wpmath0000016_14

Symplectic_basis.html

  1. 𝐞 i , 𝐟 i {\mathbf{e}}_{i},{\mathbf{f}}_{i}
  2. ω \omega
  3. ω ( 𝐞 i , 𝐞 j ) = 0 = ω ( 𝐟 i , 𝐟 j ) , ω ( 𝐞 i , 𝐟 j ) = δ i j \omega({\mathbf{e}}_{i},{\mathbf{e}}_{j})=0=\omega({\mathbf{f}}_{i},{\mathbf{f% }}_{j}),\omega({\mathbf{e}}_{i},{\mathbf{f}}_{j})=\delta_{ij}

Symplectic_category.html

  1. M × N - M\times N^{-}

Syn-copalyl-diphosphate_synthase.html

  1. \rightleftharpoons

Syn-pimara-7,15-diene_synthase.html

  1. \rightleftharpoons

Sznajd_model.html

  1. i i
  2. S i = - 1 S_{i}=-1
  3. S i = 1 S_{i}=1
  4. S i S_{i}
  5. S i + 1 S_{i+1}
  6. S i - 1 S_{i-1}
  7. S i + 2 S_{i+2}
  8. S i = S i + 1 S_{i}=S_{i+1}
  9. S i - 1 = S i S_{i-1}=S_{i}
  10. S i + 2 = S i S_{i+2}=S_{i}
  11. S i = - S i + 1 S_{i}=-S_{i+1}
  12. S i - 1 = S i + 1 S_{i-1}=S_{i+1}
  13. S i + 2 = S i S_{i+2}=S_{i}
  14. S i S_{i}
  15. S i + 1 S_{i+1}
  16. S i = S i + 1 S_{i}=S_{i+1}
  17. S i - 1 = S i S_{i-1}=S_{i}
  18. S i + 2 = S i S_{i+2}=S_{i}
  19. S i = - S i + 1 S_{i}=-S_{i+1}
  20. S i - 1 = S i S_{i-1}=S_{i}
  21. S i + 2 = S i + 1 S_{i+2}=S_{i+1}
  22. S i = 1 S_{i}=1
  23. S i = 0 S_{i}=0

Šidák_correction_for_t-test.html

  1. H 1 , , H m H_{1},...,H_{m}
  2. H n u l l : H_{null}:
  3. H i H_{i}
  4. H a l t e r n a t i v e : H_{alternative}:
  5. H i H_{i}
  6. α \alpha
  7. H n u l l H_{null}
  8. α \alpha
  9. H i H_{i}
  10. t i t_{i}
  11. t i t_{i}
  12. H n u l l H_{null}
  13. 1 - ( 1 - α ) 1 m 1-(1-\alpha)^{\frac{1}{m}}
  14. H n u l l H_{null}
  15. Y i j = μ i + ϵ i j , i = 1 , , N , j = 1 , , n , Y_{ij}=\mu_{i}+\epsilon_{ij},i=1,...,N,j=1,...,n,
  16. ϵ i 1 , , ϵ i n \epsilon_{i1},...,\epsilon_{in}
  17. ϵ 1 j , , ϵ N j \epsilon_{1j},...,\epsilon_{Nj}
  18. ϵ i j \epsilon_{ij}
  19. H n u l l : μ i = 0 , i = 1 , , N H_{null}:\mu_{i}=0,\forall i=1,...,N
  20. α \alpha
  21. t i = Y ¯ i S i / n t_{i}=\frac{\bar{Y}_{i}}{S_{i}/\sqrt{n}}
  22. Y ¯ i = 1 n j = 1 n Y i j \bar{Y}_{i}=\frac{1}{n}\sum_{j=1}^{n}Y_{ij}
  23. S i 2 = 1 n j = 1 n ( Y i j - Y ¯ i ) 2 S_{i}^{2}=\frac{1}{n}\sum_{j=1}^{n}(Y_{ij}-\bar{Y}_{i})^{2}
  24. H n u l l H_{null}
  25. 1 - ( 1 - α ) 1 / N 1-(1-\alpha)^{1/N}
  26. H n u l l H_{null}
  27. i = 1 , , N , | t i | > ζ α , N \exists i=1,...,N,|t_{i}|>\zeta_{\alpha,N}
  28. P ( | Z | > ζ α , N ) = 1 - ( 1 - α ) 1 / N P(|Z|>\zeta_{\alpha,N})=1-(1-\alpha)^{1/N}
  29. Z N ( 0 , 1 ) Z\sim N(0,1)
  30. α \alpha
  31. l e v e l = P n u l l ( reject H n u l l ) = P n u l l ( i = 1 , , N , | t i | > ζ α , N ) level=P_{null}(\,\text{reject }H_{null})=P_{null}(\exists i=1,...,N,|t_{i}|>% \zeta_{\alpha,N})
  32. = 1 - P n u l l ( i = 1 , , N | t i | ζ α , N = 1 - Π i = 1 N P n u l l ( | t i | ζ α , N ) =1-P_{null}(\forall i=1,...,N|t_{i}|\leq\zeta_{\alpha,N}=1-\Pi_{i=1}^{N}P_{% null}(|t_{i}|\leq\zeta_{\alpha,N})
  33. 1 - Π i = 1 N P ( | Z i | ζ α , N ) where Z i N ( 0 , 1 ) \rightarrow 1-\Pi_{i=1}^{N}P(|Z_{i}|\leq\zeta_{\alpha,N})\,\text{ where }Z_{i}% \sim N(0,1)
  34. = α =\alpha
  35. N N
  36. n n
  37. N ( n ) as n N(n)\rightarrow\infty\,\text{ as }n\rightarrow\infty
  38. H n u l l : all of H i are true, i = 1 , 2 , . H_{null}:\,\text{ all of }H_{i}\,\text{ are true, }i=1,2,....
  39. N ( n ) as n N(n)\rightarrow\infty\,\text{ as }n\rightarrow\infty
  40. α \alpha
  41. α \alpha
  42. N ( n ) N(n)\rightarrow\infty
  43. ϵ i j \epsilon_{ij}
  44. log N = o ( n 1 / 3 ) \log N=o(n^{1/3})
  45. α \alpha
  46. ϵ i j \epsilon_{ij}
  47. log N = o ( n 1 / 2 ) \log N=o(n^{1/2})
  48. α \alpha
  49. log N = o ( n 1 / 3 ) \log N=o(n^{1/3})
  50. ϵ i j \epsilon_{ij}
  51. t i t_{i}
  52. t i t_{i}
  53. t i t_{i}
  54. t i t_{i}
  55. t i t_{i}
  56. N ( n ) N(n)\rightarrow\infty
  57. α \alpha

T-distributed_stochastic_neighbor_embedding.html

  1. N N
  2. 𝐱 1 , , 𝐱 N \mathbf{x}_{1},\dots,\mathbf{x}_{N}
  3. p i j p_{ij}
  4. 𝐱 i \mathbf{x}_{i}
  5. 𝐱 j \mathbf{x}_{j}
  6. p j | i = exp ( - 𝐱 i - 𝐱 j 2 / 2 σ i 2 ) k i exp ( - 𝐱 i - 𝐱 k 2 / 2 σ i 2 ) , p_{j|i}=\frac{\exp(-\lVert\mathbf{x}_{i}-\mathbf{x}_{j}\rVert^{2}/2\sigma_{i}^% {2})}{\sum_{k\neq i}\exp(-\lVert\mathbf{x}_{i}-\mathbf{x}_{k}\rVert^{2}/2% \sigma_{i}^{2})},
  7. p i j = p j | i + p i | j 2 N p_{ij}=\frac{p_{j|i}+p_{i|j}}{2N}
  8. σ i \sigma_{i}
  9. σ i \sigma_{i}
  10. d d
  11. 𝐲 1 , , 𝐲 N \mathbf{y}_{1},\dots,\mathbf{y}_{N}
  12. 𝐲 i d \mathbf{y}_{i}\in\mathbb{R}^{d}
  13. p i j p_{ij}
  14. q i j q_{ij}
  15. 𝐲 i \mathbf{y}_{i}
  16. 𝐲 j \mathbf{y}_{j}
  17. q i j q_{ij}
  18. q i j = ( 1 + 𝐲 i - 𝐲 j 2 ) - 1 k l ( 1 + 𝐲 k - 𝐲 l 2 ) - 1 q_{ij}=\frac{(1+\lVert\mathbf{y}_{i}-\mathbf{y}_{j}\rVert^{2})^{-1}}{\sum_{k% \neq l}(1+\lVert\mathbf{y}_{k}-\mathbf{y}_{l}\rVert^{2})^{-1}}
  19. 𝐲 i \mathbf{y}_{i}
  20. Q Q
  21. P P
  22. K L ( P | | Q ) = i j p i j log p i j q i j KL(P||Q)=\sum_{i\neq j}p_{ij}\,\log\frac{p_{ij}}{q_{ij}}
  23. 𝐲 i \mathbf{y}_{i}

Tail_dependence.html

  1. λ l = lim q 0 P ( X 2 F 2 ( q ) X 1 F 1 ( q ) ) . \lambda_{l}=\lim_{q\rightarrow 0}\operatorname{P}(X_{2}\leq F_{2}^{\leftarrow}% (q)\mid X_{1}\leq F_{1}^{\leftarrow}(q)).
  2. λ u = lim q 1 P ( X 2 > F 2 ( q ) X 1 > F 1 ( q ) ) . \lambda_{u}=\lim_{q\rightarrow 1}\operatorname{P}(X_{2}>F_{2}^{\leftarrow}(q)% \mid X_{1}>F_{1}^{\leftarrow}(q)).

Tame_abstract_elementary_class.html

  1. \mathfrak{C}
  2. \mathfrak{C}
  3. M M
  4. M M
  5. K K
  6. κ \kappa
  7. κ \leq\kappa
  8. κ \kappa
  9. K K
  10. κ \kappa
  11. λ \lambda
  12. λ \lambda
  13. κ \kappa
  14. λ LS ( K ) + + + κ + \lambda\geq\operatorname{LS}(K)^{++}+\kappa^{+}
  15. λ \lambda
  16. μ λ \mu\geq\lambda
  17. κ \kappa
  18. λ κ \lambda\geq\kappa
  19. λ + \lambda^{+}
  20. μ \mu
  21. μ λ = μ \mu^{\lambda}=\mu
  22. κ \kappa
  23. λ \lambda
  24. κ \kappa

Tanc_function.html

  1. Tanc ( z ) = tan ( z ) z \operatorname{Tanc}(z)=\frac{\tan(z)}{z}
  2. Im ( tan ( x + i y ) x + i y ) \operatorname{Im}\left(\frac{\tan(x+iy)}{x+iy}\right)
  3. Re ( tan ( x + i y ) x + i y ) \operatorname{Re}\left(\frac{\tan\left(x+iy\right)}{x+iy}\right)
  4. | tan ( x + i y ) x + i y | \left|\frac{\tan(x+iy)}{x+iy}\right|
  5. 1 - ( tan ( z ) ) 2 z - tan ( z ) z 2 \frac{1-(\tan(z))^{2}}{z}-\frac{\tan(z)}{z^{2}}
  6. - Re ( - 1 - ( tan ( x + i y ) ) 2 x + i y + tan ( x + i y ) ( x + i y ) 2 ) -\operatorname{Re}\left(-\frac{1-(\tan(x+iy))^{2}}{x+iy}+\frac{\tan(x+iy)}{(x+% iy)^{2}}\right)
  7. - Im ( - 1 - ( tan ( x + i y ) ) 2 x + i y + tan ( x + i y ) ( x + i y ) 2 ) -\operatorname{Im}\left(-\frac{1-(\tan(x+iy))^{2}}{x+iy}+\frac{\tan(x+iy)}{(x+% iy)^{2}}\right)
  8. | - 1 - ( tan ( x + i y ) ) 2 x + i y + tan ( x + i y ) ( x + i y ) 2 | \left|-\frac{1-(\tan(x+iy))^{2}}{x+iy}+\frac{\tan(x+iy)}{(x+iy)^{2}}\right|
  9. Tanc ( z ) = 2 i KummerM ( 1 , 2 , 2 i z ) ( 2 z + π ) KummerM ( 1 , 2 , i ( 2 z + π ) ) \operatorname{Tanc}(z)={\frac{2\,i{{\rm KummerM}\left(1,\,2,\,2\,iz\right)}}{% \left(2\,z+\pi\right){{\rm KummerM}\left(1,\,2,\,i\left(2\,z+\pi\right)\right)% }}}
  10. Tanc ( z ) = 2 i HeunB ( 2 , 0 , 0 , 0 , 2 i z ) ( 2 z + π ) HeunB ( 2 , 0 , 0 , 0 , 2 ( i / 2 ) ( 2 z + π ) ) \operatorname{Tanc}(z)=\frac{2i\operatorname{HeunB}\left(2,0,0,0,\sqrt{2}\sqrt% {iz}\right)}{(2z+\pi)\operatorname{HeunB}\left(2,0,0,0,\sqrt{2}\sqrt{(i/2)(2z+% \pi)}\right)}
  11. Tanc ( z ) = WhittakerM ( 0 , 1 / 2 , 2 i z ) WhittakerM ( 0 , 1 / 2 , i ( 2 z + π ) ) z \operatorname{Tanc}(z)=\frac{{\rm WhittakerM}(0,\,1/2,\,2\,iz)}{{\rm WhittakerM% }(0,\,1/2,\,i(2z+\pi))z}
  12. Tanc z ( 1 + 1 3 z 2 + 2 15 z 4 + 17 315 z 6 + 62 2835 z 8 + 1382 155925 z 10 + 21844 6081075 z 12 + 929569 638512875 z 14 + O ( z 16 ) ) \operatorname{Tanc}z\approx\left(1+\frac{1}{3}z^{2}+\frac{2}{15}z^{4}+\frac{17% }{315}z^{6}+\frac{62}{2835}z^{8}+\frac{1382}{155925}z^{10}+\frac{21844}{608107% 5}z^{12}+\frac{929569}{638512875}z^{14}+O(z^{16})\right)
  13. 0 z tan ( x ) x d x = ( z + 1 9 z 3 + 2 75 z 5 + 17 2205 z 7 + 62 25515 z 9 + 1382 1715175 z 11 + 21844 79053975 z 13 + 929569 9577693125 z 15 + O ( z 17 ) ) \int_{0}^{z}\frac{\tan(x)}{x}\,dx=\left(z+\frac{1}{9}z^{3}+\frac{2}{75}z^{5}+% \frac{17}{2205}z^{7}+\frac{62}{25515}z^{9}+\frac{1382}{1715175}z^{11}+\frac{21% 844}{79053975}z^{13}+\frac{929569}{9577693125}z^{15}+O(z^{17})\right)
  14. 𝑇𝑎𝑖𝑛𝑐 ( z ) = ( 1 - 7 51 z 2 + 1 255 z 4 - 2 69615 z 6 + 1 34459425 z 8 ) ( 1 - 8 17 z 2 + 7 255 z 4 - 4 9945 z 6 + 1 765765 z 8 ) - 1 {\it Tainc}\left(z\right)=\left(1-{\frac{7}{51}}\,{z}^{2}+{\frac{1}{255}}\,{z}% ^{4}-{\frac{2}{69615}}\,{z}^{6}+{\frac{1}{34459425}}\,{z}^{8}\right)\left(1-{% \frac{8}{17}}\,{z}^{2}+{\frac{7}{255}}\,{z}^{4}-{\frac{4}{9945}}\,{z}^{6}+{% \frac{1}{765765}}\,{z}^{8}\right)^{-1}

Tanhc_function.html

  1. tanhc ( z ) = tanh ( z ) z \operatorname{tanhc}(z)=\frac{\tanh(z)}{z}
  2. Im ( tanh ( x + i y ) x + i y ) \operatorname{Im}\left(\frac{\tanh(x+iy)}{x+iy}\right)
  3. Re ( tanh ( x + i y ) x + i y ) \operatorname{Re}\left(\frac{\tanh\left(x+iy\right)}{x+iy}\right)
  4. | tanh ( x + i y ) x + i y | \left|\frac{\tanh(x+iy)}{x+iy}\right|
  5. 1 - tanh ( z ) ) 2 z - tanh ( z ) z 2 \frac{1-\tanh(z))^{2}}{z}-\frac{\tanh(z)}{z^{2}}
  6. - Re ( - 1 - ( tanh ( x + i y ) ) 2 x + i y + tanh ( x + i y ) ( x + i y ) 2 ) -\operatorname{Re}\left(-\frac{1-(\tanh(x+iy))^{2}}{x+iy}+\frac{\tanh(x+iy)}{(% x+iy)^{2}}\right)
  7. - Im ( - 1 - ( tanh ( x + i y ) ) 2 x + i y + tanh ( x + i y ) ( x + i y ) 2 ) -\operatorname{Im}\left(-\frac{1-(\tanh(x+iy))^{2}}{x+iy}+\frac{\tanh(x+iy)}{(% x+iy)^{2}}\right)
  8. | - 1 - ( tanh ( x + i y ) ) 2 x + i y + tanh ( x + i y ) ( x + i y ) 2 | \left|-\frac{1-(\tanh(x+iy))^{2}}{x+iy}+\frac{\tanh(x+iy)}{(x+iy)^{2}}\right|
  9. tanhc ( z ) = 2 KummerM ( 1 , 2 , 2 z ) ( 2 i z + π ) KummerM ( 1 , 2 , i π - 2 z ) e 2 z - 1 / 2 i π \operatorname{tanhc}(z)=2\,{\frac{{{\rm KummerM}\left(1,\,2,\,2\,z\right)}}{(2% \,iz+\pi){\rm KummerM}(1,\,2,\,i\pi-2\,z)e^{2\,z-1/2\,i\pi}}}
  10. tanhc ( z ) = 2 HeunB ( 2 , 0 , 0 , 0 , 2 z ) ( 2 i z + π ) HeunB ( 2 , 0 , 0 , 0 , 2 1 / 2 i π - z ) e 2 z - 1 / 2 i π \operatorname{tanhc}(z)=2\frac{\operatorname{HeunB}(2,0,0,0,\sqrt{2}\sqrt{z})}% {(2iz+\pi)\operatorname{HeunB}(2,0,0,0,\sqrt{2}\sqrt{1/2\,i\pi-z})e^{2\,z-1/2% \,i\pi}}
  11. tanhc ( z ) = i WhittakerM ( 0 , 1 / 2 , 2 z ) WhittakerM ( 0 , 1 / 2 , i π - 2 z ) z \operatorname{tanhc}(z)=\frac{i{\rm\ WhittakerM}(0,\,1/2,\,2\,z)}{{\rm WhittakerM% }(0,\,1/2,\,i\pi-2\,z)}z
  12. tanhc z ( 1 - 1 3 z 2 + 2 15 z 4 - 17 315 z 6 + 62 2835 z 8 - 1382 155925 z 10 + 21844 6081075 z 12 - 929569 638512875 z 14 + O ( z 16 ) ) \operatorname{tanhc}z\approx\left(1-\frac{1}{3}z^{2}+\frac{2}{15}z^{4}-\frac{1% 7}{315}z^{6}+\frac{62}{2835}z^{8}-\frac{1382}{155925}z^{10}+\frac{21844}{60810% 75}z^{12}-\frac{929569}{638512875}z^{14}+O(z^{16})\right)
  13. 0 z tanh ( x ) x d x = ( z - 1 9 z 3 + 2 75 z 5 - 17 2205 z 7 + 62 25515 z 9 - 1382 1715175 z 11 + O ( z 13 ) ) \int_{0}^{z}\!{\frac{\tanh\left(x\right)}{x}}{dx}=(z-{\frac{1}{9}}{z}^{3}+{% \frac{2}{75}}{z}^{5}-{\frac{17}{2205}}{z}^{7}+{\frac{62}{25515}}{z}^{9}-{\frac% {1382}{1715175}}{z}^{11}+O\left({z}^{13}\right))
  14. 𝑇𝑎𝑛ℎ𝑐 ( z ) = ( 1 + 7 51 z 2 + 1 255 z 4 + 2 69615 z 6 + 1 34459425 z 8 ) ( 1 + 8 17 z 2 + 7 255 z 4 + 4 9945 z 6 + 1 765765 z 8 ) - 1 {\it Tanhc}\left(z\right)=\left(1+{\frac{7}{51}}\,{z}^{2}+{\frac{1}{255}}\,{z}% ^{4}+{\frac{2}{69615}}\,{z}^{6}+{\frac{1}{34459425}}\,{z}^{8}\right)\left(1+{% \frac{8}{17}}\,{z}^{2}+{\frac{7}{255}}\,{z}^{4}+{\frac{4}{9945}}\,{z}^{6}+{% \frac{1}{765765}}\,{z}^{8}\right)^{-1}

Taraxerol_synthase.html

  1. \rightleftharpoons

Target_Motion_Analysis.html

  1. B R 1 BR^{1}
  2. B R 2 BR^{2}
  3. S O A 1 SOA^{1}
  4. S O A 2 SOA^{2}
  5. r 2 = S O A 2 - S O A 1 B R 1 - B R 2 r_{2}=\textstyle\frac{SOA^{2}-SOA^{1}}{BR^{1}-BR^{2}}
  6. β ( t ) = a r c t a n 2 ( V y * t + y 0 - y O B S V x * t + x 0 - x O ) \beta(t)=arctan2(\textstyle\frac{V_{y}*t+y_{0}-y^{O}BS}{V_{x}*t+x_{0}-x^{O}})
  7. N N
  8. b e t a i beta_{i}
  9. N N
  10. X = [ x 0 y 0 V x V y ] X=[x_{0}y_{0}V_{x}V_{y}]

Tau-leaping.html

  1. x ( t + τ ) = x ( t ) + τ x ( t ) x(t+\tau)=x(t)+\tau x^{\prime}(t)
  2. x ( t + τ ) = x ( t ) + P ( τ x ( t ) ) x(t+\tau)=x(t)+P(\tau x^{\prime}(t))
  3. P ( τ x ( t ) ) P(\tau x^{\prime}(t))
  4. τ x ( t ) \tau x^{\prime}(t)
  5. 𝐱 ( t ) = { X i ( t ) } \mathbf{x}(t)=\{X_{i}(t)\}
  6. E j E_{j}
  7. R j ( 𝐱 ( t ) ) R_{j}(\mathbf{x}(t))
  8. 𝐯 j \mathbf{v}_{j}
  9. i i
  10. j j
  11. 𝐱 ( t 0 ) = { X i ( t 0 ) } \mathbf{x}(t_{0})=\{X_{i}(t_{0})\}
  12. R j ( 𝐱 ( t ) ) R_{j}(\mathbf{x}(t))
  13. τ \tau
  14. E j E_{j}
  15. K j Poisson ( R j τ ) K_{j}\sim\,\text{Poisson}(R_{j}\tau)
  16. [ t , t + τ ) [t,t+\tau)
  17. 𝐱 ( t + τ ) = 𝐱 ( t ) + j K j v i j \mathbf{x}(t+\tau)=\mathbf{x}(t)+\sum_{j}K_{j}v_{ij}
  18. v i j v_{ij}
  19. X i X_{i}
  20. E j E_{j}
  21. K j K_{j}
  22. t 1 t_{1}
  23. R j R_{j}
  24. ϵ \epsilon
  25. ϵ = 0.03 \epsilon=0.03
  26. X i X_{i}
  27. ϵ / g i \epsilon/g_{i}
  28. g i g_{i}
  29. X i X_{i}
  30. g i g_{i}
  31. 2 N 2N
  32. N N
  33. X i X_{i}
  34. R j ( 𝐱 ) R_{j}(\mathbf{x})
  35. X i X_{i}
  36. R j R_{j}
  37. τ \tau
  38. X i X_{i}
  39. μ i ( 𝐱 ) = j v i j R j ( 𝐱 ) \mu_{i}(\mathbf{x})=\sum_{j}v_{ij}R_{j}(\mathbf{x})
  40. σ i 2 ( 𝐱 ) = j v i j 2 R j ( 𝐱 ) \sigma_{i}^{2}(\mathbf{x})=\sum_{j}v_{ij}^{2}R_{j}(\mathbf{x})
  41. X i X_{i}
  42. g i g_{i}
  43. τ \tau
  44. τ = min i { max { ϵ X i / g i , 1 } | μ i ( 𝐱 ) | , max { ϵ X i / g i , 1 } 2 σ i 2 ( 𝐱 ) } \tau=\min_{i}{\left\{\frac{\max{\{\epsilon X_{i}/g_{i},1\}}}{|\mu_{i}(\mathbf{% x})|},\frac{\max{\{\epsilon X_{i}/g_{i},1\}}^{2}}{\sigma_{i}^{2}(\mathbf{x})}% \right\}}
  45. τ \tau
  46. τ \tau

Taylor_Contracts_(economics).html

  1. x t x_{t}
  2. P t = x t + x t - 1 2 P_{t}=\frac{x_{t}+x_{t-1}}{2}
  3. x t * x^{*}_{t}
  4. x t * = P t + γ . Y t x^{*}_{t}=P_{t}+\gamma.Y_{t}
  5. Y t Y_{t}
  6. γ > 0 \gamma>0
  7. γ = 0 \gamma=0
  8. γ > 0 \gamma>0
  9. x t = x t * + E t x t + 1 * 2 x_{t}=\frac{x^{*}_{t}+E_{t}x^{*}_{t+1}}{2}
  10. x t E t x t + 1 * x_{t}E_{t}x^{*}_{t+1}
  11. x t + 1 * x^{*}_{t+1}
  12. M t M_{t}
  13. Y t = M t - P t Y_{t}=M_{t}-P_{t}
  14. x t * x^{*}_{t}
  15. x t = P t + E t P t + 1 + γ . ( Y t + E t Y t + 1 ) 2 x_{t}=\frac{P_{t}+E_{t}P_{t+1}+\gamma.(Y_{t}+E_{t}Y_{t+1})}{2}
  16. Y t Y_{t}
  17. x t = P t ( 1 - γ ) + E t P t + 1 ( 1 - γ ) + γ . ( M t + E t M t + 1 ) 2 x_{t}=\frac{P_{t}(1-\gamma)+E_{t}P_{t+1}(1-\gamma)+\gamma.(M_{t}+E_{t}M_{t+1})% }{2}
  18. x t x_{t}
  19. x t = A . ( x t + E t x t + 1 ) + ( 1 - 2 A ) . M t x_{t}=A.(x_{t}+E_{t}x_{t+1})+(1-2A).M_{t}
  20. A = 1 2 1 - γ 1 + γ A=\frac{1}{2}\frac{1-\gamma}{1+\gamma}
  21. M t = M t - 1 + ϵ t M_{t}=M_{t-1}+\epsilon_{t}
  22. ϵ t \epsilon_{t}
  23. x t = λ . x t - 1 + ( 1 - λ ) m t x_{t}=\lambda.x_{t-1}+(1-\lambda)m_{t}
  24. λ * \lambda^{*}
  25. λ * = 1 - γ 1 + γ \lambda^{*}=\frac{1-\gamma}{1+\gamma}
  26. λ * = 1 \lambda^{*}=1
  27. λ * < 1 \lambda^{*}<1
  28. M t M_{t}
  29. λ * \lambda^{*}
  30. x t - m t = λ . ( x t - 1 - m t ) x_{t}-m_{t}=\lambda.(x_{t-1}-m_{t})
  31. λ * \lambda^{*}
  32. λ * \lambda^{*}
  33. λ * \lambda^{*}

TDP-4-oxo-6-deoxy-alpha-D-glucose-3,4-oxoisomerase_(dTDP-3-dehydro-6-deoxy-alpha-D-galactopyranose-forming).html

  1. \rightleftharpoons

TDP-4-oxo-6-deoxy-alpha-D-glucose-3,4-oxoisomerase_(dTDP-3-dehydro-6-deoxy-alpha-D-glucopyranose-forming).html

  1. \rightleftharpoons

Teapot_Industries.html

  1. 10 10 118 10^{{10}^{118}}

Teiresias_algorithm.html

  1. O ( W L m log m + ( C m + t H ) P m a x r c ( P ) ) O\left(W^{L}m\log m+(C^{m}+t_{H})\sum_{P_{max}}{rc(P)}\right)

Template:A4_honeycombs.html

  1. A ~ 4 {\tilde{A}}_{4}

Template:A8_honeycombs.html

  1. A ~ 8 {\tilde{A}}_{8}

Template:Alcohol_metabolism_formulae.html

  1. H H H H H | | 𝖠𝖣𝖧 | | 𝖠𝖫𝖣𝖧 | H - C - C - O - H H - C - C H - C - C - O - H | | | | H H H O H O \begin{smallmatrix}&\,\text{H}&&\,\text{H}&&&&\,\text{H}&&\,\text{H}&&&\,\text% {H}&&&\\ &|&&|&&\mathsf{ADH}&&|&&|&\mathsf{ALDH}&&|&&&\\ \,\text{H}\,-&\,\text{C}&\!-&\,\text{C}&\!-\,\,\text{O}\,-\,\,\text{H}&% \xrightarrow{\qquad}&\,\text{H}\,-&\,\text{C}&\!-&\,\text{C}&\xrightarrow{% \qquad\ }&\,\text{H}\,-&\,\text{C}&\!-&\,\text{C}&\!-\,\,\text{O}\,-\,\,\text{% H}\\ &|&&|&&&&|&&\|&&&|&&\|&\\ &\,\text{H}&&\,\text{H}&&&&\,\text{H}&&\,\text{O}&&&\,\text{H}&&\,\text{O}&\\ \end{smallmatrix}

Template:Ben_Johnston's_notation.html

  1. 25 24 \frac{25}{24}
  2. 48 25 \frac{48}{25}
  3. 81 80 \frac{81}{80}
  4. 160 81 \frac{160}{81}
  5. 36 35 \frac{36}{35}
  6. 35 18 \frac{35}{18}
  7. 33 32 \frac{33}{32}
  8. 64 33 \frac{64}{33}
  9. 65 64 \frac{65}{64}
  10. 128 65 \frac{128}{65}
  11. 51 50 \frac{51}{50}
  12. 100 51 \frac{100}{51}
  13. 96 95 \frac{96}{95}
  14. 95 48 \frac{95}{48}
  15. 46 45 \frac{46}{45}
  16. 45 23 \frac{45}{23}
  17. 145 144 \frac{145}{144}
  18. 288 145 \frac{288}{145}
  19. 31 30 \frac{31}{30}
  20. 60 31 \frac{60}{31}

Template:Confusion_matrix_terms.html

  1. 𝑇𝑃𝑅 = 𝑇𝑃 / P = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑁 ) \mathit{TPR}=\mathit{TP}/P=\mathit{TP}/(\mathit{TP}+\mathit{FN})
  2. 𝑆𝑃𝐶 = 𝑇𝑁 / N = 𝑇𝑁 / ( 𝐹𝑃 + 𝑇𝑁 ) \mathit{SPC}=\mathit{TN}/N=\mathit{TN}/(\mathit{FP}+\mathit{TN})
  3. 𝑃𝑃𝑉 = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑃 ) \mathit{PPV}=\mathit{TP}/(\mathit{TP}+\mathit{FP})
  4. 𝑁𝑃𝑉 = 𝑇𝑁 / ( 𝑇𝑁 + 𝐹𝑁 ) \mathit{NPV}=\mathit{TN}/(\mathit{TN}+\mathit{FN})
  5. 𝐹𝑃𝑅 = 𝐹𝑃 / N = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑁 ) = 1 - 𝑆𝑃𝐶 \mathit{FPR}=\mathit{FP}/N=\mathit{FP}/(\mathit{FP}+\mathit{TN})=1-\mathit{SPC}
  6. 𝐹𝐷𝑅 = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑃 ) = 1 - 𝑃𝑃𝑉 \mathit{FDR}=\mathit{FP}/(\mathit{FP}+\mathit{TP})=1-\mathit{PPV}
  7. 𝐹𝑁𝑅 = 𝐹𝑁 / P = 𝐹𝑁 / ( 𝐹𝑁 + 𝑇𝑃 ) \mathit{FNR}=\mathit{FN}/P=\mathit{FN}/(\mathit{FN}+\mathit{TP})
  8. 𝐴𝐶𝐶 = ( 𝑇𝑃 + 𝑇𝑁 ) / ( P + N ) \mathit{ACC}=(\mathit{TP}+\mathit{TN})/(P+N)
  9. F1 = 2 𝑇𝑃 / ( 2 𝑇𝑃 + 𝐹𝑃 + 𝐹𝑁 ) \mathit{F1}=2\mathit{TP}/(2\mathit{TP}+\mathit{FP}+\mathit{FN})
  10. T P × T N - F P × F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) \frac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}

Template:D4_honeycombs.html

  1. D ~ 4 {\tilde{D}}_{4}

Template:D5_honeycombs.html

  1. D ~ 5 {\tilde{D}}_{5}

Template:D6_honeycombs.html

  1. D ~ 6 {\tilde{D}}_{6}
  2. B ~ 6 {\tilde{B}}_{6}
  3. C ~ 6 {\tilde{C}}_{6}

Template:D7_honeycombs.html

  1. D ~ 7 {\tilde{D}}_{7}
  2. B ~ 7 {\tilde{B}}_{7}
  3. C ~ 7 {\tilde{C}}_{7}

Template:Did_you_know_nominations::Shuttlecock_(film).html

  1. I n s e r t f o r m u l a h e r e Insertformulahere

Template:Did_you_know_nominations::Water-gas_shift_reaction.html

  1. log K eq = - 2.4198 + 0.0003855 T + 2180.6 T \log K_{\mathrm{eq}}=-2.4198+0.0003855T+\frac{2180.6}{T}
  2. log 10 K equil = 2408.1 T + 1.5350 × log 10 T - 7.452 × 10 - 5 ( T ) - 6.7753 \log_{10}K_{\mathrm{equil}}=\frac{2408.1}{T}+1.5350\times\log_{10}T-7.452% \times 10^{-5}(T)-6.7753

Template:Dynkin::testcases.html

  1. [ 2 a 12 a 21 2 ] \left[\begin{matrix}2&a_{12}\\ a_{21}&2\end{matrix}\right]
  2. [ 2 0 0 2 ] \left[\begin{smallmatrix}2&0\\ 0&2\end{smallmatrix}\right]
  3. [ 2 - 1 - 1 2 ] \left[\begin{smallmatrix}2&-1\\ -1&2\end{smallmatrix}\right]
  4. [ 2 - 2 - 1 2 ] \left[\begin{smallmatrix}2&-2\\ -1&2\end{smallmatrix}\right]
  5. A 3 {A}_{3}
  6. [ 2 - 1 - 2 2 ] \left[\begin{smallmatrix}2&-1\\ -2&2\end{smallmatrix}\right]
  7. A 3 {A}_{3}
  8. [ 2 - 1 - 3 2 ] \left[\begin{smallmatrix}2&-1\\ -3&2\end{smallmatrix}\right]
  9. D 4 {D}_{4}
  10. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-2\\ -2&2\end{smallmatrix}\right]
  11. A ~ 3 {\tilde{A}}_{3}
  12. [ 2 - 1 - 4 2 ] \left[\begin{smallmatrix}2&-1\\ -4&2\end{smallmatrix}\right]
  13. D ~ 4 {\tilde{D}}_{4}
  14. [ 2 - 1 - 5 2 ] \left[\begin{smallmatrix}2&-1\\ -5&2\end{smallmatrix}\right]
  15. [ 2 - b - a 2 ] \left[\begin{smallmatrix}2&-b\\ -a&2\end{smallmatrix}\right]
  16. [ 2 a 12 a 21 2 ] \left[\begin{matrix}2&a_{12}\\ a_{21}&2\end{matrix}\right]
  17. [ 2 0 0 2 ] \left[\begin{smallmatrix}2&0\\ 0&2\end{smallmatrix}\right]
  18. [ 2 - 1 - 1 2 ] \left[\begin{smallmatrix}2&-1\\ -1&2\end{smallmatrix}\right]
  19. [ 2 - 2 - 1 2 ] \left[\begin{smallmatrix}2&-2\\ -1&2\end{smallmatrix}\right]
  20. A 3 {A}_{3}
  21. [ 2 - 1 - 2 2 ] \left[\begin{smallmatrix}2&-1\\ -2&2\end{smallmatrix}\right]
  22. A 3 {A}_{3}
  23. [ 2 - 1 - 3 2 ] \left[\begin{smallmatrix}2&-1\\ -3&2\end{smallmatrix}\right]
  24. D 4 {D}_{4}
  25. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-2\\ -2&2\end{smallmatrix}\right]
  26. A ~ 3 {\tilde{A}}_{3}
  27. [ 2 - 1 - 4 2 ] \left[\begin{smallmatrix}2&-1\\ -4&2\end{smallmatrix}\right]
  28. D ~ 4 {\tilde{D}}_{4}
  29. [ 2 - 1 - 5 2 ] \left[\begin{smallmatrix}2&-1\\ -5&2\end{smallmatrix}\right]
  30. [ 2 - b - a 2 ] \left[\begin{smallmatrix}2&-b\\ -a&2\end{smallmatrix}\right]

Template:Infinity_symbol.html

  1. \infty

Template:Infobox_air_density.html

  1. ρ = p R specific T \rho=\frac{p}{R_{\rm specific}T}
  2. ρ = M air p R T \rho=\frac{M_{\rm air}p}{RT}

Template:Infobox_property::doc.html

  1. ρ = M air V m \rho=\frac{M_{\rm air}}{V_{\rm m}}
  2. ρ = p R specific T \rho=\frac{p}{R_{\rm specific}T}
  3. ρ = M air p R T \rho=\frac{M_{\rm air}p}{RT}

Template:My_Second_Sandbox.html

  1. % G e n e r a l t r e e n o t a t i o n 1GeneraltreenotationFigure1Generaltreenotation% scale of levels Level nnn00111222333444555666777888999101010% vertices % directed edges % labeled vertices and branches root RRRleafcapable vertexparent π⁢(V)πV\pi(V)ππ\pidirected edgedescendant VVVbranch ℬ⁢(1)ℬ1\mathcal{B}(1)depth 222three siblingsbifurcation:ℬ1⁢(3)subscriptℬ13\mathcal{B}_{1}(3), regularedge of depth 222ℬ2⁢(3)subscriptℬ23\mathcal{B}_{2}(3), irregular% period and period length periodic root PPPℬ⁢(5)ℬ5\mathcal{B}(5)ℬ⁢(6)ℬ6\mathcal{B}(6)ℬ⁢(7)ℬ7\mathcal{B}(7)ℬ⁢(8)ℬ8\mathcal{B}(8)period length 222% infinite main line infinitemainlinetree 𝒯⁢(R)𝒯R\mathcal{T}(R) \%Generaltreenotation\begin{figure}[ht]\@@toccaption{{\@tag[][]{1}% Generaltreenotation}}\@@caption{{\@tag[][:]{Figure~1}Generaltreenotation}}% \begin{picture}(10.0,12.0)(-5.0,-11.0)\% scale of levels \put(-4.0,0.5){% \makebox(0.0,0.0)[cb]{Level \(n\)}} \put(-4.0,0.0){\line(0,-1){10.0}} \put(-4.% 1,0.0){\line(1,0){0.2}}\put(-4.1,-1.0){\line(1,0){0.2}}\put(-4.1,-2.0){\line(1% ,0){0.2}}\put(-4.1,-3.0){\line(1,0){0.2}}\put(-4.1,-4.0){\line(1,0){0.2}}\put(% -4.1,-5.0){\line(1,0){0.2}}\put(-4.1,-6.0){\line(1,0){0.2}}\put(-4.1,-7.0){% \line(1,0){0.2}}\put(-4.1,-8.0){\line(1,0){0.2}}\put(-4.1,-9.0){\line(1,0){0.2% }}\put(-4.1,-10.0){\line(1,0){0.2}} \put(-4.2,0.0){\makebox(0.0,0.0)[rc]{\(0\)% }} \put(-4.2,-1.0){\makebox(0.0,0.0)[rc]{\(1\)}} \put(-4.2,-2.0){\makebox(0.0,% 0.0)[rc]{\(2\)}} \put(-4.2,-3.0){\makebox(0.0,0.0)[rc]{\(3\)}} \put(-4.2,-4.0)% {\makebox(0.0,0.0)[rc]{\(4\)}} \put(-4.2,-5.0){\makebox(0.0,0.0)[rc]{\(5\)}} % \put(-4.2,-6.0){\makebox(0.0,0.0)[rc]{\(6\)}} \put(-4.2,-7.0){\makebox(0.0,0.0% )[rc]{\(7\)}} \put(-4.2,-8.0){\makebox(0.0,0.0)[rc]{\(8\)}} \put(-4.2,-9.0){% \makebox(0.0,0.0)[rc]{\(9\)}} \put(-4.2,-10.0){\makebox(0.0,0.0)[rc]{\(10\)}} % \put(-4.0,-10.0){\vector(0,-1){1.0}} \% vertices \put(0.0,0.0){\circle*{0.2}} % \put(0.0,-1.0){\circle*{0.2}}\put(0.0,-2.0){\circle*{0.2}}\put(0.0,-3.0){% \circle*{0.2}}\put(0.0,-4.0){\circle*{0.2}}\put(0.0,-5.0){\circle*{0.2}}\put(0% .0,-6.0){\circle*{0.2}}\put(0.0,-7.0){\circle*{0.2}}\put(0.0,-8.0){\circle*{0.% 2}}\put(0.0,-9.0){\circle*{0.2}} \put(-1.0,-1.0){\circle*{0.2}} \put(-3.0,-2.0% ){\circle*{0.2}}\put(2.0,-2.0){\circle*{0.2}} \put(-3.0,-3.0){\circle*{0.2}} % \put(1.0,-3.0){\circle*{0.2}}\put(2.0,-3.0){\circle*{0.2}}\put(3.0,-3.0){% \circle*{0.2}} \put(-1.0,-4.0){\circle*{0.2}} \put(2.0,-5.0){\circle*{0.2}} % \put(-1.0,-6.0){\circle*{0.2}}\put(-1.0,-7.0){\circle*{0.2}}\put(-1.0,-8.0){% \circle*{0.2}}\put(-1.0,-9.0){\circle*{0.2}} \put(-2.0,-6.0){\circle*{0.2}}% \put(-2.0,-7.0){\circle*{0.2}}\put(-2.0,-8.0){\circle*{0.2}}\put(-2.0,-9.0){% \circle*{0.2}} \put(-3.0,-6.0){\circle*{0.2}}\put(-3.0,-8.0){\circle*{0.2}} % \put(1.0,-6.0){\circle*{0.2}}\put(1.0,-7.0){\circle*{0.2}}\put(1.0,-8.0){% \circle*{0.2}}\put(1.0,-9.0){\circle*{0.2}} \put(2.0,-6.0){\circle*{0.2}}\put(% 2.0,-7.0){\circle*{0.2}}\put(2.0,-8.0){\circle*{0.2}}\put(2.0,-9.0){\circle*{0% .2}} \put(3.0,-6.0){\circle*{0.2}}\put(3.0,-7.0){\circle*{0.2}}\put(3.0,-8.0){% \circle*{0.2}}\put(3.0,-9.0){\circle*{0.2}} \% directed edges \put(0.0,0.0){% \line(0,-1){1.0}}\put(0.0,-1.0){\line(0,-1){1.0}}\put(0.0,-2.0){\line(0,-1){1.% 0}}\put(0.0,-3.0){\line(0,-1){1.0}}\put(0.0,-4.0){\line(0,-1){1.0}}\put(0.0,-5% .0){\line(0,-1){1.0}}\put(0.0,-6.0){\line(0,-1){1.0}}\put(0.0,-7.0){\line(0,-1% ){1.0}}\put(0.0,-8.0){\line(0,-1){1.0}} \put(0.0,0.0){\line(-1,-1){1.0}} \put(% 0.0,-1.0){\line(-3,-1){3.0}} \put(-3.0,-3.0){\vector(0,1){0.9}} \put(0.0,-1.0)% {\line(2,-1){2.0}} \put(2.0,-2.0){\line(-1,-1){1.0}} \put(2.0,-2.0){\line(0,-1% ){1.0}} \put(2.0,-2.0){\line(1,-1){1.0}} \put(0.0,-3.0){\line(-1,-1){1.0}} % \put(0.0,-3.0){\line(1,-1){2.0}} \put(0.0,-5.0){\line(-1,-1){1.0}}\put(0.0,-6.% 0){\line(-1,-1){1.0}}\put(0.0,-7.0){\line(-1,-1){1.0}}\put(0.0,-8.0){\line(-1,% -1){1.0}} \put(0.0,-5.0){\line(-2,-1){2.0}}\put(0.0,-6.0){\line(-2,-1){2.0}}% \put(0.0,-7.0){\line(-2,-1){2.0}}\put(0.0,-8.0){\line(-2,-1){2.0}} \put(0.0,-5% .0){\line(-3,-1){3.0}}\put(0.0,-7.0){\line(-3,-1){3.0}} \put(0.0,-5.0){\line(1% ,-1){1.0}}\put(0.0,-6.0){\line(1,-1){1.0}}\put(0.0,-7.0){\line(1,-1){1.0}}\put% (0.0,-8.0){\line(1,-1){1.0}} \put(0.0,-5.0){\line(2,-1){2.0}}\put(0.0,-6.0){% \line(2,-1){2.0}}\put(0.0,-7.0){\line(2,-1){2.0}}\put(0.0,-8.0){\line(2,-1){2.% 0}} \put(0.0,-5.0){\line(3,-1){3.0}}\put(0.0,-6.0){\line(3,-1){3.0}}\put(0.0,-% 7.0){\line(3,-1){3.0}}\put(0.0,-8.0){\line(3,-1){3.0}} \% labeled vertices and% branches \put(-0.2,0.0){\makebox(0.0,0.0)[rc]{root \(R\)}} \put(-1.2,-1.0){% \makebox(0.0,0.0)[rc]{leaf}} \put(0.2,-1.0){\makebox(0.0,0.0)[lc]{capable % vertex}} \put(-2.7,-2.0){\makebox(0.0,0.0)[lc]{parent \(\pi(V)\)}} \put(-3.1,-% 2.5){\makebox(0.0,0.0)[rc]{\(\pi\)}} \put(-2.9,-2.5){\makebox(0.0,0.0)[lc]{% directed edge}} \put(-2.7,-3.0){\makebox(0.0,0.0)[lc]{descendant \(V\)}} \put(% 1.2,-1.5){\makebox(0.0,0.0)[lc]{branch \(\mathcal{B}(1)\)}} \put(3.5,-2.0){% \vector(0,1){1.0}} \put(3.7,-2.0){\makebox(0.0,0.0)[lc]{depth \(2\)}} \put(3.5% ,-2.0){\vector(0,-1){1.0}} \put(2.0,-3.2){\makebox(0.0,0.0)[ct]{three siblings% }} \put(-0.7,-3.5){\makebox(0.0,0.0)[rc]{bifurcation:}} \put(-1.2,-4.0){% \makebox(0.0,0.0)[rc]{\(\mathcal{B}_{1}(3)\), regular}} \put(1.2,-4.0){% \makebox(0.0,0.0)[lc]{edge of depth \(2\)}} \put(2.2,-4.7){\makebox(0.0,0.0)[% lc]{\(\mathcal{B}_{2}(3)\), irregular}} \% period and period length \put(-0.2,% -5.0){\makebox(0.0,0.0)[rc]{periodic root \(P\)}} \put(2.2,-5.5){\makebox(0.0,% 0.0)[lc]{\(\mathcal{B}(5)\)}} \put(2.2,-6.5){\makebox(0.0,0.0)[lc]{\(\mathcal{% B}(6)\)}} \put(2.2,-7.5){\makebox(0.0,0.0)[lc]{\(\mathcal{B}(7)\)}} \put(2.2,-% 8.5){\makebox(0.0,0.0)[lc]{\(\mathcal{B}(8)\)}} \put(3.5,-7.0){\vector(0,1){1.% 0}} \put(3.7,-7.0){\makebox(0.0,0.0)[lc]{period length \(2\)}} \put(3.5,-7.0){% \vector(0,-1){1.0}} \% infinite main line \put(0.0,-9.0){\vector(0,-1){2.0}} % \put(0.2,-10.0){\makebox(0.0,0.0)[lc]{infinite}} \put(0.2,-10.5){\makebox(0.0,% 0.0)[lc]{mainline}} \put(-0.2,-10.5){\makebox(0.0,0.0)[rc]{tree \(\mathcal{T}(% R)\)}} \end{picture}\end{figure}

Template:POTD_protected::2015-04-27.html

  1. | |

Template:Quasiregular_polychora_and_honeycombs.html

  1. { 3 , 3 3 } \left\{3,{3\atop 3}\right\}
  2. { 3 , 3 4 } \left\{3,{3\atop 4}\right\}
  3. { 3 , 3 5 } \left\{3,{3\atop 5}\right\}
  4. { 3 , 3 6 } \left\{3,{3\atop 6}\right\}
  5. { 4 , 3 4 } \left\{4,{3\atop 4}\right\}
  6. { 4 , 4 4 } \left\{4,{4\atop 4}\right\}

Template:Regular_and_Quasiregular_honeycombs.html

  1. { 3 , 3 3 } \left\{3,{3\atop 3}\right\}
  2. { 4 , 3 3 } \left\{4,{3\atop 3}\right\}
  3. { 5 , 3 3 } \left\{5,{3\atop 3}\right\}
  4. { 6 , 3 3 } \left\{6,{3\atop 3}\right\}

Template:Turbulence_and_Heat_Flux_modeling_for_Turbomachinery_Flows.html

  1. u y \frac{\partial u}{\partial y}

Temporal_discretization.html

  1. φ t ( x , t ) = F ( φ ) . \frac{\partial\varphi}{\partial t}(x,t)=F(\varphi).~{}
  2. φ n + 1 - φ n Δ t = F ( φ ) , \frac{\varphi^{n+1}-\varphi^{n}}{\Delta t}=F(\varphi),
  3. 3 φ n + 1 - 4 φ n + φ n - 1 2 Δ t = F ( φ ) , \frac{3\varphi^{n+1}-4\varphi^{n}+\varphi^{n-1}}{2\Delta t}=F(\varphi),
  4. φ \varphi
  5. φ n + 1 - φ n Δ t = f . F ( φ n + 1 ) + ( 1 - f ) . F ( φ n ) , \frac{\varphi^{n+1}-\varphi^{n}}{\Delta t}=f.F(\varphi^{n+1})+(1-f).F(\varphi^% {n}),
  6. t t + Δ t F ( φ ) d t = [ f . F φ t + Δ t + ( 1 - f ) . F φ t ] Δ t \int\limits_{t}^{t+\Delta t}F(\varphi)dt=[f.F_{\varphi}^{t+\Delta t}+(1-f).F_{% \varphi}^{t}]\Delta t
  7. φ \varphi
  8. φ \varphi
  9. φ \varphi
  10. φ n + 1 - φ n Δ t = F ( φ n + 1 ) , \frac{\varphi^{n+1}-\varphi^{n}}{\Delta t}=F(\varphi^{n+1}),
  11. φ n + 1 = φ n + Δ t F ( φ n + 1 ) , \varphi^{n+1}=\varphi^{n}+\Delta tF(\varphi^{n+1}),
  12. φ \varphi
  13. φ n + 1 - φ n Δ t = F ( φ n ) , \frac{\varphi^{n+1}-\varphi^{n}}{\Delta t}=F(\varphi^{n}),
  14. φ n + 1 = φ n + Δ t F ( φ n ) , \varphi^{n+1}=\varphi^{n}+\Delta tF(\varphi^{n}),

Tempotron.html

  1. V ( t ) V(t)
  2. V ( t ) = i ω i t i K ( t - t i ) + V r e s t , V(t)=\sum_{i}\omega_{i}\sum_{t_{i}}K(t-t_{i})+V_{rest},
  3. t i t_{i}
  4. ω i \omega_{i}
  5. V r e s t V_{rest}
  6. K ( t - t i ) K(t-t_{i})
  7. K ( t - t i ) = { V 0 [ e x p ( - ( t - t i ) / τ ) - e x p ( - ( t - t i ) / τ s ) ] t t i 0 t < t i K(t-t_{i})=\begin{cases}V_{0}[exp(-(t-t_{i})/\tau)-exp(-(t-t_{i})/\tau_{s})]&t% \geq t_{i}\\ 0&t<t_{i}\end{cases}
  8. τ \tau
  9. τ s \tau_{s}
  10. V 0 V_{0}
  11. V t h V_{th}
  12. \circ
  13. \bullet
  14. ω i \omega_{i}
  15. \circ
  16. Δ ω i \Delta\omega_{i}
  17. \bullet
  18. - Δ ω i -\Delta\omega_{i}
  19. Δ ω i = λ t i < t m a x K ( t m a x - t i ) . \Delta\omega_{i}=\lambda\sum_{t_{i}<t_{max}}K(t_{max}-t_{i}).
  20. t m a x t_{max}
  21. V ( t ) V(t)

Tensor_operator.html

  1. U [ R ( θ , 𝐧 ^ ) ] = exp ( - i θ 𝐧 ^ 𝐉 ^ ) U[R(\theta,\hat{\mathbf{n}})]=\exp\left(-\frac{i\theta}{\hbar}\hat{\mathbf{n}}% \cdot\widehat{\mathbf{J}}\right)
  2. J x = ( 0 0 0 0 0 - i 0 i 0 ) J y = ( 0 0 i 0 0 0 - i 0 0 ) J z = ( 0 - i 0 i 0 0 0 0 0 ) J_{x}=\begin{pmatrix}0&0&0\\ 0&0&-i\\ 0&i&0\end{pmatrix}\,\quad J_{y}=\begin{pmatrix}0&0&i\\ 0&0&0\\ -i&0&0\end{pmatrix}\,\quad J_{z}=\begin{pmatrix}0&-i&0\\ i&0&0\\ 0&0&0\end{pmatrix}
  3. R ^ = R ^ ( θ , 𝐧 ^ ) \widehat{R}=\widehat{R}(\theta,\hat{\mathbf{n}})
  4. U [ R ( θ , 𝐧 ^ ) ] = 1 - i θ 𝐧 ^ 𝐉 ^ U[R(\theta,\hat{\mathbf{n}})]=1-\frac{i\theta}{\hbar}\hat{\mathbf{n}}\cdot% \widehat{\mathbf{J}}
  5. Ω ^ \widehat{\Omega}
  6. Ω ^ = U Ω ^ U \widehat{\Omega}={U}^{\dagger}\widehat{\Omega}U
  7. U ^ ( R ) \widehat{U}(R)
  8. Ω ^ = U ( R ) Ω ^ U ( R ) = exp ( i θ 𝐧 ^ 𝐉 ^ ) Ω ^ exp ( - i θ 𝐧 ^ 𝐉 ^ ) \widehat{\Omega}={U(R)}^{\dagger}\widehat{\Omega}U(R)=\exp\left(\frac{i\theta}% {\hbar}\hat{\mathbf{n}}\cdot\widehat{\mathbf{J}}\right)\widehat{\Omega}\exp% \left(-\frac{i\theta}{\hbar}\hat{\mathbf{n}}\cdot\widehat{\mathbf{J}}\right)
  9. | j , m |j,m\rangle
  10. | ψ = m | j , m |\psi\rangle=\sum_{m}|j,m\rangle
  11. | j , m |j,m\rangle
  12. | ψ ¯ = U ( R ) | ψ |\bar{\psi}\rangle=U(R)|\psi\rangle
  13. I = m | j , m j , m | I=\sum_{m^{\prime}}|j,m^{\prime}\rangle\langle j,m^{\prime}|
  14. | ψ ¯ = I U ( R ) | ψ = m m | j , m j , m | U ( R ) | j , m |\bar{\psi}\rangle=IU(R)|\psi\rangle=\sum_{mm^{\prime}}|j,m^{\prime}\rangle% \langle j,m^{\prime}|U(R)|j,m\rangle
  15. D ( R ) m m ( j ) = j , m | U ( R ) | j , m {D(R)}^{(j)}_{m^{\prime}m}=\langle j,m^{\prime}|U(R)|j,m\rangle
  16. | ψ ¯ = m m D m m ( j ) | j , m | ψ ¯ = D ( j ) | ψ |\bar{\psi}\rangle=\sum_{mm^{\prime}}D^{(j)}_{m^{\prime}m}|j,m\rangle\quad% \Rightarrow\quad|\bar{\psi}\rangle=D^{(j)}|\psi\rangle
  17. | j , m ¯ = m D ( R ) m m ( j ) | j , m |\overline{j,m}\rangle=\sum_{m^{\prime}}{D(R)}^{(j)}_{m^{\prime}m}|j,m^{\prime}\rangle
  18. | l , m |l,m\rangle
  19. Y m ( θ , φ ) = θ , ϕ | , m = ( 2 + 1 ) 4 π ( - m ) ! ( + m ) ! P m ( cos θ ) e i m ϕ Y_{\ell}^{m}(\theta,\varphi)=\langle\theta,\phi|\ell,m\rangle=\sqrt{{(2\ell+1)% \over 4\pi}{(\ell-m)!\over(\ell+m)!}}\,P_{\ell}^{m}(\cos{\theta})\,e^{im\phi}
  20. 𝐧 ^ ( θ , ϕ ) = cos ϕ sin θ 𝐞 x + sin ϕ sin θ 𝐞 y + cos θ 𝐞 z \hat{\mathbf{n}}(\theta,\phi)=\cos\phi\sin\theta\mathbf{e}_{x}+\sin\phi\sin% \theta\mathbf{e}_{y}+\cos\theta\mathbf{e}_{z}
  21. Y l m = 𝐧 | l m Y_{l}^{m}=\langle\mathbf{n}|lm\rangle
  22. | m , l |m,l\rangle
  23. U ^ ( R ) \widehat{U}(R)
  24. | , m ¯ = m D m m ( ) U ( R - 1 ) | , m , | 𝐧 ^ ¯ = U ( R ) | 𝐧 ^ |\overline{\ell,m}\rangle=\sum_{m^{\prime}}D_{m^{\prime}m}^{(\ell)}U(R^{-1})|% \ell,m^{\prime}\rangle\,,\quad|\overline{\hat{\mathbf{n}}}\rangle=U(R)|\hat{% \mathbf{n}}\rangle
  25. 𝐀 ^ \widehat{\mathbf{A}}
  26. ψ | A ^ | ψ = ψ | A ^ | ψ \langle\psi^{\prime}|\widehat{A^{\prime}}|\psi^{\prime}\rangle=\langle\psi|% \widehat{A}|\psi\rangle
  27. | ψ |\psi\rangle
  28. | ψ = U ( R ) | ψ , ψ | |\psi^{\prime}\rangle=U(R)|\psi\rangle\,,\quad\langle\psi|
  29. ψ | = ψ | U ^ ( R ) \langle\psi^{\prime}|=\langle\psi|\widehat{U}^{\dagger}(R)
  30. ψ | U ( R ) A ^ U ( R ) | ψ = ψ | A ^ | ψ \langle\psi|U^{\dagger}(R)\widehat{A}^{\prime}U(R)|\psi\rangle=\langle\psi|% \widehat{A}|\psi\rangle
  31. | ψ |\psi\rangle
  32. U ( R ) A ^ U ( R ) = A ^ U^{\dagger}(R)\widehat{A}^{\prime}U(R)=\widehat{A}
  33. U ( R ) S ^ U ( R ) = S ^ U(R)^{\dagger}\widehat{S}U(R)=\widehat{S}
  34. [ S ^ , 𝐉 ^ ] = 0 \left[\widehat{S},\widehat{\mathbf{J}}\right]=0
  35. E ^ ψ = i t ψ \widehat{E}\psi=i\hbar\frac{\partial}{\partial t}\psi
  36. V ^ ( 𝐫 , t ) ψ = V ( 𝐫 , t ) ψ \widehat{V}(\mathbf{r},t)\psi=V(\mathbf{r},t)\psi
  37. T ^ ( 𝐫 , t ) ψ = - 2 2 m 2 ( 𝐫 , t ) \widehat{T}(\mathbf{r},t)\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}(\mathbf{r},t)
  38. 𝐋 ^ 𝐒 ^ = L ^ x S ^ x + L ^ y S ^ y + L ^ z S ^ z . \widehat{\mathbf{L}}\cdot\widehat{\mathbf{S}}=\widehat{L}_{x}\widehat{S}_{x}+% \widehat{L}_{y}\widehat{S}_{y}+\widehat{L}_{z}\widehat{S}_{z}\,.
  39. U ( R ) V ^ i U ( R ) = j R i j V ^ j {U(R)}^{\dagger}\widehat{V}_{i}U(R)=\sum_{j}R_{ij}\widehat{V}_{j}
  40. [ V ^ a , J ^ b ] = i ε a b c V ^ c \left[\widehat{V}_{a},\widehat{J}_{b}\right]=i\hbar\varepsilon_{abc}\widehat{V% }_{c}
  41. 𝐫 ^ ψ = 𝐫 ψ \widehat{\mathbf{r}}\psi=\mathbf{r}\psi
  42. 𝐩 ^ ψ = - i ψ \widehat{\mathbf{p}}\psi=-i\hbar\nabla\psi
  43. 𝐋 ^ ψ = - i 𝐫 × ψ \widehat{\mathbf{L}}\psi=-i\hbar\mathbf{r}\times\nabla\psi
  44. 𝐉 ^ = 𝐋 ^ + 𝐒 ^ . \widehat{\mathbf{J}}=\widehat{\mathbf{L}}+\widehat{\mathbf{S}}\,.
  45. ψ ¯ | V ^ a | ψ ¯ = ψ | U ( R ) V ^ a U ( R ) | ψ = b R a b ψ | V ^ b | ψ \langle\bar{\psi}|\widehat{V}_{a}|\bar{\psi}\rangle=\langle\psi|{U(R)}^{% \dagger}\widehat{V}_{a}U(R)|\psi\rangle=\sum_{b}R_{ab}\langle\psi|\widehat{V}_% {b}|\psi\rangle
  46. U ( R ) V ^ a U ( R ) = b R a b V ^ b {U(R)}^{\dagger}\widehat{V}_{a}U(R)=\sum_{b}R_{ab}\widehat{V}_{b}
  47. | ψ |ψ\rangle
  48. V + 1 = - 1 2 ( V x + i V y ) V - 1 = 1 2 ( V x - i V y ) , V 0 = V z , V_{+1}=-\frac{1}{\sqrt{2}}(V_{x}+iV_{y})\,\quad V_{-1}=\frac{1}{\sqrt{2}}(V_{x% }-iV_{y})\,,\quad V_{0}=V_{z}\,,
  49. [ J z , V q ] = q V q \left[J_{z},V_{q}\right]=qV_{q}
  50. [ J ± , V 0 ] = 2 V ± \left[J_{\pm},V_{0}\right]=\sqrt{2}V_{\pm}
  51. [ J ± , V ] = 2 V 0 \left[J_{\pm},V_{\mp}\right]=\sqrt{2}V_{0}
  52. [ J ± , V ± ] = 0 \left[J_{\pm},V_{\pm}\right]=0
  53. J ± = J x ± i J y , J_{\pm}=J_{x}\pm iJ_{y}\,,
  54. J ± 1 = 1 2 J ± , J 0 = J z J_{\pm 1}=\mp\frac{1}{\sqrt{2}}J_{\pm}\,,\quad J_{0}=J_{z}
  55. U ( R ) V ^ q U ( R ) = q D ( R ) q q ( 1 ) V ^ q {U(R)}^{\dagger}\widehat{V}_{q}U(R)=\sum_{q^{\prime}}{D(R)}^{(1)}_{qq^{\prime}% }\widehat{V}_{q^{\prime}}
  56. U ( R ) T ^ p q r U ( R ) = R p i R q j R r k T ^ i j k U(R)^{\dagger}\widehat{T}_{pqr\cdots}U(R)=R_{pi}R_{qj}R_{rk}\cdots\widehat{T}_% {ijk\cdots}
  57. U ( R ) T ^ p q U ( R ) = R p i R q j T ^ i j = R p i a ^ i R q j b ^ j U(R)^{\dagger}\widehat{T}_{pq}U(R)=R_{pi}R_{qj}\widehat{T}_{ij}=R_{pi}\widehat% {a}_{i}R_{qj}\widehat{b}_{j}
  58. 𝐓 ^ = 𝐞 i a ^ i 𝐞 j b ^ j = 𝐞 i 𝐞 j a ^ i b ^ j \hat{\mathbf{T}}=\mathbf{e}_{i}\widehat{a}_{i}\otimes\mathbf{e}_{j}\widehat{b}% _{j}=\mathbf{e}_{i}\otimes\mathbf{e}_{j}\widehat{a}_{i}\widehat{b}_{j}
  59. 𝐚 ^ = 𝐞 i a ^ i , 𝐛 ^ = 𝐞 j b ^ j \hat{\mathbf{a}}=\mathbf{e}_{i}\widehat{a}_{i}\,,\quad\hat{\mathbf{b}}=\mathbf% {e}_{j}\widehat{b}_{j}
  60. 𝐓 = 𝐓 ( 1 ) + 𝐓 ( 2 ) + 𝐓 ( 3 ) \mathbf{T}=\mathbf{T}^{(1)}+\mathbf{T}^{(2)}+\mathbf{T}^{(3)}
  61. T ^ i j ( 1 ) = a ^ k b ^ k 3 δ i j \widehat{T}^{(1)}_{ij}=\frac{\widehat{a}_{k}\widehat{b}_{k}}{3}\delta_{ij}
  62. T ^ i j ( 2 ) = 1 2 [ a ^ i b ^ j - a ^ j b ^ i ] = a ^ [ i b ^ j ] \widehat{T}^{(2)}_{ij}=\frac{1}{2}[\widehat{a}_{i}\widehat{b}_{j}-\widehat{a}_% {j}\widehat{b}_{i}]=\widehat{a}_{[i}\widehat{b}_{j]}
  63. T ^ i j ( 3 ) = 1 2 ( a ^ i b ^ j + a ^ j b ^ i ) - a ^ k b ^ k 3 δ i j = a ^ ( i b ^ j ) - T i j ( 1 ) \widehat{T}^{(3)}_{ij}=\frac{1}{2}(\widehat{a}_{i}\widehat{b}_{j}+\widehat{a}_% {j}\widehat{b}_{i})-\frac{\widehat{a}_{k}\widehat{b}_{k}}{3}\delta_{ij}=% \widehat{a}_{(i}\widehat{b}_{j)}-T^{(1)}_{ij}
  64. Q i j = α q α ( 3 r α i r α j - r α 2 δ i j ) Q_{ij}=\sum_{\alpha}q_{\alpha}(3r_{\alpha i}r_{\alpha j}-r_{\alpha}^{2}\delta_% {ij})
  65. T i j = V i W j T_{ij}=V_{i}W_{j}
  66. T i 1 i 2 j 1 j 2 = V i 1 i 2 W j 1 j 2 T_{i_{1}i_{2}\cdots j_{1}j_{2}\cdots}=V_{i_{1}i_{2}\cdots}W_{j_{1}j_{2}\cdots}
  67. T ^ ± 2 ( 2 ) = a ^ ± 1 b ^ ± 1 \widehat{T}^{(2)}_{\pm 2}=\widehat{a}_{\pm 1}\widehat{b}_{\pm 1}
  68. T ^ ± 1 ( 2 ) = 1 2 ( a ^ ± 1 b ^ 0 + a ^ 0 b ^ ± 1 ) \widehat{T}^{(2)}_{\pm 1}=\frac{1}{\sqrt{2}}\left(\widehat{a}_{\pm 1}\widehat{% b}_{0}+\widehat{a}_{0}\widehat{b}_{\pm 1}\right)
  69. T ^ 0 ( 2 ) = 1 6 ( a ^ + 1 b ^ - 1 + a ^ - 1 b ^ + 1 + 2 a ^ 0 b ^ 0 ) \widehat{T}^{(2)}_{0}=\frac{1}{\sqrt{6}}\left(\widehat{a}_{+1}\widehat{b}_{-1}% +\widehat{a}_{-1}\widehat{b}_{+1}+2\widehat{a}_{0}\widehat{b}_{0}\right)
  70. [ J a , T ^ q ( 2 ) ] = q D ( J ) q q ( 2 ) T ^ q ( 2 ) \left[J_{a},\widehat{T}^{(2)}_{q}\right]=\sum_{q^{\prime}}{D(J)}^{(2)}_{qq^{% \prime}}\widehat{T}_{q^{\prime}}^{(2)}
  71. U ( R ) T ^ q ( 2 ) U ( R ) = q D ( R ) q q ( 2 ) T ^ q ( 2 ) {U(R)}^{\dagger}\widehat{T}^{(2)}_{q}U(R)=\sum_{q^{\prime}}{D(R)}^{(2)}_{qq^{% \prime}}\widehat{T}_{q^{\prime}}^{(2)}
  72. U ( R ) T ^ q ( k ) U ( R ) = q D q q ( k ) T ^ q ( k ) {U(R)}^{\dagger}\widehat{T}_{q}^{(k)}U(R)=\sum_{q^{\prime}}D^{(k)}_{qq^{\prime% }}\widehat{T}_{q^{\prime}}^{(k)}
  73. [ J ± , T ^ k q ] = ( k q ) ( k ± q + 1 ) T ^ k q ± 1 \left[J_{\pm},\widehat{T}_{k}^{q}\right]=\hbar\sqrt{(k\mp q)(k\pm q+1)}% \widehat{T}_{k}^{q\pm 1}
  74. [ J z , T ^ k q ] = q T ^ k q \left[J_{z},\widehat{T}_{k}^{q}\right]=\hbar q\widehat{T}_{k}^{q}
  75. 𝐚 = a x 𝐞 x + a y 𝐞 y + a z 𝐞 z \mathbf{a}=a_{x}\mathbf{e}_{x}+a_{y}\mathbf{e}_{y}+a_{z}\mathbf{e}_{z}
  76. T q ( k ) = Y = k m = q ( 𝐚 ) = 𝐚 | k , q T_{q}^{(k)}=Y_{\ell=k}^{m=q}(\mathbf{a})=\langle\mathbf{a}|k,q\rangle
  77. T q ( k ) = q k 1 , k 2 , q 1 , q 2 | k , q , q 1 , q 2 A q 1 ( k 1 ) B q 2 ( k 2 ) T_{q}^{(k)}=\sum_{q}\langle k_{1},k_{2},q_{1},q_{2}|k,q,q_{1},q_{2}\rangle A_{% q_{1}}^{(k_{1})}B_{q_{2}}^{(k_{2})}
  78. L ± = L x ± i L y L_{\pm}=L_{x}\pm iL_{y}
  79. S ± = S x ± i S y S_{\pm}=S_{x}\pm iS_{y}
  80. ψ n l m ( r , θ , ϕ ) = R n l ( r ) Y l m ( θ , ϕ ) \psi_{nlm}(r,\theta,\phi)=R_{nl}(r)Y_{lm}(\theta,\phi)
  81. n l m | 𝐫 | n l m \langle n^{\prime}l^{\prime}m^{\prime}|\mathbf{r}|nlm\rangle
  82. r q = 𝐞 ^ q 𝐫 r_{q}=\hat{\mathbf{e}}_{q}\cdot\mathbf{r}
  83. r Y 11 ( θ , ϕ ) = - r 3 8 π sin ( θ ) e i ϕ = 3 4 π ( - x + i y 2 ) rY_{11}(\theta,\phi)=-r\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{i\phi}=\sqrt{\frac{% 3}{4\pi}}\left(-\frac{x+iy}{\sqrt{2}}\right)
  84. r Y 10 ( θ , ϕ ) = r 3 π cos ( θ ) = 3 4 π z rY_{10}(\theta,\phi)=r\sqrt{\frac{3}{\pi}}\cos(\theta)=\sqrt{\frac{3}{4\pi}}z
  85. r Y 1 - 1 ( θ , ϕ ) = r 3 8 π sin ( θ ) e - i ϕ = 3 4 π ( x - i y 2 ) rY_{1-1}(\theta,\phi)=r\sqrt{\frac{3}{8\pi}}\sin(\theta)e^{-i\phi}=\sqrt{\frac% {3}{4\pi}}\left(\frac{x-iy}{\sqrt{2}}\right)
  86. r Y 1 q ( θ , ϕ ) = 3 4 π r q rY_{1q}(\theta,\phi)=\sqrt{\frac{3}{4\pi}}r_{q}
  87. n l m | 𝐫 | n l m = ( 0 r 2 d r R n l * ( r ) r R n l ( r ) ) ( 4 π 3 d Ω Y l m * ( θ , ϕ ) Y 1 q ( θ , ϕ ) Y l m ( θ , ϕ ) ) \langle n^{\prime}l^{\prime}m^{\prime}|\mathbf{r}|nlm\rangle=\left(\int_{0}^{% \infty}r^{2}drR_{n^{\prime}l^{\prime}}^{*}(r)rR_{nl}(r)\right)\left(\sqrt{% \frac{4\pi}{3}}\int d\Omega Y_{l^{\prime}m^{\prime}}^{*}(\theta,\phi)Y_{1q}(% \theta,\phi)Y_{lm}(\theta,\phi)\right)
  88. l m | l 1 m q \langle l^{\prime}m^{\prime}|l1mq\rangle

Tensor_product_bundle.html

  1. Λ p T * M \Lambda^{p}T^{*}M
  2. Λ p T * M E \Lambda^{p}T^{*}M\otimes E

Terpentedienyl-diphosphate_synthase.html

  1. \rightleftharpoons

Terpentetriene_synthase.html

  1. \rightleftharpoons

Terpinolene_synthase.html

  1. \rightleftharpoons

Tetrahydrosarcinapterin_synthase.html

  1. \rightleftharpoons

Tetraprenyl-beta-curcumene_synthase.html

  1. \rightleftharpoons

Thaine's_theorem.html

  1. p p
  2. q q
  3. q q
  4. p - 1 p-1
  5. G + G^{+}
  6. F = ( ζ p + ) F=\mathbb{Q}(\zeta_{p}^{+})
  7. \mathbb{Q}
  8. E E
  9. C C
  10. C l + Cl^{+}
  11. θ [ G + ] \theta\in\mathbb{Z}[G^{+}]
  12. E / C E q E/CE^{q}
  13. C l + / C l + q Cl^{+}/Cl^{+q}

Thalianol_synthase.html

  1. \rightleftharpoons

The_Smithers_Divide.html

  1. ϕ P a S a ϵ < 0.75 \frac{\phi_{PaSa}}{\epsilon}<0.75
  2. ϕ P a S a \phi_{PaSa}
  3. ϵ \epsilon
  4. ϕ ϵ = 0.25 \frac{\phi}{\epsilon}=0.25

Thermal_center.html

  1. d 2 ( P , Q ) = K d 1 ( P , Q ) d_{2}(P,Q)=K\cdot d_{1}(P,Q)
  2. K = 1 + α Δ T K=1+\alpha\cdot\Delta T
  3. α \alpha

Thermodynamic_relations_across_normal_shocks.html

  1. M x {M_{x}}
  2. M y {M_{y}}
  3. M y 2 = 2 γ - 1 + M x 2 2 γ γ - 1 M x 2 - 1 {M^{2}_{y}=\frac{{\frac{2}{\gamma-1}}+M^{2}_{x}}{\frac{2\gamma}{\gamma-1}M^{2}% _{x}-1}}
  4. P y P x = 2 γ γ + 1 M x 2 - γ - 1 γ + 1 {\frac{P_{y}}{P_{x}}=\frac{2\gamma}{\gamma+1}M^{2}_{x}-{\frac{\gamma-1}{\gamma% +1}}}
  5. P o y P o x = ( γ + 1 2 M x 2 1 + γ - 1 2 M x 2 ) γ ( γ - 1 ) ( 2 γ γ + 1 M x 2 - γ - 1 γ + 1 ) - 1 ( γ - 1 ) {\frac{P_{o}y}{P_{o}x}}=\left(\frac{\frac{\gamma+1}{2}M^{2}_{x}}{1+\frac{% \gamma-1}{2}M^{2}_{x}}\right)^{\frac{\gamma}{\left(\gamma-1\right)}}\left(% \frac{2\gamma}{\gamma+1}M^{2}_{x}-\frac{\gamma-1}{\gamma+1}\right)^{\frac{-1}{% \left(\gamma-1\right)}}
  6. Δ S R = γ γ - 1 ln ( 2 ( γ + 1 ) M x 2 + γ - 1 γ + 1 ) + 1 γ - 1 ln ( 2 γ γ + 1 M x 2 - γ - 1 γ + 1 ) {\frac{\Delta S}{R}=\frac{\gamma}{\gamma-1}\ln\left(\frac{2}{\left(\gamma+1% \right)M^{2}_{x}}+\frac{\gamma-1}{\gamma+1}\right)+\frac{1}{\gamma-1}\ln\left(% \frac{2\gamma}{\gamma+1}M^{2}_{x}-\frac{\gamma-1}{\gamma+1}\right)}

Theta_constant.html

  1. θ a , b ( τ , z ) = ξ Z n exp [ π i ( ξ + a ) τ ( ξ + a ) t + ( ξ + a ) ( z + b ) t ] \theta_{a,b}(\tau,z)=\sum_{\xi\in Z^{n}}\exp\left[\pi{\rm{i}}(\xi+a)\tau(\xi+a% )^{t}+(\xi+a)(z+b)^{t}\right]

Thiazole_tautomerase.html

  1. \rightleftharpoons

Thickness_(graph_theory).html

  1. G G
  2. G G
  3. k k
  4. G G
  5. G G
  6. k k
  7. G G
  8. n n
  9. n + 7 6 , \left\lfloor\frac{n+7}{6}\right\rfloor,
  10. n = 9 , 10 n=9,10
  11. a b 2 ( a + b - 2 ) . \left\lceil\frac{ab}{2(a+b-2)}\right\rceil.
  12. G G
  13. G G
  14. G G
  15. n n
  16. t ( 3 n 6 ) t(3n−6)
  17. 6 t 1 6t−1
  18. D D
  19. D D
  20. G G
  21. G G

Thomas'_cyclically_symmetric_attractor.html

  1. d x d t = sin ( y ) - b x \frac{dx}{dt}=\sin(y)-bx
  2. d y d t = sin ( z ) - b y \frac{dy}{dt}=\sin(z)-by
  3. d z d t = sin ( x ) - b z \frac{dz}{dt}=\sin(x)-bz
  4. b b
  5. b b
  6. b > 1 b>1
  7. b = 1 b=1
  8. b 0.32899 b\approx 0.32899
  9. b 0.208186 b\approx 0.208186
  10. b = 0 b=0

Threo-3-hydroxy-D-aspartate_ammonia-lyase.html

  1. \rightleftharpoons

Thujopsene_synthase.html

  1. \rightleftharpoons

Tienstra_formula.html

  1. E p = K 1 E a + K 2 E b + K 3 E c K 1 + K 2 + K 3 E_{p}=\frac{K_{1}E_{a}+K_{2}E_{b}+K_{3}E_{c}}{K_{1}+K_{2}+K_{3}}
  2. N p = K 1 N a + K 2 N b + K 3 N c K 1 + K 2 + K 3 N_{p}=\frac{K_{1}N_{a}+K_{2}N_{b}+K_{3}N_{c}}{K_{1}+K_{2}+K_{3}}
  3. K 1 = 1 c o t ( A ) - c o t ( α ) K_{1}=\frac{1}{cot(A)-cot(\alpha)}
  4. K 2 = 1 c o t ( B ) - c o t ( β ) K_{2}=\frac{1}{cot(B)-cot(\beta)}
  5. K 3 = 1 c o t ( C ) - c o t ( γ ) K_{3}=\frac{1}{cot(C)-cot(\gamma)}

Time-dependent_variational_Monte_Carlo.html

  1. Ψ ( X , t ) = exp ( k a k ( t ) O k ( X ) ) \Psi(X,t)=\exp\left(\sum_{k}a_{k}(t)O_{k}(X)\right)
  2. a k ( t ) a_{k}(t)
  3. X X
  4. O k ( X ) O_{k}(X)
  5. a k ( t ) a_{k}(t)
  6. i k O k O k t c a ˙ k = O k t c , i\sum_{k^{\prime}}\langle O_{k}O_{k^{\prime}}\rangle_{t}^{c}\dot{a}_{k^{\prime% }}=\langle O_{k}\mathcal{H}\rangle_{t}^{c},
  7. \mathcal{H}
  8. A B t c = A B t - A t B t \langle AB\rangle_{t}^{c}=\langle AB\rangle_{t}-\langle A\rangle_{t}\langle B% \rangle_{t}
  9. t Ψ ( t ) | | Ψ ( t ) \langle\cdots\rangle_{t}\equiv\langle\Psi(t)|\cdots|\Psi(t)\rangle
  10. | Ψ ( X , t ) | 2 | Ψ ( X , t ) | 2 d X \frac{|\Psi(X,t)|^{2}}{\int|\Psi(X,t)|^{2}\,dX}
  11. X X
  12. t t
  13. a ( t ) a(t)

Time-series_segmentation.html

  1. s y m b o l y 1 : T = ( s y m b o l y 1 , , s y m b o l y T ) symbol{y}_{1:T}=(symbol{y}_{1},...,symbol{y}_{T})
  2. z { 1 , 2 , , n } z\in\{1,2,...,n\}
  3. t t
  4. s y m b o l y t symbol{y}_{t}
  5. s y m b o l y t P z t ( s y m b o l y t ) symbol{y}_{t}\sim P_{z_{t}}(symbol{y}_{t})

Time-varying_network.html

  1. t N t_{N}
  2. t P t_{P}
  3. t N t P t_{N}\gg t_{P}
  4. t N t P t_{N}\sim t_{P}
  5. t N t P t_{N}\ll t_{P}
  6. C C
  7. ( i , j , t ) (i,j,t)
  8. i i
  9. j j
  10. t t
  11. E E
  12. e e
  13. T e = { t 1 , , t n } T_{e}=\{t_{1},\ldots,t_{n}\}
  14. T e T_{e}
  15. e e
  16. T e = { ( t 1 , t 1 ) , , ( t n , t n ) } T_{e}=\{(t_{1},t_{1}^{\prime}),\ldots,(t_{n},t_{n}^{\prime})\}
  17. i i
  18. j j
  19. j j
  20. i i
  21. i i
  22. j j
  23. j j
  24. k k
  25. i i
  26. k k
  27. i i
  28. i i
  29. t t
  30. i i
  31. i i
  32. i i
  33. i i
  34. i i
  35. j j
  36. t t
  37. λ i , t ( j ) \lambda_{i,t}(j)
  38. i i
  39. λ i ( j ) \lambda_{i}(j)
  40. j j
  41. C C ( i , t ) = N - 1 j i λ i , t ( j ) C_{C}(i,t)=\frac{N-1}{\sum_{j\not=i}{\lambda_{i,t}(j)}}
  42. j j
  43. k k
  44. i i
  45. j j
  46. k k
  47. C B ( i , t ) = i j k ν i ( j , k ) i j k ν ( j , k ) C_{B}(i,t)=\frac{\sum_{i\not=j\not=k}{\nu_{i}(j,k)}}{\sum_{i\not=j\not=k}{\nu_% {(}j,k)}}
  48. C E ( i , t ) = 1 N - 1 j i 1 λ i , t ( j ) C_{E}(i,t)=\frac{1}{N-1}\sum_{j\not=i}{\frac{1}{\lambda_{i,t}(j)}}
  49. Δ t \Delta t
  50. τ \tau
  51. σ \sigma
  52. m m
  53. σ τ / m τ \sigma_{\tau}/m_{\tau}
  54. B = σ τ / m τ - 1 σ τ / m τ + 1 B=\frac{\sigma_{\tau}/m_{\tau}\ -1}{\sigma_{\tau}/m_{\tau}\ +1}

Time::memory::data_tradeoff_attack.html

  1. N N
  2. T T
  3. M M
  4. T M 2 = N 2 T{M^{2}}={N^{2}}
  5. 1 T N 1\leq T\leq N
  6. T M = N TM=N
  7. 1 T D 1\leq T\leq D
  8. D D
  9. N N
  10. P P
  11. T T
  12. M M
  13. D D
  14. N N
  15. N N
  16. x x
  17. y y
  18. f f
  19. N N
  20. f f
  21. f - 1 ( y ) = x {f}^{-1}(y)=x
  22. N N
  23. m × t m\times t
  24. f f
  25. m m
  26. N N
  27. t t
  28. N N
  29. m m
  30. m m
  31. m m
  32. m t mt
  33. t t
  34. m t mt
  35. t t
  36. t \sdot m t N t\sdot mt\leq N
  37. m t 2 = N m{t}^{2}=N
  38. m × t m\times t
  39. m t 2 = N m{t}^{2}=N
  40. m t / N = 1 / t mt/N=1/t
  41. t t
  42. f f
  43. f i ( x ) = h i ( f ( x ) ) {f}_{i}(x)={h}_{i}(f(x))
  44. h i {h}_{i}
  45. f ( x ) f(x)
  46. P N P\approx N
  47. M = m t M=mt
  48. t t
  49. m m
  50. f - 1 ( y ) f^{-1}(y)
  51. T = t 2 T=t^{2}
  52. t t
  53. t t
  54. f i f_{i}
  55. m t 2 = N m{t}^{2}=N
  56. T M 2 = N 2 , P = N , D = 1 T{M^{2}}={N^{2}},P=N,D=1
  57. T T
  58. M M
  59. f f
  60. N N
  61. f f
  62. T M = N TM=N
  63. N N
  64. D D
  65. N N
  66. l o g ( N ) log(N)
  67. f ( x ) = y f(x)=y
  68. x x
  69. y y
  70. N N
  71. M M
  72. x i {x}_{i}
  73. y i {y}_{i}
  74. ( x i , y i ) ({x}_{i},{y}_{i})
  75. y i {y}_{i}
  76. D + l o g ( N ) - 1 D+log(N)-1
  77. D D
  78. y 1 , y 2 , , y D , {y}_{1},{y}_{2},...,{y}_{D},
  79. l o g ( N ) log(N)
  80. y i {y}_{i}
  81. y i {y}_{i}
  82. x i {x}_{i}
  83. N N
  84. N N
  85. D M = N DM=N
  86. T = D T=D
  87. P = M P=M
  88. T M = N TM=N
  89. T T
  90. T = D T=D
  91. 1 1
  92. T M = N TM=N
  93. 1 T D 1\leq T\leq D
  94. P = M P=M
  95. N N
  96. f i f_{i}
  97. f f
  98. D D
  99. N N
  100. N N
  101. N / D N/D
  102. t t
  103. t / D t/D
  104. m m
  105. t D t\geq D
  106. M = m t / D M=mt/D
  107. t / D t/D
  108. P = N / D P=N/D
  109. T = ( t / D ) \sdot t \sdot D = t 2 T=(t/D)\sdot t\sdot D=t^{2}
  110. T M 2 D 2 = t 2 \sdot ( m 2 t 2 / D 2 ) \sdot D 2 = m 2 t 4 = N 2 TM^{2}D^{2}=t^{2}\sdot(m^{2}t^{2}/D^{2})\sdot D^{2}=m^{2}t^{4}=N^{2}
  111. D 2 T N D^{2}\leq T\leq N
  112. t D t\geq D
  113. k k
  114. k k
  115. R = 2 - k R=2^{-k}
  116. k k
  117. N = 2 n N=2^{n}
  118. n n
  119. n n
  120. R = 2 - k R=2^{-k}
  121. n - k n-k
  122. n - k n-k
  123. n - k n-k
  124. k k
  125. N R = 2 n - k NR=2^{n-k}
  126. D R 1 DR\geq 1
  127. D D
  128. D R DR
  129. N N
  130. N R NR
  131. T M 2 D 2 = N 2 TM^{2}D^{2}=N^{2}
  132. ( D R ) 2 T N R (DR)^{2}\leq T\leq NR
  133. T T
  134. ( D R ) 2 (DR)^{2}
  135. 1 1
  136. t t
  137. t R tR
  138. D R DR

Ting-Chao_Chou.html

  1. C I = ( D ) 1 ( D x ) 1 + ( D ) 2 ( D x ) 2 = ( D ) 1 ( D m ) 1 [ f a / ( 1 - f a ) ] 1 / m 1 + ( D ) 2 ( D m ) 2 [ f a / ( 1 - f a ) ] 1 / m 2 CI=\frac{(D)_{1}}{(D_{x})_{1}}+\frac{(D)_{2}}{(D_{x})_{2}}=\frac{(D)_{1}}{(D_{% m})_{1}[f_{a}/(1-f_{a})]^{1/m_{1}}}+\frac{(D)_{2}}{(D_{m})_{2}[f_{a}/(1-f_{a})% ]^{1/m_{2}}}

Tirucalladienol_synthase.html

  1. \rightleftharpoons

Toda–Smith_complex.html

  1. p p
  2. S 1 S^{1}
  3. S 1 S 1 S^{1}\to S^{1}\,
  4. z z p z\mapsto z^{p}\,
  5. p p
  6. S k S^{k}
  7. k k\in\mathbb{N}
  8. Σ S 1 Σ S 1 = : 𝕊 1 𝕊 1 \Sigma^{\infty}S^{1}\to\Sigma^{\infty}S^{1}=:\mathbb{S}^{1}\to\mathbb{S}^{1}
  9. S 𝑝 S S / p S\xrightarrow{p}S\to S/p
  10. S / p S/p
  11. H n ( X ) Z / p H^{n}(X)\simeq Z/p
  12. H ~ * ( X ) \tilde{H}^{*}(X)
  13. * n *\neq n
  14. α t \alpha_{t}
  15. β t \beta_{t}
  16. γ t \gamma_{t}
  17. V ( 0 ) k V(0)_{k}
  18. V ( 1 ) k V(1)_{k}
  19. V 2 ( k ) V_{2}(k)
  20. n n
  21. V ( n ) V(n)
  22. n - 1 , 0 , 1 , 2 , 3 , n\in-1,0,1,2,3,\ldots
  23. B P * ( V ( n ) ) := [ 𝕊 𝟘 , B P V ( n ) ] BP_{*}(V(n)):=[\mathbb{S^{0}},BP\wedge V(n)]
  24. B P * / ( p , , v n ) BP_{*}/(p,\ldots,v_{n})
  25. B P BP
  26. V ( n ) V(n)
  27. B P * ( V ( - 1 ) ) \displaystyle BP_{*}(V(-1))
  28. B P * = p [ v 1 , v 2 , ] BP_{*}=\mathbb{Z}_{p}[v_{1},v_{2},...]
  29. deg v i \deg v_{i}
  30. 2 ( p i - 1 ) 2(p^{i}-1)
  31. B P * ( S 0 ) B P * BP_{*}(S^{0})\simeq BP_{*}
  32. V ( - 1 ) V(-1)
  33. B P * ( S / p ) B P * / p BP_{*}(S/p)\simeq BP_{*}/p
  34. V ( 0 ) V(0)

Token_reconfiguration.html

  1. G G
  2. V V ( G ) V\subset V(G)
  3. n = | V | n=|V|
  4. v 1 v_{1}
  5. v 2 v_{2}
  6. v 1 v_{1}
  7. v 2 v_{2}
  8. G G
  9. V V ( G ) V^{\prime}\subset V(G)
  10. n = | V ( G ) | - 1 n=|V(G)|-1
  11. n n
  12. v 1 , v 2 , , v n v_{1},v_{2},\ldots,v_{n}
  13. U U
  14. S 1 , S 2 , , S m S_{1},S_{2},\ldots,S_{m}
  15. U U
  16. n n
  17. α = 1 / 4 , β = 1 \alpha=1/4,\beta=1
  18. n n
  19. α = 1 / 5 , β = 2 \alpha=1/5,\beta=2

Toomre's_Stability_Criterion.html

  1. c s Ω π G Σ > 1 , \frac{c_{s}\Omega}{\pi G\Sigma}>1,
  2. c s c_{s}
  3. Ω \Omega
  4. Σ \Sigma
  5. Q gas c s Ω π G Σ . Q_{\mathrm{gas}}\equiv\frac{c_{s}\Omega}{\pi G\Sigma}.
  6. Q > 1 Q>1
  7. Q star σ R κ 3.36 G Σ , Q_{\mathrm{star}}\equiv\frac{\sigma_{R}\kappa}{3.36G\Sigma},
  8. σ R \sigma_{R}
  9. κ \kappa

Topographic_Wetness_Index.html

  1. ln a t a n b \ln{a\over tanb}

Topology_of_uniform_convergence.html

  1. V W V↦W
  2. 𝒢 \mathcal{G}
  3. 𝒩 \mathcal{N}
  4. Y T Y^{T}
  5. Y T Y^{T}
  6. 𝒰 ( G , N ) = { f Y T : f ( G ) N } \mathcal{U}(G,N)=\{f\in Y^{T}:f(G)\subseteq N\}
  7. G 𝒢 G\in\mathcal{G}
  8. N 𝒩 N\in\mathcal{N}
  9. 𝒩 \mathcal{N}
  10. 𝒢 \mathcal{G}
  11. 𝒢 \mathcal{G}
  12. 𝒢 \mathcal{G}
  13. 𝒢 \mathcal{G}
  14. 𝒢 1 \mathcal{G}_{1}
  15. 𝒢 \mathcal{G}
  16. 𝒢 \mathcal{G}
  17. G 𝒢 G\in\mathcal{G}
  18. 𝒢 1 \mathcal{G}_{1}
  19. 𝒢 \mathcal{G}
  20. 𝒢 1 \mathcal{G}_{1}
  21. Y T Y^{T}
  22. 𝒢 \mathcal{G}
  23. Y T Y^{T}
  24. Y T Y^{T}
  25. Y T Y^{T}
  26. Y T Y^{T}
  27. Y T Y^{T}
  28. 𝒢 \mathcal{G}
  29. G 𝒢 G\in\mathcal{G}
  30. 𝒢 \mathcal{G}
  31. Y T Y^{T}
  32. ( p α ) (p_{\alpha})
  33. 𝒢 \mathcal{G}
  34. p G , α ( f ) = sup x G p α ( f ( x ) ) p_{G,\alpha}(f)=\sup_{x\in G}p_{\alpha}(f(x))
  35. 𝒢 \mathcal{G}
  36. α \alpha
  37. G 𝒢 G \bigcup_{G\in\mathcal{G}}G
  38. 𝒢 \mathcal{G}
  39. Y T Y^{T}
  40. G 𝒢 G \bigcup_{G\in\mathcal{G}}G
  41. Y T Y^{T}
  42. 𝒢 \mathcal{G}
  43. G 𝒢 G\in\mathcal{G}
  44. u H u ( G ) \cup_{u\in H}u(G)
  45. 𝒢 \mathcal{G}
  46. Y X Y^{X}
  47. L 𝒢 ( X , Y ) L_{\mathcal{G}}(X,Y)
  48. 𝒢 \mathcal{G}
  49. G 𝒢 G\in\mathcal{G}
  50. 𝒢 \mathcal{G}
  51. 𝒢 \mathcal{G}
  52. 𝒢 1 \mathcal{G}_{1}
  53. G 1 , G 2 𝒢 G_{1}^{\prime},G_{2}^{\prime}\in\mathcal{G^{\prime}}
  54. G 𝒢 G^{\prime}\in\mathcal{G^{\prime}}
  55. G 1 G 2 G G_{1}^{\prime}\cup G_{2}^{\prime}\subseteq G^{\prime}
  56. 𝒢 2 \mathcal{G}_{2}
  57. G 1 𝒢 G_{1}^{\prime}\in\mathcal{G^{\prime}}
  58. λ \lambda
  59. G 𝒢 G^{\prime}\in\mathcal{G^{\prime}}
  60. λ G 1 G \lambda G_{1}^{\prime}\subseteq G^{\prime}
  61. 𝒢 \mathcal{G}
  62. 𝒢 \mathcal{G}
  63. 𝒢 1 \mathcal{G}_{1}
  64. 𝒢 2 \mathcal{G}_{2}
  65. L 𝒢 ( X , Y ) L_{\mathcal{G}}(X,Y)
  66. L 𝒢 ( X , Y ) L_{\mathcal{G}}(X,Y)
  67. X 𝒢 * X^{*}_{\mathcal{G}}
  68. L 𝒢 ( X , Y ) L_{\mathcal{G}}(X,Y)
  69. L 𝒢 ( X , Y ) L_{\mathcal{G}}(X,Y)
  70. G 𝒢 G\in\mathcal{G}
  71. u H u ( G ) \cup_{u\in H}u(G)
  72. u H u - 1 ( V ) \cap_{u\in H}u^{-1}(V)
  73. G 𝒢 G\in\mathcal{G}
  74. L σ ( X , Y ) L_{\sigma}(X,Y)
  75. 𝒢 \mathcal{G}
  76. 𝒢 \mathcal{G}
  77. 𝒢 \mathcal{G}
  78. 𝒢 \mathcal{G}
  79. 𝒢 \mathcal{G}
  80. L σ ( X , Y ) L_{\sigma}(X,Y)
  81. L ( X , Y ) L(X,Y)
  82. L σ ( X , Y ) L_{\sigma}(X,Y)
  83. Y X Y^{X}
  84. F ( X , Y ) F(X,Y)
  85. Y X Y^{X}
  86. 𝒢 \mathcal{G}
  87. L γ ( X , Y ) L_{\gamma}(X,Y)
  88. 𝒢 \mathcal{G}
  89. L c ( X , Y ) L_{c}(X,Y)
  90. L c ( X , Y ) L_{c}(X,Y)
  91. L c ( X , Y ) L_{c}(X,Y)
  92. L b ( X , Y ) L_{b}(X,Y)
  93. 𝒢 \mathcal{G}
  94. L b ( X , Y ) L_{b}(X,Y)
  95. L b ( X , Y ) L_{b}(X,Y)
  96. L b ( X , Y ) L_{b}(X,Y)
  97. L b ( X , Y ) L_{b}(X,Y)
  98. \mathcal{F}
  99. L ( X , ) L(X,\mathcal{F})
  100. X * X^{*}
  101. X X^{\prime}
  102. 𝒢 \mathcal{G}
  103. Y = Y=\mathcal{F}
  104. X * X^{*}
  105. 𝒢 \mathcal{G}
  106. X 𝒢 * X^{*}_{\mathcal{G}}
  107. X 𝒢 * X^{*}_{\mathcal{G}}
  108. G G
  109. 𝒢 \mathcal{G}
  110. G := { x X * : sup x G | x , x | 1 } G^{\circ}:=\{x^{\prime}\in X^{*}:\sup_{x\in G}|\langle x^{\prime},x\rangle|% \leq 1\}
  111. F F^{\prime}
  112. X * X^{*}
  113. x X * x^{\prime}\in X^{*}
  114. 𝒢 \mathcal{G}
  115. X * X^{*}
  116. F F^{\prime}
  117. x x^{\prime}
  118. G 𝒢 G\in\mathcal{G}
  119. G G^{\circ}
  120. 𝒢 \mathcal{G}
  121. X 𝒢 * X^{*}_{\mathcal{G}}
  122. G 𝒢 G \bigcup_{G\in\mathcal{G}}G
  123. X 𝒢 * X^{*}_{\mathcal{G}}
  124. G 𝒢 G \bigcup_{G\in\mathcal{G}}G
  125. ( X 𝒢 * ) * (X^{*}_{\mathcal{G}})^{*}
  126. x X x\in X
  127. X * X^{*}
  128. x X * x , x x^{\prime}\in X^{*}\mapsto\langle x^{\prime},x\rangle
  129. X 𝒢 * X^{*}_{\mathcal{G}}
  130. X * X^{*}
  131. ( X 𝒢 * ) * (X^{*}_{\mathcal{G}})^{*}
  132. u : E F u:E\to F
  133. 𝒢 \mathcal{G}
  134. \mathcal{H}
  135. 𝒢 1 \mathcal{G}_{1}
  136. 𝒢 2 \mathcal{G}_{2}
  137. u u
  138. u t : Y * X 𝒢 * {}^{t}u:Y^{*}_{\mathcal{H}}\to X^{*}_{\mathcal{G}}
  139. G 𝒢 G\in\mathcal{G}
  140. H H\in\mathcal{H}
  141. u u
  142. X * X^{*}
  143. σ ( X * , X ) \sigma(X^{*},X)
  144. γ ( X * , X ) \gamma(X^{*},X)
  145. c ( X * , X ) c(X^{*},X)
  146. b ( X * , X ) b(X^{*},X)
  147. Y * Y^{*}
  148. σ ( Y * , Y ) \sigma(Y^{*},Y)
  149. γ ( Y * , Y ) \gamma(Y^{*},Y)
  150. c ( Y * , Y ) c(Y^{*},Y)
  151. b ( Y * , Y ) b(Y^{*},Y)
  152. \mathcal{F}
  153. 𝒢 \mathcal{G}
  154. 𝒢 1 \mathcal{G}_{1}
  155. 𝒢 2 \mathcal{G}_{2}
  156. X × X 𝒢 * X\times X^{*}_{\mathcal{G}}\to\mathcal{F}
  157. ( x , x ) x , x = x ( x ) (x,x^{\prime})\mapsto\langle x^{\prime},x\rangle=x^{\prime}(x)
  158. 𝒢 \mathcal{G}
  159. X * X^{*}
  160. b ( X * , X ) b(X^{*},X)
  161. 𝒢 \mathcal{G}
  162. 𝒢 1 \mathcal{G}_{1}
  163. 𝒢 2 \mathcal{G}_{2}
  164. 𝒢 \mathcal{G}
  165. X 𝒢 * X^{*}_{\mathcal{G}}
  166. 𝒢 \mathcal{G}
  167. X * X^{*}
  168. X * X^{*}
  169. σ ( X * , X ) \sigma(X^{*},X)
  170. X * X^{*}
  171. X σ * X^{*}_{\sigma}
  172. X σ ( X * , X ) * X^{*}_{\sigma(X^{*},X)}
  173. σ ( X * , X ) \sigma(X^{*},X)
  174. u : X Y u:X\to Y
  175. u u
  176. u u
  177. u t : Y * X * {}^{t}u:Y^{*}\to X^{*}
  178. u t {}^{t}u
  179. X σ ( X * , X ) * X^{*}_{\sigma(X^{*},X)}
  180. Z Z
  181. u : X σ * × Y σ * Z σ * u:X^{*}_{\sigma}\times Y^{*}_{\sigma}\to Z^{*}_{\sigma}
  182. u : X b * × Y b * Z b * u:X^{*}_{b}\times Y^{*}_{b}\to Z^{*}_{b}
  183. X σ ( X * , X ) * X^{*}_{\sigma(X^{*},X)}
  184. σ ( X * , X ) \sigma(X^{*},X)
  185. X * X^{*}
  186. b ( X * , X ) b(X^{*},X)
  187. σ ( X * , X ) \sigma(X^{*},X)
  188. X * X^{*}
  189. σ ( X * , X ) \sigma(X^{*},X)
  190. X ^ \hat{X}
  191. X X ^ X\neq\hat{X}
  192. σ ( X * , X ^ ) \sigma(X^{*},\hat{X})
  193. σ ( X * , X ) \sigma(X^{*},X)
  194. σ ( X * , X ) \sigma(X^{*},X)
  195. 𝒢 \mathcal{G}
  196. X * X^{*}
  197. γ ( X * , X ) \gamma(X^{*},X)
  198. X * X^{*}
  199. X γ * X^{*}_{\gamma}
  200. X γ ( X * , X ) * X^{*}_{\gamma(X^{*},X)}
  201. γ ( X * , X ) = c ( X * , X ) \gamma(X^{*},X)=c(X^{*},X)
  202. 𝒢 \mathcal{G}
  203. X * X^{*}
  204. c ( X * , X ) c(X^{*},X)
  205. X * X^{*}
  206. X c * X^{*}_{c}
  207. X c ( X * , X ) * X^{*}_{c(X^{*},X)}
  208. c ( X * , X ) c(X^{*},X)
  209. W X * W^{\prime}\subseteq X^{*}
  210. W W^{\prime}
  211. X * X^{*}
  212. W W^{\prime}
  213. c ( X * , X ) c(X^{*},X)
  214. 𝒢 \mathcal{G}
  215. X * X^{*}
  216. X * X^{*}
  217. σ ( X * , X ) \sigma(X^{*},X)
  218. 𝒢 \mathcal{G}
  219. X * X^{*}
  220. X * X^{*}
  221. τ ( X * , X ) \tau(X^{*},X)
  222. X * X^{*}
  223. X τ ( X * , X ) * X^{*}_{\tau(X^{*},X)}
  224. 𝒢 \mathcal{G}
  225. X * X^{*}
  226. X * X^{*}
  227. b ( X * , X ) b(X^{*},X)
  228. X * X^{*}
  229. X b * X^{*}_{b}
  230. X b ( X * , X ) * X^{*}_{b(X^{*},X)}
  231. X b * X^{*}_{b}
  232. X * * X^{**}
  233. ( X b * ) * = X * * (X^{*}_{b})^{*}=X^{**}
  234. b ( X * , X ) b(X^{*},X)
  235. 𝒢 \mathcal{G}
  236. X * X^{*}
  237. 𝒢 \mathcal{G}
  238. X b ( X * , X ) * X^{*}_{b(X^{*},X)}
  239. X * X^{*}
  240. x = sup x X , , x = 1 | x , x | \|x^{\prime}\|=\sup_{x\in X,,\|x\|=1}|\langle x^{\prime},x\rangle|
  241. x X * x^{\prime}\in X^{*}
  242. X k X_{k}
  243. k = 0 , 1 k=0,1\dots
  244. X b ( X * , X ) * X^{*}_{b(X^{*},X)}
  245. X k X_{k}
  246. X b ( X * , X ) * X^{*}_{b(X^{*},X)}
  247. X b ( X * , X ) * X^{*}_{b(X^{*},X)}
  248. X b ( X * , X ) * X^{*}_{b(X^{*},X)}
  249. X b ( X * , X ) * X^{*}_{b(X^{*},X)}
  250. σ ( X * , X ) \sigma(X^{*},X)
  251. b ( X * , X ) b(X^{*},X)
  252. X * X^{*}
  253. 𝒢 ′′ \mathcal{G^{\prime\prime}}
  254. X * * = ( X b * ) * X^{**}=(X^{*}_{b})^{*}
  255. X * X^{*}
  256. X * X^{*}
  257. X * * X^{**}
  258. X * * X^{**}
  259. τ ( X * , X * * ) \tau(X^{*},X^{**})
  260. X * X^{*}
  261. X τ ( X * , X * * ) * X^{*}_{\tau(X^{*},X^{**})}
  262. b ( X * , X ) b(X^{*},X)
  263. τ ( X * , X ) \tau(X^{*},X)
  264. 𝒢 \mathcal{G}
  265. X * X^{*}
  266. ( X σ * ) * (X^{*}_{\sigma})^{*}
  267. x X x\in X
  268. x X * x , x x^{\prime}\in X^{*}\mapsto\langle x^{\prime},x\rangle
  269. X σ * X^{*}_{\sigma}
  270. ( X σ * ) * (X^{*}_{\sigma})^{*}
  271. X = ( X σ * ) * X=(X^{*}_{\sigma})^{*}
  272. X σ * X^{*}_{\sigma}
  273. X σ * X^{*}_{\sigma}
  274. X σ * X^{*}_{\sigma}
  275. 𝒢 \mathcal{G^{\prime}}
  276. X σ * X^{*}_{\sigma}
  277. X 𝒢 X_{\mathcal{G^{\prime}}}
  278. G := { x X : sup x G | x , x | 1 } G^{\prime\circ}:=\{x\in X:\sup_{x^{\prime}\in G^{\prime}}|\langle x^{\prime},x% \rangle|\leq 1\}
  279. G G^{\prime}
  280. 𝒢 \mathcal{G^{\prime}}
  281. 𝒢 \mathcal{G^{\prime}}
  282. X X^{\prime}
  283. X * X^{*}
  284. σ ( X , X * ) \sigma(X,X^{*})
  285. X σ X_{\sigma}
  286. X σ ( X , X * ) X_{\sigma(X,X^{*})}
  287. u : X Y u:X\to Y
  288. u : X Y u:X\to Y
  289. u : σ ( X , X * ) σ ( Y , Y * ) u:\sigma(X,X^{*})\to\sigma(Y,Y^{*})
  290. u : X Y u:X\to Y
  291. u u
  292. 𝒢 \mathcal{G^{\prime}}
  293. X * X^{*}
  294. X * X^{*}
  295. ϵ ( X , X * ) \epsilon(X,X^{*})
  296. X ϵ X_{\epsilon}
  297. X ϵ ( X , X * ) X_{\epsilon(X,X^{*})}
  298. 𝒢 \mathcal{G^{\prime}}
  299. X * X^{*}
  300. ϵ ( X , X * ) \epsilon(X,X^{*})
  301. 𝒢 \mathcal{G^{\prime}}
  302. X * X^{*}
  303. X * X^{*}
  304. τ ( X , X * ) \tau(X,X^{*})
  305. X τ X_{\tau}
  306. X τ ( X , X * ) X_{\tau(X,X^{*})}
  307. τ ( X , X * ) \tau(X,X^{*})
  308. 𝒢 \mathcal{G}
  309. X * X^{*}
  310. b ( X , X * ) b(X,X^{*})
  311. X * X^{*}
  312. X b * X^{*}_{b}
  313. X b ( X , X * ) * X^{*}_{b(X,X^{*})}
  314. σ ( X , Y ) \sigma(X,Y)
  315. X σ ( X , Y ) X_{\sigma(X,Y)}
  316. σ ( X , Y ) \sigma(X,Y)
  317. X σ ( X , Y ) * = ( X σ ( X , Y ) ) * = Y X_{\sigma(X,Y)}^{*}=(X_{\sigma(X,Y)})^{*}=Y
  318. 𝒯 \mathcal{T}
  319. X * X^{*}
  320. X 𝒯 X_{\mathcal{T}}
  321. 𝒯 \mathcal{T}
  322. X * X^{*}
  323. σ ( X , X * ) \sigma(X,X^{*})
  324. X 𝒯 * X_{\mathcal{T}}^{*}
  325. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  326. B ( X , Y ; Z ) B(X,Y;Z)
  327. X , Y X,Y
  328. Z Z
  329. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  330. B ( X , Y ; Z ) B(X,Y;Z)
  331. 𝒢 \mathcal{G}
  332. \mathcal{H}
  333. 𝒢 × \mathcal{G}\times\mathcal{H}
  334. G 𝒢 G\in\mathcal{G}
  335. H H\in\mathcal{H}
  336. Z X × Y Z^{X\times Y}
  337. 𝒢 × \mathcal{G}\times\mathcal{H}
  338. B ( X , Y ; Z ) B(X,Y;Z)
  339. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  340. 𝒢 - \mathcal{G}-\mathcal{H}
  341. G × H G\times H
  342. 𝒢 × \mathcal{G}\times\mathcal{H}
  343. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  344. B ( X , Y ; Z ) B(X,Y;Z)
  345. b b
  346. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  347. B ( X , Y ; Z ) B(X,Y;Z)
  348. G 𝒢 G\in\mathcal{G}
  349. H H\in\mathcal{H}
  350. b ( G , H ) b(G,H)
  351. 𝒢 \mathcal{G}
  352. \mathcal{H}
  353. B ( X , Y ; Z ) B(X,Y;Z)
  354. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  355. 𝒢 \mathcal{G}
  356. \mathcal{H}
  357. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  358. ( X , Y ; Z ) \mathcal{B}(X,Y;Z)
  359. 𝒢 \mathcal{G}
  360. \mathcal{H}
  361. Z Z
  362. Z Z
  363. ( X , Y ; Z ) = B ( X , Y ; Z ) \mathcal{B}(X,Y;Z)=B(X,Y;Z)
  364. X , Y X,Y
  365. Z Z
  366. Z Z
  367. X , Y X,Y
  368. Z Z
  369. 𝒢 \mathcal{G}
  370. \mathcal{H}
  371. X * X^{*}
  372. Y * Y^{*}
  373. 𝒢 \mathcal{G}
  374. \mathcal{H}
  375. ( X b ( X * , X ) * , Y b ( X * , X ) * ; Z ) \mathcal{B}(X^{*}_{b(X^{*},X)},Y^{*}_{b(X^{*},X)};Z)
  376. ( X b ( X * , X ) * , Y b ( X * , X ) ; Z ) \mathcal{B}(X^{*}_{b(X^{*},X)},Y_{b(X^{*},X)};Z)
  377. ϵ ( X b ( X * , X ) * , Y b ( X * , X ) * ; Z ) \mathcal{B}_{\epsilon}(X^{*}_{b(X^{*},X)},Y^{*}_{b(X^{*},X)};Z)
  378. ϵ ( X b * , Y b * ; Z ) \mathcal{B}_{\epsilon}(X^{*}_{b},Y^{*}_{b};Z)
  379. ( X σ ( X * , X ) * , Y σ ( X * , X ) * ; Z ) \mathcal{B}(X^{*}_{\sigma(X^{*},X)},Y^{*}_{\sigma(X^{*},X)};Z)
  380. ( X σ * , Y σ * ; Z ) \mathcal{B}(X^{*}_{\sigma},Y^{*}_{\sigma};Z)
  381. ϵ \mathcal{B}_{\epsilon}
  382. ( X b * , Y b * ; Z ) (X^{*}_{b},Y^{*}_{b};Z)
  383. ϵ ( X σ * , Y σ * ; Z ) \mathcal{B}_{\epsilon}(X^{*}_{\sigma},Y^{*}_{\sigma};Z)
  384. ( X σ * , Y σ * ) \mathcal{B}(X^{*}_{\sigma},Y^{*}_{\sigma})
  385. ( X σ * , Y σ * ) \mathcal{B}(X^{*}_{\sigma},Y^{*}_{\sigma})
  386. L ( X σ ( X * , X ) * , Y σ ( Y * , Y ) ) L(X^{*}_{\sigma(X^{*},X)},Y_{\sigma(Y^{*},Y)})
  387. L ( X τ ( X * , X ) * , Y ) L(X^{*}_{\tau(X^{*},X)},Y)
  388. ϵ \mathcal{B}_{\epsilon}
  389. ( X σ * , Y σ * ) (X^{*}_{\sigma},Y^{*}_{\sigma})
  390. ϵ \mathcal{B}_{\epsilon}
  391. ( X σ * , Y σ * ) (X^{*}_{\sigma},Y^{*}_{\sigma})

Torsor_(algebraic_geometry).html

  1. Y × X P Y\times_{X}P
  2. Y X Y\to X
  3. Y × G Y Y\times G\to Y
  4. G X = X × G G_{X}=X\times G
  5. G X G_{X}
  6. P P
  7. { U i X } \{U_{i}\to X\}
  8. U i U_{i}
  9. G U i G_{U_{i}}
  10. 𝔾 m \mathbb{G}_{m}
  11. G L n GL_{n}
  12. L / K L/K
  13. Spec L Spec K \operatorname{Spec}L\to\operatorname{Spec}K
  14. Gal ( L / K ) \operatorname{Gal}(L/K)
  15. P ( X ) = Mor ( X , P ) P(X)=\operatorname{Mor}(X,P)
  16. s : X P s:X\to P
  17. X × G P , ( x , g ) s ( x ) g X\times G\to P,(x,g)\mapsto s(x)g
  18. { U i X } \{U_{i}\to X\}
  19. s i P ( U i ) s_{i}\in P(U_{i})
  20. s i s_{i}
  21. s i g i j = s j s_{i}g_{ij}=s_{j}
  22. U i j U_{ij}
  23. g i j G ( U i j ) g_{ij}\in G(U_{ij})
  24. s i s_{i}
  25. g i j g_{ij}
  26. H 1 ( X , G ) H^{1}(X,G)
  27. H 1 ( X , G ) H^{1}(X,G)
  28. 𝐅 q \mathbf{F}_{q}
  29. Spec 𝐅 q \operatorname{Spec}\mathbf{F}_{q}
  30. P X P\to X
  31. P × G F X P\times^{G}F\to X
  32. P × H G P\times^{H}G
  33. P × H G P^{\prime}\times^{H}G
  34. X R = X × Spec k Spec R X_{R}=X\times_{\operatorname{Spec}k}\operatorname{Spec}R
  35. R R R\to R^{\prime}
  36. P × X R X R P\times_{X_{R}}X_{R^{\prime}}
  37. deg i ( P ) \operatorname{deg}_{i}(P)
  38. Lie ( P ) \operatorname{Lie}(P)
  39. deg i ( G ) = max { deg i ( P ) | P G parabolic subgroups } \operatorname{deg}_{i}(G)=\max\{\operatorname{deg}_{i}(P)|P\subset G\,\text{ % parabolic subgroups}\}
  40. G E = Aut G ( E ) {}^{E}G=\operatorname{Aut}_{G}(E)
  41. deg i ( E ) = deg i ( G E ) \operatorname{deg}_{i}(E)=\operatorname{deg}_{i}({}^{E}G)
  42. deg i ( E ) 0 \operatorname{deg}_{i}(E)\leq 0
  43. deg i ( E ) < 0 \operatorname{deg}_{i}(E)<0

Total_position_spread.html

  1. 𝐫 ^ = i = 1 n r ^ i \mathbf{\hat{r}}=\sum_{i=1}^{n}\hat{r}_{i}
  2. x ^ = i = 1 n x ^ i y ^ = i = 1 n y ^ i z ^ = i = 1 n z ^ i \hat{x}=\sum_{i=1}^{n}\hat{x}_{i}\qquad\hat{y}=\sum_{i=1}^{n}\hat{y}_{i}\qquad% \hat{z}=\sum_{i=1}^{n}\hat{z}_{i}\qquad
  3. x ^ = i j i x ^ j a i a j \hat{x}=\sum_{ij}\left\langle i\mid\hat{x}\mid j\right\rangle a^{\dagger}_{i}a% _{j}
  4. s ^ x x = i = 1 n x ^ i 2 s ^ x y = i = 1 n x ^ y ^ s ^ x z = i = 1 n x ^ z ^ \hat{s}_{xx}=\sum_{i=1}^{n}\hat{x}_{i}^{2}\qquad\hat{s}_{xy}=\sum_{i=1}^{n}% \hat{x}\hat{y}\qquad\hat{s}_{xz}=\sum_{i=1}^{n}\hat{x}\hat{z}
  5. s ^ y y = i = 1 n y ^ i 2 s ^ y z = i = 1 n y ^ z ^ s ^ z z = i = 1 n z ^ i 2 \hat{s}_{yy}=\sum_{i=1}^{n}\hat{y}_{i}^{2}\qquad\hat{s}_{yz}=\sum_{i=1}^{n}% \hat{y}\hat{z}\qquad\hat{s}_{zz}=\sum_{i=1}^{n}\hat{z}_{i}^{2}
  6. r ^ \hat{r}
  7. S ^ x x = i = 1 n j = 1 n x ^ i x ^ j S ^ x y = i = 1 n j = 1 n x ^ i y ^ j S ^ x z = i = 1 n j = 1 n x ^ i z ^ j \hat{S}_{xx}=\sum_{i=1}^{n}\sum_{j=1}^{n}\hat{x}_{i}\hat{x}_{j}\qquad\hat{S}_{% xy}=\sum_{i=1}^{n}\sum_{j=1}^{n}\hat{x}_{i}\hat{y}_{j}\qquad\hat{S}_{xz}=\sum_% {i=1}^{n}\sum_{j=1}^{n}\hat{x}_{i}\hat{z}_{j}
  8. S ^ y y = i = 1 n j = 1 n y ^ i y ^ j S ^ y z = i = 1 n j = 1 n y ^ i z ^ j S ^ z z = i = 1 n j = 1 n z ^ i z ^ j \hat{S}_{yy}=\sum_{i=1}^{n}\sum_{j=1}^{n}\hat{y}_{i}\hat{y}_{j}\qquad\hat{S}_{% yz}=\sum_{i=1}^{n}\sum_{j=1}^{n}\hat{y}_{i}\hat{z}_{j}\qquad\hat{S}_{zz}=\sum_% {i=1}^{n}\sum_{j=1}^{n}\hat{z}_{i}\hat{z}_{j}
  9. 𝐒 a b = 𝐒 ^ a b + 𝐬 ^ a b \mathbf{S}_{ab}=\mathbf{\hat{S}}_{ab}+\mathbf{\hat{s}}_{ab}
  10. Ψ \Psi
  11. Λ = Ψ S a b Ψ c = Ψ S a b Ψ - Ψ a ^ Ψ Ψ b ^ Ψ \Lambda=\left\langle\Psi\mid S_{ab}\mid\Psi\right\rangle_{c}=\left\langle\Psi% \mid S_{ab}\mid\Psi\right\rangle-\left\langle\Psi\mid\hat{a}\mid\Psi\right% \rangle\left\langle\Psi\mid\hat{b}\mid\Psi\right\rangle
  12. 𝐫 ^ = σ = α , β 𝐫 ^ σ \mathbf{\hat{r}}=\sum_{\sigma=\alpha,\beta}\mathbf{\hat{r}_{\sigma}}
  13. 𝐑 ^ = i = 1 n σ = α , β 𝐑 ^ ( i ) 𝐧 ^ σ ( i ) \mathbf{\hat{R}}=\sum_{i=1}^{n}\sum_{\sigma=\alpha,\beta}\mathbf{\hat{R}}(i)% \mathbf{\hat{n}}_{\sigma}(i)
  14. 𝐑 ^ \mathbf{\hat{R}}
  15. α \alpha
  16. β \beta
  17. 𝐑 ^ = 𝐑 ^ α + 𝐑 ^ β \mathbf{\hat{R}}=\mathbf{\hat{R}_{\alpha}}+\mathbf{\hat{R}_{\beta}}
  18. 𝐑 ^ 2 = 𝐑 ^ α 2 + 𝐑 ^ β 2 + 𝐑 ^ α 𝐑 ^ β + 𝐑 ^ β 𝐑 ^ α \mathbf{\hat{R}}^{2}=\mathbf{\hat{R}}_{\alpha}^{2}+\mathbf{\hat{R}}_{\beta}^{2% }+\mathbf{\hat{R}}_{\alpha}\mathbf{\hat{R}}_{\beta}+\mathbf{\hat{R}}_{\beta}% \mathbf{\hat{R}}_{\alpha}
  19. 𝚲 = 𝚲 α α + 𝚲 β β + 𝚲 α β + 𝚲 β α \mathbf{\Lambda}=\mathbf{\Lambda}_{\alpha\alpha}+\mathbf{\Lambda}_{\beta\beta}% +\mathbf{\Lambda}_{\alpha\beta}+\mathbf{\Lambda}_{\beta\alpha}

Trans-2,3-dihydro-3-hydroxyanthranilate_isomerase.html

  1. \rightleftharpoons

Transfer_entropy.html

  1. X t X_{t}
  2. Y t Y_{t}
  3. t t\in\mathbb{N}
  4. T X Y = H ( Y t Y t - 1 : t - L ) - H ( Y t Y t - 1 : t - L , X t - 1 : t - L ) , T_{X\rightarrow Y}=H\left(Y_{t}\mid Y_{t-1:t-L}\right)-H\left(Y_{t}\mid Y_{t-1% :t-L},X_{t-1:t-L}\right),
  5. Y t - 1 : t - L Y_{t-1:t-L}

Transformer_utilization_factor.html

  1. T . U . F = P o d c V A r a t i n g o f t r a n s f o r m e r T.U.F=\frac{P_{odc}}{V_{A}\ rating\ of\ transformer}
  2. V A V_{A}
  3. V A = V r . m . s I ˙ r . m . s ( F o r s e c o n d a r y c o i l . ) V_{A}=V_{r.m.s}\dot{I}_{r.m.s}(For\ secondary\ coil.)

Translation_of_axes.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  2. y = y + k y=y^{\prime}+k
  3. 9 x 2 + 25 y 2 + 18 x - 100 y - 116 = 0 , 9x^{2}+25y^{2}+18x-100y-116=0,
  4. 9 ( x 2 + 2 x ) + 25 ( y 2 - 4 y ) = 116. 9(x^{2}+2x\qquad)+25(y^{2}-4y\qquad)=116.
  5. 9 ( x 2 + 2 x + 1 ) + 25 ( y 2 - 4 y + 4 ) = 116 + 9 + 100 9(x^{2}+2x+1)+25(y^{2}-4y+4)=116+9+100
  6. 9 ( x + 1 ) 2 + 25 ( y - 2 ) 2 = 225. \Leftrightarrow 9(x+1)^{2}+25(y-2)^{2}=225.
  7. x = x + 1 x^{\prime}=x+1
  8. y = y - 2. y^{\prime}=y-2.
  9. h = - 1 , k = 2. h=-1,k=2.
  10. 9 x 2 + 25 y 2 = 225. 9x^{\prime 2}+25y^{\prime 2}=225.
  11. x 2 25 + y 2 9 = 1 , \frac{x^{\prime 2}}{25}+\frac{y^{\prime 2}}{9}=1,
  12. a = 5 , b = 3 , c 2 = a 2 - b 2 = 16 , c = 4 , e = 4 5 . a=5,b=3,c^{2}=a^{2}-b^{2}=16,c=4,e=\tfrac{4}{5}.
  13. ( 0 , 0 ) (0,0)
  14. ( ± 5 , 0 ) (\pm 5,0)
  15. ( ± 4 , 0 ) . (\pm 4,0).
  16. x = x - 1 , y = y + 2 x=x^{\prime}-1,y=y^{\prime}+2
  17. ( - 1 , 2 ) (-1,2)
  18. ( 4 , 2 ) , ( - 6 , 2 ) (4,2),(-6,2)
  19. ( 3 , 2 ) , ( - 5 , 2 ) (3,2),(-5,2)
  20. 4 5 . \tfrac{4}{5}.
  21. A , B , C , , L A,B,C,\ldots,L
  22. x 2 + 4 y 2 + 3 z 2 + 2 x - 8 y + 9 z = 10. x^{2}+4y^{2}+3z^{2}+2x-8y+9z=10.
  23. x 2 + 2 x + 4 ( y 2 - 2 y ) + 3 ( z 2 + 3 z ) = 10. x^{2}+2x\qquad+4(y^{2}-2y\qquad)+3(z^{2}+3z\qquad)=10.
  24. ( x + 1 ) 2 + 4 ( y - 1 ) 2 + 3 ( z + 3 2 ) 2 = 10 + 1 + 4 + 27 4 . (x+1)^{2}+4(y-1)^{2}+3(z+\tfrac{3}{2})^{2}=10+1+4+\tfrac{27}{4}.
  25. x = x + 1 , y = y - 1 , z = z + 3 2 , x^{\prime}=x+1,\qquad y^{\prime}=y-1,\qquad z^{\prime}=z+\tfrac{3}{2},
  26. x 2 + 4 y 2 + 3 z 2 = 87 4 , x^{\prime 2}+4y^{\prime 2}+3z^{\prime 2}=\tfrac{87}{4},

Translation_operator_(quantum_mechanics).html

  1. 𝐱 \mathbf{x}
  2. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  3. 𝐱 \mathbf{x}
  4. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  5. 𝐫 \mathbf{r}
  6. ( 𝐫 + 𝐱 ) \mathbf{(r+x)}
  7. y y
  8. y y
  9. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  10. | 𝐫 |\mathbf{r}\rangle
  11. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  12. T ^ ( 𝐱 ) | 𝐫 = | 𝐫 + 𝐱 \hat{T}(\mathbf{x})|\mathbf{r}\rangle=|\mathbf{r}+\mathbf{x}\rangle
  13. ψ ( 𝐫 ) \psi(\mathbf{r})
  14. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  15. ψ ( 𝐫 ) \psi^{\prime}(\mathbf{r})
  16. ψ ( 𝐫 ) = ψ ( 𝐫 - 𝐱 ) \psi^{\prime}(\mathbf{r})=\psi(\mathbf{r}-\mathbf{x})
  17. ψ ( 𝐫 + 𝐱 ) = ψ ( 𝐫 ) \psi^{\prime}(\mathbf{r}+\mathbf{x})=\psi(\mathbf{r})
  18. | 𝐚 |\mathbf{a}\rangle
  19. ψ ( 𝐫 ) = δ ( 𝐫 - 𝐚 ) \psi(\mathbf{r})=\delta(\mathbf{r}-\mathbf{a})
  20. δ \delta
  21. T ^ ( 𝐱 ) | 𝐚 = | 𝐚 + 𝐱 \hat{T}(\mathbf{x})|\mathbf{a}\rangle=|\mathbf{a}+\mathbf{x}\rangle
  22. ψ ( 𝐫 ) = δ ( 𝐫 - ( 𝐚 + 𝐱 ) ) \psi^{\prime}(\mathbf{r})=\delta(\mathbf{r}-(\mathbf{a}+\mathbf{x}))
  23. ψ ( 𝐫 ) = ψ ( 𝐫 - 𝐱 ) \psi^{\prime}(\mathbf{r})=\psi(\mathbf{r}-\mathbf{x})
  24. 𝐫 ^ \mathbf{\hat{r}}
  25. { | 𝐫 } \left\{|\mathbf{r}\rangle\right\}
  26. { | 𝐫 } \left\{|\mathbf{r}\rangle\right\}
  27. ψ ( 𝐫 ) = 𝐫 | ψ \psi(\mathbf{r})=\langle\mathbf{r}|\psi\rangle
  28. 𝐫 ^ | ψ \mathbf{\hat{r}}|\psi\rangle
  29. { | 𝐫 } \left\{|\mathbf{r}\rangle\right\}
  30. 𝐫 ^ \mathbf{\hat{r}}
  31. | 𝐫 |\mathbf{r}\rangle
  32. 𝐫 ^ \mathbf{\hat{r}}
  33. 𝐫 \mathbf{r}
  34. 𝐫 | 𝐫 ^ | ψ = 𝐫 𝐫 | ψ = 𝐫 ψ ( 𝐫 ) \langle\mathbf{r}|\mathbf{\hat{r}}|\psi\rangle=\mathbf{r}\langle\mathbf{r}|% \psi\rangle=\mathbf{r}\psi(\mathbf{r})
  35. 𝐫 ^ \mathbf{\hat{r}}
  36. { | 𝐫 } \left\{|\mathbf{r}\rangle\right\}
  37. 𝐫 \mathbf{r}
  38. 𝐫 | T ^ ( 𝐱 ) = 𝐫 - 𝐱 | \langle\mathbf{r}|\hat{T}(\mathbf{x})=\langle\mathbf{r}-\mathbf{x}|
  39. T ^ ( 𝐱 ) | ψ \hat{T}(\mathbf{x})|\psi\rangle
  40. { | 𝐫 } \left\{|\mathbf{r}\rangle\right\}
  41. 𝐫 | T ^ ( 𝐱 ) | ψ = 𝐫 - 𝐱 | ψ = ψ ( 𝐫 - 𝐱 ) \langle\mathbf{r}|\hat{T}(\mathbf{x})|\psi\rangle=\langle\mathbf{r}-\mathbf{x}% |\psi\rangle=\psi(\mathbf{r}-\mathbf{x})
  42. p ^ x \hat{p}_{x}
  43. 𝐩 ^ \mathbf{\hat{p}}
  44. ( p ^ x , p ^ y , p ^ z ) (\hat{p}_{x},\hat{p}_{y},\hat{p}_{z})
  45. p ^ x = i lim a 0 T ^ ( a x ^ ) - 𝕀 ^ a \hat{p}_{x}=i\hbar\lim_{a\rightarrow 0}\frac{\hat{T}(a\hat{x})-\hat{\mathbb{I}% }}{a}
  46. 𝕀 ^ \hat{\mathbb{I}}
  47. x ^ \hat{x}
  48. p ^ y , p ^ z \hat{p}_{y},\hat{p}_{z}
  49. 𝐩 ^ \mathbf{\hat{p}}
  50. ψ ( 𝐫 ) \psi(\mathbf{r})
  51. 𝐩 ^ \mathbf{\hat{p}}
  52. ( p ^ ψ ) ( r ) \displaystyle(\hat{p}\psi)(r)
  53. 𝐩 ^ = - i \mathbf{\hat{p}}=-i\hbar\nabla
  54. 𝐩 ^ \mathbf{\hat{p}}
  55. 𝐩 ^ \mathbf{\hat{p}}
  56. 𝐩 ^ \mathbf{\hat{p}}
  57. 𝐩 ^ \mathbf{\hat{p}}
  58. T ^ ( 𝐱 ) \displaystyle\hat{T}(\mathbf{x})
  59. T ^ ( 𝐱 ) 1 - i 𝐱 𝐩 ^ / \hat{T}(\mathbf{x})\approx 1-i\mathbf{x}\cdot\mathbf{\hat{p}}/\hbar
  60. ψ ( 𝐫 - 𝐱 ) = T ^ ( 𝐱 ) ψ ( 𝐫 ) = exp ( - i 𝐱 𝐩 ^ ) ψ ( 𝐫 ) = ( n = 0 1 n ! ( - i 𝐱 𝐩 ^ ) n ) ψ ( 𝐫 ) = ( n = 0 1 n ! ( - 𝐱 ) n ) ψ ( 𝐫 ) = ψ ( 𝐫 ) - 𝐱 ψ ( 𝐫 ) + 1 2 ! ( 𝐱 ) 2 ψ ( 𝐫 ) - \begin{aligned}\displaystyle\psi(\mathbf{r}-\mathbf{x})&\displaystyle=\hat{T}(% \mathbf{x})\psi(\mathbf{r})\\ &\displaystyle=\exp\left(-\frac{i\mathbf{x}\cdot\mathbf{\hat{p}}}{\hbar}\right% )\psi(\mathbf{r})\\ &\displaystyle=\left(\sum_{n=0}^{\infty}\frac{1}{n!}(-\frac{i}{\hbar}\mathbf{x% }\cdot\mathbf{\hat{p}})^{n}\right)\psi(\mathbf{r})\\ &\displaystyle=\left(\sum_{n=0}^{\infty}\frac{1}{n!}(-\mathbf{x}\cdot\mathbf{% \nabla})^{n}\right)\psi(\mathbf{r})\\ &\displaystyle=\psi(\mathbf{r})-\mathbf{x}\cdot\mathbf{\nabla}\psi(\mathbf{r})% +\frac{1}{2!}(\mathbf{x}\cdot\mathbf{\nabla})^{2}\psi(\mathbf{r})-\dots\end{aligned}
  61. T ^ ( 𝐱 1 ) T ^ ( 𝐱 2 ) = T ^ ( 𝐱 1 + 𝐱 2 ) \hat{T}(\mathbf{x}_{1})\hat{T}(\mathbf{x}_{2})=\hat{T}(\mathbf{x}_{1}+\mathbf{% x}_{2})
  62. T ^ ( 𝐱 1 ) T ^ ( 𝐱 2 ) | 𝐫 = T ^ ( 𝐱 1 ) | 𝐱 2 + 𝐫 = | 𝐱 1 + 𝐱 2 + 𝐫 = T ^ ( 𝐱 1 + 𝐱 2 ) | 𝐫 \hat{T}(\mathbf{x}_{1})\hat{T}(\mathbf{x}_{2})|\mathbf{r}\rangle=\hat{T}(% \mathbf{x}_{1})|\mathbf{x}_{2}+\mathbf{r}\rangle=|\mathbf{x}_{1}+\mathbf{x}_{2% }+\mathbf{r}\rangle=\hat{T}(\mathbf{x}_{1}+\mathbf{x}_{2})|\mathbf{r}\rangle
  63. T ^ ( 𝐱 1 ) T ^ ( 𝐱 2 ) \hat{T}(\mathbf{x}_{1})\hat{T}(\mathbf{x}_{2})
  64. T ^ ( 𝐱 1 + 𝐱 2 ) \hat{T}(\mathbf{x}_{1}+\mathbf{x}_{2})
  65. ( T ^ ( 𝐱 ) ) - 1 = T ^ ( - 𝐱 ) (\hat{T}(\mathbf{x}))^{-1}=\hat{T}(-\mathbf{x})
  66. T ^ ( 0 ) = 𝕀 ^ \hat{T}(0)=\hat{\mathbb{I}}
  67. T ^ ( 𝐱 ) T ^ ( 𝐲 ) = T ^ ( 𝐲 ) T ^ ( 𝐱 ) \hat{T}(\mathbf{x})\hat{T}(\mathbf{y})=\hat{T}(\mathbf{y})\hat{T}(\mathbf{x})
  68. T ^ ( 𝐱 + 𝐲 ) \hat{T}(\mathbf{x}+\mathbf{y})
  69. ψ ( 𝐫 ) \psi(\mathbf{r})
  70. ϕ ( 𝐫 ) \phi(\mathbf{r})
  71. ψ \psi
  72. ϕ \phi
  73. d 3 𝐫 ψ * ( 𝐫 ) ϕ ( 𝐫 ) \int d^{3}\mathbf{r}\psi^{*}(\mathbf{r})\phi(\mathbf{r})
  74. T ^ ( 𝐚 ) ψ \hat{T}(\mathbf{a})\psi
  75. T ^ ( 𝐚 ) ϕ \hat{T}(\mathbf{a})\phi
  76. d 3 𝐫 ψ * ( 𝐫 - 𝐚 ) ϕ ( 𝐫 - 𝐚 ) \int d^{3}\mathbf{r}\psi^{*}(\mathbf{r}-\mathbf{a})\phi(\mathbf{r}-\mathbf{a})
  77. ( T ^ ( 𝐱 ) ) = ( T ^ ( 𝐱 ) ) - 1 . (\hat{T}(\mathbf{x}))^{\dagger}=(\hat{T}(\mathbf{x}))^{-1}.
  78. 𝐫 | T ^ ( 𝐱 ) = 𝐫 - 𝐱 | \langle\mathbf{r}|\hat{T}(\mathbf{x})=\langle\mathbf{r}-\mathbf{x}|
  79. T ^ ( 𝐱 ) | 𝐫 = | 𝐫 + 𝐱 \hat{T}(\mathbf{x})|\mathbf{r}\rangle=|\mathbf{r}+\mathbf{x}\rangle
  80. 𝐫 | T ^ ( 𝐱 ) = 𝐫 + 𝐱 | \langle\mathbf{r}|\hat{T}^{\dagger}(\mathbf{x})=\langle\mathbf{r}+\mathbf{x}|
  81. T ( 𝐱 ) = T - 1 ( 𝐱 ) = T ( - 𝐱 ) T^{\dagger}(\mathbf{x})=T^{-1}(\mathbf{x})=T(-\mathbf{x})
  82. 𝐫 | T ^ ( - 𝐱 ) = 𝐫 + 𝐱 | \langle\mathbf{r}|\hat{T}(-\mathbf{x})=\langle\mathbf{r}+\mathbf{x}|
  83. 𝐱 \mathbf{x}
  84. - 𝐱 -\mathbf{x}
  85. 𝐫 | T ^ ( 𝐱 ) = 𝐫 - 𝐱 | \langle\mathbf{r}|\hat{T}(\mathbf{x})=\langle\mathbf{r}-\mathbf{x}|
  86. 𝐱 = ( x , y , z ) \mathbf{x}=(x,y,z)
  87. T ^ ( 𝐱 ) = T ^ ( x x ^ ) T ^ ( y y ^ ) T ^ ( z z ^ ) \hat{T}(\mathbf{x})=\hat{T}(x\hat{x})\hat{T}(y\hat{y})\hat{T}(z\hat{z})
  88. x ^ , y ^ , z ^ \hat{x},\hat{y},\hat{z}
  89. | 𝐫 |\mathbf{r}\rangle
  90. 𝐫 ^ \mathbf{\hat{r}}
  91. 𝐫 \mathbf{r}
  92. T ^ ( 𝐱 ) 𝐫 ^ | 𝐫 = T ^ ( 𝐱 ) 𝐫 | 𝐫 = 𝐫 | 𝐱 + 𝐫 \hat{T}(\mathbf{x})\mathbf{\hat{r}}|\mathbf{r}\rangle=\hat{T}(\mathbf{x})% \mathbf{r}|\mathbf{r}\rangle=\mathbf{r}|\mathbf{x}+\mathbf{r}\rangle
  93. 𝐫 ^ T ^ ( 𝐱 ) | 𝐫 = 𝐫 ^ | 𝐱 + 𝐫 = ( 𝐱 + 𝐫 ) | 𝐱 + 𝐫 \mathbf{\hat{r}}\hat{T}(\mathbf{x})|\mathbf{r}\rangle=\hat{\mathbf{r}}|\mathbf% {x}+\mathbf{r}\rangle=(\mathbf{x}+\mathbf{r})|\mathbf{x}+\mathbf{r}\rangle
  94. [ 𝐫 ^ , T ^ ( 𝐱 ) ] 𝐫 ^ T ^ ( 𝐱 ) - T ^ ( 𝐱 ) 𝐫 ^ = 𝐱 T ^ ( 𝐱 ) [\mathbf{\hat{r}},\hat{T}(\mathbf{x})]\equiv\mathbf{\hat{r}}\hat{T}(\mathbf{x}% )-\hat{T}(\mathbf{x})\mathbf{\hat{r}}=\mathbf{x}\hat{T}(\mathbf{x})
  95. T ^ - 1 ( 𝐱 ) 𝐫 ^ T ^ ( 𝐱 ) = 𝐫 ^ + 𝐱 𝕀 ^ \hat{T}^{-1}(\mathbf{x})\mathbf{\hat{r}}\hat{T}(\mathbf{x})=\mathbf{\hat{r}}+% \mathbf{x}\hat{\mathbb{I}}
  96. 𝕀 ^ \hat{\mathbb{I}}
  97. T ^ ( 𝐱 ) 𝐩 ^ = 𝐩 ^ T ^ ( 𝐱 ) \hat{T}(\mathbf{x})\hat{\mathbf{p}}=\hat{\mathbf{p}}\hat{T}(\mathbf{x})
  98. 𝔗 \mathfrak{T}
  99. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  100. 𝐱 \mathbf{x}
  101. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  102. T ^ ( - 𝐱 ) \hat{T}(-\mathbf{x})
  103. T ^ ( 𝐱 1 ) ( T ^ ( 𝐱 2 ) T ^ ( 𝐱 3 ) ) = ( T ^ ( 𝐱 1 ) T ^ ( 𝐱 2 ) ) T ^ ( 𝐱 3 ) \hat{T}(\mathbf{x}_{1})\left(\hat{T}(\mathbf{x}_{2})\hat{T}(\mathbf{x}_{3})% \right)=\left(\hat{T}(\mathbf{x}_{1})\hat{T}(\mathbf{x}_{2})\right)\hat{T}(% \mathbf{x}_{3})
  104. 𝔗 \mathfrak{T}
  105. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  106. 𝐱 \mathbf{x}
  107. | ψ |\psi\rangle
  108. ψ | 𝐫 ^ | ψ \langle\psi|\mathbf{\hat{r}}|\psi\rangle
  109. 𝐫 ^ \mathbf{\hat{r}}
  110. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  111. | ψ |\psi\rangle
  112. | ψ 2 |\psi_{2}\rangle
  113. | ψ 2 |\psi_{2}\rangle
  114. | ψ |\psi\rangle
  115. | ψ 2 = T ^ ( 𝐱 ) | ψ |\psi_{2}\rangle=\hat{T}(\mathbf{x})|\psi\rangle
  116. ψ 2 | 𝐫 ^ | ψ 2 = ( ψ | T ( 𝐱 ) ) 𝐫 ^ ( T ( 𝐱 ) | ψ ) = ψ | 𝐫 ^ | ψ + 𝐱 \begin{aligned}\displaystyle\langle\psi_{2}|\hat{\mathbf{r}}|\psi_{2}\rangle&% \displaystyle=(\langle\psi|T^{\dagger}(\mathbf{x}))\hat{\mathbf{r}}(T(\mathbf{% x})|\psi\rangle)\\ &\displaystyle=\langle\psi|\hat{\mathbf{r}}|\psi\rangle+\mathbf{x}\end{aligned}
  117. ψ | ψ = 1 \langle\psi|\psi\rangle=1
  118. T ^ ( 𝐱 ) - 1 H ^ T ^ ( 𝐱 ) = H ^ \hat{T}(\mathbf{x})^{-1}\hat{H}\hat{T}(\mathbf{x})=\hat{H}
  119. [ H ^ , T ^ ( 𝐱 ) ] = 0 [\hat{H},\hat{T}(\mathbf{x})]=0
  120. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  121. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  122. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  123. 𝐩 ^ \hat{\mathbf{p}}
  124. [ H ^ , T ^ ( 𝐱 ) ] = 0 \displaystyle\left[\hat{H},\hat{T}(\mathbf{x})\right]=0
  125. V ( r j ± a ) = V ( r j ) V(r_{j}\pm a)=V(r_{j})
  126. T ^ j ( x j ) \hat{T}_{j}(x_{j})
  127. x j x_{j}
  128. T ^ j ( x j ) \hat{T}_{j}(x_{j})
  129. T ^ j ( x j ) | r j = | r j + x j \hat{T}_{j}(x_{j})|r_{j}\rangle=|r_{j}+x_{j}\rangle
  130. T ^ j ( x j ) r ^ j T ^ j ( x j ) = r ^ j + x j 𝕀 ^ \hat{T}_{j}^{\dagger}(x_{j})\hat{r}_{j}\hat{T}_{j}(x_{j})=\hat{r}_{j}+x_{j}% \hat{\mathbb{I}}
  131. 𝕀 ^ \hat{\mathbb{I}}
  132. x j x_{j}
  133. a a
  134. T ^ j ( a ) V ( r ^ j ) T ^ j ( a ) = V ( r ^ j + a 𝕀 ^ ) = V ( r ^ j ) \hat{T}_{j}^{\dagger}(a)V(\hat{r}_{j})\hat{T}_{j}(a)=V(\hat{r}_{j}+a\hat{% \mathbb{I}})=V(\hat{r}_{j})
  135. H ^ \hat{H}
  136. 𝐩 ^ \mathbf{\hat{p}}
  137. T ^ j ( a ) H ^ T ^ j ( a ) = H ^ \hat{T}_{j}^{\dagger}(a)\hat{H}\hat{T}_{j}(a)=\hat{H}
  138. V ( 𝐫 ) V(\mathbf{r})
  139. V ( 𝐫 + 𝐑 ) = V ( 𝐫 ) V(\mathbf{r+R})=V(\mathbf{r})
  140. 𝐑 \mathbf{R}
  141. 𝐑 \mathbf{R}
  142. T ^ 𝐑 \hat{T}_{\mathbf{R}}
  143. f ( 𝐫 ) f(\mathbf{r})
  144. 𝐑 \mathbf{R}
  145. T ^ 𝐑 f ( 𝐫 ) = f ( 𝐫 + 𝐑 ) \hat{T}_{\mathbf{R}}f(\mathbf{r})=f(\mathbf{r}+\mathbf{R})
  146. T ^ 𝐑 1 T ^ 𝐑 2 = T ^ 𝐑 2 T ^ 𝐑 1 = T ^ 𝐑 𝟏 + 𝐑 𝟐 \hat{T}_{\mathbf{R}_{1}}\hat{T}_{\mathbf{R}_{2}}=\hat{T}_{\mathbf{R}_{2}}\hat{% T}_{\mathbf{R}_{1}}=\hat{T}_{\mathbf{R_{1}+R_{2}}}
  147. T ^ 𝐑 H ^ = H ^ T ^ 𝐑 \hat{T}_{\mathbf{R}}\hat{H}=\hat{H}\hat{T}_{\mathbf{R}}
  148. T ^ 𝐑 \hat{T}_{\mathbf{R}}
  149. 𝐑 \mathbf{R}
  150. H ^ \hat{H}
  151. H ^ \hat{H}
  152. T ^ 𝐑 \hat{T}_{\mathbf{R}}
  153. H ^ ψ = ψ \hat{H}\psi=\mathcal{E}\psi
  154. T ^ 𝐑 ψ = c ( 𝐑 ) ψ \hat{T}_{\mathbf{R}}\psi=c(\mathbf{R})\psi
  155. c ( 𝐑 ) c(\mathbf{R})
  156. T ^ 𝐑 1 T ^ 𝐑 2 = T ^ 𝐑 2 T ^ 𝐑 1 = T ^ 𝐑 𝟏 + 𝐑 𝟐 \hat{T}_{\mathbf{R}_{1}}\hat{T}_{\mathbf{R}_{2}}=\hat{T}_{\mathbf{R}_{2}}\hat{% T}_{\mathbf{R}_{1}}=\hat{T}_{\mathbf{R_{1}+R_{2}}}
  157. T ^ 𝐑 1 T ^ 𝐑 2 ψ = c ( 𝐑 1 ) T ^ 𝐑 2 ψ = c ( 𝐑 1 ) c ( 𝐑 2 ) ψ \begin{aligned}\displaystyle\hat{T}_{\mathbf{R}_{1}}\hat{T}_{\mathbf{R}_{2}}% \psi&\displaystyle=c(\mathbf{R}_{1})\hat{T}_{\mathbf{R}_{2}}\psi\\ &\displaystyle=c(\mathbf{R}_{1})c(\mathbf{R}_{2})\psi\end{aligned}
  158. T ^ 𝐑 𝟏 + 𝐑 𝟐 ψ = c ( 𝐑 1 + 𝐑 2 ) ψ \hat{T}_{\mathbf{R_{1}+R_{2}}}\psi=c(\mathbf{R}_{1}+\mathbf{R}_{2})\psi
  159. c ( 𝐑 1 + 𝐑 2 ) = c ( 𝐑 1 ) c ( 𝐑 2 ) c(\mathbf{R}_{1}+\mathbf{R}_{2})=c(\mathbf{R}_{1})c(\mathbf{R}_{2})
  160. 𝐚 𝐢 \mathbf{a_{i}}
  161. x i x_{i}
  162. c ( 𝐚 𝐢 ) c(\mathbf{a_{i}})
  163. c ( 𝐚 𝐢 ) = e 2 π i x i c(\mathbf{a_{i}})=e^{2\pi ix_{i}}
  164. 𝐑 \mathbf{R}
  165. 𝐑 = n 1 𝐚 1 + n 2 𝐚 2 + n 3 𝐚 3 \mathbf{R}=n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}+n_{3}\mathbf{a}_{3}
  166. c ( 𝐑 ) = c ( n 1 𝐚 1 + n 2 𝐚 2 + n 3 𝐚 3 ) = c ( n 1 𝐚 1 ) c ( n 2 𝐚 2 ) c ( n 3 𝐚 3 ) = c ( 𝐚 1 ) n 1 c ( 𝐚 2 ) n 2 c ( 𝐚 3 ) n 3 \begin{aligned}\displaystyle c(\mathbf{R})&\displaystyle=c(n_{1}\mathbf{a}_{1}% +n_{2}\mathbf{a}_{2}+n_{3}\mathbf{a}_{3})\\ &\displaystyle=c(n_{1}\mathbf{a}_{1})c(n_{2}\mathbf{a}_{2})c(n_{3}\mathbf{a}_{% 3})\\ &\displaystyle=c(\mathbf{a}_{1})^{n_{1}}c(\mathbf{a}_{2})^{n_{2}}c(\mathbf{a}_% {3})^{n_{3}}\end{aligned}
  167. c ( 𝐚 𝐢 ) = e 2 π i x i c(\mathbf{a_{i}})=e^{2\pi ix_{i}}
  168. c ( 𝐑 ) = e 2 π i ( n 1 x 1 + n 2 x 2 + n 3 x 3 ) = e i 𝐤 𝐑 \begin{aligned}\displaystyle c(\mathbf{R})&\displaystyle=e^{2\pi i(n_{1}x_{1}+% n_{2}x_{2}+n_{3}x_{3})}\\ &\displaystyle=e^{i\mathbf{k}\cdot\mathbf{R}}\end{aligned}
  169. 𝐤 = 𝐛 1 x 1 + 𝐛 2 x 2 + 𝐛 3 x 3 \mathbf{k}=\mathbf{b}_{1}x_{1}+\mathbf{b}_{2}x_{2}+\mathbf{b}_{3}x_{3}
  170. 𝐛 i \mathbf{b}_{i}
  171. 𝐛 i 𝐚 j = 2 π δ i j \mathbf{b}_{i}\cdot\mathbf{a}_{j}=2\pi\delta_{ij}
  172. ψ \psi
  173. H ^ \hat{H}
  174. T ^ 𝐑 \hat{T}_{\mathbf{R}}
  175. 𝐑 \mathbf{R}
  176. ψ ( 𝐫 + 𝐑 ) = T ^ 𝐑 ψ ( 𝐫 ) = c ( 𝐑 ) ψ ( 𝐫 ) = e i 𝐤 𝐑 ψ ( 𝐫 ) \begin{aligned}\displaystyle\psi(\mathbf{r+R})&\displaystyle=\hat{T}_{\mathbf{% R}}\psi(\mathbf{r})\\ &\displaystyle=c(\mathbf{R})\psi(\mathbf{r})\\ &\displaystyle=e^{i\mathbf{k}\cdot\mathbf{R}}\psi(\mathbf{r})\end{aligned}
  177. [ T ^ ( 𝐱 ) , H ^ ] = 0 [\hat{T}(\mathbf{x}),\hat{H}]=0
  178. [ T ^ ( 𝐱 ) , U ^ ( t ) ] = 0 [\hat{T}(\mathbf{x}),\hat{U}(t)]=0
  179. U ^ ( t ) \hat{U}(t)
  180. U ^ ( t ) = exp ( - i H ^ t ) \hat{U}(t)=\exp\left(\frac{-i\hat{H}t}{\hbar}\right)
  181. p ^ \hat{p}
  182. T ^ ( 𝐱 ) \hat{T}(\mathbf{x})
  183. H ^ ( t ) \hat{H}(t)
  184. t = 0 t=0
  185. x = 0 x=0
  186. x = a x=a
  187. | ψ ( 0 ) |\psi(0)\rangle
  188. T ^ ( 𝐚 ) | ψ ( 0 ) \hat{T}\mathbf{(a)}|\psi(0)\rangle
  189. t t
  190. U ^ ( t ) | ψ ( 0 ) \hat{U}(t)|\psi(0)\rangle
  191. U ^ ( t ) T ^ ( 𝐚 ) | ψ ( 0 ) \hat{U}(t)\hat{T}\mathbf{(a)}|\psi(0)\rangle
  192. T ^ ( 𝐚 ) U ^ ( t ) | ψ ( 0 ) \hat{T}\mathbf{(a)}\hat{U}(t)|\psi(0)\rangle
  193. t = 0 t=0

Transmission_loss.html

  1. T L = 10 log 10 | W i W t | TL=10\log_{10}\left|{W_{i}\over W_{t}}\right|
  2. W i W_{i}
  3. W t W_{t}

Transmission_Loss_(duct_acoustics).html

  1. T L = L W i - L W o = 10 log 10 | S i p i + 2 2 2 S o p o 2 | = 10 log 10 | S i p i + 2 S o p o 2 | TL=L_{Wi}-L_{Wo}=10\log_{10}\left|{S_{i}p_{i+}^{2}\over 2}{2\over S_{o}p_{o}^{% 2}}\right|=10\log_{10}\left|{S_{i}p_{i+}^{2}\over S_{o}p_{o}^{2}}\right|
  2. L W i L_{Wi}
  3. L W o L_{Wo}
  4. S i , S o S_{i},S_{o}
  5. p i + p_{i+}
  6. p o p_{o}
  7. p i + p_{i+}
  8. p i - p_{i-}
  9. p o = p o + p_{o}=p_{o+}
  10. p o - = 0 p{o-}=0
  11. T L = 20 log 10 | p i + p o | TL=20\log_{10}\left|{p_{i+}\over p_{o}}\right|
  12. | p i + | \left|p_{i+}\right|
  13. | p o | \left|p_{o}\right|
  14. [ p ^ i q ^ i ] = [ A B C D ] [ p ^ o q ^ o ] \begin{bmatrix}\hat{p}_{i}\\ \hat{q}_{i}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}\hat{p}_{o}\\ \hat{q}_{o}\end{bmatrix}
  15. p ^ i \hat{p}_{i}
  16. p ^ o \hat{p}_{o}
  17. q ^ i \hat{q}_{i}
  18. q ^ o \hat{q}_{o}
  19. T L = 10 log 10 ( 1 4 | A + B S ρ c + C ρ c S + D | 2 ) TL=10\log_{10}\left({{1\over 4}\left|{A+B{S\over\rho c}+C{\rho c\over S}+D}% \right|^{2}}\right)
  20. S S
  21. ρ c \rho c
  22. [ A B C D ] = [ cos k l j ρ c S 2 sin k l j S 2 ρ c sin k l cos k l ] \begin{bmatrix}A&B\\ C&D\end{bmatrix}=\begin{bmatrix}{\cos kl}&{j{\rho c\over S_{2}}\sin kl}\\ {j{S_{2}\over\rho c}\sin kl}&{\cos kl}\end{bmatrix}
  23. T L = 10 log 10 ( 1 4 | cos k l + j S 1 S 2 sin k l + j S 2 S 1 sin k l + cos k l | 2 ) = 10 log 10 ( cos 2 k l + 1 4 ( h + 1 h ) 2 sin 2 k l ) = 10 log 10 ( 1 + 1 4 ( h - 1 h ) 2 sin 2 k l ) , \begin{aligned}\displaystyle TL&\displaystyle=10\log_{10}\left({{1\over 4}% \left|{{\cos kl}+{j{S_{1}\over S_{2}}\sin kl}+{j{S_{2}\over S_{1}}\sin kl}+{% \cos kl}}\right|^{2}}\right)\\ &\displaystyle=10\log_{10}\left({{\cos^{2}kl}+{1\over 4}\left(h+{1\over h}% \right)^{2}{\sin^{2}kl}}\right)\\ &\displaystyle=10\log_{10}\left({1+{1\over 4}\left(h-{1\over h}\right)^{2}{% \sin^{2}kl}}\right),\end{aligned}
  24. h h
  25. l l
  26. k = ω / c k=\omega/c
  27. c c
  28. l l
  29. c 2 l c\over{2l}
  30. c 4 l c\over{4l}
  31. f c = 1.84 c π D f_{c}=1.84{c\over{\pi D}}

TraPPE_force_field.html

  1. 2 {}_{2}
  2. 2 {}_{2}
  3. 2 {}_{2}
  4. 3 {}_{3}
  5. U ( r N ) = j = 1 N - 1 i = j + 1 N { 4 ϵ i j [ ( σ i j r i j ) 12 - ( σ i j r i j ) 6 ] + q i q j 4 π ϵ 0 r i j } + angles k a ( θ - θ 0 ) 2 2 + U torsion U(r^{N})=\sum_{j=1}^{N-1}\sum_{i=j+1}^{N}\biggl\{4\epsilon_{ij}\biggl[\left(% \frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)% ^{6}\biggr]+\frac{q_{i}q_{j}}{4\pi\epsilon_{0}r_{ij}}\biggr\}\ +\ \sum\text{% angles}\frac{{k\text{a}(\theta-\theta_{0})^{2}}}{2}\ +\ \ U\text{torsion}
  6. x {}_{x}
  7. σ i j \sigma_{ij}
  8. R 0 , i j R_{0,ij}
  9. σ i j = R 0 , i j / 2 1 / 6 \sigma_{ij}=R_{0,ij}/2^{1/6}
  10. ϵ i j \epsilon_{ij}

TRevPAR.html

  1. T R e v P A R = T o t a l R e v e n u e / R o o m s A v a i l a b l e TRevPAR=TotalRevenue/RoomsAvailable\,

Triangular_tiling_honeycomb.html

  1. Y ¯ \overline{Y}
  2. P P ¯ 3 {\bar{PP}}_{3}
  3. Y ¯ \overline{Y}

Tricyclene_synthase.html

  1. \rightleftharpoons

Trigonal_trapezohedral_honeycomb.html

  1. 3 ¯ \overline{3}
  2. A ~ 3 {\tilde{A}}_{3}

Trinomial_triangle.html

  1. 1 1 1 1 1 2 3 2 1 1 3 6 7 6 3 1 1 4 10 16 19 16 10 4 1 \begin{matrix}&&&&1\\ &&&1&1&1\\ &&1&2&3&2&1\\ &1&3&6&7&6&3&1\\ 1&4&10&16&19&16&10&4&1\end{matrix}
  2. k k
  3. n n
  4. ( n k ) 2 {n\choose k}_{2}
  5. n n
  6. - n -n
  7. ( n k ) 2 = ( n - k ) 2 {n\choose k}_{2}={n\choose-k}_{2}
  8. n n
  9. ( 1 + x + x 2 ) (1+x+x^{2})
  10. n n
  11. ( 1 + x + x 2 ) n = j = 0 2 n ( n j - n ) 2 x j = k = - n n ( n k ) 2 x n + k \left(1+x+x^{2}\right)^{n}=\sum_{j=0}^{2n}{n\choose j-n}_{2}x^{j}=\sum_{k=-n}^% {n}{n\choose k}_{2}x^{n+k}
  12. ( 1 + x + 1 / x ) n = k = - n n ( n k ) 2 x k \left(1+x+1/x\right)^{n}=\sum_{k=-n}^{n}{n\choose k}_{2}x^{k}
  13. ( n k ) 2 = 0 μ , ν n μ + 2 ν = n + k n ! μ ! ν ! ( n - μ - ν ) ! {n\choose k}_{2}=\sum_{\textstyle{0\leq\mu,\nu\leq n\atop\mu+2\nu=n+k}}\frac{n% !}{\mu!\,\nu!\,(n-\mu-\nu)!}
  14. n n
  15. 3 n 3^{n}
  16. ( 0 0 ) 2 = 1 {0\choose 0}_{2}=1
  17. ( n + 1 k ) 2 = ( n k - 1 ) 2 + ( n k ) 2 + ( n k + 1 ) 2 {n+1\choose k}_{2}={n\choose k-1}_{2}+{n\choose k}_{2}+{n\choose k+1}_{2}
  18. n 0 n\geq 0
  19. ( n k ) 2 = 0 {n\choose k}_{2}=0
  20. k < - n \ k<-n
  21. k > n \ k>n
  22. n n
  23. ( n 0 ) 2 = k = 0 n n ( n - 1 ) ( n - 2 k + 1 ) ( k ! ) 2 = k = 0 n ( n 2 k ) ( 2 k k ) . {n\choose 0}_{2}=\sum_{k=0}^{n}\frac{n(n-1)\cdots(n-2k+1)}{(k!)^{2}}=\sum_{k=0% }^{n}{n\choose 2k}{2k\choose k}.
  24. 1 + x + 3 x 2 + 7 x 3 + 19 x 4 + = 1 ( 1 + x ) ( 1 - 3 x ) . 1+x+3x^{2}+7x^{3}+19x^{4}+\ldots=\frac{1}{\sqrt{(1+x)(1-3x)}}.
  25. 3 ( n + 1 0 ) 2 - ( n + 2 0 ) 2 = f n ( f n + 1 ) 3{n+1\choose 0}_{2}-{n+2\choose 0}_{2}=f_{n}(f_{n}+1)
  26. 0 n 7 0\leq n\leq 7
  27. f n f_{n}
  28. n n
  29. 2 k [ ( n + 1 10 k ) 2 - ( n + 1 10 k + 1 ) 2 ] = f n ( f n + 1 ) . 2\sum_{k\in\mathbb{Z}}\left[{n+1\choose 10k}_{2}-{n+1\choose 10k+1}_{2}\right]% =f_{n}(f_{n}+1).
  30. x k x^{k}
  31. ( 1 + x + x 2 ) n \left(1+x+x^{2}\right)^{n}
  32. k k
  33. n n
  34. ( 24 12 - 24 ) 2 = ( 24 - 12 ) 2 = ( 24 12 ) 2 {24\choose 12-24}_{2}={24\choose-12}_{2}={24\choose 12}_{2}
  35. p p
  36. ( n p ) {n\choose p}
  37. k - 2 p k-2p
  38. ( n - p k - 2 p ) {n-p\choose k-2p}
  39. ( n k - n ) 2 = p = max ( 0 , k - n ) min ( n , [ k / 2 ] ) ( n p ) ( n - p k - 2 p ) {n\choose k-n}_{2}=\sum_{p=\max(0,k-n)}^{\min(n,[k/2])}{n\choose p}{n-p\choose k% -2p}
  40. 6 = ( 3 2 - 3 ) 2 = ( 3 0 ) ( 3 2 ) + ( 3 1 ) ( 2 0 ) = 1 3 + 3 1 6={3\choose 2-3}_{2}={3\choose 0}{3\choose 2}+{3\choose 1}{2\choose 0}=1\cdot 3% +3\cdot 1

Triple_exponential_moving_average.html

  1. 𝑇𝐸𝑀𝐴 = 3 * E M A - 3 * E M A ( E M A ) + E M A ( E M A ( E M A ) ) \,\textit{TEMA}={3*EMA-3*EMA(EMA)+EMA(EMA(EMA))}

TRNA_pseudouridine13_synthase.html

  1. \rightleftharpoons

TRNA_pseudouridine31_synthase.html

  1. \rightleftharpoons

TRNA_pseudouridine32_synthase.html

  1. \rightleftharpoons

TRNA_pseudouridine38::39_synthase.html

  1. \rightleftharpoons

TRNA_pseudouridine55_synthase.html

  1. \rightleftharpoons

TRNA_pseudouridine65_synthase.html

  1. \rightleftharpoons

TRNAIle-lysidine_synthase.html

  1. \rightleftharpoons

Trochoidal_wave.html

  1. H H
  2. λ \lambda
  3. c . c.
  4. X ( a , b , t ) = a + e k b k sin k ( a + c t ) , Y ( a , b , t ) = b - e k b k cos k ( a + c t ) , \begin{aligned}\displaystyle X(a,b,t)&\displaystyle=a+\frac{\mbox{e}~{}^{kb}}{% k}\sin k(a+ct),\\ \displaystyle Y(a,b,t)&\displaystyle=b-\frac{\mbox{e}~{}^{kb}}{k}\cos k(a+ct),% \end{aligned}
  5. x = X ( a , b , t ) x=X(a,b,t)
  6. y = Y ( a , b , t ) y=Y(a,b,t)
  7. ( x , y ) (x,y)
  8. t t
  9. x x
  10. y y
  11. ( a , b ) (a,b)
  12. ( x , y ) = ( a , b ) (x,y)=(a,b)
  13. c exp ( k b ) . c\,\exp(kb).
  14. k = 2 π / λ k=2\pi/\lambda
  15. λ \lambda
  16. c c
  17. x x
  18. c 2 = g k , c^{2}=\frac{g}{k},
  19. H H
  20. c c
  21. b = b s b=b_{s}
  22. b s b_{s}
  23. b s = 0 b_{s}=0
  24. H = ( 2 / k ) exp ( k b s ) . H=(2/k)\exp(kb_{s}).
  25. x x
  26. λ ; \lambda;
  27. T = λ / c = 2 π λ / g . T=\lambda/c=\sqrt{2\pi\lambda/g}.
  28. x x
  29. z z
  30. y y
  31. y = 0 y=0
  32. y y
  33. g . g.
  34. α \alpha
  35. β , \beta,
  36. t . t.
  37. ( x , y , z ) = ( α , 0 , β ) (x,y,z)=(\alpha,0,\beta)
  38. x = ξ ( α , β , t ) , x=\xi(\alpha,\beta,t),
  39. y = ζ ( α , β , t ) y=\zeta(\alpha,\beta,t)
  40. z = η ( α , β , t ) z=\eta(\alpha,\beta,t)
  41. ξ \displaystyle\xi
  42. tanh \tanh
  43. M M
  44. a m a_{m}
  45. m = 1 M {m=1\dots M}
  46. ϕ m \phi_{m}
  47. k m = ( k x , m 2 + k z , m 2 ) {k_{m}=\scriptstyle\sqrt{(k_{x,m}^{2}+k_{z,m}^{2})}}
  48. ω m \omega_{m}
  49. k m k_{m}
  50. ω m , \omega_{m},
  51. ω m 2 = g k m tanh ( k m h ) , \omega_{m}^{2}=g\,k_{m}\,\tanh\left(k_{m}\,h\right),
  52. h h
  53. h h\to\infty
  54. tanh ( k m h ) 1. {\tanh(k_{m}\,h)\to 1.}
  55. k x , m k_{x,m}
  56. k z , m k_{z,m}
  57. s y m b o l k m symbol{k}_{m}
  58. m . m.
  59. a m , k x , m , k z , m a_{m},k_{x,m},k_{z,m}
  60. ϕ m \phi_{m}
  61. m = 1 M , {m=1\dots{M},}
  62. h h
  63. ( k x , k z ) (k_{x},k_{z})
  64. a m a_{m}
  65. ϕ m \phi_{m}
  66. s y m b o l n symbol{n}
  67. × \times
  68. s y m b o l n = \partialsymbol s α × \partialsymbol s β with s y m b o l s ( α , β , t ) = ( ξ ( α , β , t ) ζ ( α , β , t ) η ( α , β , t ) ) . symbol{n}=\frac{\partialsymbol{s}}{\partial\alpha}\times\frac{\partialsymbol{s% }}{\partial\beta}\quad\,\text{with}\quad symbol{s}(\alpha,\beta,t)=\begin{% pmatrix}\xi(\alpha,\beta,t)\\ \zeta(\alpha,\beta,t)\\ \eta(\alpha,\beta,t)\end{pmatrix}.
  69. s y m b o l e n = s y m b o l n / s y m b o l n , symbol{e}_{n}=symbol{n}/\|symbol{n}\|,
  70. s y m b o l n \|symbol{n}\|
  71. s y m b o l n . symbol{n}.

Trommel_screen.html

  1. P = ( 1 - d a ) 2 Q P=(1-\dfrac{d}{a})^{2}Q\,
  2. d d
  3. a a
  4. Q Q
  5. P = ( 1 - d a ) ( 1 - d A ) Q P=(1-\dfrac{d}{a})(1-\dfrac{d}{A})Q\,
  6. a a
  7. A A
  8. V V
  9. n n
  10. σ t \sigma_{t}
  11. V ( t ) = ( 1 - P ) σ t t V(t)=(1-P)^{\sigma^{t}_{t}}\,
  12. V ( t ) = W ( t ) W ( 0 ) V(t)=\dfrac{W(t)}{W(0)}\,
  13. W ( t ) W(t)
  14. t t
  15. W ( 0 ) W(0)
  16. d W ( t ) d t = σ t l n ( 1 - P ) W ( t ) \dfrac{dW(t)}{dt}=\sigma_{t}ln(1-P)W(t)\,
  17. f ( x ) f(x)
  18. x 0 x_{0}
  19. x m x_{m}
  20. n n
  21. P ( x 0 , x m ) = x 0 x m f ( x ) ( 1 - ( 1 - p ) n ) d x P(x_{0},x_{m})=\int_{x_{0}}^{x_{m}}f(x)\cdot(1-(1-p)^{n})\,dx
  22. F ( x 0 , x m ) = x 0 x m f ( x ) d x F({x_{0}},{x_{m}})=\int\limits_{x_{0}}^{x_{m}}f(x)\ dx
  23. E ( x 0 , x m ) = P ( x 0 , x m ) F ( x 0 , x m ) E(x_{0},x_{m})=\frac{P(x_{0},x_{m})}{F(x_{0},x_{m})}
  24. α = cos - 1 ( r ω t 2 g cos β ) \alpha={\cos^{-1}}(\frac{r{\cdot}\omega^{2}_{t}}{g{\cdot}\cos\beta})
  25. r r
  26. ω t \omega_{t}
  27. g g
  28. β \beta
  29. t r = L ( 360 - 4 α + 229.2 cos α sin α ) 48 n r tan β cos α ( sin α ) 2 t_{r}=\frac{L\cdot(360-4\alpha+229.2\cdot\cos\alpha\cdot\sin\alpha)}{48\cdot{n% }\cdot{r}\cdot\tan\beta\cdot\cos\alpha\cdot(\sin\alpha)^{2}}
  30. L L
  31. n n
  32. α \alpha
  33. V V
  34. V y V_{y}
  35. V x V_{x}
  36. θ \theta
  37. V y > V x V_{y}>V_{x}
  38. V y < V x V_{y}<V_{x}
  39. V V

Truncated_rhombicosidodecahedron.html

  1. t r { 5 3 } tr\begin{Bmatrix}5\\ 3\end{Bmatrix}

Truncated_rhombicuboctahedron.html

  1. t r { 4 3 } tr\begin{Bmatrix}4\\ 3\end{Bmatrix}

Truncated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}

Tubular_pinch_effect.html

  1. v p v_{p}
  2. d T d_{T}
  3. d p d_{p}
  4. r * r^{*}
  5. w * w^{*}
  6. R e Re
  7. r r
  8. v p = 0 , 17 w * R e ( d p d T ) 2 , 84 2 r d T ( 1 - r r * ) v\text{p}=0{,}17\cdot w^{*}\cdot Re\cdot{\left(\frac{d\text{p}}{d\text{T}}% \right)}^{2{,}84}\cdot\frac{2r}{d\text{T}}\cdot\left(1-\frac{r}{r^{*}}\right)

Tunnel_field-effect_transistor.html

  1. n = 63 m V / d e c . V t h = 315 m V I o n / I o f f = V t h / n = 5 d e c . = 10 , 000 n=63mV/dec.\,\,\,V_{th}=315mV\rightarrow I_{on}/I_{off}=V_{th}/n=5\,dec.=10,000

Turán's_method.html

  1. s ν = n = 1 N b n z n ν s_{\nu}=\sum_{n=1}^{N}b_{n}z_{n}^{\nu}
  2. | z n | 1 |z_{n}|\geq 1
  3. c ( M , N ) = ( k = 0 N - 1 ( M + k k ) 2 k ) - 1 . c(M,N)=\left({\sum_{k=0}^{N-1}{\left({{M+k}\atop{k}}\right)}2^{k}}\right)^{-1}\ .
  4. ( N 2 e ( M + N ) ) N - 1 \left({\frac{N}{2e(M+N)}}\right)^{N-1}
  5. | z n | 1 |z_{n}|\leq 1
  6. | s ν | 2 ( N 8 e ( M + N ) ) N min 1 j N | n = 1 j b n | . |s_{\nu}|\geq 2\left({\frac{N}{8e(M+N)}}\right)^{N}\min_{1\leq j\leq N}\left|{% \sum_{n=1}^{j}b_{n}}\right|\ .

Turbine_inlet_air_cooling.html

  1. m = ρ V {m}={\rho}{V}
  2. m m
  3. ρ \rho
  4. V {V}
  5. V {V}
  6. ρ \rho
  7. ρ humid air = p d R d T + p v R v T = p d M d + p v M v R T \rho_{\,\mathrm{humid~{}air}}=\frac{p_{d}}{R_{d}T}+\frac{p_{v}}{R_{v}T}=\frac{% p_{d}M_{d}+p_{v}M_{v}}{RT}\,
  8. ρ humid air = \rho_{\,\mathrm{humid~{}air}}=
  9. p d = p_{d}=
  10. R d = R_{d}=
  11. T = T=
  12. p v = p_{v}=
  13. R v = R_{v}=
  14. M d = M_{d}=
  15. M v = M_{v}=
  16. R = R=

Turning_point_test.html

  1. z = T - 2 n - 4 3 16 n - 29 90 z=\frac{T-\frac{2n-4}{3}}{\sqrt{\frac{16n-29}{90}}}

Tuttminx.html

  1. 60 ! × 60 ! × 30 ! × 2 29 8 1.2325 × 10 204 \frac{60!\times 60!\times 30!\times 2^{29}}{8}\approx 1.2325\times 10^{204}

Tübingen_triangle.html

  1. φ = a b = 1 + 5 2 1.618. \varphi=\frac{a}{b}=\frac{1+\sqrt{5}}{2}\approx 1.618.

Twist_(mathematics).html

  1. X = X ( s ) X=X(s)
  2. s s
  3. X X
  4. U = U ( s ) U=U(s)
  5. X X
  6. ( X , U ) (X,U)
  7. X X
  8. X = X + ε U X^{\prime}=X+\varepsilon U
  9. T w Tw
  10. X X^{\prime}
  11. X X
  12. T w = 1 2 π ( d U d s × U ) d X d s d s , Tw=\dfrac{1}{2\pi}\int\left(\dfrac{dU}{ds}\times U\right)\cdot\dfrac{dX}{ds}ds\;,
  13. d X / d s dX/ds
  14. X X
  15. T w Tw
  16. T T
  17. N N
  18. T w = 1 2 π τ d s + [ Θ ] X 2 π = T + N , Tw=\dfrac{1}{2\pi}\int\tau\;ds+\dfrac{\left[\Theta\right]_{X}}{2\pi}=T+N\;,
  19. τ = τ ( s ) \tau=\tau(s)
  20. X X
  21. [ Θ ] X \left[\Theta\right]_{X}
  22. U U
  23. X X
  24. T w Tw
  25. U U
  26. X X
  27. T w Tw
  28. W r Wr
  29. X X
  30. L k = W r + T w Lk=Wr+Tw

Twisted_polynomial_ring.html

  1. p p
  2. τ \tau
  3. x x p x\mapsto x^{p}
  4. τ x = x p τ \tau x=x^{p}\tau
  5. x x
  6. k k
  7. p p
  8. k { τ } k\{\tau\}
  9. τ \tau
  10. k k
  11. τ x = x p τ \tau x=x^{p}\tau
  12. ( a + b τ ) ( c + d τ ) = a ( c + d τ ) + b τ ( c + d τ ) = a c + a d τ + b c p τ + b d p τ 2 (a+b\tau)(c+d\tau)=a(c+d\tau)+b\tau(c+d\tau)=ac+ad\tau+bc^{p}\tau+bd^{p}\tau^{2}
  13. k { τ } k [ x ] , a 0 + a 1 τ + + a n τ n a 0 x + a 1 x p + + a n x p n k\{\tau\}\to k[x],\quad a_{0}+a_{1}\tau+\cdots+a_{n}\tau^{n}\mapsto a_{0}x+a_{% 1}x^{p}+\cdots+a_{n}x^{p^{n}}
  14. ( a x + b x p ) ( c x + d x p ) = a ( c x + d x p ) + b ( c x + d x p ) p = a c x + a d x p + b c p x p + b d p x p 2 , (ax+bx^{p})\circ(cx+dx^{p})=a(cx+dx^{p})+b(cx+dx^{p})^{p}=acx+adx^{p}+bc^{p}x^% {p}+bd^{p}x^{p^{2}},
  15. p p
  16. ( x + y ) p = x p + y p (x+y)^{p}=x^{p}+y^{p}
  17. k k
  18. k k
  19. k k
  20. x q - x x^{q}-x
  21. q q

Two-dimensional_flow.html

  1. X - Y X-Y
  2. ( x , y , z ) (x,y,z)
  3. t t
  4. s y m b o l v ¯ ( x , y , z , t ) = v x ( x , y , z , t ) s y m b o l i ^ + v y ( x , y , z , t ) s y m b o l j ^ . \bar{symbol{v}}(x,y,z,t)=v_{x}(x,y,z,t)\hat{symbol{i}}+v_{y}(x,y,z,t)\hat{% symbol{j}}.
  5. r - θ r-\theta
  6. ( r , θ , z ) (r,\theta,z)
  7. t t
  8. s y m b o l v ¯ ( r , θ , z , t ) = v r ( r , θ , z , t ) s y m b o l r ^ + v θ ( r , θ , z , t ) s y m b o l θ ^ . \bar{symbol{v}}(r,\theta,z,t)=v_{r}(r,\theta,z,t)\hat{symbol{r}}+v_{\theta}(r,% \theta,z,t)\hat{symbol{\theta}}.
  9. X - Y X-Y
  10. s y m b o l ω ¯ = ω z s y m b o l k ^ , \bar{symbol{\omega}}=\omega_{z}\hat{symbol{k}},
  11. ω z = v y x - v x y . \omega_{z}=\frac{\partial v_{y}}{\partial x}-\frac{\partial v_{x}}{\partial y}.
  12. r - θ r-\theta
  13. s y m b o l ω ¯ = ω z s y m b o l k ^ \bar{symbol{\omega}}=\omega_{z}\hat{symbol{k}}
  14. ω z = 1 r r ( r ψ r ) + 1 r 2 2 ψ r 2 . \omega_{z}=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial\psi}{% \partial r})+\frac{1}{r^{2}}\frac{\partial^{2}\psi}{\partial r^{2}}.
  15. Q Q
  16. Z Z
  17. Q Q
  18. Q Q
  19. s y m b o l v ¯ = v r ( r ) s y m b o l r ^ . \bar{symbol{v}}=v_{r}(r)\hat{symbol{r}}.
  20. S s y m b o l v ¯ d s y m b o l S ¯ = 2 π r v r ( r ) = Q , \int\limits_{S}\bar{symbol{v}}\cdot{d\bar{symbol{S}}}=2\pi rv_{r}(r)=Q,
  21. v r ( r ) = Q 2 π r . \therefore\;\;v_{r}(r)=\frac{Q}{2\pi r}.
  22. ψ ( r , θ ) = - Q 2 π θ . \psi(r,\theta)=-\frac{Q}{2\pi}\theta.
  23. ϕ ( r , θ ) = - Q 2 π ln r . \phi(r,\theta)=-\frac{Q}{2\pi}\ln r.
  24. s y m b o l v ¯ = - v r ( r ) s y m b o l e ^ r , \bar{symbol{v}}=-v_{r}(r)\hat{symbol{e}}_{r},
  25. v r ( r ) = Q 2 π r . v_{r}(r)=\frac{Q}{2\pi r}.
  26. ψ ( r , θ ) = Q 2 π θ . \psi(r,\theta)=\frac{Q}{2\pi}\theta.
  27. ϕ ( r , θ ) = Q 2 π ln r . \phi(r,\theta)=\frac{Q}{2\pi}\ln r.
  28. inf \inf
  29. r = 0 r=0
  30. s y m b o l v = v θ s y m b o l e ^ θ , symbol{v}=v_{\theta}\hat{symbol{e}}_{\theta},
  31. v θ = K 2 π r . v_{\theta}=\frac{K}{2\pi r}.
  32. v r = 0. v_{r}=0.
  33. ψ = - K 2 π ln r . \psi=-\frac{K}{2\pi}\ln r.
  34. ϕ = - K 2 π θ . \phi=-\frac{K}{2\pi}\theta.
  35. Γ = K \Gamma=K
  36. Γ = 0 \Gamma=0
  37. Q Q
  38. d s ds
  39. Λ = Q d s 2 π . \Lambda=Q\frac{ds}{2\pi}.
  40. s y m b o l v = v r s y m b o l e ^ r + v θ s y m b o l e ^ θ , symbol{v}=v_{r}\hat{symbol{e}}_{r}+v_{\theta}\hat{symbol{e}}_{\theta},
  41. v r = - Λ r 2 cos θ , v_{r}=-\frac{\Lambda}{r^{2}}\cos\theta,
  42. v θ = - Λ r 2 sin θ . v_{\theta}=\frac{-\Lambda}{r^{2}}\sin\theta.
  43. d s 0 ds\rightarrow 0

Two-Higgs-doublet_model.html

  1. h h
  2. H H
  3. H H
  4. h h
  5. A A
  6. H ± H^{\pm}
  7. m h , m H , m A , m H ± m_{h},m_{H},m_{A},m_{H^{\pm}}
  8. tan β \tan\beta
  9. α \alpha
  10. cos ( β - α ) = 0 \cos(\beta-\alpha)=0
  11. h h
  12. Φ \Phi
  13. Φ 2 \Phi_{2}
  14. Φ 2 \Phi_{2}
  15. Φ 2 \Phi_{2}
  16. Φ 2 \Phi_{2}
  17. Φ 1 \Phi_{1}
  18. Φ 1 \Phi_{1}
  19. Φ 2 \Phi_{2}
  20. Φ 2 \Phi_{2}
  21. Φ 1 \Phi_{1}
  22. Φ 2 \Phi_{2}
  23. Φ 1 \Phi_{1}
  24. Φ 2 \Phi_{2}
  25. Φ 1 , Φ 2 \Phi_{1},\Phi_{2}
  26. Φ 1 , Φ 2 \Phi_{1},\Phi_{2}
  27. Φ 1 , Φ 2 \Phi_{1},\Phi_{2}
  28. Φ 2 \Phi_{2}

Two-photon_circular_dichroism.html

  1. Δ δ ( λ ) = δ L T P A ( λ ) - δ R T P A ( λ ) \Delta\delta(\lambda)=\delta_{L}^{TPA}(\lambda)-\delta_{R}^{TPA}(\lambda)
  2. Δ δ T P C D ( ω ) = 4 15 ( 2 π ) 3 c 0 3 ( 4 π ϵ 0 ) 2 × ω 2 f g ( 2 ω , ω 0 f , Γ ) \sdot R 0 f T P C D ( ω 0 f ) \Delta\delta^{TPCD}(\omega)=\frac{4}{15}\frac{(2\pi)^{3}}{c_{0}^{3}(4\pi% \epsilon_{0})^{2}}\times\omega^{2}\sum_{f}g(2\omega,\omega_{0f},\Gamma)\sdot R% _{0f}^{TPCD}(\omega_{0f})
  3. ω \omega
  4. ω 0 f \omega_{0f}
  5. R 0 f T P C D ( ω 0 f ) R^{TPCD}_{0f}(\omega_{0f})
  6. g ( 2 ω , 2 ω 0 f , Γ ) g(2\omega,2\omega_{0f},\Gamma)
  7. ϵ 0 \epsilon_{0}
  8. c 0 c_{0}
  9. R 0 f T P C D ( ω 0 f ) R^{TPCD}_{0f}(\omega_{0f})
  10. R 0 f T P C D ( ω 0 f ) = - b 1 B 1 T I ( ω 0 f ) - b 2 B 2 T I ( ω 0 f ) - b 3 B 3 T I ( ω 0 f ) R^{TPCD}_{0f}(\omega_{0f})=-b_{1}B^{TI}_{1}(\omega_{0f})-b_{2}B^{TI}_{2}(% \omega_{0f})-b_{3}B^{TI}_{3}(\omega_{0f})
  11. b n b_{n}
  12. b 1 = 6 b_{1}=6
  13. b 2 = - b 3 = 2 b_{2}=-b_{3}=2
  14. B 1 T I ( ω 0 f ) = 1 ω 3 ρ σ M ρ σ p , 0 f ( ω 0 f ) P ρ σ p * , 0 f ( ω 0 f ) B^{TI}_{1}(\omega_{0f})=\frac{1}{\omega^{3}}\sum_{\rho\sigma}M^{p,0f}_{\rho% \sigma}(\omega_{0f})P^{p^{*},0f}_{\rho\sigma}(\omega_{0f})
  15. B 2 T I ( ω 0 f ) = 1 2 ω 3 ρ σ T ρ σ + , 0 f ( ω 0 f ) P ρ σ p * , 0 f ( ω 0 f ) B^{TI}_{2}(\omega_{0f})=\frac{1}{2\omega^{3}}\sum_{\rho\sigma}T^{+,0f}_{\rho% \sigma}(\omega_{0f})P^{p^{*},0f}_{\rho\sigma}(\omega_{0f})
  16. B 3 T I ( ω 0 f ) = 1 ω 3 ρ σ M ρ σ p , 0 f ( ω 0 f ) P σ σ p * , 0 f ( ω 0 f ) B^{TI}_{3}(\omega_{0f})=\frac{1}{\omega^{3}}\sum_{\rho\sigma}M^{p,0f}_{\rho% \sigma}(\omega_{0f})P^{p^{*},0f}_{\sigma\sigma}(\omega_{0f})
  17. M ρ σ p , 0 f ( ω 0 f ) M^{p,0f}_{\rho\sigma}(\omega_{0f})
  18. P ρ σ p * , 0 f ( ω 0 f ) P^{p^{*},0f}_{\rho\sigma}(\omega_{0f})
  19. T ρ σ + , 0 f ( ω 0 f ) T^{+,0f}_{\rho\sigma}(\omega_{0f})

Two-ray_ground-reflection_model.html

  1. r l o s ( t ) = R e { λ G l o s 4 π × s ( t ) e - j 2 π l / λ l } r_{los}(t)=Re\left\{\frac{\lambda\sqrt{G_{los}}}{4\pi}\times\frac{s(t)e^{-j2% \pi l/\lambda}}{l}\right\}
  2. r g r ( t ) = R e { λ Γ ( θ ) G g r 4 π × s ( t - τ ) e - j 2 π ( x + x ) / λ x + x } r_{gr}(t)=Re\left\{\frac{\lambda\Gamma(\theta)\sqrt{G_{gr}}}{4\pi}\times\frac{% s(t-\tau)e^{-j2\pi(x+x^{\prime})/\lambda}}{x+x^{\prime}}\right\}
  3. s ( t ) s(t)
  4. Γ ( θ ) \Gamma(\theta)
  5. τ \tau
  6. ( x + x - l ) / c (x+x^{\prime}-l)/c
  7. Γ ( θ ) = sin θ - X sin θ + X \Gamma(\theta)=\frac{\sin\theta-X}{\sin\theta+X}
  8. X v = X_{v}=
  9. ε g - cos 2 θ ε g {\sqrt{\varepsilon_{g}-{\cos}^{2}\theta}}\over{\varepsilon_{g}}
  10. X h = ε g - cos 2 θ X_{h}=\sqrt{\varepsilon_{g}-{\cos}^{2}\theta}
  11. x + x = ( h t + h r ) 2 + d 2 x+x^{\prime}=\sqrt{(h_{t}+h_{r})^{2}+d^{2}}
  12. l = ( h t - h r ) 2 + d 2 l=\sqrt{(h_{t}-h_{r})^{2}+d^{2}}
  13. Δ d = x + x - l = ( h t + h r ) 2 + d 2 - ( h t - h r ) 2 + d 2 \Delta d=x+x^{\prime}-l=\sqrt{(h_{t}+h_{r})^{2}+d^{2}}-\sqrt{(h_{t}-h_{r})^{2}% +d^{2}}
  14. Δ ϕ = 2 π Δ d λ \Delta\phi=\frac{2\pi\Delta d}{\lambda}
  15. r l o s 2 + r g r 2 r_{los}^{2}+r_{gr}^{2}
  16. τ \tau
  17. s ( t ) = s ( t - τ ) s(t)=s(t-\tau)
  18. | s ( t ) | 2 ( λ 4 π ) 2 × ( G l o s × e - j 2 π l / λ l + Γ ( θ ) G g r e - j 2 π ( x + x ) / λ x + x ) 2 = P t ( λ 4 π ) 2 × ( G l o s l + Γ ( θ ) G g r e - j Δ ϕ x + x ) 2 |s(t)|^{2}\left({\frac{\lambda}{4\pi}}\right)^{2}\times\left(\frac{\sqrt{G_{% los}}\times e^{-j2\pi l/\lambda}}{l}+\Gamma(\theta)\sqrt{G_{gr}}\frac{e^{-j2% \pi(x+x^{\prime})/\lambda}}{x+x^{\prime}}\right)^{2}=P_{t}\left({\frac{\lambda% }{4\pi}}\right)^{2}\times\left(\frac{\sqrt{G_{los}}}{l}+\Gamma(\theta)\sqrt{G_% {gr}}\frac{e^{-j\Delta\phi}}{x+x^{\prime}}\right)^{2}
  19. P t P_{t}
  20. d d
  21. x + x - l x+x^{\prime}-l
  22. x + x - l = ( h t + h r ) 2 + d 2 - ( h t - h r ) 2 + d 2 = d ( ( h t + h r ) 2 d 2 + 1 - ( h t - h r ) 2 d 2 + 1 ) \begin{aligned}\displaystyle x+x^{\prime}-l&\displaystyle=\sqrt{(h_{t}+h_{r})^% {2}+d^{2}}-\sqrt{(h_{t}-h_{r})^{2}+d^{2}}\\ &\displaystyle=d\Bigg(\sqrt{\frac{(h_{t}+h_{r})^{2}}{d^{2}}+1}-\sqrt{\frac{(h_% {t}-h_{r})^{2}}{d^{2}}+1}\Bigg)\\ \end{aligned}
  23. 1 + x \sqrt{1+x}
  24. 1 + x = n = 0 ( - 1 ) n ( 2 n ) ! ( 1 - 2 n ) ( n ! ) 2 ( 4 n ) x n = 1 + 1 2 x - 1 8 x 2 + 1 16 x 3 - 5 128 x 4 + , \sqrt{1+x}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}x^{n}% =1+\textstyle\frac{1}{2}x-\frac{1}{8}x^{2}+\frac{1}{16}x^{3}-\frac{5}{128}x^{4% }+\dots,\!
  25. x + x - l d 2 × ( ( h t + h r ) 2 d 2 - ( h t - h r ) 2 d 2 ) = 2 h t h r d x+x^{\prime}-l\approx\frac{d}{2}\times\left(\frac{(h_{t}+h_{r})^{2}}{d^{2}}-% \frac{(h_{t}-h_{r})^{2}}{d^{2}}\right)=\frac{2h_{t}h_{r}}{d}
  26. Δ ϕ 4 π h t h r λ d \Delta\phi\approx\frac{4\pi h_{t}h_{r}}{\lambda d}
  27. d d
  28. d l x + x , Γ ( θ ) - 1 , G l o s G g r = G \begin{aligned}\displaystyle d&\displaystyle\approx l\approx x+x^{\prime},\\ \displaystyle\Gamma(\theta)&\displaystyle\approx-1,\\ \displaystyle G_{los}&\displaystyle\approx G_{gr}=G\\ \end{aligned}
  29. P r = P t ( λ G 4 π d ) 2 × ( 1 - e - j Δ ϕ ) 2 \therefore P_{r}=P_{t}\left({\frac{\lambda\sqrt{G}}{4\pi d}}\right)^{2}\times(% 1-e^{-j\Delta\phi})^{2}
  30. e - j Δ ϕ e^{-j\Delta\phi}
  31. e x = 1 + x 1 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! + = 1 + x + x 2 2 + x 3 6 + x 4 24 + x 5 120 + = n = 0 x n n ! e^{x}=1+\frac{x^{1}}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}+% \frac{x^{5}}{5!}+\cdots=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}+% \frac{x^{5}}{120}+\cdots\!=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}
  32. e - j Δ ϕ 1 + ( - j Δ ϕ ) + e^{-j\Delta\phi}\approx 1+({-j\Delta\phi})+\cdots
  33. P r P t ( λ G 4 π d ) 2 × ( 1 - ( 1 - j Δ ϕ ) ) 2 = P t ( λ G 4 π d ) 2 × ( j Δ ϕ ) 2 = P t ( λ G 4 π d ) 2 × - ( 4 π h t h r λ d ) 2 = - P t G h t 2 h r 2 d 4 \begin{aligned}\displaystyle\therefore P_{r}&\displaystyle\approx P_{t}\left({% \frac{\lambda\sqrt{G}}{4\pi d}}\right)^{2}\times(1-(1-j\Delta\phi))^{2}\\ &\displaystyle=P_{t}\left({\frac{\lambda\sqrt{G}}{4\pi d}}\right)^{2}\times(j% \Delta\phi)^{2}\\ &\displaystyle=P_{t}\left({\frac{\lambda\sqrt{G}}{4\pi d}}\right)^{2}\times-% \left(\frac{4\pi h_{t}h_{r}}{\lambda d}\right)^{2}\\ &\displaystyle=-P_{t}\frac{Gh_{t}^{2}h_{r}^{2}}{d^{4}}\end{aligned}
  34. | P r | = P t G h t 2 h r 2 d 4 |P_{r}|=P_{t}\frac{Gh_{t}^{2}h_{r}^{2}}{d^{4}}
  35. d d
  36. P r d B m = P t d B m + 10 log 10 ( G h t 2 h r 2 ) - 40 log 10 ( d ) P_{r_{dBm}}=P_{t_{dBm}}+10\log_{10}(Gh_{t}^{2}h_{r}^{2})-40\log_{10}(d)
  37. P L = P t d B m - P r d B m = 40 log 10 ( d ) - 10 log 10 ( G h t 2 h r 2 ) PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40\log_{10}(d)-10\log_{10}(Gh_{t}^{2}h_{r}^{2})
  38. P L = P T d B m - P R d B m = P L 0 + 10 γ log 10 d d 0 + X g , PL\;=P_{T_{dBm}}-P_{R_{dBm}}\;=\;PL_{0}\;+\;10\gamma\;\log_{10}\frac{d}{d_{0}}% \;+\;X_{g},
  39. P L = P t d B m - P r d B m = 40 log 10 ( d ) - 10 log 10 ( G h t 2 h r 2 ) PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40\log_{10}(d)-10\log_{10}(Gh_{t}^{2}h_{r}^{2})
  40. P L 0 = 40 log 10 ( d 0 ) - 10 log 10 ( G h t 2 h r 2 ) PL_{0}=40\log_{10}(d_{0})-10\log_{10}(Gh_{t}^{2}h_{r}^{2})
  41. X g = 0 X_{g}=0
  42. γ = 4 \gamma=4
  43. d , d 0 > d c d,d_{0}>d_{c}

Tzitzeica_equation.html

  1. u x y = exp ( u ) - exp ( - 2 * u ) . u_{xy}=\exp(u)-\exp(-2*u).
  2. w ( x , y ) = e x p ( u ( x , y ) ) w(x,y)=exp(u(x,y))
  3. w ( x , y ) y , x * w ( x , y ) - w ( x , y ) x * w ( x , y ) y - w ( x , y ) 3 + 1 = 0 w(x,y)_{y,x}*w(x,y)-w(x,y)_{x}*w(x,y)_{y}-w(x,y)^{3}+1=0
  4. u ( x , y ) = l n ( w ( x , y ) ) u(x,y)=ln(w(x,y))

UDP-2,3-diacetamido-2,3-dideoxyglucuronic_acid_2-epimerase.html

  1. \rightleftharpoons

UDP-arabinopyranose_mutase.html

  1. \rightleftharpoons

UDP-N-acetylglucosamine_4,6-dehydratase_(configuration-retaining).html

  1. \rightleftharpoons

UDP-N-acetylmuramoyl-L-alanyl-D-glutamate—D-lysine_ligase.html

  1. \rightleftharpoons

UDP-N-acetylmuramoylalanyl-D-glutamate-2,6-diamino-pimelate_ligase.html

  1. \rightleftharpoons

Ultra-Low_Fouling.html

  1. L o = n l 5 / 3 Γ 1 / 3 L_{o}=n\,l^{5/3}\,\Gamma^{1/3}
  2. L o L_{o}
  3. n n
  4. l l
  5. Γ \Gamma
  6. Δ n = d n d c * Γ h \Delta n=\frac{dn}{dc}*\frac{\Gamma}{h}
  7. h h
  8. n n
  9. c c
  10. Γ \Gamma

Ultrabright_electron.html

  1. B = lim A \zero lim Ω \zero δ I / δ A δ Ω B=\lim_{A\to\zero}\lim_{Ω\to\zero}δI/δAδΩ

Ultrasonic_pulse_velocity_test.html

  1. P u l s e V e l o c i t y = W i d t h o f s t r u c t u r e T i m e t a k e n b y p u l s e t o g o t h r o u g h Pulse\;Velocity=\frac{\;Width\;ofstructure}{Time\;taken\;by\;pulse\;to\;go\;through}

Uncial_0189.html

  1. Α Ν Ο Σ ¯ \overline{ΑΝΟΣ}
  2. Π Ν Α ¯ \overline{ΠΝΑ}
  3. Κ Υ ¯ \overline{ΚΥ}
  4. Κ Ω ¯ \overline{ΚΩ}
  5. Ι Λ Η Μ ¯ \overline{ΙΛΗΜ}
  6. Θ Ω ¯ \overline{ΘΩ}
  7. Ι Σ Η Λ ¯ \overline{ΙΣΗΛ}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}

Unified_methods_for_Computing_Incompressible_and_Compressible_flow.html

  1. ρ D V D t = - p + . τ + ρ f \rho{}\frac{DV}{Dt}=-\nabla{}p+\nabla{}.\tau{}+\rho{}f
  2. ρ [ h t + . ( h V ) ] = - D p D t + . ( k T ) + Φ \rho{}\left[\frac{\partial{}h}{\partial{}t}+\nabla{}.\left(hV\right)\right]=-% \frac{Dp}{Dt}+\nabla{}.\left(k\nabla{}T\right)+\Phi{}
  3. ρ p p t + ρ T T t + m x = 0 {\rho{}}_{p}p_{t}+{\rho{}}_{T}T_{t}+m_{x}=0
  4. ρ p = γ M r 2 / T {\rho{}}_{p}=\gamma{}M_{r}^{2}/T
  5. ρ T = - ρ T {\rho{}}_{T}=-\frac{\rho{}}{T}
  6. m t + ( u m + p ) x = 0 m_{t}+{(um+p)}_{x}=0
  7. T t + ( u T ) x + ( γ - 2 ) T u x = 0 T_{t}+{(uT)}_{x}+\left(\gamma{}-2\right)Tu_{x}=0
  8. γ M r 2 ( p j n + 1 - p j n ) - ρ j n ( T j n + 1 - T j n ) + λ T j n ( s n ρ n u n + ( 1 - s n ) m n + 1 ) | j - 1 / 2 j + 1 / 2 = 0 \gamma{}M_{r}^{2}\left(p_{j}^{n+1}-p_{j}^{n}\right)-{\rho{}}_{j}^{n}\left(T_{j% }^{n+1}-T_{j}^{n}\right)+\lambda{}T_{j}^{n}\left(s^{n}{\rho{}}^{n}u^{n}+\left(% 1-s^{n}\right)m^{n+1}\right){|{}}_{j-1/2}^{j+1/2}=0
  9. m j + 1 / 2 n + 1 - m j + 1 / 2 n + λ ( u n m n + p n + 1 / 2 ) | j j + 1 = 0 m_{j+1/2}^{n+1}-m_{j+1/2}^{n}+\lambda{}\left(u^{n}m^{n}+p^{n+1/2}\right){|{}}_% {j}^{j+1}=0
  10. T j n + 1 - T j n + λ ( u n T n ) | j - 1 2 j + 1 2 + λ ( γ - 2 ) T j n u n | j - 1 2 j + 1 2 = 0 T_{j}^{n+1}-T_{j}^{n}+\lambda{}\left(u^{n}T^{n}\right){|{}}_{j-\frac{1}{2}}^{j% +\frac{1}{2}}+\lambda{}\left(\gamma{}-2\right)T_{j}^{n}u^{n}{|{}}_{j-\frac{1}{% 2}}^{j+\frac{1}{2}}=0
  11. λ = τ / h \lambda{}=\tau{}/h
  12. p n + 1 / 2 = ( p n + p n + 1 ) / 2 p^{n+1/2}=(p^{n}+p^{n+1})/2
  13. s n = s ( M n ) s^{n}=s(M^{n})
  14. s ( M ) = 0 , M 1 / 2 , s(M)=0,M<=1/2,
  15. s ( M ) = M - 1 / 21 / 2 < | M | < 3 / 2 s(M)=M-1/21/2<|{}M|{}<3/2
  16. s ( M ) = 1 M 3 / 2 s(M)=1M>=3/2
  17. M j + 1 2 = 2 | u j + 1 2 | c j + c j + 1 M_{j+\frac{1}{2}}=\frac{2\left|{}u_{j+\frac{1}{2}}\right|{}}{c_{j}+c_{j+1}}
  18. m j + 1 / 2 * - m j + 1 2 n + λ ( u n m n + p n ) | j j + 1 = 0 m_{j+1/2}^{*}-m_{j+\frac{1}{2}}^{n}+\lambda{}\left(u^{n}m^{n}+p^{n}\right){|{}% }_{j}^{j+1}=0
  19. δ m = m n + 1 - m * \delta{}m=m^{n+1}-m^{*}
  20. δ m j + 1 2 = - ( 1 2 ) λ δ p | j j + 1 \delta{}m_{j+\frac{1}{2}}=-\left(\frac{1}{2}\right)\lambda{}\delta{}p{|{}}_{j}% ^{j+1}
  21. δ p = p n + 1 - p n \delta{}p=p^{n+1}-p^{n}
  22. m n + 1 = m * + δ m m^{n+1}=m^{*}+\delta{}m
  23. δ p : \delta{}p:
  24. γ M r 2 δ p j - ( 1 2 ) λ 2 T j n { ( 1 - s j + 1 2 n ) δ p | j j + 1 - ( 1 - s j - 1 2 n ) δ p | j - 1 j } = ρ j n T j | n n + 1 - λ T j n ( s n ρ n u n ) | j - 1 2 j + 1 2 \gamma{}M_{r}^{2}\delta{}p_{j}-\left(\frac{1}{2}\right){\lambda{}}^{2}T_{j}^{n% }\left\{\left(1-s_{j+\frac{1}{2}}^{n}\right)\delta{}p{|{}}_{j}^{j+1}-\left(1-s% _{j-\frac{1}{2}}^{n}\right)\delta{}p{|{}}_{j-1}^{j}\right\}={\rho{}}_{j}^{n}T_% {j}{|{}}_{n}^{n+1}-\lambda{}T_{j}^{n}\left(s^{n}{\rho{}}^{n}u^{n}\right){|{}}_% {j-\frac{1}{2}}^{j+\frac{1}{2}}
  25. ( 1 2 ) λ δ p | 0 1 = - λ δ m 1 / 2 = - λ ( ρ b u b ) | t n t n + 1 \left(\frac{1}{2}\right)\lambda{}\delta{}p{|{}}_{0}^{1}=-\lambda{}\delta{}m_{1% /2}=-\lambda{}({\rho{}}_{b}u_{b}){|{}}_{t_{n}}^{t_{n+1}}
  26. m J + 1 / 2 * - m J + 1 / 2 n + 2 λ ( u n m n + p n ) | J J + 1 / 2 = 0 m_{J+1/2}^{*}-m_{J+1/2}^{n}+2\lambda{}\left(u^{n}m^{n}+p^{n}\right){|{}}_{J}^{% J+1/2}=0
  27. p J + 1 / 2 n = p b ( t n ) p_{J+1/2}^{n}=p_{b}(t^{n})
  28. δ m J + 1 / 2 = - λ ( p b | t n t n + 1 - δ p j ) \delta{}m_{J+1/2}=-\lambda{}(p_{b}{|{}}_{t_{n}}^{t_{n+1}}-\delta{}p_{j})
  29. T j ( m + 1 ) - T j n + α m + 1 λ ( u n T ( m ) ) | j - 1 / 2 j + 1 / 2 + α m + 1 λ ( γ - 2 ) T j ( m ) u n | j - 1 / 2 j + 1 / 2 = 0 T_{j}^{(m+1)}-T_{j}^{n}+{\alpha{}}_{m+1}\lambda{}\left(u^{n}T^{(m)}\right){|{}% }_{j-1/2}^{j+1/2}+{\alpha{}}_{m+1}\lambda{}\left(\gamma{}-2\right)T_{j}^{(m)}u% ^{n}{|{}}_{j-1/2}^{j+1/2}=0
  30. m j + 1 2 ( m + 1 ) - m j + 1 2 n + α m + 1 λ ( u n m ( m ) + p n ) | j j + 1 = 0 m_{j+\frac{1}{2}}^{\left(m+1\right)}-m_{j+\frac{1}{2}}^{n}+{\alpha{}}_{m+1}% \lambda{}\left(u^{n}m^{\left(m\right)}+p^{n}\right){|{}}_{j}^{j+1}=0
  31. γ M r 2 δ p j - ( 1 2 ) λ 2 T j ( 4 ) { ( 1 - s j + 1 2 ( 4 ) ) δ p | j j + 1 - ( 1 - s j - 1 2 ( 4 ) ) δ p | j - 1 j } = ρ j n ( T j ( 4 ) - T j n ) - λ T j ( 4 ) ( s ( 4 ) ρ n u n ) | j - 1 2 j + 1 2 \gamma{}M_{r}^{2}\delta{}p_{j}-\left(\frac{1}{2}\right){\lambda{}}^{2}T_{j}^{% \left(4\right)}\left\{\left(1-s_{j+\frac{1}{2}}^{\left(4\right)}\right)\delta{% }p{|{}}_{j}^{j+1}-\left(1-s_{j-\frac{1}{2}}^{\left(4\right)}\right)\delta{}p{|% {}}_{j-1}^{j}\right\}={\rho{}}_{j}^{n}\left(T_{j}^{\left(4\right)}-T_{j}^{n}% \right)-\lambda{}T_{j}^{\left(4\right)}\left(s^{\left(4\right)}{\rho{}}^{n}u^{% n}\right){|{}}_{j-\frac{1}{2}}^{j+\frac{1}{2}}

Uniformly_smooth_space.html

  1. X X
  2. ϵ > 0 \epsilon>0
  3. δ > 0 \delta>0
  4. x , y X x,y\in X
  5. x = 1 \|x\|=1
  6. y δ \|y\|\leq\delta
  7. x + y + x - y 2 + ϵ y . \|x+y\|+\|x-y\|\leq 2+\epsilon\|y\|.
  8. ρ X ( t ) = sup { x + y + x - y 2 - 1 : x = 1 , y = t } . \rho_{X}(t)=\sup\Bigl\{\frac{\|x+y\|+\|x-y\|}{2}-1\,:\,\|x\|=1,\;\|y\|=t\Bigr\}.
  9. X X
  10. X * X^{*}
  11. ρ X * ( t ) = sup { t ε / 2 - δ X ( ε ) : ε [ 0 , 2 ] } , t 0 , \rho_{X^{*}}(t)=\sup\{t\varepsilon/2-\delta_{X}(\varepsilon):\varepsilon\in[0,% 2]\},\quad t\geq 0,
  12. δ ~ X ( ε ) = sup { ε t / 2 - ρ X * ( t ) : t 0 } . \tilde{\delta}_{X}(\varepsilon)=\sup\{\varepsilon t/2-\rho_{X^{*}}(t):t\geq 0\}.
  13. δ X ( ε / 2 ) δ ~ X ( ε ) δ X ( ε ) , ε [ 0 , 2 ] . \delta_{X}(\varepsilon/2)\leq\tilde{\delta}_{X}(\varepsilon)\leq\delta_{X}(% \varepsilon),\quad\varepsilon\in[0,2].
  14. lim t 0 x + t y - x t \lim_{t\to 0}\frac{\|x+ty\|-\|x\|}{t}
  15. x , y S X x,y\in S_{X}
  16. S X S_{X}
  17. X X
  18. ρ X ( t ) C t p , t > 0. \rho_{X}(t)\leq C\,t^{p},\quad t>0.
  19. δ Y ( ε ) c ε q , ε [ 0 , 2 ] . \delta_{Y}(\varepsilon)\geq c\,\varepsilon^{q},\quad\varepsilon\in[0,2].

Unital_(geometry).html

  1. = ( 2 , q 2 ) \mathcal{H}=\mathcal{H}(2,q^{2})
  2. P G ( 2 , q 2 ) PG(2,q^{2})
  3. q q
  4. \mathcal{H}
  5. = { ( x 0 , x 1 , x 2 ) : x 0 q + 1 + x 1 q + 1 + x 2 q + 1 = 0 } . \mathcal{H}=\{(x_{0},x_{1},x_{2})\colon x_{0}^{q+1}+x_{1}^{q+1}+x_{2}^{q+1}=0\}.

Universal_representation_(C*-algebra).html

  1. Φ := ρ S π ρ \Phi:=\sum_{\rho\in S}\oplus\;\pi_{\rho}
  2. H Φ H_{\Phi}
  3. ρ ¯ \overline{ρ}
  4. ρ ¯ \overline{ρ}
  5. ρ ¯ \overline{ρ}
  6. K = Φ ( A ) Φ ( A ) - E , K - = Φ ( A ) - E K=\Phi(A)\cap\Phi(A)^{-}E,K^{-}=\Phi(A)^{-}E
  7. f ( σ 1 ( a ) , σ 1 ( a * ) , σ 1 ( a a * ) , σ 1 ( a * a ) , , σ m ( a ) , σ m ( a * ) , σ m ( a a * ) , σ m ( a * a ) ) f(\sigma_{1}(a),\sigma_{1}(a^{*}),\sigma_{1}(aa^{*}),\sigma_{1}(a^{*}a),\cdots% ,\sigma_{m}(a),\sigma_{m}(a^{*}),\sigma_{m}(aa^{*}),\sigma_{m}(a^{*}a))
  8. g ( ρ 1 ( a ) , ρ 1 ( a * ) , ρ 1 ( a a * ) , ρ 1 ( a * a ) , , ρ n ( a ) , ρ n ( a * ) , ρ n ( a a * ) , ρ n ( a * a ) ) . \leq g(\rho_{1}(a),\rho_{1}(a^{*}),\rho_{1}(aa^{*}),\rho_{1}(a^{*}a),\cdots,% \rho_{n}(a),\rho_{n}(a^{*}),\rho_{n}(aa^{*}),\rho_{n}(a^{*}a)).

Universality_and_quantum_systems.html

  1. λ \lambda
  2. λ \lambda
  3. 0
  4. V 0 V_{0}
  5. V V
  6. x x
  7. V ( x ) = { 0 x < - a V ( x ) - a < x < a V 0 a < x , V(x)=\begin{cases}0&x<-a\\ V(x)&-a<x<a\\ V_{0}&a<x\end{cases},
  8. i t Ψ ( x , t ) = [ - 2 2 m 2 + V ( x , t ) ] Ψ ( x , t ) i\hbar\frac{\partial}{\partial t}\Psi(x,t)=\left[\frac{-\hbar^{2}}{2m}\nabla^{% 2}+V(x,t)\right]\Psi(x,t)
  9. Ψ ( x , t ) \Psi(x,t)
  10. e x p [ i ( p x - E t ) / ] exp[i(px-Et)/\hbar]
  11. p p
  12. E E
  13. V V
  14. E E
  15. r = 2 m ( V 0 - E ) r=\frac{\hbar}{\sqrt{2m(V_{0}-E)}}
  16. r = 2 m V 0 r=\frac{\hbar}{\sqrt{2mV_{0}}}
  17. E E
  18. 0
  19. r r
  20. a a
  21. R = 1 R=1
  22. E E
  23. λ = 2 m E \lambda=\frac{\hbar}{\sqrt{2mE}}
  24. λ \lambda
  25. a a
  26. V ( x ) V(x)
  27. V ( x ) V(x)
  28. V 0 V_{0}
  29. ψ ( x ) \psi(x)
  30. H ψ ( x ) = [ - 2 2 m d 2 d x 2 + V ( x ) ] ψ ( x ) = E ψ ( x ) H\psi(x)=\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x)\right]\psi(x)=E% \psi(x)
  31. H H
  32. \hbar
  33. m m
  34. E E
  35. V ( x ) = V 0 [ Θ ( a - x ) - Θ ( a + x ) ] V(x)=V_{0}[\Theta(a-x)-\Theta(a+x)]
  36. V 0 > 0 V_{0}>0
  37. 2 a 2a
  38. Θ ( x ) = 0 , x < 0 ; Θ ( x ) = 1 , x > 0 \Theta(x)=0,\;x<0;\;\Theta(x)=1,\;x>0
  39. y = x / a y=x/a
  40. z = E / V 0 z=E/V_{0}
  41. ρ = a 2 m E \rho=\frac{a\sqrt{2mE}}{\hbar}
  42. ρ = ρ z - 1 z \rho^{\prime}=\rho\sqrt{\frac{z-1}{z}}
  43. ψ I ( x ) = A I e i ρ y + A I I e - i ρ y y < - 1 \psi_{I}(x)=A_{I}e^{i\rho y}+A_{II}e^{-i\rho y}\quad y<-1
  44. ψ I I ( x ) = B I e i ρ y + B I I e - i ρ y | y | < 1 \psi_{II}(x)=B_{I}e^{i\rho^{\prime}y}+B_{II}e^{-i\rho^{\prime}y}\quad|y|<1
  45. ψ I I I ( x ) = C I e i ρ y + C I I e - i ρ y y > 1 \psi_{III}(x)=C_{I}e^{i\rho y}+C_{II}e^{-i\rho y}\quad y>1
  46. ψ \psi
  47. C I I = 0 C_{II}=0
  48. R = ( ρ 2 - ρ 2 ) 2 ( sin ( 2 ρ ) ) 2 4 ρ 2 ρ 2 + ( ρ 2 - ρ 2 ) 2 ( sin ( 2 ρ ) ) 2 R=\frac{(\rho^{2}-\rho^{\prime 2})^{2}(\sin(2\rho))^{2}}{4\rho^{2}\rho^{\prime 2% }+(\rho^{2}-\rho^{\prime 2})^{2}(\sin(2\rho))^{2}}
  49. 2 ρ = 2 n π 2\rho=2n\pi
  50. R = 0 R=0
  51. T = 1 T=1
  52. E n * = n 2 π 2 2 2 m a 2 E_{n}^{*}=\frac{n^{2}\pi^{2}\hbar^{2}}{2ma^{2}}
  53. r 2 r^{2}
  54. r 2 := ρ 2 - ρ 2 = 2 m V 0 a 2 2 r^{2}:=\rho^{2}-\rho^{\prime 2}=\frac{2mV_{0}a^{2}}{\hbar^{2}}
  55. V 0 V_{0}
  56. a a
  57. r r
  58. a a
  59. = 1 \hbar=1
  60. 1 / a 1/a
  61. 1 / a 2 1/a^{2}
  62. 1 / a 2 1/a^{2}
  63. V 0 a 2 V_{0}a^{2}
  64. \hbar
  65. m m
  66. 2 m V 0 a 2 / 2 2mV_{0}a^{2}/\hbar^{2}
  67. r r
  68. r r
  69. R = r 4 ( sin ( 2 ρ ) ) 2 4 ρ 2 ρ 2 + r 4 ( sin ( 2 ρ ) ) 2 R=\frac{r^{4}(\sin(2\rho))^{2}}{4\rho^{2}\rho^{\prime 2}+r^{4}(\sin(2\rho))^{2}}
  70. 2 ρ = 2 n π + δ 2\rho=2n\pi+\delta
  71. R R
  72. r 4 δ 2 16 n 4 π 4 \frac{r^{4}\delta^{2}}{16n^{4}\pi^{4}}
  73. δ 2 \delta^{2}
  74. δ \delta
  75. n n
  76. n n
  77. E 0 E_{0}
  78. R ( E 0 ) + R ( E 0 ) 1 ! ( E - E 0 ) + R ′′ ( E 0 ) 2 ! ( E - E 0 ) 2 + . R(E_{0})+\frac{R^{\prime}(E_{0})}{1!}(E-E_{0})+\frac{R^{\prime\prime}(E_{0})}{% 2!}(E-E_{0})^{2}+\cdots.
  79. E 0 E_{0}
  80. R ( E 0 ) = 0. R^{\prime}(E_{0})=0.
  81. E 0 , R ( E 0 ) = 0. E_{0},R(E_{0})=0.
  82. R ( E ) R(E)
  83. ( E - E 0 ) 2 . (E-E_{0})^{2}.
  84. δ \delta
  85. ( E - E 0 ) (E-E_{0})
  86. δ \delta
  87. R ( E ) R(E)
  88. δ 2 \delta^{2}
  89. λ d \lambda>>d

University_Ranking_by_Academic_Performance.html

  1. A I T = i = 1 23 [ ( C P P i C P P _ W o r l d i ) * A r t i c l e s i ] AIT=\sum_{i=1}^{23}\Bigg[\Bigg(\frac{CPP_{i}}{CPP\_{World_{i}}}\Bigg)*Articles% _{i}\Bigg]
  2. C I T = i = 1 23 [ ( C P P i C P P _ W o r l d i ) * C i t a t i o n s i ] CIT=\sum_{i=1}^{23}\Bigg[\Bigg(\frac{CPP_{i}}{CPP\_{World_{i}}}\Bigg)*% Citations_{i}\Bigg]

Unnormalized_KdV_equation.html

  1. u t + α * u x x x + β * u * u x = 0 u_{t}+\alpha*u_{xxx}+\beta*u*u_{x}=0

Unnormalized_modified_KdV_equation.html

  1. u t + u x x x + α * u 2 * u x = 0 u_{t}+u_{xxx}+\alpha*u^{2}*u_{x}=0

UOBYQA.html

  1. Q k Q_{k}
  2. ( n + 1 ) ( n + 2 ) / 2 (n+1)(n+2)/2
  3. Q k Q_{k}
  4. ( n + 1 ) ( n + 2 ) / 2 (n+1)(n+2)/2
  5. 2 n + 1 2n+1

Upper_bound_theorem.html

  1. f i ( Δ ( n , d ) ) = ( n i + 1 ) for 0 i < [ d 2 ] f_{i}(\Delta(n,d))={\left({{n}\atop{i+1}}\right)}\quad\textrm{for}\quad 0\leq i% <\left[\frac{d}{2}\right]
  2. ( f 0 , , f [ d 2 ] - 1 ) (f_{0},\ldots,f_{[\frac{d}{2}]-1})
  3. ( f [ d 2 ] , , f d - 1 ) (f_{[\frac{d}{2}]},\ldots,f_{d-1})
  4. f i ( Δ ) f i ( Δ ( n , d ) ) for i = 0 , 1 , , d - 1. f_{i}(\Delta)\leq f_{i}(\Delta(n,d))\quad\textrm{for}\quad i=0,1,\ldots,d-1.
  5. O ( n d / 2 ) \scriptstyle O(n^{\lfloor d/2\rfloor})
  6. h i ( Δ ) ( n - d + i - 1 i ) for 0 i < [ d 2 ] . h_{i}(\Delta)\leq{\textstyle\left({{n-d+i-1}\atop{i}}\right)}\quad\textrm{for}% \quad 0\leq i<\left[\frac{d}{2}\right].

Upside_beta.html

  1. r i r_{i}
  2. r m r_{m}
  3. i i
  4. m m
  5. u m u_{m}
  6. β + = c o v ( r i , r m | r m > u m ) v a r ( r m | r m > u m ) \beta^{+}=\frac{cov(r_{i},r_{m}|r_{m}>u_{m})}{var(r_{m}|r_{m}>u_{m})}
  7. β - \beta^{-}
  8. β + \beta^{+}
  9. i i

Upward_planar_drawing.html

  1. a < b , a < c , b < d , b < e , c < d , c < e , d < f , e < f a<b,a<c,b<d,b<e,c<d,c<e,d<f,e<f

Upwind_differencing_scheme_for_convection.html

  1. t ( ρ ϕ ) + ( ρ 𝐮 ϕ ) = ( Γ grad ϕ ) + S ϕ \frac{\partial}{\partial t}(\rho\phi)+\nabla\cdot(\rho\mathbf{u}\phi)\,=\nabla% \cdot(\Gamma\operatorname{grad}\phi)+S_{\phi}
  2. ( ρ u A ) e - ( ρ u A ) w = 0 \left(\rho uA\right)_{e}-\left(\rho uA\right)_{w}=0\,
  3. ρ \rho
  4. Γ \Gamma
  5. 𝐮 \mathbf{u}
  6. ϕ \phi
  7. S ϕ S_{\phi}
  8. e e
  9. w w
  10. F e ϕ e - F w ϕ w = D e ( ϕ E - ϕ P ) - D w ( ϕ P - ϕ W ) F_{e}\phi_{e}-F_{w}\phi_{w}\,=D_{e}(\phi_{E}-\phi_{P})-D_{w}(\phi_{P}-\phi_{W})
  11. \;
  12. F e - F w = 0 F_{e}-F_{w}\,=0
  13. \;
  14. E E
  15. W W
  16. P P
  17. F = ρ u A F\,=\rho uA
  18. \;
  19. \;
  20. D = Γ A δ x D\,=\frac{\Gamma A}{\delta x}
  21. \;
  22. P e = F D = ρ u Γ / δ x Pe\,=\frac{F}{D}\,=\frac{\rho u}{\Gamma/\delta x}
  23. \;
  24. u w > 0 u_{w}>0
  25. u e > 0 u_{e}>0
  26. F e ϕ P - F w ϕ W = D e ( ϕ E - ϕ P ) - D w ( ϕ P - ϕ W ) F_{e}\phi_{P}-F_{w}\phi_{W}\,=D_{e}(\phi_{E}-\phi_{P})-D_{w}(\phi_{P}-\phi_{W})
  27. ϕ e = ϕ P \phi_{e}\,=\phi_{P}
  28. ϕ w = ϕ W \phi_{w}\,=\phi_{W}
  29. [ ( D w + F w ) + D e + ( F e - F w ) ] ϕ P = ( D w + F w ) ϕ W + D e ϕ E ) [(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]\phi_{P}\,=(D_{w}+F_{w})\phi_{W}+D_{e}\phi_% {E})
  30. \;
  31. a P = [ ( D w + F w ) + D e + ( F e - F w ) ] a_{P}\,=[(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]
  32. \;
  33. a W = ( D w + F w ) a_{W}\,=(D_{w}+F_{w})
  34. a E = D E a_{E}\,=D_{E}
  35. u w < 0 u_{w}<0
  36. u e < 0 u_{e}<0
  37. F e ϕ E - F w ϕ P = D e ( ϕ E - ϕ P ) - D w ( ϕ P - ϕ W ) F_{e}\phi_{E}-F_{w}\phi_{P}\,=D_{e}(\phi_{E}-\phi_{P})-D_{w}(\phi_{P}-\phi_{W})
  38. ϕ W = ϕ w \phi_{W}\,=\phi_{w}
  39. ϕ P = ϕ e \phi_{P}\,=\phi_{e}
  40. [ ( D e - F e ) + D w + ( F e - F w ) ] ϕ P = D w ϕ W + ( D e - F e ) ϕ E [(D_{e}-F_{e})+D_{w}+(F_{e}-F_{w})]\phi_{P}=D_{w}\phi_{W}+(D_{e}-F_{e})\phi_{E}
  41. a W = D w a_{W}\,=D_{w}
  42. a E = D e - F e a_{E}\,=D_{e}-F_{e}
  43. a W = D w + max ( F w , 0 ) a_{W}=D_{w}+\max(F_{w},0)
  44. a E = D e + max ( 0 , - F e ) a_{E}=D_{e}+\max(0,-F_{e})

Urania_Propitia.html

  1. M = E - ϵ sin E M=E-\epsilon\cdot\sin E

Utility_functions_on_indivisible_goods.html

  1. \succ
  2. A A
  3. B B
  4. A B A\succ B
  5. A A
  6. A A
  7. A A
  8. B B
  9. A B A\succeq B
  10. u u
  11. A A
  12. u ( A ) u(A)
  13. u ( ) = 0 u(\emptyset)=0
  14. \emptyset
  15. u ( A ) > u ( B ) u(A)>u(B)
  16. A B A\succ B
  17. u ( A ) u ( B ) u(A)\geq u(B)
  18. A B A\succeq B
  19. A B A\supseteq B
  20. A B A\succeq B
  21. A B A\supseteq B
  22. u ( A ) B u(A)\geq B
  23. A A
  24. u ( A ) u(A)
  25. \emptyset
  26. A A
  27. u ( A ) = x A u ( x ) u(A)=\sum_{x\in A}u({x})
  28. u u
  29. A A
  30. B B
  31. u ( A ) + u ( B ) = u ( A B ) + u ( A B ) u(A)+u(B)=u(A\cup B)+u(A\cap B)
  32. A A
  33. u ( A ) u(A)
  34. \emptyset
  35. A A
  36. B B
  37. u ( A ) + u ( B ) u ( A B ) + u ( A B ) u(A)+u(B)\geq u(A\cup B)+u(A\cap B)
  38. u u
  39. A A
  40. B B
  41. A B A\subseteq B
  42. x B x\notin B
  43. u ( A { x } ) - u ( A ) u ( B { x } ) - u ( B ) u(A\cup\{x\})-u(A)\geq u(B\cup\{x\})-u(B)
  44. A A
  45. u ( A ) u(A)
  46. \emptyset
  47. A A
  48. B B
  49. u ( A ) + u ( B ) u ( A B ) + u ( A B ) u(A)+u(B)\leq u(A\cup B)+u(A\cap B)
  50. u u
  51. A A
  52. B B
  53. A B A\subseteq B
  54. x B x\notin B
  55. u ( A { x } ) - u ( A ) u ( B { x } ) - u ( B ) u(A\cup\{x\})-u(A)\leq u(B\cup\{x\})-u(B)
  56. A A
  57. u ( A ) u(A)
  58. \emptyset
  59. A A
  60. B B
  61. u ( A B ) u ( A ) + u ( B ) u(A\cup B)\leq u(A)+u(B)
  62. u u
  63. A A
  64. B B
  65. u ( A B ) u ( A ) + u ( B ) u(A\cup B)\geq u(A)+u(B)
  66. u u
  67. A A
  68. u ( A ) u(A)
  69. \emptyset
  70. A A
  71. u ( A ) u(A)
  72. \emptyset
  73. B B
  74. A B A\subseteq B
  75. | A | = 1 |A|=1
  76. A B A\succeq B
  77. A A
  78. u ( A ) = max x A u ( x ) u(A)=\max_{x\in A}u({x})
  79. A A
  80. B B
  81. X A X\subseteq A
  82. Y B Y\subseteq B
  83. u ( A ) + u ( B ) u ( A X Y ) + u ( A Y X ) u(A)+u(B)\leq u(A\setminus X\cup Y)+u(A\setminus Y\cup X)
  84. u u
  85. A A
  86. B B
  87. X X
  88. Y Y
  89. p p
  90. u u
  91. p p
  92. A A
  93. u ( A ) - p A u(A)-p\cdot A
  94. D ( u , p ) D(u,p)
  95. q q
  96. p p
  97. q p q\geq p
  98. A D ( u , p ) A\in D(u,p)
  99. B D ( u , q ) B\in D(u,q)
  100. B { a A | p a = q a } B\supseteq\{a\in A|p_{a}=q_{a}\}
  101. p p
  102. A D ( u , p ) A\notin D(u,p)
  103. B B
  104. u ( B ) - p B > u ( A ) - p A u(B)-p\cdot B>u(A)-p\cdot A
  105. | A B | 1 |A\setminus B|\leq 1
  106. | B A | 1 |B\setminus A|\leq 1
  107. p p
  108. A , B D ( u , p ) A,B\in D(u,p)
  109. X A X\subseteq A
  110. Y B Y\subseteq B
  111. A X Y D ( u , p ) A\setminus X\cup Y\in D(u,p)
  112. U D S N C N C = S I = G S S u b m o d u l a r S u b a d d i t i v e UD\subsetneq SNC\subsetneq NC=SI=GS\subsetneq Submodular\subsetneq Subadditive

Uzawa_iteration.html

  1. ( A B B * ) ( x 1 x 2 ) = ( b 1 b 2 ) , \begin{pmatrix}A&B\\ B^{*}&\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}=\begin{pmatrix}b_{1}\\ b_{2}\end{pmatrix},
  2. A A
  3. B * A - 1 B^{*}A^{-1}
  4. ( A B - S ) ( x 1 x 2 ) = ( b 1 b 2 - B * A - 1 b 1 ) , \begin{pmatrix}A&B\\ &-S\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}=\begin{pmatrix}b_{1}\\ b_{2}-B^{*}A^{-1}b_{1}\end{pmatrix},
  5. S := B * A - 1 B S:=B^{*}A^{-1}B
  6. S S
  7. S x 2 = B * A - 1 b 1 - b 2 Sx_{2}=B^{*}A^{-1}b_{1}-b_{2}
  8. x 2 x_{2}
  9. x 1 x_{1}
  10. A x 1 = b 1 - B x 2 . Ax_{1}=b_{1}-Bx_{2}.\,
  11. x 1 x_{1}
  12. x 2 x_{2}
  13. r 2 := B * A - 1 b 1 - b 2 - S x 2 = B * A - 1 ( b 1 - B x 2 ) - b 2 = B * x 1 - b 2 , r_{2}:=B^{*}A^{-1}b_{1}-b_{2}-Sx_{2}=B^{*}A^{-1}(b_{1}-Bx_{2})-b_{2}=B^{*}x_{1% }-b_{2},
  14. x 1 := A - 1 ( b 1 - B x 2 ) x_{1}:=A^{-1}(b_{1}-Bx_{2})
  15. x 2 x_{2}
  16. p 2 := r 2 . p_{2}:=r_{2}.\,
  17. a 2 := S p 2 = B * A - 1 B p 2 = B * p 1 a_{2}:=Sp_{2}=B^{*}A^{-1}Bp_{2}=B^{*}p_{1}
  18. p 1 := A - 1 B p 2 p_{1}:=A^{-1}Bp_{2}
  19. α := p 2 * r 2 / p 2 * a 2 \alpha:=p_{2}^{*}r_{2}/p_{2}^{*}a_{2}
  20. x 2 := x 2 + α p 2 , r 2 := r 2 - α a 2 . x_{2}:=x_{2}+\alpha p_{2},\quad r_{2}:=r_{2}-\alpha a_{2}.
  21. p 1 p_{1}
  22. x 1 := x 1 - α p 1 . x_{1}:=x_{1}-\alpha p_{1}.\,
  23. β := r 2 * a 2 / p 2 * a 2 , p 2 := r 2 - β p 2 . \beta:=r_{2}^{*}a_{2}/p_{2}^{*}a_{2},\quad p_{2}:=r_{2}-\beta p_{2}.
  24. r 2 r_{2}
  25. p 2 p_{2}
  26. r 2 r_{2}
  27. A x = b Ax=b

Uzi_Vishne.html

  1. C P 1 × T CP^{1}\times T

Vainshtein_radius.html

  1. r V = ( G M m G 4 ) 1 5 r_{V}=\left(\frac{GM}{m^{4}_{G}}\right)^{\frac{1}{5}}
  2. M M
  3. m G m_{G}
  4. r r V r\gg r_{V}

Valencene_synthase.html

  1. \rightleftharpoons

Valerena-4,7(11)-diene_synthase.html

  1. \rightleftharpoons

Van_der_Corput's_method.html

  1. n = a b e ( f ( n ) ) \sum_{n=a}^{b}e(f(n))
  2. h = 1 H | n = a b - h e ( f h ( n ) ) | b - a . \sum_{h=1}^{H}\left|{\sum_{n=a}^{b-h}e(f_{h}(n))}\right|\leq b-a\ .
  3. | n = a b e ( f ( n ) ) | b - a H . \left|{\sum_{n=a}^{b}e(f(n))}\right|\ll\frac{b-a}{\sqrt{H}}\ .
  4. g ( y ) = f ( u ( y ) ) - y u ( y ) . g(y)=f(u(y))-yu(y)\ .
  5. | n = a b e ( f ( n ) ) | 1 λ max α γ β | ν = α γ e ( g ( ν ) ) | . \left|{\sum_{n=a}^{b}e(f(n))}\right|\ll\frac{1}{\sqrt{\lambda}}\max_{\alpha% \leq\gamma\leq\beta}\left|{\sum_{\nu=\alpha}^{\gamma}e(g(\nu))}\right|\ .
  6. | f ( r + 1 ) ( x ) - ( - 1 ) r s ( s + 1 ) ( s + r ) T x - s - r | δ s ( s + 1 ) ( s + r ) T x - s - r \left|{f^{(r+1)}(x)-(-1)^{r}s(s+1)\cdots(s+r)Tx^{-s-r}}\right|\leq\delta s(s+1% )\cdots(s+r)Tx^{-s-r}
  7. | n = a b e ( f ( n ) ) | ( T N σ ) k N l \left|{\sum_{n=a}^{b}e(f(n))}\right|\ll\left({\frac{T}{N^{\sigma}}}\right)^{k}% N^{l}
  8. ( k 2 k + 2 , k + l + 1 2 k + 2 ) \left({\frac{k}{2k+2},\frac{k+l+1}{2k+2}}\right)
  9. ( l - 1 / 2 , k + 1 / 2 ) \left({l-1/2,k+1/2}\right)
  10. ζ ( 1 / 2 + i t ) t θ log t \zeta(1/2+it)\ll t^{\theta}\log t
  11. θ = ( k + l - 1 / 2 ) / 2 \theta=(k+l-1/2)/2

Van_Lamoen_circle.html

  1. A b A_{b}
  2. A c A_{c}
  3. B c B_{c}
  4. B a B_{a}
  5. C a C_{a}
  6. C b C_{b}
  7. T T
  8. T T
  9. A A
  10. B B
  11. C C
  12. T T
  13. G G
  14. M a M_{a}
  15. M b M_{b}
  16. M c M_{c}
  17. B C BC
  18. C A CA
  19. A B AB
  20. A G M c AGM_{c}
  21. B G M c BGM_{c}
  22. B G M a BGM_{a}
  23. C G M a CGM_{a}
  24. C G M b CGM_{b}
  25. A G M b AGM_{b}
  26. T T
  27. X ( 1153 ) X(1153)
  28. P P
  29. A A AA^{\prime}
  30. B B BB^{\prime}
  31. C C CC^{\prime}
  32. P P
  33. A P B APB^{\prime}
  34. A P C APC^{\prime}
  35. B P C BPC^{\prime}
  36. B P A BPA^{\prime}
  37. C P A CPA^{\prime}
  38. C P B CPB^{\prime}
  39. P P
  40. T T

Variables_sampling_plan.html

  1. X ¯ + k σ U S L \bar{X}+k\sigma\leqslant USL
  2. X ¯ - k σ L S L \bar{X}-k\sigma\geqslant LSL
  3. X ¯ \bar{X}
  4. σ \sigma
  5. k k
  6. U S L USL
  7. L S L LSL
  8. α \alpha
  9. β \beta
  10. α \alpha
  11. β \beta
  12. n n
  13. k k
  14. k = Z L Q L Z α + Z A Q L Z β Z α + Z β k=\frac{Z_{LQL}Z_{\alpha}+Z_{AQL}Z_{\beta}}{Z_{\alpha}+Z_{\beta}}
  15. Z Z
  16. n n
  17. n = ( Z α + Z β Z A Q L - Z L Q L ) 2 n=\left(\frac{Z_{\alpha}+Z_{\beta}}{Z_{AQL}-Z_{LQL}}\right)^{2}
  18. σ \sigma
  19. n = ( Z α + Z β Z A Q L - Z L Q L ) 2 ( 1 + k 2 2 ) n=\left(\frac{Z_{\alpha}+Z_{\beta}}{Z_{AQL}-Z_{LQL}}\right)^{2}\left(1+\frac{k% ^{2}}{2}\right)

Variance_function.html

  1. E [ y X ] = μ E[y\mid X]=\mu
  2. η = X B = j = 1 p X i j T B j \eta=XB=\sum_{j=1}^{p}X_{ij}^{T}B_{j}
  3. η = g ( μ ) , μ = g - 1 ( η ) \eta=g(\mu),\mu=g^{-1}(\eta)
  4. 𝑦 \,\textit{y}
  5. f ( y , θ , ϕ ) = exp ( y θ - b ( θ ) ϕ - c ( y , ϕ ) ) \operatorname{f}(y,\theta,\phi)=\exp\left(\frac{y\theta-b(\theta)}{\phi}-c(y,% \phi)\right)
  6. l ( θ , y , ϕ ) = log ( f ( y , θ , ϕ ) ) = y θ - b ( θ ) ϕ - c ( y , ϕ ) \operatorname{l}(\theta,y,\phi)=\log(\operatorname{f}(y,\theta,\phi))=\frac{y% \theta-b(\theta)}{\phi}-c(y,\phi)
  7. θ \theta
  8. ϕ \phi
  9. θ , f θ ( ) \theta,f_{\theta}()
  10. E θ [ θ log ( f θ ( y ) ) ] = 0 \operatorname{E}_{\theta}\left[\frac{\partial}{\partial\theta}\log(f_{\theta}(% y))\right]=0
  11. Var θ [ θ log ( f θ ( y ) ) ] + E θ [ 2 θ 2 log ( f θ ( y ) ) ] = 0 \operatorname{Var}_{\theta}\left[\frac{\partial}{\partial\theta}\log(f_{\theta% }(y))\right]+\operatorname{E}_{\theta}\left[\frac{\partial^{2}}{\partial\theta% ^{2}}\log(f_{\theta}(y))\right]=0
  12. 𝑦 \,\textit{y}
  13. E θ [ y ] , V a r θ [ y ] E_{\theta}[y],Var_{\theta}[y]
  14. θ \theta
  15. θ log ( f ( y , θ , ϕ ) ) = θ [ y θ - b ( θ ) ϕ - c ( y , ϕ ) ] = y - b ( θ ) ϕ \frac{\partial}{\partial\theta}\log(\operatorname{f}(y,\theta,\phi))=\frac{% \partial}{\partial\theta}\left[\frac{y\theta-b(\theta)}{\phi}-c(y,\phi)\right]% =\frac{y-b^{\prime}(\theta)}{\phi}
  16. E θ [ y - b ( θ ) ϕ ] = E θ [ y ] - b ( θ ) ϕ = 0 \operatorname{E}_{\theta}\left[\frac{y-b^{\prime}(\theta)}{\phi}\right]=\frac{% \operatorname{E}_{\theta}[y]-b^{\prime}(\theta)}{\phi}=0
  17. E θ [ y ] = b ( θ ) \operatorname{E}_{\theta}[y]=b^{\prime}(\theta)
  18. Var θ [ θ ( y θ - b ( θ ) ϕ - c ( y , ϕ ) ) ] + E θ [ 2 θ 2 ( y θ - b ( θ ) ϕ - c ( y , ϕ ) ) ] = 0 \operatorname{Var}_{\theta}\left[\frac{\partial}{\partial\theta}\left(\frac{y% \theta-b(\theta)}{\phi}-c(y,\phi)\right)\right]+\operatorname{E}_{\theta}\left% [\frac{\partial^{2}}{\partial\theta^{2}}\left(\frac{y\theta-b(\theta)}{\phi}-c% (y,\phi)\right)\right]=0
  19. Var θ [ y - b ( θ ) ϕ ] + E θ [ - b ′′ ( θ ) ϕ ] = 0 \operatorname{Var}_{\theta}\left[\frac{y-b^{\prime}(\theta)}{\phi}\right]+% \operatorname{E}_{\theta}\left[\frac{-b^{\prime\prime}(\theta)}{\phi}\right]=0
  20. Var θ [ y ] = b ′′ ( θ ) ϕ \operatorname{Var}_{\theta}\left[y\right]=b^{\prime\prime}(\theta)\phi
  21. μ \mu
  22. θ \theta
  23. μ = b ( θ ) \mu=b^{\prime}(\theta)
  24. θ = b - 1 ( μ ) \theta=b^{\prime-1}(\mu)
  25. μ \mu
  26. V ( θ ) = b ′′ ( θ ) the part of the variance that depends on θ V(\theta)=b^{\prime\prime}(\theta)\rightarrow\,\text{the part of the variance % that depends on }\theta
  27. V ( μ ) = b ′′ ( b - 1 ( μ ) ) \operatorname{V}(\mu)=b^{\prime\prime}(b^{\prime-1}(\mu))
  28. Var θ [ y ] > 0 , b ′′ ( θ ) > 0 \operatorname{Var}_{\theta}\left[y\right]>0,b^{\prime\prime}(\theta)>0
  29. b : θ μ b^{\prime}:\theta\rightarrow\mu
  30. y N ( μ , σ 2 ) y\sim N(\mu,\sigma^{2})
  31. f ( y ) = exp ( y μ - μ 2 2 σ 2 - y 2 2 σ 2 - 1 2 ln 2 π σ 2 ) f(y)=\exp\left(\frac{y\mu-\frac{\mu^{2}}{2}}{\sigma^{2}}-\frac{y^{2}}{2\sigma^% {2}}-\frac{1}{2}\ln{2\pi\sigma^{2}}\right)
  32. θ = μ , \theta=\mu,
  33. b ( θ ) = μ 2 2 , b(\theta)=\frac{\mu^{2}}{2},
  34. ϕ = σ 2 , \phi=\sigma^{2},
  35. c ( y , ϕ ) = - y 2 2 σ 2 - 1 2 ln 2 π σ 2 c(y,\phi)=-\frac{y^{2}}{2\sigma^{2}}-\frac{1}{2}\ln{2\pi\sigma^{2}}
  36. V ( μ ) V(\mu)
  37. θ \theta
  38. μ \mu
  39. V ( θ ) V(\theta)
  40. μ \mu
  41. θ = μ \theta=\mu
  42. b ( θ ) = θ = E [ y ] = μ b^{\prime}(\theta)=\theta=\operatorname{E}[y]=\mu
  43. V ( θ ) = b ′′ ( θ ) = 1 V(\theta)=b^{\prime\prime}(\theta)=1
  44. y Bernoulli ( p ) y\sim\,\text{Bernoulli}(p)
  45. f ( y ) = exp ( y ln p 1 - p + ln ( 1 - p ) ) f(y)=\exp\left(y\ln\frac{p}{1-p}+\ln(1-p)\right)
  46. θ = ln p 1 - p = \theta=\ln\frac{p}{1-p}=
  47. p = e θ 1 + e θ = p=\frac{e^{\theta}}{1+e^{\theta}}=
  48. ( θ ) (\theta)
  49. b ( θ ) = ln ( 1 + e θ ) b(\theta)=\ln(1+e^{\theta})
  50. b ( θ ) = e θ 1 + e θ = b^{\prime}(\theta)=\frac{e^{\theta}}{1+e^{\theta}}=
  51. ( θ ) = p = μ (\theta)=p=\mu
  52. b ′′ ( θ ) = e θ 1 + e θ - ( e θ 1 + e θ ) 2 b^{\prime\prime}(\theta)=\frac{e^{\theta}}{1+e^{\theta}}-\left(\frac{e^{\theta% }}{1+e^{\theta}}\right)^{2}
  53. V ( μ ) = μ ( 1 - μ ) V(\mu)=\mu(1-\mu)
  54. y Poisson ( λ ) y\sim\,\text{Poisson}(\lambda)
  55. f ( y ) = exp ( y ln λ - ln λ ) f(y)=\exp(y\ln\lambda-\ln\lambda)
  56. θ = ln λ = \theta=\ln\lambda=
  57. λ = e θ \lambda=e^{\theta}
  58. b ( θ ) = e θ b(\theta)=e^{\theta}
  59. b ( θ ) = e θ = λ = μ b^{\prime}(\theta)=e^{\theta}=\lambda=\mu
  60. b ′′ ( θ ) = e θ = μ b^{\prime\prime}(\theta)=e^{\theta}=\mu
  61. V ( μ ) = μ V(\mu)=\mu
  62. ( μ , ν ) (\mu,\nu)
  63. f μ , ν ( y ) = 1 Γ ( ν ) y ( ν y μ ) ν e ν y μ f_{\mu,\nu}(y)=\frac{1}{\Gamma(\nu)y}\left(\frac{\nu y}{\mu}\right)^{\nu}e^{% \frac{\nu y}{\mu}}
  64. f μ , ν ( y ) = exp ( - 1 μ y + ln ( 1 μ ) 1 ν + ln ( ν ν y ν - 1 Γ ( ν ) ) ) f_{\mu,\nu}(y)=\exp\left(\frac{-\frac{1}{\mu}y+\ln(\frac{1}{\mu})}{\frac{1}{% \nu}}+\ln\left(\frac{\nu^{\nu}y^{\nu-1}}{\Gamma(\nu)}\right)\right)
  65. θ = - 1 μ μ = - 1 θ \theta=\frac{-1}{\mu}\rightarrow\mu=\frac{-1}{\theta}
  66. ϕ = 1 ν \phi=\frac{1}{\nu}
  67. b ( θ ) = - l n ( - θ ) b(\theta)=-ln(-\theta)
  68. b ( θ ) = - 1 θ = - 1 - 1 μ = μ b^{\prime}(\theta)=\frac{-1}{\theta}=\frac{-1}{\frac{-1}{\mu}}=\mu
  69. b ′′ ( θ ) = 1 θ 2 = μ 2 b^{\prime\prime}(\theta)=\frac{1}{\theta^{2}}=\mu^{2}
  70. V ( μ ) = μ 2 V(\mu)=\mu^{2}
  71. β \beta
  72. Z = g ( E [ y X ] ) = X β Z=g(E[y\mid X])=X\beta
  73. ( Z - X B ) T W ( Z - X B ) (Z-XB)^{T}W(Z-XB)
  74. W n × n = [ 1 ϕ V ( μ 1 ) g ( μ 1 ) 2 0 0 0 0 1 ϕ V ( μ 2 ) g ( μ 2 ) 2 0 0 0 0 0 0 1 ϕ V ( μ n ) g ( μ n ) 2 ] , \underbrace{W}_{n\times n}=\begin{bmatrix}\frac{1}{\phi V(\mu_{1})g^{\prime}(% \mu_{1})^{2}}&0&\cdots&0&0\\ 0&\frac{1}{\phi V(\mu_{2})g^{\prime}(\mu_{2})^{2}}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&0\\ \vdots&\vdots&\vdots&\vdots&0\\ 0&\cdots&\cdots&0&\frac{1}{\phi V(\mu_{n})g^{\prime}(\mu_{n})^{2}}\end{bmatrix},
  75. ϕ , V ( μ ) , g ( μ ) \phi,V(\mu),g(\mu)
  76. ( Z - X B ) T W ( Z - X B ) (Z-XB)^{T}W(Z-XB)
  77. β \beta
  78. β r , 1 r p \beta_{r},1\leq r\leq p
  79. i = 1 n l i β r = 0 \sum_{i=1}^{n}\frac{\partial l_{i}}{\partial\beta_{r}}=0
  80. l ( θ , y , ϕ ) = log ( f ( y , θ , ϕ ) ) = y θ - b ( θ ) ϕ - c ( y , ϕ ) \operatorname{l}(\theta,y,\phi)=\log(\operatorname{f}(y,\theta,\phi))=\frac{y% \theta-b(\theta)}{\phi}-c(y,\phi)
  81. l β r = l θ θ μ μ η η β r \frac{\partial l}{\partial\beta_{r}}=\frac{\partial l}{\partial\theta}\frac{% \partial\theta}{\partial\mu}\frac{\partial\mu}{\partial\eta}\frac{\partial\eta% }{\partial\beta_{r}}
  82. η β r = x r \frac{\partial\eta}{\partial\beta_{r}}=x_{r}
  83. l θ = y - b ( θ ) ϕ = y - μ ϕ \frac{\partial l}{\partial\theta}=\frac{y-b^{\prime}(\theta)}{\phi}=\frac{y-% \mu}{\phi}
  84. θ μ = b - 1 ( μ ) μ = 1 b ′′ ( b ( μ ) ) = 1 V ( μ ) \frac{\partial\theta}{\partial\mu}=\frac{\partial b^{\prime-1}(\mu)}{\mu}=% \frac{1}{b^{\prime\prime}(b^{\prime}(\mu))}=\frac{1}{V(\mu)}
  85. l β r = y - μ ϕ V ( μ ) μ η x r \frac{\partial l}{\partial\beta_{r}}=\frac{y-\mu}{\phi V(\mu)}\frac{\partial% \mu}{\partial\eta}x_{r}
  86. η μ = g ( μ ) \frac{\partial\eta}{\partial\mu}=g^{\prime}(\mu)
  87. l β r = y - μ ϕ V ( μ ) W η μ x r \frac{\partial l}{\partial\beta_{r}}=\frac{y-\mu}{\phi V(\mu)}W\frac{\partial% \eta}{\partial\mu}x_{r}
  88. H = X T ( y - μ ) [ β s W β r ] - X T W X H=X^{T}(y-\mu)\left[\frac{\partial}{\beta_{s}}W\frac{\partial}{\beta_{r}}% \right]-X^{T}WX
  89. FI = - E [ H ] = X T W X \,\text{FI}=-E[H]=X^{T}WX
  90. β ^ \hat{\beta}
  91. β ^ N p ( β , ( X T W X ) - 1 ) \hat{\beta}\sim N_{p}(\beta,(X^{T}WX)^{-1})
  92. E [ y ] = μ = g - 1 ( η ) E[y]=\mu=g^{-1}(\eta)
  93. V ( μ ) , where the Var θ ( y ) = σ 2 V ( μ ) V(\mu)\,\text{, where the }\operatorname{Var}_{\theta}(y)=\sigma^{2}V(\mu)
  94. β \beta
  95. Q i ( μ i , y i ) = y i μ i y i - t σ 2 V ( t ) d t Q_{i}(\mu_{i},y_{i})=\int_{y_{i}}^{\mu_{i}}\frac{y_{i}-t}{\sigma^{2}V(t)}\,dt
  96. Q ( μ , y ) = i = 1 n Q i ( μ i , y i ) = i = 1 n y i μ i y - t σ 2 V ( t ) d t Q(\mu,y)=\sum_{i=1}^{n}Q_{i}(\mu_{i},y_{i})=\sum_{i=1}^{n}\int_{y_{i}}^{\mu_{i% }}\frac{y-t}{\sigma^{2}V(t)}\,dt
  97. l ( μ y ) \operatorname{l}(\mu\mid y)
  98. U = l d μ . U=\frac{\partial l}{d\mu}.
  99. U = y - μ σ 2 V ( μ ) U=\frac{y-\mu}{\sigma^{2}V(\mu)}
  100. Q μ = y - μ σ 2 V ( μ ) \frac{\partial Q}{\partial\mu}=\frac{y-\mu}{\sigma^{2}V(\mu)}
  101. E [ U ] = 0 E[U]=0
  102. Cov ( U ) + E [ U μ ] = 0. \operatorname{Cov}(U)+E\left[\frac{\partial U}{\partial\mu}\right]=0.
  103. β \beta
  104. β \beta
  105. μ = g - 1 ( η ) \mu=g^{-1}(\eta)
  106. η = X β \eta=X\beta
  107. μ = g - 1 ( X β ) . \mu=g^{-1}(X\beta).
  108. i b = - E [ U β ] i_{b}=-\operatorname{E}\left[\frac{\partial U}{\partial\beta}\right]
  109. β \beta
  110. β \beta
  111. μ = g - 1 ( X β ) \mu=g^{-1}(X\beta)
  112. β \beta
  113. β \beta
  114. Q ( β , y ) = y μ ( β ) y - t σ 2 V ( t ) d t Q(\beta,y)=\int_{y}^{\mu(\beta)}\frac{y-t}{\sigma^{2}V(t)}\,dt
  115. β \beta
  116. U j ( β j ) = β j Q ( β , y ) = i = 1 n μ i β j y i - μ i ( β j ) σ 2 V ( μ i ) U_{j}(\beta_{j})=\frac{\partial}{\partial\beta_{j}}Q(\beta,y)=\sum_{i=1}^{n}% \frac{\partial\mu_{i}}{\partial\beta_{j}}\frac{y_{i}-\mu_{i}(\beta_{j})}{% \sigma^{2}V(\mu_{i})}
  117. U ( β ) = [ U 1 ( β ) U 2 ( β ) U p ( β ) ] = D T V - 1 ( y - μ ) σ 2 U(\beta)=\begin{bmatrix}U_{1}(\beta)\\ U_{2}(\beta)\\ \vdots\\ \vdots\\ U_{p}(\beta)\end{bmatrix}=D^{T}V^{-1}\frac{(y-\mu)}{\sigma^{2}}
  118. D n × p = [ μ 1 β 1 μ 1 β p μ 2 β 1 μ 2 β p μ m β 1 μ m β p ] V n × n = diag ( V ( μ 1 ) , V ( μ 2 ) , , , V ( μ n ) ) \underbrace{D}_{n\times p}=\begin{bmatrix}\frac{\partial\mu_{1}}{\partial\beta% _{1}}&\cdots&\cdots&\frac{\partial\mu_{1}}{\partial\beta_{p}}\\ \frac{\partial\mu_{2}}{\partial\beta_{1}}&\cdots&\cdots&\frac{\partial\mu_{2}}% {\partial\beta_{p}}\\ \vdots\\ \vdots\\ \frac{\partial\mu_{m}}{\partial\beta_{1}}&\cdots&\cdots&\frac{\partial\mu_{m}}% {\partial\beta_{p}}\end{bmatrix}\underbrace{V}_{n\times n}=\operatorname{diag}% (V(\mu_{1}),V(\mu_{2}),\ldots,\ldots,V(\mu_{n}))
  119. β \beta
  120. i b = - U β = Cov ( U ( β ) ) = D T V - 1 D σ 2 i_{b}=-\frac{\partial U}{\partial\beta}=\operatorname{Cov}(U(\beta))=\frac{D^{% T}V^{-1}D}{\sigma^{2}}
  121. β \beta
  122. g ( x ) = E [ y X = x ] g(x)=\operatorname{E}[y\mid X=x]
  123. g ( x ) g(x)
  124. g v ( x ) = Var ( Y X = x ) g_{v}(x)=\operatorname{Var}(Y\mid X=x)
  125. g v ( x ) = Var ( Y X = x ) = E [ y 2 X = x ] - [ E [ y X = x ] ] 2 g_{v}(x)=\operatorname{Var}(Y\mid X=x)=\operatorname{E}[y^{2}\mid X=x]-\left[% \operatorname{E}[y\mid X=x]\right]^{2}
  126. g v ( x ) = Var ( Y X = x ) = E [ y 2 X = x ] - [ E [ y X = x ] ] 2 g_{v}(x)=\operatorname{Var}(Y\mid X=x)=\operatorname{E}[y^{2}\mid X=x]-\left[% \operatorname{E}[y\mid X=x]\right]^{2}
  127. E [ y 2 X = x ] \operatorname{E}[y^{2}\mid X=x]
  128. [ E [ y X = x ] ] 2 \left[\operatorname{E}[y\mid X=x]\right]^{2}