wpmath0000003_15

Singular_perturbation.html

  1. φ ( x ) n = 0 N δ n ( ε ) ψ n ( x ) \varphi(x)\approx\sum_{n=0}^{N}\delta_{n}(\varepsilon)\psi_{n}(x)\,
  2. ε 0 \varepsilon\to 0
  3. ε \varepsilon
  4. δ n ( ε ) \delta_{n}(\varepsilon)
  5. ε \varepsilon
  6. δ n ( ε ) = ε n \delta_{n}(\varepsilon)=\varepsilon^{n}
  7. ε \varepsilon
  8. ε \varepsilon
  9. ε u ′′ ( x ) + u ( x ) = - e - x , 0 < x < 1 u ( 0 ) = 0 , u ( 1 ) = 1. \begin{matrix}\varepsilon u^{\prime\prime}(x)+u^{\prime}(x)=-e^{-x},\ \ 0<x<1% \\ u(0)=0,\ \ u(1)=1.\end{matrix}
  10. ε = 0.1 \varepsilon=0.1
  11. ε = 0 \varepsilon=0
  12. x ˙ 1 = f 1 ( x 1 , x 2 ) + ε g 1 ( x 1 , x 2 , ε ) , \dot{x}_{1}=f_{1}(x_{1},x_{2})+\varepsilon g_{1}(x_{1},x_{2},\varepsilon),\,
  13. ε x ˙ 2 = f 2 ( x 1 , x 2 ) + ε g 2 ( x 1 , x 2 , ε ) , \varepsilon\dot{x}_{2}=f_{2}(x_{1},x_{2})+\varepsilon g_{2}(x_{1},x_{2},% \varepsilon),\,
  14. x 1 ( 0 ) = a 1 , x 2 ( 0 ) = a 2 , x_{1}(0)=a_{1},x_{2}(0)=a_{2},\,
  15. 0 < ε < < 1 0<\varepsilon<\!\!<1
  16. x 2 x_{2}
  17. x 1 x_{1}
  18. x ˙ 1 = f 1 ( x 1 , x 2 ) , \dot{x}_{1}=f_{1}(x_{1},x_{2}),\,
  19. f 2 ( x 1 , x 2 ) = 0 , f_{2}(x_{1},x_{2})=0,\,
  20. x 1 ( 0 ) = a 1 , x_{1}(0)=a_{1},\,
  21. ε \varepsilon
  22. u t = ε u x x + u f ( u ) - v g ( u ) , u_{t}=\varepsilon u_{xx}+uf(u)-vg(u),\,
  23. v t = v x x + v h ( u ) , v_{t}=v_{xx}+vh(u),\,
  24. u u
  25. v v
  26. p ( x ) = ε x 3 - x 2 + 1 p(x)=\varepsilon x^{3}-x^{2}+1
  27. ε 0 \varepsilon\to 0
  28. 1 - x 2 1-x^{2}
  29. x = ± 1 x=\pm 1
  30. x 1 / ε x\approx 1/\varepsilon\,
  31. ε = 0.1 \varepsilon=0.1
  32. ε = 0.01 \varepsilon=0.01
  33. ε = 0.001 \varepsilon=0.001
  34. ε \varepsilon

Singular_solution.html

  1. x y ( x ) + 2 y ( x ) = 0 , xy^{\prime}(x)+2y(x)=0,\,\!
  2. y ( x ) = C x - 2 . y(x)=Cx^{-2}.\,\!
  3. C C
  4. x = 0 x=0
  5. x 0 x\not=0
  6. ( x , y ( x ) ) (x,y(x))
  7. y ( x ) 2 = 4 y ( x ) . y^{\prime}(x)^{2}=4y(x).\,\!
  8. y c ( x ) = ( x - c ) 2 . y_{c}(x)=(x-c)^{2}.\,\!
  9. y s ( x ) = 0. y_{s}(x)=0.\,\!
  10. y = 0 y=0
  11. y > 0 y>0
  12. x = c x=c
  13. x x
  14. y s y_{s}
  15. y s ( x ) = 0 y_{s}(x)=0
  16. y c ( x ) = ( x - c ) 2 y_{c}(x)=(x-c)^{2}
  17. y s y_{s}
  18. y c ( x ) y_{c}(x)
  19. ( c , 0 ) (c,0)
  20. c 1 < c 2 c_{1}<c_{2}
  21. y ( x ) y(x)
  22. ( x - c 1 ) 2 (x-c_{1})^{2}
  23. x < c 1 x<c_{1}
  24. 0
  25. c 1 x c 2 c_{1}\leq x\leq c_{2}
  26. ( x - c 2 ) 2 (x-c_{2})^{2}
  27. x > c 2 x>c_{2}
  28. x = c 1 x=c_{1}
  29. x = c 2 x=c_{2}
  30. c 1 x c 2 c_{1}\leq x\leq c_{2}
  31. x = c 1 x=c_{1}
  32. x = c 2 x=c_{2}
  33. y ( x ) = 0 y(x)=0
  34. y ( x ) = x y + ( y ) 2 y(x)=x\cdot y^{\prime}+(y^{\prime})^{2}\,\!
  35. y ( x ) = x p + ( p ) 2 . y(x)=x\cdot p+(p)^{2}.\,\!
  36. p = y = p + x p + 2 p p p=y^{\prime}=p+xp^{\prime}+2pp^{\prime}\,\!
  37. 0 = ( 2 p + x ) p . 0=(2p+x)p^{\prime}.\,\!
  38. y c ( x ) = c x + c 2 y_{c}(x)=c\cdot x+c^{2}\,\!
  39. y s ( x ) = - ( 1 / 2 ) x 2 + ( - ( 1 / 2 ) x ) 2 = - ( 1 / 4 ) x 2 . y_{s}(x)=-(1/2)x^{2}+(-(1/2)x)^{2}=-(1/4)\cdot x^{2}.\,\!
  40. c x + c 2 = y c ( x ) = y s ( x ) = - ( 1 / 4 ) x 2 c\cdot x+c^{2}=y_{c}(x)=y_{s}(x)=-(1/4)\cdot x^{2}\,\!
  41. ( - 2 c , - c 2 ) (-2c,-c^{2})
  42. y c ( - 2 c ) = c y_{c}^{\prime}(-2\cdot c)=c\,\!
  43. y s ( - 2 c ) = - ( 1 / 2 ) x | x = - 2 c = c . y_{s}^{\prime}(-2\cdot c)=-(1/2)\cdot x|_{x=-2\cdot c}=c.\,\!
  44. y s ( x ) = - ( 1 / 4 ) x 2 y_{s}(x)=-(1/4)\cdot x^{2}\,\!
  45. y c ( x ) = c x + c 2 y_{c}(x)=c\cdot x+c^{2}\,\!
  46. y ( x ) = x y + ( y ) 2 . y(x)=x\cdot y^{\prime}+(y^{\prime})^{2}.\,\!

Singularity_theory.html

  1. y 2 = x 2 + x 3 y^{2}=x^{2}+x^{3}
  2. y 2 = x 3 y^{2}=x^{3}

Siphon.html

  1. v 2 2 + g y + P ρ = constant {v^{2}\over 2}+gy+{P\over\rho}=\mathrm{constant}
  2. v v\;
  3. g g\;
  4. y y\;
  5. P P\;
  6. ρ \rho\;
  7. 0 2 2 + g ( 0 ) + P atm ρ = constant {0^{2}\over 2}+g(0)+{P_{\mathrm{atm}}\over\rho}=\mathrm{constant}
  8. v A 2 2 - g d + P A ρ = constant {v_{A}^{2}\over 2}-gd+{P_{A}\over\rho}=\mathrm{constant}
  9. v B 2 2 + g h B + P B ρ = constant {v_{B}^{2}\over 2}+gh_{B}+{P_{B}\over\rho}=\mathrm{constant}
  10. v C 2 2 - g h C + P atm ρ = constant {v_{C}^{2}\over 2}-gh_{C}+{P_{\mathrm{atm}}\over\rho}=\mathrm{constant}
  11. 0 2 2 + g ( 0 ) + P atm ρ = v C 2 2 - g h C + P atm ρ {0^{2}\over 2}+g(0)+{P_{\mathrm{atm}}\over\rho}={v_{C}^{2}\over 2}-gh_{C}+{P_{% \mathrm{atm}}\over\rho}
  12. v C = 2 g h C v_{C}=\sqrt{2gh_{C}}
  13. 0 2 2 + g ( 0 ) + P atm ρ = v B 2 2 + g h B + P B ρ {0^{2}\over 2}+g(0)+{P_{\mathrm{atm}}\over\rho}={v_{B}^{2}\over 2}+gh_{B}+{P_{% B}\over\rho}
  14. v max = 2 ( P atm ρ - g h B ) v_{\mathrm{max}}=\sqrt{2\left({P_{\mathrm{atm}}\over\rho}-gh_{B}\right)}
  15. 0 2 2 + g ( 0 ) + P atm ρ = v B 2 2 + g h B + P B ρ {0^{2}\over 2}+g(0)+{P_{\mathrm{atm}}\over\rho}={v_{B}^{2}\over 2}+gh_{B}+{P_{% B}\over\rho}
  16. P atm ρ = v B 2 2 + g h B {P_{\mathrm{atm}}\over{\rho}}={v_{B}^{2}\over 2}+gh_{B}
  17. h B = P atm ρ g - v B 2 2 g . h_{B}={P_{\mathrm{atm}}\over\rho g}-{v_{B}^{2}\over 2g}.
  18. h B , max = P atm ρ g h_{B\mathrm{,max}}={P_{\mathrm{atm}}\over\rho g}

Siteswap.html

  1. A ~ n {\tilde{A}}_{n}

Skew-Hermitian.html

  1. n n
  2. n n
  3. A = ( a i , j ) 1 i , j n A=(a_{i,j})_{1\leq i,j\leq n}
  4. n n
  5. K n K^{n}
  6. A * = - A A^{*}=-A
  7. n n
  8. K n K^{n}
  9. ( | ) (\cdot|\cdot)
  10. K n K^{n}
  11. A A
  12. u , v K n u,v\in K^{n}
  13. ( A u | v ) = - ( u | A v ) . (Au|v)=-(u|Av)\,.
  14. K n K^{n}
  15. a i j = - a ¯ j i a_{ij}=-{\overline{a}}_{ji}
  16. 1 i , j n 1\leq i,j\leq n

Skew-Hermitian_matrix.html

  1. A = - A , A^{\dagger}=-A,\;
  2. \dagger
  3. a i , j = - a j , i ¯ , a_{i,j}=-\overline{a_{j,i}},
  4. [ - i 2 + i - ( 2 - i ) 0 ] \begin{bmatrix}-i&2+i\\ -(2-i)&0\end{bmatrix}
  5. C = A + B with A = 1 2 ( C + C ) and B = 1 2 ( C - C ) . C=A+B\quad\mbox{with}~{}\quad A=\frac{1}{2}(C+C^{\dagger})\quad\mbox{and}~{}% \quad B=\frac{1}{2}(C-C^{\dagger}).

Skip_list.html

  1. log 1 / p n \log_{1/p}n\,
  2. ( log 1 / p n ) / p , (\log_{1/p}n)/p,\,
  3. 𝒪 ( log n ) \mathcal{O}(\log n)\,
  4. 𝒪 ( n ) \mathcal{O}(n)
  5. 𝒪 ( log n ) \mathcal{O}(\log n)
  6. 𝒪 ( n ) \mathcal{O}(n)
  7. 𝒪 ( n log n ) \mathcal{O}(n\log n)
  8. 𝒪 ( log n ) \mathcal{O}(\log n)
  9. 𝒪 ( log n ) \mathcal{O}(\log n)
  10. 𝒪 ( n ) \mathcal{O}(n)
  11. 𝒪 ( log n ) \mathcal{O}(\log n)

Skolem_normal_form.html

  1. x P ( x ) \exists xP(x)
  2. P ( c ) P(c)
  3. c c
  4. y y
  5. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  6. f f
  7. x 1 , , x n x_{1},\ldots,x_{n}
  8. y y
  9. y \exists y
  10. f f
  11. x y z . P ( x , y , z ) \forall x\exists y\forall z.P(x,y,z)
  12. y \exists y
  13. y y
  14. f ( x ) f(x)
  15. f f
  16. y y
  17. x z . P ( x , f ( x ) , z ) \forall x\forall z.P(x,f(x),z)
  18. f ( x ) f(x)
  19. x x
  20. z z
  21. y \exists y
  22. x \forall x
  23. z \forall z
  24. x x
  25. y y
  26. z z
  27. x ( R ( g ( x ) ) y R ( x , y ) ) x ( R ( g ( x ) ) R ( x , f ( x ) ) ) \forall x\Big(R(g(x))\vee\exists yR(x,y)\Big)\iff\forall x\Big(R(g(x))\vee R(x% ,f(x))\Big)
  28. f ( x ) f(x)
  29. x x
  30. y y
  31. x x
  32. y y
  33. R ( x , y ) R(x,y)
  34. f f
  35. x x
  36. y y
  37. x x
  38. R ( x , f ( x ) ) R(x,f(x))
  39. Φ \Phi
  40. M M
  41. μ \mu
  42. x . R ( x , f ( x ) ) \forall x.R(x,f(x))
  43. M M
  44. f f
  45. x . R ( x , f ( x ) ) \forall x.R(x,f(x))
  46. f x . R ( x , f ( x ) ) \exists f\forall x.R(x,f(x))
  47. x y . R ( x , y ) \forall x\exists y.R(x,y)
  48. Φ \Phi
  49. M μ . ( M , μ Φ ) \exists M\exists\mu~{}.~{}(M,\mu\models\Phi)
  50. M M
  51. μ \mu
  52. \models
  53. Φ \Phi
  54. M M
  55. μ \mu
  56. Φ \Phi
  57. M \exists M
  58. f x . R ( x , f ( x ) ) \exists f\forall x.R(x,f(x))
  59. x . R ( x , f ( x ) ) \forall x.R(x,f(x))
  60. M \exists M
  61. F 1 = x 1 x n y R ( x 1 , , x n , y ) F_{1}=\forall x_{1}\dots\forall x_{n}\exists yR(x_{1},\dots,x_{n},y)
  62. M M
  63. x 1 , , x n x_{1},\dots,x_{n}
  64. y y
  65. R ( x 1 , , x n , y ) R(x_{1},\dots,x_{n},y)
  66. f f
  67. y = f ( x 1 , , x n ) y=f(x_{1},\dots,x_{n})
  68. F 2 = x 1 x n R ( x 1 , , x n , f ( x 1 , , x n ) ) F_{2}=\forall x_{1}\dots\forall x_{n}R(x_{1},\dots,x_{n},f(x_{1},\dots,x_{n}))
  69. f f
  70. M M
  71. F 1 F_{1}
  72. F 2 F_{2}
  73. F 2 F_{2}
  74. M M^{\prime}
  75. f f
  76. x 1 , , x n x_{1},\dots,x_{n}
  77. R ( x 1 , , x n , f ( x 1 , , x n ) ) R(x_{1},\dots,x_{n},f(x_{1},\dots,x_{n}))
  78. F 1 F_{1}
  79. x 1 , , x n x_{1},\ldots,x_{n}
  80. y = f ( x 1 , , x n ) y=f(x_{1},\dots,x_{n})
  81. f f
  82. M M^{\prime}
  83. x . Φ ( x , y 1 , , y n ) \exists x.\Phi(x,y_{1},\ldots,y_{n})
  84. x , y 1 , , y n x,y_{1},\ldots,y_{n}
  85. Φ ( x , y 1 , , y n ) \Phi(x,y_{1},\ldots,y_{n})
  86. Φ ( f ( y 1 , , y n ) , y 1 , , y n ) \Phi(f(y_{1},\ldots,y_{n}),y_{1},\ldots,y_{n})
  87. f f
  88. T T
  89. F F
  90. x 1 , , x n , y x_{1},\dots,x_{n},y
  91. T T

Slater-type_orbital.html

  1. R ( r ) = N r n - 1 e - ζ r R(r)=Nr^{n-1}e^{-\zeta r}\,
  2. ζ \zeta
  3. 0 x n e - α x d x = n ! α n + 1 . \int_{0}^{\infty}x^{n}e^{-\alpha x}dx=\frac{n!}{\alpha^{n+1}}.
  4. N 2 0 ( r n - 1 e - ζ r ) 2 r 2 d r = 1 N = ( 2 ζ ) n 2 ζ ( 2 n ) ! . N^{2}\int_{0}^{\infty}\left(r^{n-1}e^{-\zeta r}\right)^{2}r^{2}dr=1% \Longrightarrow N=(2\zeta)^{n}\sqrt{\frac{2\zeta}{(2n)!}}.
  5. Y l m ( 𝐫 ) Y_{l}^{m}(\mathbf{r})
  6. 𝐫 \mathbf{r}
  7. R ( r ) r = [ ( n - 1 ) r - ζ ] R ( r ) {\partial R(r)\over\partial r}=\left[\frac{(n-1)}{r}-\zeta\right]R(r)
  8. 2 = 1 r 2 r ( r 2 r ) \nabla^{2}={1\over r^{2}}{\partial\over\partial r}\left(r^{2}{\partial\over% \partial r}\right)
  9. ( r 2 r ) R ( r ) = [ ( n - 1 ) r - ζ r 2 ] R ( r ) \left(r^{2}{\partial\over\partial r}\right)R(r)=\left[(n-1)r-\zeta r^{2}\right% ]R(r)
  10. 2 R ( r ) = ( 1 r 2 r ) [ ( n - 1 ) r - ζ r 2 ] R ( r ) \nabla^{2}R(r)=\left({1\over r^{2}}{\partial\over\partial r}\right)\left[(n-1)% r-\zeta r^{2}\right]R(r)
  11. 2 R ( r ) = [ n ( n - 1 ) r 2 - 2 n ζ r + ζ 2 ] R ( r ) \nabla^{2}R(r)=\left[{n(n-1)\over r^{2}}-{2n\zeta\over r}+\zeta^{2}\right]R(r)
  12. χ n l m ( 𝐫 ) = r n - 1 e - ζ r Y l m ( 𝐫 ) . \chi_{nlm}({\mathbf{r}})=r^{n-1}e^{-\zeta r}Y_{l}^{m}({\mathbf{r}}).
  13. χ n l m ( 𝐤 ) = d 3 r e i 𝐤 𝐫 χ n l m ( 𝐫 ) \chi_{nlm}({\mathbf{k}})=\int d^{3}re^{i{\mathbf{k}}\cdot{\mathbf{r}}}\chi_{% nlm}({\mathbf{r}})
  14. = 4 π ( n - l ) ! ( 2 ζ ) n ( i k / ζ ) l Y l m ( 𝐤 ) s = 0 ( n - l ) / 2 ω s n l ( k 2 + ζ 2 ) n + 1 - s =4\pi(n-l)!(2\zeta)^{n}(ik/\zeta)^{l}Y_{l}^{m}({\mathbf{k}})\sum_{s=0}^{% \lfloor(n-l)/2\rfloor}\frac{\omega_{s}^{nl}}{(k^{2}+\zeta^{2})^{n+1-s}}
  15. ω \omega
  16. ω s n l ( - 1 4 ζ 2 ) s ( n - s ) ! s ! ( n - l - 2 s ) ! \omega_{s}^{nl}\equiv(-\frac{1}{4\zeta^{2}})^{s}\frac{(n-s)!}{s!(n-l-2s)!}
  17. χ n l m * ( r ) χ n l m ( r ) d 3 r = δ l l δ m m ( n + n ) ! ( ζ + ζ ) n + n + 1 \int\chi^{*}_{nlm}(r)\chi_{n^{\prime}l^{\prime}m^{\prime}}(r)d^{3}r=\delta_{ll% ^{\prime}}\delta_{mm^{\prime}}\frac{(n+n^{\prime})!}{(\zeta+\zeta^{\prime})^{n% +n^{\prime}+1}}
  18. χ n l m * ( r ) ( - 2 2 ) χ n l m ( r ) d 3 r = 1 2 δ l l δ m m 0 d r e - ( ζ + ζ ) r [ [ l ( l + 1 ) - n ( n - 1 ) ] r n + n - 2 + 2 ζ n r n + n - 1 - ζ 2 r n + n ] , \int\chi^{*}_{nlm}(r)(-\frac{\nabla^{2}}{2})\chi_{n^{\prime}l^{\prime}m^{% \prime}}(r)d^{3}r=\frac{1}{2}\delta_{ll^{\prime}}\delta_{mm^{\prime}}\int_{0}^% {\infty}dre^{-(\zeta+\zeta^{\prime})r}\left[[l^{\prime}(l^{\prime}+1)-n^{% \prime}(n^{\prime}-1)]r^{n+n^{\prime}-2}+2\zeta^{\prime}n^{\prime}r^{n+n^{% \prime}-1}-\zeta^{\prime 2}r^{n+n^{\prime}}\right],
  19. χ n l m * ( 𝐫 ) = d 3 k ( 2 π ) 3 e i 𝐤 𝐫 χ n m l * ( 𝐤 ) \chi^{*}_{nlm}({\mathbf{r}})=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i{\mathbf{k}}% \cdot{\mathbf{r}}}\chi^{*}_{nml}({\mathbf{k}})
  20. χ n l m * ( 𝐫 ) 1 | 𝐫 - 𝐫 | χ n l m ( 𝐫 ) d 3 r = 4 π d 3 k ( 2 π ) 3 χ n l m * ( 𝐤 ) 1 k 2 χ n l m ( 𝐤 ) \int\chi^{*}_{nlm}({\mathbf{r}})\frac{1}{|{\mathbf{r}}-{\mathbf{r}}^{\prime}|}% \chi_{n^{\prime}l^{\prime}m^{\prime}}({\mathbf{r}}^{\prime})d^{3}r=4\pi\int% \frac{d^{3}k}{(2\pi)^{3}}\chi^{*}_{nlm}({\mathbf{k}})\frac{1}{k^{2}}\chi_{n^{% \prime}l^{\prime}m^{\prime}}({\mathbf{k}})
  21. = 8 δ l l δ m m ( n - l ) ! ( n - l ) ! ( 2 ζ ) n ζ l ( 2 ζ ) n ζ l 0 d k k 2 l s = 0 ( n - l ) / 2 ω s n l ( k 2 + ζ 2 ) n + 1 - s s = 0 ( n - l ) / 2 ω s n l ( k 2 + ζ 2 ) n + 1 - s =8\delta_{ll^{\prime}}\delta_{mm^{\prime}}(n-l)!(n^{\prime}-l)!\frac{(2\zeta)^% {n}}{\zeta^{l}}\frac{(2\zeta^{\prime})^{n^{\prime}}}{\zeta^{\prime l}}\int_{0}% ^{\infty}dkk^{2l}\sum_{s=0}^{\lfloor(n-l)/2\rfloor}\frac{\omega_{s}^{nl}}{(k^{% 2}+\zeta^{2})^{n+1-s}}\sum_{s^{\prime}=0}^{\lfloor(n^{\prime}-l)/2\rfloor}% \frac{\omega_{s^{\prime}}^{n^{\prime}l^{\prime}}}{(k^{2}+\zeta^{\prime 2})^{n^% {\prime}+1-s^{\prime}}}

Slater_determinant.html

  1. χ ( 𝐱 ) \chi(\mathbf{x})
  2. 𝐱 \mathbf{x}
  3. 𝐱 1 \mathbf{x}_{1}
  4. 𝐱 2 \mathbf{x}_{2}
  5. Ψ ( 𝐱 1 , 𝐱 2 ) = χ 1 ( 𝐱 1 ) χ 2 ( 𝐱 2 ) . \Psi(\mathbf{x}_{1},\mathbf{x}_{2})=\chi_{1}(\mathbf{x}_{1})\chi_{2}(\mathbf{x% }_{2}).
  6. Ψ ( 𝐱 1 , 𝐱 2 ) = - Ψ ( 𝐱 2 , 𝐱 1 ) \Psi(\mathbf{x}_{1},\mathbf{x}_{2})=-\Psi(\mathbf{x}_{2},\mathbf{x}_{1})
  7. Ψ ( 𝐱 1 , 𝐱 2 ) = 1 2 { χ 1 ( 𝐱 1 ) χ 2 ( 𝐱 2 ) - χ 1 ( 𝐱 2 ) χ 2 ( 𝐱 1 ) } \Psi(\mathbf{x}_{1},\mathbf{x}_{2})=\frac{1}{\sqrt{2}}\{\chi_{1}(\mathbf{x}_{1% })\chi_{2}(\mathbf{x}_{2})-\chi_{1}(\mathbf{x}_{2})\chi_{2}(\mathbf{x}_{1})\}
  8. = 1 2 | χ 1 ( 𝐱 1 ) χ 2 ( 𝐱 1 ) χ 1 ( 𝐱 2 ) χ 2 ( 𝐱 2 ) | =\frac{1}{\sqrt{2}}\begin{vmatrix}\chi_{1}(\mathbf{x}_{1})&\chi_{2}(\mathbf{x}% _{1})\\ \chi_{1}(\mathbf{x}_{2})&\chi_{2}(\mathbf{x}_{2})\end{vmatrix}
  9. Ψ ( 𝐱 1 , 𝐱 2 , , 𝐱 N ) = 1 N ! | χ 1 ( 𝐱 1 ) χ 2 ( 𝐱 1 ) χ N ( 𝐱 1 ) χ 1 ( 𝐱 2 ) χ 2 ( 𝐱 2 ) χ N ( 𝐱 2 ) χ 1 ( 𝐱 N ) χ 2 ( 𝐱 N ) χ N ( 𝐱 N ) | | χ 1 χ 2 χ N | , \Psi(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N})=\frac{1}{\sqrt{N!}}% \left|\begin{matrix}\chi_{1}(\mathbf{x}_{1})&\chi_{2}(\mathbf{x}_{1})&\cdots&% \chi_{N}(\mathbf{x}_{1})\\ \chi_{1}(\mathbf{x}_{2})&\chi_{2}(\mathbf{x}_{2})&\cdots&\chi_{N}(\mathbf{x}_{% 2})\\ \vdots&\vdots&\ddots&\vdots\\ \chi_{1}(\mathbf{x}_{N})&\chi_{2}(\mathbf{x}_{N})&\cdots&\chi_{N}(\mathbf{x}_{% N})\end{matrix}\right|\equiv\left|\begin{matrix}\chi_{1}&\chi_{2}&\cdots&\chi_% {N}\\ \end{matrix}\right|,

Slew_rate.html

  1. SR 2 π f V pk , \mathrm{SR}\geq 2\pi fV_{\mathrm{pk}},
  2. V pk V_{\mathrm{pk}}
  3. SR = max ( | d v out ( t ) d t | ) \mathrm{SR}=\max\left(\left|\frac{dv_{\mathrm{out}}(t)}{dt}\right|\right)
  4. v out ( t ) v_{\mathrm{out}}(t)
  5. C C
  6. A 2 A_{2}
  7. SR = I sat C A 2 \mathrm{SR}=\frac{I_{\mathrm{sat}}}{C}A_{2}
  8. I sat I_{\mathrm{sat}}

Sliding_mode_control.html

  1. s = 0 s=0
  2. s = 0 s=0
  3. s = x 1 + x ˙ 1 = 0 s=x_{1}+\dot{x}_{1}=0
  4. x ˙ 1 = - x 1 \dot{x}_{1}=-x_{1}
  5. s = 0 s=0
  6. s = 0 s=0
  7. s = 0 s=0
  8. 𝐱 ˙ ( t ) = f ( 𝐱 , t ) + B ( 𝐱 , t ) 𝐮 ( t ) \dot{\mathbf{x}}(t)=f(\mathbf{x},t)+B(\mathbf{x},t)\,\mathbf{u}(t)
  9. ( 1 ) (1)\,
  10. 𝐱 ( t ) [ x 1 ( t ) x 2 ( t ) x n - 1 ( t ) x n ( t ) ] n \mathbf{x}(t)\triangleq\begin{bmatrix}x_{1}(t)\\ x_{2}(t)\\ \vdots\\ x_{n-1}(t)\\ x_{n}(t)\end{bmatrix}\in\mathbb{R}^{n}
  11. n n
  12. 𝐮 ( t ) [ u 1 ( t ) u 2 ( t ) u m - 1 ( t ) u m ( t ) ] m \mathbf{u}(t)\triangleq\begin{bmatrix}u_{1}(t)\\ u_{2}(t)\\ \vdots\\ u_{m-1}(t)\\ u_{m}(t)\end{bmatrix}\in\mathbb{R}^{m}
  13. m m
  14. f : n × n f:\mathbb{R}^{n}\times\mathbb{R}\mapsto\mathbb{R}^{n}
  15. B : n × n × m B:\mathbb{R}^{n}\times\mathbb{R}\mapsto\mathbb{R}^{n\times m}
  16. 𝐱 ( t ) \mathbf{x}(t)
  17. 𝐮 ( 𝐱 ( t ) ) \mathbf{u}(\mathbf{x}(t))
  18. 𝐱 ( t ) \mathbf{x}(t)
  19. t t
  20. 𝐮 \mathbf{u}
  21. 𝐱 = [ 0 , 0 , , 0 ] T \mathbf{x}=[0,0,\ldots,0]^{\,\text{T}}
  22. x 1 x_{1}
  23. 𝐱 \mathbf{x}
  24. 𝐮 \mathbf{u}
  25. x 1 x_{1}
  26. x 1 = 0 x_{1}=0
  27. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}
  28. σ : n m \sigma:\mathbb{R}^{n}\mapsto\mathbb{R}^{m}
  29. 𝐱 \mathbf{x}
  30. 𝐱 \mathbf{x}
  31. σ ( 𝐱 ) 0 \sigma(\mathbf{x})\neq 0
  32. σ ( 𝐱 ) = 0 \sigma(\mathbf{x})=0
  33. σ ( 𝐱 ) = 0 \sigma(\mathbf{x})=0
  34. 𝐱 ( t ) \mathbf{x}(t)
  35. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}
  36. n × m n\times m
  37. n n
  38. 𝐱 \mathbf{x}
  39. m m
  40. 𝐮 \mathbf{u}
  41. 1 k m 1\leq k\leq m
  42. n × 1 n\times 1
  43. { 𝐱 n : σ k ( 𝐱 ) = 0 } \left\{\mathbf{x}\in\mathbb{R}^{n}:\sigma_{k}(\mathbf{x})=0\right\}
  44. ( 2 ) (2)\,
  45. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  46. 𝐱 \mathbf{x}
  47. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  48. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  49. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  50. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  51. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  52. σ ˙ ( 𝐱 ) = 𝟎 \dot{\sigma}(\mathbf{x})=\mathbf{0}
  53. V ( σ ( 𝐱 ) ) = 1 2 σ T ( 𝐱 ) σ ( 𝐱 ) = 1 2 σ ( 𝐱 ) 2 2 V(\sigma(\mathbf{x}))=\frac{1}{2}\sigma^{\,\text{T}}(\mathbf{x})\sigma(\mathbf% {x})=\frac{1}{2}\|\sigma(\mathbf{x})\|_{2}^{2}
  54. ( 3 ) (3)\,
  55. \|\mathord{\cdot}\|
  56. σ ( 𝐱 ) 2 \|\sigma(\mathbf{x})\|_{2}
  57. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  58. σ T V σ σ ˙ d σ d t d V d t < 0 (i.e., d V d t < 0 ) \underbrace{\overbrace{\sigma^{\,\text{T}}}^{\tfrac{\partial V}{\partial\sigma% }}\overbrace{\dot{\sigma}}^{\tfrac{\operatorname{d}\sigma}{\operatorname{d}t}}% }_{\tfrac{\operatorname{d}V}{\operatorname{d}t}}<0\qquad\,\text{(i.e., }\tfrac% {\operatorname{d}V}{\operatorname{d}t}<0\,\text{)}
  59. σ ( 𝐱 ) = 0 \sigma(\mathbf{x})=0
  60. m = 1 m=1
  61. σ T σ ˙ < 0 \sigma^{\,\text{T}}\dot{\sigma}<0
  62. u ( 𝐱 ) u(\mathbf{x})
  63. σ \sigma
  64. σ ˙ \dot{\sigma}
  65. u ( 𝐱 ) u(\mathbf{x})
  66. σ ˙ ( 𝐱 ) \dot{\sigma}(\mathbf{x})
  67. σ ( 𝐱 ) \sigma(\mathbf{x})
  68. u ( 𝐱 ) u(\mathbf{x})
  69. σ ˙ ( 𝐱 ) \dot{\sigma}(\mathbf{x})
  70. σ ( 𝐱 ) \sigma(\mathbf{x})
  71. σ ˙ = σ 𝐱 𝐱 ˙ d 𝐱 d t = σ 𝐱 ( f ( 𝐱 , t ) + B ( 𝐱 , t ) 𝐮 ) 𝐱 ˙ \dot{\sigma}=\frac{\partial\sigma}{\partial\mathbf{x}}\overbrace{\dot{\mathbf{% x}}}^{\tfrac{\operatorname{d}\mathbf{x}}{\operatorname{d}t}}=\frac{\partial% \sigma}{\partial\mathbf{x}}\overbrace{\left(f(\mathbf{x},t)+B(\mathbf{x},t)% \mathbf{u}\right)}^{\dot{\mathbf{x}}}
  72. 𝐮 ( 𝐱 ) \mathbf{u}(\mathbf{x})
  73. σ ˙ \dot{\sigma}
  74. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  75. d V / d t \operatorname{d}V/{\operatorname{d}t}
  76. d V d t - μ ( V ) α \frac{\operatorname{d}V}{\operatorname{d}t}\leq-\mu(\sqrt{V})^{\alpha}
  77. μ > 0 \mu>0
  78. 0 < α 1 0<\alpha\leq 1
  79. V [ 0 , 1 ] V\in[0,1]
  80. d V d t - μ ( V ) α - μ V . \frac{\operatorname{d}V}{\operatorname{d}t}\leq-\mu(\sqrt{V})^{\alpha}\leq-\mu% \sqrt{V}.
  81. V ( 0 , 1 ] V\in(0,1]
  82. 1 V d V d t - μ , \frac{1}{\sqrt{V}}\frac{\operatorname{d}V}{\operatorname{d}t}\leq-\mu,
  83. d W / d t \operatorname{d}W/{\operatorname{d}t}
  84. W 2 V W\triangleq 2\sqrt{V}
  85. D + ( 2 V σ 2 W ) D + W Upper right-hand W ˙ = 1 V d V d t - μ \mathord{\underbrace{D^{+}\Bigl(\mathord{\underbrace{2\mathord{\overbrace{% \sqrt{V}}^{{}\propto\|\sigma\|_{2}}}}_{W}}\Bigr)}_{D^{+}W\,\triangleq\,% \mathord{\,\text{Upper right-hand }\dot{W}}}}=\frac{1}{\sqrt{V}}\frac{% \operatorname{d}V}{\operatorname{d}t}\leq-\mu
  86. D + D^{+}
  87. 2 V 2\sqrt{V}
  88. \propto
  89. z ( t ) = z 0 - μ t z(t)=z_{0}-\mu t
  90. z ˙ = - μ \dot{z}=-\mu
  91. z ( 0 ) = z 0 z(0)=z_{0}
  92. 2 V ( t ) V 0 - μ t 2\sqrt{V(t)}\leq V_{0}-\mu t
  93. t t
  94. V 0 \sqrt{V}\geq 0
  95. V \sqrt{V}
  96. V = 0 \sqrt{V}=0
  97. V V
  98. V = 0 V=0
  99. V \sqrt{V}
  100. 2 \|\mathord{\cdot}\|_{2}
  101. σ \sigma
  102. σ T V σ σ ˙ d σ d t d V d t - μ ( σ 2 V ) α \underbrace{\overbrace{\sigma^{\,\text{T}}}^{\tfrac{\partial V}{\partial\sigma% }}\overbrace{\dot{\sigma}}^{\tfrac{\operatorname{d}\sigma}{\operatorname{d}t}}% }_{\tfrac{\operatorname{d}V}{\operatorname{d}t}}\leq-\mu(\mathord{\overbrace{% \|\sigma\|_{2}}^{\sqrt{V}}})^{\alpha}
  103. \|\mathord{\cdot}\|
  104. σ \sigma
  105. σ σ ˙ - μ | σ | α \sigma\dot{\sigma}\leq-\mu|\sigma|^{\alpha}
  106. α = 1 \alpha=1
  107. sgn ( σ ) σ ˙ - μ \operatorname{sgn}(\sigma)\dot{\sigma}\leq-\mu
  108. sgn ( σ ) sgn ( σ ˙ ) and | σ ˙ | μ > 0 \operatorname{sgn}(\sigma)\neq\operatorname{sgn}(\dot{\sigma})\qquad\,\text{% and}\qquad|\dot{\sigma}|\geq\mu>0
  109. σ = 0 \sigma=0
  110. | σ ˙ | |\dot{\sigma}|
  111. σ \sigma
  112. 𝐱 \mathbf{x}
  113. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  114. σ ˙ \dot{\sigma}
  115. σ = 0 \sigma=0
  116. { 𝐱 n : σ ( 𝐱 ) = 𝟎 } \{\mathbf{x}\in\mathbb{R}^{n}:\sigma(\mathbf{x})=\mathbf{0}\}
  117. { 𝐱 n : σ T ( 𝐱 ) σ ˙ ( 𝐱 ) < 0 } \{\mathbf{x}\in\mathbb{R}^{n}:\sigma^{\,\text{T}}(\mathbf{x})\dot{\sigma}(% \mathbf{x})<0\}
  118. V ( σ ) V(\sigma)
  119. 𝐱 \mathbf{x}
  120. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  121. V ˙ \dot{V}
  122. σ = 0 \sigma=0
  123. σ 𝐱 B ( 𝐱 , t ) \frac{\partial\sigma}{\partial{\mathbf{x}}}B(\mathbf{x},t)
  124. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  125. σ ( 𝐱 ) \sigma(\mathbf{x})
  126. σ ˙ = 𝟎 \dot{\sigma}=\mathbf{0}
  127. 𝐱 \mathbf{x}
  128. σ ˙ ( 𝐱 ) = 0 \dot{\sigma}(\mathbf{x})=0
  129. 𝐮 ( 𝐱 ) \mathbf{u}(\mathbf{x})
  130. σ 𝐱 ( f ( 𝐱 , t ) + B ( 𝐱 , t ) 𝐮 ) 𝐱 ˙ = 𝟎 \frac{\partial\sigma}{\partial\mathbf{x}}\overbrace{\left(f(\mathbf{x},t)+B(% \mathbf{x},t)\mathbf{u}\right)}^{\dot{\mathbf{x}}}=\mathbf{0}
  131. 𝐮 = - ( σ 𝐱 B ( 𝐱 , t ) ) - 1 σ 𝐱 f ( 𝐱 , t ) \mathbf{u}=-\left(\frac{\partial\sigma}{\partial\mathbf{x}}B(\mathbf{x},t)% \right)^{-1}\frac{\partial\sigma}{\partial\mathbf{x}}f(\mathbf{x},t)
  132. 𝐮 \mathbf{u}
  133. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  134. 𝐱 ˙ = f ( 𝐱 , t ) - B ( 𝐱 , t ) ( σ 𝐱 B ( 𝐱 , t ) ) - 1 σ 𝐱 f ( 𝐱 , t ) f ( 𝐱 , t ) + B ( 𝐱 , t ) u = f ( 𝐱 , t ) ( 𝐈 - B ( 𝐱 , t ) ( σ 𝐱 B ( 𝐱 , t ) ) - 1 σ 𝐱 ) \dot{\mathbf{x}}=\overbrace{f(\mathbf{x},t)-B(\mathbf{x},t)\left(\frac{% \partial\sigma}{\partial\mathbf{x}}B(\mathbf{x},t)\right)^{-1}\frac{\partial% \sigma}{\partial\mathbf{x}}f(\mathbf{x},t)}^{f(\mathbf{x},t)+B(\mathbf{x},t)u}% =f(\mathbf{x},t)\left(\mathbf{I}-B(\mathbf{x},t)\left(\frac{\partial\sigma}{% \partial\mathbf{x}}B(\mathbf{x},t)\right)^{-1}\frac{\partial\sigma}{\partial% \mathbf{x}}\right)
  135. σ ˙ ( 𝐱 ) = 𝟎 \dot{\sigma}(\mathbf{x})=\mathbf{0}
  136. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  137. σ ˙ = 0 \dot{\sigma}=0
  138. σ ( 𝐱 ) = 𝟎 \sigma(\mathbf{x})=\mathbf{0}
  139. σ T σ ˙ < 0 \sigma^{\,\text{T}}\dot{\sigma}<0
  140. σ ˙ \dot{\sigma}
  141. x ˙ + x = 0 \dot{x}+x=0
  142. x ¨ = a ( t , x , x ˙ ) + u \ddot{x}=a(t,x,\dot{x})+u
  143. a ( ) a(\cdot)
  144. x ˙ = - x \dot{x}=-x
  145. x ˙ + x = 0 \dot{x}+x=0
  146. ( x , x ˙ ) = ( 0 , 0 ) (x,\dot{x})=(0,0)
  147. u u
  148. m = 1 m=1
  149. σ ( 𝐱 ) s 1 x 1 + s 2 x 2 + + s n - 1 x n - 1 + s n x n \sigma(\mathbf{x})\triangleq s_{1}x_{1}+s_{2}x_{2}+\cdots+s_{n-1}x_{n-1}+s_{n}% x_{n}
  150. ( 4 ) (4)\,
  151. s i > 0 s_{i}>0
  152. 1 i n 1\leq i\leq n
  153. σ ( 𝐱 ) = 0 \sigma(\mathbf{x})=0
  154. σ ˙ ( 𝐱 ) = 0 \dot{\sigma}(\mathbf{x})=0
  155. s 1 x ˙ 1 + s 2 x ˙ 2 + + s n - 1 x ˙ n - 1 + s n x ˙ n = 0 s_{1}\dot{x}_{1}+s_{2}\dot{x}_{2}+\cdots+s_{n-1}\dot{x}_{n-1}+s_{n}\dot{x}_{n}=0
  156. n - 1 n-1
  157. ( n - 1 ) (n-1)
  158. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}
  159. V ˙ ( σ ( 𝐱 ) ) = σ ( 𝐱 ) T σ 𝐱 σ ˙ ( 𝐱 ) d σ d t \dot{V}(\sigma(\mathbf{x}))=\overbrace{\sigma(\mathbf{x})^{\,\text{T}}}^{% \tfrac{\partial\sigma}{\partial\mathbf{x}}}\overbrace{\dot{\sigma}(\mathbf{x})% }^{\tfrac{\operatorname{d}\sigma}{\operatorname{d}t}}
  160. V ˙ \dot{V}
  161. V ˙ < 0 \dot{V}<0
  162. σ = 0 \mathbf{\sigma}=0
  163. u ( 𝐱 ) u(\mathbf{x})
  164. { σ ˙ < 0 if σ > 0 σ ˙ > 0 if σ < 0 \begin{cases}\dot{\sigma}<0&\,\text{if }\sigma>0\\ \dot{\sigma}>0&\,\text{if }\sigma<0\end{cases}
  165. σ σ ˙ < 0 \sigma\dot{\sigma}<0
  166. σ ˙ ( 𝐱 ) = σ ( 𝐱 ) 𝐱 𝐱 ˙ σ ˙ ( 𝐱 ) = σ ( 𝐱 ) 𝐱 ( f ( 𝐱 , t ) + B ( 𝐱 , t ) u ) 𝐱 ˙ = [ s 1 , s 2 , , s n ] σ ( 𝐱 ) 𝐱 ( f ( 𝐱 , t ) + B ( 𝐱 , t ) u ) 𝐱 ˙ ( i.e., an n × 1 vector ) \dot{\sigma}(\mathbf{x})=\overbrace{\frac{\partial{\sigma(\mathbf{x})}}{% \partial{\mathbf{x}}}\dot{\mathbf{x}}}^{\dot{\sigma}(\mathbf{x})}=\frac{% \partial{\sigma(\mathbf{x})}}{\partial{\mathbf{x}}}\overbrace{\left(f(\mathbf{% x},t)+B(\mathbf{x},t)u\right)}^{\dot{\mathbf{x}}}=\overbrace{[s_{1},s_{2},% \ldots,s_{n}]}^{\frac{\partial{\sigma(\mathbf{x})}}{\partial{\mathbf{x}}}}% \underbrace{\overbrace{\left(f(\mathbf{x},t)+B(\mathbf{x},t)u\right)}^{\dot{% \mathbf{x}}}}_{\,\text{( i.e., an }n\times 1\,\text{ vector )}}
  167. ( 5 ) (5)\,
  168. u ( 𝐱 ) u(\mathbf{x})
  169. u ( 𝐱 ) = { u + ( 𝐱 ) if σ ( 𝐱 ) > 0 u - ( 𝐱 ) if σ ( 𝐱 ) < 0 u(\mathbf{x})=\begin{cases}u^{+}(\mathbf{x})&\,\text{if }\sigma(\mathbf{x})>0% \\ u^{-}(\mathbf{x})&\,\text{if }\sigma(\mathbf{x})<0\end{cases}
  170. u + ( 𝐱 ) u^{+}(\mathbf{x})
  171. σ ˙ \dot{\sigma}
  172. 𝐱 \mathbf{x}
  173. u - ( 𝐱 ) u^{-}(\mathbf{x})
  174. σ ˙ \dot{\sigma}
  175. 𝐱 \mathbf{x}
  176. σ ( 𝐱 ) = 0 \sigma(\mathbf{x})=0
  177. σ ( 𝐱 ) = 0 \sigma(\mathbf{x})=0
  178. x ¨ = a ( t , x , x ˙ ) + u \ddot{x}=a(t,x,\dot{x})+u
  179. x 1 = x x_{1}=x
  180. x 2 = x ˙ x_{2}=\dot{x}
  181. { x ˙ 1 = x 2 x ˙ 2 = a ( t , x 1 , x 2 ) + u \begin{cases}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=a(t,x_{1},x_{2})+u\end{cases}
  182. sup { | a ( ) | } k \sup\{|a(\cdot)|\}\leq k
  183. | a | |a|
  184. k k
  185. σ ( x 1 , x 2 ) = x 1 + x 2 = x + x ˙ \sigma(x_{1},x_{2})=x_{1}+x_{2}=x+\dot{x}
  186. u ( x , x ˙ ) u(x,\dot{x})
  187. σ σ ˙ < 0 \sigma\dot{\sigma}<0
  188. σ ˙ = x ˙ 1 + x ˙ 2 = x ˙ + x ¨ = x ˙ + a ( t , x , x ˙ ) + u x ¨ \dot{\sigma}=\dot{x}_{1}+\dot{x}_{2}=\dot{x}+\ddot{x}=\dot{x}\,+\,\overbrace{a% (t,x,\dot{x})+u}^{\ddot{x}}
  189. x + x ˙ < 0 x+\dot{x}<0
  190. σ < 0 \sigma<0
  191. σ ˙ > 0 \dot{\sigma}>0
  192. u > | x ˙ + a ( t , x , x ˙ ) | u>|\dot{x}+a(t,x,\dot{x})|
  193. x + x ˙ > 0 x+\dot{x}>0
  194. σ > 0 \sigma>0
  195. σ ˙ < 0 \dot{\sigma}<0
  196. u < - | x ˙ + a ( t , x , x ˙ ) | u<-|\dot{x}+a(t,x,\dot{x})|
  197. | x ˙ | + | a ( t , x , x ˙ ) | | x ˙ + a ( t , x , x ˙ ) | |\dot{x}|+|a(t,x,\dot{x})|\geq|\dot{x}+a(t,x,\dot{x})|
  198. | a | |a|
  199. | x ˙ | + k + 1 > | x ˙ | + | a ( t , x , x ˙ ) | |\dot{x}|+k+1>|\dot{x}|+|a(t,x,\dot{x})|
  200. u ( x , x ˙ ) = { | x ˙ | + k + 1 if x + x ˙ < 0 , - ( | x ˙ | + k + 1 ) if x + x ˙ σ > 0 u(x,\dot{x})=\begin{cases}|\dot{x}|+k+1&\,\text{if }\underbrace{x+\dot{x}}<0,% \\ -\left(|\dot{x}|+k+1\right)&\,\text{if }\overbrace{x+\dot{x}}^{\sigma}>0\end{cases}
  201. u ( x , x ˙ ) = - ( | x ˙ | + k + 1 ) sgn ( x ˙ + x σ ) (i.e., tests σ > 0 ) u(x,\dot{x})=-(|\dot{x}|+k+1)\underbrace{\operatorname{sgn}(\overbrace{\dot{x}% +x}^{\sigma})}_{\,\text{(i.e., tests }\sigma>0\,\text{)}}
  202. σ ( 𝐱 ) = 0 \sigma(\mathbf{x})=0
  203. x ˙ = - x (i.e., σ ( x , x ˙ ) = x + x ˙ = 0 ) \dot{x}=-x\qquad\,\text{(i.e., }\sigma(x,\dot{x})=x+\dot{x}=0\,\text{)}
  204. ( x , x ˙ ) = ( 0 , 0 ) (x,\dot{x})=(0,0)
  205. { 𝐱 ˙ = A 𝐱 + B 𝐮 y = [ 1 0 0 ] 𝐱 = x 1 \begin{cases}\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}\\ y=\begin{bmatrix}1&0&0&\cdots&\end{bmatrix}\mathbf{x}=x_{1}\end{cases}
  206. 𝐱 ( x 1 , x 2 , , x n ) n \mathbf{x}\triangleq(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}
  207. 𝐮 ( u 1 , u 2 , , u r ) r \mathbf{u}\triangleq(u_{1},u_{2},\dots,u_{r})\in\mathbb{R}^{r}
  208. y y
  209. 𝐱 \mathbf{x}
  210. A [ a 11 A 12 A 21 A 22 ] A\triangleq\begin{bmatrix}a_{11}&A_{12}\\ A_{21}&A_{22}\end{bmatrix}
  211. a 11 a_{11}
  212. x 1 x_{1}
  213. A 21 ( n - 1 ) A_{21}\in\mathbb{R}^{(n-1)}
  214. A 22 ( n - 1 ) × ( n - 1 ) A_{22}\in\mathbb{R}^{(n-1)\times(n-1)}
  215. A 12 1 × ( n - 1 ) A_{12}\in\mathbb{R}^{1\times(n-1)}
  216. 𝐱 \mathbf{x}
  217. y = x 1 y=x_{1}
  218. 𝐱 ^ = ( x ^ 1 , x ^ 2 , , x ^ n ) n \hat{\mathbf{x}}=(\hat{x}_{1},\hat{x}_{2},\dots,\hat{x}_{n})\in\mathbb{R}^{n}
  219. n n
  220. 𝐱 ^ ˙ = A 𝐱 ^ + B 𝐮 + L v ( x ^ 1 - x 1 ) \dot{\hat{\mathbf{x}}}=A\hat{\mathbf{x}}+B\mathbf{u}+Lv(\hat{x}_{1}-x_{1})
  221. v : \R \R v:\R\mapsto\R
  222. x ^ 1 \hat{x}_{1}
  223. y = x 1 y=x_{1}
  224. L n L\in\mathbb{R}^{n}
  225. L = [ - 1 L 2 ] L=\begin{bmatrix}-1\\ L_{2}\end{bmatrix}
  226. L 2 ( n - 1 ) L_{2}\in\mathbb{R}^{(n-1)}
  227. 𝐞 = ( e 1 , e 2 , , e n ) n \mathbf{e}=(e_{1},e_{2},\dots,e_{n})\in\mathbb{R}^{n}
  228. 𝐞 = 𝐱 ^ - 𝐱 \mathbf{e}=\hat{\mathbf{x}}-\mathbf{x}
  229. { 𝐞 ˙ = 𝐱 ^ ˙ - 𝐱 ˙ = A 𝐱 ^ + B 𝐮 + L v ( x ^ 1 - x 1 ) - A 𝐱 - B 𝐮 = A ( 𝐱 ^ - 𝐱 ) + L v ( x ^ 1 - x 1 ) = A 𝐞 + L v ( e 1 ) \begin{cases}\dot{\mathbf{e}}=\dot{\hat{\mathbf{x}}}-\dot{\mathbf{x}}\\ =A\hat{\mathbf{x}}+B\mathbf{u}+Lv(\hat{x}_{1}-x_{1})-A\mathbf{x}-B\mathbf{u}\\ =A(\hat{\mathbf{x}}-\mathbf{x})+Lv(\hat{x}_{1}-x_{1})\\ =A\mathbf{e}+Lv(e_{1})\end{cases}
  230. e 1 = x ^ 1 - x 1 e_{1}=\hat{x}_{1}-x_{1}
  231. v v
  232. 0 = x ^ 1 - x 1 0=\hat{x}_{1}-x_{1}
  233. x ^ 1 \hat{x}_{1}
  234. x 1 x_{1}
  235. x ^ 1 = x 1 \hat{x}_{1}=x_{1}
  236. σ ( x ^ 1 , x ^ ) e 1 = x ^ 1 - x 1 . \sigma(\hat{x}_{1},\hat{x})\triangleq e_{1}=\hat{x}_{1}-x_{1}.
  237. σ ˙ \dot{\sigma}
  238. σ \sigma
  239. σ σ ˙ < 0 \sigma\dot{\sigma}<0
  240. 𝐱 \mathbf{x}
  241. σ ˙ = e ˙ 1 = a 11 e 1 + A 12 𝐞 2 - v ( e 1 ) = a 11 e 1 + A 12 𝐞 2 - v ( σ ) \dot{\sigma}=\dot{e}_{1}=a_{11}e_{1}+A_{12}\mathbf{e}_{2}-v(e_{1})=a_{11}e_{1}% +A_{12}\mathbf{e}_{2}-v(\sigma)
  242. 𝐞 2 ( e 2 , e 3 , , e n ) ( n - 1 ) \mathbf{e}_{2}\triangleq(e_{2},e_{3},\ldots,e_{n})\in\mathbb{R}^{(n-1)}
  243. σ σ ˙ < 0 \sigma\dot{\sigma}<0
  244. v ( σ ) = M sgn ( σ ) v(\sigma)=M\operatorname{sgn}(\sigma)
  245. M > max { | a 11 e 1 + A 12 𝐞 2 | } . M>\max\{|a_{11}e_{1}+A_{12}\mathbf{e}_{2}|\}.
  246. M M
  247. M M
  248. M M
  249. e 1 = 0 e_{1}=0
  250. x ^ 1 = x 1 \hat{x}_{1}=x_{1}
  251. e 1 e_{1}
  252. e ˙ 1 = 0 \dot{e}_{1}=0
  253. v ( σ ) v(\sigma)
  254. v eq v_{\,\text{eq}}
  255. 0 = σ ˙ = a 11 e 1 = 0 + A 12 𝐞 2 - v eq v ( σ ) = A 12 𝐞 2 - v eq . 0=\dot{\sigma}=a_{11}\mathord{\overbrace{e_{1}}^{{}=0}}+A_{12}\mathbf{e}_{2}-% \mathord{\overbrace{v_{\,\text{eq}}}^{v(\sigma)}}=A_{12}\mathbf{e}_{2}-v_{\,% \text{eq}}.
  256. v eq scalar = A 12 1 × ( n - 1 ) vector 𝐞 2 ( n - 1 ) × 1 vector . \mathord{\overbrace{v_{\,\text{eq}}}^{\,\text{scalar}}}=\mathord{\overbrace{A_% {12}}^{1\times(n-1)\,\text{ vector}}}\mathord{\overbrace{\mathbf{e}_{2}}^{(n-1% )\times 1\,\text{ vector}}}.
  257. v eq v_{\,\text{eq}}
  258. ( n - 1 ) (n-1)
  259. x 1 x_{1}
  260. A 12 A_{12}
  261. [ e ˙ 2 e ˙ 3 e ˙ n ] 𝐞 ˙ 2 = A 2 [ e 2 e 3 e n ] 𝐞 2 + L 2 v ( e 1 ) = A 2 𝐞 2 + L 2 v eq = A 2 𝐞 2 + L 2 A 12 𝐞 2 = ( A 2 + L 2 A 12 ) 𝐞 2 . \mathord{\overbrace{\begin{bmatrix}\dot{e}_{2}\\ \dot{e}_{3}\\ \vdots\\ \dot{e}_{n}\end{bmatrix}}^{\dot{\mathbf{e}}_{2}}}=A_{2}\mathord{\overbrace{% \begin{bmatrix}e_{2}\\ e_{3}\\ \vdots\\ e_{n}\end{bmatrix}}^{\mathbf{e}_{2}}}+L_{2}v(e_{1})=A_{2}\mathbf{e}_{2}+L_{2}v% _{\,\text{eq}}=A_{2}\mathbf{e}_{2}+L_{2}A_{12}\mathbf{e}_{2}=(A_{2}+L_{2}A_{12% })\mathbf{e}_{2}.
  262. 𝐞 2 \mathbf{e}_{2}
  263. ( n - 1 ) × 1 (n-1)\times 1
  264. L 2 L_{2}
  265. ( n - 1 ) × ( n - 1 ) (n-1)\times(n-1)
  266. ( A 2 + L 2 A 12 ) (A_{2}+L_{2}A_{12})
  267. 𝐞 2 \mathbf{e}_{2}
  268. A 12 A_{12}
  269. C C
  270. v eq v_{\,\text{eq}}
  271. v = M sgn ( x ^ 1 - x ) v=M\operatorname{sgn}(\hat{x}_{1}-x)
  272. v v
  273. v eq v_{\,\text{eq}}
  274. { 𝐱 ^ ˙ = A 𝐱 ^ + B 𝐮 + L M sgn ( x ^ 1 - x 1 ) = A 𝐱 ^ + B 𝐮 + [ - 1 L 2 ] M sgn ( x ^ 1 - x 1 ) = A 𝐱 ^ + B 𝐮 + [ - M L 2 M ] sgn ( x ^ 1 - x 1 ) = A 𝐱 ^ + [ B [ - M L 2 M ] ] [ 𝐮 sgn ( x ^ 1 - x 1 ) ] = A obs 𝐱 ^ + B obs 𝐮 obs \begin{cases}\dot{\hat{\mathbf{x}}}=A\hat{\mathbf{x}}+B\mathbf{u}+LM% \operatorname{sgn}(\hat{x}_{1}-x_{1})\\ =A\hat{\mathbf{x}}+B\mathbf{u}+\begin{bmatrix}-1\\ L_{2}\end{bmatrix}M\operatorname{sgn}(\hat{x}_{1}-x_{1})\\ =A\hat{\mathbf{x}}+B\mathbf{u}+\begin{bmatrix}-M\\ L_{2}M\end{bmatrix}\operatorname{sgn}(\hat{x}_{1}-x_{1})\\ =A\hat{\mathbf{x}}+\begin{bmatrix}B&\begin{bmatrix}-M\\ L_{2}M\end{bmatrix}\end{bmatrix}\begin{bmatrix}\mathbf{u}\\ \operatorname{sgn}(\hat{x}_{1}-x_{1})\end{bmatrix}\\ =A_{\,\text{obs}}\hat{\mathbf{x}}+B_{\,\text{obs}}\mathbf{u}_{\,\text{obs}}% \end{cases}
  275. A obs A A_{\,\text{obs}}\triangleq A
  276. B obs [ B [ - M L 2 M ] ] B_{\,\text{obs}}\triangleq\begin{bmatrix}B&\begin{bmatrix}-M\\ L_{2}M\end{bmatrix}\end{bmatrix}
  277. u obs [ 𝐮 sgn ( x ^ 1 - x 1 ) ] u_{\,\text{obs}}\triangleq\begin{bmatrix}\mathbf{u}\\ \operatorname{sgn}(\hat{x}_{1}-x_{1})\end{bmatrix}
  278. 𝐮 \mathbf{u}
  279. sgn ( x ^ 1 - x 1 ) \operatorname{sgn}(\hat{x}_{1}-x_{1})
  280. sgn ( x ^ 1 - x 1 ) \operatorname{sgn}(\hat{x}_{1}-x_{1})
  281. y = x 1 y=x_{1}
  282. 𝐲 = C 𝐱 \mathbf{y}=C\mathbf{x}
  283. C C
  284. 𝐲 ^ \hat{\mathbf{y}}
  285. 𝐲 \mathbf{y}
  286. σ ( 𝐱 ) 𝐲 ^ - 𝐲 = 𝟎 \sigma(\mathbf{x})\triangleq\hat{\mathbf{y}}-\mathbf{y}=\mathbf{0}

Slip_angle.html

  1. v x v_{x}
  2. v y v_{y}
  3. α \alpha
  4. α - arctan ( v y | v x | ) \alpha\triangleq-\arctan\left(\frac{v_{y}}{|v_{x}|}\right)

SM.html

  1. s m - 3 sm^{-3}

SMART_Information_Retrieval_System.html

  1. t t
  2. d d
  3. tf t , d \,\text{tf}_{t,d}
  4. tf t , d \,\text{tf}_{t,d}
  5. N d f t \tfrac{N}{df_{t}}
  6. 1 w 1 2 + w 2 2 + + w M 2 \tfrac{1}{\sqrt{w_{1}^{2}+w_{2}^{2}+...+w_{M}^{2}}}
  7. 0.5 × tf t , d max(tf t , d ) \tfrac{0.5\times\,\text{tf}_{t,d}}{\,\text{max(tf}_{t,d})}
  8. 𝐦𝐚𝐱 ( 0 , log N - d f t d f t ) \,\textbf{max}\left(0,\,\text{log}\tfrac{N-df_{t}}{df_{t}}\right)
  9. 1 / C h a r L e n g t h α , α < 1 1/CharLength^{\alpha},\alpha<1
  10. { 1 , if tf t , d > 0 0 , otherwise \begin{cases}1,&\,\text{if tf}_{t,d}>0\\ 0,&\,\text{otherwise}\end{cases}
  11. 1 + log ( tf t , d ) 1 + log ( ave t ϵ d ( tf t , d ) ) \tfrac{1+\,\text{log}(\,\text{tf}_{t,d})}{1+\,\text{log}(\,\text{ave}_{t% \epsilon d}(\,\text{tf}_{t,d}))}
  12. t , d {}_{t,d}
  13. t t
  14. d d

Smith_chart.html

  1. S n n S_{nn}\,
  2. Z 0 Z_{0}\,
  3. Y 0 Y_{0}\,
  4. Y 0 = 1 Z 0 Y_{0}=\frac{1}{Z_{0}}\,
  5. Z T Z_{T}\,
  6. z T = Z T Z 0 z_{T}=\frac{Z_{T}}{Z_{0}}\,
  7. y T = Y T Y 0 y_{T}=\frac{Y_{T}}{Y_{0}}\,
  8. Z T Z_{T}\,
  9. Z 0 Z_{0}\,
  10. V F V_{F}\,
  11. V R V_{R}\,
  12. V F = A exp ( j ω t ) exp ( γ l ) V_{F}=A\exp(j\omega t)\exp(\gamma l)\,
  13. V R = B exp ( j ω t ) exp ( - γ l ) V_{R}=B\exp(j\omega t)\exp(-\gamma l)\,
  14. exp ( j ω t ) \exp(j\omega t)\,
  15. exp ( γ l ) \exp(\gamma l)\,
  16. ω = 2 π f \omega=2\pi f\,
  17. ω \omega\,
  18. f f\,
  19. t t\,
  20. A A\,
  21. B B\,
  22. l l\,
  23. γ = α + j β \gamma=\alpha+j\beta\,
  24. α \alpha\,
  25. β \beta\,
  26. exp ( ω t ) \exp(\omega t)\,
  27. V F = A exp ( γ l ) V_{F}=A\exp(\gamma l)\,
  28. V R = B exp ( - γ l ) V_{R}=B\exp(-\gamma l)\,
  29. A A\,
  30. B B\,
  31. Γ \Gamma\,
  32. Γ = V R V F = B exp ( - γ l ) A exp ( γ l ) = C exp ( - 2 γ l ) \Gamma=\frac{V_{R}}{V_{F}}=\frac{B\exp(-\gamma l)}{A\exp(\gamma l)}=C\exp(-2% \gamma l)\,
  33. γ \gamma\,
  34. α \alpha\,
  35. α = 0 \alpha=0\,
  36. Γ = Γ L exp ( - 2 j β l ) \Gamma=\Gamma_{L}\exp(-2j\beta l)\,
  37. Γ L \Gamma_{L}\,
  38. l l\,
  39. β \beta\,
  40. β = 2 π λ \beta=\frac{2\pi}{\lambda}\,
  41. λ \lambda\,
  42. Γ = Γ L exp ( - 4 j π λ l ) \Gamma=\Gamma_{L}\exp\left(\frac{-4j\pi}{\lambda}l\right)\,
  43. V V\,
  44. I I\,
  45. V F + V R = V V_{F}+V_{R}=V\,
  46. V F - V R = Z 0 I V_{F}-V_{R}=Z_{0}I\,
  47. Γ = V R V F \Gamma=\frac{V_{R}}{V_{F}}\,
  48. z T = V Z 0 I z_{T}=\frac{V}{Z_{0}I}\,
  49. z T = 1 + Γ 1 - Γ z_{T}=\frac{1+\Gamma}{1-\Gamma}\,
  50. Γ = z T - 1 z T + 1 \Gamma=\frac{z_{T}-1}{z_{T}+1}\,
  51. Γ \Gamma\,
  52. z T z_{T}\,
  53. Γ \Gamma\,
  54. z T z_{T}\,
  55. Γ \Gamma\,
  56. Γ = B exp ( - γ l ) A exp ( γ l ) = B exp ( - j β l ) A exp ( j β l ) \Gamma=\frac{B\exp(-\gamma l)}{A\exp(\gamma l)}=\frac{B\exp(-j\beta l)}{A\exp(% j\beta l)}\,
  57. z T = 1 + Γ 1 - Γ z_{T}=\frac{1+\Gamma}{1-\Gamma}\,
  58. exp ( j θ ) = cos θ + j sin θ \exp(j\theta)=\cos\theta+j\sin\theta\,
  59. Z I N = Z 0 Z L + j Z 0 tan ( β l ) Z 0 + j Z L tan ( β l ) Z_{IN}=Z_{0}\frac{Z_{L}+jZ_{0}\tan(\beta l)}{Z_{0}+jZ_{L}\tan(\beta l)}\,
  60. Z I N Z_{IN}\,
  61. Z L Z_{L}\,
  62. Z L Z_{L}\,
  63. z T = 1 ± j 0 z_{T}=1\pm j0\,
  64. z T = ± j z_{T}=\infty\pm j\infty\,
  65. Γ \Gamma
  66. z z\,
  67. z = a + j b z=a+jb\,
  68. Γ = c + j d \Gamma=c+jd\,
  69. Γ = z - 1 z + 1 \Gamma=\frac{z-1}{z+1}\,
  70. Γ = c + j d = a 2 + b 2 - 1 ( a + 1 ) 2 + b 2 + j ( 2 b ( a + 1 ) 2 + b 2 ) \Gamma=c+jd=\frac{a^{2}+b^{2}-1}{(a+1)^{2}+b^{2}}+j\left(\frac{2b}{(a+1)^{2}+b% ^{2}}\right)\,
  71. y T = 1 z T y_{T}=\frac{1}{z_{T}}\,
  72. y T = 1 - Γ 1 + Γ y_{T}=\frac{1-\Gamma}{1+\Gamma}\,
  73. Γ = 1 - y T 1 + y T \Gamma=\frac{1-y_{T}}{1+y_{T}}\,
  74. 0.63 60 0.63\angle 60^{\circ}\,
  75. 60 \angle 60^{\circ}\,
  76. 0.63 60 0.63\angle 60^{\circ}\,
  77. 0.80 + j 1.40 0.80+j1.40\,
  78. 0.73 125 0.73\angle 125^{\circ}\,
  79. 0.20 + j 0.50 0.20+j0.50\,
  80. 0.44 - 116 0.44\angle-116^{\circ}\,
  81. 0.50 - j 0.50 0.50-j0.50\,
  82. 0.63 60 0.63\angle 60^{\circ}\,
  83. z P = 0.80 + j 1.40 z_{P}=0.80+j1.40\,
  84. y P = 0.30 - j 0.54 y_{P}=0.30-j0.54\,
  85. y T = 1 z T y_{T}=\frac{1}{z_{T}}\,
  86. z = 0.80 + j 1.40 z=0.80+j1.40\,
  87. y = 0.30 - j 0.54 y=0.30-j0.54\,
  88. z = 0.10 + j 0.22 z=0.10+j0.22\,
  89. y = 1.80 - j 3.90 y=1.80-j3.90\,
  90. Z T S Z_{TS}
  91. Z T P Z_{TP}
  92. Z T S = Z 1 + Z 2 + Z 3 + Z_{TS}=Z_{1}+Z_{2}+Z_{3}+...\,
  93. 1 Z T P = 1 Z 1 + 1 Z 2 + 1 Z 3 + \frac{1}{Z_{TP}}=\frac{1}{Z_{1}}+\frac{1}{Z_{2}}+\frac{1}{Z_{3}}+...\,
  94. Y T P = Y 1 + Y 2 + Y 3 + Y_{TP}=Y_{1}+Y_{2}+Y_{3}+...\,
  95. 1 Y T S = 1 Y 1 + 1 Y 2 + 1 Y 3 + \frac{1}{Y_{TS}}=\frac{1}{Y_{1}}+\frac{1}{Y_{2}}+\frac{1}{Y_{3}}+...\,
  96. Ω \Omega\,
  97. Z = R Z=R\,
  98. z = R Z 0 = R Y 0 z=\frac{R}{Z_{0}}=RY_{0}\,
  99. Y = G = 1 R Y=G=\frac{1}{R}\,
  100. y = g = 1 R Y 0 = Z 0 R y=g=\frac{1}{RY_{0}}=\frac{Z_{0}}{R}\,
  101. Z = j X L = j ω L Z=jX_{L}=j\omega L\,
  102. z = j x L = j ω L Z 0 = j ω L Y 0 z=jx_{L}=j\frac{\omega L}{Z_{0}}=j\omega LY_{0}\,
  103. Y = - j B L = - j ω L Y=-jB_{L}=\frac{-j}{\omega L}
  104. y = - j b L = - j ω L Y 0 = - j Z 0 ω L y=-jb_{L}=\frac{-j}{\omega LY_{0}}=\frac{-jZ_{0}}{\omega L}\,
  105. Z = - j X C = - j ω C Z=-jX_{C}=\frac{-j}{\omega C}\,
  106. z = - j x C = - j ω C Z 0 = - j Y 0 ω C z=-jx_{C}=\frac{-j}{\omega CZ_{0}}=\frac{-jY_{0}}{\omega C}\,
  107. Y = j B C = j ω C Y=jB_{C}=j\omega C\,
  108. y = j b C = j ω C Y 0 = j ω C Z 0 y=jb_{C}=j\frac{\omega C}{Y_{0}}=j\omega CZ_{0}\,
  109. Z 0 = 50 Ω Z_{0}=50\ \Omega
  110. Ω \Omega
  111. Z L = j ω L = j 2 π f L = j 32.7 Ω Z_{L}=j\omega L=j2\pi fL=j32.7\ \Omega\,
  112. Z T Z_{T}
  113. Z T = 17.5 + j 32.7 Ω Z_{T}=17.5+j32.7\ \Omega\,
  114. z T z_{T}
  115. z T = Z T Z 0 = 0.35 + j 0.65 z_{T}=\frac{Z_{T}}{Z_{0}}=0.35+j0.65\,
  116. L 1 = 0.098 λ L_{1}=0.098\lambda\,
  117. z P 21 = 1.00 + j 1.52 z_{P21}=1.00+j1.52\,
  118. L 2 = 0.177 λ L_{2}=0.177\lambda\,
  119. L 2 - L 1 = 0.177 λ - 0.098 λ = 0.079 λ L_{2}-L_{1}=0.177\lambda-0.098\lambda=0.079\lambda\,
  120. λ = c f \lambda=\frac{c}{f}\,
  121. c c\,
  122. f f\,
  123. λ = 375 mm \lambda=375\ \mathrm{mm}\,
  124. z m a t c h z_{match}\,
  125. z m a t c h = - j ( 1.52 ) , z_{match}=-j(1.52),\!
  126. C m C_{m}\,
  127. z m a t c h = - j 1.52 = - j ω C m Z 0 = - j 2 π f C m Z 0 z_{match}=-j1.52=\frac{-j}{\omega C_{m}Z_{0}}=\frac{-j}{2\pi fC_{m}Z_{0}}\,
  128. C m = 1 ( 1.52 ) ω Z 0 = 1 ( 1.52 ) ( 2 π f ) Z 0 C_{m}=\frac{1}{(1.52)\omega Z_{0}}=\frac{1}{(1.52)(2\pi f)Z_{0}}
  129. C m = 2.6 pF C_{m}=2.6\ \mathrm{pF}\,
  130. y Q 20 = 0.65 - j 1.20 y_{Q20}=0.65-j1.20\,
  131. L 3 = 0.152 λ L_{3}=0.152\lambda\,
  132. y Q 21 = 1.00 + j 1.52 y_{Q21}=1.00+j1.52\,
  133. L 2 + L 3 = 0.177 λ + 0.152 λ = 0.329 λ L_{2}+L_{3}=0.177\lambda+0.152\lambda=0.329\lambda\,
  134. y m a t c h y_{match}
  135. y m a t c h = - j 1.52 y_{match}=-j1.52\,
  136. L m L_{m}\,
  137. - j 1.52 = - j ω L m Y 0 = - j Z 0 2 π f L m -j1.52=\frac{-j}{\omega L_{m}Y_{0}}=\frac{-jZ_{0}}{2\pi fL_{m}}\,
  138. L m = 6.5 nH L_{m}=6.5\ \mathrm{nH}\,
  139. Z 0 = 50 Ω Z_{0}=50\ \Omega\,
  140. R 1 = 50 Ω R_{1}=50\ \Omega\,
  141. R 1 = 50 Ω R_{1}=50\ \Omega\,
  142. O P 1 O\rightarrow P_{1}\,
  143. Z Z\,
  144. - j 0.80 -j0.80\,
  145. - j 0.80 = - j ω C 1 Z 0 -j0.80=\frac{-j}{\omega C_{1}Z_{0}}\,
  146. C 1 = 40 pF C_{1}=40\ \mathrm{pF}\,
  147. Q 1 Q 2 Q_{1}\rightarrow Q_{2}\,
  148. Y Y\,
  149. - j 1.49 -j1.49\,
  150. - j 1.49 = - j ω L 1 Y 0 -j1.49=\frac{-j}{\omega L_{1}Y_{0}}\,
  151. L 1 = 53 nH L_{1}=53\ \mathrm{nH}\,
  152. P 2 P 3 P_{2}\rightarrow P_{3}\,
  153. - j 0.23 -j0.23\,
  154. - j 0.23 = - j ω C 2 Z 0 -j0.23=\frac{-j}{\omega C_{2}Z_{0}}\,
  155. C 2 = 138 pF C_{2}=138\ \mathrm{pF}\,
  156. Q 3 O Q_{3}\rightarrow O\,
  157. + j 1.14 +j1.14\,
  158. + j 1.14 = j ω C 3 Y 0 +j1.14=\frac{j\omega C_{3}}{Y_{0}}\,
  159. C 3 = 36 pF C_{3}=36\ \mathrm{pF}\,

Snefru.html

  1. 2 88.5 2^{88.5}

Sobel_operator.html

  1. 𝐆 x = [ - 1 0 + 1 - 2 0 + 2 - 1 0 + 1 ] * 𝐀 and 𝐆 y = [ - 1 - 2 - 1 0 0 0 + 1 + 2 + 1 ] * 𝐀 \mathbf{G}_{x}=\begin{bmatrix}-1&0&+1\\ -2&0&+2\\ -1&0&+1\end{bmatrix}*\mathbf{A}\quad\mbox{and}~{}\quad\mathbf{G}_{y}=\begin{% bmatrix}-1&-2&-1\\ 0&0&0\\ +1&+2&+1\end{bmatrix}*\mathbf{A}
  2. * *
  3. 𝐆 𝐱 \mathbf{G_{x}}
  4. [ - 1 0 + 1 - 2 0 + 2 - 1 0 + 1 ] = [ 1 2 1 ] [ - 1 0 + 1 ] \begin{bmatrix}-1&0&+1\\ -2&0&+2\\ -1&0&+1\end{bmatrix}=\begin{bmatrix}1\\ 2\\ 1\end{bmatrix}\begin{bmatrix}-1&0&+1\end{bmatrix}
  5. 𝐆 = 𝐆 x 2 + 𝐆 y 2 \mathbf{G}=\sqrt{{\mathbf{G}_{x}}^{2}+{\mathbf{G}_{y}}^{2}}
  6. 𝚯 = atan2 ( 𝐆 y , 𝐆 x ) \mathbf{\Theta}=\operatorname{atan2}\left({\mathbf{G}_{y},\mathbf{G}_{x}}\right)
  7. h ( - 1 ) = 1 , h ( 0 ) = 2 , h ( 1 ) = 1 h(-1)=1,h(0)=2,h(1)=1
  8. h ( - 1 ) = 1 , h ( 0 ) = 0 , h ( 1 ) = - 1 h^{\prime}(-1)=1,h^{\prime}(0)=0,h^{\prime}(1)=-1
  9. x , y , z , t { 0 , - 1 , 1 } x,y,z,t\in\left\{0,-1,1\right\}
  10. h x ( x ) = h ( x ) ; h_{x}^{\prime}(x)=h^{\prime}(x);
  11. h x ( x , y ) = h ( x ) h ( y ) h_{x}^{\prime}(x,y)=h^{\prime}(x)h(y)
  12. h x ( x , y , z ) = h ( x ) h ( y ) h ( z ) h_{x}^{\prime}(x,y,z)=h^{\prime}(x)h(y)h(z)
  13. h x ( x , y , z , t ) = h ( x ) h ( y ) h ( z ) h ( t ) h_{x}^{\prime}(x,y,z,t)=h^{\prime}(x)h(y)h(z)h(t)
  14. h z ( : , : , - 1 ) = [ + 1 + 2 + 1 + 2 + 4 + 2 + 1 + 2 + 1 ] h z ( : , : , 0 ) = [ 0 0 0 0 0 0 0 0 0 ] h z ( : , : , 1 ) = [ - 1 - 2 - 1 - 2 - 4 - 2 - 1 - 2 - 1 ] h_{z}^{\prime}(:,:,-1)=\begin{bmatrix}+1&+2&+1\\ +2&+4&+2\\ +1&+2&+1\end{bmatrix}\quad h_{z}^{\prime}(:,:,0)=\begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix}\quad h_{z}^{\prime}(:,:,1)=\begin{bmatrix}-1&-2&-1\\ -2&-4&-2\\ -1&-2&-1\end{bmatrix}
  15. [ 1 0 - 1 2 0 - 2 1 0 - 1 ] = [ 1 2 1 ] [ 1 0 - 1 ] = [ 1 1 ] * [ 1 1 ] [ 1 - 1 ] * [ 1 1 ] \begin{bmatrix}1&0&-1\\ 2&0&-2\\ 1&0&-1\end{bmatrix}=\begin{bmatrix}1\\ 2\\ 1\end{bmatrix}\begin{bmatrix}1&0&-1\end{bmatrix}=\begin{bmatrix}1\\ 1\end{bmatrix}*\begin{bmatrix}1\\ 1\end{bmatrix}\begin{bmatrix}1&-1\end{bmatrix}*\begin{bmatrix}1&1\end{bmatrix}
  16. [ 1 2 1 0 0 0 - 1 - 2 - 1 ] = [ 1 0 - 1 ] [ 1 2 1 ] = [ 1 1 ] * [ 1 - 1 ] [ 1 1 ] * [ 1 1 ] \begin{bmatrix}\ \ 1&\ \ 2&\ \ 1\\ \ \ 0&\ \ 0&\ \ 0\\ -1&-2&-1\end{bmatrix}=\begin{bmatrix}\ \ 1\\ \ \ 0\\ -1\end{bmatrix}\begin{bmatrix}1&2&1\end{bmatrix}=\begin{bmatrix}1\\ 1\end{bmatrix}*\begin{bmatrix}\ \ 1\\ -1\end{bmatrix}\begin{bmatrix}1&1\end{bmatrix}*\begin{bmatrix}1&1\end{bmatrix}
  17. 𝐆 x = [ 1 2 1 ] * ( [ 1 0 - 1 ] * 𝐀 ) and 𝐆 y = [ 1 0 - 1 ] * ( [ 1 2 1 ] * 𝐀 ) \mathbf{G}_{x}=\begin{bmatrix}1\\ 2\\ 1\end{bmatrix}*\left(\begin{bmatrix}1&0&-1\end{bmatrix}*\mathbf{A}\right)\quad% \mbox{and}~{}\quad\mathbf{G}_{y}=\begin{bmatrix}\ \ 1\\ \ \ 0\\ -1\end{bmatrix}*\left(\begin{bmatrix}1&2&1\end{bmatrix}*\mathbf{A}\right)
  18. [ + 3 + 10 + 3 0 0 0 - 3 - 10 - 3 ] [ + 3 0 - 3 + 10 0 - 10 + 3 0 - 3 ] \begin{bmatrix}+3&+10&+3\\ 0&0&0\\ -3&-10&-3\end{bmatrix}\ \ \ \ \ \ \ \ \ \begin{bmatrix}+3&0&-3\\ +10&0&-10\\ +3&0&-3\end{bmatrix}
  19. [ 3 10 3 ] = [ 1 3 ] * [ 3 1 ] \begin{bmatrix}3&10&3\end{bmatrix}=\begin{bmatrix}1&3\end{bmatrix}*\begin{% bmatrix}3&1\end{bmatrix}

Sobolev_space.html

  1. Ω u D α φ d x = ( - 1 ) | α | Ω φ D α u d x , \int_{\Omega}uD^{\alpha}\varphi\;dx=(-1)^{|\alpha|}\int_{\Omega}\varphi D^{% \alpha}u\;dx,
  2. D α f = | α | f x 1 α 1 x n α n , D^{\alpha}f=\frac{\partial^{|\alpha|}f}{\partial x_{1}^{\alpha_{1}}\dots% \partial x_{n}^{\alpha_{n}}},
  3. Ω u D α φ d x = ( - 1 ) | α | Ω φ v d x , φ C c ( Ω ) , \int_{\Omega}uD^{\alpha}\varphi\;dx=(-1)^{|\alpha|}\int_{\Omega}\varphi v\;dx,% \ \ \ \ \varphi\in C_{c}^{\infty}(\Omega),
  4. u ( x ) = { 1 + x if - 1 < x < 0 10 if x = 0 1 - x if 0 < x < 1 0 otherwise u(x)=\begin{cases}1+x&\,\text{if }-1<x<0\\ 10&\,\text{if }x=0\\ 1-x&\,\text{if }0<x<1\\ 0&\,\text{otherwise}\end{cases}
  5. v ( x ) = { 1 if - 1 < x < 0 - 1 if 0 < x < 1 0 otherwise v(x)=\begin{cases}1&\,\text{if }-1<x<0\\ -1&\,\text{if }0<x<1\\ 0&\,\text{otherwise}\end{cases}
  6. u ( x ) u(x)
  7. W 1 , p W^{1,p}
  8. 𝐑 \mathbf{R}
  9. f f
  10. f f
  11. k k
  12. p ( 1 p + ) p(1≤p≤+∞)
  13. ( k 1 ) (k−1)
  14. f f
  15. f k , p = ( i = 0 k f ( i ) p p ) 1 p = ( i = 0 k | f ( i ) ( t ) | p d t ) 1 p . \|f\|_{k,p}=\left(\sum_{i=0}^{k}\left\|f^{(i)}\right\|_{p}^{p}\right)^{\frac{1% }{p}}=\left(\sum_{i=0}^{k}\int\left|f^{(i)}(t)\right|^{p}\,dt\right)^{\frac{1}% {p}}.
  16. f ( k ) p + f p \left\|f^{(k)}\right\|_{p}+\|f\|_{p}
  17. p = 2 p=2
  18. p = 2 p=2
  19. H k ( 𝕋 ) = { f L 2 ( 𝕋 ) : n = - ( 1 + n 2 + n 4 + + n 2 k ) | f ^ ( n ) | 2 < } H^{k}({\mathbb{T}})=\left\{f\in L^{2}({\mathbb{T}}):\sum_{n=-\infty}^{\infty}% \left(1+n^{2}+n^{4}+\dots+n^{2k}\right)\left|\widehat{f}(n)\right|^{2}<\infty\right\}
  20. f ^ \widehat{f}
  21. f f
  22. f k , 2 2 = n = - ( 1 + | n | 2 ) k | f ^ ( n ) | 2 . \|f\|^{2}_{k,2}=\sum_{n=-\infty}^{\infty}\left(1+|n|^{2}\right)^{k}\left|% \widehat{f}(n)\right|^{2}.
  23. u , v H k = i = 0 k D i u , D i v L 2 . \langle u,v\rangle_{H^{k}}=\sum_{i=0}^{k}\left\langle D^{i}u,D^{i}v\right% \rangle_{L^{2}}.
  24. ( 0 , 1 ) (0,1)
  25. I I
  26. I I
  27. Ω Ω
  28. k k
  29. 1 p + 1≤p≤+∞
  30. f f
  31. Ω Ω
  32. α α
  33. | α | k |α|≤k
  34. f ( α ) = | α | f x 1 α 1 x n α n f^{(\alpha)}=\frac{\partial^{|\alpha|}f}{\partial x_{1}^{\alpha_{1}}\dots% \partial x_{n}^{\alpha_{n}}}
  35. f ( α ) L p < . \left\|f^{(\alpha)}\right\|_{L^{p}}<\infty.
  36. W k , p ( Ω ) = { u L p ( Ω ) : D α u L p ( Ω ) | α | k } . W^{k,p}(\Omega)=\left\{u\in L^{p}(\Omega):D^{\alpha}u\in L^{p}(\Omega)\,\,% \forall|\alpha|\leq k\right\}.
  37. k k
  38. u W k , p ( Ω ) := { ( | α | k D α u L p ( Ω ) p ) 1 p , 1 p < + ; max | α | k D α u L ( Ω ) , p = + ; \|u\|_{W^{k,p}(\Omega)}:=\begin{cases}\left(\sum_{|\alpha|\leq k}\left\|D^{% \alpha}u\right\|_{L^{p}(\Omega)}^{p}\right)^{\frac{1}{p}},&1\leq p<+\infty;\\ \max_{|\alpha|\leq k}\left\|D^{\alpha}u\right\|_{L^{\infty}(\Omega)},&p=+% \infty;\end{cases}
  39. u W k , p ( Ω ) := { | α | k D α u L p ( Ω ) , 1 p < + ; | α | k D α u L ( Ω ) , p = + . \|u\|^{\prime}_{W^{k,p}(\Omega)}:=\begin{cases}\sum_{|\alpha|\leq k}\left\|D^{% \alpha}u\right\|_{L^{p}(\Omega)},&1\leq p<+\infty;\\ \sum_{|\alpha|\leq k}\left\|D^{\alpha}u\right\|_{L^{\infty}(\Omega)},&p=+% \infty.\end{cases}
  40. W k , 2 ( Ω ) \|\cdot\|_{W^{k,2}(\Omega)}
  41. p p
  42. Ω Ω
  43. u m - u W k , p ( Ω ) 0. \left\|u_{m}-u\right\|_{W^{k,p}(\Omega)}\to 0.
  44. f ( x ) = | x | - α W k , p ( 𝐁 n ) α < n p - k . f(x)=|x|^{-\alpha}\in W^{k,p}(\mathbf{B}^{n})\ \Leftrightarrow\ \alpha<\tfrac{% n}{p}-k.
  45. 1 p + 1≤p≤+∞
  46. f f
  47. f ∇f
  48. f f
  49. f f
  50. | f | |∇f|
  51. f f
  52. f f
  53. p > n p>n
  54. γ = 1 n / p γ=1−n/p
  55. p = + p=+∞
  56. Ω Ω
  57. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  58. Ω Ω
  59. f H 1 = ( Ω ( | f | 2 + | f | 2 ) ) 1 2 . \|f\|_{H^{1}}=\left(\int_{\Omega}\left(|f|^{2}+|\nabla f|^{2}\right)\right)^{% \frac{1}{2}}.
  60. Ω Ω
  61. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  62. n = 1 n=1
  63. Ω = ( a , b ) Ω=(a,b)
  64. H [ u s u , u p = 1 , u b = 0 ] ( a , b ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](a,b)
  65. a a , b aa,b
  66. f ( x ) = a x f ( t ) d t , x [ a , b ] f(x)=\int_{a}^{x}f^{\prime}(t)\,\mathrm{d}t,\qquad x\in[a,b]
  67. f f′
  68. f ( b ) = f ( a ) = 0 f(b)=f(a)=0
  69. Ω Ω
  70. C = C ( Ω ) C=C(Ω)
  71. Ω | f | 2 C 2 Ω | f | 2 , f H 0 1 ( Ω ) . \int_{\Omega}|f|^{2}\leq C^{2}\,\int_{\Omega}|\nabla f|^{2},\quad f\in H^{1}_{% 0}(\Omega).
  72. Ω Ω
  73. H [ u s u , u p = 1 , u b = 0 ] ( Ω ) H[u^{\prime}su^{\prime},u^{\prime}p=1^{\prime},u^{\prime}b=0^{\prime}](Ω)
  74. W k , p ( n ) = H k , p ( n ) := { f L p ( n ) : - 1 [ ( 1 + | ξ | 2 ) k 2 f ] L p ( n ) } W^{k,p}(\mathbb{R}^{n})=H^{k,p}(\mathbb{R}^{n}):=\left\{f\in L^{p}(\mathbb{R}^% {n}):\mathcal{F}^{-1}[(1+|\xi|^{2})^{\frac{k}{2}}\mathcal{F}f]\in L^{p}(% \mathbb{R}^{n})\right\}
  75. f H k , p ( n ) := - 1 [ ( 1 + | ξ | 2 ) k 2 f ] L p ( n ) \|f\|_{H^{k,p}(\mathbb{R}^{n})}:=\left\|\mathcal{F}^{-1}[\left(1+|\xi|^{2}% \right)^{\frac{k}{2}}\mathcal{F}f]\right\|_{L^{p}(\mathbb{R}^{n})}
  76. H s , p ( n ) := { f L p ( n ) : - 1 [ ( 1 + | ξ | 2 ) s 2 f ] L p ( n ) } H^{s,p}(\mathbb{R}^{n}):=\left\{f\in L^{p}(\mathbb{R}^{n}):\mathcal{F}^{-1}[% \left(1+|\xi|^{2}\right)^{\frac{s}{2}}\mathcal{F}f]\in L^{p}(\mathbb{R}^{n})\right\}
  77. f H s , p ( Ω ) := inf { g H s , p ( n ) : g H s , p ( n ) , g | Ω = f } \|f\|_{H^{s,p}(\Omega)}:=\inf\left\{\|g\|_{H^{s,p}(\mathbb{R}^{n})}:g\in H^{s,% p}(\mathbb{R}^{n}),g|_{\Omega}=f\right\}
  78. s > 0 s>0
  79. θ = s - s ( 0 , 1 ) \theta=s-\lfloor s\rfloor\in(0,1)
  80. W s , p ( Ω ) := { f W s , p ( Ω ) : sup | α | = s [ D α f ] θ , p , Ω < } W^{s,p}(\Omega):=\left\{f\in W^{\lfloor s\rfloor,p}(\Omega):\sup_{|\alpha|=% \lfloor s\rfloor}[D^{\alpha}f]_{\theta,p,\Omega}<\infty\right\}
  81. f W s , p ( Ω ) := f W s , p ( Ω ) + sup | α | = s [ D α f ] θ , p , Ω \|f\|_{W^{s,p}(\Omega)}:=\|f\|_{W^{\lfloor s\rfloor,p}(\Omega)}+\sup_{|\alpha|% =\lfloor s\rfloor}[D^{\alpha}f]_{\theta,p,\Omega}
  82. W k + 1 , p ( Ω ) W s , p ( Ω ) W s , p ( Ω ) W k , p ( Ω ) , k s s k + 1 W^{k+1,p}(\Omega)\hookrightarrow W^{s^{\prime},p}(\Omega)\hookrightarrow W^{s,% p}(\Omega)\hookrightarrow W^{k,p}(\Omega),\quad k\leq s\leq s^{\prime}\leq k+1
  83. W s , p ( Ω ) = ( W k , p ( Ω ) , W k + 1 , p ( Ω ) ) θ , p , k , s ( k , k + 1 ) , θ = s - s W^{s,p}(\Omega)=\left(W^{k,p}(\Omega),W^{k+1,p}(\Omega)\right)_{\theta,p},% \quad k\in\mathbb{N},s\in(k,k+1),\theta=s-\lfloor s\rfloor
  84. u | Ω u|_{\partial\Omega}
  85. T : W 1 , p ( Ω ) L p ( Ω ) T:W^{1,p}(\Omega)\to L^{p}(\partial\Omega)
  86. T u = u | Ω u W 1 , p ( Ω ) C ( Ω ¯ ) T u L p ( Ω ) c ( p , Ω ) u W 1 , p ( Ω ) u W 1 , p ( Ω ) . \begin{aligned}\displaystyle Tu&\displaystyle=u|_{\partial\Omega}&&% \displaystyle u\in W^{1,p}(\Omega)\cap C(\overline{\Omega})\\ \displaystyle\left\|Tu\right\|_{L^{p}(\partial\Omega)}&\displaystyle\leq c(p,% \Omega)\|u\|_{W^{1,p}(\Omega)}&&\displaystyle u\in W^{1,p}(\Omega).\end{aligned}
  87. W 0 1 , p ( Ω ) = { u W 1 , p ( Ω ) : T u = 0 } , W_{0}^{1,p}(\Omega)=\left\{u\in W^{1,p}(\Omega):Tu=0\right\},
  88. W 0 1 , p ( Ω ) := { u W 1 , p ( Ω ) : { u m } m = 1 C c ( Ω ) , such that u m u in W 1 , p ( Ω ) } . W_{0}^{1,p}(\Omega):=\left\{u\in W^{1,p}(\Omega):\exists\{u_{m}\}_{m=1}^{% \infty}\subset C_{c}^{\infty}(\Omega),\ \textrm{such}\ \textrm{that}\ u_{m}\to u% \ \textrm{in}\ W^{1,p}(\Omega)\right\}.
  89. W k , p ( X ) W^{k,p}(X)
  90. W k , p ( n ) W^{k,p}({\mathbb{R}}^{n})
  91. H s ( X ) H^{s}(X)
  92. H s ( X ) H^{s}(X)
  93. H s ( X ) H^{s}(X)
  94. H s ( n ) H^{s}(\mathbb{R}^{n})
  95. H s ( X ) H^{s}(X)
  96. H s ( X ) H^{s}(X)
  97. H 0 s ( X ) H^{s}_{0}(X)
  98. H s ( X ) H^{s}(X)
  99. C c ( X ) C^{\infty}_{c}(X)
  100. H s ( X ) H^{s}(X)
  101. ( u , d u d n , , d k u d n k ) | G \left.\left(u,\frac{du}{dn},\dots,\frac{d^{k}u}{dn^{k}}\right)\right|_{G}
  102. H 0 s H^{s}_{0}
  103. u H 0 s ( X ) u\in H^{s}_{0}(X)
  104. u ~ L 2 ( n ) \tilde{u}\in L^{2}({\mathbb{R}}^{n})
  105. u ~ ( x ) = u ( x ) if x X , 0 otherwise. \tilde{u}(x)=u(x)\;\textrm{ if }\;x\in X,0\;\textrm{ otherwise.}
  106. u ~ \tilde{u}
  107. H s ( n ) H^{s}({\mathbb{R}}^{n})
  108. E f := { f on Ω , 0 otherwise Ef:=\begin{cases}f&\textrm{on}\ \Omega,\\ 0&\textrm{otherwise}\end{cases}
  109. E f L p ( n ) = f L p ( Ω ) . \left\|Ef\right\|_{L^{p}(\mathbb{R}^{n})}=\left\|f\right\|_{L^{p}(\Omega)}.
  110. E : W 1 , p ( Ω ) W 1 , p ( n ) , E:W^{1,p}(\Omega)\rightarrow W^{1,p}(\mathbb{R}^{n}),
  111. E u W 1 , p ( n ) C u W 1 , p ( Ω ) . \left\|Eu\right\|_{W^{1,p}(\mathbb{R}^{n})}\leq C\left\|u\right\|_{W^{1,p}(% \Omega)}.
  112. W k , p W^{k,p}
  113. W k , W^{k,\infty}
  114. W m , W^{m,\infty}

Sociable_number.html

  1. 1264460 1264460
  2. = 2 2 5 17 3719 =2^{2}\cdot 5\cdot 17\cdot 3719
  3. 1547860 1547860
  4. = 2 2 5 193 401 =2^{2}\cdot 5\cdot 193\cdot 401
  5. 1727636 1727636
  6. = 2 2 521 829 =2^{2}\cdot 521\cdot 829
  7. 1305184 1305184
  8. = 2 5 40787 =2^{5}\cdot 40787

Solar_radius.html

  1. 1 R 1\,R_{\odot}
  2. 1 R = 6.955 × 10 5 km 1\,R_{\odot}=6.955\times 10^{5}\hbox{ km}

Solar_thermal_energy.html

  1. E = h ν E=h\nu
  2. ν \nu

Solenoidal_vector_field.html

  1. 𝐯 = 0. \nabla\cdot\mathbf{v}=0.\,
  2. 𝐯 = × 𝐀 \mathbf{v}=\nabla\times\mathbf{A}
  3. 𝐯 = ( × 𝐀 ) = 0. \nabla\cdot\mathbf{v}=\nabla\cdot(\nabla\times\mathbf{A})=0.
  4. 𝐯 = × 𝐀 . \mathbf{v}=\nabla\times\mathbf{A}.
  5. d 𝐒 d\mathbf{S}
  6. ρ e = 0 \rho_{e}=0
  7. ρ e t = 0 \frac{\partial\rho_{e}}{\partial t}=0

Solid_modeling.html

  1. f = a x + b y + c z + d f=ax+by+cz+d
  2. f ( p ) = 0 f(p)=0
  3. f ( p ) > 0 f(p)>0
  4. f ( p ) < 0 f(p)<0

Solid_of_revolution.html

  1. f ( x ) f(x)
  2. g ( x ) g(x)
  3. x = a x=a
  4. x = b x=b
  5. V = π a b | f ( x ) 2 - g ( x ) 2 | d x V=\pi\int_{a}^{b}|f(x)^{2}-g(x)^{2}|\,dx
  6. V = π a b f ( x ) 2 d x ( 1 ) V=\pi\int_{a}^{b}f(x)^{2}\,dx\qquad(1)
  7. f ( y ) f(y)
  8. g ( y ) g(y)
  9. g ( y ) = 0 g(y)=0
  10. π ( R 2 - r 2 ) \pi(R^{2}-r^{2})
  11. π f ( y ) 2 d y \pi f(y)^{2}dy
  12. f ( x ) f(x)
  13. g ( x ) g(x)
  14. x = a x=a
  15. x = b x=b
  16. V = 2 π a b x | f ( x ) - g ( x ) | d x V=2\pi\int_{a}^{b}x|f(x)-g(x)|\,dx
  17. V = 2 π a b x | f ( x ) | d x V=2\pi\int_{a}^{b}x|f(x)|\,dx
  18. [ f ( x ) - g ( x ) ] [f(x)-g(x)]
  19. 2 π r h 2\pi rh
  20. [ f ( x ) - g ( x ) ] [f(x)-g(x)]
  21. ( x ( t ) , y ( t ) ) (x(t),y(t))
  22. [ a , b ] [a,b]
  23. V x = a b π y 2 d x d t d t V_{x}=\int_{a}^{b}\,\pi\,y^{2}\,\frac{dx}{dt}\,dt
  24. V y = a b π x 2 d y d t d t . V_{y}=\int_{a}^{b}\pi\,\,x^{2}\,\frac{dy}{dt}\,dt.
  25. A x = a b 2 π y ( d x d t ) 2 + ( d y d t ) 2 d t A_{x}=\int_{a}^{b}2\pi y\,\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}% {dt}\right)^{2}}\,dt
  26. A y = a b 2 π x ( d x d t ) 2 + ( d y d t ) 2 d t A_{y}=\int_{a}^{b}2\pi x\,\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}% {dt}\right)^{2}}\,dt

Solomonoff's_theory_of_inductive_inference.html

  1. ϵ \epsilon
  2. ϵ \epsilon

Solution_set.html

  1. { f i } \{f_{i}\}
  2. R R
  3. R R
  4. { x R : i I , f i ( x ) = 0 } . \{x\in R:\forall i\in I,f_{i}(x)=0\}.
  5. x = 0 x=0
  6. f f
  7. ( x j ) j J {(x_{j})}_{j\in J}
  8. ( X j ) j J {(X_{j})}_{j\in J}
  9. x ( k ) x^{(k)}
  10. ( x j ( k ) ) j J j J X j {(x^{(k)}_{j})}_{j\in J}\in\prod_{j\in J}X_{j}
  11. ( x j ) j J {(x_{j})}_{j\in J}
  12. x ( k ) x^{(k)}
  13. ( x , y ) 2 (x,y)\in\mathbb{R}^{2}
  14. x x\in\mathbb{R}
  15. E = { x 4 } E=\{\sqrt{x}\leq 4\}
  16. x x\in\mathbb{R}
  17. x \sqrt{x}
  18. E = { exp ( i x ) = 1 } E=\{\exp(ix)=1\}
  19. x x\in\mathbb{C}

Sorting_network.html

  1. x x
  2. y y
  3. x = min ( x , y ) x^{\prime}=\min(x,y)
  4. y = max ( x , y ) y^{\prime}=\max(x,y)
  5. 2 n - 3 2n-3
  6. n ! n!
  7. n n
  8. n n
  9. n ! n!
  10. n n
  11. f f
  12. x x
  13. y y
  14. f ( x ) f(x)
  15. f ( y ) f(y)
  16. m i n ( f ( x ) , f ( y ) ) = f ( m i n ( x , y ) ) min(f(x),f(y))=f(min(x,y))
  17. m a x ( f ( x ) , f ( y ) ) = f ( m a x ( x , y ) ) max(f(x),f(y))=f(max(x,y))
  18. f ( x ) = { 1 if x > a i 0 otherwise. f(x)=\begin{cases}1&\mbox{if }~{}x>a_{i}\\ 0&\mbox{otherwise.}\end{cases}
  19. O ( l o g < s u p > 2 n ) O(log<sup>2n)

Sound_intensity.html

  1. 𝐈 = p 𝐯 \mathbf{I}=p\mathbf{v}
  2. 𝐈 = 1 T 0 T p ( t ) 𝐯 ( t ) d t . \langle\mathbf{I}\rangle=\frac{1}{T}\int_{0}^{T}p(t)\mathbf{v}(t)\,\mathrm{d}t.
  3. I = P A , I=\frac{P}{A},
  4. I = c w , I=cw,
  5. I ( r ) = P A ( r ) = P 4 π r 2 , I(r)=\frac{P}{A(r)}=\frac{P}{4\pi r^{2}},
  6. I ( r ) 1 r 2 . I(r)\propto\frac{1}{r^{2}}.
  7. L I = 1 2 ln ( I I 0 ) Np = log 10 ( I I 0 ) B = 10 log 10 ( I I 0 ) dB , L_{I}=\frac{1}{2}\ln\!\left(\frac{I}{I_{0}}\right)\!~{}\mathrm{Np}=\log_{10}\!% \left(\frac{I}{I_{0}}\right)\!~{}\mathrm{B}=10\log_{10}\!\left(\frac{I}{I_{0}}% \right)\!~{}\mathrm{dB},
  8. I 0 = 1 pW / m 2 . I_{0}=1~{}\mathrm{pW/m^{2}}.
  9. I p 2 . I\propto p^{2}.
  10. I I 0 = p 2 p 0 2 , \frac{I}{I_{0}}=\frac{p^{2}}{p_{0}^{2}},
  11. p v = z 0 , \frac{p}{v}=z_{0},
  12. I 0 = p 0 2 I p 2 = p 0 2 p v p 2 = p 0 2 z 0 . I_{0}=\frac{p_{0}^{2}I}{p^{2}}=\frac{p_{0}^{2}pv}{p^{2}}=\frac{p_{0}^{2}}{z_{0% }}.

Sound_power.html

  1. P = 𝐟 𝐯 = A p 𝐮 𝐯 = A p v P=\mathbf{f}\cdot\mathbf{v}=Ap\,\mathbf{u}\cdot\mathbf{v}=Apv
  2. P = A p 2 ρ c cos θ , P=\frac{Ap^{2}}{\rho c}\cos\theta,
  3. P = A I , P=AI,
  4. P = A c w , P=Acw,
  5. L W = 1 2 ln ( P P 0 ) Np = log 10 ( P P 0 ) B = 10 log 10 ( P P 0 ) dB , L_{W}=\frac{1}{2}\ln\!\left(\frac{P}{P_{0}}\right)\!~{}\mathrm{Np}=\log_{10}\!% \left(\frac{P}{P_{0}}\right)\!~{}\mathrm{B}=10\log_{10}\!\left(\frac{P}{P_{0}}% \right)\!~{}\mathrm{dB},
  6. P 0 = 1 pW . P_{0}=1~{}\mathrm{pW}.
  7. P 0 = A 0 I 0 , P_{0}=A_{0}I_{0},
  8. L W = L p + 10 log 10 ( 4 π r 2 A 0 ) dB , L_{W}=L_{p}+10\log_{10}\!\left(\frac{4\pi r^{2}}{A_{0}}\right)\!~{}\mathrm{dB},
  9. L W \displaystyle L_{W}
  10. z 0 = p v , z_{0}=\frac{p}{v},
  11. A = 4 π r 2 , A=4\pi r^{2},
  12. I = p v = p 2 z 0 , I=pv=\frac{p^{2}}{z_{0}},
  13. L W = 1 2 ln ( p 2 p 0 2 ) + 1 2 ln ( 4 π r 2 A 0 ) = ln ( p p 0 ) + 1 2 ln ( 4 π r 2 A 0 ) = L p + 10 log 10 ( 4 π r 2 A 0 ) dB . \begin{aligned}\displaystyle L_{W}&\displaystyle=\frac{1}{2}\ln\!\left(\frac{p% ^{2}}{p_{0}^{2}}\right)+\frac{1}{2}\ln\!\left(\frac{4\pi r^{2}}{A_{0}}\right)% \\ &\displaystyle=\ln\!\left(\frac{p}{p_{0}}\right)+\frac{1}{2}\ln\!\left(\frac{4% \pi r^{2}}{A_{0}}\right)\\ &\displaystyle=L_{p}+10\log_{10}\!\left(\frac{4\pi r^{2}}{A_{0}}\right)\!~{}% \mathrm{dB}.\end{aligned}

Sound_pressure.html

  1. p total = p stat + p , p_{\mathrm{total}}=p_{\mathrm{stat}}+p,
  2. 𝐈 = p 𝐯 , \mathbf{I}=p\mathbf{v},
  3. Z ( s ) = p ^ ( s ) Q ^ ( s ) , Z(s)=\frac{\hat{p}(s)}{\hat{Q}(s)},
  4. p ^ ( s ) \hat{p}(s)
  5. Q ^ ( s ) \hat{Q}(s)
  6. z ( s ) = p ^ ( s ) v ^ ( s ) , z(s)=\frac{\hat{p}(s)}{\hat{v}(s)},
  7. p ^ ( s ) \hat{p}(s)
  8. v ^ ( s ) \hat{v}(s)
  9. ξ ( 𝐫 , t ) = ξ m cos ( 𝐤 𝐫 - ω t + φ ξ , 0 ) , \xi(\mathbf{r},\,t)=\xi_{\mathrm{m}}\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+% \varphi_{\xi,0}),
  10. φ ξ , 0 \varphi_{\xi,0}
  11. v ( 𝐫 , t ) = ξ t ( 𝐫 , t ) = ω ξ m cos ( 𝐤 𝐫 - ω t + φ ξ , 0 + π 2 ) = v m cos ( 𝐤 𝐫 - ω t + φ v , 0 ) , v(\mathbf{r},\,t)=\frac{\partial\xi}{\partial t}(\mathbf{r},\,t)=\omega\xi_{% \mathrm{m}}\cos\!\left(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi_{\xi,0}+% \frac{\pi}{2}\right)=v_{\mathrm{m}}\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+% \varphi_{v,0}),
  12. p ( 𝐫 , t ) = - ρ c 2 ξ x ( 𝐫 , t ) = ρ c 2 k x ξ m cos ( 𝐤 𝐫 - ω t + φ ξ , 0 + π 2 ) = p m cos ( 𝐤 𝐫 - ω t + φ p , 0 ) , p(\mathbf{r},\,t)=-\rho c^{2}\frac{\partial\xi}{\partial x}(\mathbf{r},\,t)=% \rho c^{2}k_{x}\xi_{\mathrm{m}}\cos\!\left(\mathbf{k}\cdot\mathbf{r}-\omega t+% \varphi_{\xi,0}+\frac{\pi}{2}\right)=p_{\mathrm{m}}\cos(\mathbf{k}\cdot\mathbf% {r}-\omega t+\varphi_{p,0}),
  13. φ v , 0 \varphi_{v,0}
  14. φ p , 0 \varphi_{p,0}
  15. v ^ ( 𝐫 , s ) = v m s cos φ v , 0 - ω sin φ v , 0 s 2 + ω 2 , \hat{v}(\mathbf{r},\,s)=v_{\mathrm{m}}\frac{s\cos\varphi_{v,0}-\omega\sin% \varphi_{v,0}}{s^{2}+\omega^{2}},
  16. p ^ ( 𝐫 , s ) = p m s cos φ p , 0 - ω sin φ p , 0 s 2 + ω 2 . \hat{p}(\mathbf{r},\,s)=p_{\mathrm{m}}\frac{s\cos\varphi_{p,0}-\omega\sin% \varphi_{p,0}}{s^{2}+\omega^{2}}.
  17. φ v , 0 = φ p , 0 \varphi_{v,0}=\varphi_{p,0}
  18. z m ( 𝐫 , s ) = | z ( 𝐫 , s ) | = | p ^ ( 𝐫 , s ) v ^ ( 𝐫 , s ) | = p m v m = ρ c 2 k x ω . z_{\mathrm{m}}(\mathbf{r},\,s)=|z(\mathbf{r},\,s)|=\left|\frac{\hat{p}(\mathbf% {r},\,s)}{\hat{v}(\mathbf{r},\,s)}\right|=\frac{p_{\mathrm{m}}}{v_{\mathrm{m}}% }=\frac{\rho c^{2}k_{x}}{\omega}.
  19. ξ m = v m ω , \xi_{\mathrm{m}}=\frac{v_{\mathrm{m}}}{\omega},
  20. ξ m = p m ω z m ( 𝐫 , s ) . \xi_{\mathrm{m}}=\frac{p_{\mathrm{m}}}{\omega z_{\mathrm{m}}(\mathbf{r},\,s)}.
  21. p ( r ) 1 r . p(r)\propto\frac{1}{r}.
  22. p 2 = r 1 r 2 p 1 . p_{2}=\frac{r_{1}}{r_{2}}\,p_{1}.
  23. I ( r ) 1 r 2 . I(r)\propto\frac{1}{r^{2}}.
  24. I ( r ) = p ( r ) v ( r ) = p ( r ) [ p * z - 1 ] ( r ) p 2 ( r ) , I(r)=p(r)v(r)=p(r)[p*z^{-1}](r)\propto p^{2}(r),
  25. * *
  26. p ( r ) 1 r . p(r)\propto\frac{1}{r}.
  27. L p = ln ( p p 0 ) Np = 2 log 10 ( p p 0 ) B = 20 log 10 ( p p 0 ) dB , L_{p}=\ln\!\left(\frac{p}{p_{0}}\right)\!~{}\mathrm{Np}=2\log_{10}\!\left(% \frac{p}{p_{0}}\right)\!~{}\mathrm{B}=20\log_{10}\!\left(\frac{p}{p_{0}}\right% )\!~{}\mathrm{dB},
  28. p 0 = 20 μ Pa , p_{0}=20~{}\mathrm{\mu Pa},
  29. L p 2 = L p 1 + 20 log 10 ( r 1 r 2 ) dB . L_{p_{2}}=L_{p_{1}}+20\log_{10}\!\left(\frac{r_{1}}{r_{2}}\right)\!~{}\mathrm{% dB}.
  30. L Σ = 10 log 10 ( p 1 2 + p 2 2 + + p n 2 p 0 2 ) dB = 10 log 10 [ ( p 1 p 0 ) 2 + ( p 2 p 0 ) 2 + + ( p n p 0 ) 2 ] dB . L_{\Sigma}=10\log_{10}\!\left(\frac{{p_{1}}^{2}+{p_{2}}^{2}+\ldots+{p_{n}}^{2}% }{{p_{0}}^{2}}\right)\!~{}\mathrm{dB}=10\log_{10}\!\left[\left(\frac{p_{1}}{p_% {0}}\right)^{2}+\left(\frac{p_{2}}{p_{0}}\right)^{2}+\ldots+\left(\frac{p_{n}}% {p_{0}}\right)^{2}\right]\!~{}\mathrm{dB}.
  31. ( p i p 0 ) 2 = 10 L i 10 dB , i = 1 , 2 , , n , \left(\frac{p_{i}}{p_{0}}\right)^{2}=10^{\frac{L_{i}}{10\,\mathrm{dB}}},\quad i% =1,\,2,\,\ldots,\,n,
  32. L Σ = 10 log 10 ( 10 L 1 10 dB + 10 L 2 10 dB + + 10 L n 10 dB ) dB . L_{\Sigma}=10\log_{10}\!\left(10^{\frac{L_{1}}{10\,\mathrm{dB}}}+10^{\frac{L_{% 2}}{10\,\mathrm{dB}}}+\ldots+10^{\frac{L_{n}}{10\,\mathrm{dB}}}\right)\!~{}% \mathrm{dB}.
  33. S P L {}_{SPL}

Space-filling_curve.html

  1. 𝒞 \scriptstyle\mathcal{C}
  2. 𝟐 \scriptstyle\mathbf{2}^{\mathbb{N}}
  3. h \scriptstyle h
  4. 𝒞 \scriptstyle\mathcal{C}
  5. [ 0 , 1 ] \scriptstyle[0,\,1]
  6. H \scriptstyle H
  7. 𝒞 × 𝒞 \scriptstyle\mathcal{C}\;\times\;\mathcal{C}
  8. [ 0 , 1 ] × [ 0 , 1 ] \scriptstyle[0,\,1]\;\times\;[0,\,1]
  9. H ( x , y ) = ( h ( x ) , h ( y ) ) . H(x,y)=(h(x),h(y)).\,
  10. 𝒞 × 𝒞 \scriptstyle\mathcal{C}\times\mathcal{C}
  11. g \scriptstyle g
  12. 𝒞 × 𝒞 \scriptstyle\mathcal{C}\;\times\;\mathcal{C}
  13. f \scriptstyle f
  14. H \scriptstyle H
  15. g \scriptstyle g
  16. f \scriptstyle f
  17. f \scriptstyle f
  18. F \scriptstyle F
  19. [ 0 , 1 ] \scriptstyle[0,\,1]
  20. f \scriptstyle f
  21. f \scriptstyle f
  22. ( a , b ) \scriptstyle(a,\,b)
  23. F \scriptstyle F
  24. ( a , b ) \scriptstyle(a,\,b)
  25. f ( a ) \scriptstyle f(a)
  26. f ( b ) \scriptstyle f(b)

Spanning_tree.html

  1. K p , q K_{p,q}
  2. t ( G ) = p q - 1 q p - 1 t(G)=p^{q-1}q^{p-1}
  3. Q n Q_{n}
  4. t ( G ) = 2 2 n - n - 1 k = 2 n k < m t p l > ( n k ) t(G)=2^{2^{n}-n-1}\prod_{k=2}^{n}k^{<}mtpl>{{n\choose k}}

Sparse_matrix.html

  1. ( 11 22 0 0 0 0 0 0 33 44 0 0 0 0 0 0 55 66 77 0 0 0 0 0 0 0 88 0 0 0 0 0 0 0 99 ) \left(\begin{smallmatrix}11&22&0&0&0&0&0\\ 0&33&44&0&0&0&0\\ 0&0&55&66&77&0&0\\ 0&0&0&0&0&88&0\\ 0&0&0&0&0&0&99\\ \end{smallmatrix}\right)
  2. i i
  3. j j
  4. i i
  5. j j
  6. m × n m×n
  7. m × n m×n
  8. ( r o w , c o l u m n ) (row,column)
  9. ( r o w , c o l u m n , v a l u e ) (row,column,value)
  10. m × n m×n
  11. 𝐌 \mathbf{M}
  12. ( A , I A , J A ) (A,IA,JA)
  13. N N Z NNZ
  14. A A
  15. N N Z NNZ
  16. I A IA
  17. m + 1 m+1
  18. A A
  19. N N Z NNZ
  20. I A i i IAii
  21. A A
  22. i i
  23. i i
  24. A I A A i i AIAAii
  25. A I A A i i + 1 11 AIAAii+1−11
  26. I A m m IAmm
  27. A A
  28. J A JA
  29. 𝐌 \mathbf{M}
  30. A A
  31. N N Z NNZ
  32. ( 0 0 0 0 5 8 0 0 0 0 3 0 0 6 0 0 ) \begin{pmatrix}0&0&0&0\\ 5&8&0&0\\ 0&0&3&0\\ 0&6&0&0\\ \end{pmatrix}
  33. 4 × 4 4×4
  34. J A JA
  35. 5 5
  36. A A
  37. 0
  38. 8 8
  39. 6 6
  40. 1 1
  41. 3 3
  42. 2 2
  43. ( 10 20 0 0 0 0 0 30 0 40 0 0 0 0 50 60 70 0 0 0 0 0 0 80 ) \begin{pmatrix}10&20&0&0&0&0\\ 0&30&0&40&0&0\\ 0&0&50&60&70&0\\ 0&0&0&0&0&80\\ \end{pmatrix}
  44. 4 × 6 4×6
  45. I A IA
  46. A A
  47. J A JA
  48. I A IA
  49. N N Z NNZ
  50. ( v a l , c o l i n d , r o w p t r ) (val,col_{i}nd,row_{p}tr)
  51. v a l val
  52. c o l i n d col_{i}nd
  53. r o w p t r row_{p}tr
  54. ( v a l , r o w i n d , c o l p t r ) (val,row_{i}nd,col_{p}tr)
  55. v a l val
  56. r o w i n d row_{i}nd
  57. c o l p t r col_{p}tr
  58. v a l val
  59. 𝐀 \mathbf{A}
  60. p p
  61. i > j + p i>j+p
  62. p p
  63. 1 1
  64. 1 1
  65. ( X X X X X X X X X X X X X X X X X X X X X X X ) \left(\begin{smallmatrix}X&X&X&\cdot&\cdot&\cdot&\cdot&\\ X&X&\cdot&X&X&\cdot&\cdot&\\ X&\cdot&X&\cdot&X&\cdot&\cdot&\\ \cdot&X&\cdot&X&\cdot&X&\cdot&\\ \cdot&X&X&\cdot&X&X&X&\\ \cdot&\cdot&\cdot&X&X&X&\cdot&\\ \cdot&\cdot&\cdot&\cdot&X&\cdot&X&\\ \end{smallmatrix}\right)
  66. 𝐀 \mathbf{A}
  67. 𝐀 \mathbf{A}′
  68. n × n n×n
  69. n n
  70. A x i Ax_{i}
  71. A A

Spearman–Brown_prediction_formula.html

  1. ρ x x * {\rho}^{*}_{xx^{\prime}}
  2. ρ x x * = N ρ x x 1 + ( N - 1 ) ρ x x {\rho}^{*}_{xx^{\prime}}=\frac{N{\rho}_{xx^{\prime}}}{1+(N-1){\rho}_{xx^{% \prime}}}
  3. ρ x x {\rho}_{xx^{\prime}}
  4. N = ρ x x * ( 1 - ρ x x ) ρ x x ( 1 - ρ x x * ) N=\frac{{\rho}^{*}_{xx^{\prime}}(1-{\rho}_{xx^{\prime}})}{{\rho}_{xx^{\prime}}% (1-{\rho}^{*}_{xx^{\prime}})}

Special_number_field_sieve.html

  1. n n
  2. exp ( ( 1 + o ( 1 ) ) ( 32 9 log n ) 1 / 3 ( log log n ) 2 / 3 ) = L n [ 1 / 3 , ( 32 / 9 ) 1 / 3 ] \exp\left(\left(1+o(1)\right)\left(\tfrac{32}{9}\log n\right)^{1/3}\left(\log% \log n\right)^{2/3}\right)=L_{n}\left[1/3,(32/9)^{1/3}\right]
  3. N max N_{\max}
  4. N max N_{\max}
  5. ( 3 log N log log N ) 1 / 3 \left(3\frac{\log N}{\log\log N}\right)^{1/3}
  6. f ( x ) 0 ( mod N ) f(x)\equiv 0\;\;(\mathop{{\rm mod}}N)
  7. a x + b 0 ( mod N ) ax+b\equiv 0\;\;(\mathop{{\rm mod}}N)
  8. N 1 / d N^{1/d}
  9. a b ± 1 a^{b}\pm 1
  10. 3 480 + 3 0 ( mod 3 479 + 1 ) 3^{480}+3\equiv 0\;\;(\mathop{{\rm mod}}3^{479}+1)
  11. F 709 F_{709}
  12. n 5 + 10 n 3 + 10 n 2 + 10 n + 3 n^{5}+10n^{3}+10n^{2}+10n+3
  13. F 142 x - F 141 = 0 F_{142}x-F_{141}=0

Species_diversity.html

  1. D q = 1 i = 1 S p i p i q - 1 q - 1 {}^{q}\!D={1\over\sqrt[q-1]{{\sum_{i=1}^{S}p_{i}p_{i}^{q-1}}}}
  2. p i p_{i}
  3. D q = ( i = 1 S p i q ) 1 / ( 1 - q ) {}^{q}\!D=\left({\sum_{i=1}^{S}p_{i}^{q}}\right)^{1/(1-q)}
  4. p i p_{i}
  5. lim q 1 D q = exp ( - i = 1 S p i ln p i ) \lim_{q\rightarrow 1}{}^{q}\!D=\exp\left(-\sum_{i=1}^{S}p_{i}\ln p_{i}\right)
  6. p i p_{i}
  7. p i p_{i}
  8. p i p_{i}

Specific_activity.html

  1. N = N 0 ( 1 2 ) t T 1 / 2 N=N_{0}\left(\frac{1}{2}\right)^{t\over T_{1/2}}
  2. ln ( N ) = ln ( N 0 ) + ( t T 1 / 2 ) ln ( 1 2 ) \ln(N)=\ln(N_{0})+\left(\frac{t}{T_{1/2}}\right)\ln\left(\frac{1}{2}\right)
  3. 1 N d N d t = ln ( 1 2 ) T 1 / 2 \frac{1}{N}\frac{dN}{dt}=\frac{\ln\left(\frac{1}{2}\right)}{T_{1/2}}
  4. d N d t = N ln ( 1 2 ) T 1 / 2 \frac{dN}{dt}=\frac{N\ln\left(\frac{1}{2}\right)}{T_{1/2}}
  5. d N d t = - 0.693 N T 1 / 2 \frac{dN}{dt}=\frac{-0.693\,N}{T_{1/2}}
  6. T 1 / 2 = - 0.693 N d N d t T_{1/2}=\frac{-0.693\,N}{\frac{dN}{dt}}
  7. T 1 / 2 = - 0.693 ( 6.9 × 10 21 ) - 3200 s - 1 = 1.5 × 10 18 s or 47 billion years T_{1/2}=\frac{-0.693(6.9\times 10^{21})}{-3200\,\text{ s}^{-1}}=1.5\times 10^{% 18}\,\text{ s or 47 billion years}
  8. - d N d t = λ N -\frac{dN}{dt}=\lambda N
  9. N N A [ mol ] × m [ g mol - 1 ] \frac{N}{N_{A}}[\,\text{mol}]\times{m}[\text{g }\,\text{mol}^{-1}]
  10. a [ Bq/g ] = λ N M N / N A = λ N A M a[\text{Bq/g}]=\frac{\lambda N}{MN/N_{A}}=\frac{\lambda N_{A}}{M}
  11. λ = l n 2 T 1 / 2 {\lambda}=\frac{ln2}{T_{1/2}}
  12. a = l n 2 × N A T 1 / 2 × M a=\frac{ln2\times{N_{A}}}{T_{1/2}\times{M}}
  13. a [ Bq/g ] 4.17 × 10 23 [ mol - 1 ] T 1 / 2 [ s ] × M [ g mol - 1 ] a[\text{Bq/g}]\simeq\frac{4.17\times 10^{23}[\,\text{mol}^{-1}]}{T_{1/2}[s]% \times M[\text{g }\,\text{mol}^{-1}]}
  14. a [ Bq/g ] = l n 2 × N A T 1 / 2 [ s ] × M [ g mol - 1 ] = l n 2 × N A T 1 / 2 [ y e a r ] × 365 × 24 × 60 × 60 × M [ g mol - 1 ] 1.32 × 10 16 [ mol - 1 ] T 1 / 2 [ y e a r ] × M [ g mol - 1 ] a[\text{Bq/g}]=\frac{ln2\times{N_{A}}}{T_{1/2}[s]\times{M[\text{g }\,\text{mol% }^{-1}]}}=\frac{ln2\times{N_{A}}}{T_{1/2}[year]\times 365\times 24\times 60% \times 60\times M[\text{g }\,\text{mol}^{-1}]}\simeq\frac{1.32\times 10^{16}[% \,\text{mol}^{-1}]}{T_{1/2}[year]\times M[\text{g }\,\text{mol}^{-1}]}
  15. a R a [ Bq/g ] = 1.32 × 10 16 [ mol - 1 ] 1600 [ y e a r ] × 226 [ g mol - 1 ] 3.7 × 10 10 [ Bq/g ] a_{Ra}[\text{Bq/g}]=\frac{1.32\times 10^{16}[\,\text{mol}^{-1}]}{1600[year]% \times 226[\text{g }\,\text{mol}^{-1}]}\simeq{3.7}\times 10^{10}[\text{Bq/g}]
  16. a T h [ Bq/g ] = 1.32 × 10 16 [ mol - 1 ] 1.405 × 10 10 [ y e a r ] × 232 [ g mol - 1 ] 4.059 × 10 3 [ Bq/g ] a_{Th}[\text{Bq/g}]=\frac{1.32\times 10^{16}[\,\text{mol}^{-1}]}{1.405\times 1% 0^{10}[year]\times 232[\text{g }\,\text{mol}^{-1}]}\simeq{4.059}\times 10^{3}[% \text{Bq/g}]
  17. a K [ Bq/g ] = 1.32 × 10 16 [ mol - 1 ] 1.251 × 10 9 [ y e a r ] × 40 [ g mol - 1 ] 2.63789 × 10 5 [ Bq/g ] a_{K}[\text{Bq/g}]=\frac{1.32\times 10^{16}[\,\text{mol}^{-1}]}{1.251\times 10% ^{9}[year]\times 40[\text{g }\,\text{mol}^{-1}]}\simeq{2.63789}\times 10^{5}[% \text{Bq/g}]

Specific_gravity.html

  1. S G true = ρ sample ρ H 2 O SG\text{true}=\frac{\rho\text{sample}}{\rho_{\rm H_{2}O}}
  2. ρ sample \rho\text{sample}\,
  3. ρ H 2 O \rho_{\rm H_{2}O}
  4. S G apparent = W A sample W A H 2 O SG\text{apparent}=\frac{W_{A\text{sample}}}{W_{A_{\rm H_{2}O}}}
  5. W A sample W_{A\text{sample}}
  6. W A H 2 O W_{A_{\rm H_{2}O}}
  7. S G true = ρ sample ρ H 2 O = ( m sample / V ) ( m H 2 O / V ) = m sample m H 2 O g g = W V sample W V H 2 O SG\text{true}=\frac{\rho\text{sample}}{\rho_{\rm H_{2}O}}=\frac{(m\text{sample% }/V)}{(m_{\rm H_{2}O}/V)}=\frac{m\text{sample}}{m_{\rm H_{2}O}}\frac{g}{g}=% \frac{W_{V\text{sample}}}{W_{V_{\rm H_{2}O}}}
  8. g g
  9. V V
  10. ρ sample {\rho\text{sample}}
  11. ρ H 2 O \rho_{\rm H_{2}O}
  12. W V W_{V}
  13. ( T s / T r ) (T\text{s}/T\text{r})
  14. T s T\text{s}
  15. T r T\text{r}
  16. S G H 2 O = 1.000000 SG_{\rm H_{2}O}=1.000000
  17. S G H 2 O = 0.998203 / 0.999840 = 0.998363 SG_{\rm H_{2}O}=0.998203/0.999840=0.998363
  18. ρ H 2 O \rho_{\rm H_{2}O}
  19. ρ substance = S G × ρ H 2 O . {\rho\text{substance}}=SG\times\rho_{\rm H_{2}O}.
  20. V V
  21. F b = g ( m b - ρ a m b ρ b ) F_{b}=g(m_{b}-\rho_{a}{m_{b}\over\rho_{b}})
  22. m b m_{b}
  23. g g
  24. ρ a \rho_{a}
  25. ρ b \rho_{b}
  26. F w = g ( m b - ρ a m b ρ b + V ρ w - V ρ a ) . F_{w}=g(m_{b}-\rho_{a}{m_{b}\over\rho_{b}}+V\rho_{w}-V\rho_{a}).
  27. F w , n = g V ( ρ w - ρ a ) F_{w,n}=gV(\rho_{w}-\rho_{a})
  28. F s , n = g V ( ρ s - ρ a ) F_{s,n}=gV(\rho_{s}-\rho_{a})
  29. ρ s \rho_{s}
  30. S G A = g V ( ρ s - ρ a ) g V ( ρ w - ρ a ) = ( ρ s - ρ a ) ( ρ w - ρ a ) . SG_{A}={gV(\rho_{s}-\rho_{a})\over gV(\rho_{w}-\rho_{a})}={(\rho_{s}-\rho_{a})% \over(\rho_{w}-\rho_{a})}.
  31. S G A SG_{A}
  32. S G V SG_{V}
  33. ρ s ρ w \rho_{s}\over\rho_{w}
  34. S G A = ρ s ρ w - ρ a ρ w 1 - ρ a ρ w = S G V - ρ a ρ w 1 - ρ a ρ w SG_{A}={{\rho_{s}\over\rho_{w}}-{\rho_{a}\over\rho_{w}}\over 1-{\rho_{a}\over% \rho_{w}}}={SG_{V}-{\rho_{a}\over\rho_{w}}\over 1-{\rho_{a}\over\rho_{w}}}
  35. S G V = S G A - ρ a ρ w ( S G A - 1 ) . SG_{V}=SG_{A}-{\rho_{a}\over\rho_{w}}(SG_{A}-1).

Spectral_graph_theory.html

  1. h ( G ) = min 0 < | S | n 2 | ( S ) | | S | , h(G)=\min_{0<|S|\leq\frac{n}{2}}\frac{|\partial(S)|}{|S|},
  2. 1 2 ( d - λ 2 ) h ( G ) 2 d ( d - λ 2 ) . \tfrac{1}{2}(d-\lambda_{2})\leq h(G)\leq\sqrt{2d(d-\lambda_{2})}.

Spectral_radius.html

  1. ρ ( A ) ρ(A)
  2. ρ ( A ) = max { | λ 1 | , , | λ n | } . \rho(A)=\max\left\{|\lambda_{1}|,\cdots,|\lambda_{n}|\right\}.
  3. ρ ( A ) ρ(A)
  4. [ u ! ! ] [ u ! ! ] [u^{\prime}!!^{\prime}]⋅[u^{\prime}!!^{\prime}]
  5. k 𝐍 k∈\mathbf{N}
  6. ρ ( A ) A k 1 k . \rho(A)\leq\|A^{k}\|^{\frac{1}{k}}.
  7. ( 𝐯 , λ ) (\mathbf{v},λ)
  8. | λ | k 𝐯 = λ k 𝐯 = A k 𝐯 A k 𝐯 |\lambda|^{k}\|\mathbf{v}\|=\|\lambda^{k}\mathbf{v}\|=\|A^{k}\mathbf{v}\|\leq% \|A^{k}\|\cdot\|\mathbf{v}\|
  9. 𝐯 0 \mathbf{v}≠0
  10. | λ | k A k |\lambda|^{k}\leq\|A^{k}\|
  11. ρ ( A ) A k 1 k . \rho(A)\leq\|A^{k}\|^{\frac{1}{k}}.
  12. ρ ( A ) ρ(A)
  13. lim k A k = 0. \lim_{k\to\infty}A^{k}=0.
  14. ρ ( A ) > 1 ρ(A)>1
  15. [ u ! ! ] A < s u p > k [ u ! ! ] [u^{\prime}!!^{\prime}]A<sup>k[u^{\prime}!!^{\prime}]
  16. ( 𝐯 , λ ) (\mathbf{v},λ)
  17. 0 \displaystyle 0
  18. 𝐯 0 \mathbf{v}≠0
  19. lim k λ k = 0 \lim_{k\to\infty}\lambda^{k}=0
  20. 1 1
  21. V V
  22. J J
  23. A = V J V - 1 A=VJV^{-1}
  24. J = [ J m 1 ( λ 1 ) 0 0 0 0 J m 2 ( λ 2 ) 0 0 0 0 J m s - 1 ( λ s - 1 ) 0 0 0 J m s ( λ s ) ] J=\begin{bmatrix}J_{m_{1}}(\lambda_{1})&0&0&\cdots&0\\ 0&J_{m_{2}}(\lambda_{2})&0&\cdots&0\\ \vdots&\cdots&\ddots&\cdots&\vdots\\ 0&\cdots&0&J_{m_{s-1}}(\lambda_{s-1})&0\\ 0&\cdots&\cdots&0&J_{m_{s}}(\lambda_{s})\end{bmatrix}
  25. J m i ( λ i ) = [ λ i 1 0 0 0 λ i 1 0 0 0 λ i 1 0 0 0 λ i ] 𝐂 m i × m i , 1 i s . J_{m_{i}}(\lambda_{i})=\begin{bmatrix}\lambda_{i}&1&0&\cdots&0\\ 0&\lambda_{i}&1&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&\cdots&\lambda_{i}&1\\ 0&0&\cdots&0&\lambda_{i}\end{bmatrix}\in\mathbf{C}^{m_{i}\times m_{i}},1\leq i% \leq s.
  26. A k = V J k V - 1 A^{k}=VJ^{k}V^{-1}
  27. J J
  28. J k = [ J m 1 k ( λ 1 ) 0 0 0 0 J m 2 k ( λ 2 ) 0 0 0 0 J m s - 1 k ( λ s - 1 ) 0 0 0 J m s k ( λ s ) ] J^{k}=\begin{bmatrix}J_{m_{1}}^{k}(\lambda_{1})&0&0&\cdots&0\\ 0&J_{m_{2}}^{k}(\lambda_{2})&0&\cdots&0\\ \vdots&\cdots&\ddots&\cdots&\vdots\\ 0&\cdots&0&J_{m_{s-1}}^{k}(\lambda_{s-1})&0\\ 0&\cdots&\cdots&0&J_{m_{s}}^{k}(\lambda_{s})\end{bmatrix}
  29. k k
  30. m i × m i m_{i}\times m_{i}
  31. k m i - 1 k\geq m_{i}-1
  32. J m i k ( λ i ) = [ λ i k ( k 1 ) λ i k - 1 ( k 2 ) λ i k - 2 ( k m i - 1 ) λ i k - m i + 1 0 λ i k ( k 1 ) λ i k - 1 ( k m i - 2 ) λ i k - m i + 2 0 0 λ i k ( k 1 ) λ i k - 1 0 0 0 λ i k ] J_{m_{i}}^{k}(\lambda_{i})=\begin{bmatrix}\lambda_{i}^{k}&{k\choose 1}\lambda_% {i}^{k-1}&{k\choose 2}\lambda_{i}^{k-2}&\cdots&{k\choose m_{i}-1}\lambda_{i}^{% k-m_{i}+1}\\ 0&\lambda_{i}^{k}&{k\choose 1}\lambda_{i}^{k-1}&\cdots&{k\choose m_{i}-2}% \lambda_{i}^{k-m_{i}+2}\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&\cdots&\lambda_{i}^{k}&{k\choose 1}\lambda_{i}^{k-1}\\ 0&0&\cdots&0&\lambda_{i}^{k}\end{bmatrix}
  33. ρ ( A ) < 1 \rho(A)<1
  34. i i
  35. | λ i | < 1 |\lambda_{i}|<1
  36. i i
  37. lim k J m i k = 0 \lim_{k\to\infty}J_{m_{i}}^{k}=0
  38. lim k J k = 0. \lim_{k\to\infty}J^{k}=0.
  39. lim k A k = lim k V J k V - 1 = V ( lim k J k ) V - 1 = 0 \lim_{k\to\infty}A^{k}=\lim_{k\to\infty}VJ^{k}V^{-1}=V\left(\lim_{k\to\infty}J% ^{k}\right)V^{-1}=0
  40. ρ ( A ) > 1 \rho(A)>1
  41. J J
  42. [ u ! ! ] [ u ! ! ] , [u^{\prime}!!^{\prime}]⋅[u^{\prime}!!^{\prime}],
  43. ρ ( A ) = lim k A k 1 k . \rho(A)=\lim_{k\to\infty}\left\|A^{k}\right\|^{\frac{1}{k}}.
  44. ε > 0 ε>0
  45. A ± = 1 ρ ( A ) ± ε A . A_{\pm}=\frac{1}{\rho(A)\pm\varepsilon}A.
  46. ρ ( A ± ) = ρ ( A ) ρ ( A ) ± ε , ρ ( A + ) < 1 < ρ ( A - ) . \rho\left(A_{\pm}\right)=\frac{\rho(A)}{\rho(A)\pm\varepsilon},\qquad\rho(A_{+% })<1<\rho(A_{-}).
  47. lim k A + k = 0. \lim_{k\to\infty}A_{+}^{k}=0.
  48. k N + A + k < 1 k N + A k < ( ρ ( A ) + ε ) k k N + A k 1 k < ρ ( A ) + ε . \begin{aligned}\displaystyle\forall k\geq N_{+}\quad\left\|A_{+}^{k}\right\|<1% &\displaystyle\Rightarrow\qquad\forall k\geq N_{+}\quad\left\|A^{k}\right\|<(% \rho(A)+\varepsilon)^{k}\\ &\displaystyle\Rightarrow\qquad\forall k\geq N_{+}\quad\left\|A^{k}\right\|^{% \frac{1}{k}}<\rho(A)+\varepsilon.\end{aligned}
  49. A - k \|A_{-}^{k}\|
  50. k N - A - k > 1 k N - A k > ( ρ ( A ) - ε ) k k N - A k 1 k > ρ ( A ) - ε . \begin{aligned}\displaystyle\forall k\geq N_{-}\quad\left\|A_{-}^{k}\right\|>1% &\displaystyle\Rightarrow\qquad\forall k\geq N_{-}\quad\left\|A^{k}\right\|>(% \rho(A)-\varepsilon)^{k}\\ &\displaystyle\Rightarrow\qquad\forall k\geq N_{-}\quad\left\|A^{k}\right\|^{% \frac{1}{k}}>\rho(A)-\varepsilon.\end{aligned}
  51. ε > 0 , N 𝐍 , k N ρ ( A ) - ε < A k 1 k < ρ ( A ) + ε \forall\varepsilon>0,\exists N\in\mathbf{N},\forall k\geq N\quad\rho(A)-% \varepsilon<\left\|A^{k}\right\|^{\frac{1}{k}}<\rho(A)+\varepsilon
  52. lim k A k 1 k = ρ ( A ) . \lim_{k\to\infty}\left\|A^{k}\right\|^{\frac{1}{k}}=\rho(A).
  53. ρ ( A 1 A n ) ρ ( A 1 ) ρ ( A n ) . \rho(A_{1}\cdots A_{n})\leq\rho(A_{1})\cdots\rho(A_{n}).
  54. ε > 0 , N 𝐍 , k N ρ ( A ) A k 1 k < ρ ( A ) + ε \forall\varepsilon>0,\exists N\in\mathbf{N},\forall k\geq N\quad\rho(A)\leq\|A% ^{k}\|^{\frac{1}{k}}<\rho(A)+\varepsilon
  55. lim k A k 1 k = ρ ( A ) + . \lim_{k\to\infty}\left\|A^{k}\right\|^{\frac{1}{k}}=\rho(A)^{+}.
  56. A = [ 9 - 1 2 - 2 8 4 1 1 8 ] A=\begin{bmatrix}9&-1&2\\ -2&8&4\\ 1&1&8\end{bmatrix}
  57. 5 , 10 , 10 5,10,10
  58. ρ ( A ) = 10 ρ(A)=10
  59. A k 1 k \|A^{k}\|^{\frac{1}{k}}
  60. . 1 = . \|.\|_{1}=\|.\|_{\infty}
  61. . 1 = . \|.\|_{1}=\|.\|_{\infty}
  62. . F \|.\|_{F}
  63. . 2 \|.\|_{2}
  64. \vdots
  65. \vdots
  66. \vdots
  67. \vdots
  68. \vdots
  69. \vdots
  70. \vdots
  71. \vdots
  72. \vdots
  73. \vdots
  74. \vdots
  75. \vdots
  76. \vdots
  77. \vdots
  78. \vdots
  79. \vdots
  80. \vdots
  81. \vdots
  82. \vdots
  83. \vdots
  84. \vdots
  85. \vdots
  86. \vdots
  87. \vdots
  88. A A
  89. ρ ( A ) = lim k A k 1 k . \rho(A)=\lim_{k\to\infty}\|A^{k}\|^{\frac{1}{k}}.
  90. C C
  91. C C
  92. G G
  93. 2 ( G ) = { f : V ( G ) 𝐑 : v V ( G ) f ( v ) 2 < } . \ell^{2}(G)=\left\{f:V(G)\to\mathbf{R}\ :\ \sum\nolimits_{v\in V(G)}\left\|f(v% )^{2}\right\|<\infty\right\}.
  94. γ γ
  95. G G
  96. { γ : 2 ( G ) 2 ( G ) ( γ f ) ( v ) = ( u , v ) E ( G ) f ( u ) \begin{cases}\gamma:\ell^{2}(G)\to\ell^{2}(G)\\ (\gamma f)(v)=\sum_{(u,v)\in E(G)}f(u)\end{cases}
  97. G G
  98. γ γ

Spectral_space.html

  1. \circ
  2. \circ
  3. \circ
  4. \circ
  5. \circ
  6. \circ
  7. \circ
  8. \circ

Spectral_theory.html

  1. R ζ = ( ζ I - T ) - 1 . R_{\zeta}=\left(\zeta I-T\right)^{-1}.
  2. T T - 1 = T - 1 T = I . TT^{-1}=T^{-1}T=I.
  3. L = | k 1 b 1 | , L=|k_{1}\rangle\langle b_{1}|,
  4. b 1 | \langle b_{1}|
  5. | k 1 |k_{1}\rangle
  6. | f |f\rangle
  7. ( x 1 , x 2 , x 3 , ) (x_{1},x_{2},x_{3},\dots)
  8. f ( x ) = x , f f(x)=\langle x,f\rangle
  9. f 2 = f , f = f , x x , f d x = f * ( x ) f ( x ) d x \|f\|^{2}=\langle f,f\rangle=\int\langle f,x\rangle\langle x,f\rangle\,dx=\int f% ^{*}(x)f(x)\,dx
  10. L | f = | k 1 b 1 | f L|f\rangle=|k_{1}\rangle\langle b_{1}|f\rangle
  11. | k 1 |k_{1}\rangle
  12. b 1 | f \langle b_{1}|f\rangle
  13. L = λ 1 | e 1 f 1 | + λ 2 | e 2 f 2 | + λ 3 | e 3 f 3 | + , L=\lambda_{1}|e_{1}\rangle\langle f_{1}|+\lambda_{2}|e_{2}\rangle\langle f_{2}% |+\lambda_{3}|e_{3}\rangle\langle f_{3}|+\dots,
  14. { λ i } \{\,\lambda_{i}\,\}
  15. { | e i } \{\,|e_{i}\rangle\,\}
  16. { f i | } \{\,\langle f_{i}|\,\}
  17. f i | e j = δ i j \langle f_{i}|e_{j}\rangle=\delta_{ij}
  18. { λ i } \{\,\lambda_{i}\,\}
  19. { | e i } \{\,|e_{i}\rangle\,\}
  20. I = i = 1 n | e i f i | I=\sum_{i=1}^{n}|e_{i}\rangle\langle f_{i}|
  21. | e i |e_{i}\rangle
  22. f i | \langle f_{i}|
  23. f i | e j = δ i j . \langle f_{i}|e_{j}\rangle=\delta_{ij}.
  24. I k = I I^{k}=I\,
  25. | ψ |\psi\rangle
  26. I | ψ = | ψ = i = 1 n | e i f i | ψ = i = 1 n c i | e i I|\psi\rangle=|\psi\rangle=\sum_{i=1}^{n}|e_{i}\rangle\langle f_{i}|\psi% \rangle=\sum_{i=1}^{n}\ c_{i}|e_{i}\rangle
  27. c i = f i | ψ c_{i}=\langle f_{i}|\psi\rangle
  28. O | ψ = | h O|\psi\rangle=|h\rangle
  29. O | ψ = i = 1 n c i ( O | e i ) = i = 1 n | e i f i | h , O|\psi\rangle=\sum_{i=1}^{n}c_{i}\left(O|e_{i}\rangle\right)=\sum_{i=1}^{n}|e_% {i}\rangle\langle f_{i}|h\rangle,
  30. f j | O | ψ = i = 1 n c i f j | O | e i = i = 1 n f j | e i f i | h = f j | h , j \langle f_{j}|O|\psi\rangle=\sum_{i=1}^{n}c_{i}\langle f_{j}|O|e_{i}\rangle=% \sum_{i=1}^{n}\langle f_{j}|e_{i}\rangle\langle f_{i}|h\rangle=\langle f_{j}|h% \rangle,\quad\forall j
  31. f j | h \langle f_{j}|h\rangle
  32. O j i = f j | O | e i O_{ji}=\langle f_{j}|O|e_{i}\rangle
  33. L | e i = λ i | e i ; L|e_{i}\rangle=\lambda_{i}|e_{i}\rangle\,;
  34. L I = L = i = 1 n L | e i f i | = i = 1 n λ i | e i f i | . LI=L=\sum_{i=1}^{n}L|e_{i}\rangle\langle f_{i}|=\sum_{i=1}^{n}\lambda_{i}|e_{i% }\rangle\langle f_{i}|.
  35. R = ( λ I - L ) - 1 , R=(\lambda I-L)^{-1},\,
  36. φ \varphi
  37. R | φ = ( λ I - L ) - 1 | φ = i = 1 n 1 λ - λ i | e i f i | φ . R|\varphi\rangle=(\lambda I-L)^{-1}|\varphi\rangle=\sum_{i=1}^{n}\frac{1}{% \lambda-\lambda_{i}}|e_{i}\rangle\langle f_{i}|\varphi\rangle.
  38. 1 2 π i C R | φ d λ = - i = 1 n | e i f i | φ = - | φ , \frac{1}{2\pi i}\oint_{C}R|\varphi\rangle d\lambda=-\sum_{i=1}^{n}|e_{i}% \rangle\langle f_{i}|\varphi\rangle=-|\varphi\rangle,
  39. x , φ = φ ( x 1 , x 2 , ) . \langle x,\varphi\rangle=\varphi(x_{1},x_{2},...).
  40. x , y = δ ( x - y ) , \langle x,y\rangle=\delta(x-y),
  41. x , φ = x , y y , φ d y . \langle x,\varphi\rangle=\int\langle x,y\rangle\langle y,\varphi\rangle dy.
  42. x , 1 2 π i C φ λ I - L d λ \displaystyle\left\langle x,\frac{1}{2\pi i}\oint_{C}\frac{\varphi}{\lambda I-% L}d\lambda\right\rangle
  43. G ( x , y ; λ ) \displaystyle G(x,y;\lambda)
  44. 1 2 π i C G ( x , y ; λ ) d λ = - i = 1 n x , e i f i , y = - x , y = - δ ( x - y ) . \frac{1}{2\pi i}\oint_{C}G(x,y;\lambda)d\lambda=-\sum_{i=1}^{n}\langle x,e_{i}% \rangle\langle f_{i},y\rangle=-\langle x,y\rangle=-\delta(x-y).
  45. ( O - λ I ) | ψ = | h ; (O-\lambda I)|\psi\rangle=|h\rangle;
  46. x , ( O - λ I ) y y , ψ d y = h ( x ) . \int\langle x,(O-\lambda I)y\rangle\langle y,\psi\rangle dy=h(x).
  47. y , G ( λ ) z = y , ( O - λ I ) - 1 z = G ( y , z ; λ ) , \langle y,G(\lambda)z\rangle=\left\langle y,(O-\lambda I)^{-1}z\right\rangle=G% (y,z;\lambda),
  48. x , ( O - λ I ) y y , G ( λ ) z d y = x , ( O - λ I ) y y , ( O - λ I ) - 1 z d y = x , z = δ ( x - z ) . \int\langle x,(O-\lambda I)y\rangle\langle y,G(\lambda)z\rangle dy=\int\langle x% ,(O-\lambda I)y\rangle\left\langle y,(O-\lambda I)^{-1}z\right\rangle dy=% \langle x,z\rangle=\delta(x-z).
  49. x , ( O - λ I ) y G ( y , z ; λ ) d y = δ ( x - z ) . \int\langle x,(O-\lambda I)y\rangle G(y,z;\lambda)dy=\delta(x-z).
  50. d z h ( z ) d y x , ( O - λ I ) y G ( y , z ; λ ) = d y x , ( O - λ I ) y d z h ( z ) G ( y , z ; λ ) = h ( x ) , \int dzh(z)\int dy\langle x,(O-\lambda I)y\rangle G(y,z;\lambda)=\int dy% \langle x,(O-\lambda I)y\rangle\int dzh(z)G(y,z;\lambda)=h(x),
  51. ψ ( x ) = h ( z ) G ( x , z ; λ ) d z . \psi(x)=\int h(z)G(x,z;\lambda)dz.
  52. G ( x , z ; λ ) = i = 1 n e i ( x ) f i * ( z ) λ - λ i . G(x,z;\lambda)=\sum_{i=1}^{n}\frac{e_{i}(x)f_{i}^{*}(z)}{\lambda-\lambda_{i}}.
  53. λ 1 λ 2 λ n \lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}
  54. v i v_{i}
  55. x = i v i T x v i x=\sum_{i}\ v_{i}^{T}xv_{i}
  56. v j T i v i T x v i v_{j}^{T}\sum_{i}v_{i}^{T}xv_{i}
  57. = i v i T x v j T v i =\sum_{i}v_{i}^{T}xv_{j}^{T}v_{i}
  58. = ( v j T x ) v j T v j =(v_{j}^{T}x)v_{j}^{T}v_{j}
  59. = v j T x =v_{j}^{T}x
  60. x T M x x^{T}Mx
  61. = ( i ( v i T x ) v i ) T M ( j ( v j T x ) v j ) =(\sum_{i}(v_{i}^{T}x)v_{i})^{T}M(\sum_{j}(v_{j}^{T}x)v_{j})
  62. = ( i ( v i T x ) v i T ) ( j ( v j T x ) v j λ j ) =(\sum_{i}(v_{i}^{T}x)v_{i}^{T})(\sum_{j}(v_{j}^{T}x)v_{j}\lambda_{j})
  63. = i , j ( v i T x ) v i T ( v j T x ) v j λ j =\sum_{i,j}(v_{i}^{T}x)v_{i}^{T}(v_{j}^{T}x)v_{j}\lambda_{j}
  64. = j ( v j T x ) ( v j T x ) λ j =\sum_{j}(v_{j}^{T}x)(v_{j}^{T}x)\lambda_{j}
  65. = j ( v j T x ) 2 λ j λ n j ( v j T x ) 2 =\sum_{j}(v_{j}^{T}x)^{2}\lambda_{j}\leq\lambda_{n}\sum_{j}(v_{j}^{T}x)^{2}
  66. = λ n x T x =\lambda_{n}x^{T}x
  67. x T M x x T x λ n \frac{x^{T}Mx}{x^{T}x}\leq\lambda_{n}
  68. λ n \lambda_{n}

Speedometer.html

  1. Percentage error = 100 × ( 1 - new diameter / standard diameter ) \mbox{Percentage error}=100\times(1-\mbox{new diameter}~{}/\mbox{standard % diameter}~{})
  2. Diameter in millimetres = 2 × T × A / 100 + W × 25.4 \mbox{Diameter in millimetres}=2\times T\times A/100+W\times 25.4
  3. Diameter in inches = T × A / 1270 + W \mbox{Diameter in inches}=T\times A/1270+W

Sperner's_lemma.html

  1. 𝒜 = A 1 A 2 A n + 1 . \mathcal{A}=A_{1}A_{2}\ldots A_{n+1}.
  2. 𝒜 \mathcal{A}
  3. A i 1 A i 2 A i k + 1 A_{i_{1}}A_{i_{2}}\ldots A_{i_{k+1}}
  4. i 1 , i 2 , , i k + 1 . i_{1},i_{2},\ldots,i_{k+1}.

Sphere_eversion.html

  1. f : S 2 \R 3 f\colon S^{2}\to\R^{3}
  2. f t : S 2 \R 3 f_{t}\colon S^{2}\to\R^{3}
  3. S 2 S^{2}\,
  4. \R 3 \R^{3}

Sphere_packing.html

  1. π 3 2 0.74048. \frac{\pi}{3\sqrt{2}}\simeq 0.74048.
  2. π 6 0.5236. \frac{\pi}{6}\simeq 0.5236.
  3. π 3 3 0.6046 \frac{\pi}{3\sqrt{3}}\simeq 0.6046
  4. π 3 16 0.3401 \frac{\pi\sqrt{3}}{16}\simeq 0.3401
  5. × 10 30 \times 10^{−}30
  6. c n 2 - n cn2^{-n}
  7. n 2 n\geq 2

Spin_group.html

  1. 1 Z 2 Spin ( n ) SO ( n ) 1. 1\to\mathrm{Z}_{2}\to\operatorname{Spin}(n)\to\operatorname{SO}(n)\to 1.
  2. n ( n 1 ) / 2 n(n− 1)/2
  3. z z
  4. B 1 A 1 . B_{1}\cong A_{1}.
  5. D 2 A 1 × A 1 . D_{2}\cong A_{1}\times A_{1}.
  6. B 2 C 2 . B_{2}\cong C_{2}.
  7. D 3 A 3 . D_{3}\cong A_{3}.
  8. n n
  9. n n
  10. p + q > 2 p+q>2
  11. π 1 ( G ) Z ( G ) , \pi_{1}(G)\subset\operatorname{Z}(G^{\prime}),
  12. 𝔤 \mathfrak{g}
  13. 𝔤 \mathfrak{g}
  14. π 1 ( Spin ( p , q ) ) = { { 0 } ( p , q ) = ( 1 , 1 ) or ( 1 , 0 ) { 0 } p > 2 , q = 0 , 1 𝐙 ( p , q ) = ( 2 , 0 ) or ( 2 , 1 ) 𝐙 × 𝐙 ( p , q ) = ( 2 , 2 ) 𝐙 p > 2 , q = 2 Z 2 p , q > 2 \pi_{1}(\mbox{Spin}~{}(p,q))=\begin{cases}\{0\}&(p,q)=(1,1)\mbox{ or }~{}(1,0)% \\ \{0\}&p>2,q=0,1\\ \mathbf{Z}&(p,q)=(2,0)\mbox{ or }~{}(2,1)\\ \mathbf{Z}\times\mathbf{Z}&(p,q)=(2,2)\\ \mathbf{Z}&p>2,q=2\\ \mathrm{Z}_{2}&p,q>2\\ \end{cases}
  15. Z ( Spin ( n , 𝐂 ) ) = { Z 2 n = 2 k + 1 Z 4 n = 4 k + 2 Z 2 Z 2 n = 4 k Z ( Spin ( p , q ) ) = { Z 2 n = 2 k + 1 , Z 2 n = 2 k , and p , q odd Z 4 n = 2 k , and p , q even \begin{aligned}\displaystyle\operatorname{Z}(\operatorname{Spin}(n,\mathbf{C})% )&\displaystyle=\begin{cases}\mathrm{Z}_{2}&n=2k+1\\ \mathrm{Z}_{4}&n=4k+2\\ \mathrm{Z}_{2}\oplus\mathrm{Z}_{2}&n=4k\\ \end{cases}\\ \displaystyle\operatorname{Z}(\operatorname{Spin}(p,q))&\displaystyle=\begin{% cases}\mathrm{Z}_{2}&n=2k+1,\\ \mathrm{Z}_{2}&n=2k,\,\text{ and }p,q\,\text{ odd}\\ \mathrm{Z}_{4}&n=2k,\,\text{ and }p,q\,\text{ even}\\ \end{cases}\end{aligned}
  16. 𝔰 𝔬 ( n , 𝐑 ) . \mathfrak{so}(n,\mathbf{R}).
  17. C 2 × G \mathrm{C}_{2}\times G
  18. C 2 k + 1 \mathrm{C}_{2k+1}
  19. C 4 k + 2 C 2 k + 1 × C 2 , \mathrm{C}_{4k+2}\cong\mathrm{C}_{2k+1}\times\mathrm{C}_{2},
  20. C 2 k + 1 < Spin ( n ) \mathrm{C}_{2k+1}<\operatorname{Spin}(n)
  21. C 2 k + 1 < SO ( n ) . \mathrm{C}_{2k+1}<\operatorname{SO}(n).
  22. 2 A n A n , 2\cdot A_{n}\to A_{n},
  23. 1 Z 2 Spin ( n ) SO ( n ) × U ( 1 ) 1. 1\to\mathrm{Z}_{2}\to{\mathrm{Spin}}^{\mathbb{C}}(n)\to{\mathrm{SO}}(n)\times{% \mathrm{U}}(1)\to 1.

Spinor_bundle.html

  1. n n
  2. ( M , g ) , (M,g),\,
  3. π 𝐒 : 𝐒 M \pi_{\mathbf{S}}\colon{\mathbf{S}}\to M\,
  4. π 𝐏 : 𝐏 M \pi_{\mathbf{P}}\colon{\mathbf{P}}\to M\,
  5. M M
  6. Spin ( n ) {\mathrm{Spin}}(n)\,
  7. Δ n . \Delta_{n}.\,
  8. 𝐒 {\mathbf{S}}\,
  9. ( 𝐏 , F 𝐏 ) ({\mathbf{P}},F_{\mathbf{P}})
  10. ( M , g ) , (M,g),\,
  11. F S O ( M ) M \mathrm{F}_{SO}(M)\to M
  12. ρ : Spin ( n ) SO ( n ) . \rho\colon{\mathrm{Spin}}(n)\to{\mathrm{SO}}(n).\,
  13. 𝐒 {\mathbf{S}}\,
  14. 𝐒 = 𝐏 × κ Δ n {\mathbf{S}}={\mathbf{P}}\times_{\kappa}\Delta_{n}\,
  15. 𝐏 {\mathbf{P}}
  16. κ : Spin ( n ) U ( Δ n ) , \kappa\colon{\mathrm{Spin}}(n)\to{\mathrm{U}}(\Delta_{n}),\,
  17. U ( 𝐖 ) {\mathrm{U}}({\mathbf{W}})\,
  18. 𝐖 . {\mathbf{W}}.\,
  19. κ \kappa
  20. Spin ( n ) {\mathrm{Spin}}(n)

Spin–statistics_theorem.html

  1. ψ ( x , y ) ϕ ( x ) ϕ ( y ) d x d y \iint\psi(x,y)\phi(x)\phi(y)\,dx\,dy
  2. ϕ \phi
  3. ψ ( x , y ) \psi(x,y)
  4. ψ ( x , y ) \psi(x,y)
  5. x y x\neq y
  6. ϕ ( x ) ϕ ( y ) = ϕ ( y ) ϕ ( x ) \phi(x)\phi(y)=\phi(y)\phi(x)
  7. ψ \psi
  8. ψ ( x , y ) = ψ ( y , x ) \psi(x,y)=\psi(y,x)
  9. ϕ \phi
  10. ϕ ( x ) ϕ ( y ) = - ϕ ( y ) ϕ ( x ) , \phi(x)\phi(y)=-\phi(y)\phi(x),
  11. ψ \psi
  12. ψ ( x , y ) = - ψ ( y , x ) \psi(x,y)=-\psi(y,x)
  13. R ( π ) ϕ ( x ) ϕ ( - x ) , R(\pi)\phi(x)\phi(-x),
  14. ϕ \phi
  15. ϕ \phi
  16. x x
  17. - x -x
  18. π \pi
  19. π \pi
  20. x x
  21. - x -x
  22. π \pi
  23. R ( 2 π ) ϕ ( - x ) R ( π ) ϕ ( x ) , R(2\pi)\phi(-x)R(\pi)\phi(x),
  24. ϕ ( - x ) R ( π ) ϕ ( x ) \phi(-x)R(\pi)\phi(x)
  25. - ϕ ( - x ) R ( π ) ϕ ( x ) -\phi(-x)R(\pi)\phi(x)
  26. ± ϕ ( - x ) R ( π ) ϕ ( x ) \pm\phi(-x)R(\pi)\phi(x)
  27. x x
  28. - x -x
  29. R ( π ) ϕ ( x ) ϕ ( - x ) = { ϕ ( - x ) R ( π ) ϕ ( x ) for integral spins , - ϕ ( - x ) R ( π ) ϕ ( x ) for half-integral spins . R(\pi)\phi(x)\phi(-x)=\begin{cases}\phi(-x)R(\pi)\phi(x)&\,\text{ for integral% spins},\\ -\phi(-x)R(\pi)\phi(x)&\,\text{ for half-integral spins}.\end{cases}
  30. ϕ ( - x ) ϕ ( x ) . \phi(-x)\phi(x).
  31. x x
  32. - x -x
  33. 0 | ϕ ( - x ) ϕ ( x ) | ψ . \langle 0|\phi(-x)\phi(x)|\psi\rangle.
  34. | ψ |\psi\rangle
  35. G ( x ) = 0 | ϕ ( - x ) ϕ ( x ) | 0 . G(x)=\langle 0|\phi(-x)\phi(x)|0\rangle.
  36. G ( x ) G(x)
  37. G ( - x ) G(-x)
  38. 0 | R ϕ ( x ) ϕ ( - x ) | 0 \langle 0|R\phi(x)\phi(-x)|0\rangle
  39. 0 | R R ϕ ( x ) R ϕ ( - x ) | 0 = ± 0 | ϕ ( - x ) R ϕ ( x ) | 0 \langle 0|RR\phi(x)R\phi(-x)|0\rangle=\pm\langle 0|\phi(-x)R\phi(x)|0\rangle
  40. 0 | ( R ϕ ( x ) ϕ ( y ) - ϕ ( y ) R ϕ ( x ) ) | 0 = 0 \langle 0|(R\phi(x)\phi(y)-\phi(y)R\phi(x))|0\rangle=0\,
  41. 0 | R ϕ ( x ) ϕ ( y ) + ϕ ( y ) R ϕ ( x ) | 0 = 0 \langle 0|R\phi(x)\phi(y)+\phi(y)R\phi(x)|0\rangle=0\,

Spline_(mathematics).html

  1. S ( t ) = { ( t + 1 ) 2 - 1 - 2 t < 0 1 - ( t - 1 ) 2 0 t 2 S(t)=\begin{cases}(t+1)^{2}-1&-2\leq t<0\\ 1-(t-1)^{2}&0\leq t\leq 2\end{cases}
  2. S ( 0 ) = 2 S^{\prime}(0)=2
  3. S ( t ) = | t | 3 S(t)=\left|t\right|^{3}
  4. S ( t ) = { t 3 t 0 - t 3 t < 0 S(t)=\begin{cases}t^{3}&t\geq 0\\ -t^{3}&t<0\end{cases}
  5. S ( 0 ) = 0 S^{\prime}(0)=\ 0
  6. S ′′ ( 0 ) = 0 S^{\prime\prime}(0)=\ 0
  7. f X ( x ) = { 1 4 ( x + 2 ) 3 - 2 x - 1 1 4 ( 3 | x | 3 - 6 x 2 + 4 ) - 1 x 1 1 4 ( 2 - x ) 3 1 x 2 f_{X}(x)=\begin{cases}\frac{1}{4}(x+2)^{3}&-2\leq x\leq-1\\ \frac{1}{4}\left(3|x|^{3}-6x^{2}+4\right)&-1\leq x\leq 1\\ \frac{1}{4}(2-x)^{3}&1\leq x\leq 2\end{cases}
  8. S : [ a , b ] S:[a,b]\to\mathbb{R}
  9. [ t i - 1 , t i ] [t_{i-1},t_{i}]
  10. a = t 0 < t 1 < < t k - 1 < t k = b a=t_{0}<t_{1}<\cdots<t_{k-1}<t_{k}=b
  11. P i : [ t i - 1 , t i ] P_{i}:[t_{i-1},t_{i}]\to\mathbb{R}
  12. S ( t ) = P 1 ( t ) , t 0 t < t 1 , S(t)=P_{1}(t)\mbox{ , }~{}t_{0}\leq t<t_{1},
  13. S ( t ) = P 2 ( t ) , t 1 t < t 2 , S(t)=P_{2}(t)\mbox{ , }~{}t_{1}\leq t<t_{2},
  14. \vdots
  15. S ( t ) = P k ( t ) , t k - 1 t t k . S(t)=P_{k}(t)\mbox{ , }~{}t_{k-1}\leq t\leq t_{k}.
  16. P i ( t ) P_{i}(t)
  17. t i t_{i}
  18. i = 1 , , k - 1 i=1,\dots,k-1
  19. j = 0 , , n - 1 j=0,\dots,n-1
  20. P i ( j ) ( t i ) = P i + 1 ( j ) ( t i ) P_{i}^{(j)}(t_{i})=P_{i+1}^{(j)}(t_{i})

Spontaneous_fission.html

  1. Z 2 / A 47. \hbox{Z}^{2}/\hbox{A}\geq 47.
  2. Z 2 / A \hbox{Z}^{2}/\hbox{A}

Spurious_relationship.html

  1. y = a 0 + a 1 x 1 + a 2 x 2 + + a k x k + e y=a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{k}x_{k}+e
  2. y y
  3. x j x_{j}
  4. e e
  5. x j x_{j}
  6. a j a_{j}
  7. a j = 0 a_{j}=0
  8. a j 0 a_{j}\neq 0
  9. x j x_{j}
  10. a j = 0 a_{j}=0
  11. x j x_{j}
  12. a j 0 a_{j}\neq 0
  13. x j x_{j}
  14. x j x_{j}
  15. x j x_{j}

Square_pyramidal_number.html

  1. n × n n×n
  2. 1 , 5 , 14 , 30 , 55 , 91 , 140 , 1,5,14,30,55,91,140,
  3. 204 , 285 , 385 , 506 , 650 , 819 204,285,385,506,650,819
  4. P n = k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6 = 2 n 3 + 3 n 2 + n 6 . P_{n}=\sum_{k=1}^{n}k^{2}=\frac{n(n+1)(2n+1)}{6}=\frac{2n^{3}+3n^{2}+n}{6}.
  5. L ( P , t ) L(P,t)
  6. P P
  7. P P
  8. t t
  9. P n = ( n + 2 3 ) + ( n + 1 3 ) . P_{n}={{n+2}\choose 3}+{{n+1}\choose 3}.
  10. P n = 1 4 ( 2 n + 2 3 ) . P_{n}=\frac{1}{4}{\left({{2n+2}\atop{3}}\right)}.
  11. n n
  12. n 1 n−1
  13. P n = n ( n + 1 2 ) - ( n + 1 3 ) . P_{n}=n{\left({{n+1}\atop{2}}\right)}-{\left({{n+1}\atop{3}}\right)}.
  14. 1 1
  15. 4900 4900
  16. 70 70
  17. 24 24
  18. ( 1 , 1 ) (1,1)
  19. ( 2 , 1 ) (2,1)
  20. n n
  21. n n
  22. 1 × 1 1×1
  23. 2 × 2 2×2
  24. 2 × 2 2×2
  25. k × k k×k
  26. ( 1 k n ) (1≤k≤n)
  27. k × k k×k
  28. n × n n×n
  29. n 2 + ( n - 1 ) 2 + ( n - 2 ) 2 + ( n - 3 ) 2 + + 1 2 = n ( n + 1 ) ( 2 n + 1 ) 6 . n^{2}+(n-1)^{2}+(n-2)^{2}+(n-3)^{2}+\ldots+1^{2}=\frac{n(n+1)(2n+1)}{6}.
  30. k k
  31. ( k 1 ) (k−1)
  32. 2 k 1 2k−1
  33. 0 1 4 9 16 25 ( n - 1 ) 2 n 2 1 3 5 7 9 2 n - 1 \begin{array}[]{ccccccccccccccc}0&&1&&4&&9&&16&&25&\ldots&(n-1)^{2}&&n^{2}\\ &1&&3&&5&&7&&9&&\ldots&&2n-1&\end{array}
  34. n 2 = i = 1 n 2 i - 1 n^{2}=\sum_{i=1}^{n}2i-1
  35. n n
  36. n n
  37. 1 2 = 1 2 2 = 1 3 3 2 = 1 3 5 4 2 = 1 3 5 7 5 2 = 1 3 5 7 9 ( n - 1 ) 2 = 1 2 n - 3 n 2 = 1 2 n - 3 2 n - 1 \begin{array}[]{rcccccccc}\scriptstyle 1^{2}\scriptstyle=&1&&&&&&&\\ \scriptstyle 2^{2}\scriptstyle=&1&3&&&&&&\\ \scriptstyle 3^{2}\scriptstyle=&1&3&5&&&&&\\ \scriptstyle 4^{2}\scriptstyle=&1&3&5&7&&&&\\ \scriptstyle 5^{2}\scriptstyle=&1&3&5&7&9&&&\\ \vdots&\vdots&&&&&\ddots&&\\ \scriptstyle(n-1)^{2}\scriptstyle=&1&\cdots&&&&\cdots&\scriptstyle 2n-3&\\ \scriptstyle n^{2}\scriptstyle=&1&\cdots&&&&\cdots&\scriptstyle 2n-3&% \scriptstyle 2n-1\end{array}
  38. 2 n - 1 2 n - 3 2 n - 3 9 9 7 7 7 5 5 5 5 3 3 3 3 3 1 1 1 1 1 1 = n 2 = ( n - 1 ) 2 = 5 2 = 4 2 = 3 2 = 2 2 = 1 2 \begin{array}[]{cccccccc}\scriptstyle 2n-1&&&&&&\\ \scriptstyle 2n-3&\scriptstyle 2n-3&&&&&\\ \vdots&&\ddots&&&&\\ 9&\cdots&\cdots&9&&&&\\ 7&\cdots&\cdots&7&7&&&\\ 5&\cdots&\cdots&5&5&5\\ 3&\cdots&\cdots&3&3&3&3\\ 1&\cdots&\cdots&1&1&1&1&1\\ \hline\scriptstyle=n^{2}&\scriptstyle=(n-1)^{2}&\cdots&\scriptstyle=5^{2}&% \scriptstyle=4^{2}&\scriptstyle=3^{2}&\scriptstyle=2^{2}&\scriptstyle=1^{2}% \end{array}
  39. 1 3 1 5 3 1 7 5 3 1 9 7 5 3 1 2 n - 3 1 2 n - 1 2 n - 3 1 = n 2 = ( n - 1 ) 2 = 5 2 = 4 2 = 3 2 = 2 2 = 1 2 \begin{array}[]{cccccccc}1&&&&&&&\\ 3&1&&&&&&\\ 5&3&1&&&&&\\ 7&5&3&1&&&&\\ 9&7&5&3&1&&&\\ \vdots&&&&&\ddots&&\\ \scriptstyle 2n-3&\cdots&&&&\cdots&1&\\ \scriptstyle 2n-1&\scriptstyle 2n-3&&&&\cdots&&1\\ \hline\scriptstyle=n^{2}&\scriptstyle=(n-1)^{2}&\cdots&\scriptstyle=5^{2}&% \scriptstyle=4^{2}&\scriptstyle=3^{2}&\scriptstyle=2^{2}&\scriptstyle=1^{2}% \end{array}
  40. 2 n + 1 2n+1
  41. 2 n 1 + 1 + 1 = 2 n + 1 2n−1+1+1=2n+1
  42. 1 + 2 + + n = n ( n + 1 ) 2 1+2+\ldots+n=\tfrac{n(n+1)}{2}
  43. n ( n + 1 ) ( 2 n + 1 ) 2 \tfrac{n(n+1)(2n+1)}{2}
  44. n n
  45. P n = n ( n + 1 ) ( 2 n + 1 ) 6 P_{n}=\frac{n(n+1)(2n+1)}{6}

Square_root_of_2.html

  1. 2 \sqrt{2}
  2. 1 + 1 2 + 1 2 + 1 2 + 1 2 + 1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}
  3. 2 \sqrt{2}
  4. 2 \sqrt{2}
  5. 1 + 24 60 + 51 60 2 + 10 60 3 = 30547 21600 = 1.41421 296 ¯ . 1+\frac{24}{60}+\frac{51}{60^{2}}+\frac{10}{60^{3}}=\frac{30547}{21600}=1.4142% 1\overline{296}.
  6. 1 + 1 3 + 1 3 4 - 1 3 4 34 = 577 408 = 1.41421 56862745098039 ¯ . 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}=\frac{577}{408}=1.% 41421\overline{56862745098039}.
  7. 2 \sqrt{2}
  8. 2 \sqrt{2}
  9. a 0 > 0 a_{0}>0
  10. a n + 1 = a n + 2 a n 2 = a n 2 + 1 a n . a_{n+1}=\frac{a_{n}+\frac{2}{a_{n}}}{2}=\frac{a_{n}}{2}+\frac{1}{a_{n}}.
  11. 2 \sqrt{2}
  12. 2 \sqrt{2}
  13. π \pi
  14. 2 \sqrt{2}
  15. p ( x ) p(x)
  16. p ( x ) p(x)
  17. p ( x ) = x 2 - 2 p(x)=x^{2}-2
  18. 2 \sqrt{2}
  19. 2 \sqrt{2}
  20. 2 \sqrt{2}
  21. 2 \sqrt{2}
  22. 2 \sqrt{2}
  23. 2 \sqrt{2}
  24. a / b a/b
  25. a a
  26. b b
  27. a 2 / b 2 = 2 a^{2}/b^{2}=2
  28. a 2 = 2 b 2 a^{2}=2b^{2}
  29. ( a / b ) n = a n / b n (a/b)^{n}=a^{n}/b^{n}
  30. a 2 a^{2}
  31. 2 b 2 2b^{2}
  32. 2 b 2 2b^{2}
  33. a a
  34. a a
  35. k k
  36. a = 2 k a=2k
  37. 2 k 2k
  38. a a
  39. 2 b 2 = ( 2 k ) 2 2b^{2}=(2k)^{2}
  40. 2 b 2 = 4 k 2 2b^{2}=4k^{2}
  41. b 2 = 2 k 2 b^{2}=2k^{2}
  42. 2 k 2 2k^{2}
  43. 2 k 2 = b 2 2k^{2}=b^{2}
  44. b 2 b^{2}
  45. b b
  46. a a
  47. b b
  48. a / b a/b
  49. 2 \sqrt{2}
  50. 2 \sqrt{2}
  51. 2 \sqrt{2}
  52. 2 \sqrt{2}
  53. 2 = a / b \sqrt{2}=a/b
  54. 2 \sqrt{2}
  55. 2 \sqrt{2}
  56. 2 > 2 > 1 2>\sqrt{2}>1
  57. 1 > 2 - 1 > 0 1>\sqrt{2}-1>0
  58. 2 \sqrt{2}
  59. n 0 n\neq 0
  60. m / n = 2 m/n=\sqrt{2}
  61. m = n 2 m=n\sqrt{2}
  62. m 2 = 2 n m\sqrt{2}=2n
  63. n 2 n\sqrt{2}
  64. 2 = m n = m ( 2 - 1 ) n ( 2 - 1 ) = 2 n - m m - n \sqrt{2}=\frac{m}{n}=\frac{m(\sqrt{2}-1)}{n(\sqrt{2}-1)}=\frac{2n-m}{m-n}
  65. 1 > 2 - 1 > 0 1>\sqrt{2}-1>0
  66. n > n ( 2 - 1 ) = m - n > 0 n>n(\sqrt{2}-1)=m-n>0
  67. 2 \sqrt{2}
  68. 2 \sqrt{2}
  69. 2 \sqrt{2}
  70. m / n = 2 m/n=\sqrt{2}
  71. 2 \sqrt{2}
  72. 2 \sqrt{2}
  73. 2 \sqrt{2}
  74. 2 \sqrt{2}
  75. α + \alpha\in\mathbb{R}^{+}
  76. p 1 , p 2 , , q 1 , q 2 , p_{1},p_{2},\dots,q_{1},q_{2},\ldots\in\mathbb{N}
  77. | α q n - p n | 0 \left|\alpha q_{n}-p_{n}\right|\neq 0
  78. n n\in\mathbb{N}
  79. lim n p n = lim n q n = \lim_{n\rightarrow\infty}p_{n}=\lim_{n\rightarrow\infty}q_{n}=\infty\,
  80. lim n | α q n - p n | = 0. \lim_{n\rightarrow\infty}\left|\alpha q_{n}-p_{n}\right|=0.\,
  81. α = a / b \alpha=a/b
  82. a , b + a,b\in\mathbb{N}^{+}
  83. n n
  84. 0 < | α q n - p n | < 1 b 0<\left|\alpha q_{n}-p_{n}\right|<\frac{1}{b}
  85. 0 < | a q n / b - p n | < 1 b 0<\left|aq_{n}/\ \!b-p_{n}\right|<\frac{1}{b}
  86. 0 < | a q n - b p n | < 1 0<\left|aq_{n}-bp_{n}\right|<1\,
  87. a q n - b p n aq_{n}-bp_{n}
  88. α \alpha
  89. 2 \sqrt{2}
  90. p 1 = q 1 = 1 p_{1}=q_{1}=1
  91. p n + 1 = p n 2 + 2 q n 2 p_{n+1}=p_{n}^{2}+2q_{n}^{2}\,
  92. q n + 1 = 2 p n q n q_{n+1}=2p_{n}q_{n}\,\!
  93. n n\in\mathbb{N}
  94. 0 < | 2 q n - p n | < 1 2 2 n - 1 0<\left|\sqrt{2}q_{n}-p_{n}\right|<\frac{1}{2^{2^{n-1}}}
  95. n n\in\mathbb{N}
  96. n = 1 n=1
  97. 0 < | 2 q 1 - p 1 | < 1 2 0<\left|\sqrt{2}q_{1}-p_{1}\right|<\frac{1}{2}
  98. n + 1 n+1
  99. 0 < | 2 q n - p n | 2 < 1 2 2 n 0<\left|\sqrt{2}q_{n}-p_{n}\right|^{2}<\frac{1}{2^{2^{n}}}
  100. 0 < | 2 ( 2 p n q n ) - ( p n 2 + 2 q n 2 ) | < 1 2 2 n 0<\left|\sqrt{2}(2p_{n}q_{n})-(p_{n}^{2}+2q_{n}^{2})\right|<\frac{1}{2^{2^{n}}}
  101. 0 < | 2 q n + 1 - p n + 1 | < 1 2 2 n . 0<\left|\sqrt{2}q_{n+1}-p_{n+1}\right|<\frac{1}{2^{2^{n}}}.
  102. 2 \sqrt{2}
  103. | 2 b 2 - a 2 | 1 |2b^{2}-a^{2}|\geq 1
  104. | 2 - a b | = | 2 b 2 - a 2 | b 2 ( 2 + a / b ) 1 b 2 ( 2 + a / b ) 1 3 b 2 , \left|\sqrt{2}-\frac{a}{b}\right|=\frac{|2b^{2}-a^{2}|}{b^{2}(\sqrt{2}+a/b)}% \geq\frac{1}{b^{2}(\sqrt{2}+a/b)}\geq\frac{1}{3b^{2}},
  105. a b 3 - 2 \tfrac{a}{b}\leq 3-\sqrt{2}
  106. 1 3 b 2 \frac{1}{3b^{2}}
  107. | 2 - a / b | |\sqrt{2}-a/b|
  108. 2 \sqrt{2}
  109. 2 \sqrt{2}
  110. ( 2 2 , 2 2 ) . \left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right).
  111. 1 2 2 = 1 2 = 1 2 = cos ( 45 ) = sin ( 45 ) . \tfrac{1}{2}\sqrt{2}=\sqrt{\tfrac{1}{2}}=\frac{1}{\sqrt{2}}=\cos(45^{\circ})=% \sin(45^{\circ}).
  112. 1 2 - 1 = 2 + 1 \!\ {1\over{\sqrt{2}-1}}=\sqrt{2}+1
  113. ( 2 + 1 ) ( 2 - 1 ) = 2 - 1 = 1. (\sqrt{2}+1)(\sqrt{2}-1)=2-1=1.
  114. i + i i i and - i - i - i - i \frac{\sqrt{i}+i\sqrt{i}}{i}\,\text{ and }\frac{\sqrt{-i}-i\sqrt{-i}}{-i}
  115. 2 ( 2 ( 2 ( ) ) ) = 2. \sqrt{2}^{(\sqrt{2}^{(\sqrt{2}^{(\ \cdot^{\cdot^{\cdot})))}}}}=2.
  116. 2 m 2 - 2 + 2 + + 2 π as m 2^{m}\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}\to\pi\,\text{ as }m\to\infty\,
  117. sin ( 5 5 8 ) = 1 2 2 - 2 + 2 + 2 ; \sin(5\tfrac{5}{8}^{\circ})=\frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}};
  118. sin ( 11 1 4 ) = 1 2 2 - 2 + 2 ; \sin(11\tfrac{1}{4}^{\circ})=\frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2}}};
  119. sin ( 16 7 8 ) = 1 2 2 - 2 + 2 - 2 ; \sin(16\tfrac{7}{8}^{\circ})=\frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}};
  120. sin ( 22 1 2 ) = 1 2 2 - 2 ; \sin(22\tfrac{1}{2}^{\circ})=\frac{1}{2}\sqrt{2-\sqrt{2}};
  121. sin ( 28 1 8 ) = 1 2 2 - 2 - 2 - 2 ; \sin(28\tfrac{1}{8}^{\circ})=\frac{1}{2}\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}};
  122. sin ( 33 3 4 ) = 1 2 2 - 2 - 2 ; \sin(33\tfrac{3}{4}^{\circ})=\frac{1}{2}\sqrt{2-\sqrt{2-\sqrt{2}}};
  123. sin ( 39 3 8 ) = 1 2 2 - 2 - 2 + 2 ; \sin(39\tfrac{3}{8}^{\circ})=\frac{1}{2}\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}};
  124. sin ( 45 ) = 1 2 2 ; \sin(45^{\circ})=\frac{1}{2}\sqrt{2};
  125. sin ( 50 5 8 ) = 1 2 2 + 2 - 2 + 2 ; \sin(50\tfrac{5}{8}^{\circ})=\frac{1}{2}\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}};
  126. sin ( 56 1 4 ) = 1 2 2 + 2 - 2 ; \sin(56\tfrac{1}{4}^{\circ})=\frac{1}{2}\sqrt{2+\sqrt{2-\sqrt{2}}};
  127. sin ( 61 7 8 ) = 1 2 2 + 2 - 2 - 2 ; \sin(61\tfrac{7}{8}^{\circ})=\frac{1}{2}\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}};
  128. sin ( 67 1 2 ) = 1 2 2 + 2 ; \sin(67\tfrac{1}{2}^{\circ})=\frac{1}{2}\sqrt{2+\sqrt{2}};
  129. sin ( 73 1 8 ) = 1 2 2 + 2 + 2 - 2 ; \sin(73\tfrac{1}{8}^{\circ})=\frac{1}{2}\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}};
  130. sin ( 78 3 4 ) = 1 2 2 + 2 + 2 ; \sin(78\tfrac{3}{4}^{\circ})=\frac{1}{2}\sqrt{2+\sqrt{2+\sqrt{2}}};
  131. sin ( 84 3 8 ) = 1 2 2 + 2 + 2 + 2 . \sin(84\tfrac{3}{8}^{\circ})=\frac{1}{2}\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}.
  132. 2 \sqrt{2}
  133. cos ( π / 4 ) = sin ( π / 4 ) = 1 / 2 \cos(\pi/4)=\sin(\pi/4)=1/\sqrt{2}
  134. 1 2 = k = 0 ( 1 - 1 ( 4 k + 2 ) 2 ) = ( 1 - 1 4 ) ( 1 - 1 36 ) ( 1 - 1 100 ) \frac{1}{\sqrt{2}}=\prod_{k=0}^{\infty}\left(1-\frac{1}{(4k+2)^{2}}\right)=% \left(1-\frac{1}{4}\right)\left(1-\frac{1}{36}\right)\left(1-\frac{1}{100}% \right)\cdots
  135. 2 = k = 0 ( 4 k + 2 ) 2 ( 4 k + 1 ) ( 4 k + 3 ) = ( 2 2 1 3 ) ( 6 6 5 7 ) ( 10 10 9 11 ) ( 14 14 13 15 ) \sqrt{2}=\prod_{k=0}^{\infty}\frac{(4k+2)^{2}}{(4k+1)(4k+3)}=\left(\frac{2% \cdot 2}{1\cdot 3}\right)\left(\frac{6\cdot 6}{5\cdot 7}\right)\left(\frac{10% \cdot 10}{9\cdot 11}\right)\left(\frac{14\cdot 14}{13\cdot 15}\right)\cdots
  136. 2 = k = 0 ( 1 + 1 4 k + 1 ) ( 1 - 1 4 k + 3 ) = ( 1 + 1 1 ) ( 1 - 1 3 ) ( 1 + 1 5 ) ( 1 - 1 7 ) . \sqrt{2}=\prod_{k=0}^{\infty}\left(1+\frac{1}{4k+1}\right)\left(1-\frac{1}{4k+% 3}\right)=\left(1+\frac{1}{1}\right)\left(1-\frac{1}{3}\right)\left(1+\frac{1}% {5}\right)\left(1-\frac{1}{7}\right)\cdots.
  137. cos ( π / 4 ) \cos(\pi/4)
  138. 1 2 = k = 0 ( - 1 ) k ( π 4 ) 2 k ( 2 k ) ! . \frac{1}{\sqrt{2}}=\sum_{k=0}^{\infty}\frac{(-1)^{k}\left(\frac{\pi}{4}\right)% ^{2k}}{(2k)!}.
  139. ( 1 + x ) \sqrt{(1+x)}
  140. x = 1 x=1
  141. n ! ! n!!
  142. 2 = k = 0 ( - 1 ) k + 1 ( 2 k - 3 ) ! ! ( 2 k ) ! ! = 1 + 1 2 - 1 2 4 + 1 3 2 4 6 - 1 3 5 2 4 6 8 + . \sqrt{2}=\sum_{k=0}^{\infty}(-1)^{k+1}\frac{(2k-3)!!}{(2k)!!}=1+\frac{1}{2}-% \frac{1}{2\cdot 4}+\frac{1\cdot 3}{2\cdot 4\cdot 6}-\frac{1\cdot 3\cdot 5}{2% \cdot 4\cdot 6\cdot 8}+\cdots.
  143. 2 = k = 0 ( 2 k + 1 ) ! ( k ! ) 2 2 3 k + 1 = 1 2 + 3 8 + 15 64 + 35 256 + 315 4096 + 693 16384 + . \sqrt{2}=\sum_{k=0}^{\infty}\frac{(2k+1)!}{(k!)^{2}2^{3k+1}}=\frac{1}{2}+\frac% {3}{8}+\frac{15}{64}+\frac{35}{256}+\frac{315}{4096}+\frac{693}{16384}+\cdots.
  144. 2 \sqrt{2}
  145. π 2 \pi\sqrt{2}
  146. 2 ln ( 1 + 2 ) \sqrt{2}\ln(1+\sqrt{2})
  147. 2 = 1 + 1 2 + 1 2 + 1 2 + 1 2 + . \!\ \sqrt{2}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}.
  148. 1 2 q 2 2 \scriptstyle{\frac{1}{2q^{2}\sqrt{2}}}
  149. 1 / 2 = 2 / 2 = sin 45 = cos 45 = 0.70710678118654752440084436210484903928... 1/\sqrt{2}=\sqrt{2}/2=\sin 45^{\circ}=\cos 45^{\circ}=0.7071067811865475244008% 4436210484903928...
  150. 2 \sqrt{2}
  151. 1 + 2 1+\sqrt{2}
  152. 2 \sqrt{2}
  153. 2 \sqrt{2}

Square_triangular_number.html

  1. N k = s k 2 = t k ( t k + 1 ) 2 . N_{k}=s_{k}^{2}=\frac{t_{k}(t_{k}+1)}{2}.
  2. N = n ( n + 1 ) 2 N=\frac{n(n+1)}{2}
  3. n n
  4. n = 8 N + 1 - 1 2 . n=\frac{\sqrt{8N+1}-1}{2}.
  5. N N
  6. 8 N + 1 8N+1
  7. N 2 N^{2}
  8. 8 N 2 + 1 8N^{2}+1
  9. x x
  10. y y
  11. x 2 - 8 y 2 = 1 x^{2}-8y^{2}=1
  12. ( x 0 , y 0 ) (x_{0},y_{0})
  13. ( x k , y k ) (x_{k},y_{k})
  14. x k + 1 = 2 x k x 1 - x k - 1 x_{k+1}=2x_{k}x_{1}-x_{k-1}
  15. y k + 1 = 2 y k x 1 - y k - 1 y_{k+1}=2y_{k}x_{1}-y_{k-1}
  16. ( x k , y k ) (x_{k},y_{k})
  17. s k = y k , t k = x k - 1 2 , s_{k}=y_{k},t_{k}=\frac{x_{k}-1}{2},
  18. N k = y k 2 . N_{k}=y_{k}^{2}.
  19. N k = ( ( 3 + 2 2 ) k - ( 3 - 2 2 ) k 4 2 ) 2 . N_{k}=\left(\frac{(3+2\sqrt{2})^{k}-(3-2\sqrt{2})^{k}}{4\sqrt{2}}\right)^{2}.
  20. N k \displaystyle N_{k}
  21. s k = ( 3 + 2 2 ) k - ( 3 - 2 2 ) k 4 2 s_{k}=\frac{(3+2\sqrt{2})^{k}-(3-2\sqrt{2})^{k}}{4\sqrt{2}}
  22. t k = ( 3 + 2 2 ) k + ( 3 - 2 2 ) k - 2 4 . t_{k}=\frac{(3+2\sqrt{2})^{k}+(3-2\sqrt{2})^{k}-2}{4}.
  23. t ( t + 1 ) 2 = s 2 . \frac{t(t+1)}{2}=s^{2}.
  24. ( 2 t + 1 ) 2 = 8 s 2 + 1 , (2t+1)^{2}=8s^{2}+1,
  25. x 2 - 2 y 2 = 1 x^{2}-2y^{2}=1
  26. x = P 2 k + P 2 k - 1 , y = P 2 k ; x=P_{2k}+P_{2k-1},\quad y=P_{2k};
  27. s k = P 2 k 2 , t k = P 2 k + P 2 k - 1 - 1 2 , N k = ( P 2 k 2 ) 2 . s_{k}=\frac{P_{2k}}{2},\quad t_{k}=\frac{P_{2k}+P_{2k-1}-1}{2},\quad N_{k}=% \left(\frac{P_{2k}}{2}\right)^{2}.
  28. N k = 34 N k - 1 - N k - 2 + 2 , with N 0 = 0 and N 1 = 1. N_{k}=34N_{k-1}-N_{k-2}+2,\,\text{ with }N_{0}=0\,\text{ and }N_{1}=1.
  29. N k = ( 6 N k - 1 - N k - 2 ) 2 , with N 0 = 0 and N 1 = 1. N_{k}=\left(6\sqrt{N_{k-1}}-\sqrt{N_{k-2}}\right)^{2},\,\text{ with }N_{0}=0\,% \text{ and }N_{1}=1.
  30. s k = 6 s k - 1 - s k - 2 , with s 0 = 0 and s 1 = 1 ; s_{k}=6s_{k-1}-s_{k-2},\,\text{ with }s_{0}=0\,\text{ and }s_{1}=1;
  31. t k = 6 t k - 1 - t k - 2 + 2 , with t 0 = 0 and t 1 = 1. t_{k}=6t_{k-1}-t_{k-2}+2,\,\text{ with }t_{0}=0\,\text{ and }t_{1}=1.
  32. ( 4 n ( n + 1 ) ) ( 4 n ( n + 1 ) + 1 ) 2 = 2 2 n ( n + 1 ) 2 ( 2 n + 1 ) 2 . \frac{\bigl(4n(n+1)\bigr)\bigl(4n(n+1)+1\bigr)}{2}=2^{2}\,\frac{n(n+1)}{2}\,(2% n+1)^{2}.
  33. t k t_{k}
  34. 49 = 7 2 = 2 * 5 2 - 1 , 288 = 17 2 - 1 = 2 * 12 2 49=7^{2}=2*5^{2}-1,288=17^{2}-1=2*12^{2}
  35. 1681 = 41 2 = 2 * 29 2 - 1. 1681=41^{2}=2*29^{2}-1.
  36. s k : 5 * 7 = 35 , 12 * 17 = 204 , s_{k}:5*7=35,12*17=204,
  37. 29 * 41 = 1189. 29*41=1189.
  38. N k - N k - 1 = s 2 k - 1 : 36 - 1 = 35 , 1225 - 36 = 1189 , N_{k}-N_{k-1}=s_{2k-1}:36-1=35,1225-36=1189,
  39. 41616 - 1225 = 40391. 41616-1225=40391.
  40. 1 + z ( 1 - z ) ( z 2 - 34 z + 1 ) = 1 + 36 z + 1225 z 2 + . \frac{1+z}{(1-z)(z^{2}-34z+1)}=1+36z+1225z^{2}+\cdots.
  41. k k
  42. t k / s k t_{k}/s_{k}
  43. 2 1.41421 \sqrt{2}\approx 1.41421
  44. ( 1 + 2 ) 4 = 17 + 12 2 33.97056 (1+\sqrt{2})^{4}=17+12\sqrt{2}\approx 33.97056
  45. k k
  46. k k
  47. N k N_{k}
  48. s k s_{k}
  49. t k t_{k}
  50. t k / s k t_{k}/s_{k}
  51. N k / N k - 1 N_{k}/N_{k-1}
  52. 0
  53. 0
  54. 0
  55. 0
  56. 1 1
  57. 1 1
  58. 1 1
  59. 1 1
  60. 1 1
  61. 2 2
  62. 36 36
  63. 6 6
  64. 8 8
  65. 1.33333 1.33333
  66. 36 36
  67. 3 3
  68. 1 225 1\,225
  69. 35 35
  70. 49 49
  71. 1.4 1.4
  72. 34.02778 34.02778
  73. 4 4
  74. 41 616 41\,616
  75. 204 204
  76. 288 288
  77. 1.41176 1.41176
  78. 33.97224 33.97224
  79. 5 5
  80. 1 413 721 1\,413\,721
  81. 1 189 1\,189
  82. 1 681 1\,681
  83. 1.41379 1.41379
  84. 33.97061 33.97061
  85. 6 6
  86. 48 024 900 48\,024\,900
  87. 6 930 6\,930
  88. 9 800 9\,800
  89. 1.41414 1.41414
  90. 33.97056 33.97056
  91. 7 7
  92. 1 631 432 881 1\,631\,432\,881
  93. 40 391 40\,391
  94. 57 121 57\,121
  95. 1.41420 1.41420
  96. 33.97056 33.97056

Squeeze_theorem.html

  1. π \pi
  2. g ( x ) f ( x ) h ( x ) g(x)\leq f(x)\leq h(x)\,
  3. lim x a g ( x ) = lim x a h ( x ) = L . \lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L.\,
  4. lim x a f ( x ) = L . \lim_{x\to a}f(x)=L.
  5. L = lim x a g ( x ) lim inf x a f ( x ) lim sup x a f ( x ) lim x a h ( x ) = L , L=\lim_{x\to a}g(x)\leq\liminf_{x\to a}f(x)\leq\limsup_{x\to a}f(x)\leq\lim_{x% \to a}h(x)=L,
  6. lim x a g ( x ) = L \lim_{x\to a}g(x)=L
  7. ε > 0 δ 1 > 0 : x ( 0 < | x - a | < δ 1 - ε < g ( x ) - L < ε ) . ( 1 ) \forall\varepsilon>0\ \exists\ \delta_{1}>0:\forall x\ (0<|x-a|<\delta_{1}\ % \Rightarrow\ -\varepsilon<g(x)-L<\varepsilon).\qquad(1)
  8. lim x a h ( x ) = L \lim_{x\to a}h(x)=L
  9. ε > 0 δ 2 > 0 : x ( 0 < | x - a | < δ 2 - ε < h ( x ) - L < ε ) , ( 2 ) \forall\varepsilon>0\ \exists\ \delta_{2}>0:\forall x\ (0<|x-a|<\delta_{2}\ % \Rightarrow\ -\varepsilon<h(x)-L<\varepsilon),\qquad(2)
  10. g ( x ) f ( x ) h ( x ) g(x)\leq f(x)\leq h(x)\,
  11. g ( x ) - L f ( x ) - L h ( x ) - L g(x)-L\leq f(x)-L\leq h(x)-L\,
  12. δ := min { δ 1 , δ 2 } \delta:=\min\left\{\delta_{1},\delta_{2}\right\}
  13. | x - a | < δ |x-a|<\delta
  14. - ε < g ( x ) - L f ( x ) - L h ( x ) - L < ε , -\varepsilon<g(x)-L\leq f(x)-L\leq h(x)-L\ <\varepsilon,
  15. - ε < f ( x ) - L < ε -\varepsilon<f(x)-L<\varepsilon
  16. \blacksquare
  17. lim x 0 x 2 sin ( 1 x ) \lim_{x\to 0}x^{2}\sin(\tfrac{1}{x})
  18. lim x a ( f ( x ) g ( x ) ) = lim x a f ( x ) lim x a g ( x ) , \lim_{x\to a}(f(x)\cdot g(x))=\lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x),
  19. lim x 0 sin ( 1 x ) \lim_{x\to 0}\sin(\tfrac{1}{x})
  20. - 1 sin ( 1 x ) 1. -1\leq\sin(\tfrac{1}{x})\leq 1.\,
  21. - x 2 x 2 sin ( 1 x ) x 2 -x^{2}\leq x^{2}\sin(\tfrac{1}{x})\leq x^{2}\,
  22. lim x 0 - x 2 = lim x 0 x 2 = 0 \lim_{x\to 0}-x^{2}=\lim_{x\to 0}x^{2}=0
  23. lim x 0 x 2 sin ( 1 x ) \lim_{x\to 0}x^{2}\sin(\tfrac{1}{x})
  24. lim x 0 sin x x = 1 , \displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1,
  25. cos x < sin x x < 1 \cos x<\frac{\sin x}{x}<1
  26. d d θ tan θ = sec 2 θ \frac{d}{d\theta}\tan\theta=\sec^{2}\theta
  27. sec 2 θ Δ θ 2 , \frac{\sec^{2}\theta\,\Delta\theta}{2},
  28. sec 2 ( θ + Δ θ ) Δ θ 2 . \frac{\sec^{2}(\theta+\Delta\theta)\,\Delta\theta}{2}.
  29. tan ( θ + Δ θ ) - tan ( θ ) 2 . \frac{\tan(\theta+\Delta\theta)-\tan(\theta)}{2}.
  30. sec 2 θ Δ θ 2 tan ( θ + Δ θ ) - tan ( θ ) 2 sec 2 ( θ + Δ θ ) Δ θ 2 \frac{\sec^{2}\theta\,\Delta\theta}{2}\leq\frac{\tan(\theta+\Delta\theta)-\tan% (\theta)}{2}\leq\frac{\sec^{2}(\theta+\Delta\theta)\,\Delta\theta}{2}
  31. sec 2 θ tan ( θ + Δ θ ) - tan ( θ ) Δ θ sec 2 ( θ + Δ θ ) , \sec^{2}\theta\leq\frac{\tan(\theta+\Delta\theta)-\tan(\theta)}{\Delta\theta}% \leq\sec^{2}(\theta+\Delta\theta),
  32. lim ( x , y ) ( 0 , 0 ) x 2 y x 2 + y 2 \lim_{(x,y)\to(0,0)}\frac{x^{2}y}{x^{2}+y^{2}}
  33. 0 x 2 x 2 + y 2 1 0\leq\frac{x^{2}}{x^{2}+y^{2}}\leq 1
  34. - | y | y | y | -\left|y\right|\leq y\leq\left|y\right|
  35. - | y | x 2 y x 2 + y 2 | y | -\left|y\right|\leq\frac{x^{2}y}{x^{2}+y^{2}}\leq\left|y\right|
  36. lim ( x , y ) ( 0 , 0 ) - | y | = 0 \lim_{(x,y)\to(0,0)}-\left|y\right|=0
  37. lim ( x , y ) ( 0 , 0 ) | y | = 0 \lim_{(x,y)\to(0,0)}\left|y\right|=0
  38. 0 lim ( x , y ) ( 0 , 0 ) x 2 y x 2 + y 2 0 0\leq\lim_{(x,y)\to(0,0)}\frac{x^{2}y}{x^{2}+y^{2}}\leq 0
  39. lim ( x , y ) ( 0 , 0 ) x 2 y x 2 + y 2 = 0 \lim_{(x,y)\to(0,0)}\frac{x^{2}y}{x^{2}+y^{2}}=0

St._Petersburg_paradox.html

  1. E = 1 2 2 + 1 4 4 + 1 8 8 + 1 16 16 + E=\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 4+\frac{1}{8}\cdot 8+\frac{1}{16}\cdot 1% 6+\cdots
  2. = 1 + 1 + 1 + 1 + =1+1+1+1+\cdots
  3. = . =\infty\,.
  4. E ( U ) = k = 1 ( ln ( w + 2 k - 1 - c ) - ln ( w ) ) 2 k < . E(U)=\sum_{k=1}^{\infty}\frac{(\ln(w+2^{k-1}-c)-\ln(w))}{2^{k}}<\infty\,.
  5. e 2 k e^{2^{k}}
  6. E \displaystyle E
  7. g ¯ ( w , c ) = k = 1 p k ln ( w - c + D k w ) \bar{g}(w,c)=\sum_{k=1}^{\infty}p_{k}\ln\left(\frac{w-c+D_{k}}{w}\right)
  8. D k D_{k}
  9. k k
  10. p k p_{k}
  11. w w
  12. c c
  13. D k = 2 k - 1 D_{k}=2^{k-1}
  14. p k = 2 - k p_{k}=2^{-k}
  15. g ¯ \bar{g}
  16. w w
  17. c c
  18. g ¯ ( w , c ) > 0. \bar{g}(w,c)>0.
  19. w - c + D k = 0 , w-c+D_{k}=0,
  20. k k
  21. g ¯ \bar{g}
  22. g ¯ < 0 \bar{g}<0
  23. D 1 D_{1}
  24. c max = w + D 1 , c_{\mathrm{max}}=w+D_{1},
  25. c max c_{\mathrm{max}}
  26. w w

Standard_basis.html

  1. 𝐞 x = ( 1 , 0 ) , 𝐞 y = ( 0 , 1 ) , \mathbf{e}_{x}=(1,0),\quad\mathbf{e}_{y}=(0,1),
  2. 𝐞 x = ( 1 , 0 , 0 ) , 𝐞 y = ( 0 , 1 , 0 ) , 𝐞 z = ( 0 , 0 , 1 ) . \mathbf{e}_{x}=(1,0,0),\quad\mathbf{e}_{y}=(0,1,0),\quad\mathbf{e}_{z}=(0,0,1).
  3. v x 𝐞 x + v y 𝐞 y + v z 𝐞 z , v_{x}\,\mathbf{e}_{x}+v_{y}\,\mathbf{e}_{y}+v_{z}\,\mathbf{e}_{z},
  4. n n
  5. { 𝐞 i : 1 i n } , \{\mathbf{e}_{i}:1\leq i\leq n\},
  6. i i
  7. m × n \mathcal{M}_{m\times n}
  8. 𝐞 11 = ( 1 0 0 0 ) , 𝐞 12 = ( 0 1 0 0 ) , 𝐞 21 = ( 0 0 1 0 ) , 𝐞 22 = ( 0 0 0 1 ) . \mathbf{e}_{11}=\begin{pmatrix}1&0\\ 0&0\end{pmatrix},\quad\mathbf{e}_{12}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix},\quad\mathbf{e}_{21}=\begin{pmatrix}0&0\\ 1&0\end{pmatrix},\quad\mathbf{e}_{22}=\begin{pmatrix}0&0\\ 0&1\end{pmatrix}.
  9. v 1 = ( 3 2 , 1 2 ) v_{1}=\left({\sqrt{3}\over 2},{1\over 2}\right)\,
  10. v 2 = ( 1 2 , - 3 2 ) v_{2}=\left({1\over 2},{-\sqrt{3}\over 2}\right)\,
  11. ( e i ) i I = ( ( δ i j ) j I ) i I {(e_{i})}_{i\in I}=((\delta_{ij})_{j\in I})_{i\in I}
  12. I I
  13. δ i j \delta_{ij}
  14. R ( I ) R^{(I)}
  15. f = ( f i ) f=(f_{i})

Standard_enthalpy_of_reaction.html

  1. Δ H r = B v B Δ H f ( B ) \Delta H_{\mathrm{r}}^{\ominus}=\sum_{B}{v_{B}\Delta H_{\mathrm{f}}^{\ominus}(% B)}
  2. Δ E = Q v \Delta E=Q_{v}
  3. Δ E = E p r o d u c t s - E r e a c t a n t s \Delta E=\sum E_{products}-\sum E_{reactants}
  4. Q p = Δ E + W Q_{p}=\Delta E+W
  5. Q p = Δ H = Δ E + P Δ V Q_{p}=\Delta H=\Delta E+P\Delta V
  6. Q p = ( E p - E r ) + P ( V p - V r ) Q_{p}=\left(\sum E_{p}-\sum E_{r}\right)+P\left(V_{p}-V_{r}\right)
  7. Q p = ( E p + P V p ) - ( E r + P V r ) Q_{p}=\left(\sum E_{p}+PV_{p}\right)-\left(\sum E_{r}+PV_{r}\right)
  8. H = E + P V H=E+PV
  9. Q p = H p - H r = Δ H Q_{p}=\sum H_{p}-\sum H_{r}=\Delta H
  10. Δ H \Delta H

Standard_error.html

  1. SE x ¯ = s n \,\text{SE}_{\bar{x}}\ =\frac{s}{\sqrt{n}}
  2. SD x ¯ = σ n \,\text{SD}_{\bar{x}}\ =\frac{\sigma}{\sqrt{n}}
  3. X 1 , X 2 , , X n X_{1},X_{2},\ldots,X_{n}
  4. n n
  5. μ \mu
  6. σ \sigma
  7. T = ( X 1 + X 2 + + X n ) T=(X_{1}+X_{2}+\cdots+X_{n})
  8. n σ 2 . n\sigma^{2}.
  9. T / n T/n
  10. 1 n 2 n σ 2 = σ 2 n . \frac{1}{n^{2}}n\sigma^{2}=\frac{\sigma^{2}}{n}.
  11. T / n T/n
  12. σ / n \sigma/{\sqrt{n}}
  13. T / n T/n
  14. x ¯ \bar{x}
  15. x ¯ \bar{x}
  16. S E SE
  17. = x ¯ + ( SE × 1.96 ) , =\bar{x}+(\,\text{SE}\times 1.96),
  18. = x ¯ - ( SE × 1.96 ) . =\bar{x}-(\,\text{SE}\times 1.96).
  19. FPC = N - n N - 1 \,\text{FPC}=\sqrt{\frac{N-n}{N-1}}
  20. f = 1 + ρ 1 - ρ , f=\sqrt{\frac{1+\rho}{1-\rho}},

Standard_hydrogen_electrode.html

  1. E = R T F ln a H + ( p H 2 / p 0 ) 1 / 2 E={RT\over F}\ln{a_{H^{+}}\over(p_{H_{2}}/p^{0})^{1/2}}
  2. E = - 2.303 R T F p H - R T 2 F ln p H 2 / p 0 E=-{2.303RT\over F}pH-{RT\over 2F}\ln{p_{H_{2}}/p^{0}}

Stark_effect.html

  1. E int = ρ ( 𝐫 ) V ( 𝐫 ) d 𝐫 3 E_{\mathrm{int}}=\int\rho(\mathbf{r})V(\mathbf{r})d\mathbf{r}^{3}
  2. V ( 𝐫 ) = V ( 𝟎 ) - i = 1 3 r i F i V(\mathbf{r})=V(\mathbf{0})-\sum_{i=1}^{3}r_{i}F_{i}
  3. F i - ( V r i ) | 𝟎 F_{i}\equiv-\left.\left(\frac{\partial V}{\partial r_{i}}\right)\right|_{% \mathbf{0}}
  4. E int = - i = 1 3 F i ρ ( 𝐫 ) r i d 𝐫 - i = 1 3 F i μ i = - 𝐅 s y m b o l μ E_{\mathrm{int}}=-\sum_{i=1}^{3}F_{i}\int\rho(\mathbf{r})r_{i}d\mathbf{r}% \equiv-\sum_{i=1}^{3}F_{i}\mu_{i}=-\mathbf{F}\cdot symbol{\mu}
  5. s y m b o l μ j = 1 N q j 𝐫 j symbol{\mu}\equiv\sum_{j=1}^{N}q_{j}\mathbf{r}_{j}
  6. V int = - 𝐅 s y m b o l μ . V_{\mathrm{int}}=-\mathbf{F}\cdot symbol{\mu}.
  7. ψ 1 0 , , ψ g 0 \psi^{0}_{1},\ldots,\psi^{0}_{g}
  8. ( 𝐕 int ) k l = ψ k 0 | V int | ψ l 0 = - 𝐅 ψ k 0 | s y m b o l μ | ψ l 0 , k , l = 1 , , g . (\mathbf{V}_{\mathrm{int}})_{kl}=\langle\psi^{0}_{k}|V_{\mathrm{int}}|\psi^{0}% _{l}\rangle=-\mathbf{F}\cdot\langle\psi^{0}_{k}|symbol{\mu}|\psi^{0}_{l}% \rangle,\qquad k,l=1,\ldots,g.
  9. s y m b o l μ symbol{\mu}
  10. E ( 1 ) = - 𝐅 ψ 1 0 | s y m b o l μ | ψ 1 0 = - 𝐅 s y m b o l μ . E^{(1)}=-\mathbf{F}\cdot\langle\psi^{0}_{1}|symbol{\mu}|\psi^{0}_{1}\rangle=-% \mathbf{F}\cdot\langle symbol{\mu}\rangle.
  11. ψ i 0 \psi^{0}_{i}
  12. n 2 = = 0 n - 1 ( 2 + 1 ) , n^{2}=\sum_{\ell=0}^{n-1}(2\ell+1),
  13. \ell
  14. \ell
  15. 16 = 1 + 3 + 5 + 7 n = 4 contains s p d f . 16=1+3+5+7\;\;\Longrightarrow\;\;n=4\;\hbox{contains}\;s\oplus p\oplus d\oplus f.
  16. \ell
  17. \ell
  18. | J K M = ( D M K J ) * with M , K = - J , - J + 1 , , J |JKM\rangle=(D^{J}_{MK})^{*}\quad\mathrm{with}\quad M,K=-J,-J+1,\dots,J
  19. H ( 0 ) ψ k 0 = E k ( 0 ) ψ k 0 , k = 0 , 1 , , E 0 ( 0 ) < E 1 ( 0 ) E 2 ( 0 ) , H^{(0)}\psi^{0}_{k}=E^{(0)}_{k}\psi^{0}_{k},\quad k=0,1,\ldots,\quad E^{(0)}_{% 0}<E^{(0)}_{1}\leq E^{(0)}_{2},\dots
  20. E ( 2 ) = k > 0 ψ 0 0 | V int | ψ k 0 ψ k 0 | V int | ψ 0 0 E 0 ( 0 ) - E k ( 0 ) = - 1 2 i , j = 1 3 F i α i j F j E^{(2)}=\sum_{k>0}\frac{\langle\psi^{0}_{0}|V_{\mathrm{int}}|\psi^{0}_{k}% \rangle\langle\psi^{0}_{k}|V_{\mathrm{int}}|\psi^{0}_{0}\rangle}{E^{(0)}_{0}-E% ^{(0)}_{k}}=-\frac{1}{2}\sum_{i,j=1}^{3}F_{i}\alpha_{ij}F_{j}
  21. α i j - 2 k > 0 ψ 0 0 | μ i | ψ k 0 ψ k 0 | μ j | ψ 0 0 E 0 ( 0 ) - E k ( 0 ) . \alpha_{ij}\equiv-2\sum_{k>0}\frac{\langle\psi^{0}_{0}|\mu_{i}|\psi^{0}_{k}% \rangle\langle\psi^{0}_{k}|\mu_{j}|\psi^{0}_{0}\rangle}{E^{(0)}_{0}-E^{(0)}_{k% }}.
  22. α i j = α 0 δ i j E ( 2 ) = - 1 2 α 0 F 2 , \alpha_{ij}=\alpha_{0}\delta_{ij}\Longrightarrow E^{(2)}=-\frac{1}{2}\alpha_{0% }F^{2},

Stark–Heegner_theorem.html

  1. d { - 1 , - 2 , - 3 , - 7 , - 11 , - 19 , - 43 , - 67 , - 163 } . d\in\{\,-1,-2,-3,-7,-11,-19,-43,-67,-163\,\}.
  2. D = - 3 , - 4 , - 7 , - 8 , - 11 , - 19 , - 43 , - 67 , - 163 , D=-3,-4,-7,-8,-11,-19,-43,-67,-163,\,

State-space_representation.html

  1. p p
  2. q q
  3. q × p q\times p
  4. n n
  5. p p
  6. q q
  7. n n
  8. 𝐱 ˙ ( t ) = A ( t ) 𝐱 ( t ) + B ( t ) 𝐮 ( t ) \dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)+B(t)\mathbf{u}(t)
  9. 𝐲 ( t ) = C ( t ) 𝐱 ( t ) + D ( t ) 𝐮 ( t ) \mathbf{y}(t)=C(t)\mathbf{x}(t)+D(t)\mathbf{u}(t)
  10. 𝐱 ( ) \mathbf{x}(\cdot)
  11. 𝐱 ( t ) n \mathbf{x}(t)\in\mathbb{R}^{n}
  12. 𝐲 ( ) \mathbf{y}(\cdot)
  13. 𝐲 ( t ) q \mathbf{y}(t)\in\mathbb{R}^{q}
  14. 𝐮 ( ) \mathbf{u}(\cdot)
  15. 𝐮 ( t ) p \mathbf{u}(t)\in\mathbb{R}^{p}
  16. A ( ) A(\cdot)
  17. dim [ A ( ) ] = n × n \operatorname{dim}[A(\cdot)]=n\times n
  18. B ( ) B(\cdot)
  19. dim [ B ( ) ] = n × p \operatorname{dim}[B(\cdot)]=n\times p
  20. C ( ) C(\cdot)
  21. dim [ C ( ) ] = q × n \operatorname{dim}[C(\cdot)]=q\times n
  22. D ( ) D(\cdot)
  23. D ( ) D(\cdot)
  24. dim [ D ( ) ] = q × p \operatorname{dim}[D(\cdot)]=q\times p
  25. 𝐱 ˙ ( t ) := d d t 𝐱 ( t ) \dot{\mathbf{x}}(t):=\frac{\operatorname{d}}{\operatorname{d}t}\mathbf{x}(t)
  26. t t
  27. t t\in\mathbb{R}
  28. t t\in\mathbb{Z}
  29. k k
  30. t t
  31. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B 𝐮 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t)
  32. 𝐲 ( t ) = C 𝐱 ( t ) + D 𝐮 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)
  33. 𝐱 ˙ ( t ) = 𝐀 ( t ) 𝐱 ( t ) + 𝐁 ( t ) 𝐮 ( t ) \dot{\mathbf{x}}(t)=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t)
  34. 𝐲 ( t ) = 𝐂 ( t ) 𝐱 ( t ) + 𝐃 ( t ) 𝐮 ( t ) \mathbf{y}(t)=\mathbf{C}(t)\mathbf{x}(t)+\mathbf{D}(t)\mathbf{u}(t)
  35. 𝐱 ( k + 1 ) = A 𝐱 ( k ) + B 𝐮 ( k ) \mathbf{x}(k+1)=A\mathbf{x}(k)+B\mathbf{u}(k)
  36. 𝐲 ( k ) = C 𝐱 ( k ) + D 𝐮 ( k ) \mathbf{y}(k)=C\mathbf{x}(k)+D\mathbf{u}(k)
  37. 𝐱 ( k + 1 ) = 𝐀 ( k ) 𝐱 ( k ) + 𝐁 ( k ) 𝐮 ( k ) \mathbf{x}(k+1)=\mathbf{A}(k)\mathbf{x}(k)+\mathbf{B}(k)\mathbf{u}(k)
  38. 𝐲 ( k ) = 𝐂 ( k ) 𝐱 ( k ) + 𝐃 ( k ) 𝐮 ( k ) \mathbf{y}(k)=\mathbf{C}(k)\mathbf{x}(k)+\mathbf{D}(k)\mathbf{u}(k)
  39. s 𝐗 ( s ) = A 𝐗 ( s ) + B 𝐔 ( s ) s\mathbf{X}(s)=A\mathbf{X}(s)+B\mathbf{U}(s)
  40. 𝐘 ( s ) = C 𝐗 ( s ) + D 𝐔 ( s ) \mathbf{Y}(s)=C\mathbf{X}(s)+D\mathbf{U}(s)
  41. z 𝐗 ( z ) = A 𝐗 ( z ) + B 𝐔 ( z ) z\mathbf{X}(z)=A\mathbf{X}(z)+B\mathbf{U}(z)
  42. 𝐘 ( z ) = C 𝐗 ( z ) + D 𝐔 ( z ) \mathbf{Y}(z)=C\mathbf{X}(z)+D\mathbf{U}(z)
  43. 𝐆 ( s ) = k ( s - z 1 ) ( s - z 2 ) ( s - z 3 ) ( s - p 1 ) ( s - p 2 ) ( s - p 3 ) ( s - p 4 ) . \,\textbf{G}(s)=k\frac{(s-z_{1})(s-z_{2})(s-z_{3})}{(s-p_{1})(s-p_{2})(s-p_{3}% )(s-p_{4})}.\,
  44. s I - A sI-A
  45. λ ( s ) = | s I - A | . \mathbf{\lambda}(s)=|sI-A|.\,
  46. 𝐆 ( s ) \,\textbf{G}(s)
  47. rank [ B A B A 2 B A n - 1 B ] = n , \operatorname{rank}\begin{bmatrix}B&AB&A^{2}B&\dots&A^{n-1}B\end{bmatrix}=n,\,
  48. rank [ C C A C A n - 1 ] = n . \operatorname{rank}\begin{bmatrix}C\\ CA\\ \vdots\\ CA^{n-1}\end{bmatrix}=n.\,
  49. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B 𝐮 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t)
  50. s 𝐗 ( s ) - X ( 0 ) = A 𝐗 ( s ) + B 𝐔 ( s ) . s\mathbf{X}(s)-X(0)=A\mathbf{X}(s)+B\mathbf{U}(s).\,
  51. 𝐗 ( s ) \mathbf{X}(s)
  52. ( s 𝐈 - A ) 𝐗 ( s ) = X ( 0 ) + B 𝐔 ( s ) , (s\mathbf{I}-A)\mathbf{X}(s)=X(0)+B\mathbf{U}(s),\,
  53. 𝐗 ( s ) = ( s 𝐈 - A ) - 1 X ( 0 ) + ( s 𝐈 - A ) - 1 B 𝐔 ( s ) . \mathbf{X}(s)=(s\mathbf{I}-A)^{-1}X(0)+(s\mathbf{I}-A)^{-1}B\mathbf{U}(s).\,
  54. 𝐗 ( s ) \mathbf{X}(s)
  55. 𝐘 ( s ) = C 𝐗 ( s ) + D 𝐔 ( s ) , \mathbf{Y}(s)=C\mathbf{X}(s)+D\mathbf{U}(s),
  56. 𝐘 ( s ) = C ( ( s 𝐈 - A ) - 1 B 𝐔 ( s ) ) + D 𝐔 ( s ) . \mathbf{Y}(s)=C((s\mathbf{I}-A)^{-1}B\mathbf{U}(s))+D\mathbf{U}(s).\,
  57. 𝐆 ( s ) \mathbf{G}(s)
  58. 𝐆 ( s ) = 𝐘 ( s ) / 𝐔 ( s ) \mathbf{G}(s)=\mathbf{Y}(s)/\mathbf{U}(s)
  59. 𝐘 ( s ) \mathbf{Y}(s)
  60. 𝐔 ( s ) \mathbf{U}(s)
  61. 𝐆 ( s ) = C ( s 𝐈 - A ) - 1 B + D . \mathbf{G}(s)=C(s\mathbf{I}-A)^{-1}B+D.\,
  62. 𝐆 ( s ) \mathbf{G}(s)
  63. q q
  64. p p
  65. q p qp
  66. q q
  67. 𝐆 ( s ) = n 1 s 3 + n 2 s 2 + n 3 s + n 4 s 4 + d 1 s 3 + d 2 s 2 + d 3 s + d 4 . \,\textbf{G}(s)=\frac{n_{1}s^{3}+n_{2}s^{2}+n_{3}s+n_{4}}{s^{4}+d_{1}s^{3}+d_{% 2}s^{2}+d_{3}s+d_{4}}.\,
  68. 𝐱 ˙ ( t ) = [ - d 1 - d 2 - d 3 - d 4 1 0 0 0 0 1 0 0 0 0 1 0 ] 𝐱 ( t ) + [ 1 0 0 0 ] 𝐮 ( t ) \dot{\,\textbf{x}}(t)=\begin{bmatrix}-d_{1}&-d_{2}&-d_{3}&-d_{4}\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{bmatrix}\,\textbf{x}(t)+\begin{bmatrix}1\\ 0\\ 0\\ 0\\ \end{bmatrix}\,\textbf{u}(t)
  69. 𝐲 ( t ) = [ n 1 n 2 n 3 n 4 ] 𝐱 ( t ) . \,\textbf{y}(t)=\begin{bmatrix}n_{1}&n_{2}&n_{3}&n_{4}\end{bmatrix}\,\textbf{x% }(t).\,
  70. 𝐱 ˙ ( t ) = [ - d 1 1 0 0 - d 2 0 1 0 - d 3 0 0 1 - d 4 0 0 0 ] 𝐱 ( t ) + [ n 1 n 2 n 3 n 4 ] 𝐮 ( t ) \dot{\,\textbf{x}}(t)=\begin{bmatrix}-d_{1}&1&0&0\\ -d_{2}&0&1&0\\ -d_{3}&0&0&1\\ -d_{4}&0&0&0\end{bmatrix}\,\textbf{x}(t)+\begin{bmatrix}n_{1}\\ n_{2}\\ n_{3}\\ n_{4}\end{bmatrix}\,\textbf{u}(t)
  71. 𝐲 ( t ) = [ 1 0 0 0 ] 𝐱 ( t ) . \,\textbf{y}(t)=\begin{bmatrix}1&0&0&0\end{bmatrix}\,\textbf{x}(t).\,
  72. 𝐆 ( s ) = 𝐆 SP ( s ) + 𝐆 ( ) . \,\textbf{G}(s)=\,\textbf{G}_{\mathrm{SP}}(s)+\,\textbf{G}(\infty).\,
  73. 𝐲 ( t ) = 𝐆 ( ) 𝐮 ( t ) \,\textbf{y}(t)=\,\textbf{G}(\infty)\,\textbf{u}(t)
  74. 𝐆 ( s ) = s 2 + 3 s + 3 s 2 + 2 s + 1 = s + 2 s 2 + 2 s + 1 + 1 \,\textbf{G}(s)=\frac{s^{2}+3s+3}{s^{2}+2s+1}=\frac{s+2}{s^{2}+2s+1}+1
  75. 𝐱 ˙ ( t ) = [ - 2 - 1 1 0 ] 𝐱 ( t ) + [ 1 0 ] 𝐮 ( t ) \dot{\,\textbf{x}}(t)=\begin{bmatrix}-2&-1\\ 1&0\\ \end{bmatrix}\,\textbf{x}(t)+\begin{bmatrix}1\\ 0\end{bmatrix}\,\textbf{u}(t)
  76. 𝐲 ( t ) = [ 1 2 ] 𝐱 ( t ) + [ 1 ] 𝐮 ( t ) \,\textbf{y}(t)=\begin{bmatrix}1&2\end{bmatrix}\,\textbf{x}(t)+\begin{bmatrix}% 1\end{bmatrix}\,\textbf{u}(t)
  77. 𝐆 ( ) \,\textbf{G}(\infty)
  78. 𝐮 ( t ) = K 𝐲 ( t ) \mathbf{u}(t)=K\mathbf{y}(t)
  79. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B 𝐮 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t)
  80. 𝐲 ( t ) = C 𝐱 ( t ) + D 𝐮 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)
  81. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B K 𝐲 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+BK\mathbf{y}(t)
  82. 𝐲 ( t ) = C 𝐱 ( t ) + D K 𝐲 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)+DK\mathbf{y}(t)
  83. 𝐲 ( t ) \mathbf{y}(t)
  84. 𝐱 ˙ ( t ) = ( A + B K ( I - D K ) - 1 C ) 𝐱 ( t ) \dot{\mathbf{x}}(t)=\left(A+BK\left(I-DK\right)^{-1}C\right)\mathbf{x}(t)
  85. 𝐲 ( t ) = ( I - D K ) - 1 C 𝐱 ( t ) \mathbf{y}(t)=\left(I-DK\right)^{-1}C\mathbf{x}(t)
  86. ( A + B K ( I - D K ) - 1 C ) \left(A+BK\left(I-DK\right)^{-1}C\right)
  87. 𝐱 ˙ ( t ) = ( A + B K ) 𝐱 ( t ) \dot{\mathbf{x}}(t)=\left(A+BK\right)\mathbf{x}(t)
  88. 𝐲 ( t ) = 𝐱 ( t ) \mathbf{y}(t)=\mathbf{x}(t)
  89. A + B K A+BK
  90. r ( t ) r(t)
  91. 𝐮 ( t ) = - K 𝐲 ( t ) + 𝐫 ( t ) \mathbf{u}(t)=-K\mathbf{y}(t)+\mathbf{r}(t)
  92. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B 𝐮 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t)
  93. 𝐲 ( t ) = C 𝐱 ( t ) + D 𝐮 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)
  94. 𝐱 ˙ ( t ) = A 𝐱 ( t ) - B K 𝐲 ( t ) + B 𝐫 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)-BK\mathbf{y}(t)+B\mathbf{r}(t)
  95. 𝐲 ( t ) = C 𝐱 ( t ) - D K 𝐲 ( t ) + D 𝐫 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)-DK\mathbf{y}(t)+D\mathbf{r}(t)
  96. 𝐲 ( t ) \mathbf{y}(t)
  97. 𝐱 ˙ ( t ) = ( A - B K ( I + D K ) - 1 C ) 𝐱 ( t ) + B ( I - K ( I + D K ) - 1 D ) 𝐫 ( t ) \dot{\mathbf{x}}(t)=\left(A-BK\left(I+DK\right)^{-1}C\right)\mathbf{x}(t)+B% \left(I-K\left(I+DK\right)^{-1}D\right)\mathbf{r}(t)
  98. 𝐲 ( t ) = ( I + D K ) - 1 C 𝐱 ( t ) + ( I + D K ) - 1 D 𝐫 ( t ) \mathbf{y}(t)=\left(I+DK\right)^{-1}C\mathbf{x}(t)+\left(I+DK\right)^{-1}D% \mathbf{r}(t)
  99. 𝐱 ˙ ( t ) = ( A - B K C ) 𝐱 ( t ) + B 𝐫 ( t ) \dot{\mathbf{x}}(t)=\left(A-BKC\right)\mathbf{x}(t)+B\mathbf{r}(t)
  100. 𝐲 ( t ) = C 𝐱 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)
  101. m y ¨ ( t ) = u ( t ) - b y ˙ ( t ) - k y ( t ) m\ddot{y}(t)=u(t)-b\dot{y}(t)-ky(t)
  102. y ( t ) y(t)
  103. y ˙ ( t ) \dot{y}(t)
  104. y ¨ ( t ) \ddot{y}(t)
  105. u ( t ) u(t)
  106. b b
  107. k k
  108. m m
  109. [ 𝐱 ˙ 𝟏 ( t ) 𝐱 ˙ 𝟐 ( t ) ] = [ 0 1 - k m - b m ] [ 𝐱 𝟏 ( t ) 𝐱 𝟐 ( t ) ] + [ 0 1 m ] 𝐮 ( t ) \left[\begin{matrix}\mathbf{\dot{x}_{1}}(t)\\ \mathbf{\dot{x}_{2}}(t)\end{matrix}\right]=\left[\begin{matrix}0&1\\ -\frac{k}{m}&-\frac{b}{m}\end{matrix}\right]\left[\begin{matrix}\mathbf{x_{1}}% (t)\\ \mathbf{x_{2}}(t)\end{matrix}\right]+\left[\begin{matrix}0\\ \frac{1}{m}\end{matrix}\right]\mathbf{u}(t)
  110. 𝐲 ( t ) = [ 1 0 ] [ 𝐱 𝟏 ( t ) 𝐱 𝟐 ( t ) ] \mathbf{y}(t)=\left[\begin{matrix}1&0\end{matrix}\right]\left[\begin{matrix}% \mathbf{x_{1}}(t)\\ \mathbf{x_{2}}(t)\end{matrix}\right]
  111. x 1 ( t ) x_{1}(t)
  112. x 2 ( t ) = x ˙ 1 ( t ) x_{2}(t)=\dot{x}_{1}(t)
  113. x ˙ 2 ( t ) = x ¨ 1 ( t ) \dot{x}_{2}(t)=\ddot{x}_{1}(t)
  114. 𝐲 ( t ) \mathbf{y}(t)
  115. [ B A B ] = [ [ 0 1 m ] [ 0 1 - k m - b m ] [ 0 1 m ] ] = [ 0 1 m 1 m - b m 2 ] \left[\begin{matrix}B&AB\end{matrix}\right]=\left[\begin{matrix}\left[\begin{% matrix}0\\ \frac{1}{m}\end{matrix}\right]&\left[\begin{matrix}0&1\\ -\frac{k}{m}&-\frac{b}{m}\end{matrix}\right]\left[\begin{matrix}0\\ \frac{1}{m}\end{matrix}\right]\end{matrix}\right]=\left[\begin{matrix}0&\frac{% 1}{m}\\ \frac{1}{m}&-\frac{b}{m^{2}}\end{matrix}\right]
  116. b b
  117. m m
  118. [ C C A ] = [ [ 1 0 ] [ 1 0 ] [ 0 1 - k m - b m ] ] = [ 1 0 0 1 ] \left[\begin{matrix}C\\ CA\end{matrix}\right]=\left[\begin{matrix}\left[\begin{matrix}1&0\end{matrix}% \right]\\ \left[\begin{matrix}1&0\end{matrix}\right]\left[\begin{matrix}0&1\\ -\frac{k}{m}&-\frac{b}{m}\end{matrix}\right]\end{matrix}\right]=\left[\begin{% matrix}1&0\\ 0&1\end{matrix}\right]
  119. 𝐱 ˙ ( t ) = 𝐟 ( t , x ( t ) , u ( t ) ) \mathbf{\dot{x}}(t)=\mathbf{f}(t,x(t),u(t))
  120. 𝐲 ( t ) = 𝐡 ( t , x ( t ) , u ( t ) ) \mathbf{y}(t)=\mathbf{h}(t,x(t),u(t))
  121. f ( , , ) f(\cdot,\cdot,\cdot)
  122. u ( t ) u(t)
  123. m 2 θ ¨ ( t ) = - m g sin θ ( t ) - k θ ˙ ( t ) m\ell^{2}\ddot{\theta}(t)=-m\ell g\sin\theta(t)-k\ell\dot{\theta}(t)
  124. θ ( t ) \theta(t)
  125. m m
  126. g g
  127. k k
  128. \ell
  129. m m
  130. x 1 ˙ ( t ) = x 2 ( t ) \dot{x_{1}}(t)=x_{2}(t)
  131. x 2 ˙ ( t ) = - g sin x 1 ( t ) - k m x 2 ( t ) \dot{x_{2}}(t)=-\frac{g}{\ell}\sin{x_{1}}(t)-\frac{k}{m\ell}{x_{2}}(t)
  132. x 1 ( t ) = θ ( t ) x_{1}(t)=\theta(t)
  133. x 2 ( t ) = x 1 ˙ ( t ) x_{2}(t)=\dot{x_{1}}(t)
  134. x 2 ˙ = x 1 ¨ \dot{x_{2}}=\ddot{x_{1}}
  135. 𝐱 ˙ ( t ) = ( x 1 ˙ ( t ) x 2 ˙ ( t ) ) = 𝐟 ( t , x ( t ) ) = ( x 2 ( t ) - g sin x 1 ( t ) - k m x 2 ( t ) ) . \dot{\mathbf{x}}(t)=\left(\begin{matrix}\dot{x_{1}}(t)\\ \dot{x_{2}}(t)\end{matrix}\right)=\mathbf{f}(t,x(t))=\left(\begin{matrix}x_{2}% (t)\\ -\frac{g}{\ell}\sin{x_{1}}(t)-\frac{k}{m\ell}{x_{2}}(t)\end{matrix}\right).
  136. x ˙ = 0 \dot{x}=0
  137. ( x 1 x 2 ) = ( n π 0 ) \left(\begin{matrix}x_{1}\\ x_{2}\end{matrix}\right)=\left(\begin{matrix}n\pi\\ 0\end{matrix}\right)

State_(functional_analysis).html

  1. g ( ) = d μ g(\cdot)=\int\cdot\;d\mu
  2. μ = μ + - μ - , \mu=\mu_{+}-\mu_{-},\;
  3. τ \tau
  4. H α H_{\alpha}
  5. H H
  6. τ ( H α ) \tau(H_{\alpha})\;
  7. τ ( H ) \tau(H)\;
  8. τ \tau
  9. τ ( A B ) = τ ( B A ) . \tau(AB)=\tau(BA)\;.

State_function.html

  1. D D
  2. D = 2 D=2
  3. P ( t ) P(t)
  4. V ( t ) V(t)
  5. t 0 t_{0}
  6. t 1 t_{1}
  7. t 0 t_{0}
  8. t 1 t_{1}
  9. W ( t 0 , t 1 ) = 𝚜𝚝𝚊𝚝𝚎 0 𝚜𝚝𝚊𝚝𝚎 1 P d V = t 0 t 1 P ( t ) d V ( t ) d t d t . W(t_{0},t_{1})=\int_{\mathtt{state}_{0}}^{\mathtt{state}_{1}}P\,dV=\int_{t_{0}% }^{t_{1}}P(t)\frac{dV(t)}{dt}\,dt.
  10. P ( t ) P(t)
  11. V ( t ) V(t)
  12. t t
  13. V d P VdP
  14. Φ ( t 0 , t 1 ) = t 0 t 1 P d V d t d t + t 0 t 1 V d P d t d t = t 0 t 1 d ( P V ) d t d t = P ( t 1 ) V ( t 1 ) - P ( t 0 ) V ( t 0 ) . \Phi(t_{0},t_{1})=\int_{t_{0}}^{t_{1}}P\frac{dV}{dt}\,dt+\int_{t_{0}}^{t_{1}}V% \frac{dP}{dt}\,dt=\int_{t_{0}}^{t_{1}}\frac{d(PV)}{dt}\,dt=P(t_{1})V(t_{1})-P(% t_{0})V(t_{0}).
  15. P ( t ) V ( t ) P(t)V(t)
  16. P ( t ) V ( t ) P(t)V(t)
  17. P V PV
  18. d Φ d\Phi
  19. Φ ( t 1 ) - Φ ( t 0 ) \Phi(t_{1})-\Phi(t_{0})
  20. δ W = P d V \delta W=PdV
  21. P V PV
  22. W W

State_space.html

  1. ( 64 32 ) {\textstyle\left({{64}\atop{32}}\right)}

State_space_search.html

  1. S : < S , A , A c t i o n ( s ) , R e s u l t ( s , a ) , C o s t ( s , a ) Align g t ; S:<S,A,Action(s),Result(s,a),Cost(s,a)&gt;
  2. S S
  3. A A
  4. A c t i o n ( s ) Action(s)
  5. R e s u l t ( s , a ) Result(s,a)
  6. a a
  7. s s
  8. C o s t ( s , a ) Cost(s,a)
  9. a a
  10. s s

Static_single_assignment_form.html

  1. \leftarrow

Stationary_process.html

  1. { X t } \left\{X_{t}\right\}
  2. F X ( x t 1 + τ , , x t k + τ ) F_{X}(x_{t_{1}+\tau},\ldots,x_{t_{k}+\tau})
  3. { X t } \left\{X_{t}\right\}
  4. t 1 + τ , , t k + τ t_{1}+\tau,\ldots,t_{k}+\tau
  5. { X t } \left\{X_{t}\right\}
  6. k k
  7. τ \tau
  8. t 1 , , t k t_{1},\ldots,t_{k}
  9. F X ( x t 1 + τ , , x t k + τ ) = F X ( x t 1 , , x t k ) . F_{X}(x_{t_{1}+\tau},\ldots,x_{t_{k}+\tau})=F_{X}(x_{t_{1}},\ldots,x_{t_{k}}).
  10. τ \tau
  11. F X ( ) F_{X}(\cdot)
  12. F X F_{X}
  13. X t = Y for all t . X_{t}=Y\qquad\,\text{ for all }t.
  14. X t = cos ( t + Y ) for t . X_{t}=\cos(t+Y)\quad\,\text{ for }t\in\mathbb{R}.
  15. 𝔼 [ x ( t ) ] = m x ( t ) = m x ( t + τ ) for all τ \mathbb{E}[x(t)]=m_{x}(t)=m_{x}(t+\tau)\,\,\,\text{ for all }\,\tau\in\mathbb{R}
  16. 𝔼 [ ( x ( t 1 ) - m x ( t 1 ) ) ( x ( t 2 ) - m x ( t 2 ) ) ] = C x ( t 1 , t 2 ) = C x ( t 1 + ( - t 2 ) , t 2 + ( - t 2 ) ) = C x ( t 1 - t 2 , 0 ) . \mathbb{E}[(x(t_{1})-m_{x}(t_{1}))(x(t_{2})-m_{x}(t_{2}))]=C_{x}(t_{1},t_{2})=% C_{x}(t_{1}+(-t_{2}),t_{2}+(-t_{2}))=C_{x}(t_{1}-t_{2},0).
  17. t 1 t_{1}
  18. t 2 t_{2}
  19. C x ( t 1 - t 2 , 0 ) \,\!C_{x}(t_{1}-t_{2},0)\,
  20. C x ( τ ) where τ = t 1 - t 2 . C_{x}(\tau)\,\!\mbox{ where }~{}\tau=t_{1}-t_{2}.
  21. τ = t 1 - t 2 \tau=t_{1}-t_{2}
  22. R x ( t 1 , t 2 ) = R x ( t 1 - t 2 ) . \,\!R_{x}(t_{1},t_{2})=R_{x}(t_{1}-t_{2}).
  23. x ( t ) = e - 2 π i λ t d ω λ , x(t)=\int e^{-2\pi i\lambda\cdot t}d\omega_{\lambda},

Steiner_tree_problem.html

  1. R R
  2. G m G_{m}
  3. G G
  4. G m G_{m}
  5. G G
  6. G m G_{m}
  7. | V | * ( | V | - 1 ) / 2 = O ( | V | 2 ) |V|*(|V|-1)/2=O(|V|^{2})
  8. G M G_{M}
  9. O p t G M Opt_{G_{M}}
  10. O p t G Opt_{G}
  11. G G
  12. 2 - 2 / | R | 2-2/|R|
  13. | R | |R|
  14. ln ( 4 ) + ϵ 1.39 \ln(4)+\epsilon\leq 1.39
  15. 2 3 \frac{2}{\sqrt{3}}
  16. 3 2 \frac{3}{2}

Step_response.html

  1. A F B = A O L 1 + β A O L , A_{FB}=\frac{A_{OL}}{1+\beta A_{OL}},
  2. A O L = A 0 1 + j ω τ , A_{OL}=\frac{A_{0}}{1+j\omega\tau},
  3. S O L ( t ) = A 0 ( 1 - e - t / τ ) S_{OL}(t)=A_{0}(1-e^{-t/\tau})
  4. A F B = A 0 1 + β A 0 A_{FB}=\frac{A_{0}}{1+\beta A_{0}}
  5. 1 1 + j ω τ 1 + β A 0 . \ \frac{1}{1+j\omega\frac{\tau}{1+\beta A_{0}}}.
  6. S F B ( t ) = A 0 1 + β A 0 ( 1 - e - t ( 1 + β A 0 ) / τ ) S_{FB}(t)=\frac{A_{0}}{1+\beta A_{0}}(1-e^{-t(1+\beta A_{0})/\tau})
  7. A O L = A 0 ( 1 + j ω τ 1 ) ( 1 + j ω τ 2 ) , A_{OL}=\frac{A_{0}}{(1+j\omega\tau_{1})(1+j\omega\tau_{2})},
  8. A F B = A 0 1 + β A 0 A_{FB}=\frac{A_{0}}{1+\beta A_{0}}
  9. 1 1 + j ω τ 1 + τ 2 1 + β A 0 + ( j ω ) 2 τ 1 τ 2 1 + β A 0 . \ \frac{1}{1+j\omega\frac{\tau_{1}+\tau_{2}}{1+\beta A_{0}}+(j\omega)^{2}\frac% {\tau_{1}\tau_{2}}{1+\beta A_{0}}}.
  10. A F B = A 0 τ 1 τ 2 A_{FB}=\frac{A_{0}}{\tau_{1}\tau_{2}}
  11. 1 s 2 + s ( 1 τ 1 + 1 τ 2 ) + 1 + β A 0 τ 1 τ 2 \frac{1}{s^{2}+s\left(\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}\right)+\frac{1+% \beta A_{0}}{\tau_{1}\tau_{2}}}
  12. 2 s = - ( 1 τ 1 + 1 τ 2 ) 2s=-\left(\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}\right)
  13. ± ( 1 τ 1 - 1 τ 2 ) 2 - 4 β A 0 τ 1 τ 2 , \pm\sqrt{\left(\frac{1}{\tau_{1}}-\frac{1}{\tau_{2}}\right)^{2}-\frac{4\beta A% _{0}}{\tau_{1}\tau_{2}}},
  14. s ± = - ρ ± j μ , s_{\pm}=-\rho\pm j\mu,\,
  15. ρ = 1 2 ( 1 τ 1 + 1 τ 2 ) , \rho=\frac{1}{2}\left(\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}\right),
  16. μ = 1 2 4 β A 0 τ 1 τ 2 - ( 1 τ 1 - 1 τ 2 ) 2 . \mu=\frac{1}{2}\sqrt{\frac{4\beta A_{0}}{\tau_{1}\tau_{2}}-\left(\frac{1}{\tau% _{1}}-\frac{1}{\tau_{2}}\right)^{2}}.
  17. | s | = | s ± | = ρ 2 + μ 2 , |s|=|s_{\pm}|=\sqrt{\rho^{2}+\mu^{2}},
  18. cos ϕ = ρ | s | , sin ϕ = μ | s | . \cos\phi=\frac{\rho}{|s|},\sin\phi=\frac{\mu}{|s|}.
  19. e - ρ t sin ( μ t ) and e^{-\rho t}\sin(\mu t)\quad\,\text{and}\quad
  20. e - ρ t cos ( μ t ) , e^{-\rho t}\cos(\mu t),
  21. S ( t ) = ( A 0 1 + β A 0 ) ( 1 - e - ρ t sin ( μ t + ϕ ) sin ( ϕ ) ) , S(t)=\left(\frac{A_{0}}{1+\beta A_{0}}\right)\left(1-e^{-\rho t}\ \frac{\sin% \left(\mu t+\phi\right)}{\sin(\phi)}\right)\ ,
  22. S ( t ) = 1 - e - ρ t sin ( μ t + ϕ ) sin ( ϕ ) S(t)=1-e^{-\rho t}\ \frac{\sin\left(\mu t+\phi\right)}{\sin(\phi)}
  23. S max = 1 + exp ( - π ρ μ ) . S_{\max}=1+\exp\left(-\pi\frac{\rho}{\mu}\right).
  24. 4 β A 0 τ 1 τ 2 = ( 1 τ 1 - 1 τ 2 ) 2 . \frac{4\beta A_{0}}{\tau_{1}\tau_{2}}=\left(\frac{1}{\tau_{1}}-\frac{1}{\tau_{% 2}}\right)^{2}.
  25. x = β A 0 + β A 0 + 1 . x=\sqrt{\beta A_{0}}+\sqrt{\beta A_{0}+1}.\,
  26. τ 1 τ 2 = 4 β A 0 . \frac{\tau_{1}}{\tau_{2}}=4\beta A_{0}.
  27. τ 1 τ 2 = α β A 0 , \frac{\tau_{1}}{\tau_{2}}=\alpha\beta A_{0},
  28. S ( t ) 1 + Δ , S(t)\leq 1+\Delta,\,
  29. Δ = e - ρ t S or t S = ln ( 1 Δ ) ρ = τ 2 2 ln ( 1 Δ ) 1 + τ 2 τ 1 2 τ 2 ln ( 1 Δ ) , \Delta=e^{-\rho t_{S}}\,\text{ or }t_{S}=\frac{\ln\left(\frac{1}{\Delta}\right% )}{\rho}=\tau_{2}\frac{2\ln\left(\frac{1}{\Delta}\right)}{1+\frac{\tau_{2}}{% \tau_{1}}}\approx 2\tau_{2}\ln\left(\frac{1}{\Delta}\right),
  30. | β A OL ( f 0 db ) | = 1. |\beta A\text{OL}(f\text{0 db})|=1.
  31. f 0 dB = β A 0 f 1 . f\text{0 dB}=\beta A_{0}f_{1}.\,
  32. ϕ m = 180 - arctan ( f 0 dB / f 1 ) - arctan ( f 0 dB / f 2 ) . \phi_{m}=180^{\circ}-\arctan(f\text{0 dB}/f_{1})-\arctan(f\text{0 dB}/f_{2}).
  33. ϕ m = 90 - arctan ( f 0 dB / f 2 ) \phi_{m}=90^{\circ}-\arctan(f\text{0 dB}/f_{2})\,
  34. = 90 - arctan ( β A 0 f 1 α β A 0 f 1 ) =90^{\circ}-\arctan\left(\frac{\beta A_{0}f_{1}}{\alpha\beta A_{0}f_{1}}\right)
  35. = 90 - arctan ( 1 α ) = arctan ( α ) . =90^{\circ}-\arctan\left(\frac{1}{\alpha}\right)=\arctan\left(\alpha\right).
  36. 𝔖 \scriptstyle\mathfrak{S}
  37. t T \scriptstyle t\in T
  38. \scriptstylesymbol x | t M \scriptstylesymbol{x}|_{t}\in M
  39. t t\,
  40. Φ : T × M M \scriptstyle\Phi:T\times M\longrightarrow M
  41. Φ ( 0 , s y m b o l x ) = s y m b o l x 0 M \scriptstyle\Phi(0,symbol{x})=symbol{x}_{0}\in M
  42. H ( t ) \scriptstyle H(t)\,
  43. s y m b o l x | t = Φ { H ( t ) } ( t , s y m b o l x 0 ) . symbol{x}|_{t}=\Phi_{\{H(t)\}\left(t,{symbol{x}_{0}}\right)}.\,
  44. < m t p l > S S \scriptstyle\mathfrak{<}mtpl>{{S}}\ \equiv\ S
  45. a ( t ) = h * H ( t ) = H * h ( t ) = - + h ( τ ) H ( t - τ ) d τ = - t h ( τ ) d τ . a(t)={h*H}(t)={H*h}(t)=\int_{-\infty}^{+\infty}h(\tau)H(t-\tau)\,d\tau=\int_{-% \infty}^{t}h(\tau)\,d\tau.

Sterilization_(microbiology).html

  1. N 0 N_{0}
  2. 10 - 1 10^{-1}
  3. N N
  4. t t
  5. N N 0 = 10 ( - t D ) \frac{N}{N_{0}}=10^{\left(-\frac{t}{D}\right)}

Stiffness.html

  1. k = F δ k=\frac{F}{\delta}
  2. k = M θ k=\frac{M}{\theta}
  3. k = A E L k=\frac{AE}{L}
  4. k = G J L k=\frac{GJ}{L}
  5. k : N m r a d k:\frac{N\cdot m}{rad}

Stochastic_programming.html

  1. min x X { g ( x ) = f ( x ) + E [ Q ( x , ξ ) ] } \min_{x\in X}\{g(x)=f(x)+E[Q(x,\xi)]\}
  2. Q ( x , ξ ) Q(x,\xi)
  3. min y { q ( y , ξ ) | T ( ξ ) x + W ( ξ ) y = h ( ξ ) } \min_{y}\{q(y,\xi)|T(\xi)x+W(\xi)y=h(\xi)\}
  4. min x n g ( x ) = c T x + E [ Q ( x , ξ ) ] subject to A x = b x 0 \begin{array}[]{llr}\min\limits_{x\in\mathbb{R}^{n}}&g(x)=c^{T}x+E[Q(x,\xi)]&% \\ \,\text{subject to}&Ax=b&\\ &x\geq 0&\end{array}
  5. Q ( x , ξ ) Q(x,\xi)
  6. min y m q ( ξ ) T y subject to T ( ξ ) x + W ( ξ ) y = h ( ξ ) y 0 \begin{array}[]{llr}\min\limits_{y\in\mathbb{R}^{m}}&q(\xi)^{T}y&\\ \,\text{subject to}&T(\xi)x+W(\xi)y=h(\xi)&\\ &y\geq 0&\end{array}
  7. x n x\in\mathbb{R}^{n}
  8. y m y\in\mathbb{R}^{m}
  9. ξ ( q , T , W , h ) \xi(q,T,W,h)
  10. x x
  11. ξ \xi
  12. ξ \xi
  13. c T x c^{T}x
  14. W y Wy
  15. T x h Tx\leq h
  16. q T y q^{T}y
  17. ξ \xi
  18. ξ \xi
  19. ξ \xi
  20. ξ \xi
  21. ξ 1 , , ξ K \xi_{1},\dots,\xi_{K}
  22. p 1 , , p K p_{1},\dots,p_{K}
  23. E [ Q ( x , ξ ) ] = k = 1 K p k Q ( x , ξ k ) E[Q(x,\xi)]=\sum\limits_{k=1}^{K}p_{k}Q(x,\xi_{k})
  24. ξ \xi
  25. k t h k^{th}
  26. k t h k^{th}
  27. Minimize f T x + g T y + h k T z k subject to T x + U y = r V k y + W k z k = s k x , y , z k 0 \begin{array}[]{lccccccc}\,\text{Minimize}&f^{T}x&+&g^{T}y&+&h_{k}^{T}z_{k}&&% \\ \,\text{subject to}&Tx&+&Uy&&&=&r\\ &&&V_{k}y&+&W_{k}z_{k}&=&s_{k}\\ &x&,&y&,&z_{k}&\geq&0\end{array}
  28. x x
  29. y y
  30. z k z_{k}
  31. T x + U y = r Tx+Uy=r
  32. k t h k^{th}
  33. k t h k^{th}
  34. p k p_{k}
  35. k = 1 , , K k=1,\dots,K
  36. Minimize f T x + g T y + p 1 h 1 T z 1 + p 2 h 2 T z 2 + + p K h K T z K subject to T x + U y = r V 1 y + W 1 z 1 = s 1 V 2 y + W 2 z 2 = s 2 V K y + W K z K = s K x , y , z 1 , z 2 , , z K 0 \begin{array}[]{lccccccccccccc}\,\text{Minimize}&f^{T}x&+&g^{T}y&+&p_{1}h_{1}^% {T}z_{1}&+&p_{2}h_{2}^{T}z_{2}&+&\cdots&+&p_{K}h_{K}^{T}z_{K}&&\\ \,\text{subject to}&Tx&+&Uy&&&&&&&&&=&r\\ &&&V_{1}y&+&W_{1}z_{1}&&&&&&&=&s_{1}\\ &&&V_{2}y&&&+&W_{2}z_{2}&&&&&=&s_{2}\\ &&&\vdots&&&&&&\ddots&&&&\vdots\\ &&&V_{K}y&&&&&&&+&W_{K}z_{K}&=&s_{K}\\ &x&,&y&,&z_{1}&,&z_{2}&,&\ldots&,&z_{K}&\geq&0\\ \end{array}
  37. z k z_{k}
  38. k k
  39. x x
  40. y y
  41. x x
  42. y y
  43. x * x^{*}
  44. ξ \xi
  45. d d
  46. K = 3 d K=3^{d}
  47. d d
  48. ξ \xi
  49. ξ 1 , ξ 2 , , ξ N \xi^{1},\xi^{2},\dots,\xi^{N}
  50. N N
  51. ξ \xi
  52. q ( x ) = E [ Q ( x , ξ ) ] q(x)=E[Q(x,\xi)]
  53. q ^ N ( x ) = 1 N j = 1 N Q ( x , ξ j ) \hat{q}_{N}(x)=\frac{1}{N}\sum_{j=1}^{N}Q(x,\xi^{j})
  54. g ^ N ( x ) = min x n c T x + 1 N j = 1 N Q ( x , ξ j ) subject to A x = b x 0 \begin{array}[]{rlrrr}\hat{g}_{N}(x)=&\min\limits_{x\in\mathbb{R}^{n}}&c^{T}x+% \frac{1}{N}\sum_{j=1}^{N}Q(x,\xi^{j})&\\ &\,\text{subject to}&Ax&=&b\\ &&x&\geq&0\end{array}
  55. ξ 1 , ξ 2 , , ξ N \xi^{1},\xi^{2},\dots,\xi^{N}
  56. ξ j \xi^{j}
  57. j = 1 , , N j=1,\dots,N
  58. p j = 1 N p_{j}=\frac{1}{N}
  59. min x X { g ( x ) = f ( x ) + E [ Q ( x , ξ ) ] } \min\limits_{x\in X}\{g(x)=f(x)+E[Q(x,\xi)]\}
  60. X X
  61. n \mathbb{R}^{n}
  62. ξ \xi
  63. P P
  64. Ξ d \Xi\subset\mathbb{R}^{d}
  65. Q : X × Ξ Q:X\times\Xi\rightarrow\mathbb{R}
  66. Q ( x , ξ ) Q(x,\xi)
  67. g ( x ) g(x)
  68. x X x\in X
  69. x X x\in X
  70. Q ( x , ξ ) Q(x,\xi)
  71. ξ 1 , , ξ N \xi^{1},\dots,\xi^{N}
  72. N N
  73. ξ \xi
  74. N N
  75. ξ \xi
  76. min x X { g ^ N ( x ) = f ( x ) + 1 N j = 1 N Q ( x , ξ j ) } \min\limits_{x\in X}\{\hat{g}_{N}(x)=f(x)+\frac{1}{N}\sum_{j=1}^{N}Q(x,\xi^{j})\}
  77. 1 N j = 1 N Q ( x , ξ j ) \frac{1}{N}\sum_{j=1}^{N}Q(x,\xi^{j})
  78. E [ Q ( x , ξ ) ] E[Q(x,\xi)]
  79. N N\rightarrow\infty
  80. E [ g ^ N ( x ) ] = g ( x ) E[\hat{g}_{N}(x)]=g(x)
  81. g ^ N ( x ) \hat{g}_{N}(x)
  82. g ( x ) g(x)
  83. N N\rightarrow\infty
  84. X X
  85. ϑ * \vartheta^{*}
  86. S * S^{*}
  87. ϑ ^ N \hat{\vartheta}_{N}
  88. S ^ N \hat{S}_{N}
  89. g : X g:X\rightarrow\mathbb{R}
  90. g ^ N : X \hat{g}_{N}:X\rightarrow\mathbb{R}
  91. x ¯ X \overline{x}\in X
  92. { x N } X \{x_{N}\}\subset X
  93. x ¯ \overline{x}
  94. g ^ N ( x N ) \hat{g}_{N}(x_{N})
  95. g ( x ¯ ) g(\overline{x})
  96. f ( ) f(\cdot)
  97. X X
  98. g ^ N ( ) \hat{g}_{N}(\cdot)
  99. g ( ) g(\cdot)
  100. X X
  101. g ^ N ( x ) \hat{g}_{N}(x)
  102. g ( x ) g(x)
  103. N N\rightarrow\infty
  104. X X
  105. ϑ ^ N \hat{\vartheta}_{N}
  106. ϑ * \vartheta^{*}
  107. N N\rightarrow\infty
  108. C n C\subset\mathbb{R}^{n}
  109. S S
  110. C C
  111. g ( x ) g(x)
  112. C C
  113. g ^ N ( x ) \hat{g}_{N}(x)
  114. g ( x ) g(x)
  115. N N\rightarrow\infty
  116. x C x\in C
  117. N N
  118. S ^ N \hat{S}_{N}
  119. S ^ N C \hat{S}_{N}\subset C
  120. ϑ ^ N ϑ * \hat{\vartheta}_{N}\rightarrow\vartheta^{*}
  121. 𝔻 ( S * , S ^ N ) 0 \mathbb{D}(S^{*},\hat{S}_{N})\rightarrow 0
  122. N N\rightarrow\infty
  123. 𝔻 ( A , B ) \mathbb{D}(A,B)
  124. A A
  125. B B
  126. 𝔻 ( A , B ) := sup x A { inf x B x - x } \mathbb{D}(A,B):=\sup_{x\in A}\{\inf_{x^{\prime}\in B}\|x-x^{\prime}\|\}
  127. X X
  128. min x X N g ^ N ( x ) \min_{x\in X_{N}}\hat{g}_{N}(x)
  129. X N X_{N}
  130. n \mathbb{R}^{n}
  131. C n C\subset\mathbb{R}^{n}
  132. S S
  133. C C
  134. g ( x ) g(x)
  135. C C
  136. g ^ N ( x ) \hat{g}_{N}(x)
  137. g ( x ) g(x)
  138. N N\rightarrow\infty
  139. x C x\in C
  140. N N
  141. S ^ N \hat{S}_{N}
  142. S ^ N C \hat{S}_{N}\subset C
  143. x N X N x_{N}\in X_{N}
  144. x N x_{N}
  145. x x
  146. x X x\in X
  147. x S * x\in S^{*}
  148. x N X N x_{N}\in X_{N}
  149. x N x x_{N}\rightarrow x
  150. ϑ ^ N ϑ * \hat{\vartheta}_{N}\rightarrow\vartheta^{*}
  151. 𝔻 ( S * , S ^ N ) 0 \mathbb{D}(S^{*},\hat{S}_{N})\rightarrow 0
  152. N N\rightarrow\infty
  153. ξ 1 , , ξ N \xi^{1},\dots,\xi^{N}
  154. x X x\in X
  155. g ^ N ( x ) \hat{g}_{N}(x)
  156. g ( x ) g(x)
  157. 1 N σ 2 ( x ) \frac{1}{N}\sigma^{2}(x)
  158. σ 2 ( x ) := V a r [ Q ( x , ξ ) ] \sigma^{2}(x):=Var[Q(x,\xi)]
  159. N [ g ^ N - g ( x ) ] 𝒟 Y x \sqrt{N}[\hat{g}_{N}-g(x)]\xrightarrow{\mathcal{D}}Y_{x}
  160. 𝒟 \xrightarrow{\mathcal{D}}
  161. Y x Y_{x}
  162. 0
  163. σ 2 ( x ) \sigma^{2}(x)
  164. 𝒩 ( 0 , σ 2 ( 0 ) ) \mathcal{N}(0,\sigma^{2}(0))
  165. g ^ N ( x ) \hat{g}_{N}(x)
  166. N N
  167. g ^ N ( x ) \hat{g}_{N}(x)
  168. g ( x ) g(x)
  169. 1 N σ 2 ( x ) \frac{1}{N}\sigma^{2}(x)
  170. 100 ( 1 - α ) 100(1-\alpha)
  171. f ( x ) f(x)
  172. [ g ^ N ( x ) - z α / 2 σ ^ ( x ) N , g ^ N ( x ) + z α / 2 σ ^ ( x ) N ] \left[\hat{g}_{N}(x)-z_{\alpha/2}\frac{\hat{\sigma}(x)}{\sqrt{N}},\hat{g}_{N}(% x)+z_{\alpha/2}\frac{\hat{\sigma}(x)}{\sqrt{N}}\right]
  173. z α / 2 := Φ - 1 ( 1 - α / 2 ) z_{\alpha/2}:=\Phi^{-1}(1-\alpha/2)
  174. Φ ( ) \Phi(\cdot)
  175. σ ^ 2 ( x ) := 1 N - 1 j = 1 N [ Q ( x , ξ j ) - 1 N j = 1 N Q ( x , ξ j ) ] 2 \hat{\sigma}^{2}(x):=\frac{1}{N-1}\sum_{j=1}^{N}\left[Q(x,\xi^{j})-\frac{1}{N}% \sum_{j=1}^{N}Q(x,\xi^{j})\right]^{2}
  176. σ 2 ( x ) \sigma^{2}(x)
  177. g ( x ) g(x)
  178. O ( N ) O(\sqrt{N})
  179. t = 0 t=0
  180. W 0 W_{0}
  181. n n
  182. t = 1 , , T - 1 t=1,\dots,T-1
  183. t t
  184. W t W_{t}
  185. n n
  186. x 0 = ( x 10 , , x n 0 ) x_{0}=(x_{10},\dots,x_{n0})
  187. x i 0 x_{i0}
  188. i = 1 n x i 0 = W 0 \sum_{i=1}^{n}x_{i0}=W_{0}
  189. ξ t = ( ξ 1 t , , ξ n t ) \xi_{t}=(\xi_{1t},\dots,\xi_{nt})
  190. t = 1 , , T t=1,\dots,T
  191. ξ 1 , , ξ T \xi_{1},\dots,\xi_{T}
  192. t = 1 t=1
  193. x 1 = ( x 11 , , x n 1 ) x_{1}=(x_{11},\dots,x_{n1})
  194. t = 1 t=1
  195. ξ 1 \xi_{1}
  196. x 1 = x 1 ( ξ 1 ) x_{1}=x_{1}(\xi_{1})
  197. t t
  198. x t = ( x 1 t , , x n t ) x_{t}=(x_{1t},\dots,x_{nt})
  199. x t = x t ( ξ [ t ] ) x_{t}=x_{t}(\xi_{[t]})
  200. ξ [ t ] = ( ξ 1 , , ξ t ) \xi_{[t]}=(\xi_{1},\dots,\xi_{t})
  201. t t
  202. x t = x t ( ξ [ t ] ) x_{t}=x_{t}(\xi_{[t]})
  203. t = 0 , , T - 1 t=0,\dots,T-1
  204. x 0 x_{0}
  205. x i t ( ξ [ t ] ) 0 x_{it}(\xi_{[t]})\geq 0
  206. i = 1 , , n i=1,\dots,n
  207. t = 0 , , T - 1 t=0,\dots,T-1
  208. i = 1 n x i t ( ξ [ t ] ) = W t , \sum_{i=1}^{n}x_{it}(\xi_{[t]})=W_{t},
  209. t = 1 , , T t=1,\dots,T
  210. W t W_{t}
  211. W t = i = 1 n ξ i t x i , t - 1 ( ξ [ t - 1 ] ) , W_{t}=\sum_{i=1}^{n}\xi_{it}x_{i,t-1}(\xi_{[t-1]}),
  212. t t
  213. max E [ U ( W T ) ] . \max E[U(W_{T})].
  214. t = 0 t=0
  215. t = T - 1 t=T-1
  216. ξ 1 , , ξ T \xi_{1},\dots,\xi_{T}
  217. 2 n T 2^{nT}
  218. t = T - 1 t=T-1
  219. ξ [ T - 1 ] = ( ξ 1 , , ξ T - 1 ) \xi_{[T-1]}=(\xi_{1},\dots,\xi_{T-1})
  220. x T - 2 x_{T-2}
  221. max x T - 1 E [ U ( W T ) | ξ [ T - 1 ] ] subject to W T = i = 1 n ξ i T x i , T - 1 i = 1 n x i , T - 1 = W T - 1 x T - 1 0 \begin{array}[]{lrclr}\max\limits_{x_{T-1}}&E[U(W_{T})|\xi_{[T-1]}]&\\ \,\text{subject to}&W_{T}&=&\sum_{i=1}^{n}\xi_{iT}x_{i,T-1}\\ &\sum_{i=1}^{n}x_{i,T-1}&=&W_{T-1}\\ &x_{T-1}&\geq&0\end{array}
  222. E [ U ( W T ) | ξ [ T - 1 ] ] E[U(W_{T})|\xi_{[T-1]}]
  223. U ( W T ) U(W_{T})
  224. ξ [ T - 1 ] \xi_{[T-1]}
  225. W T - 1 W_{T-1}
  226. ξ [ T - 1 ] \xi_{[T-1]}
  227. Q T - 1 ( W T - 1 , ξ [ T - 1 ] ) Q_{T-1}(W_{T-1},\xi_{[T-1]})
  228. t = T - 2 , , 1 t=T-2,\dots,1
  229. max x t E [ Q t + 1 ( W t + 1 , ξ [ t + 1 ] ) | ξ [ t ] ] subject to W t + 1 = i = 1 n ξ i , t + 1 x i , t i = 1 n x i , t = W t x t 0 \begin{array}[]{lrclr}\max\limits_{x_{t}}&E[Q_{t+1}(W_{t+1},\xi_{[t+1]})|\xi_{% [t]}]&\\ \,\text{subject to}&W_{t+1}&=&\sum_{i=1}^{n}\xi_{i,t+1}x_{i,t}\\ &\sum_{i=1}^{n}x_{i,t}&=&W_{t}\\ &x_{t}&\geq&0\end{array}
  230. Q t ( W t , ξ [ t ] ) Q_{t}(W_{t},\xi_{[t]})
  231. t = 0 t=0
  232. max x 0 E [ Q 1 ( W 1 , ξ [ 1 ] ) ] subject to W 1 = i = 1 n ξ i , 1 x i 0 i = 1 n x i 0 = W 0 x 0 0 \begin{array}[]{lrclr}\max\limits_{x_{0}}&E[Q_{1}(W_{1},\xi_{[1]})]&\\ \,\text{subject to}&W_{1}&=&\sum_{i=1}^{n}\xi_{i,1}x_{i0}\\ &\sum_{i=1}^{n}x_{i0}&=&W_{0}\\ &x_{0}&\geq&0\end{array}
  233. ξ t \xi_{t}
  234. ξ t \xi_{t}
  235. ξ t \xi_{t}
  236. ξ 1 , , ξ t - 1 \xi_{1},\dots,\xi_{t-1}
  237. t = 2 , , T t=2,\dots,T
  238. Q t ( W t ) Q_{t}(W_{t})
  239. t = 1 , , T - 1 t=1,\dots,T-1
  240. ξ [ t ] \xi_{[t]}
  241. Q T - 1 ( W T - 1 ) Q_{T-1}(W_{T-1})
  242. max x T - 1 E [ U ( W T ) ] subject to W T = i = 1 n ξ i T x i , T - 1 i = 1 n x i , T - 1 = W T - 1 x T - 1 0 \begin{array}[]{lrclr}\max\limits_{x_{T-1}}&E[U(W_{T})]&\\ \,\text{subject to}&W_{T}&=&\sum_{i=1}^{n}\xi_{iT}x_{i,T-1}\\ &\sum_{i=1}^{n}x_{i,T-1}&=&W_{T-1}\\ &x_{T-1}&\geq&0\end{array}
  243. Q t ( W t ) Q_{t}(W_{t})
  244. max x t E [ Q t + 1 ( W t + 1 ) ] subject to W t + 1 = i = 1 n ξ i , t + 1 x i , t i = 1 n x i , t = W t x t 0 \begin{array}[]{lrclr}\max\limits_{x_{t}}&E[Q_{t+1}(W_{t+1})]&\\ \,\text{subject to}&W_{t+1}&=&\sum_{i=1}^{n}\xi_{i,t+1}x_{i,t}\\ &\sum_{i=1}^{n}x_{i,t}&=&W_{t}\\ &x_{t}&\geq&0\end{array}
  245. t = T - 2 , , 1 t=T-2,\dots,1

Stochastic_tunneling.html

  1. Δ E \Delta E
  2. min ( 1 ; exp ( - β Δ E ) ) \min\left(1;\exp\left(-\beta\cdot\Delta E\right)\right)
  3. β \beta
  4. f S T U N := 1 - exp ( - γ ( E ( x ) - E o ) ) f_{STUN}:=1-\exp\left(-\gamma\cdot\left(E(x)-E_{o}\right)\right)
  5. E o E_{o}
  6. f S T U N f_{STUN}
  7. E E
  8. min ( 1 ; exp ( - β Δ f S T U N ) ) \min\left(1;\exp\left(-\beta\cdot\Delta f_{STUN}\right)\right)
  9. γ \gamma

Stone's_representation_theorem_for_Boolean_algebras.html

  1. { x S ( B ) b x } , \{x\in S(B)\mid b\in x\},

Stone_duality.html

  1. x S = { x s : s S } , x\wedge\bigvee S=\bigvee\{\,x\wedge s:s\in S\,\},

Stoneham_number.html

  1. α b , c = n = c k > 1 1 b n n = k = 1 1 b c k c k \alpha_{b,c}=\sum_{n=c^{k}>1}\frac{1}{b^{n}n}=\sum_{k=1}^{\infty}\frac{1}{b^{c% ^{k}}c^{k}}

Strain_gauge.html

  1. G F GF
  2. G F = Δ R / R G ϵ GF=\frac{\Delta R/R_{G}}{\epsilon}
  3. Δ R \Delta R
  4. R G R_{G}
  5. ϵ \epsilon
  6. v v
  7. v = B V G F ϵ 4 v=\frac{BV\cdot GF\cdot\epsilon}{4}
  8. B V BV

Strangeness.html

  1. S = - ( n s - n s ¯ ) S=-(n_{s}-n_{\bar{s}})

Strategy_dynamics.html

  1. \bigtriangleup
  2. \bigtriangleup

Stratification_(mathematics).html

  1. Q 1 Q n ¬ Q n + 1 ¬ Q n + m P Q_{1}\wedge\dots\wedge Q_{n}\wedge\neg Q_{n+1}\wedge\dots\wedge\neg Q_{n+m}\rightarrow P
  2. S ( P ) S ( Q ) S(P)\geq S(Q)
  3. S ( P ) > S ( Q ) S(P)>S(Q)
  4. ϕ \phi
  5. σ \sigma
  6. ϕ \phi
  7. x y x\in y
  8. ϕ \phi
  9. σ ( x ) + 1 = σ ( y ) \sigma(x)+1=\sigma(y)
  10. x = y x=y
  11. ϕ \phi
  12. σ ( x ) = σ ( y ) \sigma(x)=\sigma(y)
  13. { x ϕ } \{x\mid\phi\}
  14. σ \sigma
  15. σ \sigma
  16. ( ι x . ϕ ) (\iota x.\phi)
  17. ϕ \phi
  18. σ \sigma

Stress–strain_analysis.html

  1. maximum allowable stress = ultimate tensile strength factor of safety \,\text{maximum allowable stress}=\frac{\,\text{ultimate tensile strength}}{\,% \text{factor of safety}}
  2. σ e = P A o \sigma_{\mathrm{e}}=\tfrac{P}{A_{o}}
  3. σ true = ( 1 + ε e ) ( σ e ) \sigma_{\mathrm{true}}=(1+\varepsilon_{\mathrm{e}})(\sigma_{\mathrm{e}})\,\!
  4. ε e \varepsilon_{\mathrm{e}}\,\!
  5. σ e \sigma_{\mathrm{e}}\,\!
  6. ε true = ln ( 1 + ε e ) \varepsilon_{\mathrm{true}}=\ln(1+\varepsilon_{\mathrm{e}})\,\!
  7. σ n \sigma_{\mathrm{n}}\,\!
  8. τ n \tau_{\mathrm{n}}\,\!
  9. 𝐧 \mathbf{n}\,\!
  10. ( n 1 , n 2 , n 3 ) \left(n_{1},n_{2},n_{3}\right)\,\!
  11. σ n \sigma_{\mathrm{n}}\,\!
  12. ( σ 11 , σ 22 , σ 33 , σ 12 , σ 23 , σ 13 ) (\sigma_{11},\sigma_{22},\sigma_{33},\sigma_{12},\sigma_{23},\sigma_{13})\,\!
  13. ( σ 1 , σ 2 , σ 3 ) (\sigma_{1},\sigma_{2},\sigma_{3})\,\!
  14. σ 1 \sigma_{1}\,\!