wpmath0000004_13

S_transform.html

  1. S x ( t , f ) = - x ( τ ) | f | e - π ( t - τ ) 2 f 2 e - j 2 π f τ d τ S_{x}(t,f)=\int_{-\infty}^{\infty}x(\tau)|f|e^{-\pi(t-\tau)^{2}f^{2}}e^{-j2\pi f% \tau}\,d\tau
  2. x ( τ ) = - [ - S x ( t , f ) d t ] e j 2 π f τ d f x(\tau)=\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}S_{x}(t,f)\,dt% \right]\,e^{j2\pi f\tau}\,df
  3. ( x ( τ ) e - j 2 π f τ ) (x(\tau)e^{-j2\pi f\tau})
  4. ( | f | e - π t 2 f 2 ) (|f|e^{-\pi t^{2}f^{2}})
  5. ( x ( τ ) e - j 2 π f τ ) (x(\tau)e^{-j2\pi f\tau})
  6. ( | f | e - π t 2 f 2 ) (|f|e^{-\pi t^{2}f^{2}})
  7. S x ( t , f ) = - X ( f + α ) e - π α 2 / f 2 e j 2 π α t d α S_{x}(t,f)=\int_{-\infty}^{\infty}X(f+\alpha)\,e^{-\pi\alpha^{2}/f^{2}}\,e^{j2% \pi\alpha t}\,d\alpha
  8. t = n Δ T f = m Δ F α = p Δ F t=n\Delta_{T}\,\,f=m\Delta_{F}\,\,\alpha=p\Delta_{F}
  9. Δ T \Delta_{T}
  10. Δ F \Delta_{F}
  11. S x ( n Δ T , m Δ F ) = p = 0 N - 1 X [ ( p + m ) Δ F ] e - π p 2 m 2 e j 2 p n N S_{x}(n\Delta_{T}\,,m\Delta_{F})=\sum_{p=0}^{N-1}X[(p+m)\,\Delta_{F}]\,e^{-\pi% \frac{p^{2}}{m^{2}}}\,e^{\frac{j2pn}{N}}
  12. X [ p Δ F ] X[p\Delta_{F}]\,
  13. e - π p 2 m 2 e^{-\pi\frac{p^{2}}{m^{2}}}
  14. f = m Δ F f=m\Delta_{F}
  15. X [ p Δ F ] X[p\Delta_{F}]
  16. X [ ( p + m ) Δ F ] X[(p+m)\Delta_{F}]
  17. B [ m , p ] = X [ ( p + m ) Δ F ] e - π p 2 m 2 B[m,p]=X[(p+m)\Delta_{F}]\cdot e^{-\pi\frac{p^{2}}{m^{2}}}
  18. B [ m , p ] B[m,p]
  19. ( e - π ( t - τ ) 2 ) (e^{-\pi(t-\tau)^{2}})

Saddle_point.html

  1. z = x 2 - y 2 z=x^{2}-y^{2}
  2. ( 0 , 0 ) (0,0)
  3. [ 2 0 0 - 2 ] \begin{bmatrix}2&0\\ 0&-2\\ \end{bmatrix}
  4. ( 0 , 0 ) (0,0)
  5. z = x 4 - y 4 , z=x^{4}-y^{4},

Sahlqvist_formula.html

  1. p \Box\cdots\Box p
  2. i p \Box^{i}p
  3. 0 i < ω 0\leq i<\omega
  4. \Diamond
  5. \Box
  6. p p p\rightarrow\Diamond p
  7. x R x x \forall x\;Rxx
  8. p p p\rightarrow\Box\Diamond p
  9. x y [ R x y R y x ] \forall x\forall y[Rxy\rightarrow Ryx]
  10. p p \Diamond\Diamond p\rightarrow\Diamond p
  11. p p \Box p\rightarrow\Box\Box p
  12. x y z [ ( R x y R y z ) R x z ] \forall x\forall y\forall z[(Rxy\land Ryz)\rightarrow Rxz]
  13. p p \Diamond p\rightarrow\Diamond\Diamond p
  14. p p \Box\Box p\rightarrow\Box p
  15. x y [ R x y z ( R x z R z y ) ] \forall x\forall y[Rxy\rightarrow\exists z(Rxz\land Rzy)]
  16. p p \Box p\rightarrow\Diamond p
  17. x y R x y \forall x\exists y\;Rxy
  18. p p \Diamond\Box p\rightarrow\Box\Diamond p
  19. x x 1 z 0 [ R x x 1 R x z 0 z 1 ( R x 1 z 1 R z 0 z 1 ) ] \forall x\forall x_{1}\forall z_{0}[Rxx_{1}\land Rxz_{0}\rightarrow\exists z_{% 1}(Rx_{1}z_{1}\land Rz_{0}z_{1})]
  20. p p \Box\Diamond p\rightarrow\Diamond\Box p
  21. ( p p ) p \Box(\Box p\rightarrow p)\rightarrow\Box p
  22. ( p p ) ( q q ) (\Box\Diamond p\rightarrow\Diamond\Box p)\land(\Diamond\Diamond q\rightarrow% \Diamond q)
  23. x [ y ( R x y z [ R y z ] ) y ( R x y z [ R y z z = y ] ) ] \forall x[\forall y(Rxy\rightarrow\exists z[Ryz])\rightarrow\exists y(Rxy% \wedge\forall z[Ryz\rightarrow z=y])]

Salem_number.html

  1. P ( x ) = x 10 + x 9 - x 7 - x 6 - x 5 - x 4 - x 3 + x + 1 , P(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1,
  2. Q ( x ) = x 12 - x 7 - x 6 - x 5 + 1 , Q(x)=x^{12}-x^{7}-x^{6}-x^{5}+1,
  3. x 630 - 1 = ( x 315 - 1 ) ( x 210 - 1 ) ( x 126 - 1 ) 2 ( x 90 - 1 ) ( x 3 - 1 ) 3 ( x 2 - 1 ) 5 ( x - 1 ) 3 ( x 35 - 1 ) ( x 15 - 1 ) 2 ( x 14 - 1 ) 2 ( x 5 - 1 ) 6 x 68 x^{630}-1=\frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^{2}(x^{90}-1)(x^{3}-1)^{3}(x% ^{2}-1)^{5}(x-1)^{3}}{(x^{35}-1)(x^{15}-1)^{2}(x^{14}-1)^{2}(x^{5}-1)^{6}\,x^{% 68}}
  4. x 3 - x - 1 , x^{3}-x-1,
  5. x n P ( x ) = ± P * ( x ) x^{n}P(x)=\pm P^{*}(x)\,
  6. x n ( x 3 - x - 1 ) = - ( x 3 + x 2 - 1 ) x^{n}(x^{3}-x-1)=-(x^{3}+x^{2}-1)
  7. ( x - 1 ) ( x 10 + x 9 - x 7 - x 6 - x 5 - x 4 - x 3 + x + 1 ) = 0 (x-1)(x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1)=0
  8. x ( x 3 - x - 1 ) 1 / n = ± ( x 3 + x 2 - 1 ) 1 / n x(x^{3}-x-1)^{1/n}=\pm(x^{3}+x^{2}-1)^{1/n}
  9. x n ( x 4 - x 3 - 1 ) = - ( x 4 + x - 1 ) x^{n}(x^{4}-x^{3}-1)=-(x^{4}+x-1)
  10. ( x - 1 ) ( x 10 - x 6 - x 5 - x 4 + 1 ) = 0 (x-1)(x^{10}-x^{6}-x^{5}-x^{4}+1)=0

Samples_per_inch.html

  1. v S P I * h S P I * a r e a * c o l o r d e p t h e n c o d i n g 8 \tfrac{vSPI\ *\ hSPI\ *\ area\ *\ color~{}depth~{}encoding}{8}

Sand_casting.html

  1. N a 2 O ( S i O 2 ) + C O 2 N a 2 C O 3 + 2 S i O 2 + H e a t Na_{2}O(SiO_{2})+CO_{2}\rightleftharpoons Na_{2}CO_{3}+2SiO_{2}+Heat

Sangaku.html

  1. 1 r middle = 1 r left + 1 r right . \frac{1}{\sqrt{r\text{middle}}}=\frac{1}{\sqrt{r\text{left}}}+\frac{1}{\sqrt{r% \text{right}}}.

Sard's_theorem.html

  1. f : n m f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}
  2. C k C^{k}
  3. k k
  4. k max { n - m + 1 , 1 } k\geq\max\{n-m+1,1\}
  5. X X
  6. f , f,
  7. x n x\in\mathbb{R}^{n}
  8. f f
  9. < m <m
  10. f ( X ) f(X)
  11. m \mathbb{R}^{m}
  12. X X
  13. f f
  14. n \mathbb{R}^{n}
  15. m \mathbb{R}^{m}
  16. M M
  17. N N
  18. m m
  19. n n
  20. X X
  21. C k C^{k}
  22. f : N M f:N\rightarrow M
  23. d f : T N T M df:TN\rightarrow TM
  24. m m
  25. k max { n - m + 1 , 1 } k\geq\max\{n-m+1,1\}
  26. X X
  27. M M
  28. m = 1 m=1
  29. f : M N f:M\rightarrow N
  30. C k C^{k}
  31. k max { n - m + 1 , 1 } k\geq\max\{n-m+1,1\}
  32. A r M A_{r}\subseteq M
  33. x M x\in M
  34. d f x df_{x}
  35. r r
  36. f ( A r ) f(A_{r})
  37. f ( A r ) f(A_{r})
  38. f ( A r ) f(A_{r})

Savitch's_theorem.html

  1. f Ω ( log ( n ) ) f\in\Omega(\log(n))
  2. NSPACE ( f ( n ) ) DSPACE ( ( f ( n ) ) 2 ) . \,\text{NSPACE}\left(f\left(n\right)\right)\subseteq\,\text{DSPACE}\left(\left% (f\left(n\right)\right)^{2}\right).
  3. O ( ( log n ) 2 ) O\left((\log n)^{2}\right)
  4. O ( ( log n ) 2 ) O\left((\log n)^{2}\right)
  5. L NSPACE ( f ( n ) ) L\in\,\text{NSPACE}\left(f\left(n\right)\right)
  6. O ( 2 f ( n ) ) O\left(2^{f(n)}\right)
  7. L L
  8. x { 0 , 1 } n x\in\{0,1\}^{n}
  9. x x
  10. x L x\in L
  11. L L
  12. DSPACE ( ( f ( n ) ) 2 ) \,\text{DSPACE}\left(\left(f\left(n\right)\right)^{2}\right)
  13. DSPACE ( ( log n ) 2 ) \,\text{DSPACE}\left(\left(\log n\right)^{2}\right)

Scale_factor_(cosmology).html

  1. t t
  2. t 0 t_{0}
  3. d ( t ) = a ( t ) d 0 , d(t)=a(t)d_{0},\,
  4. d ( t ) d(t)
  5. t t
  6. d 0 d_{0}
  7. t 0 t_{0}
  8. a ( t ) a(t)
  9. a ( t 0 ) = 1 a(t_{0})=1
  10. t t
  11. t 0 t_{0}
  12. 13.798 ± 0.037 Gyr 13.798\pm 0.037\,\mathrm{Gyr}
  13. a a
  14. a ( t 0 ) a(t_{0})
  15. 1 1
  16. H a ˙ ( t ) a ( t ) H\equiv{\dot{a}(t)\over a(t)}
  17. d ( t ) = d 0 a ( t ) d(t)=d_{0}a(t)
  18. d ˙ ( t ) = d 0 a ˙ ( t ) \dot{d}(t)=d_{0}\dot{a}(t)
  19. d 0 = d ( t ) a ( t ) d_{0}=\frac{d(t)}{a(t)}
  20. d ˙ ( t ) = d ( t ) a ˙ ( t ) a ( t ) \dot{d}(t)=\frac{d(t)\dot{a}(t)}{a(t)}
  21. d ˙ ( t ) = H d ( t ) \dot{d}(t)=Hd(t)
  22. a ¨ ( t ) \ddot{a}(t)
  23. a ˙ ( t ) \dot{a}(t)
  24. d ˙ ( t ) \dot{d}(t)
  25. a ( t ) = 1 1 + z a(t)=\frac{1}{1+z}

Scale_invariance.html

  1. f ( x ) f(x)
  2. x x
  3. f ( λ x ) f(\lambda x)
  4. λ \lambda
  5. f ( x ) f(x)
  6. f ( λ x ) = λ Δ f ( x ) f(\lambda x)=\lambda^{\Delta}f(x)
  7. Δ \Delta
  8. λ \lambda
  9. f ( x ) = x n f(x)=x^{n}
  10. Δ = n \Delta=n
  11. f ( λ x ) = ( λ x ) n = λ n f ( x ) . f(\lambda x)=(\lambda x)^{n}=\lambda^{n}f(x).
  12. θ = 1 b ln ( r / a ) . \theta=\frac{1}{b}\ln(r/a).
  13. λ \lambda
  14. θ ( λ r ) \theta(\lambda r)
  15. θ ( r ) \theta(r)
  16. λ \lambda
  17. Δ = 1 \Delta=1
  18. λ = 1 / 3 n \lambda=1/3^{n}
  19. P ( f ) P(f)
  20. f f
  21. P ( f ) = λ - Δ P ( λ f ) P(f)=\lambda^{-\Delta}P(\lambda f)
  22. Δ = 0 \Delta=0
  23. Δ = - 1 \Delta=-1
  24. Δ = - 2 \Delta=-2
  25. var ( Y ) = a [ E ( Y ) ] p \,\text{var}\,(Y)=a[\,\text{E}\,(Y)]^{p}
  26. φ \varphi
  27. φ ( x ) \varphi(x)
  28. x λ x , x\rightarrow\lambda x,
  29. φ λ - Δ φ . \varphi\rightarrow\lambda^{-\Delta}\varphi.
  30. Δ \Delta
  31. φ ( x ) \varphi(x)
  32. λ Δ φ ( λ x ) \lambda^{\Delta}\varphi(\lambda x)
  33. φ ( x ) \varphi(x)
  34. φ ( x ) = λ - Δ φ ( λ x ) \varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)
  35. Δ \Delta
  36. 𝐄 ( 𝐱 , t ) \mathbf{E}(\mathbf{x},t)
  37. 𝐁 ( 𝐱 , t ) \mathbf{B}(\mathbf{x},t)
  38. 2 𝐄 = 1 c 2 2 𝐄 t 2 \nabla^{2}\mathbf{E}=\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial t^{% 2}}
  39. 2 𝐁 = 1 c 2 2 𝐁 t 2 \nabla^{2}\mathbf{B}=\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{B}}{\partial t^{% 2}}
  40. x λ x , x\rightarrow\lambda x,
  41. t λ t . t\rightarrow\lambda t.
  42. 𝐄 ( 𝐱 , t ) \mathbf{E}(\mathbf{x},t)
  43. 𝐁 ( 𝐱 , t ) \mathbf{B}(\mathbf{x},t)
  44. 𝐄 ( λ 𝐱 , λ t ) \mathbf{E}(\lambda\mathbf{x},\lambda t)
  45. 𝐁 ( λ 𝐱 , λ t ) \mathbf{B}(\lambda\mathbf{x},\lambda t)
  46. φ ( 𝐱 , t ) φ(\mathbf{x},t)
  47. t t
  48. 1 c 2 2 φ t 2 - 2 φ = 0 , \frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}-\nabla^{2}\varphi=0,
  49. x λ x , x\rightarrow\lambda x,
  50. t λ t . t\rightarrow\lambda t.
  51. m 2 φ \propto m^{2}\varphi
  52. m m
  53. L = m c , L=\frac{\hbar}{mc},
  54. Δ Δ
  55. φ φ
  56. Δ = D - 2 2 , \Delta=\frac{D-2}{2},
  57. D D
  58. φ φ
  59. D D
  60. 1 c 2 2 φ t 2 - 2 φ + g φ 3 = 0. \frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}-\nabla^{2}\varphi+g% \varphi^{3}=0.
  61. φ φ
  62. φ φ
  63. D D
  64. Δ Δ
  65. x λ x , x\rightarrow\lambda x,
  66. t λ t , t\rightarrow\lambda t,
  67. φ λ - 1 φ . \varphi\rightarrow\lambda^{-1}\varphi.
  68. g g
  69. φ φ
  70. D D
  71. D = 4 D=4
  72. Δ \Delta
  73. D D
  74. D D
  75. D D
  76. r r
  77. G ( r ) 1 r D - 2 + η , G(r)\propto\frac{1}{r^{D-2+\eta}},
  78. η \eta
  79. G ( r ) G(r)
  80. ϕ ( 0 ) ϕ ( r ) 1 r D - 2 + η . \langle\phi(0)\phi(r)\rangle\propto\frac{1}{r^{D-2+\eta}}.
  81. η η
  82. Δ = D - 2 2 \Delta=\frac{D-2}{2}
  83. Δ = D - 2 + η 2 , \Delta=\frac{D-2+\eta}{2},
  84. D D
  85. D 4 ε D≡4−ε
  86. η η
  87. η = ϵ 2 54 + O ( ϵ 3 ) \eta=\frac{\epsilon^{2}}{54}+O(\epsilon^{3})
  88. ε ε
  89. η η
  90. η η
  91. η D = 2 = 1 4 \eta_{{}_{D=2}}=\frac{1}{4}
  92. 𝐮 ( 𝐱 , t ) \mathbf{u}(\mathbf{x},t)
  93. ρ ( 𝐱 , t ) \rho(\mathbf{x},t)
  94. P ( 𝐱 , t ) P(\mathbf{x},t)
  95. ρ 𝐮 t + ρ 𝐮 𝐮 = - P + μ ( 2 𝐮 + 1 3 ( 𝐮 ) ) \rho\frac{\partial\mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla\mathbf{u}=% -\nabla P+\mu\left(\nabla^{2}\mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot% \mathbf{u}\right)\right)
  96. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{u}\right)=0
  97. μ \mu
  98. P = c s 2 ρ , P=c_{s}^{2}\rho,
  99. c s c_{s}
  100. x λ x , x\rightarrow\lambda x,
  101. t λ t , t\rightarrow\lambda t,
  102. ρ λ - 1 ρ , \rho\rightarrow\lambda^{-1}\rho,
  103. 𝐮 𝐮 . \mathbf{u}\rightarrow\mathbf{u}.
  104. 𝐮 ( 𝐱 , t ) \mathbf{u}(\mathbf{x},t)
  105. ρ ( 𝐱 , t ) \rho(\mathbf{x},t)
  106. 𝐮 ( λ 𝐱 , λ t ) \mathbf{u}(\lambda\mathbf{x},\lambda t)
  107. λ ρ ( λ 𝐱 , λ t ) \lambda\rho(\lambda\mathbf{x},\lambda t)

Scale_parameter.html

  1. F ( x ; s , θ ) = F ( x / s ; 1 , θ ) , F(x;s,\theta)=F(x/s;1,\theta),\!
  2. f s ( x ) = f ( x / s ) / s , f_{s}(x)=f(x/s)/s,\!
  3. f s f_{s}
  4. g ( x ) = x / s g(x)=x/s
  5. f s ( x ) = f ( x / s ) × 1 / s = f ( g ( x ) ) × g ( x ) . f_{s}(x)=f(x/s)\times 1/s=f(g(x))\times g^{\prime}(x).\!
  6. 1 = - f ( x ) d x = g ( - ) g ( ) f ( x ) d x . 1=\int_{-\infty}^{\infty}f(x)\,dx=\int_{g(-\infty)}^{g(\infty)}f(x)\,dx.\!
  7. 1 = - f ( g ( x ) ) × g ( x ) d x = - f s ( x ) d x . 1=\int_{-\infty}^{\infty}f(g(x))\times g^{\prime}(x)\,dx=\int_{-\infty}^{% \infty}f_{s}(x)\,dx.\!
  8. f s f_{s}
  9. f ( x ; β ) = 1 β e - x / β , x 0 f(x;\beta)=\frac{1}{\beta}e^{-x/\beta},\;x\geq 0
  10. f ( x ; λ ) = λ e - λ x , x 0. f(x;\lambda)=\lambda e^{-\lambda x},\;x\geq 0.
  11. μ \mu
  12. σ \sigma
  13. σ 2 \sigma^{2}
  14. θ \theta
  15. 1 / Φ - 1 ( 3 / 4 ) 1.4826 , 1/\Phi^{-1}(3/4)\approx 1.4826,

Scaling_(geometry).html

  1. S v = [ v x 0 0 0 v y 0 0 0 v z ] . S_{v}=\begin{bmatrix}v_{x}&0&0\\ 0&v_{y}&0\\ 0&0&v_{z}\\ \end{bmatrix}.
  2. S v p = [ v x 0 0 0 v y 0 0 0 v z ] [ p x p y p z ] = [ v x p x v y p y v z p z ] . S_{v}p=\begin{bmatrix}v_{x}&0&0\\ 0&v_{y}&0\\ 0&0&v_{z}\\ \end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\end{bmatrix}=\begin{bmatrix}v_{x}p_{x}\\ v_{y}p_{y}\\ v_{z}p_{z}\end{bmatrix}.
  3. n n
  4. n \mathbb{R}^{n}
  5. v v
  6. v v
  7. v v
  8. v v
  9. v I vI
  10. v 1 , v 2 , v n v_{1},v_{2},\ldots v_{n}
  11. i i
  12. v i v_{i}
  13. S v = [ v x 0 0 0 0 v y 0 0 0 0 v z 0 0 0 0 1 ] . S_{v}=\begin{bmatrix}v_{x}&0&0&0\\ 0&v_{y}&0&0\\ 0&0&v_{z}&0\\ 0&0&0&1\end{bmatrix}.
  14. S v p = [ v x 0 0 0 0 v y 0 0 0 0 v z 0 0 0 0 1 ] [ p x p y p z 1 ] = [ v x p x v y p y v z p z 1 ] . S_{v}p=\begin{bmatrix}v_{x}&0&0&0\\ 0&v_{y}&0&0\\ 0&0&v_{z}&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ 1\end{bmatrix}=\begin{bmatrix}v_{x}p_{x}\\ v_{y}p_{y}\\ v_{z}p_{z}\\ 1\end{bmatrix}.
  15. S v = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 s ] . S_{v}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&\frac{1}{s}\end{bmatrix}.
  16. S v p = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 s ] [ p x p y p z 1 ] = [ p x p y p z 1 s ] S_{v}p=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&\frac{1}{s}\end{bmatrix}\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ 1\end{bmatrix}=\begin{bmatrix}p_{x}\\ p_{y}\\ p_{z}\\ \frac{1}{s}\end{bmatrix}
  17. [ s p x s p y s p z 1 ] . \begin{bmatrix}sp_{x}\\ sp_{y}\\ sp_{z}\\ 1\end{bmatrix}.

Scattering_amplitude.html

  1. ψ ( 𝐫 ) = e i k z + f ( θ ) e i k r r , \psi(\mathbf{r})=e^{ikz}+f(\theta)\frac{e^{ikr}}{r}\;,
  2. 𝐫 ( x , y , z ) \mathbf{r}\equiv(x,y,z)
  3. r | 𝐫 | r\equiv|\mathbf{r}|
  4. e i k z e^{ikz}
  5. k k
  6. z z
  7. e i k r / r e^{ikr}/r
  8. θ \theta
  9. f ( θ ) f(\theta)
  10. d σ d Ω = | f ( θ ) | 2 . \frac{d\sigma}{d\Omega}=|f(\theta)|^{2}\;.
  11. f = = 0 ( 2 + 1 ) f P ( cos θ ) f=\sum_{\ell=0}^{\infty}(2\ell+1)f_{\ell}P_{\ell}(\cos\theta)
  12. = e 2 i δ =e^{2i\delta_{\ell}}
  13. f = S - 1 2 i k = e 2 i δ - 1 2 i k = e i δ sin δ k = 1 k cot δ - i k . f_{\ell}=\frac{S_{\ell}-1}{2ik}=\frac{e^{2i\delta_{\ell}}-1}{2ik}=\frac{e^{i% \delta_{\ell}}\sin\delta_{\ell}}{k}=\frac{1}{k\cot\delta_{\ell}-ik}\;.
  14. d σ d Ω = | f ( θ ) | 2 = 1 k 2 | = 0 ( 2 + 1 ) e i δ sin δ P ( cos θ ) | 2 \frac{d\sigma}{d\Omega}=|f(\theta)|^{2}=\frac{1}{k^{2}}\left|\sum_{\ell=0}^{% \infty}(2\ell+1)e^{i\delta_{\ell}}\sin\delta_{\ell}P_{\ell}(\cos\theta)\right|% ^{2}
  15. σ = 2 π 0 π d σ d Ω sin θ d θ = 4 π k Im f ( 0 ) \sigma=2\pi\int_{0}^{\pi}\frac{d\sigma}{d\Omega}\sin\theta\,d\theta=\frac{4\pi% }{k}\operatorname{Im}f(0)
  16. I m f ( 0 ) Imf(0)
  17. f ( 0 ) f(0)
  18. r 0 r_{0}
  19. b b

Schauder_fixed_point_theorem.html

  1. K K
  2. V V
  3. T T
  4. K K
  5. T ( K ) T(K)
  6. K K
  7. T T
  8. T T
  9. X X
  10. { x X : x = λ T x for some 0 λ 1 } \{x\in X:x=\lambda Tx\mbox{ for some }~{}0\leq\lambda\leq 1\}
  11. T T

Scheimpflug_principle.html

  1. J = f sin θ . J=\frac{f}{\sin\theta}.
  2. v f = sin θ [ 1 tan ( ψ - θ ) + 1 tan θ ] ; \frac{v^{\prime}}{f}=\sin\theta\left[\frac{1}{\tan\left(\psi-\theta\right)}+% \frac{1}{\tan\theta}\right]\,;
  3. tan ψ = v v cos θ - f sin θ . \tan\psi=\frac{v^{\prime}}{v^{\prime}\cos\theta-f}\sin\theta.
  4. tan ψ = u f sin θ . \tan\psi=\frac{u^{\prime}}{f}\sin\theta.
  5. 1 u + 1 v = 1 f , \frac{1}{u}+\frac{1}{v}=\frac{1}{f},
  6. y x = f v u u h J , y_{x}=\frac{f}{v^{\prime}}\frac{u^{\prime}}{u}_{\mathrm{h}}J\,,
  7. y x = N c f ( 1 tan θ - 1 tan ψ ) u , y_{x}=\frac{Nc}{f}\left(\frac{1}{\tan\theta}-\frac{1}{\tan\psi}\right)u^{% \prime}\,,
  8. y x u u h J , y_{x}\approx\frac{u^{\prime}}{u}_{\mathrm{h}}J\,,
  9. y x N c f u tan θ . y_{x}\approx\frac{Nc}{f}\frac{u^{\prime}}{\tan\theta}\,.
  10. 2 J = 2 f sin θ = 2 × 90 mm sin 8 = 1293 mm . 2J=2\frac{f}{\sin\theta}=2\times\frac{90\,\text{ mm}}{\sin 8^{\circ}}=1293% \text{ mm}\,.
  11. f 2 N c = 90 2 2.8 × 0.03 = 96.4 m . \frac{f^{2}}{Nc}=\frac{90^{2}}{2.8\times 0.03}=96.4\text{ m}\,.
  12. y u = a u + b y_{u}=au+b
  13. 1 u + 1 v = 1 f ; \frac{1}{u}+\frac{1}{v}=\frac{1}{f}\,;
  14. u = v f v - f , u=\frac{vf}{v-f}\,,
  15. y u = a v f v - f + b y_{u}=a\,\frac{vf}{v-f}+b
  16. m = y v y u ; m=\frac{y_{v}}{y_{u}}\,;
  17. m = - v u = - v - f f m=-\frac{v}{u}=-\frac{v-f}{f}
  18. y v = m y u = - v - f f ( a v f v - f + b ) = - ( a v + v f b - b ) , \begin{aligned}\displaystyle y_{v}&\displaystyle=my_{u}\\ &\displaystyle=-\frac{v-f}{f}\left(a\,\frac{vf}{v-f}+b\right)\\ &\displaystyle=-\left(av+\frac{v}{f}b-b\right)\,,\end{aligned}
  19. y v = - ( a + b f ) v + b y_{v}=-\left(a+\frac{b}{f}\right)v+b
  20. tan ψ = u + v S , \tan\psi=\frac{u^{\prime}+v^{\prime}}{S}\,,
  21. tan θ = v S ; \tan\theta=\frac{v^{\prime}}{S}\,;
  22. tan ψ = u + v v tan θ = ( u v + 1 ) tan θ . \tan\psi=\frac{u^{\prime}+v^{\prime}}{v^{\prime}}\tan\theta=\left(\frac{u^{% \prime}}{v^{\prime}}+1\right)\tan\theta\,.
  23. 1 u + 1 v = 1 u cos θ + 1 v cos θ = 1 f . \frac{1}{u}+\frac{1}{v}=\frac{1}{u^{\prime}\cos\theta}+\frac{1}{v^{\prime}\cos% \theta}=\frac{1}{f}\,.
  24. u = v f v cos θ - f ; u^{\prime}=\frac{v^{\prime}f}{v^{\prime}\cos\theta-f}\,;
  25. tan ψ = ( f v cos θ - f + 1 ) tan θ = f + v cos θ - f v cos θ - f tan θ , \tan\psi=\left(\frac{f}{v^{\prime}\cos\theta-f}+1\right)\tan\theta=\frac{f+v^{% \prime}\cos\theta-f}{v^{\prime}\cos\theta-f}\tan\theta\,,
  26. tan ψ = v v cos θ - f sin θ . \tan\psi=\frac{v^{\prime}}{v^{\prime}\cos\theta-f}\sin\theta\,.
  27. tan ψ = u f sin θ . \tan\psi=\frac{u^{\prime}}{f}\sin\theta\,.
  28. u f = u f 1 cos θ = m + 1 m 1 cos θ , \frac{u^{\prime}}{f}=\frac{u}{f}\frac{1}{\cos\theta}=\frac{m+1}{m}\frac{1}{% \cos\theta}\,,
  29. tan ψ = m + 1 m tan θ . \tan\psi=\frac{m+1}{m}\tan\theta\,.
  30. tan ψ = u J ; \tan\psi=\frac{u^{\prime}}{J}\,;
  31. sin θ = f J . \sin\theta=\frac{f}{J}\,.
  32. sin θ = d J , \sin\theta=\frac{d}{J}\,,
  33. tan θ = v S , \tan\theta=\frac{v^{\prime}}{S}\,,

Schnorr_group.html

  1. p × \mathbb{Z}_{p}^{\times}
  2. p p
  3. p p
  4. p p
  5. q q
  6. r r
  7. p = q r + 1 p=qr+1\;
  8. p p
  9. q q
  10. h h
  11. 1 < h < p 1<h<p
  12. h r 1 ( mod p ) h^{r}\not\equiv 1\;(\,\text{mod}\;p)
  13. g = h r mod p g=h^{r}\,\text{ mod }p
  14. p × \mathbb{Z}_{p}^{\times}
  15. q q
  16. p p
  17. q q
  18. x x
  19. 0 < x < p 0<x<p
  20. x q 1 ( mod p ) x^{q}\equiv 1\;(\,\text{mod }p)
  21. 1 1

Schreier's_lemma.html

  1. H H
  2. G G
  3. S S
  4. S \scriptstyle\langle S\rangle
  5. R R
  6. H H
  7. G G
  8. R R
  9. G H \ G G\to H\backslash G
  10. H \ G H\backslash G
  11. H H
  12. G G
  13. g g
  14. G G
  15. g ¯ \overline{g}
  16. R R
  17. H g Hg
  18. g H g ¯ . g\in H\overline{g}.
  19. H H
  20. { r s ( r s ¯ ) - 1 | r R , s S } \{rs(\overline{rs})^{-1}|r\in R,s\in S\}
  21. 3 = { e , ( 1 2 3 ) , ( 1 3 2 ) } \mathbb{Z}_{3}=\{e,(1\ 2\ 3),(1\ 3\ 2)\}
  22. S 3 = { e , ( 1 2 ) , ( 1 3 ) , ( 2 3 ) , ( 1 2 3 ) , ( 1 3 2 ) } S_{3}=\{e,(1\ 2),(1\ 3),(2\ 3),(1\ 2\ 3),(1\ 3\ 2)\}
  23. e e
  24. \scriptstyle\langle
  25. \scriptstyle\rangle
  26. t 1 s 1 = ( 1 2 ) , so t 1 s 1 ¯ = ( 1 2 ) t 1 s 2 = ( 1 2 3 ) , so t 1 s 2 ¯ = e t 2 s 1 = e , so t 2 s 1 ¯ = e t 2 s 2 = ( 2 3 ) , so t 2 s 2 ¯ = ( 1 2 ) . \begin{matrix}t_{1}s_{1}=(1\ 2),&\quad\,\text{so}&\overline{t_{1}s_{1}}=(1\ 2)% \\ t_{1}s_{2}=(1\ 2\ 3),&\quad\,\text{so}&\overline{t_{1}s_{2}}=e\\ t_{2}s_{1}=e,&\quad\,\text{so}&\overline{t_{2}s_{1}}=e\\ t_{2}s_{2}=(2\ 3),&\quad\,\text{so}&\overline{t_{2}s_{2}}=(1\ 2).\\ \end{matrix}
  27. t 1 s 1 t 1 s 1 ¯ - 1 = e t_{1}s_{1}\overline{t_{1}s_{1}}^{-1}=e
  28. t 1 s 2 t 1 s 2 ¯ - 1 = ( 1 2 3 ) t_{1}s_{2}\overline{t_{1}s_{2}}^{-1}=(1\ 2\ 3)
  29. t 2 s 1 t 2 s 1 ¯ - 1 = e t_{2}s_{1}\overline{t_{2}s_{1}}^{-1}=e
  30. t 2 s 2 t 2 s 2 ¯ - 1 = ( 1 2 3 ) . t_{2}s_{2}\overline{t_{2}s_{2}}^{-1}=(1\ 2\ 3).

Schreier_refinement_theorem.html

  1. / ( 2 ) × S 3 \mathbb{Z}/(2)\times S_{3}
  2. S 3 S_{3}
  3. { [ 0 ] } × { id } / ( 2 ) × { id } / ( 2 ) × S 3 , \{[0]\}\times\{\operatorname{id}\}\;\triangleleft\;\mathbb{Z}/(2)\times\{% \operatorname{id}\}\;\triangleleft\;\mathbb{Z}/(2)\times S_{3},
  4. { [ 0 ] } × { id } { [ 0 ] } × S 3 / ( 2 ) × S 3 . \{[0]\}\times\{\operatorname{id}\}\;\triangleleft\;\{[0]\}\times S_{3}\;% \triangleleft\;\mathbb{Z}/(2)\times S_{3}.
  5. S 3 S_{3}
  6. A 3 A_{3}
  7. { [ 0 ] } × { id } / ( 2 ) × { id } / ( 2 ) × A 3 / ( 2 ) × S 3 \{[0]\}\times\{\operatorname{id}\}\;\triangleleft\;\mathbb{Z}/(2)\times\{% \operatorname{id}\}\;\triangleleft\;\mathbb{Z}/(2)\times A_{3}\;\triangleleft% \;\mathbb{Z}/(2)\times S_{3}
  8. ( / ( 2 ) , A 3 , / ( 2 ) ) (\mathbb{Z}/(2),A_{3},\mathbb{Z}/(2))
  9. { [ 0 ] } × { id } { [ 0 ] } × A 3 { [ 0 ] } × S 3 / ( 2 ) × S 3 \{[0]\}\times\{\operatorname{id}\}\;\triangleleft\;\{[0]\}\times A_{3}\;% \triangleleft\;\{[0]\}\times S_{3}\;\triangleleft\;\mathbb{Z}/(2)\times S_{3}
  10. ( A 3 , / ( 2 ) , / ( 2 ) ) (A_{3},\mathbb{Z}/(2),\mathbb{Z}/(2))

Schreier–Sims_algorithm.html

  1. G S n G\leq S_{n}
  2. t t
  3. O ( n 2 log 3 | G | + t n log | G | ) O(n^{2}\log^{3}|G|+tn\log|G|)
  4. O ( n 2 log | G | + t n ) O(n^{2}\log|G|+tn)
  5. O ( n 3 log 3 | G | + t n 2 log | G | ) O(n^{3}\log^{3}|G|+tn^{2}\log|G|)
  6. O ( n log 2 | G | + t n ) O(n\log^{2}|G|+tn)
  7. O ( n log n log 4 | G | + t n log | G | ) O(n\log n\log^{4}|G|+tn\log|G|)
  8. O ( n log | G | + t n ) O(n\log|G|+tn)

Schrödinger_method.html

  1. X 1 , , X n X_{1},\dots,X_{n}\,
  2. X ( 1 ) , , X ( n ) X_{(1)},\dots,X_{(n)}\,
  3. P ( A N = n ) P(A\mid N=n)\,
  4. P λ ( A ) = n = 0 P ( A N = n ) P ( N = n ) = n = 0 P ( A N = n ) λ n e - λ n ! , P_{\lambda}(A)=\sum_{n=0}^{\infty}P(A\mid N=n)P(N=n)=\sum_{n=0}^{\infty}P(A% \mid N=n){\lambda^{n}e^{-\lambda}\over n!},
  5. e λ P λ ( A ) = n = 0 P ( A N = n ) λ n n ! . e^{\lambda}\,P_{\lambda}(A)=\sum_{n=0}^{\infty}P(A\mid N=n){\lambda^{n}\over n% !}.
  6. P ( A N = n ) = [ d n d λ n ( e λ P λ ( A ) ) ] λ = 0 . P(A\mid N=n)=\left[{d^{n}\over d\lambda^{n}}\left(e^{\lambda}\,P_{\lambda}(A)% \right)\right]_{\lambda=0}.

Schuler_tuning.html

  1. T = 2 π L g 2 π 6378100 9.81 5066 seconds 84.4 minutes T=2\pi\sqrt{\frac{L}{g}}\approx 2\pi\sqrt{\frac{6378100}{9.81}}\approx 5066\ % \,\text{seconds}\approx 84.4\ \,\text{minutes}
  2. θ ˙ = v / R \dot{\theta}=v/R
  3. θ ¨ = a / R \ddot{\theta}=a/R

Schulze_method.html

  1. d [ V , W ] d[V,W]
  2. V V
  3. W W
  4. X X
  5. Y Y
  6. p p
  7. C ( 1 ) , , C ( n ) C(1),...,C(n)
  8. C ( 1 ) = X C(1)=X
  9. C ( n ) = Y C(n)=Y
  10. i = 1 , , ( n - 1 ) : d [ C ( i ) , C ( i + 1 ) ] > d [ C ( i + 1 ) , C ( i ) ] i=1,...,(n-1):d[C(i),C(i+1)]>d[C(i+1),C(i)]
  11. i = 1 , , ( n - 1 ) : d [ C ( i ) , C ( i + 1 ) ] p i=1,...,(n-1):d[C(i),C(i+1)]\,\text{≥}p
  12. p [ A , B ] p[A,B]
  13. A A
  14. B B
  15. A A
  16. B B
  17. A A
  18. B B
  19. p [ A , B ] = 0 p[A,B]=0
  20. D D
  21. E E
  22. p [ D , E ] > p [ E , D ] p[D,E]>p[E,D]
  23. D D
  24. p [ D , E ] p [ E , D ] p[D,E]\,\text{≥}p[E,D]
  25. E E
  26. p [ X , Y ] > p [ Y , X ] p[X,Y]>p[Y,X]
  27. p [ Y , Z ] > p [ Z , Y ] p[Y,Z]>p[Z,Y]
  28. p [ X , Z ] > p [ Z , X ] p[X,Z]>p[Z,X]
  29. D D
  30. p [ D , E ] p [ E , D ] p[D,E]\,\text{≥}p[E,D]
  31. E E
  32. A C B E D ACBED
  33. A > C > B > E > D A>C>B>E>D
  34. A D E C B ADECB
  35. B E D A C BEDAC
  36. C A B E D CABED
  37. C A E B D CAEBD
  38. C B A D E CBADE
  39. D C E B A DCEBA
  40. E B A D C EBADC
  41. A A
  42. B B
  43. 5 + 5 + 3 + 7 = 20 5+5+3+7=20
  44. A A
  45. B B
  46. 8 + 2 + 7 + 8 = 25 8+2+7+8=25
  47. B B
  48. A A
  49. d [ A , B ] = 20 d[A,B]=20
  50. d [ B , A ] = 25 d[B,A]=25

Schur_multiplier.html

  1. H 2 ( G , 𝐙 ) ( R [ F , F ] ) / [ F , R ] H_{2}(G,\mathbf{Z})\cong(R\cap[F,F])/[F,R]
  2. H 2 ( G , 𝐙 ) ( H 2 ( G , 𝐂 × ) ) * H_{2}(G,\mathbf{Z})\cong\bigl(H^{2}(G,\mathbf{C}^{\times})\bigr)^{*}
  3. ( H 2 ( G , 𝐂 × ) ) * H 2 ( G , 𝐂 × ) . \bigl(H^{2}(G,\mathbf{C}^{\times})\bigr)^{*}\cong H^{2}(G,\mathbf{C}^{\times}).

Schwinger's_quantum_action_principle.html

  1. | A |A\rangle
  2. | B |B\rangle
  3. δ B | A = i B | δ S | A , \delta\langle B|A\rangle=i\langle B|\delta S|A\rangle,
  4. exp ( i S ) \exp(iS)
  5. | A |A\rangle
  6. | B |B\rangle

Schwinger–Dyson_equation.html

  1. | ψ |\psi\rangle
  2. ψ | 𝒯 { δ δ ϕ F [ ϕ ] } | ψ = - i ψ | 𝒯 { F [ ϕ ] δ δ ϕ S [ ϕ ] } | ψ \left\langle\psi\left|\mathcal{T}\left\{\frac{\delta}{\delta\phi}F[\phi]\right% \}\right|\psi\right\rangle=-i\left\langle\psi\left|\mathcal{T}\left\{F[\phi]% \frac{\delta}{\delta\phi}S[\phi]\right\}\right|\psi\right\rangle
  3. 𝒯 \mathcal{T}
  4. ρ ( 𝒯 { δ δ ϕ F [ ϕ ] } ) = - i ρ ( 𝒯 { F [ ϕ ] δ δ ϕ S [ ϕ ] } ) . \rho\left(\mathcal{T}\left\{\frac{\delta}{\delta\phi}F[\phi]\right\}\right)=-i% \rho\left(\mathcal{T}\left\{F[\phi]\frac{\delta}{\delta\phi}S[\phi]\right\}% \right).
  5. ψ | 𝒯 { F ϕ j } | ψ = ψ | 𝒯 { i F , i D i j - F S i n t , i D i j } | ψ . \langle\psi|\mathcal{T}\{F\phi^{j}\}|\psi\rangle=\langle\psi|\mathcal{T}\{iF_{% ,i}D^{ij}-FS_{int,i}D^{ij}\}|\psi\rangle.
  6. F [ ϕ ] = k 1 x 1 k 1 ϕ ( x 1 ) k n x n k n ϕ ( x n ) F[\phi]=\frac{\partial^{k_{1}}}{\partial x_{1}^{k_{1}}}\phi(x_{1})\cdots\frac{% \partial^{k_{n}}}{\partial x_{n}^{k_{n}}}\phi(x_{n})
  7. F [ - i δ δ J ] G [ J ] = ( - i ) n k 1 x 1 k 1 δ δ J ( x 1 ) k n x n k n δ δ J ( x n ) G [ J ] . F\left[-i\frac{\delta}{\delta J}\right]G[J]=(-i)^{n}\frac{\partial^{k_{1}}}{% \partial x_{1}^{k_{1}}}\frac{\delta}{\delta J(x_{1})}\cdots\frac{\partial^{k_{% n}}}{\partial x_{n}^{k_{n}}}\frac{\delta}{\delta J(x_{n})}G[J].
  8. δ n Z δ J ( x 1 ) δ J ( x n ) [ 0 ] = i n Z [ 0 ] ϕ ( x 1 ) ϕ ( x n ) \frac{\delta^{n}Z}{\delta J(x_{1})\cdots\delta J(x_{n})}[0]=i^{n}Z[0]\langle% \phi(x_{1})\cdots\phi(x_{n})\rangle
  9. δ 𝒮 δ ϕ ( x ) [ ϕ ] + J ( x ) J = 0 {\left\langle\frac{\delta\mathcal{S}}{\delta\phi(x)}\left[\phi\right]+J(x)% \right\rangle}_{J}=0
  10. δ S δ ϕ ( x ) [ - i δ δ J ] Z [ J ] + J ( x ) Z [ J ] = 0. \frac{\delta S}{\delta\phi(x)}\left[-i\frac{\delta}{\delta J}\right]Z[J]+J(x)Z% [J]=0.
  11. S [ ϕ ] = d d x ( 1 2 μ ϕ ( x ) μ ϕ ( x ) - 1 2 m 2 ϕ ( x ) 2 - λ 4 ! ϕ ( x ) 4 ) S[\phi]=\int d^{d}x\left(\frac{1}{2}\partial^{\mu}\phi(x)\partial_{\mu}\phi(x)% -\frac{1}{2}m^{2}\phi(x)^{2}-\frac{\lambda}{4!}\phi(x)^{4}\right)
  12. δ S δ ϕ ( x ) = - μ μ ϕ ( x ) - m 2 ϕ ( x ) - λ 3 ! ϕ ( x ) 3 \frac{\delta S}{\delta\phi(x)}=-\partial_{\mu}\partial^{\mu}\phi(x)-m^{2}\phi(% x)-\frac{\lambda}{3!}\phi(x)^{3}
  13. i μ μ δ δ J ( x ) Z [ J ] + i m 2 δ δ J ( x ) Z [ J ] - i λ 3 ! δ 3 δ J ( x ) 3 Z [ J ] + J ( x ) Z [ J ] = 0 i\partial_{\mu}\partial^{\mu}\frac{\delta}{\delta J(x)}Z[J]+im^{2}\frac{\delta% }{\delta J(x)}Z[J]-\frac{i\lambda}{3!}\frac{\delta^{3}}{\delta J(x)^{3}}Z[J]+J% (x)Z[J]=0
  14. δ 3 δ J ( x ) 3 \frac{\delta^{3}}{\delta J(x)^{3}}
  15. δ 3 δ J ( x 1 ) δ J ( x 2 ) δ J ( x 3 ) Z [ J ] \frac{\delta^{3}}{\delta J(x_{1})\delta J(x_{2})\delta J(x_{3})}Z[J]
  16. - μ μ - m 2 -\partial^{\mu}\partial_{\mu}-m^{2}
  17. ψ | 𝒯 { ϕ ( x 0 ) ϕ ( x 1 ) } | ψ = i D ( x 0 , x 1 ) + λ 3 ! d d x 2 D ( x 0 , x 2 ) ψ | 𝒯 { ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x 2 ) ϕ ( x 2 ) } | ψ \langle\psi|\mathcal{T}\{\phi(x_{0})\phi(x_{1})\}|\psi\rangle=iD(x_{0},x_{1})+% \frac{\lambda}{3!}\int d^{d}x_{2}D(x_{0},x_{2})\langle\psi|\mathcal{T}\{\phi(x% _{1})\phi(x_{2})\phi(x_{2})\phi(x_{2})\}|\psi\rangle
  18. ψ | 𝒯 { ϕ ( x 0 ) ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x 3 ) } | ψ = i D ( x 0 , x 1 ) ψ | 𝒯 { ϕ ( x 2 ) ϕ ( x 3 ) } | ψ + i D ( x 0 , x 2 ) ψ | 𝒯 { ϕ ( x 1 ) ϕ ( x 3 ) } | ψ + i D ( x 0 , x 3 ) ψ | 𝒯 { ϕ ( x 1 ) ϕ ( x 2 ) } | ψ + λ 3 ! d d x 4 D ( x 0 , x 4 ) ψ | 𝒯 { ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x 3 ) ϕ ( x 4 ) ϕ ( x 4 ) ϕ ( x 4 ) } | ψ \begin{aligned}\displaystyle\langle\psi|\mathcal{T}\{\phi(x_{0})\phi(x_{1})% \phi(x_{2})\phi(x_{3})\}|\psi\rangle=&\displaystyle iD(x_{0},x_{1})\langle\psi% |\mathcal{T}\{\phi(x_{2})\phi(x_{3})\}|\psi\rangle+iD(x_{0},x_{2})\langle\psi|% \mathcal{T}\{\phi(x_{1})\phi(x_{3})\}|\psi\rangle\\ &\displaystyle+iD(x_{0},x_{3})\langle\psi|\mathcal{T}\{\phi(x_{1})\phi(x_{2})% \}|\psi\rangle\\ &\displaystyle+\frac{\lambda}{3!}\int d^{d}x_{4}D(x_{0},x_{4})\langle\psi|% \mathcal{T}\{\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\phi(x_{4})\phi(x_{4}% )\}|\psi\rangle\end{aligned}

Science_of_Logic.html

  1. k = y x . k={y\over x}.
  2. y = k x ; y=kx;\,
  3. x = y / k . x=y/k.\,
  4. k = x y k=xy\,
  5. y = k x . y={k\over x}.
  6. y = k x . y=k^{x}.\,
  7. v = d t . v={d\over t}.
  8. d = a t 2 d=at^{2}\,
  9. d 3 = a t 2 . d^{3}=at^{2}\,.

Science_of_value.html

  1. 0 \aleph_{0}
  2. 1 \aleph_{1}
  3. 1 \aleph_{1}
  4. 1 2 \frac{{}_{1}}{\aleph_{2}}
  5. 1 1 \frac{{}_{1}}{\aleph_{1}}

Scintillation_(physics).html

  1. ψ ( x ) = ψ ( x + l ) \psi(x)=\psi(x+l)\,
  2. ψ 0 \displaystyle\psi_{0}

Scorer's_function.html

  1. y ′′ ( x ) - x y ( x ) = 1 π y^{\prime\prime}(x)-x\ y(x)=\frac{1}{\pi}
  2. Gi ( x ) = 1 π 0 sin ( t 3 3 + x t ) d t , \mathrm{Gi}(x)=\frac{1}{\pi}\int_{0}^{\infty}\sin\left(\frac{t^{3}}{3}+xt% \right)\,dt,
  3. Hi ( x ) = 1 π 0 exp ( - t 3 3 + x t ) d t . \mathrm{Hi}(x)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-\frac{t^{3}}{3}+xt% \right)\,dt.
  4. Gi ( x ) \displaystyle\mathrm{Gi}(x)
  5. Gi ( z ) = 1 π 0 sin ( u z + 1 3 u 3 ) d u {\rm Gi}(z)=\frac{1}{\pi}\int^{\infty}_{0}{\rm sin}\left(uz+\frac{1}{3}u^{3}% \right)du

Scott_continuity.html

  1. f : P Q f:P\rightarrow Q
  2. f [ D ] = f ( D ) \sqcup f[D]=f(\sqcup D)

Scott_domain.html

  1. \bot
  2. x D x\in D
  3. x y x\sqsubseteq y
  4. y y
  5. x x
  6. X \sqcup X
  7. X D X\subseteq D
  8. X X
  9. X X
  10. X \sqcap X
  11. X X
  12. X X
  13. \bot

Search_tree.html

  1. 2 a ( b + 1 ) 2 2\leq a\leq\frac{(b+1)}{2}

Secant_method.html

  1. x n = x n - 1 - f ( x n - 1 ) x n - 1 - x n - 2 f ( x n - 1 ) - f ( x n - 2 ) = x n - 2 f ( x n - 1 ) - x n - 1 f ( x n - 2 ) f ( x n - 1 ) - f ( x n - 2 ) x_{n}=x_{n-1}-f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}=\frac{x_% {n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}
  2. y = f ( x 1 ) - f ( x 0 ) x 1 - x 0 ( x - x 1 ) + f ( x 1 ) y=\frac{f(x_{1})-f(x_{0})}{x_{1}-x_{0}}(x-x_{1})+f(x_{1})
  3. x x
  4. y = 0 y=0
  5. x x
  6. 0 = f ( x 1 ) - f ( x 0 ) x 1 - x 0 ( x - x 1 ) + f ( x 1 ) 0=\frac{f(x_{1})-f(x_{0})}{x_{1}-x_{0}}(x-x_{1})+f(x_{1})
  7. x = x 1 - f ( x 1 ) x 1 - x 0 f ( x 1 ) - f ( x 0 ) x=x_{1}-f(x_{1})\frac{x_{1}-x_{0}}{f(x_{1})-f(x_{0})}
  8. x x
  9. x 2 = x 1 - f ( x 1 ) x 1 - x 0 f ( x 1 ) - f ( x 0 ) x_{2}=x_{1}-f(x_{1})\frac{x_{1}-x_{0}}{f(x_{1})-f(x_{0})}
  10. x 3 = x 2 - f ( x 2 ) x 2 - x 1 f ( x 2 ) - f ( x 1 ) x_{3}=x_{2}-f(x_{2})\frac{x_{2}-x_{1}}{f(x_{2})-f(x_{1})}
  11. x n = x n - 1 - f ( x n - 1 ) x n - 1 - x n - 2 f ( x n - 1 ) - f ( x n - 2 ) x_{n}=x_{n-1}-f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}
  12. x n x_{n}
  13. f f
  14. x 0 x_{0}
  15. x 1 x_{1}
  16. α = 1 + 5 2 1.618 \alpha=\frac{1+\sqrt{5}}{2}\approx 1.618
  17. f f
  18. [ x 0 , x 1 ] [~{}x_{0},~{}x_{1}~{}]
  19. f f
  20. f = 0 f^{\prime}=0
  21. x n - 1 x_{n-1}
  22. x n x_{n}
  23. x n x_{n}
  24. x k x_{k}
  25. f ( x k ) f(x_{k})
  26. f ( x n ) f(x_{n})
  27. x n = x n - 1 - f ( x n - 1 ) f ( x n - 1 ) x_{n}=x_{n-1}-\frac{f(x_{n-1})}{f^{\prime}(x_{n-1})}
  28. f ( x n - 1 ) f ( x n - 1 ) - f ( x n - 2 ) x n - 1 - x n - 2 f^{\prime}(x_{n-1})\approx\frac{f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}
  29. f f
  30. f f^{\prime}
  31. f f
  32. f f
  33. f ( x ) = x < s u p > 2 612 f(x)=x<sup>2−612

Second-countable_space.html

  1. T T
  2. 𝒰 = { U i } i = 1 \mathcal{U}=\{U_{i}\}_{i=1}^{\infty}
  3. T T
  4. T T
  5. 𝒰 \mathcal{U}
  6. X = [ 0 , 1 ] [ 2 , 3 ] [ 4 , 5 ] [ 2 k , 2 k + 1 ] X=[0,1]\cup[2,3]\cup[4,5]\cup\cdots\cup[2k,2k+1]\cup\cdots

Second_derivative_test.html

  1. f ′′ ( x ) < 0 \ f^{\prime\prime}(x)<0
  2. f \ f
  3. x \ x
  4. f ′′ ( x ) > 0 \ f^{\prime\prime}(x)>0
  5. f \ f
  6. x \ x
  7. f ′′ ( x ) = 0 \ f^{\prime\prime}(x)=0
  8. f ′′ ( x ) > 0 f^{\prime\prime}(x)>0
  9. f ′′ ( x ) < 0 f^{\prime\prime}(x)<0
  10. f ( x ) = 0 f^{\prime}(x)=0
  11. 0 < f ′′ ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 f ( x + h ) - 0 h = lim h 0 f ( x + h ) h . 0<f^{\prime\prime}(x)=\lim_{h\to 0}\frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}=% \lim_{h\to 0}\frac{f^{\prime}(x+h)-0}{h}=\lim_{h\to 0}\frac{f^{\prime}(x+h)}{h}.
  12. f ( x + h ) h > 0 \frac{f^{\prime}(x+h)}{h}>0
  13. f ( x + h ) < 0 f^{\prime}(x+h)<0
  14. f f
  15. x x
  16. f ′′ ( x ) > 0 \ f^{\prime\prime}(x)>0
  17. f ′′ ( x ) < 0 \ f^{\prime\prime}(x)<0
  18. f ( x ) = x 4 \ f(x)=x^{4}
  19. x = 0 \ x=0

Second_fundamental_form.html

  1. I I \mathrm{I\!I}
  2. z = L x 2 2 + M x y + N y 2 2 + higher order terms , z=L\frac{x^{2}}{2}+Mxy+N\frac{y^{2}}{2}+\mathrm{\scriptstyle{{\ }higher{\ }% order{\ }terms}},
  3. L d x 2 + 2 M d x d y + N d y 2 . L\,\,\text{d}x^{2}+2M\,\,\text{d}x\,\,\text{d}y+N\,\,\text{d}y^{2}.\,
  4. 𝐧 = 𝐫 u × 𝐫 v | 𝐫 u × 𝐫 v | . \mathbf{n}=\frac{\mathbf{r}_{u}\times\mathbf{r}_{v}}{|\mathbf{r}_{u}\times% \mathbf{r}_{v}|}.
  5. I I = L d u 2 + 2 M d u d v + N d v 2 , \mathrm{I\!I}=L\,\,\text{d}u^{2}+2M\,\,\text{d}u\,\,\text{d}v+N\,\,\text{d}v^{% 2},\,
  6. [ L M M N ] . \begin{bmatrix}L&M\\ M&N\end{bmatrix}.
  7. L = 𝐫 u u 𝐧 , M = 𝐫 u v 𝐧 , N = 𝐫 v v 𝐧 . L=\mathbf{r}_{uu}\cdot\mathbf{n},\quad M=\mathbf{r}_{uv}\cdot\mathbf{n},\quad N% =\mathbf{r}_{vv}\cdot\mathbf{n}.
  8. 𝐧 = 𝐫 1 × 𝐫 2 | 𝐫 1 × 𝐫 2 | . \mathbf{n}=\frac{\mathbf{r}_{1}\times\mathbf{r}_{2}}{|\mathbf{r}_{1}\times% \mathbf{r}_{2}|}.
  9. I I = b α β d u α d u β . \mathrm{I\!I}=b_{\alpha\beta}\,\,\text{d}u^{\alpha}\,\,\text{d}u^{\beta}.\,
  10. b α β = r α β γ n γ . b_{\alpha\beta}=r_{\alpha\beta}^{\ \ \gamma}n_{\gamma}.
  11. I I ( v , w ) = - d ν ( v ) , w ν \mathrm{I\!I}(v,w)=-\langle d\nu(v),w\rangle\nu
  12. ν \nu
  13. d ν d\nu
  14. ν \nu
  15. S S
  16. I I ( v , w ) = S ( v ) , w n = - v n , w n = n , v w n , \mathrm{I}\!\mathrm{I}(v,w)=\langle S(v),w\rangle n=-\langle\nabla_{v}n,w% \rangle n=\langle n,\nabla_{v}w\rangle n,
  17. v w \nabla_{v}w
  18. n n
  19. n n
  20. I I ( v , w ) = ( v w ) , \mathrm{I\!I}(v,w)=(\nabla_{v}w)^{\bot},
  21. ( v w ) (\nabla_{v}w)^{\bot}
  22. v w \nabla_{v}w
  23. R ( u , v ) w , z = I I ( u , z ) , I I ( v , w ) - I I ( u , w ) , I I ( v , z ) . \langle R(u,v)w,z\rangle=\langle\mathrm{I}\!\mathrm{I}(u,z),\mathrm{I}\!% \mathrm{I}(v,w)\rangle-\langle\mathrm{I}\!\mathrm{I}(u,w),\mathrm{I}\!\mathrm{% I}(v,z)\rangle.
  24. N N
  25. M , g M,g
  26. R N R_{N}
  27. N N
  28. R M R_{M}
  29. M M
  30. R N ( u , v ) w , z = R M ( u , v ) w , z + I I ( u , z ) , I I ( v , w ) - I I ( u , w ) , I I ( v , z ) . \langle R_{N}(u,v)w,z\rangle=\langle R_{M}(u,v)w,z\rangle+\langle\mathrm{I}\!% \mathrm{I}(u,z),\mathrm{I}\!\mathrm{I}(v,w)\rangle-\langle\mathrm{I}\!\mathrm{% I}(u,w),\mathrm{I}\!\mathrm{I}(v,z)\rangle.

Sedimentation.html

  1. v t e r m v_{term}
  2. F d r a g = f v t e r m F_{drag}=fv_{term}
  3. F app = q E app F_{\mathrm{app}}=qE_{\mathrm{app}}
  4. s = def q / f s\ \stackrel{\mathrm{def}}{=}\ q/f
  5. m b / f m_{b}/f
  6. m b m_{b}
  7. m b / f m_{b}/f
  8. m b m_{b}

Seebeck_coefficient.html

  1. S = - Δ V Δ T S=-{\Delta V\over\Delta T}
  2. 𝐉 = - σ s y m b o l V - σ S s y m b o l T \mathbf{J}=-\sigma symbol\nabla V-\sigma Ssymbol\nabla T
  3. 𝐉 \scriptstyle\mathbf{J}
  4. σ \scriptstyle\sigma
  5. \scriptstylesymbol V \scriptstylesymbol\nabla V
  6. \scriptstylesymbol T \scriptstylesymbol\nabla T
  7. 𝐉 = 0 \scriptstyle\mathbf{J}=0
  8. \scriptstylesymbol V = - S s y m b o l T \scriptstylesymbol\nabla V=-Ssymbol\nabla T
  9. S = - V l e f t - V r i g h t T l e f t - T r i g h t S=-\frac{V_{left}-V_{right}}{T_{left}-T_{right}}
  10. Π \scriptstyle\Pi
  11. S = Π T , S=\frac{\Pi}{T},
  12. T T
  13. 𝒦 \scriptstyle\mathcal{K}
  14. S = 𝒦 T d T . S=\int\frac{\mathcal{K}}{T}\,dT.
  15. S = 0 \scriptstyle S=0
  16. S A B = S B - S A = Δ V B Δ T - Δ V A Δ T S_{AB}=S_{B}-S_{A}={\Delta V_{B}\over\Delta T}-{\Delta V_{A}\over\Delta T}
  17. μ \mu
  18. S ( T ) = 0 T μ ( T ) T d T S(T)=\int_{0}^{T}{\mu(T^{\prime})\over T^{\prime}}dT^{\prime}
  19. μ \mu
  20. S ( T ) S(T)
  21. μ / T \mu/T
  22. 𝐉 = 0 \scriptstyle\mathbf{J}=0
  23. σ \scriptstyle\sigma
  24. 𝐉 = - \sigmasymbol V \scriptstyle\mathbf{J}=-\sigmasymbol\nabla V
  25. 𝐉 = - σ S s y m b o l T \scriptstyle\mathbf{J}=-\sigma Ssymbol\nabla T
  26. σ S \scriptstyle\sigma S
  27. σ = c ( E ) ( - d f ( E ) d E ) d E , \sigma=\int c(E)\Bigg(-\frac{df(E)}{dE}\Bigg)\,dE,
  28. σ S = k B - e E - μ k B T c ( E ) ( - d f ( E ) d E ) d E \sigma S=\frac{k_{\rm B}}{-e}\int\frac{E-\mu}{k_{\rm B}T}c(E)\Bigg(-\frac{df(E% )}{dE}\Bigg)\,dE
  29. c ( E ) \scriptstyle c(E)
  30. f ( E ) \scriptstyle f(E)
  31. - d f ( E ) d E = 1 4 k T sech 2 E - μ 2 k T \scriptstyle-\frac{df(E)}{dE}=\frac{1}{4kT}\operatorname{sech}^{2}\tfrac{E-\mu% }{2kT}
  32. μ \scriptstyle\mu
  33. 3.5 k T \scriptstyle 3.5kT
  34. σ \scriptstyle\sigma
  35. c ( E ) = e 2 D ( E ) ν ( E ) \scriptstyle c(E)=e^{2}D(E)\nu(E)
  36. D ( E ) \scriptstyle D(E)
  37. ν ( E ) \scriptstyle\nu(E)
  38. c ( E ) \scriptstyle c(E)
  39. E μ ± k T E\approx\mu\pm kT
  40. c ( E ) = c ( μ ) + c ( μ ) ( E - μ ) + O [ ( E - μ ) 2 ] \scriptstyle c(E)=c(\mu)+c^{\prime}(\mu)(E-\mu)+O[(E-\mu)^{2}]
  41. S metal = π 2 k 2 T - 3 e c ( μ ) c ( μ ) + O [ ( k T ) 3 ] , σ metal = c ( μ ) + O [ ( k T ) 2 ] . S_{\rm metal}=\frac{\pi^{2}k^{2}T}{-3e}\frac{c^{\prime}(\mu)}{c(\mu)}+O[(kT)^{% 3}],\quad\sigma_{\rm metal}=c(\mu)+O[(kT)^{2}].
  42. c ( μ ) / c ( μ ) \scriptstyle c^{\prime}(\mu)/c(\mu)
  43. 1 / ( k T F ) \scriptstyle 1/(kT_{\rm F})
  44. T F T_{\rm F}
  45. S Fermi gas π 2 k - 3 e T / T F \scriptstyle S_{\rm Fermi~{}gas}\approx\tfrac{\pi^{2}k}{-3e}T/T_{\rm F}
  46. c ( μ ) / c ( μ ) \scriptstyle c^{\prime}(\mu)/c(\mu)
  47. σ metal \scriptstyle\sigma_{\rm metal}
  48. c ( μ ) / c ( μ ) \scriptstyle c^{\prime}(\mu)/c(\mu)
  49. E C - μ k T \scriptstyle E_{\rm C}-\mu\gg kT
  50. E C \scriptstyle E_{\rm C}
  51. - d f ( E ) d E 1 k T e - ( E - μ ) / ( k T ) \scriptstyle-\frac{df(E)}{dE}\approx\tfrac{1}{kT}e^{-(E-\mu)/(kT)}
  52. c ( E ) = A C ( E - E C ) a C \scriptstyle c(E)=A_{\rm C}(E-E_{\rm C})^{a_{\rm C}}
  53. A C \scriptstyle A_{\rm C}
  54. a C \scriptstyle a_{\rm C}
  55. S C = k - e [ E C - μ k T + a C + 1 ] , σ C = A C ( k T ) a C e - E C - μ k T Γ ( a C + 1 ) . S_{\rm C}=\frac{k}{-e}\Big[\frac{E_{\rm C}-\mu}{kT}+a_{\rm C}+1\Big],\quad% \sigma_{\rm C}=A_{\rm C}(kT)^{a_{\rm C}}e^{-\frac{E_{\rm C}-\mu}{kT}}\Gamma(a_% {\rm C}+1).
  56. μ - E V k T \scriptstyle\mu-E_{\rm V}\gg kT
  57. c ( E ) = A V ( E V - E ) a V \scriptstyle c(E)=A_{\rm V}(E_{\rm V}-E)^{a_{\rm V}}
  58. S V = k e [ μ - E V k T + a V + 1 ] , σ V = A V ( k T ) a V e - μ - E V k T Γ ( a V + 1 ) . S_{\rm V}=\frac{k}{e}\Big[\frac{\mu-E_{\rm V}}{kT}+a_{\rm V}+1\Big],\quad% \sigma_{\rm V}=A_{\rm V}(kT)^{a_{\rm V}}e^{-\frac{\mu-E_{\rm V}}{kT}}\Gamma(a_% {\rm V}+1).
  59. a C \scriptstyle a_{\rm C}
  60. a V \scriptstyle a_{\rm V}
  61. S semi = σ C S C + σ V S V σ C + σ V , σ semi = σ C + σ V S_{\rm semi}=\frac{\sigma_{\rm C}S_{\rm C}+\sigma_{\rm V}S_{\rm V}}{\sigma_{% \rm C}+\sigma_{\rm V}},\quad\sigma_{\rm semi}=\sigma_{\rm C}+\sigma_{\rm V}
  62. σ S 2 \scriptstyle\sigma S^{2}
  63. T 1 5 θ D T\approx{1\over 5}\theta_{\mathrm{D}}
  64. θ D \scriptstyle\theta_{D}

Seismic_refraction.html

  1. i c 0 = a s i n ( V 0 V 1 ) i_{c_{0}}=asin\left({V_{0}\over V_{1}}\right)
  2. T = 2 h 0 c o s ( i c 0 ) V 0 + X V 1 = T 0 1 + X V 1 T={2h_{0}cos(i_{c_{0}})\over V_{0}}+{X\over V_{1}}=T0_{1}+{X\over V_{1}}
  3. h 0 = T 0 1 V 0 2 c o s ( i c ) h_{0}={T0_{1}V_{0}\over 2cos(i_{c})}
  4. h 0 = X c r o s s 1 2 V 1 - V 0 V 1 + V 0 h_{0}={X_{cross_{1}}\over 2}\sqrt{{V_{1}-V_{0}\over V_{1}+V_{0}}}
  5. h n = V n c o s ( i n ) ( T 0 n + 1 2 - j = 0 n - 1 h j 1 V j 2 - 1 V j + 1 2 ) h_{n}={V_{n}\over cos(i_{n})}\left({T0_{n+1}\over 2}-\sum_{j=0}^{n-1}{h_{j}% \sqrt{{1\over V_{j}^{2}}-{1\over V_{j+1}^{2}}}}\right)

Selberg_trace_formula.html

  1. G G
  2. G G
  3. Γ Γ
  4. G G
  5. Γ Γ
  6. 𝐙 \mathbf{Z}
  7. G = 𝐑 G=\mathbf{R}
  8. G / Γ G/Γ
  9. G G
  10. S L ( 2 , 𝐑 ) SL(2,\mathbf{R})
  11. Γ Γ
  12. S S
  13. S S
  14. Γ Γ
  15. X X
  16. Γ \ 𝐇 , \Gamma\backslash\mathbf{H},
  17. Γ Γ
  18. P S L ( 2 , 𝐑 ) PSL(2,\mathbf{R})
  19. 𝐇 \mathbf{H}
  20. Γ Γ
  21. 𝐇 \mathbf{H}
  22. Γ Γ
  23. X X
  24. 0 = μ 0 < μ 1 μ 2 0=\mu_{0}<\mu_{1}\leq\mu_{2}\leq\cdots
  25. Γ Γ
  26. u u
  27. { u ( γ z ) = u ( z ) , γ Γ y 2 ( u x x + u y y ) + μ n u = 0. \begin{cases}u(\gamma z)=u(z),\qquad\forall\gamma\in\Gamma\\ y^{2}\left(u_{xx}+u_{yy}\right)+\mu_{n}u=0.\end{cases}
  28. μ = s ( 1 - s ) , s = 1 2 + i r \mu=s(1-s),\qquad s=\tfrac{1}{2}+ir
  29. r n , n 0. r_{n},n\geq 0.
  30. n = 0 h ( r n ) = μ ( F ) 4 π - r h ( r ) tanh ( π r ) d r + { T } log N ( T 0 ) N ( T ) 1 2 - N ( T ) - 1 2 g ( log N ( T ) ) . \sum_{n=0}^{\infty}h(r_{n})=\frac{\mu(F)}{4\pi}\int_{-\infty}^{\infty}r\,h(r)% \tanh(\pi r)dr+\sum_{\{T\}}\frac{\log N(T_{0})}{N(T)^{\frac{1}{2}}-N(T)^{-% \frac{1}{2}}}g\left(\log N(T)\right).
  31. Γ Γ
  32. h h
  33. | I m ( r ) | 1 2 + δ |Im(r)|≤\frac{1}{2}+δ
  34. h ( r ) = h ( r ) h(−r)=h(r)
  35. δ δ
  36. M M
  37. | h ( r ) | M ( 1 + | Re ( r ) | - 2 - δ ) . |h(r)|\leq M\left(1+|\,\text{Re}(r)|^{-2-\delta}\right).
  38. g g
  39. h h
  40. h ( r ) = - g ( u ) e i r u d u . h(r)=\int_{-\infty}^{\infty}g(u)e^{iru}du.

Self-stabilization.html

  1. k k

Self-verifying_theories.html

  1. Π 2 0 \Pi^{0}_{2}
  2. ( x , y ) ( z ) multiply ( x , y , z ) . (\forall x,y)\ (\exists z)\ {\rm multiply}(x,y,z).
  3. multiply {\rm multiply}
  4. z / y = x z/y=x
  5. Π 1 0 \Pi^{0}_{1}

Self_number.html

  1. C k = 8 10 k - 1 + C k - 1 + 8 C_{k}=8\cdot 10^{k-1}+C_{k-1}+8
  2. C k = 2 j + C k - 1 + 1 C_{k}=2^{j}+C_{k-1}+1\,
  3. C k = ( b - 2 ) b k - 1 + C k - 1 + ( b - 2 ) C_{k}=(b-2)b^{k-1}+C_{k-1}+(b-2)\,
  4. D R * ( n ) = { D R ( n ) 2 , if D R ( n ) is even D R ( n ) + 9 2 , if D R ( n ) is odd DR*(n)=\begin{cases}\frac{DR(n)}{2},&\mbox{if }~{}DR(n)\mbox{ is even}\\ \frac{DR(n)+9}{2},&\mbox{if }~{}DR(n)\mbox{ is odd}\end{cases}
  5. D R ( n ) = { 9 , if S O D ( n ) mod 9 = 0 S O D ( n ) mod 9 , otherwise = ( n - 1 ) mod 9 + 1 \begin{aligned}\displaystyle DR(n)&\displaystyle{}=\begin{cases}9,&\mbox{if }~% {}SOD(n)\mod 9=0\\ SOD(n)\mod 9,&\mbox{ otherwise}\end{cases}\\ &\displaystyle{}=(n-1)\mod 9+1\end{aligned}
  6. S O D ( n ) is the sum of all digits in n SOD(n)\mbox{ is the sum of all digits in }~{}n
  7. d ( n ) is the number of digits in n d(n)\mbox{ is the number of digits in }~{}n
  8. 1959 = [ 1 , 8 , 39 ] 40 1959=[1,8,39]_{40}
  9. 1967 = [ 1 , 4 , 35 ] 42 1967=[1,4,35]_{42}
  10. 1971 = [ 1 , 0 , 35 ] 44 1971=[1,0,35]_{44}
  11. 1928 = [ 43 , 36 ] 44 1928=[43,36]_{44}
  12. 1926 = [ 41 , 40 ] 46 1926=[41,40]_{46}
  13. 1959 = [ 39 , 9 ] 50 1959=[39,9]_{50}
  14. 1947 = [ 37 , 23 ] 52 1947=[37,23]_{52}
  15. 1931 = [ 35 , 41 ] 54 1931=[35,41]_{54}
  16. 1966 = [ 35 , 6 ] 56 1966=[35,6]_{56}
  17. 1944 = [ 33 , 30 ] 58 1944=[33,30]_{58}
  18. 1918 = [ 31 , 58 ] 60 1918=[31,58]_{60}

Semantic_security.html

  1. m m
  2. ( p k , s k ) (pk,sk)
  3. G e n ( 1 n ) Gen(1^{n})
  4. p k pk
  5. m 0 m_{0}
  6. m 1 m_{1}
  7. b 0 , 1 b\in{0,1}
  8. m b m_{b}
  9. c c
  10. 1 / 2 1/2
  11. m 0 m_{0}
  12. m 1 m_{1}
  13. c c

Semi-empirical_mass_formula.html

  1. m = Z m p + N m n - E B c 2 m=Zm_{p}+Nm_{n}-\frac{E_{B}}{c^{2}}
  2. m p m_{p}
  3. m n m_{n}
  4. E B E_{B}
  5. E B = a V A - a S A 2 / 3 - a C Z 2 A 1 / 3 - a A ( A - 2 Z ) 2 A - δ ( A , Z ) E_{B}=a_{V}A-a_{S}A^{2/3}-a_{C}\frac{Z^{2}}{A^{1/3}}-a_{A}\frac{(A-2Z)^{2}}{A}% -\delta(A,Z)
  6. a V a_{V}
  7. a S a_{S}
  8. a C a_{C}
  9. a A a_{A}
  10. δ ( A , Z ) \delta(A,Z)
  11. a V A a_{V}A
  12. A ( A - 1 ) 2 \frac{A(A-1)}{2}
  13. A 2 A^{2}
  14. a V a_{V}
  15. E b E_{b}
  16. A A
  17. 3 5 A ϵ F {3\over 5}A\epsilon_{F}
  18. ϵ F \epsilon_{F}
  19. a V a_{V}
  20. E b - 3 5 ϵ F 17 MeV E_{b}-{3\over 5}\epsilon_{F}\sim 17\;\mathrm{MeV}
  21. a S A 2 / 3 a_{S}A^{2/3}
  22. A 1 / 3 A^{1/3}
  23. A 2 / 3 A^{2/3}
  24. A 2 / 3 A^{2/3}
  25. a S a_{S}
  26. a V a_{V}
  27. a C Z ( Z - 1 ) A 1 / 3 a_{C}\frac{Z(Z-1)}{A^{1/3}}
  28. a C Z 2 A 1 / 3 a_{C}\frac{Z^{2}}{A^{1/3}}
  29. E = 3 5 ( 1 4 π ϵ 0 ) Q 2 R E=\frac{3}{5}\left(\frac{1}{4\pi\epsilon_{0}}\right)\frac{Q^{2}}{R}
  30. Z e Ze
  31. A 1 / 3 A^{1/3}
  32. Z 2 Z^{2}
  33. Z ( Z - 1 ) Z(Z-1)
  34. a C a_{C}
  35. R r 0 A 1 3 . R\approx r_{0}A^{\frac{1}{3}}.
  36. Q = Z e Q=Ze
  37. Z 2 Z ( Z - 1 ) . Z^{2}\approx Z(Z-1)\ .
  38. E = 3 5 ( 1 4 π ϵ 0 ) Q 2 R = 3 5 ( 1 4 π ϵ 0 ) ( Z e ) 2 ( r 0 A 1 3 ) = 3 e 2 Z 2 20 π ϵ 0 r 0 A 1 3 3 e 2 Z ( Z - 1 ) 20 π ϵ 0 r 0 A 1 3 = a C Z ( Z - 1 ) A 1 / 3 E=\frac{3}{5}\left(\frac{1}{4\pi\epsilon_{0}}\right)\frac{Q^{2}}{R}=\frac{3}{5% }\left(\frac{1}{4\pi\epsilon_{0}}\right)\frac{(Ze)^{2}}{(r_{0}A^{\frac{1}{3}})% }=\frac{3e^{2}Z^{2}}{20\pi\epsilon_{0}r_{0}A^{\frac{1}{3}}}\approx\frac{3e^{2}% Z(Z-1)}{20\pi\epsilon_{0}r_{0}A^{\frac{1}{3}}}=a_{C}\frac{Z(Z-1)}{A^{1/3}}
  39. E = 3 e 2 Z ( Z - 1 ) 20 π ϵ 0 r 0 A 1 3 E=\frac{3e^{2}Z(Z-1)}{20\pi\epsilon_{0}r_{0}A^{\frac{1}{3}}}
  40. a C = 3 e 2 20 π ϵ 0 r 0 a_{C}=\frac{3e^{2}}{20\pi\epsilon_{0}r_{0}}
  41. a C a_{C}
  42. a C = 3 5 ( c α r 0 ) = 3 5 ( R P r 0 ) α m p c 2 a_{C}=\frac{3}{5}\left(\frac{\hbar c\alpha}{r_{0}}\right)=\frac{3}{5}\left(% \frac{R_{P}}{r_{0}}\right)\alpha m_{p}c^{2}
  43. α \alpha
  44. r 0 A 1 / 3 r_{0}A^{1/3}
  45. r 0 r_{0}
  46. R P R_{P}
  47. m p m_{p}
  48. a C a_{C}
  49. a C = 0.691 MeV a_{C}=0.691\;\,\text{MeV}
  50. a A ( A - 2 Z ) 2 A a_{A}\frac{(A-2Z)^{2}}{A}
  51. 4 a A ( ( A / 2 ) - Z ) 2 A {4}a_{A}\frac{((A/2)-Z)^{2}}{A}
  52. A = N + Z A=N+Z
  53. ( N - Z ) (N-Z)
  54. ( A - 2 Z ) (A-2Z)
  55. E k = 3 5 ( N p ϵ F p + N n ϵ F n ) E_{k}={3\over 5}(N_{p}{\epsilon_{F}}_{p}+N_{n}{\epsilon_{F}}_{n})
  56. N p N_{p}
  57. N n N_{n}
  58. ϵ F p {\epsilon_{F}}_{p}
  59. ϵ F n {\epsilon_{F}}_{n}
  60. N p 2 / 3 {N_{p}}^{2/3}
  61. N n 2 / 3 {N_{n}}^{2/3}
  62. E k = C ( N p 5 / 3 + N n 5 / 3 ) E_{k}=C(N_{p}^{5/3}+N_{n}^{5/3})
  63. N n - N p N_{n}-N_{p}
  64. E k = C 2 2 / 3 ( ( N p + N n ) 5 / 3 + 5 9 ( N n - N p ) 2 ( N p + N n ) 1 / 3 ) + O ( ( N n - N p ) 2 ) . E_{k}={C\over 2^{2/3}}\left((N_{p}+N_{n})^{5/3}+{5\over 9}{(N_{n}-N_{p})^{2}% \over(N_{p}+N_{n})^{1/3}}\right)+O((N_{n}-N_{p})^{2}).
  65. ϵ F ϵ F p = ϵ F n \epsilon_{F}\equiv{\epsilon_{F}}_{p}={\epsilon_{F}}_{n}
  66. 3 5 ( N p + N n ) 2 / 3 {3\over 5}(N_{p}+N_{n})^{2/3}
  67. E k = 3 5 ϵ F ( N p + N n ) + 1 3 ϵ F ( N n - N p ) 2 ( N p + N n ) + O ( ( N n - N p ) 4 ) = 3 5 ϵ F A + 1 3 ϵ F ( A - 2 Z ) 2 A + O ( ( A - 2 Z ) 4 ) . E_{k}={3\over 5}\epsilon_{F}(N_{p}+N_{n})+{1\over 3}\epsilon_{F}{(N_{n}-N_{p})% ^{2}\over(N_{p}+N_{n})}+O((N_{n}-N_{p})^{4})={3\over 5}\epsilon_{F}A+{1\over 3% }\epsilon_{F}{(A-2Z)^{2}\over A}+O((A-2Z)^{4}).
  68. ϵ F \epsilon_{F}
  69. a A a_{A}
  70. | N - Z | |N-Z|
  71. ( A - 2 Z ) 2 (A-2Z)^{2}
  72. | N - Z | |N-Z|
  73. δ ( A , Z ) \delta(A,Z)
  74. δ ( A , Z ) = { + δ 0 Z , N even ( A even ) 0 A odd - δ 0 Z , N odd ( A even ) \delta(A,Z)=\begin{cases}+\delta_{0}&Z,N\mbox{ even }~{}(A\mbox{ even}~{})\\ 0&A\mbox{ odd}\\ -\delta_{0}&Z,N\mbox{ odd }~{}(A\mbox{ even}~{})\end{cases}
  75. δ 0 = a P A 1 / 2 . \delta_{0}=\frac{a_{P}}{A^{1/2}}.
  76. A - 1 / 2 A^{-1/2}
  77. A - 1 A^{-1}
  78. a V a_{V}
  79. a S a_{S}
  80. a C a_{C}
  81. a A a_{A}
  82. a P a_{P}
  83. δ \delta
  84. δ \delta
  85. δ \delta
  86. N / Z 1 + a C 2 a A A 2 / 3 . N/Z\approx 1+\frac{a_{C}}{2a_{A}}A^{2/3}.

Semi-local_ring.html

  1. / m \mathbb{Z}/m\mathbb{Z}
  2. m m
  3. / m \mathbb{Z}/m\mathbb{Z}
  4. i = 1 n F i \bigoplus_{i=1}^{n}{F_{i}}
  5. R / i = 1 n m i i = 1 n R / m i R/\bigcap_{i=1}^{n}m_{i}\cong\bigoplus_{i=1}^{n}R/m_{i}\,

Semi-minor_axis.html

  1. a a
  2. e e
  3. l l
  4. b = a 1 - e 2 b=a\sqrt{1-e^{2}}\,\!
  5. a l = b 2 al=b^{2}\,\!
  6. r m a x r_{max}
  7. r m i n r_{min}
  8. b = r m a x r m i n . b=\sqrt{r_{max}r_{min}}.
  9. 2 b = ( p + q ) 2 - f 2 2b=\sqrt{(p+q)^{2}-f^{2}}
  10. x 2 a 2 - y 2 b 2 = 1. \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1.
  11. b = a e 2 - 1 . b=a\sqrt{e^{2}-1}.

Semiclassical_gravity.html

  1. G μ ν G_{\mu\nu}
  2. T μ ν T_{\mu\nu}
  3. G μ ν = 8 π G c 4 T ^ μ ν ψ G_{\mu\nu}=\frac{8\pi G}{c^{4}}\left\langle\hat{T}_{\mu\nu}\right\rangle_{\psi}
  4. ψ \psi
  5. d d x - g R 2 \int d^{d}x\,\sqrt{-g}R^{2}
  6. d d x - g R μ ν R μ ν \int d^{d}x\,\sqrt{-g}R^{\mu\nu}R_{\mu\nu}
  7. d d x - g R μ ν ρ σ R μ ν ρ σ \int d^{d}x\,\sqrt{-g}R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}
  8. 1 2 ( | M at A + | M at B ) \frac{1}{\sqrt{2}}\left(\left|M\,\text{ at }A\right\rangle+\left|M\,\text{ at % }B\right\rangle\right)

Semidefinite_embedding.html

  1. X X\,\!
  2. Y Y\,\!
  3. i , j i,j\,\!
  4. | X i - X j | 2 = | Y i - Y j | 2 |X_{i}-X_{j}|^{2}=|Y_{i}-Y_{j}|^{2}\,\!
  5. G , K G,K\,\!
  6. X X\,\!
  7. Y Y\,\!
  8. G i j = X i X j , K i j = Y i Y j G_{ij}=X_{i}\cdot X_{j},K_{ij}=Y_{i}\cdot Y_{j}\,\!
  9. i , j i,j\,\!
  10. G , K G,K\,\!
  11. G i i + G j j - G i j - G j i = K i i + K j j - K i j - K j i G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji}\,\!
  12. Y Y\,\!
  13. i Y i = 0 ( i Y i ) 2 = 0 i , j Y i Y j = 0 i , j K i j = 0 \sum_{i}Y_{i}=0\Leftrightarrow(\sum_{i}Y_{i})^{2}=0\Rightarrow\sum_{i,j}Y_{i}Y% _{j}=0\Rightarrow\sum_{i,j}K_{ij}=0
  14. T ( Y ) = 1 2 N i , j | Y i - Y j | 2 T(Y)=\dfrac{1}{2N}\sum_{i,j}|Y_{i}-Y_{j}|^{2}
  15. τ = m a x { η i j | Y i - Y j | 2 } \tau=max\{\eta_{ij}|Y_{i}-Y_{j}|^{2}\}\,\!
  16. η i j = 1 \eta_{ij}=1\,\!
  17. η i j = 0 \eta_{ij}=0\,\!
  18. | Y i - Y j | 2 N τ |Y_{i}-Y_{j}|^{2}\leq N\tau\,\!
  19. T ( Y ) = 1 2 N i , j | Y i - Y j | 2 1 2 N i , j ( N τ ) 2 = N 3 τ 2 2 T(Y)=\dfrac{1}{2N}\sum_{i,j}|Y_{i}-Y_{j}|^{2}\leq\dfrac{1}{2N}\sum_{i,j}(N\tau% )^{2}=\dfrac{N^{3}\tau^{2}}{2}\,\!
  20. T ( Y ) = 1 2 N i , j | Y i - Y j | 2 = 1 2 N ( i , j ( Y i 2 + Y j 2 - Y i Y j - Y j Y i ) = 1 2 N ( i , j Y i 2 + i , j Y j 2 - i , j Y i Y j - i , j Y j Y i ) = 1 2 N ( i , j Y i 2 + i , j Y j 2 - 0 - 0 ) = 1 N ( i Y i 2 ) = 1 N ( T r ( K ) ) \begin{aligned}\displaystyle T(Y)&\displaystyle{}=\dfrac{1}{2N}\sum_{i,j}|Y_{i% }-Y_{j}|^{2}\\ &\displaystyle{}=\dfrac{1}{2N}(\sum_{i,j}(Y_{i}^{2}+Y_{j}^{2}-Y_{i}\cdot Y_{j}% -Y_{j}\cdot Y_{i})\\ &\displaystyle{}=\dfrac{1}{2N}(\sum_{i,j}Y_{i}^{2}+\sum_{i,j}Y_{j}^{2}-\sum_{i% ,j}Y_{i}\cdot Y_{j}-\sum_{i,j}Y_{j}\cdot Y_{i})\\ &\displaystyle{}=\dfrac{1}{2N}(\sum_{i,j}Y_{i}^{2}+\sum_{i,j}Y_{j}^{2}-0-0)\\ &\displaystyle{}=\dfrac{1}{N}(\sum_{i}Y_{i}^{2})=\dfrac{1}{N}(Tr(K))\\ \end{aligned}\,\!
  21. T r ( K ) Tr(K)\,\!
  22. K 0 K\succeq 0\,\!
  23. i , j \forall i,j\,\!
  24. η i j = 1 , G i i + G j j - G i j - G j i = K i i + K j j - K i j - K j i \eta_{ij}=1,G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji}\,\!
  25. K K\,\!
  26. Y Y\,\!
  27. K i j = α = 1 N ( λ α V α i V α j ) K_{ij}=\sum_{\alpha=1}^{N}(\lambda_{\alpha}V_{\alpha i}V_{\alpha j})\,\!
  28. V α i V_{\alpha i}\,\!
  29. V α V_{\alpha}\,\!
  30. λ α \lambda_{\alpha}\,\!
  31. α \alpha\,\!
  32. Y i Y_{i}\,\!
  33. λ α V α i \sqrt{\lambda_{\alpha}}V_{\alpha i}\,\!

Semigroup_action.html

  1. T T
  2. T : S × X X T\colon S\times X\to X
  3. S S
  4. T ( s , x ) T(s,x)
  5. s x s\cdot x
  6. s s
  7. S S
  8. T s : X X T_{s}\colon X\to X
  9. X X
  10. T s ( x ) = T ( s , x ) . T_{s}(x)=T(s,x).
  11. S S
  12. T T
  13. T s * t = T s T t . T_{s*t}=T_{s}\circ T_{t}.
  14. s T s s\mapsto T_{s}
  15. c u r r y ( T ) : S ( X X ) curry(T):S\to(X\to X)
  16. c u r r y curry
  17. S ( X X ) S\to(X\to X)
  18. c u r r y ( T ) ( s * t ) = c u r r y ( T ) ( s ) c u r r y ( T ) ( t ) . curry(T)(s*t)=curry(T)(s)\circ curry(T)(t).
  19. T T
  20. S S
  21. X X
  22. c u r r y ( T ) curry(T)
  23. S S
  24. X X
  25. F : X X F\colon X\to X^{\prime}
  26. F ( s x ) = s F ( x ) F(sx)=sF(x)
  27. s S s\in S
  28. x X x\in X
  29. Hom S ( X , X ) \mathrm{Hom}_{S}(X,X^{\prime})
  30. S S
  31. X X
  32. T T
  33. S S
  34. X X
  35. T ( s , x ) = s ( x ) T(s,x)=s(x)
  36. s S , x X s\in S,x\in X
  37. c u r r y ( T ) curry(T)
  38. T T
  39. S S
  40. X X
  41. S = { T s | s S } S^{\prime}=\{T_{s}|s\in S\}
  42. S S
  43. S S^{\prime}
  44. c u r r y ( T ) curry(T)
  45. c u r r y ( T ) curry(T)
  46. f f
  47. c u r r y ( T ) curry(T)
  48. f f
  49. S S
  50. S S^{\prime}
  51. T T
  52. S S^{\prime}
  53. T T^{\prime}
  54. S S^{\prime}
  55. X X
  56. T ( f ( s ) , x ) = T ( s , x ) T^{\prime}(f(s),x)=T(s,x)
  57. s S , x X s\in S,x\in X
  58. T T
  59. T T^{\prime}
  60. f f
  61. T T
  62. T : Σ × X X T\colon\Sigma\times X\to X
  63. T v w = T w T v . T_{vw}=T_{w}\circ T_{v}.

Semilattice.html

  1. S , \langle S,\land\rangle
  2. S , \langle S,\land\rangle
  3. 𝒮 \mathcal{S}
  4. ( , 0 ) (\vee,0)
  5. 𝒜 \mathcal{A}
  6. S S
  7. Id S \operatorname{Id}\ S
  8. ( , 0 ) (\vee,0)
  9. f : S T f\colon S\to T
  10. ( , 0 ) (\vee,0)
  11. Id f : Id S Id T \operatorname{Id}\ f\colon\operatorname{Id}\ S\to\operatorname{Id}\ T
  12. I I
  13. S S
  14. T T
  15. f ( I ) f(I)
  16. Id : 𝒮 𝒜 \operatorname{Id}\colon\mathcal{S}\to\mathcal{A}
  17. A A
  18. ( , 0 ) (\vee,0)
  19. K ( A ) K(A)
  20. A A
  21. f : A B f\colon A\to B
  22. K ( f ) : K ( A ) K ( B ) K(f)\colon K(A)\to K(B)
  23. K : 𝒜 𝒮 K\colon\mathcal{A}\to\mathcal{S}
  24. ( Id , K ) (\operatorname{Id},K)
  25. 𝒮 \mathcal{S}
  26. 𝒜 \mathcal{A}
  27. \vee

Semiperimeter.html

  1. s = a + b + c 2 . s=\frac{a+b+c}{2}.
  2. s = | A B | + | A B | = | A B | + | A B | = | A C | + | A C | s=|AB|+|A^{\prime}B|=|AB|+|AB^{\prime}|=|AC|+|A^{\prime}C|
  3. = | A C | + | A C | = | B C | + | B C | = | B C | + | B C | . =|AC|+|AC^{\prime}|=|BC|+|B^{\prime}C|=|BC|+|BC^{\prime}|.
  4. K = r s . K=rs.
  5. K = s ( s - a ) ( s - b ) ( s - c ) . K=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}.
  6. R = a b c 4 s ( s - a ) ( s - b ) ( s - c ) . R=\frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}.
  7. r = ( s - a ) ( s - b ) ( s - c ) s . r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.
  8. t a = 2 b c s ( s - a ) b + c . t_{a}=\frac{2\sqrt{bcs(s-a)}}{b+c}.
  9. ( s - a ) ( s - b ) (s-a)(s-b)
  10. s = a + b + c + d 2 . s=\frac{a+b+c+d}{2}.
  11. K = r s . K=rs.
  12. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) . K=\sqrt{\left(s-a\right)\left(s-b\right)\left(s-c\right)\left(s-d\right)}.
  13. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) - a b c d cos 2 ( α + γ 2 ) , K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cdot\cos^{2}\left(\frac{\alpha+\gamma}{2}% \right)},
  14. α \alpha\,
  15. γ \gamma\,

Semisimple_module.html

  1. 3 2 3\Rightarrow 2
  2. x M x\in M
  3. P P
  4. x P x\notin P
  5. P P
  6. F k M F\otimes_{k}M
  7. F k A F\otimes_{k}A
  8. M i M_{i}
  9. i M i \bigoplus_{i}M_{i}
  10. k [ G ] k[G]
  11. M n 1 ( D 1 ) × M n 2 ( D 2 ) × × M n r ( D r ) M_{n_{1}}(D_{1})\times M_{n_{2}}(D_{2})\times\dots\times M_{n_{r}}(D_{r})
  12. D i D_{i}
  13. n i n_{i}
  14. M n ( D ) M_{n}(D)
  15. M ( K ) M_{\infty}(K)

Sensor_array.html

  1. Δ t i = ( i - 1 ) d cos θ c , i = 1 , 2 , , M ( 1 ) \Delta t_{i}=\frac{(i-1)d\cos\theta}{c},i=1,2,...,M\ \ (1)
  2. y = 1 M i = 1 M s y m b o l x i ( t - Δ t i ) ( 2 ) y=\frac{1}{M}\sum_{i=1}^{M}symbolx_{i}(t-\Delta t_{i})\ \ (2)
  3. y = 1 M i = 1 M [ s y m b o l w i s y m b o l x i ( t - Δ t i ) ] ( 3 ) y=\frac{1}{M}\sum_{i=1}^{M}[symbolw_{i}symbolx_{i}(t-\Delta t_{i})]\ \ (3)
  4. ω τ \omega\tau
  5. ± π \pm\pi
  6. θ \theta
  7. [ - π 2 , π 2 ] [-\frac{\pi}{2},\frac{\pi}{2}]
  8. d λ / 2 d\leq\lambda/2
  9. θ ^ [ 0 , π ] \hat{\theta}\in[0,\pi]
  10. s y m b o l x ( t ) = x 1 ( t ) [ 1 e - j ω Δ t e - j ω ( M - 1 ) Δ t ] T symbolx(t)=x_{1}(t)\begin{bmatrix}1&e^{-j\omega\Delta t}&\cdots&e^{-j\omega(M-% 1)\Delta t}\end{bmatrix}^{T}
  11. x 1 ( t ) x_{1}(t)
  12. s y m b o l R = E { s y m b o l x ( t ) s y m b o l x T ( t ) } symbolR=E\{symbolx(t)symbolx^{T}(t)\}
  13. s y m b o l R = s y m b o l V s y m b o l S s y m b o l V H + σ 2 s y m b o l I ( 4 ) symbolR=symbolVsymbolSsymbolV^{H}+\sigma^{2}symbolI\ \ (4)
  14. σ 2 \sigma^{2}
  15. s y m b o l I symbolI
  16. s y m b o l V symbolV
  17. s y m b o l V = [ s y m b o l v 1 s y m b o l v k ] T symbolV=\begin{bmatrix}symbolv_{1}&\cdots&symbolv_{k}\end{bmatrix}^{T}
  18. s y m b o l v i = [ 1 e - j ω Δ t i e - j ω ( M - 1 ) Δ t i ] T symbolv_{i}=\begin{bmatrix}1&e^{-j\omega\Delta t_{i}}&\cdots&e^{-j\omega(M-1)% \Delta t_{i}}\end{bmatrix}^{T}
  19. P ^ B a r t l e t t ( θ ) = s y m b o l v H s y m b o l R s y m b o l v ( 5 ) \hat{P}_{Bartlett}(\theta)=symbolv^{H}symbolRsymbolv\ \ (5)
  20. P ^ C a p o n ( θ ) = 1 s y m b o l v H s y m b o l R - 1 s y m b o l v ( 6 ) \hat{P}_{Capon}(\theta)=\frac{1}{symbolv^{H}symbolR^{-1}symbolv}\ \ (6)
  21. s y m b o l R = s y m b o l U s s y m b o l Λ s s y m b o l U s H + s y m b o l U n s y m b o l Λ n s y m b o l U n H ( 7 ) symbolR=symbolU_{s}symbol\Lambda_{s}symbolU_{s}^{H}+symbolU_{n}symbol\Lambda_{% n}symbolU_{n}^{H}\ \ (7)
  22. P ^ M U S I C ( θ ) = 1 s y m b o l v H s y m b o l U n s y m b o l U n H s y m b o l v ( 8 ) \hat{P}_{MUSIC}(\theta)=\frac{1}{symbolv^{H}symbolU_{n}symbolU_{n}^{H}symbolv}% \ \ (8)
  23. L M L ( θ ) = s y m b o l R ^ - s y m b o l R F 2 = s y m b o l R ^ - ( s y m b o l V s y m b o l S s y m b o l V H + σ 2 s y m b o l I ) F 2 ( 9 ) L_{ML}(\theta)=\|\hat{symbolR}-symbolR\|_{F}^{2}=\|\hat{symbolR}-(% symbolVsymbolSsymbolV^{H}+\sigma^{2}symbolI)\|_{F}^{2}\ \ (9)
  24. F \|\cdot\|_{F}
  25. θ \theta
  26. s y m b o l V symbolV
  27. x n + 1 = x n - f ( x n ) f ( x n ) ( 10 ) x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\ \ (10)
  28. x 0 x_{0}

Separable_sigma_algebra.html

  1. \mathcal{F}
  2. ρ ( A , B ) = μ ( A B ) \rho(A,B)=\mu(A\triangle B)
  3. A , B A,B\in\mathcal{F}
  4. μ \mu
  5. \triangle

Sequence_logo.html

  1. i i
  2. R i = log 2 ( 20 ) - ( H i + e n ) R_{i}=\log_{2}(20)-(H_{i}+e_{n})
  3. R i = 2 - ( H i + e n ) R_{i}=2-(H_{i}+e_{n})
  4. H i H_{i}
  5. i i
  6. H i = - f a , i × log 2 f a , i H_{i}=-\sum f_{a,i}\times\log_{2}f_{a,i}
  7. f a , i f_{a,i}
  8. a a
  9. i i
  10. e n e_{n}
  11. n n
  12. a a
  13. i i
  14. height = f a , i × R i \,\text{height}=f_{a,i}\times R_{i}
  15. e n e_{n}
  16. e n = 1 ln 2 × s - 1 2 n e_{n}=\frac{1}{\ln{2}}\times\frac{s-1}{2n}
  17. s s
  18. n n

Series-parallel_networks_problem.html

  1. a n a_{n}
  2. b n b_{n}
  3. a n = { 1 , if n is 1 2 b n , otherwise a_{n}=\left\{\begin{matrix}1,&\mbox{if }~{}n\mbox{ is 1}\\ 2b_{n},&\mbox{otherwise}\end{matrix}\right.
  4. b n b_{n}
  5. n n
  6. { p i } \{p_{i}\}
  7. i i p i = n . \sum_{i}{ip_{i}}=n.
  8. p i p_{i}
  9. i ( b i + p i - 1 p i ) . \prod_{i}{{b_{i}+p_{i}-1}\choose{p_{i}}}.
  10. b n = p i i ( b i + p i - 1 p i ) b_{n}=\sum_{p_{i}}{\prod_{i}{{b_{i}+p_{i}-1}\choose{p_{i}}}}
  11. p i p_{i}
  12. b n b_{n}
  13. a n a_{n}
  14. a n a_{n}
  15. b n b_{n}

Sesquilinear_form.html

  1. V V
  2. V × V 𝐂 V×V→\mathbf{C}
  3. K K
  4. K K
  5. R R
  6. R R
  7. V V
  8. φ : V × V 𝐂 φ:V×V→\mathbf{C}
  9. φ ( x + y , z + w ) = φ ( x , z ) + φ ( x , w ) + φ ( y , z ) + φ ( y , w ) φ ( a x , b y ) = a ¯ b φ ( x , y ) \begin{aligned}&\displaystyle\varphi(x+y,z+w)=\varphi(x,z)+\varphi(x,w)+% \varphi(y,z)+\varphi(y,w)\\ &\displaystyle\varphi(ax,by)=\overline{a}b\,\varphi(x,y)\end{aligned}
  10. x , y , z , w V x,y,z,w∈V
  11. a , b 𝐂 a,b∈\mathbf{C}
  12. a ¯ \overline{a}
  13. a a
  14. V ¯ × V 𝐂 \overline{V}\times V\to\mathbf{C}
  15. V ¯ \overline{V}
  16. V V
  17. V ¯ V 𝐂 . \overline{V}\otimes V\to\mathbf{C}.
  18. z z
  19. V V
  20. w φ ( z , w ) w↦φ(z,w)
  21. V V
  22. w φ ( w , z ) w↦φ(w,z)
  23. V V
  24. φ φ
  25. V V
  26. ψ ψ
  27. ψ ( w , z ) = φ ( z , w ) ¯ . \psi(w,z)=\overline{\varphi(z,w)}.
  28. ψ ψ
  29. φ φ
  30. φ φ
  31. φ φ
  32. B B
  33. S S
  34. h : V × V 𝐂 h:V×V→\mathbf{C}
  35. h ( w , z ) = h ( z , w ) ¯ . h(w,z)=\overline{h(z,w)}.
  36. w , z = i = 1 n w i ¯ z i . \langle w,z\rangle=\sum_{i=1}^{n}\overline{w_{i}}z_{i}.
  37. ( V , h ) (V,h)
  38. V V
  39. V V
  40. 𝐇 \mathbf{H}
  41. w w
  42. 𝐰 \mathbf{w}
  43. z z
  44. 𝐳 \mathbf{z}
  45. h ( w , z ) = 𝐰 ¯ T 𝐇𝐳 . h(w,z)={\overline{\mathbf{w}}}^{\mathrm{T}}\mathbf{Hz}.
  46. 𝐇 \mathbf{H}
  47. Q ( z ) = h ( z , z ) Q(z)=h(z,z)
  48. z V z∈V
  49. s : V × V 𝐂 s:V×V→\mathbf{C}
  50. s ( w , z ) = - s ( z , w ) ¯ . s(w,z)=-\overline{s(z,w)}.
  51. i i
  52. V V
  53. V V
  54. s s
  55. 𝐒 \mathbf{S}
  56. w w
  57. 𝐰 \mathbf{w}
  58. z z
  59. 𝐳 \mathbf{z}
  60. s ( w , z ) = 𝐰 ¯ T 𝐒𝐳 . s(w,z)={\overline{\mathbf{w}}}^{\mathrm{T}}\mathbf{S\mathbf{z}}.
  61. Q ( z ) = s ( z , z ) Q(z)=s(z,z)
  62. V V
  63. F F
  64. σ σ
  65. φ : V × V F φ:V×V→F
  66. φ ( x + y , z + w ) = φ ( x , z ) + φ ( x , w ) + φ ( y , z ) + φ ( y , w ) φ ( c x , d y ) = c d σ φ ( x , y ) = c σ ( d ) φ ( x , y ) \begin{aligned}&\displaystyle\varphi(x+y,z+w)=\varphi(x,z)+\varphi(x,w)+% \varphi(y,z)+\varphi(y,w)\\ &\displaystyle\varphi(cx,dy)=cd^{\sigma}\varphi(x,y)=c\,\sigma(d)\,\varphi(x,y% )\end{aligned}
  67. x , y , z , w V x,y,z,w∈V
  68. c , d F c,d∈F
  69. σ = i d σ=id
  70. φ φ
  71. x , y V x,y∈V
  72. φ ( x , y ) = 0 φ(x,y)=0
  73. φ ( y , x ) = 0 φ(y,x)=0
  74. φ φ
  75. σ σ
  76. φ ( x , y ) = φ ( y , x ) σ \varphi(x,y)=\varphi(y,x)^{\sigma}
  77. x , y V x,y∈V
  78. φ ( x , x ) φ(x,x)
  79. σ σ
  80. σ = i d σ=id
  81. V V
  82. φ φ
  83. V V
  84. φ ( x , y ) = x M φ y σ T . \varphi(x,y)=xM_{\varphi}y^{\sigma\rm T}.
  85. F F
  86. V × V * F V\times V^{*}\to F
  87. V V
  88. V V
  89. q q
  90. φ φ
  91. φ ( x , y ) = x 1 y 1 + q x 2 y 2 + q x 3 y 3 . q \varphi(x,y)=x_{1}y_{1}{}^{q}+x_{2}y_{2}{}^{q}+x_{3}y_{3}{}^{q}.
  92. F F
  93. φ φ
  94. σ σ
  95. G G
  96. δ δ
  97. S , T S,T
  98. G G
  99. φ φ
  100. φ ( x , y ) = 0 φ(x,y)=0
  101. y y
  102. V V
  103. x = 0 x=0
  104. R R
  105. V V
  106. R R
  107. σ σ
  108. R R
  109. φ : V × V R φ:V×V→R
  110. φ ( x + y , z + w ) = φ ( x , z ) + φ ( x , w ) + φ ( y , z ) + φ ( y , w ) φ ( c x , d y ) = c φ ( x , y ) d σ \begin{aligned}&\displaystyle\varphi(x+y,z+w)=\varphi(x,z)+\varphi(x,w)+% \varphi(y,z)+\varphi(y,w)\\ &\displaystyle\varphi(cx,dy)=c\varphi(x,y)d^{\sigma}\end{aligned}
  111. x , y , z , w V x,y,z,w∈V
  112. c , d R c,d∈R
  113. σ σ
  114. σ ( s t ) = σ ( t ) σ ( s ) σ(st)=σ(t)σ(s)
  115. s , t s,t
  116. R R
  117. σ = i d σ=id
  118. R R
  119. φ φ
  120. R R
  121. R R
  122. V V
  123. σ : R R σ:R→R
  124. R R < s u p > o p R→R<sup>op

Set_cover_problem.html

  1. { 1 , 2 , , m } \{1,2,...,m\}
  2. S S
  3. n n
  4. S S
  5. U = { 1 , 2 , 3 , 4 , 5 } U=\{1,2,3,4,5\}
  6. S = { { 1 , 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 4 , 5 } } S=\{\{1,2,3\},\{2,4\},\{3,4\},\{4,5\}\}
  7. S S
  8. U U
  9. { { 1 , 2 , 3 } , { 4 , 5 } } \{\{1,2,3\},\{4,5\}\}
  10. 𝒰 \mathcal{U}
  11. 𝒮 \mathcal{S}
  12. 𝒰 \mathcal{U}
  13. 𝒞 𝒮 \mathcal{C}\subseteq\mathcal{S}
  14. 𝒰 \mathcal{U}
  15. ( 𝒰 , 𝒮 ) (\mathcal{U},\mathcal{S})
  16. k k
  17. k k
  18. ( 𝒰 , 𝒮 ) (\mathcal{U},\mathcal{S})
  19. S 𝒮 x S \sum_{S\in\mathcal{S}}x_{S}
  20. S : e S x S 1 \sum_{S\colon e\in S}x_{S}\geqslant 1
  21. e 𝒰 e\in\mathcal{U}
  22. x S { 0 , 1 } x_{S}\in\{0,1\}
  23. S 𝒮 S\in\mathcal{S}
  24. log n \scriptstyle\log n
  25. log n \scriptstyle\log n
  26. n \scriptstyle n
  27. H ( s ) H(s)
  28. s s
  29. H ( n ) H(n)
  30. n n
  31. H ( n ) = k = 1 n 1 k ln n + 1 H(n)=\sum_{k=1}^{n}\frac{1}{k}\leq\ln{n}+1
  32. H ( s ) H(s^{\prime})
  33. s s^{\prime}
  34. S S
  35. c ln m c\ln{m}
  36. c > 0 c>0
  37. log 2 ( n ) / 2 \log_{2}(n)/2
  38. n = 2 ( k + 1 ) - 2 n=2^{(k+1)}-2
  39. k k
  40. S 1 , , S k S_{1},\ldots,S_{k}
  41. 2 , 4 , 8 , , 2 k 2,4,8,\ldots,2^{k}
  42. T 0 , T 1 T_{0},T_{1}
  43. S i S_{i}
  44. S k , , S 1 S_{k},\ldots,S_{1}
  45. T 0 T_{0}
  46. T 1 T_{1}
  47. k = 3 k=3
  48. n n
  49. 1 2 log 2 n 0.72 ln n \tfrac{1}{2}\log_{2}{n}\approx 0.72\ln{n}
  50. ( 1 - o ( 1 ) ) ln n \bigl(1-o(1)\bigr)\cdot\ln{n}
  51. c ln n c\cdot\ln{n}
  52. c c
  53. \not=
  54. c c
  55. ( 1 - o ( 1 ) ) ln n \bigl(1-o(1)\bigr)\cdot\ln{n}
  56. = =

Set_of_uniqueness.html

  1. n = - c ( n ) e i n t \sum_{n=-\infty}^{\infty}c(n)e^{int}
  2. t E t\notin E
  3. c ( n ) = 0 2 π f ( t ) e - i n t d t c(n)=\int_{0}^{2\pi}f(t)e^{-int}\,dt
  4. lim n S ^ ( n ) = 0 \lim_{n\to\infty}\widehat{S}(n)=0
  5. S ^ ( n ) \hat{S}(n)

Sethi–Ullman_algorithm.html

  1. a * b + a * c = a * ( b + c ) a*b+a*c=a*(b+c)
  2. a = ( b + c + f * g ) * ( d + 3 ) a=(b+c+f*g)*(d+3)
  3. ( b + c + f * g ) * ( d + 3 ) (b+c+f*g)*(d+3)
  4. ( b + c + f * g ) * ( d + 3 ) (b+c+f*g)*(d+3)

Shallow_donor.html

  1. 3 k b T 3k_{b}T

Shapiro_delay.html

  1. Δ t = - 2 G M c 3 log ( 1 - 𝐑 𝐱 ) \Delta t=-\frac{2GM}{c^{3}}\log(1-\mathbf{R}\cdot\mathbf{x})
  2. Δ x = - R s log ( 1 - 𝐑 𝐱 ) , \Delta x=-R_{s}\log(1-\mathbf{R}\cdot\mathbf{x}),
  3. R s R_{s}
  4. Δ t = - ( 1 + γ ) R s 2 c log ( 1 - 𝐑 𝐱 ) , \Delta t=-(1+\gamma)\frac{R_{s}}{2c}\log(1-\mathbf{R}\cdot\mathbf{x}),
  5. γ = 0 \gamma=0

Sharpe_ratio.html

  1. S a = E [ R a - R b ] σ a = E [ R a - R b ] var [ R a - R b ] , S_{a}=\frac{E[R_{a}-R_{b}]}{\sigma_{a}}=\frac{E[R_{a}-R_{b}]}{\sqrt{\mathrm{% var}[R_{a}-R_{b}]}},
  2. R a R_{a}
  3. R b R_{b}
  4. E [ R a - R b ] E[R_{a}-R_{b}]
  5. σ {\sigma}
  6. S = E [ R - R f ] var [ R ] . S=\frac{E[R-R_{f}]}{\sqrt{\mathrm{var}[R]}}.
  7. S = E [ R - R b ] var [ R - R b ] . S=\frac{E[R-R_{b}]}{\sqrt{\mathrm{var}[R-R_{b}]}}.
  8. var [ R - R f ] = var [ R ] . \sqrt{\mathrm{var}[R-R_{f}]}=\sqrt{\mathrm{var}[R]}.
  9. R - R f = 0.15 R-R_{f}=0.15
  10. σ = 0.10 \sigma=0.10

Sheaf_cohomology.html

  1. 𝒜 , , 𝒞 \mathcal{A},\mathcal{B},\mathcal{C}
  2. 0 𝒜 ϕ ψ 𝒞 0 0\ \rightarrow\mathcal{A}\ \stackrel{\phi}{\rightarrow}\ \mathcal{B}\ % \stackrel{\psi}{\rightarrow}\ \mathcal{C}\ \rightarrow\ 0
  3. ϕ \phi
  4. ψ \psi
  5. Im ϕ = Ker ψ \,\text{Im}\phi=\,\text{Ker}\psi
  6. ϕ \phi
  7. 𝒞 / 𝒜 \mathcal{C}\cong\mathcal{B}/\mathcal{A}
  8. 0 Γ ( 𝒜 , X ) ϕ * Γ ( , X ) ψ * Γ ( 𝒞 , X ) 0\ \rightarrow\ \Gamma(\mathcal{A},X)\ \stackrel{\phi_{*}}{\rightarrow}\ % \Gamma(\mathcal{B},X)\ \stackrel{\psi_{*}}{\rightarrow}\ \Gamma(\mathcal{C},X)
  9. ψ * \psi_{*}
  10. \mathcal{F}
  11. Γ X : ( X ) . \Gamma_{X}:\mathcal{F}\mapsto\mathcal{F}(X).
  12. H i ( X , ) , i 0. H^{i}(X,\mathcal{F}),i\geq 0.
  13. χ ( ) \chi(\mathcal{F})
  14. \mathcal{F}
  15. χ ( ) := i 𝐙 0 + ( - 1 ) i rank ( H i ( X , ) ) . \chi(\mathcal{F}):=\sum_{i\in\mathbf{Z}_{0}^{+}}(-1)^{i}\,{\rm rank}\,(H^{i}(X% ,\mathcal{F})).
  16. i N i\geq N
  17. N N

Shear_mapping.html

  1. ( x , y ) (x,y)
  2. ( x + 2 y , y ) (x+2y,y)
  3. x x
  4. y y
  5. n n
  6. n \mathbb{R}^{n}
  7. n \mathbb{R}^{n}
  8. n n
  9. 2 = × \mathbb{R}^{2}=\mathbb{R}\times\mathbb{R}
  10. ( x , y ) (x,y)
  11. ( x + m y , y ) (x+my,y)
  12. m m
  13. y y
  14. x x
  15. x x
  16. m > 0 m>0
  17. m < 0 m<0
  18. x x
  19. x x
  20. x x
  21. 1 / m 1/m
  22. m m
  23. φ \varphi
  24. ( x y ) = ( x + m y y ) = ( 1 m 0 1 ) ( x y ) . \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}x+my\\ y\end{pmatrix}=\begin{pmatrix}1&m\\ 0&1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}.
  25. y y
  26. x x
  27. y y
  28. ( x y ) = ( x m x + y ) = ( 1 0 m 1 ) ( x y ) . \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}x\\ mx+y\end{pmatrix}=\begin{pmatrix}1&0\\ m&1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}.
  29. y y
  30. m m
  31. y y
  32. φ \varphi
  33. m m
  34. ( I M 0 I ) \begin{pmatrix}I&M\\ 0&I\end{pmatrix}

Shift-share_analysis.html

  1. e i t + n - e i t = N S i + I M i + R S i e_{i}^{t+n}-e_{i}^{t}=NS_{i}+IM_{i}+RS_{i}
  2. N S i = e i t ( G ) NS_{i}=e_{i}^{t}\left(G\right)
  3. I M i = e i t ( G i - G ) IM_{i}=e_{i}^{t}\left(G_{i}-G\right)
  4. R S i = e i t ( g i - G i ) RS_{i}=e_{i}^{t}\left(g_{i}-G_{i}\right)
  5. e i t + n - e i t = N S i + I M i + R S i e_{i}^{t+n}-e_{i}^{t}=NS_{i}+IM_{i}+RS_{i}
  6. N S i = k = t + 1 t + n [ e i k - 1 ( G k ) ] NS_{i}=\sum_{k=t+1}^{t+n}\left[e_{i}^{k-1}\left(G^{k}\right)\right]
  7. I M i = k = t + 1 t + n [ e i k - 1 ( G i k - G k ) ] IM_{i}=\sum_{k=t+1}^{t+n}\left[e_{i}^{k-1}\left(G_{i}^{k}-G^{k}\right)\right]
  8. R S i = k = t + 1 t + n [ e i k - 1 ( g i k - G i k ) ] RS_{i}=\sum_{k=t+1}^{t+n}\left[e_{i}^{k-1}\left(g_{i}^{k}-G_{i}^{k}\right)\right]
  9. e i t + n - e i t = N S i + I M i + R S i + A L i e_{i}^{t+n}-e_{i}^{t}=NS_{i}+IM_{i}+RS_{i}+AL_{i}
  10. h i t = e t × E i t E t h_{i}^{t}=e^{t}\times{E_{i}^{t}\over E^{t}}
  11. N S i = e i t ( G ) NS_{i}=e_{i}^{t}\left(G\right)
  12. I M i = e i t ( G i - G ) IM_{i}=e_{i}^{t}\left(G_{i}-G\right)
  13. R S i = h i t ( g i - G i ) RS_{i}=h_{i}^{t}\left(g_{i}-G_{i}\right)
  14. A L i = ( e i t - h i t ) ( g i - G i ) AL_{i}=\left(e_{i}^{t}-h_{i}^{t}\right)\left(g_{i}-G_{i}\right)
  15. e i t + n - e i t = N S E i + N S D i + I M E i + I M D i + R G E i + R G D i + R I E i + R I D i e_{i}^{t+n}-e_{i}^{t}=NSE_{i}+NSD_{i}+IME_{i}+IMD_{i}+RGE_{i}+RGD_{i}+RIE_{i}+% RID_{i}
  16. N S i = N S E i + N S D i NS_{i}=NSE_{i}+NSD_{i}
  17. I M i = I M E i + I M D i IM_{i}=IME_{i}+IMD_{i}
  18. R S i = R G E i + R G D i + R I E i + R I D i RS_{i}=RGE_{i}+RGD_{i}+RIE_{i}+RID_{i}
  19. h i t = e t × E i t E t h_{i}^{t}=e^{t}\times{E_{i}^{t}\over E^{t}}
  20. N S E i = h i t × G NSE_{i}=h_{i}^{t}\times G
  21. N S D i = ( e i t - h i t ) × G NSD_{i}=\left(e_{i}^{t}-h_{i}^{t}\right)\times G
  22. I M E i = h i t × ( G i - G ) IME_{i}=h_{i}^{t}\times\left(G_{i}-G\right)
  23. I M D i = ( e i t - h i t ) × ( G i - G ) IMD_{i}=\left(e_{i}^{t}-h_{i}^{t}\right)\times\left(G_{i}-G\right)
  24. R G E i = h i t × ( g - G ) RGE_{i}=h_{i}^{t}\times\left(g-G\right)
  25. R G D i = ( e i t - h i t ) × ( g - G ) RGD_{i}=\left(e_{i}^{t}-h_{i}^{t}\right)\times\left(g-G\right)
  26. R I E i = h i t × ( g i - g - G i + G ) RIE_{i}=h_{i}^{t}\times\left(g_{i}-g-G_{i}+G\right)
  27. R I D i = ( e i t - h i t ) × ( g i - g - G i + G ) RID_{i}=\left(e_{i}^{t}-h_{i}^{t}\right)\times\left(g_{i}-g-G_{i}+G\right)
  28. < v a r > G <var>G

Shoe_size.html

  1. 2 / 3 {2}/{3}
  2. 2 / 3 {2}/{3}
  3. 1 / 4 {1}/{4}
  4. 1 / 3 {1}/{3}
  5. 1 / 8 {1}/{8}
  6. 1 / 3 {1}/{3}
  7. 1 / 6 {1}/{6}
  8. 102 / 3 10{2}/{3}
  9. 3 / 16 {3}/{16}
  10. 1 / 3 {1}/{3}
  11. 131 / 2 13{1}/{2}
  12. 81 / 2 8{1}/{2}
  13. 251 / 2 25{1}/{2}
  14. child shoe size = 3 × last length in inches - 12 \mbox{child shoe size}~{}=3\times\mbox{last length in inches}~{}-12
  15. 82 / 3 8{2}/{3}
  16. adult shoe size = 3 × last length in inches - 25 \mbox{adult shoe size}~{}=3\times\mbox{last length in inches}~{}-25
  17. 61 / 2 6{1}/{2}
  18. 71 / 2 7{1}/{2}
  19. male shoe size = 3 × last length in inches - 24 \mbox{male shoe size}~{}=3\times\mbox{last length in inches}~{}-24
  20. female shoe size (common) = 3 × last length in inches - 22 1 2 \mbox{female shoe size (common)}~{}=3\times\mbox{last length in inches}~{}-22% \frac{1}{2}
  21. female shoe size (FIA) = 3 × last length in inches - 23 \mbox{female shoe size (FIA)}~{}=3\times\mbox{last length in inches}~{}-23
  22. 121 / 3 12{1}/{3}
  23. child shoe size = 3 × last length in inches - 11 2 3 \mbox{child shoe size}~{}=3\times\mbox{last length in inches}~{}-11\frac{2}{3}
  24. 5 / 9 {5}/{9}
  25. 8 / 9 {8}/{9}
  26. 2 / 9 {2}/{9}
  27. 5 / 9 {5}/{9}
  28. 8 / 9 {8}/{9}
  29. 2 / 9 {2}/{9}
  30. 5 / 9 {5}/{9}
  31. 5 / 6 {5}/{6}
  32. 8 / 9 {8}/{9}
  33. 1 / 6 {1}/{6}
  34. 2 / 9 {2}/{9}
  35. 1 / 3 {1}/{3}
  36. 1 / 2 {1}/{2}
  37. 2 / 3 {2}/{3}
  38. 5 / 6 {5}/{6}
  39. 1 / 6 {1}/{6}
  40. 1 / 3 {1}/{3}
  41. 1 / 2 {1}/{2}
  42. 2 / 3 {2}/{3}
  43. 5 / 6 {5}/{6}
  44. 1 / 3 {1}/{3}
  45. male shoe size (Brannock) = 3 × foot length in inches - 22 \mbox{male shoe size (Brannock)}~{}=3\times\mbox{foot length in inches}~{}-22
  46. female shoe size (Brannock) = 3 × foot length in inches - 21 \mbox{female shoe size (Brannock)}~{}=3\times\mbox{foot length in inches}~{}-21
  47. shoe size ( Paris points ) = 3 2 × last length ( cm ) \mathrm{shoe~{}size~{}({Paris~{}points})={\frac{3}{2}}\times{last~{}length}% \left({cm}\right)}
  48. shoe size ( Paris points ) = 3 2 × [ foot length ( cm ) + 1.5 cm ] \mathrm{shoe~{}size~{}({Paris~{}points})={\frac{3}{2}}\times{{\left[~{}foot~{}% length\left({cm}\right)+1.5~{}{cm}~{}\right]}}}
  49. adult shoe size = 3 × last length in inches - 25.5 \mbox{adult shoe size}~{}=3\times\mbox{last length in inches}~{}-25.5

Shotgun_shell.html

  1. d n = 7000 1501.339 × n 3 d_{n}=\sqrt[3]{\frac{7000}{1501.339\times n}}
  2. 1 / 256 {1}/{256}
  3. 1 / 16 {1}/{16}

Side-channel_attack.html

  1. d d
  2. e e
  3. m m
  4. y d y^{d}
  5. y y
  6. r r
  7. e e
  8. r e r^{e}
  9. y r e y\cdot r^{e}
  10. ( y r e ) d = y d r e d = y d r {(y\cdot r^{e})}^{d}=y^{d}\cdot r^{e\cdot d}=y^{d}\cdot r
  11. r r
  12. m m
  13. r r
  14. y d y^{d}

Sierpiński_curve.html

  1. n n\rightarrow\infty
  2. n n\rightarrow\infty
  3. 2 2
  4. S n S_{n}
  5. l n = 2 3 ( 1 + 2 ) 2 n - 1 3 ( 2 - 2 ) 1 2 n l_{n}={2\over 3}(1+\sqrt{2})2^{n}-{1\over 3}(2-\sqrt{2}){1\over 2^{n}}
  6. n n
  7. n n\rightarrow\infty
  8. S n S_{n}
  9. 5 / 12 5/12\,

Sieve_theory.html

  1. N ϵ N^{\epsilon}
  2. ϵ \epsilon
  3. N 1 / 2 N^{1/2}
  4. a 2 + b 4 a^{2}+b^{4}

Sigma_additivity.html

  1. μ \mu
  2. 𝒜 \scriptstyle\mathcal{A}
  3. μ \mu
  4. 𝒜 \scriptstyle\mathcal{A}
  5. μ ( A B ) = μ ( A ) + μ ( B ) . \mu(A\cup B)=\mu(A)+\mu(B).\,
  6. μ ( n = 1 N A n ) = n = 1 N μ ( A n ) \mu\left(\bigcup_{n=1}^{N}A_{n}\right)=\sum_{n=1}^{N}\mu(A_{n})
  7. A 1 , A 2 , , A N A_{1},A_{2},\dots,A_{N}
  8. 𝒜 \scriptstyle\mathcal{A}
  9. 𝒜 \scriptstyle\mathcal{A}
  10. A 1 , A 2 , , A n , A_{1},A_{2},\dots,A_{n},\dots
  11. 𝒜 \scriptstyle\mathcal{A}
  12. μ ( n = 1 A n ) = n = 1 μ ( A n ) \mu\left(\bigcup_{n=1}^{\infty}A_{n}\right)=\sum_{n=1}^{\infty}\mu(A_{n})
  13. , ,
  14. 𝒜 \scriptstyle\mathcal{A}
  15. 𝒢 \scriptstyle\mathcal{G}
  16. 𝒜 \scriptstyle\mathcal{A}
  17. μ ( 𝒢 ) = sup G 𝒢 μ ( G ) \mu\left(\bigcup\mathcal{G}\right)=\sup_{G\in\mathcal{G}}\mu(G)
  18. , ,
  19. μ ( A ) = { 1 if 0 A 0 if 0 A . \mu(A)=\begin{cases}1&\mbox{ if }~{}0\in A\\ 0&\mbox{ if }~{}0\notin A.\end{cases}
  20. A 1 , A 2 , , A n , A_{1},A_{2},\dots,A_{n},\dots
  21. μ ( n = 1 A n ) = n = 1 μ ( A n ) \mu\left(\bigcup_{n=1}^{\infty}A_{n}\right)=\sum_{n=1}^{\infty}\mu(A_{n})
  22. μ ( A ) = lim k 1 k λ ( A ( 0 , k ) ) , \mu(A)=\lim_{k\to\infty}\frac{1}{k}\cdot\lambda\left(A\cap\left(0,k\right)% \right),
  23. A n = [ n , n + 1 ) A_{n}=\left[n,n+1\right)

Signal_strength.html

  1. L \scriptstyle{L}
  2. λ / 2 \scriptstyle{\lambda/2}
  3. E θ ( r ) = - j I 2 π ε c r cos ( π 2 cos θ ) sin θ e j ( ω t - k r ) E_{\theta}(r)={-jI_{\circ}\over 2\pi\varepsilon_{\circ}c\,r}{\cos\left(% \scriptstyle{\pi\over 2}\cos\theta\right)\over\sin\theta}e^{j\left(\omega t-kr% \right)}
  4. θ \scriptstyle{\theta}
  5. I \scriptstyle{I_{\circ}}
  6. ε = 8.85 × 10 - 12 F / m \scriptstyle{\varepsilon_{\circ}\,=\,8.85\times 10^{-12}\,F/m}
  7. c = 3 × 10 8 m / S \scriptstyle{c\,=\,3\times 10^{8}\,m/S}
  8. r \scriptstyle{r}
  9. θ = π / 2 \scriptstyle{\theta\,=\,\pi/2}
  10. | E π / 2 ( r ) | = I 2 π ε c r . |E_{\pi/2}(r)|={I_{\circ}\over 2\pi\varepsilon_{\circ}c\,r}\,.
  11. I = 2 π ε c r | E π / 2 ( r ) | . I_{\circ}=2\pi\varepsilon_{\circ}c\,r|E_{\pi/2}(r)|\,.
  12. P a v g = 1 2 R a I 2 {P_{avg}={1\over 2}R_{a}\,I_{\circ}^{2}}
  13. R a = 73.13 Ω \scriptstyle{R_{a}=73.13\,\Omega}
  14. I \scriptstyle{I_{\circ}}
  15. P a v g \scriptstyle{P_{avg}}
  16. | E π / 2 ( r ) | = 1 π ε c r P a v g 2 R a = 9.91 r P a v g ( L = λ / 2 ) . |E_{\pi/2}(r)|\,=\,{1\over\pi\varepsilon_{\circ}c\,r}\sqrt{{P_{avg}\over 2R_{a% }}}\,=\,{9.91\over r}\sqrt{P_{avg}}\quad(L=\lambda/2)\,.
  17. L λ / 2 \scriptstyle{L\ll\lambda/2}
  18. E θ ( r ) = - j I sin ( θ ) 4 ε c r ( L λ ) e j ( ω t - k r ) , R a = 20 π 2 ( L λ ) 2 . E_{\theta}(r)={-jI_{\circ}\sin(\theta)\over 4\varepsilon_{\circ}c\,r}\left({L% \over\lambda}\right)e^{j\left(\omega t-kr\right)}\,,\quad R_{a}=20\pi^{2}\left% ({L\over\lambda}\right)^{2}.
  19. | E π / 2 ( r ) | = 1 π ε c r P a v g 160 = 9.48 r P a v g ( L λ / 2 ) . |E_{\pi/2}(r)|\,=\,{1\over\pi\varepsilon_{\circ}c\,r}\sqrt{{P_{avg}\over 160}}% \,=\,{9.48\over r}\sqrt{P_{avg}}\quad(L\ll\lambda/2)\,.
  20. d B m e = - 113.0 - 40.0 l o g 10 ( r / R ) dBm_{e}=-113.0-40.0\ log_{10}(r/R)
  21. d B m e = - 113.0 - 10.0 γ l o g 10 ( r / R ) dBm_{e}=-113.0-10.0\ \gamma\ log_{10}(r/R)
  22. d B m e = d B m 0 - 10.0 γ l o g 10 ( r / r 0 ) dBm_{e}=dBm_{0}-10.0\ \gamma\ log_{10}(r/r_{0})
  23. l o g 10 ( R / r ) log_{10}(R/r)
  24. R e = a v g [ r 10 ( d B m m + 113.0 ) / 40.0 ] R_{e}=avg\ [\ r\ 10^{(dBm_{m}+113.0)/40.0}\ ]

Signature_(topology).html

  1. α p β q = ( - 1 ) p q ( β q α p ) \alpha^{p}\smile\beta^{q}=(-1)^{pq}(\beta^{q}\smile\alpha^{p})
  2. L 4 k , L^{4k},
  3. L 4 k , L_{4k},
  4. 𝐙 / 2 \mathbf{Z}/2
  5. L 4 k + 2 L_{4k+2}
  6. L 4 k + 1 L^{4k+1}
  7. d = 4 k + 2 = 2 ( 2 k + 1 ) d=4k+2=2(2k+1)

Signed-digit_representation.html

  1. - k -k
  2. ( b - 1 ) - k (b-1)-k
  3. k = b 2 k=\left\lfloor\frac{b}{2}\right\rfloor
  4. b = 3 b=3
  5. b = 2 b=2

SIGSALY.html

  1. 3 - 5 - 2 , - 2 + 6 4 ( mod 6 ) 3-5\equiv-2,\ -2+6\equiv 4\;\;(\mathop{{\rm mod}}6)\,
  2. 4 + 5 9 , 9 - 6 3 ( mod 6 ) 4+5\equiv 9,\ 9-6\equiv 3\;\;(\mathop{{\rm mod}}6)\,

Silver_chloride.html

  1. 1.77 * 10 - 10 \sqrt{1.77*10^{-10}}

Silver_ratio.html

  1. 2 + 1 2 + 1 2 + 1 2 + 1 2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}
  2. 1 + 2 {1+\sqrt{2}}
  3. 2 a + b a = a b δ S . \frac{2a+b}{a}=\frac{a}{b}\equiv\delta_{S}\,.
  4. δ S = 2 + 1 2 + 1 2 + 1 2 + . \delta_{S}=2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}\,.
  5. δ S - 1 = 1 δ S - 2 = [ 0 ; 2 , 2 , 2 , 2 , 2 , ] 0.41421 \delta_{S}^{-1}=1\delta_{S}-2=[0;2,2,2,2,2,\dots]\approx 0.41421
  6. δ S 0 = 0 δ S + 1 = [ 1 ] = 1 \delta_{S}^{0}\!\ \!\ =0\delta_{S}+1=[1]=1
  7. δ S 1 = 1 δ S + 0 = [ 2 ; 2 , 2 , 2 , 2 , 2 , ] 2.41421 \delta_{S}^{1}\!\ \!\ =1\delta_{S}+0=[2;2,2,2,2,2,\dots]\approx 2.41421
  8. δ S 2 = 2 δ S + 1 = [ 5 ; 1 , 4 , 1 , 4 , 1 , ] 5.82842 \delta_{S}^{2}\!\ \!\ =2\delta_{S}+1=[5;1,4,1,4,1,\dots]\approx 5.82842
  9. δ S 3 = 5 δ S + 2 = [ 14 ; 14 , 14 , 14 , ] 14.07107 \delta_{S}^{3}\!\ \!\ =5\delta_{S}+2=[14;14,14,14,\dots]\approx 14.07107
  10. δ S 4 = 12 δ S + 5 = [ 33 ; 1 , 32 , 1 , 32 , ] 33.97056 \delta_{S}^{4}\!\ \!\ =12\delta_{S}+5=[33;1,32,1,32,\dots]\approx 33.97056
  11. δ S n = K n δ S + K n - 1 \delta_{S}^{n}=K_{n}\delta_{S}+K_{n-1}
  12. K n = 2 K n - 1 + K n - 2 K_{n}=2K_{n-1}+K_{n-2}
  13. δ S 5 = 29 δ S + 12 = [ 82 ; 82 , 82 , 82 , ] 82.01219 \!\ \delta_{S}^{5}=29\delta_{S}+12=[82;82,82,82,\dots]\approx 82.01219
  14. K n = 2 K n - 1 + K n - 2 \!\ K_{n}=2K_{n-1}+K_{n-2}
  15. K n = 1 2 2 ( δ S n + 1 - ( 2 - δ S ) n + 1 ) \!\ K_{n}=\frac{1}{2\sqrt{2}}{(\delta_{S}^{n+1}-{(2-\delta_{S})}^{n+1})}
  16. sin 1 8 π = cos 3 8 π = 1 2 2 - 2 = 1 2 δ s - 1 \textstyle\sin\tfrac{1}{8}\pi=\cos\tfrac{3}{8}\pi=\frac{1}{2}\sqrt{2-\sqrt{2}}% =\sqrt{\tfrac{1}{2}\delta_{s}^{-1}}
  17. cos 1 8 π = sin 3 8 π = 1 2 2 + 2 = 1 2 δ s \textstyle\cos\tfrac{1}{8}\pi=\sin\tfrac{3}{8}\pi=\frac{1}{2}\sqrt{2+\sqrt{2}}% =\sqrt{\tfrac{1}{2}\delta_{s}}
  18. tan 1 8 π = cot 3 8 π = 2 - 1 = δ s - 1 \textstyle\tan\tfrac{1}{8}\pi=\cot\tfrac{3}{8}\pi=\sqrt{2}-1=\delta_{s}^{-1}
  19. cot 1 8 π = tan 3 8 π = 2 + 1 = δ s \textstyle\cot\tfrac{1}{8}\pi=\tan\tfrac{3}{8}\pi=\sqrt{2}+1=\delta_{s}
  20. A = 2 a 2 cot 1 8 π = 2 ( 1 + 2 ) a 2 4.828427 a 2 . A=\textstyle 2a^{2}\cot\tfrac{1}{8}\pi=2(1+\sqrt{2})a^{2}\simeq 4.828427a^{2}.
  21. 2 \sqrt{2}
  22. 2 \sqrt{2}
  23. 2 \sqrt{2}
  24. 2 \sqrt{2}
  25. 2 \sqrt{2}
  26. 2 \sqrt{2}

Similarity_matrix.html

  1. ( i , j ) (i,j)
  2. i i
  3. j j
  4. e - || s 1 - s 2 || 2 / 2 σ 2 e^{-||s_{1}-s_{2}||^{2}/2\sigma^{2}}

Similitude_(model).html

  1. ρ \rho
  2. μ \mu
  3. R e R_{e}
  4. C p C_{p}
  5. R e = ( ρ V L μ ) V model = V application × ( ρ a ρ m ) × ( L a L m ) × ( μ m μ a ) C p = ( 2 Δ p ρ V 2 ) , F = Δ p L 2 F application = F model × ( ρ a ρ m ) × ( V a V m ) 2 × ( L a L m ) 2 . \begin{aligned}&\displaystyle R_{e}=\left(\frac{\rho VL}{\mu}\right)&% \displaystyle\longrightarrow&\displaystyle V\text{model}=V\text{application}% \times\left(\frac{\rho_{a}}{\rho_{m}}\right)\times\left(\frac{L_{a}}{L_{m}}% \right)\times\left(\frac{\mu_{m}}{\mu_{a}}\right)\\ &\displaystyle C_{p}=\left(\frac{2\Delta p}{\rho V^{2}}\right),F=\Delta pL^{2}% &\displaystyle\longrightarrow&\displaystyle F\text{application}=F\text{model}% \times\left(\frac{\rho_{a}}{\rho_{m}}\right)\times\left(\frac{V_{a}}{V_{m}}% \right)^{2}\times\left(\frac{L_{a}}{L_{m}}\right)^{2}.\end{aligned}
  6. p p
  7. F F
  8. p p
  9. F / L 2 F/L^{2}
  10. V model = V application × 21.9 V\text{model}=V\text{application}\times 21.9
  11. F m o d e l F_{model}
  12. F a p p l i c a t i o n F_{application}
  13. F application = F model × 3.44 F\text{application}=F\text{model}\times 3.44
  14. P P
  15. P [ W ] = F application × V application = F model [ N ] × 17.2 m / s P[\mathrm{W}]=F\text{application}\times V\text{application}=F\text{model}[% \mathrm{N}]\times 17.2\ \mathrm{m/s}

Simple_public-key_infrastructure.html

  1. K K
  2. N N

Simplex_category.html

  1. Δ \Delta
  2. Δ \Delta
  3. [ n ] = { 0 , 1 , , n } [n]=\{0,1,\dots,n\}
  4. [ n ] [n]
  5. n + 1 n+1
  6. Δ \Delta
  7. Δ \Delta
  8. Δ \Delta
  9. Δ + \Delta_{+}
  10. Δ + = Δ [ - 1 ] \Delta_{+}=\Delta\cup[-1]
  11. [ - 1 ] = [-1]=\emptyset
  12. Δ + \Delta_{+}
  13. Δ + \Delta_{+}
  14. [ - 1 ] [-1]
  15. Δ \Delta
  16. Δ + \Delta_{+}
  17. [ 0 ] [0]
  18. Δ + op \Delta_{+}\text{op}

Simplicial_homology.html

  1. H 1 H_{1}
  2. i = 1 N c i σ i \sum_{i=1}^{N}c_{i}\sigma_{i}\,
  3. ( v 0 , v 1 ) = - ( v 1 , v 0 ) . (v_{0},v_{1})=-(v_{1},v_{0}).
  4. k : C k C k - 1 \partial_{k}:C_{k}\rightarrow C_{k-1}
  5. k ( σ ) = i = 0 k ( - 1 ) i ( v 0 , , v i ^ , , v k ) , \partial_{k}(\sigma)=\sum_{i=0}^{k}(-1)^{i}(v_{0},\dots,\widehat{v_{i}},\dots,% v_{k}),
  6. ( v 0 , , v i ^ , , v k ) (v_{0},\dots,\widehat{v_{i}},\dots,v_{k})
  7. Z k = ker k Z_{k}=\ker\partial_{k}
  8. B k = im k + 1 B_{k}=\operatorname{im}\partial_{k+1}
  9. ( C k , k ) (C_{k},\partial_{k})
  10. H k ( S ) = Z k / B k . H_{k}(S)=Z_{k}/B_{k}\,.
  11. β k = rank ( H k ( S ) ) \beta_{k}={\rm rank}(H_{k}(S))\,
  12. ( v 0 , v 1 ) = ( v 1 ) - ( v 0 ) \partial(v_{0},v_{1})=(v_{1})-(v_{0})
  13. ( v 0 , v 2 ) = ( v 2 ) - ( v 0 ) \partial(v_{0},v_{2})=(v_{2})-(v_{0})
  14. ( v 1 , v 2 ) = ( v 2 ) - ( v 1 ) \partial(v_{1},v_{2})=(v_{2})-(v_{1})
  15. f ( ( v 0 , , v k ) ) = ( f ( v 0 ) , , f ( v k ) ) f((v_{0},\ldots,v_{k}))=(f(v_{0}),...,f(v_{k}))

Simplicial_set.html

  1. δ 0 , , δ n : [ n - 1 ] [ n ] \delta^{0},\ldots,\delta^{n}\colon[n-1]\to[n]
  2. δ i \delta^{i}
  3. [ n - 1 ] [ n ] [n-1]\to[n]
  4. i i
  5. d 0 , , d n d_{0},\ldots,d_{n}
  6. σ 0 , , σ n : [ n + 1 ] [ n ] \sigma^{0},\ldots,\sigma^{n}\colon[n+1]\to[n]
  7. σ i \sigma^{i}
  8. [ n + 1 ] [ n ] [n+1]\to[n]
  9. i i
  10. s 0 , , s n s_{0},\ldots,s_{n}
  11. | Δ n | = { ( x 0 , , x n ) n + 1 : 0 x i 1 , x i = 1 } . |\Delta^{n}|=\{(x_{0},\dots,x_{n})\in\mathbb{R}^{n+1}:0\leq x_{i}\leq 1,\sum x% _{i}=1\}.
  12. Δ n = Δ op ( 𝐧 , - ) \Delta^{n}=\Delta^{\mathrm{op}}(\mathbf{n},-)
  13. Nat ( Δ op ( 𝐧 , - ) , X ) X ( 𝐧 ) \mathrm{Nat}(\Delta^{\mathrm{op}}(\mathbf{n},-),X)\cong X(\mathbf{n})
  14. Δ X \Delta\downarrow{X}
  15. Δ X \Delta\downarrow{X}
  16. X lim Δ n X Δ n X\cong\underrightarrow{\lim}_{\Delta^{n}\to X}\Delta^{n}
  17. X × Y X\times Y
  18. | X | × K e | Y | |X|\times_{Ke}|Y|
  19. × K e \times_{Ke}
  20. K K
  21. G G
  22. B G BG
  23. G G
  24. Ω B G \Omega BG
  25. B G BG
  26. G G
  27. Ω 2 B ( B G ) \Omega^{2}B(BG)
  28. G G
  29. G G
  30. X X
  31. X X
  32. K K

Simultaneous_localization_and_mapping.html

  1. o t o_{t}
  2. t t
  3. x t x_{t}
  4. m t m_{t}
  5. P ( m t , x t | o 1 : t ) P(m_{t},x_{t}|o_{1:t})
  6. P ( x t | x t - 1 ) P(x_{t}|x_{t-1})
  7. P ( x t | o 1 : t , m t ) = m t - 1 P ( o t | x t , m t ) x t - 1 P ( x t | x t - 1 ) P ( x t - 1 | m t , o 1 : t - 1 ) / Z P(x_{t}|o_{1:t},m_{t})=\sum_{m_{t-1}}P(o_{t}|x_{t},m_{t})\sum_{x_{t-1}}P(x_{t}% |x_{t-1})P(x_{t-1}|m_{t},o_{1:t-1})/Z
  8. P ( m t | x t , o 1 : t ) = x t m t P ( m t | x t , m t - 1 , o t ) P ( m t - 1 , x t | o 1 : t - 1 , m t - 1 ) P(m_{t}|x_{t},o_{1:t})=\sum_{x_{t}}\sum_{m_{t}}P(m_{t}|x_{t},m_{t-1},o_{t})P(m% _{t-1},x_{t}|o_{1:t-1},m_{t-1})
  9. P ( m t | x t , m t - 1 , o t ) P(m_{t}|x_{t},m_{t-1},o_{t})
  10. o t o_{t}
  11. x t x_{t}
  12. P ( o t | x t ) P(o_{t}|x_{t})
  13. P ( x t | x t - 1 ) P(x_{t}|x_{t-1})

Singlet_state.html

  1. S 2 \vec{S}^{2}
  2. 2 ( 1 / 2 ) ( 1 / 2 + 1 ) = ( 3 / 4 ) 2 \hbar^{2}\,(1/2)\,(1/2+1)=(3/4)\,\hbar^{2}
  3. ( S 1 + S 2 ) 2 \left(\vec{S}_{1}+\vec{S}_{2}\right)^{2}
  4. S 1 \vec{S}_{1}
  5. S 2 \vec{S}_{2}
  6. 1 2 ( | - | ) \frac{1}{\sqrt{2}}\left(\left|\uparrow\downarrow\right\rangle-\left|\downarrow% \uparrow\right\rangle\right)

Singular_point_of_an_algebraic_variety.html

  1. ( x - x 0 ) F x ( x 0 , y 0 ) + ( y - y 0 ) F y ( x 0 , y 0 ) , (x-x_{0})F^{\prime}_{x}(x_{0},y_{0})+(y-y_{0})F^{\prime}_{y}(x_{0},y_{0}),
  2. y 3 + 2 x 2 y - x 4 = 0 y^{3}+2x^{2}y-x^{4}=0

Singular_value.html

  1. A * A = U | Λ | U * \sqrt{A^{*}A}=U|\Lambda|U^{*}
  2. s n ( T ) = inf { T - L : L is an operator of finite rank < n } . s_{n}(T)=\inf\big\{\,\|T-L\|:L\,\text{ is an operator of finite rank }<n\,\big\}.

Skellam_distribution.html

  1. 1 / ( μ 1 + μ 2 ) 1/(\mu_{1}+\mu_{2})\,
  2. e - ( μ 1 + μ 2 ) + μ 1 e t + μ 2 e - t e^{-(\mu_{1}+\mu_{2})+\mu_{1}e^{t}+\mu_{2}e^{-t}}
  3. e - ( μ 1 + μ 2 ) + μ 1 e i t + μ 2 e - i t e^{-(\mu_{1}+\mu_{2})+\mu_{1}e^{it}+\mu_{2}e^{-it}}
  4. n 1 - n 2 n_{1}-n_{2}
  5. N 1 N_{1}
  6. N 2 N_{2}
  7. μ 1 \mu_{1}
  8. μ 2 \mu_{2}
  9. k = n 1 - n 2 k=n_{1}-n_{2}
  10. μ 1 \mu_{1}
  11. μ 2 \mu_{2}
  12. f ( k ; μ 1 , μ 2 ) = e - ( μ 1 + μ 2 ) ( μ 1 μ 2 ) k / 2 I k ( 2 μ 1 μ 2 ) f(k;\mu_{1},\mu_{2})=e^{-(\mu_{1}+\mu_{2})}\left({\mu_{1}\over\mu_{2}}\right)^% {k/2}I_{k}(2\sqrt{\mu_{1}\mu_{2}})
  13. f ( n ; μ ) = μ n n ! e - μ . f(n;\mu)={\mu^{n}\over n!}e^{-\mu}.\,
  14. n 0 n\geq 0
  15. k = n 1 - n 2 k=n_{1}-n_{2}
  16. f ( k ; μ 1 , μ 2 ) = n = - f ( k + n ; μ 1 ) f ( n ; μ 2 ) f(k;\mu_{1},\mu_{2})=\sum_{n=-\infty}^{\infty}\!f(k\!+\!n;\mu_{1})f(n;\mu_{2})
  17. = e - ( μ 1 + μ 2 ) n = m a x ( 0 , - k ) μ 1 k + n μ 2 n n ! ( k + n ) ! =e^{-(\mu_{1}+\mu_{2})}\sum_{n=max(0,-k)}^{\infty}{{\mu_{1}^{k+n}\mu_{2}^{n}}% \over{n!(k+n)!}}
  18. ( f ( n < 0 ; μ ) = 0 ) (f(n<0;\mu)=0)
  19. n 0 n>=0
  20. n + k 0 n+k>=0
  21. f ( k ; μ 1 , μ 2 ) f ( - k ; μ 1 , μ 2 ) = ( μ 1 μ 2 ) k \frac{f(k;\mu_{1},\mu_{2})}{f(-k;\mu_{1},\mu_{2})}=\left(\frac{\mu_{1}}{\mu_{2% }}\right)^{k}
  22. f ( k ; μ 1 , μ 2 ) = e - ( μ 1 + μ 2 ) ( μ 1 μ 2 ) k / 2 I | k | ( 2 μ 1 μ 2 ) f(k;\mu_{1},\mu_{2})=e^{-(\mu_{1}+\mu_{2})}\left({\mu_{1}\over\mu_{2}}\right)^% {k/2}I_{|k|}(2\sqrt{\mu_{1}\mu_{2}})
  23. μ 1 = μ 2 ( = μ ) \mu_{1}=\mu_{2}(=\mu)
  24. f ( k ; μ , μ ) = e - 2 μ I | k | ( 2 μ ) . f\left(k;\mu,\mu\right)=e^{-2\mu}I_{|k|}(2\mu).
  25. μ 2 = 0 \mu_{2}=0
  26. k = - f ( k ; μ 1 , μ 2 ) = 1. \sum_{k=-\infty}^{\infty}f(k;\mu_{1},\mu_{2})=1.
  27. G ( t ; μ ) = e μ ( t - 1 ) . G\left(t;\mu\right)=e^{\mu(t-1)}.
  28. G ( t ; μ 1 , μ 2 ) G(t;\mu_{1},\mu_{2})
  29. G ( t ; μ 1 , μ 2 ) = k = 0 f ( k ; μ 1 , μ 2 ) t k G(t;\mu_{1},\mu_{2})=\sum_{k=0}^{\infty}f(k;\mu_{1},\mu_{2})t^{k}
  30. = G ( t ; μ 1 ) G ( 1 / t ; μ 2 ) =G\left(t;\mu_{1}\right)G\left(1/t;\mu_{2}\right)\,
  31. = e - ( μ 1 + μ 2 ) + μ 1 t + μ 2 / t . =e^{-(\mu_{1}+\mu_{2})+\mu_{1}t+\mu_{2}/t}.
  32. M ( t ; μ 1 , μ 2 ) = G ( e t ; μ 1 , μ 2 ) M\left(t;\mu_{1},\mu_{2}\right)=G(e^{t};\mu_{1},\mu_{2})
  33. = k = 0 t k k ! m k =\sum_{k=0}^{\infty}{t^{k}\over k!}\,m_{k}
  34. Δ = def μ 1 - μ 2 \Delta\ \stackrel{\mathrm{def}}{=}\ \mu_{1}-\mu_{2}\,
  35. μ = def ( μ 1 + μ 2 ) / 2. \mu\ \stackrel{\mathrm{def}}{=}\ (\mu_{1}+\mu_{2})/2.\,
  36. m 1 = Δ m_{1}=\left.\Delta\right.\,
  37. m 2 = 2 μ + Δ 2 m_{2}=\left.2\mu+\Delta^{2}\right.\,
  38. m 3 = Δ ( 1 + 6 μ + Δ 2 ) m_{3}=\left.\Delta(1+6\mu+\Delta^{2})\right.\,
  39. M 2 = 2 μ , M_{2}=\left.2\mu\right.,\,
  40. M 3 = Δ , M_{3}=\left.\Delta\right.,\,
  41. M 4 = 2 μ + 12 μ 2 . M_{4}=\left.2\mu+12\mu^{2}\right..\,
  42. E ( n ) = Δ \left.\right.E(n)=\Delta\,
  43. σ 2 = 2 μ \sigma^{2}=\left.2\mu\right.\,
  44. γ 1 = Δ / ( 2 μ ) 3 / 2 \gamma_{1}=\left.\Delta/(2\mu)^{3/2}\right.\,
  45. γ 2 = 1 / 2 μ . \gamma_{2}=\left.1/2\mu\right..\,
  46. K ( t ; μ 1 , μ 2 ) = def ln ( M ( t ; μ 1 , μ 2 ) ) = k = 0 t k k ! κ k K(t;\mu_{1},\mu_{2})\ \stackrel{\mathrm{def}}{=}\ \ln(M(t;\mu_{1},\mu_{2}))=% \sum_{k=0}^{\infty}{t^{k}\over k!}\,\kappa_{k}
  47. κ 2 k = 2 μ \kappa_{2k}=\left.2\mu\right.
  48. κ 2 k + 1 = Δ . \kappa_{2k+1}=\left.\Delta\right..
  49. f ( k ; μ , μ ) 1 4 π μ [ 1 + n = 1 ( - 1 ) n { 4 k 2 - 1 2 } { 4 k 2 - 3 2 } { 4 k 2 - ( 2 n - 1 ) 2 } n ! 2 3 n ( 2 μ ) n ] f(k;\mu,\mu)\sim{1\over\sqrt{4\pi\mu}}\left[1+\sum_{n=1}^{\infty}(-1)^{n}{\{4k% ^{2}-1^{2}\}\{4k^{2}-3^{2}\}\cdots\{4k^{2}-(2n-1)^{2}\}\over n!\,2^{3n}\,(2\mu% )^{n}}\right]
  50. f ( k ; μ , μ ) e - k 2 / 4 μ 4 π μ . f(k;\mu,\mu)\sim{e^{-k^{2}/4\mu}\over\sqrt{4\pi\mu}}.
  51. { - μ 1 P ( k ) + μ 2 P ( k + 2 ) + ( k + 1 ) P ( k + 1 ) = 0 , P ( 0 ) = e 0 - μ 1 - μ 2 F ~ 1 ( ; 1 ; μ 1 μ 2 ) , P ( 1 ) = e - μ 1 - μ 2 μ 1 F ~ 1 0 ( ; 2 ; μ 1 μ 2 ) } \left\{-\mu_{1}P(k)+\mu_{2}P(k+2)+(k+1)P(k+1)=0,P(0)=e^{-\mu_{1}-\mu_{2}}\,_{0% }\tilde{F}_{1}\left(;1;\mu_{1}\mu_{2}\right),P(1)=e^{-\mu_{1}-\mu_{2}}\mu_{1}% \,{}_{0}\tilde{F}_{1}\left(;2;\mu_{1}\mu_{2}\right)\right\}
  52. X S k e l l a m ( μ 1 , μ 2 ) X\sim Skellam(\mu_{1},\mu_{2})
  53. μ 1 < μ 2 \mu_{1}<\mu_{2}
  54. exp ( - ( μ 1 - μ 2 ) 2 ) ( μ 1 + μ 2 ) 2 - e - ( μ 1 + μ 2 ) 2 μ 1 μ 2 - e - ( μ 1 + μ 2 ) 4 μ 1 μ 2 P ( X 0 ) exp ( - ( μ 1 - μ 2 ) 2 ) \frac{\exp(-(\sqrt{\mu_{1}}-\sqrt{\mu_{2}})^{2})}{(\mu_{1}+\mu_{2})^{2}}-\frac% {e^{-(\mu_{1}+\mu_{2})}}{2\sqrt{\mu_{1}\mu_{2}}}-\frac{e^{-(\mu_{1}+\mu_{2})}}% {4\mu_{1}\mu_{2}}\leq P(X\geq 0)\leq\exp(-(\sqrt{\mu_{1}}-\sqrt{\mu_{2}})^{2})

Skip_reentry.html

  1. γ F = - γ E \gamma_{F}=-\gamma_{E}
  2. V F V E = e 2 γ E L / D \frac{V_{F}}{V_{E}}=e^{\frac{2\gamma_{E}}{L/D}}

Skolem–Noether_theorem.html

  1. B = M n ( k ) = End k ( k n ) B=\operatorname{M}_{n}(k)=\operatorname{End}_{k}(k^{n})
  2. k n k^{n}
  3. V f , V g V_{f},V_{g}
  4. V f , V g V_{f},V_{g}
  5. b : V g V f b:V_{g}\to V_{f}
  6. M n ( k ) = B \operatorname{M}_{n}(k)=B
  7. B B op B\otimes B^{\,\text{op}}
  8. ( f 1 ) ( a z ) = b ( g 1 ) ( a z ) b - 1 (f\otimes 1)(a\otimes z)=b(g\otimes 1)(a\otimes z)b^{-1}
  9. a A a\in A
  10. z B op z\in B^{\,\text{op}}
  11. a = 1 a=1
  12. 1 z = b ( 1 z ) b - 1 1\otimes z=b(1\otimes z)b^{-1}
  13. Z B B op ( k B op ) = B k Z_{B\otimes B^{\,\text{op}}}(k\otimes B^{\,\text{op}})=B\otimes k
  14. b = b 1 b=b^{\prime}\otimes 1
  15. z = 1 z=1
  16. f ( a ) = b g ( a ) b - 1 f(a)=b^{\prime}g(a){b^{\prime-1}}

SL_(complexity).html

  1. L SL NL \mathrm{L}\subseteq\mathrm{SL}\subseteq\mathrm{NL}

Slerp.html

  1. Slerp ( p 0 , p 1 ; t ) = sin [ ( 1 - t ) Ω ] sin Ω p 0 + sin [ t Ω ] sin Ω p 1 . \mathrm{Slerp}(p_{0},p_{1};t)=\frac{\sin{[(1-t)\Omega}]}{\sin\Omega}p_{0}+% \frac{\sin[t\Omega]}{\sin\Omega}p_{1}.
  2. Slerp ( p 0 , p 1 ; t ) = ( 1 - t ) p 0 + t p 1 . \mathrm{Slerp}(p_{0},p_{1};t)=(1-t)p_{0}+tp_{1}.\,\!
  3. e q = 1 + q + q 2 2 + q 3 6 + + q n n ! + . e^{q}=1+q+\frac{q^{2}}{2}+\frac{q^{3}}{6}+\cdots+\frac{q^{n}}{n!}+\cdots.
  4. Slerp ( q 0 , q 1 , t ) \displaystyle\mathrm{Slerp}(q_{0},q_{1},t)

Slip_(aerodynamics).html

  1. β \beta
  2. β \beta
  3. β \beta
  4. β \beta

Sloped_armour.html

  1. T L = T N c o s ( θ ) T_{L}=\frac{T_{N}}{cos(\theta)}
  2. T L T_{L}
  3. T N T_{N}
  4. θ \theta
  5. α \alpha
  6. α \alpha
  7. α \alpha
  8. E d / E k = s i n 2 ( α ) E_{d}/E_{k}={sin^{2}(\alpha)}
  9. E d E_{d}
  10. E k E_{k}
  11. α \alpha
  12. α \alpha
  13. α \alpha
  14. α \alpha
  15. α \alpha
  16. α \alpha

Slug_(mass).html

  1. 1 slug = 1 lb F s 2 ft 1 lb F = 1 slug ft s 2 1\,\,\text{slug}=1\,\frac{\,\text{lb}_{F}\cdot\,\text{s}^{2}}{\,\text{ft}}% \qquad\Longleftrightarrow\qquad 1\,\,\text{lb}_{F}=1\,\frac{\,\text{slug}\cdot% \,\text{ft}}{\,\text{s}^{2}}

Slurry.html

  1. ϕ s l = ρ s ( ρ s l - ρ l ) ρ s l ( ρ s - ρ l ) \phi_{sl}=\frac{\rho_{s}(\rho_{sl}-\rho_{l})}{\rho_{sl}(\rho_{s}-\rho_{l})}
  2. ϕ s l \phi_{sl}
  3. ρ s \rho_{s}
  4. ρ s l \rho_{sl}
  5. ρ l \rho_{l}
  6. S G w a t e r SG_{water}
  7. ϕ s l = ρ s ( ρ s l - 1 ) ρ s l ( ρ s - 1 ) \phi_{sl}=\frac{\rho_{s}(\rho_{sl}-1)}{\rho_{sl}(\rho_{s}-1)}
  8. ϕ s l = M s M s l \phi_{sl}=\frac{M_{s}}{M_{sl}}
  9. M s l = M s ϕ s l M_{sl}=\frac{M_{s}}{\phi_{sl}}
  10. M s + M l = M s ϕ s l M_{s}+M_{l}=\frac{M_{s}}{\phi_{sl}}
  11. M l = M s ϕ s l - M s M_{l}=\frac{M_{s}}{\phi_{sl}}-M_{s}
  12. M l = 1 - ϕ s l ϕ s l M s M_{l}=\frac{1-\phi_{sl}}{\phi_{sl}}M_{s}
  13. ϕ s l \phi_{sl}
  14. M s M_{s}
  15. M s l M_{sl}
  16. M l M_{l}
  17. ϕ s l , m = M s M s l \phi_{sl,m}=\frac{M_{s}}{M_{sl}}
  18. ϕ s l , v = V s V s l \phi_{sl,v}=\frac{V_{s}}{V_{sl}}
  19. ϕ s l , v = M s S G s M s S G s + M l 1 \phi_{sl,v}=\frac{\frac{M_{s}}{SG_{s}}}{\frac{M_{s}}{SG_{s}}+\frac{M_{l}}{1}}
  20. ϕ s l , v = M s M s + M l S G s \phi_{sl,v}=\frac{M_{s}}{M_{s}+M_{l}SG_{s}}
  21. ϕ s l , v = 1 1 + M l S G s M s \phi_{sl,v}=\frac{1}{1+\frac{M_{l}SG_{s}}{M_{s}}}
  22. ϕ s l , v = 1 1 + M l S G s ϕ s l , m M s M s M s + M l \phi_{sl,v}=\frac{1}{1+\frac{M_{l}SG_{s}}{\phi_{sl,m}M_{s}}\frac{M_{s}}{M_{s}+% M_{l}}}
  23. ϕ s l , v = 1 1 + S G s ϕ s l , m M l M s + M l \phi_{sl,v}=\frac{1}{1+\frac{SG_{s}}{\phi_{sl,m}}\frac{M_{l}}{M_{s}+M_{l}}}
  24. ϕ s l , m = M s M s + M l = 1 - M l M s + M l \phi_{sl,m}=\frac{M_{s}}{M_{s}+M_{l}}=1-\frac{M_{l}}{M_{s}+M_{l}}
  25. ϕ s l , v = 1 1 + S G s ( 1 ϕ s l , m - 1 ) \phi_{sl,v}=\frac{1}{1+SG_{s}(\frac{1}{\phi_{sl,m}}-1)}
  26. ϕ s l , v \phi_{sl,v}
  27. ϕ s l , m \phi_{sl,m}
  28. M s M_{s}
  29. M s l M_{sl}
  30. M l M_{l}
  31. S G s SG_{s}

Smash_product.html

  1. X Y = ( X × Y ) / ( X Y ) . X\wedge Y=(X\times Y)/(X\vee Y).\,
  2. Σ X X S 1 . \Sigma X\cong X\wedge S^{1}.\,
  3. Σ k X X S k . \Sigma^{k}X\cong X\wedge S^{k}.\,
  4. X Y Y X , ( X Y ) Z X ( Y Z ) . \begin{aligned}\displaystyle X\wedge Y&\displaystyle\cong Y\wedge X,\\ \displaystyle(X\wedge Y)\wedge Z&\displaystyle\cong X\wedge(Y\wedge Z).\end{aligned}
  5. Hom ( X A , Y ) Hom ( X , Hom ( A , Y ) ) . \mathrm{Hom}(X\otimes A,Y)\cong\mathrm{Hom}(X,\mathrm{Hom}(A,Y)).
  6. Hom ( X A , Y ) Hom ( X , Hom ( A , Y ) ) \mathrm{Hom}(X\wedge A,Y)\cong\mathrm{Hom}(X,\mathrm{Hom}(A,Y))
  7. Hom ( Σ X , Y ) Hom ( X , Ω Y ) . \mathrm{Hom}(\Sigma X,Y)\cong\mathrm{Hom}(X,\Omega Y).

Smith_normal_form.html

  1. m × m m\times m
  2. n × n n\times n
  3. ( α 1 0 0 0 0 α 2 0 0 0 0 0 α r 0 0 0 ) . \begin{pmatrix}\alpha_{1}&0&0&&\cdots&&0\\ 0&\alpha_{2}&0&&\cdots&&0\\ 0&0&\ddots&&&&0\\ \vdots&&&\alpha_{r}&&&\vdots\\ &&&&0&&\\ &&&&&\ddots&\\ 0&&&\cdots&&&0\end{pmatrix}.
  4. α i \alpha_{i}
  5. α i α i + 1 1 i < r \alpha_{i}\mid\alpha_{i+1}\;\forall\;1\leq i<r
  6. α i \alpha_{i}
  7. α i = d i ( A ) d i - 1 ( A ) , \alpha_{i}=\frac{d_{i}(A)}{d_{i-1}(A)},
  8. d i ( A ) d_{i}(A)
  9. i × i i\times i
  10. R n R^{n}
  11. R m R^{m}
  12. S : R m R m S:R^{m}\to R^{m}
  13. T : R n R n T:R^{n}\to R^{n}
  14. S A T S\cdot A\cdot T
  15. A = S A T A^{\prime}=S^{\prime}\cdot A\cdot T^{\prime}
  16. A , S , T A^{\prime},S^{\prime},T^{\prime}
  17. a t , j t 0 a_{t,j_{t}}\neq 0
  18. a k , j t 0 a_{k,j_{t}}\neq 0
  19. t t
  20. a t , j t 0 a_{t,j_{t}}\neq 0
  21. a t , j t a k , j t a_{t,j_{t}}\nmid a_{k,j_{t}}
  22. β = gcd ( a t , j t , a k , j t ) \beta=\gcd\left(a_{t,j_{t}},a_{k,j_{t}}\right)
  23. a t , j t σ + a k , j t τ = β . a_{t,j_{t}}\cdot\sigma+a_{k,j_{t}}\cdot\tau=\beta.
  24. α = a t , j t / β \alpha=a_{t,j_{t}}/\beta
  25. γ = a k , j t / β \gamma=a_{k,j_{t}}/\beta
  26. σ α + τ γ = 1 , \sigma\cdot\alpha+\tau\cdot\gamma=1,
  27. L 0 = ( σ τ - γ α ) L_{0}=\begin{pmatrix}\sigma&\tau\\ -\gamma&\alpha\\ \end{pmatrix}
  28. ( α - τ γ σ ) . \begin{pmatrix}\alpha&-\tau\\ \gamma&\sigma\\ \end{pmatrix}.
  29. L 0 L_{0}
  30. a t , j t a_{t,j_{t}}
  31. δ ( β ) < δ ( a t , j t ) \delta(\beta)<\delta(a_{t,j_{t}})
  32. m × n m\times n
  33. j 1 < < j r j_{1}<\ldots<j_{r}
  34. r min ( m , n ) r\leq\min(m,n)
  35. ( l , j l ) (l,j_{l})
  36. ( i , i ) (i,i)
  37. 1 i r 1\leq i\leq r
  38. α i \alpha_{i}
  39. ( i , i ) (i,i)
  40. i < r i<r
  41. α i α i + 1 \alpha_{i}\nmid\alpha_{i+1}
  42. i i
  43. i + 1 i+1
  44. i + 1 i+1
  45. i i
  46. α i + 1 \alpha_{i+1}
  47. α i \alpha_{i}
  48. ( i , i ) (i,i)
  49. ( i , i ) (i,i)
  50. β = gcd ( α i , α i + 1 ) \beta=\gcd(\alpha_{i},\alpha_{i+1})
  51. ( i + 1 , i + 1 ) (i+1,i+1)
  52. α i , α i + 1 \alpha_{i},\alpha_{i+1}
  53. δ ( α 1 ) + + δ ( α r ) \delta(\alpha_{1})+\cdots+\delta(\alpha_{r})
  54. r × r r\times r
  55. j = 1 r ( r - j ) δ ( α j ) . \sum_{j=1}^{r}(r-j)\delta(\alpha_{j}).
  56. α 1 α 2 α r \alpha_{1}\mid\alpha_{2}\mid\cdots\mid\alpha_{r}
  57. m × m m\times m
  58. n × n n\times n
  59. ( 2 4 4 - 6 6 12 10 - 4 - 16 ) \begin{pmatrix}2&4&4\\ -6&6&12\\ 10&-4&-16\end{pmatrix}
  60. ( 2 0 0 - 6 18 24 10 - 24 - 36 ) ( 2 0 0 0 18 24 0 - 24 - 36 ) \to\begin{pmatrix}2&0&0\\ -6&18&24\\ 10&-24&-36\end{pmatrix}\to\begin{pmatrix}2&0&0\\ 0&18&24\\ 0&-24&-36\end{pmatrix}
  61. ( 2 0 0 0 18 24 0 - 6 - 12 ) ( 2 0 0 0 6 12 0 18 24 ) \to\begin{pmatrix}2&0&0\\ 0&18&24\\ 0&-6&-12\end{pmatrix}\to\begin{pmatrix}2&0&0\\ 0&6&12\\ 0&18&24\end{pmatrix}
  62. ( 2 0 0 0 6 12 0 0 - 12 ) ( 2 0 0 0 6 0 0 0 12 ) \to\begin{pmatrix}2&0&0\\ 0&6&12\\ 0&0&-12\end{pmatrix}\to\begin{pmatrix}2&0&0\\ 0&6&0\\ 0&0&12\end{pmatrix}
  63. ( 2 0 0 0 6 0 0 0 12 ) \begin{pmatrix}2&0&0\\ 0&6&0\\ 0&0&12\end{pmatrix}
  64. x I - A xI-A
  65. x I - B xI-B
  66. A = [ 1 2 0 1 ] , SNF ( x I - A ) = [ 1 0 0 ( x - 1 ) 2 ] B = [ 3 - 4 1 - 1 ] , SNF ( x I - B ) = [ 1 0 0 ( x - 1 ) 2 ] C = [ 1 0 1 2 ] , SNF ( x I - C ) = [ 1 0 0 ( x - 1 ) ( x - 2 ) ] . \begin{aligned}\displaystyle A&\displaystyle{}=\begin{bmatrix}1&2\\ 0&1\end{bmatrix},&&\displaystyle\mbox{SNF}~{}(xI-A)=\begin{bmatrix}1&0\\ 0&(x-1)^{2}\end{bmatrix}\\ \displaystyle B&\displaystyle{}=\begin{bmatrix}3&-4\\ 1&-1\end{bmatrix},&&\displaystyle\mbox{SNF}~{}(xI-B)=\begin{bmatrix}1&0\\ 0&(x-1)^{2}\end{bmatrix}\\ \displaystyle C&\displaystyle{}=\begin{bmatrix}1&0\\ 1&2\end{bmatrix},&&\displaystyle\mbox{SNF}~{}(xI-C)=\begin{bmatrix}1&0\\ 0&(x-1)(x-2)\end{bmatrix}.\end{aligned}

Smith–Volterra–Cantor_set.html

  1. [ 0 , 3 8 ] [ 5 8 , 1 ] . \left[0,\frac{3}{8}\right]\cup\left[\frac{5}{8},1\right].
  2. [ 0 , 5 32 ] [ 7 32 , 3 8 ] [ 5 8 , 25 32 ] [ 27 32 , 1 ] . \left[0,\frac{5}{32}\right]\cup\left[\frac{7}{32},\frac{3}{8}\right]\cup\left[% \frac{5}{8},\frac{25}{32}\right]\cup\left[\frac{27}{32},1\right].
  3. n = 0 2 n 2 2 n + 2 = 1 4 + 1 8 + 1 16 + = 1 2 \sum_{n=0}^{\infty}\frac{2^{n}}{2^{2n+2}}=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}% +\cdots=\frac{1}{2}\,

Smooth_number.html

  1. Ψ ( x , y ) \scriptstyle\Psi(x,y)
  2. Ψ ( x , B ) \scriptstyle\Psi(x,B)
  3. Ψ ( x , B ) 1 π ( B ) ! p B log x log p . \Psi(x,B)\sim\frac{1}{\pi(B)!}\prod_{p\leq B}\frac{\log x}{\log p}.
  4. π ( B ) \scriptstyle{\pi(B)}
  5. B \scriptstyle B
  6. Ψ ( x , y ) = x ρ ( u ) + O ( x log y ) \Psi(x,y)=x\cdot\rho(u)+O\left(\frac{x}{\log y}\right)
  7. ρ ( u ) \scriptstyle\rho(u)
  8. p ν \scriptstyle p^{\nu}
  9. p ν B . p^{\nu}\leq B.\,
  10. 3 2 = 9 5 3^{2}=9\nleq 5
  11. 2 3 > 5 2^{3}>5

SO(8).html

  1. n > 2 n>2
  2. ( ± 1 , ± 1 , 0 , 0 ) (\pm 1,\pm 1,0,0)
  3. ( ± 1 , 0 , ± 1 , 0 ) (\pm 1,0,\pm 1,0)
  4. ( ± 1 , 0 , 0 , ± 1 ) (\pm 1,0,0,\pm 1)
  5. ( 0 , ± 1 , ± 1 , 0 ) (0,\pm 1,\pm 1,0)
  6. ( 0 , ± 1 , 0 , ± 1 ) (0,\pm 1,0,\pm 1)
  7. ( 0 , 0 , ± 1 , ± 1 ) (0,0,\pm 1,\pm 1)
  8. ( 2 - 1 - 1 - 1 - 1 2 0 0 - 1 0 2 0 - 1 0 0 2 ) \begin{pmatrix}2&-1&-1&-1\\ -1&2&0&0\\ -1&0&2&0\\ -1&0&0&2\end{pmatrix}

Soft_error.html

  1. 1 , 000 , 000 , 000 24 × 365.25 {\frac{1{,}000{,}000{,}000}{24\times 365.25}}

Softening.html

  1. Φ = - 1 r 2 + ϵ 2 , \Phi=-\frac{1}{\sqrt{r^{2}+\epsilon^{2}}},
  2. ϵ \epsilon

Solar_azimuth_angle.html

  1. sin ϕ s = - sin h cos δ cos θ s \,\sin\phi_{\mathrm{s}}=\frac{-\sin h\cos\delta}{\cos\theta_{\mathrm{s}}}
  2. cos ϕ s = sin δ cos Φ - cos h cos δ sin Φ cos θ s \,\cos\phi_{\mathrm{s}}=\frac{\sin\delta\cos\Phi-\cos h\cos\delta\sin\Phi}{% \cos\theta_{\mathrm{s}}}
  3. cos ϕ s = sin δ - sin θ s sin Φ cos θ s cos Φ \,\cos\phi_{\mathrm{s}}=\frac{\sin\delta-\sin\theta_{\mathrm{s}}\sin\Phi}{\cos% \theta_{\mathrm{s}}\cos\Phi}
  4. ϕ s \,\phi_{\mathrm{s}}
  5. θ s \,\theta_{\mathrm{s}}
  6. h \,h
  7. δ \,\delta
  8. Φ \,\Phi

Solar_zenith_angle.html

  1. cos θ s = sin φ sin δ + cos φ cos δ cos h \cos\theta_{s}=\sin\varphi\sin\delta+\cos\varphi\cos\delta\cos h
  2. sin α s = cos h cos δ cos φ + sin δ sin φ \sin\alpha_{\mathrm{s}}=\cos h\cos\delta\cos\varphi+\sin\delta\sin\varphi
  3. cos h 0 = - tan ϕ tan δ . \cos h_{0}=-\tan\phi\tan\delta.
  4. cos θ s ¯ = - h 0 h 0 Q cos θ s d h - h 0 h 0 Q d h \overline{\cos\theta_{s}}=\frac{\int_{-h_{0}}^{h_{0}}Q\cos\theta_{s}\,\text{d}% h}{\int_{-h_{0}}^{h_{0}}Q\,\text{d}h}

Solid_oxide_fuel_cell.html

  1. 1 2 O 2 ( g ) + 2 e + V o O o × \frac{1}{2}\mathrm{O_{2}(g)}+2\mathrm{e^{\prime}}+{V}^{\bullet\bullet}_{o}% \longrightarrow{O}^{\times}_{o}
  2. V = E 0 - i R ω - η c a t h o d e - η a n o d e {V}={E}_{0}-{iR}_{\omega}-{\eta}_{cathode}-{\eta}_{anode}
  3. E 0 {E}_{0}
  4. η c a t h o d e {\eta}_{cathode}
  5. η a n o d e {\eta}_{anode}
  6. E S O F C = E m a x - i m a x η f r 1 r 1 r 2 ( 1 - η f ) + 1 E_{SOFC}=\frac{E_{max}-i_{max}\cdot\eta_{f}\cdot r_{1}}{\frac{r_{1}}{r_{2}}% \cdot\left(1-\eta_{f}\right)+1}
  7. E S O F C E_{SOFC}
  8. E m a x E_{max}
  9. i m a x i_{max}
  10. η f \eta_{f}
  11. r 1 r_{1}
  12. r 2 r_{2}
  13. r 1 = δ σ r_{1}=\frac{\delta}{\sigma}
  14. δ \delta
  15. σ \sigma
  16. σ = σ 0 e - E R T \sigma=\sigma_{0}\cdot e^{\frac{-E}{R\cdot T}}
  17. σ 0 \sigma_{0}
  18. E E
  19. T T
  20. R R
  21. η a c t = R T β z F × l n ( i i 0 ) {\eta}_{act}=\frac{RT}{{\beta}zF}\times ln\left(\frac{i}{{i}_{0}}\right)
  22. R R
  23. T 0 {T}_{0}
  24. β {\beta}
  25. z z
  26. F F
  27. i i
  28. i 0 i_{0}

Solovay–Strassen_primality_test.html

  1. a ( p - 1 ) / 2 ( a p ) ( mod p ) a^{(p-1)/2}\equiv\left(\frac{a}{p}\right)\;\;(\mathop{{\rm mod}}p)
  2. ( a p ) \left(\tfrac{a}{p}\right)
  3. ( a n ) \left(\tfrac{a}{n}\right)
  4. a ( n - 1 ) / 2 ( a n ) ( mod n ) a^{(n-1)/2}\equiv\left(\frac{a}{n}\right)\;\;(\mathop{{\rm mod}}n)
  5. a ( / n ) * a\in(\mathbb{Z}/n\mathbb{Z})^{*}
  6. ( a n ) (\tfrac{a}{n})
  7. ( 47 221 ) (\tfrac{47}{221})
  8. ( a n ) (\tfrac{a}{n})
  9. ( 2 221 ) (\tfrac{2}{221})
  10. ( a n ) \left(\tfrac{a}{n}\right)
  11. a ( n - 1 ) / 2 x ( mod n ) a^{(n-1)/2}\not\equiv x\;\;(\mathop{{\rm mod}}n)
  12. ( a a i ) ( n - 1 ) / 2 = a ( n - 1 ) / 2 a i ( n - 1 ) / 2 = a ( n - 1 ) / 2 ( a i n ) ( a n ) ( a i n ) ( mod n ) . (a\cdot a_{i})^{(n-1)/2}=a^{(n-1)/2}\cdot a_{i}^{(n-1)/2}=a^{(n-1)/2}\cdot% \left(\frac{a_{i}}{n}\right)\not\equiv\left(\frac{a}{n}\right)\left(\frac{a_{i% }}{n}\right)\;\;(\mathop{{\rm mod}}n).
  13. ( a n ) ( a i n ) = ( a a i n ) , \left(\frac{a}{n}\right)\left(\frac{a_{i}}{n}\right)=\left(\frac{a\cdot a_{i}}% {n}\right),
  14. ( a a i ) ( n - 1 ) / 2 ( a a i n ) ( mod n ) . (a\cdot a_{i})^{(n-1)/2}\not\equiv\left(\frac{a\cdot a_{i}}{n}\right)\;\;(% \mathop{{\rm mod}}n).
  15. 2 - k exp ( - ( 1 + o ( 1 ) ) log x log log log x log log x ) 2^{-k}\exp\left(-(1+o(1))\frac{\log x\,\log\log\log x}{\log\log x}\right)

Sorgenfrey_plane.html

  1. \mathbb{R}
  2. 𝕊 \mathbb{S}
  3. 𝕊 \mathbb{S}
  4. 𝕊 \mathbb{S}
  5. Δ = { ( x , - x ) x } \Delta=\{(x,-x)\mid x\in\mathbb{R}\}
  6. 𝕊 \mathbb{S}
  7. K = { ( x , - x ) x } K=\{(x,-x)\mid x\in\mathbb{Q}\}
  8. Δ K \Delta\setminus K
  9. 𝕊 \mathbb{S}

Sortino_ratio.html

  1. S S
  2. S = R - T D R S=\frac{R-T}{DR}
  3. R R
  4. T T
  5. D R DR
  6. S S
  7. D R = - T ( T - r ) 2 f ( r ) d r DR=\sqrt{\int_{-\infty}^{T}(T-r)^{2}f(r)\,dr}
  8. D R DR
  9. D R 2 DR^{2}
  10. T T
  11. r r
  12. f ( r ) f(r)
  13. f ( r ) f(r)

Sound_energy_density.html

  1. w = p v c w=\frac{pv}{c}
  2. w = I c , w=\frac{I}{c},
  3. w = P A c , w=\frac{P}{Ac},

Source-synchronous.html

  1. T c l o c k > T s e t u p + T k o + T s k e w T_{clock}>T_{setup}+T_{ko}+T_{skew}
  2. T k o T_{ko}
  3. T s k e w T_{skew}
  4. T c l o c k T_{clock}

Space_form.html

  1. M n M^{n}
  2. K = - 1 K=-1
  3. H n H^{n}
  4. K = 0 K=0
  5. R n R^{n}
  6. K = + 1 K=+1
  7. S n S^{n}
  8. R n + 1 R^{n+1}
  9. H n H^{n}
  10. M K M_{K}
  11. K K
  12. K < 0 K<0
  13. S n S^{n}
  14. M K M_{K}
  15. K K
  16. K > 0 K>0
  17. M M
  18. K K
  19. M K M_{K}
  20. Γ \Gamma
  21. M K M_{K}
  22. M M
  23. π 1 ( M ) \pi_{1}(M)
  24. Γ \Gamma
  25. R n R^{n}
  26. H 2 H^{2}
  27. H 3 H^{3}

Space_hierarchy_theorem.html

  1. SPACE ( o ( f ( n ) ) ) SPACE ( f ( n ) ) \operatorname{SPACE}\left(o(f(n))\right)\subsetneq\operatorname{SPACE}(f(n))
  2. o o
  3. f : f:\mathbb{N}\longrightarrow\mathbb{N}
  4. f ( n ) log n f(n)\geq\log~{}n
  5. f ( n ) f(n)
  6. O ( f ( n ) ) O(f(n))
  7. 1 n 1^{n}
  8. 1 n 1^{n}
  9. n n
  10. 1 1
  11. f : f:\mathbb{N}\longrightarrow\mathbb{N}
  12. L L
  13. O ( f ( n ) ) O(f(n))
  14. o ( f ( n ) ) o(f(n))
  15. O ( f ( n ) ) O(f(n))
  16. o ( f ( n ) ) o(f(n))
  17. L L
  18. L = { ( M , 10 k ) : M does not accept ( M , 10 k ) using space f ( | M , 10 k | ) } L=\{~{}(\langle M\rangle,10^{k}):M\mbox{ does not accept }~{}(\langle M\rangle% ,10^{k})\mbox{ using space }~{}\leq f(|\langle M\rangle,10^{k}|)~{}\}
  19. M M
  20. o ( f ( n ) ) o(f(n))
  21. L L
  22. M M
  23. ( M , 10 k ) (\langle M\rangle,10^{k})
  24. L L
  25. x x
  26. f ( | x | ) f(|x|)
  27. f ( | x | ) f(|x|)
  28. f ( | x | ) f(|x|)
  29. x x
  30. M , 10 k \langle M\rangle,10^{k}
  31. M M
  32. M M
  33. x x
  34. 2 f ( | x | ) 2^{f(|x|)}
  35. f ( | x | ) f(|x|)
  36. f ( | x | ) f(|x|)
  37. 2 f ( | x | ) 2^{f(|x|)}
  38. M M
  39. x x
  40. 2 f ( | x | ) 2^{f(|x|)}
  41. M M
  42. x x
  43. M M
  44. O ( f ( x ) ) O(f(x))
  45. M M
  46. x x
  47. L L
  48. o ( f ( n ) ) o(f(n))
  49. L L
  50. o ( f ( n ) ) o(f(n))
  51. L ¯ \overline{L}
  52. M ¯ \overline{M}
  53. o ( f ( n ) ) o(f(n))
  54. w = ( M ¯ , 10 k ) w=(\langle\overline{M}\rangle,10^{k})
  55. L L
  56. M M
  57. M ¯ \overline{M}
  58. w w
  59. w w
  60. L L
  61. w = ( M ¯ , 10 k ) w=(\langle\overline{M}\rangle,10^{k})
  62. L L
  63. M M
  64. M ¯ \overline{M}
  65. w w
  66. w w
  67. L L
  68. f 1 f_{1}
  69. f 2 : f_{2}:\mathbb{N}\longrightarrow\mathbb{N}
  70. f 1 f_{1}
  71. f 2 f_{2}
  72. f 2 f_{2}
  73. f 1 f_{1}
  74. \subsetneq
  75. f 2 f_{2}
  76. n k n^{k}
  77. k 1 < k 2 k_{1}<k_{2}
  78. n k 1 n^{k_{1}}
  79. \subsetneq
  80. n k 2 n^{k_{2}}
  81. a 1 < a 2 , a_{1}<a_{2},
  82. n a 1 n^{a_{1}}
  83. \subsetneq
  84. n a 2 n^{a_{2}}
  85. \subsetneq
  86. \subseteq
  87. log 2 n \log^{2}n
  88. log 2 n ) \log^{2}n)\subsetneq
  89. n n
  90. \notin
  91. \subsetneq
  92. \subsetneq

Spaceship_(cellular_automaton).html

  1. c / 4 c/4
  2. c / 2 c/2
  3. ( x , y ) (x,y)
  4. n n
  5. v v
  6. v = max ( | x | , | y | ) n c v=\frac{\max\left(|x|,|y|\right)}{n}\,c

Space–time_tradeoff.html

  1. 2 n + 1 2^{n+1}
  2. O ( 2 n ) O(2^{n})
  3. 2 2 n 2^{2n}
  4. O ( 1 ) O(1)

Span_of_control.html

  1. n . n.
  2. n ( n - 1 ) n(n-1)
  3. n ( 2 n / 2 - 1 ) n(2^{n}/2-1)

Sparse_array.html

  1. v a l i val_{i}
  2. p o s i pos_{i}

Sparse_grid.html

  1. l l
  2. N l N_{l}
  3. r r
  4. r r
  5. d d
  6. | E l | = O ( N l - r d ) |E_{l}|=O(N_{l}^{-\frac{r}{d}})
  7. Q ( 1 ) Q^{(1)}
  8. d d
  9. Q ( d ) Q^{(d)}
  10. f f
  11. Q l ( d ) f = ( i = 1 l ( Q i ( 1 ) - Q i - 1 ( 1 ) ) Q l - i + 1 ( d - 1 ) ) f Q_{l}^{(d)}f=\left(\sum_{i=1}^{l}\left(Q_{i}^{(1)}-Q_{i-1}^{(1)}\right)\otimes Q% _{l-i+1}^{(d-1)}\right)f
  12. Q Q
  13. 1 - d 1-d
  14. i i
  15. O ( 2 i ) O(2^{i})
  16. r r
  17. | E l | = O ( N l - r ( log N l ) ( d - 1 ) ( r + 1 ) ) |E_{l}|=O\left(N_{l}^{-r}\left(\log N_{l}\right)^{(d-1)(r+1)}\right)

Specific_absorption_rate.html

  1. SAR = 1 V sample σ ( 𝐫 ) | 𝐄 ( 𝐫 ) | 2 ρ ( 𝐫 ) d 𝐫 \,\text{SAR}=\frac{1}{V}\int_{\textrm{sample}}\frac{\sigma(\mathbf{r})|\mathbf% {E}(\mathbf{r})|^{2}}{\rho(\mathbf{r})}d\mathbf{r}
  2. σ \sigma
  3. E E
  4. ρ \rho
  5. V V

Specific_orbital_energy.html

  1. ϵ \epsilon\,\!
  2. ϵ p \epsilon_{p}\,\!
  3. ϵ k \epsilon_{k}\,\!
  4. ϵ = ϵ k + ϵ p \epsilon=\epsilon_{k}+\epsilon_{p}\!
  5. ϵ = v 2 2 - μ r = - 1 2 μ 2 h 2 ( 1 - e 2 ) = - μ 2 a \epsilon={v^{2}\over{2}}-{\mu\over{r}}=-{1\over{2}}{\mu^{2}\over{h^{2}}}\left(% 1-e^{2}\right)=-\frac{\mu}{2a}
  6. v v\,\!
  7. r r\,\!
  8. μ = G ( m 1 + m 2 ) \mu={G}(m_{1}+m_{2})\,\!
  9. h h\,\!
  10. e e\,\!
  11. a a\,\!
  12. ϵ = - μ 2 a \epsilon=-{\mu\over{2a}}\,\!
  13. μ = G ( m 1 + m 2 ) \mu={G}(m_{1}+m_{2})\,\!
  14. a {a}\,\!
  15. h 2 = μ p = μ a ( 1 - e 2 ) {h^{2}}=\mu p=\mu a(1-e^{2})\,\!
  16. ϵ = v 2 2 - μ r \epsilon={v^{2}\over{2}}-{\mu\over{r}}
  17. v p 2 = h 2 < m t p l > r p 2 = h 2 a 2 ( 1 - e ) 2 = μ a ( 1 - e 2 ) a 2 ( 1 - e ) 2 = μ ( 1 - e 2 ) a ( 1 - e ) 2 {v_{p}^{2}}={h^{2}\over<mtpl>{{r_{p}}}^{2}}={h^{2}\over{{a^{2}(1-e)}}^{2}}={% \mu a(1-e^{2})\over{{a^{2}(1-e)}}^{2}}={\mu(1-e^{2})\over{{a(1-e)}}^{2}}\,\!
  18. ϵ = μ a [ ( 1 - e 2 ) < m t p l > 2 ( 1 - e ) 2 - 1 ( 1 - e ) ] = μ a [ ( 1 - e ) ( 1 + e ) 2 ( 1 - e ) 2 - 1 ( 1 - e ) ] = μ a [ ( 1 + e ) 2 ( 1 - e ) - 2 2 ( 1 - e ) ] = μ a [ e - 1 2 ( 1 - e ) ] \epsilon={\mu\over{a}}{\left[{(1-e^{2})\over<mtpl>{{2(1-e)}}^{2}}-{1\over{(1-e% )}}\right]}={\mu\over{a}}{\left[{{(1-e)(1+e)}\over{{2(1-e)}}^{2}}-{1\over{(1-e% )}}\right]}={\mu\over{a}}{\left[{(1+e)\over{{2(1-e)}}}-{2\over{2(1-e)}}\right]% }={\mu\over{a}}{\left[{{e-1}\over{2(1-e)}}\right]}\,\!
  19. ϵ = - μ 2 a \epsilon=-{\mu\over{2a}}\,\!
  20. ϵ = 0 . \epsilon=0\,\!.
  21. ϵ = μ 2 a . \epsilon={\mu\over{2a}}\,\!.
  22. C 3 C_{3}\,\!
  23. v v_{\infty}\,\!
  24. 2 ϵ = C 3 = v 2 . 2\epsilon=C_{3}=v_{\infty}^{2}\,\!.
  25. 𝐫 \mathbf{r}\,\!
  26. 𝐯 \mathbf{v}\,\!
  27. μ \mu\,\!
  28. μ 2 a 2 \frac{\mu}{2a^{2}}\,\!
  29. μ = G ( m 1 + m 2 ) \mu={G}(m_{1}+m_{2})\,\!
  30. a a\,\!
  31. - μ 2 a + μ R = μ ( 2 a - R ) 2 a R . \ -\frac{\mu}{2a}+\frac{\mu}{R}=\frac{\mu(2a-R)}{2aR}.
  32. R / 2 R/2
  33. ( 2 a - R ) g (2a-R)g
  34. 2 a - R 2a-R
  35. μ = G M \mu=GM\,\!
  36. ϵ = ϵ k + ϵ p = v 2 2 - μ r = \epsilon=\epsilon_{k}+\epsilon_{p}={v^{2}\over{2}}-{\mu\over{r}}=
  37. v = v_{\infty}=\,\!
  38. 𝐯 𝐚 \mathbf{v}\cdot\mathbf{a}
  39. 𝐯 ( 𝐚 - 𝐠 ) \mathbf{v}\cdot(\mathbf{a}-\mathbf{g})
  40. 𝐯 𝐠 \mathbf{v}\cdot\mathbf{g}
  41. 𝐯 𝐚 | 𝐚 | \frac{\mathbf{v\cdot a}}{|\mathbf{a}|}
  42. Δ ϵ = v d ( Δ v ) = v a d t \Delta\epsilon=\int v\,d(\Delta v)=\int v\,adt

Specific_relative_angular_momentum.html

  1. 𝐡 \mathbf{h}\,\!
  2. 𝐫 \mathbf{r}\,\!
  3. 𝐯 \mathbf{v}\,\!
  4. 𝐡 = 𝐫 × 𝐯 = 𝐋 μ \mathbf{h}=\mathbf{r}\times\mathbf{v}={\mathbf{L}\over\mu}
  5. 𝐫 \mathbf{r}\,\!
  6. 𝐯 \mathbf{v}\,\!
  7. 𝐋 = 𝐋 𝟏 + 𝐋 𝟐 \mathbf{L}=\mathbf{L_{1}}+\mathbf{L_{2}}\,
  8. μ \mu\,
  9. 𝐡 \mathbf{h}\,\!
  10. 𝐡 \mathbf{h}\,\!
  11. 𝐡 \mathbf{h}\,\!
  12. h h\,\!
  13. h = 𝐡 h=\left\|\mathbf{h}\right\|
  14. h h\,\!
  15. h = 2 π a b 2 π a 3 G ( M + m ) = b G ( M + m ) a = a ( 1 - e 2 ) G ( M + m ) = p G ( M + m ) h=\frac{2\pi ab}{2\pi\sqrt{\frac{a^{3}}{G(M\!+\!m)}}}=b\sqrt{\frac{G(M\!+\!m)}% {a}}=\sqrt{a(1-e^{2})G(M\!+\!m)}=\sqrt{pG(M\!+\!m)}
  16. a a\,
  17. b b\,
  18. p p\,
  19. G G\,
  20. M M\,
  21. m m\,

Specific_weight.html

  1. γ = ρ g \gamma=\rho\,g
  2. γ \gamma
  3. ρ \rho
  4. g g
  5. γ = ( 1 + w ) G s γ w 1 + e \gamma=\frac{(1+w)G_{s}\gamma_{w}}{1+e}
  6. γ \gamma
  7. γ w \gamma_{w}
  8. γ d = G s γ w 1 + e = γ 1 + w \gamma_{d}=\frac{G_{s}\gamma_{w}}{1+e}=\frac{\gamma}{1+w}
  9. γ \gamma
  10. γ d \gamma_{d}
  11. γ w \gamma_{w}
  12. γ s = ( G s + e ) γ w 1 + e \gamma_{s}=\frac{(G_{s}+e)\gamma_{w}}{1+e}
  13. γ s \gamma_{s}
  14. γ w \gamma_{w}
  15. γ = γ s - γ w \gamma^{{}^{\prime}}=\gamma_{s}-\gamma_{w}
  16. γ \gamma^{{}^{\prime}}
  17. γ s \gamma_{s}
  18. γ w \gamma_{w}

Spectral_sequence.html

  1. E p , q r E^{r}_{p,q}
  2. E r p , q E^{p,q}_{r}
  3. 0 = B 0 B 1 B 2 B r Z r Z 2 Z 1 Z 0 = E 0 0=B_{0}\subset B_{1}\subset B_{2}\subset\dots\subset B_{r}\subset\dots\subset Z% _{r}\subset\dots\subset Z_{2}\subset Z_{1}\subset Z_{0}=E_{0}
  4. E r Z r - 1 / B r - 1 E_{r}\simeq Z_{r-1}/B_{r-1}
  5. Z 0 = E 0 , B 0 = 0 Z_{0}=E_{0},B_{0}=0
  6. Z r , B r Z_{r},B_{r}
  7. Z r / B r - 1 , B r / B r - 1 Z_{r}/B_{r-1},B_{r}/B_{r-1}
  8. E r d r E r . E_{r}\overset{d_{r}}{\to}E_{r}.
  9. Z = r Z r , B = r B r Z_{\infty}=\cap_{r}Z_{r},B_{\infty}=\cup_{r}B_{r}
  10. E = Z / B E_{\infty}=Z_{\infty}/B_{\infty}
  11. E E_{\infty}
  12. E r p , q E_{r}^{p,q}
  13. E p , q = F p H p + q / F p - 1 H p + q E^{\infty}_{p,q}=F_{p}H_{p+q}/F_{p-1}H_{p+q}
  14. E p , q r E^{r}_{p,q}
  15. E p , q 2 = 0 E^{2}_{p,q}=0
  16. E = E 2 E^{\infty}=E^{2}
  17. 0 = F - 1 H n F 0 H n F n H n = H n 0=F_{-1}H_{n}\subset F_{0}H_{n}\subset\dots\subset F_{n}H_{n}=H_{n}
  18. E p , q = F p H p + q / F p - 1 H p + q E^{\infty}_{p,q}=F_{p}H_{p+q}/F_{p-1}H_{p+q}
  19. F 0 H n = E 0 , n 2 F_{0}H_{n}=E^{2}_{0,n}
  20. F 1 H n / F 0 H n = E 1 , n - 1 2 F_{1}H_{n}/F_{0}H_{n}=E^{2}_{1,n-1}
  21. F 2 H n / F 1 H n = 0 F_{2}H_{n}/F_{1}H_{n}=0
  22. F 3 H n / F 2 H n = 0 F_{3}H_{n}/F_{2}H_{n}=0
  23. 0 E 0 , n 2 H n E 1 , n - 1 2 0 0\to E^{2}_{0,n}\to H_{n}\to E^{2}_{1,n-1}\to 0
  24. E p , q r E^{r}_{p,q}
  25. E p , 0 3 = ker ( d : E p , 0 2 E p - 2 , 1 2 ) E^{3}_{p,0}=\operatorname{ker}(d:E^{2}_{p,0}\to E^{2}_{p-2,1})
  26. E p , 1 3 = coker ( d : E p + 2 , 0 2 E p , 1 2 ) E^{3}_{p,1}=\operatorname{coker}(d:E^{2}_{p+2,0}\to E^{2}_{p,1})
  27. 0 E p , 0 E p , 0 2 𝑑 E p - 2 , 1 2 E p - 2 , 1 0 0\to E^{\infty}_{p,0}\to E^{2}_{p,0}\overset{d}{\to}E^{2}_{p-2,1}\to E^{\infty% }_{p-2,1}\to 0
  28. F p - 2 H p / F p - 3 H p = E p - 2 , 2 = 0 F_{p-2}H_{p}/F_{p-3}H_{p}=E^{\infty}_{p-2,2}=0
  29. F p - 3 H p / F p - 4 H p = 0 F_{p-3}H_{p}/F_{p-4}H_{p}=0
  30. 0 E p - 1 , 1 H p E p , 0 0 0\to E^{\infty}_{p-1,1}\to H_{p}\to E^{\infty}_{p,0}\to 0
  31. H p + 1 E p + 1 , 0 2 𝑑 E p - 1 , 1 2 H p E p , 0 2 𝑑 E p - 2 , 1 2 H p - 1 . \cdots\to H_{p+1}\to E^{2}_{p+1,0}\overset{d}{\to}E^{2}_{p-1,1}\to H_{p}\to E^% {2}_{p,0}\overset{d}{\to}E^{2}_{p-2,1}\to H_{p-1}\to\dots.
  32. F 𝑖 E 𝑝 S n F\overset{i}{\to}E\overset{p}{\to}S^{n}
  33. E p , q 2 = H p ( S n ; H q ( F ) ) H p + q ( E ) E^{2}_{p,q}=H_{p}(S^{n};H_{q}(F))\Rightarrow H_{p+q}(E)
  34. E p , q = F p H p + q ( E ) / F p - 1 H p + q ( E ) E^{\infty}_{p,q}=F_{p}H_{p+q}(E)/F_{p-1}H_{p+q}(E)
  35. H p ( S n ) H_{p}(S^{n})
  36. E p , q 2 E^{2}_{p,q}
  37. E p , q 2 = H q ( F ) E^{2}_{p,q}=H_{q}(F)
  38. E = E n + 1 , E n = E 2 E^{\infty}=E^{n+1},E^{n}=E^{2}
  39. E n + 1 E^{n+1}
  40. 0 E n , q - n E n , q - n n 𝑑 E 0 , q - 1 n E 0 , q - 1 0. 0\to E^{\infty}_{n,q-n}\to E^{n}_{n,q-n}\overset{d}{\to}E^{n}_{0,q-1}\to E^{% \infty}_{0,q-1}\to 0.
  41. H = H ( E ) H=H(E)
  42. F 1 H q / F 0 H q = E 1 , q - 1 = 0 F_{1}H_{q}/F_{0}H_{q}=E^{\infty}_{1,q-1}=0
  43. E n , q - n = F n H q / F 0 H q E^{\infty}_{n,q-n}=F_{n}H_{q}/F_{0}H_{q}
  44. F n H q = H q F_{n}H_{q}=H_{q}
  45. 0 E 0 , q H q E n , q - n 0. 0\to E^{\infty}_{0,q}\to H_{q}\to E^{\infty}_{n,q-n}\to 0.
  46. H q ( F ) i * H q ( E ) H q - n ( F ) 𝑑 H q - 1 ( F ) i * H q - 1 ( E ) H q - n - 1 ( F ) \dots\to H_{q}(F)\overset{i_{*}}{\to}H_{q}(E)\to H_{q-n}(F)\overset{d}{\to}H_{% q-1}(F)\overset{i_{*}}{\to}H_{q-1}(E)\to H_{q-n-1}(F)\to\dots
  47. E r p , q E_{r}^{p,q}
  48. 0 = F n + 1 H n F n H n F 0 H n = H n 0=F^{n+1}H^{n}\subset F^{n}H^{n}\subset\dots\subset F^{0}H^{n}=H^{n}
  49. E p , q = F p H p + q / F p + 1 H p + q . E_{\infty}^{p,q}=F^{p}H^{p+q}/F^{p+1}H^{p+q}.
  50. E p , q 2 E^{2}_{p,q}
  51. 0 E 0 , 1 E 2 0 , 1 𝑑 E 2 2 , 0 E 2 , 0 0. 0\to E^{0,1}_{\infty}\to E^{0,1}_{2}\overset{d}{\to}E^{2,0}_{2}\to E^{2,0}_{% \infty}\to 0.
  52. E 1 , 0 = E 2 1 , 0 E_{\infty}^{1,0}=E_{2}^{1,0}
  53. F 2 H 1 = 0 , F^{2}H^{1}=0,
  54. 0 E 2 1 , 0 H 1 E 0 , 1 0 0\to E_{2}^{1,0}\to H^{1}\to E^{0,1}_{\infty}\to 0
  55. F 3 H 2 = 0 F^{3}H^{2}=0
  56. E 2 , 0 H 2 E^{2,0}_{\infty}\subset H^{2}
  57. 0 E 2 1 , 0 H 1 E 2 0 , 1 𝑑 E 2 2 , 0 H 2 . 0\to E^{1,0}_{2}\to H^{1}\to E^{0,1}_{2}\overset{d}{\to}E^{2,0}_{2}\to H^{2}.
  58. E p , q r E^{r}_{p,q}
  59. E p , q r = 0 E^{r}_{p,q}=0
  60. E p , 0 r E p , 0 r - 1 E p , 0 3 E p , 0 2 E^{r}_{p,0}\to E^{r-1}_{p,0}\to\dots\to E^{3}_{p,0}\to E^{2}_{p,0}
  61. E p , q r = 0 E^{r}_{p,q}=0
  62. τ : E p , 0 2 E 0 , p - 1 2 \tau:E^{2}_{p,0}\to E^{2}_{0,p-1}
  63. E p , 0 2 E p , 0 p 𝑑 E 0 , p - 1 p E 0 , p - 1 2 E^{2}_{p,0}\to E^{p}_{p,0}\overset{d}{\to}E^{p}_{0,p-1}\to E^{2}_{0,p-1}
  64. E r p , q E_{r}^{p,q}
  65. E p , q r = 0 E^{r}_{p,q}=0
  66. E p , q r = 0 E^{r}_{p,q}=0
  67. τ : E 2 0 , q - 1 E 2 q , 0 \tau:E_{2}^{0,q-1}\to E_{2}^{q,0}
  68. d : E q 0 , q - 1 E q q , 0 d:E_{q}^{0,q-1}\to E_{q}^{q,0}
  69. E r p , q E^{p,q}_{r}
  70. E r E_{r}
  71. E r + 1 E_{r+1}
  72. E r E_{r}
  73. F E B F\to E\to B
  74. E E_{\infty}
  75. Z - 1 p , q = Z 0 p , q = F p C p + q Z_{-1}^{p,q}=Z_{0}^{p,q}=F^{p}C^{p+q}
  76. B 0 p , q = 0 B_{0}^{p,q}=0
  77. E 0 p , q = Z 0 p , q B 0 p , q + Z - 1 p + 1 , q - 1 = F p C p + q F p + 1 C p + q E_{0}^{p,q}=\frac{Z_{0}^{p,q}}{B_{0}^{p,q}+Z_{-1}^{p+1,q-1}}=\frac{F^{p}C^{p+q% }}{F^{p+1}C^{p+q}}
  78. E 0 = p , q Z E 0 p , q E_{0}=\bigoplus_{p,q\in{Z}}E_{0}^{p,q}
  79. Z ¯ 1 p , q = ker d 0 p , q : E 0 p , q E 0 p , q + 1 = ker d 0 p , q : F p C p + q / F p + 1 C p + q F p C p + q + 1 / F p + 1 C p + q + 1 \bar{Z}_{1}^{p,q}=\ker d_{0}^{p,q}:E_{0}^{p,q}\rightarrow E_{0}^{p,q+1}=\ker d% _{0}^{p,q}:F^{p}C^{p+q}/F^{p+1}C^{p+q}\rightarrow F^{p}C^{p+q+1}/F^{p+1}C^{p+q% +1}
  80. B ¯ 1 p , q = im d 0 p , q - 1 : E 0 p , q - 1 E 0 p , q = im d 0 p , q - 1 : F p C p + q - 1 / F p + 1 C p + q - 1 F p C p + q / F p + 1 C p + q \bar{B}_{1}^{p,q}=\mbox{im }~{}d_{0}^{p,q-1}:E_{0}^{p,q-1}\rightarrow E_{0}^{p% ,q}=\mbox{im }~{}d_{0}^{p,q-1}:F^{p}C^{p+q-1}/F^{p+1}C^{p+q-1}\rightarrow F^{p% }C^{p+q}/F^{p+1}C^{p+q}
  81. E 1 p , q = Z ¯ 1 p , q B ¯ 1 p , q = ker d 0 p , q : E 0 p , q E 0 p , q + 1 im d 0 p , q - 1 : E 0 p , q - 1 E 0 p , q E_{1}^{p,q}=\frac{\bar{Z}_{1}^{p,q}}{\bar{B}_{1}^{p,q}}=\frac{\ker d_{0}^{p,q}% :E_{0}^{p,q}\rightarrow E_{0}^{p,q+1}}{\mbox{im }~{}d_{0}^{p,q-1}:E_{0}^{p,q-1% }\rightarrow E_{0}^{p,q}}
  82. E 1 = p , q Z E 1 p , q = p , q Z Z ¯ 1 p , q B ¯ 1 p , q E_{1}=\bigoplus_{p,q\in{Z}}E_{1}^{p,q}=\bigoplus_{p,q\in{Z}}\frac{\bar{Z}_{1}^% {p,q}}{\bar{B}_{1}^{p,q}}
  83. Z ¯ 1 p , q \bar{Z}_{1}^{p,q}
  84. B ¯ 1 p , q \bar{B}_{1}^{p,q}
  85. E 0 p , q E_{0}^{p,q}
  86. Z 1 p , q = ker d 0 p , q : F p C p + q C p + q + 1 / F p + 1 C p + q + 1 Z_{1}^{p,q}=\ker d_{0}^{p,q}:F^{p}C^{p+q}\rightarrow C^{p+q+1}/F^{p+1}C^{p+q+1}
  87. B 1 p , q = ( im d 0 p , q - 1 : F p C p + q - 1 C p + q ) F p C p + q B_{1}^{p,q}=(\mbox{im }~{}d_{0}^{p,q-1}:F^{p}C^{p+q-1}\rightarrow C^{p+q})\cap F% ^{p}C^{p+q}
  88. E 1 p , q = Z 1 p , q B 1 p , q + Z 0 p + 1 , q - 1 . E_{1}^{p,q}=\frac{Z_{1}^{p,q}}{B_{1}^{p,q}+Z_{0}^{p+1,q-1}}.
  89. Z 1 p , q Z_{1}^{p,q}
  90. B 1 p , q B_{1}^{p,q}
  91. Z r p , q Z_{r}^{p,q}
  92. B r p , q B_{r}^{p,q}
  93. Z r p , q = ker d 0 p , q : F p C p + q C p + q + 1 / F p + r C p + q + 1 Z_{r}^{p,q}=\ker d_{0}^{p,q}:F^{p}C^{p+q}\rightarrow C^{p+q+1}/F^{p+r}C^{p+q+1}
  94. B r p , q = ( im d 0 p - r + 1 , q + r - 2 : F p - r + 1 C p + q - 1 C p + q ) F p C p + q B_{r}^{p,q}=(\mbox{im }~{}d_{0}^{p-r+1,q+r-2}:F^{p-r+1}C^{p+q-1}\rightarrow C^% {p+q})\cap F^{p}C^{p+q}
  95. E r p , q = Z r p , q B r p , q + Z r - 1 p + 1 , q - 1 E_{r}^{p,q}=\frac{Z_{r}^{p,q}}{B_{r}^{p,q}+Z_{r-1}^{p+1,q-1}}
  96. B r p , q = d 0 p , q ( Z r - 1 p - r + 1 , q + r - 2 ) . B_{r}^{p,q}=d_{0}^{p,q}(Z_{r-1}^{p-r+1,q+r-2}).
  97. d r p , q : E r p , q E r p + r , q - r + 1 d_{r}^{p,q}:E_{r}^{p,q}\rightarrow E_{r}^{p+r,q-r+1}
  98. C p + q C^{p+q}
  99. Z r p , q Z_{r}^{p,q}
  100. H i I ( H j I I ( C , ) ) H^{I}_{i}(H^{II}_{j}(C_{\bullet,\bullet}))
  101. H j I I ( H i I ( C , ) ) H^{II}_{j}(H^{I}_{i}(C_{\bullet,\bullet}))
  102. ( C i , j I ) p = { 0 if i < p C i , j if i p (C_{i,j}^{I})_{p}=\begin{cases}0&\,\text{if }i<p\\ C_{i,j}&\,\text{if }i\geq p\end{cases}
  103. ( C i , j I I ) p = { 0 if j < p C i , j if j p (C_{i,j}^{II})_{p}=\begin{cases}0&\,\text{if }j<p\\ C_{i,j}&\,\text{if }j\geq p\end{cases}
  104. i + j = n C i , j \bigoplus_{i+j=n}C_{i,j}
  105. T n ( C , ) p I = i + j = n i > p - 1 C i , j T_{n}(C_{\bullet,\bullet})^{I}_{p}=\bigoplus_{i+j=n\atop i>p-1}C_{i,j}
  106. T n ( C , ) p I I = i + j = n j > p - 1 C i , j T_{n}(C_{\bullet,\bullet})^{II}_{p}=\bigoplus_{i+j=n\atop j>p-1}C_{i,j}
  107. E p , q 0 I = T n ( C , ) p I / T n ( C , ) p + 1 I = i + j = n i > p - 1 C i , j / i + j = n i > p C i , j = C p , q , {}^{I}E^{0}_{p,q}=T_{n}(C_{\bullet,\bullet})^{I}_{p}/T_{n}(C_{\bullet,\bullet}% )^{I}_{p+1}=\bigoplus_{i+j=n\atop i>p-1}C_{i,j}\Big/\bigoplus_{i+j=n\atop i>p}% C_{i,j}=C_{p,q},
  108. d p , q I + d p , q I I : T n ( C , ) p I / T n ( C , ) p + 1 I = C p , q T n - 1 ( C , ) p I / T n - 1 ( C , ) p + 1 I = C p , q - 1 d^{I}_{p,q}+d^{II}_{p,q}:T_{n}(C_{\bullet,\bullet})^{I}_{p}/T_{n}(C_{\bullet,% \bullet})^{I}_{p+1}=C_{p,q}\rightarrow T_{n-1}(C_{\bullet,\bullet})^{I}_{p}/T_% {n-1}(C_{\bullet,\bullet})^{I}_{p+1}=C_{p,q-1}
  109. E p , q 1 I = H q I I ( C p , ) . {}^{I}E^{1}_{p,q}=H^{II}_{q}(C_{p,\bullet}).
  110. d p , q I + d p , q I I : H q I I ( C p , ) H q I I ( C p + 1 , ) d^{I}_{p,q}+d^{II}_{p,q}:H^{II}_{q}(C_{p,\bullet})\rightarrow H^{II}_{q}(C_{p+% 1,\bullet})
  111. E p , q 2 I = H p I ( H q I I ( C , ) ) . {}^{I}E^{2}_{p,q}=H^{I}_{p}(H^{II}_{q}(C_{\bullet,\bullet})).
  112. E p , q 2 I I = H q I I ( H p I ( C , ) ) . {}^{II}E^{2}_{p,q}=H^{II}_{q}(H^{I}_{p}(C_{\bullet,\bullet})).
  113. E r p , q E_{r}^{p,q}
  114. E p , q E_{\infty}^{p,q}
  115. d r p - r , q + r - 1 d_{r}^{p-r,q+r-1}
  116. d r p , q d_{r}^{p,q}
  117. E r p , q E_{r}^{p,q}
  118. E p , q E_{\infty}^{p,q}
  119. E r p , q p E p , q E_{r}^{p,q}\Rightarrow_{p}E_{\infty}^{p,q}
  120. E 2 p , q E_{2}^{p,q}
  121. E E_{\infty}
  122. E n E_{\infty}^{n}
  123. F E n F^{\bullet}E_{\infty}^{n}
  124. E p , q = gr E p + q p = F p E p + q / F p + 1 E p + q E_{\infty}^{p,q}=\mbox{gr}~{}_{p}E_{\infty}^{p+q}=F^{p}E_{\infty}^{p+q}/F^{p+1% }E_{\infty}^{p+q}
  125. E r p , q p E n E_{r}^{p,q}\Rightarrow_{p}E_{\infty}^{n}
  126. E r p , q E_{r}^{p,q}
  127. E p , q E_{\infty}^{p,q}
  128. E r p , q E_{r}^{p,q}
  129. E r p , q E_{r}^{p,q}
  130. E r p , q E_{r}^{p,q}
  131. Z 0 p , q \supe Z 1 p , q \supe Z 2 p , q \supe \supe B 2 p , q \supe B 1 p , q \supe B 0 p , q Z_{0}^{p,q}\supe Z_{1}^{p,q}\supe Z_{2}^{p,q}\supe\cdots\supe B_{2}^{p,q}\supe B% _{1}^{p,q}\supe B_{0}^{p,q}
  132. Z p , q = r = 0 Z r p , q , Z_{\infty}^{p,q}=\bigcap_{r=0}^{\infty}Z_{r}^{p,q},
  133. B p , q = r = 0 B r p , q , B_{\infty}^{p,q}=\bigcup_{r=0}^{\infty}B_{r}^{p,q},
  134. E p , q = Z p , q B p , q + Z p + 1 , q - 1 . E_{\infty}^{p,q}=\frac{Z_{\infty}^{p,q}}{B_{\infty}^{p,q}+Z_{\infty}^{p+1,q-1}}.
  135. E p , q E_{\infty}^{p,q}
  136. Z p , q = r = 0 Z r p , q = r = 0 ker ( F p C p + q C p + q + 1 / F p + r C p + q + 1 ) Z_{\infty}^{p,q}=\bigcap_{r=0}^{\infty}Z_{r}^{p,q}=\bigcap_{r=0}^{\infty}\ker(% F^{p}C^{p+q}\rightarrow C^{p+q+1}/F^{p+r}C^{p+q+1})
  137. B p , q = r = 0 B r p , q = r = 0 ( im d p , q - r : F p - r C p + q - 1 C p + q ) F p C p + q B_{\infty}^{p,q}=\bigcup_{r=0}^{\infty}B_{r}^{p,q}=\bigcup_{r=0}^{\infty}(% \mbox{im }~{}d^{p,q-r}:F^{p-r}C^{p+q-1}\rightarrow C^{p+q})\cap F^{p}C^{p+q}
  138. Z p , q Z_{\infty}^{p,q}
  139. Z p , q = ker ( F p C p + q C p + q + 1 ) Z_{\infty}^{p,q}=\ker(F^{p}C^{p+q}\rightarrow C^{p+q+1})
  140. B p , q B_{\infty}^{p,q}
  141. B p , q = im ( C p + q - 1 C p + q ) F p C p + q B_{\infty}^{p,q}=\,\text{im }(C^{p+q-1}\rightarrow C^{p+q})\cap F^{p}C^{p+q}
  142. E p , q = gr H p + q p ( C \bull ) E_{\infty}^{p,q}=\mbox{gr}~{}_{p}H^{p+q}(C^{\bull})
  143. E r p , q p H p + q ( C \bull ) E_{r}^{p,q}\Rightarrow_{p}H^{p+q}(C^{\bull})
  144. F 0 B n = B n F^{0}B^{n}=B^{n}
  145. F 1 B n = A n F^{1}B^{n}=A^{n}
  146. F 2 B n = 0 F^{2}B^{n}=0
  147. E 0 p , q = F p B p + q F p + 1 B p + q = { 0 if p < 0 or p > 1 C q if p = 0 A q + 1 if p = 1 E^{p,q}_{0}=\frac{F^{p}B^{p+q}}{F^{p+1}B^{p+q}}=\begin{cases}0&\,\text{if }p<0% \,\text{ or }p>1\\ C^{q}&\,\text{if }p=0\\ A^{q+1}&\,\text{if }p=1\end{cases}
  148. E 1 p , q = { 0 if p < 0 or p > 1 H q ( C \bull ) if p = 0 H q + 1 ( A \bull ) if p = 1 E^{p,q}_{1}=\begin{cases}0&\,\text{if }p<0\,\text{ or }p>1\\ H^{q}(C^{\bull})&\,\text{if }p=0\\ H^{q+1}(A^{\bull})&\,\text{if }p=1\end{cases}
  149. H q ( B \bull ) H q ( C \bull ) H q + 1 ( A \bull ) H q + 1 ( B \bull ) \cdots\rightarrow H^{q}(B^{\bull})\rightarrow H^{q}(C^{\bull})\rightarrow H^{q% +1}(A^{\bull})\rightarrow H^{q+1}(B^{\bull})\rightarrow\cdots
  150. E 2 p , q gr p H p + q ( B \bull ) = { 0 if p < 0 or p > 1 H q ( B \bull ) / H q ( A \bull ) if p = 0 im H q + 1 f \bull : H q + 1 ( A \bull ) H q + 1 ( B \bull ) if p = 1 E^{p,q}_{2}\cong\,\text{gr}_{p}H^{p+q}(B^{\bull})=\begin{cases}0&\,\text{if }p% <0\,\text{ or }p>1\\ H^{q}(B^{\bull})/H^{q}(A^{\bull})&\,\text{if }p=0\\ \,\text{im }H^{q+1}f^{\bull}:H^{q+1}(A^{\bull})\rightarrow H^{q+1}(B^{\bull})&% \,\text{if }p=1\end{cases}
  151. H q ( B \bull ) / H q ( A \bull ) ker d 0 , q 1 : H q ( C \bull ) H q + 1 ( A \bull ) H^{q}(B^{\bull})/H^{q}(A^{\bull})\cong\ker d^{1}_{0,q}:H^{q}(C^{\bull})% \rightarrow H^{q+1}(A^{\bull})
  152. im H q + 1 f \bull : H q + 1 ( A \bull ) H q + 1 ( B \bull ) H q + 1 ( A \bull ) / ( im d 0 , q 1 : H q ( C \bull ) H q + 1 ( A \bull ) ) \,\text{im }H^{q+1}f^{\bull}:H^{q+1}(A^{\bull})\rightarrow H^{q+1}(B^{\bull})% \cong H^{q+1}(A^{\bull})/(\mbox{im }~{}d^{1}_{0,q}:H^{q}(C^{\bull})\rightarrow H% ^{q+1}(A^{\bull}))
  153. H p I ( H q I I ( C \bull , \bull ) ) p H p + q ( T ( C \bull , \bull ) ) H^{I}_{p}(H^{II}_{q}(C_{\bull,\bull}))\Rightarrow_{p}H^{p+q}(T(C_{\bull,\bull}))
  154. H q I I ( H p I ( C \bull , \bull ) ) q H p + q ( T ( C \bull , \bull ) ) H^{II}_{q}(H^{I}_{p}(C_{\bull,\bull}))\Rightarrow_{q}H^{p+q}(T(C_{\bull,\bull}))
  155. H p I ( H q I I ( P \bull Q \bull ) ) = H p I ( P \bull H q I I ( Q \bull ) ) H^{I}_{p}(H^{II}_{q}(P_{\bull}\otimes Q_{\bull}))=H^{I}_{p}(P_{\bull}\otimes H% ^{II}_{q}(Q_{\bull}))
  156. H q I I ( H p I ( P \bull Q \bull ) ) = H q I I ( H p I ( P \bull ) Q \bull ) H^{II}_{q}(H^{I}_{p}(P_{\bull}\otimes Q_{\bull}))=H^{II}_{q}(H^{I}_{p}(P_{% \bull})\otimes Q_{\bull})
  157. H p I ( P \bull N ) = Tor ( M , N ) p H^{I}_{p}(P_{\bull}\otimes N)=\mbox{Tor}~{}_{p}(M,N)
  158. H q I I ( M Q \bull ) = Tor ( N , M ) q H^{II}_{q}(M\otimes Q_{\bull})=\mbox{Tor}~{}_{q}(N,M)
  159. E p , q 2 E^{2}_{p,q}
  160. Tor ( M , N ) p E p = H p ( T ( C \bull , \bull ) ) \mbox{Tor}~{}_{p}(M,N)\cong E^{\infty}_{p}=H_{p}(T(C_{\bull,\bull}))
  161. Tor ( N , M ) q E q = H q ( T ( C \bull , \bull ) ) \mbox{Tor}~{}_{q}(N,M)\cong E^{\infty}_{q}=H_{q}(T(C_{\bull,\bull}))

Spectrum_(topology).html

  1. X n X_{n}
  2. S 1 X n X n + 1 S^{1}\wedge X_{n}\to X_{n+1}
  3. E := { E n } n E:=\{E_{n}\}_{n\in\mathbb{N}}
  4. Σ E n E n + 1 \Sigma E_{n}\to E_{n+1}
  5. Σ E n \Sigma E_{n}
  6. E n + 1 E_{n+1}
  7. H n ( X ; A ) H^{n}(X;A)
  8. H n ( X ; A ) H^{n}(X;A)
  9. K 0 ( X ) K^{0}(X)
  10. K 1 ( X ) K^{1}(X)
  11. × B U \mathbb{Z}\times BU
  12. U U
  13. U U
  14. B U BU
  15. K 2 n ( X ) K 0 ( X ) K^{2n}(X)\cong K^{0}(X)
  16. K 2 n + 1 ( X ) K 1 ( X ) K^{2n+1}(X)\cong K^{1}(X)
  17. × B U \mathbb{Z}\times BU
  18. U U
  19. X n = S n X X_{n}=S^{n}\wedge X
  20. 𝕊 \mathbb{S}
  21. X n Ω X n + 1 X_{n}\to\Omega X_{n+1}
  22. S 0 X S^{0}\to X
  23. X n X_{n}
  24. π k ( X ) = colim n π n + k ( X n ) \pi_{k}(X)=\operatorname{colim}_{n}\pi_{n+k}(X_{n})
  25. π k ( 𝕊 ) \pi_{k}(\mathbb{S})
  26. 𝕊 \mathbb{S}
  27. π k \pi_{k}
  28. E n E_{n}
  29. F n F_{n}
  30. E j E_{j}
  31. E j + 1 E_{j+1}
  32. f : E F f:E\to F
  33. G G
  34. E E
  35. F F
  36. Y Y
  37. Σ n Y \Sigma^{n}Y
  38. E E
  39. X X
  40. ( E X ) n = E n X (E\wedge X)_{n}=E_{n}\wedge X
  41. ( E I + ) F (E\wedge I^{+})\to F
  42. I + I^{+}
  43. [ 0 , 1 ] { * } [0,1]\sqcup\{*\}
  44. * *
  45. ( Σ E ) n = E n + 1 (\Sigma E)_{n}=E_{n+1}
  46. ( Σ - 1 E ) n = E n - 1 (\Sigma^{-1}E)_{n}=E_{n-1}
  47. X Y Y C X ( Y C X ) C Y Σ X X\rightarrow Y\rightarrow Y\cup CX\rightarrow(Y\cup CX)\cup CY\cong\Sigma X
  48. π n E = [ Σ n S , E ] \displaystyle\pi_{n}E=[\Sigma^{n}S,E]
  49. S S
  50. [ X , Y ] [X,Y]
  51. X X
  52. Y Y
  53. E n X = π n ( E X ) = [ Σ n S , E X ] E_{n}X=\pi_{n}(E\wedge X)=[\Sigma^{n}S,E\wedge X]
  54. E n X = [ Σ - n X , E ] . \displaystyle E^{n}X=[\Sigma^{-n}X,E].
  55. X X

Spectrum_of_a_C*-algebra.html

  1. X ¯ = { ρ Prim ( A ) : ρ π X π } . \overline{X}=\left\{\rho\in\operatorname{Prim}(A):\rho\supseteq\bigcap_{\pi\in X% }\pi\right\}.
  2. X ¯ ¯ = X ¯ , \overline{\overline{X}}=\overline{X},
  3. k : A ^ Prim ( A ) . \operatorname{k}:\hat{A}\to\operatorname{Prim}(A).
  4. I : X Prim ( C ( X ) ) . \operatorname{I}:X\cong\operatorname{Prim}(\operatorname{C}(X)).
  5. I ( x ) = { f C ( X ) : f ( x ) = 0 } . \operatorname{I}(x)=\{f\in\operatorname{C}(X):f(x)=0\}.
  6. A ^ Prim ( A ) . \hat{A}\cong\operatorname{Prim}(A).
  7. A e min ( A ) A e , A\cong\bigoplus_{e\in\operatorname{min}(A)}Ae,
  8. A ^ Prim ( A ) . \hat{A}\cong\operatorname{Prim}(A).
  9. f ξ ( x ) = ξ π ( x ) ξ f_{\xi}(x)=\langle\xi\mid\pi(x)\xi\rangle
  10. κ : PureState ( A ) A ^ \kappa:\operatorname{PureState}(A)\to\hat{A}
  11. π i ( x ) ξ η π ( x ) ξ η ξ , η H n x A . \langle\pi_{i}(x)\xi\mid\eta\rangle\to\langle\pi(x)\xi\mid\eta\rangle\quad% \forall\xi,\eta\in H_{n}\ x\in A.
  12. π i ( x ) ξ π ( x ) ξ normwise ξ H n x A . \pi_{i}(x)\xi\to\pi(x)\xi\quad\mbox{ normwise }~{}\forall\xi\in H_{n}\ x\in A.

Specular_reflection.html

  1. θ i = θ r \theta_{i}=\theta_{r}
  2. θ i \theta_{i}
  3. 𝐝 ^ i \mathbf{\hat{d}}_{\mathrm{i}}
  4. 𝐝 ^ n , \mathbf{\hat{d}}_{\mathrm{n}},
  5. 𝐝 ^ s \mathbf{\hat{d}}_{\mathrm{s}}
  6. 𝐝 ^ s = 2 ( 𝐝 ^ n 𝐝 ^ i ) 𝐝 ^ n - 𝐝 ^ i , \mathbf{\hat{d}}_{\mathrm{s}}=2\left(\mathbf{\hat{d}}_{\mathrm{n}}\cdot\mathbf% {\hat{d}}_{\mathrm{i}}\right)\mathbf{\hat{d}}_{\mathrm{n}}-\mathbf{\hat{d}}_{% \mathrm{i}},
  7. 𝐝 ^ n 𝐝 ^ i \mathbf{\hat{d}}_{\mathrm{n}}\cdot\mathbf{\hat{d}}_{\mathrm{i}}
  8. 𝐝 ^ s = 𝐑 𝐝 ^ i , \mathbf{\hat{d}}_{\mathrm{s}}=\mathbf{R}\;\mathbf{\hat{d}}_{\mathrm{i}},
  9. 𝐑 \mathbf{R}
  10. 𝐑 = 2 𝐝 ^ n 𝐝 ^ n T - 𝐈 ; \mathbf{R}=2\mathbf{\hat{d}}_{\mathrm{n}}\mathbf{\hat{d}}_{\mathrm{n}}^{% \mathrm{T}}-\mathbf{I};
  11. T \mathrm{T}
  12. 𝐈 \mathbf{I}

Speedcubing.html

  1. R - 1 R^{-1}

Sperner's_theorem.html

  1. ( n n / 2 ) {\textstyle\left({{n}\atop{\lfloor n/2\rfloor}}\right)}
  2. | S | ( n n / 2 ) . |S|\leq{\left({{n}\atop{\lfloor n/2\rfloor}}\right)}.
  3. ( n n / 2 ) {\left({{n}\atop{\lfloor n/2\rfloor}}\right)}
  4. ( n n / 2 ) ( n k ) {n\choose\lfloor{n/2}\rfloor}\geq{n\choose k}
  5. s k ( n n / 2 ) s k ( n k ) . {s_{k}\over{n\choose\lfloor{n/2}\rfloor}}\leq{s_{k}\over{n\choose k}}.
  6. k = 0 n s k ( n n / 2 ) k = 0 n s k ( n k ) 1 , \sum_{k=0}^{n}{s_{k}\over{n\choose\lfloor{n/2}\rfloor}}\leq\sum_{k=0}^{n}{s_{k% }\over{n\choose k}}\leq 1,
  7. | S | = k = 0 n s k ( n n / 2 ) . |S|=\sum_{k=0}^{n}s_{k}\leq{n\choose\lfloor{n/2}\rfloor}.
  8. 𝒫 ( E ) , \mathcal{P}(E),
  9. { S 0 , S 1 , , S r } 𝒫 ( E ) \{S_{0},S_{1},\dots,S_{r}\}\subseteq\mathcal{P}(E)
  10. S 0 S 1 S r S_{0}\subset S_{1}\subset\dots\subset S_{r}
  11. ( n i ) {\left({{n}\atop{i}}\right)}
  12. 𝒫 ( E ) p \mathcal{P}(E)^{p}
  13. ( S 1 , , S p ) (S_{1},\dots,S_{p})
  14. ( T 1 , , T p ) , (T_{1},\dots,T_{p}),
  15. S i T i S_{i}\subseteq T_{i}
  16. ( S 1 , , S p ) (S_{1},\dots,S_{p})
  17. S 1 , , S p S_{1},\dots,S_{p}
  18. ( n n 1 n 2 n p ) , {\left({{n}\atop{n_{1}\ n_{2}\ \dots\ n_{p}}}\right)},
  19. S 2 = E S 1 S_{2}=E\setminus S_{1}
  20. S 1 S_{1}
  21. i = 1 , 2 , , p - 1 i=1,2,\dots,p-1
  22. r p - 1 r^{p-1}
  23. ( p , F q ) \mathcal{L}(p,F_{q})
  24. ( p , F q ) \mathcal{L}(p,F_{q})
  25. [ d + 1 k ] ; \begin{bmatrix}d+1\\ k\end{bmatrix};
  26. ( p , F q ) \mathcal{L}(p,F_{q})
  27. ( A 1 , , A p ) (A_{1},\ldots,A_{p})
  28. d - p + 1 d-p+1
  29. i = 1 , 2 , , p - 1 , i=1,2,\dots,p-1,
  30. r p - 1 r^{p-1}
  31. [ d + 1 n 1 n 2 n p ] q s 2 ( n 1 , , n p ) , \begin{bmatrix}d+1\\ n_{1}\ n_{2}\ \dots\ n_{p}\end{bmatrix}q^{s_{2}(n_{1},\ldots,n_{p})},
  32. [ d + 1 n 1 n 2 n p ] \begin{bmatrix}d+1\\ n_{1}\ n_{2}\ \dots\ n_{p}\end{bmatrix}
  33. d + 1 = n 1 + + n p d+1=n_{1}+\cdots+n_{p}
  34. [ d + 1 n 1 ] [ d + 1 - n 1 n 2 ] [ d + 1 - ( n 1 + + n p - 1 ) n p ] \begin{bmatrix}d+1\\ n_{1}\end{bmatrix}\begin{bmatrix}d+1-n_{1}\\ n_{2}\end{bmatrix}\cdots\begin{bmatrix}d+1-(n_{1}+\cdots+n_{p-1})\\ n_{p}\end{bmatrix}
  35. s 2 ( n 1 , , n p ) := n 1 n 2 + n 1 n 3 + n 2 n 3 + n 1 n 4 + + n p - 1 n p , s_{2}(n_{1},\ldots,n_{p}):=n_{1}n_{2}+n_{1}n_{3}+n_{2}n_{3}+n_{1}n_{4}+\cdots+% n_{p-1}n_{p},
  36. n 1 , n 2 , , n p . n_{1},n_{2},\dots,n_{p}.

Sperner_family.html

  1. | S | ( n n / 2 ) . |S|\leq{\left({{n}\atop{\lfloor n/2\rfloor}}\right)}.
  2. k = 0 n a k < m t p l > ( n k ) 1. \sum_{k=0}^{n}\frac{a_{k}}{<}mtpl>{{n\choose k}}\leq 1.
  3. ( V , E ) (V,E)
  4. A B A\not\subseteq B
  5. A , B E A,B\in E
  6. A B A\neq B
  7. H = ( V , E ) H=(V,E)
  8. b ( H ) b(H)
  9. B V B\subseteq V
  10. B A B\cap A\neq\varnothing
  11. A E A\in E
  12. b ( b ( H ) ) = H b(b(H))=H
  13. ν ( H ) \nu(H)
  14. τ ( H ) \tau(H)
  15. b ( H ) b(H)
  16. ν ( H ) τ ( H ) \nu(H)\leq\tau(H)
  17. H = ( V ( G ) , E ( G ) ) H=(V(G),E(G))
  18. b ( H ) b(H)
  19. ν ( H ) \nu(H)
  20. τ ( H ) \tau(H)
  21. ν ( H ) = τ ( H ) \nu(H)=\tau(H)
  22. s , t V ( G ) s,t\in V(G)
  23. H = ( V , E ) H=(V,E)
  24. V = E ( G ) V=E(G)
  25. b ( H ) b(H)
  26. ν ( H ) \nu(H)
  27. τ ( H ) \tau(H)
  28. ν ( H ) = τ ( H ) \nu(H)=\tau(H)
  29. E ( G ) E(G)
  30. b ( H ) b(H)
  31. H = ( V , E ) H=(V,E)
  32. v V v\in V
  33. H v H\setminus v
  34. V { v } V\setminus\{v\}
  35. A E A\in E
  36. H / v = b ( b ( H ) v ) H/v=b(b(H)\setminus v)