wpmath0000005_10

Preferred_number.html

  1. R ( i , b ) = 10 i b R(i,b)=10^{\frac{i}{b}}
  2. f ( n ) = k × 2 n 12 . f(n)=k\times 2^{\frac{n}{12}}.

Pregeometry_(model_theory).html

  1. ( X , cl ) (X,\,\text{cl})\,
  2. cl : đ’« ( X ) → đ’« ( X ) \,\text{cl}:\mathcal{P}(X)\to\mathcal{P}(X)\,
  3. a , b ∈ X a,b\in X\,
  4. A , B , C ⊆ X A,B,C\subseteq X\,
  5. cl : ( đ’« ( X ) , ⊆ ) → ( đ’« ( X ) , ⊆ ) \,\text{cl}:(\mathcal{P}(X),\subseteq)\to(\mathcal{P}(X),\subseteq)\,
  6. id \,\text{id}\,
  7. A ⊆ B A\subseteq B\,
  8. A ⊆ cl ( A ) ⊆ cl ( B ) A\subseteq\,\text{cl}(A)\subseteq\,\text{cl}(B)\,
  9. a ∈ cl ( A ) a\in\,\text{cl}(A)\,
  10. F ⊆ A F\subseteq A\,
  11. a ∈ cl ( F ) a\in\,\text{cl}(F)\,
  12. b ∈ cl ( C âˆȘ { a } ) ∖ cl ( C ) b\in\,\text{cl}(C\cup\{a\})\smallsetminus\,\text{cl}(C)\,
  13. a ∈ cl ( C âˆȘ { b } ) a\in\,\text{cl}(C\cup\{b\})
  14. a ∈ cl ( C âˆȘ { b } ) ∖ cl ( C ) a\in\,\text{cl}(C\cup\{b\})\smallsetminus\,\text{cl}(C)\,
  15. A , B ⊂ S A,B\subset S
  16. A A
  17. B B
  18. a ∉ cl ( ( A ∖ { a } ) âˆȘ B ) a\notin\,\text{cl}((A\setminus\{a\})\cup B)
  19. a ∈ A a\in A
  20. A 0 ⊂ A A_{0}\subset A
  21. A A
  22. B B
  23. B B
  24. A ⊂ cl ( A 0 âˆȘ B ) A\subset\,\text{cl}(A_{0}\cup B)
  25. A A
  26. B B
  27. dim B A = | A 0 | \,\text{dim}_{B}A=|A_{0}|
  28. A , B A,B
  29. C C
  30. dim B âˆȘ C = dim C A â€Č \,\text{dim}_{B\cup C}=\dim_{C}A^{\prime}
  31. A â€Č A^{\prime}
  32. A A
  33. S S
  34. σ : 2 S → 2 S \sigma:2^{S}\to 2^{S}
  35. σ cl ( X ) = cl ( X ) σ \sigma\,\text{cl}(X)=\,\text{cl}(X)\sigma
  36. X ⊂ S X\subset S
  37. S S
  38. X ⊂ S X\subset S
  39. a , b ∈ S ∖ X a,b\in S\setminus X
  40. S S
  41. a a
  42. b b
  43. X X
  44. ( S , cl ) (S,\,\text{cl})
  45. ( S â€Č , cl â€Č ) (S^{\prime},\,\text{cl}^{\prime})
  46. S â€Č = { cl ( a ) ∣ a ∈ S ∖ cl ( ∅ ) } S^{\prime}=\{\,\text{cl}(a)\mid a\in S\setminus\,\text{cl}(\emptyset)\}
  47. X ⊂ S X\subset S
  48. cl â€Č ( { cl ( a ) ∣ a ∈ X = { cl â€Č ( b ) ∣ b ∈ cl X } \,\text{cl}^{\prime}(\{\,\text{cl}(a)\mid a\in X=\{\,\text{cl}^{\prime}(b)\mid b% \in\,\text{cl}X\}
  49. A ⊂ S A\subset S
  50. S S
  51. ( S , cl A ) (S,\,\text{cl}_{A})
  52. cl A ( X ) = cl ( X âˆȘ A ) \,\text{cl}_{A}(X)=\,\text{cl}(X\cup A)
  53. ( S , cl ) (S,\,\text{cl})
  54. cl ( X ) = ⋃ { cl ( a ) ∣ a ∈ X } \,\text{cl}(X)=\bigcup\{\,\text{cl}(a)\mid a\in X\}
  55. X , Y ⊂ S X,Y\subset S
  56. dim ( X âˆȘ Y ) = dim ( X ) + dim ( Y ) - dim ( X ∩ Y ) \,\text{dim}(X\cup Y)=\,\text{dim}(X)+\,\text{dim}(Y)-\,\text{dim}(X\cap Y)
  57. X X
  58. Y Y
  59. X ∩ Y X\cap Y
  60. S S
  61. a ∈ S ∖ cl ∅ a\in S\setminus\,\text{cl}\emptyset
  62. S S
  63. b b
  64. S S
  65. a , b ∈ S a,b\in S
  66. A ⊂ S A\subset S
  67. dim { a , b } = 2 \,\text{dim}\{a,b\}=2
  68. dim A { a , b } ≀ 1 \,\text{dim}_{A}\{a,b\}\leq 1
  69. ( cl { a , b } ∩ cl ( A ) ) ∖ cl ∅ ≠ ∅ (\,\text{cl}\{a,b\}\cap\,\text{cl}(A))\setminus\,\text{cl}\emptyset\neq\emptyset
  70. S S
  71. cl ( A ) = A \,\text{cl}(A)=A
  72. F F
  73. V V
  74. Îș \kappa
  75. F F
  76. V V
  77. V V
  78. F F
  79. V V
  80. 2 2
  81. Îș \kappa
  82. F F
  83. ( Îș - 1 ) (\kappa-1)
  84. F F
  85. V V
  86. Îș \kappa
  87. F F
  88. ( Îș + 1 ) (\kappa+1)
  89. X X
  90. Y Y
  91. k k
  92. tr.deg ( k ) ≄ ω \,\text{tr.deg}(k)\geq\omega

Preload_(cardiology).html

  1. L V E D P ⋅ L V E D R 2 h \frac{LVEDP\cdot LVEDR}{2h}
  2. ( p r e s s u r e ) × ( r a d i u s ) 2 × ( w a l l t h i c k n e s s ) \frac{(pressure)\times(radius)}{2\times(wall\ thickness)}

Pressure_sensor.html

  1. h = ( 1 - ( P / P ref ) 0.190284 ) × 145366.45 ft h=(1-(P/P_{\mathrm{ref}})^{0.190284})\times 145366.45\mathrm{ft}
  2. P = ρ g h P=\rho gh
  3. V out = P × K × V s actual V s ideal V_{\mathrm{out}}={P\times K\times Vs_{\mathrm{actual}}\over Vs_{\mathrm{ideal}}}
  4. V out V_{\mathrm{out}}
  5. P P
  6. K K
  7. V s actual Vs_{\mathrm{actual}}
  8. V s ideal Vs_{\mathrm{ideal}}
  9. P = V out × V s ideal K × V s actual P={V_{\mathrm{out}}\times Vs_{\mathrm{ideal}}\over K\times Vs_{\mathrm{actual}}}

Pretzel_link.html

  1. ( p 1 , p 2 , 
 , p n ) (p_{1},\,p_{2},\dots,\,p_{n})
  2. p 1 p_{1}
  3. p 2 p_{2}
  4. p n p_{n}
  5. ( p 1 , p 2 , 
 , p n ) (p_{1},p_{2},\dots,p_{n})
  6. n n
  7. p i p_{i}
  8. p i p_{i}
  9. ( p 1 , p 2 , 
 , p n ) (p_{1},\,p_{2},\dots,\,p_{n})
  10. p i p_{i}
  11. ( - p 1 , - p 2 , 
 , - p n ) (-p_{1},-p_{2},\dots,-p_{n})
  12. ( p 1 , p 2 , 
 , p n ) (p_{1},\,p_{2},\dots,\,p_{n})
  13. ( p 1 , p 2 , 
 , p n ) (p_{1},\,p_{2},\dots,\,p_{n})
  14. ( p 2 , p 3 , 
 , p n , p 1 ) (p_{2},\,p_{3},\dots,\,p_{n},\,p_{1})
  15. ( p 1 , p 2 , 
 , p n ) (p_{1},\,p_{2},\dots,\,p_{n})
  16. ( p k , p k + 1 , 
 , p n , p 1 , p 2 , 
 , p k - 1 ) (p_{k},\,p_{k+1},\dots,\,p_{n},\,p_{1},\,p_{2},\dots,\,p_{k-1})
  17. ( p 1 , p 2 , 
 , p n ) (p_{1},\,p_{2},\,\dots,\,p_{n})
  18. ( p n , p n - 1 , 
 , p 2 , p 1 ) (p_{n},\,p_{n-1},\dots,\,p_{2},\,p_{1})

Prewitt_operator.html

  1. 𝐀 \mathbf{A}
  2. 𝐆 đ± \mathbf{G_{x}}
  3. 𝐆 đČ \mathbf{G_{y}}
  4. 𝐆 đ± = [ - 1 0 + 1 - 1 0 + 1 - 1 0 + 1 ] * 𝐀 and 𝐆 đČ = [ - 1 - 1 - 1 0 0 0 + 1 + 1 + 1 ] * 𝐀 \mathbf{G_{x}}=\begin{bmatrix}-1&0&+1\\ -1&0&+1\\ -1&0&+1\end{bmatrix}*\mathbf{A}\quad\mbox{and}~{}\quad\mathbf{G_{y}}=\begin{% bmatrix}-1&-1&-1\\ 0&0&0\\ +1&+1&+1\end{bmatrix}*\mathbf{A}
  5. * *
  6. 𝐆 đ± \mathbf{G_{x}}
  7. [ - 1 0 + 1 - 1 0 + 1 - 1 0 + 1 ] = [ 1 1 1 ] [ - 1 0 1 ] \begin{bmatrix}-1&0&+1\\ -1&0&+1\\ -1&0&+1\end{bmatrix}=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}\begin{bmatrix}-1&0&1\end{bmatrix}
  8. 𝐆 = 𝐆 x 2 + 𝐆 y 2 \mathbf{G}=\sqrt{{\mathbf{G}_{x}}^{2}+{\mathbf{G}_{y}}^{2}}
  9. 𝚯 = atan2 ( 𝐆 y , 𝐆 x ) \mathbf{\Theta}=\operatorname{atan2}\left({\mathbf{G}_{y},\mathbf{G}_{x}}\right)

Price_index.html

  1. C C
  2. C C
  3. t t
  4. ∑ c ∈ C ( p c , t ⋅ q c , t ) \sum_{c\,\in\,C}(p_{c,t}\cdot q_{c,t})
  5. p c , t p_{c,t}\,
  6. c c
  7. t t
  8. q c , t q_{c,t}\,
  9. c c
  10. t t
  11. t 0 t_{0}
  12. t n t_{n}
  13. q c , t n = q c = q c , t 0 ∀ c q_{c,t_{n}}=q_{c}=q_{c,t_{0}}\,\forall c
  14. P = ∑ ( p c , t n ⋅ q c ) ∑ ( p c , t 0 ⋅ q c ) P=\frac{\sum(p_{c,t_{n}}\cdot q_{c})}{\sum(p_{c,t_{0}}\cdot q_{c})}
  15. P = ∑ ( p c , t n ⋅ q c , t n ) ∑ ( p c , t 0 ⋅ q c , t 0 ) P=\frac{\sum(p_{c,t_{n}}\cdot q_{c,t_{n}})}{\sum(p_{c,t_{0}}\cdot q_{c,t_{0}})}
  16. t 0 t_{0}
  17. t n t_{n}
  18. P P
  19. t 0 t_{0}
  20. t n t_{n}
  21. P P
  22. P P
  23. P P
  24. P P = ∑ ( p c , t n ⋅ q c , t n ) ∑ ( p c , t 0 ⋅ q c , t n ) P_{P}=\frac{\sum(p_{c,t_{n}}\cdot q_{c,t_{n}})}{\sum(p_{c,t_{0}}\cdot q_{c,t_{% n}})}
  25. P L = ∑ ( p c , t n ⋅ q c , t 0 ) ∑ ( p c , t 0 ⋅ q c , t 0 ) P_{L}=\frac{\sum(p_{c,t_{n}}\cdot q_{c,t_{0}})}{\sum(p_{c,t_{0}}\cdot q_{c,t_{% 0}})}
  26. P P
  27. t 0 t_{0}
  28. t n t_{n}
  29. c c
  30. P M E = ∑ [ p c , t n ⋅ 1 2 ⋅ ( q c , t 0 + q c , t n ) ] ∑ [ p c , t 0 ⋅ 1 2 ⋅ ( q c , t 0 + q c , t n ) ] = ∑ [ p c , t n ⋅ ( q c , t 0 + q c , t n ) ] ∑ [ p c , t 0 ⋅ ( q c , t 0 + q c , t n ) ] P_{ME}=\frac{\sum[p_{c,t_{n}}\cdot\frac{1}{2}\cdot(q_{c,t_{0}}+q_{c,t_{n}})]}{% \sum[p_{c,t_{0}}\cdot\frac{1}{2}\cdot(q_{c,t_{0}}+q_{c,t_{n}})]}=\frac{\sum[p_% {c,t_{n}}\cdot(q_{c,t_{0}}+q_{c,t_{n}})]}{\sum[p_{c,t_{0}}\cdot(q_{c,t_{0}}+q_% {c,t_{n}})]}
  31. P P P_{P}
  32. P L P_{L}
  33. P F = P P ⋅ P L P_{F}=\sqrt{P_{P}\cdot P_{L}}
  34. E c , t 0 E_{c,t_{0}}
  35. E c , t 0 = p c , t 0 ⋅ q c , t 0 E_{c,t_{0}}=p_{c,t_{0}}\cdot q_{c,t_{0}}
  36. E c , t 0 p c , t 0 = q c , t 0 \frac{E_{c,t_{0}}}{p_{c,t_{0}}}=q_{c,t_{0}}
  37. P L = ∑ ( p c , t n ⋅ q c , t 0 ) ∑ ( p c , t 0 ⋅ q c , t 0 ) = ∑ ( p c , t n ⋅ E c , t 0 p c , t 0 ) ∑ E c , t 0 = ∑ ( p c , t n p c , t 0 ⋅ E c , t 0 ) ∑ E c , t 0 P_{L}=\frac{\sum(p_{c,t_{n}}\cdot q_{c,t_{0}})}{\sum(p_{c,t_{0}}\cdot q_{c,t_{% 0}})}=\frac{\sum(p_{c,t_{n}}\cdot\frac{E_{c,t_{0}}}{p_{c,t_{0}}})}{\sum E_{c,t% _{0}}}=\frac{\sum(\frac{p_{c,t_{n}}}{p_{c,t_{0}}}\cdot E_{c,t_{0}})}{\sum E_{c% ,t_{0}}}
  38. t n t_{n}
  39. t 0 t_{0}
  40. P t n = ∑ ( p c , t 1 ⋅ q c , t 0 ) ∑ ( p c , t 0 ⋅ q c , t 0 ) × ∑ ( p c , t 2 ⋅ q c , t 1 ) ∑ ( p c , t 1 ⋅ q c , t 1 ) × ⋯ × ∑ ( p c , t n ⋅ q c , t n - 1 ) ∑ ( p c , t n - 1 ⋅ q c , t n - 1 ) P_{t_{n}}=\frac{\sum(p_{c,t_{1}}\cdot q_{c,t_{0}})}{\sum(p_{c,t_{0}}\cdot q_{c% ,t_{0}})}\times\frac{\sum(p_{c,t_{2}}\cdot q_{c,t_{1}})}{\sum(p_{c,t_{1}}\cdot q% _{c,t_{1}})}\times\cdots\times\frac{\sum(p_{c,t_{n}}\cdot q_{c,t_{n-1}})}{\sum% (p_{c,t_{n-1}}\cdot q_{c,t_{n-1}})}
  41. ∑ ( p c , t n ⋅ q c , t n - 1 ) ∑ ( p c , t n - 1 ⋅ q c , t n - 1 ) \frac{\sum(p_{c,t_{n}}\cdot q_{c,t_{n-1}})}{\sum(p_{c,t_{n-1}}\cdot q_{c,t_{n-% 1}})}
  42. t n - 1 t_{n-1}
  43. t n t_{n}
  44. t 0 t_{0}
  45. t 0 t_{0}
  46. I ( P t 0 , P t m , Q t 0 , Q t m ) I(P_{t_{0}},P_{t_{m}},Q_{t_{0}},Q_{t_{m}})
  47. P t 0 P_{t_{0}}
  48. P t m P_{t_{m}}
  49. Q t 0 Q_{t_{0}}
  50. Q t m Q_{t_{m}}
  51. I ( p t m , p t n , α ⋅ q t m , ÎČ â‹… q t n ) = 1 ∀ ( α , ÎČ ) ∈ ( 0 , ∞ ) 2 I(p_{t_{m}},p_{t_{n}},\alpha\cdot q_{t_{m}},\beta\cdot q_{t_{n}})=1~{}~{}% \forall(\alpha,\beta)\in(0,\infty)^{2}
  52. α \alpha
  53. ÎČ \beta
  54. I ( p t m , α ⋅ p t n , q t m , q t n ) = α ⋅ I ( p t m , p t n , q t m , q t n ) I(p_{t_{m}},\alpha\cdot p_{t_{n}},q_{t_{m}},q_{t_{n}})=\alpha\cdot I(p_{t_{m}}% ,p_{t_{n}},q_{t_{m}},q_{t_{n}})
  55. I ( α ⋅ p t m , α ⋅ p t n , ÎČ â‹… q t m , Îł ⋅ q t n ) = I ( p t m , p t n , q t m , q t n ) ∀ ( α , ÎČ , Îł ) ∈ ( 0 , ∞ ) 3 I(\alpha\cdot p_{t_{m}},\alpha\cdot p_{t_{n}},\beta\cdot q_{t_{m}},\gamma\cdot q% _{t_{n}})=I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})~{}~{}\forall(\alpha,\beta% ,\gamma)\in(0,\infty)^{3}
  56. I ( p t n , p t m , q t n , q t m ) = 1 I ( p t m , p t n , q t m , q t n ) I(p_{t_{n}},p_{t_{m}},q_{t_{n}},q_{t_{m}})=\frac{1}{I(p_{t_{m}},p_{t_{n}},q_{t% _{m}},q_{t_{n}})}
  57. I ( p t m , p t n , q t m , q t n ) ≀ I ( p t m , p t r , q t m , q t r ) ⇐ p t n ≀ p t r I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})\leq I(p_{t_{m}},p_{t_{r}},q_{t_{m}}% ,q_{t_{r}})~{}~{}\Leftarrow~{}~{}p_{t_{n}}\leq p_{t_{r}}
  58. I ( p t m , p t n , q t m , q t n ) ⋅ I ( p t n , p t r , q t n , q t r ) = I ( p t m , p t r , q t m , q t r ) ⇐ t m ≀ t n ≀ t r I(p_{t_{m}},p_{t_{n}},q_{t_{m}},q_{t_{n}})\cdot I(p_{t_{n}},p_{t_{r}},q_{t_{n}% },q_{t_{r}})=I(p_{t_{m}},p_{t_{r}},q_{t_{m}},q_{t_{r}})~{}~{}\Leftarrow~{}~{}t% _{m}\leq t_{n}\leq t_{r}
  59. t m t_{m}
  60. t n t_{n}
  61. t r t_{r}
  62. t m t_{m}
  63. t n t_{n}
  64. t n t_{n}
  65. t r t_{r}
  66. t m t_{m}
  67. t r t_{r}
  68. P ( M ) t P(M)_{t}
  69. P ( N ) t + 1 P(N)_{t+1}
  70. P ( N ) t + 1 {P(N)_{t+1}}
  71. P ( N ) t {P(N)_{t}}
  72. P ( N ) t + 1 P(N)_{t+1}
  73. P ( M ) t P(M)_{t}

Price_level.html

  1. C C
  2. C C
  3. t t
  4. ∑ c ∈ C ( p c , t ⋅ q c , t ) = ∑ c ∈ C [ ( P t ⋅ p c , t â€Č ) ⋅ q c , t ] = P t ⋅ ∑ c ∈ C ( p c , t â€Č ⋅ q c , t ) \sum_{c\,\in\,C}(p_{c,t}\cdot q_{c,t})=\sum_{c\,\in\,C}[(P_{t}\cdot p^{\prime}% _{c,t})\cdot q_{c,t}]=P_{t}\cdot\sum_{c\,\in\,C}(p^{\prime}_{c,t}\cdot q_{c,t})
  5. q c , t q_{c,t}\,
  6. c c
  7. t t
  8. p c , t p_{c,t}\,
  9. c c
  10. t t
  11. p c , t â€Č p^{\prime}_{c,t}
  12. c c
  13. t t
  14. P t P_{t}
  15. t t
  16. P t 1 - P t 0 t 1 - t 0 \frac{P_{t_{1}}-P_{t_{0}}}{t_{1}-t_{0}}
  17. ( G D P ) t 1 P t 1 - ( G D P ) t 0 P t 0 \frac{(GDP)_{t_{1}}}{P_{t_{1}}}-\frac{(GDP)_{t_{0}}}{P_{t_{0}}}

Primary_decomposition.html

  1. â„€ [ - 5 ] , \mathbb{Z}[\sqrt{-5}],
  2. 6 = 2 ⋅ 3 = ( 1 + - 5 ) ( 1 - - 5 ) . 6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5}).
  3. I = Q 1 ∩ ⋯ ∩ Q n I=Q_{1}\cap\cdots\cap Q_{n}
  4. Q i Q_{i}
  5. Q 1 ∩ 
 ∩ Q i ^ ∩ 
 ∩ Q n ⊈ Q i Q_{1}\cap\dots\cap\widehat{Q_{i}}\cap\dots\cap Q_{n}\nsubseteq Q_{i}
  6. Q i \sqrt{Q_{i}}
  7. â„€ \mathbb{Z}
  8. n = ± p 1 d 1 ⋯ p r d r n=\pm p_{1}^{d_{1}}\cdots p_{r}^{d_{r}}
  9. ( n ) ⊂ â„€ (n)\subset\mathbb{Z}
  10. ( n ) = ( p 1 d 1 ) ∩ ⋯ ∩ ( p r d r ) . (n)=(p_{1}^{d_{1}})\cap\cdots\cap(p_{r}^{d_{r}}).
  11. M / N M/N
  12. N = 0 N=0
  13. 0 = ∩ Q i ⇔ ∅ = Ass ( ∩ Q i ) = ∩ Ass ( Q i ) 0=\cap Q_{i}\Leftrightarrow\emptyset=\operatorname{Ass}(\cap Q_{i})=\cap% \operatorname{Ass}(Q_{i})
  14. Q i Q_{i}
  15. P ∉ Ass ( Q ) P\not\in\operatorname{Ass}(Q)
  16. { N ⊆ M | P ∉ Ass ( N ) } \{N\subseteq M|P\not\in\operatorname{Ass}(N)\}
  17. P â€Č ≠ P P^{\prime}\neq P
  18. M / Q M/Q
  19. R / P â€Č ≃ Q â€Č / Q R/P^{\prime}\simeq Q^{\prime}/Q
  20. P ∉ Ass ( Q ) ⊂ Ass ( Q â€Č ) P\not\in\operatorname{Ass}(Q)\subset\operatorname{Ass}(Q^{\prime})
  21. Q i Q_{i}

Prime_constant.html

  1. ρ \rho
  2. n n
  3. n n
  4. ρ \rho
  5. ρ = ∑ p 1 2 p = ∑ n = 1 ∞ χ ℙ ( n ) 2 n \rho=\sum_{p}\frac{1}{2^{p}}=\sum_{n=1}^{\infty}\frac{\chi_{\mathbb{P}}(n)}{2^% {n}}
  6. p p
  7. χ ℙ \chi_{\mathbb{P}}
  8. ρ = 0.414682509851111660248109622 
 \rho=0.414682509851111660248109622\ldots
  9. ρ = 0.011010100010100010100010000 
 2 \rho=0.011010100010100010100010000\ldots_{2}
  10. ρ \rho
  11. k k
  12. ρ \rho
  13. r k r_{k}
  14. ρ \rho
  15. N N
  16. k k
  17. r n = r n + i k r_{n}=r_{n+ik}
  18. n > N n>N
  19. i ∈ ℕ i\in\mathbb{N}
  20. p > N p>N
  21. r p = 1 r_{p}=1
  22. r p = r p + i k r_{p}=r_{p+ik}
  23. i ∈ ℕ i\in\mathbb{N}
  24. i = p i=p
  25. r p + i ⋅ k = r p + p ⋅ k = r p ( k + 1 ) = 0 r_{p+i\cdot k}=r_{p+p\cdot k}=r_{p(k+1)}=0
  26. p ( k + 1 ) p(k+1)
  27. k + 1 ≄ 2 k+1\geq 2
  28. r p ≠ r p ( k + 1 ) r_{p}\neq r_{p(k+1)}
  29. ρ \rho

Prime_model.html

  1. P P
  2. M M
  3. M M
  4. P P
  5. L L
  6. Îș \kappa
  7. T T
  8. L , L,
  9. T T
  10. max ( Îș , â„” 0 ) ; \max(\kappa,\aleph_{0});
  11. T T
  12. ⟹ ℕ , S ⟩ \langle{\mathbb{N}},S\rangle
  13. ⟹ ℕ + â„€ , S ⟩ , \langle{\mathbb{N}}+{\mathbb{Z}},S\rangle,
  14. ⟹ ℕ , S ⟩ \langle{\mathbb{N}},S\rangle

Prime_Obsession.html

  1. π ( N ) ≈ N l o g ( N ) \pi(N)\approx\frac{N}{log(N)}
  2. ζ ( s ) = 1 + 1 2 s + 1 3 s + 1 4 s + ⋯ \zeta(s)=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\cdots
  3. ζ ( s ) = ∏ p p r i m e 1 1 - p - s = ∑ n = 1 ∞ 1 n s \zeta(s)=\prod_{p\ prime}\frac{1}{1-{p^{-s}}}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}

Prime_power.html

  1. ϕ ( p n ) = p n - 1 ϕ ( p ) = p n - 1 ( p - 1 ) = p n - p n - 1 = p n ( 1 - 1 p ) , \phi(p^{n})=p^{n-1}\phi(p)=p^{n-1}(p-1)=p^{n}-p^{n-1}=p^{n}\left(1-\frac{1}{p}% \right),
  2. σ 0 ( p n ) = ∑ j = 0 n p 0 ⋅ j = ∑ j = 0 n 1 = n + 1 , \sigma_{0}(p^{n})=\sum_{j=0}^{n}p^{0\cdot j}=\sum_{j=0}^{n}1=n+1,
  3. σ 1 ( p n ) = ∑ j = 0 n p 1 ⋅ j = ∑ j = 0 n p j = p n + 1 - 1 p - 1 . \sigma_{1}(p^{n})=\sum_{j=0}^{n}p^{1\cdot j}=\sum_{j=0}^{n}p^{j}=\frac{p^{n+1}% -1}{p-1}.

Principal_part.html

  1. z = a z=a
  2. f ( z ) = ∑ k = - ∞ ∞ a k ( z - a ) k f(z)=\sum_{k=-\infty}^{\infty}a_{k}(z-a)^{k}
  3. ∑ k = - ∞ - 1 a k ( z - a ) k \sum_{k=-\infty}^{-1}a_{k}(z-a)^{k}
  4. f f
  5. a a
  6. f ( z ) f(z)
  7. a a
  8. Δ y Δ x = f â€Č ( x ) + Δ \frac{\Delta y}{\Delta x}=f^{\prime}(x)+\varepsilon
  9. Δ y = f â€Č ( x ) Δ x + Δ Δ x = d y + Δ Δ x \Delta y=f^{\prime}(x)\Delta x+\varepsilon\Delta x=dy+\varepsilon\Delta x

Pro-p_group.html

  1. G G
  2. N ◁ G N\triangleleft G
  3. G / N G/N
  4. ℚ p \mathbb{Q}_{p}
  5. r r
  6. r r
  7. â„€ p = lim ← â„€ / p n â„€ . \mathbb{Z}_{p}=\displaystyle\lim_{\leftarrow}\mathbb{Z}/p^{n}\mathbb{Z}.
  8. G L n ( â„€ p ) \ GL_{n}(\mathbb{Z}_{p})
  9. â„€ p \ \mathbb{Z}_{p}
  10. p â„€ p \ p\mathbb{Z}_{p}
  11. G L n ( â„€ p ) \ GL_{n}(\mathbb{Z}_{p})

Probit.html

  1. Ί ( z ) \Phi(z)
  2. Ί - 1 ( p ) \Phi^{-1}(p)
  3. Ί ( - 1.96 ) = 0.025 = 1 - Ί ( 1.96 ) . \Phi(-1.96)=0.025=1-\Phi(1.96).\,\!
  4. probit ( 0.025 ) = - 1.96 = - probit ( 0.975 ) \operatorname{probit}(0.025)=-1.96=-\operatorname{probit}(0.975)
  5. Ί ( probit ( p ) ) = p \Phi(\operatorname{probit}(p))=p
  6. probit ( Ί ( z ) ) = z . \operatorname{probit}(\Phi(z))=z.
  7. probit ( p ) = 2 erf - 1 ( 2 p - 1 ) . \operatorname{probit}(p)=\sqrt{2}\,\operatorname{erf}^{-1}(2p-1).
  8. w ( p ) w(p)
  9. d w d p = 1 f ( w ) \frac{dw}{dp}=\frac{1}{f(w)}
  10. f ( w ) f(w)
  11. w w
  12. d w d p = 2 π e w 2 2 \frac{dw}{dp}=\sqrt{2\pi}\ e^{\frac{w^{2}}{2}}
  13. d 2 w d p 2 = w ( d w d p ) 2 \frac{d^{2}w}{dp^{2}}=w\left(\frac{dw}{dp}\right)^{2}
  14. w ( 1 / 2 ) = 0 , w\left(1/2\right)=0,
  15. w â€Č ( 1 / 2 ) = 2 π . w^{\prime}\left(1/2\right)=\sqrt{2\pi}.
  16. w ( p ) = π 2 ∑ k = 0 ∞ d k ( 2 k + 1 ) ( 2 p - 1 ) ( 2 k + 1 ) w(p)=\sqrt{\frac{\pi}{2}}\sum_{k=0}^{\infty}\frac{d_{k}}{(2k+1)}(2p-1)^{(2k+1)}
  17. d k d_{k}
  18. d k + 1 = π 4 ∑ j = 0 k d j d k - j ( j + 1 ) ( 2 j + 1 ) d_{k+1}=\frac{\pi}{4}\sum_{j=0}^{k}\frac{d_{j}d_{k-j}}{(j+1)(2j+1)}
  19. d 0 = 1 d_{0}=1
  20. d k + 1 / d k → 1 d_{k+1}/d_{k}\rightarrow 1
  21. k → ∞ k\rightarrow\infty
  22. logit ( x ) \operatorname{logit}(x)
  23. Ί - 1 ( x ) / π 8 \Phi^{-1}(x)/\sqrt{\frac{\pi}{8}}
  24. logit ( p ) = log ( p 1 - p ) . \operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right).

Probit_model.html

  1. Pr ( Y = 1 ∣ X ) = Ί ( X â€Č ÎČ ) , \Pr(Y=1\mid X)=\Phi(X^{\prime}\beta),
  2. Y ∗ = X â€Č ÎČ + Δ , Y^{\ast}=X^{\prime}\beta+\varepsilon,\,
  3. Y = { 1 if Y ∗ > 0 i.e. - Δ < X â€Č ÎČ , 0 otherwise. Y=\begin{cases}1&\,\text{if }Y^{\ast}>0\ \,\text{ i.e. }-\varepsilon<X^{\prime% }\beta,\\ 0&\,\text{otherwise.}\end{cases}
  4. Pr ( Y = 1 ∣ X ) = Pr ( Y ∗ > 0 ) = Pr ( X â€Č ÎČ + Δ > 0 ) = Pr ( Δ > - X â€Č ÎČ ) = Pr ( Δ < X â€Č ÎČ ) (by symmetry of the normal dist) = Ί ( X â€Č ÎČ ) \begin{aligned}\displaystyle\Pr(Y=1\mid X)&\displaystyle=\Pr(Y^{\ast}>0)=\Pr(X% ^{\prime}\beta+\varepsilon>0)\\ &\displaystyle=\Pr(\varepsilon>-X^{\prime}\beta)\\ &\displaystyle=\Pr(\varepsilon<X^{\prime}\beta)\quad\,\text{(by symmetry of % the normal dist)}\\ &\displaystyle=\Phi(X^{\prime}\beta)\end{aligned}
  5. { y i , x i } i = 1 n \{y_{i},x_{i}\}_{i=1}^{n}
  6. ln ℒ ( ÎČ ) = ∑ i = 1 n ( y i ln Ί ( x i â€Č ÎČ ) + ( 1 - y i ) ln ( 1 - Ί ( x i â€Č ÎČ ) ) ) \ln\mathcal{L}(\beta)=\sum_{i=1}^{n}\bigg(y_{i}\ln\Phi(x_{i}^{\prime}\beta)+(1% -y_{i})\ln\!\big(1-\Phi(x_{i}^{\prime}\beta)\big)\bigg)
  7. ÎČ ^ \hat{\beta}
  8. ÎČ ^ \hat{\beta}
  9. n ( ÎČ ^ - ÎČ ) → 𝑑 đ’© ( 0 , Ω - 1 ) , \sqrt{n}(\hat{\beta}-\beta)\ \xrightarrow{d}\ \mathcal{N}(0,\,\Omega^{-1}),
  10. Ω = E [ φ 2 ( X â€Č ÎČ ) Ί ( X â€Č ÎČ ) ( 1 - Ί ( X â€Č ÎČ ) ) X X â€Č ] , Ω ^ = 1 n ∑ i = 1 n φ 2 ( x i â€Č ÎČ ^ ) Ί ( x i â€Č ÎČ ^ ) ( 1 - Ί ( x i â€Č ÎČ ^ ) ) x i x i â€Č \Omega=\operatorname{E}\bigg[\frac{\varphi^{2}(X^{\prime}\beta)}{\Phi(X^{% \prime}\beta)(1-\Phi(X^{\prime}\beta))}XX^{\prime}\bigg],\qquad\hat{\Omega}=% \frac{1}{n}\sum_{i=1}^{n}\frac{\varphi^{2}(x^{\prime}_{i}\hat{\beta})}{\Phi(x^% {\prime}_{i}\hat{\beta})(1-\Phi(x^{\prime}_{i}\hat{\beta}))}x_{i}x^{\prime}_{i}
  11. y i y_{i}
  12. x i x_{i}
  13. { y i , x i } i = 1 n \{y_{i},x_{i}\}_{i=1}^{n}
  14. { x ( 1 ) , 
 , x ( T ) } \{x_{(1)},\ldots,x_{(T)}\}
  15. n t n_{t}
  16. x i = x ( t ) , x_{i}=x_{(t)},
  17. r t r_{t}
  18. y i = 1 y_{i}=1
  19. t , lim n → ∞ n t / n = c t > 0 t,\lim_{n\rightarrow\infty}n_{t}/n=c_{t}>0
  20. p ^ t = r t / n t \hat{p}_{t}=r_{t}/n_{t}
  21. σ ^ t 2 = 1 n t p ^ t ( 1 - p ^ t ) φ 2 ( Ί - 1 ( p ^ t ) ) \hat{\sigma}_{t}^{2}=\frac{1}{n_{t}}\frac{\hat{p}_{t}(1-\hat{p}_{t})}{\varphi^% {2}\big(\Phi^{-1}(\hat{p}_{t})\big)}
  22. Ί - 1 ( p ^ t ) \Phi^{-1}(\hat{p}_{t})
  23. x ( t ) x_{(t)}
  24. σ ^ t - 2 \hat{\sigma}_{t}^{-2}
  25. ÎČ ^ = ( ∑ t = 1 T σ ^ t - 2 x ( t ) x ( t ) â€Č ) - 1 ∑ t = 1 T σ ^ t - 2 x ( t ) Ί - 1 ( p ^ t ) \hat{\beta}=\Bigg(\sum_{t=1}^{T}\hat{\sigma}_{t}^{-2}x_{(t)}x^{\prime}_{(t)}% \Bigg)^{-1}\sum_{t=1}^{T}\hat{\sigma}_{t}^{-2}x_{(t)}\Phi^{-1}(\hat{p}_{t})
  26. r t r_{t}
  27. n t n_{t}
  28. x ( t ) x_{(t)}
  29. s y m b o l ÎČ âˆŒ đ’© ( 𝐛 0 , 𝐁 0 ) y i ∗ ∣ đ± i , s y m b o l ÎČ âˆŒ đ’© ( đ± i â€Č s y m b o l ÎČ , 1 ) y i = { 1 if y i ∗ > 0 0 otherwise \begin{aligned}\displaystyle symbol\beta&\displaystyle\sim\mathcal{N}(\mathbf{% b}_{0},\mathbf{B}_{0})\\ \displaystyle y_{i}^{\ast}\mid\mathbf{x}_{i},symbol\beta&\displaystyle\sim% \mathcal{N}(\mathbf{x}^{\prime}_{i}symbol\beta,1)\\ \displaystyle y_{i}&\displaystyle=\begin{cases}1&\,\text{if }y_{i}^{\ast}>0\\ 0&\,\text{otherwise}\end{cases}\end{aligned}
  30. 𝐁 \displaystyle\mathbf{B}
  31. [ y i ∗ < 0 ] [y_{i}^{\ast}<0]
  32. ℐ ( y i ∗ < 0 ) \mathcal{I}(y_{i}^{\ast}<0)
  33. đ± i â€Č s y m b o l ÎČ \mathbf{x}^{\prime}_{i}symbol\beta

Proca_action.html

  1. ℒ = - 1 16 π ( ∂ ÎŒ A Μ - ∂ Μ A ÎŒ ) ( ∂ ÎŒ A Μ - ∂ Μ A ÎŒ ) + m 2 c 2 8 π ℏ 2 A Μ A Μ . \mathcal{L}=-\frac{1}{16\pi}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})(% \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})+\frac{m^{2}c^{2}}{8\pi\hbar^{2}}A% ^{\nu}A_{\nu}.
  2. ∂ ÎŒ ( ∂ ÎŒ A Μ - ∂ Μ A ÎŒ ) + ( m c ℏ ) 2 A Μ = 0 \partial_{\mu}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})+\left(\frac{mc}{% \hbar}\right)^{2}A^{\nu}=0
  3. [ ∂ ÎŒ ∂ ÎŒ + ( m c ℏ ) 2 ] A Μ = 0 \left[\partial_{\mu}\partial^{\mu}+\left(\frac{mc}{\hbar}\right)^{2}\right]A^{% \nu}=0
  4. ∂ ÎŒ A ÎŒ = 0 \partial_{\mu}A^{\mu}=0\!
  5. □ ϕ - ∂ ∂ t ( 1 c 2 ∂ ϕ ∂ t + ∇ ⋅ 𝐀 ) = - ( m c ℏ ) 2 ϕ \Box\phi-\frac{\partial}{\partial t}\left(\frac{1}{c^{2}}\frac{\partial\phi}{% \partial t}+\nabla\cdot\mathbf{A}\right)=-\left(\frac{mc}{\hbar}\right)^{2}\phi\!
  6. □ 𝐀 + ∇ ( 1 c 2 ∂ ϕ ∂ t + ∇ ⋅ 𝐀 ) = - ( m c ℏ ) 2 𝐀 \Box\mathbf{A}+\nabla\left(\frac{1}{c^{2}}\frac{\partial\phi}{\partial t}+% \nabla\cdot\mathbf{A}\right)=-\left(\frac{mc}{\hbar}\right)^{2}\mathbf{A}\!
  7. □ \Box
  8. A ÎŒ → A ÎŒ - ∂ ÎŒ f A^{\mu}\rightarrow A^{\mu}-\partial^{\mu}f

Progressive_function.html

  1. supp f ^ ⊆ ℝ + . \mathop{\rm supp}\hat{f}\subseteq\mathbb{R}_{+}.
  2. supp f ^ ⊆ ℝ - . \mathop{\rm supp}\hat{f}\subseteq\mathbb{R}_{-}.
  3. H + 2 ( R ) H^{2}_{+}(R)
  4. f ( t ) = ∫ 0 ∞ e 2 π i s t f ^ ( s ) d s f(t)=\int_{0}^{\infty}e^{2\pi ist}\hat{f}(s)\,ds
  5. { t + i u : t , u ∈ R , u ≄ 0 } \{t+iu:t,u\in R,u\geq 0\}
  6. f ( t + i u ) = ∫ 0 ∞ e 2 π i s ( t + i u ) f ^ ( s ) d s = ∫ 0 ∞ e 2 π i s t e - 2 π s u f ^ ( s ) d s . f(t+iu)=\int_{0}^{\infty}e^{2\pi is(t+iu)}\hat{f}(s)\,ds=\int_{0}^{\infty}e^{2% \pi ist}e^{-2\pi su}\hat{f}(s)\,ds.
  7. { t + i u : t , u ∈ R , u ≀ 0 } \{t+iu:t,u\in R,u\leq 0\}

Project_Cyclops.html

  1. > 10 9 >10^{9}

Projection-slice_theorem.html

  1. F 1 P 1 = S 1 F 2 F_{1}P_{1}=S_{1}F_{2}\,
  2. F m P m = S m F N . F_{m}P_{m}=S_{m}F_{N}.\,
  3. p ( x ) = ∫ - ∞ ∞ f ( x , y ) d y . p(x)=\int_{-\infty}^{\infty}f(x,y)\,dy.
  4. f ( x , y ) f(x,y)
  5. F ( k x , k y ) = ∫ - ∞ ∞ ∫ - ∞ ∞ f ( x , y ) e - 2 π i ( x k x + y k y ) d x d y . F(k_{x},k_{y})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\,e^{-2\pi i% (xk_{x}+yk_{y})}\,dxdy.
  6. s ( k x ) s(k_{x})
  7. s ( k x ) = F ( k x , 0 ) = ∫ - ∞ ∞ ∫ - ∞ ∞ f ( x , y ) e - 2 π i x k x d x d y s(k_{x})=F(k_{x},0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\,e^{-% 2\pi ixk_{x}}\,dxdy
  8. = ∫ - ∞ ∞ [ ∫ - ∞ ∞ f ( x , y ) d y ] e - 2 π i x k x d x =\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}f(x,y)\,dy\right]\,e^{-2% \pi ixk_{x}}dx
  9. = ∫ - ∞ ∞ p ( x ) e - 2 π i x k x d x =\int_{-\infty}^{\infty}p(x)\,e^{-2\pi ixk_{x}}dx
  10. F 1 A 1 = H F_{1}A_{1}=H\,

Projective_hierarchy.html

  1. A A
  2. X X
  3. s y m b o l ÎŁ n 1 symbol{\Sigma}^{1}_{n}
  4. n n
  5. A A
  6. s y m b o l ÎŁ 1 1 symbol{\Sigma}^{1}_{1}
  7. A A
  8. s y m b o l Π n 1 symbol{\Pi}^{1}_{n}
  9. A A
  10. X ∖ A X\setminus A
  11. s y m b o l ÎŁ n 1 symbol{\Sigma}^{1}_{n}
  12. s y m b o l ÎŁ n + 1 1 symbol{\Sigma}^{1}_{n+1}
  13. Y Y
  14. s y m b o l Π n 1 symbol{\Pi}^{1}_{n}
  15. C ⊆ X × Y C\subseteq X\times Y
  16. A A
  17. C C
  18. A = { x ∈ X ∣ ∃ y ∈ Y ( x , y ) ∈ C } A=\{x\in X\mid\exists y\in Y(x,y)\in C\}
  19. Y Y
  20. ÎŁ \Sigma
  21. Π \Pi
  22. s y m b o l ÎŁ symbol{\Sigma}
  23. s y m b o l Π symbol{\Pi}
  24. s y m b o l ÎŁ n 1 symbol{\Sigma}^{1}_{n}
  25. ÎŁ n 1 \Sigma^{1}_{n}
  26. s y m b o l ÎŁ n 1 symbol{\Sigma}^{1}_{n}
  27. ÎŁ n 1 , A \Sigma^{1,A}_{n}
  28. s y m b o l Π n 1 symbol{\Pi}^{1}_{n}

Projective_line_over_a_ring.html

  1. ( c 0 0 c ) \begin{pmatrix}c&0\\ 0&c\end{pmatrix}
  2. U ( a , 1 ) ( 0 1 1 0 ) = U ( 1 , a ) ∌ U ( a - 1 , 1 ) . U(a,1)\begin{pmatrix}0&1\\ 1&0\end{pmatrix}=U(1,a)\thicksim U(a^{-1},1).
  3. ( u 0 0 1 ) ( 0 1 1 0 ) ( v 0 0 1 ) ( 0 1 1 0 ) = ( u 0 0 v ) . \begin{pmatrix}u&0\\ 0&1\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}v&0\\ 0&1\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}=\begin{pmatrix}u&0\\ 0&v\end{pmatrix}.
  4. U ( a , 1 ) ( v 0 0 u ) = U ( a v , u ) ∌ U ( u - 1 a v , 1 ) . U(a,1)\begin{pmatrix}v&0\\ 0&u\end{pmatrix}=U(av,u)\thicksim U(u^{-1}av,1).
  5. U ( z , 1 ) ( a c b d ) = U ( z a + b , z c + d ) ∌ U ( ( z c + d ) - 1 ( z a + b ) , 1 ) . U(z,1)\begin{pmatrix}a&c\\ b&d\end{pmatrix}=U(za+b,zc+d)\thicksim U((zc+d)^{-1}(za+b),1).
  6. U F ( x , 1 ) ↩ U A ( x , 1 ) , U F ( 1 , 0 ) ↩ U A ( 1 , 0 ) . U_{F}(x,1)\mapsto U_{A}(x,1),\quad U_{F}(1,0)\mapsto U_{A}(1,0).
  7. A ⊕ A A\oplus A
  8. A ⊕ A A\oplus A
  9. ( 0 1 1 0 ) , ( 1 0 t 1 ) , ( u 0 0 1 ) \begin{pmatrix}0&1\\ 1&0\end{pmatrix},\begin{pmatrix}1&0\\ t&1\end{pmatrix},\begin{pmatrix}u&0\\ 0&1\end{pmatrix}
  10. 1 b - c + 1 c - a \frac{1}{b-c}+\frac{1}{c-a}
  11. b - a ( b - c ) ( c - a ) = ( b - c ) + ( c - a ) ( b - c ) ( c - a ) \frac{b-a}{(b-c)(c-a)}=\frac{(b-c)+(c-a)}{(b-c)(c-a)}
  12. ( b - c ) - 1 + ( c - a ) - 1 (b-c)^{-1}+(c-a)^{-1}
  13. p = ( b - c ) - 1 + ( c - a ) - 1 p=(b-c)^{-1}+(c-a)^{-1}

Promise_problem.html

  1. L ⊆ { 0 , 1 } * L\subseteq\{0,1\}^{*}
  2. L L
  3. L L
  4. L YES L_{\,\text{YES}}
  5. L NO L_{\,\text{NO}}
  6. L YES ∩ L NO = ∅ L_{\,\text{YES}}\cap L_{\,\text{NO}}=\varnothing
  7. L YES L_{\,\text{YES}}
  8. L NO L_{\,\text{NO}}
  9. L YES âˆȘ L NO L_{\,\text{YES}}\cup L_{\,\text{NO}}
  10. { 0 , 1 } * \{0,1\}^{*}

Prompt_criticality.html

  1. N ( t ) = N 0 e k t / T N(t)=N_{0}e^{kt/T}\,

Prompt_neutron.html

  1. ÎČ = precursor atoms prompt neutrons + precursor atoms . \beta=\frac{\mbox{precursor atoms}~{}}{\mbox{prompt neutrons}~{}+\mbox{% precursor atoms}~{}}.
  2. D N F = delayed neutrons prompt neutrons + delayed neutrons . DNF=\frac{\mbox{delayed neutrons}~{}}{\mbox{prompt neutrons}~{}+\mbox{delayed % neutrons}~{}}.

Proofs_involving_the_Laplace–Beltrami_operator.html

  1. ∫ M d f ( X ) ω = - ∫ M f div X ω \int_{M}df(X)\;\omega=-\int_{M}f\,\operatorname{div}X\;\omega
  2. ∫ M ( f div ( X ) + X ( f ) ) ω = ∫ M ( f ℒ X + ℒ X ( f ) ) ω \int_{M}(f\mathrm{div}(X)+X(f))\omega=\int_{M}(f\mathcal{L}_{X}+\mathcal{L}_{X% }(f))\omega
  3. = ∫ M ℒ X f ω = ∫ M d Îč X f ω = ∫ ∂ M Îč X f ω =\int_{M}\mathcal{L}_{X}f\omega=\int_{M}\mathrm{d}\iota_{X}f\omega=\int_{% \partial M}\iota_{X}f\omega
  4. Δ f = d ή f + ή d f = ή d f = ή ∂ i f d x i \Delta f=\mathrm{d}\delta f+\delta\,\mathrm{d}f=\delta\,\mathrm{d}f=\delta\,% \partial_{i}f\,\mathrm{d}x^{i}
  5. = - * d * ∂ i f d x i = - * d ( Δ i J | g | ∂ i f d x J ) =-*\mathrm{d}{*\partial_{i}f\,\mathrm{d}x^{i}}=-*\mathrm{d}(\varepsilon_{iJ}% \sqrt{|g|}\partial^{i}f\,\mathrm{d}x^{J})
  6. = - * Δ i J ∂ j ( | g | ∂ i f ) d x j ∧ d x J = - * 1 | g | ∂ i ( | g | ∂ i f ) vol n =-*\varepsilon_{iJ}\,\partial_{j}(\sqrt{|g|}\partial^{i}f)\,\mathrm{d}x^{j}% \wedge\mathrm{d}x^{J}=-*\frac{1}{\sqrt{|g|}}\,\partial_{i}(\sqrt{|g|}\,% \partial^{i}f)\mathrm{vol}_{n}
  7. = - 1 | g | ∂ i ( | g | ∂ i f ) , =-\frac{1}{\sqrt{|g|}}\,\partial_{i}(\sqrt{|g|}\,\partial^{i}f),
  8. Δ ( f h ) = f Δ h + 2 ∂ i f ∂ i h + h Δ f . \Delta(fh)=f\,\Delta h+2\partial_{i}f\,\partial^{i}h+h\,\Delta f.
  9. Δ ( f h ) = ή d f h = ή ( f d h + h d f ) = * d ( f * d h ) + * d ( h * d f ) \Delta(fh)=\delta\,\mathrm{d}fh=\delta(f\,\mathrm{d}h+h\,\mathrm{d}f)=*\mathrm% {d}(f{*\mathrm{d}h})+*\mathrm{d}(h{*\mathrm{d}f})\;
  10. = * ( f d * d h + d f ∧ * d h + d h ∧ * d f + h d * d f ) =*(f\,\mathrm{d}*\mathrm{d}h+\mathrm{d}f\wedge*\mathrm{d}h+\mathrm{d}h\wedge*% \mathrm{d}f+h\,\mathrm{d}*\mathrm{d}f)
  11. = f * d * d h + * ( d f ∧ * d h + d h ∧ * d f ) + h * d * d f =f*\mathrm{d}*\mathrm{d}h+*(\mathrm{d}f\wedge*\mathrm{d}h+\mathrm{d}h\wedge*% \mathrm{d}f)+h*\mathrm{d}*\mathrm{d}f
  12. = f Δ h =f\,\Delta h
  13. + * ( ∂ i f d x i ∧ Δ j J | g | ∂ j h d x J + ∂ i h d x i ∧ Δ j J | g | ∂ j f d x J ) {}+*(\partial_{i}f\,\mathrm{d}x^{i}\wedge\varepsilon_{jJ}\sqrt{|g|}\partial^{j% }h\,\mathrm{d}x^{J}+\partial_{i}h\,\mathrm{d}x^{i}\wedge\varepsilon_{jJ}\sqrt{% |g|}\partial^{j}f\,\mathrm{d}x^{J})
  14. + h Δ f {}+h\,\Delta f
  15. = f Δ h + ( ∂ i f ∂ i h + ∂ i h ∂ i f ) * vol n + h Δ f =f\,\Delta h+(\partial_{i}f\,\partial^{i}h+\partial_{i}h\,\partial^{i}f){*% \mathrm{vol}_{n}}+h\,\Delta f
  16. = f Δ h + 2 ∂ i f ∂ i h + h Δ f =f\,\Delta h+2\partial_{i}f\,\partial^{i}h+h\,\Delta f

Proofs_of_Fermat's_theorem_on_sums_of_two_squares.html

  1. p = x 2 + y 2 p=x^{2}+y^{2}
  2. ( a 2 + b 2 ) ( p 2 + q 2 ) = ( a p + b q ) 2 + ( a q - b p ) 2 (a^{2}+b^{2})(p^{2}+q^{2})=(ap+bq)^{2}+(aq-bp)^{2}\,
  3. a 2 + b 2 a^{2}+b^{2}
  4. p 2 + q 2 p^{2}+q^{2}
  5. p 2 + q 2 p^{2}+q^{2}
  6. ( p b - a q ) ( p b + a q ) = p 2 b 2 - a 2 q 2 = p 2 ( a 2 + b 2 ) - a 2 ( p 2 + q 2 ) . (pb-aq)(pb+aq)=p^{2}b^{2}-a^{2}q^{2}=p^{2}(a^{2}+b^{2})-a^{2}(p^{2}+q^{2}).
  7. p 2 + q 2 p^{2}+q^{2}
  8. p b - a q pb-aq
  9. ( a 2 + b 2 ) ( p 2 + q 2 ) = ( a p + b q ) 2 + ( a q - b p ) 2 (a^{2}+b^{2})(p^{2}+q^{2})=(ap+bq)^{2}+(aq-bp)^{2}\,
  10. p 2 + q 2 p^{2}+q^{2}
  11. ( a p + b q ) 2 (ap+bq)^{2}
  12. p 2 + q 2 p^{2}+q^{2}
  13. ( p 2 + q 2 ) 2 (p^{2}+q^{2})^{2}
  14. a 2 + b 2 p 2 + q 2 = ( a p + b q p 2 + q 2 ) 2 + ( a q - b p p 2 + q 2 ) 2 \frac{a^{2}+b^{2}}{p^{2}+q^{2}}=\left(\frac{ap+bq}{p^{2}+q^{2}}\right)^{2}+% \left(\frac{aq-bp}{p^{2}+q^{2}}\right)^{2}
  15. p 2 + q 2 p^{2}+q^{2}
  16. p b + a q pb+aq
  17. ( a 2 + b 2 ) ( q 2 + p 2 ) = ( a q + b p ) 2 + ( a p - b q ) 2 (a^{2}+b^{2})(q^{2}+p^{2})=(aq+bp)^{2}+(ap-bq)^{2}\,
  18. x x
  19. a 2 + b 2 a^{2}+b^{2}
  20. p 1 p 2 ⋯ p n p_{1}p_{2}\cdots p_{n}
  21. a 2 + b 2 = x p 1 p 2 ⋯ p n a^{2}+b^{2}=xp_{1}p_{2}\cdots p_{n}
  22. p i p_{i}
  23. a 2 + b 2 a^{2}+b^{2}
  24. p 1 p_{1}
  25. p 2 p_{2}
  26. x x
  27. x x
  28. x x
  29. p i p_{i}
  30. a a
  31. b b
  32. a 2 + b 2 a^{2}+b^{2}
  33. x x
  34. a 2 + b 2 a^{2}+b^{2}
  35. a = m x ± c , b = n x ± d a=mx\pm c,\qquad b=nx\pm d
  36. c c
  37. d d
  38. x x
  39. a 2 + b 2 = m 2 x 2 ± 2 m x c + c 2 + n 2 x 2 ± 2 n x d + d 2 = A x + ( c 2 + d 2 ) . a^{2}+b^{2}=m^{2}x^{2}\pm 2mxc+c^{2}+n^{2}x^{2}\pm 2nxd+d^{2}=Ax+(c^{2}+d^{2}).
  40. c 2 + d 2 c^{2}+d^{2}
  41. x x
  42. c 2 + d 2 = y x c^{2}+d^{2}=yx
  43. c c
  44. d d
  45. x x
  46. x x
  47. a a
  48. b b
  49. y y
  50. c 2 + d 2 c^{2}+d^{2}
  51. e 2 + f 2 = z x e^{2}+f^{2}=zx
  52. e e
  53. f f
  54. z z
  55. x x
  56. z x = e 2 + f 2 ≀ c 2 + d 2 ≀ ( x 2 ) 2 + ( x 2 ) 2 = 1 2 x 2 . zx=e^{2}+f^{2}\leq c^{2}+d^{2}\leq\left(\frac{x}{2}\right)^{2}+\left(\frac{x}{% 2}\right)^{2}=\frac{1}{2}x^{2}.
  57. c c
  58. d d
  59. e e
  60. f f
  61. x x
  62. z z
  63. w w
  64. x x
  65. w w
  66. x x
  67. 4 n + 1 4n+1
  68. p = 4 n + 1 p=4n+1
  69. 1 , 2 4 n , 3 4 n , 
 , ( 4 n ) 4 n 1,2^{4n},3^{4n},\dots,(4n)^{4n}
  70. p p
  71. 2 4 n - 1 , 3 4 n - 2 4 n , 
 , ( 4 n ) 4 n - ( 4 n - 1 ) 4 n 2^{4n}-1,3^{4n}-2^{4n},\dots,(4n)^{4n}-(4n-1)^{4n}
  72. p p
  73. a 4 n - b 4 n = ( a 2 n + b 2 n ) ( a 2 n - b 2 n ) . a^{4n}-b^{4n}=\left(a^{2n}+b^{2n}\right)\left(a^{2n}-b^{2n}\right).
  74. p p
  75. 4 n - 1 4n-1
  76. p p
  77. a a
  78. b b
  79. 1 1
  80. p p
  81. 4 n - 1 4n-1
  82. 2 2 n - 1 , 3 2 n - 2 2 n , 
 , ( 4 n ) 2 n - ( 4 n - 1 ) 2 n 2^{2n}-1,3^{2n}-2^{2n},\dots,(4n)^{2n}-(4n-1)^{2n}
  83. 4 n - 2 4n-2
  84. 4 n - 3 4n-3
  85. k k
  86. 1 k , 2 k , 3 k , 
 1^{k},2^{k},3^{k},\dots
  87. k ! k!
  88. 2 n 2n
  89. ( 2 n ) ! (2n)!
  90. p p
  91. p p
  92. p p
  93. a x 2 + 2 b x y + c y 2 ax^{2}+2bxy+cy^{2}
  94. a , b , c a,b,c
  95. n n
  96. x , y x,y
  97. n = a x 2 + 2 b x y + c y 2 n=ax^{2}+2bxy+cy^{2}
  98. p p
  99. x 2 + y 2 x^{2}+y^{2}
  100. a = c = 1 a=c=1
  101. b = 0 b=0
  102. p p
  103. 1 1
  104. 4 4
  105. b 2 - a c b^{2}-ac
  106. x y xy
  107. b 2 - 4 a c b^{2}-4ac
  108. x 2 + y 2 x^{2}+y^{2}
  109. - 1 -1
  110. a x 2 + 2 b x y + c y 2 ax^{2}+2bxy+cy^{2}
  111. r x â€Č 2 + 2 s x â€Č y â€Č + t y â€Č 2 rx^{\prime 2}+2sx^{\prime}y^{\prime}+ty^{\prime 2}
  112. x = α x â€Č + ÎČ y â€Č x=\alpha x^{\prime}+\beta y^{\prime}
  113. y = Îł x â€Č + ÎŽ y â€Č y=\gamma x^{\prime}+\delta y^{\prime}
  114. α ÎŽ - ÎČ Îł = ± 1 \alpha\delta-\beta\gamma=\pm 1
  115. p p
  116. m m
  117. p p
  118. m 2 + 1 m^{2}+1
  119. p x 2 + 2 m x y + ( m 2 + 1 p ) y 2 , px^{2}+2mxy+\left(\frac{m^{2}+1}{p}\right)y^{2},
  120. z p - 1 - 1 = z 4 n - 1 = ( z 2 n - 1 ) ( z 2 n + 1 ) z^{p-1}-1=z^{4n}-1=(z^{2n}-1)(z^{2n}+1)
  121. z 2 n - 1 ≡ 0 ( mod p ) z^{2n}-1\equiv 0\;\;(\mathop{{\rm mod}}p)
  122. z 2 n - 1 z^{2n}-1
  123. z 4 n - 1 = ( z 2 n - 1 ) ( z 2 n + 1 ) z^{4n}-1=(z^{2n}-1)(z^{2n}+1)
  124. z 2 n + 1 z^{2n}+1
  125. m = z n m=z^{n}
  126. p p
  127. i p - 1 = ( - 1 ) p - 1 2 i^{p-1}=(-1)^{\frac{p-1}{2}}
  128. ω p = ( x + y i ) p ≡ x p + y p i p ≡ x + ( - 1 ) p - 1 2 y i ( mod p ) , \omega^{p}=(x+yi)^{p}\equiv x^{p}+y^{p}i^{p}\equiv x+(-1)^{\frac{p-1}{2}}yi\;% \;(\mathop{{\rm mod}}p),
  129. p = 4 n + 1 p=4n+1
  130. ( p ) (p)
  131. N ( x + i y ) = x 2 + y 2 N(x+iy)=x^{2}+y^{2}
  132. α \alpha
  133. N ( p ) = p 2 N(p)=p^{2}
  134. p = N ( α ) = N ( a + b i ) = a 2 + b 2 p=N(\alpha)=N(a+bi)=a^{2}+b^{2}
  135. p = 4 n + 1 p=4n+1
  136. m 2 + 1 m^{2}+1
  137. m + i m+i
  138. m - i m-i
  139. m 2 + 1 m^{2}+1
  140. p p
  141. p 2 = N ( p ) p^{2}=N(p)
  142. p = ( x + y i ) ( x - y i ) p=(x+yi)(x-yi)
  143. x x
  144. y y
  145. p = x 2 + y 2 p=x^{2}+y^{2}
  146. ( x , y , z ) ↩ { ( x + 2 z , z , y - x - z ) , if x < y - z ( 2 y - x , y , x - y + z ) , if y - z < x < 2 y ( x - 2 y , x - y + z , y ) , if x > 2 y (x,y,z)\mapsto\begin{cases}(x+2z,~{}z,~{}y-x-z),\quad\textrm{if}\,\,\,x<y-z\\ (2y-x,~{}y,~{}x-y+z),\quad\textrm{if}\,\,\,y-z<x<2y\\ (x-2y,~{}x-y+z,~{}y),\quad\textrm{if}\,\,\,x>2y\end{cases}

Propargyl.html

  1. ≡ \equiv

Proper_transfer_function.html

  1. 𝐆 ( s ) = 𝐍 ( s ) 𝐃 ( s ) = s 4 + n 1 s 3 + n 2 s 2 + n 3 s + n 4 s 4 + d 1 s 3 + d 2 s 2 + d 3 s + d 4 \,\textbf{G}(s)=\frac{\,\textbf{N}(s)}{\,\textbf{D}(s)}=\frac{s^{4}+n_{1}s^{3}% +n_{2}s^{2}+n_{3}s+n_{4}}{s^{4}+d_{1}s^{3}+d_{2}s^{2}+d_{3}s+d_{4}}
  2. deg ( 𝐍 ( s ) ) = 4 ≀ deg ( 𝐃 ( s ) ) = 4 \deg(\,\textbf{N}(s))=4\leq\deg(\,\textbf{D}(s))=4
  3. deg ( 𝐍 ( s ) ) = 4 = deg ( 𝐃 ( s ) ) = 4 \deg(\,\textbf{N}(s))=4=\deg(\,\textbf{D}(s))=4
  4. deg ( 𝐍 ( s ) ) = 4 ≼ deg ( 𝐃 ( s ) ) = 4 \deg(\,\textbf{N}(s))=4\nless\deg(\,\textbf{D}(s))=4
  5. 𝐆 ( s ) = 𝐍 ( s ) 𝐃 ( s ) = s 4 + n 1 s 3 + n 2 s 2 + n 3 s + n 4 d 1 s 3 + d 2 s 2 + d 3 s + d 4 \,\textbf{G}(s)=\frac{\,\textbf{N}(s)}{\,\textbf{D}(s)}=\frac{s^{4}+n_{1}s^{3}% +n_{2}s^{2}+n_{3}s+n_{4}}{d_{1}s^{3}+d_{2}s^{2}+d_{3}s+d_{4}}
  6. deg ( 𝐍 ( s ) ) = 4 ≰ deg ( 𝐃 ( s ) ) = 3 \deg(\,\textbf{N}(s))=4\nleq\deg(\,\textbf{D}(s))=3
  7. 𝐆 ( s ) = 𝐍 ( s ) 𝐃 ( s ) = n 1 s 3 + n 2 s 2 + n 3 s + n 4 s 4 + d 1 s 3 + d 2 s 2 + d 3 s + d 4 \,\textbf{G}(s)=\frac{\,\textbf{N}(s)}{\,\textbf{D}(s)}=\frac{n_{1}s^{3}+n_{2}% s^{2}+n_{3}s+n_{4}}{s^{4}+d_{1}s^{3}+d_{2}s^{2}+d_{3}s+d_{4}}
  8. deg ( 𝐍 ( s ) ) = 3 < deg ( 𝐃 ( s ) ) = 4 \deg(\,\textbf{N}(s))=3<\deg(\,\textbf{D}(s))=4
  9. | 𝐆 ( ± j ∞ ) | < ∞ |\,\textbf{G}(\pm j\infty)|<\infty
  10. 𝐆 ( ± j ∞ ) = 0 \,\textbf{G}(\pm j\infty)=0

ProPhoto_RGB_color_space.html

  1. X R O M M â€Č = { 0 ; X R O M M < 0.0 16 X R O M M I M A X ; 0.0 ≀ X R O M M < E t ( X R O M M ) 1 / 1.8 I M A X ; E t ≀ X R O M M < 1.0 I M A X ; X R O M M ≄ 1.0 X^{\prime}_{ROMM}=\begin{cases}0;&X_{ROMM}<0.0\\ 16X_{ROMM}I_{MAX};&0.0\leq X_{ROMM}<E_{t}\\ (X_{ROMM})^{1/1.8}I_{MAX};&E_{t}\leq X_{ROMM}<1.0\\ I_{MAX};&X_{ROMM}\geq 1.0\end{cases}
  2. X = R , G , o r B X=R,G,orB
  3. I M A X I_{MAX}
  4. E t = 16 1.8 / ( 1 - 1.8 ) = 0.001953 E_{t}=16^{1.8/(1-1.8)}=0.001953

Propositional_formula.html

  1. α \alpha
  2.  α \lnot\alpha
  3. ∧ , ∹ , → , ↔ \land,\lor,\to,\leftrightarrow
  4. α \alpha
  5. ÎČ \beta
  6. ( α → ÎČ ) (\alpha\to\beta)
  7. ↔ \leftrightarrow
  8. ÂŹ \lnot
  9. ÂŹ \lnot
  10. ∧ \land
  11. √ \lor
  12. → \to
  13. ↔ \leftrightarrow
  14. ¬ , ∧ , ∹ , → , ↔ \lnot,\land,\lor,\to,\leftrightarrow
  15. (  α ) (\lnot\alpha)
  16. α \alpha
  17. ( α □ ÎČ ) (\alpha\,\Box\,\beta)
  18. α \alpha
  19. ÎČ \beta
  20. □ \Box
  21. ∧ , ∹ , → , ↔ \land,\lor,\to,\leftrightarrow
  22. ℰ ¬ ( z ) \mathcal{E}_{\lnot}(z)
  23. ( ÂŹ z ) (\lnot z)
  24. ℰ ∧ ( y , z ) \mathcal{E}_{\land}(y,z)
  25. ( y ∧ x ) (y\land x)
  26. ℰ ∹ \mathcal{E}_{\lor}
  27. ℰ → \mathcal{E}_{\to}
  28. ℰ ↔ \mathcal{E}_{\leftrightarrow}
  29. ( ÂŹ (\lnot
  30. { ∧ ,  } \{\land,\lnot\}
  31. { √ ,  } \{\lor,\lnot\}
  32. { → , ¬ } \{\to,\lnot\}
  33. { ∧ , √ } \{\land,\lor\}
  34. ⊀ \top
  35. ⊄ \bot
  36. ⊀ ≡ ( a | ( a | a ) ) \top\equiv(a|(a|a))
  37. ⊄ ≡ ( ⊀ | ⊀ ) \bot\equiv(\top|\top)
  38. ⊄ \bot
  39. ⊀ \top

Prosthaphaeresis.html

  1. cos a = cos b cos c + sin b sin c cos α \cos a=\cos b\cos c+\sin b\sin c\cos\alpha
  2. sin b sin α = sin a sin ÎČ \sin b\sin\alpha=\sin a\sin\beta
  3. sin a sin b = cos ( a - b ) - cos ( a + b ) 2 \sin a\sin b=\frac{\cos(a-b)-\cos(a+b)}{2}
  4. cos a cos b = cos ( a - b ) + cos ( a + b ) 2 \cos a\cos b=\frac{\cos(a-b)+\cos(a+b)}{2}
  5. sin a cos b = sin ( a + b ) + sin ( a - b ) 2 \sin a\cos b=\frac{\sin(a+b)+\sin(a-b)}{2}
  6. cos a sin b = sin ( a + b ) - sin ( a - b ) 2 \cos a\sin b=\frac{\sin(a+b)-\sin(a-b)}{2}
  7. e i x = cos x + i sin x e^{ix}=\cos x+i\sin x
  8. sin a + sin b = 2 sin ( a + b 2 ) cos ( a - b 2 ) \sin a+\sin b=2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)
  9. sin a - sin b = 2 cos ( a + b 2 ) sin ( a - b 2 ) \sin a-\sin b=2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)
  10. cos a + cos b = 2 cos ( a + b 2 ) cos ( a - b 2 ) \cos a+\cos b=2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)
  11. cos a - cos b = - 2 sin ( a + b 2 ) sin ( a - b 2 ) \cos a-\cos b=-2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)

Protein_design.html

  1. min E T = ∑ i [ E i ( r i ) + ∑ i ≠ j E i j ( r i , r j ) ] \min E_{T}=\sum_{i}\Big[E_{i}(r_{i})+\sum_{i\neq j}E_{ij}(r_{i},r_{j})\Big]\,
  2. E ( r i â€Č ) + ∑ j ≠ i min r j E ( r i â€Č , r j ) > E ( r i ) + ∑ j ≠ i max r j E ( r i , r j ) E(r^{\prime}_{i})+\sum_{j\neq i}\min_{r_{j}}E(r^{\prime}_{i},r_{j})>E(r_{i})+% \sum_{j\neq i}\max_{r_{j}}E(r_{i},r_{j})
  3. g = ∑ i = 1 d ( E ( r i ) + ∑ j = i + 1 d E ( r i , r j ) ) g=\sum_{i=1}^{d}(E(r_{i})+\sum_{j=i+1}^{d}E(r_{i},r_{j}))
  4. h = ∑ j = d + 1 n [ min r j ( E ( r j ) + ∑ i = 1 d E ( r i , r j ) + ∑ k = j + 1 n min r k E ( r j , r k ) ) ] h=\sum_{j=d+1}^{n}[\min_{r_{j}}(E(r_{j})+\sum_{i=1}^{d}E(r_{i},r_{j})+\sum_{k=% j+1}^{n}\min_{r_{k}}E(r_{j},r_{k}))]
  5. min ∑ i ∑ r i E i ( r i ) q i ( r i ) + ∑ j ≠ i ∑ r j E i j ( r i , r j ) q i j ( r i , r j ) \ \min\sum_{i}\sum_{r_{i}}E_{i}(r_{i})q_{i}(r_{i})+\sum_{j\neq i}\sum_{r_{j}}E% _{ij}(r_{i},r_{j})q_{ij}(r_{i},r_{j})\,
  6. ∑ r i q i ( r i ) = 1 , ∀ i \sum_{r_{i}}q_{i}(r_{i})=1,\ \forall i
  7. ∑ r j q i j ( r i , r j ) = q i ( r i ) , ∀ i , r i , j \sum_{r_{j}}q_{ij}(r_{i},r_{j})=q_{i}(r_{i}),\forall i,r_{i},j
  8. q i , q i j ∈ { 0 , 1 } q_{i},q_{ij}\in\{0,1\}
  9. p = e - ÎČ ( E new - E old ) ) , p=e^{-\beta(E_{\,\text{new}}-E_{\,\text{old}}))},
  10. m i → j ( r j ) = max r i ( e - E i ( r i ) - E i j ( r i , r j ) T ) ∏ k ∈ N ( i ) \ j m k → i ( r i ) m_{i\to j}(r_{j})=\max_{r_{i}}\Big(e^{\frac{-E_{i}(r_{i})-E_{ij}(r_{i},r_{j})}% {T}}\Big)\prod_{k\in N(i)\backslash j}m_{k\to i(r_{i})}
  11. Δ G = E P L - E P - E L \Delta_{G}=E_{PL}-E_{P}-E_{L}
  12. K * = ∑ x ∈ P L e - E ( x ) / R T ∑ x ∈ P e - E ( x ) / R T ∑ x ∈ L e - E ( x ) / R T K^{*}=\frac{\sum\limits_{x\in PL}e^{-E(x)/RT}}{\sum\limits_{x\in P}e^{-E(x)/RT% }\sum\limits_{x\in L}e^{-E(x)/RT}}

Protein_efficiency_ratio.html

  1. P E R = G a i n i n b o d y m a s s ( g ) P r o t e i n i n t a k e ( g ) PER\,=\frac{Gain\ in\ body\ mass(g)}{Protein\ intake(g)}

Proximity_search_(text).html

  1. n - 1 n-1

Prüfer_rank.html

  1. G G
  2. sup { d ( H ) | H ≀ G } \sup\{d(H)|H\leq G\}
  3. d ( H ) d(H)
  4. H / Ί ( H ) H/\Phi(H)
  5. Ί ( H ) \Phi(H)
  6. H H
  7. H H
  8. H H
  9. d ( H ) d(H)
  10. H H

Prüfer_sequence.html

  1. K n K_{n}
  2. d i d_{i}
  3. i i
  4. ( n - 2 d 1 - 1 , d 2 - 1 , 
 , d n - 1 ) = ( n - 2 ) ! ( d 1 - 1 ) ! ( d 2 - 1 ) ! ⋯ ( d n - 1 ) ! . {\left({{n-2}\atop{d_{1}-1,\,d_{2}-1,\,\dots,\,d_{n}-1}}\right)}=\frac{(n-2)!}% {(d_{1}-1)!(d_{2}-1)!\cdots(d_{n}-1)!}.
  5. i i
  6. ( d i - 1 ) (d_{i}-1)
  7. n 1 n 2 - 1 n 2 n 1 - 1 n_{1}^{n_{2}-1}n_{2}^{n_{1}-1}

Pseudogroup.html

  1. f â€Č ∘ f : f - 1 ( V ∩ U â€Č ) → f â€Č ( V ∩ U â€Č ) f^{\prime}\circ f\colon f^{-1}(V\cap U^{\prime})\to f^{\prime}(V\cap U^{\prime})

Pseudorandom_binary_sequence.html

  1. a 0 , 
 , a N - 1 a_{0},\ldots,a_{N-1}
  2. N N
  3. a j ∈ { 0 , 1 } a_{j}\in\{0,1\}
  4. j = 0 , 1 , 
 , N - 1 j=0,1,...,N-1
  5. m = ∑ a j m=\sum a_{j}
  6. N - m N-m
  7. C ( v ) = ∑ j = 0 N - 1 a j a j + v C(v)=\sum_{j=0}^{N-1}a_{j}a_{j+v}
  8. C ( v ) = { m , if v ≡ 0 ( mod N ) m c , otherwise C(v)=\begin{cases}m,\mbox{ if }~{}v\equiv 0\;\;(\mbox{mod}~{}N)\\ \\ mc,\mbox{ otherwise }\end{cases}
  9. c = m - 1 N - 1 c=\frac{m-1}{N-1}
  10. a j a_{j}
  11. N N
  12. N = 2 k - 1 N=2^{k}-1
  13. x 7 + x 6 + 1 x^{7}+x^{6}+1
  14. x 15 + x 14 + 1 x^{15}+x^{14}+1
  15. x 23 + x 18 + 1 x^{23}+x^{18}+1
  16. x 31 + x 28 + 1 x^{31}+x^{28}+1

Pseudorandom_generator.html

  1. 𝒜 = { A : { 0 , 1 } n → { 0 , 1 } * } \mathcal{A}=\{A:\{0,1\}^{n}\to\{0,1\}^{*}\}
  2. G : { 0 , 1 } ℓ → { 0 , 1 } n G:\{0,1\}^{\ell}\to\{0,1\}^{n}
  3. ℓ ≀ n \ell\leq n
  4. 𝒜 \mathcal{A}
  5. Ï” \epsilon
  6. A A
  7. 𝒜 \mathcal{A}
  8. A ( G ( U ℓ ) ) A(G(U_{\ell}))
  9. A ( U n ) A(U_{n})
  10. Ï” \epsilon
  11. U k U_{k}
  12. { 0 , 1 } k \{0,1\}^{k}
  13. ℓ \ell
  14. n - ℓ n-\ell
  15. ( 𝒜 n ) n ∈ đ’© (\mathcal{A}_{n})_{n\in\mathcal{N}}
  16. Ï” ( n ) \epsilon(n)
  17. ( G n ) n ∈ đ’© (G_{n})_{n\in\mathcal{N}}
  18. G n : { 0 , 1 } ℓ ( n ) → { 0 , 1 } n G_{n}:\{0,1\}^{\ell(n)}\to\{0,1\}^{n}
  19. 𝒜 n \mathcal{A}_{n}
  20. Ï” ( n ) \epsilon(n)
  21. ℓ ( n ) \ell(n)
  22. 𝒜 \mathcal{A}
  23. 𝒜 \mathcal{A}
  24. 𝒜 \mathcal{A}
  25. 𝒜 \mathcal{A}
  26. O ( log n ) O(\log n)
  27. O ( log 2 n ) O(\log^{2}n)
  28. O ( log 2 n ) O(\log^{2}n)
  29. O ( log n ) O(\log n)
  30. O ( log 1.5 n ) O(\log^{1.5}n)
  31. đ”œ \mathbb{F}
  32. ℓ = log n + O ( log ( Ï” - 1 ) ) \ell=\log n+O(\log(\epsilon^{-1}))
  33. d d
  34. d d
  35. ℓ = d ⋅ log n + O ( 2 d ⋅ log ( Ï” - 1 ) ) \ell=d\cdot\log n+O(2^{d}\cdot\log(\epsilon^{-1}))

Pseudorange.html

  1. Δ t \Delta t
  2. 2 \textstyle{\sqrt{2}}

Pseudotensor.html

  1. P ^ j 1 
 j p i 1 
 i q = ( - 1 ) A A i 1 ⋯ k 1 A i q B l 1 k q ⋯ j 1 B l p P l 1 
 l p k 1 
 k q j p \hat{P}^{i_{1}\ldots i_{q}}_{\,j_{1}\ldots j_{p}}=(-1)^{A}A^{i_{1}}{}_{k_{1}}% \cdots A^{i_{q}}{}_{k_{q}}B^{l_{1}}{}_{j_{1}}\cdots B^{l_{p}}{}_{j_{p}}P^{k_{1% }\ldots k_{q}}_{l_{1}\ldots l_{p}}
  2. P ^ j 1 
 j p i 1 
 i q , P l 1 
 l p k 1 
 k q \hat{P}^{i_{1}\ldots i_{q}}_{\,j_{1}\ldots j_{p}},P^{k_{1}\ldots k_{q}}_{l_{1}% \ldots l_{p}}
  3. A i q k q A^{i_{q}}{}_{k_{q}}
  4. B l p j p B^{l_{p}}{}_{j_{p}}
  5. ( - 1 ) A = sign ( det ( A i q ) k q ) = ± 1 (-1)^{A}=\mathrm{sign}(\det(A^{i_{q}}{}_{k_{q}}))=\pm{1}

Psi_function.html

  1. ψ ( α ) \psi(\alpha)
  2. ψ ( n ) \psi(n)
  3. ψ ( x ) \psi(x)
  4. ψ m ( z ) \psi^{m}(z)
  5. ψ ( z ) \psi(z)
  6. ψ 1 ( z ) \psi^{1}(z)

Ptolemy's_theorem.html

  1. | A C ¯ | ⋅ | B D ¯ | = | A B ¯ | ⋅ | C D ¯ | + | B C ¯ | ⋅ | A D ¯ | |\overline{AC}|\cdot|\overline{BD}|=|\overline{AB}|\cdot|\overline{CD}|+|% \overline{BC}|\cdot|\overline{AD}|
  2. q s = p s + r s ⇒ q = p + r . qs=ps+rs\Rightarrow q=p+r.
  3. a a
  4. a 2 a\sqrt{2}
  5. φ = b a = 1 + 5 2 . \varphi={b\over a}={{1+\sqrt{5}}\over 2}.
  6. a d = 2 b c ad=2bc\;
  7. ⇒ a d = 2 φ a c \Rightarrow ad=2\varphi ac
  8. φ \varphi
  9. ⇒ c = d 2 φ . \Rightarrow c=\frac{d}{2\varphi}.
  10. a = b φ = b ( φ - 1 ) . a=\frac{b}{\varphi}=b(\varphi-1).
  11. A B AB
  12. B C BC
  13. C D CD
  14. α \alpha
  15. ÎČ \beta
  16. Îł \gamma
  17. R R
  18. A B = 2 R sin α AB=2R\sin\alpha
  19. B C = 2 R sin ÎČ BC=2R\sin\beta
  20. C D = 2 R sin Îł CD=2R\sin\gamma
  21. A D = 2 R sin ( α + ÎČ + Îł ) AD=2R\sin(\alpha+\beta+\gamma)
  22. A C = 2 R sin ( α + ÎČ ) AC=2R\sin(\alpha+\beta)
  23. B D = 2 R sin ( ÎČ + Îł ) BD=2R\sin(\beta+\gamma)
  24. sin ( α + ÎČ ) sin ( ÎČ + Îł ) = sin α sin Îł + sin ÎČ sin ( α + ÎČ + Îł ) \sin(\alpha+\beta)\sin(\beta+\gamma)=\sin\alpha\sin\gamma+\sin\beta\sin(\alpha% +\beta+\gamma)
  25. 4 R 2 4R^{2}
  26. sin ( x + y ) = sin x cos y + cos x sin y \sin(x+y)=\sin{x}\cos{y}+\cos{x}\sin{y}
  27. cos ( x + y ) = cos x cos y - sin x sin y \cos(x+y)=\cos{x}\cos{y}-\sin{x}\sin{y}
  28. sin α sin ÎČ cos ÎČ cos Îł + sin α cos 2 ÎČ sin Îł + cos α sin 2 ÎČ cos Îł + cos α sin ÎČ cos ÎČ sin Îł \sin\alpha\sin\beta\cos\beta\cos\gamma+\sin\alpha\cos^{2}\beta\sin\gamma+\cos% \alpha\sin^{2}\beta\cos\gamma+\cos\alpha\sin\beta\cos\beta\sin\gamma
  29. S 1 , S 2 , S 3 , S 4 S_{1},S_{2},S_{3},S_{4}\;
  30. Ξ 1 , Ξ 2 , Ξ 3 \theta_{1},\theta_{2},\theta_{3}\,
  31. Ξ 4 \theta_{4}\,
  32. sin Ξ 1 sin Ξ 3 + sin Ξ 2 sin Ξ 4 = sin ( Ξ 3 + Ξ 2 ) sin ( Ξ 1 + Ξ 2 ) \sin\theta_{1}\sin\theta_{3}+\sin\theta_{2}\sin\theta_{4}=\sin(\theta_{3}+% \theta_{2})\sin(\theta_{1}+\theta_{2})\;
  33. Ξ 1 , Ξ 2 , Ξ 3 \theta_{1},\theta_{2},\theta_{3}\,
  34. Ξ 4 \theta_{4}\,
  35. Ξ 1 + Ξ 2 + Ξ 3 + Ξ 4 = 180 ∘ \theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=180^{\circ}\,
  36. Ξ 1 = Ξ 3 \theta_{1}=\theta_{3}\;
  37. Ξ 2 = Ξ 4 \theta_{2}=\theta_{4}\;
  38. Ξ 1 + Ξ 2 = Ξ 3 + Ξ 4 = 90 ∘ \theta_{1}+\theta_{2}=\theta_{3}+\theta_{4}=90^{\circ}\;
  39. sin Ξ 1 sin Ξ 3 + sin Ξ 2 sin Ξ 4 = sin ( Ξ 3 + Ξ 2 ) sin ( Ξ 3 + Ξ 4 ) \sin\theta_{1}\sin\theta_{3}+\sin\theta_{2}\sin\theta_{4}=\sin(\theta_{3}+% \theta_{2})\sin(\theta_{3}+\theta_{4})\,
  40. sin 2 Ξ 1 + sin 2 Ξ 2 = sin 2 ( Ξ 1 + Ξ 2 ) \sin^{2}\theta_{1}+\sin^{2}\theta_{2}=\sin^{2}(\theta_{1}+\theta_{2})\,
  41. sin 2 Ξ 1 + cos 2 Ξ 1 = 1 \sin^{2}\theta_{1}+\cos^{2}\theta_{1}=1\,
  42. Ξ 2 = Ξ 4 \theta_{2}=\theta_{4}\;
  43. 2 x 2x
  44. x = S 2 cos ( Ξ 2 + Ξ 3 ) x=S_{2}\cos(\theta_{2}+\theta_{3})\;
  45. S 1 S 3 + S 2 S 4 = A C ¯ ⋅ B D ¯ ⇒ S 1 S 3 + S 2 2 = A C ¯ 2 ⇒ S 1 [ S 1 - 2 S 2 cos ( ξ 2 + ξ 3 ) ] + S 2 2 = A C ¯ 2 ⇒ S 1 2 + S 2 2 - 2 S 1 S 2 cos ( ξ 2 + ξ 3 ) = A C ¯ 2 \begin{array}[]{lcl}\\ S_{1}S_{3}+S_{2}S_{4}={\overline{AC}}\cdot{\overline{BD}}\\ \Rightarrow S_{1}S_{3}+{S_{2}}^{2}={\overline{AC}}^{2}\\ \Rightarrow S_{1}[S_{1}-2{S_{2}}\cos(\theta_{2}+\theta_{3})]+{S_{2}}^{2}={% \overline{AC}}^{2}\\ \Rightarrow{S_{1}}^{2}+{S_{2}}^{2}-2{S_{1}}{S_{2}}\cos(\theta_{2}+\theta_{3})=% {\overline{AC}}^{2}\\ \end{array}
  46. Ξ 1 + Ξ 2 = Ξ 3 + Ξ 4 = 90 ∘ . \theta_{1}+\theta_{2}=\theta_{3}+\theta_{4}=90^{\circ}.\;
  47. sin Ξ 1 sin Ξ 3 + sin Ξ 2 sin Ξ 4 = sin ( Ξ 3 + Ξ 2 ) sin ( Ξ 3 + Ξ 4 ) \sin\theta_{1}\sin\theta_{3}+\sin\theta_{2}\sin\theta_{4}=\sin(\theta_{3}+% \theta_{2})\sin(\theta_{3}+\theta_{4})
  48. cos ξ 2 sin ξ 3 + sin ξ 2 cos ξ 3 = sin ( ξ 3 + ξ 2 ) × 1 \cos\theta_{2}\sin\theta_{3}+\sin\theta_{2}\cos\theta_{3}=\sin(\theta_{3}+% \theta_{2})\times 1\,
  49. Ξ 1 = 90 ∘ \theta_{1}=90^{\circ}
  50. Ξ 2 + ( Ξ 3 + Ξ 4 ) = 90 ∘ \theta_{2}+(\theta_{3}+\theta_{4})=90^{\circ}
  51. sin Ξ 1 sin Ξ 3 + sin Ξ 2 sin Ξ 4 = sin ( Ξ 3 + Ξ 2 ) sin ( Ξ 3 + Ξ 4 ) \sin\theta_{1}\sin\theta_{3}+\sin\theta_{2}\sin\theta_{4}=\sin(\theta_{3}+% \theta_{2})\sin(\theta_{3}+\theta_{4})\,
  52. sin Ξ 3 + sin Ξ 2 cos ( Ξ 2 + Ξ 3 ) = sin ( Ξ 3 + Ξ 2 ) cos Ξ 2 \sin\theta_{3}+\sin\theta_{2}\cos(\theta_{2}+\theta_{3})=\sin(\theta_{3}+% \theta_{2})\cos\theta_{2}\,
  53. sin Ξ 3 = sin ( Ξ 3 + Ξ 2 ) cos Ξ 2 - cos ( Ξ 2 + Ξ 3 ) sin Ξ 2 \sin\theta_{3}=\sin(\theta_{3}+\theta_{2})\cos\theta_{2}-\cos(\theta_{2}+% \theta_{3})\sin\theta_{2}\,
  54. Ξ 3 = 90 ∘ \theta_{3}=90^{\circ}
  55. Ξ 1 + ( Ξ 2 + Ξ 4 ) = 90 ∘ \theta_{1}+(\theta_{2}+\theta_{4})=90^{\circ}
  56. sin Ξ 1 sin Ξ 3 + sin Ξ 2 sin Ξ 4 = sin ( Ξ 3 + Ξ 2 ) sin ( Ξ 3 + Ξ 4 ) \sin\theta_{1}\sin\theta_{3}+\sin\theta_{2}\sin\theta_{4}=\sin(\theta_{3}+% \theta_{2})\sin(\theta_{3}+\theta_{4})\,
  57. cos ( Ξ 2 + Ξ 4 ) + sin Ξ 2 sin Ξ 4 = cos Ξ 2 cos Ξ 4 \cos(\theta_{2}+\theta_{4})+\sin\theta_{2}\sin\theta_{4}=\cos\theta_{2}\cos% \theta_{4}\,
  58. cos ( Ξ 2 + Ξ 4 ) = cos Ξ 2 cos Ξ 4 - sin Ξ 2 sin Ξ 4 \cos(\theta_{2}+\theta_{4})=\cos\theta_{2}\cos\theta_{4}-\sin\theta_{2}\sin% \theta_{4}\,
  59. A B ÂŻ ⋅ C D ÂŻ + B C ÂŻ ⋅ D A ÂŻ ≄ A C ÂŻ ⋅ B D ÂŻ \overline{AB}\cdot\overline{CD}+\overline{BC}\cdot\overline{DA}\geq\overline{% AC}\cdot\overline{BD}

Pullback_bundle.html

  1. f * E = { ( b â€Č , e ) ∈ B â€Č × E ∣ f ( b â€Č ) = π ( e ) } ⊂ B â€Č × E f^{*}E=\{(b^{\prime},e)\in B^{\prime}\times E\mid f(b^{\prime})=\pi(e)\}% \subset B^{\prime}\times E
  2. π â€Č ( b â€Č , e ) = b â€Č . \pi^{\prime}(b^{\prime},e)=b^{\prime}.\,
  3. f ~ : f * E → E \tilde{f}\colon f^{*}E\to E
  4. f ∗ E ⟶ f ~ E π â€Č ↓ ↓ π B â€Č ⟶ f B \begin{array}[]{ccc}f^{\ast}E&\stackrel{\tilde{f}}{\longrightarrow}&E\\ {\pi}^{\prime}\downarrow&&\downarrow\pi\\ B^{\prime}&\stackrel{f}{\longrightarrow}&B\end{array}
  5. ψ ( b â€Č , e ) = ( b â€Č , proj ( φ ( e ) ) 2 ) . \psi(b^{\prime},e)=(b^{\prime},\mbox{proj}~{}_{2}(\varphi(e))).\,
  6. f ~ \tilde{f}
  7. f * s = s ∘ f f^{*}s=s\circ f
  8. f * t i j = t i j ∘ f . f^{*}t_{ij}=t_{ij}\circ f.
  9. ( x , e ) ⋅ g = ( x , e ⋅ g ) (x,e)\cdot g=(x,e\cdot g)
  10. f ~ \tilde{f}

Pulmonary_surfactant.html

  1. P = 2 Îł r P=\frac{2\gamma}{r}

Pulsar.html

  1. DM = ∫ 0 D n e ( s ) d s , \mathrm{DM}=\int_{0}^{D}n_{e}(s)ds,
  2. D D
  3. n e n_{e}

Pulse-Doppler_radar.html

  1. Doppler Frequency = ( 2 × Transmit Frequency × Range Velocity C ) \,\text{Doppler Frequency}=\left(\frac{2\times\,\text{Transmit Frequency}% \times\,\text{Range Velocity}}{C}\right)
  2. I = I 0 sin ( 4 π ( x 0 + v Δ t ) λ ) = I 0 sin ( Θ 0 + Δ Θ ) I=I_{0}\sin\left(\frac{4\pi(x_{0}+v\Delta t)}{\lambda}\right)=I_{0}\sin\left(% \Theta_{0}+\Delta\Theta\right)
  3. x 0 = distance radar to target λ = radar wavelength Δ t = time between two pulses \begin{aligned}\displaystyle x_{0}&\displaystyle=\,\text{distance radar to % target}\\ \displaystyle\lambda&\displaystyle=\,\text{radar wavelength}\\ \displaystyle\Delta t&\displaystyle=\,\text{time between two pulses}\end{aligned}
  4. Δ Θ = ( 4 π v Δ t λ ) \Delta\Theta=\left(\frac{4\pi v\Delta t}{\lambda}\right)
  5. v = target speed = λ Δ Θ 4 π Δ t v=\,\text{target speed}=\frac{\lambda\Delta\Theta}{4\pi\Delta t}
  6. Δ Θ \ \Delta\Theta
  7. | ( Doppler Frequency × C 2 × Transmit Frequency ) | > Velocity Threshold \left|\left(\frac{\,\text{Doppler Frequency}\times C}{2\times\,\text{Transmit % Frequency}}\right)\right|>\,\text{Velocity Threshold}
  8. | ( Doppler Frequency × C 2 × Transmit Frequency ) - Ground Speed × cos ( Θ ) | > Velocity Threshold \left|\left(\frac{\,\text{Doppler Frequency}\times C}{2\times\,\text{Transmit % Frequency}}\right)-\,\text{Ground Speed}\times\cos\left(\Theta\right)\right|>% \,\text{Velocity Threshold}
  9. Θ \Theta
  10. Range Resolution = ( C PRF × ( Number of Samples Between Transmit Pulses ) ) \,\text{Range Resolution}=\left(\frac{C}{\,\text{PRF}\times(\,\text{Number of % Samples Between Transmit Pulses})}\right)
  11. Velocity Resolution = ( C × PRF ( Transmit Frequency ) × ( Filter Size In Transmit Pulses ) ) \,\text{Velocity Resolution}=\left(\frac{C\times\,\text{PRF}}{(\,\text{% Transmit Frequency})\times(\,\text{Filter Size In Transmit Pulses})}\right)
  12. T = ( 1 e S C V / 20 × S × P R F ) T=\left(\frac{1}{e^{SCV/20}\times S\times PRF}\right)
  13. Dynamic Range = Smaller of { Carrier Power Noise Power Transmit Noise, where bandwidth is PRF Filter Size 2 ( Sample Bits + Filter Size ) Receiver Dynamic Range \,\text{Dynamic Range}=\,\text{Smaller of}\begin{cases}\tfrac{\,\text{Carrier % Power}}{\,\text{Noise Power}}&\,\text{Transmit Noise, where bandwidth is}\,% \tfrac{\,\text{PRF}}{\,\text{Filter Size}}\\ \\ 2^{\left(}\,\text{Sample Bits + Filter Size}\right)&\,\text{Receiver Dynamic % Range}\end{cases}
  14. Subclutter Visibility = ( Dynamic Range CFAR Detection Threshold ) \,\text{Subclutter Visibility}=\left(\tfrac{\,\text{Dynamic Range}}{\,\text{% CFAR Detection Threshold}}\right)
  15. Target Power > ( Clutter Power Subclutter Visibility ) \,\text{Target Power}>\left(\tfrac{\,\text{Clutter Power}}{\,\text{Subclutter % Visibility}}\right)
  16. R = ( P t G t A r σ F D 2 16 π 2 K b T B N ) 1 / 4 R=\left(\frac{P_{t}\ G_{t}\ A_{r}\ \sigma F\ D^{2}}{16\ \pi^{2}\ K_{b}\ T\ B\ % N}\right)^{1/4}

Pulse_wave.html

  1. τ \tau
  2. T T
  3. τ τ
  4. f ( t ) = τ T + ∑ n = 1 ∞ 2 n π sin ( π n τ T ) cos ( 2 π n T t ) f(t)=\frac{\tau}{T}+\sum_{n=1}^{\infty}\frac{2}{n\pi}\sin\left(\frac{\pi n\tau% }{T}\right)\cos\left(\frac{2\pi n}{T}t\right)
  5. t = 0 t=0
  6. t t
  7. t - τ / 2 t-τ/2

Pushforward_(differential).html

  1. d φ x : 𝐑 m → 𝐑 n . \mathrm{d}\varphi_{x}:\mathbf{R}^{m}\to\mathbf{R}^{n}\ .
  2. d φ x : T x M → T φ ( x ) N \mathrm{d}\varphi_{x}:T_{x}M\to T_{\varphi(x)}N\,
  3. d φ x ( Îł â€Č ( 0 ) ) = ( φ ∘ Îł ) â€Č ( 0 ) . \mathrm{d}\varphi_{x}(\gamma^{\prime}(0))=(\varphi\circ\gamma)^{\prime}(0).
  4. d φ x ( X ) ( f ) = X ( f ∘ φ ) . \mathrm{d}\varphi_{x}(X)(f)=X(f\circ\varphi).
  5. φ ^ : U → V \widehat{\varphi}:U\to V
  6. d φ x ( ∂ ∂ u a ) = ∂ φ ^ b ∂ u a ∂ ∂ v b , \mathrm{d}\varphi_{x}\left(\frac{\partial}{\partial u^{a}}\right)=\frac{% \partial\widehat{\varphi}^{b}}{\partial u^{a}}\frac{\partial}{\partial v^{b}},
  7. ( d φ x ) a b = ∂ φ ^ b ∂ u a . (\mathrm{d}\varphi_{x})_{a}^{\;b}=\frac{\partial\widehat{\varphi}^{b}}{% \partial u^{a}}.
  8. D φ x , ( φ * ) x , φ â€Č ( x ) . D\varphi_{x},\;(\varphi_{*})_{x},\;\varphi^{\prime}(x).
  9. Y x = φ * ( X φ - 1 ( x ) ) . Y_{x}=\varphi_{*}(X_{\varphi^{-1}(x)}).

Q-analog.html

  1. lim q → 1 1 - q n 1 - q = n \lim_{q\rightarrow 1}\frac{1-q^{n}}{1-q}=n
  2. [ n ] q = 1 - q n 1 - q = 1 + q + q 2 + 
 + q n - 1 . [n]_{q}=\frac{1-q^{n}}{1-q}=1+q+q^{2}+\ldots+q^{n-1}.
  3. [ n ] q ! \displaystyle\big[n]_{q}!
  4. ∑ w ∈ S n q inv ( w ) = [ n ] q ! . \sum_{w\in S_{n}}q^{\,\text{inv}(w)}=[n]_{q}!.
  5. q → 1 q\rightarrow 1
  6. [ n ] q ! = ( q ; q ) n ( 1 - q ) n . [n]_{q}!=\frac{(q;q)_{n}}{(1-q)^{n}}.
  7. ( n k ) q = [ n ] q ! [ n - k ] q ! [ k ] q ! . {\left({{n}\atop{k}}\right)}_{q}=\frac{[n]_{q}!}{[n-k]_{q}![k]_{q}!}.
  8. e q x = ∑ n = 0 ∞ x n [ n ] q ! . e_{q}^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{[n]_{q}!}.
  9. ( n k ) q . {\left({{n}\atop{k}}\right)}_{q}.
  10. ( n k ) , {\left({{n}\atop{k}}\right)},

Q-ball.html

  1. ϕ \phi
  2. V ( ϕ ) V(\phi)
  3. V free ( ϕ ) = m 2 | ϕ | 2 V_{\rm free}(\phi)=m^{2}|\phi|^{2}
  4. - λ | ϕ | 4 -\lambda|\phi|^{4}
  5. ϕ \phi
  6. ϕ \phi
  7. V ( ϕ ) < V free ( ϕ ) V(\phi)<V_{\rm free}(\phi)
  8. ϕ \phi
  9. U ( 1 ) U(1)
  10. U ( 1 ) U(1)
  11. E ω = E + ω [ Q - 1 2 i ∫ d 3 x ( ϕ * ∂ t ϕ - ϕ ∂ t ϕ * ) ] , E_{\omega}=E+\omega\left[Q-\frac{1}{2i}\int d^{3}x(\phi^{*}\partial_{t}\phi-% \phi\partial_{t}\phi^{*})\right],
  12. E = ∫ d 3 x [ 1 2 ϕ ˙ 2 + 1 2 | ∇ ϕ | 2 + U ( ϕ , ϕ * ) ] , E=\int d^{3}x\left[\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}|\nabla\phi|^{2}+U(% \phi,\phi^{*})\right],
  13. ω \omega
  14. E ω E_{\omega}
  15. E ω = ∫ d 3 x [ 1 2 | ϕ ˙ - i ω ϕ | 2 + 1 2 | ∇ ϕ | 2 + U ^ ω ( ϕ , ϕ * ) ] E_{\omega}=\int d^{3}x\left[\frac{1}{2}|\dot{\phi}-i\omega\phi|^{2}+\frac{1}{2% }|\nabla\phi|^{2}+\hat{U}_{\omega}(\phi,\phi^{*})\right]
  16. U ^ ω = U - 1 2 ω 2 ϕ 2 \hat{U}_{\omega}=U-\frac{1}{2}\omega^{2}\phi^{2}
  17. ϕ ( r → , t ) = ϕ 0 ( r → ) e i ω t . \phi(\vec{r},t)=\phi_{0}(\vec{r})e^{i\omega t}.
  18. ω \omega
  19. ϕ * ϕ \phi^{*}\phi
  20. m 2 ϕ * ϕ m^{2}\phi^{*}\phi
  21. m m
  22. ϕ 0 ( r ) = Ξ ( R - r ) ϕ 0 . \phi_{0}(r)=\theta(R-r)\phi_{0}.
  23. Q = ω ϕ 0 2 V Q=\omega\phi_{0}^{2}V
  24. ω \omega
  25. E = 1 2 Q 2 ϕ 0 2 V + U ( ϕ 0 ) V . E=\frac{1}{2}\frac{Q^{2}}{\phi_{0}^{2}V}+U(\phi_{0})V.
  26. V V
  27. V = Q 2 2 U ( ϕ 0 ) ϕ 0 2 . V=\sqrt{\frac{Q^{2}}{2U(\phi_{0})\phi_{0}^{2}}}.
  28. E = 2 U ( ϕ 0 ) ϕ 0 2 Q . E=\sqrt{\frac{2U(\phi_{0})}{\phi_{0}^{2}}}~{}Q.
  29. ϕ 0 \phi_{0}
  30. m i n = 2 U ( ϕ ) ϕ 2 , min=\frac{2U(\phi)}{\phi^{2}},
  31. ϕ > 0 \phi>0
  32. M ( Q ) = ω 0 Q . M(Q)=\omega_{0}Q.
  33. ϕ \phi

Q-learning.html

  1. a ∈ A a\in A
  2. Δ t \Delta t
  3. γ Δ t \gamma^{\Delta t}
  4. Îł \gamma
  5. 0 ≀ Îł ≀ 1 0\leq\gamma\leq 1
  6. Îł \gamma
  7. Δ t \Delta t
  8. Q : S × A → ℝ Q:S\times A\to\mathbb{R}
  9. Q t + 1 ( s t , a t ) = Q t ( s t , a t ) ⏟ old value + α t ( s t , a t ) ⏟ learning rate ⋅ ( R t + 1 ⏟ reward + Îł ⏟ discount factor max a Q t ( s t + 1 , a ) ⏟ estimate of optimal future value ⏞ learned value - Q t ( s t , a t ) ⏟ old value ) Q_{t+1}(s_{t},a_{t})=\underbrace{Q_{t}(s_{t},a_{t})}_{\rm old~{}value}+% \underbrace{\alpha_{t}(s_{t},a_{t})}_{\rm learning~{}rate}\cdot\left(% \overbrace{\underbrace{R_{t+1}}_{\rm reward}+\underbrace{\gamma}_{\rm discount% ~{}factor}\underbrace{\max_{a}Q_{t}(s_{t+1},a)}_{\rm estimate~{}of~{}optimal~{% }future~{}value}}^{\rm learned~{}value}-\underbrace{Q_{t}(s_{t},a_{t})}_{\rm old% ~{}value}\right)
  10. R t + 1 R_{t+1}
  11. a t a_{t}
  12. s t s_{t}
  13. α t ( s , a ) \alpha_{t}(s,a)
  14. 0 < α ≀ 1 0<\alpha\leq 1
  15. s t + 1 s_{t+1}
  16. s f s_{f}
  17. Q ( s f , a ) Q(s_{f},a)
  18. Q ( s f , a ) Q(s_{f},a)
  19. α t ( s , a ) = 1 \alpha_{t}(s,a)=1
  20. α t ( s , a ) = 0.1 \alpha_{t}(s,a)=0.1
  21. t t
  22. Îł Îł
  23. Îł = 1 Îł=1
  24. Q 0 Q_{0}
  25. r r
  26. Q Q
  27. Îł \gamma

QBD_(electronics).html

  1. Q b d = ∫ 0 t b d i ( t ) d t Q_{bd}=\int_{0}^{t_{bd}}i(t)\,dt
  2. t b d t_{bd}

QCD_vacuum.html

  1. ⟹ ψ ÂŻ i ψ i ⟩ \langle\overline{\psi}_{i}\psi_{i}\rangle
  2. L I = N ÂŻ Îł 5 π N L_{I}=\bar{N}\gamma_{5}\pi N\,
  3. g N ÂŻ Îł ÎŒ ∂ ÎŒ π N g\bar{N}\gamma^{\mu}\partial_{\mu}\pi N\,
  4. π → π + C \pi\rightarrow\pi+C\,
  5. g π N N F π = G A M N g_{\pi NN}F_{\pi}=G_{A}M_{N}\,
  6. G A G_{A}
  7. J ÎŒ | 0 ⟩ = k ÎŒ | π ⟩ , J_{\mu}|0\rangle=k_{\mu}|\pi\rangle\,,
  8. k Ό \!k_{\mu}
  9. ∂ ÎŒ J ÎŒ | 0 ⟩ = k ÎŒ k ÎŒ | π ⟩ = m π 2 | π ⟩ = 0 . \partial_{\mu}J^{\mu}|0\rangle=k^{\mu}k_{\mu}|\pi\rangle=m_{\pi}^{2}|\pi% \rangle=0\,.
  10. m π 2 = 0 m_{\pi}^{2}=0
  11. k ÎŒ ⟹ N ( p ) | π ( k ) N ( p â€Č ) ⟩ = ⟹ N ( p ) | J ÎŒ | N ( p â€Č ) ⟩ . k_{\mu}\langle N(p)|\pi(k)N(p^{\prime})\rangle=\langle N(p)|J_{\mu}|N(p^{% \prime})\rangle\,.
  12. ⟹ N | J ÎŒ | N ⟩ ⟹ e | J ÎŒ | Μ ⟩ \langle N|J^{\mu}|N\rangle\langle e|J_{\mu}|\nu\rangle\,
  13. ⟹ ( g G ) 2 ⟩ = def ⟹ g 2 G ÎŒ Μ G ÎŒ Μ ⟩ ≃ 0.5 GeV 4 \langle(gG)^{2}\rangle\ \stackrel{\mathrm{def}}{=}\ \langle g^{2}G_{\mu\nu}G^{% \mu\nu}\rangle\simeq 0.5{\rm\ GeV}^{4}
  14. ⟹ ψ ÂŻ ψ ⟩ ≃ ( - 0.23 ) 3 GeV 3 \langle\overline{\psi}\psi\rangle\simeq(-0.23)^{3}{\rm\ GeV}^{3}
  15. ⟹ ( g G ) 4 ⟩ ≃ 5 : 10 ⟹ ( g G ) 2 ⟩ 2 \langle(gG)^{4}\rangle\simeq 5:10\langle(gG)^{2}\rangle^{2}
  16. ψ ÂŻ Îł 5 Îł ÎŒ ψ \overline{\psi}\gamma_{5}\gamma_{\mu}\psi
  17. ⟹ ψ ÂŻ Îł 0 ψ ⟩ \langle\overline{\psi}\gamma_{0}\psi\rangle
  18. g A g_{A}

Quadratic_assignment_problem.html

  1. ∑ a , b ∈ P w ( a , b ) ⋅ d ( f ( a ) , f ( b ) ) \sum_{a,b\in P}w(a,b)\cdot d(f(a),f(b))
  2. ∑ a , b ∈ P w a , b d f ( a ) , f ( b ) \sum_{a,b\in P}w_{a,b}d_{f(a),f(b)}
  3. min X ∈ Π n trace ( W X D X T ) \min_{X\in\Pi_{n}}\operatorname{trace}(WXDX^{T})
  4. Π n \Pi_{n}

Quadratic_classifier.html

  1. đ± 𝐓 đ€đ± + 𝐛 𝐓 đ± + c \mathbf{x^{T}Ax}+\mathbf{b^{T}x}+c
  2. y ∈ { 0 , 1 } y\in\{0,1\}
  3. Ό y = 0 , Ό y = 1 \mu_{y=0},\mu_{y=1}
  4. ÎŁ y = 0 , ÎŁ y = 1 \Sigma_{y=0},\Sigma_{y=1}
  5. 2 π | ÎŁ y = 1 | - 1 exp ( - 1 2 ( x - ÎŒ y = 1 ) T ÎŁ y = 1 - 1 ( x - ÎŒ y = 1 ) ) 2 π | ÎŁ y = 0 | - 1 exp ( - 1 2 ( x - ÎŒ y = 0 ) T ÎŁ y = 0 - 1 ( x - ÎŒ y = 0 ) ) < t \frac{\sqrt{2\pi|\Sigma_{y=1}|}^{-1}\exp\left(-\frac{1}{2}(x-\mu_{y=1})^{T}% \Sigma_{y=1}^{-1}(x-\mu_{y=1})\right)}{\sqrt{2\pi|\Sigma_{y=0}|}^{-1}\exp\left% (-\frac{1}{2}(x-\mu_{y=0})^{T}\Sigma_{y=0}^{-1}(x-\mu_{y=0})\right)}<t
  6. [ x 1 , x 2 , x 3 ] [x_{1},\;x_{2},\;x_{3}]
  7. [ x 1 , x 2 , x 3 , x 1 2 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 , x 3 2 ] [x_{1},\;x_{2},\;x_{3},\;x_{1}^{2},\;x_{1}x_{2},\;x_{1}x_{3},\;x_{2}^{2},\;x_{% 2}x_{3},\;x_{3}^{2}]
  8. x 1 2 + x 2 2 + x 3 2 
 \;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\;\ldots\;
  9. x 1 2 x 2 2 , x 1 2 x 3 2 
 \;x_{1}^{2}x_{2}^{2},\;x_{1}^{2}x_{3}^{2}\;\ldots\;

Quadratic_probing.html

  1. H + 1 , H + 2 , H + 3 , H + 4 , 
 , H + k H+1,H+2,H+3,H+4,...,H+k
  2. H + 1 2 , H + 2 2 , H + 3 2 , H + 4 2 , 
 , H + k 2 H+1^{2},H+2^{2},H+3^{2},H+4^{2},...,H+k^{2}
  3. h ( k , i ) = ( h ( k ) + c 1 i + c 2 i 2 ) ( mod m ) h(k,i)=(h(k)+c_{1}i+c_{2}i^{2})\;\;(\mathop{{\rm mod}}m)
  4. h ( k , i ) = ( h ( k ) + i + i 2 ) ( mod m ) h(k,i)=(h(k)+i+i^{2})\;\;(\mathop{{\rm mod}}m)
  5. h ( k ) , h ( k ) + 2 , h ( k ) + 6 , 
 h(k),h(k)+2,h(k)+6,...
  6. h ( k ) , h ( k ) + 1 , h ( k ) + 3 , h ( k ) + 6 , 
 h(k),h(k)+1,h(k)+3,h(k)+6,...

Quadrature_(mathematics).html

  1. x = a b x=\sqrt{ab}

Quadrupole.html

  1. Q x x + Q y y + Q z z = 0 Q_{xx}+Q_{yy}+Q_{zz}=0
  2. q l q_{l}
  3. m l m_{l}
  4. r l → = ( r x l , r y l , r z l ) \vec{r_{l}}=(r_{xl},r_{yl},r_{zl})
  5. Q i j = ∑ l q l ( 3 r i l r j l - ∄ r l → ∄ 2 ÎŽ i j ) Q_{ij}=\sum_{l}q_{l}(3r_{il}r_{jl}-\|\vec{r_{l}}\|^{2}\delta_{ij})
  6. i , j i,j
  7. x , y , z x,y,z
  8. ÎŽ i j \delta_{ij}
  9. ρ ( x , y , z ) \rho(x,y,z)
  10. Q i j = ∫ ρ ( 3 r i r j - ∄ r → ∄ 2 ÎŽ i j ) d 3 r Q_{ij}=\int\,\rho(3r_{i}r_{j}-\|\vec{r}\|^{2}\delta_{ij})\,d^{3}{r}
  11. 1 / r 1/r
  12. V q ( 𝐑 ) = k | 𝐑 | 3 ∑ i , j 1 2 Q i j n i n j , V_{q}(\mathbf{R})=\frac{k}{|\mathbf{R}|^{3}}\sum_{i,j}\frac{1}{2}Q_{ij}\,n_{i}% n_{j}\ ,
  13. k k
  14. n i , n j n_{i},n_{j}
  15. V q ( 𝐑 ) = 1 4 π Ï” 0 1 | 𝐑 | 3 ∑ i , j 1 2 Q i j n i n j , V_{q}(\mathbf{R})=\frac{1}{4\pi\epsilon_{0}}\frac{1}{|\mathbf{R}|^{3}}\sum_{i,% j}\frac{1}{2}Q_{ij}\,n_{i}n_{j}\ ,
  16. Ï” 0 \epsilon_{0}
  17. Q i j Q_{ij}
  18. lim a → 0 ; a 3 ⋅ Q → const . \lim_{a\to 0;\,a^{3}\cdot Q\to\rm{const.}}
  19. V q ( 𝐑 ) = - G 1 2 1 | 𝐑 | 3 ∑ i , j Q i j n i n j . V_{q}(\mathbf{R})=-G\frac{1}{2}\frac{1}{|\mathbf{R}|^{3}}\sum_{i,j}Q_{ij}\,n_{% i}n_{j}\ .
  20. 1 | 𝐑 | 3 \frac{1}{|\mathbf{R}|^{3}}
  21. Q i j = M ( 3 x i x j - ÎŽ i j ) Q_{ij}=M(3x_{i}x_{j}-\delta_{ij})
  22. x i x_{i}

Quake_engine.html

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

Quantitative_feedback_theory.html

  1. đ’« ( s ) = { ∏ i ( s + z i ) ∏ j ( s + p j ) , ∀ z i ∈ [ z i , m i n , z i , m a x ] , p j ∈ [ p j , m i n , p j , m a x ] } \mathcal{P}(s)=\left\{\dfrac{\prod_{i}(s+z_{i})}{\prod_{j}(s+p_{j})},\forall z% _{i}\in[z_{i,min},z_{i,max}],p_{j}\in[p_{j,min},p_{j,max}]\right\}
  2. L 0 ( s ) = G ( s ) P 0 ( s ) L_{0}(s)=G(s)P_{0}(s)
  3. L 0 ( s ) L_{0}(s)
  4. G ( s ) G(s)
  5. F ( s ) F(s)

Quantization_(linguistics).html

  1. U U
  2. F F
  3. p p
  4. U U
  5. < p <_{p}
  6. ( ∀ F ⊆ U p ) ( Q U A ( F ) ⇔ ( ∀ x , y ) ( F ( x ) ∧ F ( y ) ⇒ ¬ x < p y ) ) (\forall F\subseteq U_{p})(QUA(F)\iff(\forall x,y)(F(x)\wedge F(y)\Rightarrow% \neg x<_{p}y))

Quantum_correlation.html

  1. N + + - N + - - N - + + N - - N t o t a l \frac{N_{++}-N_{+-}-N_{-+}+N_{--}}{N_{total}}
  2. N + + + N + - + N - + + N - - N_{++}+N_{+-}+N_{-+}+N_{--}
  3. Q C ( a , b ) = ∫ d λ ρ ( λ ) A ( a , λ ) B ( b , λ ) QC(a,b)=\int d\lambda\rho(\lambda)A(a,\lambda)B(b,\lambda)

Quantum_sort.html

  1. Ω ( n log n ) \Omega(n\log n)

Quantum_wire.html

  1. R = ρ l A R=\rho{l\over A}
  2. ρ \rho
  3. l l
  4. A A
  5. 2 e 2 / h 2e^{2}/h
  6. e e
  7. h h
  8. 4 e 2 / h 4e^{2}/h

Quark_model.html

  1. 3 / 2 {3}/{2}
  2. 𝟑 ¯ \overline{\mathbf{3}}
  3. 𝟑 ⊗ 𝟑 ¯ = 𝟖 ⊕ 𝟏 \mathbf{3}\otimes\mathbf{\overline{3}}=\mathbf{8}\oplus\mathbf{1}
  4. L L
  5. S S
  6. 𝟑 ⊗ 𝟑 ⊗ 𝟑 = 𝟏𝟎 S ⊕ 𝟖 M ⊕ 𝟖 M ⊕ 𝟏 A \mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3}=\mathbf{10}_{S}\oplus\mathbf{8}_{% M}\oplus\mathbf{8}_{M}\oplus\mathbf{1}_{A}
  7. 𝟔 ⊗ 𝟔 ⊗ 𝟔 = 𝟓𝟔 S ⊕ 𝟕𝟎 M ⊕ 𝟕𝟎 M ⊕ 𝟐𝟎 A . \mathbf{6}\otimes\mathbf{6}\otimes\mathbf{6}=\mathbf{56}_{S}\oplus\mathbf{70}_% {M}\oplus\mathbf{70}_{M}\oplus\mathbf{20}_{A}~{}.
  8. 𝟓𝟔 = 𝟏𝟎 3 2 ⊕ 𝟖 1 2 , \mathbf{56}=\mathbf{10}^{\frac{3}{2}}\oplus\mathbf{8}^{\frac{1}{2}}~{},
  9. 1 / 2 {1}/{2}
  10. 3 / 2 {3}/{2}
  11. 3 / 2 {3}/{2}

Quarkonium.html

  1. D 0 D ÂŻ * 0 D^{0}\bar{D}^{*0}
  2. V ( r ) = - a r + b r V(r)=-\frac{a}{r}+br
  3. r r
  4. a a
  5. b b
  6. a / r a/r
  7. 1 / r 1/r
  8. b r br
  9. a a
  10. b b

Quarter_period.html

  1. K ( m ) = ∫ 0 π 2 d Ξ 1 - m sin 2 Ξ K(m)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-m\sin^{2}\theta}}
  2. i K â€Č ( m ) = i K ( 1 - m ) . {\rm{i}}K^{\prime}(m)={\rm{i}}K(1-m).\,
  3. sn u {\rm{sn}}u\,
  4. cn u {\rm{cn}}u\,
  5. 4 K 4K\,
  6. 4 i K â€Č 4{\rm{i}}K^{\prime}\,
  7. k 2 = m k^{2}=m\,
  8. K ( k ) K(k)\,
  9. K ( m ) K(m)\,
  10. k k\,
  11. m m\,
  12. m m\,
  13. m 1 = 1 - m m_{1}=1-m\,
  14. k k\,
  15. k â€Č k^{\prime}\,
  16. k â€Č 2 = m 1 {k^{\prime}}^{2}=m_{1}\,\!
  17. α \alpha\,\!
  18. k = sin α k=\sin\alpha\,\!
  19. π 2 - α \frac{\pi}{2}-\alpha\,\!
  20. m 1 = sin 2 ( π 2 - α ) = cos 2 α . m_{1}=\sin^{2}\left(\frac{\pi}{2}-\alpha\right)=\cos^{2}\alpha.\,\!
  21. k = ns ( K + i K â€Č ) k=\textrm{ns}(K+{\rm{i}}K^{\prime})\,\!
  22. k â€Č = dn K k^{\prime}=\textrm{dn}K\,
  23. q q\,
  24. q = e - π K â€Č K . q=e^{-\frac{\pi K^{\prime}}{K}}.\,
  25. q 1 = e - π K K â€Č . q_{1}=e^{-\frac{\pi K}{K^{\prime}}}.\,
  26. K = π 2 + 2 π ∑ n = 1 ∞ q n 1 + q 2 n . K=\frac{\pi}{2}+2\pi\sum_{n=1}^{\infty}\frac{q^{n}}{1+q^{2n}}.\,

Quartic_interaction.html

  1. φ \varphi
  2. φ \varphi
  3. λ 4 ! φ 4 \frac{\lambda}{4!}\varphi^{4}
  4. λ \lambda
  5. ℒ ( φ ) = 1 2 [ ∂ ÎŒ φ ∂ ÎŒ φ - m 2 φ 2 ] - 1 4 λ φ 4 . \mathcal{L}(\varphi)=\frac{1}{2}[\partial^{\mu}\varphi\partial_{\mu}\varphi-m^% {2}\varphi^{2}]-\frac{1}{4}\lambda\varphi^{4}.
  6. φ \varphi
  7. - φ -\varphi
  8. φ 1 \varphi_{1}
  9. φ 2 \varphi_{2}
  10. ℒ ( φ 1 , φ 2 ) = 1 2 [ ∂ ÎŒ φ 1 ∂ ÎŒ φ 1 - m 2 φ 1 2 ] + 1 2 [ ∂ ÎŒ φ 2 ∂ ÎŒ φ 2 - m 2 φ 2 2 ] - 1 4 λ ( φ 1 2 + φ 2 2 ) 2 , \mathcal{L}(\varphi_{1},\varphi_{2})=\frac{1}{2}[\partial_{\mu}\varphi_{1}% \partial^{\mu}\varphi_{1}-m^{2}\varphi_{1}^{2}]+\frac{1}{2}[\partial_{\mu}% \varphi_{2}\partial^{\mu}\varphi_{2}-m^{2}\varphi_{2}^{2}]-\frac{1}{4}\lambda(% \varphi_{1}^{2}+\varphi_{2}^{2})^{2},
  11. ϕ \phi
  12. ϕ ≡ 1 2 ( φ 1 + i φ 2 ) , \phi\equiv\frac{1}{\sqrt{2}}(\varphi_{1}+i\varphi_{2}),
  13. ϕ * ≡ 1 2 ( φ 1 - i φ 2 ) . \phi^{*}\equiv\frac{1}{\sqrt{2}}(\varphi_{1}-i\varphi_{2}).
  14. ℒ ( ϕ ) = ∂ ÎŒ ϕ * ∂ ÎŒ ϕ - m 2 ϕ * ϕ - λ ( ϕ * ϕ ) 2 , \mathcal{L}(\phi)=\partial^{\mu}\phi^{*}\partial_{\mu}\phi-m^{2}\phi^{*}\phi-% \lambda(\phi^{*}\phi)^{2},
  15. φ 1 , φ 2 \varphi_{1},\varphi_{2}
  16. ϕ \phi
  17. N N
  18. φ 4 \varphi^{4}
  19. ℒ ( φ 1 , 
 , φ N ) = 1 2 [ ∂ ÎŒ φ a ∂ ÎŒ φ a - m 2 φ a φ a ] - 1 4 λ ( φ a φ a ) 2 , a = 1 , 
 , N . \mathcal{L}(\varphi_{1},...,\varphi_{N})=\frac{1}{2}[\partial^{\mu}\varphi_{a}% \partial_{\mu}\varphi_{a}-m^{2}\varphi_{a}\varphi_{a}]-\frac{1}{4}\lambda(% \varphi_{a}\varphi_{a})^{2},\quad a=1,...,N.
  20. λ \lambda
  21. ϕ 4 \phi^{4}
  22. ⟹ Ω | 𝒯 { ϕ ( x 1 ) ⋯ ϕ ( x n ) } | Ω ⟩ = ∫ 𝒟 ϕ ϕ ( x 1 ) ⋯ ϕ ( x n ) e i ∫ d 4 x ( 1 2 ∂ ÎŒ ϕ ∂ ÎŒ ϕ - m 2 2 ϕ 2 - λ 4 ! ϕ 4 ) ∫ 𝒟 ϕ e i ∫ d 4 x ( 1 2 ∂ ÎŒ ϕ ∂ ÎŒ ϕ - m 2 2 ϕ 2 - λ 4 ! ϕ 4 ) . \langle\Omega|\mathcal{T}\{{\phi}(x_{1})\cdots{\phi}(x_{n})\}|\Omega\rangle=% \frac{\int\mathcal{D}\phi\phi(x_{1})\cdots\phi(x_{n})e^{i\int d^{4}x\left({1% \over 2}\partial^{\mu}\phi\partial_{\mu}\phi-{m^{2}\over 2}\phi^{2}-{\lambda% \over 4!}\phi^{4}\right)}}{\int\mathcal{D}\phi e^{i\int d^{4}x\left({1\over 2}% \partial^{\mu}\phi\partial_{\mu}\phi-{m^{2}\over 2}\phi^{2}-{\lambda\over 4!}% \phi^{4}\right)}}.
  23. Z [ J ] = ∫ 𝒟 ϕ e i ∫ d 4 x ( 1 2 ∂ ÎŒ ϕ ∂ ÎŒ ϕ - m 2 2 ϕ 2 - λ 4 ! ϕ 4 + J ϕ ) = Z [ 0 ] ∑ n = 0 ∞ 1 n ! ⟹ Ω | 𝒯 { ϕ ( x 1 ) ⋯ ϕ ( x n ) } | Ω ⟩ . Z[J]=\int\mathcal{D}\phi e^{i\int d^{4}x\left({1\over 2}\partial^{\mu}\phi% \partial_{\mu}\phi-{m^{2}\over 2}\phi^{2}-{\lambda\over 4!}\phi^{4}+J\phi% \right)}=Z[0]\sum_{n=0}^{\infty}\frac{1}{n!}\langle\Omega|\mathcal{T}\{{\phi}(% x_{1})\cdots{\phi}(x_{n})\}|\Omega\rangle.
  24. Z [ J ] = ∫ 𝒟 ϕ e - ∫ d 4 x ( 1 2 ( ∇ ϕ ) 2 + m 2 2 ϕ 2 + λ 4 ! ϕ 4 + J ϕ ) . Z[J]=\int\mathcal{D}\phi e^{-\int d^{4}x\left({1\over 2}(\nabla\phi)^{2}+{m^{2% }\over 2}\phi^{2}+{\lambda\over 4!}\phi^{4}+J\phi\right)}.
  25. Z ~ [ J ~ ] = ∫ 𝒟 ϕ ~ e - ∫ d 4 p ( 1 2 ( p 2 + m 2 ) ϕ ~ 2 - J ~ ϕ ~ + λ 4 ! ∫ d 4 p 1 d 4 p 2 d 4 p 3 ÎŽ ( p - p 1 - p 2 - p 3 ) ϕ ~ ( p ) ϕ ~ ( p 1 ) ϕ ~ ( p 2 ) ϕ ~ ( p 3 ) ) . \tilde{Z}[\tilde{J}]=\int\mathcal{D}\tilde{\phi}e^{-\int d^{4}p\left({1\over 2% }(p^{2}+m^{2})\tilde{\phi}^{2}-\tilde{J}\tilde{\phi}+{\lambda\over 4!}{\int d^% {4}p_{1}d^{4}p_{2}d^{4}p_{3}\delta(p-p_{1}-p_{2}-p_{3})\tilde{\phi}(p)\tilde{% \phi}(p_{1})\tilde{\phi}(p_{2})\tilde{\phi}(p_{3})}\right)}.
  26. ÎŽ ( x ) \delta(x)
  27. Z ~ [ J ~ ] = ∫ 𝒟 ϕ ~ ∏ p [ e - ( p 2 + m 2 ) ϕ ~ 2 / 2 e - λ / 4 ! ∫ d 4 p 1 d 4 p 2 d 4 p 3 ÎŽ ( p - p 1 - p 2 - p 3 ) ϕ ~ ( p ) ϕ ~ ( p 1 ) ϕ ~ ( p 2 ) ϕ ~ ( p 3 ) e J ~ ϕ ~ ] . \tilde{Z}[\tilde{J}]=\int\mathcal{D}\tilde{\phi}\prod_{p}\left[e^{-(p^{2}+m^{2% })\tilde{\phi}^{2}/2}e^{-\lambda/4!\int d^{4}p_{1}d^{4}p_{2}d^{4}p_{3}\delta(p% -p_{1}-p_{2}-p_{3})\tilde{\phi}(p)\tilde{\phi}(p_{1})\tilde{\phi}(p_{2})\tilde% {\phi}(p_{3})}e^{\tilde{J}\tilde{\phi}}\right].
  28. ϕ ~ ( p ) \tilde{\phi}(p)
  29. Z ~ [ 0 ] \tilde{Z}[0]
  30. - i λ -i\lambda
  31. φ \varphi
  32. ℒ ( φ ) = 1 2 ( ∂ φ ) 2 + 1 2 ÎŒ 2 φ 2 - 1 4 λ φ 4 ≡ 1 2 ( ∂ φ ) 2 - V ( φ ) , \mathcal{L}(\varphi)=\frac{1}{2}(\partial\varphi)^{2}+\frac{1}{2}\mu^{2}% \varphi^{2}-\frac{1}{4}\lambda\varphi^{4}\equiv\frac{1}{2}(\partial\varphi)^{2% }-V(\varphi),
  33. Ό 2 > 0 \mu^{2}>0
  34. V ( φ ) ≡ - 1 2 ÎŒ 2 φ 2 + 1 4 λ φ 4 . V(\varphi)\equiv-\frac{1}{2}\mu^{2}\varphi^{2}+\frac{1}{4}\lambda\varphi^{4}.
  35. φ \varphi
  36. V â€Č ( φ 0 ) = 0 âŸș φ 0 2 ≡ v 2 = ÎŒ 2 λ . V^{\prime}(\varphi_{0})=0\Longleftrightarrow\varphi_{0}^{2}\equiv v^{2}=\frac{% \mu^{2}}{\lambda}.
  37. φ ( x ) = v + σ ( x ) , \varphi(x)=v+\sigma(x),
  38. ℒ ( φ ) = - ÎŒ 4 4 λ ⏟ unimportant constant + 1 2 [ ( ∂ σ ) 2 - ( 2 ÎŒ ) 2 σ 2 ] ⏟ massive scalar field + ( - λ v σ 3 - λ 4 σ 4 ) ⏟ self-interactions . \mathcal{L}(\varphi)=\underbrace{-\frac{\mu^{4}}{4\lambda}}_{\,\text{% unimportant constant}}+\underbrace{\frac{1}{2}[(\partial\sigma)^{2}-(\sqrt{2}% \mu)^{2}\sigma^{2}]}_{\,\text{massive scalar field}}+\underbrace{(-\lambda v% \sigma^{3}-\frac{\lambda}{4}\sigma^{4})}_{\,\text{self-interactions}}.
  39. σ \sigma
  40. Z 2 Z_{2}
  41. φ → - φ \varphi\rightarrow-\varphi
  42. ⟹ Ω | φ | Ω ⟩ = ± 6 ÎŒ 2 λ \langle\Omega|\varphi|\Omega\rangle=\pm\sqrt{\frac{6\mu^{2}}{\lambda}}
  43. | Ω ± ⟩ |\Omega_{\pm}\rangle
  44. ⟹ Ω ± | φ | Ω ± ⟩ = ± 6 ÎŒ 2 λ . \langle\Omega_{\pm}|\varphi|\Omega_{\pm}\rangle=\pm\sqrt{\frac{6\mu^{2}}{% \lambda}}.
  45. Z 2 Z_{2}
  46. φ → - φ \varphi\rightarrow-\varphi
  47. | Ω + ⟩ ↔ | Ω - ⟩ |\Omega_{+}\rangle\leftrightarrow|\Omega_{-}\rangle
  48. Z 2 Z_{2}
  49. σ → - σ - 2 v . \sigma\rightarrow-\sigma-2v.
  50. ∂ 2 φ + ÎŒ 2 φ + λ φ 3 = 0 \partial^{2}\varphi+\mu^{2}\varphi+\lambda\varphi^{3}=0
  51. Ό = 0 \mu=0
  52. φ ( x ) = ± ÎŒ ( 2 λ ) 1 4 sn ( p ⋅ x + Ξ , - 1 ) , \varphi(x)=\pm\mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x+% \theta,-1),
  53. sn \,\rm sn\!
  54. Ό , Ξ \,\mu,\theta\!
  55. p 2 = Ό 2 ( λ 2 ) 1 2 . p^{2}=\mu^{2}\left(\frac{\lambda}{2}\right)^{1\over 2}.
  56. φ ( x ) = ± 2 ÎŒ 4 ÎŒ 0 2 + ÎŒ 0 4 + 2 λ ÎŒ 4 sn ( p ⋅ x + Ξ , - ÎŒ 0 2 + ÎŒ 0 4 + 2 λ ÎŒ 4 - ÎŒ 0 2 - ÎŒ 0 4 + 2 λ ÎŒ 4 ) \varphi(x)=\pm\sqrt{\frac{2\mu^{4}}{\mu_{0}^{2}+\sqrt{\mu_{0}^{4}+2\lambda\mu^% {4}}}}{\rm sn}\left(p\cdot x+\theta,\sqrt{\frac{-\mu_{0}^{2}+\sqrt{\mu_{0}^{4}% +2\lambda\mu^{4}}}{-\mu_{0}^{2}-\sqrt{\mu_{0}^{4}+2\lambda\mu^{4}}}}\right)
  57. p 2 = Ό 0 2 + λ Ό 4 Ό 0 2 + Ό 0 4 + 2 λ Ό 4 . p^{2}=\mu_{0}^{2}+\frac{\lambda\mu^{4}}{\mu_{0}^{2}+\sqrt{\mu_{0}^{4}+2\lambda% \mu^{4}}}.
  58. φ ( x ) = ± v ⋅ dn ( p ⋅ x + Ξ , i ) , \varphi(x)=\pm v\cdot{\rm dn}(p\cdot x+\theta,i),
  59. v = 2 Ό 0 2 3 λ v=\sqrt{\frac{2\mu_{0}^{2}}{3\lambda}}
  60. p 2 = λ v 2 2 . p^{2}=\frac{\lambda v^{2}}{2}.
  61. dn \,{\rm dn}\!
  62. φ = φ ( Ο ) \varphi=\varphi(\xi)
  63. Ο = p ⋅ x \xi=p\cdot x
  64. p p

Quartz_crystal_microbalance.html

  1. u 0 = 4 ( n π ) 2 d Q U el u_{0}=\frac{4}{\left(n\pi\right)^{2}}dQU_{\mathrm{el}}
  2. Δ f * f f = i π Z q Z L \frac{\Delta f^{*}}{f_{f}}=\frac{i}{\pi Z_{q}}Z_{L}
  3. Z L = - i Z q tan ( π Δ f f f ) Z_{L}=-iZ_{q}\tan\left(\pi\frac{\Delta f}{f_{f}}\right)
  4. Δ f f f = 1 π Z q 2 ω u 0 ⟹ σ ( t ) cos ( ω t ) ⟩ t \frac{\Delta f}{f_{f}}=\frac{1}{\pi Z_{q}}\,\frac{2}{\omega u_{0}}\left\langle% \sigma\left(t\right)\cos\left(\omega t\right)\right\rangle_{t}
  5. Δ ( w / 2 ) f f = 1 π Z q 2 ω u 0 ⟹ σ ( t ) sin ( ω t ) ⟩ t \frac{\Delta(w/2)}{f_{f}}=\frac{1}{\pi Z_{q}}\,\frac{2}{\omega u_{0}}\left% \langle\sigma\left(t\right)\sin\left(\omega t\right)\right\rangle_{t}
  6. Δ f * f f = i π Z q σ u ˙ = i π Z q Z ac = i π Z q ρ i ω η \frac{\Delta f^{*}}{f_{f}}=\frac{i}{\pi Z_{q}}\,\frac{\sigma}{\dot{u}}=\frac{i% }{\pi Z_{q}}Z_{\mathrm{ac}}=\frac{i}{\pi Z_{q}}\sqrt{\rho i\omega\eta}
  7. = 1 π Z q - 1 + i 2 ρ ω ( η â€Č - i η â€Čâ€Č ) = i π Z q ρ ( G â€Č + i G â€Čâ€Č ) =\frac{1}{\pi Z_{q}}\,\frac{-1+i}{\sqrt{2}}\sqrt{\rho\omega\left(\eta^{\prime}% -i\eta^{\prime\prime}\right)}=\frac{i}{\pi Z_{q}}\sqrt{\rho\left(G^{\prime}+iG% ^{\prime\prime}\right)}
  8. η â€Č = - π Z q 2 ρ Liq f Δ f Δ ( w / 2 ) f f 2 \eta^{\prime}=-\frac{\pi Z_{q}^{2}}{\rho_{\mathrm{Liq}}\,f}\,\frac{\Delta f% \Delta\left(w/2\right)}{f_{f}^{2}}
  9. η â€Čâ€Č = 1 2 π Z q 2 ρ Liq f ( ( Δ ( w / 2 ) ) 2 - Δ f 2 ) f f 2 \eta^{\prime\prime}=\frac{1}{2}\frac{\pi Z_{q}^{2}}{\rho_{\mathrm{Liq}}\,f}\,% \frac{\left(\left(\Delta\left(w/2\right)\right)^{2}-\Delta f^{2}\right)}{f_{f}% ^{2}}
  10. Δ f * f f ≈ i π Z q - ω 2 u 0 m % F i ω u 0 = - 2 f Z q m F \frac{\Delta f^{*}}{f_{f}}\approx\frac{i}{\pi Z_{q}}\frac{-\omega^{2}u_{0}m_{% \%\mathrm{F}}}{i\omega u_{0}}=-\frac{2\,f}{Z_{q}}m_{\mathrm{F}}
  11. tan ( π Δ f f f ) = - Z F Z q tan ( k F d F ) \tan\left(\frac{\pi\Delta f}{f_{f}}\right)=\frac{-Z_{\mathrm{F}}}{Z_{q}}\tan% \left(k_{\mathrm{F}}d_{\mathrm{F}}\right)
  12. Δ f = - f f π ( arctan Z F Z q tan ( 2 π f Z F m F ) ) \Delta f=-\frac{f_{f}}{\pi}\left(\arctan\frac{Z_{\mathrm{F}}}{Z_{q}}\tan\left(% \frac{2\pi f}{Z_{\mathrm{F}}}m_{\mathrm{F}}\right)\right)
  13. Δ f * f f = - 1 π Z q Z F tan ( k F d F ) \frac{\Delta f^{*}}{f_{f}}=\frac{-1}{\pi Z_{q}}Z_{\mathrm{F}}\tan\left(k_{% \mathrm{F}}d_{\mathrm{F}}\right)
  14. Δ f * f f = - Z F π Z q Z F tan ( k F d F ) - i Z Liq Z F + i Z Liq tan ( k F d F ) \frac{\Delta f^{*}}{f_{f}}=\frac{-Z_{\mathrm{F}}}{\pi Z_{q}}\frac{Z_{\mathrm{F% }}\tan\left(k_{\mathrm{F}}d_{\mathrm{F}}\right)-iZ_{\mathrm{Liq}}}{Z_{\mathrm{% F}}+iZ_{\mathrm{Liq}}\tan\left(k_{\mathrm{F}}d_{\mathrm{F}}\right)}
  15. Δ f * f f = - m F π Z q ( 1 - Z Liq 2 Z F 2 ) = - m F π Z q ( 1 - J F Z Liq 2 ρ F ) \frac{\Delta f^{*}}{f_{f}}=\frac{-m_{\mathrm{F}}}{\pi Z_{q}}\left(1-\frac{Z_{% \mathrm{Liq}}^{2}}{Z_{\mathrm{F}}^{2}}\right)=\frac{-m_{\mathrm{F}}}{\pi Z_{q}% }\left(1-J_{\mathrm{F}}\frac{Z_{\mathrm{Liq}}^{2}}{\rho_{\mathrm{F}}}\right)
  16. Δ ( ω / 2 ) - Δ f ≈ η ω J F â€Č \frac{\Delta\left(\omega/2\right)}{-\Delta f}\approx\eta\omega J_{F}^{\,\prime}
  17. Δ ( ω / 2 ) = 8 3 ρ F Z q f f 4 m F 3 n 3 π 2 J â€Čâ€Č \Delta\left(\omega/2\right)=\frac{8}{3\rho_{\mathrm{F}}Z_{q}}f_{f}^{\,4}m_{% \mathrm{F}}^{3}n^{3}\pi^{2}J^{\prime\prime}
  18. Δ f * f f = N S π Z q Îș S * ω \frac{\Delta f^{*}}{f_{f}}=\frac{N_{S}}{\pi Z_{q}}\frac{\kappa_{S}^{*}}{\omega}

Quasi-projective_variety.html

  1. U U
  2. U ÂŻ \bar{U}
  3. X = 𝔾 1 - 0 X=\mathbb{A}^{1}-0
  4. x y - 1 xy-1

Quasisimple_group.html

  1. n ≄ 5. n\geq 5.

Quaternary_ammonium_cation.html

  1. m q / z q m^{q}/z^{q}
  2. ( m + z ) / z (m+z)/z
  3. m q / 2 m^{q}/2
  4. m q / 1 m^{q}/1

Qutrit.html

  1. | 0 ⟩ |0\rangle
  2. | 1 ⟩ |1\rangle
  3. | 2 ⟩ |2\rangle
  4. | ψ ⟩ = α | 0 ⟩ + ÎČ | 1 ⟩ + Îł | 2 ⟩ |\psi\rangle=\alpha|0\rangle+\beta|1\rangle+\gamma|2\rangle
  5. | α | 2 + | ÎČ | 2 + | Îł | 2 = 1 |\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}=1\,
  6. H 2 H_{2}
  7. H 3 H_{3}

R::K_selection_theory.html

  1. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)

Rademacher_distribution.html

  1. f ( k ) = { 1 / 2 if k = - 1 , 1 / 2 if k = + 1 , 0 otherwise. f(k)=\left\{\begin{matrix}1/2&\mbox{if }k=-1,\\ 1/2&\mbox{if }k=+1,\\ 0&\mbox{otherwise.}\end{matrix}\right.
  2. f ( k ) = 1 2 ( ÎŽ ( k - 1 ) + ÎŽ ( k + 1 ) ) . f(k)=\frac{1}{2}\left(\delta\left(k-1\right)+\delta\left(k+1\right)\right).
  3. Pr ( | ∑ i = 1 n X i n | ≀ 1 ) ≄ 0.5. \Pr\Bigl(\Bigl|\frac{\sum_{i=1}^{n}X_{i}}{\sqrt{n}}\Bigr|\leq 1\Bigr)\geq 0.5.
  4. Pr ( ∑ i X i a i > t || a || 2 ) ≀ e - t 2 2 \Pr(\sum_{i}X_{i}a_{i}>t||a||_{2})\leq e^{-\frac{t^{2}}{2}}
  5. P r ( | | Y | | > s t ) ≀ [ 1 c P r ( | | Y | | > t ) ] c s 2 Pr(||Y||>st)\leq[\frac{1}{c}Pr(||Y||>t)]^{cs^{2}}
  6. c 1 [ ∑ | a i | 2 ] 1 2 ≀ ( E [ | ∑ a i X i | p ] ) 1 p ≀ c 2 [ ∑ | a i | 2 ] 1 2 c_{1}[\sum{|a_{i}|^{2}}]^{\frac{1}{2}}\leq(E[|\sum{a_{i}X_{i}}|^{p}])^{\frac{1% }{p}}\leq c_{2}[\sum{|a_{i}|^{2}}]^{\frac{1}{2}}
  7. c 2 ≀ c 1 p c_{2}\leq c_{1}\sqrt{p}
  8. X + 1 2 \frac{X+1}{2}

Radiant_intensity.html

  1. I e , Ω = ∂ Ί e ∂ Ω , I_{\mathrm{e},\Omega}=\frac{\partial\Phi_{\mathrm{e}}}{\partial\Omega},
  2. I e , Ω , Μ = ∂ I e , Ω ∂ Μ , I_{\mathrm{e},\Omega,\nu}=\frac{\partial I_{\mathrm{e},\Omega}}{\partial\nu},
  3. I e , Ω , λ = ∂ I e , Ω ∂ λ , I_{\mathrm{e},\Omega,\lambda}=\frac{\partial I_{\mathrm{e},\Omega}}{\partial% \lambda},
  4. I e , Ω = E e ( r ) r 2 , I_{\mathrm{e},\Omega}=E_{\mathrm{e}}(r)\,r^{2},

Radiation_length.html

  1. 1 / e {1}/{e}
  2. 7 / 9 {7}/{9}
  3. X 0 = 716.4 ⋅ A Z ( Z + 1 ) ln 287 Z g ⋅ cm - 2 = 1432.8 ⋅ A Z ( Z + 1 ) ( 11.319 - ln Z ) g ⋅ cm - 2 X_{0}=\frac{716.4\cdot A}{Z(Z+1)\ln{\frac{287}{\sqrt{Z}}}}\;\mathrm{g}\cdot% \mathrm{cm}^{-2}=\frac{1432.8\cdot A}{Z(Z+1)(11.319-\ln{Z})}\;\mathrm{g}\cdot% \mathrm{cm}^{-2}
  4. Z Z
  5. A A
  6. 1 X 0 = 4 α N A Z ( Z + 1 ) r e 2 log ( 183 Z - 1 / 3 ) A \frac{1}{X_{0}}=\frac{4\alpha N_{A}Z(Z+1)r_{e}^{2}\log(183Z^{-1/3})}{A}

Radiation_zone.html

  1. d T ( r ) d r = - 3 Îș ( r ) ρ ( r ) L ( r ) ( 4 π r 2 ) ( 16 σ ) T 3 ( r ) \frac{\,\text{d}T(r)}{\,\text{d}r}\ =\ -\frac{3\kappa(r)\rho(r)L(r)}{(4\pi r^{% 2})(16\sigma)T^{3}(r)}

Radical_ion.html

  1. M ∙ + M^{\bullet+}
  2. M ∙ - M^{\bullet-}
  3. M + ∙ M^{+\bullet}
  4. M - ∙ M^{-\bullet}

Radical_of_an_integer.html

  1. rad ( n ) = ∏ p ∣ n p prime p \displaystyle\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\,\text{ prime}}p
  2. 504 = 2 3 ⋅ 3 2 ⋅ 7 504=2^{3}\cdot 3^{2}\cdot 7
  3. rad ( 504 ) = 2 ⋅ 3 ⋅ 7 = 42 \mathrm{rad}(504)=2\cdot 3\cdot 7=42
  4. rad \mathrm{rad}
  5. rad t \mathrm{rad}_{t}
  6. rad t ( p e ) = p min ( e , t - 1 ) \mathrm{rad}_{t}(p^{e})=p^{\mathrm{min}(e,t-1)}
  7. c < K Δ rad ( a b c ) 1 + Δ c<K_{\varepsilon}\,\operatorname{rad}(abc)^{1+\varepsilon}
  8. â„€ / n â„€ \mathbb{Z}/n\mathbb{Z}

Radical_polymerization.html

  1. 1 x n = ( 1 x n ) o + k t r [ s o l v e n t ] k p [ m o n o m e r ] \frac{1}{x_{n}}=\left(\frac{1}{x_{n}}\right)_{o}+\frac{k_{tr}[solvent]}{k_{p}[% monomer]}
  2. v i = d [ M ⋅ ] / d t = 2 k d f [ I ] v_{i}={\operatorname{d}[M\cdot]/\operatorname{d}t}=2k_{d}f[I]
  3. v p = k p [ M ] [ M ⋅ ] v_{p}=k_{p}[M][M\cdot]
  4. v t = - d [ M ⋅ ] / d t = 2 k t [ M ⋅ ] 2 v_{t}={-\operatorname{d}[M\cdot]/\operatorname{d}t}=2k_{t}[M\cdot]^{2}
  5. [ M ⋅ ] = ( k d [ I ] f k t ) 1 / 2 [M\cdot]=\left(\frac{k_{d}[I]f}{k_{t}}\right)^{1/2}
  6. r a t e = k p ( f k d k t ) 1 / 2 [ I ] 1 / 2 [ M ] rate={k_{p}}\left(\frac{fk_{d}}{k_{t}}\right)^{1/2}[I]^{1/2}[M]
  7. v = R p R d = k p [ M ] [ M ⋅ ] 2 f k d [ I ] = k p [ M ] 2 ( f k d k t [ I ] ) 1 / 2 \ v=\frac{R_{p}}{R_{d}}=\frac{k_{p}[M][M\cdot]}{2fk_{d}[I]}=\frac{k_{p}[M]}{2(% fk_{d}k_{t}[I])^{1/2}}
  8. P n = v P_{n}=v
  9. P n = 2 v P_{n}=2v
  10. P n = 2 1 + ÎŽ v P_{n}=\frac{2}{1+\delta}v
  11. 1 P n = 2 k t , d + k t , c k p 2 [ M ] 2 R p + C M + C S [ S ] [ M ] + C I [ I ] [ M ] + C P [ P ] [ M ] + C T [ T ] [ M ] \frac{1}{P_{n}}=\frac{2k_{t,d}+k_{t,c}}{{k_{p}}^{2}[M]^{2}}R_{p}+C_{M}+C_{S}% \frac{[S]}{[M]}+C_{I}\frac{[I]}{[M]}+C_{P}\frac{[P]}{[M]}+C_{T}\frac{[T]}{[M]}
  12. C M = k t r M k p C_{M}=\frac{k^{M}_{tr}}{k_{p}}
  13. C S = k t r S k p C_{S}=\frac{k^{S}_{tr}}{k_{p}}
  14. C I = k t r I k p C_{I}=\frac{k^{I}_{tr}}{k_{p}}
  15. C P = k t r P k p C_{P}=\frac{k^{P}_{tr}}{k_{p}}
  16. C T = k t r T k p C_{T}=\frac{k^{T}_{tr}}{k_{p}}
  17. k 12 = Q 1 Q 2 e x p ( - e 1 e 2 ) k_{12}=Q_{1}Q_{2}exp(-e_{1}e_{2})
  18. r 1 = Q 1 / Q 2 e x p ( - e 1 ( e 1 - e 2 ) ) r_{1}=Q_{1}/Q_{2}exp(-e_{1}(e_{1}-e_{2}))

Radius.html

  1. d ≐ 2 r ⇒ r = d 2 . d\doteq 2r\quad\Rightarrow\quad r=\frac{d}{2}.
  2. r = C 2 π . r=\frac{C}{2\pi}.
  3. r = C τ . r=\frac{C}{\tau}.
  4. τ {\tau}
  5. 2 π 2{\pi}
  6. r = A π r=\sqrt{\frac{A}{\pi}}
  7. r = | P 1 - P 3 | 2 sin Ξ , r=\frac{|P_{1}-P_{3}|}{2\sin\theta},
  8. ∠ P 1 P 2 P 3 . \angle P_{1}P_{2}P_{3}.
  9. ( x 1 , y 1 ) (x_{1},y_{1})
  10. ( x 2 , y 2 ) (x_{2},y_{2})
  11. ( x 3 , y 3 ) (x_{3},y_{3})
  12. r = ( ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 ) ( ( x 2 - x 3 ) 2 + ( y 2 - y 3 ) 2 ) ( ( x 3 - x 1 ) 2 + ( y 3 - y 1 ) 2 ) 2 | x 1 y 2 + x 2 y 3 + x 3 y 1 - x 1 y 3 - x 2 y 1 - x 3 y 2 | . r={\frac{\sqrt{\left(\left({\it x_{2}}-{\it x_{1}}\right)^{2}+\left({\it y_{2}% }-{\it y_{1}}\right)^{2}\right)\left(\left({\it x_{2}}-{\it x_{3}}\right)^{2}+% \left({\it y_{2}}-{\it y_{3}}\right)^{2}\right)\left(\left({\it x_{3}}-{\it x_% {1}}\right)^{2}+\left({\it y_{3}}-{\it y_{1}}\right)^{2}\right)}}{2\left|{\it x% _{1}}\,{\it y_{2}}+{\it x_{2}}\,{\it y_{3}}+{\it x_{3}}\,{\it y_{1}}-{\it x_{1% }}\,{\it y_{3}}-{\it x_{2}}\,{\it y_{1}}-{\it x_{3}}\,{\it y_{2}}\right|}}.
  13. r = R n s r=R_{n}\,s
  14. R n = 1 / ( 2 sin π n ) : R_{n}=1/\left(2\sin\frac{\pi}{n}\right):
  15. n R n n R n 2 0.50000000 10 1.6180340 - 3 0.5773503 - 11 1.7747328 - 4 0.7071068 - 12 1.9318517 - 5 0.8506508 + 13 2.0892907 + 6 1.00000000 14 2.2469796 + 7 1.1523824 + 15 2.4048672 - 8 1.3065630 - 16 2.5629154 + 9 1.4619022 + 17 2.7210956 - \begin{array}[]{r|ccr|c}n&R_{n}&&n&R_{n}\\ \hline 2&0.50000000&&10&1.6180340-\\ 3&0.5773503-&&11&1.7747328-\\ 4&0.7071068-&&12&1.9318517-\\ 5&0.8506508+&&13&2.0892907+\\ 6&1.00000000&&14&2.2469796+\\ 7&1.1523824+&&15&2.4048672-\\ 8&1.3065630-&&16&2.5629154+\\ 9&1.4619022+&&17&2.7210956-\end{array}
  16. r = s 2 d . r=\frac{s}{2}\sqrt{d}.

Rado's_theorem_(Ramsey_theory).html

  1. A đ± = 𝟎 A\mathbf{x}=\mathbf{0}
  2. A A
  3. r r
  4. r r
  5. A đ± = 𝟎 A\mathbf{x}=\mathbf{0}
  6. s i = Σ j ∈ C i c j s_{i}=\Sigma_{j\in C_{i}}c_{j}
  7. x T = ∑ i ∈ T x { i } , x_{T}=\sum_{i\in T}x_{\{i\}},

Radon's_theorem.html

  1. { x 1 , x 2 , 
 , x d + 2 } ⊂ 𝐑 d \{x_{1},x_{2},\dots,x_{d+2}\}\subset\mathbf{R}^{d}
  2. ∑ i = 1 d + 2 a i x i = 0 , ∑ i = 1 d + 2 a i = 0 , \sum_{i=1}^{d+2}a_{i}x_{i}=0,\quad\sum_{i=1}^{d+2}a_{i}=0,
  3. p = ∑ i ∈ I a i A x i = ∑ j ∈ J - a j A x j , p=\sum_{i\in I}\frac{a_{i}}{A}x_{i}=\sum_{j\in J}\frac{-a_{j}}{A}x_{j},
  4. A = ∑ i ∈ I a i = - ∑ j ∈ J a j . A=\sum_{i\in I}a_{i}=-\sum_{j\in J}a_{j}.
  5. ( d + 1 ) ( r - 1 ) + 1 (d+1)(r-1)+1

Rafael_Bombelli.html

  1. n \sqrt{n}
  2. n = ( a ± r ) 2 = a 2 ± 2 a r + r 2 n=(a\pm r)^{2}=a^{2}\pm 2ar+r^{2}
  3. 0 < r < 1 0<r<1
  4. r = | n - a 2 | 2 a ± r r=\frac{|n-a^{2}|}{2a\pm r}
  5. r r
  6. a ± | n - a 2 | 2 a ± | n - a 2 | 2 a ± | n - a 2 | 2 a ± ⋯ a\pm\frac{|n-a^{2}|}{2a\pm\frac{|n-a^{2}|}{2a\pm\frac{|n-a^{2}|}{2a\pm\cdots}}}
  7. r r
  8. a a
  9. n n
  10. 13 \sqrt{13}
  11. 3 2 3 , 3 3 5 , 3 20 33 , 3 66 109 , 3 109 180 , 3 720 1189 , ⋯ 3\frac{2}{3},\ 3\frac{3}{5},\ 3\frac{20}{33},\ 3\frac{66}{109},\ 3\frac{109}{1% 80},\ 3\frac{720}{1189},\ \cdots
  12. 265 153 < 3 < 1351 780 \frac{265}{153}<\sqrt{3}<\frac{1351}{780}
  13. π \pi
  14. r r

Ragtime_progression.html

  1. 3 6 {}^{6}_{3}

Ram_pressure.html

  1. P = ρ v 2 P=\rho v^{2}
  2. P P
  3. ρ \rho
  4. v v

Ramanujan_tau_function.html

  1. τ : ℕ → â„€ \tau:\mathbb{N}\to\mathbb{Z}
  2. ∑ n ≄ 1 τ ( n ) q n = q ∏ n ≄ 1 ( 1 - q n ) 24 = η ( z ) 24 = Δ ( z ) , \sum_{n\geq 1}\tau(n)q^{n}=q\prod_{n\geq 1}(1-q^{n})^{24}=\eta(z)^{24}=\Delta(% z),
  3. q = exp ( 2 π i z ) q=\exp(2\pi iz)
  4. ℑ z > 0 \Im z>0
  5. η \eta
  6. Δ ( z ) \Delta(z)
  7. | τ ( n ) | |\tau(n)|
  8. n n
  9. τ ( n ) \tau(n)
  10. τ ( n ) \tau(n)
  11. τ ( m n ) = τ ( m ) τ ( n ) \tau(mn)=\tau(m)\tau(n)
  12. gcd ( m , n ) = 1 \gcd(m,n)=1
  13. τ ( n ) \tau(n)
  14. τ ( p r + 1 ) = τ ( p ) τ ( p r ) - p 11 τ ( p r - 1 ) \tau(p^{r+1})=\tau(p)\tau(p^{r})-p^{11}\tau(p^{r-1})
  15. | τ ( p ) | ≀ 2 p 11 / 2 |\tau(p)|\leq 2p^{11/2}
  16. τ ( n ) ≡ σ 11 ( n ) mod 2 11 for n ≡ 1 mod 8 \tau(n)\equiv\sigma_{11}(n)\ \bmod\ 2^{11}\,\text{ for }n\equiv 1\ \bmod\ 8
  17. τ ( n ) ≡ 1217 σ 11 ( n ) mod 2 13 for n ≡ 3 mod 8 \tau(n)\equiv 1217\sigma_{11}(n)\ \bmod\ 2^{13}\,\text{ for }n\equiv 3\ \bmod\ 8
  18. τ ( n ) ≡ 1537 σ 11 ( n ) mod 2 12 for n ≡ 5 mod 8 \tau(n)\equiv 1537\sigma_{11}(n)\ \bmod\ 2^{12}\,\text{ for }n\equiv 5\ \bmod\ 8
  19. τ ( n ) ≡ 705 σ 11 ( n ) mod 2 14 for n ≡ 7 mod 8 \tau(n)\equiv 705\sigma_{11}(n)\ \bmod\ 2^{14}\,\text{ for }n\equiv 7\ \bmod\ 8
  20. τ ( n ) ≡ n - 610 σ 1231 ( n ) mod 3 6 for n ≡ 1 mod 3 \tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{6}\,\text{ for }n\equiv 1\ % \bmod\ 3
  21. τ ( n ) ≡ n - 610 σ 1231 ( n ) mod 3 7 for n ≡ 2 mod 3 \tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{7}\,\text{ for }n\equiv 2\ % \bmod\ 3
  22. τ ( n ) ≡ n - 30 σ 71 ( n ) mod 5 3 for n ≱ 0 mod 5 \tau(n)\equiv n^{-30}\sigma_{71}(n)\ \bmod\ 5^{3}\,\text{ for }n\not\equiv 0\ % \bmod\ 5
  23. τ ( n ) ≡ n σ 9 ( n ) mod 7 for n ≡ 0 , 1 , 2 , 4 mod 7 \tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7\,\text{ for }n\equiv 0,1,2,4\ \bmod\ 7
  24. τ ( n ) ≡ n σ 9 ( n ) mod 7 2 for n ≡ 3 , 5 , 6 mod 7 \tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7^{2}\,\text{ for }n\equiv 3,5,6\ \bmod\ 7
  25. τ ( n ) ≡ σ 11 ( n ) mod 691. \tau(n)\equiv\sigma_{11}(n)\ \bmod\ 691.
  26. τ ( p ) ≡ 0 mod 23 if ( p 23 ) = - 1 \tau(p)\equiv 0\ \bmod\ 23\,\text{ if }\left(\frac{p}{23}\right)=-1
  27. τ ( p ) ≡ σ 11 ( p ) mod 23 2 if p is of the form a 2 + 23 b 2 \tau(p)\equiv\sigma_{11}(p)\ \bmod\ 23^{2}\,\text{ if }p\,\text{ is of the % form }a^{2}+23b^{2}
  28. τ ( p ) ≡ - 1 mod 23 otherwise . \tau(p)\equiv-1\ \bmod\ 23\,\text{ otherwise}.
  29. f f
  30. k k
  31. a ( n ) a(n)
  32. f f
  33. p p
  34. a ( p ) ≠ 0 mod p a(p)\neq 0\bmod p
  35. a ( n ) mod p a(n)\bmod p
  36. n n
  37. p p
  38. a ( p ) mod p a(p)\bmod p
  39. p p
  40. a ( p ) = 0 a(p)=0
  41. 0 mod p 0\bmod p
  42. f f
  43. > 2 >2
  44. a ( p ) ≠ 0 a(p)\neq 0
  45. p p
  46. p p
  47. p p
  48. a ( p ) = 0 a(p)=0
  49. p p
  50. p p
  51. a ( p ) = 0 mod p a(p)=0\bmod p
  52. p p
  53. τ ( p ) \tau(p)
  54. 12 12
  55. p p
  56. τ ( p ) = 0 mod p \tau(p)=0\bmod p
  57. p = 7758337633 p=7758337633
  58. τ ( p ) ≡ 0 mod p \tau(p)\equiv 0\bmod p
  59. p = 2 , 3 , 5 , 7 , 2411 , p=2,3,5,7,2411,
  60. 7758337633 7758337633
  61. 10 10 10^{10}
  62. τ ( n ) ≠ 0 \tau(n)\neq 0
  63. n n
  64. n < 214928639999 n<214928639999
  65. n n
  66. 10 15 10^{15}

Ramsey_problem.html

  1. ( p 1 , 
 , p N ) \left(p_{1},\ldots,p_{N}\right)
  2. C ( z 1 , z 2 , 
 , z N ) = C ( 𝐳 ) C(z_{1},z_{2},\ldots,z_{N})=C(\mathbf{z})
  3. z n z_{n}
  4. p n p_{n}
  5. z n ( p n ) , z_{n}\left(p_{n}\right),
  6. p n ( z ) . p_{n}(z).
  7. R ( đ© , 𝐳 ) = ∑ n p n z n ( p n ) . R\left(\mathbf{p,z}\right)=\sum_{n}p_{n}z_{n}(p_{n}).
  8. W ( đ© , 𝐳 ) = ∑ n ( ∫ 0 z n ( p n ) p n ( z ) d z ) - C ( 𝐳 ) . W\left(\mathbf{p,z}\right)=\sum_{n}\left(\int\limits_{0}^{z_{n}(p_{n})}p_{n}(z% )dz\right)-C\left(\mathbf{z}\right).
  9. W ( đ© , 𝐳 ) W\left(\mathbf{p,z}\right)
  10. Π = R - C \Pi=R-C
  11. Π * \Pi^{*}
  12. R ( đ© , 𝐳 ) - C ( 𝐳 ) = Π * R(\mathbf{p,z})-C(\mathbf{z})=\Pi^{*}
  13. 𝐳 \mathbf{z}
  14. p n - C n ( 𝐳 ) = - λ ( ∂ R ∂ z n - C n ( 𝐳 ) ) p_{n}-C_{n}\left(\mathbf{z}\right)=-\lambda\left(\frac{\partial R}{\partial z_% {n}}-C_{n}\left(\mathbf{z}\right)\right)
  15. = - λ ( p n ( 1 + z n p n ∂ p n ∂ z n ) - C n ( 𝐳 ) ) =-\lambda\left(p_{n}\left(1+\frac{z_{n}}{p_{n}}\frac{\partial p_{n}}{\partial z% _{n}}\right)-C_{n}\left(\mathbf{z}\right)\right)
  16. λ \lambda
  17. p n p_{n}
  18. p n - C n ( 𝐳 ) p n = k Δ n \frac{p_{n}-C_{n}\left(\mathbf{z}\right)}{p_{n}}=\frac{k}{\varepsilon_{n}}
  19. k = λ 1 + λ k=\frac{\lambda}{1+\lambda}
  20. Δ n = - ∂ z n ∂ p n p n z n \varepsilon_{n}=-\frac{\partial z_{n}}{\partial p_{n}}\frac{p_{n}}{z_{n}}
  21. n . n.
  22. n n
  23. k = 1 k=1
  24. λ = 1 \lambda=1
  25. Π * = R - C \Pi^{*}=R-C

Random_forest.html

  1. X X
  2. Y Y
  3. b b
  4. B B
  5. n n
  6. X X
  7. Y Y
  8. x x
  9. x x
  10. f ^ = 1 B ∑ b = 1 B f ^ b ( x â€Č ) \hat{f}=\frac{1}{B}\sum_{b=1}^{B}\hat{f}_{b}(x^{\prime})
  11. B B
  12. B B
  13. x ᔹ xᔹ
  14. x ᔹ xᔹ
  15. B B
  16. p p
  17. p \sqrt{p}
  18. 𝒟 n = { ( X i , Y i ) } i = 1 n \mathcal{D}_{n}=\{(X_{i},Y_{i})\}_{i=1}^{n}
  19. j j
  20. j j
  21. j j
  22. k k
  23. k k
  24. { ( x i , y i ) } i = 1 n \{(x_{i},y_{i})\}_{i=1}^{n}
  25. y ^ \hat{y}
  26. x x
  27. W W
  28. y ^ = ∑ i = 1 n W ( x i , x â€Č ) y i . \hat{y}=\sum_{i=1}^{n}W(x_{i},x^{\prime})\,y_{i}.
  29. W ( x i , x â€Č ) W(x_{i},x^{\prime})
  30. i i
  31. x x
  32. x x
  33. k k
  34. W ( x i , x â€Č ) = 1 k W(x_{i},x^{\prime})=\frac{1}{k}
  35. k k
  36. x x
  37. W ( x i , x â€Č ) W(x_{i},x^{\prime})
  38. x x
  39. m m
  40. W j W_{j}
  41. y ^ = 1 m ∑ j = 1 m ∑ i = 1 n W j ( x i , x â€Č ) y i = ∑ i = 1 n ( 1 m ∑ j = 1 m W j ( x i , x â€Č ) ) y i . \hat{y}=\frac{1}{m}\sum_{j=1}^{m}\sum_{i=1}^{n}W_{j}(x_{i},x^{\prime})\,y_{i}=% \sum_{i=1}^{n}\left(\frac{1}{m}\sum_{j=1}^{m}W_{j}(x_{i},x^{\prime})\right)\,y% _{i}.
  42. x x
  43. x i x_{i}
  44. x x
  45. x x

Random_matrix.html

  1. 1 Z GUE ( n ) e - n 2 tr H 2 \frac{1}{Z_{\,\text{GUE}(n)}}e^{-\frac{n}{2}\mathrm{tr}H^{2}}
  2. π \pi
  3. 1 Z GOE ( n ) e - n 4 tr H 2 \frac{1}{Z_{\,\text{GOE}(n)}}e^{-\frac{n}{4}\mathrm{tr}H^{2}}
  4. 1 Z GSE ( n ) e - n tr H 2 \frac{1}{Z_{\,\text{GSE}(n)}}e^{-n\mathrm{tr}H^{2}}\,
  5. 1 Z ÎČ , n ∏ k = 1 n e - ÎČ n 4 λ k 2 ∏ i < j | λ j - λ i | ÎČ , ( 1 ) \frac{1}{Z_{\beta,n}}\prod_{k=1}^{n}e^{-\frac{\beta n}{4}\lambda_{k}^{2}}\prod% _{i<j}\left|\lambda_{j}-\lambda_{i}\right|^{\beta}~{},\quad(1)
  6. ÎČ \beta
  7. λ j = λ i \lambda_{j}=\lambda_{i}
  8. λ 1 < 
 < λ n < λ n + 1 < 
 \lambda_{1}<\ldots<\lambda_{n}<\lambda_{n+1}<\ldots
  9. s = ( λ n + 1 - λ n ) / ⟹ s ⟩ s=(\lambda_{n+1}-\lambda_{n})/\langle s\rangle
  10. ⟹ s ⟩ = ⟹ λ n + 1 - λ n ⟩ \langle s\rangle=\langle\lambda_{n+1}-\lambda_{n}\rangle
  11. p 1 ( s ) = π 2 s e - π 4 s 2 p_{1}(s)=\frac{\pi}{2}s\,\mathrm{e}^{-\frac{\pi}{4}s^{2}}
  12. ÎČ = 1 \beta=1
  13. p 2 ( s ) = 32 π 2 s 2 e - 4 π s 2 p_{2}(s)=\frac{32}{\pi^{2}}s^{2}\mathrm{e}^{-\frac{4}{\pi}s^{2}}
  14. ÎČ = 2 \beta=2
  15. p 4 ( s ) = 2 18 3 6 π 3 s 4 e - 64 9 π s 2 p_{4}(s)=\frac{2^{18}}{3^{6}\pi^{3}}s^{4}\mathrm{e}^{-\frac{64}{9\pi}s^{2}}
  16. ÎČ = 4 \beta=4
  17. p ÎČ ( s ) p_{\beta}(s)
  18. ∫ 0 ∞ d s p ÎČ ( s ) = 1 \int_{0}^{\infty}ds\,p_{\beta}(s)=1
  19. ∫ 0 ∞ d s s p ÎČ ( s ) = 1 , \int_{0}^{\infty}ds\,s\,p_{\beta}(s)=1,
  20. ÎČ = 1 , 2 , 4 \beta=1,2,4
  21. H n = ( H n ( i , j ) ) i , j = 1 n \textstyle H_{n}=(H_{n}(i,j))_{i,j=1}^{n}
  22. { H n ( i , j ) , 1 ≀ i ≀ j ≀ n } \left\{H_{n}(i,j)~{},\,1\leq i\leq j\leq n\right\}
  23. { H n ( i , j ) , 1 ≀ i < j ≀ n } \left\{H_{n}(i,j)~{},\,1\leq i<j\leq n\right\}
  24. 1 Z n e - n tr V ( H ) , \textstyle\frac{1}{Z_{n}}e^{-n\mathrm{tr}V(H)}~{},
  25. ÎŒ H ( A ) = 1 n # { eigenvalues of H in A } = N 1 A , H , A ⊂ ℝ . \mu_{H}(A)=\frac{1}{n}\,\#\left\{\,\text{eigenvalues of }H\,\text{ in }A\right% \}=N_{1_{A},H},\quad A\subset\mathbb{R}.
  26. Ό H \mu_{H}
  27. N f , H - ∫ f ( λ ) d N ( λ ) σ f , n ⟶ đ· N ( 0 , 1 ) \frac{N_{f,H}-\int f(\lambda)\,dN(\lambda)}{\sigma_{f,n}}\overset{D}{% \longrightarrow}N(0,1)
  28. λ 0 \lambda_{0}
  29. N ( λ ) N(\lambda)
  30. Ξ ( λ 0 ) = ∑ j ÎŽ ( ⋅ - n ρ ( λ 0 ) ( λ j - λ 0 ) ) , \Xi(\lambda_{0})=\sum_{j}\delta\Big({\cdot}-n\rho(\lambda_{0})(\lambda_{j}-% \lambda_{0})\Big)~{},
  31. λ j \lambda_{j}
  32. Ξ ( λ 0 ) \Xi(\lambda_{0})
  33. λ 0 \lambda_{0}
  34. Ξ ( λ 0 ) \Xi(\lambda_{0})
  35. K ( x , y ) = sin π ( x - y ) π ( x - y ) K(x,y)=\frac{\sin\pi(x-y)}{\pi(x-y)}
  36. Ξ ( λ 0 ) \Xi(\lambda_{0})
  37. n → ∞ n\to\infty
  38. λ 0 \lambda_{0}

Random_minimal_spanning_tree.html

  1. â„€ N 2 . \mathbb{Z}^{2}_{N}.

Random_password_generator.html

  1. 2 n 2^{n}
  2. H = L log 2 N = L log N log 2 H=L\,\log_{2}N=L{\log N\over\log 2}

Random_permutation.html

  1. ( 1 2 3 ⋯ n x 1 x 2 x 3 ⋯ x n ) , \begin{pmatrix}1&2&3&\cdots&n\\ x_{1}&x_{2}&x_{3}&\cdots&x_{n}\\ \end{pmatrix},

Random_walk_hypothesis.html

  1. X t = Ό + X t - 1 + ϔ t X_{t}=\mu+X_{t-1}+\epsilon_{t}\,
  2. X t X_{t}
  3. t t
  4. Ό \mu
  5. Ï” t \epsilon_{t}
  6. đ”Œ [ Ï” t ] = 0 \mathbb{E}[\epsilon_{t}]=0
  7. đ”Œ [ Ï” t Ï” τ ] = 0 \mathbb{E}[\epsilon_{t}\epsilon_{\tau}]=0
  8. τ ≠ t \tau\neq t
  9. ( X t - X t + τ ) (X_{t}-X_{t+\tau})
  10. τ \tau
  11. Ï” t \epsilon_{t}

Range_problem.html

  1. f ( x ) ⋅ ( 1 - 1 1 - g ( | x | ) ) f(x)\cdot\left(1-\frac{1}{1-g(|x|)}\right)
  2. f ( x ) ⋅ ( 1 - 1 1 + g ( | x | ) ) f(x)\cdot\left(1-\frac{1}{1+g(|x|)}\right)

Rarita–Schwinger_equation.html

  1. ( Ï” ÎŒ Îș ρ Μ Îł 5 Îł Îș ∂ ρ - i m σ ÎŒ Μ ) ψ Μ = 0 \left(\epsilon^{\mu\kappa\rho\nu}\gamma_{5}\gamma_{\kappa}\partial_{\rho}-im% \sigma^{\mu\nu}\right)\psi_{\nu}=0
  2. Ï” ÎŒ Îș ρ Μ \epsilon^{\mu\kappa\rho\nu}
  3. Îł 5 \gamma_{5}
  4. Îł Μ \gamma_{\nu}
  5. m m
  6. σ ÎŒ Μ ≡ i 2 [ Îł ÎŒ , Îł Μ ] \sigma^{\mu\nu}\equiv\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]
  7. ψ Μ \psi_{\nu}
  8. ( 1 2 , 1 2 ) ⊗ ( ( 1 2 , 0 ) ⊕ ( 0 , 1 2 ) ) (\frac{1}{2},\frac{1}{2})⊗((\frac{1}{2},0)⊕(0,\frac{1}{2}))
  9. ( 1 , 1 2 ) ⊕ ( 1 2 , 1 ) (1,\frac{1}{2})⊕(\frac{1}{2},1)
  10. ℒ = - 1 2 ψ ÂŻ ÎŒ ( Ï” ÎŒ Îș ρ Μ Îł 5 Îł Îș ∂ ρ - i m σ ÎŒ Μ ) ψ Μ \mathcal{L}=-\tfrac{1}{2}\;\bar{\psi}_{\mu}\left(\epsilon^{\mu\kappa\rho\nu}% \gamma_{5}\gamma_{\kappa}\partial_{\rho}-im\sigma^{\mu\nu}\right)\psi_{\nu}
  11. ψ ÎŒ → ψ ÎŒ + ∂ ÎŒ Ï” \psi_{\mu}\rightarrow\psi_{\mu}+\partial_{\mu}\epsilon
  12. Ï” ≡ Ï” α \epsilon\equiv\epsilon_{\alpha}
  13. ℒ R S = ψ ÂŻ ÎŒ Îł ÎŒ Μ ρ ∂ Μ ψ ρ , \mathcal{L}_{RS}=\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho},
  14. ψ ÎŒ \psi_{\mu}
  15. Îł ÎŒ Μ ρ ≡ 1 3 ! Îł [ ÎŒ Îł Μ Îł ρ ] . \gamma^{\mu\nu\rho}\equiv\frac{1}{3!}\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}.
  16. ψ ÎŒ \psi_{\mu}
  17. ÎŽ ℒ R S = ÎŽ ψ ÂŻ ÎŒ Îł ÎŒ Μ ρ ∂ Μ ψ ρ + ψ ÂŻ ÎŒ Îł ÎŒ Μ ρ ∂ Μ ÎŽ ψ ρ = ÎŽ ψ ÂŻ ÎŒ Îł ÎŒ Μ ρ ∂ Μ ψ ρ - ∂ Μ ψ ÂŻ ÎŒ Îł ÎŒ Μ ρ ÎŽ ψ ρ + boundary terms \delta\mathcal{L}_{RS}=\delta\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}% \psi_{\rho}+\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\delta\psi_{\rho}% =\delta\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho}-\partial_{% \nu}\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\delta\psi_{\rho}+\,\text{ boundary terms}
  18. ÎŽ ℒ R S = 2 ÎŽ ψ ÂŻ ÎŒ Îł ÎŒ Μ ρ ∂ Μ ψ ρ , \delta\mathcal{L}_{RS}=2\delta\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu% }\psi_{\rho},
  19. ή ℒ R S = 0 \delta\mathcal{L}_{RS}=0
  20. Îł ÎŒ Μ ρ ∂ Μ ψ ρ = 0. \gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho}=0.
  21. ∂ ÎŒ → D ÎŒ = ∂ ÎŒ - i e A ÎŒ \partial_{\mu}\rightarrow D_{\mu}=\partial_{\mu}-ieA_{\mu}

Rate-determining_step.html

  1. d [ N O 3 ] d t = r 1 - r 2 - r - 1 ≈ 0 \frac{{d}{[NO_{3}]}}{dt}=r_{1}-r_{2}-r_{-1}\approx 0
  2. K 1 = [ N O ] [ N O 3 ] [ N O 2 ] 2 K_{1}=\frac{{[NO]}{[NO_{3}]}}{{[NO_{2}]}^{2}}
  3. [ N O 3 ] = K 1 < m t p l > [ N O 2 ] 2 [ N O ] [NO_{3}]=K_{1}\frac{<}{m}tpl>{{[NO_{2}]^{2}}}{{[NO]}}
  4. r = r 2 = k 2 [ N O 3 ] [ C O ] = k 2 K 1 < m t p l > [ N O 2 ] 2 [ C O ] [ N O ] r=r_{2}=k_{2}[NO_{3}][CO]=k_{2}K_{1}\frac{<}{m}tpl>{{[NO_{2}]^{2}[CO]}}{{[NO]}}
  5. v = k < m t p l > [ C l 2 ] [ H 2 C 2 O 4 ] [ H + ] 2 [ C l - ] v=k\frac{<}{m}tpl>{{[Cl_{2}][H_{2}C_{2}O_{4}]}}{{[H^{+}]^{2}[Cl^{-}]}}
  6. ⇌ \rightleftharpoons
  7. ⇌ \rightleftharpoons

Rate_equation.html

  1. r = k [ A ] x [ B ] y r\;=\;k[\mathrm{A}]^{x}[\mathrm{B}]^{y}
  2. r = k [ A ] [ B ] r\;=\;k[\mathrm{A}][\mathrm{B}]
  3. r = k \ r=k
  4. r = - d [ A ] d t = k r=-\frac{d[A]}{dt}=k
  5. [ A ] t = - k t + [ A ] 0 \ [A]_{t}=-kt+[A]_{0}
  6. [ A ] t \ [A]_{t}
  7. [ A ] 0 \ [A]_{0}
  8. [ A ] t \ [A]_{t}
  9. - k -k
  10. t 1 2 = [ A ] 0 2 k \ t_{\frac{1}{2}}=\frac{[A]_{0}}{2k}
  11. 2 N H 3 ( g ) → 3 H 2 ( g ) + N 2 ( g ) 2NH_{3}(g)\rightarrow\;3H_{2}(g)+N_{2}(g)
  12. - d [ A ] d t ≡ r = k [ A ] \frac{-d[A]}{dt}\equiv r=k[A]
  13. ln [ A ] = - k t + ln [ A ] 0 \ \ln{[A]}=-kt+\ln{[A]_{0}}
  14. ln [ A ] \ln{[A]}
  15. - k -k
  16. t 1 2 = ln ( 2 ) k \ t_{\frac{1}{2}}=\frac{\ln{(2)}}{k}
  17. H O 2 ( l ) 2 → H O 2 ( l ) + 1 2 O ( g ) 2 \mbox{H}~{}_{2}\mbox{O}~{}_{2}(l)\rightarrow\;\mbox{H}~{}_{2}\mbox{O}~{}(l)+% \frac{1}{2}\mbox{O}~{}_{2}(g)
  18. SO Cl 2 ( l ) 2 → SO ( g ) 2 + Cl ( g ) 2 \mbox{SO}~{}_{2}\mbox{Cl}~{}_{2}(l)\rightarrow\;\mbox{SO}~{}_{2}(g)+\mbox{Cl}~% {}_{2}(g)
  19. 2 N O 2 ( g ) 5 → 4 NO ( g ) 2 + O ( g ) 2 2\mbox{N}~{}_{2}\mbox{O}~{}_{5}(g)\rightarrow\;4\mbox{NO}~{}_{2}(g)+\mbox{O}~{% }_{2}(g)
  20. ln [ A ] = - k t + ln [ A ] 0 \ \ln{[A]}=-kt+\ln{[A]_{0}}
  21. A = A 0 e - k t A=A_{0}e^{-kt}\,
  22. A = A 0 ( e - k Δ t p ) n A=A_{0}\left(e^{-k\Delta t_{p}}\right)^{n}
  23. Δ t p \Delta t_{p}
  24. n n
  25. f R P f_{RP}
  26. A n A n - 1 = f R P = e - k Δ t p \frac{A_{n}}{A_{n-1}}=f_{RP}=e^{-k\Delta t_{p}}
  27. n n
  28. A A 0 ≡ A n A 0 = ( e - k Δ t p ) n = ( f R P ) n = ( 1 - f B P ) n \frac{A}{A_{0}}\equiv\frac{A_{n}}{A_{0}}=\left(e^{-k\Delta t_{p}}\right)^{n}=% \left(f_{RP}\right)^{n}=\left(1-f_{BP}\right)^{n}
  29. f B P f_{BP}
  30. Δ t p = ln ( 2 ) k \Delta t_{p}=\frac{\ln\left(2\right)}{k}
  31. r a v g , n = - Δ A Δ t p = A n - 1 - A n Δ t p r_{avg,n}=-\frac{\Delta A}{\Delta t_{p}}=\frac{A_{n-1}-A_{n}}{\Delta t_{p}}
  32. A n = A n - 1 - r a v g , n Δ t p A_{n}=A_{n-1}-r_{avg,n}\Delta t_{p}
  33. f B P = 1 - A n A n - 1 f_{BP}=1-\frac{A_{n}}{A_{n-1}}
  34. f B P = r a v g , n Δ t p A n - 1 f_{BP}=\frac{r_{avg,n}\Delta t_{p}}{A_{n-1}}
  35. A n = A n - 1 ( 1 - r a v g , n Δ t p A n - 1 ) A_{n}=A_{n-1}\left(1-\frac{r_{avg,n}\Delta t_{p}}{A_{n-1}}\right)
  36. - d [ A ] d t = k [ A ] 2 \ -\frac{d[A]}{dt}=k[A]^{2}
  37. - d [ A ] d t = k [ A ] [ B ] \ -\frac{d[A]}{dt}=k[A][B]
  38. - d [ A ] d t = 2 k [ A ] 2 \ -\frac{d[A]}{dt}=2k[A]^{2}
  39. 1 [ A ] = 1 [ A ] 0 + k t \frac{1}{[A]}=\frac{1}{[A]_{0}}+kt
  40. [ A ] [ B ] = [ A ] 0 [ B ] 0 e ( [ A ] 0 - [ B ] 0 ) k t \frac{[A]}{[B]}=\frac{[A]_{0}}{[B]_{0}}e^{([A]_{0}-[B]_{0})kt}
  41. t 1 2 = 1 k [ A ] 0 \ t_{\frac{1}{2}}=\frac{1}{k[A]_{0}}
  42. ln r = ln k + 2 ln [ A ] \ln{}r=\ln{}k+2\ln\left[A\right]
  43. 2 NO ( g ) 2 → 2 NO ( g ) + O ( g ) 2 2\mbox{NO}~{}_{2}(g)\rightarrow\;2\mbox{NO}~{}(g)+\mbox{O}~{}_{2}(g)
  44. r = k [ A ] [ B ] = k â€Č [ A ] \ r=k[A][B]=k^{\prime}[A]
  45. k â€Č = k [ B ] 0 k^{\prime}=k[B]_{0}
  46. k â€Č k^{\prime}
  47. k â€Č k^{\prime}
  48. k k
  49. - d [ A ] d t = k -\frac{d[A]}{dt}=k
  50. - d [ A ] d t = k [ A ] -\frac{d[A]}{dt}=k[A]
  51. - d [ A ] d t = k [ A ] 2 -\frac{d[A]}{dt}=k[A]^{2}
  52. - d [ A ] d t = k [ A ] n -\frac{d[A]}{dt}=k[A]^{n}
  53. [ A ] = [ A ] 0 - k t \ [A]=[A]_{0}-kt
  54. [ A ] = [ A ] 0 e - k t \ [A]=[A]_{0}e^{-kt}
  55. 1 [ A ] = 1 [ A ] 0 + k t \frac{1}{[A]}=\frac{1}{[A]_{0}}+kt
  56. 1 [ A ] n - 1 = 1 [ A ] 0 n - 1 + ( n - 1 ) k t \frac{1}{[A]^{n-1}}=\frac{1}{{[A]_{0}}^{n-1}}+(n-1)kt
  57. M s \rm\frac{M}{s}
  58. 1 s \rm\frac{1}{s}
  59. 1 M ⋅ s \rm\frac{1}{M\cdot s}
  60. 1 M n - 1 ⋅ s \frac{1}{{\rm M}^{n-1}\cdot\rm s}
  61. [ A ] vs. t [A]\ \mbox{vs.}~{}\ t
  62. ln ( [ A ] ) vs. t \ln([A])\ \mbox{vs.}~{}\ t
  63. 1 [ A ] vs. t \frac{1}{[A]}\ \mbox{vs.}~{}\ t
  64. 1 [ A ] n - 1 vs. t \frac{1}{[A]^{n-1}}\ \mbox{vs.}~{}\ t
  65. t 1 / 2 = [ A ] 0 2 k t_{1/2}=\frac{[A]_{0}}{2k}
  66. t 1 / 2 = ln ( 2 ) k t_{1/2}=\frac{\ln(2)}{k}
  67. t 1 / 2 = 1 k [ A ] 0 t_{1/2}=\frac{1}{k[A]_{0}}
  68. t 1 / 2 = lim x → n 2 x - 1 - 1 ( x - 1 ) k [ A ] 0 x - 1 t_{1/2}=\lim_{x\to n}\frac{2^{x-1}-1}{(x-1)k{[A]_{0}}^{x-1}}
  69. s A + t B ⇌ u X + v Y \ sA+tB\rightleftharpoons uX+vY
  70. r = k 1 [ A ] s [ B ] t - k 2 [ X ] u [ Y ] v r={k_{1}[A]^{s}[B]^{t}}-{k_{2}[X]^{u}[Y]^{v}}\,
  71. k 1 [ A ] s [ B ] t = k 2 [ X ] u [ Y ] v {k_{1}[A]^{s}[B]^{t}=k_{2}[X]^{u}[Y]^{v}}\,
  72. K = [ X ] u [ Y ] v [ A ] s [ B ] t = k 1 k 2 K=\frac{[X]^{u}[Y]^{v}}{[A]^{s}[B]^{t}}=\frac{k_{1}}{k_{2}}
  73. A ⇌ B A\rightleftharpoons B
  74. [ A ] 0 [A]_{0}
  75. K = def k f k b = [ B ] e [ A ] e K\ \stackrel{\mathrm{def}}{=}\ \frac{k_{f}}{k_{b}}=\frac{\left[B\right]_{e}}{% \left[A\right]_{e}}
  76. [ A ] e [A]_{e}
  77. [ B ] e [B]_{e}
  78. [ A ] t [A]_{t}
  79. [ B ] t [B]_{t}
  80. [ A ] t = [ A ] 0 - [ B ] t \ [A]_{t}=[A]_{0}-[B]_{t}
  81. [ B ] 0 [B]_{0}
  82. [ A ] e = [ A ] 0 - [ B ] e \ [A]_{e}=[A]_{0}-[B]_{e}
  83. [ B ] e = x = k f k f + k b [ A ] 0 \ [B]_{e}=x=\frac{k_{f}}{k_{f}+k_{b}}[A]_{0}
  84. [ A ] e = [ A ] 0 - x = k b k f + k b [ A ] 0 \ [A]_{e}=[A]_{0}-x=\frac{k_{b}}{k_{f}+k_{b}}[A]_{0}
  85. r = k 1 [ A ] s [ B ] t - k 2 [ X ] u [ Y ] v r={k_{1}[A]^{s}[B]^{t}}-{k_{2}[X]^{u}[Y]^{v}}\,
  86. - d [ A ] d t = k f [ A ] t - k b [ B ] t -\frac{d[A]}{dt}={k_{f}[A]_{t}}-{k_{b}[B]_{t}}\,
  87. [ A ] t [A]_{t}
  88. x e x_{e}
  89. - d [ A ] d t = k f [ A ] t - k b [ B ] t -\frac{d[A]}{dt}={k_{f}[A]_{t}}-{k_{b}[B]_{t}}\,
  90. - d x d t = k f x - k b [ B ] t -\frac{dx}{dt}={k_{f}x}-{k_{b}[B]_{t}}\,
  91. - d x d t = k f x - k b ( [ A ] 0 - x ) -\frac{dx}{dt}={k_{f}x}-{k_{b}([A]_{0}-x)}\,
  92. - d x d t = ( k f + k b ) x - k b [ A ] 0 -\frac{dx}{dt}={(k_{f}+k_{b})x}-{k_{b}[A]_{0}}\,
  93. k f + k b = k b [ A ] 0 x e k_{f}+k_{b}={k_{b}\frac{[A]_{0}}{x_{e}}}
  94. d x d t = k b [ A ] 0 x e ( x e - x ) \ \frac{dx}{dt}=\frac{k_{b}[A]_{0}}{x_{e}}(x_{e}-x)
  95. ln ( [ A ] 0 - [ A ] e [ A ] t - [ A ] e ) = ( k f + k b ) t \ln\left(\frac{[A]_{0}-[A]_{e}}{[A]_{t}-[A]_{e}}\right)=(k_{f}+k_{b})t
  96. [ A ] = [ A ] 0 1 k f + k b ( k b + k f e - ( k f + k b ) t ) + [ B ] 0 k b k f + k b ( 1 - e - ( k f + k b ) t ) \left[A\right]=\left[A\right]_{0}\frac{1}{k_{f}+k_{b}}\left(k_{b}+k_{f}e^{-% \left(k_{f}+k_{b}\right)t}\right)+\left[B\right]_{0}\frac{k_{b}}{k_{f}+k_{b}}% \left(1-e^{-\left(k_{f}+k_{b}\right)t}\right)
  97. [ B ] = [ A ] 0 k f k f + k b ( 1 - e - ( k f + k b ) t ) + [ B ] 0 1 k f + k b ( k f + k b e - ( k f + k b ) t ) \left[B\right]=\left[A\right]_{0}\frac{k_{f}}{k_{f}+k_{b}}\left(1-e^{-\left(k_% {f}+k_{b}\right)t}\right)+\left[B\right]_{0}\frac{1}{k_{f}+k_{b}}\left(k_{f}+k% _{b}e^{-\left(k_{f}+k_{b}\right)t}\right)
  98. k 1 k_{1}
  99. k 2 k_{2}
  100. A → B → C A\rightarrow\;B\rightarrow\;C
  101. d [ A ] d t = - k 1 [ A ] \frac{d[A]}{dt}=-k_{1}[A]
  102. d [ B ] d t = k 1 [ A ] - k 2 [ B ] \frac{d[B]}{dt}=k_{1}[A]-k_{2}[B]
  103. d [ C ] d t = k 2 [ B ] \frac{d[C]}{dt}=k_{2}[B]
  104. [ A ] = [ A ] 0 e - k 1 t [A]=[A]_{0}e^{-k_{1}t}
  105. [ B ] = { [ A ] 0 k 1 k 2 - k 1 ( e - k 1 t - e - k 2 t ) + [ B ] 0 e - k 2 t k 1 ≠ k 2 [ A ] 0 k 1 t e - k 1 t + [ B ] 0 e - k 1 t otherwise \left[B\right]=\left\{\begin{array}[]{*{35}l}\left[A\right]_{0}\frac{k_{1}}{k_% {2}-k_{1}}\left(e^{-k_{1}t}-e^{-k_{2}t}\right)+\left[B\right]_{0}e^{-k_{2}t}&k% _{1}\neq k_{2}\\ \left[A\right]_{0}k_{1}te^{-k_{1}t}+\left[B\right]_{0}e^{-k_{1}t}&\,\text{% otherwise}\\ \end{array}\right.
  106. [ C ] = { [ A ] 0 ( 1 + k 1 e - k 2 t - k 2 e - k 1 t k 2 - k 1 ) + [ B ] 0 ( 1 - e - k 2 t ) + [ C ] 0 k 1 ≠ k 2 [ A ] 0 ( 1 - e - k 1 t - k 1 t e - k 1 t ) + [ B ] 0 ( 1 - e - k 1 t ) + [ C ] 0 otherwise \left[C\right]=\left\{\begin{array}[]{*{35}l}\left[A\right]_{0}\left(1+\frac{k% _{1}e^{-k_{2}t}-k_{2}e^{-k_{1}t}}{k_{2}-k_{1}}\right)+\left[B\right]_{0}\left(% 1-e^{-k_{2}t}\right)+\left[C\right]_{0}&k_{1}\neq k_{2}\\ \left[A\right]_{0}\left(1-e^{-k_{1}t}-k_{1}te^{-k_{1}t}\right)+\left[B\right]_% {0}\left(1-e^{-k_{1}t}\right)+\left[C\right]_{0}&\,\text{otherwise}\\ \end{array}\right.
  107. A → B A\rightarrow\;B
  108. A → C A\rightarrow\;C
  109. k 1 k_{1}
  110. k 2 k_{2}
  111. - d [ A ] d t = ( k 1 + k 2 ) [ A ] -\frac{d[A]}{dt}=(k_{1}+k_{2})[A]
  112. d [ B ] d t = k 1 [ A ] \frac{d[B]}{dt}=k_{1}[A]
  113. d [ C ] d t = k 2 [ A ] \frac{d[C]}{dt}=k_{2}[A]
  114. [ A ] = [ A ] 0 e - ( k 1 + k 2 ) t \ [A]=[A]_{0}e^{-(k_{1}+k_{2})t}
  115. [ B ] = k 1 k 1 + k 2 [ A ] 0 ( 1 - e - ( k 1 + k 2 ) t ) [B]=\frac{k_{1}}{k_{1}+k_{2}}[A]_{0}(1-e^{-(k_{1}+k_{2})t})
  116. [ C ] = k 2 k 1 + k 2 [ A ] 0 ( 1 - e - ( k 1 + k 2 ) t ) [C]=\frac{k_{2}}{k_{1}+k_{2}}[A]_{0}(1-e^{-(k_{1}+k_{2})t})
  117. [ B ] [ C ] = k 1 k 2 \frac{[B]}{[C]}=\frac{k_{1}}{k_{2}}
  118. A + H 2 O → B A+H_{2}O\rightarrow\ B
  119. A + R → C A+R\rightarrow\ C
  120. d [ B ] d t = k 1 [ A ] [ H 2 O ] = k 1 â€Č [ A ] \frac{d[B]}{dt}=k_{1}[A][H_{2}O]=k_{1}^{\prime}[A]
  121. d [ C ] d t = k 2 [ A ] [ R ] \frac{d[C]}{dt}=k_{2}[A][R]
  122. k 1 â€Č k_{1}^{\prime}
  123. [ C ] = [ R ] 0 [ 1 - e - k 2 k 1 â€Č [ A ] 0 ( 1 - e - k 1 â€Č t ) ] [C]=[R]_{0}\left[1-e^{-\frac{k_{2}}{k_{1}^{\prime}}[A]_{0}(1-e^{-k_{1}^{\prime% }t})}\right]
  124. l n [ R ] 0 [ R ] 0 - [ C ] = k 2 [ A ] 0 k 1 â€Č ( 1 - e - k 1 â€Č t ) ln\frac{[R]_{0}}{[R]_{0}-[C]}=\frac{k_{2}[A]_{0}}{k_{1}^{\prime}}(1-e^{-k_{1}^% {\prime}t})
  125. [ B ] = - k 1 â€Č k 2 l n ( 1 - [ C ] [ R ] 0 ) [B]=-\frac{k_{1}^{\prime}}{k_{2}}ln\left(1-\frac{[C]}{[R]_{0}}\right)
  126. [ A ] 0 - [ C ] ≈ [ A ] 0 [A]_{0}-[C]\approx\;[A]_{0}
  127. N N
  128. R R
  129. j j
  130. s 1 j X 1 + s 2 j X 2 
 + s N j X N → k j r 1 j X 1 + r 2 j X 2 + 
 + r N j X N , s_{1j}X_{1}+s_{2j}X_{2}\ldots+s_{Nj}X_{N}\xrightarrow{k_{j}}\ r_{1j}X_{1}+\ r_% {2j}X_{2}+\ldots+r_{Nj}X_{N},
  131. ∑ i = 1 N s i j X i → k j ∑ i = 1 N r i j X i . \sum_{i=1}^{N}s_{ij}X_{i}\xrightarrow{k_{j}}\sum_{i=1}^{N}\ r_{ij}X_{i}.
  132. j j
  133. R R
  134. X i X_{i}
  135. i i
  136. k j k_{j}
  137. j j
  138. s i j s_{ij}
  139. r i j r_{ij}
  140. f j ( [ X → ] ) = k j ∏ z = 1 N [ X z ] s z j f_{j}([\vec{X}])=k_{j}\prod_{z=1}^{N}[X_{z}]^{s_{zj}}
  141. [ X → ] = ( [ X 1 ] , [ X 2 ] , 
 , [ X N ] ) [\vec{X}]=([X_{1}],[X_{2}],...,[X_{N}])
  142. s z j = 0 s_{zj}=0
  143. z z
  144. s z j = 1 s_{zj}=1
  145. z z
  146. s z j = 1 s_{zj}=1
  147. z z
  148. s z j = 2 s_{zj}=2
  149. z z
  150. S i j = r i j - s i j , S_{ij}=r_{ij}-s_{ij},
  151. i i
  152. j j
  153. d [ X i ] d t = ∑ j = 1 R S i j f j ( [ X → ] ) . \frac{d[X_{i}]}{dt}=\sum_{j=1}^{R}S_{ij}f_{j}([\vec{X}]).
  154. d [ X i ] d t = 0 \frac{d[X_{i}]}{dt}=0
  155. S i j S_{ij}
  156. f j f_{j}
  157. N N
  158. t t
  159. X 1 ( t ) X_{1}(t)
  160. X N ( t ) X_{N}(t)
  161. X i X_{i}
  162. X j X_{j}
  163. k i j k_{ij}
  164. K K
  165. k i j k_{ij}
  166. X ( t ) = ( X 1 ( t ) , X 2 ( t ) , 
 , X N ( t ) ) T X(t)=(X_{1}(t),X_{2}(t),...,X_{N}(t))^{T}
  167. J = ( 1 , 1 , 1 , 
 , 1 ) T J=(1,1,1,...,1)^{T}
  168. I I
  169. N N
  170. N N
  171. D i a g Diag
  172. ℒ - 1 \displaystyle\mathcal{L}^{-1}
  173. s s
  174. t t
  175. X ( t ) X(t)
  176. X ( t ) = ℒ - 1 [ ( s I + D i a g ( K J ) - K T ) - 1 X ( 0 ) ] X(t)=\displaystyle\mathcal{L}^{-1}[(sI+Diag(KJ)-K^{T})^{-1}X(0)]
  177. t t

Rayleigh–Taylor_instability.html

  1. U ( x , z ) = W ( x , z ) = 0 , U(x,z)=W(x,z)=0,\,
  2. 𝐠 = - g 𝐳 ^ . \,\textbf{g}=-g\hat{\,\textbf{z}}.\,
  3. z = 0 z=0\,
  4. ρ G \rho_{G}\,
  5. ρ L \rho_{L}\,
  6. exp ( Îł t ) , with Îł = 𝒜 g α and 𝒜 = ρ heavy - ρ light ρ heavy + ρ light , \exp(\gamma\,t)\;,\qquad\,\text{with}\quad\gamma={\sqrt{\mathcal{A}g\alpha}}% \quad\,\text{and}\quad\mathcal{A}=\frac{\rho_{\,\text{heavy}}-\rho_{\,\text{% light}}}{\rho_{\,\text{heavy}}+\rho_{\,\text{light}}},\,
  7. Îł \gamma\,
  8. α \alpha\,
  9. 𝒜 \mathcal{A}\,
  10. ( u â€Č ( x , z , t ) , w â€Č ( x , z , t ) ) . (u^{\prime}(x,z,t),w^{\prime}(x,z,t)).\,
  11. 𝐼 â€Č = ( u â€Č ( x , z , t ) , w â€Č ( x , z , t ) ) = ( ψ z , - ψ x ) , \,\textbf{u}^{\prime}=(u^{\prime}(x,z,t),w^{\prime}(x,z,t))=(\psi_{z},-\psi_{x% }),\,
  12. ∇ × 𝐼 â€Č = 0 \nabla\times\,\textbf{u}^{\prime}=0\,
  13. ∇ 2 ψ = 0. \nabla^{2}\psi=0.\,
  14. ψ ( x , z , t ) = e i α ( x - c t ) Κ ( z ) , \psi\left(x,z,t\right)=e^{i\alpha\left(x-ct\right)}\Psi\left(z\right),\,
  15. α \alpha\,
  16. ( D 2 - α 2 ) Κ j = 0 , D = d d z , j = L , G . \left(D^{2}-\alpha^{2}\right)\Psi_{j}=0,\,\,\,\ D=\frac{d}{dz},\,\,\,\ j=L,G.\,
  17. - ∞ < z ≀ 0 -\infty<z\leq 0\,
  18. 0 ≀ z < ∞ 0\leq z<\infty\,
  19. w i â€Č w^{\prime}_{i}\,
  20. z = ± ∞ . z=\pm\infty.\,
  21. w L â€Č = 0 w_{L}^{\prime}=0\,
  22. z = - ∞ z=-\infty\,
  23. w G â€Č = 0 w_{G}^{\prime}=0\,
  24. z = ∞ z=\infty\,
  25. ι L ( - ∞ ) = 0 , ι G ( ∞ ) = 0. \Psi_{L}\left(-\infty\right)=0,\qquad\Psi_{G}\left(\infty\right)=0.\,
  26. z = η ( x , t ) z=\eta\left(x,t\right)\,
  27. z = η z=\eta
  28. w L â€Č = w G â€Č w^{\prime}_{L}=w^{\prime}_{G}\,
  29. Κ L ( η ) = Κ G ( η ) . \Psi_{L}\left(\eta\right)=\Psi_{G}\left(\eta\right).\,
  30. z = 0 z=0\,
  31. Κ L ( 0 ) = Κ G ( 0 ) + H.O.T. , \Psi_{L}\left(0\right)=\Psi_{G}\left(0\right)+\,\text{H.O.T.},\,
  32. z = η ( x , t ) z=\eta\left(x,t\right)\,
  33. ∂ η ∂ t + u â€Č ∂ η ∂ x = w â€Č ( η ) . \frac{\partial\eta}{\partial t}+u^{\prime}\frac{\partial\eta}{\partial x}=w^{% \prime}\left(\eta\right).\,
  34. ∂ η ∂ t = w â€Č ( 0 ) , \frac{\partial\eta}{\partial t}=w^{\prime}\left(0\right),\,
  35. w â€Č ( η ) w^{\prime}\left(\eta\right)\,
  36. z = 0 z=0\,
  37. c η = Κ c\eta=\Psi\,
  38. z = η z=\eta
  39. p G ( z = η ) - p L ( z = η ) = σ Îș , p_{G}\left(z=\eta\right)-p_{L}\left(z=\eta\right)=\sigma\kappa,\,
  40. Îș = ∇ 2 η = η x x . \kappa=\nabla^{2}\eta=\eta_{xx}.\,
  41. p G ( z = η ) - p L ( z = η ) = σ η x x . p_{G}\left(z=\eta\right)-p_{L}\left(z=\eta\right)=\sigma\eta_{xx}.\,
  42. [ P G ( η ) + p G â€Č ( 0 ) ] - [ P L ( η ) + p L â€Č ( 0 ) ] = σ η x x . \left[P_{G}\left(\eta\right)+p^{\prime}_{G}\left(0\right)\right]-\left[P_{L}% \left(\eta\right)+p^{\prime}_{L}\left(0\right)\right]=\sigma\eta_{xx}.\,
  43. P L = - ρ L g z + p 0 , P G = - ρ G g z + p 0 , P_{L}=-\rho_{L}gz+p_{0},\qquad P_{G}=-\rho_{G}gz+p_{0},\,
  44. p G â€Č - p L â€Č = g η ( ρ G - ρ L ) + σ η x x , on z = 0. p^{\prime}_{G}-p^{\prime}_{L}=g\eta\left(\rho_{G}-\rho_{L}\right)+\sigma\eta_{% xx},\qquad\,\text{on }z=0.\,
  45. ∂ u i â€Č ∂ t = - 1 ρ i ∂ p i â€Č ∂ x \frac{\partial u_{i}^{\prime}}{\partial t}=-\frac{1}{\rho_{i}}\frac{\partial p% _{i}^{\prime}}{\partial x}\,
  46. i = L , G , i=L,G,\,
  47. p i â€Č = ρ i c D Κ i , i = L , G . p_{i}^{\prime}=\rho_{i}cD\Psi_{i},\qquad i=L,G.\,
  48. p G â€Č - p L â€Č p^{\prime}_{G}-p^{\prime}_{L}
  49. c ( ρ G D Κ G - ρ L D Κ L ) = g η ( ρ G - ρ L ) + σ η x x . c\left(\rho_{G}D\Psi_{G}-\rho_{L}D\Psi_{L}\right)=g\eta\left(\rho_{G}-\rho_{L}% \right)+\sigma\eta_{xx}.\,
  50. c η = Κ c\eta=\Psi\,
  51. c 2 ( ρ G D Κ G - ρ L D Κ L ) = g Κ ( ρ G - ρ L ) - σ α 2 Κ , c^{2}\left(\rho_{G}D\Psi_{G}-\rho_{L}D\Psi_{L}\right)=g\Psi\left(\rho_{G}-\rho% _{L}\right)-\sigma\alpha^{2}\Psi,\,
  52. Κ \Psi\,
  53. Κ L = Κ G \Psi_{L}=\Psi_{G}\,
  54. z = 0. z=0.\,
  55. ( D 2 - α 2 ) Κ i = 0 , \left(D^{2}-\alpha^{2}\right)\Psi_{i}=0,\,
  56. Κ ( ± ∞ ) \Psi\left(\pm\infty\right)\,
  57. Κ L = A L e α z , Κ G = A G e - α z . \Psi_{L}=A_{L}e^{\alpha z},\qquad\Psi_{G}=A_{G}e^{-\alpha z}.\,
  58. Κ L = Κ G \Psi_{L}=\Psi_{G}\,
  59. z = 0 z=0\,
  60. A L = A G = A . A_{L}=A_{G}=A.\,
  61. c 2 ( ρ G D Κ G - ρ L D Κ L ) = g Κ ( ρ G - ρ L ) - σ α 2 Κ . c^{2}\left(\rho_{G}D\Psi_{G}-\rho_{L}D\Psi_{L}\right)=g\Psi\left(\rho_{G}-\rho% _{L}\right)-\sigma\alpha^{2}\Psi.\,
  62. A c 2 α ( - ρ G - ρ L ) = A g ( ρ G - ρ L ) - σ α 2 A . Ac^{2}\alpha\left(-\rho_{G}-\rho_{L}\right)=Ag\left(\rho_{G}-\rho_{L}\right)-% \sigma\alpha^{2}A.\,
  63. c 2 = g α ρ L - ρ G ρ L + ρ G + σ α ρ L + ρ G . c^{2}=\frac{g}{\alpha}\frac{\rho_{L}-\rho_{G}}{\rho_{L}+\rho_{G}}+\frac{\sigma% \alpha}{\rho_{L}+\rho_{G}}.\,
  64. c 2 = g α ρ L - ρ G ρ L + ρ G , σ = 0 , c^{2}=\frac{g}{\alpha}\frac{\rho_{L}-\rho_{G}}{\rho_{L}+\rho_{G}},\qquad\sigma% =0,\,
  65. ρ G < ρ L \rho_{G}<\rho_{L}\,
  66. c 2 > 0 c^{2}>0\,
  67. ρ G > ρ L \rho_{G}>\rho_{L}\,
  68. c 2 < 0 c^{2}<0\,
  69. c 2 < 0 c^{2}<0\,
  70. c = ± i g 𝒜 α , 𝒜 = ρ G - ρ L ρ G + ρ L , c=\pm i\sqrt{\frac{g\mathcal{A}}{\alpha}},\qquad\mathcal{A}=\frac{\rho_{G}-% \rho_{L}}{\rho_{G}+\rho_{L}},\,
  71. 𝒜 \mathcal{A}\,
  72. Κ ( x , z , t ) = A e - α | z | exp [ i α ( x - c t ) ] = A exp ( α g 𝒜 ~ α t ) exp ( i α x - α | z | ) \Psi\left(x,z,t\right)=Ae^{-\alpha|z|}\exp\left[i\alpha\left(x-ct\right)\right% ]=A\exp\left(\alpha\sqrt{\frac{g\tilde{\mathcal{A}}}{\alpha}}t\right)\exp\left% (i\alpha x-\alpha|z|\right)\,
  73. c η = Κ . c\eta=\Psi.\,
  74. B = A / c . B=A/c.\,
  75. z = η ( x , t ) , z=\eta(x,t),\,
  76. η ( x , 0 ) = ℜ { B exp ( i α x ) } , \eta(x,0)=\Re\left\{B\,\exp\left(i\alpha x\right)\right\},\,
  77. η = ℜ { B exp ( 𝒜 g α t ) exp ( i α x ) } \eta=\Re\left\{B\,\exp\left(\sqrt{\mathcal{A}g\alpha}\,t\right)\exp\left(i% \alpha x\right)\right\}\,
  78. ℜ { ⋅ } \Re\left\{\cdot\right\}\,

Reaching_definition.html

  1. S S
  2. REACH in [ S ] = ⋃ p ∈ p r e d [ S ] REACH out [ p ] {\rm REACH}_{\rm in}[S]=\bigcup_{p\in pred[S]}{\rm REACH}_{\rm out}[p]
  3. REACH out [ S ] = GEN [ S ] âˆȘ ( REACH in [ S ] - KILL [ S ] ) {\rm REACH}_{\rm out}[S]={\rm GEN}[S]\cup({\rm REACH}_{\rm in}[S]-{\rm KILL}[S])
  4. S S
  5. S S
  6. p r e d [ S ] pred[S]
  7. p r e d [ S ] pred[S]
  8. S S
  9. S S
  10. S S
  11. S S
  12. GEN {\rm GEN}
  13. KILL {\rm KILL}
  14. GEN [ d : y ← f ( x 1 , ⋯ , x n ) ] = { d } {\rm GEN}[d:y\leftarrow f(x_{1},\cdots,x_{n})]=\{d\}
  15. KILL [ d : y ← f ( x 1 , ⋯ , x n ) ] = DEFS [ y ] - { d } {\rm KILL}[d:y\leftarrow f(x_{1},\cdots,x_{n})]={\rm DEFS}[y]-\{d\}
  16. DEFS [ y ] {\rm DEFS}[y]
  17. y y
  18. d d

Reaction_quotient.html

  1. Q r = { S t } σ { T t } τ { A t } α { B t } ÎČ Q_{r}=\frac{\left\{S_{t}\right\}^{\sigma}\left\{T_{t}\right\}^{\tau}}{\left\{A% _{t}\right\}^{\alpha}\left\{B_{t}\right\}^{\beta}}
  2. Q r = ∏ j a j Μ j ( t ) ∏ i a i Μ i ( t ) Q_{r}=\frac{\prod_{j}a_{j}^{\nu_{j}}(t)}{\prod_{i}a_{i}^{\nu_{i}}(t)}

Reaction_rate_constant.html

  1. λ \lambda
  2. a A + b B → c C aA+bB\rightarrow cC
  3. r = k ( T ) [ A ] m [ B ] n r=k(T)[A]^{m}[B]^{n}
  4. k = A e - E a R T k=Ae^{\frac{-E_{a}}{RT}}
  5. r = A e - E a R T [ A ] m [ B ] n , r=Ae^{\frac{-E_{a}}{RT}}[A]^{m}[B]^{n},
  6. k = k S D ⋅ α R S S D k=k_{SD}\cdot\alpha^{SD}_{RS}
  7. α R S S D \alpha^{SD}_{RS}
  8. k S D k_{SD}

Real_coordinate_space.html

  1. n n
  2. n n
  3. n n
  4. n n
  5. n n
  6. n n
  7. 𝐑 \mathbf{R}
  8. n n
  9. n n
  10. 𝐑 \mathbf{R}
  11. đ± = ( x 1 , x 2 , 
 , x n ) \mathbf{x}=(x_{1},x_{2},\ldots,x_{n})
  12. n n
  13. n n
  14. n n
  15. n n
  16. n − 1 n−1
  17. n n
  18. n n
  19. n = 2 n=2
  20. F ( t ) = f ( g 1 ( t ) , g 2 ( t ) ) , F(t)=f(g_{1}(t),g_{2}(t)),
  21. F F
  22. f f
  23. F F
  24. n n
  25. đ± + đČ = ( x 1 + y 1 , x 2 + y 2 , 
 , x n + y n ) \mathbf{x}+\mathbf{y}=(x_{1}+y_{1},x_{2}+y_{2},\ldots,x_{n}+y_{n})
  26. α đ± = ( α x 1 , α x 2 , 
 , α x n ) . \alpha\mathbf{x}=(\alpha x_{1},\alpha x_{2},\ldots,\alpha x_{n}).
  27. 𝟎 = ( 0 , 0 , 
 , 0 ) \mathbf{0}=(0,0,\ldots,0)
  28. đ± \mathbf{x}
  29. - đ± = ( - x 1 , - x 2 , 
 , - x n ) . -\mathbf{x}=(-x_{1},-x_{2},\ldots,-x_{n}).
  30. n n
  31. đ± = [ x 1 x 2 ⋼ x n ] \mathbf{x}=\begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{bmatrix}
  32. đ± = [ x 1 x 2 
 x n ] . \mathbf{x}=\begin{bmatrix}x_{1}&x_{2}&\dots&x_{n}\end{bmatrix}.
  33. n × 1 n× 1
  34. 1 × n 1 ×n
  35. m × n m×n
  36. ( A đ± ) k = ∑ l = 1 n A k l x l (A{\mathbf{x}})_{k}=\sum\limits_{l=1}^{n}A_{kl}x_{l}
  37. m m
  38. 𝐞 1 \displaystyle\mathbf{e}_{1}
  39. đ± = ∑ i = 1 n x i 𝐞 i . \mathbf{x}=\sum_{i=1}^{n}x_{i}\mathbf{e}_{i}.
  40. n n
  41. n n
  42. n n
  43. n n
  44. đ± ⋅ đČ = ∑ i = 1 n x i y i = x 1 y 1 + x 2 y 2 + ⋯ + x n y n \mathbf{x}\cdot\mathbf{y}=\sum_{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+% \cdots+x_{n}y_{n}
  45. | đ± | = |\mathbf{x}|=
  46. đ± ⋅ đ± \sqrt{\mathbf{x}}{⋅}{\mathbf{x}}
  47. d ( đ± , đČ ) = ∄ đ± - đČ ∄ = ∑ i = 1 n ( x i - y i ) 2 d(\mathbf{x},\mathbf{y})=\|\mathbf{x}-\mathbf{y}\|=\sqrt{\sum_{i=1}^{n}(x_{i}-% y_{i})^{2}}
  48. n n
  49. n n
  50. n n
  51. q q
  52. q ( đ± − đČ ) \sqrt{q}{(}{\mathbf{x}}{−}{\mathbf{y}}{)}
  53. \exist C 1 > 0 , \exist C 2 > 0 , ∀ đ± , đČ ∈ ℝ n : C 1 d ( đ± , đČ ) ≀ q ( đ± - đČ ) ≀ C 2 d ( đ± , đČ ) . \exist C_{1}>0,\ \exist C_{2}>0,\ \forall\mathbf{x},\mathbf{y}\in\mathbb{R}^{n% }:C_{1}d(\mathbf{x},\mathbf{y})\leq\sqrt{q(\mathbf{x}-\mathbf{y})}\leq C_{2}d(% \mathbf{x},\mathbf{y}).
  54. q ( đ± − đČ ) \sqrt{q}{(}{\mathbf{x}}{−}{\mathbf{y}}{)}
  55. M ( đ± − đČ ) M(\mathbf{x}−\mathbf{y})
  56. M M
  57. n n
  58. m m
  59. n n
  60. n n
  61. n n
  62. n n
  63. n n
  64. n n
  65. [ 0 , 1 ] [0,1]
  66. n n
  67. 2 n 2n
  68. 0 ≀ x 1 ≀ 1 ⋼ 0 ≀ x n ≀ 1 \displaystyle\begin{matrix}0\leq x_{1}\leq 1\\ \vdots\\ 0\leq x_{n}\leq 1\end{matrix}
  69. [ 0 , 1 ] [0,1]
  70. < t d > x 1 | ≀ 1 ⋼ < / t d > < t d > x n | ≀ 1 < c o d e > ( f o r < / c o d e > < m a t h > [ − 1 , 1 ] \displaystyle\begin{matrix}<td>x_{1}|\leq 1\\ \vdots\\ </td><td>x_{n}|\leq 1\end{matrix}<code> (for </code><math>[−1,1]
  71. k k
  72. k k
  73. n n
  74. ∑ k = 1 n | x k | ≀ 1 , \sum\limits_{k=1}^{n}|x_{k}|\leq 1\,,
  75. n n
  76. ( 0 , 0 , 
 , 0 ) (0, 0, 
 , 0)
  77. n n
  78. n + 1 n+1
  79. 0 ≀ x 1 ⋼ 0 ≀ x n ∑ k = 1 n x k ≀ 1 \begin{matrix}0\leq x_{1}\\ \vdots\\ 0\leq x_{n}\\ \sum\limits_{k=1}^{n}x_{k}\leq 1\end{matrix}
  80. 𝐑 < s u p > n \mathbf{R}<sup>n
  81. 𝐑 < s u p > n \mathbf{R}<sup>n

Real_estate_bubble.html

  1. House P/E ratio = House price Rent - Expenses \mbox{House P/E ratio}~{}=\frac{\mbox{House price}~{}}{\mbox{Rent}~{}-\mbox{% Expenses}~{}}
  2. House Price-Rent ratio = House price Monthly Rent × 12 \mbox{House Price-Rent ratio}~{}=\frac{\mbox{House price}~{}}{\mbox{Monthly % Rent}~{}\times 12}
  3. Gross Rental Yield = Monthly Rent × 12 House Price × 100 % \mbox{Gross Rental Yield}~{}=\frac{\mbox{Monthly Rent}~{}\times 12}{\mbox{% House Price}~{}}\times 100\%

Real_gas.html

  1. R T = ( P + a V m 2 ) ( V m - b ) RT=\left(P+\frac{a}{V_{m}^{2}}\right)(V_{m}-b)
  2. a = 27 R 2 T c 2 64 P c a=\frac{27R^{2}T_{c}^{2}}{64P_{c}}
  3. b = R T c 8 P c b=\frac{RT_{c}}{8P_{c}}
  4. R T = P ( V m - b ) + a V m ( V m + b ) T 1 2 ( V m - b ) RT=P(V_{m}-b)+\frac{a}{V_{m}(V_{m}+b)T^{\frac{1}{2}}}(V_{m}-b)
  5. a = 0.4275 R 2 T c 2.5 P c a=0.4275\frac{R^{2}T_{c}^{2.5}}{P_{c}}
  6. b = 0.0867 R T c P c b=0.0867\frac{RT_{c}}{P_{c}}
  7. P = R T V m - b - a T V m 2 P=\frac{RT}{V_{m}-b}-\frac{a}{TV_{m}^{2}}
  8. P = R T V m [ 1 + 9 P / P c 128 T / T c ( 1 - 6 ( T / T c ) 2 ) ] P=\frac{RT}{V_{m}}\left[1+\frac{9P/P_{c}}{128T/T_{c}}\left(1-\frac{6}{(T/T_{c}% )^{2}}\right)\right]
  9. P = R T exp ( - a V m R T ) V m - b P=RT\frac{\exp{(\frac{-a}{V_{m}RT})}}{V_{m}-b}
  10. R T = ( P + a T ( V m + c ) 2 ) ( V m - b ) RT=\left(P+\frac{a}{T(V_{m}+c)^{2}}\right)(V_{m}-b)
  11. a = 27 R 2 T c 3 64 P c a=\frac{27R^{2}T_{c}^{3}}{64P_{c}}
  12. b = V c - R T c 4 P c b=V_{c}-\frac{RT_{c}}{4P_{c}}
  13. c = 3 R T c 8 P c - V c c=\frac{3RT_{c}}{8P_{c}}-V_{c}
  14. P V m = R T ( 1 + B ( T ) V m + C ( T ) V m 2 + D ( T ) V m 3 + 
 ) PV_{m}=RT\left(1+\frac{B(T)}{V_{m}}+\frac{C(T)}{V_{m}^{2}}+\frac{D(T)}{V_{m}^{% 3}}+...\right)
  15. P V m = R T ( 1 + B â€Č ( T ) P + C â€Č ( T ) P 2 + D â€Č ( T ) P 3 + 
 ) PV_{m}=RT\left(1+\frac{B^{\prime}(T)}{P}+\frac{C^{\prime}(T)}{P^{2}}+\frac{D^{% \prime}(T)}{P^{3}}+...\right)
  16. P = R T V m - b - a ( T ) V m ( V m + b ) + b ( V m - b ) P=\frac{RT}{V_{m}-b}-\frac{a(T)}{V_{m}(V_{m}+b)+b(V_{m}-b)}
  17. R T = ( P + a T V m ( V m - b ) - c T 2 V m 3 ) ( V m - b ) RT=\left(P+\frac{a}{TV_{m}(V_{m}-b)}-\frac{c}{T^{2}V_{m}^{3}}\right)(V_{m}-b)
  18. a = 6 P c T c V c 2 a=6P_{c}T_{c}V_{c}^{2}
  19. b = V c 4 b=\frac{V_{c}}{4}
  20. c = 4 P c T c 2 V c 3 c=4P_{c}T_{c}^{2}V_{c}^{3}
  21. P = R T v 2 ( 1 - c v T 3 ) ( v + B ) - A v 2 P=\frac{RT}{v^{2}}\left(1-\frac{c}{vT^{3}}\right)(v+B)-\frac{A}{v^{2}}
  22. A = A 0 ( 1 - a v ) A=A_{0}\left(1-\frac{a}{v}\right)
  23. B = B 0 ( 1 - b v ) B=B_{0}\left(1-\frac{b}{v}\right)
  24. m 3 K m o l \frac{m^{3}}{Kmol}
  25. k P a . m 3 K m o l . K \frac{kPa.m^{3}}{Kmol.K}
  26. P = R T d + d 2 ( R T ( B + b d ) - ( A + a d - a α d 4 ) - 1 T 2 [ C - c d ( 1 + γ d 2 ) exp ( - γ d 2 ) ] ) P=RTd+d^{2}\left(RT(B+bd)-(A+ad-a{\alpha}d^{4})-\frac{1}{T^{2}}[C-cd(1+{\gamma% }d^{2})\exp(-{\gamma}d^{2})]\right)

Recoil_temperature.html

  1. T r e c o i l = ℏ 2 k 2 2 m k B T_{recoil}=\frac{\hbar^{2}k^{2}}{2mk_{B}}
  2. p = ℏ k p=\hbar k
  3. k k
  4. m m
  5. k B k_{B}
  6. ℏ \hbar

Rectangular_function.html

  1. rect ( t ) = Π ( t ) = { 0 if | t | > 1 2 1 2 if | t | = 1 2 1 if | t | < 1 2 . \mathrm{rect}(t)=\Pi(t)=\begin{cases}0&\mbox{if }~{}|t|>\frac{1}{2}\\ \frac{1}{2}&\mbox{if }~{}|t|=\frac{1}{2}\\ 1&\mbox{if }~{}|t|<\frac{1}{2}.\\ \end{cases}
  2. rect ( ± 1 2 ) \mathrm{rect}(\pm\tfrac{1}{2})
  3. rect ( t - X Y ) = u ( t - ( X - Y / 2 ) ) - u ( t - ( X + Y / 2 ) ) = u ( t - X + Y / 2 ) - u ( t - X - Y / 2 ) \operatorname{rect}\left(\frac{t-X}{Y}\right)=u(t-(X-Y/2))-u(t-(X+Y/2))=u(t-X+% Y/2)-u(t-X-Y/2)
  4. ∫ - ∞ ∞ rect ( t ) ⋅ e - i 2 π f t d t = sin ( π f ) π f = sinc ( π f ) , \int_{-\infty}^{\infty}\mathrm{rect}(t)\cdot e^{-i2\pi ft}\,dt=\frac{\sin(\pi f% )}{\pi f}=\mathrm{sinc}(\pi f),\,
  5. 1 2 π ∫ - ∞ ∞ rect ( t ) ⋅ e - i ω t d t = 1 2 π ⋅ sin ( ω / 2 ) ω / 2 = 1 2 π sinc ( ω / 2 ) , \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{rect}(t)\cdot e^{-i\omega t% }\,dt=\frac{1}{\sqrt{2\pi}}\cdot\frac{\mathrm{sin}\left(\omega/2\right)}{% \omega/2}=\frac{1}{\sqrt{2\pi}}\mathrm{sinc}\left(\omega/2\right),\,
  6. sinc \mathrm{sinc}
  7. tri = rect * rect . \mathrm{tri}=\mathrm{rect}*\mathrm{rect}.\,
  8. a , b = - 1 2 , 1 2 a,b=-\frac{1}{2},\frac{1}{2}
  9. φ ( k ) = sin ( k / 2 ) k / 2 , \varphi(k)=\frac{\sin(k/2)}{k/2},\,
  10. M ( k ) = sinh ( k / 2 ) k / 2 , M(k)=\frac{\mathrm{sinh}(k/2)}{k/2},\,
  11. sinh ( t ) \mathrm{sinh}(t)
  12. Π ( t ) = lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 \Pi(t)=\lim_{n\rightarrow\infty,n\in\mathbb{(}Z)}\frac{1}{(2t)^{2n}+1}
  13. | t | < 1 2 |t|<\frac{1}{2}
  14. ( 2 t ) 2 n (2t)^{2n}
  15. n n
  16. 2 t < 1 2t<1
  17. ( 2 t ) 2 n (2t)^{2n}
  18. n n
  19. lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = 1 0 + 1 = 1 , | t | < 1 2 \lim_{n\rightarrow\infty,n\in\mathbb{(}Z)}\frac{1}{(2t)^{2n}+1}=\frac{1}{0+1}=% 1,|t|<\frac{1}{2}
  20. | t | > 1 2 |t|>\frac{1}{2}
  21. ( 2 t ) 2 n (2t)^{2n}
  22. n n
  23. 2 t > 1 2t>1
  24. ( 2 t ) 2 n (2t)^{2n}
  25. n n
  26. lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = 1 + ∞ + 1 = 0 , | t | > 1 2 \lim_{n\rightarrow\infty,n\in\mathbb{(}Z)}\frac{1}{(2t)^{2n}+1}=\frac{1}{+% \infty+1}=0,|t|>\frac{1}{2}
  27. | t | = 1 2 |t|=\frac{1}{2}
  28. lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = lim n → ∞ , n ∈ ( Z ) 1 1 2 n + 1 = 1 1 + 1 = 1 2 \lim_{n\rightarrow\infty,n\in\mathbb{(}Z)}\frac{1}{(2t)^{2n}+1}=\lim_{n% \rightarrow\infty,n\in\mathbb{(}Z)}\frac{1}{1^{2n}+1}=\frac{1}{1+1}=\frac{1}{2}
  29. ∮ rect ( t ) = Π ( t ) = lim n → ∞ , n ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = { 0 if | t | > 1 2 1 2 if | t | = 1 2 1 if | t | < 1 2 . \therefore\mathrm{rect}(t)=\Pi(t)=\lim_{n\rightarrow\infty,n\in\mathbb{(}Z)}% \frac{1}{(2t)^{2n}+1}=\begin{cases}0&\mbox{if }~{}|t|>\frac{1}{2}\\ \frac{1}{2}&\mbox{if }~{}|t|=\frac{1}{2}\\ 1&\mbox{if }~{}|t|<\frac{1}{2}.\\ \end{cases}

Recurrent_neural_network.html

  1. i i
  2. y i y_{i}
  3. τ i y ˙ i = - y i + σ ( ∑ j = 1 n w j i y j - Θ j ) + I i ( t ) \tau_{i}\dot{y}_{i}=-y_{i}+\sigma(\sum_{j=1}^{n}w_{ji}y_{j}-\Theta_{j})+I_{i}(t)
  4. τ i \tau_{i}
  5. y i y_{i}
  6. y ˙ i \dot{y}_{i}
  7. w j i w{}_{ji}
  8. σ ( x ) \sigma(x)
  9. σ ( x ) = 1 / ( 1 + e - x ) \sigma(x)=1/(1+e^{-x})
  10. y j y_{j}
  11. Θ j \Theta_{j}
  12. I i ( t ) I_{i}(t)
  13. w i j k w{}_{ijk}
  14. w i j w{}_{ij}

Recursive_least_squares_filter.html

  1. x ( n ) = ∑ k = 0 q b n ( k ) d ( n - k ) + v ( n ) x(n)=\sum_{k=0}^{q}b_{n}(k)d(n-k)+v(n)
  2. v ( n ) v(n)
  3. d ( n ) d(n)
  4. p + 1 p+1
  5. 𝐰 \mathbf{w}
  6. d ^ ( n ) = ∑ k = 0 p w n ( k ) x ( n - k ) = 𝐰 n T đ± n \hat{d}(n)=\sum_{k=0}^{p}w_{n}(k)x(n-k)=\mathbf{w}_{n}^{\mathit{T}}\mathbf{x}_% {n}
  7. đ± n = [ x ( n ) x ( n - 1 ) 
 x ( n - p ) ] T \mathbf{x}_{n}=[x(n)\quad x(n-1)\quad\ldots\quad x(n-p)]^{T}
  8. p + 1 p+1
  9. x ( n ) x(n)
  10. 𝐰 \mathbf{w}
  11. 𝐰 n \mathbf{w}_{n}
  12. 𝐰 n + 1 \mathbf{w}_{n+1}
  13. 𝐰 n \mathbf{w}_{n}
  14. C C
  15. 𝐰 n \mathbf{w}_{n}
  16. e ( n ) e(n)
  17. d ( n ) d(n)
  18. d ^ ( n ) \hat{d}(n)
  19. e ( n ) = d ( n ) - d ^ ( n ) e(n)=d(n)-\hat{d}(n)
  20. C C
  21. C ( 𝐰 n ) = ∑ i = 0 n λ n - i e 2 ( i ) C(\mathbf{w}_{n})=\sum_{i=0}^{n}\lambda^{n-i}e^{2}(i)
  22. 0 < λ ≀ 1 0<\lambda\leq 1
  23. k k
  24. 𝐰 n \mathbf{w}_{n}
  25. ∂ C ( 𝐰 n ) ∂ w n ( k ) = ∑ i = 0 n 2 λ n - i e ( i ) ∂ e ( i ) ∂ w n ( k ) = - ∑ i = 0 n 2 λ n - i e ( i ) x ( i - k ) = 0 k = 0 , 1 , ⋯ , p \frac{\partial C(\mathbf{w}_{n})}{\partial w_{n}(k)}=\sum_{i=0}^{n}\,2\lambda^% {n-i}e(i)\,\frac{\partial e(i)}{\partial w_{n}(k)}={-}\sum_{i=0}^{n}\,2\lambda% ^{n-i}e(i)\,x(i-k)=0\qquad k=0,1,\cdots,p
  26. e ( n ) e(n)
  27. ∑ i = 0 n λ n - i [ d ( i ) - ∑ l = 0 p w n ( l ) x ( i - l ) ] x ( i - k ) = 0 k = 0 , 1 , ⋯ , p \sum_{i=0}^{n}\lambda^{n-i}\left[d(i)-\sum_{l=0}^{p}w_{n}(l)x(i-l)\right]x(i-k% )=0\qquad k=0,1,\cdots,p
  28. ∑ l = 0 p w n ( l ) [ ∑ i = 0 n λ n - i x ( i - l ) x ( i - k ) ] = ∑ i = 0 n λ n - i d ( i ) x ( i - k ) k = 0 , 1 , ⋯ , p \sum_{l=0}^{p}w_{n}(l)\left[\sum_{i=0}^{n}\lambda^{n-i}\,x(i-l)x(i-k)\right]=% \sum_{i=0}^{n}\lambda^{n-i}d(i)x(i-k)\qquad k=0,1,\cdots,p
  29. 𝐑 x ( n ) 𝐰 n = đ« d x ( n ) \mathbf{R}_{x}(n)\,\mathbf{w}_{n}=\mathbf{r}_{dx}(n)
  30. 𝐑 x ( n ) \mathbf{R}_{x}(n)
  31. x ( n ) x(n)
  32. đ« d x ( n ) \mathbf{r}_{dx}(n)
  33. d ( n ) d(n)
  34. x ( n ) x(n)
  35. 𝐰 n = 𝐑 x - 1 ( n ) đ« d x ( n ) \mathbf{w}_{n}=\mathbf{R}_{x}^{-1}(n)\,\mathbf{r}_{dx}(n)
  36. λ \lambda
  37. λ \lambda
  38. λ = 1 \lambda=1
  39. λ \lambda
  40. 𝐰 n = 𝐰 n - 1 + Δ 𝐰 n - 1 \mathbf{w}_{n}=\mathbf{w}_{n-1}+\Delta\mathbf{w}_{n-1}
  41. Δ 𝐰 n - 1 \Delta\mathbf{w}_{n-1}
  42. n - 1 {n-1}
  43. đ« d x ( n ) \mathbf{r}_{dx}(n)
  44. đ« d x ( n - 1 ) \mathbf{r}_{dx}(n-1)
  45. đ« d x ( n ) \mathbf{r}_{dx}(n)
  46. = ∑ i = 0 n λ n - i d ( i ) đ± ( i ) =\sum_{i=0}^{n}\lambda^{n-i}d(i)\mathbf{x}(i)
  47. = ∑ i = 0 n - 1 λ n - i d ( i ) đ± ( i ) + λ 0 d ( n ) đ± ( n ) =\sum_{i=0}^{n-1}\lambda^{n-i}d(i)\mathbf{x}(i)+\lambda^{0}d(n)\mathbf{x}(n)
  48. = λ đ« d x ( n - 1 ) + d ( n ) đ± ( n ) =\lambda\mathbf{r}_{dx}(n-1)+d(n)\mathbf{x}(n)
  49. đ± ( i ) \mathbf{x}(i)
  50. p + 1 {p+1}
  51. đ± ( i ) = [ x ( i ) , x ( i - 1 ) , 
 , x ( i - p ) ] T \mathbf{x}(i)=[x(i),x(i-1),\dots,x(i-p)]^{T}
  52. 𝐑 x ( n ) \mathbf{R}_{x}(n)
  53. 𝐑 x ( n - 1 ) \mathbf{R}_{x}(n-1)
  54. 𝐑 x ( n ) \mathbf{R}_{x}(n)
  55. = ∑ i = 0 n λ n - i đ± ( i ) đ± T ( i ) =\sum_{i=0}^{n}\lambda^{n-i}\mathbf{x}(i)\mathbf{x}^{T}(i)
  56. = λ 𝐑 x ( n - 1 ) + đ± ( n ) đ± T ( n ) =\lambda\mathbf{R}_{x}(n-1)+\mathbf{x}(n)\mathbf{x}^{T}(n)
  57. A A
  58. = λ 𝐑 x ( n - 1 ) =\lambda\mathbf{R}_{x}(n-1)
  59. ( p + 1 ) (p+1)
  60. ( p + 1 ) (p+1)
  61. U U
  62. = đ± ( n ) =\mathbf{x}(n)
  63. ( p + 1 ) (p+1)
  64. V V
  65. = đ± T ( n ) =\mathbf{x}^{T}(n)
  66. ( p + 1 ) (p+1)
  67. C C
  68. = 𝐈 1 =\mathbf{I}_{1}
  69. 𝐑 x - 1 ( n ) \mathbf{R}_{x}^{-1}(n)
  70. = =
  71. [ λ 𝐑 x ( n - 1 ) + đ± ( n ) đ± T ( n ) ] - 1 \left[\lambda\mathbf{R}_{x}(n-1)+\mathbf{x}(n)\mathbf{x}^{T}(n)\right]^{-1}
  72. = =
  73. λ - 1 𝐑 x - 1 ( n - 1 ) \lambda^{-1}\mathbf{R}_{x}^{-1}(n-1)
  74. - λ - 1 𝐑 x - 1 ( n - 1 ) đ± ( n ) -\lambda^{-1}\mathbf{R}_{x}^{-1}(n-1)\mathbf{x}(n)
  75. { 1 + đ± T ( n ) λ - 1 𝐑 x - 1 ( n - 1 ) đ± ( n ) } - 1 đ± T ( n ) λ - 1 𝐑 x - 1 ( n - 1 ) \left\{1+\mathbf{x}^{T}(n)\lambda^{-1}\mathbf{R}_{x}^{-1}(n-1)\mathbf{x}(n)% \right\}^{-1}\mathbf{x}^{T}(n)\lambda^{-1}\mathbf{R}_{x}^{-1}(n-1)
  76. 𝐏 ( n ) \mathbf{P}(n)
  77. = 𝐑 x - 1 ( n ) =\mathbf{R}_{x}^{-1}(n)
  78. = λ - 1 𝐏 ( n - 1 ) - 𝐠 ( n ) đ± T ( n ) λ - 1 𝐏 ( n - 1 ) =\lambda^{-1}\mathbf{P}(n-1)-\mathbf{g}(n)\mathbf{x}^{T}(n)\lambda^{-1}\mathbf% {P}(n-1)
  79. g ( n ) g(n)
  80. 𝐠 ( n ) \mathbf{g}(n)
  81. = λ - 1 𝐏 ( n - 1 ) đ± ( n ) { 1 + đ± T ( n ) λ - 1 𝐏 ( n - 1 ) đ± ( n ) } - 1 =\lambda^{-1}\mathbf{P}(n-1)\mathbf{x}(n)\left\{1+\mathbf{x}^{T}(n)\lambda^{-1% }\mathbf{P}(n-1)\mathbf{x}(n)\right\}^{-1}
  82. = 𝐏 ( n - 1 ) đ± ( n ) { λ + đ± T ( n ) 𝐏 ( n - 1 ) đ± ( n ) } - 1 =\mathbf{P}(n-1)\mathbf{x}(n)\left\{\lambda+\mathbf{x}^{T}(n)\mathbf{P}(n-1)% \mathbf{x}(n)\right\}^{-1}
  83. 𝐠 ( n ) \mathbf{g}(n)
  84. 𝐠 ( n ) { 1 + đ± T ( n ) λ - 1 𝐏 ( n - 1 ) đ± ( n ) } \mathbf{g}(n)\left\{1+\mathbf{x}^{T}(n)\lambda^{-1}\mathbf{P}(n-1)\mathbf{x}(n% )\right\}
  85. = λ - 1 𝐏 ( n - 1 ) đ± ( n ) =\lambda^{-1}\mathbf{P}(n-1)\mathbf{x}(n)
  86. 𝐠 ( n ) + 𝐠 ( n ) đ± T ( n ) λ - 1 𝐏 ( n - 1 ) đ± ( n ) \mathbf{g}(n)+\mathbf{g}(n)\mathbf{x}^{T}(n)\lambda^{-1}\mathbf{P}(n-1)\mathbf% {x}(n)
  87. = λ - 1 𝐏 ( n - 1 ) đ± ( n ) =\lambda^{-1}\mathbf{P}(n-1)\mathbf{x}(n)
  88. 𝐠 ( n ) \mathbf{g}(n)
  89. = λ - 1 𝐏 ( n - 1 ) đ± ( n ) - 𝐠 ( n ) đ± T ( n ) λ - 1 𝐏 ( n - 1 ) đ± ( n ) =\lambda^{-1}\mathbf{P}(n-1)\mathbf{x}(n)-\mathbf{g}(n)\mathbf{x}^{T}(n)% \lambda^{-1}\mathbf{P}(n-1)\mathbf{x}(n)
  90. = λ - 1 [ 𝐏 ( n - 1 ) - 𝐠 ( n ) đ± T ( n ) 𝐏 ( n - 1 ) ] đ± ( n ) =\lambda^{-1}\left[\mathbf{P}(n-1)-\mathbf{g}(n)\mathbf{x}^{T}(n)\mathbf{P}(n-% 1)\right]\mathbf{x}(n)
  91. 𝐏 ( n ) \mathbf{P}(n)
  92. 𝐠 ( n ) = 𝐏 ( n ) đ± ( n ) \mathbf{g}(n)=\mathbf{P}(n)\mathbf{x}(n)
  93. 𝐰 n \mathbf{w}_{n}
  94. = 𝐏 ( n ) đ« d x ( n ) =\mathbf{P}(n)\,\mathbf{r}_{dx}(n)
  95. = λ 𝐏 ( n ) đ« d x ( n - 1 ) + d ( n ) 𝐏 ( n ) đ± ( n ) =\lambda\mathbf{P}(n)\,\mathbf{r}_{dx}(n-1)+d(n)\mathbf{P}(n)\,\mathbf{x}(n)
  96. đ« d x ( n ) \mathbf{r}_{dx}(n)
  97. 𝐏 ( n ) \mathbf{P}(n)
  98. 𝐠 ( n ) \mathbf{g}(n)
  99. 𝐰 n \mathbf{w}_{n}
  100. = λ [ λ - 1 𝐏 ( n - 1 ) - 𝐠 ( n ) đ± T ( n ) λ - 1 𝐏 ( n - 1 ) ] đ« d x ( n - 1 ) + d ( n ) 𝐠 ( n ) =\lambda\left[\lambda^{-1}\mathbf{P}(n-1)-\mathbf{g}(n)\mathbf{x}^{T}(n)% \lambda^{-1}\mathbf{P}(n-1)\right]\mathbf{r}_{dx}(n-1)+d(n)\mathbf{g}(n)
  101. = 𝐏 ( n - 1 ) đ« d x ( n - 1 ) - 𝐠 ( n ) đ± T ( n ) 𝐏 ( n - 1 ) đ« d x ( n - 1 ) + d ( n ) 𝐠 ( n ) =\mathbf{P}(n-1)\mathbf{r}_{dx}(n-1)-\mathbf{g}(n)\mathbf{x}^{T}(n)\mathbf{P}(% n-1)\mathbf{r}_{dx}(n-1)+d(n)\mathbf{g}(n)
  102. = 𝐏 ( n - 1 ) đ« d x ( n - 1 ) + 𝐠 ( n ) [ d ( n ) - đ± T ( n ) 𝐏 ( n - 1 ) đ« d x ( n - 1 ) ] =\mathbf{P}(n-1)\mathbf{r}_{dx}(n-1)+\mathbf{g}(n)\left[d(n)-\mathbf{x}^{T}(n)% \mathbf{P}(n-1)\mathbf{r}_{dx}(n-1)\right]
  103. 𝐰 n - 1 = 𝐏 ( n - 1 ) đ« d x ( n - 1 ) \mathbf{w}_{n-1}=\mathbf{P}(n-1)\mathbf{r}_{dx}(n-1)
  104. 𝐰 n \mathbf{w}_{n}
  105. = 𝐰 n - 1 + 𝐠 ( n ) [ d ( n ) - đ± T ( n ) 𝐰 n - 1 ] =\mathbf{w}_{n-1}+\mathbf{g}(n)\left[d(n)-\mathbf{x}^{T}(n)\mathbf{w}_{n-1}\right]
  106. = 𝐰 n - 1 + 𝐠 ( n ) α ( n ) =\mathbf{w}_{n-1}+\mathbf{g}(n)\alpha(n)
  107. α ( n ) = d ( n ) - đ± T ( n ) 𝐰 n - 1 \alpha(n)=d(n)-\mathbf{x}^{T}(n)\mathbf{w}_{n-1}
  108. e ( n ) = d ( n ) - đ± T ( n ) 𝐰 n e(n)=d(n)-\mathbf{x}^{T}(n)\mathbf{w}_{n}
  109. Δ 𝐰 n - 1 = 𝐠 ( n ) α ( n ) \Delta\mathbf{w}_{n-1}=\mathbf{g}(n)\alpha(n)
  110. λ \lambda
  111. p = p=
  112. λ = \lambda=
  113. ÎŽ = \delta=
  114. 𝐏 ( 0 ) \mathbf{P}(0)
  115. 𝐰 ( n ) = 0 \mathbf{w}(n)=0
  116. x ( k ) = 0 , k = - p , 
 , - 1 x(k)=0,k=-p,\dots,-1
  117. d ( k ) = 0 , k = - p , 
 , - 1 d(k)=0,k=-p,\dots,-1
  118. 𝐏 ( 0 ) = ή - 1 I \mathbf{P}(0)=\delta^{-1}I
  119. I I
  120. p + 1 p+1
  121. n = 1 , 2 , 
 n=1,2,\dots
  122. đ± ( n ) = [ x ( n ) x ( n - 1 ) ⋼ x ( n - p ) ] \mathbf{x}(n)=\left[\begin{matrix}x(n)\\ x(n-1)\\ \vdots\\ x(n-p)\end{matrix}\right]
  123. α ( n ) = d ( n ) - đ± T ( n ) 𝐰 ( n - 1 ) \alpha(n)=d(n)-\mathbf{x}^{T}(n)\mathbf{w}(n-1)
  124. 𝐠 ( n ) = 𝐏 ( n - 1 ) đ± * ( n ) { λ + đ± T ( n ) 𝐏 ( n - 1 ) đ± * ( n ) } - 1 \mathbf{g}(n)=\mathbf{P}(n-1)\mathbf{x}^{*}(n)\left\{\lambda+\mathbf{x}^{T}(n)% \mathbf{P}(n-1)\mathbf{x}^{*}(n)\right\}^{-1}
  125. 𝐏 ( n ) = λ - 1 𝐏 ( n - 1 ) - 𝐠 ( n ) đ± T ( n ) λ - 1 𝐏 ( n - 1 ) \mathbf{P}(n)=\lambda^{-1}\mathbf{P}(n-1)-\mathbf{g}(n)\mathbf{x}^{T}(n)% \lambda^{-1}\mathbf{P}(n-1)
  126. 𝐰 ( n ) = 𝐰 ( n - 1 ) + α ( n ) 𝐠 ( n ) \mathbf{w}(n)=\mathbf{w}(n-1)+\,\alpha(n)\mathbf{g}(n)
  127. P P
  128. d ( k ) d(k)\,\!
  129. d ( k ) = x ( k ) d(k)=x(k)\,\!
  130. x ( k - 1 ) x(k-1)\,\!
  131. d ( k ) = x ( k - i - 1 ) d(k)=x(k-i-1)\,\!
  132. x ( k ) x(k)\,\!
  133. Îș f ( k , i ) \kappa_{f}(k,i)\,\!
  134. Îș b ( k , i ) \kappa_{b}(k,i)\,\!
  135. e f ( k , i ) e_{f}(k,i)\,\!
  136. e b ( k , i ) e_{b}(k,i)\,\!
  137. Ο b m i n d ( k , i ) \xi^{d}_{b_{min}}(k,i)\,\!
  138. Ο f m i n d ( k , i ) \xi^{d}_{f_{min}}(k,i)\,\!
  139. Îł ( k , i ) \gamma(k,i)\,\!
  140. v i ( k ) v_{i}(k)\,\!
  141. Ï” \epsilon\,\!
  142. ÎŽ ( - 1 , i ) = ÎŽ D ( - 1 , i ) = 0 \delta(-1,i)=\delta_{D}(-1,i)=0\,\!
  143. Ο b m i n d ( - 1 , i ) = Ο f m i n d ( - 1 , i ) = ϔ \xi^{d}_{b_{min}}(-1,i)=\xi^{d}_{f_{min}}(-1,i)=\epsilon
  144. Îł ( - 1 , i ) = 1 \gamma(-1,i)=1\,\!
  145. e b ( - 1 , i ) = 0 e_{b}(-1,i)=0\,\!
  146. Îł ( k , 0 ) = 1 \gamma(k,0)=1\,\!
  147. e b ( k , 0 ) = e f ( k , 0 ) = x ( k ) e_{b}(k,0)=e_{f}(k,0)=x(k)\,\!
  148. Ο b m i n d ( k , 0 ) = Ο f m i n d ( k , 0 ) = x 2 ( k ) + λ Ο f m i n d ( k - 1 , 0 ) \xi^{d}_{b_{min}}(k,0)=\xi^{d}_{f_{min}}(k,0)=x^{2}(k)+\lambda\xi^{d}_{f_{min}% }(k-1,0)\,\!
  149. e ( k , 0 ) = d ( k ) e(k,0)=d(k)\,\!
  150. Ύ ( k , i ) = λ Ύ ( k - 1 , i ) + e b ( k - 1 , i ) e f ( k , i ) γ ( k - 1 , i ) \delta(k,i)=\lambda\delta(k-1,i)+\frac{e_{b}(k-1,i)e_{f}(k,i)}{\gamma(k-1,i)}
  151. Îł ( k , i + 1 ) = Îł ( k , i ) - e b 2 ( k , i ) Ο b m i n d ( k , i ) \gamma(k,i+1)=\gamma(k,i)-\frac{e_{b}^{2}(k,i)}{\xi^{d}_{b_{min}}(k,i)}
  152. Îș b ( k , i ) = ÎŽ ( k , i ) Ο f m i n d ( k , i ) \kappa_{b}(k,i)=\frac{\delta(k,i)}{\xi^{d}_{f_{min}}(k,i)}
  153. Îș f ( k , i ) = ÎŽ ( k , i ) Ο b m i n d ( k - 1 , i ) \kappa_{f}(k,i)=\frac{\delta(k,i)}{\xi^{d}_{b_{min}}(k-1,i)}
  154. e b ( k , i + 1 ) = e b ( k - 1 , i ) - Îș b ( k , i ) e f ( k , i ) e_{b}(k,i+1)=e_{b}(k-1,i)-\kappa_{b}(k,i)e_{f}(k,i)\,\!
  155. e f ( k , i + 1 ) = e f ( k , i ) - Îș f ( k , i ) e b ( k - 1 , i ) e_{f}(k,i+1)=e_{f}(k,i)-\kappa_{f}(k,i)e_{b}(k-1,i)\,\!
  156. Ο b m i n d ( k , i + 1 ) = Ο b m i n d ( k - 1 , i ) - ÎŽ ( k , i ) Îș b ( k , i ) \xi^{d}_{b_{min}}(k,i+1)=\xi^{d}_{b_{min}}(k-1,i)-\delta(k,i)\kappa_{b}(k,i)
  157. Ο f m i n d ( k , i + 1 ) = Ο f m i n d ( k , i ) - ÎŽ ( k , i ) Îș f ( k , i ) \xi^{d}_{f_{min}}(k,i+1)=\xi^{d}_{f_{min}}(k,i)-\delta(k,i)\kappa_{f}(k,i)
  158. Ύ D ( k , i ) = λ Ύ D ( k - 1 , i ) + e ( k , i ) e b ( k , i ) γ ( k , i ) \delta_{D}(k,i)=\lambda\delta_{D}(k-1,i)+\frac{e(k,i)e_{b}(k,i)}{\gamma(k,i)}
  159. v i ( k ) = Ύ D ( k , i ) Ο b m i n d ( k , i ) v_{i}(k)=\frac{\delta_{D}(k,i)}{\xi^{d}_{b_{min}}(k,i)}
  160. e ( k , i + 1 ) = e ( k , i ) - v i ( k ) e b ( k , i ) e(k,i+1)=e(k,i)-v_{i}(k)e_{b}(k,i)\,\!
  161. ÎŽ ÂŻ ( - 1 , i ) = 0 \overline{\delta}(-1,i)=0\,\!
  162. ÎŽ ÂŻ D ( - 1 , i ) = 0 \overline{\delta}_{D}(-1,i)=0\,\!
  163. e ÂŻ b ( - 1 , i ) = 0 \overline{e}_{b}(-1,i)=0\,\!
  164. σ x 2 ( - 1 ) = λ σ d 2 ( - 1 ) = Ï” \sigma_{x}^{2}(-1)=\lambda\sigma_{d}^{2}(-1)=\epsilon\,\!
  165. σ x 2 ( k ) = λ σ x 2 ( k - 1 ) + x 2 ( k ) \sigma_{x}^{2}(k)=\lambda\sigma_{x}^{2}(k-1)+x^{2}(k)\,\!
  166. σ d 2 ( k ) = λ σ d 2 ( k - 1 ) + d 2 ( k ) \sigma_{d}^{2}(k)=\lambda\sigma_{d}^{2}(k-1)+d^{2}(k)\,\!
  167. e ÂŻ b ( k , 0 ) = e ÂŻ f ( k , 0 ) = x ( k ) σ x ( k ) \overline{e}_{b}(k,0)=\overline{e}_{f}(k,0)=\frac{x(k)}{\sigma_{x}(k)}\,\!
  168. e ÂŻ ( k , 0 ) = d ( k ) σ d ( k ) \overline{e}(k,0)=\frac{d(k)}{\sigma_{d}(k)}\,\!
  169. ÎŽ ÂŻ ( k , i ) = ÎŽ ( k - 1 , i ) ( 1 - e ÂŻ b 2 ( k - 1 , i ) ) ( 1 - e ÂŻ f 2 ( k , i ) ) + e ÂŻ b ( k - 1 , i ) e ÂŻ f ( k , i ) \overline{\delta}(k,i)=\delta(k-1,i)\sqrt{(1-\overline{e}_{b}^{2}(k-1,i))(1-% \overline{e}_{f}^{2}(k,i))}+\overline{e}_{b}(k-1,i)\overline{e}_{f}(k,i)
  170. e ÂŻ b ( k , i + 1 ) = e ÂŻ b ( k - 1 , i ) - ÎŽ ÂŻ ( k , i ) e ÂŻ f ( k , i ) ( 1 - ÎŽ ÂŻ 2 ( k , i ) ) ( 1 - e ÂŻ f 2 ( k , i ) ) \overline{e}_{b}(k,i+1)=\frac{\overline{e}_{b}(k-1,i)-\overline{\delta}(k,i)% \overline{e}_{f}(k,i)}{\sqrt{(1-\overline{\delta}^{2}(k,i))(1-\overline{e}_{f}% ^{2}(k,i))}}
  171. e ÂŻ f ( k , i + 1 ) = e ÂŻ f ( k , i ) - ÎŽ ÂŻ ( k , i ) e ÂŻ b ( k - 1 , i ) ( 1 - ÎŽ ÂŻ 2 ( k , i ) ) ( 1 - e ÂŻ b 2 ( k - 1 , i ) ) \overline{e}_{f}(k,i+1)=\frac{\overline{e}_{f}(k,i)-\overline{\delta}(k,i)% \overline{e}_{b}(k-1,i)}{\sqrt{(1-\overline{\delta}^{2}(k,i))(1-\overline{e}_{% b}^{2}(k-1,i))}}
  172. ÎŽ ÂŻ D ( k , i ) = ÎŽ ÂŻ D ( k - 1 , i ) ( 1 - e ÂŻ b 2 ( k , i ) ) ( 1 - e ÂŻ 2 ( k , i ) ) + e ÂŻ ( k , i ) e ÂŻ b ( k , i ) \overline{\delta}_{D}(k,i)=\overline{\delta}_{D}(k-1,i)\sqrt{(1-\overline{e}_{% b}^{2}(k,i))(1-\overline{e}^{2}(k,i))}+\overline{e}(k,i)\overline{e}_{b}(k,i)
  173. e ÂŻ ( k , i + 1 ) = 1 ( 1 - e ÂŻ b 2 ( k , i ) ) ( 1 - ÎŽ ÂŻ D 2 ( k , i ) ) [ e ÂŻ ( k , i ) - ÎŽ ÂŻ D ( k , i ) e ÂŻ b ( k , i ) ] \overline{e}(k,i+1)=\frac{1}{\sqrt{(1-\overline{e}_{b}^{2}(k,i))(1-\overline{% \delta}_{D}^{2}(k,i))}}[\overline{e}(k,i)-\overline{\delta}_{D}(k,i)\overline{% e}_{b}(k,i)]

Red_drum.html

  1. W = a L b W=aL^{b}\!\,
  2. S p r i n g : W = 0.000005297 L 3.110 Spring:W=0.000005297L^{3.110}\!\,
  3. F a l l : W = 0.000015241 L 2.94 Fall:W=0.000015241L^{2.94}\!\,

Reduced_form.html

  1. Δ \varepsilon
  2. f ( Y , X , Δ ) = 0 f(Y,X,\varepsilon)=0
  3. Y = g ( X , Δ ) Y=g(X,\varepsilon)
  4. Q = a S + b S P Q=a_{S}+b_{S}P\,
  5. Q = a D + b D P Q=a_{D}+b_{D}P\,
  6. a S + b S P = a D + b D P a_{S}+b_{S}P=a_{D}+b_{D}P
  7. P ( b S - b D ) = a D - a S P(b_{S}-b_{D})=a_{D}-a_{S}
  8. P = ( a D - a S ) / ( b S - b D ) P=(a_{D}-a_{S})/(b_{S}-b_{D})
  9. π 2 \pi_{2}
  10. π 1 \pi_{1}
  11. Q = π 1 Q=\pi_{1}\,
  12. P = π 2 P=\pi_{2}\,
  13. π \pi
  14. π 1 = ( a D b S - a S b D ) / ( b S - b D ) \pi_{1}=(a_{D}b_{S}-a_{S}b_{D})/(b_{S}-b_{D})\,
  15. π 2 = ( a D - a S ) / ( b S - b D ) \pi_{2}=(a_{D}-a_{S})/(b_{S}-b_{D})\,
  16. Q = a S + b S P Q=a_{S}+b_{S}P\,
  17. Q = a D + b D P + c Z Q=a_{D}+b_{D}P+cZ\,
  18. Q = π 10 + π 11 Z Q=\pi_{10}+\pi_{11}Z\,
  19. P = π 20 + π 21 Z P=\pi_{20}+\pi_{21}Z\,
  20. π \pi
  21. a S a_{S}
  22. b S b_{S}
  23. π 10 \pi_{10}
  24. π 11 \pi_{11}
  25. π 20 \pi_{20}
  26. π 21 \pi_{21}
  27. a S = ( π 10 π 21 - π 11 π 20 ) / π 21 a_{S}=(\pi_{10}\pi_{21}-\pi_{11}\pi_{20})/\pi_{21}
  28. b S = π 11 / π 21 b_{S}=\pi_{11}/\pi_{21}
  29. A y = B x Ay=Bx\,
  30. y = Π x y=\Pi x\,
  31. Π = A - 1 B \Pi=A^{-1}B\,

Reduction_of_the_structure_group.html

  1. G G
  2. G G
  3. H H
  4. G G
  5. H H
  6. H → G H\to G
  7. B H B_{H}
  8. B H × H G B_{H}\times^{H}G
  9. G L + < G L GL^{+}<GL
  10. S L < G L SL<GL
  11. S L → G L + SL\to GL^{+}
  12. S L ± < G L SL^{\pm}<GL
  13. O ( n ) < G L ( n ) O(n)<GL(n)
  14. O ( n ) O(n)
  15. O ( 1 , n - 1 ) < G L ( n ) O(1,n-1)<GL(n)
  16. G L ( n , 𝐂 ) < G L ( 2 n , 𝐑 ) GL(n,\mathbf{C})<GL(2n,\mathbf{R})
  17. G L ( n , 𝐇 ) ⋅ S p ( 1 ) < G L ( 4 n , 𝐑 ) GL(n,\mathbf{H})\cdot Sp(1)<GL(4n,\mathbf{R})
  18. G L ( n , 𝐇 ) GL(n,\mathbf{H})
  19. 𝐇 n ≅ 𝐑 4 n \mathbf{H}^{n}\cong\mathbf{R}^{4n}
  20. 𝐇 n \mathbf{H}^{n}
  21. Spin ( n ) → SO ( n ) \mbox{Spin}~{}(n)\to\mbox{SO}~{}(n)
  22. G L ( k ) × G L ( n - k ) < G L ( n ) GL(k)\times GL(n-k)<GL(n)
  23. σ \sigma

Redundancy_(engineering).html

  1. p = ∏ i = 1 n p i {p}=\prod_{i=1}^{n}p_{i}
  2. n n
  3. p i p_{i}
  4. p p

Redundancy_(information_theory).html

  1. r = lim n → ∞ 1 n H ( M 1 , M 2 , 
 M n ) , r=\lim_{n\to\infty}\frac{1}{n}H(M_{1},M_{2},\dots M_{n}),
  2. H ( M ) H(M)
  3. R = log | 𝕄 | , R=\log|\mathbb{M}|,\,
  4. D = R - r , D=R-r,\,
  5. D R \frac{D}{R}
  6. R : r R:r
  7. r R , \frac{r}{R},
  8. r R + D R = 1 \frac{r}{R}+\frac{D}{R}=1
  9. n n
  10. L ( M n ) L(M^{n})\,\!
  11. L ( M n ) / n L(M^{n})/n\,\!
  12. n r nr\,\!
  13. r r\,\!
  14. L ( M n ) / n - r L(M^{n})/n-r\,\!
  15. n n\,\!
  16. L ( M n ) - n r L(M^{n})-nr\,\!

Reed–Muller_code.html

  1. R = 1 n {R=\tfrac{1}{n}}
  2. R = r + 1 n R=\tfrac{r+1}{n}
  3. d min = n 2 d_{\min}=\tfrac{n}{2}
  4. X = đ”œ 2 d = { x 1 , 
 , x 2 d } . X=\mathbb{F}_{2}^{d}=\{x_{1},\ldots,x_{2^{d}}\}.
  5. đ”œ 2 d \mathbb{F}_{2}^{d}
  6. đ”œ 2 n \mathbb{F}_{2}^{n}
  7. 𝕀 A ∈ đ”œ 2 n \mathbb{I}_{A}\in\mathbb{F}_{2}^{n}
  8. A ⊂ X A\subset X
  9. ( 𝕀 A ) i = { 1 if x i ∈ A 0 otherwise \left(\mathbb{I}_{A}\right)_{i}=\begin{cases}1&\mbox{ if }~{}x_{i}\in A\\ 0&\mbox{ otherwise}\\ \end{cases}
  10. đ”œ 2 n \mathbb{F}_{2}^{n}
  11. w ∧ z = ( w 1 ⋅ z 1 , 
 , w n ⋅ z n ) , w\wedge z=(w_{1}\cdot z_{1},\ldots,w_{n}\cdot z_{n}),
  12. w = ( w 1 , w 2 , 
 , w n ) w=(w_{1},w_{2},\ldots,w_{n})
  13. z = ( z 1 , z 2 , 
 , z n ) z=(z_{1},z_{2},\ldots,z_{n})
  14. đ”œ 2 n \mathbb{F}_{2}^{n}
  15. ⋅ \cdot
  16. đ”œ 2 \mathbb{F}_{2}
  17. đ”œ 2 d \mathbb{F}_{2}^{d}
  18. đ”œ 2 \mathbb{F}_{2}
  19. ( đ”œ 2 ) d = { ( y d , 
 , y 1 ) | y i ∈ đ”œ 2 } (\mathbb{F}_{2})^{d}=\{(y_{d},\ldots,y_{1})|y_{i}\in\mathbb{F}_{2}\}
  20. đ”œ 2 n \mathbb{F}_{2}^{n}
  21. v i = 𝕀 H i v_{i}=\mathbb{I}_{H_{i}}
  22. ( đ”œ 2 ) d (\mathbb{F}_{2})^{d}
  23. H i = { y ∈ ( đ”œ 2 ) d ∣ y i = 0 } H_{i}=\{y\in(\mathbb{F}_{2})^{d}\mid y_{i}=0\}
  24. v 0 , v 1 , 
 , v d , 
 , ( v i ∧ v j ) , 
 ( v i ∧ v j 
 ∧ v r ) {v_{0},v_{1},\ldots,v_{d},\ldots,(v_{i}\wedge v_{j}),\ldots(v_{i}\wedge v_{j}% \ldots\wedge v_{r})}
  25. X = đ”œ 2 3 = { ( 0 , 0 , 0 ) , ( 0 , 0 , 1 ) , 
 , ( 1 , 1 , 1 ) } , X=\mathbb{F}_{2}^{3}=\{(0,0,0),(0,0,1),\ldots,(1,1,1)\},
  26. v 0 = ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) v 1 = ( 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 ) v 2 = ( 1 , 1 , 0 , 0 , 1 , 1 , 0 , 0 ) v 3 = ( 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 ) . \begin{matrix}v_{0}&=&(1,1,1,1,1,1,1,1)\\ v_{1}&=&(1,0,1,0,1,0,1,0)\\ v_{2}&=&(1,1,0,0,1,1,0,0)\\ v_{3}&=&(1,1,1,1,0,0,0,0).\\ \end{matrix}
  27. { v 0 , v 1 , v 2 , v 3 } , \{v_{0},v_{1},v_{2},v_{3}\},\,
  28. ( 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0 ) \begin{pmatrix}1&1&1&1&1&1&1&1\\ 1&0&1&0&1&0&1&0\\ 1&1&0&0&1&1&0&0\\ 1&1&1&1&0&0&0&0\\ \end{pmatrix}
  29. { v 0 , v 1 , v 2 , v 3 , v 1 ∧ v 2 , v 1 ∧ v 3 , v 2 ∧ v 3 } \{v_{0},v_{1},v_{2},v_{3},v_{1}\wedge v_{2},v_{1}\wedge v_{3},v_{2}\wedge v_{3}\}
  30. ( 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 ) \begin{pmatrix}1&1&1&1&1&1&1&1\\ 1&0&1&0&1&0&1&0\\ 1&1&0&0&1&1&0&0\\ 1&1&1&1&0&0&0&0\\ 1&0&0&0&1&0&0&0\\ 1&0&1&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ \end{pmatrix}
  31. đ”œ 2 n \mathbb{F}_{2}^{n}
  32. ∑ s = 0 r ( d s ) . \sum_{s=0}^{r}{d\choose s}.
  33. ∑ s = 0 d ( d s ) = 2 d = n \sum_{s=0}^{d}{d\choose s}=2^{d}=n
  34. đ”œ 2 n \mathbb{F}_{2}^{n}
  35. đ”œ 2 n \mathbb{F}_{2}^{n}
  36. y i = { v i if x i = 0 1 + v i if x i = 1 y_{i}=\begin{cases}v_{i}&\mbox{ if }~{}x_{i}=0\\ 1+v_{i}&\mbox{ if }~{}x_{i}=1\\ \end{cases}
  37. 𝕀 { x } = y i ∧ 
 ∧ y d \mathbb{I}_{\{x\}}=y_{i}\wedge\ldots\wedge y_{d}
  38. 𝕀 { x } ∈ RM(d,d) \mathbb{I}_{\{x\}}\in\mbox{ RM(d,d)}~{}
  39. { 𝕀 { x } ∣ x ∈ X } \{\mathbb{I}_{\{x\}}\mid x\in X\}
  40. đ”œ 2 n \mathbb{F}_{2}^{n}
  41. đ”œ 2 n \mathbb{F}_{2}^{n}
  42. đ”œ 2 n \mathbb{F}_{2}^{n}
  43. min { 2 w ( C 1 ) , w ( C 2 ) } \min\{2w(C_{1}),w(C_{2})\}
  44. min { 2 × 2 d - 1 - r , 2 d - r } = 2 d - r . \min\{2\times 2^{d-1-r},2^{d-r}\}=2^{d-r}.
  45. m ≄ 0 m\geq 0
  46. 0 ≀ r ≀ m 0\leq r\leq m
  47. 2 m , 2 m , 1 2^{m},2^{m},1
  48. 2 m , 0 , ∞ 2^{m},0,\infty
  49. R M ( r , m ) = { ( 𝐼 , 𝐼 + 𝐯 ) | 𝐼 ∈ R M ( r , m - 1 ) , 𝐯 ∈ R M ( r - 1 , m - 1 ) } . RM(r,m)=\{(\mathbf{u},\mathbf{u}+\mathbf{v})|\mathbf{u}\in RM(r,m-1),\mathbf{v% }\in RM(r-1,m-1)\}.
  50. k ( r , m ) = k ( r , m - 1 ) + k ( r - 1 , m - 1 ) k(r,m)=k(r,m-1)+k(r-1,m-1)
  51. d = 2 m - r d=2^{m-r}
  52. r ≄ 0 r\geq 0
  53. đ”œ \mathbb{F}
  54. m m
  55. d d
  56. m m
  57. d d
  58. ( m + d m ) {m+d\choose m}
  59. đ”œ \mathbb{F}
  60. | đ”œ | m |\mathbb{F}|^{m}
  61. m m
  62. f f
  63. d d
  64. đ”œ \mathbb{F}
  65. ( m + d m ) {m+d\choose m}
  66. f f
  67. f f
  68. x ∈ đ”œ m x\in\mathbb{F}^{m}
  69. x ∈ đ”œ m x\in\mathbb{F}^{m}
  70. f ( x ) f(x)
  71. 2 m , 2 m , 1 2^{m},2^{m},1
  72. 2 m , 2 m - 1 , 2 2^{m},2^{m}-1,2
  73. 2 m , 2 m - m - 1 , 4 2^{m},2^{m}-m-1,4
  74. ∞ \infty
  75. ∞ \infty
  76. ∞ \infty
  77. 2 m , m + 1 , 2 m - 1 2^{m},m+1,2^{m-1}
  78. ∞ \infty
  79. ∞ \infty
  80. 2 m , 1 , 2 m 2^{m},1,2^{m}
  81. ∞ \infty
  82. 2 m , 0 , ∞ 2^{m},0,\infty

Reference_class_problem.html

  1. X X
  2. F F
  3. G G
  4. I I
  5. F F
  6. I I
  7. G G
  8. X X
  9. F F
  10. G G
  11. I I
  12. F F

Refinement_(computing).html

  1. f : ℕ → { n : ℕ | n > 5 } f:\mathbb{N}\rightarrow\{n:\mathbb{N}|n>5\}

Regge_theory.html

  1. z z
  2. A ( z ) ∝ z l ( E 2 ) A(z)\propto z^{l(E^{2})}
  3. l ( E 2 ) l(E^{2})
  4. E E
  5. s = E 2 s=E^{2}
  6. l ( s ) l(s)
  7. z z
  8. z z
  9. t t
  10. t t
  11. s s
  12. t t
  13. A ( z ) ∝ s l ( t ) A(z)\propto s^{l(t)}
  14. t t
  15. l ( s ) = k s l(s)=ks
  16. k k

Regularization_(mathematics).html

  1. L ₁ L₁
  2. L ₂ L₂
  3. E ( X , Y ) E(X,Y)
  4. E ( X , Y ) + α ‖ w ‖ E(X,Y)+α‖w‖
  5. w w
  6. L ₁ L₁
  7. L ₂ L₂
  8. L ₂ L₂
  9. L ₁ L₁
  10. L ₁ L₁
  11. ∄ Y - X ÎČ âˆ„ 2 \|Y-X\beta\|_{2}
  12. ∄ ÎČ âˆ„ 0 \|\beta\|_{0}
  13. ∄ Y - X ÎČ âˆ„ 2 \|Y-X\beta\|_{2}
  14. ∄ ÎČ âˆ„ 2 \|\beta\|_{2}
  15. ∄ Y - X ÎČ âˆ„ 2 \|Y-X\beta\|_{2}
  16. ∄ ÎČ âˆ„ 1 \|\beta\|_{1}
  17. ∄ Y - X ÎČ âˆ„ 2 \|Y-X\beta\|_{2}
  18. λ ∄ ÎČ âˆ„ 1 \lambda\|\beta\|_{1}
  19. ∄ Y - X ÎČ âˆ„ 2 \|Y-X\beta\|_{2}
  20. λ ∄ ∇ ÎČ âˆ„ 1 \lambda\|\nabla\beta\|_{1}
  21. ∄ Y - X ÎČ âˆ„ 2 \|Y-X\beta\|_{2}
  22. λ ∄ ∇ ÎČ âˆ„ 0 \lambda\|\nabla\beta\|_{0}
  23. ∄ Y - X ÎČ âˆ„ 1 \|Y-X\beta\|_{1}
  24. ∄ ÎČ âˆ„ 1 \|\beta\|_{1}
  25. ∄ X ⊀ ( Y - X ÎČ ) ∄ ∞ \|X^{\top}(Y-X\beta)\|_{\infty}
  26. ∄ ÎČ âˆ„ 1 \|\beta\|_{1}
  27. ∄ Y - X ÎČ âˆ„ 2 \|Y-X\beta\|_{2}
  28. ∑ i = 1 p λ i | ÎČ | ( i ) \sum_{i=1}^{p}\lambda_{i}|\beta|_{(i)}

Regularization_(physics).html

  1. Ï” \epsilon
  2. Ï” → 0 \epsilon\to 0
  3. 1 / Ï” 1/\epsilon
  4. Ï” → 0 \epsilon\to 0
  5. 1 / Ï” 1/\epsilon
  6. Ï” ≫ 1 / Λ \epsilon\gg 1/\Lambda
  7. 1 / Λ â€Č 1/\Lambda^{\prime}
  8. 1 / Λ â‰Ș Ï” â‰Ș 1 / Λ â€Č 1/\Lambda\ll\epsilon\ll 1/\Lambda^{\prime}

Reilly's_law_of_retail_gravitation.html

  1. d A d B = P A P B \frac{d_{A}}{d_{B}}=\sqrt{\frac{P_{A}}{P_{B}}}
  2. d A d_{A}
  3. d B d_{B}
  4. P A / P B P_{A}/P_{B}
  5. d = D 1 + P B / P A d=\frac{D}{1+\sqrt{P_{B}/P_{A}}}

Related_rates.html

  1. x = 6 x=6
  2. h = 10 h=10
  3. d x d t = 3 \frac{dx}{dt}=3
  4. d h d t = 0 \frac{dh}{dt}=0
  5. d y d t = ? \frac{dy}{dt}=?
  6. x 2 + y 2 = h 2 x^{2}+y^{2}=h^{2}\,
  7. d d t ( x 2 + y 2 ) = d d t ( h 2 ) \frac{d}{dt}(x^{2}+y^{2})=\frac{d}{dt}(h^{2})
  8. d d t ( x 2 ) + d d t ( y 2 ) = d d t ( h 2 ) \frac{d}{dt}(x^{2})+\frac{d}{dt}(y^{2})=\frac{d}{dt}(h^{2})
  9. ( 2 x ) d x d t + ( 2 y ) d y d t = ( 2 h ) d h d t (2x)\frac{dx}{dt}+(2y)\frac{dy}{dt}=(2h)\frac{dh}{dt}
  10. x d x d t + y d y d t = h d h d t x\frac{dx}{dt}+y\frac{dy}{dt}=h\frac{dh}{dt}
  11. d y d t = h d h d t - x d x d t y . \frac{dy}{dt}=\frac{h\frac{dh}{dt}-x\frac{dx}{dt}}{y}.
  12. d y d t = h d h d t - x d x d t y . \frac{dy}{dt}=\frac{h\frac{dh}{dt}-x\frac{dx}{dt}}{y}.
  13. d y d t = 10 × 0 - 6 × 3 y = - 18 y . \frac{dy}{dt}=\frac{10\times 0-6\times 3}{y}=-\frac{18}{y}.
  14. x 2 + y 2 = h 2 x^{2}+y^{2}=h^{2}
  15. 6 2 + y 2 = 10 2 6^{2}+y^{2}=10^{2}
  16. y = 8 y=8
  17. - 18 y = - 18 8 = - 9 4 -\frac{18}{y}=-\frac{18}{8}=-\frac{9}{4}
  18. 9 / 4 {9}/{4}
  19. c = ( x 2 + y 2 ) 1 / 2 c=(x^{2}+y^{2})^{1/2}
  20. d c d t = d d t ( x 2 + y 2 ) 1 / 2 \frac{dc}{dt}=\frac{d}{dt}(x^{2}+y^{2})^{1/2}
  21. = 1 2 ( x 2 + y 2 ) - 1 / 2 d d t ( x 2 + y 2 ) =\frac{1}{2}(x^{2}+y^{2})^{-1/2}\frac{d}{dt}(x^{2}+y^{2})
  22. = 1 2 ( x 2 + y 2 ) - 1 / 2 [ d d t ( x 2 ) + d d t ( y 2 ) ] =\frac{1}{2}(x^{2}+y^{2})^{-1/2}\left[\frac{d}{dt}(x^{2})+\frac{d}{dt}(y^{2})\right]
  23. = 1 2 ( x 2 + y 2 ) - 1 / 2 [ 2 x d x d t + 2 y d y d t ] =\frac{1}{2}(x^{2}+y^{2})^{-1/2}\left[2x\frac{dx}{dt}+2y\frac{dy}{dt}\right]
  24. = x d x d t + y d y d t x 2 + y 2 =\frac{x\frac{dx}{dt}+y\frac{dy}{dt}}{\sqrt{x^{2}+y^{2}}}
  25. d c d t = 4 m i ⋅ ( - 80 m i / h r ) + 3 m i ⋅ ( 60 ) m i / h r ( 4 m i ) 2 + ( 3 m i ) 2 = - 320 m i 2 / h r + 180 m i 2 / h r 5 m i = - 140 m i 2 / h r 5 m i = - 28 m i / h r \begin{aligned}\displaystyle\frac{dc}{dt}&\displaystyle=\frac{4mi\cdot(-80mi/% hr)+3mi\cdot(60)mi/hr}{\sqrt{(4mi)^{2}+(3mi)^{2}}}\\ &\displaystyle=\frac{-320mi^{2}/hr+180mi^{2}/hr}{5mi}\\ &\displaystyle=\frac{-140mi^{2}/hr}{5mi}\\ &\displaystyle=-28mi/hr\\ \end{aligned}
  26. Ί B = B A cos ( Ξ ) , \Phi_{B}=BA\cos(\theta),
  27. ℰ \mathcal{E}
  28. Ί B \Phi_{B}
  29. ℰ = - d Ω B d t , \mathcal{E}=-{{d\Phi_{B}}\over dt},
  30. Ί B \Phi_{B}
  31. ℰ = - d Ω B d t = B A sin ξ d ξ d t \mathcal{E}=-\frac{d\Phi_{B}}{dt}=BA\sin{\theta}\frac{d\theta}{dt}
  32. ℰ = ω B A sin ω t \mathcal{E}=\omega BA\sin{\omega t}

Relational_operator.html

  1. E = { x < y y > x x ≱ y y ≰ x E=\begin{cases}x<y\\ y>x\\ x\ngeq y\\ y\nleq x\end{cases}

Relative_homology.html

  1. A ⊂ X A\subset X
  2. 0 → C ∙ ( A ) → C ∙ ( X ) → C ∙ ( X ) / C ∙ ( A ) → 0 0\to C_{\bullet}(A)\to C_{\bullet}(X)\to C_{\bullet}(X)/C_{\bullet}(A)\to 0
  3. C ∙ ( X ) C_{\bullet}(X)
  4. C ∙ ( X ) C_{\bullet}(X)
  5. C ∙ ( A ) C_{\bullet}(A)
  6. H n ( X , A ) = H n ( C ∙ ( X ) / C ∙ ( A ) ) . H_{n}(X,A)=H_{n}(C_{\bullet}(X)/C_{\bullet}(A)).
  7. ⋯ → H n ( A ) → H n ( X ) → H n ( X , A ) → ή H n - 1 ( A ) → ⋯ . \cdots\to H_{n}(A)\to H_{n}(X)\to H_{n}(X,A)\stackrel{\delta}{\to}H_{n-1}(A)% \to\cdots.
  8. χ ( X , Y ) = ∑ j = 0 n ( - 1 ) j rank H j ( X , Y ) . \chi(X,Y)=\sum_{j=0}^{n}(-1)^{j}\;\mbox{rank}~{}\;H_{j}(X,Y).
  9. χ ( X , Z ) = χ ( X , Y ) + χ ( Y , Z ) . \chi(X,Z)=\chi(X,Y)+\chi(Y,Z).\,
  10. C ∙ C_{\bullet}
  11. C ∙ : T o p 2 → 𝒞 𝒞 C_{\bullet}:{Top}^{2}\to\mathcal{CC}
  12. 𝒞 𝒞 \mathcal{CC}
  13. X / A X/A
  14. A A
  15. X X
  16. A A
  17. A A
  18. H ~ n ( X / A ) \tilde{H}_{n}(X/A)
  19. H n ( X , A ) H_{n}(X,A)
  20. S n S^{n}
  21. S n = D n / S n - 1 S^{n}=D^{n}/S^{n-1}
  22. ⋯ → H n ( D n ) → H n ( D n , S n - 1 ) → H n - 1 ( S n - 1 ) → H n - 1 ( D n ) → ⋯ . \cdots\to H_{n}(D^{n})\rightarrow H_{n}(D^{n},S^{n-1})\rightarrow H_{n-1}(S^{n% -1})\rightarrow H_{n-1}(D^{n})\to\cdots.
  23. 0 → H n ( D n , S n - 1 ) → H n - 1 ( S n - 1 ) → 0. 0\rightarrow H_{n}(D^{n},S^{n-1})\rightarrow H_{n-1}(S^{n-1})\rightarrow 0.
  24. H n ( D n , S n - 1 ) ≅ H n - 1 ( S n - 1 ) H_{n}(D^{n},S^{n-1})\cong H_{n-1}(S^{n-1})
  25. H n ( D n , S n - 1 ) ≅ â„€ H_{n}(D^{n},S^{n-1})\cong\mathbb{Z}
  26. S n - 1 S^{n-1}
  27. D n D^{n}
  28. H n ( D n , S n - 1 ) ≅ H n ( S n ) ≅ â„€ . H_{n}(D^{n},S^{n-1})\cong H_{n}(S^{n})\cong\mathbb{Z}.

Relative_thermal_resistance.html

  1. R T R = ( ( p 1 - p 2 ) × 10 6 ) / 8 RTR=((p_{1}-p_{2})\times 10^{6})/8
  2. p 2 p_{2}
  3. p 1 p_{1}

Relativistic_Breit–Wigner_distribution.html

  1. f ( E ) = k ( E 2 - M 2 ) 2 + M 2 Γ 2 , f(E)=\frac{k}{\left(E^{2}-M^{2}\right)^{2}+M^{2}\Gamma^{2}}~{},
  2. k k
  3. k = 2 2 M Γ Îł π M 2 + Îł k=\frac{2\sqrt{2}M\Gamma\gamma}{\pi\sqrt{M^{2}+\gamma}}~{}~{}~{}~{}
  4. γ = M 2 ( M 2 + Γ 2 ) . ~{}~{}~{}~{}\gamma=\sqrt{M^{2}\left(M^{2}+\Gamma^{2}\right)}~{}.
  5. E E
  6. M M
  7. Γ Γ
  8. E E
  9. f ( E ) f(E)
  10. E E
  11. M M
  12. | E − M | = Γ / 2 |E−M|=Γ/2
  13. M ≫ Γ M≫Γ
  14. f f
  15. Γ Γ
  16. Γ Γ
  17. f f
  18. Γ Γ
  19. E E
  20. Γ Γ
  21. M M
  22. M M
  23. Γ Γ
  24. E E
  25. E E
  26. M M
  27. p p
  28. k ( E 2 - M 2 ) + i M Γ . \frac{\sqrt{k}}{\left(E^{2}-M^{2}\right)+iM\Gamma}~{}.
  29. s s
  30. p p
  31. E E
  32. { f â€Č ( E ) ( ( E 2 - M 2 ) 2 + Γ 2 M 2 ) - 4 E f ( E ) ( M - E ) ( E + M ) = 0 f ( M ) = k Γ 2 M 2 } \left\{\begin{array}[]{l}f^{\prime}(\,\text{E})\left(\left(\,\text{E}^{2}-M^{2% }\right)^{2}+\Gamma^{2}M^{2}\right)-4\,\text{E}f(\,\text{E})(M-\,\text{E})(\,% \text{E}+M)=0\\ f(M)=\frac{k}{\Gamma^{2}M^{2}}\end{array}\right\}

Relativistic_mechanics.html

  1. s y m b o l 𝐔 = d s y m b o l 𝐗 d τ = ( c d t d τ , d đ± d τ ) symbol{\mathbf{U}}=\frac{dsymbol{\mathbf{X}}}{d\tau}=\left(\frac{cdt}{d\tau},% \frac{d\mathbf{x}}{d\tau}\right)
  2. s y m b o l 𝐗 = ( c t , đ± ) symbol{\mathbf{X}}=(ct,\mathbf{x})
  3. d τ d t = 1 Îł ( 𝐯 ) \frac{d\tau}{dt}=\frac{1}{\gamma(\mathbf{v})}
  4. γ ( 𝐯 ) = 1 1 - 𝐯 ⋅ 𝐯 / c 2 ⇌ γ ( v ) = 1 1 - ( v / c ) 2 . \gamma(\mathbf{v})=\frac{1}{\sqrt{1-\mathbf{v}\cdot\mathbf{v}/c^{2}}}\,% \rightleftharpoons\,\gamma(v)=\frac{1}{\sqrt{1-(v/c)^{2}}}.
  5. s y m b o l 𝐔 = γ ( 𝐯 ) ( c , 𝐯 ) symbol{\mathbf{U}}=\gamma(\mathbf{v})(c,\mathbf{v})
  6. s y m b o l 𝐏 = m 0 s y m b o l 𝐔 = ( E / c , đ© ) symbol{\mathbf{P}}=m_{0}symbol{\mathbf{U}}=(E/c,\mathbf{p})
  7. E = Îł ( 𝐯 ) m 0 c 2 đ© = Îł ( 𝐯 ) m 0 𝐯 \begin{aligned}\displaystyle E&\displaystyle=\gamma(\mathbf{v})m_{0}c^{2}\\ \displaystyle\mathbf{p}&\displaystyle=\gamma(\mathbf{v})m_{0}\mathbf{v}\end{aligned}
  8. m = γ ( 𝐯 ) m 0 m=\gamma(\mathbf{v})m_{0}
  9. E 2 - ( p c ) 2 = ( m 0 c 2 ) 2 E^{2}-(pc)^{2}=(m_{0}c^{2})^{2}\,
  10. đ© c 2 = E 𝐯 . \mathbf{p}c^{2}=E\mathbf{v}\,.
  11. m 0 tot = E tot 2 - ( p tot c ) 2 c 2 {m_{0}}\text{tot}=\frac{\sqrt{E\text{tot}^{2}-(p\text{tot}c)^{2}}}{c^{2}}
  12. E = p c E=pc
  13. E 2 - đ© ⋅ đ© c 2 = m 0 2 c 4 E^{2}-\mathbf{p}\cdot\mathbf{p}c^{2}=m_{0}^{2}c^{4}
  14. m 0 , system = ∑ n E n / c 2 m_{0,\,{\rm system}}=\sum_{n}E_{n}/c^{2}
  15. 𝐍 = m ( đ± - t 𝐯 ) \mathbf{N}=m\left(\mathbf{x}-t\mathbf{v}\right)
  16. 𝐋 = đ± × đ© \mathbf{L}=\mathbf{x}\times\mathbf{p}
  17. 𝐌 = 𝐗 ∧ 𝐏 \mathbf{M}=\mathbf{X}\wedge\mathbf{P}
  18. 𝐅 = d đ© d t \mathbf{F}=\frac{d\mathbf{p}}{dt}
  19. 𝐅 = Îł ( 𝐯 ) 3 m 0 𝐚 ∄ + Îł ( 𝐯 ) m 0 𝐚 ⟂ \mathbf{F}=\gamma(\mathbf{v})^{3}m_{0}\,\mathbf{a}_{\parallel}+\gamma(\mathbf{% v})m_{0}\,\mathbf{a}_{\perp}
  20. 𝐅 = m 0 d ( γ ( 𝐯 ) 𝐯 ) d t = m 0 ( d γ ( 𝐯 ) d t 𝐯 + γ ( 𝐯 ) d 𝐯 d t ) . \mathbf{F}=m_{0}\frac{d(\gamma(\mathbf{v})\,\mathbf{v})}{dt}=m_{0}\left(\frac{% d\gamma(\mathbf{v})}{dt}\,\mathbf{v}+\gamma(\mathbf{v})\frac{d\mathbf{v}}{dt}% \right).
  21. 𝐅 = γ ( 𝐯 ) 3 m 0 c 2 ( 𝐯 ⋅ 𝐚 ) 𝐯 + γ ( 𝐯 ) m 0 𝐚 . \mathbf{F}=\frac{\gamma(\mathbf{v})^{3}m_{0}}{c^{2}}\left(\mathbf{v}\cdot% \mathbf{a}\right)\,\mathbf{v}+\gamma(\mathbf{v})m_{0}\,\mathbf{a}.
  22. 𝐚 = 𝐚 ∄ + 𝐚 ⟂ , 𝐯 ⋅ 𝐚 ⟂ = 0 , 𝐯 ⋅ 𝐚 = 𝐯 ⋅ 𝐚 ∄ , \mathbf{a}=\mathbf{a}_{\parallel}+\mathbf{a}_{\perp}\,,\quad\mathbf{v}\cdot% \mathbf{a}_{\perp}=0\,,\quad\mathbf{v}\cdot\mathbf{a}=\mathbf{v}\cdot\mathbf{a% }_{\parallel}\,,
  23. 𝐅 = Îł ( 𝐯 ) 3 m 0 c 2 ( 𝐯 ⋅ 𝐚 ∄ ) 𝐯 + Îł ( 𝐯 ) m 0 ( 𝐚 ⟂ + 𝐚 ∄ ) . \mathbf{F}=\frac{\gamma(\mathbf{v})^{3}m_{0}}{c^{2}}\left(\mathbf{v}\cdot% \mathbf{a}_{\parallel}\right)\,\mathbf{v}+\gamma(\mathbf{v})m_{0}\,(\mathbf{a}% _{\perp}+\mathbf{a}_{\parallel})\,.
  24. ( 𝐯 ⋅ 𝐚 ) 𝐯 = v 2 𝐚 ∄ (\mathbf{v}\cdot\mathbf{a})\mathbf{v}=v^{2}\mathbf{a}_{\parallel}
  25. 𝐅 = Îł ( 𝐯 ) 3 m 0 v 2 c 2 𝐚 ∄ + Îł ( 𝐯 ) m 0 ( 𝐚 ∄ + 𝐚 ⟂ ) = Îł ( 𝐯 ) 3 m 0 ( v 2 c 2 + 1 Îł ( 𝐯 ) 2 ) 𝐚 ∄ + Îł ( 𝐯 ) m 0 𝐚 ⟂ = Îł ( 𝐯 ) 3 m 0 ( v 2 c 2 + 1 - v 2 c 2 ) 𝐚 ∄ + Îł ( 𝐯 ) m 0 𝐚 ⟂ = Îł ( 𝐯 ) 3 m 0 𝐚 ∄ + Îł ( 𝐯 ) m 0 𝐚 ⟂ \begin{aligned}\displaystyle\mathbf{F}&\displaystyle=\frac{\gamma(\mathbf{v})^% {3}m_{0}v^{2}}{c^{2}}\,\mathbf{a}_{\parallel}+\gamma(\mathbf{v})m_{0}\,(% \mathbf{a}_{\parallel}+\mathbf{a}_{\perp})\\ &\displaystyle=\gamma(\mathbf{v})^{3}m_{0}\left(\frac{v^{2}}{c^{2}}+\frac{1}{% \gamma(\mathbf{v})^{2}}\right)\mathbf{a}_{\parallel}+\gamma(\mathbf{v})m_{0}\,% \mathbf{a}_{\perp}\\ &\displaystyle=\gamma(\mathbf{v})^{3}m_{0}\left(\frac{v^{2}}{c^{2}}+1-\frac{v^% {2}}{c^{2}}\right)\mathbf{a}_{\parallel}+\gamma(\mathbf{v})m_{0}\,\mathbf{a}_{% \perp}\\ &\displaystyle=\gamma(\mathbf{v})^{3}m_{0}\,\mathbf{a}_{\parallel}+\gamma(% \mathbf{v})m_{0}\,\mathbf{a}_{\perp}\end{aligned}\,
  26. 𝐚 = 1 m 0 γ ( 𝐯 ) ( 𝐅 - ( 𝐯 ⋅ 𝐅 ) 𝐯 c 2 ) . \mathbf{a}=\frac{1}{m_{0}\gamma(\mathbf{v})}\left(\mathbf{F}-\frac{(\mathbf{v}% \cdot\mathbf{F})\mathbf{v}}{c^{2}}\right)\,.
  27. s y m b o l Γ = d 𝐌 d τ = 𝐗 ∧ 𝐅 symbol{\Gamma}=\frac{d\mathbf{M}}{d\tau}=\mathbf{X}\wedge\mathbf{F}
  28. Γ α ÎČ = X α F ÎČ - X ÎČ F α \Gamma_{\alpha\beta}=X_{\alpha}F_{\beta}-X_{\beta}F_{\alpha}
  29. Δ K = W = [ γ 1 - γ 0 ] m 0 c 2 . \displaystyle\Delta K=W=[\gamma_{1}-\gamma_{0}]m_{0}c^{2}.
  30. Δ K = W = ∫ đ± 0 đ± 1 𝐅 ⋅ d đ± = ∫ t 0 t 1 d d t ( Îł m 0 𝐯 ) ⋅ 𝐯 d t = Îł m 0 𝐯 ⋅ 𝐯 | t 0 t 1 - ∫ t 0 t 1 Îł m 0 𝐯 ⋅ d 𝐯 d t d t = Îł m 0 v 2 | t 0 t 1 - m 0 ∫ v 0 v 1 Îł v d v = m 0 ( Îł v 2 | t 0 t 1 - c 2 ∫ v 0 v 1 2 v / c 2 2 1 - v 2 / c 2 d v ) = m 0 ( v 2 1 - v 2 / c 2 + c 2 1 - v 2 / c 2 ) | t 0 t 1 = m 0 c 2 1 - v 2 / c 2 | t 0 t 1 = Îł m 0 c 2 | t 0 t 1 = Îł 1 m 0 c 2 - Îł 0 m c 2 . \begin{aligned}\displaystyle\Delta K=W&\displaystyle=\int_{\mathbf{x}_{0}}^{% \mathbf{x}_{1}}\mathbf{F}\cdot d\mathbf{x}\\ &\displaystyle=\int_{t_{0}}^{t_{1}}\frac{d}{dt}(\gamma m_{0}\mathbf{v})\cdot% \mathbf{v}dt\\ &\displaystyle=\left.\gamma m_{0}\mathbf{v}\cdot\mathbf{v}\right|^{t_{1}}_{t_{% 0}}-\int_{t_{0}}^{t_{1}}\gamma m_{0}\mathbf{v}\cdot\frac{d\mathbf{v}}{dt}dt\\ &\displaystyle=\left.\gamma m_{0}v^{2}\right|^{t_{1}}_{t_{0}}-m_{0}\int_{v_{0}% }^{v_{1}}\gamma v\,dv\\ &\displaystyle=m_{0}\left(\left.\gamma v^{2}\right|^{t_{1}}_{t_{0}}-c^{2}\int_% {v_{0}}^{v_{1}}\frac{2v/c^{2}}{2\sqrt{1-v^{2}/c^{2}}}\,dv\right)\\ &\displaystyle=\left.m_{0}\left(\frac{v^{2}}{\sqrt{1-v^{2}/c^{2}}}+c^{2}\sqrt{% 1-v^{2}/c^{2}}\right)\right|^{t_{1}}_{t_{0}}\\ &\displaystyle=\left.\frac{m_{0}c^{2}}{\sqrt{1-v^{2}/c^{2}}}\right|^{t_{1}}_{t% _{0}}\\ &\displaystyle=\left.{\gamma m_{0}c^{2}}\right|^{t_{1}}_{t_{0}}\\ &\displaystyle=\gamma_{1}m_{0}c^{2}-\gamma_{0}mc^{2}.\end{aligned}
  31. K = [ Îł ( v ) - 1 ] m 0 c 2 , K=[\gamma(v)-1]m_{0}c^{2}\,,
  32. E - m 0 c 2 = 1 2 m 0 v 2 + 3 8 m 0 v 4 c 2 + 5 16 m 0 v 6 c 4 + ⋯ ; E-m_{0}c^{2}=\frac{1}{2}m_{0}v^{2}+\frac{3}{8}\frac{m_{0}v^{4}}{c^{2}}+\frac{5% }{16}\frac{m_{0}v^{6}}{c^{4}}+\cdots;
  33. đ© = m 0 𝐯 + 1 2 m 0 v 2 𝐯 c 2 + 3 8 m 0 v 4 𝐯 c 4 + 5 16 m 0 v 6 𝐯 c 6 + ⋯ . \mathbf{p}=m_{0}\mathbf{v}+\frac{1}{2}\frac{m_{0}v^{2}\mathbf{v}}{c^{2}}+\frac% {3}{8}\frac{m_{0}v^{4}\mathbf{v}}{c^{4}}+\frac{5}{16}\frac{m_{0}v^{6}\mathbf{v% }}{c^{6}}+\cdots.

Relativistic_plasma.html

  1. Îł \gamma
  2. Îł > 2 \gamma>2
  3. Îł > 2 \gamma>2

Relativistic_quantum_chemistry.html

  1. m r e l = m e 1 - ( v e / c ) 2 m_{rel}=\frac{m_{e}}{\sqrt{1-(v_{e}/c)^{2}}}
  2. m e , v e , c \displaystyle m_{e},v_{e},c
  3. a 0 \displaystyle a_{0}
  4. a 0 = ℏ m e c α a_{0}=\frac{\hbar}{m_{e}c\alpha}
  5. ℏ \hbar
  6. a r e l = ℏ 1 - ( v e / c ) 2 m e c α a_{rel}=\frac{\hbar\sqrt{1-(v_{e}/c)^{2}}}{m_{e}c\alpha}
  7. a r e l a 0 = 1 - ( v e / c ) 2 \frac{a_{rel}}{a_{0}}=\sqrt{1-(v_{e}/c)^{2}}
  8. L â€Č = L 1 - v 2 / c 2 L^{\prime}=L\,\sqrt{1-v^{2}/c^{2}}
  9. a r e l = a 0 1 - v 2 / c 2 a_{rel}=a_{0}\,\sqrt{1-v^{2}/c^{2}}
  10. r = n 2 ℏ 2 4 π Ï” 0 m e Z e 2 r=\frac{n^{2}\hbar^{2}4\pi\epsilon_{0}}{m_{e}Ze^{2}}
  11. n n
  12. m v e r = n ℏ mv_{e}r=n\hbar
  13. v v
  14. r = m v e r n ℏ 4 π Ï” 0 m Z e 2 r=\frac{mv_{e}rn\hbar 4\pi\epsilon_{0}}{mZe^{2}}
  15. 1 = v e n ℏ 4 π Ï” 0 Z e 2 1=\frac{v_{e}n\hbar 4\pi\epsilon_{0}}{Ze^{2}}
  16. v e = Z e 2 n ℏ 4 π Ï” 0 v_{e}=\frac{Ze^{2}}{n\hbar 4\pi\epsilon_{0}}
  17. v e = Z n v_{e}=\frac{Z}{n}
  18. a r e l a 0 = 1 - ( Z n c ) 2 \frac{a_{rel}}{a_{0}}=\sqrt{1-\left(\frac{Z}{nc}\right)^{2}}
  19. n n
  20. Z Z
  21. a r e l a 0 < 1 \frac{a_{rel}}{a_{0}}<1

Relaxation_(physics).html

  1. m d 2 y d t 2 + Îł d y d t + k y = 0 m\frac{d^{2}y}{dt^{2}}+\gamma\frac{dy}{dt}+ky=0
  2. y ( t ) = A e - t / T cos ( Ό t - Ύ ) y(t)=Ae^{-t/T}\cos(\mu t-\delta)
  3. V ( t ) = V 0 e - t R C , V(t)=V_{0}e^{-\frac{t}{RC}}\ ,
  4. τ = R C \tau=RC
  5. T r = 0.34 σ 3 G 2 m ρ ln Λ T_{r}={0.34\sigma^{3}\over G^{2}m\rho\ln\Lambda}
  6. ≈ 0.95 × 10 10 ( σ 200 km s - 1 ) 3 ( ρ 10 6 M ⊙ pc - 3 ) - 1 ( m ⋆ M ⊙ ) - 1 ( ln Λ 15 ) - 1 yr \approx 0.95\times 10^{10}\!\left({\sigma\over 200\,\mathrm{km\,s}^{-1}}\right% )^{\!3}\!\!\left({\rho\over 10^{6}\,M_{\odot}\,\mathrm{pc}^{-3}}\right)^{\!-1}% \!\!\left({m_{\star}\over M_{\odot}}\right)^{\!-1}\!\!\left({\ln\Lambda\over 1% 5}\right)^{\!-1}\!\mathrm{yr}

Reliability_engineering.html

  1. R ( t ) = P r { T > t } = ∫ t ∞ f ( x ) d x R(t)=Pr\{T>t\}=\int_{t}^{\infty}f(x)\,dx\ \!
  2. f ( x ) f(x)\!
  3. t t
  4. t t\!