emptyset
  16. C = Δ α < κ C α C=\Delta_{\alpha<\kappa}C_{\alpha}
  17. C C
  18. α S C \alpha\in S\cap C
  19. α C β \alpha\in C_{\beta}
  20. β < α \beta<\alpha
  21. β < α \beta<\alpha
  22. α f - 1 ( β ) \alpha\in f^{-1}(\beta)
  23. f ( α ) α f(\alpha)\geq\alpha
  24. T T
  25. f : T T f:T\rightarrow T
  26. f ( t ) < t f(t)<t
  27. T T
  28. t T t\in T
  29. S T S\subset T
  30. f f

FOIL_method.html

  1. ( a + b ) ( c + d ) = a c first + a d outside + b c inside + b d last (a+b)(c+d)=\underbrace{ac}_{\mathrm{first}}+\underbrace{ad}_{\mathrm{outside}}% +\underbrace{bc}_{\mathrm{inside}}+\underbrace{bd}_{\mathrm{last}}
  2. a a
  3. b b
  4. ( x + 3 ) ( x + 5 ) = x x + x 5 + 3 x + 3 5 = x 2 + 5 x + 3 x + 15 = x 2 + 8 x + 15 \begin{aligned}\displaystyle(x+3)(x+5)&\displaystyle=\,x\cdot x\,+\,x\cdot 5\,% +\,3\cdot x\,+\,3\cdot 5\\ &\displaystyle=\,x^{2}+5x+3x+15\\ &\displaystyle=\,x^{2}+8x+15\end{aligned}
  5. ( 2 x - 3 ) ( 3 x - 4 ) = ( 2 x ) ( 3 x ) + ( 2 x ) ( - 4 ) + ( - 3 ) ( 3 x ) + ( - 3 ) ( - 4 ) = 6 x 2 - 8 x - 9 x + 12 = 6 x 2 - 17 x + 12 \begin{aligned}\displaystyle(2x-3)(3x-4)&\displaystyle=(2x)(3x)+(2x)(-4)+(-3)(% 3x)+(-3)(-4)\\ &\displaystyle=6x^{2}-8x-9x+12\\ &\displaystyle=6x^{2}-17x+12\end{aligned}
  6. ( a + b ) ( c + d ) \displaystyle(a+b)(c+d)
  7. ( c + d ) (c+d)
  8. × c d a a c a d b b c b d \begin{matrix}\times&c&d\\ a&ac&ad\\ b&bc&bd\end{matrix}
  9. ( a x + b ) ( c x + d ) , (ax+b)(cx+d),
  10. × c x d a x a c x 2 a d x b b c x b d \begin{matrix}\times&cx&d\\ ax&acx^{2}&adx\\ b&bcx&bd\end{matrix}
  11. ( a x + b ) ( c x + d ) = a c x 2 + ( a d + b c ) x + b d . (ax+b)(cx+d)=acx^{2}+(ad+bc)x+bd.
  12. × w x y z a a w a x a y a z b b w b x b y b z c c w c x c y c z \begin{matrix}\times&w&x&y&z\\ a&aw&ax&ay&az\\ b&bw&bx&by&bz\\ c&cw&cx&cy&cz\end{matrix}
  13. ( a + b + c ) ( w + x + y + z ) = a w + a x + a y + a z + b w + b x + b y + b z + c w + c x + c y + c z . \begin{aligned}\displaystyle(a+b+c)(w+x+y+z)=&\displaystyle aw+ax+ay+az\\ &\displaystyle{}+bw+bx+by+bz\\ &\displaystyle{}+cw+cx+cy+cz.\end{aligned}
  14. ( a x 2 + b x + c ) ( d x 3 + e x 2 + f x + g ) , (ax^{2}+bx+c)(dx^{3}+ex^{2}+fx+g),
  15. × d e f g a a d a e a f a g b b d b e b f b g c c d c e c f c g \begin{matrix}\times&d&e&f&g\\ a&ad&ae&af&ag\\ b&bd&be&bf&bg\\ c&cd&ce&cf&cg\end{matrix}
  16. ( a x 2 \displaystyle(ax^{2}
  17. ( a + b + c + d ) ( x + y + z + w ) = ( ( a + b ) + ( c + d ) ) ( ( x + y ) + ( z + w ) ) = ( a + b ) ( x + y ) + ( a + b ) ( z + w ) + ( c + d ) ( x + y ) + ( c + d ) ( z + w ) = a x + a y + b x + b y + a z + a w + b z + b w + c x + c y + d x + d y + c z + c w + d z + d w . \begin{aligned}\displaystyle(a+b+c+d)(x+y+z+w)=&\displaystyle\,((a+b)+(c+d))((% x+y)+(z+w))\\ \displaystyle=&\displaystyle\,(a+b)(x+y)+(a+b)(z+w)\\ &\displaystyle\,{}+(c+d)(x+y)+(c+d)(z+w)\\ \displaystyle=&\displaystyle\,ax+ay+bx+by+az+aw+bz+bw\\ &\displaystyle\,{}+cx+cy+dx+dy+cz+cw+dz+dw.\end{aligned}
  18. ( a + b + c + d ) ( x + y + z + w ) = ( a + ( b + c + d ) ) ( x + y + z + w ) = a ( x + y + z + w ) + ( b + c + d ) ( x + y + z + w ) = a ( x + y + z + w ) + ( b + ( c + d ) ) ( x + y + z + w ) = a ( x + y + z + w ) + b ( x + y + z + w ) + ( c + d ) ( x + y + z + w ) = a ( x + y + z + w ) + b ( x + y + z + w ) + c ( x + y + z + w ) + d ( x + y + z + w ) = a x + a y + a z + a w + b x + b y + b z + b w + c x + c y + c z + c w + d x + d y + d z + d w . \begin{aligned}\displaystyle(a+b+c+d)(x+y+z+w)&\displaystyle=(a+(b+c+d))(x+y+z% +w)\\ &\displaystyle=a(x+y+z+w)+(b+c+d)(x+y+z+w)\\ &\displaystyle=a(x+y+z+w)+(b+(c+d))(x+y+z+w)\\ &\displaystyle=a(x+y+z+w)+b(x+y+z+w)\\ &\displaystyle\qquad+(c+d)(x+y+z+w)\\ &\displaystyle=a(x+y+z+w)+b(x+y+z+w)\\ &\displaystyle\qquad+c(x+y+z+w)+d(x+y+z+w)\\ &\displaystyle=ax+ay+az+aw+bx+by+bz+bw\\ &\displaystyle\qquad+cx+cy+cz+cw+dx+dy+dz+dw.\end{aligned}

Foot-lambert.html

  1. L v = E v × R L_{\mathrm{v}}=E_{\mathrm{v}}\times R
  2. L v L_{\mathrm{v}}
  3. E v E_{\mathrm{v}}
  4. R R
  5. R = 0.18 R=0.18

Force_density.html

  1. 𝐟 = - P \mathbf{f}=-\nabla P
  2. d 𝐅 = 𝐟 d V d\mathbf{F}=\mathbf{f}dV
  3. ( F ) = f ( r ) d 3 r \mathbf{(}F)=\int f(r)d^{3}r
  4. ( f ) = p E + J c × B , \mathbf{(}f)=pE+\frac{J}{c}\times B,

Foreign_exchange_option.html

  1. x 1 / x x\mapsto 1/x
  2. x 1 / x x\mapsto 1/x
  3. r d r_{d}
  4. r f r_{f}
  5. c = S 0 e - r f T 𝒩 ( d 1 ) - K e - r d T 𝒩 ( d 2 ) c=S_{0}e^{-r_{f}T}\mathcal{N}(d_{1})-Ke^{-r_{d}T}\mathcal{N}(d_{2})
  6. p = K e - r d T 𝒩 ( - d 2 ) - S 0 e - r f T 𝒩 ( - d 1 ) p=Ke^{-r_{d}T}\mathcal{N}(-d_{2})-S_{0}e^{-r_{f}T}\mathcal{N}(-d_{1})
  7. d 1 = ln ( S 0 / K ) + ( r d - r f + σ 2 / 2 ) T σ T d_{1}=\frac{\ln(S_{0}/K)+(r_{d}-r_{f}+\sigma^{2}/2)T}{\sigma\sqrt{T}}
  8. d 2 = d 1 - σ T d_{2}=d_{1}-\sigma\sqrt{T}
  9. S 0 S_{0}
  10. K K
  11. N N
  12. r d r_{d}
  13. r f r_{f}
  14. T T
  15. σ \sigma

Formal_charge.html

  1. F C = V - ( N + B 2 ) FC=V-\left(N+\frac{B}{2}\right)

Forward_price.html

  1. F = S 0 e ( r - q ) T - i = 1 N D i e ( r - q ) ( T - t i ) F=S_{0}e^{(r-q)T}-\sum_{i=1}^{N}D_{i}e^{(r-q)(T-t_{i})}\,
  2. F F
  3. T T
  4. e x e^{x}
  5. r r
  6. q q
  7. S 0 S_{0}
  8. D i D_{i}
  9. t i t_{i}
  10. 0 < t i < T . 0<t_{i}<T.
  11. K = C + S - F K=C+S-F\,
  12. C = S ( e r T - 1 ) C=S(e^{rT}-1)\,
  13. e r T = 1 + j e^{rT}=1+j\,
  14. F = c 1 e r ( T - t 1 ) + + c n e r ( T - t n ) F=c_{1}e^{r(T-t_{1})}+\cdots+c_{n}e^{r(T-t_{n})}
  15. K = ( S - I ) e r T . K=(S-I)e^{rT}.\,

Forward_rate.html

  1. r t 1 , t 2 = 1 d 2 - d 1 ( 1 + r 2 d 2 1 + r 1 d 1 - 1 ) r_{t_{1},t_{2}}=\frac{1}{d_{2}-d_{1}}\left(\frac{1+r_{2}d_{2}}{1+r_{1}d_{1}}-1\right)
  2. r t 1 , t 2 = ( ( 1 + r 2 ) d 2 ( 1 + r 1 ) d 1 ) 1 d 2 - d 1 - 1 r_{t_{1},t_{2}}=\left(\frac{(1+r_{2})^{d_{2}}}{(1+r_{1})^{d_{1}}}\right)^{% \frac{1}{d_{2}-d_{1}}}-1
  3. r t 1 , t 2 = r 2 d 2 - r 1 d 1 d 2 - d 1 r_{t_{1},t_{2}}=\frac{r_{2}d_{2}-r_{1}d_{1}}{d_{2}-d_{1}}
  4. r t 1 , t 2 r_{t_{1},t_{2}}
  5. t 1 t_{1}
  6. t 2 t_{2}
  7. d 1 d_{1}
  8. t 1 t_{1}
  9. d 2 d_{2}
  10. t 2 t_{2}
  11. r 1 r_{1}
  12. ( 0 , t 1 ) (0,t_{1})
  13. r 2 r_{2}
  14. ( 0 , t 2 ) (0,t_{2})
  15. ( t 1 , t 2 ) (t_{1},t_{2})
  16. r 1 r_{1}
  17. ( 0 , t 1 ) (0,t_{1})
  18. r 2 r_{2}
  19. ( 0 , t 2 ) (0,t_{2})
  20. r t 1 , t 2 r_{t_{1},t_{2}}
  21. ( t 1 , t 2 ) (t_{1},t_{2})
  22. r 1 r_{1}
  23. ( 0 , t 1 ) (0,t_{1})
  24. r t 1 , t 2 r_{t_{1},t_{2}}
  25. ( t 1 , t 2 ) (t_{1},t_{2})
  26. r 2 r_{2}
  27. ( 0 , t 2 ) (0,t_{2})
  28. ( 1 + r 1 ) d 1 ( 1 + r t 1 , t 2 ) d 2 - d 1 = ( 1 + r 2 ) d 2 (1+r_{1})^{d_{1}}(1+r_{t_{1},t_{2}})^{d_{2}-d_{1}}=(1+r_{2})^{d_{2}}
  29. r t 1 , t 2 r_{t_{1},t_{2}}

Four-bar_linkage.html

  1. T 1 = g + h - a - b , T 2 = b + g - a - h , T 3 = b + h - a - g . T_{1}=g+h-a-b,T_{2}=b+g-a-h,T_{3}=b+h-a-g.
  2. T 1 T_{1}
  3. T 2 T_{2}
  4. T 3 T_{3}
  5. Q = Time of slower stroke Time of quicker stroke 1 Q=\frac{\,\text{Time of slower stroke}}{\,\text{Time of quicker stroke}}\geq 1
  6. Δ t cycle = Time of slower stroke + Time of quicker stroke \Delta t\text{cycle}=\,\text{Time of slower stroke}+\,\text{Time of quicker stroke}
  7. ω crank = ( Δ t cycle ) - 1 \omega\text{crank}=(\Delta t\text{cycle})^{-1}
  8. Q = 180 + β 180 - β Q=\frac{180^{\circ}+\beta}{180^{\circ}-\beta}
  9. β = 180 × Q - 1 Q + 1 \beta=180^{\circ}\times\frac{Q-1}{Q+1}
  10. Δ R ΔR
  11. 1 2 \tfrac{1}{2}
  12. Δ t Δt
  13. Δ R ΔR
  14. 1 4 \tfrac{1}{4}
  15. a a
  16. ( Δ t ) 2 (Δt)^{2}
  17. ( Δ R < s u b > 4 ) m a x (ΔR<sub>4)_{max}

Four-current.html

  1. η μ ν \eta_{\mu\nu}
  2. J α = ( c ρ , j 1 , j 2 , j 3 ) = ( c ρ , 𝐣 ) J^{\alpha}=\left(c\rho,j^{1},j^{2},j^{3}\right)=\left(c\rho,\mathbf{j}\right)
  3. J α = ρ 0 U α = ρ 1 - u 2 c 2 U α J^{\alpha}=\rho_{0}U^{\alpha}=\rho\sqrt{1-\frac{u^{2}}{c^{2}}}U^{\alpha}
  4. J α x α = ρ t + 𝐣 = 0 \dfrac{\partial J^{\alpha}}{\partial x^{\alpha}}=\frac{\partial\rho}{\partial t% }+\nabla\cdot\mathbf{j}=0
  5. / x α \partial/\partial x^{\alpha}
  6. J α = ; α 0 J^{\alpha}{}_{;\alpha}=0\,
  7. A α = μ 0 J α \Box A^{\alpha}=\mu_{0}J^{\alpha}
  8. \Box
  9. β F α β = μ 0 J α \partial_{\beta}F^{\alpha\beta}=\mu_{0}J^{\alpha}
  10. 𝒟 μ ν = 1 μ 0 g μ α F α β g β ν - g \mathcal{D}^{\mu\nu}\,=\,\frac{1}{\mu_{0}}\,g^{\mu\alpha}\,F_{\alpha\beta}\,g^% {\beta\nu}\,\sqrt{-g}\,
  11. J μ = ν 𝒟 μ ν J^{\mu}=\partial_{\nu}\mathcal{D}^{\mu\nu}

Four-dimensional_space.html

  1. 𝐚 = ( a 1 a 2 a 3 a 4 ) . \mathbf{a}=\begin{pmatrix}a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\end{pmatrix}.
  2. 𝐞 1 = ( 1 0 0 0 ) ; 𝐞 2 = ( 0 1 0 0 ) ; 𝐞 3 = ( 0 0 1 0 ) ; 𝐞 4 = ( 0 0 0 1 ) , \mathbf{e}_{1}=\begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix};\mathbf{e}_{2}=\begin{pmatrix}0\\ 1\\ 0\\ 0\end{pmatrix};\mathbf{e}_{3}=\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix};\mathbf{e}_{4}=\begin{pmatrix}0\\ 0\\ 0\\ 1\end{pmatrix},
  3. 𝐚 = a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 + a 4 𝐞 4 . \mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}+a_{4}% \mathbf{e}_{4}.
  4. 𝐚 𝐛 = a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 . \mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}.
  5. | 𝐚 | = 𝐚 𝐚 = a 1 2 + a 2 2 + a 3 2 + a 4 2 , \left|\mathbf{a}\right|=\sqrt{\mathbf{a}\cdot\mathbf{a}}=\sqrt{{a_{1}}^{2}+{a_% {2}}^{2}+{a_{3}}^{2}+{a_{4}}^{2}},
  6. θ = arccos 𝐚 𝐛 | 𝐚 | | 𝐛 | . \theta=\arccos{\frac{\mathbf{a}\cdot\mathbf{b}}{\left|\mathbf{a}\right|\left|% \mathbf{b}\right|}}.
  7. 𝐚 𝐛 = a 1 b 1 + a 2 b 2 + a 3 b 3 - a 4 b 4 . \mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}-a_{4}b_{4}.
  8. b 4 b_{4}
  9. 𝐚 𝐛 = ( a 1 b 2 - a 2 b 1 ) 𝐞 12 + ( a 1 b 3 - a 3 b 1 ) 𝐞 13 + ( a 1 b 4 - a 4 b 1 ) 𝐞 14 + ( a 2 b 3 - a 3 b 2 ) 𝐞 23 \displaystyle\mathbf{a}\wedge\mathbf{b}=(a_{1}b_{2}-a_{2}b_{1})\mathbf{e}_{12}% +(a_{1}b_{3}-a_{3}b_{1})\mathbf{e}_{13}+(a_{1}b_{4}-a_{4}b_{1})\mathbf{e}_{14}% +(a_{2}b_{3}-a_{3}b_{2})\mathbf{e}_{23}
  10. 𝐕 = 1 2 π 2 R 4 \mathbf{V}=\begin{matrix}\frac{1}{2}\end{matrix}\pi^{2}R^{4}
  11. C = 2 π r C=2\pi r
  12. A = 4 π r 2 A=4\pi r^{2}
  13. V = 6 π r 3 V=6\pi r^{3}
  14. V = 8 π r 3 V=8\pi r^{3}
  15. V = 2 π 2 r 3 V=2\pi^{2}r^{3}

Four-frequency.html

  1. N a = ( ν , ν 𝐧 ) N^{a}=\left(\nu,\nu\mathbf{n}\right)
  2. ν \nu
  3. 𝐧 \mathbf{n}
  4. V V
  5. 1 c η ( N a , V ) \tfrac{1}{c}\eta(N^{a},V)
  6. η \eta
  7. K a = ( ω c , 𝐤 ) K^{a}=\left(\frac{\omega}{c},\mathbf{k}\right)
  8. ω = 2 π ν \omega=2\pi\nu
  9. c c
  10. 𝐤 = 2 π λ 𝐧 \mathbf{k}=\frac{2\pi}{\lambda}\mathbf{n}
  11. λ \lambda
  12. c = ν λ c=\nu\lambda
  13. K a = 2 π c N a K^{a}=\frac{2\pi}{c}N^{a}

Four-gradient.html

  1. x α = ( 1 c t , ) = α = , α \dfrac{\partial}{\partial x^{\alpha}}=\left(\frac{1}{c}\frac{\partial}{% \partial t},\nabla\right)=\partial_{\alpha}={}_{,\alpha}
  2. , α {}_{,\alpha}
  3. x α x^{\alpha}
  4. α = g α β β = ( 1 c t , - ) \partial^{\alpha}\ =g^{\alpha\beta}\partial_{\beta}=\left(\frac{1}{c}\frac{% \partial}{\partial t},-\nabla\right)
  5. α \partial_{\alpha}
  6. \Box
  7. D D = α α = 1 c 2 2 t 2 - 2 D\cdot D=\partial_{\alpha}\partial^{\alpha}=\frac{1}{c^{2}}\frac{\partial^{2}}% {\partial t^{2}}-\nabla^{2}
  8. \Box
  9. 2 \Box^{2}
  10. \Box
  11. α = ( t , ) \partial^{\alpha}\ =\left(\frac{\partial}{\partial t},\nabla\right)
  12. α = ( 1 c t , - ) \partial^{\alpha}\ =\left(\frac{1}{c}\frac{\partial}{\partial t},-\nabla\right)

Four-wave_mixing.html

  1. ± f 1 ± f 2 ± f 3 \pm f_{1}\pm f_{2}\pm f_{3}
  2. f i j k = f i + f j - f k , where i , j k f_{ijk}=f_{i}+f_{j}-f_{k},\mathrm{where}\,i,j\neq k
  3. f 0 = f 1 + f 1 - f 2 f_{0}=f_{1}+f_{1}-f_{2}

Fraction_(mathematics).html

  1. 1 2 \tfrac{1}{2}
  2. 3 4 \tfrac{3}{4}
  3. 3 / 2 {3}/{2}
  4. 2 5 \tfrac{2}{5}
  5. 7 3 \tfrac{7}{3}
  6. a b \tfrac{a}{b}
  7. 1 2 \tfrac{1}{2}
  8. - 8 5 -\tfrac{8}{5}
  9. - 8 5 \tfrac{-8}{5}
  10. 8 - 5 \tfrac{8}{-5}
  11. 2 + 3 4 = 2 3 4 2+\frac{3}{4}=2\tfrac{3}{4}
  12. a b c a\tfrac{b}{c}
  13. a b c = a × b c a\tfrac{b}{c}=a\times\tfrac{b}{c}
  14. a b c a\tfrac{b}{c}
  15. a × b c a\times\tfrac{b}{c}
  16. a b c a\cdot\tfrac{b}{c}
  17. a ( b c ) a(\tfrac{b}{c})
  18. 2 3 4 2\tfrac{3}{4}
  19. 2 + 3 4 2+\tfrac{3}{4}
  20. 2 = 8 4 2=\tfrac{8}{4}
  21. 2 3 4 = 8 4 + 3 4 = 11 4 2\tfrac{3}{4}=\tfrac{8}{4}+\tfrac{3}{4}=\tfrac{11}{4}
  22. 11 4 \tfrac{11}{4}
  23. 11 4 = 2 3 4 \tfrac{11}{4}=2\tfrac{3}{4}
  24. - 2 3 4 -2\tfrac{3}{4}
  25. - ( 2 + 3 4 ) = - 2 - 3 4 -(2+\tfrac{3}{4})=-2-\tfrac{3}{4}
  26. 3 7 \tfrac{3}{7}
  27. 7 3 \tfrac{7}{3}
  28. 17 1 \tfrac{17}{1}
  29. 1 17 \tfrac{1}{17}
  30. 1 2 1 3 \frac{\tfrac{1}{2}}{\tfrac{1}{3}}
  31. 12 3 4 26 \frac{12\tfrac{3}{4}}{26}
  32. 1 2 1 3 = 1 2 × 3 1 = 3 2 = 1 1 2 \frac{\tfrac{1}{2}}{\tfrac{1}{3}}=\tfrac{1}{2}\times\tfrac{3}{1}=\tfrac{3}{2}=% 1\tfrac{1}{2}
  33. 12 3 4 26 = 12 3 4 1 26 = 12 4 + 3 4 1 26 = 51 4 1 26 = 51 104 \frac{12\tfrac{3}{4}}{26}=12\tfrac{3}{4}\cdot\tfrac{1}{26}=\tfrac{12\cdot 4+3}% {4}\cdot\tfrac{1}{26}=\tfrac{51}{4}\cdot\tfrac{1}{26}=\tfrac{51}{104}
  34. 3 2 5 = 3 2 × 1 5 = 3 10 \frac{\tfrac{3}{2}}{5}=\tfrac{3}{2}\times\tfrac{1}{5}=\tfrac{3}{10}
  35. 8 1 3 = 8 × 3 1 = 24 \frac{8}{\tfrac{1}{3}}=8\times\tfrac{3}{1}=24
  36. 3 4 \tfrac{3}{4}
  37. 5 7 \tfrac{5}{7}
  38. 3 4 × 5 7 = 15 28 \tfrac{3}{4}\times\tfrac{5}{7}=\tfrac{15}{28}
  39. 3 75 100 3\tfrac{75}{100}
  40. 1 7 \tfrac{1}{7}
  41. 1 2 + 1 3 \tfrac{1}{2}+\tfrac{1}{3}
  42. 1 2 \tfrac{1}{2}
  43. 2 3 \tfrac{2}{3}
  44. 3 4 \tfrac{3}{4}
  45. 5 7 \tfrac{5}{7}
  46. 1 2 + 1 6 + 1 21 . \tfrac{1}{2}+\tfrac{1}{6}+\tfrac{1}{21}.
  47. 13 17 \tfrac{13}{17}
  48. 1 2 + 1 4 + 1 68 \tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{68}
  49. 1 3 + 1 4 + 1 6 + 1 68 \tfrac{1}{3}+\tfrac{1}{4}+\tfrac{1}{6}+\tfrac{1}{68}
  50. 1 8 \tfrac{1}{8}
  51. n n
  52. n n = 1 \tfrac{n}{n}=1
  53. n n \tfrac{n}{n}
  54. 1 2 \tfrac{1}{2}
  55. 2 4 \tfrac{2}{4}
  56. 1 2 \tfrac{1}{2}
  57. 2 4 \tfrac{2}{4}
  58. 1 2 \tfrac{1}{2}
  59. 3 9 \tfrac{3}{9}
  60. 3 8 \tfrac{3}{8}
  61. 5 10 \tfrac{5}{10}
  62. 1 2 \tfrac{1}{2}
  63. 10 20 \tfrac{10}{20}
  64. 50 100 \tfrac{50}{100}
  65. 63 462 \tfrac{63}{462}
  66. 63 462 = 63 ÷ 21 462 ÷ 21 = 3 22 \tfrac{63}{462}=\tfrac{63\div 21}{462\div 21}=\tfrac{3}{22}
  67. 3 4 > 2 4 \tfrac{3}{4}>\tfrac{2}{4}
  68. a b \tfrac{a}{b}
  69. c d \tfrac{c}{d}
  70. a d b d \tfrac{ad}{bd}
  71. b c b d \tfrac{bc}{bd}
  72. 2 3 \tfrac{2}{3}
  73. 1 2 \tfrac{1}{2}
  74. 4 6 > 3 6 \tfrac{4}{6}>\tfrac{3}{6}
  75. 5 18 \tfrac{5}{18}
  76. 4 17 \tfrac{4}{17}
  77. 5 × 17 18 × 17 \tfrac{5\times 17}{18\times 17}
  78. 4 × 18 17 × 18 \tfrac{4\times 18}{17\times 18}
  79. 5 18 > 4 17 \tfrac{5}{18}>\tfrac{4}{17}
  80. 2 4 + 3 4 = 5 4 = 1 1 4 \tfrac{2}{4}+\tfrac{3}{4}=\tfrac{5}{4}=1\tfrac{1}{4}
  81. 1 2 \tfrac{1}{2}
  82. 1 4 \tfrac{1}{4}
  83. 1 4 + 1 3 = 1 * 3 4 * 3 + 1 * 4 3 * 4 = 3 12 + 4 12 = 7 12 \tfrac{1}{4}\ +\tfrac{1}{3}=\tfrac{1*3}{4*3}\ +\tfrac{1*4}{3*4}=\tfrac{3}{12}% \ +\tfrac{4}{12}=\tfrac{7}{12}
  84. 3 5 + 2 3 \tfrac{3}{5}+\tfrac{2}{3}
  85. 3 5 \tfrac{3}{5}
  86. 3 5 × 3 3 = 9 15 \tfrac{3}{5}\times\tfrac{3}{3}=\tfrac{9}{15}
  87. 3 3 \tfrac{3}{3}
  88. 3 3 \tfrac{3}{3}
  89. 2 3 \tfrac{2}{3}
  90. 2 3 × 5 5 = 10 15 \tfrac{2}{3}\times\tfrac{5}{5}=\tfrac{10}{15}
  91. 3 5 + 2 3 \tfrac{3}{5}+\tfrac{2}{3}
  92. 9 15 + 10 15 = 19 15 = 1 4 15 \tfrac{9}{15}+\tfrac{10}{15}=\tfrac{19}{15}=1\tfrac{4}{15}
  93. a b + c d = a d + c b b d \tfrac{a}{b}+\tfrac{c}{d}=\tfrac{ad+cb}{bd}
  94. a b + c d + e f = a ( d f ) + c ( b f ) + e ( b d ) b d f \tfrac{a}{b}+\tfrac{c}{d}+\tfrac{e}{f}=\tfrac{a(df)+c(bf)+e(bd)}{bdf}
  95. 3 4 \tfrac{3}{4}
  96. 5 12 \tfrac{5}{12}
  97. 3 4 + 5 12 = 9 12 + 5 12 = 14 12 = 7 6 = 1 1 6 \tfrac{3}{4}+\tfrac{5}{12}=\tfrac{9}{12}+\tfrac{5}{12}=\tfrac{14}{12}=\tfrac{7% }{6}=1\tfrac{1}{6}
  98. 2 3 - 1 2 = 4 6 - 3 6 = 1 6 \tfrac{2}{3}-\tfrac{1}{2}=\tfrac{4}{6}-\tfrac{3}{6}=\tfrac{1}{6}
  99. 2 3 × 3 4 = 6 12 \tfrac{2}{3}\times\tfrac{3}{4}=\tfrac{6}{12}
  100. 2 3 × 3 4 = \cancel 2 1 \cancel 3 1 × \cancel 3 1 \cancel 4 2 = 1 1 × 1 2 = 1 2 \tfrac{2}{3}\times\tfrac{3}{4}=\tfrac{\cancel{2}^{~{}1}}{\cancel{3}^{~{}1}}% \times\tfrac{\cancel{3}^{~{}1}}{\cancel{4}^{~{}2}}=\tfrac{1}{1}\times\tfrac{1}% {2}=\tfrac{1}{2}
  101. 6 × 3 4 = 6 1 × 3 4 = 18 4 6\times\tfrac{3}{4}=\tfrac{6}{1}\times\tfrac{3}{4}=\tfrac{18}{4}
  102. 3 × 2 3 4 = 3 × ( 8 4 + 3 4 ) = 3 × 11 4 = 33 4 = 8 1 4 3\times 2\tfrac{3}{4}=3\times\left(\tfrac{8}{4}+\tfrac{3}{4}\right)=3\times% \tfrac{11}{4}=\tfrac{33}{4}=8\tfrac{1}{4}
  103. 2 3 4 2\tfrac{3}{4}
  104. 8 4 + 3 4 \tfrac{8}{4}+\tfrac{3}{4}
  105. 8 1 4 8\tfrac{1}{4}
  106. 10 3 ÷ 5 \tfrac{10}{3}\div 5
  107. 2 3 \tfrac{2}{3}
  108. 10 3 5 = 10 15 \tfrac{10}{3\cdot 5}=\tfrac{10}{15}
  109. 2 3 \tfrac{2}{3}
  110. 1 2 ÷ 3 4 = 1 2 × 4 3 = 1 4 2 3 = 2 3 \tfrac{1}{2}\div\tfrac{3}{4}=\tfrac{1}{2}\times\tfrac{4}{3}=\tfrac{1\cdot 4}{2% \cdot 3}=\tfrac{2}{3}
  111. 789 ¯ \overline{789}
  112. 5 ¯ \overline{5}
  113. 62 ¯ \overline{62}
  114. 264 ¯ \overline{264}
  115. 6291 ¯ \overline{6291}
  116. 5 ¯ \overline{5}
  117. 392 ¯ \overline{392}
  118. 12 ¯ \overline{12}
  119. 987 ¯ \overline{987}
  120. 987 ¯ \overline{987}
  121. 987 ¯ \overline{987}
  122. 987 ¯ \overline{987}
  123. 987 ¯ \overline{987}
  124. 987 ¯ \overline{987}
  125. 987 ¯ \overline{987}
  126. ( a , b ) + ( c , d ) = ( a d + b c , b d ) (a,b)+(c,d)=(ad+bc,bd)\,
  127. ( a , b ) - ( c , d ) = ( a d - b c , b d ) (a,b)-(c,d)=(ad-bc,bd)\,
  128. ( a , b ) ( c , d ) = ( a c , b d ) (a,b)\cdot(c,d)=(ac,bd)
  129. ( a , b ) ÷ ( c , d ) = ( a d , b c ) (a,b)\div(c,d)=(ad,bc)
  130. ( a , b ) (a,b)
  131. ( c , d ) (c,d)
  132. a d = b c ad=bc
  133. 3 x x 2 + 2 x - 3 \frac{3x}{x^{2}+2x-3}
  134. x + 2 x 2 - 3 \frac{\sqrt{x+2}}{x^{2}-3}
  135. 3 x x 2 + 2 x - 3 \frac{3x}{x^{2}+2x-3}
  136. x + 2 x 2 - 3 \frac{\sqrt{x+2}}{x^{2}-3}
  137. 1 + 1 x 1 - 1 x \frac{1+\tfrac{1}{x}}{1-\tfrac{1}{x}}
  138. π 2 \textstyle{\tfrac{\pi}{2}}
  139. 2 x x 2 - 1 \textstyle{2x\over x^{2}-1}
  140. 1 x + 1 \textstyle{1\over x+1}
  141. 1 x - 1 \textstyle{1\over x-1}
  142. 3 7 = 3 7 7 7 = 3 7 7 \frac{3}{\sqrt{7}}=\frac{3}{\sqrt{7}}\cdot\frac{\sqrt{7}}{\sqrt{7}}=\frac{3% \sqrt{7}}{7}
  143. 3 3 - 2 5 = 3 3 - 2 5 3 + 2 5 3 + 2 5 = 3 ( 3 + 2 5 ) 3 2 - ( 2 5 ) 2 = 3 ( 3 + 2 5 ) 9 - 20 = - 9 + 6 5 11 \frac{3}{3-2\sqrt{5}}=\frac{3}{3-2\sqrt{5}}\cdot\frac{3+2\sqrt{5}}{3+2\sqrt{5}% }=\frac{3(3+2\sqrt{5})}{{3}^{2}-(2\sqrt{5})^{2}}=\frac{3(3+2\sqrt{5})}{9-20}=-% \frac{9+6\sqrt{5}}{11}
  144. 3 3 + 2 5 = 3 3 + 2 5 3 - 2 5 3 - 2 5 = 3 ( 3 - 2 5 ) 3 2 - ( 2 5 ) 2 = 3 ( 3 - 2 5 ) 9 - 20 = - 9 - 6 5 11 \frac{3}{3+2\sqrt{5}}=\frac{3}{3+2\sqrt{5}}\cdot\frac{3-2\sqrt{5}}{3-2\sqrt{5}% }=\frac{3(3-2\sqrt{5})}{{3}^{2}-(2\sqrt{5})^{2}}=\frac{3(3-2\sqrt{5})}{9-20}=-% \frac{9-6\sqrt{5}}{11}
  145. 1 2 \tfrac{1}{2}
  146. 1 2 \frac{1}{2}
  147. 3 1 5 3 \frac{3\quad 1}{5\quad 3}
  148. a b \tfrac{a}{b}
  149. a a
  150. b b

Fractional-order_control.html

  1. 1 s λ \frac{1}{s^{\lambda}}
  2. G I ( s ) {G_{I}}(s)

Fractional-order_integrator.html

  1. 𝔻 t q a ( f ( x ) ) {}_{a}\mathbb{D}^{q}_{t}\left(f(x)\right)
  2. q = - 1 2 , q=-\frac{1}{2},

Fractional_Brownian_motion.html

  1. E [ B H ( t ) B H ( s ) ] = 1 2 ( | t | 2 H + | s | 2 H - | t - s | 2 H ) , E[B_{H}(t)B_{H}(s)]=\tfrac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),
  2. X H ( t ) = 1 Γ ( H + 1 / 2 ) 0 t ( t - s ) H - 1 / 2 d B ( s ) X^{H}(t)=\frac{1}{\Gamma(H+1/2)}\int_{0}^{t}(t-s)^{H-1/2}\,dB(s)
  3. B H ( t ) = B H ( 0 ) + 1 Γ ( H + 1 / 2 ) { - 0 [ ( t - s ) H - 1 / 2 - ( - s ) H - 1 / 2 ] d B ( s ) + 0 t ( t - s ) H - 1 / 2 d B ( s ) } B_{H}(t)=B_{H}(0)+\frac{1}{\Gamma(H+1/2)}\left\{\int_{-\infty}^{0}\left[(t-s)^% {H-1/2}-(-s)^{H-1/2}\right]\,dB(s)+\int_{0}^{t}(t-s)^{H-1/2}\,dB(s)\right\}
  4. B H ( a t ) | a | H B H ( t ) . B_{H}(at)\sim|a|^{H}B_{H}(t).
  5. B H ( t ) - B H ( s ) B H ( t - s ) . B_{H}(t)-B_{H}(s)\;\sim\;B_{H}(t-s).
  6. n = 1 E [ B H ( 1 ) ( B H ( n + 1 ) - B H ( n ) ) ] = . \sum_{n=1}^{\infty}E[B_{H}(1)(B_{H}(n+1)-B_{H}(n))]=\infty.
  7. | B H ( t ) - B H ( s ) | c | t - s | H - ε |B_{H}(t)-B_{H}(s)|\leq c|t-s|^{H-\varepsilon}
  8. n n
  9. O ( n 3 ) O(n^{3})
  10. t 1 , , t n t_{1},\ldots,t_{n}
  11. Γ = ( R ( t i , t j ) , i , j = 1 , , n ) \Gamma=\bigl(R(t_{i},\,t_{j}),i,j=1,\ldots,\,n\bigr)
  12. R ( t , s ) = ( s 2 H + t 2 H - | t - s | 2 H ) / 2 \,R(t,s)=(s^{2H}+t^{2H}-|t-s|^{2H})/2
  13. Σ \,\Sigma
  14. Γ \,\Gamma
  15. Σ 2 = Γ \,\Sigma^{2}=\Gamma
  16. Σ \,\Sigma
  17. Γ \,\Gamma
  18. v \,v
  19. u = Σ v \,u=\Sigma v
  20. u \,u
  21. Σ \,\Sigma
  22. Γ \,\Gamma
  23. Γ \,\Gamma
  24. λ i \,\lambda_{i}
  25. Γ \,\Gamma
  26. λ i 0 \,\lambda_{i}\geq 0
  27. i = 1 , , n i=1,\dots,n
  28. Λ \,\Lambda
  29. Λ i j = λ i δ i j \Lambda_{ij}=\lambda_{i}\,\delta_{ij}
  30. δ i j \delta_{ij}
  31. Λ 1 / 2 \Lambda^{1/2}
  32. λ i 1 / 2 \lambda_{i}^{1/2}
  33. Λ i j 1 / 2 = λ i 1 / 2 δ i j \Lambda_{ij}^{1/2}=\lambda_{i}^{1/2}\,\delta_{ij}
  34. λ i 0 \lambda_{i}\geq 0
  35. v i \,v_{i}
  36. λ i \,\lambda_{i}
  37. P \,P
  38. i i
  39. v i \,v_{i}
  40. P \,P
  41. Σ = P Λ 1 / 2 P - 1 \Sigma=P\,\Lambda^{1/2}\,P^{-1}
  42. Γ = P Λ P - 1 \Gamma=P\,\Lambda\,P^{-1}
  43. B H ( t ) = 0 t K H ( t , s ) d B ( s ) B_{H}(t)=\int_{0}^{t}K_{H}(t,s)\,dB(s)
  44. K H ( t , s ) = ( t - s ) H - 1 2 Γ ( H + 1 2 ) 2 F 1 ( H - 1 2 ; 1 2 - H ; H + 1 2 ; 1 - t s ) . K_{H}(t,s)=\frac{(t-s)^{H-\frac{1}{2}}}{\Gamma(H+\frac{1}{2})}\;_{2}F_{1}\left% (H-\frac{1}{2};\,\frac{1}{2}-H;\;H+\frac{1}{2};\,1-\frac{t}{s}\right).
  45. F 1 2 {}_{2}F_{1}
  46. 0 = t 0 < t 1 < < t n = T 0=t_{0}<t_{1}<\cdots<t_{n}=T
  47. ( δ B 1 , , δ B n ) (\delta B_{1},\ldots,\delta B_{n})
  48. t j t_{j}
  49. B H ( t j ) = n T i = 0 j - 1 t i t i + 1 K H ( t j , s ) d s δ B i . B_{H}(t_{j})=\frac{n}{T}\sum_{i=0}^{j-1}\int_{t_{i}}^{t_{i+1}}K_{H}(t_{j},\,s)% \,ds\ \delta B_{i}.

Fractional_quantum_Hall_effect.html

  1. e 2 / h e^{2}/h
  2. ν = 1 / m \nu=1/m
  3. m m
  4. ν = p / q , \nu=p/q,
  5. 1 3 , 2 5 , 3 7 , etc., {1\over 3},{2\over 5},{3\over 7},\mbox{etc.,}~{}
  6. 2 3 , 3 5 , 4 7 , etc. {2\over 3},{3\over 5},{4\over 7},\mbox{etc.}~{}
  7. 1 / q 1/q
  8. e * = e q e^{*}={e\over q}
  9. θ = π q \theta={\pi\over q}
  10. e i θ e^{i\theta}
  11. ν = 1 / q \nu=1/q
  12. ν = 2 / 5 \nu=2/5
  13. 2 / 7 2/7
  14. ν = 1 / 3 \nu=1/3

Franck–Condon_principle.html

  1. | ϵ v |\epsilon v\rangle
  2. | ϵ v |\epsilon^{\prime}v^{\prime}\rangle
  3. s y m b o l μ = s y m b o l μ e + s y m b o l μ N = - e i s y m b o l r i + e j Z j s y m b o l R j . symbol{\mu}=symbol{\mu}_{e}+symbol{\mu}_{N}=-e\sum\limits_{i}symbol{r}_{i}+e% \sum\limits_{j}Z_{j}symbol{R}_{j}.
  4. P = ψ | s y m b o l μ | ψ = ψ * s y m b o l μ ψ d τ , P=\left\langle\psi^{\prime}\right|symbol{\mu}\left|\psi\right\rangle=\int{\psi% ^{\prime*}}symbol{\mu}\psi\,d\tau,
  5. ψ \psi
  6. ψ \psi^{\prime}
  7. ψ = ψ e ψ v ψ s . \psi=\psi_{e}\psi_{v}\psi_{s}.
  8. P = ψ e ψ v ψ s | s y m b o l μ | ψ e ψ v ψ s = ψ e * ψ v * ψ s * ( s y m b o l μ e + s y m b o l μ N ) ψ e ψ v ψ s d τ P=\left\langle\psi_{e}^{\prime}\psi_{v}^{\prime}\psi_{s}^{\prime}\right|symbol% {\mu}\left|\psi_{e}\psi_{v}\psi_{s}\right\rangle=\int\psi_{e}^{\prime*}\psi_{v% }^{\prime*}\psi_{s}^{\prime*}(symbol{\mu}_{e}+symbol{\mu}_{N})\psi_{e}\psi_{v}% \psi_{s}\,d\tau
  9. = ψ e * ψ v * ψ s * s y m b o l μ e ψ e ψ v ψ s d τ + ψ e * ψ v * ψ s * s y m b o l μ N ψ e ψ v ψ s d τ =\int\psi_{e}^{\prime*}\psi_{v}^{\prime*}\psi_{s}^{\prime*}symbol{\mu}_{e}\psi% _{e}\psi_{v}\psi_{s}\,d\tau+\int\psi_{e}^{\prime*}\psi_{v}^{\prime*}\psi_{s}^{% \prime*}symbol{\mu}_{N}\psi_{e}\psi_{v}\psi_{s}\,d\tau
  10. = ψ v * ψ v d τ n Franck–Condon factor ψ e * s y m b o l μ e ψ e d τ e orbital selection rule ψ s * ψ s d τ s spin selection rule + ψ e * ψ e d τ e 0 ψ v * s y m b o l μ N ψ v d τ v ψ s * ψ s d τ s . =\underbrace{\int\psi_{v}^{\prime*}\psi_{v}\,d\tau_{n}}_{\displaystyle{\,\text% {Franck--Condon}\atop\,\text{factor}}}\underbrace{\int\psi_{e}^{\prime*}symbol% {\mu}_{e}\psi_{e}\,d\tau_{e}}_{\displaystyle{\,\text{orbital}\atop\,\text{% selection rule}}}\underbrace{\int\psi_{s}^{\prime*}\psi_{s}\,d\tau_{s}}_{% \displaystyle{\,\text{spin}\atop\,\text{selection rule}}}+\underbrace{\int\psi% _{e}^{\prime*}\psi_{e}\,d\tau_{e}}_{\displaystyle 0}\int\psi_{v}^{\prime*}% symbol{\mu}_{N}\psi_{v}\,d\tau_{v}\int\psi_{s}^{\prime*}\psi_{s}\,d\tau_{s}.
  11. ψ v * ψ e * s y m b o l μ e ψ e ψ v d τ e d τ n ψ v * ψ v d τ n ψ e * s y m b o l μ e ψ e d τ e . \iint\psi_{v}^{\prime*}\psi_{e}^{\prime*}symbol{\mu}_{e}\psi_{e}\psi_{v}\,d% \tau_{e}d\tau_{n}\approx\int\psi_{v}^{\prime*}\psi_{v}\,d\tau_{n}\int\psi_{e}^% {\prime*}symbol{\mu}_{e}\psi_{e}\,d\tau_{e}.
  12. ψ e * s y m b o l μ e ψ e d τ e \int\psi_{e}^{\prime*}symbol{\mu}_{e}\psi_{e}\,d\tau_{e}
  13. ψ e \psi_{e}
  14. ψ e \psi^{\prime}_{e}
  15. q i q_{i}
  16. Ω i \hbar\Omega_{i}

Frank_Harary.html

  1. * *
  2. * *
  3. * *
  4. = =

Fredholm's_theorem.html

  1. ( row M ) = ker M . (\operatorname{row}M)^{\bot}=\ker M.
  2. ( col M ) = ker M * . (\operatorname{col}M)^{\bot}=\ker M^{*}.
  3. K ( x , y ) K(x,y)
  4. a b K ( x , y ) ϕ ( y ) d y = λ ϕ ( x ) \int_{a}^{b}K(x,y)\phi(y)\,dy=\lambda\phi(x)
  5. a b ψ ( x ) K ( x , y ) ¯ d x = λ ¯ ψ ( y ) . \int_{a}^{b}\psi(x)\overline{K(x,y)}\,dx=\overline{\lambda}\psi(y).
  6. λ ¯ \overline{\lambda}
  7. λ \lambda
  8. K ( x , y ) ¯ \overline{K(x,y)}
  9. λ \lambda
  10. ψ ( x ) = ϕ ( x ) = 0 \psi(x)=\phi(x)=0
  11. ϕ 1 ( x ) , , ϕ n ( x ) \phi_{1}(x),\cdots,\phi_{n}(x)
  12. ψ 1 ( y ) , , ψ n ( y ) \psi_{1}(y),\cdots,\psi_{n}(y)
  13. K ( x , y ) K(x,y)
  14. [ a , b ] × [ a , b ] [a,b]\times[a,b]
  15. λ ϕ ( x ) - a b K ( x , y ) ϕ ( y ) d y = f ( x ) . \lambda\phi(x)-\int_{a}^{b}K(x,y)\phi(y)\,dy=f(x).
  16. f ( x ) f(x)
  17. { ψ n ( x ) } \{\psi_{n}(x)\}
  18. a b ψ n ( x ) ¯ f ( x ) d x = 0 \int_{a}^{b}\overline{\psi_{n}(x)}f(x)\,dx=0
  19. ψ n ( x ) ¯ \overline{\psi_{n}(x)}
  20. ψ n ( x ) \psi_{n}(x)
  21. λ ψ ( y ) ¯ - a b ψ ( x ) ¯ K ( x , y ) d x = 0. \lambda\overline{\psi(y)}-\int_{a}^{b}\overline{\psi(x)}K(x,y)\,dx=0.
  22. K ( x , y ) K(x,y)
  23. [ a , b ] × [ a , b ] [a,b]\times[a,b]

Fredkin_gate.html

  1. [ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] \begin{bmatrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&1\\ \end{bmatrix}
  2. C ¯ \overline{C}
  3. C ¯ \overline{C}
  4. I 2 = 0 I_{2}=0
  5. O 2 = C AND I 1 O_{2}=C\operatorname{AND}I_{1}
  6. I 1 = 0 I_{1}=0
  7. I 2 = 1 I_{2}=1
  8. O 2 = NOT C O_{2}=\operatorname{NOT}C

Free-fall_time.html

  1. m m
  2. R R
  3. M M
  4. R / 2 R/2
  5. R / 2 R/2
  6. t orbit = 2 π G ( M + m ) ( R 2 ) 3 / 2 = π R 3 / 2 2 G ( M + m ) . t_{\,\text{orbit}}=\frac{2\pi}{\sqrt{G(M+m)}}\left(\frac{R}{2}\right)^{3/2}=% \frac{\pi R^{3/2}}{\sqrt{2G(M+m)}}.
  7. R / 2 R/2
  8. M M
  9. R R
  10. M M
  11. R R
  12. R / 2 R/2
  13. R R
  14. M M
  15. t ff = t orbit / 2 = π 2 R 3 / 2 2 G ( M + m ) t_{\,\text{ff}}=t_{\,\text{orbit}}/2=\frac{\pi}{2}\frac{R^{3/2}}{\sqrt{2G(M+m)}}
  16. t orbit t_{\,\text{orbit}}
  17. P / 32 P/\sqrt{32}
  18. M M
  19. ρ \rho
  20. ρ = 3 M 4 π R 3 \rho=\frac{3M}{4\pi R^{3}}
  21. ( 4 / 3 ) π R 3 . {(4/3)\pi R^{3}}.
  22. R R
  23. R R
  24. R R
  25. M M
  26. R R
  27. t ff = 3 π 32 G ρ 0.5427 1 G ρ 66430 s 1 ρ t_{\,\text{ff}}=\sqrt{\frac{3\pi}{32G\rho}}\simeq 0.5427\frac{1}{\sqrt{G\rho}}% \simeq 66430\,{\rm s}\frac{1}{\sqrt{\rho}}
  28. M m M\gg m
  29. t ff 35 min ρ g cm 3 . t_{\,\text{ff}}\simeq\frac{35\,\mbox{min}~{}}{\sqrt{\rho}}\cdot\sqrt{\frac{% \mbox{g}~{}}{\mbox{cm}~{}^{3}}}.
  30. 4 3 π \frac{4}{3\pi}

Free-radical_halogenation.html

  1. C H 4 + C l 2 u v l i g h t C H 3 C l + H C l CH_{4}+Cl_{2}\xrightarrow{uv\ light}CH_{3}Cl+HCl

Free_field.html

  1. μ μ ϕ + m 2 ϕ = 0 \partial^{\mu}\partial_{\mu}\phi+m^{2}\phi=0
  2. { ϕ ( x ) , ϕ ( y ) } = Δ ( x ; y ) \{\phi(x),\phi(y)\}=\Delta(x;y)
  3. [ ϕ [ f ] , ϕ [ g ] ] = i Δ [ f , g ] [\phi[f],\phi[g]]=i\Delta[f,g]\,
  4. 𝒯 { [ ( ( μ μ + m 2 ) ϕ ) [ f ] , ϕ [ g ] ] } = - i d d x f ( x ) g ( x ) \mathcal{T}\{[((\partial^{\mu}\partial_{\mu}+m^{2})\phi)[f],\phi[g]]\}=-i\int d% ^{d}xf(x)g(x)
  5. 𝒯 \mathcal{T}
  6. [ ϕ [ f ] , ϕ [ g ] ] = 0 [\phi[f],\phi[g]]=0

Free_logic.html

  1. x A x A \forall xA\rightarrow\exists xA
  2. x A A ( r / x ) \forall xA\rightarrow A(r/x)
  3. A r x A x Ar\rightarrow\exists xAx
  4. x ( F x G x ) x F x x ( F x G x ) \forall x(Fx\rightarrow Gx)\land\exists xFx\rightarrow\exists x(Fx\land Gx)
  5. x A E ! t x A \forall xA\land E!t\rightarrow\exists xA
  6. x F x ( x ( E ! F x ) ) \exists xFx\rightarrow(\exists x(E!Fx))
  7. F y ( E ! y x F x ) Fy\rightarrow(E!y\rightarrow\exists xFx)

Free_particle.html

  1. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}
  2. E = 1 2 m v 2 E=\frac{1}{2}mv^{2}
  3. - 2 2 m 2 ψ ( 𝐫 , t ) = i t ψ ( 𝐫 , t ) -\frac{\hbar^{2}}{2m}\nabla^{2}\ \psi(\mathbf{r},t)=i\hbar\frac{\partial}{% \partial t}\psi(\mathbf{r},t)
  4. ψ ( 𝐫 , t ) = A e i ( 𝐤 𝐫 - ω t ) = A e i ( 𝐩 𝐫 - E t ) / \psi(\mathbf{r},t)=Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}=Ae^{i(\mathbf{p}% \cdot\mathbf{r}-Et)/\hbar}
  5. Δ p x Δ x 2 , Δ E Δ t \Delta p_{x}\Delta x\geq\frac{\hbar}{2},\quad\Delta E\Delta t\geq\hbar
  6. 𝐩 = 𝐤 , E = ω \mathbf{p}=\hbar\mathbf{k},\quad E=\hbar\omega
  7. E = T 2 k 2 2 m = ω E=T\,\rightarrow\,\frac{\hbar^{2}k^{2}}{2m}=\hbar\omega
  8. ρ ( 𝐫 , t ) = ψ * ( 𝐫 , t ) ψ ( 𝐫 , t ) = | ψ ( 𝐫 , t ) | 2 \rho(\mathbf{r},t)=\psi^{*}(\mathbf{r},t)\psi(\mathbf{r},t)=|\psi(\mathbf{r},t% )|^{2}
  9. all space | ψ ( 𝐫 , t ) | 2 d 3 𝐫 = 1 \int_{\mathrm{all\,space}}|\psi(\mathbf{r},t)|^{2}d^{3}\mathbf{r}=1
  10. ψ ( 𝐫 , t ) = 1 ( 2 π ) 3 all 𝐩 space A ( 𝐩 ) e i ( 𝐩 𝐫 - E t ) / d 3 𝐩 = 1 ( 2 π ) 3 all 𝐤 space A ( 𝐤 ) e i ( 𝐤 𝐫 - ω t ) d 3 𝐤 \psi(\mathbf{r},t)=\frac{1}{(\sqrt{2\pi}\hbar)^{3}}\int_{\mathrm{all\,\,% \textbf{p}\,space}}A(\mathbf{p})e^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar}d^{3}% \mathbf{p}=\frac{1}{(\sqrt{2\pi})^{3}}\int_{\mathrm{all\,\,\textbf{k}\,space}}% A(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}d^{3}\mathbf{k}
  11. E = E ( 𝐩 ) = 𝐩 2 2 m E=E(\mathbf{p})=\frac{\mathbf{p}^{2}}{2m}
  12. ω = ω ( 𝐤 ) = 𝐤 2 2 m \omega=\omega(\mathbf{k})=\frac{\hbar\mathbf{k}^{2}}{2m}
  13. A ( 𝐩 ) = A ( 𝐤 ) A(\mathbf{p})=A(\hbar\mathbf{k})
  14. A ( 𝐤 ) A(\mathbf{k})
  15. A ^ ( 𝐤 ) = A ( 𝐤 ) \hat{A}(\mathbf{k})=A(\hbar\mathbf{k})
  16. A A
  17. A ^ \hat{A}
  18. 𝐩 = ψ | - i | ψ = all space ψ * ( 𝐫 , t ) ( - i ) ψ ( 𝐫 , t ) d 3 𝐫 = 𝐤 \langle\mathbf{p}\rangle=\left\langle\psi\left|-i\hbar\nabla\right|\psi\right% \rangle=\int_{\mathrm{all\,space}}\psi^{*}(\mathbf{r},t)(-i\hbar\nabla)\psi(% \mathbf{r},t)d^{3}\mathbf{r}=\hbar\mathbf{k}
  19. 𝐩 = all space ψ * ( 𝐫 , t ) ( - i ) ψ ( 𝐫 , t ) d 3 𝐫 = all 𝐤 space 𝐤 | A ( 𝐤 ) | 2 d 3 𝐤 \langle\mathbf{p}\rangle=\int_{\mathrm{all\,space}}\psi^{*}(\mathbf{r},t)(-i% \hbar\nabla)\psi(\mathbf{r},t)d^{3}\mathbf{r}=\int_{\mathrm{all\,\,\textbf{k}% \,space}}\hbar\mathbf{k}|A(\mathbf{k})|^{2}d^{3}\mathbf{k}
  20. E = ψ | i t | ψ = all space ψ * ( 𝐫 , t ) ( i t ) ψ ( 𝐫 , t ) d 3 𝐫 = ω \langle E\rangle=\left\langle\psi\left|i\hbar\frac{\partial}{\partial t}\right% |\psi\right\rangle=\int_{\mathrm{all\,space}}\psi^{*}(\mathbf{r},t)\left(i% \hbar\frac{\partial}{\partial t}\right)\psi(\mathbf{r},t)d^{3}\mathbf{r}=\hbar\omega
  21. E = 𝐩 2 2 m . \langle E\rangle=\frac{\langle\mathbf{p}\rangle^{2}}{2m}.
  22. E = p 2 2 m \langle E\rangle=\frac{\langle p^{2}\rangle}{2m}
  23. v g = d ω d k v_{g}=\frac{d\omega}{dk}
  24. v p = ω k = E p = p 2 m = v 2 v_{p}=\frac{\omega}{k}=\frac{E}{p}=\frac{p}{2m}=\frac{v}{2}

Free_regular_set.html

  1. x X x\in X
  2. g ( U ) U = g(U)\cap U=\varnothing
  3. g G g\in G
  4. Ω = Ω ( G ) \Omega=\Omega(G)
  5. Ω \Omega
  6. Ω / G \Omega/G
  7. Ω ( Γ ) = { τ H : | τ | > 1 , | τ + τ ¯ | < 1 } \Omega(\Gamma)=\{\tau\in H:|\tau|>1,|\tau+\overline{\tau}|<1\}
  8. Γ \Gamma

Freedman–Diaconis_rule.html

  1. Bin size = 2 IQR ( x ) n - 1 / 3 \,\text{Bin size}=2\,\,\text{IQR}(x)n^{-1/3}\;
  2. IQR ( x ) \scriptstyle\operatorname{IQR}(x)\;
  3. n \scriptstyle n\;
  4. x . \scriptstyle x.\;
  5. 1 + log 2 n \scriptstyle 1+\log_{2}n

Frequency_multiplier.html

  1. x ( t ) = A sin ( 2 π f t ) x(t)=A\sin(2\pi ft)\,
  2. x ( t ) = k = - c k e i 2 π k f t . x(t)=\sum_{k=-\infty}^{\infty}c_{k}e^{i2\pi kft}.
  3. c k = 1 2 π 0 T x ( t ) e - i 2 π k t / T d t c_{k}=\frac{1}{2\pi}\int_{0}^{T}x(t)\,e^{-i2\pi kt/T}\,dt

Friction_stir_welding.html

  1. Q total = 2 3 π P μ ω ( R shoulder 3 - R pin 3 ) Q\text{total}={2\over 3}\pi P\mu\omega\left(R\text{shoulder}^{3}-R\text{pin}^{% 3}\right)
  2. Q total = 2 3 π τ ω ( R shoulder 3 - R pin 3 ) Q\text{total}={2\over 3}\pi\tau\omega\left(R\text{shoulder}^{3}-R\text{pin}^{3% }\right)

Friedrich_Hasenöhrl.html

  1. m e m = 4 3 E e m c 2 m_{em}=\frac{4}{3}\cdot\frac{E_{em}}{c^{2}}
  2. μ = 8 3 E 0 𝔅 2 \mu=\frac{8}{3}\frac{E_{0}}{\mathfrak{B}^{2}}
  3. 𝔅 \mathfrak{B}
  4. m = 8 3 h ε 0 c 2 m=\frac{8}{3}\cdot\frac{h\,\varepsilon_{0}}{c^{2}}
  5. m = 4 3 h ε 0 c 2 m=\frac{4}{3}\cdot\frac{h\,\varepsilon_{0}}{c^{2}}
  6. E = m c 2 E=mc^{2}
  7. m = E / c 2 m=E/c^{2}
  8. E = m c 2 \displaystyle{E=mc^{2}}

Friis_formulas_for_noise.html

  1. F t o t a l = F 1 + F 2 - 1 G 1 + F 3 - 1 G 1 G 2 + F 4 - 1 G 1 G 2 G 3 + + F n - 1 G 1 G 2 G n - 1 F_{total}=F_{1}+\frac{F_{2}-1}{G_{1}}+\frac{F_{3}-1}{G_{1}G_{2}}+\frac{F_{4}-1% }{G_{1}G_{2}G_{3}}+...+\frac{F_{n}-1}{G_{1}G_{2}...G_{n-1}}
  2. F i F_{i}
  3. G i G_{i}
  4. F r e c e i v e r = F L N A + ( F r e s t - 1 ) G L N A F_{receiver}=F_{LNA}+\frac{(F_{rest}-1)}{G_{LNA}}
  5. F r e s t F_{rest}
  6. F r e c e i v e r F_{receiver}
  7. F L N A F_{LNA}
  8. T e q = T 1 + T 2 G 1 + T 3 G 1 G 2 + T_{eq}=T_{1}+\frac{T_{2}}{G_{1}}+\frac{T_{3}}{G_{1}G_{2}}+...

Frobenius_algebra.html

  1. ( A , μ , η , δ , ε ) (A,\mu,\eta,\delta,\varepsilon)
  2. ( C , , I ) (C,\otimes,I)
  3. μ : A A A , η : I A , δ : A A A and ε : A I \mu:A\otimes A\to A,\qquad\eta:I\to A,\qquad\delta:A\to A\otimes A\qquad% \mathrm{and}\qquad\varepsilon:A\to I
  4. ( A , μ , η ) (A,\mu,\eta)\,
  5. ( A , δ , ε ) (A,\delta,\varepsilon)
  6. μ δ = Id A \mu\circ\delta=\mathrm{Id}_{A}
  7. { x i } i = 1 n \{x_{i}\}^{n}_{i=1}
  8. { y i } i = 1 n \{y_{i}\}^{n}_{i=1}
  9. i = 1 n E ( a x i ) y i = a = i = 1 n x i E ( y i a ) \sum_{i=1}^{n}E(ax_{i})y_{i}=a=\sum_{i=1}^{n}x_{i}E(y_{i}a)
  10. x i , y i x_{i},y_{i}
  11. x i , y i x_{i},y_{i}
  12. E ( g G n g g ) = h H n h h for n g k E\left(\sum_{g\in G}n_{g}g\right)=\sum_{h\in H}n_{h}h\ \ \ \,\text{ for }n_{g}\in k
  13. E ( g i - 1 g j ) = δ i j 1 E(g_{i}^{-1}g_{j})=\delta_{ij}1
  14. x i = g i , y i = g i - 1 x_{i}=g_{i},y_{i}=g_{i}^{-1}
  15. i = 1 n g i E ( g i - 1 g G n g g ) = i h H n g i h g i h = g G n g g \sum_{i=1}^{n}g_{i}E(g_{i}^{-1}\sum_{g\in G}n_{g}g)=\sum_{i}\sum_{h\in H}n_{g_% {i}h}g_{i}h=\sum_{g\in G}n_{g}g
  16. H g 1 - 1 , , H g n - 1 Hg_{1}^{-1},\ldots,Hg_{n}^{-1}
  17. B = { x A | g G , g ( x ) = x } . B=\{x\in A|\forall g\in G,g(x)=x\}.
  18. { a i } i = 1 n , { b i } i = 1 n \{a_{i}\}_{i=1}^{n},\{b_{i}\}_{i=1}^{n}
  19. g G : i = 1 n a i g ( b i ) = δ g , 1 G 1 A \forall g\in G:\ \ \sum_{i=1}^{n}a_{i}g(b_{i})=\delta_{g,1_{G}}1_{A}
  20. g G : i = 1 n g ( a i ) b i = δ g , 1 G 1 A . \forall g\in G:\ \ \sum_{i=1}^{n}g(a_{i})b_{i}=\delta_{g,1_{G}}1_{A}.
  21. E ( a ) = g G g ( a ) E(a)=\sum_{g\in G}g(a)
  22. x A : i = 1 n E ( x a i ) b i = x = i = 1 n a i E ( b i x ) . \forall x\in A:\ \ \sum_{i=1}^{n}E(xa_{i})b_{i}=x=\sum_{i=1}^{n}a_{i}E(b_{i}x).
  23. e = i = 1 n a i B b i e=\sum_{i=1}^{n}a_{i}\otimes_{B}b_{i}
  24. i = 1 n a i b i = 1 \sum_{i=1}^{n}a_{i}b_{i}=1
  25. a B 1 = g G t g g ( a ) a\otimes_{B}1=\sum_{g\in G}t_{g}g(a)
  26. t g = i = 1 n a i B g ( b i ) t_{g}=\sum_{i=1}^{n}a_{i}\otimes_{B}g(b_{i})

Frobenius_group.html

  1. G = K H G=K\rtimes H
  2. x a x + b x\mapsto ax+b
  3. a 0 a\neq 0
  4. G = K H G=K\rtimes H

Froth_flotation.html

  1. γ l v cos θ = ( γ s v - γ s l ) \gamma_{lv}\,\text{cos}\theta=(\gamma_{sv}-\gamma_{sl})
  2. R R
  3. R = N c ( π 4 ) ( d p + d b ) 2 H c R=\frac{N_{c}}{\left(\tfrac{\pi}{4}\right)\left(d_{p}+d_{b}\right)^{2}Hc}
  4. N c = P N c i N_{c}=PN_{c}^{i}
  5. P P
  6. N c i N_{c}^{i}
  7. d p d_{p}
  8. d b d_{b}
  9. H H
  10. c c
  11. F F
  12. C C
  13. T T
  14. c c
  15. t t
  16. f f
  17. F C \tfrac{F}{C}
  18. F C = c - t f - t \frac{F}{C}=\frac{c-t}{f-t}
  19. \Chi R \Chi_{R}
  20. \Chi R = 100 ( c f ) ( f - t c - t ) \Chi_{R}=100\left(\frac{c}{f}\right)\left(\frac{f-t}{c-t}\right)
  21. \Chi L \Chi_{L}
  22. \Chi L = 100 - \Chi R \Chi_{L}=100-\Chi_{R}
  23. ( \Chi W ) \left(\Chi_{W}\right)
  24. \Chi W = 100 ( C F ) = 100 f - t c - t \Chi_{W}=100\left(\frac{C}{F}\right)=100\frac{f-t}{c-t}

Fuchsian_model.html

  1. π 1 ( R ) \pi_{1}(R)
  2. S L ( 2 , ) SL(2,\mathbb{R})
  3. f : R f:\mathbb{H}\rightarrow R
  4. R h = / Γ R^{h}=\mathbb{H}/\Gamma
  5. R h R^{h}
  6. ρ : G P S L ( 2 , ) \rho:G\rightarrow PSL(2,\mathbb{R})
  7. ρ ( G ) \rho(G)
  8. A ( G ) = { ρ : ρ defined as above } A(G)=\{\rho:\rho\mbox{ defined as above }~{}\}
  9. ρ A ( G ) \rho\in A(G)
  10. h γ h - 1 = ρ ( γ ) h\circ\gamma\circ h^{-1}=\rho(\gamma)
  11. γ G \gamma\in G

Fulkerson_Prize.html

  1. O ( log n ) O(\log n)
  2. O ( log n ) O(\sqrt{\log n})

Function_composition_(computer_science).html

  1. f f
  2. g g
  3. z = f ( y ) z=f(y)
  4. y = g ( x ) y=g(x)
  5. y = g ( x ) y=g(x)
  6. y y
  7. z = f ( y ) z=f(y)

Function_of_a_real_variable.html

  1. f : X f:X\rightarrow\mathbb{R}
  2. V : X V:X\rightarrow\mathbb{R}
  3. X = { x : 0 x } X=\{x\in\mathbb{R}\,:\,0\leq x\}
  4. f ( x ) = x f(x)=\sqrt{x}
  5. f ( x ) f(x)
  6. f f
  7. f f
  8. ( x ) r (x)\mapsto r
  9. r f : ( x ) r f ( x ) rf:(x)\mapsto rf(x)
  10. f + g : ( x ) f ( x ) + g ( x ) f+g:(x)\mapsto f(x)+g(x)
  11. f g : ( x ) f ( x ) g ( x ) f\,g:(x)\mapsto f(x)\,g(x)
  12. 1 / f : ( x ) 1 / f ( x ) , 1/f:(x)\mapsto 1/f(x),
  13. d ( x , y ) = | x - y | d(x,y)=|x-y|
  14. a a
  15. | f ( x ) - f ( a ) | < ϵ |f(x)-f(a)|<\epsilon
  16. x x
  17. d ( x , a ) < φ . d(x,a)<\varphi.
  18. a a
  19. f ( a ) . f(a).
  20. L = lim x a f ( x ) , L=\lim_{x\rightarrow a}f(x),
  21. | f ( x ) - L | < ε |f(x)-L|<\varepsilon
  22. d ( x , a ) < δ . d(x,a)<\delta.
  23. f ( a ) = lim x a f ( x ) . f(a)=\lim_{x\rightarrow a}f(x).
  24. y 1 = f 1 ( x ) , y 2 = f 2 ( x ) , , y n = f n ( x ) y_{1}=f_{1}(x)\,,\quad y_{2}=f_{2}(x)\,,\ldots,y_{n}=f_{n}(x)
  25. 𝐲 = ( y 1 , y 2 , , y n ) = [ f 1 ( x ) , f 2 ( x ) , , f n ( x ) ] \mathbf{y}=(y_{1},y_{2},\ldots,y_{n})=[f_{1}(x),f_{2}(x),\ldots,f_{n}(x)]
  26. d 𝐲 d x = ( d y 1 d x , d y 2 d x , , d y n d x ) \frac{d\mathbf{y}}{dx}=\left(\frac{dy_{1}}{dx},\frac{dy_{2}}{dx},\ldots,\frac{% dy_{n}}{dx}\right)
  27. a b 𝐲 ( x ) d 𝐫 = a b 𝐲 ( x ) d 𝐫 ( x ) d x d x \int_{a}^{b}\mathbf{y}(x)\cdot d\mathbf{r}=\int_{a}^{b}\mathbf{y}(x)\cdot\frac% {d\mathbf{r}(x)}{dx}dx
  28. ϕ : 2 { 0 } \phi:\mathbb{R}^{2}\rightarrow\{0\}
  29. ϕ ( x , y ) = 0 \phi(x,y)=0
  30. y = f ( x ) y=f(x)
  31. ϕ ( x , y ) = y - f ( x ) = 0 \phi(x,y)=y-f(x)=0
  32. r 1 : \displaystyle r_{1}:\mathbb{R}\rightarrow\mathbb{R}
  33. 𝐫 : n , 𝐫 = 𝐫 ( t ) \mathbf{r}:\mathbb{R}\rightarrow\mathbb{R}^{n}\,,\quad\mathbf{r}=\mathbf{r}(t)
  34. 𝐫 ( t ) = [ r 1 ( t ) , r 2 ( t ) , , r n ( t ) ] \mathbf{r}(t)=[r_{1}(t),r_{2}(t),\ldots,r_{n}(t)]
  35. r 1 ( t ) - a 1 d r 1 ( t ) / d t = r 2 ( t ) - a 2 d r 2 ( t ) / d t = = r n ( t ) - a n d r n ( t ) / d t \frac{r_{1}(t)-a_{1}}{dr_{1}(t)/dt}=\frac{r_{2}(t)-a_{2}}{dr_{2}(t)/dt}=\cdots% =\frac{r_{n}(t)-a_{n}}{dr_{n}(t)/dt}
  36. ( p 1 - a 1 ) d r 1 ( t ) d t + ( p 2 - a 2 ) d r 2 ( t ) d t + + ( p n - a n ) d r n ( t ) d t = 0 (p_{1}-a_{1})\frac{dr_{1}(t)}{dt}+(p_{2}-a_{2})\frac{dr_{2}(t)}{dt}+\cdots+(p_% {n}-a_{n})\frac{dr_{n}(t)}{dt}=0
  37. ( 𝐩 - 𝐚 ) d 𝐫 ( t ) d t = 0 (\mathbf{p}-\mathbf{a})\cdot\frac{d\mathbf{r}(t)}{dt}=0
  38. R ( θ ) = [ cos θ - sin θ sin θ cos θ ] R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
  39. Λ ( β ) = [ 1 1 - β 2 - β 1 - β 2 0 0 - β 1 - β 2 1 1 - β 2 0 0 0 0 1 0 0 0 0 1 ] \Lambda(\beta)=\begin{bmatrix}\frac{1}{\sqrt{1-\beta^{2}}}&-\frac{\beta}{\sqrt% {1-\beta^{2}}}&0&0\\ -\frac{\beta}{\sqrt{1-\beta^{2}}}&\frac{1}{\sqrt{1-\beta^{2}}}&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}
  40. i t Ψ = H ^ Ψ i\hbar\frac{\partial}{\partial t}\Psi=\hat{H}\Psi
  41. f ( x ) f(x)
  42. f ( x ) = g ( x ) + i h ( x ) , f(x)=g(x)+ih(x),
  43. g g
  44. h h

Functional_(mathematics).html

  1. x 0 f ( x 0 ) x_{0}\mapsto f(x_{0})
  2. x 0 x_{0}
  3. f f
  4. f f ( x 0 ) f\mapsto f(x_{0})
  5. x 0 x_{0}
  6. f I [ f ] = Ω H ( f ( x ) , f ( x ) , ) μ ( d x ) f\mapsto I[f]=\int_{\Omega}H(f(x),f^{\prime}(x),\ldots)\;\mu(\mbox{d}~{}x)
  7. f x 0 x 1 f ( x ) d x f\mapsto\int_{x_{0}}^{x_{1}}f(x)\;\mathrm{d}x
  8. f ( | f | p d x ) 1 / p f\mapsto\left(\int|f|^{p}\;\mathrm{d}x\right)^{1/p}
  9. f x 0 x 1 1 + | f ( x ) | 2 d x f\mapsto\int_{x_{0}}^{x_{1}}\sqrt{1+|f^{\prime}(x)|^{2}}\;\mathrm{d}x
  10. x \vec{x}
  11. X X
  12. y \vec{y}
  13. x y \vec{x}\cdot\vec{y}
  14. x , y \langle\vec{x},\vec{y}\rangle
  15. X X
  16. X X
  17. F ( y ) = x 0 x 1 y ( x ) d x F(y)=\int_{x_{0}}^{x_{1}}y(x)\;\mathrm{d}x
  18. F ( y ) = x 0 x 1 y ( x ) d x x 0 x 1 ( 1 + [ y ( x ) ] 2 ) d x F(y)=\frac{\int_{x_{0}}^{x_{1}}y(x)\;\mathrm{d}x}{\int_{x_{0}}^{x_{1}}(1+[y(x)% ]^{2})\;\mathrm{d}x}
  19. F = G F=G
  20. f f
  21. f ( x + y ) = f ( x ) + f ( y ) f\left(x+y\right)=f\left(x\right)+f\left(y\right)

Fundamental_pair_of_periods.html

  1. ω 1 , ω 2 \Complex \omega_{1},\omega_{2}\in\Complex
  2. 2 \mathbb{R}^{2}
  3. Λ = { m ω 1 + n ω 2 | m , n } \Lambda=\{m\omega_{1}+n\omega_{2}\,\,|\,\,m,n\in\mathbb{Z}\}
  4. ω 1 \omega_{1}
  5. ω 2 \omega_{2}
  6. ( ω 1 , ω 2 ) (\omega_{1},\omega_{2})
  7. ( α 1 , α 2 ) (\alpha_{1},\alpha_{2})
  8. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}
  9. ( α 1 α 2 ) = ( a b c d ) ( ω 1 ω 2 ) , \begin{pmatrix}\alpha_{1}\\ \alpha_{2}\end{pmatrix}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}\omega_{1}\\ \omega_{2}\end{pmatrix},
  10. α 1 = a ω 1 + b ω 2 \alpha_{1}=a\omega_{1}+b\omega_{2}\,
  11. α 2 = c ω 1 + d ω 2 . \alpha_{2}=c\omega_{1}+d\omega_{2}.\,
  12. S L ( 2 , ) SL(2,\mathbb{Z})
  13. 2 \mathbb{Z}^{2}
  14. z z\in\mathbb{C}
  15. z = p + m ω 1 + n ω 2 z=p+m\omega_{1}+n\omega_{2}
  16. \Complex / Λ \Complex/\Lambda
  17. U = { z H : | z | > 1 , | Re ( z ) | < 1 2 } . U=\left\{z\in H:\left|z\right|>1,\,\left|\,\mbox{Re}~{}(z)\,\right|<\tfrac{1}{% 2}\right\}.
  18. D = U { z H : | z | 1 , Re ( z ) = - 1 2 } { z H : | z | = 1 , Re ( z ) 0 } . D=U\cup\left\{z\in H:\left|z\right|\geq 1,\,\mbox{Re}~{}(z)=-\tfrac{1}{2}% \right\}\cup\left\{z\in H:\left|z\right|=1,\,\mbox{Re}~{}(z)\leq 0\right\}.
  19. ( ω 1 , ω 2 ) (\omega_{1},\omega_{2})
  20. ( - ω 1 , - ω 2 ) (-\omega_{1},-\omega_{2})
  21. τ = i \tau=i
  22. ( i ω 1 , i ω 2 ) (i\omega_{1},i\omega_{2})
  23. ( ω 1 , ω 2 ) (\omega_{1},\omega_{2})
  24. ( τ ω 1 , τ ω 2 ) (\tau\omega_{1},\tau\omega_{2})
  25. ( τ 2 ω 1 , τ 2 ω 2 ) (\tau^{2}\omega_{1},\tau^{2}\omega_{2})
  26. τ = i \tau=i

Fundamental_polygon.html

  1. A A - 1 AA^{-1}
  2. A B B - 1 A - 1 ABB^{-1}A^{-1}
  3. A A AA
  4. A B A B ABAB
  5. A B A B - 1 ABAB^{-1}
  6. A A B B AABB
  7. A B A - 1 B - 1 ABA^{-1}B^{-1}
  8. A B C A - 1 B - 1 C - 1 ABCA^{-1}B^{-1}C^{-1}
  9. 2 \mathbb{R}^{2}
  10. A A
  11. A ( x , y ) = ( x + 1 , y ) A(x,y)=(x+1,y)
  12. B ( x , y ) = ( x , y + 1 ) B(x,y)=(x,y+1)
  13. A , B A,B
  14. Γ = 2 \Gamma=\mathbb{Z}^{2}
  15. T = 2 / 2 T=\mathbb{R}^{2}/\mathbb{Z}^{2}
  16. A , B A,B
  17. A B A - 1 B - 1 = 1 ABA^{-1}B^{-1}=1
  18. A B C A - 1 B - 1 C - 1 = 1 ABCA^{-1}B^{-1}C^{-1}=1
  19. Γ \Gamma
  20. PSL ( 2 , ) \operatorname{PSL}(2,\mathbb{R})
  21. A 1 B 1 A 1 - 1 B 1 - 1 A 2 B 2 A 2 - 1 B 2 - 1 A n B n A n - 1 B n - 1 = 1 A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A% _{n}^{-1}B_{n}^{-1}=1
  22. A 1 A 1 A 2 A 2 A n A n A_{1}A_{1}A_{2}A_{2}\cdots A_{n}A_{n}
  23. A 1 B 1 A 1 - 1 B 1 - 1 A 2 B 2 A 2 - 1 B 2 - 1 A n B n A n - 1 B n = 1 A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A% _{n}^{-1}B_{n}=1
  24. B n B_{n}
  25. A 1 B 1 A 1 - 1 B 1 - 1 A 2 B 2 A 2 - 1 B 2 - 1 A n B n A n - 1 B n - 1 C 2 = 1 A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A% _{n}^{-1}B_{n}^{-1}C^{2}=1
  26. z 0 z_{0}
  27. F = { z : d ( z , z 0 ) < d ( z , g z 0 ) g Γ , g 1 } F=\{z\in\mathbb{H}:d(z,z_{0})<d(z,gz_{0})\;\;\forall g\in\Gamma,g\neq 1\}
  28. H = g Γ g F ¯ H=\cup_{g\in\Gamma}\,g\overline{F}
  29. F ¯ \overline{F}
  30. A A
  31. A - 1 A^{-1}
  32. A 1 B 1 A 1 - 1 B 1 - 1 A 2 B 2 A 2 - 1 B 2 - 1 A n B n A n - 1 B n - 1 A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A% _{n}^{-1}B_{n}^{-1}
  33. π 1 ( / Γ ) \pi_{1}(\mathbb{H}/\Gamma)
  34. π 1 ( / Γ ) \pi_{1}(\mathbb{H}/\Gamma)
  35. A 1 , B 1 , A 2 , B 2 , A n , B n A_{1},B_{1},A_{2},B_{2},\cdots A_{n},B_{n}
  36. A 1 B 1 A 1 - 1 B 1 - 1 A 2 B 2 A 2 - 1 B 2 - 1 A n B n A n - 1 B n - 1 = 1 A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A% _{n}^{-1}B_{n}^{-1}=1
  37. 4 π ( n - 1 ) 4\pi(n-1)
  38. Γ \Gamma
  39. n n
  40. 2 n 2n
  41. a k a_{k}
  42. a k = ( cos k α - sin k α sin k α cos k α ) ( e p 0 0 e - p ) ( cos k α sin k α - sin k α cos k α ) a_{k}=\left(\begin{matrix}\cos k\alpha&-\sin k\alpha\\ \sin k\alpha&\cos k\alpha\end{matrix}\right)\left(\begin{matrix}e^{p}&0\\ 0&e^{-p}\end{matrix}\right)\left(\begin{matrix}\cos k\alpha&\sin k\alpha\\ -\sin k\alpha&\cos k\alpha\end{matrix}\right)
  43. 0 k < 2 n 0\leq k<2n
  44. α = π 4 n ( 2 n - 1 ) \alpha=\frac{\pi}{4n}\left(2n-1\right)
  45. β = π 4 n \beta=\frac{\pi}{4n}
  46. p = ln cos β + cos 2 β sin β p=\ln\frac{\cos\beta+\sqrt{\cos 2\beta}}{\sin\beta}
  47. a 0 a 1 a 2 n - 1 a 0 - 1 a 1 - 1 a 2 n - 1 - 1 = 1 a_{0}a_{1}\cdots a_{2n-1}a^{-1}_{0}a^{-1}_{1}\cdots a^{-1}_{2n-1}=1

Fundamental_solution.html

  1. L L
  2. δ ( x ) δ(x)
  3. F F
  4. L F = δ ( x ) . LF=δ(x).
  5. F F
  6. L f = s i n ( x ) Lf=sin(x)
  7. L = d 2 d x 2 L=\frac{d^{2}}{dx^{2}}
  8. L F = δ ( x ) LF=δ(x)
  9. d 2 d x 2 F ( x ) = δ ( x ) . \frac{d^{2}}{dx^{2}}F(x)=\delta(x)~{}.
  10. H H
  11. d d x H ( x ) = δ ( x ) , \frac{d}{dx}H(x)=\delta(x)~{},
  12. d d x F ( x ) = H ( x ) + C . \frac{d}{dx}F(x)=H(x)+C~{}.
  13. C C
  14. C C
  15. F ( x ) = x H ( x ) - 1 2 x = 1 2 | x | . F(x)=xH(x)-\frac{1}{2}x=\frac{1}{2}|x|~{}.
  16. L L
  17. d 2 d x 2 f ( x ) = sin ( x ) . \frac{d^{2}}{dx^{2}}f(x)=\sin(x)~{}.
  18. x x
  19. f ( x ) = - 1 2 | x - y | sin ( y ) d y . f(x)=\int_{-\infty}^{\infty}\frac{1}{2}|x-y|\sin(y)dy~{}.
  20. f ( x ) = s i n x f(x)=−sinx
  21. x x
  22. f f
  23. d 2 d x 2 f ( x ) = I ( x ) , \frac{d^{2}}{dx^{2}}f(x)=I(x)~{},
  24. I I
  25. I F I∗F
  26. I I
  27. F F
  28. g g
  29. F g F∗g
  30. L f = g ( x ) Lf=g(x)
  31. F g F∗g
  32. L ( F g ) = g L(F∗g)=g
  33. L L
  34. L ( F * g ) = ( L F ) * g , L(F*g)=(LF)*g~{},
  35. L L
  36. F F
  37. δ * g . \delta*g~{}.
  38. g ( x ) g(x)
  39. L ( F * g ) = ( L F ) * g = δ ( x ) * g ( x ) = - δ ( x - y ) g ( y ) d y = g ( x ) . L(F*g)=(LF)*g=\delta(x)*g(x)=\int_{-\infty}^{\infty}\delta(x-y)g(y)dy=g(x)~{}.
  40. F F
  41. F g F∗g
  42. L f = g ( x ) Lf=g(x)
  43. [ - Δ ] Φ ( 𝐱 , 𝐱 ) = δ ( 𝐱 - 𝐱 ) [-\Delta]\Phi(\mathbf{x},\mathbf{x}^{\prime})=\delta(\mathbf{x}-\mathbf{x}^{% \prime})
  44. Φ 2 D ( 𝐱 , 𝐱 ) = - 1 2 π ln | 𝐱 - 𝐱 | , Φ 3 D ( 𝐱 , 𝐱 ) = 1 4 π | 𝐱 - 𝐱 | . \Phi_{2D}(\mathbf{x},\mathbf{x}^{\prime})=-\frac{1}{2\pi}\ln|\mathbf{x}-% \mathbf{x}^{\prime}|,\quad\Phi_{3D}(\mathbf{x},\mathbf{x}^{\prime})=\frac{1}{4% \pi|\mathbf{x}-\mathbf{x}^{\prime}|}~{}.
  45. k k
  46. [ - Δ + k 2 ] Φ ( 𝐱 , 𝐱 ) = δ ( 𝐱 - 𝐱 ) , [-\Delta+k^{2}]\Phi(\mathbf{x},\mathbf{x}^{\prime})=\delta(\mathbf{x}-\mathbf{% x}^{\prime})~{},
  47. Φ 2 D ( 𝐱 , 𝐱 ) = 1 2 π K 0 ( k | 𝐱 - 𝐱 | ) , Φ 3 D ( 𝐱 , 𝐱 ) = 1 4 π | 𝐱 - 𝐱 | exp ( - k | 𝐱 - 𝐱 | ) . \Phi_{2D}(\mathbf{x},\mathbf{x}^{\prime})=\frac{1}{2\pi}K_{0}(k|\mathbf{x}-% \mathbf{x}^{\prime}|),\quad\Phi_{3D}(\mathbf{x},\mathbf{x}^{\prime})=\frac{1}{% 4\pi|\mathbf{x}-\mathbf{x}^{\prime}|}\exp(-k|\mathbf{x}-\mathbf{x}^{\prime}|)~% {}.
  48. [ - Δ 2 ] Φ ( 𝐱 , 𝐱 ) = δ ( 𝐱 - 𝐱 ) [-\Delta^{2}]\Phi(\mathbf{x},\mathbf{x}^{\prime})=\delta(\mathbf{x}-\mathbf{x}% ^{\prime})
  49. Φ 2 D ( 𝐱 , 𝐱 ) = - | 𝐱 - 𝐱 | 2 8 π ( ln | 𝐱 - 𝐱 | - 1 ) , Φ 3 D ( 𝐱 , 𝐱 ) = | 𝐱 - 𝐱 | 8 π . \Phi_{2D}(\mathbf{x},\mathbf{x}^{\prime})=-\frac{|\mathbf{x}-\mathbf{x}^{% \prime}|^{2}}{8\pi}(\ln|\mathbf{x}-\mathbf{x}^{\prime}|-1),\quad\Phi_{3D}(% \mathbf{x},\mathbf{x}^{\prime})=\frac{|\mathbf{x}-\mathbf{x}^{\prime}|}{8\pi}~% {}.

Fundamental_theorem_of_Galois_theory.html

  1. { \{
  2. } \}
  3. { \{
  4. } \}
  5. K = 𝐐 ( 2 , 3 ) = Q 𝐐 ( 2 ) ( ( 3 ) K=\mathbf{Q}(\sqrt{2},\sqrt{3})=Q\mathbf{Q}(\sqrt{2})((\sqrt{3})
  6. K K
  7. 2 \sqrt{2}
  8. 3 \sqrt{3}
  9. K K
  10. ( a + b 2 ) + ( c + d 2 ) 3 (a+b\sqrt{2})+(c+d\sqrt{2})\sqrt{3}
  11. a a
  12. b b
  13. c c
  14. d d
  15. G = G a l ( K / 𝐐 ) G=Gal(K/\mathbf{Q})
  16. K K
  17. a a
  18. 2 \sqrt{2}
  19. 2 \sqrt{2}
  20. 2 –\sqrt{2}
  21. 3 \sqrt{3}
  22. 3 \sqrt{3}
  23. 3 –\sqrt{3}
  24. f f
  25. 2 \sqrt{2}
  26. 2 –\sqrt{2}
  27. f ( ( a + b 2 ) + ( c + d 2 ) 3 ) = ( a - b 2 ) + ( c - d 2 ) 3 = a - b 2 + c 3 - d 6 , f\left((a+b\sqrt{2})+(c+d\sqrt{2})\sqrt{3}\right)=(a-b\sqrt{2})+(c-d\sqrt{2})% \sqrt{3}=a-b\sqrt{2}+c\sqrt{3}-d\sqrt{6},
  28. g g
  29. 3 \sqrt{3}
  30. 3 –\sqrt{3}
  31. g ( ( a + b 2 ) + ( c + d 2 ) 3 ) = ( a + b 2 ) - ( c + d 2 ) 3 = a + b 2 - c 3 - d 6 . g\left((a+b\sqrt{2})+(c+d\sqrt{2})\sqrt{3}\right)=(a+b\sqrt{2})-(c+d\sqrt{2})% \sqrt{3}=a+b\sqrt{2}-c\sqrt{3}-d\sqrt{6}.
  32. K K
  33. e e
  34. f f
  35. g g
  36. ( f g ) ( ( a + b 2 ) + ( c + d 2 ) 3 ) = ( a - b 2 ) - ( c - d 2 ) 3 = a - b 2 - c 3 + d 6 . (fg)\left((a+b\sqrt{2})+(c+d\sqrt{2})\sqrt{3}\right)=(a-b\sqrt{2})-(c-d\sqrt{2% })\sqrt{3}=a-b\sqrt{2}-c\sqrt{3}+d\sqrt{6}.
  37. G = { 1 , f , g , f g } , G=\left\{1,f,g,fg\right\},
  38. G G
  39. K K
  40. K K
  41. G G
  42. 𝐐 \mathbf{Q}
  43. 𝐐 ( 3 ) \mathbf{Q}(\sqrt{3})
  44. f f
  45. 3 \sqrt{3}
  46. 𝐐 ( 2 ) \mathbf{Q}(\sqrt{2})
  47. g g
  48. 2 \sqrt{2}
  49. 𝐐 ( 6 ) \mathbf{Q}(\sqrt{6})
  50. f g fg
  51. 6 \sqrt{6}
  52. ω = - 1 2 + i 3 2 . \omega=\frac{-1}{2}+i\frac{\sqrt{3}}{2}.
  53. f ( θ ) = ω θ , f ( ω ) = ω , f(\theta)=\omega\theta,\quad f(\omega)=\omega,
  54. g ( θ ) = θ , g ( ω ) = ω 2 , g(\theta)=\theta,\quad g(\omega)=\omega^{2},
  55. G = { 1 , f , f 2 , g , g f , g f 2 } . G=\left\{1,f,f^{2},g,gf,gf^{2}\right\}.
  56. E = ( λ ) E=\mathbb{Q}(\lambda)
  57. λ \lambda
  58. G = { λ , 1 1 - λ , λ - 1 λ , 1 λ , λ λ - 1 , 1 - λ } Aut ( E ) G=\left\{{\lambda,\frac{1}{1-\lambda},\frac{\lambda-1}{\lambda},\frac{1}{% \lambda},\frac{\lambda}{\lambda-1},1-\lambda}\right\}\subset{\rm Aut}(E)
  59. S 3 S_{3}
  60. F F
  61. G G
  62. Gal ( E / F ) = G {\rm Gal}(E/F)=G
  63. H H
  64. G G
  65. P ( T ) := h H ( T - h ) E [ T ] P(T):=\prod_{h\in H}(T-h)\in E[T]
  66. H H
  67. E / F E/F
  68. H = { λ , 1 - λ } H=\{\lambda,1-\lambda\}
  69. ( λ ( 1 - λ ) ) \mathbb{Q}(\lambda(1-\lambda))
  70. H = { λ , 1 / λ } H=\{\lambda,1/\lambda\}
  71. ( λ + 1 / λ ) \mathbb{Q}(\lambda+1/\lambda)
  72. F F
  73. G G
  74. ( j ) \mathbb{Q}(j)
  75. j j
  76. P 1 ( ) P^{1}(\mathbb{C})
  77. ( x ) \mathbb{C}(x)

Fundamental_theorems_of_welfare_economics.html

  1. w i w_{i}
  2. Σ i w i = p ω + Σ j p y j * \Sigma_{i}w_{i}=p\cdot\omega+\Sigma_{j}p\cdot y^{*}_{j}
  3. ω \omega
  4. y j * y^{*}_{j}
  5. x i > i x i * x_{i}>_{i}x^{*}_{i}
  6. p x i > w i p\cdot x_{i}>w_{i}
  7. x i * x^{*}_{i}
  8. x i i x i * x_{i}\geq_{i}x^{*}_{i}
  9. p x i w i p\cdot x_{i}\geq w_{i}
  10. x i i x i * x_{i}\geq_{i}x^{*}_{i}
  11. p x i < w i p\cdot x_{i}<w_{i}
  12. x i x^{\prime}_{i}
  13. x i x_{i}
  14. x i * x^{*}_{i}
  15. x i * x^{*}_{i}
  16. ( x , y ) (x,y)
  17. ( x * , y * ) (x^{*},y^{*})
  18. x i i x i * x_{i}\geq_{i}x^{*}_{i}
  19. x i > i x i * x_{i}>_{i}x^{*}_{i}
  20. p x i w i p\cdot x_{i}\geq w_{i}
  21. p x i > w i p\cdot x_{i}>w_{i}
  22. Σ i p x i > Σ i w i = p ω + Σ j p y j * \Sigma_{i}p\cdot x_{i}>\Sigma_{i}w_{i}=p\cdot\omega+\Sigma_{j}p\cdot y^{*}_{j}
  23. y * y^{*}
  24. Σ j p y j * Σ j p y j \Sigma_{j}p\cdot y^{*}_{j}\geq\Sigma_{j}p\cdot y_{j}
  25. Σ i p x i > p ω + Σ j p y j \Sigma_{i}p\cdot x_{i}>p\cdot\omega+\Sigma_{j}p\cdot y_{j}
  26. ( x , y ) (x,y)
  27. ( x * , y * ) (x^{*},y^{*})
  28. Y j Y_{j}
  29. i \geq_{i}
  30. ( x * , y * ) (x^{*},y^{*})
  31. Σ i w i = p ω + Σ j p y j * \Sigma_{i}w_{i}=p\cdot\omega+\Sigma_{j}p\cdot y^{*}_{j}
  32. ω \omega
  33. y j * y^{*}_{j}
  34. p y j p y j * p\cdot y_{j}\leq p\cdot y_{j}^{*}
  35. y j Y j y_{j}\in Y_{j}
  36. y j * y_{j}^{*}
  37. x i > i x i * x_{i}>_{i}x_{i}^{*}
  38. p x i w i p\cdot x_{i}\geq w_{i}
  39. x i x_{i}
  40. x i * x_{i}^{*}
  41. x i * x_{i}^{*}
  42. Σ i x i * = ω + Σ j y j * \Sigma_{i}x_{i}^{*}=\omega+\Sigma_{j}y_{j}^{*}
  43. p x i w i p\cdot x_{i}\geq w_{i}
  44. V i V_{i}
  45. x i * x_{i}^{*}
  46. V i V_{i}
  47. V i V_{i}
  48. i \geq_{i}
  49. V i V_{i}
  50. Y + { ω } Y+\{\omega\}
  51. Y i Y_{i}
  52. Y i Y_{i}
  53. Y + { ω } Y+\{\omega\}
  54. ( x * , y * ) (x^{*},y^{*})
  55. ( x * , y * ) (x^{*},y^{*})
  56. p 0 p\neq 0
  57. p z r p\cdot z\geq r
  58. z V z\in V
  59. p z r p\cdot z\leq r
  60. z Y + { ω } z\in Y+\{\omega\}
  61. x i i x i * x_{i}\geq_{i}x_{i}^{*}
  62. p ( Σ i x i ) r p\cdot(\Sigma_{i}x_{i})\geq r
  63. x i x^{\prime}_{i}
  64. x i x_{i}
  65. x i * x_{i}^{*}
  66. V i V_{i}
  67. p ( Σ i x i ) r p\cdot(\Sigma_{i}x^{\prime}_{i})\geq r
  68. x i x i x^{\prime}_{i}\rightarrow x_{i}
  69. p ( Σ i x i ) r p\cdot(\Sigma_{i}x_{i})\geq r
  70. x i x_{i}
  71. x i * x_{i}^{*}
  72. p ( Σ i x i * ) r p\cdot(\Sigma_{i}x_{i}^{*})\geq r
  73. Σ i x i * Y + { ω } \Sigma_{i}x_{i}^{*}\in Y+\{\omega\}
  74. p ( Σ i x i * ) r p\cdot(\Sigma_{i}x_{i}^{*})\leq r
  75. p ( Σ i x i * ) = r p\cdot(\Sigma_{i}x_{i}^{*})=r
  76. ( x * , y * , p ) (x^{*},y^{*},p)
  77. p ( Σ i x i * ) = r p\cdot(\Sigma_{i}x_{i}^{*})=r
  78. Σ i x i * = ω + Σ j y j * \Sigma_{i}x_{i}^{*}=\omega+\Sigma_{j}y_{j}^{*}
  79. p ( ω + y j + Σ h y h * ) r = p ( ω + y j * + Σ h y h * ) p\cdot(\omega+y_{j}+\Sigma_{h}y_{h}^{*})\leq r=p\cdot(\omega+y_{j}^{*}+\Sigma_% {h}y_{h}^{*})
  80. h j h\neq j
  81. p y j p y j * p\cdot y_{j}\leq p\cdot y_{j}^{*}
  82. p ( x i + Σ k x k * ) r = p ( x i * + Σ k x k * ) p\cdot(x_{i}+\Sigma_{k}x_{k}^{*})\geq r=p\cdot(x_{i}^{*}+\Sigma_{k}x_{k}^{*})
  83. k i k\neq i
  84. p x i p x i * p\cdot x_{i}\geq p\cdot x_{i}^{*}
  85. w i = p x i * w_{i}=p\cdot x_{i}^{*}
  86. x i > i x i * x_{i}>_{i}x_{i}^{*}
  87. p x i w i p\cdot x_{i}\geq w_{i}
  88. x i > i x i * x_{i}>_{i}x_{i}^{*}
  89. p x i > w i p\cdot x_{i}>w_{i}
  90. X i X_{i}
  91. i \geq_{i}
  92. x i x^{\prime}_{i}
  93. x i X i x^{\prime}_{i}\in X_{i}
  94. p x i < w i p\cdot x^{\prime}_{i}<w_{i}
  95. x i > i x i * x_{i}>_{i}x_{i}^{*}
  96. p x i = w i p\cdot x_{i}=w_{i}
  97. x i x_{i}
  98. X i X_{i}
  99. x i ′′ = α x i + ( 1 - α ) x i X i x^{\prime\prime}_{i}=\alpha x_{i}+(1-\alpha)x^{\prime}_{i}\in X_{i}
  100. p x i ′′ < w i p\cdot x^{\prime\prime}_{i}<w_{i}
  101. i \geq_{i}
  102. α \alpha
  103. α x i + ( 1 - α ) x i > i x i * \alpha x_{i}+(1-\alpha)x^{\prime}_{i}>_{i}x_{i}^{*}
  104. x i * x_{i}^{*}
  105. w i w_{i}
  106. x i x^{\prime}_{i}
  107. w i w_{i}

Fusion_energy_gain_factor.html

  1. P h e a t = η h e a t f r e c i r c η e l e c ( 1 - f c h ) P f u s P_{heat}=\eta_{heat}\cdot f_{recirc}\cdot\eta_{elec}\cdot(1-f_{ch})\cdot P_{fus}
  2. Q P f u s P h e a t = 1 η h e a t f r e c i r c η e l e c ( 1 - f c h ) Q\equiv\frac{P_{fus}}{P_{heat}}=\frac{1}{\eta_{heat}\cdot f_{recirc}\cdot\eta_% {elec}\cdot(1-f_{ch})}

Fuzz_testing.html

  1. O ( c n ) O(c^{n})
  2. c > 1 c>1

Fuzzy_measure_theory.html

  1. 𝐗 \mathbf{X}
  2. 𝒞 \mathcal{C}
  3. 𝐗 \mathbf{X}
  4. E , F 𝒞 E,F\in\mathcal{C}
  5. g : 𝒞 g:\mathcal{C}\to\mathbb{R}
  6. 𝒞 g ( ) = 0 \emptyset\in\mathcal{C}\Rightarrow g(\emptyset)=0
  7. E F g ( E ) g ( F ) E\subseteq F\Rightarrow g(E)\leq g(F)
  8. g ( 𝐗 ) = 1 g(\mathbf{X})=1
  9. E , F 𝒞 E,F\in\mathcal{C}
  10. g ( E F ) = g ( E ) + g ( F ) . g(E\cup F)=g(E)+g(F).
  11. E F = E\cap F=\emptyset
  12. g ( E F ) + g ( E F ) g ( E ) + g ( F ) g(E\cup F)+g(E\cap F)\geq g(E)+g(F)
  13. g ( E F ) + g ( E F ) g ( E ) + g ( F ) g(E\cup F)+g(E\cap F)\leq g(E)+g(F)
  14. g ( E F ) g ( E ) + g ( F ) g(E\cup F)\geq g(E)+g(F)
  15. E F = E\cap F=\emptyset
  16. g ( E F ) g ( E ) + g ( F ) g(E\cup F)\leq g(E)+g(F)
  17. E F = E\cap F=\emptyset
  18. | E | = | F | |E|=|F|
  19. g ( E ) = g ( F ) g(E)=g(F)
  20. g ( E ) = 0 g(E)=0
  21. g ( E ) = 1 g(E)=1
  22. E , F X E,F\subseteq X
  23. M ( E ) = F E ( - 1 ) | E \ F | g ( F ) . M(E)=\sum_{F\subseteq E}(-1)^{|E\backslash F|}g(F).
  24. M ( ) = 0 M(\emptyset)=0
  25. F E | i F M ( F ) 0 \sum_{F\subseteq E|i\in F}M(F)\geq 0
  26. E 𝐗 E\subseteq\mathbf{X}
  27. i E i\in E
  28. E 𝐗 M ( E ) = 1. \sum_{E\subseteq\mathbf{X}}M(E)=1.
  29. g ( E ) = F E M ( F ) , E 𝐗 . g(E)=\sum_{F\subseteq E}M(F),\forall E\subseteq\mathbf{X}.
  30. g ( E ) = i E g ( { i } ) g(E)=\sum_{i\in E}g(\{i\})
  31. λ \lambda
  32. λ \lambda
  33. 𝐗 = { x 1 , , x n } \mathbf{X}=\left\{x_{1},\dots,x_{n}\right\}
  34. λ ( - 1 , + ) \lambda\in(-1,+\infty)
  35. λ \lambda
  36. g : 2 X [ 0 , 1 ] g:2^{X}\to[0,1]
  37. g ( X ) = 1 g(X)=1
  38. A , B 𝐗 A,B\subseteq\mathbf{X}
  39. A , B 2 𝐗 A,B\in 2^{\mathbf{X}}
  40. A B = A\cap B=\emptyset
  41. g ( A B ) = g ( A ) + g ( B ) + λ g ( A ) g ( B ) g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B)
  42. { x i } \left\{x_{i}\right\}
  43. g i = g ( { x i } ) g_{i}=g(\left\{x_{i}\right\})
  44. λ \lambda
  45. λ + 1 = i = 1 n ( 1 + λ g i ) \lambda+1=\prod_{i=1}^{n}(1+\lambda g_{i})
  46. λ \lambda
  47. E X E\subseteq X
  48. | E | = k |E|=k
  49. 1 k | 𝐗 | 1\leq k\leq|\mathbf{X}|
  50. M ( E ) = 0 M(E)=0
  51. | E | > k |E|>k
  52. E 𝐗 E\subseteq\mathbf{X}
  53. M ( F ) 0 M(F)\neq 0
  54. | 𝐗 | = n |\mathbf{X}|=n
  55. i , , n X i,\dots,n\in X
  56. ϕ ( i ) = E 𝐗 \ { i } ( n - | E | - 1 ) ! | E | ! n ! [ g ( E { i } ) - g ( E ) ] . \phi(i)=\sum_{E\subseteq\mathbf{X}\backslash\{i\}}\frac{(n-|E|-1)!|E|!}{n!}[g(% E\cup\{i\})-g(E)].
  57. ϕ ( g ) = ( ψ ( 1 ) , , ψ ( n ) ) . \mathbf{\phi}(g)=(\psi(1),\dots,\psi(n)).