wpmath0000007_3

Debris_flow.html

  1. 1 - γ 1-\gamma
  2. γ \gamma
  3. γ \gamma
  4. γ = 1 \gamma=1
  5. γ = 0 \gamma=0

Debt_service_coverage_ratio.html

  1. DSCR = Net Operating Income / Debt Services \,\text{DSCR}=\,\text{Net Operating Income}/\,\text{Debt Services}
  2. Net Operating Income = Net Income + Amortization and Depreciation + Interest Expense + Other Non-cash Items \,\text{Net Operating Income}=\,\text{Net Income}+\,\text{Amortization and % Depreciation}+\,\text{Interest Expense}+\,\text{Other Non-cash Items}
  3. Debt Services = Principal Repayment + Interest Payments + Lease Payments \,\text{Debt Services}=\,\text{Principal Repayment}+\,\text{Interest Payments}% +\,\text{Lease Payments}

Debye_frequency.html

  1. N / V N/V
  2. ν m = ( 3 N 4 π V ) 1 / 3 v s \nu_{m}=\left(\frac{3N}{4\pi V}\right)^{1/3}v_{s}

Debye–Waller_factor.html

  1. DWF = exp ( i 𝐪 𝐮 ) 2 \,\text{DWF}=\left\langle\exp\left(i\mathbf{q}\cdot\mathbf{u}\right)\right% \rangle^{2}
  2. 𝐪 𝐮 \mathbf{q}\cdot\mathbf{u}
  3. DWF = exp ( - [ 𝐪 𝐮 ] 2 ) \,\text{DWF}=\exp\left(-\langle[\mathbf{q}\cdot\mathbf{u}]^{2}\rangle\right)
  4. DWF = exp ( - q 2 u 2 / 3 ) \,\text{DWF}=\exp\left(-q^{2}\langle u^{2}\rangle/3\right)
  5. u 2 \langle u^{2}\rangle
  6. U U
  7. U = u 2 U=\langle u^{2}\rangle
  8. λ \lambda
  9. 2 θ 2\theta
  10. q = 4 π sin ( θ ) λ q=\frac{4\pi\sin(\theta)}{\lambda}
  11. B = 8 π 2 u 2 B=8\pi^{2}\langle u^{2}\rangle

Decimal_degrees.html

  1. DD = D + M 60 + S 3600 \mathrm{DD}=\mathrm{D}+\frac{\mathrm{M}}{60}+\frac{\mathrm{S}}{3600}
  2. DD = { D = trunc ( DD ) M = trunc ( | DD | × 60 ) mod 60 S = ( | DD | × 3600 ) mod 60 \mathrm{DD}=\begin{cases}\mathrm{D}&=\operatorname{trunc}(\mathrm{DD})\\ \mathrm{M}&=\operatorname{trunc}(|\mathrm{DD}|\times 60)\,\bmod\,60\\ \mathrm{S}&=\left(|\mathrm{DD}|\times 3600\right)\,\bmod\,60\end{cases}

Decimal_representation.html

  1. r = i = 0 a i 10 i r=\sum_{i=0}^{\infty}\frac{a_{i}}{10^{i}}
  2. r = a 0 . a 1 a 2 a 3 . r=a_{0}.a_{1}a_{2}a_{3}\dots.\,
  3. r = lim n i = 0 n a i 10 i r=\lim_{n\to\infty}\sum_{i=0}^{n}\frac{a_{i}}{10^{i}}
  4. x 0 x\geq 0
  5. n 1 n\geq 1
  6. r n = a 0 . a 1 a 2 a n r_{n}=a_{0}.a_{1}a_{2}\cdots a_{n}
  7. r n x < r n + 1 10 n . r_{n}\leq x<r_{n}+\frac{1}{10^{n}}.\,
  8. r n = p 10 n r_{n}=\textstyle\frac{p}{10^{n}}
  9. p = 10 n x p=\lfloor 10^{n}x\rfloor
  10. p 10 n x < p + 1 p\leq 10^{n}x<p+1
  11. 10 n 10^{n}
  12. r n r_{n}
  13. x = i = 0 n a i 10 i = i = 0 n 10 n - i a i / 10 n x=\sum_{i=0}^{n}\frac{a_{i}}{10^{i}}=\sum_{i=0}^{n}10^{n-i}a_{i}/10^{n}
  14. x = p 2 n 5 m = 2 m 5 n p 2 n + m 5 n + m = 2 m 5 n p 10 n + m x=\frac{p}{2^{n}5^{m}}=\frac{2^{m}5^{n}p}{2^{n+m}5^{n+m}}=\frac{2^{m}5^{n}p}{1% 0^{n+m}}
  15. p 10 k \textstyle\frac{p}{10^{k}}
  16. p = i = 0 n 10 i a i p=\sum_{i=0}^{n}10^{i}a_{i}
  17. x = i = 0 n 10 n - i a i / 10 n = i = 0 n a i 10 i x=\sum_{i=0}^{n}10^{n-i}a_{i}/10^{n}=\sum_{i=0}^{n}\frac{a_{i}}{10^{i}}

Decomposition_method_(constraint_satisfaction).html

  1. { x , y , z } \{x,y,z\}
  2. { u , x , z } \{u,x,z\}
  3. { y , w } \{y,w\}
  4. y y
  5. ( x , y , z ) (x,y,z)
  6. ( y , w ) (y,w)
  7. ( x , y , z ) (x,y,z)
  8. ( y , w ) (y,w)
  9. x i x_{i}
  10. x j x_{j}
  11. x j x_{j}
  12. x i x_{i}
  13. x i x_{i}
  14. H H
  15. H H
  16. H H
  17. H H
  18. H H
  19. H H
  20. H H
  21. F F
  22. H H
  23. F F
  24. H H
  25. H H
  26. { C ( a , b ) , c , d , e } \{C(a,b),c,d,e\}
  27. D ( c , d , e , f ) D(c,d,e,f)
  28. { C , D } \{C,D\}
  29. { a , b , c , d , e } \{a,b,c,d,e\}
  30. n n
  31. n n
  32. n n
  33. 1 1
  34. n n
  35. n + k n+k
  36. k k
  37. i i
  38. j j
  39. i i
  40. j j
  41. X i X_{i}
  42. X j X_{j}
  43. i i
  44. X j X_{j}
  45. X i X j X_{i}\cap X_{j}
  46. X j X_{j}
  47. i i
  48. X i X j X_{i}\cap X_{j}
  49. n n
  50. d n d^{n}
  51. d d
  52. n 1 n_{1}
  53. n 2 n_{2}
  54. N 1 N_{1}
  55. N 2 N_{2}
  56. N 1 N 2 = N_{1}\cap N_{2}=\emptyset
  57. n 1 n_{1}
  58. n 2 n_{2}
  59. N 1 N 2 N_{1}\cup N_{2}
  60. k k
  61. k k

Defense-Independent_ERA.html

  1. dIBB = 0.9926 IBB \rm{dIBB=0.9926\ IBB}\,
  2. dHB = HB ( BFP - dIBB BFP - IBB ) \rm{dHB=HB\left(\frac{BFP-dIBB}{BFP-IBB}\right)}\,
  3. dBB = ( BB - IBB ) ( BFP - dIBB - dHB BFP - IBB - HB ) + IBB \rm{dBB=(BB-IBB)\left(\frac{BFP-dIBB-dHB}{BFP-IBB-HB}\right)+IBB}\,
  4. dK = K ( BFP - dIBB - dHB BFP - IBB - HB ) \rm{dK=K\left(\frac{BFP-dIBB-dHB}{BFP-IBB-HB}\right)}\,
  5. dHR = HR ( BFP - dHB - dBB - dK BFP - HB - BB - K ) \rm{dHR=HR\left(\frac{BFP-dHB-dBB-dK}{BFP-HB-BB-K}\right)}\,

Defined_daily_dose.html

  1. D r u g u s a g e ( D D D s ) = I t e m s i s s u e d × A m o u n t o f d r u g p e r i t e m D D D Drug\ usage(DDDs)=\frac{Items\ issued\times Amount\ of\ drug\ per\ item}{DDD}
  2. D r u g u s a g e ( D D D s ) = 24 × 500 m g 3 g = 4 Drug\ usage(DDDs)=\frac{24\times 500mg}{3g}=4

Definite_clause_grammar.html

  1. a n b n c n a^{n}b^{n}c^{n}
  2. a n b n c n a^{n}b^{n}c^{n}
  3. n n

Defocus_aberration.html

  1. a ( 2 ρ 2 - 1 ) a(2\rho^{2}-1)
  2. a a

Degen's_eight-square_identity.html

  1. ( a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 + a 7 2 + a 8 2 ) ( b 1 2 + b 2 2 + b 3 2 + b 4 2 + b 5 2 + b 6 2 + b 7 2 + b 8 2 ) = (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{% 2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2}+b_{5}^{2}+b_{6}^{2}+b_{7}^{2}+b_{8% }^{2})=\,
  2. ( a 1 b 1 - a 2 b 2 - a 3 b 3 - a 4 b 4 - a 5 b 5 - a 6 b 6 - a 7 b 7 - a 8 b 8 ) 2 + (a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4}-a_{5}b_{5}-a_{6}b_{6}-a_{7}b_{7}-% a_{8}b_{8})^{2}+\,
  3. ( a 1 b 2 + a 2 b 1 + a 3 b 4 - a 4 b 3 + a 5 b 6 - a 6 b 5 - a 7 b 8 + a 8 b 7 ) 2 + (a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3}+a_{5}b_{6}-a_{6}b_{5}-a_{7}b_{8}+% a_{8}b_{7})^{2}+\,
  4. ( a 1 b 3 - a 2 b 4 + a 3 b 1 + a 4 b 2 + a 5 b 7 + a 6 b 8 - a 7 b 5 - a 8 b 6 ) 2 + (a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2}+a_{5}b_{7}+a_{6}b_{8}-a_{7}b_{5}-% a_{8}b_{6})^{2}+\,
  5. ( a 1 b 4 + a 2 b 3 - a 3 b 2 + a 4 b 1 + a 5 b 8 - a 6 b 7 + a 7 b 6 - a 8 b 5 ) 2 + (a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1}+a_{5}b_{8}-a_{6}b_{7}+a_{7}b_{6}-% a_{8}b_{5})^{2}+\,
  6. ( a 1 b 5 - a 2 b 6 - a 3 b 7 - a 4 b 8 + a 5 b 1 + a 6 b 2 + a 7 b 3 + a 8 b 4 ) 2 + (a_{1}b_{5}-a_{2}b_{6}-a_{3}b_{7}-a_{4}b_{8}+a_{5}b_{1}+a_{6}b_{2}+a_{7}b_{3}+% a_{8}b_{4})^{2}+\,
  7. ( a 1 b 6 + a 2 b 5 - a 3 b 8 + a 4 b 7 - a 5 b 2 + a 6 b 1 - a 7 b 4 + a 8 b 3 ) 2 + (a_{1}b_{6}+a_{2}b_{5}-a_{3}b_{8}+a_{4}b_{7}-a_{5}b_{2}+a_{6}b_{1}-a_{7}b_{4}+% a_{8}b_{3})^{2}+\,
  8. ( a 1 b 7 + a 2 b 8 + a 3 b 5 - a 4 b 6 - a 5 b 3 + a 6 b 4 + a 7 b 1 - a 8 b 2 ) 2 + (a_{1}b_{7}+a_{2}b_{8}+a_{3}b_{5}-a_{4}b_{6}-a_{5}b_{3}+a_{6}b_{4}+a_{7}b_{1}-% a_{8}b_{2})^{2}+\,
  9. ( a 1 b 8 - a 2 b 7 + a 3 b 6 + a 4 b 5 - a 5 b 4 - a 6 b 3 + a 7 b 2 + a 8 b 1 ) 2 (a_{1}b_{8}-a_{2}b_{7}+a_{3}b_{6}+a_{4}b_{5}-a_{5}b_{4}-a_{6}b_{3}+a_{7}b_{2}+% a_{8}b_{1})^{2}\,
  10. a b = a b \|ab\|=\|a\|\|b\|
  11. ( a 1 2 + a 2 2 + a 3 2 + a 4 2 ) ( b 1 2 + b 2 2 + b 3 2 + b 4 2 ) = (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^% {2})=\,
  12. ( a 1 b 1 - a 2 b 2 - a 3 b 3 - a 4 b 4 ) 2 + (a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4})^{2}+\,
  13. ( a 1 b 2 + a 2 b 1 + a 3 b 4 - a 4 b 3 ) 2 + (a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3})^{2}+\,
  14. ( a 1 b 3 - a 2 b 4 + a 3 b 1 + a 4 b 2 ) 2 + (a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2})^{2}+\,
  15. ( a 1 b 4 + a 2 b 3 - a 3 b 2 + a 4 b 1 ) 2 (a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1})^{2}\,
  16. z i z_{i}
  17. x i , y i x_{i},y_{i}
  18. ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) ( y 1 2 + y 2 2 + y 3 2 + y 4 2 + y 5 2 + y 6 2 + y 7 2 + y 8 2 ) = z 1 2 + z 2 2 + z 3 2 + z 4 2 + z 5 2 + z 6 2 + z 7 2 + z 8 2 (x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{% 2})(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}+y_{5}^{2}+y_{6}^{2}+y_{7}^{2}+y_{8% }^{2})=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}+z_{5}^{2}+z_{6}^{2}+z_{7}^{2}+z% _{8}^{2}
  19. z 1 = x 1 y 1 - x 2 y 2 - x 3 y 3 - x 4 y 4 + u 1 y 5 - u 2 y 6 - u 3 y 7 - u 4 y 8 z_{1}=x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}+u_{1}y_{5}-u_{2}y_{6}-u_{3}y% _{7}-u_{4}y_{8}
  20. z 2 = x 2 y 1 + x 1 y 2 + x 4 y 3 - x 3 y 4 + u 2 y 5 + u 1 y 6 + u 4 y 7 - u 3 y 8 z_{2}=x_{2}y_{1}+x_{1}y_{2}+x_{4}y_{3}-x_{3}y_{4}+u_{2}y_{5}+u_{1}y_{6}+u_{4}y% _{7}-u_{3}y_{8}
  21. z 3 = x 3 y 1 - x 4 y 2 + x 1 y 3 + x 2 y 4 + u 3 y 5 - u 4 y 6 + u 1 y 7 + u 2 y 8 z_{3}=x_{3}y_{1}-x_{4}y_{2}+x_{1}y_{3}+x_{2}y_{4}+u_{3}y_{5}-u_{4}y_{6}+u_{1}y% _{7}+u_{2}y_{8}
  22. z 4 = x 4 y 1 + x 3 y 2 - x 2 y 3 + x 1 y 4 + u 4 y 5 + u 3 y 6 - u 2 y 7 + u 1 y 8 z_{4}=x_{4}y_{1}+x_{3}y_{2}-x_{2}y_{3}+x_{1}y_{4}+u_{4}y_{5}+u_{3}y_{6}-u_{2}y% _{7}+u_{1}y_{8}
  23. z 5 = x 5 y 1 - x 6 y 2 - x 7 y 3 - x 8 y 4 + x 1 y 5 - x 2 y 6 - x 3 y 7 - x 4 y 8 z_{5}=x_{5}y_{1}-x_{6}y_{2}-x_{7}y_{3}-x_{8}y_{4}+x_{1}y_{5}-x_{2}y_{6}-x_{3}y% _{7}-x_{4}y_{8}
  24. z 6 = x 6 y 1 + x 5 y 2 + x 8 y 3 - x 7 y 4 + x 2 y 5 + x 1 y 6 + x 4 y 7 - x 3 y 8 z_{6}=x_{6}y_{1}+x_{5}y_{2}+x_{8}y_{3}-x_{7}y_{4}+x_{2}y_{5}+x_{1}y_{6}+x_{4}y% _{7}-x_{3}y_{8}
  25. z 7 = x 7 y 1 - x 8 y 2 + x 5 y 3 + x 6 y 4 + x 3 y 5 - x 4 y 6 + x 1 y 7 + x 2 y 8 z_{7}=x_{7}y_{1}-x_{8}y_{2}+x_{5}y_{3}+x_{6}y_{4}+x_{3}y_{5}-x_{4}y_{6}+x_{1}y% _{7}+x_{2}y_{8}
  26. z 8 = x 8 y 1 + x 7 y 2 - x 6 y 3 + x 5 y 4 + x 4 y 5 + x 3 y 6 - x 2 y 7 + x 1 y 8 z_{8}=x_{8}y_{1}+x_{7}y_{2}-x_{6}y_{3}+x_{5}y_{4}+x_{4}y_{5}+x_{3}y_{6}-x_{2}y% _{7}+x_{1}y_{8}
  27. u 1 = ( a x 1 2 + x 2 2 + x 3 2 + x 4 2 ) x 5 - 2 x 1 ( b x 1 x 5 + x 2 x 6 + x 3 x 7 + x 4 x 8 ) c u_{1}=\frac{(ax_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})x_{5}-2x_{1}(bx_{1}x_{5}% +x_{2}x_{6}+x_{3}x_{7}+x_{4}x_{8})}{c}
  28. u 2 = ( x 1 2 + a x 2 2 + x 3 2 + x 4 2 ) x 6 - 2 x 2 ( x 1 x 5 + b x 2 x 6 + x 3 x 7 + x 4 x 8 ) c u_{2}=\frac{(x_{1}^{2}+ax_{2}^{2}+x_{3}^{2}+x_{4}^{2})x_{6}-2x_{2}(x_{1}x_{5}+% bx_{2}x_{6}+x_{3}x_{7}+x_{4}x_{8})}{c}
  29. u 3 = ( x 1 2 + x 2 2 + a x 3 2 + x 4 2 ) x 7 - 2 x 3 ( x 1 x 5 + x 2 x 6 + b x 3 x 7 + x 4 x 8 ) c u_{3}=\frac{(x_{1}^{2}+x_{2}^{2}+ax_{3}^{2}+x_{4}^{2})x_{7}-2x_{3}(x_{1}x_{5}+% x_{2}x_{6}+bx_{3}x_{7}+x_{4}x_{8})}{c}
  30. u 4 = ( x 1 2 + x 2 2 + x 3 2 + a x 4 2 ) x 8 - 2 x 4 ( x 1 x 5 + x 2 x 6 + x 3 x 7 + b x 4 x 8 ) c u_{4}=\frac{(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+ax_{4}^{2})x_{8}-2x_{4}(x_{1}x_{5}+% x_{2}x_{6}+x_{3}x_{7}+bx_{4}x_{8})}{c}
  31. a = - 1 , b = 0 , c = x 1 2 + x 2 2 + x 3 2 + x 4 2 a=-1,\;\;b=0,\;\;c=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}
  32. u i u_{i}
  33. u 1 2 + u 2 2 + u 3 2 + u 4 2 = x 5 2 + x 6 2 + x 7 2 + x 8 2 u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}

Degree_of_a_field_extension.html

  1. [ M : K ] = [ M : L ] [ L : K ] . [M:K]=[M:L]\cdot[L:K].
  2. x = n = 1 e a n w n = a 1 w 1 + + a e w e . x=\sum_{n=1}^{e}a_{n}w_{n}=a_{1}w_{1}+\cdots+a_{e}w_{e}.
  3. a n = m = 1 d b m , n u m = b 1 , n u 1 + + b d , n u d . a_{n}=\sum_{m=1}^{d}b_{m,n}u_{m}=b_{1,n}u_{1}+\cdots+b_{d,n}u_{d}.
  4. x = n = 1 e ( m = 1 d b m , n u m ) w n = n = 1 e m = 1 d b m , n ( u m w n ) , x=\sum_{n=1}^{e}\left(\sum_{m=1}^{d}b_{m,n}u_{m}\right)w_{n}=\sum_{n=1}^{e}% \sum_{m=1}^{d}b_{m,n}(u_{m}w_{n}),
  5. 0 = n = 1 e m = 1 d b m , n ( u m w n ) 0=\sum_{n=1}^{e}\sum_{m=1}^{d}b_{m,n}(u_{m}w_{n})
  6. 0 = n = 1 e ( m = 1 d b m , n u m ) w n , 0=\sum_{n=1}^{e}\left(\sum_{m=1}^{d}b_{m,n}u_{m}\right)w_{n},
  7. 0 = m = 1 d b m , n u m 0=\sum_{m=1}^{d}b_{m,n}u_{m}

Degree_of_unsaturation.html

  1. D U = 1 + 1 2 n i ( v i - 2 ) DU=1+\frac{1}{2}\sum n_{i}(v_{i}-2)
  2. n i n_{i}
  3. v i v_{i}
  4. D o u b l e B o n d E q u i v a l e n t = ( a + 1 ) - b - c + f 2 Double\ Bond\ Equivalent=(a+1)-\frac{b-c+f}{2}
  5. R i n g s + π B o n d s = C - H 2 - X 2 + N 2 + 1 Rings+\pi\ Bonds=C-\frac{H}{2}-\frac{X}{2}+\frac{N}{2}+1\,

Delay_differential_equation.html

  1. x ( t ) n x(t)\in\mathbb{R}^{n}
  2. d dt x ( t ) = f ( t , x ( t ) , x t ) , \frac{\rm d}{{\rm d}t}x(t)=f(t,x(t),x_{t}),
  3. x t = { x ( τ ) : τ t } x_{t}=\{x(\tau):\tau\leq t\}
  4. f f
  5. × n × C 1 ( , n ) \mathbb{R}\times\mathbb{R}^{n}\times C^{1}(\mathbb{R},\mathbb{R}^{n})
  6. n . \mathbb{R}^{n}.\,
  7. d dt x ( t ) = f ( t , x ( t ) , - 0 x ( t + τ ) d μ ( τ ) ) \frac{\rm d}{{\rm d}t}x(t)=f\left(t,x(t),\int_{-\infty}^{0}x(t+\tau)\,{\rm d}% \mu(\tau)\right)
  8. d dt x ( t ) = f ( t , x ( t ) , x ( t - τ 1 ) , , x ( t - τ m ) ) \frac{\rm d}{{\rm d}t}x(t)=f(t,x(t),x(t-\tau_{1}),\ldots,x(t-\tau_{m}))
  9. τ 1 > > τ m 0 \tau_{1}>\cdots>\tau_{m}\geq 0
  10. d dt x ( t ) = A 0 x ( t ) + A 1 x ( t - τ 1 ) + + A m x ( t - τ m ) \frac{\rm d}{{\rm d}t}x(t)=A_{0}x(t)+A_{1}x(t-\tau_{1})+\cdots+A_{m}x(t-\tau_{% m})
  11. A 0 , , A m n × n A_{0},\ldots,A_{m}\in\mathbb{R}^{n\times n}
  12. d dt x ( t ) = a x ( t ) + b x ( λ t ) , \frac{\rm d}{{\rm d}t}x(t)=ax(t)+bx(\lambda t),
  13. d dt x ( t ) = f ( x ( t ) , x ( t - τ ) ) \frac{\rm d}{{\rm d}t}x(t)=f(x(t),x(t-\tau))
  14. ϕ : [ - τ , 0 ] n \phi\colon[-\tau,0]\rightarrow\mathbb{R}^{n}
  15. [ 0 , τ ] [0,\tau]
  16. ψ ( t ) \psi(t)
  17. d dt ψ ( t ) = f ( ψ ( t ) , ϕ ( t - τ ) ) \frac{\rm d}{{\rm d}t}\psi(t)=f(\psi(t),\phi(t-\tau))
  18. ψ ( 0 ) = ϕ ( 0 ) \psi(0)=\phi(0)
  19. f ( x ( t ) , x ( t - τ ) ) = a x ( t - τ ) f(x(t),x(t-\tau))=ax(t-\tau)
  20. ϕ ( t ) = 1 \phi(t)=1
  21. x ( t ) = x ( 0 ) + s = 0 t d dt x ( s ) d s = 1 + a s = 0 t ϕ ( s - τ ) d s x(t)=x(0)+\int_{s=0}^{t}\frac{\rm d}{{\rm d}t}x(s)\,{\rm d}s=1+a\int_{s=0}^{t}% \phi(s-\tau)\,{\rm d}s
  22. x ( t ) = a t + 1 x(t)=at+1
  23. x ( 0 ) = ϕ ( 0 ) = 1 x(0)=\phi(0)=1
  24. t [ τ , 2 τ ] t\in[\tau,2\tau]
  25. x ( t ) = x ( τ ) + s = τ t d dt x ( s ) d s = ( a τ + 1 ) + a s = τ t a ( s - τ ) + 1 d s = ( a τ + 1 ) + a s = 0 t - τ a s + 1 d s x(t)=x(\tau)+\int_{s=\tau}^{t}\frac{\rm d}{{\rm d}t}x(s)\,{\rm d}s=(a\tau+1)+a% \int_{s=\tau}^{t}a(s-\tau)+1\,{\rm d}s=(a\tau+1)+a\int_{s=0}^{t-\tau}as+1\,{% \rm d}s
  26. x ( t ) = ( a τ + 1 ) + a ( t - τ ) ( a ( t - τ ) 2 + 1 ) x(t)=(a\tau+1)+a(t-\tau)\left(\frac{a(t-\tau)}{2}+1\right)
  27. d dt x ( t ) = f ( t , x ( t ) , - 0 x ( t + τ ) e λ τ d τ ) . \frac{\rm d}{{\rm d}t}x(t)=f\left(t,x(t),\int_{-\infty}^{0}x(t+\tau)e^{\lambda% \tau}\,{\rm d}\tau\right).
  28. y ( t ) = - 0 x ( t + τ ) e λ τ d τ y(t)=\int_{-\infty}^{0}x(t+\tau)e^{\lambda\tau}\,{\rm d}\tau
  29. d dt x ( t ) = f ( t , x , y ) , d dt y ( t ) = x - λ y . \frac{\rm d}{{\rm d}t}x(t)=f(t,x,y),\quad\frac{\rm d}{{\rm d}t}y(t)=x-\lambda y.
  30. d dt x ( t ) = f ( t , x ( t ) , - 0 x ( t + τ ) cos ( α τ + β ) d τ ) \frac{\rm d}{{\rm d}t}x(t)=f\left(t,x(t),\int_{-\infty}^{0}x(t+\tau)\cos(% \alpha\tau+\beta)\,{\rm d}\tau\right)
  31. d dt x ( t ) = f ( t , x , y ) , d dt y ( t ) = cos ( β ) x + α z , d dt z ( t ) = sin ( β ) x - α y , \frac{\rm d}{{\rm d}t}x(t)=f(t,x,y),\quad\frac{\rm d}{{\rm d}t}y(t)=\cos(\beta% )x+\alpha z,\quad\frac{\rm d}{{\rm d}t}z(t)=\sin(\beta)x-\alpha y,
  32. y = - 0 x ( t + τ ) cos ( α τ + β ) d τ , z = - 0 x ( t + τ ) sin ( α τ + β ) d τ . y=\int_{-\infty}^{0}x(t+\tau)\cos(\alpha\tau+\beta)\,{\rm d}\tau,\quad z=\int_% {-\infty}^{0}x(t+\tau)\sin(\alpha\tau+\beta)\,{\rm d}\tau.
  33. d dt x ( t ) = A 0 x ( t ) + A 1 x ( t - τ 1 ) + + A m x ( t - τ m ) \frac{\rm d}{{\rm d}t}x(t)=A_{0}x(t)+A_{1}x(t-\tau_{1})+\cdots+A_{m}x(t-\tau_{% m})
  34. det ( - λ I + A 0 + A 1 e - τ 1 λ + + A m e - τ m λ ) = 0 {\rm det}(-\lambda I+A_{0}+A_{1}e^{-\tau_{1}\lambda}+\cdots+A_{m}e^{-\tau_{m}% \lambda})=0
  35. d dt x ( t ) = - x ( t - 1 ) . \frac{\rm d}{{\rm d}t}x(t)=-x(t-1).
  36. - λ - e - λ = 0. -\lambda-e^{-\lambda}=0.\,
  37. λ = W k ( - 1 ) \lambda=W_{k}(-1)

Delay_spread.html

  1. A c ( τ ) A_{c}(\tau)
  2. τ ¯ = 0 τ A c ( τ ) d τ 0 A c ( τ ) d τ \overline{\tau}=\frac{\int_{0}^{\infty}\tau A_{c}(\tau)d\tau}{\int_{0}^{\infty% }A_{c}(\tau)d\tau}
  3. τ rms = 0 ( τ - τ ¯ ) 2 A c ( τ ) d τ 0 A c ( τ ) d τ \tau_{\,\text{rms}}=\sqrt{\frac{\int_{0}^{\infty}(\tau-\overline{\tau})^{2}A_{% c}(\tau)d\tau}{\int_{0}^{\infty}A_{c}(\tau)d\tau}}

Delta.html

  1. Δ A B C \Delta ABC
  2. Δ n 0 \Delta^{0}_{n}
  3. Δ n 1 \Delta^{1}_{n}
  4. Δ i P \Delta^{P}_{i}
  5. \nabla
  6. δ \delta
  7. δ i j \delta_{ij}
  8. δ ( x - y ) \delta(x-y)
  9. = Δ \overset{\Delta}{=}
  10. Δ % \Delta\%

Demand.html

  1. Q = a P c Q=aP^{c}
  2. c 0 c\leq 0

Denotational_semantics_of_the_Actor_model.html

  1. x 0 x 1 x 2 x 3 x_{0}\leq x_{1}\leq x_{2}\leq x_{3}\leq...
  2. R S P [ S + ( S × R ) ] R\approx S\rightarrow P[S+(S\times R)]
  3. x 0 x 1 x 2 x 3 x_{0}\leq x_{1}\leq x_{2}\leq x_{3}\leq...
  4. x 0 x 1 x 2 x 3 x_{0}\leq x_{1}\leq x_{2}\leq x_{3}\leq...

Density_contrast.html

  1. Δ = ϱ ϱ \Delta=\frac{\partial\varrho}{\varrho}

Density_on_a_manifold.html

  1. μ ( A v 1 , , A v n ) = | det A | μ ( v 1 , , v n ) , A GL ( V ) . \mu(Av_{1},\ldots,Av_{n})=\left|\det A\right|\mu(v_{1},\ldots,v_{n}),\quad A% \in\operatorname{GL}(V).
  2. | ω | |ω|
  3. | ω | ( v 1 , , v n ) := | ω ( v 1 , , v n ) | . |\omega|(v_{1},\ldots,v_{n}):=|\omega(v_{1},\ldots,v_{n})|.
  4. o ( A v 1 , , A v n ) = sign ( det A ) o ( v 1 , , v n ) , A GL ( V ) o(Av_{1},\ldots,Av_{n})=\operatorname{sign}(\det A)o(v_{1},\ldots,v_{n}),\quad A% \in\operatorname{GL}(V)
  5. o ( v 1 , , v n ) | ω | ( v 1 , , v n ) = ω ( v 1 , , v n ) , o(v_{1},\ldots,v_{n})|\omega|(v_{1},\ldots,v_{n})=\omega(v_{1},\ldots,v_{n}),
  6. ω ( v 1 , , v n ) = o ( v 1 , , v n ) μ ( v 1 , , v n ) . \omega(v_{1},\ldots,v_{n})=o(v_{1},\ldots,v_{n})\mu(v_{1},\ldots,v_{n}).
  7. Or ( V ) Vol ( V ) = n V * , Vol ( V ) = Or ( V ) n V * . \operatorname{Or}(V)\otimes\operatorname{Vol}(V)=\bigwedge^{n}V^{*},\quad% \operatorname{Vol}(V)=\operatorname{Or}(V)\otimes\bigwedge^{n}V^{*}.
  8. μ ( A v 1 , , A v n ) = | det A | s μ ( v 1 , , v n ) , A GL ( V ) . \mu(Av_{1},\ldots,Av_{n})=\left|\det A\right|^{s}\mu(v_{1},\ldots,v_{n}),\quad A% \in\operatorname{GL}(V).
  9. | ω | s ( v 1 , , v n ) := | ω ( v 1 , , v n ) | s . |\omega|^{s}(v_{1},\ldots,v_{n}):=|\omega(v_{1},\ldots,v_{n})|^{s}.
  10. μ ( v 1 , , v n ) := μ 1 ( v 1 , , v n ) μ 2 ( v 1 , , v n ) . \mu(v_{1},\ldots,v_{n}):=\mu_{1}(v_{1},\ldots,v_{n})\mu_{2}(v_{1},\ldots,v_{n}).
  11. Vol s 1 ( V ) Vol s 2 ( V ) = Vol s 1 + s 2 ( V ) . \operatorname{Vol}^{s_{1}}(V)\otimes\operatorname{Vol}^{s_{2}}(V)=% \operatorname{Vol}^{s_{1}+s_{2}}(V).
  12. ρ ( A ) = | det A | - s , A GL ( n ) \rho(A)=\left|\det A\right|^{-s},\quad A\in\operatorname{GL}(n)
  13. | Λ | M s = | Λ | s ( T M ) . \left|\Lambda\right|^{s}_{M}=\left|\Lambda\right|^{s}(TM).
  14. | Λ | M s \left|\Lambda\right|^{s}_{M}
  15. t α : | Λ | M s | U α ϕ α ( U α ) × t_{\alpha}:\left|\Lambda\right|^{s}_{M}|_{U_{\alpha}}\to\phi_{\alpha}(U_{% \alpha})\times\mathbb{R}
  16. t α β = | det ( d ϕ α d ϕ β - 1 ) | - s . t_{\alpha\beta}=\left|\det(d\phi_{\alpha}\circ d\phi_{\beta}^{-1})\right|^{-s}.
  17. U α f = ϕ α ( U α ) t α f ϕ α - 1 d μ \int_{U_{\alpha}}f=\int_{\phi_{\alpha}(U_{\alpha})}t_{\alpha}\circ f\circ\phi_% {\alpha}^{-1}d\mu
  18. | Λ | M 1 |\Lambda|^{1}_{M}
  19. | ϕ | p = ( | ϕ | p ) 1 / p < |\phi|_{p}=(\int|\phi|^{p})^{1/p}<\infty
  20. L p ( M ) L^{p}(M)
  21. ρ ( A ) = | det A | - s / n . \rho(A)=\left|\det A\right|^{-s/n}.
  22. | Λ | M s |\Lambda|^{s}_{M}
  23. | Λ | M - s |\Lambda|^{-s}_{M}

Dependency_graph.html

  1. R S × S R\subseteq S\times S
  2. ( a , b ) R (a,b)\in R
  3. G = ( S , T ) G=(S,T)
  4. T R T\subseteq R
  5. S = A , B , C , D S={A,B,C,D}
  6. R = ( A , B ) , ( A , C ) , ( B , D ) R={(A,B),(A,C),(B,D)}
  7. n : S n:S\rightarrow\mathbb{N}
  8. n ( a ) < n ( b ) ( a , b ) R n(a)<n(b)\Rightarrow(a,b)\notin R
  9. a , b S a,b\in S

Derivation_of_self_inductance.html

  1. M i j = μ 0 4 π C i C j 𝐝𝐬 i 𝐝𝐬 j | 𝐑 i j | M_{ij}=\frac{\mu_{0}}{4\pi}\oint_{C_{i}}\oint_{C_{j}}\frac{\mathbf{ds}_{i}% \cdot\mathbf{ds}_{j}}{|\mathbf{R}_{ij}|}
  2. Φ i = S i 𝐁 𝐝𝐚 = S i ( × 𝐀 ) 𝐝𝐚 = C i 𝐀 𝐝𝐬 = C i ( j μ 0 I j 4 π C j 𝐝𝐬 j | 𝐑 | ) 𝐝𝐬 i \Phi_{i}=\int_{S_{i}}\mathbf{B}\cdot\mathbf{da}=\int_{S_{i}}(\nabla\times% \mathbf{A})\cdot\mathbf{da}=\oint_{C_{i}}\mathbf{A}\cdot\mathbf{ds}=\oint_{C_{% i}}\left(\sum_{j}\frac{\mu_{0}I_{j}}{4\pi}\oint_{C_{j}}\frac{\mathbf{ds}_{j}}{% |\mathbf{R}|}\right)\cdot\mathbf{ds}_{i}
  3. Φ i \Phi_{i}\ \,
  4. M i j = def Φ i I j = μ 0 4 π C i C j 𝐝𝐬 i 𝐝𝐬 j | 𝐑 i j | M_{ij}\ \stackrel{\mathrm{def}}{=}\ \frac{\Phi_{i}}{I_{j}}=\frac{\mu_{0}}{4\pi% }\oint_{C_{i}}\oint_{C_{j}}\frac{\mathbf{ds}_{i}\cdot\mathbf{ds}_{j}}{|\mathbf% {R}_{ij}|}
  5. M = M i i = μ 0 4 π ( C C 𝐝𝐬 𝐝𝐬 | 𝐑 s s | ) | 𝐑 | > a / 2 + μ 0 2 π l Y + O ( μ 0 a ) . M=M_{ii}=\frac{\mu_{0}}{4\pi}\left(\oint_{C}\oint_{C^{\prime}}\frac{\mathbf{ds% }\cdot\mathbf{ds}^{\prime}}{|\mathbf{R}_{ss^{\prime}}|}\right)_{|\mathbf{R}|>a% /2}+\frac{\mu_{0}}{2\pi}lY+O\left(\mu_{0}a\right).

Describing_function.html

  1. H ( A , j ω ) H(A,\,j\omega)
  2. ω \omega
  3. H ( j ω ) H(j\omega)
  4. H ( A , j ω ) H(A,\,j\omega)
  5. ω \omega

Descriptivist_theory_of_names.html

  1. x ( K ( x ) y ( K ( y ) x = y ) B ( x ) ) \exists x(K(x)\land\forall y(K(y)\rightarrow x=y)\land B(x))

Deterministic_pushdown_automaton.html

  1. M = ( Q , Σ , Γ , q 0 , Z 0 , A , δ ) M=(Q\,,\Sigma\,,\Gamma\,,q_{0}\,,Z_{0}\,,A\,,\delta\,)
  2. Q Q\,
  3. Σ \Sigma\,
  4. Γ \Gamma\,
  5. q 0 Q q_{0}\,\in Q\,
  6. Z 0 Γ Z_{0}\,\in\Gamma\,
  7. A Q A\,\subseteq Q\,
  8. A A
  9. δ \delta\,
  10. δ : ( Q × ( Σ { ε } ) × Γ ) 𝒫 ( Q × Γ * ) \delta\colon(Q\,\times(\Sigma\,\cup\left\{\varepsilon\,\right\})\times\Gamma\,% )\longrightarrow\mathcal{P}(Q\times\Gamma^{*})
  11. * *
  12. Γ * \Gamma^{*}
  13. ε \varepsilon
  14. Γ \Gamma
  15. ε \varepsilon
  16. 𝒫 ( X ) \mathcal{P}(X)
  17. X X
  18. q Q , a Σ { ε } , x Γ q\in Q,a\in\Sigma\cup\left\{\varepsilon\right\},x\in\Gamma
  19. δ ( q , a , x ) \delta(q,a,x)\,
  20. q Q , x Γ q\in Q,x\in\Gamma
  21. δ ( q , ε , x ) \delta(q,\varepsilon,x)\not=\emptyset\,
  22. δ ( q , a , x ) = \delta\left(q,a,x\right)=\emptyset
  23. a Σ . a\in\Sigma.
  24. L ( A ) L(A)
  25. A A
  26. L ( A ) L(A)
  27. L ( A ) L(A)

Diaeresis_(diacritic).html

  1. a ˙ = d a d t {\dot{a}}={\mathrm{d}a\over\mathrm{d}t}
  2. a ¨ = d 2 a d t 2 {\ddot{a}}={\mathrm{d}^{2}a\over\mathrm{d}t^{2}}
  3. f ( x ) = d d x f ( x ) f^{\prime}(x)={\mathrm{d}\over\mathrm{d}x}f(x)
  4. f ′′ ( x ) = d 2 d x 2 f ( x ) f^{\prime\prime}(x)={\mathrm{d}^{2}\over\mathrm{d}x^{2}}f(x)

Diagonal_functor.html

  1. a a
  2. 𝒞 \mathcal{C}
  3. a × a a\times a
  4. δ a : a a × a \delta_{a}:a\rightarrow a\times a
  5. π k δ a = i d a \pi_{k}\circ\delta_{a}=id_{a}
  6. k { 1 , 2 } k\in\{1,2\}
  7. π k \pi_{k}
  8. k k
  9. x x
  10. a a
  11. δ a ( x ) = x , x \delta_{a}(x)=\langle x,x\rangle
  12. x x
  13. 2 \mathbb{R}\rightarrow\mathbb{R}^{2}
  14. y = x y=x
  15. X X^{\infty}
  16. X X
  17. 𝒞 𝒞 × 𝒞 \mathcal{C}\rightarrow\mathcal{C}\times\mathcal{C}
  18. Δ ( a ) = a , a \Delta(a)=\langle a,a\rangle
  19. 𝒞 \mathcal{C}
  20. a × b a\times b
  21. Δ \Delta
  22. a , b \langle a,b\rangle
  23. 𝒞 𝒥 \mathcal{C}^{\mathcal{J}}
  24. 𝒥 \mathcal{J}
  25. a a
  26. 𝒞 \mathcal{C}
  27. a a
  28. Δ ( a ) 𝒞 𝒥 \Delta(a)\in\mathcal{C}^{\mathcal{J}}
  29. Δ : 𝒞 𝒞 𝒥 \Delta:\mathcal{C}\rightarrow\mathcal{C}^{\mathcal{J}}
  30. 𝒞 \mathcal{C}
  31. Δ ( a ) \Delta(a)
  32. f : a b f:a\rightarrow b
  33. 𝒞 \mathcal{C}
  34. η \eta
  35. 𝒞 𝒥 \mathcal{C}^{\mathcal{J}}
  36. η j = f \eta_{j}=f
  37. 𝒥 \mathcal{J}
  38. 𝒞 𝒞 × 𝒞 \mathcal{C}\rightarrow\mathcal{C}\times\mathcal{C}
  39. : 𝒥 𝒞 \mathcal{F}:\mathcal{J}\rightarrow\mathcal{C}
  40. Δ \Delta
  41. \mathcal{F}
  42. F Δ F\rightarrow\Delta
  43. 𝒥 \mathcal{J}
  44. 𝒞 \mathcal{C}
  45. 𝒞 \mathcal{C}
  46. 𝒞 𝒥 \mathcal{C}^{\mathcal{J}}
  47. 𝒞 \mathcal{C}
  48. 𝒞 𝒞 × 𝒞 \mathcal{C}\rightarrow\mathcal{C}\times\mathcal{C}

Diamond-square_algorithm.html

  1. [ 0 2 4 8 ] \begin{bmatrix}0&2\\ 4&8\\ \end{bmatrix}
  2. [ 0 ( 0 + 2 ) / 2 ( 0 + 4 ) / 2 ( 0 + 2 + 4 + 8 ) / 4 ] = [ 0 1 2 3.5 ] \begin{bmatrix}0&(0+2)/2\\ (0+4)/2&(0+2+4+8)/4\\ \end{bmatrix}=\begin{bmatrix}0&1\\ 2&3.5\\ \end{bmatrix}

Diamonds_as_an_investment.html

  1. P r i c e = W 2 * C Price=W^{2}*C

Dielectric_resonator.html

  1. ε r \varepsilon_{r}
  2. T E 01 n TE_{01n}
  3. T E 01 n TE_{01n}
  4. f G H z = 34 a ε r ( a L + 3.45 ) f_{GHz}=\frac{34}{a\sqrt{\varepsilon_{r}}}\left(\frac{a}{L}+3.45\right)
  5. a a
  6. L L
  7. a a
  8. L L
  9. f G H z f_{GHz}
  10. 0.5 < a L < 2 0.5<\frac{a}{L}<2
  11. 30 < ε r < 50 30<\varepsilon_{r}<50
  12. T E 01 n TE_{01n}

Difference_hierarchy.html

  1. { A : C , D Γ ( A = C D ) } \{A:\exists C,D\in\Gamma(A=C\setminus D)\}
  2. { A : C , D , E Γ ( A = C ( D E ) ) } \{A:\exists C,D,E\in\Gamma(A=C\setminus(D\setminus E))\}

Difference_in_differences.html

  1. y i s t = γ s + λ t + δ D s t + ϵ i s t y_{ist}~{}=~{}\gamma_{s}+\lambda_{t}+\delta D_{st}+\epsilon_{ist}
  2. y i s t y_{ist}
  3. i i
  4. s s
  5. t t
  6. s s
  7. t t
  8. γ s \gamma_{s}
  9. λ t \lambda_{t}
  10. s s
  11. t t
  12. D s t D_{st}
  13. δ \delta
  14. ϵ i s t \epsilon_{ist}
  15. y ¯ s t = 1 n i = 1 n y i s t \overline{y}_{st}~{}=~{}\frac{1}{n}\sum_{i=1}^{n}y_{ist}
  16. γ ¯ s = 1 n i = 1 n γ s = γ s \overline{\gamma}_{s}~{}=~{}\frac{1}{n}\sum_{i=1}^{n}\gamma_{s}~{}=~{}\gamma_{s}
  17. λ ¯ t = 1 n i = 1 n λ t = λ t \overline{\lambda}_{t}~{}=~{}\frac{1}{n}\sum_{i=1}^{n}\lambda_{t}~{}=~{}% \lambda_{t}
  18. D ¯ s t = 1 n i = 1 n D s t = D s t \overline{D}_{st}~{}=~{}\frac{1}{n}\sum_{i=1}^{n}D_{st}~{}=~{}D_{st}
  19. ϵ ¯ s t = 1 n i = 1 n ϵ i s t \overline{\epsilon}_{st}~{}=~{}\frac{1}{n}\sum_{i=1}^{n}\epsilon_{ist}
  20. s = 1 , 2 s=1,2
  21. t = 1 , 2 t=1,2
  22. ( y ¯ 11 - y ¯ 12 ) - ( y ¯ 21 - y ¯ 22 ) (\overline{y}_{11}-\overline{y}_{12})-(\overline{y}_{21}-\overline{y}_{22})
  23. = [ ( γ 1 + λ 1 + δ D 11 + ϵ ¯ 11 ) - ( γ 1 + λ 2 + δ D 12 + ϵ ¯ 12 ) ] - [ ( γ 2 + λ 1 + δ D 21 + ϵ ¯ 21 ) - ( γ 2 + λ 2 + δ D 22 + ϵ ¯ 22 ) ] =\left[(\gamma_{1}+\lambda_{1}+\delta D_{11}+\overline{\epsilon}_{11})-(\gamma% _{1}+\lambda_{2}+\delta D_{12}+\overline{\epsilon}_{12})\right]-\left[(\gamma_% {2}+\lambda_{1}+\delta D_{21}+\overline{\epsilon}_{21})-(\gamma_{2}+\lambda_{2% }+\delta D_{22}+\overline{\epsilon}_{22})\right]
  24. = δ ( D 11 - D 12 ) + δ ( D 22 - D 21 ) + ϵ ¯ 11 - ϵ ¯ 12 + ϵ ¯ 22 - ϵ ¯ 21 =\delta(D_{11}-D_{12})+\delta(D_{22}-D_{21})+\overline{\epsilon}_{11}-% \overline{\epsilon}_{12}+\overline{\epsilon}_{22}-\overline{\epsilon}_{21}
  25. E [ ( y ¯ 11 - y ¯ 12 ) - ( y ¯ 21 - y ¯ 22 ) ] = δ ( D 11 - D 12 ) + δ ( D 22 - D 21 ) E\left[(\overline{y}_{11}-\overline{y}_{12})-(\overline{y}_{21}-\overline{y}_{% 22})\right]~{}=~{}\delta(D_{11}-D_{12})+\delta(D_{22}-D_{21})
  26. D 22 = 1 D_{22}=1
  27. D 11 = D 12 = D 21 = 0 D_{11}=D_{12}=D_{21}=0
  28. δ ^ = ( y ¯ 11 - y ¯ 12 ) - ( y ¯ 21 - y ¯ 22 ) \hat{\delta}~{}=~{}(\overline{y}_{11}-\overline{y}_{12})-(\overline{y}_{21}-% \overline{y}_{22})
  29. D s t D_{st}
  30. λ 2 - λ 1 \lambda_{2}-\lambda_{1}
  31. s = 1 s=1
  32. s = 2 s=2
  33. λ s t : λ 22 - λ 21 λ 12 - λ 11 \lambda_{st}~{}:~{}\lambda_{22}-\lambda_{21}\neq\lambda_{12}-\lambda_{11}
  34. y s t y_{st}
  35. s = 2 s=2
  36. s = 1 s=1
  37. t = 2 t=2
  38. y 22 y_{22}
  39. y 12 y_{12}
  40. y 12 - y 22 y_{12}-y_{22}
  41. t = 1 t=1
  42. y 21 y_{21}
  43. y 11 y_{11}
  44. y 11 - y 21 y_{11}-y_{21}
  45. y 21 - y 22 y_{21}-y_{22}
  46. y 11 - y 12 y_{11}-y_{12}
  47. ( y 11 - y 21 ) - ( y 12 - y 22 ) (y_{11}-y_{21})-(y_{12}-y_{22})
  48. y = β 0 + β 1 T + β 2 S + β 3 ( T S ) + ε y~{}=~{}\beta_{0}+\beta_{1}T+\beta_{2}S+\beta_{3}(T\cdot S)+\varepsilon
  49. T T
  50. t = 2 t=2
  51. S S
  52. s = 2 s=2
  53. ( T S ) (T\cdot S)
  54. S = T = 1 S=T=1
  55. β ^ 0 = ( y | T = 0 , S = 0 ) \hat{\beta}_{0}~{}=~{}(y~{}|~{}T=0,~{}S=0)
  56. β ^ 1 = ( y | T = 1 , S = 0 ) - ( y | T = 0 , S = 0 ) \hat{\beta}_{1}~{}=~{}(y~{}|~{}T=1,~{}S=0)-(y~{}|~{}T=0,~{}S=0)
  57. β ^ 2 = ( y | T = 0 , S = 1 ) - ( y | T = 0 , S = 0 ) \hat{\beta}_{2}~{}=~{}(y~{}|~{}T=0,~{}S=1)-(y~{}|~{}T=0,~{}S=0)
  58. β ^ 3 = [ ( y | T = 1 , S = 1 ) - ( y | T = 0 , S = 1 ) ] - [ ( y | T = 1 , S = 0 ) - ( y | T = 0 , S = 0 ) ] \hat{\beta}_{3}~{}=~{}[(y~{}|~{}T=1,~{}S=1)-(y~{}|~{}T=0,~{}S=1)]-[(y~{}|~{}T=% 1,~{}S=0)-(y~{}|~{}T=0,~{}S=0)]
  59. β ^ 3 = ( y 11 - y 21 ) - ( y 12 - y 22 ) \hat{\beta}_{3}~{}=~{}(y_{11}-y_{21})-(y_{12}-y_{22})

Difference_of_Gaussians.html

  1. I : { 𝕏 n } { 𝕐 m } I:\{\mathbb{X}\subseteq\mathbb{R}^{n}\}\rightarrow\{\mathbb{Y}\subseteq\mathbb% {R}^{m}\}
  2. I I
  3. Γ σ 1 , σ 2 : { 𝕏 n } { } \Gamma_{\sigma_{1},\sigma_{2}}:\{\mathbb{X}\subseteq\mathbb{R}^{n}\}% \rightarrow\{\mathbb{Z}\subseteq\mathbb{R}\}
  4. I I
  5. σ 2 2 \sigma^{2}_{2}
  6. I I
  7. σ 1 2 \sigma^{2}_{1}
  8. σ 2 > σ 1 \sigma_{2}>\sigma_{1}
  9. Γ \Gamma
  10. Γ σ 1 , σ 2 ( x ) = I * 1 σ 1 2 π e - ( x 2 ) / ( 2 σ 1 2 ) - I * 1 σ 2 2 π e - ( x 2 ) / ( 2 σ 2 2 ) . \Gamma_{\sigma_{1},\sigma_{2}}(x)=I*\frac{1}{\sigma_{1}\sqrt{2\pi}}\,e^{-(x^{2% })/(2\sigma^{2}_{1})}-I*\frac{1}{\sigma_{2}\sqrt{2\pi}}\,e^{-(x^{2})/(2\sigma_% {2}^{2})}.
  11. Γ σ , K σ ( x , y ) = I * 1 2 π σ 2 e - ( x 2 + y 2 ) / ( 2 σ 2 ) - I * 1 2 π K 2 σ 2 e - ( x 2 + y 2 ) / ( 2 K 2 σ 2 ) \Gamma_{\sigma,K\sigma}(x,y)=I*\frac{1}{2\pi\sigma^{2}}e^{-(x^{2}+y^{2})/(2% \sigma^{2})}-I*\frac{1}{2\pi K^{2}\sigma^{2}}e^{-(x^{2}+y^{2})/(2K^{2}\sigma^{% 2})}
  12. Γ σ , K σ ( x , y ) = I * ( 1 2 π σ 2 e - ( x 2 + y 2 ) / ( 2 σ 2 ) - 1 2 π K 2 σ 2 e - ( x 2 + y 2 ) / ( 2 K 2 σ 2 ) ) \Gamma_{\sigma,K\sigma}(x,y)=I*(\frac{1}{2\pi\sigma^{2}}e^{-(x^{2}+y^{2})/(2% \sigma^{2})}-\frac{1}{2\pi K^{2}\sigma^{2}}e^{-(x^{2}+y^{2})/(2K^{2}\sigma^{2}% )})

Difference_polynomials.html

  1. p n ( z ) = z n ( z - β n - 1 n - 1 ) p_{n}(z)=\frac{z}{n}{{z-\beta n-1}\choose{n-1}}
  2. ( z n ) {z\choose n}
  3. β = 0 \beta=0
  4. p n ( z ) p_{n}(z)
  5. p n ( z ) = ( z n ) = z ( z - 1 ) ( z - n + 1 ) n ! . p_{n}(z)={z\choose n}=\frac{z(z-1)\cdots(z-n+1)}{n!}.
  6. β = 1 \beta=1
  7. β = - 1 / 2 \beta=-1/2
  8. f ( z ) f(z)
  9. n ( f ) = Δ n f ( β n ) \mathcal{L}_{n}(f)=\Delta^{n}f(\beta n)
  10. Δ \Delta
  11. f ( z ) = n = 0 p n ( z ) n ( f ) . f(z)=\sum_{n=0}^{\infty}p_{n}(z)\mathcal{L}_{n}(f).
  12. e z t = n = 0 p n ( z ) [ ( e t - 1 ) e β t ] n . e^{zt}=\sum_{n=0}^{\infty}p_{n}(z)\left[\left(e^{t}-1\right)e^{\beta t}\right]% ^{n}.
  13. K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = n = 0 p n ( z ) w n K(z,w)=A(w)\Psi(zg(w))=\sum_{n=0}^{\infty}p_{n}(z)w^{n}
  14. A ( w ) = 1 A(w)=1
  15. Ψ ( x ) = e x \Psi(x)=e^{x}
  16. g ( w ) = t g(w)=t
  17. w = ( e t - 1 ) e β t w=(e^{t}-1)e^{\beta t}

Differential_coding.html

  1. x i x_{i}
  2. y i y_{i}
  3. y i = y i - 1 x i , ( 1 ) y_{i}=y_{i-1}\oplus x_{i},\qquad(1)
  4. x i = y i y i - 1 . ( 2 ) x_{i}=y_{i}\oplus y_{i-1}.\qquad(2)
  5. \oplus{}
  6. x i x_{i}
  7. y i y_{i}
  8. y i - 1 y_{i-1}
  9. ( 1 ) (1)
  10. ( 2 ) (2)
  11. h ( k ) = { 1 , if k 0 0 , if k < 0 h(k)=\begin{cases}1,&\mbox{if }~{}k\geq 0\\ 0,&\mbox{if }~{}k<0\end{cases}
  12. H ( z ) = 1 1 - z - 1 . \ H(z)=\frac{1}{1-z^{-1}}.
  13. h ( k ) = { 1 , if k = 0 - 1 , if k = 1 0 , otherwise h(k)=\begin{cases}1,&\mbox{if }~{}k=0\\ -1,&\mbox{if }~{}k=1\\ 0,&\mbox{otherwise}\end{cases}
  14. H ( z ) = 1 - z - 1 . \ H(z)=1-z^{-1}.
  15. y i - 1 x i = y i y_{i-1}\oplus x_{i}=y_{i}
  16. u = F ( y , x ) u=F(y,x)
  17. u 0 = F ( y 0 , x ) u_{0}=F(y_{0},x)
  18. y 0 y_{0}
  19. u 0 u_{0}
  20. y i y_{i}
  21. x i x_{i}
  22. x i + 1 x_{i+1}
  23. x i = y i y i - 1 x_{i}=y_{i}\oplus y_{i-1}
  24. x i + 1 = y i + 1 y i x_{i+1}=y_{i+1}\oplus y_{i}

Differential_entropy.html

  1. 𝕏 \mathbb{X}
  2. h ( X ) = - 𝕏 f ( x ) log f ( x ) d x h(X)=-\int_{\mathbb{X}}f(x)\log f(x)\,dx
  3. h ( Q ) = 0 1 log Q ( p ) d p h(Q)=\int_{0}^{1}\log Q^{\prime}(p)\,dp
  4. 0 1 2 - 2 log ( 2 ) d x = - log ( 2 ) \int_{0}^{\frac{1}{2}}-2\log(2)\,dx=-\log(2)\,
  5. h ( X 1 , , X n ) = i = 1 n h ( X i | X 1 , , X i - 1 ) h ( X i ) h(X_{1},\ldots,X_{n})=\sum_{i=1}^{n}h(X_{i}|X_{1},\ldots,X_{i-1})\leq\sum h(X_% {i})
  6. h ( 𝐘 ) h ( 𝐗 ) + f ( x ) log | m x | d x h(\mathbf{Y})\leq h(\mathbf{X})+\int f(x)\log\left|\frac{\partial m}{\partial x% }\right|dx
  7. | m x | \left|\frac{\partial m}{\partial x}\right|
  8. h ( 𝐗 ) 1 2 log [ ( 2 π e ) n det K ] h(\mathbf{X})\leq\frac{1}{2}\log[(2\pi e)^{n}\det{K}]
  9. 0 D K L ( f | | g ) = - f ( x ) log ( f ( x ) g ( x ) ) d x = - h ( f ) - - f ( x ) log ( g ( x ) ) d x . 0\leq D_{KL}(f||g)=\int_{-\infty}^{\infty}f(x)\log\left(\frac{f(x)}{g(x)}% \right)dx=-h(f)-\int_{-\infty}^{\infty}f(x)\log(g(x))dx.
  10. - f ( x ) log ( g ( x ) ) d x = - f ( x ) log ( 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 ) d x = - f ( x ) log 1 2 π σ 2 d x + log ( e ) - f ( x ) ( - ( x - μ ) 2 2 σ 2 ) d x = - 1 2 log ( 2 π σ 2 ) - log ( e ) σ 2 2 σ 2 = - 1 2 ( log ( 2 π σ 2 ) + log ( e ) ) = - 1 2 log ( 2 π e σ 2 ) = - h ( g ) \begin{aligned}\displaystyle\int_{-\infty}^{\infty}f(x)\log(g(x))dx&% \displaystyle=\int_{-\infty}^{\infty}f(x)\log\left(\frac{1}{\sqrt{2\pi\sigma^{% 2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}\right)dx\\ &\displaystyle=\int_{-\infty}^{\infty}f(x)\log\frac{1}{\sqrt{2\pi\sigma^{2}}}% dx+\log(e)\int_{-\infty}^{\infty}f(x)\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}% \right)dx\\ &\displaystyle=-\tfrac{1}{2}\log(2\pi\sigma^{2})-\log(e)\frac{\sigma^{2}}{2% \sigma^{2}}\\ &\displaystyle=-\tfrac{1}{2}\left(\log(2\pi\sigma^{2})+\log(e)\right)\\ &\displaystyle=-\tfrac{1}{2}\log(2\pi e\sigma^{2})\\ &\displaystyle=-h(g)\end{aligned}
  11. h ( g ) - h ( f ) 0 h(g)-h(f)\geq 0\!
  12. L = - g ( x ) ln ( g ( x ) ) d x - λ 0 ( 1 - - g ( x ) d x ) - λ ( σ 2 - - g ( x ) ( x - μ ) 2 d x ) L=\int_{-\infty}^{\infty}g(x)\ln(g(x))\,dx-\lambda_{0}\left(1-\int_{-\infty}^{% \infty}g(x)\,dx\right)-\lambda\left(\sigma^{2}-\int_{-\infty}^{\infty}g(x)(x-% \mu)^{2}\,dx\right)
  13. ( 1 = - g ( x ) d x ) \left(1=\int_{-\infty}^{\infty}g(x)\,dx\right)
  14. ( σ 2 = - g ( x ) ( x - μ ) 2 d x ) \left(\sigma^{2}=\int_{-\infty}^{\infty}g(x)(x-\mu)^{2}\,dx\right)
  15. 0 = δ L = - δ g ( x ) ( ln ( g ( x ) ) + 1 + λ 0 + λ ( x - μ ) 2 ) d x 0=\delta L=\int_{-\infty}^{\infty}\delta g(x)\left(\ln(g(x))+1+\lambda_{0}+% \lambda(x-\mu)^{2}\right)\,dx
  16. g ( x ) = e - λ 0 - 1 - λ ( x - μ ) 2 g(x)=e^{-\lambda_{0}-1-\lambda(x-\mu)^{2}}
  17. g ( x ) = 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 g(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}
  18. f ( x ) = λ e - λ x for x 0. f(x)=\lambda e^{-\lambda x}\mbox{ for }~{}x\geq 0.
  19. h e ( X ) h_{e}(X)\,
  20. = - 0 λ e - λ x log ( λ e - λ x ) d x =-\int_{0}^{\infty}\lambda e^{-\lambda x}\log(\lambda e^{-\lambda x})\,dx
  21. = - ( 0 ( log λ ) λ e - λ x d x + 0 ( - λ x ) λ e - λ x d x ) =-\left(\int_{0}^{\infty}(\log\lambda)\lambda e^{-\lambda x}\,dx+\int_{0}^{% \infty}(-\lambda x)\lambda e^{-\lambda x}\,dx\right)
  22. = - log λ 0 f ( x ) d x + λ E [ X ] =-\log\lambda\int_{0}^{\infty}f(x)\,dx+\lambda E[X]
  23. = - log λ + 1 . =-\log\lambda+1\,.
  24. h e ( X ) h_{e}(X)
  25. h ( X ) h(X)
  26. Γ ( x ) = 0 e - t t x - 1 d t \Gamma(x)=\int_{0}^{\infty}e^{-t}t^{x-1}dt
  27. ψ ( x ) = d d x ln Γ ( x ) = Γ ( x ) Γ ( x ) \psi(x)=\frac{d}{dx}\ln\Gamma(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}
  28. B ( p , q ) = Γ ( p ) Γ ( q ) Γ ( p + q ) B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}
  29. f ( x ) = 1 b - a f(x)=\frac{1}{b-a}
  30. ln ( b - a ) \ln(b-a)\,
  31. [ a , b ] [a,b]\,
  32. f ( x ) = 1 2 π σ 2 exp ( - ( x - μ ) 2 2 σ 2 ) f(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)
  33. ln ( σ 2 π e ) \ln\left(\sigma\sqrt{2\,\pi\,e}\right)
  34. ( - , ) (-\infty,\infty)\,
  35. f ( x ) = λ exp ( - λ x ) f(x)=\lambda\exp\left(-\lambda x\right)
  36. 1 - ln λ 1-\ln\lambda\,
  37. [ 0 , ) [0,\infty)\,
  38. f ( x ) = x σ 2 exp ( - x 2 2 σ 2 ) f(x)=\frac{x}{\sigma^{2}}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)
  39. 1 + ln σ 2 + γ E 2 1+\ln\frac{\sigma}{\sqrt{2}}+\frac{\gamma_{E}}{2}
  40. [ 0 , ) [0,\infty)\,
  41. f ( x ) = x α - 1 ( 1 - x ) β - 1 B ( α , β ) f(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}
  42. 0 x 1 0\leq x\leq 1
  43. ln B ( α , β ) - ( α - 1 ) [ ψ ( α ) - ψ ( α + β ) ] \ln B(\alpha,\beta)-(\alpha-1)[\psi(\alpha)-\psi(\alpha+\beta)]\,
  44. - ( β - 1 ) [ ψ ( β ) - ψ ( α + β ) ] -(\beta-1)[\psi(\beta)-\psi(\alpha+\beta)]\,
  45. [ 0 , 1 ] [0,1]\,
  46. f ( x ) = γ π 1 γ 2 + x 2 f(x)=\frac{\gamma}{\pi}\frac{1}{\gamma^{2}+x^{2}}
  47. ln ( 4 π γ ) \ln(4\pi\gamma)\,
  48. ( - , ) (-\infty,\infty)\,
  49. f ( x ) = 2 2 k / 2 Γ ( k / 2 ) x k - 1 exp ( - x 2 2 ) f(x)=\frac{2}{2^{k/2}\Gamma(k/2)}x^{k-1}\exp\left(-\frac{x^{2}}{2}\right)
  50. ln Γ ( k / 2 ) 2 - k - 1 2 ψ ( k 2 ) + k 2 \ln{\frac{\Gamma(k/2)}{\sqrt{2}}}-\frac{k-1}{2}\psi\left(\frac{k}{2}\right)+% \frac{k}{2}
  51. [ 0 , ) [0,\infty)\,
  52. f ( x ) = 1 2 k / 2 Γ ( k / 2 ) x k 2 - 1 exp ( - x 2 ) f(x)=\frac{1}{2^{k/2}\Gamma(k/2)}x^{\frac{k}{2}\!-\!1}\exp\left(-\frac{x}{2}\right)
  53. ln 2 Γ ( k 2 ) - ( 1 - k 2 ) ψ ( k 2 ) + k 2 \ln 2\Gamma\left(\frac{k}{2}\right)-\left(1-\frac{k}{2}\right)\psi\left(\frac{% k}{2}\right)+\frac{k}{2}
  54. [ 0 , ) [0,\infty)\,
  55. f ( x ) = λ k ( k - 1 ) ! x k - 1 exp ( - λ x ) f(x)=\frac{\lambda^{k}}{(k-1)!}x^{k-1}\exp(-\lambda x)
  56. ( 1 - k ) ψ ( k ) + ln Γ ( k ) λ + k (1-k)\psi(k)+\ln\frac{\Gamma(k)}{\lambda}+k
  57. [ 0 , ) [0,\infty)\,
  58. f ( x ) = n 1 n 1 2 n 2 n 2 2 B ( n 1 2 , n 2 2 ) x n 1 2 - 1 ( n 2 + n 1 x ) n 1 + n 2 2 f(x)=\frac{n_{1}^{\frac{n_{1}}{2}}n_{2}^{\frac{n_{2}}{2}}}{B(\frac{n_{1}}{2},% \frac{n_{2}}{2})}\frac{x^{\frac{n_{1}}{2}-1}}{(n_{2}+n_{1}x)^{\frac{n_{1}+n2}{% 2}}}
  59. ln n 1 n 2 B ( n 1 2 , n 2 2 ) + ( 1 - n 1 2 ) ψ ( n 1 2 ) - \ln\frac{n_{1}}{n_{2}}B\left(\frac{n_{1}}{2},\frac{n_{2}}{2}\right)+\left(1-% \frac{n_{1}}{2}\right)\psi\left(\frac{n_{1}}{2}\right)-
  60. ( 1 + n 2 2 ) ψ ( n 2 2 ) + n 1 + n 2 2 ψ ( n 1 + n 2 2 ) \left(1+\frac{n_{2}}{2}\right)\psi\left(\frac{n_{2}}{2}\right)+\frac{n_{1}+n_{% 2}}{2}\psi\left(\frac{n_{1}\!+\!n_{2}}{2}\right)
  61. [ 0 , ) [0,\infty)\,
  62. f ( x ) = x k - 1 exp ( - x θ ) θ k Γ ( k ) f(x)=\frac{x^{k-1}\exp(-\frac{x}{\theta})}{\theta^{k}\Gamma(k)}
  63. ln ( θ Γ ( k ) ) + ( 1 - k ) ψ ( k ) + k \ln(\theta\Gamma(k))+(1-k)\psi(k)+k\,
  64. [ 0 , ) [0,\infty)\,
  65. f ( x ) = 1 2 b exp ( - | x - μ | b ) f(x)=\frac{1}{2b}\exp\left(-\frac{|x-\mu|}{b}\right)
  66. 1 + ln ( 2 b ) 1+\ln(2b)\,
  67. ( - , ) (-\infty,\infty)\,
  68. f ( x ) = e - x ( 1 + e - x ) 2 f(x)=\frac{e^{-x}}{(1+e^{-x})^{2}}
  69. 2 2\,
  70. ( - , ) (-\infty,\infty)\,
  71. f ( x ) = 1 σ x 2 π exp ( - ( ln x - μ ) 2 2 σ 2 ) f(x)=\frac{1}{\sigma x\sqrt{2\pi}}\exp\left(-\frac{(\ln x-\mu)^{2}}{2\sigma^{2% }}\right)
  72. μ + 1 2 ln ( 2 π e σ 2 ) \mu+\frac{1}{2}\ln(2\pi e\sigma^{2})
  73. [ 0 , ) [0,\infty)\,
  74. f ( x ) = 1 a 3 2 π x 2 exp ( - x 2 2 a 2 ) f(x)=\frac{1}{a^{3}}\sqrt{\frac{2}{\pi}}\,x^{2}\exp\left(-\frac{x^{2}}{2a^{2}}\right)
  75. ln ( a 2 π ) + γ E - 1 2 \ln(a\sqrt{2\pi})+\gamma_{E}-\frac{1}{2}
  76. [ 0 , ) [0,\infty)\,
  77. f ( x ) = 2 β α 2 Γ ( α 2 ) x α - 1 exp ( - β x 2 ) f(x)=\frac{2\beta^{\frac{\alpha}{2}}}{\Gamma(\frac{\alpha}{2})}x^{\alpha-1}% \exp(-\beta x^{2})
  78. ln Γ ( α / 2 ) 2 β 1 2 - α - 1 2 ψ ( α 2 ) + α 2 \ln{\frac{\Gamma(\alpha/2)}{2\beta^{\frac{1}{2}}}}-\frac{\alpha-1}{2}\psi\left% (\frac{\alpha}{2}\right)+\frac{\alpha}{2}
  79. ( - , ) (-\infty,\infty)\,
  80. f ( x ) = α x m α x α + 1 f(x)=\frac{\alpha x_{m}^{\alpha}}{x^{\alpha+1}}
  81. ln x m α + 1 + 1 α \ln\frac{x_{m}}{\alpha}+1+\frac{1}{\alpha}
  82. [ x m , ) [x_{m},\infty)\,
  83. f ( x ) = ( 1 + x 2 / ν ) - ν + 1 2 ν B ( 1 2 , ν 2 ) f(x)=\frac{(1+x^{2}/\nu)^{-\frac{\nu+1}{2}}}{\sqrt{\nu}B(\frac{1}{2},\frac{\nu% }{2})}
  84. ν + 1 2 ( ψ ( ν + 1 2 ) - ψ ( ν 2 ) ) + ln ν B ( 1 2 , ν 2 ) \frac{\nu\!+\!1}{2}\left(\psi\left(\frac{\nu\!+\!1}{2}\right)\!-\!\psi\left(% \frac{\nu}{2}\right)\right)\!+\!\ln\sqrt{\nu}B\left(\frac{1}{2},\frac{\nu}{2}\right)
  85. ( - , ) (-\infty,\infty)\,
  86. f ( x ) = { 2 ( x - a ) ( b - a ) ( c - a ) for a x c , 2 ( b - x ) ( b - a ) ( b - c ) for c < x b , f(x)=\begin{cases}\frac{2(x-a)}{(b-a)(c-a)}&\mathrm{for\ }a\leq x\leq c,\\ \frac{2(b-x)}{(b-a)(b-c)}&\mathrm{for\ }c<x\leq b,\\ \end{cases}
  87. 1 2 + ln b - a 2 \frac{1}{2}+\ln\frac{b-a}{2}
  88. [ 0 , 1 ] [0,1]\,
  89. f ( x ) = k λ k x k - 1 exp ( - x k λ k ) f(x)=\frac{k}{\lambda^{k}}x^{k-1}\exp\left(-\frac{x^{k}}{\lambda^{k}}\right)
  90. ( k - 1 ) γ E k + ln λ k + 1 \frac{(k-1)\gamma_{E}}{k}+\ln\frac{\lambda}{k}+1
  91. [ 0 , ) [0,\infty)\,
  92. f X ( x ) = f_{X}(\vec{x})=
  93. exp ( - 1 2 ( x - μ ) Σ - 1 ( x - μ ) ) ( 2 π ) N / 2 | Σ | 1 / 2 \frac{\exp\left(-\frac{1}{2}(\vec{x}-\vec{\mu})^{\top}\Sigma^{-1}\cdot(\vec{x}% -\vec{\mu})\right)}{(2\pi)^{N/2}\left|\Sigma\right|^{1/2}}
  94. 1 2 ln { ( 2 π e ) N det ( Σ ) } \frac{1}{2}\ln\{(2\pi e)^{N}\det(\Sigma)\}
  95. ( - , ) (-\vec{\infty},\vec{\infty})\,
  96. D ( p | | m ) = p ( x ) log p ( x ) m ( x ) d x . D(p||m)=\int p(x)\log\frac{p(x)}{m(x)}\,dx.
  97. H h = - i h f ( i h ) log ( f ( i h ) ) - h f ( i h ) log ( h ) . H_{h}=-\sum_{i}hf(ih)\log(f(ih))-\sum hf(ih)\log(h).

Digital_comparator.html

  1. A = A 3 A 2 A 1 A 0 A=A_{3}A_{2}A_{1}A_{0}
  2. B = B 3 B 2 B 1 B 0 B=B_{3}B_{2}B_{1}B_{0}
  3. A 3 = B 3 A_{3}=B_{3}
  4. A 2 = B 2 A_{2}=B_{2}
  5. A 1 = B 1 A_{1}=B_{1}
  6. A 0 = B 0 A_{0}=B_{0}
  7. A i A_{i}
  8. B i B_{i}
  9. x i = A i B i + A ¯ i B ¯ i x_{i}=A_{i}B_{i}+\overline{A}_{i}\overline{B}_{i}
  10. x i x_{i}
  11. A i A_{i}
  12. B i B_{i}
  13. x i x_{i}
  14. ( A = B ) = x 3 x 2 x 1 x 0 (A=B)=x_{3}x_{2}x_{1}x_{0}
  15. ( A > B ) = A 3 B ¯ 3 + x 3 A 2 B ¯ 2 + x 3 x 2 A 1 B ¯ 1 + x 3 x 2 x 1 A 0 B ¯ 0 (A>B)=A_{3}\overline{B}_{3}+x_{3}A_{2}\overline{B}_{2}+x_{3}x_{2}A_{1}% \overline{B}_{1}+x_{3}x_{2}x_{1}A_{0}\overline{B}_{0}
  16. ( A < B ) = A ¯ 3 B 3 + x 3 A ¯ 2 B 2 + x 3 x 2 A ¯ 1 B 1 + x 3 x 2 x 1 A ¯ 0 B 0 (A<B)=\overline{A}_{3}B_{3}+x_{3}\overline{A}_{2}B_{2}+x_{3}x_{2}\overline{A}_% {1}B_{1}+x_{3}x_{2}x_{1}\overline{A}_{0}B_{0}

Dilation_(operator_theory).html

  1. P H V | H = T P_{H}\;V|_{H}=T
  2. P H P_{H}
  3. X X
  4. X \partial X
  5. σ ( V ) X \sigma(V)\in\partial X
  6. P H f ( V ) | H = f ( T ) P_{H}\;f(V)|_{H}=f(T)
  7. D T = ( I - T * T ) 1 2 D_{T}=(I-T^{*}T)^{\frac{1}{2}}
  8. H H H\oplus H
  9. V = [ T D T * D T - T * ] . V=\begin{bmatrix}T&D_{T^{*}}\\ \ D_{T}&-T^{*}\end{bmatrix}.
  10. T D T = D T * T . TD_{T}=D_{T^{*}}T.
  11. T = cos θ T=\cos\theta
  12. V = [ cos θ sin θ sin θ - cos θ ] . V=\begin{bmatrix}\cos\theta&\sin\theta\\ \ \sin\theta&-\cos\theta\end{bmatrix}.
  13. T 2 = P H V 2 | H T^{2}=P_{H}\;V^{2}|_{H}
  14. ( X ) \mathcal{R}(X)
  15. X X
  16. X \partial X

Dilution_(equation).html

  1. C 1 × V 1 = C 2 × V 2 C_{1}\times V_{1}=C_{2}\times V_{2}
  2. C 1 = C_{1}=
  3. V 1 = V_{1}=
  4. C 2 = C_{2}=
  5. V 2 = V_{2}=
  6. D t = [ V Q ] l n [ C i n i t i a l C e n d i n g ] \ D_{t}=\left[\frac{V}{Q}\right]\cdot ln\left[\frac{C_{initial}}{C_{ending}}% \right]\quad
  7. l n [ C e n d i n g C i n i t i a l ] = - Q V ( t e n d i n g - t i n i t i a l ) \ ln\left[\frac{C_{ending}}{C_{initial}}\right]\quad={-}\frac{Q}{V}\cdot(t_{% ending}-t_{initial})\quad
  8. t i n i t i a l = 0 t_{initial}=0
  9. D t = D_{t}=
  10. Q Q
  11. V = V=
  12. Q = Q=
  13. C i n i t i a l = C_{initial}=
  14. C e n d i n g = C_{ending}=
  15. d C d t = G - Q C V {dC\over dt}=\frac{G-Q^{\prime}C}{V}
  16. Q = Q K Q^{\prime}=\frac{Q}{K}
  17. C = C=
  18. G = G=
  19. V = V=
  20. Q = Q^{\prime}=

Dimensional_reduction.html

  1. ϕ \phi
  2. ϕ n = A n cos ( 2 π n x L ) \phi_{n}=A_{n}\cos\left(\frac{2\pi nx}{L}\right)
  3. ϕ \phi

Dinatural_transformation.html

  1. α \alpha
  2. S , T : C op × C X , S,T:C^{\mathrm{op}}\times C\to X,
  3. α : S ¨ T , \alpha:S\ddot{\to}T,
  4. α c : S ( c , c ) T ( c , c ) \alpha_{c}:S(c,c)\to T(c,c)
  5. f : c c f:c\to c^{\prime}

Diode_modelling.html

  1. I I
  2. V D V_{D}
  3. I = I S ( e V D / ( n V T ) - 1 ) I=I_{S}\left(e^{V_{D}/(nV_{T})}-1\right)
  4. I S I_{S}
  5. V D V_{D}
  6. V T V_{T}
  7. V T V_{T}
  8. k T / q kT/q
  9. n n
  10. n n
  11. V D n V T V_{D}\gg nV_{T}
  12. I I S e V D / ( n V T ) I\approx I_{S}\cdot e^{V_{D}/(nV_{T})}
  13. I = V S - V D R I=\frac{V_{S}-V_{D}}{R}
  14. I I
  15. V D V_{D}
  16. V S V_{S}
  17. I I
  18. V S V_{S}
  19. V D V_{D}
  20. W W
  21. f ( w ) = w e w f(w)=we^{w}
  22. w = W ( f ) w=W(f)
  23. w w
  24. w = I S R n V T ( I I S + 1 ) w=\frac{I_{S}R}{nV_{T}}\left(\frac{I}{I_{S}}+1\right)
  25. I / I S = e V D / n V T - 1 I/I_{S}=e^{V_{D}/nV_{T}}-1
  26. w e w = I S R n V T e V D / n V T e I S R n V T ( I I S + 1 ) we^{w}=\frac{I_{S}R}{nV_{T}}e^{V_{D}/nV_{T}}e^{\frac{I_{S}R}{nV_{T}}\left(% \frac{I}{I_{S}}+1\right)}
  27. V D = V S - I R V_{D}=V_{S}-IR
  28. w e w = I S R n V T e V S / n V T e - I R n V T e I R I S n V T I S e I S R / n V T we^{w}=\frac{I_{S}R}{nV_{T}}e^{V_{S}/nV_{T}}e^{\frac{-IR}{nV_{T}}}e^{\frac{IRI% _{S}}{nV_{T}I_{S}}}e^{I_{S}R/nV_{T}}
  29. w e w = I S R n V T e ( V s + I s R ) / ( n V T ) we^{w}=\frac{I_{S}R}{nV_{T}}e^{(V_{s}+I_{s}R)/(nV_{T})}
  30. W W
  31. w = W ( I S R n V T e ( V s + I s R ) / ( n V T ) ) w=W\left(\frac{I_{S}R}{nV_{T}}e^{(V_{s}+I_{s}R)/(nV_{T})}\right)
  32. I s R V S I_{s}R\ll V_{S}
  33. I / I S 1 I/I_{S}\gg 1
  34. I n V T R W ( I S R n V T e V s / ( n V T ) ) I\approx\frac{nV_{T}}{R}W\left(\frac{I_{S}R}{nV_{T}}e^{V_{s}/(nV_{T})}\right)
  35. W ( x ) W(x)
  36. W ( x ) = ln x - ln ln x + o ( 1 ) W(x)=\ln x-\ln\ln x+o(1)
  37. I S R n V T e V s / ( n V T ) \frac{I_{S}R}{nV_{T}}e^{V_{s}/(nV_{T})}
  38. V D V_{D}
  39. V S V_{S}
  40. I S I_{S}
  41. I I S + 1 = e V D / ( n V T ) \frac{I}{I_{S}}+1=e^{V_{D}/(nV_{T})}
  42. V D n V T = ln ( I I S + 1 ) \frac{V_{D}}{nV_{T}}=\ln\left(\frac{I}{I_{S}}+1\right)
  43. I I
  44. V D V_{D}
  45. I I
  46. I I
  47. V D n V T = ln ( ( V S - V D ) / R I S + 1 ) \frac{V_{D}}{nV_{T}}=\ln\left(\frac{(V_{S}-V_{D})/R}{I_{S}}+1\right)
  48. V D = n V T ln ( V S - V D R I S + 1 ) V_{D}=nV_{T}\ln\left(\frac{V_{S}-V_{D}}{RI_{S}}+1\right)
  49. V S V_{S}
  50. V D V_{D}
  51. V D V_{D}
  52. V D V_{D}
  53. V D V_{D}
  54. V D V_{D}
  55. I I
  56. V D = 600 mV V_{D}=600\,\mathrm{mV}
  57. V t V_{t}
  58. I = I S ( e V D / ( n V T ) - 1 ) I=I_{S}\left(e^{V_{D}/(n\cdot V_{T})}-1\right)\Leftrightarrow
  59. ln ( 1 + I I S ) = V D n V T \ln\left(1+\frac{I}{I_{S}}\right)=\frac{V_{D}}{n\cdot V_{T}}\Leftrightarrow
  60. V D = n V T ln ( 1 + I I S ) n V T ln ( I I S ) V_{D}=n\cdot V_{T}\ln\left(1+\frac{I}{I_{S}}\right)\approx n\cdot V_{T}\ln% \left(\frac{I}{I_{S}}\right)\Leftrightarrow
  61. V D n V T ln 10 log 10 ( I I S ) V_{D}\approx n\cdot V_{T}\cdot\ln{10}\cdot\log_{10}{\left(\frac{I}{I_{S}}% \right)}
  62. n = 1 n=1
  63. T = 25 C T=25C
  64. V D 0.05916 log 10 ( I I S ) V_{D}\approx 0.05916\cdot\log_{10}{\left(\frac{I}{I_{S}}\right)}
  65. I S = 10 - 12 I_{S}=10^{-12}
  66. I S = 10 - 6 I_{S}=10^{-6}
  67. V D V_{D}
  68. I I S \frac{I}{I_{S}}
  69. V D 0.53 V V_{D}\approx 0.53V
  70. V D 0.18 V V_{D}\approx 0.18V
  71. V D 0.65 V V_{D}\approx 0.65V
  72. V D 0.30 V V_{D}\approx 0.30V
  73. r D r_{D}
  74. I Q I_{Q}
  75. V Q V_{Q}
  76. g D g_{D}
  77. g D = d I d V | Q = I s V T e V Q / V T I Q V T g_{D}=\frac{dI}{dV}\Big|_{Q}=\frac{I_{s}}{V_{T}}e^{V_{Q}/V_{T}}\approx\frac{I_% {Q}}{V_{T}}
  78. I Q I_{Q}
  79. V T 25 mV V_{T}\approx 25\,\mathrm{mV}
  80. V Q / V T V_{Q}/V_{T}
  81. r D r_{D}
  82. r D = n V T I Q r_{D}=\frac{n\cdot V_{T}}{I_{Q}}
  83. I Q I_{Q}
  84. Q = I Q τ F + Q J Q=I_{Q}\tau_{F}+Q_{J}
  85. τ F \tau_{F}
  86. I Q I_{Q}
  87. C D = d Q d V Q = d I Q d V Q τ F + d Q J d V Q I Q V T τ F + C J C_{D}=\frac{dQ}{dV_{Q}}=\frac{dI_{Q}}{dV_{Q}}\tau_{F}+\frac{dQ_{J}}{dV_{Q}}% \approx\frac{I_{Q}}{V_{T}}\tau_{F}+C_{J}
  88. C J = d Q J d V Q C_{J}=\begin{matrix}\frac{dQ_{J}}{dV_{Q}}\end{matrix}
  89. V D / ( k T / q ) V_{D}/(kT/q)
  90. I S I_{S}

Direct-ethanol_fuel_cell.html

  1. C 2 H 5 OH + 3 H 2 O 12 H + + 12 e - + 2 CO 2 \mathrm{C_{2}H_{5}OH+3\ H_{2}O\to 12\ H^{+}+12\ e^{-}+2\ CO_{2}}
  2. 3 O 2 + 12 H + + 12 e - 6 H 2 O \mathrm{3\ O_{2}+12\ H^{+}+12\ e^{-}\to 6\ H_{2}O}
  3. C 2 H 5 OH + 3 O 2 3 H 2 O + 2 CO 2 \mathrm{C_{2}H_{5}OH+3\ O_{2}\to 3\ H_{2}O+2\ CO_{2}}

Direct_numerical_simulation.html

  1. η \eta
  2. η = ( ν 3 / ε ) 1 / 4 \eta=(\nu^{3}/\varepsilon)^{1/4}
  3. N h > L , Nh>L,\,
  4. h η , h\leq\eta,\,
  5. ε u 3 / L , \varepsilon\approx{u^{\prime}}^{3}/L,
  6. N 3 N^{3}
  7. N 3 Re 9 / 4 = Re 2.25 N^{3}\geq\mathrm{Re}^{9/4}=\mathrm{Re}^{2.25}
  8. Re = u L ν . \mathrm{Re}=\frac{u^{\prime}L}{\nu}.
  9. C = u Δ t h < 1 C=\frac{u^{\prime}\Delta t}{h}<1
  10. τ \tau
  11. τ = L u . \tau=\frac{L}{u^{\prime}}.
  12. η \eta
  13. L / ( C η ) L/(C\eta)
  14. L η Re 3 / 4 , \frac{L}{\eta}\sim\mathrm{Re}^{3/4},

Direct_product_of_groups.html

  1. G G
  2. H H
  3. G × H G×H
  4. G H G⊕H
  5. G G
  6. H H
  7. G × H G×H
  8. G × H G×H
  9. ( g , h ) (g,h)
  10. g G g∈G
  11. h H h∈H
  12. G × H G×H
  13. G G
  14. H H
  15. G × H G×H
  16. ( g < s u b > 1 , h 1 ) · ( g 2 , h 2 ) = ( g 1 · g 2 , h 1 · h 2 ) (g<sub>1,h_{1})·(g_{2},h_{2})=(g_{1}·g_{2},h_{1}·h_{2})
  17. G × H G×H
  18. G G
  19. H H
  20. ( g , h ) (g,h)
  21. G × H G×H
  22. g g
  23. G G
  24. h h
  25. H H
  26. 𝐑 \mathbf{R}
  27. 𝐑 × 𝐑 \mathbf{R}×\mathbf{R}
  28. ( x , y ) (x,y)
  29. ( x < s u b > 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) (x<sub>1,y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})
  30. G G
  31. H H
  32. G G
  33. 1 1
  34. a a
  35. 1 1
  36. 1 1
  37. a a
  38. a a
  39. a a
  40. 1 1
  41. H H
  42. 1 1
  43. b b
  44. 1 1
  45. 1 1
  46. b b
  47. b b
  48. b b
  49. 1 1
  50. G × H G×H
  51. G × H G×H
  52. ( 1 , 1 ) (1,1)
  53. ( a , 1 ) (a,1)
  54. ( 1 , b ) (1,b)
  55. ( a , b ) (a,b)
  56. ( 1 , 1 ) (1,1)
  57. ( 1 , 1 ) (1,1)
  58. ( a , 1 ) (a,1)
  59. ( 1 , b ) (1,b)
  60. ( a , b ) (a,b)
  61. ( a , 1 ) (a,1)
  62. ( a , 1 ) (a,1)
  63. ( 1 , 1 ) (1,1)
  64. ( a , b ) (a,b)
  65. ( 1 , b ) (1,b)
  66. ( 1 , b ) (1,b)
  67. ( 1 , b ) (1,b)
  68. ( a , b ) (a,b)
  69. ( 1 , 1 ) (1,1)
  70. ( a , 1 ) (a,1)
  71. ( a , b ) (a,b)
  72. ( a , b ) (a,b)
  73. ( 1 , b ) (1,b)
  74. ( a , 1 ) (a,1)
  75. ( 1 , 1 ) (1,1)
  76. G × H G×H
  77. G G
  78. H H
  79. G × H G×H
  80. G G
  81. H H
  82. ( g , h ) (g,h)
  83. g g
  84. h h
  85. ( g , h ) (g,h)
  86. l c m lcm
  87. g g
  88. h h
  89. g g
  90. h h
  91. ( g , h ) (g,h)
  92. g g
  93. h h
  94. G G
  95. H H
  96. G × H G×H
  97. m m
  98. n n
  99. G G
  100. H H
  101. P = G × H P=G×H
  102. P P
  103. P P
  104. G G
  105. H H
  106. G G
  107. H H
  108. P P
  109. G G
  110. H H
  111. P P
  112. G G′
  113. H H′
  114. G G
  115. H H
  116. G H G∩H
  117. P P
  118. G G
  119. H H
  120. G G
  121. H H
  122. P P
  123. P P
  124. G G
  125. H H
  126. P P
  127. G G
  128. H H
  129. P P
  130. G G
  131. H H
  132. G G
  133. H H
  134. P P
  135. g g , h gg,h
  136. g g
  137. G G
  138. h h
  139. H H
  140. V V
  141. V V
  142. 1 1
  143. a a
  144. b b
  145. c c
  146. 1 1
  147. 1 1
  148. a a
  149. b b
  150. c c
  151. a a
  152. a a
  153. 1 1
  154. c c
  155. b b
  156. b b
  157. b b
  158. c c
  159. 1 1
  160. a a
  161. c c
  162. c c
  163. b b
  164. a a
  165. 1 1
  166. V V
  167. a 〈a〉
  168. m n mn
  169. m m
  170. n n
  171. m m
  172. n n
  173. a 〈a〉
  174. 𝐓 \mathbf{T}
  175. G L ( n , 𝐑 ) GL(n,\mathbf{R})
  176. S L ( n , 𝐑 ) SL(n,\mathbf{R})
  177. O ( n , 𝐑 ) O(n,\mathbf{R})
  178. S O ( n , 𝐑 ) SO(n,\mathbf{R})
  179. I I
  180. I I
  181. I −I
  182. n n
  183. 4 n 4n
  184. G × H G×H
  185. G G
  186. H H
  187. G G
  188. H H
  189. G × H G×H
  190. H H
  191. G G
  192. G × H G×H
  193. G G
  194. H H
  195. G × H G×H
  196. G × H G×H
  197. G G
  198. H H
  199. G × H G×H
  200. P P
  201. ƒ : P G × H ƒ:P→G×H
  202. ƒ ƒ
  203. A A
  204. G G
  205. B B
  206. H H
  207. A × B A×B
  208. G × H G×H
  209. G G
  210. G × H G×H
  211. G ×
  212. H H
  213. A A
  214. B B
  215. A × B A×B
  216. G × H G×H
  217. ( G × H ) / ( A × B ) (G×H)/(A×B)
  218. G / A G/A
  219. H / B H/B
  220. ( G × H ) / ( A × B ) (G×H)/(A×B)
  221. ( G / A ) × ( H / B ) (G/A)×(H/B)
  222. G × H G×H
  223. G G
  224. H H
  225. G G
  226. G × G G×G
  227. Δ Δ
  228. ( g , g ) : g G (g,g):g∈G
  229. G G
  230. G G
  231. H H
  232. G × H G×H
  233. G G
  234. H H
  235. G × H G×H
  236. G G
  237. H H
  238. ( g , h ) G × H (g,h)∈G×H
  239. ( g , h ) (g,h)
  240. g g
  241. h h
  242. G × H G×H
  243. G G
  244. H H
  245. Z ( G × H ) Z(G×H)
  246. Z ( G ) × Z ( H ) Z(G)×Z(H)
  247. α α
  248. G G
  249. β β
  250. H H
  251. α × β : G × H G × H α×β:G×H→G×H
  252. G × H G×H
  253. A u t ( G × H ) Aut(G×H)
  254. A u t ( G ) × A u t ( H ) Aut(G)×Aut(H)
  255. G × H G×H
  256. A u t ( G ) × A u t ( H ) Aut(G)×Aut(H)
  257. A u t ( G × H ) Aut(G×H)
  258. G G
  259. σ σ
  260. G × G G×G
  261. 𝐙 × 𝐙 \mathbf{Z}×\mathbf{Z}
  262. G L ( 2 , 𝐙 ) GL(2,\mathbf{Z})
  263. 2 × 2 2×2
  264. ± 1 ±1
  265. G × H G×H
  266. 2 × 2 2×2
  267. [ α β γ δ ] \begin{bmatrix}\alpha&\beta\\ \gamma&\delta\end{bmatrix}
  268. α α
  269. G G
  270. δ δ
  271. H H
  272. β : H G β:H→G
  273. γ : G H γ:G→H
  274. α α
  275. β β
  276. γ γ
  277. δ δ
  278. i = 1 n G i = G 1 × G 2 × × G n \prod_{i=1}^{n}G_{i}\;=\;G_{1}\times G_{2}\times\cdots\times G_{n}
  279. G < s u b > 1 × · · · × G n G<sub>1×···×G_{n}
  280. G < s u b > 1 × · · · × G n G<sub>1×···×G_{n}
  281. G < s u b > 1 , G 2 , G<sub>1,G_{2},...

Direct_shear_test.html

  1. ϕ \phi

Directional_symmetry_(time_series).html

  1. t t
  2. t i t_{i}
  3. i = 1 , , n i=1,\ldots,n
  4. t ^ i \hat{t}_{i}
  5. DS ( t , t ^ ) = 100 n - 1 i = 2 n d i , \operatorname{DS}(t,\hat{t})=\frac{100}{n-1}\sum_{i=2}^{n}d_{i},
  6. d i = { 1 , if ( t i - t i - 1 ) ( t ^ i - t ^ i - 1 ) 0 0 , otherwise . d_{i}=\begin{cases}1,&\,\text{if }(t_{i}-t_{i-1})(\hat{t}_{i}-\hat{t}_{i-1})% \geq 0\\ 0,&\,\text{otherwise}.\end{cases}
  7. D S = 100 % DS=100\%

Directivity.html

  1. D ( θ , ϕ ) D(\theta,\phi)
  2. U ( θ , ϕ ) U(\theta,\phi)
  3. D ( θ , ϕ ) = U ( θ , ϕ ) P tot / ( 4 π ) D(\theta,\phi)=\frac{U(\theta,\phi)}{P_{\mathrm{tot}}/\left(4\pi\right)}
  4. θ \theta
  5. ϕ \phi
  6. U ( θ , ϕ ) U(\theta,\phi)
  7. P tot P_{\mathrm{tot}}
  8. U ( θ , ϕ ) U(\theta,\phi)
  9. P tot P_{\mathrm{tot}}
  10. P tot = ϕ = 0 ϕ = 2 π θ = 0 θ = π U sin θ d θ d ϕ ; P_{\mathrm{tot}}=\int_{\phi=0}^{\phi=2\pi}\int_{\theta=0}^{\theta=\pi}U\sin% \theta\,d\theta\,d\phi;
  11. P tot P_{\mathrm{tot}}
  12. U ( θ , ϕ ) U(\theta,\phi)
  13. P tot / 4 π P_{\mathrm{tot}}/4\pi
  14. ( θ , ϕ ) (\theta,\phi)
  15. D \displaystyle D
  16. Ω A \Omega_{A}
  17. D = 4 π Ω A D=\frac{4\pi}{\Omega_{A}}
  18. D 4 π ( 180 π ) 2 Θ 1 d Θ 2 d = 41253 Θ 1 d Θ 2 d D\approx 4\pi\frac{(\frac{180}{\pi})^{2}}{\Theta_{1d}\Theta_{2d}}=\frac{41253}% {\Theta_{1d}\Theta_{2d}}
  19. Θ 1 d \Theta_{1d}
  20. Θ 2 d \Theta_{2d}
  21. D 32400 Θ 1 d Θ 2 d D\approx\frac{32400}{\Theta_{1d}\Theta_{2d}}
  22. D D
  23. D d B = 10 log 10 [ D D r e f e r e n c e ] D_{dB}=10\cdot\log_{10}\left[\frac{D}{D_{reference}}\right]
  24. D d B i = 10 log 10 [ D ] D_{dBi}=10\cdot\log_{10}\left[D\right]
  25. D d B d = 10 log 10 [ D 1.64 ] D_{dBd}=10\cdot\log_{10}\left[\frac{D}{1.64}\right]

Disability-adjusted_life_year.html

  1. DALY = YLL + YLD \mathrm{DALY}=\mathrm{YLL}+\mathrm{YLD}
  2. W = 0.1658 Y e - 0.04 Y W=0.1658Ye^{-0.04Y}
  3. Y Y
  4. W W

Disc_permeameter.html

  1. Q * = 2 π a + 4 Q*=2\pi a+4
  2. a = α r / 2 a=\alpha r/2
  3. α \alpha
  4. K ( h ) = K s e α h K(h)=K_{s}\mathrm{e}^{\alpha h}
  5. q = α ϕ 0 + 4 ϕ 0 π r q_{\infty}=\alpha\phi_{0}+\frac{4\phi_{0}}{\pi_{r}}

Discontinuous_linear_map.html

  1. f ( x ) = i = 1 n x i f ( e i ) , f(x)=\sum^{n}_{i=1}x_{i}f(e_{i}),
  2. f ( x ) = i = 1 n x i f ( e i ) i = 1 n | x i | f ( e i ) . \|f(x)\|=\left\|\sum^{n}_{i=1}x_{i}f(e_{i})\right\|\leq\sum^{n}_{i=1}|x_{i}|\|% f(e_{i})\|.
  3. M = sup i { f ( e i ) } , M=\sup_{i}\{\|f(e_{i})\|\},
  4. i = 1 n | x i | C x \sum^{n}_{i=1}|x_{i}|\leq C\|x\|
  5. f ( x ) ( i = 1 n | x i | ) M C M x . \|f(x)\|\leq\left(\sum^{n}_{i=1}|x_{i}|\right)M\leq CM\|x\|.
  6. f = sup x [ 0 , 1 ] | f ( x ) | . \|f\|=\sup_{x\in[0,1]}|f(x)|.
  7. T ( f ) = f ( 0 ) T(f)=f^{\prime}(0)\,
  8. f n ( x ) = sin ( n 2 x ) n f_{n}(x)=\frac{\sin(n^{2}x)}{n}
  9. T ( f n ) = n 2 cos ( n 2 0 ) n = n T(f_{n})=\frac{n^{2}\cos(n^{2}\cdot 0)}{n}=n\to\infty
  10. T ( f n ) T ( 0 ) = 0 T(f_{n})\to T(0)=0
  11. T ( e n ) = n e n T(e_{n})=n\|e_{n}\|\,
  12. T T
  13. X X
  14. Y Y
  15. Dom ( T ) \operatorname{Dom}(T)
  16. T : Dom ( T ) X Y T:\operatorname{Dom}(T)\subseteq X\to Y
  17. Dom ( T ) \operatorname{Dom}(T)
  18. Γ ( T ) \Gamma(T)
  19. T T
  20. Γ ( T ) ¯ \overline{\Gamma(T)}
  21. Γ ( T ) ¯ \overline{\Gamma(T)}
  22. T ¯ \overline{T}
  23. T T
  24. T ¯ \overline{T}
  25. T T

Discrete_choice.html

  1. P n i P r o b ( Person n chooses alternative i ) = G ( x n i , x n j , j i , s n , β ) , P_{ni}\equiv Prob(\,\text{Person }n\,\text{ chooses alternative }i)=G(x_{ni},% \;x_{nj},\;j\neq i,\;s_{n},\;\beta),
  2. x n i \scriptstyle x_{ni}
  3. x n j , j i \scriptstyle x_{nj},\;j\neq i
  4. s n s_{n}
  5. β \beta
  6. n : j = 1 J P n j = 1 , \scriptstyle\forall n:\;\sum_{j=1}^{J}P_{nj}=1,
  7. = 1 N n = 1 N P n i , ={1\over N}{\sum_{n=1}^{N}P_{ni}},
  8. y n i = { 1 , if U n i > U n j , j i , 0 , otherwise y_{ni}=\begin{cases}1,&\,\text{if}\quad U_{ni}>U_{nj},\quad j\neq i,\\ 0,&\,\text{otherwise}\end{cases}
  9. U n i = β z n i + ε n i U_{ni}=\beta z_{ni}+\varepsilon_{ni}
  10. z n i \textstyle z_{ni}
  11. z n i = z ( x n i , s n ) \textstyle z_{ni}=z(x_{ni},\,s_{n})
  12. β \textstyle\beta
  13. ε n i \varepsilon_{ni}
  14. P n i = P r o b ( y n i = 1 ) = P r o b ( U n i > U n j , j i ) = P r o b ( β z n i + ε n i > β z n j + ε n j , j i ) = P r o b ( ε n j - ε n i < β z n i - β z n j , j i ) \begin{aligned}\displaystyle P_{ni}&\displaystyle=Prob(\,y_{ni}=1\,)=Prob(\,U_% {ni}>U_{nj},\quad j\not=i\,)\\ &\displaystyle=Prob(\,\beta z_{ni}+\varepsilon_{ni}>\beta z_{nj}+\varepsilon_{% nj},\;j\neq i\,)\\ &\displaystyle=Prob(\,\varepsilon_{nj}-\varepsilon_{ni}<\beta z_{ni}-\beta z_{% nj},\;j\neq i\,)\end{aligned}
  15. j i : β z n i - β z n j , \textstyle\forall j\neq i:\beta z_{ni}-\beta z_{nj},\;
  16. P n i \displaystyle P_{ni}
  17. U n = β s n + ε n U_{n}=\beta s_{n}+\varepsilon_{n}
  18. y n = { 1 , i f U n > 0 , 0 , i f U n 0 y_{n}=\begin{cases}1,&if\,U_{n}>0,\\ 0,&if\,U_{n}\leq 0\end{cases}
  19. P r o b ( y n = 1 ) = 1 1 + e x p ( - β s n ) Prob(y_{n}=1)={1\over 1+exp(-\beta s_{n})}
  20. y n = { 1 , i f U n > 0 , 0 , i f U n 0 y_{n}=\begin{cases}1,&if\,U_{n}>0,\\ 0,&if\,U_{n}\leq 0\end{cases}
  21. P r o b ( y n = 1 ) = Φ ( β s n ) Prob(y_{n}=1)=\textstyle\Phi(\beta s_{n})
  22. ε n 1 , ε n 2 \varepsilon_{n1},\;\varepsilon_{n2}\sim
  23. P n 1 = e x p ( β z n 1 ) ( e x p ( β z n 1 ) + e x p ( β z n 2 ) ) P_{n1}={exp(\beta z_{n1})\over(exp(\beta z_{n1})+exp(\beta z_{n2}))}
  24. P n 1 = 1 ( 1 + e x p ( - β ( z n 1 - z n 2 ) ) P_{n1}={1\over(1+exp(-\beta(z_{n1}-z_{n2}))}
  25. P n 1 = Φ ( β ( z n 1 - z n 2 ) ) , P_{n1}=\textstyle\Phi(\beta(z_{n1}-z_{n2})),
  26. β i = 0 \scriptstyle\beta_{i}=0
  27. β 1 = 0 \scriptstyle\beta_{1}=0
  28. P n i = e x p ( β i s n ) j = 1 J e x p ( β j s n ) , P_{ni}={exp(\beta_{i}s_{n})\over\sum_{j=1}^{J}exp(\beta_{j}s_{n})},
  29. P n i = e x p ( β z n i ) j = 1 J e x p ( β z n j ) , P_{ni}={exp(\beta z_{ni})\over\sum_{j=1}^{J}exp(\beta z_{nj})},
  30. d j k \scriptstyle d_{j}^{k}
  31. d j k = { 1 , i f j = k , 0 , o t h e r w i s e \scriptstyle d_{j}^{k}=\begin{cases}\scriptstyle 1,&\scriptstyle if\,j=k,\\ \scriptstyle 0,&\scriptstyle otherwise\end{cases}
  32. w n j k = s n d j k \scriptstyle w_{nj}^{k}=s_{n}\,d_{j}^{k}
  33. z n j = { w n j 1 , w n j 2 , , w n j J } \scriptstyle z_{nj}=\{w^{1}_{nj},w^{2}_{nj},\ldots,w^{J}_{nj}\}
  34. β = { β 1 , β 2 , , β J } \scriptstyle\beta=\{\beta_{1},\beta_{2},\ldots,\beta_{J}\}
  35. ε n ( ε n 1 , , ε n J ) N ( 0 , Ω ) , \scriptstyle\varepsilon_{n}\equiv(\varepsilon_{n1},\ldots,\varepsilon_{nJ})% \sim N(0,\Omega),
  36. P n i \displaystyle P_{ni}
  37. ϕ ( ε n | Ω ) \scriptstyle\phi(\varepsilon_{n}|\Omega)
  38. Ω \scriptstyle\Omega
  39. Ω \scriptstyle\Omega
  40. β f ( β | θ ) \scriptstyle\beta\;\sim f(\beta|\theta)
  41. f \it f
  42. θ \scriptstyle\theta
  43. P n i = β L n i ( β ) f ( β | θ ) d β , P_{ni}=\int_{\beta}L_{ni}(\beta)\,f(\beta|\theta)\,d\beta,
  44. L n i ( β ) = e x p ( β z n i ) j = 1 J e x p ( β z n j ) L_{ni}(\beta)={exp(\beta z_{ni})\over{\sum_{j=1}^{J}exp(\beta z_{nj})}}
  45. β , \scriptstyle\beta,
  46. J J
  47. P r o b ( r a n k i n g 1 , 2 , , J ) = e x p ( β z 1 ) j = 1 J e x p ( β z n j ) e x p ( β z 2 ) j = 2 J e x p ( β z n j ) e x p ( β z J - 1 ) j = J - 1 J e x p ( β z n j ) Prob(ranking\;1,2,\ldots,J)={exp(\beta z_{1})\over\sum_{j=1}^{J}exp(\beta z_{% nj})}{exp(\beta z_{2})\over\sum_{j=2}^{J}exp(\beta z_{nj})}\ldots{exp(\beta z_% {J-1})\over\sum_{j=J-1}^{J}exp(\beta z_{nj})}
  48. U n = β z n + ε , ε U_{n}=\beta z_{n}+\varepsilon,\;\varepsilon\sim
  49. P r o b ( c h o o s i n g 1 ) \displaystyle Prob(choosing\,1)
  50. P r o b ( c h o o s i n g 2 ) \displaystyle Prob(choosing\,2)
  51. P r o b ( c h o o s i n g 5 ) \displaystyle Prob(choosing\,5)

Discretionary_policy.html

  1. M V = Y MV=Y
  2. m + v = y , m+v=y,
  3. σ y 2 = σ m 2 + σ v 2 + 2 ρ m v σ m σ v , \sigma_{y}^{2}=\sigma_{m}^{2}+\sigma_{v}^{2}+2\rho_{mv}\sigma_{m}\sigma_{v},
  4. σ \sigma
  5. ρ \rho
  6. σ m \sigma_{m}
  7. σ y 2 \sigma_{y}^{2}
  8. σ v 2 . \sigma_{v}^{2}.
  9. σ y 2 < σ v 2 \sigma_{y}^{2}<\sigma_{v}^{2}
  10. ρ m v < - σ m 2 σ v . \rho_{mv}<\frac{-\sigma_{m}}{2\sigma_{v}}.

Disjunctive_sequence.html

  1. 0 1 00 01 10 11 000 001 0\ 1\ 00\ 01\ 10\ 11\ 000\ 001\ldots
  2. 0 0 1 1 0 2 00 0 4 01 0 8 10 0 16 11 0 32 000 0 64 0\ 0^{1}\ 1\ 0^{2}\ 00\ 0^{4}\ 01\ 0^{8}\ 10\ 0^{16}\ 11\ 0^{32}\ 000\ 0^{64}\ldots

Disk_encryption_theory.html

  1. C i = E K ( C i - 1 P i ) C_{i}=E_{K}(C_{i-1}\oplus P_{i})
  2. C - 1 \scriptstyle C_{-1}
  3. b 1 \scriptstyle b_{1}
  4. b 2 \scriptstyle b_{2}
  5. b 1 I V 1 = b 2 I V 2 \scriptstyle b_{1}\,\oplus\,IV_{1}\;=\;b_{2}\,\oplus\,IV_{2}
  6. b 1 \scriptstyle b_{1}
  7. b 2 \scriptstyle b_{2}
  8. I V ( SN ) = E s ( SN ) , where s = hash ( K ) IV(\textrm{SN})=E_{s}(\textrm{SN}),\quad\textrm{where}\;\;s=\textrm{hash}(K)
  9. E K \scriptstyle E_{K}
  10. K \scriptstyle K
  11. E K T \scriptstyle E_{K}^{T}
  12. K \scriptstyle K
  13. T \scriptstyle T
  14. K \scriptstyle K
  15. F \scriptstyle F
  16. K \scriptstyle K
  17. F \scriptstyle F
  18. P \scriptstyle P
  19. I \scriptstyle I
  20. C \displaystyle C
  21. \scriptstyle\otimes
  22. \scriptstyle\oplus
  23. GF ( 2 128 ) \scriptstyle\,\text{GF}\left(2^{128}\right)
  24. F I = F ( I 0 δ ) = F I 0 F δ \scriptstyle F\,\otimes\,I\;=\;F\,\otimes\,(I_{0}\,\oplus\,\delta)\;=\;F\,% \otimes\,I_{0}\,\oplus\,F\,\otimes\,\delta
  25. F δ \scriptstyle F\,\otimes\,\delta
  26. δ \scriptstyle\delta
  27. C \scriptstyle C
  28. X \displaystyle X
  29. P \scriptstyle P
  30. I \scriptstyle I
  31. α \scriptstyle\alpha
  32. GF ( 2 128 ) \scriptstyle\,\text{GF}\left(2^{128}\right)
  33. x \scriptstyle x
  34. j \scriptstyle j
  35. 2 20 2^{20}
  36. C - 1 = E A ( I ) \scriptstyle C_{-1}\;=\;E_{A}(I)
  37. 2 ( C 0 C k - 1 ) \scriptstyle 2(C^{\prime}_{0}\,\oplus\,C^{\prime}_{k-1})
  38. P 0 \scriptstyle P_{0}
  39. L = E K ( 0 ) \scriptstyle L\;=\;E_{K}(0)
  40. P i = E K ( P i 2 i L ) \scriptstyle P^{\prime}_{i}\;=\;E_{K}(P_{i}\,\oplus\,2^{i}L)
  41. M = M P M C \scriptstyle M\;=\;M_{P}\,\oplus\,M_{C}
  42. M P = I P i \scriptstyle M_{P}\;=\;I\,\oplus\,\bigoplus P^{\prime}_{i}
  43. M C = E K ( M P ) \scriptstyle M_{C}\;=\;E_{K}(M_{P})
  44. C i = P i 2 i M \scriptstyle C^{\prime}_{i}\;=\;P^{\prime}_{i}\,\oplus\,2^{i}M
  45. i = 1 , , k - 1 \scriptstyle i\;=\;1,\,\ldots,\,k-1
  46. C 0 = M C I i = 1 k - 1 C i \scriptstyle C^{\prime}_{0}\;=\;M_{C}\,\oplus\,I\,\oplus\,\bigoplus_{i=1}^{k-1% }C^{\prime}_{i}
  47. C i = E K ( C i ) 2 i L \scriptstyle C_{i}\;=\;E_{K}(C^{\prime}_{i})\,\oplus\,2^{i}L
  48. i = 0 , , k - 1 \scriptstyle i\;=\;0,\,\ldots,\,k-1
  49. K \scriptstyle K

Disposal_tax_effect.html

  1. S V > U C C SV>UCC
  2. S V < U C C SV<UCC

Dissociative_recombination.html

  1. C H 3 + + e - C H 2 + H CH_{3}^{+}+e^{-}\rightarrow CH_{2}+H

Distributed_algorithm.html

  1. v v
  2. v v
  3. v v
  4. v v
  5. v v
  6. v v

Distributed_Bragg_reflector.html

  1. R R
  2. R = [ n o ( n 2 ) 2 N - n s ( n 1 ) 2 N n o ( n 2 ) 2 N + n s ( n 1 ) 2 N ] 2 , R=\left[\frac{n_{o}(n_{2})^{2N}-n_{s}(n_{1})^{2N}}{n_{o}(n_{2})^{2N}+n_{s}(n_{% 1})^{2N}}\right]^{2},
  3. n o , n 1 , n 2 n_{o},\ n_{1},\ n_{2}
  4. n s n_{s}\,
  5. N N
  6. Δ f 0 \Delta f_{0}
  7. Δ f 0 f 0 = 4 π arcsin ( n 2 - n 1 n 2 + n 1 ) , \frac{\Delta f_{0}}{f_{0}}=\frac{4}{\pi}\arcsin\left(\frac{n_{2}-n_{1}}{n_{2}+% n_{1}}\right),
  8. f o f_{o}

Distributed_constraint_optimization.html

  1. A , V , 𝔇 , f , α , η \langle A,V,\mathfrak{D},f,\alpha,\eta\rangle
  2. A A
  3. V V
  4. { v 1 , v 2 , , v | V | } \{v_{1},v_{2},\cdots,v_{|V|}\}
  5. 𝔇 \mathfrak{D}
  6. { D 1 , D 2 , , D | V | } \{D_{1},D_{2},\ldots,D_{|V|}\}
  7. D 𝔇 D\in\mathfrak{D}
  8. f f
  9. f : S 𝔓 ( V ) v i S ( { v i } × D i ) { } f:\bigcup_{S\in\mathfrak{P}(V)}\sum{v_{i}\in S}\left(\{v_{i}\}\times D_{i}% \right)\rightarrow\mathbb{N}\cup\{\infty\}
  10. α \alpha
  11. α : V A \alpha:V\rightarrow A
  12. α ( v i ) a j \alpha(v_{i})\mapsto a_{j}
  13. a j a_{j}
  14. v i v_{i}
  15. α \alpha
  16. η \eta
  17. f f
  18. η ( f ) s S 𝔓 ( V ) v i S ( { v i } × D i ) f ( s ) \eta(f)\mapsto\sum_{s\in\bigcup_{S\in\mathfrak{P}(V)}\sum{v_{i}\in S}\left(\{v% _{i}\}\times D_{i}\right)}f(s)
  19. η ( f ) \eta(f)
  20. t : V ( D 𝔇 ) { } . t:V\rightarrow(D\in\mathfrak{D})\cup\{\emptyset\}.
  21. t ( v i ) t(v_{i})\mapsto\emptyset
  22. α ( v i ) \alpha(v_{i})
  23. v i v_{i}
  24. t t
  25. f f
  26. G = N , E G=\langle N,E\rangle
  27. C C
  28. n N n\in N
  29. c C c\in C
  30. | C | |C|
  31. n i N n_{i}\in N
  32. v i V v_{i}\in V
  33. D i = C D_{i}=C
  34. n i , n j E \langle n_{i},n_{j}\rangle\in E
  35. ( c C : f ( v i , c , v j , c ) 1 ) . (\forall c\in C:f(\langle v_{i},c\rangle,\langle v_{j},c\rangle)\mapsto 1).
  36. η ( f ) \eta(f)
  37. I I
  38. K K
  39. s : I s:I\leftarrow\mathbb{N}
  40. c : K c:K\leftarrow\mathbb{N}
  41. i I i\in I
  42. v i V v_{i}\in V
  43. D i = K D_{i}=K
  44. t t
  45. f ( t ) k K { 0 r ( t , k ) c ( k ) , r ( t , k ) - c ( k ) otherwise , f(t)\mapsto\sum_{k\in K}\begin{cases}0&r(t,k)\leq c(k),\\ r(t,k)-c(k)&\,\text{otherwise},\end{cases}
  46. r ( t , k ) r(t,k)
  47. r ( t , k ) = v i t - 1 ( k ) s ( i ) . r(t,k)=\sum_{v_{i}\in t^{-1}(k)}s(i).
  48. 𝔓 ( 𝔙 ) \mathfrak{P}(\mathfrak{V})
  49. V V
  50. × \times
  51. \sum

Distributed_parameter_system.html

  1. A A\,
  2. B B\,
  3. C C\,
  4. D D\,
  5. x ( k + 1 ) = A x ( k ) + B u ( k ) x(k+1)=Ax(k)+Bu(k)\,
  6. y ( k ) = C x ( k ) + D u ( k ) y(k)=Cx(k)+Du(k)\,
  7. x x\,
  8. u u\,
  9. y y\,
  10. x ˙ ( t ) = A x ( t ) + B u ( t ) \dot{x}(t)=Ax(t)+Bu(t)\,
  11. y ( t ) = C x ( t ) + D u ( t ) y(t)=Cx(t)+Du(t)\,
  12. t > 0 t>0
  13. ξ [ 0 , 1 ] \xi\in[0,1]
  14. t w ( t , ξ ) = - ξ w ( t , ξ ) + u ( t ) , \frac{\partial}{\partial t}w(t,\xi)=-\frac{\partial}{\partial\xi}w(t,\xi)+u(t),
  15. w ( 0 , ξ ) = w 0 ( ξ ) , w(0,\xi)=w_{0}(\xi),
  16. w ( t , 0 ) = 0 , w(t,0)=0,
  17. y ( t ) = 0 1 w ( t , ξ ) d ξ , y(t)=\int_{0}^{1}w(t,\xi)\,d\xi,
  18. A x = - x , D ( A ) = { x X : x absolutely continuous , x L 2 ( 0 , 1 ) and x ( 0 ) = 0 } . Ax=-x^{\prime},~{}~{}~{}D(A)=\left\{x\in X:x\,\text{ absolutely continuous },x% ^{\prime}\in L^{2}(0,1)\,\text{ and }x(0)=0\right\}.
  19. B u = u , C x = 0 1 x ( ξ ) d ξ , D = 0. Bu=u,~{}~{}~{}Cx=\int_{0}^{1}x(\xi)\,d\xi,~{}~{}~{}D=0.
  20. w ˙ ( t ) = w ( t ) + w ( t - τ ) + u ( t ) , \dot{w}(t)=w(t)+w(t-\tau)+u(t),
  21. y ( t ) = w ( t ) , y(t)=w(t),
  22. A ( r f ) = ( r + f ( - τ ) f ) , D ( A ) = { ( r f ) X : f absolutely continuous , f L 2 ( [ - τ , 0 ] ) and r = f ( 0 ) } . A\begin{pmatrix}r\\ f\end{pmatrix}=\begin{pmatrix}r+f(-\tau)\\ f^{\prime}\end{pmatrix},~{}~{}~{}D(A)=\left\{\begin{pmatrix}r\\ f\end{pmatrix}\in X:f\,\text{ absolutely continuous },f^{\prime}\in L^{2}([-% \tau,0])\,\text{ and }r=f(0)\right\}.
  23. B u = ( u 0 ) , C ( r f ) = r , D = 0. Bu=\begin{pmatrix}u\\ 0\end{pmatrix},~{}~{}~{}C\begin{pmatrix}r\\ f\end{pmatrix}=r,~{}~{}~{}D=0.
  24. D + k = 0 C A k B z k D+\sum_{k=0}^{\infty}CA^{k}Bz^{k}
  25. D + C z ( I - z A ) - 1 B D+Cz(I-zA)^{-1}B
  26. D + C ( s I - A ) - 1 B D+C(sI-A)^{-1}B
  27. w 0 w_{0}
  28. s W ( s , ξ ) = - d d ξ W ( s , ξ ) + U ( s ) , sW(s,\xi)=-\frac{d}{d\xi}W(s,\xi)+U(s),
  29. W ( s , 0 ) = 0 , W(s,0)=0,
  30. Y ( s ) = 0 1 W ( s , ξ ) d ξ . Y(s)=\int_{0}^{1}W(s,\xi)\,d\xi.
  31. ξ \xi
  32. W ( s , ξ ) = U ( s ) ( 1 - e - s ξ ) / s W(s,\xi)=U(s)(1-e^{-s\xi})/s
  33. Y ( s ) = U ( s ) ( e - s + s - 1 ) / s 2 Y(s)=U(s)(e^{-s}+s-1)/s^{2}
  34. ( e - s + s - 1 ) / s 2 (e^{-s}+s-1)/s^{2}
  35. 1 / ( s - 1 - e - s ) 1/(s-1-e^{-s})
  36. Φ n \Phi_{n}
  37. Φ n u = k = 0 n A k B u k \Phi_{n}u=\sum_{k=0}^{n}A^{k}Bu_{k}
  38. Φ n u \Phi_{n}u
  39. Φ n \Phi_{n}
  40. Φ n \Phi_{n}
  41. Φ n \Phi_{n}
  42. Φ t \Phi_{t}
  43. 0 t e A s B u ( s ) d s \int_{0}^{t}{\rm e}^{As}Bu(s)\,ds
  44. Φ n \Phi_{n}
  45. Φ t \Phi_{t}
  46. Φ t \Phi_{t}
  47. Φ t \Phi_{t}
  48. Φ t \Phi_{t}
  49. e A t {\rm e}^{At}
  50. Ψ n \Psi_{n}
  51. ( Ψ n x ) k = C A k x (\Psi_{n}x)_{k}=CA^{k}x
  52. Ψ n x \Psi_{n}x
  53. Ψ n x k n x \|\Psi_{n}x\|\geq k_{n}\|x\|
  54. Ψ n \Psi_{n}
  55. Ψ n x k n A n x \|\Psi_{n}x\|\geq k_{n}\|A^{n}x\|
  56. Ψ t \Psi_{t}
  57. ( Ψ t ) ( s ) = C e A s x (\Psi_{t})(s)=C{\rm e}^{As}x
  58. Ψ n \Psi_{n}
  59. Ψ t \Psi_{t}
  60. Ψ t x k t x \|\Psi_{t}x\|\geq k_{t}\|x\|
  61. Ψ t \Psi_{t}
  62. Ψ t x k t e A t x \|\Psi_{t}x\|\geq k_{t}\|{\rm e}^{At}x\|
  63. Φ \Phi
  64. Ψ \Psi

Distribution_(pharmacology).html

  1. V d = A b C p Vd=\frac{Ab}{Cp}\,
  2. A b Ab
  3. C p Cp
  4. A b Ab
  5. V d Vd
  6. C p Cp
  7. C p Cp
  8. V d Vd
  9. C p Cp
  10. V d Vd
  11. D c = V d . C p D a . B Dc=\frac{Vd.Cp}{Da.B}

Distribution_constant.html

  1. [ A ] o r g [ A ] a q [A]_{org}\propto[A]_{aq}
  2. ( K D ) A = ( a A ) o r g ( a A ) a q [ A ] o r g [ A ] a q (K_{D})_{A}={(aA)_{org}\over(a_{A})_{aq}}\approx{[A]_{org}\over[A]_{aq}}
  3. [ A ] i = ( V a q V o r g K D + V a q ) i [ A ] 0 [A]_{i}=({V_{aq}\over V_{org}K_{D}+V_{aq}})^{i}[A]_{0}

Diversification_(finance).html

  1. x x
  2. y y
  3. σ x 2 \sigma^{2}_{x}
  4. σ y 2 \sigma^{2}_{y}
  5. q q
  6. 1 - q 1-q
  7. q x + ( 1 - q ) y qx+(1-q)y
  8. x x
  9. y y
  10. v a r ( q x + ( 1 - q ) y ) = q 2 σ x 2 + ( 1 - q ) 2 σ y 2 var(qx+(1-q)y)=q^{2}\sigma^{2}_{x}+(1-q)^{2}\sigma^{2}_{y}
  11. q q
  12. q = σ y 2 / [ σ x 2 + σ y 2 ] q=\sigma^{2}_{y}/[\sigma^{2}_{x}+\sigma^{2}_{y}]
  13. 0
  14. 1 1
  15. q q
  16. σ x 2 σ y 2 / [ σ x 2 + σ y 2 ] \sigma^{2}_{x}\sigma^{2}_{y}/[\sigma^{2}_{x}+\sigma^{2}_{y}]
  17. q = 1 q=1
  18. q = 0 q=0
  19. σ x 2 \sigma^{2}_{x}
  20. σ y 2 \sigma^{2}_{y}
  21. x x
  22. y y
  23. n n
  24. σ x 2 \sigma^{2}_{x}
  25. 1 / n 1/n
  26. v a r [ ( 1 / n ) x 1 + ( 1 / n ) x 2 + + ( 1 / n ) x n ] var[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]
  27. n ( 1 / n 2 ) σ x 2 n(1/n^{2})\sigma^{2}_{x}
  28. σ x 2 / n \sigma^{2}_{x}/n
  29. n n
  30. x 1 + x 2 + + x n x_{1}+x_{2}+\dots+x_{n}
  31. ( 1 / n ) x 1 + ( 1 / n ) x 2 + + ( 1 / n ) x n , (1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},
  32. v a r [ x 1 + x 2 + + x n ] = σ x 2 + σ x 2 + + σ x 2 = n σ x 2 , var[x_{1}+x_{2}+\dots+x_{n}]=\sigma^{2}_{x}+\sigma^{2}_{x}+\dots+\sigma^{2}_{x% }=n\sigma^{2}_{x},
  33. 𝔼 [ R P ] = i = 1 n x i 𝔼 [ R i ] \mathbb{E}[R_{P}]=\sum^{n}_{i=1}x_{i}\mathbb{E}[R_{i}]
  34. x i x_{i}
  35. i i
  36. Var ( R P ) σ P 2 = 𝔼 [ R P - 𝔼 [ R P ] ] 2 \underbrace{\,\text{Var}(R_{P})}_{\equiv\sigma^{2}_{P}}=\mathbb{E}[R_{P}-% \mathbb{E}[R_{P}]]^{2}
  37. 𝔼 [ R P ] \mathbb{E}[R_{P}]
  38. σ P 2 = 𝔼 [ i = 1 n x i R i - i = 1 n x i 𝔼 [ R i ] ] 2 \sigma^{2}_{P}=\mathbb{E}\left[\sum^{n}_{i=1}x_{i}R_{i}-\sum^{n}_{i=1}x_{i}% \mathbb{E}[R_{i}]\right]^{2}
  39. σ P 2 = 𝔼 [ i = 1 n x i ( R i - 𝔼 [ R i ] ) ] 2 \sigma^{2}_{P}=\mathbb{E}\left[\sum^{n}_{i=1}x_{i}(R_{i}-\mathbb{E}[R_{i}])% \right]^{2}
  40. σ P 2 = 𝔼 [ i = 1 n j = 1 n x i x j ( R i - 𝔼 [ R i ] ) ( R j - 𝔼 [ R j ] ) ] \sigma^{2}_{P}=\mathbb{E}\left[\sum^{n}_{i=1}\sum^{n}_{j=1}x_{i}x_{j}(R_{i}-% \mathbb{E}[R_{i}])(R_{j}-\mathbb{E}[R_{j}])\right]
  41. σ P 2 = 𝔼 [ i = 1 n x i 2 ( R i - 𝔼 [ R i ] ) 2 + i = 1 n j = 1 , i j n x i x j ( R i - 𝔼 [ R i ] ) ( R j - 𝔼 [ R j ] ) ] \sigma_{P}^{2}=\mathbb{E}\left[\sum_{i=1}^{n}x_{i}^{2}(R_{i}-\mathbb{E}[R_{i}]% )^{2}+\sum_{i=1}^{n}\sum_{j=1,i\neq j}^{n}x_{i}x_{j}(R_{i}-\mathbb{E}[R_{i}])(% R_{j}-\mathbb{E}[R_{j}])\right]
  42. σ P 2 = i = 1 n x i 2 𝔼 [ R i - 𝔼 [ R i ] ] 2 σ i 2 + i = 1 n j = 1 , i j n x i x j 𝔼 [ ( R i - 𝔼 [ R i ] ) ( R j - 𝔼 [ R j ] ) ] σ i j \sigma_{P}^{2}=\sum_{i=1}^{n}x_{i}^{2}\underbrace{\mathbb{E}\left[R_{i}-% \mathbb{E}[R_{i}]\right]^{2}}_{\equiv\sigma_{i}^{2}}+\sum_{i=1}^{n}\sum_{j=1,i% \neq j}^{n}x_{i}x_{j}\underbrace{\mathbb{E}\left[(R_{i}-\mathbb{E}[R_{i}])(R_{% j}-\mathbb{E}[R_{j}])\right]}_{\equiv\sigma_{ij}}
  43. σ P 2 = i = 1 n x i 2 σ i 2 + i = 1 n j = 1 , i j n x i x j σ i j \sigma^{2}_{P}=\sum^{n}_{i=1}x^{2}_{i}\sigma^{2}_{i}+\sum^{n}_{i=1}\sum^{n}_{j% =1,i\neq j}x_{i}x_{j}\sigma_{ij}
  44. σ i 2 \sigma^{2}_{i}
  45. i i
  46. σ i j \sigma_{ij}
  47. i i
  48. j j
  49. x i = x j = 1 n , i , j x_{i}=x_{j}=\frac{1}{n},\forall i,j
  50. σ P 2 = n 1 n 2 σ i 2 + n ( n - 1 ) 1 n 1 n σ ¯ i j \sigma^{2}_{P}=n\frac{1}{n^{2}}\sigma^{2}_{i}+n(n-1)\frac{1}{n}\frac{1}{n}\bar% {\sigma}_{ij}
  51. σ ¯ i j \bar{\sigma}_{ij}
  52. σ i j \sigma_{ij}
  53. i j i\neq j
  54. σ P 2 = 1 n σ i 2 + n - 1 n σ ¯ i j \sigma^{2}_{P}=\frac{1}{n}\sigma^{2}_{i}+\frac{n-1}{n}\bar{\sigma}_{ij}
  55. lim n σ P 2 = σ ¯ i j \lim_{n\rightarrow\infty}\sigma^{2}_{P}=\bar{\sigma}_{ij}

Division_algorithm.html

  1. N / D = ( Q , R ) N/D=(Q,R)
  2. P j + 1 = R × P j - q n - ( j + 1 ) × D P_{j+1}=R\times P_{j}-q_{n-(j+1)}\times D\,
  3. Q = 111 1 ¯ 1 1 ¯ 1 1 ¯ Q=111\bar{1}1\bar{1}1\bar{1}
  4. N = 00010101 N=00010101\,
  5. N ¯ = 11101011 \bar{N}=11101011
  6. P = 11101010 P=11101010\,
  7. P P\,
  8. N ¯ \bar{N}
  9. Q = 11010101 Q=11010101\,
  10. D D
  11. N N
  12. X 0 X_{0}
  13. 1 / D 1/D
  14. D D
  15. X 1 , X 2 , , X S X_{1},X_{2},\ldots,X_{S}
  16. Q = N X S Q=NX_{S}
  17. D D
  18. f ( X ) f(X)
  19. X = 1 / D X=1/D
  20. f ( X ) = D X - 1 f(X)=DX-1
  21. D D
  22. f ( X ) f(X)
  23. f ( X ) = ( 1 / X ) - D f(X)=(1/X)-D
  24. X i + 1 = X i - f ( X i ) f ( X i ) = X i - 1 / X i - D - 1 / X i 2 = X i + X i ( 1 - D X i ) = X i ( 2 - D X i ) , X_{i+1}=X_{i}-{f(X_{i})\over f^{\prime}(X_{i})}=X_{i}-{1/X_{i}-D\over-1/X_{i}^% {2}}=X_{i}+X_{i}(1-DX_{i})=X_{i}(2-DX_{i}),
  25. X i X_{i}
  26. X i + 1 = X i + X i ( 1 - D X i ) X_{i+1}=X_{i}+X_{i}(1-DX_{i})
  27. X i + 1 = X i ( 2 - D X i ) X_{i+1}=X_{i}(2-DX_{i})
  28. X i X_{i}
  29. ( 2 - D X i ) (2-DX_{i})
  30. X i X_{i}
  31. ( 1 - D X i ) (1-DX_{i})
  32. ϵ i = D X i - 1 \epsilon_{i}=DX_{i}-1\,
  33. ϵ i + 1 \displaystyle\epsilon_{i+1}
  34. X 0 X_{0}
  35. X 0 = T 1 + T 2 D 1 D X_{0}=T_{1}+T_{2}D\approx\frac{1}{D}\,
  36. [ 0.5 , 1 ] [0.5,1]
  37. X 0 = 48 17 - 32 17 D . X_{0}={48\over 17}-{32\over 17}D.\,
  38. | ( T 1 + T 2 D - 1 / D ) / ( 1 / D ) | = | D ( T 1 + T 2 D ) - 1 | |(T_{1}+T_{2}D-1/D)/(1/D)|=|D(T_{1}+T_{2}D)-1|
  39. F ( D ) = D ( T 1 + T 2 D ) - 1 F(D)=D(T_{1}+T_{2}D)-1
  40. F ( D ) F(D)
  41. F ( D ) = 0 F^{\prime}(D)=0
  42. D = - T 1 / ( 2 T 2 ) D=-T_{1}/(2T_{2})
  43. F ( 1 / 2 ) = F ( 1 ) = - F ( - T 1 / ( 2 T 2 ) ) F(1/2)=F(1)=-F(-T_{1}/(2T_{2}))
  44. T 1 = 48 / 17 T_{1}=48/17
  45. T 2 = - 32 / 17 T_{2}=-32/17
  46. F ( 1 ) = 1 / 17 F(1)=1/17
  47. | ϵ 0 | 1 17 0.059. |\epsilon_{0}|\leq{1\over 17}\approx 0.059.\,
  48. S = log 2 P + 1 log 2 17 S=\left\lceil\log_{2}\frac{P+1}{\log_{2}17}\right\rceil\,
  49. P P\,
  50. Q = N D F 1 F 1 F 2 F 2 F F . Q=\frac{N}{D}\frac{F_{1}}{F_{1}}\frac{F_{2}}{F_{2}}\frac{F_{\ldots}}{F_{\ldots% }}.
  51. F i + 1 = 2 - D i . F_{i+1}=2-D_{i}.
  52. N i + 1 D i + 1 = N i D i F i + 1 F i + 1 . \frac{N_{i+1}}{D_{i+1}}=\frac{N_{i}}{D_{i}}\frac{F_{i+1}}{F_{i+1}}.
  53. Q = N k Q=N_{k}
  54. D ( 1 2 , 1 ] D\in(\tfrac{1}{2},1]
  55. D = 1 - x D=1-x
  56. F i = 1 + x 2 i F_{i}=1+x^{2^{i}}
  57. N 1 - x = N ( 1 + x ) 1 - x 2 = N ( 1 + x ) ( 1 + x 2 ) 1 - x 4 = = Q = N = N ( 1 + x ) ( 1 + x 2 ) ( 1 + x 2 ( n - 1 ) ) D = 1 - x 2 n 1 \frac{N}{1-x}=\frac{N\cdot(1+x)}{1-x^{2}}=\frac{N\cdot(1+x)\cdot(1+x^{2})}{1-x% ^{4}}=\cdots=Q^{\prime}=\frac{N^{\prime}=N\cdot(1+x)\cdot(1+x^{2})\cdot\cdot% \cdot(1+x^{2^{(n-1)}})}{D^{\prime}=1-x^{2^{n}}\approx 1}
  58. n n
  59. ( x [ 0 , 1 2 ) ) (x\in[0,\tfrac{1}{2}))
  60. 1 - x 2 n 1-x^{2^{n}}
  61. 1 1
  62. ϵ n = Q - N Q = x 2 n \epsilon_{n}=\frac{Q^{\prime}-N^{\prime}}{Q^{\prime}}=x^{2^{n}}
  63. 2 - 2 n 2^{-2^{n}}
  64. x = 1 2 x={1\over 2}
  65. 2 n 2^{n}

DO-242A.html

  1. NAG v \mathrm{NAG}_{v}

Dolbear's_law.html

  1. T F = 50 + ( N 60 - 40 4 ) . T_{F}=50+\left(\frac{N_{60}-40}{4}\right).
  2. N 15 N_{15}
  3. T F = 40 + N 15 \,T_{F}=40+N_{15}
  4. T C = 10 + ( N 60 - 40 7 ) . T_{C}=10+\left(\frac{N_{60}-40}{7}\right).
  5. N 8 N_{8}
  6. T C = 5 + N 8 \,T_{C}=5+N_{8}

Dolev–Yao_model.html

  1. x x
  2. E x E_{x}
  3. D x D_{x}
  4. D x E x = E x D x = 1 D_{x}E_{x}=E_{x}D_{x}=1
  5. E x ( M ) E_{x}(M)
  6. M M

Dollar_cost_averaging.html

  1. r = p F p ~ P - 1 , r=\frac{p_{F}}{\tilde{p}_{P}}-1,
  2. p F p_{F}
  3. p ~ P \tilde{p}_{P}
  4. p ~ P \tilde{p}_{P}

Domain_of_holomorphy.html

  1. Ω \Omega
  2. n {\mathbb{C}}^{n}
  3. U Ω U\subset\Omega
  4. V n V\subset{\mathbb{C}}^{n}
  5. V V
  6. V Ω V\not\subset\Omega
  7. U Ω V U\subset\Omega\cap V
  8. f f
  9. Ω \Omega
  10. g g
  11. V V
  12. f = g f=g
  13. U U
  14. n = 1 n=1
  15. n 2 n\geq 2
  16. Ω \Omega
  17. Ω \Omega
  18. Ω \Omega
  19. Ω \Omega
  20. Ω \Omega
  21. S n Ω S_{n}\subseteq\Omega
  22. S n S , S n Γ S_{n}\rightarrow S,\partial S_{n}\rightarrow\Gamma
  23. Γ \Gamma
  24. S Ω S\subseteq\Omega
  25. Ω \partial\Omega
  26. Ω \Omega
  27. x Ω x\in\partial\Omega
  28. U U
  29. x x
  30. f f
  31. U Ω U\cap\Omega
  32. f f
  33. x x
  34. 1 2 , 3 4 , 1 4 , 3 5 1\Leftrightarrow 2,3\Leftrightarrow 4,1\Rightarrow 4,3\Rightarrow 5
  35. 1 3 1\Rightarrow 3
  36. 5 1 5\Rightarrow 1
  37. ¯ \bar{\partial}
  38. Ω 1 , , Ω n \Omega_{1},\dots,\Omega_{n}
  39. Ω = j = 1 n Ω j \Omega=\bigcap_{j=1}^{n}\Omega_{j}
  40. Ω 1 Ω 2 \Omega_{1}\subseteq\Omega_{2}\subseteq\dots
  41. Ω = n = 1 Ω n \Omega=\bigcup_{n=1}^{\infty}\Omega_{n}
  42. Ω 1 \Omega_{1}
  43. Ω 2 \Omega_{2}
  44. Ω 1 × Ω 2 \Omega_{1}\times\Omega_{2}

Dominating_decision_rule.html

  1. δ 1 \delta_{1}
  2. δ 2 \delta_{2}
  3. R ( θ , δ ) R(\theta,\delta)
  4. δ \delta
  5. θ \theta
  6. δ 1 \delta_{1}
  7. δ 2 \delta_{2}
  8. R ( θ , δ 1 ) R ( θ , δ 2 ) R(\theta,\delta_{1})\leq R(\theta,\delta_{2})
  9. θ \theta
  10. θ \theta

Donsker's_theorem.html

  1. G n ( x ) = n ( F n ( x ) - F ( x ) ) G_{n}(x)=\sqrt{n}(F_{n}(x)-F(x))\,
  2. 𝒟 ( - , ) \mathcal{D}(-\infty,\infty)
  3. cov [ G ( s ) , G ( t ) ] = E [ G ( s ) G ( t ) ] = min { F ( s ) , F ( t ) } - F ( s ) F ( t ) . \operatorname{cov}[G(s),G(t)]=E[G(s)G(t)]=\min\{F(s),F(t)\}-F(s)F(t).\,
  4. sup t G n ( t ) \scriptstyle\sup_{t}G_{n}(t)
  5. sup t | G n ( t ) | \scriptstyle\sup_{t}|G_{n}(t)|
  6. 𝒟 [ 0 , 1 ] \mathcal{D}[0,1]
  7. G n - B n \|G_{n}-B_{n}\|_{\infty}

Doob_martingale.html

  1. X = X 1 , X 2 , , X n \vec{X}=X_{1},X_{2},...,X_{n}
  2. A A
  3. f : A n f:A^{n}\to\mathbb{R}
  4. B i = E X i + 1 , X i + 2 , , X n [ f ( X ) | X 1 , X 2 , X i ] B_{i}=E_{X_{i+1},X_{i+2},...,X_{n}}[f(\vec{X})|X_{1},X_{2},...X_{i}]
  5. X i + 1 , X i + 2 , , X n , X_{i+1},X_{i+2},...,X_{n},
  6. X 1 , X 2 , X i X_{1},X_{2},...X_{i}
  7. B i B_{i}
  8. X i X_{i}
  9. B i {B_{i}}
  10. | B i + 1 - B i | |B_{i+1}-B_{i}|
  11. f ( X ) f(\vec{X})
  12. E [ f ( X ) ] = B 0 . E[f(\vec{X})]=B_{0}.
  13. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  14. f f
  15. sup x 1 , x 2 , , x n , x ^ i | f ( x 1 , x 2 , , x n ) - f ( x 1 , x 2 , , x i - 1 , x ^ i , x i + 1 , , x n ) | c i for 1 i n . \sup_{x_{1},x_{2},\dots,x_{n},\hat{x}_{i}}|f(x_{1},x_{2},\dots,x_{n})-f(x_{1},% x_{2},\dots,x_{i-1},\hat{x}_{i},x_{i+1},\dots,x_{n})|\leq c_{i}\qquad\,\text{% for}\quad 1\leq i\leq n\;.
  16. i i
  17. x i x_{i}
  18. f f
  19. c i c_{i}
  20. | B i + 1 - B i | c i |B_{i+1}-B_{i}|\leq c_{i}
  21. ε > 0 \varepsilon>0
  22. Pr { f ( X 1 , X 2 , , X n ) - E [ f ( X 1 , X 2 , , X n ) ] ε } exp ( - 2 ε 2 i = 1 n c i 2 ) \Pr\left\{f(X_{1},X_{2},\dots,X_{n})-E[f(X_{1},X_{2},\dots,X_{n})]\geq% \varepsilon\right\}\leq\exp\left(-\frac{2\varepsilon^{2}}{\sum_{i=1}^{n}c_{i}^% {2}}\right)
  23. Pr { E [ f ( X 1 , X 2 , , X n ) ] - f ( X 1 , X 2 , , X n ) ε } exp ( - 2 ε 2 i = 1 n c i 2 ) \Pr\left\{E[f(X_{1},X_{2},\dots,X_{n})]-f(X_{1},X_{2},\dots,X_{n})\geq% \varepsilon\right\}\leq\exp\left(-\frac{2\varepsilon^{2}}{\sum_{i=1}^{n}c_{i}^% {2}}\right)
  24. Pr { | E [ f ( X 1 , X 2 , , X n ) ] - f ( X 1 , X 2 , , X n ) | ε } 2 exp ( - 2 ε 2 i = 1 n c i 2 ) . \Pr\left\{|E[f(X_{1},X_{2},\dots,X_{n})]-f(X_{1},X_{2},\dots,X_{n})|\geq% \varepsilon\right\}\leq 2\exp\left(-\frac{2\varepsilon^{2}}{\sum_{i=1}^{n}c_{i% }^{2}}\right).\;

Doob–Meyer_decomposition_theorem.html

  1. Z Z
  2. Z 0 = 0 Z_{0}=0
  3. { Z T T a finite valued stopping time } \{Z_{T}\mid\,\text{T a finite valued stopping time}\}
  4. Z Z
  5. Z 0 = 0 Z_{0}=0
  6. A A
  7. A 0 = 0 A_{0}=0
  8. M t = Z t - A t M_{t}=Z_{t}-A_{t}

Doppler_cooling.html

  1. v 2 \langle v^{2}\rangle
  2. γ \gamma
  3. T Doppler = γ / 2 k B T_{\mathrm{Doppler}}=\hbar\gamma/2k_{B}
  4. k B k_{B}
  5. \hbar

Double-clad_fiber.html

  1. R ( ϕ ) R^{\prime}(\phi)
  2. R ( ϕ ) > 0 R^{\prime}(\phi)>0
  3. F = 0.8 F=0.8
  4. F = 0.9 F=0.9
  5. 1 - exp ( - F π r 2 S α L ) , 1-\exp\left(-F\frac{\pi r^{2}}{S}\alpha L\right),
  6. S S
  7. r r
  8. α \alpha
  9. L L
  10. F F
  11. 0 < F < 1 0<F<1
  12. F F
  13. F F
  14. F F
  15. F F

Double_auction.html

  1. b = s = p [ B , S ] . b=s=p\in[B,S].
  2. 1 - 1 / k 1-1/k
  3. p b k + ( 1 - p ) max ( b k + 1 , s k ) pb_{k}+(1-p)\max{(b_{k+1},s_{k})}
  4. p s k + ( 1 - p ) min ( s k + 1 , b k ) ps_{k}+(1-p)\min{(s_{k+1},b_{k})}
  5. max ( b k + 1 , s k ) \max{(b_{k+1},s_{k})}
  6. min ( s k + 1 , b k ) \min{(s_{k+1},b_{k})}

Double_subscript_notation.html

  1. V CB V_{\mathrm{CB}}
  2. I CE I_{\mathrm{CE}}

Doubly_periodic_function.html

  1. f ( z + u ) = f ( z + v ) = f ( z ) f(z+u)=f(z+v)=f(z)\,

Drag_area.html

  1. C d A , C_{d}A,
  2. A A
  3. C d C_{d}
  4. F d = 1 2 ρ C d A v 2 . F_{d}=\frac{1}{2}\ \rho\ C_{d}A\ v^{2}.
  5. F d F_{d}
  6. ρ \rho
  7. v v

Dual_wavelet.html

  1. ψ L 2 ( ) \psi\in L^{2}(\mathbb{R})
  2. { ψ j k } \{\psi_{jk}\}
  3. ψ j k ( x ) = 2 j / 2 ψ ( 2 j x - k ) \psi_{jk}(x)=2^{j/2}\psi(2^{j}x-k)
  4. j , k j,k\in\mathbb{Z}
  5. { ψ j k } \{\psi_{jk}\}
  6. L 2 ( ) L^{2}(\mathbb{R})
  7. 0 < A B < 0<A\leq B<\infty
  8. A c j k l 2 2 j k = - c j k ψ j k L 2 2 B c j k l 2 2 A\|c_{jk}\|^{2}_{l^{2}}\leq\bigg\|\sum_{jk=-\infty}^{\infty}c_{jk}\psi_{jk}% \bigg\|^{2}_{L^{2}}\leq B\|c_{jk}\|^{2}_{l^{2}}\,
  9. { c j k } \{c_{jk}\}
  10. l 2 \|\cdot\|_{l^{2}}
  11. c j k l 2 2 = j k = - | c j k | 2 \|c_{jk}\|^{2}_{l^{2}}=\sum_{jk=-\infty}^{\infty}|c_{jk}|^{2}
  12. L 2 \|\cdot\|_{L^{2}}
  13. L 2 ( ) L^{2}(\mathbb{R})
  14. f L 2 2 = - | f ( x ) | 2 d x \|f\|^{2}_{L^{2}}=\int_{-\infty}^{\infty}|f(x)|^{2}dx
  15. ψ j k \psi^{jk}
  16. ψ j k | ψ l m = δ j l δ k m \langle\psi^{jk}|\psi_{lm}\rangle=\delta_{jl}\delta_{km}
  17. δ j k \delta_{jk}
  18. f | g \langle f|g\rangle
  19. L 2 ( ) L^{2}(\mathbb{R})
  20. f ( x ) = j k ψ j k | f ψ j k ( x ) f(x)=\sum_{jk}\langle\psi^{jk}|f\rangle\psi_{jk}(x)
  21. ψ ~ L 2 ( ) \tilde{\psi}\in L^{2}(\mathbb{R})
  22. ψ ~ j k = ψ j k \tilde{\psi}_{jk}=\psi^{jk}
  23. ψ ~ \tilde{\psi}
  24. ψ = ψ ~ \psi=\tilde{\psi}
  25. ϕ \phi
  26. ψ ( x ) = ϕ ( x ) + z ϕ ( 2 x ) \psi(x)=\phi(x)+z\phi(2x)

Duality_(optimization).html

  1. ( X , X * ) \left(X,X^{*}\right)
  2. ( Y , Y * ) \left(Y,Y^{*}\right)
  3. f : X { + } f:X\to\mathbb{R}\cup\{+\infty\}
  4. x ^ \hat{x}
  5. f ( x ^ ) = inf x X f ( x ) . f(\hat{x})=\inf_{x\in X}f(x).\,
  6. f ( x ^ ) f(\hat{x})
  7. f f
  8. f f
  9. f ~ = f + I constraints \tilde{f}=f+I_{\mathrm{constraints}}
  10. I I
  11. F : X × Y { + } F:X\times Y\to\mathbb{R}\cup\{+\infty\}
  12. F ( x , 0 ) = f ~ ( x ) F(x,0)=\tilde{f}(x)
  13. sup y * Y * - F * ( 0 , y * ) inf x X F ( x , 0 ) , \sup_{y^{*}\in Y^{*}}-F^{*}(0,y^{*})\leq\inf_{x\in X}F(x,0),\,
  14. F * F^{*}
  15. sup \sup
  16. d * d^{*}
  17. p * p^{*}
  18. p * - d * p^{*}-d^{*}
  19. minimize f 0 ( x ) subject to f i ( x ) 0 , i { 1 , , m } h i ( x ) = 0 , i { 1 , , p } \begin{aligned}\displaystyle\,\text{minimize }&\displaystyle f_{0}(x)\\ \displaystyle\,\text{subject to }&\displaystyle f_{i}(x)\leq 0,\ i\in\left\{1,% \dots,m\right\}\\ &\displaystyle h_{i}(x)=0,\ i\in\left\{1,\dots,p\right\}\end{aligned}
  20. 𝒟 n \mathcal{D}\subset\mathbb{R}^{n}
  21. Λ : n × m × p \Lambda:\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}\to\mathbb{R}
  22. Λ ( x , λ , ν ) = f 0 ( x ) + i = 1 m λ i f i ( x ) + i = 1 p ν i h i ( x ) . \Lambda(x,\lambda,\nu)=f_{0}(x)+\sum_{i=1}^{m}\lambda_{i}f_{i}(x)+\sum_{i=1}^{% p}\nu_{i}h_{i}(x).
  23. λ \lambda
  24. ν \nu
  25. g : m × p g:\mathbb{R}^{m}\times\mathbb{R}^{p}\to\mathbb{R}
  26. g ( λ , ν ) = inf x 𝒟 Λ ( x , λ , ν ) = inf x 𝒟 ( f 0 ( x ) + i = 1 m λ i f i ( x ) + i = 1 p ν i h i ( x ) ) . g(\lambda,\nu)=\inf_{x\in\mathcal{D}}\Lambda(x,\lambda,\nu)=\inf_{x\in\mathcal% {D}}\left(f_{0}(x)+\sum_{i=1}^{m}\lambda_{i}f_{i}(x)+\sum_{i=1}^{p}\nu_{i}h_{i% }(x)\right).
  27. p * p^{*}
  28. λ 0 \lambda\geq 0
  29. ν \nu
  30. g ( λ , ν ) p * g(\lambda,\nu)\leq p^{*}
  31. d * = max λ 0 , ν g ( λ , ν ) = inf f 0 = p * d^{*}=\max_{\lambda\geq 0,\nu}g(\lambda,\nu)=\inf f_{0}=p^{*}
  32. minimize 𝑥 \displaystyle\underset{x}{\operatorname{minimize}}
  33. maximize 𝑢 \displaystyle\underset{u}{\operatorname{maximize}}
  34. f f
  35. g 1 , , g m g_{1},\cdots,g_{m}
  36. maximize x , u \displaystyle\underset{x,u}{\operatorname{maximize}}
  37. ( u , x ) (u,x)
  38. f ( x ) + j = 1 m u j g j ( x ) \nabla f(x)+\sum_{j=1}^{m}u_{j}\nabla g_{j}(x)

Duhamel's_principle.html

  1. { u t ( x , t ) - Δ u ( x , t ) = 0 ( x , t ) 𝐑 n × ( 0 , ) u ( x , 0 ) = g ( x ) x 𝐑 n \begin{cases}u_{t}(x,t)-\Delta u(x,t)=0&(x,t)\in\mathbf{R}^{n}\times(0,\infty)% \\ u(x,0)=g(x)&x\in\mathbf{R}^{n}\end{cases}
  2. { u t ( x , t ) - Δ u ( x , t ) = f ( x , t ) ( x , t ) 𝐑 n × ( 0 , ) u ( x , 0 ) = 0 x 𝐑 n \begin{cases}u_{t}(x,t)-\Delta u(x,t)=f(x,t)&(x,t)\in\mathbf{R}^{n}\times(0,% \infty)\\ u(x,0)=0&x\in\mathbf{R}^{n}\end{cases}
  3. u : D × ( 0 , ) 𝐑 u:D\times(0,\infty)\to\mathbf{R}
  4. { u t ( x , t ) - L u ( x , t ) = f ( x , t ) ( x , t ) D × ( 0 , ) u | D = 0 u ( x , 0 ) = 0 x D , \begin{cases}u_{t}(x,t)-Lu(x,t)=f(x,t)&(x,t)\in D\times(0,\infty)\\ u|_{\partial D}=0&\\ u(x,0)=0&x\in D,\end{cases}
  5. u ( x , t ) = 0 t ( P s f ) ( x , t ) d s u(x,t)=\int_{0}^{t}(P^{s}f)(x,t)\,ds
  6. { u t - L u = 0 ( x , t ) D × ( s , ) u | D = 0 u ( x , s ) = f ( x , s ) x D . \begin{cases}u_{t}-Lu=0&(x,t)\in D\times(s,\infty)\\ u|_{\partial D}=0&\\ u(x,s)=f(x,s)&x\in D.\end{cases}
  7. 2 u t 2 - c 2 2 u x 2 = f ( x , t ) . \frac{\partial^{2}u}{\partial t^{2}}-c^{2}\frac{\partial^{2}u}{\partial x^{2}}% =f(x,t).\,
  8. u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = v 0 ( x ) . u(x,0)=u_{0}(x),\qquad\frac{\partial u}{\partial t}(x,0)=v_{0}(x).\,
  9. u ( x , T ) = u T ( x ) , u t ( x , T ) = v T ( x ) . u(x,T)=u_{T}(x),\qquad\frac{\partial u}{\partial t}(x,T)=v_{T}(x).\,
  10. 2 u t 2 - c 2 2 u x 2 = 0 \frac{\partial^{2}u}{\partial t^{2}}-c^{2}\frac{\partial^{2}u}{\partial x^{2}}=0
  11. u ( x , T ) = 0 , u t ( x , T ) = f ( x , T ) d T . u(x,T)=0,\qquad\frac{\partial u}{\partial t}(x,T)=f(x,T)dT.
  12. ( 1 2 c x - c ( t - T ) x + c ( t - T ) f ( ξ , T ) d ξ ) d T \left(\frac{1}{2c}\int_{x-c(t-T)}^{x+c(t-T)}f(\xi,T)\,d\xi\right)\,dT
  13. P T f ( x , t ) P^{T}f(x,t)
  14. u ( x , t ) = u 0 ( x ) + 1 2 c 0 t x - c ( t - T ) x + c ( t - T ) f ( ξ , T ) d ξ d T . u(x,t)=u_{0}(x)+\frac{1}{2c}\int_{0}^{t}\int_{x-c(t-T)}^{x+c(t-T)}f(\xi,T)\,d% \xi\,dT.\,
  15. P ( t ) u ( t ) = F ( t ) P(\partial_{t})u(t)=F(t)\,
  16. t j u ( 0 ) = 0 , 0 j m - 1 \partial_{t}^{j}u(0)=0,\;0\leq j\leq m-1
  17. P ( t ) := a m t m + + a 1 t + a 0 , a m 0. P(\partial_{t}):=a_{m}\partial_{t}^{m}+\cdots+a_{1}\partial_{t}+a_{0},\;a_{m}% \neq 0.
  18. P ( t ) G = 0 , t j G ( 0 ) = 0 , 0 j m - 2 , t m - 1 G ( 0 ) = 1 / a m . P(\partial_{t})G=0,\;\partial^{j}_{t}G(0)=0,\quad 0\leq j\leq m-2,\;\partial_{% t}^{m-1}G(0)=1/a_{m}.
  19. H = G χ [ 0 , ) H=G\chi_{[0,\infty)}
  20. χ [ 0 , ) \chi_{[0,\infty)}
  21. [ 0 , ) [0,\infty)
  22. P ( t ) H = δ P(\partial_{t})H=\delta
  23. u ( t ) = ( H F ) ( t ) u(t)=(H\ast F)(t)
  24. = 0 G ( τ ) F ( t - τ ) d τ =\int_{0}^{\infty}G(\tau)F(t-\tau)\,d\tau
  25. = - t G ( t - τ ) F ( τ ) d τ =\int_{-\infty}^{t}G(t-\tau)F(\tau)\,d\tau
  26. P ( t , D x ) u ( t , x ) = F ( t , x ) P(\partial_{t},D_{x})u(t,x)=F(t,x)\,
  27. D x = 1 i x . D_{x}=\frac{1}{i}\frac{\partial}{\partial x}.\,
  28. P ( t , ξ ) u ^ ( t , ξ ) = F ^ ( t , ξ ) . P(\partial_{t},\xi)\hat{u}(t,\xi)=\hat{F}(t,\xi).
  29. P ( t , ξ ) P(\partial_{t},\xi)
  30. a m a_{m}
  31. P ( t , ξ ) P(\partial_{t},\xi)
  32. ξ \xi
  33. G ( t , ξ ) G(t,\xi)
  34. P ( t , ξ ) G ( t , ξ ) = 0 , t j G ( 0 , ξ ) = 0 for 0 j m - 2 , t m - 1 G ( 0 , ξ ) = 1 / a m . P(\partial_{t},\xi)G(t,\xi)=0,\;\partial^{j}_{t}G(0,\xi)=0\;\mbox{ for }~{}0% \leq j\leq m-2,\;\partial_{t}^{m-1}G(0,\xi)=1/a_{m}.
  35. H ( t , ξ ) = G ( t , ξ ) χ [ 0 , ) ( t ) H(t,\xi)=G(t,\xi)\chi_{[0,\infty)}(t)
  36. P ( t , ξ ) H ( t , ξ ) = δ ( t ) P(\partial_{t},\xi)H(t,\xi)=\delta(t)
  37. u ^ ( t , ξ ) = ( H ( , ξ ) F ^ ( , ξ ) ) ( t ) \hat{u}(t,\xi)=(H(\cdot,\xi)\ast\hat{F}(\cdot,\xi))(t)
  38. = 0 G ( τ , ξ ) F ( t - τ , ξ ) d τ =\int_{0}^{\infty}G(\tau,\xi)F(t-\tau,\xi)\,d\tau
  39. = - t G ( t - τ , ξ ) F ( τ , ξ ) d τ =\int_{-\infty}^{t}G(t-\tau,\xi)F(\tau,\xi)\,d\tau

Durand–Kerner_method.html

  1. P = x - f ( x ) ( x - Q ) ( x - R ) ( x - S ) . P=x-\frac{f(x)}{(x-Q)(x-R)(x-S)}.
  2. x 1 := x 0 - f ( x 0 ) ( x 0 - Q ) ( x 0 - R ) ( x 0 - S ) , x_{1}:=x_{0}-\frac{f(x_{0})}{(x_{0}-Q)(x_{0}-R)(x_{0}-S)},
  3. x k + 1 := x k - f ( x k ) ( x k - q ) ( x k - r ) ( x k - s ) , x_{k+1}:=x_{k}-\frac{f(x_{k})}{(x_{k}-q)(x_{k}-r)(x_{k}-s)},
  4. P - f ( P ) ( P - q ) ( P - r ) ( P - s ) = P - 0 = P . P-\frac{f(P)}{(P-q)(P-r)(P-s)}=P-0=P.
  5. p n = p n - 1 - f ( p n - 1 ) ( p n - 1 - q n - 1 ) ( p n - 1 - r n - 1 ) ( p n - 1 - s n - 1 ) ; p_{n}=p_{n-1}-\frac{f(p_{n-1})}{(p_{n-1}-q_{n-1})(p_{n-1}-r_{n-1})(p_{n-1}-s_{% n-1})};
  6. q n = q n - 1 - f ( q n - 1 ) ( q n - 1 - p n ) ( q n - 1 - r n - 1 ) ( q n - 1 - s n - 1 ) ; q_{n}=q_{n-1}-\frac{f(q_{n-1})}{(q_{n-1}-p_{n})(q_{n-1}-r_{n-1})(q_{n-1}-s_{n-% 1})};
  7. r n = r n - 1 - f ( r n - 1 ) ( r n - 1 - p n ) ( r n - 1 - q n ) ( r n - 1 - s n - 1 ) ; r_{n}=r_{n-1}-\frac{f(r_{n-1})}{(r_{n-1}-p_{n})(r_{n-1}-q_{n})(r_{n-1}-s_{n-1}% )};
  8. s n = s n - 1 - f ( s n - 1 ) ( s n - 1 - p n ) ( s n - 1 - q n ) ( s n - 1 - r n ) . s_{n}=s_{n-1}-\frac{f(s_{n-1})}{(s_{n-1}-p_{n})(s_{n-1}-q_{n})(s_{n-1}-r_{n})}.
  9. 1 + max ( | a | , | b | , | c | , | d | ) 1+\max(|a|,|b|,|c|,|d|)
  10. g z ( X ) = ( X - z 1 ) ( X - z n ) . g_{\vec{z}}(X)=(X-z_{1})\cdots(X-z_{n}).
  11. g z ( X ) = X n + g n - 1 ( z ) X n - 1 + + g 0 ( z ) . g_{\vec{z}}(X)=X^{n}+g_{n-1}(\vec{z})X^{n-1}+\cdots+g_{0}(\vec{z}).
  12. α 1 ( z ) , , α n ( z ) \alpha_{1}(\vec{z}),\dots,\alpha_{n}(\vec{z})
  13. f ( X ) = X n + c n - 1 X n - 1 + + c 0 f(X)=X^{n}+c_{n-1}X^{n-1}+\cdots+c_{0}
  14. ( c n - 1 , , c 0 ) (c_{n-1},\dots,c_{0})
  15. c 0 = g 0 ( z ) = ( - 1 ) n α n ( z ) = ( - 1 ) n z 1 z n c 1 = g 1 ( z ) = ( - 1 ) n - 1 α n - 1 ( z ) c n - 1 = g n - 1 ( z ) = - α 1 ( z ) = - ( z 1 + z 2 + + z n ) . \begin{matrix}c_{0}&=&g_{0}(\vec{z})&=&(-1)^{n}\alpha_{n}(\vec{z})&=&(-1)^{n}z% _{1}\cdots z_{n}\\ c_{1}&=&g_{1}(\vec{z})&=&(-1)^{n-1}\alpha_{n-1}(\vec{z})\\ &\vdots&\\ c_{n-1}&=&g_{n-1}(\vec{z})&=&-\alpha_{1}(\vec{z})&=&-(z_{1}+z_{2}+\cdots+z_{n}% ).\end{matrix}
  16. g z ( X ) = f ( X ) g_{\vec{z}}(X)=f(X)
  17. z \vec{z}
  18. w \vec{w}
  19. g z + w ( X ) = f ( X ) g_{\vec{z}+\vec{w}}(X)=f(X)
  20. f ( X ) - g z ( X ) = k = 1 n g z ( X ) z k w k = - k = 1 n w k j k ( X - z j ) . f(X)-g_{\vec{z}}(X)=\sum_{k=1}^{n}\frac{\partial g_{\vec{z}}(X)}{\partial z_{k% }}w_{k}=-\sum_{k=1}^{n}w_{k}\prod_{j\neq k}(X-z_{j}).
  21. z 1 , , z n z_{1},\dots,z_{n}
  22. [ X ] n - 1 \mathbb{C}[X]_{n-1}
  23. w \vec{w}
  24. w \vec{w}
  25. - k = 1 n w k j k ( X - z j ) = f ( X ) - j = 1 n ( X - z j ) -\sum_{k=1}^{n}w_{k}\prod_{j\neq k}(X-z_{j})=f(X)-\prod_{j=1}^{n}(X-z_{j})
  26. X = z k X=z_{k}
  27. - w k j k ( z k - z j ) = - w k g z ( z k ) = f ( z k ) -w_{k}\prod_{j\neq k}(z_{k}-z_{j})=-w_{k}g_{\vec{z}}^{\prime}(z_{k})=f(z_{k})
  28. w k = - f ( z k ) j k ( z k - z j ) . w_{k}=-\frac{f(z_{k})}{\prod_{j\neq k}(z_{k}-z_{j})}.
  29. b k ( X ) = 1 j n , j k ( X - z j ) , k = 1 , , n , b_{k}(X)=\prod_{1\leq j\leq n,\;j\neq k}(X-z_{j}),\quad k=1,\dots,n,
  30. z 1 , , z n z_{1},\dots,z_{n}\in\mathbb{C}
  31. L k ( X ) = b k ( X ) b k ( z k ) L_{k}(X)=\frac{b_{k}(X)}{b_{k}(z_{k})}
  32. X b k ( X ) mod f ( X ) = X b k ( X ) - f ( X ) X\cdot b_{k}(X)\mod f(X)=X\cdot b_{k}(X)-f(X)
  33. = j = 1 n ( z j b k ( z j ) - f ( z j ) ) b j ( X ) b j ( z j ) =\sum_{j=1}^{n}\Big(z_{j}\cdot b_{k}(z_{j})-f(z_{j})\Big)\cdot\frac{b_{j}(X)}{% b_{j}(z_{j})}
  34. = z k b k ( X ) + j = 1 n w j b j ( X ) =z_{k}\cdot b_{k}(X)+\sum_{j=1}^{n}w_{j}\cdot b_{j}(X)
  35. w j = - f ( z j ) b j ( z j ) w_{j}=-\frac{f(z_{j})}{b_{j}(z_{j})}
  36. A = diag ( z 1 , , z n ) + ( 1 1 ) ( w 1 , , w n ) . A=\mathrm{diag}(z_{1},\dots,z_{n})+\begin{pmatrix}1\\ \vdots\\ 1\end{pmatrix}\cdot\left(w_{1},\dots,w_{n}\right).
  37. D ( a k , k , r k ) D(a_{k,k},r_{k})
  38. r k = j k | a j , k | r_{k}=\sum_{j\neq k}\big|a_{j,k}\big|
  39. a k , k = z k + w k a_{k,k}=z_{k}+w_{k}
  40. r k = ( n - 1 ) | w k | r_{k}=(n-1)\left|w_{k}\right|
  41. z 1 , , z n z_{1},\dots,z_{n}\in\mathbb{C}
  42. T A T - 1 TAT^{-1}
  43. z k z_{k}
  44. z k z_{k}
  45. z k + w k z_{k}+w_{k}
  46. O ( | w k | 2 ) O(|w_{k}|^{2})
  47. z = ( z 1 , , z n ) \vec{z}=(z_{1},\dots,z_{n})
  48. z \vec{z}
  49. w = ( w 1 , , w n ) \vec{w}=(w_{1},\dots,w_{n})
  50. max 1 k n | w k | 1 5 n min 1 j < k n | z k - z j | , \max_{1\leq k\leq n}\big|w_{k}\big|\leq\frac{1}{5n}\min_{1\leq j<k\leq n}\big|% z_{k}-z_{j}\big|,
  51. D ( z k + w k , ( n - 1 ) | w k | ) \textstyle D\left(z_{k}+w_{k},(n-1)|w_{k}|\right)
  52. D ( z k + w k , 1 4 | w k | ) k = 1 , , n , \textstyle D\left(z_{k}+w_{k},\frac{1}{4}|w_{k}|\right)\qquad k=1,\dots,n,

Durbin–Watson_statistic.html

  1. d = t = 2 T ( e t - e t - 1 ) 2 t = 1 T e t 2 , d={\sum_{t=2}^{T}(e_{t}-e_{t-1})^{2}\over{\sum_{t=1}^{T}e_{t}^{2}}},
  2. 𝐗 \mathbf{X}
  3. d d
  4. d d
  5. i = 1 n - k ν i ξ i 2 i = 1 n - k ξ i 2 , \frac{\sum_{i=1}^{n-k}\nu_{i}\xi_{i}^{2}}{\sum_{i=1}^{n-k}\xi_{i}^{2}},
  6. ξ i \xi_{i}
  7. ν i \nu_{i}
  8. ( 𝐈 - 𝐗 ( 𝐗 T 𝐗 ) - 1 𝐗 T ) 𝐀 , (\mathbf{I}-\mathbf{X}(\mathbf{X}^{T}\mathbf{X})^{-1}\mathbf{X}^{T})\mathbf{A},
  9. 𝐀 \mathbf{A}
  10. d d
  11. d = 𝐞 T 𝐀𝐞 . d=\mathbf{e}^{T}\mathbf{A}\mathbf{e}.
  12. h = ( 1 - 1 2 d ) T 1 - T Var ^ ( β ^ 1 ) , h=\left(1-\frac{1}{2}d\right)\sqrt{\frac{T}{1-T\cdot\widehat{\operatorname{Var% }}(\widehat{\beta}_{1}\,)}},
  13. V a r ^ ( β ^ 1 ) \widehat{Var}(\widehat{\beta}_{1})
  14. T V a r ^ ( β ^ 1 ) < 1. T\cdot\widehat{Var}(\widehat{\beta}_{1})<1.\,
  15. d p d = i = 1 N t = 2 T ( e i , t - e i , t - 1 ) 2 i = 1 N t = 1 T e i , t 2 . d_{pd}=\frac{\sum_{i=1}^{N}\sum_{t=2}^{T}(e_{i,t}-e_{i,t-1})^{2}}{\sum_{i=1}^{% N}\sum_{t=1}^{T}e_{i,t}^{2}}.

Durbin–Wu–Hausman_test.html

  1. H = ( b 1 - b 0 ) ( Var ( b 0 ) - Var ( b 1 ) ) ( b 1 - b 0 ) , H=(b_{1}-b_{0})^{\prime}\big(\operatorname{Var}(b_{0})-\operatorname{Var}(b_{1% })\big)^{\dagger}(b_{1}-b_{0}),

Dusty_plasma.html

  1. m d v d t = 𝐅 𝐋 + 𝐅 𝐆 + 𝐅 𝐏 + 𝐅 𝐃 + 𝐅 𝐓 m\frac{dv}{dt}=\mathbf{F_{L}}+\mathbf{F_{G}}+\mathbf{F_{P}}+\mathbf{F_{D}}+% \mathbf{F_{T}}
  2. F L = q ( 𝐄 + 𝐯 c × 𝐁 ) F_{L}=q\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right)
  3. 𝐅 𝐠 \mathbf{F_{g}}
  4. 𝐅 𝐏 \mathbf{F_{P}}
  5. F P = π r d 2 c I 𝐞 𝐢 ^ F_{P}=\frac{\pi r_{d}^{2}}{c}I\mathbf{\hat{e_{i}}}
  6. 𝐞 𝐢 ^ \mathbf{\hat{e_{i}}}
  7. I I
  8. r d r_{d}
  9. 𝐅 𝐋 \mathbf{F_{L}}
  10. 𝐅 𝐆 \mathbf{F_{G}}

Dyall_Hamiltonian.html

  1. ^ D = ^ i D + ^ v D + C \hat{\mathcal{H}}^{D}=\hat{\mathcal{H}}^{D}_{i}+\hat{\mathcal{H}}^{D}_{v}+C
  2. ^ i D = i core ϵ i E i i + r virt ϵ r E r r \hat{\mathcal{H}}^{D}_{i}=\sum_{i}^{\rm core}\epsilon_{i}E_{ii}+\sum_{r}^{\rm virt% }\epsilon_{r}E_{rr}
  3. ^ v D = a b act h a b eff E a b + 1 2 a b c d act a b | c d ( E a c E b d - δ b c E a d ) \hat{\mathcal{H}}^{D}_{v}=\sum_{ab}^{\rm act}h_{ab}^{\rm eff}E_{ab}+\frac{1}{2% }\sum_{abcd}^{\rm act}\left\langle ab\left.\right|cd\right\rangle\left(E_{ac}E% _{bd}-\delta_{bc}E_{ad}\right)
  4. C = 2 i core h i i + i j core ( 2 i j | i j - i j | j i ) - 2 i core ϵ i C=2\sum_{i}^{\rm core}h_{ii}+\sum_{ij}^{\rm core}\left(2\left\langle ij\left.% \right|ij\right\rangle-\left\langle ij\left.\right|ji\right\rangle\right)-2% \sum_{i}^{\rm core}\epsilon_{i}
  5. h a b eff = h a b + j ( 2 a j | b j - a j | j b ) h_{ab}^{\rm eff}=h_{ab}+\sum_{j}\left(2\left\langle aj\left.\right|bj\right% \rangle-\left\langle aj\left.\right|jb\right\rangle\right)
  6. i , j , i,j,\ldots
  7. a , b , a,b,\ldots
  8. r , s , r,s,\ldots
  9. ϵ i \epsilon_{i}
  10. ϵ r \epsilon_{r}
  11. E m n E_{mn}
  12. a m α a n α + a m β a n β a^{\dagger}_{m\alpha}a_{n\alpha}+a^{\dagger}_{m\beta}a_{n\beta}
  13. S 2 S^{2}
  14. S z S_{z}

Dynamic_factor.html

  1. X t = Λ t F t + e t , X_{t}=\Lambda_{t}F_{t}+e_{t},
  2. F t = ( f t , , f t - q ) F_{t}=(f^{\top}_{t},\dots,f^{\top}_{t-q})
  3. T × N T\times N
  4. X t X_{t}
  5. Λ t \Lambda_{t}
  6. e t e_{t}
  7. f t f_{t}
  8. X t X_{t}

Dynamic_financial_analysis.html

  1. d r t = a ( b - r t ) d t + σ r t d W t dr_{t}=a(b-r_{t})\,dt+\sigma\sqrt{r_{t}}\,dW_{t}
  2. r t r_{t}
  3. σ \sigma\,
  4. a ( b - r t ) a(b-r_{t})
  5. σ r t \sigma\sqrt{r_{t}}\,

Dynamic_logic_(digital_electronics).html

  1. O u t = A B ¯ Out=\overline{AB}

Dynamic_modulus.html

  1. π / 2 \pi/2
  2. ε = ε 0 sin ( t ω ) \varepsilon=\varepsilon_{0}\sin(t\omega)
  3. σ = σ 0 sin ( t ω + δ ) \sigma=\sigma_{0}\sin(t\omega+\delta)\,
  4. ω = 2 π f \omega=2\pi f
  5. f f
  6. t t
  7. δ \delta
  8. E = σ 0 ε 0 cos δ E^{\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\cos\delta
  9. E ′′ = σ 0 ε 0 sin δ E^{\prime\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\sin\delta
  10. G G^{\prime}
  11. G ′′ G^{\prime\prime}
  12. E * E^{*}
  13. G * G^{*}
  14. E * = E + i E ′′ E^{*}=E^{\prime}+iE^{\prime\prime}\,
  15. G * = G + i G ′′ G^{*}=G^{\prime}+iG^{\prime\prime}\,
  16. i i

Dynamization.html

  1. P P
  2. M M
  3. S S
  4. P ( M , S ) P(M,S)
  5. P P
  6. S S
  7. S i S_{i}
  8. + +
  9. P ( M , S ) = P ( M , S 0 ) + P ( M , S 1 ) + + P ( M , S n ) P(M,S)=P(M,S_{0})+P(M,S_{1})+\dots+P(M,S_{n})
  10. n n
  11. 2 i * n i 2^{i}*n_{i}
  12. n i n_{i}
  13. i i
  14. n n
  15. n n
  16. i i
  17. log 2 ( n ) \log_{2}\left(n\right)
  18. O ( log ( n ) ) O(\log\left(n\right))
  19. P S ( n ) P_{S}\left(n\right)\,\!
  20. Q S ( n ) Q_{S}\left(n\right)\,\!
  21. Q D ( n ) Q_{D}\left(n\right)\,\!
  22. I ¯ \overline{I}
  23. Q D ( n ) = O ( Q S ( n ) log ( n ) ) Q_{D}\left(n\right)=O(Q_{S}\left(n\right)\log\left(n\right))\,\!
  24. I ¯ = O ( ( P S ( n ) / n ) log ( n ) ) \overline{I}=O(\left(P_{S}\left(n\right)/n\right)\log\left(n\right))
  25. Q S ( n ) Q_{S}\left(n\right)
  26. Q D ( n ) = O ( Q S ( n ) ) Q_{D}\left(n\right)=O\left(Q_{S}\left(n\right)\right)

E0.html

  1. ε 0 = ω ε 0 \varepsilon_{0}=\omega^{\varepsilon_{0}}

Easton's_theorem.html

  1. κ < cf ( 2 κ ) \kappa<\operatorname{cf}(2^{\kappa})\,
  2. κ < λ \kappa<\lambda\,
  3. 2 κ 2 λ 2^{\kappa}\leq 2^{\lambda}\,
  4. G ( α ) \aleph_{G(\alpha)}
  5. α \aleph_{\alpha}
  6. α \aleph_{\alpha}
  7. 2 α = G ( α ) 2^{\aleph_{\alpha}}=\aleph_{G(\alpha)}\,
  8. α \alpha
  9. 2 λ 2^{\lambda}
  10. λ \lambda

EC50.html

  1. Y = B o t t o m + T o p - B o t t o m 1 + ( X E C 50 ) - Hillcoefficient Y=Bottom+\frac{Top-Bottom}{1+(\frac{X}{EC_{50}})^{\mathrm{-Hillcoefficient}}}

Economic_dispatch.html

  1. W W
  2. n n
  3. k k
  4. k k
  5. min I k ( - W ) = min I k { k = 1 n C k ( I k ) } \min_{I_{k}}\;(-W)=\min_{I_{k}}\;\left\{\sum_{k=1}^{n}C_{k}(I_{k})\right\}
  6. k = 1 n I k = L ( I 1 , I 2 , , I n - 1 ) \sum_{k=1}^{n}I_{k}=L(I_{1},I_{2},\dots,I_{n-1})
  7. L L
  8. L L
  9. n n
  10. m m
  11. F l ( I 1 , I 2 , , I n - 1 ) F l m a x l = 1 , , m F_{l}(I_{1},I_{2},\dots,I_{n-1})\leq F_{l}^{max}\qquad l=1,\dots,m
  12. l l
  13. = k = 1 n C k ( I k ) + π [ L ( I 1 , I 2 , , I n - 1 ) - k = 1 n I k ] + l = 1 m μ l [ F l m a x - F l ( I 1 , I 2 , , I n - 1 ) ] \mathcal{L}=\sum_{k=1}^{n}C_{k}(I_{k})+\pi\left[L(I_{1},I_{2},\dots,I_{n-1})-% \sum_{k=1}^{n}I_{k}\right]+\sum_{l=1}^{m}\mu_{l}\left[F_{l}^{max}-F_{l}(I_{1},% I_{2},\dots,I_{n-1})\right]
  14. I k = 0 k = 1 , , n {\partial\mathcal{L}\over\partial I_{k}}=0\qquad k=1,\dots,n
  15. π = 0 {\partial\mathcal{L}\over\partial\pi}=0
  16. μ l = 0 l = 1 , , m {\partial\mathcal{L}\over\partial\mu_{l}}=0\qquad l=1,\dots,m
  17. μ l [ F l m a x - F l ( I 1 , I 2 , , I n - 1 ) ] = 0 μ l 0 k = 1 , , n \mu_{l}\cdot\left[F_{l}^{max}-F_{l}(I_{1},I_{2},\dots,I_{n-1})\right]=0\quad% \mu_{l}\geq 0\quad k=1,\dots,n

Economic_production_quantity.html

  1. x = D P x=\frac{D}{P}
  2. Q 2 F ( 1 - x ) \frac{Q}{2}F(1-x)
  3. Q 2 \frac{Q}{2}
  4. D Q K \frac{D}{Q}K
  5. D Q \frac{D}{Q}
  6. 1 2 F D ( 1 - x ) * T \frac{1}{2}FD(1-x)*T
  7. x ( T ) = 1 2 F D ( 1 - x ) T + K T x(T)=\frac{1}{2}FD(1-x)T+\frac{K}{T}

Edge_cover.html

  1. ρ ( G ) \rho(G)

Edwin_Joseph_Cohn.html

  1. log K p = - α S [ S ] + β S \log K_{p}=-\alpha_{S}[S]+\beta_{S}
  2. K p K_{p}
  3. α S \alpha_{S}
  4. β S \beta_{S}

Effective_descriptive_set_theory.html

  1. Σ n 0 \Sigma^{0}_{n}
  2. Π n 0 \Pi^{0}_{n}
  3. ϕ \phi
  4. ϕ \phi
  5. Σ 0 0 \Sigma^{0}_{0}
  6. Π 0 0 \Pi^{0}_{0}
  7. Σ n 0 \Sigma^{0}_{n}
  8. Π n 0 \Pi^{0}_{n}
  9. ϕ \phi
  10. n 1 n 2 n k ψ \exists n_{1}\exists n_{2}\cdots\exists n_{k}\psi
  11. ψ \psi
  12. Π n 0 \Pi^{0}_{n}
  13. ϕ \phi
  14. Σ n + 1 0 \Sigma^{0}_{n+1}
  15. ϕ \phi
  16. n 1 n 2 n k ψ \forall n_{1}\forall n_{2}\cdots\forall n_{k}\psi
  17. ψ \psi
  18. Σ n 0 \Sigma^{0}_{n}
  19. ϕ \phi
  20. Π n + 1 0 \Pi^{0}_{n+1}

Effective_potential.html

  1. U eff U\text{eff}
  2. U eff ( 𝐫 ) = L 2 2 m r 2 + U ( 𝐫 ) U\text{eff}(\mathbf{r})=\frac{L^{2}}{2mr^{2}}+U(\mathbf{r})
  3. 𝐅 eff \displaystyle\mathbf{F}\text{eff}
  4. 𝐫 ^ \hat{\mathbf{r}}
  5. U eff E U\text{eff}\leq E
  6. r r
  7. r 0 r_{0}
  8. d U eff d r = 0 \frac{dU\text{eff}}{dr}=0
  9. r 0 r_{0}
  10. U eff U\text{eff}
  11. U effmax U\text{eff}\text{max}
  12. ω = U eff ′′ m \omega=\sqrt{\frac{U\text{eff}^{\prime\prime}}{m}}
  13. r r
  14. E = 1 2 m ( r ˙ 2 + r 2 ϕ ˙ 2 ) - G m M r , E=\frac{1}{2}m\left(\dot{r}^{2}+r^{2}\dot{\phi}^{2}\right)-\frac{GmM}{r},
  15. L = m r 2 ϕ ˙ L=mr^{2}\dot{\phi}\,
  16. r ˙ \dot{r}
  17. ϕ ˙ \dot{\phi}
  18. m r ˙ 2 = 2 E - L 2 m r 2 + 2 G m M r = 2 E - 1 r 2 ( L 2 m - 2 G m M r ) , m\dot{r}^{2}=2E-\frac{L^{2}}{mr^{2}}+\frac{2GmM}{r}=2E-\frac{1}{r^{2}}\left(% \frac{L^{2}}{m}-2GmMr\right),
  19. 1 2 m r ˙ 2 = E - U eff ( r ) , \frac{1}{2}m\dot{r}^{2}=E-U\text{eff}(r),
  20. U eff ( r ) = L 2 2 m r 2 - G m M r U\text{eff}(r)=\frac{L^{2}}{2mr^{2}}-\frac{GmM}{r}

Effective_renal_plasma_flow.html

  1. e R P F = R P F × e x t r a c t i o n r a t i o eRPF=RPF\times extractionratio

Effective_stress.html

  1. σ = σ - u \sigma^{\prime}=\sigma-u\,
  2. σ = H soil γ soil u = H w γ w \begin{aligned}\displaystyle\sigma&\displaystyle=H_{\mathrm{soil}}\,\gamma_{% \mathrm{soil}}\\ \displaystyle u&\displaystyle=H_{\mathrm{w}}\,\gamma_{\mathrm{w}}\end{aligned}

Eigenspinor.html

  1. S x S_{x}
  2. S y S_{y}
  3. S z S_{z}
  4. χ + = [ 1 0 ] \chi_{+}=\begin{bmatrix}1\\ 0\\ \end{bmatrix}
  5. χ - = [ 0 1 ] \chi_{-}=\begin{bmatrix}0\\ 1\\ \end{bmatrix}
  6. χ + \chi_{+}
  7. χ - \chi_{-}
  8. S z S_{z}
  9. χ + z = [ 1 0 ] \chi_{+}^{z}=\begin{bmatrix}1\\ 0\\ \end{bmatrix}
  10. χ - z = [ 0 1 ] \chi_{-}^{z}=\begin{bmatrix}0\\ 1\\ \end{bmatrix}
  11. S x S_{x}
  12. χ + x = 1 2 [ 1 1 ] \chi_{+}^{x}={1\over\sqrt{2}}\begin{bmatrix}1\\ 1\\ \end{bmatrix}
  13. χ - x = 1 2 [ 1 - 1 ] \chi_{-}^{x}={1\over\sqrt{2}}\begin{bmatrix}1\\ -1\\ \end{bmatrix}
  14. S y S_{y}
  15. χ + y = 1 2 [ 1 i ] \chi_{+}^{y}={1\over\sqrt{2}}\begin{bmatrix}1\\ i\\ \end{bmatrix}
  16. χ - y = 1 2 [ i 1 ] \chi_{-}^{y}={1\over\sqrt{2}}\begin{bmatrix}i\\ 1\\ \end{bmatrix}
  17. χ + = [ cos ( θ / 2 ) e i φ sin ( θ / 2 ) ] \chi_{+}=\begin{bmatrix}\cos(\theta/2)\\ e^{i\varphi}\sin(\theta/2)\\ \end{bmatrix}
  18. χ - = [ - e - i φ sin ( θ / 2 ) cos ( θ / 2 ) ] \chi_{-}=\begin{bmatrix}-e^{-i\varphi}\sin(\theta/2)\\ \cos(\theta/2)\\ \end{bmatrix}
  19. χ = 1 5 [ 1 2 ] \chi={1\over\sqrt{5}}\begin{bmatrix}1\\ 2\\ \end{bmatrix}
  20. c + = [ 1 0 ] * χ = 1 5 c_{+}=\begin{bmatrix}1\ 0\\ \end{bmatrix}*\chi={1\over\sqrt{5}}
  21. P + = 1 5 P_{+}={1\over 5}

Eikonal_equation.html

  1. | u ( x ) | = F ( x ) , x Ω |\nabla u(x)|=F(x),\ x\in\Omega
  2. u | Ω = 0 u|_{\partial\Omega}=0
  3. Ω \Omega
  4. n \mathbb{R}^{n}
  5. F ( x ) F(x)
  6. \nabla
  7. F ( x ) F(x)
  8. u ( x ) u(x)
  9. Ω \partial\Omega
  10. x x
  11. Ω , \Omega,
  12. F ( x ) F(x)
  13. x x
  14. F = 1 F=1
  15. Ω \partial\Omega
  16. E = - V E=-\nabla V
  17. E E
  18. V V
  19. H ( x , u ( x ) ) = 0 H(x,\nabla u(x))=0
  20. u ( 0 , x ) = u 0 ( x ) , for x = ( x 1 , x ) u(0,x^{\prime})=u_{0}(x^{\prime}),\,\text{ for }x=(x_{1},x^{\prime})
  21. x = ( 0 , x ) x=(0,x^{\prime})
  22. x 1 x_{1}
  23. t . t.
  24. c ( x ) 2 | x u ( x , t ) | 2 = | t u ( x , t ) | 2 c(x)^{2}|\nabla_{x}u(x,t)|^{2}=|\partial_{t}u(x,t)|^{2}
  25. p 1 H ( x , p ) 0 \partial_{p_{1}}H(x,p)\neq 0
  26. x = ( 0 , x ) x=(0,x^{\prime})
  27. H ( x , ξ ( x ) ) = 0 H(x,\xi(x))=0
  28. ξ ( x ) = u ( x ) , x H \xi(x)=\nabla u(x),x\in H
  29. ξ \xi
  30. x ˙ ( s ) = ξ H ( x ( s ) , ξ ( s ) ) , ξ ˙ ( s ) = - x H ( x ( s ) , ξ ( s ) ) . \dot{x}(s)=\nabla_{\xi}H(x(s),\xi(s)),\;\;\;\;\dot{\xi}(s)=-\nabla_{x}H(x(s),% \xi(s)).
  31. x ( 0 ) = x 0 , ξ ( x ( 0 ) ) = u ( x ( 0 ) ) . x(0)=x_{0},\;\;\;\;\xi(x(0))=\nabla u(x(0)).
  32. u u
  33. u ( x ) \nabla u(x)
  34. x = ( 0 , x ) x=(0,x^{\prime})
  35. H H
  36. 0 s < s 1 0\leq s<s_{1}
  37. x = ( 0 , x ) x=(0,x^{\prime})
  38. ξ \xi
  39. s H ( x ( s ) , ξ ( s ) ) = 0 \partial_{s}H(x(s),\xi(s))=0
  40. H = 0 H=0
  41. u u
  42. u = ξ \nabla u=\xi
  43. s s
  44. ( u ) ( x ( s ) ) = ξ ( x ( s ) ) . (\nabla u)(x(s))=\xi(x(s)).
  45. u ( x ) u(x)
  46. d d s u ( x ( s ) ) = u ( x ( s ) ) x ˙ ( s ) = ξ H ξ \frac{d}{ds}u(x(s))=\nabla u(x(s))\cdot\dot{x}(s)=\xi\cdot\frac{\partial H}{% \partial\xi}
  47. u ( x ( t ) ) = u ( x ( 0 ) ) + 0 t ξ ( x ( s ) ) x ˙ ( s ) d s . u(x(t))=u(x(0))+\int_{0}^{t}\xi(x(s))\cdot\dot{x}(s)\,ds.
  48. u u
  49. x ( t ) x(t)
  50. u u
  51. ξ ( x ( t ) ) = ξ ( x ( 0 ) ) - 0 s x H ( x ( s ) , ξ ( x ( s ) ) ) d s . \xi(x(t))=\xi(x(0))-\int_{0}^{s}\nabla_{x}H(x(s),\xi(x(s)))\,ds.
  52. ξ \xi
  53. u u
  54. ξ \xi
  55. ξ \xi
  56. ξ ( x ( 0 ) ) = u ( x ( 0 ) ) \xi(x(0))=\nabla u(x(0))
  57. 2 x k x j H = 2 x j x k H . \frac{\partial^{2}}{\partial x_{k}\,\partial x_{j}}H=\frac{\partial^{2}}{% \partial x_{j}\,\partial x_{k}}H.

Einstein_aether_theory.html

  1. S = 1 16 π G N d 4 x - g S=\frac{1}{16\pi G_{N}}\int d^{4}x\sqrt{-g}\mathcal{L}
  2. = - R - K m n a b a u m b u n - λ ( g a b u a u b - 1 ) . \mathcal{L}=-R-K^{ab}_{mn}\nabla_{a}u^{m}\nabla_{b}u^{n}-\lambda(g_{ab}u^{a}u^% {b}-1).
  3. \nabla
  4. K m n a b = c 1 g a b g m n + c 2 δ m a δ n b + c 3 δ n a δ m b + c 4 u a u b g m n . K^{ab}_{mn}=c_{1}g^{ab}g_{mn}+c_{2}\delta^{a}_{m}\delta^{b}_{n}+c_{3}\delta^{a% }_{n}\delta^{b}_{m}+c_{4}u^{a}u^{b}g_{mn}.

Einthoven's_triangle.html

  1. I = L A - R A I=LA-RA
  2. I I = L L - R A II=LL-RA
  3. I I I = L L - L A III=LL-LA

Einzel_lens.html

  1. Δ v r = q E r ( r , z ) m v z d z , \Delta v_{r}=\int\frac{qE_{r}(r,z)}{mv_{z}}dz,
  2. E r ( r , z ) E_{r}(r,z)
  3. m m
  4. v z v_{z}

Ekman_layer.html

  1. - f v = - 1 ρ o p x + K m 2 u z 2 , f u = - 1 ρ o p y + K m 2 v z 2 , 0 = - 1 ρ o p z , \begin{aligned}\displaystyle-fv&\displaystyle=-\frac{1}{\rho_{o}}\frac{% \partial p}{\partial x}+K_{m}\frac{\partial^{2}u}{\partial z^{2}},\\ \displaystyle fu&\displaystyle=-\frac{1}{\rho_{o}}\frac{\partial p}{\partial y% }+K_{m}\frac{\partial^{2}v}{\partial z^{2}},\\ \displaystyle 0&\displaystyle=-\frac{1}{\rho_{o}}\frac{\partial p}{\partial z}% ,\end{aligned}
  2. u \ u
  3. v \ v
  4. x \ x
  5. y \ y
  6. f \ f
  7. K m \ K_{m}
  8. p p
  9. at z = 0 : A u z = τ x and A v z = τ y , \,\text{at }z=0:\quad A\frac{\partial u}{\partial z}=\tau^{x}\quad\,\text{and}% \quad A\frac{\partial v}{\partial z}=\tau^{y},
  10. τ x \ \tau^{x}
  11. τ y \ \tau^{y}
  12. τ \ \tau
  13. u g \ u_{g}
  14. v g \ v_{g}
  15. x \ x
  16. y \ y
  17. z : u u g , v v g . \ z\to\infty:u\to u_{g},v\to v_{g}.
  18. u \displaystyle u
  19. - z -z
  20. w = 1 f ρ o [ - ( τ x x + τ y y ) e z / d sin ( z / d ) + ( τ y x - τ x y ) ( 1 - e z / d cos ( z / d ) ) ] . w=\frac{1}{f\rho_{o}}\left[-\left(\frac{\partial\tau^{x}}{\partial x}+\frac{% \partial\tau^{y}}{\partial y}\right)e^{z/d}\sin(z/d)+\left(\frac{\partial\tau^% {y}}{\partial x}-\frac{\partial\tau^{x}}{\partial y}\right)(1-e^{z/d}\cos(z/d)% )\right].

Elastic_energy.html

  1. U = 1 2 k Δ x 2 U=\frac{1}{2}k\Delta x^{2}\,
  2. d U = - P d V , dU=-P\,dV\ ,
  3. k = - F r L - L o k=-\tfrac{F_{r}}{L-L_{o}}
  4. ( L - L o ) x (L-L_{o})\rightarrow x\,
  5. F r = - k x F_{r}=\,-k\,x
  6. F a x = F a x . \vec{F_{a}}\cdot\vec{x}=F_{a}\,x.
  7. U = 0 L - L o k x d x = 1 2 k ( L - L o ) 2 U=\int_{0}^{L-L_{o}}{k\ x\ dx}=\tfrac{1}{2}k(L-L_{o})^{2}
  8. Δ l \Delta l
  9. U e = Y A 0 Δ l l 0 d l = Y A 0 Δ l 2 2 l 0 U_{e}=\int{\frac{YA_{0}\Delta l}{l_{0}}}\,dl=\frac{YA_{0}{\Delta l}^{2}}{2l_{0}}
  10. U e A 0 l 0 = Y Δ l 2 2 l 0 2 = 1 2 Y ε 2 \frac{U_{e}}{A_{0}l_{0}}=\frac{Y{\Delta l}^{2}}{2l_{0}^{2}}=\frac{1}{2}Y{% \varepsilon}^{2}
  11. ε = Δ l l 0 \varepsilon=\frac{\Delta l}{l_{0}}
  12. f ( ε i j ) = 1 2 λ ε i i 2 + μ ε i j 2 f(\varepsilon_{ij})=\frac{1}{2}\lambda\varepsilon_{ii}^{2}+\mu\varepsilon_{ij}% ^{2}
  13. σ i j = ( f ε i j ) T , \sigma_{ij}=\left(\frac{\partial f}{\partial\varepsilon_{ij}}\right)_{T},
  14. f = 1 2 ε i j σ i j . f=\frac{1}{2}\varepsilon_{ij}\sigma_{ij}.
  15. U = 1 2 C i j k l ε i j ε k l U=\frac{1}{2}C_{ijkl}\varepsilon_{ij}\varepsilon_{kl}
  16. C i j k l C_{ijkl}
  17. ε i j \varepsilon_{ij}
  18. C i j k l C_{ijkl}
  19. C i j k l = λ δ i j δ k l + μ ( δ i k δ j l + δ i l δ j k ) C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jl}+\delta_{il}% \delta_{jk})
  20. λ \lambda
  21. μ \mu
  22. δ i j \delta_{ij}
  23. ε i j = 1 2 ( i u j + j u i ) \varepsilon_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i})
  24. u i u_{i}
  25. i t h i^{th}
  26. j \partial_{j}
  27. j t h j^{th}
  28. ε j j = j u j \varepsilon_{jj}=\partial_{j}u_{j}
  29. j j

Electric_power.html

  1. P = work done per unit time = V Q t = V I P=\,\text{work done per unit time}=\frac{VQ}{t}=VI\,
  2. P = I V = I 2 R = V 2 R , P=IV=I^{2}R=\frac{V^{2}}{R},
  3. P = 1 2 V p I p cos θ = V r m s I r m s cos θ P={1\over 2}V_{p}I_{p}\cos\theta=V_{rms}I_{rms}\cos\theta\,
  4. (apparent power) = 2 (real power) + 2 (reactive power) 2 \mbox{(apparent power)}~{}^{2}=\mbox{(real power)}~{}^{2}+\mbox{(reactive % power)}~{}^{2}
  5. (real power) = (apparent power) cos θ \mbox{(real power)}~{}=\mbox{(apparent power)}\cos\theta
  6. (reactive power) = (apparent power) sin θ \mbox{(reactive power)}~{}=\mbox{(apparent power)}\sin\theta
  7. P = S ( 𝐄 × 𝐇 ) 𝐝𝐀 . P=\int_{S}(\mathbf{E}\times\mathbf{H})\cdot\mathbf{dA}.\,

Electrical_characteristics_of_dynamic_loudspeakers.html

  1. Z nom = 1.15 Z min Z_{\mathrm{nom}}=1.15\cdot Z_{\mathrm{min}}

Electrical_load.html

  1. V S V_{S}
  2. V S V_{S}
  3. R S R_{S}
  4. V S V_{S}
  5. V O U T = V S R L R L + R S V_{OUT}=V_{S}\cdot\frac{R_{L}}{R_{L}+R_{S}}

Electrical_resonance.html

  1. ω = 1 L C \omega=\frac{1}{\sqrt{LC}}

Electrodialysis.html

  1. ξ = z F Q f ( C i n l e t d - C o u t l e t d ) N I \xi=\frac{zFQ_{f}(C_{inlet}^{d}-C_{outlet}^{d})}{NI}
  2. ξ \xi
  3. z z
  4. F F
  5. Q f Q_{f}
  6. C i n l e t d C_{inlet}^{d}
  7. C o u t l e t d C_{outlet}^{d}
  8. N N
  9. I I

Electromagnetic_mass.html

  1. E e m E_{em}
  2. m e m m_{em}
  3. E e m = 1 2 e 2 a , m e m = 2 3 e 2 a c 2 E_{em}=\frac{1}{2}\frac{e^{2}}{a},\qquad m_{em}=\frac{2}{3}\frac{e^{2}}{ac^{2}}
  4. e e
  5. a a
  6. m e m = 4 3 E e m c 2 m_{em}=\frac{4}{3}\frac{E_{em}}{c^{2}}
  7. M M
  8. G G
  9. r r
  10. G 4 3 E e m c 2 M r G\frac{\frac{4}{3}\frac{E_{em}}{c^{2}}M}{r}
  11. E e m v = E e m [ 1 β ln 1 + β 1 - β - 1 ] , β = v c , E_{em}^{v}=E_{em}\left[\frac{1}{\beta}\ln\frac{1+\beta}{1-\beta}-1\right],% \qquad\beta=\frac{v}{c},
  12. m L = 3 4 m e m 1 β 2 [ - 1 β 2 ln ( 1 + β 1 - β ) + 2 1 - β 2 ] m_{L}=\frac{3}{4}\cdot m_{em}\cdot\frac{1}{\beta^{2}}\left[-\frac{1}{\beta^{2}% }\ln\left(\frac{1+\beta}{1-\beta}\right)+\frac{2}{1-\beta^{2}}\right]
  13. m T = 3 4 m e m 1 β 2 [ ( 1 + β 2 2 β ) ln ( 1 + β 1 - β ) - 1 ] m_{T}=\frac{3}{4}\cdot m_{em}\cdot\frac{1}{\beta^{2}}\left[\left(\frac{1+\beta% ^{2}}{2\beta}\right)\ln\left(\frac{1+\beta}{1-\beta}\right)-1\right]
  14. k 3 ε k^{3}\varepsilon
  15. k ε k\varepsilon
  16. k = 1 - v 2 / c 2 k=\sqrt{1-v^{2}/c^{2}}
  17. ε \varepsilon
  18. ε \varepsilon
  19. m L = m e m ( 1 - v 2 c 2 ) 3 , m T = m e m 1 - v 2 c 2 m_{L}=\frac{m_{em}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}% =\frac{m_{em}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  20. m L = m e m ( 1 - 1 3 v 2 c 2 ) ( 1 - v 2 c 2 ) 8 / 3 , m T = m e m ( 1 - v 2 c 2 ) 2 / 3 m_{L}=\frac{m_{em}\left(1-\frac{1}{3}\frac{v^{2}}{c^{2}}\right)}{\left(\sqrt{1% -\frac{v^{2}}{c^{2}}}\right)^{8/3}},\quad m_{T}=\frac{m_{em}}{\left(\sqrt{1-% \frac{v^{2}}{c^{2}}}\right)^{2/3}}
  21. m e m = ( 4 / 3 ) E e m / c 2 m_{em}=(4/3)E_{em}/c^{2}
  22. m e s = E e m / c 2 m_{es}=E_{em}/c^{2}
  23. E p E_{p}
  24. E e m E_{em}
  25. E t o t E_{tot}
  26. E t o t c 2 = E e m + E p c 2 = E e m + E e m 3 c 2 = 4 3 E e m c 2 = 4 3 m e s = m e m \frac{E_{tot}}{c^{2}}=\frac{E_{em}+E_{p}}{c^{2}}=\frac{E_{em}+\frac{E_{em}}{3}% }{c^{2}}=\frac{4}{3}\frac{E_{em}}{c^{2}}=\frac{4}{3}m_{es}=m_{em}
  27. E e m / c 2 E_{em}/c^{2}
  28. m e m = E e m / c 2 m_{em}=E_{em}/c^{2}
  29. E e m / c 2 E_{em}/c^{2}
  30. E e m / c E_{em}/c
  31. m = ( 8 / 3 ) E / c 2 m=(8/3)E/c^{2}
  32. m = ( 4 / 3 ) E / c 2 m=(4/3)E/c^{2}
  33. E = m c 2 E=mc^{2}
  34. F = d p / d t \vec{F}=\mathrm{d}\vec{p}/\mathrm{d}t
  35. M = m 0 1 - v 2 c 2 , m 0 = E c 2 , M=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\qquad m_{0}=\frac{E}{c^{2}},
  36. m e m = E e m / c 2 m_{em}=E_{em}/c^{2}
  37. m t o t = E t o t / c 2 m_{tot}=E_{tot}/c^{2}
  38. m e m = E e m / c 2 m_{em}=E_{em}/c^{2}

Electromagnetic_reverberation_chamber.html

  1. s s
  2. e e
  3. E \vec{E}
  4. H \vec{H}
  5. E T , H T E_{T},\,H_{T}
  6. E R , H R E_{R},\,H_{R}
  7. Z 0 = | E | | H | = 120 π Ω Z_{0}=\frac{|\vec{E}|}{|\vec{H}|}=120\cdot\pi\,\Omega
  8. η Tx \eta_{\rm Tx}
  9. η Rx \eta_{\rm Rx}
  10. P fwd , P bwd P_{\rm fwd},\,P_{\rm bwd}
  11. Q Q
  12. X N s {}_{s}\langle X\rangle_{N}
  13. X X
  14. N N
  15. X N e {}_{e}\langle X\rangle_{N}
  16. X X
  17. N N
  18. X \langle X\rangle
  19. X \langle X\rangle_{\infty}
  20. X N s {}_{s}\lceil X\rceil_{N}
  21. X X
  22. N N
  23. X N e {}_{e}\lceil X\rceil_{N}
  24. X X
  25. N N
  26. X \lceil X\rceil
  27. X \lceil X\rceil_{\infty}
  28. s ( X ) N {}_{s}\!\dagger\!(X)_{N}
  29. e ( X ) N {}_{e}\!\dagger\!(X)_{N}
  30. f m n p f_{mnp}
  31. f m n p = c 2 ( m l ) 2 + ( n w ) 2 + ( p h ) 2 , f_{mnp}=\frac{c}{2}\sqrt{\left(\frac{m}{l}\right)^{2}+\left(\frac{n}{w}\right)% ^{2}+\left(\frac{p}{h}\right)^{2}},
  32. c c
  33. l l
  34. w w
  35. h h
  36. m m
  37. n n
  38. p p
  39. f f
  40. N ( f ) N(f)
  41. T E m n p TE_{mnp}
  42. T M m n p TM_{mnp}
  43. ( x , y , z ) (x,y,z)
  44. H z = 0 H_{z}=0
  45. E x = - 1 j ω ϵ k x k z cos k x x sin k y y sin k z z E_{x}=-\frac{1}{j\omega\epsilon}k_{x}k_{z}\cos k_{x}x\sin k_{y}y\sin k_{z}z
  46. E y = - 1 j ω ϵ k y k z sin k x x cos k y y sin k z z E_{y}=-\frac{1}{j\omega\epsilon}k_{y}k_{z}\sin k_{x}x\cos k_{y}y\sin k_{z}z
  47. E z = 1 j ω ϵ k x y 2 sin k x x sin k y y cos k z z E_{z}=\frac{1}{j\omega\epsilon}k_{xy}^{2}\sin k_{x}x\sin k_{y}y\cos k_{z}z
  48. H x = k y sin k x x cos k y y cos k z z H_{x}=k_{y}\sin k_{x}x\cos k_{y}y\cos k_{z}z
  49. H y = - k x cos k x x sin k y y cos k z z H_{y}=-k_{x}\cos k_{x}x\sin k_{y}y\cos k_{z}z
  50. k r 2 = k x 2 + k y 2 + k z 2 , k x = m π l , k y = n π w , k z = p π h k x y 2 = k x 2 + k y 2 k_{r}^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2},\,k_{x}=\frac{m\pi}{l},\,k_{y}=\frac{n% \pi}{w},\,k_{z}=\frac{p\pi}{h}\,k_{xy}^{2}=k_{x}^{2}+k_{y}^{2}
  51. E z = 0 E_{z}=0
  52. E x = k y cos k x x sin k y y sin k z z E_{x}=k_{y}\cos k_{x}x\sin k_{y}y\sin k_{z}z
  53. E y = - k x sin k x x cos k y y sin k z z E_{y}=-k_{x}\sin k_{x}x\cos k_{y}y\sin k_{z}z
  54. H x = - 1 j ω μ k x k z sin k x x cos k y y cos k z z H_{x}=-\frac{1}{j\omega\mu}k_{x}k_{z}\sin k_{x}x\cos k_{y}y\cos k_{z}z
  55. H y = - 1 j ω μ k y k z cos k x x sin k y y cos k z z H_{y}=-\frac{1}{j\omega\mu}k_{y}k_{z}\cos k_{x}x\sin k_{y}y\cos k_{z}z
  56. H z = 1 j ω μ k x y 2 cos k x x cos k y y sin k z z H_{z}=\frac{1}{j\omega\mu}k_{xy}^{2}\cos k_{x}x\cos k_{y}y\sin k_{z}z
  57. N ( f ) N(f)
  58. N ¯ ( f ) \overline{N}(f)
  59. N ¯ ( f ) = 8 π 3 l w h ( f c ) 3 - ( l + w + h ) f c + 1 2 . \overline{N}(f)=\frac{8\pi}{3}lwh\left(\frac{f}{c}\right)^{3}-(l+w+h)\frac{f}{% c}+\frac{1}{2}.
  60. N ¯ ( f ) \overline{N}(f)
  61. n ¯ ( f ) \overline{n}(f)
  62. n ¯ ( f ) = d N ¯ ( f ) d f = 8 π c l w h ( f c ) 2 - ( l + w + h ) 1 c . \overline{n}(f)=\frac{d\overline{N}(f)}{df}=\frac{8\pi}{c}lwh\left(\frac{f}{c}% \right)^{2}-(l+w+h)\frac{1}{c}.
  63. Δ f \Delta f
  64. N ¯ Δ f ( f ) \overline{N}_{\Delta f}(f)
  65. N ¯ Δ f ( f ) = f - Δ f / 2 f + Δ f / 2 n ¯ ( f ) d f = N ¯ ( f + Δ f / 2 ) - N ¯ ( f - Δ f / 2 ) 8 π l w h c 3 f 2 Δ f \begin{matrix}\overline{N}_{\Delta f}(f)&=&\int_{f-\Delta f/2}^{f+\Delta f/2}% \overline{n}(f)df\\ &=&\overline{N}(f+\Delta f/2)-\overline{N}(f-\Delta f/2)\\ &\simeq&\frac{8\pi lwh}{c^{3}}\cdot f^{2}\cdot\Delta f\end{matrix}
  66. Q = ω maximum stored energy average power loss = ω W s P l , Q=\omega\frac{\rm maximum\;stored\;energy}{\rm average\;power\;loss}=\omega% \frac{W_{s}}{P_{l}},
  67. ω = 2 π f \omega=2\pi f
  68. W s W_{s}
  69. W s = ϵ 2 V | E | 2 d V = μ 2 V | H | 2 d V . W_{s}=\frac{\epsilon}{2}\iiint_{V}|\vec{E}|^{2}dV=\frac{\mu}{2}\iiint_{V}|\vec% {H}|^{2}dV.
  70. σ \sigma
  71. μ \mu
  72. R s R_{s}
  73. R s = 1 σ δ s = π μ f σ , R_{s}=\frac{1}{\sigma\delta_{s}}=\sqrt{\frac{\pi\mu f}{\sigma}},
  74. δ s = 1 / π μ σ f \delta_{s}=1/\sqrt{\pi\mu\sigma f}
  75. P l P_{l}
  76. P l = R s 2 S | H | 2 d S . P_{l}=\frac{R_{s}}{2}\iint_{S}|\vec{H}|^{2}dS.
  77. Q TE mnp = Z 0 l w h 4 R s k x y 2 k r 3 ζ l h ( k x y 4 + k x 2 k z 2 ) + ξ w h ( k x y 4 + k y 2 k z 2 ) + l w k x y 2 k z 2 Q_{\rm TE_{mnp}}=\frac{Z_{0}lwh}{4R_{s}}\frac{k_{xy}^{2}k_{r}^{3}}{\zeta lh% \left(k_{xy}^{4}+k_{x}^{2}k_{z}^{2}\right)+\xi wh\left(k_{xy}^{4}+k_{y}^{2}k_{% z}^{2}\right)+lwk_{xy}^{2}k_{z}^{2}}
  78. ζ = { 1 if n 0 1 / 2 if n = 0 , ξ = { 1 if m 0 1 / 2 if m = 0 \zeta=\begin{cases}1&\mbox{if }~{}n\neq 0\\ 1/2&\mbox{if }~{}n=0\end{cases},\quad\xi=\begin{cases}1&\mbox{if }~{}m\neq 0\\ 1/2&\mbox{if }~{}m=0\end{cases}
  79. Q TM mnp = Z 0 l w h 4 R s k x y 2 k r w ( γ l + h ) k x 2 + l ( γ w + h ) k y 2 Q_{\rm TM_{mnp}}=\frac{Z_{0}lwh}{4R_{s}}\frac{k_{xy}^{2}k_{r}}{w(\gamma l+h)k_% {x}^{2}+l(\gamma w+h)k_{y}^{2}}
  80. γ = { 1 if p 0 1 / 2 if p = 0 \gamma=\begin{cases}1&\mbox{if }~{}p\neq 0\\ 1/2&\mbox{if }~{}p=0\end{cases}
  81. Q s ~ \tilde{Q_{s}}
  82. 1 Q s ~ = 1 Q m n p k k r k r + Δ k \frac{1}{\tilde{Q_{s}}}=\langle\frac{1}{Q_{mnp}}\rangle_{k\leq k_{r}\leq k_{r}% +\Delta k}
  83. Q s ~ = 3 2 V S δ s 1 1 + 3 c 16 f ( 1 / l + 1 / w + 1 / h ) \tilde{Q_{s}}=\frac{3}{2}\frac{V}{S\delta_{s}}\frac{1}{1+\frac{3c}{16f}\left(1% /l+1/w+1/h\right)}
  84. Q s ~ \tilde{Q_{s}}
  85. Q a Q_{a}
  86. Q a = 16 π 2 V f 3 c 3 N a , Q_{a}=\frac{16\pi^{2}Vf^{3}}{c^{3}N_{a}},
  87. N a N_{a}
  88. 1 Q = i 1 Q i \frac{1}{Q}=\sum_{i}\frac{1}{Q_{i}}
  89. BW Q {\rm BW}_{Q}
  90. BW Q {\rm BW}_{Q}
  91. BW Q = f Q {\rm BW}_{Q}=\frac{f}{Q}
  92. N ¯ Δ f ( f ) \overline{N}_{\Delta f}(f)
  93. BW Q {\rm BW}_{Q}
  94. M ( f ) = 8 π V f 3 c 3 Q . M(f)=\frac{8\pi Vf^{3}}{c^{3}Q}.
  95. τ \tau
  96. τ = Q 2 π f . \tau=\frac{Q}{2\pi f}.

Electromagnetic_stress–energy_tensor.html

  1. T μ ν = 1 μ 0 [ F μ α F ν - α 1 4 η μ ν F α β F α β ] . T^{\mu\nu}=\frac{1}{\mu_{0}}\left[F^{\mu\alpha}F^{\nu}{}_{\alpha}-\frac{1}{4}% \eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right]\,.
  2. F μ ν F^{\mu\nu}
  3. η μ ν \eta_{\mu\nu}
  4. T μ ν T^{\mu\nu}
  5. T μ ν = [ 1 2 ( ϵ 0 E 2 + 1 μ 0 B 2 ) S x / c S y / c S z / c S x / c - σ x x - σ xy - σ xz S y / c - σ y x - σ yy - σ yz S z / c - σ z x - σ zy - σ zz ] , T^{\mu\nu}=\begin{bmatrix}\frac{1}{2}\left(\epsilon_{0}E^{2}+\frac{1}{\mu_{0}}% B^{2}\right)&S\text{x}/c&S\text{y}/c&S\text{z}/c\\ S\text{x}/c&-\sigma_{xx}&-\sigma\text{xy}&-\sigma\text{xz}\\ S\text{y}/c&-\sigma_{yx}&-\sigma\text{yy}&-\sigma\text{yz}\\ S\text{z}/c&-\sigma_{zx}&-\sigma\text{zy}&-\sigma\text{zz}\end{bmatrix},
  6. S = 1 μ 0 E × B , {S}=\frac{1}{\mu_{0}}{E}\times{B},
  7. σ i j = ϵ 0 E i E j + 1 < m t p l > μ 0 B i B j - 1 2 ( ϵ 0 E 2 + 1 μ 0 B 2 ) δ i j \sigma_{ij}=\epsilon_{0}E_{i}E_{j}+\frac{1}{<}mtpl>{{\mu_{0}}}B_{i}B_{j}-\frac% {1}{2}\left(\epsilon_{0}E^{2}+\frac{1}{\mu_{0}}B^{2}\right)\delta_{ij}
  8. T μ ν T^{\mu\nu}
  9. ϵ 0 = 1 4 π , μ 0 = 4 π \epsilon_{0}=\frac{1}{4\pi},\quad\mu_{0}=4\pi\,
  10. T μ ν = 1 4 π [ F μ α F ν - α 1 4 η μ ν F α β F α β ] . T^{\mu\nu}=\frac{1}{4\pi}[F^{\mu\alpha}F^{\nu}{}_{\alpha}-\frac{1}{4}\eta^{\mu% \nu}F_{\alpha\beta}F^{\alpha\beta}]\,.
  11. T μ ν = [ 1 8 π ( E 2 + B 2 ) S x / c S y / c S z / c S x / c - σ xx - σ xy - σ xz S y / c - σ yx - σ yy - σ yz S z / c - σ zx - σ zy - σ zz ] T^{\mu\nu}=\begin{bmatrix}\frac{1}{8\pi}(E^{2}+B^{2})&S\text{x}/c&S\text{y}/c&% S\text{z}/c\\ S_{x}/c&-\sigma\text{xx}&-\sigma\text{xy}&-\sigma\text{xz}\\ S\text{y}/c&-\sigma\text{yx}&-\sigma\text{yy}&-\sigma\text{yz}\\ S\text{z}/c&-\sigma\text{zx}&-\sigma\text{zy}&-\sigma\text{zz}\end{bmatrix}
  12. S = c 4 π E × B . {S}=\frac{c}{4\pi}{E}\times{B}.
  13. T μ ν T^{\mu\nu}\!
  14. P μ P^{\mu}\!
  15. x ν x^{\nu}
  16. T μ ν = T ν μ T^{\mu\nu}=T^{\nu\mu}
  17. T ν α T^{\nu}{}_{\alpha}
  18. T α = α 0 T^{\alpha}{}_{\alpha}=0
  19. T 00 0 T^{00}\geq 0
  20. ν T μ ν + η μ ρ f ρ = 0 \partial_{\nu}T^{\mu\nu}+\eta^{\mu\rho}\,f_{\rho}=0\,
  21. f ρ f_{\rho}
  22. u em t + S + J E = 0 \frac{\partial u_{\mathrm{em}}}{\partial t}+{\nabla}\cdot{S}+{J}\cdot{E}=0\,
  23. p em t - σ + ρ E + J × B = 0 \frac{\partial{p}_{\mathrm{em}}}{\partial t}-{\nabla}\cdot\sigma+\rho{E}+{J}% \times{B}=0\,
  24. u em = ϵ 0 2 E 2 + 1 2 μ 0 B 2 u_{\mathrm{em}}=\frac{\epsilon_{0}}{2}E^{2}+\frac{1}{2\mu_{0}}B^{2}\,
  25. p em = S c 2 {p}_{\mathrm{em}}={{S}\over{c^{2}}}

Electron_electric_dipole_moment.html

  1. U = 𝐝 e 𝐄 . U=\mathbf{d}_{\rm e}\cdot\mathbf{E}.

Electron_liquid.html

  1. r s r_{s}
  2. r s = ( 3 / ( 4 π n ) ) 1 / 3 r_{s}=(3/(4\pi n))^{1/3}
  3. r s r_{s}
  4. 1 / r s 2 1/r_{s}^{2}
  5. 1 / r s 1/r_{s}
  6. r s r_{s}
  7. r s < 1 r_{s}<1
  8. r s r_{s}
  9. r s r_{s}
  10. r s r_{s}
  11. r s r_{s}
  12. r s > 1 r_{s}>1
  13. 2 < r s < 6 2<r_{s}<6
  14. 1 / r 1/r
  15. e x p ( - r / r T F ) / r exp(-r/r_{TF})/r
  16. r T F r_{TF}
  17. E F E_{F}
  18. E F E_{F}

Electron_scattering.html

  1. 𝐅 = q 𝐄 + q 𝐯 × 𝐁 \mathbf{F}=q\mathbf{E}+q\mathbf{v}\times\mathbf{B}
  2. 𝐅 = q [ - ϕ - d 𝐀 d t + ( 𝐀 𝐯 ) ] \mathbf{F}=q[-\nabla\phi-\frac{d\mathbf{A}}{dt}+\nabla(\mathbf{A}\cdot\mathbf{% v})]
  3. ϕ \phi
  4. m d 𝐯 d t = q 𝐄 + q 𝐯 × 𝐁 m\frac{d\mathbf{v}}{dt}=q\mathbf{E}+q\mathbf{v}\times\mathbf{B}
  5. m d γ 𝐯 d t = q 𝐄 + q 𝐯 × 𝐁 m\frac{d\gamma\mathbf{v}}{dt}=q\mathbf{E}+q\mathbf{v}\times\mathbf{B}
  6. γ ( v ) 1 1 - v 2 / c 2 \gamma(v)\equiv\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  7. s y m b o l r 21 = s y m b o l r 1 - r 2 symbol{r_{21}}=symbol{r_{1}-r_{2}}
  8. s y m b o l r ^ 21 = s y m b o l r 21 / | s y m b o l r 21 | symbol{\hat{r}_{21}}={symbol{r_{21}}/|symbol{r_{21}}|}
  9. F = k | q 1 q 2 | r 2 = | q 1 q 2 | 4 π ϵ 0 r 2 F=k\frac{|q_{1}q_{2}|}{r^{2}}=\frac{|q_{1}q_{2}|}{4\pi\epsilon_{0}r^{2}}
  10. 𝐅 = k q 1 q 2 | 𝐫 | 2 𝐫 ^ = q 1 q 2 4 π ϵ 0 | 𝐫 | 2 𝐫 ^ \mathbf{F}=k\frac{q_{1}q_{2}}{|\mathbf{r}|^{2}}\mathbf{\hat{r}}=\frac{q_{1}q_{% 2}}{4\pi\epsilon_{0}|\mathbf{r}|^{2}}\mathbf{\hat{r}}
  11. k = 1 4 π ϵ 0 8.988 × 10 9 N m 2 C - 2 k=\frac{1}{4\pi\epsilon_{0}}\approx 8.988\times 10^{9}Nm^{2}{C^{-}}^{2}
  12. ϵ 0 8.854 × 10 - 12 C 2 N - 1 m - 2 \epsilon_{0}\approx 8.854\times 10^{-12}C^{2}N^{-1}m^{-2}
  13. F = k | q 1 q 2 | ϵ r r 2 F=k\frac{|q_{1}q_{2}|}{\epsilon_{r}r^{2}}
  14. E = h ν = h f E=h\nu=hf
  15. λ f - λ i = h m e c ( 1 - cos θ ) \lambda_{f}-\lambda_{i}=\frac{h}{m_{e}c}(1-\cos\theta)
  16. 𝐚 𝐯 \mathbf{a}\perp\mathbf{v}
  17. P = 2 K e 2 3 c 2 a 2 P=\frac{2Ke^{2}}{3c^{2}}a^{2}
  18. a n o n - r e l a t i v i s t i c = v 2 r a r e l a t i v i s t i c = 1 m d p d τ = 1 m γ d ( γ m v ) d t = γ 2 d v d t = γ 2 v 2 r a_{non-relativistic}=\frac{v^{2}}{r}\rightarrow a_{relativistic}=\frac{1}{m}% \frac{dp}{d\tau}=\frac{1}{m}\gamma\frac{d(\gamma mv)}{dt}=\gamma^{2}\frac{dv}{% dt}=\gamma^{2}\frac{v^{2}}{r}
  19. P = 2 K e 2 3 c 2 ( γ 2 v 2 r ) 2 = 2 K e 2 3 c 2 γ 4 v 4 r 2 P=\frac{2Ke^{2}}{3c^{2}}(\frac{\gamma^{2}v^{2}}{r})^{2}=\frac{2Ke^{2}}{3c^{2}}% \frac{\gamma^{4}v^{4}}{r^{2}}

Elementary_abelian_group.html

  1. ( / p ) n e 1 , , e n e i p = 1 , e i e j = e j e i (\mathbb{Z}/p\mathbb{Z})^{n}\cong\langle e_{1},\ldots,e_{n}\mid e_{i}^{p}=1,\ % e_{i}e_{j}=e_{j}e_{i}\rangle
  2. \cong
  3. \cong
  4. \cong
  5. \overset{\sim}{\to}

Elementary_divisors.html

  1. R R
  2. M M
  3. R R
  4. M R r i = 1 l R / ( q i ) with r , l 0 M\cong R^{r}\oplus\bigoplus_{i=1}^{l}R/(q_{i})\qquad\,\text{with }r,l\geq 0
  5. ( q i ) (q_{i})
  6. q i q_{i}
  7. q i = p i r i q_{i}=p_{i}^{r_{i}}
  8. r r
  9. M M
  10. r r
  11. p p
  12. k k
  13. p < s u p > k p<sup>k

Ell_(disambiguation).html

  1. \ell

Ellingham_diagram.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 3 {1}/{3}
  4. 1 / 3 {1}/{3}
  5. 1 / 2 {1}/{2}
  6. 1 / 2 {1}/{2}
  7. 4 3 C r ( s ) + O 2 ( g ) 2 3 C r 2 O 3 {4\over 3}Cr_{(s)}+O_{2(g)}\rightarrow{2\over 3}Cr_{2}O_{3}
  8. 4 3 A l ( s ) + O 2 ( g ) 2 3 A l 2 O 3 {4\over 3}Al_{(s)}+O_{2(g)}\rightarrow{2\over 3}Al_{2}O_{3}
  9. 2 3 C r 2 O 3 ( s ) + 4 3 A l ( s ) 2 3 A l 2 O 3 + 4 3 C r {2\over 3}Cr_{2}O_{3(s)}+{4\over 3}Al_{(s)}\rightarrow{2\over 3}Al_{2}O_{3}+{4% \over 3}Cr
  10. Δ G 0 = - 287 k J \Delta G^{0}=-287kJ

Ellipsoid_method.html

  1. f 0 ( x ) : n f_{0}(x):\mathbb{R}^{n}\to\mathbb{R}
  2. x x
  3. f i ( x ) 0 f_{i}(x)\leq 0
  4. f i \ f_{i}
  5. h i ( x ) = 0 \ h_{i}(x)=0
  6. ( 0 ) n \mathcal{E}^{(0)}\subset\mathbb{R}^{n}
  7. ( 0 ) = { z n : ( z - x 0 ) T P ( 0 ) - 1 ( z - x 0 ) 1 } \mathcal{E}^{(0)}=\left\{z\in\mathbb{R}^{n}:(z-x_{0})^{T}P_{(0)}^{-1}(z-x_{0})% \leq 1\right\}
  8. x * \ x^{*}
  9. P 0 P\succ 0
  10. x 0 x_{0}
  11. \mathcal{E}
  12. f f
  13. g g
  14. f f
  15. k k
  16. x ( k ) x^{(k)}
  17. ( k ) = { x n : ( x - x ( k ) ) T P ( k ) - 1 ( x - x ( k ) ) 1 } \mathcal{E}^{(k)}=\left\{x\in\mathbb{R}^{n}:(x-x^{(k)})^{T}P_{(k)}^{-1}(x-x^{(% k)})\leq 1\right\}
  18. g ( k + 1 ) n g^{(k+1)}\in\mathbb{R}^{n}
  19. g ( k + 1 ) T ( x * - x ( k ) ) 0 g^{(k+1)T}(x^{*}-x^{(k)})\leq 0
  20. x * ( k ) { z : g ( k + 1 ) T ( z - x ( k ) ) 0 } . x^{*}\in\mathcal{E}^{(k)}\cap\{z:g^{(k+1)T}(z-x^{(k)})\leq 0\}.
  21. ( k + 1 ) \mathcal{E}^{(k+1)}
  22. x ( k + 1 ) x^{(k+1)}
  23. x ( k + 1 ) = x ( k ) - 1 n + 1 P ( k ) g ~ ( k + 1 ) x^{(k+1)}=x^{(k)}-\frac{1}{n+1}P_{(k)}\tilde{g}^{(k+1)}
  24. P ( k + 1 ) = n 2 n 2 - 1 ( P ( k ) - 2 n + 1 P ( k ) g ~ ( k + 1 ) g ~ ( k + 1 ) T P ( k ) ) P_{(k+1)}=\frac{n^{2}}{n^{2}-1}\left(P_{(k)}-\frac{2}{n+1}P_{(k)}\tilde{g}^{(k% +1)}\tilde{g}^{(k+1)T}P_{(k)}\right)
  25. g ~ ( k + 1 ) = ( 1 / g ( k + 1 ) T P g ( k + 1 ) ) g ( k + 1 ) \tilde{g}^{(k+1)}=(1/\sqrt{g^{(k+1)T}Pg^{(k+1)}})g^{(k+1)}
  26. g ( k ) T P ( k ) g ( k ) ϵ f ( x ( k ) ) - f ( x * ) ϵ . \sqrt{g^{(k)T}P_{(k)}g^{(k)}}\leq\epsilon\Rightarrow f(x^{(k)})-f(x^{*})\leq\epsilon.
  27. k k
  28. x ( k ) x^{(k)}
  29. ( k ) \mathcal{E}^{(k)}
  30. f best ( k ) f_{\rm{best}}^{(k)}
  31. x ( k ) x^{(k)}
  32. x ( k ) x^{(k)}
  33. g 0 g_{0}
  34. g 0 T ( x * - x ( k ) ) + f 0 ( x ( k ) ) - f best ( k ) 0 g_{0}^{T}(x^{*}-x^{(k)})+f_{0}(x^{(k)})-f_{\rm{best}}^{(k)}\leq 0
  35. x ( k ) x^{(k)}
  36. j j
  37. g j g_{j}
  38. f j f_{j}
  39. g j T ( z - x ( k ) ) + f j ( x ( k ) ) 0 g_{j}^{T}(z-x^{(k)})+f_{j}(x^{(k)})\leq 0
  40. z z

Ellipsoidal_coordinates.html

  1. ( λ , μ , ν ) (\lambda,\mu,\nu)
  2. ( x , y , z ) (x,y,z)
  3. ( λ , μ , ν ) (\lambda,\mu,\nu)
  4. x 2 = ( a 2 + λ ) ( a 2 + μ ) ( a 2 + ν ) ( a 2 - b 2 ) ( a 2 - c 2 ) x^{2}=\frac{\left(a^{2}+\lambda\right)\left(a^{2}+\mu\right)\left(a^{2}+\nu% \right)}{\left(a^{2}-b^{2}\right)\left(a^{2}-c^{2}\right)}
  5. y 2 = ( b 2 + λ ) ( b 2 + μ ) ( b 2 + ν ) ( b 2 - a 2 ) ( b 2 - c 2 ) y^{2}=\frac{\left(b^{2}+\lambda\right)\left(b^{2}+\mu\right)\left(b^{2}+\nu% \right)}{\left(b^{2}-a^{2}\right)\left(b^{2}-c^{2}\right)}
  6. z 2 = ( c 2 + λ ) ( c 2 + μ ) ( c 2 + ν ) ( c 2 - b 2 ) ( c 2 - a 2 ) z^{2}=\frac{\left(c^{2}+\lambda\right)\left(c^{2}+\mu\right)\left(c^{2}+\nu% \right)}{\left(c^{2}-b^{2}\right)\left(c^{2}-a^{2}\right)}
  7. - λ < c 2 < - μ < b 2 < - ν < a 2 . -\lambda<c^{2}<-\mu<b^{2}<-\nu<a^{2}.
  8. λ \lambda
  9. x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1 , \frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}+\frac{z^{2}}{c^{2}+% \lambda}=1,
  10. μ \mu
  11. x 2 a 2 + μ + y 2 b 2 + μ + z 2 c 2 + μ = 1 , \frac{x^{2}}{a^{2}+\mu}+\frac{y^{2}}{b^{2}+\mu}+\frac{z^{2}}{c^{2}+\mu}=1,
  12. ν \nu
  13. x 2 a 2 + ν + y 2 b 2 + ν + z 2 c 2 + ν = 1 \frac{x^{2}}{a^{2}+\nu}+\frac{y^{2}}{b^{2}+\nu}+\frac{z^{2}}{c^{2}+\nu}=1
  14. S ( σ ) = def ( a 2 + σ ) ( b 2 + σ ) ( c 2 + σ ) S(\sigma)\ \stackrel{\mathrm{def}}{=}\ \left(a^{2}+\sigma\right)\left(b^{2}+% \sigma\right)\left(c^{2}+\sigma\right)
  15. σ \sigma
  16. ( λ , μ , ν ) (\lambda,\mu,\nu)
  17. h λ = 1 2 ( λ - μ ) ( λ - ν ) S ( λ ) h_{\lambda}=\frac{1}{2}\sqrt{\frac{\left(\lambda-\mu\right)\left(\lambda-\nu% \right)}{S(\lambda)}}
  18. h μ = 1 2 ( μ - λ ) ( μ - ν ) S ( μ ) h_{\mu}=\frac{1}{2}\sqrt{\frac{\left(\mu-\lambda\right)\left(\mu-\nu\right)}{S% (\mu)}}
  19. h ν = 1 2 ( ν - λ ) ( ν - μ ) S ( ν ) h_{\nu}=\frac{1}{2}\sqrt{\frac{\left(\nu-\lambda\right)\left(\nu-\mu\right)}{S% (\nu)}}
  20. d V = ( λ - μ ) ( λ - ν ) ( μ - ν ) 8 - S ( λ ) S ( μ ) S ( ν ) d λ d μ d ν dV=\frac{\left(\lambda-\mu\right)\left(\lambda-\nu\right)\left(\mu-\nu\right)}% {8\sqrt{-S(\lambda)S(\mu)S(\nu)}}\ d\lambda d\mu d\nu
  21. 2 Φ = 4 S ( λ ) ( λ - μ ) ( λ - ν ) λ [ S ( λ ) Φ λ ] + \nabla^{2}\Phi=\frac{4\sqrt{S(\lambda)}}{\left(\lambda-\mu\right)\left(\lambda% -\nu\right)}\frac{\partial}{\partial\lambda}\left[\sqrt{S(\lambda)}\frac{% \partial\Phi}{\partial\lambda}\right]\ +
  22. 4 S ( μ ) ( μ - λ ) ( μ - ν ) μ [ S ( μ ) Φ μ ] + 4 S ( ν ) ( ν - λ ) ( ν - μ ) ν [ S ( ν ) Φ ν ] \frac{4\sqrt{S(\mu)}}{\left(\mu-\lambda\right)\left(\mu-\nu\right)}\frac{% \partial}{\partial\mu}\left[\sqrt{S(\mu)}\frac{\partial\Phi}{\partial\mu}% \right]\ +\ \frac{4\sqrt{S(\nu)}}{\left(\nu-\lambda\right)\left(\nu-\mu\right)% }\frac{\partial}{\partial\nu}\left[\sqrt{S(\nu)}\frac{\partial\Phi}{\partial% \nu}\right]
  23. 𝐅 \nabla\cdot\mathbf{F}
  24. × 𝐅 \nabla\times\mathbf{F}
  25. ( λ , μ , ν ) (\lambda,\mu,\nu)

Elliptic_coordinate_system.html

  1. F 1 F_{1}
  2. F 2 F_{2}
  3. - a -a
  4. + a +a
  5. x x
  6. ( μ , ν ) (\mu,\nu)
  7. x = a cosh μ cos ν x=a\ \cosh\mu\ \cos\nu
  8. y = a sinh μ sin ν y=a\ \sinh\mu\ \sin\nu
  9. μ \mu
  10. ν [ 0 , 2 π ] . \nu\in[0,2\pi].
  11. x + i y = a cosh ( μ + i ν ) x+iy=a\ \cosh(\mu+i\nu)
  12. x 2 a 2 cosh 2 μ + y 2 a 2 sinh 2 μ = cos 2 ν + sin 2 ν = 1 \frac{x^{2}}{a^{2}\cosh^{2}\mu}+\frac{y^{2}}{a^{2}\sinh^{2}\mu}=\cos^{2}\nu+% \sin^{2}\nu=1
  13. μ \mu
  14. x 2 a 2 cos 2 ν - y 2 a 2 sin 2 ν = cosh 2 μ - sinh 2 μ = 1 \frac{x^{2}}{a^{2}\cos^{2}\nu}-\frac{y^{2}}{a^{2}\sin^{2}\nu}=\cosh^{2}\mu-% \sinh^{2}\mu=1
  15. ν \nu
  16. ( μ , ν ) (\mu,\nu)
  17. h μ = h ν = a sinh 2 μ + sin 2 ν = a cosh 2 μ - cos 2 ν . h_{\mu}=h_{\nu}=a\sqrt{\sinh^{2}\mu+\sin^{2}\nu}=a\sqrt{\cosh^{2}\mu-\cos^{2}% \nu}.
  18. h μ = h ν = a 1 2 ( cosh 2 μ - cos 2 ν ) . h_{\mu}=h_{\nu}=a\sqrt{\frac{1}{2}(\cosh 2\mu-\cos 2\nu}).
  19. d A = h μ h ν d μ d ν = a 2 ( sinh 2 μ + sin 2 ν ) d μ d ν = a 2 ( cosh 2 μ - cos 2 ν ) d μ d ν = a 2 2 ( cosh 2 μ - cos 2 ν ) d μ d ν dA=h_{\mu}h_{\nu}d\mu d\nu=a^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)d\mu d\nu% =a^{2}\left(\cosh^{2}\mu-\cos^{2}\nu\right)d\mu d\nu=\frac{a^{2}}{2}\left(% \cosh 2\mu-\cos 2\nu\right)d\mu d\nu
  20. 2 Φ = 1 a 2 ( sinh 2 μ + sin 2 ν ) ( 2 Φ μ 2 + 2 Φ ν 2 ) = 1 a 2 ( cosh 2 μ - cos 2 ν ) ( 2 Φ μ 2 + 2 Φ ν 2 ) = 2 a 2 ( cosh 2 μ - cos 2 ν ) ( 2 Φ μ 2 + 2 Φ ν 2 ) . \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)}\left(\frac% {\partial^{2}\Phi}{\partial\mu^{2}}+\frac{\partial^{2}\Phi}{\partial\nu^{2}}% \right)=\frac{1}{a^{2}\left(\cosh^{2}\mu-\cos^{2}\nu\right)}\left(\frac{% \partial^{2}\Phi}{\partial\mu^{2}}+\frac{\partial^{2}\Phi}{\partial\nu^{2}}% \right)=\frac{2}{a^{2}\left(\cosh 2\mu-\cos 2\nu\right)}\left(\frac{\partial^{% 2}\Phi}{\partial\mu^{2}}+\frac{\partial^{2}\Phi}{\partial\nu^{2}}\right).
  21. 𝐅 \nabla\cdot\mathbf{F}
  22. × 𝐅 \nabla\times\mathbf{F}
  23. ( μ , ν ) (\mu,\nu)
  24. ( σ , τ ) (\sigma,\tau)
  25. σ = cosh μ \sigma=\cosh\mu
  26. τ = cos ν \tau=\cos\nu
  27. σ \sigma
  28. τ \tau
  29. τ \tau
  30. σ \sigma
  31. ( σ , τ ) (\sigma,\tau)
  32. F 1 F_{1}
  33. F 2 F_{2}
  34. d 1 + d 2 d_{1}+d_{2}
  35. 2 a σ 2a\sigma
  36. d 1 - d 2 d_{1}-d_{2}
  37. 2 a τ 2a\tau
  38. F 1 F_{1}
  39. a ( σ + τ ) a(\sigma+\tau)
  40. F 2 F_{2}
  41. a ( σ - τ ) a(\sigma-\tau)
  42. F 1 F_{1}
  43. F 2 F_{2}
  44. x = - a x=-a
  45. x = + a x=+a
  46. ( σ , τ ) (\sigma,\tau)
  47. x = a σ τ x=a\left.\sigma\right.\tau
  48. y 2 = a 2 ( σ 2 - 1 ) ( 1 - τ 2 ) . y^{2}=a^{2}\left(\sigma^{2}-1\right)\left(1-\tau^{2}\right).
  49. ( σ , τ ) (\sigma,\tau)
  50. h σ = a σ 2 - τ 2 σ 2 - 1 h_{\sigma}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{\sigma^{2}-1}}
  51. h τ = a σ 2 - τ 2 1 - τ 2 . h_{\tau}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{1-\tau^{2}}}.
  52. d A = a 2 σ 2 - τ 2 ( σ 2 - 1 ) ( 1 - τ 2 ) d σ d τ dA=a^{2}\frac{\sigma^{2}-\tau^{2}}{\sqrt{\left(\sigma^{2}-1\right)\left(1-\tau% ^{2}\right)}}d\sigma d\tau
  53. 2 Φ = 1 a 2 ( σ 2 - τ 2 ) [ σ 2 - 1 σ ( σ 2 - 1 Φ σ ) + 1 - τ 2 τ ( 1 - τ 2 Φ τ ) ] . \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sigma^{2}-\tau^{2}\right)}\left[\sqrt{% \sigma^{2}-1}\frac{\partial}{\partial\sigma}\left(\sqrt{\sigma^{2}-1}\frac{% \partial\Phi}{\partial\sigma}\right)+\sqrt{1-\tau^{2}}\frac{\partial}{\partial% \tau}\left(\sqrt{1-\tau^{2}}\frac{\partial\Phi}{\partial\tau}\right)\right].
  54. 𝐅 \nabla\cdot\mathbf{F}
  55. × 𝐅 \nabla\times\mathbf{F}
  56. ( σ , τ ) (\sigma,\tau)
  57. z z
  58. x x
  59. y y
  60. 𝐩 \mathbf{p}
  61. 𝐪 \mathbf{q}
  62. 𝐫 = 𝐩 + 𝐪 \mathbf{r}=\mathbf{p}+\mathbf{q}
  63. | 𝐩 | \left|\mathbf{p}\right|
  64. | 𝐪 | \left|\mathbf{q}\right|
  65. 𝐫 \mathbf{r}
  66. x x
  67. 𝐫 = 2 a 𝐱 ^ \mathbf{r}=2a\mathbf{\hat{x}}
  68. 𝐫 \mathbf{r}
  69. 𝐩 \mathbf{p}
  70. 𝐪 \mathbf{q}

Elliptic_curve_Diffie–Hellman.html

  1. ( p , a , b , G , n , h ) (p,a,b,G,n,h)
  2. ( m , f ( x ) , a , b , G , n , h ) (m,f(x),a,b,G,n,h)
  3. d d
  4. [ 1 , n - 1 ] [1,n-1]
  5. Q Q
  6. Q = d G Q=dG
  7. G G
  8. d d
  9. ( d A , Q A ) (d_{A},Q_{A})
  10. ( d B , Q B ) (d_{B},Q_{B})
  11. ( x k , y k ) = d A Q B (x_{k},y_{k})=d_{A}Q_{B}
  12. ( x k , y k ) = d B Q A (x_{k},y_{k})=d_{B}Q_{A}
  13. x k x_{k}
  14. x k x_{k}
  15. d A Q B = d A d B G = d B d A G = d B Q A d_{A}Q_{B}=d_{A}d_{B}G=d_{B}d_{A}G=d_{B}Q_{A}

Elliptic_cylindrical_coordinates.html

  1. z z
  2. F 1 F_{1}
  3. F 2 F_{2}
  4. - a -a
  5. + a +a
  6. x x
  7. ( μ , ν , z ) (\mu,\nu,z)
  8. x = a cosh μ cos ν x=a\ \cosh\mu\ \cos\nu
  9. y = a sinh μ sin ν y=a\ \sinh\mu\ \sin\nu
  10. z = z z=z\!
  11. μ \mu
  12. ν [ 0 , 2 π ) \nu\in[0,2\pi)
  13. x 2 a 2 cosh 2 μ + y 2 a 2 sinh 2 μ = cos 2 ν + sin 2 ν = 1 \frac{x^{2}}{a^{2}\cosh^{2}\mu}+\frac{y^{2}}{a^{2}\sinh^{2}\mu}=\cos^{2}\nu+% \sin^{2}\nu=1
  14. μ \mu
  15. x 2 a 2 cos 2 ν - y 2 a 2 sin 2 ν = cosh 2 μ - sinh 2 μ = 1 \frac{x^{2}}{a^{2}\cos^{2}\nu}-\frac{y^{2}}{a^{2}\sin^{2}\nu}=\cosh^{2}\mu-% \sinh^{2}\mu=1
  16. ν \nu
  17. μ \mu
  18. ν \nu
  19. h μ = h ν = a sinh 2 μ + sin 2 ν h_{\mu}=h_{\nu}=a\sqrt{\sinh^{2}\mu+\sin^{2}\nu}
  20. h z = 1 h_{z}=1
  21. d V = a 2 ( sinh 2 μ + sin 2 ν ) d μ d ν d z dV=a^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)d\mu d\nu dz
  22. 2 Φ = 1 a 2 ( sinh 2 μ + sin 2 ν ) ( 2 Φ μ 2 + 2 Φ ν 2 ) + 2 Φ z 2 \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)}\left(\frac% {\partial^{2}\Phi}{\partial\mu^{2}}+\frac{\partial^{2}\Phi}{\partial\nu^{2}}% \right)+\frac{\partial^{2}\Phi}{\partial z^{2}}
  23. 𝐅 \nabla\cdot\mathbf{F}
  24. × 𝐅 \nabla\times\mathbf{F}
  25. ( μ , ν , z ) (\mu,\nu,z)
  26. ( σ , τ , z ) (\sigma,\tau,z)
  27. σ = cosh μ \sigma=\cosh\mu
  28. τ = cos ν \tau=\cos\nu
  29. σ \sigma
  30. τ \tau
  31. τ \tau
  32. σ \sigma
  33. ( σ , τ , z ) (\sigma,\tau,z)
  34. F 1 F_{1}
  35. F 2 F_{2}
  36. d 1 + d 2 d_{1}+d_{2}
  37. 2 a σ 2a\sigma
  38. d 1 - d 2 d_{1}-d_{2}
  39. 2 a τ 2a\tau
  40. F 1 F_{1}
  41. a ( σ + τ ) a(\sigma+\tau)
  42. F 2 F_{2}
  43. a ( σ - τ ) a(\sigma-\tau)
  44. F 1 F_{1}
  45. F 2 F_{2}
  46. x = - a x=-a
  47. x = + a x=+a
  48. x = a σ τ x=a\sigma\tau\!
  49. y 2 = a 2 ( σ 2 - 1 ) ( 1 - τ 2 ) y^{2}=a^{2}\left(\sigma^{2}-1\right)\left(1-\tau^{2}\right)
  50. ( σ , τ , z ) (\sigma,\tau,z)
  51. h σ = a σ 2 - τ 2 σ 2 - 1 h_{\sigma}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{\sigma^{2}-1}}
  52. h τ = a σ 2 - τ 2 1 - τ 2 h_{\tau}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{1-\tau^{2}}}
  53. h z = 1 h_{z}=1
  54. d V = a 2 σ 2 - τ 2 ( σ 2 - 1 ) ( 1 - τ 2 ) d σ d τ d z dV=a^{2}\frac{\sigma^{2}-\tau^{2}}{\sqrt{\left(\sigma^{2}-1\right)\left(1-\tau% ^{2}\right)}}d\sigma d\tau dz
  55. 2 Φ = 1 a 2 ( σ 2 - τ 2 ) [ σ 2 - 1 σ ( σ 2 - 1 Φ σ ) + 1 - τ 2 τ ( 1 - τ 2 Φ τ ) ] + 2 Φ z 2 \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sigma^{2}-\tau^{2}\right)}\left[\sqrt{% \sigma^{2}-1}\frac{\partial}{\partial\sigma}\left(\sqrt{\sigma^{2}-1}\frac{% \partial\Phi}{\partial\sigma}\right)+\sqrt{1-\tau^{2}}\frac{\partial}{\partial% \tau}\left(\sqrt{1-\tau^{2}}\frac{\partial\Phi}{\partial\tau}\right)\right]+% \frac{\partial^{2}\Phi}{\partial z^{2}}
  56. 𝐅 \nabla\cdot\mathbf{F}
  57. × 𝐅 \nabla\times\mathbf{F}
  58. ( σ , τ ) (\sigma,\tau)
  59. 2 a 2a
  60. 𝐩 \mathbf{p}
  61. 𝐪 \mathbf{q}
  62. 𝐫 = 𝐩 + 𝐪 \mathbf{r}=\mathbf{p}+\mathbf{q}
  63. | 𝐩 | \left|\mathbf{p}\right|
  64. | 𝐪 | \left|\mathbf{q}\right|
  65. 𝐫 \mathbf{r}
  66. x x
  67. 𝐫 = 2 a 𝐱 ^ \mathbf{r}=2a\mathbf{\hat{x}}
  68. 𝐫 \mathbf{r}
  69. 𝐩 \mathbf{p}
  70. 𝐪 \mathbf{q}