wpmath0000004_12

Pythagorean_trigonometric_identity.html

  1. sin 2 θ + cos 2 θ = 1. \sin^{2}\theta+\cos^{2}\theta=1.\!
  2. sin θ = opposite hypotenuse = b c \sin\theta=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{b}{c}
  3. cos θ = adjacent hypotenuse = a c . \cos\theta=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}=\frac{a}{c}\ .
  4. opposite 2 + adjacent 2 hypotenuse 2 \frac{\mathrm{opposite}^{2}+\mathrm{adjacent}^{2}}{\mathrm{hypotenuse}^{2}}
  5. sin 2 θ + cos 2 θ = sin 2 ( t + 1 2 π ) + cos 2 ( t + 1 2 π ) = cos 2 t + sin 2 t = 1. \sin^{2}\theta+\cos^{2}\theta=\sin^{2}\left(t+\frac{1}{2}\pi\right)+\cos^{2}% \left(t+\frac{1}{2}\pi\right)=\cos^{2}t+\sin^{2}t=1.
  6. i = 0 n ( 2 n 2 i ) - i = 0 n - 1 ( 2 n 2 i + 1 ) = j = 0 2 n ( - 1 ) j ( 2 n j ) = ( 1 - 1 ) 2 n = 0 \sum_{i=0}^{n}{2n\choose 2i}-\sum_{i=0}^{n-1}{2n\choose 2i+1}=\sum_{j=0}^{2n}(% -1)^{j}{2n\choose j}=(1-1)^{2n}=0
  7. sin 2 x + cos 2 x = 1 , \sin^{2}x+\cos^{2}x=1\ ,
  8. y ′′ + y = 0 y^{\prime\prime}+y=0\,
  9. z = sin 2 x + cos 2 x z=\sin^{2}x+\cos^{2}x\,
  10. d d x z = 2 sin x cos x + 2 cos x ( - sin x ) = 0 , \frac{d}{dx}z=2\sin x\ \cos x+2\cos x\ (-\sin x)=0\ ,

QT_interval.html

  1. Q T c B = Q T R R QTcB={QT\over\sqrt{RR}}
  2. Q T c F = Q T R R 3 QTcF={QT\over\sqrt[3]{RR}}
  3. Q T c L = Q T + 0.154 ( 1 - R R ) QTcL={QT+0.154(1-RR)}

Quadratic_residuosity_problem.html

  1. a a
  2. N N
  3. a a
  4. N N
  5. N = p 1 p 2 N=p_{1}p_{2}
  6. p 1 p_{1}
  7. p 2 p_{2}
  8. a a
  9. N N
  10. a a
  11. T T
  12. a a
  13. T T
  14. b b
  15. a b 2 ( mod T ) a\equiv b^{2}\;\;(\mathop{{\rm mod}}T)
  16. T = p T=p
  17. ( a p ) = { 1 if a is a quadratic residue modulo p and a 0 ( mod p ) , - 1 if a is a quadratic non-residue modulo p , 0 if a 0 ( mod p ) . \left(\frac{a}{p}\right)=\begin{cases}1&\,\text{ if }a\,\text{ is a quadratic % residue modulo }p\,\text{ and }a\not\equiv 0\;\;(\mathop{{\rm mod}}p),\\ -1&\,\text{ if }a\,\text{ is a quadratic non-residue modulo }p,\\ 0&\,\text{ if }a\equiv 0\;\;(\mathop{{\rm mod}}p).\end{cases}
  18. ( a p ) = 1 \big(\tfrac{a}{p}\big)=1
  19. ( p - 1 ) / 2 (p-1)/2
  20. 1 , , p - 1 1,\ldots,p-1
  21. - 1 -1
  22. N = p 1 p 2 N=p_{1}p_{2}
  23. p 1 p_{1}
  24. p 2 p_{2}
  25. a a
  26. N N
  27. a a
  28. p 1 p_{1}
  29. p 2 p_{2}
  30. p 1 p_{1}
  31. p 2 p_{2}
  32. ( a p 1 ) \big(\tfrac{a}{p_{1}}\big)
  33. ( a p 2 ) \big(\tfrac{a}{p_{2}}\big)
  34. ( a N ) = ( a p 1 ) ( a p 2 ) \left(\frac{a}{N}\right)=\left(\frac{a}{p_{1}}\right)\left(\frac{a}{p_{2}}\right)
  35. ( a N ) \big(\tfrac{a}{N}\big)
  36. a a
  37. N N
  38. ( a N ) = - 1 \big(\tfrac{a}{N}\big)=-1
  39. a a
  40. p 1 p_{1}
  41. p 2 p_{2}
  42. ( a N ) = 1 \big(\tfrac{a}{N}\big)=1
  43. a a
  44. p 1 p_{1}
  45. p 2 p_{2}
  46. p 1 p_{1}
  47. p 2 p_{2}
  48. ( a N ) = 1 \big(\tfrac{a}{N}\big)=1
  49. a a
  50. N = p 1 p 2 N=p_{1}p_{2}
  51. p 1 p_{1}
  52. p 2 p_{2}
  53. ( a N ) = 1 \big(\tfrac{a}{N}\big)=1
  54. a a
  55. N N
  56. a a
  57. 0 , , N - 1 0,\ldots,N-1
  58. ( a N ) = 1 \big(\tfrac{a}{N}\big)=1
  59. a a
  60. N N
  61. a { 1 , , p 1 - 1 } a\in\{1,\ldots,p_{1}-1\}
  62. ( a p 1 ) = 1 \big(\tfrac{a}{p_{1}}\big)=1
  63. ( a p 1 ) = - 1 \big(\tfrac{a}{p_{1}}\big)=-1
  64. a { 1 , , N - 1 } p 1 a\in\{1,\ldots,N-1\}\setminus p_{1}\mathbb{Z}
  65. p 2 p_{2}
  66. ( / N ) × (\mathbb{Z}/N\mathbb{Z})^{\times}
  67. ( a p 1 ) \big(\tfrac{a}{p_{1}}\big)
  68. ( a p 2 ) \big(\tfrac{a}{p_{2}}\big)
  69. a a
  70. ( a p 1 ) = ( a p 1 ) = 1 \big(\tfrac{a}{p_{1}}\big)=\big(\tfrac{a}{p_{1}}\big)=1
  71. ( a p 1 ) = ( a p 1 ) = - 1 \big(\tfrac{a}{p_{1}}\big)=\big(\tfrac{a}{p_{1}}\big)=-1
  72. a a

Quantale.html

  1. x * ( i I y i ) = i I ( x * y i ) x*(\bigvee_{i\in I}{y_{i}})=\bigvee_{i\in I}(x*y_{i})
  2. ( i I y i ) * x = i I ( y i * x ) (\bigvee_{i\in I}{y_{i}})*{x}=\bigvee_{i\in I}(y_{i}*x)
  3. ( x y ) = y x (xy)^{\circ}=y^{\circ}x^{\circ}
  4. ( i I x i ) = i I ( x i ) . \biggl(\bigvee_{i\in I}{x_{i}}\biggr)^{\circ}=\bigvee_{i\in I}(x_{i}^{\circ}).
  5. f ( x y ) = f ( x ) f ( y ) f(xy)=f(x)f(y)
  6. f ( i I x i ) = i I f ( x i ) f\biggl(\bigvee_{i\in I}{x_{i}}\biggl)=\bigvee_{i\in I}f(x_{i})

Quantity_adjustment.html

  1. d P d t = k ( Q D - Q S ) , \frac{dP}{dt}=k(QD-QS),
  2. d Q S d t = k ( D P - S P ) , \frac{dQS}{dt}=k(DP-SP),

Quantity_theory_of_money.html

  1. M V T = i ( p i q i ) = 𝐩 T 𝐪 M\cdot V_{T}=\sum_{i}(p_{i}\cdot q_{i})=\mathbf{p}^{\mathrm{T}}\mathbf{q}
  2. M M\,
  3. V T V_{T}\,
  4. p i p_{i}\,
  5. q i q_{i}\,
  6. 𝐩 \mathbf{p}
  7. p i p_{i}\,
  8. 𝐪 \mathbf{q}
  9. q i q_{i}\,
  10. M V T = P T T M\cdot V_{T}=P_{T}\cdot T
  11. P T P_{T}
  12. T T
  13. M V = P Q M\cdot V=P\cdot Q
  14. V V
  15. Q Q
  16. M M
  17. Q Q
  18. P P
  19. P Q P\cdot Q
  20. P P
  21. Q Q
  22. P Q P\cdot Q
  23. M M
  24. V V
  25. P Q P\cdot Q
  26. P Y P\cdot Y
  27. M 𝑑 = 𝑘 P Y M^{\,\textit{d}}=\,\textit{k}\cdot P\cdot Y
  28. M 𝑑 = M M^{\,\textit{d}}=M
  29. Y Y
  30. M 1 k = P Y M\cdot\frac{1}{k}=P\cdot Y
  31. P Q PQ
  32. P P
  33. M M
  34. ( 1 ) P Q = f ( M + ) (1)PQ={f}(\overset{+}{M})
  35. ( 2 ) P = g ( M + ) (2)P={g}(\overset{+}{M})
  36. Q Q
  37. P P
  38. Q Q
  39. M M

Quantization_(image_processing).html

  1. [ - 415 - 33 - 58 35 58 - 51 - 15 - 12 5 - 34 49 18 27 1 - 5 3 - 46 14 80 - 35 - 50 19 7 - 18 - 53 21 34 - 20 2 34 36 12 9 - 2 9 - 5 - 32 - 15 45 37 - 8 15 - 16 7 - 8 11 4 7 19 - 28 - 2 - 26 - 2 7 - 44 - 21 18 25 - 12 - 44 35 48 - 37 - 3 ] \begin{bmatrix}-415&-33&-58&35&58&-51&-15&-12\\ 5&-34&49&18&27&1&-5&3\\ -46&14&80&-35&-50&19&7&-18\\ -53&21&34&-20&2&34&36&12\\ 9&-2&9&-5&-32&-15&45&37\\ -8&15&-16&7&-8&11&4&7\\ 19&-28&-2&-26&-2&7&-44&-21\\ 18&25&-12&-44&35&48&-37&-3\end{bmatrix}
  2. [ 16 11 10 16 24 40 51 61 12 12 14 19 26 58 60 55 14 13 16 24 40 57 69 56 14 17 22 29 51 87 80 62 18 22 37 56 68 109 103 77 24 35 55 64 81 104 113 92 49 64 78 87 103 121 120 101 72 92 95 98 112 100 103 99 ] \begin{bmatrix}16&11&10&16&24&40&51&61\\ 12&12&14&19&26&58&60&55\\ 14&13&16&24&40&57&69&56\\ 14&17&22&29&51&87&80&62\\ 18&22&37&56&68&109&103&77\\ 24&35&55&64&81&104&113&92\\ 49&64&78&87&103&121&120&101\\ 72&92&95&98&112&100&103&99\end{bmatrix}
  3. [ - 26 - 3 - 6 2 2 - 1 0 0 0 - 3 4 1 1 0 0 0 - 3 1 5 - 1 - 1 0 0 0 - 4 1 2 - 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] \begin{bmatrix}-26&-3&-6&2&2&-1&0&0\\ 0&-3&4&1&1&0&0&0\\ -3&1&5&-1&-1&0&0&0\\ -4&1&2&-1&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{bmatrix}
  4. round ( - 415 16 ) = round ( - 25.9375 ) = - 26 \mathrm{round}\left(\frac{-415}{16}\right)=\mathrm{round}\left(-25.9375\right)% =-26

Quantum_channel.html

  1. H A H_{A}
  2. H B H_{B}
  3. L ( H A ) L(H_{A})
  4. H A H_{A}
  5. H A H_{A}
  6. H B H_{B}
  7. I n Φ I_{n}\otimes\Phi
  8. I n Φ I_{n}\otimes\Phi
  9. Φ : L ( H A ) L ( H B ) \Phi:L(H_{A})\rightarrow L(H_{B})
  10. A , Φ ( ρ ) = Φ * ( A ) , ρ . \langle A,\Phi(\rho)\rangle=\langle\Phi^{*}(A),\rho\rangle.
  11. Ψ : L ( H B ) L ( H A ) \Psi:L(H_{B})\rightarrow L(H_{A})
  12. Ψ : 𝒜 . \Psi:\mathcal{B}\rightarrow\mathcal{A}.
  13. n \mathbb{R}^{n}
  14. \mathcal{B}
  15. Ψ : L ( H B ) C ( X ) L ( H A ) . \Psi:L(H_{B})\otimes C(X)\rightarrow L(H_{A}).
  16. L ( H B ) C ( X ) L(H_{B})\otimes C(X)
  17. 𝒜 \mathcal{A}
  18. ρ U ρ U * , \rho\rightarrow U\rho\;U^{*},
  19. U = e - i H t / U=e^{-iHt/\hbar}
  20. A U * A U . A\rightarrow U^{*}AU.
  21. H A H B . H_{A}\otimes H_{B}.
  22. ρ H A H B , \rho\in H_{A}\otimes H_{B},
  23. ρ A = Tr B ρ . \rho^{A}=\operatorname{Tr}_{B}\;\rho.
  24. A A I B , A\rightarrow A\otimes I_{B},
  25. f i f_{i}\in\mathbb{C}
  26. F i F_{i}
  27. F i F_{i}
  28. F i = I \sum F_{i}=I
  29. f = [ f 1 f n ] C ( X ) f=\begin{bmatrix}f_{1}\\ \vdots\\ f_{n}\end{bmatrix}\in C(X)
  30. Ψ ( f ) = i f i F i . \;\Psi(f)=\sum_{i}f_{i}F_{i}.
  31. Ψ ( ρ ) = [ F 1 , ρ F n , ρ ] \Psi(\rho)=\begin{bmatrix}\langle F_{1},\rho\rangle\\ \vdots\\ \langle F_{n},\rho\rangle\end{bmatrix}
  32. Ψ ( ρ ) = [ ρ ( F 1 ) ρ ( F n ) ] . \Psi(\rho)=\begin{bmatrix}\rho(F_{1})\\ \vdots\\ \rho(F_{n})\end{bmatrix}.
  33. { F 1 , , F n } \{F_{1},\cdots,F_{n}\}
  34. ρ L ( H ) \rho\in L(H)
  35. C ( X ) L ( H ) C(X)\otimes L(H)
  36. Φ ( ρ ) = [ ρ ( F 1 ) F 1 ρ ( F n ) F n ] . \Phi(\rho)=\begin{bmatrix}\rho(F_{1})\cdot F_{1}\\ \vdots\\ \rho(F_{n})\cdot F_{n}\end{bmatrix}.
  37. f = [ f 1 f n ] C ( X ) . f=\begin{bmatrix}f_{1}\\ \vdots\\ f_{n}\end{bmatrix}\in C(X).
  38. Ψ ( f A ) = [ f 1 Ψ 1 ( A ) f n Ψ n ( A ) ] \Psi(f\otimes A)=\begin{bmatrix}f_{1}\Psi_{1}(A)\\ \vdots\\ f_{n}\Psi_{n}(A)\end{bmatrix}
  39. Ψ i \Psi_{i}
  40. F i = M i 2 F_{i}=M_{i}^{2}
  41. Ψ i ( A ) = M i A M i \;\Psi_{i}(A)=M_{i}AM_{i}
  42. Ψ ( f I ) \Psi(f\otimes I)
  43. Ψ ~ ( A ) = i Ψ i ( A ) = i M i A M i {\tilde{\Psi}}(A)=\sum_{i}\Psi_{i}(A)=\sum_{i}M_{i}AM_{i}
  44. Φ 1 ( ρ ) = [ ρ ( F 1 ) ρ ( F n ) ] . \;\Phi_{1}(\rho)=\begin{bmatrix}\rho(F_{1})\\ \vdots\\ \rho(F_{n})\end{bmatrix}.
  45. Φ 2 ( [ ρ ( F 1 ) ρ ( F n ) ] ) = i ρ ( F i ) R i . \Phi_{2}(\begin{bmatrix}\rho(F_{1})\\ \vdots\\ \rho(F_{n})\end{bmatrix})=\sum_{i}\rho(F_{i})R_{i}.
  46. Φ ( ρ ) = Φ 2 Φ 1 ( ρ ) = i ρ ( F i ) R i . \Phi(\rho)=\Phi_{2}\circ\Phi_{1}(\rho)=\sum_{i}\rho(F_{i})R_{i}.
  47. Φ * = Φ 1 * Φ 2 * \Phi^{*}=\Phi_{1}^{*}\circ\Phi_{2}^{*}
  48. Φ * ( A ) = i R i ( A ) F i . \;\Phi^{*}(A)=\sum_{i}R_{i}(A)F_{i}.
  49. Ψ : n × n m × m . \Psi:\mathbb{C}^{n\times n}\rightarrow\mathbb{C}^{m\times m}.
  50. Ψ ( A ) = i = 1 N K i A K i * \Psi(A)=\sum_{i=1}^{N}K_{i}AK_{i}^{*}
  51. Φ - Λ = sup { ( Φ - Λ ) ( A ) | A 1 } . \|\Phi-\Lambda\|=\sup\{\|(\Phi-\Lambda)(A)\|\;|\;\|A\|\leq 1\}.
  52. Φ I n \|\Phi\otimes I_{n}\|
  53. n . n\rightarrow\infty.
  54. Φ c b = sup n Φ I n . \|\Phi\|_{cb}=\sup_{n}\|\Phi\otimes I_{n}\|.
  55. Ψ : 1 𝒜 1 \Psi:\mathcal{B}_{1}\rightarrow\mathcal{A}_{1}
  56. Ψ i d : 2 𝒜 2 \Psi_{id}:\mathcal{B}_{2}\rightarrow\mathcal{A}_{2}
  57. Ψ ^ = D Φ E : 2 𝒜 2 {\hat{\Psi}}=D\circ\Phi\circ E:\mathcal{B}_{2}\rightarrow\mathcal{A}_{2}
  58. Δ ( Ψ ^ , Ψ i d ) = inf E , D Ψ ^ - Ψ i d c b \Delta({\hat{\Psi}},\Psi_{id})=\inf_{E,D}\|{\hat{\Psi}}-\Psi_{id}\|_{cb}
  59. Ψ i d n = Ψ i d Ψ i d . \Psi_{id}^{\otimes n}=\Psi_{id}\otimes\cdots\otimes\Psi_{id}.
  60. \otimes
  61. Ψ i d \Psi_{id}
  62. Ψ ^ m {\hat{\Psi}}^{\otimes m}
  63. Δ ( Ψ ^ m , Ψ i d n ) \Delta({\hat{\Psi}}^{\otimes m},\Psi_{id}^{\otimes n})
  64. Ψ \Psi
  65. Ψ i d \Psi_{id}
  66. { n α } , { m α } \{n_{\alpha}\},\{m_{\alpha}\}\subset\mathbb{N}
  67. m α m_{\alpha}\rightarrow\infty
  68. lim sup α ( n α / m α ) < r \lim\sup_{\alpha}(n_{\alpha}/m_{\alpha})<r
  69. lim α Δ ( Ψ ^ m α , Ψ i d n α ) = 0. \lim_{\alpha}\Delta({\hat{\Psi}}^{\otimes m_{\alpha}},\Psi_{id}^{\otimes n_{% \alpha}})=0.
  70. { n α } \{n_{\alpha}\}
  71. Ψ \Psi
  72. Ψ i d \Psi_{id}
  73. C ( Ψ , Ψ i d ) \;C(\Psi,\Psi_{id})
  74. \mathcal{B}
  75. Ψ i d \Psi_{id}
  76. I I_{\mathcal{B}}
  77. n × n \mathbb{C}^{n\times n}
  78. n × n \mathbb{C}^{n\times n}
  79. m \mathbb{C}^{m}
  80. m \mathbb{C}^{m}
  81. n × n \mathbb{C}^{n\times n}
  82. C ( m , n × n ) = 0. C(\mathbb{C}^{m},\mathbb{C}^{n\times n})=0.
  83. C ( m , n ) = C ( m × m , n × n ) = = C ( m × m , n ) = = log n log m . C(\mathbb{C}^{m},\mathbb{C}^{n})=C(\mathbb{C}^{m\times m},\mathbb{C}^{n\times n% })==C(\mathbb{C}^{m\times m},\mathbb{C}^{n})==\frac{\log n}{\log m}.
  84. C ( Ψ , 2 ) , C(\Psi,\mathbb{C}^{2}),
  85. 2 \mathbb{C}^{2}
  86. C ( Ψ , 2 × 2 ) , C(\Psi,\mathbb{C}^{2\times 2}),
  87. 2 × 2 \mathbb{C}^{2\times 2}

Quantum_circuit.html

  1. H QB ( n ) = 2 ( { 0 , 1 } n ) . H_{\operatorname{QB}(n)}=\ell^{2}(\{0,1\}^{n}).
  2. | x 1 , x 2 , , x n |x_{1},x_{2},\cdots,x_{n}\rangle\quad
  3. W f ( | x 1 , x 2 , , x n ) = | f ( x 1 , x 2 , , x n ) . W_{f}(|x_{1},x_{2},\cdots,x_{n}\rangle)=|f(x_{1},x_{2},\cdots,x_{n})\rangle.
  4. U θ = [ e i θ 0 0 1 ] , U_{\theta}=\begin{bmatrix}e^{i\theta}&0\\ 0&1\end{bmatrix},
  5. U θ | 0 = e i θ | 0 U θ | 1 = | 1 . U_{\theta}|0\rangle=e^{i\theta}|0\rangle\quad U_{\theta}|1\rangle=|1\rangle.
  6. f ( x 1 , , x n , 0 , , 0 ) = ( y 1 , , y n , 0 , , 0 ) f(x_{1},\ldots,x_{n},\underbrace{0,\dots,0})=(y_{1},\ldots,y_{n},\underbrace{0% ,\ldots,0})
  7. ( y 1 , , y n ) = h ( x 1 , , x n ) (y_{1},\ldots,y_{n})=h(x_{1},\ldots,x_{n})
  8. H QB ( r ) H QB ( r ) . H_{\operatorname{QB}(r)}\rightarrow H_{\operatorname{QB}(r)}.
  9. | x , 0 = | x 1 , x 2 , , x m , 0 , , 0 , |\vec{x},0\rangle=|x_{1},x_{2},\cdots,x_{m},\underbrace{0,\dots,0}\rangle,
  10. Tr ( | | x , 0 x , 0 | - S | ) δ . \operatorname{Tr}\left(\big||\vec{x},0\rangle\langle\vec{x},0|-S\big|\right)% \leq\delta.
  11. I = y Y E y . I=\sum_{y\in Y}\operatorname{E}_{y}.
  12. Pr { y } = Tr ( S E y ) . \operatorname{Pr}\{y\}=\operatorname{Tr}(S\operatorname{E}_{y}).
  13. x , 0 | U * E F ( x ) U | x , 0 = E F ( x ) U ( | x , 0 ) | U ( | x , 0 ) 1 - ϵ . \left\langle\vec{x},0\big|U^{*}\operatorname{E}_{F(x)}U\big|\vec{x},0\right% \rangle=\left\langle\operatorname{E}_{F(x)}U(|\vec{x},0\rangle)\big|U(|\vec{x}% ,0\rangle)\right\rangle\geq 1-\epsilon.
  14. | Tr ( S U * E F ( x ) U ) - x , 0 | U * E F ( x ) U | x , 0 | Tr ( | | x , 0 x , 0 | - S | ) U * E F ( x ) U δ \left|\operatorname{Tr}(SU^{*}\operatorname{E}_{F(x)}U)-\left\langle\vec{x},0% \big|U^{*}\operatorname{E}_{F(x)}U\big|\vec{x},0\right\rangle\right|\leq% \operatorname{Tr}(\big||\vec{x},0\rangle\langle\vec{x},0|-S\big|)\|U^{*}% \operatorname{E}_{F(x)}U\|\leq\delta
  15. Tr ( S U * E F ( x ) U ) 1 - ϵ - δ . \operatorname{Tr}(SU^{*}\operatorname{E}_{F(x)}U)\geq 1-\epsilon-\delta.
  16. 1 - e - 2 γ 2 k , 1-e^{-2\gamma^{2}k},

Quantum_efficiency.html

  1. EQE = electrons/sec photons/sec = current / (charge of one electron) ( total power of photons ) / ( energy of one photon ) \,\text{EQE}=\frac{\,\text{electrons/sec}}{\,\text{photons/sec}}=\frac{\,\text% {current}/\,\text{(charge of one electron)}}{(\,\text{total power of photons})% /(\,\text{energy of one photon})}
  2. IQE = electrons/sec absorbed photons/sec = EQE 1-Reflection-Transmission \,\text{IQE}=\frac{\,\text{electrons/sec}}{\,\text{absorbed photons/sec}}=% \frac{\,\text{EQE}}{\,\text{1-Reflection-Transmission}}
  3. Q E λ = R λ λ × h c e R λ λ × ( 1240 W nm / A ) QE_{\lambda}=\frac{R_{\lambda}}{\lambda}\times\frac{hc}{e}\approx\frac{R_{% \lambda}}{\lambda}{\times}(1240\;{\rm W}\cdot{\rm nm/A})
  4. Q E λ = η = N e N ν QE_{\lambda}=\eta=\frac{N_{e}}{N_{\nu}}
  5. N e N_{e}
  6. N ν N_{\nu}
  7. N ν t = Φ o λ h c \frac{N_{\nu}}{t}=\Phi_{o}\frac{\lambda}{hc}
  8. N e t = Φ ξ λ h c \frac{N_{e}}{t}=\Phi_{\xi}\frac{\lambda}{hc}
  9. Φ o \Phi_{o}
  10. Φ ξ \Phi_{\xi}

Quantum_error_correction.html

  1. | ψ = α 0 | 0 + α 1 | 1 |\psi\rangle=\alpha_{0}|0\rangle+\alpha_{1}|1\rangle
  2. | 0 |0\rangle
  3. | ψ = α 0 | 000 + α 1 | 111 . |\psi^{\prime}\rangle=\alpha_{0}|000\rangle+\alpha_{1}|111\rangle.
  4. E bit E_{\,\text{bit}}
  5. | ψ r = α 0 | 100 + α 1 | 011 |\psi^{\prime}_{r}\rangle=\alpha_{0}|100\rangle+\alpha_{1}|011\rangle
  6. P 0 = | 000 000 | + | 111 111 | P_{0}=|000\rangle\langle 000|+|111\rangle\langle 111|
  7. P 1 = | 100 100 | + | 011 011 | P_{1}=|100\rangle\langle 100|+|011\rangle\langle 011|
  8. P 2 = | 010 010 | + | 101 101 | P_{2}=|010\rangle\langle 010|+|101\rangle\langle 101|
  9. P 3 = | 001 001 | + | 110 110 | P_{3}=|001\rangle\langle 001|+|110\rangle\langle 110|
  10. ψ r | P 0 | ψ r = 0 \langle\psi^{\prime}_{r}|P_{0}|\psi^{\prime}_{r}\rangle=0
  11. ψ r | P 1 | ψ r = 1 \langle\psi^{\prime}_{r}|P_{1}|\psi^{\prime}_{r}\rangle=1
  12. ψ r | P 2 | ψ r = 0 \langle\psi^{\prime}_{r}|P_{2}|\psi^{\prime}_{r}\rangle=0
  13. ψ r | P 3 | ψ r = 0 \langle\psi^{\prime}_{r}|P_{3}|\psi^{\prime}_{r}\rangle=0
  14. P 1 P_{1}
  15. | 0 |0\rangle
  16. | 1 |1\rangle
  17. | - = ( | 0 - | 1 ) / 2 |-\rangle=(|0\rangle-|1\rangle)/\sqrt{2}
  18. | + = ( | 0 + | 1 ) / 2 . |+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}.
  19. | ψ = α 0 | + + α 1 | - |\psi\rangle=\alpha_{0}|+\rangle+\alpha_{1}|-\rangle
  20. | ψ = α 0 | + + + + α 1 | - - - . |\psi^{\prime}\rangle=\alpha_{0}|{+}{+}{+}\rangle+\alpha_{1}|{-}{-}{-}\rangle.
  21. E phase E\text{phase}
  22. | ψ |\psi\rangle
  23. E phase E\text{phase}
  24. E E
  25. | ψ = α 0 | 0 + α 1 | 1 |\psi\rangle=\alpha_{0}|0\rangle+\alpha_{1}|1\rangle
  26. | ψ = α 0 | 0 S + α 1 | 1 S |\psi^{\prime}\rangle=\alpha_{0}|0_{S}\rangle+\alpha_{1}|1_{S}\rangle
  27. | 0 S = 1 2 2 ( | 000 + | 111 ) ( | 000 + | 111 ) ( | 000 + | 111 ) |0_{S}\rangle=\frac{1}{2\sqrt{2}}(|000\rangle+|111\rangle)\otimes(|000\rangle+% |111\rangle)\otimes(|000\rangle+|111\rangle)
  28. | 1 S = 1 2 2 ( | 000 - | 111 ) ( | 000 - | 111 ) ( | 000 - | 111 ) |1_{S}\rangle=\frac{1}{2\sqrt{2}}(|000\rangle-|111\rangle)\otimes(|000\rangle-% |111\rangle)\otimes(|000\rangle-|111\rangle)
  29. | ψ |\psi\rangle
  30. U U
  31. U = c 0 I + c 1 σ x + c 2 σ y + c 3 σ z U=c_{0}I+c_{1}\sigma_{x}+c_{2}\sigma_{y}+c_{3}\sigma_{z}
  32. c 0 c_{0}
  33. c 1 c_{1}
  34. c 2 c_{2}
  35. c 3 c_{3}
  36. σ x = ( 0 1 1 0 ) ; \sigma_{x}=\biggl(\begin{matrix}0&1\\ 1&0\end{matrix}\biggr);
  37. σ y = ( 0 - i i 0 ) ; \sigma_{y}=\biggl(\begin{matrix}0&-i\\ i&0\end{matrix}\biggr);
  38. σ z = ( 1 0 0 - 1 ) \sigma_{z}=\biggl(\begin{matrix}1&0\\ 0&-1\end{matrix}\biggr)
  39. U = σ x U=\sigma_{x}
  40. U = σ z U=\sigma_{z}
  41. U = i σ y U=i\sigma_{y}
  42. \mathcal{E}
  43. 𝒞 \mathcal{C}\subseteq\mathcal{H}
  44. \mathcal{H}
  45. \mathcal{R}
  46. ( ) ( ρ ) = ρ ρ = P 𝒞 ρ P 𝒞 , (\mathcal{R}\circ\mathcal{E})(\rho)=\rho\quad\forall\rho=P_{\mathcal{C}}\rho P% _{\mathcal{C}},
  47. P 𝒞 P_{\mathcal{C}}
  48. 𝒞 \mathcal{C}
  49. \mathcal{R}

Quantum_gate.html

  1. v 0 | 0 + v 1 | 1 [ v 0 v 1 ] v_{0}|0\rangle+v_{1}|1\rangle\rightarrow\begin{bmatrix}v_{0}\\ v_{1}\end{bmatrix}
  2. v 00 | 00 + v 01 | 01 + v 10 | 10 + v 11 | 11 [ v 00 v 01 v 10 v 11 ] v_{00}|00\rangle+v_{01}|01\rangle+v_{10}|10\rangle+v_{11}|11\rangle\rightarrow% \begin{bmatrix}v_{00}\\ v_{01}\\ v_{10}\\ v_{11}\end{bmatrix}
  3. | a b |ab\rangle
  4. | 0 |0\rangle
  5. | 0 + | 1 2 \frac{|0\rangle+|1\rangle}{\sqrt{2}}
  6. | 1 |1\rangle
  7. | 0 - | 1 2 \frac{|0\rangle-|1\rangle}{\sqrt{2}}
  8. π \pi
  9. ( x ^ + z ^ ) / 2 (\hat{x}+\hat{z})/\sqrt{2}
  10. H = 1 2 [ 1 1 1 - 1 ] H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}
  11. H H * = I HH^{*}=I
  12. | 0 |0\rangle
  13. | 1 |1\rangle
  14. | 0 |0\rangle
  15. | 1 |1\rangle
  16. | 1 |1\rangle
  17. | 0 |0\rangle
  18. X = [ 0 1 1 0 ] X=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}
  19. | 0 |0\rangle
  20. i | 1 i|1\rangle
  21. | 1 |1\rangle
  22. - i | 0 -i|0\rangle
  23. Y = [ 0 - i i 0 ] Y=\begin{bmatrix}0&-i\\ i&0\end{bmatrix}
  24. | 0 |0\rangle
  25. | 1 |1\rangle
  26. - | 1 -|1\rangle
  27. Z = [ 1 0 0 - 1 ] Z=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  28. | 0 |0\rangle
  29. | 1 |1\rangle
  30. e i ϕ | 1 e^{i\phi}|1\rangle
  31. | 0 |0\rangle
  32. | 1 |1\rangle
  33. ϕ \phi
  34. R ϕ = [ 1 0 0 e i ϕ ] R_{\phi}=\begin{bmatrix}1&0\\ 0&e^{i\phi}\end{bmatrix}
  35. π 8 \frac{\pi}{8}
  36. π 4 \frac{\pi}{4}
  37. π 2 \frac{\pi}{2}
  38. | 00 |00\rangle
  39. | 01 |01\rangle
  40. | 10 |10\rangle
  41. | 11 |11\rangle
  42. SWAP = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] \mbox{SWAP}~{}=\begin{bmatrix}1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\end{bmatrix}
  43. SWAP = [ 1 0 0 0 0 1 2 ( 1 + i ) 1 2 ( 1 - i ) 0 0 1 2 ( 1 - i ) 1 2 ( 1 + i ) 0 0 0 0 1 ] \sqrt{\mbox{SWAP}}~{}=\begin{bmatrix}1&0&0&0\\ 0&\frac{1}{2}(1+i)&\frac{1}{2}(1-i)&0\\ 0&\frac{1}{2}(1-i)&\frac{1}{2}(1+i)&0\\ 0&0&0&1\end{bmatrix}
  44. | 1 |1\rangle
  45. CNOT = [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] \mbox{CNOT}~{}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0\end{bmatrix}
  46. U = [ x 00 x 01 x 10 x 11 ] U=\begin{bmatrix}x_{00}&x_{01}\\ x_{10}&x_{11}\end{bmatrix}
  47. | 00 | 00 |00\rangle\mapsto|00\rangle
  48. | 01 | 01 |01\rangle\mapsto|01\rangle
  49. | 10 | 1 U | 0 = | 1 ( x 00 | 0 + x 10 | 1 ) |10\rangle\mapsto|1\rangle U|0\rangle=|1\rangle\left(x_{00}|0\rangle+x_{10}|1% \rangle\right)
  50. | 11 | 1 U | 1 = | 1 ( x 01 | 0 + x 11 | 1 ) |11\rangle\mapsto|1\rangle U|1\rangle=|1\rangle\left(x_{01}|0\rangle+x_{11}|1% \rangle\right)
  51. C ( U ) = [ 1 0 0 0 0 1 0 0 0 0 x 00 x 01 0 0 x 10 x 11 ] \mbox{C}~{}(U)=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&x_{00}&x_{01}\\ 0&0&x_{10}&x_{11}\end{bmatrix}
  52. | 1 |1\rangle
  53. [ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 ] \begin{bmatrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&1&0\\ \end{bmatrix}
  54. | a , b , c |a,b,c\rangle
  55. | a , b , c a b |a,b,c\oplus ab\rangle
  56. [ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] \begin{bmatrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&1\\ \end{bmatrix}
  57. H H
  58. π / 8 \pi/8
  59. R ( π / 4 ) R(\pi/4)
  60. D ( θ ) D(\theta)
  61. | a , b , c { i cos ( θ ) | a , b , c + sin ( θ ) | a , b , 1 - c for a = b = 1 | a , b , c otherwise. |a,b,c\rangle\mapsto\begin{cases}i\cos(\theta)|a,b,c\rangle+\sin(\theta)|a,b,1% -c\rangle&\mbox{for }~{}a=b=1\\ |a,b,c\rangle&\mbox{otherwise.}\end{cases}
  62. D ( π 2 ) D(\begin{matrix}\frac{\pi}{2}\end{matrix})

Quantum_geometry.html

  1. | ψ = ψ ( 𝐱 , t ) | 𝐱 , t d 3 𝐱 |\psi\rangle=\int\psi(\mathbf{x},t)|\mathbf{x},t\rangle\mathrm{d}^{3}\mathbf{x}
  2. 𝐱 = ( x 1 , x 2 , x 3 ) \mathbf{x}=(x^{1},x^{2},x^{3})
  3. d 3 𝐱 = d x 1 d x 2 d x 3 \mathrm{d}^{3}\mathbf{x}=\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x% ^{3}
  4. | ψ = ψ ( x 1 , x 2 , x 3 , t ) | x 1 , x 2 , x 3 , t d x 1 d x 2 d x 3 |\psi\rangle=\int\psi(x^{1},x^{2},x^{3},t)|x^{1},x^{2},x^{3},t\rangle\mathrm{d% }x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}
  5. χ | ψ = χ * ψ d 3 𝐱 \langle\chi|\psi\rangle=\int\chi^{*}\psi\mathrm{d}^{3}\mathbf{x}
  6. χ | ψ = χ * ψ d x 1 d x 2 d x 3 \langle\chi|\psi\rangle=\int\chi^{*}\psi\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}% \wedge\mathrm{d}x^{3}
  7. ψ | ψ = R ψ * ψ d x 1 d x 2 d x 3 \langle\psi|\psi\rangle=\int_{R}\psi^{*}\psi\mathrm{d}x^{1}\wedge\mathrm{d}x^{% 2}\wedge\mathrm{d}x^{3}

Quantum_logic.html

  1. a = a a=a^{\perp\perp}
  2. \cup
  3. 1 = b b 1=b\cup b^{\perp}
  4. a ( a b ) = a a\cup(a^{\perp}\cup b)^{\perp}=a
  5. 1 = ( a b ) ( a b ) 1=(a^{\perp}\cup b^{\perp})^{\perp}\cup(a\cup b)^{\perp}
  6. a = b a=b
  7. LUB ( { E i } ) = i = 1 E i . \operatorname{LUB}(\{E_{i}\})=\bigcup_{i=1}^{\infty}E_{i}.
  8. f ( A ) = f ( λ ) d E ( λ ) . f(A)=\int_{\mathbb{R}}f(\lambda)\,d\operatorname{E}(\lambda).
  9. I = p q I=p\vee q
  10. 0 = p q 0=p\wedge q
  11. φ ( i = 1 S i ) = i = 1 φ ( S i ) . \varphi\left(\bigcup_{i=1}^{\infty}S_{i}\right)=\sum_{i=1}^{\infty}\varphi(S_{% i}).
  12. P ( i = 1 E i ) = i = 1 P ( E i ) . \operatorname{P}\!\left(\sum_{i=1}^{\infty}E_{i}\right)=\sum_{i=1}^{\infty}% \operatorname{P}(E_{i}).
  13. P ( E ) = Tr ( S E ) \operatorname{P}(E)=\operatorname{Tr}(SE)
  14. α ( i = 1 E i ) = i = 1 α ( E i ) \alpha\!\left(\sum_{i=1}^{\infty}E_{i}\right)=\sum_{i=1}^{\infty}\alpha(E_{i})
  15. Tr ( α * ( S ) E ) = Tr ( S α ( E ) ) . \operatorname{Tr}(\alpha^{*}(S)E)=\operatorname{Tr}(S\alpha(E)).
  16. α * ( r 1 S 1 + r 2 S 2 ) = r 1 α * ( S 1 ) + r 2 α * ( S 2 ) \alpha^{*}(r_{1}S_{1}+r_{2}S_{2})=r_{1}\alpha^{*}(S_{1})+r_{2}\alpha^{*}(S_{2})\quad
  17. β ( S ) = U S U * \beta(S)=USU^{*}\,
  18. F s , t ( S ) = U s - t S U s - t * \operatorname{F}_{s,t}(S)=U_{s-t}SU_{s-t}^{*}
  19. U t + s = σ ( t , s ) U t U s U_{t+s}=\sigma(t,s)U_{t}U_{s}
  20. Tr ( S E ) = E ψ | ψ \operatorname{Tr}(SE)=\langle E\psi|\psi\rangle
  21. S = | ψ ψ | , S=|\psi\rangle\langle\psi|,
  22. Tr ( S E ) = ψ | E | ψ . \operatorname{Tr}(SE)=\langle\psi|E|\psi\rangle.
  23. 1 Tr ( E S ) E S E . \frac{1}{\operatorname{Tr}(ES)}ESE.\,
  24. 1 Tr ( ( I - E ) S ) ( I - E ) S ( I - E ) . \frac{1}{\operatorname{Tr}((I-E)S)}(I-E)S(I-E).\,
  25. M E ( S ) = E S E + ( I - E ) S ( I - E ) . \operatorname{M}_{E}(S)=ESE+(I-E)S(I-E).\,

Quantum_operation.html

  1. [ S 11 S 1 n S n 1 S n n ] \begin{bmatrix}S_{11}&\cdots&S_{1n}\\ \vdots&\ddots&\vdots\\ S_{n1}&\cdots&S_{nn}\end{bmatrix}
  2. [ Φ ( S 11 ) Φ ( S 1 n ) Φ ( S n 1 ) Φ ( S n n ) ] \begin{bmatrix}\Phi(S_{11})&\cdots&\Phi(S_{1n})\\ \vdots&\ddots&\vdots\\ \Phi(S_{n1})&\cdots&\Phi(S_{nn})\end{bmatrix}
  3. Φ I n \Phi\otimes I_{n}
  4. I n I_{n}
  5. n × n n\times n
  6. { B i } 1 i n m \{B_{i}\}_{1\leq i\leq nm}
  7. Φ ( S ) = i B i * S B i . \Phi(S)=\sum_{i}B^{*}_{i}SB_{i}.
  8. i B i B i * 1. \sum_{i}B_{i}B^{*}_{i}\leq 1.
  9. { B i } \{B_{i}\}
  10. { B i } \{B_{i}\}
  11. ( u i j ) i j \;(u_{ij})_{ij}
  12. C i = j u i j B j . C_{i}=\sum_{j}u_{ij}B_{j}.\quad
  13. Tr A i A j δ i j {\rm Tr}A^{\dagger}_{i}A_{j}\sim\delta_{ij}
  14. α t ( E ) = U t * E U t . \alpha_{t}(E)=U^{*}_{t}EU_{t}.
  15. Tr ( β t ( S ) E ) = Tr ( S α - t ( E ) ) = Tr ( S U t E U t * ) = Tr ( U t * S U t E ) . \operatorname{Tr}(\beta_{t}(S)E)=\operatorname{Tr}(S\alpha_{-t}(E))=% \operatorname{Tr}(SU_{t}EU^{*}_{t})=\operatorname{Tr}(U^{*}_{t}SU_{t}E).
  16. g E = U g E U g * . g\cdot E=U_{g}EU_{g}^{*}.\quad
  17. S E S E + ( I - E ) S ( I - E ) . S\mapsto ESE+(I-E)S(I-E).
  18. A = λ λ E A ( λ ) A=\sum_{\lambda}\lambda\operatorname{E}_{A}(\lambda)
  19. Pr ( λ ) = Tr ( S E A ( λ ) ) . \operatorname{Pr}(\lambda)=\operatorname{Tr}(S\operatorname{E}_{A}(\lambda)).
  20. S λ E A ( λ ) S E A ( λ ) . S\mapsto\sum_{\lambda}\operatorname{E}_{A}(\lambda)S\operatorname{E}_{A}(% \lambda)\ .

Quantum_phase_transition.html

  1. C = U / T C=\partial U/\partial T
  2. ξ | ϵ | - ν = ( | T - T c | T c ) - ν \xi\propto|\epsilon|^{-\nu}\,\,=\left(\frac{|T-T_{c}|}{T_{c}}\right)^{-\nu}
  3. τ c ξ z | ϵ | - ν z , \tau_{c}\propto\xi^{z}\propto|\epsilon|^{-\nu z},
  4. ϵ = T - T c T c \epsilon=\frac{T-T_{c}}{T_{c}}

Quantum_statistical_mechanics.html

  1. 𝔼 ( X ) = λ d D X ( λ ) \mathbb{E}(X)=\int_{\mathbb{R}}\lambda\,d\,\operatorname{D}_{X}(\lambda)
  2. E A ( U ) = U λ d E ( λ ) , \operatorname{E}_{A}(U)=\int_{U}\lambda d\operatorname{E}(\lambda),
  3. D A ( U ) = Tr ( E A ( U ) S ) . \operatorname{D}_{A}(U)=\operatorname{Tr}(\operatorname{E}_{A}(U)S).
  4. 𝔼 ( A ) = λ d D A ( λ ) . \mathbb{E}(A)=\int_{\mathbb{R}}\lambda\,d\,\operatorname{D}_{A}(\lambda).
  5. 𝔼 ( A ) = Tr ( A S ) = Tr ( S A ) . \mathbb{E}(A)=\operatorname{Tr}(AS)=\operatorname{Tr}(SA).
  6. 𝔼 ( A ) = ψ | A | ψ . \mathbb{E}(A)=\langle\psi|A|\psi\rangle.
  7. Tr ( A ) = m m | A | m . \operatorname{Tr}(A)=\sum_{m}\langle m|A|m\rangle.
  8. H ( S ) = - Tr ( S log 2 S ) \operatorname{H}(S)=-\operatorname{Tr}(S\log_{2}S)
  9. [ λ 1 0 0 0 λ 2 0 0 0 λ n ] \begin{bmatrix}\lambda_{1}&0&\cdots&0&\cdots\\ 0&\lambda_{2}&\cdots&0&\cdots\\ &&\cdots&\\ 0&0&\cdots&\lambda_{n}&\cdots\\ &&\cdots&\cdots\end{bmatrix}
  10. H ( S ) = - i λ i log 2 λ i . \operatorname{H}(S)=-\sum_{i}\lambda_{i}\log_{2}\lambda_{i}.
  11. 0 log 2 0 = 0 \;0\log_{2}0=0
  12. T = [ 1 2 ( log 2 2 ) 2 0 0 0 1 3 ( log 2 3 ) 2 0 0 0 1 n ( log 2 n ) 2 ] T=\begin{bmatrix}\frac{1}{2(\log_{2}2)^{2}}&0&\cdots&0&\cdots\\ 0&\frac{1}{3(\log_{2}3)^{2}}&\cdots&0&\cdots\\ &&\cdots&\\ 0&0&\cdots&\frac{1}{n(\log_{2}n)^{2}}&\cdots\\ &&\cdots&\cdots\end{bmatrix}
  13. [ 1 n 0 0 0 1 n 0 0 0 1 n ] \begin{bmatrix}\frac{1}{n}&0&\cdots&0\\ 0&\frac{1}{n}&\dots&0\\ &&\cdots&\\ 0&0&\cdots&\frac{1}{n}\end{bmatrix}
  14. S = | ψ ψ | , S=|\psi\rangle\langle\psi|,
  15. E n E_{n}
  16. S = e - β H Tr ( e - β H ) . S=\frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})}.
  17. Tr ( S H ) = E \operatorname{Tr}(SH)=E
  18. Tr ( e - β H ) = n e - β E n = Z ( β ) \operatorname{Tr}(e^{-\beta H})=\sum_{n}e^{-\beta E_{n}}=Z(\beta)
  19. E m E_{m}
  20. e - β E m n e - β E n . \frac{e^{-\beta E_{m}}}{\sum_{n}e^{-\beta E_{n}}}.
  21. ρ = e - β ( H + μ 1 N 1 + μ 2 N 2 + ) Tr ( e - β ( H + μ 1 N 1 + μ 2 N 2 + ) ) . \rho=\frac{e^{-\beta(H+\mu_{1}N_{1}+\mu_{2}N_{2}+\cdots)}}{\operatorname{Tr}(e% ^{-\beta(H+\mu_{1}N_{1}+\mu_{2}N_{2}+\cdots)})}.
  22. 𝒵 ( β , μ 1 , μ 2 , ) = Tr ( e - β ( H + μ 1 N 1 + μ 2 N 2 + ) ) \mathcal{Z}(\beta,\mu_{1},\mu_{2},\cdots)=\operatorname{Tr}(e^{-\beta(H+\mu_{1% }N_{1}+\mu_{2}N_{2}+\cdots)})

Quantum_tunnelling.html

  1. U ( x ) = 8 e - 0.25 x 2 U(x)=8e^{-0.25x^{2}}
  2. H ( x , p ) = p 2 / 2 + U ( x ) H(x,p)=p^{2}/2+U(x)
  3. - 2 2 m d 2 d x 2 Ψ ( x ) + V ( x ) Ψ ( x ) = E Ψ ( x ) -\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\Psi(x)+V(x)\Psi(x)=E\Psi(x)
  4. d 2 d x 2 Ψ ( x ) = 2 m 2 ( V ( x ) - E ) Ψ ( x ) 2 m 2 M ( x ) Ψ ( x ) , \frac{d^{2}}{dx^{2}}\Psi(x)=\frac{2m}{\hbar^{2}}\left(V(x)-E\right)\Psi(x)% \equiv\frac{2m}{\hbar^{2}}M(x)\Psi(x),
  5. \hbar
  6. d 2 d x 2 Ψ ( x ) = 2 m 2 M ( x ) Ψ ( x ) = - k 2 Ψ ( x ) , where k 2 = - 2 m 2 M . \frac{d^{2}}{dx^{2}}\Psi(x)=\frac{2m}{\hbar^{2}}M(x)\Psi(x)=-k^{2}\Psi(x),\;\;% \;\;\;\;\mathrm{where}\;\;\;k^{2}=-\frac{2m}{\hbar^{2}}M.
  7. d 2 d x 2 Ψ ( x ) = 2 m 2 M ( x ) Ψ ( x ) = κ 2 Ψ ( x ) , where κ 2 = 2 m 2 M . \frac{d^{2}}{dx^{2}}\Psi(x)=\frac{2m}{\hbar^{2}}M(x)\Psi(x)={\kappa}^{2}\Psi(x% ),\;\;\;\;\;\;\mathrm{where}\;\;\;{\kappa}^{2}=\frac{2m}{\hbar^{2}}M.
  8. Ψ ( x ) = e Φ ( x ) \Psi(x)=e^{\Phi(x)}
  9. Φ ′′ ( x ) + Φ ( x ) 2 = 2 m 2 ( V ( x ) - E ) . \Phi^{\prime\prime}(x)+\Phi^{\prime}(x)^{2}=\frac{2m}{\hbar^{2}}\left(V(x)-E% \right).
  10. Φ ( x ) \Phi^{\prime}(x)
  11. Φ ( x ) = A ( x ) + i B ( x ) \Phi^{\prime}(x)=A(x)+iB(x)
  12. A ( x ) + A ( x ) 2 - B ( x ) 2 = 2 m 2 ( V ( x ) - E ) A^{\prime}(x)+A(x)^{2}-B(x)^{2}=\frac{2m}{\hbar^{2}}\left(V(x)-E\right)
  13. \hbar
  14. - 1 \hbar^{-1}
  15. A ( x ) = 1 k = 0 k A k ( x ) A(x)=\frac{1}{\hbar}\sum_{k=0}^{\infty}\hbar^{k}A_{k}(x)
  16. B ( x ) = 1 k = 0 k B k ( x ) B(x)=\frac{1}{\hbar}\sum_{k=0}^{\infty}\hbar^{k}B_{k}(x)
  17. A 0 ( x ) 2 - B 0 ( x ) 2 = 2 m ( V ( x ) - E ) A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-E\right)
  18. A 0 ( x ) B 0 ( x ) = 0 A_{0}(x)B_{0}(x)=0
  19. A 0 ( x ) = 0 A_{0}(x)=0
  20. B 0 ( x ) = ± 2 m ( E - V ( x ) ) B_{0}(x)=\pm\sqrt{2m\left(E-V(x)\right)}
  21. Ψ ( x ) C e i d x 2 m 2 ( E - V ( x ) ) + θ 2 m 2 ( E - V ( x ) ) 4 \Psi(x)\approx C\frac{e^{i\int dx\sqrt{\frac{2m}{\hbar^{2}}\left(E-V(x)\right)% }+\theta}}{\sqrt[4]{\frac{2m}{\hbar^{2}}\left(E-V(x)\right)}}
  22. B 0 ( x ) = 0 B_{0}(x)=0
  23. A 0 ( x ) = ± 2 m ( V ( x ) - E ) A_{0}(x)=\pm\sqrt{2m\left(V(x)-E\right)}
  24. Ψ ( x ) C + e + d x 2 m 2 ( V ( x ) - E ) + C - e - d x 2 m 2 ( V ( x ) - E ) 2 m 2 ( V ( x ) - E ) 4 \Psi(x)\approx\frac{C_{+}e^{+\int dx\sqrt{\frac{2m}{\hbar^{2}}\left(V(x)-E% \right)}}+C_{-}e^{-\int dx\sqrt{\frac{2m}{\hbar^{2}}\left(V(x)-E\right)}}}{% \sqrt[4]{\frac{2m}{\hbar^{2}}\left(V(x)-E\right)}}
  25. E = V ( x ) E=V(x)
  26. x 1 x_{1}
  27. 2 m 2 ( V ( x ) - E ) \frac{2m}{\hbar^{2}}\left(V(x)-E\right)
  28. x 1 x_{1}
  29. 2 m 2 ( V ( x ) - E ) = v 1 ( x - x 1 ) + v 2 ( x - x 1 ) 2 + \frac{2m}{\hbar^{2}}\left(V(x)-E\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots
  30. 2 m 2 ( V ( x ) - E ) = v 1 ( x - x 1 ) \frac{2m}{\hbar^{2}}\left(V(x)-E\right)=v_{1}(x-x_{1})
  31. x 1 x_{1}
  32. d 2 d x 2 Ψ ( x ) = v 1 ( x - x 1 ) Ψ ( x ) \frac{d^{2}}{dx^{2}}\Psi(x)=v_{1}(x-x_{1})\Psi(x)
  33. Ψ ( x ) = C A A i ( v 1 3 ( x - x 1 ) ) + C B B i ( v 1 3 ( x - x 1 ) ) \Psi(x)=C_{A}Ai\left(\sqrt[3]{v_{1}}(x-x_{1})\right)+C_{B}Bi\left(\sqrt[3]{v_{% 1}}(x-x_{1})\right)
  34. C , θ C,\theta
  35. C + , C - C_{+},C_{-}
  36. C + = 1 2 C cos ( θ - π 4 ) C_{+}=\frac{1}{2}C\cos{\left(\theta-\frac{\pi}{4}\right)}
  37. C - = - C sin ( θ - π 4 ) C_{-}=-C\sin{\left(\theta-\frac{\pi}{4}\right)}
  38. T ( E ) = e - 2 x 1 x 2 d x 2 m 2 [ V ( x ) - E ] T(E)=e^{-2\int_{x_{1}}^{x_{2}}\mathrm{d}x\sqrt{\frac{2m}{\hbar^{2}}\left[V(x)-% E\right]}}
  39. x 1 , x 2 x_{1},x_{2}
  40. T ( E ) = e - 2 2 m 2 ( V 0 - E ) ( x 2 - x 1 ) = V ~ 0 - ( x 2 - x 1 ) T(E)=e^{-2\sqrt{\frac{2m}{\hbar^{2}}(V_{0}-E)}(x_{2}-x_{1})}=\tilde{V}_{0}^{-(% x_{2}-x_{1})}

Quartic.html

  1. x 4 x^{4}

Quasi-continuous_function.html

  1. X X
  2. f : X f:X\rightarrow\mathbb{R}
  3. x X x\in X
  4. ϵ > 0 \epsilon>0
  5. U U
  6. x x
  7. G U G\subset U
  8. | f ( x ) - f ( y ) | < ϵ y G |f(x)-f(y)|<\epsilon\;\;\;\;\forall y\in G
  9. x G x\in G
  10. f : X f:X\rightarrow\mathbb{R}
  11. f f
  12. f : X f:X\rightarrow\mathbb{R}
  13. g : X g:X\rightarrow\mathbb{R}
  14. f + g f+g
  15. f : f:\mathbb{R}\rightarrow\mathbb{R}
  16. f ( x ) = 0 f(x)=0
  17. x 0 x\leq 0
  18. f ( x ) = 1 f(x)=1
  19. x > 0 x>0
  20. G U G\subset U
  21. y < 0 y G y<0\;\forall y\in G
  22. | f ( 0 ) - f ( y ) | = 0 y G |f(0)-f(y)|=0\;\forall y\in G

Quasidihedral_group.html

  1. r , s r 2 n - 1 = s 2 = 1 , s r s = r 2 n - 2 - 1 \langle r,s\mid r^{2^{n-1}}=s^{2}=1,srs=r^{2^{n-2}-1}\rangle\,\!
  2. r , s r 2 n - 1 = s 2 = 1 , s r s = r 2 n - 2 + 1 \langle r,s\mid r^{2^{n-1}}=s^{2}=1,srs=r^{2^{n-2}+1}\rangle\,\!
  3. / 2 n - 1 \mathbb{Z}/2^{n-1}\mathbb{Z}
  4. 2 n - 1 - 1 2^{n-1}-1
  5. 2 n - 2 - 1 2^{n-2}-1
  6. 2 n - 2 + 1 2^{n-2}+1

Quasinormal_mode.html

  1. ψ ( t ) e - ω ′′ t cos ω t \psi(t)\approx e^{-\omega^{\prime\prime}t}\cos\omega^{\prime}t
  2. ψ ( t ) \psi\left(t\right)
  3. ω \omega^{\prime}
  4. ω ′′ \omega^{\prime\prime}
  5. ω = ( ω , ω ′′ ) \omega=\left(\omega^{\prime},\omega^{\prime\prime}\right)
  6. ψ ( t ) Re ( e i ω t ) \psi\left(t\right)\approx\operatorname{Re}(e^{i\omega t})
  7. ω = ω + i ω ′′ \omega=\omega^{\prime}+i\omega^{\prime\prime}
  8. ω \mathbf{\omega}
  9. log | ψ ( t ) | \log\left|\psi(t)\right|

Quaternionic_representation.html

  1. j : V V j\colon V\to V\,
  2. j 2 = - 1. j^{2}=-1.\,
  3. ρ ( g h ) = ρ ( g ) ρ ( h ) for all g , h G . \rho(gh)=\rho(g)\rho(h)\,\text{ for all }g,h\in G.\,
  4. j : V V j\colon V\to V\,
  5. j 2 = + 1. j^{2}=+1.\,
  6. ρ ( g ) ρ ( g ) = 𝟏 \rho(g)^{\dagger}\rho(g)=\mathbf{1}\,

Quiver_diagram.html

  1. ( M , N ¯ ) (M,\bar{N})

Quotient_space_(linear_algebra).html

  1. V = U W V=U\oplus W
  2. 0 U V V / U 0. 0\to U\to V\to V/U\to 0.\,
  3. codim ( U ) = dim ( V / U ) = dim ( V ) - dim ( U ) . \mathrm{codim}(U)=\dim(V/U)=\dim(V)-\dim(U).
  4. [ x ] X / M = inf m M x - m X . \|[x]\|_{X/M}=\inf_{m\in M}\|x-m\|_{X}.
  5. q α ( [ x ] ) = inf x [ x ] p α ( x ) . q_{\alpha}([x])=\inf_{x\in[x]}p_{\alpha}(x).

R-tree.html

  1. M M
  2. l = number of objects / capacity l=\lceil\,\text{number of objects}/\,\text{capacity}\rceil
  3. s = l 1 / d s=\lceil l^{1/d}\rceil
  4. s s

R._H._Bing.html

  1. 3 \mathbb{R}^{3}
  2. 3 \mathbb{R}^{3}
  3. B B
  4. B × B\times\mathbb{R}
  5. 4 \mathbb{R}^{4}

Racks_and_quandles.html

  1. R \scriptstyle\mathrm{R}\,\!
  2. \scriptstyle\triangleright
  3. a , b , c R \scriptstyle a,\,b,\,c\;\in\;\mathrm{R}
  4. a ( b c ) = ( a b ) ( a c ) a\triangleright(b\triangleright c)=(a\triangleright b)\triangleright(a% \triangleright c)
  5. a , b R \scriptstyle a,\,b\;\in\;\mathrm{R}
  6. c R \scriptstyle c\;\in\;\mathrm{R}
  7. a c = b a\triangleright c=b
  8. c R \scriptstyle c\;\in\;\mathrm{R}
  9. a c = b \scriptstyle a\,\triangleright\,c\;=\;b
  10. b a \scriptstyle b\,\triangleleft\,a
  11. a c = b c = b a a\triangleright c=b\iff c=b\triangleleft a
  12. a ( b a ) = b a\triangleright(b\triangleleft a)=b
  13. ( a b ) a = b (a\triangleright b)\triangleleft a=b
  14. R \scriptstyle\mathrm{R}\,\!
  15. \scriptstyle\triangleright
  16. \scriptstyle\triangleleft
  17. a , b , c R \scriptstyle a,\,b,\,c\;\in\;\mathrm{R}
  18. a ( b c ) = ( a b ) ( a c ) a\triangleright(b\triangleright c)=(a\triangleright b)\triangleright(a% \triangleright c)
  19. ( c b ) a = ( c a ) ( b a ) (c\triangleleft b)\triangleleft a=(c\triangleleft a)\triangleleft(b% \triangleleft a)
  20. ( a b ) a = b (a\triangleright b)\triangleleft a=b
  21. a ( b a ) = b a\triangleright(b\triangleleft a)=b
  22. a R \scriptstyle a\;\in\;\mathrm{R}
  23. a b \scriptstyle a\,\triangleright\,b
  24. b a \scriptstyle b\,\triangleleft\,a
  25. \scriptstyle\triangleright
  26. \scriptstyle\triangleleft
  27. a b = b a a\triangleright b={}^{a}b
  28. b a = b a b\triangleleft a=b^{a}
  29. b a = b a b\triangleleft a=b\star a
  30. a ( b c ) \displaystyle a\triangleright(b\triangleleft c)
  31. Q \scriptstyle\mathrm{Q}\,
  32. a Q \scriptstyle\forall a\;\in\;\mathrm{Q}
  33. a a = a a\triangleright a=a
  34. a a = a a\triangleleft a=a
  35. a b \displaystyle a\triangleright b
  36. Q \scriptstyle\mathrm{Q}\,
  37. Q \scriptstyle\mathrm{Q}\,
  38. A \scriptstyle\mathrm{A}
  39. [ t , t - 1 ] \scriptstyle\mathbb{Z}\left[t,\,t^{-1}\right]
  40. A \scriptstyle\mathrm{A}
  41. a b = t a + ( 1 - t ) b a\triangleright b=ta+(1-t)b
  42. Q \scriptstyle\mathrm{Q}
  43. a , b Q \scriptstyle a,\,b\;\in\;\mathrm{Q}
  44. a ( a b ) = b a\triangleright(a\triangleright b)=b
  45. ( b a ) a = b (b\triangleleft a)\triangleleft a=b
  46. a b \scriptstyle a\,\triangleright\,b
  47. b \scriptstyle b
  48. a \scriptstyle a

Radar_cross-section.html

  1. σ 0 = R C S i A i \sigma^{0}=\left\langle{{RCS_{i}}\over{A_{i}}}\right\rangle
  2. P r = P t G t 4 π r 2 σ 1 4 π r 2 A eff P_{r}={{P_{t}G_{t}}\over{4\pi r^{2}}}\sigma{{1}\over{4\pi r^{2}}}A_{\mathrm{% eff}}
  3. P t P_{t}
  4. G t G_{t}
  5. r r
  6. σ \sigma
  7. A eff A_{\mathrm{eff}}
  8. P r P_{r}
  9. P t G t 4 π r 2 {{P_{t}G_{t}}\over{4\pi r^{2}}}
  10. σ \sigma
  11. P t G t 4 π r 2 σ {{P_{t}G_{t}}\over{4\pi r^{2}}}\sigma
  12. 1 4 π r 2 {{1}\over{4\pi r^{2}}}
  13. P t G t 4 π r 2 σ 1 4 π r 2 {{P_{t}G_{t}}\over{4\pi r^{2}}}\sigma{{1}\over{4\pi r^{2}}}
  14. A eff A_{\mathrm{eff}}
  15. P r / P t P_{r}/P_{t}
  16. σ = lim r 4 π r 2 S s S i \sigma=\lim_{r\to\infty}4\pi r^{2}\frac{S_{s}}{S_{i}}
  17. σ \sigma
  18. S i S_{i}
  19. S s S_{s}
  20. r r
  21. σ = lim r 4 π r 2 | E s | 2 | E i | 2 \sigma=\lim_{r\to\infty}4\pi r^{2}\frac{|E_{s}|^{2}}{|E_{i}|^{2}}
  22. E s E_{s}
  23. E i E_{i}

Radical_initiator.html

  1. \overrightarrow{\leftarrow}

Radix.html

  1. ( x ) y , (x)_{y},
  2. ( 100 ) 10 (100)_{10}
  3. ( 100 ) 2 (100)_{2}
  4. 3 × 13 2 + 9 × 13 1 + 8 × 13 0 3\times 13^{2}+9\times 13^{1}+8\times 13^{0}
  5. d 1 d n d_{1}\ldots d_{n}
  6. d 1 b n - 1 + d 2 b n - 2 + + d n b 0 d_{1}b^{n-1}+d_{2}b^{n-2}+\cdots+d_{n}b^{0}
  7. 0 d i < b 0\leq d_{i}<b

Rain_fade.html

  1. A 2 A 1 = ( b 2 b 1 ) 1 - 1.12 10 - 3 b 2 / b 1 ( b 1 A 1 ) 0.55 \frac{A_{2}}{A_{1}}=\left(\frac{b_{2}}{b_{1}}\right)^{1-1.12\cdot 10^{-3}\sqrt% {b_{2}/b_{1}}(b_{1}A_{1})^{0.55}}
  2. b i = f i 2 1 + 10 - 4 f i 2 b_{i}=\frac{f_{i}^{2}}{1+10^{-4}f_{i}^{2}}

Rainflow-counting_algorithm.html

  1. B ( N 1 N f 1 + N 2 N f 2 + + N k N f k ) = 1 B(\frac{N_{1}}{N_{f1}}+\frac{N_{2}}{N_{f2}}+...+\frac{N_{k}}{N_{fk}})=1
  2. N f = 1 2 ( σ a σ f 1 1 - σ m σ u ) 1 b N_{f}=\frac{1}{2}(\frac{\sigma_{a}}{\sigma^{\prime}_{f}}\frac{1}{1-\frac{% \sigma_{m}}{\sigma_{u}}})^{\frac{1}{b}}

Ramanujan_graph.html

  1. K n , n K_{n,n}
  2. G G
  3. d d
  4. n n
  5. λ 0 λ 1 λ n - 1 \lambda_{0}\geq\lambda_{1}\geq\ldots\geq\lambda_{n-1}
  6. G G
  7. G G
  8. d d
  9. d = λ 0 > λ 1 d=\lambda_{0}>\lambda_{1}
  10. λ n - 1 - d \geq\ldots\geq\lambda_{n-1}\geq-d
  11. λ i \lambda_{i}
  12. | λ i | < d |\lambda_{i}|<d
  13. λ ( G ) = max | λ i | < d | λ i | . \lambda(G)=\max_{|\lambda_{i}|<d}|\lambda_{i}|.\,
  14. d d
  15. G G
  16. λ ( G ) 2 d - 1 \lambda(G)\leq 2\sqrt{d-1}
  17. d d
  18. n n
  19. d d
  20. n n
  21. λ ( G ) \lambda(G)
  22. G G
  23. d d
  24. m m
  25. λ 1 2 d - 1 - 2 d - 1 - 1 m / 2 . \lambda_{1}\geq 2\sqrt{d-1}-\frac{2\sqrt{d-1}-1}{\lfloor m/2\rfloor}.
  26. G G
  27. d d
  28. | λ 1 | < d |\lambda_{1}|<d
  29. λ ( G ) λ 1 \lambda(G)\geq\lambda_{1}
  30. 𝒢 n d \mathcal{G}_{n}^{d}
  31. d d
  32. G G
  33. n n
  34. 𝒢 n d \mathcal{G}_{n}^{d}
  35. d d
  36. n n
  37. lim n inf G 𝒢 n d λ ( G ) 2 d - 1 . \lim_{n\to\infty}\inf_{G\in\mathcal{G}_{n}^{d}}\lambda(G)\geq 2\sqrt{d-1}.
  38. d d
  39. d d

Ramanujan–Petersson_conjecture.html

  1. τ ( n ) τ(n)
  2. Δ ( z ) Δ(z)
  3. 12 12
  4. Δ ( z ) = n > 0 τ ( n ) q n = q n > 0 ( 1 - q n ) 24 = q - 24 q 2 + 252 q 3 - 1472 q 4 + 4830 q 5 - , \Delta(z)=\sum_{n>0}\tau(n)q^{n}=q\prod_{n>0}\left(1-q^{n}\right)^{24}=q-24q^{% 2}+252q^{3}-1472q^{4}+4830q^{5}-\cdots,
  5. q = e 2 π i z q=e^{2\pi iz}
  6. | τ ( p ) | 2 p 11 2 , |\tau(p)|\leq 2p^{\frac{11}{2}},
  7. p p
  8. L ( s , a ) = p ( 1 + a ( p ) p s + a ( p 2 ) p 2 s ) L(s,a)=\prod_{p}\left(1+\frac{a(p)}{p^{s}}+\frac{a(p^{2})}{p^{2s}}\cdots\right)
  9. L ( s , a ) = p ( 1 - a ( p ) p s ) - 1 . L(s,a)=\prod_{p}\left(1-\frac{a(p)}{p^{s}}\right)^{-1}.
  10. L ( s , τ ) = p ( 1 - τ ( p ) p s + 1 p 2 s - 11 ) - 1 , L(s,\tau)=\prod_{p}\left(1-\frac{\tau(p)}{p^{s}}+\frac{1}{p^{2s-11}}\right)^{-% 1},
  11. τ ( p ) τ(p)
  12. 1 p 2 s - 11 \frac{1}{p^{2s-11}}
  13. τ τ
  14. τ τ
  15. p p
  16. j j
  17. 𝐍 \mathbf{N}
  18. 1 - τ ( p ) u + p 11 u 2 1-\tau(p)u+p^{11}u^{2}
  19. α α
  20. β β
  21. Re ( α ) = Re ( β ) = p 11 2 , \,\text{Re}(\alpha)=\,\text{Re}(\beta)=p^{\frac{11}{2}},
  22. τ ( n ) τ(n)
  23. ε > 0 ε>0
  24. O ( n 11 2 + ε ) . O\left(n^{\frac{11}{2}+\varepsilon}\right).
  25. Γ Γ
  26. S L ( 2 , 𝐙 ) SL(2,\mathbf{Z})
  27. f ( z ) = n = 0 a n q n q = e 2 π i z , f(z)=\sum^{\infty}_{n=0}a_{n}q^{n}\qquad q=e^{2\pi iz},
  28. φ ( s ) = n = 1 a n n s . \varphi(s)=\sum^{\infty}_{n=1}\frac{a_{n}}{n^{s}}.
  29. f ( z ) f(z)
  30. k 2 k≥2
  31. Γ Γ
  32. φ ( s ) φ(s)
  33. R e ( s ) > k Re(s)>k
  34. f f
  35. k k
  36. ( s k ) φ ( s ) (s−k)φ(s)
  37. R ( k - s ) = ( - 1 ) k 2 R ( s ) ; R(k-s)=(-1)^{\frac{k}{2}}R(s);
  38. f f
  39. φ φ
  40. x > 0 x>0
  41. g ( x ) g(x)
  42. R ( s ) R(s)
  43. R ( s ) = 0 g ( x ) x s - 1 d x g ( x ) = 1 2 π i Re ( s ) = σ 0 R ( s ) x - s d s . R(s)=\int^{\infty}_{0}g(x)x^{s-1}dx\Leftrightarrow g(x)=\frac{1}{2\pi i}\int_{% \,\text{Re}(s)=\sigma_{0}}R(s)x^{-s}ds.
  44. S L ( 2 , 𝐙 ) SL(2,\mathbf{Z})
  45. k 3 k≥3
  46. ( k 1 ) / 2 (k−1)/2
  47. k k
  48. k = 1 k=1
  49. G L ( 2 ) GL(2)
  50. U ( 2 , 1 ) U(2,1)
  51. S p ( 4 ) Sp(4)
  52. G L ( n ) GL(n)
  53. G L ( n ) GL(n)
  54. π = π v . \pi=\bigotimes\pi_{v}.
  55. τ 1 , v τ d , v . \tau_{1,v}\otimes\cdots\otimes\tau_{d,v}.
  56. τ i , v \tau_{i,v}
  57. v v
  58. τ i 0 , v | det | v σ i , v \tau_{i_{0},v}\otimes|\det|_{v}^{\sigma_{i,v}}
  59. τ i 0 , v \tau_{i_{0},v}
  60. n 2 n≥2
  61. δ 0 δ≥0
  62. max i | σ i , v | δ . \max_{i}\left|\sigma_{i,v}\right|\leq\delta.
  63. δ = 0 δ=0
  64. δ 1 / 2 δ≤1/2
  65. G L ( n ) GL(n)
  66. δ 1 / 2 ( n < s u p > 2 + 1 ) 1 δ≡1/2−(n<sup>2+1)^{−1}

Randall–Sundrum_model.html

  1. d s 2 = 1 k 2 y 2 ( d y 2 + η μ ν d x μ d x ν ) \mathrm{d}s^{2}={1\over k^{2}y^{2}}(\mathrm{d}y^{2}+\eta_{\mu\nu}\,\mathrm{d}x% ^{\mu}\,\mathrm{d}x^{\nu})
  2. 0 1 k 1 W k 0\leq{1\over k}\leq{1\over Wk}
  3. φ = def - π ln ( k y ) ln ( W ) , \varphi\ \stackrel{\mathrm{def}}{=}\ -{\pi\ln(ky)\over\ln(W)},
  4. 0 φ π 0\leq\varphi\leq\pi
  5. d s 2 = ( ln ( W ) π k ) 2 d φ 2 + e 2 ln ( W ) φ π η μ ν d x μ d x ν . \mathrm{d}s^{2}=\left({\ln(W)\over\pi k}\right)^{2}\,\mathrm{d}\varphi^{2}+e^{% 2\ln(W)\varphi\over\pi}\eta_{\mu\nu}\,\mathrm{d}x^{\mu}\,\mathrm{d}x^{\nu}.

Rankine_cycle.html

  1. Q ˙ \dot{Q}
  2. m ˙ \dot{m}
  3. W ˙ \dot{W}
  4. η t h e r m \eta_{therm}
  5. η p u m p , η t u r b \eta_{pump},\eta_{turb}
  6. h 1 , h 2 , h 3 , h 4 h_{1},h_{2},h_{3},h_{4}
  7. h 4 s h_{4s}
  8. p 1 , p 2 p_{1},p_{2}
  9. η t h e r m = W ˙ t h e r m a l - W ˙ Q ˙ i n W ˙ t u r b Q ˙ i n . \eta_{therm}=\frac{\dot{W}_{thermal}-\dot{W}}{\dot{Q}_{in}}\approx\frac{\dot{W% }_{turb}}{\dot{Q}_{in}}.
  10. η t h e r m \eta_{therm}
  11. Q ˙ i n m ˙ = h 3 - h 2 \frac{\dot{Q}_{in}}{\dot{m}}=h_{3}-h_{2}
  12. Q ˙ o u t m ˙ = h 4 - h 1 \frac{\dot{Q}_{out}}{\dot{m}}=h_{4}-h_{1}
  13. W ˙ p u m p m ˙ = h 2 - h 1 \frac{\dot{W}_{pump}}{\dot{m}}=h_{2}-h_{1}
  14. W ˙ t u r b i n e m ˙ = h 3 - h 4 \frac{\dot{W}_{turbine}}{\dot{m}}=h_{3}-h_{4}
  15. W ˙ p u m p m ˙ = h 2 - h 1 v 1 Δ p η p u m p = v 1 ( p 2 - p 1 ) η p u m p \frac{\dot{W}_{pump}}{\dot{m}}=h_{2}-h_{1}\approx\frac{v_{1}\Delta p}{\eta_{% pump}}=\frac{v_{1}(p_{2}-p_{1})}{\eta_{pump}}
  16. W ˙ t u r b i n e m ˙ = h 3 - h 4 ( h 3 - h 4 ) η t h e r m a l \frac{\dot{W}_{turbine}}{\dot{m}}=h_{3}-h_{4}\approx(h_{3}-h_{4})\eta_{thermal}
  17. ( T ¯ 𝑖𝑛 = 2 3 T d Q Q 𝑖𝑛 ) \left(\bar{T}_{\mathit{in}}=\frac{\int_{2}^{3}T\,dQ}{Q_{\mathit{in}}}\right)

RANSAC.html

  1. t t
  2. d d
  3. k k
  4. p p
  5. n n
  6. p p
  7. w w
  8. w w
  9. w w
  10. n n
  11. w n w^{n}
  12. 1 - w n 1-w^{n}
  13. n n
  14. k k
  15. n n
  16. 1 - p 1-p
  17. 1 - p = ( 1 - w n ) k 1-p=(1-w^{n})^{k}
  18. k = log ( 1 - p ) log ( 1 - w n ) k=\frac{\log(1-p)}{\log(1-w^{n})}
  19. n n
  20. k k
  21. k k
  22. k k
  23. S D ( k ) = 1 - w n w n SD(k)=\frac{\sqrt{1-w^{n}}}{w^{n}}

Rare_Earth_hypothesis.html

  1. N N
  2. N = N * n e f g f p f p m f i f c f l f m f j f m e N=N^{*}\cdot n_{e}\cdot f_{g}\cdot f_{p}\cdot f_{pm}\cdot f_{i}\cdot f_{c}% \cdot f_{l}\cdot f_{m}\cdot f_{j}\cdot f_{me}
  3. n e n_{e}
  4. n e n_{e}
  5. N * n e = 5 10 11 N^{*}\cdot n_{e}=5\cdot 10^{11}
  6. N N
  7. N N
  8. f g f_{g}
  9. f p f_{p}
  10. f p m f_{pm}
  11. f i f_{i}
  12. f c f_{c}
  13. f l f_{l}
  14. f m f_{m}
  15. f j f_{j}
  16. f m e f_{me}

Rate_of_convergence.html

  1. lim k | x k + 1 - L | | x k - L | = μ . \lim_{k\to\infty}\frac{|x_{k+1}-L|}{|x_{k}-L|}=\mu.
  2. lim k | x k + 2 - x k + 1 | | x k + 1 - x k | = 1 , \lim_{k\to\infty}\frac{|x_{k+2}-x_{k+1}|}{|x_{k+1}-x_{k}|}=1,
  3. lim k | x k + 1 - L | | x k - L | q = μ | μ > 0. \lim_{k\to\infty}\frac{|x_{k+1}-L|}{|x_{k}-L|^{q}}=\mu\,\big|\;\mu>0.
  4. | x k - L | ε k for all k , |x_{k}-L|\leq\varepsilon_{k}\quad\mbox{for all }~{}k,
  5. a 0 \displaystyle a_{0}
  6. x n x_{n}
  7. | x n - L | < C n - p for all n . |x_{n}-L|<Cn^{-p}\,\text{ for all }n.

Rational_pricing.html

  1. C t C_{t}\,
  2. r t r_{t}\,
  3. P 0 = t = 1 T C t ( 1 + r t ) t P_{0}=\sum_{t=1}^{T}\frac{C_{t}}{(1+r_{t})^{t}}
  4. P 0 = t = 1 T C ( t ) × P ( t ) P_{0}=\sum_{t=1}^{T}C(t)\times P(t)
  5. F ( t ) F(t)\,
  6. S ( t ) S(t)\,
  7. t t\,
  8. T T\,
  9. r r\,
  10. F ( t ) = S ( t ) × ( 1 + r ) ( T - t ) F(t)=S(t)\times(1+r)^{(T-t)}\,
  11. m a x max
  12. max \max
  13. max \max
  14. S = p × S u + ( 1 - p ) × S d 1 + r = p × u × S + ( 1 - p ) × d × S 1 + r p = ( 1 + r ) - d u - d \begin{aligned}\displaystyle S&\displaystyle=\frac{p\times S_{u}+(1-p)\times S% _{d}}{1+r}\\ &\displaystyle=\frac{p\times u\times S+(1-p)\times d\times S}{1+r}\\ \displaystyle\Rightarrow p&\displaystyle=\frac{(1+r)-d}{u-d}\\ \end{aligned}
  15. C = p × C u + ( 1 - p ) × C d 1 + r = p × max ( S u - k , 0 ) + ( 1 - p ) × max ( S d - k , 0 ) 1 + r \begin{aligned}\displaystyle C&\displaystyle=\frac{p\times C_{u}+(1-p)\times C% _{d}}{1+r}\\ &\displaystyle=\frac{p\times\max(S_{u}-k,0)+(1-p)\times\max(S_{d}-k,0)}{1+r}\\ \end{aligned}
  16. E ( r j ) = r f + b j 1 F 1 + b j 2 F 2 + + b j n F n + ϵ j E\left(r_{j}\right)=r_{f}+b_{j1}F_{1}+b_{j2}F_{2}+...+b_{jn}F_{n}+\epsilon_{j}
  17. E ( r j ) E(r_{j})
  18. r f r_{f}
  19. F k F_{k}
  20. b j k b_{jk}
  21. k k
  22. ϵ j \epsilon_{j}

Rational_variety.html

  1. K ( U 1 , , U d ) , K(U_{1},\dots,U_{d}),
  2. { U 1 , , U d } \{U_{1},\dots,U_{d}\}
  3. K [ X 1 , , X n ] K[X_{1},\dots,X_{n}]
  4. K ( U 1 , , U d ) K(U_{1},\dots,U_{d})
  5. f i ( g 1 / g 0 , , g n / g 0 ) = 0. f_{i}(g_{1}/g_{0},\ldots,g_{n}/g_{0})=0.
  6. x i = g i g 0 ( u 1 , , u d ) x_{i}=\frac{g_{i}}{g_{0}}(u_{1},\ldots,u_{d})
  7. K ( U 1 , , U d ) K(U_{1},\dots,U_{d})
  8. K L K\subset L
  9. L L
  10. K K
  11. K K
  12. L L
  13. K K
  14. { y 1 , , y n } \{y_{1},\dots,y_{n}\}
  15. G G
  16. L L
  17. L G L^{G}
  18. K L G K\subset L^{G}

RC_oscillator.html

  1. f = 1 2 π R C f=\frac{1}{2\pi RC}
  2. x = 2 x=2

Real_closed_field.html

  1. F ( - 1 ) F(\sqrt{-1})
  2. alg \mathbb{R}_{\mathrm{alg}}
  3. ( 2 ) \mathbb{Q}(\sqrt{2})
  4. alg × alg \mathbb{R}_{\mathrm{alg}}\times\mathbb{R}_{\mathrm{alg}}
  5. ( 2 ) \mathbb{Q}(\sqrt{2})
  6. ( 2 ) \mathbb{Q}(\sqrt{2})
  7. \mathbb{R}
  8. alg \mathbb{R}_{\mathrm{alg}}
  9. 2 2 n 2^{2^{\cdot^{\cdot^{\cdot^{n}}}}}
  10. 2 2 Ω ( n ) 2^{2^{\Omega(n)}}
  11. 2 2 Ω ( n ) 2^{2^{\Omega(n)}}
  12. α \aleph_{\alpha}
  13. α \aleph_{\alpha}
  14. α \aleph_{\alpha}
  15. / 𝐌 \mathbb{R}^{\mathbb{N}}/{\mathbf{M}}
  16. \mathbb{R}
  17. β \aleph_{\beta}
  18. ( ( G ) ) \mathbb{R}((G))
  19. 1 \aleph_{1}
  20. 1 \aleph_{1}
  21. 2 \aleph_{2}
  22. 2 \aleph_{2}
  23. 1 \aleph_{1}
  24. 1 \aleph_{1}
  25. 0 \aleph_{0}
  26. 1 \aleph_{1}
  27. 0 \aleph_{0}

Real_projective_space.html

  1. ( - 1 ) p (-1)^{p}
  2. π 1 ( 𝐑𝐏 n ) \pi_{1}(\mathbf{RP}^{n})
  3. ( - 1 ) n + 1 (-1)^{n+1}
  4. 𝐑𝐏 n = 𝐑𝐏 n - 1 f D n . \mathbf{RP}^{n}=\mathbf{RP}^{n-1}\cup_{f}D^{n}.
  5. [ * : 0 : 0 : : 0 ] [*:0:0:\dots:0]
  6. [ * : * : 0 : : 0 ] [*:*:0:\dots:0]
  7. \vdots
  8. [ * : * : * : : * ] . [*:*:*:\dots:*].
  9. g ( x 1 , , x n + 1 ) = 1 n + 1 i | x i | 2 . g(x_{1},\cdots,x_{n+1})=\sum_{1}^{n+1}i\cdot|x_{i}|^{2}.
  10. 𝐙 2 S n 𝐑𝐏 n . \mathbf{Z}_{2}\to S^{n}\to\mathbf{RP}^{n}.
  11. S 0 S n 𝐑𝐏 n S^{0}\to S^{n}\to\mathbf{RP}^{n}
  12. O ( 1 ) S n 𝐑𝐏 n O(1)\to S^{n}\to\mathbf{RP}^{n}
  13. π i ( 𝐑𝐏 n ) = { 0 i = 0 𝐙 i = 1 , n = 1 𝐙 / 2 𝐙 i = 1 , n > 1 π i ( S n ) i > 1 , n > 0. \pi_{i}(\mathbf{RP}^{n})=\begin{cases}0&i=0\\ \mathbf{Z}&i=1,n=1\\ \mathbf{Z}/2\mathbf{Z}&i=1,n>1\\ \pi_{i}(S^{n})&i>1,n>0.\end{cases}
  14. deg ( d k ) = 1 + ( - 1 ) k . \mathrm{deg}(d_{k})=1+(-1)^{k}.
  15. H i ( 𝐑𝐏 n ) = { 𝐙 i = 0 or i = n odd, 𝐙 / 2 𝐙 0 < i < n , i odd, 0 else. H_{i}(\mathbf{RP}^{n})=\begin{cases}\mathbf{Z}&i=0\mbox{ or }~{}i=n\mbox{ odd,% }\\ \mathbf{Z}/2\mathbf{Z}&0<i<n,\ i\ \mbox{odd,}\\ 0&\mbox{else.}\end{cases}
  16. 𝐑𝐏 := lim n 𝐑𝐏 n . \mathbf{RP}^{\infty}:=\lim_{n}\mathbf{RP}^{n}.
  17. S S^{\infty}
  18. H q ( 𝐑𝐏 ; 𝐙 / 2 ) = 𝐙 / 2 H_{q}(\mathbf{RP}^{\infty};\mathbf{Z}/2)=\mathbf{Z}/2
  19. H * ( 𝐑𝐏 ; 𝐙 / 2 𝐙 ) = 𝐙 / 2 𝐙 [ w 1 ] , H^{*}(\mathbf{RP}^{\infty};\mathbf{Z}/2\mathbf{Z})=\mathbf{Z}/2\mathbf{Z}[w_{1% }],
  20. w 1 w_{1}
  21. 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}
  22. w 1 w_{1}

Real_representation.html

  1. j : V V j\colon V\to V\,
  2. j 2 = + 1. j^{2}=+1.\,
  3. j : V V j\colon V\to V\,
  4. j 2 = - 1. j^{2}=-1.\,
  5. g G χ ( g 2 ) d μ \int_{g\in G}\chi(g^{2})\,d\mu
  6. 1 | G | g G χ ( g 2 ) . {1\over|G|}\sum_{g\in G}\chi(g^{2}).

Rearrangement_inequality.html

  1. x n y 1 + + x 1 y n x σ ( 1 ) y 1 + + x σ ( n ) y n x 1 y 1 + + x n y n x_{n}y_{1}+\cdots+x_{1}y_{n}\leq x_{\sigma(1)}y_{1}+\cdots+x_{\sigma(n)}y_{n}% \leq x_{1}y_{1}+\cdots+x_{n}y_{n}
  2. x 1 x n and y 1 y n x_{1}\leq\cdots\leq x_{n}\quad\,\text{and}\quad y_{1}\leq\cdots\leq y_{n}
  3. x σ ( 1 ) , , x σ ( n ) x_{\sigma(1)},\dots,x_{\sigma(n)}\,
  4. x 1 < < x n and y 1 < < y n , x_{1}<\cdots<x_{n}\quad\,\text{and}\quad y_{1}<\cdots<y_{n},
  5. - x n - x 1 . -x_{n}\leq\cdots\leq-x_{1}.
  6. x σ ( 1 ) y 1 + + x σ ( n ) y n x_{\sigma(1)}y_{1}+\cdots+x_{\sigma(n)}y_{n}
  7. j < k y j y k and j < σ ( j ) x j x σ ( j ) . ( 1 ) j<k\Rightarrow y_{j}\leq y_{k}\qquad\,\text{and}\qquad j<\sigma(j)\Rightarrow x% _{j}\leq x_{\sigma(j)}.\quad(1)
  8. 0 ( x σ ( j ) - x j ) ( y k - y j ) . ( 2 ) 0\leq(x_{\sigma(j)}-x_{j})(y_{k}-y_{j}).\quad(2)
  9. x σ ( j ) y j + x j y k x j y j + x σ ( j ) y k , ( 3 ) x_{\sigma(j)}y_{j}+x_{j}y_{k}\leq x_{j}y_{j}+x_{\sigma(j)}y_{k}\,,\quad(3)
  10. τ ( i ) := { i for i { 1 , , j } , σ ( j ) for i = k , σ ( i ) for i { j + 1 , , n } { k } , \tau(i):=\begin{cases}i&\,\text{for }i\in\{1,\ldots,j\},\\ \sigma(j)&\,\text{for }i=k,\\ \sigma(i)&\,\text{for }i\in\{j+1,\ldots,n\}\setminus\{k\},\end{cases}
  11. x 1 < < x n and y 1 < < y n , x_{1}<\cdots<x_{n}\quad\,\text{and}\quad y_{1}<\cdots<y_{n},

Receiver_operating_characteristic.html

  1. - -\infty
  2. + +\infty
  3. T T
  4. P 1 ( T ) P_{1}(T)
  5. P 0 ( T ) P_{0}(T)
  6. FPR ( T ) = T P 0 ( T ) d T \mbox{FPR}~{}(T)=\int_{T}^{\infty}P_{0}(T^{\prime})\,dT^{\prime}
  7. TPR ( T ) = T P 1 ( T ) d T \mbox{TPR}~{}(T)=\int_{T}^{\infty}P_{1}(T^{\prime})\,dT^{\prime}
  8. A = - y ( T ) x ( T ) d T = - TPR ( T ) FPR ( T ) d T = - TPR ( T ) P 0 ( T ) d T = TPR A=\int_{\infty}^{-\infty}y(T)x^{\prime}(T)\,dT=\int_{\infty}^{-\infty}\mbox{% TPR}~{}(T)\mbox{FPR}~{}^{\prime}(T)\,dT=\int_{-\infty}^{\infty}\mbox{TPR}~{}(T% )P_{0}(T)\,dT=\langle\mbox{TPR}~{}\rangle
  9. G 1 G_{1}
  10. G 1 = 2 AUC - 1 G_{1}=2\mbox{AUC}~{}-1
  11. G 1 = 1 - k = 1 n ( X k - X k - 1 ) ( Y k + Y k - 1 ) G_{1}=1-\sum_{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})
  12. c ( c - 1 ) c(c-1)
  13. c c

Receptive_field.html

  1. r e c e p t i v e f i e l d = c e n t e r + s u r r o u n d receptive\ field=center+surround

Reciprocal_lattice.html

  1. e i 𝐊 𝐫 = cos ( 𝐊 𝐫 ) + i sin ( 𝐊 𝐫 ) e^{i\mathbf{K}\cdot\mathbf{r}}=\cos{(\mathbf{K}\cdot\mathbf{r})}+i\sin{(% \mathbf{K}\cdot\mathbf{r})}
  2. e i 𝐊 ( 𝐫 + 𝐑 ) = e i 𝐊 𝐫 e^{i\mathbf{K}\cdot\mathbf{(r+R)}}=e^{i\mathbf{K}\cdot\mathbf{r}}
  3. e i 𝐊 𝐫 e i 𝐊 𝐑 = e i 𝐊 𝐫 \therefore e^{i\mathbf{K}\cdot\mathbf{r}}e^{i\mathbf{K}\cdot\mathbf{R}}=e^{i% \mathbf{K}\cdot\mathbf{r}}
  4. e i 𝐊 𝐑 = 1 \Rightarrow e^{i\mathbf{K}\cdot\mathbf{R}}=1
  5. ( 𝐚 𝟏 , 𝐚 𝟐 ) (\mathbf{a_{1}},\mathbf{a_{2}})
  6. 𝐛 𝟏 = 2 π ( x ^ y ^ - y ^ x ^ ) 𝐚 𝟐 𝐚 𝟏 ( x ^ y ^ - y ^ x ^ ) 𝐚 𝟐 \mathbf{b_{1}}=2\pi\frac{(\hat{x}\otimes\hat{y}-\hat{y}\otimes\hat{x})\mathbf{% a_{2}}}{\mathbf{a_{1}}\cdot(\hat{x}\otimes\hat{y}-\hat{y}\otimes\hat{x})% \mathbf{a_{2}}}
  7. 𝐛 𝟐 = 2 π ( y ^ x ^ - x ^ y ^ ) 𝐚 𝟏 𝐚 𝟐 ( y ^ x ^ - x ^ y ^ ) 𝐚 𝟏 \mathbf{b_{2}}=2\pi\frac{(\hat{y}\otimes\hat{x}-\hat{x}\otimes\hat{y})\mathbf{% a_{1}}}{\mathbf{a_{2}}\cdot(\hat{y}\otimes\hat{x}-\hat{x}\otimes\hat{y})% \mathbf{a_{1}}}
  8. \otimes
  9. x ^ \hat{x}
  10. y ^ \hat{y}
  11. ( 𝐚 𝟏 , 𝐚 𝟐 , 𝐚 𝟑 ) (\mathbf{a_{1}},\mathbf{a_{2}},\mathbf{a_{3}})
  12. 𝐛 𝟏 = 2 π 𝐚 𝟐 × 𝐚 𝟑 𝐚 𝟏 ( 𝐚 𝟐 × 𝐚 𝟑 ) \mathbf{b_{1}}=2\pi\frac{\mathbf{a_{2}}\times\mathbf{a_{3}}}{\mathbf{a_{1}}% \cdot(\mathbf{a_{2}}\times\mathbf{a_{3}})}
  13. 𝐛 𝟐 = 2 π 𝐚 𝟑 × 𝐚 𝟏 𝐚 𝟐 ( 𝐚 𝟑 × 𝐚 𝟏 ) \mathbf{b_{2}}=2\pi\frac{\mathbf{a_{3}}\times\mathbf{a_{1}}}{\mathbf{a_{2}}% \cdot(\mathbf{a_{3}}\times\mathbf{a_{1}})}
  14. 𝐛 𝟑 = 2 π 𝐚 𝟏 × 𝐚 𝟐 𝐚 𝟑 ( 𝐚 𝟏 × 𝐚 𝟐 ) \mathbf{b_{3}}=2\pi\frac{\mathbf{a_{1}}\times\mathbf{a_{2}}}{\mathbf{a_{3}}% \cdot(\mathbf{a_{1}}\times\mathbf{a_{2}})}
  15. [ 𝐛 𝟏 𝐛 𝟐 𝐛 𝟑 ] T = 2 π [ 𝐚 𝟏 𝐚 𝟐 𝐚 𝟑 ] - 1 . \left[\mathbf{b_{1}}\mathbf{b_{2}}\mathbf{b_{3}}\right]^{T}=2\pi\left[\mathbf{% a_{1}}\mathbf{a_{2}}\mathbf{a_{3}}\right]^{-1}.
  16. 2 π 2\pi
  17. e 2 π i 𝐊 𝐑 = 1 e^{2\pi i\mathbf{K}\cdot\mathbf{R}}=1
  18. 𝐛 𝟏 = 𝐚 𝟐 × 𝐚 𝟑 𝐚 𝟏 ( 𝐚 𝟐 × 𝐚 𝟑 ) \mathbf{b_{1}}=\frac{\mathbf{a_{2}}\times\mathbf{a_{3}}}{\mathbf{a_{1}}\cdot(% \mathbf{a_{2}}\times\mathbf{a_{3}})}
  19. 𝐛 𝟏 \mathbf{b_{1}}
  20. 𝐚 𝟏 \mathbf{a_{1}}
  21. 𝐚 𝟐 × 𝐚 𝟑 \mathbf{a_{2}}\times\mathbf{a_{3}}
  22. 2 π 2\pi
  23. a a
  24. 2 π a \begin{matrix}\frac{2\pi}{a}\end{matrix}
  25. 1 a \begin{matrix}\frac{1}{a}\end{matrix}
  26. ( 𝐚 𝟏 , 𝐚 𝟐 , 𝐚 𝟑 ) (\mathbf{a_{1}},\mathbf{a_{2}},\mathbf{a_{3}})
  27. ( 𝐛 𝟏 , 𝐛 𝟐 , 𝐛 𝟑 ) (\mathbf{b_{1}},\mathbf{b_{2}},\mathbf{b_{3}})
  28. 2 π c \begin{matrix}\frac{2\pi}{c}\end{matrix}
  29. 4 π a 3 \begin{matrix}\frac{4\pi}{a\sqrt{3}}\end{matrix}
  30. e i 𝐊 𝟏 ( 𝐑 ) = 1 e^{i\mathbf{K_{1}}\cdot\mathbf{(R)}}=1
  31. e i 𝐊 𝟐 ( 𝐑 ) = 1 e^{i\mathbf{K_{2}}\cdot\mathbf{(R)}}=1
  32. 𝐊 𝟏 \plusmn 𝐊 𝟐 \mathbf{K_{1}}\plusmn\mathbf{K_{2}}
  33. e i ( 𝐊 𝟏 + 𝐊 𝟐 ) ( 𝐑 ) = e i 𝐊 𝟏 𝐑 e i 𝐊 𝟐 𝐑 = 1 1 = 1 e^{i\mathbf{(K_{1}+K_{2})}\cdot\mathbf{(R)}}=e^{i\mathbf{K_{1}}\cdot\mathbf{R}% }\cdot e^{i\mathbf{K_{2}}\cdot\mathbf{R}}=1\cdot 1=1
  34. e i ( 𝐊 𝟏 - 𝐊 𝟐 ) ( 𝐑 ) = e i 𝐊 𝟏 𝐑 / e i 𝐊 𝟐 𝐑 = 1 e^{i\mathbf{(K_{1}-K_{2})}\cdot\mathbf{(R)}}=e^{i\mathbf{K_{1}}\cdot\mathbf{R}% }/e^{i\mathbf{K_{2}}\cdot\mathbf{R}}=1
  35. 𝐊 = k 1 𝐛 𝟏 + k 2 𝐛 𝟐 + k 3 𝐛 𝟑 \mathbf{K}=k_{1}\mathbf{b_{1}}+k_{2}\mathbf{b_{2}}+k_{3}\mathbf{b_{3}}
  36. 𝐛 𝟏 \mathbf{b_{1}}
  37. 𝐛 𝐢 𝐚 𝐣 = 2 π δ i j \mathbf{b_{i}}\cdot\mathbf{a_{j}}=2\pi\delta_{ij}
  38. δ i j \delta_{ij}
  39. 𝐑 = n 1 𝐚 𝟏 + n 2 𝐚 𝟐 + n 3 𝐚 𝟑 \mathbf{R}=n_{1}\mathbf{a_{1}}+n_{2}\mathbf{a_{2}}+n_{3}\mathbf{a_{3}}
  40. 𝐊 𝐑 = 2 π ( k 1 n 1 + k 2 n 2 + k 3 n 3 ) \mathbf{K}\cdot\mathbf{R}=2\pi(k_{1}n_{1}+k_{2}n_{2}+k_{3}n_{3})
  41. 𝐊 \mathbf{K}
  42. e i 𝐊 𝐑 = 1 e^{i\mathbf{K}\cdot\mathbf{R}}=1
  43. 𝐊 𝐑 \mathbf{K}\cdot\mathbf{R}
  44. 2 π 2\pi
  45. n i n_{i}\in\mathbb{Z}
  46. k i k_{i}\in\mathbb{Z}
  47. 𝐊 \mathbf{K}
  48. 𝐆 \mathbf{G}
  49. e i 𝐆 𝐊 = 1 e^{i\mathbf{G}\cdot\mathbf{K}}=1
  50. 𝐊 \mathbf{K}
  51. 𝐆 \mathbf{G}
  52. 𝐑 \mathbf{R}
  53. e i 𝐑 𝐊 = 1 e^{i\mathbf{R}\cdot\mathbf{K}}=1
  54. F [ g ] = j = 1 N f j [ g ] e 2 π i g r j . F[\vec{g}]=\sum_{j=1}^{N}f_{j}[\vec{g}]e^{2\pi i\vec{g}\cdot\vec{r}_{j}}.
  55. F h k l = j = 1 m f j [ g h k l ] e 2 π i ( h u j + k v j + l w j ) F_{hkl}=\sum_{j=1}^{m}f_{j}[g_{hkl}]e^{2\pi i(hu_{j}+kv_{j}+lw_{j})}
  56. I [ g ] = j = 1 N k = 1 N f j [ g ] f k [ g ] e 2 π i g r j k . I[\vec{g}]=\sum_{j=1}^{N}\sum_{k=1}^{N}f_{j}[\vec{g}]f_{k}[\vec{g}]e^{2\pi i% \vec{g}\cdot\vec{r}_{jk}}.
  57. A = B ( B T B ) - 1 A=B(B^{T}B)^{-1}
  58. k \vec{k}

Reciprocal_polynomial.html

  1. p ( x ) = a 0 + a 1 x + a 2 x 2 + + a n x n , p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n},\,\!
  2. p * ( x ) = a n + a n - 1 x + + a 0 x n = x n p ( x - 1 ) . p^{*}(x)=a_{n}+a_{n-1}x+\cdots+a_{0}x^{n}=x^{n}p(x^{-1}).
  3. p ( z ) = a 0 + a 1 z + a 2 z 2 + + a n z n , p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots+a_{n}z^{n},\,\!
  4. p ( z ) = a n ¯ + a n - 1 ¯ z + + a 0 ¯ z n = z n p ( z ¯ - 1 ) ¯ , p^{\dagger}(z)=\overline{a_{n}}+\overline{a_{n-1}}z+\cdots+\overline{a_{0}}z^{% n}=z^{n}\overline{p(\bar{z}^{-1})},
  5. a i ¯ \overline{a_{i}}
  6. a i a_{i}\,\!
  7. p ( x ) p * ( x ) p(x)\equiv p^{*}(x)
  8. p ( x ) p ( x ) p(x)\equiv p^{\dagger}(x)
  9. p ( x ) = ω p ( x ) p(x)=\omega p^{\dagger}(x)
  10. z 0 1 z_{0}\neq 1
  11. z 0 n p ( 1 / z 0 ¯ ) ¯ = z 0 n p ( z 0 ) ¯ = z 0 n 0 ¯ = 0. z_{0}^{n}\overline{p(1/\bar{z_{0}})}=z_{0}^{n}\overline{p(z_{0})}=z_{0}^{n}% \bar{0}=0.
  12. z n p ( z ¯ - 1 ) ¯ z^{n}\overline{p(\bar{z}^{-1})}
  13. c p ( z ) = z n p ( z ¯ - 1 ) ¯ cp(z)=z^{n}\overline{p(\bar{z}^{-1})}
  14. c a i = a n - i ¯ = a n - i ca_{i}=\overline{a_{n-i}}=a_{n-i}
  15. Φ n \Phi_{n}
  16. n > 1 n>1
  17. x 11 ± 1 x^{11}\pm 1
  18. x 13 ± 1 x^{13}\pm 1
  19. x 15 ± 1 x^{15}\pm 1
  20. x 21 ± 1 x^{21}\pm 1
  21. ϕ \phi

Reciprocity_(photography).html

  1. I × t 0.86 I\times t^{0.86}
  2. E = I t p E=It^{p}
  3. = ( t + 1 ) ( p - 1 ) =(t+1)^{(p-1)}
  4. I t / ψ It/\psi
  5. ψ = 1 2 [ ( I / I 0 ) a + ( I / I 0 ) - a ] \psi=\frac{1}{2}[(I/I_{0})^{a}+(I/I_{0})^{-a}]

Record_linkage.html

  1. α ( a ) \alpha(a)
  2. β ( b ) \beta(b)
  3. K K
  4. M = { ( a , b ) ; a = b ; a A ; b B } M=\left\{(a,b);a=b;a\in A;b\in B\right\}
  5. M M
  6. U U
  7. U = { ( a , b ) ; a b ; a A , b B } U=\{(a,b);a\neq b;a\in A,b\in B\}
  8. γ \gamma
  9. γ [ α ( a ) , β ( b ) ] = { γ 1 [ α ( a ) , β ( b ) ] , , γ K [ α ( a ) , β ( b ) ] } \gamma\left[\alpha(a),\beta(b)\right]=\{\gamma^{1}\left[\alpha(a),\beta(b)% \right],...,\gamma^{K}\left[\alpha(a),\beta(b)\right]\}
  10. K K
  11. γ \gamma
  12. ( a , b ) M (a,b)\in M
  13. ( a , b ) U (a,b)\in U
  14. m ( γ ) = P { γ [ α ( a ) , β ( b ) ] | ( a , b ) M } = ( a , b ) M P { γ [ α ( a ) , β ( b ) ] } P [ ( a , b ) | M ] m(\gamma)=P\left\{\gamma\left[\alpha(a),\beta(b)\right]|(a,b)\in M\right\}=% \sum_{(a,b)\in M}P\left\{\gamma\left[\alpha(a),\beta(b)\right]\right\}\cdot P% \left[(a,b)|M\right]
  15. u ( γ ) = P { γ [ α ( a ) , β ( b ) ] | ( a , b ) U } = ( a , b ) U P { γ [ α ( a ) , β ( b ) ] } P [ ( a , b ) | U ] , u(\gamma)=P\left\{\gamma\left[\alpha(a),\beta(b)\right]|(a,b)\in U\right\}=% \sum_{(a,b)\in U}P\left\{\gamma\left[\alpha(a),\beta(b)\right]\right\}\cdot P% \left[(a,b)|U\right],

Rectangle_method.html

  1. ( a , b ) (a,b)
  2. N N
  3. h = ( b - a ) / N h=(b-a)/N
  4. x x
  5. N N
  6. a b f ( x ) d x h n = 0 N - 1 f ( x n ) \int_{a}^{b}f(x)\,dx\approx h\sum_{n=0}^{N-1}f(x_{n})
  7. h = ( b - a ) / N h=(b-a)/N
  8. x n = a + n h x_{n}=a+nh
  9. x n x_{n}
  10. x n x_{n}
  11. n n\to\infty
  12. f f
  13. ( a , b ) (a,b)
  14. i i^{\prime}
  15. n n
  16. f f
  17. ( a , a + Δ ) (a,a+\Delta)
  18. E i Δ 3 24 f ′′ ( ξ ) E_{i}\leq\frac{\Delta^{3}}{24}\,f^{\prime\prime}(\xi)
  19. ξ \xi
  20. ( a , a + Δ ) (a,a+\Delta)
  21. n n
  22. Δ \Delta
  23. n = 1 , 2 , 3 , n=1,2,3,\ldots
  24. n + 1 n+1
  25. E n Δ 3 24 f ′′ ( ξ ) E\leq\frac{n\Delta^{3}}{24}f^{\prime\prime}(\xi)
  26. n Δ = b - a n\Delta=b-a
  27. E ( b - a ) Δ 2 24 f ′′ ( ξ ) E\leq\frac{(b-a)\Delta^{2}}{24}f^{\prime\prime}(\xi)
  28. ξ \xi
  29. ( a , b ) (a,b)

Recursive_definition.html

  1. 0 + a = a 0+a=a
  2. ( 1 + n ) + a = 1 + ( n + a ) (1+n)+a=1+(n+a)
  3. 0 a = 0 0a=0
  4. ( 1 + n ) a = a + n a (1+n)a=a+na
  5. a 0 = 1 a^{0}=1
  6. a 1 + n = a a n a^{1+n}=aa^{n}
  7. ( a 0 ) = 1 {\left({{a}\atop{0}}\right)}=1
  8. ( 1 + a 1 + n ) = ( 1 + a ) ( a n ) 1 + n {\left({{1+a}\atop{1+n}}\right)}=\frac{(1+a){\left({{a}\atop{n}}\right)}}{1+n}

Reduction_(complexity).html

  1. a × b = ( ( a + b ) 2 - a 2 - b 2 ) 2 a\times b=\frac{\left(\left(a+b\right)^{2}-a^{2}-b^{2}\right)}{2}
  2. 2 \sqrt{2}
  3. f F . x . x A f ( x ) B \exists f\in F\mbox{ . }~{}\forall x\in\mathbb{N}\mbox{ . }~{}x\in A% \Leftrightarrow f(x)\in B
  4. A F B A\leq_{F}B
  5. s S . A P ( N ) . A s A S \forall s\in S\mbox{ . }~{}\forall A\in P(N)\mbox{ . }~{}A\leq s\Rightarrow A\in S
  6. s S . s A \forall s\in S\mbox{ . }~{}s\leq A

Reduction_(mathematics).html

  1. [ K 11 K 12 K 21 K 22 ] [ x 1 x 2 ] = [ F 1 F 2 ] \begin{bmatrix}K_{11}&K_{12}\\ K_{21}&K_{22}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}=\begin{bmatrix}F_{1}\\ F_{2}\end{bmatrix}
  2. [ K 11 , r e d u c e d ] [ x 1 ] = [ F 1 ] \begin{bmatrix}K_{11,reduced}\end{bmatrix}\begin{bmatrix}x_{1}\end{bmatrix}=% \begin{bmatrix}F_{1}\end{bmatrix}
  3. K 11 x 1 + K 12 x 2 = F 1 (Eq. 1) K_{11}x_{1}+K_{12}x_{2}=F_{1}\quad\,\text{(Eq. 1)}
  4. K 21 x 1 + K 22 x 2 = 0. (Eq. 2) K_{21}x_{1}+K_{22}x_{2}=0.\quad\,\text{(Eq. 2)}
  5. x 2 x_{2}
  6. K 22 K_{22}
  7. - K 22 - 1 K 21 x 1 = x 2 . -K_{22}^{-1}K_{21}x_{1}=x_{2}.
  8. K 11 x 1 - K 12 K 22 - 1 K 21 x 1 = F 1 . K_{11}x_{1}-K_{12}K_{22}^{-1}K_{21}x_{1}=F_{1}.
  9. K 11 , r e d u c e d = K 11 - K 12 K 22 - 1 K 21 . K_{11,reduced}=K_{11}-K_{12}K_{22}^{-1}K_{21}.

Reductive_group.html

  1. k k
  2. n n
  3. k k

Reductive_Lie_algebra.html

  1. 𝔤 = 𝔰 𝔞 ; \mathfrak{g}=\mathfrak{s}\oplus\mathfrak{a};
  2. 𝔤 𝔩 n \mathfrak{gl}_{n}
  3. n × n n\times n
  4. 𝔤 𝔩 ( V ) . \mathfrak{gl}(V).
  5. 𝔤 𝔩 n = 𝔰 𝔩 n 𝔨 , \mathfrak{gl}_{n}=\mathfrak{sl}_{n}\oplus\mathfrak{k},
  6. 𝔤 \mathfrak{g}
  7. 𝔤 \mathfrak{g}
  8. 𝔤 \mathfrak{g}
  9. 𝔤 \mathfrak{g}
  10. 𝔯 ( 𝔤 ) = 𝔷 ( 𝔤 ) . \mathfrak{r}(\mathfrak{g})=\mathfrak{z}(\mathfrak{g}).
  11. 𝔤 \mathfrak{g}
  12. 𝔰 0 \mathfrak{s}_{0}
  13. 𝔷 ( 𝔤 ) : \mathfrak{z}(\mathfrak{g}):
  14. 𝔤 = 𝔰 0 𝔷 ( 𝔤 ) . \mathfrak{g}=\mathfrak{s}_{0}\oplus\mathfrak{z}(\mathfrak{g}).
  15. 𝔤 \mathfrak{g}
  16. 𝔰 \mathfrak{s}
  17. 𝔞 \mathfrak{a}
  18. 𝔤 = 𝔰 𝔞 . \mathfrak{g}=\mathfrak{s}\oplus\mathfrak{a}.
  19. 𝔤 \mathfrak{g}
  20. 𝔤 = 𝔤 i . \mathfrak{g}=\textstyle{\sum\mathfrak{g}_{i}}.
  21. 𝔰 𝔞 \mathfrak{s}\oplus\mathfrak{a}
  22. 𝔞 , \mathfrak{a},
  23. 𝔤 = 𝔰 0 𝔷 ( 𝔤 ) . \mathfrak{g}=\mathfrak{s}_{0}\oplus\mathfrak{z}(\mathfrak{g}).
  24. 𝔨 \mathfrak{k}

Reeh–Schlieder_theorem.html

  1. 𝒜 ( 𝒪 ) \mathcal{A}(\mathcal{O})
  2. 𝒪 \mathcal{O}

Reflection_seismology.html

  1. Z = V ρ Z=V\rho
  2. R R
  3. R = ( Z 1 - Z 0 ) ( Z 1 + Z 0 ) R=\frac{(Z_{1}-Z_{0})}{(Z_{1}+Z_{0})}
  4. Z 0 Z_{0}
  5. Z 1 Z_{1}
  6. T = 2 Z 0 Z 1 ( Z 1 + Z 0 ) T=\frac{2Z_{0}Z_{1}}{(Z_{1}+Z_{0})}
  7. 1 - R 2 = ( Z 1 + Z 0 ) 2 - ( Z 1 - Z 0 ) 2 ( Z 1 + Z 0 ) 2 = 4 Z 0 Z 1 ( Z 1 + Z 0 ) 2 = T 2 1-R^{2}=\frac{(Z_{1}+Z_{0})^{2}-(Z_{1}-Z_{0})^{2}}{(Z_{1}+Z_{0})^{2}}=\frac{4Z% _{0}Z_{1}}{(Z_{1}+Z_{0})^{2}}=T^{2}
  8. R ( θ ) = R ( 0 ) + G sin 2 θ R(\theta)=R(0)+G\sin^{2}\theta
  9. R ( 0 ) R(0)
  10. G G
  11. ( θ ) (\theta)
  12. ( θ ) (\theta)
  13. t t
  14. t = 2 d V t=2\frac{d}{V}
  15. d d
  16. V V

Regression_analysis.html

  1. Y f ( 𝐗 , s y m b o l β ) Y\approx f(\mathbf{X},symbol{\beta})
  2. y i y_{i}
  3. n n
  4. x i x_{i}
  5. β 0 \beta_{0}
  6. β 1 \beta_{1}
  7. y i = β 0 + β 1 x i + ε i , i = 1 , , n . y_{i}=\beta_{0}+\beta_{1}x_{i}+\varepsilon_{i},\quad i=1,\dots,n.\!
  8. y i = β 0 + β 1 x i + β 2 x i 2 + ε i , i = 1 , , n . y_{i}=\beta_{0}+\beta_{1}x_{i}+\beta_{2}x_{i}^{2}+\varepsilon_{i},\ i=1,\dots,% n.\!
  9. x i x_{i}
  10. β 0 \beta_{0}
  11. β 1 \beta_{1}
  12. β 2 . \beta_{2}.
  13. ε i \varepsilon_{i}
  14. i i
  15. y i ^ = β ^ 0 + β ^ 1 x i . \widehat{y_{i}}=\widehat{\beta}_{0}+\widehat{\beta}_{1}x_{i}.
  16. e i = y i - y ^ i e_{i}=y_{i}-\widehat{y}_{i}
  17. y i ^ \widehat{y_{i}}
  18. y i y_{i}
  19. S S E = i = 1 n e i 2 . SSE=\sum_{i=1}^{n}e_{i}^{2}.\,
  20. β ^ 0 , β ^ 1 \widehat{\beta}_{0},\widehat{\beta}_{1}
  21. β 1 ^ = ( x i - x ¯ ) ( y i - y ¯ ) ( x i - x ¯ ) 2 and β 0 ^ = y ¯ - β 1 ^ x ¯ \widehat{\beta_{1}}=\frac{\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum(x_{i}-\bar{% x})^{2}}\,\text{ and }\hat{\beta_{0}}=\bar{y}-\widehat{\beta_{1}}\bar{x}
  22. x ¯ \bar{x}
  23. x x
  24. y ¯ \bar{y}
  25. y y
  26. σ ^ ε 2 = S S E n - 2 . \hat{\sigma}^{2}_{\varepsilon}=\frac{SSE}{n-2}.\,
  27. σ ^ β 0 = σ ^ ε 1 n + x ¯ 2 ( x i - x ¯ ) 2 \hat{\sigma}_{\beta_{0}}=\hat{\sigma}_{\varepsilon}\sqrt{\frac{1}{n}+\frac{% \bar{x}^{2}}{\sum(x_{i}-\bar{x})^{2}}}
  28. σ ^ β 1 = σ ^ ε 1 ( x i - x ¯ ) 2 . \hat{\sigma}_{\beta_{1}}=\hat{\sigma}_{\varepsilon}\sqrt{\frac{1}{\sum(x_{i}-% \bar{x})^{2}}}.
  29. y i = β 1 x i 1 + β 2 x i 2 + + β p x i p + ε i , y_{i}=\beta_{1}x_{i1}+\beta_{2}x_{i2}+\cdots+\beta_{p}x_{ip}+\varepsilon_{i},\,
  30. β 1 \beta_{1}
  31. ε i = y i - β ^ 1 x i 1 - - β ^ p x i p . \varepsilon_{i}=y_{i}-\hat{\beta}_{1}x_{i1}-\cdots-\hat{\beta}_{p}x_{ip}.
  32. i = 1 n k = 1 p X i j X i k β ^ k = i = 1 n X i j y i , j = 1 , , p . \sum_{i=1}^{n}\sum_{k=1}^{p}X_{ij}X_{ik}\hat{\beta}_{k}=\sum_{i=1}^{n}X_{ij}y_% {i},\ j=1,\dots,p.\,
  33. ( 𝐗 𝐗 ) 𝐬𝐲𝐦𝐛𝐨𝐥 β ^ = 𝐗 𝐘 , \mathbf{(X^{\top}X)\hat{symbol{\beta}}={}X^{\top}Y},\,
  34. β ^ \hat{\beta}
  35. β ^ j \hat{\beta}_{j}
  36. β ^ \hat{\beta}
  37. 𝐬𝐲𝐦𝐛𝐨𝐥 β ^ = ( 𝐗 𝐗 ) - 𝟏 𝐗 𝐘 . \mathbf{\hat{symbol{\beta}}={}(X^{\top}X)^{-1}X^{\top}Y}.\,
  38. N = m n N=m^{n}
  39. N N
  40. n n
  41. m m
  42. N N
  43. m m
  44. log 1000 log 5 = 4.29 \frac{\log{1000}}{\log{5}}=4.29

Reissner–Nordström_metric.html

  1. d s 2 = ( 1 - r S r + r Q 2 r 2 ) c 2 d t 2 - ( 1 - r S r + r Q 2 r 2 ) - 1 d r 2 - r 2 d Ω ( 2 ) 2 , ds^{2}=\left(1-\frac{r_{\mathrm{S}}}{r}+\frac{r_{Q}^{2}}{r^{2}}\right)c^{2}\,% dt^{2}-\left(1-\frac{r_{\mathrm{S}}}{r}+\frac{r_{Q}^{2}}{r^{2}}\right)^{-1}dr^% {2}-r^{2}\,d\Omega^{2}_{(2)},
  2. d Ω ( 2 ) 2 \textstyle d\Omega^{2}_{(2)}
  3. d Ω ( 2 ) 2 = d θ 2 + s i n 2 θ d ϕ 2 d\Omega^{2}_{(2)}=d\theta^{2}+sin^{2}\theta d\phi^{2}
  4. r s = 2 G M c 2 , r_{s}=\frac{2GM}{c^{2}},
  5. r Q 2 = Q 2 G 4 π ε 0 c 4 . r_{Q}^{2}=\frac{Q^{2}G}{4\pi\varepsilon_{0}c^{4}}.
  6. 0 = 1 / g r r = 1 - r S r + r Q 2 r 2 . 0=1/g^{rr}=1-\frac{r_{\mathrm{S}}}{r}+\frac{r_{Q}^{2}}{r^{2}}.
  7. r ± = 1 2 ( r s ± r s 2 - 4 r Q 2 ) . r_{\pm}=\frac{1}{2}\left(r_{s}\pm\sqrt{r_{s}^{2}-4r_{Q}^{2}}\right).
  8. A α = ( Q / r , 0 , 0 , 0 ) . A_{\alpha}=\left(Q/r,0,0,0\right).

Relative_angular_momentum.html

  1. 𝐇 𝟐 / 𝟏 \mathbf{H_{{2}/{1}}}\,\!
  2. m 2 m_{2}\,\!
  3. m 1 m_{1}\,\!
  4. m 2 m_{2}\,\!
  5. 𝐇 𝟐 / 𝟏 = 𝐫 × m 2 𝐯 \mathbf{H_{{2}/{1}}}=\mathbf{r}\times m_{2}\mathbf{v}\,\!
  6. 𝐫 \mathbf{r}\,\!
  7. 𝐯 \mathbf{v}\,\!
  8. m 2 m_{2}\,\!
  9. 𝐇 𝟐 / 𝟏 \mathbf{H_{{2}/{1}}}\,\!
  10. 𝐡 \mathbf{h}\,\!
  11. 𝐡 = 𝐇 𝟐 / 𝟏 / m 2 \mathbf{h}=\mathbf{H_{{2}/{1}}}/m_{2}\,\!
  12. m 2 m_{2}\,\!

Relativistic_beaming.html

  1. log ν = 3 \log\nu=3
  2. log ν = 7 \log\nu=7
  3. log S \log S
  4. log ν \log\nu
  5. S o = S e D p , S_{o}=S_{e}D^{p}\,,
  6. p = 3 - α . p=3-\alpha\,.
  7. D D
  8. c c
  9. D 2 . D^{2}\,.
  10. D 1 . D^{1}\,.
  11. 1 D α . \frac{1}{D^{\alpha}}\,.
  12. S ν 3 \frac{S}{\nu^{3}}
  13. θ \theta\,\!
  14. v j v_{j}\,\!
  15. S e S_{e}\,\!
  16. S o S_{o}\,\!
  17. α \alpha\,\!
  18. S ν α S\propto\nu^{\alpha}\,\!
  19. c = 2.9979 × 10 8 c\,\!=2.9979\times 10^{8}
  20. β = v j c \beta=\frac{v_{j}}{c}
  21. Γ = 1 1 - β 2 \Gamma=\frac{1}{\sqrt{1-\beta^{2}}}
  22. γ \gamma\,\!
  23. D = 1 Γ ( 1 - β cos θ ) D=\frac{1}{\Gamma(1-\beta\cos\theta)}

Relativistic_Euler_equations.html

  1. T μ ν T^{\mu\nu}
  2. μ T μ ν = 0. \nabla_{\mu}T^{\mu\nu}=0.
  3. T μ ν = ( e + p ) u μ u ν + p η μ ν . T^{\mu\nu}\,=(e+p)u^{\mu}u^{\nu}+p\eta^{\mu\nu}.
  4. e e
  5. p p
  6. u μ u^{\mu}
  7. η μ ν \eta^{\mu\nu}
  8. ( - , + , + , + ) (-,+,+,+)
  9. n n
  10. μ ( n u μ ) = 0. \nabla_{\mu}(nu^{\mu})=0.
  11. e e
  12. e = ρ c 2 + ρ e C e=\rho c^{2}+\rho e^{C}
  13. e C e^{C}
  14. S S
  15. S 2 = c 2 p e | adiabatic . S^{2}=c^{2}\left.\frac{\partial p}{\partial e}\right|_{\rm adiabatic}.
  16. e = ρ ( c 2 + e C ) e=\rho(c^{2}+e^{C})
  17. ρ \rho
  18. e / c 2 e/c^{2}

Relativistic_Heavy_Ion_Collider.html

  1. d n / d ϕ 1 + 2 v 2 ( p T ) cos 2 ϕ dn/d\phi\propto 1+2v_{2}(p_{\mathrm{T}})\cos 2\phi
  2. ϕ \phi
  3. Q s 2 N part / 2 Q_{s}^{2}\propto\langle N_{\mathrm{part}}\rangle/2
  4. n ch / A 1 / α s ( Q s 2 ) n_{\mathrm{ch}}/A\propto 1/\alpha_{s}(Q_{s}^{2})
  5. μ B \mu_{B}

Replicator_equation.html

  1. x i ˙ = x i [ f i ( x ) - ϕ ( x ) ] , ϕ ( x ) = j = 1 n x j f j ( x ) \dot{x_{i}}=x_{i}[f_{i}(x)-\phi(x)],\quad\phi(x)=\sum_{j=1}^{n}{x_{j}f_{j}(x)}
  2. x i x_{i}
  3. i i
  4. x = ( x 1 , , x n ) x=(x_{1},\ldots,x_{n})
  5. f i ( x ) f_{i}(x)
  6. i i
  7. ϕ ( x ) \phi(x)
  8. n n
  9. x x
  10. x i ˙ = x i ( ( A x ) i - x T A x ) , \dot{x_{i}}=x_{i}\left(\left(Ax\right)_{i}-x^{T}Ax\right),
  11. A A
  12. ( A x ) i \left(Ax\right)_{i}
  13. x T A x x^{T}Ax
  14. n n
  15. n - 1 n-1
  16. x i = y i 1 + j = 1 n - 1 y j i = 1 , , n - 1 x_{i}=\frac{y_{i}}{1+\sum_{j=1}^{n-1}{y_{j}}}\quad i=1,\ldots,n-1
  17. x n = 1 1 + j = 1 n - 1 y j , x_{n}=\frac{1}{1+\sum_{j=1}^{n-1}{y_{j}}},
  18. y i y_{i}
  19. x i ˙ = j = 1 n x j f j ( x ) Q j i - ϕ ( x ) x i , \dot{x_{i}}=\sum_{j=1}^{n}{x_{j}f_{j}(x)Q_{ji}}-\phi(x)x_{i},
  20. Q Q
  21. j j
  22. i i

Representable_functor.html

  1. u = Φ A ( id A ) . u=\Phi_{A}(\mathrm{id}_{A}).\,
  2. Φ X ( f ) = ( F f ) ( u ) \Phi_{X}(f)=(Ff)(u)\,
  3. Φ 1 - 1 Φ 2 = Hom ( φ , - ) \Phi_{1}^{-1}\circ\Phi_{2}=\mathrm{Hom}(\varphi,-)
  4. ( F φ ) u 1 = u 2 . (F\varphi)u_{1}=u_{2}.
  5. Φ X , Y : Hom 𝒟 ( F X , Y ) Hom 𝒞 ( X , G Y ) \Phi_{X,Y}\colon\mathrm{Hom}_{\mathcal{D}}(FX,Y)\to\mathrm{Hom}_{\mathcal{C}}(% X,GY)

Representation_of_a_Lie_superalgebra.html

  1. ( c 1 A + c 2 B ) X = c 1 A X + c 2 B X (c_{1}A+c_{2}B)\cdot X=c_{1}A\cdot X+c_{2}B\cdot X\,
  2. A ( c 1 X + c 2 Y ) = c 1 A X + c 2 A Y A\cdot(c_{1}X+c_{2}Y)=c_{1}A\cdot X+c_{2}A\cdot Y\,
  3. ( - 1 ) A X = ( - 1 ) A ( - 1 ) X (-1)^{A\cdot X}=(-1)^{A}(-1)^{X}\,
  4. [ A , B ] X = A ( B X ) - ( - 1 ) A B B ( A X ) . [A,B]\cdot X=A\cdot(B\cdot X)-(-1)^{AB}B\cdot(A\cdot X).\,

Representation_theory_of_diffeomorphism_groups.html

  1. C ( M ) C^{\infty}(M)
  2. I x ( M ) I_{x}(M)
  3. I x n ( M ) I_{x}^{n}(M)
  4. I x n ( M ) I_{x}^{n}(M)
  5. I x ( M ) / I x n ( M ) I_{x}(M)/I_{x}^{n}(M)
  6. G L + ( n , ) + × S L ( n , ) GL^{+}(n,\mathbb{R})\cong\mathbb{R}^{+}\times SL(n,\mathbb{R})

Representation_theory_of_finite_groups.html

  1. m = [ - 1 0 0 1 ] m=\begin{bmatrix}-1&0\\ 0&1\end{bmatrix}
  2. n = [ 0 1 1 0 ] n=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}
  3. G ^ , \widehat{G},
  4. ( ρ × σ ) ( g , h ) = ρ ( g ) σ ( h ) . (\rho\times\sigma)(g,h)=\rho(g)\cdot\sigma(h).
  5. ρ : x [ 0 1 - 1 0 ] . \rho:x\mapsto\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.
  6. [ 0 1 - 1 0 ] \left[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right]
  7. f : V ¯ V 𝐂 [ G ] ¯ f^{\prime}:\bar{V}\otimes V\rightarrow\overline{\mathbf{C}[G]}
  8. f ′′ : V ¯ V 𝐂 [ G ] f^{\prime\prime}:\bar{V}\otimes V\rightarrow\mathbf{C}[G]
  9. f ′′ ( x y ) = g G x , ρ ( g ) [ y ] g f^{\prime\prime}(x\otimes y)=\sum_{g\in G}\langle x,\rho(g)[y]\rangle g
  10. V ¯ \bar{V}
  11. g G f ( g ) h g k - 1 = g G f ( h - 1 g k ) g \sum_{g\in G}f(g)hgk^{-1}=\sum_{g\in G}f(h^{-1}gk)g
  12. f ′′ ( X Y ) = g G X , ρ ( g ) [ Y ] g = g G X , Y g - 1 g f^{\prime\prime}(X\otimes Y)=\sum_{g\in G}\langle X,\rho(g)[Y]\rangle g=\sum_{% g\in G}\langle X,Yg^{-1}\rangle g
  13. = g G X , g - 1 g Y = g G X , g - 1 ( g , e ) [ Y ] =\sum_{g\in G}\langle X,g^{-1}\rangle gY=\sum_{g\in G}\langle X,g^{-1}\rangle(% g,e)[Y]
  14. B ¯ B \bar{B}\otimes B
  15. V V ¯ G V \sum_{V}\bar{V}\otimes_{G}V
  16. χ ρ ( g ) = Tr ( ρ ( g ) ) \chi_{\rho}(g)=\mathrm{Tr}(\rho(g))\,
  17. Tr ( ρ ( g h g - 1 ) ) = Tr ( ρ ( g ) ρ ( h ) ρ ( g ) - 1 ) = Tr ( ρ ( h ) ) \mathrm{Tr}(\rho(ghg^{-1}))=\mathrm{Tr}(\rho(g)\rho(h)\rho(g)^{-1})=\mathrm{Tr% }(\rho(h))
  18. i = 1 p n i χ V i ( g ) = | G | δ g e \sum_{i=1}^{p}n_{i}\chi_{V_{i}}(g)=|G|\delta_{ge}
  19. i = 1 p χ V ¯ i ( g ) χ V i ( h ) = i = 1 p χ V i ( g - 1 ) χ V i ( h ) = i = 1 p χ V i ( g ) * χ V i ( h ) \sum_{i=1}^{p}\chi_{\bar{V}_{i}}(g)\chi_{V_{i}}(h)=\sum_{i=1}^{p}\chi_{V_{i}}(% g^{-1})\chi_{V_{i}}(h)=\sum_{i=1}^{p}\chi_{V_{i}}(g)^{*}\chi_{V_{i}}(h)
  20. h G Δ C [ G ] ( ( g , h k h - 1 ) ) \sum_{h\in G}\Delta_{C[G]}((g,hkh^{-1}))
  21. | G | | C i | δ i j = k = 1 p χ V k ( C i ) * χ V k ( C j ) \frac{|G|}{|C_{i}|}\delta_{ij}=\sum_{k=1}^{p}\chi_{V_{k}}(C_{i})^{*}\chi_{V_{k% }}(C_{j})
  22. g G ρ ( g ) \sum_{g\in G}\rho(g)
  23. g G g \sum_{g\in G}g
  24. g G ρ ( g ) \sum_{g\in G}\rho(g)
  25. V ¯ i V j \bar{V}_{i}\otimes V_{j}
  26. χ V ¯ i V j ( g ) = χ V ¯ i ( g ) χ V j ( g ) = χ V i ( g ) * χ V k ( g ) \chi_{\bar{V}_{i}\otimes V_{j}}(g)=\chi_{\bar{V}_{i}}(g)\chi_{V_{j}}(g)=\chi_{% V_{i}}(g)^{*}\chi_{V_{k}}(g)
  27. g G χ V i ( g ) * χ V j ( g ) = k | C k | χ V i ( C k ) * χ V j ( C k ) = | G | δ i j . \sum_{g\in G}\chi_{V_{i}}(g)^{*}\chi_{V_{j}}(g)=\sum_{k}|C_{k}|\chi_{V_{i}}(C_% {k})^{*}\chi_{V_{j}}(C_{k})=|G|\delta_{ij}.

Representation_theory_of_Hopf_algebras.html

  1. h v = Δ h ( v ( 1 ) v ( 2 ) ) = h ( 1 ) v ( 1 ) h ( 2 ) v ( 2 ) , hv=\Delta h(v_{(1)}\otimes v_{(2)})=h_{(1)}v_{(1)}\otimes h_{(2)}v_{(2)},
  2. Δ ( a b ) ( v ( 1 ) v ( 2 ) ) = ( a b ) v = a [ b [ v ] ] = Δ a [ Δ b ( v ( 1 ) v ( 2 ) ) ] = ( Δ a ) ( Δ b ) ( v ( 1 ) v ( 2 ) ) . \Delta(ab)(v_{(1)}\otimes v_{(2)})=(ab)v=a[b[v]]=\Delta a[\Delta b(v_{(1)}% \otimes v_{(2)})]=(\Delta a)(\Delta b)(v_{(1)}\otimes v_{(2)}).
  3. V 1 ( V 2 V 3 ) V_{1}\otimes(V_{2}\otimes V_{3})
  4. ( V 1 V 2 ) V 3 (V_{1}\otimes V_{2})\otimes V_{3}
  5. V 1 ( V 2 V 3 ) = ( V 1 V 2 ) V 3 V_{1}\otimes(V_{2}\otimes V_{3})=(V_{1}\otimes V_{2})\otimes V_{3}
  6. ( ( id Δ ) Δ h ) ( v ( 1 ) v ( 2 ) v ( 3 ) ) = h ( 1 ) v ( 1 ) h ( 2 ) ( 1 ) v ( 2 ) h ( 2 ) ( 2 ) v ( 3 ) = h v = ( ( Δ id ) Δ h ) ( v ( 1 ) v ( 2 ) v ( 3 ) ) . ((\operatorname{id}\otimes\Delta)\Delta h)(v_{(1)}\otimes v_{(2)}\otimes v_{(3% )})=h_{(1)}v_{(1)}\otimes h_{(2)(1)}v_{(2)}\otimes h_{(2)(2)}v_{(3)}=hv=((% \Delta\otimes\operatorname{id})\Delta h)(v_{(1)}\otimes v_{(2)}\otimes v_{(3)}).
  7. ( id Δ ) Δ A = ( Δ id ) Δ A . (\operatorname{id}\otimes\Delta)\Delta A=(\Delta\otimes\operatorname{id})% \Delta A.
  8. ( ε ( h ( 1 ) ) h ( 2 ) ) c v = h ( 1 ) c h ( 2 ) v = h ( c v ) = h ( c v ) = ( h ( 1 ) ε ( h ( 2 ) ) ) c v . (\varepsilon(h_{(1)})h_{(2)})cv=h_{(1)}c\otimes h_{(2)}v=h(c\otimes v)=h(cv)=(% h_{(1)}\varepsilon(h_{(2)}))cv.
  9. ε ( h ( 1 ) ) h ( 2 ) = h = h ( 1 ) ε ( h ( 2 ) ) \varepsilon(h_{(1)})h_{(2)}=h=h_{(1)}\varepsilon(h_{(2)})
  10. y , S ( h ) x = h y , x . \langle y,S(h)x\rangle=\langle hy,x\rangle.
  11. , \langle\cdot,\cdot\rangle
  12. φ : V V * ε H \varphi:V\otimes V^{*}\rightarrow\varepsilon_{H}
  13. φ ( h ( x y ) ) = φ ( x S ( h ( 1 ) ) h ( 2 ) y ) = φ ( S ( h ( 2 ) ) h ( 1 ) x y ) = h φ ( x y ) = ε ( h ) φ ( x y ) , \varphi\left(h(x\otimes y)\right)=\varphi\left(x\otimes S(h_{(1)})h_{(2)}y% \right)=\varphi\left(S(h_{(2)})h_{(1)}x\otimes y\right)=h\varphi(x\otimes y)=% \varepsilon(h)\varphi(x\otimes y),
  14. S ( h ( 1 ) ) h ( 2 ) = ε ( h ) = h ( 1 ) S ( h ( 2 ) ) S(h_{(1)})h_{(2)}=\varepsilon(h)=h_{(1)}S(h_{(2)})

Representation_theory_of_the_Poincaré_group.html

  1. ψ \psi
  2. γ μ \gamma_{\mu}
  3. ψ | ϕ = ψ ¯ ϕ = ψ γ 0 ϕ \langle\psi|\phi\rangle=\bar{\psi}\phi=\psi^{\dagger}\gamma_{0}\phi

Reproducing_kernel_Hilbert_space.html

  1. L x : f f ( x ) f H . L_{x}:f\mapsto f(x)\,\text{ }\forall f\in H.
  2. L x L_{x}
  3. f f
  4. H H
  5. L x L_{x}
  6. f f
  7. H H
  8. L x [ f ] = f ( x ) M f H f H . L_{x}[f]=f(x)\leq M\|f\|_{H}\,\text{ }\forall f\in H.\,
  9. f f
  10. K x K_{x}
  11. K x K_{x}
  12. f ( x ) = L x ( f ) = f , K x f H . f(x)=L_{x}(f)=\langle f,\ K_{x}\rangle\quad\forall f\in H.
  13. K x K_{x}
  14. K x ( y ) = K y , K x H . K_{x}(y)=\langle K_{y},\ K_{x}\rangle_{H}.
  15. K : X × X K:X\times X\to\mathbb{R}
  16. K ( x , y ) = K x ( y ) . K(x,y)=K_{x}(y).
  17. K : X × X K:X\times X\to\mathbb{R}
  18. i , j = 1 n c i c j K ( x i , x j ) 0 \sum_{i,j=1}^{n}c_{i}c_{j}K(x_{i},x_{j})\geq 0
  19. n , x 1 , , x n X , and c 1 , , c n . n\in\mathbb{N},x_{1},\dots,x_{n}\in X,\,\text{ and }c_{1},\dots,c_{n}\in% \mathbb{R}.
  20. H H
  21. H = { f L 2 ( ) | supp ( F ) [ - a , a ] , a < } H=\{f\in L_{2}(\mathbb{R})|\operatorname{supp}(F)\subset[-a,a],a<\infty\}
  22. F ( ω ) = f ( x ) e - i ω x d x F(\omega)=\int f(x)e^{-i\omega x}dx
  23. f f
  24. f H f\in H
  25. f ( x ) = 1 2 π - a a ϕ ( ω ) e i x ω d ω f(x)=\frac{1}{2\pi}\int_{-a}^{a}\phi(\omega)e^{ix\omega}d\omega
  26. ϕ L 2 [ - a , a ] \phi\in L_{2}[-a,a]
  27. | f ( x ) | 1 2 π - a a 1 d ω 1 2 π - a a | ϕ ( ω ) | 2 d ω = a π f . |f(x)|\leq\sqrt{\frac{1}{2\pi}\int_{-a}^{a}1d\omega}\sqrt{\frac{1}{2\pi}\int_{% -a}^{a}|\phi(\omega)|^{2}d\omega}=\sqrt{\frac{a}{\pi}}\|f\|.
  28. H H
  29. H H
  30. K x K_{x}
  31. K x ( y ) = a π sinc ( a ( y - x ) ) = sin ( a ( y - x ) ) π ( y - x ) K_{x}(y)=\frac{a}{\pi}\operatorname{sinc}(a(y-x))=\frac{\sin(a(y-x))}{\pi(y-x)}
  32. K x K_{x}
  33. K x K_{x}
  34. δ ( - x ) \delta(\cdot-x)
  35. j = 1 n b j K y j , i = 1 m a i K x i = i = 1 m j = 1 n a i b j K ( y j , x i ) . \left\langle\sum_{j=1}^{n}b_{j}K_{y_{j}},\sum_{i=1}^{m}a_{i}K_{x_{i}}\right% \rangle=\sum_{i=1}^{m}\sum_{j=1}^{n}{a_{i}}b_{j}K(y_{j},x_{i}).
  36. f ( x ) = i = 1 a i K x i ( x ) f(x)=\sum_{i=1}^{\infty}a_{i}K_{x_{i}}(x)
  37. i = 1 a i 2 K ( x i , x i ) < \sum_{i=1}^{\infty}a_{i}^{2}K(x_{i},x_{i})<\infty
  38. f , K x = i = 1 a i K x i , K x = i = 1 a i K ( x i , x ) = f ( x ) . \langle f,K_{x}\rangle=\left\langle\sum_{i=1}^{\infty}a_{i}K_{x_{i}},K_{x}% \right\rangle=\sum_{i=1}^{\infty}a_{i}K(x_{i},x)=f(x).
  39. K x , K y H = K ( x , y ) = K x , K y G . \langle K_{x},K_{y}\rangle_{H}=K(x,y)=\langle K_{x},K_{y}\rangle_{G}.\,
  40. , H = , G \langle\cdot,\cdot\rangle_{H}=\langle\cdot,\cdot\rangle_{G}
  41. K K
  42. X X
  43. μ \mu
  44. K : X × X K:X\times X\to\mathbb{R}
  45. T K : L 2 ( X ) L 2 ( X ) T_{K}:L_{2}(X)\rightarrow L_{2}(X)
  46. [ T K f ] ( ) = X K ( , t ) f ( t ) d μ ( t ) [T_{K}f](\cdot)=\int_{X}K(\cdot,t)f(t)\,d\mu(t)
  47. L 2 ( X ) L_{2}(X)
  48. μ \mu
  49. T K T_{K}
  50. K K
  51. K K
  52. T K T_{K}
  53. K K
  54. T k T_{k}
  55. ( σ i ) i 0 (\sigma_{i})_{i}\geq 0
  56. lim i σ i = 0 \lim_{i\to\infty}\sigma_{i}=0
  57. T K ϕ i ( x ) = σ i ϕ i ( x ) T_{K}\phi_{i}(x)=\sigma_{i}\phi_{i}(x)
  58. { ϕ i } \{\phi_{i}\}
  59. L 2 ( X ) L_{2}(X)
  60. T k T_{k}
  61. σ i > 0 i \sigma_{i}>0\,\text{ }\forall i
  62. T k T_{k}
  63. C ( X ) C(X)
  64. ϕ i C ( X ) i \phi_{i}\in C(X)\,\text{ }\forall i
  65. K K
  66. K ( x , y ) = j = 1 σ j ϕ j ( x ) ϕ j ( y ) K(x,y)=\sum_{j=1}^{\infty}\sigma_{j}\,\phi_{j}(x)\,\phi_{j}(y)
  67. x , y x,y
  68. X X
  69. lim n sup u , v | K ( u , v ) - j = 1 n σ j ϕ j ( u ) ϕ j ( v ) | = 0. \lim_{n\to\infty}\sup_{u,v}|K(u,v)-\sum_{j=1}^{n}\sigma_{j}\,\phi_{j}(u)\,\phi% _{j}(v)|=0.
  70. K K
  71. H H
  72. K K
  73. H = { f L 2 ( X ) | i = 1 f , ϕ i 2 σ i < } H=\left\{f{\in}L_{2}(X)\mathrel{\Bigg|}\sum_{i=1}^{\infty}\frac{\left\langle f% ,\phi_{i}\right\rangle^{2}}{\sigma_{i}}<\infty\right\}
  74. H H
  75. f , g H = i = 1 f , ϕ i L 2 g , ϕ i L 2 σ i . \left\langle f,g\right\rangle_{H}=\sum_{i=1}^{\infty}\frac{\left\langle f,\phi% _{i}\right\rangle_{L_{2}}\left\langle g,\phi_{i}\right\rangle_{L_{2}}}{\sigma_% {i}}.
  76. φ : X F \varphi:X\rightarrow F
  77. F F
  78. K K
  79. F F
  80. F = H F=H
  81. φ ( x ) = K x \varphi(x)=K_{x}
  82. x X x\in X
  83. F = 2 F=\ell^{2}
  84. φ ( x ) = ( σ i ϕ i ( x ) ) i \varphi(x)=(\sqrt{\sigma_{i}}\phi_{i}(x))_{i}
  85. H H
  86. H φ = { f : X | w F , f ( x ) = w , φ ( x ) F , x X } . H_{\varphi}=\{f:X\to\mathbb{R}|\exists w\in F,f(x)=\langle w,\varphi(x)\rangle% _{F},\forall\,\text{ }x\in X\}.
  87. H φ H_{\varphi}
  88. || f || φ = inf { || w || F : w F , f ( x ) = w , φ ( x ) F , x X } . ||f||_{\varphi}=\,\text{inf}\{||w||_{F}:w\in F,f(x)=\langle w,\varphi(x)% \rangle_{F},\forall\,\text{ }x\in X\}.
  89. H φ H_{\varphi}
  90. K ( x , y ) = φ ( x ) , φ ( y ) K(x,y)=\langle\varphi(x),\varphi(y)\rangle
  91. ( X i ) i = 1 p (X_{i})_{i=1}^{p}
  92. ( K i ) i = 1 p (K_{i})_{i=1}^{p}
  93. ( X i ) i = 1 p (X_{i})_{i=1}^{p}
  94. K ( ( x 1 , , x p ) , ( y 1 , , y p ) ) = K 1 ( x 1 , y 1 ) K p ( x p , y p ) K((x_{1},\dots,x_{p}),(y_{1},\dots,y_{p}))=K_{1}(x_{1},y_{1})\dots K_{p}(x_{p}% ,y_{p})
  95. X = X 1 × × X p X=X_{1}\times\dots\times X_{p}
  96. X 0 X X_{0}\subset X
  97. K K
  98. X 0 × X 0 X_{0}\times X_{0}
  99. K K
  100. K ( x , x ) = 1 K(x,x)=1
  101. x X x\in X
  102. d K ( x , y ) = || K x - K y || H 2 = 2 ( 1 - K ( x , y ) ) x X d_{K}(x,y)=||K_{x}-K_{y}||_{H}^{2}=2(1-K(x,y))\,\text{ }\forall x\in X
  103. K ( x , y ) 2 K ( x , x ) K ( y , y ) x , y X . K(x,y)^{2}\leq K(x,x)K(y,y)\,\text{ }\forall x,y\in X.
  104. K K
  105. x , y X x,y\in X
  106. K ( x , y ) K(x,y)
  107. x , y X x,y\in X
  108. K ( x , y ) K(x,y)
  109. { K x | x X } \{K_{x}|x\in X\}
  110. H H
  111. K ( x , y ) = x , y K(x,y)=\langle x,y\rangle
  112. K ( x , y ) = ( α x , y + 1 ) d , α , d K(x,y)=(\alpha\langle x,y\rangle+1)^{d},\alpha{\in}\mathbb{R},d{\in}\mathbb{N}
  113. K ( x , y ) = K ( x - y ) K(x,y)=K(\|x-y\|)
  114. K ( x , y ) = e - x - y 2 2 σ 2 , σ > 0 K(x,y)=e^{-\frac{\|x-y\|^{2}}{2\sigma^{2}}},\sigma>0
  115. K ( x , y ) = e - x - y σ , σ > 0 K(x,y)=e^{-\frac{\|x-y\|}{\sigma}},\sigma>0
  116. K ( x , y ) = 1 π 1 ( 1 - x y ¯ ) 2 . K(x,y)=\frac{1}{\pi}\frac{1}{(1-x\overline{y})^{2}}.
  117. f f
  118. L 2 ( ) L^{2}(\mathbb{R})
  119. π \pi
  120. K ( x , y ) = sin π ( x - y ) π ( x - y ) . K(x,y)=\frac{\sin\pi(x-y)}{\pi(x-y)}.
  121. Γ \Gamma
  122. x , y x,y
  123. X X
  124. f : X T f:X\to\mathbb{R}^{T}
  125. c T c\in\mathbb{R}^{T}
  126. x X x\in X
  127. Γ x c ( y ) = Γ ( x , y ) c H for y X \Gamma_{x}c(y)=\Gamma(x,y)c\in H\,\text{ for }y\in X
  128. < f , Γ x c > H = f ( x ) c . <f,\Gamma_{x}c>_{H}=f(x)^{\intercal}c.
  129. { Γ x c : x X , c T } \{\Gamma_{x}c:x\in X,c\in\mathbb{R}^{T}\}
  130. H H
  131. Λ = { 1 , , T } \Lambda=\{1,\dots,T\}
  132. X × Λ X\times\Lambda
  133. γ : X × Λ × X × Λ . \gamma:X\times\Lambda\times X\times\Lambda\to\mathbb{R}.
  134. { γ ( x , t ) : x X , t Λ } \{\gamma_{(x,t)}:x\in X,t\in\Lambda\}
  135. γ ( x , t ) ( y , s ) = γ ( ( x , t ) , ( y , s ) ) \ \gamma_{(x,t)}(y,s)=\gamma((x,t),(y,s))
  136. ( x , t ) , ( y , s ) X × Λ (x,t),(y,s)\in X\times\Lambda
  137. Γ ( x , y ) ( t , s ) = γ ( ( x , t ) , ( y , s ) ) . \Gamma(x,y)_{(t,s)}=\gamma((x,t),(y,s)).
  138. D : H Γ H γ D:H_{\Gamma}\to H_{\gamma}
  139. ( D f ) ( x , t ) = < f ( x ) , e t > T (Df)(x,t)=<f(x),e_{t}>_{\mathbb{R}^{T}}
  140. e t e_{t}
  141. t t h t^{th}
  142. T \mathbb{R}^{T}
  143. D D
  144. H Γ H_{\Gamma}
  145. H γ H_{\gamma}
  146. T T
  147. γ ( ( x , t ) , ( y , s ) ) = K ( x , y ) K T ( t , s ) \gamma((x,t),(y,s))=K(x,y)K_{T}(t,s)
  148. x , y x,y
  149. X X
  150. t , s t,s
  151. T T

Reserve_requirement.html

  1. M 1 = M B * m M_{1}=MB*m\,
  2. M 1 M_{1}
  3. m = ( 1 + c ) ( c + R ) = 1 + C D C D + R m=\frac{(1+c)}{(c+R)}=\frac{1+\frac{C}{D}}{\frac{C}{D}+R}
  4. M 1 M_{1}
  5. c = c=
  6. R = R=

Residue_numeral_system.html

  1. C = A ± B mod M C=A\pm B\mod M
  2. c i = a i ± b i mod m i c_{i}=a_{i}\pm b_{i}\mod m_{i}
  3. C = A B mod M , C=A\cdot B\mod M,
  4. c i = a i b i mod m i c_{i}=a_{i}\cdot b_{i}\mod m_{i}
  5. b i 0 b_{i}\not=0
  6. C = A B - 1 mod M C=A\cdot B^{-1}\mod M
  7. c i = a i b i - 1 mod m i c_{i}=a_{i}\cdot b_{i}^{-1}\mod m_{i}
  8. B - 1 B^{-1}
  9. b i - 1 b_{i}^{-1}
  10. b i b_{i}
  11. m i m_{i}
  12. X = Y Z X=Y\cdot Z
  13. m 1 = 2 , m 2 = 3 , m 3 = 5 , m_{1}=2,m_{2}=3,m_{3}=5,\dots
  14. Y > m N Y>m_{N}
  15. Z > m N Z>m_{N}
  16. x i = y i z i x_{i}=y_{i}\cdot z_{i}
  17. x i 0 x_{i}\not=0
  18. y i = 0 y_{i}=0
  19. z i = 0 z_{i}=0
  20. i = 1 N ( m i - 1 ) = M i = 1 N ( 1 - 1 m i ) \prod_{i=1}^{N}(m_{i}-1)=M\prod_{i=1}^{N}\left(1-\frac{1}{m_{i}}\right)
  21. m i m_{i}
  22. { x 1 , x 2 , x 3 , , x n } \{x_{1},x_{2},x_{3},\ldots,x_{n}\}
  23. X = i = 1 N x i M i - 1 = x 1 + m 1 ( x 2 + m 2 ( + m N - 1 x N ) ) , X=\sum_{i=1}^{N}x_{i}M_{i-1}=x_{1}+m_{1}(x_{2}+m_{2}(\cdots+m_{N-1}x_{N})% \cdots),
  24. M 0 = 1 , M i = j = 1 i m j M_{0}=1,M_{i}=\prod_{j=1}^{i}m_{j}
  25. i > 0 i>0
  26. 0 x i < m i . 0\leq x_{i}<m_{i}.

Resistance_distance.html

  1. Ω i , j := Γ i , i + Γ j , j - Γ i , j - Γ j , i \Omega_{i,j}:=\Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i}\,
  2. Ω i , j = 0. \Omega_{i,j}=0.\,
  3. Ω i , j = Ω j , i = Γ i , i + Γ j , j - 2 Γ i , j \Omega_{i,j}=\Omega_{j,i}=\Gamma_{i,i}+\Gamma_{j,j}-2\Gamma_{i,j}\,
  4. i , j V ( L M L ) i , j Ω i , j = - 2 tr ( M L ) \sum_{i,j\in V}(LML)_{i,j}\Omega_{i,j}=-2\operatorname{tr}(ML)\,
  5. ( i , j ) E Ω i , j = N - 1 \sum_{(i,j)\in E}\Omega_{i,j}=N-1
  6. i < j V Ω i , j = N k = 1 N - 1 λ k - 1 \sum_{i<j\in V}\Omega_{i,j}=N\sum_{k=1}^{N-1}\lambda_{k}^{-1}
  7. λ k \lambda_{k}
  8. Ω i , j = { | { t : t T , e i , j t } | | T | , ( i , j ) E | T - T | | T | , ( i , j ) E \Omega_{i,j}=\begin{cases}\frac{\left|\{t:t\in T,e_{i,j}\in t\}\right|}{\left|% T\right|},&(i,j)\in E\\ \frac{\left|T^{\prime}-T\right|}{\left|T\right|},&(i,j)\not\in E\end{cases}
  9. T T^{\prime}
  10. G = ( V , E + e i , j ) G^{\prime}=(V,E+e_{i,j})
  11. L L
  12. Γ \Gamma
  13. K K
  14. Γ = K K T \Gamma=KK^{T}
  15. Ω i , j = Γ i , i + Γ j , j - Γ i , j - Γ j , i = K i K i T + K j K j T - K i K j T - K j K i T = ( K i - K j ) 2 \Omega_{i,j}=\Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i}=K_{i}K_{i}^{T% }+K_{j}K_{j}^{T}-K_{i}K_{j}^{T}-K_{j}K_{i}^{T}=(K_{i}-K_{j})^{2}
  16. K K
  17. n + 1 n+1
  18. i i
  19. n + 1 n+1
  20. i = 1 , 2 , 3 , n , i=1,2,3,...n,
  21. i i
  22. i + 1 i+1
  23. i = 1 , 2 , 3 , , n - 1. i=1,2,3,...,n-1.
  24. n + 1 n+1
  25. i { 1 , 2 , 3 , , n } i\in\{1,2,3,...,n\}
  26. F 2 ( n - i ) + 1 F 2 i - 1 F 2 n \frac{F_{2(n-i)+1}F_{2i-1}}{F_{2n}}
  27. F j F_{j}
  28. j j
  29. j 0 j\geq 0

Resolved_sideband_cooling.html

  1. 10 14 δ ν / ν 10^{14}\delta\nu/\nu
  2. ω = ω 0 - ν \omega=\omega_{0}-\nu
  3. ω 0 \omega_{0}
  4. ν \nu
  5. | g , n | e , n - 1 |g,n\rangle\rightarrow|e,n-1\rangle
  6. | a , m |a,m\rangle
  7. | e , n - 1 | g , n - 1 . |e,n-1\rangle\rightarrow|g,n-1\rangle.
  8. | g , 0 |g,0\rangle
  9. 2 π c / ω 0 2\pi c/\omega_{0}
  10. H = H H O + H A L H=H_{HO}+H_{AL}
  11. H H O = ν ( n + 1 2 ) H_{HO}=\hbar\nu\left(n+\frac{1}{2}\right)
  12. H A L = - Δ | e e | + Ω 2 ( | e g | e i 𝐤 𝐫 + | g e | e - i 𝐤 𝐫 ) H_{AL}=-\hbar\Delta\left|e\right\rangle\left\langle e\right|+\hbar\frac{\Omega% }{2}\left(\left|e\right\rangle\left\langle g\right|e^{i\mathbf{k}\cdot\mathbf{% r}}+\left|g\right\rangle\left\langle e\right|e^{-i\mathbf{k}\cdot\mathbf{r}}\right)
  13. n n
  14. ν \nu
  15. Ω \Omega
  16. Δ \Delta
  17. ω 0 \omega_{0}
  18. 𝐤 \mathbf{k}
  19. m , n m,n
  20. | m | e i 𝐤 𝐫 | n | 2 \left|\left\langle m\right|e^{i\mathbf{k}\cdot\mathbf{r}}\left|n\right\rangle% \right|^{2}
  21. n n
  22. { | g , n , | e , n } \{\left|g,n\right\rangle,\left|e,n\right\rangle\}
  23. | m | e i 𝐤 𝐫 | n | \left|\left\langle m\right|e^{i\mathbf{k}\cdot\mathbf{r}}\left|n\right\rangle\right|
  24. ω 0 \omega_{0}
  25. Γ ν \Gamma\ll\nu
  26. ω 0 - q ν , q { 1 , 2 , 3 , . . } \omega_{0}-q\nu,q\in\{1,2,3,..\}
  27. | g , n \left|g,n\right\rangle
  28. | e , n - q \left|e,n-q\right\rangle
  29. ω 0 \omega_{0}
  30. q q
  31. n ¯ = R q 1 / q 1 - R q 1 / q \bar{n}=\frac{R_{q}^{1/q}}{1-R_{q}^{1/q}}
  32. R q R_{q}
  33. q q
  34. n ¯ ( Γ / ν ) 2 1 \bar{n}\approx(\Gamma/\nu)^{2}\ll 1
  35. n ¯ \bar{n}
  36. σ - \sigma^{-}
  37. σ - \sigma^{-}
  38. 10 6 10^{6}

Resting_potential.html

  1. E m = g K + g t o t E K + + g N a + g t o t E N a + + g C l - g t o t E C l - E_{m}=\frac{g_{K^{+}}}{g_{tot}}E_{K^{+}}+\frac{g_{Na^{+}}}{g_{tot}}E_{Na^{+}}+% \frac{g_{Cl^{-}}}{g_{tot}}E_{Cl^{-}}
  2. E e q , K + = R T z F ln [ K + ] o [ K + ] i , E_{eq,K^{+}}=\frac{RT}{zF}\ln\frac{[K^{+}]_{o}}{[K^{+}]_{i}},
  3. E e q , K + = 61.54 m V log [ K + ] o [ K + ] i , E_{eq,K^{+}}=61.54mV\log\frac{[K^{+}]_{o}}{[K^{+}]_{i}},
  4. E m = R T F ln ( P N a + [ N a + ] o + P K + [ K + ] o + P C l - [ C l - ] i P N a + [ N a + ] i + P K + [ K + ] i + P C l - [ C l - ] o ) E_{m}=\frac{RT}{F}\ln{\left(\frac{P_{Na^{+}}[Na^{+}]_{o}+P_{K^{+}}[K^{+}]_{o}+% P_{Cl^{-}}[Cl^{-}]_{i}}{P_{Na^{+}}[Na^{+}]_{i}+P_{K^{+}}[K^{+}]_{i}+P_{Cl^{-}}% [Cl^{-}]_{o}}\right)}
  5. E m = g K + E e q , K + + g N a + E e q , N a + + g C l - E e q , C l - g K + + g N a + + g C l - E_{m}=\frac{g_{K^{+}}E_{eq,K^{+}}+g_{Na^{+}}E_{eq,Na^{+}}+g_{Cl^{-}}E_{eq,Cl^{% -}}}{g_{K^{+}}+g_{Na^{+}}+g_{Cl^{-}}}
  6. E m = g K + g t o t E e q , K + + g N a + g t o t E e q , N a + + g C l - g t o t E e q , C l - E_{m}=\frac{g_{K^{+}}}{g_{tot}}E_{eq,K^{+}}+\frac{g_{Na^{+}}}{g_{tot}}E_{eq,Na% ^{+}}+\frac{g_{Cl^{-}}}{g_{tot}}E_{eq,Cl^{-}}

Restricted_representation.html

  1. 𝐟 : f 1 f 2 f N \mathbf{f}\,\colon\,f_{1}\geq f_{2}\geq\cdots\geq f_{N}
  2. Tr π 𝐟 ( U ) = det z j f i + N - i i < j ( z i - z j ) . \mathrm{Tr}\,\pi_{\mathbf{f}}(U)={\mathrm{det}\,z_{j}^{f_{i}+N-i}\over\prod_{i% <j}(z_{i}-z_{j})}.
  3. π 𝐟 | U ( N - 1 ) = f 1 g 1 f 2 g 2 f N - 1 g N - 1 f N π 𝐠 \pi_{\mathbf{f}}|_{U(N-1)}=\bigoplus_{f_{1}\geq g_{1}\geq f_{2}\geq g_{2}\geq% \cdots\geq f_{N-1}\geq g_{N-1}\geq f_{N}}\pi_{\mathbf{g}}
  4. ( q 1 , , q N ) ( r 1 , , r N ) = r i * q i (q_{1},\ldots,q_{N})\cdot(r_{1},\ldots,r_{N})=\sum r_{i}^{*}q_{i}
  5. q i j = ( α i j β i j - β ¯ i j α ¯ i j ) , q_{ij}=\begin{pmatrix}\alpha_{ij}&\beta_{ij}\\ -\overline{\beta}_{ij}&\overline{\alpha}_{ij}\end{pmatrix},
  6. q i = ( z i 0 0 z ¯ i ) , q_{i}=\begin{pmatrix}z_{i}&0\\ 0&\overline{z}_{i}\end{pmatrix},
  7. 𝐟 : f 1 f 2 f N 0 \mathbf{f}\,\colon\,f_{1}\geq f_{2}\geq\cdots\geq f_{N}\geq 0
  8. Tr σ 𝐟 ( U ) = det z j f i + N - i + 1 - z j - f i - N + i - 1 ( z i - z i - 1 ) i < j ( z i + z i - 1 - z j - z j - 1 ) . \mathrm{Tr}\,\sigma_{\mathbf{f}}(U)={\mathrm{det}\,z_{j}^{f_{i}+N-i+1}-z_{j}^{% -f_{i}-N+i-1}\over\prod(z_{i}-z_{i}^{-1})\cdot\prod_{i<j}(z_{i}+z_{i}^{-1}-z_{% j}-z_{j}^{-1})}.
  9. σ 𝐟 | Sp ( N - 1 ) = f i g i f i + 2 m ( 𝐟 , 𝐠 ) σ 𝐠 \sigma_{\mathbf{f}}|_{\mathrm{Sp}(N-1)}=\bigoplus_{f_{i}\geq g_{i}\geq f_{i+2}% }m(\mathbf{f},\mathbf{g})\sigma_{\mathbf{g}}
  10. m ( 𝐟 , 𝐠 ) = i = 1 N ( a i - b i + 1 ) m(\mathbf{f},\mathbf{g})=\prod_{i=1}^{N}(a_{i}-b_{i}+1)
  11. a 1 b 1 a 2 b 2 a N b N = 0 a_{1}\geq b_{1}\geq a_{2}\geq b_{2}\geq\cdots\geq a_{N}\geq b_{N}=0
  12. f 1 f 2 f N 0 Tr Π 𝐟 , 0 ( z 1 , z 1 - 1 , , z N , z N - 1 ) Tr π 𝐟 ( t 1 , , t N ) = f 1 f 2 f N 0 Tr σ 𝐟 ( z 1 , , z N ) Tr π 𝐟 ( t 1 , , t N ) i < j ( 1 - z i z j ) - 1 , \sum_{f_{1}\geq f_{2}\geq f_{N}\geq 0}\mathrm{Tr}\Pi_{\mathbf{f},0}(z_{1},z_{1% }^{-1},\ldots,z_{N},z_{N}^{-1})\cdot\mathrm{Tr}\pi_{\mathbf{f}}(t_{1},\ldots,t% _{N})=\sum_{f_{1}\geq f_{2}\geq f_{N}\geq 0}\mathrm{Tr}\sigma_{\mathbf{f}}(z_{% 1},\ldots,z_{N})\cdot\mathrm{Tr}\pi_{\mathbf{f}}(t_{1},\ldots,t_{N})\cdot\prod% _{i<j}(1-z_{i}z_{j})^{-1},
  13. i < j ( 1 - z i z j ) - 1 = f 2 i - 1 = f 2 i Tr π f ( z 1 , , z N ) , \prod_{i<j}(1-z_{i}z_{j})^{-1}=\sum_{f_{2i-1}=f_{2i}}\mathrm{Tr}\,\pi_{f}(z_{1% },\ldots,z_{N}),
  14. Π 𝐟 , 0 | Sp ( N ) = 𝐡 , 𝐠 , g 2 i - 1 = g 2 i M ( 𝐠 , 𝐡 ; 𝐟 ) σ 𝐡 \Pi_{\mathbf{f},0}|_{\mathrm{Sp}(N)}=\bigoplus_{\mathbf{h},\,\,\mathbf{g},\,\,% g_{2i-1}=g_{2i}}M(\mathbf{g},\mathbf{h};\mathbf{f})\sigma_{\mathbf{h}}
  15. \otimes
  16. Π 𝐟 | Sp ( N ) = 𝐡 , 𝐠 , g 2 i - 1 = g 2 i M N ( 𝐠 , 𝐡 ; 𝐟 ) σ 𝐡 \Pi_{\mathbf{f}}|_{\mathrm{Sp}(N)}=\bigoplus_{\mathbf{h},\,\,\mathbf{g},\,\,g_% {2i-1}=g_{2i}}M_{N}(\mathbf{g},\mathbf{h};\mathbf{f})\sigma_{\mathbf{h}}
  17. f 1 f 2 f n - 1 | f n | f_{1}\geq f_{2}\geq\cdots\geq f_{n-1}\geq|f_{n}|
  18. f 1 f 2 f n 0 f_{1}\geq f_{2}\geq\cdots\geq f_{n}\geq 0
  19. Tr π 𝐟 ( U ) = det ( z j f i + n - i + z j - f i - n + i ) i < j ( z i + z i - 1 - z j - z j - 1 ) \mathrm{Tr}\,\pi_{\mathbf{f}}(U)={\mathrm{det}\,(z_{j}^{f_{i}+n-i}+z_{j}^{-f_{% i}-n+i})\over\prod_{i<j}(z_{i}+z_{i}^{-1}-z_{j}-z_{j}^{-1})}
  20. Tr π 𝐟 ( U ) = det ( z j f i + 1 / 2 + n - i - z j - f i - 1 / 2 - n + i ) i < j ( z i + z i - 1 - z j - z j - 1 ) k ( z k 1 / 2 - z k - 1 / 2 ) \mathrm{Tr}\,\pi_{\mathbf{f}}(U)={\mathrm{det}\,(z_{j}^{f_{i}+1/2+n-i}-z_{j}^{% -f_{i}-1/2-n+i})\over\prod_{i<j}(z_{i}+z_{i}^{-1}-z_{j}-z_{j}^{-1})\cdot\prod_% {k}(z_{k}^{1/2}-z_{k}^{-1/2})}
  21. π 𝐟 | S O ( 2 n ) = f 1 g 1 f 2 g 2 f n - 1 g n - 1 f n | g n | π 𝐠 \pi_{\mathbf{f}}|_{SO(2n)}=\bigoplus_{f_{1}\geq g_{1}\geq f_{2}\geq g_{2}\geq% \cdots\geq f_{n-1}\geq g_{n-1}\geq f_{n}\geq|g_{n}|}\pi_{\mathbf{g}}
  22. π 𝐟 | S O ( 2 n - 1 ) = f 1 g 1 f 2 g 2 f n - 1 g n - 1 | f n | π 𝐠 \pi_{\mathbf{f}}|_{SO(2n-1)}=\bigoplus_{f_{1}\geq g_{1}\geq f_{2}\geq g_{2}% \geq\cdots\geq f_{n-1}\geq g_{n-1}\geq|f_{n}|}\pi_{\mathbf{g}}
  23. 𝔤 𝔩 \mathfrak{gl}
  24. \rightarrow
  25. 𝔤 𝔩 \mathfrak{gl}
  26. 𝔤 𝔩 \mathfrak{gl}
  27. 𝔤 𝔩 \mathfrak{gl}
  28. 𝔤 𝔩 \mathfrak{gl}

Return_on_capital.html

  1. R O C = Net Operating Profit - Adjusted Taxes Book Value Of Debt + Book Value Of Equity - Cash ROC=\frac{\textrm{Net Operating Profit}-\textrm{Adjusted Taxes}}{\textrm{Book % Value Of Debt}+\textrm{Book Value Of Equity}-\textrm{Cash}}
  2. R O I C = Net Operating Profit - Adjusted Taxes Invested Capital ROIC=\frac{\textrm{Net Operating Profit}-\textrm{Adjusted Taxes}}{\textrm{% Invested Capital}}

Return_period.html

  1. n + 1 m {n+1\over m}
  2. 1 / 10 = 0.1 1/10=0.1
  3. P r ( t ) = ( μ t ) r r ! e - μ t P_{r}(t)={(\mu t)^{r}\over r!}e^{-\mu t}
  4. r r
  5. t t
  6. μ \mu
  7. μ \mu
  8. P r ( t ) = ( μ t ) r r ! e - μ t P_{r}(t)={(\mu t)^{r}\over r!}e^{-\mu t}
  9. P 0 ( 10 ) = ( 0.0042 × 10 ) 0 0 ! e - 0.0042 × 10 = 0.958 P_{0}(10)={(0.0042\times 10)^{0}\over 0!}e^{-0.0042\times 10}=0.958
  10. μ \mu
  11. P r ( X = r ) = ( n r ) μ r ( 1 - μ ) n - r P_{r}(X=r)={n\choose r}\mu^{r}(1-\mu)^{n-r}
  12. n , μ 0 n\rightarrow\infty,\mu\rightarrow 0
  13. n μ λ n\mu\rightarrow\lambda
  14. n ! ( n - r ) ! r ! μ r ( 1 - μ ) n - r e - λ λ r r ! . \frac{n!}{(n-r)!r!}\mu^{r}(1-\mu)^{n-r}\rightarrow e^{-\lambda}\frac{\lambda^{% r}}{r!}.
  15. μ = 1 / T = m n + 1 \mu=1/T={m\over n+1}
  16. p = 1 100 = 0.01 p={1\over 100}=0.01
  17. P = ( 10 1 ) × 0.01 1 × 0.99 9 P={\left({{10}\atop{1}}\right)}\times 0.01^{1}\times 0.99^{9}
  18. 10 × 0.01 × 0.914 \approx 10\times 0.01\times 0.914
  19. 0.0914 \approx 0.0914\,
  20. R ¯ = 1 - ( 1 - 1 T ) n = 1 - ( 1 - p ( X x T ) ) n \overline{R}=1-(1-{1\over T})^{n}=1-(1-p(X\geq{x_{T}}))^{n}
  21. 1 T = p ( X x T ) {1\over T}=p(X\geq{x_{T}})

Returns_to_scale.html

  1. F ( K , L ) \ F(K,L)
  2. F ( a K , a L ) = a F ( K , L ) \ F(aK,aL)=aF(K,L)
  3. F ( a K , a L ) > a F ( K , L ) , \ F(aK,aL)>aF(K,L),
  4. F ( a K , a L ) < a F ( K , L ) \ F(aK,aL)<aF(K,L)
  5. T \ T
  6. T \ T
  7. a T = T , a > 0 \ aT=T,\forall a>0
  8. T \ T
  9. F ( K , L ) = A K b L 1 - b \ F(K,L)=AK^{b}L^{1-b}
  10. A > 0 A>0
  11. 0 < b < 1 0<b<1
  12. F ( a K , a L ) = A ( a K ) b ( a L ) 1 - b = A a b a 1 - b K b L 1 - b = a A K b L 1 - b = a F ( K , L ) . \ F(aK,aL)=A(aK)^{b}(aL)^{1-b}=Aa^{b}a^{1-b}K^{b}L^{1-b}=aAK^{b}L^{1-b}=aF(K,L).
  13. F ( K , L ) = A K b L c \ F(K,L)=AK^{b}L^{c}
  14. 0 < c < 1 , 0<c<1,
  15. a F ( K , L ) aF(K,L)

Reynolds_decomposition.html

  1. u \scriptstyle u
  2. u ( x , y , z , t ) = u ( x , y , z ) ¯ + u ( x , y , z , t ) u(x,y,z,t)=\overline{u(x,y,z)}+u^{\prime}(x,y,z,t)\,
  3. u ¯ \scriptstyle\overline{u}
  4. u \scriptstyle u\,
  5. u u^{\prime}\,

Rhombic_triacontahedron.html

  1. S = a 2 12 5 26.8328 a 2 S=a^{2}\cdot 12\sqrt{5}\approx 26.8328\cdot a^{2}
  2. V = a 3 4 5 + 2 5 12.3107 a 3 V=a^{3}\cdot 4\sqrt{5+2\sqrt{5}}\approx 12.3107\cdot a^{3}
  3. r i = a φ 2 1 + φ 2 = a 1 + 2 5 1.37638 a r_{i}=a\cdot\frac{\varphi^{2}}{\sqrt{1+\varphi^{2}}}=a\cdot\sqrt{1+\frac{2}{% \sqrt{5}}}\approx 1.37638\cdot a
  4. r m = a ( 1 + 1 5 ) 1.44721 a r_{m}=a\cdot\left(1+\frac{1}{\sqrt{5}{}}\right)\approx 1.44721\cdot a

Ribet's_theorem.html

  1. ρ f , p ρ g , p . \rho_{f,p}\simeq\rho_{g,p}.
  2. \mathbb{Q}
  3. E : y 2 + x y + y = x 3 - 663204 x + 206441595 E:y^{2}+xy+y=x^{3}-663204x+206441595
  4. a p + b p = c p , a^{p}+b^{p}=c^{p},
  5. y 2 = x ( x - a p ) ( x + b p ) . y^{2}=x(x-a^{p})(x+b^{p}).
  6. \mathbb{Q}
  7. \mathbb{Q}

Richard_P._Brent.html

  1. π \pi
  2. π \pi
  3. γ \gamma
  4. γ \gamma
  5. x 6972593 + x 3037958 + 1. x^{6972593}+x^{3037958}+1.
  6. x 43112609 + x 3569337 + 1. x^{43112609}+x^{3569337}+1.

Richardson_extrapolation.html

  1. A * A^{*}
  2. A ( h ) A(h)
  3. h h
  4. A ( h ) = A + C h n + o ( h n + 1 ) A(h)=A^{\ast}+Ch^{n}+o(h^{n+1})\;
  5. R ( h , k ) := k n A ( h ) - A ( k h ) k n - 1 R(h,k):=\frac{k^{n}A(h)-A(k\,h)}{k^{n}-1}
  6. R ( h , k ) = k n ( A * + C h n + o ( h n + 1 ) ) - ( A * + C k n h n + o ( h n + 1 ) ) k n - 1 = A * + o ( h n + 1 ) . R(h,k)=\frac{k^{n}(A^{*}+Ch^{n}+o(h^{n+1}))-(A^{*}+Ck^{n}h^{n}+o(h^{n+1}))}{k^% {n}-1}=A^{*}+o(h^{n+1}).
  7. R ( h , k ) R(h,k)
  8. o ( h n + 1 ) o(h^{n+1})
  9. A ( h ) A(h)
  10. A - A ( h ) = a 0 h k 0 + a 1 h k 1 + a 2 h k 2 + A-A(h)=a_{0}h^{k_{0}}+a_{1}h^{k_{1}}+a_{2}h^{k_{2}}+\cdots
  11. A = A ( h ) + a 0 h k 0 + a 1 h k 1 + a 2 h k 2 + A=A(h)+a_{0}h^{k_{0}}+a_{1}h^{k_{1}}+a_{2}h^{k_{2}}+\cdots
  12. A = A ( h ) + a 0 h k 0 + O ( h k 1 ) . A=A(h)+a_{0}h^{k_{0}}+O(h^{k_{1}}).\,\!
  13. A = A ( h ) + a 0 h k 0 + O ( h k 1 ) A=A(h)+a_{0}h^{k_{0}}+O(h^{k_{1}})\,\!
  14. A = A ( h t ) + a 0 ( h t ) k 0 + O ( h k 1 ) . A=A\!\left(\frac{h}{t}\right)+a_{0}\left(\frac{h}{t}\right)^{k_{0}}+O(h^{k_{1}% }).
  15. ( t k 0 - 1 ) A = t k 0 A ( h t ) - A ( h ) + O ( h k 1 ) (t^{k_{0}}-1)A=t^{k_{0}}A\left(\frac{h}{t}\right)-A(h)+O(h^{k_{1}})
  16. A = t k 0 A ( h t ) - A ( h ) t k 0 - 1 + O ( h k 1 ) . A=\frac{t^{k_{0}}A\left(\frac{h}{t}\right)-A(h)}{t^{k_{0}}-1}+O(h^{k_{1}}).
  17. A i + 1 ( h ) = t k i A i ( h t ) - A i ( h ) t k i - 1 A_{i+1}(h)=\frac{t^{k_{i}}A_{i}\left(\frac{h}{t}\right)-A_{i}(h)}{t^{k_{i}}-1}
  18. A = A i + 1 ( h ) + O ( h k i + 1 ) A=A_{i+1}(h)+O(h^{k_{i+1}})
  19. A 0 = A ( h ) A_{0}=A(h)
  20. A = t k 0 A ( h t ) - A ( h ) t k 0 - 1 + O ( h k 1 ) = s k 0 A ( h s ) - A ( h ) s k 0 - 1 + O ( h k 1 ) A=\frac{t^{k_{0}}A\left(\frac{h}{t}\right)-A(h)}{t^{k_{0}}-1}+O(h^{k_{1}})=% \frac{s^{k_{0}}A\left(\frac{h}{s}\right)-A(h)}{s^{k_{0}}-1}+O(h^{k_{1}})
  21. A ( h t ) + A ( h t ) - A ( h ) t k 0 - 1 A ( h s ) + A ( h s ) - A ( h ) s k 0 - 1 A\left(\frac{h}{t}\right)+\frac{A\left(\frac{h}{t}\right)-A(h)}{t^{k_{0}}-1}% \approx A\left(\frac{h}{s}\right)+\frac{A\left(\frac{h}{s}\right)-A(h)}{s^{k_{% 0}}-1}
  22. f ( x + h ) = f ( x ) + f ( x ) h + f ′′ ( x ) 2 h 2 + f(x+h)=f(x)+f^{\prime}(x)h+\frac{f^{\prime\prime}(x)}{2}h^{2}+\cdots
  23. f ( x ) = f ( x + h ) - f ( x ) h - f ′′ ( x ) 2 h + . f^{\prime}(x)=\frac{f(x+h)-f(x)}{h}-\frac{f^{\prime\prime}(x)}{2}h+\cdots.
  24. A 0 ( h ) = f ( x + h ) - f ( x ) h A_{0}(h)=\frac{f(x+h)-f(x)}{h}
  25. A = 2 A 0 ( h 2 ) - A 0 ( h ) + O ( h 2 ) . A=2A_{0}\!\left(\frac{h}{2}\right)-A_{0}(h)+O(h^{2}).
  26. A 1 ( h ) = 2 A 0 ( h 2 ) - A 0 ( h ) A_{1}(h)=2A_{0}\!\left(\frac{h}{2}\right)-A_{0}(h)
  27. A = 4 A 1 ( h 2 ) - A 1 ( h ) 3 + O ( h 3 ) . A=\frac{4A_{1}\!\left(\frac{h}{2}\right)-A_{1}(h)}{3}+O(h^{3}).
  28. y ( t ) = - y 2 y^{\prime}(t)=-y^{2}
  29. y ( 0 ) = 1 y(0)=1
  30. h h
  31. t = 2 t=2
  32. 4 = 2 2 = t 2 4=2^{2}=t^{2}
  33. y ( 5 ) y(5)
  34. 1 5 + 1 = 1 6 = 0.1666... \frac{1}{5+1}=\frac{1}{6}=0.1666...
  35. y ( t ) = 1 1 + t y(t)=\frac{1}{1+t}

Riesz–Thorin_theorem.html

  1. f f
  2. f p θ f p 0 1 - θ f p 1 θ . \|f\|_{p_{\theta}}\leq\|f\|_{p_{0}}^{1-\theta}\|f\|_{p_{1}}^{\theta}.
  3. 0 , 0,∞∞
  4. f f
  5. \mathcal{F}
  6. ( f 1 + f 2 ) = L 1 ( f 1 ) + L 2 ( f 2 ) \mathcal{F}(f_{1}+f_{2})=\mathcal{F}_{L^{1}}(f_{1})+\mathcal{F}_{L^{2}}(f_{2})
  7. L 1 : L 1 ( 𝐑 d ) L ( 𝐑 d ) , \mathcal{F}_{L^{1}}:L^{1}(\mathbf{R}^{d})\to L^{\infty}(\mathbf{R}^{d}),
  8. L 2 : L 2 ( 𝐑 d ) L 2 ( 𝐑 d ) . \mathcal{F}_{L^{2}}:L^{2}(\mathbf{R}^{d})\to L^{2}(\mathbf{R}^{d}).
  9. L 1 \mathcal{F}_{L^{1}}
  10. L 2 \mathcal{F}_{L^{2}}
  11. σ σ
  12. T T
  13. T T
  14. ( 1 p , 1 q ) (\frac{1}{p},\frac{1}{q})
  15. 0 , 11 × 0 , 11 0,11×0,11
  16. T T
  17. ( p , q ) (p,q)
  18. T T
  19. T f q θ = sup g p θ 1 | ( T f ) g d μ 2 | . \|Tf\|_{q_{\theta}}=\sup_{\|g\|_{p_{\theta}}\leq 1}\left|\int(Tf)g\,d\mu_{2}% \right|.
  20. f f
  21. g g
  22. z z
  23. 𝐂 \mathbf{C}
  24. ϕ ( z ) = ( T f z ) g z d μ 2 \phi(z)=\int\left(Tf_{z}\right)g_{z}\,d\mu_{2}
  25. z = θ z=θ
  26. ( T f ) g . \int(Tf)g.
  27. Φ Φ
  28. R e ( z ) = 0 Re(z)=0
  29. R e ( z ) = 1 Re(z)=1
  30. Φ Φ
  31. R e ( z ) = θ Re(z)=θ
  32. z = θ z=θ
  33. T T
  34. φ ( z ) = ( T z f z ) g z d μ 2 , \varphi(z)=\int(T_{z}f_{z})g_{z}\,d\mu_{2},
  35. ( Ω < s u b > 1 , Σ 1 , μ 1 ) (Ω<sub>1,Σ_{1},μ_{1})
  36. z ( T z f ) g d μ 2 z\mapsto\int(T_{z}f)g\,d\mu_{2}
  37. S ¯ \overline{S}
  38. S S
  39. f f
  40. g g
  41. k < π k<π
  42. sup z S e - k | Im ( z ) | log | ( T z f ) g μ 2 | < \sup_{z\in S}e^{-k|\,\text{Im}(z)|}\log\left|\int(T_{z}f)g\,\mu_{2}\right|<\infty
  43. T < s u b > z T<sub>z
  44. T < s u b > z T<sub>z
  45. sup Re ( z ) = 0 , 1 e - k | Im ( z ) | log T z < \sup_{\,\text{Re}(z)=0,1}e^{-k|\,\text{Im}(z)|}\log\left\|T_{z}\right\|<\infty
  46. k < π k<π
  47. f * g p f 1 g p . \|f*g\|_{p}\leq\|f\|_{1}\|g\|_{p}.
  48. S S
  49. g g
  50. S S
  51. g g
  52. S S
  53. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  54. f * g s f r g p \|f*g\|_{s}\leq\|f\|_{r}\|g\|_{p}
  55. 1 r + 1 p = 1 + 1 s . \frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.
  56. f : 𝐑 𝐂 f:\mathbf{R}→\mathbf{C}
  57. f ( x ) = 1 π p . v . - f ( x - t ) t d t = ( 1 π p . v . 1 t f ) ( x ) , \mathcal{H}f(x)=\frac{1}{\pi}\,\mathrm{p.v.}\int_{-\infty}^{\infty}\frac{f(x-t% )}{t}\,dt=\left(\frac{1}{\pi}\,\mathrm{p.v.}\frac{1}{t}\ast f\right)(x),
  58. f ^ ( ξ ) = - i sgn ( ξ ) f ^ ( ξ ) . \widehat{\mathcal{H}f}(\xi)=-i\,\mathrm{sgn}(\xi)\hat{f}(\xi).
  59. ( f ) 2 = f 2 + 2 ( f f ) (\mathcal{H}f)^{2}=f^{2}+2\mathcal{H}(f\mathcal{H}f)
  60. f : 𝐑 𝐂 f:\mathbf{R}→\mathbf{C}
  61. f p A p f p \|\mathcal{H}f\|_{p}\leq A_{p}\|f\|_{p}
  62. 𝐂 \mathbf{C}
  63. μ ( { x : T f ( x ) > α } ) ( C p , q f p α ) q , \mu\left(\{x:Tf(x)>\alpha\}\right)\leq\left(\frac{C_{p,q}\|f\|_{p}}{\alpha}% \right)^{q},
  64. A 1 1 , A M . \|A\|_{\ell_{1}\to\ell_{1}},\|A\|_{\ell_{\infty}\to\ell_{\infty}}\leq M.
  65. A X X M \|A\|_{X\to X}\leq M
  66. X X
  67. ( x i ) X (x_{i})\in X
  68. ( ε i ) { - 1 , 1 } (\varepsilon_{i})\in\{-1,1\}^{\infty}
  69. ( ε i x i ) X = ( x i ) X \|(\varepsilon_{i}x_{i})\|_{X}=\|(x_{i})\|_{X}
  70. L < s u p > p 0 L p 1 L<sup>p_{0}∩L^{p_{1}}
  71. z z
  72. L < s u p > 1 L<sup>1

Rings_of_Jupiter.html

  1. n ( r ) = A × r - q n(r)=A\times r^{-q}
  2. A A
  3. τ \scriptstyle\tau
  4. τ l = 4.7 × 10 - 6 \scriptstyle\tau_{l}\,=\,4.7\times 10^{-6}
  5. τ s = 1.3 × 10 - 6 \scriptstyle\tau_{s}=1.3\times 10^{-6}
  6. τ s 10 - 6 \scriptstyle\tau_{s}\,\sim\,10^{-6}

Rise_time.html

  1. t r t_{r}\,
  2. f L f_{L}\,
  3. f H f_{H}\,
  4. h ( t ) h(t)\,
  5. H ( ω ) H(\omega)\,
  6. B W = f H - f L BW=f_{H}-f_{L}\,
  7. f L f_{L}
  8. f H f_{H}
  9. B W f H BW\cong f_{H}\,
  10. f L = 0 f H = B W f_{L}=0\,\Leftrightarrow\,f_{H}=BW
  11. V 0 V_{0}
  12. | H ( ω ) | = e - ω 2 σ 2 |H(\omega)|=e^{-\frac{\omega^{2}}{\sigma^{2}}}
  13. σ > 0 \sigma>0
  14. f H = σ 2 π 3 20 l n 10 0.0935 σ f_{H}=\frac{\sigma}{2\pi}\sqrt{\frac{3}{20}ln10}\cong 0.0935\sigma
  15. - 1 { H } ( t ) = h ( t ) = 1 2 π - + e - ω 2 σ 2 e i ω t d ω = σ 2 π e - 1 4 σ 2 t 2 \mathcal{F}^{-1}\{H\}(t)=h(t)=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}{e^% {-\frac{\omega^{2}}{\sigma^{2}}}e^{i\omega t}}d\omega=\frac{\sigma}{2\sqrt{\pi% }}e^{-\frac{1}{4}\sigma^{2}t^{2}}
  16. V ( t ) = V 0 H * h ( t ) = V 0 π - σ t 2 e - τ 2 d τ = V 0 2 [ 1 + erf ( σ t 2 ) ] V ( t ) V 0 = 1 2 [ 1 + erf ( σ t 2 ) ] V(t)=V_{0}{H*h}(t)=\frac{V_{0}}{\sqrt{\pi}}\int\limits_{-\infty}^{\frac{\sigma t% }{2}}e^{-\tau^{2}}d\tau=\frac{V_{0}}{2}\left[1+\mathrm{erf}\left(\frac{\sigma t% }{2}\right)\right]\Leftrightarrow\frac{V(t)}{V_{0}}=\frac{1}{2}\left[1+\mathrm% {erf}\left(\frac{\sigma t}{2}\right)\right]
  17. 0.1 = 1 2 [ 1 + erf ( σ t 1 2 ) ] 0.9 = 1 2 [ 1 + erf ( σ t 2 2 ) ] 0.1=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\sigma t_{1}}{2}\right)\right]% \qquad 0.9=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\sigma t_{2}}{2}\right)\right]
  18. t = - t 1 = t 2 t=-t_{1}=t_{2}
  19. t r = t 2 - t 1 = 2 t t_{r}=t_{2}-t_{1}=2t
  20. t r = 4 σ erf - 1 ( 0.8 ) 0.3394 f H t_{r}=\frac{4}{\sigma}{\mathrm{erf}^{-1}(0.8)}\cong\frac{0.3394}{f_{H}}
  21. t r 0.34 B W B W t r 0.34 t_{r}\cong\frac{0.34}{BW}\quad\Longleftrightarrow\quad BW\cdot t_{r}\cong 0.34
  22. τ = R C \tau=RC
  23. t r 2.197 τ t_{r}\cong 2.197\tau\,
  24. V 0 V_{0}
  25. V ( t ) = V 0 ( 1 - e - t τ ) V(t)=V_{0}\left(1-e^{-\frac{t}{\tau}}\right)
  26. V ( t ) V 0 = ( 1 - e - t τ ) \frac{V(t)}{V_{0}}=\left(1-e^{-\frac{t}{\tau}}\right)
  27. V ( t ) V 0 - 1 = - e - t τ \frac{V(t)}{V_{0}}-1=-e^{-\frac{t}{\tau}}
  28. 1 - V ( t ) V 0 = e - t τ 1-\frac{V(t)}{V_{0}}=e^{-\frac{t}{\tau}}
  29. ln ( 1 - V ( t ) V 0 ) = - t τ \ln\left(1-\frac{V(t)}{V_{0}}\right)=-\frac{t}{\tau}
  30. t = - τ ln ( 1 - V ( t ) V 0 ) t=-\tau\;\ln\left(1-\frac{V(t)}{V_{0}}\right)
  31. V ( t ) V 0 = 0.1 \frac{V(t)}{V_{0}}=0.1
  32. V ( t ) V 0 = 0.9 \frac{V(t)}{V_{0}}=0.9
  33. t 1 = - τ ln ( 1 - 0.1 ) = - τ ln ( 0.9 ) = - τ ln ( 9 10 ) = τ ln ( 10 9 ) = τ ( ln 10 - ln 9 ) t_{1}=-\tau\;\ln\left(1-0.1\right)=-\tau\;\ln\left(0.9\right)=-\tau\;\ln\left(% \frac{9}{10}\right)=\tau\;\ln\left(\frac{10}{9}\right)=\tau({\ln 10}-{\ln 9})
  34. t 2 = τ ln 10 t_{2}=\tau\ln{10}\,
  35. t 1 t_{1}
  36. t 2 t_{2}
  37. t r = t 2 - t 1 = τ ln 9 τ 2.197 t_{r}=t_{2}-t_{1}=\tau\cdot\ln 9\cong\tau\cdot 2.197
  38. τ = R C = 1 2 π f H \tau=RC=\frac{1}{2\pi f_{H}}
  39. t r 2.197 2 π f H 0.349 f H t_{r}\cong\frac{2.197}{2\pi f_{H}}\cong\frac{0.349}{f_{H}}
  40. t r 0.35 B W B W t r 0.35 t_{r}\cong\frac{0.35}{BW}\quad\Longleftrightarrow\quad BW\cdot t_{r}\cong 0.35
  41. t r t_{r}
  42. t r 1.386 τ t r 0.22 B W t_{r}\cong 1.386\tau\quad\Longleftrightarrow\quad t_{r}\cong\frac{0.22}{BW}
  43. n n
  44. t r i \scriptstyle{t_{r_{i}}}
  45. t r S \scriptstyle{t_{r_{S}}}
  46. t r O \scriptstyle{t_{r_{O}}}
  47. t r O = t r S 2 + t r 1 2 + + t r n 2 t_{r_{O}}=\sqrt{t_{r_{S}}^{2}+t_{r_{1}}^{2}+\dots+t_{r_{n}}^{2}}
  48. 2.2 R C 2.2RC
  49. t r ω 0 = 2.230 ζ 2 - 0.078 ζ + 1.12 t_{r}\cdot\omega_{0}=2.230\zeta^{2}-0.078\zeta+1.12\,
  50. t r ω 0 = 1 1 - ζ 2 ( π - tan - 1 ( 1 - ζ 2 ζ ) ) t_{r}\cdot\omega_{0}=\frac{1}{\sqrt{1-\zeta^{2}}}\left(\pi-\tan^{-1}\left({% \frac{\sqrt{1-\zeta^{2}}}{\zeta}}\right)\right)

Risk_factor.html

  1. R i s k = number of persons experiencing event (food poisoning) number of persons exposed to risk factor (food) Risk=\frac{\mbox{number of persons experiencing event (food poisoning)}~{}}{% \mbox{number of persons exposed to risk factor (food)}~{}}

Robert_Manning_(engineer).html

  1. V = 32 [ R S ( 1 + R 1 / 3 ) ] 1 / 2 V=32\left[RS\left(1+R^{1/3}\right)\right]^{1/2}
  2. V = C R x S 1 / 2 V=CR^{x}S^{1/2}
  3. x x
  4. V = C R 2 / 3 S 1 / 2 V=CR^{2/3}S^{1/2}
  5. V = C ( g S ) 1 / 2 [ R 1 / 2 + ( 0.22 m 1 / 2 ) ( R - 0.15 m ) ] V=C(gS)^{1/2}\left[R^{1/2}+\left(\dfrac{0.22}{m^{1/2}}\right)\left(R-0.15m% \right)\right]
  6. m m
  7. C C
  8. V = ( 1 n ) R 2 / 3 S 1 / 2 V=\left(\dfrac{1}{n}\right)R^{2/3}S^{1/2}
  9. C C
  10. n n
  11. n n
  12. K K
  13. C C
  14. n n

Rock_magnetism.html

  1. T 1 T_{1}
  2. T 2 T_{2}
  3. T B T_{B}

Rodrigues'_formula.html

  1. P n P_{n}
  2. P n ( x ) = 1 2 n n ! d n d x n [ ( x 2 - 1 ) n ] . P_{n}(x)={1\over 2^{n}n!}{d^{n}\over dx^{n}}\left[(x^{2}-1)^{n}\right].
  3. L n ( x ) = e x n ! d n d x n ( e - x x n ) = 1 n ! ( d d x - 1 ) n x n , L_{n}(x)=\frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}\left(e^{-x}x^{n}\right)=\frac{1}% {n!}\left(\frac{d}{dx}-1\right)^{n}x^{n},
  4. H n ( x ) = ( - 1 ) n e x 2 d n d x n e - x 2 = ( 2 x - d d x ) n 1 H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}=\left(2x-\frac{d}{dx}% \right)^{n}\cdot 1

Rolling.html

  1. 𝐯 = 𝐫 \timessymbol ω \mathbf{v}=\mathbf{r}\timessymbol{\omega}
  2. 𝐫 \mathbf{r}
  3. s y m b o l ω symbol{\omega}
  4. K rolling = K translation + K rotation K\text{rolling}=K\text{translation}+K\text{rotation}
  5. v c.o.m. = r ω v\text{c.o.m.}=r\omega
  6. a = r α a=r\alpha
  7. a = F net m = r α = r τ I a=\frac{F\text{net}}{m}=r\alpha=\frac{r\tau}{I}
  8. F net = F external 1 + I m r 2 = F external 1 + ( r gyr. r ) 2 F\text{net}=\frac{F_{\,\text{external}}}{1+\frac{I}{mr^{2}}}=\frac{F_{\,\text{% external}}}{1+\left(\frac{r\text{gyr.}}{r}\right)^{2}}
  9. a = F external m + I r 2 a=\frac{F_{\,\text{external}}}{m+\frac{I}{r^{2}}}
  10. F friction F_{\,\text{friction}}
  11. F external F\text{external}
  12. ( 1 ) (1)
  13. r α = a r\alpha=a
  14. α \alpha
  15. a a
  16. F friction F\text{friction}
  17. r τ I \displaystyle r\frac{\tau}{I}
  18. F friction F_{\,\text{friction}}
  19. ( 1 ) (1)
  20. F net \displaystyle F_{\,\text{net}}
  21. F net F\text{net}
  22. a a
  23. a \displaystyle a
  24. F net F\text{net}
  25. r gyr. = I m r\text{gyr.}=\sqrt{\frac{I}{m}}
  26. r gyr. 2 = I m r\text{gyr.}^{2}=\frac{I}{m}
  27. I m r 2 = ( I m ) r 2 = r gyr. 2 r 2 = ( r gyr. r ) 2 \begin{aligned}\displaystyle\frac{I}{mr^{2}}&\displaystyle=\frac{\left(\frac{I% }{m}\right)}{r^{2}}\\ &\displaystyle=\frac{r\text{gyr.}^{2}}{r^{2}}\\ &\displaystyle=\left(\frac{r\text{gyr.}}{r}\right)^{2}\\ \end{aligned}
  28. F net F\text{net}
  29. F net = F external 1 + ( r gyr. r ) 2 F\text{net}=\frac{F_{\,\text{external}}}{1+\left(\frac{r\text{gyr.}}{r}\right)% ^{2}}
  30. I r 2 \tfrac{I}{r^{2}}
  31. I I
  32. r r
  33. I r 2 \tfrac{I}{r^{2}}
  34. m + I r 2 m+\tfrac{I}{r^{2}}
  35. 1 / ( 1 + I m r 2 ) 1/\left(1+\tfrac{I}{mr^{2}}\right)
  36. I m r 2 \tfrac{I}{mr^{2}}
  37. ( r gyr. r ) 2 \left(\tfrac{r\text{gyr.}}{r}\right)^{2}
  38. r gyr. r\text{gyr.}
  39. a = g sin ( θ ) 1 + ( r gyr. r ) 2 a=\frac{g\sin\left(\theta\right)}{1+\left(\tfrac{r\text{gyr.}}{r}\right)^{2}}
  40. m m
  41. r gyr. r \tfrac{r\text{gyr.}}{r}

Romberg's_method.html

  1. a b f ( x ) d x \int_{a}^{b}f(x)\,dx
  2. R ( 0 , 0 ) = 1 2 ( b - a ) ( f ( a ) + f ( b ) ) R(0,0)=\frac{1}{2}(b-a)(f(a)+f(b))
  3. R ( n , 0 ) = 1 2 R ( n - 1 , 0 ) + h n k = 1 2 n - 1 f ( a + ( 2 k - 1 ) h n ) R(n,0)=\frac{1}{2}R(n-1,0)+h_{n}\sum_{k=1}^{2^{n-1}}f(a+(2k-1)h_{n})
  4. R ( n , m ) = R ( n , m - 1 ) + 1 4 m - 1 ( R ( n , m - 1 ) - R ( n - 1 , m - 1 ) ) R(n,m)=R(n,m-1)+\frac{1}{4^{m}-1}(R(n,m-1)-R(n-1,m-1))
  5. R ( n , m ) = 1 4 m - 1 ( 4 m R ( n , m - 1 ) - R ( n - 1 , m - 1 ) ) R(n,m)=\frac{1}{4^{m}-1}(4^{m}R(n,m-1)-R(n-1,m-1))
  6. n m n\geq m\,
  7. m 1 m\geq 1\,
  8. h n = b - a 2 n . h_{n}=\frac{b-a}{2^{n}}.
  9. O ( h n 2 m + 2 ) . O\left(h_{n}^{2m+2}\right).\,

Roothaan_equations.html

  1. 𝐅𝐂 = 𝐒𝐂 ϵ \mathbf{F}\mathbf{C}=\mathbf{S}\mathbf{C}\mathbf{\epsilon}
  2. ϵ \epsilon

Rose_(mathematics).html

  1. r = cos ( k θ ) \!\,r=\cos(k\theta)
  2. x = cos ( k t ) cos ( t ) \!\,x=\cos(kt)\cos(t)
  3. y = cos ( k t ) sin ( t ) \!\,y=\cos(kt)\sin(t)
  4. sin ( k θ ) = cos ( k θ - π 2 ) = cos ( k ( θ - π 2 k ) ) \sin(k\theta)=\cos\left(k\theta-\frac{\pi}{2}\right)=\cos\left(k\left(\theta-% \frac{\pi}{2k}\right)\right)
  5. θ \theta
  6. r = sin ( k θ ) \,r=\sin(k\theta)
  7. r = cos ( k θ ) \,r=\cos(k\theta)
  8. r = a cos ( k θ ) r=a\cos(k\theta)\,
  9. 1 2 0 2 π ( a cos ( k θ ) ) 2 d θ = a 2 2 ( π + sin ( 4 k π ) 4 k ) = π a 2 2 \frac{1}{2}\int_{0}^{2\pi}(a\cos(k\theta))^{2}\,d\theta=\frac{a^{2}}{2}\left(% \pi+\frac{\sin(4k\pi)}{4k}\right)=\frac{\pi a^{2}}{2}
  10. 1 2 0 π ( a cos ( k θ ) ) 2 d θ = a 2 2 ( π 2 + sin ( 2 k π ) 4 k ) = π a 2 4 \frac{1}{2}\int_{0}^{\pi}(a\cos(k\theta))^{2}\,d\theta=\frac{a^{2}}{2}\left(% \frac{\pi}{2}+\frac{\sin(2k\pi)}{4k}\right)=\frac{\pi a^{2}}{4}
  11. r = a sin ( k θ ) r=a\sin(k\theta)\,
  12. π \pi
  13. 2 \sqrt{2}
  14. r = cos ( k θ ) + c \!\,r=\cos(k\theta)+c

Rossby_parameter.html

  1. β \beta
  2. β \beta
  3. β = f y = 1 a d d ϕ ( 2 ω s i n ϕ ) = 2 ω c o s ϕ a \beta=\frac{\partial f}{\partial y}=\frac{1}{a}\frac{d}{d\phi}(2\omega sin\phi% )=\frac{2\omega cos\phi}{a}
  4. f f
  5. ϕ \phi
  6. ω \omega
  7. a a

Rotary_variable_differential_transformer.html

  1. V 1 V_{1}
  2. V 2 V_{2}
  3. θ = G ( V 1 - V 2 V 1 + V 2 ) \theta\ =G\cdot\ \left(\frac{V_{1}-V_{2}}{V_{1}+V_{2}}\right)
  4. G G
  5. V 2 = V 1 ± G θ V_{2}=V_{1}\pm\ G\cdot\ \theta
  6. V 1 - V 2 V_{1}-V_{2}
  7. Δ V = 2 G θ \Delta\ V=2\cdot\ G\cdot\ \theta
  8. C = V = 2 V 0 C=\sum\ V=2\cdot\ V_{0}
  9. V = G θ V=G\cdot\ \theta

Rotation_matrix.html

  1. R = [ cos θ - sin θ sin θ cos θ ] R=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
  2. θ θ
  3. R R
  4. R R
  5. R R
  6. R R
  7. d e t R = 1 detR=1
  8. n n
  9. S O ( n ) SO(n)
  10. n n
  11. O ( n ) O(n)
  12. θ θ
  13. x x
  14. R ( θ ) = [ cos θ - sin θ sin θ cos θ ] R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
  15. [ x y ] = [ cos θ - sin θ sin θ cos θ ] [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\\ \end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}\begin{bmatrix}x\\ y\\ \end{bmatrix}
  16. x = x cos θ - y sin θ x^{\prime}=x\cos\theta-y\sin\theta\,
  17. y = x sin θ + y cos θ y^{\prime}=x\sin\theta+y\cos\theta\,
  18. θ θ
  19. θ θ
  20. R ( - θ ) = [ cos θ sin θ - sin θ cos θ ] R(-\theta)=\begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{bmatrix}\,
  21. θ θ
  22. R ( θ ) R(θ)
  23. x x
  24. y y
  25. R ( θ ) R(θ)
  26. R ( 90 ) = [ 0 - 1 1 0 ] (90° counterclockwise rotation) , R ( 180 ) = [ - 1 0 0 - 1 ] (180° rotation in either direction – a half-turn) , R ( 270 ) = [ 0 1 - 1 0 ] (270° counter-clockwise rotation, the same as a 90° clockwise rotation) . \begin{aligned}\displaystyle R(90^{\circ})&\displaystyle=\begin{bmatrix}0&-1\\ 1&0\\ \end{bmatrix}&\displaystyle\,\text{(90° counterclockwise rotation)},\\ \displaystyle R(180^{\circ})&\displaystyle=\begin{bmatrix}-1&0\\ 0&-1\\ \end{bmatrix}&\displaystyle\,\text{(180° rotation in either direction – a half% -turn)},\\ \displaystyle R(270^{\circ})&\displaystyle=\begin{bmatrix}0&1\\ -1&0\\ \end{bmatrix}&\displaystyle\,\text{(270° counter-clockwise rotation, the same % as a 90° clockwise rotation)}.\end{aligned}
  27. R x ( θ ) = [ 1 0 0 0 cos θ - sin θ 0 sin θ cos θ ] R y ( θ ) = [ cos θ 0 sin θ 0 1 0 - sin θ 0 cos θ ] R z ( θ ) = [ cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ] \begin{aligned}\displaystyle R_{x}(\theta)&\displaystyle=\begin{bmatrix}1&0&0% \\ 0&\cos\theta&-\sin\theta\\ 0&\sin\theta&\cos\theta\\ \end{bmatrix}\\ \displaystyle R_{y}(\theta)&\displaystyle=\begin{bmatrix}\cos\theta&0&\sin% \theta\\ 0&1&0\\ -\sin\theta&0&\cos\theta\\ \end{bmatrix}\\ \displaystyle R_{z}(\theta)&\displaystyle=\begin{bmatrix}\cos\theta&-\sin% \theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\\ \end{bmatrix}\end{aligned}
  28. θ θ
  29. ( 1 , 0 , 0 ) (1,0,0)
  30. R z ( 90 ) [ 1 0 0 ] = [ cos 90 - sin 90 0 sin 90 cos 90 0 0 0 1 ] [ 1 0 0 ] = [ 0 - 1 0 1 0 0 0 0 1 ] [ 1 0 0 ] = [ 0 1 0 ] R_{z}(90^{\circ})\begin{bmatrix}1\\ 0\\ 0\\ \end{bmatrix}=\begin{bmatrix}\cos 90^{\circ}&-\sin 90^{\circ}&0\\ \sin 90^{\circ}&\cos 90^{\circ}&0\\ 0&0&1\\ \end{bmatrix}\begin{bmatrix}1\\ 0\\ 0\\ \end{bmatrix}=\begin{bmatrix}0&-1&0\\ 1&0&0\\ 0&0&1\\ \end{bmatrix}\begin{bmatrix}1\\ 0\\ 0\\ \end{bmatrix}=\begin{bmatrix}0\\ 1\\ 0\\ \end{bmatrix}
  31. R = R z ( α ) R y ( β ) R x ( γ ) R=R_{z}(\alpha)\,R_{y}(\beta)\,R_{x}(\gamma)\,\!
  32. α , β α,β
  33. γ γ
  34. α , β , γ α,β,γ
  35. z , y , x z,y,x
  36. R = R y ( γ ) R x ( β ) R y ( α ) R=R_{y}(\gamma)\,R_{x}(\beta)\,R_{y}(\alpha)\,\!
  37. α , β , γ α,β,γ
  38. y , x , y y,x,y
  39. 3 × 3 3×3
  40. R R
  41. 𝐮 \mathbf{u}
  42. R 𝐮 = 𝐮 , R\,\textbf{u}=\,\textbf{u}~{},
  43. 𝐮 \mathbf{u}
  44. 𝐮 \mathbf{u}
  45. 𝐮 \mathbf{u}
  46. R = I R=I
  47. R 𝐮 = I 𝐮 ( R - I ) 𝐮 = 0 , R\,\textbf{u}=I\,\textbf{u}\quad\Rightarrow\quad(R-I)\,\textbf{u}=0~{},
  48. 𝐮 \mathbf{u}
  49. R I R−I
  50. 𝐮 \mathbf{u}
  51. R R
  52. λ = 1 λ=1
  53. 𝐯 \mathbf{v}
  54. 𝐯 \mathbf{v}
  55. R 𝐯 R\mathbf{v}
  56. θ θ
  57. Tr ( R ) = 1 + 2 cos θ . \operatorname{Tr}(R)=1+2\cos\theta~{}.
  58. R = [ cos θ + u x 2 ( 1 - cos θ ) u x u y ( 1 - cos θ ) - u z sin θ u x u z ( 1 - cos θ ) + u y sin θ u y u x ( 1 - cos θ ) + u z sin θ cos θ + u y 2 ( 1 - cos θ ) u y u z ( 1 - cos θ ) - u x sin θ u z u x ( 1 - cos θ ) - u y sin θ u z u y ( 1 - cos θ ) + u x sin θ cos θ + u z 2 ( 1 - cos θ ) ] . R=\begin{bmatrix}\cos\theta+u_{x}^{2}\left(1-\cos\theta\right)&u_{x}u_{y}\left% (1-\cos\theta\right)-u_{z}\sin\theta&u_{x}u_{z}\left(1-\cos\theta\right)+u_{y}% \sin\theta\\ u_{y}u_{x}\left(1-\cos\theta\right)+u_{z}\sin\theta&\cos\theta+u_{y}^{2}\left(% 1-\cos\theta\right)&u_{y}u_{z}\left(1-\cos\theta\right)-u_{x}\sin\theta\\ u_{z}u_{x}\left(1-\cos\theta\right)-u_{y}\sin\theta&u_{z}u_{y}\left(1-\cos% \theta\right)+u_{x}\sin\theta&\cos\theta+u_{z}^{2}\left(1-\cos\theta\right)% \end{bmatrix}.
  59. R = cos θ 𝐈 + sin θ [ 𝐮 ] × + ( 1 - cos θ ) 𝐮 𝐮 , R=\cos\theta\mathbf{I}+\sin\theta[\mathbf{u}]_{\times}+(1-\cos\theta)\mathbf{u% }\otimes\mathbf{u}~{},
  60. [ 𝐮 ] × [\mathbf{u}]_{\times}
  61. \otimes
  62. 𝐮 𝐮 = [ u x 2 u x u y u x u z u x u y u y 2 u y u z u x u z u y u z u z 2 ] , [ 𝐮 ] × = [ 0 - u z u y u z 0 - u x - u y u x 0 ] . \mathbf{u}\otimes\mathbf{u}=\begin{bmatrix}u_{x}^{2}&u_{x}u_{y}&u_{x}u_{z}\\ u_{x}u_{y}&u_{y}^{2}&u_{y}u_{z}\\ u_{x}u_{z}&u_{y}u_{z}&u_{z}^{2}\end{bmatrix},\qquad[\mathbf{u}]_{\times}=% \begin{bmatrix}0&-u_{z}&u_{y}\\ u_{z}&0&-u_{x}\\ -u_{y}&u_{x}&0\end{bmatrix}.
  63. R T = R - 1 R^{T}=R^{-1}
  64. det R = ± 1 \det R=\pm 1
  65. det R = 1 \det R=1
  66. det R = - 1 \det R=-1
  67. e ± i θ k e^{\pm i\theta_{k}}
  68. e ± i θ e^{\pm i\theta}
  69. 2 cos ( θ ) 2\cos(\theta)
  70. θ \theta
  71. e ± i θ e^{\pm i\theta}
  72. θ \theta
  73. 1 + 2 cos ( θ ) 1+2\cos(\theta)
  74. e ± i θ e^{\pm i\theta}
  75. e ± i ϕ e^{\pm i\phi}
  76. 2 ( cos ( θ ) + cos ( ϕ ) ) 2(\cos(\theta)+\cos(\phi))
  77. θ \theta
  78. ϕ \phi
  79. θ = ϕ \theta=\phi
  80. e ± i θ e^{\pm i\theta}
  81. θ \theta
  82. 4 cos ( θ ) 4\cos(\theta)
  83. Q = [ 0 1 - 1 0 ] Q=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}
  84. M = [ 0.936 0.352 0.352 - 0.936 ] M=\begin{bmatrix}0.936&0.352\\ 0.352&-0.936\end{bmatrix}
  85. Q = [ 1 0 0 0 3 2 1 2 0 - 1 2 3 2 ] Q=\begin{bmatrix}1&0&0\\ 0&\frac{\sqrt{3}}{2}&\frac{1}{2}\\ 0&-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}
  86. Q = [ 0.36 0.48 - 0.8 - 0.8 0.60 0 0.48 0.64 0.60 ] Q=\begin{bmatrix}0.36&0.48&-0.8\\ -0.8&0.60&0\\ 0.48&0.64&0.60\end{bmatrix}
  87. P = [ 0 0 1 1 0 0 0 1 0 ] P=\begin{bmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{bmatrix}
  88. M = [ 3 - 4 1 5 3 - 7 - 9 2 6 ] M=\begin{bmatrix}3&-4&1\\ 5&3&-7\\ -9&2&6\end{bmatrix}
  89. M = [ 0.5 - 0.1 0.7 0.1 0.5 - 0.5 - 0.7 0.5 0.5 - 0.5 - 0.7 - 0.1 ] M=\begin{bmatrix}0.5&-0.1&0.7\\ 0.1&0.5&-0.5\\ -0.7&0.5&0.5\\ -0.5&-0.7&-0.1\end{bmatrix}
  90. Q = [ - 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ] Q=\begin{bmatrix}-1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{bmatrix}
  91. Q = [ 0 - 1 0 0 0 1 0 0 0 0 0 0 - 1 0 0 0 0 0 - 1 0 0 0 0 0 1 ] Q=\begin{bmatrix}0&-1&0&0&0\\ 1&0&0&0&0\\ 0&0&-1&0&0\\ 0&0&0&-1&0\\ 0&0&0&0&1\end{bmatrix}
  92. 𝐩 \mathbf{p}
  93. O O
  94. P P
  95. P P
  96. d 2 ( O , P ) = p 2 = r = 1 n p r 2 d^{2}(O,P)=\|{p}\|^{2}=\sum_{r=1}^{n}p_{r}^{2}
  97. p 2 = [ p 1 p n ] [ p 1 p n ] = p T p . \|{p}\|^{2}=\begin{bmatrix}p_{1}\cdots p_{n}\end{bmatrix}\begin{bmatrix}p_{1}% \\ \vdots\\ p_{n}\end{bmatrix}={p}^{T}{p}.
  98. Q 𝐩 Q\mathbf{p}
  99. p T p = ( Q p ) T ( Q p ) , {p}^{T}{p}=(Q{p})^{T}(Q{p}),\,\!
  100. p T I p = ( p T Q T ) ( Q p ) = p T ( Q T Q ) p . \begin{aligned}\displaystyle{p}^{T}I{p}&\displaystyle{}=({p}^{T}Q^{T})(Q{p})\\ &\displaystyle{}={p}^{T}(Q^{T}Q){p}.\end{aligned}
  101. 𝐩 \mathbf{p}
  102. Q Q
  103. Q T Q = I . Q^{T}Q=I.\,\!
  104. det Q = + 1. \det Q=+1.\,\!
  105. ( Q T ) T ( Q T ) \displaystyle(Q^{T})^{T}(Q^{T})
  106. ( Q 1 Q 2 ) T ( Q 1 Q 2 ) \displaystyle(Q_{1}Q_{2})^{T}(Q_{1}Q_{2})
  107. n n
  108. 2 2
  109. n × n n×n
  110. Q 1 = [ 0 - 1 0 1 0 0 0 0 1 ] Q 2 = [ 0 0 1 0 1 0 - 1 0 0 ] Q 1 Q 2 = [ 0 - 1 0 0 0 1 - 1 0 0 ] Q 2 Q 1 = [ 0 0 1 1 0 0 0 1 0 ] . \begin{aligned}\displaystyle Q_{1}&\displaystyle{}=\begin{bmatrix}0&-1&0\\ 1&0&0\\ 0&0&1\end{bmatrix}&\displaystyle Q_{2}&\displaystyle{}=\begin{bmatrix}0&0&1\\ 0&1&0\\ -1&0&0\end{bmatrix}\\ \displaystyle Q_{1}Q_{2}&\displaystyle{}=\begin{bmatrix}0&-1&0\\ 0&0&1\\ -1&0&0\end{bmatrix}&\displaystyle Q_{2}Q_{1}&\displaystyle{}=\begin{bmatrix}0&% 0&1\\ 1&0&0\\ 0&1&0\end{bmatrix}.\end{aligned}
  111. n × n n×n
  112. n > 2 n> 2
  113. n × n n×n
  114. P P
  115. C S CS
  116. P P
  117. P P
  118. 𝐯 \mathbf{v}
  119. P P
  120. P P
  121. 𝐯 \mathbf{v}
  122. C S CS
  123. C S CS
  124. P P
  125. 𝐯 \mathbf{v}
  126. R ( θ ) = [ cos θ - sin θ sin θ cos θ ] R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
  127. 𝐯 \mathbf{v}
  128. θ θ
  129. C S CS
  130. P P
  131. 𝐯 \mathbf{v}
  132. 𝐰 \mathbf{w}
  133. ( R 𝐯 ) (R\mathbf{v})
  134. ( 𝐰 R ) (\mathbf{w}R)
  135. R 𝐯 R\mathbf{v}
  136. 𝐰 R \mathbf{w}R
  137. P P
  138. Q = [ 0.36 0.48 - 0.8 - 0.8 0.60 0 0.48 0.64 0.60 ] . Q=\begin{bmatrix}0.36&0.48&-0.8\\ -0.8&0.60&0\\ 0.48&0.64&0.60\end{bmatrix}.
  139. Q v = λ v , Q{v}=\lambda{v},\,\!
  140. 0 = ( λ I - Q ) v . {0}=(\lambda I-Q){v}.\,\!
  141. 0 = det ( λ I - Q ) = λ 3 - 39 25 λ 2 + 39 25 λ - 1 = ( λ - 1 ) ( λ 2 - 14 25 λ + 1 ) . \begin{aligned}\displaystyle 0&\displaystyle{}=\det(\lambda I-Q)\\ &\displaystyle{}=\lambda^{3}-\tfrac{39}{25}\lambda^{2}+\tfrac{39}{25}\lambda-1% \\ &\displaystyle{}=(\lambda-1)(\lambda^{2}-\tfrac{14}{25}\lambda+1).\end{aligned}
  142. Q = [ a - b b a ] Q=\begin{bmatrix}a&-b\\ b&a\end{bmatrix}
  143. [ a b c ] . \begin{bmatrix}a\\ b\\ c\end{bmatrix}.
  144. [ r 0 c ] , \begin{bmatrix}r\\ 0\\ c\end{bmatrix},
  145. Q x z Q x y Q = [ 1 0 0 0 0 ] . Q_{xz}Q_{xy}Q=\begin{bmatrix}1&0&0\\ 0&\ast&\ast\\ 0&\ast&\ast\end{bmatrix}.
  146. Q y z Q x z Q x y Q = [ 1 0 0 0 1 0 0 0 1 ] , Q_{yz}Q_{xz}Q_{xy}Q=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix},
  147. Q = Q x y - 1 Q x z - 1 Q y z - 1 . Q=Q_{xy}^{-1}Q_{xz}^{-1}Q_{yz}^{-1}.
  148. k = 1 n - 1 k = n ( n - 1 ) 2 \sum_{k=1}^{n-1}k=\frac{n(n-1)}{2}\,\!
  149. Q 3 × 3 = [ cos θ sin θ \color C a d e t B l u e 0 - sin θ cos θ \color C a d e t B l u e 0 \color C a d e t B l u e 0 \color C a d e t B l u e 0 \color C a d e t B l u e 1 ] Q_{3\times 3}=\begin{bmatrix}\cos\theta&\sin\theta&{\color{CadetBlue}0}\\ -\sin\theta&\cos\theta&{\color{CadetBlue}0}\\ {\color{CadetBlue}0}&{\color{CadetBlue}0}&{\color{CadetBlue}1}\end{bmatrix}
  150. Q 2 × 2 = [ cos θ sin θ - sin θ cos θ ] , Q_{2\times 2}=\begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix},
  151. Q 3 × 3 = [ Q 2 × 2 0 0 T 1 ] . Q_{3\times 3}=\left[\begin{matrix}Q_{2\times 2}&{0}\\ {0}^{T}&1\end{matrix}\right].
  152. Q x ( θ ) = [ 1 0 0 0 cos θ sin θ 0 - sin θ cos θ ] , Q_{{x}}(\theta)=\begin{bmatrix}1&0&0\\ 0&\cos\theta&\sin\theta\\ 0&-\sin\theta&\cos\theta\end{bmatrix},
  153. Q y ( θ ) = [ cos θ 0 - sin θ 0 1 0 sin θ 0 cos θ ] , Q_{{y}}(\theta)=\begin{bmatrix}\cos\theta&0&-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\end{bmatrix},
  154. Q z ( θ ) = [ cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ] , Q_{{z}}(\theta)=\begin{bmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix},
  155. Q u ( θ ) = [ 0 - z y z 0 - x - y x 0 ] sin θ + ( I - u u T ) cos θ + u u T = [ ( 1 - x 2 ) c θ + x 2 - z s θ - x y c θ + x y y s θ - x z c θ + x z z s θ - x y c θ + x y ( 1 - y 2 ) c θ + y 2 - x s θ - y z c θ + y z - y s θ - x z c θ + x z x s θ - y z c θ + y z ( 1 - z 2 ) c θ + z 2 ] = [ x 2 ( 1 - c θ ) + c θ x y ( 1 - c θ ) - z s θ x z ( 1 - c θ ) + y s θ x y ( 1 - c θ ) + z s θ y 2 ( 1 - c θ ) + c θ y z ( 1 - c θ ) - x s θ x z ( 1 - c θ ) - y s θ y z ( 1 - c θ ) + x s θ z 2 ( 1 - c θ ) + c θ ] , \begin{aligned}\displaystyle Q_{{u}}(\theta)&\displaystyle{}=\begin{bmatrix}0&% -z&y\\ z&0&-x\\ -y&x&0\end{bmatrix}\sin\theta+(I-{u}{u}^{T})\cos\theta+{u}{u}^{T}\\ &\displaystyle{}=\begin{bmatrix}(1-x^{2})c_{\theta}+x^{2}&-zs_{\theta}-xyc_{% \theta}+xy&ys_{\theta}-xzc_{\theta}+xz\\ zs_{\theta}-xyc_{\theta}+xy&(1-y^{2})c_{\theta}+y^{2}&-xs_{\theta}-yzc_{\theta% }+yz\\ -ys_{\theta}-xzc_{\theta}+xz&xs_{\theta}-yzc_{\theta}+yz&(1-z^{2})c_{\theta}+z% ^{2}\end{bmatrix}\\ &\displaystyle{}=\begin{bmatrix}x^{2}(1-c_{\theta})+c_{\theta}&xy(1-c_{\theta}% )-zs_{\theta}&xz(1-c_{\theta})+ys_{\theta}\\ xy(1-c_{\theta})+zs_{\theta}&y^{2}(1-c_{\theta})+c_{\theta}&yz(1-c_{\theta})-% xs_{\theta}\\ xz(1-c_{\theta})-ys_{\theta}&yz(1-c_{\theta})+xs_{\theta}&z^{2}(1-c_{\theta})+% c_{\theta}\end{bmatrix},\end{aligned}
  156. S O ( n ) S O ( n + 1 ) S n , SO(n)\hookrightarrow SO(n+1)\to S^{n},\,\!
  157. A ( I + A ) ( I - A ) - 1 , A\mapsto(I+A)(I-A)^{-1},\,\!
  158. [ 0 - z y z 0 - x - y x 0 ] 1 1 + x 2 + y 2 + z 2 [ 1 + x 2 - y 2 - z 2 2 x y - 2 z 2 y + 2 x z 2 x y + 2 z 1 - x 2 + y 2 - z 2 2 y z - 2 x 2 x z - 2 y 2 x + 2 y z 1 - x 2 - y 2 + z 2 ] . \begin{aligned}&\displaystyle\begin{bmatrix}0&-z&y\\ z&0&-x\\ -y&x&0\end{bmatrix}\mapsto\\ &\displaystyle\quad\frac{1}{1+x^{2}+y^{2}+z^{2}}\begin{bmatrix}1+x^{2}-y^{2}-z% ^{2}&2xy-2z&2y+2xz\\ 2xy+2z&1-x^{2}+y^{2}-z^{2}&2yz-2x\\ 2xz-2y&2x+2yz&1-x^{2}-y^{2}+z^{2}\end{bmatrix}.\end{aligned}
  159. R ( θ ) \displaystyle R(\theta)
  160. n × n n×n
  161. n n
  162. n n
  163. S O ( 4 ) SO(4)
  164. 𝐬𝐨 ( n ) \mathbf{so}(n)
  165. S O ( n ) SO(n)
  166. 𝔰 𝔬 ( n ) = 𝔬 ( n ) = { X M n ( ) | X = - X T } , \mathfrak{so}(n)=\mathfrak{o}(n)=\left\{X\in M_{n}(\mathbb{R})|X=-X^{T}\right\},
  167. n n
  168. 𝐨 ( n ) \mathbf{o}(n)
  169. O ( n ) O(n)
  170. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  171. L x = [ 0 0 0 0 0 - 1 0 1 0 ] , L y = [ 0 0 1 0 0 0 - 1 0 0 ] , L z = [ 0 - 1 0 1 0 0 0 0 0 ] . L_{{x}}=\left[\begin{smallmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{smallmatrix}\right],\quad L_{{y}}=\left[\begin{smallmatrix}0&0&1\\ 0&0&0\\ -1&0&0\end{smallmatrix}\right],\quad L_{{z}}=\left[\begin{smallmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{smallmatrix}\right].
  172. A A
  173. A A
  174. 3 × 3 3×3
  175. A 𝐬𝐨 ( 3 ) A∈\mathbf{so}(3)
  176. ω = θ 𝐮 \mathbf{ω}=θ\mathbf{u}
  177. 𝐮 = ( x , y , z ) \mathbf{u}=(x,y,z)
  178. 𝐮 \mathbf{u}
  179. A A
  180. 𝐮 \mathbf{u}
  181. e x p ( A ) exp(A)
  182. θ = θ / 2 + θ / 2 θ={θ}/{2}+{θ}/{2}
  183. exp ( A ) = exp ( θ ( s y m b o l u L ) ) = exp ( [ 0 - z θ y θ z θ 0 - x θ - y θ x θ 0 ] ) = s y m b o l I + 2 cos θ 2 sin θ 2 s y m b o l u L + 2 sin 2 θ 2 ( s y m b o l u L ) 2 , \begin{aligned}\displaystyle\exp(A)&\displaystyle{}=\exp(\theta(symbol{u\cdot L% }))=\exp\left(\left[\begin{smallmatrix}0&-z\theta&y\theta\\ z\theta&0&-x\theta\\ -y\theta&x\theta&0\end{smallmatrix}\right]\right)=symbol{I}+2\cos\frac{\theta}% {2}\sin\frac{\theta}{2}~{}symbol{u\cdot L}+2\sin^{2}\frac{\theta}{2}~{}(symbol% {u\cdot L})^{2},\end{aligned}
  184. c = c o s θ / 2 c=cos{θ}/{2}
  185. s = s i n θ / 2 s=sin{θ}/{2}
  186. 𝐮 \mathbf{u}
  187. θ θ
  188. X X
  189. Y Y
  190. Z = C ( X , Y ) = X + Y + 1 2 [ X , Y ] + 1 12 [ X , [ X , Y ] ] - 1 12 [ Y , [ X , Y ] ] + . Z=C(X,Y)=X+Y+\tfrac{1}{2}[X,Y]+\tfrac{1}{12}[X,[X,Y]]-\tfrac{1}{12}[Y,[X,Y]]+% \cdots~{}.
  191. 3 × 3 3×3
  192. Z = α X + β Y + γ [ X , Y ] , Z=\alpha X+\beta Y+\gamma[X,Y],
  193. n × n n×n
  194. n × n n×n
  195. S O ( n ) SO(n)
  196. S p i n ( n ) Spin(n)
  197. 2 × 2 2×2
  198. S O ( 2 ) SO(2)
  199. S p i n ( 3 ) Spin(3)
  200. 3 × 3 3×3
  201. S p i n ( 3 ) Spin(3)
  202. 4 × 4 4×4
  203. 2 × 2 2×2
  204. S U ( 2 ) SU(2)
  205. { q : || q || = 1 } w + i x + j y + k z [ 1 - 2 y 2 - 2 z 2 2 x y - 2 z w 2 x z + 2 y w 2 x y + 2 z w 1 - 2 x 2 - 2 z 2 2 y z - 2 x w 2 x z - 2 y w 2 y z + 2 x w 1 - 2 x 2 - 2 y 2 ] S O ( 3 ) , \mathbb{H}\supset\{q\in\mathbb{H}:||q||=1\}\ni w+{i}x+{j}y+{k}z\mapsto\left[% \begin{smallmatrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\ 2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\ 2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{smallmatrix}\right]\in SO(3),
  206. SU ( 2 ) [ α β - β ¯ α ¯ ] [ 1 2 ( α 2 - β 2 + α 2 ¯ - β 2 ¯ ) i 2 ( - α 2 - β 2 + α 2 ¯ + β 2 ¯ ) - α β - α ¯ β ¯ i 2 ( α 2 - β 2 - α 2 ¯ + β 2 ¯ ) i 2 ( α 2 + β 2 + α 2 ¯ + β 2 ¯ ) - i ( + α β - α ¯ β ¯ ) α β ¯ + α ¯ β i ( - α β ¯ + α ¯ β ) α α ¯ - β β ¯ ] S O ( 3 ) . \mathrm{SU}(2)\ni\left[\begin{smallmatrix}\alpha&\beta\\ -\overline{\beta}&\overline{\alpha}\end{smallmatrix}\right]\mapsto\left[\begin% {smallmatrix}\frac{1}{2}(\alpha^{2}-\beta^{2}+\overline{\alpha^{2}}-\overline{% \beta^{2}})&\frac{i}{2}(-\alpha^{2}-\beta^{2}+\overline{\alpha^{2}}+\overline{% \beta^{2}})&-\alpha\beta-\overline{\alpha}\overline{\beta}\\ \frac{i}{2}(\alpha^{2}-\beta^{2}-\overline{\alpha^{2}}+\overline{\beta^{2}})&% \frac{i}{2}(\alpha^{2}+\beta^{2}+\overline{\alpha^{2}}+\overline{\beta^{2}})&-% i(+\alpha\beta-\overline{\alpha}\overline{\beta})\\ \alpha\overline{\beta}+\overline{\alpha}\beta&i(-\alpha\overline{\beta}+% \overline{\alpha}\beta)&\alpha\overline{\alpha}-\beta\overline{\beta}\end{% smallmatrix}\right]\in SO(3).
  207. S O ( n ) SO(n)
  208. n > 2 n>2
  209. S O ( 3 ) SO(3)
  210. S U ( 2 ) SU(2)
  211. I + A d θ , I+A\,d\theta~{},
  212. d θ
  213. A S O ( n ) A∈SO(n)
  214. d L x = [ 1 0 0 0 1 - d θ 0 d θ 1 ] . dL_{x}=\left[\begin{smallmatrix}1&0&0\\ 0&1&-d\theta\\ 0&d\theta&1\end{smallmatrix}\right].
  215. 𝐪 = w + x 𝐢 + y 𝐣 + z 𝐤 \mathbf{q}=w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}
  216. Q = [ 1 - 2 y 2 - 2 z 2 2 x y - 2 z w 2 x z + 2 y w 2 x y + 2 z w 1 - 2 x 2 - 2 z 2 2 y z - 2 x w 2 x z - 2 y w 2 y z + 2 x w 1 - 2 x 2 - 2 y 2 ] . Q=\begin{bmatrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\ 2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\ 2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{bmatrix}.
  217. copysign ( x , y ) = sign ( y ) | x | . \operatorname{copysign}(x,y)=\operatorname{sign}(y)\;|x|.
  218. K = 1 3 [ Q x x - Q y y - Q z z Q y x + Q x y Q z x + Q x z Q y z - Q z y Q y x + Q x y Q y y - Q x x - Q z z Q z y + Q y z Q z x - Q x z Q z x + Q x z Q z y + Q y z Q z z - Q x x - Q y y Q x y - Q y x Q y z - Q z y Q z x - Q x z Q x y - Q y x Q x x + Q y y + Q z z ] , K=\frac{1}{3}\begin{bmatrix}Q_{xx}-Q_{yy}-Q_{zz}&Q_{yx}+Q_{xy}&Q_{zx}+Q_{xz}&Q% _{yz}-Q_{zy}\\ Q_{yx}+Q_{xy}&Q_{yy}-Q_{xx}-Q_{zz}&Q_{zy}+Q_{yz}&Q_{zx}-Q_{xz}\\ Q_{zx}+Q_{xz}&Q_{zy}+Q_{yz}&Q_{zz}-Q_{xx}-Q_{yy}&Q_{xy}-Q_{yx}\\ Q_{yz}-Q_{zy}&Q_{zx}-Q_{xz}&Q_{xy}-Q_{yx}&Q_{xx}+Q_{yy}+Q_{zz}\end{bmatrix},
  219. ( Q x x - M x x ) 2 + ( Q x y - M x y ) 2 + ( Q y x - M y x ) 2 + ( Q y y - M y y ) 2 + ( Q x x 2 + Q y x 2 - 1 ) Y x x + ( Q x y 2 + Q y y 2 - 1 ) Y y y + 2 ( Q x x Q x y + Q y x Q y y ) Y x y . \begin{aligned}&\displaystyle\scriptstyle{(Q_{xx}-M_{xx})^{2}+(Q_{xy}-M_{xy})^% {2}}\\ &\displaystyle\scriptstyle{{}+(Q_{yx}-M_{yx})^{2}+(Q_{yy}-M_{yy})^{2}}\\ &\displaystyle\scriptstyle{{}+(Q_{xx}^{2}+Q_{yx}^{2}-1)Y_{xx}+(Q_{xy}^{2}+Q_{% yy}^{2}-1)Y_{yy}}\\ &\displaystyle\scriptstyle{{}+2(Q_{xx}Q_{xy}+Q_{yx}Q_{yy})Y_{xy}.}\end{aligned}
  220. 2 [ Q x x - M x x + Q x x Y x x + Q x y Y x y Q x y - M x y + Q x x Y x y + Q x y Y y y Q y x - M y x + Q y x Y x x + Q y y Y x y Q y y - M y y + Q y x Y x y + Q y y Y y y ] \scriptstyle{2\begin{bmatrix}\scriptstyle{Q_{xx}-M_{xx}+Q_{xx}Y_{xx}+Q_{xy}Y_{% xy}}&\scriptstyle{Q_{xy}-M_{xy}+Q_{xx}Y_{xy}+Q_{xy}Y_{yy}}\\ \scriptstyle{Q_{yx}-M_{yx}+Q_{yx}Y_{xx}+Q_{yy}Y_{xy}}&\scriptstyle{Q_{yy}-M_{% yy}+Q_{yx}Y_{xy}+Q_{yy}Y_{yy}}\end{bmatrix}}
  221. 0 = 2 ( Q - M ) + 2 Q Y , 0=2(Q-M)+2QY,\,\!
  222. M = Q ( I + Y ) = Q S , M=Q(I+Y)=QS,\,\!
  223. S 2 = ( Q T M ) T ( Q T M ) = M T Q Q T M = M T M S^{2}=(Q^{T}M)^{T}(Q^{T}M)=M^{T}QQ^{T}M=M^{T}M\,\!
  224. c = cos θ s = sin θ C = 1 - c \begin{aligned}\displaystyle c&\displaystyle=&\displaystyle\cos\theta\\ \displaystyle s&\displaystyle=&\displaystyle\sin\theta\\ \displaystyle C&\displaystyle=&\displaystyle 1-c\end{aligned}
  225. Q ( θ ) = [ x x C + c x y C - z s x z C + y s y x C + z s y y C + c y z C - x s z x C - y s z y C + x s z z C + c ] Q(\theta)=\begin{bmatrix}xxC+c&xyC-zs&xzC+ys\\ yxC+zs&yyC+c&yzC-xs\\ zxC-ys&zyC+xs&zzC+c\end{bmatrix}
  226. x \displaystyle x
  227. Q ( θ 1 , θ 2 , θ 3 ) = Q x ( θ 1 ) Q y ( θ 2 ) Q z ( θ 3 ) , Q(\theta_{1},\theta_{2},\theta_{3})=Q_{{x}}(\theta_{1})Q_{{y}}(\theta_{2})Q_{{% z}}(\theta_{3}),\,\!
  228. Q ( θ 1 , θ 2 , θ 3 ) = Q z ( θ 3 ) Q y ( θ 2 ) Q x ( θ 1 ) . Q(\theta_{1},\theta_{2},\theta_{3})=Q_{{z}}(\theta_{3})Q_{{y}}(\theta_{2})Q_{{% x}}(\theta_{1}).\,\!
  229. θ θ
  230. θ θ
  231. 𝐮 𝐮 = ( [ 𝐮 ] × ) 2 - 𝐈 \mathbf{u}\otimes\mathbf{u}=([\mathbf{u}]_{\times})^{2}-{\mathbf{I}}
  232. 𝐑 = 𝐈 + ( sin θ ) [ 𝐮 ] × + ( 1 - cos θ ) ( [ 𝐮 ] × ) 2 . \mathbf{R}=\mathbf{I}+(\sin\theta)[\mathbf{u}]_{\times}+(1-\cos\theta)([% \mathbf{u}]_{\times})^{2}~{}.
  233. e 2 A - I + A I - A = - 2 3 A 3 + O ( A 4 ) . e^{2A}-\frac{I+A}{I-A}=-\frac{2}{3}A^{3}+\mathrm{O}(A^{4})~{}.
  234. A A
  235. e x p ( 2 a r c t a n h A ) exp(2arctanhA)
  236. || X || + || Y || < l o g 2 ||X||+||Y||<log2

Rotation_operator_(quantum_mechanics).html

  1. | α R = D ( R ) | α |\alpha\rangle_{R}=D(R)|\alpha\rangle
  2. D ( 𝐧 ^ , ϕ ) = exp ( - i ϕ 𝐧 ^ 𝐉 ) D(\mathbf{\hat{n}},\phi)=\exp\left(-i\phi\frac{\mathbf{\hat{n}}\cdot\mathbf{J}% }{\hbar}\right)
  3. 𝐧 ^ \mathbf{\hat{n}}
  4. 𝐉 \mathbf{J}
  5. R ( z , θ ) \,\mbox{R}~{}(z,\theta)
  6. z \,z
  7. θ \,\theta
  8. T ( a ) \,\mbox{T}~{}(a)
  9. | x |x\rangle
  10. T ( a ) | x = | x + a \mbox{T}~{}(a)|x\rangle=|x+a\rangle
  11. T ( 0 ) = 1 \,\mbox{T}~{}(0)=1
  12. T ( a ) T ( d a ) | x = T ( a ) | x + d a = | x + a + d a = T ( a + d a ) | x \,\mbox{T}~{}(a)\mbox{T}~{}(da)|x\rangle=\mbox{T}~{}(a)|x+da\rangle=|x+a+da% \rangle=\mbox{T}~{}(a+da)|x\rangle\Rightarrow
  13. T ( a ) T ( d a ) = T ( a + d a ) \,\mbox{T}~{}(a)\mbox{T}~{}(da)=\mbox{T}~{}(a+da)
  14. T ( d a ) = T ( 0 ) + d T ( 0 ) d a d a + = 1 - i h p x d a \,\mbox{T}~{}(da)=\mbox{T}~{}(0)+\frac{d\mbox{T}~{}(0)}{da}da+...=1-\frac{i}{h% }\ p_{x}\ da
  15. p x = i h d T ( 0 ) d a \,p_{x}=ih\frac{d\mbox{T}~{}(0)}{da}
  16. T ( a + d a ) = T ( a ) T ( d a ) = T ( a ) ( 1 - i h p x d a ) \,\mbox{T}~{}(a+da)=\mbox{T}~{}(a)\mbox{T}~{}(da)=\mbox{T}~{}(a)\left(1-\frac{% i}{h}p_{x}da\right)\Rightarrow
  17. [ T ( a + d a ) - T ( a ) ] / d a = d T d a = - i h p x T ( a ) \,[\mbox{T}~{}(a+da)-\mbox{T}~{}(a)]/da=\frac{d\mbox{T}~{}}{da}=-\frac{i}{h}p_% {x}\mbox{T}~{}(a)
  18. T ( a ) = exp ( - i h p x a ) \,\mbox{T}~{}(a)=\mbox{exp}~{}\left(-\frac{i}{h}p_{x}a\right)
  19. H \,H
  20. x \,x
  21. p x \,p_{x}
  22. [ p x , H ] = 0 \,[p_{x},H]=0
  23. [ H , T ( a ) ] = 0 \,[H,\mbox{T}~{}(a)]=0
  24. l = r × p \,l=r\times p
  25. r \,r
  26. p \,p
  27. d t \,dt
  28. x = r cos ( t + d t ) = x - y d t + \,x^{\prime}=r\cos(t+dt)=x-ydt+...
  29. y = r sin ( t + d t ) = y + x d t + \,y^{\prime}=r\sin(t+dt)=y+xdt+...
  30. R ( z , d t ) | r \,\mbox{R}~{}(z,dt)|r\rangle
  31. = R ( z , d t ) | x , y , z =\mbox{R}~{}(z,dt)|x,y,z\rangle
  32. = | x - y d t , y + x d t , z =|x-ydt,y+xdt,z\rangle
  33. = T ( - y d t ) x T ( x d t ) y | x , y , z =\mbox{T}~{}_{x}(-ydt)\mbox{T}~{}_{y}(xdt)|x,y,z\rangle
  34. = T ( - y d t ) x T ( x d t ) y | r =\mbox{T}~{}_{x}(-ydt)\mbox{T}~{}_{y}(xdt)|r\rangle
  35. R ( z , d t ) = T ( - y d t ) x T ( x d t ) y \,\mbox{R}~{}(z,dt)=\mbox{T}~{}_{x}(-ydt)\mbox{T}~{}_{y}(xdt)
  36. T k ( a ) = exp ( - i h p k a ) \,T_{k}(a)=\exp\left(-\frac{i}{h}\ p_{k}\ a\right)
  37. k = x , y \,k=x,y
  38. R ( z , d t ) = exp [ - i h ( x p y - y p x ) d t ] \,\mbox{R}~{}(z,dt)=\exp\left[-\frac{i}{h}\ (xp_{y}-yp_{x})dt\right]
  39. = exp ( - i h l z d t ) = 1 - i h l z d t + =\exp\left(-\frac{i}{h}\ l_{z}dt\right)=1-\frac{i}{h}l_{z}dt+...
  40. t \,t
  41. R ( z , 0 ) = 1 \mbox{R}~{}(z,0)=1
  42. R ( z , t + d t ) = R ( z , t ) R ( z , d t ) \,\mbox{R}~{}(z,t+dt)=\mbox{R}~{}(z,t)\mbox{R}~{}(z,dt)\Rightarrow
  43. [ R ( z , t + d t ) - R ( z , t ) ] / d t = d R / d t \,[\mbox{R}~{}(z,t+dt)-\mbox{R}~{}(z,t)]/dt=d\mbox{R}~{}/dt
  44. = R ( z , t ) [ R ( z , d t ) - 1 ] / d t \,=\mbox{R}~{}(z,t)[\mbox{R}~{}(z,dt)-1]/dt
  45. = - i h l z R ( z , t ) \,=-\frac{i}{h}l_{z}\mbox{R}~{}(z,t)\Rightarrow
  46. R ( z , t ) = exp ( - i h t l z ) \,\mbox{R}~{}(z,t)=\exp\left(-\frac{i}{h}\ t\ l_{z}\right)
  47. H \,H
  48. [ l z , H ] = 0 \,[l_{z},H]=0
  49. [ R ( z , t ) , H ] = 0 \,[\mbox{R}~{}(z,t),H]=0
  50. l z \,l_{z}
  51. S y = h 2 σ y \,S_{y}=\frac{h}{2}\sigma_{y}
  52. D ( y , t ) = exp ( - i t 2 σ y ) \,\mbox{D}~{}(y,t)=\exp\left(-i\frac{t}{2}\sigma_{y}\right)
  53. A \,A
  54. A = P A P - 1 \,A^{\prime}=PAP^{-1}
  55. P \,P
  56. b \,b
  57. c \,c
  58. t \,t
  59. S b \,S_{b}
  60. S c \,S_{c}
  61. S c = D ( y , t ) S b D ( y , t ) - 1 \,S_{c}=\mbox{D}~{}(y,t)S_{b}\mbox{D}~{}^{-1}(y,t)
  62. S b | b + = 2 | b + \,S_{b}|b+\rangle=\frac{\hbar}{2}|b+\rangle
  63. S c | c + = 2 | c + \,S_{c}|c+\rangle=\frac{\hbar}{2}|c+\rangle
  64. | b + \,|b+\rangle
  65. | c + \,|c+\rangle
  66. 2 | c + = S c | c + = D ( y , t ) S b D ( y , t ) - 1 | c + \,\frac{\hbar}{2}|c+\rangle=S_{c}|c+\rangle=\mbox{D}~{}(y,t)S_{b}\mbox{D}~{}^{% -1}(y,t)|c+\rangle\Rightarrow
  67. S b D ( y , t ) - 1 | c + = 2 D ( y , t ) - 1 | c + \,S_{b}\mbox{D}~{}^{-1}(y,t)|c+\rangle=\frac{\hbar}{2}\mbox{D}~{}^{-1}(y,t)|c+\rangle
  68. S b | b + = 2 | b + \,S_{b}|b+\rangle=\frac{\hbar}{2}|b+\rangle
  69. | b + = D - 1 ( y , t ) | c + \,|b+\rangle=D^{-1}(y,t)|c+\rangle
  70. | c + \,|c+\rangle
  71. t \,t
  72. | b + \,|b+\rangle

Rotational_speed.html

  1. ω c y c \omega_{cyc}
  2. ω c y c \omega_{cyc}
  3. v = 2 π r ω c y c v=2\pi r\omega_{cyc}
  4. v = r ω r a d v=r\omega_{rad}
  5. ω c y c = v / 2 π r \omega_{cyc}=v/2\pi r
  6. ω r a d = v / r \omega_{rad}=v/r
  7. ω c y c = ω r a d / 2 π \omega_{cyc}=\omega_{rad}/2\pi\,
  8. ω c y c = ω d e g / 360 \omega_{cyc}=\omega_{deg}/360\,
  9. ω c y c \omega_{cyc}\,
  10. ω r a d \omega_{rad}\,
  11. ω d e g \omega_{deg}\,

Rotational_symmetry.html

  1. m = 3 m=3
  2. 3 / 7 {3}/{7}
  3. n 2 n≥2
  4. 1 3 3 \frac{1}{3}\sqrt{3}
  5. 1 2 2 \frac{1}{2}\sqrt{2}

Rotational_transition.html

  1. [ - 2 2 μ R 2 R ( R 2 R ) + Φ s | N 2 | Φ s 2 μ R 2 + E s ( R ) - E ] F s ( 𝐑 ) = 0 \left[-\frac{\hbar^{2}}{2\mu R^{2}}\frac{\partial}{\partial R}\left(R^{2}\frac% {\partial}{\partial R}\right)+\frac{\langle\Phi_{s}|N^{2}|\Phi_{s}\rangle}{2% \mu R^{2}}+E_{s}(R)-E\right]F_{s}(\mathbf{R})=0
  2. N 2 = - 2 [ 1 sin Θ Θ ( sin Θ Θ ) + 1 sin 2 Θ 2 Φ 2 ] N^{2}=-\hbar^{2}\left[\frac{1}{\sin\Theta}\frac{\partial}{\partial\Theta}\left% (\sin\Theta\frac{\partial}{\partial\Theta}\right)+\frac{1}{\sin^{2}\Theta}% \frac{\partial^{2}}{\partial\Phi^{2}}\right]
  3. Ψ s = F s ( 𝐑 ) Φ s ( 𝐑 , 𝐫 1 , 𝐫 2 , . , 𝐫 N ) \Psi_{s}=F_{s}(\mathbf{R})\Phi_{s}(\mathbf{R},\mathbf{r}_{1},\mathbf{r}_{2},..% ..,\mathbf{r}_{N})
  4. 𝐍 = 𝐉 - 𝐋 \mathbf{N}=\mathbf{J}-\mathbf{L}
  5. 𝐍 = 𝐑 × 𝐏 \mathbf{N}=\mathbf{R}\times\mathbf{P}
  6. J z = L z J_{z}=L_{z}
  7. J 2 Ψ s = J ( J + 1 ) 2 Ψ s J^{2}\Psi_{s}=J(J+1)\hbar^{2}\Psi_{s}
  8. J z Ψ s = M j Ψ s J_{z}\Psi_{s}=M_{j}\hbar\Psi_{s}
  9. L z Φ s = ± Λ Φ s L_{z}\Phi_{s}=\pm\Lambda\hbar\Phi_{s}
  10. J = Λ , Λ + 1 , Λ + 2 , J=\Lambda,\Lambda+1,\Lambda+2,......
  11. Ψ s | L x | Ψ s = L x = 0 \langle\Psi_{s}|L_{x}|\Psi_{s}\rangle=\langle L_{x}\rangle=0
  12. Ψ s | L y | Ψ s = L y = 0 \langle\Psi_{s}|L_{y}|\Psi_{s}\rangle=\langle L_{y}\rangle=0
  13. 𝐉 . 𝐋 = J z L z = L z 2 \langle\mathbf{J}.\mathbf{L}\rangle=\langle J_{z}L_{z}\rangle=\langle{L_{z}}^{% 2}\rangle
  14. Φ s | N 2 | Φ s F s ( 𝐑 ) = Φ s | J 2 + L 2 - 2 𝐉 . 𝐋 | Φ s F s ( 𝐑 ) = 2 [ J ( J + 1 ) - Λ 2 ] F s ( 𝐑 ) + Φ s | L x 2 + L y 2 | Φ s F s ( 𝐑 ) \langle\Phi_{s}|N^{2}|\Phi_{s}\rangle F_{s}(\mathbf{R})=\langle\Phi_{s}|J^{2}+% L^{2}-2\mathbf{J}.\mathbf{L}|\Phi_{s}\rangle F_{s}(\mathbf{R})=\hbar^{2}[J(J+1% )-\Lambda^{2}]F_{s}(\mathbf{R})+\langle\Phi_{s}|{L_{x}}^{2}+{L_{y}}^{2}|\Phi_{% s}\rangle F_{s}(\mathbf{R})
  15. - 2 2 μ R 2 [ R ( R 2 R ) - J ( J + 1 ) ] F s ( 𝐑 ) + [ E s ( R ) - E ] F s ( 𝐑 ) = 0 -\frac{\hbar^{2}}{2\mu R^{2}}\left[\frac{\partial}{\partial R}\left(R^{2}\frac% {\partial}{\partial R}\right)-J(J+1)\right]F_{s}(\mathbf{R})+[{E^{\prime}}_{s}% (R)-E]F_{s}(\mathbf{R})=0
  16. E s ( R ) = E s ( R ) - Λ 2 2 2 μ R 2 + 1 2 μ R 2 Φ s | L x 2 + L y 2 | Φ s {E^{\prime}}_{s}(R)=E_{s}(R)-\frac{\Lambda^{2}\hbar^{2}}{2\mu R^{2}}+\frac{1}{% 2\mu R^{2}}\langle\Phi_{s}|{L_{x}}^{2}+{L_{y}}^{2}|\Phi_{s}\rangle
  17. E r = 2 2 μ R 0 2 J ( J + 1 ) = 2 2 I 0 J ( J + 1 ) = B J ( J + 1 ) E_{r}=\frac{\hbar^{2}}{2\mu{R_{0}}^{2}}J(J+1)=\frac{\hbar^{2}}{2I_{0}}J(J+1)=% BJ(J+1)
  18. J = Λ , Λ + 1 , Λ + 2 , J=\Lambda,\Lambda+1,\Lambda+2,......
  19. ω = E r ( J + 1 ) - E r ( J ) = 2 B ( J + 1 ) \hbar\omega=E_{r}(J+1)-E_{r}(J)=2B(J+1)

Rouché's_theorem.html

  1. K \partial K
  2. f , g ( G ) f,\,g\in\mathcal{H}(G)
  3. K K
  4. | f ( z ) - g ( z ) | < | f ( z ) | + | g ( z ) | ( z K ) |f(z)-g(z)|<|f(z)|+|g(z)|\qquad\left(z\in\partial K\right)
  5. K \partial K
  6. f ( z ) := f ( z ) + g ( z ) f(z):=f(z)+g(z)
  7. g ( z ) := f ( z ) g(z):=f(z)
  8. z 5 + 3 z 3 + 7 z^{5}+3z^{3}+7
  9. | z | < 2 |z|<2
  10. | 3 z 3 + 7 | < 32 = | z 5 | |3z^{3}+7|<32=|z^{5}|
  11. | z | = 2 |z|=2
  12. z 5 z^{5}
  13. z 2 + 2 a z + b 2 z^{2}+2az+b^{2}
  14. a > b > 0 a>b>0
  15. - a ± a 2 - b 2 -a\pm\sqrt{a^{2}-b^{2}}
  16. | z 2 + b 2 | 2 b 2 < 2 a | z | |z^{2}+b^{2}|\leq 2b^{2}<2a|z|
  17. | z | = b |z|=b
  18. | z | < b |z|<b
  19. a + a 2 - b 2 a+\sqrt{a^{2}-b^{2}}
  20. a - a 2 - b 2 a-\sqrt{a^{2}-b^{2}}
  21. p ( z ) = a 0 + a 1 z + a 2 z 2 + + a n z n , a n 0 p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots+a_{n}z^{n},\quad a_{n}\neq 0\,
  22. R > 0 R>0
  23. | a 0 + a 1 z + + a n - 1 z n - 1 | j = 0 n - 1 | a j | R n - 1 < | a n | R n = | a n z n | for | z | = R . |a_{0}+a_{1}z+\cdots+a_{n-1}z^{n-1}|\leq\sum_{j=0}^{n-1}|a_{j}|R^{n-1}<|a_{n}|% R^{n}=|a_{n}z^{n}|\,\text{ for }|z|=R.
  24. a n z n a_{n}z^{n}
  25. n n
  26. | z | < R |z|<R
  27. R > 0 R>0
  28. p p
  29. C : [ 0 , 1 ] C\colon[0,1]\to\mathbb{C}
  30. K \partial K
  31. K \partial K
  32. 1 2 π i C f ( z ) f ( z ) d z = 1 2 π i f C d z z = Ind f C ( 0 ) , \frac{1}{2\pi i}\oint_{C}\frac{f^{\prime}(z)}{f(z)}\,dz=\frac{1}{2\pi i}\oint_% {f\circ C}\frac{dz}{z}=\mathrm{Ind}_{f\circ C}(0),
  33. f C f\circ C
  34. H t ( x ) = ( 1 - t ) f ( C ( x ) ) + t g ( C ( x ) ) H_{t}(x)=(1-t)f(C(x))+tg(C(x))
  35. f C f\circ C
  36. g C g\circ C
  37. I ( t ) = Ind H t ( 0 ) = 1 2 π i H t d z z I(t)=\mathrm{Ind}_{H_{t}}(0)=\frac{1}{2\pi i}\oint_{H_{t}}\frac{dz}{z}
  38. N f ( K ) = Ind f C ( 0 ) = Ind g C ( 0 ) = N g ( K ) . N_{f}(K)=\mathrm{Ind}_{f\circ C}(0)=\mathrm{Ind}_{g\circ C}(0)=N_{g}(K).

Roulette_(curve).html

  1. r , f : r,f:\mathbb{R}\to\mathbb{C}
  2. r r
  3. f f
  4. r ( 0 ) = f ( 0 ) r(0)=f(0)
  5. r ( 0 ) = f ( 0 ) r^{\prime}(0)=f^{\prime}(0)
  6. | r ( t ) | = | f ( t ) | 0 |r^{\prime}(t)|=|f^{\prime}(t)|\neq 0
  7. t t
  8. p p\in\mathbb{C}
  9. r r
  10. f f
  11. t f ( t ) + ( p - r ( t ) ) f ( t ) r ( t ) . t\mapsto f(t)+(p-r(t)){f^{\prime}(t)\over r^{\prime}(t)}.
  12. f ( t ) = t + i ( cosh ( t ) - 1 ) r ( t ) = sinh ( t ) f(t)=t+i(\cosh(t)-1)\qquad r(t)=\sinh(t)
  13. f ( t ) = 1 + i sinh ( t ) r ( t ) = cosh ( t ) . f^{\prime}(t)=1+i\sinh(t)\qquad r^{\prime}(t)=\cosh(t).
  14. | f ( t ) | |f^{\prime}(t)|\,
  15. = 1 2 + sinh 2 ( t ) =\sqrt{1^{2}+\sinh^{2}(t)}
  16. = cosh 2 ( t ) =\sqrt{\cosh^{2}(t)}
  17. = | r ( t ) | . =|r^{\prime}(t)|.\,
  18. f ( t ) + ( p - r ( t ) ) f ( t ) r ( t ) = t - i + p - sinh ( t ) + i ( 1 + p sinh ( t ) ) cosh ( t ) = t - i + ( p + i ) 1 + i sinh ( t ) cosh ( t ) . f(t)+(p-r(t)){f^{\prime}(t)\over r^{\prime}(t)}=t-i+{p-\sinh(t)+i(1+p\sinh(t))% \over\cosh(t)}=t-i+(p+i){1+i\sinh(t)\over\cosh(t)}.

Round-robin_tournament.html

  1. n n
  2. n 2 ( n - 1 ) \begin{matrix}\frac{n}{2}\end{matrix}(n-1)
  3. n n
  4. ( n - 1 ) (n-1)
  5. n 2 \begin{matrix}\frac{n}{2}\end{matrix}
  6. n n
  7. n n
  8. n - 1 2 \begin{matrix}\frac{n-1}{2}\end{matrix}
  9. ( n - 1 ) (n-1)
  10. n 2 \begin{matrix}\frac{n}{2}\end{matrix}
  11. n 2 \begin{matrix}\frac{n}{2}\end{matrix}
  12. n n
  13. ( n - 1 ) (n-1)
  14. ( n 2 - 1 ) (\begin{matrix}\frac{n}{2}\end{matrix}-1)
  15. n / 2 n/2
  16. n - 1 n-1
  17. n n
  18. n n
  19. n n

Row_vector.html

  1. 𝐱 = [ x 1 x 2 x m ] . \mathbf{x}=\begin{bmatrix}x_{1}&x_{2}&\dots&x_{m}\end{bmatrix}.
  2. [ x 1 x 2 x m ] T = [ x 1 x 2 x m ] . \begin{bmatrix}x_{1}\;x_{2}\;\dots\;x_{m}\end{bmatrix}^{\rm T}=\begin{bmatrix}% x_{1}\\ x_{2}\\ \vdots\\ x_{m}\end{bmatrix}.
  3. 𝐱 = [ x 1 , x 2 , , x m ] . \mathbf{x}=\begin{bmatrix}x_{1},x_{2},\dots,x_{m}\end{bmatrix}.
  4. 𝐚 𝐛 = [ a 1 a 2 a 3 ] [ b 1 b 2 b 3 ] . \mathbf{a}\cdot\mathbf{b}=\begin{bmatrix}a_{1}&a_{2}&a_{3}\end{bmatrix}\begin{% bmatrix}b_{1}\\ b_{2}\\ b_{3}\end{bmatrix}.
  5. α ( S T ) = ( α S ) T = β T = γ \alpha(ST)=(\alpha S)T=\beta T=\gamma

Rudvalis_group.html

  1. × 10 1 1 \times 10^{1}1
  2. ( 1 + i ) (1+i)

Ruffini's_rule.html

  1. P ( x ) = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}
  2. Q ( x ) = x - r Q(x)=x-r\,\!
  3. R ( x ) = b n - 1 x n - 1 + b n - 2 x n - 2 + + b 1 x + b 0 R(x)=b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+\cdots+b_{1}x+b_{0}
  4. P ( x ) = 2 x 3 + 3 x 2 - 4 P(x)=2x^{3}+3x^{2}-4\,\!
  5. Q ( x ) = x + 1. Q(x)=x+1.\,\!
  6. Q ( x ) = x + 1 = x - ( - 1 ) . Q(x)=x+1=x-(-1).\,\!
  7. P ( x ) = Q ( x ) R ( x ) + s P(x)=Q(x)R(x)+s\,\!
  8. R ( x ) = 2 x 2 + x - 1 R(x)=2x^{2}+x-1\,\!
  9. s = - 3 ; 2 x 3 + 3 x 2 - 4 = ( 2 x 2 + x - 1 ) ( x + 1 ) - 3 s=-3;\quad\Rightarrow 2x^{3}+3x^{2}-4=(2x^{2}+x-1)(x+1)-3\!
  10. P ( x ) = x 3 + 2 x 2 - x - 2 = 0 , P(x)=x^{3}+2x^{2}-x-2=0\,\!,
  11. Possible roots: { + 1 , - 1 , + 2 , - 2 } . \mbox{Possible roots:}~{}\left\{+1,-1,+2,-2\right\}.
  12. x 1 = + 1 x_{1}=+1\,\!
  13. x 2 = - 1 x_{2}=-1\,\!
  14. x 3 = - 2 x_{3}=-2\,\!
  15. x 1 = - 1 x_{1}=-1\,\!
  16. x 2 = + 1 x_{2}=+1\,\!
  17. x 3 = - 2 x_{3}=-2\,\!
  18. P ( x ) = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}\,\!
  19. R = { roots of P ( x ) } R=\left\{\mbox{roots of }~{}P(x)\in\mathbb{Q}\right\}\,\!
  20. R ( x ) = a n ( x - r ) for all r R . R(x)=a_{n}{\prod(x-r)}\mbox{ for all }~{}r\in R.\,\!
  21. S ( x ) = P ( x ) R ( x ) S(x)=\frac{P(x)}{R(x)}\,\!
  22. P ( x ) = R ( x ) S ( x ) . P(x)=R(x)\cdot S(x).\,\!
  23. P ( x ) = x 3 + 2 x 2 - x - 2. P(x)=x^{3}+2x^{2}-x-2.\,\!
  24. R = { + 1 , - 1 , - 2 } . R=\left\{+1,-1,-2\right\}.\,\!
  25. R ( x ) = 1 ( x - 1 ) ( x + 1 ) ( x + 2 ) . R(x)=1(x-1)(x+1)(x+2).\,\!
  26. S ( x ) = 1. S(x)=1.\,\!
  27. P ( x ) = ( x - 1 ) ( x + 1 ) ( x + 2 ) . P(x)=(x-1)(x+1)(x+2).\,\!
  28. P ( x ) = 2 x 4 - 3 x 3 + x 2 - 2 x - 8. P(x)=2x^{4}-3x^{3}+x^{2}-2x-8.\,\!
  29. R = { - 1 , + 2 } . R=\left\{-1,+2\right\}.\,\!
  30. R ( x ) = ( x + 1 ) ( x - 2 ) . R(x)=(x+1)(x-2).\,\!
  31. S ( x ) = 2 x 2 - x + 4. S(x)=2x^{2}-x+4.\,\!
  32. S ( x ) 1 S(x){\neq}1
  33. P ( x ) = ( x + 1 ) ( x - 2 ) ( 2 x 2 - x + 4 ) . P(x)=(x+1)(x-2)(2x^{2}-x+4).\,\!
  34. P ( x ) = 2 x 4 - 3 x 3 + x 2 - 2 x - 8. P(x)=2x^{4}-3x^{3}+x^{2}-2x-8.\,\!
  35. P ( x ) = ( x + 1 ) ( x - 2 ) ( 2 x 2 - x + 4 ) . P(x)=(x+1)(x-2)(2x^{2}-x+4).\,\!
  36. 2 x 2 - x + 4 = 0. {2x^{2}-x+4}=0.\,\!
  37. x = - b ± b 2 - 4 a c 2 a = 1 ± ( - 1 ) 2 - 4 2 4 2 2 = 1 ± - 31 4 x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{1\pm\sqrt{(-1)^{2}-4\cdot 2\cdot 4}}{% 2\cdot 2}=\frac{1\pm\sqrt{-31}}{4}\,\!
  38. x 1 = 1 + - 31 4 x_{1}=\frac{1+\sqrt{-31}}{4}\,\!
  39. x 2 = 1 - - 31 4 . x_{2}=\frac{1-\sqrt{-31}}{4}.\,\!
  40. P ( x ) = 2 ( x + 1 ) ( x - 2 ) ( x - 1 + i 31 4 ) ( x - 1 - i 31 4 ) . P(x)=2(x+1)(x-2)(x-\frac{1+i\sqrt{31}}{4})(x-\frac{1-i\sqrt{31}}{4}).\,\!

Rule_of_succession.html

  1. P ( X n + 1 = 1 X 1 + + X n = s ) = s + 1 n + 2 . P(X_{n+1}=1\mid X_{1}+\cdots+X_{n}=s)={s+1\over n+2}.
  2. P ( X n + 1 = 1 X 1 + + X n = s ) = s n . P^{\prime}(X_{n+1}=1\mid X_{1}+\cdots+X_{n}=s)={s\over n}.
  3. s = 0 s=0
  4. s = n s=n
  5. P P
  6. P P^{\prime}
  7. L ( p ) = P ( X 1 = x 1 , , X n = x n p ) = i = 1 n p x i ( 1 - p ) 1 - x i = p s ( 1 - p ) n - s L(p)=P(X_{1}=x_{1},\ldots,X_{n}=x_{n}\mid p)=\prod_{i=1}^{n}p^{x_{i}}(1-p)^{1-% x_{i}}=p^{s}(1-p)^{n-s}
  8. f ( p ) = ( n + 1 ) ! s ! ( n - s ) ! p s ( 1 - p ) n - s . f(p)={(n+1)!\over s!(n-s)!}p^{s}(1-p)^{n-s}.
  9. 0 1 p f ( p ) d p = s + 1 n + 2 . \int_{0}^{1}pf(p)\,dp={s+1\over n+2}.
  10. P ( X n + 1 = 1 X i = x i for i = 1 , , n ) = s + 1 n + 2 . P(X_{n+1}=1\mid X_{i}=x_{i}\,\text{ for }i=1,\dots,n)={s+1\over n+2}.
  11. P ( X n + 1 = 1 X i = x i for i = 1 , , n ) = s n . P^{\prime}(X_{n+1}=1\mid X_{i}=x_{i}\,\text{ for }i=1,\dots,n)={s\over n}.
  12. Hyp ( s | N , n , S ) \mathrm{Hyp}(s|N,n,S)
  13. Bin ( r | n , p ) \mathrm{Bin}(r|n,p)
  14. N , S N,S\rightarrow\infty
  15. p = S N p={S\over N}
  16. S S
  17. N N
  18. 1 p ( 1 - p ) {1\over p(1-p)}
  19. 1 S ( N - S ) {1\over S(N-S)}
  20. 1 S N - 1 1\leq S\leq N-1
  21. N N
  22. p p
  23. S S
  24. N N
  25. S S
  26. P ( S | N , n , s ) 1 S ( N - S ) ( S s ) ( N - S n - s ) S ! ( N - S ) ! S ( N - S ) ( S - s ) ! ( N - S - [ n - s ] ) ! P(S|N,n,s)\propto{1\over S(N-S)}{S\choose s}{N-S\choose n-s}\propto{S!(N-S)!% \over S(N-S)(S-s)!(N-S-[n-s])!}
  27. P ( S | N , n , s = 0 ) ( N - S - 1 ) ! S ( N - S - n ) ! = j = 1 n - 1 ( N - S - j ) S P(S|N,n,s=0)\propto{(N-S-1)!\over S(N-S-n)!}={\prod_{j=1}^{n-1}(N-S-j)\over S}
  28. P ( S | N , n , s = 0 ) = j = 1 n - 1 ( N - S - j ) S R = 1 N - n j = 1 n - 1 ( N - R - j ) R P(S|N,n,s=0)={\prod_{j=1}^{n-1}(N-S-j)\over S\sum_{R=1}^{N-n}{\prod_{j=1}^{n-1% }(N-R-j)\over R}}
  29. p = S N p={S\over N}
  30. E ( S N | n , s = 0 , N ) = 1 N S = 1 N - n S P ( S | N , n = 1 , s = 0 ) = 1 N S = 1 N - n j = 1 n - 1 ( N - S - j ) R = 1 N - n j = 1 n - 1 ( N - R - j ) R E\left({S\over N}|n,s=0,N\right)={1\over N}\sum_{S=1}^{N-n}SP(S|N,n=1,s=0)={1% \over N}{\sum_{S=1}^{N-n}\prod_{j=1}^{n-1}(N-S-j)\over\sum_{R=1}^{N-n}{\prod_{% j=1}^{n-1}(N-R-j)\over R}}
  31. j = 1 n - 1 ( N - R - j ) ( N - R ) n - 1 \prod_{j=1}^{n-1}(N-R-j)\approx(N-R)^{n-1}
  32. S = 1 N - n j = 1 n - 1 ( N - S - j ) 1 N - n ( N - S ) n - 1 d S = ( N - 1 ) n - n n n N n n \sum_{S=1}^{N-n}\prod_{j=1}^{n-1}(N-S-j)\approx\int_{1}^{N-n}(N-S)^{n-1}\,dS={% (N-1)^{n}-n^{n}\over n}\approx{N^{n}\over n}
  33. R = 1 N - n j = 1 n - 1 ( N - R - j ) R \displaystyle\sum_{R=1}^{N-n}{\prod_{j=1}^{n-1}(N-R-j)\over R}
  34. E ( S N | n , s = 0 , N ) 1 N N n n N n - 1 ln ( N ) = 1 n [ ln ( N ) ] = log 10 ( e ) n [ log 10 ( N ) ] = 0.434294 n [ log 10 ( N ) ] E\left({S\over N}|n,s=0,N\right)\approx{1\over N}{{N^{n}\over n}\over N^{n-1}% \ln(N)}={1\over n[\ln(N)]}={\log_{10}(e)\over n[\log_{10}(N)]}={0.434294\over n% [\log_{10}(N)]}
  35. E ( S N n , s = 0 , N = 10 k ) 0.434294 n k E\left({S\over N}\mid n,s=0,N=10^{k}\right)\approx{0.434294\over nk}
  36. p = s n p={s\over n}
  37. f ( p 1 , , p m n 1 , , n m , I ) = { Γ ( i = 1 m ( n i + 1 ) ) i = 1 m Γ ( n i + 1 ) p 1 n 1 p m n m , i = 1 m p i = 1 0 otherwise. f(p_{1},\ldots,p_{m}\mid n_{1},\ldots,n_{m},I)=\begin{cases}{\displaystyle% \frac{\Gamma\left(\sum_{i=1}^{m}(n_{i}+1)\right)}{\prod_{i=1}^{m}\Gamma(n_{i}+% 1)}p_{1}^{n_{1}}\cdots p_{m}^{n_{m}}},&\sum_{i=1}^{m}p_{i}=1\\ \\ 0&\,\text{otherwise.}\end{cases}
  38. P ( A i | n 1 , , n m , I m ) = n i + 1 n + m . P(A_{i}|n_{1},\ldots,n_{m},I_{m})={n_{i}+1\over n+m}.
  39. P ( success | n 1 , , n m , I m ) = s + c n + m , P(\,\text{success}|n_{1},\ldots,n_{m},I_{m})={s+c\over n+m},
  40. P r ( data | something else , I ) Pr(\,\text{data}|\,\text{something else},I)
  41. s + 0.5 n + 1 \frac{s+0.5}{n+1}

Ruled_surface.html

  1. S ( t , u ) = p ( t ) + u r ( t ) S(t,u)=p(t)+ur(t)
  2. S ( t , u ) S(t,u)
  3. p ( t ) p(t)
  4. r ( t ) r(t)
  5. p ( t ) = ( cos ( 2 t ) , sin ( 2 t ) , 0 ) r ( t ) = ( cos t cos 2 t , cos t sin 2 t , sin t ) \begin{aligned}\displaystyle p(t)&\displaystyle=(\cos(2t),\sin(2t),0)\\ \displaystyle r(t)&\displaystyle=(\cos t\cos 2t,\cos t\sin 2t,\sin t)\end{aligned}
  6. S ( t , u ) = ( 1 - u ) p ( t ) + u q ( t ) S(t,u)=(1-u)p(t)+uq(t)
  7. p p
  8. q q
  9. p ( t ) p(t)
  10. q ( t ) q(t)

Rylands_Library_Papyrus_P52.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 \mathfrak{P}
  25. 𝔓 \mathfrak{P}
  26. 𝔓 \mathfrak{P}
  27. 𝔓 \mathfrak{P}
  28. 𝔓 \mathfrak{P}
  29. 𝔓 \mathfrak{P}
  30. 𝔓 \mathfrak{P}
  31. 𝔓 \mathfrak{P}
  32. 𝔓 \mathfrak{P}
  33. 𝔓 \mathfrak{P}
  34. 𝔓 \mathfrak{P}
  35. 𝔓 \mathfrak{P}
  36. 𝔓 \mathfrak{P}
  37. 𝔓 \mathfrak{P}
  38. 𝔓 \mathfrak{P}
  39. 𝔓 \mathfrak{P}
  40. 𝔓 \mathfrak{P}
  41. 𝔓 \mathfrak{P}
  42. 𝔓 \mathfrak{P}
  43. 𝔓 \mathfrak{P}
  44. 𝔓 \mathfrak{P}
  45. 𝔓 \mathfrak{P}
  46. 𝔓 \mathfrak{P}
  47. 𝔓 \mathfrak{P}
  48. 𝔓 \mathfrak{P}
  49. 𝔓 \mathfrak{P}
  50. 𝔓 \mathfrak{P}
  51. 𝔓 \mathfrak{P}
  52. 𝔓 \mathfrak{P}
  53. 𝔓 \mathfrak{P}
  54. 𝔓 \mathfrak{P}
  55. 𝔓 \mathfrak{P}
  56. 𝔓 \mathfrak{P}
  57. 𝔓 \mathfrak{P}
  58. 𝔓 \mathfrak{P}
  59. 𝔓 \mathfrak{P}
  60. 𝔓 \mathfrak{P}
  61. 𝔓 \mathfrak{P}
  62. 𝔓 \mathfrak{P}
  63. 𝔓 \mathfrak{P}
  64. 𝔓 \mathfrak{P}
  65. 𝔓 \mathfrak{P}
  66. 𝔓 \mathfrak{P}
  67. 𝔓 \mathfrak{P}
  68. 𝔓 \mathfrak{P}
  69. 𝔓 \mathfrak{P}
  70. 𝔓 \mathfrak{P}
  71. 𝔓 \mathfrak{P}
  72. 𝔓 \mathfrak{P}
  73. 𝔓 \mathfrak{P}
  74. 𝔓 \mathfrak{P}
  75. 𝔓 \mathfrak{P}
  76. 𝔓 \mathfrak{P}
  77. 𝔓 \mathfrak{P}
  78. 𝔓 \mathfrak{P}
  79. 𝔓 \mathfrak{P}
  80. 𝔓 \mathfrak{P}
  81. 𝔓 \mathfrak{P}
  82. 𝔓 \mathfrak{P}
  83. 𝔓 \mathfrak{P}
  84. 𝔓 \mathfrak{P}
  85. 𝔓 \mathfrak{P}
  86. 𝔓 \mathfrak{P}
  87. 𝔓 \mathfrak{P}