wpmath0000001_6

Design_of_experiments.html

  1. θ 1 , , θ 8 . \theta_{1},\dots,\theta_{8}.\,
  2. left pan right pan 1st weighing: 1 2 3 4 5 6 7 8 (empty) 2nd: 1 2 3 8 4 5 6 7 3rd: 1 4 5 8 2 3 6 7 4th: 1 6 7 8 2 3 4 5 5th: 2 4 6 8 1 3 5 7 6th: 2 5 7 8 1 3 4 6 7th: 3 4 7 8 1 2 5 6 8th: 3 5 6 8 1 2 4 7 \begin{matrix}&\mbox{left pan}&\mbox{right pan}\\ \mbox{1st weighing:}&1\ 2\ 3\ 4\ 5\ 6\ 7\ 8&\,\text{(empty)}\\ \mbox{2nd:}&1\ 2\ 3\ 8&4\ 5\ 6\ 7\\ \mbox{3rd:}&1\ 4\ 5\ 8&2\ 3\ 6\ 7\\ \mbox{4th:}&1\ 6\ 7\ 8&2\ 3\ 4\ 5\\ \mbox{5th:}&2\ 4\ 6\ 8&1\ 3\ 5\ 7\\ \mbox{6th:}&2\ 5\ 7\ 8&1\ 3\ 4\ 6\\ \mbox{7th:}&3\ 4\ 7\ 8&1\ 2\ 5\ 6\\ \mbox{8th:}&3\ 5\ 6\ 8&1\ 2\ 4\ 7\end{matrix}
  3. θ ^ 1 = Y 1 + Y 2 + Y 3 + Y 4 - Y 5 - Y 6 - Y 7 - Y 8 8 . \widehat{\theta}_{1}=\frac{Y_{1}+Y_{2}+Y_{3}+Y_{4}-Y_{5}-Y_{6}-Y_{7}-Y_{8}}{8}.
  4. θ ^ 2 = Y 1 + Y 2 - Y 3 - Y 4 + Y 5 + Y 6 - Y 7 - Y 8 8 . \widehat{\theta}_{2}=\frac{Y_{1}+Y_{2}-Y_{3}-Y_{4}+Y_{5}+Y_{6}-Y_{7}-Y_{8}}{8}.

Determinant.html

  1. | A | = | a b c d e f g h i | |A|=\begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix}
  2. = a | e f h i | - b | d f g i | + c | d e g h | =a\begin{vmatrix}e&f\\ h&i\end{vmatrix}-b\begin{vmatrix}d&f\\ g&i\end{vmatrix}+c\begin{vmatrix}d&e\\ g&h\end{vmatrix}
  3. = a e i + b f g + c d h - c e g - b d i - a f h . =aei+bfg+cdh-ceg-bdi-afh.
  4. | a b c d e f g h i j k l m n o p | \begin{vmatrix}a&b&c&d\\ e&f&g&h\\ i&j&k&l\\ m&n&o&p\end{vmatrix}
  5. = a | f g h j k l n o p | =a\begin{vmatrix}f&g&h\\ j&k&l\\ n&o&p\end{vmatrix}
  6. - b | e g h i k l m o p | -b\begin{vmatrix}e&g&h\\ i&k&l\\ m&o&p\end{vmatrix}
  7. + c | e f h i j l m n p | +c\begin{vmatrix}e&f&h\\ i&j&l\\ m&n&p\end{vmatrix}
  8. - d | e f g i j k m n o | . -d\begin{vmatrix}e&f&g\\ i&j&k\\ m&n&o\end{vmatrix}.
  9. A = [ a 1 , a 2 , , a n ] A=\begin{bmatrix}a_{1},&a_{2},&\ldots,&a_{n}\end{bmatrix}
  10. a j a_{j}
  11. det [ a 1 , , b a j + c v , , a n ] = b det ( A ) + c det [ a 1 , , v , , a n ] det [ a 1 , , a j , a j + 1 , , a n ] = - det [ a 1 , , a j + 1 , a j , , a n ] det ( I ) = 1 \begin{aligned}&\displaystyle\det\begin{bmatrix}a_{1},&\ldots,&ba_{j}+cv,&% \ldots,a_{n}\end{bmatrix}=b\det(A)+c\det\begin{bmatrix}a_{1},&\ldots,&v,&% \ldots,a_{n}\end{bmatrix}\\ &\displaystyle\det\begin{bmatrix}a_{1},&\ldots,&a_{j},&a_{j+1},&\ldots,a_{n}% \end{bmatrix}=-\det\begin{bmatrix}a_{1},&\ldots,&a_{j+1},&a_{j},&\ldots,a_{n}% \end{bmatrix}\\ &\displaystyle\det(I)=1\end{aligned}
  12. A = [ a 1 , 1 a 1 , 2 a 1 , n a 2 , 1 a 2 , 2 a 2 , n a n , 1 a n , 2 a n , n ] . A=\begin{bmatrix}a_{1,1}&a_{1,2}&\dots&a_{1,n}\\ a_{2,1}&a_{2,2}&\dots&a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n,1}&a_{n,2}&\dots&a_{n,n}\end{bmatrix}.\,
  13. | a 1 , 1 a 1 , 2 a 1 , n a 2 , 1 a 2 , 2 a 2 , n a n , 1 a n , 2 a n , n | . \begin{vmatrix}a_{1,1}&a_{1,2}&\dots&a_{1,n}\\ a_{2,1}&a_{2,2}&\dots&a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n,1}&a_{n,2}&\dots&a_{n,n}\end{vmatrix}.\,
  14. | a b c d | = a d - b c . \begin{vmatrix}a&b\\ c&d\end{vmatrix}=ad-bc.
  15. | a b c d e f g h i | = a | e f h i | - b | d f g i | + c | d e g h | = a ( e i - f h ) - b ( d i - f g ) + c ( d h - e g ) = a e i + b f g + c d h - c e g - b d i - a f h . \begin{aligned}\displaystyle\begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix}&\displaystyle=a\begin{vmatrix}e&f\\ h&i\end{vmatrix}-b\begin{vmatrix}d&f\\ g&i\end{vmatrix}+c\begin{vmatrix}d&e\\ g&h\end{vmatrix}\\ &\displaystyle=a(ei-fh)-b(di-fg)+c(dh-eg)\\ &\displaystyle=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}
  16. det ( A ) = σ S n sgn ( σ ) i = 1 n a i , σ i . \det(A)=\sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}a_{i,% \sigma_{i}}.
  17. n ! n!
  18. i = 1 n a i , σ i \prod_{i=1}^{n}a_{i,\sigma_{i}}
  19. a 1 , σ 1 a 2 , σ 2 a n , σ n . a_{1,\sigma_{1}}\cdot a_{2,\sigma_{2}}\cdots a_{n,\sigma_{n}}.
  20. σ S n sgn ( σ ) i = 1 n a i , σ i = sgn ( [ 1 , 2 , 3 ] ) i = 1 n a i , [ 1 , 2 , 3 ] i + sgn ( [ 1 , 3 , 2 ] ) i = 1 n a i , [ 1 , 3 , 2 ] i + sgn ( [ 2 , 1 , 3 ] ) i = 1 n a i , [ 2 , 1 , 3 ] i + sgn ( [ 2 , 3 , 1 ] ) i = 1 n a i , [ 2 , 3 , 1 ] i + sgn ( [ 3 , 1 , 2 ] ) i = 1 n a i , [ 3 , 1 , 2 ] i + sgn ( [ 3 , 2 , 1 ] ) i = 1 n a i , [ 3 , 2 , 1 ] i = i = 1 n a i , [ 1 , 2 , 3 ] i - i = 1 n a i , [ 1 , 3 , 2 ] i - i = 1 n a i , [ 2 , 1 , 3 ] i + i = 1 n a i , [ 2 , 3 , 1 ] i + i = 1 n a i , [ 3 , 1 , 2 ] i - i = 1 n a i , [ 3 , 2 , 1 ] i = a 1 , 1 a 2 , 2 a 3 , 3 - a 1 , 1 a 2 , 3 a 3 , 2 - a 1 , 2 a 2 , 1 a 3 , 3 + a 1 , 2 a 2 , 3 a 3 , 1 + a 1 , 3 a 2 , 1 a 3 , 2 - a 1 , 3 a 2 , 2 a 3 , 1 . \begin{aligned}\displaystyle\sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)% \prod_{i=1}^{n}a_{i,\sigma_{i}}&\displaystyle=\operatorname{sgn}([1,2,3])\prod% _{i=1}^{n}a_{i,[1,2,3]_{i}}+\operatorname{sgn}([1,3,2])\prod_{i=1}^{n}a_{i,[1,% 3,2]_{i}}+\operatorname{sgn}([2,1,3])\prod_{i=1}^{n}a_{i,[2,1,3]_{i}}\\ &\displaystyle+\operatorname{sgn}([2,3,1])\prod_{i=1}^{n}a_{i,[2,3,1]_{i}}+% \operatorname{sgn}([3,1,2])\prod_{i=1}^{n}a_{i,[3,1,2]_{i}}+\operatorname{sgn}% ([3,2,1])\prod_{i=1}^{n}a_{i,[3,2,1]_{i}}\\ &\displaystyle=\prod_{i=1}^{n}a_{i,[1,2,3]_{i}}-\prod_{i=1}^{n}a_{i,[1,3,2]_{i% }}-\prod_{i=1}^{n}a_{i,[2,1,3]_{i}}+\prod_{i=1}^{n}a_{i,[2,3,1]_{i}}+\prod_{i=% 1}^{n}a_{i,[3,1,2]_{i}}-\prod_{i=1}^{n}a_{i,[3,2,1]_{i}}\\ &\displaystyle=a_{1,1}a_{2,2}a_{3,3}-a_{1,1}a_{2,3}a_{3,2}-a_{1,2}a_{2,1}a_{3,% 3}+a_{1,2}a_{2,3}a_{3,1}\\ &\displaystyle\qquad+a_{1,3}a_{2,1}a_{3,2}-a_{1,3}a_{2,2}a_{3,1}.\end{aligned}
  21. ε i 1 , , i n \varepsilon_{i_{1},\cdots,i_{n}}
  22. ε σ ( 1 ) , , σ ( n ) = sgn ( σ ) \varepsilon_{\sigma(1),\cdots,\sigma(n)}=\operatorname{sgn}(\sigma)
  23. ε i 1 , , i n = 0 \varepsilon_{i_{1},\cdots,i_{n}}=0
  24. σ ( j ) = i j \sigma(j)=i_{j}
  25. j = 1 , , n j=1,\ldots,n
  26. det ( A ) = i 1 , i 2 , , i n = 1 n ε i 1 i n a 1 , i 1 a n , i n , \det(A)=\sum_{i_{1},i_{2},\ldots,i_{n}=1}^{n}\varepsilon_{i_{1}\cdots i_{n}}a_% {1,i_{1}}\cdots a_{n,i_{n}},
  27. det ( A ) = 1 n ! ε i 1 i n ε j 1 j n a i 1 j 1 a i n j n , \det(A)=\frac{1}{n!}\sum\varepsilon_{i_{1}\cdots i_{n}}\varepsilon_{j_{1}% \cdots j_{n}}a_{i_{1}j_{1}}\cdots a_{i_{n}j_{n}},
  28. det ( I n ) = 1 \det(I_{n})=1
  29. det ( A T ) = det ( A ) . \det(A^{\rm T})=\det(A).
  30. det ( A - 1 ) = 1 det ( A ) = det ( A ) - 1 . \det(A^{-1})=\frac{1}{\det(A)}=\det(A)^{-1}.
  31. det ( A B ) = det ( A ) det ( B ) . \det(AB)=\det(A)\det(B).
  32. det ( c A ) = c n det ( A ) \det(cA)=c^{n}\det(A)
  33. det ( A ) = a 1 , 1 a 2 , 2 a n , n = i = 1 n a i , i . \det(A)=a_{1,1}a_{2,2}\cdots a_{n,n}=\prod_{i=1}^{n}a_{i,i}.
  34. A = [ - 2 2 - 3 - 1 1 3 2 0 - 1 ] A=\begin{bmatrix}-2&2&-3\\ -1&1&3\\ 2&0&-1\end{bmatrix}
  35. B = [ - 2 2 - 3 0 0 4.5 2 0 - 1 ] , C = [ - 2 2 - 3 0 0 4.5 0 2 - 4 ] , D = [ - 2 2 - 3 0 2 - 4 0 0 4.5 ] . B=\begin{bmatrix}-2&2&-3\\ 0&0&4.5\\ 2&0&-1\end{bmatrix},\quad C=\begin{bmatrix}-2&2&-3\\ 0&0&4.5\\ 0&2&-4\end{bmatrix},\quad D=\begin{bmatrix}-2&2&-3\\ 0&2&-4\\ 0&0&4.5\end{bmatrix}.
  36. det ( A B ) = det ( A ) det ( B ) . \det(AB)=\det(A)\det(B).
  37. det ( A - 1 ) = 1 det ( A ) . \det(A^{-1})=\frac{1}{\det(A)}.
  38. det ( A ) = j = 1 n ( - 1 ) i + j a i , j M i , j = i = 1 n ( - 1 ) i + j a i , j M i , j . \det(A)=\sum_{j=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j}=\sum_{i=1}^{n}(-1)^{i+j}a_{i,j}% M_{i,j}.
  39. A = [ - 2 2 - 3 - 1 1 3 2 0 - 1 ] , A=\begin{bmatrix}-2&2&-3\\ -1&1&3\\ 2&0&-1\end{bmatrix}\,,
  40. det ( A ) \det(A)\,
  41. = =\,
  42. ( - 1 ) 1 + 2 2 | - 1 3 2 - 1 | + ( - 1 ) 2 + 2 1 | - 2 - 3 2 - 1 | + ( - 1 ) 3 + 2 0 | - 2 - 3 - 1 3 | (-1)^{1+2}\cdot 2\cdot\begin{vmatrix}-1&3\\ 2&-1\end{vmatrix}+(-1)^{2+2}\cdot 1\cdot\begin{vmatrix}-2&-3\\ 2&-1\end{vmatrix}+(-1)^{3+2}\cdot 0\cdot\begin{vmatrix}-2&-3\\ -1&3\end{vmatrix}
  43. = =\,
  44. ( - 2 ) ( ( - 1 ) ( - 1 ) - 2 3 ) + 1 ( ( - 2 ) ( - 1 ) - 2 ( - 3 ) ) (-2)\cdot((-1)\cdot(-1)-2\cdot 3)+1\cdot((-2)\cdot(-1)-2\cdot(-3))
  45. = =\,
  46. ( - 2 ) ( - 5 ) + 8 = 18. (-2)\cdot(-5)+8=18.\,
  47. ( adj ( A ) ) i , j = ( - 1 ) i + j M j , i . (\operatorname{adj}(A))_{i,j}=(-1)^{i+j}M_{j,i}.\,
  48. ( det A ) I = A adj A = ( adj A ) A . (\operatorname{det}A)I=A\,\operatorname{adj}A=(\operatorname{adj}A)\,A.
  49. det ( I m + A B ) = det ( I n + B A ) \det(I_{\mathit{m}}+AB)=\det(I_{\mathit{n}}+BA)
  50. det ( I m + c r ) = 1 + r c \det(I_{\mathit{m}}+cr)=1+rc
  51. det ( X + A B ) = det ( X ) det ( I n + B X - 1 A ) \det(X+AB)=\det(X)\det(I_{\mathit{n}}+BX^{-1}A)
  52. det ( X + c r ) = det ( X ) ( 1 + r X - 1 c ) = det ( X ) + r adj ( X ) c \det(X+cr)=\det(X)(1+rX^{-1}c)=\det(X)+r\,\mathrm{adj}(X)\,c
  53. A A
  54. n × n n\times n
  55. λ 1 \lambda_{1}
  56. λ 2 \lambda_{2}
  57. λ n \lambda_{n}
  58. μ \mu
  59. μ \mu
  60. A A
  61. det ( A ) = i = 1 n λ i = λ 1 λ 2 λ n \operatorname{det}(A)=\prod_{i=1}^{n}\lambda_{i}=\lambda_{1}\lambda_{2}\cdots% \lambda_{n}
  62. det ( A - x I ) = 0 , \det(A-xI)=0,\,
  63. A k := [ a 1 , 1 a 1 , 2 a 1 , k a 2 , 1 a 2 , 2 a 2 , k a k , 1 a k , 2 a k , k ] A_{k}:=\begin{bmatrix}a_{1,1}&a_{1,2}&\dots&a_{1,k}\\ a_{2,1}&a_{2,2}&\dots&a_{2,k}\\ \vdots&\vdots&\ddots&\vdots\\ a_{k,1}&a_{k,2}&\dots&a_{k,k}\end{bmatrix}
  64. det ( exp ( A ) ) = exp ( tr ( A ) ) \det(\exp(A))=\exp(\operatorname{tr}(A))\,
  65. tr ( A ) = log ( det ( exp ( A ) ) ) . \operatorname{tr}(A)=\log(\det(\exp(A))).\,
  66. exp ( L ) = A \exp(L)=A\,
  67. det ( A ) = exp ( tr ( L ) ) . \det(A)=\exp(\operatorname{tr}(L)).\,
  68. det ( A ) = ( ( tr A ) 2 - tr ( A 2 ) ) / 2 , \det(A)=\bigl((\operatorname{tr}A)^{2}-\operatorname{tr}(A^{2})\bigr)/2,
  69. det ( A ) = ( ( tr A ) 3 - 3 tr A tr ( A 2 ) + 2 tr ( A 3 ) ) / 6 , \det(A)=\Bigl((\operatorname{tr}A)^{3}-3\operatorname{tr}A~{}\operatorname{tr}% (A^{2})+2\operatorname{tr}(A^{3})\Bigr)/6,
  70. det ( A ) = ( ( tr A ) 4 - 6 tr ( A 2 ) ( tr A ) 2 + 3 ( tr ( A 2 ) ) 2 + 8 tr ( A 3 ) tr A - 6 tr ( A 4 ) ) / 24. \det(A)=\Bigl((\operatorname{tr}A)^{4}-6\operatorname{tr}(A^{2})(\operatorname% {tr}A)^{2}+3(\operatorname{tr}(A^{2}))^{2}+8\operatorname{tr}(A^{3})~{}% \operatorname{tr}A-6\operatorname{tr}(A^{4})\Bigr)/24.
  71. det ( A ) = k 1 , k 2 , , k n l = 1 n ( - 1 ) k l + 1 l k l k l ! tr ( A l ) k l , \det(A)=\sum_{k_{1},k_{2},\ldots,k_{n}}\prod_{l=1}^{n}\frac{(-1)^{k_{l}+1}}{l^% {k_{l}}k_{l}!}\operatorname{tr}(A^{l})^{k_{l}},
  72. l = 1 n l k l = n . \sum_{l=1}^{n}lk_{l}=n.
  73. ( A B ) J I = K A K I B J K , tr ( A ) = I A I I . (AB)^{I}_{J}=\sum_{K}A^{I}_{K}B^{K}_{J},\operatorname{tr}(A)=\sum_{I}A^{I}_{I}.
  74. A B ( 0 , 1 ) A\in B(0,1)
  75. det ( I + A ) = k = 0 1 k ! ( - j = 1 ( - 1 ) j j tr ( A j ) ) k , \begin{aligned}\displaystyle\det(I+A)=\sum_{k=0}^{\infty}\frac{1}{k!}\left(-% \sum_{j=1}^{\infty}\frac{(-1)^{j}}{j}\operatorname{tr}(A^{j})\right)^{k}\,,% \end{aligned}
  76. tr ( I - A - 1 ) log det ( A ) tr ( A - I ) \begin{aligned}\displaystyle\operatorname{tr}(I-A^{-1})\leq\log\det(A)\leq% \operatorname{tr}(A-I)\end{aligned}
  77. A = I A=I
  78. A x = b Ax=b\,
  79. x i = det ( A i ) det ( A ) i = 1 , 2 , 3 , , n x_{i}=\frac{\det(A_{i})}{\det(A)}\qquad i=1,2,3,\ldots,n\,
  80. det ( A i ) = det [ a 1 , , b , , a n ] = j = 1 n x j det [ a 1 , , a i - 1 , a j , a i + 1 , , a n ] = x i det ( A ) \det(A_{i})=\det\begin{bmatrix}a_{1},&\ldots,&b,&\ldots,&a_{n}\end{bmatrix}=% \sum_{j=1}^{n}x_{j}\det\begin{bmatrix}a_{1},&\ldots,a_{i-1},&a_{j},&a_{i+1},&% \ldots,&a_{n}\end{bmatrix}=x_{i}\det(A)
  81. a j a_{j}
  82. A adj ( A ) = adj ( A ) A = det ( A ) I n . A\,\operatorname{adj}(A)=\operatorname{adj}(A)\,A=\det(A)\,I_{n}.
  83. det ( A 0 C D ) = det ( A ) det ( D ) = det ( A B 0 D ) . \det\begin{pmatrix}A&0\\ C&D\end{pmatrix}=\det(A)\det(D)=\det\begin{pmatrix}A&B\\ 0&D\end{pmatrix}.
  84. ( A 0 C D ) = ( A 0 C I m ) ( I n 0 0 D ) . \begin{pmatrix}A&0\\ C&D\end{pmatrix}=\begin{pmatrix}A&0\\ C&I_{m}\end{pmatrix}\begin{pmatrix}I_{n}&0\\ 0&D\end{pmatrix}.
  85. det ( A B C D ) = det ( A ) det ( D - C A - 1 B ) . \det\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\det(A)\det(D-CA^{-1}B).
  86. ( A B C D ) = ( A 0 C I m ) ( I n A - 1 B 0 D - C A - 1 B ) . \begin{pmatrix}A&B\\ C&D\end{pmatrix}=\begin{pmatrix}A&0\\ C&I_{m}\end{pmatrix}\begin{pmatrix}I_{n}&A^{-1}B\\ 0&D-CA^{-1}B\end{pmatrix}.
  87. det ( D ) \det(D)
  88. det ( A B C D ) = det ( D ) det ( A - B D - 1 C ) . \det\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\det(D)\det(A-BD^{-1}C).
  89. det ( A B C D ) = det ( A D - B C ) . \det\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\det(AD-BC).
  90. det ( A B B A ) = det ( A - B ) det ( A + B ) . \det\begin{pmatrix}A&B\\ B&A\end{pmatrix}=\det(A-B)\det(A+B).
  91. det ( A B C D ) = ( D - 1 ) det ( A ) + det ( A - B C ) = ( D + 1 ) det A - det ( A + B C ) . \det\begin{pmatrix}A&B\\ C&D\end{pmatrix}=(D-1)\det(A)+\det(A-BC)=(D+1)\det{A}-\det(A+BC)\,.
  92. d det ( A ) d α = tr ( adj ( A ) d A d α ) . \frac{\mathrm{d}\det(A)}{\mathrm{d}\alpha}=\operatorname{tr}\left(% \operatorname{adj}(A)\frac{\mathrm{d}A}{\mathrm{d}\alpha}\right).
  93. d det ( A ) d α = det ( A ) tr ( A - 1 d A d α ) . \frac{\mathrm{d}\det(A)}{\mathrm{d}\alpha}=\det(A)\operatorname{tr}\left(A^{-1% }\frac{\mathrm{d}A}{\mathrm{d}\alpha}\right).
  94. det ( A ) A i j = adj ( A ) j i = det ( A ) ( A - 1 ) j i . \frac{\partial\det(A)}{\partial A_{ij}}=\operatorname{adj}(A)_{ji}=\det(A)(A^{% -1})_{ji}.
  95. det ( A + ϵ X ) - det ( A ) = tr ( adj ( A ) X ) ϵ + O ( ϵ 2 ) = det ( A ) tr ( A - 1 X ) ϵ + O ( ϵ 2 ) \det(A+\epsilon X)-\det(A)=\operatorname{tr}(\operatorname{adj}(A)X)\epsilon+O% (\epsilon^{2})=\det(A)\operatorname{tr}(A^{-1}X)\epsilon+O(\epsilon^{2})
  96. A = I A=I
  97. det ( I + ϵ X ) = 1 + tr ( X ) ϵ + O ( ϵ 2 ) . \det(I+\epsilon X)=1+\operatorname{tr}(X)\epsilon+O(\epsilon^{2}).
  98. A = [ 𝐚 𝐛 𝐜 ] A=\begin{bmatrix}\mathbf{a}&\mathbf{b}&\mathbf{c}\end{bmatrix}
  99. 𝐚 det ( A ) \displaystyle\nabla_{\mathbf{a}}\det(A)
  100. det ( A ) = det ( X ) - 1 det ( B ) det ( X ) = det ( B ) det ( X ) - 1 det ( X ) = det ( B ) . \det(A)=\det(X)^{-1}\det(B)\det(X)=\det(B)\det(X)^{-1}\det(X)=\det(B).
  101. T : V V T:V\rightarrow V\,
  102. Λ n A : Λ n V Λ n V \Lambda^{n}A:\Lambda^{n}V\rightarrow\Lambda^{n}V
  103. v 1 v 2 v n A v 1 A v 2 A v n . v_{1}\wedge v_{2}\wedge\dots\wedge v_{n}\mapsto Av_{1}\wedge Av_{2}\wedge\dots% \wedge Av_{n}.
  104. ( Λ n A ) ( v 1 v n ) = det ( A ) v 1 v n . (\Lambda^{n}A)(v_{1}\wedge\dots\wedge v_{n})=\det(A)\cdot v_{1}\wedge\dots% \wedge v_{n}.
  105. D : M n ( K ) K D:M_{n}(K)\to K\,
  106. D ( v 1 , , v i - 1 , a v i + b w , v i + 1 , , v n ) = a D ( v 1 , , v i - 1 , v i , v i + 1 , , v n ) + b D ( v 1 , , v i - 1 , w , v i + 1 , , v n ) D(v_{1},\dots,v_{i-1},av_{i}+bw,v_{i+1},\dots,v_{n})=aD(v_{1},\dots,v_{i-1},v_% {i},v_{i+1},\dots,v_{n})+bD(v_{1},\dots,v_{i-1},w,v_{i+1},\dots,v_{n})\,
  107. F ( M ) = F ( I ) D ( M ) . F(M)=F(I)D(M).
  108. GL n ( R ) R × , \mathrm{GL}_{n}(R)\rightarrow R^{\times},\,
  109. f ( det ( ( a i , j ) ) ) = det ( ( f ( a i , j ) ) ) f(\det((a_{i,j})))=\det((f(a_{i,j})))
  110. det : GL n 𝔾 m . \det:\mathrm{GL}_{n}\rightarrow\mathbb{G}_{m}.
  111. det ( I + A ) = exp ( tr ( log ( I + A ) ) ) . \det(I+A)=\exp(\operatorname{tr}(\log(I+A))).\,
  112. A = P L U . A=PLU.\,
  113. ε \varepsilon
  114. det ( A ) = ε det ( L ) det ( U ) , \det(A)=\varepsilon\det(L)\cdot\det(U),\,
  115. det ( A ) = ε det ( U ) . \det(A)=\varepsilon\det(U).
  116. W ( f 1 , , f n ) ( x ) = | f 1 ( x ) f 2 ( x ) f n ( x ) f 1 ( x ) f 2 ( x ) f n ( x ) f 1 ( n - 1 ) ( x ) f 2 ( n - 1 ) ( x ) f n ( n - 1 ) ( x ) | . W(f_{1},\ldots,f_{n})(x)=\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots&f_{n}(x)\\ f_{1}^{\prime}(x)&f_{2}^{\prime}(x)&\cdots&f_{n}^{\prime}(x)\\ \vdots&\vdots&\ddots&\vdots\\ f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots&f_{n}^{(n-1)}(x)\end{vmatrix}.
  117. volume ( f ( S ) ) = det ( A T A ) × volume ( S ) . \operatorname{volume}(f(S))=\sqrt{\det(A^{\mathrm{T}}A)}\times\operatorname{% volume}(S).
  118. f : 𝐑 n 𝐑 n , f:\mathbf{R}^{n}\rightarrow\mathbf{R}^{n},
  119. D ( f ) = ( f i x j ) 1 i , j n . D(f)=\left(\frac{\partial f_{i}}{\partial x_{j}}\right)_{1\leq i,j\leq n}.\,
  120. f ( U ) ϕ ( 𝐯 ) d 𝐯 = U ϕ ( f ( 𝐮 ) ) | det ( D f ) ( 𝐮 ) | d 𝐮 . \int_{f(U)}\phi(\mathbf{v})\,d\mathbf{v}=\int_{U}\phi(f(\mathbf{u}))\left|\det% (\operatorname{D}f)(\mathbf{u})\right|\,d\mathbf{u}.
  121. | 1 1 1 x 1 x 2 x 3 x 1 2 x 2 2 x 3 2 | = ( x 3 - x 2 ) ( x 3 - x 1 ) ( x 2 - x 1 ) . \left|\begin{array}[]{ccc}1&1&1\\ x_{1}&x_{2}&x_{3}\\ x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\end{array}\right|=\left(x_{3}-x_{2}\right)\left(% x_{3}-x_{1}\right)\left(x_{2}-x_{1}\right).
  122. | 1 1 1 1 x 1 x 2 x 3 x n x 1 2 x 2 2 x 3 2 x n 2 x 1 n - 1 x 2 n - 1 x 3 n - 1 x n n - 1 | = 1 i < j n ( x j - x i ) , \left|\begin{array}[]{ccccc}1&1&1&\cdots&1\\ x_{1}&x_{2}&x_{3}&\cdots&x_{n}\\ x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&\cdots&x_{n}^{2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{1}^{n-1}&x_{2}^{n-1}&x_{3}^{n-1}&\cdots&x_{n}^{n-1}\end{array}\right|=\prod% _{1\leq i<j\leq n}\left(x_{j}-x_{i}\right),
  123. | x 1 x 2 x 2 x 1 | = ( x 1 + x 2 ) ( x 1 - x 2 ) . \left|\begin{array}[]{cc}x_{1}&x_{2}\\ x_{2}&x_{1}\end{array}\right|=\left(x_{1}+x_{2}\right)\left(x_{1}-x_{2}\right).
  124. | x 1 x 2 x 3 x 3 x 1 x 2 x 2 x 3 x 1 | = ( x 1 + x 2 + x 3 ) ( x 1 + ω x 2 + ω 2 x 3 ) ( x 1 + ω 2 x 2 + ω x 3 ) , \left|\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\\ x_{3}&x_{1}&x_{2}\\ x_{2}&x_{3}&x_{1}\end{array}\right|=\left(x_{1}+x_{2}+x_{3}\right)\left(x_{1}+% \omega x_{2}+\omega^{2}x_{3}\right)\left(x_{1}+\omega^{2}x_{2}+\omega x_{3}% \right),
  125. | x 1 x 2 x 3 x n x n x 1 x 2 x n - 1 x n - 1 x n x 1 x n - 2 x 2 x 3 x 4 x 1 | = j = 1 n ( x 1 + x 2 ω j + x 3 ω j 2 + + x n ω j n - 1 ) , \left|\begin{array}[]{ccccc}x_{1}&x_{2}&x_{3}&\cdots&x_{n}\\ x_{n}&x_{1}&x_{2}&\cdots&x_{n-1}\\ x_{n-1}&x_{n}&x_{1}&\cdots&x_{n-2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{2}&x_{3}&x_{4}&\cdots&x_{1}\end{array}\right|=\prod_{j=1}^{n}\left(x_{1}+x_% {2}\omega_{j}+x_{3}\omega_{j}^{2}+\ldots+x_{n}\omega_{j}^{n-1}\right),
  126. a b = a b | 1 0 0 1 | = a | 1 0 0 b | = | a 0 0 b | = b | a 0 0 1 | = b a | 1 0 0 1 | = b a , ab=ab\left|\begin{matrix}1&0\\ 0&1\end{matrix}\right|=a\left|\begin{matrix}1&0\\ 0&b\end{matrix}\right|=\left|\begin{matrix}a&0\\ 0&b\end{matrix}\right|=b\left|\begin{matrix}a&0\\ 0&1\end{matrix}\right|=ba\left|\begin{matrix}1&0\\ 0&1\end{matrix}\right|=ba,

Determinism.html

  1. i ψ ( x , t ) t = - 2 2 m 2 ψ ( x , t ) x 2 + V ( x ) ψ i\hbar\frac{\partial\psi(x,t)}{\partial t}=-\frac{\hbar^{2}}{2m}\frac{\partial% ^{2}\psi(x,t)}{\partial x^{2}}+V(x)\psi

Deuterium.html

  1. 1 2 ( | - | ) . \frac{1}{\sqrt{2}}\Big(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle% \Big).
  2. ( | 1 2 ( | + | ) | ) \left(\begin{array}[]{ll}|\uparrow\uparrow\rangle\\ \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)\\ |\downarrow\downarrow\rangle\end{array}\right)
  3. μ = 1 ( j + 1 ) ( l , s ) , j , m j = j | μ j | ( l , s ) , j , m j = j \mu={1\over(j+1)}\langle(l,s),j,m_{j}=j|\overrightarrow{\mu}\cdot% \overrightarrow{j}|(l,s),j,m_{j}=j\rangle
  4. μ = g ( l ) l + g ( s ) s \overrightarrow{\mu}=g^{(l)}\overrightarrow{l}+g^{(s)}\overrightarrow{s}
  5. l \overrightarrow{l}
  6. s \overrightarrow{s}
  7. μ = 1 ( j + 1 ) ( l , s ) , j , m j = j | ( 1 2 l g ( l ) p + 1 2 s ( g ( s ) p + g ( s ) n ) ) j | ( l , s ) , j , m j = j \mu={1\over(j+1)}\langle(l,s),j,m_{j}=j|\left({1\over 2}\overrightarrow{l}{g^{% (l)}}_{p}+{1\over 2}\overrightarrow{s}({g^{(s)}}_{p}+{g^{(s)}}_{n})\right)% \cdot\overrightarrow{j}|(l,s),j,m_{j}=j\rangle
  8. μ = 1 4 ( j + 1 ) [ ( g ( s ) p + g ( s ) n ) ( j ( j + 1 ) - l ( l + 1 ) + s ( s + 1 ) ) + ( j ( j + 1 ) + l ( l + 1 ) - s ( s + 1 ) ) ] \mu={1\over 4(j+1)}\left[({g^{(s)}}_{p}+{g^{(s)}}_{n})\big(j(j+1)-l(l+1)+s(s+1% )\big)+\big(j(j+1)+l(l+1)-s(s+1)\big)\right]
  9. μ = 1 2 ( g ( s ) p + g ( s ) n ) = 0.879 \mu={1\over 2}({g^{(s)}}_{p}+{g^{(s)}}_{n})=0.879
  10. μ = - 1 4 ( g ( s ) p + g ( s ) n ) + 3 4 = 0.310 \mu=-{1\over 4}({g^{(s)}}_{p}+{g^{(s)}}_{n})+{3\over 4}=0.310

Dew_point.html

  1. γ ( T , R H ) \displaystyle\gamma(T,R\!H)
  2. P s ( T ) = 100 R H P a ( T ) = a exp ( b T c + T ) ; P a ( T ) = R H 100 P s ( T ) = a exp ( γ ( T , R H ) ) , P s ( T w ) - B P mb 0.00066 [ 1 + ( 0.00115 T w ) ] ( T - T w ) ; T dp = c ln ( P a ( T ) / a ) b - ln ( P a ( T ) / a ) ; \begin{aligned}\displaystyle P_{s}(T)&\displaystyle=\frac{100}{R\!H}P\text{a}(% T)=a\exp\left(\frac{bT}{c+T}\right);\\ \displaystyle P\text{a}(T)&\displaystyle=\frac{R\!H}{100}P_{s}(T)=a\exp(\gamma% (T,R\!H)),\\ &\displaystyle\approx P_{s}(T\text{w})-B\!P\text{mb}0.00066\left[1+(0.00115T% \text{w}\right)]\left(T-T\text{w}\right);\\ \displaystyle T\text{dp}&\displaystyle=\frac{c\ln(P\text{a}(T)/a)}{b-\ln(P% \text{a}(T)/a)};\end{aligned}
  3. P s : m ( T ) \displaystyle P_{s:m}(T)
  4. a = 6.1121 millibar ; b = 18.678 ; c = 257.14 C ; d = 234.5 C . \scriptstyle{a=6.1121\ \mathrm{millibar};\quad\;b=18.678;\quad\;c=257.14^{% \circ}\mathrm{C};\quad\;d=234.5^{\circ}\mathrm{C}.}
  5. a = 6.112 millibar ; b = 17.67 ; c = 243.5 C ; \begin{aligned}\displaystyle a&\displaystyle=6.112\ \mathrm{millibar};\quad\;b% &\displaystyle=17.67;\quad\;c&\displaystyle=243.5^{\circ}\mathrm{C};\end{aligned}
  6. a = 6.112 millibar ; b = 17.62 ; c = 243.12 C : - 45 C T + 60 C ( ± 0.35 C ) \scriptstyle{a=6.112\ \mathrm{millibar};\quad\;b=17.62;\quad\;c=243.12^{\circ}% \mathrm{C}:\quad-45^{\circ}\mathrm{C}\leq T\leq+60^{\circ}\mathrm{C}\quad(\pm 0% .35^{\circ}\mathrm{C})}
  7. a = 6.105 millibar ; b = 17.27 ; c = 237.7 C : 0 C T + 60 C ( ± 0.4 C ) \scriptstyle{a=6.105\ \mathrm{millibar};\quad\;b=17.27;\quad\;c=237.7^{\circ}% \mathrm{C}:\quad 0^{\circ}\mathrm{C}\leq T\leq+60^{\circ}\mathrm{C}\quad(\pm 0% .4^{\circ}\mathrm{C})}
  8. a = 6.1121 millibar ; b = 17.368 ; c = 238.88 C : 0 C T + 50 C ( 0.05 % ) \scriptstyle{a=6.1121\ \mathrm{millibar};\quad\;b=17.368;\quad\;c=238.88^{% \circ}\mathrm{C}:\quad\quad\!0^{\circ}\mathrm{C}\leq T\leq+50^{\circ}\mathrm{C% }\;\;(\leq 0.05\%)}
  9. a = 6.1121 millibar ; b = 17.966 ; c = 247.15 C : - 40 C T 0 C ( 0.06 % ) \scriptstyle{a=6.1121\ \mathrm{millibar};\quad\;b=17.966;\quad\;c=247.15^{% \circ}\mathrm{C}:\quad-40^{\circ}\mathrm{C}\leq T\leq 0^{\circ}\mathrm{C}\quad% \!\;\;(\leq 0.06\%)}
  10. T d p T - 100 - R H 5 ; T_{dp}\approx T-\frac{100-R\!H}{5};
  11. R H 100 - 5 ( T - T d p ) ; R\!H\approx 100-5(T-T_{dp});\,
  12. T d p : f T f - 9 25 ( 100 - R H ) ; T_{dp:f}\approx T_{f}-\frac{9}{25}(100-R\!H);
  13. R H 100 - 25 9 ( T f - T d p : f ) ; R\!H\approx 100-\frac{25}{9}(T_{f}-T_{dp:f});

Diamagnetism.html

  1. I = - Z e 2 B 4 π m . I=-\frac{Ze^{2}B}{4\pi m}.
  2. π ρ 2 \scriptstyle\pi\left\langle\rho^{2}\right\rangle
  3. ρ 2 \scriptstyle\left\langle\rho^{2}\right\rangle
  4. μ = - Z e 2 B 4 m ρ 2 . \mu=-\frac{Ze^{2}B}{4m}\langle\rho^{2}\rangle.
  5. x 2 = y 2 = z 2 = 1 3 r 2 \scriptstyle\left\langle x^{2}\right\rangle\;=\;\left\langle y^{2}\right% \rangle\;=\;\left\langle z^{2}\right\rangle\;=\;\frac{1}{3}\left\langle r^{2}\right\rangle
  6. r 2 \scriptstyle\left\langle r^{2}\right\rangle
  7. ρ 2 = x 2 + y 2 = 2 3 r 2 \scriptstyle\left\langle\rho^{2}\right\rangle\;=\;\left\langle x^{2}\right% \rangle\;+\;\left\langle y^{2}\right\rangle\;=\;\frac{2}{3}\left\langle r^{2}\right\rangle
  8. N N
  9. χ = μ 0 N μ B = - μ 0 N Z e 2 6 m r 2 . \chi=\frac{\mu_{0}N\mu}{B}=-\frac{\mu_{0}NZe^{2}}{6m}\langle r^{2}\rangle.

Diameter.html

  1. d = 2 r r = d 2 . d=2r\quad\Rightarrow\quad r=\frac{d}{2}.

Diatomic_molecule.html

  1. X X
  2. A A
  3. B B
  4. C C
  5. Λ 2 S + 1 ( v ) {}^{2S+1}\Lambda(v)
  6. S S
  7. Λ \Lambda
  8. v v
  9. Λ \Lambda
  10. Σ \Sigma
  11. Π \Pi
  12. Δ \Delta
  13. v = 0 v=0
  14. Λ \Lambda
  15. T 0 T_{0}
  16. X 1 Σ g + X^{1}\Sigma_{g}^{+}
  17. A 3 Σ u + A^{3}\Sigma_{u}^{+}
  18. B 3 Π g B^{3}\Pi_{g}
  19. W 3 Δ u W^{3}\Delta_{u}
  20. B Σ u - 3 B^{\prime}{}^{3}\Sigma_{u}^{-}
  21. a Σ u - 1 a^{\prime}{}^{1}\Sigma_{u}^{-}
  22. a 1 Π g a^{1}\Pi_{g}
  23. w 1 Δ u w^{1}\Delta_{u}
  24. A A
  25. X X
  26. J J
  27. J J
  28. R R
  29. Δ J = + 1 \Delta J=+1
  30. P P
  31. Δ J = - 1 \Delta J=-1
  32. Q Q
  33. Δ J = 0 \Delta J=0
  34. E t r a n s = 1 2 m v 2 E_{trans}=\frac{1}{2}mv^{2}
  35. m m
  36. v v
  37. E r o t = L 2 2 I E_{rot}=\frac{L^{2}}{2I}\,
  38. L L\,
  39. I I\,
  40. L 2 = l ( l + 1 ) 2 L^{2}=l(l+1)\hbar^{2}\,
  41. l l
  42. \hbar
  43. I = μ r 0 2 I=\mu r_{0}^{2}\,
  44. μ \mu\,
  45. r 0 r_{0}\,
  46. E r o t = l ( l + 1 ) 2 2 μ r 0 2 l = 0 , 1 , 2 , E_{rot}=\frac{l(l+1)\hbar^{2}}{2\mu r_{0}^{2}}\ \ \ \ \ l=0,1,2,...\,
  47. E v i b = ( n + 1 2 ) ω n = 0 , 1 , 2 , . E_{vib}=\left(n+\frac{1}{2}\right)\hbar\omega\ \ \ \ \ n=0,1,2,....\,
  48. n n
  49. \hbar
  50. ω \omega

Dielectric.html

  1. 𝐏 = ε 0 χ e 𝐄 , {\mathbf{P}}=\varepsilon_{0}\chi_{e}{\mathbf{E}},
  2. ε 0 \,\varepsilon_{0}
  3. ε r \,\varepsilon_{r}
  4. χ e = ε r - 1. \chi_{e}\ =\varepsilon_{r}-1.
  5. χ e = 0. \chi_{e}\ =0.
  6. 𝐃 = ε 0 𝐄 + 𝐏 = ε 0 ( 1 + χ e ) 𝐄 = ε r ε 0 𝐄 . \mathbf{D}\ =\ \varepsilon_{0}\mathbf{E}+\mathbf{P}\ =\ \varepsilon_{0}(1+\chi% _{e})\mathbf{E}\ =\ \varepsilon_{r}\varepsilon_{0}\mathbf{E}.
  7. 𝐏 ( t ) = ε 0 - t χ e ( t - t ) 𝐄 ( t ) d t . \mathbf{P}(t)=\varepsilon_{0}\int_{-\infty}^{t}\chi_{e}(t-t^{\prime})\mathbf{E% }(t^{\prime})\,dt^{\prime}.
  8. χ e ( Δ t ) \chi_{e}(\Delta t)
  9. χ e ( Δ t ) = 0 \chi_{e}(\Delta t)=0
  10. Δ t < 0 \Delta t<0
  11. χ e ( Δ t ) = χ e δ ( Δ t ) \chi_{e}(\Delta t)=\chi_{e}\delta(\Delta t)
  12. 𝐏 ( ω ) = ε 0 χ e ( ω ) 𝐄 ( ω ) . \mathbf{P}(\omega)=\varepsilon_{0}\chi_{e}(\omega)\mathbf{E}(\omega).
  13. χ e ( Δ t ) = 0 \chi_{e}(\Delta t)=0
  14. Δ t < 0 \Delta t<0
  15. χ e ( ω ) \chi_{e}(\omega)
  16. 𝐌 = 𝐅 ( 𝐄 ) \mathbf{M}=\mathbf{F}(\mathbf{E})
  17. ε \varepsilon\,\!
  18. ω \omega
  19. ε ^ ( ω ) = ε + Δ ε 1 + i ω τ , \hat{\varepsilon}(\omega)=\varepsilon_{\infty}+\frac{\Delta\varepsilon}{1+i% \omega\tau},
  20. ε \varepsilon_{\infty}
  21. Δ ε = ε s - ε \Delta\varepsilon=\varepsilon_{s}-\varepsilon_{\infty}
  22. ε s \varepsilon_{s}
  23. τ \tau
  24. d d
  25. σ ε = ε V d \sigma_{\varepsilon}=\varepsilon\frac{V}{d}
  26. c = σ ε V = ε d c=\frac{\sigma_{\varepsilon}}{V}=\frac{\varepsilon}{d}

Dielectric_strength.html

  1. 1 V/m = 2.54 × 10 - 5 V/mil 1\,\text{ V/m}=2.54\times 10^{-5}\,\text{ V/mil}
  2. 1 V/mil = 3.94 × 10 4 V/m 1\,\text{ V/mil}=3.94\times 10^{4}\,\text{ V/m}

Diesel_cycle.html

  1. V 2 V_{2}
  2. V 3 V_{3}
  3. p p
  4. v v
  5. W i n W_{in}
  6. Q i n Q_{in}
  7. W o u t W_{out}
  8. Q o u t Q_{out}
  9. W i n W_{in}
  10. Q i n Q_{in}
  11. W o u t W_{out}
  12. Q o u t Q_{out}
  13. Q i n Q_{in}
  14. Q o u t Q_{out}
  15. W o u t W_{out}
  16. η t h = 1 - 1 r γ - 1 ( α γ - 1 γ ( α - 1 ) ) \eta_{th}=1-\frac{1}{r^{\gamma-1}}\left(\frac{\alpha^{\gamma}-1}{\gamma(\alpha% -1)}\right)
  17. η t h \eta_{th}
  18. α \alpha
  19. V 3 V 2 \frac{V_{3}}{V_{2}}
  20. r r
  21. V 1 V 2 \frac{V_{1}}{V_{2}}
  22. γ \gamma
  23. T 2 T 1 = ( V 1 V 2 ) γ - 1 = r γ - 1 \frac{T_{2}}{T_{1}}={\left(\frac{V_{1}}{V_{2}}\right)^{\gamma-1}}=r^{\gamma-1}
  24. T 2 = T 1 r γ - 1 \displaystyle{T_{2}}={T_{1}}r^{\gamma-1}
  25. V 3 V 2 = T 3 T 2 \frac{V_{3}}{V_{2}}=\frac{T_{3}}{T_{2}}
  26. α = ( T 3 T 1 ) ( 1 r γ - 1 ) \alpha=\left(\frac{T_{3}}{T_{1}}\right)\left(\frac{1}{r^{\gamma-1}}\right)
  27. T 3 T_{3}
  28. p 3 p_{3}
  29. T 1 T_{1}
  30. η o t t o , t h = 1 - 1 r γ - 1 \eta_{otto,th}=1-\frac{1}{r^{\gamma-1}}
  31. r r

Diffeomorphism.html

  1. g | U X = f | U X g_{|U\cap X}=f_{|U\cap X}
  2. { f : 𝐑 2 { ( 0 , 0 ) } 𝐑 2 { ( 0 , 0 ) } ( x , y ) ( x 2 - y 2 , 2 x y ) \begin{cases}f:\mathbf{R}^{2}\setminus\{(0,0)\}\to\mathbf{R}^{2}\setminus\{(0,% 0)\}\\ (x,y)\mapsto(x^{2}-y^{2},2xy)\end{cases}
  3. det D f x = 4 ( x 2 + y 2 ) 0 \det Df_{x}=4(x^{2}+y^{2})\neq 0
  4. D f x : T x U T f ( x ) V Df_{x}:T_{x}U\to T_{f(x)}V
  5. f i / x j \partial f_{i}/\partial x_{j}
  6. f ( x , y ) = ( x 2 + y 3 , x 2 - y 3 ) . f(x,y)=\left(x^{2}+y^{3},x^{2}-y^{3}\right).
  7. J f = ( 2 x 3 y 2 2 x - 3 y 2 ) . J_{f}=\begin{pmatrix}2x&3y^{2}\\ 2x&-3y^{2}\end{pmatrix}.
  8. g ( x , y ) = ( a 0 + a 1 , 0 x + a 0 , 1 y + , b 0 + b 1 , 0 x + b 0 , 1 y + ) g(x,y)=\left(a_{0}+a_{1,0}x+a_{0,1}y+\cdots,\ b_{0}+b_{1,0}x+b_{0,1}y+\cdots\right)
  9. a i , j a_{i,j}
  10. b i , j b_{i,j}
  11. J g ( 0 , 0 ) = ( a 1 , 0 a 0 , 1 b 1 , 0 b 0 , 1 ) . J_{g}(0,0)=\begin{pmatrix}a_{1,0}&a_{0,1}\\ b_{1,0}&b_{0,1}\end{pmatrix}.
  12. a 1 , 0 b 0 , 1 - a 0 , 1 b 1 , 0 0 , a_{1,0}b_{0,1}-a_{0,1}b_{1,0}\neq 0,
  13. h ( x , y ) = ( sin ( x 2 + y 2 ) , cos ( x 2 + y 2 ) ) . h(x,y)=\left(\sin(x^{2}+y^{2}),\cos(x^{2}+y^{2})\right).
  14. J h = ( 2 x cos ( x 2 + y 2 ) 2 y cos ( x 2 + y 2 ) - 2 x sin ( x 2 + y 2 ) - 2 y sin ( x 2 + y 2 ) ) . J_{h}=\begin{pmatrix}2x\cos(x^{2}+y^{2})&2y\cos(x^{2}+y^{2})\\ -2x\sin(x^{2}+y^{2})&-2y\sin(x^{2}+y^{2})\end{pmatrix}.
  15. f ( x , y ) = ( u , v ) . f(x,y)=(u,v).
  16. d u = u x d x + u y d y , du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy,
  17. ( d u , d v ) = ( d x , d y ) D f (du,dv)=(dx,dy)Df
  18. d K ( f , g ) = sup x K d ( f ( x ) , g ( x ) ) + 1 p r sup x K D p f ( x ) - D p g ( x ) d_{K}(f,g)=\sup\nolimits_{x\in K}d(f(x),g(x))+\sum\nolimits_{1\leq p\leq r}% \sup\nolimits_{x\in K}\left\|D^{p}f(x)-D^{p}g(x)\right\|
  19. d ( f , g ) = n 2 - n d K n ( f , g ) 1 + d K n ( f , g ) . d(f,g)=\sum\nolimits_{n}2^{-n}\frac{d_{K_{n}}(f,g)}{1+d_{K_{n}}(f,g)}.
  20. x μ x μ + ε h μ ( x ) x^{\mu}\to x^{\mu}+\varepsilon h^{\mu}(x)
  21. L h = h μ ( x ) x μ . L_{h}=h^{\mu}(x)\frac{\partial}{\partial x_{\mu}}.
  22. \mapsto

Difference_engine.html

  1. ( n - 1 ) (n-1)
  2. n n
  3. n + 1 n+1
  4. n - 1 n-1
  5. p ( x ) = 2 x 2 - 3 x + 2 p(x)=2x^{2}-3x+2\,
  6. 1 0 1_{0}
  7. f ( 0 ) f(0)
  8. 2 0 2_{0}
  9. f ( 1 ) f(1)
  10. f ( 0 ) f(0)
  11. f ( x ) = a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}\,
  12. 1 0 1_{0}
  13. 2 0 2_{0}
  14. 3 0 3_{0}
  15. 4 0 4_{0}
  16. 5 0 5_{0}
  17. 6 0 6_{0}
  18. ...
  19. ± 1 \pm 1
  20. n = 0 f ( n ) ( 0 ) n ! x n \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\ x^{n}
  21. a n f ( n ) ( 0 ) n ! a_{n}\equiv\frac{f^{(n)}(0)}{n!}

Differential_calculus.html

  1. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}
  2. x x
  3. y y
  4. y y
  5. x x
  6. x x
  7. y y
  8. y = f ( x ) y=f(x)
  9. f ( x ) f(x)
  10. m m
  11. b b
  12. y = m x + b y=mx+b
  13. m m
  14. m = change in y change in x = Δ y Δ x , m=\frac{\,\text{change in }y}{\,\text{change in }x}=\frac{\Delta y}{\Delta x},
  15. Δ Δ
  16. Δ y = m Δ x Δy=mΔx
  17. f f
  18. x = a x=a
  19. f f
  20. a a
  21. f ( a ) f′(a)
  22. f f
  23. a a
  24. f f
  25. a a
  26. f f
  27. a a
  28. a a
  29. f f
  30. a a
  31. f f
  32. a a
  33. f ( x ) f′(x)
  34. d y d x \frac{dy}{dx}
  35. f ( x ) = d y d x = 2 x f′(x)=\frac{dy}{dx}=2x
  36. x x
  37. y y
  38. f f
  39. x x
  40. f f
  41. x x
  42. f f
  43. f f
  44. x x
  45. y y
  46. f f
  47. f f
  48. y ∂\frac{y}{∂}
  49. f f
  50. f f
  51. x x
  52. f f
  53. f f
  54. x x
  55. f f
  56. x x
  57. x x
  58. f f
  59. f f
  60. x x
  61. x x
  62. x x
  63. x x
  64. x = 0 x=0
  65. x = 0 x=0
  66. x ( t ) = - 16 t 2 + 16 t + 32 , x(t)=-16t^{2}+16t+32,\,\!
  67. x ˙ ( t ) = x ( t ) = - 32 t + 16 , \dot{x}(t)=x^{\prime}(t)=-32t+16,\,\!
  68. x ¨ ( t ) = x ′′ ( t ) = - 32 , \ddot{x}(t)=x^{\prime\prime}(t)=-32,\,\!
  69. F ( t ) = m d 2 x d t 2 . F(t)=m\frac{d^{2}x}{dt^{2}}.
  70. u t = α 2 u x 2 . \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}}.
  71. u ( x , t ) u(x,t)
  72. x x
  73. t t
  74. α α
  75. f ( x ) f(x)
  76. a a
  77. b b
  78. ( a , f ( a ) ) (a,f(a))
  79. ( b , f ( b ) ) (b,f(b))
  80. f f
  81. c c
  82. a a
  83. b b
  84. f ( c ) = f ( b ) - f ( a ) b - a . f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}.
  85. f f
  86. f f
  87. f f
  88. f ( x ) f(x)
  89. x < s u b > 0 x<sub>0

Differential_geometry.html

  1. α \alpha
  2. H p = ker α p T p M . H_{p}=\ker\alpha_{p}\subset T_{p}M.
  3. α \alpha
  4. α ( d α ) n \alpha\wedge(d\alpha)^{n}
  5. M M
  6. J : T M T M J:TM\rightarrow TM
  7. J 2 = - 1. J^{2}=-1.\,
  8. N J = 0 N_{J}=0
  9. N J N_{J}
  10. J J
  11. g ( J X , J Y ) = g ( X , Y ) g(JX,JY)=g(X,Y)\,
  12. ω J , g ( X , Y ) := g ( J X , Y ) \omega_{J,g}(X,Y):=g(JX,Y)\,
  13. N J = 0 and d ω = 0 N_{J}=0\mbox{ and }~{}d\omega=0\,
  14. J = 0 \nabla J=0\,
  15. \nabla
  16. g g
  17. ( J , g ) (J,g)

Diffie–Hellman_key_exchange.html

  1. p p
  2. p p
  3. g g
  4. a a
  5. b b
  6. c c
  7. g a g^{a}
  8. ( g a ) b = g a b (g^{a})^{b}=g^{ab}
  9. ( g a b ) c = g a b c (g^{ab})^{c}=g^{abc}
  10. g b g^{b}
  11. ( g b ) c = g b c (g^{b})^{c}=g^{bc}
  12. ( g b c ) a = g b c a = g a b c (g^{bc})^{a}=g^{bca}=g^{abc}
  13. g c g^{c}
  14. ( g c ) a = g c a (g^{c})^{a}=g^{ca}
  15. ( g c a ) b = g c a b = g a b c (g^{ca})^{b}=g^{cab}=g^{abc}
  16. g a g^{a}
  17. g b g^{b}
  18. g c g^{c}
  19. g a b g^{ab}
  20. g a c g^{ac}
  21. g b c g^{bc}
  22. g a b c g^{abc}
  23. g g
  24. N - 1 N-1
  25. N N
  26. N N
  27. N N
  28. N N
  29. N N
  30. N N
  31. N N
  32. log 2 ( N ) + 1 \log_{2}(N)+1
  33. g a b c d g^{abcd}
  34. g e f g h g^{efgh}
  35. g e f g h a b g^{efghab}
  36. g e f g h c d g^{efghcd}
  37. g e f g h c d a g^{efghcda}
  38. g e f g h c d b g^{efghcdb}
  39. g e f g h c d b a = g a b c d e f g h g^{efghcdba}=g^{abcdefgh}
  40. g e f g h c d a b = g a b c d e f g h g^{efghcdab}=g^{abcdefgh}
  41. g a b c d g^{abcd}
  42. g a b c d e f g h g^{abcdefgh}
  43. ( g a mod p , g , p ) (g^{a}\bmod{p},g,p)
  44. g b mod p g^{b}\bmod p
  45. ( g a ) b mod p (g^{a})^{b}\bmod{p}

Diffraction.html

  1. d sin ( θ ) 2 \frac{d\sin(\theta)}{2}
  2. d sin θ min = λ d\,\sin\theta\text{min}=\lambda
  3. θ min \theta\text{min}
  4. λ \lambda
  5. d sin θ n = n λ d\,\sin\theta_{n}=n\lambda
  6. I ( θ ) = I 0 sinc 2 ( d π λ sin θ ) I(\theta)=I_{0}\,\operatorname{sinc}^{2}\left(\frac{d\pi}{\lambda}\sin\theta\right)
  7. I ( θ ) I(\theta)
  8. I 0 I_{0}
  9. d ( sin θ m + sin θ i ) = m λ . d\left(\sin{\theta_{m}}+\sin{\theta_{i}}\right)=m\lambda.
  10. I ( θ ) = I 0 ( 2 J 1 ( k a sin θ ) k a sin θ ) 2 I(\theta)=I_{0}\left(\frac{2J_{1}(ka\sin\theta)}{ka\sin\theta}\right)^{2}
  11. ψ \psi
  12. 2 ψ + k 2 ψ = δ ( r ) \nabla^{2}\psi+k^{2}\psi=\delta(r)
  13. δ ( r ) \delta(r)
  14. 2 ψ = 1 r 2 r 2 ( r ψ ) \nabla^{2}\psi=\frac{1}{r}\frac{\partial^{2}}{\partial r^{2}}(r\psi)
  15. e - i ω t e^{-i\omega t}
  16. ψ ( r ) = e i k r 4 π r \psi(r)=\frac{e^{ikr}}{4\pi r}
  17. r r^{\prime}
  18. r r
  19. ψ ( r | r ) = e i k | r - r | 4 π | r - r | \psi(r|r^{\prime})=\frac{e^{ik|r-r^{\prime}|}}{4\pi|r-r^{\prime}|}
  20. Ψ ( r ) aperture E i n c ( x , y ) e i k | r - r | 4 π | r - r | d x d y , \Psi(r)\propto\int\!\!\!\int_{\mathrm{aperture}}E_{inc}(x^{\prime},y^{\prime})% ~{}\frac{e^{ik|r-r^{\prime}|}}{4\pi|r-r^{\prime}|}\,dx^{\prime}\,dy^{\prime},
  21. r = x x ^ + y y ^ {r}^{\prime}=x^{\prime}{\hat{x}}+y^{\prime}{\hat{y}}
  22. ψ ( r | r ) = e i k | r - r | 4 π | r - r | \psi(r|r^{\prime})=\frac{e^{ik|r-r^{\prime}|}}{4\pi|r-r^{\prime}|}
  23. ψ ( r | r ) = e i k r 4 π r e - i k ( r r ^ ) \psi({r}|{r}^{\prime})=\frac{e^{ikr}}{4\pi r}e^{-ik({r}^{\prime}\cdot{\hat{r}})}
  24. Ψ ( r ) e i k r 4 π r aperture E i n c ( x , y ) e - i k ( r r ^ ) d x d y , \Psi(r)\propto\frac{e^{ikr}}{4\pi r}\int\!\!\!\int_{\mathrm{aperture}}E_{inc}(% x^{\prime},y^{\prime})e^{-ik({r}^{\prime}\cdot{\hat{r}})}\,dx^{\prime}\,dy^{% \prime},
  25. r = x x ^ + y y ^ {r}^{\prime}=x^{\prime}{\hat{x}}+y^{\prime}{\hat{y}}
  26. r ^ = sin θ cos ϕ x ^ + sin θ sin ϕ y ^ + cos θ z ^ {\hat{r}}=\sin\theta\cos\phi{\hat{x}}+\sin\theta~{}\sin\phi~{}{\hat{y}}+\cos% \theta{\hat{z}}
  27. Ψ ( r ) e i k r 4 π r aperture E i n c ( x , y ) e - i k sin θ ( cos ϕ x + sin ϕ y ) d x d y \Psi(r)\propto\frac{e^{ikr}}{4\pi r}\int\!\!\!\int_{\mathrm{aperture}}E_{inc}(% x^{\prime},y^{\prime})e^{-ik\sin\theta(\cos\phi x^{\prime}+\sin\phi y^{\prime}% )}\,dx^{\prime}\,dy^{\prime}
  28. k x = k sin θ cos ϕ k_{x}=k\sin\theta\cos\phi\,\!
  29. k y = k sin θ sin ϕ k_{y}=k\sin\theta\sin\phi\,\!
  30. Ψ ( r ) e i k r 4 π r aperture E i n c ( x , y ) e - i ( k x x + k y y ) d x d y , \Psi(r)\propto\frac{e^{ikr}}{4\pi r}\int\!\!\!\int_{\mathrm{aperture}}E_{inc}(% x^{\prime},y^{\prime})e^{-i(k_{x}x^{\prime}+k_{y}y^{\prime})}\,dx^{\prime}\,dy% ^{\prime},
  31. d = 1.22 λ N , d=1.22\lambda N,\,
  32. sin θ = 1.22 λ D , \sin\theta=1.22\frac{\lambda}{D},\,
  33. λ = h p \lambda=\frac{h}{p}\,
  34. m λ = 2 d sin θ m\lambda=2d\sin\theta\,

Diffraction_grating.html

  1. d sin θ m = m λ . d\sin\theta_{m}=m\lambda.
  2. d ( sin θ i + sin θ m ) = m λ . d(\sin\theta_{i}+\sin\theta_{m})=m\lambda.
  3. θ m = arcsin ( m λ d - sin θ i ) . \theta_{m}=\arcsin\!\left(\frac{m\lambda}{d}-\sin\theta_{i}\right)\!.

Digital_filter.html

  1. H ( z ) = B ( z ) A ( z ) = b 0 + b 1 z - 1 + b 2 z - 2 + + b N z - N 1 + a 1 z - 1 + a 2 z - 2 + + a M z - M H(z)=\frac{B(z)}{A(z)}=\frac{{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots+b_{N}z^{-N}% }}{{1+a_{1}z^{-1}+a_{2}z^{-2}+\cdots+a_{M}z^{-M}}}
  2. h [ k ] h[k]
  3. h k h_{k}
  4. x 0 = 1 x_{0}=1
  5. x k = 0 x_{k}=0
  6. k 0 k\neq 0
  7. y n = k = 0 n - 1 h k x n - k \ y_{n}=\sum_{k=0}^{n-1}h_{k}x_{n-k}
  8. m = 0 M - 1 a m y n - m = k = 0 n - 1 b k x n - k \ \sum_{m=0}^{M-1}a_{m}y_{n-m}=\sum_{k=0}^{n-1}b_{k}x_{n-k}
  9. H ( z ) = ( z + 1 ) 2 ( z - 1 2 ) ( z + 3 4 ) H(z)=\frac{(z+1)^{2}}{(z-\frac{1}{2})(z+\frac{3}{4})}
  10. H ( z ) = z 2 + 2 z + 1 z 2 + 1 4 z - 3 8 H(z)=\frac{z^{2}+2z+1}{z^{2}+\frac{1}{4}z-\frac{3}{8}}
  11. z z
  12. H ( z ) = 1 + 2 z - 1 + z - 2 1 + 1 4 z - 1 - 3 8 z - 2 = Y ( z ) X ( z ) H(z)=\frac{1+2z^{-1}+z^{-2}}{1+\frac{1}{4}z^{-1}-\frac{3}{8}z^{-2}}=\frac{Y(z)% }{X(z)}
  13. a k a_{k}
  14. b k b_{k}
  15. y [ n ] = - k = 1 M a k y [ n - k ] + k = 0 N b k x [ n - k ] y[n]=-\sum_{k=1}^{M}a_{k}y[n-k]+\sum_{k=0}^{N}b_{k}x[n-k]
  16. Y ( z ) X ( z ) = 1 + 2 z - 1 + z - 2 1 + 1 4 z - 1 - 3 8 z - 2 \frac{Y(z)}{X(z)}=\frac{1+2z^{-1}+z^{-2}}{1+\frac{1}{4}z^{-1}-\frac{3}{8}z^{-2}}
  17. ( 1 + 1 4 z - 1 - 3 8 z - 2 ) Y ( z ) = ( 1 + 2 z - 1 + z - 2 ) X ( z ) \Rightarrow(1+\frac{1}{4}z^{-1}-\frac{3}{8}z^{-2})Y(z)=(1+2z^{-1}+z^{-2})X(z)
  18. y [ n ] + 1 4 y [ n - 1 ] - 3 8 y [ n - 2 ] = x [ n ] + 2 x [ n - 1 ] + x [ n - 2 ] \Rightarrow y[n]+\frac{1}{4}y[n-1]-\frac{3}{8}y[n-2]=x[n]+2x[n-1]+x[n-2]
  19. y [ n ] y[n]
  20. y [ n ] = - 1 4 y [ n - 1 ] + 3 8 y [ n - 2 ] + x [ n ] + 2 x [ n - 1 ] + x [ n - 2 ] y[n]=-\frac{1}{4}y[n-1]+\frac{3}{8}y[n-2]+x[n]+2x[n-1]+x[n-2]
  21. y [ n ] y[n]
  22. y [ n - p ] y[n-p]
  23. x [ n ] x[n]
  24. x [ n - p ] x[n-p]
  25. a x + b x + c ax+bx+c
  26. x ( a + b ) + c x(a+b)+c
  27. z - 1 z^{-1}
  28. ( N + 1 ) 2 (N+1)^{2}
  29. 4 N - 1 4N-1
  30. 4 N - 1 4N-1

Digital_signal_processing.html

  1. T s N = T k T_{s}N=Tk
  2. z = r e j ω z=re^{j\omega}
  3. ω = 2 π F \omega=2\pi F
  4. F F

Digital_Signature_Algorithm.html

  1. H H
  2. m m
  3. k k
  4. 0 < k < q 0<k<q
  5. r = ( g k mod p ) mod q r=\left(g^{k}\bmod\,p\right)\bmod\,q
  6. r = 0 r=0
  7. k k
  8. s = k - 1 ( H ( m ) + x r ) mod q s=k^{-1}\left(H\left(m\right)+xr\right)\bmod\,q
  9. s = 0 s=0
  10. k k
  11. ( r , s ) \left(r,s\right)
  12. k - 1 mod q k^{-1}\bmod\,q
  13. k q - 2 mod q k^{q-2}\bmod\,q
  14. 0 < r < q 0<r<q
  15. 0 < s < q 0<s<q
  16. w = s - 1 mod q w=s^{-1}\bmod\,q
  17. u 1 = H ( m ) w mod q u_{1}=H\left(m\right)\cdot w\,\bmod\,q
  18. u 2 = r w mod q u_{2}=r\cdot w\,\bmod\,q
  19. v = ( ( g u 1 y u 2 ) mod p ) mod q v=\left(\left(g^{u_{1}}y^{u_{2}}\right)\bmod\,p\right)\bmod\,q
  20. v = r v=r
  21. s = k - 1 ( H ( m ) + x r ) mod q s=k^{-1}(H(m)+xr)\bmod\,q
  22. k H ( m ) s - 1 + x r s - 1 H ( m ) w + x r w ( mod q ) \begin{aligned}\displaystyle k&\displaystyle\equiv H(m)s^{-1}+xrs^{-1}\\ &\displaystyle\equiv H(m)w+xrw\;\;(\mathop{{\rm mod}}q)\end{aligned}
  23. g k g H ( m ) w g x r w g H ( m ) w y r w g u 1 y u 2 ( mod p ) \begin{aligned}\displaystyle g^{k}&\displaystyle\equiv g^{H(m)w}g^{xrw}\\ &\displaystyle\equiv g^{H(m)w}y^{rw}\\ &\displaystyle\equiv g^{u1}y^{u2}\;\;(\mathop{{\rm mod}}p)\end{aligned}
  24. r = ( g k mod p ) mod q = ( g u 1 y u 2 mod p ) mod q = v \begin{aligned}\displaystyle r&\displaystyle=(g^{k}\bmod\,p)\bmod\,q\\ &\displaystyle=(g^{u1}y^{u2}\bmod\,p)\bmod\,q\\ &\displaystyle=v\end{aligned}

Dijkstra's_algorithm.html

  1. O ( | V | 2 ) O(|V|^{2})
  2. | V | |V|
  3. O ( | E | + | V | log | V | ) O(|E|+|V|\log|V|)
  4. | E | |E|
  5. E E
  6. V V
  7. | E | |E|
  8. | V | |V|
  9. Q Q
  10. | E | = O ( | V | 2 ) |E|=O(|V|^{2})
  11. | E | |E|
  12. Q Q
  13. O ( | E | T dk + | V | T em ) O(|E|\cdot T_{\mathrm{dk}}+|V|\cdot T_{\mathrm{em}})
  14. T dk T_{\mathrm{dk}}
  15. T em T_{\mathrm{em}}
  16. Q Q
  17. Q Q
  18. Q Q
  19. O ( | E | + | V | 2 ) = O ( | V | 2 ) O(|E|+|V|^{2})=O(|V|^{2})
  20. | V | 2 |V|^{2}
  21. Q Q
  22. Θ ( ( | E | + | V | ) log | V | ) \Theta((|E|+|V|)\log|V|)
  23. Θ ( | E | log | V | ) \Theta(|E|\log|V|)
  24. O ( | E | + | V | log | V | ) O(|E|+|V|\log|V|)
  25. O ( | V | log ( | E | / | V | ) ) O(|V|\log(|E|/|V|))
  26. O ( | E | + | V | log | E | | V | log | V | ) O(|E|+|V|\log\frac{|E|}{|V|}\log|V|)
  27. ε ε
  28. O ( | E | log log | C | ) O(|E|\log\log|C|)
  29. O ( | E | + | V | log | C | ) O(|E|+|V|\sqrt{\log|C|})
  30. O ( | E | log log | V | ) O(|E|\log\log|V|)
  31. O ( | E | + | V | min { ( log | V | ) 1 / 3 + ϵ , ( log | C | ) 1 / 4 + ϵ } ) O(|E|+|V|\min\{(\log|V|)^{1/3+\epsilon},(\log|C|)^{1/4+\epsilon}\})
  32. O ( | E | + | V | ) O(|E|+|V|)
  33. O ( | V | + | E | ) O(|V|+|E|)

Dimension.html

  1. n n
  2. n n
  3. n n
  4. ε ε
  5. n n
  6. n n
  7. n = 3 n=3
  8. 4 4
  9. V 0 V 1 V d V_{0}\subsetneq V_{1}\subsetneq\ldots\subsetneq V_{d}
  10. \subsetneq
  11. 𝒫 0 𝒫 1 𝒫 n \mathcal{P}_{0}\subsetneq\mathcal{P}_{1}\subsetneq\ldots\subsetneq\mathcal{P}_% {n}
  12. X X
  13. X X
  14. n + 1 n+1
  15. X = n X=n
  16. X X
  17. n n
  18. X X
  19. X = X=∞
  20. X X
  21. X = 1 X=−1
  22. X X
  23. n + 1 n+1
  24. n n
  25. n n

Dimension_(vector_space).html

  1. { ( 1 0 0 ) , ( 0 1 0 ) , ( 0 0 1 ) } \left\{\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}\right\}
  2. tr id 𝐑 2 = tr ( 1 0 0 1 ) = 1 + 1 = 2. \operatorname{tr}\ \operatorname{id}_{\mathbf{R}^{2}}=\operatorname{tr}\left(% \begin{smallmatrix}1&0\\ 0&1\end{smallmatrix}\right)=1+1=2.
  3. η : K A \eta\colon K\to A
  4. ϵ : A K \epsilon\colon A\to K
  5. ϵ η : K K \epsilon\circ\eta\colon K\to K
  6. ϵ := 1 n tr \epsilon:=\textstyle{\frac{1}{n}}\operatorname{tr}
  7. χ : G K , \chi\colon G\to K,
  8. 1 G 1\in G
  9. χ ( 1 G ) = tr I V = dim V . \chi(1_{G})=\operatorname{tr}\ I_{V}=\dim V.
  10. χ ( g ) \chi(g)

Dimensional_analysis.html

  1. 10 \cancel mile 1 \cancel hour × 1609 meter 1 \cancel mile × 1 \cancel hour 3600 second = 4.47 meter second . \frac{10\ \cancel{\,\text{mile}}}{1\ \cancel{\,\text{hour}}}\times\frac{1609\,% \text{ meter}}{1\ \cancel{\,\text{mile}}}\times\frac{1\ \cancel{\,\text{hour}}% }{3600\,\text{ second}}=4.47\ \frac{\,\text{meter}}{\,\text{second}}.
  2. 10 \cancel m 3 NOx 10 6 \cancel m 3 gas × 20 \cancel m 3 gas 1 \cancel minute × 60 \cancel minute 1 hour × 1 \cancel kgmol NOx 22.414 \cancel m 3 NOx × 46 \cancel kg NOx 1 \cancel kgmol NOx × 1000 g 1 \cancel kg = 24.63 g NOx hour \frac{10\ \cancel{\,\text{m}^{3}\,\text{ NOx}}}{10^{6}\ \cancel{\,\text{m}^{3}% \,\text{ gas}}}\times\frac{20\ \cancel{\,\text{m}^{3}\,\text{ gas}}}{1\ % \cancel{\,\text{minute}}}\times\frac{60\ \cancel{\,\text{minute}}}{1\,\text{ % hour}}\times\frac{1\ \cancel{\,\text{kgmol NOx}}}{22.414\ \cancel{\,\text{m}^{% 3}\,\text{ NOx}}}\times\frac{46\ \cancel{\,\text{kg}}\,\text{ NOx}}{1\ \cancel% {\,\text{kgmol NOx}}}\times\frac{1000\,\text{ g}}{1\ \cancel{\,\text{kg}}}=24.% 63\ \frac{\,\text{g NOx}}{\,\text{hour}}
  3. Pa m 3 = \cancel mol 1 × Pa m 3 \cancel mol \cancel K × \cancel K 1 \,\text{Pa m}^{3}=\frac{\cancel{\,\text{mol}}}{1}\times\frac{\,\text{Pa m}^{3}% }{\cancel{\,\text{mol}}\ \cancel{\,\text{K}}}\times\frac{\cancel{\,\text{K}}}{1}
  4. x a x + b x\mapsto ax+b
  5. x a x x\mapsto ax
  6. x n , x^{n},
  7. ( n - 1 ) (n-1)
  8. x n - 1 . x^{n-1}.
  9. C n r n , C_{n}r^{n},
  10. C n . C_{n}.
  11. X = i = 1 m ( π i ) k i . X=\prod_{i=1}^{m}(\pi_{i})^{k_{i}}\,.
  12. f ( π 1 , π 2 , , π m ) = 0 . f(\pi_{1},\pi_{2},...,\pi_{m})=0\,.
  13. 1 2 ( - 32 foot second 2 ) t 2 + ( 500 foot second ) t . \frac{1}{2}\cdot\left(-32\frac{\,\text{foot}}{\,\text{second}^{2}}\right)\cdot t% ^{2}+\left(500\frac{\,\text{foot}}{\,\text{second}}\right)\cdot t.
  14. 1 2 ( - 32 foot second 2 ) ( 0.01 minute ) 2 \displaystyle{}\qquad\frac{1}{2}\cdot\left(-32\frac{\,\text{foot}}{\,\text{% second}^{2}}\right)\cdot(0.01\,\text{ minute})^{2}
  15. Z = n × [ Z ] = n [ Z ] Z=n\times[Z]=n[Z]
  16. 1 ft = 0.3048 m 1\ \mbox{ft}~{}=0.3048\ \mbox{m}~{}
  17. 1 = 0.3048 m 1 ft . 1=\frac{0.3048\ \mbox{m}~{}}{1\ \mbox{ft}~{}}.
  18. 0.3048 m ft 0.3048\ \frac{\mbox{m}~{}}{\mbox{ft}~{}}
  19. T T
  20. m m
  21. k k
  22. g g
  23. T T
  24. T T
  25. m m
  26. k k
  27. g g
  28. T T
  29. m m
  30. k k
  31. g g
  32. G 1 G_{1}
  33. T 2 k / m T^{2}k/m
  34. G 1 = C G_{1}=C
  35. C C
  36. g g
  37. g g
  38. k k
  39. m m
  40. T T
  41. g g
  42. g g
  43. g g
  44. T = κ m / k T=\kappa\sqrt{m/k}
  45. C \sqrt{C}
  46. g g
  47. π 1 = E / A s π 2 = / A . \begin{aligned}\displaystyle\pi_{1}&\displaystyle=E/As\\ \displaystyle\pi_{2}&\displaystyle=\ell/A.\end{aligned}
  48. F ( E / A s , / A ) = 0 , F(E/As,\ell/A)=0,
  49. E = A s f ( / A ) , E=Asf(\ell/A),
  50. V y V\text{y}
  51. V x V\text{x}
  52. V x V\text{x}
  53. V y V\text{y}
  54. R V x a V y b g c . R\propto V\text{x}^{a}\,V\text{y}^{b}\,g^{c}.\,
  55. 𝖫 = ( 𝖫 / 𝖳 ) a + b ( 𝖫 / 𝖳 2 ) c \mathsf{L}=(\mathsf{L}/\mathsf{T})^{a+b}(\mathsf{L}/\mathsf{T}^{2})^{c}\,
  56. a + b + c = 1 a+b+c=1
  57. a + b + 2 c = 0 a+b+2c=0
  58. V x V\text{x}
  59. V y V\text{y}
  60. 𝖫 x = ( 𝖫 x / 𝖳 ) a ( 𝖫 y / T ) b ( 𝖫 y / 𝖳 2 ) c \mathsf{L}\text{x}=(\mathsf{L}\text{x}/\mathsf{T})^{a}\,(\mathsf{L}\text{y}/T)% ^{b}(\mathsf{L}\text{y}/\mathsf{T}^{2})^{c}\,
  61. a = 1 a=1
  62. b = 1 b=1
  63. c = - 1 c=-1
  64. m ˙ \dot{m}
  65. p x p\text{x}
  66. ρ \rho
  67. η \eta
  68. r r
  69. π 1 = m ˙ / η r \pi_{1}=\dot{m}/\eta r
  70. π 2 = p x ρ r 5 / m ˙ 2 \pi_{2}=p\text{x}\rho r^{5}/\dot{m}^{2}
  71. C = π 1 π 2 a = ( m ˙ η r ) ( p x ρ r 5 m ˙ 2 ) a C=\pi_{1}\pi_{2}^{a}=\left(\frac{\dot{m}}{\eta r}\right)\left(\frac{p\text{x}% \rho r^{5}}{\dot{m}^{2}}\right)^{a}
  72. M i M\text{i}
  73. M s M\text{s}
  74. C = p x ρ r 4 η m ˙ C=\frac{p\text{x}\rho r^{4}}{\eta\dot{m}}
  75. π / 8 \pi/8
  76. 𝟏 𝟎 𝟏 x 𝟏 y 𝟏 z 𝟏 𝟎 1 0 1 x 1 y 1 z 𝟏 x 1 x 1 0 1 z 1 y 𝟏 y 1 y 1 z 1 0 1 x 𝟏 z 1 z 1 y 1 x 1 0 \begin{array}[]{c|cccc}&\mathbf{1_{0}}&\mathbf{1\text{x}}&\mathbf{1\text{y}}&% \mathbf{1\text{z}}\\ \hline\mathbf{1_{0}}&1_{0}&1\text{x}&1\text{y}&1\text{z}\\ \mathbf{1\text{x}}&1\text{x}&1_{0}&1\text{z}&1\text{y}\\ \mathbf{1\text{y}}&1\text{y}&1\text{z}&1_{0}&1\text{x}\\ \mathbf{1\text{z}}&1\text{z}&1\text{y}&1\text{x}&1_{0}\end{array}
  77. sin ( θ + π / 2 ) = cos ( θ ) \sin(\theta+\pi/2)=\cos(\theta)
  78. sin ( a 1 z + b 1 z ) = sin ( a 1 z ) cos ( b 1 z ) + sin ( b 1 z ) cos ( a 1 z ) , \sin(a\,1\text{z}+b\,1\text{z})=\sin(a\,1\text{z})\cos(b\,1\text{z})+\sin(b\,1% \text{z})\cos(a\,1\text{z}),
  79. a = θ a=\theta
  80. b = π / 2 b=\pi/2
  81. sin ( θ 1 z + ( π / 2 ) 1 z ) = 1 z cos ( θ 1 z ) \sin(\theta\,1\text{z}+(\pi/2)\,1\text{z})=1\text{z}\cos(\theta\,1\text{z})
  82. e i θ e^{i\theta}
  83. R = g a v b θ c which means L 1 x ( 𝖫 1 y 𝖳 2 ) a ( 𝖫 𝖳 ) b 1 𝗓 c . R=g^{a}\,v^{b}\,\theta^{c}\,\text{ which means }L\,1\text{x}\sim\left(\frac{% \mathsf{L}\,1\text{y}}{\mathsf{T}^{2}}\right)^{a}\left(\frac{\mathsf{L}}{% \mathsf{T}}\right)^{b}\,1_{\mathsf{z}}^{c}.\,
  84. κ \kappa
  85. ξ \xi
  86. 1 / ξ d \sim 1/\xi^{d}
  87. d d
  88. c c\rightarrow\infty
  89. 0 \hbar\rightarrow 0
  90. G 0 G\rightarrow 0
  91. F d Fd
  92. S / t P t S/t\equiv Pt
  93. m v 2 p v p 2 / m mv^{2}\equiv pv\equiv p^{2}/m
  94. I ω 2 L ω L 2 / I I\omega^{2}\equiv L\omega\equiv L^{2}/I
  95. p V n R T k B T T S pV\equiv nRT\equiv k_{B}T\equiv TS
  96. I A t S A t IAt\equiv SAt
  97. q ϕ q\phi
  98. ϵ E 2 V B 2 V / μ \epsilon E^{2}V\equiv B^{2}V/\mu
  99. p E m B I A pE\equiv mB\equiv IA
  100. m v F t mv\equiv Ft
  101. S / r L / r S/r\equiv L/r
  102. m v 2 m\sqrt{\langle v^{2}\rangle}
  103. v 2 \sqrt{\langle v^{2}\rangle}
  104. ρ V v \rho Vv
  105. q A qA
  106. m a p / t ma\equiv p/t
  107. T δ S / δ r T\delta S/\delta r
  108. ρ V v \rho Vv
  109. E q B q v Eq\equiv Bqv
  110. M = E / v 2 , L = S v / E , t = S / E M=E/v^{2},\quad L=Sv/E,\quad t=S/E
  111. 𝖤 n = 𝖬 p 𝖫 q 𝖳 r = 𝖤 p - q - r \mathsf{E}^{n}=\mathsf{M}^{p}\mathsf{L}^{q}\mathsf{T}^{r}=\mathsf{E}^{p-q-r}

Dimensionless_quantity.html

  1. π \pi
  2. e e
  3. φ φ
  4. V = n d - 1 n F - n C V=\frac{n_{d}-1}{n_{F}-n_{C}}
  5. γ \gamma
  6. γ = < m t p l > a x \gamma=\frac{<}{m}tpl>{{a}}{{x}}
  7. α \alpha
  8. α = ( 1 - D ) α ¯ ( θ i ) + D α ¯ ¯ \alpha=(1-D)\bar{\alpha}(\theta_{i})+D\bar{\bar{\alpha}}
  9. Ar = g L 3 ρ ( ρ - ρ ) μ 2 \mathrm{Ar}=\frac{gL^{3}\rho_{\ell}(\rho-\rho_{\ell})}{\mu^{2}}
  10. α \alpha
  11. α = E a R T \alpha=\frac{E_{a}}{RT}
  12. A = ρ 1 - ρ 2 ρ 1 + ρ 2 \mathrm{A}=\frac{\rho_{1}-\rho_{2}}{\rho_{1}+\rho_{2}}
  13. Ba = ρ d 2 λ 1 / 2 γ μ \mathrm{Ba}=\frac{\rho d^{2}\lambda^{1/2}\gamma}{\mu}
  14. Be = Δ P L 2 μ α \mathrm{Be}=\frac{\Delta PL^{2}}{\mu\alpha}
  15. Be = S ˙ gen , Δ T S ˙ gen , Δ T + S ˙ gen , Δ p \mathrm{Be}=\frac{\dot{S}^{\prime}_{\mathrm{gen},\,\Delta T}}{\dot{S}^{\prime}% _{\mathrm{gen},\,\Delta T}+\dot{S}^{\prime}_{\mathrm{gen},\,\Delta p}}
  16. Bm = τ y L μ V \mathrm{Bm}=\frac{\tau_{y}L}{\mu V}
  17. Bi = h L C k b \mathrm{Bi}=\frac{hL_{C}}{k_{b}}
  18. B = u ρ μ ( 1 - ϵ ) D \mathrm{B}=\frac{u\rho}{\mu(1-\epsilon)D}
  19. Bo = v L / 𝒟 = Re Sc \mathrm{Bo}=vL/\mathcal{D}=\mathrm{Re}\,\mathrm{Sc}
  20. Bo = ρ a L 2 γ \mathrm{Bo}=\frac{\rho aL^{2}}{\gamma}
  21. Br = μ U 2 κ ( T w - T 0 ) \mathrm{Br}=\frac{\mu U^{2}}{\kappa(T_{w}-T_{0})}
  22. N BK = u μ k rw σ \mathrm{N}_{\mathrm{BK}}=\frac{u\mu}{k_{\mathrm{rw}}\sigma}
  23. Ca = μ V γ \mathrm{Ca}=\frac{\mu V}{\gamma}
  24. Q = B 0 2 d 2 μ 0 ρ ν λ \mathrm{Q}=\frac{{B_{0}}^{2}d^{2}}{\mu_{0}\rho\nu\lambda}
  25. μ k \mu_{k}
  26. μ s \mu_{s}
  27. R 2 R^{2}
  28. σ μ \frac{\sigma}{\mu}
  29. σ μ \frac{\sigma}{\mu}
  30. 𝔼 [ ( X - μ X ) ( Y - μ Y ) ] σ X σ Y \frac{{\mathbb{E}}[(X-\mu_{X})(Y-\mu_{Y})]}{\sigma_{X}\sigma_{Y}}
  31. k = 1 n ( x k - x ¯ ) ( y k - y ¯ ) k = 1 n ( x k - x ¯ ) 2 k = 1 n ( y k - y ¯ ) 2 \frac{\sum_{k=1}^{n}(x_{k}-\bar{x})(y_{k}-\bar{y})}{\sqrt{\sum_{k=1}^{n}(x_{k}% -\bar{x})^{2}\sum_{k=1}^{n}(y_{k}-\bar{y})^{2}}}
  32. x ¯ = k = 1 n x k / n \bar{x}=\sum_{k=1}^{n}x_{k}/n
  33. y ¯ \bar{y}
  34. C = u Δ t Δ x C=\frac{u\,\Delta t}{\Delta x}
  35. Da = k τ \mathrm{Da}=k\tau
  36. ζ \zeta
  37. ζ = c 2 k m \zeta=\frac{c}{2\sqrt{km}}
  38. Da = K d 2 \mathrm{Da}=\frac{K}{d^{2}}
  39. D = ρ V d μ ( d 2 R ) 1 / 2 \mathrm{D}=\frac{\rho Vd}{\mu}\left(\frac{d}{2R}\right)^{1/2}
  40. De = t c t p \mathrm{De}=\frac{t_{\mathrm{c}}}{t_{\mathrm{p}}}
  41. c d = 2 F d ρ v 2 A , c_{\mathrm{d}}=\dfrac{2F_{\mathrm{d}}}{\rho v^{2}A}\,,
  42. Du = κ σ K m a \mathrm{Du}=\frac{\kappa^{\sigma}}{{K_{m}}a}
  43. Ec = V 2 c p Δ T \mathrm{Ec}=\frac{V^{2}}{c_{p}\Delta T}
  44. Ek = ν 2 D 2 Ω sin φ \mathrm{Ek}=\frac{\nu}{2D^{2}\Omega\sin\varphi}
  45. E x , y = ln ( x ) ln ( y ) = x y y x E_{x,y}=\frac{\partial\ln(x)}{\partial\ln(y)}=\frac{\partial x}{\partial y}% \frac{y}{x}
  46. Eo = Δ ρ g L 2 σ \mathrm{Eo}=\frac{\Delta\rho\,g\,L^{2}}{\sigma}
  47. Er = μ v L K \mathrm{Er}=\frac{\mu vL}{K}
  48. Eu = Δ p ρ V 2 \mathrm{Eu}=\frac{\Delta{}p}{\rho V^{2}}
  49. e = n = 0 1 n ! 2.71828 e=\displaystyle\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}\approx 2.71828
  50. Θ r \Theta_{r}
  51. Θ r = c p ( T - T e ) U e 2 / 2 \Theta_{r}=\frac{c_{p}(T-T_{e})}{U_{e}^{2}/2}
  52. α \alpha
  53. δ \delta
  54. α 2.50290 , \alpha\approx 2.50290,
  55. δ 4.66920 \ \delta\approx 4.66920
  56. α \alpha
  57. α = e 2 2 ε 0 h c \alpha=\frac{e^{2}}{2\varepsilon_{0}hc}
  58. f = < m t p l > D f=\frac{<}{m}tpl>{{\ell}}{{D}}
  59. γ \gamma
  60. γ = Y r 2 κ \gamma=\frac{Yr^{2}}{\kappa}
  61. Fo = α t L 2 \mathrm{Fo}=\frac{\alpha t}{L^{2}}
  62. F = a 2 L λ \mathit{F}=\frac{a^{2}}{L\lambda}
  63. Fr = v g \mathrm{Fr}=\frac{v}{\sqrt{g\ell}}
  64. Ga = g L 3 ν 2 \mathrm{Ga}=\frac{g\,L^{3}}{\nu^{2}}
  65. φ \varphi
  66. φ = 1 + 5 2 1.61803 \varphi=\frac{1+\sqrt{5}}{2}\approx 1.61803
  67. G = U e θ ν ( θ R ) 1 / 2 \mathrm{G}=\frac{U_{e}\theta}{\nu}\left(\frac{\theta}{R}\right)^{1/2}
  68. Gz = D H L Re Pr \mathrm{Gz}={D_{H}\over L}\mathrm{Re}\,\mathrm{Pr}
  69. Gr L = g β ( T s - T ) L 3 ν 2 \mathrm{Gr}_{L}=\frac{g\beta(T_{s}-T_{\infty})L^{3}}{\nu^{2}}
  70. α G \alpha_{G}
  71. α G = G m e 2 c \alpha_{G}=\frac{Gm_{e}^{2}}{\hbar c}
  72. Ha = N A0 N A0 phys \mathrm{Ha}=\frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}}
  73. Hg = - 1 ρ d p d x L 3 ν 2 \mathrm{Hg}=-\frac{1}{\rho}\frac{\mathrm{d}p}{\mathrm{d}x}\frac{L^{3}}{\nu^{2}}
  74. i = d h d l = h 2 - h 1 length i=\frac{\mathrm{d}h}{\mathrm{d}l}=\frac{h_{2}-h_{1}}{\mathrm{length}}
  75. Ir = tan α H / L 0 \mathrm{Ir}=\frac{\tan\alpha}{\sqrt{H/L_{0}}}
  76. Ja = c p ( T s - T sat ) Δ H f \mathrm{Ja}=\frac{c_{p}(T_{\mathrm{s}}-T_{\mathrm{sat}})}{\Delta H_{\mathrm{f}}}
  77. Ka = k t c \mathrm{Ka}=kt_{c}
  78. K C = V T L \mathrm{K_{C}}=\frac{V\,T}{L}
  79. Kn = λ L \mathrm{Kn}=\frac{\lambda}{L}
  80. Ku = U h ρ g 1 / 2 ( σ g ( ρ l - ρ g ) ) 1 / 4 \mathrm{Ku}=\frac{U_{h}\rho_{g}^{1/2}}{\left({\sigma g(\rho_{l}-\rho_{g})}% \right)^{1/4}}
  81. La = σ ρ L μ 2 \mathrm{La}=\frac{\sigma\rho L}{\mu^{2}}
  82. Le = α D = Sc Pr \mathrm{Le}=\frac{\alpha}{D}=\frac{\mathrm{Sc}}{\mathrm{Pr}}
  83. C L = L q S C_{\mathrm{L}}=\frac{L}{q\,S}
  84. χ \chi
  85. χ = m m g ρ g ρ \chi=\frac{m_{\ell}}{m_{g}}\sqrt{\frac{\rho_{g}}{\rho_{\ell}}}
  86. S = μ 0 L V A η S=\frac{\mu_{0}LV_{A}}{\eta}
  87. M = < m t p l > v v sound \mathrm{M}=\frac{<}{m}tpl>{{v}}{{v_{\mathrm{sound}}}}
  88. R m = U L η \mathrm{R}_{\mathrm{m}}=\frac{UL}{\eta}
  89. Mg = - d σ d T L Δ T η α \mathrm{Mg}=-{\frac{\mathrm{d}\sigma}{\mathrm{d}T}}\frac{L\Delta T}{\eta\alpha}
  90. Mo = g μ c 4 Δ ρ ρ c 2 σ 3 \mathrm{Mo}=\frac{g\mu_{c}^{4}\,\Delta\rho}{\rho_{c}^{2}\sigma^{3}}
  91. Nu = h d k \mathrm{Nu}=\frac{hd}{k}
  92. Oh = μ ρ σ L = We Re \mathrm{Oh}=\frac{\mu}{\sqrt{\rho\sigma L}}=\frac{\sqrt{\mathrm{We}}}{\mathrm{% Re}}
  93. Pe = d u ρ c p k = Re Pr \mathrm{Pe}=\frac{du\rho c_{p}}{k}=\mathrm{Re}\,\mathrm{Pr}
  94. N P = Restoring force Adhesive force N_{\mathrm{P}}=\frac{\,\text{Restoring force}}{\,\text{Adhesive force}}
  95. K = < m t p l > I I 0 2 β 3 γ 3 ( 1 - γ 2 f e ) {K}=\frac{<}{m}tpl>{{I}}{{I_{0}}}\,\frac{{2}}{{\beta}^{3}{\gamma}^{3}}(1-% \gamma^{2}f_{e})
  96. pH \mathrm{pH}
  97. pH = - log 10 ( a H + ) = log 10 ( 1 a H + ) \mathrm{pH}=-\log_{10}(a_{\textrm{H}^{+}})=\log_{10}\left(\frac{1}{a_{\textrm{% H}^{+}}}\right)
  98. π \pi
  99. π = C d 3.14159 \pi=\frac{C}{d}\approx 3.14159
  100. ν \nu
  101. ν = - d ε trans d ε axial \nu=-\frac{\mathrm{d}\varepsilon_{\mathrm{trans}}}{\mathrm{d}\varepsilon_{% \mathrm{axial}}}
  102. ϕ \phi
  103. ϕ = V V V T \phi=\frac{V_{\mathrm{V}}}{V_{\mathrm{T}}}
  104. N p = P ρ n 3 d 5 N_{p}={P\over\rho n^{3}d^{5}}
  105. Pr = ν α = c p μ k \mathrm{Pr}=\frac{\nu}{\alpha}=\frac{c_{p}\mu}{k}
  106. β = - Δ H r D T A e C A S λ e T s \beta=\frac{-\Delta H_{r}D_{TA}^{e}C_{AS}}{\lambda^{e}T_{s}}
  107. C p = p - p 1 2 ρ V 2 C_{p}={p-p_{\infty}\over\frac{1}{2}\rho_{\infty}V_{\infty}^{2}}
  108. Q = 2 π f r Energy Stored Power Loss Q=2\pi f_{r}\frac{\,\text{Energy Stored}}{\,\text{Power Loss}}
  109. arc length / radius \,\text{arc length}/\,\text{radius}
  110. Ra x = g β ν α ( T s - T ) x 3 \mathrm{Ra}_{x}=\frac{g\beta}{\nu\alpha}(T_{s}-T_{\infty})x^{3}
  111. n = c v n=\frac{c}{v}
  112. R D = ρ substance ρ reference RD=\frac{\rho_{\mathrm{substance}}}{\rho_{\mathrm{reference}}}
  113. μ r \mu_{r}
  114. μ r = μ μ 0 \mu_{r}=\frac{\mu}{\mu_{0}}
  115. ε r \varepsilon_{r}
  116. ε r = C x C 0 \varepsilon_{r}=\frac{C_{x}}{C_{0}}
  117. Re = v L ρ μ \mathrm{Re}=\frac{vL\rho}{\mu}
  118. Ri = g h u 2 = 1 Fr 2 \mathrm{Ri}=\frac{gh}{u^{2}}=\frac{1}{\mathrm{Fr}^{2}}
  119. C r r = F N f C_{rr}=\frac{F}{N_{f}}
  120. Ro = f L 2 ν = St Re \mathrm{Ro}={fL^{2}\over\nu}=\mathrm{St}\,\mathrm{Re}
  121. Ro = U L f \mathrm{Ro}=\frac{U}{Lf}
  122. P = w s κ u * \mathrm{P}=\frac{w_{s}}{\kappa u_{*}}
  123. Sc = ν D \mathrm{Sc}=\frac{\nu}{D}
  124. H = δ * θ H=\frac{\delta^{*}}{\theta}
  125. Sh = K L D \mathrm{Sh}=\frac{KL}{D}
  126. τ * \tau_{*}
  127. θ \theta
  128. τ = τ ( ρ s - ρ ) g D \tau_{\ast}=\frac{\tau}{(\rho_{s}-\rho)gD}
  129. S = ( r c ) 2 μ N P \mathrm{S}=\left(\frac{r}{c}\right)^{2}\frac{\mu N}{P}
  130. St = h c p ρ V = Nu Re Pr \mathrm{St}=\frac{h}{c_{p}\rho V}=\frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}}
  131. Ste = c p Δ T L \mathrm{Ste}=\frac{c_{p}\Delta T}{L}
  132. Stk = τ U o d c \mathrm{Stk}=\frac{\tau U_{o}}{d_{c}}
  133. ϵ \epsilon
  134. ϵ = F X - 1 \epsilon=\cfrac{\partial{F}}{\partial{X}}-1
  135. St = ω L v \mathrm{St}={\omega L\over v}
  136. N = B 2 L c σ ρ U = Ha 2 Re \mathrm{N}=\frac{B^{2}L_{c}\sigma}{\rho U}=\frac{\mathrm{Ha}^{2}}{\mathrm{Re}}
  137. Ta = 4 Ω 2 R 4 ν 2 \mathrm{Ta}=\frac{4\Omega^{2}R^{4}}{\nu^{2}}
  138. T = I I 0 T=\frac{I}{I_{0}}
  139. U = H λ 2 h 3 \mathrm{U}=\frac{H\,\lambda^{2}}{h^{3}}
  140. Va = ϕ Pr Da \mathrm{Va}=\frac{\phi\,\mathrm{Pr}}{\mathrm{Da}}
  141. ϕ \phi
  142. i = 1 + α ( n - 1 ) i=1+\alpha(n-1)
  143. j * = R ( ω ρ μ ) 1 2 j^{*}=R\left(\frac{\omega\rho}{\mu}\right)^{\frac{1}{2}}
  144. Wea = w w H 100 \mathrm{Wea}=\frac{w}{w_{\mathrm{H}}}100
  145. We = ρ v 2 l σ \mathrm{We}=\frac{\rho v^{2}l}{\sigma}
  146. Wi = γ ˙ λ \mathrm{Wi}=\dot{\gamma}\lambda
  147. α \alpha
  148. α = R ( ω ρ μ ) 1 2 \alpha=R\left(\frac{\omega\rho}{\mu}\right)^{\frac{1}{2}}

Diode.html

  1. I = I S ( e V D / ( n V T ) - 1 ) , I=I_{\mathrm{S}}\left(e^{V_{\mathrm{D}}/(nV_{\mathrm{T}})}-1\right),\,
  2. V T = k T q , V_{\mathrm{T}}=\frac{kT}{q}\,,
  3. I = I S e V D / ( n V T ) I=I_{\mathrm{S}}e^{V_{\mathrm{D}}/(nV_{\mathrm{T}})}

Diophantine_equation.html

  1. a x + b y = 1 ax+by=1\,
  2. w 3 + x 3 = y 3 + z 3 w^{3}+x^{3}=y^{3}+z^{3}\,
  3. 12 3 + 1 3 = 9 3 + 10 3 = 1729 12^{3}+1^{3}=9^{3}+10^{3}=1729
  4. x n + y n = z n x^{n}+y^{n}=z^{n}\,
  5. x 2 - n y 2 = ± 1 x^{2}-ny^{2}=\pm 1\,
  6. 4 n = 1 x + 1 y + 1 z \frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}
  7. x 4 + y 4 + z 4 = w 4 x^{4}+y^{4}+z^{4}=w^{4}
  8. a ( x + k v ) + b ( y - k u ) = a x + b y + k ( a v - b u ) = a x + b y + k ( u d v - v d u ) = a x + b y a(x+kv)+b(y-ku)=ax+by+k(av-bu)=ax+by+k(udv-vdu)=ax+by
  9. x \displaystyle x
  10. A X = C , A\,X=C,
  11. A A
  12. X X
  13. C C
  14. A A
  15. U U
  16. V V
  17. B = [ b i , j ] = U A V B=\left[b_{i,j}\right]=UAV
  18. B ( V - 1 X ) = U C . B\,(V^{-1}X)=UC.
  19. V - 1 X V^{-1}X
  20. D = U C , D=UC,
  21. x x
  22. x = V y x=Vy
  23. y y
  24. B y = D By=D
  25. V [ d 1 b 1 , 1 d k b k , k h k + 1 h n ] , V\,\left[\begin{array}[]{c}\frac{d_{1}}{b_{1,1}}\\ \vdots\\ \frac{d_{k}}{b_{k,k}}\\ h_{k+1}\\ \vdots\\ h_{n}\end{array}\right]\,,
  26. N = A 2 + 2 B 2 + 3 C 2 + 4 D 2 + 5 E 2 + , N=A^{2}+2B^{2}+3C^{2}+4D^{2}+5E^{2}+...,
  27. N = A 2 + 4 B 2 + 9 C 2 + 16 D 2 + 25 E 2 + , N=A^{2}+4B^{2}+9C^{2}+16D^{2}+25E^{2}+...,

Diophantus.html

  1. x x
  2. x = x 6 + x 12 + x 7 + 5 + x 2 + 4 x=\frac{x}{6}+\frac{x}{12}+\frac{x}{7}+5+\frac{x}{2}+4
  3. x x
  4. a n + b n = c n a^{n}+b^{n}=c^{n}
  5. a 3 - b 3 = c 3 + d 3 . a^{3}-b^{3}=c^{3}+d^{3}.
  6. a x 2 + b x = c ax^{2}+bx=c
  7. a x 2 = b x + c ax^{2}=bx+c
  8. a x 2 + c = b x ax^{2}+c=bx
  9. a , b , c a,b,c
  10. 4 = 4 x + 20 4=4x+20
  11. ( 12 + 6 n ) / ( n 2 - 3 ) (12+6n)/(n^{2}-3)

Dipole.html

  1. 𝔭 = i = 1 N q i 𝐫 i . \mathfrak{p}=\sum_{i=1}^{N}\,q_{i}\,\mathbf{r}_{i}\,.
  2. 𝔭 - 1 = - 𝔭 , \mathfrak{I}\;\mathfrak{p}\;\mathfrak{I}^{-1}=-\mathfrak{p},
  3. 𝔭 \stackrel{\mathfrak{p}}{}
  4. \stackrel{\mathfrak{I}}{}\,
  5. 𝔭 = S | 𝔭 | S , \langle\mathfrak{p}\rangle=\langle\,S\,|\mathfrak{p}|\,S\,\rangle,
  6. | S |\,S\,\rangle
  7. | S = ± | S \mathfrak{I}\,|\,S\,\rangle=\pm|\,S\,\rangle
  8. 𝔭 = - 1 S | 𝔭 | - 1 S = S | 𝔭 - 1 | S = - 𝔭 \langle\mathfrak{p}\rangle=\langle\,\mathfrak{I}^{-1}\,S\,|\mathfrak{p}|\,% \mathfrak{I}^{-1}\,S\,\rangle=\langle\,S\,|\mathfrak{I}\,\mathfrak{p}\,% \mathfrak{I}^{-1}|\,S\,\rangle=-\langle\mathfrak{p}\rangle
  9. \mathfrak{I}\,
  10. - 1 = * \mathfrak{I}^{-1}=\mathfrak{I}^{*}\,
  11. * \mathfrak{I}^{*}\,
  12. * * = \mathfrak{I}^{**}=\mathfrak{I}\,
  13. 𝔭 = 0. \langle\mathfrak{p}\rangle=0.
  14. B ( m , r , λ ) = μ 0 4 π m r 3 1 + 3 sin 2 λ , B(m,r,\lambda)=\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\sqrt{1+3\sin^{2}\lambda}\,,
  15. λ = arcsin ( z z 2 + ρ 2 ) \lambda=\arcsin\left(\frac{z}{\sqrt{z^{2}+\rho^{2}}}\right)
  16. B ( ρ , z ) = μ 0 m 4 π ( z 2 + ρ 2 ) 3 / 2 1 + 3 z 2 z 2 + ρ 2 B(\rho,z)=\frac{\mu_{0}m}{4\pi(z^{2}+\rho^{2})^{3/2}}\sqrt{1+\frac{3z^{2}}{z^{% 2}+\rho^{2}}}
  17. 𝐁 ( 𝐦 , 𝐫 ) = μ 0 4 π ( 3 ( 𝐦 𝐫 ^ ) 𝐫 ^ - 𝐦 r 3 ) + 2 μ 0 3 𝐦 δ 3 ( 𝐫 ) \mathbf{B}(\mathbf{m},\mathbf{r})=\frac{\mu_{0}}{4\pi}\left(\frac{3(\mathbf{m}% \cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}}{r^{3}}\right)+\frac{2\mu_{0% }}{3}\mathbf{m}\delta^{3}(\mathbf{r})
  18. 𝐫 ^ = 𝐫 / r \hat{\mathbf{r}}=\mathbf{r}/r
  19. 𝐀 ( 𝐫 ) = μ 0 4 π 𝐦 × 𝐫 ^ r 2 \mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\frac{\mathbf{m}\times\hat{\mathbf{% r}}}{r^{2}}
  20. Φ ( 𝐫 ) = 1 4 π ε 0 𝐩 𝐫 ^ r 2 \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon_{0}}\,\frac{\mathbf{p}\cdot\hat{% \mathbf{r}}}{r^{2}}
  21. 𝐫 ^ \hat{\mathbf{r}}
  22. 𝐄 = - Φ = 1 4 π ϵ 0 ( 3 ( 𝐩 𝐫 ^ ) 𝐫 ^ - 𝐩 r 3 ) - 1 3 ϵ 0 𝐩 δ 3 ( 𝐫 ) \mathbf{E}=-\nabla\Phi=\frac{1}{4\pi\epsilon_{0}}\left(\frac{3(\mathbf{p}\cdot% \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{r^{3}}\right)-\frac{1}{3\epsilon% _{0}}\mathbf{p}\delta^{3}(\mathbf{r})
  23. s y m b o l τ = 𝐩 × 𝐄 symbol{\tau}=\mathbf{p}\times\mathbf{E}
  24. s y m b o l τ = 𝐦 × 𝐁 symbol{\tau}=\mathbf{m}\times\mathbf{B}
  25. U = - 𝐩 𝐄 U=-\mathbf{p}\cdot\mathbf{E}
  26. U = - 𝐦 𝐁 U=-\mathbf{m}\cdot\mathbf{B}
  27. p 0 p_{0}
  28. z ^ \hat{z}
  29. 𝐩 ( 𝐫 , t ) = 𝐩 ( 𝐫 ) e - i ω t = p 0 𝐳 ^ e - i ω t . \mathbf{p}(\mathbf{r},t)=\mathbf{p}(\mathbf{r})e^{-i\omega t}=p_{0}\hat{% \mathbf{z}}e^{-i\omega t}.
  30. 𝐄 = 1 4 π ε 0 { ω 2 c 2 r ( 𝐫 ^ × 𝐩 ) × 𝐫 ^ + ( 1 r 3 - i ω c r 2 ) [ 3 𝐫 ^ ( 𝐫 ^ 𝐩 ) - 𝐩 ] } e i ω r / c e - i ω t \mathbf{E}=\frac{1}{4\pi\varepsilon_{0}}\left\{\frac{\omega^{2}}{c^{2}r}(\hat{% \mathbf{r}}\times\mathbf{p})\times\hat{\mathbf{r}}+\left(\frac{1}{r^{3}}-\frac% {i\omega}{cr^{2}}\right)\left[3\hat{\mathbf{r}}(\hat{\mathbf{r}}\cdot\mathbf{p% })-\mathbf{p}\right]\right\}e^{i\omega r/c}e^{-i\omega t}
  31. 𝐁 = ω 2 4 π ε 0 c 3 𝐫 ^ × 𝐩 ( 1 - c i ω r ) e i ω r / c r e - i ω t . \mathbf{B}=\frac{\omega^{2}}{4\pi\varepsilon_{0}c^{3}}\hat{\mathbf{r}}\times% \mathbf{p}\left(1-\frac{c}{i\omega r}\right)\frac{e^{i\omega r/c}}{r}e^{-i% \omega t}.
  32. r ω / c 1 \scriptstyle r\omega/c\gg 1
  33. 𝐁 = ω 2 4 π ε 0 c 3 ( 𝐫 ^ × 𝐩 ) e i ω ( r / c - t ) r = ω 2 μ 0 p 0 4 π c ( 𝐫 ^ × 𝐳 ^ ) e i ω ( r / c - t ) r = - ω 2 μ 0 p 0 4 π c sin θ e i ω ( r / c - t ) r ϕ ^ \mathbf{B}=\frac{\omega^{2}}{4\pi\varepsilon_{0}c^{3}}(\hat{\mathbf{r}}\times% \mathbf{p})\frac{e^{i\omega(r/c-t)}}{r}=\frac{\omega^{2}\mu_{0}p_{0}}{4\pi c}(% \hat{\mathbf{r}}\times\hat{\mathbf{z}})\frac{e^{i\omega(r/c-t)}}{r}=-\frac{% \omega^{2}\mu_{0}p_{0}}{4\pi c}\sin\theta\frac{e^{i\omega(r/c-t)}}{r}\mathbf{% \hat{\phi}}
  34. 𝐄 = c 𝐁 × 𝐫 ^ = - ω 2 μ 0 p 0 4 π sin θ ( ϕ ^ × 𝐫 ^ ) e i ω ( r / c - t ) r = - ω 2 μ 0 p 0 4 π sin θ e i ω ( r / c - t ) r θ ^ . \mathbf{E}=c\mathbf{B}\times\hat{\mathbf{r}}=-\frac{\omega^{2}\mu_{0}p_{0}}{4% \pi}\sin\theta(\hat{\phi}\times\mathbf{\hat{r}})\frac{e^{i\omega(r/c-t)}}{r}=-% \frac{\omega^{2}\mu_{0}p_{0}}{4\pi}\sin\theta\frac{e^{i\omega(r/c-t)}}{r}\hat{% \theta}.
  35. 𝐒 = ( μ 0 p 0 2 ω 4 32 π 2 c ) sin 2 θ r 2 𝐫 ^ \langle\mathbf{S}\rangle=\bigg(\frac{\mu_{0}p_{0}^{2}\omega^{4}}{32\pi^{2}c}% \bigg)\frac{\sin^{2}\theta}{r^{2}}\mathbf{\hat{r}}
  36. sin θ \sin\theta
  37. l = 1 l=1
  38. P = μ 0 ω 4 p 0 2 12 π c . P=\frac{\mu_{0}\omega^{4}p_{0}^{2}}{12\pi c}.

Dirac_delta_function.html

  1. δ δ
  2. f ( x ) = 1 2 π - d α f ( α ) - d p cos ( p x - p α ) , f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\ \ d\alpha f(\alpha)\ \int_{-\infty% }^{\infty}dp\ \cos(px-p\alpha)\ ,
  3. δ ( x - α ) = 1 2 π - d p cos ( p x - p α ) . \delta(x-\alpha)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dp\ \cos(px-p\alpha)\ .
  4. f ( x ) = 1 2 π - e i p x ( - e - i p α f ( α ) d α ) d p . f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\ e^{ipx}\left(\int_{-\infty}^{% \infty}e^{-ip\alpha}f(\alpha)\ d\alpha\right)\ dp.
  5. f ( x ) \displaystyle f(x)
  6. δ ( x - α ) = 1 2 π - e i p ( x - α ) d p . \delta(x-\alpha)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ip(x-\alpha)}\ dp\ .
  7. δ ( x ) = { + , x = 0 0 , x 0 \delta(x)=\begin{cases}+\infty,&x=0\\ 0,&x\neq 0\end{cases}
  8. - δ ( x ) d x = 1. \int_{-\infty}^{\infty}\delta(x)\,dx=1.
  9. - f ( x ) δ { d x } = f ( 0 ) \int_{-\infty}^{\infty}f(x)\,\delta\{dx\}=f(0)
  10. - f ( x ) δ ( x ) d x = f ( 0 ) \int_{-\infty}^{\infty}f(x)\delta(x)\,dx=f(0)
  11. H ( x ) = { 1 if x 0 0 if x < 0. H(x)=\begin{cases}1&\,\text{if }x\geq 0\\ 0&\,\text{if }x<0.\end{cases}
  12. H ( x ) = 𝐑 𝟏 ( - , x ] ( t ) δ { d t } = δ ( - , x ] . H(x)=\int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta\{dt\}=\delta(-\infty% ,x].
  13. - f ( x ) δ { d x } = - f ( x ) d H ( x ) . \int_{-\infty}^{\infty}f(x)\delta\{dx\}=\int_{-\infty}^{\infty}f(x)\,dH(x).
  14. | S [ ϕ ] | C N k = 0 M N sup x [ - N , N ] | ϕ ( k ) ( x ) | . |S[\phi]|\leq C_{N}\sum_{k=0}^{M_{N}}\sup_{x\in[-N,N]}|\phi^{(k)}(x)|.
  15. δ [ ϕ ] = - - ϕ ( x ) H ( x ) d x . \delta[\phi]=-\int_{-\infty}^{\infty}\phi^{\prime}(x)H(x)\,dx.
  16. - ϕ ( x ) H ( x ) d x = - ϕ ( x ) δ ( x ) d x , \int_{-\infty}^{\infty}\phi(x)H^{\prime}(x)\,dx=\int_{-\infty}^{\infty}\phi(x)% \delta(x)\,dx,
  17. - - ϕ ( x ) H ( x ) d x = - ϕ ( x ) d H ( x ) . -\int_{-\infty}^{\infty}\phi^{\prime}(x)H(x)\,dx=\int_{-\infty}^{\infty}\phi(x% )\,dH(x).
  18. 𝐑 n f ( 𝐱 ) δ { d 𝐱 } = f ( 𝟎 ) \int_{\mathbf{R}^{n}}f(\mathbf{x})\delta\{d\mathbf{x}\}=f(\mathbf{0})
  19. δ ( 𝐱 ) = δ ( x 1 ) δ ( x 2 ) δ ( x n ) . \delta(\mathbf{x})=\delta(x_{1})\delta(x_{2})\dots\delta(x_{n}).
  20. δ x 0 ( A ) = { 1 if x 0 A 0 if x 0 A \delta_{x_{0}}(A)=\begin{cases}1&\,\text{if }x_{0}\in A\\ 0&\,\text{if }x_{0}\notin A\end{cases}
  21. δ x 0 [ ϕ ] = ϕ ( x 0 ) \delta_{x_{0}}[\phi]=\phi(x_{0})
  22. x 0 δ x 0 x_{0}\mapsto\delta_{x_{0}}
  23. - δ ( α x ) d x = - δ ( u ) d u | α | = 1 | α | \int_{-\infty}^{\infty}\delta(\alpha x)\,dx=\int_{-\infty}^{\infty}\delta(u)\,% \frac{du}{|\alpha|}=\frac{1}{|\alpha|}
  24. δ ( α x ) = δ ( x ) | α | . \delta(\alpha x)=\frac{\delta(x)}{|\alpha|}.
  25. δ ( - x ) = δ ( x ) \delta(-x)=\delta(x)
  26. x δ ( x ) = 0. x\delta(x)=0.
  27. f ( x ) = g ( x ) + c δ ( x ) f(x)=g(x)+c\delta(x)
  28. - f ( t ) δ ( t - T ) d t = f ( T ) . \int_{-\infty}^{\infty}f(t)\delta(t-T)\,dt=f(T).
  29. ( f ( t ) * δ ( t - T ) ) (f(t)*\delta(t-T))\,
  30. = def - f ( τ ) δ ( t - T - τ ) d τ \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}f(\tau)\delta(t-T-\tau)\,d\tau
  31. = - f ( τ ) δ ( τ - ( t - T ) ) d τ =\int\limits_{-\infty}^{\infty}f(\tau)\delta(\tau-(t-T))\,d\tau
  32. δ ( - x ) = δ ( x ) \delta(-x)=\delta(x)
  33. = f ( t - T ) . =f(t-T).\,
  34. - δ ( ξ - x ) δ ( x - η ) d x = δ ( ξ - η ) . \int_{-\infty}^{\infty}\delta(\xi-x)\delta(x-\eta)\,dx=\delta(\xi-\eta).
  35. 𝐑 δ ( g ( x ) ) f ( g ( x ) ) | g ( x ) | d x = g ( 𝐑 ) δ ( u ) f ( u ) d u \int_{\mathbf{R}}\delta\bigl(g(x)\bigr)f\bigl(g(x)\bigr)|g^{\prime}(x)|\,dx=% \int_{g(\mathbf{R})}\delta(u)f(u)\,du
  36. δ g \delta\circ g
  37. δ ( g ( x ) ) = δ ( x - x 0 ) | g ( x 0 ) | . \delta(g(x))=\frac{\delta(x-x_{0})}{|g^{\prime}(x_{0})|}.
  38. δ ( g ( x ) ) = i δ ( x - x i ) | g ( x i ) | \delta(g(x))=\sum_{i}\frac{\delta(x-x_{i})}{|g^{\prime}(x_{i})|}
  39. δ ( x 2 - α 2 ) = 1 2 | α | [ δ ( x + α ) + δ ( x - α ) ] . \delta\left(x^{2}-\alpha^{2}\right)=\frac{1}{2|\alpha|}\Big[\delta\left(x+% \alpha\right)+\delta\left(x-\alpha\right)\Big].
  40. - f ( x ) δ ( g ( x ) ) d x = i f ( x i ) | g ( x i ) | . \int_{-\infty}^{\infty}f(x)\,\delta(g(x))\,dx=\sum_{i}\frac{f(x_{i})}{|g^{% \prime}(x_{i})|}.
  41. δ ( α 𝐱 ) = | α | - n δ ( 𝐱 ) \delta(\alpha\mathbf{x})=|\alpha|^{-n}\delta(\mathbf{x})
  42. δ ( ρ 𝐱 ) = δ ( 𝐱 ) . \delta(\rho\mathbf{x})=\delta(\mathbf{x}).
  43. 𝐑 n δ ( g ( 𝐱 ) ) f ( g ( 𝐱 ) ) | det g ( 𝐱 ) | d 𝐱 = g ( 𝐑 n ) δ ( 𝐮 ) f ( 𝐮 ) d 𝐮 \int_{\mathbf{R}^{n}}\delta(g(\mathbf{x}))\,f(g(\mathbf{x}))\,|\det g^{\prime}% (\mathbf{x})|\,d\mathbf{x}=\int_{g(\mathbf{R}^{n})}\delta(\mathbf{u})f(\mathbf% {u})\,d\mathbf{u}
  44. 𝐑 n f ( 𝐱 ) δ ( g ( 𝐱 ) ) d 𝐱 = g - 1 ( 0 ) f ( 𝐱 ) | g | d σ ( 𝐱 ) \int_{\mathbf{R}^{n}}f(\mathbf{x})\,\delta(g(\mathbf{x}))\,d\mathbf{x}=\int_{g% ^{-1}(0)}\frac{f(\mathbf{x})}{|\mathbf{\nabla}g|}\,d\sigma(\mathbf{x})
  45. δ S [ g ] = S g ( 𝐬 ) d σ ( 𝐬 ) \delta_{S}[g]=\int_{S}g(\mathbf{s})\,d\sigma(\mathbf{s})
  46. - 𝐑 n g ( 𝐱 ) 1 D ( 𝐱 ) n d 𝐱 = S g ( 𝐬 ) d σ ( 𝐬 ) , -\int_{\mathbf{R}^{n}}g(\mathbf{x})\,\frac{\partial 1_{D}(\mathbf{x})}{% \partial n}\;d\mathbf{x}=\int_{S}\,g(\mathbf{s})\;d\sigma(\mathbf{s}),
  47. δ ^ ( ξ ) = - e - 2 π i x ξ δ ( x ) d x = 1. \hat{\delta}(\xi)=\int_{-\infty}^{\infty}e^{-2\pi ix\xi}\delta(x)\,dx=1.
  48. , \langle\cdot,\cdot\rangle
  49. δ ^ \hat{\delta}
  50. δ ^ , ϕ = δ , ϕ ^ \langle\hat{\delta},\phi\rangle=\langle\delta,\hat{\phi}\rangle
  51. δ ^ = 1. \hat{\delta}=1.
  52. S * δ = S . S*\delta=S.\,
  53. - 1 e 2 π i x ξ d ξ = δ ( x ) \int_{-\infty}^{\infty}1\cdot e^{2\pi ix\xi}\,d\xi=\delta(x)
  54. 1 , f = f ( 0 ) = δ , f \langle 1,f^{\vee}\rangle=f(0)=\langle\delta,f\rangle
  55. - e i 2 π ξ 1 t [ e i 2 π ξ 2 t ] * d t = - e - i 2 π ( ξ 2 - ξ 1 ) t d t = δ ( ξ 2 - ξ 1 ) . \int_{-\infty}^{\infty}e^{i2\pi\xi_{1}t}\left[e^{i2\pi\xi_{2}t}\right]^{*}\,dt% =\int_{-\infty}^{\infty}e^{-i2\pi(\xi_{2}-\xi_{1})t}\,dt=\delta(\xi_{2}-\xi_{1% }).
  56. f ( t ) = e i 2 π ξ 1 t f(t)=e^{i2\pi\xi_{1}t}
  57. f ^ ( ξ 2 ) = δ ( ξ 1 - ξ 2 ) \hat{f}(\xi_{2})=\delta(\xi_{1}-\xi_{2})
  58. 0 δ ( t - a ) e - s t d t = e - s a . \int_{0}^{\infty}\delta(t-a)e^{-st}\,dt=e^{-sa}.
  59. δ [ φ ] = - δ [ φ ] = - φ ( 0 ) . \delta^{\prime}[\varphi]=-\delta[\varphi^{\prime}]=-\varphi^{\prime}(0).
  60. - δ ( x ) φ ( x ) d x = - - δ ( x ) φ ( x ) d x . \int_{-\infty}^{\infty}\delta^{\prime}(x)\varphi(x)\,dx=-\int_{-\infty}^{% \infty}\delta(x)\varphi^{\prime}(x)\,dx.
  61. δ ( k ) [ φ ] = ( - 1 ) k φ ( k ) ( 0 ) . \delta^{(k)}[\varphi]=(-1)^{k}\varphi^{(k)}(0).
  62. δ ( x ) = lim h 0 δ ( x + h ) - δ ( x ) h . \delta^{\prime}(x)=\lim_{h\to 0}\frac{\delta(x+h)-\delta(x)}{h}.
  63. δ = lim h 0 1 h ( τ h δ - δ ) \delta^{\prime}=\lim_{h\to 0}\frac{1}{h}(\tau_{h}\delta-\delta)
  64. ( τ h S ) [ φ ] = S [ τ - h φ ] . (\tau_{h}S)[\varphi]=S[\tau_{-h}\varphi].
  65. d d x δ ( - x ) = d d x δ ( x ) \frac{d}{dx}\delta(-x)=\frac{d}{dx}\delta(x)
  66. δ ( - x ) = - δ ( x ) \delta^{\prime}(-x)=-\delta^{\prime}(x)
  67. x δ ( x ) = - δ ( x ) . x\delta^{\prime}(x)=-\delta(x).
  68. δ * f = δ * f = f , \delta^{\prime}*f=\delta*f^{\prime}=f^{\prime},
  69. δ a [ ϕ ] = ϕ ( a ) \delta_{a}[\phi]=\phi(a)
  70. α δ a , φ = ( - 1 ) | α | δ a , α φ = ( - 1 ) | α | α φ ( x ) | x = a for all φ S ( U ) . \left\langle\partial^{\alpha}\delta_{a},\varphi\right\rangle=(-1)^{|\alpha|}% \left\langle\delta_{a},\partial^{\alpha}\varphi\right\rangle=\left.(-1)^{|% \alpha|}\partial^{\alpha}\varphi(x)\right|_{x=a}\mbox{ for all }~{}\varphi\in S% (U).
  71. S = | α | m c α α δ a . S=\sum_{|\alpha|\leq m}c_{\alpha}\partial^{\alpha}\delta_{a}.
  72. δ ( x ) = lim ε 0 + η ε ( x ) , \delta(x)=\lim_{\varepsilon\to 0^{+}}\eta_{\varepsilon}(x),\,
  73. lim ε 0 + - η ε ( x ) f ( x ) d x = f ( 0 ) \lim_{\varepsilon\to 0^{+}}\int_{-\infty}^{\infty}\eta_{\varepsilon}(x)f(x)\,% dx=f(0)
  74. η ε ( x ) = ε - 1 η ( x ε ) . \eta_{\varepsilon}(x)=\varepsilon^{-1}\eta\left(\frac{x}{\varepsilon}\right).
  75. η ε ( x ) = ε - n η ( x ε ) . \eta_{\varepsilon}(x)=\varepsilon^{-n}\eta\left(\frac{x}{\varepsilon}\right).
  76. f * η ε f as ε 0. f*\eta_{\varepsilon}\to f\quad\rm{as\ }\varepsilon\to 0.
  77. η ( x ) = { e - 1 1 - | x | 2 if | x | < 1 0 if | x | 1. \eta(x)=\begin{cases}e^{-\frac{1}{1-|x|^{2}}}&\,\text{ if }|x|<1\\ 0&\,\text{ if }|x|\geq 1.\end{cases}
  78. η ε ( x ) = ε - 1 max ( 1 - | x ε | , 0 ) \eta_{\varepsilon}(x)=\varepsilon^{-1}\max\left(1-|\frac{x}{\varepsilon}|,0\right)
  79. η ε ( x ) = 1 ε rect ( x ε ) = { 1 ε , - ε 2 < x < ε 2 0 , otherwise . \eta_{\varepsilon}(x)=\frac{1}{\varepsilon}\ \textrm{rect}\left(\frac{x}{% \varepsilon}\right)=\begin{cases}\frac{1}{\varepsilon},&-\frac{\varepsilon}{2}% <x<\frac{\varepsilon}{2}\\ 0,&\,\text{otherwise}.\end{cases}
  80. η ε ( x ) = { 2 π ε 2 ε 2 - x 2 , - ε < x < ε 0 , otherwise \eta_{\varepsilon}(x)=\begin{cases}\frac{2}{\pi\varepsilon^{2}}\sqrt{% \varepsilon^{2}-x^{2}},&-\varepsilon<x<\varepsilon\\ 0,&\,\text{otherwise}\end{cases}
  81. η ε * η δ = η ε + δ \eta_{\varepsilon}*\eta_{\delta}=\eta_{\varepsilon+\delta}
  82. { t η ( t , x ) = A η ( t , x ) , t > 0 lim t 0 + η ( t , x ) = δ ( x ) \begin{cases}\frac{\partial}{\partial t}\eta(t,x)=A\eta(t,x),\quad t>0\\ \displaystyle\lim_{t\to 0^{+}}\eta(t,x)=\delta(x)\end{cases}
  83. η ε ( x ) = 1 2 π ε e - x 2 2 ε \eta_{\varepsilon}(x)=\frac{1}{\sqrt{2\pi\varepsilon}}\mathrm{e}^{-\frac{x^{2}% }{2\varepsilon}}
  84. u t = 1 2 2 u x 2 . \frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^{2}u}{\partial x^{2}}.
  85. η ε = 1 ( 2 π ε ) n / 2 e - x x 2 ε , \eta_{\varepsilon}=\frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x% }{2\varepsilon}},
  86. η ε ( x ) = 1 π ε ε 2 + x 2 = - e 2 π i ξ x - | ε ξ | d ξ \eta_{\varepsilon}(x)=\frac{1}{\pi}\frac{\varepsilon}{\varepsilon^{2}+x^{2}}=% \int_{-\infty}^{\infty}\mathrm{e}^{2\pi\mathrm{i}\xi x-|\varepsilon\xi|}\;d\xi
  87. u t = - ( - 2 x 2 ) 1 2 u ( t , x ) \frac{\partial u}{\partial t}=-\left(-\frac{\partial^{2}}{\partial x^{2}}% \right)^{\frac{1}{2}}u(t,x)
  88. [ ( - 2 x 2 ) 1 2 f ] ( ξ ) = | 2 π ξ | f ( ξ ) . \mathcal{F}\left[\left(-\frac{\partial^{2}}{\partial x^{2}}\right)^{\frac{1}{2% }}f\right](\xi)=|2\pi\xi|\mathcal{F}f(\xi).
  89. ε - 1 3 Ai ( x ε - 1 3 ) . \varepsilon^{-\frac{1}{3}}\operatorname{Ai}\left(x\varepsilon^{-\frac{1}{3}}% \right).
  90. c - 2 2 u t 2 - Δ u = 0 u = 0 , u t = δ for t = 0. \begin{aligned}\displaystyle c^{-2}\frac{\partial^{2}u}{\partial t^{2}}-\Delta u% &\displaystyle=0\\ \displaystyle u=0,\quad\frac{\partial u}{\partial t}=\delta&\displaystyle% \qquad\,\text{for }t=0.\end{aligned}
  91. η ε ( x ) = 1 π x sin ( x ε ) = 1 2 π - 1 ε 1 ε cos ( k x ) d k \eta_{\varepsilon}(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=% \frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}}\cos(kx)\;dk
  92. η ε ( x ) = 1 ε J 1 ε ( x + 1 ε ) . \eta_{\varepsilon}(x)=\frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}}\left(% \frac{x+1}{\varepsilon}\right).
  93. L [ u ] = f , L[u]=f,\,
  94. L [ u ] = δ . L[u]=\delta.\,
  95. L [ u ] = h L[u]=h\,
  96. h = h ( x ξ ) h=h(x\cdot\xi)
  97. g ( s ) = Re [ - s k log ( - i s ) k ! ( 2 π i ) n ] = { | s | k 4 k ! ( 2 π i ) n - 1 n odd - | s | k log | s | k ! ( 2 π i ) n n even. g(s)=\operatorname{Re}\left[\frac{-s^{k}\log(-is)}{k!(2\pi i)^{n}}\right]=% \begin{cases}\frac{|s|^{k}}{4k!(2\pi i)^{n-1}}&n\,\text{ odd}\\ &\\ -\frac{|s|^{k}\log|s|}{k!(2\pi i)^{n}}&n\,\text{ even.}\end{cases}
  98. δ ( x ) = Δ x n + k 2 S n - 1 g ( x ξ ) d ω ξ . \delta(x)=\Delta_{x}^{\frac{n+k}{2}}\int_{S^{n-1}}g(x\cdot\xi)\,d\omega_{\xi}.
  99. φ ( x ) = 𝐑 n φ ( y ) d y Δ x n + k 2 S n - 1 g ( ( x - y ) ξ ) d ω ξ . \varphi(x)=\int_{\mathbf{R}^{n}}\varphi(y)\,dy\,\Delta_{x}^{\frac{n+k}{2}}\int% _{S^{n-1}}g((x-y)\cdot\xi)\,d\omega_{\xi}.
  100. c n Δ x n + 1 2 S n - 1 φ ( y ) | ( y - x ) ξ | d ω ξ d y = c n Δ x n + 1 2 S n - 1 d ω ξ - | p | R φ ( ξ , p + x ξ ) d p c_{n}\Delta^{\frac{n+1}{2}}_{x}\int\int_{S^{n-1}}\varphi(y)|(y-x)\cdot\xi|\,d% \omega_{\xi}\,dy=c_{n}\Delta^{\frac{n+1}{2}}_{x}\int_{S^{n-1}}\,d\omega_{\xi}% \int_{-\infty}^{\infty}|p|R\varphi(\xi,p+x\cdot\xi)\,dp
  101. R φ ( ξ , p ) = x ξ = p f ( x ) d n - 1 x . R\varphi(\xi,p)=\int_{x\cdot\xi=p}f(x)\,d^{n-1}x.
  102. δ ( x ) = ( n - 1 ) ! ( 2 π i ) n S n - 1 ( x ξ ) - n d ω ξ \delta(x)=\frac{(n-1)!}{(2\pi i)^{n}}\int_{S^{n-1}}(x\cdot\xi)^{-n}\,d\omega_{\xi}
  103. δ ( x ) = 1 2 ( 2 π i ) n - 1 S n - 1 δ ( n - 1 ) ( x ξ ) d ω ξ \delta(x)=\frac{1}{2(2\pi i)^{n-1}}\int_{S^{n-1}}\delta^{(n-1)}(x\cdot\xi)\,d% \omega_{\xi}
  104. D N ( x ) = n = - N N e i n x = sin ( ( N + 1 2 ) x ) sin ( x / 2 ) . D_{N}(x)=\sum_{n=-N}^{N}e^{inx}=\frac{\sin\left((N+\tfrac{1}{2})x\right)}{\sin% (x/2)}.
  105. s N ( f ) ( x ) = D N * f ( x ) = n = - N N a n e i n x s_{N}(f)(x)=D_{N}*f(x)=\sum_{n=-N}^{N}a_{n}e^{inx}
  106. a n = 1 2 π - π π f ( y ) e - i n y d y . a_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(y)e^{-iny}\,dy.
  107. s N ( f ) ( 0 ) = 𝐑 D N ( x ) f ( x ) d x 2 π f ( 0 ) s_{N}(f)(0)=\int_{\mathbf{R}}D_{N}(x)f(x)\,dx\to 2\pi f(0)
  108. δ ( x ) = 1 2 π n = - e i n x \delta(x)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}e^{inx}
  109. F N ( x ) = 1 N n = 0 N - 1 D n ( x ) = 1 N ( sin N x 2 sin x 2 ) 2 . F_{N}(x)=\frac{1}{N}\sum_{n=0}^{N-1}D_{n}(x)=\frac{1}{N}\left(\frac{\sin\frac{% Nx}{2}}{\sin\frac{x}{2}}\right)^{2}.
  110. 𝐑 F N ( x ) f ( x ) d x 2 π f ( 0 ) \int_{\mathbf{R}}F_{N}(x)f(x)\,dx\to 2\pi f(0)
  111. f H 1 2 = - | f ^ ( ξ ) | 2 ( 1 + | ξ | 2 ) d ξ < \|f\|_{H^{1}}^{2}=\int_{-\infty}^{\infty}|\hat{f}(\xi)|^{2}(1+|\xi|^{2})\,d\xi<\infty
  112. δ [ f ] = | f ( 0 ) | < C f | H 1 . \delta[f]=|f(0)|<C\|f\|_{H^{1}}.
  113. f ( z ) = 1 2 π i D f ( ζ ) d ζ ζ - z , z D f(z)=\frac{1}{2\pi i}\oint_{\partial D}\frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D
  114. δ z [ f ] = f ( z ) = 1 2 π i D f ( ζ ) d ζ ζ - z . \delta_{z}[f]=f(z)=\frac{1}{2\pi i}\oint_{\partial D}\frac{f(\zeta)\,d\zeta}{% \zeta-z}.
  115. f = n = 1 α n φ n . f=\sum_{n=1}^{\infty}\alpha_{n}\varphi_{n}.
  116. α n = φ n , f , \alpha_{n}=\langle\varphi_{n},f\rangle,
  117. α n = φ n f , \alpha_{n}=\varphi_{n}^{\dagger}f,
  118. f = n = 1 φ n ( φ n f ) . f=\sum_{n=1}^{\infty}\varphi_{n}\left(\varphi_{n}^{\dagger}f\right).
  119. I = n = 1 φ n φ n , I=\sum_{n=1}^{\infty}\varphi_{n}\varphi_{n}^{\dagger},
  120. φ n φ n , \varphi_{n}\varphi_{n}^{\dagger},
  121. f ( x ) = n = 1 D ( φ n ( x ) φ n * ( ξ ) ) f ( ξ ) d ξ . f(x)=\sum_{n=1}^{\infty}\int_{D}\,\left(\varphi_{n}(x)\varphi_{n}^{*}(\xi)% \right)f(\xi)\,d\xi.
  122. f ( x ) = δ ( x - ξ ) f ( ξ ) d ξ , f(x)=\int\,\delta(x-\xi)f(\xi)\,d\xi,
  123. δ ( x - ξ ) = n = 1 φ n ( x ) φ n * ( ξ ) . \delta(x-\xi)=\sum_{n=1}^{\infty}\varphi_{n}(x)\varphi_{n}^{*}(\xi).
  124. F ( x ) δ α ( x ) = F ( 0 ) \int F(x)\delta_{\alpha}(x)=F(0)
  125. F ( x ) δ α ( x ) = F ( 0 ) \int F(x)\delta_{\alpha}(x)=F(0)
  126. Δ ( x ) = n = - δ ( x - n ) , \Delta(x)=\sum_{n=-\infty}^{\infty}\delta(x-n),
  127. ( f * Δ ) ( x ) = n = - f ( x - n ) . (f*\Delta)(x)=\sum_{n=-\infty}^{\infty}f(x-n).
  128. ( f * Δ ) = f ^ Δ ^ = f ^ Δ (f*\Delta)^{\wedge}=\hat{f}\widehat{\Delta}=\hat{f}\Delta
  129. p . v . 1 x , ϕ = lim ε 0 + | x | > ε ϕ ( x ) x d x . \left\langle\operatorname{p.v.}\frac{1}{x},\phi\right\rangle=\lim_{\varepsilon% \to 0^{+}}\int_{|x|>\varepsilon}\frac{\phi(x)}{x}\,dx.
  130. lim ε 0 + 1 x ± i ε = p . v . 1 x i π δ ( x ) , \lim_{\varepsilon\to 0^{+}}\frac{1}{x\pm i\varepsilon}=\operatorname{p.v.}% \frac{1}{x}\mp i\pi\delta(x),
  131. lim ε 0 + - f ( x ) x ± i ε d x = i π f ( 0 ) + lim ε 0 + | x | > ε f ( x ) x d x . \lim_{\varepsilon\to 0^{+}}\int_{-\infty}^{\infty}\frac{f(x)}{x\pm i% \varepsilon}\,dx=\mp i\pi f(0)+\lim_{\varepsilon\to 0^{+}}\int_{|x|>% \varepsilon}\frac{f(x)}{x}\,dx.
  132. δ i j = { 1 i = j 0 i j \delta_{ij}=\begin{cases}1&i=j\\ 0&i\not=j\end{cases}
  133. ( a i ) i 𝐙 (a_{i})_{i\in\mathbf{Z}}
  134. i = - a i δ i k = a k . \sum_{i=-\infty}^{\infty}a_{i}\delta_{ik}=a_{k}.
  135. - f ( x ) δ ( x - x 0 ) d x = f ( x 0 ) . \int_{-\infty}^{\infty}f(x)\delta(x-x_{0})\,dx=f(x_{0}).
  136. f ( x ) = i = 1 n p i δ ( x - x i ) . f(x)=\sum_{i=1}^{n}p_{i}\delta(x-x_{i}).
  137. f ( x ) = 0.6 1 2 π e - x 2 2 + 0.4 δ ( x - 3.5 ) . f(x)=0.6\,\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}+0.4\,\delta(x-3.5).
  138. ( x , t ) = 0 t δ ( x - B ( s ) ) d s \ell(x,t)=\int_{0}^{t}\delta(x-B(s))\,ds
  139. ( x , t ) = lim ε 0 + 1 2 ε 0 t 𝟏 [ x - ε , x + ε ] ( B ( s ) ) d s \ell(x,t)=\lim_{\varepsilon\to 0^{+}}\frac{1}{2\varepsilon}\int_{0}^{t}\mathbf% {1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds
  140. ϕ n | ϕ m = δ n m \langle\phi_{n}|\phi_{m}\rangle=\delta_{nm}
  141. ψ = c n ϕ n , \psi=\sum c_{n}\phi_{n},
  142. c n = ϕ n | ψ c_{n}=\langle\phi_{n}|\psi\rangle
  143. I = | ϕ n ϕ n | . I=\sum|\phi_{n}\rangle\langle\phi_{n}|.
  144. ϕ y ( x ) = δ ( x - y ) . \phi_{y}(x)=\delta(x-y).\;
  145. ϕ y = | y \phi_{y}=|y\rangle
  146. P ϕ y = y ϕ y . P\phi_{y}=y\phi_{y}.\;
  147. ϕ y , ϕ y = δ ( y - y ) \langle\phi_{y},\phi_{y^{\prime}}\rangle=\delta(y-y^{\prime})
  148. ψ ( x ) = Ω c ( y ) ϕ y ( x ) d y \psi(x)=\int_{\Omega}c(y)\phi_{y}(x)\,dy
  149. c ( y ) = ψ , ϕ y . c(y)=\langle\psi,\phi_{y}\rangle.
  150. I = Ω | ϕ y ϕ y | d y I=\int_{\Omega}|\phi_{y}\rangle\,\langle\phi_{y}|\,dy
  151. m d 2 ξ d t 2 + k ξ = I δ ( t ) , m\frac{\mathrm{d}^{2}\xi}{\mathrm{d}t^{2}}+k\xi=I\delta(t),
  152. E I d 4 w d x 4 = q ( x ) , EI\frac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}=q(x),\,
  153. q ( x ) = F δ ( x - x 0 ) . q(x)=F\delta(x-x_{0}).\,
  154. q ( x ) = lim d 0 ( F δ ( x ) - F δ ( x - d ) ) = lim d 0 ( M d δ ( x ) - M d δ ( x - d ) ) = M lim d 0 δ ( x ) - δ ( x - d ) d = M δ ( x ) . \begin{aligned}\displaystyle q(x)&\displaystyle=\lim_{d\to 0}\Big(F\delta(x)-F% \delta(x-d)\Big)\\ &\displaystyle=\lim_{d\to 0}\left(\frac{M}{d}\delta(x)-\frac{M}{d}\delta(x-d)% \right)\\ &\displaystyle=M\lim_{d\to 0}\frac{\delta(x)-\delta(x-d)}{d}\\ &\displaystyle=M\delta^{\prime}(x).\end{aligned}

Dirac_equation.html

  1. ψ = ψ ( x , t ) ψ=ψ(x,t)
  2. m m
  3. x , t x,t
  4. c c
  5. ħ ħ
  6. 2 π
  7. β β
  8. ψ ψ
  9. ψ ψ
  10. β β
  11. α i 2 = β 2 = I 4 \alpha_{i}^{2}=\beta^{2}=I_{4}
  12. i i
  13. j j
  14. α i α j + α j α i = 0 \alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}=0
  15. α i β + β α i = 0 \alpha_{i}\beta+\beta\alpha_{i}=0
  16. - 2 2 m 2 ϕ = i t ϕ . -\frac{\hbar^{2}}{2m}\nabla^{2}\phi=i\hbar\frac{\partial}{\partial t}\phi.
  17. E 2 c 2 - p 2 = m 2 c 2 \frac{E^{2}}{c^{2}}-p^{2}=m^{2}c^{2}
  18. m m
  19. ( - 1 c 2 2 t 2 + 2 ) ϕ = m 2 c 2 2 ϕ \left(-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}+\nabla^{2}\right)% \phi=\frac{m^{2}c^{2}}{\hbar^{2}}\phi
  20. ϕ ϕ
  21. ρ = ϕ * ϕ \rho=\phi^{*}\phi\,
  22. J = - i 2 m ( ϕ * ϕ - ϕ ϕ * ) J=-\frac{i\hbar}{2m}(\phi^{*}\nabla\phi-\phi\nabla\phi^{*})
  23. J + ρ t = 0. \nabla\cdot J+\frac{\partial\rho}{\partial t}=0.
  24. ρ = i 2 m ( ψ * t ψ - ψ t ψ * ) . \rho=\frac{i\hbar}{2m}(\psi^{*}\partial_{t}\psi-\psi\partial_{t}\psi^{*}).
  25. J μ = i 2 m ( ψ * μ ψ - ψ μ ψ * ) J^{\mu}=\frac{i\hbar}{2m}(\psi^{*}\partial^{\mu}\psi-\psi\partial^{\mu}\psi^{*})
  26. ψ ψ
  27. E = c p 2 + m 2 c 2 , E=c\sqrt{p^{2}+m^{2}c^{2}}\,,
  28. p p
  29. 2 - 1 c 2 2 t 2 = ( A x + B y + C z + i c D t ) ( A x + B y + C z + i c D t ) . \nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}=\left(A\partial_% {x}+B\partial_{y}+C\partial_{z}+\frac{i}{c}D\partial_{t}\right)\left(A\partial% _{x}+B\partial_{y}+C\partial_{z}+\frac{i}{c}D\partial_{t}\right).
  30. A B + B A = 0 , AB+BA=0,\;\ldots
  31. A 2 = B 2 = = 1. A^{2}=B^{2}=\ldots=1.\,
  32. A A
  33. B B
  34. C C
  35. D D
  36. ( A x + B y + C z + i c D t ) ψ = κ ψ \left(A\partial_{x}+B\partial_{y}+C\partial_{z}+\frac{i}{c}D\partial_{t}\right% )\psi=\kappa\psi
  37. κ κ
  38. ( 2 - 1 c 2 t 2 ) ψ = κ 2 ψ . \left(\nabla^{2}-\frac{1}{c^{2}}\partial_{t}^{2}\right)\psi=\kappa^{2}\psi.
  39. κ = m c / ħ κ=mc/ħ
  40. ( A x + B y + C z + i c D t - m c ) ψ = 0. \left(A\partial_{x}+B\partial_{y}+C\partial_{z}+\frac{i}{c}D\partial_{t}-\frac% {mc}{\hbar}\right)\psi=0.
  41. A = i β α 1 , B = i β α 2 , C = i β α 3 , D = β , A=i\beta\alpha_{1},B=i\beta\alpha_{2},C=i\beta\alpha_{3},D=\beta\,,
  42. γ 0 = β \gamma^{0}=\beta\,
  43. γ k = γ 0 α k . \gamma^{k}=\gamma^{0}\alpha^{k}.\,
  44. μ = 0 , 1 , 2 , 3 μ=0,1,2,3
  45. γ 0 = ( I 2 0 0 - I 2 ) , γ 1 = ( 0 σ x - σ x 0 ) , γ 2 = ( 0 σ y - σ y 0 ) , γ 3 = ( 0 σ z - σ z 0 ) . \gamma^{0}=\left(\begin{array}[]{cccc}I_{2}&0\\ 0&-I_{2}\end{array}\right),\gamma^{1}=\left(\begin{array}[]{cccc}0&\sigma_{x}% \\ -\sigma_{x}&0\end{array}\right),\gamma^{2}=\left(\begin{array}[]{cccc}0&\sigma% _{y}\\ -\sigma_{y}&0\end{array}\right),\gamma^{3}=\left(\begin{array}[]{cccc}0&\sigma% _{z}\\ -\sigma_{z}&0\end{array}\right).\,
  46. { γ μ , γ ν } = 2 η μ ν \{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}\,
  47. { a , b } = a b + b a \{a,b\}=ab+ba
  48. ( + ) (+−−−)
  49. P op ψ = m c ψ . P_{\mathrm{op}}\psi=mc\psi.\,
  50. / {\partial\!\!\!\big/}
  51. i / ψ - m c ψ = 0 i\hbar{\partial\!\!\!\big/}\psi-mc\psi=0
  52. ħ = c = 1 ħ=c=1
  53. γ μ = S - 1 γ μ S . \gamma^{\mu\prime}=S^{-1}\gamma^{\mu}S.
  54. S S
  55. γ μ = U γ μ U . \gamma^{\mu\prime}=U^{\dagger}\gamma^{\mu}U.
  56. U U
  57. ( i U γ μ U μ - m ) ψ ( x , t ) = 0 (iU^{\dagger}\gamma^{\mu}U\partial_{\mu}^{\prime}-m)\psi(x^{\prime},t^{\prime}% )=0
  58. U ( i γ μ μ - m ) U ψ ( x , t ) = 0. U^{\dagger}(i\gamma^{\mu}\partial_{\mu}^{\prime}-m)U\psi(x^{\prime},t^{\prime}% )=0.
  59. ψ = U ψ \psi^{\prime}=U\psi
  60. ( i γ μ μ - m ) ψ ( x , t ) = 0. (i\gamma^{\mu}\partial_{\mu}^{\prime}-m)\psi^{\prime}(x^{\prime},t^{\prime})=0.
  61. V = 1 4 ! ϵ μ ν α β γ μ γ ν γ α γ β . V=\frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{% \alpha}\gamma^{\beta}.
  62. g \sqrt{g}
  63. g g
  64. V = i γ 0 γ 1 γ 2 γ 3 . V=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}.
  65. γ 5 = ( 0 I 2 I 2 0 ) . \gamma_{5}=\begin{pmatrix}0&I_{2}\\ I_{2}&0\end{pmatrix}.
  66. γ 5 γ μ + γ μ γ 5 = 0 \gamma^{5}\gamma^{\mu}+\gamma^{\mu}\gamma^{5}=0
  67. ψ ¯ = ψ γ 0 \bar{\psi}=\psi^{\dagger}\gamma^{0}
  68. ψ ψ
  69. ( γ μ ) γ 0 = γ 0 γ μ (\gamma^{\mu})^{\dagger}\gamma^{0}=\gamma^{0}\gamma^{\mu}\,
  70. ψ ¯ ( - i γ μ μ - m ) = 0 \bar{\psi}(-i\gamma^{\mu}\partial_{\mu}-m)=0\,
  71. ψ ¯ \overline{ψ}
  72. ψ ψ
  73. μ ( ψ ¯ γ μ ψ ) = 0. \partial_{\mu}\left(\bar{\psi}\gamma^{\mu}\psi\right)=0.
  74. J 0 = ψ ¯ γ 0 ψ = ψ ψ . J^{0}=\bar{\psi}\gamma^{0}\psi=\psi^{\dagger}\psi.
  75. N N
  76. 1 / 2 {1}/{2}
  77. H = 1 2 m ( σ ( p - e c A ) ) 2 + e ϕ . H=\frac{1}{2m}\left(\sigma\cdot\left(p-\frac{e}{c}A\right)\right)^{2}+e\phi.
  78. A A
  79. φ φ
  80. H = 1 2 m ( p - e c A ) 2 + e ϕ - e 2 m c σ B . H=\frac{1}{2m}\left(p-\frac{e}{c}A\right)^{2}+e\phi-\frac{e\hbar}{2mc}\sigma% \cdot B.
  81. ( i γ μ ( μ + i e A μ ) - m ) ψ = 0 (i\gamma^{\mu}(\partial_{\mu}+ieA_{\mu})-m)\psi=0\,
  82. i i
  83. ( ( m c 2 - E + e ϕ ) c σ ( p - e c A ) - c σ ( p - e c A ) ( m c 2 + E - e ϕ ) ) ( ψ + ψ - ) = ( 0 0 ) . \begin{pmatrix}(mc^{2}-E+e\phi)&c\sigma\cdot\left(p-\frac{e}{c}A\right)\\ -c\sigma\cdot\left(p-\frac{e}{c}A\right)&\left(mc^{2}+E-e\phi\right)\end{% pmatrix}\begin{pmatrix}\psi_{+}\\ \psi_{-}\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}.
  84. ( E - e ϕ ) ψ + - c σ ( p - e c A ) ψ - = m c 2 ψ + (E-e\phi)\psi_{+}-c\sigma\cdot\left(p-\frac{e}{c}A\right)\psi_{-}=mc^{2}\psi_{+}
  85. - ( E - e ϕ ) ψ - + c σ ( p - e c A ) ψ + = m c 2 ψ - -(E-e\phi)\psi_{-}+c\sigma\cdot\left(p-\frac{e}{c}A\right)\psi_{+}=mc^{2}\psi_% {-}
  86. E - e ϕ m c 2 E-e\phi\approx mc^{2}
  87. p m v p\approx mv
  88. ψ - 1 2 m c σ ( p - e c A ) ψ + \psi_{-}\approx\frac{1}{2mc}\sigma\cdot\left(p-\frac{e}{c}A\right)\psi_{+}
  89. v / c v/c
  90. ( E - m c 2 ) ψ + = 1 2 m [ σ ( p - e c A ) ] 2 ψ + + e ϕ ψ + (E-mc^{2})\psi_{+}=\frac{1}{2m}\left[\sigma\cdot\left(p-\frac{e}{c}A\right)% \right]^{2}\psi_{+}+e\phi\psi_{+}
  91. m 0 m→0
  92. = i c ψ ¯ γ μ μ ψ - m c 2 ψ ¯ ψ \mathcal{L}=i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^{2}% \overline{\psi}\psi
  93. ψ ψ
  94. ψ ¯ \overline{ψ}
  95. H = γ 0 [ m c 2 + c γ k ( p k - q c A k ) ] + q A 0 . H=\gamma^{0}\left[mc^{2}+c\gamma^{k}\left(p_{k}-\frac{q}{c}A_{k}\right)\right]% +qA^{0}.
  96. k = 1 , 2 , 3 k=1,2,3
  97. A = 0 A=0
  98. H = c ( p - q c A ) 2 + m 2 c 2 + q A 0 . H=c\sqrt{\left(p-\frac{q}{c}A\right)^{2}+m^{2}c^{2}}+qA^{0}.
  99. E E

Direct_product.html

  1. \mathbb{R}
  2. × \mathbb{R}\times\mathbb{R}
  3. { ( x , y ) | x , y } \{(x,y)|x,y\in\mathbb{R}\}
  4. \mathbb{R}
  5. × \mathbb{R}\times\mathbb{R}
  6. { ( x , y ) | x , y } \{(x,y)|x,y\in\mathbb{R}\}
  7. × \mathbb{R}\times\mathbb{R}
  8. ( a , b ) + ( c , d ) = ( a + c , b + d ) (a,b)+(c,d)=(a+c,b+d)
  9. \mathbb{R}
  10. × \mathbb{R}\times\mathbb{R}
  11. { ( x , y ) | x , y } \{(x,y)|x,y\in\mathbb{R}\}
  12. ( a , b ) + ( c , d ) = ( a + c , b + d ) (a,b)+(c,d)=(a+c,b+d)
  13. ( a , b ) ( c , d ) = ( a c , b d ) (a,b)(c,d)=(ac,bd)
  14. \mathbb{R}
  15. × \mathbb{R}\times\mathbb{R}
  16. { ( x , y ) | x , y } \{(x,y)|x,y\in\mathbb{R}\}
  17. ( 1 , 0 ) (1,0)
  18. × × × \mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}
  19. × × × \mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\cdots
  20. G H G\oplus H
  21. π 1 : G × H G by π 1 ( g , h ) = g \pi_{1}:G\times H\to G\quad\,\text{by}\quad\pi_{1}(g,h)=g
  22. π 2 : G × H H by π 2 ( g , h ) = h \pi_{2}:G\times H\to H\quad\,\text{by}\quad\pi_{2}(g,h)=h
  23. f i = π i f f_{i}=\pi_{i}\circ f
  24. i = 1 n X i \prod_{i=1}^{n}X_{i}
  25. i = 1 n X i \bigoplus_{i=1}^{n}X_{i}
  26. X = i = 1 X=\prod_{i=1}^{\infty}\mathbb{R}
  27. Y = i = 1 Y=\bigoplus_{i=1}^{\infty}\mathbb{R}
  28. i I X i . \prod_{i\in I}X_{i}.
  29. = { U 1 × × U n | U i open in X i } . \mathcal{B}=\{U_{1}\times\cdots\times U_{n}\ |\ U_{i}\ \mathrm{open\ in}\ X_{i% }\}.
  30. = { i I U i | ( j 1 , , j n ) ( U j i open in X j i ) and ( i j 1 , , j n ) ( U i = X i ) } . \mathcal{B}=\left\{\prod_{i\in I}U_{i}\ \Big|\ (\exists j_{1},\ldots,j_{n})(U_% {j_{i}}\ \mathrm{open\ in}\ X_{j_{i}})\ \mathrm{and}\ (\forall i\neq j_{1},% \ldots,j_{n})(U_{i}=X_{i})\right\}.
  31. π i : G G i by π i ( g ) = g i \pi_{i}\colon G\to G_{i}\quad\mathrm{by}\quad\pi_{i}(g)=g_{i}
  32. ( g j ) j I (g_{j})_{j\in I}
  33. A , B X A,B\subset X
  34. A × B X A\times B\cong X

Direct_sum_of_modules.html

  1. V W V\oplus W
  2. G H G\oplus H
  3. ( α i ) (\alpha_{i})
  4. α i M i \alpha_{i}\in M_{i}
  5. α i = 0 \alpha_{i}=0
  6. i I i\in I
  7. M i M_{i}
  8. ( α + β ) i = α i + β i (\alpha+\beta)_{i}=\alpha_{i}+\beta_{i}
  9. r ( α ) i = ( r α ) i r(\alpha)_{i}=(r\alpha)_{i}
  10. i I M i . \bigoplus_{i\in I}M_{i}.
  11. ( α i ) (\alpha_{i})
  12. Σ α i \Sigma\alpha_{i}
  13. Σ α i \Sigma^{\prime}\alpha_{i}
  14. Hom R ( i I M i , L ) i I Hom R ( M i , L ) . \operatorname{Hom}_{R}\biggl(\bigoplus_{i\in I}M_{i},L\biggr)\cong\prod_{i\in I% }\operatorname{Hom}_{R}\left(M_{i},L\right).
  15. τ - 1 ( β ) ( α ) = i I β ( i ) ( α ( i ) ) \tau^{-1}(\beta)(\alpha)=\sum_{i\in I}\beta(i)(\alpha(i))
  16. p k : A 1 A n A k p_{k}:A_{1}\oplus\cdots\oplus A_{n}\to A_{k}
  17. i k : A k A 1 A n i_{k}:A_{k}\mapsto A_{1}\oplus\cdots\oplus A_{n}
  18. i 1 p 1 + + i n p n i_{1}\circ p_{1}+\cdots+i_{n}\circ p_{n}
  19. p k i l p_{k}\circ i_{l}
  20. j i : M i k I M k j_{i}:M_{i}\rightarrow\bigoplus_{k\in I}M_{k}
  21. f : i I M i M f:\bigoplus_{i\in I}M_{i}\rightarrow M
  22. ( x 1 + y 1 ) ( x 2 + y 2 ) = ( x 1 x 2 + y 1 y 2 ) . (x_{1}+y_{1})(x_{2}+y_{2})=(x_{1}x_{2}+y_{1}y_{2}).
  23. R R R\oplus R
  24. C C C\oplus C
  25. H H H\oplus H
  26. λ ( x y ) = λ x λ y \lambda(x\oplus y)=\lambda x\oplus\lambda y
  27. λ ( x , y ) = ( λ x , y ) = ( x , λ y ) \lambda(x,y)=(\lambda x,y)=(x,\lambda y)\!
  28. R 2 , 2 C , 2 H {}^{2}R,\ ^{2}C,\ ^{2}H\!
  29. i I x ( i ) X i < . \sum_{i\in I}\|x(i)\|_{X_{i}}<\infty.
  30. A B A\oplus B
  31. \ell^{\infty}
  32. B ( ( x i ) , ( y i ) ) = i I b i ( x i , y i ) B\left({\left({x_{i}}\right),\left({y_{i}}\right)}\right)=\sum_{i\in I}b_{i}% \left({x_{i},y_{i}}\right)
  33. ( x 1 , , x n ) , ( y 1 , , y n ) = x 1 , y 1 + + x n , y n . \langle(x_{1},...,x_{n}),(y_{1},...,y_{n})\rangle=\langle x_{1},y_{1}\rangle+.% ..+\langle x_{n},y_{n}\rangle.
  34. i α ( i ) 2 < . \sum_{i}\left\|\alpha_{(i)}\right\|^{2}<\infty.
  35. α , β = i α i , β i . \langle\alpha,\beta\rangle=\sum_{i}\langle\alpha_{i},\beta_{i}\rangle.
  36. a = i a i 2 \left\|a\right\|=\sqrt{\sum_{i}\left\|a_{i}\right\|^{2}}

Directed_set.html

  1. × \times
  2. × \times
  3. \cap
  4. \cap
  5. \cap
  6. \cap

Discounted_cash_flow.html

  1. D C F = C F 1 ( 1 + r ) 1 + C F 2 ( 1 + r ) 2 + + C F n ( 1 + r ) n DCF=\frac{CF_{1}}{(1+r)^{1}}+\frac{CF_{2}}{(1+r)^{2}}+\cdots+\frac{CF_{n}}{(1+% r)^{n}}
  2. F V = D C F ( 1 + r ) n FV=DCF\cdot(1+r)^{n}
  3. D P V = F V ( 1 + r ) n DPV=\frac{FV}{(1+r)^{n}}
  4. D P V = t = 0 N F V t ( 1 + r ) t DPV=\sum_{t=0}^{N}\frac{FV_{t}}{(1+r)^{t}}
  5. D P V = 0 T F V ( t ) e - λ t d t , DPV=\int_{0}^{T}FV(t)\,e^{-\lambda t}dt\,,
  6. F V ( t ) FV(t)
  7. λ = l o g ( 1 + r ) \lambda=log(1+r)

Discounting.html

  1. P ( 1 + r ) t P(1+r)^{t}
  2. Discount = P ( 1 + r ) t - P \,\text{Discount}=P(1+r)^{t}-P
  3. P = F ( 1 + r ) t P=\frac{F}{(1+r)^{t}}
  4. PV = $ 100 ( 1 + 0.12 ) 5 = $ 56.74. {\rm PV}=\frac{\$100}{(1+0.12)^{5}}=\$56.74.
  5. D F ( T ) = 1 ( 1 + r T ) DF(T)=\frac{1}{(1+rT)}
  6. D F ( T ) = 1 ( 1 + r ) T DF(T)=\frac{1}{(1+r)^{T}}
  7. D F ( T ) = 1 ( 1 + r 360 ) 360 T DF(T)=\frac{1}{(1+\frac{r}{360})^{360T}}
  8. D F ( T ) = 1 ( 1 + r 365 ) 365 T DF(T)=\frac{1}{(1+\frac{r}{365})^{365T}}
  9. D F ( T ) = e - r T DF(T)=e^{-rT}\,

Discrete_cosine_transform.html

  1. N × N N\times N
  2. N N
  3. ( 0 , 0 ) (0,0)
  4. f ( x ) f(x)
  5. x x
  6. x x
  7. f ( x ) f(x)
  8. f : N N f:\mathbb{R}^{N}\to\mathbb{R}^{N}
  9. \mathbb{R}
  10. X k = 1 2 ( x 0 + ( - 1 ) k x N - 1 ) + n = 1 N - 2 x n cos [ π N - 1 n k ] k = 0 , , N - 1. X_{k}=\frac{1}{2}(x_{0}+(-1)^{k}x_{N-1})+\sum_{n=1}^{N-2}x_{n}\cos\left[\frac{% \pi}{N-1}nk\right]\quad\quad k=0,\dots,N-1.
  11. 2 / ( N - 1 ) \sqrt{2/(N-1)}
  12. 2 N - 2 2N-2
  13. X k = n = 0 N - 1 x n cos [ π N ( n + 1 2 ) k ] k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}\cos\left[\frac{\pi}{N}\left(n+\frac{1}{2}\right)k% \right]\quad\quad k=0,\dots,N-1.
  14. 4 N 4N
  15. 4 N 4N
  16. y n y_{n}
  17. y 2 n = 0 y_{2n}=0
  18. y 2 n + 1 = x n y_{2n+1}=x_{n}
  19. 0 n < N 0\leq n<N
  20. y 2 N = 0 y_{2N}=0
  21. y 4 N - n = y n y_{4N-n}=y_{n}
  22. 0 < n < 2 N 0<n<2N
  23. 2 / N \sqrt{2/N}
  24. X k = 1 2 x 0 + n = 1 N - 1 x n cos [ π N n ( k + 1 2 ) ] k = 0 , , N - 1. X_{k}=\frac{1}{2}x_{0}+\sum_{n=1}^{N-1}x_{n}\cos\left[\frac{\pi}{N}n\left(k+% \frac{1}{2}\right)\right]\quad\quad k=0,\dots,N-1.
  25. 2 / N \sqrt{2/N}
  26. X k = n = 0 N - 1 x n cos [ π N ( n + 1 2 ) ( k + 1 2 ) ] k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}\cos\left[\frac{\pi}{N}\left(n+\frac{1}{2}\right)% \left(k+\frac{1}{2}\right)\right]\quad\quad k=0,\dots,N-1.
  27. 2 / N \sqrt{2/N}
  28. 2 / N \sqrt{2/N}
  29. X k 1 , k 2 \displaystyle X_{k_{1},k_{2}}
  30. N 1 = N 2 = 8 N_{1}=N_{2}=8
  31. 8 × 8 8\times 8
  32. 4 N 4N
  33. 4 N 4N
  34. N N
  35. N N
  36. O ( N ) O(N)
  37. N N
  38. 2 N log 2 N - N + 2 2N\log_{2}N-N+2
  39. N N
  40. [ 6.1917 - 0.3411 1.2418 0.1492 0.1583 0.2742 - 0.0724 0.0561 0.2205 0.0214 0.4503 0.3947 - 0.7846 - 0.4391 0.1001 - 0.2554 1.0423 0.2214 - 1.0017 - 0.2720 0.0789 - 0.1952 0.2801 0.4713 - 0.2340 - 0.0392 - 0.2617 - 0.2866 0.6351 0.3501 - 0.1433 0.3550 0.2750 0.0226 0.1229 0.2183 - 0.2583 - 0.0742 - 0.2042 - 0.5906 0.0653 0.0428 - 0.4721 - 0.2905 0.4745 0.2875 - 0.0284 - 0.1311 0.3169 0.0541 - 0.1033 - 0.0225 - 0.0056 0.1017 - 0.1650 - 0.1500 - 0.2970 - 0.0627 0.1960 0.0644 - 0.1136 - 0.1031 0.1887 0.1444 ] \begin{bmatrix}6.1917&-0.3411&1.2418&0.1492&0.1583&0.2742&-0.0724&0.0561\\ 0.2205&0.0214&0.4503&0.3947&-0.7846&-0.4391&0.1001&-0.2554\\ 1.0423&0.2214&-1.0017&-0.2720&0.0789&-0.1952&0.2801&0.4713\\ -0.2340&-0.0392&-0.2617&-0.2866&0.6351&0.3501&-0.1433&0.3550\\ 0.2750&0.0226&0.1229&0.2183&-0.2583&-0.0742&-0.2042&-0.5906\\ 0.0653&0.0428&-0.4721&-0.2905&0.4745&0.2875&-0.0284&-0.1311\\ 0.3169&0.0541&-0.1033&-0.0225&-0.0056&0.1017&-0.1650&-0.1500\\ -0.2970&-0.0627&0.1960&0.0644&-0.1136&-0.1031&0.1887&0.1444\\ \end{bmatrix}
  41. 2 N log 2 N - N + 2 2N\log_{2}N-N+2
  42. 2 \sqrt{2}

Discrete_Fourier_transform.html

  1. x 0 , x 1 , , x N - 1 x_{0},x_{1},\ldots,x_{N-1}
  2. X k X_{k}
  3. x n x_{n}
  4. | X k | / N = Re ( X k ) 2 + Im ( X k ) 2 / N |X_{k}|/N=\sqrt{\operatorname{Re}(X_{k})^{2}+\operatorname{Im}(X_{k})^{2}}/N
  5. arg ( X k ) = atan2 ( Im ( X k ) , Re ( X k ) ) = - i ln ( X k | X k | ) , \arg(X_{k})=\operatorname{atan2}\big(\operatorname{Im}(X_{k}),\operatorname{Re% }(X_{k})\big)=-i\operatorname{ln}\left(\frac{X_{k}}{|X_{k}|}\right),
  6. \mathcal{F}
  7. 𝐗 = { 𝐱 } \mathbf{X}=\mathcal{F}\left\{\mathbf{x}\right\}
  8. ( 𝐱 ) \mathcal{F}\left(\mathbf{x}\right)
  9. 𝐱 \mathcal{F}\mathbf{x}
  10. x n = 1 N k = 0 N - 1 X k e i 2 π k n / N , n x_{n}=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}\cdot e^{i2\pi kn/N},\quad n\in\mathbb{Z}\,
  11. n [ 0 , N - 1 ] , \scriptstyle n\ \in\ [0,\ N-1],
  12. 1 / N \scriptstyle\sqrt{1/N}
  13. : N N \mathcal{F}\colon\mathbb{C}^{N}\to\mathbb{C}^{N}
  14. \mathbb{C}
  15. u k = [ e 2 π i N k n | n = 0 , 1 , , N - 1 ] T u_{k}=\left[e^{\frac{2\pi i}{N}kn}\;|\;n=0,1,\ldots,N-1\right]^{T}
  16. u k T u k * = n = 0 N - 1 ( e 2 π i N k n ) ( e 2 π i N ( - k ) n ) = n = 0 N - 1 e 2 π i N ( k - k ) n = N δ k k u^{T}_{k}u_{k^{\prime}}^{*}=\sum_{n=0}^{N-1}\left(e^{\frac{2\pi i}{N}kn}\right% )\left(e^{\frac{2\pi i}{N}(-k^{\prime})n}\right)=\sum_{n=0}^{N-1}e^{\frac{2\pi i% }{N}(k-k^{\prime})n}=N~{}\delta_{kk^{\prime}}
  17. δ k k ~{}\delta_{kk^{\prime}}
  18. k = k k=k^{\prime}
  19. n = 0 N - 1 x n y n * = 1 N k = 0 N - 1 X k Y k * \sum_{n=0}^{N-1}x_{n}y^{*}_{n}=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}Y^{*}_{k}
  20. n = 0 N - 1 | x n | 2 = 1 N k = 0 N - 1 | X k | 2 . \sum_{n=0}^{N-1}|x_{n}|^{2}=\frac{1}{N}\sum_{k=0}^{N-1}|X_{k}|^{2}.
  21. X k + N = def n = 0 N - 1 x n e - 2 π i N ( k + N ) n = n = 0 N - 1 x n e - 2 π i N k n e - 2 π i n 1 = n = 0 N - 1 x n e - 2 π i N k n = X k . X_{k+N}\ \stackrel{\mathrm{def}}{=}\ \sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}% (k+N)n}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}kn}\underbrace{e^{-2\pi in}}_% {1}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}kn}=X_{k}.
  22. x n x_{n}
  23. e 2 π i N n m e^{\frac{2\pi i}{N}nm}
  24. X k X_{k}
  25. X k X_{k}
  26. X k - m X_{k-m}
  27. x n x_{n}
  28. X k X_{k}
  29. { x n } \{x_{n}\}
  30. ( { x n } ) k = X k \mathcal{F}(\{x_{n}\})_{k}=X_{k}
  31. ( { x n e 2 π i N n m } ) k = X k - m \mathcal{F}(\{x_{n}\cdot e^{\frac{2\pi i}{N}nm}\})_{k}=X_{k-m}
  32. ( { x n - m } ) k = X k e - 2 π i N k m \mathcal{F}(\{x_{n-m}\})_{k}=X_{k}\cdot e^{-\frac{2\pi i}{N}km}
  33. - 1 { 𝐗 𝐘 } n = l = 0 N - 1 x l ( y N ) n - l = def ( 𝐱 * 𝐲 𝐍 ) n , \mathcal{F}^{-1}\left\{\mathbf{X\cdot Y}\right\}_{n}\ =\sum_{l=0}^{N-1}x_{l}% \cdot(y_{N})_{n-l}\ \ \stackrel{\mathrm{def}}{=}\ \ (\mathbf{x*y_{N}})_{n}\ ,
  34. 𝐱 \mathbf{x}
  35. 𝐲 \mathbf{y}
  36. ( 𝐲 𝐍 ) n = def p = - y ( n - p N ) = y n ( m o d N ) . (\mathbf{y_{N}})_{n}\ \stackrel{\mathrm{def}}{=}\ \sum_{p=-\infty}^{\infty}y_{% (n-pN)}=y_{n(modN)}.\,
  37. 𝐱 \mathbf{x}
  38. 𝐲 𝐍 \mathbf{y_{N}}
  39. - 1 { 𝐗 * 𝐘 } n = l = 0 N - 1 x l * ( y N ) n + l = def ( 𝐱 𝐲 𝐍 ) n . \mathcal{F}^{-1}\left\{\mathbf{X^{*}\cdot Y}\right\}_{n}=\sum_{l=0}^{N-1}x_{l}% ^{*}\cdot(y_{N})_{n+l}\ \ \stackrel{\mathrm{def}}{=}\ \ (\mathbf{x\star y_{N}}% )_{n}\ .
  40. 𝐱 * 𝐲 . \mathbf{x*y}.
  41. 𝐱 * 𝐲 \mathbf{x*y}
  42. 𝐱 \mathbf{x}
  43. 𝐲 \mathbf{y}
  44. O ( N 2 ) \scriptstyle O(N^{2})
  45. O ( N log N ) \scriptstyle O(N\log N)
  46. { 𝐱 𝐲 } k = def n = 0 N - 1 x n y n e - 2 π i N k n \mathcal{F}\left\{\mathbf{x\cdot y}\right\}_{k}\ \stackrel{\mathrm{def}}{=}% \sum_{n=0}^{N-1}x_{n}\cdot y_{n}\cdot e^{-\frac{2\pi i}{N}kn}
  47. = 1 N ( 𝐗 * 𝐘 𝐍 ) k , =\frac{1}{N}(\mathbf{X*Y_{N}})_{k},\,
  48. 𝐗 \mathbf{X}
  49. 𝐘 \mathbf{Y}
  50. p ( t ) = 1 N [ X 0 + X 1 e 2 π i t + + X N / 2 - 1 e ( N / 2 - 1 ) 2 π i t + X N / 2 cos ( N π t ) + X N / 2 + 1 e ( - N / 2 + 1 ) 2 π i t + + X N - 1 e - 2 π i t ] p(t)=\frac{1}{N}\left[X_{0}+X_{1}e^{2\pi it}+\cdots+X_{N/2-1}e^{(N/2-1)2\pi it% }+X_{N/2}\cos(N\pi t)+X_{N/2+1}e^{(-N/2+1)2\pi it}+\cdots+X_{N-1}e^{-2\pi it}\right]
  51. p ( t ) = 1 N [ X 0 + X 1 e 2 π i t + + X N / 2 e N / 2 2 π i t + X N / 2 + 1 e - N / 2 2 π i t + + X N - 1 e - 2 π i t ] p(t)=\frac{1}{N}\left[X_{0}+X_{1}e^{2\pi it}+\cdots+X_{\lfloor N/2\rfloor}e^{% \lfloor N/2\rfloor 2\pi it}+X_{\lfloor N/2\rfloor+1}e^{-\lfloor N/2\rfloor 2% \pi it}+\cdots+X_{N-1}e^{-2\pi it}\right]
  52. p ( n / N ) = x n p(n/N)=x_{n}
  53. n = 0 , , N - 1 n=0,\ldots,N-1
  54. X N / 2 N cos ( N π t ) \frac{X_{N/2}}{N}\cos(N\pi t)
  55. e - i t e^{-it}
  56. e i ( N - 1 ) t e^{i(N-1)t}
  57. x n x_{n}
  58. x n x_{n}
  59. p ( t ) p(t)
  60. N - 1 N-1
  61. - N / 2 -N/2
  62. + N / 2 +N/2
  63. x n x_{n}
  64. 𝐅 = [ ω N 0 0 ω N 0 1 ω N 0 ( N - 1 ) ω N 1 0 ω N 1 1 ω N 1 ( N - 1 ) ω N ( N - 1 ) 0 ω N ( N - 1 ) 1 ω N ( N - 1 ) ( N - 1 ) ] \mathbf{F}=\begin{bmatrix}\omega_{N}^{0\cdot 0}&\omega_{N}^{0\cdot 1}&\ldots&% \omega_{N}^{0\cdot(N-1)}\\ \omega_{N}^{1\cdot 0}&\omega_{N}^{1\cdot 1}&\ldots&\omega_{N}^{1\cdot(N-1)}\\ \vdots&\vdots&\ddots&\vdots\\ \omega_{N}^{(N-1)\cdot 0}&\omega_{N}^{(N-1)\cdot 1}&\ldots&\omega_{N}^{(N-1)% \cdot(N-1)}\\ \end{bmatrix}
  65. ω N = e - 2 π i / N \omega_{N}=e^{-2\pi i/N}\,
  66. 𝐅 - 1 = 1 N 𝐅 * \mathbf{F}^{-1}=\frac{1}{N}\mathbf{F}^{*}
  67. 1 / N 1/\sqrt{N}
  68. 𝐔 = 𝐅 / N \mathbf{U}=\mathbf{F}/\sqrt{N}
  69. 𝐔 - 1 = 𝐔 * \mathbf{U}^{-1}=\mathbf{U}^{*}
  70. | det ( 𝐔 ) | = 1 |\det(\mathbf{U})|=1
  71. ± 1 \pm 1
  72. ± i \pm i
  73. m = 0 N - 1 U k m U m n * = δ k n \sum_{m=0}^{N-1}U_{km}U_{mn}^{*}=\delta_{kn}
  74. X k = n = 0 N - 1 U k n x n X_{k}=\sum_{n=0}^{N-1}U_{kn}x_{n}
  75. n = 0 N - 1 x n y n * = k = 0 N - 1 X k Y k * \sum_{n=0}^{N-1}x_{n}y_{n}^{*}=\sum_{k=0}^{N-1}X_{k}Y_{k}^{*}
  76. 𝐱 = 𝐲 \mathbf{x}=\mathbf{y}
  77. n = 0 N - 1 | x n | 2 = k = 0 N - 1 | X k | 2 \sum_{n=0}^{N-1}|x_{n}|^{2}=\sum_{k=0}^{N-1}|X_{k}|^{2}
  78. F F
  79. - 1 ( { x n } ) = ( { x N - n } ) / N \mathcal{F}^{-1}(\{x_{n}\})=\mathcal{F}(\{x_{N-n}\})/N
  80. n = 0 n=0
  81. x N - 0 = x 0 x_{N-0}=x_{0}
  82. - 1 ( 𝐱 ) = ( 𝐱 * ) * / N \mathcal{F}^{-1}(\mathbf{x})=\mathcal{F}(\mathbf{x}^{*})^{*}/N
  83. x n x_{n}
  84. x n x_{n}
  85. x n = a + b i x_{n}=a+bi
  86. x n x_{n}
  87. b + a i b+ai
  88. x n x_{n}
  89. i x n * ix_{n}^{*}
  90. - 1 ( 𝐱 ) = swap ( ( swap ( 𝐱 ) ) ) / N \mathcal{F}^{-1}(\mathbf{x})=\textrm{swap}(\mathcal{F}(\textrm{swap}(\mathbf{x% })))/N
  91. T ( 𝐱 ) = ( 𝐱 * ) / N T(\mathbf{x})=\mathcal{F}(\mathbf{x}^{*})/\sqrt{N}
  92. T ( T ( 𝐱 ) ) = 𝐱 T(T(\mathbf{x}))=\mathbf{x}
  93. H ( 𝐱 ) = ( ( 1 + i ) 𝐱 * ) / 2 N H(\mathbf{x})=\mathcal{F}((1+i)\mathbf{x}^{*})/\sqrt{2N}
  94. ( 1 + i ) (1+i)
  95. H ( H ( 𝐱 ) ) H(H(\mathbf{x}))
  96. 𝐱 \mathbf{x}
  97. H ( 𝐱 ) H(\mathbf{x})
  98. 𝐔 \mathbf{U}
  99. 𝐔 m , n = 1 N ω N ( m - 1 ) ( n - 1 ) = 1 N e - 2 π i N ( m - 1 ) ( n - 1 ) . \mathbf{U}_{m,n}=\frac{1}{\sqrt{N}}\omega_{N}^{(m-1)(n-1)}=\frac{1}{\sqrt{N}}e% ^{-\frac{2\pi i}{N}(m-1)(n-1)}.
  100. 𝐔 4 = 𝐈 . \mathbf{U}^{4}=\mathbf{I}.
  101. 𝐔 \mathbf{U}
  102. 𝐔 \mathbf{U}
  103. λ \lambda
  104. λ 4 = 1. \lambda^{4}=1.
  105. 𝐔 \mathbf{U}
  106. λ \lambda
  107. N × N N\times N
  108. 𝐔 \mathbf{U}
  109. det ( λ I - 𝐔 ) = ( λ - 1 ) N + 4 4 ( λ + 1 ) N + 2 4 ( λ + i ) N + 1 4 ( λ - i ) N - 1 4 . \det(\lambda I-\mathbf{U})=(\lambda-1)^{\left\lfloor\tfrac{N+4}{4}\right% \rfloor}(\lambda+1)^{\left\lfloor\tfrac{N+2}{4}\right\rfloor}(\lambda+i)^{% \left\lfloor\tfrac{N+1}{4}\right\rfloor}(\lambda-i)^{\left\lfloor\tfrac{N-1}{4% }\right\rfloor}.
  110. F ( m ) = k exp ( - π ( m + N k ) 2 N ) F(m)=\sum_{k\in\mathbb{Z}}\exp\left(-\frac{\pi\cdot(m+N\cdot k)^{2}}{N}\right)
  111. F ( m ) = s = K + 1 L [ cos ( 2 π N m ) - cos ( 2 π N s ) ] F(m)=\prod_{s=K+1}^{L}\left[\cos\left(\frac{2\pi}{N}m\right)-\cos\left(\frac{2% \pi}{N}s\right)\right]
  112. F ( m ) = sin ( 2 π N m ) s = K + 1 L - 1 [ cos ( 2 π N m ) - cos ( 2 π N s ) ] F(m)=\sin\left(\frac{2\pi}{N}m\right)\prod_{s=K+1}^{L-1}\left[\cos\left(\frac{% 2\pi}{N}m\right)-\cos\left(\frac{2\pi}{N}s\right)\right]
  113. n = 0 N - 1 | X n | 2 = 1 , \sum_{n=0}^{N-1}|X_{n}|^{2}=1~{},
  114. P n = | X n | 2 P_{n}=|X_{n}|^{2}
  115. n n
  116. Q m = N | x m | 2 . Q_{m}=N|x_{m}|^{2}~{}.
  117. P ( x ) P(x)
  118. Q ( k ) Q(k)
  119. D 0 ( X ) D 0 ( x ) 1 16 π 2 D_{0}(X)D_{0}(x)\geq\frac{1}{16\pi^{2}}
  120. D 0 ( X ) D_{0}(X)
  121. D 0 ( x ) D_{0}(x)
  122. | X | 2 |X|^{2}
  123. | x | 2 |x|^{2}
  124. H ( X ) = - n = 0 N - 1 P n ln P n H(X)=-\sum_{n=0}^{N-1}P_{n}\ln P_{n}
  125. H ( x ) = - m = 0 N - 1 Q m ln Q m , H(x)=-\sum_{m=0}^{N-1}Q_{m}\ln Q_{m}~{},
  126. H ( X ) + H ( x ) ln ( N ) . H(X)+H(x)\geq\ln(N)~{}.
  127. P n P_{n}
  128. A A
  129. A A
  130. N N
  131. Q m Q_{m}
  132. B = N / A B=N/A
  133. x 0 \|x\|_{0}
  134. X 0 \|X\|_{0}
  135. x 0 , x 1 , , x N - 1 x_{0},x_{1},\ldots,x_{N-1}
  136. X 0 , X 1 , , X N - 1 X_{0},X_{1},\ldots,X_{N-1}
  137. N x 0 X 0 . N\leq\|x\|_{0}\cdot\|X\|_{0}.
  138. 2 N x 0 + X 0 2\sqrt{N}\leq\|x\|_{0}+\|X\|_{0}
  139. x 0 , , x N - 1 x_{0},\ldots,x_{N-1}
  140. X N - k X - k = X k * , X_{N-k}\equiv X_{-k}=X_{k}^{*},
  141. X * X^{*}\,
  142. X k = n = 0 N - 1 x n e - 2 π i N ( k + b ) ( n + a ) k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}(k+b)(n+a)}\quad\quad k=0,\dots% ,N-1.
  143. 1 / 2 1/2
  144. a = 1 / 2 a=1/2
  145. X k + N = - X k X_{k+N}=-X_{k}
  146. b = 1 / 2 b=1/2
  147. a = b = 1 / 2 a=b=1/2
  148. a = b = - ( N - 1 ) / 2 a=b=-(N-1)/2
  149. x n x_{n}
  150. x n 1 , n 2 , , n d x_{n_{1},n_{2},\dots,n_{d}}
  151. n = 0 , 1 , , N - 1 n_{\ell}=0,1,\dots,N_{\ell}-1
  152. \ell
  153. 1 , 2 , , d 1,2,\dots,d
  154. X k 1 , k 2 , , k d = n 1 = 0 N 1 - 1 ( ω N 1 k 1 n 1 n 2 = 0 N 2 - 1 ( ω N 2 k 2 n 2 n d = 0 N d - 1 ω N d k d n d x n 1 , n 2 , , n d ) ) , X_{k_{1},k_{2},\dots,k_{d}}=\sum_{n_{1}=0}^{N_{1}-1}\left(\omega_{N_{1}}^{~{}k% _{1}n_{1}}\sum_{n_{2}=0}^{N_{2}-1}\left(\omega_{N_{2}}^{~{}k_{2}n_{2}}\cdots% \sum_{n_{d}=0}^{N_{d}-1}\omega_{N_{d}}^{~{}k_{d}n_{d}}\cdot x_{n_{1},n_{2},% \dots,n_{d}}\right)\right)\,,
  155. ω N = exp ( - 2 π i / N ) \omega_{N_{\ell}}=\exp(-2\pi i/N_{\ell})
  156. k = 0 , 1 , , N - 1 k_{\ell}=0,1,\dots,N_{\ell}-1
  157. 𝐧 = ( n 1 , n 2 , , n d ) \mathbf{n}=(n_{1},n_{2},\dots,n_{d})
  158. 𝐤 = ( k 1 , k 2 , , k d ) \mathbf{k}=(k_{1},k_{2},\dots,k_{d})
  159. 𝐍 - 1 \mathbf{N}-1
  160. 𝐍 - 1 = ( N 1 - 1 , N 2 - 1 , , N d - 1 ) \mathbf{N}-1=(N_{1}-1,N_{2}-1,\dots,N_{d}-1)
  161. X 𝐤 = 𝐧 = 𝟎 𝐍 - 1 e - 2 π i 𝐤 ( 𝐧 / 𝐍 ) x 𝐧 , X_{\mathbf{k}}=\sum_{\mathbf{n}=\mathbf{0}}^{\mathbf{N}-1}e^{-2\pi i\mathbf{k}% \cdot(\mathbf{n}/\mathbf{N})}x_{\mathbf{n}}\,,
  162. 𝐧 / 𝐍 \mathbf{n}/\mathbf{N}
  163. 𝐧 / 𝐍 = ( n 1 / N 1 , , n d / N d ) \mathbf{n}/\mathbf{N}=(n_{1}/N_{1},\dots,n_{d}/N_{d})
  164. x 𝐧 = 1 = 1 d N 𝐤 = 𝟎 𝐍 - 1 e 2 π i 𝐧 ( 𝐤 / 𝐍 ) X 𝐤 . x_{\mathbf{n}}=\frac{1}{\prod_{\ell=1}^{d}N_{\ell}}\sum_{\mathbf{k}=\mathbf{0}% }^{\mathbf{N}-1}e^{2\pi i\mathbf{n}\cdot(\mathbf{k}/\mathbf{N})}X_{\mathbf{k}}\,.
  165. x n x_{n}
  166. 𝐤 / 𝐍 \mathbf{k}/\mathbf{N}
  167. X 𝐤 X_{\mathbf{k}}
  168. x n 1 , n 2 x_{n_{1},n_{2}}
  169. N 1 N_{1}
  170. n 2 n_{2}
  171. y n 1 , k 2 y_{n_{1},k_{2}}
  172. N 2 N_{2}
  173. n 1 n_{1}
  174. X k 1 , k 2 X_{k_{1},k_{2}}
  175. x n 1 , n 2 , , n d x_{n_{1},n_{2},\dots,n_{d}}
  176. X k 1 , k 2 , , k d = X N 1 - k 1 , N 2 - k 2 , , N d - k d * , X_{k_{1},k_{2},\dots,k_{d}}=X_{N_{1}-k_{1},N_{2}-k_{2},\dots,N_{d}-k_{d}}^{*},
  177. \ell
  178. N N_{\ell}
  179. = 1 , 2 , , d \ell=1,2,\ldots,d
  180. { x n } \{x_{n}\}\,
  181. x ( t ) x(t)\,
  182. 𝐜 = 𝐚 * 𝐛 \mathbf{c}=\mathbf{a}*\mathbf{b}
  183. * *\,
  184. c n = m = 0 d - 1 a m b n - m mod d n = 0 , 1 , d - 1 c_{n}=\sum_{m=0}^{d-1}a_{m}b_{n-m\ \mathrm{mod}\ d}\qquad\qquad\qquad n=0,1% \dots,d-1
  185. ( 𝐜 ) = ( 𝐚 ) ( 𝐛 ) \mathcal{F}(\mathbf{c})=\mathcal{F}(\mathbf{a})\mathcal{F}(\mathbf{b})
  186. 𝐜 = - 1 ( ( 𝐚 ) ( 𝐛 ) ) . \mathbf{c}=\mathcal{F}^{-1}(\mathcal{F}(\mathbf{a})\mathcal{F}(\mathbf{b})).
  187. x n = 1 N k = 0 N - 1 X k e i 2 π k n / N x_{n}=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}e^{i2\pi kn/N}
  188. X k = n = 0 N - 1 x n e - i 2 π k n / N X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-i2\pi kn/N}
  189. x n e i 2 π n / N x_{n}e^{i2\pi n\ell/N}\,
  190. X k - X_{k-\ell}\,
  191. x n - x_{n-\ell}\,
  192. X k e - i 2 π k / N X_{k}e^{-i2\pi k\ell/N}\,
  193. x n x_{n}\in\mathbb{R}
  194. X k = X N - k * X_{k}=X_{N-k}^{*}\,
  195. a n a^{n}\,
  196. { N if a = e i 2 π k / N 1 - a N 1 - a e - i 2 π k / N otherwise \left\{\begin{matrix}N&\mbox{if }~{}a=e^{i2\pi k/N}\\ \frac{1-a^{N}}{1-a\,e^{-i2\pi k/N}}&\mbox{otherwise}\end{matrix}\right.
  197. ( N - 1 n ) {N-1\choose n}\,
  198. ( 1 + e - i 2 π k / N ) N - 1 \left(1+e^{-i2\pi k/N}\right)^{N-1}\,
  199. { 1 W if 2 n < W or 2 ( N - n ) < W 0 otherwise \left\{\begin{matrix}\frac{1}{W}&\mbox{if }~{}2n<W\mbox{ or }~{}2(N-n)<W\\ 0&\mbox{otherwise}\end{matrix}\right.
  200. { 1 if k = 0 sin ( π W k N ) W sin ( π k N ) otherwise \left\{\begin{matrix}1&\mbox{if }~{}k=0\\ \frac{\sin\left(\frac{\pi Wk}{N}\right)}{W\sin\left(\frac{\pi k}{N}\right)}&% \mbox{otherwise}\end{matrix}\right.
  201. x n x_{n}
  202. X k X_{k}
  203. X k X_{k}
  204. j exp ( - π c N ( n + N j ) 2 ) \sum_{j\in\mathbb{Z}}\exp\left(-\frac{\pi}{cN}\cdot(n+N\cdot j)^{2}\right)
  205. c N j exp ( - π c N ( k + N j ) 2 ) \sqrt{cN}\cdot\sum_{j\in\mathbb{Z}}\exp\left(-\frac{\pi c}{N}\cdot(k+N\cdot j)% ^{2}\right)
  206. c > 0 c>0
  207. c c
  208. 1 c \frac{1}{c}
  209. c c
  210. e - 2 π i N e^{-\frac{2\pi i}{N}}
  211. ω N \omega_{N}
  212. W N W_{N}
  213. ω N N = 1 \omega_{N}^{N}=1
  214. { 0 , 1 , , N 1 - 1 } × × { 0 , 1 , , N d - 1 } . \{0,1,\ldots,N_{1}-1\}\times\cdots\times\{0,1,\ldots,N_{d}-1\}\to\mathbb{C}.
  215. ω \omega
  216. ω = e - 2 π i / N \omega=e^{-2\pi i/N}
  217. X k = n = 0 N - 1 x n ω k n X_{k}=\sum_{n=0}^{N-1}x_{n}\cdot\omega^{kn}

Discrete_mathematics.html

  1. V ( x - c ) Spec K [ x ] = 𝔸 1 V(x-c)\subset\operatorname{Spec}K[x]=\mathbb{A}^{1}
  2. K K
  3. Spec K [ x ] / ( x - c ) Spec K \operatorname{Spec}K[x]/(x-c)\cong\operatorname{Spec}K
  4. Spec K [ x ] ( x - c ) \operatorname{Spec}K[x]_{(x-c)}

Discrete_space.html

  1. ρ \rho
  2. ρ ( x , y ) = { 1 if x y , 0 if x = y \rho(x,y)=\left\{\begin{matrix}1&\mbox{if}~{}\ x\neq y,\\ 0&\mbox{if}~{}\ x=y\end{matrix}\right.
  3. x , y X x,y\in X
  4. ( X , ρ ) (X,\rho)
  5. ( X , d ) (X,d)
  6. S X S\subseteq X
  7. x S x\in S
  8. δ > 0 \delta>0
  9. x x
  10. d ( x , y ) > δ d(x,y)>\delta
  11. y S { x } y\in S\setminus\{x\}
  12. ( X , d ) (X,d)
  13. S X S\subseteq X
  14. x , y S x,y\in S
  15. d ( x , y ) d(x,y)
  16. ( E , d ) (E,d)
  17. r > 0 r>0
  18. x , y E x,y\in E
  19. x = y x=y
  20. d ( x , y ) > r d(x,y)>r

Discriminant.html

  1. a x 2 + b x + c ax^{2}+bx+c\,
  2. Δ = b 2 - 4 a c . \Delta=\,b^{2}-4ac.
  3. Δ = b 2 c 2 - 4 a c 3 - 4 b 3 d - 27 a 2 d 2 + 18 a b c d . \Delta=\,b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd.
  4. Δ = a n 2 n - 2 i < j ( r i - r j ) 2 = ( - 1 ) n ( n - 1 ) / 2 a n 2 n - 2 i j ( r i - r j ) \Delta=a_{n}^{2n-2}\prod_{i<j}{(r_{i}-r_{j})^{2}}=(-1)^{n(n-1)/2}a_{n}^{2n-2}% \prod_{i\neq j}{(r_{i}-r_{j})}
  5. a n a_{n}
  6. r 1 , , r n r_{1},...,r_{n}
  7. a n 2 n - 2 a_{n}^{2n-2}
  8. x 3 + b x 2 + c x + d x^{3}+bx^{2}+cx+d
  9. x 4 + c x 2 + d x + e x^{4}+cx^{2}+dx+e
  10. a x 2 + b x + c \displaystyle ax^{2}+bx+c
  11. Δ = b 2 - 4 a c . \Delta=b^{2}-4ac.\,
  12. a x 3 + b x 2 + c x + d \displaystyle ax^{3}+bx^{2}+cx+d
  13. Δ = b 2 c 2 - 4 a c 3 - 4 b 3 d - 27 a 2 d 2 + 18 a b c d . \Delta=b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd.\,
  14. a x 4 + b x 3 + c x 2 + d x + e \displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e
  15. Δ = 256 a 3 e 3 - 192 a 2 b d e 2 - 128 a 2 c 2 e 2 + 144 a 2 c d 2 e - 27 a 2 d 4 + 144 a b 2 c e 2 - 6 a b 2 d 2 e \Delta=256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2% }d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e
  16. - 80 a b c 2 d e + 18 a b c d 3 + 16 a c 4 e - 4 a c 3 d 2 - 27 b 4 e 2 + 18 b 3 c d e - 4 b 3 d 3 - 4 b 2 c 3 e + b 2 c 2 d 2 . -80abc^{2}de+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d% ^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}.
  17. ( n 2 ) = n ( n - 1 ) 2 \textstyle{\left({{n}\atop{2}}\right)}=\frac{n(n-1)}{2}
  18. P = a 0 x n + a 1 x n - 1 + + a n . P=a_{0}x^{n}+a_{1}x_{n-1}+\cdots+a_{n}.
  19. a i a_{i}
  20. a i a_{i}
  21. a 0 i 0 , a n i n a_{0}^{i_{0}}\cdots,a_{n}^{i_{n}}
  22. i 0 + i 1 + + i n = 2 n - 2 i_{0}+i_{1}+\cdots+i_{n}=2n-2
  23. 0 i 0 + 1 i 1 + + n i n = n ( n - 1 ) 0\,i_{0}+1\,i_{1}+\cdots+n\,i_{n}=n(n-1)
  24. a x 2 + b x + c ax^{2}+bx+c
  25. [ i 0 , i 1 , i 2 ] , [i_{0},i_{1},i_{2}],
  26. a x 3 + b x 2 + c x + d ax^{3}+bx^{2}+cx+d
  27. a 2 d 2 = a a d d \displaystyle a^{2}d^{2}=aadd
  28. a x 4 + b x 3 + c x 2 + d x + e ax^{4}+bx^{3}+cx^{2}+dx+e
  29. ( i 0 , , i 4 ) = ( 0 , 1 , 4 , 1 , 0 ) (i_{0},\ldots,i_{4})=(0,1,4,1,0)
  30. 0 + 1 + 4 + 1 + 0 = 6 0+1+4+1+0=6
  31. 1 1 + 2 4 + 3 1 = 12 1\cdot 1+2\cdot 4+3\cdot 1=12
  32. b c 4 d bc^{4}d
  33. p ( x ) = a x 2 + b x + c \ p(x)=ax^{2}+bx+c
  34. Δ = b 2 - 4 a c , \Delta=b^{2}-4ac,\,
  35. x 1 , 2 = - b ± Δ 2 a = - b ± b 2 - 4 a c 2 a x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}
  36. x 1 = x 2 = - b 2 a x_{1}=x_{2}=-\frac{b}{2a}
  37. z 1 , 2 = - b ± i - Δ 2 a = - b ± i 4 a c - b 2 2 a . z_{1,2}=\frac{-b\pm i\sqrt{-\Delta}}{2a}=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}.
  38. ( x - r ) 2 = x 2 - 2 r x + r 2 . (x-r)^{2}=x^{2}-2rx+r^{2}.
  39. ( - 2 r ) 2 = 4 ( r 2 ) , (-2r)^{2}=4(r^{2}),
  40. b 2 = 4 c , b^{2}=4c,
  41. r = - b / 2. r=-b/2.
  42. b 2 - 4 a c . b^{2}-4ac.
  43. p ( x ) p(x)
  44. p ( x ) , p^{\prime}(x),
  45. D ( p ) D(p)
  46. R ( p , p ) R(p,p^{\prime})
  47. R ( p , p ) R(p,p^{\prime})
  48. p ( x ) = a n x n + a n - 1 x n - 1 + a n - 2 x n - 2 + + a 1 x + a 0 p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_{1}x+a_{0}
  49. R ( p , p ) = | a n a n - 1 a n - 2 a 1 a 0 0 0 0 a n a n - 1 a n - 2 a 1 a 0 0 0 0 0 a n a n - 1 a n - 2 a 1 a 0 n a n ( n - 1 ) a n - 1 ( n - 2 ) a n - 2 a 1 0 0 0 n a n ( n - 1 ) a n - 1 ( n - 2 ) a n - 2 a 1 0 0 0 0 0 n a n ( n - 1 ) a n - 1 ( n - 2 ) a n - 2 a 1 | . R(p,p^{\prime})=\left|\begin{matrix}&a_{n}&a_{n-1}&a_{n-2}&\ldots&a_{1}&a_{0}&% 0\ldots&\ldots&0\\ &0&a_{n}&a_{n-1}&a_{n-2}&\ldots&a_{1}&a_{0}&0\ldots&0\\ &\vdots&&&&&&&&\vdots\\ &0&\ldots&0&a_{n}&a_{n-1}&a_{n-2}&\ldots&a_{1}&a_{0}\\ &na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots&a_{1}&0&\ldots&\ldots&0\\ &0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots&a_{1}&0&\ldots&0\\ &\vdots&&&&&&&&\vdots\\ &0&0&\ldots&0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots&a_{1}\\ \end{matrix}\right|.
  50. D ( p ) D(p)
  51. p ( x ) p(x)
  52. D ( p ) = ( - 1 ) 1 2 n ( n - 1 ) 1 a n R ( p , p ) . D(p)=(-1)^{\frac{1}{2}n(n-1)}\frac{1}{a_{n}}R(p,p^{\prime}).\,
  53. | a 4 a 3 a 2 a 1 a 0 0 0 0 a 4 a 3 a 2 a 1 a 0 0 0 0 a 4 a 3 a 2 a 1 a 0 4 a 4 3 a 3 2 a 2 1 a 1 0 0 0 0 4 a 4 3 a 3 2 a 2 1 a 1 0 0 0 0 4 a 4 3 a 3 2 a 2 1 a 1 0 0 0 0 4 a 4 3 a 3 2 a 2 1 a 1 | . \begin{vmatrix}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0&0\\ &0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0\\ &0&0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\ &4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0&0\\ &0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0\\ &0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0\\ &0&0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\ \end{vmatrix}.
  54. a 4 a_{4}
  55. a n 2 n - 2 i < j ( r i - r j ) 2 a_{n}^{2n-2}\prod_{i<j}{(r_{i}-r_{j})^{2}}
  56. p ( x ) = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 = a n ( x - r 1 ) ( x - r 2 ) ( x - r n ) . \begin{matrix}p(x)&=&a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{1}x+a_{0}\\ &=&a_{n}(x-r_{1})(x-r_{2})\ldots(x-r_{n}).\end{matrix}
  57. a n a_{n}
  58. a n a_{n}
  59. 0 k n 4 0\leq k\leq\frac{n}{4}
  60. A x 2 + B x y + C y 2 + D x + E y + F = 0 , Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,\,
  61. B 2 - 4 A C , B^{2}-4AC,\,
  62. A T S A , A^{T}SA,
  63. ( det A ) 2 det S , (\det A)^{2}\det S,
  64. K n K^{n}
  65. a 1 x 1 2 + + a n x n 2 . a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}.
  66. i = 1 n a i L i 2 \sum_{i=1}^{n}a_{i}L_{i}^{2}
  67. n 0 > 0 n_{0}>0
  68. ( - 1 ) n - . (-1)^{n_{-}}.
  69. a x 2 + b x + c ax^{2}+bx+c
  70. a x 2 + b x y + c y 2 ax^{2}+bxy+cy^{2}
  71. [ a b / 2 b / 2 c ] . \begin{bmatrix}a&b/2\\ b/2&c\end{bmatrix}.
  72. a c - ( b / 2 ) 2 = a c - b 2 / 4 ac-(b/2)^{2}=ac-b^{2}/4
  73. b 2 - 4 a c b^{2}-4ac
  74. Λ n \Lambda_{n}
  75. Λ n \Lambda_{n}

Disjoint_sets.html

  1. A B A\cap B

Disjunction_elimination.html

  1. P P
  2. Q Q
  3. R R
  4. Q Q
  5. P P
  6. R R
  7. Q Q
  8. P Q , R Q , P R Q \frac{P\to Q,R\to Q,PR}{\therefore Q}
  9. P Q P\to Q
  10. R Q R\to Q
  11. P R PR
  12. Q Q
  13. ( P Q ) , ( R Q ) , ( P R ) Q (P\to Q),(R\to Q),(PR)\vdash Q
  14. \vdash
  15. Q Q
  16. P Q P\to Q
  17. R Q R\to Q
  18. P R PR
  19. ( ( ( P Q ) and ( R Q ) ) and ( P R ) ) Q (((P\to Q)\and(R\to Q))\and(PR))\to Q
  20. P P
  21. Q Q
  22. R R

Disjunction_introduction.html

  1. P P Q \frac{P}{\therefore PQ}
  2. P P
  3. P Q PQ
  4. P ( P Q ) P\vdash(PQ)
  5. \vdash
  6. P Q PQ
  7. P P
  8. P ( P Q ) P\to(PQ)
  9. P P
  10. Q Q

Disjunctive_syllogism.html

  1. P Q , ¬ P Q \frac{PQ,\neg P}{\therefore Q}
  2. P Q PQ
  3. ¬ P \neg P
  4. Q Q
  5. P Q , ¬ P Q P\lor Q,\lnot P\vdash Q
  6. \vdash
  7. Q Q
  8. P Q P\lor Q
  9. ¬ P \lnot P
  10. ( ( P Q ) and ¬ P ) Q ((PQ)\and\neg P)\to Q
  11. P P
  12. Q Q

Disperser.html

  1. A { 0 , 1 } m A\subseteq\{0,1\}^{m}
  2. P r U m [ A ] > 1 - ϵ Pr_{U_{m}}[A]>1-\epsilon
  3. ( k , ϵ ) (k,\epsilon)
  4. D i s : { 0 , 1 } n × { 0 , 1 } d { 0 , 1 } m Dis:\{0,1\}^{n}\times\{0,1\}^{d}\rightarrow\{0,1\}^{m}
  5. X X
  6. { 0 , 1 } n \{0,1\}^{n}
  7. H ( X ) k H_{\infty}(X)\geq k
  8. D i s ( X , U d ) Dis(X,U_{d})
  9. ( 1 - ϵ ) 2 m (1-\epsilon)2^{m}

Dissociation_constant.html

  1. K d K_{d}
  2. A x B y x A + y B \mathrm{A}_{x}\mathrm{B}_{y}\rightleftharpoons x\mathrm{A}+y\mathrm{B}
  3. A x B y \mathrm{A}_{x}\mathrm{B}_{y}
  4. K d = [ A ] x [ B ] y [ A x B y ] K_{d}=\frac{[A]^{x}\cdot[B]^{y}}{[A_{x}B_{y}]}
  5. [ A ] = K d [A]=K_{d}
  6. [ B ] = [ A B ] [B]=[AB]
  7. [ A B ] / ( [ B ] + [ A B ] ) = 1 / 2 [AB]/([B]+[AB])=1/2
  8. [ A ] 0 = [ A ] + [ A B ] [A]_{0}=[A]+[AB]
  9. [ B ] 0 = [ B ] + [ A B ] [B]_{0}=[B]+[AB]
  10. [ A ] 0 = K d [ A B ] [ B ] + [ A B ] [A]_{0}=K_{d}\frac{[AB]}{[B]}+[AB]
  11. [ A B ] = [ A ] 0 [ B ] K d + [ B ] = [ B ] 0 [ A ] K d + [ A ] [AB]=\frac{[A]_{0}[B]}{K_{d}+[B]}=\frac{[B]_{0}[A]}{K_{d}+[A]}
  12. L L
  13. M M
  14. L L
  15. n n
  16. [ L ] b o u n d [L]_{bound}
  17. [ L ] b o u n d = n [ M ] 0 [ L ] K d + [ L ] [L]_{bound}=\frac{n[M]_{0}[L]}{K_{d}+[L]}
  18. [ L ] b o u n d [ L M ] [L]_{bound}\neq[LM]
  19. [ L ] b o u n d = [ L M ] + 2 [ L 2 M ] + 3 [ L 3 M ] + + n [ L n M ] [L]_{bound}=[LM]+2[L_{2}M]+3[L_{3}M]+...+n[L_{n}M]
  20. [ L ] + [ M ] [ L M ] K 1 = [ L ] [ M ] [ L M ] [ L M ] = [ L ] [ M ] K 1 [L]+[M]\rightleftharpoons[LM]\qquad K^{\prime}_{1}=\frac{[L][M]}{[LM]}\qquad[% LM]=\frac{[L][M]}{K^{\prime}_{1}}
  21. [ L ] + [ L M ] [ L 2 M ] K 2 = [ L ] [ L M ] [ L 2 M ] [ L 2 M ] = [ L ] 2 [ M ] K 1 K 2 [L]+[LM]\rightleftharpoons[L_{2}M]\qquad K^{\prime}_{2}=\frac{[L][LM]}{[L_{2}M% ]}\qquad[L_{2}M]=\frac{[L]^{2}[M]}{K^{\prime}_{1}K^{\prime}_{2}}
  22. [ L ] + [ L 2 M ] [ L 3 M ] K 3 = [ L ] [ L 2 M ] [ L 3 M ] [ L 3 M ] = [ L ] 3 [ M ] K 1 K 2 K 3 [L]+[L_{2}M]\rightleftharpoons[L_{3}M]\qquad K^{\prime}_{3}=\frac{[L][L_{2}M]}% {[L_{3}M]}\qquad[L_{3}M]=\frac{[L]^{3}[M]}{K^{\prime}_{1}K^{\prime}_{2}K^{% \prime}_{3}}
  23. \vdots
  24. [ L ] + [ L n - 1 M ] [ L n M ] K n = [ L ] [ L n - 1 M ] [ L n M ] [ L n M ] = [ L ] n [ M ] K 1 K 2 K 3 K n [L]+[L_{n-1}M]\rightleftharpoons[L_{n}M]\qquad K^{\prime}_{n}=\frac{[L][L_{n-1% }M]}{[L_{n}M]}\qquad[L_{n}M]=\frac{[L]^{n}[M]}{K^{\prime}_{1}K^{\prime}_{2}K^{% \prime}_{3}\cdots K^{\prime}_{n}}
  25. r r
  26. r = [ L ] b o u n d [ M ] 0 = [ L M ] + 2 [ L 2 M ] + 3 [ L 3 M ] + + n [ L n M ] [ M ] + [ L M ] + [ L 2 M ] + [ L 3 M ] + + [ L n M ] = i = 1 n ( i [ L ] i j = 1 i K j ) 1 + i = 1 n ( [ L ] i j = 1 i K j ) r=\frac{[L]_{bound}}{[M]_{0}}=\frac{[LM]+2[L_{2}M]+3[L_{3}M]+...+n[L_{n}M]}{[M% ]+[LM]+[L_{2}M]+[L_{3}M]+...+[L_{n}M]}=\frac{\sum_{i=1}^{n}\left(\frac{i[L]^{i% }}{\prod_{j=1}^{i}K_{j}^{\prime}}\right)}{1+\sum_{i=1}^{n}\left(\frac{[L]^{i}}% {\prod_{j=1}^{i}K_{j}^{\prime}}\right)}
  27. K i = K d 1 n - i + 1 K_{i}^{\prime}=K_{d}\frac{1}{n-i+1}
  28. r = i = 1 n i ( j = 1 i n - j + 1 j ) ( [ L ] K d ) i 1 + i = 1 n ( j = 1 i n - j + 1 j ) ( [ L ] K d ) i = i = 1 n i ( n i ) ( [ L ] K d ) i 1 + i = 1 n ( n i ) ( [ L ] K d ) i r=\frac{\sum_{i=1}^{n}i\left(\prod_{j=1}^{i}\frac{n-j+1}{j}\right)\left(\frac{% [L]}{K_{d}}\right)^{i}}{1+\sum_{i=1}^{n}\left(\prod_{j=1}^{i}\frac{n-j+1}{j}% \right)\left(\frac{[L]}{K_{d}}\right)^{i}}=\frac{\sum_{i=1}^{n}i{\left({{n}% \atop{i}}\right)}\left(\frac{[L]}{K_{d}}\right)^{i}}{1+\sum_{i=1}^{n}{\left({{% n}\atop{i}}\right)}\left(\frac{[L]}{K_{d}}\right)^{i}}
  29. ( n i ) = n ! ( n - i ) ! i ! {\left({{n}\atop{i}}\right)}=\frac{n!}{(n-i)!i!}
  30. r = n ( [ L ] K d ) ( 1 + [ L ] K d ) n - 1 ( 1 + [ L ] K d ) n = n ( [ L ] K d ) ( 1 + [ L ] K d ) = n [ L ] K d + [ L ] = [ L ] b o u n d [ M ] 0 r=\frac{n\left(\frac{[L]}{K_{d}}\right)\left(1+\frac{[L]}{K_{d}}\right)^{n-1}}% {\left(1+\frac{[L]}{K_{d}}\right)^{n}}=\frac{n\left(\frac{[L]}{K_{d}}\right)}{% \left(1+\frac{[L]}{K_{d}}\right)}=\frac{n[L]}{K_{d}+[L]}=\frac{[L]_{bound}}{[M% ]_{0}}
  31. L \mathrm{L}
  32. P \mathrm{P}
  33. LP \mathrm{LP}
  34. L + P LP \mathrm{L}+\mathrm{P}\rightleftharpoons\mathrm{LP}
  35. K d = [ L ] [ P ] [ LP ] K_{d}=\frac{\left[\mathrm{L}\right]\left[\mathrm{P}\right]}{\left[\mathrm{LP}% \right]}
  36. [ P ] [\mathrm{P}]
  37. [ L ] [\mathrm{L}]
  38. [ LP ] [\mathrm{LP}]
  39. L \mathrm{L}
  40. [ C ] [\mathrm{C}]
  41. [ P ] [\mathrm{P}]
  42. Ab + Ag AbAg \,\text{Ab}+\,\text{Ag}\rightleftharpoons\,\text{AbAg}
  43. K a = [ AbAg ] [ Ab ] [ Ag ] = 1 K d K_{a}=\frac{\left[\,\text{AbAg}\right]}{\left[\,\text{Ab}\right]\left[\,\text{% Ag}\right]}=\frac{1}{K_{d}}
  44. K a = k forward k back = on-rate off-rate K_{a}=\frac{k\text{forward}}{k\text{back}}=\frac{\mbox{on-rate}~{}}{\mbox{off-% rate}~{}}
  45. K a K_{a}
  46. K a K_{a}
  47. p K a = - log 10 K a \mathrm{p}K_{a}=-\log_{10}{K_{a}}
  48. p K \mathrm{p}K
  49. H 3 B H + + H 2 B - K 1 = [ H + ] [ H 2 B - ] [ H 3 B ] p K 1 = - log K 1 H_{3}B\rightleftharpoons\ H^{+}+H_{2}B^{-}\qquad K_{1}={[H^{+}]\cdot[H_{2}B^{-% }]\over[H_{3}B]}\qquad pK_{1}=-\log K_{1}
  50. H 2 B - H + + H B - 2 K 2 = [ H + ] [ H B - 2 ] [ H 2 B - ] p K 2 = - log K 2 H_{2}B^{-}\rightleftharpoons\ H^{+}+HB^{-2}\qquad K_{2}={[H^{+}]\cdot[HB^{-2}]% \over[H_{2}B^{-}]}\qquad pK_{2}=-\log K_{2}
  51. H B - 2 H + + B - 3 K 3 = [ H + ] [ B - 3 ] [ H B - 2 ] p K 3 = - log K 3 HB^{-2}\rightleftharpoons\ H^{+}+B^{-3}\qquad K_{3}={[H^{+}]\cdot[B^{-3}]\over% [HB^{-2}]}\qquad pK_{3}=-\log K_{3}
  52. K w = [ H ] + [ OH ] - K_{w}=[\mbox{H}~{}^{+}][\mbox{OH}~{}^{-}]
  53. [ H O 2 ] \left[\mbox{H}~{}_{2}\mbox{O}~{}\right]

Distance.html

  1. d = ( Δ x ) 2 + ( Δ y ) 2 = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 . d=\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{% 2}}.\,
  2. d = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 . d=\sqrt{(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}=\sqrt{(x_{2}-x_{1})^{2}+% (y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}.
  3. = i = 1 n | x i - y i | =\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|
  4. = ( i = 1 n | x i - y i | 2 ) 1 / 2 =\left(\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|^{2}\right)^{1/2}
  5. = ( i = 1 n | x i - y i | p ) 1 / p =\left(\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|^{p}\right)^{1/p}
  6. = lim p ( i = 1 n | x i - y i | p ) 1 / p =\lim_{p\to\infty}\left(\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|^{p}\right)^{1/p}
  7. = max ( | x 1 - y 1 | , | x 2 - y 2 | , , | x n - y n | ) . =\max\left(|x_{1}-y_{1}|,|x_{2}-y_{2}|,\ldots,|x_{n}-y_{n}|\right).
  8. A = r ( 0 ) A=\vec{r}(0)
  9. B = r ( T ) B=\vec{r}(T)
  10. D = 0 T ( r ( t ) t ) 2 d t D=\int_{0}^{T}\sqrt{\left({\partial\vec{r}(t)\over\partial t}\right)^{2}}\,dt
  11. r ( t ) \vec{r}(t)
  12. r = r * r=r^{*}
  13. r * r^{*}
  14. g a b g_{ab}
  15. g a c r ˙ c g a b r ˙ b \sqrt{g^{ac}\dot{r}_{c}g_{ab}\dot{r}^{b}}
  16. 𝒟 = 0 L 0 T { ( r ( s , t ) t ) 2 + λ [ ( r ( s , t ) s ) 2 - 1 ] } d s d t \mathcal{D}=\int_{0}^{L}\int_{0}^{T}\left\{\sqrt{\left({\partial\vec{r}(s,t)% \over\partial t}\right)^{2}}+\lambda\left[\sqrt{\left({\partial\vec{r}(s,t)% \over\partial s}\right)^{2}}-1\right]\right\}\,ds\,dt
  17. s s
  18. t t
  19. r ( s , t = t i ) \vec{r}(s,t=t_{i})
  20. t i t_{i}
  21. s s
  22. r ( s = S , t ) \vec{r}(s=S,t)
  23. r ( s , 0 ) \vec{r}(s,0)
  24. r ( s , T ) \vec{r}(s,T)
  25. λ \lambda
  26. x T C x = 0 x\text{T}Cx=0
  27. x x^{\prime}
  28. x T C x x^{\prime\,}\text{T}Cx^{\prime}
  29. d ( A , B ) = inf x A , y B d ( x , y ) . d(A,B)=\inf_{x\in A,y\in B}d(x,y).

Distinct.html

  1. x 2 - 3 x + 2 = 0 x^{2}-3x+2=0
  2. ( x - 1 ) ( x - 2 ) = 0 (x-1)(x-2)=0
  3. x 2 - 2 x + 1 = 0 x^{2}-2x+1=0
  4. ( x - 1 ) ( x - 1 ) = 0 (x-1)(x-1)=0

Distortion.html

  1. y ( t ) y(t)
  2. x x
  3. y ( t ) = F ( x ( t ) ) y(t)=F(x(t))
  4. y ( t ) = A x ( t - T ) y(t)=A\cdot x(t-T)

Distribution_(mathematics).html

  1. T , φ \langle T,\varphi\rangle
  2. δ , φ = φ ( 0 ) , \left\langle\delta,\varphi\right\rangle=\varphi(0),
  3. T f , φ = 𝐑 f ( x ) φ ( x ) d x for φ D ( 𝐑 ) . \left\langle T_{f},\varphi\right\rangle=\int_{\mathbf{R}}f(x)\varphi(x)\,dx% \qquad\,\text{for}\quad\varphi\in D(\mathbf{R}).
  4. R μ , φ = 𝐑 φ d μ for φ D ( 𝐑 ) . \left\langle R_{\mu},\varphi\right\rangle=\int_{\mathbf{R}}\varphi\,d\mu\qquad% \,\text{for}\quad\varphi\in D(\mathbf{R}).
  5. 𝐑 φ d P = φ ( 0 ) , \int_{\mathbf{R}}\varphi\,dP=\varphi(0),
  6. T f = T f T^{\prime}_{f}=T_{f^{\prime}}
  7. f , φ = 𝐑 f φ d x = [ f ( x ) φ ( x ) ] - - 𝐑 f φ d x = - f , φ \left\langle f^{\prime},\varphi\right\rangle=\int_{\mathbf{R}}{}{f^{\prime}% \varphi\,dx}=\left[f(x)\varphi(x)\right]_{-\infty}^{\infty}-\int_{\mathbf{R}}{% }{f\varphi^{\prime}\,dx}=-\left\langle f,\varphi^{\prime}\right\rangle
  8. T , φ = - T , φ . \left\langle T^{\prime},\varphi\right\rangle=-\left\langle T,\varphi^{\prime}% \right\rangle.
  9. δ , φ = φ ( 0 ) . \left\langle\delta,\varphi\right\rangle=\varphi(0).
  10. H , φ = - H , φ = - - H ( x ) φ ( x ) d x = - 0 φ ( x ) d x = φ ( 0 ) - φ ( ) = φ ( 0 ) = δ , φ , \left\langle H^{\prime},\varphi\right\rangle=-\left\langle H,\varphi^{\prime}% \right\rangle=-\int_{-\infty}^{\infty}H(x)\varphi^{\prime}(x)\,dx=-\int_{0}^{% \infty}\varphi^{\prime}(x)\,dx=\varphi(0)-\varphi(\infty)=\varphi(0)=\left% \langle\delta,\varphi\right\rangle,
  11. δ , φ = - φ ( 0 ) . \langle\delta^{\prime},\varphi\rangle=-\varphi^{\prime}(0).
  12. k supp ( φ k ) K . \bigcup\nolimits_{k}\operatorname{supp}(\varphi_{k})\subset K.
  13. α φ k \partial^{\alpha}\varphi_{k}
  14. α φ \partial^{\alpha}\varphi
  15. X = i X i X=\bigcup\nolimits_{i}X_{i}
  16. ι i \iota_{i}
  17. U ¯ \overline{U}
  18. D ( U ) = i D K i \mathrm{D}(U)=\bigcup\nolimits_{i}\mathrm{D}_{K_{i}}
  19. φ α = max x K i | α φ | , \|\varphi\|_{\alpha}=\max_{x\in K_{i}}\left|\partial^{\alpha}\varphi\right|,
  20. φ α , K i = max x K i | α φ | . \|\varphi\|_{\alpha,K_{i}}=\max_{x\in K_{i}}\left|\partial^{\alpha}\varphi% \right|.
  21. T ( c 1 φ 1 + c 2 φ 2 ) = c 1 T ( φ 1 ) + c 2 T ( φ 2 ) T(c_{1}\varphi_{1}+c_{2}\varphi_{2})=c_{1}T(\varphi_{1})+c_{2}T(\varphi_{2})
  22. lim k T ( φ k ) = T ( lim k φ k ) \lim_{k\to\infty}T(\varphi_{k})=T\left(\lim_{k\to\infty}\varphi_{k}\right)
  23. | T ( φ ) | C K sup K | α φ | |T(\varphi)|\leq C_{K}\sup_{K}|\partial^{\alpha}\varphi|
  24. { D ( U ) × D ( U ) 𝐑 ( T , φ ) T , φ , \begin{cases}\mathrm{D}^{\prime}(U)\times\mathrm{D}(U)\to\mathbf{R}\\ (T,\varphi)\mapsto\langle T,\varphi\rangle,\end{cases}
  25. \langle
  26. \rangle
  27. T k , φ T , φ \langle T_{k},\varphi\rangle\to\langle T,\varphi\rangle
  28. f k ( x ) = { k if 0 x 1 / k 0 otherwise f_{k}(x)=\begin{cases}k&\,\text{if}\ 0\leq x\leq 1/k\\ 0&\,\text{otherwise}\end{cases}
  29. T k , φ = k 0 1 / k φ ( x ) d x φ ( 0 ) = δ , φ \langle T_{k},\varphi\rangle=k\int_{0}^{1/k}\varphi(x)\,dx\to\varphi(0)=% \langle\delta,\varphi\rangle
  30. T f , φ = U f φ d x . \langle T_{f},\varphi\rangle=\int_{U}f\varphi\,dx.
  31. f , φ = T f , φ . \langle f,\varphi\rangle=\langle T_{f},\varphi\rangle.
  32. μ , φ \langle\mu,\varphi\rangle
  33. φ n , ψ T , ψ \langle\varphi_{n},\psi\rangle\to\langle T,\psi\rangle
  34. U A φ ( x ) ψ ( x ) d x = U φ ( x ) A t ψ ( x ) d x for all φ , ψ D ( U ) . \int_{U}A\varphi(x)\cdot\psi(x)\,dx=\int_{U}\varphi(x)\cdot A^{t}\psi(x)\,dx% \qquad\,\text{for all}\ \varphi,\psi\in D(U).
  35. A T , φ = T , A t φ for all φ D ( U ) . \langle AT,\varphi\rangle=\langle T,A^{t}\varphi\rangle\qquad\,\text{for all}% \ \varphi\in D(U).
  36. A φ = φ x k . A\varphi=\frac{\partial\varphi}{\partial x_{k}}.
  37. U φ x k ψ d x = - U φ ψ x k d x , \int_{U}\frac{\partial\varphi}{\partial x_{k}}\psi\,dx=-\int_{U}\varphi\frac{% \partial\psi}{\partial x_{k}}\,dx,
  38. T x k , φ = - T , φ x k for all φ D ( U ) . \left\langle\frac{\partial T}{\partial x_{k}},\varphi\right\rangle=-\left% \langle T,\frac{\partial\varphi}{\partial x_{k}}\right\rangle\qquad\,\text{for% all}\ \varphi\in D(U).
  39. α T , φ = ( - 1 ) | α | T , α φ for all φ D ( U ) . \left\langle\partial^{\alpha}T,\varphi\right\rangle=(-1)^{|\alpha|}\left% \langle T,\partial^{\alpha}\varphi\right\rangle\mbox{ for all }~{}\varphi\in% \mathrm{D}(U).
  40. m T , φ = T , m φ for all φ D ( U ) . \langle mT,\varphi\rangle=\langle T,m\varphi\rangle\qquad\,\text{for all}\ % \varphi\in D(U).
  41. U M φ ( x ) ψ ( x ) d x = U m ( x ) φ ( x ) ψ ( x ) d x = U φ ( x ) m ( x ) ψ ( x ) d x = U φ ( x ) M ψ ( x ) d x , \int_{U}M\varphi(x)\cdot\psi(x)\,dx=\int_{U}m(x)\varphi(x)\cdot\psi(x)\,dx=% \int_{U}\varphi(x)\cdot m(x)\psi(x)\,dx=\int_{U}\varphi(x)\cdot M\psi(x)\,dx,
  42. m δ = m ( 0 ) δ - m δ = m ( 0 ) δ - m ( 0 ) δ . m\delta^{\prime}=m(0)\delta^{\prime}-m^{\prime}\delta=m(0)\delta^{\prime}-m^{% \prime}(0)\delta.\,
  43. P T = | α | k p α α T , PT=\sum\nolimits_{|\alpha|\leq k}p_{\alpha}\partial^{\alpha}T,
  44. | α | k p α α T , φ = T , | α | k ( - 1 ) | α | α ( p α φ ) . \left\langle\sum\nolimits_{|\alpha|\leq k}p_{\alpha}\partial^{\alpha}T,\varphi% \right\rangle=\left\langle T,\sum\nolimits_{|\alpha|\leq k}(-1)^{|\alpha|}% \partial^{\alpha}(p_{\alpha}\varphi)\right\rangle.
  45. T F D ( V ) . T\circ F\in\mathrm{D}^{\prime}(V).
  46. F : T F T = T F . F^{\sharp}:T\mapsto F^{\sharp}T=T\circ F.
  47. V φ F ( x ) ψ ( x ) d x = U φ ( x ) ψ ( F - 1 ( x ) ) | det d F - 1 ( x ) | d x . \int_{V}\varphi\circ F(x)\psi(x)\,dx=\int_{U}\varphi(x)\psi\left(F^{-1}(x)% \right)\left|\det dF^{-1}(x)\right|\,dx.
  48. F T , φ = T , | det d ( F - 1 ) | φ F - 1 . \left\langle F^{\sharp}T,\varphi\right\rangle=\left\langle T,\left|\det d(F^{-% 1})\right|\varphi\circ F^{-1}\right\rangle.
  49. ρ V U T , φ = T , E V U φ \langle\rho_{VU}T,\varphi\rangle=\langle T,E_{VU}\varphi\rangle
  50. T ( x ) = n = 1 n δ ( x - 1 n ) T(x)=\sum_{n=1}^{\infty}n\,\delta\left(x-\frac{1}{n}\right)
  51. T , φ = 0 \langle T,\varphi\rangle=0
  52. supp T = U { V ρ V U T = 0 } . \operatorname{supp}\,T=U\setminus\bigcup\left\{V\mid\rho_{VU}T=0\right\}.
  53. p α , β ( φ ) = sup x 𝐑 n | x α D β φ ( x ) | p_{\alpha,\beta}(\varphi)=\sup_{x\in\mathbf{R}^{n}}|x^{\alpha}D^{\beta}\varphi% (x)|
  54. p α , β ( φ ) < . p_{\alpha,\beta}(\varphi)<\infty.
  55. lim m T ( φ m ) = 0. \lim_{m\to\infty}T(\varphi_{m})=0.
  56. lim m p α , β ( φ m ) = 0 \lim_{m\to\infty}p_{\alpha,\beta}(\varphi_{m})=0
  57. | x | n exp ( - x 2 ) \propto|x|^{n}\cdot\exp(-x^{2})
  58. F d T d x = i x F T F\dfrac{dT}{dx}=ixFT
  59. F ( ψ T ) = F ψ * F T F(\psi T)=F\psi*FT\,
  60. { C f : D ( 𝐑 n ) D ( 𝐑 n ) C f : g f * g \begin{cases}C_{f}:\mathrm{D}(\mathbf{R}^{n})\to\mathrm{D}(\mathbf{R}^{n})\\ C_{f}:g\mapsto f*g\end{cases}
  61. C f g , φ = 𝐑 n φ ( x ) 𝐑 n f ( x - y ) g ( y ) d y d x = g , C f ~ φ \left\langle C_{f}g,\varphi\right\rangle=\int_{\mathbf{R}^{n}}\varphi(x)\int_{% \mathbf{R}^{n}}f(x-y)g(y)\,dy\,dx=\left\langle g,C_{\widetilde{f}}\varphi\right\rangle
  62. f ~ ( x ) = f ( - x ) \scriptstyle{\widetilde{f}(x)=f(-x)}
  63. f * T , φ = T , f ~ * φ \langle f*T,\varphi\rangle=\left\langle T,\widetilde{f}*\varphi\right\rangle
  64. τ x φ ( y ) = φ ( y - x ) \tau_{x}\varphi(y)=\varphi(y-x)
  65. ( f * T ) ( x ) = T , τ x f ~ . (f*T)(x)=\left\langle T,\tau_{x}\widetilde{f}\right\rangle.
  66. ch ( f * T ) = ch f + ch T \operatorname{ch}(f*T)=\operatorname{ch}f+\operatorname{ch}T
  67. S * ( T * φ ) = ( S * T ) * φ S*(T*\varphi)=(S*T)*\varphi
  68. ψ ( x ) = T , τ - x φ . \psi(x)=\langle T,\tau_{-x}\varphi\rangle.
  69. S * T , φ = S , ψ . \langle S*T,\varphi\rangle=\langle S,\psi\rangle.
  70. α ( S * T ) = ( α S ) * T = S * ( α T ) . \partial^{\alpha}(S*T)=(\partial^{\alpha}S)*T=S*(\partial^{\alpha}T).
  71. f , φ C | α | N , | β | M sup x 𝐑 n | x α D β φ ( x ) | = C | α | N , | β | M p α , β ( φ ) . \langle f,\varphi\rangle\leq C\sum\nolimits_{|\alpha|\leq N,|\beta|\leq M}\sup% _{x\in\mathbf{R}^{n}}\left|x^{\alpha}D^{\beta}\varphi(x)\right|=C\sum\nolimits% _{|\alpha|\leq N,|\beta|\leq M}p_{\alpha,\beta}(\varphi).
  72. f = D α F . f=D^{\alpha}F.\,
  73. f = | α | m a α D α ( τ P δ ) f=\sum\nolimits_{|\alpha|\leq m}a_{\alpha}D^{\alpha}(\tau_{P}\delta)
  74. T = α D α g α \displaystyle T=\sum\nolimits_{\alpha}D^{\alpha}g_{\alpha}
  75. ( p . v . 1 x ) [ ϕ ] = lim ϵ 0 + | x | ϵ ϕ ( x ) x d x \left(\operatorname{p.v.}\frac{1}{x}\right)[\phi]=\lim_{\epsilon\to 0^{+}}\int% _{|x|\geq\epsilon}\frac{\phi(x)}{x}\,dx
  76. ( δ × x ) × p . v . 1 x = 0 \left(\delta\times x\right)\times\operatorname{p.v.}\frac{1}{x}=0
  77. δ × ( x × p . v . 1 x ) = δ \delta\times\left(x\times\operatorname{p.v.}\frac{1}{x}\right)=\delta

Disulfide_bond.html

  1. χ s s \chi_{ss}
  2. C β - S γ - S γ - C β C^{\beta}-S^{\gamma}-S^{\gamma}-C^{\beta}
  3. i = n ! ( n - 2 p ) ! p ! 2 p i=\frac{n!}{(n-2p)!\ p!\ 2^{p}}
  4. i = n ! ( n - 2 p ) ! p ! 2 p = n ! ( n - 2 ( n 2 ) ) ! ( n 2 ) ! 2 ( n 2 ) = n ! ( n 2 ) ! 2 ( n 2 ) i=\frac{n!}{(n-2p)!\ p!\ 2^{p}}=\frac{n!}{(n-2(\frac{n}{2}))!\ {(\frac{n}{2})}% !\ {2^{(\frac{n}{2})}}}=\frac{n!}{{(\frac{n}{2})}!\ {2^{(\frac{n}{2})}}}
  5. i = ( n - 1 ) ( n - 3 ) ( n - 5 ) 1 i=(n-1)\ (n-3)\ (n-5)\ ...1
  6. i = ( 8 - 1 ) ( 8 - 3 ) ( 8 - 5 ) ( 8 - 7 ) = 7 5 3 1 = 105 i=(8-1)\ (8-3)\ (8-5)\ (8-7)=7\ \cdot\ 5\ \cdot\ 3\ \cdot\ 1=105

Divergence.html

  1. div 𝐅 ( p ) = lim V { p } S ( V ) 𝐅 𝐧 | V | d S \operatorname{div}\,\mathbf{F}(p)=\lim_{V\rightarrow\{p\}}\iint_{S(V)}{\mathbf% {F}\cdot\mathbf{n}\over|V|}\;dS
  2. div 𝐅 = 𝐅 = U x + V y + W z . \operatorname{div}\,\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial U}{% \partial x}+\frac{\partial V}{\partial y}+\frac{\partial W}{\partial z}.
  3. ϵ ¯ ¯ \underline{\underline{\epsilon}}
  4. div ( ϵ ¯ ¯ ) = [ ϵ x x x + ϵ x y y + ϵ x z z ϵ y x x + ϵ y y y + ϵ y z z ϵ z x x + ϵ z y y + ϵ z z z ] \overrightarrow{\operatorname{div}}\,(\mathbf{\underline{\underline{\epsilon}}% })=\begin{bmatrix}\frac{\partial\epsilon_{xx}}{\partial x}+\frac{\partial% \epsilon_{xy}}{\partial y}+\frac{\partial\epsilon_{xz}}{\partial z}\\ \frac{\partial\epsilon_{yx}}{\partial x}+\frac{\partial\epsilon_{yy}}{\partial y% }+\frac{\partial\epsilon_{yz}}{\partial z}\\ \frac{\partial\epsilon_{zx}}{\partial x}+\frac{\partial\epsilon_{zy}}{\partial y% }+\frac{\partial\epsilon_{zz}}{\partial z}\end{bmatrix}
  5. 𝐅 = 𝐞 r F r + 𝐞 θ F θ + 𝐞 z F z , \mathbf{F}=\mathbf{e}_{r}F_{r}+\mathbf{e}_{\theta}F_{\theta}+\mathbf{e}_{z}F_{% z},
  6. div 𝐅 = 𝐅 = 1 r r ( r F r ) + 1 r F θ θ + F z z . \operatorname{div}\,\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{1}{r}\frac{\partial% }{\partial r}(rF_{r})+\frac{1}{r}\frac{\partial F_{\theta}}{\partial\theta}+% \frac{\partial F_{z}}{\partial z}\,.
  7. θ \theta
  8. ϕ \phi
  9. div 𝐅 = 𝐅 = 1 r 2 r ( r 2 F r ) + 1 r sin θ θ ( sin θ F θ ) + 1 r sin θ F ϕ ϕ . \operatorname{div}\,\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{1}{r^{2}}\frac{% \partial}{\partial r}(r^{2}F_{r})+\frac{1}{r\sin\theta}\frac{\partial}{% \partial\theta}(\sin\theta\,F_{\theta})+\frac{1}{r\sin\theta}\frac{\partial F_% {\phi}}{\partial\phi}.
  10. 3 {\mathbb{R}}^{3}
  11. 𝐄 = - Φ ( 𝐫 ) , \mathbf{E}=-\nabla\Phi(\mathbf{r})\,,
  12. Φ ( 𝐫 ) = 3 d 3 𝐫 div 𝐯 ( 𝐫 ) 4 π | 𝐫 - 𝐫 | . \Phi(\mathbf{r})=\int_{\mathbb{R}^{3}}\,{\rm d}^{3}\mathbf{r}^{\prime}\;\frac{% \operatorname{div}\,\mathbf{v}(\mathbf{r}^{\prime})}{4\pi|\mathbf{r}-\mathbf{r% }^{\prime}|}\,.
  13. div ( a 𝐅 + b 𝐆 ) = a div ( 𝐅 ) + b div ( 𝐆 ) \operatorname{div}(a\mathbf{F}+b\mathbf{G})=a\;\operatorname{div}(\mathbf{F})+% b\;\operatorname{div}(\mathbf{G})
  14. φ \varphi
  15. div ( φ 𝐅 ) = grad ( φ ) 𝐅 + φ div ( 𝐅 ) , \operatorname{div}(\varphi\mathbf{F})=\operatorname{grad}(\varphi)\cdot\mathbf% {F}+\varphi\;\operatorname{div}(\mathbf{F}),
  16. ( φ 𝐅 ) = ( φ ) 𝐅 + φ ( 𝐅 ) . \nabla\cdot(\varphi\mathbf{F})=(\nabla\varphi)\cdot\mathbf{F}+\varphi\;(\nabla% \cdot\mathbf{F}).
  17. div ( 𝐅 × 𝐆 ) = curl ( 𝐅 ) 𝐆 - 𝐅 curl ( 𝐆 ) , \operatorname{div}(\mathbf{F}\times\mathbf{G})=\operatorname{curl}(\mathbf{F})% \cdot\mathbf{G}\;-\;\mathbf{F}\cdot\operatorname{curl}(\mathbf{G}),
  18. ( 𝐅 × 𝐆 ) = ( × 𝐅 ) 𝐆 - 𝐅 ( × 𝐆 ) . \nabla\cdot(\mathbf{F}\times\mathbf{G})=(\nabla\times\mathbf{F})\cdot\mathbf{G% }-\mathbf{F}\cdot(\nabla\times\mathbf{G}).
  19. div ( φ ) = Δ φ . \operatorname{div}(\nabla\varphi)=\Delta\varphi.
  20. ( × 𝐅 ) = 0 \nabla\cdot(\nabla\times\mathbf{F})=0
  21. { scalar fields on U } \{\mbox{scalar fields on }~{}U\}\;
  22. { vector fields on U } \to\{\mbox{vector fields on }~{}U\}\;
  23. { vector fields on U } \to\{\mbox{vector fields on }~{}U\}\;
  24. { scalar fields on U } \to\{\mbox{scalar fields on }~{}U\}\;
  25. j = F 1 d y d z + F 2 d z d x + F 3 d x d y j=F_{1}\ dy\wedge dz+F_{2}\ dz\wedge dx+F_{3}\ dx\wedge dy
  26. ρ = 1 d x d y d z \rho=1dx\wedge dy\wedge dz
  27. d j dj
  28. d j = ( F 1 x + F 2 y + F 3 z ) d x d y d z = ( 𝐅 ) ρ dj=\left(\frac{\partial F_{1}}{\partial x}+\frac{\partial F_{2}}{\partial y}+% \frac{\partial F_{3}}{\partial z}\right)dx\wedge dy\wedge dz=(\nabla\cdot% \mathbf{F})\rho
  29. 𝐅 = d 𝐅 \nabla\cdot\mathbf{F}=\star{\mathrm{d}}{\star{\mathbf{F}^{\flat}}}
  30. \flat
  31. \star
  32. 𝐅 = ( F 1 , F 2 , , F n ) , \mathbf{F}=(F_{1},F_{2},\dots,F_{n}),
  33. 𝐱 = ( x 1 , x 2 , , x n ) \mathbf{x}=(x_{1},x_{2},\dots,x_{n})
  34. d 𝐱 = ( d x 1 , d x 2 , , d x n ) d\mathbf{x}=(dx_{1},dx_{2},\dots,dx_{n})
  35. div 𝐅 = 𝐅 = F 1 x 1 + F 2 x 2 + + F n x n . \operatorname{div}\,\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial F_{1}}{% \partial x_{1}}+\frac{\partial F_{2}}{\partial x_{2}}+\cdots+\frac{\partial F_% {n}}{\partial x_{n}}.
  36. ( φ 𝐅 ) = ( φ ) 𝐅 + φ ( 𝐅 ) . \nabla\cdot(\varphi\mathbf{F})=(\nabla\varphi)\cdot\mathbf{F}+\varphi\;(\nabla% \cdot\mathbf{F}).
  37. φ \varphi
  38. μ \mu
  39. 3 \mathbb{R}^{3}
  40. j = i X μ j=i_{X}\mu
  41. μ \mu
  42. d j = div ( X ) μ dj=\operatorname{div}(X)\mu
  43. X μ = div ( X ) μ \mathcal{L}_{X}\mu=\operatorname{div}(X)\mu
  44. \nabla
  45. div ( X ) = X = X ; a a \operatorname{div}(X)=\nabla\cdot X=X^{a}_{;a}
  46. X \nabla X
  47. div ( X ) = 1 det g a ( det g X a ) \operatorname{div}(X)=\frac{1}{\sqrt{\operatorname{det}g}}\partial_{a}(\sqrt{% \operatorname{det}g}X^{a})
  48. g g
  49. a \partial_{a}
  50. x a x^{a}
  51. F μ F^{\mu}
  52. 𝐅 = μ F μ \nabla\cdot\mathbf{F}=\nabla_{\mu}F^{\mu}
  53. μ \nabla_{\mu}
  54. ( div T ) ( Y 1 , , Y q - 1 ) = trace ( X # ( T ) ( X , , Y 1 , , Y q - 1 ) ) (\operatorname{div}T)(Y_{1},...,Y_{q-1})=\operatorname{trace}(X\mapsto\#(% \nabla T)(X,\cdot,Y_{1},...,Y_{q-1}))

Division_(mathematics).html

  1. 20 ÷ 4 = 5 20\div 4=5
  2. a a
  3. b = c , a / b = c , or a b = c . b=c,\quad a/b=c,\quad\,\text{or}\quad\frac{a}{b}=c.
  4. a / b a/b
  5. a b , \tfrac{a}{b},
  6. a b \frac{a}{b}
  7. a / b a/b\,
  8. b \ a b\backslash a
  9. a / b {a}/{b}
  10. a ÷ b a\div b
  11. b ) a b)~{}a
  12. b ) a ¯ b\overline{)a}
  13. a + b c = ( a + b ) ÷ c = a c + b c \frac{a+b}{c}=(a+b)\div c=\frac{a}{c}+\frac{b}{c}
  14. ( a + b ) × c = a × c + b × c (a+b)\times c=a\times c+b\times c
  15. a b + c = a ÷ ( b + c ) a b + a c \frac{a}{b+c}=a\div(b+c)\neq\frac{a}{b}+\frac{a}{c}
  16. 26 11 2 36 100 . \tfrac{26}{11}\simeq 2\tfrac{36}{100}.
  17. 26 11 . \tfrac{26}{11}.
  18. 26 11 \tfrac{26}{11}
  19. 26 11 = 2 remainder 4. \tfrac{26}{11}=2\mbox{ remainder }~{}4.
  20. 26 11 = 2. \tfrac{26}{11}=2.
  21. p / q r / s = p q × s r = p s q r . {p/q\over r/s}={p\over q}\times{s\over r}={ps\over qr}.
  22. p + i q r + i s = ( p + i q ) ( r - i s ) ( r + i s ) ( r - i s ) = p r + q s + i ( q r - p s ) r 2 + s 2 = p r + q s r 2 + s 2 + i q r - p s r 2 + s 2 . {p+iq\over r+is}={(p+iq)(r-is)\over(r+is)(r-is)}={pr+qs+i(qr-ps)\over r^{2}+s^% {2}}={pr+qs\over r^{2}+s^{2}}+i{qr-ps\over r^{2}+s^{2}}.
  23. p e i q r e i s = p e i q e - i s r e i s e - i s = p r e i ( q - s ) . {pe^{iq}\over re^{is}}={pe^{iq}e^{-is}\over re^{is}e^{-is}}={p\over r}e^{i(q-s% )}.
  24. a b {a\over b}
  25. a 1 b a\cdot{1\over b}
  26. a b - 1 a\cdot b^{-1}
  27. b b
  28. b - 1 b^{-1}
  29. b b - 1 = b - 1 b = 1 bb^{-1}=b^{-1}b=1
  30. 1 1
  31. a b = a c ab=ac
  32. b a = c a ba=ca
  33. ( f g ) = f g - f g g 2 . {\left(\frac{f}{g}\right)}^{\prime}=\frac{f^{\prime}g-fg^{\prime}}{g^{2}}.

Division_ring.html

  1. σ : \sigma:\mathbb{C}\rightarrow\mathbb{C}
  2. \mathbb{C}
  3. ( ( z , σ ) ) \mathbb{C}((z,\sigma))
  4. z z
  5. α \alpha\in\mathbb{C}
  6. z i α := σ i ( α ) z i z^{i}\alpha:=\sigma^{i}(\alpha)z^{i}
  7. i i\in\mathbb{Z}
  8. σ \sigma
  9. σ = i d σ=id
  10. F F
  11. F F
  12. σ \sigma

Divisor.html

  1. n n
  2. n n
  3. n n
  4. m m
  5. n n
  6. m m
  7. n n
  8. m m
  9. n n
  10. n n
  11. m m
  12. m n , m\mid n,
  13. k k
  14. m k = n mk=n
  15. 0 0 0\mid 0
  16. m 0 m\neq 0
  17. 0 0 0\mid 0
  18. 7 × 6 = 42 7\times 6=42
  19. 7 42 7\mid 42
  20. 5 0 5\mid 0
  21. 5 × 0 = 0 5\times 0=0
  22. A = { 1 , 2 , 3 , 4 , 5 , 6 , 10 , 12 , 15 , 20 , 30 , 60 } A=\{1,2,3,4,5,6,10,12,15,20,30,60\}
  23. a b a\mid b
  24. b c b\mid c
  25. a c a\mid c
  26. a b a\mid b
  27. b a b\mid a
  28. a = b a=b
  29. a = - b a=-b
  30. a b a\mid b
  31. a c a\mid c
  32. a ( b + c ) a\mid(b+c)
  33. a ( b - c ) a\mid(b-c)
  34. a b a\mid b
  35. c b c\mid b
  36. ( a + c ) b (a+c)\mid b
  37. 2 6 2\mid 6
  38. 3 6 3\mid 6
  39. a b c a\mid bc
  40. ( a , b ) = 1 (a,b)=1
  41. a c a\mid c
  42. p p
  43. p a b p\mid ab
  44. p a p\mid a
  45. p b p\mid b
  46. n n
  47. n n
  48. n n
  49. n n
  50. n n
  51. n > 1 n>1
  52. n n
  53. n n
  54. n n
  55. n n
  56. n n
  57. n n
  58. d ( n ) d(n)
  59. m m
  60. n n
  61. d ( m n ) = d ( m ) × d ( n ) d(mn)=d(m)\times d(n)
  62. d ( 42 ) = 8 = 2 × 2 × 2 = d ( 2 ) × d ( 3 ) × d ( 7 ) d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)
  63. m m
  64. n n
  65. d ( m n ) = d ( m ) × d ( n ) d(mn)=d(m)\times d(n)
  66. n n
  67. σ ( n ) \sigma(n)
  68. σ ( 42 ) = 96 = 3 × 4 × 8 = σ ( 2 ) × σ ( 3 ) × σ ( 7 ) = 1 + 2 + 3 + 6 + 7 + 14 + 21 + 42 \sigma(42)=96=3\times 4\times 8=\sigma(2)\times\sigma(3)\times\sigma(7)=1+2+3+% 6+7+14+21+42
  69. n n
  70. n = p 1 ν 1 p 2 ν 2 p k ν k n=p_{1}^{\nu_{1}}\,p_{2}^{\nu_{2}}\cdots p_{k}^{\nu_{k}}
  71. n n
  72. d ( n ) = ( ν 1 + 1 ) ( ν 2 + 1 ) ( ν k + 1 ) , d(n)=(\nu_{1}+1)(\nu_{2}+1)\cdots(\nu_{k}+1),
  73. p 1 μ 1 p 2 μ 2 p k μ k p_{1}^{\mu_{1}}\,p_{2}^{\mu_{2}}\cdots p_{k}^{\mu_{k}}
  74. 0 μ i ν i 0\leq\mu_{i}\leq\nu_{i}
  75. 1 i k . 1\leq i\leq k.
  76. n n
  77. d ( n ) < 2 n d(n)<2\sqrt{n}
  78. d ( 1 ) + d ( 2 ) + + d ( n ) = n ln n + ( 2 γ - 1 ) n + O ( n ) . d(1)+d(2)+\cdots+d(n)=n\ln n+(2\gamma-1)n+O(\sqrt{n}).
  79. γ \gamma
  80. ln n \ln n
  81. 0 0 0\mid 0
  82. \mathbb{N}
  83. \mathbb{Z}
  84. a b , a c b = j a , c = k a b + c = ( j + k ) a a ( b + c ) a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b+c=(j+k)a\Rightarrow a% \mid(b+c)
  85. a b , a c b = j a , c = k a b - c = ( j - k ) a a ( b - c ) a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b-c=(j-k)a\Rightarrow a% \mid(b-c)

Domain_of_a_function.html

  1. f ( x ) = 1 / x f(x)=1/x
  2. f ( x ) = { 1 / x x 0 0 x = 0 f(x)=\begin{cases}1/x&x\not=0\\ 0&x=0\end{cases}
  3. \mathbb{R}
  4. x x x\mapsto\sqrt{x}
  5. x 0 x\geq 0
  6. x x x\mapsto\sqrt{x}
  7. tan x = sin x cos x \tan x=\frac{\sin x}{\cos x}
  8. x π 2 + k π , k = 0 , ± 1 , ± 2 , x\neq\frac{\pi}{2}+k\pi,k=0,\pm 1,\pm 2,\ldots

Dominance_(genetics).html

  1. p 2 + 2 p q + q 2 = 1 p^{2}+2pq+q^{2}=1
  2. p + q = 1 p+q=1
  3. q = [ u r a d i c a l , u l e s s t h a n v a r > f ( a a ) ] q=[u^{\prime}radical^{\prime},u^{\prime}\\ lessthanvar>f(aa)^{\prime}]

Dominoes.html

  1. ( n + 1 ) ( n + 2 ) 2 \frac{(n+1)(n+2)}{2}
  2. n n

Doppler_effect.html

  1. f f
  2. f 0 f\text{0}
  3. f = ( c + v r c + v s ) f 0 f=\left(\frac{c+v\text{r}}{c+v\text{s}}\right)f_{0}\,
  4. c c\;
  5. v r v\text{r}\,
  6. v s v\text{s}\,
  7. v s v\text{s}\,
  8. v r v\text{r}\,
  9. f f
  10. f 0 f\text{0}
  11. f = ( 1 + Δ v c ) f 0 f=\left(1+\frac{\Delta v}{c}\right)f_{0}
  12. Δ f = Δ v c f 0 \Delta f=\frac{\Delta v}{c}f_{0}
  13. Δ f = f - f 0 \Delta f=f-f_{0}\,
  14. Δ v = v r - v s \Delta v=v\text{r}-v\text{s}\,
  15. f = ( c + v r c + v s ) f 0 f=\left(\frac{c+v\text{r}}{c+v\text{s}}\right)f_{0}\,
  16. c c
  17. f = ( 1 + v r c 1 + v s c ) f 0 = ( 1 + v r c ) ( 1 1 + v s c ) f 0 f=\left(\frac{1+\frac{v\text{r}}{c}}{1+\frac{v\text{s}}{c}}\right)f_{0}=\left(% 1+\frac{v\text{r}}{c}\right)\left(\frac{1}{1+\frac{v\text{s}}{c}}\right)f_{0}\,
  18. v s c 1 \frac{v\text{s}}{c}\ll 1
  19. 1 1 + v s c 1 - v s c \frac{1}{1+\frac{v\text{s}}{c}}\approx 1-\frac{v\text{s}}{c}
  20. f 0 f\text{0}
  21. \,\text{--}
  22. f f
  23. f = ( c c + v s ) f 0 f=\left(\frac{c}{c+v\text{s}}\right)f_{0}
  24. \,\text{--}
  25. \,\text{--}
  26. f = ( c + v r c ) f 0 f=\left(\frac{c+v\text{r}}{c}\right)f_{0}
  27. f = ( c + v r c + v s ) f 0 f=\left(\frac{c+v\text{r}}{c+v\text{s}}\right)f_{0}
  28. v radial = v s cos θ v\text{radial}=v\text{s}\cdot\cos{\theta}
  29. θ \theta
  30. Δ v \Delta v
  31. Δ f = 2 Δ v c f 0 \Delta f=\frac{2\Delta v}{c}f_{0}

Double-precision_floating-point_format.html

  1. e e
  2. ( - 1 ) sign ( 1. b 51 b 50 b 0 ) 2 × 2 e - 1023 (-1)^{\,\text{sign}}(1.b_{51}b_{50}...b_{0})_{2}\times 2^{e-1023}
  3. ( - 1 ) sign ( 1 + i = 1 52 b 52 - i 2 - i ) × 2 e - 1023 (-1)^{\,\text{sign}}\left(1+\sum_{i=1}^{52}b_{52-i}2^{-i}\right)\times 2^{e-10% 23}
  4. ( - 1 ) sign × 2 exponent - exponent bias × 1. mantissa (-1)^{\,\text{sign}}\times 2^{\,\text{exponent}-\,\text{exponent bias}}\times 1% .\,\text{mantissa}
  5. ( - 1 ) sign × 2 1 - exponent bias × 0. mantissa (-1)^{\,\text{sign}}\times 2^{1-\,\text{exponent bias}}\times 0.\,\text{mantissa}

Double-sideband_suppressed-carrier_transmission.html

  1. V m cos ( ω m t ) Message × V c cos ( ω c t ) Carrier = V m V c 2 [ cos ( ( ω m + ω c ) t ) + cos ( ( ω m - ω c ) t ) ] Modulated Signal \underbrace{V_{m}\cos\left(\omega_{m}t\right)}_{\mbox{Message}~{}}\times% \underbrace{V_{c}\cos\left(\omega_{c}t\right)}_{\mbox{Carrier}~{}}=\underbrace% {\frac{V_{m}V_{c}}{2}\left[\cos\left(\left(\omega_{m}+\omega_{c}\right)t\right% )+\cos\left(\left(\omega_{m}-\omega_{c}\right)t\right)\right]}_{\mbox{% Modulated Signal}~{}}
  2. V m V c 2 [ cos ( ( ω m + ω c ) t ) + cos ( ( ω m - ω c ) t ) ] Modulated Signal × V c cos ( ω c t ) Carrier \overbrace{\frac{V_{m}V_{c}}{2}\left[\cos\left(\left(\omega_{m}+\omega_{c}% \right)t\right)+\cos\left(\left(\omega_{m}-\omega_{c}\right)t\right)\right]}^{% \mbox{Modulated Signal}~{}}\times\overbrace{V^{\prime}_{c}\cos\left(\omega_{c}% t\right)}^{\mbox{Carrier}~{}}
  3. = ( 1 2 V c V c ) V m cos ( ω m t ) original message + 1 2 V c V c V m [ cos ( ( ω m + 2 ω c ) t ) + cos ( ( ω m - 2 ω c ) t ) ] =\left(\frac{1}{2}V_{c}V^{\prime}_{c}\right)\underbrace{V_{m}\cos(\omega_{m}t)% }_{\,\text{original message}}+\frac{1}{2}V_{c}V^{\prime}_{c}V_{m}\left[\cos((% \omega_{m}+2\omega_{c})t)+\cos((\omega_{m}-2\omega_{c})t)\right]
  4. ω c ω m \omega_{c}\gg\omega_{m}
  5. f ( t ) f(t)
  6. V c cos ( ω c t ) V_{c}\cos(\omega_{c}t)
  7. V c cos [ ( ω c + Δ ω ) t + θ ] V^{\prime}_{c}\cos\left[(\omega_{c}+\Delta\omega)t+\theta\right]
  8. f ( t ) × V c cos ( ω c t ) × V c cos [ ( ω c + Δ ω ) t + θ ] f(t)\times V_{c}\cos(\omega_{c}t)\times V^{\prime}_{c}\cos\left[(\omega_{c}+% \Delta\omega)t+\theta\right]
  9. = 1 2 V c V c f ( t ) cos ( Δ ω t + θ ) + 1 2 V c V c f ( t ) cos [ ( 2 ω c + Δ ω ) t + θ ] =\frac{1}{2}V_{c}V^{\prime}_{c}f(t)\cos\left(\Delta\omega\cdot t+\theta\right)% +\frac{1}{2}V_{c}V^{\prime}_{c}f(t)\cos\left[(2\omega_{c}+\Delta\omega)t+% \theta\right]
  10. After low pass filter 1 2 V c V c f ( t ) cos ( Δ ω t + θ ) \xrightarrow{\,\text{After low pass filter}}\frac{1}{2}V_{c}V^{\prime}_{c}f(t)% \cos\left(\Delta\omega\cdot t+\theta\right)
  11. cos ( Δ ω t + θ ) \cos\left(\Delta\omega\cdot t+\theta\right)
  12. θ \theta
  13. Δ ω t \Delta\omega\cdot t
  14. s ( t ) = 1 2 cos ( 2 π 800 t ) - 1 2 cos ( 2 π 1200 t ) s(t)=\frac{1}{2}\cos\left(2\pi 800t\right)-\frac{1}{2}\cos\left(2\pi 1200t\right)
  15. c ( t ) = cos ( 2 π 5000 t ) c(t)=\cos\left(2\pi 5000t\right)
  16. x ( t ) = cos ( 2 π 5000 t ) Carrier × [ 1 2 cos ( 2 π 800 t ) - 1 2 cos ( 2 π 1200 t ) ] Message Signal x(t)=\underbrace{\cos\left(2\pi 5000t\right)}_{\mbox{Carrier}}~{}\times% \underbrace{\left[\frac{1}{2}\cos\left(2\pi 800t\right)-\frac{1}{2}\cos\left(2% \pi 1200t\right)\right]}_{\mbox{Message Signal}}~{}

Double-slit_experiment.html

  1. θ θ
  2. d sin θ d θ d\sin\theta\approx d\theta
  3. d θ n = n λ , n = 0 , 1 , 2 , ~{}d\theta_{n}=n\lambda,~{}n=0,1,2,\ldots
  4. θ f λ / d \theta_{f}\approx\lambda/d
  5. z z
  6. w = z θ f = z λ / d ~{}w=z\theta_{f}=z\lambda/d
  7. d d
  8. λ λ
  9. z z
  10. b b
  11. I ( θ ) cos 2 [ π d sin θ λ ] sinc 2 [ π b sin θ λ ] \begin{aligned}\displaystyle I(\theta)&\displaystyle\propto\cos^{2}\left[{% \frac{\pi d\sin\theta}{\lambda}}\right]~{}\mathrm{sinc}^{2}\left[\frac{\pi b% \sin\theta}{\lambda}\right]\end{aligned}
  12. s i n c sinc
  13. c o s cos
  14. s i n c sinc
  15. A path ( x , y , z , t ) = e i S ( x , y , z , t ) A_{\,\text{path}}(x,y,z,t)=e^{iS(x,y,z,t)}
  16. p ( x , y , z , t ) | all paths e i S ( x , y , z , t ) | 2 p(x,y,z,t)\propto\left|\int_{\,\text{all paths}}e^{iS(x,y,z,t)}\right|^{2}
  17. all space p ( x , y , z , t ) d V = 1 \iiint_{\,\text{all space}}p(x,y,z,t)\,\mathrm{d}V=1

Double_pendulum.html

  1. \ell
  2. m m
  3. I = 1 12 m 2 \textstyle I=\frac{1}{12}m\ell^{2}
  4. x 1 = 2 sin θ 1 , x_{1}=\frac{\ell}{2}\sin\theta_{1},
  5. y 1 = - 2 cos θ 1 y_{1}=-\frac{\ell}{2}\cos\theta_{1}
  6. x 2 = ( sin θ 1 + 1 2 sin θ 2 ) , x_{2}=\ell\left(\sin\theta_{1}+\frac{1}{2}\sin\theta_{2}\right),
  7. y 2 = - ( cos θ 1 + 1 2 cos θ 2 ) . y_{2}=-\ell\left(\cos\theta_{1}+\frac{1}{2}\cos\theta_{2}\right).
  8. L = Kinetic Energy - Potential Energy = 1 2 m ( v 1 2 + v 2 2 ) + 1 2 I ( θ ˙ 1 2 + θ ˙ 2 2 ) - m g ( y 1 + y 2 ) = 1 2 m ( x ˙ 1 2 + y ˙ 1 2 + x ˙ 2 2 + y ˙ 2 2 ) + 1 2 I ( θ ˙ 1 2 + θ ˙ 2 2 ) - m g ( y 1 + y 2 ) \begin{aligned}\displaystyle L&\displaystyle=\mathrm{Kinetic~{}Energy}-\mathrm% {Potential~{}Energy}\\ &\displaystyle=\frac{1}{2}m\left(v_{1}^{2}+v_{2}^{2}\right)+\frac{1}{2}I\left(% {\dot{\theta}_{1}}^{2}+{\dot{\theta}_{2}}^{2}\right)-mg\left(y_{1}+y_{2}\right% )\\ &\displaystyle=\frac{1}{2}m\left({\dot{x}_{1}}^{2}+{\dot{y}_{1}}^{2}+{\dot{x}_% {2}}^{2}+{\dot{y}_{2}}^{2}\right)+\frac{1}{2}I\left({\dot{\theta}_{1}}^{2}+{% \dot{\theta}_{2}}^{2}\right)-mg\left(y_{1}+y_{2}\right)\end{aligned}
  9. L = 1 6 m 2 [ θ ˙ 2 2 + 4 θ ˙ 1 2 + 3 θ ˙ 1 θ ˙ 2 cos ( θ 1 - θ 2 ) ] + 1 2 m g ( 3 cos θ 1 + cos θ 2 ) . L=\frac{1}{6}m\ell^{2}\left[{\dot{\theta}_{2}}^{2}+4{\dot{\theta}_{1}}^{2}+3{% \dot{\theta}_{1}}{\dot{\theta}_{2}}\cos(\theta_{1}-\theta_{2})\right]+\frac{1}% {2}mg\ell\left(3\cos\theta_{1}+\cos\theta_{2}\right).
  10. p θ 1 = L θ ˙ 1 = 1 6 m 2 [ 8 θ ˙ 1 + 3 θ ˙ 2 cos ( θ 1 - θ 2 ) ] p_{\theta_{1}}=\frac{\partial L}{\partial{\dot{\theta}_{1}}}=\frac{1}{6}m\ell^% {2}\left[8{\dot{\theta}_{1}}+3{\dot{\theta}_{2}}\cos(\theta_{1}-\theta_{2})\right]
  11. p θ 2 = L θ ˙ 2 = 1 6 m 2 [ 2 θ ˙ 2 + 3 θ ˙ 1 cos ( θ 1 - θ 2 ) ] . p_{\theta_{2}}=\frac{\partial L}{\partial{\dot{\theta}_{2}}}=\frac{1}{6}m\ell^% {2}\left[2{\dot{\theta}_{2}}+3{\dot{\theta}_{1}}\cos(\theta_{1}-\theta_{2})% \right].
  12. θ ˙ 1 = 6 m 2 2 p θ 1 - 3 cos ( θ 1 - θ 2 ) p θ 2 16 - 9 cos 2 ( θ 1 - θ 2 ) {\dot{\theta}_{1}}=\frac{6}{m\ell^{2}}\frac{2p_{\theta_{1}}-3\cos(\theta_{1}-% \theta_{2})p_{\theta_{2}}}{16-9\cos^{2}(\theta_{1}-\theta_{2})}
  13. θ ˙ 2 = 6 m 2 8 p θ 2 - 3 cos ( θ 1 - θ 2 ) p θ 1 16 - 9 cos 2 ( θ 1 - θ 2 ) . {\dot{\theta}_{2}}=\frac{6}{m\ell^{2}}\frac{8p_{\theta_{2}}-3\cos(\theta_{1}-% \theta_{2})p_{\theta_{1}}}{16-9\cos^{2}(\theta_{1}-\theta_{2})}.
  14. p ˙ θ 1 = L θ 1 = - 1 2 m 2 [ θ ˙ 1 θ ˙ 2 sin ( θ 1 - θ 2 ) + 3 g sin θ 1 ] {\dot{p}_{\theta_{1}}}=\frac{\partial L}{\partial\theta_{1}}=-\frac{1}{2}m\ell% ^{2}\left[{\dot{\theta}_{1}}{\dot{\theta}_{2}}\sin(\theta_{1}-\theta_{2})+3% \frac{g}{\ell}\sin\theta_{1}\right]
  15. p ˙ θ 2 = L θ 2 = - 1 2 m 2 [ - θ ˙ 1 θ ˙ 2 sin ( θ 1 - θ 2 ) + g sin θ 2 ] . {\dot{p}_{\theta_{2}}}=\frac{\partial L}{\partial\theta_{2}}=-\frac{1}{2}m\ell% ^{2}\left[-{\dot{\theta}_{1}}{\dot{\theta}_{2}}\sin(\theta_{1}-\theta_{2})+% \frac{g}{\ell}\sin\theta_{2}\right].
  16. 10 / g 10\sqrt{\ell/g}
  17. 100 / g 100\sqrt{\ell/g}
  18. 1000 / g 1000\sqrt{\ell/g}
  19. 10000 / g 10000\sqrt{\ell/g}
  20. 10000 / g 10000\sqrt{\ell/g}
  21. 3 cos θ 1 + cos θ 2 = 2. 3\cos\theta_{1}+\cos\theta_{2}=2.\,
  22. 3 cos θ 1 + cos θ 2 > 2 , 3\cos\theta_{1}+\cos\theta_{2}>2,\,

Dow_Jones_Industrial_Average.html

  1. DJIA = p d \,\text{DJIA}={\sum p\over d}
  2. DJIA = p old d old = p new d new . \,\text{DJIA}={\sum p\text{old}\over d\text{old}}={\sum p\text{new}\over d% \text{new}}.

Drake_equation.html

  1. N = R f p n e f f i f c L N=R_{\ast}\cdot f_{p}\cdot n_{e}\cdot f_{\ell}\cdot f_{i}\cdot f_{c}\cdot L
  2. N = R f p n e f f i f c L N=R_{\ast}\cdot f_{p}\cdot n_{e}\cdot f_{\ell}\cdot f_{i}\cdot f_{c}\cdot L
  3. f , f i , f c , f_{\ell},f_{i},f_{c},
  4. L L
  5. R f p n e f f i R^{\ast}\cdot f_{p}\cdot n_{e}\cdot f_{\ell}\cdot f_{i}
  6. f i f_{i}
  7. n e n_{e}
  8. f c f_{c}
  9. L L
  10. F f 2 ( M g E ) - C 1 R i 1 M = L / S o Ff^{2}(MgE)-C^{1}Ri^{1}~{}\cdot~{}M=L/So
  11. N = R f p n e f f i f c L f d N=R_{\ast}\cdot f_{p}\cdot n_{e}\cdot f_{\ell}\cdot f_{i}\cdot f_{c}\cdot L% \cdot f_{d}

Draw_(poker).html

  1. P 1 P_{1}
  2. P 1 = outs unseen cards P_{1}=\frac{\mathrm{outs}}{\mathrm{unseen}\,\,\mathrm{cards}}
  3. P 2 P_{2}
  4. P 2 = 1 - non outs unseen cards × non outs - 1 unseen cards - 1 P_{2}=1-\frac{\mathrm{non}\,\mathrm{outs}}{\mathrm{unseen}\,\,\mathrm{cards}}% \times\frac{\mathrm{non}\,\mathrm{outs}-1}{\mathrm{unseen}\,\,\mathrm{cards}-1}
  5. n o n o u t s = unseen cards - outs nonouts={\mathrm{unseen}\,\,\mathrm{cards}}-\mathrm{outs}
  6. P r r P_{rr}
  7. P r r = outs unseen cards × outs - 1 unseen cards - 1 P_{rr}=\frac{\mathrm{outs}}{\mathrm{unseen}\,\,\mathrm{cards}}\times\frac{% \mathrm{outs}-1}{\mathrm{unseen}\,\,\mathrm{cards}-1}

Drop_(liquid).html

  1. γ \gamma
  2. F γ = π d γ \,F_{\gamma}=\pi d\gamma
  3. F g = m g F_{g}=mg
  4. F γ sin α F_{\gamma}\sin\alpha
  5. m g = π d γ sin α \,mg=\pi d\gamma\sin\alpha
  6. γ \gamma
  7. m g = π d γ \,mg=\pi d\gamma

Dual_number.html

  1. ε = ( 0 1 0 0 ) and a + b ε = ( a b 0 a ) \varepsilon=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\quad\,\text{and}\quad a+b\varepsilon=\begin{pmatrix}a&b\\ 0&a\end{pmatrix}
  2. exp ( b ε ) = ( n = 0 ( b ε ) n / n ! ) = 1 + b ε \exp(b\varepsilon)=\left(\sum^{\infty}_{n=0}(b\varepsilon)^{n}/n!\right)=1+b\varepsilon\!
  3. ( t , x ) = ( t , x ) ( 1 v 0 1 ) , (t^{\prime},x^{\prime})=(t,x)\begin{pmatrix}1&v\\ 0&1\end{pmatrix}\ ,
  4. t = t , x = v t + x , \ \ t^{\prime}=t,\ \ x^{\prime}=vt+x,\!
  5. x 1 = x , y 1 = v x + y x_{1}=x,\ \ y_{1}=vx+y\
  6. x = x 1 = v / 2 a , y = y 1 + v 2 / 4 a x^{\prime}=x_{1}=v/2a,\ \ y^{\prime}=y_{1}+v^{2}/4a
  7. 1 \mathbb{R}^{1}
  8. P ( a + b ε ) = \displaystyle P(a+b\varepsilon)=
  9. P P^{\prime}
  10. P P
  11. f ( a + b ε ) = n = 0 f ( n ) ( a ) b n ε n n ! = f ( a ) + b f ( a ) ε f(a+b\varepsilon)=\sum_{n=0}^{\infty}{{f^{(n)}(a)b^{n}\varepsilon^{n}}\over{n!% }}=f(a)+bf^{\prime}(a)\varepsilon
  12. a + b ε c + d ε {a+b\varepsilon\over c+d\varepsilon}
  13. = ( a + b ε ) ( c - d ε ) ( c + d ε ) ( c - d ε ) = a c - a d ε + b c ε - b d ε 2 ( c 2 + c d ε - c d ε - d 2 ε 2 ) = a c - a d ε + b c ε - 0 c 2 - 0 ={(a+b\varepsilon)(c-d\varepsilon)\over(c+d\varepsilon)(c-d\varepsilon)}={ac-% ad\varepsilon+bc\varepsilon-bd\varepsilon^{2}\over(c^{2}+cd\varepsilon-cd% \varepsilon-d^{2}\varepsilon^{2})}={ac-ad\varepsilon+bc\varepsilon-0\over c^{2% }-0}
  14. = a c + ε ( b c - a d ) c 2 ={ac+\varepsilon(bc-ad)\over c^{2}}
  15. = a c + ( b c - a d ) c 2 ε ={a\over c}+{(bc-ad)\over c^{2}}\varepsilon
  16. a + b ε = ( x + y ε ) d ε = x d ε + 0 {a+b\varepsilon=(x+y\varepsilon)d\varepsilon}={xd\varepsilon+0}
  17. b d + y ε {b\over d}+{y\varepsilon}

Dual_polyhedron.html

  1. x 2 + y 2 + z 2 = r 2 , x^{2}+y^{2}+z^{2}=r^{2},
  2. ( x 0 , y 0 , z 0 ) (x_{0},y_{0},z_{0})
  3. x 0 x + y 0 y + z 0 z = r 2 x_{0}x+y_{0}y+z_{0}z=r^{2}
  4. r 0 r_{0}
  5. r 1 r_{1}
  6. r 2 r_{2}
  7. r 1 . r 2 = r 0 2 r_{1}.r_{2}=r_{0}^{2}

Dual_space.html

  1. ( φ + ψ ) ( x ) = φ ( x ) + ψ ( x ) \displaystyle(\varphi+\psi)(x)=\varphi(x)+\psi(x)
  2. 𝐞 i ( c 1 𝐞 1 + + c n 𝐞 n ) = c i , i = 1 , , n \mathbf{e}^{i}(c^{1}\mathbf{e}_{1}+\cdots+c^{n}\mathbf{e}_{n})=c^{i},\quad i=1% ,\ldots,n
  3. 𝐞 i ( 𝐞 j ) = δ j i \mathbf{e}^{i}(\mathbf{e}_{j})=\delta^{i}_{j}
  4. δ j i \delta^{i}_{j}
  5. α A f α 𝐞 α \sum_{\alpha\in A}f_{\alpha}\mathbf{e}_{\alpha}
  6. T ( α A f α 𝐞 α ) = α A f α T ( e α ) = α A f α θ α . T\biggl(\sum_{\alpha\in A}f_{\alpha}\mathbf{e}_{\alpha}\biggr)=\sum_{\alpha\in A% }f_{\alpha}T(e_{\alpha})=\sum_{\alpha\in A}f_{\alpha}\theta_{\alpha}.
  7. V ( F A ) 0 α A F . V\cong(F^{A})_{0}\cong\bigoplus_{\alpha\in A}{F}.
  8. V * ( α A F ) * α A F * α A F F A V^{*}\cong\biggl(\bigoplus_{\alpha\in A}F\biggr)^{*}\cong\prod_{\alpha\in A}F^% {*}\cong\prod_{\alpha\in A}F\cong F^{A}
  9. v v , v\mapsto\langle v,\cdot\rangle
  10. Φ , : V V * \Phi_{\langle\cdot,\cdot\rangle}:V\to V^{*}
  11. [ Φ , ( v ) , w ] = v , w . [\Phi_{\langle\cdot,\cdot\rangle}(v),w]=\langle v,w\rangle.
  12. v , w Φ = ( Φ ( v ) ) ( w ) = [ Φ ( v ) , w ] . \langle v,w\rangle_{\Phi}=(\Phi(v))(w)=[\Phi(v),w].\,
  13. Φ , : V V * ¯ . \Phi_{\langle\cdot,\cdot\rangle}:V\to\overline{V^{*}}.
  14. f ( α v ) = α ¯ f ( v ) . f(\alpha v)=\overline{\alpha}f(v).
  15. f * ( φ ) = φ f f^{*}(\varphi)=\varphi\circ f\,
  16. [ f * ( φ ) , v ] = [ φ , f ( v ) ] , [f^{*}(\varphi),\,v]=[\varphi,\,f(v)],
  17. 0 T o S o V * . 0\subset T^{o}\subset S^{o}\subset V^{*}.
  18. ( A B ) o A o + B o , (A\cap B)^{o}\supseteq A^{o}+B^{o},
  19. ( i I A i ) o = i I A i o . \left(\bigcup_{i\in I}A_{i}\right)^{o}=\bigcap_{i\in I}A_{i}^{o}.
  20. ( A + B ) o = A o B o . (A+B)^{o}=A^{o}\cap B^{o}.\,
  21. W o o = W W^{oo}=W\,
  22. ( V / W ) * W o . (V/W)^{*}\cong W^{o}.
  23. 𝔽 = {\mathbb{F}}={\mathbb{C}}
  24. {\mathbb{R}}
  25. V * V^{*}
  26. V V^{\prime}
  27. V V
  28. V V^{\prime}
  29. φ : V 𝔽 \varphi:V\to{\mathbb{F}}
  30. V V^{\prime}
  31. V V
  32. 𝒜 {\mathcal{A}}
  33. V V
  34. V V
  35. 𝒜 {\mathcal{A}}
  36. φ A = sup x A | φ ( x ) | , \|\varphi\|_{A}=\sup_{x\in A}|\varphi(x)|,
  37. φ \varphi
  38. V V
  39. A A
  40. 𝒜 {\mathcal{A}}
  41. φ i \varphi_{i}
  42. φ \varphi
  43. V V^{\prime}
  44. A 𝒜 φ i - φ A = sup x A | φ i ( x ) - φ ( x ) | i 0. \forall A\in{\mathcal{A}}\qquad\|\varphi_{i}-\varphi\|_{A}=\sup_{x\in A}|% \varphi_{i}(x)-\varphi(x)|\underset{i\to\infty}{\longrightarrow}0.
  45. 𝒜 {\mathcal{A}}
  46. x x
  47. V V
  48. A 𝒜 A\in{\mathcal{A}}
  49. x V A 𝒜 x A , \forall x\in V\qquad\exists A\in{\mathcal{A}}\qquad x\in A,
  50. A 𝒜 A\in{\mathcal{A}}
  51. B 𝒜 B\in{\mathcal{A}}
  52. C 𝒜 C\in{\mathcal{A}}
  53. A , B 𝒜 C 𝒜 A B C , \forall A,B\in{\mathcal{A}}\qquad\exists C\in{\mathcal{A}}\qquad A\cup B% \subseteq C,
  54. 𝒜 {\mathcal{A}}
  55. A 𝒜 λ 𝔽 λ A 𝒜 , \forall A\in{\mathcal{A}}\qquad\forall\lambda\in{\mathbb{F}}\qquad\lambda\cdot A% \in{\mathcal{A}},
  56. V V^{\prime}
  57. U A = { φ V : | | φ | | A < 1 } , A 𝒜 , U_{A}=\{\varphi\in V^{\prime}:\quad||\varphi||_{A}<1\},\qquad A\in{\mathcal{A}},
  58. V V^{\prime}
  59. V V
  60. 𝒜 {\mathcal{A}}
  61. V V
  62. V V
  63. V V^{\prime}
  64. φ = sup x 1 | φ ( x ) | . \|\varphi\|=\sup_{\|x\|\leq 1}|\varphi(x)|.
  65. V V^{\prime}
  66. V V
  67. 𝒜 {\mathcal{A}}
  68. V V
  69. V V^{\prime}
  70. V V
  71. 𝒜 {\mathcal{A}}
  72. V V
  73. V V^{\prime}
  74. 𝐚 p = ( n = 0 | a n | p ) 1 / p \|\mathbf{a}\|_{p}=\left(\sum_{n=0}^{\infty}|a_{n}|^{p}\right)^{1/p}
  75. T ( φ ) = φ T , φ W . T^{\prime}(\varphi)=\varphi\circ T,\quad\varphi\in W^{\prime}.\,
  76. ( U T ) = T U . (U\circ T)^{\prime}=T^{\prime}\circ U^{\prime}.\,
  77. i V T * = T i V . i_{V}\circ T^{*}=T^{\prime}\circ i_{V}.\,
  78. W = { φ V : W ker φ } . W^{\perp}=\{\varphi\in V^{\prime}:W\subset\ker\varphi\}.\,
  79. ker ( j ) = W \ker(j^{\prime})=W^{\perp}
  80. Ψ ( x ) ( φ ) = φ ( x ) , x V , φ V . \Psi(x)(\varphi)=\varphi(x),\quad x\in V,\ \varphi\in V^{\prime}.\,
  81. φ V φ ( x ) , x V , \varphi\in V^{\prime}\mapsto\varphi(x),\quad x\in V,\,

Duodecimal.html

  1. 1 / 2 {1}/{2}
  2. 1 / 3 {1}/{3}
  3. 2 / 3 {2}/{3}
  4. 1 / 4 {1}/{4}
  5. 3 / 4 {3}/{4}
  6. 3 ¯ \overline{3}
  7. 5 ¯ \overline{5}
  8. 3 ¯ \overline{3}
  9. 5 ¯ \overline{5}
  10. 8 ¯ \overline{8}
  11. 8 ¯ \overline{8}
  12. 4972 ¯ \overline{4972}
  13. 0 Ɛ 62 68781 Ɛ 05915343 ¯ \overline{0Ɛ62ᘔ68781Ɛ05915343ᘔ}
  14. 4972 ¯ \overline{4972}
  15. 0 Ɛ 62 68781 Ɛ 05915343 ¯ \overline{0Ɛ62ᘔ68781Ɛ05915343ᘔ}
  16. 43 0 Ɛ 62 68781 Ɛ 059153 ¯ \overline{43ᘔ0Ɛ62ᘔ68781Ɛ059153}
  17. 43 0 Ɛ 62 68781 Ɛ 059153 ¯ \overline{43ᘔ0Ɛ62ᘔ68781Ɛ059153}
  18. 3 ¯ \overline{3}
  19. 4 ¯ \overline{4}
  20. 6 ¯ \overline{6}
  21. 8 ¯ \overline{8}
  22. 3 ¯ \overline{3}
  23. 3 ¯ \overline{3}
  24. 7 ¯ \overline{7}
  25. 6 ¯ \overline{6}
  26. 2 ¯ \overline{2}
  27. 6 ¯ \overline{6}
  28. 3 ¯ \overline{3}
  29. 1 ¯ \overline{1}
  30. 6 ¯ \overline{6}
  31. 5 ¯ \overline{5}
  32. 3 ¯ \overline{3}
  33. 4 ¯ \overline{4}
  34. 6 ¯ \overline{6}
  35. 8 ¯ \overline{8}
  36. 2497 ¯ \overline{2497}
  37. 15343 0 Ɛ 62 68781 Ɛ 059 ¯ \overline{15343ᘔ0Ɛ62ᘔ68781Ɛ059}
  38. 2497 ¯ \overline{2497}
  39. 2 68781 Ɛ 05915343 0 Ɛ 6 ¯ \overline{2ᘔ68781Ɛ05915343ᘔ0Ɛ6}
  40. 7249 ¯ \overline{7249}
  41. 43 0 Ɛ 62 68781 Ɛ 059153 ¯ \overline{43ᘔ0Ɛ62ᘔ68781Ɛ059153}
  42. 4972 ¯ \overline{4972}
  43. 05915343 0 Ɛ 62 68781 Ɛ ¯ \overline{05915343ᘔ0Ɛ62ᘔ68781Ɛ}
  44. 7249 ¯ \overline{7249}
  45. 7249 ¯ \overline{7249}
  46. 8781 Ɛ 05915343 0 Ɛ 62 6 ¯ \overline{8781Ɛ05915343ᘔ0Ɛ62ᘔ6}
  47. 4972 ¯ \overline{4972}
  48. 0 Ɛ 62 68781 Ɛ 05915343 ¯ \overline{ᘔ0Ɛ62ᘔ68781Ɛ05915343}
  49. 9724 ¯ \overline{9724}
  50. 0 Ɛ 62 68781 Ɛ 05915343 ¯ \overline{0Ɛ62ᘔ68781Ɛ05915343ᘔ}
  51. 9724 ¯ \overline{9724}
  52. 0 Ɛ 62 68781 Ɛ 05915343 ¯ \overline{0Ɛ62ᘔ68781Ɛ05915343ᘔ}
  53. 3 ¯ \overline{3}
  54. 2497 ¯ \overline{2497}
  55. 6 ¯ \overline{6}
  56. 142857 ¯ \overline{142857}
  57. 186 35 ¯ \overline{186ᘔ35}
  58. 1 ¯ \overline{1}
  59. 05915343 0 < b r > Ɛ 62 68781 Ɛ ¯ \overline{05915343ᘔ0<br>Ɛ62ᘔ68781Ɛ}
  60. 1 ¯ \overline{1}
  61. 2497 ¯ \overline{2497}
  62. 09 ¯ \overline{09}
  63. 1 ¯ \overline{1}
  64. 3 ¯ \overline{3}
  65. 012345679 ¯ \overline{012345679}
  66. 15343 0 Ɛ 6 < b r > 2 68781 Ɛ 059 ¯ \overline{15343ᘔ0Ɛ6<br>2ᘔ68781Ɛ059}
  67. 1 / 2 {1}/{2}
  68. 1 / 3 {1}/{3}
  69. 1 / 4 {1}/{4}
  70. 1 / 6 {1}/{6}
  71. 1 / 8 {1}/{8}
  72. 1 / 9 {1}/{9}
  73. 1 / 10 {1}/{10}
  74. 1 / 5 {1}/{5}
  75. 1 / 7 {1}/{7}
  76. 1 / {1}/{ᘔ}
  77. 1 / Ɛ {1}/{Ɛ}
  78. 1 / 11 {1}/{11}
  79. 1 / 12 {1}/{12}
  80. 5 / 8 {5}/{8}
  81. 5 / 8 {5}/{8}
  82. 5 / 8 {5}/{8}
  83. 5 / 8 {5}/{8}
  84. 576 / 9 {576}/{9}
  85. 810 / 9 {810}/{9}
  86. 400 / 9 {400}/{9}
  87. 576 / 9 {576}/{9}
  88. 1 / 8 {1}/{8}
  89. 1 / ( 2 × 2 × 2 ) {1}/{(2×2×2)}
  90. 1 / 20 {1}/{20}
  91. 1 / ( 2 × 2 × 5 ) {1}/{(2×2×5)}
  92. 1 / 500 {1}/{500}
  93. 1 / ( 2 × 2 × 5 × 5 × 5 ) {1}/{(2×2×5×5×5)}
  94. 1 / 3 {1}/{3}
  95. 1 / 7 {1}/{7}
  96. 1 / 8 {1}/{8}
  97. 1 / 20 {1}/{20}
  98. 1 / 500 {1}/{500}
  99. 1 / 3 {1}/{3}
  100. 1 / 7 {1}/{7}
  101. 3 ¯ \overline{3}
  102. 2497 ¯ \overline{2497}
  103. 6 ¯ \overline{6}
  104. 142857 ¯ \overline{142857}
  105. 186 35 ¯ \overline{186ᘔ35}
  106. 1 ¯ \overline{1}
  107. 2497 ¯ \overline{2497}
  108. 09 ¯ \overline{09}
  109. 1 ¯ \overline{1}
  110. 3 ¯ \overline{3}
  111. 076923 ¯ \overline{076923}
  112. 0 Ɛ ¯ \overline{0Ɛ}
  113. 714285 ¯ \overline{714285}
  114. 35186 ¯ \overline{ᘔ35186}
  115. 6 ¯ \overline{6}
  116. 9724 ¯ \overline{9724}
  117. 0588235294117647 ¯ \overline{0588235294117647}
  118. 08579214 Ɛ 36429 7 ¯ \overline{08579214Ɛ36429ᘔ7}
  119. 5 ¯ \overline{5}
  120. 052631578947368421 ¯ \overline{052631578947368421}
  121. 076 Ɛ 45 ¯ \overline{076Ɛ45}
  122. 7249 ¯ \overline{7249}
  123. 047619 ¯ \overline{047619}
  124. 6 3518 ¯ \overline{6ᘔ3518}
  125. 45 ¯ \overline{45}
  126. 6 ¯ \overline{6}
  127. 0434782608695652173913 ¯ \overline{0434782608695652173913}
  128. 06316948421 ¯ \overline{06316948421}
  129. 6 ¯ \overline{6}
  130. 05915343 0 Ɛ 62 68781 Ɛ ¯ \overline{05915343ᘔ0Ɛ62ᘔ68781Ɛ}
  131. 384615 ¯ \overline{384615}
  132. 56 ¯ \overline{56}
  133. 037 ¯ \overline{037}
  134. 571428 ¯ \overline{571428}
  135. 5186 3 ¯ \overline{5186ᘔ3}
  136. 0344827586206896551724137931 ¯ \overline{0344827586206896551724137931}
  137. 04 Ɛ 7 ¯ \overline{04Ɛ7}
  138. 3 ¯ \overline{3}
  139. 4972 ¯ \overline{4972}
  140. 032258064516129 ¯ \overline{032258064516129}
  141. 0478 093598166 Ɛ 74311 Ɛ 28623 55 ¯ \overline{0478ᘔᘔ093598166Ɛ74311Ɛ28623ᘔ55}
  142. 03 ¯ \overline{03}
  143. 4 ¯ \overline{4}
  144. 2941176470588235 ¯ \overline{2941176470588235}
  145. 429 708579214 Ɛ 36 ¯ \overline{429ᘔ708579214Ɛ36}
  146. 285714 ¯ \overline{285714}
  147. 0414559 Ɛ 3931 ¯ \overline{0414559Ɛ3931}
  148. 7 ¯ \overline{7}
  149. 1 + 5 2 \scriptstyle\frac{1+\sqrt{5}}{2}
  150. x x\!
  151. x x\!
  152. x x\!

Duty_cycle.html

  1. D = T P × 100 % D=\frac{T}{P}\times 100\%
  2. D D
  3. T T
  4. P P

Dyadic_rational.html

  1. a 2 b \frac{a}{2^{b}}
  2. a 2 b + c 2 d = 2 d - b a + c 2 d ( d b ) \frac{a}{2^{b}}+\frac{c}{2^{d}}=\frac{2^{d-b}a+c}{2^{d}}\quad(d\geq b)
  3. a 2 b - c 2 d = 2 d - b a - c 2 d ( d b ) \frac{a}{2^{b}}-\frac{c}{2^{d}}=\frac{2^{d-b}a-c}{2^{d}}\quad(d\geq b)
  4. a 2 b - c 2 d = a - 2 b - d c 2 b ( d < b ) \frac{a}{2^{b}}-\frac{c}{2^{d}}=\frac{a-2^{b-d}c}{2^{b}}\quad(d<b)
  5. a 2 b × c 2 d = a × c 2 b + d . \frac{a}{2^{b}}\times\frac{c}{2^{d}}=\frac{a\times c}{2^{b+d}}.
  6. 2 i x / 2 i \lfloor 2^{i}x\rfloor/2^{i}
  7. lim { 2 - i i = 0 , 1 , 2 , } \underrightarrow{\lim}\left\{2^{-i}\mathbb{Z}\mid i=0,1,2,\dots\right\}
  8. ζ ζ 2 . \zeta\mapsto\zeta^{2}.

Dynamic_range.html

  1. 20 × log 10 ( 5 V 10 μ V ) = 20 × log 10 ( 500000 ) = 20 × 5.7 = 114 dB 20\times\log_{10}\left(\frac{5V}{10\mu V}\right)=20\times\log_{10}(500000)=20% \times 5.7=114\,\mathrm{dB}
  2. DR ADC = 20 × log 10 ( 2 Q 1 ) = ( 6.02 Q ) dB \mathrm{DR_{ADC}}=20\times\log_{10}\left(\frac{2^{Q}}{1}\right)=\left(6.02% \cdot Q\right)\ \mathrm{dB}\,\!
  3. SNR ADC = ( 1.76 + 6.02 Q ) dB \mathrm{SNR_{ADC}}=\left(1.76+6.02\cdot Q\right)\ \mathrm{dB}\,\!

Dynamical_system.html

  1. x ˙ = v ( x ) . \dot{x}=v(x).\,
  2. G ( x , x ˙ ) = 0 G(x,\dot{x})=0\,
  3. x ˙ = ϕ ( x ) = A x + b , \dot{x}=\phi(x)=Ax+b,\,
  4. Φ t ( x 1 ) = x 1 + b t . \Phi^{t}(x_{1})=x_{1}+bt.\,
  5. Φ t ( x 0 ) = e t A x 0 . \Phi^{t}(x_{0})=e^{tA}x_{0}.\,
  6. x n + 1 = A x n + b , x_{n+1}=Ax_{n}+b,\,
  7. h - 1 F h ( x ) = J x . h^{-1}\circ F\circ h(x)=J\cdot x.\,
  8. vol ( A ) = vol ( Φ t ( A ) ) . \mathrm{vol}(A)=\mathrm{vol}(\Phi^{t}(A)).\,
  9. ( U t a ) ( x ) = a ( Φ - t ( x ) ) . (U^{t}a)(x)=a(\Phi^{-t}(x)).\,
  10. , f , 𝒯 \langle\mathcal{M},f,\mathcal{T}\rangle
  11. \mathcal{M}
  12. 𝒯 \mathcal{T}
  13. t 𝒯 t\in\mathcal{T}
  14. 𝒯 \mathcal{T}
  15. 𝒯 \mathcal{T}
  16. τ - 1 σ Σ \tau^{-1}\sigma\in\Sigma
  17. μ ( τ - 1 σ ) = μ ( σ ) \mu(\tau^{-1}\sigma)=\mu(\sigma)
  18. τ n = τ τ τ \tau^{n}=\tau\circ\tau\circ\cdots\circ\tau