wpmath0000014_9

Nondominant_seventh_chord.html

  1. 7 m {}_{m}^{7}
  2. 7 M {}_{M}^{7}
  3. 7 m {}_{m}^{7}
  4. 7 M {}_{M}^{7}

Nonparametric_skew.html

  1. S = μ - ν σ S=\frac{\mu-\nu}{\sigma}
  2. S ( a X + b ) = sign ( a ) S ( X ) S(aX+b)=\operatorname{sign}(a)\,S(X)
  3. 2 ( p q ) 1 / 2 ( p + q ) 1 / 2 \frac{2(pq)^{1/2}}{(p+q)^{1/2}}
  4. p = Pr ( X > E ( X ) ) p=\Pr(X>\operatorname{E}(X))
  5. q = Pr ( X < E ( X ) ) , q=\Pr(X<\operatorname{E}(X)),
  6. | μ - ν 0 | E ( | X - ν 0 | ) E ( | X - μ | ) σ , |\mu-\nu_{0}|\leq\operatorname{E}(|X-\nu_{0}|)\leq\operatorname{E}(|X-\mu|)% \leq\sigma,
  7. | μ - x q | σ max ( ( 1 - q ) q , q ( 1 - q ) ) \frac{|\mu-x_{q}|}{\sigma}\leq\max\left(\sqrt{\frac{(1-q)}{q}},\sqrt{\frac{q}{% (1-q)}}\right)
  8. μ - σ 1 - q q x q μ + σ q 1 - q \mu-\sigma\sqrt{\frac{1-q}{q}}\leq x_{q}\leq\mu+\sigma\sqrt{\frac{q}{1-q}}
  9. μ - 1 2 q E | X - μ | x q μ + 1 ( 2 - 2 q ) E | X - μ | \mu-\frac{1}{2q}\operatorname{E}|X-\mu|\leq x_{q}\leq\mu+\frac{1}{(2-2q)}% \operatorname{E}|X-\mu|
  10. Pr ( X = a ) = q \Pr(X=a)=q
  11. Pr ( X = b ) = 1 - q \Pr(X=b)=1-q
  12. | m - x r | s max [ ( n - 1 ) ( r - 1 ) n ( n - r + 1 ) , ( n - 1 ) ( n - r ) n r ] \frac{|m-x_{r}|}{s}\leq\,\text{max}\left[\sqrt{\frac{(n-1)(r-1)}{n(n-r+1)}},% \sqrt{\frac{(n-1)(n-r)}{nr}}\right]
  13. | m - a | s n 2 - n n 2 = n - 1 n \frac{|m-a|}{s}\leq\sqrt{\frac{n^{2}-n}{n^{2}}}=\sqrt{\frac{n-1}{n}}
  14. D = n ( m - a ) s D=\frac{n(m-a)}{s}
  15. 2 n ( m - a s ) \sqrt{2n}\left(\frac{m-a}{s}\right)
  16. γ 1 = 2 ( m - a ) , \gamma_{1}=2(m-a),
  17. J = 1 n π 2 | X i - a | J=\frac{1}{n}\sqrt{\frac{\pi}{2}}\sum{|X_{i}-a|}
  18. T = m - a J . T=\frac{m-a}{J}.
  19. S = 1 3 ( α - 2 β ) ( α + β + 1 ) 1 / 2 ( α + β - 2 / 3 ) ( α β ) 1 / 2 S=\frac{1}{3}\frac{(\alpha-2\beta)(\alpha+\beta+1)^{1/2}}{(\alpha+\beta-2/3)(% \alpha\beta)^{1/2}}
  20. S = 2 β 2 ( 4 + 5 α 2 ) S=\frac{2}{\beta^{2}(4+5\alpha^{2})}
  21. - 4 3 S 4 3 \frac{-4}{3}\leq S\leq\frac{4}{3}
  22. S 1 - ( 1 - 2 k ) 3 2 S\approx\frac{1-(1-\frac{2}{k})^{3}}{2}
  23. S = 1 - log e ( 2 ) 0.31 S=1-\log_{e}(2)\approx 0.31
  24. S = 1 - log e ( 2 ) 0.31 S=1-\log_{e}(2)\approx 0.31
  25. S = - p o l y l o g ( 2 , 1 - p ) + ln ( 1 + p ) ln p - [ 2 p o l y l o g ( 3 , 1 - p ) + p o l y l o g 2 ( 2 , 1 - p ) ] S=-\frac{polylog(2,1-p)+\ln(1+\sqrt{p})\ln p}{\sqrt{-[2polylog(3,1-p)+polylog^% {2}(2,1-p)]}}
  26. 0 S 1 - log e ( 2 ) 0\leq S\leq 1-\log_{e}(2)
  27. S = n - 3 / 2 n - 4 n - 2 + O ( n - 5 / 2 ) S=n^{-3/2}\sqrt{\frac{n-4}{n-2}}+O(n^{-5/2})
  28. S = Γ ( 1 - 1 α ) - 1 α log e ( 2 ) Γ ( 1 - 2 α ) - ( Γ ( 1 - 1 α ) ) 2 S=\frac{\Gamma\left(1-\frac{1}{\alpha}\right)-\frac{1}{\sqrt{\alpha}\log_{e}(2% )}}{\sqrt{}}{\Gamma\left(1-\frac{2}{\alpha}\right)-\left(\Gamma\left(1-\frac{1% }{\alpha}\right)\right)^{2}}
  29. S β 3 α + 0.2 S\approx\frac{\beta}{3\alpha+0.2}
  30. S = - exp ( - k 2 2 ) - 1 exp ( k 2 2 ) - 1 S=-\frac{\exp(\frac{-k^{2}}{2})-1}{\sqrt{\exp(\frac{k^{2}}{2})-1}}
  31. S = ( 2 k - 1 k - 2 k ) ( 1 - 2 k ) 0.5 S=\left(\frac{2^{k}-1}{k}-2^{k}\right)(1-2k)^{0.5}
  32. 6 [ γ + log e ( log e ( 2 ) ) ] π 0.1643 \frac{\sqrt{6}[\gamma+\log_{e}(\log_{e}(2))]}{\pi}\approx 0.1643
  33. S 2 - 0.6745 π π - 2 0.36279 S\approx\frac{\sqrt{2}-0.6745\sqrt{\pi}}{\sqrt{\pi-2}}\approx 0.36279
  34. π \pi
  35. S = b - sin ( b ) b tan ( b ) - b 2 S=\frac{b-\sin(b)}{\sqrt{b\tan(b)-b^{2}}}
  36. S = 1 ( e σ 2 2 + 1 ) ( e μ + σ 2 ) S=\frac{1}{(e^{\frac{\sigma^{2}}{2}}+1)(e^{\mu+\sigma^{2}})}
  37. S [ log e ( log e ( 2 ) ) - 0.5772 ] 6 π - 0.1643 S\approx\frac{[\log_{e}(\log_{e}(2))-0.5772]\sqrt{6}}{\pi}\approx-0.1643
  38. S = ( α - 1 ) ( α - 2 ) ( 1 - ( α - 1 ) ( 2 1 / α - 1 ) ) α 1 / 2 S=\frac{(\alpha-1)(\alpha-2)(1-(\alpha-1)(2^{1/\alpha}-1))}{\alpha^{1/2}}
  39. S 2 - 1.5382 Γ ( 3 2 ) 2 ( Γ ( 5 2 ) - Γ ( 3 2 ) ) 0.0854 S\approx\frac{\sqrt{2}-1.5382\Gamma(\frac{3}{2})}{\sqrt{2(\Gamma(\frac{5}{2})-% \Gamma(\frac{3}{2}))}}\approx 0.0854
  40. S = - 1 S=-1
  41. S = ( α - 2 1 / α [ α - 1 ] ) ( α - 2 α ) 1 / 2 , S=(\alpha-2^{1/\alpha}[\alpha-1])(\frac{\alpha-2}{\alpha})^{1/2},
  42. - log e ( 2 ) λ 1 2 S 1 3 λ 1 2 \frac{-\log_{e}(2)}{\lambda^{\frac{1}{2}}}\leq S\leq\frac{1}{3\lambda^{\frac{1% }{2}}}
  43. S = 2 4 - π [ ( π 2 ) 0.5 - log e ( 4 ) ] 0.1251 S=\sqrt{\frac{2}{4-\pi}}[(\frac{\pi}{2})^{0.5}-\log_{e}(4)]\approx 0.1251
  44. S = Γ ( 1 + 1 / k ) - log e ( 2 ) 1 / k ( Γ ( 1 + 2 / k ) - Γ ( 1 + 1 / k ) ) 1 / 2 , S=\frac{\Gamma(1+1/k)-\log_{e}(2)^{1/k}}{(\Gamma(1+2/k)-\Gamma(1+1/k))^{1/2}},
  45. μ - θ σ , \frac{\mu-\theta}{\sigma},
  46. 3 ( μ - ν ) σ , \frac{3(\mu-\nu)}{\sigma},
  47. ν - θ = 2 ( μ - ν ) \nu-\theta=2(\mu-\nu)
  48. θ = 3 ν - 2 μ , \theta=3\nu-2\mu,
  49. μ - θ = α ( μ - ν ) \mu-\theta=\alpha(\mu-\nu)
  50. σ ω 2 σ \sigma\leq\omega\leq 2\sigma
  51. | ν - μ | 3 4 ω , |\nu-\mu|\leq\sqrt{\frac{3}{4}}\omega,
  52. θ ν μ \theta\leq\nu\leq\mu
  53. μ ν θ \mu\leq\nu\leq\theta
  54. | θ - μ | σ 3 , \frac{|\theta-\mu|}{\sigma}\leq\sqrt{3},
  55. | ν - μ | σ 0.6 , \frac{|\nu-\mu|}{\sigma}\leq\sqrt{0.6},
  56. | θ - ν | σ 3 , \frac{|\theta-\nu|}{\sigma}\leq\sqrt{3},
  57. θ ν μ , \theta\leq\nu\leq\mu,
  58. F ( ν - x ) + F ( ν + x ) 1 for all x , F(\nu-x)+F(\nu+x)\geq 1\,\text{ for all }x,
  59. S K 2 = Q 3 + Q 1 - 2 Q 2 Q 3 - Q 1 SK_{2}=\frac{Q_{3}+Q_{1}-2Q_{2}}{Q_{3}-Q_{1}}
  60. S K = F - 1 ( 1 - α ) + F - 1 ( α ) - 2 Q 2 Q 3 - Q 1 SK=\frac{F^{-1}(1-\alpha)+F^{-1}(\alpha)-2Q_{2}}{Q_{3}-Q_{1}}
  61. α \alpha
  62. α \alpha
  63. S K 3 = μ - Q 2 E | y - Q 2 | SK_{3}=\frac{\mu-Q_{2}}{E|y-Q_{2}|}

Norm_(abelian_group).html

  1. ν : G \scriptstyle\nu\colon G\to\mathbb{R}
  2. ν ( g ) > 0 if g 0 \nu(g)>0\,\,\mathrm{if}\,\,g\neq 0
  3. ν ( g + h ) ν ( g ) + ν ( h ) \nu(g+h)\leq\nu(g)+\nu(h)
  4. ν ( m g ) = | m | ν ( g ) if m \nu(mg)=|m|\nu(g)\,\,\mathrm{if}\,\,m\in\mathbb{Z}

Norm_group.html

  1. N L / K ( L × ) N_{L/K}(L^{\times})
  2. L / K L/K
  3. K × K^{\times}
  4. K × K^{\times}

Normal-inverse-Wishart_distribution.html

  1. s y m b o l μ | s y m b o l μ 0 , λ , s y m b o l Σ 𝒩 ( s y m b o l μ | s y m b o l μ 0 , 1 λ s y m b o l Σ ) symbol\mu|symbol\mu_{0},\lambda,symbol\Sigma\sim\mathcal{N}\left(symbol\mu\Big% |symbol\mu_{0},\frac{1}{\lambda}symbol\Sigma\right)
  2. s y m b o l μ 0 symbol\mu_{0}
  3. 1 λ s y m b o l Σ \tfrac{1}{\lambda}symbol\Sigma
  4. s y m b o l Σ | s y m b o l Ψ , ν 𝒲 - 1 ( s y m b o l Σ | s y m b o l Ψ , ν ) symbol\Sigma|symbol\Psi,\nu\sim\mathcal{W}^{-1}(symbol\Sigma|symbol\Psi,\nu)
  5. ( s y m b o l μ , s y m b o l Σ ) (symbol\mu,symbol\Sigma)
  6. ( s y m b o l μ , s y m b o l Σ ) NIW ( s y m b o l μ 0 , λ , s y m b o l Ψ , ν ) . (symbol\mu,symbol\Sigma)\sim\mathrm{NIW}(symbol\mu_{0},\lambda,symbol\Psi,\nu).
  7. f ( s y m b o l μ , s y m b o l Σ | s y m b o l μ 0 , λ , s y m b o l Ψ , ν ) = 𝒩 ( s y m b o l μ | s y m b o l μ 0 , 1 λ s y m b o l Σ ) 𝒲 - 1 ( s y m b o l Σ | s y m b o l Ψ , ν ) f(symbol\mu,symbol\Sigma|symbol\mu_{0},\lambda,symbol\Psi,\nu)=\mathcal{N}% \left(symbol\mu\Big|symbol\mu_{0},\frac{1}{\lambda}symbol\Sigma\right)\mathcal% {W}^{-1}(symbol\Sigma|symbol\Psi,\nu)
  8. s y m b o l Σ symbol\Sigma
  9. s y m b o l μ symbol\mu
  10. s y m b o l Σ symbol\Sigma
  11. s y m b o l μ symbol\mu
  12. s y m b o l Σ symbol\Sigma
  13. s y m b o l Ψ symbol\Psi
  14. ν \nu
  15. s y m b o l μ symbol\mu
  16. s y m b o l μ 0 symbol\mu_{0}
  17. s y m b o l 1 λ s y m b o l Σ symbol\tfrac{1}{\lambda}symbol\Sigma
  18. ( s y m b o l μ , s y m b o l Σ ) NIW ( s y m b o l μ 0 , λ , s y m b o l Ψ , ν ) (symbol\mu,symbol\Sigma)\sim\mathrm{NIW}(symbol\mu_{0},\lambda,symbol\Psi,\nu)
  19. ( s y m b o l μ , s y m b o l Σ - 1 ) NW ( s y m b o l μ 0 , λ , s y m b o l Ψ - 1 , ν ) (symbol\mu,symbol\Sigma^{-1})\sim\mathrm{NW}(symbol\mu_{0},\lambda,symbol\Psi^% {-1},\nu)

Normal-Wishart_distribution.html

  1. s y m b o l μ | s y m b o l μ 0 , λ , s y m b o l Λ 𝒩 ( s y m b o l μ | s y m b o l μ 0 , ( \lambdasymbol Λ ) - 1 ) symbol\mu|symbol\mu_{0},\lambda,symbol\Lambda\sim\mathcal{N}(symbol\mu|symbol% \mu_{0},(\lambdasymbol\Lambda)^{-1})
  2. s y m b o l μ 0 symbol\mu_{0}
  3. ( \lambdasymbol Λ ) - 1 (\lambdasymbol\Lambda)^{-1}
  4. s y m b o l Λ | 𝐖 , ν 𝒲 ( s y m b o l Λ | 𝐖 , ν ) symbol\Lambda|\mathbf{W},\nu\sim\mathcal{W}(symbol\Lambda|\mathbf{W},\nu)
  5. ( s y m b o l μ , s y m b o l Λ ) (symbol\mu,symbol\Lambda)
  6. ( s y m b o l μ , s y m b o l Λ ) NW ( s y m b o l μ 0 , λ , 𝐖 , ν ) . (symbol\mu,symbol\Lambda)\sim\mathrm{NW}(symbol\mu_{0},\lambda,\mathbf{W},\nu).
  7. f ( s y m b o l μ , s y m b o l Λ | s y m b o l μ 0 , λ , 𝐖 , ν ) = 𝒩 ( s y m b o l μ | s y m b o l μ 0 , ( \lambdasymbol Λ ) - 1 ) 𝒲 ( s y m b o l Λ | 𝐖 , ν ) f(symbol\mu,symbol\Lambda|symbol\mu_{0},\lambda,\mathbf{W},\nu)=\mathcal{N}(% symbol\mu|symbol\mu_{0},(\lambdasymbol\Lambda)^{-1})\ \mathcal{W}(symbol% \Lambda|\mathbf{W},\nu)
  8. s y m b o l Λ symbol\Lambda
  9. s y m b o l μ symbol\mu
  10. s y m b o l Λ symbol\Lambda
  11. s y m b o l μ symbol\mu
  12. s y m b o l Λ symbol\Lambda
  13. 𝐖 \mathbf{W}
  14. ν \nu
  15. s y m b o l μ symbol\mu
  16. s y m b o l μ 0 symbol\mu_{0}
  17. ( \lambdasymbol Λ ) - 1 (\lambdasymbol\Lambda)^{-1}

Normalized_Google_distance.html

  1. NGD ( x , y ) = max { log f ( x ) , log f ( y ) } - log f ( x , y ) log M - min { log f ( x ) , log f ( y ) } \operatorname{NGD}(x,y)=\frac{\max\{\log f(x),\log f(y)\}-\log f(x,y)}{\log M-% \min\{\log f(x),\log f(y)\}}

North–South_model.html

  1. θ \theta
  2. θ = p r i c e o f p r i m a r y p r o d u c t s p r i c e o f m a n u f a c t u r e s \theta=\frac{price\,of\,primary\,products}{price\,of\,manufactures}
  3. θ \theta
  4. θ \theta

Nottingham_group.html

  1. f = t + n = 2 a n t n f=t+\sum_{n=2}^{\infty}a_{n}t^{n}
  2. g g
  3. g f = f ( g ) = g + n = 2 a n g n gf=f(g)=g+\sum_{n=2}^{\infty}a_{n}g^{n}

Novikov–Veselov_equation.html

  1. v = v ( x 1 , x 2 , t ) , v=v(x_{1},x_{2},t),
  2. w = w ( x 1 , x 2 , t ) w=w(x_{1},x_{2},t)
  3. \Re
  4. z = 1 2 ( x 1 - i x 2 ) , z ¯ = 1 2 ( x 1 + i x 2 ) . \partial_{z}=\frac{1}{2}(\partial_{x_{1}}-i\partial_{x_{2}}),\quad\partial_{% \bar{z}}=\frac{1}{2}(\partial_{x_{1}}+i\partial_{x_{2}}).
  5. v v
  6. w w
  7. v v
  8. E E
  9. L ψ = E ψ , L = - Δ + v ( x , t ) , Δ = x 1 2 + x 2 2 . L\psi=E\psi,\quad L=-\Delta+v(x,t),\quad\Delta=\partial_{x_{1}}^{2}+\partial_{% x_{2}}^{2}.
  10. v v
  11. w w
  12. v = v ( x 1 , t ) v=v(x_{1},t)
  13. w = w ( x 1 , t ) w=w(x_{1},t)
  14. E ± E\to\pm\infty
  15. L = - x 2 + v ( x , t ) L=-\partial_{x}^{2}+v(x,t)
  16. A = x 3 + 3 4 ( v ( x , t ) x + x v ( x , t ) ) A=\partial_{x}^{3}+\frac{3}{4}(v(x,t)\partial_{x}+\partial_{x}v(x,t))
  17. [ , ] [\cdot,\cdot]
  18. L ψ = λ ψ , ψ t = A ψ \begin{aligned}&\displaystyle L\psi=\lambda\psi,\\ &\displaystyle\psi_{t}=A\psi\end{aligned}
  19. λ \lambda
  20. L L
  21. A A
  22. B B
  23. L L
  24. L ψ = λ ψ , ψ t = A ψ \begin{aligned}&\displaystyle L\psi=\lambda\psi,\\ &\displaystyle\psi_{t}=A\psi\end{aligned}
  25. λ \lambda
  26. L L

Numeric_precision_in_Microsoft_Excel.html

  1. Σ ( x - x ¯ ) 2 n = Σ [ x - ( Σ x ) / n ] 2 n , \sqrt{\frac{\Sigma(x-\bar{x})^{2}}{n}}=\sqrt{\frac{\Sigma\left[x-\left(\Sigma x% \right)/n\right]^{2}}{n}}\ ,
  2. n Σ x 2 - ( Σ x ) 2 n 2 . \sqrt{\frac{n\Sigma x^{2}-\left(\Sigma x\right)^{2}}{n^{2}}}\ .
  3. A 1 := 28.552 A1:=28.552
  4. A 2 := 27.399 A2:=27.399
  5. A 3 := 26.246 A3:=26.246
  6. B 1 := A 1 - A 2 B1:=A1-A2
  7. B 2 := A 2 - A 3 B2:=A2-A3
  8. B 1 B1
  9. B 2 B2
  10. 1.1530 1.1530
  11. C 1 C1
  12. B 1 - B 2 B1-B2
  13. C 1 C1
  14. 0
  15. - 3.55271 E - 15 -3.55271E-15
  16. a x 2 + b x + c = 0 . ax^{2}+bx+c=0\ .
  17. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
  18. b 2 - 4 a c = b 1 - 4 a c b 2 b ( 1 - 2 a c b 2 + 2 a 2 c 2 b 4 + ) . \sqrt{b^{2}-4ac}=b\ \sqrt{1-\frac{4ac}{b^{2}}}\approx b\left(1-\frac{2ac}{b^{2% }}+\frac{2a^{2}c^{2}}{b^{4}}+\cdots\right).
  19. b - b 2 - 4 a c b ( 2 a c b 2 - 2 a 2 c 2 b 4 + ) , b-\sqrt{b^{2}-4ac}\approx b\left(\frac{2ac}{b^{2}}-\frac{2a^{2}c^{2}}{b^{4}}+% \cdots\right),
  20. ε = 2 a c b , \varepsilon=\frac{2ac}{b},
  21. b - b 2 - 4 a c b - b + ε . b-\sqrt{b^{2}-4ac}\approx b-b+\varepsilon.
  22. ( x - r 1 ) ( x - r 2 ) = x 2 - ( r 1 + r 2 ) x + r 1 r 2 = 0. \left(x-r_{1}\right)\left(x-r_{2}\right)=x^{2}-\left(r_{1}+r_{2}\right)x+r_{1}% \ r_{2}=0.
  23. r 1 - b a , r_{1}\approx-\frac{b}{a},
  24. r 1 r 2 = c a . r_{1}\ r_{2}=\frac{c}{a}.
  25. r 2 = c a r 1 - c b . r_{2}=\frac{c}{a\ r_{1}}\approx-\frac{c}{b}.

Numerical_3-dimensional_matching.html

  1. X X
  2. Y Y
  3. Z Z
  4. k k
  5. b b
  6. M M
  7. X × Y × Z X\times Y\times Z
  8. X X
  9. Y Y
  10. Z Z
  11. ( x , y , z ) (x,y,z)
  12. x + y + z = b x+y+z=b
  13. X = { 3 , 4 , 4 } X=\{3,4,4\}
  14. Y = { 1 , 4 , 6 } Y=\{1,4,6\}
  15. Z = { 1 , 2 , 5 } Z=\{1,2,5\}
  16. b = 10 b=10
  17. { ( 3 , 6 , 1 ) , ( 4 , 4 , 2 ) , ( 4 , 1 , 5 ) } \{(3,6,1),(4,4,2),(4,1,5)\}
  18. b = 10 b=10
  19. { ( 3 , 6 , 1 ) , ( 3 , 4 , 2 ) , ( 4 , 1 , 5 ) } \{(3,6,1),(3,4,2),(4,1,5)\}
  20. 4 X 4\in X
  21. 3 X 3\in X
  22. b b
  23. 3 + 4 + 2 = 9 b = 10 3+4+2=9\neq b=10
  24. b = 11 b=11
  25. X X
  26. Y Y
  27. Z Z
  28. 30 30
  29. k b = 33 k\cdot b=33

O-minimal_theory.html

  1. X = X 0 I 1 I r . X=X_{0}\cup I_{1}\cup\ldots\cup I_{r}.

Oldroyd-B_model.html

  1. 𝐓 + λ 1 𝐓 = 2 η 0 ( 𝐃 + λ 2 𝐃 ) \mathbf{T}+\lambda_{1}\stackrel{\nabla}{\mathbf{T}}=2\eta_{0}(\mathbf{D}+% \lambda_{2}\stackrel{\nabla}{\mathbf{D}})
  2. 𝐓 \mathbf{T}
  3. λ 1 \lambda_{1}
  4. λ 2 \lambda_{2}
  5. η s η 0 λ 1 \frac{\eta_{s}}{\eta_{0}}\lambda_{1}
  6. 𝐓 \stackrel{\nabla}{\mathbf{T}}
  7. 𝐓 = t 𝐓 + 𝐯 𝐓 - ( ( 𝐯 ) T 𝐓 + 𝐓 ( 𝐯 ) ) \stackrel{\nabla}{\mathbf{T}}=\frac{\partial}{\partial t}\mathbf{T}+\mathbf{v}% \cdot\nabla\mathbf{T}-((\nabla\mathbf{v})^{T}\cdot\mathbf{T}+\mathbf{T}\cdot(% \nabla\mathbf{v}))
  8. 𝐯 \mathbf{v}
  9. η 0 \eta_{0}
  10. η 0 = η s + η p \eta_{0}=\eta_{s}+\eta_{p}
  11. 𝐃 \mathbf{D}
  12. 𝐃 = 1 2 [ s y m b o l 𝐯 + ( s y m b o l 𝐯 ) T ] \mathbf{D}=\frac{1}{2}\left[symbol\nabla\mathbf{v}+(symbol\nabla\mathbf{v})^{T% }\right]
  13. 𝐓 = 2 η s 𝐃 + τ \mathbf{T}=2\eta_{s}\mathbf{D}+\mathbf{\tau}
  14. τ + λ 1 τ = 2 η p 𝐃 \mathbf{\tau}+\lambda_{1}\stackrel{\nabla}{\mathbf{\tau}}=2\eta_{p}\mathbf{D}

Olog.html

  1. 𝒞 \mathcal{C}
  2. 𝒞 \mathcal{C}
  3. 𝒞 \mathcal{C}
  4. 𝐒𝐞𝐭 \,\textbf{Set}
  5. 𝐒𝐞𝐭 \,\textbf{Set}
  6. 𝒞 \mathcal{C}_{\mathbb{P}}
  7. 𝐒𝐞𝐭 \,\textbf{Set}
  8. 𝐒𝐞𝐭 \,\textbf{Set}
  9. 𝐒𝐞𝐭 \,\textbf{Set}
  10. 𝐒𝐞𝐭 \,\textbf{Set}
  11. ( , η , μ ) (\mathbb{P},\eta,\mu)
  12. 𝐒𝐞𝐭 \,\textbf{Set}
  13. A O b ( 𝐒𝐞𝐭 ) A\in Ob(\,\textbf{Set})
  14. ( A ) \mathbb{P}(A)
  15. η \eta
  16. a A a\in A
  17. { a } \{a\}
  18. μ \mu
  19. f : A B f:A\to B
  20. 𝒞 \mathcal{C}_{\mathbb{P}}
  21. a A a\in A
  22. b B b\in B
  23. ( a , b ) R (a,b)\in R
  24. b f ( a ) b\in f(a)
  25. 𝒞 \mathcal{C}_{\mathbb{P}}
  26. 𝒞 \mathcal{C}_{\mathbb{P}}
  27. 𝒞 \mathcal{C}
  28. 𝒞 \mathcal{C}
  29. T T
  30. 𝒞 \mathcal{C}
  31. 𝒞 \mathcal{C}
  32. I : 𝒞 𝐒𝐞𝐭 I:\mathcal{C}\to\,\textbf{Set}
  33. 𝒞 \mathcal{C}
  34. 𝒟 \mathcal{D}
  35. I : 𝒞 𝐒𝐞𝐭 I:\mathcal{C}\to\,\textbf{Set}
  36. J : 𝒟 𝐒𝐞𝐭 J:\mathcal{D}\to\,\textbf{Set}
  37. F : 𝒞 𝒟 F:\mathcal{C}\to\mathcal{D}
  38. F F
  39. m : I F * J m:I\to F^{*}J
  40. 𝒞 \mathcal{C}
  41. 𝒟 \mathcal{D}
  42. F F
  43. 𝐒𝐞𝐭 \,\textbf{Set}
  44. 𝒞 \mathcal{C}
  45. 𝒟 \mathcal{D}
  46. F F
  47. X 𝒞 X\in\mathcal{C}
  48. I ( X ) = J ( F ( X ) ) I(X)=J(F(X))
  49. m m
  50. F F
  51. 𝒞 \mathcal{C}
  52. 𝒟 \mathcal{D}
  53. \mathcal{B}
  54. 𝒞 \mathcal{C}
  55. 𝒟 \mathcal{D}
  56. F 𝒞 : 𝒞 F_{\mathcal{C}}:\mathcal{B}\to\mathcal{C}
  57. F 𝒟 : 𝒟 F_{\mathcal{D}}:\mathcal{B}\to\mathcal{D}
  58. 𝐒𝐞𝐭 \,\textbf{Set}
  59. m m
  60. w w
  61. ( m , w ) (m,w)
  62. m m
  63. w w

Omnitruncated_5-simplex_honeycomb.html

  1. A ~ 5 {\tilde{A}}_{5}
  2. 5 * {}^{*}_{5}
  3. 5 6 {}^{6}_{5}
  4. A ~ 5 {\tilde{A}}_{5}
  5. C ~ 3 {\tilde{C}}_{3}

Omnitruncated_simplectic_honeycomb.html

  1. A ~ n {\tilde{A}}_{n}
  2. A ~ 1 + {\tilde{A}}_{1+}
  3. A ~ 1 {\tilde{A}}_{1}
  4. A ~ 2 {\tilde{A}}_{2}
  5. A ~ 3 {\tilde{A}}_{3}
  6. A ~ 4 {\tilde{A}}_{4}
  7. A ~ 5 {\tilde{A}}_{5}
  8. A ~ 6 {\tilde{A}}_{6}
  9. A ~ 7 {\tilde{A}}_{7}
  10. A ~ 8 {\tilde{A}}_{8}
  11. A ~ 3 {\tilde{A}}_{3}
  12. A ~ 5 {\tilde{A}}_{5}
  13. A ~ 7 {\tilde{A}}_{7}
  14. A ~ 9 {\tilde{A}}_{9}
  15. C ~ 2 {\tilde{C}}_{2}
  16. C ~ 3 {\tilde{C}}_{3}
  17. C ~ 4 {\tilde{C}}_{4}
  18. C ~ 5 {\tilde{C}}_{5}

On_Physical_Lines_of_Force.html

  1. × 𝐁 = μ 0 𝐉 + μ 0 ϵ 0 t 𝐄 Maxwell s term \mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}+\underbrace{\mu_{0}\epsilon_% {0}\frac{\partial}{\partial t}\mathbf{E}}_{\mathrm{Maxwell^{\prime}s\ term}}

On_shell_renormalization_scheme.html

  1. 0 | T ( ψ ( x ) ψ ¯ ( 0 ) ) | 0 = i S F ( x ) = d 4 p ( 2 π ) 4 i e - i p x p / - m + i ϵ \langle 0|T(\psi(x)\bar{\psi}(0))|0\rangle=iS_{F}(x)=\int\frac{d^{4}p}{(2\pi)^% {4}}\frac{ie^{-ip\cdot x}}{p\!\!\!/-m+i\epsilon}
  2. T T
  3. | 0 |0\rangle
  4. ψ ( x ) \psi(x)
  5. ψ ¯ ( x ) \bar{\psi}(x)
  6. e e
  7. | Ω |\Omega\rangle
  8. Ω | T ( ψ ( x ) ψ ¯ ( 0 ) ) | Ω = d 4 p ( 2 π ) 4 i Z 2 e - i p x p / - m r + i ϵ \langle\Omega|T(\psi(x)\bar{\psi}(0))|\Omega\rangle=\int\frac{d^{4}p}{(2\pi)^{% 4}}\frac{iZ_{2}e^{-ip\cdot x}}{p\!\!\!/-m_{r}+i\epsilon}
  9. m r m_{r}
  10. Z 2 Z_{2}
  11. e 0 e\rightarrow 0
  12. m r m m_{r}\rightarrow m
  13. Z 2 1 Z_{2}\rightarrow 1
  14. m r m_{r}
  15. Z 2 Z_{2}
  16. e e
  17. = c = 1 \hbar=c=1
  18. e = 4 π α 0.3 e=\sqrt{4\pi\alpha}\simeq 0.3
  19. α \alpha
  20. Z 2 = 1 + δ 2 Z_{2}=1+\delta_{2}
  21. m r = m + δ m m_{r}=m+\delta m
  22. e e
  23. Σ ( p ) \Sigma(p)
  24. Ω | T ( ψ ( x ) ψ ¯ ( 0 ) ) | Ω = d 4 p ( 2 π ) 4 i e - i p x p / - m - Σ ( p ) + i ϵ \langle\Omega|T(\psi(x)\bar{\psi}(0))|\Omega\rangle=\int\frac{d^{4}p}{(2\pi)^{% 4}}\frac{ie^{-ip\cdot x}}{p\!\!\!/-m-\Sigma(p)+i\epsilon}
  25. e e
  26. m r m_{r}
  27. e e
  28. Π ( q 2 ) \Pi(q^{2})
  29. η μ ν \eta^{\mu\nu}
  30. Ω | T ( A μ ( x ) A ν ( 0 ) ) | Ω = d 4 q ( 2 π ) 4 - i η μ ν e - i p x q 2 ( 1 - Π ( q 2 ) ) + i ϵ = d 4 q ( 2 π ) 4 - i Z 3 η μ ν e - i p x q 2 + i ϵ \langle\Omega|T(A^{\mu}(x)A^{\nu}(0))|\Omega\rangle=\int\frac{d^{4}q}{(2\pi)^{% 4}}\frac{-i\eta^{\mu\nu}e^{-ip\cdot x}}{q^{2}(1-\Pi(q^{2}))+i\epsilon}=\int% \frac{d^{4}q}{(2\pi)^{4}}\frac{-iZ_{3}\eta^{\mu\nu}e^{-ip\cdot x}}{q^{2}+i\epsilon}
  31. δ 3 = Z 3 - 1 \delta_{3}=Z_{3}-1
  32. q q
  33. q 2 0 q^{2}\rightarrow 0
  34. - i η μ ν e - i p x q 2 ( 1 - Π ( q 2 ) ) + i ϵ - i η μ ν e - i p x q 2 \frac{-i\eta^{\mu\nu}e^{-ip\cdot x}}{q^{2}(1-\Pi(q^{2}))+i\epsilon}\sim\frac{-% i\eta^{\mu\nu}e^{-ip\cdot x}}{q^{2}}
  35. δ 3 \delta_{3}
  36. Π ( 0 ) \Pi(0)
  37. e r e_{r}
  38. δ 1 \delta_{1}
  39. δ 2 \delta_{2}
  40. Z i Z_{i}
  41. = - 1 4 F μ ν F μ ν + ψ ¯ ( i - m ) ψ + e ψ ¯ γ μ ψ A μ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\partial-m)\psi+e\bar% {\psi}\gamma^{\mu}\psi A_{\mu}
  42. F μ ν F_{\mu\nu}
  43. ψ \psi
  44. ψ , A , m \psi,\;A,\;m
  45. e e
  46. ψ = Z 2 ψ r A = Z 3 A r m = m r + δ m e = Z 1 Z 2 Z 3 e r w i t h Z i = 1 + δ i \psi=\sqrt{Z_{2}}\psi_{r}\;\;\;\;\;A=\sqrt{Z_{3}}A_{r}\;\;\;\;\;m=m_{r}+\delta m% \;\;\;\;\;e=\frac{Z_{1}}{Z_{2}\sqrt{Z_{3}}}e_{r}\;\;\;\;\;with\;\;\;\;\;Z_{i}=% 1+\delta_{i}
  47. δ i \delta_{i}
  48. = - 1 4 Z 3 F μ ν , r F r μ ν + Z 2 ψ ¯ r ( i - m r ) ψ r - ψ ¯ r δ m ψ r + Z 1 e r ψ ¯ r γ μ ψ r A μ , r \mathcal{L}=-\frac{1}{4}Z_{3}F_{\mu\nu,r}F^{\mu\nu}_{r}+Z_{2}\bar{\psi}_{r}(i% \partial-m_{r})\psi_{r}-\bar{\psi}_{r}\delta m\psi_{r}+Z_{1}e_{r}\bar{\psi}_{r% }\gamma^{\mu}\psi_{r}A_{\mu,r}
  49. ψ \psi
  50. e ψ ¯ γ μ ψ A μ e\bar{\psi}\gamma^{\mu}\psi A_{\mu}

Op_amp_integrator.html

  1. v i n R 1 \frac{v_{in}}{R_{1}}
  2. i 1 = I B + i F i_{\,\text{1}}=I_{\,\text{B}}+i_{\,\text{F}}
  3. I B = 0 I_{\,\text{B}}=0
  4. i 1 = i F i_{\,\text{1}}=i_{\,\text{F}}
  5. I C = C d V c d t I_{\,\text{C}}=C\frac{dV_{\,\text{c}}}{dt}
  6. v in - v 2 R 1 = C F d ( v 2 - v o ) d t \frac{v_{\,\text{in}}-v_{\,\text{2}}}{R_{\,\text{1}}}=C_{\,\text{F}}\frac{d(v_% {\,\text{2}}-v_{\,\text{o}})}{dt}
  7. v 2 = v 1 = 0 v_{2}=v_{1}=0
  8. v in R 1 = - C F d v o d t \frac{v_{\,\text{in}}}{R_{\,\text{1}}}=-C_{\,\text{F}}\frac{dv_{\,\text{o}}}{dt}
  9. 0 t v in R 1 d t = - 0 t C F d v o d t d t \int_{0}^{t}\frac{v_{\,\text{in}}}{R_{\,\text{1}}}\ dt\ =-\int_{0}^{t}C_{\,% \text{F}}\frac{dv_{\,\text{o}}}{dt}\,dt
  10. v o = - 1 R 1 C F 0 t v in d t v_{\,\text{o}}=-\frac{1}{R_{\,\text{1}}C_{\,\text{F}}}\int_{0}^{t}v_{\,\text{% in}}\,dt
  11. I B I_{B}
  12. v in = 0 v_{\,\text{in}}=0
  13. I B I_{B}
  14. v in v_{\,\text{in}}
  15. R on = R 1 || R f || R L R_{\,\text{on}}=R_{1}||R_{f}||R_{L}
  16. V E = ( R f R 1 + 1 ) V I O S V\text{E}=\left(\frac{R\text{f}}{R_{1}}+1\right)V_{IOS}
  17. R F R_{F}
  18. V E = ( R f R 1 + 1 ) ( V I O S + I B I ( R f R 1 ) ) V\text{E}=\left(\frac{R\text{f}}{R_{1}}+1\right)\left(V_{IOS}+I_{BI}\left(R% \text{f}\parallel R_{1}\right)\right)
  19. V I O S V_{IOS}
  20. I B I I_{BI}
  21. R f R 1 R_{f}\parallel R_{1}
  22. f b f\text{b}
  23. f b = 1 2 π R 1 C F f_{\,\text{b}}=\frac{1}{{2\pi}{R_{\,\text{1}}}{C_{\,\text{F}}}}
  24. f a f\text{a}
  25. f a = 1 2 π R F C F f_{\,\text{a}}=\frac{1}{{2\pi}{R_{\,\text{F}}}{C_{\,\text{F}}}}
  26. [ f a , f b ] \left[f\text{a},f\text{b}\right]
  27. f a < f b f\text{a}<f\text{b}
  28. R F C F R\text{F}C\text{F}
  29. R 1 C F R_{1}C\text{F}

Oppermann's_conjecture.html

  1. g n < p n g_{n}<\sqrt{p_{n}}\,

Opposite_group.html

  1. G G
  2. * *
  3. G G
  4. G o p G^{op}
  5. G G
  6. \mathbin{\ast^{\prime}}
  7. g 1 , g 2 = g 2 * g 1 g_{1}\mathbin{\ast^{\prime}}g_{2}=g_{2}*g_{1}
  8. G G
  9. G G
  10. φ : G G o p \varphi:G\to G^{op}
  11. φ ( x ) = x - 1 \varphi(x)=x^{-1}
  12. ψ : G G \psi:G\to G
  13. ψ : G G o p \psi^{\prime}:G\to G^{op}
  14. ψ ( g ) = ψ ( g ) \psi^{\prime}(g)=\psi(g)
  15. ψ ( g * h ) = ψ ( g * h ) = ψ ( h ) * ψ ( g ) = ψ ( g ) , ψ ( h ) = ψ ( g ) , ψ ( h ) . \psi^{\prime}(g*h)=\psi(g*h)=\psi(h)*\psi(g)=\psi(g)\mathbin{\ast^{\prime}}% \psi(h)=\psi^{\prime}(g)\mathbin{\ast^{\prime}}\psi^{\prime}(h).
  16. X X
  17. ρ : G Aut ( X ) \rho:G\to\mathrm{Aut}(X)
  18. ρ o p : G o p Aut ( X ) \rho^{op}:G^{op}\to\mathrm{Aut}(X)
  19. ρ o p ( g ) x = ρ ( g ) x \rho^{op}(g)x=\rho(g)x
  20. g o p x = x g g^{op}x=xg

Optical_conductivity.html

  1. 𝐉 \mathbf{J}
  2. σ \sigma
  3. 𝐄 \mathbf{E}
  4. 𝐉 ( ω ) = σ ( ω ) 𝐄 ( ω ) \mathbf{J}(\omega)=\sigma(\omega)\mathbf{E}(\omega)
  5. ε \varepsilon
  6. 𝐃 ( ω ) = ε ( ω ) 𝐄 ( ω ) \mathbf{D}(\omega)=\varepsilon(\omega)\mathbf{E}(\omega)
  7. ε ( ω ) = ε 0 + i σ ( ω ) ω \varepsilon(\omega)=\varepsilon_{0}+\frac{i\sigma(\omega)}{\omega}
  8. ε 0 \varepsilon_{0}
  9. i i

Optical_depth_(astrophysics).html

  1. τ \tau
  2. z z
  3. τ \tau
  4. τ 0 z ( α ) d z = σ N \tau\equiv\int_{0}^{z}(\alpha)dz=\sigma N
  5. α \alpha
  6. α \alpha
  7. α \alpha
  8. α = e 4 π κ λ 0 \alpha=e^{\frac{4\pi\kappa}{\lambda_{0}}}
  9. κ \kappa
  10. λ 0 \lambda_{0}
  11. λ 0 \lambda_{0}
  12. τ \tau
  13. τ \tau
  14. α \alpha
  15. z = 0 z=0
  16. z = z z=z
  17. τ = 2 3 \tau=\frac{2}{3}
  18. H - H^{-}
  19. T 4 = 3 4 T e 4 ( τ + 2 3 ) T^{4}=\frac{3}{4}T_{e}^{4}\left(\tau+\frac{2}{3}\right)
  20. T e T_{e}
  21. τ \tau
  22. τ \tau
  23. α \alpha
  24. T 4 = 3 4 T e 4 ( 0 z ( α ) d z + 2 3 ) T^{4}=\frac{3}{4}T_{e}^{4}\left(\int_{0}^{z}(\alpha)dz+\frac{2}{3}\right)
  25. τ \tau
  26. α \alpha

Optical_overheating_protection.html

  1. θ c = S i n - 1 ( n 1 n 2 ) , n 1 n 2 1 \theta_{c}=Sin^{-1}(\frac{n_{1}}{n_{2}}),\;\frac{n_{1}}{n_{2}}\leq 1

Optical_unit.html

  1. u axial = 8 π n λ sin 2 ( α 2 ) z u_{\mathrm{axial}}=\frac{8\pi n}{\lambda}\sin^{2}(\frac{\alpha}{2})z
  2. v radial = 2 π λ n sin α M tot r v_{\mathrm{radial}}=\frac{2\pi}{\lambda}\frac{n\sin\alpha}{M_{\mathrm{tot}}}r
  3. n n
  4. λ \lambda
  5. α \alpha
  6. M tot M_{\mathrm{tot}}
  7. M M
  8. n * sin α n*\sin\alpha

Optimal_estimation.html

  1. 𝐀 x = y \mathbf{A}\vec{x}=\vec{y}
  2. x \vec{x}
  3. y \vec{y}
  4. P ( y | x ) P(\vec{y}|\vec{x})
  5. P ( y | x ) = 1 ( 2 π ) m n / 2 | s y m b o l S y | exp [ - 1 2 ( s y m b o l A x - y ) T s y m b o l S y - 1 ( s y m b o l A x - y ) ] P(\vec{y}|\vec{x})=\frac{1}{(2\pi)^{mn/2}|symbol{S_{y}}|}\exp\left[-\frac{1}{2% }(symbol{A}\vec{x}-\vec{y})^{T}symbol{S_{y}}^{-1}(symbol{A}\vec{x}-\vec{y})\right]
  6. x \vec{x}
  7. y \vec{y}
  8. s y m b o l A symbol{A}
  9. s y m b o l S y symbol{S_{y}}
  10. y \vec{y}
  11. x \vec{x}
  12. P ( x ) = 1 ( 2 π ) m / 2 | s y m b o l S x a | exp [ - 1 2 ( x - x a ^ ) T s y m b o l S x a - 1 ( x - x a ^ ) ] P(\vec{x})=\frac{1}{(2\pi)^{m/2}|symbol{S_{x_{a}}}|}\exp\left[-\frac{1}{2}(% \vec{x}-\widehat{x_{a}})^{T}symbol{S_{x_{a}}}^{-1}(\vec{x}-\widehat{x_{a}})\right]
  13. P ( x ) P(\vec{x})
  14. x a ^ \widehat{x_{a}}
  15. x \vec{x}
  16. s y m b o l S x a symbol{S_{x_{a}}}
  17. P ( x | y ) P(\vec{x}|\vec{y})
  18. P ( x | y ) = 1 ( 2 π ) m n / 2 | s y m b o l S x | exp [ - 1 2 ( x - x ^ ) T s y m b o l S x - 1 ( x - x ^ ) ] P(\vec{x}|\vec{y})=\frac{1}{(2\pi)^{mn/2}|symbol{S_{x}}|}\exp\left[-\frac{1}{2% }(\vec{x}-\widehat{x})^{T}symbol{S_{x}}^{-1}(\vec{x}-\widehat{x})\right]
  19. P ( y ) P(\vec{y})
  20. x \vec{x}
  21. x \vec{x}
  22. x ^ \widehat{x}
  23. P ( x | y ) P(\vec{x}|\vec{y})
  24. P ( y | x ) P ( x ) P(\vec{y}|\vec{x})P(\vec{x})
  25. s y m b o l S x = ( s y m b o l A T s y m b o l S y - 1 s y m b o l A + s y m b o l S x a - 1 ) - 1 symbol{S_{x}}=(symbol{A}^{T}symbol{S_{y}^{-1}}symbol{A}+symbol{S_{x_{a}}^{-1}}% )^{-1}
  26. x ^ = x a ^ + s y m b o l S x s y m b o l A T s y m b o l S y - 1 ( y - s y m b o l A x a ^ ) \widehat{x}=\widehat{x_{a}}+symbol{S_{x}}symbol{A}^{T}symbol{S_{y}}^{-1}(\vec{% y}-symbol{A}\widehat{x_{a}})
  27. s y m b o l R = ( s y m b o l A T s y m b o l S y - 1 s y m b o l A + s y m b o l S x a - 1 ) - 1 s y m b o l A T s y m b o l S y - 1 s y m b o l A symbol{R}=(symbol{A}^{T}symbol{S_{y}}^{-1}symbol{A}+symbol{S_{x_{a}}}^{-1})^{-% 1}symbol{A}^{T}symbol{S_{y}}^{-1}symbol{A}

Orbital-free_density_functional_theory.html

  1. | ϕ i |\phi_{i}\rangle
  2. E k i n e t i c = - 1 2 ϕ i | 2 | ϕ i E_{kinetic}=-\frac{1}{2}\langle\phi_{i}|\nabla^{2}|\phi_{i}\rangle
  3. E T F = 3 10 ( 3 π 2 ) 2 3 [ n ( r ) ] 5 3 d 3 r E_{TF}=\frac{3}{10}\left(3\pi^{2}\right)^{\frac{2}{3}}\int{\left[n\left(\vec{r% }\right)\right]^{\frac{5}{3}}d^{3}r}

Orbital_angular_momentum_of_light.html

  1. 𝐏 \mathbf{P}
  2. 𝐋 e = 𝐫 × 𝐏 \mathbf{L}_{e}=\mathbf{r}\times\mathbf{P}
  3. m m
  4. m = 0 m=0
  5. m = ± 1 m=\pm 1
  6. m m
  7. λ \lambda
  8. | m | 2 |m|\geqslant 2
  9. | m | |m|
  10. | m | λ |m|\lambda
  11. m m
  12. m m
  13. m m\hbar
  14. 𝐋 = ϵ 0 i = x , y , z ( E i ( 𝐫 × ) A i ) d 3 𝐫 , \mathbf{L}=\epsilon_{0}\sum_{i=x,y,z}\int\left(E^{i}\left(\mathbf{r}\times% \mathbf{\nabla}\right)A^{i}\right)d^{3}\mathbf{r},
  15. 𝐄 \mathbf{E}
  16. 𝐀 \mathbf{A}
  17. ϵ 0 \epsilon_{0}
  18. i i
  19. 𝐋 = ϵ 0 2 i ω i = x , y , z ( E i ( 𝐫 × ) E i ) d 3 𝐫 . \mathbf{L}=\frac{\epsilon_{0}}{2i\omega}\sum_{i=x,y,z}\int\left({E^{i}}^{\ast}% \left(\mathbf{r}\times\mathbf{\nabla}\right)E^{i}\right)d^{3}\mathbf{r}.
  20. 𝐋 z = m . \mathbf{L}_{z}=m\hbar.
  21. 𝐫 | m e i m ϕ . \langle\mathbf{r}|m\rangle\propto e^{im\phi}.
  22. ϕ \phi
  23. l = ± 1 l=\pm 1
  24. l l
  25. l l
  26. s = l λ / ( n - 1 ) s=l\lambda/(n-1)
  27. n n
  28. l = 0 l=0
  29. l = 1 l=1
  30. l > 1 l>1
  31. l l
  32. l = - 3 l=-3
  33. l = 3 l=3

Orbital_magnetization.html

  1. 𝐦 orb = 1 2 d 3 𝐫 𝐫 × 𝐉 ( 𝐫 ) \mathbf{m}_{\rm orb}=\frac{1}{2}\int d^{3}\mathbf{r}\,\mathbf{r}\times\mathbf{% J}(\mathbf{r})
  2. 𝐦 orb = - e 2 m e Ψ | 𝐋 | Ψ \mathbf{m}_{\rm orb}=\frac{-e}{2m_{e}}\langle\Psi|\mathbf{L}|\Psi\rangle
  3. 𝐦 = 𝐦 orb + 𝐦 spin \mathbf{m}=\mathbf{m}_{\rm orb}+\mathbf{m}_{\rm spin}
  4. 𝐦 spin = - g s μ B Ψ | 𝐒 | Ψ \mathbf{m}_{\rm spin}=\frac{-g_{s}\mu_{\rm B}}{\hbar}\,\langle\Psi|\mathbf{S}|\Psi\rangle
  5. 𝐌 orb = 1 V j V 𝐦 orb , j . \mathbf{M}_{\rm orb}=\frac{1}{V}\sum_{j\in V}\mathbf{m}_{{\rm orb},j}\;.
  6. 𝐌 orb = 1 2 V V d 3 𝐫 𝐫 × 𝐉 ( 𝐫 ) \mathbf{M}_{\rm orb}=\frac{1}{2V}\int_{V}d^{3}\mathbf{r}\,\mathbf{r}\times% \mathbf{J}(\mathbf{r})
  7. 𝐌 orb = - e 2 m e n BZ d 3 k ( 2 π ) 3 ψ n 𝐤 | 𝐫 × 𝐩 | ψ n 𝐤 , \mathbf{M}_{\rm orb}=\frac{-e}{2m_{e}}\sum_{n}\int_{\rm BZ}\frac{d^{3}k}{(2\pi% )^{3}}\,\langle\psi_{n\mathbf{k}}|\mathbf{r}\times\mathbf{p}|\psi_{n\mathbf{k}% }\rangle\,,
  8. 𝐌 orb = e 2 n BZ d 3 k ( 2 π ) 3 f n 𝐤 Im u n 𝐤 𝐤 | × ( H 𝐤 + E n 𝐤 - 2 μ ) | u n 𝐤 𝐤 , \mathbf{M}_{\rm orb}=\frac{e}{2\hbar}\sum_{n}\int_{\rm BZ}\frac{d^{3}k}{(2\pi)% ^{3}}\,f_{n\mathbf{k}}\;{\rm Im}\;\langle\frac{\partial u_{n\mathbf{k}}}{% \partial{\mathbf{k}}}|\times(H_{\mathbf{k}}+E_{n\mathbf{k}}-2\mu)|\frac{% \partial u_{n\mathbf{k}}}{\partial{\mathbf{k}}}\rangle,
  9. H 𝐤 = e i 𝐤 𝐫 H e - i 𝐤 𝐫 H_{\mathbf{k}}=e^{i\mathbf{k}\cdot\mathbf{r}}He^{-i\mathbf{k}\cdot\mathbf{r}}
  10. u n 𝐤 ( 𝐫 ) = e - i 𝐤 𝐫 ψ n 𝐤 ( 𝐫 ) u_{n\mathbf{k}}(\mathbf{r})=e^{-i\mathbf{k}\cdot\mathbf{r}}\psi_{n\mathbf{k}}(% \mathbf{r})
  11. H 𝐤 | u n 𝐤 = E n 𝐤 | u n 𝐤 . H_{\mathbf{k}}|u_{n\mathbf{k}}\rangle=E_{n\mathbf{k}}|u_{n\mathbf{k}}\rangle\;.
  12. γ = 1 + M orb ( M spin + M orb ) \gamma=1+\frac{M_{\mathrm{orb}}}{(M_{\mathrm{spin}}+M_{\mathrm{orb}})}

Orbital_overlap.html

  1. 𝐒 AB = Ψ A * Ψ B d V , \mathbf{S}_{\mathrm{AB}}=\int\Psi_{\mathrm{A}}^{*}\Psi_{\mathrm{B}}\,dV,
  2. 𝐒 j k = b j | b k = Ψ j * Ψ k d τ \mathbf{S}_{jk}=\left\langle b_{j}|b_{k}\right\rangle=\int\Psi_{j}^{*}\Psi_{k}% \,d\tau
  3. | b j \left|b_{j}\right\rangle
  4. Ψ j \Psi_{j}
  5. Ψ j ( x ) = x | b j \Psi_{j}(x)=\left\langle x|b_{j}\right\rangle

Orbital_perturbation_analysis_(spacecraft).html

  1. g ( x 1 , x 2 , x 3 , v 1 , v 2 , v 3 ) g(x_{1},x_{2},x_{3},v_{1},v_{2},v_{3})\,
  2. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}\,
  3. v 1 , v 2 , v 3 v_{1},v_{2},v_{3}\,
  4. g g
  5. g ˙ = g x 1 v 1 + g x 2 v 2 + g x 3 v 3 + g v 1 f 1 + g v 2 f 2 + g v 3 f 3 \dot{g}\ =\ \frac{\partial g}{\partial x_{1}}\ v_{1}\ +\ \frac{\partial g}{% \partial x_{2}}\ v_{2}\ +\frac{\partial g}{\partial x_{3}}\ v_{3}\ +\ \frac{% \partial g}{\partial v_{1}}\ f_{1}\ +\ \frac{\partial g}{\partial v_{2}}\ f_{2% }\ +\ \frac{\partial g}{\partial v_{3}}\ f_{3}
  6. f 1 , f 2 , f 3 f_{1}\ ,\ f_{2}\ ,\ f_{3}
  7. g g
  8. ( f 1 , f 2 , f 3 ) = - μ r 3 ( x 1 , x 2 , x 3 ) (f_{1}\ ,\ f_{2}\ ,\ f_{3})\ =\ -\frac{\mu}{r^{3}}\ (x_{1}\ ,\ x_{2}\ ,\ x_{3})
  9. g ˙ = 0 \dot{g}\ =\ 0\,
  10. ( h 1 , h 2 , h 3 ) (h_{1}\ ,\ h_{2}\ ,\ h_{3})
  11. ( f 1 , f 2 , f 3 ) = - μ r 3 ( x 1 , x 2 , x 3 ) + ( h 1 , h 2 , h 3 ) (f_{1}\ ,\ f_{2}\ ,\ f_{3})\ =\ -\frac{\mu}{r^{3}}\ (x_{1}\ ,\ x_{2}\ ,\ x_{3}% )\ +\ (h_{1}\ ,\ h_{2}\ ,\ h_{3})
  12. g ˙ = g v 1 h 1 + g v 2 h 2 + g v 3 h 3 \dot{g}\ =\frac{\partial g}{\partial v_{1}}\ h_{1}\ +\ \frac{\partial g}{% \partial v_{2}}\ h_{2}\ +\ \frac{\partial g}{\partial v_{3}}\ h_{3}
  13. g g\,
  14. t = t 1 t=t_{1}\,
  15. t = t 2 t=t_{2}\,
  16. Δ g = t 1 t 2 ( g v 1 h 1 + g v 2 h 2 + g v 3 h 3 ) d t \Delta g\ =\ \int\limits_{t_{1}}^{t_{2}}\left(\frac{\partial g}{\partial v_{1}% }\ h_{1}\ +\ \frac{\partial g}{\partial v_{2}}\ h_{2}\ +\ \frac{\partial g}{% \partial v_{3}}\ h_{3}\right)dt
  17. ( h 1 , h 2 , h 3 ) (h_{1}\ ,\ h_{2}\ ,\ h_{3})\,
  18. Δ g \Delta g\,
  19. x 1 ( t ) , x 2 ( t ) , x 3 ( t ) x_{1}(t),x_{2}(t),x_{3}(t)\,
  20. Δ g \Delta g\,
  21. θ \theta\,
  22. Δ g = 0 2 π ( g v 1 h 1 + g v 2 h 2 + g v 3 h 3 ) r 2 μ p d θ \Delta g\ =\ \int\limits_{0}^{2\pi}\left(\frac{\partial g}{\partial v_{1}}\ h_% {1}\ +\ \frac{\partial g}{\partial v_{2}}\ h_{2}\ +\ \frac{\partial g}{% \partial v_{3}}\ h_{3}\right)\frac{r^{2}}{\sqrt{\mu p}}d\theta
  23. r ( θ ) = p 1 + e cos θ r(\theta)=\frac{p}{1+e\cos\theta}\,
  24. H = r 2 θ ˙ = μ p H\ =\ r^{2}\ \dot{\theta}\ =\ \sqrt{\mu p}\,
  25. p p\,
  26. r = p r\ =\ p
  27. Δ g = P 2 π 0 2 π ( g v 1 h 1 + g v 2 h 2 + g v 3 h 3 ) d θ \Delta g\ =\ \frac{P}{2\pi}\ \int\limits_{0}^{2\pi}\left(\frac{\partial g}{% \partial v_{1}}\ h_{1}\ +\ \frac{\partial g}{\partial v_{2}}\ h_{2}\ +\ \frac{% \partial g}{\partial v_{3}}\ h_{3}\right)d\theta
  28. P = 2 π r r μ P\ =\ 2\pi\ r\ \sqrt{\frac{r}{\mu}}\,
  29. g = V 2 2 - μ r g=\frac{V^{2}}{2}-\frac{\mu}{r}
  30. V V\,
  31. g = - μ 2 a g\ =\ -\frac{\mu}{2\cdot a}
  32. h ¯ \bar{h}\,
  33. V ¯ \bar{V}\,
  34. Δ g = 0 2 π V ¯ h ¯ r 2 μ p d θ \Delta g\ =\ \int\limits_{0}^{2\pi}\bar{V}\bar{h}\frac{r^{2}}{\sqrt{\mu p}}d\theta
  35. Δ g = P 2 π 0 2 π V ¯ h ¯ d θ \Delta g\ =\ \frac{P}{2\pi}\ \int\limits_{0}^{2\pi}\bar{V}\bar{h}d\theta
  36. Δ g \Delta g\,
  37. g g\,
  38. a a\,
  39. P = 2 π a a μ P\ =\ 2\pi\ a\ \sqrt{\frac{a}{\mu}}\,
  40. g ^ \hat{g}\,
  41. h ^ \hat{h}\,
  42. ω \omega\,
  43. g ^ \hat{g}\,
  44. h ^ \hat{h}\,
  45. θ \theta\,
  46. θ = u - ω \theta=u-\omega\,
  47. Δ z ^ \Delta\hat{z}\,
  48. z ^ \hat{z}\,
  49. Δ z ^ = 0 2 π f z V t ( g ^ cos u + h ^ sin u ) r 2 μ p d u × z ^ = 1 μ p [ g ^ 0 2 π f z r 3 cos u d u + h ^ 0 2 π f z r 3 sin u d u ] × z ^ \Delta\hat{z}\ =\ \int\limits_{0}^{2\pi}\frac{f_{z}}{V_{t}}(\hat{g}\cos u+\hat% {h}\sin u)\frac{r^{2}}{\sqrt{\mu p}}du\quad\times\ \hat{z}=\ \frac{1}{\mu p}% \left[\hat{g}\int\limits_{0}^{2\pi}f_{z}r^{3}\cos u\ du+\ \hat{h}\int\limits_{% 0}^{2\pi}f_{z}r^{3}\sin u\ du\right]\quad\times\ \hat{z}
  50. f z f_{z}\,
  51. z ^ \hat{z}\,
  52. V t = μ p ( 1 + e cos θ ) V_{t}=\sqrt{\frac{\mu}{p}}\ (1+e\ \cos\theta)\,
  53. r = p 1 + e cos θ r=\frac{p}{1+e\ \cos\theta}\,
  54. Δ z ^ = r 2 μ [ g ^ 0 2 π f z cos u d u + h ^ 0 2 π f z sin u d u ] × z ^ \Delta\hat{z}\ =\ \frac{r^{2}}{\mu}\left[\hat{g}\int\limits_{0}^{2\pi}f_{z}% \cos u\ du+\ \hat{h}\int\limits_{0}^{2\pi}f_{z}\sin u\ du\right]\quad\times\ % \hat{z}
  55. F z ^ F\ \hat{z}\,
  56. sin u \sin u\,
  57. P = 2 π r r μ P\ =\ 2\pi\ r\ \sqrt{\frac{r}{\mu}}\,
  58. Δ z ^ = r 2 μ [ 2 F 0 π sin u d u ] h ^ × z ^ = r 2 μ 4 F g ^ \Delta\hat{z}\ =\ \frac{r^{2}}{\mu}\left[\ 2\ F\int\limits_{0}^{\pi}\sin u\ du% \right]\quad\hat{h}\times\hat{z}=\ \frac{r^{2}}{\mu}\ 4\ F\ \quad\hat{g}
  59. Δ z ^ P \frac{\Delta\hat{z}}{P}\,
  60. Δ z ^ P = 2 π F V g ^ \frac{\Delta\hat{z}}{P}=\ \frac{2}{\pi}\ \frac{F}{V}\ \hat{g}
  61. V = μ r , V\ =\ \sqrt{\frac{\mu}{r}},
  62. e ¯ = ( V t - V 0 ) r ^ - V r t ^ V 0 \bar{e}=\frac{(V_{t}-V_{0})\cdot\hat{r}-V_{r}\cdot\hat{t}}{V_{0}}
  63. V 0 = μ p V_{0}=\sqrt{\frac{\mu}{p}}
  64. p = ( r V t ) 2 μ p=\frac{{(r\cdot V_{t})}^{2}}{\mu}
  65. e ¯ V r = - 1 V 0 t ^ \frac{\partial\bar{e}}{\partial V_{r}}=-\frac{1}{V_{0}}\hat{t}
  66. e ¯ V t = 1 V 0 ( 2 r ^ - V r V t t ^ ) \frac{\partial\bar{e}}{\partial V_{t}}=\frac{1}{V_{0}}\left(2\ \hat{r}-\frac{V% _{r}}{V_{t}}\ \hat{t}\right)
  67. V r = μ p e sin θ V_{r}=\sqrt{\frac{\mu}{p}}\cdot e\cdot\sin\theta
  68. V t = μ p ( 1 + e cos θ ) V_{t}=\sqrt{\frac{\mu}{p}}\cdot(1+e\cdot\cos\theta)
  69. r ^ \hat{r}
  70. t ^ \hat{t}
  71. e ¯ V z = - V r V 0 t ^ V z \frac{\partial\bar{e}}{\partial V_{z}}=-\frac{V_{r}}{V_{0}}\ \frac{\partial% \hat{t}}{\partial V_{z}}
  72. t ^ V z = 1 V t z ^ \frac{\partial\hat{t}}{\partial V_{z}}=\frac{1}{V_{t}}\ \hat{z}
  73. Δ e ¯ = 1 V 0 0 2 π ( - t ^ f r + ( 2 r ^ - V r V t t ^ ) f t ) r 2 μ p d u = 1 μ 0 2 π ( - t ^ f r + ( 2 r ^ - V r V t t ^ ) f t ) r 2 d u \begin{aligned}\displaystyle\Delta\bar{e}\ =&\displaystyle\frac{1}{V_{0}}\ % \int\limits_{0}^{2\pi}\left(-\hat{t}\ f_{r}\ +\ \left(2\ \hat{r}-\frac{V_{r}}{% V_{t}}\ \hat{t}\right)\ f_{t}\right)\frac{r^{2}}{\sqrt{\mu p}}du\ =\\ &\displaystyle\frac{1}{\mu}\ \int\limits_{0}^{2\pi}\left(-\hat{t}\ f_{r}\ +\ % \left(2\ \hat{r}-\frac{V_{r}}{V_{t}}\ \hat{t}\right)\ f_{t}\right)r^{2}du\end{aligned}
  74. e ¯ + Δ e ¯ \bar{e}+\Delta\bar{e}
  75. z ^ + Δ z ^ \hat{z}+\Delta\hat{z}
  76. r r\,
  77. V 0 = μ r V_{0}\ =\ \sqrt{\frac{\mu}{r}}
  78. k ^ \hat{k}\,
  79. l ^ \hat{l}\,
  80. k ^ \hat{k}\,
  81. F F\,
  82. r ^ = cos ( u ) k ^ + sin ( u ) l ^ \hat{r}=\cos(u)\ \hat{k}\ +\ \sin(u)\ \hat{l}\,
  83. t ^ = - sin ( u ) k ^ + cos ( u ) l ^ \hat{t}=-\sin(u)\ \hat{k}\ +\ \cos(u)\ \hat{l}\,
  84. F r = - cos ( u ) F F_{r}=-\cos(u)\ F\,
  85. F t = sin ( u ) F F_{t}=\sin(u)\ F\,
  86. u u\,
  87. r ^ \hat{r}\,
  88. k ^ \hat{k}\,
  89. l ^ \hat{l}\,
  90. Δ e ^ = P 2 π 1 V 0 0 2 π ( ( - sin ( u ) k ^ + cos ( u ) l ^ ) F cos ( u ) + 2 ( cos ( u ) k ^ + sin ( u ) l ^ ) F sin ( u ) ) d u = P 3 2 1 V 0 F l ^ \begin{aligned}\displaystyle\Delta\hat{e}&\displaystyle=\ \frac{P}{2\pi}\ % \frac{1}{V_{0}}\ \int\limits_{0}^{2\pi}\left((-\sin(u)\ \hat{k}\ +\ \cos(u)\ % \hat{l})\ F\ \cos(u)\ +\ 2\ (\cos(u)\ \hat{k}\ +\ \sin(u)\ \hat{l})\ F\ \sin(u% )\right)\ du\\ &\displaystyle=P\ \frac{3}{2}\ \frac{1}{V_{0}}\ \ F\ \hat{l}\end{aligned}
  91. l ^ \hat{l}\,
  92. P P\,
  93. 3 2 F V 0 \frac{3}{2}\ \frac{F}{V_{0}}\,
  94. F r r ^ F_{r}\ \hat{r}\,
  95. F λ λ ^ F_{\lambda}\ \hat{\lambda}\,
  96. λ ^ \hat{\lambda}\,
  97. r ^ \hat{r}\,
  98. λ ^ \hat{\lambda}\,
  99. t ^ \hat{t}\,
  100. r ^ \hat{r}\,
  101. z ^ \hat{z}\,
  102. F λ F_{\lambda}
  103. F λ λ ^ F_{\lambda}\ \hat{\lambda}\,
  104. F t t ^ F_{t}\ \hat{t}
  105. F z z ^ F_{z}\ \hat{z}
  106. a ^ , b ^ , n ^ \hat{a}\ ,\ \hat{b}\ ,\ \hat{n}\,
  107. n ^ \hat{n}\,
  108. a ^ , b ^ \hat{a}\ ,\ \hat{b}\,
  109. a ^ \hat{a}\,
  110. r ^ , t ^ , z ^ \hat{r}\ ,\ \hat{t}\ ,\ \hat{z}\,
  111. t ^ , z ^ , \hat{t}\ ,\ \hat{z},
  112. a ^ , b ^ , n ^ \hat{a}\ ,\ \hat{b}\ ,\ \hat{n}\,
  113. r a = cos u r_{a}=\cos u\,
  114. r b = cos i sin u r_{b}=\cos i\ \sin u\,
  115. r n = sin i sin u r_{n}=\sin i\ \sin u\,
  116. t a = - sin u t_{a}=-\sin u\,
  117. t b = cos i cos u t_{b}=\cos i\ \cos u\,
  118. t n = sin i cos u t_{n}=\sin i\ \cos u\,
  119. z a = 0 z_{a}=0\,
  120. z b = - sin i z_{b}=-\sin i\,
  121. z n = cos i z_{n}=\cos i\,
  122. u u\,
  123. r ^ \hat{r}\,
  124. g ^ = a ^ \hat{g}=\hat{a}\,
  125. h ^ = cos i b ^ + sin i n ^ \hat{h}=\cos i\ \hat{b}\ +\ \sin i\ \hat{n}\,
  126. sin λ = r n = sin i sin u \sin\lambda=\ r_{n}\ =\ \sin i\ \sin u\,
  127. λ \lambda\,
  128. r ^ \hat{r}\,
  129. f r = J 2 1 r 4 3 2 ( 3 sin 2 i sin 2 u - 1 ) f_{r}=J_{2}\ \frac{1}{r^{4}}\ \frac{3}{2}\ \left(3\ \sin^{2}i\ \sin^{2}u\ -\ 1\right)
  130. n ^ \hat{n}\,
  131. t ^ , z ^ , \hat{t}\ ,\ \hat{z},
  132. sin i cos u t ^ + cos i z ^ \sin i\ \cos u\ \hat{t}\ +\ \cos i\ \hat{z}\,
  133. cos λ λ ^ \cos\lambda\ \hat{\lambda}\,
  134. λ ^ \hat{\lambda}\,
  135. λ ^ \hat{\lambda}
  136. f λ λ ^ = - J 2 1 r 4 3 sin λ ( sin i cos u t ^ + cos i z ^ ) = - J 2 1 r 4 3 sin i sin u ( sin i cos u t ^ + cos i z ^ ) f_{\lambda}\ \hat{\lambda}\ =\ -J_{2}\ \frac{1}{r^{4}}\ 3\ \sin\lambda\ (\sin i% \ \cos u\ \hat{t}\ +\ \cos i\ \hat{z})=\ -J_{2}\ \frac{1}{r^{4}}\ 3\ \sin i\ % \sin u\ (\sin i\ \cos u\ \hat{t}\ +\ \cos i\ \hat{z})\,
  137. f t = - J 2 1 r 4 3 sin 2 i sin u cos u f_{t}=\ -J_{2}\ \frac{1}{r^{4}}\ 3\ \sin^{2}i\ \sin u\ \cos u
  138. f z = - J 2 1 r 4 3 sin i cos i sin u f_{z}=\ -J_{2}\ \frac{1}{r^{4}}\ 3\ \sin i\ \cos i\ \sin u
  139. Δ z ^ = - J 2 3 sin i cos i μ p 2 [ g ^ 0 2 π p r sin u cos u d u + h ^ 0 2 π p r sin 2 u d u ] × z ^ \Delta\hat{z}\ =\ -J_{2}\ \frac{3\ \sin i\ \cos i}{\mu p^{2}}\left[\hat{g}\int% \limits_{0}^{2\pi}\frac{p}{r}\ \sin u\ \cos u\ du+\ \hat{h}\int\limits_{0}^{2% \pi}\frac{p}{r}\ \sin^{2}u\ du\right]\quad\times\ \hat{z}
  140. p r \frac{p}{r}\,
  141. p r = 1 + e cos ( u - ω ) = 1 + e cos u cos ω + e sin u sin ω \frac{p}{r}\ =\ 1\ +\ e\ \cos(u-\omega)\ =\ 1\ +\ e\ \cos u\ \cos\omega\ +\ e% \ \sin u\ \sin\omega\,
  142. e e\,
  143. ω \omega\,
  144. 0 2 π cos m u sin n u d u \int\limits_{0}^{2\pi}\cos^{m}u\ \sin^{n}u\ du\,
  145. n n\,
  146. m m\,
  147. Δ z ^ = - 2 π J 2 μ p 2 3 2 sin i cos i h ^ × z ^ \Delta\hat{z}\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \sin i\ \cos i% \ \quad\hat{h}\times\hat{z}
  148. n ^ = cos i z ^ + sin i h ^ \hat{n}\ =\ \cos i\ \hat{z}\ +\sin i\ \hat{h}
  149. Δ z ^ = - 2 π J 2 μ p 2 3 2 cos i n ^ × z ^ \Delta\hat{z}\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \cos i\ \quad% \hat{n}\times\hat{z}
  150. n ^ \hat{n}
  151. z ^ \hat{z}\,
  152. n ^ \hat{n}
  153. - 2 π J 2 μ p 2 3 2 cos i -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \cos i\,
  154. Δ Ω = - 2 π J 2 μ p 2 3 2 cos i \Delta\Omega\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \cos i
  155. i i\,
  156. Ω \Omega\,
  157. Δ e ¯ = J 2 μ p 2 0 2 π ( - t ^ ( p r ) 2 3 2 ( 3 sin 2 i sin 2 u - 1 ) - ( 2 r ^ - V r V t t ^ ) ( p r ) 2 3 sin 2 i cos u sin u ) d u \Delta\bar{e}\ =\ \frac{J_{2}}{\mu\ p^{2}}\ \int\limits_{0}^{2\pi}\left(-\hat{% t}\ \left(\frac{p}{r}\right)^{2}\ \frac{3}{2}\ \left(3\ \sin^{2}i\ \sin^{2}u\ % -\ 1\right)\ -\ \left(2\ \hat{r}-\frac{V_{r}}{V_{t}}\ \hat{t}\right)\ \left(% \frac{p}{r}\right)^{2}\ 3\ \sin^{2}i\cos u\ \sin u\right)du
  158. e ¯ + Δ e ¯ \bar{e}+\Delta\bar{e}
  159. z ^ + Δ z ^ \hat{z}+\Delta\hat{z}
  160. Δ z ^ \Delta\hat{z}\,
  161. g ^ = a ^ \hat{g}=\hat{a}\,
  162. h ^ = cos i b ^ + sin i n ^ \hat{h}=\cos i\ \hat{b}\ +\ \sin i\ \hat{n}\,
  163. r ^ = cos u g ^ + sin u h ^ \hat{r}=\cos u\ \hat{g}\ +\ \sin u\ \hat{h}\,
  164. t ^ = - sin u g ^ + cos u h ^ \hat{t}=-\sin u\ \hat{g}\ +\ \cos u\ \hat{h}\,
  165. p r = 1 + e cos θ = 1 + e g cos u + e h sin u \frac{p}{r}\ =\ 1+e\cdot\cos\theta\ =\ 1+e_{g}\cdot\cos u+e_{h}\cdot\sin u
  166. V r V t = e g sin u - e h cos u p r \frac{V_{r}}{V_{t}}=\frac{e_{g}\cdot\sin u\ -\ e_{h}\cdot\cos u}{\frac{p}{r}}
  167. e g = e cos ω e_{g}=\ e\ \cos\omega
  168. e h = e sin ω e_{h}=\ e\ \sin\omega
  169. g ^ , h ^ \hat{g}\ ,\ \hat{h}\,
  170. Δ e ¯ = - 2 π J 2 μ p 2 3 2 ( 3 2 sin 2 i - 1 ) ( - e h g ^ + e g h ^ ) = - 2 π J 2 μ p 2 3 2 ( 3 2 sin 2 i - 1 ) z ^ × e ¯ \Delta\bar{e}\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\left(\frac{3}{2% }\ \sin^{2}i\ -\ 1\right)\ \left(-e_{h}\hat{g}\ +\ e_{g}\hat{h}\right)\ =\ -2% \pi\ \frac{J_{2}}{\mu\ p^{2}}\frac{3}{2}\left(\frac{3}{2}\ \sin^{2}i\ -\ 1% \right)\ \hat{z}\ \times\ \bar{e}
  171. e ¯ \bar{e}\,
  172. e e\,
  173. Δ e ¯ \Delta\bar{e}\ \,
  174. e ¯ \bar{e}\ \,
  175. Δ ω 1 = - 2 π J 2 μ p 2 3 2 ( 3 2 sin 2 i - 1 ) \Delta\omega_{1}\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\left(\frac{3% }{2}\ \sin^{2}i\ -\ 1\right)\,
  176. Δ ω 2 = - cos i Δ Ω = 2 π J 2 μ p 2 3 2 cos 2 i \Delta\omega_{2}\ =\ -\cos i\ \Delta\Omega\ =\ 2\pi\ \frac{J_{2}}{\mu\ p^{2}}% \ \frac{3}{2}\ \cos^{2}i\,
  177. Δ ω = Δ ω 1 + Δ ω 2 = - 2 π J 2 μ p 2 3 ( 5 4 sin 2 i - 1 ) \Delta\omega\ =\Delta\omega_{1}\ +\ \Delta\omega_{2}\ =\ \ -2\pi\ \frac{J_{2}}% {\mu\ p^{2}}\ 3\left(\frac{5}{4}\ \sin^{2}i\ -\ 1\right)
  178. e g , e h e_{g},e_{h}\,
  179. g ^ , h ^ \hat{g},\hat{h}\,
  180. ( Δ e g , Δ e h ) = - 2 π J 2 μ p 2 3 2 ( 3 2 sin 2 i - 1 ) ( - e h , e g ) + 2 π J 2 μ p 2 3 2 cos 2 i ( - e h , e g ) = - 2 π J 2 μ p 2 3 ( 5 4 sin 2 i - 1 ) ( - e h , e g ) \begin{aligned}&\displaystyle(\Delta e_{g},\Delta e_{h})\ =\\ &\displaystyle-2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\left(\frac{3}{2}\ % \sin^{2}i\ -\ 1\right)\ (-e_{h},e_{g})\ +\ 2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ % \frac{3}{2}\ \cos^{2}i\ (-e_{h},e_{g})=\\ &\displaystyle-2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ 3\left(\frac{5}{4}\ \sin^{2}i\ % -\ 1\right)\ (-e_{h},e_{g})\end{aligned}
  181. Δ ω \Delta\omega\,
  182. sin 2 i = 4 5 \sin^{2}i\ =\frac{4}{5}\,
  183. 0 2 π ( - t ^ ( p r ) 2 3 2 ( 3 sin 2 i sin 2 u - 1 ) - ( 2 r ^ - V r V t t ^ ) ( p r ) 2 3 sin 2 i cos u sin u ) d u \int\limits_{0}^{2\pi}\left(-\hat{t}\ \left(\frac{p}{r}\right)^{2}\ \frac{3}{2% }\ \left(3\ \sin^{2}i\ \sin^{2}u\ -\ 1\right)\ -\ \left(2\ \hat{r}-\frac{V_{r}% }{V_{t}}\ \hat{t}\right)\ \left(\frac{p}{r}\right)^{2}\ 3\ \sin^{2}i\cos u\ % \sin u\right)du
  184. r ^ = cos u G ^ + sin u H ^ \hat{r}=\cos u\ \hat{G}\ +\ \sin u\ \hat{H}\,
  185. t ^ = - sin u G ^ + cos u H ^ \hat{t}=-\sin u\ \hat{G}\ +\ \cos u\ \hat{H}\,
  186. p r = 1 + e g cos u + e h sin u \frac{p}{r}\ =\ 1\ +\ e_{g}\ \cos u\ +\ e_{h}\ \sin u
  187. V r V t = e g sin u - e h cos u p r \frac{V_{r}}{V_{t}}\ =\ \frac{e_{g}\ \sin u\ -\ e_{h}\ \cos u}{\frac{p}{r}}
  188. - 2 π 3 2 ( 3 2 sin 2 i - 1 ) ( - e h G ^ + e g H ^ ) -2\pi\ \frac{3}{2}\left(\frac{3}{2}\ \sin^{2}i\ -\ 1\right)\ \left(-e_{h}\hat{% G}\ +\ e_{g}\hat{H}\right)
  189. 0 2 π - t g ( p r ) 2 3 2 3 sin 2 i sin 2 u d u = 9 2 sin 2 i 0 2 π ( 1 + e g cos u + e h sin u ) 2 sin 3 u d u = 9 sin 2 i e h 0 2 π sin 4 u d u = 2 π 27 8 sin 2 i e h \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}-t_{g}\ \left(\frac{p}{r}% \right)^{2}\ \frac{3}{2}\ 3\ \sin^{2}i\ \sin^{2}u\ du\ =\ \frac{9}{2}\ \sin^{2% }i\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_{g}\ \cos u\ +\ e_{h}\ \sin u\right)% ^{2}\ \ \sin^{3}u\ du\ =\\ &\displaystyle 9\ \sin^{2}i\ e_{h}\ \int\limits_{0}^{2\pi}\sin^{4}u\ du\ =\ 2% \pi\frac{27}{8}\ \sin^{2}i\ e_{h}\end{aligned}
  190. 0 2 π - t h ( p r ) 2 3 2 3 sin 2 i sin 2 u d u = - 9 2 sin 2 i 0 2 π ( 1 + e g cos u + e h sin u ) 2 sin 2 u cos u d u = - 9 sin 2 i e g 0 2 π sin 2 u cos 2 u d u = - 2 π 9 8 sin 2 i e g \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}-t_{h}\ \left(\frac{p}{r}% \right)^{2}\ \frac{3}{2}\ 3\ \sin^{2}i\ \sin^{2}u\ du\ =\ -\frac{9}{2}\ \sin^{% 2}i\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_{g}\ \cos u\ +\ e_{h}\ \sin u\right% )^{2}\ \ \sin^{2}u\ \cos u\ du\ =\\ &\displaystyle-9\ \sin^{2}i\ e_{g}\ \int\limits_{0}^{2\pi}\sin^{2}u\ \cos^{2}u% \ du\ =\ -2\pi\frac{9}{8}\ \sin^{2}i\ e_{g}\end{aligned}
  191. 0 2 π t g ( p r ) 2 3 2 d u = - 3 2 0 2 π ( 1 + e g cos u + e h sin u ) 2 sin u d u = - 3 e h 0 2 π sin 2 u d u = - 2 π 3 2 e h \int\limits_{0}^{2\pi}t_{g}\ \left(\frac{p}{r}\right)^{2}\ \frac{3}{2}\ du\ =% \ -\frac{3}{2}\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_{g}\ \cos u\ +\ e_{h}\ % \sin u\right)^{2}\ \ \sin u\ du\ =\ -3\ e_{h}\ \int\limits_{0}^{2\pi}\sin^{2}u% \ du\ =\ -2\pi\frac{3}{2}\ e_{h}
  192. 0 2 π t h ( p r ) 2 3 2 d u = 3 2 0 2 π ( 1 + e g cos u + e h sin u ) 2 cos u d u = 3 e g 0 2 π cos 2 u d u = 2 π 3 2 e g \int\limits_{0}^{2\pi}t_{h}\ \left(\frac{p}{r}\right)^{2}\ \frac{3}{2}\ du\ =% \ \frac{3}{2}\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_{g}\ \cos u\ +\ e_{h}\ % \sin u\right)^{2}\ \ \cos u\ du\ =\ 3\ e_{g}\ \int\limits_{0}^{2\pi}\cos^{2}u% \ du\ =\ 2\pi\frac{3}{2}\ e_{g}
  193. - 0 2 π 2 r g ( p r ) 2 3 sin 2 i cos u sin u d u = - 6 sin 2 i 0 2 π ( 1 + e g cos u + e h sin u ) 2 cos 2 u sin u d u = - 12 sin 2 i e h 0 2 π cos 2 u sin 2 u d u = - 2 π 3 2 sin 2 i e h \begin{aligned}&\displaystyle-\int\limits_{0}^{2\pi}\ 2\ r_{g}\ \left(\frac{p}% {r}\right)^{2}\ 3\ \sin^{2}i\cos u\ \sin u\ du\ =\ -6\ \sin^{2}i\int\limits_{0% }^{2\pi}\ \left(1\ +\ e_{g}\ \cos u\ +\ e_{h}\ \sin u\right)^{2}\ \cos^{2}u\ % \sin u\ du\ =\\ &\displaystyle-12\ \sin^{2}i\ e_{h}\int\limits_{0}^{2\pi}\ \cos^{2}u\ \sin^{2}% u\ du\ =\ -2\pi\frac{3}{2}\ \sin^{2}i\ e_{h}\end{aligned}
  194. - 0 2 π 2 r h ( p r ) 2 3 sin 2 i cos u sin u d u = - 6 sin 2 i 0 2 π ( 1 + e g cos u + e h sin u ) 2 cos u sin 2 u d u = - 12 sin 2 i e g 0 2 π sin 2 u cos 2 u d u = - 2 π 3 2 sin 2 i e g \begin{aligned}&\displaystyle-\int\limits_{0}^{2\pi}\ 2\ r_{h}\ \left(\frac{p}% {r}\right)^{2}\ 3\ \sin^{2}i\cos u\ \sin u\ du\ =\ -6\ \sin^{2}i\int\limits_{0% }^{2\pi}\ \left(1\ +\ e_{g}\ \cos u\ +\ e_{h}\ \sin u\right)^{2}\ \cos u\ \sin% ^{2}u\ du\ =\\ &\displaystyle-12\ \sin^{2}i\ e_{g}\int\limits_{0}^{2\pi}\ \sin^{2}u\cos^{2}u% \ du\ =\ -2\pi\ \frac{3}{2}\ \sin^{2}i\ e_{g}\end{aligned}
  195. 0 2 π t g V r V t ( p r ) 2 3 sin i cos 2 u sin u d u = - 3 sin 2 i 0 2 π ( e g sin u - e h cos u ) p r cos u sin 2 u d u = - 3 sin 2 i 0 2 π ( e g sin u - e h cos u ) ( 1 + e g cos u + e h sin u ) cos u sin 2 u d u = 3 sin 2 i e h 0 2 π cos 2 u sin 2 u d u = 2 π 3 8 sin 2 i e h \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}t_{g}\ \frac{V_{r}}{V_{t}}% \ \left(\frac{p}{r}\right)^{2}\ 3\ \sin i\cos^{2}u\ \sin u\ du\ =-3\ \sin^{2}i% \int\limits_{0}^{2\pi}(e_{g}\ \sin u\ -\ e_{h}\ \cos u)\ \frac{p}{r}\ \cos u\ % \sin^{2}u\ du\ =\\ &\displaystyle-3\ \sin^{2}i\int\limits_{0}^{2\pi}(e_{g}\ \sin u\ -\ e_{h}\ % \cos u)\ (1\ +\ e_{g}\ \cos u\ +\ e_{h}\ \sin u)\ \cos u\ \sin^{2}u\ du\ =\\ &\displaystyle 3\ \sin^{2}i\ e_{h}\int\limits_{0}^{2\pi}\ \ \cos^{2}u\ \sin^{2% }u\ du\ =\ 2\pi\frac{3}{8}\sin^{2}i\ e_{h}\end{aligned}
  196. 0 2 π t h V r V t ( p r ) 2 3 sin 2 i cos u sin u d u = 3 sin 2 i 0 2 π ( e g sin u - e h cos u ) p r cos 2 u sin u d u = 3 sin 2 i 0 2 π ( e g sin u - e h cos u ) ( 1 + e g cos u + e h sin u ) cos 2 u sin u d u = 3 sin 2 i e g 0 2 π cos 2 u sin 2 u d u = 2 π 3 8 sin 2 i e g \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}t_{h}\ \frac{V_{r}}{V_{t}}% \ \left(\frac{p}{r}\right)^{2}\ 3\ \sin^{2}i\cos u\ \sin u\ du\ =3\ \sin^{2}i% \int\limits_{0}^{2\pi}(e_{g}\ \sin u\ -\ e_{h}\ \cos u)\ \frac{p}{r}\ \cos^{2}% u\ \sin u\ du\ =\\ &\displaystyle 3\ \sin^{2}i\int\limits_{0}^{2\pi}(e_{g}\ \sin u\ -\ e_{h}\ % \cos u)\ (1\ +\ e_{g}\ \cos u\ +\ e_{h}\ \sin u)\ \cos^{2}u\ \sin u\ du\ =\\ &\displaystyle 3\ \sin^{2}i\ e_{g}\int\limits_{0}^{2\pi}\ \ \cos^{2}u\ \sin^{2% }u\ du\ =\ 2\pi\frac{3}{8}\sin^{2}i\ e_{g}\end{aligned}
  197. 2 π 27 8 sin 2 i e h - 2 π 3 2 e h - 2 π 3 2 sin 2 i e h + 2 π 3 8 sin 2 i e h = 2 π 3 2 ( 3 2 sin 2 i - 1 ) e h 2\pi\frac{27}{8}\ \sin^{2}i\ e_{h}\ -\ 2\pi\frac{3}{2}\ e_{h}\ -\ 2\pi\frac{3}% {2}\ \sin^{2}i\ e_{h}\ +\ 2\pi\frac{3}{8}\sin^{2}i\ e_{h}\ =\ 2\pi\ \frac{3}{2% }\left(\frac{3}{2}\ \sin^{2}i\ -\ 1\right)\ e_{h}
  198. - 2 π 9 8 sin 2 i e g + 2 π 3 2 e g - 2 π 3 2 sin 2 i e g + 2 π 3 8 sin 2 i e g = - 2 π 3 2 ( 3 2 sin 2 i - 1 ) e g -2\pi\frac{9}{8}\ \sin^{2}i\ e_{g}\ +\ 2\pi\frac{3}{2}\ e_{g}\ -\ 2\pi\ \frac{% 3}{2}\ \sin^{2}i\ e_{g}\ +\ 2\pi\frac{3}{8}\sin^{2}i\ e_{g}\ =\ -2\pi\ \frac{3% }{2}\left(\frac{3}{2}\ \sin^{2}i\ -\ 1\right)\ e_{g}

Orbital_stability.html

  1. u ( x , t ) = e - i ω t ϕ ( x ) u(x,t)=e^{-i\omega t}\phi(x)\,
  2. ϕ ( x ) \phi(x)\,
  3. e - i ω t ϕ ( x ) e^{-i\omega t}\phi(x)\,
  4. i d u d t = A ( u ) , u ( t ) X , t \R , i\frac{du}{dt}=A(u),\qquad u(t)\in X,\quad t\in\R,
  5. X X\,
  6. \C \C\,
  7. A : X X A\,:X\to X
  8. U ( 1 ) \mathrm{U}(1)\,
  9. A ( e i s u ) = e i s A ( u ) A(e^{is}u)=e^{is}A(u)\,
  10. u X u\in X\,
  11. s \R s\in\R\,
  12. ω ϕ = A ( ϕ ) \omega\phi=A(\phi)\,
  13. u ( t ) = e - i ω t ϕ u(t)=e^{-i\omega t}\phi\,
  14. e - i ω t ϕ e^{-i\omega t}\phi\,
  15. ϵ > 0 \epsilon>0\,
  16. δ > 0 \delta>0\,
  17. v 0 X v_{0}\in X
  18. ϕ - v 0 X < δ \|\phi-v_{0}\|_{X}<\delta\,
  19. v ( t ) v(t)\,
  20. t 0 t\geq 0
  21. v ( 0 ) = v 0 v(0)=v_{0}\,
  22. sup t 0 inf s \R v ( t ) - e i s ϕ X < ϵ . \sup_{t\geq 0}\inf_{s\in\R}\|v(t)-e^{is}\phi\|_{X}<\epsilon.
  23. e - i ω t ϕ ω ( x ) e^{-i\omega t}\phi_{\omega}(x)\,
  24. i t u = - 2 x 2 u + g ( | u | 2 ) u , u ( x , t ) \C , x \R , t \R , i\frac{\partial}{\partial t}u=-\frac{\partial^{2}}{\partial x\,^{2}}u+g(|u|^{2% })u,\qquad u(x,t)\in\C,\quad x\in\R,\quad t\in\R,
  25. g g\,
  26. d d ω Q ( ϕ ω ) < 0 , \frac{d}{d\omega}Q(\phi_{\omega})<0,
  27. Q ( u ) = 1 2 \R | u ( x , t ) | 2 d x Q(u)=\frac{1}{2}\int_{\R}|u(x,t)|^{2}\,dx
  28. u ( x , t ) u(x,t)\,
  29. u ( x , t ) u(x,t)\,
  30. d d ω Q ( ω ) < 0 \frac{d}{d\omega}Q(\omega)<0
  31. ω \omega\,
  32. e - i ω t ϕ ω ( x ) e^{-i\omega t}\phi_{\omega}(x)\,
  33. L ( u ) = E ( u ) - ω Q ( u ) + Γ ( Q ( u ) - Q ( ϕ ω ) ) 2 L(u)=E(u)-\omega Q(u)+\Gamma(Q(u)-Q(\phi_{\omega}))^{2}\,
  34. E ( u ) = 1 2 \R ( | u x | 2 + G ( | u | 2 ) ) d x E(u)=\frac{1}{2}\int_{\R}\left(|\frac{\partial u}{\partial x}|^{2}+G(|u|^{2})% \right)\,dx
  35. u ( x , t ) u(x,t)\,
  36. G ( y ) = 0 y g ( z ) d z G(y)=\int_{0}^{y}g(z)\,dz
  37. g g\,
  38. Γ > 0 \Gamma>0\,

Orchard-planting_problem.html

  1. ( n 2 ) / ( 3 2 ) = n 2 - n 6 . \left\lfloor{\left({{n}\atop{2}}\right)}\Big/{\left({{3}\atop{2}}\right)}% \right\rfloor=\left\lfloor\frac{n^{2}-n}{6}\right\rfloor.
  2. ( n 2 ) - 6 n / 13 3 = n 2 6 - 25 n 78 . \left\lfloor\frac{{\left({{n}\atop{2}}\right)}-6n/13}{3}\right\rfloor=\left% \lfloor\frac{n^{2}}{6}-\frac{25n}{78}\right\rfloor.

Order_of_reaction.html

  1. r = k [ A ] x [ B ] y r\;=\;k[\mathrm{A}]^{x}[\mathrm{B}]^{y}...
  2. r = k K 1 K 2 C A C B ( 1 + K 1 C A + K 2 C B ) 2 . r=k\frac{K_{1}K_{2}C_{A}C_{B}}{(1+K_{1}C_{A}+K_{2}C_{B})^{2}}.\,
  3. ln r = ln k + x ln [ A ] + y ln [ B ] + \ln r=\ln k+x\ln[A]+y\ln[B]+...
  4. ln r = x ln [ A ] + constant \ln r=x\ln[A]+\textrm{constant}
  5. ln r \ln r
  6. ln [ A ] \ln[A]
  7. ln [ A ] = - k t + ln [ A ] 0 \ \ln{[A]}=-kt+\ln{[A]_{0}}
  8. ln [ A ] \ln{[A]}
  9. k k
  10. r = k [ A ] α [ B ] β r=k\cdot[{\rm A}]^{\alpha}\cdot[{\rm B}]^{\beta}
  11. r = k [ A ] α r=k^{\prime}\cdot[{\rm A}]^{\alpha}
  12. k = k [ B ] β k^{\prime}=k\cdot[{\rm B}]^{\beta}
  13. r = k [ A ] 2 r=k[A]^{2}\,
  14. r = k [ A ] [ B ] r=k[A][B]\,
  15. r = k [ N O 2 ] 2 r=k[NO_{2}]^{2}\,
  16. r = k [ A ] [ B ] r=k[A][B]\,
  17. r = k [ A ] [ B ] = k [ A ] r=k[A][B]=k^{\prime}[A]\,
  18. k = k [ B ] k^{\prime}=k[B]\,
  19. r = k [ s u c r o s e ] r=k[sucrose]\,
  20. r = k [ s u c r o s e ] [ H + ] [ H 2 O ] r=k[sucrose][H^{+}][H_{2}O]\,
  21. d [ C H 3 ] d t = k i [ C H 3 C H O ] - k t [ C H 3 ] 2 = 0 \frac{d[\cdot CH_{3}]}{dt}=k_{i}[CH_{3}CHO]-k_{t}[\cdot CH_{3}]^{2}=0
  22. [ C H 3 ] [ C H 3 C H O ] 1 / 2 [\cdot CH_{3}]\quad\propto\quad[CH_{3}CHO]^{1/2}
  23. v = d [ C H 4 ] d t = k p [ C H 3 ] [ C H 3 C H O ] [ C H 3 C H O ] 3 / 2 v=\frac{d[CH_{4}]}{dt}=k_{p}[\cdot CH_{3}][CH_{3}CHO]\quad\propto\quad[CH_{3}% CHO]^{3/2}
  24. r = k 1 [ A ] + k 2 [ A ] 2 r=k_{1}[A]+k_{2}[A]^{2}
  25. r = [ F e ( C N ) 6 ] 2 - k α + k β [ F e ( C N ) 6 ] 2 - r=\frac{[Fe(CN)_{6}]^{2-}}{k_{\alpha}+k_{\beta}[Fe(CN)_{6}]^{2-}}
  26. r = k [ O 3 ] 2 [ O 2 ] r=k\frac{[O_{3}]^{2}}{[O_{2}]}\,

Ordinary_differential_equation.html

  1. m d 2 x ( t ) d t 2 = F ( x ( t ) ) , m\frac{\mathrm{d}^{2}x(t)}{\mathrm{d}t^{2}}=F(x(t)),\,
  2. F ( x , y , y , y ( n - 1 ) ) = y ( n ) F\left(x,y,y^{\prime},\cdots y^{(n-1)}\right)=y^{(n)}
  3. F ( x , y , y , y ′′ , , y ( n ) ) = 0 F\left(x,y,y^{\prime},y^{\prime\prime},\ \cdots,\ y^{(n)}\right)=0
  4. y ( n ) = i = 0 n - 1 a i ( x ) y ( i ) + r ( x ) y^{(n)}=\sum_{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)
  5. 𝐲 ( n ) = 𝐅 ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n - 1 ) ) \mathbf{y}^{(n)}=\mathbf{F}\left(x,\mathbf{y},\mathbf{y}^{\prime},\mathbf{y}^{% \prime\prime},\cdots\mathbf{y}^{(n-1)}\right)
  6. ( y 1 ( n ) y 2 ( n ) y m ( n ) ) = ( f 1 ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n - 1 ) ) f 2 ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n - 1 ) ) f m ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n - 1 ) ) ) \begin{pmatrix}y_{1}^{(n)}\\ y_{2}^{(n)}\\ \vdots\\ y_{m}^{(n)}\end{pmatrix}=\begin{pmatrix}f_{1}\left(x,\mathbf{y},\mathbf{y}^{% \prime},\mathbf{y}^{\prime\prime},\cdots\mathbf{y}^{(n-1)}\right)\\ f_{2}\left(x,\mathbf{y},\mathbf{y}^{\prime},\mathbf{y}^{\prime\prime},\cdots% \mathbf{y}^{(n-1)}\right)\\ \vdots\\ f_{m}\left(x,\mathbf{y},\mathbf{y}^{\prime},\mathbf{y}^{\prime\prime},\cdots% \mathbf{y}^{(n-1)}\right)\\ \end{pmatrix}
  7. 𝐅 ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n ) ) = s y m b o l 0 \mathbf{F}\left(x,\mathbf{y},\mathbf{y}^{\prime},\mathbf{y}^{\prime\prime},% \cdots\mathbf{y}^{(n)}\right)=symbol{0}
  8. ( f 1 ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n ) ) f 2 ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n ) ) f m ( x , 𝐲 , 𝐲 , 𝐲 ′′ , 𝐲 ( n ) ) ) = ( 0 0 0 ) \begin{pmatrix}f_{1}(x,\mathbf{y},\mathbf{y}^{\prime},\mathbf{y}^{\prime\prime% },\cdots\mathbf{y}^{(n)})\\ f_{2}(x,\mathbf{y},\mathbf{y}^{\prime},\mathbf{y}^{\prime\prime},\cdots\mathbf% {y}^{(n)})\\ \vdots\\ f_{m}(x,\mathbf{y},\mathbf{y}^{\prime},\mathbf{y}^{\prime\prime},\cdots\mathbf% {y}^{(n)})\\ \end{pmatrix}=\begin{pmatrix}0\\ 0\\ \vdots\\ 0\\ \end{pmatrix}
  9. 𝐅 ( x , 𝐲 , 𝐲 ) = s y m b o l 0 \mathbf{F}\left(x,\mathbf{y},\mathbf{y}^{\prime}\right)=symbol{0}
  10. 𝐅 ( x , 𝐮 , 𝐯 ) 𝐯 \frac{\partial\mathbf{F}(x,\mathbf{u},\mathbf{v})}{\partial\mathbf{v}}
  11. F ( x , y , y , , y ( n ) ) = 0 F\left(x,y,y^{\prime},\cdots,y^{(n)}\right)=0
  12. F ( x , u , u , , u ( n ) ) = 0 x I . F(x,u,u^{\prime},\ \cdots,\ u^{(n)})=0\quad x\in I.
  13. u ( x ) = v ( x ) x I . u(x)=v(x)\quad x\in I.\,
  14. y = F ( x , y ) , y 0 = y ( x 0 ) y^{\prime}=F(x,y)\,,\quad y_{0}=y(x_{0})
  15. R = [ x 0 - a , x 0 + a ] × [ y 0 - b , y 0 + b ] R=[x_{0}-a,x_{0}+a]\times[y_{0}-b,y_{0}+b]
  16. I = [ x 0 - h , x 0 + h ] [ x 0 - a , x 0 + a ] I=[x_{0}-h,x_{0}+h]\subset[x_{0}-a,x_{0}+a]
  17. I m a x = ( x - , x + ) , x ± , x 0 I m a x I_{max}=(x_{-},x_{+}),x_{\pm}\in\mathbb{R},x_{0}\in I_{max}
  18. x ± ± x_{\pm}\nrightarrow\pm\infty
  19. lim sup x x ± y ( x ) \limsup_{x\to x_{\pm}}\|y(x)\|\rightarrow\infty
  20. lim x x ± Ω ¯ \lim_{x\to x_{\pm}}\in\partial\bar{\Omega}
  21. Ω ¯ \partial\bar{\Omega}
  22. y = y 2 y^{\prime}=y^{2}
  23. y ( x ) = y 0 ( x 0 - x ) y 0 + 1 y(x)=\frac{y_{0}}{(x_{0}-x)y_{0}+1}
  24. { y 0 = 0 ( - , x 0 + 1 y 0 ) y 0 > 0 ( x 0 + 1 y 0 , + ) y 0 < 0 \begin{cases}\mathbb{R}&y_{0}=0\\ (-\infty,x_{0}+\frac{1}{y_{0}})&y_{0}>0\\ (x_{0}+\frac{1}{y_{0}},+\infty)&y_{0}<0\end{cases}
  25. 𝐑 ( x 0 + 1 / y 0 ) \mathbf{R}\smallsetminus(x_{0}+1/y_{0})
  26. lim x x ± y ( x ) , \lim_{x\to x_{\pm}}\|y(x)\|\rightarrow\infty\,,
  27. F ( x , y , y , y ′′ , , y ( n - 1 ) ) = y ( n ) F\left(x,y,y^{\prime},y^{\prime\prime},\ \cdots,\ y^{(n-1)}\right)=y^{(n)}
  28. y i = y ( i - 1 ) . y_{i}=y^{(i-1)}.\!
  29. y 1 = y 2 y 2 = y 3 y n - 1 = y n y n = F ( x , y 1 , , y n ) . \begin{array}[]{rcl}y_{1}^{\prime}&=&y_{2}\\ y_{2}^{\prime}&=&y_{3}\\ &\vdots&\\ y_{n-1}^{\prime}&=&y_{n}\\ y_{n}^{\prime}&=&F(x,y_{1},\cdots,y_{n}).\end{array}
  30. 𝐲 = 𝐅 ( x , 𝐲 ) \mathbf{y}^{\prime}=\mathbf{F}(x,\mathbf{y})
  31. 𝐲 = ( y 1 , , y n ) , 𝐅 ( x , y 1 , , y n ) = ( y 2 , , y n , F ( x , y 1 , , y n ) ) . \mathbf{y}=(y_{1},\cdots,y_{n}),\quad\mathbf{F}(x,y_{1},\cdots,y_{n})=(y_{2},% \cdots,y_{n},F(x,y_{1},\cdots,y_{n})).
  32. P 1 ( x ) Q 1 ( y ) + P 2 ( x ) Q 2 ( y ) d y d x = 0 P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y)\,\frac{dy}{dx}=0\,\!
  33. P 1 ( x ) Q 1 ( y ) d x + P 2 ( x ) Q 2 ( y ) d y = 0 P_{1}(x)Q_{1}(y)\,dx+P_{2}(x)Q_{2}(y)\,dy=0\,\!
  34. x P 1 ( λ ) P 2 ( λ ) d λ + y Q 2 ( λ ) Q 1 ( λ ) d λ = C \int^{x}\frac{P_{1}(\lambda)}{P_{2}(\lambda)}\,d\lambda+\int^{y}\frac{Q_{2}(% \lambda)}{Q_{1}(\lambda)}\,d\lambda=C\,\!
  35. d y d x = F ( x ) \frac{dy}{dx}=F(x)\,\!
  36. d y = F ( x ) d x dy=F(x)\,dx\,\!
  37. y = x F ( λ ) d λ + C y=\int^{x}F(\lambda)\,d\lambda+C\,\!
  38. d y d x = F ( y ) \frac{dy}{dx}=F(y)\,\!
  39. d y = F ( y ) d x dy=F(y)\,dx\,\!
  40. x = y d λ F ( λ ) + C x=\int^{y}\frac{d\lambda}{F(\lambda)}+C\,\!
  41. P ( y ) d y d x + Q ( x ) = 0 P(y)\frac{dy}{dx}+Q(x)=0\,\!
  42. P ( y ) d y + Q ( x ) d x = 0 P(y)\,dy+Q(x)\,dx=0\,\!
  43. y P ( λ ) d λ + x Q ( λ ) d λ = C \int^{y}P(\lambda)\,{d\lambda}+\int^{x}Q(\lambda)\,d\lambda=C\,\!
  44. d y d x = F ( y x ) \frac{dy}{dx}=F\left(\frac{y}{x}\right)\,\!
  45. ln ( C x ) = y / x d λ F ( λ ) - λ \ln(Cx)=\int^{y/x}\frac{d\lambda}{F(\lambda)-\lambda}\,\!
  46. y M ( x y ) + x N ( x y ) d y d x = 0 yM(xy)+xN(xy)\,\frac{dy}{dx}=0\,\!
  47. y M ( x y ) d x + x N ( x y ) d y = 0 yM(xy)\,dx+xN(xy)\,dy=0\,\!
  48. ln ( C x ) = x y N ( λ ) d λ λ [ N ( λ ) - M ( λ ) ] \ln(Cx)=\int^{xy}\frac{N(\lambda)\,d\lambda}{\lambda[N(\lambda)-M(\lambda)]}\,\!
  49. M ( x , y ) d y d x + N ( x , y ) = 0 M(x,y)\frac{dy}{dx}+N(x,y)=0\,\!
  50. M ( x , y ) d y + N ( x , y ) d x = 0 M(x,y)\,dy+N(x,y)\,dx=0\,\!
  51. M x = N y \frac{\partial M}{\partial x}=\frac{\partial N}{\partial y}\,\!
  52. F ( x , y ) = y M ( x , λ ) d λ + x N ( λ , y ) d λ + Y ( y ) + X ( x ) = C \begin{aligned}\displaystyle F(x,y)&\displaystyle=\int^{y}M(x,\lambda)\,d% \lambda+\int^{x}N(\lambda,y)\,d\lambda\\ &\displaystyle+Y(y)+X(x)=C\end{aligned}\,\!
  53. M ( x , y ) d y d x + N ( x , y ) = 0 M(x,y)\frac{dy}{dx}+N(x,y)=0\,\!
  54. M ( x , y ) d y + N ( x , y ) d x = 0 M(x,y)\,dy+N(x,y)\,dx=0\,\!
  55. M x N y \frac{\partial M}{\partial x}\neq\frac{\partial N}{\partial y}\,\!
  56. ( μ M ) x = ( μ N ) y \frac{\partial(\mu M)}{\partial x}=\frac{\partial(\mu N)}{\partial y}\,\!
  57. F ( x , y ) = y μ ( x , λ ) M ( x , λ ) d λ + x μ ( λ , y ) N ( λ , y ) d λ + Y ( y ) + X ( x ) = C \begin{aligned}\displaystyle F(x,y)&\displaystyle=\int^{y}\mu(x,\lambda)M(x,% \lambda)\,d\lambda+\int^{x}\mu(\lambda,y)N(\lambda,y)\,d\lambda\\ &\displaystyle+Y(y)+X(x)=C\\ \end{aligned}\,\!
  58. d 2 y d x 2 = F ( y ) \frac{d^{2}y}{dx^{2}}=F(y)\,\!
  59. 2 d y d x d 2 y d x 2 = d d x ( d y d x ) 2 2\frac{dy}{dx}\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)^{2}\,\!
  60. x = ± y d λ 2 λ F ( ϵ ) d ϵ + C 1 + C 2 x=\pm\int^{y}\frac{d\lambda}{\sqrt{2\int^{\lambda}F(\epsilon)\,d\epsilon+C_{1}% }}+C_{2}\,\!
  61. d y d x + P ( x ) y = Q ( x ) \frac{dy}{dx}+P(x)y=Q(x)\,\!
  62. e x P ( λ ) d λ e^{\int^{x}P(\lambda)\,d\lambda}
  63. y = e - x P ( λ ) d λ [ x e λ P ( ϵ ) d ϵ Q ( λ ) d λ + C ] y=e^{-\int^{x}P(\lambda)\,d\lambda}\left[\int^{x}e^{\int^{\lambda}P(\epsilon)% \,d\epsilon}Q(\lambda)\,{d\lambda}+C\right]
  64. d 2 y d x 2 + b d y d x + c y = r ( x ) \frac{d^{2}y}{dx^{2}}+b\frac{dy}{dx}+cy=r(x)\,\!
  65. e α j x e^{\alpha_{j}x}
  66. y = y c + y p y=y_{c}+y_{p}
  67. y c = C 1 e ( - b + b 2 - 4 c ) x 2 + C 2 e - ( b + b 2 - 4 c ) x 2 y_{c}=C_{1}e^{\left(-b+\sqrt{b^{2}-4c}\right)\frac{x}{2}}+C_{2}e^{-\left(b+% \sqrt{b^{2}-4c}\right)\frac{x}{2}}\,\!
  68. y c = ( C 1 x + C 2 ) e - b x / 2 y_{c}=(C_{1}x+C_{2})e^{-bx/2}\,\!
  69. j = 0 n b j d j y d x j = r ( x ) \sum_{j=0}^{n}b_{j}\frac{d^{j}y}{dx^{j}}=r(x)\,\!
  70. e α j x e^{\alpha_{j}x}
  71. y = y c + y p y=y_{c}+y_{p}
  72. j = 1 n ( α - α j ) = 0 \prod_{j=1}^{n}\left(\alpha-\alpha_{j}\right)=0\,\!
  73. y c = j = 1 n C j e α j x y_{c}=\sum_{j=1}^{n}C_{j}e^{\alpha_{j}x}\,\!
  74. y c = j = 1 n ( = 1 k j C x - 1 ) e α j x y_{c}=\sum_{j=1}^{n}\left(\sum_{\ell=1}^{k_{j}}C_{\ell}x^{\ell-1}\right)e^{% \alpha_{j}x}\,\!
  75. C j e α j x = C j e χ j x cos ( γ j x + ϕ j ) C_{j}e^{\alpha_{j}x}=C_{j}e^{\chi_{j}x}\cos(\gamma_{j}x+\phi_{j})\,\!

Organic_photorefractive_materials.html

  1. n = N c e - ( E c - E F ) k B T n=N_{c}\frac{e^{-(E_{c}-E_{F})}}{k_{B}T}
  2. h = N v e - ( E c - E F ) k B T h=N_{v}\frac{e^{-(E_{c}-E_{F})}}{k_{B}T}
  3. n = N c e - ( E c - E F n ) k B T n=N_{c}\frac{e^{-(E_{c}-E_{Fn})}}{k_{B}T}
  4. h = N v e - ( E F p - E v ) k B T h=N_{v}\frac{e^{-(E_{Fp}-E_{v})}}{k_{B}T}
  5. σ d = e ( N d - N d + ) β μ τ \sigma_{d}=e\left(N_{d}-N_{d}^{+}\right)\beta\mu\tau
  6. σ p h = e ( N d - N d + ) s I μ τ h ν \sigma_{ph}=e\left(N_{d}-N_{d}^{+}\right)\frac{sI\mu\tau}{h\nu}
  7. R 1 2 ε 1 + R 2 2 ε 2 + R 3 2 ε 3 = 1 \frac{R_{1}^{2}}{\varepsilon_{1}}+\frac{R_{2}^{2}}{\varepsilon_{2}}+\frac{R_{3% }^{2}}{\varepsilon_{3}}=1
  8. Δ R i 2 = j = 1 3 r i j E j \Delta R_{i}^{2}=\sum_{j=1}^{3}r_{ij}E_{j}

Organoxenon_compound.html

  1. 2 C 6 F 5 XeF + Cd ( C 6 F 5 ) 2 Xe ( C 6 F 5 ) 2 + CdF 2 \rm\ 2C_{6}F_{5}XeF+Cd(C_{6}F_{5})_{2}\rightarrow Xe(C_{6}F_{5})_{2}+CdF_{2}\downarrow
  2. C 6 F 5 XeF + Me 3 SiCN C 6 F 5 XeCN + Me 3 SiF \rm\ C_{6}F_{5}XeF+Me_{3}SiCN\rightarrow C_{6}F_{5}XeCN+Me_{3}SiF
  3. C 6 F 5 XeF + Cd ( 2 , 4 , 6 - C 6 H 2 F 3 ) 2 2 , 4 , 6 - C 6 H 2 F 3 XeC 6 F 5 + CdF 2 \rm\ C_{6}F_{5}XeF+Cd(2,4,6-C_{6}H_{2}F_{3})_{2}\rightarrow 2,4,6-C_{6}H_{2}F_% {3}XeC_{6}F_{5}+CdF_{2}\downarrow
  4. XeF 4 + C 6 F 5 BF 2 CH 2 Cl 2 - 55 o C [ C 6 F 5 XeF 2 ] + BF 4 - \rm\ XeF_{4}+C_{6}F_{5}BF_{2}\xrightarrow[-55^{o}C]{CH_{2}Cl_{2}}[C_{6}F_{5}% XeF_{2}]^{+}BF_{4}^{-}

Orlicz–Pettis_theorem.html

  1. { x n } \left\{{{x}_{n}}\right\}
  2. x n \sum{{{x}_{n}}}

Orthocentric_tetrahedron.html

  1. A B 2 + C D 2 = A C 2 + B D 2 = A D 2 + B C 2 . \displaystyle AB^{2}+CD^{2}=AC^{2}+BD^{2}=AD^{2}+BC^{2}.
  2. V = 1 6 4 ( c 2 + d 2 ) s ( s - a ) ( s - b ) ( s - c ) - a 2 b 2 c 2 V=\frac{1}{6}\sqrt{4(c^{2}+d^{2})s(s-a)(s-b)(s-c)-a^{2}b^{2}c^{2}}
  3. s = 1 2 ( a + b + c ) s=\tfrac{1}{2}(a+b+c)

Orthodiagonal_quadrilateral.html

  1. a 2 + c 2 = b 2 + d 2 . \displaystyle a^{2}+c^{2}=b^{2}+d^{2}.
  2. P A B + P B A + P C D + P D C = π \angle PAB+\angle PBA+\angle PCD+\angle PDC=\pi
  3. m 1 2 + m 3 2 = m 2 2 + m 4 2 m_{1}^{2}+m_{3}^{2}=m_{2}^{2}+m_{4}^{2}
  4. R 1 2 + R 3 2 = R 2 2 + R 4 2 R_{1}^{2}+R_{3}^{2}=R_{2}^{2}+R_{4}^{2}
  5. 1 h 1 2 + 1 h 3 2 = 1 h 2 2 + 1 h 4 2 \frac{1}{h_{1}^{2}}+\frac{1}{h_{3}^{2}}=\frac{1}{h_{2}^{2}}+\frac{1}{h_{4}^{2}}
  6. a + c = b + d a+c=b+d
  7. a 2 + c 2 = b 2 + d 2 a^{2}+c^{2}=b^{2}+d^{2}
  8. R 1 + R 3 = R 2 + R 4 R_{1}+R_{3}=R_{2}+R_{4}
  9. R 1 2 + R 3 2 = R 2 2 + R 4 2 R_{1}^{2}+R_{3}^{2}=R_{2}^{2}+R_{4}^{2}
  10. 1 h 1 + 1 h 3 = 1 h 2 + 1 h 4 \frac{1}{h_{1}}+\frac{1}{h_{3}}=\frac{1}{h_{2}}+\frac{1}{h_{4}}
  11. 1 h 1 2 + 1 h 3 2 = 1 h 2 2 + 1 h 4 2 \frac{1}{h_{1}^{2}}+\frac{1}{h_{3}^{2}}=\frac{1}{h_{2}^{2}}+\frac{1}{h_{4}^{2}}
  12. K = p q 2 . K=\frac{p\cdot q}{2}.
  13. D 2 = p 1 2 + p 2 2 + q 1 2 + q 2 2 = a 2 + c 2 = b 2 + d 2 D^{2}=p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}=a^{2}+c^{2}=b^{2}+d^{2}
  14. R = 1 2 p 1 2 + p 2 2 + q 1 2 + q 2 2 R=\tfrac{1}{2}\sqrt{p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}}
  15. R = 1 2 a 2 + c 2 = 1 2 b 2 + d 2 . R=\tfrac{1}{2}\sqrt{a^{2}+c^{2}}=\tfrac{1}{2}\sqrt{b^{2}+d^{2}}.
  16. a 2 + b 2 + c 2 + d 2 = 8 R 2 . a^{2}+b^{2}+c^{2}+d^{2}=8R^{2}.
  17. R = p 2 + q 2 + 4 x 2 8 . R=\sqrt{\frac{p^{2}+q^{2}+4x^{2}}{8}}.
  18. K = 1 2 ( a c + b d ) . K=\tfrac{1}{2}(ac+bd).

Orthogonal_polynomials.html

  1. f ( x ) d α ( x ) \int f(x)\;d\alpha(x)
  2. f , g = f ( x ) g ( x ) d α ( x ) . \langle f,g\rangle=\int f(x)g(x)\;d\alpha(x).
  3. deg P n = n , P m , P n = 0 for m n . \deg P_{n}=n~{},\quad\langle P_{m},\,P_{n}\rangle=0\quad\,\text{for}\quad m% \neq n~{}.
  4. P n , P n = 1 , \langle P_{n},P_{n}\rangle=1~{},
  5. d α ( x ) = W ( x ) d x d\alpha(x)=W(x)\,dx
  6. W : [ x 1 , x 2 ] W:[x_{1},x_{2}]\to\mathbb{R}
  7. f , g = x 1 x 2 f ( x ) g ( x ) W ( x ) d x . \langle f,g\rangle=\int_{x_{1}}^{x_{2}}f(x)g(x)W(x)\;dx.
  8. m n = x n d α ( x ) m_{n}=\int x^{n}\,d\alpha(x)
  9. P n ( x ) = c n det [ m 0 m 1 m 2 m n m 1 m 2 m 3 m n + 1 m n - 1 m n m n + 1 m 2 n - 1 1 x x 2 x n ] , P_{n}(x)=c_{n}\,\det\begin{bmatrix}m_{0}&m_{1}&m_{2}&\cdots&m_{n}\\ m_{1}&m_{2}&m_{3}&\cdots&m_{n+1}\\ &&\vdots&&\\ m_{n-1}&m_{n}&m_{n+1}&\cdots&m_{2n-1}\\ 1&x&x^{2}&\cdots&x^{n}\end{bmatrix}~{},
  10. P n ( x ) = ( A n x + B n ) P n - 1 ( x ) + C n P n - 2 ( x ) . P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)~{}.

Orthogonal_polynomials_on_the_unit_circle.html

  1. Φ n + 1 ( z ) = z Φ n ( z ) - α ¯ n Φ n * ( z ) \Phi_{n+1}(z)=z\Phi_{n}(z)-\overline{\alpha}_{n}\Phi_{n}^{*}(z)
  2. Φ n * ( z ) = z n Φ n ( 1 / z ¯ ) ¯ \Phi_{n}^{*}(z)=z^{n}\overline{\Phi_{n}(1/\overline{z})}
  3. 1 + z f ( z ) 1 - z f ( z ) = F ( z ) = e i θ + z e i θ - z d μ . \frac{1+zf(z)}{1-zf(z)}=F(z)=\int\frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu.
  4. ( 1 - | α j | 2 ) = exp ( 0 2 π log ( w ( θ ) ) d θ / 2 π ) \prod(1-|\alpha_{j}|^{2})=\exp\big(\int_{0}^{2\pi}\log(w(\theta))d\theta/2\pi\big)

Orthogonal_symmetric_Lie_algebra.html

  1. ( 𝔤 , s ) (\mathfrak{g},s)
  2. 𝔤 \mathfrak{g}
  3. s s
  4. 𝔤 \mathfrak{g}
  5. 2 2
  6. 𝔲 \mathfrak{u}
  7. 𝔲 \mathfrak{u}
  8. 𝔲 \mathfrak{u}
  9. 𝔤 \mathfrak{g}
  10. s s

Orthoptic_(geometry).html

  1. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  2. x 2 + y 2 = a 2 + b 2 x^{2}+y^{2}=a^{2}+b^{2}
  3. x 2 a 2 - y 2 b 2 = 1 , a > b , \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1,\,a>b,
  4. x 2 + y 2 = a 2 - b 2 x^{2}+y^{2}=a^{2}-b^{2}
  5. a b a\leq b
  6. x 2 / 3 + y 2 / 3 = 1 x^{2/3}+y^{2/3}=1
  7. r = 1 2 cos ( 2 φ ) , 0 φ < 2 π , r=\tfrac{1}{\sqrt{2}}\cos(2\varphi),\ 0\leq\varphi<2\pi,
  8. P 1 P 2 ¯ \overline{P_{1}P_{2}}
  9. P 1 , P 2 P_{1},P_{2}
  10. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  11. c ± ( m ) = ( - m a 2 ± m 2 a 2 + b 2 , b 2 ± m 2 a 2 + b 2 ) , m \R , \vec{c}_{\pm}(m)=\left(-\frac{ma^{2}}{\pm\sqrt{m^{2}a^{2}+b^{2}}},\frac{b^{2}}% {\pm\sqrt{m^{2}a^{2}+b^{2}}}\right),\,m\in\R,
  12. m m
  13. c + ( m ) \vec{c}_{+}(m)
  14. c - ( m ) \vec{c}_{-}(m)
  15. ( ± a , 0 ) (\pm a,0)
  16. ( 0 , ± b ) (0,\pm b)
  17. ( ± a , ± b ) (\pm a,\pm b)
  18. x 2 + y 2 = a 2 + b 2 x^{2}+y^{2}=a^{2}+b^{2}
  19. c ± ( m ) \vec{c}_{\pm}(m)
  20. y = m x ± m 2 a 2 + b 2 . y=mx\pm\sqrt{m^{2}a^{2}+b^{2}}.
  21. ( x 0 , y 0 ) (x_{0},y_{0})
  22. y 0 = m x 0 ± m 2 a 2 + b 2 y_{0}=mx_{0}\pm\sqrt{m^{2}a^{2}+b^{2}}
  23. m 2 - 2 x 0 y 0 x 0 2 - a 2 m + y 0 2 - b 2 x 0 2 - a 2 = 0 , m^{2}-\frac{2x_{0}y_{0}}{x_{0}^{2}-a^{2}}m+\frac{y_{0}^{2}-b^{2}}{x_{0}^{2}-a^% {2}}=0,
  24. m 1 , m 2 m_{1},m_{2}
  25. ( x 0 , y 0 ) (x_{0},y_{0})
  26. ( x 0 , y 0 ) (x_{0},y_{0})
  27. m 1 m 2 = - 1 = y 0 2 - b 2 x 0 2 - a 2 m_{1}m_{2}=-1=\frac{y_{0}^{2}-b^{2}}{x_{0}^{2}-a^{2}}
  28. x 0 2 + y 0 2 = a 2 + b 2 x_{0}^{2}+y_{0}^{2}=a^{2}+b^{2}
  29. x 2 + y 2 = a 2 + b 2 x^{2}+y^{2}=a^{2}+b^{2}
  30. b 2 b^{2}
  31. - b 2 -b^{2}
  32. m m
  33. | m | > b / a |m|>b/a
  34. x 2 + y 2 = a 2 - b 2 , a > b x^{2}+y^{2}=a^{2}-b^{2},\ a>b
  35. c ( t ) = ( cos 3 t , sin 3 t ) , 0 t < 2 π \vec{c}(t)=(\cos^{3}t,\sin^{3}t),\;0\leq t<2\pi
  36. c ˙ ( t ) c ˙ ( t + α ) = 0 \vec{\dot{c}}(t)\cdot\vec{\dot{c}}(t+\alpha)=0
  37. α \alpha
  38. c ˙ ( t ) \vec{\dot{c}}(t)
  39. t t
  40. α = ± π 2 \alpha=\pm\tfrac{\pi}{2}
  41. c ( t ) \vec{c}(t)
  42. c ( t + π 2 ) \vec{c}(t+\tfrac{\pi}{2})
  43. y = - tan t ( x - cos 3 t ) + sin 3 t , y = 1 tan t ( x + sin 3 t ) + cos 3 t y=-\tan t(x-\cos^{3}t)+\sin^{3}t,\ y=\tfrac{1}{\tan t}(x+\sin^{3}t)+\cos^{3}t
  44. x = sin t cos t ( sin t - cos t ) x=\sin t\cos t(\sin t-\cos t)
  45. y = sin t cos t ( sin t + cos t ) y=\sin t\cos t(\sin t+\cos t)
  46. t t
  47. 2 ( x 2 + y 2 ) 3 - ( x 2 - y 2 ) 2 = 0. 2(x^{2}+y^{2})^{3}-(x^{2}-y^{2})^{2}=0.
  48. φ = t - 5 4 π \varphi=t-\tfrac{5}{4}\pi
  49. x = 1 2 cos ( 2 φ ) cos φ , y = 1 2 cos ( 2 φ ) sin φ x=\tfrac{1}{\sqrt{2}}\cos(2\varphi)\,\cos\varphi,\ \ y=\tfrac{1}{\sqrt{2}}\cos% (2\varphi)\,\sin\varphi
  50. r = 1 2 cos ( 2 φ ) , 0 φ < 2 π r=\frac{1}{\sqrt{2}}\cos(2\varphi),\ 0\leq\varphi<2\pi
  51. α 90 \alpha\neq 90^{\circ}
  52. α \alpha
  53. α \alpha
  54. y = a x 2 y=ax^{2}
  55. x 2 - tan 2 α ( y + 1 4 a ) 2 - y a = 0. x^{2}-\tan^{2}\alpha\left(y+\frac{1}{4a}\right)^{2}-\frac{y}{a}=0.
  56. α , 180 - α \alpha,180^{\circ}\!-\!\alpha
  57. α \alpha
  58. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  59. tan 2 α ( x 2 + y 2 - a 2 - b 2 ) 2 = 4 ( a 2 y 2 + b 2 x 2 - a 2 b 2 ) \tan^{2}\alpha\;(x^{2}+y^{2}-a^{2}-b^{2})^{2}=4(a^{2}y^{2}+b^{2}x^{2}-a^{2}b^{% 2})
  60. α \alpha
  61. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  62. tan 2 α ( x 2 + y 2 - a 2 + b 2 ) 2 = 4 ( a 2 y 2 - b 2 x 2 + a 2 b 2 ) . \tan^{2}\alpha\;(x^{2}+y^{2}-a^{2}+b^{2})^{2}=4(a^{2}y^{2}-b^{2}x^{2}+a^{2}b^{% 2}).
  63. y = a x 2 y=ax^{2}
  64. m = 2 a x m=2ax
  65. c ( m ) = ( m 2 a , m 2 4 a ) , m \R . \vec{c}(m)=\left(\frac{m}{2a},\frac{m^{2}}{4a}\right),\,m\in\R.
  66. m m
  67. y = m x - m 2 4 a . y=mx-\frac{m^{2}}{4a}.
  68. ( x 0 , y 0 ) (x_{0},y_{0})
  69. y 0 = m x 0 - m 2 4 a y_{0}=mx_{0}-\frac{m^{2}}{4a}
  70. m 1 , m 2 m_{1},m_{2}
  71. ( x 0 , y 0 ) (x_{0},y_{0})
  72. m 2 - 4 a x 0 m + 4 a y 0 = 0. m^{2}-4ax_{0}m+4ay_{0}=0.
  73. α \alpha
  74. 180 - α 180^{\circ}-\alpha
  75. tan 2 α = ( m 1 - m 2 1 + m 1 m 2 ) 2 \tan^{2}\alpha=\left(\frac{m_{1}-m_{2}}{1+m_{1}m_{2}}\right)^{2}
  76. m , m,
  77. m 1 , m 2 m_{1},m_{2}
  78. x 0 2 - tan 2 α ( y 0 + 1 4 a ) 2 - y 0 a = 0. x_{0}^{2}-\tan^{2}\alpha\left(y_{0}+\frac{1}{4a}\right)^{2}-\frac{y_{0}}{a}=0.
  79. α \alpha
  80. 180 - α 180^{\circ}-\alpha
  81. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  82. m 2 - 2 x 0 y 0 x 0 2 - a 2 m + y 0 2 - b 2 x 0 2 - a 2 = 0. m^{2}-\frac{2x_{0}y_{0}}{x_{0}^{2}-a^{2}}m+\frac{y_{0}^{2}-b^{2}}{x_{0}^{2}-a^% {2}}=0.
  83. m 1 , m 2 m_{1},m_{2}
  84. tan 2 α = ( m 1 - m 2 1 + m 1 m 2 ) 2 \tan^{2}\alpha=\left(\tfrac{m_{1}-m_{2}}{1+m_{1}m_{2}}\right)^{2}
  85. tan 2 α ( x 0 2 + y 0 2 - a 2 - b 2 ) 2 = 4 ( a 2 y 0 2 + b 2 x 0 2 - a 2 b 2 ) . \tan^{2}\alpha\;(x_{0}^{2}+y_{0}^{2}-a^{2}-b^{2})^{2}=4(a^{2}y_{0}^{2}+b^{2}x_% {0}^{2}-a^{2}b^{2}).
  86. b 2 b^{2}
  87. - b 2 -b^{2}

Oscillator_representation.html

  1. ( α β β ¯ α ¯ ) \begin{pmatrix}\alpha&\beta\\ \overline{\beta}&\overline{\alpha}\end{pmatrix}
  2. | α | 2 - | β | 2 = 1. |\alpha|^{2}-|\beta|^{2}=1.
  3. C = ( 1 i i 1 ) C=\begin{pmatrix}1&i\\ i&1\end{pmatrix}
  4. G = C G 1 C - 1 , G=CG_{1}C^{-1},
  5. J = ( 0 1 - 1 0 ) J=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}
  6. ( a 0 b a - 1 ) \displaystyle{\begin{pmatrix}a&0\\ b&a^{-1}\end{pmatrix}}
  7. v = ( 0 1 ) \displaystyle{v=\begin{pmatrix}0\\ 1\end{pmatrix}}
  8. 𝔤 \mathfrak{g}
  9. ( i x w w ¯ - i x ) \begin{pmatrix}ix&w\\ \overline{w}&-ix\end{pmatrix}
  10. σ ( g ) = M g ¯ M - 1 , \displaystyle{\sigma(g)=M\overline{g}M^{-1},}
  11. M = ( 0 1 1 0 ) , M=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},
  12. σ ( a b c d ) = ( d ¯ c ¯ b ¯ a ¯ ) . \sigma\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}\overline{d}&\overline{c}\\ \overline{b}&\overline{a}\end{pmatrix}.
  13. 𝔤 c \mathfrak{g}_{c}
  14. 𝔤 c \mathfrak{g}_{c}
  15. L 0 = ( 1 2 0 0 - 1 2 ) , L - 1 = ( 0 1 0 0 ) , L 1 = ( 0 0 - 1 0 ) . L_{0}=\begin{pmatrix}{1\over 2}&0\\ 0&-{1\over 2}\end{pmatrix},\quad L_{-1}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix},\quad L_{1}=\begin{pmatrix}0&0\\ -1&0\end{pmatrix}.
  16. [ L m , L n ] = ( m - n ) L m + n . \displaystyle{[L_{m},L_{n}]=(m-n)L_{m+n}.}
  17. 𝔤 c = 𝔤 i 𝔤 , \mathfrak{g}_{c}=\mathfrak{g}\oplus i\mathfrak{g},
  18. 𝔤 \mathfrak{g}
  19. i 𝔤 i\mathfrak{g}
  20. i 𝔤 i\mathfrak{g}
  21. X = ( x w - w ¯ - x ) . X=\begin{pmatrix}x&w\\ -\overline{w}&-x\end{pmatrix}.
  22. - det X = x 2 - | w | 2 . -\det X=x^{2}-|w|^{2}.
  23. i 𝔤 i\mathfrak{g}
  24. det X < 0. \det X<0.
  25. g ( D ¯ ) D . g(\overline{D})\subset D.
  26. H = G exp ( C ) = exp ( C ) G . H=G\cdot\exp(C)=\exp(C)\cdot G.
  27. Y = ( - y 0 0 y ) Y=\begin{pmatrix}-y&0\\ 0&y\end{pmatrix}
  28. y = x 2 - | w | 2 > 0. y=\sqrt{x^{2}-|w|^{2}}>0.
  29. e - Y g e^{-Y}g
  30. H ¯ = { g S L ( 2 , 𝐂 ) : g D D } = G exp C ¯ = exp C ¯ G . \overline{H}=\left\{g\in SL(2,{\mathbf{C}})\ :\ gD\subseteq D\right\}=G\cdot% \exp{\overline{C}}=\exp{\overline{C}}\cdot G.
  31. H = G A + G , H=GA_{+}G,
  32. A + = { ( e - y 0 0 e y ) : y > 0 } . A_{+}=\left\{\begin{pmatrix}e^{-y}&0\\ 0&e^{y}\end{pmatrix}\ :\ y>0\right\}.
  33. ( a b c d ) \displaystyle{\begin{pmatrix}a&b\\ c&d\end{pmatrix}}
  34. ( a ¯ b ¯ c ¯ d ¯ ) , ( a - c - b d ) , \begin{pmatrix}\overline{a}&\overline{b}\\ \overline{c}&\overline{d}\end{pmatrix},\quad\begin{pmatrix}a&-c\\ -b&d\end{pmatrix},
  35. ( a ¯ - c ¯ - b ¯ d ¯ ) . \displaystyle{\begin{pmatrix}\overline{a}&-\overline{c}\\ -\overline{b}&\overline{d}\end{pmatrix}.}
  36. M = ( a 0 b a - 1 ) M=\begin{pmatrix}a&0\\ b&a^{-1}\end{pmatrix}
  37. | a | < 1 \displaystyle{|a|<1}
  38. | b | < | a | - 1 - | a | . \displaystyle{|b|<|a|^{-1}-|a|.}
  39. M ( x 0 0 x - 1 ) = ( x 0 b a - 1 ( x - x - 1 ) x - 1 ) M , \displaystyle{M\begin{pmatrix}x&0\\ 0&x^{-1}\end{pmatrix}=\begin{pmatrix}x&0\\ ba^{-1}(x-x^{-1})&x^{-1}\end{pmatrix}M,}
  40. M S ( t ) = S 0 ( t ) M , \displaystyle{MS(t)=S_{0}(t)M,}
  41. S ( t ) N = N S 0 ( t ) , \displaystyle{S(t)N=NS_{0}(t),}
  42. Z = ( z 0 w - z ) . Z=\begin{pmatrix}z&0\\ w&-z\end{pmatrix}.
  43. z < 0 \Re z<0
  44. | z | > 1 2 | w | . |\Re z|>{1\over 2}|w|.
  45. exp Z = ( e z 0 f ( z ) w e - z ) , f ( z ) = sinh z z . \displaystyle{\exp Z=\begin{pmatrix}e^{z}&0\\ f(z)w&e^{-z}\end{pmatrix},\qquad f(z)={\sinh z\over z}.}
  46. b = ( 1 - δ ) ( | a | - 1 - | a | ) , \displaystyle{b=(1-\delta)(|a|^{-1}-|a|),}
  47. ( 1 - δ ) 2 = | α + α - 1 | | α | + | α - 1 | , \displaystyle{(1-\delta)^{2}={|\alpha+\alpha^{-1}|\over|\alpha|+|\alpha^{-1}|}},
  48. ( a 0 b a - 1 ) \begin{pmatrix}a&0\\ b&a^{-1}\end{pmatrix}
  49. ( α 0 β α - 1 ) . \begin{pmatrix}\alpha&0\\ \beta&\alpha^{-1}\end{pmatrix}.
  50. | β | = b | α + α - 1 | = | α | - 1 - | α | 1 - δ > | α | - 1 - | α | . \displaystyle{|\beta|={b\over|\alpha+\alpha^{-1}|}={|\alpha|^{-1}-|\alpha|% \over 1-\delta}>|\alpha|^{-1}-|\alpha|.}
  51. H ¯ \overline{H}
  52. H ¯ \overline{H}
  53. H ¯ \overline{H}
  54. H ¯ \overline{H}
  55. H ¯ \overline{H}
  56. g = ( a b c d ) , \displaystyle{g=\begin{pmatrix}a&b\\ c&d\end{pmatrix},}
  57. g + = ( a ¯ - c ¯ - b ¯ d ¯ ) \displaystyle{g^{+}=\begin{pmatrix}\overline{a}&-\overline{c}\\ -\overline{b}&\overline{d}\end{pmatrix}}
  58. g = ( d ¯ - b ¯ - c ¯ a ¯ ) \displaystyle{g^{\dagger}=\begin{pmatrix}\overline{d}&-\overline{b}\\ -\overline{c}&\overline{a}\end{pmatrix}}
  59. 𝒮 \mathcal{S}
  60. 𝒮 \mathcal{S}
  61. P f ( x ) = i f ( x ) , Q f ( x ) = x f ( x ) . \displaystyle{Pf(x)=if^{\prime}(x),\qquad Qf(x)=xf(x).}
  62. P Q - Q P = i I . \displaystyle{PQ-QP=iI.}
  63. 𝒮 \mathcal{S}
  64. 𝒮 \mathcal{S}
  65. U ( s ) f ( x ) = f ( x - s ) , V ( t ) f ( x ) = e i x t f ( x ) . \displaystyle{U(s)f(x)=f(x-s),\qquad V(t)f(x)=e^{ixt}f(x).}
  66. d d s U ( s ) f = i P U ( s ) f , d d t V ( t ) f = i Q V ( t ) f \displaystyle{{d\over ds}U(s)f=iPU(s)f,\qquad{d\over dt}V(t)f=iQV(t)f}
  67. 𝒮 \mathcal{S}
  68. U ( s ) = e i P s , V ( t ) = e i Q t . \displaystyle{U(s)=e^{iPs},\qquad V(t)=e^{iQt}.}
  69. U ( s ) V ( t ) = e - i s t V ( t ) U ( s ) . \displaystyle{U(s)V(t)=e^{-ist}V(t)U(s).}
  70. 𝒮 \mathcal{S}
  71. f ^ ( ξ ) = 1 2 π - f ( x ) e - i x ξ d x . \displaystyle{\widehat{f}(\xi)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e% ^{-ix\xi}\,dx.}
  72. 𝒮 \mathcal{S}
  73. H 0 ( x ) = e - x 2 / 2 2 π \displaystyle{H_{0}(x)={e^{-x^{2}/2}\over\sqrt{2\pi}}}
  74. P f ^ = - Q f ^ , Q f ^ = P f ^ . \widehat{Pf}=-Q\widehat{f},\qquad\widehat{Qf}=P\widehat{f}.
  75. 𝒮 \mathcal{S}
  76. T f ( x ) = f ^ ^ ( - x ) \displaystyle{Tf(x)=\widehat{\widehat{f}}(-x)}
  77. T H 0 = H 0 \displaystyle{TH_{0}=H_{0}}
  78. g ( x ) = f ( x ) - f ( a ) H 0 ( x ) / H 0 ( a ) x - a \displaystyle{g(x)={f(x)-f(a)H_{0}(x)/H_{0}(a)\over x-a}}
  79. 𝒮 \mathcal{S}
  80. T ( x - a ) g | x = a = ( x - a ) T g | x = a = 0 \displaystyle{T(x-a)g|_{x=a}=(x-a)Tg|_{x=a}=0}
  81. T f ( a ) = f ( a ) . \displaystyle{Tf(a)=f(a).}
  82. f ( x ) = 1 2 π - f ^ ( ξ ) e i x ξ d ξ \displaystyle{f(x)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}\widehat{f}(\xi)e% ^{ix\xi}\,d\xi}
  83. 𝒮 \mathcal{S}
  84. - f ( x ) g ^ ( x ) d x = 1 2 π f ( x ) g ( ξ ) e - i x ξ d x d ξ = - f ^ ( ξ ) g ( ξ ) d ξ . \displaystyle{\int_{-\infty}^{\infty}f(x)\widehat{g}(x)\,dx={1\over\sqrt{2\pi}% }\iint f(x)g(\xi)e^{-ix\xi}\,dxd\xi=\int_{-\infty}^{\infty}\widehat{f}(\xi)g(% \xi)\,d\xi.}
  85. ( f ^ , g ^ ) = ( f , g ) \displaystyle{(\widehat{f},\widehat{g})=(f,g)}
  86. 𝒮 \mathcal{S}
  87. \mathcal{H}
  88. U ( s ) V ( t ) = e - i s t V ( t ) U ( s ) . \displaystyle{U(s)V(t)=e^{-ist}V(t)U(s).}
  89. 𝒮 ( 𝐑 × 𝐑 ) \mathcal{S}(\mathbf{R}\times\mathbf{R})
  90. F ( x , y ) = 1 2 π - f ( t , y ) e - i t x d t \displaystyle{F^{\vee}(x,y)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t,y)e^% {-itx}\,dt}
  91. \mathcal{H}
  92. T ( F ) = F ( x , x + y ) U ( x ) V ( y ) d x d y . \displaystyle{T(F)=\iint F^{\vee}(x,x+y)U(x)V(y)\,dxdy.}
  93. T ( F ) T ( G ) = T ( F G ) , T ( F ) * = T ( F * ) , \displaystyle{T(F)T(G)=T(F\star G),\qquad T(F)^{*}=T(F^{*}),}
  94. F G ( x , y ) = F ( x , z ) G ( z , y ) d z , F * ( x , y ) = F ( y , x ) ¯ . \displaystyle{F\star G(x,y)=\int F(x,z)G(z,y)\,dz,\qquad F^{*}(x,y)=\overline{% F(y,x)}.}
  95. \mathcal{H}
  96. U ( f ) = U ( s ) f ( s ) d s , V ( g ) = V ( t ) g ( t ) d t \displaystyle{U(f)=\int U(s)f(s)\,ds,\qquad V(g)=\int V(t)g(t)\,dt}
  97. U ( f ) , V ( g ) 1 \displaystyle{\|U(f)\|,\,\|V(g)\|\leq 1}
  98. \mathcal{H}
  99. T ( F ) f ( x ) = F ( x , y ) f ( y ) d y . T(F)f(x)=\int F(x,y)f(y)\,dy.
  100. \mathcal{H}
  101. W ( x , y ) = e i x y / 2 U ( x ) V ( y ) . \displaystyle{W(x,y)=e^{ixy/2}U(x)V(y).}
  102. W ( x 1 , y 1 ) W ( x 2 , y 2 ) = e i ( x 1 y 2 - y 1 x 2 ) W ( x 1 + x 2 , y 1 + y 2 ) , \displaystyle{W(x_{1},y_{1})W(x_{2},y_{2})=e^{i(x_{1}y_{2}-y_{1}x_{2})}W(x_{1}% +x_{2},y_{1}+y_{2}),}
  103. ω ( z 1 , z 2 ) = e i B ( z 1 , z 2 ) , \displaystyle{\omega(z_{1},z_{2})=e^{iB(z_{1},z_{2})},}
  104. z = x + i y = ( x , y ) , \displaystyle{z=x+iy=(x,y),}
  105. B ( z 1 , z 2 ) = x 1 y 2 - y 1 x 2 = z 1 z 2 ¯ . \displaystyle{B(z_{1},z_{2})=x_{1}y_{2}-y_{1}x_{2}=\Im\,z_{1}\overline{z_{2}}.}
  106. π ( g ) W ( z ) π ( g ) * = W ( g ( z ) ) . \displaystyle{\pi(g)W(z)\pi(g)^{*}=W(g(z)).}
  107. K ( x , y ) U ( x ) V ( y ) d x d y , \displaystyle{\iint K(x,y)U(x)V(y)\,dxdy,}
  108. π ( g h ) = ± π ( g ) π ( h ) , \displaystyle{\pi(gh)=\pm\pi(g)\pi(h),}
  109. g 1 = ( a 0 0 a - 1 ) , g 2 = ( 1 0 b 1 ) , g 3 = ( 0 1 - 1 0 ) , \displaystyle{g_{1}=\begin{pmatrix}a&0\\ 0&a^{-1}\end{pmatrix},\,\,g_{2}=\begin{pmatrix}1&0\\ b&1\end{pmatrix},\,\,g_{3}=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},}
  110. π ( g 1 ) f ( x ) = ± a - 1 / 2 f ( a - 1 x ) , π ( g 2 ) f ( x ) = ± e - i b x 2 f ( x ) , π ( g 3 ) f ( x ) = ± e i π / 8 f ^ ( x ) \displaystyle{\pi(g_{1})f(x)=\pm a^{-1/2}f(a^{-1}x),\,\,\pi(g_{2})f(x)=\pm e^{% -ibx^{2}}f(x),\,\,\pi(g_{3})f(x)=\pm e^{i\pi/8}\widehat{f}(x)}
  111. g 3 2 = g 1 ( - 1 ) , g 3 g 1 ( a ) g 3 - 1 = g 1 ( a - 1 ) , g 1 ( a ) g 2 ( b ) g 1 ( a ) - 1 = g 2 ( a - 2 b ) , g 1 ( a ) = g 3 g 2 ( a - 1 ) g 3 g 2 ( a ) g 3 g 2 ( a - 1 ) . \displaystyle{g_{3}^{2}=g_{1}(-1),\,\,g_{3}g_{1}(a)g_{3}^{-1}=g_{1}(a^{-1}),\,% \,g_{1}(a)g_{2}(b)g_{1}(a)^{-1}=g_{2}(a^{-2}b),\,\,g_{1}(a)=g_{3}g_{2}(a^{-1})% g_{3}g_{2}(a)g_{3}g_{2}(a^{-1}).}
  112. g = ( a b c d ) , \displaystyle{g=\begin{pmatrix}a&b\\ c&d\end{pmatrix},}
  113. d g ( z ) d z = 1 ( c z + d ) 2 . \displaystyle{{dg(z)\over dz}={1\over(cz+d)^{2}}.}
  114. m ( g , z ) = c z + d \displaystyle{m(g,z)=cz+d}
  115. m ( g h , z ) = m ( g , h z ) m ( h , z ) . \displaystyle{m(gh,z)=m(g,hz)m(h,z).}
  116. M p ( 2 , 𝐑 ) = { ( g , G ) : G ( z ) 2 = m ( g , z ) } . \displaystyle{Mp(2,\mathbf{R})=\{(g,G):\,G(z)^{2}=m(g,z)\}.}
  117. ( g , G ) ( h , H ) = ( g h , K ) , \displaystyle{(g,G)\cdot(h,H)=(gh,K),}
  118. K ( z ) = G ( h z ) H ( z ) . \displaystyle{K(z)=G(hz)H(z).}
  119. z > 0 \Im z>0
  120. f z ( x ) = e i z x 2 / 2 \displaystyle{f_{z}(x)=e^{izx^{2}/2}}
  121. f i ( x ) = e - x 2 / 2 , \displaystyle{f_{i}(x)=e^{-x^{2}/2},}
  122. π ( ( g t ) - 1 ) f z ( x ) = ± m ( g , z ) - 1 / 2 f g z ( x ) . \displaystyle{\pi((g^{t})^{-1})f_{z}(x)=\pm m(g,z)^{-1/2}f_{gz}(x).}
  123. π ( ( g t ) - 1 ) f z ( x ) = m ( g , z ) - 1 / 2 f g z ( x ) . \displaystyle{\pi((g^{t})^{-1})f_{z}(x)=m(g,z)^{-1/2}f_{gz}(x).}
  124. τ ( u , v , w ) \tau(u,v,w)
  125. Q ( a , b , c ) = a b B ( u , v ) + b c B ( v , w ) + c a B ( w , u ) . \displaystyle{Q(a,b,c)=abB(u,v)+bcB(v,w)+caB(w,u).}
  126. τ ( v , w , z ) - τ ( u , w , z ) + τ ( u , v , z ) - τ ( u , v , w ) = 0 \displaystyle{\tau(v,w,z)-\tau(u,w,z)+\tau(u,v,z)-\tau(u,v,w)=0}
  127. Ω ( g , h ) = exp - π i 4 τ ( u 0 , g u 0 , g h u 0 ) \displaystyle{\Omega(g,h)=\exp-{\pi i\over 4}\tau(u_{0},gu_{0},ghu_{0})}
  128. b ( g ) = f ( u 0 , g u 0 ) , \displaystyle{b(g)=f(u_{0},gu_{0}),}
  129. Ω ( g , h ) 2 = b ( g h ) b ( g ) - 1 b ( h ) - 1 . \displaystyle{\Omega(g,h)^{2}=b(gh)b(g)^{-1}b(h)^{-1}.}
  130. π ( g h ) = ω ( g , h ) π ( g ) π ( h ) \displaystyle{\pi(gh)=\omega(g,h)\pi(g)\pi(h)}
  131. ω ( g , h ) = Ω ( g , h ) β ( g h ) - 1 β ( g ) β ( h ) , \displaystyle{\omega(g,h)=\Omega(g,h)\beta(gh)^{-1}\beta(g)\beta(h),}
  132. β ( g ) 2 = b ( g ) . \displaystyle{\beta(g)^{2}=b(g).}
  133. \mathcal{F}
  134. 1 π | f ( z ) | 2 e - | z | 2 d x d y \displaystyle{{1\over\pi}\iint_{\mathbb{C}}|f(z)|^{2}e^{-|z|^{2}}\,dxdy}
  135. ( f 1 , f 2 ) = 1 π f 1 ( z ) f 2 ( z ) ¯ e - | z | 2 d x d y . \displaystyle{(f_{1},f_{2})={1\over\pi}\iint_{\mathbb{C}}f_{1}(z)\overline{f_{% 2}(z)}e^{-|z|^{2}}\,dxdy.}
  136. \mathcal{F}
  137. e n ( z ) = z n n ! \displaystyle{e_{n}(z)={z^{n}\over\sqrt{n!}}}
  138. \mathcal{F}
  139. | f ( z ) | = | n 0 a n z n | f e | z | 2 / 2 , \displaystyle{|f(z)|=\left|\sum_{n\geq 0}a_{n}z^{n}\right|\leq\|f\|e^{|z|^{2}/% 2},}
  140. \mathcal{F}
  141. f ( a ) = ( f , E a ) \displaystyle{f(a)=(f,E_{a})}
  142. E a ( z ) = n 0 ( E a , e n ) z n n ! = n 0 z n a ¯ n n ! = e z a ¯ . \displaystyle{E_{a}(z)=\sum_{n\geq 0}{(E_{a},e_{n})z^{n}\over\sqrt{n!}}=\sum_{% n\geq 0}{z^{n}\overline{a}^{n}\over n!}=e^{z\overline{a}}.}
  143. \mathcal{F}
  144. \mathcal{F}
  145. W ( z ) f ( w ) = e - | z | 2 / 2 e w z ¯ f ( w - z ) . \displaystyle{W_{\mathcal{F}}(z)f(w)=e^{-|z|^{2}/2}e^{w\overline{z}}f(w-z).}
  146. W ( z 1 ) W ( z 2 ) = e - i z 1 z 2 ¯ W ( z 1 + z 2 ) , \displaystyle{W_{\mathcal{F}}(z_{1})W_{\mathcal{F}}(z_{2})=e^{-i\Im z_{1}% \overline{z_{2}}}W_{\mathcal{F}}(z_{1}+z_{2}),}
  147. W ( a ) E 0 = e - | a | 2 / 2 E a . \displaystyle{W_{\mathcal{F}}(a)E_{0}=e^{-|a|^{2}/2}E_{a}.}
  148. W W_{\mathcal{F}}
  149. \mathcal{F}
  150. f ( z ) = ( P E 0 , E z ) = e | z | 2 ( P E 0 , W ( z ) E 0 ) = ( P E - z , E 0 ) = f ( - z ) ¯ . \displaystyle{f(z)=(PE_{0},E_{z})=e^{|z|^{2}}(PE_{0},W_{\mathcal{F}}(z)E_{0})=% (PE_{-z},E_{0})=\overline{f(-z)}.}
  151. P E 0 = λ E 0 , \displaystyle{PE_{0}=\lambda E_{0}},
  152. 𝒰 \mathcal{U}
  153. \mathcal{F}
  154. 𝒰 \mathcal{U}
  155. W ( a ) 𝒰 = 𝒰 W ( a ) \displaystyle{W_{\mathcal{F}}(a)\mathcal{U}=\mathcal{U}W(a)}
  156. 𝒰 H 0 = E 0 . \displaystyle{\mathcal{U}H_{0}=E_{0}.}
  157. 𝒰 f ( z ) = ( 𝒰 f , E z ) = ( f , 𝒰 * E z ) = e - | z | 2 ( f , 𝒰 * W ( z ) E 0 ) = e - | z | 2 ( W ( - z ) f , H 0 ) , \displaystyle{\mathcal{U}f(z)=(\mathcal{U}f,E_{z})=(f,\mathcal{U}^{*}E_{z})=e^% {-|z|^{2}}(f,\mathcal{U}^{*}W_{\mathcal{F}}(z)E_{0})=e^{-|z|^{2}}(W(-z)f,H_{0}% ),}
  158. 𝒰 f ( z ) = 1 2 π - e - ( x 2 + y 2 ) e - 2 i x y f ( t + x ) e - t 2 / 2 d t = 1 2 π - B ( z , t ) f ( t ) d t , \displaystyle{\mathcal{U}f(z)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-(x% ^{2}+y^{2})}e^{-2ixy}f(t+x)e^{-t^{2}/2}\,dt={1\over\sqrt{2\pi}}\int_{-\infty}^% {\infty}B(z,t)f(t)\,dt,}
  159. B ( z , t ) = exp [ - z 2 - t 2 / 2 + z t ] . \displaystyle{B(z,t)=\exp\,[-z^{2}-t^{2}/2+zt].}
  160. 𝒰 \mathcal{U}
  161. 𝒰 \mathcal{U}
  162. 𝒰 * F ( t ) = 1 π 𝐂 B ( z ¯ , t ) F ( z ) d x d y . \displaystyle{\mathcal{U}^{*}F(t)={1\over\pi}\iint_{\mathbf{C}}B(\overline{z},% t)F(z)\,dxdy.}
  163. g = ( α β β ¯ α ¯ ) \displaystyle{g=\begin{pmatrix}\alpha&\beta\\ \overline{\beta}&\overline{\alpha}\end{pmatrix}}
  164. γ 2 = α . \displaystyle{\gamma^{2}=\alpha.}
  165. γ = γ 1 γ 2 ( 1 + β 1 β 2 ¯ α 1 α 2 ) 1 / 2 , \displaystyle{\gamma=\gamma_{1}\gamma_{2}\left(1+{\beta_{1}\overline{\beta_{2}% }\over\alpha_{1}\alpha_{2}}\right)^{1/2},}
  166. g z = α z + β z ¯ . \displaystyle{g\cdot z=\alpha z+\beta\overline{z}.}
  167. \mathcal{F}
  168. K T ( a , b ) = ( T E b ¯ , E a ) , \displaystyle{K_{T}(a,b)=(TE_{\overline{b}},E_{a}),}
  169. \mathcal{F}
  170. T F ( a ) = ( T F , E a ) = ( F , T * E a ) = 1 π 𝐂 F ( z ) ( T * E a , E z ) ¯ e - | z | 2 d x d y = 1 π 𝐂 K T ( a , z ¯ ) F ( z ) e - | z | 2 d x d y . TF(a)=(TF,E_{a})=(F,T^{*}E_{a})=\frac{1}{\pi}\iint_{\mathbf{C}}F(z)\overline{(% T^{*}E_{a},E_{z})}e^{-|z|^{2}}\,dxdy=\frac{1}{\pi}\iint_{\mathbf{C}}K_{T}(a,% \overline{z})F(z)e^{-|z|^{2}}\,dxdy.
  171. K S ( a , z ) = C exp 1 2 α ( β ¯ z 2 + 2 a z - β a 2 ) \displaystyle{K_{S}(a,z)=C\cdot\exp\,{1\over 2\alpha}(\overline{\beta}z^{2}+2% az-\beta a^{2})}
  172. C = γ - 1 \displaystyle{C=\gamma^{-1}}
  173. F w ( z ) = e w z 2 / 2 . \displaystyle{F_{w}(z)=e^{wz^{2}/2}.}
  174. \mathcal{F}
  175. π ( g , γ ) F w = ( α ¯ + β ¯ w ) - 1 2 F g w = 1 γ ¯ ( 1 + β ¯ α ¯ w ) - 1 / 2 F g w , \pi(g,\gamma)F_{w}=(\overline{\alpha}+\overline{\beta}w)^{-\frac{1}{2}}F_{gw}=% \frac{1}{\overline{\gamma}}\left(1+{\overline{\beta}\over\overline{\alpha}}w% \right)^{-1/2}F_{gw},
  176. g w = α w + β β ¯ w + α ¯ . \displaystyle{gw={\alpha w+\beta\over\overline{\beta}w+\overline{\alpha}}.}
  177. \mathcal{F}
  178. π ( g , γ ) [ z F w ] ( z ) = ( α ¯ + β ¯ w ) - 3 / 2 z F g w ( z ) . \displaystyle{\pi(g,\gamma)[zF_{w}](z)=(\overline{\alpha}+\overline{\beta}w)^{% -3/2}zF_{gw}(z).}
  179. ( π ( g , γ ) 1 , 1 ) = γ - 1 . \displaystyle{(\pi(g,\gamma)1,1)=\gamma^{-1}.}
  180. \mathcal{F}
  181. 1 2 π | F + ( z ) | 2 ( 1 - | z | 2 ) - 1 / 2 d x d y + 2 π | F + ( z ) | 2 ( 1 - | z | 2 ) 1 2 d x d y \displaystyle{{1\over 2\pi}\iint|F_{+}(z)|^{2}(1-|z|^{2})^{-1/2}\,dxdy+{2\over% \pi}\iint|F^{\prime}_{+}(z)|^{2}(1-|z|^{2})^{\frac{1}{2}}\,dxdy}
  182. 1 2 π | F - ( z ) | 2 ( 1 - | z | 2 ) - 1 / 2 d x d y \displaystyle{{1\over 2\pi}\iint|F_{-}(z)|^{2}(1-|z|^{2})^{-1/2}\,dxdy}
  183. π ± ( g - 1 ) F ± ( z ) = ( β ¯ z + α ¯ ) - 1 ± 1 / 2 F ± ( g z ) \displaystyle{\pi_{\pm}(g^{-1})F_{\pm}(z)=(\overline{\beta}z+\overline{\alpha}% )^{-1\pm 1/2}F_{\pm}(gz)}
  184. ( α β β ¯ α ¯ ) . \displaystyle{\begin{pmatrix}\alpha&\beta\\ \overline{\beta}&\overline{\alpha}\end{pmatrix}.}
  185. π ± ( g ) F ± ( z ) = ( - β ¯ z + α ) - 1 ± 1 / 2 F ± ( α ¯ z - β - β ¯ z + α ) . \displaystyle{\pi_{\pm}(g)F_{\pm}(z)=(-\overline{\beta}z+\alpha)^{-1\pm 1/2}F_% {\pm}\left({\overline{\alpha}z-\beta\over-\overline{\beta}z+\alpha}\right).}
  186. 𝔤 \mathfrak{g}
  187. \mathcal{F}
  188. U + ( F ) ( w ) = 1 π 𝐂 F ( z ) e 1 2 w z ¯ 2 e - | z | 2 d x d y , U_{+}(F)(w)=\frac{1}{\pi}\iint_{\mathbf{C}}F(z)e^{\frac{1}{2}w\overline{z}^{2}% }e^{-|z|^{2}}\,dxdy,
  189. U - ( F ) ( w ) = 1 π 𝐂 F ( z ) z ¯ e 1 2 w z ¯ 2 e - | z | 2 d x d y , U_{-}(F)(w)=\frac{1}{\pi}\iint_{\mathbf{C}}F(z)\overline{z}e^{\frac{1}{2}w% \overline{z}^{2}}e^{-|z|^{2}}\,dxdy,
  190. \mathcal{H}
  191. f ( x ) = p ( x ) e - x 2 2 , f(x)=p(x)e^{-\frac{x^{2}}{2}},
  192. X = Q - i P = d d x + x , Y = Q + i P = - d d x + x \displaystyle{X=Q-iP={d\over dx}+x,\,\,\,\,Y=Q+iP=-{d\over dx}+x}
  193. \mathcal{H}
  194. X = Y * , X Y - Y X = 2 I . \displaystyle{X=Y^{*},\,\,\,XY-YX=2I.}
  195. X Y = P 2 + Q 2 + I = D + I . \displaystyle{XY=P^{2}+Q^{2}+I=D+I.}
  196. F n ( x ) = Y n e - x 2 / 2 \displaystyle{F_{n}(x)=Y^{n}e^{-x^{2}/2}}
  197. D F n = ( 2 n + 1 ) F n . \displaystyle{DF_{n}=(2n+1)F_{n}.}
  198. X Y n - Y n X = 2 n Y n - 1 \displaystyle{XY^{n}-Y^{n}X=2nY^{n-1}}
  199. D F n = ( X Y - I ) Y n F 0 = ( Y n + 1 X + ( 2 n + 1 ) Y n ) F 0 = ( 2 n + 1 ) F n , \displaystyle{DF_{n}=(XY-I)Y^{n}F_{0}=(Y^{n+1}X+(2n+1)Y^{n})F_{0}=(2n+1)F_{n},}
  200. F n 2 2 = 2 n n ! π , \displaystyle{\|F_{n}\|^{2}_{2}=2^{n}n!\sqrt{\pi},}
  201. ( F n , F n ) = ( X Y n F 0 , Y n - 1 F 0 ) = 2 n ( F n - 1 , F n - 1 ) . \displaystyle{(F_{n},F_{n})=(XY^{n}F_{0},Y^{n-1}F_{0})=2n(F_{n-1},F_{n-1}).}
  202. H n ( x ) = F n - 1 F n ( x ) = p n ( x ) e - x 2 / 2 . \displaystyle{H_{n}(x)=\|F_{n}\|^{-1}F_{n}(x)=p_{n}(x)e^{-x^{2}/2}.}
  203. D = - d 2 d x 2 + x 2 . \displaystyle{D=-{d^{2}\over dx^{2}}+x^{2}.}
  204. A = 1 2 Y = 1 2 ( - d d x + x ) , A * = 1 2 X = 1 2 ( d d x + x ) . \displaystyle{A={1\over\sqrt{2}}Y={1\over\sqrt{2}}\left(-{d\over dx}+x\right),% \,\,\,A^{*}={1\over\sqrt{2}}X={1\over\sqrt{2}}\left({d\over dx}+x\right).}
  205. A A * - A * A = I . \displaystyle{AA^{*}-A^{*}A=I.}
  206. \mathcal{H}
  207. \mathcal{H}
  208. A = z , A * = z . A=\frac{\partial}{\partial z},\qquad A^{*}=z.
  209. D = 2 A * A + I . \displaystyle{D=2A^{*}A+I.}
  210. L 0 = 1 2 A * A = 1 2 z z \displaystyle{L_{0}={1\over 2}A^{*}A={1\over 2}z{\partial\over\partial z}}
  211. L 0 z n = n 2 z n . \displaystyle{L_{0}z^{n}={n\over 2}z^{n}.}
  212. [ L 0 , A ] = - 1 2 A , [ L 0 , A * ] = 1 2 A * , \displaystyle{[L_{0},A]=-{1\over 2}A,\,\,\,[L_{0},A^{*}]={1\over 2}A^{*},}
  213. L - 1 = 1 2 A 2 , L 1 = 1 2 A * 2 , \displaystyle{L_{-1}={1\over 2}A^{2},\,\,\,L_{1}={1\over 2}A^{*2},}
  214. [ L - 1 , A ] = 0 , [ L - 1 , A * ] = A , [ L 1 , A ] = - A * , [ L 1 , A * ] = 0. \displaystyle{[L_{-1},A]=0,\,\,\,[L_{-1},A^{*}]=A,\,\,\,[L_{1},A]=-A^{*},\,\,% \,[L_{1},A^{*}]=0.}
  215. [ L m , L n ] = ( m - n ) L m + n \displaystyle{[L_{m},L_{n}]=(m-n)L_{m+n}}
  216. F n ( x ) = ( x - d d x ) n e - x 2 / 2 = ( - 1 ) n e x 2 / 2 d n e - x 2 d x n = ( 2 n x n + ) e - x 2 / 2 . \displaystyle{F_{n}(x)=\left(x-{d\over dx}\right)^{n}e^{-x^{2}/2}=(-1)^{n}e^{x% ^{2}/2}{d^{n}e^{-x^{2}}\over dx^{n}}=(2^{n}x^{n}+\cdots)e^{-x^{2}/2}.}
  217. \mathcal{H}
  218. e - D t H n = e - ( 2 n + 1 ) t H n . \displaystyle{e^{-Dt}H_{n}=e^{-(2n+1)t}H_{n}.}
  219. K t ( x , y ) n 0 e - ( 2 n + 1 ) t H n ( x ) H n ( y ) = ( 4 π t ) - 1 2 ( 2 t sinh 2 t ) 1 2 exp ( - 1 4 t [ 2 t tanh 2 t ( x 2 + y 2 ) - 2 t sinh 2 t ( 2 x y ) ] ) . K_{t}(x,y)\equiv\sum_{n\geq 0}e^{-(2n+1)t}H_{n}(x)H_{n}(y)=(4\pi t)^{-{1\over 2% }}\left({2t\over\sinh 2t}\right)^{1\over 2}\exp\left(-{1\over 4t}\left[{2t% \over\tanh 2t}(x^{2}+y^{2})-{2t\over\sinh 2t}(2xy)\right]\right).
  220. n 0 s n H n ( x ) H n ( y ) = 1 π ( 1 - s 2 ) exp 4 x y s - ( 1 + s 2 ) ( x 2 + y 2 ) 2 ( 1 - s 2 ) . \displaystyle{\sum_{n\geq 0}s^{n}H_{n}(x)H_{n}(y)={1\over\sqrt{\pi(1-s^{2})}}% \exp{4xys-(1+s^{2})(x^{2}+y^{2})\over 2(1-s^{2})}.}
  221. F σ , x ( z ) n 0 σ n e n ( z ) H n ( x ) = π - 1 4 e - x 2 2 n 0 ( - z ) n σ n 2 n n ! d n e x 2 d x n = π - 1 4 exp ( - x 2 2 + 2 x z σ - z 2 σ 2 2 ) . F_{\sigma,x}(z)\equiv\sum_{n\geq 0}\sigma^{n}e_{n}(z)H_{n}(x)=\pi^{-{1\over 4}% }e^{-\frac{x^{2}}{2}}\sum_{n\geq 0}{(-z)^{n}\sigma^{n}\over 2^{n}n!}{d^{n}e^{x% ^{2}}\over dx^{n}}=\pi^{-\frac{1}{4}}\exp(-{x^{2}\over 2}+\sqrt{2}xz\sigma-{z^% {2}\sigma^{2}\over 2}).
  222. n 0 s n H n ( x ) H n ( y ) = ( F σ , x , F σ , y ) , \displaystyle{\sum_{n\geq 0}s^{n}H_{n}(x)H_{n}(y)=(F_{\sigma,x},F_{\sigma,y})_% {\mathcal{F}},}
  223. K t ( x , y ) f ( y ) d y \displaystyle{\int K_{t}(x,y)f(y)\,dy}
  224. H n ^ = ( - i ) n H n , \displaystyle{\widehat{H_{n}}=(-i)^{n}H_{n},}
  225. P = i d d x , Q = x \displaystyle{P=i{d\over dx},\,\,\,Q=x}
  226. 𝒮 \mathcal{S}
  227. D = P 2 + Q 2 = - d 2 d x 2 + x 2 \displaystyle{D=P^{2}+Q^{2}=-{d^{2}\over dx^{2}}+x^{2}}
  228. 𝒮 \mathcal{S}
  229. f ( s ) 2 = n 0 | a n | 2 ( 1 + 2 n ) s , \displaystyle{\|f\|_{(s)}^{2}=\sum_{n\geq 0}|a_{n}|^{2}(1+2n)^{s},}
  230. f = a n H n \displaystyle{f=\sum a_{n}H_{n}}
  231. f ( s ) 2 = ( D s f , f ) , ( f 1 , f 2 ) ( s ) = ( D s f 1 , f 2 ) . \displaystyle{\|f\|_{(s)}^{2}=(D^{s}f,f),\,\,\,(f_{1},f_{2})_{(s)}=(D^{s}f_{1}% ,f_{2}).}
  232. f 1 , f 2 = f 1 f 2 d x . \displaystyle{\langle f_{1},f_{2}\rangle=\int f_{1}f_{2}\,dx.}
  233. ( a P + b Q ) f ( s ) ( | a | + | b | ) C s f ( s + 1 2 ) \displaystyle{\|(aP+bQ)f\|_{(s)}\leq(|a|+|b|)C_{s}\|f\|_{(s+{1\over 2})}}
  234. R f ( s ) C s f ( s + 1 ) . \displaystyle{\|Rf\|_{(s)}\leq C_{s}^{\prime}\|f\|_{(s+1)}.}
  235. | f ( x ) | C s , k f ( s + k ) ( 1 + x 2 ) - k \displaystyle{|f(x)|\leq C_{s,k}\|f\|_{(s+k)}(1+x^{2})^{-k}}
  236. f ( x ) = 1 2 π - f ^ ( t ) e i t x d t \displaystyle{f(x)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}\widehat{f}(t)e^{% itx}\,dt}
  237. | f ( x ) | C ( | f ^ ( t ) | 2 ( 1 + t 2 ) s d t ) 1 2 = C ( ( I + Q 2 ) s f ^ , f ^ ) 1 2 C f ^ ( s ) = C f ( s ) . \displaystyle{|f(x)|\leq C\left(\int|\widehat{f}(t)|^{2}(1+t^{2})^{s}\,dt% \right)^{1\over 2}=C((I+Q^{2})^{s}\widehat{f},\widehat{f})^{1\over 2}\leq C^{% \prime}\|\widehat{f}\|_{(s)}=C^{\prime}\|f\|_{(s)}.}
  238. 𝒮 \mathcal{S}
  239. 𝒮 \mathcal{S}
  240. g D g - 1 = D + A \displaystyle{gDg^{-1}=D+A}
  241. π ( g ) f ( s ) 2 = | ( ( D + A ) s f , f ) | ( D + A ) s f ( - s ) f ( s ) C f ( s ) 2 . \displaystyle{\|\pi(g)f\|^{2}_{(s)}=|((D+A)^{s}f,f)|\leq\|(D+A)^{s}f\|_{(-s)}% \cdot\|f\|_{(s)}\leq C\|f\|_{(s)}^{2}.}
  242. d d s U ( s ) f = i P U ( s ) f , d d t V ( t ) f = i Q V ( t ) f , \displaystyle{{d\over ds}U(s)f=iPU(s)f,\,\,\,{d\over dt}V(t)f=iQV(t)f,}
  243. Φ ( z ) = W ( z ) u \displaystyle{\Phi(z)=W(z)u}
  244. 𝒮 \mathcal{S}
  245. 𝒮 \mathcal{S}
  246. 𝒮 \mathcal{S}
  247. 𝒮 \mathcal{S}
  248. Π ( f ) = - f ( t ) Π ( t ) d t , \displaystyle{\Pi(f)=\int_{-\infty}^{\infty}f(t)\Pi(t)\,dt,}
  249. f ε ( x ) = 1 2 π ε e - x 2 / 2 ε \displaystyle{f_{\varepsilon}(x)={1\over\sqrt{2\pi\varepsilon}}e^{-x^{2}/2% \varepsilon}}
  250. Φ ( t ) = Π ( t ) v \displaystyle{\Phi(t)=\Pi(t)v}
  251. Π ( t ) = e i t D . \displaystyle{\Pi(t)=e^{it\sqrt{D}}.}
  252. Π ( t ) H n = e i ( 2 n + 1 ) 1 2 t H n . \displaystyle{\Pi(t)H_{n}=e^{i(2n+1)^{1\over 2}t}H_{n}.}
  253. v = e - ε D ξ . \displaystyle{v=e^{-\varepsilon D}\xi.}
  254. n 0 r n D n 2 v n ! < \displaystyle{\sum_{n\geq 0}{r^{n}\|D^{n\over 2}v\|\over n!}<\infty}
  255. F ( s , t ) = U ( s ) V ( t ) v \displaystyle{F(s,t)=U(s)V(t)v}
  256. m , n 0 1 m ! n ! z m w n P m Q n v C k 0 ( | z | + | w | ) k k ! D k 2 v < . \displaystyle{\left\|\sum_{m,n\geq 0}{1\over m!n!}z^{m}w^{n}P^{m}Q^{n}v\right% \|\leq C\sum_{k\geq 0}{(|z|+|w|)^{k}\over k!}\|D^{k\over 2}v\|<\infty.}
  257. W ( z , w ) = e - i z w / 2 U ( z ) V ( w ) \displaystyle{W(z,w)=e^{-izw/2}U(z)V(w)}
  258. e - i z w / 2 F ( z , w ) = W ( z , w ) v . \displaystyle{e^{-izw/2}F(z,w)=W(z,w)v.}
  259. W ( z 1 , w 1 ) W ( z 2 , w 2 ) = e i ( z 1 w 2 - w 1 z 2 ) W ( z 1 + z 2 , w 1 + w 2 ) . \displaystyle{W(z_{1},w_{1})W(z_{2},w_{2})=e^{i(z_{1}w_{2}-w_{1}z_{2})}W(z_{1}% +z_{2},w_{1}+w_{2}).}
  260. π ( g ) W ( u ) π ( g ) * = W ( g u ) , \displaystyle{\pi(g)W(u)\pi(g)^{*}=W(gu),}
  261. W ( z , w ) * = W ( - z ¯ , - w ¯ ) . \displaystyle{W(z,w)^{*}=W(-\overline{z},-\overline{w}).}
  262. H ¯ \overline{H}
  263. g = ( a b c d ) \displaystyle{g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}}
  264. ( - b ¯ z + d ¯ ) 1 2 \displaystyle{\left(-\overline{b}z+\overline{d}\right)^{1\over 2}}
  265. T W ( u ) = W ( g u ) T , T * W ( u ) = W ( g u ) T * \displaystyle{TW(u)=W(g\cdot u)T,\,\,\,T^{*}W(u)=W(g^{\dagger}\cdot u)T^{*}}
  266. S 0 ( t ) = e - t L 0 \displaystyle{S_{0}(t)=e^{-tL_{0}}}
  267. t > 0 \Re\,t>0
  268. H ¯ = G exp C ¯ \displaystyle{\overline{H}=G\cdot\exp\overline{C}}
  269. t > 0 \Re t>0
  270. ( a b c d ) , \displaystyle{\begin{pmatrix}a&b\\ c&d\end{pmatrix},}
  271. π ( g , γ ) f ( w ) = 1 π 𝐂 K ( w , z ¯ ) f ( z ) e - | z | 2 d x d y , \displaystyle{\pi(g,\gamma)f(w)={1\over\pi}\iint_{\mathbf{C}}K(w,\overline{z})% f(z)e^{-|z|^{2}}\,dxdy,}
  272. K ( w , z ) = γ - 1 exp 1 2 a ( c z 2 + 2 w z - b w 2 ) \displaystyle{K(w,z)=\gamma^{-1}\cdot\exp\,{1\over 2a}(cz^{2}+2wz-bw^{2})}
  273. K ( w , z ) = C exp 1 2 ( p z 2 + 2 q w z + r w 2 ) \displaystyle{K(w,z)=C\cdot\exp\,{1\over 2}(pz^{2}+2qwz+rw^{2})}
  274. ( p q q r ) \displaystyle{\begin{pmatrix}p&q\\ q&r\end{pmatrix}}
  275. π ± ( g ) F ± ( z ) = ( - b ¯ z + d ¯ ) - 1 ± 1 / 2 F ± ( a ¯ z - c ¯ - b ¯ z + d ¯ ) . \displaystyle{\pi_{\pm}(g)F_{\pm}(z)=(-\overline{b}z+\overline{d})^{-1\pm 1/2}% F_{\pm}\left({\overline{a}z-\overline{c}\over-\overline{b}z+\overline{d}}% \right).}
  276. Z = ( A B B D ) \displaystyle{Z=\begin{pmatrix}A&B\\ B&D\end{pmatrix}}
  277. K Z ( x , y ) = e - ( A x 2 + 2 B x y + D y 2 ) . \displaystyle{K_{Z}(x,y)=e^{-(Ax^{2}+2Bxy+Dy^{2})}.}
  278. T Z f ( x ) = - K Z ( x , y ) f ( y ) d y \displaystyle{T_{Z}f(x)=\int_{-\infty}^{\infty}K_{Z}(x,y)f(y)\,dy}
  279. T Z 1 T Z 2 = ( D 1 + A 2 ) - 1 / 2 T Z 3 \displaystyle{T_{Z_{1}}T_{Z_{2}}=(D_{1}+A_{2})^{-1/2}T_{Z_{3}}}
  280. Z 3 = ( A 1 - B 1 2 ( D 1 + A 2 ) - 1 - B 1 B 2 ( D 1 + A 2 ) - 1 - B 1 B 2 ( D 1 + A 2 ) - 1 D 2 - B 2 2 ( D 1 + A 2 ) - 1 ) . \displaystyle{Z_{3}=\begin{pmatrix}A_{1}-B_{1}^{2}(D_{1}+A_{2})^{-1}&-B_{1}B_{% 2}(D_{1}+A_{2})^{-1}\\ -B_{1}B_{2}(D_{1}+A_{2})^{-1}&D_{2}-B_{2}^{2}(D_{1}+A_{2})^{-1}\end{pmatrix}.}
  281. T Z * = T Z + \displaystyle{T_{Z}^{*}=T_{Z^{+}}}
  282. Z + = ( D ¯ B ¯ B ¯ A ¯ ) . \displaystyle{Z^{+}=\begin{pmatrix}\overline{D}&\overline{B}\\ \overline{B}&\overline{A}\end{pmatrix}.}
  283. t > 0 \Re\,t>0
  284. e - t ( P 2 + Q 2 ) = ( cosech 2 t ) 1 2 T Z ( t ) \displaystyle{e^{-t(P^{2}+Q^{2})}=(\mathrm{cosech}\,2t)^{1\over 2}\cdot T_{Z(t% )}}
  285. Z ( t ) = ( coth 2 t - cosech 2 t - cosech 2 t coth 2 t ) . \displaystyle{Z(t)=\begin{pmatrix}\coth 2t&-\mathrm{cosech}\,2t\\ -\mathrm{cosech}\,2t&\coth 2t\end{pmatrix}.}
  286. S Z = B 1 2 T Z . \displaystyle{S_{Z}=B^{1\over 2}\cdot T_{Z}.}
  287. g = ( - D B - 1 D A B - 1 - B B - 1 - A B - 1 ) \displaystyle{g=\begin{pmatrix}-DB^{-1}&DAB^{-1}-B\\ B^{-1}&-AB^{-1}\end{pmatrix}}
  288. S Z < 1. \displaystyle{\|S_{Z}\|<1.}
  289. S Z 1 S Z 2 = ± S Z 3 . \displaystyle{S_{Z_{1}}S_{Z_{2}}=\pm S_{Z_{3}}.}
  290. ψ ( a ) = 1 2 π a ^ ( x , y ) W ( x , y ) d x d y . \displaystyle{\psi(a)={1\over 2\pi}\int\widehat{a}(x,y)W(x,y)\,dxdy.}
  291. ψ ( a ) f ( x ) = K ( x , y ) f ( y ) d y , \displaystyle{\psi(a)f(x)=\int K(x,y)f(y)\,dy,}
  292. K ( x , y ) = a ( t , x + y 2 ) e i ( x - y ) t d t . \displaystyle{K(x,y)=\int a(t,{x+y\over 2})e^{i(x-y)t}\,dt.}
  293. W ( F ) = 1 2 π F ( z ) W ( z ) d x d y , \displaystyle{W(F)={1\over 2\pi}\int F(z)W(z)\,dxdy,}
  294. W ( F ) W ( G ) = W ( F G ) , \displaystyle{W(F)W(G)=W(F\star G),}
  295. F G ( z ) = 1 2 π F ( z 1 ) G ( z 2 - z 1 ) e i ( x 1 y 2 - y 1 x 2 ) d x 1 d y 1 . \displaystyle{F\star G(z)={1\over 2\pi}\int F(z_{1})G(z_{2}-z_{1})e^{i(x_{1}y_% {2}-y_{1}x_{2})}\,dx_{1}dy_{1}.}
  296. ψ ( a ) ψ ( b ) = ψ ( a b ) , \displaystyle{\psi(a)\psi(b)=\psi(a\circ b),}
  297. a b = n 0 i n n ! ( 2 x 1 y 2 - 2 y 1 x 2 ) n a b | diagonal . a\circ b=\sum_{n\geq 0}{i^{n}\over n!}\left({\partial^{2}\over\partial x_{1}% \partial y_{2}}-{\partial^{2}\over\partial y_{1}\partial x_{2}}\right)^{n}a% \otimes b|_{\mathrm{diagonal}}.
  298. | α a ( z ) | C α ( 1 + | z | ) m - | α | \displaystyle{|\partial^{\alpha}a(z)|\leq C_{\alpha}(1+|z|)^{m-|\alpha|}}
  299. a ^ = χ a ^ + ( 1 - χ ) a ^ = T + S , \displaystyle{\widehat{a}=\chi\widehat{a}+(1-\chi)\widehat{a}=T+S,}
  300. Ψ m Ψ m Ψ m + n , [ Ψ m , Ψ n ] Ψ m + n - 2 . \displaystyle{\Psi^{m}\cdot\Psi^{m}\subseteq\Psi^{m+n},\,\,\,\,[\Psi^{m},\Psi^% {n}]\subseteq\Psi^{m+n-2}.}
  301. 𝒮 \mathcal{S}
  302. Tr ψ ( a ) = a \displaystyle{\mathrm{Tr}\,\psi(a)=\int a}
  303. T a , b v = ( v , b ) a , \displaystyle{T_{a,b}v=(v,b)a,}
  304. T W ( z ) a , b = e | z | 2 / 2 [ W ( z ) T a , E 0 W ( z ) - 1 T E 0 , b ] , \displaystyle{T_{W(z)a,b}=e^{|z|^{2}/2}[W(z)T_{a,E_{0}}W(z)^{-1}T_{E_{0},b}],}
  305. a b \|a\|\cdot\|b\|
  306. W ( F ) e R 2 / 2 F ^ . \displaystyle{\|W(F)\|\leq e^{R^{2}/2}\|\widehat{F}\|_{\infty}.}
  307. S = ψ ( a ) \displaystyle{S=\psi(a)}
  308. a ( z ) = 1 | z | 2 + 1 . \displaystyle{a(z)={1\over|z|^{2}+1}.}
  309. A S + S R + S R 2 + \displaystyle{A\sim S+SR+SR^{2}+\cdots}
  310. D - 1 = A - D - 1 T \displaystyle{D^{-1}=A-D^{-1}T}
  311. ( a , b ) = Q ( a ) Q ( b ) Q ( a + b ) - 1 \displaystyle{(a,b)=Q(a)Q(b)Q(a+b)^{-1}}
  312. V = 2 ( A ) , \displaystyle{V=\ell^{2}(A),}
  313. ( f , g ) = x A f ( x ) g ( x ) ¯ . \displaystyle{(f,g)=\sum_{x\in A}f(x)\overline{g(x)}.}
  314. U ( x ) f ( t ) = f ( t - x ) , V ( y ) f ( t ) = ( y , t ) f ( t ) \displaystyle{U(x)f(t)=f(t-x),\,\,\,V(y)f(t)=(y,t)f(t)}
  315. U ( x ) V ( y ) = ( x , y ) V ( y ) U ( x ) . \displaystyle{U(x)V(y)=(x,y)V(y)U(x).}
  316. W ( z ) = U ( x ) V ( y ) . \displaystyle{W(z)=U(x)V(y).}
  317. W ( z 1 ) W ( z 2 ) = B ( z 1 , z 2 ) W ( z 2 ) W ( z 1 ) , \displaystyle{W(z_{1})W(z_{2})=B(z_{1},z_{2})W(z_{2})W(z_{1}),}
  318. B ( z 1 , z 2 ) = ( x 1 , y 2 ) ( x 2 , y 1 ) - 1 , \displaystyle{B(z_{1},z_{2})=(x_{1},y_{2})(x_{2},y_{1})^{-1},}
  319. W ( z 1 ) W ( z 2 ) = B ( z 1 , z 2 ) W ( z 2 ) W ( z 1 ) , \displaystyle{W^{\prime}(z_{1})W^{\prime}(z_{2})=B(z_{1},z_{2})W^{\prime}(z_{2% })W^{\prime}(z_{1}),}
  320. W ( z ) = λ ( z ) U W ( z ) U * , \displaystyle{W^{\prime}(z)=\lambda(z)UW(z)U^{*},}
  321. W ( g z ) = λ g ( z ) π ( g ) W ( z ) π ( g ) * . \displaystyle{W(gz)=\lambda_{g}(z)\pi(g)W(z)\pi(g)^{*}.}
  322. S = ( 0 1 - 1 0 ) , R = ( 1 0 1 1 ) . \displaystyle{S=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},\qquad R=\begin{pmatrix}1&0\\ 1&1\end{pmatrix}.}
  323. S 2 = Z , ( S R ) 3 = Z , Z 2 = I . \displaystyle{S^{2}=Z,\,\,\,(SR)^{3}=Z,\,\,\,Z^{2}=I.}
  324. π ( S ) f ( t ) = | A | - 1 2 x A ( - x , t ) f ( x ) \displaystyle{\pi(S)f(t)=|A|^{-\frac{1}{2}}\sum_{x\in A}(-x,t)f(x)}
  325. π ( Z ) f ( t ) = f ( - t ) , \displaystyle{\pi(Z)f(t)=f(-t),}
  326. π ( R ) f ( t ) = Q ( t ) - 1 f ( t ) . \displaystyle{\pi(R)f(t)=Q(t)^{-1}f(t).}
  327. ( π ( S ) π ( R ) ) 3 = μ π ( Z ) , \displaystyle{(\pi(S)\pi(R))^{3}=\mu\pi(Z),}
  328. μ = | A | - 1 / 2 x A Q ( x ) . \displaystyle{\mu=|A|^{-1/2}\sum_{x\in A}Q(x).}
  329. M p ( 2 , 𝐑 ) = { ( ( a b c d ) , G ) : G ( τ ) 2 = c τ + d , τ 𝐇 } , Mp(2,\mathbf{R})=\left\{\left(\begin{pmatrix}a&b\\ c&d\end{pmatrix},G\right)\ :\ G(\tau)^{2}=c\tau+d,\tau\in\mathbf{H}\right\},
  330. f τ ( x ) = e 1 2 i τ x 2 f_{\tau}(x)=e^{\frac{1}{2}i\tau x^{2}}
  331. π ( ( g t ) - 1 ) f τ = ( c τ + d ) - 1 / 2 f g τ . \pi((g^{t})^{-1})f_{\tau}=(c\tau+d)^{-1/2}f_{g\tau}.
  332. 𝒮 \mathcal{S}
  333. 𝒮 \mathcal{S}^{\prime}
  334. 2 ( A ) \ell^{2}(A)
  335. M = 2 π m 𝐙 . \displaystyle{M=\sqrt{2\pi m}\cdot\mathbf{Z}.}
  336. Ψ b = x M δ x + b \displaystyle{\Psi_{b}=\sum\nolimits_{x\in M}\delta_{x+b}}
  337. M 1 = 1 2 m M M . M_{1}={1\over 2m}M\supset M.
  338. 𝒮 \mathcal{S}^{\prime}
  339. U ( b ) Ψ b = Ψ b + b , V ( b ) Ψ b = e - i m b b Ψ b . U(b)\Psi_{b^{\prime}}=\Psi_{b+b^{\prime}},\qquad V(b)\Psi_{b^{\prime}}=e^{-% imbb^{\prime}}\Psi_{b^{\prime}}.
  340. R ( Ψ b ) = e π i m b 2 Ψ b . R(\Psi_{b})=e^{\pi imb^{2}}\Psi_{b}.
  341. 𝒮 \mathcal{S}
  342. F ( t ) = x M f ( x + t ) . F(t)=\sum\nolimits_{x\in M}f(x+t).
  343. F ( t + a ) = F ( t ) . F(t+a)=F(t).
  344. F ( 0 ) = n 𝐙 c n F(0)=\sum\nolimits_{n\in\mathbf{Z}}c_{n}
  345. c n = a - 1 0 a F ( t ) e - 2 π i n t a d t = a - 1 - f ( t ) e - 2 π i n t a d t = 2 π a f ^ ( 2 π n a ) . c_{n}=a^{-1}\int_{0}^{a}F(t)e^{-\frac{2\pi int}{a}}\,dt=a^{-1}\int_{-\infty}^{% \infty}f(t)e^{-\frac{2\pi int}{a}}\,dt={\sqrt{2\pi}\over a}\widehat{f}\left(% \tfrac{2\pi n}{a}\right).
  346. n 𝐙 f ( n a ) = 2 π a n 𝐙 f ^ ( 2 π n a ) , \sum\nolimits_{n\in\mathbf{Z}}f(na)=\frac{\sqrt{2\pi}}{a}\sum\nolimits_{n\in% \mathbf{Z}}\widehat{f}\left(\tfrac{2\pi n}{a}\right),
  347. S ( Ψ b ) = ( 2 m ) - 1 2 b M 1 / M e - i m b b Ψ b , S(\Psi_{b})=(2m)^{-\frac{1}{2}}\sum\nolimits_{b^{\prime}\in M_{1}/M}e^{-imbb^{% \prime}}\Psi_{b^{\prime}},
  348. b ( n ) = 2 π n 2 m b(n)=\frac{\sqrt{2\pi}n}{2m}
  349. Θ m , n ( τ , z ) = ( W ( z ) f τ , Ψ b ( n ) ) . \Theta_{m,n}(\tau,z)=(W(z)f_{\tau},\Psi_{b(n)}).
  350. q = e 2 π i τ , u = e π i z q=e^{2\pi i\tau},\qquad u=e^{\pi iz}
  351. Θ n , m ( τ , z + a ) = Θ n , m ( τ , z ) , a 𝐙 \Theta_{n,m}(\tau,z+a)=\Theta_{n,m}(\tau,z),\qquad a\in\mathbf{Z}
  352. Θ n , m ( τ , z + b τ ) = q - b 2 u - b Θ n , m ( τ , z ) , b 𝐙 \Theta_{n,m}(\tau,z+b\tau)=q^{-b^{2}}u^{-b}\Theta_{n,m}(\tau,z),\qquad b\in% \mathbf{Z}
  353. Θ n , m ( τ + 1 , z ) = e π i n 2 m Θ n , m ( τ , z ) \Theta_{n,m}(\tau+1,z)=e^{\frac{\pi in^{2}}{m}}\Theta_{n,m}(\tau,z)
  354. Θ n , m ( - 1 τ , z τ ) = τ 1 2 e - i π 8 ( 2 m ) - 1 2 n 𝐙 / 2 m 𝐙 e - π i n n m Θ n , m ( τ , z ) \Theta_{n,m}(-\tfrac{1}{\tau},\tfrac{z}{\tau})=\tau^{\frac{1}{2}}e^{-\frac{i% \pi}{8}}(2m)^{-\frac{1}{2}}\sum\nolimits_{n^{\prime}\in\mathbf{Z}/2m\mathbf{Z}% }e^{-\frac{\pi inn^{\prime}}{m}}\Theta_{n^{\prime},m}(\tau,z)
  355. ( π ( S ) π ( R ) ) 3 = π ( J ) \displaystyle{(\pi(S)\pi(R))^{3}=\pi(J)}
  356. μ = 1 2 ( e i π 4 + e - i π 4 ) = 1. \mu=\frac{1}{\sqrt{2}}\left(e^{\frac{i\pi}{4}}+e^{-\frac{i\pi}{4}}\right)=1.
  357. x 𝐙 / 2 m 𝐙 e π i x 2 / 2 m = m ( 1 + i ) . \displaystyle{\sum_{x\in\mathbf{Z}/2m\mathbf{Z}}e^{\pi ix^{2}/2m}=\sqrt{m}(1+i% ).}
  358. G ( c , m ) = x 𝐙 / m 𝐙 e 2 π i c x 2 / m . \displaystyle{G(c,m)=\sum_{x\in\mathbf{Z}/m\mathbf{Z}}e^{2\pi icx^{2}/m}.}
  359. G ( c , p ) = ( c p ) G ( 1 , p ) \displaystyle{G(c,p)=\left({c\over p}\right)G(1,p)}
  360. ( c p ) \left({c\over p}\right)
  361. G ( 1 , p q ) / G ( 1 , p ) G ( 1 , q ) = ( p q ) ( q p ) . \displaystyle{G(1,pq)/G(1,p)G(1,q)=\left({p\over q}\right)\left({q\over p}% \right)}.
  362. ( p q ) ( q p ) = ( - 1 ) ( p - 1 ) ( q - 1 ) 4 . \displaystyle{\left({p\over q}\right)\left({q\over p}\right)=(-1)^{\frac{(p-1)% (q-1)}{4}}.}
  363. L 2 ( 𝐑 n ) = L 2 ( 𝐑 ) n . \displaystyle{L^{2}({\mathbf{R}}^{n})=L^{2}({\mathbf{R}})^{\otimes n}.}
  364. 𝒮 \mathcal{S}
  365. 𝒮 \mathcal{S}
  366. U ( s ) f ( x ) = f ( x - s ) , V ( t ) f ( t x ) = e i x t f ( x ) . \displaystyle{U(s)f(x)=f(x-s),\qquad V(t)f(tx)=e^{ix\cdot t}f(x).}
  367. U ( s ) V ( t ) = e - i s t V ( t ) U ( s ) . \displaystyle{U(s)V(t)=e^{-is\cdot t}V(t)U(s).}
  368. 𝒮 \mathcal{S}
  369. f ^ ( t ) = 1 ( 2 π ) n / 2 𝐑 n f ( x ) e - i x t d x . \displaystyle{\widehat{f}(t)={1\over(2\pi)^{n/2}}\int_{{\mathbf{R}}^{n}}f(x)e^% {-ix\cdot t}\,dx.}
  370. f ( x ) = 1 ( 2 π ) n / 2 𝐑 n f ^ ( t ) e i x t d t \displaystyle{f(x)={1\over(2\pi)^{n/2}}\int_{{\mathbf{R}}^{n}}\widehat{f}(t)e^% {ix\cdot t}\,dt}
  371. 𝒮 \mathcal{S}
  372. W ( x , y ) = e i x y / 2 U ( x ) V ( y ) . \displaystyle{W(x,y)=e^{ix\cdot y/2}U(x)V(y).}
  373. W ( x 1 , y 1 ) W ( x 2 , y 2 ) = e i ( x 1 y 2 - y 1 x 2 ) W ( x 1 + x 2 , y 1 + y 2 ) , \displaystyle{W(x_{1},y_{1})W(x_{2},y_{2})=e^{i(x_{1}\cdot y_{2}-y_{1}\cdot x_% {2})}W(x_{1}+x_{2},y_{1}+y_{2}),}
  374. ω ( z 1 , z 2 ) = e i B ( z 1 , z 2 ) , \displaystyle{\omega(z_{1},z_{2})=e^{iB(z_{1},z_{2})},}
  375. z = x + i y = ( x , y ) , \displaystyle{z=x+iy=(x,y),}
  376. B ( z 1 , z 2 ) = x 1 y 2 - y 1 x 2 = z 1 z 2 ¯ . \displaystyle{B(z_{1},z_{2})=x_{1}\cdot y_{2}-y_{1}\cdot x_{2}=\Im\,z_{1}\cdot% \overline{z_{2}}.}
  377. π ( g ) W ( z ) π ( g ) * = W ( g ( z ) ) . \displaystyle{\pi(g)W(z)\pi(g)^{*}=W(g(z)).}
  378. g J g t = J , \displaystyle{gJg^{t}=J,}
  379. J = ( 0 - I I 0 ) . \displaystyle{J=\begin{pmatrix}0&-I\\ I&0\end{pmatrix}.}
  380. g 1 = ( A 0 0 ( A t ) - 1 ) , g 2 = ( I 0 B I ) , g 3 = ( 0 I - I 0 ) , \displaystyle{g_{1}=\begin{pmatrix}A&0\\ 0&(A^{t})^{-1}\end{pmatrix},\,\,g_{2}=\begin{pmatrix}I&0\\ B&I\end{pmatrix},\,\,g_{3}=\begin{pmatrix}0&I\\ -I&0\end{pmatrix},}
  381. π ( g 1 ) f ( x ) = ± det ( A ) - 1 / 2 f ( A - 1 x ) , π ( g 2 ) f ( x ) = ± e - i x t B x f ( x ) , π ( g 3 ) f ( x ) = ± e i n π / 8 f ^ ( x ) \displaystyle{\pi(g_{1})f(x)=\pm\det(A)^{-1/2}f(A^{-1}x),\,\,\pi(g_{2})f(x)=% \pm e^{-ix^{t}Bx}f(x),\,\,\pi(g_{3})f(x)=\pm e^{in\pi/8}\widehat{f}(x)}
  382. g Z = ( A Z + B ) ( C Z + D ) - 1 \displaystyle{gZ=(AZ+B)(CZ+D)^{-1}}
  383. g = ( A B C D ) . \displaystyle{g=\begin{pmatrix}A&B\\ C&D\end{pmatrix}.}
  384. m ( g , z ) = det ( C Z + D ) \displaystyle{m(g,z)=\det(CZ+D)}
  385. m ( g h , Z ) = m ( g , h Z ) m ( h , Z ) . \displaystyle{m(gh,Z)=m(g,hZ)m(h,Z).}
  386. M p ( 2 , 𝐑 ) = { ( g , G ) : G ( Z ) 2 = m ( g , Z ) } \displaystyle{Mp(2,\mathbf{R})=\{(g,G):\,G(Z)^{2}=m(g,Z)\}}
  387. Z > 0 \Im Z>0
  388. f z ( x ) = e i x t Z x / 2 \displaystyle{f_{z}(x)=e^{ix^{t}Zx/2}}
  389. f i I ( x ) = e - x x / 2 . \displaystyle{f_{iI}(x)=e^{-x\cdot x/2}.}
  390. π ( ( g t ) - 1 ) f Z ( x ) = m ( g , Z ) - 1 / 2 f g Z ( x ) \displaystyle{\pi((g^{t})^{-1})f_{Z}(x)=m(g,Z)^{-1/2}f_{gZ}(x)}
  391. n \mathcal{F}_{n}
  392. 1 π n 𝐂 n | f ( z ) | 2 e - | z | 2 d x d y \displaystyle{{1\over\pi^{n}}\int_{{\mathbf{C}}^{n}}|f(z)|^{2}e^{-|z|^{2}}\,dx% \cdot dy}
  393. ( f 1 , f 2 ) = 1 π n 𝐂 n f 1 ( z ) f 2 ( z ) ¯ e - | z | 2 d x d y . \displaystyle{(f_{1},f_{2})={1\over\pi^{n}}\int_{{\mathbf{C}}^{n}}f_{1}(z)% \overline{f_{2}(z)}e^{-|z|^{2}}\,dx\cdot dy.}
  394. e α ( z ) = z α α ! \displaystyle{e_{\alpha}(z)={z^{\alpha}\over\sqrt{\alpha!}}}
  395. n {\mathcal{F}}_{n}
  396. W n ( z ) f ( w ) = e - | z | 2 e w z ¯ f ( w - z ) . \displaystyle{W_{{\mathcal{F}}_{n}}(z)f(w)=e^{-|z|^{2}}e^{w\overline{z}}f(w-z).}
  397. 𝒰 \mathcal{U}
  398. n {\mathcal{F}}_{n}
  399. 𝒰 f ( z ) = 1 ( 2 π ) n / 2 B ( z , t ) f ( t ) d t , \displaystyle{\mathcal{U}f(z)={1\over(2\pi)^{n/2}}\int B(z,t)f(t)\,dt,}
  400. B ( z , t ) = exp [ - z z - t t / 2 + z t ] . \displaystyle{B(z,t)=\exp\,[-z\cdot z-t\cdot t/2+z\cdot t].}
  401. 𝒰 * {\mathcal{U}}^{*}
  402. 𝒰 * F ( t ) = 1 π n 𝐂 n B ( z ¯ , t ) F ( z ) d x d y . \displaystyle{\mathcal{U}^{*}F(t)={1\over\pi^{n}}\int_{{\mathbf{C}}^{n}}B(% \overline{z},t)F(z)\,dx\cdot dy.}
  403. Δ n = i = 1 n - 2 x i 2 + x i 2 . \displaystyle{\Delta_{n}=\sum_{i=1}^{n}-{\partial^{2}\over\partial x_{i}^{2}}+% x_{i}^{2}.}
  404. n \mathcal{F}_{n}
  405. C = 1 2 ( I i I I - i I ) \displaystyle{C={1\over\sqrt{2}}\begin{pmatrix}I&iI\\ I&-iI\end{pmatrix}}
  406. g = ( A B B ¯ A ¯ ) \displaystyle{g=\begin{pmatrix}A&B\\ \overline{B}&\overline{A}\end{pmatrix}}
  407. A A * - B B * = I , A B t = B A t ; \displaystyle{AA^{*}-BB^{*}=I,\,\,\,AB^{t}=BA^{t};}
  408. g K g * = K , \displaystyle{gKg^{*}=K,}
  409. K = ( I 0 0 - I ) . \displaystyle{K=\begin{pmatrix}I&0\\ 0&-I\end{pmatrix}.}
  410. W = ( Z - i I ) ( Z + i I ) - 1 . \displaystyle{W=(Z-iI)(Z+iI)^{-1}.}
  411. g W = ( A W + B ) ( B ¯ W + A ¯ ) - 1 \displaystyle{gW=(AW+B)(\overline{B}W+\overline{A})^{-1}}
  412. f W ( z ) = e z t W z / 2 . \displaystyle{f_{W}(z)=e^{z^{t}Wz/2}.}
  413. ( f W 1 , f W 2 ) = det ( 1 - W 1 W 2 ¯ ) - 1 / 2 . \displaystyle{(f_{W_{1}},f_{W_{2}})=\det(1-W_{1}\overline{W_{2}})^{-1/2}.}
  414. π ( g ) f W = det ( A ¯ + B ¯ W ) - 1 / 2 f g W . \displaystyle{\pi(g)f_{W}=\det(\overline{A}+\overline{B}W)^{-1/2}f_{gW}.}
  415. n + \mathcal{F}_{n}^{+}
  416. n + 1 + = n + n - \displaystyle{\mathcal{F}_{n+1}^{+}=\mathcal{F}_{n}^{+}\oplus\mathcal{F}_{n}^{% -}}
  417. n \mathcal{F}_{n}
  418. | i , j ( S T v i , v j ) | i v i 2 , \displaystyle\left|\sum_{i,j}(STv_{i},v_{j})\right|\leq\|\sum_{i}v_{i}\|^{2},
  419. T ( f ) = P m ( f ) P \displaystyle{T(f)=Pm(f)P}
  420. T ( f ) T ( f - 1 ) - I , T ( f - 1 ) T ( f ) - I \displaystyle{T(f)T(f^{-1})-I,\qquad T(f^{-1})T(f)-I}
  421. ind T ( f ) = dim ker T ( f ) - dim ker T ( f ) * . \displaystyle{\mathrm{ind}\,T(f)=\dim\ker T(f)-\dim\ker T(f)^{*}.}
  422. ind T ( f ) = Tr ( I - T ( f - 1 ) T ( f ) ) n - Tr ( I - T ( f ) T ( f - 1 ) ) n . \displaystyle{\mathrm{ind}\,T(f)=\mathrm{Tr}\,(I-T(f^{-1})T(f))^{n}-\mathrm{Tr% }\,(I-T(f)T(f^{-1}))^{n}.}
  423. T j = T ( z j ) \displaystyle{T_{j}=T(z_{j})}
  424. ( P j + i Q j ) Δ - 1 / 2 . \displaystyle{(P_{j}+iQ_{j})\Delta^{-1/2}.}
  425. k 0 S k ( H ) . \displaystyle{\bigoplus_{k\geq 0}S^{k}(H).}
  426. ( x 1 x k , y 1 y k ) = k ! i = 1 k ( x i , y i ) . \displaystyle{(x_{1}\otimes\cdots\otimes x_{k},y_{1}\otimes\cdots\otimes y_{k}% )=k!\cdot\prod_{i=1}^{k}(x_{i},y_{i}).}
  427. e ξ = k 0 ( k ! ) - 1 ξ k . \displaystyle{e^{\xi}=\sum_{k\geq 0}(k!)^{-1}\xi^{\otimes k}.}
  428. ( e ξ , e η ) = e ( ξ , η ) . \displaystyle{(e^{\xi},e^{\eta})=e^{(\xi,\eta)}.}
  429. S ( H 1 H 2 ) = S ( H 1 ) S ( H 2 ) , e x 1 x 2 = e x 1 e x 2 . S(H_{1}\oplus H_{2})=S(H_{1})\otimes S(H_{2}),\qquad e^{x_{1}\oplus x_{2}}=e^{% x_{1}}\otimes e^{x_{2}}.
  430. ( v ξ , v η ) = e ( ξ , η ) \displaystyle{(v_{\xi},v_{\eta})=e^{(\xi,\eta)}}
  431. v ξ = U ( e ξ ) . \displaystyle{v_{\xi}=U(e^{\xi}).}
  432. W ( x ) e y = e - x 2 / 2 e - ( x , y ) e x + y . \displaystyle{W(x)e^{y}=e^{-\|x\|^{2}/2}e^{-(x,y)}e^{x+y}.}
  433. W ( x ) W ( y ) = e - i 2 ( x , y ) W ( x + y ) . \displaystyle{W(x)W(y)=e^{-{i\over 2}\Im(x,y)}W(x+y).}
  434. W ( x ) W ( y ) = e - i ( x , y ) W ( y ) W ( x ) . \displaystyle{W(x)W(y)=e^{-i\Im(x,y)}W(y)W(x).}
  435. B ( x , y ) = - ( x , y ) \displaystyle{B(x,y)=-\Im(x,y)}
  436. ( x , y ) 𝐑 = ( x , y ) . \displaystyle{(x,y)_{\mathbf{R}}=\Re(x,y).}
  437. J ( x ) = i x \displaystyle{J(x)=ix}
  438. B ( x , y ) = - ( J x , y ) 𝐑 . \displaystyle{B(x,y)=-(Jx,y)_{\mathbf{R}}.}
  439. T J T t = J = T t J T . \displaystyle{TJT^{t}=J=T^{t}JT.}
  440. π ( T ) W ( x ) π ( T ) * = W ( T x ) . \displaystyle{\pi(T)W(x)\pi(T)^{*}=W(Tx).}
  441. g = ( A B B ¯ A ¯ ) \displaystyle{g=\begin{pmatrix}A&B\\ \overline{B}&\overline{A}\end{pmatrix}}
  442. g Z = ( A W + B ) ( B ¯ W + A ¯ ) - 1 . \displaystyle{gZ=(AW+B)(\overline{B}W+\overline{A})^{-1}.}
  443. ( f W 1 , f W 2 ) = det ( I - W 2 * W 1 ) - 1 / 2 . \displaystyle{(f_{W_{1}},f_{W_{2}})=\det(I-W_{2}^{*}W_{1})^{-1/2}.}
  444. e W = det ( I - W * W ) 1 / 4 f W \displaystyle{e_{W}=\det(I-W^{*}W)^{1/4}f_{W}}
  445. π ( g ) e W = μ ( det ( I + A ¯ - 1 B ¯ W ) - 1 / 2 ) e g W , \displaystyle{\pi(g)e_{W}=\mu(\det(I+\overline{A}^{-1}\overline{B}W)^{-1/2})e_% {gW},}
  446. ω ( g 1 , g 2 ) = μ [ det ( A 3 ( A 1 A 2 ) - 1 ) - 1 / 2 ] . \displaystyle{\omega(g_{1},g_{2})=\mu[\det(A_{3}(A_{1}A_{2})^{-1})^{-1/2}].}
  447. f ( θ ) = n 0 a n e i n θ \displaystyle{f(\theta)=\sum_{n\neq 0}a_{n}e^{in\theta}}
  448. n 0 | n | | a n | 2 < . \displaystyle{\sum_{n\neq 0}|n||a_{n}|^{2}<\infty.}
  449. ( a n e i n θ , b m e i m θ ) = n 0 | n | a n b n ¯ . \left(\sum a_{n}e^{in\theta},\sum b_{m}e^{im\theta}\right)=\sum_{n\neq 0}|n|a_% {n}\overline{b_{n}}.
  450. J sin ( n θ ) = cos ( n θ ) , J cos ( n θ ) = - sin ( n θ ) J\sin(n\theta)=\cos(n\theta),\qquad J\cos(n\theta)=-\sin(n\theta)
  451. J n 0 a n e i n θ = n 0 i sign ( n ) a n e i n θ , J\sum_{n\neq 0}a_{n}e^{in\theta}=\sum_{n\neq 0}i\,\mathrm{sign}(n)a_{n}e^{in% \theta},
  452. B ( f , g ) = S 1 f d g . \displaystyle{B(f,g)=\int_{S^{1}}fdg.}
  453. T φ f ( θ ) = f ( φ - 1 ( θ ) ) - 1 2 π 0 2 π f ( φ - 1 ( θ ) ) d θ , \displaystyle{T_{\varphi}f(\theta)=f(\varphi^{-1}(\theta))-{1\over 2\pi}\int_{% 0}^{2\pi}f(\varphi^{-1}(\theta))\,d\theta,}
  454. L n = - π ( i e i n θ d d θ ) \displaystyle{L_{n}=-\pi\left(ie^{in\theta}{d\over d\theta}\right)}
  455. [ L m , L n ] = ( m - n ) L m + n + m 3 - m 12 δ m + n , 0 . \displaystyle{[L_{m},L_{n}]=(m-n)L_{m+n}+{m^{3}-m\over 12}\delta_{m+n,0}.}

Oseen's_approximation.html

  1. a a
  2. μ \mu\,
  3. U U
  4. u 1 = u + u 1 , u 2 = u 2 , u 3 = u 3 . u_{1}=u+u_{1}^{^{\prime}},\qquad u_{2}=u_{2}^{\prime},\qquad u_{3}=u_{3}^{% \prime}.
  5. u u 1 x 1 = - 1 ρ p x 1 + ν 2 u i ( i = 1 , 2 , 3 ) . u{\partial u_{1}^{\prime}\over\partial x_{1}}=-{1\over\rho}{\partial p\over% \partial x_{1}}+\nu\nabla^{2}u_{i}^{\prime}\qquad\left({i=1,2,3}\right).
  6. x x
  7. ( 2 - U 2 v x ) χ = G ( x ) = 0 (\nabla^{2}-{U\over 2v}{\partial\over\partial x})\chi=G(x)=0
  8. G ( x ) G(x)
  9. x x
  10. U v u x = 2 u {U\over v}{\partial u^{\prime}\over\partial x}=\nabla^{2}u^{\prime}
  11. χ \chi
  12. e - U x 2 v χ = C e - U N R 2 v N R e^{-Ux\over 2v}\chi={{Ce^{-UN_{R}\over 2v}}\over N_{R}}
  13. x x
  14. φ = A 0 N R + A 1 x 1 N R + A 2 2 x 2 1 N R + \varphi={A_{0}\over N_{R}}+A_{1}{\partial\over\partial x}{1\over N_{R}}+A_{2}{% \partial^{2}\over\partial x^{2}}{1\over N_{R}}+\ldots
  15. C = - 3 2 U a , A 0 = - 3 2 v a , A 1 = 1 4 U a 3 , etc. C=-{3\over 2}Ua,\ A_{0}=-{3\over 2}va,\ A_{1}={1\over 4}Ua^{3}\ \,\text{, etc.}
  16. C d = 24 N R ( 1 + 3 16 N R ) C_{d}={24\over N_{R}}(1+{3\over 16}N_{R})
  17. F = 6 π μ a u ( 1 + 3 8 N R ) , F=6\pi\,\mu\,au\left(1+{3\over 8}N_{R}\right),
  18. N R N_{R}
  19. F F
  20. u u
  21. a a
  22. μ \mu\,
  23. 1 + ( 3 8 ) N R . 1+\left({3\over 8}\right)N_{R}.
  24. u = 0 \triangledown u^{\prime}~{}=0
  25. u u = - p + ν 2 u , u\triangledown u^{\prime}~{}=-\triangledown p+\nu\triangledown^{2}u^{\prime},
  26. u y = u cos θ ( 1 + a 3 2 r 3 - 3 a 2 r ) u_{y}=u\cos\theta\left({1+{a^{3}\over 2r^{3}}-{3a\over 2r}}\right)
  27. u z = - u sin θ ( 1 - a 3 4 r 3 - 3 a 4 r ) . u_{z}=-u\sin\theta\left({1-{a^{3}\over 4r^{3}}-{3a\over 4r}}\right).
  28. r a {r\over a}
  29. 2 u = O ( a 3 r 3 ) . \triangledown^{2}u^{\prime}=O\left({a^{3}\over r^{3}}\right).
  30. u u z 1 O ( a 2 r 2 ) . u{\partial u^{\prime}\over\partial z_{1}}\sim O\left({a^{2}\over r^{2}}\right).
  31. u u z 1 ν 2 u = O ( r a ) . u{{\partial u^{\prime}\over\partial z_{1}}\over{\nu\triangledown^{2}u^{\prime}% }}=O\left({r\over a}\right).
  32. r a {r\over a}
  33. 2 ζ = 0. \triangledown^{2}\zeta\,=0.
  34. ζ \zeta\,
  35. r a . {r\over a}.
  36. F = 6 π μ a U ( 1 + 3 8 N R + 9 40 N R 2 ln N R + 𝒪 ( N R 2 ) ) . F=6\pi\,\mu\,aU\left(1+{3\over 8}N_{R}+{9\over 40}N_{R}^{2}\ln N_{R}+\mathcal{% O}(N_{R}^{2})\right).

P-adic_exponential_function.html

  1. exp ( z ) = n = 0 z n n ! . \exp(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}.
  2. exp p ( z ) = n = 0 z n n ! . \exp_{p}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}.
  3. | z | p < p - 1 / ( p - 1 ) . |z|_{p}<p^{-1/(p-1)}.
  4. log ( 1 + x ) = n = 1 ( - 1 ) n + 1 x n n , \log(1+x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{n}}{n},
  5. 𝐂 p × {\mathbf{C}}_{p}^{×}
  6. 𝐂 p × {\mathbf{C}}_{p}^{×}
  7. 𝐂 p × {\mathbf{C}}_{p}^{×}
  8. 𝐂 p × {\mathbf{C}}_{p}^{×}

P-adic_gamma_function.html

  1. Γ p ( s ) = ( - 1 ) s 0 < i < s , p i i \Gamma_{p}(s)=(-1)^{s}\prod_{0<i<s,\ p\nmid i}i

P-adic_quantum_mechanics.html

  1. 𝔸 = { x , x 2 , x 3 , x 5 } \mathbb{A}=\{x_{\infty},x_{2},x_{3},x_{5}...\}
  2. x x_{\infty}
  3. x p x_{p}
  4. p \mathbb{Q}_{p}
  5. x x_{\infty}
  6. x p x_{p}
  7. p \mathbb{Z}_{p}
  8. 𝔸 * = { x 𝔸 where x * and | x p | = 1 for all but finitely many primes. } \mathbb{A}^{*}=\{x\in\mathbb{A}\,\text{ where }x_{\infty}\in\mathbb{R}^{*}\,% \text{ and }|x_{p}|=1\,\text{ for all but finitely many primes.}\}
  9. | r | p | r | p = 1 |r|_{\infty}\prod_{p}|r|_{p}=1
  10. r r
  11. | r | p = p - e |r|_{p}=p^{-e}
  12. r = 12 = 2 2 × 3 r=12=2^{2}\times 3
  13. | 12 | 2 = 2 - 2 = 1 / 4 | 12 | 3 = 3 - 1 = 1 / 3 | 12 | 5 = 5 0 = 1 | 12 | 7 = 7 0 = 1 . \begin{aligned}\displaystyle|12|_{2}&\displaystyle=2^{-2}=1/4\\ \displaystyle|12|_{3}&\displaystyle=3^{-1}=1/3\\ \displaystyle|12|_{5}&\displaystyle=5^{0}=1\\ \displaystyle|12|_{7}&\displaystyle=7^{0}=1\\ \displaystyle\vdots&\displaystyle\ \ \ \ \ \ \ \ \vdots\end{aligned}\ .
  14. | 12 | × p : p r i m e | 12 | p = 12 1 / 4 1 / 3 1 = 1 |12|_{\infty}\times\prod_{p:prime}|12|_{p}=12\ \cdot\ 1/4\ \cdot 1/3\ \cdot\ 1% \ \cdots=1
  15. d x = d x d x 2 d x 3 d x 5 and d x * = d x * d x * 2 d x * 3 d x * 5 dx=dx_{\infty}\,dx_{2}\,dx_{3}\,dx_{5}\cdots\,\text{ and }dx^{*}={dx^{*}}_{% \infty}{dx^{*}}_{2}{dx^{*}}_{3}{dx^{*}}_{5}\cdots
  16. χ ( a x 2 + b x ) d x p p χ p ( a x p 2 + b x p ) d x p = 1 \int_{\mathbb{Q}_{\infty}}\chi_{\infty}(a{x_{\infty}}^{2}+bx_{\infty})dx_{% \infty}\prod_{p}\int_{\mathbb{Q}_{p}}\chi_{p}(a{x_{p}}^{2}+bx_{p})dx_{p}=1
  17. χ \,\chi
  18. χ ( x ) = χ ( x ) p χ p ( x p ) e - 2 π i x p e 2 π i { x p } p \chi(x)=\chi_{\infty}(x_{\infty})\prod_{p}\chi_{p}(x_{p})\rightarrow e^{-2\pi ix% _{\infty}}\prod_{p}e^{2\pi i\{x_{p}\}_{p}}
  19. { x p } p \,\{x_{p}\}_{p}
  20. x p \,x_{p}
  21. / n e 2 π i / n \mathbb{Z}/n\mathbb{Z}\rightarrow e^{2\pi i/n}
  22. K A ( x ′′ , t ′′ ; x , t ) = α x α , t α x α ′′ , t α ′′ χ α ( - 1 / h t α t α ′′ L ( q ˙ α , q α , t α ) d t α ) D q α \,K_{A}(x^{{}^{\prime\prime}},t^{{}^{\prime\prime}};x^{{}^{\prime}},t^{{}^{% \prime}})=\prod_{\alpha}\int_{x^{{}^{\prime}}_{\alpha},t^{{}^{\prime}}_{\alpha% }}^{x^{{}^{\prime\prime}}_{\alpha},t^{{}^{\prime\prime}}_{\alpha}}\chi_{\alpha% }(-1/h\int_{t^{{}^{\prime}}_{\alpha}}^{t^{{}^{\prime\prime}}_{\alpha}}L(\dot{q% }_{\alpha},q_{\alpha},t_{\alpha})dt_{\alpha})Dq_{\alpha}
  23. U ( t ) ψ α ( x ) = χ ( E α ( t ) ) ψ α ( x ) \,U(t)\psi_{\alpha}(x)=\chi(E_{\alpha}(t))\psi_{\alpha}(x)
  24. U U\,
  25. ψ α \,\psi_{\alpha}
  26. E α \,E_{\alpha}
  27. α \,\alpha
  28. χ \,\chi
  29. ψ 0 ( x ) = 2 1 / 4 e - π x 2 p Ω ( | x p | p ) \psi_{0}(x)=2^{1/4}e^{-\pi{x_{\infty}}^{2}}\prod_{p}\Omega(|x_{p}|_{p})
  30. Ω ( | x p | p ) \,\Omega(|x_{p}|_{p})
  31. | x p | p \,{|x_{p}|}_{p}
  32. Φ ( α ) = 2 Γ ( α / 2 ) π - α / 2 ζ ( α ) \Phi(\alpha)=\sqrt{2}\,\Gamma(\alpha/2)\pi^{-\alpha/2}\zeta(\alpha)
  33. Γ \,\Gamma
  34. ζ \,\zeta
  35. Φ ( α ) = Φ ( 1 - α ) \,\Phi(\alpha)={\Phi}^{\prime}(1-\alpha)
  36. Γ ( α / 2 ) π - α / 2 ζ ( α ) = Γ ( ( 1 - α ) / 2 ) π ( α - 1 ) / 2 ζ ( 1 - α ) \,\Gamma(\alpha/2)\pi^{-\alpha/2}\zeta(\alpha)=\Gamma((1-\alpha)/2)\pi^{(% \alpha-1)/2}\zeta(1-\alpha)
  37. Z ( T ) = n = 1 exp ( - E 0 log n k B T ) = n = 1 1 n s = ζ ( s ) Z(T)=\sum_{n=1}^{\infty}\exp\left(\frac{-E_{0}\log n}{k_{B}T}\right)=\sum_{n=1% }^{\infty}\frac{1}{n^{s}}=\zeta(s)
  38. A ( a , b ) p A p ( a , b ) = 1 A_{\infty}(a,b)\prod_{p}A_{p}(a,b)=1

P-nuclei.html

  1. γ \gamma
  2. ( p , γ ) (p,\gamma)
  3. ( γ , n ) (\gamma,n)
  4. ( γ , p ) (\gamma,p)
  5. ( p , γ ) (p,\gamma)
  6. γ \gamma
  7. ( γ , n ) (\gamma,n)
  8. ( γ , α ) (\gamma,\alpha)
  9. ( γ , p ) (\gamma,p)
  10. ν \nu
  11. ( p , γ ) (p,\gamma)
  12. ( p , γ ) (p,\gamma)
  13. 10 28 10^{28}
  14. ( n , p ) (n,p)
  15. ( n , p ) (n,p)
  16. ν ¯ e + p e + + n \bar{\nu}_{e}+p\rightarrow e^{+}+n
  17. A < 100 A<100
  18. 150 A 165 150\leq A\leq 165
  19. ν e + n e - + p \nu_{e}+n\rightarrow e^{-}+p
  20. ( α , n ) (\alpha,n)
  21. ( α , n ) (\alpha,n)
  22. ( α , n ) (\alpha,n)
  23. 10 5 - 10 6 10^{5}-10^{6}

Pachinko_allocation.html

  1. P ( 𝐃 | α ) = d P ( d | α ) P(\mathbf{D}|\alpha)=\prod_{d}P(d|\alpha)

Paczyński–Wiita_potential.html

  1. Φ P W ( r ) = - G M r - r g \Phi_{PW}(r)=-\frac{GM}{r-r_{g}}
  2. r r
  3. G G
  4. M M
  5. r g = 2 G M / c 2 r_{g}=2GM/c^{2}
  6. c c

Padding_argument.html

  1. EXP NEXP \mathrm{EXP}\subseteq\mathrm{NEXP}
  2. NEXP EXP \mathrm{NEXP}\subseteq\mathrm{EXP}
  3. 2 n c 2^{n^{c}}
  4. L = { x 1 2 | x | c x L } , L^{\prime}=\{x1^{2^{|x|^{c}}}\mid x\in L\},
  5. L L^{\prime}
  6. L L^{\prime}
  7. x x^{\prime}
  8. x = x 1 2 | x | c x^{\prime}=x1^{2^{|x|^{c}}}
  9. 2 | x | c 2^{|x|^{c}}
  10. x x^{\prime}
  11. L L^{\prime}
  12. L L^{\prime}
  13. x x
  14. D M ( x 1 2 | x | c ) DM(x1^{2^{|x|^{c}}})
  15. x x
  16. 1 d 1^{d}

Padding_oracle_attack.html

  1. C 1 , C 2 , C 3 C_{1},C_{2},C_{3}
  2. P 2 P_{2}
  3. C 3 C_{3}
  4. n n
  5. P i = D K ( C i ) C i - 1 , C 0 = I V . P_{i}=D_{K}(C_{i})\oplus C_{i-1},C_{0}=IV.
  6. C 1 C_{1}
  7. ( I V , C 1 , C 2 ) (IV,C_{1},C_{2})
  8. P 1 P_{1}
  9. P 2 P_{2}
  10. P 2 P_{2}
  11. b - 1 b_{-1}
  12. C 1 C_{1}
  13. b - 1 = b - 1 z - 1 0x01 b_{-1}=b_{-1}\oplus z_{-1}\oplus\,\text{0x01}
  14. z - 1 z_{-1}
  15. P 2 P_{2}
  16. z - 1 z_{-1}
  17. P 2 P_{2}
  18. P 2 P_{2}
  19. z - 1 z_{-1}
  20. z - 1 z_{-1}
  21. P 2 P_{2}
  22. P 2 P_{2}
  23. C 1 C_{1}
  24. b - 1 = b - 1 z - 1 0x02 b_{-1}=b_{-1}\oplus z_{-1}\oplus\,\text{0x02}
  25. b - 2 = b - 2 z - 2 0x02 b_{-2}=b_{-2}\oplus z_{-2}\oplus\,\text{0x02}
  26. z - 2 z_{-2}
  27. z - 3 z_{-3}
  28. P 2 P_{2}

Pairwise_sorting_network.html

  1. n ( log n ) ( log n - 1 ) / 4 + n - 1 n(\log n)(\log n-1)/4+n-1
  2. ( log n ) ( log n + 1 ) / 2 (\log n)(\log n+1)/2

Palm_calculus.html

  1. P 0 ( ) P^{0}(\cdot)
  2. E 0 [ ] E^{0}[\cdot]
  3. L = λ W L=\lambda W
  4. λ \lambda

Palm–Khintchine_theorem.html

  1. { N i ( t ) , t 0 } , i = 1 , 2 , , m \{N_{i}(t),t\geq 0\},i=1,2,...,m
  2. { N ( t ) , t > 0 } \{N(t),t>0\}
  3. X 2 j m X_{2jm}
  4. j j
  5. N j m ( t ) N_{jm}(t)
  6. j j
  7. F j m ( t ) = P ( X 2 j m t ) F_{jm}(t)=P(X_{2jm}\leq t)
  8. λ j m = 1 / ( E ( ( X 2 j m ) ) ) \lambda_{jm}=1/(E((X_{2jm)}))
  9. m m
  10. λ 1 m + λ 2 m + + λ m m = λ < \lambda_{1m}+\lambda_{2m}+...+\lambda_{mm}=\lambda<\infty
  11. ϵ > 0 \epsilon>0
  12. t > 0 t>0
  13. m m
  14. F j m ( t ) < ϵ F_{jm}(t)<\epsilon
  15. j j
  16. N 0 m ( t ) = N 1 m ( t ) + N 2 m ( t ) + + N m m ( t ) N_{0m}(t)=N_{1m}(t)+N_{2m}(t)+...+N_{mm}(t)
  17. m m
  18. \infty

Parabolic_geometry_(differential_geometry).html

  1. Q 2 ( p + q ) - 1 = S U ( p , q ) / P p + q Q^{2(p+q)-1}=SU(p,q)/P\subseteq\mathbb{C}^{p+q}
  2. P P
  3. S P ( n ) / P SP(n)/P
  4. P P
  5. 2 n \mathbb{R}^{2n}

Parallel-plate_flow_chamber.html

  1. τ \tau
  2. ρ ( V x t + V x V x x + V y V x y + V z V x z = P x + μ ( 2 V x x 2 + 2 V x y 2 + 2 V x z 2 ) + ρ g x \rho(\frac{\partial V_{x}}{\partial t}+V_{x}\frac{\partial V_{x}}{\partial x}+% V_{y}\frac{\partial V_{x}}{\partial y}+V_{z}\frac{\partial V_{x}}{\partial z}=% \frac{\partial P}{\partial x}+\mu(\frac{\partial^{2}V_{x}}{\partial x^{2}}+% \frac{\partial^{2}V_{x}}{\partial y^{2}}+\frac{\partial^{2}V_{x}}{\partial z^{% 2}})+\rho g_{x}
  3. - P x = μ ( 2 V x y 2 ) -\frac{\partial P}{\partial x}=\mu(\frac{\partial^{2}V_{x}}{\partial y^{2}})
  4. - P x y = μ ( V x y ) + C -\frac{\partial P}{\partial x}y=\mu(\frac{\partial V_{x}}{\partial y})+C
  5. V x = 1 2 μ P x ( H 2 - y 2 ) V_{x}=\frac{1}{2\mu}\frac{\partial P}{\partial x}(H^{2}-y^{2})
  6. Q = A V x d A = W - H H 1 2 μ P x ( H 2 - y 2 ) d y Q=\int\int_{A}V_{x}dA=W\int_{-H}^{H}\frac{1}{2\mu}\frac{\partial P}{\partial x% }(H^{2}-y^{2})dy
  7. Q = 2 W H 3 3 μ P x Q=\frac{2WH^{3}}{3\mu}\frac{\partial P}{\partial x}
  8. τ = - μ V x y = P x H = 3 Q μ 2 W H 2 \tau=-\mu\frac{\partial V_{x}}{\partial y}=\frac{\partial P}{\partial x}H=% \frac{3Q\mu}{2WH^{2}}

Parallel_axis_theorem.html

  1. m m
  2. z z
  3. z z′
  4. d d
  5. I I
  6. I = I cm + m d 2 . I=I_{\mathrm{cm}}+md^{2}.
  7. d d
  8. z z
  9. z z′
  10. I cm = ( x 2 + y 2 ) d m . I_{\mathrm{cm}}=\int(x^{2}+y^{2})\,dm.
  11. z z′
  12. r r
  13. I = [ ( x + r ) 2 + y 2 ] d m I=\int\left[(x+r)^{2}+y^{2}\right]\,dm
  14. I = ( x 2 + y 2 ) d m + r 2 d m + 2 r x d m . I=\int(x^{2}+y^{2})\,dm+r^{2}\int dm+2r\int x\,dm.
  15. I = I cm + m r 2 . I=I_{\mathrm{cm}}+mr^{2}.
  16. J i j = I i j + m ( | 𝐑 | 2 δ i j - R i R j ) , J_{ij}=I_{ij}+m\left(|\mathbf{R}|^{2}\delta_{ij}-R_{i}R_{j}\right),
  17. 𝐑 = R 1 𝐱 ^ + R 2 𝐲 ^ + R 3 𝐳 ^ \mathbf{R}=R_{1}\mathbf{\hat{x}}+R_{2}\mathbf{\hat{y}}+R_{3}\mathbf{\hat{z}}\!
  18. i = j i=j
  19. 𝐉 = 𝐈 + m [ ( 𝐑 𝐑 ) 𝐄 3 - 𝐑 𝐑 ] , \mathbf{J}=\mathbf{I}+m\left[\left(\mathbf{R}\cdot\mathbf{R}\right)\mathbf{E}_% {3}-\mathbf{R}\otimes\mathbf{R}\right],
  20. \otimes
  21. I z = I x + A r 2 , I_{z}=I_{x}+Ar^{2},
  22. A A
  23. r r
  24. z z
  25. V ρ ( 𝐫 ) ( 𝐫 - 𝐑 ) d V = 0 , \int_{V}\rho(\mathbf{r})(\mathbf{r}-\mathbf{R})\,dV=0,
  26. I S = V ρ ( 𝐫 ) ( 𝐫 - 𝐒 ) ( 𝐫 - 𝐒 ) d V , I_{S}=\int_{V}\rho(\mathbf{r})(\mathbf{r}-\mathbf{S})\cdot(\mathbf{r}-\mathbf{% S})\,dV,
  27. I S \displaystyle I_{S}
  28. I S = I R + M d 2 , I_{S}=I_{R}+Md^{2},\,
  29. [ I S ] = - i = 1 n m i [ r i - S ] [ r i - S ] , [I_{S}]=-\sum_{i=1}^{n}m_{i}[r_{i}-S][r_{i}-S],
  30. [ r i - S ] 𝐲 = ( 𝐫 i - 𝐒 ) × 𝐲 , [r_{i}-S]\mathbf{y}=(\mathbf{r}_{i}-\mathbf{S})\times\mathbf{y},
  31. 𝐑 = ( 𝐑 - 𝐒 ) + 𝐒 = 𝐝 + 𝐒 , \mathbf{R}=(\mathbf{R}-\mathbf{S})+\mathbf{S}=\mathbf{d}+\mathbf{S},
  32. [ I S ] = - i = 1 n m i [ r i - R + d ] [ r i - R + d ] . [I_{S}]=-\sum_{i=1}^{n}m_{i}[r_{i}-R+d][r_{i}-R+d].
  33. [ I S ] = ( - i = 1 n m i [ r i - R ] [ r i - R ] ) + ( - i = 1 n m i [ r i - R ] ) [ d ] + [ d ] ( - i = 1 n m i [ r i - R ] ) + ( - i = 1 n m i ) [ d ] [ d ] . [I_{S}]=\left(-\sum_{i=1}^{n}m_{i}[r_{i}-R][r_{i}-R]\right)+\left(-\sum_{i=1}^% {n}m_{i}[r_{i}-R]\right)[d]+[d]\left(-\sum_{i=1}^{n}m_{i}[r_{i}-R]\right)+% \left(-\sum_{i=1}^{n}m_{i}\right)[d][d].
  34. i = 1 n m i ( 𝐫 i - 𝐑 ) = 0. \sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R})=0.
  35. [ I S ] = [ I R ] - M [ d ] 2 , [I_{S}]=[I_{R}]-M[d]^{2},
  36. - [ R ] [ R ] = - [ 0 - z y z 0 - x - y x 0 ] 2 = [ y 2 + z 2 - x y - x z - y x x 2 + z 2 - y z - z x - z y x 2 + y 2 ] . -[R][R]=-\begin{bmatrix}0&-z&y\\ z&0&-x\\ -y&x&0\end{bmatrix}^{2}=\begin{bmatrix}y^{2}+z^{2}&-xy&-xz\\ -yx&x^{2}+z^{2}&-yz\\ -zx&-zy&x^{2}+y^{2}\end{bmatrix}.
  37. - [ R ] 2 = | 𝐑 | 2 [ E 3 ] - [ 𝐑𝐑 T ] = [ x 2 + y 2 + z 2 0 0 0 x 2 + y 2 + z 2 0 0 0 x 2 + y 2 + z 2 ] - [ x 2 x y x z y x y 2 y z z x z y z 2 ] , -[R]^{2}=|\mathbf{R}|^{2}[E_{3}]-[\mathbf{R}\mathbf{R}^{T}]=\begin{bmatrix}x^{% 2}+y^{2}+z^{2}&0&0\\ 0&x^{2}+y^{2}+z^{2}&0\\ 0&0&x^{2}+y^{2}+z^{2}\end{bmatrix}-\begin{bmatrix}x^{2}&xy&xz\\ yx&y^{2}&yz\\ zx&zy&z^{2}\end{bmatrix},
  38. | 𝐑 | 2 = 𝐑 𝐑 = tr [ 𝐑𝐑 T ] , |\mathbf{R}|^{2}=\mathbf{R}\cdot\mathbf{R}=\operatorname{tr}[\mathbf{R}\mathbf% {R}^{T}],

Parallelization_(mathematics).html

  1. M M\,
  2. M M\,
  3. M M\,
  4. { X 1 , , X n } \{X_{1},\dots,X_{n}\}
  5. M M\,
  6. p M p\in M\,
  7. { X 1 ( p ) , , X n ( p ) } \{X_{1}(p),\dots,X_{n}(p)\}
  8. T p M T_{p}M\,
  9. T p M T_{p}M\,
  10. p p\,
  11. T M TM\,
  12. M M\,
  13. ϕ : T M M × n \phi\colon TM\longrightarrow M\times{\mathbb{R}^{n}}\,
  14. ϕ \phi\,
  15. τ M : T M M \tau_{M}\colon TM\longrightarrow M\,
  16. p M p\in M\,
  17. T p M T_{p}M\,
  18. ϕ p : T p M n \phi_{p}\colon T_{p}M\rightarrow{\mathbb{R}^{n}}\,
  19. M M\,
  20. τ M : T M M \tau_{M}\colon TM\longrightarrow M\,
  21. M M\,
  22. n {\mathbb{R}^{n}}\,
  23. n {\mathbb{R}^{n}}\,
  24. T M TM\,
  25. M × n M\times{\mathbb{R}^{n}}\,
  26. M M\,

Parametric_process_(optics).html

  1. 𝐏 = ε 0 χ 𝐄 = ( n r + i n i ) 2 𝐄 , {\mathbf{P}}=\varepsilon_{0}\chi{\mathbf{E}}=(n_{r}+in_{i})^{2}{\mathbf{E}},
  2. χ ( 3 ) \chi^{(3)}
  3. χ ( 3 ) \chi^{(3)}

Paraproduct.html

  1. Λ \Lambda
  2. ( f , g ) (f,g)
  3. f g = Λ ( f , g ) + Λ ( g , f ) . fg=\Lambda(f,g)+\Lambda(g,f).
  4. f f
  5. h h
  6. h ( 0 ) = 0 h(0)=0
  7. h ( f ) = Λ ( f , h ( f ) ) h(f)=\Lambda(f,h^{\prime}(f))

Parity_progression_ratios.html

  1. a x a_{x}
  2. a x = (women with at least x + 1 children ever borne) / (women with at least x children ever born) a_{x}=\,\text{(women with at least }x+1\text{ children ever borne) }/\,\text{ % (women with at least }x\,\text{ children ever born)}
  3. a 2 a_{2}
  4. a 0 a_{0}

Partial_cube.html

  1. Θ \Theta
  2. e = { x , y } e=\{x,y\}
  3. f = { u , v } f=\{u,v\}
  4. Θ \Theta
  5. e Θ f e\mathrel{\Theta}f
  6. d ( x , u ) + d ( y , v ) d ( x , v ) + d ( y , u ) d(x,u)+d(y,v)\not=d(x,v)+d(y,u)
  7. Θ \Theta
  8. O ( n 2 ) O(n^{2})
  9. n n
  10. O ( n m ) O(nm)
  11. O ( n 2 ) O(n^{2})
  12. n n
  13. n - 1 n-1

Partial_group_algebra.html

  1. par ( 4 ) \mathbb{C}_{\,\text{par}}\left(\mathbb{Z}_{4}\right)
  2. M 2 ( ) M 3 ( ) \mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}% \oplus\mathbb{C}\oplus\mathbb{C}\oplus M_{2}\left(\mathbb{C}\right)\oplus M_{3% }\left(\mathbb{C}\right)

Partial_permutation.html

  1. n n
  2. i = 0 n i ! ( n i ) 2 \sum_{i=0}^{n}i!{\left({{n}\atop{i}}\right)}^{2}
  3. P ( n ) = 2 n P ( n - 1 ) - ( n - 1 ) 2 P ( n - 2 ) . P(n)=2nP(n-1)-(n-1)^{2}P(n-2).
  4. P ( n - 1 ) P(n-1)
  5. P ( n - 1 ) P(n-1)
  6. ( n - 1 ) P ( n - 1 ) (n-1)P(n-1)
  7. ( n - 1 ) P ( n - 1 ) (n-1)P(n-1)
  8. - ( n - 1 ) 2 P ( n - 2 ) -(n-1)^{2}P(n-2)

Partial_wave_analysis.html

  1. k k
  2. ψ in ( 𝐫 ) = e i k z = = 0 ( 2 + 1 ) i j ( k r ) P ( cos θ ) \psi_{\,\text{in}}(\mathbf{r})=e^{ikz}=\sum_{\ell=0}^{\infty}(2\ell+1)i^{\ell}% j_{\ell}(kr)P_{\ell}(\cos\theta)
  3. z z
  4. j ( k r ) = 1 2 ( h ( 1 ) ( k r ) + h ( 2 ) ( k r ) ) j_{\ell}(kr)=\frac{1}{2}\left(h_{\ell}^{(1)}(kr)+h_{\ell}^{(2)}(kr)\right)
  5. r r
  6. u ( r ) r r i k 2 π ( h ( 1 ) ( k r ) + S h ( 2 ) ( k r ) ) \frac{u_{\ell}(r)}{r}\stackrel{r\to\infty}{\longrightarrow}\frac{i^{\ell}k}{% \sqrt{2\pi}}\left(h_{\ell}^{(1)}(kr)+S_{\ell}h_{\ell}^{(2)}(kr)\right)
  7. S = e 2 i δ S_{\ell}=e^{2i\delta_{\ell}}
  8. ψ ( 𝐫 ) r = 0 ( 2 + 1 ) i h ( 1 ) ( k r ) + S h ( 2 ) ( k r ) 2 P ( cos θ ) \psi(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}\sum_{\ell=0}^{\infty}(2% \ell+1)i^{\ell}\frac{h_{\ell}^{(1)}(kr)+S_{\ell}h_{\ell}^{(2)}(kr)}{2}P_{\ell}% (\cos\theta)
  9. ψ out ( 𝐫 ) r = 0 ( 2 + 1 ) i S - 1 2 h ( 2 ) ( k r ) P ( cos θ ) \psi_{\,\text{out}}(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}\sum_{% \ell=0}^{\infty}(2\ell+1)i^{\ell}\frac{S_{\ell}-1}{2}h_{\ell}^{(2)}(kr)P_{\ell% }(\cos\theta)
  10. ψ out ( 𝐫 ) r e i k r r = 0 ( 2 + 1 ) S - 1 2 i k P ( cos θ ) \psi_{\,\text{out}}(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}\frac{e^{% ikr}}{r}\sum_{\ell=0}^{\infty}(2\ell+1)\frac{S_{\ell}-1}{2ik}P_{\ell}(\cos\theta)
  11. f ( θ , φ ) f(θ,φ)
  12. ψ out ( 𝐫 ) r e i k r r f ( θ , ϕ ) \psi_{\,\text{out}}(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}\frac{e^{% ikr}}{r}f(\theta,\phi)
  13. f ( θ , φ ) = = 0 ( 2 + 1 ) S - 1 2 i k P ( cos θ ) = = 0 ( 2 + 1 ) e i δ sin δ k P ( cos θ ) f(\theta,\varphi)=\sum_{\ell=0}^{\infty}(2\ell+1)\frac{S_{\ell}-1}{2ik}P_{\ell% }(\cos\theta)=\sum_{\ell=0}^{\infty}(2\ell+1)\frac{e^{i\delta}\sin\delta}{k}P_% {\ell}(\cos\theta)
  14. d σ d Ω = | f ( θ , φ ) | 2 = 1 k 2 | = 0 ( 2 + 1 ) e i δ sin δ P ( cos θ ) | 2 \frac{d\sigma}{d\Omega}=|f(\theta,\varphi)|^{2}=\frac{1}{k^{2}}\left|\sum_{% \ell=0}^{\infty}(2\ell+1)e^{i\delta_{\ell}}\sin\delta_{\ell}P_{\ell}(\cos% \theta)\right|^{2}

Particle-laden_flows.html

  1. ρ u i d t + ρ u i u j x j = - P x i + τ i j x j + S i , \frac{\partial\rho u_{i}}{dt}+\frac{\partial\rho u_{i}u_{j}}{\partial x_{j}}=-% \frac{\partial P}{\partial x_{i}}+\frac{\partial\tau_{ij}}{\partial x_{j}}+S_{% i},
  2. S i S_{i}
  3. d v i d t = 1 τ p ( u i - v i ) , \frac{dv_{i}}{dt}=\frac{1}{\tau_{p}}(u_{i}-v_{i}),
  4. u i u_{i}
  5. v i v_{i}
  6. τ p \tau_{p}
  7. τ p = ρ p d p 2 18 μ . \tau_{p}=\frac{\rho_{p}d_{p}^{2}}{18\mu}.
  8. d p d_{p}
  9. ρ p \rho_{p}
  10. μ \mu
  11. τ p = ρ p d p 2 18 μ ( 1 + 0.15 R e p 0.687 ) - 1 , \tau_{p}=\frac{\rho_{p}d_{p}^{2}}{18\mu}(1+0.15Re_{p}^{0.687})^{-1},
  12. R e p Re_{p}
  13. R e p = | u - v | d p ν . Re_{p}=\frac{|\vec{u}-\vec{v}|d_{p}}{\nu}.

Party-list_representation_in_the_House_of_Representatives_of_the_Philippines.html

  1. T P s = 1 if g 0.02 TP_{s}=1~{}\mbox{if}~{}~{}g\ >=0.02
  2. T P s = 2 if g 0.04 TP_{s}=2~{}\mbox{if}~{}~{}g\ >=0.04
  3. T P s = 3 if g 0.06 TP_{s}=3~{}\mbox{if}~{}~{}g\ >=0.06
  4. S = ( PV TP ) × T P s \mathrm{S}=(\frac{\mathrm{PV}}{\mathrm{TP}})\times{TP_{s}}
  5. S = ( D 0.8 ) × 0.2 S=\left(\frac{D}{0.8}\right)\times 0.2
  6. g = P V g=\frac{P}{V}
  7. R 1 = 1 if g 0.02 R_{1}=1~{}\mbox{if}~{}~{}g\ >=0.02
  8. R 2 = ( S - T 1 ) × g R_{2}=(S-T_{1})\times g
  9. R 2 {R_{2}}
  10. T 1 {T_{1}}
  11. ( R 1 ) ({R_{1}})
  12. R 2 {R_{2}}
  13. R 2 {R_{2}}
  14. T 3 = ( S - T 1 - T 2 ) T_{3}=(S-T_{1}-T_{2})
  15. T 3 {T_{3}}
  16. T 1 {T_{1}}
  17. T 2 {T_{2}}
  18. R 1 = 1 since 0.0519 0.02 R_{1}=1~{}\mbox{since}~{}~{}0.0519\ >=0.02
  19. R 2 = ( 57 - 12 ) × 0.0519 R_{2}=(57-12)\times 0.0519
  20. R 2 = 45 × 0.0519 R_{2}=45\times 0.0519
  21. R 2 = 2.3397 R_{2}=2.3397
  22. R 2 = 2 R_{2}=2
  23. S = 1 + 2 = 3 S=1+2=3
  24. R 1 = 1 since 0.0362 0.02 R_{1}=1~{}\mbox{since}~{}~{}0.0362\ >=0.02
  25. R 2 = ( 57 - 12 ) × 0.0362 R_{2}=(57-12)\times 0.0362
  26. R 2 = 45 × 0.0362 R_{2}=45\times 0.0362
  27. R 2 = 1.6303 R_{2}=1.6303
  28. R 2 = 1 R_{2}=1
  29. S = 1 + 1 = 2 S=1+1=2
  30. R 0 = 0 since 0.0078 0.02 R_{0}=0~{}\mbox{since}~{}~{}0.0078\ >=0.02
  31. R 2 = ( 57 - 35 ) × 0.0078 R_{2}=(57-35)\times 0.0078
  32. R 2 = 45 × 0.0078 R_{2}=45\times 0.0078
  33. R 2 = 0.3489 R_{2}=0.3489
  34. R 2 = 0 R_{2}=0
  35. S = 0 + 0 = 0 S=0+0=0

Passivity_(engineering).html

  1. E A ( x ) = sup x T 0 0 T - v ( t ) , i ( t ) d t E_{A}(x)=\sup_{x\to T\geq 0}\int_{0}^{T}-\langle v(t),i(t)\rangle\,\mathord{% \operatorname{d}}t
  2. v ( t ) , i ( t ) \langle v(t),i(t)\rangle

Patent_visualisation.html

  1. W e i g h t = T e r m F r e q u e n c y D o c u m e n t F r e q u e n c y = F r e q u e n c y o f t h e w o r d o r e x p r e s s i o n i n t h e T e x t S e a N u m b e r o f d o c u m e n t s c o n t a i n i n g t h e e x p r e s s i o n o r w o r d Weight=\frac{Term\ Frequency}{Document\ Frequency}=\frac{Frequency\ of\ % theword\ or\ expression\ in\ the\ Text\ Sea}{Number\ of\ documents\ containing% \ the\ expression\ or\ word}

PCOLA-SOQ.html

  1. Component test coverage = # of components tested Total # of components \,\text{Component test coverage}=\frac{\#\,\text{ of components tested}}{\,% \text{Total }\#\,\text{ of components}}
  2. PCBA shorts test coverage = # of accessible nodes # of PCBA nodes \,\text{PCBA shorts test coverage}=\frac{\#\,\text{ of accessible nodes}}{\#\,% \text{ of PCBA nodes}}

Pearl_vortex.html

  1. r r
  2. r 2 r^{2}
  3. l o g ( 1 / r ) log(1/r)
  4. v v
  5. Λ = 2 λ 2 \Lambda=2\lambda^{2}
  6. d d
  7. d d
  8. λ \lambda
  9. r r

Pedal_equation.html

  1. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}
  2. p = x f x + y f y ( f x ) 2 + ( f y ) 2 . p=\frac{x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}}{\sqrt{(% \frac{\partial f}{\partial x})^{2}+(\frac{\partial f}{\partial y})^{2}}}.
  3. p = g z ( g x ) 2 + ( g y ) 2 p=\frac{\frac{\partial g}{\partial z}}{\sqrt{(\frac{\partial g}{\partial x})^{% 2}+(\frac{\partial g}{\partial y})^{2}}}
  4. p = r sin ψ p=r\sin\psi\,
  5. r = d r d θ tan ψ . r=\frac{dr}{d\theta}\tan\psi.
  6. r n = a n sin ( n θ ) r^{n}=a^{n}\sin(n\theta)\,
  7. ψ = n θ \psi=n\theta
  8. p a n = r n + 1 . pa^{n}=r^{n+1}.
  9. 1 / 2 {1}/{2}
  10. 1 / 2 {1}/{2}
  11. x ( θ ) = ( a + b ) cos θ - b cos ( a + b b θ ) x(\theta)=(a+b)\cos\theta-b\cos\left(\frac{a+b}{b}\theta\right)
  12. y ( θ ) = ( a + b ) sin θ - b sin ( a + b b θ ) , y(\theta)=(a+b)\sin\theta-b\sin\left(\frac{a+b}{b}\theta\right),
  13. r 2 = a 2 + 4 ( a + b ) b ( a + 2 b ) 2 p 2 r^{2}=a^{2}+\frac{4(a+b)b}{(a+2b)^{2}}p^{2}
  14. p 2 = A ( r 2 - a 2 ) p^{2}=A(r^{2}-a^{2})
  15. A = ( a + 2 b ) 2 4 ( a + b ) b . A=\frac{(a+2b)^{2}}{4(a+b)b}.
  16. a / n {a}/{n}
  17. 1 / 2 {1}/{2}
  18. p 2 = 9 8 ( r 2 - a 2 ) p^{2}=\frac{9}{8}(r^{2}-a^{2})
  19. 2 / 3 {2}/{3}
  20. p 2 = 4 3 ( r 2 - a 2 ) p^{2}=\frac{4}{3}(r^{2}-a^{2})
  21. 3 / 2 {3}/{2}
  22. p 2 = - 1 8 ( r 2 - a 2 ) p^{2}=-\frac{1}{8}(r^{2}-a^{2})
  23. 4 / 3 {4}/{3}
  24. p 2 = - 1 3 ( r 2 - a 2 ) p^{2}=-\frac{1}{3}(r^{2}-a^{2})
  25. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  26. a 2 b 2 p 2 + r 2 = a 2 + b 2 \frac{a^{2}b^{2}}{p^{2}}+r^{2}=a^{2}+b^{2}
  27. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  28. - a 2 b 2 p 2 + r 2 = a 2 - b 2 -\frac{a^{2}b^{2}}{p^{2}}+r^{2}=a^{2}-b^{2}
  29. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  30. b 2 p 2 = 2 a r - 1 \frac{b^{2}}{p^{2}}=\frac{2a}{r}-1
  31. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  32. b 2 p 2 = 2 a r + 1 \frac{b^{2}}{p^{2}}=\frac{2a}{r}+1
  33. r = a e θ cot α r=ae^{\theta\cot\alpha}\,
  34. p = r sin α p=r\sin\alpha

Peeling_theorem.html

  1. γ \gamma
  2. ( M , g a b ) (M,g_{ab})
  3. λ \lambda
  4. λ \lambda
  5. C a b c d = C a b c d ( 1 ) λ + C a b c d ( 2 ) λ 2 + C a b c d ( 3 ) λ 3 + C a b c d ( 4 ) λ 4 + O ( 1 λ 5 ) C_{abcd}=\frac{C^{(1)}_{abcd}}{\lambda}+\frac{C^{(2)}_{abcd}}{\lambda^{2}}+% \frac{C^{(3)}_{abcd}}{\lambda^{3}}+\frac{C^{(4)}_{abcd}}{\lambda^{4}}+O\left(% \frac{1}{\lambda^{5}}\right)
  6. C a b c d C_{abcd}
  7. C a b c d ( 1 ) C^{(1)}_{abcd}
  8. C a b c d ( 2 ) C^{(2)}_{abcd}
  9. C a b c d ( 3 ) C^{(3)}_{abcd}
  10. C a b c d ( 4 ) C^{(4)}_{abcd}

Penalized_present_value.html

  1. P P V = μ - t σ PPV=\mu-t\sigma
  2. μ = a + b 2 \mu\ =\frac{a+b}{2}
  3. σ = b - a 6 \sigma\ =\frac{b-a}{6}
  4. P P V = a + b 2 - 1.5 b - a 6 = 0.25 b + 0.75 a PPV=\frac{a+b}{2}-1.5\frac{b-a}{6}=0.25b+0.75a

Pentagonal_polytope.html

  1. H 1 H_{1}
  2. H 2 H_{2}
  3. H 3 H_{3}
  4. H 4 H_{4}
  5. H ¯ 4 {\bar{H}}_{4}
  6. H 1 H_{1}
  7. H 2 H_{2}
  8. H 3 H_{3}
  9. H 4 H_{4}
  10. H ¯ 4 {\bar{H}}_{4}

Penzhin_Tidal_Power_Plant_Project.html

  1. H H e a d H_{Head}
  2. E = [ ρ s w S B a s i n ( H T i d e - H H e a d ) ] g H H e a d E=[\rho_{sw}\cdot S_{Basin}\cdot(H_{Tide}-H_{Head})]\cdot g\cdot H_{Head}
  3. E E
  4. ρ s w \rho_{sw}
  5. S B a s i n S_{Basin}
  6. H T i d e H_{Tide}
  7. g g
  8. H H e a d H_{Head}
  9. H T i d e H_{Tide}
  10. H H e a d H_{Head}
  11. H H e a d = 5 H_{Head}=5
  12. H H e a d H_{Head}
  13. × 10 1 8 \times 10^{1}8
  14. × 10 1 8 \times 10^{1}8

Percobaltate.html

  1. 4 Co 3 O 4 + 18 N a 2 O + 7 O 2 12 N a 3 CoO 4 \rm\ 4Co_{3}O_{4}+18Na_{2}O+7O_{2}\rightarrow 12Na_{3}CoO_{4}

Peregrine_soliton.html

  1. i ψ τ + 1 2 2 ψ ξ 2 + | ψ | 2 ψ = 0 i\frac{\partial\psi}{\partial\tau}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial% \xi^{2}}+|\psi|^{2}\psi=0
  2. ξ \xi
  3. τ \tau
  4. ψ ( ξ , τ ) \psi(\xi,\tau)
  5. ψ ( ξ , τ ) = [ 1 - 4 ( 1 + 2 i τ ) 1 + 4 ξ 2 + 4 τ 2 ] e i τ \psi(\xi,\tau)=\left[1-\frac{4(1+2i\tau)}{1+4\xi^{2}+4\tau^{2}}\right]e^{i\tau}
  6. ξ = 0 \xi=0
  7. τ = 0 \tau=0
  8. η \eta
  9. ψ ~ ( η , τ ) = 1 2 π ψ ( ξ , τ ) e i η ξ d ξ = 2 π e i τ [ 1 + 2 i τ 1 + 4 τ 2 exp ( - | η | 2 1 + 4 τ 2 ) - δ ( η ) ] \tilde{\psi}(\eta,\tau)=\frac{1}{\sqrt{2\pi}}\int{\psi(\xi,\tau)e^{i\eta\xi}d% \xi}=\sqrt{2\pi}e^{i\tau}\left[\frac{1+2i\tau}{\sqrt{1+4\tau^{2}}}\exp\left(-% \frac{|\eta|}{2}\sqrt{1+4\tau^{2}}\right)-\delta(\eta)\right]
  10. δ \delta
  11. | ψ ~ ( η , τ ) | = 2 π exp ( - | η | 2 1 + 4 τ 2 ) . |\tilde{\psi}(\eta,\tau)|=\sqrt{2\pi}\exp\left(-\frac{|\eta|}{2}\sqrt{1+4\tau^% {2}}\right).
  12. τ \tau
  13. τ = 0 \tau=0
  14. i ψ z - β 2 2 2 ψ t 2 + γ | ψ | 2 ψ = 0 i\frac{\partial\psi}{\partial z}-\frac{\beta_{2}}{2}\frac{\partial^{2}\psi}{% \partial t^{2}}+\gamma|\psi|^{2}\psi=0
  15. β 2 \beta_{2}
  16. β 2 < 0 \beta_{2}<0
  17. γ \gamma
  18. z z
  19. t t
  20. ψ ( z , t ) = P 0 [ 1 - 4 ( 1 + 2 i z L N L ) 1 + 4 ( t T 0 ) 2 + 4 ( z L N L ) 2 ] e i z L N L \psi(z,t)=\sqrt{P_{0}}\left[1-\frac{4\left(1+2i\dfrac{z}{L_{NL}}\right)}{1+4% \left(\dfrac{t}{T_{0}}\right)^{2}+4\left(\dfrac{z}{L_{NL}}\right)^{2}}\right]e% ^{\dfrac{iz}{L_{NL}}}
  21. L N L L_{NL}
  22. L N L = 1 γ P 0 L_{NL}=\dfrac{1}{\gamma P_{0}}
  23. T 0 T_{0}
  24. T 0 = 1 β 2 L N L T_{0}=\dfrac{1}{\sqrt{\beta_{2}L_{NL}}}
  25. P 0 P_{0}

Perfect_matrix.html

  1. [ 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 ] . \begin{bmatrix}1&1&0&0&0\\ 0&1&1&0&0\\ 0&0&1&1&0\\ 0&0&0&1&1\\ 1&0&0&0&1\end{bmatrix}.

Periodic_summation.html

  1. f P f_{P}
  2. f f
  3. f P ( x ) = n = - f ( x + n T ) = n = - f ( x - n T ) . f_{P}(x)=\sum_{n=-\infty}^{\infty}f(x+nT)=\sum_{n=-\infty}^{\infty}f(x-nT).
  4. f P f_{P}
  5. f f
  6. 1 / T . \scriptstyle 1/T.
  7. f , f,
  8. f , f,
  9. T {\triangle\!\triangle\!\triangle}_{T}
  10. T f {\triangle\!\triangle\!\triangle}_{T}f
  11. ( T f ) ( t ) = def k = - f ( t - k T ) ({\triangle\!\triangle\!\triangle}_{T}f)(t)\,\stackrel{\mathrm{def}}{=}\,\sum_% {k=-\infty}^{\infty}\,f(t-kT)
  12. f ( t ) f(t)
  13. | t | |t|
  14. T T
  15. T f {\triangle\!\triangle\!\triangle}_{T}f
  16. T f {\triangle\!\triangle\!\triangle}_{T}f
  17. f f
  18. f f
  19. f f
  20. T T
  21. / ( P ) \mathbb{R}/(P\mathbb{Z})
  22. φ P : / ( P ) \varphi_{P}:\mathbb{R}/(P\mathbb{Z})\to\mathbb{R}
  23. φ P ( x ) = τ x f ( τ ) \varphi_{P}(x)=\sum_{\tau\in x}f(\tau)
  24. φ P \varphi_{P}
  25. P P

Periodic_systems_of_small_molecules.html

  1. ( Li Be Na Mg ) ( Li Be Na Mg ) = ( Li 2 LiBe BeLi Be 2 LiNa LiMg BeNa BeMg NaLi NaBe MgLi MgBe Na 2 NaMg MgNa Mg 2 ) \begin{pmatrix}\rm Li&\rm Be\\ \rm Na&\rm Mg\end{pmatrix}\otimes\begin{pmatrix}\rm Li&\rm Be\\ \rm Na&\rm Mg\end{pmatrix}=\begin{pmatrix}\rm Li_{2}&\rm LiBe&\rm BeLi&\rm Be_% {2}\\ \rm LiNa&\rm LiMg&\rm BeNa&\rm BeMg\\ \rm NaLi&\rm NaBe&\rm MgLi&\rm MgBe\\ \rm Na_{2}&\rm NaMg&\rm MgNa&\rm Mg_{2}\\ \end{pmatrix}

Permutation_representation_(disambiguation).html

  1. G G
  2. G G
  3. G G
  4. X X
  5. G G
  6. X X
  7. ρ : G Sym ( X ) . \rho\colon G\to\operatorname{Sym}(X).
  8. ρ ( G ) \sub Sym ( X ) \rho(G)\sub\operatorname{Sym}(X)
  9. G G
  10. X X
  11. G G
  12. X X
  13. G × X X . G\times X\to X.
  14. G G
  15. n n
  16. G G
  17. G G
  18. ρ : G GL n ( K ) \rho\colon G\to\operatorname{GL}_{n}(K)
  19. g G g\in G
  20. K K
  21. G G
  22. K n K^{n}
  23. G G
  24. G G

Perrin_number.html

  1. G ( P ( n ) ; x ) = 3 - x 2 1 - x 2 - x 3 . G(P(n);x)=\frac{3-x^{2}}{1-x^{2}-x^{3}}.
  2. ( 0 1 0 0 0 1 1 1 0 ) n ( 3 0 2 ) = ( P ( n ) P ( n + 1 ) P ( n + 2 ) ) \begin{pmatrix}0&1&0\\ 0&0&1\\ 1&1&0\end{pmatrix}^{n}\begin{pmatrix}3\\ 0\\ 2\end{pmatrix}=\begin{pmatrix}P\left(n\right)\\ P\left(n+1\right)\\ P\left(n+2\right)\end{pmatrix}
  3. x 3 - x - 1 = 0. x^{3}-x-1=0.
  4. P ( n ) = p n + q n + r n . P\left(n\right)={p^{n}}+{q^{n}}+{r^{n}}.
  5. P ( n ) p n P\left(n\right)\approx{p^{n}}
  6. G ( n - 1 ) = p - 1 p n + q - 1 q n + r - 1 r n G ( n ) = p n + q n + r n G ( n + 1 ) = p p n + q q n + r r n \begin{matrix}G(n-1)&=&p^{-1}p^{n}+&q^{-1}q^{n}+&r^{-1}r^{n}\\ G(n)&=&p^{n}+&q^{n}+&r^{n}\\ G(n+1)&=&pp^{n}+&qq^{n}+&rr^{n}\end{matrix}
  7. x 3 - x - 1 x^{3}-x-1
  8. p n , q n , r n p^{n},q^{n},r^{n}
  9. u = G ( n - 1 ) , v = G ( n ) , w = G ( n + 1 ) u=G(n-1),v=G(n),w=G(n+1)
  10. 23 G ( 2 n - 1 ) = 4 u 2 + 3 v 2 + 9 w 2 + 18 u v - 12 u w - 4 v w 23 G ( 2 n ) = - 6 u 2 + 7 v 2 - 2 w 2 - 4 u v + 18 u w + 6 v w 23 G ( 2 n + 1 ) = 9 u 2 + v 2 + 3 w 2 + 6 u v - 4 u w + 14 v w 23 G ( 3 n - 1 ) = ( - 4 u 3 + 2 v 3 - w 3 + 9 ( u v 2 + v w 2 + w u 2 ) + 3 v 2 w + 6 u v w ) 23 G ( 3 n ) = ( 3 u 3 + 2 v 3 + 3 w 3 - 3 ( u v 2 + u w 2 + v w 2 + v u 2 ) + 6 v 2 w + 18 u v w ) 23 G ( 3 n + 1 ) = ( v 3 - w 3 + 6 u v 2 + 9 u w 2 + 6 v w 2 + 9 v u 2 - 3 w u 2 + 6 w v 2 - 6 u v w ) \begin{matrix}23G(2n-1)&=&4u^{2}+3v^{2}+9w^{2}+18uv-12uw-4vw\\ 23G(2n)&=&-6u^{2}+7v^{2}-2w^{2}-4uv+18uw+6vw\\ 23G(2n+1)&=&9u^{2}+v^{2}+3w^{2}+6uv-4uw+14vw\\ 23G(3n-1)&=&\left(-4u^{3}+2v^{3}-w^{3}+9(uv^{2}+vw^{2}+wu^{2})+3v^{2}w+6uvw% \right)\\ 23G(3n)&=&\left(3u^{3}+2v^{3}+3w^{3}-3(uv^{2}+uw^{2}+vw^{2}+vu^{2})+6v^{2}w+18% uvw\right)\\ 23G(3n+1)&=&\left(v^{3}-w^{3}+6uv^{2}+9uw^{2}+6vw^{2}+9vu^{2}-3wu^{2}+6wv^{2}-% 6uvw\right)\end{matrix}
  11. O ( log n ) O(\log n)

Perry_Robertson_formula.html

  1. σ m = 1 2 ( f y + σ e ( 1 + θ ) - ( f y + σ e ( 1 + θ ) ) 2 - 4 f y σ e ) \sigma_{m}=\frac{1}{2}\left(f_{y}+\sigma_{e}\left(1+\theta\right)-\sqrt{\left(% f_{y}+\sigma_{e}\left(1+\theta\right)\right)^{2}-4f_{y}\sigma_{e}}\right)
  2. θ = w o , 1 c i 2 \theta=\frac{w_{o,1}c}{i^{2}}
  3. σ m \sigma_{m}
  4. f y f_{y}
  5. σ e \sigma_{e}
  6. w o , 1 w_{o,1}
  7. c c
  8. i i
  9. θ = 0.003 λ \theta=0.003\lambda
  10. λ \lambda

Persistent_homology.html

  1. f : K f:K\rightarrow\mathbb{R}
  2. f ( σ ) f ( τ ) f(\sigma)\leq f(\tau)
  3. σ \sigma
  4. τ \tau
  5. K K
  6. a a\in\mathbb{R}
  7. K ( a ) = f - 1 ( - , a ] K(a)=f^{-1}(-\infty,a]
  8. f f
  9. K K
  10. = K 0 K 1 K n = K \emptyset=K_{0}\subseteq K_{1}\subseteq\ldots\subseteq K_{n}=K
  11. 0 i j n 0\leq i\leq j\leq n
  12. K i K j K_{i}\hookrightarrow K_{j}
  13. f p i , j : H p ( K i ) H p ( K j ) f_{p}^{i,j}:H_{p}(K_{i})\rightarrow H_{p}(K_{j})
  14. p p
  15. p t h p^{th}
  16. p t h p^{th}
  17. β p i , j \beta_{p}^{i,j}
  18. p = 0 p=0

Perturbation_function.html

  1. ( X , X * ) \left(X,X^{*}\right)
  2. ( Y , Y * ) \left(Y,Y^{*}\right)
  3. f : X { + } f:X\to\mathbb{R}\cup\{+\infty\}
  4. inf x X f ( x ) . \inf_{x\in X}f(x).\,
  5. f f
  6. f = f + I constraints f=f+I_{\mathrm{constraints}}
  7. I I
  8. F : X × Y { + } F:X\times Y\to\mathbb{R}\cup\{+\infty\}
  9. F ( x , 0 ) = f ( x ) F(x,0)=f(x)
  10. sup y * Y * - F * ( 0 , y * ) inf x X F ( x , 0 ) , \sup_{y^{*}\in Y^{*}}-F^{*}(0,y^{*})\leq\inf_{x\in X}F(x,0),
  11. F * F^{*}
  12. 0 core ( Pr Y ( dom F ) ) 0\in\operatorname{core}(\operatorname{Pr}_{Y}(\operatorname{dom}F))
  13. core \operatorname{core}
  14. Pr Y \operatorname{Pr}_{Y}
  15. Pr Y ( x , y ) = y \operatorname{Pr}_{Y}(x,y)=y
  16. ( X , X * ) (X,X^{*})
  17. ( Y , Y * ) (Y,Y^{*})
  18. L : X × Y * { + } L:X\times Y^{*}\to\mathbb{R}\cup\{+\infty\}
  19. L ( x , - y * ) = inf y Y { F ( x , y ) - y * ( y ) } . L(x,-y^{*})=\inf_{y\in Y}\left\{F(x,y)-y^{*}(y)\right\}.
  20. sup y * Y * - F * ( 0 , y * ) = sup y * Y * inf x X L ( x , y * ) inf x X sup y * Y * L ( x , y * ) = inf x X F ( x , 0 ) . \sup_{y^{*}\in Y^{*}}-F^{*}(0,y^{*})=\sup_{y^{*}\in Y^{*}}\inf_{x\in X}L(x,y^{% *})\leq\inf_{x\in X}\sup_{y^{*}\in Y^{*}}L(x,y^{*})=\inf_{x\in X}F(x,0).
  21. inf x : g ( x ) 0 f ( x ) = inf x X f ~ ( x ) \inf_{x:g(x)\leq 0}f(x)=\inf_{x\in X}\tilde{f}(x)
  22. f ~ ( x ) = f ( x ) + I + d ( - g ( x ) ) \tilde{f}(x)=f(x)+I_{\mathbb{R}^{d}_{+}}(-g(x))
  23. inf x : g ( x ) y f ( x ) \inf_{x:g(x)\leq y}f(x)
  24. F ( x , y ) = f ( x ) + I + d ( y - g ( x ) ) F(x,y)=f(x)+I_{\mathbb{R}^{d}_{+}}(y-g(x))
  25. L ( x , y * ) = { f ( x ) + y * ( g ( x ) ) if y * + d - else L(x,y^{*})=\begin{cases}f(x)+y^{*}(g(x))&\,\text{if }y^{*}\in\mathbb{R}^{d}_{+% }\\ -\infty&\,\text{else}\end{cases}
  26. ( X , X * ) (X,X^{*})
  27. ( Y , Y * ) (Y,Y^{*})
  28. T : X Y T:X\to Y
  29. T * : Y * X * T^{*}:Y^{*}\to X^{*}
  30. f ( x ) f(x)
  31. f ( x ) = J ( x , T x ) f(x)=J(x,Tx)
  32. J : X × Y { + } J:X\times Y\to\mathbb{R}\cup\{+\infty\}
  33. F ( x , y ) = J ( x , T x - y ) F(x,y)=J(x,Tx-y)
  34. f ( x ) + g ( T x ) f(x)+g(Tx)
  35. F ( x , y ) = f ( x ) + g ( T x - y ) F(x,y)=f(x)+g(Tx-y)

Peters_polynomials.html

  1. s n ( x ) t n / n ! = ( 1 + t ) x ( 1 + ( 1 + t ) λ ) - μ \displaystyle\sum s_{n}(x)t^{n}/n!=\frac{(1+t)^{x}}{(1+(1+t)^{\lambda})^{-\mu}}

Petroleum_naphtha.html

  1. \Rightarrow
  2. \Rightarrow

Petrov–Galerkin_method.html

  1. a ( x ) d u d x + b ( x ) d 2 u d x 2 = f ( x ) , x ( 0 , L ) & u ( 0 ) = u o , d u d x | x = L = u L a(x)\dfrac{\mathrm{d}u}{\mathrm{d}x}+b(x)\dfrac{\mathrm{d}^{2}u}{\mathrm{d}x^{% 2}}=f(x),\quad x\in(0,L)\quad\&\quad u(0)=u_{o},\left.\dfrac{\mathrm{d}u}{% \mathrm{d}x}\right|_{x=L}=u_{L}^{\prime}
  2. v ( x ) v(x)
  3. 0 L a ( x ) v ( x ) d u d x d x - 0 L b ( x ) d v d x d u d x d x + [ b ( x ) v d u d x ] 0 L = 0 L v ( x ) f ( x ) d x \int_{0}^{L}a(x)v(x)\dfrac{\mathrm{d}u}{\mathrm{d}x}\mathrm{d}x-\int_{0}^{L}b(% x)\dfrac{\mathrm{d}v}{\mathrm{d}x}\dfrac{\mathrm{d}u}{\mathrm{d}x}\mathrm{d}x+% \left[b(x)v\dfrac{\mathrm{d}u}{\mathrm{d}x}\right]_{0}^{L}=\int_{0}^{L}v(x)f(x% )\,\mathrm{d}x
  4. H 0 1 H_{0}^{1}
  5. H 0 1 H_{0}^{1}
  6. L 2 L^{2}

Petzval_field_curvature.html

  1. i n i + 1 - n i r i n i + 1 n i , \sum_{i}\frac{n_{i+1}-n_{i}}{r_{i}n_{i+1}n_{i}},
  2. r i r_{i}

PFD_allowance.html

  1. T std = T n * ( 1 + A pfd ) T\text{std}=T_{n}*(1+A\text{pfd})
  2. T std = T\text{std}=
  3. T n = T_{n}=
  4. A pfd = A\text{pfd}=

Phase-contrast_X-ray_imaging.html

  1. n = 1 - δ + i β n=1-\delta+i\beta
  2. Ψ ( z ) = E 0 e i k z \Psi(z)=E_{0}e^{ikz}
  3. Ψ ( z ) = E 0 e i n k z = E 0 e i ( 1 - δ ) k z e - β k z \Psi(z)=E_{0}e^{inkz}=E_{0}e^{i(1-\delta)kz}e^{-\beta kz}
  4. < v a r > δ k z < v a r > <var>δkz<var>
  5. Φ ( z ) = 2 π λ 0 z δ ( z ) d z \Phi(z)=\frac{2\pi}{\lambda}\int_{0}^{z}\!\delta(z^{\prime})\,\mathrm{d}z^{\prime}
  6. β = ρ a σ a 2 k \beta=\frac{\rho_{a}\sigma_{a}}{2k}
  7. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  8. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  9. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  10. δ = ρ a p k \delta=\frac{\rho_{a}p}{k}
  11. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  12. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  13. p = 2 π Z r 0 k p=\frac{2\pi Zr_{0}}{k}
  14. δ = ρ a p k = 2 π ρ a Z r 0 k 2 \delta=\frac{\rho_{a}p}{k}=\frac{2\pi\rho_{a}Zr_{0}}{k^{2}}
  15. β = ρ a σ a 2 k = 0.01 [ barn ] ρ a k 0 3 ( Z k ) 4 \beta=\frac{\rho_{a}\sigma_{a}}{2k}=0.01[\,\text{barn}]\rho_{a}k_{0}^{3}\left(% \frac{Z}{k}\right)^{4}
  16. β k - 4 \beta\propto k^{-4}
  17. δ k - 2 \delta\propto k^{-2}
  18. π π
  19. μ μ
  20. μ μ
  21. Δ α = 1 k ϕ ( x ) x \Delta\alpha=\frac{1}{k}\frac{\partial\phi(x)}{\partial x}
  22. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  23. Δ α = 1 k ϕ ( x ) x \Delta\alpha=\frac{1}{k}\frac{\partial\phi(x)}{\partial x}
  24. Δ α = 1 k ϕ ( x ) x \Delta\alpha=\frac{1}{k}\frac{\partial\phi(x)}{\partial x}

Phase_detector_characteristic.html

  1. f 1 ( θ 1 ( t ) ) f^{1}(\theta^{1}(t))
  2. f 2 ( θ 2 ( t ) ) f^{2}(\theta^{2}(t))
  3. f 1 ( θ ) f^{1}(\theta)
  4. f 2 ( θ ) f^{2}(\theta)
  5. θ 1 , 2 ( t ) \theta^{1,2}(t)
  6. g ( t ) g(t)
  7. f 1 , 2 ( θ ) f^{1,2}(\theta)
  8. θ 1 , 2 ( t ) \theta^{1,2}(t)
  9. ϕ ( θ ) \phi(\theta)
  10. g ( t ) = 0 t f 1 ( θ 1 ( t ) ) f 2 ( θ 2 ( t ) ) d t g(t)=\int\limits_{0}^{t}f^{1}(\theta^{1}(t))f^{2}(\theta^{2}(t))dt
  11. G ( t ) = 0 t φ ( θ 1 ( t ) - θ 2 ( t ) ) d t G(t)=\int\limits_{0}^{t}\varphi(\theta^{1}(t)-\theta^{2}(t))dt
  12. g ( t ) - G ( t ) 0 g(t)-G(t)\approx 0
  13. f 1 ( θ ) = sin ( θ ) , f^{1}(\theta)=\sin(\theta),
  14. f 2 ( θ ) = cos ( θ ) f^{2}(\theta)=\cos(\theta)
  15. sin ( θ 1 ( t ) ) cos ( θ 2 ( t ) ) = 1 2 sin ( θ 1 ( t ) + θ 2 ( t ) ) + 1 2 sin ( θ 1 ( t ) - θ 2 ( t ) ) \sin(\theta^{1}(t))\cos(\theta^{2}(t))=\frac{1}{2}\sin(\theta^{1}(t)+\theta^{2% }(t))+\frac{1}{2}\sin(\theta^{1}(t)-\theta^{2}(t))
  16. sin ( θ 1 ( t ) + θ 2 ( t ) ) \sin(\theta^{1}(t)+\theta^{2}(t))
  17. sin ( θ 1 ( t ) - θ 2 ( t ) ) \sin(\theta^{1}(t)-\theta^{2}(t))
  18. φ ( θ ) = 1 2 sin ( θ ) . \varphi(\theta)=\frac{1}{2}\sin(\theta).
  19. f 1 ( t ) = sgn ( sin ( θ 1 ( t ) ) ) f^{1}(t)=\operatorname{sgn}(\sin(\theta^{1}(t)))
  20. f 2 ( t ) = sgn ( cos ( θ 2 ( t ) ) ) f^{2}(t)=\operatorname{sgn}(\cos(\theta^{2}(t)))
  21. φ ( θ ) = { 1 + 2 θ π , if θ [ - π , 0 ] , 1 - 2 θ π , if θ [ 0 , π ] . \varphi(\theta)=\begin{cases}1+\frac{2\theta}{\pi},&\,\text{if }\theta\in[-\pi% ,0],\\ 1-\frac{2\theta}{\pi},&\,\text{if }\theta\in[0,\pi].\\ \end{cases}
  22. f 1 ( θ ) f^{1}(\theta)
  23. f 2 ( θ ) f^{2}(\theta)
  24. a i p = 1 π - π π f p ( x ) sin ( i x ) d x , a^{p}_{i}=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f^{p}(x)\sin(ix)dx,
  25. b i p = 1 π - π π f p ( x ) cos ( i x ) d x , b^{p}_{i}=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f^{p}(x)\cos(ix)dx,
  26. c i p = 1 π - π π f p ( x ) d x , p = 1 , 2 c^{p}_{i}=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f^{p}(x)dx,p=1,2
  27. f 1 ( θ ) f^{1}(\theta)
  28. f 2 ( θ ) f^{2}(\theta)
  29. φ ( θ ) = c 1 c 2 + 1 2 l = 1 ( ( a l 1 a l 2 + b l 1 b l 2 ) cos ( l θ ) + ( a l 1 b l 2 - b l 1 a l 2 ) sin ( l θ ) ) . \varphi(\theta)=c^{1}c^{2}+\frac{1}{2}\sum\limits_{l=1}^{\infty}\bigg((a^{1}_{% l}a^{2}_{l}+b^{1}_{l}b^{2}_{l})\cos(l\theta)+(a^{1}_{l}b^{2}_{l}-b^{1}_{l}a^{2% }_{l})\sin(l\theta)\bigg).
  30. φ ( θ ) \varphi(\theta)
  31. \mathbb{R}
  32. f 1 , 2 ( θ ) f^{1,2}(\theta)
  33. φ ( θ ) \varphi(\theta)

Phase_line_(mathematics).html

  1. f ( y ) f(y)
  2. d y d t = f ( y ) \tfrac{dy}{dt}=f(y)
  3. n n
  4. y y
  5. f ( y ) = 0 f(y)=0
  6. f ( y ) f(y)
  7. f ( y ) = 0 f(y)=0
  8. y y
  9. f ( y ) > 0 f(y)>0
  10. y y
  11. y y
  12. f ( y ) < 0 f(y)<0
  13. y y
  14. n n

Philippine_House_of_Representatives_elections,_2001.html

  1. Seats = ( 802 , 060 1 , 708 , 253 ) 3 = 1.41 \mathrm{Seats}=(\frac{\mathrm{802,060}}{\mathrm{1,708,253}}){3}={1.41}

Phoenix_network_coordinates.html

  1. N N
  2. N × N N\times N
  3. D X Y T D\approx XY^{T}
  4. X X
  5. Y Y
  6. N × d N\times d

Phone_hacking.html

  1. 10 4 10^{4}

Phonological_opacity.html

  1. A B / C ¯ D A\rightarrow B/C\underline{\quad}D
  2. C ¯ D C\underline{\quad}D
  3. C ¯ D C\underline{\quad}D
  4. C ¯ D C\underline{\quad}D

Phonon_noise.html

  1. k B T k_{B}T
  2. k B k_{B}
  3. T T
  4. N E P = 4 k B T 2 G , \ NEP=\sqrt{4k_{B}T^{2}G},
  5. W / Hz \mathrm{W/\sqrt{Hz}}
  6. δ E \delta E
  7. δ E = k B T 2 C , \ \delta E=\sqrt{k_{B}T^{2}C},

Photoconductive_atomic_force_microscopy.html

  1. f = - k d f=-kd
  2. f o = 1 2 π k m o f_{o}=\frac{1}{2\pi}\sqrt{\frac{k}{m_{o}}}
  3. f = - d V d r = 24 ε o r o [ 2 r o r 12 - r o r 6 ] f={\operatorname{-d}V\over\operatorname{d}r}={24\varepsilon_{o}\over r_{o}}% \left[{2}{r_{o}\over r}^{12}-{r_{o}\over r}^{6}\right]
  4. I = A e f f ( q 2 m o 8 π h m e f f ) ( 1 t ( E 2 ) ) ( β 2 V 2 ϕ d 2 ) e ( ( ( 8 π ) ( 2 m e f f q ) 1 2 ( 3 h ) ) ( ν ( E ) ) ( d β V ) ( ϕ 1 3 ) ) I=A_{eff}\left(\frac{q^{2}m_{o}}{8\pi hm_{eff}}\right)\left(\frac{1}{t\left(E^% {2}\right)}\right)\left(\frac{\beta^{2}V^{2}}{\phi d^{2}}\right)e^{\left(}% \left(\frac{\left(8\pi\right)\left(2m_{eff}q\right)^{\frac{1}{2}}}{\left(3h% \right)}\right)\left(\nu\left(E\right)\right)\left(\frac{d}{\beta V}\right)% \left(\phi^{\frac{1}{3}}\right)\right)
  5. L - Δ L = ( R - r ) Θ L-\Delta L=\left(R-r\right)\Theta
  6. L + Δ L = ( R + r ) Θ L+\Delta L=\left(R+r\right)\Theta
  7. Δ L = E d 31 = ( d 31 L t ) U x \Delta L=Ed_{31}=\left(\frac{d_{31}L}{t}\right)U_{x}
  8. Θ = L R = ( d 31 L t r ) U x \Theta=\frac{L}{R}=\left(\frac{d_{31}L}{t_{r}}\right)U_{x}
  9. d x = ( R + χ ) ( 1 - c o s Θ ) + ( D s s + D s p ) U x dx=(R+\chi)\left(1-cos\Theta\right)+\left(D_{ss}+D_{sp}\right)U_{x}
  10. d z = ( ( R + χ ) s i n Θ - L ) + ( D s s + D s p ) ( c o s Θ - 1 ) dz=\left(\left(R+\chi\right)sin\Theta-L\right)+\left(D_{ss}+D_{sp}\right)\left% (cos\Theta-1\right)
  11. J = 8 9 ε o ε r μ V 3 L 3 J=\frac{8}{9}\varepsilon_{o}\varepsilon_{r}\mu\frac{V^{3}}{L^{3}}

Photoelectron_photoion_coincidence_spectroscopy.html

  1. E int ion = E int neutral + h ν - I E ad E_{\,\text{int}}^{\,\text{ion}}=E_{\,\text{int}}^{\,\text{neutral}}+h\nu-IE_{% \,\text{ad}}
  2. \rightarrow

Photon_energy.html

  1. E = h c λ E=\frac{hc}{\lambda}
  2. E ( e V ) = 1.2398 λ ( μ m ) E(eV)=\frac{1.2398}{\mathrm{\lambda}({\mu}m)}
  3. c λ = f \frac{c}{\lambda}=f
  4. E = h f E=hf

Photon_rocket.html

  1. v = c ( m i m f ) 2 - 1 ( m i m f ) 2 + 1 v=c\frac{\left(\frac{m_{i}}{m_{f}}\right)^{2}-1}{\left(\frac{m_{i}}{m_{f}}% \right)^{2}+1}
  2. m i m_{i}
  3. m f m_{f}
  4. γ = 1 2 ( m i m f + m f m i ) \gamma=\frac{1}{2}\left(\frac{m_{i}}{m_{f}}+\frac{m_{f}}{m_{i}}\right)
  5. P i P_{i}
  6. P f P_{f}
  7. P ph P_{\,\text{ph}}
  8. P ph = P i - P f P_{\,\text{ph}}=P_{i}-P_{f}
  9. P ph 2 = P i 2 + P f 2 - 2 P i P f P_{\,\text{ph}}^{2}=P_{i}^{2}+P_{f}^{2}-2P_{i}\cdot P_{f}
  10. P ph 2 = 0 P_{\,\text{ph}}^{2}=0
  11. 0 = m i 2 + m f 2 - 2 m i m f γ 0=m_{i}^{2}+m_{f}^{2}-2m_{i}m_{f}\gamma
  12. γ = 1 2 ( m i m f + m f m i ) \gamma=\frac{1}{2}\left(\frac{m_{i}}{m_{f}}+\frac{m_{f}}{m_{i}}\right)

Photon_surface.html

  1. { r = r p s } \{r=r_{ps}\}
  2. r o r_{o}
  3. r o r p s . r_{o}\rightarrow r_{ps}.
  4. g = - β ( r ) d t 2 - α ( r ) d r 2 - σ ( r ) r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , g=-\beta\left(r\right)dt^{2}-\alpha(r)dr^{2}-\sigma(r)r^{2}(d\theta^{2}+\sin^{% 2}\theta d\phi^{2}),
  5. 2 σ ( r ) β + r d σ ( r ) d r β ( r ) - r d β ( r ) d r σ ( r ) = 0. 2\sigma(r)\beta+r\frac{d\sigma(r)}{dr}\beta(r)-r\frac{d\beta(r)}{dr}\sigma(r)=0.
  6. γ {\gamma}
  7. γ ˙ {\dot{\gamma}}

Physics_of_failure.html

  1. M T T F = A ( J - n ) e E a k T MTTF=A(J^{-n})e^{\frac{Ea}{kT}}
  2. T T F = A 0 ( R H ) - 2.7 f ( V ) exp ( E a K B T ) TTF=A_{0}(RH)^{-2.7}f(V)\exp\left(\frac{E_{a}}{K_{B}T}\right)
  3. N f ( 50 % ) = 1 2 [ 2 ϵ f Δ D ] - 1 c Δ D ( leadless ) = [ F L D Δ ( α Δ T ) h ] N_{f}(50\%)=\frac{1}{2}\left[\frac{2\epsilon^{\prime}_{f}}{\Delta D}\right]^{% \frac{-1}{c}}\quad\Delta D(\,\text{leadless})=\left[\frac{FL_{D}\Delta(\alpha% \Delta T)}{h}\right]
  4. Z 0 = 9.8 × 3 π / 2 × P S D × f n × Q f n 2 Z c = 0.00022 B c h r L Z_{0}=\frac{9.8\times 3\sqrt{\pi/2\times PSD\times f_{n}\times Q}}{f_{n}^{2}}% \quad Z_{c}=\frac{0.00022B}{chr\sqrt{L}}
  5. σ = ( α E - α C u ) Δ T A E E E E C u A E E E + A C u E C u , for σ S Y \sigma=\frac{(\alpha_{E}-\alpha_{Cu})\Delta TA_{E}E_{E}E_{Cu}}{A_{E}E_{E}+A_{% Cu}E_{Cu}},\quad\,\text{for }\sigma\leq S_{Y}
  6. N f - 0.6 D f 0.75 + 0.9 S u E [ exp ( D f ) 0.36 ] 0.1785 log 10 5 N f - Δ ϵ = 0 N_{f}^{-0.6}D_{f}^{0.75}+0.9\frac{S_{u}}{E}\left[\frac{\exp(D_{f})}{0.36}% \right]^{0.1785\log\frac{10^{5}}{N_{f}}}-\Delta\epsilon=0

Physiological_cross-sectional_area.html

  1. PCSA = muscle volume fiber length = muscle mass ρ fiber length , \,\text{PCSA}={\,\text{muscle volume}\over\,\text{fiber length}}={\,\text{% muscle mass}\over{\rho\cdot\,\text{fiber length}}},
  2. ρ = muscle mass muscle volume . \rho={\,\text{muscle mass}\over\,\text{muscle volume}}.
  3. Total force = PCSA Specific tension \,\text{Total force}=\,\text{PCSA}\cdot\,\text{Specific tension}
  4. Muscle force = Total force cos Φ \,\text{Muscle force}=\,\text{Total force}\cdot\cos\Phi
  5. PCSA 2 = muscle volume cos Φ fiber length = muscle mass cos Φ ρ fiber length , \,\text{PCSA}_{2}={\,\text{muscle volume}\cdot\cos\Phi\over\,\text{fiber % length}}={\,\text{muscle mass}\cdot\cos\Phi\over{\rho\cdot\,\text{fiber length% }}},
  6. PCSA 2 = PCSA cos Φ . \,\text{PCSA}_{2}=\,\text{PCSA}\cdot\cos\Phi.
  7. cos Φ \cos\Phi
  8. Muscle force = PCSA 2 Specific tension \,\text{Muscle force}=\,\text{PCSA}_{2}\cdot\,\text{Specific tension}

PI_curve.html

  1. P = P max ( 1 - e - α I / P max ) e - β I / P max P=P_{\max}(1-e^{-\alpha I/P_{\max}})e^{-\beta I/P_{\max}}\,

Picard–Lefschetz_theory.html

  1. γ \gamma
  2. w i ( γ ) = γ + ( - 1 ) ( k + 1 ) ( k + 2 ) / 2 γ , δ i δ i w_{i}(\gamma)=\gamma+(-1)^{(k+1)(k+2)/2}\langle\gamma,\delta_{i}\rangle\delta_% {i}

Pickands–Balkema–de_Haan_theorem.html

  1. F F
  2. X X
  3. F u F_{u}
  4. X X
  5. u u
  6. F u ( y ) = P ( X - u y | X > u ) = F ( u + y ) - F ( u ) 1 - F ( u ) F_{u}(y)=P(X-u\leq y|X>u)=\frac{F(u+y)-F(u)}{1-F(u)}\,
  7. 0 y x F - u 0\leq y\leq x_{F}-u
  8. x F x_{F}
  9. F F
  10. F u F_{u}
  11. u u
  12. ( X 1 , X 2 , ) (X_{1},X_{2},\ldots)
  13. F u F_{u}
  14. F F
  15. u u
  16. F u F_{u}
  17. F u ( y ) G k , σ ( y ) , as u F_{u}(y)\rightarrow G_{k,\sigma}(y),\,\text{ as }u\rightarrow\infty
  18. G k , σ ( y ) = 1 - ( 1 + k y / σ ) - 1 / k G_{k,\sigma}(y)=1-(1+ky/\sigma)^{-1/k}
  19. k 0 k\neq 0
  20. G k , σ ( y ) = 1 - e - y / σ G_{k,\sigma}(y)=1-e^{-y/\sigma}
  21. k = 0. k=0.
  22. [ 0 , σ ] [0,\sigma]

Pidduck_polynomials.html

  1. n s n ( x ) n ! t n = ( 1 + t 1 - t ) x ( 1 - t ) - 1 \displaystyle\sum_{n}\frac{s_{n}(x)}{n!}t^{n}=\left(\frac{1+t}{1-t}\right)^{x}% (1-t)^{-1}

Pinched_torus.html

  1. f ( x , y ) = ( g ( x , y ) cos x , g ( x , y ) sin x , sin ( x 2 ) sin y ) f(x,y)=\left(g(x,y)\cos x,g(x,y)\sin x,\sin\!\left(\frac{x}{2}\right)\sin y\right)
  2. H 0 ( P , \Z ) \Z , H 1 ( P , \Z ) \Z , and H 2 ( P , \Z ) \Z . H_{0}(P,\Z)\cong\Z,\ H_{1}(P,\Z)\cong\Z,\ \,\text{and}\ H_{2}(P,\Z)\cong\Z.
  3. H 0 ( P , \Z ) \Z , H 1 ( P , \Z ) \Z , and H 2 ( P , \Z ) \Z . H^{0}(P,\Z)\cong\Z,\ H^{1}(P,\Z)\cong\Z,\ \,\text{and}\ H^{2}(P,\Z)\cong\Z.

Pincherle_polynomials.html

  1. ( 1 - 3 x t + t 3 ) - 1 / 2 = n = 0 P n ( x ) t n \displaystyle(1-3xt+t^{3})^{-1/2}=\sum^{\infty}_{n=0}P_{n}(x)t^{n}

Pitch_constellation.html

  1. fP5 fT = 2 ( 7 / 12 ) 1.49821 3 2 {\,\text{f}\text{P5}\over\,\text{f}\text{T}}=2^{(7/12)}\approx 1.49821\approx{% 3\over 2}

Pitch_interval.html

  1. ip x , y = y - x \operatorname{ip}\langle x,y\rangle=y-x
  2. ip y , x = x - y \operatorname{ip}\langle y,x\rangle=x-y
  3. ip ( x , y ) = | y - x | \operatorname{ip}(x,y)=|y-x|
  4. i x , y = y - x \operatorname{i}\langle x,y\rangle=y-x

Placzek_transient.html

  1. Δ E m a x = ( 1 - α ) E \Delta E_{max}=(1-\alpha)E
  2. α = ( A - 1 A + 1 ) 2 \alpha=(\frac{A-1}{A+1})^{2}
  3. α E \alpha E
  4. E m i n = α E E_{min}=\alpha E
  5. E m i n E_{min}
  6. α E \alpha E
  7. α E \alpha E
  8. ( 1 - α ) E (1-\alpha)E

Plancherel_measure.html

  1. G G
  2. G G
  3. S n S_{n}
  4. G G
  5. G G^{\wedge}
  6. G G^{\wedge}
  7. μ ( π ) = ( dim π ) 2 | G | , \mu(\pi)=\frac{(\mathrm{dim}\,\pi)^{2}}{|G|},
  8. π G \pi\in G^{\wedge}
  9. dim π \mathrm{dim}\pi
  10. π \pi
  11. S n S_{n}
  12. S n S_{n}
  13. n n
  14. S n S_{n}^{\wedge}
  15. n n
  16. λ \lambda
  17. f λ f^{\lambda}
  18. λ \lambda
  19. μ ( λ ) = ( f λ ) 2 n ! . \mu(\lambda)=\frac{(f^{\lambda})^{2}}{n!}.
  20. λ n ( f λ ) 2 = n ! , \sum_{\lambda\vdash n}(f^{\lambda})^{2}=n!,
  21. σ \sigma
  22. S n S_{n}
  23. L ( σ ) L(\sigma)
  24. σ \sigma
  25. S n S_{n}
  26. λ \lambda
  27. σ \sigma
  28. L ( σ ) = λ 1 , L(\sigma)=\lambda_{1},\,
  29. λ 1 \lambda_{1}
  30. λ \lambda
  31. λ \lambda
  32. S n S_{n}
  33. L ( σ ) L(\sigma)
  34. λ 1 \lambda_{1}
  35. λ \lambda
  36. S n S_{n}
  37. S n S_{n}
  38. n n
  39. L ( σ ) L(\sigma)
  40. n n\rightarrow\infty
  41. 𝒫 * \mathcal{P}^{*}
  42. θ > 0 \theta>0
  43. θ \theta
  44. 𝒫 * \mathcal{P}^{*}
  45. μ θ ( λ ) = e - θ θ | λ | ( f λ ) 2 ( | λ | ! ) 2 , \mu_{\theta}(\lambda)=e^{-\theta}\frac{\theta^{|\lambda|}(f^{\lambda})^{2}}{(|% \lambda|!)^{2}},
  46. λ 𝒫 * \lambda\in\mathcal{P}^{*}
  47. λ ( 1 ) = ( 1 ) , λ ( 2 ) , λ ( 3 ) , , \lambda^{(1)}=(1),~{}\lambda^{(2)},~{}\lambda^{(3)},~{}\ldots,
  48. λ ( n ) \lambda^{(n)}
  49. n n
  50. λ ( n ) \lambda^{(n)}
  51. λ ( n - 1 ) \lambda^{(n-1)}
  52. p ( ν , λ ) = ( λ ( n ) = λ | λ ( n - 1 ) = ν ) = f λ n f ν , p(\nu,\lambda)=\mathbb{P}(\lambda^{(n)}=\lambda~{}|~{}\lambda^{(n-1)}=\nu)=% \frac{f^{\lambda}}{nf^{\nu}},
  53. ν \nu
  54. λ \lambda
  55. λ ( n ) \lambda^{(n)}
  56. S n S_{n}

Planck_units.html

  1. F = G m 1 m 2 r 2 , F=G\frac{m_{1}m_{2}}{r^{2}},
  2. F F P = ( m 1 m P ) ( m 2 m P ) ( r l P ) 2 . \frac{F}{F\text{P}}=\frac{\left(\dfrac{m_{1}}{m\text{P}}\right)\left(\dfrac{m_% {2}}{m\text{P}}\right)}{\left(\dfrac{r}{l\text{P}}\right)^{2}}.
  3. F = m 1 m 2 r 2 . F=\frac{m_{1}m_{2}}{r^{2}}\ .
  4. l P = c t P l\text{P}=c\ t\text{P}
  5. F P = m P l P t P 2 = G m P 2 l P 2 F\text{P}=\frac{m\text{P}l\text{P}}{t\text{P}^{2}}=G\ \frac{m\text{P}^{2}}{l% \text{P}^{2}}
  6. E P = m P l P 2 t P 2 = 1 t P E\text{P}=\frac{m\text{P}l\text{P}^{2}}{t\text{P}^{2}}=\hbar\ \frac{1}{t\text{% P}}
  7. F P = m P l P t P 2 = 1 4 π ε 0 q P 2 l P 2 F\text{P}=\frac{m\text{P}l\text{P}}{t\text{P}^{2}}=\frac{1}{4\pi\varepsilon_{0% }}\ \frac{q\text{P}^{2}}{l\text{P}^{2}}
  8. E P = m P l P 2 t P 2 = k B T P . E\text{P}=\frac{m\text{P}l\text{P}^{2}}{t\text{P}^{2}}=k\text{B}\ T\text{P}.
  9. l P 2 = G c 3 l\text{P}^{2}=\frac{\hbar G}{c^{3}}
  10. l P 3 = ( G c 3 ) 3 2 = ( G ) 3 c 9 l\text{P}^{3}=\left(\frac{\hbar G}{c^{3}}\right)^{\frac{3}{2}}=\sqrt{\frac{(% \hbar G)^{3}}{c^{9}}}
  11. m P c = l P = c 3 G m\text{P}c=\frac{\hbar}{l\text{P}}=\sqrt{\frac{\hbar c^{3}}{G}}
  12. E P = m P c 2 = t P = c 5 G E\text{P}=m\text{P}c^{2}=\frac{\hbar}{t\text{P}}=\sqrt{\frac{\hbar c^{5}}{G}}
  13. F P = E P l P = l P t P = c 4 G F\text{P}=\frac{E\text{P}}{l\text{P}}=\frac{\hbar}{l\text{P}t\text{P}}=\frac{c% ^{4}}{G}
  14. P P = E P t P = t P 2 = c 5 G P\text{P}=\frac{E\text{P}}{t\text{P}}=\frac{\hbar}{t\text{P}^{2}}=\frac{c^{5}}% {G}
  15. ρ P = m P l P 3 = t P l P 5 = c 5 G 2 \rho\text{P}=\frac{m\text{P}}{l\text{P}^{3}}=\frac{\hbar t\text{P}}{l\text{P}^% {5}}=\frac{c^{5}}{\hbar G^{2}}
  16. ρ E P = E P l P 3 = c 7 G 2 \rho^{E}\text{P}=\frac{E\text{P}}{l\text{P}^{3}}=\frac{c^{7}}{\hbar G^{2}}
  17. I P = ρ E P c = P P l P 2 = c 8 G 2 I\text{P}=\rho^{E}\text{P}c=\frac{P\text{P}}{l\text{P}^{2}}=\frac{c^{8}}{\hbar G% ^{2}}
  18. ω P = 1 t P = c 5 G \omega\text{P}=\frac{1}{t\text{P}}=\sqrt{\frac{c^{5}}{\hbar G}}
  19. p P = F P l P 2 = l P 3 t P = c 7 G 2 p\text{P}=\frac{F\text{P}}{l\text{P}^{2}}=\frac{\hbar}{l\text{P}^{3}t\text{P}}% =\frac{c^{7}}{\hbar G^{2}}
  20. I P = q P t P = 4 π ϵ 0 c 6 G I\text{P}=\frac{q\text{P}}{t\text{P}}=\sqrt{\frac{4\pi\epsilon_{0}c^{6}}{G}}
  21. V P = E P q P = t P q P = c 4 4 π ϵ 0 G V\text{P}=\frac{E\text{P}}{q\text{P}}=\frac{\hbar}{t\text{P}q\text{P}}=\sqrt{% \frac{c^{4}}{4\pi\epsilon_{0}G}}
  22. Z P = V P I P = q P 2 = 1 4 π ϵ 0 c = Z 0 4 π Z\text{P}=\frac{V\text{P}}{I\text{P}}=\frac{\hbar}{q\text{P}^{2}}=\frac{1}{4% \pi\epsilon_{0}c}=\frac{Z_{0}}{4\pi}
  23. F = - G m 1 m 2 r 2 F=-G\frac{m_{1}m_{2}}{r^{2}}
  24. F = - m 1 m 2 r 2 F=-\frac{m_{1}m_{2}}{r^{2}}
  25. G μ ν = 8 π G c 4 T μ ν {G_{\mu\nu}=8\pi{G\over c^{4}}T_{\mu\nu}}
  26. G μ ν = 8 π T μ ν {G_{\mu\nu}=8\pi T_{\mu\nu}}
  27. E = m c 2 {E=mc^{2}}
  28. E = m {E=m}
  29. E 2 = m 2 c 4 + p 2 c 2 E^{2}=m^{2}c^{4}+p^{2}c^{2}\;
  30. E 2 = m 2 + p 2 E^{2}=m^{2}+p^{2}\;
  31. E = 1 2 k B T {E=\tfrac{1}{2}k\text{B}T}
  32. E = 1 2 T {E=\tfrac{1}{2}T}
  33. S = k B ln Ω {S=k\text{B}\ln\Omega}
  34. S = ln Ω {S=\ln\Omega}
  35. E = ω {E=\hbar\omega}
  36. E = ω {E=\omega}
  37. I ( ω , T ) = ω 3 4 π 3 c 2 1 e ω k B T - 1 I(\omega,T)=\frac{\hbar\omega^{3}}{4\pi^{3}c^{2}}~{}\frac{1}{e^{\frac{\hbar% \omega}{k\text{B}T}}-1}
  38. I ( ω , T ) = ω 3 4 π 3 1 e ω / T - 1 I(\omega,T)=\frac{\omega^{3}}{4\pi^{3}}~{}\frac{1}{e^{\omega/T}-1}
  39. σ = π 2 k B 4 60 3 c 2 \sigma=\frac{\pi^{2}k\text{B}^{4}}{60\hbar^{3}c^{2}}
  40. σ = π 2 / 60 \ \sigma=\pi^{2}/60
  41. S BH = A BH k B c 3 4 G = 4 π G k B m 2 BH c S\text{BH}=\frac{A\text{BH}k\text{B}c^{3}}{4G\hbar}=\frac{4\pi Gk\text{B}m^{2}% \text{BH}}{\hbar c}
  42. S BH = A BH / 4 = 4 π m 2 BH S\text{BH}=A\text{BH}/4=4\pi m^{2}\text{BH}
  43. - 2 2 m 2 ψ ( 𝐫 , t ) + V ( 𝐫 ) ψ ( 𝐫 , t ) = i ψ ˙ ( 𝐫 , t ) -\frac{\hbar^{2}}{2m}\nabla^{2}\psi(\mathbf{r},t)+V(\mathbf{r})\psi(\mathbf{r}% ,t)=i\hbar\dot{\psi}(\mathbf{r},t)
  44. - 1 2 m 2 ψ ( 𝐫 , t ) + V ( 𝐫 ) ψ ( 𝐫 , t ) = i ψ ˙ ( 𝐫 , t ) -\frac{1}{2m}\nabla^{2}\psi(\mathbf{r},t)+V(\mathbf{r})\psi(\mathbf{r},t)=i% \dot{\psi}(\mathbf{r},t)
  45. H | ψ t = i | ψ t / t H\left|\psi_{t}\right\rangle=i\hbar\partial\left|\psi_{t}\right\rangle/\partial t
  46. H | ψ t = i | ψ t / t H\left|\psi_{t}\right\rangle=i\partial\left|\psi_{t}\right\rangle/\partial t
  47. ( i γ μ μ - m c ) ψ = 0 \ (i\hbar\gamma^{\mu}\partial_{\mu}-mc)\psi=0
  48. ( i γ μ μ - m ) ψ = 0 \ (i\gamma^{\mu}\partial_{\mu}-m)\psi=0
  49. F = 1 4 π ϵ 0 q 1 q 2 r 2 F=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}
  50. F = q 1 q 2 r 2 F=\frac{q_{1}q_{2}}{r^{2}}
  51. 𝐄 = 1 ϵ 0 ρ \nabla\cdot\mathbf{E}=\frac{1}{\epsilon_{0}}\rho
  52. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  53. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  54. × 𝐁 = 1 c 2 ( 1 ϵ 0 𝐉 + 𝐄 t ) \nabla\times\mathbf{B}=\frac{1}{c^{2}}\left(\frac{1}{\epsilon_{0}}\mathbf{J}+% \frac{\partial\mathbf{E}}{\partial t}\right)
  55. 𝐄 = 4 π ρ \nabla\cdot\mathbf{E}=4\pi\rho
  56. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  57. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  58. × 𝐁 = 4 π 𝐉 + 𝐄 t \nabla\times\mathbf{B}=4\pi\mathbf{J}+\frac{\partial\mathbf{E}}{\partial t}
  59. 8 π \sqrt{8π}
  60. 1 / [ u g a p s , u 299 , u 792 , u 458 ] {1}/{[u^{\prime}gaps^{\prime},u^{\prime}299^{\prime},u^{\prime}792^{\prime},u^% {\prime}458^{\prime}]}
  61. e = α q P 0.085424543 q P e=\sqrt{\alpha}\cdot q_{\,\text{P}}\approx 0.085424543\cdot q_{\,\text{P}}\,
  62. α {\alpha}
  63. α = ( e q P ) 2 = e 2 4 π ε 0 c = 1 137.03599911 \alpha=\left(\frac{e}{q_{\,\text{P}}}\right)^{2}=\frac{e^{2}}{4\pi\varepsilon_% {0}\hbar c}=\frac{1}{137.03599911}
  64. 2 2 \scriptstyle 2\sqrt{2}
  65. a 0 = 4 π ϵ 0 2 m e e 2 = m P m e α l P . a_{0}=\frac{4\pi\epsilon_{0}\hbar^{2}}{m_{e}e^{2}}=\frac{m\text{P}}{m_{e}% \alpha}l\text{P}.
  66. 2 2 \scriptstyle 2\sqrt{2}
  67. 2 2 \scriptstyle 2\sqrt{2}
  68. 2 2 \scriptstyle 2\sqrt{2}
  69. 4 2 \scriptstyle 4\sqrt{2}
  70. 4 2 \scriptstyle 4\sqrt{2}
  71. c 2 4 2 2 2 \scriptstyle\frac{c}{2}\frac{4\sqrt{2}}{2\sqrt{2}}

Plane_wave_expansion.html

  1. e i 𝐤 𝐫 = = 0 ( 2 + 1 ) i j ( k r ) P ( 𝐤 ^ 𝐫 ^ ) e^{i\mathbf{k}\cdot\mathbf{r}}=\sum_{\ell=0}^{\infty}(2\ell+1)i^{\ell}j_{\ell}% (kr)P_{\ell}(\hat{\mathbf{k}}\cdot\hat{\mathbf{r}})
  2. i i
  3. 𝐤 \mathbf{k}
  4. k k
  5. 𝐫 \mathbf{r}
  6. r r
  7. $^$
  8. 𝐤 \mathbf{k}
  9. e i k r cos θ = = 0 ( 2 + 1 ) i j ( k r ) P ( cos θ ) e^{ikr\cos\theta}=\sum_{\ell=0}^{\infty}(2\ell+1)i^{\ell}j_{\ell}(kr)P_{\ell}(% \cos\theta)
  10. θ θ
  11. 𝐫 \mathbf{r}
  12. e i 𝐤 𝐫 = 4 π = 0 m = - i j ( k r ) Y m ( 𝐤 ^ ) * Y m ( 𝐫 ^ ) e^{i\mathbf{k}\cdot\mathbf{r}}=4\pi\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}% i^{\ell}j_{\ell}(kr)Y_{\ell}^{m}{}^{*}(\hat{\mathbf{k}})Y_{\ell}^{m}(\hat{% \mathbf{r}})
  13. Y < s u b > m Y<sub>ℓ^{m}

Plant_cover.html

  1. c k c_{k}
  2. c k = y k n c_{k}=\frac{y_{k}}{n}
  3. y k y_{k}

Plasmonic_nanoparticles.html

  1. σ s c a t t = 8 π 3 k 4 R 6 | ε p a r t i c l e - ε m e d i u m ε p a r t i c l e + 2 ε m e d i u m | 2 {{\sigma}_{scatt}}=\frac{8\pi}{3}{{k}^{4}}{{R}^{6}}{{\left|\frac{{{\varepsilon% }_{particle}}-{{\varepsilon}_{medium}}}{{{\varepsilon}_{particle}}+2{{% \varepsilon}_{medium}}}\right|}^{2}}
  2. σ a b s = 4 π k R 3 I m | ε p a r t i c l e - ε m e d i u m ε p a r t i c l e + 2 ε m e d i u m | {{\sigma}_{abs}}=4\pi k{{R}^{3}}Im\left|\frac{{{\varepsilon}_{particle}}-{{% \varepsilon}_{medium}}}{{{\varepsilon}_{particle}}+2{{\varepsilon}_{medium}}}\right|
  3. ε m e d i u m {{\varepsilon}_{medium}}
  4. ε p a r t i c l e {{\varepsilon}_{particle}}
  5. ε p a r t i c l e = 1 - ω p 2 ω 2 {{\varepsilon}_{particle}}=1-\frac{\omega_{p}^{2}}{{{\omega}^{2}}}
  6. ω p {{\omega}_{p}}
  7. ε p a r t i c l e + 2 ε m e d i u m 0 {{\varepsilon}_{particle}}+2{{\varepsilon}_{medium}}\approx 0

Playfair's_axiom.html

  1. \ell
  2. \ell
  3. \ell

Poincaré_complex.html

  1. ρ : H k ( C ) H n ( C ) H n - k ( C ) , where ρ ( x y ) = x y , \rho:H^{k}(C)\otimes H_{n}(C)\to H_{n-k}(C),\ \,\text{where}\ \ \rho(x\otimes y% )=x\frown y,
  2. \scriptstyle\frown
  3. ( μ ) : H k ( C ) H n - k ( C ) (\frown\mu):H^{k}(C)\to H_{n-k}(C)

Poincaré_disk_model.html

  1. d ( p , q ) = log | a q | | p b | | a p | | q b | d(p,q)=\log\frac{\left|aq\right|\,\left|pb\right|}{\left|ap\right|\,\left|qb% \right|}
  2. δ ( u , v ) = 2 u - v 2 ( 1 - u 2 ) ( 1 - v 2 ) , \delta(u,v)=2\frac{\left\|u-v\right\|^{2}}{(1-\left\|u\right\|^{2})(1-\left\|v% \right\|^{2})}\,,
  3. \lVert\cdot\rVert
  4. d ( u , v ) = arcosh ( 1 + δ ( u , v ) ) . d(u,v)=\operatorname{arcosh}(1+\delta(u,v))\,.
  5. d s 2 = 4 i d x i 2 ( 1 - i x i 2 ) 2 = 4 d 𝐱 2 ( 1 - 𝐱 2 ) 2 ds^{2}=4\frac{\sum_{i}dx_{i}^{2}}{(1-\sum_{i}x_{i}^{2})^{2}}=\frac{4\left\|d% \mathbf{x}\right\|^{2}}{\bigl(1-\left\|\mathbf{x}\right\|^{2}\bigr)^{2}}
  6. u u
  7. s = 2 u 1 + u u . s=\frac{2u}{1+u\cdot u}.
  8. s s
  9. u = s 1 + 1 - s s = ( 1 - 1 - s s ) s s s . u=\frac{s}{1+\sqrt{1-s\cdot s}}=\frac{\left(1-\sqrt{1-s\cdot s}\right)s}{s% \cdot s}.
  10. y i = x i 1 + t y_{i}=\frac{x_{i}}{1+t}
  11. ( t , x i ) = ( 1 + y i 2 , 2 y i ) 1 - y i 2 . (t,x_{i})=\frac{\left(1+\sum{y_{i}^{2}},\,2y_{i}\right)}{1-\sum{y_{i}^{2}}}\,.
  12. x 2 + y 2 + a x + b y + 1 = 0 , x^{2}+y^{2}+ax+by+1=0\,,
  13. x 2 + y 2 + u 2 ( v 1 2 + v 2 2 ) - v 2 ( u 1 2 + u 2 2 ) + u 2 - v 2 u 1 v 2 - u 2 v 1 x + v 1 ( u 1 2 + u 2 2 ) - u 1 ( v 1 2 + v 2 2 ) + v 1 - u 1 u 1 v 2 - u 2 v 1 y + 1 = 0 . \begin{aligned}&\displaystyle{}x^{2}+y^{2}+\frac{u_{2}(v_{1}^{2}+v_{2}^{2})-v_% {2}(u_{1}^{2}+u_{2}^{2})+u_{2}-v_{2}}{u_{1}v_{2}-u_{2}v_{1}}x\\ &\displaystyle{}\quad+\frac{v_{1}(u_{1}^{2}+u_{2}^{2})-u_{1}(v_{1}^{2}+v_{2}^{% 2})+v_{1}-u_{1}}{u_{1}v_{2}-u_{2}v_{1}}y+1=0\,.\end{aligned}
  14. x 2 + y 2 + 2 ( u 2 - v 2 ) u 1 v 2 - u 2 v 1 x - 2 ( u 1 - v 1 ) u 1 v 2 - u 2 v 1 y + 1 = 0 . x^{2}+y^{2}+\frac{2(u_{2}-v_{2})}{u_{1}v_{2}-u_{2}v_{1}}x-\frac{2(u_{1}-v_{1})% }{u_{1}v_{2}-u_{2}v_{1}}y+1=0\,.
  15. cos ( θ ) = u s . \cos(\theta)=u\cdot s\,.
  16. \wedge
  17. cos 2 ( θ ) = P 2 Q R , \cos^{2}(\theta)=\frac{P^{2}}{QR},
  18. P = u ( s - t ) , P=u\cdot(s-t)\,,
  19. Q = u u , Q=u\cdot u\,,
  20. R = ( s - t ) ( s - t ) - ( s t ) ( s t ) . R=(s-t)\cdot(s-t)-(s\wedge t)\cdot(s\wedge t)\,.
  21. cos 2 ( θ ) = P 2 Q R , \cos^{2}(\theta)=\frac{P^{2}}{QR}\,,
  22. P = ( u - v ) ( s - t ) - ( u v ) ( s t ) , P=(u-v)\cdot(s-t)-(u\wedge v)\cdot(s\wedge t)\,,
  23. Q = ( u - v ) ( u - v ) - ( u v ) ( u v ) , Q=(u-v)\cdot(u-v)-(u\wedge v)\cdot(u\wedge v)\,,
  24. R = ( s - t ) ( s - t ) - ( s t ) ( s t ) . R=(s-t)\cdot(s-t)-(s\wedge t)\cdot(s\wedge t)\,.
  25. P = ( u - v ) ( s - t ) + ( u t ) ( v s ) - ( u s ) ( v t ) . P=(u-v)\cdot(s-t)+(u\cdot t)(v\cdot s)-(u\cdot s)(v\cdot t)\,.
  26. Q = ( 1 - u v ) 2 , Q=(1-u\cdot v)^{2}\,,
  27. R = ( 1 - s t ) 2 . R=(1-s\cdot t)^{2}\,.
  28. 𝐯 \mathbf{v}
  29. 𝐱 \mathbf{x}
  30. ( 1 + 2 𝐯 𝐱 + | 𝐱 | 2 ) 𝐯 + ( 1 - | 𝐯 | 2 ) 𝐱 1 + 2 𝐯 𝐱 + | 𝐯 | 2 | 𝐱 | 2 . \frac{(1+2\mathbf{v}\cdot\mathbf{x}+\left|\mathbf{x}\right|^{2})\mathbf{v}+(1-% \left|\mathbf{v}\right|^{2})\mathbf{x}}{1+2\mathbf{v}\cdot\mathbf{x}+\left|% \mathbf{v}\right|^{2}\left|\mathbf{x}\right|^{2}}.

Poincaré–Lelong_equation.html

  1. i ¯ u = ρ i\partial\overline{\partial}u=\rho

Poisson_binomial_distribution.html

  1. p 1 , p 2 , , p n p_{1},p_{2},\dots,p_{n}
  2. p 1 = p 2 = = p n p_{1}=p_{2}=\cdots=p_{n}
  3. μ = i = 1 n p i \mu=\sum\limits_{i=1}^{n}p_{i}
  4. σ 2 = i = 1 n ( 1 - p i ) p i \sigma^{2}=\sum\limits_{i=1}^{n}(1-p_{i})p_{i}
  5. μ \mu
  6. Pr ( K = k ) = A F k i A p i j A c ( 1 - p j ) \Pr(K=k)=\sum\limits_{A\in F_{k}}\prod\limits_{i\in A}p_{i}\prod\limits_{j\in A% ^{c}}(1-p_{j})
  7. F k F_{k}
  8. F 2 = { { 1 , 2 } , { 1 , 3 } , { 2 , 3 } } F_{2}=\left\{\{1,2\},\{1,3\},\{2,3\}\right\}
  9. A c A^{c}
  10. A A
  11. A c = { 1 , 2 , 3 , , n } A A^{c}=\{1,2,3,\dots,n\}\setminus A
  12. F k F_{k}
  13. n ! / ( ( n - k ) ! k ! ) n!/((n-k)!k!)
  14. F 15 F_{15}
  15. Pr ( K = k ) \Pr(K=k)
  16. Pr ( K = k ) = { i = 1 n ( 1 - p i ) k = 0 1 k i = 1 k ( - 1 ) i - 1 Pr ( K = k - i ) T ( i ) k > 0 \Pr(K=k)=\begin{cases}\prod\limits_{i=1}^{n}(1-p_{i})&k=0\\ \frac{1}{k}\sum\limits_{i=1}^{k}(-1)^{i-1}\Pr(K=k-i)T(i)&k>0\\ \end{cases}
  17. T ( i ) = j = 1 n ( p j 1 - p j ) i . T(i)=\sum\limits_{j=1}^{n}\left(\frac{p_{j}}{1-p_{j}}\right)^{i}.
  18. n n
  19. Pr ( K = k ) = 1 n + 1 l = 0 n C - l k m = 1 n ( 1 + ( C l - 1 ) p m ) \Pr(K=k)=\frac{1}{n+1}\sum\limits_{l=0}^{n}C^{-lk}\prod\limits_{m=1}^{n}\left(% 1+(C^{l}-1)p_{m}\right)
  20. C = exp ( 2 i π n + 1 ) C=\exp\left(\frac{2i\pi}{n+1}\right)
  21. i = - 1 i=\sqrt{-1}
  22. p 1 , p 2 , , p n p_{1},p_{2},\dots,p_{n}

Polarization_mixing.html

  1. [ I Q U V ] = [ | E v | 2 + | E h | 2 | E v | 2 - | E h | 2 2 R e < E v E h * > 2 I m < E v E h * > ] , \left[\begin{array}[]{c}I\\ Q\\ U\\ V\end{array}\right]=\left[\begin{array}[]{c}|E_{v}|^{2}+|E_{h}|^{2}\\ |E_{v}|^{2}-|E_{h}|^{2}\\ 2\mathrm{Re}<E_{v}E_{h}^{*}>\\ 2\mathrm{Im}<E_{v}E_{h}^{*}>\end{array}\right],
  2. θ \theta
  3. θ eff \theta_{\mathrm{eff}}
  4. α \alpha
  5. 𝐯 ^ = ( sin θ , 0 , cos θ ) . \mathbf{\hat{v}}=(\sin\theta,~{}0,~{}\cos\theta).
  6. 𝐧 ^ \mathbf{\hat{n}}
  7. 𝐧 ^ = ( cos ψ sin μ , sin ψ cos μ , cos μ ) , \mathbf{\hat{n}}=(\cos\psi\sin\mu,~{}\sin\psi\cos\mu,~{}\cos\mu),
  8. μ \mu
  9. ψ \psi
  10. θ e f f = cos - 1 ( 𝐧 ^ 𝐯 ^ ) , \theta_{eff}=\cos^{-1}(\mathbf{\hat{n}}\cdot\mathbf{\hat{v}}),
  11. α = sgn ( 𝐧 ^ 𝐣 ^ ) cos - 1 ( 𝐣 ^ 𝐧 ^ × 𝐯 ^ | 𝐧 ^ × 𝐯 ^ | ) , \alpha=\mathrm{sgn}(\mathbf{\hat{n}}\cdot\mathbf{\hat{j}})\cos^{-1}\left(\frac% {\mathbf{\hat{j}}\cdot\mathbf{\hat{n}}\times\mathbf{\hat{v}}}{|\mathbf{\hat{n}% }\times\mathbf{\hat{v}}|}\right),
  12. 𝐣 ^ \mathbf{\hat{j}}
  13. α \alpha
  14. α \alpha
  15. e U = e v 2 - e h 2 sin ( 2 α ) e_{U}=\sqrt{e_{v}^{2}-e_{h}^{2}}\sin(2\alpha)
  16. U = e U T U=e_{U}T

Polyelectrolyte_adsorption.html

  1. c b c_{b}
  2. Γ = 0 c ( x ) d x | y s | 3 / 2 λ B a p 1 / 2 \Gamma=\int_{0}^{\infty}\!c(x)\,dx\,\approx\frac{\left|y_{s}\right|^{3/2}}{{% \lambda_{B}}{a}{p^{1/2}}}
  3. y s y_{s}
  4. y s = e ψ s k B T y_{s}=\frac{{e}{\psi_{s}}}{{k_{B}}{T}}\,
  5. λ B \lambda_{B}
  6. λ B = e 2 4 π ε 0 ε r k B T , \lambda_{B}=\frac{e^{2}}{4\pi\varepsilon_{0}\varepsilon_{r}\ k_{B}T},

Polymer_adsorption.html

  1. c o s θ = 1 - β ( γ L - γ c ) \ cos\theta=1-\beta(\gamma_{L}-\gamma_{c})
  2. θ \theta
  3. γ L \gamma_{L}
  4. γ c \gamma_{c}
  5. d G = 0 dG=0
  6. γ L cos θ C = γ SV - γ SL - π e \gamma_{L}\cos\theta_{\mathrm{C}}\,=\gamma_{\mathrm{SV}}-\gamma_{\mathrm{SL}}-% \pi_{\mathrm{e}}
  7. γ L \gamma_{L}
  8. θ C \theta_{\mathrm{C}}
  9. γ SV \gamma_{\mathrm{SV}}
  10. γ SL \gamma_{\mathrm{SL}}
  11. π e \pi_{\mathrm{e}}
  12. γ SL = γ S + γ L - 2 γ S , d γ S , d - 2 γ S , p γ S , p \gamma_{\mathrm{SL}}=\gamma_{\mathrm{S}}+\gamma_{\mathrm{L}}-2\sqrt{\gamma_{% \mathrm{S,d}}\gamma_{\mathrm{S,d}}}-2\sqrt{\gamma_{\mathrm{S,p}}\gamma_{% \mathrm{S,p}}}
  13. γ S \gamma_{\mathrm{S}}
  14. γ L \gamma_{\mathrm{L}}
  15. γ S , d \gamma_{\mathrm{S,d}}
  16. γ S , p \gamma_{\mathrm{S,p}}
  17. γ p o l y m e r = i = 1 n f i γ i \gamma\!_{polymer}=\sum_{i=1}^{n}f_{i}\gamma\!_{i}
  18. γ p o l y m e r \gamma\!_{polymer}
  19. f i f_{i}
  20. i t h i^{th}
  21. γ i \gamma\!_{i}
  22. i t h i^{th}
  23. cos θ o b s = i = 1 n f i cos θ i \cos\theta\ obs=\sum_{i=1}^{n}f_{i}\cos\theta\!_{i}
  24. f i f_{i}
  25. i t h i^{th}
  26. θ i \theta_{i}\!
  27. i t h i^{th}
  28. cos θ o b s = f cos θ 1 + ( 1 - f ) c o s θ 2 \cos\theta\ obs=\ f\cos\theta\!_{1}+(1-f)cos\theta\!_{2}
  29. θ o b s \theta obs
  30. f f
  31. ( 1 - f ) (1-f)
  32. θ 1 \theta\!_{1}
  33. θ 2 \theta\!_{2}
  34. A + S A - S A+S\leftrightharpoons A-S
  35. A A
  36. S S
  37. A - S A-S
  38. k a d = [ A - S ] [ A ] [ B ] k_{ad}=\frac{[A-S]}{[A][B]}
  39. θ \theta
  40. θ = k a d [ A ] k a d [ A ] + 1 \theta\ =\frac{k_{ad}[A]}{k_{ad}[A]+1}
  41. θ \theta
  42. k a d k_{ad}
  43. Δ G a d \Delta G_{ad}
  44. Δ G a d = - R T l n ( K a d ) \Delta G_{ad}=-RTln(K_{ad})
  45. Δ G a d \Delta G_{ad}
  46. Δ G a d = Δ G p + Δ G c \Delta G_{ad}=\Delta G_{p}+\Delta G_{c}
  47. Q = - Δ H a d * N Q=-\Delta H_{ad}*N
  48. Q Q
  49. Δ H a d \Delta H_{ad}
  50. N N
  51. Δ S a d = Δ H a d T \Delta S_{ad}=\frac{\Delta H_{ad}}{T}
  52. Δ S a d \Delta S_{ad}
  53. T T