wpmath0000013_5

Gennadi_Sardanashvily.html

  1. \mathbb{R}

Gent_(hyperelastic_model).html

  1. I m I_{m}
  2. W = - μ J m 2 ln ( 1 - I 1 - 3 J m ) W=-\cfrac{\mu J_{m}}{2}\ln\left(1-\cfrac{I_{1}-3}{J_{m}}\right)
  3. μ \mu
  4. J m = I m - 3 J_{m}=I_{m}-3
  5. I m I_{m}\rightarrow\infty
  6. W = μ 2 x ln [ 1 - ( I 1 - 3 ) x ] ; x := 1 J m W=\cfrac{\mu}{2x}\ln\left[1-(I_{1}-3)x\right]~{};~{}~{}x:=\cfrac{1}{J_{m}}
  7. ln [ 1 - ( I 1 - 3 ) x ] \ln\left[1-(I_{1}-3)x\right]
  8. x = 0 x=0
  9. x 0 x\rightarrow 0
  10. W = μ 2 ( I 1 - 3 ) W=\cfrac{\mu}{2}(I_{1}-3)
  11. W = - μ J m 2 ln ( 1 - I 1 - 3 J m ) + κ 2 ( J 2 - 1 2 - ln J ) 4 W=-\cfrac{\mu J_{m}}{2}\ln\left(1-\cfrac{I_{1}-3}{J_{m}}\right)+\cfrac{\kappa}% {2}\left(\cfrac{J^{2}-1}{2}-\ln J\right)^{4}
  12. J = det ( s y m b o l F ) J=\det(symbol{F})
  13. κ \kappa
  14. s y m b o l F symbol{F}
  15. W = C 0 ln ( 1 - I 1 - 3 J m ) W=C_{0}\ln\left(1-\cfrac{I_{1}-3}{J_{m}}\right)
  16. 2 W I 1 ( 3 ) = μ 2\cfrac{\partial W}{\partial I_{1}}(3)=\mu
  17. μ \mu
  18. I 1 = 3 ( λ i = λ j = 1 ) I_{1}=3(\lambda_{i}=\lambda_{j}=1)
  19. W I 1 = - C 0 J m \cfrac{\partial W}{\partial I_{1}}=-\cfrac{C_{0}}{J_{m}}
  20. - 2 C 0 J m = μ C 0 = - μ J m 2 -\cfrac{2C_{0}}{J_{m}}=\mu\,\qquad\implies\qquad C_{0}=-\cfrac{\mu J_{m}}{2}
  21. J m 1 J_{m}\gg 1
  22. s y m b o l σ = - p s y m b o l 1 + 2 W I 1 s y m b o l B = - p s y m b o l 1 + μ J m J m - I 1 + 3 s y m b o l B symbol{\sigma}=-p~{}symbol{\mathit{1}}+2~{}\cfrac{\partial W}{\partial I_{1}}~% {}symbol{B}=-p~{}symbol{\mathit{1}}+\cfrac{\mu J_{m}}{J_{m}-I_{1}+3}~{}symbol{B}
  23. 𝐧 1 \mathbf{n}_{1}
  24. λ 1 = λ , λ 2 = λ 3 \lambda_{1}=\lambda,~{}\lambda_{2}=\lambda_{3}
  25. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  26. λ 2 2 = λ 3 2 = 1 / λ \lambda_{2}^{2}=\lambda_{3}^{2}=1/\lambda
  27. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 2 λ . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\cfrac{2}{% \lambda}~{}.
  28. s y m b o l B = λ 2 𝐧 1 𝐧 1 + 1 λ ( 𝐧 2 𝐧 2 + 𝐧 3 𝐧 3 ) . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\cfrac{1}{\lambda}% ~{}(\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\mathbf{n}_{3}\otimes\mathbf{n}_{3})~{}.
  29. σ 11 = - p + λ 2 μ J m J m - I 1 + 3 ; σ 22 = - p + μ J m λ ( J m - I 1 + 3 ) = σ 33 . \sigma_{11}=-p+\cfrac{\lambda^{2}\mu J_{m}}{J_{m}-I_{1}+3}~{};~{}~{}\sigma_{22% }=-p+\cfrac{\mu J_{m}}{\lambda(J_{m}-I_{1}+3)}=\sigma_{33}~{}.
  30. σ 22 = σ 33 = 0 \sigma_{22}=\sigma_{33}=0
  31. p = μ J m λ ( J m - I 1 + 3 ) . p=\cfrac{\mu J_{m}}{\lambda(J_{m}-I_{1}+3)}~{}.
  32. σ 11 = ( λ 2 - 1 λ ) ( μ J m J m - I 1 + 3 ) . \sigma_{11}=\left(\lambda^{2}-\cfrac{1}{\lambda}\right)\left(\cfrac{\mu J_{m}}% {J_{m}-I_{1}+3}\right)~{}.
  33. λ - 1 \lambda-1\,
  34. T 11 = σ 11 / λ = ( λ - 1 λ 2 ) ( μ J m J m - I 1 + 3 ) . T_{11}=\sigma_{11}/\lambda=\left(\lambda-\cfrac{1}{\lambda^{2}}\right)\left(% \cfrac{\mu J_{m}}{J_{m}-I_{1}+3}\right)~{}.
  35. 𝐧 1 \mathbf{n}_{1}
  36. 𝐧 2 \mathbf{n}_{2}
  37. λ 1 = λ 2 = λ \lambda_{1}=\lambda_{2}=\lambda\,
  38. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  39. λ 3 = 1 / λ 2 \lambda_{3}=1/\lambda^{2}\,
  40. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = 2 λ 2 + 1 λ 4 . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=2~{}\lambda^{2}+\cfrac{1% }{\lambda^{4}}~{}.
  41. s y m b o l B = λ 2 𝐧 1 𝐧 1 + λ 2 𝐧 2 𝐧 2 + 1 λ 4 𝐧 3 𝐧 3 . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\lambda^{2}~{}% \mathbf{n}_{2}\otimes\mathbf{n}_{2}+\cfrac{1}{\lambda^{4}}~{}\mathbf{n}_{3}% \otimes\mathbf{n}_{3}~{}.
  42. σ 11 = ( λ 2 - 1 λ 4 ) ( μ J m J m - I 1 + 3 ) = σ 22 . \sigma_{11}=\left(\lambda^{2}-\cfrac{1}{\lambda^{4}}\right)\left(\cfrac{\mu J_% {m}}{J_{m}-I_{1}+3}\right)=\sigma_{22}~{}.
  43. λ - 1 \lambda-1\,
  44. T 11 = σ 11 λ = ( λ - 1 λ 5 ) ( μ J m J m - I 1 + 3 ) = T 22 . T_{11}=\cfrac{\sigma_{11}}{\lambda}=\left(\lambda-\cfrac{1}{\lambda^{5}}\right% )\left(\cfrac{\mu J_{m}}{J_{m}-I_{1}+3}\right)=T_{22}~{}.
  45. 𝐧 1 \mathbf{n}_{1}
  46. 𝐧 3 \mathbf{n}_{3}
  47. λ 1 = λ , λ 3 = 1 \lambda_{1}=\lambda,~{}\lambda_{3}=1
  48. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  49. λ 2 = 1 / λ \lambda_{2}=1/\lambda\,
  50. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 1 λ 2 + 1 . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\cfrac{1}{% \lambda^{2}}+1~{}.
  51. s y m b o l B = λ 2 𝐧 1 𝐧 1 + 1 λ 2 𝐧 2 𝐧 2 + 𝐧 3 𝐧 3 . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\cfrac{1}{\lambda^% {2}}~{}\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\mathbf{n}_{3}\otimes\mathbf{n}_{3}% ~{}.
  52. σ 11 = ( λ 2 - 1 λ 2 ) ( μ J m J m - I 1 + 3 ) ; σ 22 = 0 ; σ 33 = ( 1 - 1 λ 2 ) ( μ J m J m - I 1 + 3 ) . \sigma_{11}=\left(\lambda^{2}-\cfrac{1}{\lambda^{2}}\right)\left(\cfrac{\mu J_% {m}}{J_{m}-I_{1}+3}\right)~{};~{}~{}\sigma_{22}=0~{};~{}~{}\sigma_{33}=\left(1% -\cfrac{1}{\lambda^{2}}\right)\left(\cfrac{\mu J_{m}}{J_{m}-I_{1}+3}\right)~{}.
  53. λ - 1 \lambda-1\,
  54. T 11 = σ 11 λ = ( λ - 1 λ 3 ) ( μ J m J m - I 1 + 3 ) . T_{11}=\cfrac{\sigma_{11}}{\lambda}=\left(\lambda-\cfrac{1}{\lambda^{3}}\right% )\left(\cfrac{\mu J_{m}}{J_{m}-I_{1}+3}\right)~{}.
  55. s y m b o l F = s y m b o l 1 + γ 𝐞 1 𝐞 2 symbol{F}=symbol{1}+\gamma~{}\mathbf{e}_{1}\otimes\mathbf{e}_{2}
  56. 𝐞 1 , 𝐞 2 \mathbf{e}_{1},\mathbf{e}_{2}
  57. γ = λ - 1 λ ; λ 1 = λ ; λ 2 = 1 λ ; λ 3 = 1 \gamma=\lambda-\cfrac{1}{\lambda}~{};~{}~{}\lambda_{1}=\lambda~{};~{}~{}% \lambda_{2}=\cfrac{1}{\lambda}~{};~{}~{}\lambda_{3}=1
  58. s y m b o l F = [ 1 γ 0 0 1 0 0 0 1 ] ; s y m b o l B = s y m b o l F \cdotsymbol F T = [ 1 + γ 2 γ 0 γ 1 0 0 0 1 ] symbol{F}=\begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}~{};~{}~{}symbol{B}=symbol{F}\cdotsymbol{F}^{T}=\begin{% bmatrix}1+\gamma^{2}&\gamma&0\\ \gamma&1&0\\ 0&0&1\end{bmatrix}
  59. I 1 = tr ( s y m b o l B ) = 3 + γ 2 I_{1}=\mathrm{tr}(symbol{B})=3+\gamma^{2}
  60. s y m b o l σ = - p s y m b o l 1 + μ J m J m - γ 2 s y m b o l B symbol{\sigma}=-p~{}symbol{\mathit{1}}+\cfrac{\mu J_{m}}{J_{m}-\gamma^{2}}~{}% symbol{B}
  61. s y m b o l σ = [ - p + μ J m ( 1 + γ 2 ) J m - γ 2 μ J m γ J m - γ 2 0 μ J m γ J m - γ 2 - p + μ J m J m - γ 2 0 0 0 - p + μ J m J m - γ 2 ] symbol{\sigma}=\begin{bmatrix}-p+\cfrac{\mu J_{m}(1+\gamma^{2})}{J_{m}-\gamma^% {2}}&\cfrac{\mu J_{m}\gamma}{J_{m}-\gamma^{2}}&0\\ \cfrac{\mu J_{m}\gamma}{J_{m}-\gamma^{2}}&-p+\cfrac{\mu J_{m}}{J_{m}-\gamma^{2% }}&0\\ 0&0&-p+\cfrac{\mu J_{m}}{J_{m}-\gamma^{2}}\end{bmatrix}

Genus–degree_formula.html

  1. C 2 C\subset\mathbb{P}^{2}
  2. g = 1 2 ( d - 1 ) ( d - 2 ) . g=\frac{1}{2}(d-1)(d-2).\,
  3. 1 2 r ( r - 1 ) \scriptstyle\frac{1}{2}r(r-1)
  4. H H
  5. n \mathbb{P}^{n}
  6. g = ( d - 1 n ) , g={\left({{d-1}\atop{n}}\right)},\,
  7. ( d - 1 n ) {\textstyle\left({{d-1}\atop{n}}\right)}

Geometric_design_of_roads.html

  1. L = A S 2 120 + 3.5 S L=\frac{AS^{2}}{120+3.5S}
  2. L = 2 S - 120 + 3.5 S A L=2S-\frac{120+3.5S}{A}
  3. L = A S 2 400 + 3.5 S L=\frac{AS^{2}}{400+3.5S}
  4. L = 2 S - 400 + 3.5 S A L=2S-\frac{400+3.5S}{A}
  5. L = 2 S - 200 ( h 1 + h 2 ) 2 A L=2S-\frac{200(\sqrt{h_{1}}+\sqrt{h_{2}})^{2}}{A}
  6. T = R tan ( Δ 2 ) T=R\tan\left(\frac{\Delta}{2}\right)
  7. C = 2 R sin ( Δ 2 ) C=2R\sin\left(\frac{\Delta}{2}\right)
  8. L = R π Δ 180 L=R\pi\frac{\Delta}{180}
  9. M = R ( 1 - cos ( Δ 2 ) ) M=R\left(1-\cos\left(\frac{\Delta}{2}\right)\right)
  10. E = R ( 1 cos ( Δ 2 ) - 1 ) E=R\left(\frac{1}{\cos\left(\frac{\Delta}{2}\right)}-1\right)
  11. M = R ( 1 - cos ( 28.65 S R ) ) M=R\left(1-\cos\left(\frac{28.65S}{R}\right)\right)
  12. R = u 2 15 ( e + f s ) R=\frac{u^{2}}{15(e+f_{s})}
  13. A M F = 1.55 L c + 80.2 R - .012 S 1.55 L c AMF=\frac{1.55L_{c}+\frac{80.2}{R}-.012S}{1.55L_{c}}
  14. D C S D D_{CSD}
  15. t g t_{g}
  16. V D S V_{DS}
  17. D C S D = V D S t g D_{CSD}=V_{DS}t_{g}

Geometric_progression.html

  1. a , a r , a r 2 , a r 3 , a r 4 , a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots
  2. a n = a r n - 1 . a_{n}=a\,r^{n-1}.
  3. a n = r a n - 1 a_{n}=r\,a_{n-1}
  4. n 1. n\geq 1.
  5. b 2 = a c b^{2}=ac
  6. 2 + 10 + 50 + 250 = 2 + 2 × 5 + 2 × 5 2 + 2 × 5 3 . 2+10+50+250=2+2\times 5+2\times 5^{2}+2\times 5^{3}.\,
  7. a ( 1 - r m ) 1 - r \frac{a(1-r^{m})}{1-r}
  8. 2 + 10 + 50 + 250 = 2 ( 1 - 5 4 ) 1 - 5 = - 1248 - 4 = 312. 2+10+50+250=\frac{2(1-5^{4})}{1-5}=\frac{-1248}{-4}=312.
  9. - 2 π + 4 π 2 - 8 π 3 = - 2 π + ( - 2 π ) 2 + ( - 2 π ) 3 = - 2 π ( 1 - ( - 2 π ) 3 ) 1 - ( - 2 π ) = - 2 π ( 1 + 8 π 3 ) 1 + 2 π - 214.855. -2\pi+4\pi^{2}-8\pi^{3}=-2\pi+(-2\pi)^{2}+(-2\pi)^{3}=\frac{-2\pi(1-(-2\pi)^{3% })}{1-(-2\pi)}=\frac{-2\pi(1+8\pi^{3})}{1+2\pi}\approx-214.855.
  10. k = 1 n a r k - 1 = a r 0 + a r 1 + a r 2 + a r 3 + + a r n - 1 . \sum_{k=1}^{n}ar^{k-1}=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots+ar^{n-1}.\,
  11. ( 1 - r ) k = 1 n a r k - 1 = ( 1 - r ) ( a r 0 + a r 1 + a r 2 + a r 3 + + a r n - 1 ) = a r 0 + a r 1 + a r 2 + a r 3 + + a r n - 1 - a r 1 - a r 2 - a r 3 - - a r n - 1 - a r n = a - a r n \begin{aligned}\displaystyle(1-r)\sum_{k=1}^{n}ar^{k-1}&\displaystyle=(1-r)(ar% ^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots+ar^{n-1})\\ &\displaystyle=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots+ar^{n-1}-ar^{1}-ar^{2}-ar^{3% }-\cdots-ar^{n-1}-ar^{n}\\ &\displaystyle=a-ar^{n}\end{aligned}
  12. k = 1 n a r k - 1 = a ( 1 - r n ) 1 - r . \sum_{k=1}^{n}ar^{k-1}=\frac{a(1-r^{n})}{1-r}.
  13. k = m n a r k = a ( r m - r n + 1 ) 1 - r . \sum_{k=m}^{n}ar^{k}=\frac{a(r^{m}-r^{n+1})}{1-r}.
  14. k = 0 n k s r k . \sum_{k=0}^{n}k^{s}r^{k}.
  15. d d r k = 0 n r k = k = 1 n k r k - 1 = 1 - r n + 1 ( 1 - r ) 2 - ( n + 1 ) r n 1 - r . \frac{d}{dr}\sum_{k=0}^{n}r^{k}=\sum_{k=1}^{n}kr^{k-1}=\frac{1-r^{n+1}}{(1-r)^% {2}}-\frac{(n+1)r^{n}}{1-r}.
  16. ( 1 - r 2 ) k = 0 n a r 2 k = a - a r 2 n + 2 . (1-r^{2})\sum_{k=0}^{n}ar^{2k}=a-ar^{2n+2}.
  17. k = 0 n a r 2 k = a ( 1 - r 2 n + 2 ) 1 - r 2 . \sum_{k=0}^{n}ar^{2k}=\frac{a(1-r^{2n+2})}{1-r^{2}}.
  18. ( 1 - r 2 ) k = 0 n a r 2 k + 1 = a r - a r 2 n + 3 (1-r^{2})\sum_{k=0}^{n}ar^{2k+1}=ar-ar^{2n+3}
  19. k = 0 n a r 2 k + 1 = a r ( 1 - r 2 n + 2 ) 1 - r 2 . \sum_{k=0}^{n}ar^{2k+1}=\frac{ar(1-r^{2n+2})}{1-r^{2}}.
  20. | r | |r|
  21. r n + 1 0 as n when | r | < 1. r^{n+1}\to 0\mbox{ as }~{}n\to\infty\mbox{ when }~{}|r|<1.
  22. k = 0 a r k = a 1 - r - 0 = a 1 - r \sum_{k=0}^{\infty}ar^{k}=\frac{a}{1-r}-0=\frac{a}{1-r}
  23. r r
  24. k = 0 a r 2 k = a 1 - r 2 \sum_{k=0}^{\infty}ar^{2k}=\frac{a}{1-r^{2}}
  25. k = 0 a r 2 k + 1 = a r 1 - r 2 \sum_{k=0}^{\infty}ar^{2k+1}=\frac{ar}{1-r^{2}}
  26. k = m a r k = a r m 1 - r \sum_{k=m}^{\infty}ar^{k}=\frac{ar^{m}}{1-r}
  27. | r | |r|
  28. | r | |r|
  29. 1 2 + 1 4 + 1 8 + 1 16 + = 1 / 2 1 - ( + 1 / 2 ) = 1. \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)}=1.
  30. 1 2 - 1 4 + 1 8 - 1 16 + = 1 / 2 1 - ( - 1 / 2 ) = 1 3 . \frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)}=% \frac{1}{3}.
  31. k = 0 sin ( k x ) r k = r sin ( x ) 1 + r 2 - 2 r cos ( x ) \sum_{k=0}^{\infty}\frac{\sin(kx)}{r^{k}}=\frac{r\sin(x)}{1+r^{2}-2r\cos(x)}
  32. sin ( k x ) = e i k x - e - i k x 2 i , \sin(kx)=\frac{e^{ikx}-e^{-ikx}}{2i},
  33. k = 0 sin ( k x ) r k = 1 2 i [ k = 0 ( e i x r ) k - k = 0 ( e - i x r ) k ] \sum_{k=0}^{\infty}\frac{\sin(kx)}{r^{k}}=\frac{1}{2i}\left[\sum_{k=0}^{\infty% }\left(\frac{e^{ix}}{r}\right)^{k}-\sum_{k=0}^{\infty}\left(\frac{e^{-ix}}{r}% \right)^{k}\right]
  34. i = 0 n a r i = ( a 0 a n ) n + 1 \prod_{i=0}^{n}ar^{i}=\left(\sqrt{a_{0}\cdot a_{n}}\right)^{n+1}
  35. a , r > 0 a,r>0
  36. P = a a r a r 2 a r n - 1 a r n P=a\cdot ar\cdot ar^{2}\cdots ar^{n-1}\cdot ar^{n}
  37. P = a n + 1 r 1 + 2 + 3 + + ( n - 1 ) + ( n ) P=a^{n+1}r^{1+2+3+\cdots+(n-1)+(n)}
  38. P = a n + 1 r n ( n + 1 ) 2 P=a^{n+1}r^{\frac{n(n+1)}{2}}
  39. P = ( a r n 2 ) n + 1 P=(ar^{\frac{n}{2}})^{n+1}
  40. P 2 = ( a 2 r n ) n + 1 = ( a a r n ) n + 1 P^{2}=(a^{2}r^{n})^{n+1}=(a\cdot ar^{n})^{n+1}
  41. P 2 = ( a 0 a n ) n + 1 P^{2}=(a_{0}\cdot a_{n})^{n+1}
  42. P = ( a 0 a n ) n + 1 2 P=(a_{0}\cdot a_{n})^{\frac{n+1}{2}}

Geometry_of_roots_of_real_polynomials.html

  1. y = a ( x - h ) 2 + k , y=a(x-h)^{2}+k,\,
  2. 5 ± 3 i . 5\pm 3i.\,

Gewirtz_graph.html

  1. ( x - 10 ) ( x - 2 ) 35 ( x + 4 ) 20 . (x-10)(x-2)^{35}(x+4)^{20}.\,

Giambelli's_formula.html

  1. σ λ = det ( σ λ i + j - i ) 1 i , j r \displaystyle\sigma_{\lambda}=\det(\sigma_{\lambda_{i}+j-i})_{1\leq i,j\leq r}

Gieseking_manifold.html

  1. π / 3 \pi/3

Ginzburg_criterion.html

  1. ϕ \phi
  2. ( δ ϕ ) 2 ( ϕ ) 2 \displaystyle\mathcal{\langle}(\delta\phi)^{2}\rangle\quad{\ll}\quad\langle(% \phi)^{2}\rangle
  3. T T\rightarrow\infty

Gipps'_model.html

  1. τ \tau
  2. a n ( t + τ ) = l n [ v n - 1 ( t ) - v n ( t ) ] k [ x n - 1 ( t ) - x n ( t ) ] m a_{n}(t+\tau)=l_{n}\frac{\left[v_{n-1}(t)-v_{n}(t)\right]^{k}}{\left[x_{n-1}(t% )-x_{n}(t)\right]^{m}}
  3. a n a_{n}
  4. n n
  5. b n b_{n}
  6. n n
  7. ( b n < 0 ) (b_{n}<0)
  8. s n s_{n}
  9. n n
  10. V n V_{n}
  11. n n
  12. x n ( t ) x_{n}(t)
  13. n n
  14. t t
  15. v n ( t ) v_{n}(t)
  16. n n
  17. t t
  18. τ \tau
  19. v n ( t + τ ) v n ( t ) + 2.5 a n τ ( 1 - v n / V n ) ( 0.025 + v n ( t ) / V n ) 1 / 2 v_{n}(t+\tau)\leq v_{n}(t)+2.5a_{n}\tau(1-v_{n}/V_{n})(0.025+v_{n}(t)/V_{n})^{% 1/2}
  20. x n - 1 = x n - 1 ( t ) - v n - 1 ( t ) 2 / 2 b n - 1 x_{n-1}^{\ast}=x_{n-1}(t)-v_{n-1}(t)^{2}/2b_{n-1}
  21. n - 1 n-1
  22. x n - 1 x_{n-1}^{\ast}
  23. x n x_{n}^{\ast}
  24. x n = x n ( t ) + [ v n ( t ) + v n ( t + τ ) ] τ / 2 - v n ( t + τ ) 2 / 2 b n x_{n}^{\ast}=x_{n}(t)+\left[v_{n}(t)+v_{n}(t+\tau)\right]\tau/2-v_{n}(t+\tau)^% {2}/2b_{n}
  25. t + τ t+\tau
  26. x n - 1 x_{n-1}^{\ast}
  27. s n - 1 s_{n-1}
  28. x n x_{n}^{\ast}
  29. θ \theta
  30. v n ( t + τ ) v_{n}(t+\tau)
  31. x n - 1 ( t ) - v n - 1 ( t ) 2 / 2 b n - 1 - s n - 1 x n ( t ) + [ v n ( t ) - v n ( t + τ ) ] τ / 2 - v n ( t + τ ) θ - v n ( t + τ ) 2 / 2 b n x_{n-1}(t)-v_{n-1}(t)^{2}/2b_{n-1}-s_{n-1}\geq x_{n}(t)+\left[v_{n}(t)-v_{n}(t% +\tau)\right]\tau/2-v_{n}(t+\tau)\theta-v_{n}(t+\tau)^{2}/2b_{n}
  32. b n - 1 b_{n-1}
  33. b ^ \hat{b}
  34. - v n ( t + τ ) 2 / 2 b n + v n ( t + τ ) ( τ / 2 + θ ) - [ x n - 1 ( t ) - s n - 1 - x n ( t ) ] + v n ( t ) τ / 2 + v n - 1 ( t ) 2 / 2 b ^ 0 -v_{n}(t+\tau)^{2}/2b_{n}+v_{n}(t+\tau)(\tau/2+\theta)-\left[x_{n-1}(t)-s_{n-1% }-x_{n}(t)\right]+v_{n}(t)\tau/2+v_{n-1}(t)^{2}/2\hat{b}\leq 0
  35. θ \theta
  36. τ / 2 \tau/2
  37. v n ( t + τ ) b n τ + b n 2 τ 2 - b n ( 2 [ x n - 1 ( t ) - s n - 1 - x n ( t ) ] - v n ( t ) τ - v n - 1 ( t ) 2 / b ^ ) v_{n}(t+\tau)\leq b_{n}\tau+\sqrt{b_{n}^{2}\tau^{2}-b_{n}\left(2\left[x_{n-1}(% t)-s_{n-1}-x_{n}(t)\right]-v_{n}(t)\tau-v_{n-1}(t)^{2}/\hat{b}\right)}
  38. v n ( t + τ ) = min { v n ( t ) + 2.5 a n τ ( 1 - v n ( t ) / V n ) ( 0.025 + v n ( t ) / V n ) 1 / 2 , v_{n}(t+\tau)=\mbox{min}~{}\left\{v_{n}(t)+2.5a_{n}\tau(1-v_{n}(t)/V_{n})\left% (0.025+v_{n}\left(t\right)/V_{n}\right)^{1/2}\right.,
  39. b n τ + b n 2 τ 2 - b n [ 2 [ x n - 1 ( t ) - s n - 1 - x n ( t ) ] - v n ( t ) τ - v n - 1 ( t ) 2 / b ] } \left.b_{n}\tau+\sqrt{b_{n}^{2}\tau^{2}-b_{n}\left[2\left[x_{n-1}(t)-s_{n-1}-x% _{n}(t)\right]-v_{n}(t)\tau-v_{n-1}(t)^{2}/b\right]}\right\}

Giusto_Bellavitis.html

  1. A B C D . AB\bumpeq CD.
  2. A B + B C A C . AB+BC\bumpeq AC.

GJMS_operator.html

  1. L k : E [ k - n / 2 ] E [ - k - n / 2 ] . L_{k}:E[k-n/2]\to E[-k-n/2].

Glass_transition.html

  1. T g T 0 c as d T d t 0. T_{g}\to T_{0c}\,\text{ as }\frac{dT}{dt}\to 0.

Glicksberg's_theorem.html

  1. A × B A\times B
  2. sup f inf g K d f d g = inf g sup f K d f d g \sup_{f}\inf_{g}\int\int K\,df\,dg=\inf_{g}\sup_{f}\int\int K\,df\,dg

Global_index_grammar.html

  1. A α A\to\alpha
  2. α \alpha
  3. A 𝑓 α A\xrightarrow[f]{}\alpha
  4. α \alpha
  5. A + f x α A\xrightarrow[+f]{}x\alpha
  6. x α x\alpha
  7. A - f α A\xrightarrow[-f]{}\alpha
  8. α \alpha
  9. α \alpha
  10. a b X d [ f f g ] \frac{abXd}{[ffg]}
  11. { w w + : w { a , b } * } \{ww^{+}:w\in\{a,b\}^{*}\}
  12. S A S | B S | C | ϵ S\to AS~{}|~{}BS~{}|~{}C~{}|~{}\epsilon
  13. C R C | L C\to RC~{}|~{}L
  14. R - f R A R\xrightarrow[-f]{}RA
  15. R - g R B R\xrightarrow[-g]{}RB
  16. R # ϵ R\xrightarrow[\#]{}\epsilon
  17. A + f a A\xrightarrow[+f]{}a
  18. B + g b B\xrightarrow[+g]{}b
  19. L - f L a | a L\xrightarrow[-f]{}La~{}|~{}a
  20. L - g L b | b L\xrightarrow[-g]{}Lb~{}|~{}b
  21. S [ # ] A S [ # ] a S [ # f ] a B S [ # f ] a b S [ # f g ] a b C [ # f g ] a b R C [ # f g ] a b R B C [ # f ] \frac{S}{[\#]}\to\frac{AS}{[\#]}\to\frac{aS}{[\#f]}\to\frac{aBS}{[\#f]}\to% \frac{abS}{[\#fg]}\to\frac{abC}{[\#fg]}\to\frac{abRC}{[\#fg]}\to\frac{abRBC}{[% \#f]}
  22. a b R A B C [ # ] a b A B C [ # ] a b a B C [ # f ] a b a b C [ # f g ] a b a b L [ # f g ] a b a b L b [ # f ] a b a b a b [ # ] \frac{abRABC}{[\#]}\to\frac{abABC}{[\#]}\to\frac{abaBC}{[\#f]}\to\frac{ababC}{% [\#fg]}\to\frac{ababL}{[\#fg]}\to\frac{ababLb}{[\#f]}\to\frac{ababab}{[\#]}
  23. { p ( a n b n c n ) : n 1 } \{p(a^{n}b^{n}c^{n}):n\geq 1\}
  24. { a m b n c m d n : m , n 1 } \{a^{m}b^{n}c^{m}d^{n}:m,n\geq 1\}
  25. S \displaystyle S
  26. S [ # ] \displaystyle\frac{S}{[\#]}

Glossary_of_algebraic_groups.html

  1. 𝔾 m {\mathbb{G}}_{m}
  2. SO n {\rm SO}_{n}

Glossary_of_Lie_algebras.html

  1. 𝔤 \mathfrak{g}
  2. F F
  3. a , b F , x , y , z 𝔤 \forall a,b\in F,x,y,z\in\mathfrak{g}
  4. [ a x + b y , z ] = a [ x , z ] + b [ y , z ] [ax+by,z]=a[x,z]+b[y,z]
  5. [ x , x ] = 0 [x,x]=0
  6. [ [ x , y ] , z ] + [ [ y , z ] , x ] + [ [ z , x ] , y ] = 0 [[x,y],z]+[[y,z],x]+[[z,x],y]=0
  7. A A
  8. [ x , y ] = x y - y x [x,y]=xy-yx
  9. x , y x,y
  10. x , y A \forall x,y\in A
  11. ϕ : 𝔤 1 𝔤 2 \phi:\mathfrak{g}_{1}\to\mathfrak{g}_{2}
  12. ϕ ( [ x , y ] ) = [ ϕ ( x ) , ϕ ( y ) ] x , y 𝔤 1 . \phi([x,y])=[\phi(x),\phi(y)]\,\forall x,y\in\mathfrak{g}_{1}.
  13. x 𝔤 x\in\mathfrak{g}
  14. ad x \textrm{ad}_{x}
  15. ad x : 𝔤 𝔤 y [ x , y ] \begin{aligned}\displaystyle\textrm{ad}_{x}:&\displaystyle\mathfrak{g}\to% \mathfrak{g}\\ &\displaystyle y\mapsto[x,y]\end{aligned}
  16. ad x \textrm{ad}_{x}
  17. ad : 𝔤 𝔤 𝔩 ( 𝔤 ) x ad x \begin{aligned}\displaystyle\textrm{ad}:&\displaystyle\mathfrak{g}\to\mathfrak% {gl}(\mathfrak{g})\\ &\displaystyle x\mapsto\mathrm{ad}_{x}\end{aligned}
  18. ad : 𝔤 End ( 𝔤 ) \textrm{ad}:\mathfrak{g}\to\textrm{End}(\mathfrak{g})
  19. ad x ( [ y , z ] ) = [ ad x ( y ) , z ] + [ y , ad x ( z ) ] \textrm{ad}_{x}([y,z])=[\textrm{ad}_{x}(y),z]+[y,\textrm{ad}_{x}(z)]
  20. 𝔤 \mathfrak{g^{\prime}}
  21. 𝔤 \mathfrak{g}
  22. 𝔤 \mathfrak{g}
  23. [ 𝔤 , 𝔤 ] 𝔤 . [\mathfrak{g^{\prime}},\mathfrak{g^{\prime}}]\subseteq\mathfrak{g^{\prime}}.
  24. 𝔤 \mathfrak{g^{\prime}}
  25. 𝔤 \mathfrak{g}
  26. 𝔤 \mathfrak{g}
  27. [ 𝔤 , 𝔤 ] 𝔤 . [\mathfrak{g^{\prime}},\mathfrak{g}]\subseteq\mathfrak{g^{\prime}}.
  28. 𝔤 \mathfrak{g}
  29. [ 𝔤 , 𝔤 ] [\mathfrak{g},\mathfrak{g}]
  30. K K
  31. 𝔤 \mathfrak{g}
  32. N 𝔤 ( K ) := { x 𝔤 | [ x , K ] K } N_{\mathfrak{g}}(K):=\{x\in\mathfrak{g}|[x,K]\subseteq K\}
  33. X X
  34. 𝔤 \mathfrak{g}
  35. C 𝔤 ( X ) := { x 𝔤 | [ x , X ] = { 0 } } C_{\mathfrak{g}}(X):=\{x\in\mathfrak{g}|[x,X]=\{0\}\}
  36. Z ( L ) := { x 𝔤 | [ x , 𝔤 ] = 0 } Z(L):=\{x\in\mathfrak{g}|[x,\mathfrak{g}]=0\}
  37. Rad ( 𝔤 ) \textrm{Rad}(\mathfrak{g})
  38. 𝔤 \mathfrak{g}
  39. L L
  40. C N ( L ) = { 0 } C^{N}(L)=\{0\}
  41. N N
  42. C N ( L ) = { 0 } C^{N}(L)=\{0\}
  43. N N
  44. { 0 } \{0\}
  45. C N ( L ) = L C_{N}(L)=L
  46. L L
  47. L = I 1 I 2 I 3 I n = { 0 } L=I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq\cdots\supseteq I_{n}=\{0\}
  48. [ L , I k ] I k + 1 [L,I_{k}]\subseteq I_{k+1}
  49. L L
  50. L = I 1 I 2 I 3 I n = { 0 } L=I_{1}\supseteq I_{2}\supseteq I_{3}\cdots\supseteq I_{n}=\{0\}
  51. I k / I k + 1 Z ( L / I k + 1 ) I_{k}/I_{k+1}\subseteq Z(L/I_{k+1})
  52. ad x \textrm{ad}\,x
  53. x L \forall x\in L
  54. ad L \textrm{ad}\,L
  55. L L
  56. L L
  57. x L x\in L
  58. L L
  59. a d x ad_{x}
  60. ad x \textrm{ad}_{x}
  61. 𝔤 𝔩 𝔤 \mathfrak{gl}_{\mathfrak{g}}
  62. N + , ( ad x ) N = 0 \exists N\in\mathbb{Z}^{+},(\textrm{ad}_{x})^{N}=0
  63. ( ad x ) N y = [ x [ x [ x [ x , y ] ] = 0 y L (\textrm{ad}_{x})^{N}y=[x[x\ldots[x[x,y]\ldots]=0\ \forall y\in L
  64. L L
  65. C 0 ( L ) = L , C 1 ( L ) = [ L , L ] , C n + 1 ( L ) = [ L , C n ( L ) ] C^{0}(L)=L,\,C^{1}(L)=[L,L],\,C^{n+1}(L)=[L,C^{n}(L)]
  66. L L
  67. C 0 ( L ) = L , C 1 ( L ) = Z ( L ) C_{0}(L)=L,\,C_{1}(L)=Z(L)
  68. C n + 1 ( L ) = π n - 1 ( Z ( L / C n ( L ) ) ) C_{n+1}(L)=\pi_{n}^{-1}(Z(L/C_{n}(L)))
  69. π i \pi_{i}
  70. L L / C n ( L ) L\to L/C_{n}(L)
  71. L L
  72. L ( N ) = 0 L^{(N)}=0
  73. N N
  74. { 0 } \{0\}
  75. L L
  76. L = I 1 I 2 I 3 I n = { 0 } L=I_{1}\supseteq I_{2}\supseteq I_{3}\cdots\supseteq I_{n}=\{0\}
  77. [ I k , I k ] I k + 1 [I_{k},I_{k}]\subseteq I_{k+1}
  78. L L
  79. L L
  80. I I
  81. L L
  82. L / I , I L/I,I
  83. L L
  84. L ( 0 ) = L , L ( 1 ) = [ L , L ] , L ( n + 1 ) = [ L ( n ) , L ( n ) ] L^{(0)}=L,\,L^{(1)}=[L,L],\,L^{(n+1)}=[L^{(n)},L^{(n)}]
  85. { 0 } \{0\}
  86. { 0 } \{0\}
  87. 𝔤 \mathfrak{g}
  88. 0
  89. V V
  90. 𝔤 \mathfrak{g}
  91. V V
  92. 𝔤 \mathfrak{g}
  93. V V
  94. 𝔤 \mathfrak{g}
  95. 𝔤 \mathfrak{g}
  96. κ ( x , y ) := Tr ( ad x ad y ) x , y 𝔤 \kappa(x,y):=\textrm{Tr}(\textrm{ad}\,x\,\textrm{ad}\,y)\ \forall x,y\in% \mathfrak{g}
  97. 𝔤 \mathfrak{g}
  98. κ ( 𝔤 , [ 𝔤 , 𝔤 ] ) = 0 \kappa(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0
  99. κ ( , ) \kappa(\cdot,\cdot)
  100. 𝔤 \mathfrak{g}
  101. 𝔤 \mathfrak{g}
  102. F F
  103. κ ( , ) \kappa(\cdot,\cdot)
  104. ( , ) (\cdot,\cdot)
  105. < , > <\cdot,\cdot>
  106. < β , α ( β , α ) ( α , α ) α , β E <\beta,\alpha>=\frac{(\beta,\alpha)}{(\alpha,\alpha)}\,\forall\alpha,\beta\in E
  107. 𝔥 \mathfrak{h}
  108. 𝔤 \mathfrak{g}
  109. N 𝔤 ( 𝔥 ) = 𝔥 N_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}
  110. 𝔤 \mathfrak{g}
  111. 𝔥 \mathfrak{h}
  112. 𝔤 \mathfrak{g}
  113. α 𝔥 * \alpha\in\mathfrak{h}^{*}
  114. 𝔤 α := { x 𝔤 | [ h , x ] = α ( h ) x h 𝔥 } \mathfrak{g_{\alpha}}:=\{x\in\mathfrak{g}|[h,x]=\alpha(h)x\,\forall h\in% \mathfrak{h}\}
  115. 𝔤 \mathfrak{g}
  116. 𝔤 α { 0 } \mathfrak{g_{\alpha}}\neq\{0\}
  117. Φ \Phi
  118. Φ \Phi
  119. E E
  120. Φ \Phi
  121. span ( Φ ) = E \textrm{span}(\Phi)=E
  122. 0 Φ 0\notin\Phi
  123. α Φ \alpha\in\Phi
  124. c c\in\mathbb{R}
  125. c α Φ c\alpha\in\Phi
  126. c = ± 1 c=\pm 1
  127. α , β Φ \alpha,\beta\in\Phi
  128. < α , β Align g t ; <\alpha,\beta&gt;
  129. α , β Φ \alpha,\beta\in\Phi
  130. S α ( β ) Φ S_{\alpha}(\beta)\in\Phi
  131. S α S_{\alpha}
  132. α \alpha
  133. S α ( x ) = x - < x , α > α S_{\alpha}(x)=x-<x,\alpha>\alpha
  134. Φ \Phi
  135. ( < α i , α j > ) i , j = 1 n (<\alpha_{i},\alpha_{j}>)_{i,j=1}^{n}
  136. Δ = { α 1 α n } \Delta=\{\alpha_{1}\ldots\alpha_{n}\}
  137. Φ \Phi
  138. Δ \Delta
  139. Φ \Phi
  140. Δ \Delta
  141. E E
  142. Φ \Phi
  143. Δ \Delta
  144. λ , μ E , λ > μ λ - μ > 0 k 1 , k 2 , , k n + , α 1 , α 2 , , α n Δ , λ - μ = i k i α i \forall\lambda,\mu\in E,\,\,\lambda>\mu\iff\lambda-\mu>0\iff\,\,\,\exists k_{1% },k_{2},...,k_{n}\in\mathbb{Z}^{+},\,\alpha_{1},\alpha_{2},...,\alpha_{n}\in% \Delta,\,\,\lambda-\mu=\sum_{i}k_{i}\alpha_{i}
  145. Φ \Phi
  146. γ E \gamma\in E
  147. ( γ , α ) 0 γ Φ (\gamma,\alpha)\neq 0\,\forall\gamma\in\Phi
  148. Δ \Delta
  149. Φ \Phi
  150. γ E \gamma\in E
  151. ( γ , α ) > 0 γ Δ (\gamma,\alpha)>0\,\forall\gamma\in\Delta
  152. γ \gamma
  153. Δ ( γ ) \Delta(\gamma)
  154. Δ = Δ ( γ ) \Delta=\Delta(\gamma)
  155. Φ + ( γ ) = { α Φ | ( α , γ ) > 0 } \Phi^{+}(\gamma)=\{\alpha\in\Phi|(\alpha,\gamma)>0\}
  156. α \alpha
  157. Φ + ( γ ) \Phi^{+}(\gamma)
  158. α = α + α ′′ \alpha=\alpha^{\prime}+\alpha^{\prime\prime}
  159. α , α ′′ Φ + ( γ ) \alpha^{\prime},\alpha^{\prime\prime}\in\Phi^{+}(\gamma)
  160. Δ ( γ ) \Delta(\gamma)
  161. Φ + ( γ ) \Phi^{+}(\gamma)
  162. Φ \Phi
  163. Δ \Delta
  164. Φ \Phi
  165. Δ \Delta
  166. Φ \Phi
  167. Δ \Delta
  168. Φ \Phi
  169. Δ \Delta
  170. Φ \Phi
  171. E E
  172. Φ \Phi
  173. Φ \Phi
  174. α v = 2 α ( α , α ) \alpha^{v}=\frac{2\alpha}{(\alpha,\alpha)}
  175. Φ v = { α v | α Φ } \Phi^{v}=\{\alpha^{v}|\alpha\in\Phi\}
  176. Φ v \Phi^{v}
  177. Φ \Phi
  178. < λ , α > α Φ <\lambda,\alpha>\in\mathbb{Z}\,\forall\alpha\in\Phi
  179. < λ , α > + <\lambda,\alpha>\in\mathbb{Z}^{+}
  180. α Φ \alpha\in\Phi
  181. Δ = { α 1 , α 2 , , α n } \Delta=\{\alpha_{1},\alpha_{2},...,\alpha_{n}\}
  182. E E
  183. α 1 v , α 2 v , , α n v Φ v \alpha_{1}^{v},\alpha_{2}^{v},...,\alpha_{n}^{v}\in\Phi^{v}
  184. E E
  185. λ 1 , λ 2 , , λ n \lambda_{1},\lambda_{2},...,\lambda_{n}
  186. ( λ i , α j v ) = δ i j (\lambda_{i},\alpha_{j}^{v})=\delta_{ij}
  187. 𝔤 \mathfrak{g}
  188. V V
  189. 𝔤 × V V , ( x , v ) x v \mathfrak{g}\times V\to V,\,(x,v)\mapsto xv
  190. a , b F , x , y 𝔤 , v , w V \forall a,b\in F,x,y\in\mathfrak{g},v,w\in V
  191. ( a x + b y ) v = a ( x v ) + b ( y v ) (ax+by)v=a(xv)+b(yv)
  192. x ( a v + b w ) = a ( x v ) + b ( x w ) x(av+bw)=a(xv)+b(xw)
  193. [ x , y ] v = x ( y v ) - y ( x v ) [x,y]v=x(yv)-y(xv)
  194. V V
  195. 𝔤 \mathfrak{g}
  196. V , 𝔤 V,\mathfrak{g}
  197. F F
  198. 𝔤 \mathfrak{g}
  199. 𝔤 𝔤 𝔩 V \mathfrak{g}\to\mathfrak{gl}_{V}
  200. 𝔤 \mathfrak{g}
  201. V V
  202. 𝔤 \mathfrak{g}
  203. W V W\subset V
  204. V V
  205. π : 𝔤 𝔤 𝔩 V \pi:\mathfrak{g}\to\mathfrak{gl}_{V}
  206. π \pi
  207. 𝔤 \mathfrak{g}
  208. 𝔤 𝔤 𝔩 V \mathfrak{g}\to\mathfrak{gl}_{V}
  209. 𝔤 \mathfrak{g}
  210. V V
  211. 𝔤 \mathfrak{g}
  212. 𝔤 \mathfrak{g}
  213. ϕ \phi
  214. ϕ ( x v ) = x ϕ ( v ) x 𝔤 , v V \phi(xv)=x\phi(v)\forall x\in\mathfrak{g},v\in V
  215. 𝔤 \mathfrak{g}
  216. 𝔤 \mathfrak{g}
  217. V V
  218. x v = 0 x 𝔤 , v V xv=0\forall x\in\mathfrak{g},v\in V
  219. 𝔤 𝔤 𝔩 V \mathfrak{g}\to\mathfrak{gl}_{V}
  220. 𝔤 \mathfrak{g}
  221. 𝔤 𝔩 ( n , F ) \mathfrak{gl}(n,F)
  222. 𝔰 𝔩 ( n , F ) , 𝔬 ( 2 l , F ) , 𝔱 ( n , F ) \mathfrak{sl}(n,F),\mathfrak{o}(2l,F),\mathfrak{t}(n,F)
  223. 𝔤 𝔤 𝔩 ( n , F ) \mathfrak{g}\to\mathfrak{gl}(n,F)
  224. F n F^{n}
  225. ad : 𝔤 gl 𝔤 x ad x \begin{aligned}\displaystyle\textrm{ad}:&\displaystyle\mathfrak{g}\to\textrm{% gl}_{\mathfrak{g}}\\ &\displaystyle x\mapsto\textrm{ad}_{x}\end{aligned}
  226. 𝔤 \mathfrak{g}
  227. 𝔤 \mathfrak{g}
  228. 𝔤 \mathfrak{g}
  229. 𝔤 \mathfrak{g}
  230. x ( v + W ) = x v + W x 𝔤 , v V x(v+W)=xv+W\,\forall x\in\mathfrak{g},v\in V
  231. 𝔤 \mathfrak{g}
  232. ϕ V V \phi V\to V
  233. 𝔤 \mathfrak{g}
  234. ϕ = λ 1 V \phi=\lambda 1_{V}
  235. λ F \lambda\in F
  236. ϕ : L 𝔤 𝔩 ( V ) \phi\colon L\to\mathfrak{gl}(V)
  237. θ End ( V ) \theta\in\textrm{End}(V)
  238. θ ϕ ( x ) = ϕ ( x ) θ \theta\phi(x)=\phi(x)\theta
  239. θ = λ 1 V \theta=\lambda 1_{V}
  240. λ F \lambda\in F
  241. [ H α i , H α j ] = 0 [H_{\alpha_{i}},H_{\alpha_{j}}]=0
  242. [ H α i , E α j ] = A i j E α j [H_{\alpha_{i}},E_{\alpha_{j}}]=A_{ij}E_{\alpha_{j}}
  243. [ E α i , E α j ] = H α j [E_{\alpha_{i}},E_{\alpha_{j}}]=H_{\alpha_{j}}
  244. [ E β , E γ ] = ± ( p + 1 ) E β + γ [E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}
  245. g l ( n , F ) gl(n,F)
  246. 𝔤 𝔩 V \mathfrak{gl}_{V}
  247. 𝔤 𝔩 ( n , F ) \mathfrak{gl}(n,F)
  248. A l ( l 1 ) A_{l}\ (l\geq 1)
  249. l 2 + 2 l l^{2}+2l
  250. 𝔰 𝔩 ( l + 1 , F ) = { x 𝔤 𝔩 ( l + 1 , F ) | T r ( x ) = 0 } \mathfrak{sl}(l+1,F)=\{x\in\mathfrak{gl}(l+1,F)|Tr(x)=0\}
  251. B l ( l 1 ) B_{l}\ (l\geq 1)
  252. 2 l 2 + l 2l^{2}+l
  253. 𝔬 ( 2 l + 1 , F ) = { x 𝔤 𝔩 ( 2 l + 1 , F ) | s x = - x t s , s = ( 1 0 0 0 0 I l 0 I l 0 ) } \mathfrak{o}(2l+1,F)=\{x\in\mathfrak{gl}(2l+1,F)|sx=-x^{t}s,s=\begin{pmatrix}1% &0&0\\ 0&0&I_{l}\\ 0&I_{l}&0\end{pmatrix}\}
  254. C l ( l 2 ) C_{l}\ (l\geq 2)
  255. 2 l 2 - l 2l^{2}-l
  256. 𝔰 𝔭 ( 2 l , F ) = { x 𝔤 𝔩 ( 2 l , F ) | s x = - x t s , s = ( 0 I l - I l 0 ) } \mathfrak{sp}(2l,F)=\{x\in\mathfrak{gl}(2l,F)|sx=-x^{t}s,s=\begin{pmatrix}0&I_% {l}\\ -I_{l}&0\end{pmatrix}\}
  257. D l ( l 1 ) D_{l}(l\geq 1)
  258. 2 l 2 + l 2l^{2}+l
  259. 𝔬 ( 2 l , F ) = { x 𝔤 𝔩 ( 2 l , F ) | s x = - x t s , s = ( 0 I l I l 0 ) } \mathfrak{o}(2l,F)=\{x\in\mathfrak{gl}(2l,F)|sx=-x^{t}s,s=\begin{pmatrix}0&I_{% l}\\ I_{l}&0\end{pmatrix}\}

Glossary_of_module_theory.html

  1. M M
  2. R R
  3. ( M , + ) (M,+)
  4. R × M M R\times M\to M
  5. r , s R , m , n M \forall r,s\in R,m,n\in M
  6. r ( m + n ) = r m + r n r(m+n)=rm+rn
  7. r ( s m ) = ( r s ) m r(sm)=(rs)m
  8. 1 R m = m 1_{R}\,m=m
  9. M M
  10. R R
  11. ( M , + ) (M,+)
  12. M × R M M\times R\to M
  13. r , s R , m , n M \forall r,s\in R,m,n\in M
  14. ( m + n ) r = m r + n r (m+n)r=mr+nr
  15. ( m s ) r = r ( s m ) (ms)r=r(sm)
  16. m 1 R = m m1_{R}=m
  17. M M
  18. R op R^{\textrm{op}}
  19. R R
  20. M M
  21. S S
  22. R R
  23. ( S , R ) (S,R)
  24. s ( m r ) = ( s m ) r s S , r R , m M s(mr)=(sm)r\,\forall s\in S,r\in R,m\in M
  25. M M
  26. R R
  27. N N
  28. M M
  29. R N N RN\subseteq N
  30. R R
  31. R R
  32. M 1 , M 2 M_{1},M_{2}
  33. ϕ : M 1 M 2 \phi:M_{1}\to M_{2}
  34. R R
  35. r ϕ ( m ) = ϕ ( r m ) r R , m M 1 r\phi(m)=\phi(rm)\,\forall r\in R,m\in M_{1}
  36. R R
  37. M M
  38. N N
  39. M / N M/N
  40. R R
  41. r ( m + N ) = r m + N r R , m M r(m+N)=rm+N\,\forall r\in R,m\in M
  42. R R
  43. M M
  44. Ann ( M ) := { r R | r m = 0 m M } \textrm{Ann}(M):=\{r\in R|rm=0\,\forall m\in M\}
  45. R R
  46. m M m\in M
  47. Ann ( m ) := { r R | r m = 0 } \textrm{Ann}(m):=\{r\in R|rm=0\}
  48. M M
  49. x 1 , , x n x_{1},...,x_{n}
  50. M M
  51. M M
  52. R R
  53. R R
  54. M M
  55. M M
  56. R R
  57. P P
  58. R R
  59. g : P M g:P\to M
  60. R R
  61. f : N M f:N\to M
  62. R R
  63. h : P N h:P\to N
  64. f h = g f\circ h=g
  65. Hom R ( P , - ) \textrm{Hom}_{R}(P,-)
  66. M M
  67. 0 L L P 0 0\to L\to L^{\prime}\to P\to 0
  68. M M
  69. R R
  70. Q Q
  71. R R
  72. g : X Q g:X\to Q
  73. R R
  74. f : X Y f:X\to Y
  75. R R
  76. h : Y Q h:Y\to Q
  77. f h = g f\circ h=g
  78. Hom R ( - , I ) \textrm{Hom}_{R}(-,I)
  79. I I
  80. 0 I L L 0 0\to I\to L\to L^{\prime}\to 0
  81. R R
  82. F F
  83. - R F -\otimes_{R}F
  84. M M
  85. r R r\in R
  86. M M
  87. r x 0 rx\neq 0
  88. Ann ( M ) \textrm{Ann}(M)
  89. M M
  90. A = n A n A=\bigoplus_{n\in\mathbb{N}}A_{n}
  91. M M
  92. i M i \bigoplus_{i\in\mathbb{N}}M_{i}
  93. A i M j M i + j A_{i}M_{j}\subseteq M_{i+j}

GNSS_positioning_calculation.html

  1. t ~ i \scriptstyle\tilde{t}_{i}
  2. i = 1 , 2 , 3 , 4 , . . , n \scriptstyle i\;=\;1,\,2,\,3,\,4,\,..,\,n
  3. s y m b o l r i ( t ) \scriptstyle symbol{r}_{i}(t)
  4. δ t clock,sv , i ( t ) \scriptstyle\delta t_{\,\text{clock,sv},i}(t)
  5. t i \scriptstyle t_{i}
  6. t ~ i = t i + δ t clock , i ( t i ) \scriptstyle\tilde{t}_{i}\;=\;t_{i}\,+\,\delta t_{\,\text{clock},i}(t_{i})
  7. δ t clock , i ( t i ) = δ t clock,sv , i ( t i ) + δ t orbit-relativ , i ( s y m b o l r i , s y m b o l r ˙ i ) \scriptstyle\delta t_{\,\text{clock},i}(t_{i})\;=\;\delta t_{\,\text{clock,sv}% ,i}(t_{i})\,+\,\delta t_{\,\text{orbit-relativ},\,i}(symbol{r}_{i},\,\dot{% symbol{r}}_{i})
  8. δ t orbit-relativ , i ( s y m b o l r i , s y m b o l r ˙ i ) \scriptstyle\delta t_{\,\text{orbit-relativ},i}(symbol{r}_{i},\,\dot{symbol{r}% }_{i})
  9. t i \scriptstyle t_{i}
  10. s y m b o l r i = s y m b o l r i ( t i ) \scriptstyle symbol{r}_{i}\;=\;symbol{r}_{i}(t_{i})
  11. s y m b o l r ˙ i = s y m b o l r ˙ i ( t i ) \scriptstyle\dot{symbol{r}}_{i}\;=\;\dot{symbol{r}}_{i}(t_{i})
  12. r ( s y m b o l r A , s y m b o l r B ) \scriptstyle r(symbol{r}_{A},\,symbol{r}_{B})
  13. \scriptstylesymbol r A \scriptstylesymbol{r}_{A}
  14. \scriptstylesymbol r B \scriptstylesymbol{r}_{B}
  15. s y m b o l r rec \scriptstyle symbol{r}_{\,\text{rec}}
  16. t rec \scriptstyle t_{\,\text{rec}}
  17. r ( s y m b o l r i , s y m b o l r rec ) / c + ( t i - t rec ) = 0 \scriptstyle r(symbol{r}_{i},\,symbol{r}_{\,\text{rec}})/c\,+\,(t_{i}-t_{\,% \text{rec}})\;=\;0
  18. c \scriptstyle c
  19. - ( t i - t rec ) \scriptstyle-(t_{i}\,-\,t_{\,\text{rec}})
  20. r ( s y m b o l r i , s y m b o l r rec ) / c + ( t i - t rec ) + δ t atmos , i - δ t meas-err , i = 0 \scriptstyle r(symbol{r}_{i},\,symbol{r}_{\,\text{rec}})/c\,+\,(t_{i}\,-\,t_{% \,\text{rec}})\,+\,\delta t_{\,\text{atmos},i}\,-\,\delta t_{\,\text{meas-err}% ,i}\;=\;0
  21. δ t atmos , i \scriptstyle\delta t_{\,\text{atmos},i}
  22. δ t meas-err , i \scriptstyle\delta t_{\,\text{meas-err},i}
  23. ( s y m b o l r ^ rec , t ^ rec ) = arg min ϕ ( s y m b o l r rec , t rec ) \scriptstyle(\hat{symbol{r}}_{\,\text{rec}},\,\hat{t}_{\,\text{rec}})\;=\;\arg% \min\phi(symbol{r}_{\,\text{rec}},\,t_{\,\text{rec}})
  24. ϕ ( s y m b o l r rec , t rec ) = i = 1 n ( δ t meas-err , i / σ δ t meas-err , i ) 2 \scriptstyle\phi(symbol{r}_{\,\text{rec}},\,t_{\,\text{rec}})\;=\;\sum_{i=1}^{% n}(\delta t_{\,\text{meas-err},i}/\sigma_{\delta t_{\,\text{meas-err},i}})^{2}
  25. δ t meas-err , i \scriptstyle\delta t_{\,\text{meas-err},i}
  26. s y m b o l r rec \scriptstyle symbol{r}_{\,\text{rec}}
  27. t rec \scriptstyle t_{\,\text{rec}}
  28. s y m b o l r rec \scriptstyle symbol{r}_{\,\text{rec}}
  29. t rec \scriptstyle t_{\,\text{rec}}
  30. exp ( - 1 2 ϕ ( s y m b o l r rec , t rec ) ) \scriptstyle\exp(-\frac{1}{2}\phi(symbol{r}_{\,\text{rec}},\,t_{\,\text{rec}}))
  31. ( s y m b o l r ^ rec , t ^ rec ) \scriptstyle(\hat{symbol{r}}_{\,\text{rec}},\,\hat{t}_{\,\text{rec}})
  32. s y m b o l r rec \scriptstyle symbol{r}_{\,\text{rec}}
  33. - exp ( - 1 2 ϕ ( s y m b o l r rec , t rec ) ) d t rec \scriptstyle\int_{-\infty}^{\infty}\exp(-\frac{1}{2}\phi(symbol{r}_{\,\text{% rec}},\,t_{\,\text{rec}}))\,dt_{\,\text{rec}}
  34. ( s y m b o l r ^ rec , t ^ rec ) \scriptstyle(\hat{symbol{r}}_{\,\text{rec}},\,\hat{t}_{\,\text{rec}})
  35. { Δ t i ( t i , E i ) t i + δ t clock , i ( t i , E i ) - t ~ i = 0 , Δ M i ( t i , E i ) M i ( t i ) - ( E i - e i sin E i ) = 0 , \scriptstyle\begin{cases}\scriptstyle\Delta t_{i}(t_{i},\,E_{i})\;\triangleq\;% t_{i}\,+\,\delta t_{\,\text{clock},i}(t_{i},\,E_{i})\,-\,\tilde{t}_{i}\;=\;0,% \\ \scriptstyle\Delta M_{i}(t_{i},\,E_{i})\;\triangleq\;M_{i}(t_{i})\,-\,(E_{i}\,% -\,e_{i}\sin E_{i})\;=\;0,\end{cases}
  36. E i \scriptstyle E_{i}
  37. i i
  38. M i \scriptstyle M_{i}
  39. e i \scriptstyle e_{i}
  40. δ t clock , i ( t i , E i ) = δ t clock,sv , i ( t i ) + δ t orbit-relativ , i ( E i ) \scriptstyle\delta t_{\,\text{clock},i}(t_{i},\,E_{i})\;=\;\delta t_{\,\text{% clock,sv},i}(t_{i})\,+\,\delta t_{\,\text{orbit-relativ},i}(E_{i})
  41. t i \scriptstyle t_{i}
  42. E i \scriptstyle E_{i}
  43. ( t i E i ) ( t i E i ) - ( 1 0 M ˙ i ( t i ) 1 - e i cos E i - 1 1 - e i cos E i ) ( Δ t i Δ M i ) \scriptscriptstyle\begin{pmatrix}t_{i}\\ E_{i}\\ \end{pmatrix}\leftarrow\begin{pmatrix}t_{i}\\ E_{i}\\ \end{pmatrix}-\begin{pmatrix}1&&0\\ \frac{\dot{M}_{i}(t_{i})}{1-e_{i}\cos E_{i}}&&-\frac{1}{1-e_{i}\cos E_{i}}\\ \end{pmatrix}\begin{pmatrix}\Delta t_{i}\\ \Delta M_{i}\\ \end{pmatrix}
  44. δ t clock,sv , i ( t ) \scriptstyle\delta t_{\,\text{clock,sv},i}(t)
  45. δ t clock , i ( t ) \scriptstyle\delta t_{\,\text{clock},i}(t)
  46. r ~ i = - c ( t ~ i - t ~ rec ) \scriptstyle\tilde{r}_{i}\;=\;-c(\tilde{t}_{i}\,-\,\tilde{t}_{\,\text{rec}})
  47. t ~ rec \scriptstyle\tilde{t}_{\,\text{rec}}
  48. δ t clock,rec = t ~ rec - t rec \scriptstyle\delta t_{\,\text{clock,rec}}\;=\;\tilde{t}_{\,\text{rec}}\,-\,t_{% \,\text{rec}}
  49. r ~ i \scriptstyle\tilde{r}_{i}
  50. t ~ rec \scriptstyle\tilde{t}_{\,\text{rec}}
  51. - ( t i - t rec ) = r ~ i / c + δ t clock , i - δ t clock,rec \scriptstyle-(t_{i}-t_{\,\text{rec}})\;=\;\tilde{r}_{i}/c\,+\,\delta t_{\,% \text{clock},i}\,-\,\delta t_{\,\text{clock,rec}}
  52. r ( s y m b o l r i , s y m b o l r rec ) = | Ω E ( t i - t rec ) s y m b o l r i , ECEF - s y m b o l r rec,ECEF | \scriptstyle r(symbol{r}_{i},\,symbol{r}_{\,\text{rec}})\;=\;|\Omega_{\,\text{% E}}(t_{i}\,-\,t_{\,\text{rec}})symbol{r}_{i,\,\text{ECEF}}\,-\,symbol{r}_{\,% \text{rec,ECEF}}|
  53. Ω E \scriptstyle\Omega_{\,\text{E}}
  54. Ω E ( t i - t rec ) = Ω E ( δ t clock,rec ) Ω E ( - r ~ i / c - δ t clock , i ) \scriptstyle\Omega_{\,\text{E}}(t_{i}\,-\,t_{\,\text{rec}})\;=\;\Omega_{\,% \text{E}}(\delta t_{\,\text{clock,rec}})\Omega_{\,\text{E}}(-\tilde{r}_{i}/c\,% -\,\delta t_{\,\text{clock},i})
  55. s y m b o l r rec,ECEF \scriptstyle symbol{r}_{\,\text{rec,ECEF}}
  56. s y m b o l e i , rec,ECEF = - r ( s y m b o l r i , s y m b o l r rec ) s y m b o l r rec,ECEF \scriptstyle symbol{e}_{i,\,\text{rec,ECEF}}\;=\;-\frac{\partial r(symbol{r}_{% i},\,symbol{r}_{\,\text{rec}})}{\partial symbol{r}_{\,\text{rec,ECEF}}}
  57. s y m b o l r rec,ECEF \scriptstyle symbol{r}_{\,\text{rec,ECEF}}
  58. δ t clock,rec \scriptstyle\delta t_{\,\text{clock,rec}}
  59. r ( s y m b o l r A , s y m b o l r B ) = | s y m b o l r A - s y m b o l r B | = ( x A - x B ) 2 + ( y A - y B ) 2 + ( z A - z B ) 2 \scriptstyle r(symbol{r}_{A},\,symbol{r}_{B})=|symbol{r}_{A}-symbol{r}_{B}|=% \sqrt{(x_{A}-x_{B})^{2}+(y_{A}-y_{B})^{2}+(z_{A}-z_{B})^{2}}
  60. \scriptstylesymbol r A = ( x A , y A , z A ) \scriptstylesymbol{r}_{A}=(x_{A},y_{A},z_{A})
  61. \scriptstylesymbol r B = ( x B , y B , z B ) \scriptstylesymbol{r}_{B}=(x_{B},y_{B},z_{B})

Goldner–Harary_graph.html

  1. - ( x - 1 ) 2 x 2 ( x + 2 ) 3 ( x 2 - 3 ) ( x 2 - 4 x - 9 ) -(x-1)^{2}x^{2}(x+2)^{3}(x^{2}-3)(x^{2}-4x-9)

Good_quantum_number.html

  1. H H
  2. O O
  3. O | q j = q j | q j O|q_{j}\rangle=q_{j}|q_{j}\rangle
  4. q j q_{j}
  5. | q j |q_{j}\rangle
  6. O O
  7. O | q j = O k c k ( 0 ) | e k = q j | q j O|q_{j}\rangle=O\sum_{k}c_{k}(0)|e_{k}\rangle=q_{j}|q_{j}\rangle
  8. O k c k ( 0 ) exp ( - i e k t / ) | e k = q j k c k ( 0 ) exp ( - i e k t / ) | e k O\sum_{k}c_{k}(0)\exp(-ie_{k}t/\hbar)\,|e_{k}\rangle=q_{j}\sum_{k}c_{k}(0)\exp% (-ie_{k}t/\hbar)\,|e_{k}\rangle
  9. | q j |q_{j}\rangle
  10. q q
  11. e k e_{k}
  12. O O
  13. H H
  14. [ O , H ] = 0 [O,\,H]=0
  15. | ψ 0 |\psi_{0}\rangle
  16. O O
  17. O | ψ 0 = q j | ψ 0 O|\psi_{0}\rangle=q_{j}|\psi_{0}\rangle
  18. O | ψ t = O T ( t ) | ψ 0 O|\psi_{t}\rangle=O\,T(t)\,|\psi_{0}\rangle
  19. = O e - i t H / | ψ 0 =Oe^{-itH/\hbar}|\psi_{0}\rangle
  20. = O n = 0 1 n ! ( - i H t / ) n | ψ 0 =O\sum_{n=0}^{\infty}\frac{1}{n!}(-iHt/\hbar)^{n}|\psi_{0}\rangle
  21. = n = 0 1 n ! ( - i H t / ) n O | ψ 0 =\sum_{n=0}^{\infty}\frac{1}{n!}(-iHt/\hbar)^{n}O|\psi_{0}\rangle
  22. = q j | ψ t =q_{j}|\psi_{t}\rangle
  23. d d t A ( t ) = A ( t ) t + 1 i [ A ( t ) , H ] \frac{d}{dt}\langle A(t)\rangle=\left\langle\frac{\partial A(t)}{\partial t}% \right\rangle+\frac{1}{i\hbar}\langle[A(t),H]\rangle
  24. A A
  25. H H
  26. A A
  27. l l
  28. s s
  29. L L
  30. S S
  31. H H
  32. J = L + S J=L+S
  33. J J
  34. H H
  35. σ 2 = ( A - A ) 2 \sigma^{2}=\langle(A-\langle A\rangle)^{2}\rangle
  36. = ψ | ( A - a ) 2 ψ =\langle\psi|(A-a)^{2}\psi\rangle
  37. A = a \langle A\rangle=a
  38. = ( A - a ) ψ | ( A - a ) ψ = 0 =\langle(A-a)\psi|(A-a)\psi\rangle=0
  39. d d t A ( t ) = A ( t ) t + 1 i [ A ( t ) , H ) ] = 0 \frac{d}{dt}\langle A(t)\rangle=\left\langle\frac{\partial A(t)}{\partial t}% \right\rangle+\frac{1}{i\hbar}\langle[A(t),H)]\rangle=0
  40. A A
  41. d A d t = A t + { A , H } \frac{dA}{dt}=\frac{\partial A}{\partial t}+\{A,H\}
  42. A A
  43. H ^ | Ψ = E Ψ | Ψ \hat{H}|\Psi\rangle=E_{\Psi}|\Psi\rangle
  44. | Ψ |\Psi\rangle
  45. H ^ \hat{H}
  46. E Ψ E_{\Psi}
  47. | Ψ |\Psi\rangle
  48. i t | Ψ = E Ψ | Ψ i\hbar\frac{\partial}{\partial t}|\Psi\rangle=E_{\Psi}|\Psi\rangle
  49. | Ψ ( t ) = e - i E Ψ t / | Ψ ( 0 ) |\Psi(t)\rangle=e^{-iE_{\Psi}t/\hbar}|\Psi(0)\rangle
  50. z z
  51. l , j , m l , m s , m j l,j,m\text{l},m_{s},m_{j}
  52. s s
  53. l , j , m l , m s , m j l,j,m\text{l},m_{s},m_{j}
  54. L 2 , J 2 , L z , J z L^{2},J^{2},L_{z},J_{z}
  55. n , l , m l , m s n,l,m\text{l},m_{s}
  56. n , l , j , m j n,l,j,m\text{j}
  57. Δ H SO = - s y m b o l μ \cdotsymbol B . \Delta H\text{SO}=-symbol{\mu}\cdotsymbol{B}.
  58. Δ H SO \Delta H\text{SO}
  59. s y m b o l L symbol{L}
  60. s y m b o l S symbol{S}
  61. s y m b o l J symbol{J}
  62. l , j , m l , m s l,j,m\text{l},m_{s}
  63. l , j , m j l,j,m\text{j}
  64. Δ H SO = β 2 ( j ( j + 1 ) - l ( l + 1 ) - s ( s + 1 ) ) \Delta H\text{SO}={\beta\over 2}(j(j+1)-l(l+1)-s(s+1))
  65. β = β ( n , l ) = Z 4 μ 0 4 π 4 g s μ B 2 1 n 3 a 0 3 l ( l + 1 / 2 ) ( l + 1 ) \beta=\beta(n,l)=Z^{4}{\mu_{0}\over 4{\pi}^{4}}g\text{s}\mu\text{B}^{2}{1\over n% ^{3}a_{0}^{3}l(l+1/2)(l+1)}
  66. l , s , j l,s,j

Gordan's_lemma.html

  1. σ \sigma
  2. u 1 , , u r u_{1},\dots,u_{r}
  3. σ = { x u i , x 0 , 1 i r } . \sigma=\{x\mid\langle u_{i},x\rangle\geq 0,1\leq i\leq r\}.
  4. u i u_{i}
  5. σ \sigma^{\vee}
  6. u i u_{i}
  7. σ C \sigma\subset C^{\vee}
  8. S σ = σ d , S_{\sigma}=\sigma^{\vee}\cap\mathbb{Z}^{d},
  9. x = i n i u i + i r i u i x=\sum_{i}n_{i}u_{i}+\sum_{i}r_{i}u_{i}
  10. n i n_{i}
  11. 0 r i 1 0\leq r_{i}\leq 1
  12. S σ S_{\sigma}
  13. [ S ] \mathbb{C}[S]
  14. \mathbb{C}
  15. d \mathbb{Z}^{d}
  16. S + = S { x x , v 0 } S^{+}=S\cap\{x\mid\langle x,v\rangle\geq 0\}
  17. A = [ S ] A=\mathbb{C}[S]
  18. χ a , a S \chi^{a},\,a\in S
  19. \mathbb{Z}
  20. A n = span { χ a a S , a , v = n } A_{n}=\operatorname{span}\{\chi^{a}\mid a\in S,\langle a,v\rangle=n\}
  21. [ S + ] = 0 A n \mathbb{C}[S^{+}]=\oplus_{0}^{\infty}A_{n}
  22. A 0 A_{0}
  23. S 0 = S { x x , v = 0 } S_{0}=S\cap\{x\mid\langle x,v\rangle=0\}
  24. A 0 = [ S 0 ] A_{0}=\mathbb{C}[S_{0}]
  25. S + S^{+}
  26. \mathbb{Z}
  27. A + = 0 A n A^{+}=\oplus_{0}^{\infty}A_{n}
  28. A 0 A_{0}
  29. f i s f_{i}^{\prime}s
  30. f = i g i f i f=\sum_{i}g_{i}f_{i}
  31. g i g_{i}
  32. g i g_{i}
  33. A n A_{n}
  34. A 0 A_{0}
  35. N i N_{i}
  36. A n A_{n}
  37. A n A_{n}
  38. N i A N_{i}A
  39. N i = N i A A n . N_{i}=N_{i}A\cap A_{n}.
  40. A + A^{+}
  41. A 0 A_{0}

Gosset_graph.html

  1. ( x - 27 ) ( x - 9 ) 7 ( x + 1 ) 27 ( x + 3 ) 21 . (x-27)(x-9)^{7}(x+1)^{27}(x+3)^{21}.\,

Graded_manifold.html

  1. ( n , m ) (n,m)
  2. ( Z , A ) (Z,A)
  3. Z Z
  4. n n
  5. A A
  6. C Z C^{\infty}_{Z}
  7. m m
  8. C Z C^{\infty}_{Z}
  9. Z Z
  10. A A
  11. ( Z , A ) (Z,A)
  12. Z Z
  13. ( Z , A ) (Z,A)
  14. A A
  15. ( Z , A ) (Z,A)
  16. C ( Z ) C^{\infty}(Z)
  17. A ( Z ) A(Z)
  18. ( Z , A ) (Z,A)
  19. ( Z , A ) (Z,A)
  20. E Z E\to Z
  21. m m
  22. V V
  23. A A
  24. ( Z , A ) (Z,A)
  25. Λ ( E ) \Lambda(E)
  26. E E
  27. Λ ( V ) \Lambda(V)
  28. Z Z
  29. C ( Z ) C^{\infty}(Z)
  30. Z Z
  31. C ( Z ) C^{\infty}(Z)
  32. ( U ; z A , y a ) (U;z^{A},y^{a})
  33. E Z E\to Z
  34. ( U ; z A , c a ) (U;z^{A},c^{a})
  35. ( Z , A ) (Z,A)
  36. { c a } \{c^{a}\}
  37. E E
  38. Λ ( V ) \Lambda(V)
  39. f = k = 0 m 1 k ! f a 1 a k ( z ) c a 1 c a k f=\sum_{k=0}^{m}\frac{1}{k!}f_{a_{1}\ldots a_{k}}(z)c^{a_{1}}\cdots c^{a_{k}}
  40. f a 1 a k ( z ) f_{a_{1}\cdots a_{k}}(z)
  41. U U
  42. c a c^{a}
  43. Λ ( V ) \Lambda(V)
  44. ( Z , A ) (Z,A)
  45. A ( Z ) A(Z)
  46. ( Z , A ) (Z,A)
  47. A ( Z ) \partial A(Z)
  48. [ u , u ] = u u - ( - 1 ) [ u ] [ u ] u u [u,u^{\prime}]=u\cdot u^{\prime}-(-1)^{[u][u^{\prime}]}u^{\prime}\cdot u
  49. [ u ] [u]
  50. u A ( Z ) u\in\partial A(Z)
  51. u = u A A + u a c a u=u^{A}\partial_{A}+u^{a}\frac{\partial}{\partial c^{a}}
  52. f f
  53. u ( f a 1 a k c a 1 c a k ) = u A A ( f a 1 a k ) c a 1 c a k + i u a i ( - 1 ) i - 1 f a 1 a k c a 1 c a i - 1 c a i + 1 c a k u(f_{a_{1}\ldots a_{k}}c^{a_{1}}\cdots c^{a_{k}})=u^{A}\partial_{A}(f_{a_{1}% \ldots a_{k}})c^{a_{1}}\cdots c^{a_{k}}+\sum_{i}u^{a_{i}}(-1)^{i-1}f_{a_{1}% \ldots a_{k}}c^{a_{1}}\cdots c^{a_{i-1}}c^{a_{i+1}}\cdots c^{a_{k}}
  54. A ( Z ) A(Z)
  55. A ( Z ) \partial A(Z)
  56. O 1 ( Z ) O^{1}(Z)
  57. ϕ = ϕ A d z A + ϕ a d c a \phi=\phi_{A}dz^{A}+\phi_{a}dc^{a}
  58. A ( Z ) \partial A(Z)
  59. O 1 ( Z ) O^{1}(Z)
  60. u ϕ = u A ϕ A + ( - 1 ) [ ϕ a ] u a ϕ a u\rfloor\phi=u^{A}\phi_{A}+(-1)^{[\phi_{a}]}u^{a}\phi_{a}
  61. d z A d c i = - d c i d z A , d c i d c j = d c j d c i dz^{A}\wedge dc^{i}=-dc^{i}\wedge dz^{A},\qquad dc^{i}\wedge dc^{j}=dc^{j}% \wedge dc^{i}
  62. O * ( Z ) O^{*}(Z)
  63. ϕ ϕ = ( - 1 ) | ϕ | | ϕ | + [ ϕ ] [ ϕ ] ϕ ϕ \phi\wedge\phi^{\prime}=(-1)^{|\phi||\phi^{\prime}|+[\phi][\phi^{\prime}]}\phi% ^{\prime}\wedge\phi
  64. | ϕ | |\phi|
  65. ϕ \phi
  66. O * ( Z ) O^{*}(Z)
  67. d ϕ = d z A A ϕ + d c a c a ϕ d\phi=dz^{A}\wedge\partial_{A}\phi+dc^{a}\wedge\frac{\partial}{\partial c^{a}}\phi
  68. A \partial_{A}
  69. / c a \partial/\partial c^{a}
  70. d z A dz^{A}
  71. d c a dc^{a}
  72. d ( ϕ ϕ ) = d ( ϕ ) ϕ + ( - 1 ) | ϕ | ϕ d ϕ d(\phi\wedge\phi^{\prime})=d(\phi)\wedge\phi^{\prime}+(-1)^{|\phi|}\phi\wedge d% \phi^{\prime}

Gradient_boosting.html

  1. F F
  2. y ^ = F ( x ) \hat{y}=F(x)
  3. ( y ^ - y ) 2 (\hat{y}-y)^{2}
  4. y y
  5. 1 m M 1\leq m\leq M
  6. F m F_{m}
  7. y y
  8. F m F_{m}
  9. h h
  10. F m + 1 ( x ) = F m ( x ) + h ( x ) F_{m+1}(x)=F_{m}(x)+h(x)
  11. h h
  12. h h
  13. F m + 1 = F m ( x ) + h ( x ) = y F_{m+1}=F_{m}(x)+h(x)=y
  14. h ( x ) = y - F m ( x ) h(x)=y-F_{m}(x)
  15. h h
  16. y - F m ( x ) y-F_{m}(x)
  17. F m + 1 F_{m+1}
  18. F m F_{m}
  19. y - F ( x ) y-F(x)
  20. 1 2 ( y - F ( x ) ) 2 \frac{1}{2}(y-F(x))^{2}
  21. y y
  22. x x
  23. P ( x , y ) P(x,y)
  24. { ( x 1 , y 1 ) , , ( x n , y n ) } \{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  25. x x
  26. y y
  27. F ^ ( x ) \hat{F}(x)
  28. F F
  29. L ( y , F ( x ) ) L(y,F(x))
  30. F * = arg min 𝐹 𝔼 x , y [ L ( y , F ( x ) ) ] F^{*}=\underset{F}{\arg\min}\,\mathbb{E}_{x,y}[L(y,F(x))]
  31. F ^ ( x ) \hat{F}(x)
  32. h ( x ) hᵢ(x)
  33. F ( x ) = i = 1 M γ i h i ( x ) + const F(x)=\sum_{i=1}^{M}\gamma_{i}h_{i}(x)+\mbox{const}~{}
  34. F ^ ( x ) \hat{F}(x)
  35. F 0 ( x ) \!F_{0}(x)
  36. F 0 ( x ) = arg min 𝛾 i = 1 n L ( y i , γ ) F_{0}(x)=\underset{\gamma}{\arg\min}\sum_{i=1}^{n}L(y_{i},\gamma)
  37. F m ( x ) = F m - 1 ( x ) + arg min f i = 1 n L ( y i , F m - 1 ( x i ) + f ( x i ) ) F_{m}(x)=F_{m-1}(x)+\underset{f\in\mathcal{H}}{\operatorname{arg\,min}}\sum_{i% =1}^{n}L(y_{i},F_{m-1}(x_{i})+f(x_{i}))
  38. f f
  39. f f
  40. L L
  41. f f
  42. L ( y , f ) L(y,f)
  43. f f
  44. f ( x 1 ) , , f ( x n ) \!f(x_{1}),\ldots,f(x_{n})
  45. F m ( x ) = F m - 1 ( x ) - γ m i = 1 n f L ( y i , F m - 1 ( x i ) ) , F_{m}(x)=F_{m-1}(x)-\gamma_{m}\sum_{i=1}^{n}\nabla_{f}L(y_{i},F_{m-1}(x_{i})),
  46. γ m = arg min 𝛾 i = 1 n L ( y i , F m - 1 ( x i ) - γ L ( y i , F m - 1 ( x i ) ) f ( x i ) ) . \gamma_{m}=\underset{\gamma}{\arg\min}\sum_{i=1}^{n}L\left(y_{i},F_{m-1}(x_{i}% )-\gamma\frac{\partial L(y_{i},F_{m-1}(x_{i}))}{\partial f(x_{i})}\right).
  47. f f
  48. L L
  49. f f
  50. γ γ
  51. { ( x i , y i ) } i = 1 n , \!\{(x_{i},y_{i})\}_{i=1}^{n},
  52. L ( y , F ( x ) ) , \!L(y,F(x)),
  53. M . \!M.
  54. F 0 ( x ) = arg min 𝛾 i = 1 n L ( y i , γ ) . F_{0}(x)=\underset{\gamma}{\arg\min}\sum_{i=1}^{n}L(y_{i},\gamma).
  55. m m
  56. M M
  57. r i m = - [ L ( y i , F ( x i ) ) F ( x i ) ] F ( x ) = F m - 1 ( x ) for i = 1 , , n . r_{im}=-\left[\frac{\partial L(y_{i},F(x_{i}))}{\partial F(x_{i})}\right]_{F(x% )=F_{m-1}(x)}\quad\mbox{for }~{}i=1,\ldots,n.
  58. h m ( x ) \!h_{m}(x)
  59. { ( x i , r i m ) } i = 1 n \{(x_{i},r_{im})\}_{i=1}^{n}
  60. γ m \!\gamma_{m}
  61. γ m = arg min 𝛾 i = 1 n L ( y i , F m - 1 ( x i ) + γ h m ( x i ) ) . \gamma_{m}=\underset{\gamma}{\operatorname{arg\,min}}\sum_{i=1}^{n}L\left(y_{i% },F_{m-1}(x_{i})+\gamma h_{m}(x_{i})\right).
  62. F m ( x ) = F m - 1 ( x ) + γ m h m ( x ) . F_{m}(x)=F_{m-1}(x)+\gamma_{m}h_{m}(x).
  63. F M ( x ) . F_{M}(x).
  64. h m ( x ) \!h_{m}(x)
  65. J \!J
  66. J \!J
  67. R 1 m , , R J m \!R_{1m},\ldots,R_{Jm}
  68. h m ( x ) \!h_{m}(x)
  69. h m ( x ) = j = 1 J b j m I ( x R j m ) , h_{m}(x)=\sum_{j=1}^{J}b_{jm}I(x\in R_{jm}),
  70. b j m \!b_{jm}
  71. R j m \!R_{jm}
  72. b j m \!b_{jm}
  73. γ m \!\gamma_{m}
  74. F m ( x ) = F m - 1 ( x ) + γ m h m ( x ) , γ m = arg min 𝛾 i = 1 n L ( y i , F m - 1 ( x i ) + γ h m ( x i ) ) . F_{m}(x)=F_{m-1}(x)+\gamma_{m}h_{m}(x),\quad\gamma_{m}=\underset{\gamma}{% \operatorname{arg\,min}}\sum_{i=1}^{n}L(y_{i},F_{m-1}(x_{i})+\gamma h_{m}(x_{i% })).
  75. γ j m \!\gamma_{jm}
  76. γ m \!\gamma_{m}
  77. b j m \!b_{jm}
  78. F m ( x ) = F m - 1 ( x ) + j = 1 J γ j m I ( x R j m ) , γ j m = arg min 𝛾 x i R j m L ( y i , F m - 1 ( x i ) + γ h m ( x i ) ) . F_{m}(x)=F_{m-1}(x)+\sum_{j=1}^{J}\gamma_{jm}I(x\in R_{jm}),\quad\gamma_{jm}=% \underset{\gamma}{\operatorname{arg\,min}}\sum_{x_{i}\in R_{jm}}L(y_{i},F_{m-1% }(x_{i})+\gamma h_{m}(x_{i})).
  79. J \!J
  80. J = 2 \!J=2
  81. J = 3 \!J=3
  82. 4 J 8 4\leq J\leq 8
  83. J J
  84. J = 2 J=2
  85. J > 10 J>10
  86. F m ( x ) = F m - 1 ( x ) + ν γ m h m ( x ) , 0 < ν 1 , F_{m}(x)=F_{m-1}(x)+\nu\cdot\gamma_{m}h_{m}(x),\quad 0<\nu\leq 1,
  87. ν \nu
  88. ν < 0.1 \nu<0.1
  89. ν = 1 \nu=1
  90. 0.5 f 0.8 \!0.5\leq f\leq 0.8
  91. b j m b_{jm}
  92. R j m R_{jm}
  93. R j m R_{jm}

Gradient_pattern_analysis.html

  1. N V = M 2 N_{V}=M^{2}
  2. M 2 M^{2}
  3. M 2 M^{2}
  4. G A = N C - N V N V G_{A}=\frac{N_{C}-N_{V}}{N_{V}}
  5. N V > 0 N_{V}>0
  6. N C N_{C}
  7. N C > N V N_{C}>N_{V}
  8. G A G_{A}
  9. G A G_{A}

Granny_knot_(mathematics).html

  1. Δ ( t ) = ( t - 1 + t - 1 ) 2 , \Delta(t)=(t-1+t^{-1})^{2},
  2. ( z ) = ( z 2 + 1 ) 2 . \nabla(z)=(z^{2}+1)^{2}.
  3. V ( q ) = ( q - 1 + q - 3 - q - 4 ) 2 = q - 2 + 2 q - 4 - 2 q - 5 + q - 6 - 2 q - 7 + q - 8 . V(q)=(q^{-1}+q^{-3}-q^{-4})^{2}=q^{-2}+2q^{-4}-2q^{-5}+q^{-6}-2q^{-7}+q^{-8}.
  4. x , y , z x y x = y x y , x z x = z x z . \langle x,y,z\mid xyx=yxy,xzx=zxz\rangle.

Graph_algebra.html

  1. D = ( V , E ) D=(V,E)
  2. 0
  3. V V
  4. D D
  5. V { 0 } V\cup\{0\}
  6. x y = x xy=x
  7. x , y V , ( x , y ) E x,y\in V,(x,y)\in E
  8. x y = 0 xy=0
  9. x , y V { 0 } , ( x , y ) E x,y\in V\cup\{0\},(x,y)\notin E

GRAph_ALigner_(GRAAL).html

  1. G G
  2. H H
  3. ( x , y ) (x,y)
  4. x x
  5. G G
  6. y y
  7. H H
  8. G ( V , E ) G(V,E)\!
  9. H ( U , F ) H(U,F)
  10. v v
  11. u u
  12. H H
  13. d e g ( v ) deg(v)
  14. v v
  15. G G
  16. m a x d e g ( G ) max_{deg(G)}
  17. G G
  18. S ( v , u ) S(v,u)
  19. v v
  20. u u
  21. α \alpha
  22. 1 - α 1-\alpha
  23. v v
  24. u u
  25. C ( v , u ) = 2 - ( ( 1 - α ) × d e g ( v ) + d e g ( u ) m a x _ d e g ( G ) + m a x _ d e g ( H ) + α × S ( v , u ) ) C(v,u)=2-((1-\alpha)\times\frac{deg(v)+deg(u)}{max\_deg(G)+max\_deg(H)}+\alpha% \times S(v,u))
  26. 0
  27. v v
  28. u u
  29. 2 2
  30. ( v , u ) (v,u)
  31. v V v\in V
  32. u U u\in U
  33. v v
  34. u u
  35. r r
  36. v v
  37. S G ( v , r ) = { x V : d ( v , x ) = r } S_{G}(v,r)=\{x\in V:d(v,x)=r\}
  38. r r
  39. v v
  40. d ( v , x ) d(v,x)
  41. v v
  42. x x
  43. ( v , u ) : v S G ( v , r ) (v^{\prime},u^{\prime}):v^{\prime}\in S_{G}(v,r)
  44. u S H ( u , r ) u^{\prime}\in S_{H}(u,r)
  45. ( v , u ) (v,u)
  46. ( G p , H p ) (G^{p},H^{p})
  47. p = 1 , 2 , p=1,2,
  48. 3 3
  49. G p G^{p}
  50. G p = ( V , E p ) G^{p}=(V,E^{p})
  51. G G
  52. ( v , x ) E p (v,x)\in E^{p}
  53. v v
  54. x x
  55. G G
  56. p p
  57. d G ( v , x ) p d_{G}(v,x)\leq p
  58. G 1 = G G^{1}=G
  59. G p G^{p}
  60. p > 1 p>1
  61. p p
  62. G G
  63. H H

Graph_bandwidth.html

  1. max { | f ( v i ) - f ( v j ) | : v i v j E } \max\{\,|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}
  2. max { w i j | f ( v i ) - f ( v j ) | : v i v j E } \max\{\,w_{ij}|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}
  3. φ ( G ) \varphi(G)
  4. φ ( K n ) = n - 1 \varphi(K_{n})=n-1
  5. φ ( K m , n ) = ( m - 1 ) / 2 + n \varphi(K_{m,n})=\lfloor(m-1)/2\rfloor+n
  6. m n 1 , m\geq n\geq 1,
  7. S k = K k , 1 S_{k}=K_{k,1}
  8. φ ( S k ) = ( k - 1 ) / 2 + 1 \varphi(S_{k})=\lfloor(k-1)/2\rfloor+1
  9. Q n Q_{n}
  10. 2 n 2^{n}
  11. φ ( Q n ) = m = 0 n - 1 ( m m / 2 ) . \varphi(Q_{n})=\sum_{m=0}^{n-1}{\left({{m}\atop{\lfloor m/2\rfloor}}\right)}.
  12. P m × P n P_{m}\times P_{n}
  13. m m
  14. n n
  15. φ ( G ) χ ( G ) - 1 ; \varphi(G)\geq\chi(G)-1;
  16. ( n - 1 ) / diam ( G ) φ ( G ) n - diam ( G ) , \lceil(n-1)/\operatorname{diam}(G)\rceil\leq\varphi(G)\leq n-\operatorname{% diam}(G),
  17. n n
  18. G G
  19. φ ( T ) 5 n log Δ n . \varphi(T)\leq\frac{5n}{\log_{\Delta}n}.
  20. φ ( G ) 20 n log Δ n . \varphi(G)\leq\frac{20n}{\log_{\Delta}n}.

Graph_equation.html

  1. K 2 K_{2}
  2. K 3 K_{3}
  3. K 4 K_{4}

Graphical_models_for_protein_structure.html

  1. X u X v | X V { u , v } if { u , v } E . X_{u}\perp\!\!\!\perp X_{v}|X_{V\setminus\{u,v\}}\quad\,\text{if }\{u,v\}% \notin E.
  2. p ( X = x | Θ ) = p ( X b = x b ) p ( X s = x s | X b , Θ ) , p(X=x|\Theta)=p(X_{b}=x_{b})p(X_{s}=x_{s}|X_{b},\Theta),\,
  3. Θ \Theta
  4. Θ \Theta
  5. p ( X s = x s | X b = x b ) = 1 Z c C ( G ) Φ c ( x s c , x b c ) p(X_{s}=x_{s}|X_{b}=x_{b})=\frac{1}{Z}\prod_{c\in C(G)}\Phi_{c}(x_{s}^{c},x_{b% }^{c})
  6. Φ \Phi
  7. Φ \Phi
  8. Φ ( x s i p , x b j q ) = exp ( - E ( x s i p , x b j q ) / K B T ) \Phi(x_{s}^{i_{p}},x_{b}^{j_{q}})=\exp(-E(x_{s}^{i_{p}},x_{b}^{j_{q}})/K_{B}T)
  9. E ( x s i p , x b j q ) E(x_{s}^{i_{p}},x_{b}^{j_{q}})
  10. X i s X_{i}^{s}
  11. X j s X_{j}^{s}
  12. k B k_{B}
  13. G = E - T S G=E-TS
  14. G = x p ( x ) E ( x ) - T x p ( x ) ln ( p ( x ) ) G=\sum_{x}p(x)E(x)-T\sum_{x}p(x)\ln(p(x))\,
  15. Θ = [ θ 1 , θ 2 , , θ n ] \Theta=[\theta_{1},\theta_{2},\dots,\theta_{n}]
  16. n n
  17. n n
  18. f ( Θ = D ) f(\Theta=D)
  19. f ( Θ = D ) = 1 Z exp { - 1 2 ( D - μ ) T Σ - 1 ( D - μ ) } f(\Theta=D)=\frac{1}{Z}\exp\left\{-\frac{1}{2}(D-\mu)^{T}\Sigma^{-1}(D-\mu)\right\}
  20. Z = ( 2 π ) n / 2 | Σ | 1 / 2 Z=(2\pi)^{n/2}|\Sigma|^{1/2}
  21. μ \mu
  22. Σ \Sigma
  23. μ \mu
  24. Σ - 1 \Sigma^{-1}
  25. Σ - 1 \Sigma^{-1}
  26. μ \mu
  27. Σ - 1 \Sigma^{-1}

Graphlets.html

  1. i i
  2. i { 1 , , 29 } i\in\{1,\ldots,29\}
  3. T ( G ) = i = 1 29 N i ( G ) T(G)=\sum_{i=1}^{29}N_{i}(G)
  4. D ( G , H ) = i = 1 29 | F i ( G ) - F i ( H ) | D(G,H)=\sum_{i=1}^{29}|F_{i}(G)-F_{i}(H)|
  5. F i ( G ) = - log ( N i ( G ) / T ( G ) ) F_{i}(G)=-\log(N_{i}(G)/T(G))
  6. i = 0 , 1 , , 29 i=0,1,...,29
  7. S G j ( k ) = d G j ( k ) k S_{G}^{j}(k)=\frac{d_{G}^{j}(k)}{k}
  8. T G j = k = 1 S G j ( k ) T_{G}^{j}=\sum_{k=1}^{\infty}S_{G}^{j}(k)
  9. N G j ( k ) = S G j ( k ) T G j N_{G}^{j}(k)=\frac{S_{G}^{j}(k)}{T_{G}^{j}}
  10. D j ( G , H ) = 1 2 ( k = 1 [ N G j ( k ) - N H j ( k ) ] 2 ) 1 2 D^{j}(G,H)=\frac{1}{\sqrt{2}}(\sum_{k=1}^{\infty}[N_{G}^{j}(k)-N_{H}^{j}(k)]^{% 2})^{\frac{1}{2}}
  11. A j ( G , H ) = 1 - D j ( G , H ) A^{j}(G,H)=1-D^{j}(G,H)
  12. j { 0 , 1 , , 72 } j\in\{0,1,\ldots,72\}
  13. A a r i t h ( G , H ) = 1 73 j = 0 72 A j ( G , H ) A_{arith}(G,H)=\frac{1}{73}\sum_{j=0}^{72}A^{j}(G,H)
  14. A g e o ( G , H ) = ( j = 0 72 A j ( G , H ) ) 1 73 A_{geo}(G,H)=\left(\prod_{j=0}^{72}A^{j}(G,H)\right)^{\frac{1}{73}}
  15. D i ( u , v ) = w i × | l o g ( u i + 1 ) - l o g ( v i + 1 ) | l o g ( m a x { u i , v i } + 2 ) D_{i}(u,v)=w_{i}\times\frac{|log(u_{i}+1)-log(v_{i}+1)|}{log(max\{u_{i},v_{i}% \}+2)}
  16. D ( u , v ) = i = 0 72 D i i = 0 72 w i D(u,v)=\frac{\sum_{i=0}^{72}D_{i}}{\sum_{i=0}^{72}w_{i}}
  17. S ( u , v ) = 1 - D ( u , v ) S(u,v)=1-D(u,v)

Gravitoelectromagnetism.html

  1. 𝐄 g = - 4 π G ρ g \nabla\cdot\mathbf{E}\text{g}=-4\pi G\rho\text{g}
  2. 𝐄 = ρ ϵ 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}}
  3. 𝐁 g = 0 \nabla\cdot\mathbf{B}\text{g}=0
  4. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  5. × 𝐄 g = - 𝐁 g t \nabla\times\mathbf{E}\text{g}=-\frac{\partial\mathbf{B}\text{g}}{\partial t}
  6. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  7. × 𝐁 g = - 4 π G c 2 𝐉 g + 1 c 2 𝐄 g t \nabla\times\mathbf{B}\text{g}=-\frac{4\pi G}{c^{2}}\mathbf{J}\text{g}+\frac{1% }{c^{2}}\frac{\partial\mathbf{E}\text{g}}{\partial t}
  8. × 𝐁 = 1 ϵ 0 c 2 𝐉 + 1 c 2 𝐄 t \nabla\times\mathbf{B}=\frac{1}{\epsilon_{0}c^{2}}\mathbf{J}+\frac{1}{c^{2}}% \frac{\partial\mathbf{E}}{\partial t}
  9. 𝐅 g = m ( 𝐄 g + 4 𝐯 × 𝐁 g ) \mathbf{F\text{g}}=m\left(\mathbf{E}\text{g}\ +\ 4\mathbf{v}\times\mathbf{B}% \text{g}\right)
  10. 𝐅 e = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F\text{e}}=q\left(\mathbf{E}\ +\ \mathbf{v}\times\mathbf{B}\right)
  11. 𝒮 g = - c 2 4 π G 𝐄 g × 4 𝐁 g \mathcal{S}\text{g}=-\frac{c^{2}}{4\pi G}\mathbf{E}\text{g}\times 4\mathbf{B}% \text{g}
  12. 𝒮 = c 2 ε 0 𝐄 × 𝐁 \mathcal{S}=c^{2}\varepsilon_{0}\mathbf{E}\times\mathbf{B}
  13. 𝐁 g = G 2 c 2 𝐋 - 3 ( 𝐋 𝐫 / r ) 𝐫 / r r 3 , \mathbf{B}\text{g}=\frac{G}{2c^{2}}\frac{\mathbf{L}-3(\mathbf{L}\cdot\mathbf{r% }/r)\mathbf{r}/r}{r^{3}},
  14. 𝐁 g = G 2 c 2 𝐋 r 3 , \mathbf{B}\text{g}=\frac{G}{2c^{2}}\frac{\mathbf{L}}{r^{3}},
  15. L = I ball ω = 2 m r 2 5 2 π T L=I\text{ball}\omega=\frac{2mr^{2}}{5}\frac{2\pi}{T}
  16. I ball = 2 m r 2 5 I\text{ball}=\frac{2mr^{2}}{5}
  17. ω \omega
  18. B g, Earth = G 5 c 2 m r 2 π T = 2 π r g 5 c 2 T , B\text{g, Earth}=\frac{G}{5c^{2}}\frac{m}{r}\frac{2\pi}{T}=\frac{2\pi rg}{5c^{% 2}T},
  19. g = G m r 2 g=G\frac{m}{r^{2}}
  20. B g = 2 π G m 5 r c 2 T B\text{g}=\frac{2\pi Gm}{5rc^{2}T}

Gravitomagnetic_clock_effect.html

  1. T Kep = 2 π a 3 / G M T_{\rm Kep}=2\pi\sqrt{a^{3}/GM}
  2. T = T Kep + T Gvm = T Kep ± S / M c 2 , T=T_{\rm Kep}+T_{\rm Gvm}=T_{\rm Kep}\pm{S}/{Mc^{2}},

Gray's_paradox.html

  1. 𝐏 m u s c l e + 𝐏 t h r u s t = 𝐏 d r a g + 𝐏 d e f o r m a t i o n \mathbf{P}_{muscle}+\mathbf{P}_{thrust}=\mathbf{P}_{drag}+\mathbf{P}_{deformation}
  2. 𝐏 m u s c l e = 𝐏 d e f o r m a t i o n and 𝐏 t h r u s t = 𝐏 d r a g \mathbf{P}_{muscle}=\mathbf{P}_{deformation}\text{ and }\mathbf{P}_{thrust}=% \mathbf{P}_{drag}

Gregorian_calendar.html

  1. 97 400 \tfrac{97}{400}
  2. D = Y / 100 - Y / 400 - 2 D=\left\lfloor{Y/100}\right\rfloor-\left\lfloor{Y/400}\right\rfloor-2
  3. D D
  4. Y Y
  5. x \left\lfloor{x}\right\rfloor
  6. 49928114 70499183 \tfrac{49928114}{70499183}

Gregory's_series.html

  1. 0 x d u 1 + u 2 = arctan x = x - x 3 3 + x 5 5 - x 7 7 + \int_{0}^{x}\,\frac{du}{1+u^{2}}=\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-% \frac{x^{7}}{7}+\cdots

Gregory–Laflamme_instability.html

  1. D 5 D\geq 5
  2. 5 D 13 5\leq D\leq 13

Grey_atmosphere.html

  1. α ν \alpha_{\nu}
  2. α ν \alpha_{\nu}
  3. ν \nu
  4. α ν α \alpha_{\nu}\longrightarrow\alpha
  5. z z
  6. z z
  7. d s ds
  8. θ \theta
  9. d s = d z c o s θ ds=\frac{dz}{cos\theta}
  10. d τ = - α d s d\tau=-\alpha ds
  11. α \alpha
  12. d I d s = j - α I \frac{dI}{ds}=j-\alpha I
  13. I I
  14. j j
  15. d s ds
  16. - α -\alpha
  17. μ d I d τ = I - S \mu\frac{dI}{d\tau}=I-S
  18. S S
  19. e - τ / μ e^{-\tau/\mu}
  20. d d τ ( I e - τ / μ ) \frac{d}{d\tau}(Ie^{-\tau/\mu})
  21. τ \tau
  22. I ( τ , μ ) = e τ μ μ τ S e - τ μ d τ I(\tau,\mu)=\frac{e^{\frac{\tau}{\mu}}}{\mu}\int^{\infty}_{\tau}Se^{-\frac{% \tau}{\mu}}d\tau
  23. τ [ τ , ) \tau\in[\tau,\infty)
  24. μ [ 0 , 1 ] \mu\in[0,1]
  25. S S
  26. U U
  27. F F
  28. P P
  29. U = 2 π c - 1 + 1 I d μ U=\frac{2\pi}{c}\int^{+1}_{-1}Id\mu
  30. F = 2 π - 1 + 1 I μ d μ F=2\pi\int^{+1}_{-1}I\mu d\mu
  31. P = 2 π c - 1 + 1 I μ 2 d μ P=\frac{2\pi}{c}\int^{+1}_{-1}I\mu^{2}d\mu
  32. J = 1 2 - 1 + 1 I d μ J=\frac{1}{2}\int^{+1}_{-1}Id\mu
  33. μ \mu
  34. 1 4 π d F d τ = J - S \frac{1}{4\pi}\frac{dF}{d\tau}=J-S
  35. μ 2 \frac{\mu}{2}
  36. μ \mu
  37. d P d τ = F c \frac{dP}{d\tau}=\frac{F}{c}
  38. J = c 4 π U J=\frac{c}{4\pi}U
  39. d F d τ = 0 J = S \frac{dF}{d\tau}=0\iff J=S
  40. d P d τ \frac{dP}{d\tau}
  41. P = F c ( τ + κ ) P=\frac{F}{c}(\tau+\kappa)
  42. κ \kappa
  43. P = 1 3 U = 4 π 3 c J P=\frac{1}{3}U=\frac{4\pi}{3c}J
  44. J = 3 F 4 π ( τ + κ ) J=\frac{3F}{4\pi}(\tau+\kappa)
  45. S = 3 F 4 π ( τ + κ ) S=\frac{3F}{4\pi}(\tau+\kappa)
  46. I ( τ = 0 , μ ) = 3 F 4 π e τ / μ μ 0 ( τ + κ ) e - τ / μ d τ = 3 F 4 π ( μ + κ ) I(\tau=0,\mu)=\frac{3F}{4\pi}\frac{e^{\tau/\mu}}{\mu}\int^{\infty}_{0}(\tau+% \kappa)e^{-\tau/\mu}d\tau=\frac{3F}{4\pi}(\mu+\kappa)
  47. τ \tau
  48. F = 2 π 0 + 1 I μ d μ = 3 F 2 0 + 1 ( μ 2 + κ μ ) d μ = 3 F 2 ( 1 3 + κ 2 ) F=2\pi\int^{+1}_{0}I\mu d\mu=\frac{3F}{2}\int^{+1}_{0}(\mu^{2}+\kappa\mu)d\mu=% \frac{3F}{2}(\frac{1}{3}+\frac{\kappa}{2})
  49. κ = 2 3 \kappa=\frac{2}{3}
  50. S ( τ ) = 3 F 4 π ( τ + 2 3 ) S(\tau)=\frac{3F}{4\pi}(\tau+\frac{2}{3})
  51. H H
  52. H ( τ ) = H H(\tau)=H
  53. K K
  54. K ( τ ) = τ H + 2 / 3 H = 1 / 3 J ( τ ) K(\tau)=\tau H+2/3H=1/3J(\tau)
  55. T e f f T_{eff}
  56. H H
  57. T T
  58. T 4 = T e f f 4 3 4 ( τ + 2 3 ) T^{4}=T_{eff}^{4}\frac{3}{4}\left(\tau+\frac{2}{3}\right)
  59. T e f f T_{eff}
  60. T T
  61. τ = 2 / 3 \tau=2/3
  62. 0.841 T e f f 0.841T_{eff}

Griesmer_bound.html

  1. n i = 0 k - 1 d 2 i . n\geq\sum_{i=0}^{k-1}\left\lceil\frac{d}{2^{i}}\right\rceil.
  2. N ( k , d ) N(k,d)
  3. N ( k , d ) i = 0 k - 1 d 2 i N(k,d)\geq\sum_{i=0}^{k-1}\left\lceil\frac{d}{2^{i}}\right\rceil
  4. G = [ 1 1 0 0 G ] G=\begin{bmatrix}1&\dots&1&0&\dots&0\\ \ast&\ast&\ast&&G^{\prime}&\\ \end{bmatrix}
  5. k = k - 1 k^{\prime}=k-1
  6. n = N ( k , d ) - d n^{\prime}=N(k,d)-d
  7. u C u\in C^{\prime}
  8. w ( u ) = d w(u)=d^{\prime}
  9. v ( F 2 ) d v\in(F_{2})^{d}
  10. ( v | u ) C (v|u)\in C
  11. w ( v ) + w ( u ) = w ( v | u ) d w(v)+w(u)=w(v|u)\geq d
  12. ( v | u ) + r C (v|u)+r\in C
  13. r C r\in C
  14. C C
  15. w ( ( v | u ) + r ) d w((v|u)+r)\geq d
  16. w ( ( v | u ) + r ) = w ( ( ( 1 , 1 , , 1 ) + v ) | u ) = d - w ( v ) + w ( u ) w((v|u)+r)=w(((1,1,...,1)+v)|u)=d-w(v)+w(u)
  17. d - w ( v ) + w ( u ) d d-w(v)+w(u)\geq d
  18. w ( v ) + w ( u ) d w(v)+w(u)\geq d
  19. d + 2 w ( u ) 2 d d+2w(u)\geq 2d
  20. w ( u ) = d w(u)=d^{\prime}
  21. d d / 2 d^{\prime}\geq d/2
  22. n N ( k - 1 , d / 2 ) n^{\prime}\geq N(k-1,d/2)
  23. n N ( k - 1 , d / 2 ) n^{\prime}\geq\left\lceil N(k-1,d/2)\right\rceil
  24. N ( k , d ) N ( k - 1 , d / 2 ) + d N(k,d)\geq\left\lceil N(k-1,d/2)\right\rceil+d
  25. N ( k , d ) i = 0 k - 1 d 2 i N(k,d)\geq\sum_{i=0}^{k-1}\left\lceil\frac{d}{2^{i}}\right\rceil
  26. a / 2 k - 1 2 = a 2 k \left\lceil\frac{\left\lceil a/2^{k-1}\right\rceil}{2}\right\rceil=\left\lceil% \frac{a}{2^{k}}\right\rceil
  27. 𝔽 q \mathbb{F}_{q}
  28. n i = 0 k - 1 d q i . n\geq\sum_{i=0}^{k-1}\left\lceil\frac{d}{q^{i}}\right\rceil.

Grinberg's_theorem.html

  1. k 3 ( k - 2 ) ( f k - g k ) = 0. \sum_{k\geq 3}(k-2)(f_{k}-g_{k})=0.

Group_delay_dispersion.html

  1. D 2 ( ω ) = - T g d ω = d 2 ϕ d ω 2 D_{2}(\omega)=-\frac{\partial T_{g}}{d\omega}=\frac{d^{2}\phi}{d\omega^{2}}
  2. D D
  3. D 2 ( ω ) = - 2 π c λ 2 D t o t D_{2}(\omega)=-\frac{2\pi c}{\lambda^{2}}D_{tot}

Group_testing.html

  1. n n
  2. n n
  3. I n p u t : Input:
  4. n n
  5. d d
  6. 𝐱 = ( x 1 , x 2 , , x n ) \mathbf{x}=(x_{1},x_{2},...,x_{n})
  7. x i = 1 x_{i}=1
  8. i i
  9. x i = 0 x_{i}=0
  10. 𝐱 \mathbf{x}
  11. 1 s 1^{\prime}s
  12. x x
  13. | x | d |x|\leq d
  14. | x | |x|
  15. 𝐱 \mathbf{x}
  16. 1 s 1^{\prime}s
  17. S S
  18. [ n ] [n]
  19. S [ n ] S\subseteq[n]
  20. A ( S ) = { 1 , if k S x k 1 0 , otherwise. A(S)=\begin{cases}1,\mbox{ if }~{}\displaystyle\sum_{k\in S}x_{k}\geq 1\\ 0,\mbox{ otherwise.}\end{cases}
  21. O R OR
  22. A ( S ) = i S x i A(S)=\displaystyle\bigvee_{i\in S}x_{i}
  23. 𝐱 \mathbf{x}
  24. 𝐱 \mathbf{x}
  25. n n
  26. m m
  27. t ( d , n ) : t(d,n):
  28. n n
  29. d d
  30. t ( d , n ) t(d,n)
  31. 1 t ( d , n ) n 1\leq t(d,n)\leq n
  32. t a ( d , n ) : t^{a}(d,n):
  33. n n
  34. d d
  35. t a ( d , n ) : t^{a}(d,n):
  36. S [ n ] S\subseteq[n]
  37. χ i { 0 , 1 } n \chi_{i}\in\{0,1\}^{n}
  38. i S χ s ( i ) = 1 i\in S\Leftrightarrow\chi_{s}(i)=1
  39. M M
  40. t × n t\times n
  41. χ i \chi_{i}
  42. 𝐱 \mathbf{x}
  43. 𝐫 \mathbf{r}
  44. t t
  45. t - t-
  46. S i [ n ] ( 1 i t ) S_{i}\subseteq[n](1\leq i\leq t)
  47. χ i \chi_{i}
  48. i i
  49. i t h i^{th}
  50. M M
  51. m i , j m_{i,j}
  52. i t h i^{th}
  53. j S j\in S
  54. \bigwedge
  55. \bigvee
  56. M × 𝐱 = 𝐫 M\times\mathbf{x}=\mathbf{r}
  57. 𝐫 \mathbf{r}
  58. 𝐫 \mathbf{r}
  59. i i
  60. 1 1
  61. M M
  62. 𝐱 \mathbf{x}
  63. M = ( m 1 , 1 m 1 , n m t , 1 m t , n ) M=\begin{pmatrix}m_{1,1}\cdots m_{1,n}\\ \vdots\ddots\vdots\\ m_{t,1}\cdots m_{t,n}\end{pmatrix}
  64. 𝐱 = ( x 1 x n ) 𝐫 = ( r 1 r t ) \mathbf{x}=\begin{pmatrix}x_{1}\\ \vdots\\ x_{n}\end{pmatrix}\mathbf{r}=\begin{pmatrix}r_{1}\\ \vdots\\ r_{t}\end{pmatrix}
  65. t a ( d , n ) t^{a}(d,n)
  66. t ( d , n ) t(d,n)
  67. 1 t a ( d , n ) t ( d , n ) n 1\leq t^{a}(d,n)\leq t(d,n)\leq n
  68. t a ( d , n ) t ( d , n ) t^{a}(d,n)\leq t(d,n)
  69. t a ( d , n ) t^{a}(d,n)
  70. t t
  71. 𝐱 \mathbf{x}
  72. 𝐲 \mathbf{y}
  73. | 𝐱 | , | 𝐲 | d |\mathbf{x}|,|\mathbf{y}|\leq d
  74. 𝐫 ( 𝐱 ) 𝐫 ( 𝐲 ) \mathbf{r(x)}\neq\mathbf{r(y)}
  75. 𝐫 ( 𝐱 ) \mathbf{r(x)}
  76. 𝐱 \mathbf{x}
  77. 𝐱 \mathbf{x}
  78. 𝐲 \mathbf{y}
  79. d d
  80. n n
  81. V o l 2 ( d , n ) Vol_{2}(d,n)
  82. t t
  83. 2 t 2^{t}
  84. 2 t V o l 2 ( d , n ) 2^{t}\geq Vol_{2}(d,n)
  85. log \log
  86. t log { V o l 2 ( d , n ) } t\geq\log\{Vol_{2}(d,n)\}
  87. V o l 2 ( d , n ) ( n d ) ( n d ) d Vol_{2}(d,n)\geq{n\choose d}\geq(\frac{n}{d})^{d}
  88. d log n d d\log{\frac{n}{d}}
  89. t a ( d , n ) d log n d t^{a}(d,n)\geq d\log\frac{n}{d}
  90. t a ( d , n ) t^{a}(d,n)
  91. t a ( d , n ) O ( d log n ) t^{a}(d,n)\leq O(d\log{n})
  92. d d
  93. d d
  94. O ( log n ) O(\log{n})
  95. i i
  96. x i = 1 x_{i}=1
  97. [ n ] [n]
  98. 1 1
  99. 1 1
  100. 2 log n 2\lceil\log{n}\rceil
  101. log n + 1 \lceil\log{n}\rceil+1
  102. 1 1
  103. i t h i^{th}
  104. d d
  105. O ( d log n ) O(d\log{n})
  106. t ( 1 , n ) t(1,n)
  107. t ( 1 , n ) log n t(1,n)\leq\lceil\log{n}\rceil
  108. d = 1 d=1
  109. 𝐫 \mathbf{r}
  110. i i
  111. i i
  112. log n \lceil\log{n}\rceil
  113. i i
  114. H m H_{m}
  115. [ 2 m - 1 , 2 m - m - 1 , 3 ] [2^{m}-1,2^{m}-m-1,3]
  116. t ( d , n ) t(d,n)
  117. d = 1 d=1
  118. d d
  119. Ω ( d 2 log d log n ) t ( d , n ) \Omega(\frac{d^{2}}{\log{d}}\log{n})\leq t(d,n)
  120. t ( d , n ) 𝒪 ( d 2 log n ) t(d,n)\leq\mathcal{O}(d^{2}\log{n})
  121. t ( d , n ) 𝒪 ( d 2 log 2 n ) t(d,n)\leq\mathcal{O}(d^{2}\log^{2}{n})
  122. t ( d , n ) t(d,n)
  123. d log d \frac{d}{\log{d}}
  124. t a ( d , n ) t^{a}(d,n)
  125. 1 log d \frac{1}{\log{d}}

Grouser.html

  1. H = b l c ( 1 + 2 h b ) + W tan ϕ ( 1 + 0.64 [ ( h b ) cot - 1 ( h b ) ] ) H=blc\left(1+\frac{2h}{b}\right)+W\tan\phi\left(1+0.64\left[\left(\frac{h}{b}% \right)\cot^{-1}\left(\frac{h}{b}\right)\right]\right)
  2. H = H=
  3. b = b=
  4. l = l=
  5. c = c=
  6. h = h=
  7. W = W=
  8. ϕ \phi
  9. = =

Grunwald–Winstein_equation.html

  1. log k x , s o l k x , 80 % E t O H = m Y \log\frac{k_{x,sol}}{k_{x,80\%EtOH}}=mY
  2. log k X k H = ρ σ X \log\frac{k_{X}}{k_{H}}=\rho\sigma_{X}
  3. log k t - B u C l , s o l k t - B u C l , 80 % E t O H = Y \log\frac{k_{t-BuCl,sol}}{k_{t-BuCl,80\%EtOH}}=Y
  4. < V A R > k t - B u C l , 80 % E t O H <VAR>k_{t-BuCl,80\%EtOH}

Guitar_wiring.html

  1. f = ω 2 π = 1 2 π L C f={\omega\over 2\pi}={1\over{2\pi\sqrt{LC}}}

Gunnar_Källén.html

  1. λ ( x , y , z ) x 2 + y 2 + z 2 - 2 x y - 2 y z - 2 z x \lambda(x,y,z)\equiv x^{2}+y^{2}+z^{2}-2xy-2yz-2zx

H-factor.html

  1. H = 0 t exp ( 43.2 - 16115 / T ) d t H=\int_{0}^{t}\exp\left(43.2-16115/T\right)\,dt\,

H-stable_potential.html

  1. x 1 , x 2 , R ν x_{1},x_{2},\ldots\in R^{\nu}
  2. x i x_{i}
  3. x j x_{j}
  4. ϕ ( x i - x j ) \phi(x_{i}-x_{j})\,
  5. ϕ ( x ) \phi(x)
  6. ϕ ( x ) \phi(x)
  7. B > 0 B>0
  8. n 1 n\geq 1
  9. x 1 , x 2 , , x n R ν x_{1},x_{2},\ldots,x_{n}\in R^{\nu}
  10. V n ( x 1 , x 2 , x n ) := i < j = 1 n ϕ ( x i - x j ) - B n V_{n}(x_{1},x_{2},\ldots x_{n}):=\sum_{i<j=1}^{n}\phi(x_{i}-x_{j})\geq-Bn\,
  11. ϕ ( 0 ) < \phi(0)<\infty
  12. n 1 n\geq 1
  13. x 1 , x 2 , x n R ν x_{1},x_{2},\ldots x_{n}\in R^{\nu}
  14. i , j = 1 n ϕ ( x i - x j ) 0 \sum_{i,j=1}^{n}\phi(x_{i}-x_{j})\geq 0
  15. ϕ ( x ) \phi(x)
  16. B B
  17. ϕ ( 0 ) 2 \frac{\phi(0)}{2}
  18. ϕ ( x ) \phi(x)
  19. Λ \Lambda
  20. β > 0 \beta>0
  21. z > 0 z>0
  22. n 1 z n n ! Λ n d x 1 d x n exp [ - β V n ( x 1 , x 2 , x n ) ] \sum_{n\geq 1}\frac{z^{n}}{n!}\int_{\Lambda^{n}}\!dx_{1}\cdots dx_{n}\;\exp[-% \beta V_{n}(x_{1},x_{2},\ldots x_{n})]
  23. Ξ ( β , z , Λ ) := 1 + n 1 z n n ! Λ n d x 1 d x n exp [ - β V n ( x 1 , x 2 , x n ) ] \Xi(\beta,z,\Lambda):=1+\sum_{n\geq 1}\frac{z^{n}}{n!}\int_{\Lambda^{n}}\!dx_{% 1}\cdots dx_{n}\;\exp[-\beta V_{n}(x_{1},x_{2},\ldots x_{n})]
  24. ϕ ( x ) = 1 4 π | x | \phi(x)=\frac{1}{4\pi|x|}
  25. V n ( x 1 , , x n ) = i < j ϕ ( x i - x j ) V_{n}(x_{1},\ldots,x_{n})=\sum_{i<j}\phi(x_{i}-x_{j})
  26. B = 0 B=0
  27. ϕ ( x ) - 1 2 π ln m | x | for x 0 \phi(x)\sim-\frac{1}{2\pi}\ln{m|x|}\qquad{\rm for}\quad x\sim 0
  28. H n ( q ¯ , x ¯ ) = i < j q i q j ϕ ( x i - x j ) H_{n}(\underline{q},\underline{x})=\sum_{i<j}q_{i}q_{j}\phi(x_{i}-x_{j})
  29. q j q_{j}
  30. j j
  31. H n ( q ¯ , x ¯ ) H_{n}(\underline{q},\underline{x})
  32. n = 2 n=2
  33. q 1 q 2 = 1 q_{1}q_{2}=1
  34. inf x 1 , x 2 ϕ ( x 1 - x 2 ) = - \inf_{x_{1},x_{2}}\phi(x_{1}-x_{2})=-\infty
  35. β < 4 π \beta<4\pi
  36. B : E 0 N > - B , \exists B:\frac{E_{0}}{N}>-B,\,

Hadwiger_conjecture_(combinatorial_geometry).html

  1. K i = 1 2 n s i K + v i . K\subseteq\bigcup_{i=1}^{2^{n}}s_{i}K+v_{i}.
  2. 4 n ( 5 n ln n ) . \displaystyle 4^{n}(5n\ln n).

Half-side_formula.html

  1. tan ( a 2 ) \displaystyle\tan\left(\frac{a}{2}\right)
  2. S = 1 2 ( α + β + γ ) S=\frac{1}{2}(\alpha+\beta+\gamma)
  3. R = - cos S cos ( S - α ) cos ( S - β ) cos ( S - γ ) . R=\sqrt{\frac{-\cos S}{\cos(S-\alpha)\cos(S-\beta)\cos(S-\gamma)}}.

Half_time_(physics).html

  1. Q ( t ) = Q f ( 1 - e - λ t ) Q(t)=Q_{\mathrm{f}}(1-e^{-\lambda t})\,

Hamming_scheme.html

  1. X = n X=\mathcal{F}^{n}
  2. n n
  3. x x
  4. y n y\in\mathcal{F}^{n}
  5. i i
  6. i i
  7. v v
  8. X X
  9. x x
  10. y y
  11. i i
  12. x x
  13. y y
  14. i i
  15. k k
  16. i i
  17. j j
  18. c i j k c_{ijk}
  19. i , j , k i,j,k
  20. c i i 0 = v i c_{ii0}=v_{i}
  21. i i
  22. v i v_{i}
  23. R i R_{i}
  24. c i j k c_{ijk}
  25. c i j k = { ( k i - j + k 2 ) ( n - k i + j - k 2 ) , if i + j - k is even, 0 , if i + j - k is odd. c_{ijk}=\begin{cases}{\displaystyle\left({{k}\atop{\frac{i-j+k}{2}}}\right)}{% \displaystyle\left({{n-k}\atop{\frac{i+j-k}{2}}}\right)},&\,\text{if }i+j-k\,% \text{ is even,}\\ \;\;\;\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;\;\;\;,&\,\text{if }i+j-k\,\text{ is odd.% }\end{cases}
  26. v = | X | = 2 n v=\left|X\right|=2^{n}
  27. v i = ( n i ) v_{i}={\left({{n}\atop{i}}\right)}
  28. 2 n × 2 n 2^{n}\times 2^{n}
  29. x n x\in\mathcal{F}^{n}
  30. ( x , y ) \left(x,y\right)
  31. D k D_{k}
  32. 1 1
  33. d H ( x , y ) = k d_{H}(x,y)=k

Hand_of_Eris.html

  1. \rightarrow\!\leftarrow
  2. \Rarr \Larr \Rarr\!\Larr

Hang_Seng_Index.html

  1. Current Index = [P(t) × IS × FAF × CF] [P(t-1) × IS × FAF × CF] × Yesterday’s Closing Index . \textrm{Current~{}Index}=\frac{\sum\textrm{[}\textrm{P(t)}\times\textrm{IS}% \times\textrm{FAF}\times\textrm{CF}\textrm{]}}{\sum\textrm{[}\textrm{P(t-1)}% \times\textrm{IS}\times\textrm{FAF}\times\textrm{CF}\textrm{]}}\times\textrm{% Yesterday's~{}Closing~{}Index}.

Hans_Werner_Ballmann.html

  1. L 2 L^{2}

Hansen's_problem.html

  1. ϕ + ψ = β 1 + β 2 \phi+\psi=\beta_{1}+\beta_{2}
  2. A B P 2 B = sin α 2 sin ϕ \frac{AB}{P_{2}B}=\frac{\sin\alpha_{2}}{\sin\phi}
  3. P 2 B P 1 P 2 = sin β 1 sin δ \frac{P_{2}B}{P_{1}P_{2}}=\frac{\sin\beta_{1}}{\sin\delta}
  4. A B P 1 P 2 = sin α 2 sin β 1 sin ϕ sin δ \frac{AB}{P_{1}P_{2}}=\frac{\sin\alpha_{2}\sin\beta_{1}}{\sin\phi\sin\delta}
  5. A B P 1 P 2 = sin α 1 sin β 2 sin ψ sin γ \frac{AB}{P_{1}P_{2}}=\frac{\sin\alpha_{1}\sin\beta_{2}}{\sin\psi\sin\gamma}
  6. sin ϕ sin ψ = sin γ sin α 2 sin β 1 sin δ sin α 1 sin β 2 = k \frac{\sin\phi}{\sin\psi}=\frac{\sin\gamma\sin\alpha_{2}\sin\beta_{1}}{\sin% \delta\sin\alpha_{1}\sin\beta_{2}}=k
  7. tan ϕ - ψ 2 = k - 1 k + 1 tan ϕ + ψ 2 \tan\frac{\phi-\psi}{2}=\frac{k-1}{k+1}\tan\frac{\phi+\psi}{2}
  8. ϕ \phi
  9. ψ \psi
  10. A B P 1 P 2 \frac{AB}{P_{1}P_{2}}
  11. γ = π - α 1 - β 1 - β 2 , δ = π - α 2 - β 1 - β 2 \gamma=\pi-\alpha_{1}-\beta_{1}-\beta_{2},\quad\delta=\pi-\alpha_{2}-\beta_{1}% -\beta_{2}
  12. k = sin γ sin α 2 sin β 1 sin δ sin α 1 sin β 2 k=\frac{\sin\gamma\sin\alpha_{2}\sin\beta_{1}}{\sin\delta\sin\alpha_{1}\sin% \beta_{2}}
  13. s = β 1 + β 2 s=\beta_{1}+\beta_{2}
  14. d = 2 arctan [ k - 1 k + 1 tan ( s / 2 ) ] d=2\arctan\left[\frac{k-1}{k+1}\tan(s/2)\right]
  15. ϕ = ( s + d ) / 2 \phi=(s+d)/2
  16. ψ = ( s - d ) / 2. \psi=(s-d)/2.
  17. P 1 P 2 = A B sin ϕ sin δ sin α 2 sin β 1 , P_{1}P_{2}=AB\frac{\sin\phi\sin\delta}{\sin\alpha_{2}\sin\beta_{1}},
  18. P 1 P 2 = A B sin ψ sin γ sin α 1 sin β 2 . P_{1}P_{2}=AB\frac{\sin\psi\sin\gamma}{\sin\alpha_{1}\sin\beta_{2}}.

Haploid-relative-risk.html

  1. H R R = P 1 1 - P 1 * 1 - P 2 P 2 HRR=\frac{P_{1}}{1-P_{1}}*\frac{1-P_{2}}{P_{2}}
  2. H R R = a b * d c HRR=\frac{a^{^{\prime}}}{b^{^{\prime}}}*\frac{d^{^{\prime}}}{c^{^{\prime}}}

Hardy's_paradox.html

  1. | e - | v - + i | w - 2 i | c - . |e^{-}\rangle\to\frac{|v^{-}\rangle+i|w^{-}\rangle}{\sqrt{2}}\to i|c^{-}\rangle.
  2. | w + | w - | γ | γ |w^{+}\rangle|w^{-}\rangle\to|\gamma\rangle|\gamma\rangle
  3. 1 2 ( | v + | v - + i | v + | w - + i | w + | v - - | γ | γ ) . \frac{1}{2}\left(|v^{+}\rangle|v^{-}\rangle+i|v^{+}\rangle|w^{-}\rangle+i|w^{+% }\rangle|v^{-}\rangle-|\gamma\rangle|\gamma\rangle\right).
  4. | v + i | w 2 \frac{|v\rangle+i|w\rangle}{\sqrt{2}}
  5. | v - i | w 2 \frac{|v\rangle-i|w\rangle}{\sqrt{2}}
  6. | e + e - 1 4 ( - 3 | c + | c - + i | c + | d - + i | d + | c - - | d + | d - - 2 | γ | γ ) . |e^{+}e^{-}\rangle\to\frac{1}{4}\left(-3|c^{+}\rangle|c^{-}\rangle+i|c^{+}% \rangle|d^{-}\rangle+i|d^{+}\rangle|c^{-}\rangle-|d^{+}\rangle|d^{-}\rangle-2|% \gamma\rangle|\gamma\rangle\right).
  7. | v + - i | w + 2 | v - - i | w - 2 = 1 2 ( | v + | v - - i | v + | w - - i | w + | v - - | w + | w - ) . \frac{|v^{+}\rangle-i|w^{+}\rangle}{\sqrt{2}}\frac{|v^{-}\rangle-i|w^{-}% \rangle}{\sqrt{2}}=\frac{1}{2}\left(|v^{+}\rangle|v^{-}\rangle-i|v^{+}\rangle|% w^{-}\rangle-i|w^{+}\rangle|v^{-}\rangle-|w^{+}\rangle|w^{-}\rangle\right).
  8. | d + | d - |d^{+}\rangle|d^{-}\rangle

Hardy_hierarchy.html

  1. h α + 1 ( n ) = h α ( n + 1 ) , h_{\alpha+1}(n)=h_{\alpha}(n+1),\,
  2. h α ( n ) = h α [ n ] ( n ) h_{\alpha}(n)=h_{\alpha[n]}(n)\,\!
  3. H α H_{\alpha}\,\!

Hardy–Littlewood_tauberian_theorem.html

  1. n = 0 a n e - n y 1 y \sum_{n=0}^{\infty}a_{n}e^{-ny}\sim\frac{1}{y}
  2. k = 0 n a k n \sum_{k=0}^{n}a_{k}\sim n
  3. n = 0 a n x n 1 1 - x . \sum_{n=0}^{\infty}a_{n}x^{n}\sim\frac{1}{1-x}.
  4. k = 0 n a k n . \sum_{k=0}^{n}a_{k}\sim n.
  5. n = 0 a n e - n y 1 y \sum_{n=0}^{\infty}a_{n}e^{-ny}\sim\frac{1}{y}
  6. k = 0 n a k n . \sum_{k=0}^{n}a_{k}\sim n.
  7. ω ( s ) = 0 e - s t d F ( t ) . \omega(s)=\int_{0}^{\infty}e^{-st}\,dF(t).
  8. ω ( s ) C s - ρ , as s 0 \omega(s)\sim Cs^{-\rho},\quad\rm{as\ }s\to 0
  9. F ( t ) C Γ ( ρ + 1 ) t ρ , as t . F(t)\sim\frac{C}{\Gamma(\rho+1)}t^{\rho},\quad\rm{as\ }t\to\infty.
  10. k = 0 n a k \textstyle{\sum_{k=0}^{n}a_{k}}
  11. L ( t x ) L ( x ) 1 , x \frac{L(tx)}{L(x)}\to 1,\quad x\to\infty
  12. ω ( s ) s - ρ L ( s - 1 ) , as s 0 \omega(s)\sim s^{-\rho}L(s^{-1}),\quad\rm{as\ }s\to 0
  13. F ( t ) 1 Γ ( ρ + 1 ) t ρ L ( t ) , as t . F(t)\sim\frac{1}{\Gamma(\rho+1)}t^{\rho}L(t),\quad\rm{as\ }t\to\infty.
  14. lim x 1 ( 1 - x ) a n x n g ( x n ) = 0 1 g ( t ) d t \lim_{x\rightarrow 1}(1-x)\sum a_{n}x^{n}g(x^{n})=\int_{0}^{1}g(t)dt
  15. 1 ( 1 + x ) 2 ( 1 - x ) = 1 - x + 2 x 2 - 2 x 3 + 3 x 4 - 3 x 5 + \frac{1}{(1+x)^{2}(1-x)}=1-x+2x^{2}-2x^{3}+3x^{4}-3x^{5}+\cdots
  16. a n x n s , \sum a_{n}x^{n}\to s,
  17. a n = s . \sum a_{n}=s.
  18. n = 2 Λ ( n ) e - n y 1 y , \sum_{n=2}^{\infty}\Lambda(n)e^{-ny}\sim\frac{1}{y},
  19. n x Λ ( n ) x , \sum_{n\leq x}\Lambda(n)\sim x,

Harmonic_progression_(mathematics).html

  1. 1 / a , 1 a + d , 1 a + 2 d , 1 a + 3 d , , 1 a + k d , 1/a,\ \frac{1}{a+d}\ ,\frac{1}{a+2d}\ ,\frac{1}{a+3d}\ ,\cdots,\frac{1}{a+kd},
  2. x y + z \frac{x}{y+z}
  3. x y y + z y \frac{\frac{x}{y}}{\frac{y+z}{y}}
  4. x y = a \frac{x}{y}=a
  5. z y = k d \frac{z}{y}=kd
  6. 12 5 \tfrac{12}{5}
  7. 12 1 + n \tfrac{12}{1+n}
  8. 30 7 \tfrac{30}{7}
  9. 10 1 - 2 n 3 \tfrac{10}{1-\tfrac{2n}{3}}

Harmonious_set.html

  1. χ : Λ d 𝐓 , χ Hom ( Λ d , 𝐓 ) . \chi:\Lambda_{d}\to\mathbf{T},\quad\chi\in\operatorname{Hom}(\Lambda_{d},% \mathbf{T}).
  2. sup Λ | χ ( λ ) - ξ ( λ ) | ϵ , χ Hom ( Λ d , 𝐓 ) , ξ G ^ . \sup_{\Lambda}|\chi(\lambda)-\xi(\lambda)|\leq\epsilon,\quad\chi\in% \operatorname{Hom}(\Lambda_{d},\mathbf{T}),\xi\in\hat{G}.
  3. sup Λ | χ ( λ ) - 1 | ϵ , χ G ^ . \sup_{\Lambda}|\chi(\lambda)-1|\leq\epsilon,\quad\chi\in\hat{G}.
  4. M ϵ + K ϵ = G ^ . M_{\epsilon}+K_{\epsilon}=\hat{G}.

Harries_graph.html

  1. ( x - 3 ) ( x - 1 ) 4 ( x + 1 ) 4 ( x + 3 ) ( x 2 - 6 ) ( x 2 - 2 ) ( x 4 - 6 x 2 + 2 ) 5 ( x 4 - 6 x 2 + 3 ) 4 ( x 4 - 6 x 2 + 6 ) 5 . (x-3)(x-1)^{4}(x+1)^{4}(x+3)(x^{2}-6)(x^{2}-2)(x^{4}-6x^{2}+2)^{5}(x^{4}-6x^{2% }+3)^{4}(x^{4}-6x^{2}+6)^{5}.\,

Harries–Wong_graph.html

  1. ( x - 3 ) ( x - 1 ) 4 ( x + 1 ) 4 ( x + 3 ) ( x 2 - 6 ) ( x 2 - 2 ) ( x 4 - 6 x 2 + 2 ) 5 ( x 4 - 6 x 2 + 3 ) 4 ( x 4 - 6 x 2 + 6 ) 5 . (x-3)(x-1)^{4}(x+1)^{4}(x+3)(x^{2}-6)(x^{2}-2)(x^{4}-6x^{2}+2)^{5}(x^{4}-6x^{2% }+3)^{4}(x^{4}-6x^{2}+6)^{5}.\,

Harrop_formula.html

  1. A B A\wedge B
  2. A A
  3. B B
  4. ¬ F \neg F
  5. F F
  6. F A F\rightarrow A
  7. A A
  8. F F
  9. x . A \forall x.A
  10. A A
  11. A ¬ ¬ A . A\leftrightarrow\neg\neg A.
  12. r r
  13. r ( t 1 , , t n ) r(t_{1},...,t_{n})
  14. A B A\wedge B
  15. A A
  16. B B
  17. x . A \forall x.A
  18. A A
  19. G A G\rightarrow A
  20. A A
  21. G G
  22. A B A\wedge B
  23. A A
  24. B B
  25. A B A\vee B
  26. A A
  27. B B
  28. x . A \forall x.A
  29. A A
  30. x . A \exists x.A
  31. A A
  32. H A H\rightarrow A
  33. A A
  34. H H

Hartmann_number.html

  1. Ha = B L σ μ \mathrm{Ha}=BL\sqrt{\frac{\sigma}{\mu}}

Hartogs'_extension_theorem.html

  1. n > 1 n>1
  2. H ε = { z = ( z 1 , z 2 ) Δ 2 : | z 1 | < ε or 1 - ε < | z 2 | } H_{\varepsilon}=\{z=(z_{1},z_{2})\in\Delta^{2}:|z_{1}|<\varepsilon\ \ \,\text{% or}\ \ 1-\varepsilon<|z_{2}|\}
  3. Δ 2 = { z ; | z 1 | < 1 , | z 2 | < 1 } \Delta^{2}=\{z\in\mathbb{Z};|z_{1}|<1,|z_{2}|<1\}
  4. 0 < ε < 1 0<\varepsilon<1
  5. f f
  6. H ε H_{\varepsilon}
  7. Δ 2 \Delta^{2}
  8. F F
  9. Δ 2 \Delta^{2}
  10. F = f F=f
  11. H ε H_{\varepsilon}
  12. F F
  13. f f
  14. G \K G\K
  15. G G
  16. n 2 n≥2
  17. K K
  18. G G
  19. G \K G\K
  20. f f
  21. G G
  22. n = 1 n=1
  23. 𝐂 \mathbf{C}
  24. n n

Hasse–Arf_theorem.html

  1. 𝒪 \scriptstyle{\mathcal{O}}
  2. G s ( L / K ) = { σ G : v L ( σ a - a ) s + 1 for all a 𝒪 } . G_{s}(L/K)=\{\sigma\in G\,:\,v_{L}(\sigma a-a)\geq s+1\,\text{ for all }a\in% \mathcal{O}\}.
  3. η L / K ( s ) = 0 s d x | G 0 : G x | . \eta_{L/K}(s)=\int_{0}^{s}\frac{dx}{|G_{0}:G_{x}|}.
  4. p n p^{n}
  5. p p
  6. G ( i ) G(i)
  7. G G
  8. p n - i p^{n-i}
  9. i 0 , i 1 , , i n - 1 i_{0},i_{1},...,i_{n-1}
  10. G 0 = = G i 0 = G = G 0 = = G i 0 G_{0}=\cdots=G_{i_{0}}=G=G^{0}=\cdots=G^{i_{0}}
  11. G i 0 + 1 = = G i 0 + p i 1 = G ( 1 ) = G i 0 + 1 = = G i 0 + i 1 G_{i_{0}+1}=\cdots=G_{i_{0}+pi_{1}}=G(1)=G^{i_{0}+1}=\cdots=G^{i_{0}+i_{1}}
  12. G i 0 + p i 1 + 1 = = G i 0 + p i 1 + p 2 i 2 = G ( 2 ) = G i 0 + i 1 + 1 G_{i_{0}+pi_{1}+1}=\cdots=G_{i_{0}+pi_{1}+p^{2}i_{2}}=G(2)=G^{i_{0}+i_{1}+1}
  13. G i 0 + p i 1 + + p n - 1 i n - 1 + 1 = 1 = G i 0 + + i n - 1 + 1 . G_{i_{0}+pi_{1}+\cdots+p^{n-1}i_{n-1}+1}=1=G^{i_{0}+\cdots+i_{n-1}+1}.

Hausdorff–Young_inequality.html

  1. f f
  2. f ^ ( n ) = 1 2 π 0 2 π e - i n x f ( x ) d x , n = 0 , ± 1 , ± 2 , . \widehat{f}(n)=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-inx}f(x)\,dx,\quad n=0,\pm 1,% \pm 2,\dots.
  3. L 2 L^{2}
  4. 2 \ell^{2}
  5. | ( T f ) ( n ) | = | f ^ ( n ) | = | 1 2 π 0 2 π e - i n t f ( t ) d t | 1 2 π 0 2 π | f ( t ) | d t |(Tf)(n)|=|\widehat{f}(n)|=\left|\frac{1}{2\pi}\int_{0}^{2\pi}e^{-int}f(t)\,dt% \right|\leq\frac{1}{2\pi}\int_{0}^{2\pi}|f(t)|\,dt
  6. L 1 L^{1}
  7. \ell^{\infty}
  8. q \ell^{q}
  9. 1 p + 1 q = 1. \frac{1}{p}+\frac{1}{q}=1.
  10. ( n = - | f ^ ( n ) | q ) 1 / q ( 1 2 π 0 2 π | f ( t ) | p d t ) 1 / p . \left(\sum_{n=-\infty}^{\infty}|\widehat{f}(n)|^{q}\right)^{1/q}\leq\left(% \frac{1}{2\pi}\int_{0}^{2\pi}|f(t)|^{p}\,dt\right)^{1/p}.
  11. L p L^{p}
  12. 2 \ell^{2}
  13. f L p f\in L^{p}
  14. 1 < p 2 1<p\leq 2
  15. f ^ L q p 1 / 2 p q - 1 / 2 q f L p \|\hat{f}\|_{L^{q}}\leq p^{1/2p}q^{-1/2q}\|f\|_{L^{p}}
  16. q = p / ( p - 1 ) q=p/(p-1)
  17. p p

Hájek–Le_Cam_convolution_theorem.html

  1. I q ( θ ) - 1 = q ˙ ( θ ) I - 1 ( θ ) q ˙ ( θ ) I_{q(\theta)}^{-1}=\dot{q}(\theta)I^{-1}(\theta)\dot{q}(\theta)^{\prime}
  2. ψ q ( θ ) = q ˙ ( θ ) I - 1 ( θ ) ˙ ( θ ) \psi_{q(\theta)}=\dot{q}(\theta)I^{-1}(\theta)\dot{\ell}(\theta)
  3. ˙ ( θ ) \scriptstyle\dot{\ell}(\theta)
  4. Z θ 𝒩 ( 0 , I q ( θ ) - 1 ) \scriptstyle Z_{\theta}\,\sim\,\mathcal{N}(0,\,I^{-1}_{q(\theta)})
  5. n ( T n - q ( θ ) ) 𝑑 Z θ + Δ θ , \sqrt{n}(T_{n}-q(\theta))\ \xrightarrow{d}\ Z_{\theta}+\Delta_{\theta},
  6. ( n ( T n - q ( θ ) ) - 1 n i = 1 n ψ q ( θ ) ( x i ) 1 n i = 1 n ψ q ( θ ) ( x i ) ) 𝑑 ( Δ θ Z θ ) . \begin{pmatrix}\sqrt{n}(T_{n}-q(\theta))-\tfrac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi% _{q(\theta)}(x_{i})\\ \tfrac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi_{q(\theta)}(x_{i})\end{pmatrix}\ % \xrightarrow{d}\ \begin{pmatrix}\Delta_{\theta}\\ Z_{\theta}\end{pmatrix}.

Head_grammar.html

  1. A a b c A\to abc
  2. A ( a b c , 0 ) A\to(abc,0)
  3. A a ^ b c A\to\widehat{a}bc
  4. α x ^ β \alpha\widehat{x}\beta
  5. γ y ^ δ \gamma\widehat{y}\delta
  6. w ( α x ^ β , γ y ^ δ ) = α x γ y ^ δ β w(\alpha\widehat{x}\beta,\gamma\widehat{y}\delta)=\alpha x\gamma\widehat{y}\delta\beta
  7. α x ^ β \alpha\widehat{x}\beta
  8. γ y ^ δ \gamma\widehat{y}\delta
  9. ζ z ^ η \zeta\widehat{z}\eta
  10. c 1 , 0 ( α x ^ β ) = α x ^ β c_{1,0}(\alpha\widehat{x}\beta)=\alpha\widehat{x}\beta
  11. c 2 , 0 ( α x ^ β , γ y ^ δ ) = α x ^ β γ y δ c_{2,0}(\alpha\widehat{x}\beta,\gamma\widehat{y}\delta)=\alpha\widehat{x}\beta% \gamma y\delta
  12. c 2 , 1 ( α x ^ β , γ y ^ δ ) = α x β γ y ^ δ c_{2,1}(\alpha\widehat{x}\beta,\gamma\widehat{y}\delta)=\alpha x\beta\gamma% \widehat{y}\delta
  13. c 3 , 0 ( α x ^ β , γ y ^ δ , ζ z ^ η ) = α x ^ β γ y δ ζ z η c_{3,0}(\alpha\widehat{x}\beta,\gamma\widehat{y}\delta,\zeta\widehat{z}\eta)=% \alpha\widehat{x}\beta\gamma y\delta\zeta z\eta
  14. c 3 , 1 ( α x ^ β , γ y ^ δ , ζ z ^ η ) = α x β γ y ^ δ ζ z η c_{3,1}(\alpha\widehat{x}\beta,\gamma\widehat{y}\delta,\zeta\widehat{z}\eta)=% \alpha x\beta\gamma\widehat{y}\delta\zeta z\eta
  15. c 3 , 2 ( α x ^ β , γ y ^ δ , ζ z ^ η ) = α x β γ y δ ζ z ^ η c_{3,2}(\alpha\widehat{x}\beta,\gamma\widehat{y}\delta,\zeta\widehat{z}\eta)=% \alpha x\beta\gamma y\delta\zeta\widehat{z}\eta
  16. c m , n : 0 n < m c_{m,n}:0\leq n<m
  17. X w ( α , β ) X\to w(\alpha,\beta)
  18. X c m , n ( α , β , ) X\to c_{m,n}(\alpha,\beta,...)
  19. α \alpha
  20. β \beta
  21. { a n b n c n d n : n 0 } \{a^{n}b^{n}c^{n}d^{n}:n\geq 0\}
  22. S c 1 , 0 ( ϵ ^ ) S\to c_{1,0}(\widehat{\epsilon})
  23. S c 3 , 1 ( a ^ , T , d ^ ) S\to c_{3,1}(\widehat{a},T,\widehat{d})
  24. T w ( S , b ^ c ) T\to w(S,\widehat{b}c)
  25. S S
  26. c 3 , 1 ( a ^ , T , d ^ ) c_{3,1}(\widehat{a},T,\widehat{d})
  27. c 3 , 1 ( a ^ , w ( S , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(S,\widehat{b}c),\widehat{d})
  28. c 3 , 1 ( a ^ , w ( c 1 , 0 ( ϵ ^ ) , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(c_{1,0}(\widehat{\epsilon}),\widehat{b}c),\widehat{d})
  29. c 3 , 1 ( a ^ , w ( ϵ ^ , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(\widehat{\epsilon},\widehat{b}c),\widehat{d})
  30. c 3 , 1 ( a ^ , b ^ c , d ^ ) c_{3,1}(\widehat{a},\widehat{b}c,\widehat{d})
  31. a b ^ c d a\widehat{b}cd
  32. S S
  33. c 3 , 1 ( a ^ , T , d ^ ) c_{3,1}(\widehat{a},T,\widehat{d})
  34. c 3 , 1 ( a ^ , w ( S , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(S,\widehat{b}c),\widehat{d})
  35. c 3 , 1 ( a ^ , w ( c 3 , 1 ( a ^ , T , d ^ ) , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(c_{3,1}(\widehat{a},T,\widehat{d}),\widehat{b}c),% \widehat{d})
  36. c 3 , 1 ( a ^ , w ( c 3 , 1 ( a ^ , w ( S , b ^ c ) , d ^ ) , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(c_{3,1}(\widehat{a},w(S,\widehat{b}c),\widehat{d}),% \widehat{b}c),\widehat{d})
  37. c 3 , 1 ( a ^ , w ( c 3 , 1 ( a ^ , w ( c 1 , 0 ( ϵ ^ ) , b ^ c ) , d ^ ) , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(c_{3,1}(\widehat{a},w(c_{1,0}(\widehat{\epsilon}),% \widehat{b}c),\widehat{d}),\widehat{b}c),\widehat{d})
  38. c 3 , 1 ( a ^ , w ( c 3 , 1 ( a ^ , w ( ϵ ^ , b ^ c ) , d ^ ) , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(c_{3,1}(\widehat{a},w(\widehat{\epsilon},\widehat{b}c),% \widehat{d}),\widehat{b}c),\widehat{d})
  39. c 3 , 1 ( a ^ , w ( c 3 , 1 ( a ^ , b ^ c , d ^ ) , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(c_{3,1}(\widehat{a},\widehat{b}c,\widehat{d}),\widehat{b% }c),\widehat{d})
  40. c 3 , 1 ( a ^ , w ( a b ^ c d , b ^ c ) , d ^ ) c_{3,1}(\widehat{a},w(a\widehat{b}cd,\widehat{b}c),\widehat{d})
  41. c 3 , 1 ( a ^ , a b b ^ c c d , d ^ ) c_{3,1}(\widehat{a},ab\widehat{b}ccd,\widehat{d})
  42. a a b b ^ c c d d aab\widehat{b}ccdd

Head_injury_criterion.html

  1. 𝐻𝐼𝐶 = { [ 1 t 2 - t 1 t 1 t 2 a ( t ) d t ] 2.5 ( t 2 - t 1 ) } m a x \mathit{H}\mathit{I}\mathit{C}=\bigg\{\Big[\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{% t_{2}}a(t)dt\Big]^{2.5}\left(t_{2}-t_{1}\right)\bigg\}_{max}

Heat_generation_in_integrated_circuits.html

  1. ( κ T ) + g = ρ C T t \nabla\left(\kappa\nabla T\right)+g=\rho C\frac{\partial T}{\partial t}
  2. κ \kappa
  3. ρ \rho
  4. C C
  5. k = κ ρ C k=\frac{\kappa}{\rho C}\,
  6. g g

Heine–Stieltjes_polynomials.html

  1. d 2 S d z 2 + ( j = 1 N γ j z - a j ) d S d z + V ( z ) j = 1 N ( z - a j ) S = 0 \frac{d^{2}S}{dz^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_{j}}{z-a_{j}}\right)% \frac{dS}{dz}+\frac{V(z)}{\prod_{j=1}^{N}(z-a_{j})}S=0

Heisenberg's_entryway_to_matrix_mechanics.html

  1. Q P - P Q = i h 2 π {QP-PQ=\frac{ih}{2\pi}}
  2. C ( n , n - b ) = a A ( n , n - a ) B ( n - a , n - b ) C(n,n-b)=\sum_{a}\,A(n,n-a)B(n-a,n-b)
  3. Y ( n , n - b ) = a p ( n , n - a ) q ( n - a , n - b ) Y(n,n-b)=\sum_{a}\,p(n,n-a)q(n-a,n-b)
  4. Q P - P Q = i h 2 π {QP-PQ=\frac{ih}{2\pi}}

Heisler_chart.html

  1. T ( x , t ) - T T i - T = n = 0 [ 4 sin λ n 2 λ n + sin 2 λ n e - λ n 2 α t L 2 cos λ n x L ] \frac{T(x,t)-T_{\infty}}{T_{i}-T_{\infty}}=\sum_{n=0}^{\infty}{\left[\frac{4% \sin{\lambda_{n}}}{2\lambda_{n}+\sin{2\lambda_{n}}}e^{-\lambda_{n}^{2}\frac{% \alpha t}{L^{2}}}\cos{\frac{\lambda_{n}x}{L}}\right]}
  2. = T ( 0 , t ) - T T i - T =\frac{T(0,t)-T_{\infty}}{T_{i}-T_{\infty}}
  3. = T ( x , t ) - T T ( 0 , t ) - T =\frac{T(x,t)-T_{\infty}}{T(0,t)-T_{\infty}}

Helical_boundary_conditions.html

  1. ( i ± 1 ) mod N (i\pm 1)\mod N
  2. ( i ± L ) mod N (i\pm L)\mod N
  3. \ldots
  4. ( i ± L d - 1 ) mod N (i\pm L^{d-1})\mod N

Helical_resonator.html

  1. Q = 35.9 d f Q=35.9\cdot d\cdot\sqrt{f}
  2. Z o = 136190 d f Z_{o}=\frac{136190}{d\cdot f}
  3. h = 1.5 d h=1.5\cdot d
  4. Z o Z_{o}

Help:Musical_symbols.html

  1. 4 ^ \sharp\hat{4}

Help:Wiki_markup.html

  1. x x
  2. sin 2 π x + ln e \sin 2\pi x+\ln e
  3. s i n 2 Align p i ; x + l n e sin2&pi;x+lne
  4. 2 x × 4 y ÷ 6 z + 8 - y z 2 = 0 2x\times 4y\div 6z+8-\frac{y}{z^{2}}=0
  5. sin 2 π x + ln e \sin 2\pi x+\ln e
  6. s i n 2 π x + l n e sin2πx+lne

Hendrik_van_der_Bijl.html

  1. g m = μ r p g_{m}={\mu\over r_{p}}

Herglotz–Zagier_function.html

  1. F ( x ) = n = 1 { Γ ( n x ) Γ ( n x ) - log ( n x ) } 1 n . F(x)=\sum^{\infty}_{n=1}\left\{\frac{\Gamma^{\prime}(nx)}{\Gamma(nx)}-\log(nx)% \right\}\frac{1}{n}.

Herschel_graph.html

  1. - x 3 ( x 2 - 11 ) ( x 2 - 3 ) ( x 2 - 2 ) 2 -x^{3}(x^{2}-11)(x^{2}-3)(x^{2}-2)^{2}

Hervé_Jacquet.html

  1. G L ( 2 ) GL(2)
  2. G L ( 2 ) GL(2)
  3. G L ( 2 ) GL(2)
  4. G L ( n ) GL(n)
  5. G L ( n ) GL(n)
  6. G L ( m ) GL(m)
  7. G L ( n ) GL(n)

Herz–Schur_multiplier.html

  1. Ψ : g G μ g λ g g G ψ ( g ) μ g λ g . \Psi:~{}\sum\limits_{g\in G}\mu_{g}\lambda_{g}\mapsto\sum\limits_{g\in G}\psi(% g)\mu_{g}\lambda_{g}.

Hessenberg_variety.html

  1. h : { 1 , 2 , , n } { 1 , 2 , , n } h:\{1,2,\ldots,n\}\rightarrow\{1,2,\ldots,n\}
  2. h ( i + 1 ) max ( i , h ( i ) ) for all 1 i n - 1. h(i+1)\geq\,\text{max }(i,h(i))\,\text{ for all}1\leq i\leq n-1.
  3. h ( 1 , 2 , 3 , 4 , 5 ) = ( 2 , 3 , 3 , 4 , 5 ) h(1,2,3,4,5)=(2,3,3,4,5)\,
  4. X : \C n \C n , X:\C^{n}\rightarrow\C^{n},\,
  5. F F_{\bullet}
  6. X F i F ( h ( i ) ) X\cdot F_{i}\subseteq F_{(h(i))}
  7. F ( h ( i ) ) F_{(h(i))}
  8. h ( i ) h(i)
  9. F F_{\bullet}
  10. ( X , h ) = { F X F i F ( h i ) for 1 i n } \mathcal{H}(X,h)=\{F_{\bullet}\mid XF_{i}\subset F_{(h_{i})}\,\text{ for }1% \leq i\leq n\}
  11. h h
  12. h ( i ) = i + 1 h(i)=i+1
  13. i = 1 , 2 , , n - 1 i=1,2,\dots,n-1
  14. h ( i ) = i h(i)=i
  15. i i
  16. ρ \rho

Hidden_Markov_random_field.html

  1. Y i Y_{i}
  2. i S i\in S
  3. Y i Y_{i}
  4. X i X_{i}
  5. i S i\in S
  6. N i N_{i}
  7. X i X_{i}
  8. X i X_{i}
  9. X j X_{j}
  10. X i X_{i}
  11. X i X_{i}
  12. Y i Y_{i}

High-valent_iron.html

  1. ( L ) Fe n + OIPh ( L ) Fe n + 2 O + IPh \mathrm{(L)Fe^{n}+OIPh\longrightarrow(L)Fe^{n+2}O+IPh}
  2. ( L ) Fe n N 3 ( L ) Fe n + 2 N + N 2 \mathrm{(L)Fe^{n}N_{3}\longrightarrow(L)Fe^{n+2}N+N_{2}}

Higher-dimensional_gamma_matrices.html

  1. d d
  2. η = η a b = diag ( + 1 , - 1 , , - 1 ) , \eta=\parallel\eta_{ab}\parallel=\,\text{diag}(+1,-1,\dots,-1)~{},
  3. a , b = 0 , 1 , , d 1 a,b=0,1,...,d−1
  4. d = N = 4 d=N=4
  5. d d
  6. N × N N×N
  7. Γ i , i = 0 , , d - 1 \Gamma_{i},\ i=0,\ldots,d-1
  8. { Γ a , Γ b } = Γ a Γ b + Γ b Γ a = 2 η a b I N , \{\Gamma_{a}~{},~{}\Gamma_{b}\}=\Gamma_{a}\Gamma_{b}+\Gamma_{b}\Gamma_{a}=2% \eta_{ab}I_{N}~{},
  9. N N
  10. N N
  11. d d
  12. d d
  13. Γ 0 = + Γ 0 , Γ i = - Γ i ( i = 1 , , d - 1 ) . \Gamma_{0}^{\dagger}=+\Gamma_{0}~{},~{}\Gamma_{i}^{\dagger}=-\Gamma_{i}~{}(i=1% ,\dots,d-1)~{}.
  14. C ( + ) Γ a C ( + ) - 1 = + Γ a T C_{(+)}\Gamma_{a}C_{(+)}^{-1}=+\Gamma_{a}^{T}
  15. C ( - ) Γ a C ( - ) - 1 = - Γ a T . C_{(-)}\Gamma_{a}C_{(-)}^{-1}=-\Gamma_{a}^{T}~{}.
  16. C ± C_{\pm}
  17. C ( + ) * = C ( + ) C^{*}_{(+)}=C_{(+)}
  18. C ( - ) * = C ( - ) C^{*}_{(-)}=C_{(-)}
  19. 2 2
  20. C ( + ) T = C ( + ) ; C ( + ) 2 = 1 C^{T}_{(+)}=C_{(+)};~{}~{}~{}C^{2}_{(+)}=1
  21. C ( - ) T = - C ( - ) ; C ( - ) 2 = - 1 C^{T}_{(-)}=-C_{(-)};~{}~{}~{}C^{2}_{(-)}=-1
  22. 3 3
  23. C ( - ) T = - C ( - ) ; C ( - ) 2 = - 1 C^{T}_{(-)}=-C_{(-)};~{}~{}~{}C^{2}_{(-)}=-1
  24. 4 4
  25. C ( + ) T = - C ( + ) ; C ( + ) 2 = - 1 C^{T}_{(+)}=-C_{(+)};~{}~{}~{}C^{2}_{(+)}=-1
  26. C ( - ) T = - C ( - ) ; C ( - ) 2 = - 1 C^{T}_{(-)}=-C_{(-)};~{}~{}~{}C^{2}_{(-)}=-1
  27. 5 5
  28. C ( + ) T = - C ( + ) ; C ( + ) 2 = - 1 C^{T}_{(+)}=-C_{(+)};~{}~{}~{}C^{2}_{(+)}=-1
  29. 6 6
  30. C ( + ) T = - C ( + ) ; C ( + ) 2 = - 1 C^{T}_{(+)}=-C_{(+)};~{}~{}~{}C^{2}_{(+)}=-1
  31. C ( - ) T = C ( - ) ; C ( - ) 2 = 1 C^{T}_{(-)}=C_{(-)};~{}~{}~{}C^{2}_{(-)}=1
  32. 7 7
  33. C ( - ) T = C ( - ) ; C ( - ) 2 = 1 C^{T}_{(-)}=C_{(-)};~{}~{}~{}C^{2}_{(-)}=1
  34. 8 8
  35. C ( + ) T = C ( + ) ; C ( + ) 2 = 1 C^{T}_{(+)}=C_{(+)};~{}~{}~{}C^{2}_{(+)}=1
  36. C ( - ) T = C ( - ) ; C ( - ) 2 = 1 C^{T}_{(-)}=C_{(-)};~{}~{}~{}C^{2}_{(-)}=1
  37. 9 9
  38. C ( + ) T = C ( + ) ; C ( + ) 2 = 1 C^{T}_{(+)}=C_{(+)};~{}~{}~{}C^{2}_{(+)}=1
  39. 10 10
  40. C ( + ) T = C ( + ) ; C ( + ) 2 = 1 C^{T}_{(+)}=C_{(+)};~{}~{}~{}C^{2}_{(+)}=1
  41. C ( - ) T = - C ( - ) ; C ( - ) 2 = - 1 C^{T}_{(-)}=-C_{(-)};~{}~{}~{}C^{2}_{(-)}=-1
  42. 11 11
  43. C ( - ) T = - C ( - ) ; C ( - ) 2 = - 1 C^{T}_{(-)}=-C_{(-)};~{}~{}~{}C^{2}_{(-)}=-1
  44. Γ a b c = Γ a Γ b Γ c \Gamma_{abc\ldots}=\Gamma_{a}\cdot\Gamma_{b}\cdot\Gamma_{c}\cdots
  45. Γ a \Gamma_{a}
  46. n n
  47. d d
  48. Γ a 1 a n = 1 n ! π S n ϵ ( π ) Γ a π ( 1 ) Γ a π ( n ) , \Gamma_{a_{1}\dots a_{n}}=\frac{1}{n!}\sum_{\pi\in S_{n}}\epsilon(\pi)\Gamma_{% a_{\pi(1)}}\cdots\Gamma_{a_{\pi(n)}}~{},
  49. π π
  50. n n
  51. ϵ ϵ
  52. N N
  53. N N
  54. N N
  55. d ( d 1 ) / 2 d(d−1)/2
  56. d d
  57. Γ chir = i d / 2 - 1 Γ 0 Γ 1 Γ d - 1 , \Gamma\text{chir}=i^{d/2-1}\Gamma_{0}\Gamma_{1}\dots\Gamma_{d-1}~{},
  58. Γ Γ
  59. Γ Γ
  60. ( C Γ a 1 a n ) T = + ( C Γ a 1 a n ) ; (C\Gamma_{a_{1}\dots a_{n}})^{T}=+(C\Gamma_{a_{1}\dots a_{n}})~{};
  61. C C
  62. C ( + ) C_{(+)}
  63. C ( - ) C_{(-)}
  64. C ( + ) C_{(+)}
  65. C ( - ) C_{(-)}
  66. d d
  67. 3 3
  68. C ( - ) C_{(-)}
  69. γ a \gamma_{a}
  70. I 2 I_{2}
  71. 4 4
  72. C ( - ) C_{(-)}
  73. γ a , γ a 1 a 2 \gamma_{a}~{},~{}\gamma_{a_{1}a_{2}}
  74. I 4 , γ chir , γ chir γ a I_{4}~{},~{}\gamma\text{chir}~{},~{}\gamma\text{chir}\gamma_{a}
  75. 5 5
  76. C ( + ) C_{(+)}
  77. Γ a 1 a 2 \Gamma_{a_{1}a_{2}}
  78. I 4 , Γ a I_{4}~{},~{}\Gamma_{a}
  79. 6 6
  80. C ( - ) C_{(-)}
  81. I 8 , Γ chir Γ a 1 a 2 , Γ a 1 a 2 a 3 I_{8}~{},~{}\Gamma\text{chir}\Gamma_{a_{1}a_{2}}~{},~{}\Gamma_{a_{1}a_{2}a_{3}}
  82. Γ a , Γ chir , Γ chir Γ a , Γ a 1 a 2 \Gamma_{a}~{},~{}\Gamma\text{chir}~{},~{}\Gamma\text{chir}\Gamma_{a}~{},~{}% \Gamma_{a_{1}a_{2}}
  83. 7 7
  84. C ( - ) C_{(-)}
  85. I 8 , Γ a 1 a 2 a 3 I_{8}~{},~{}\Gamma_{a_{1}a_{2}a_{3}}
  86. Γ a , Γ a 1 a 2 \Gamma_{a}~{},~{}\Gamma_{a_{1}a_{2}}
  87. 8 8
  88. C ( + ) C_{(+)}
  89. I 16 , Γ a , Γ chir , Γ chir Γ a 1 a 2 a 3 , Γ a 1 a 4 I_{16}~{},~{}\Gamma_{a}~{},~{}\Gamma\text{chir}~{},~{}\Gamma\text{chir}\Gamma_% {a_{1}a_{2}a_{3}}~{},~{}\Gamma_{a_{1}\dots a_{4}}
  90. Γ chir Γ a , Γ a 1 a 2 , Γ chir Γ a 1 a 2 , Γ a 1 a 2 a 3 \Gamma\text{chir}\Gamma_{a}~{},~{}\Gamma_{a_{1}a_{2}}~{},~{}\Gamma\text{chir}% \Gamma_{a_{1}a_{2}}~{},~{}\Gamma_{a_{1}a_{2}a_{3}}
  91. 9 9
  92. C ( + ) C_{(+)}
  93. I 16 , Γ a , Γ a 1 a 4 , Γ a 1 a 5 I_{16}~{},~{}\Gamma_{a}~{},~{}\Gamma_{a_{1}\dots a_{4}}~{},~{}\Gamma_{a_{1}% \dots a_{5}}
  94. Γ a 1 a 2 , Γ a 1 a 2 a 3 \Gamma_{a_{1}a_{2}}~{},~{}\Gamma_{a_{1}a_{2}a_{3}}
  95. 10 10
  96. C ( - ) C_{(-)}
  97. Γ a , Γ chir , Γ chir Γ a , Γ a 1 a 2 , Γ chir Γ a 1 a 4 , Γ a 1 a 5 \Gamma_{a}~{},~{}\Gamma\text{chir}~{},~{}\Gamma\text{chir}\Gamma_{a}~{},~{}% \Gamma_{a_{1}a_{2}}~{},~{}\Gamma\text{chir}\Gamma_{a_{1}\dots a_{4}}~{},~{}% \Gamma_{a_{1}\dots a_{5}}
  98. I 32 , Γ chir Γ a 1 a 2 , Γ a 1 a 2 a 3 , Γ a 1 a 4 , Γ chir Γ a 1 a 2 a 3 I_{32}~{},~{}\Gamma\text{chir}\Gamma_{a_{1}a_{2}}~{},~{}\Gamma_{a_{1}a_{2}a_{3% }}~{},~{}\Gamma_{a_{1}\dots a_{4}}~{},~{}\Gamma\text{chir}\Gamma_{a_{1}a_{2}a_% {3}}
  99. 11 11
  100. C ( - ) C_{(-)}
  101. Γ a , Γ a 1 a 2 , Γ a 1 a 5 \Gamma_{a}~{},~{}\Gamma_{a_{1}a_{2}}~{},~{}\Gamma_{a_{1}\dots a_{5}}
  102. I 32 , Γ a 1 a 2 a 3 , Γ a 1 a 4 I_{32}~{},~{}\Gamma_{a_{1}a_{2}a_{3}}~{},~{}\Gamma_{a_{1}\dots a_{4}}
  103. Γ Γ
  104. d d
  105. k k
  106. k k
  107. γ 0 = σ 1 , γ 1 = - i σ 2 \gamma_{0}=\sigma_{1}~{},~{}\gamma_{1}=-i\sigma_{2}
  108. C ( + ) = σ 1 = C ( + ) * = s ( 2 , + ) C ( + ) T = s ( 2 , + ) C ( + ) - 1 s ( 2 , + ) = + 1 C_{(+)}=\sigma_{1}=C_{(+)}^{*}=s_{(2,+)}C_{(+)}^{T}=s_{(2,+)}C_{(+)}^{-1}% \qquad s_{(2,+)}=+1
  109. C ( - ) = i σ 2 = C ( - ) * = s ( 2 , - ) C ( - ) T = s ( 2 , - ) C ( - ) - 1 s ( 2 , - ) = - 1 . C_{(-)}=i\sigma_{2}=C_{(-)}^{*}=s_{(2,-)}C_{(-)}^{T}=s_{(2,-)}C_{(-)}^{-1}% \qquad s_{(2,-)}=-1~{}.
  110. γ γ
  111. γ chir = γ 0 γ 1 = σ 3 = γ chir . \gamma\text{chir}=\gamma_{0}\gamma_{1}=\sigma_{3}=\gamma\text{chir}^{\dagger}~% {}.
  112. C C
  113. d d
  114. a = 0 , , d 1 a=0,...,d−1
  115. c c
  116. d d
  117. Γ a = γ a σ 3 ( a = 0 , , d - 1 ) , Γ d = I ( i σ 1 ) , Γ d + 1 = I ( i σ 2 ) . \Gamma_{a^{\prime}}=\gamma_{a^{\prime}}\otimes\sigma_{3}~{}~{}~{}(a^{\prime}=0% ,\dots,d-1)~{},~{}~{}~{}\Gamma_{d}=I\otimes(i\sigma_{1})~{},~{}~{}~{}\Gamma_{d% +1}=I\otimes(i\sigma_{2})~{}.
  118. C ( + ) = c ( - ) σ 1 , C ( - ) = c ( + ) ( i σ 2 ) , C_{(+)}=c_{(-)}\otimes\sigma_{1}~{},\qquad C_{(-)}=c_{(+)}\otimes(i\sigma_{2})% ~{},
  119. C ( + ) = C ( + ) * = s ( d + 2 , + ) C ( + ) T = s ( d + 2 , + ) C ( + ) - 1 s ( d + 2 , + ) = s ( d , - ) C_{(+)}=C_{(+)}^{*}=s_{(d+2,+)}C_{(+)}^{T}=s_{(d+2,+)}C_{(+)}^{-1}\qquad s_{(d% +2,+)}=s_{(d,-)}
  120. C ( - ) = C ( - ) * = s ( d + 2 , - ) C ( - ) T = s ( d + 2 , - ) C ( - ) - 1 s ( d + 2 , - ) = - s ( d , + ) . C_{(-)}=C_{(-)}^{*}=s_{(d+2,-)}C_{(-)}^{T}=s_{(d+2,-)}C_{(-)}^{-1}\qquad s_{(d% +2,-)}=-s_{(d,+)}~{}.
  121. d d
  122. s s
  123. s s
  124. s s
  125. d = 8 k d=8k
  126. d = 8 k + 2 d=8k+2
  127. d = 8 k + 4 d=8k+4
  128. d = 8 k + 6 d=8k+6
  129. s ( d , + ) s_{(d,+)}
  130. s ( d , - ) s_{(d,-)}
  131. d d
  132. Γ chir = α d + 2 Γ 0 Γ 1 Γ d + 1 = γ chir σ 3 , α d = i d / 2 - 1 , \Gamma\text{chir}=\alpha_{d+2}\Gamma_{0}\Gamma_{1}\dots\Gamma_{d+1}=\gamma% \text{chir}\otimes\sigma_{3}~{},~{}~{}~{}~{}~{}~{}\alpha_{d}=i^{d/2-1}~{},
  133. C ( ± ) Γ chir C ( ± ) - 1 = β d + 2 Γ chir T β d = ( - ) d ( d - 1 ) / 2 . C_{(\pm)}\Gamma\text{chir}C_{(\pm)}^{-1}=\beta_{d+2}\Gamma\text{chir}^{T}~{}~{% }~{}~{}~{}~{}~{}~{}\beta_{d}=(-)^{d(d-1)/2}~{}.
  134. d d
  135. i i
  136. C ( + ) C_{(+)}
  137. C ( - ) C_{(-)}
  138. Γ Γ
  139. C ( s ) Γ chir C ( s ) - 1 = β d Γ chir T = s Γ chir T . C_{(s)}\Gamma\text{chir}C_{(s)}^{-1}=\beta_{d}\Gamma\text{chir}^{T}=s\Gamma% \text{chir}^{T}~{}.
  140. d d

Higher-dimensional_supergravity.html

  1. 𝒩 \mathcal{N}
  2. 𝒩 = 2 \mathcal{N}=2
  3. 𝒩 \mathcal{N}
  4. 2 1 + 2 d - k 2 2^{1+\lfloor{\frac{2d-k}{2}}\rfloor}
  5. x \lfloor x\rfloor
  6. - 2 k 2 ( mod 8 ) -2\leq k\leq 2\;\;(\mathop{{\rm mod}}8)
  7. 3 k 5 3\leq k\leq 5
  8. 2 n 2^{\lfloor n\rfloor}
  9. Γ 5 \Gamma_{5}
  10. Γ 5 \Gamma_{5}
  11. ± ( - 1 ) - k / 2 \pm(-1)^{-k/2}
  12. Γ 5 \Gamma_{5}
  13. Γ 5 \Gamma_{5}
  14. ± ( - 1 ) - k / 2 \pm(-1)^{-k/2}
  15. 𝒩 = ( 𝒩 L , 𝒩 R ) \mathcal{N}=(\mathcal{N}_{L},\mathcal{N}_{R})
  16. 𝒩 L \mathcal{N}_{L}
  17. 𝒩 R \mathcal{N}_{R}
  18. 𝐙 2 \mathbf{Z}_{2}
  19. 𝒩 \mathcal{N}
  20. 𝒩 \mathcal{N}
  21. 𝒩 \mathcal{N}
  22. C 2 k - 1 C_{2k-1}
  23. ρ \rho
  24. d d C 2 k - 1 = ρ . ddC_{2k-1}=\rho.\,\,\,
  25. M 10 × S 1 M^{10}\times S^{1}\,
  26. M 10 M^{10}\,
  27. G 0 G_{0}
  28. C 1 C_{1}\,
  29. C 3 C_{3}\,
  30. 𝒩 = ( 1 , 1 ) \mathcal{N}=(1,1)
  31. 𝒩 = 1 \mathcal{N}=1
  32. 𝒩 = 1 \mathcal{N}=1
  33. 𝔰 𝔬 ( 32 ) \mathfrak{so}(32)
  34. 𝔢 8 𝔢 8 \mathfrak{e}_{8}\oplus\mathfrak{e}_{8}
  35. 𝔢 8 248 𝔲 ( 1 ) \mathfrak{e}_{8}\oplus 248\mathfrak{u}(1)
  36. 496 𝔲 ( 1 ) 496\mathfrak{u}(1)
  37. L = + 1 2 κ 2 e R - 1 2 e ψ ¯ M Γ M N P D N [ 1 2 ( ω - ω ¯ ) ] ψ P + 1 48 e F M N P Q 2 + 2 κ 384 e ( ψ ¯ M Γ M N P Q R S ψ S + 12 ψ ¯ N Γ P Q ψ R ) ( F + F ¯ ) N P Q R + 2 κ 3456 ε M 1 M 11 F M 1 M 4 F M 5 M 8 A M 9 M 10 M 11 \begin{array}[]{rcl}L&=&+\frac{1}{2\kappa^{2}}eR-\frac{1}{2}e\overline{\psi}_{% M}\Gamma^{MNP}D_{N}[\frac{1}{2}(\omega-\overline{\omega})]\psi_{P}\\ &&+\frac{1}{48}eF^{2}_{MNPQ}+\frac{\sqrt{2}\kappa}{384}e(\overline{\psi}_{M}% \Gamma^{MNPQRS}\psi_{S}\\ &&+12\overline{\psi}^{N}\Gamma^{PQ}\psi^{R})(F+\overline{F})_{NPQR}+\frac{% \sqrt{2}\kappa}{3456}\varepsilon^{M_{1}\dots M_{11}}F_{M_{1}\dots M_{4}}F_{M_{% 5}\dots M_{8}}A_{M_{9}M_{10}M_{11}}\end{array}
  38. e M A , ψ M , A M N P e^{A}_{M},\psi_{M},A_{MNP}

Highly_structured_ring_spectrum.html

  1. A A_{\infty}
  2. A A_{\infty}
  3. E E_{\infty}
  4. E E_{\infty}
  5. E E_{\infty}
  6. \infty
  7. \infty
  8. Ω X × Ω X Ω X \Omega X\times\Omega X\to\Omega X
  9. A A_{\infty}
  10. A A_{\infty}
  11. A A_{\infty}
  12. E E_{\infty}
  13. A A_{\infty}
  14. E E_{\infty}
  15. Ω 2 X \Omega^{2}X
  16. \infty
  17. E E_{\infty}
  18. A A_{\infty}
  19. E E_{\infty}
  20. A A_{\infty}
  21. E E_{\infty}
  22. ( X 0 , X 1 , ) (X_{0},X_{1},\dots)
  23. Σ X i X i + 1 \Sigma X_{i}\to X_{i+1}
  24. ( S 0 , S 1 , ) (S^{0},S^{1},\dots)
  25. ( X 0 , X 1 , ) (X_{0},X_{1},\dots)
  26. X n X_{n}
  27. ( X 0 , X 1 , ) (X_{0},X_{1},\dots)
  28. X n X_{n}
  29. Σ X i X i + 1 \Sigma X_{i}\to X_{i+1}
  30. \wedge
  31. X n X m X n + m , X_{n}\wedge X_{m}\to X_{n+m},
  32. A A_{\infty}
  33. E E_{\infty}
  34. A A_{\infty}
  35. E E_{\infty}
  36. E E_{\infty}
  37. S n S m S n + m S^{n}\wedge S^{m}\to S^{n+m}
  38. E E_{\infty}
  39. E E_{\infty}
  40. A A_{\infty}
  41. E 4 E_{4}
  42. E 4 E_{4}
  43. E E_{\infty}
  44. E E_{\infty}
  45. M U B P MU\to BP
  46. E 2 E_{2}
  47. E 4 E_{4}
  48. E E_{\infty}
  49. E E_{\infty}
  50. E E_{\infty}
  51. E E_{\infty}

Hilbert_metric.html

  1. d ( A , B ) = log ( | Y A | | Y B | | X B | | X A | ) . d(A,B)=\log\left(\frac{|YA|}{|YB|}\frac{|XB|}{|XA|}\right).
  2. K \leq_{K}
  3. M ( v / w ) = inf { λ : v K λ w } , m ( v / w ) = sup { μ : μ w K v } . M(v/w)=\inf\{\lambda:v\leq_{K}\lambda w\},\quad m(v/w)=\sup\{\mu:\mu w\leq_{K}% v\}.
  4. d ( v , w ) = log M ( v / w ) m ( v / w ) . d(v,w)=\log\frac{M(v/w)}{m(v/w)}.
  5. K = { ( t , t x ) : t , x Ω } , K=\{(t,tx):t\in\mathbb{R},x\in\Omega\},

Hilbert–Samuel_function.html

  1. M M
  2. A A
  3. I I
  4. A A
  5. χ M I : \chi_{M}^{I}:\mathbb{N}\rightarrow\mathbb{N}
  6. n n\in\mathbb{N}
  7. χ M I ( n ) = ( M / I n M ) \chi_{M}^{I}(n)=\ell(M/I^{n}M)
  8. \ell
  9. A A
  10. gr I ( M ) \operatorname{gr}_{I}(M)
  11. χ M I ( n ) = i = 0 n H ( gr I ( M ) , i ) . \chi_{M}^{I}(n)=\sum_{i=0}^{n}H(\operatorname{gr}_{I}(M),i).
  12. n n
  13. dim ( gr I ( M ) ) \dim(\operatorname{gr}_{I}(M))
  14. k [ [ x , y ] ] k[[x,y]]
  15. χ ( 1 ) = 1 , χ ( 2 ) = 3 , χ ( 3 ) = 5 , χ ( 4 ) = 6 and χ ( k ) = 6 for k > 4. \chi(1)=1,\quad\chi(2)=3,\quad\chi(3)=5,\quad\chi(4)=6\,\text{ and }\chi(k)=6% \,\text{ for }k>4.
  16. P I , M P_{I,M}
  17. R / I n R/I^{n}
  18. 0 ( I n M M ) / I n M M / I n M M / I n M M ′′ / I n M ′′ 0 , 0\to(I^{n}M\cap M^{\prime})/I^{n}M^{\prime}\to M^{\prime}/I^{n}M^{\prime}\to M% /I^{n}M\to M^{\prime\prime}/I^{n}M^{\prime\prime}\to 0,
  19. χ M I ( n - 1 ) = χ M I ( n - 1 ) + χ M ′′ I ( n - 1 ) - ( ( I n M M ) / I n M ) \chi_{M}^{I}(n-1)=\chi_{M^{\prime}}^{I}(n-1)+\chi_{M^{\prime\prime}}^{I}(n-1)-% \ell((I^{n}M\cap M^{\prime})/I^{n}M^{\prime})
  20. I n M M = I n - k ( ( I k M ) M ) I n - k M . I^{n}M\cap M^{\prime}=I^{n-k}((I^{k}M)\cap M^{\prime})\subset I^{n-k}M^{\prime}.
  21. ( ( I n M M ) / I n M ) χ M I ( n - 1 ) - χ M I ( n - k - 1 ) \ell((I^{n}M\cap M^{\prime})/I^{n}M^{\prime})\leq\chi^{I}_{M^{\prime}}(n-1)-% \chi^{I}_{M^{\prime}}(n-k-1)

Histogram_matching.html

  1. F 1 ( ) F_{1}()\,
  2. F 2 ( ) F_{2}()\,
  3. G 1 [ 0 , 255 ] G_{1}\in[0,255]
  4. G 2 G_{2}\,
  5. F 1 ( G 1 ) = F 2 ( G 2 ) F_{1}(G_{1})=F_{2}(G_{2})\,
  6. M ( G 1 ) = G 2 M(G_{1})=G_{2}\,
  7. M ( ) M()\,
  8. P = { p i } i = 1 k P=\{p_{i}\}_{i=1}^{k}
  9. Q = { q i } i = 1 k Q=\{q_{i}\}_{i=1}^{k}
  10. M M
  11. m i n M k d ( M ( p k ) , q k ) min_{M}\ \sum_{k}d(M(p_{k}),q_{k})
  12. d ( . , . ) d(.,.)

History_of_centrifugal_and_centripetal_forces.html

  1. r ¨ = - k / r 2 + l 2 / r 3 \ddot{r}=-k/r^{2}+l^{2}/r^{3}

History_of_macroeconomic_thought.html

  1. M V = P Q M\cdot V=P\cdot Q

Hitchin_system.html

  1. H 1 ( E n d ( F ) ) , H^{1}(End(F)),
  2. Φ H 0 ( E n d ( F ) K ) , \Phi\in H^{0}(End(F)\otimes K),
  3. ( F , Φ ) (F,\Phi)
  4. T r ( Φ k ) , k = 1 , , r a n k ( G ) Tr(\Phi^{k})\ ,\ \ \ \ k=1,...,rank(G)
  5. H 0 ( K k ) , H^{0}(K^{\otimes k}),
  6. ( F , Φ ) (F,\Phi)

Hitchin–Thorpe_inequality.html

  1. χ ( M ) 3 2 | τ ( M ) | , \chi(M)\geq\frac{3}{2}|\tau(M)|,
  2. χ ( M ) \chi(M)
  3. M M
  4. τ ( M ) \tau(M)
  5. M M
  6. ( M , g ) (M,g)
  7. χ ( M ) = 3 2 | τ ( M ) | , \chi(M)=\frac{3}{2}|\tau(M)|,
  8. ( M , g ) (M,g)
  9. χ ( M ) > 3 2 | τ ( M ) | . \chi(M)>\frac{3}{2}|\tau(M)|.

Hodge_bundle.html

  1. g \mathcal{M}_{g}
  2. g \mathcal{M}_{g}
  3. g \mathcal{M}_{g}
  4. π : 𝒞 g g \pi:\mathcal{C}_{g}\rightarrow\mathcal{M}_{g}
  5. Λ g = π * ω g . \Lambda_{g}=\pi_{*}\omega_{g}.\,

Hoeffding's_independence_test.html

  1. H = ( F 12 - F 1 F 2 ) 2 d F 12 H=\int(F_{12}-F_{1}F_{2})^{2}\,dF_{12}\!
  2. F 12 F_{12}
  3. F 1 F_{1}
  4. F 2 F_{2}
  5. H H
  6. H H
  7. F 12 F_{12}
  8. F 12 = F 1 F 2 F_{12}=F_{1}F_{2}

Hoeffding's_lemma.html

  1. 𝐄 [ e λ X ] exp ( λ 2 ( b - a ) 2 8 ) . \mathbf{E}\left[e^{\lambda X}\right]\leq\exp\left(\frac{\lambda^{2}(b-a)^{2}}{% 8}\right).
  2. e λ x e^{\lambda x}
  3. e λ x b - x b - a e λ a + x - a b - a e λ b a x b e^{\lambda x}\leq\frac{b-x}{b-a}e^{\lambda a}+\frac{x-a}{b-a}e^{\lambda b}% \qquad\forall a\leq x\leq b
  4. 𝐄 [ e λ X ] b - E X b - a e λ a + E X - a b - a e λ b . \mathbf{E}\left[e^{\lambda X}\right]\leq\frac{b-EX}{b-a}e^{\lambda a}+\frac{EX% -a}{b-a}e^{\lambda b}.
  5. h = λ ( b - a ) h=\lambda(b-a)
  6. p = - a b - a p=\frac{-a}{b-a}
  7. L ( h ) = - h p + ln ( 1 - p + p e h ) L(h)=-hp+\ln(1-p+pe^{h})
  8. b - E X b - a e λ a + E X - a b - a e λ b = e L ( h ) \frac{b-EX}{b-a}e^{\lambda a}+\frac{EX-a}{b-a}e^{\lambda b}=e^{L(h)}
  9. E X = 0 EX=0
  10. L ( h ) L(h)
  11. L ( 0 ) = L ( 0 ) = 0 and L ′′ ( h ) 1 4 L(0)=L^{{}^{\prime}}(0)=0\,\text{ and }L^{{}^{\prime\prime}}(h)\leq\frac{1}{4}
  12. L ( h ) 1 8 h 2 = 1 8 λ 2 ( b - a ) 2 L(h)\leq\frac{1}{8}h^{2}=\frac{1}{8}\lambda^{2}(b-a)^{2}
  13. 𝐄 [ e λ X ] e 1 8 λ 2 ( b - a ) 2 \mathbf{E}\left[e^{\lambda X}\right]\leq e^{\frac{1}{8}\lambda^{2}(b-a)^{2}}

Hoffman_graph.html

  1. ( x - 4 ) ( x - 2 ) 4 x 6 ( x + 2 ) 4 ( x + 4 ) (x-4)(x-2)^{4}x^{6}(x+2)^{4}(x+4)

Holmgren's_uniqueness_theorem.html

  1. α = { α 1 , , α n } 𝒩 0 n , \alpha=\{\alpha_{1},\dots,\alpha_{n}\}\in\mathcal{N}_{0}^{n},
  2. 𝒩 0 \mathcal{N}_{0}
  3. | α | = α 1 + + α n |\alpha|=\alpha_{1}+\cdots+\alpha_{n}
  4. x α = ( x 1 ) α 1 ( x n ) α n \partial_{x}^{\alpha}=\left(\frac{\partial}{\partial x_{1}}\right)^{\alpha_{1}% }\cdots\left(\frac{\partial}{\partial x_{n}}\right)^{\alpha_{n}}\,
  5. Ω \Omega\,
  6. \R n \R^{n}\,
  7. Σ \Sigma\,
  8. Ω \Omega\,
  9. Ω + \Omega_{+}\,
  10. Ω - \Omega_{-}\,
  11. Ω \Omega\,
  12. Σ \Sigma\,
  13. Ω = Ω - Σ Ω + \Omega=\Omega_{-}\cup\Sigma\cup\Omega_{+}\,
  14. P = | α | m A α ( x ) x α P=\sum_{|\alpha|\leq m}A_{\alpha}(x)\partial_{x}^{\alpha}\,
  15. Σ \Sigma\,
  16. P P\,
  17. Char P N * Σ = \mathop{\rm Char}P\cap N^{*}\Sigma=\emptyset
  18. Char P = { ( x , ξ ) T * \R n \ 0 : σ p ( P ) ( x , ξ ) = 0 } , with σ p ( x , ξ ) = | α | = m i | α | A α ( x ) ξ α \mathop{\rm Char}P=\{(x,\xi)\subset T^{*}\R^{n}\backslash 0:\sigma_{p}(P)(x,% \xi)=0\},\,\text{ with }\sigma_{p}(x,\xi)=\sum_{|\alpha|=m}i^{|\alpha|}A_{% \alpha}(x)\xi^{\alpha}\,
  19. P P\,
  20. N * Σ N^{*}\Sigma\,
  21. Σ \Sigma\,
  22. N * Σ = { ( x , ξ ) T * \R n : x Σ , ξ | T x Σ = 0 } N^{*}\Sigma=\{(x,\xi)\in T^{*}\R^{n}:x\in\Sigma,\,\xi|_{T_{x}\Sigma}=0\}\,
  23. u u\,
  24. Ω \Omega\,
  25. P u = 0 Pu=0\,
  26. Ω \Omega\,
  27. u u\,
  28. Ω - \Omega_{-}\,
  29. Σ \Sigma\,
  30. t m u = F ( t , x , x α t k u ) , α 𝒩 0 n , k 𝒩 0 , | α | + k m , k m - 1 , \partial_{t}^{m}u=F(t,x,\partial_{x}^{\alpha}\,\partial_{t}^{k}u),\quad\alpha% \in\mathcal{N}_{0}^{n},\quad k\in\mathcal{N}_{0},\quad|\alpha|+k\leq m,\quad k% \leq m-1,
  31. t k u | t = 0 = ϕ k ( x ) , 0 k m - 1 , \partial_{t}^{k}u|_{t=0}=\phi_{k}(x),\qquad 0\leq k\leq m-1,
  32. F ( t , x , z ) F(t,x,z)\,
  33. t = 0 , x = 0 , z = 0 t=0,x=0,z=0\,
  34. ϕ k ( x ) \phi_{k}(x)\,
  35. x = 0 x=0\,
  36. u ( t , x ) u(t,x)\,
  37. ( t , x ) = ( 0 , 0 ) ( \R × \R n ) (t,x)=(0,0)\in(\R\times\R^{n})\,
  38. F ( t , x , z ) F(t,x,z)\,
  39. z z\,
  40. t m u = F ( t , x , x α t k u ) = α 𝒩 0 n , 0 k m - 1 , | α | + k m A α , k ( t , x ) x α t k u , \partial_{t}^{m}u=F(t,x,\partial_{x}^{\alpha}\,\partial_{t}^{k}u)=\sum_{\alpha% \in\mathcal{N}_{0}^{n},0\leq k\leq m-1,|\alpha|+k\leq m}A_{\alpha,k}(t,x)\,% \partial_{x}^{\alpha}\,\partial_{t}^{k}u,\,
  41. u u\,

Holt_graph.html

  1. ( x 3 - 6 x + 2 ) 6 ( x + 2 ) 4 ( x - 1 ) 4 ( x - 4 ) . (x^{3}-6x+2)^{6}(x+2)^{4}(x-1)^{4}(x-4).

Homogeneous_coordinate_ring.html

  1. d = 0 H 0 ( V , L d ) \bigoplus_{d=0}^{\infty}H^{0}(V,L^{d})

Homogeneous_distribution.html

  1. S ( t x ) = t m S ( x ) S(tx)=t^{m}S(x)\,
  2. μ t : x x / t \mu_{t}:x\mapsto x/t
  3. S [ t - n φ μ t ] = t m S [ φ ] S[t^{-n}\varphi\circ\mu_{t}]=t^{m}S[\varphi]
  4. S x i \frac{\partial S}{\partial x_{i}}
  5. i = 1 n x i S x i = α S . \sum_{i=1}^{n}x_{i}\frac{\partial S}{\partial x_{i}}=\alpha S.
  6. δ ( t x ) φ ( x ) d x = δ ( y ) φ ( y / t ) d y t = t - 1 φ ( 0 ) \int_{\mathbb{R}}\delta(tx)\varphi(x)\,dx=\int_{\mathbb{R}}\delta(y)\varphi(y/% t)\,\frac{dy}{t}=t^{-1}\varphi(0)
  7. x + α = { x α if x > 0 0 otherwise x_{+}^{\alpha}=\begin{cases}x^{\alpha}&\,\text{if }x>0\\ 0&\,\text{otherwise}\end{cases}
  8. x - α = ( - x ) + α x_{-}^{\alpha}=(-x)_{+}^{\alpha}
  9. | x | α = x + α + x - α |x|^{\alpha}=x_{+}^{\alpha}+x_{-}^{\alpha}
  10. x + α x_{+}^{\alpha}
  11. α x + α \alpha\mapsto x_{+}^{\alpha}
  12. x + α [ φ μ t ] = t α + 1 x + α [ φ ] x_{+}^{\alpha}[\varphi\circ\mu_{t}]=t^{\alpha+1}x_{+}^{\alpha}[\varphi]
  13. d d x x + α = α x + α - 1 \frac{d}{dx}x_{+}^{\alpha}=\alpha x_{+}^{\alpha-1}
  14. x x + α = x + α + 1 xx_{+}^{\alpha}=x_{+}^{\alpha+1}
  15. χ + α = x + α Γ ( 1 + α ) . \chi_{+}^{\alpha}=\frac{x_{+}^{\alpha}}{\Gamma(1+\alpha)}.
  16. χ + - k = δ ( k - 1 ) . \chi_{+}^{-k}=\delta^{(k-1)}.
  17. χ + a \chi_{+}^{a}
  18. d d x χ + α = χ + α - 1 \frac{d}{dx}\chi_{+}^{\alpha}=\chi_{+}^{\alpha-1}
  19. x χ + α = α χ + α + 1 . x\chi_{+}^{\alpha}=\alpha\chi_{+}^{\alpha+1}.
  20. x ¯ k \underline{x}^{k}
  21. x ¯ - k \underline{x}^{-k}
  22. x ¯ - k = ( - 1 ) k - 1 ( k - 1 ) ! d k d x k log | x | . \underline{x}^{-k}=\frac{(-1)^{k-1}}{(k-1)!}\frac{d^{k}}{dx^{k}}\log|x|.
  23. d d x x ¯ - k = - k x ¯ - k - 1 \frac{d}{dx}\underline{x}^{-k}=-k\underline{x}^{-k-1}
  24. x x ¯ - k = x ¯ - k + 1 , if k > 1. x\underline{x}^{-k}=\underline{x}^{-k+1},\quad\,\text{if }k>1.
  25. x ¯ - k = - ϕ ( x ) - j = 0 k - 1 x j ϕ ( j ) ( 0 ) / j ! x k d x , \underline{x}^{-k}=\int_{-\infty}^{\infty}\frac{\phi(x)-\sum_{j=0}^{k-1}x^{j}% \phi^{(j)}(0)/j!}{x^{k}}\,dx,
  26. ( x + i 0 ) α = lim ϵ 0 ( x + i ϵ ) α . (x+i0)^{\alpha}=\lim_{\epsilon\downarrow 0}(x+i\epsilon)^{\alpha}.
  27. ( x + i 0 ) α [ φ ] = lim ϵ 0 ( x + i ϵ ) α φ ( x ) d x . (x+i0)^{\alpha}[\varphi]=\lim_{\epsilon\downarrow 0}\int_{\mathbb{R}}(x+i% \epsilon)^{\alpha}\varphi(x)\,dx.
  28. ( x - i 0 ) α = lim ϵ 0 ( x - i ϵ ) α (x-i0)^{\alpha}=\lim_{\epsilon\downarrow 0}(x-i\epsilon)^{\alpha}
  29. ( x ± i 0 ) α = x + α + e ± i π α x - α , (x\pm i0)^{\alpha}=x_{+}^{\alpha}+e^{\pm i\pi\alpha}x_{-}^{\alpha},
  30. d d x ( x ± i 0 ) α = α ( x ± i 0 ) α - 1 . \frac{d}{dx}(x\pm i0)^{\alpha}=\alpha(x\pm i0)^{\alpha-1}.
  31. ( x ± i 0 ) - k = x + - k + ( - 1 ) k x - - k ± π i δ ( k - 1 ) ( k - 1 ) ! , (x\pm i0)^{-k}=x_{+}^{-k}+(-1)^{k}x_{-}^{-k}\pm\pi i\frac{\delta^{(k-1)}}{(k-1% )!},
  32. x ¯ - k \underline{x}^{-k}
  33. ( x + i 0 ) - k + ( x - i 0 ) - k 2 = x ¯ - k . \frac{(x+i0)^{-k}+(x-i0)^{-k}}{2}=\underline{x}^{-k}.
  34. ( x + i 0 ) - k - ( x - i 0 ) - k = 2 π i δ ( k - 1 ) ( k - 1 ) ! , (x+i0)^{-k}-(x-i0)^{-k}=2\pi i\frac{\delta^{(k-1)}}{(k-1)!},
  35. S = a x + α + b x - α S=ax_{+}^{\alpha}+bx_{-}^{\alpha}
  36. a x ¯ - k + b δ ( k - 1 ) . a\underline{x}^{-k}+b\delta^{(k-1)}.

Homological_integration.html

  1. T , α . \langle T,\alpha\rangle.
  2. d : Ω k - 1 Ω k d:\Omega^{k-1}\to\Omega^{k}
  3. : D k D k - 1 \partial:D^{k}\to D^{k-1}
  4. T , α = T , d α \langle\partial T,\alpha\rangle=\langle T,d\alpha\rangle

Homotopy_analysis_method.html

  1. 𝒩 [ u ( x ) ] = 0 \mathcal{N}[u(x)]=0
  2. 𝒩 \mathcal{N}
  3. \mathcal{L}
  4. ( 1 - q ) [ U ( x ; q ) - u 0 ( x ; q ) ] = c 0 q 𝒩 [ U ( x ; q ) ] , (1-q)\mathcal{L}[U(x;q)-u_{0}(x;q)]=c_{0}\,q\,\mathcal{N}[U(x;q)],
  5. [ U ( x ; q ) - u 0 ( x ; q ) ] = 0 , \mathcal{L}[U(x;q)-u_{0}(x;q)]=0,
  6. 𝒩 [ u ( x ) ] = 0 \mathcal{N}[u(x)]=0
  7. U ( x ; q ) = u 0 ( x ) + m = 1 u m ( x ) q m . U(x;q)=u_{0}(x)+\sum_{m=1}^{\infty}u_{m}(x)\,q^{m}.
  8. u ( x ) = u 0 ( x ) + m = 1 u m ( x ) . u(x)=u_{0}(x)+\sum_{m=1}^{\infty}u_{m}(x).
  9. [ u m ( x ) - χ m u m - 1 ( x ) ] = c 0 R m [ u 0 , u 1 , , u m - 1 ] , \mathcal{L}[u_{m}(x)-\chi_{m}u_{m-1}(x)]=c_{0}\,R_{m}[u_{0},u_{1},\cdots,u_{m-% 1}],
  10. χ 1 = 0 \chi_{1}=0
  11. χ k = 1 \chi_{k}=1
  12. \mathcal{L}

Horizontal_form.html

  1. α \alpha
  2. α ( v 1 , , v r ) = 0 \alpha(v_{1},...,v_{r})=0
  3. v 1 , v r v_{1},v_{r}

Horton_graph.html

  1. ( x - 3 ) ( x - 1 ) 14 x 4 ( x + 1 ) 14 ( x + 3 ) ( x 2 - 5 ) 3 ( x 2 - 3 ) 11 ( x 2 - x - 3 ) ( x 2 + x - 3 ) (x-3)(x-1)^{14}x^{4}(x+1)^{14}(x+3)(x^{2}-5)^{3}(x^{2}-3)^{11}(x^{2}-x-3)(x^{2% }+x-3)
  2. ( x 10 - 23 x 8 + 188 x 6 - 644 x 4 + 803 x 2 - 101 ) 2 (x^{10}-23x^{8}+188x^{6}-644x^{4}+803x^{2}-101)^{2}
  3. ( x 10 - 20 x 8 + 143 x 6 - 437 x 4 + 500 x 2 - 59 ) (x^{10}-20x^{8}+143x^{6}-437x^{4}+500x^{2}-59)

Hosmer–Lemeshow_test.html

  1. H = g = 1 G ( O g - E g ) 2 N g π g ( 1 - π g ) . H=\sum_{g=1}^{G}\frac{(O_{g}-E_{g})^{2}}{N_{g}\pi_{g}(1-\pi_{g})}.\,\!
  2. χ 2 \chi^{2}

Huber_loss.html

  1. f f
  2. L δ ( a ) = { 1 2 a 2 for | a | δ , δ ( | a | - 1 2 δ ) , otherwise. L_{\delta}(a)=\begin{cases}\frac{1}{2}{a^{2}}&\,\text{for }|a|\leq\delta,\\ \delta(|a|-\frac{1}{2}\delta),&\,\text{otherwise.}\end{cases}
  3. a a
  4. | a | = δ |a|=\delta
  5. a a
  6. a = y - f ( x ) a=y-f(x)
  7. L δ ( y , f ( x ) ) = { 1 2 ( y - f ( x ) ) 2 for | y - f ( x ) | δ , δ | y - f ( x ) | - 1 2 δ 2 otherwise. L_{\delta}(y,f(x))=\begin{cases}\frac{1}{2}(y-f(x))^{2}&\textrm{for }|y-f(x)|% \leq\delta,\\ \delta\,|y-f(x)|-\frac{1}{2}\delta^{2}&\textrm{otherwise.}\end{cases}
  8. L ( a ) = a 2 L(a)=a^{2}
  9. L ( a ) = | a | L(a)=|a|
  10. a = 0 a=0
  11. [ - 1 + 1 ] [-1+1]
  12. a a
  13. i = 1 n L ( a i ) \sum_{i=1}^{n}L(a_{i})
  14. a = 0 a=0
  15. a = - δ a=-\delta
  16. a = δ a=\delta
  17. L δ ( a ) = δ 2 ( 1 + ( a / δ ) 2 - 1 ) . L_{\delta}(a)=\delta^{2}(\sqrt{1+(a/\delta)^{2}}-1).
  18. a 2 / 2 a^{2}/2
  19. a a
  20. δ \delta
  21. a a
  22. f ( x ) f(x)
  23. y { + 1 , - 1 } y\in\{+1,-1\}
  24. L ( y , f ( x ) ) = { max ( 0 , 1 - y f ( x ) ) 2 for y f ( x ) - 1 , - 4 y f ( x ) otherwise. L(y,f(x))=\begin{cases}\max(0,1-y\,f(x))^{2}&\textrm{for }\,\,y\,f(x)\geq-1,\\ -4y\,f(x)&\textrm{otherwise.}\end{cases}
  25. max ( 0 , 1 - y f ( x ) ) \max(0,1-y\,f(x))
  26. L L

Hudde's_rules.html

  1. a 0 x n + a 1 x n - 1 + + a n - 1 x + a n = 0 a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}=0\,
  2. b 0 , b 1 , , b n - 1 , b n b_{0},b_{1},\dots,b_{n-1},b_{n}
  3. a 0 b 0 x n + a 1 b 1 x n - 1 + + a n - 1 b n - 1 x + a n b n = 0. a_{0}b_{0}x^{n}+a_{1}b_{1}x^{n-1}+\cdots+a_{n-1}b_{n-1}x+a_{n}b_{n}=0.\,
  4. a 0 x n + a 1 x n - 1 + + a n - 1 x + a n a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}\,
  5. n a 0 x n + ( n - 1 ) a 1 x n - 1 + + 2 a n - 2 x 2 + a n - 1 x = 0 na_{0}x^{n}+(n-1)a_{1}x^{n-1}+\cdots+2a_{n-2}x^{2}+a_{n-1}x=0\,

Human_mitochondrial_molecular_clock.html

  1. r a t e = S N P s ( 2 T C H L C A 16553 ) rate=\frac{SNPs}{(2T_{CHLCA}16553)}

Hunt–McIlroy_algorithm.html

  1. P i j = { 0 if i = 0 or j = 0 1 + P i - 1 , j - 1 if A i = B j m a x ( P i - 1 , j , P i , j - 1 ) if A i B j P_{ij}=\begin{cases}0&\mbox{ if }~{}\ i=0\mbox{ or }~{}j=0\\ 1+P_{i-1,j-1}&\mbox{ if }~{}A_{i}=B_{j}\\ max(P_{i-1,j},P_{i,j-1})&\mbox{ if }~{}A_{i}\neq B_{j}\\ \end{cases}
  2. O ( m n ) O(mn)
  3. O ( m n log m ) O(mn\log m)
  4. O ( m n ) O(mn)

Hurwitz's_theorem_(complex_analysis).html

  1. f n ( z ) = z - 1 + 1 n , z f_{n}(z)=z-1+\frac{1}{n},\qquad z\in\mathbb{C}
  2. f k ( z ) f k ( z ) f ( z ) f ( z ) . \frac{f_{k}^{\prime}(z)}{f_{k}(z)}\to\frac{f^{\prime}(z)}{f(z)}.
  3. m = 1 2 π i | z - z 0 | = ρ f ( z ) f ( z ) d z = lim k 1 2 π i | z - z 0 | = ρ f k ( z ) f k ( z ) d z = lim k N k m=\frac{1}{2\pi i}\int_{|z-z_{0}|=\rho}\frac{f^{\prime}(z)}{f(z)}\,dz=\lim_{k% \to\infty}\frac{1}{2\pi i}\int_{|z-z_{0}|=\rho}\frac{f^{\prime}_{k}(z)}{f_{k}(% z)}\,dz=\lim_{k\to\infty}N_{k}

Hurwitz's_theorem_(composition_algebras).html

  1. q ( x ) q(x)
  2. z ( x , y ) z(x,y)
  3. q ( x ) q ( y ) = q ( z ( x , y ) ) q(x)q(y)=q(z(x,y))
  4. A A
  5. q q
  6. q ( a b ) = q ( a ) q ( b ) q(a b)=q(a)q(b)
  7. q q
  8. ( a , b ) = 1 2 q q ( a + b ) q ( a ) q ( b ) (a,b)=\frac{1}{2}qq(a+b)−q(a)−q(b)
  9. A A
  10. A A
  11. a a
  12. A A
  13. a * = - a + 2 ( a , 1 ) 1 , L ( a ) b = a b , R ( a ) b = b a . \displaystyle{a^{*}=-a+2(a,1)1,\,\,\,L(a)b=ab,\,\,\,R(a)b=ba.}
  14. ( a b ) * = b * a * (a b)*=b*a*
  15. L ( a * ) = L ( a ) * L(a*)=L(a)*
  16. R ( a * ) = R ( a ) * R(a*)=R(a)*
  17. R e ( a b ) = R e ( b a ) Re(a b)=Re(b a)
  18. R e x = ( x + x * ) / 2 = ( x , 1 ) 1 Rex=(x+x*)/2=(x, 1)1
  19. R e ( a b ) c = R e a ( b c ) Re(a b)c=Rea(b c)
  20. A A
  21. ( a b , a b ) = ( a , a ) ( b , b ) (a b,a b)=(a,a)(b,b)
  22. 2 ( a , b ) ( c , d ) = ( a c , b d ) + ( a d , b c ) . \displaystyle{2(a,b)(c,d)=(ac,bd)+(ad,bc).}
  23. b = 1 b=1
  24. d = 1 d=1
  25. L ( a * ) = L ( a ) * L(a*)=L(a)*
  26. R ( c * ) = R ( c ) * R(c*)=R(c)*
  27. R e ( a b ) = ( a b , 1 ) 1 = ( a , b * ) 1 = ( b a , 1 ) 1 = R e ( b a ) Re(a b)=(a b, 1)1=(a,b*)1=(b a, 1)1=Re(b a)
  28. R e ( a b ) c = ( ( a b ) c , 1 ) 1 = ( a b , c * ) 1 = ( b , a * c * ) 1 = ( b c , a * ) 1 = ( a ( b c ) , 1 ) 1 = R e a ( b c ) Re(a b)c=((a b)c,1)1=(a b,c*)1=(b,a*c*)1=(bc,a*)1=(a(bc),1)1=Rea(b c)
  29. ( ( a b ) * , c ) = ( a b , c * ) = ( b , a * c * ) = ( 1 , b * ( a * c * ) ) = ( 1 , ( b * a * ) c * ) = ( b * a * , c ) ((ab)*,c)=(ab,c*)=(b,a*c*)=(1,b*(a*c*))=(1,(b*a*)c*)=(b*a*,c)
  30. ( a b ) * = b * a * (ab)*=b*a*
  31. a a
  32. a * a*
  33. a * a*
  34. L ( a * ) L ( a ) = L ( a * a ) L(a*)L(a)=L(a*a)
  35. 𝐑 \mathbf{R}
  36. 𝐂 \mathbf{C}
  37. 𝐇 \mathbf{H}
  38. 𝐑 𝐂 𝐇 \mathbf{R}⊂\mathbf{C}⊂\mathbf{H}
  39. A A
  40. B B
  41. j j
  42. A A
  43. B B
  44. ( j , 1 ) = 0 (j, 1)=0
  45. j * = j j*=−j
  46. C C
  47. B B
  48. j j
  49. C = B B j , ( a + b j ) * = a * - b j , ( a + b j ) ( c + d j ) = ( a c - d * b ) + ( b c * + d a ) j . \displaystyle{C=B\oplus Bj,\,\,\,(a+bj)^{*}=a^{*}-bj,\,\,\,(a+bj)(c+dj)=(ac-d^% {*}b)+(bc^{*}+da)j.}
  50. B B
  51. B j B j
  52. j j
  53. B B
  54. a a
  55. B B
  56. j a = a * j j a=a*j
  57. 0 = 2 ( j , a * ) = j a a * j 0=2(j,a*)=j a−a*j
  58. B B j B⊕B j
  59. B j = j B Bj=j B
  60. B j B j
  61. ( b j ) * = b j (b j)*=−b j
  62. b ( c j ) = ( c b ) j b(c j)=(c b)j
  63. ( b , j ) = 0 (b,j)=0
  64. x x
  65. A A
  66. ( b ( c j ) , x ) = ( b ( j x ) , j ( c j ) = ( b ( j x ) , c * ) = ( c b , ( j x ) * ) = ( ( c b ) j , x * ) = ( ( c b ) j , x ) (b(c j),x)=(b(j x),j(c j)=−(b(j x),c*)=−(c b, (j x)*)=−((c b)j,x*)=((c b)j,x)
  67. ( j c ) b = j ( b c ) (j c)b=j(b c)
  68. ( b j ) ( c j ) = c * b (b j)(c j)=−c*b
  69. ( b , c j ) (b,c j)
  70. x x
  71. A A
  72. ( ( b j ) ( c j ) , x ) = ( ( c j ) x * , b j ) = ( b x * , ( c j ) j ) = ( c * b , x ) ((b j)(c j),x)=−((c j)x*,b j)=(b x*, (c j)j)=−(c*b,x)
  73. C C
  74. a + b j a+b j
  75. c + d j c+d j
  76. ( a 2 + b 2 ) ( c 2 + d 2 ) = a c - d * b 2 + b c * + d a 2 , \displaystyle{(\|a\|^{2}+\|b\|^{2})(\|c\|^{2}+\|d\|^{2})=\|ac-d^{*}b\|^{2}+\|% bc^{*}+da\|^{2},}
  77. ( a c , d * b ) = ( b c * , d a ) . \displaystyle{(ac,d^{*}b)=(bc^{*},da).}
  78. d ( a c ) = ( d a ) c d(a c)=(d a)c
  79. B B
  80. 𝐑 \mathbf{R}
  81. 𝐂 \mathbf{C}
  82. 𝐂 \mathbf{C}
  83. 𝐇 \mathbf{H}
  84. 𝐎 = 𝐇 𝐇 \mathbf{O}=\mathbf{H}⊕\mathbf{H}
  85. 𝐉 = ( 0 , 1 ) \mathbf{J}=(0, 1)
  86. A A
  87. 𝐑 \mathbf{R}
  88. 𝐑 \mathbf{R}
  89. 𝐂 \mathbf{C}
  90. 𝐂 \mathbf{C}
  91. 𝐇 \mathbf{H}
  92. 𝐎 \mathbf{O}
  93. 𝐎 \mathbf{O}
  94. 𝐇 \mathbf{H}
  95. a ( b j ) = ( b a ) j ( a b ) j a(b j)=(b a)j≠(a b)j
  96. 𝐎 \mathbf{O}
  97. N N
  98. A A
  99. L ( a ) L(a)
  100. ( a , 1 ) = 0 (a, 1)=0
  101. a a
  102. L ( a ) L(a)
  103. I −I
  104. N N
  105. A A
  106. N N
  107. N N
  108. N 1 N−1
  109. N N
  110. N N
  111. U i 2 = - I , U i U j = - U j U i ( i j ) . \displaystyle{U_{i}^{2}=-I,\,\,\,U_{i}U_{j}=-U_{j}U_{i}\,\,(i\neq j).}
  112. N 1 N−1
  113. N N
  114. ( x 1 2 + + x N 2 ) ( y 1 2 + + y N 2 ) = z 1 2 + + z N 2 , \displaystyle{(x_{1}^{2}+\cdots+x_{N}^{2})(y_{1}^{2}+\cdots+y_{N}^{2})=z_{1}^{% 2}+\cdots+z_{N}^{2},}
  115. x x
  116. y y
  117. z i = j = 1 N a i j ( x ) y j \displaystyle{z_{i}=\sum_{j=1}^{N}a_{ij}(x)y_{j}}
  118. x x
  119. T ( x ) T ( x ) t = x 1 2 + + x N 2 . \displaystyle{T(x)T(x)^{t}=x_{1}^{2}+\cdots+x_{N}^{2}.}
  120. T ( x ) = T 1 x 1 + + T N x N , \displaystyle{T(x)=T_{1}x_{1}+\cdots+T_{N}x_{N},}
  121. T i T j t + T j T i t = 2 δ i j I . \displaystyle{T_{i}T^{t}_{j}+T_{j}T_{i}^{t}=2\delta_{ij}I.}
  122. V i 2 = - I , V i V j = - V j V i ( i j ) . \displaystyle{V_{i}^{2}=-I,\,\,\,V_{i}V_{j}=-V_{j}V_{i}\,\,(i\neq j).}
  123. I −I
  124. N N
  125. G G
  126. v i 2 = ε , v i v j = ε v j v i ( i j ) , \displaystyle{v_{i}^{2}=\varepsilon,\,\,\,v_{i}v_{j}=\varepsilon v_{j}v_{i}\,% \,(i\neq j),}
  127. ε ε
  128. G G , G GG,G
  129. ε ε
  130. N N
  131. N N
  132. ε γ εγ
  133. g g
  134. G G
  135. g g
  136. ε g ε g
  137. N N
  138. N N
  139. G G
  140. N N
  141. | G | |G|
  142. | G | |G|
  143. N N
  144. N N
  145. G G
  146. N ≤N
  147. N N
  148. N = 6 N=6
  149. A A
  150. n n
  151. n n
  152. A A
  153. ( x i j ) * = ( x j i * ) . \displaystyle{(x_{ij})^{*}=(x_{ji}^{*}).}
  154. T r ( X ) Tr(X)
  155. X X
  156. Tr 𝐑 X Y = Tr 𝐑 Y X , Tr 𝐑 ( X Y ) Z = Tr 𝐑 X ( Y Z ) . \displaystyle{\mathrm{Tr}_{\mathbf{R}}\,XY=\mathrm{Tr}_{\mathbf{R}}\,YX,\,\,\,% \mathrm{Tr}_{\mathbf{R}}\,(XY)Z=\mathrm{Tr}_{\mathbf{R}}\,X(YZ).}
  157. n = 1 n=1
  158. A A
  159. [ a , b , c ] = a ( b c ) - ( a b ) c . \displaystyle{[a,b,c]=a(bc)-(ab)c.}
  160. A A
  161. A A
  162. a a , a , b = 00 aa,a,b=00
  163. b b , a , a = 00 bb,a,a=00
  164. a a
  165. b b
  166. c c
  167. 𝐑 \mathbf{R}
  168. a a , b , c = 00 aa,b,c=00
  169. X X
  170. [ X , X 2 ] = a I , \displaystyle{[X,X^{2}]=aI,}
  171. a a
  172. A A
  173. y i j = k , [ x i k , x k , x j ] . \displaystyle{y_{ij}=\sum_{k,\ell}[x_{ik},x_{k\ell},x_{\ell j}].}
  174. X X
  175. Y Y
  176. Y Y
  177. X X
  178. Y Y
  179. X Y = 1 2 ( X Y + Y X ) X∘Y=\frac{1}{2}(X Y+Y X)
  180. A A
  181. n 3 n≥3
  182. A A
  183. n = 3 n=3
  184. ( Z X , Y ) = ( X , Z Y ) (Z∘X,Y)=(X,Z∘Y)
  185. L ( X ) L(X)
  186. L ( X ) Y = X Y L(X)Y=X∘Y
  187. [ L ( X ) , L ( X 2 ) ] = 0. \displaystyle{[L(X),L(X^{2})]=0.}
  188. A A
  189. X Y = 1 2 ( X Y + Y X ) X∘Y=\frac{1}{2}(X Y+Y X)
  190. A = 𝐎 A=\mathbf{O}
  191. n = 3 n=3
  192. T T
  193. T r T = 0 TrT=0
  194. D ( X ) = T X - X T \displaystyle{D(X)=TX-XT}
  195. Tr ( T ( X ( X 2 ) ) - T ( X 2 ( X ) ) ) = Tr T ( a I ) = Tr ( T ) a = 0 , \displaystyle{\mathrm{Tr}(T(X(X^{2}))-T(X^{2}(X)))=\mathrm{Tr}\,T(aI)=\mathrm{% Tr}(T)a=0,}
  196. ( D ( X ) , X 2 ) = 0. \displaystyle{(D(X),X^{2})=0.}
  197. ( D ( X ) , Y Z ) + ( D ( Y ) , Z X ) + ( D ( Z ) , X Y ) = 0. \displaystyle{(D(X),Y\circ Z)+(D(Y),Z\circ X)+(D(Z),X\circ Y)=0.}
  198. Z = 1 Z=1
  199. D D
  200. D ( X Y ) = D ( X ) Y + X D ( Y ) D(X∘Y)=D(X)∘Y+X∘D(Y)
  201. A A
  202. n n
  203. K K
  204. E = H < s u b > n ( A ) E=H<sub>n(A)

Hurwitz's_theorem_(number_theory).html

  1. | ξ - m n | < 1 5 n 2 . \left|\xi-\frac{m}{n}\right|<\frac{1}{\sqrt{5}\,n^{2}}.
  2. 5 \scriptstyle\sqrt{5}
  3. 5 \scriptstyle\sqrt{5}
  4. A > 5 \scriptstyle A>\sqrt{5}
  5. ξ = ( 1 + 5 ) / 2 \scriptstyle\xi=(1+\sqrt{5})/2

Hurwitz_determinant.html

  1. P ( λ ) = a 0 λ n + a 1 λ n - 1 + + a n - 1 λ + a n P(\lambda)=a_{0}\lambda^{n}+a_{1}\lambda^{n-1}+\cdots+a_{n-1}\lambda+a_{n}
  2. a i a_{i}
  3. i = 0 , 1 , , n i=0,1,\ldots,n
  4. H = ( a 1 a 3 a 5 0 0 0 a 0 a 2 a 4 0 a 1 a 3 a 0 a 2 0 0 a 1 a n a 0 a n - 1 0 0 a n - 2 a n a n - 3 a n - 1 0 0 0 0 a n - 4 a n - 2 a n ) . H=\begin{pmatrix}a_{1}&a_{3}&a_{5}&\dots&\dots&\dots&0&0&0\\ a_{0}&a_{2}&a_{4}&&&&\vdots&\vdots&\vdots\\ 0&a_{1}&a_{3}&&&&\vdots&\vdots&\vdots\\ \vdots&a_{0}&a_{2}&\ddots&&&0&\vdots&\vdots\\ \vdots&0&a_{1}&&\ddots&&a_{n}&\vdots&\vdots\\ \vdots&\vdots&a_{0}&&&\ddots&a_{n-1}&0&\vdots\\ \vdots&\vdots&0&&&&a_{n-2}&a_{n}&\vdots\\ \vdots&\vdots&\vdots&&&&a_{n-3}&a_{n-1}&0\\ 0&0&0&\dots&\dots&\dots&a_{n-4}&a_{n-2}&a_{n}\end{pmatrix}.

Hypercompact_stellar_system.html

  1. R = G M V k 2 R=\frac{GM}{V_{k}^{2}}
  2. M M
  3. G G
  4. R R
  5. V < s u b > k V<sub>k

Hypercomplex_analysis.html

  1. T = i = 1 N λ i E i \textstyle T=\sum_{i=1}^{N}\lambda_{i}E_{i}
  2. f , f ( T ) = i = 1 N f ( λ i ) E i . \textstyle f,\quad f(T)=\sum_{i=1}^{N}f(\lambda_{i})E_{i}.

Hyperfinite_set.html

  1. K = k 1 , k 2 , , k n K={k_{1},k_{2},\dots,k_{n}}
  2. e i θ e^{i\theta}
  3. u n , n = 1 , 2 , \langle u_{n},n=1,2,\ldots\rangle
  4. [ u n ] [u_{n}]
  5. [ A n ] [A_{n}]
  6. A n \langle A_{n}\rangle
  7. A n , n = 1 , 2 , A_{n}\subset\mathbb{R},n=1,2,\ldots

Hyperoctahedral_group.html

  1. S 2 S n S_{2}\wr S_{n}
  2. S n S_{n}
  3. D i h 4 Dih_{4}
  4. S 2 S 2 \cong S_{2}\wr S_{2}
  5. S 4 × S 2 S_{4}\times S_{2}
  6. S 2 S 3 \cong S_{2}\wr S_{3}
  7. S 2 S 4 S_{2}\wr S_{4}
  8. S 2 S 5 S_{2}\wr S_{5}
  9. S 2 S 6 S_{2}\wr S_{6}
  10. S 2 S n S_{2}\wr S_{n}
  11. { ± 1 } \{\pm 1\}
  12. C n { ± 1 } C_{n}\to\{\pm 1\}
  13. D n . D_{n}.
  14. H 1 ( C n , 𝐙 ) = { 0 n = 0 𝐙 / 2 n = 1 𝐙 / 2 × 𝐙 / 2 n 2 . H_{1}(C_{n},\mathbf{Z})=\begin{cases}0&n=0\\ \mathbf{Z}/2&n=1\\ \mathbf{Z}/2\times\mathbf{Z}/2&n\geq 2\end{cases}.
  15. - 1 -1
  16. n 1 n\geq 1
  17. S n S_{n}
  18. n 2 n\geq 2
  19. - 1 { ± 1 } , -1\in\{\pm 1\},
  20. { ± 1 } \{\pm 1\}
  21. - 1 -1
  22. H 2 ( C n , 𝐙 ) = { 0 n = 0 , 1 𝐙 / 2 n = 2 ( 𝐙 / 2 ) 2 n = 3 ( 𝐙 / 2 ) 3 n 4 . H_{2}(C_{n},\mathbf{Z})=\begin{cases}0&n=0,1\\ \mathbf{Z}/2&n=2\\ (\mathbf{Z}/2)^{2}&n=3\\ (\mathbf{Z}/2)^{3}&n\geq 4\end{cases}.

Hyponormal_operator.html

  1. 0 < p 1 0<p\leq 1
  2. ( T * T ) p ( T T * ) p (T^{*}T)^{p}\geq(TT^{*})^{p}
  3. ( T * T ) p - ( T T * ) p (T^{*}T)^{p}-(TT^{*})^{p}
  4. p = 1 p=1
  5. p = 1 / 2 p=1/2
  6. log ( T * T ) log ( T T * ) . \log(T^{*}T)\geq\log(TT^{*}).

Hyron_Spinrad.html

  1. \approx

I-bundle.html

  1. S 1 \scriptstyle S^{1}
  2. S 1 × I \scriptstyle S^{1}\times I
  3. K \scriptstyle K
  4. K × I \scriptstyle K\times I

I-spline.html

  1. I i ( x | k , t ) = L x M i ( u | k , t ) d u , I_{i}(x|k,t)=\int_{L}^{x}M_{i}(u|k,t)du,
  2. I i ( x | k , t ) = m = i j ( t m + k + 1 - t m ) M m ( x | k + 1 , t ) / ( k + 1 ) . I_{i}(x|k,t)=\sum_{m=i}^{j}(t_{m+k+1}-t_{m})M_{m}(x|k+1,t)/(k+1).

IBAF_World_Rankings.html

  1. P o i n t s d i f f e r e n c e = 14 N o o f t e a m s - 4 Points\ difference\ =\frac{14}{No\ of\ teams\ -\ 4}

Icositetragon.html

  1. A = 6 t 2 cot π 24 = 6 t 2 ( 2 + 2 + 3 + 6 ) . A=6t^{2}\cot\frac{\pi}{24}={6}t^{2}(2+\sqrt{2}+\sqrt{3}+\sqrt{6}).

Idealizer.html

  1. 𝕀 S ( T ) = { s S s T T and T s T } \mathbb{I}_{S}(T)=\{s\in S\mid sT\subseteq T\,\text{ and }Ts\subseteq T\}
  2. 𝕀 R ( A ) \mathbb{I}_{R}(A)
  3. { r L [ r , S ] S } \{r\in L\mid[r,S]\subseteq S\}
  4. 𝕀 R ( T ) = { r R r T T } \mathbb{I}_{R}(T)=\{r\in R\mid rT\subseteq T\}
  5. 𝕀 R ( L ) = { r R L r L } \mathbb{I}_{R}(L)=\{r\in R\mid Lr\subseteq L\}
  6. ( A : B ) := { r R B r A } (A:B):=\{r\in R\mid Br\subseteq A\}
  7. 𝕀 R ( B ) = ( B : B ) \mathbb{I}_{R}(B)=(B:B)

Identifiability.html

  1. P θ 1 = P θ 2 θ 1 = θ 2 for all θ 1 , θ 2 Θ . P_{\theta_{1}}=P_{\theta_{2}}\quad\Rightarrow\quad\theta_{1}=\theta_{2}\quad\ % \,\text{for all }\theta_{1},\theta_{2}\in\Theta.
  2. 1 T t = 1 T 𝟏 { X t A } a . s . Pr [ X t A ] , \frac{1}{T}\sum_{t=1}^{T}\mathbf{1}_{\{X_{t}\in A\}}\ \xrightarrow{a.s.}\ % \operatorname{Pr}[X_{t}\in A],
  3. 𝒫 = { f θ ( x ) = 1 2 π σ e - 1 2 σ 2 ( x - μ ) 2 | θ = ( μ , σ ) : μ , σ > 0 } . \mathcal{P}=\Big\{\ f_{\theta}(x)=\tfrac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2% \sigma^{2}}(x-\mu)^{2}}\ \Big|\ \theta=(\mu,\sigma):\mu\in\mathbb{R},\,\sigma% \!>0\ \Big\}.
  4. f θ 1 = f θ 2 \displaystyle f_{\theta_{1}}=f_{\theta_{2}}
  5. y = β x + ε , E [ ε | x ] = 0 y=\beta^{\prime}x+\varepsilon,\quad\operatorname{E}[\,\varepsilon|x\,]=0
  6. { y = β x * + ε , x = x * + η , \begin{cases}y=\beta x^{*}+\varepsilon,\\ x=x^{*}+\eta,\end{cases}

Iman_Maleki.html

  1. × \scriptstyle\times
  2. × \scriptstyle\times
  3. × \scriptstyle\times
  4. × \scriptstyle\times

Indefinite_product.html

  1. Q ( f ( x ) ) = f ( x + 1 ) f ( x ) Q(f(x))=\frac{f(x+1)}{f(x)}
  2. Q ( x f ( x ) ) = f ( x ) . Q(\prod_{x}f(x))=f(x)\,.
  3. x f ( x ) = F ( x ) \prod_{x}f(x)=F(x)\,
  4. F ( x + 1 ) F ( x ) = f ( x ) . \frac{F(x+1)}{F(x)}=f(x)\,.
  5. T T\,
  6. f ( x ) f(x)\,
  7. x f ( T x ) = C f ( T x ) x - 1 \prod_{x}f(Tx)=Cf(Tx)^{x-1}\,
  8. x f ( x ) = exp ( x ln f ( x ) ) \prod_{x}f(x)=\exp\left(\sum_{x}\ln f(x)\right)\,
  9. k = 1 n f ( k ) \prod_{k=1}^{n}f(k)
  10. x f ( x ) g ( x ) = x f ( x ) x g ( x ) \prod_{x}f(x)g(x)=\prod_{x}f(x)\prod_{x}g(x)\,
  11. x f ( x ) a = ( x f ( x ) ) a \prod_{x}f(x)^{a}=\left(\prod_{x}f(x)\right)^{a}\,
  12. x a f ( x ) = a x f ( x ) \prod_{x}a^{f(x)}=a^{\sum_{x}f(x)}\,
  13. x f ( x ) \prod_{x}f(x)\,
  14. x a = C a x \prod_{x}a=Ca^{x}\,
  15. x x = C Γ ( x ) \prod_{x}x=C\,\Gamma(x)\,
  16. x x + 1 x = C x \prod_{x}\frac{x+1}{x}=Cx
  17. x x + a x = C Γ ( x + a ) Γ ( x ) \prod_{x}\frac{x+a}{x}=\frac{C\,\Gamma(x+a)}{\Gamma(x)}
  18. x x a = C Γ ( x ) a \prod_{x}x^{a}=C\,\Gamma(x)^{a}\,
  19. x a x = C a x Γ ( x ) \prod_{x}ax=Ca^{x}\Gamma(x)\,
  20. x a x = C a x 2 ( x - 1 ) \prod_{x}a^{x}=Ca^{\frac{x}{2}(x-1)}\,
  21. x a 1 x = C a Γ ( x ) Γ ( x ) \prod_{x}a^{\frac{1}{x}}=Ca^{\frac{\Gamma^{\prime}(x)}{\Gamma(x)}}\,
  22. x x x = C e ζ ( - 1 , x ) - ζ ( - 1 ) = C e ψ ( - 2 ) ( z ) + z 2 - z 2 - z 2 ln ( 2 π ) = C K ( x ) \prod_{x}x^{x}=C\,e^{\zeta^{\prime}(-1,x)-\zeta^{\prime}(-1)}=C\,e^{\psi^{(-2)% }(z)+\frac{z^{2}-z}{2}-\frac{z}{2}\ln(2\pi)}=C\,\operatorname{K}(x)\,
  23. x Γ ( x ) = C Γ ( x ) x - 1 K ( x ) = C Γ ( x ) x - 1 e z 2 ln ( 2 π ) - z 2 - z 2 - ψ ( - 2 ) ( z ) = C G ( x ) \prod_{x}\Gamma(x)=\frac{C\,\Gamma(x)^{x-1}}{\operatorname{K}(x)}=C\,\Gamma(x)% ^{x-1}e^{\frac{z}{2}\ln(2\pi)-\frac{z^{2}-z}{2}-\psi^{(-2)}(z)}=C\,% \operatorname{G}(x)\,
  24. x sexp a ( x ) = C ( sexp a ( x ) ) sexp a ( x ) ( ln a ) x \prod_{x}\operatorname{sexp}_{a}(x)=\frac{C\,(\operatorname{sexp}_{a}(x))^{% \prime}}{\operatorname{sexp}_{a}(x)(\ln a)^{x}}\,
  25. x x + a = C Γ ( x + a ) \prod_{x}x+a=C\,\Gamma(x+a)\,
  26. x a x + b = C a x Γ ( x + b a ) \prod_{x}ax+b=C\,a^{x}\Gamma\left(x+\frac{b}{a}\right)\,
  27. x a x 2 + b x = C a x Γ ( x ) Γ ( x + b a ) \prod_{x}ax^{2}+bx=C\,a^{x}\Gamma(x)\Gamma\left(x+\frac{b}{a}\right)\,
  28. x x 2 + 1 = C Γ ( x - i ) Γ ( x + i ) \prod_{x}x^{2}+1=C\,\Gamma(x-i)\Gamma(x+i)
  29. x x + 1 x = C Γ ( x - i ) Γ ( x + i ) Γ ( x ) \prod_{x}x+\frac{1}{x}=\frac{C\,\Gamma(x-i)\Gamma(x+i)}{\Gamma(x)}
  30. x csc x sin ( x + 1 ) = C sin x \prod_{x}\csc x\sin(x+1)=C\sin x\,
  31. x sec x cos ( x + 1 ) = C cos x \prod_{x}\sec x\cos(x+1)=C\cos x\,
  32. x cot x tan ( x + 1 ) = C tan x \prod_{x}\cot x\tan(x+1)=C\tan x\,
  33. x tan x cot ( x + 1 ) = C cot x \prod_{x}\tan x\cot(x+1)=C\cot x\,

Indefinite_sum.html

  1. x \sum_{x}\,
  2. Δ - 1 \Delta^{-1}\,
  3. Δ \Delta\,
  4. Δ x f ( x ) = f ( x ) . \Delta\sum_{x}f(x)=f(x)\,.
  5. x f ( x ) = F ( x ) \sum_{x}f(x)=F(x)\,
  6. F ( x + 1 ) - F ( x ) = f ( x ) . F(x+1)-F(x)=f(x)\,.
  7. k = a b f ( k ) = Δ - 1 f ( b + 1 ) - Δ - 1 f ( a ) \sum_{k=a}^{b}f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)
  8. x f ( x ) = 0 x f ( t ) d t + k = 1 c k Δ k - 1 f ( x ) k ! + C \sum_{x}f(x)=\int_{0}^{x}f(t)dt+\sum_{k=1}^{\infty}\frac{c_{k}\Delta^{k-1}f(x)% }{k!}+C
  9. c k = 0 1 Γ ( x + 1 ) Γ ( x - k + 1 ) d x c_{k}=\int_{0}^{1}\frac{\Gamma(x+1)}{\Gamma(x-k+1)}dx
  10. x f ( x ) = - k = 1 Δ k - 1 f ( x ) k ! ( - x ) k + C \sum_{x}f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_{k}+C
  11. ( x ) k = Γ ( x + 1 ) Γ ( x - k + 1 ) (x)_{k}=\frac{\Gamma(x+1)}{\Gamma(x-k+1)}
  12. x f ( x ) = n = 1 f ( n - 1 ) ( 0 ) n ! B n ( x ) + C , \sum_{x}f(x)=\sum_{n=1}^{\infty}\frac{f^{(n-1)}(0)}{n!}B_{n}(x)+C\,,
  13. lim x + f ( x ) = 0 , \lim_{x\to{+\infty}}f(x)=0,
  14. x f ( x ) = n = 0 ( f ( n ) - f ( n + x ) ) + C . \sum_{x}f(x)=\sum_{n=0}^{\infty}\left(f(n)-f(n+x)\right)+C.
  15. x f ( x ) = 0 x f ( t ) d t - 1 2 f ( x ) + k = 1 B 2 k ( 2 k ) ! f ( 2 k - 1 ) ( x ) + C \sum_{x}f(x)=\int_{0}^{x}f(t)dt-\frac{1}{2}f(x)+\sum_{k=1}^{\infty}\frac{B_{2k% }}{(2k)!}f^{(2k-1)}(x)+C
  16. F ( x ) = x f ( x ) + C F(x)=\sum_{x}f(x)+C
  17. 0 1 F ( x ) d x = 0 \int_{0}^{1}F(x)dx=0
  18. 1 2 F ( x ) d x = 0 \int_{1}^{2}F(x)dx=0
  19. x 1 f ( x ) = - f ( 0 ) - F ( 0 ) \sum_{x\geq 1}^{\Re}f(x)=-f(0)-F(0)
  20. x 1 f ( x ) = - F ( 1 ) \sum_{x\geq 1}^{\Re}f(x)=-F(1)
  21. x f ( x ) Δ g ( x ) = f ( x ) g ( x ) - x ( g ( x ) + Δ g ( x ) ) Δ f ( x ) \sum_{x}f(x)\Delta g(x)=f(x)g(x)-\sum_{x}(g(x)+\Delta g(x))\Delta f(x)\,
  22. x f ( x ) Δ g ( x ) + x g ( x ) Δ f ( x ) = f ( x ) g ( x ) - x Δ f ( x ) Δ g ( x ) \sum_{x}f(x)\Delta g(x)+\sum_{x}g(x)\Delta f(x)=f(x)g(x)-\sum_{x}\Delta f(x)% \Delta g(x)\,
  23. i = a b f ( i ) Δ g ( i ) = f ( b + 1 ) g ( b + 1 ) - f ( a ) g ( a ) - i = a b g ( i + 1 ) Δ f ( i ) \sum_{i=a}^{b}f(i)\Delta g(i)=f(b+1)g(b+1)-f(a)g(a)-\sum_{i=a}^{b}g(i+1)\Delta f% (i)
  24. T T\,
  25. f ( x ) f(x)\,
  26. x f ( T x ) = x f ( T x ) + C \sum_{x}f(Tx)=xf(Tx)+C\,
  27. T T\,
  28. f ( x ) f(x)\,
  29. f ( x + T ) = - f ( x ) f(x+T)=-f(x)
  30. x f ( T x ) = - 1 2 f ( T x ) + C \sum_{x}f(Tx)=-\frac{1}{2}f(Tx)+C\,
  31. k = 1 n f ( k ) \sum_{k=1}^{n}f(k)
  32. F ( x + 1 ) - F ( x ) = f ( x + 1 ) F(x+1)-F(x)=f(x+1)\,
  33. \nabla
  34. x a = a x + C \sum_{x}a=ax+C\,
  35. x x = x 2 2 - x 2 + C \sum_{x}x=\frac{x^{2}}{2}-\frac{x}{2}+C
  36. x x a = B a + 1 ( x ) a + 1 + C , a - \sum_{x}x^{a}=\frac{B_{a+1}(x)}{a+1}+C,\,a\notin\mathbb{Z}^{-}
  37. B a ( x ) = - a ζ ( - a + 1 , x ) B_{a}(x)=-a\zeta(-a+1,x)\,
  38. x x a = ( - 1 ) a - 1 ψ ( - a - 1 ) ( x ) Γ ( - a ) + C , a - \sum_{x}x^{a}=\frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)}+C,\,a\in\mathbb{Z}% ^{-}
  39. ψ ( n ) ( x ) \psi^{(n)}(x)
  40. x 1 x = ψ ( x ) + C \sum_{x}\frac{1}{x}=\psi(x)+C
  41. ψ ( x ) \psi(x)
  42. x a x = a x a - 1 + C \sum_{x}a^{x}=\frac{a^{x}}{a-1}+C\,
  43. x 2 x = 2 x + C \sum_{x}2^{x}=2^{x}+C\,
  44. x log b x = log b Γ ( x ) + C \sum_{x}\log_{b}x=\log_{b}\Gamma(x)+C\,
  45. x log b a x = log b ( a x - 1 Γ ( x ) ) + C \sum_{x}\log_{b}ax=\log_{b}(a^{x-1}\Gamma(x))+C\,
  46. x sinh a x = 1 2 csch ( a 2 ) cosh ( a 2 - a x ) + C \sum_{x}\sinh ax=\frac{1}{2}\operatorname{csch}\left(\frac{a}{2}\right)\cosh% \left(\frac{a}{2}-ax\right)+C\,
  47. x cosh a x = 1 2 coth ( a 2 ) sinh a x - 1 2 cosh a x + C \sum_{x}\cosh ax=\frac{1}{2}\coth\left(\frac{a}{2}\right)\sinh ax-\frac{1}{2}% \cosh ax+C\,
  48. x tanh a x = 1 a ψ e a ( x - i π 2 a ) + 1 a ψ e a ( x + i π 2 a ) - x + C \sum_{x}\tanh ax=\frac{1}{a}\psi_{e^{a}}\left(x-\frac{i\pi}{2a}\right)+\frac{1% }{a}\psi_{e^{a}}\left(x+\frac{i\pi}{2a}\right)-x+C
  49. ψ q ( x ) \psi_{q}(x)
  50. x sin a x = - 1 2 csc ( a 2 ) cos ( a 2 - a x ) + C , a n π \sum_{x}\sin ax=-\frac{1}{2}\csc\left(\frac{a}{2}\right)\cos\left(\frac{a}{2}-% ax\right)+C\,,\,\,a\neq n\pi
  51. x cos a x = 1 2 cot ( a 2 ) sin a x - 1 2 cos a x + C , a n π \sum_{x}\cos ax=\frac{1}{2}\cot\left(\frac{a}{2}\right)\sin ax-\frac{1}{2}\cos ax% +C\,,\,\,a\neq n\pi
  52. x sin 2 a x = x 2 + 1 4 csc ( a ) sin ( a - 2 a x ) + C , a n π 2 \sum_{x}\sin^{2}ax=\frac{x}{2}+\frac{1}{4}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq% \frac{n\pi}{2}
  53. x cos 2 a x = x 2 - 1 4 csc ( a ) sin ( a - 2 a x ) + C , a n π 2 \sum_{x}\cos^{2}ax=\frac{x}{2}-\frac{1}{4}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq% \frac{n\pi}{2}
  54. x tan a x = i x - 1 a ψ e 2 i a ( x - π 2 a ) + C , a n π 2 \sum_{x}\tan ax=ix-\frac{1}{a}\psi_{e^{2ia}}\left(x-\frac{\pi}{2a}\right)+C\,,% \,\,a\neq\frac{n\pi}{2}
  55. ψ q ( x ) \psi_{q}(x)
  56. x tan x = i x - ψ e 2 i ( x + π 2 ) + C = - k = 1 ( ψ ( k π - π 2 + 1 - z ) + ψ ( k π - π 2 + z ) - ψ ( k π - π 2 + 1 ) - ψ ( k π - π 2 ) ) + C \sum_{x}\tan x=ix-\psi_{e^{2i}}\left(x+\frac{\pi}{2}\right)+C=-\sum_{k=1}^{% \infty}\left(\psi\left(k\pi-\frac{\pi}{2}+1-z\right)+\psi\left(k\pi-\frac{\pi}% {2}+z\right)-\psi\left(k\pi-\frac{\pi}{2}+1\right)-\psi\left(k\pi-\frac{\pi}{2% }\right)\right)+C\,
  57. x cot a x = - i x - i ψ e 2 i a ( x ) a + C , a n π 2 \sum_{x}\cot ax=-ix-\frac{i\psi_{e^{2ia}}(x)}{a}+C\,,\,\,a\neq\frac{n\pi}{2}
  58. x artanh a x = 1 2 ln ( ( - 1 ) x Γ ( - 1 a ) Γ ( x + 1 a ) Γ ( 1 a ) Γ ( x - 1 a ) ) + C \sum_{x}\operatorname{artanh}\,ax=\frac{1}{2}\ln\left(\frac{(-1)^{x}\Gamma% \left(-\frac{1}{a}\right)\Gamma\left(x+\frac{1}{a}\right)}{\Gamma\left(\frac{1% }{a}\right)\Gamma\left(x-\frac{1}{a}\right)}\right)+C
  59. x arctan a x = i 2 ln ( ( - 1 ) x Γ ( - i a ) Γ ( x + i a ) Γ ( i a ) Γ ( x - i a ) ) + C \sum_{x}\arctan ax=\frac{i}{2}\ln\left(\frac{(-1)^{x}\Gamma(\frac{-i}{a})% \Gamma(x+\frac{i}{a})}{\Gamma(\frac{i}{a})\Gamma(x-\frac{i}{a})}\right)+C
  60. x ψ ( x ) = ( x - 1 ) ψ ( x ) - x + C \sum_{x}\psi(x)=(x-1)\psi(x)-x+C\,
  61. x Γ ( x ) = ( - 1 ) x + 1 Γ ( x ) Γ ( 1 - x , - 1 ) e + C \sum_{x}\Gamma(x)=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}{e}+C
  62. Γ ( s , x ) \Gamma(s,x)
  63. x ( x ) a = ( x ) a + 1 a + 1 + C \sum_{x}(x)_{a}=\frac{(x)_{a+1}}{a+1}+C
  64. ( x ) a (x)_{a}
  65. x sexp a ( x ) = ln a ( sexp a ( x ) ) ( ln a ) x + C \sum_{x}\operatorname{sexp}_{a}(x)=\ln_{a}\frac{(\operatorname{sexp}_{a}(x))^{% \prime}}{(\ln a)^{x}}+C\,

Index_of_a_Lie_algebra.html

  1. ξ 𝔤 * \xi\in\mathfrak{g}^{*}
  2. ind 𝔤 := min ξ 𝔤 * dim 𝔤 ξ . \mathrm{ind}\,\mathfrak{g}:=\min\limits_{\xi\in\mathfrak{g}^{*}}\mathrm{dim}\,% \mathfrak{g}_{\xi}.
  3. K ξ : 𝔤 𝔤 𝕂 : ( X , Y ) ξ ( [ X , Y ] ) K_{\xi}\colon\mathfrak{g\otimes g}\to\mathbb{K}:(X,Y)\mapsto\xi([X,Y])