wpmath0000003_0

10000_(number).html

  1. 5 6 5^{6}
  2. 2 14 2^{14}
  3. 7 5 7^{5}
  4. 140 2 140^{2}
  5. 3 9 3^{9}

1000_(number).html

  1. 2 10 2^{10}
  2. 33 2 33^{2}
  3. 34 2 34^{2}
  4. 35 2 35^{2}
  5. 12 2 + 33 2 12^{2}+33^{2}
  6. 26 2 + 27 2 26^{2}+27^{2}
  7. 7 2 + 8 2 + + 16 2 7^{2}+8^{2}+\cdots+16^{2}
  8. 38 2 38^{2}
  9. n 2 + n + 41 n^{2}+n+41
  10. 42 2 42^{2}
  11. 43 2 43^{2}
  12. 44 2 44^{2}

101_(number).html

  1. 3 n - 1 3n-1

105_(number).html

  1. n - 2 k n-2^{k}
  2. 0 < k < l o g 2 ( n ) 0<k<log_{2}(n)
  3. k = 8 k=8
  4. x n - 1 x^{n}-1
  5. ± 1 \pm 1
  6. ± 1 \pm 1

107_(number).html

  1. 2 p - 1 2^{p}-1

108_(number).html

  1. 1 1 2 2 3 3 1^{1}\cdot 2^{2}\cdot 3^{3}
  2. 2 sin ( 108 2 ) = ϕ 2\sin\left(\frac{108^{\circ}}{2}\right)=\phi

110_(number).html

  1. 110 = 5 2 + 6 2 + 7 2 110=5^{2}+6^{2}+7^{2}

112_(number).html

  1. 11 + 13 + 17 + 19 + 23 + 29 11+13+17+19+23+29

113_(number).html

  1. 3 n - 1 3n-1

115_(number).html

  1. σ ( 115 ) = 1 + 5 + 23 + 115 = 144 = 12 2 . \sigma(115)=1+5+23+115=144=12^{2}.
  2. 115 2 115 - 1 = 4 776 913 109 852 041 418 248 056 622 882 488 319 , 115\cdot 2^{115}-1=4\;776\;913\;109\;852\;041\;418\;248\;056\;622\;882\;488\;3% 19,

121_(number).html

  1. 1 + p + p 2 + p 3 + p 4 1+p+p^{2}+p^{3}+p^{4}

126_(number).html

  1. ( 9 4 ) {\textstyle\left({{9}\atop{4}}\right)}
  2. x = a b c + a b d + a c d + b c d , x=abc+abd+acd+bcd,

127_(number).html

  1. p = ( x 3 - y 3 ) / ( x - y ) p=(x^{3}-y^{3})/(x-y)
  2. x = y + 1 x=y+1

137_(number).html

  1. 3 n - 1 3n-1

142857_(number).html

  1. 0. 142857 ¯ 0.\overline{142857}
  2. 142857 ¯ \overline{142857}
  3. 142857 ¯ \overline{142857}
  4. 285714 ¯ \overline{285714}
  5. 428571 ¯ \overline{428571}
  6. 571428 ¯ \overline{571428}
  7. 714285 ¯ \overline{714285}
  8. 857142 ¯ \overline{857142}
  9. 999999 ¯ \overline{999999}
  10. 142857 ¯ \overline{142857}
  11. 285714 ¯ \overline{285714}
  12. 1 / 7 = 0.142857142857142857 = 0.14 + 0.0028 + 0.000056 + 0.00000112 + 0.0000000224 + 0.000000000448 + 0.00000000000896 + = 14 100 + 28 100 2 + 56 100 3 + 112 100 4 + 224 100 5 + + 7 × 2 N 100 N + = ( 7 50 + 7 50 2 + 7 50 3 + 7 50 4 + 7 50 5 + + 7 50 N + ) = k = 1 7 50 k \begin{aligned}\displaystyle 1/7&\displaystyle=0.142857142857142857\ldots\\ &\displaystyle=0.14+0.0028+0.000056+0.00000112+0.0000000224+0.000000000448+0.0% 0000000000896+\cdots\\ &\displaystyle=\frac{14}{100}+\frac{28}{100^{2}}+\frac{56}{100^{3}}+\frac{112}% {100^{4}}+\frac{224}{100^{5}}+\cdots+\frac{7\times 2^{N}}{100^{N}}+\cdots\\ &\displaystyle=\left(\frac{7}{50}+\frac{7}{50^{2}}+\frac{7}{50^{3}}+\frac{7}{5% 0^{4}}+\frac{7}{50^{5}}+\cdots+\frac{7}{50^{N}}+\cdots\right)\\ &\displaystyle=\sum_{k=1}^{\infty}\frac{7}{50^{k}}\end{aligned}

144_(number).html

  1. 144 5 = 27 5 + 84 5 + 110 5 + 133 5 144^{5}=27^{5}+84^{5}+110^{5}+133^{5}

153_(number).html

  1. 1 ! + 2 ! + 3 ! + 4 ! + 5 ! 1!+2!+3!+4!+5!
  2. 153 = 1 3 + 5 3 + 3 3 153=1^{3}+5^{3}+3^{3}
  3. 8 3 + 4 3 \displaystyle 8^{3}+4^{3}

169_(number).html

  1. 8 3 - 7 3 . 8^{3}-7^{3}.

175_(number).html

  1. 175 = 1 1 + 7 2 + 5 3 175=1^{1}+7^{2}+5^{3}

1766_in_science.html

  1. π \pi

2-satisfiability.html

  1. ( x 0 x 2 ) ( x 0 ¬ x 3 ) ( x 1 ¬ x 3 ) ( x 1 ¬ x 4 ) (x_{0}\lor x_{2})\land(x_{0}\lor\lnot x_{3})\land(x_{1}\lor\lnot x_{3})\land(x% _{1}\lor\lnot x_{4})\land
  2. ( x 2 ¬ x 4 ) ( x 0 ¬ x 5 ) ( x 1 ¬ x 5 ) ( x 2 ¬ x 5 ) (x_{2}\lor\lnot x_{4})\land{}(x_{0}\lor\lnot x_{5})\land(x_{1}\lor\lnot x_{5})% \land(x_{2}\lor\lnot x_{5})\land
  3. ( x 3 x 6 ) ( x 4 x 6 ) ( x 5 x 6 ) . (x_{3}\lor x_{6})\land(x_{4}\lor x_{6})\land(x_{5}\lor x_{6}).
  4. ( x 0 ¬ x 3 ) ( ¬ x 0 ¬ x 3 ) ( x 3 x 0 ) . (x_{0}\lor\lnot x_{3})\;\equiv\;(\lnot x_{0}\Rightarrow\lnot x_{3})\;\equiv\;(% x_{3}\Rightarrow x_{0}).
  5. ( a b ) (a\lor b)
  6. ( ¬ b ¬ c ) (\lnot b\lor\lnot c)
  7. ( a ¬ c ) (a\lor\lnot c)
  8. ¬ a b \lnot a\Rightarrow b
  9. b ¬ c b\Rightarrow\lnot c
  10. ¬ a ¬ c \lnot a\Rightarrow\lnot c
  11. ( x x ) (x\lor x)
  12. ( ¬ x ¬ x ) (\lnot x\lor\lnot x)
  13. x x
  14. ( x x ) (x\lor x)
  15. ( ¬ x ¬ x ) (\lnot x\lor\lnot x)
  16. ( x x ) (x\lor x)
  17. ( ¬ x ¬ x ) (\lnot x\lor\lnot x)
  18. x x
  19. ( x x ) (x\lor x)
  20. ( ¬ x ¬ x ) (\lnot x\lor\lnot x)
  21. n n
  22. O ( n ) O(n)
  23. i i
  24. j j
  25. u v uv
  26. u u
  27. i i
  28. v v
  29. j j
  30. x x
  31. m m
  32. n n
  33. O ( 1.246 n ) O(1.246^{n})
  34. min { ( 3 - cos θ ) - 1 ( 2 + ( 2 / π ) θ ) : π / 2 θ π } = 0.943... \min\left\{(3-\cos\theta)^{-1}(2+(2/\pi)\theta)\,:\,\pi/2\leq\theta\leq\pi% \right\}=0.943...
  35. n n
  36. k k
  37. k k
  38. G G
  39. k k
  40. u v uv
  41. u v u∨v
  42. k k
  43. k k
  44. f ( k ) · n < s u p > O ( 1 ) f(k)·n<sup>O(1)

2000_(number).html

  1. 7 4 - 7 3 - 7 2 7^{4}-7^{3}-7^{2}
  2. 45 2 45^{2}
  3. 21 2 + 22 2 + 23 2 + 24 2 21^{2}+22^{2}+23^{2}+24^{2}
  4. 25 2 + 26 2 + 27 2 25^{2}+26^{2}+27^{2}
  5. 46 2 46^{2}
  6. 3 7 3^{7}
  7. 13 3 13^{3}
  8. 47 2 47^{2}
  9. 48 2 48^{2}
  10. 7 4 7^{4}
  11. 49 2 49^{2}
  12. 50 2 50^{2}
  13. 51 2 51^{2}
  14. 52 2 52^{2}
  15. 14 3 14^{3}
  16. 53 2 53^{2}
  17. 54 2 54^{2}

24_(number).html

  1. 2 3 q 2^{3}q
  2. q q

256_(number).html

  1. ( ( 2 2 ) 2 ) 2 ((2^{2})^{2})^{2}

25_(number).html

  1. 1 4 \frac{1}{4}
  2. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  3. i = 1 n x i x i + 1 + x i + 2 < n 2 \sum_{i=1}^{n}\frac{x_{i}}{x_{i+1}+x_{i+2}}<\frac{n}{2}
  4. x n + 1 = x 1 , x n + 2 = x 2 x_{n+1}=x_{1},x_{n+2}=x_{2}
  5. 10 n * Z + ( 2 Z + 1 ) 10^{n}*Z+(2Z+1)

26_(number).html

  1. 25 = 5 2 25=5^{2}
  2. 27 = 3 3 27=3^{3}

28_(number).html

  1. 2 2 q 2^{2}q

29_(number).html

  1. 3 n - 1 3n-1
  2. x 2 + y 2 + z 2 = 3 x y z x^{2}+y^{2}+z^{2}=3xyz

300_(number).html

  1. ( 11 4 ) {\textstyle\left({{11}\atop{4}}\right)}

30_(number).html

  1. 2 3 r 2\cdot 3\cdot r

37_(number).html

  1. p = x 3 - y 3 x - y ( x = y + 1 ) . p=\frac{x^{3}-y^{3}}{x-y}\qquad\left(x=y+1\right).

39_(number).html

  1. 3 1 + 3 2 + 3 3 3^{1}+3^{2}+3^{3}

3SUM.html

  1. n n
  2. O ( n 2 ) O(n^{2})
  3. Ω ( n r / 2 ) \Omega(n^{\lceil r/2\rceil})
  4. Ω ( n 2 ) \Omega(n^{2})
  5. O ( n 2 / ( log n / log log n ) 2 / 3 ) O(n^{2}/({\log n}/{\log\log n})^{2/3})
  6. O ( n 2 - Ω ( 1 ) ) O(n^{2-\Omega(1)})
  7. [ - N , , N ] [-N,\dots,N]
  8. O ( n + N log N ) O(n+N\log N)
  9. S S
  10. S + S S+S
  11. - S -S
  12. S [ 0.. n - 1 ] S[0..n-1]
  13. O ( n 2 ) O(n^{2})
  14. S [ i ] S[i]
  15. i i
  16. j j
  17. - ( S [ i ] + S [ j ] ) -(S[i]+S[j])
  18. O ( n 2 ) O(n^{2})
  19. a X , b Y , c Z a∈X,b∈Y,c∈Z
  20. a + b + c = 0 a+b+c=0
  21. X i i X X i i * 10 + 11 Xii←XXii*10+11
  22. Y i i Y Y i i * 10 + 22 Yii←YYii*10+22
  23. Z i i Z Z i i * 10 - 33 Zii←ZZii*10-33
  24. a S , b b S , c c S S a∈S,bb∈S,cc∈SS
  25. a + b b + c c = = 0 a+bb+cc==0
  26. a ( a - 1 ) / 10 , b ( b b - 2 ) / 10 , c ( c c + 3 ) / 100 a←(a-1)/10,b←(bb-2)/10,c←(cc+3)/100
  27. a X , b Y , c Z a∈X,b∈Y,c∈Z
  28. S [ k ] = S [ i ] + S [ j ] S[k]=S[i]+S[j]
  29. S [ i + j ] = S [ i ] + S [ j ] S[i+j]=S[i]+S[j]
  30. T [ i ] = 2 n S [ i ] + i T[i]=2nS[i]+i
  31. S [ i + j ] = S [ i ] + S [ j ] S[i+j]=S[i]+S[j]
  32. T [ i + j ] = 2 n S [ i + j ] + i + j = ( 2 n S [ i ] + i ) + ( 2 n S [ j ] + j ) = T [ i ] + T [ j ] T[i+j]=2nS[i+j]+i+j=(2nS[i]+i)+(2nS[j]+j)=T[i]+T[j]
  33. T [ k ] = T [ i ] + T [ j ] T[k]=T[i]+T[j]
  34. 2 n S [ k ] + k = 2 n ( S [ i ] + S [ j ] ) + ( i + j ) 2nS[k]+k=2n(S[i]+S[j])+(i+j)
  35. i + j < 2 n i+j<2n
  36. S [ k ] = S [ i ] + S [ j ] S[k]=S[i]+S[j]
  37. k = i + j k=i+j
  38. h ( x + y ) = h ( x ) + h ( y ) h(x+y)=h(x)+h(y)
  39. T [ h ( x ) ] = x T[h(x)]=x
  40. z = x + y z=x+y
  41. T [ h ( z ) ] = T [ h ( x ) ] + T [ h ( y ) ] T[h(z)]=T[h(x)]+T[h(y)]
  42. h ( z ) = h ( x ) + h ( y ) h(z)=h(x)+h(y)
  43. ( 1 / R ) 3 (1/R)^{3}
  44. R 3 R^{3}
  45. h ( x + y ) = h ( x ) + h ( y ) h(x+y)=h(x)+h(y)
  46. h ( x + y ) = h ( x ) + h ( y ) + 1 h(x+y)=h(x)+h(y)+1
  47. x S x\in S
  48. T [ h ( x ) ] T[h(x)]
  49. T [ h ( x ) ] - 1 T[h(x)]-1
  50. ( 2 R ) 3 (2R)^{3}
  51. Θ ( n 2 ) \Theta(n^{2})
  52. X X
  53. Y Y
  54. n n
  55. n n
  56. x + y x+y
  57. x X , y Y x∈X,y∈Y

4000_(number).html

  1. 2 12 2^{12}
  2. 64 2 64^{2}
  3. 2 3 + 16 3 = 9 3 + 15 3 2^{3}+16^{3}=9^{3}+15^{3}
  4. 65 2 65^{2}
  5. 66 2 66^{2}
  6. 67 2 67^{2}
  7. 68 2 68^{2}
  8. 48 2 + 49 2 = 17 2 + 18 2 + + 26 2 48^{2}+49^{2}=17^{2}+18^{2}+\cdots+26^{2}
  9. 69 2 69^{2}
  10. 70 2 70^{2}

400_(number).html

  1. ( 11 5 ) {\textstyle\left({{11}\atop{5}}\right)}

40_Eridani.html

  1. h = 1 d * a h={1\over d}*a
  2. m = M v + 5 ( ( log 10 5.04 ) - 1 ) = 3.36 \begin{smallmatrix}m\ =\ M_{v}\ +\ 5\cdot((\log_{10}\ 5.04)\ -\ 1)\ =\ 3.36% \end{smallmatrix}

43_(number).html

  1. i = 0 5 1 i ! = 163 60 = 2 + 43 60 , \sum_{i=0}^{5}\frac{1}{i!}=\frac{163}{60}=2+\frac{43}{60},

44_(number).html

  1. φ ( 44 ) = 20 \varphi(44)=20
  2. φ ( 69 ) = 44. \varphi(69)=44.

47_(number).html

  1. 3 n - 1 3n-1

5-cell.html

  1. ( 1 10 , 1 6 , 1 3 , ± 1 ) \left(\frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  2. ( 1 10 , 1 6 , - 2 3 , 0 ) \left(\frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  3. ( 1 10 , - 3 2 , 0 , 0 ) \left(\frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0\right)
  4. ( - 2 2 5 , 0 , 0 , 0 ) \left(-2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right)
  5. ( 1 , 1 , 1 , - 1 / 5 ) \left(1,1,1,-1/\sqrt{5}\right)
  6. ( 1 , - 1 , - 1 , - 1 / 5 ) \left(1,-1,-1,-1/\sqrt{5}\right)
  7. ( - 1 , 1 , - 1 , - 1 / 5 ) \left(-1,1,-1,-1/\sqrt{5}\right)
  8. ( - 1 , - 1 , 1 , - 1 / 5 ) \left(-1,-1,1,-1/\sqrt{5}\right)
  9. ( 0 , 0 , 0 , 5 - 1 / 5 ) \left(0,0,0,\sqrt{5}-1/\sqrt{5}\right)

5000_(number).html

  1. 71 2 71^{2}
  2. 72 2 72^{2}
  3. 22 2 + 23 2 + + 29 2 = 20 2 + 21 2 + + 28 2 22^{2}+23^{2}+\cdots+29^{2}=20^{2}+21^{2}+\cdots+28^{2}
  4. 5280 = - j ( 1 2 ( 1 + i 67 ) ) 3 . 5280=-\sqrt[3]{j\left({\scriptstyle\frac{1}{2}}\left(1+i\sqrt{67}\,\right)% \right)}.
  5. 73 2 73^{2}
  6. 74 2 74^{2}
  7. 75 2 75^{2}
  8. 76 2 76^{2}
  9. p + 2 a 2 p+2a^{2}
  10. 77 2 77^{2}

500_(number).html

  1. 34 n 3 + 51 n 2 + 27 n + 5 34n^{3}+51n^{2}+27n+5
  2. n = 2 n=2
  3. ζ ( 3 ) \zeta(3)

54_(number).html

  1. 7 2 + 2 2 + 1 2 = 6 2 + 2 ( 3 2 ) = 2 ( 5 2 ) + 2 2 = 54 7^{2}+2^{2}+1^{2}=6^{2}+2(3^{2})=2(5^{2})+2^{2}=54

555_timer_IC.html

  1. R E S E T ¯ \overline{RESET}
  2. t = R C ln ( 3 ) 1.1 R C t=RC\ln(3)\approx 1.1RC
  3. f = 1 ln ( 2 ) C ( R 1 + 2 R 2 ) f=\frac{1}{\ln(2)\cdot C\cdot(R_{1}+2R_{2})}
  4. high = ln ( 2 ) ( R 1 + R 2 ) C \mathrm{high}=\ln(2)\cdot(R_{1}+R_{2})\cdot C
  5. low = ln ( 2 ) R 2 C \mathrm{low}=\ln(2)\cdot R_{2}\cdot C
  6. V c c 2 R 1 \frac{V_{cc}^{2}}{R_{1}}
  7. R 1 R_{1}
  8. high = R 1 C ln ( 2 V cc - 3 V diode V cc - 3 V diode ) \mathrm{high}=R_{1}C\cdot\ln\left(\frac{2V_{\textrm{cc}}-3V_{\textrm{diode}}}{% V_{\textrm{cc}}-3V_{\textrm{diode}}}\right)

59_(number).html

  1. 3 n - 1 3n-1

600-cell.html

  1. π / 5 \pi/5
  2. π / 3 \pi/3
  3. 2 π / 5 2\pi/5
  4. π / 2 \pi/2
  5. 3 π / 5 3\pi/5
  6. 2 π / 3 2\pi/3
  7. 4 π / 5 4\pi/5
  8. π \pi

6000_(number).html

  1. 78 2 78^{2}
  2. 79 2 79^{2}
  3. 80 2 80^{2}
  4. 3 8 3^{8}
  5. 81 2 81^{2}
  6. 82 2 82^{2}
  7. 83 2 83^{2}

61_(number).html

  1. p = ( x 3 - y 3 ) / ( x - y ) , x = y + 1 p=(x^{3}-y^{3})/(x-y),x=y+1

64_(number).html

  1. 3 3 , 3 3 3 3 , 3\uparrow\uparrow\uparrow\uparrow 3,3\uparrow^{3\uparrow\uparrow\uparrow% \uparrow 3}3,\ldots

65_(number).html

  1. [ 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 ] . \begin{bmatrix}17&24&1&8&15\\ 23&5&7&14&16\\ 4&6&13&20&22\\ 10&12&19&21&3\\ 11&18&25&2&9\end{bmatrix}.

68_(number).html

  1. 2 2 ( 2 2 2 + 1 ) 2^{2}\cdot(2^{2^{2}}+1)
  2. 68 6 2 + 8 2 = 100 1 2 + 0 2 + 0 2 = 1. 68\to 6^{2}+8^{2}=100\to 1^{2}+0^{2}+0^{2}=1.

6_Hebe.html

  1. ν 6 \nu_{6}\,\!

7000_(number).html

  1. 84 2 84^{2}
  2. 85 2 85^{2}
  3. 36 2 + 37 2 + 38 2 + 39 2 + 40 2 = 41 2 + 42 2 + 43 2 + 44 2 36^{2}+37^{2}+38^{2}+39^{2}+40^{2}=41^{2}+42^{2}+43^{2}+44^{2}
  4. 86 2 86^{2}
  5. 87 2 87^{2}
  6. 88 2 88^{2}
  7. 6 5 6^{5}
  8. 89 2 89^{2}

700_(number).html

  1. ( 13 4 ) {\textstyle\left({{13}\atop{4}}\right)}
  2. 1 3 + 2 3 + 3 3 + 4 3 + 5 3 + 6 3 + 7 3 1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}+7^{3}
  3. ( 12 5 ) {\textstyle\left({{12}\atop{5}}\right)}

79_(number).html

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

8000_(number).html

  1. 90 2 90^{2}
  2. 8 4 + 2 4 + 0 4 + 8 4 = 8208 8^{4}+2^{4}+0^{4}+8^{4}=8208
  3. 91 2 91^{2}
  4. 92 2 92^{2}
  5. 93 2 93^{2}
  6. 88 2 + 33 2 88^{2}+33^{2}
  7. 94 2 94^{2}

81_(number).html

  1. 1 ( b - 1 ) 2 = 0. 012 ( b - 4 ) ( b - 3 ) ( b - 1 ) ¯ , \frac{1}{\left(b-1\right)^{2}}=0.\overline{012\cdots(b-4)(b-3)(b-1)},

83_(number).html

  1. 3 n - 1 3n-1

84_(number).html

  1. g g
  2. g g

89_(number).html

  1. 3 n - 1 3n-1
  2. 1 89 = n = 1 F ( n ) × 10 - ( n + 1 ) = 0.011235955 . \frac{1}{89}=\sum_{n=1}^{\infty}{F(n)\times 10^{-(n+1)}}=0.011235955\dots\ .

900_(number).html

  1. ( 12 6 ) {\textstyle\left({{12}\atop{6}}\right)}

99_Bottles_of_Beer.html

  1. O ( log N ) O(\log N)

Abelian_and_tauberian_theorems.html

  1. d N = c 1 + c 2 + + c N N . d_{N}=\frac{c_{1}+c_{2}+\cdots+c_{N}}{N}.

Ablation.html

  1. Intensity ( W / cm 2 ) = average power ( W ) focal spot area ( cm 2 ) \,\text{Intensity }(\mathrm{W}/\mathrm{cm}^{2})=\frac{\,\text{average power }(% \mathrm{W})}{\,\text{focal spot area }(\mathrm{cm}^{2})}
  2. Peak intensity ( W / cm 2 ) = peak power ( W ) focal spot area ( cm 2 ) \,\text{Peak intensity }(\mathrm{W}/\mathrm{cm}^{2})=\frac{\,\text{peak power % }(\mathrm{W})}{\,\text{focal spot area }(\mathrm{cm}^{2})}
  3. Fluence ( J / cm 2 ) = laser pulse energy ( J ) focal spot area ( cm 2 ) \,\text{Fluence }(\mathrm{J}/\mathrm{cm}^{2})=\frac{\,\text{laser pulse energy% }(\mathrm{J})}{\,\text{focal spot area }(\mathrm{cm}^{2})}
  4. Peak power ( W ) = pulse energy ( J ) pulse duration ( s ) \,\text{Peak power }(\mathrm{W})=\frac{\,\text{pulse energy }(\mathrm{J})}{\,% \text{pulse duration }(\mathrm{s})}

Absorbance.html

  1. A = log 10 ( Φ e i Φ e t ) = - log 10 T , A=\log_{10}\!\left(\frac{\Phi_{\mathrm{e}}^{\mathrm{i}}}{\Phi_{\mathrm{e}}^{% \mathrm{t}}}\right)=-\log_{10}T,
  2. A = τ ln 10 , A=\frac{\tau}{\ln 10},
  3. A ν = log 10 ( Φ e , ν i Φ e , ν t ) = - log 10 T ν , A_{\nu}=\log_{10}\!\left(\frac{\Phi_{\mathrm{e},\nu}^{\mathrm{i}}}{\Phi_{% \mathrm{e},\nu}^{\mathrm{t}}}\right)=-\log_{10}T_{\nu},
  4. A λ = log 10 ( Φ e , λ i Φ e , λ t ) = - log 10 T λ , A_{\lambda}=\log_{10}\!\left(\frac{\Phi_{\mathrm{e},\lambda}^{\mathrm{i}}}{% \Phi_{\mathrm{e},\lambda}^{\mathrm{t}}}\right)=-\log_{10}T_{\lambda},
  5. A ν = τ ν ln 10 , A_{\nu}=\frac{\tau_{\nu}}{\ln 10},
  6. A λ = τ λ ln 10 , A_{\lambda}=\frac{\tau_{\lambda}}{\ln 10},
  7. Φ e t + Φ e att = Φ e i + Φ e e , \Phi_{\mathrm{e}}^{\mathrm{t}}+\Phi_{\mathrm{e}}^{\mathrm{att}}=\Phi_{\mathrm{% e}}^{\mathrm{i}}+\Phi_{\mathrm{e}}^{\mathrm{e}},
  8. T + A T T = 1 + E , T+ATT=1+E,
  9. T = 10 - A , T=10^{-A},
  10. A T T = 1 - 10 - A + E A + E A , if A 1 and E A . ATT=1-10^{-A}+E\approx A+E\approx A,\quad\,\text{if}\ A\ll 1\ \,\text{and}\ E% \ll A.
  11. A = 0 l a ( z ) d z , A=\int_{0}^{l}a(z)\,\mathrm{d}z,
  12. A = a l . A=al.
  13. A = 0 l ε c ( z ) d z , A=\int_{0}^{l}\varepsilon c(z)\,\mathrm{d}z,
  14. A = ε c l . A=\varepsilon cl.
  15. A λ = log 10 ( Φ e , λ i Φ e , λ t ) . A_{\lambda}=\log_{10}\!\left(\frac{\Phi_{\mathrm{e},\lambda}^{\mathrm{i}}}{% \Phi_{\mathrm{e},\lambda}^{\mathrm{t}}}\right)\!.
  16. S N = 7 3 A + 1 , SN=\frac{7}{3}A+1,
  17. S N = 7 3 ( - log 10 T ) + 1 , SN=\frac{7}{3}(-\log_{10}T)+1,

Absorbed_dose.html

  1. D T ¯ = T D ( x , y , z ) ρ ( x , y , z ) d V T ρ ( x , y , z ) d V \bar{D_{T}}=\frac{\int_{T}D(x,y,z)\rho(x,y,z)dV}{\int_{T}\rho(x,y,z)dV}
  2. D T ¯ \bar{D_{T}}
  3. T T
  4. D ( x , y , z ) D(x,y,z)
  5. ρ ( x , y , z ) \rho(x,y,z)
  6. V V

Abstract_nonsense.html

  1. f : M K ( , 2 ) f:M\to K(\mathbb{Z},2)

Abundant_number.html

  1. A ( k ) A(k)
  2. ϵ > 0 \epsilon>0
  3. ( 1 - ϵ ) ( k ln k ) 2 - ϵ < ln A ( k ) < ( 1 + ϵ ) ( k ln k ) 2 + ϵ (1-\epsilon)(k\ln k)^{2-\epsilon}<\ln A(k)<(1+\epsilon)(k\ln k)^{2+\epsilon}

Accelerator_physics.html

  1. x ( s ) \scriptstyle x(s)
  2. d 2 d s 2 x ( s ) + k ( s ) x ( s ) = 1 R Δ p p \frac{d^{2}}{ds^{2}}\,x(s)+k(s)\,x(s)=\frac{1}{R}\,\frac{\Delta p}{p}
  3. k ( s ) \scriptstyle k(s)
  4. Δ p / p \scriptstyle\Delta p/p
  5. R \scriptstyle R
  6. s \scriptstyle s

Acetazolamide.html

  1. H 2 CO 3 H 2 O + CO 2 \textrm{H}_{2}\textrm{CO}_{3}\rightleftharpoons\textrm{H}_{2}\textrm{O}+% \textrm{CO}_{2}
  2. H 2 CO 3 HCO 3 - + H + \textrm{H}_{2}\textrm{CO}_{3}\rightleftharpoons\textrm{HCO}_{3}^{-}+\textrm{H}% ^{+}

Acoustic_impedance.html

  1. p ( t ) = [ R * Q ] ( t ) , p(t)=[R*Q](t),
  2. Q ( t ) = [ G * p ] ( t ) , Q(t)=[G*p](t),
  3. * *
  4. Z ( s ) = def [ R ] ( s ) = [ p ] ( s ) [ Q ] ( s ) , Z(s)\stackrel{\mathrm{def}}{{}={}}\mathcal{L}[R](s)=\frac{\mathcal{L}[p](s)}{% \mathcal{L}[Q](s)},
  5. Z ( ω ) = def [ R ] ( ω ) = [ p ] ( ω ) [ Q ] ( ω ) , Z(\omega)\stackrel{\mathrm{def}}{{}={}}\mathcal{F}[R](\omega)=\frac{\mathcal{F% }[p](\omega)}{\mathcal{F}[Q](\omega)},
  6. Z ( t ) = def R a ( t ) = 1 2 [ p a * ( Q - 1 ) a ] ( t ) , Z(t)\stackrel{\mathrm{def}}{{}={}}R_{\mathrm{a}}(t)=\frac{1}{2}\!\left[p_{% \mathrm{a}}*\left(Q^{-1}\right)_{\mathrm{a}}\right]\!(t),
  7. \mathcal{L}
  8. \mathcal{F}
  9. Z ( s ) = R ( s ) + i X ( s ) , Z(s)=R(s)+iX(s),
  10. Z ( ω ) = R ( ω ) + i X ( ω ) , Z(\omega)=R(\omega)+iX(\omega),
  11. Z ( t ) = R ( t ) + i X ( t ) , Z(t)=R(t)+iX(t),
  12. X ( s ) = X L ( s ) - X C ( s ) , X(s)=X_{L}(s)-X_{C}(s),
  13. X ( ω ) = X L ( ω ) - X C ( ω ) , X(\omega)=X_{L}(\omega)-X_{C}(\omega),
  14. X ( t ) = X L ( t ) - X C ( t ) . X(t)=X_{L}(t)-X_{C}(t).
  15. Y ( s ) = def [ G ] ( s ) = 1 Z ( s ) = [ Q ] ( s ) [ p ] ( s ) , Y(s)\stackrel{\mathrm{def}}{{}={}}\mathcal{L}[G](s)=\frac{1}{Z(s)}=\frac{% \mathcal{L}[Q](s)}{\mathcal{L}[p](s)},
  16. Y ( ω ) = def [ G ] ( ω ) = 1 Z ( ω ) = [ Q ] ( ω ) [ p ] ( ω ) , Y(\omega)\stackrel{\mathrm{def}}{{}={}}\mathcal{F}[G](\omega)=\frac{1}{Z(% \omega)}=\frac{\mathcal{F}[Q](\omega)}{\mathcal{F}[p](\omega)},
  17. Y ( t ) = def G a ( t ) = Z - 1 ( t ) = 1 2 [ Q a * ( p - 1 ) a ] ( t ) , Y(t)\stackrel{\mathrm{def}}{{}={}}G_{\mathrm{a}}(t)=Z^{-1}(t)=\frac{1}{2}\!% \left[Q_{\mathrm{a}}*\left(p^{-1}\right)_{\mathrm{a}}\right]\!(t),
  18. Y ( s ) = G ( s ) + i B ( s ) , Y(s)=G(s)+iB(s),
  19. Y ( ω ) = G ( ω ) + i B ( ω ) , Y(\omega)=G(\omega)+iB(\omega),
  20. Y ( t ) = G ( t ) + i B ( t ) , Y(t)=G(t)+iB(t),
  21. p ( t ) = [ r * v ] ( t ) , p(t)=[r*v](t),
  22. v ( t ) = [ g * p ] ( t ) , v(t)=[g*p](t),
  23. z ( s ) = def [ r ] ( s ) = [ p ] ( s ) [ v ] ( s ) , z(s)\stackrel{\mathrm{def}}{{}={}}\mathcal{L}[r](s)=\frac{\mathcal{L}[p](s)}{% \mathcal{L}[v](s)},
  24. z ( ω ) = def [ r ] ( ω ) = [ p ] ( ω ) [ v ] ( ω ) , z(\omega)\stackrel{\mathrm{def}}{{}={}}\mathcal{F}[r](\omega)=\frac{\mathcal{F% }[p](\omega)}{\mathcal{F}[v](\omega)},
  25. z ( t ) = def r a ( t ) = 1 2 [ p a * ( v - 1 ) a ] ( t ) , z(t)\stackrel{\mathrm{def}}{{}={}}r_{\mathrm{a}}(t)=\frac{1}{2}\!\left[p_{% \mathrm{a}}*\left(v^{-1}\right)_{\mathrm{a}}\right]\!(t),
  26. z ( s ) = r ( s ) + i x ( s ) , z(s)=r(s)+ix(s),
  27. z ( ω ) = r ( ω ) + i x ( ω ) , z(\omega)=r(\omega)+ix(\omega),
  28. z ( t ) = r ( t ) + i x ( t ) , z(t)=r(t)+ix(t),
  29. x ( s ) = x L ( s ) - x C ( s ) , x(s)=x_{L}(s)-x_{C}(s),
  30. x ( ω ) = x L ( ω ) - x C ( ω ) , x(\omega)=x_{L}(\omega)-x_{C}(\omega),
  31. x ( t ) = x L ( t ) - x C ( t ) . x(t)=x_{L}(t)-x_{C}(t).
  32. y ( s ) = def [ g ] ( s ) = 1 z ( s ) = [ v ] ( s ) [ p ] ( s ) , y(s)\stackrel{\mathrm{def}}{{}={}}\mathcal{L}[g](s)=\frac{1}{z(s)}=\frac{% \mathcal{L}[v](s)}{\mathcal{L}[p](s)},
  33. y ( ω ) = def [ g ] ( ω ) = 1 z ( ω ) = [ v ] ( ω ) [ p ] ( ω ) , y(\omega)\stackrel{\mathrm{def}}{{}={}}\mathcal{F}[g](\omega)=\frac{1}{z(% \omega)}=\frac{\mathcal{F}[v](\omega)}{\mathcal{F}[p](\omega)},
  34. y ( t ) = def g a ( t ) = z - 1 ( t ) = 1 2 [ v a * ( p - 1 ) a ] ( t ) , y(t)\stackrel{\mathrm{def}}{{}={}}g_{\mathrm{a}}(t)=z^{-1}(t)=\frac{1}{2}\!% \left[v_{\mathrm{a}}*\left(p^{-1}\right)_{\mathrm{a}}\right]\!(t),
  35. y ( s ) = g ( s ) + i b ( s ) , y(s)=g(s)+ib(s),
  36. y ( ω ) = g ( ω ) + i b ( ω ) , y(\omega)=g(\omega)+ib(\omega),
  37. y ( t ) = g ( t ) + i b ( t ) , y(t)=g(t)+ib(t),
  38. Q = d V d t = A d x d t = A v . Q=\frac{\mathrm{d}V}{\mathrm{d}t}=A\frac{\mathrm{d}x}{\mathrm{d}t}=Av.
  39. Z ( s ) = [ p ] ( s ) [ Q ] ( s ) = [ p ] ( s ) A [ v ] ( s ) = z ( s ) A , Z(s)=\frac{\mathcal{L}[p](s)}{\mathcal{L}[Q](s)}=\frac{\mathcal{L}[p](s)}{A% \mathcal{L}[v](s)}=\frac{z(s)}{A},
  40. Z ( ω ) = [ p ] ( ω ) [ Q ] ( ω ) = [ p ] ( ω ) A [ v ] ( ω ) = z ( ω ) A , Z(\omega)=\frac{\mathcal{F}[p](\omega)}{\mathcal{F}[Q](\omega)}=\frac{\mathcal% {F}[p](\omega)}{A\mathcal{F}[v](\omega)}=\frac{z(\omega)}{A},
  41. Z ( t ) = 1 2 [ p a * ( Q - 1 ) a ] ( t ) = 1 2 [ p a * ( v - 1 A ) a ] ( t ) = z ( t ) A . Z(t)=\frac{1}{2}\!\left[p_{\mathrm{a}}*\left(Q^{-1}\right)_{\mathrm{a}}\right]% \!(t)=\frac{1}{2}\!\left[p_{\mathrm{a}}*\left(\frac{v^{-1}}{A}\right)_{\mathrm% {a}}\right]\!(t)=\frac{z(t)}{A}.
  42. p = - ρ c 2 ξ x , p=-\rho c^{2}\frac{\partial\xi}{\partial x},
  43. ρ 2 ξ t 2 = - p x . \rho\frac{\partial^{2}\xi}{\partial t^{2}}=-\frac{\partial p}{\partial x}.
  44. 2 ξ t 2 = c 2 ξ 2 x 2 . \frac{\partial^{2}\xi}{\partial t^{2}}=c^{2}\frac{\partial\xi^{2}}{\partial x^% {2}}.
  45. ξ ( 𝐫 , t ) = ξ ( x , t ) \xi(\mathbf{r},\,t)=\xi(x,\,t)
  46. ξ ( 𝐫 , t ) = f ( x - c t ) + g ( x + c t ) \xi(\mathbf{r},\,t)=f(x-ct)+g(x+ct)
  47. v ( 𝐫 , t ) = ξ t ( 𝐫 , t ) = - c [ f ( x - c t ) - g ( x + c t ) ] , v(\mathbf{r},\,t)=\frac{\partial\xi}{\partial t}(\mathbf{r},\,t)=-c\big[f^{% \prime}(x-ct)-g^{\prime}(x+ct)\big],
  48. p ( 𝐫 , t ) = - ρ c 2 ξ x ( 𝐫 , t ) = - ρ c 2 [ f ( x - c t ) + g ( x + c t ) ] . p(\mathbf{r},\,t)=-\rho c^{2}\frac{\partial\xi}{\partial x}(\mathbf{r},\,t)=-% \rho c^{2}\big[f^{\prime}(x-ct)+g^{\prime}(x+ct)\big].
  49. { p ( 𝐫 , t ) = - ρ c 2 f ( x - c t ) v ( 𝐫 , t ) = - c f ( x - c t ) \begin{cases}p(\mathbf{r},\,t)=-\rho c^{2}\,f^{\prime}(x-ct)\\ v(\mathbf{r},\,t)=-c\,f^{\prime}(x-ct)\end{cases}
  50. { p ( 𝐫 , t ) = - ρ c 2 g ( x + c t ) v ( 𝐫 , t ) = c g ( x + c t ) . \begin{cases}p(\mathbf{r},\,t)=-\rho c^{2}\,g^{\prime}(x+ct)\\ v(\mathbf{r},\,t)=c\,g^{\prime}(x+ct).\end{cases}
  51. z ( 𝐫 , s ) = [ p ] ( 𝐫 , s ) [ v ] ( 𝐫 , s ) = ± ρ c , z(\mathbf{r},\,s)=\frac{\mathcal{L}[p](\mathbf{r},\,s)}{\mathcal{L}[v](\mathbf% {r},\,s)}=\pm\rho c,
  52. z ( 𝐫 , ω ) = [ p ] ( 𝐫 , ω ) [ v ] ( 𝐫 , ω ) = ± ρ c , z(\mathbf{r},\,\omega)=\frac{\mathcal{F}[p](\mathbf{r},\,\omega)}{\mathcal{F}[% v](\mathbf{r},\,\omega)}=\pm\rho c,
  53. z ( 𝐫 , t ) = 1 2 [ p a * ( v - 1 ) a ] ( 𝐫 , t ) = ± ρ c . z(\mathbf{r},\,t)=\frac{1}{2}\!\left[p_{\mathrm{a}}*\left(v^{-1}\right)_{% \mathrm{a}}\right]\!(\mathbf{r},\,t)=\pm\rho c.
  54. z 0 = ρ c . z_{0}=\rho c.
  55. p ( 𝐫 , t ) v ( 𝐫 , t ) = ± ρ c = ± z 0 . \frac{p(\mathbf{r},\,t)}{v(\mathbf{r},\,t)}=\pm\rho c=\pm z_{0}.
  56. Z ( 𝐫 , s ) = ± ρ c A , Z(\mathbf{r},\,s)=\pm\frac{\rho c}{A},
  57. Z ( 𝐫 , ω ) = ± ρ c A , Z(\mathbf{r},\,\omega)=\pm\frac{\rho c}{A},
  58. Z ( 𝐫 , t ) = ± ρ c A . Z(\mathbf{r},\,t)=\pm\frac{\rho c}{A}.
  59. Z 0 = ρ c A . Z_{0}=\frac{\rho c}{A}.
  60. p ( 𝐫 , t ) Q ( 𝐫 , t ) = ± ρ c A = ± Z 0 . \frac{p(\mathbf{r},\,t)}{Q(\mathbf{r},\,t)}=\pm\frac{\rho c}{A}=\pm Z_{0}.

Active_and_passive_transformation.html

  1. R = ( cos θ - sin θ sin θ cos θ ) , R=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix},
  2. 𝐯 = R 𝐯 = ( cos θ - sin θ sin θ cos θ ) ( v 1 v 2 ) . \mathbf{v^{\prime}}=R\mathbf{v}=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}v^{1}\\ v^{2}\end{pmatrix}.
  3. 𝐯 = v a 𝐞 a = v a R 𝐞 a . \mathbf{v}=v^{a}\mathbf{e}_{a}=v^{\prime a}R\mathbf{e}_{a}.
  4. v a = ( R - 1 ) b a v b v^{\prime a}=(R^{-1})_{b}^{a}v^{b}
  5. 𝐯 = v a 𝐞 a = v b ( R - 1 ) b a R a c 𝐞 c = v b δ b c 𝐞 c = v b 𝐞 b . \mathbf{v}=v^{\prime a}\mathbf{e}^{\prime}_{a}=v^{b}(R^{-1})_{b}^{a}R_{a}^{c}% \mathbf{e}_{c}=v^{b}\delta^{c}_{b}\mathbf{e}_{c}=v^{b}\mathbf{e}_{b}.

Adaptation_(eye).html

  1. Rhodopsin Retinal + Opsin \mathrm{Rhodopsin\rightleftharpoons\ Retinal+Opsin}

Adder_(electronics).html

  1. sum = 2 × C o u t + S \mathrm{sum}=2\times C_{out}+S
  2. S = A B C i n S=A\oplus B\oplus C_{in}
  3. C o u t = ( A B ) + ( C i n ( A B ) ) C_{out}=(A\cdot B)+(C_{in}\cdot(A\oplus B))
  4. s s
  5. T F A = 2 T X O R = 2 3 D = 6 D T_{FA}=2\cdot T_{XOR}=2\cdot 3D=6D
  6. T c = 2 D T_{c}=2D
  7. T C R A ( n ) = T H A + ( n - 1 ) T c + T s = T F A + ( n - 1 ) T c = 6 D + ( n - 1 ) 2 D = ( n + 2 ) 2 D T_{CRA}(n)=T_{HA}+(n-1)\cdot T_{c}+T_{s}=T_{FA}+(n-1)\cdot T_{c}=6D+(n-1)\cdot 2% D=(n+2)\cdot 2D
  8. T C R A [ 0 : c o u t ] = T H A + n T c = 3 D + n 2 D T_{CRA_{[0:c_{out}]}}=T_{HA}+n\cdot T_{c}=3D+n\cdot 2D
  9. T C R A [ c 0 : c n ] ( n ) = n T c = n 2 D T_{CRA_{[c_{0}:c_{n}]}}(n)=n\cdot T_{c}=n\cdot 2D

Addition_chain.html

  1. l ( n ) l(n)
  2. log 2 ( n ) + log 2 ( ν ( n ) ) - 2.13 l ( n ) log 2 ( n ) + log 2 ( n ) ( 1 + o ( 1 ) ) / log 2 ( log 2 ( n ) ) \log_{2}(n)+\log_{2}(\nu(n))-2.13\leq l(n)\leq\log_{2}(n)+\log_{2}(n)(1+o(1))/% \log_{2}(\log_{2}(n))
  3. ν ( n ) \nu(n)

Adjacency_list.html

  1. | V | 2 / 8 |V|^{2}/8
  2. 8 | E | 8|E|
  3. | V | 2 |V|^{2}
  4. d = | E | / | V | 2 d=|E|/|V|^{2}
  5. 8 | E | > | V | 2 / 8 8|E|>|V|^{2}/8
  6. d > 1 / 64 d>1/64
  7. O ( | V | ) O(|V|)

Affine_cipher.html

  1. ( a x + b ) mod ( 26 ) (ax+b)\mod(26)
  2. b b
  3. m m
  4. 0.. m - 1 0..m-1
  5. E ( x ) = ( a x + b ) mod m , \mbox{E}~{}(x)=(ax+b)\mod{m},
  6. m m
  7. a a
  8. b b
  9. a a
  10. a a
  11. m m
  12. D ( x ) = a - 1 ( x - b ) mod m , \mbox{D}~{}(x)=a^{-1}(x-b)\mod{m},
  13. a - 1 a^{-1}
  14. a a
  15. m m
  16. 1 = a a - 1 mod m . 1=aa^{-1}\mod{m}.
  17. a a
  18. a a
  19. m m
  20. a a
  21. D ( E ( x ) ) = a - 1 ( E ( x ) - b ) mod m = a - 1 ( ( ( a x + b ) mod m ) - b ) mod m = a - 1 ( a x + b - b ) mod m = a - 1 a x mod m = x mod m . \begin{aligned}\displaystyle\mbox{D}~{}(\mbox{E}~{}(x))&\displaystyle=a^{-1}(% \mbox{E}~{}(x)-b)\mod{m}\\ &\displaystyle=a^{-1}(((ax+b)\mod{m})-b)\mod{m}\\ &\displaystyle=a^{-1}(ax+b-b)\mod{m}\\ &\displaystyle=a^{-1}ax\mod{m}\\ &\displaystyle=x\mod{m}.\end{aligned}
  22. a = 1 a=1
  23. m = 26 m=26
  24. a a
  25. a a
  26. b b
  27. a a
  28. m m
  29. a a
  30. b b
  31. m m
  32. a a
  33. a a
  34. b b
  35. a a
  36. y = E ( x ) = ( 5 x + 8 ) ( mod 26 ) y=E(x)=(5x+8)\;\;(\mathop{{\rm mod}}26)
  37. ( 5 x + 8 ) (5x+8)
  38. ( 5 x + 8 ) (5x+8)
  39. ( 5 x + 8 ) (5x+8)
  40. 5 x + 8 5x+8
  41. ( 5 x + 8 ) ( mod 26 ) (5x+8)\;\;(\mathop{{\rm mod}}26)
  42. 5 x + 8 5x+8
  43. ( 5 x + 8 ) ( mod 26 ) (5x+8)\;\;(\mathop{{\rm mod}}26)
  44. D ( y ) = 21 ( y - 8 ) mod 26 \mbox{D}~{}(y)=21(y-8)\mbox{ mod }~{}26
  45. a - 1 a^{-1}
  46. b b
  47. m m
  48. 21 ( y - 8 ) 21(y-8)

Affine_connection.html

  1. C ( M , T M ) × C ( M , T M ) C ( M , T M ) ( X , Y ) X Y , \begin{matrix}C^{\infty}(M,TM)\times C^{\infty}(M,TM)&\rightarrow&C^{\infty}(M% ,TM)\\ (X,Y)&\mapsto&\nabla_{X}Y,\end{matrix}
  2. f X Y = f X Y \nabla_{fX}Y=f\nabla_{X}Y
  3. X ( f Y ) = d f ( X ) Y + f X Y \nabla_{X}(fY)=\mathrm{d}f(X)Y+f\nabla_{X}Y
  4. γ ˙ ( t ) X = 0 \nabla_{\dot{\gamma}(t)}X=0
  5. X γ ( a ) = ξ . X_{\gamma(a)}=\xi.
  6. γ ~ \tilde{\gamma}
  7. T a x A x . T_{a_{x}}A_{x}.
  8. ( * ) { d p = θ 1 e 1 + + θ n e n d e i = ω i 1 e 1 + + ω i n e n , i = 1 , 2 , , n (*)\left\{\begin{matrix}\mathrm{d}{p}&=\theta^{1}{e}_{1}+\cdots+\theta^{n}{e}_% {n}&\\ \mathrm{d}{e}_{i}&=\omega^{1}_{i}{e}_{1}+\cdots+\omega^{n}_{i}{e}_{n},&\quad i% =1,2,\ldots,n\end{matrix}\right.
  9. p ( γ ( t + δ t ) ) - p ( γ ( t ) ) = ( θ 1 ( γ ( t ) ) e 1 + + θ n ( γ ( t ) ) e n ) δ t e i ( γ ( t + δ t ) ) - e i ( γ ( t ) ) = ( ω i 1 ( γ ( t ) ) e 1 + + ω i n ( γ ( t ) ) e n ) δ t . \left.\begin{matrix}p(\gamma(t+\delta t))-p(\gamma(t))&=\bigl(\theta^{1}(% \gamma^{\prime}(t)){e}_{1}+\cdots+\theta^{n}(\gamma^{\prime}(t)){e}_{n}\bigr)% \mathrm{\delta}t&\\ {e}_{i}(\gamma(t+\delta t))-{e}_{i}(\gamma(t))&=\bigl(\omega^{1}_{i}(\gamma^{% \prime}(t)){e}_{1}+\cdots+\omega^{n}_{i}(\gamma^{\prime}(t)){e}_{n}\bigr)% \delta t.\end{matrix}\right.
  10. ϕ ( p + v ) = α ( p ) + T ( v ) \phi(p+v)=\alpha(p)+T(v)\,
  11. π ( p ; 𝐞 1 , , 𝐞 n ) = p \pi(p;\mathbf{e}_{1},\dots,\mathbf{e}_{n})=p
  12. ϵ i ( p ; 𝐞 1 , , 𝐞 n ) = 𝐞 i . \epsilon_{i}(p;\mathbf{e}_{1},\dots,\mathbf{e}_{n})=\mathbf{e}_{i}.
  13. d π = θ 1 ε 1 + + θ n ε n d ε i = ω i 1 ε 1 + + ω i n ε n \begin{matrix}\mathrm{d}\pi&=\theta^{1}\varepsilon_{1}+\cdots+\theta^{n}% \varepsilon_{n}\\ \mathrm{d}\varepsilon_{i}&=\omega^{1}_{i}\varepsilon_{1}+\cdots+\omega^{n}_{i}% \varepsilon_{n}\end{matrix}
  14. d θ j - i ω i j θ i \displaystyle\mathrm{d}\theta^{j}-\sum_{i}\omega^{j}_{i}\wedge\theta^{i}
  15. d η + 1 2 [ η η ] = 0 , \mathrm{d}\eta+\tfrac{1}{2}[\eta\wedge\eta]=0,
  16. d θ + ω θ \mathrm{d}\theta+\omega\wedge\theta
  17. d ω + ω ω , \mathrm{d}\omega+\omega\wedge\omega,
  18. T ( X , Y ) = X Y - Y X - [ X , Y ] . T^{\nabla}(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y].
  19. R X , Y Z = X Y Z - Y X Z - [ X , Y ] Z . R^{\nabla}_{X,Y}Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z.
  20. γ ˙ \dot{\gamma}
  21. τ t s γ ˙ ( s ) = γ ˙ ( t ) \tau_{t}^{s}\dot{\gamma}(s)=\dot{\gamma}(t)
  22. γ ˙ ( t ) γ ˙ ( t ) = 0 \nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0
  23. γ ˙ ( 0 ) = X \dot{\gamma}(0)=X
  24. γ ˙ γ ˙ = k γ ˙ \nabla_{\dot{\gamma}}\dot{\gamma}=k\dot{\gamma}
  25. C ˙ t = τ t 0 x ˙ t , C 0 = 0. \dot{C}_{t}=\tau_{t}^{0}\dot{x}_{t},\quad C_{0}=0.
  26. , \langle,\rangle
  27. Y x , x = 0 , x 𝐒 2 . \langle Y_{x},x\rangle=0,\quad\forall x\in\mathbf{S}^{2}.
  28. ( X Y ) x = d Y x ( X ) + X x , Y x x (\nabla_{X}Y)_{x}=\mathrm{d}Y_{x}(X)+\langle X_{x},Y_{x}\rangle x\,
  29. ( X Y ) x , x = 0 ( 1 ) . \langle(\nabla_{X}Y)_{x},x\rangle=0\qquad(1).
  30. { f : 𝐒 2 𝐑 x Y x , x . \begin{cases}f:\mathbf{S}^{2}\to\mathbf{R}\\ x\longmapsto\langle Y_{x},x\rangle.\end{cases}
  31. d f x ( X ) = ( d Y ) x ( X ) , x + Y x , X x = 0. \mathrm{d}f_{x}(X)=\langle(\mathrm{d}Y)_{x}(X),x\rangle+\langle Y_{x},X_{x}% \rangle=0.\,
  32. \Box

Affine_variety.html

  1. k n k^{n}
  2. k [ x 1 , , x n ] / I k[x_{1},\ldots,x_{n}]/I
  3. H i ( X , F ) = 0 H^{i}(X,F)=0
  4. i > 0 i>0
  5. f 1 , , f k f_{1},\ldots,f_{k}
  6. V ( f 1 , , f k ) = { ( a 1 , , a n ) k n | f 1 ( a 1 , , a n ) = = f k ( a 1 , , a n ) = 0 } . V(f_{1},\ldots,f_{k})=\left\{(a_{1},\ldots,a_{n})\in k^{n}\;|\;f_{1}(a_{1},% \ldots,a_{n})=\ldots=f_{k}(a_{1},\ldots,a_{n})=0\right\}.
  7. R = k [ x 1 , , x n ] / f 1 , , f k , R=k[x_{1},\ldots,x_{n}]/\langle f_{1},\ldots,f_{k}\rangle,
  8. ( a 1 , , a n ) x 1 - a 1 ¯ , , x n - a n ¯ , (a_{1},\ldots,a_{n})\mapsto\langle\overline{x_{1}-a_{1}},\ldots,\overline{x_{n% }-a_{n}}\rangle,
  9. x i - a i ¯ \overline{x_{i}-a_{i}}
  10. x i - a i . x_{i}-a_{i}.
  11. Spec ( R ) , \operatorname{Spec}(R),
  12. f ¯ R \overline{f}\in R
  13. f k [ x 1 , , x n ] , f\in k[x_{1},\ldots,x_{n}],
  14. f f
  15. f ¯ \overline{f}
  16. X = spec A , Y = spec B X=\operatorname{spec}A,Y=\operatorname{spec}B
  17. k [ t 1 , , t n ] k[t_{1},\dots,t_{n}]
  18. ϕ : B A \phi:B\to A
  19. ϕ # : X Y \phi^{\#}:X\to Y
  20. 𝔪 ϕ - 1 ( 𝔪 ) \mathfrak{m}\mapsto\phi^{-1}(\mathfrak{m})
  21. X Y X\to Y
  22. ϕ # \phi^{\#}
  23. Y = k n Y=k^{n}
  24. ϕ \phi
  25. Y k n Y\subset k^{n}
  26. V ( I ) = { 𝔪 X I 𝔪 } V(I)=\{\mathfrak{m}\in X\mid I\subset\mathfrak{m}\}
  27. D ( f ) D(f)
  28. spec ( A [ f - 1 ] ) \operatorname{spec}(A[f^{-1}])
  29. 𝐀 n \mathbf{A}^{n}
  30. X = spec A , A = k [ x 1 , , x n ] / ( f 1 , , f r ) X=\operatorname{spec}A,A=k[x_{1},\dots,x_{n}]/(f_{1},\dots,f_{r})
  31. k n k^{n}
  32. i = 1 n f j x i ( a 1 , , a n ) ( x i - a i ) = 0 , j = 1 , , r \sum_{i=1}^{n}{\partial f_{j}\over\partial{x_{i}}}(a_{1},\dots,a_{n})(x_{i}-a_% {i})=0,\quad j=1,\dots,r
  33. x = ( a 1 , , a n ) . x=(a_{1},\dots,a_{n}).

Affirmative_conclusion_from_a_negative_premise.html

  1. A B = A\cap B=\emptyset
  2. B C = B\cap C=\emptyset

Aggregate_demand.html

  1. A D = C + I + G + ( X - M ) AD=C+I+G+(X-M)
  2. C C
  3. a c + b c ( Y - T ) a_{c}+b_{c}(Y-T)
  4. Y Y
  5. T T
  6. I I
  7. G G
  8. N X = X - M NX=X-M
  9. X X
  10. M M
  11. a m + b m ( Y - T ) a_{m}+b_{m}(Y-T)
  12. C C
  13. C = a + M P C × ( Y - T ) C=a+MPC\times(Y-T)
  14. a a
  15. M P C MPC
  16. ( Y - T ) (Y-T)
  17. I I
  18. I p I_{p}
  19. I I
  20. i i
  21. I ( Y , i ) I(Y,i)
  22. G G
  23. N X NX
  24. X - M X-M
  25. D D
  26. A D AD
  27. C + I p + G + ( X - M ) C+I_{p}+G+(X-M)
  28. Y d = C + I p + G + N X Y^{d}=C+I_{p}+G+NX
  29. P P
  30. Y d Y^{d}
  31. P P
  32. A D AD
  33. P P
  34. M s P \frac{M^{s}}{P}
  35. A D AD
  36. P P
  37. A D AD
  38. Y Y
  39. P P
  40. Y * Y^{*}
  41. A D AD
  42. Y * Y^{*}
  43. Y * Y^{*}
  44. A S AS
  45. A S AS
  46. Y Y
  47. A S AS

Agulhas_Current.html

  1. \cdot

Airspeed.html

  1. V c = A 0 5 [ ( q c P 0 + 1 ) 2 7 - 1 ] V_{c}=A_{0}\sqrt{5\Bigg[\bigg(\frac{q_{c}}{P_{0}}+1\bigg)^{\frac{2}{7}}-1\Bigg]}
  2. V c V_{c}\,
  3. q c q_{c}\,
  4. P 0 P_{0}\,
  5. A 0 A_{0}\,
  6. P 0 P_{0}
  7. A 0 A_{0}
  8. ( V t ) (V_{t})
  9. ( V g ) (V_{g})
  10. V t = V g - V w V_{t}\ =\ V_{g}-V_{w}
  11. V w V_{w}
  12. V t = a 0 M T T 0 V_{t}\ =\ a_{0}\cdot M\sqrt{\frac{T}{T_{0}}}
  13. a 0 a_{0}
  14. M M
  15. T T
  16. T 0 T_{0}
  17. V t = a 0 5 [ ( q c P + 1 ) 2 7 - 1 ] T T 0 V_{t}\ =\ a_{0}\cdot\sqrt{5\left[\left(\frac{q_{c}}{P}+1\right)^{\frac{2}{7}}-% 1\right]\cdot\frac{T}{T_{0}}}
  18. a 0 a_{0}
  19. q c q_{c}
  20. P P
  21. T T
  22. T 0 T_{0}
  23. 1 2 ρ V 2 = q = 1 2 ρ 0 V e 2 \frac{1}{2}\rho V^{2}=q=\frac{1}{2}\rho_{0}V_{e}^{2}
  24. V V e = ρ 0 ρ \frac{V}{V_{e}}=\sqrt{\frac{\rho_{0}}{\rho}}
  25. ρ \rho\,
  26. ρ ρ 0 = p T 0 p 0 T \frac{\rho}{\rho_{0}}=\frac{p\,T_{0}}{p_{0}\,T}
  27. p p\,
  28. p 0 p_{0}\,
  29. T T\,
  30. T 0 T_{0}\,

Airy_disk.html

  1. sin θ 1.22 λ d \sin\theta\approx 1.22\frac{\lambda}{d}
  2. θ 1.22 λ d \theta\approx 1.22\frac{\lambda}{d}
  3. s = 2.76 a s=\frac{2.76}{a}
  4. sin θ = 1.22 λ d \sin\theta=1.22\ \frac{\lambda}{d}
  5. x f = 1.22 λ d \frac{x}{f}=1.22\ \frac{\lambda}{d}
  6. x x
  7. f f
  8. x = 1.22 λ f d x=1.22\ \frac{\lambda f}{d}
  9. f d \frac{f}{d}
  10. x x
  11. a a
  12. λ \lambda
  13. R > a 2 / λ R>a^{2}/\lambda
  14. I ( θ ) = I 0 ( 2 J 1 ( k a sin θ ) k a sin θ ) 2 = I 0 ( 2 J 1 ( x ) x ) 2 I(\theta)=I_{0}\left(\frac{2J_{1}(ka\sin\theta)}{ka\sin\theta}\right)^{2}=I_{0% }\left(\frac{2J_{1}(x)}{x}\right)^{2}
  15. I 0 I_{0}
  16. J 1 J_{1}
  17. k = 2 π / λ k={2\pi}/{\lambda}
  18. a a
  19. θ \theta
  20. x = k a sin θ = 2 π a λ q R = π q λ N x=ka\sin\theta=\frac{2\pi a}{\lambda}\frac{q}{R}=\frac{\pi q}{\lambda N}
  21. N = R / d N=R/d
  22. θ 0 \theta\rightarrow 0
  23. x 0 x\rightarrow 0
  24. I ( 0 ) = I 0 I(0)=I_{0}
  25. J 1 ( x ) J_{1}(x)
  26. x = k a sin θ 3.8317 , 7.0156 , 10.1735 , 13.3237 , 16.4706 x=ka\sin\theta\approx 3.8317,7.0156,10.1735,13.3237,16.4706\dots
  27. k a sin θ = 3.8317 ka\sin{\theta}=3.8317\dots
  28. sin θ 3.83 k a = 3.83 λ 2 π a = 1.22 λ 2 a = 1.22 λ d \sin\theta\approx\frac{3.83}{ka}=\frac{3.83\lambda}{2\pi a}=1.22\frac{\lambda}% {2a}=1.22\frac{\lambda}{d}
  29. q 1 q_{1}
  30. θ \theta
  31. q 1 = R sin θ 1.22 R λ d = 1.22 λ N q_{1}=R\sin\theta\approx 1.22{R}\frac{\lambda}{d}=1.22\lambda N
  32. J 1 ( x ) = x / 2 2 J_{1}(x)={x}/{2\sqrt{2}}
  33. x = 1.61633 x=1.61633\dots
  34. J 1 ( x ) = x / 2 e J_{1}(x)={x}/{2e}
  35. x = 2.58383 x=2.58383\dots
  36. x = 5.13562 x=5.13562\dots
  37. I 0 I_{0}
  38. P 0 P_{0}
  39. I 0 = \Epsilon A 2 A 2 2 R 2 = P 0 A λ 2 R 2 I_{0}=\frac{\Epsilon_{A}^{2}A^{2}}{2R^{2}}=\frac{P_{0}A}{\lambda^{2}R^{2}}
  40. \Epsilon \Epsilon
  41. A = π a 2 A=\pi a^{2}
  42. I 0 = ( P 0 A ) / ( λ 2 f 2 ) I_{0}=(P_{0}A)/(\lambda^{2}f^{2})
  43. I ( θ ) I(\theta)
  44. P ( θ ) = P 0 [ 1 - J 0 2 ( k a sin θ ) - J 1 2 ( k a sin θ ) ] P(\theta)=P_{0}[1-J_{0}^{2}(ka\sin\theta)-J_{1}^{2}(ka\sin\theta)]
  45. J 0 J_{0}
  46. J 1 J_{1}
  47. J 1 ( k a sin θ ) = 0 J_{1}(ka\sin\theta)=0
  48. λ \lambda
  49. I ( q ) I 0 exp ( - q 2 2 σ 2 ) , I(q)\approx I^{\prime}_{0}\exp\left(\frac{-q^{2}}{2\sigma^{2}}\right)\ ,
  50. I 0 I^{\prime}_{0}
  51. q q
  52. σ \sigma
  53. I 0 = I 0 I^{\prime}_{0}=I_{0}
  54. σ \sigma
  55. σ 0.42 λ N , \sigma\approx 0.42\lambda N\ ,
  56. σ 0.45 λ N . \sigma\approx 0.45\lambda N\ .
  57. σ \sigma
  58. 0.42 λ N 0.42\lambda N
  59. 1.22 λ N 1.22\lambda N
  60. I ( θ ) = I 0 ( 1 - ϵ 2 ) 2 ( 2 J 1 ( x ) x - 2 ϵ J 1 ( ϵ x ) x ) 2 I(\theta)=\frac{I_{0}}{(1-\epsilon^{2})^{2}}\left(\frac{2J_{1}(x)}{x}-\frac{2% \epsilon J_{1}(\epsilon x)}{x}\right)^{2}
  61. ϵ \epsilon
  62. ( 0 ϵ < 1 ) \left(0\leq\epsilon<1\right)
  63. x = k a sin ( θ ) π R λ N x=ka\sin(\theta)\approx\frac{\pi R}{\lambda N}
  64. R R
  65. λ \lambda
  66. N N
  67. R R
  68. E ( R ) = 1 ( 1 - ϵ 2 ) ( 1 - J 0 2 ( x ) - J 1 2 ( x ) + ϵ 2 [ 1 - J 0 2 ( ϵ x ) - J 1 2 ( ϵ x ) ] - 4 ϵ 0 x J 1 ( t ) J 1 ( ϵ t ) t d t ) E(R)=\frac{1}{(1-\epsilon^{2})}\left(1-J_{0}^{2}(x)-J_{1}^{2}(x)+\epsilon^{2}% \left[1-J_{0}^{2}(\epsilon x)-J_{1}^{2}(\epsilon x)\right]-4\epsilon\int_{0}^{% x}\frac{J_{1}(t)J_{1}(\epsilon t)}{t}\,dt\right)
  69. ϵ 0 \epsilon\rightarrow 0
  70. I 0 , A i r y = ( P 0 A ) / ( λ 2 f 2 ) I_{0,Airy}=(P_{0}A)/(\lambda^{2}f^{2})
  71. P 0 P_{0}
  72. A = π D 2 / 4 A=\pi D^{2}/4
  73. D D
  74. λ \lambda
  75. f f
  76. 1 / e 2 1/e^{2}
  77. I 0 , Airy I_{0,\mathrm{Airy}}

Airy_function.html

  1. d 2 y d x 2 - x y = 0 , \frac{d^{2}y}{dx^{2}}-xy=0,\,\!
  2. Ai ( x ) = 1 π 0 cos ( t 3 3 + x t ) d t 1 π lim b 0 b cos ( t 3 3 + x t ) d t , \mathrm{Ai}(x)=\frac{1}{\pi}\int_{0}^{\infty}\cos\left(\tfrac{t^{3}}{3}+xt% \right)\,dt\equiv\frac{1}{\pi}\lim_{b\to\infty}\int_{0}^{b}\cos\left(\tfrac{t^% {3}}{3}+xt\right)\,dt,
  3. y ′′ - x y = 0. y^{\prime\prime}-xy=0.
  4. Bi ( x ) = 1 π 0 [ exp ( - t 3 3 + x t ) + sin ( t 3 3 + x t ) ] d t . \mathrm{Bi}(x)=\frac{1}{\pi}\int_{0}^{\infty}\left[\exp\left(-\tfrac{t^{3}}{3}% +xt\right)+\sin\left(\tfrac{t^{3}}{3}+xt\right)\,\right]dt.
  5. Ai ( 0 ) = 1 3 2 3 Γ ( 2 3 ) , Ai ( 0 ) = - 1 3 1 3 Γ ( 1 3 ) , Bi ( 0 ) = 1 3 1 6 Γ ( 2 3 ) , Bi ( 0 ) = 3 1 6 Γ ( 1 3 ) . \begin{aligned}\displaystyle\mathrm{Ai}(0)&\displaystyle{}=\frac{1}{3^{\frac{2% }{3}}\Gamma(\tfrac{2}{3})},&\displaystyle\quad\mathrm{Ai}^{\prime}(0)&% \displaystyle{}=-\frac{1}{3^{\frac{1}{3}}\Gamma(\tfrac{1}{3})},\\ \displaystyle\mathrm{Bi}(0)&\displaystyle{}=\frac{1}{3^{\frac{1}{6}}\Gamma(% \tfrac{2}{3})},&\displaystyle\quad\mathrm{Bi}^{\prime}(0)&\displaystyle{}=% \frac{3^{\frac{1}{6}}}{\Gamma(\tfrac{1}{3})}.\end{aligned}
  6. - Ai ( t + x ) Ai ( t + y ) d t = δ ( x - y ) \int_{-\infty}^{\infty}\mathrm{Ai}(t+x)\mathrm{Ai}(t+y)dt=\delta(x-y)
  7. Ai ( z ) e - 2 3 z 3 2 2 π z 1 4 [ n = 0 ( - 1 ) n Γ ( n + 5 6 ) Γ ( n + 1 6 ) ( 3 4 ) n 2 π n ! z 3 n / 2 ] . \mathrm{Ai}(z)\sim\dfrac{e^{-\frac{2}{3}z^{\frac{3}{2}}}}{2\sqrt{\pi}\,z^{% \frac{1}{4}}}\left[\sum_{n=0}^{\infty}\dfrac{(-1)^{n}\Gamma(n+\frac{5}{6})% \Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^{n}}{2\pi n!z^{3n/2}}\right].
  8. Ai ( - z ) sin ( 2 3 z 3 2 + π 4 ) π z 1 4 Bi ( - z ) cos ( 2 3 z 3 2 + π 4 ) π z 1 4 . \begin{aligned}\displaystyle\mathrm{Ai}(-z)&\displaystyle{}\sim\frac{\sin\left% (\frac{2}{3}z^{\frac{3}{2}}+\frac{\pi}{4}\right)}{\sqrt{\pi}\,z^{\frac{1}{4}}}% \\ \displaystyle\mathrm{Bi}(-z)&\displaystyle{}\sim\frac{\cos\left(\frac{2}{3}z^{% \frac{3}{2}}+\frac{\pi}{4}\right)}{\sqrt{\pi}\,z^{\frac{1}{4}}}.\end{aligned}
  9. Ai ( z ) = 1 2 π i C exp ( t 3 3 - z t ) d t , \mathrm{Ai}(z)=\frac{1}{2\pi i}\int_{C}\exp\left(\tfrac{t^{3}}{3}-zt\right)\,dt,
  10. [ Ai ( x + i y ) ] \Re\left[\mathrm{Ai}(x+iy)\right]
  11. [ Ai ( x + i y ) ] \Im\left[\mathrm{Ai}(x+iy)\right]
  12. | Ai ( x + i y ) | |\mathrm{Ai}(x+iy)|\,
  13. arg [ Ai ( x + i y ) ] \mathrm{arg}\left[\mathrm{Ai}(x+iy)\right]\,
  14. [ Bi ( x + i y ) ] \Re\left[\mathrm{Bi}(x+iy)\right]
  15. [ Bi ( x + i y ) ] \Im\left[\mathrm{Bi}(x+iy)\right]
  16. | Bi ( x + i y ) | |\mathrm{Bi}(x+iy)|\,
  17. arg [ Bi ( x + i y ) ] \mathrm{arg}\left[\mathrm{Bi}(x+iy)\right]\,
  18. Ai ( x ) \displaystyle\mathrm{Ai}(x)
  19. x 2 y ′′ + x y - ( x 2 + 1 9 ) y = 0. x^{2}y^{\prime\prime}+xy^{\prime}-\left(x^{2}+\tfrac{1}{9}\right)y=0.
  20. Ai ( x ) = - x π 3 K 2 3 ( 2 3 x 3 2 ) . \mathrm{Ai^{\prime}}(x)=-\frac{x}{\pi\sqrt{3}}\,K_{\frac{2}{3}}\left(\tfrac{2}% {3}x^{\frac{3}{2}}\right).
  21. Ai ( - x ) \displaystyle\mathrm{Ai}(-x)
  22. x 2 y ′′ + x y + ( x 2 - 1 9 ) y = 0. x^{2}y^{\prime\prime}+xy^{\prime}+\left(x^{2}-\tfrac{1}{9}\right)y=0.
  23. Gi ( x ) \displaystyle\mathrm{Gi}(x)
  24. ( Ai ) ( k ) := - Ai ( x ) e - 2 π i k x d x = e i 3 ( 2 π k ) 3 . \mathcal{F}(\mathrm{Ai})(k):=\int_{-\infty}^{\infty}\mathrm{Ai}(x)\ e^{-2\pi ikx% }\,dx=e^{\frac{i}{3}(2\pi k)^{3}}.
  25. T e = 1 1 + F sin 2 ( δ 2 ) , T_{e}=\frac{1}{1+F\sin^{2}(\frac{\delta}{2})},
  26. F = 4 R < m t p l > ( 1 - R ) 2 F=\frac{4R}{<}mtpl>{{(1-R)^{2}}}

AKS_primality_test.html

  1. ( x - a ) n ( x n - a ) ( mod n ) ( 1 ) (x-a)^{n}\equiv(x^{n}-a)\;\;(\mathop{{\rm mod}}n)\qquad(1)
  2. ( x - a ) n (x-a)^{n}
  3. ( n k ) 0 ( mod n ) {n\choose k}\equiv 0\;\;(\mathop{{\rm mod}}n)
  4. 0 < k < n 0<k<n
  5. ( x - a ) n ( x n - a ) ( mod ( n , x r - 1 ) ) ( 2 ) (x-a)^{n}\equiv(x^{n}-a)\;\;(\mathop{{\rm mod}}(n,x^{r}-1))\qquad(2)
  6. ( x - a ) n - ( x n - a ) = n f + ( x r - 1 ) g ( 3 ) (x-a)^{n}-(x^{n}-a)=nf+(x^{r}-1)g\qquad(3)
  7. O ~ ( log 12 ( n ) ) \tilde{O}(\log^{12}(n))
  8. O ~ ( log 6 ( n ) ) \tilde{O}(\log^{6}(n))
  9. O ~ ( log 10.5 ( n ) ) \tilde{O}(\log^{10.5}(n))
  10. O ~ ( log 7.5 ( n ) ) \tilde{O}(\log^{7.5}(n))
  11. O ~ ( log 6 ( n ) ) \tilde{O}(\log^{6}(n))
  12. O ~ ( log 12 ( n ) ) \tilde{O}(\log^{12}(n))
  13. O ~ ( log 3 ( n ) ) \tilde{O}(\log^{3}(n))
  14. φ ( r ) log 2 ( n ) \scriptstyle\lfloor\scriptstyle{\sqrt{\varphi(r)}\log_{2}(n)}\scriptstyle\rfloor
  15. φ ( r ) \scriptstyle\varphi(r)
  16. \scriptstyle\lfloor
  17. \scriptstyle\rfloor
  18. φ ( r ) log ( n ) \scriptstyle\lfloor\scriptstyle{\sqrt{\varphi(r)}\log(n)}\scriptstyle\rfloor
  19. \scriptstyle\lfloor
  20. φ ( 29 ) \scriptstyle\sqrt{\varphi(29)}\rfloor

Alcohol_by_volume.html

  1. A B V × 0.78924 = A B W × density of beverage at 20°C in g/ml ABV\times 0.78924=ABW\times\,\text{density of beverage at 20°C in g/ml}
  2. A B V = ( Starting SG - Final SG ) / 7.36 ABV=(\mathrm{Starting~{}SG}-\mathrm{Final~{}SG})/7.36
  3. A B V = 1.05 0.79 ( Starting SG - Final SG Final SG ) × 100 ABV=\frac{1.05}{0.79}\left(\frac{\mathrm{Starting~{}SG}-\mathrm{Final~{}SG}}{% \mathrm{Final~{}SG}}\right)\times 100
  4. A B V = 131 ( Starting SG - Final SG ) ABV=131\left(\mathrm{Starting~{}SG}-\mathrm{Final~{}SG}\right)

Alexandrov_topology.html

  1. 𝐗 = X , \mathbf{X}=\langle X,\leq\rangle
  2. τ \tau
  3. τ = { G X : x , y X x G x y y G , } \tau=\{\,G\subseteq X:\forall x,y\in X\ \ x\in G\ \land\ x\leq y\ \rightarrow% \ y\in G,\}
  4. 𝐓 ( 𝐗 ) = X , τ \mathbf{T}(\mathbf{X})=\langle X,\tau\rangle
  5. { S X : x , y X x S y x y S , } \{\,S\subseteq X:\forall x,y\in X\ \ x\in S\ \land\ y\leq x\ \rightarrow\ y\in S,\}

Alfréd_Rényi.html

  1. K K
  2. K K

Algebra_(ring_theory).html

  1. [ , ] : A × A A [\cdot,\cdot]:A\times A\to A
  2. [ α x + β y , z ] = α [ x , z ] + β [ y , z ] , [ z , α x + β y ] = α [ z , x ] + β [ z , y ] [\alpha x+\beta y,z]=\alpha[x,z]+\beta[y,z],\quad[z,\alpha x+\beta y]=\alpha[z% ,x]+\beta[z,y]
  3. α \alpha
  4. β \beta
  5. q = w + x i + y j + z k q=w+xi+yj+zk\!
  6. f : R A f:R\to A
  7. λ : A B \lambda:A\to B
  8. ρ : A B \rho:A\to B
  9. ρ \rho
  10. \mathbb{Z}

Algebra_homomorphism.html

  1. F : A B F:A\rightarrow B
  2. F : A B F:A\rightarrow B
  3. f f\,
  4. f ^ \hat{f}\,
  5. f ^ ( t ) = f ( t ) \hat{f}(t)=f(t)\,
  6. f f ^ f\mapsto\hat{f}\,
  7. p ( x ) = t K ( x - t ) . p(x)=\prod\limits_{t\in K}(x-t).\,
  8. p ( t ) = 0 p(t)=0\,
  9. p ^ = 0 \hat{p}=0\,
  10. f ^ = 0 \hat{f}=0\,
  11. f = 0 f=0\,
  12. deg f = n \deg f=n\,
  13. t 0 , t 1 , , t n t_{0},t_{1},\dots,t_{n}\,
  14. f ( t i ) = 0 f(t_{i})=0\,
  15. 0 i n 0\leq i\leq n
  16. f = 0 f=0\,
  17. f f ^ f\mapsto\hat{f}\,
  18. A = B A=B

Algebraic_equation.html

  1. P = Q P=Q
  2. x 5 - 3 x + 1 = 0 x^{5}-3x+1=0
  3. y 4 + x y 2 = x 3 3 - x y 2 + y 2 - 1 7 y^{4}+\frac{xy}{2}=\frac{x^{3}}{3}-xy^{2}+y^{2}-\frac{1}{7}
  4. x = 1 + 5 2 x=\frac{1+\sqrt{5}}{2}
  5. x 2 - x - 1 = 0 x^{2}-x-1=0
  6. P = Q P=Q
  7. P - Q = 0 P-Q=0
  8. y 4 + x y 2 = x 3 3 - x y 2 + y 2 - 1 7 y^{4}+\frac{xy}{2}=\frac{x^{3}}{3}-xy^{2}+y^{2}-\frac{1}{7}
  9. 42 y 4 + 21 x y - 14 x 3 + 42 x y 2 - 42 y 2 + 6 = 0. 42y^{4}+21xy-14x^{3}+42xy^{2}-42y^{2}+6=0.
  10. e T x 2 + 1 T x y + sin ( T ) z - 2 = 0 e^{T}x^{2}+\frac{1}{T}xy+\sin(T)z-2=0

Algebraic_function_field.html

  1. k ( X ) ( X 3 ) k(X)(\sqrt{X^{3}})
  2. k ( X ) k(X)
  3. k ( Y ) ( Y 2 3 ) k(Y)(\sqrt[3]{Y^{2}})
  4. k ( Y ) k(Y)

Algebraic_geometry_and_analytic_geometry.html

  1. \mathcal{F}
  2. an \mathcal{F}\text{an}
  3. \mathcal{F}
  4. 𝒢 \mathcal{G}
  5. Hom 𝒪 X ( , 𝒢 ) Hom 𝒪 X an ( an , 𝒢 an ) \,\text{Hom}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})\rightarrow\,\text{Hom}% _{\mathcal{O}^{\,\text{an}}_{X}}(\mathcal{F}^{\,\text{an}},\mathcal{G}^{\,% \text{an}})
  6. 𝒪 X \mathcal{O}_{X}
  7. 𝒪 X an \mathcal{O}_{X}^{\,\text{an}}
  8. \mathcal{F}
  9. an \mathcal{F}\text{an}
  10. 𝒪 X an \mathcal{O}^{\,\text{an}}_{X}
  11. \mathcal{F}
  12. ε q : H q ( X , ) H q ( X a n , a n ) \varepsilon_{q}\ :\ H^{q}(X,\mathcal{F})\rightarrow H^{q}(X^{an},\mathcal{F}^{% an})
  13. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  14. 𝒪 X an \mathcal{O}_{X}^{\mathrm{an}}
  15. ( X an , 𝒪 X an ) (X^{\mathrm{an}},\mathcal{O}_{X}^{\mathrm{an}})
  16. ( X an , 𝒪 X an ) (X^{\mathrm{an}},\mathcal{O}_{X}^{\mathrm{an}})
  17. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  18. 𝒪 X an ( U ) \mathcal{O}_{X}^{\mathrm{an}}(U)
  19. \mathcal{F}
  20. an \mathcal{F}^{\mathrm{an}}
  21. 𝒪 X \mathcal{O}_{X}
  22. λ X * : ( λ X ) * an \lambda_{X}^{*}:\mathcal{F}\rightarrow(\lambda_{X})_{*}\mathcal{F}^{\mathrm{an}}
  23. an \mathcal{F}^{\mathrm{an}}
  24. λ X - 1 λ X - 1 𝒪 X 𝒪 X an \lambda_{X}^{-1}\mathcal{F}\otimes_{\lambda_{X}^{-1}\mathcal{O}_{X}}\mathcal{O% }_{X}^{\mathrm{an}}
  25. an \mathcal{F}\mapsto\mathcal{F}^{\mathrm{an}}
  26. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  27. ( X an , 𝒪 X an ) (X^{\mathrm{an}},\mathcal{O}_{X}^{\mathrm{an}})
  28. \mathcal{F}
  29. ( f * ) an f * an an (f_{*}\mathcal{F})^{\mathrm{an}}\rightarrow f_{*}^{\mathrm{an}}\mathcal{F}^{% \mathrm{an}}
  30. ( R i f * ) an R i f * an an (R^{i}f_{*}\mathcal{F})^{\mathrm{an}}\cong R^{i}f_{*}^{\mathrm{an}}\mathcal{F}% ^{\mathrm{an}}
  31. , 𝒢 \mathcal{F},\mathcal{G}
  32. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  33. f : an 𝒢 an f:\mathcal{F}^{\mathrm{an}}\rightarrow\mathcal{G}^{\mathrm{an}}
  34. 𝒪 X an \mathcal{O}_{X}^{\mathrm{an}}
  35. 𝒪 X \mathcal{O}_{X}
  36. φ : 𝒢 \varphi:\mathcal{F}\rightarrow\mathcal{G}
  37. \mathcal{R}
  38. 𝒪 X an \mathcal{O}_{X}^{\mathrm{an}}
  39. \mathcal{F}
  40. 𝒪 X \mathcal{O}_{X}
  41. an \mathcal{F}^{\mathrm{an}}\cong\mathcal{R}

Algebraic_K-theory.html

  1. 0 V V V ′′ 0 , 0\to V^{\prime}\to V\to V^{\prime\prime}\to 0,
  2. F × 𝐙 F × / x ( 1 - x ) : x F { 0 , 1 } . F^{\times}\otimes_{\mathbf{Z}}F^{\times}/\langle x\otimes(1-x)\colon x\in F% \setminus\{0,1\}\rangle.
  3. 𝒦 n \mathcal{K}_{n}
  4. H 2 ( X , 𝒦 2 ) H^{2}(X,\mathcal{K}_{2})
  5. H p ( X , 𝒦 p ) CH p ( X ) H^{p}(X,\mathcal{K}_{p})\cong\operatorname{CH}^{p}(X)
  6. K ~ 0 ( A ) = 𝔭 prime ideal of A Ker dim 𝔭 , \tilde{K}_{0}\left(A\right)=\bigcap\limits_{\mathfrak{p}\,\text{ prime ideal % of }A}\mathrm{Ker}\dim_{\mathfrak{p}},
  7. dim 𝔭 : K 0 ( A ) 𝐙 \dim_{\mathfrak{p}}:K_{0}\left(A\right)\to\mathbf{Z}
  8. A 𝔭 A_{\mathfrak{p}}
  9. M 𝔭 M_{\mathfrak{p}}
  10. K ~ 0 ( A ) \tilde{K}_{0}\left(A\right)
  11. K ~ 0 ( A ) = Pic A . \tilde{K}_{0}(A)=\operatorname{Pic}A.
  12. D ( A , I ) = { ( x , y ) A × A : x - y I } . D(A,I)=\{(x,y)\in A\times A:x-y\in I\}\ .
  13. K 0 ( A , I ) = ker ( K 0 ( D ( A , I ) ) K 0 ( A ) ) . K_{0}(A,I)=\ker\left({K_{0}(D(A,I))\rightarrow K_{0}(A)}\right)\ .
  14. K 1 ( A ) = GL ( A ) ab = GL ( A ) / [ GL ( A ) , GL ( A ) ] K_{1}(A)=\operatorname{GL}(A)^{\mbox{ab}~{}}=\operatorname{GL}(A)/[% \operatorname{GL}(A),\operatorname{GL}(A)]
  15. GL ( A ) = colim GL ( n , A ) \operatorname{GL}(A)=\operatorname{colim}\operatorname{GL}(n,A)
  16. K 1 ( A , I ) = ker ( K 1 ( D ( A , I ) ) K 1 ( A ) ) . K_{1}(A,I)=\ker\left({K_{1}(D(A,I))\rightarrow K_{1}(A)}\right)\ .
  17. K 1 ( A , I ) K 1 ( A ) K 1 ( A / I ) K 0 ( A , I ) K 0 ( A ) K 0 ( A / I ) . K_{1}(A,I)\rightarrow K_{1}(A)\rightarrow K_{1}(A/I)\rightarrow K_{0}(A,I)% \rightarrow K_{0}(A)\rightarrow K_{0}(A/I)\ .
  18. 1 S K 1 ( A ) K 1 ( A ) A * 1 , 1\to SK_{1}(A)\to K_{1}(A)\to A^{*}\to 1,
  19. 1 SL ( A ) GL ( A ) A * 1. 1\to\operatorname{SL}(A)\to\operatorname{GL}(A)\to A^{*}\to 1.
  20. φ : St ( A ) GL ( A ) , \varphi\colon\operatorname{St}(A)\to\mathrm{GL}(A),
  21. K 2 ( 𝐐 ) = ( 𝐙 / 4 ) * × p odd prime ( 𝐙 / p ) * K_{2}(\mathbf{Q})=(\mathbf{Z}/4)^{*}\times\prod_{p\,\text{ odd prime}}(\mathbf% {Z}/p)^{*}
  22. K 2 ( k ) = k × 𝐙 k × / a ( 1 - a ) a 0 , 1 . K_{2}(k)=k^{\times}\otimes_{\mathbf{Z}}k^{\times}/\langle a\otimes(1-a)\mid a% \not=0,1\rangle.
  23. K 2 F 𝐩 K 1 A / 𝐩 K 1 A K 1 F 𝐩 K 0 A / 𝐩 K 0 A K 0 F 0 K_{2}F\rightarrow\oplus_{\mathbf{p}}K_{1}A/{\mathbf{p}}\rightarrow K_{1}A% \rightarrow K_{1}F\rightarrow\oplus_{\mathbf{p}}K_{0}A/{\mathbf{p}}\rightarrow K% _{0}A\rightarrow K_{0}F\rightarrow 0
  24. K 2 ( A ) K 2 ( A / I ) K 1 ( A , I ) K 1 ( A ) . K_{2}(A)\rightarrow K_{2}(A/I)\rightarrow K_{1}(A,I)\rightarrow K_{1}(A)\cdots\ .
  25. x y x - 1 y - 1 xyx^{-1}y^{-1}
  26. K * M ( k ) := T * ( k × ) / ( a ( 1 - a ) ) , K^{M}_{*}(k):=T^{*}(k^{\times})/(a\otimes(1-a)),
  27. { a ( 1 - a ) : a 0 , 1 } . \left\{a\otimes(1-a):\ a\neq 0,1\right\}.
  28. K m × K n K m + n K_{m}\times K_{n}\rightarrow K_{m+n}
  29. K * M ( F ) K^{M}_{*}(F)
  30. a 1 a n a_{1}\otimes\cdots\otimes a_{n}
  31. K n M ( k ) K^{M}_{n}(k)
  32. { a 1 , , a n } \{a_{1},\ldots,a_{n}\}
  33. : k * H 1 ( k , μ m ) \partial:k^{*}\rightarrow H^{1}(k,\mu_{m})
  34. μ m \mu_{m}
  35. n : k * × × k * H n ( k , μ m n ) \partial^{n}:k^{*}\times\cdots\times k^{*}\rightarrow H^{n}\left({k,\mu_{m}^{% \otimes n}}\right)
  36. n \partial^{n}
  37. K n M ( k ) K^{M}_{n}(k)
  38. K n ( R ) = π n ( B G L ( R ) + ) , K_{n}(R)=\pi_{n}(BGL(R)^{+}),
  39. K n ( R ) = π n ( B G L ( R ) + × K 0 ( R ) ) K_{n}(R)=\pi_{n}(BGL(R)^{+}\times K_{0}(R))
  40. M N M ′′ , M^{\prime}\longleftarrow N\longrightarrow M^{\prime\prime},
  41. K i ( P ) = π i + 1 ( BQ P , 0 ) K_{i}(P)=\pi_{i+1}(\mathrm{BQ}P,0)

Algebraic_torus.html

  1. X ( T ) X^{\bullet}(T)
  2. X ( T ) X_{\bullet}(T)
  3. X ( T ) × X ( T ) X^{\bullet}(T)\times X_{\bullet}(T)\to\mathbb{Z}
  4. ( f , g ) deg ( f g ) (f,g)\mapsto\,\text{deg}(f\circ g)
  5. D ( M ) S ( X ) := Hom ( M , 𝔾 m , S ( X ) ) . D(M)_{S}(X):=\mathrm{Hom}(M,\mathbb{G}_{m,S}(X)).
  6. X ( Res L / K T ) Ind G L G K X ( T ) . X^{\bullet}(\mathrm{Res}_{L/K}T)\cong\mathrm{Ind}_{G_{L}}^{G_{K}}X^{\bullet}(T).
  7. H 1 ( S , G L n ( ) ) H^{1}(S,GL_{n}(\mathbb{Z}))
  8. H 1 ( G K , G L n ( ) ) H^{1}(G_{K},GL_{n}(\mathbb{Z}))
  9. H 1 ( S , Hom ( X ( T 1 ) , X ( T 2 ) ) ) H^{1}(S,\mathrm{Hom}_{\mathbb{Z}}(X^{\bullet}(T_{1}),X^{\bullet}(T_{2})))
  10. 𝒪 S \mathcal{O}_{S}
  11. H 1 ( G k , X ( T ) ) E x t 1 ( T , 𝔾 m ) H^{1}(G_{k},X^{\bullet}(T))\cong Ext^{1}(T,\mathbb{G}_{m})

Algebraically_compact_module.html

  1. j J r i j x j = m i \sum_{j\in J}r_{ij}x_{j}=m_{i}

Almost_complex_manifold.html

  1. J i j = - δ i , j - 1 J_{ij}=-\delta_{i,j-1}
  2. J i j = δ i , j + 1 J_{ij}=\delta_{i,j+1}
  3. Ω r ( M ) 𝐂 = p + q = r Ω ( p , q ) ( M ) . \Omega^{r}(M)^{\mathbf{C}}=\bigoplus_{p+q=r}\Omega^{(p,q)}(M).\,
  4. = π p + 1 , q d \partial=\pi_{p+1,q}\circ d
  5. ¯ = π p , q + 1 d \overline{\partial}=\pi_{p,q+1}\circ d
  6. \partial
  7. ¯ \overline{\partial}
  8. d = r + s = p + q + 1 π r , s d = + ¯ + . d=\sum_{r+s=p+q+1}\pi_{r,s}\circ d=\partial+\overline{\partial}+\cdots.
  9. z μ = x μ + i y μ z^{\mu}=x^{\mu}+iy^{\mu}
  10. J x μ = y μ J y μ = - x μ J\frac{\partial}{\partial x^{\mu}}=\frac{\partial}{\partial y^{\mu}}\qquad J% \frac{\partial}{\partial y^{\mu}}=-\frac{\partial}{\partial x^{\mu}}
  11. J z μ = i z μ J z ¯ μ = - i z ¯ μ . J\frac{\partial}{\partial z^{\mu}}=i\frac{\partial}{\partial z^{\mu}}\qquad J% \frac{\partial}{\partial\bar{z}^{\mu}}=-i\frac{\partial}{\partial\bar{z}^{\mu}}.
  12. N A ( X , Y ) = - A 2 [ X , Y ] + A ( [ A X , Y ] + [ X , A Y ] ) - [ A X , A Y ] . N_{A}(X,Y)=-A^{2}[X,Y]+A([AX,Y]+[X,AY])-[AX,AY].\,
  13. - ( N A ) i j k = A i m m A j k - A j m m A i k - A m k ( i A j m - j A i m ) . -(N_{A})_{ij}^{k}=A_{i}^{m}\partial_{m}A^{k}_{j}-A_{j}^{m}\partial_{m}A^{k}_{i% }-A^{k}_{m}(\partial_{i}A^{m}_{j}-\partial_{j}A^{m}_{i}).
  14. d = + ¯ d=\partial+\bar{\partial}
  15. ¯ 2 = 0. \bar{\partial}^{2}=0.

Almost_disjoint_sets.html

  1. | A B | < . \left|A\cap B\right|<\infty.
  2. A i A j | A i A j | < . A_{i}\neq A_{j}\quad\Rightarrow\quad\left|A_{i}\cap A_{j}\right|<\infty.
  3. i I A i < . \bigcap_{i\in I}A_{i}<\infty.
  4. { { 1 , 2 , 3 , } , { 2 , 3 , 4 , } , { 3 , 4 , 5 , } , } \{\{1,2,3,\ldots\},\{2,3,4,\ldots\},\{3,4,5,\ldots\},\ldots\}
  5. | A B | < κ . \left|A\cap B\right|<\kappa.
  6. κ = 0 \kappa=\aleph_{0}
  7. m ( A B ) = 0. m(A\cap B)=0.

Almost_perfect_number.html

  1. b + 3 < a < m / 2 b+3<a<\sqrt{m/2}

Almost_periodic_function.html

  1. e ( σ + i t ) log n e^{(\sigma+it)\log n}\,
  2. n σ e ( log n ) i t . n^{\sigma}e^{(\log n)it}.\,
  3. || f || = sup x | f ( x ) | ||f||_{\infty}=\sup_{x}|f(x)|
  4. | f ( t + T ) - f ( t ) | < ε . \left|f(t+T)-f(t)\right|<\varepsilon.
  5. || f || S , r , p = sup x ( 1 r x x + r | f ( s ) | p d s ) 1 / p ||f||_{S,r,p}=\sup_{x}\left({1\over r}\int_{x}^{x+r}|f(s)|^{p}\,ds\right)^{1/p}
  6. || f || W , p = lim r || f || S , r , p ||f||_{W,p}=\lim_{r\mapsto\infty}||f||_{S,r,p}
  7. || f || B , p = lim sup x ( 1 2 x - x x | f ( s ) | p d s ) 1 / p ||f||_{B,p}=\limsup_{x\to\infty}\left({1\over 2x}\int_{-x}^{x}|f(s)|^{p}\,ds% \right)^{1/p}
  8. a n e i λ n t \sum a_{n}e^{i\lambda_{n}t}
  9. x ( t ) x(t)
  10. T T
  11. x ( t ) = x ( t + T ) x(t)=x(t+T)
  12. | x ( t ) - x ( t + T ) | = 0 for all t . \left|x(t)-x(t+T)\right|=0\,\text{ for all }t.
  13. x ( t ) = 1 2 a 0 + n = 1 [ a n cos ( 2 π n f 0 t ) - b n sin ( 2 π n f 0 t ) ] x(t)=\frac{1}{2}a_{0}+\sum_{n=1}^{\infty}\left[a_{n}\cos(2\pi nf_{0}t)-b_{n}% \sin(2\pi nf_{0}t)\right]
  14. x ( t ) = 1 2 a 0 + n = 1 [ r n cos ( 2 π n f 0 t + φ n ) ] x(t)=\frac{1}{2}a_{0}+\sum_{n=1}^{\infty}\left[r_{n}\cos(2\pi nf_{0}t+\varphi_% {n})\right]
  15. f 0 = 1 T f_{0}=\frac{1}{T}
  16. a n = r n cos ( φ n ) = 2 T t 0 t 0 + T x ( t ) cos ( 2 π n f 0 t ) d t a_{n}=r_{n}\cos\left(\varphi_{n}\right)=\frac{2}{T}\int_{t_{0}}^{t_{0}+T}x(t)% \cos(2\pi nf_{0}t)\,dt
  17. b n = r n sin ( φ n ) = - 2 T t 0 t 0 + T x ( t ) sin ( 2 π n f 0 t ) d t b_{n}=r_{n}\sin\left(\varphi_{n}\right)=-\frac{2}{T}\int_{t_{0}}^{t_{0}+T}x(t)% \sin(2\pi nf_{0}t)\,dt
  18. t 0 t_{0}
  19. - < t 0 < + -\infty<t_{0}<+\infty
  20. f 0 f_{0}
  21. a n a_{n}
  22. b n b_{n}
  23. r n r_{n}
  24. φ n \varphi_{n}
  25. x ( t ) x(t)
  26. x ( t ) x ( t + T ( t ) ) x(t)\approx x\left(t+T(t)\right)
  27. | x ( t ) - x ( t + T ( t ) ) | < ε \left|x(t)-x\left(t+T(t)\right)\right|<\varepsilon
  28. 0 < ϵ x = x 2 ¯ = lim τ 1 τ - τ / 2 τ / 2 x 2 ( t ) d t . 0<\epsilon\ll\left\|x\right\|=\sqrt{\overline{x^{2}}}=\sqrt{\lim_{\tau\to% \infty}\frac{1}{\tau}\int_{-\tau/2}^{\tau/2}x^{2}(t)\,dt}.
  29. x ( t ) = 1 2 a 0 ( t ) + n = 1 [ a n ( t ) cos ( 2 π n 0 t f 0 ( τ ) d τ ) - b n ( t ) sin ( 2 π n 0 t f 0 ( τ ) d τ ) ] x(t)=\frac{1}{2}a_{0}(t)\ +\ \sum_{n=1}^{\infty}\left[a_{n}(t)\cos\left(2\pi n% \int_{0}^{t}f_{0}(\tau)\,d\tau\right)-b_{n}(t)\sin\left(2\pi n\int_{0}^{t}f_{0% }(\tau)\,d\tau\right)\right]
  30. x ( t ) = 1 2 a 0 ( t ) + n = 1 [ r n ( t ) cos ( 2 π n 0 t f 0 ( τ ) d τ + φ n ( t ) ) ] x(t)=\frac{1}{2}a_{0}(t)\ +\ \sum_{n=1}^{\infty}\left[r_{n}(t)\cos\left(2\pi n% \int_{0}^{t}f_{0}(\tau)\,d\tau+\varphi_{n}(t)\right)\right]
  31. x ( t ) = 1 2 a 0 ( t ) + n = 1 [ r n ( t ) cos ( 2 π 0 t f n ( τ ) d τ + φ n ( 0 ) ) ] x(t)=\frac{1}{2}a_{0}(t)\ +\ \sum_{n=1}^{\infty}\left[r_{n}(t)\cos\left(2\pi% \int_{0}^{t}f_{n}(\tau)\,d\tau+\varphi_{n}(0)\right)\right]
  32. f 0 ( t ) = 1 T ( t ) f_{0}(t)=\frac{1}{T(t)}
  33. a n ( t ) = r n ( t ) cos ( φ n ( t ) ) a_{n}(t)=r_{n}(t)\cos\left(\varphi_{n}(t)\right)
  34. b n ( t ) = r n ( t ) sin ( φ n ( t ) ) b_{n}(t)=r_{n}(t)\sin\left(\varphi_{n}(t)\right)
  35. f n ( t ) = n f 0 ( t ) + 1 2 π φ n ( t ) . f_{n}(t)=nf_{0}(t)+\frac{1}{2\pi}\varphi_{n}^{\prime}(t).\,
  36. f 0 ( t ) f_{0}(t)
  37. f n ( t ) f_{n}(t)
  38. a n ( t ) a_{n}(t)
  39. b n ( t ) b_{n}(t)
  40. r n ( t ) r_{n}(t)
  41. φ n ( t ) \varphi_{n}(t)
  42. x ( t ) x(t)
  43. f n ( t ) f_{n}(t)
  44. φ n ( t ) \varphi_{n}(t)
  45. φ n ( t ) \varphi_{n}^{\prime}(t)
  46. n f 0 ( t ) nf_{0}(t)
  47. φ n ( t ) \varphi_{n}(t)
  48. x ( t ) x(t)

Almost_prime.html

  1. Ω ( n ) := a i if n = p i a i . \Omega(n):=\sum a_{i}\qquad\mbox{if}~{}\qquad n=\prod p_{i}^{a_{i}}.
  2. π k ( n ) ( n log n ) ( log log n ) k - 1 ( k - 1 ) ! , \pi_{k}(n)\sim\left(\frac{n}{\log n}\right)\frac{(\log\log n)^{k-1}}{(k-1)!},

Almost_surely.html

  1. ( Ω , , P ) (\Omega,\mathcal{F},P)
  2. E E\in\mathcal{F}
  3. P [ E ] = 1 P[E]=1
  4. E E
  5. E E
  6. P [ E C ] = 0 P[E^{C}]=0
  7. E E
  8. \mathcal{F}
  9. E C E^{C}
  10. N N\in\mathcal{F}
  11. P [ N ] = 0 P[N]=0
  12. P P
  13. E E
  14. P P
  15. [ P ] [P]
  16. ( { H , T } , 2 { H , T } , ) (\{H,T\},2^{\{H,T\}},\mathbb{P})
  17. { ω = H } \{\omega=H\}
  18. { ω = T } \{\omega=T\}
  19. [ ω = H ] = p ( 0 , 1 ) \mathbb{P}[\omega=H]=p\in(0,1)
  20. [ ω = T ] = 1 - p \mathbb{P}[\omega=T]=1-p
  21. { X i ( ω ) } i \{X_{i}(\omega)\}_{i\in\mathbb{N}}
  22. X i ( ω ) = ω i X_{i}(\omega)=\omega_{i}
  23. X i X_{i}
  24. i i
  25. { H , H , H , } \{H,H,H,\dots\}
  26. n n
  27. [ X i = H , i = 1 , 2 , , n ] = ( [ X 1 = H ] ) n = p n \mathbb{P}[X_{i}=H,\ i=1,2,\dots,n]=\left(\mathbb{P}[X_{1}=H]\right)^{n}=p^{n}
  28. n n\rightarrow\infty
  29. p ( 0 , 1 ) p\in(0,1)
  30. p p
  31. lim n 1 n = 0 \scriptstyle\lim\limits_{n\to\infty}\frac{1}{n}=0
  32. p 1 , 000 , 000 0 p^{1,000,000}\neq 0
  33. 1 - p 1 , 000 , 000 1-p^{1,000,000}
  34. ( 1 + ϵ ) ln n n \tfrac{(1+\epsilon)\ln n}{n}

Alternating_series.html

  1. n = 0 ( - 1 ) n a n \sum_{n=0}^{\infty}(-1)^{n}\,a_{n}
  2. n = 1 ( - 1 ) n - 1 a n \sum_{n=1}^{\infty}(-1)^{n-1}\,a_{n}
  3. n = 1 ( - 1 ) n + 1 n x n = ln ( 1 + x ) . \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}x^{n}\;=\;\ln(1+x).
  4. sin x = n = 0 ( - 1 ) n x 2 n + 1 ( 2 n + 1 ) ! \sin x=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}
  5. cos x = n = 0 ( - 1 ) n x 2 n ( 2 n ) ! . \cos x=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n)!}.
  6. J α ( x ) = m = 0 ( - 1 ) m m ! Γ ( m + α + 1 ) ( x 2 ) 2 m + α J_{\alpha}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!\,\Gamma(m+\alpha+1)}{\left% (\frac{x}{2}\right)}^{2m+\alpha}
  7. η ( s ) = n = 1 ( - 1 ) n - 1 n s = 1 1 s - 1 2 s + 1 3 s - 1 4 s + \eta(s)=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^% {s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots
  8. a n a_{n}
  9. m m
  10. m < n m<n
  11. S m - S n < a m S_{m}-S_{n}<a_{m}
  12. S m - S n \displaystyle S_{m}-S_{n}
  13. a n a_{n}
  14. - ( a m - a m + 1 ) -(a_{m}-a_{m+1})
  15. S m - S n a m S_{m}-S_{n}\leq a_{m}
  16. - a m S m - S n -a_{m}\leq S_{m}-S_{n}
  17. a m a_{m}
  18. 0
  19. S m S_{m}
  20. m m
  21. n n
  22. a n a_{n}
  23. | k = 0 ( - 1 ) k a k - k = 0 m ( - 1 ) k a k | | a m + 1 | . \left|\sum_{k=0}^{\infty}(-1)^{k}\,a_{k}\,-\,\sum_{k=0}^{m}\,(-1)^{k}\,a_{k}% \right|\leq|a_{m+1}|.
  24. a n \sum a_{n}
  25. | a n | \sum|a_{n}|
  26. a n \sum a_{n}
  27. | a n | \sum|a_{n}|
  28. 2 | a n | \sum 2|a_{n}|
  29. 0 a n + | a n | 2 | a n | 0\leq a_{n}+|a_{n}|\leq 2|a_{n}|
  30. ( a n + | a n | ) \sum(a_{n}+|a_{n}|)
  31. a n \sum a_{n}
  32. a n = ( a n + | a n | ) - | a n | \sum a_{n}=\sum(a_{n}+|a_{n}|)-\sum|a_{n}|
  33. n = 1 1 n , \sum_{n=1}^{\infty}\frac{1}{n},\!
  34. n = 1 ( - 1 ) n + 1 n , \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n},\!
  35. ln ( 2 ) = n = 1 ( - 1 ) n + 1 n = 1 - 1 2 + 1 3 - 1 4 + . \ln(2)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac% {1}{4}+\cdots.
  36. 1 2 ln ( 2 ) \frac{1}{2}\ln(2)
  37. ( 1 - 1 2 ) - 1 4 + ( 1 3 - 1 6 ) - 1 8 + ( 1 5 - 1 10 ) - 1 12 + \displaystyle{}\quad\left(1-\frac{1}{2}\right)-\frac{1}{4}+\left(\frac{1}{3}-% \frac{1}{6}\right)-\frac{1}{8}+\left(\frac{1}{5}-\frac{1}{10}\right)-\frac{1}{% 12}+\cdots

Amagat's_law.html

  1. N v ( T , p ) = i = 1 K N i v i ( T , p ) . N\cdot v(T,p)=\sum_{i=1}^{K}N_{i}\cdot v_{i}(T,p).
  2. V 1 , V 2 , , V n V_{1},V_{2},\dots,V_{n}
  3. V V
  4. V = V 1 + V 2 + V 3 + + V n = i V i V=V_{1}+V_{2}+V_{3}+\dots+V_{n}=\sum_{i}V_{i}
  5. B ( T ) = X 1 B 1 + X 2 B 2 + X 1 X 2 B 1 , 2 B(T)=X_{1}B_{1}+X_{2}B_{2}+X_{1}X_{2}B_{1,2}
  6. B 1 , 2 = 0 B_{1,2}=0
  7. B 1 , 2 = ( B 1 + B 2 ) / 2 B_{1,2}=(B_{1}+B_{2})/2

Analog_signal_processing.html

  1. y ( t ) = ( x * h ) ( t ) = a b x ( τ ) h ( t - τ ) d τ y(t)=(x*h)(t)=\int_{a}^{b}x(\tau)h(t-\tau)\,d\tau
  2. - | x ( t ) | d t < \int^{\infty}_{-\infty}|x(t)|\,dt<\infty
  3. X ( j ω ) = - x ( t ) e - j ω t d t X(j\omega)=\int^{\infty}_{-\infty}x(t)e^{-j\omega t}\,dt
  4. x ( t ) = 1 2 π - X ( j ω ) e j ω t d ω x(t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}X(j\omega)e^{j\omega t}\,d\omega
  5. X ( s ) = 0 - x ( t ) e - s t d t X(s)=\int^{\infty}_{0^{-}}x(t)e^{-st}\,dt
  6. x ( t ) = 1 2 π - X ( s ) e s t d s x(t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}X(s)e^{st}\,ds
  7. u ( t ) = - t δ ( s ) d s \mathrm{u}(t)=\int_{-\infty}^{t}\delta(s)ds

Analysis_of_covariance.html

  1. y i j = μ + τ i + B ( x i j - x i ¯ ) + ϵ i j y_{ij}=\mu+\tau_{i}+B(x_{ij}-\overline{x_{i}})+\epsilon_{ij}
  2. y i j y_{ij}
  3. μ \mu
  4. τ i \tau_{i}
  5. x i j x_{ij}
  6. x i ¯ \overline{x_{i}}
  7. ϵ i j \epsilon_{ij}
  8. ( i a τ i = 0 ) . \left(\sum_{i}^{a}\tau_{i}=0\right).
  9. 𝑀𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑀𝑆𝑤𝑖𝑡ℎ𝑖𝑛 \frac{\mathit{MSbetween}}{\mathit{MSwithin}}
  10. ϵ i j \epsilon_{ij}
  11. N ( 0 , σ 2 ) N(0,\sigma^{2})

Analytic_set.html

  1. X X
  2. Y Y
  3. B X × Y B\subseteq X\times Y
  4. A A
  5. B B
  6. A = { x X | ( y Y ) x , y B } . A=\{x\in X|(\exists y\in Y)\langle x,y\rangle\in B\}.
  7. X X
  8. ω × ω \omega\times\omega
  9. 2 × ω 2\times\omega
  10. s y m b o l Σ 1 1 symbol{\Sigma}^{1}_{1}
  11. Σ 1 1 \Sigma^{1}_{1}
  12. s y m b o l Π 1 1 symbol{\Pi}^{1}_{1}
  13. s y m b o l Δ 1 1 = s y m b o l Σ 1 1 s y m b o l Π 1 1 symbol{\Delta}^{1}_{1}=symbol{\Sigma}^{1}_{1}\cap symbol{\Pi}^{1}_{1}

Analytical_hierarchy.html

  1. \mathbb{N}
  2. \mathbb{N}
  3. \mathbb{N}
  4. Σ 0 1 = Π 0 1 = Δ 0 1 \Sigma^{1}_{0}=\Pi^{1}_{0}=\Delta^{1}_{0}
  5. Σ n + 1 1 \Sigma^{1}_{n+1}
  6. X 1 X k ψ \exists X_{1}\cdots\exists X_{k}\psi
  7. ψ \psi
  8. Π n 1 \Pi^{1}_{n}
  9. Π n + 1 1 \Pi^{1}_{n+1}
  10. X 1 X k ψ \forall X_{1}\cdots\forall X_{k}\psi
  11. ψ \psi
  12. Σ n 1 \Sigma^{1}_{n}
  13. Σ n 1 \Sigma^{1}_{n}
  14. Π n 1 \Pi^{1}_{n}
  15. n n
  16. Σ n 1 \Sigma^{1}_{n}
  17. Π n 1 \Pi^{1}_{n}
  18. n n
  19. Σ n 1 \Sigma^{1}_{n}
  20. Π n 1 \Pi^{1}_{n}
  21. n n
  22. Σ m 1 \Sigma^{1}_{m}
  23. Π m 1 \Pi^{1}_{m}
  24. m m
  25. n n
  26. Σ n 1 \Sigma^{1}_{n}
  27. Σ n 1 \Sigma^{1}_{n}
  28. Π n 1 \Pi^{1}_{n}
  29. Π n 1 \Pi^{1}_{n}
  30. Σ n 1 \Sigma^{1}_{n}
  31. Π n 1 \Pi^{1}_{n}
  32. Δ n 1 \Delta^{1}_{n}
  33. Δ 1 1 \Delta^{1}_{1}
  34. Σ n 1 \Sigma^{1}_{n}
  35. Σ n 1 \Sigma^{1}_{n}
  36. Π n 1 \Pi^{1}_{n}
  37. Π n 1 \Pi^{1}_{n}
  38. Σ n 1 \Sigma^{1}_{n}
  39. Π n 1 \Pi^{1}_{n}
  40. Δ n 1 \Delta^{1}_{n}
  41. ω \omega
  42. ω \omega
  43. Σ n 1 \Sigma^{1}_{n}
  44. Π n 1 \Pi^{1}_{n}
  45. Δ n 1 \Delta^{1}_{n}
  46. Σ n 1 , A \Sigma^{1,A}_{n}
  47. Π n 1 , A \Pi^{1,A}_{n}
  48. Y Y
  49. Σ n 1 , Y \Sigma^{1,Y}_{n}
  50. Σ n 1 , A \Sigma^{1,A}_{n}
  51. A A
  52. Y Y
  53. Π n 1 , Y \Pi^{1,Y}_{n}
  54. Δ n 1 , Y \Delta^{1,Y}_{n}
  55. Σ n 1 , Y \Sigma^{1,Y}_{n}
  56. Π n 1 , Y \Pi^{1,Y}_{n}
  57. Π 1 1 \Pi^{1}_{1}
  58. Σ 1 1 \Sigma^{1}_{1}
  59. ω \omega
  60. Π 1 1 \Pi^{1}_{1}
  61. Σ 1 1 \Sigma^{1}_{1}
  62. Σ 1 1 , Y \Sigma^{1,Y}_{1}
  63. Y Y
  64. Δ 2 1 \Delta^{1}_{2}
  65. Δ 2 1 \Delta^{1}_{2}
  66. n n
  67. Π n 1 Σ n + 1 1 \Pi^{1}_{n}\subset\Sigma^{1}_{n+1}
  68. Π n 1 Π n + 1 1 \Pi^{1}_{n}\subset\Pi^{1}_{n+1}
  69. Σ n 1 Π n + 1 1 \Sigma^{1}_{n}\subset\Pi^{1}_{n+1}
  70. Σ n 1 Σ n + 1 1 \Sigma^{1}_{n}\subset\Sigma^{1}_{n+1}
  71. Σ n 1 \Sigma^{1}_{n}

Analytical_mechanics.html

  1. 𝐪 = ( q 1 , q 2 , q N ) \mathbf{q}=(q_{1},q_{2},\cdots q_{N})
  2. d 𝐪 d t = ( d q 1 d t , d q 2 d t , d q N d t ) 𝐪 ˙ = ( q ˙ 1 , q ˙ 2 , q ˙ N ) \frac{d\mathbf{q}}{dt}=\left(\frac{dq_{1}}{dt},\frac{dq_{2}}{dt},\cdots\frac{% dq_{N}}{dt}\right)\equiv\mathbf{\dot{q}}=(\dot{q}_{1},\dot{q}_{2},\cdots\dot{q% }_{N})
  3. δ W = s y m b o l 𝒬 δ 𝐪 = 0 , \delta W=symbol{\mathcal{Q}}\cdot\delta\mathbf{q}=0\,,
  4. s y m b o l 𝒬 = ( 𝒬 1 , 𝒬 2 , 𝒬 N ) symbol{\mathcal{Q}}=(\mathcal{Q}_{1},\mathcal{Q}_{2},\cdots\mathcal{Q}_{N})
  5. s y m b o l 𝒬 = d d t ( T 𝐪 ˙ ) - T 𝐪 , symbol{\mathcal{Q}}=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{% \partial\mathbf{\dot{q}}}\right)-\frac{\partial T}{\partial\mathbf{q}}\,,
  6. 𝐪 = ( q 1 , q 2 , q N ) \frac{\partial}{\partial\mathbf{q}}=\left(\frac{\partial}{\partial q_{1}},% \frac{\partial}{\partial q_{2}},\cdots\frac{\partial}{\partial q_{N}}\right)
  7. 𝐫 = 𝐫 ( 𝐪 ( t ) , t ) \mathbf{r}=\mathbf{r}(\mathbf{q}(t),t)
  8. L ( 𝐪 , 𝐪 ˙ , t ) = T ( 𝐪 , 𝐪 ˙ , t ) - V ( 𝐪 , 𝐪 ˙ , t ) L(\mathbf{q},\mathbf{\dot{q}},t)=T(\mathbf{q},\mathbf{\dot{q}},t)-V(\mathbf{q}% ,\mathbf{\dot{q}},t)
  9. d d t ( L 𝐪 ˙ ) = L 𝐪 , \frac{d}{dt}\left(\frac{\partial L}{\partial\mathbf{\dot{q}}}\right)=\frac{% \partial L}{\partial\mathbf{q}}\,,
  10. 𝒞 = { 𝐪 N } , \mathcal{C}=\{\mathbf{q}\in\mathbb{R}^{N}\}\,,
  11. N \mathbb{R}^{N}
  12. { 𝐪 ( t ) N : t 0 , t } 𝒞 , \{\mathbf{q}(t)\in\mathbb{R}^{N}\,:\,t\geq 0,t\in\mathbb{R}\}\subseteq\mathcal% {C}\,,
  13. 𝐩 = L 𝐪 ˙ = ( L q ˙ 1 , L q ˙ 2 , L q ˙ N ) = ( p 1 , p 2 p N ) , \mathbf{p}=\frac{\partial L}{\partial\mathbf{\dot{q}}}=\left(\frac{\partial L}% {\partial\dot{q}_{1}},\frac{\partial L}{\partial\dot{q}_{2}},\cdots\frac{% \partial L}{\partial\dot{q}_{N}}\right)=(p_{1},p_{2}\cdots p_{N})\,,
  14. H ( 𝐪 , 𝐩 , t ) = 𝐩 𝐪 ˙ - L ( 𝐪 , 𝐪 ˙ , t ) H(\mathbf{q},\mathbf{p},t)=\mathbf{p}\cdot\mathbf{\dot{q}}-L(\mathbf{q},% \mathbf{\dot{q}},t)
  15. 𝐩 ˙ = - H 𝐪 , 𝐪 ˙ = + H 𝐩 , \mathbf{\dot{p}}=-\frac{\partial H}{\partial\mathbf{q}}\,,\quad\mathbf{\dot{q}% }=+\frac{\partial H}{\partial\mathbf{p}}\,,
  16. d H d t = - L t , \frac{dH}{dt}=-\frac{\partial L}{\partial t}\,,
  17. 𝐩 ˙ = s y m b o l 𝒬 . \mathbf{\dot{p}}=symbol{\mathcal{Q}}\,.
  18. = { 𝐩 N } . \mathcal{M}=\{\mathbf{p}\in\mathbb{R}^{N}\}\,.
  19. 𝒫 = 𝒞 × = { ( 𝐪 , 𝐩 ) 2 N } , \mathcal{P}=\mathcal{C}\times\mathcal{M}=\{(\mathbf{q},\mathbf{p})\in\mathbb{R% }^{2N}\}\,,
  20. { ( 𝐪 ( t ) , 𝐩 ( t ) ) 2 N : t 0 , t } 𝒫 , \{(\mathbf{q}(t),\mathbf{p}(t))\in\mathbb{R}^{2N}\,:\,t\geq 0,t\in\mathbb{R}\}% \subseteq\mathcal{P}\,,
  21. { A , B } { A , B } 𝐪 , 𝐩 \displaystyle\{A,B\}\equiv\{A,B\}_{\mathbf{q},\mathbf{p}}
  22. d A d t = { A , H } + A t . \frac{dA}{dt}=\{A,H\}+\frac{\partial A}{\partial t}\,.
  23. { A , B } 1 i [ A ^ , B ^ ] . \{A,B\}\rightarrow\frac{1}{i\hbar}[\hat{A},\hat{B}]\,.
  24. L = L + d d t F ( 𝐪 , t ) , L^{\prime}=L+\frac{d}{dt}F(\mathbf{q},t)\,,
  25. K = H + t G ( 𝐪 , 𝐩 , t ) , K=H+\frac{\partial}{\partial t}G(\mathbf{q},\mathbf{p},t)\,,
  26. L q j = 0 d p j d t = d d t L q ˙ j = 0 \frac{\partial L}{\partial q_{j}}=0\,\rightarrow\,\frac{dp_{j}}{dt}=\frac{d}{% dt}\frac{\partial L}{\partial\dot{q}_{j}}=0
  27. T ( ( λ q ˙ i ) 2 , ( λ q ˙ j λ q ˙ k ) , 𝐪 ) = λ 2 T ( ( q ˙ i ) 2 , q ˙ j q ˙ k , 𝐪 ) , L ( 𝐪 , 𝐪 ˙ ) , T((\lambda\dot{q}_{i})^{2},(\lambda\dot{q}_{j}\lambda\dot{q}_{k}),\mathbf{q})=% \lambda^{2}T((\dot{q}_{i})^{2},\dot{q}_{j}\dot{q}_{k},\mathbf{q})\,,\quad L(% \mathbf{q},\mathbf{\dot{q}})\,,
  28. H = T + V = E . H=T+V=E\,.
  29. 𝒮 = t 1 t 2 L ( 𝐪 , 𝐪 ˙ , t ) d t . \mathcal{S}=\int_{t_{1}}^{t_{2}}L(\mathbf{q},\mathbf{\dot{q}},t)dt\,.
  30. δ 𝒮 = δ t 1 t 2 L ( 𝐪 , 𝐪 ˙ , t ) d t = 0 , \delta\mathcal{S}=\delta\int_{t_{1}}^{t_{2}}L(\mathbf{q},\mathbf{\dot{q}},t)dt% =0\,,
  31. 𝒞 \mathcal{C}
  32. 𝒞 \mathcal{C}
  33. K ( 𝐐 , 𝐏 , t ) = H ( 𝐪 , 𝐩 , t ) + t G 1 ( 𝐪 , 𝐐 , t ) \displaystyle K(\mathbf{Q},\mathbf{P},t)=H(\mathbf{q},\mathbf{p},t)+\frac{% \partial}{\partial t}G_{1}(\mathbf{q},\mathbf{Q},t)
  34. 𝐏 ˙ = - K 𝐐 , 𝐐 ˙ = + K 𝐏 , \mathbf{\dot{P}}=-\frac{\partial K}{\partial\mathbf{Q}}\,,\quad\mathbf{\dot{Q}% }=+\frac{\partial K}{\partial\mathbf{P}}\,,
  35. { Q i , P i } = 1 \{Q_{i},P_{i}\}=1\,
  36. 𝒮 \mathcal{S}
  37. G 2 ( 𝐪 , t ) = 𝒮 ( 𝐪 , t ) + C , G_{2}(\mathbf{q},t)=\mathcal{S}(\mathbf{q},t)+C\,,
  38. 𝐩 = 𝒮 𝐪 \mathbf{p}=\frac{\partial\mathcal{S}}{\partial\mathbf{q}}
  39. H = - 𝒮 t H=-\frac{\partial\mathcal{S}}{\partial t}
  40. H = H ( 𝐪 , 𝐩 , t ) = H ( 𝐪 , 𝒮 𝐪 , t ) H=H(\mathbf{q},\mathbf{p},t)=H\left(\mathbf{q},\frac{\partial\mathcal{S}}{% \partial\mathbf{q}},t\right)
  41. W ( 𝐪 ) = 𝒮 ( 𝐪 , t ) + E t W(\mathbf{q})=\mathcal{S}(\mathbf{q},t)+Et
  42. R = 𝐩 𝐪 ˙ - L ( 𝐪 , 𝐩 , s y m b o l ζ , s y m b o l ζ ˙ ) , R=\mathbf{p}\cdot\mathbf{\dot{q}}-L(\mathbf{q},\mathbf{p},symbol{\zeta},\dot{% symbol{\zeta}})\,,
  43. 𝐪 ˙ = + R 𝐩 , 𝐩 ˙ = - R 𝐪 , \dot{\mathbf{q}}=+\frac{\partial R}{\partial\mathbf{p}}\,,\quad\dot{\mathbf{p}% }=-\frac{\partial R}{\partial\mathbf{q}}\,,
  44. d d t R s y m b o l ζ ˙ = R s y m b o l ζ . \frac{d}{dt}\frac{\partial R}{\partial\dot{symbol{\zeta}}}=\frac{\partial R}{% \partial symbol{\zeta}}\,.
  45. α r = q ¨ r = d 2 q r d t 2 , \alpha_{r}=\ddot{q}_{r}=\frac{d^{2}q_{r}}{dt^{2}}\,,
  46. 𝒬 r = S α r , S = 1 2 k = 1 N m k 𝐚 k 2 , \mathcal{Q}_{r}=\frac{\partial S}{\partial\alpha_{r}}\,,\quad S=\frac{1}{2}% \sum_{k=1}^{N}m_{k}\mathbf{a}_{k}^{2}\,,
  47. 𝐚 k = 𝐫 ¨ k = d 2 𝐫 k d t 2 \mathbf{a}_{k}=\ddot{\mathbf{r}}_{k}=\frac{d^{2}\mathbf{r}_{k}}{dt^{2}}
  48. = ( ϕ 1 , ϕ 2 , ϕ 1 , ϕ 2 , ϕ 1 / t , ϕ 2 / t , 𝐫 , t ) . \mathcal{L}=\mathcal{L}(\phi_{1},\phi_{2},\ldots\nabla\phi_{1},\nabla\phi_{2},% \ldots\partial\phi_{1}/\partial t,\partial\phi_{2}/\partial t,\ldots\mathbf{r}% ,t)\,.
  49. μ ( ( μ ϕ i ) ) = ϕ i , \partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi_{i}% )}\right)=\frac{\partial\mathcal{L}}{\partial\phi_{i}}\,,
  50. L = 𝒱 d V . L=\int_{\mathcal{V}}\mathcal{L}\,dV\,.
  51. π i ( 𝐫 , t ) = ϕ ˙ i ϕ ˙ i ϕ i t . \pi_{i}(\mathbf{r},t)=\frac{\partial\mathcal{L}}{\partial\dot{\phi}_{i}}\,% \quad\dot{\phi}_{i}\equiv\frac{\partial\phi_{i}}{\partial t}.
  52. \mathcal{H}
  53. ( ϕ 1 , ϕ 2 , , π 1 , π 2 , , 𝐫 , t ) = i = 1 N ϕ ˙ i ( 𝐫 , t ) π i ( 𝐫 , t ) - . \mathcal{H}(\phi_{1},\phi_{2},\ldots,\pi_{1},\pi_{2},\ldots,\mathbf{r},t)=\sum% _{i=1}^{N}\dot{\phi}_{i}(\mathbf{r},t)\pi_{i}(\mathbf{r},t)-\mathcal{L}\,.
  54. ϕ ˙ i = + δ δ π i , π ˙ i = - δ δ ϕ i , \dot{\phi}_{i}=+\frac{\delta\mathcal{H}}{\delta\pi_{i}}\,,\quad\dot{\pi}_{i}=-% \frac{\delta\mathcal{H}}{\delta\phi_{i}}\,,
  55. δ δ ϕ i = ϕ i - μ ( μ ϕ i ) \frac{\delta}{\delta\phi_{i}}=\frac{\partial}{\partial\phi_{i}}-\partial_{\mu}% \frac{\partial}{\partial(\partial_{\mu}\phi_{i})}
  56. H = 𝒱 d V . H=\int_{\mathcal{V}}\mathcal{H}\,dV\,.
  57. X ( 𝐚 ) X(\mathbf{a})
  58. 𝐫 𝐫 + 𝐚 \mathbf{r}\rightarrow\mathbf{r}+\mathbf{a}
  59. 𝐩 𝐩 \mathbf{p}\rightarrow\mathbf{p}
  60. U ( t 0 ) U(t_{0})
  61. 𝐫 ( t ) 𝐫 ( t + t 0 ) \mathbf{r}(t)\rightarrow\mathbf{r}(t+t_{0})
  62. 𝐩 ( t ) 𝐩 ( t + t 0 ) \mathbf{p}(t)\rightarrow\mathbf{p}(t+t_{0})
  63. R ( 𝐧 ^ , θ ) R(\mathbf{\hat{n}},\theta)
  64. 𝐫 R ( 𝐧 ^ , θ ) 𝐫 \mathbf{r}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{r}
  65. 𝐩 R ( 𝐧 ^ , θ ) 𝐩 \mathbf{p}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{p}
  66. G ( 𝐯 ) G(\mathbf{v})
  67. 𝐫 𝐫 + 𝐯 t \mathbf{r}\rightarrow\mathbf{r}+\mathbf{v}t
  68. 𝐩 𝐩 + m 𝐯 \mathbf{p}\rightarrow\mathbf{p}+m\mathbf{v}
  69. P P
  70. 𝐫 - 𝐫 \mathbf{r}\rightarrow-\mathbf{r}
  71. 𝐩 - 𝐩 \mathbf{p}\rightarrow-\mathbf{p}
  72. T T
  73. 𝐫 𝐫 ( - t ) \mathbf{r}\rightarrow\mathbf{r}(-t)
  74. 𝐩 - 𝐩 ( - t ) \mathbf{p}\rightarrow-\mathbf{p}(-t)
  75. L [ q ( s , t ) , q ˙ ( s , t ) ] = L [ q ( t ) , q ˙ ( t ) ] L[q(s,t),\dot{q}(s,t)]=L[q(t),\dot{q}(t)]

Analyticity_of_holomorphic_functions.html

  1. f ( z ) = n = 0 c n ( z - a ) n f(z)=\sum_{n=0}^{\infty}c_{n}(z-a)^{n}
  2. 1 w - z . \frac{1}{w-z}.
  3. f ( z ) \displaystyle f(z)
  4. f ( z ) = n = 0 ( z - a ) n 1 2 π i C f ( w ) ( w - a ) n + 1 d w . f(z)=\sum_{n=0}^{\infty}(z-a)^{n}{1\over 2\pi i}\int_{C}{f(w)\over(w-a)^{n+1}}% \,\mathrm{d}w.
  5. f ( z ) = n = 0 c n ( z - a ) n f(z)=\sum_{n=0}^{\infty}c_{n}(z-a)^{n}
  6. 1 ( w - z ) n + 1 \frac{1}{(w-z)^{n+1}}
  7. f ( n ) ( a ) = n ! 2 π i C f ( w ) ( w - a ) n + 1 d w . f^{(n)}(a)={n!\over 2\pi i}\int_{C}{f(w)\over(w-a)^{n+1}}\,dw.

Angle_of_repose.html

  1. tan ( θ ) μ s \tan{(\theta)}\approx\mu_{\mathrm{s}}\,

Ankeny–Artin–Chowla_congruence.html

  1. ε = t + u d 2 \varepsilon=\frac{t+u\sqrt{d}}{2}
  2. h t u ( mod p ) \frac{ht}{u}\;\;(\mathop{{\rm mod}}p)\;
  3. - 2 m h t u 0 < k < d χ ( k ) k k / p ( mod p ) -2{mht\over u}\equiv\sum_{0<k<d}{\chi(k)\over k}\lfloor{k/p}\rfloor\;\;(% \mathop{{\rm mod}}p)
  4. m = d p m=\frac{d}{p}\;
  5. χ \chi\;
  6. x \lfloor x\rfloor
  7. u t h B ( p - 1 ) / 2 ( mod p ) {u\over t}h\equiv B_{(p-1)/2}\;\;(\mathop{{\rm mod}}p)

Annihilator_(ring_theory).html

  1. Ann R ( S ) = { r R s S , r s = 0 } \mathrm{Ann}_{R}(S)=\{r\in R\mid\forall s\in S,rs=0\}
  2. . Ann R ( S ) \ell.\mathrm{Ann}_{R}(S)\,
  3. r . Ann R ( S ) r.\mathrm{Ann}_{R}(S)\,
  4. r ¯ m := r m \overline{r}m:=rm\,
  5. . Ann R ( S ) \ell.\mathrm{Ann}_{R}(S)\,
  6. 𝒜 \mathcal{LA}\,
  7. 𝒜 \mathcal{RA}\,
  8. 𝒜 \mathcal{LA}\,
  9. 𝒜 \mathcal{RA}\,
  10. 𝒜 \mathcal{RA}\,
  11. 𝒜 \mathcal{LA}\,
  12. 𝒜 \mathcal{LA}\,
  13. F : M × N P F\colon M\times N\to P
  14. S M S\subset M
  15. N N
  16. S S
  17. Ann ( S ) := { n N s S , F ( s , n ) = 0 } . \operatorname{Ann}(S):=\{n\in N\mid\forall s\in S,F(s,n)=0\}.
  18. T N T\subset N
  19. M M
  20. M M
  21. N N
  22. Span ( S ) Ann ( Ann ( S ) ) \operatorname{Span}(S)\leq\operatorname{Ann}(\operatorname{Ann}(S))
  23. Ann ( Ann ( Ann ( S ) ) ) = Ann ( S ) \operatorname{Ann}(\operatorname{Ann}(\operatorname{Ann}(S)))=\operatorname{% Ann}(S)
  24. V × V K V\times V\to K
  25. D S = x S , x 0 Ann R ( x ) . D_{S}=\bigcup_{x\in S,\,x\neq 0}{\mathrm{Ann}_{R}\,(x)}.
  26. D R D_{R}

Annulus_(mathematics).html

  1. R R
  2. r r
  3. A = π R 2 - π r 2 = π ( R 2 - r 2 ) . A=\pi R^{2}-\pi r^{2}=\pi(R^{2}-r^{2})\,.
  4. A = π ( R 2 - r 2 ) = π d 2 . A=\pi\left(R^{2}-r^{2}\right)=\pi d^{2}\,.
  5. d ρ
  6. 2 π ρ d ρ 2πρdρ
  7. A = r R 2 π ρ d ρ = π ( R 2 - r 2 ) . A=\int_{r}^{R}2\pi\rho\,d\rho=\pi\left(R^{2}-r^{2}\right).
  8. θ θ
  9. θ θ
  10. A = θ 2 ( R 2 - r 2 ) A=\frac{\theta}{2}\left(R^{2}-r^{2}\right)
  11. a n n ( a ; r , R ) ann(a;r,R)
  12. r < | z - a | < R . r<|z-a|<R.\,
  13. r r
  14. 0
  15. R R
  16. a a
  17. r / R r/R
  18. a n n ( a ; r , R ) ann(a;r,R)
  19. 1 1
  20. z z - a R . z\mapsto\frac{z-a}{R}.
  21. r / R < 1 r/R<1

Ant_colony_optimization_algorithms.html

  1. x x
  2. y y
  3. k k
  4. A k ( x ) A_{k}(x)
  5. k k
  6. p x y k p_{xy}^{k}
  7. x x
  8. y y
  9. η x y \eta_{xy}
  10. τ x y \tau_{xy}
  11. k k
  12. x x
  13. y y
  14. p x y k = ( τ x y α ) ( η x y β ) z allowed x ( τ x z α ) ( η x z β ) p_{xy}^{k}=\frac{(\tau_{xy}^{\alpha})(\eta_{xy}^{\beta})}{\sum_{z\in\mathrm{% allowed}_{x}}(\tau_{xz}^{\alpha})(\eta_{xz}^{\beta})}
  15. τ x y \tau_{xy}
  16. x x
  17. y y
  18. α \alpha
  19. τ x y \tau_{xy}
  20. η x y \eta_{xy}
  21. x y xy
  22. 1 / d x y 1/d_{xy}
  23. d d
  24. β \beta
  25. η x y \eta_{xy}
  26. τ x z \tau_{xz}
  27. η x z \eta_{xz}
  28. τ x y ( 1 - ρ ) τ x y + k Δ τ x y k \tau_{xy}\leftarrow(1-\rho)\tau_{xy}+\sum_{k}\Delta\tau^{k}_{xy}
  29. τ x y \tau_{xy}
  30. x y xy
  31. ρ \rho
  32. Δ τ x y k \Delta\tau^{k}_{xy}
  33. k k
  34. Δ τ x y k = { Q / L k if ant k uses curve x y in its tour 0 otherwise \Delta\tau^{k}_{xy}=\begin{cases}Q/L_{k}&\mbox{if ant }~{}k\mbox{ uses curve }% ~{}xy\mbox{ in its tour}\\ 0&\mbox{otherwise}\end{cases}
  35. L k L_{k}
  36. k k
  37. Q Q
  38. K K
  39. I M 1 M 2 I_{M_{1}M_{2}}
  40. K = ( M 1 * M 2 ) 1 2 K=(M_{1}*M_{2})^{\tfrac{1}{2}}
  41. τ ( i , j ) \tau_{(i,j)}
  42. η ( i , j ) = 1 Z * V c * I ( i , j ) \eta_{(i,j)}=\tfrac{1}{Z}*Vc*I_{(i,j)}
  43. I I
  44. M 1 * M 2 M_{1}*M_{2}
  45. Z = i = 1 : M 1 j = 1 : M 2 V c ( I i , j ) Z=\sum_{i=1:M_{1}}\sum_{j=1:M_{2}}Vc(I_{i,j})
  46. V c ( I i , j ) = f ( | I ( i - 2 , j - 1 ) - I ( i + 2 , j + 1 ) | + | I ( i - 2 , j + 1 ) - I ( i + 2 , j - 1 ) | + | I ( i - 1 , j - 2 ) - I ( i + 1 , j + 2 ) | + | I ( i - 1 , j - 1 ) - I ( i + 1 , j + 1 ) | + | I ( i - 1 , j ) - I ( i + 1 , j ) | + | I ( i - 1 , j + 1 ) - I ( i - 1 , j - 1 ) | + | I ( i - 1 , j + 2 ) - I ( i - 1 , j - 2 ) | + | I ( i , j - 1 ) - I ( i , j + 1 ) | \begin{aligned}\displaystyle Vc(I_{i,j})=&\displaystyle f(\left|I_{(i-2,j-1)}-% I_{(i+2,j+1)}\right|+\left|I_{(i-2,j+1)}-I_{(i+2,j-1)}\right|\\ &\displaystyle+\left|I_{(i-1,j-2)}-I_{(i+1,j+2)}\right|+\left|I_{(i-1,j-1)}-I_% {(i+1,j+1)}\right|\\ &\displaystyle+\left|I_{(i-1,j)}-I_{(i+1,j)}\right|+\left|I_{(i-1,j+1)}-I_{(i-% 1,j-1)}\right|\\ &\displaystyle+\left|I_{(i-1,j+2)}-I_{(i-1,j-2)}\right|+\left|I_{(i,j-1)}-I_{(% i,j+1)}\right|\end{aligned}
  47. f ( ) f(\cdot)
  48. f ( x ) = λ x , for x ≥ 0; (1) f(x)=\lambda x,\quad\,\text{for x ≥ 0; (1)}
  49. f ( x ) = λ x 2 , for x ≥ 0; (2) f(x)=\lambda x^{2},\quad\,\text{for x ≥ 0; (2)}
  50. f ( x ) = { sin ( π x 2 λ ) , for 0 ≤ x ≤ λ ; (3) 0 , else f(x)=\begin{cases}\sin(\frac{\pi x}{2\lambda}),&\,\text{for 0 ≤ x ≤}\lambda\,% \text{; (3)}\\ 0,&\,\text{else}\end{cases}
  51. f ( x ) = { π x sin ( π x 2 λ ) , for 0 ≤ x ≤ λ ; (4) 0 , else f(x)=\begin{cases}\pi x\sin(\frac{\pi x}{2\lambda}),&\,\text{for 0 ≤ x ≤}% \lambda\,\text{; (4)}\\ 0,&\,\text{else}\end{cases}
  52. λ \lambda
  53. P x , y P_{x,y}
  54. τ ( x , y ) \tau_{(x,y)}
  55. τ n e w ( 1 - ψ ) τ o l d + ψ τ 0 \tau_{new}\leftarrow(1-\psi)\tau_{old}+\psi\tau_{0}
  56. ψ \psi
  57. 0 < τ < 1 0<\tau<1

Antenna_tuner.html

  1. X L = ( R s o u r c e + j X s o u r c e ) ( ( R s o u r c e + j X s o u r c e ) - ( R l o a d + j X l o a d ) ) X_{L}=\sqrt{\Big(R_{source}+jX_{source}\Big)\Big((R_{source}+jX_{source})-(R_{% load}+jX_{load})\Big)}
  2. X C = ( R l o a d + j X l o a d ) ( R s o u r c e + j X s o u r c e ) ( R l o a d + j X l o a d ) - ( R s o u r c e + j X s o u r c e ) X_{C}=(R_{load}+jX_{load})\sqrt{\frac{(R_{source}+jX_{source})}{(R_{load}+jX_{% load})-(R_{source}+jX_{source})}}
  3. X L = ( 50 ) ( 50 - 1000 ) = ( - 47500 ) = j 217.94 O h m s X_{L}=\sqrt{(50)(50-1000)}=\sqrt{(-47500)}=j\,217.94\ Ohms
  4. X C = 1000 50 ( 1000 - 50 ) = 1000 × 0.2294 O h m s = 229.4 O h m s X_{C}=1000\sqrt{\frac{50}{(1000-50)}}=1000\,\times\,0.2294\ Ohms=229.4\ Ohms
  5. X C = 1 2 π f C X_{C}=\frac{1}{2\pi fC}
  6. 2 π f X C = 1 C 2\pi fX_{C}=\frac{1}{C}
  7. 1 2 π f X C = C = 24.78 p F \frac{1}{2\pi fX_{C}}=C=24.78\ pF
  8. X L = 2 π f L X_{L}=2\pi fL\!
  9. L = X L 2 π f = 1.239 μ H L=\frac{X_{L}}{2\pi f}=1.239\ \mu H
  10. Z = R L 2 + X C 2 = 1020 Ω Z=\sqrt{R_{L}^{2}\ +\ X_{C}^{2}}=1020\,\Omega
  11. ( θ ) = tan - 1 ( X C R L ) = 11.31 (\theta)=\tan^{-1}\ (\frac{X_{C}}{R_{L}})=11.31^{\circ}
  12. X C = 1 Y sin θ X_{C}^{^{\prime}}=\frac{1}{Y\sin\ \theta}
  13. R L = 1 Y cos θ = 1040 Ω R_{L}^{\prime}=\frac{1}{Y\cos\ \theta}=1040\ \Omega
  14. Q Q
  15. Q Q
  16. Q Q

Antichain.html

  1. L A = { x y A s.t. x y } . L_{A}=\{x\mid\exists y\in A\mbox{ s.t. }~{}x\leq y\}.
  2. A B = { x A B y A B s.t. x < y } . A\vee B=\{x\in A\cup B\mid\not\exists y\in A\cup B\mbox{ s.t. }~{}x<y\}.
  3. A B = { x L A L B y L A L B s.t. x < y } . A\wedge B=\{x\in L_{A}\cap L_{B}\mid\not\exists y\in L_{A}\cap L_{B}\mbox{ s.t% . }~{}x<y\}.

Antifreeze_protein.html

  1. [ 01 1 ¯ 2 ] [01\overline{1}2]

Anyon.html

  1. | ψ 1 ψ 2 = ± | ψ 2 ψ 1 \left|\psi_{1}\psi_{2}\right\rangle=\pm\left|\psi_{2}\psi_{1}\right\rangle
  2. | ψ 1 ψ 2 = e i θ | ψ 2 ψ 1 , \left|\psi_{1}\psi_{2}\right\rangle=e^{i\,\theta}\left|\psi_{2}\psi_{1}\right% \rangle\,,
  3. e i θ = e 2 i π s = ( - 1 ) 2 s e^{i\,\theta}=e^{2\,i\pi s}=(-1)^{2\,s}
  4. | ψ 1 ψ 2 = ( - 1 ) 2 s | ψ 2 ψ 1 . \left|\psi_{1}\psi_{2}\right\rangle=(-1)^{2\,s}\left|\psi_{2}\psi_{1}\right\rangle.

Anyonic_Lie_algebra.html

  1. L L
  2. \mathbb{C}
  3. [ - , - ] [-,-]
  4. ε : L \varepsilon\colon L\to\mathbb{C}
  5. Δ : L L L \Delta\colon L\to L\otimes L
  6. ε ( [ X , Y ] ) = ε ( X ) ε ( Y ) \varepsilon([X,Y])=\varepsilon(X)\varepsilon(Y)

Approximation.html

  1. \approx
  2. π 3.14 \pi\approx 3.14
  3. \simeq
  4. f ( n ) 3 n 2 f(n)\simeq 3n^{2}
  5. π 3.14 \pi\simeq 3.14
  6. \sim
  7. f ( n ) f(n)
  8. f ( n ) n 2 f(n)\sim n^{2}
  9. \cong
  10. Δ A B C Δ A B C \Delta ABC\cong\Delta A^{\prime}B^{\prime}C^{\prime}

Approximation_algorithm.html

  1. { OPT f ( x ) ρ OPT , if ρ > 1 ; ρ OPT f ( x ) OPT , if ρ < 1. \begin{cases}\mathrm{OPT}\leq f(x)\leq\rho\mathrm{OPT},\qquad\mbox{if }~{}\rho% >1;\\ \rho\mathrm{OPT}\leq f(x)\leq\mathrm{OPT},\qquad\mbox{if }~{}\rho<1.\end{cases}
  2. ( OPT - c ) f ( x ) ( OPT + c ) . (\mathrm{OPT}-c)\leq f(x)\leq(\mathrm{OPT}+c).
  3. R ( x , y ) = max ( O P T f ( y ) , f ( y ) O P T ) , R(x,y)=\max\left(\frac{OPT}{f(y)},\frac{f(y)}{OPT}\right),
  4. r = ρ - 1 r=\rho^{-1}
  5. \Rho A \Rho_{A}
  6. R A ( x ) R_{A}(x)
  7. \Rho A = inf { r 1 R A ( x ) r , x } . \Rho_{A}=\inf\{r\geq 1\mid R_{A}(x)\leq r,\forall x\}.
  8. \Rho A \Rho_{A}
  9. R A R_{A}^{\infty}
  10. R A = inf { r 1 n + , R A ( x ) r , x , | x | n } . R_{A}^{\infty}=\inf\{r\geq 1\mid\exists n\in\mathbb{Z}^{+},R_{A}(x)\leq r,% \forall x,|x|\geq n\}.
  11. f ( x ) f(x)\geq
  12. r - 1 OPT r^{-1}\mathrm{OPT}
  13. ρ OPT \rho\mathrm{OPT}
  14. ( 1 - c ) OPT (1-c)\mathrm{OPT}
  15. ( 1 - c ) OPT (1-c)\mathrm{OPT}
  16. ( 1 - c ) OPT + c WORST (1-c)\mathrm{OPT}+c\mathrm{WORST}
  17. OPT - c \mathrm{OPT}-c
  18. f ( x ) f(x)\leq
  19. r OPT r\mathrm{OPT}
  20. ρ OPT \rho\mathrm{OPT}
  21. ( 1 + c ) OPT (1+c)\mathrm{OPT}
  22. ( 1 - c ) - 1 OPT (1-c)^{-1}\mathrm{OPT}
  23. ( 1 - c ) - 1 OPT + c WORST (1-c)^{-1}\mathrm{OPT}+c\mathrm{WORST}
  24. OPT + c \mathrm{OPT}+c

Approximation_property.html

  1. 2 \ell^{2}
  2. p \ell^{p}
  3. p 2 p\neq 2
  4. c 0 c_{0}
  5. X X
  6. K X K\subset X
  7. ε > 0 \varepsilon>0
  8. T : X X T\colon X\to X
  9. T x - x ε \|Tx-x\|\leq\varepsilon
  10. x K x\in K
  11. X X
  12. 1 λ < 1\leq\lambda<\infty
  13. λ \lambda
  14. λ \lambda
  15. K X K\subset X
  16. ε > 0 \varepsilon>0
  17. T : X X T\colon X\to X
  18. T x - x ε \|Tx-x\|\leq\varepsilon
  19. x K x\in K
  20. T λ \|T\|\leq\lambda
  21. λ \lambda
  22. λ \lambda
  23. T T
  24. p \ell^{p}

Apriori_algorithm.html

  1. C C
  2. C C
  3. k k
  4. k - 1 k-1
  5. k k
  6. T T
  7. ϵ \epsilon
  8. T T
  9. C k C_{k}
  10. k k
  11. c o u n t [ c ] count[c]
  12. c c
  13. Apriori ( T , ϵ ) L 1 { large 1 - itemsets } k 2 𝐰𝐡𝐢𝐥𝐞 L k - 1 C k { a { b } a L k - 1 b a } - { c { s s c | s | = k - 1 } L k - 1 } 𝐟𝐨𝐫 transactions t T C t { c c C k c t } 𝐟𝐨𝐫 candidates c C t 𝑐𝑜𝑢𝑛𝑡 [ c ] 𝑐𝑜𝑢𝑛𝑡 [ c ] + 1 L k { c c C k 𝑐𝑜𝑢𝑛𝑡 [ c ] ϵ } k k + 1 𝐫𝐞𝐭𝐮𝐫𝐧 k L k \begin{aligned}&\displaystyle\mathrm{Apriori}(T,\epsilon)\\ &\displaystyle\qquad L_{1}\leftarrow\{\mathrm{large~{}1-itemsets}\}\\ &\displaystyle\qquad k\leftarrow 2\\ &\displaystyle\qquad\mathrm{\,\textbf{while}}~{}L_{k-1}\neq\ \emptyset\\ &\displaystyle\qquad\qquad C_{k}\leftarrow\{a\cup\{b\}\mid a\in L_{k-1}\land b% \not\in a\}-\{c\mid\{s\mid s\subseteq c\land|s|=k-1\}\nsubseteq L_{k-1}\}\\ &\displaystyle\qquad\qquad\mathrm{\,\textbf{for}~{}transactions}~{}t\in T\\ &\displaystyle\qquad\qquad\qquad C_{t}\leftarrow\{c\mid c\in C_{k}\land c% \subseteq t\}\\ &\displaystyle\qquad\qquad\qquad\mathrm{\,\textbf{for}~{}candidates}~{}c\in C_% {t}\\ &\displaystyle\qquad\qquad\qquad\qquad\mathit{count}[c]\leftarrow\mathit{count% }[c]+1\\ &\displaystyle\qquad\qquad L_{k}\leftarrow\{c\mid c\in C_{k}\land~{}\mathit{% count}[c]\geq\epsilon\}\\ &\displaystyle\qquad\qquad k\leftarrow k+1\\ &\displaystyle\qquad\mathrm{\,\textbf{return}}~{}\bigcup_{k}L_{k}\end{aligned}
  14. 2 | S | - 1 2^{|S|}-1

Arbitrary-precision_arithmetic.html

  1. \mathbb{R}
  2. \mathbb{Z}

Archimedes'_principle.html

  1. apparent immersed weight = weight of object - weight of displaced fluid \,\text{apparent immersed weight}=\,\text{weight of object}-\,\text{weight of % displaced fluid}\,
  2. density of object density of fluid = weight weight of displaced fluid \frac{\,\text{density of object}}{\,\text{density of fluid}}=\frac{\,\text{% weight}}{\,\text{weight of displaced fluid}}
  3. density of object density of fluid = weight weight - apparent immersed weight . \frac{\text{density of object}}{\,\text{density of fluid}}=\frac{\,\text{% weight}}{\,\text{weight}-\,\text{apparent immersed weight}}.\,

Archimedes_number.html

  1. Ar = g L 3 ρ ( ρ - ρ ) μ 2 \mathrm{Ar}=\frac{gL^{3}\rho_{\ell}(\rho-\rho_{\ell})}{\mu^{2}}
  2. kg / m 3 {\rm kg/m}^{3}
  3. kg / m 3 {\rm kg/m}^{3}
  4. μ \mu
  5. kg / ms {\rm kg/ms}
  6. β \beta
  7. T T
  8. Ar = Ri Re 2 \mathrm{Ar}=\mathrm{Ri}\,\mathrm{Re}^{2}

Arg_max.html

  1. arg max x f ( x ) := { x y : f ( y ) f ( x ) } . \operatorname*{arg\,max}_{x}f(x):=\{x\mid\forall y:f(y)\leq f(x)\}.
  2. arg max x ( 1 - | x | ) = { 0 } \operatorname*{arg\,max}_{x}(1-|x|)=\{0\}
  3. max x f ( x ) \max_{x}f(x)
  4. { f ( x ) y : f ( y ) f ( x ) } . \{f(x)\mid\forall y:f(y)\leq f(x)\}.
  5. arg max x f ( x ) = { x f ( x ) = M } = : f - 1 ( M ) \operatorname*{arg\,max}_{x}\,f(x)=\{x\mid f(x)=M\}=:f^{-1}(M)
  6. arg max x ( x ( 10 - x ) ) = 5 \operatorname*{arg\,max}_{x\in\mathbb{R}}(x(10-x))=5
  7. arg max x [ 0 , 4 π ] cos ( x ) = { 0 , 2 π , 4 π } \operatorname*{arg\,max}_{x\in[0,4\pi]}\cos(x)=\{0,2\pi,4\pi\}
  8. { 0 , 2 π , - 2 π , 4 π , } . \{0,2\pi,-2\pi,4\pi,\dots\}.
  9. arg max x x 3 \operatorname*{arg\,max}_{x\in\mathbb{R}}x^{3}
  10. x 3 x^{3}
  11. arg min x f ( x ) \operatorname*{arg\,min}_{x}\,f(x)
  12. x ( 10 - x ) = 25 - ( x - 5 ) 2 25 x(10-x)=25-(x-5)^{2}\leq 25
  13. x - 5 = 0 x-5=0

Arima_Yoriyuki.html

  1. π \pi
  2. π 2 \pi^{2}
  3. π 428224593349304 136308121570117 = 3.14159265358979323846264338327 ( 569... ) \pi\approx\frac{428224593349304}{136308121570117}=3.14159265358979323846264338% 327(569...)

Armstrong_oscillator.html

  1. f = 1 2 π L C f=\frac{1}{2\pi\sqrt{LC}}\,

Arthur_Compton.html

  1. λ - λ = h m e c ( 1 - cos θ ) , \lambda^{\prime}-\lambda=\frac{h}{m_{e}c}(1-\cos{\theta}),
  2. λ \lambda
  3. λ \lambda^{\prime}
  4. h h
  5. m e m_{e}
  6. c c
  7. θ \theta
  8. h / m < s u b > e c {h}/{m<sub>ec}

Artificial_neuron.html

  1. y k = φ ( j = 0 m w k j x j ) y_{k}=\varphi\left(\sum_{j=0}^{m}w_{kj}x_{j}\right)
  2. φ \varphi
  3. u = i = 1 n w i x i u=\sum_{i=1}^{n}w_{i}x_{i}
  4. y = { 1 if u θ 0 if u < θ y=\begin{cases}1&\,\text{if }u\geq\theta\\ 0&\,\text{if }u<\theta\end{cases}

Artinian_module.html

  1. / \mathbb{Q}/\mathbb{Z}
  2. [ 1 / p ] / \mathbb{Z}[1/p]/\mathbb{Z}
  3. ( p ) \mathbb{Z}(p^{\infty})
  4. \mathbb{Z}
  5. 1 / p 1 / p 2 1 / p 3 \langle 1/p\rangle\subset\langle 1/p^{2}\rangle\subset\langle 1/p^{3}\rangle\subset\cdots
  6. ( p ) \mathbb{Z}(p^{\infty})
  7. / \mathbb{Q}/\mathbb{Z}
  8. 1 / n 1 1 / n 2 1 / n 3 \langle 1/n_{1}\rangle\supseteq\langle 1/n_{2}\rangle\supseteq\langle 1/n_{3}% \rangle\supseteq\cdots
  9. n 1 , n 2 , n 3 , n_{1},n_{2},n_{3},\ldots
  10. 1 / n i + 1 1 / n i \langle 1/n_{i+1}\rangle\subseteq\langle 1/n_{i}\rangle
  11. n i + 1 n_{i+1}
  12. n i n_{i}
  13. n 1 , n 2 , n 3 , n_{1},n_{2},n_{3},\ldots
  14. ( p ) \mathbb{Z}(p^{\infty})

Artinian_ring.html

  1. / n \mathbb{Z}/n\mathbb{Z}
  2. k [ t ] / ( t n ) k[t]/(t^{n})
  3. A / I A/I
  4. n 1 n\geq 1
  5. M n ( R ) M_{n}(R)
  6. \mathbb{Z}
  7. Spec A \operatorname{Spec}A
  8. Spec A \operatorname{Spec}A
  9. A I AI
  10. A I = A AI=A
  11. a i A a_{i}\in A
  12. 1 a 1 I + + a k I 1\in a_{1}I+\cdots+a_{k}I
  13. I k A , ( y 1 , , y k ) a 1 y 1 + + a k y k . I^{\oplus k}\to A,\,(y_{1},\dots,y_{k})\mapsto a_{1}y_{1}+\cdots+a_{k}y_{k}.
  14. a 1 y 1 = a 2 y 2 + + a k y k a_{1}y_{1}=a_{2}y_{2}+\cdots+a_{k}y_{k}
  15. y 1 y_{1}
  16. y 1 A = I y_{1}A=I
  17. a 1 I = a 1 y 1 A a 2 I + + a k I a_{1}I=a_{1}y_{1}A\subset a_{2}I+\cdots+a_{k}I
  18. I k A I^{\oplus k}\simeq A
  19. A End A ( A ) M k ( End A ( I ) ) A\simeq\operatorname{End}_{A}(A)\simeq M_{k}(\operatorname{End}_{A}(I))

Artin–Wedderburn_theorem.html

  1. k k
  2. M n i ( D i ) \prod M_{n_{i}}(D_{i})
  3. n i n_{i}
  4. D i D_{i}
  5. k k
  6. M n i ( D i ) M_{n_{i}}(D_{i})
  7. n i × n i n_{i}\times n_{i}
  8. D i D_{i}

Arzelà–Ascoli_theorem.html

  1. I = a a , b I=aa,b
  2. | f n ( x ) | M \left|f_{n}(x)\right|\leq M
  3. x a a , b x∈aa,b
  4. ε > 0 ε>0
  5. δ > 0 δ>0
  6. | f n ( x ) - f n ( y ) | < ε \left|f_{n}(x)-f_{n}(y)\right|<\varepsilon
  7. a a , b aa,b
  8. x x
  9. y y
  10. | f n ( x ) - f n ( y ) | K | x - y | , \left|f_{n}(x)-f_{n}(y)\right|\leq K|x-y|,
  11. n n
  12. ε > 0 ε>0
  13. δ = ε 2 K δ=ε\frac{2}{K}
  14. a a , b aa,b
  15. a a , b aa,b
  16. a a , b aa,b
  17. K K
  18. | f n ( x ) - f n ( y ) | K | x - y | \left|f_{n}(x)-f_{n}(y)\right|\leq K|x-y|
  19. x , y a a , b x,y∈aa,b
  20. a a , b aa,b
  21. K K
  22. 𝐅 \mathbf{F}
  23. f f
  24. a a , b aa,b
  25. α α
  26. M M
  27. | f ( x ) - f ( y ) | M | x - y | α , x , y [ a , b ] \left|f(x)-f(y)\right|\leq M\,|x-y|^{\alpha},\qquad x,y\in[a,b]
  28. C ( a a , b ) ) C(aa,b))
  29. C ( a a , b ) ) C(aa,b))
  30. X X
  31. X X
  32. d d
  33. 𝐑 \mathbf{R}
  34. d d
  35. I = a a , b 𝐑 I=aa,b⊂\mathbf{R}
  36. f : I 𝐑 f:I→\mathbf{R}
  37. { f n 1 } { f n 2 } \left\{f_{n_{1}}\right\}\supseteq\left\{f_{n_{2}}\right\}\supseteq\cdots
  38. ε > 0 ε>0
  39. | f n ( x k ) - f m ( x k ) | < ε 3 , n , m N . |f_{n}(x_{k})-f_{m}(x_{k})|<\tfrac{\varepsilon}{3},\qquad n,m\geq N.
  40. | f ( s ) - f ( t ) | < ε 3 |f(s)-f(t)|<\tfrac{\varepsilon}{3}
  41. 1 j J 1≤j≤J
  42. 1 k K 1≤k≤K
  43. | f n ( t ) - f m ( t ) | | f n ( t ) - f n ( x k ) | + | f n ( x k ) - f m ( x k ) | + | f m ( x k ) - f m ( t ) | < ε 3 + ε 3 + ε 3 \begin{aligned}\displaystyle\left|f_{n}(t)-f_{m}(t)\right|&\displaystyle\leq% \left|f_{n}(t)-f_{n}(x_{k})\right|+|f_{n}(x_{k})-f_{m}(x_{k})|+|f_{m}(x_{k})-f% _{m}(t)|\\ &\displaystyle<\tfrac{\varepsilon}{3}+\tfrac{\varepsilon}{3}+\tfrac{% \varepsilon}{3}\end{aligned}
  44. 𝐅 C ( X ) \mathbf{F}⊂C(X)
  45. ε > 0 ε>0
  46. y U x , f 𝐅 : | f ( y ) - f ( x ) | < ε . \forall y\in U_{x},\forall f\in\mathbf{F}:\qquad|f(y)-f(x)|<\varepsilon.
  47. 𝐅 C ( X , 𝐑 ) \mathbf{F}⊂C(X,\mathbf{R})
  48. sup { | f ( x ) | : f 𝐅 } < . \sup\{|f(x)|:f\in\mathbf{F}\}<\infty.
  49. 𝐅 C ( X , Y ) \mathbf{F}⊂C(X,Y)
  50. N ( ε , U ) = { f | osc U f < ε } . N(\varepsilon,U)=\{f|\operatorname{osc}_{U}f<\varepsilon\}.
  51. g g
  52. p p
  53. 0 , 11 0,11
  54. G G
  55. 0 , 11 0,11
  56. G ( x ) = 0 x g ( t ) d t . G(x)=\int_{0}^{x}g(t)\,\mathrm{d}t.
  57. 𝐅 \mathbf{F}
  58. G G
  59. g g
  60. q q
  61. p p
  62. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  63. 𝐅 \mathbf{F}
  64. α = 1 q α=\frac{1}{q}
  65. M = 1 M=1
  66. 𝐅 \mathbf{F}
  67. C ( 0 , 11 ) ) C(0,11))
  68. g G g→G
  69. T T
  70. C ( 0 , 11 ) ) C(0,11))
  71. C ( 0 , 11 ) ) C(0,11))
  72. T T
  73. p = 2 p=2
  74. H 0 1 ( Ω ) H^{1}_{0}(\Omega)
  75. Ω Ω
  76. T T
  77. X X
  78. Y Y
  79. T ( B ) T(B)
  80. B B
  81. X X
  82. K K
  83. Y Y
  84. Y Y
  85. K K
  86. 𝐅 \mathbf{F}
  87. K K
  88. K K
  89. T * ( y n k * ) T^{*}(y^{*}_{n_{k}})
  90. X < s u p > X<sup> ∗

Asian_option.html

  1. P ( T ) = max ( A ( 0 , T ) - K , 0 ) , P(T)=\,\text{max}\left(A(0,T)-K,0\right),
  2. P ( T ) = max ( K - A ( 0 , T ) , 0 ) . P(T)=\,\text{max}\left(K-A(0,T),0\right).
  3. P ( T ) = max ( S ( T ) - k A ( 0 , T ) , 0 ) , P(T)=\,\text{max}\left(S(T)-kA(0,T),0\right),
  4. P ( T ) = max ( k A ( 0 , T ) - S ( T ) , 0 ) . P(T)=\,\text{max}\left(kA(0,T)-S(T),0\right).
  5. A A
  6. A ( 0 , T ) = 1 T 0 T S ( t ) d t . A(0,T)=\frac{1}{T}\int_{0}^{T}S(t)dt.
  7. t 1 , t 2 , , t n t_{1},t_{2},\dots,t_{n}
  8. A ( 0 , T ) = 1 N i = 1 N S ( t i ) . A(0,T)=\frac{1}{N}\sum_{i=1}^{N}S(t_{i}).
  9. A ( 0 , T ) = exp ( 1 T 0 T ln ( S ( t ) ) d t ) . A(0,T)=\exp\left(\frac{1}{T}\int_{0}^{T}\ln(S(t))dt\right).

Associated_bundle.html

  1. G G
  2. F 1 F_{1}
  3. F 2 F_{2}
  4. G G
  5. G G
  6. 2 \mathbb{Z}_{2}
  7. F F
  8. \mathbb{R}
  9. [ - 1 , 1 ] [-1,\ 1]
  10. { - 1 , 1 } \{-1,\ 1\}
  11. G G
  12. x - x x\ \rightarrow\ -x
  13. [ - 1 , 1 ] × I [-1,\ 1]\times I
  14. [ - 1 , 1 ] × J [-1,\ 1]\times J
  15. [ - 1 , 1 ] [-1,\ 1]
  16. { - 1 , 1 } \{-1,\ 1\}
  17. [ - 1 , 1 ] [-1,\ 1]
  18. F F
  19. G G
  20. G G
  21. F 1 F_{1}
  22. F 2 F_{2}
  23. ( p , f ) g = ( p g , ρ ( g - 1 ) f ) . (p,f)\cdot g=(p\cdot g,\rho(g^{-1})f)\,.
  24. [ p g , f ] = [ p , ρ ( g ) f ] for all g G . [p\cdot g,f]=[p,\rho(g)f]\mbox{ for all }~{}g\in G.
  25. G G
  26. B B
  27. H H
  28. C C
  29. G G
  30. B B
  31. B B
  32. H H

Association_rule_learning.html

  1. { onions , potatoes } { burger } \{\mathrm{onions,potatoes}\}\Rightarrow\{\mathrm{burger}\}
  2. I = { i 1 , i 2 , , i n } I=\{i_{1},i_{2},\ldots,i_{n}\}
  3. n n
  4. D = { t 1 , t 2 , , t m } D=\{t_{1},t_{2},\ldots,t_{m}\}
  5. D D
  6. I I
  7. X Y X\Rightarrow Y
  8. X , Y I X,Y\subseteq I
  9. X Y = X\cap Y=\emptyset
  10. X X
  11. Y Y
  12. X X
  13. Y Y
  14. I = { milk , bread , butter , beer , diapers } I=\{\mathrm{milk,bread,butter,beer,diapers}\}
  15. { butter , bread } { milk } \{\mathrm{butter,bread}\}\Rightarrow\{\mathrm{milk}\}
  16. X X
  17. X Y X\Rightarrow Y
  18. T T
  19. X X
  20. T T
  21. X X
  22. supp ( X ) \mathrm{supp}(X)
  23. { milk , bread , butter } \{\mathrm{milk,bread,butter}\}
  24. 1 / 5 = 0.2 1/5=0.2
  25. supp ( ) \mathrm{supp}()
  26. X Y X\Rightarrow Y
  27. T T
  28. X X
  29. Y Y
  30. conf ( X Y ) = supp ( X Y ) / supp ( X ) \mathrm{conf}(X\Rightarrow Y)=\mathrm{supp}(X\cup Y)/\mathrm{supp}(X)
  31. { butter , bread } { milk } \{\mathrm{butter,bread}\}\Rightarrow\{\mathrm{milk}\}
  32. 0.2 / 0.2 = 1.0 0.2/0.2=1.0
  33. s u p p ( X Y ) supp(X\cup Y)
  34. s u p p ( X Y ) supp(X\cup Y)
  35. P ( E X E Y ) P(E_{X}\cap E_{Y})
  36. E X E_{X}
  37. E Y E_{Y}
  38. X X
  39. Y Y
  40. P ( E Y | E X ) P(E_{Y}|E_{X})
  41. lift ( X Y ) = supp ( X Y ) supp ( X ) × supp ( Y ) \mathrm{lift}(X\Rightarrow Y)=\frac{\mathrm{supp}(X\cup Y)}{\mathrm{supp}(X)% \times\mathrm{supp}(Y)}
  42. { milk , bread } { butter } \{\mathrm{milk,bread}\}\Rightarrow\{\mathrm{butter}\}
  43. 0.2 0.4 × 0.4 = 1.25 \frac{0.2}{0.4\times 0.4}=1.25
  44. conv ( X Y ) = 1 - supp ( Y ) 1 - conf ( X Y ) \mathrm{conv}(X\Rightarrow Y)=\frac{1-\mathrm{supp}(Y)}{1-\mathrm{conf}(X% \Rightarrow Y)}
  45. { milk , bread } { butter } \{\mathrm{milk,bread}\}\Rightarrow\{\mathrm{butter}\}
  46. 1 - 0.4 1 - .5 = 1.2 \frac{1-0.4}{1-.5}=1.2
  47. { milk , bread } { butter } \{\mathrm{milk,bread}\}\Rightarrow\{\mathrm{butter}\}
  48. I I
  49. 2 n - 1 2^{n}-1
  50. n n
  51. I I

Atiyah–Singer_index_theorem.html

  1. ¯ + ¯ * \overline{\partial}+\overline{\partial}^{*}
  2. ¯ \overline{\partial}
  3. D Δ := ( 𝐝 + 𝐝 * ) 2 D\equiv\Delta:=(\mathbf{d}+\mathbf{d^{*}})^{2}

Attribute_grammar.html

  1. G = < V n , V t , P , S Align g t ; G=<V_{n},V_{t},P,S&gt;
  2. V n V_{n}
  3. V t V_{t}
  4. P P
  5. S S
  6. A . a A.a
  7. A α P A\rightarrow\alpha\in P
  8. α = α 1 α n , i , 1 i n : α i ( V n V t ) \alpha=\alpha_{1}\ldots\alpha_{n},\forall_{i,1\leq i\leq n}:\alpha_{i}\in(V_{n% }\cup V_{t})
  9. { α j 1 , , α j m } { α 1 , , α n } \{\alpha_{j1},\ldots,\alpha_{jm}\}\subseteq\{\alpha_{1},\ldots,\alpha_{n}\}
  10. A . a = f ( α j 1 . a 1 , , α j m . a m ) A.a=f(\alpha_{j1}.a_{1},\ldots,\alpha_{jm}.a_{m})

Augmented_sixth_chord.html

  1. + 6 {}^{+6}
  2. 6 {}^{6}
  3. 6 {}^{6}
  4. + 6 {}^{+6}
  5. 4 3 {}_{3}^{4}
  6. + 6 {}^{+6}
  7. 6 5 {}_{5}^{6}
  8. 6 4 {}_{4}^{6}
  9. 4 3 {}_{3}^{4}
  10. 6 5 {}_{5}^{6}

Aureus.html

  1. 1 40 \tfrac{1}{40}
  2. 1 100 \tfrac{1}{100}
  3. 1 45 \tfrac{1}{45}
  4. 1 50 \tfrac{1}{50}

Auto_magma_object.html

  1. : ( X , ) × ( X , ) ( X , ) \top^{\prime}\colon(X,\top)\times(X,\top)\to(X,\top)
  2. \top^{\prime}
  3. ( x y ) ( u z ) = ( x u ) ( y z ) (x\top^{\prime}y)\top(u\top^{\prime}z)=(x\top u)\top^{\prime}(y\top z)
  4. ( x y ) ( u z ) = ( x u ) ( y z ) (x\top y)\top(u\top z)=(x\top u)\top(y\top z)

Automatic_label_placement.html

  1. exp - Δ E T \exp\frac{-\Delta E}{T}
  2. Δ E \Delta E
  3. T T

Autoregressive_conditional_heteroskedasticity.html

  1. ϵ t ~{}\epsilon_{t}~{}
  2. ϵ t ~{}\epsilon_{t}~{}
  3. z t z_{t}
  4. σ t \sigma_{t}
  5. ϵ t = σ t z t ~{}\epsilon_{t}=\sigma_{t}z_{t}~{}
  6. z t z_{t}
  7. σ t 2 \sigma_{t}^{2}
  8. σ t 2 = α 0 + α 1 ϵ t - 1 2 + + α q ϵ t - q 2 = α 0 + i = 1 q α i ϵ t - i 2 \sigma_{t}^{2}=\alpha_{0}+\alpha_{1}\epsilon_{t-1}^{2}+\cdots+\alpha_{q}% \epsilon_{t-q}^{2}=\alpha_{0}+\sum_{i=1}^{q}\alpha_{i}\epsilon_{t-i}^{2}
  9. α 0 > 0 ~{}\alpha_{0}>0~{}
  10. α i 0 , i > 0 \alpha_{i}\geq 0,~{}i>0
  11. y t = a 0 + a 1 y t - 1 + + a q y t - q + ϵ t = a 0 + i = 1 q a i y t - i + ϵ t y_{t}=a_{0}+a_{1}y_{t-1}+\cdots+a_{q}y_{t-q}+\epsilon_{t}=a_{0}+\sum_{i=1}^{q}% a_{i}y_{t-i}+\epsilon_{t}
  12. ϵ ^ 2 \hat{\epsilon}^{2}
  13. ϵ ^ t 2 = α ^ 0 + i = 1 q α ^ i ϵ ^ t - i 2 \hat{\epsilon}_{t}^{2}=\hat{\alpha}_{0}+\sum_{i=1}^{q}\hat{\alpha}_{i}\hat{% \epsilon}_{t-i}^{2}
  14. α i = 0 \alpha_{i}=0
  15. i = 1 , , q i=1,\cdots,q
  16. α i \alpha_{i}
  17. χ 2 \chi^{2}
  18. T T^{\prime}
  19. T = T - q T^{\prime}=T-q
  20. σ 2 ~{}\sigma^{2}
  21. ϵ 2 ~{}\epsilon^{2}
  22. σ t 2 = α 0 + α 1 ϵ t - 1 2 + + α q ϵ t - q 2 + β 1 σ t - 1 2 + + β p σ t - p 2 = α 0 + i = 1 q α i ϵ t - i 2 + i = 1 p β i σ t - i 2 \sigma_{t}^{2}=\alpha_{0}+\alpha_{1}\epsilon_{t-1}^{2}+\cdots+\alpha_{q}% \epsilon_{t-q}^{2}+\beta_{1}\sigma_{t-1}^{2}+\cdots+\beta_{p}\sigma_{t-p}^{2}=% \alpha_{0}+\sum_{i=1}^{q}\alpha_{i}\epsilon_{t-i}^{2}+\sum_{i=1}^{p}\beta_{i}% \sigma_{t-i}^{2}
  23. y t = a 0 + a 1 y t - 1 + + a q y t - q + ϵ t = a 0 + i = 1 q a i y t - i + ϵ t y_{t}=a_{0}+a_{1}y_{t-1}+\cdots+a_{q}y_{t-q}+\epsilon_{t}=a_{0}+\sum_{i=1}^{q}% a_{i}y_{t-i}+\epsilon_{t}
  24. ϵ 2 \epsilon^{2}
  25. ρ = t = i + 1 T ( ϵ ^ t 2 - σ ^ t 2 ) ( ϵ ^ t - 1 2 - σ ^ t - 1 2 ) t = 1 T ( ϵ ^ t 2 - σ ^ t 2 ) 2 \rho={{\sum^{T}_{t=i+1}(\hat{\epsilon}^{2}_{t}-\hat{\sigma}^{2}_{t})(\hat{% \epsilon}^{2}_{t-1}-\hat{\sigma}^{2}_{t-1})}\over{\sum^{T}_{t=1}(\hat{\epsilon% }^{2}_{t}-\hat{\sigma}^{2}_{t})^{2}}}
  26. ρ ( i ) \rho(i)
  27. 1 / T 1/\sqrt{T}
  28. χ 2 \chi^{2}
  29. ϵ t 2 \epsilon^{2}_{t}
  30. σ t 2 = ω + α ( ϵ t - 1 - θ σ t - 1 ) 2 + β σ t - 1 2 ~{}\sigma_{t}^{2}=~{}\omega+~{}\alpha(~{}\epsilon_{t-1}-~{}\theta~{}\sigma_{t-% 1})^{2}+~{}\beta~{}\sigma_{t-1}^{2}
  31. α , β 0 ; ω > 0 ~{}\alpha,~{}\beta\geq 0;~{}\omega>0
  32. θ ~{}\theta
  33. i = 1 p β i + i = 1 q α i = 1 \sum^{p}_{i=1}~{}\beta_{i}+\sum_{i=1}^{q}~{}\alpha_{i}=1
  34. log σ t 2 = ω + k = 1 q β k g ( Z t - k ) + k = 1 p α k log σ t - k 2 \log\sigma_{t}^{2}=\omega+\sum_{k=1}^{q}\beta_{k}g(Z_{t-k})+\sum_{k=1}^{p}% \alpha_{k}\log\sigma_{t-k}^{2}
  35. g ( Z t ) = θ Z t + λ ( | Z t | - E ( | Z t | ) ) g(Z_{t})=\theta Z_{t}+\lambda(|Z_{t}|-E(|Z_{t}|))
  36. σ t 2 \sigma_{t}^{2}
  37. ω \omega
  38. β \beta
  39. α \alpha
  40. θ \theta
  41. λ \lambda
  42. Z t Z_{t}
  43. g ( Z t ) g(Z_{t})
  44. Z t Z_{t}
  45. log σ t 2 \log\sigma_{t}^{2}
  46. y t = β x t + λ σ t + ϵ t y_{t}=~{}\beta x_{t}+~{}\lambda~{}\sigma_{t}+~{}\epsilon_{t}
  47. ϵ t ~{}\epsilon_{t}
  48. ϵ t = σ t × z t ~{}\epsilon_{t}=~{}\sigma_{t}~{}\times z_{t}
  49. σ t ~{}\sigma_{t}
  50. ϵ t = σ t z t ~{}\epsilon_{t}=~{}\sigma_{t}z_{t}
  51. z t z_{t}
  52. σ t 2 = K + α ϵ t - 1 2 + β σ t - 1 2 + ϕ ϵ t - 1 ~{}\sigma_{t}^{2}=K+~{}\alpha~{}\epsilon_{t-1}^{2}+~{}\beta~{}\sigma_{t-1}^{2}% +~{}\phi~{}\epsilon_{t-1}
  53. ϵ t = σ t z t ~{}\epsilon_{t}=~{}\sigma_{t}z_{t}
  54. z t z_{t}
  55. σ t 2 = K + δ σ t - 1 2 + α ϵ t - 1 2 + ϕ ϵ t - 1 2 I t - 1 ~{}\sigma_{t}^{2}=K+~{}\delta~{}\sigma_{t-1}^{2}+~{}\alpha~{}\epsilon_{t-1}^{2% }+~{}\phi~{}\epsilon_{t-1}^{2}I_{t-1}
  56. I t - 1 = 0 I_{t-1}=0
  57. ϵ t - 1 0 ~{}\epsilon_{t-1}\geq 0
  58. I t - 1 = 1 I_{t-1}=1
  59. ϵ t - 1 < 0 ~{}\epsilon_{t-1}<0
  60. σ t = K + δ σ t - 1 + α 1 + ϵ t - 1 + + α 1 - ϵ t - 1 - ~{}\sigma_{t}=K+~{}\delta~{}\sigma_{t-1}+~{}\alpha_{1}^{+}~{}\epsilon_{t-1}^{+% }+~{}\alpha_{1}^{-}~{}\epsilon_{t-1}^{-}
  61. ϵ t - 1 + = ϵ t - 1 ~{}\epsilon_{t-1}^{+}=~{}\epsilon_{t-1}
  62. ϵ t - 1 > 0 ~{}\epsilon_{t-1}>0
  63. ϵ t - 1 + = 0 ~{}\epsilon_{t-1}^{+}=0
  64. ϵ t - 1 0 ~{}\epsilon_{t-1}\leq 0
  65. ϵ t - 1 - = ϵ t - 1 ~{}\epsilon_{t-1}^{-}=~{}\epsilon_{t-1}
  66. ϵ t - 1 0 ~{}\epsilon_{t-1}\leq 0
  67. ϵ t - 1 - = 0 ~{}\epsilon_{t-1}^{-}=0
  68. ϵ t - 1 > 0 ~{}\epsilon_{t-1}>0
  69. ϵ t = σ t z t , \epsilon_{t}=\sigma_{t}z_{t},
  70. σ t 2 = α 0 + α 1 ϵ t - 1 2 + β 1 σ t - 1 2 = α 0 + α 1 σ t - 1 2 z t - 1 2 + β 1 σ t - 1 2 , \sigma_{t}^{2}=\alpha_{0}+\alpha_{1}\epsilon^{2}_{t-1}+\beta_{1}\sigma^{2}_{t-% 1}=\alpha_{0}+\alpha_{1}\sigma_{t-1}^{2}z_{t-1}^{2}+\beta_{1}\sigma^{2}_{t-1},
  71. z t z_{t}
  72. d L t \mathrm{d}L_{t}
  73. ( L t ) t 0 (L_{t})_{t\geq 0}
  74. z t 2 z^{2}_{t}
  75. d [ L , L ] t d \mathrm{d}[L,L]^{\mathrm{d}}_{t}
  76. [ L , L ] t d = s [ 0 , t ] ( Δ L t ) 2 , t 0 , [L,L]^{\mathrm{d}}_{t}=\sum_{s\in[0,t]}(\Delta L_{t})^{2},\quad t\geq 0,
  77. L L
  78. d G t = σ t - d L t , \mathrm{d}G_{t}=\sigma_{t-}\,\mathrm{d}L_{t},
  79. d σ t 2 = ( β - η σ t 2 ) d t + φ σ t - 2 d [ L , L ] t d , \mathrm{d}\sigma_{t}^{2}=(\beta-\eta\sigma^{2}_{t})\,\mathrm{d}t+\varphi\sigma% _{t-}^{2}\,\mathrm{d}[L,L]^{\mathrm{d}}_{t},
  80. β \beta
  81. η \eta
  82. φ \varphi
  83. α 0 \alpha_{0}
  84. α 1 \alpha_{1}
  85. β 1 \beta_{1}
  86. ( G 0 , σ 0 2 ) (G_{0},\sigma^{2}_{0})
  87. ( G t , σ t 2 ) t 0 (G_{t},\sigma^{2}_{t})_{t\geq 0}

Average_absolute_deviation.html

  1. 1 n i = 1 n | x i - m ( X ) | . \frac{1}{n}\sum_{i=1}^{n}|x_{i}-m(X)|.
  2. m ( X ) m(X)
  3. m ( X ) m(X)
  4. | 2 - 5 | + | 2 - 5 | + | 3 - 5 | + | 4 - 5 | + | 14 - 5 | 5 = 3.6 \frac{|2-5|+|2-5|+|3-5|+|4-5|+|14-5|}{5}=3.6
  5. | 2 - 3 | + | 2 - 3 | + | 3 - 3 | + | 4 - 3 | + | 14 - 3 | 5 = 2.8 \frac{|2-3|+|2-3|+|3-3|+|4-3|+|14-3|}{5}=2.8
  6. | 2 - 2 | + | 2 - 2 | + | 3 - 2 | + | 4 - 2 | + | 14 - 2 | 5 = 3.0 \frac{|2-2|+|2-2|+|3-2|+|4-2|+|14-2|}{5}=3.0
  7. φ ( 𝔼 [ X ] ) 𝔼 [ φ ( X ) ] \varphi\left(\mathbb{E}[X]\right)\leq\mathbb{E}\left[\varphi(X)\right]
  8. 𝔼 ( | x - μ | ) 2 𝔼 ( | x - μ | 2 ) \mathbb{E}\left(|x-\mu\right|)^{2}\leq\mathbb{E}\left(|x-\mu|^{2}\right)
  9. 𝔼 ( | x - μ | ) 2 Var ( x ) \mathbb{E}\left(|x-\mu\right|)^{2}\leq\operatorname{Var}(x)
  10. 𝔼 ( | x - μ | ) Var ( x ) \mathbb{E}\left(|x-\mu\right|)\leq\sqrt{\operatorname{Var}(x)}
  11. 2 / π = 0.79788456 \sqrt{2/\pi}=0.79788456\ldots
  12. w = E | X | E ( X 2 ) = 2 π . w=\frac{E|X|}{\sqrt{E(X^{2})}}=\sqrt{\frac{2}{\pi}}.
  13. w n [ 0 , 1 ] w_{n}\in[0,1]
  14. D m = E | X - μ | = 2 Cov ( X , I O ) D_{m}=E|X-\mu|=2\operatorname{Cov}(X,I_{O})
  15. I O I_{O}
  16. 𝐈 O := { 1 if x > μ , 0 otherwise . \mathbf{I}_{O}:=\begin{cases}1&\,\text{if }x>\mu,\\ 0&\,\text{otherwise}.\end{cases}
  17. D med = E | X - median | D\text{med}=E|X-\,\text{median}|
  18. D med = σ 2 / π D\text{med}=\sigma\sqrt{2/\pi}
  19. D med D mean D\text{med}\leq D\text{mean}
  20. D med = E | X - median | = 2 Cov ( X , I O ) D\text{med}=E|X-\,\text{median}|=2\operatorname{Cov}(X,I_{O})
  21. 𝐈 O := { 1 if x > median , 0 otherwise . \mathbf{I}_{O}:=\begin{cases}1&\,\text{if }x>\,\text{median},\\ 0&\,\text{otherwise}.\end{cases}
  22. m ( X ) = max ( X ) m(X)=\max(X)
  23. max ( X ) \max(X)

Average_cost.html

  1. A C = T C Q AC=\frac{TC}{Q}

Axiom_(computer_algebra_system).html

  1. 4 × 4 4\times 4

Axion.html

  1. Θ ¯ \overline{Θ}
  2. Θ ¯ \overline{Θ}
  3. Θ ¯ \overline{Θ}
  4. Θ ¯ \overline{Θ}
  5. Θ ¯ \overline{Θ}
  6. Θ ¯ \overline{Θ}
  7. 0.05 G e V / c m 3 0.05GeV/cm^{3}
  8. ( 0.3 ± 0.1 ) G e V / c m 3 (0.3\pm 0.1)GeV/cm^{3}

Azimuthal_quantum_number.html

  1. 𝐋 2 Ψ = 2 ( + 1 ) Ψ \mathbf{L}^{2}\Psi=\hbar^{2}{\ell(\ell+1)}\Psi
  2. Ψ \Psi
  3. ȷ \vec{\jmath}
  4. 1 \vec{\ell_{1}}
  5. 2 \vec{\ell_{2}}
  6. ȷ = 1 + 2 \vec{\jmath}=\vec{\ell_{1}}+\vec{\ell_{2}}
  7. j j
  8. | 1 - 2 | |\ell_{1}-\ell_{2}|
  9. 1 + 2 \ell_{1}+\ell_{2}
  10. 1 \ell_{1}
  11. 2 \ell_{2}
  12. J = L + S \vec{J}=\vec{L}+\vec{S}
  13. [ J i , J j ] = i ϵ i j k J k [J_{i},J_{j}]=i\hbar\epsilon_{ijk}J_{k}
  14. [ J i , J 2 ] = 0 \left[J_{i},J^{2}\right]=0
  15. Ψ \Psi
  16. 𝐉 2 Ψ = 2 j ( j + 1 ) Ψ \mathbf{J}^{2}\Psi=\hbar^{2}{j(j+1)}\Psi
  17. 𝐉 z Ψ = m j Ψ \mathbf{J}_{z}\Psi=\hbar{m_{j}}\Psi

Azuma's_inequality.html

  1. | X k - X k - 1 | < c k , |X_{k}-X_{k-1}|<c_{k},\,
  2. P ( X N - X 0 t ) exp ( - t 2 2 k = 1 N c k 2 ) . P(X_{N}-X_{0}\geq t)\leq\exp\left({-t^{2}\over 2\sum_{k=1}^{N}c_{k}^{2}}\right).
  3. P ( X N - X 0 - t ) exp ( - t 2 2 k = 1 N c k 2 ) . P(X_{N}-X_{0}\leq-t)\leq\exp\left({-t^{2}\over 2\sum_{k=1}^{N}c_{k}^{2}}\right).
  4. P ( | X N - X 0 | t ) 2 exp ( - t 2 2 k = 1 N c k 2 ) . P(|X_{N}-X_{0}|\geq t)\leq 2\exp\left({-t^{2}\over 2\sum_{k=1}^{N}c_{k}^{2}}% \right).
  5. X i = j = 1 i F j X_{i}=\sum_{j=1}^{i}F_{j}
  6. Pr [ X N > t ] exp ( - t 2 2 N ) . \Pr[X_{N}>t]\leq\exp\left(\frac{-t^{2}}{2N}\right).

Ba_space.html

  1. b a ( Σ ) ba(\Sigma)
  2. Σ \Sigma
  3. Σ \Sigma
  4. ν = | ν | ( X ) . \|\nu\|=|\nu|(X).
  5. c a ( Σ ) ca(\Sigma)
  6. b a ( Σ ) ba(\Sigma)
  7. r c a ( X ) rca(X)
  8. c a ( Σ ) ca(\Sigma)
  9. c a ( Σ ) ca(\Sigma)
  10. b a ( Σ ) ba(\Sigma)
  11. r c a ( X ) rca(X)
  12. c a ( Σ ) ca(\Sigma)
  13. Σ \Sigma
  14. b a ( Σ ) ba(\Sigma)
  15. b a ba
  16. μ ( A ) = ζ ( 1 A ) \mu(A)=\zeta\left(1_{A}\right)
  17. N μ := { f B ( Σ ) : f = 0 μ -almost everywhere } . N_{\mu}:=\{f\in B(\Sigma):f=0\ \mu\,\text{-almost everywhere}\}.
  18. N μ = { σ b a ( Σ ) : μ ( A ) = 0 σ ( A ) = 0 for any A Σ } , N_{\mu}^{\perp}=\{\sigma\in ba(\Sigma):\mu(A)=0\Rightarrow\sigma(A)=0\,\text{ % for any }A\in\Sigma\},
  19. L 1 ( μ ) L 1 ( μ ) * * = L ( μ ) * L^{1}(\mu)\subset L^{1}(\mu)^{**}=L^{\infty}(\mu)^{*}

Baby-step_giant-step.html

  1. G G
  2. n n
  3. α \alpha
  4. β \beta
  5. x x
  6. α x = β . \alpha^{x}=\beta\,.
  7. x x
  8. x = i m + j x=im+j
  9. m = n m=\left\lceil\sqrt{n}\right\rceil
  10. 0 i < m 0\leq i<m
  11. 0 j < m 0\leq j<m
  12. β ( α - m ) i = α j . \beta(\alpha^{-m})^{i}=\alpha^{j}\,.
  13. α j \alpha^{j}
  14. j j
  15. m m
  16. i i
  17. j j
  18. α j \alpha^{j}
  19. α x = β \alpha^{x}=\beta

Baby_Monster_group.html

  1. × 10 3 3 \times 10^{3}3
  2. T 2 A ( τ ) T_{2A}(\tau)
  3. j 2 A ( τ ) = T 2 A ( τ ) + 104 = ( ( η ( τ ) η ( 2 τ ) ) 12 + 2 6 ( η ( 2 τ ) η ( τ ) ) 12 ) 2 = 1 q + 104 + 4372 q + 96256 q 2 + 1240002 q 3 + 10698752 q 4 \begin{aligned}\displaystyle j_{2A}(\tau)&\displaystyle=T_{2A}(\tau)+104\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{12}+2^{6}\big(% \tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{12}\Big)^{2}\\ &\displaystyle=\frac{1}{q}+104+4372q+96256q^{2}+1240002q^{3}+10698752q^{4}% \dots\end{aligned}

Balance_of_payments.html

  1. current account + broadly defined capital account + balancing item = 0. \,\text{current account}+\,\text{ broadly defined capital account}+\,\text{% balancing item}=0.\,
  2. B O P s u r p l u s = current account surplus + narrowly defined capital account surplus . BOPsurplus=\,\text{current account surplus}+\,\text{narrowly defined capital % account surplus}.\,
  3. current account + financial account + capital account + balancing item = 0. \,\text{current account}\,+\,\,\text{financial account}\,+\,\,\text{capital % account}\,+\,\,\text{balancing item}\,=\,0.\,
  4. CA = NS - NI \,\text{CA}=\,\text{NS}-\,\text{NI}\,

Ball_valve.html

  1. A 1 V 1 = A 2 V 2 A_{1}V_{1}=A_{2}V_{2}

Balmer_series.html

  1. 3 2 \scriptstyle 3\rightarrow 2
  2. Z = 1 Z=1
  3. n n
  4. \infty
  5. λ = B ( n 2 n 2 - m 2 ) = B ( n 2 n 2 - 2 2 ) \lambda\ =B\left(\frac{n^{2}}{n^{2}-m^{2}}\right)=B\left(\frac{n^{2}}{n^{2}-2^% {2}}\right)
  6. λ \lambda
  7. 1 λ = 4 B ( 1 2 2 - 1 n 2 ) = R H ( 1 2 2 - 1 n 2 ) for n = 3 , 4 , 5 , \frac{1}{\lambda}=\frac{4}{B}\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right)=R_{% \mathrm{H}}\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right)\quad\mathrm{for~{}}n=3% ,4,5,...
  8. 4 / B 4/B
  9. 4 / ( 3.6450682 * 10 - 7 4/(3.6450682*10^{-7}
  10. ) )

Banach–Alaoglu_theorem.html

  1. ρ ( x , y ) = n = 1 2 - n | x - y , x n | 1 + | x - y , x n | \rho(x,y)=\sum_{n=1}^{\infty}\,2^{-n}\,\frac{\left|\langle x-y,x_{n}\rangle% \right|}{1+\left|\langle x-y,x_{n}\rangle\right|}
  2. , \langle\cdot,\cdot\rangle
  3. F : X * F:X^{*}\to{\mathbb{R}}
  4. x 1 , x 2 , X * x_{1},x_{2},\ldots\in X^{*}
  5. X = C 0 ( ) X=C_{0}({\mathbb{R}})
  6. D x = { z : | z | x } , D_{x}=\{z\in\mathbb{C}:\left|z\right|\leq\|x\|\},
  7. D = Π x X D x . D=\Pi_{x\in X}D_{x}.
  8. f B 1 ( X * ) ( f ( x ) ) x X D . f\in B_{1}\left(X^{*}\right)\mapsto(f(x))_{x\in X}\in D.
  9. ( f α ( x ) ) x X ( λ x ) x X (f_{\alpha}(x))_{x\in X}\rightarrow(\lambda_{x})_{x\in X}
  10. g ( x ) = λ x g(x)=\lambda_{x}\,
  11. f n k g d μ f g d μ \int f_{n_{k}}g\,d\mu\to\int fg\,d\mu

Banach–Mazur_game.html

  1. Y Y
  2. X Y X\subset Y
  3. W W
  4. Y Y
  5. W W
  6. Y Y
  7. W W
  8. M B ( X , Y , W ) MB(X,Y,W)
  9. P 1 P_{1}
  10. P 2 P_{2}
  11. W 0 W_{0}
  12. W 1 W_{1}
  13. \cdots
  14. W W
  15. W 0 W 1 W_{0}\supset W_{1}\supset\cdots
  16. P 1 P_{1}
  17. X ( n < ω W n ) X\cap(\cap_{n<\omega}W_{n})\neq\emptyset
  18. P 2 M B ( X , Y , W ) P_{2}\uparrow MB(X,Y,W)
  19. X X
  20. Y Y
  21. Y Y
  22. P 1 M S ( X , Y , W ) P_{1}\uparrow MS(X,Y,W)
  23. X X
  24. Y Y
  25. X X
  26. Y Y
  27. M B ( X , Y , W ) MB(X,Y,W)
  28. P 2 P_{2}
  29. B M ( X ) BM(X)
  30. M B ( X , Y , W ) MB(X,Y,W)
  31. X = Y X=Y
  32. W W
  33. X X
  34. P 2 P_{2}
  35. ( W 0 , W 1 , ) (W_{0},W_{1},\cdots)
  36. n < ω W n \cap_{n<\omega}W_{n}\neq\emptyset
  37. X X
  38. P 2 P_{2}
  39. B M ( X ) BM(X)
  40. P 2 P_{2}
  41. B M ( X ) BM(X)
  42. P 2 P_{2}
  43. B M ( X ) BM(X)
  44. P 2 P_{2}
  45. P 1 P_{1}
  46. X X
  47. α \alpha
  48. P 2 P_{2}
  49. B M ( X ) BM(X)
  50. X X
  51. P 1 P_{1}
  52. B M ( X ) BM(X)
  53. α \alpha
  54. M B ( X , J ) MB(X,J)
  55. Y = J Y=J
  56. [ 0 , 1 ] [0,1]
  57. W W
  58. [ a , b ] [a,b]
  59. [ 0 , 1 ] [0,1]
  60. J 0 , J 1 , J_{0},J_{1},\cdots
  61. J J
  62. J 0 J 1 J_{0}\supset J_{1}\supset\cdots
  63. P 1 P_{1}
  64. X ( n < ω J n ) X\cap(\cap_{n<\omega}J_{n})\neq\emptyset
  65. P 2 P_{2}
  66. X ( n < ω J n ) = X\cap(\cap_{n<\omega}J_{n})=\emptyset
  67. X X
  68. P 2 P_{2}
  69. X X
  70. P 2 P_{2}
  71. X X
  72. X X
  73. Y Y
  74. P 2 P_{2}
  75. P 2 P_{2}
  76. X X
  77. Y Y
  78. Y Y
  79. X X
  80. x 1 , x 2 , x_{1},x_{2},\cdots
  81. W 1 W_{1}
  82. P 1 P_{1}
  83. U 1 U_{1}
  84. W 1 W_{1}
  85. U 1 { x 1 } U_{1}\setminus\{x_{1}\}
  86. Y Y
  87. P 2 P_{2}
  88. W 2 W_{2}
  89. W W
  90. P 1 P_{1}
  91. W 3 W_{3}
  92. W 2 W_{2}
  93. P 2 P_{2}
  94. W 4 W 3 W_{4}\subset W_{3}
  95. x 2 x_{2}
  96. x n x_{n}
  97. W 2 n W_{2n}
  98. W n W_{n}
  99. X X
  100. Y Y
  101. Y = { a , b , c } Y=\{a,b,c\}
  102. W W
  103. Y Y
  104. P 2 P_{2}
  105. X = { a } X=\{a\}
  106. Y Y
  107. W = { Y } W=\{Y\}
  108. X X
  109. Y Y
  110. W W
  111. Y Y
  112. X X
  113. Y Y
  114. P 2 P_{2}
  115. X X
  116. P 1 P_{1}
  117. X X
  118. P 1 P_{1}
  119. W i W W_{i}\in W
  120. X W i X\cap W_{i}
  121. W i W_{i}
  122. Y Y
  123. [ 0 , 1 ] [0,1]
  124. W W
  125. [ a , b ] [a,b]
  126. [ 0 , 1 ] [0,1]

Bandlimiting.html

  1. x ( t ) = sin ( 2 π f t + θ ) x(t)=\sin(2\pi ft+\theta)
  2. f s = 1 T > 2 f f_{s}=\frac{1}{T}>2f
  3. x ( n T ) x(nT)
  4. n n
  5. x ( t ) x(t)
  6. x ( t ) x(t)
  7. X ( f ) X(f)
  8. x ( t ) x(t)
  9. B B
  10. R N = 2 B R_{N}=2B\,
  11. x ( t ) x(t)
  12. x [ n ] = def x ( n T ) = x ( n f s ) x[n]\ \stackrel{\mathrm{def}}{=}\ x(nT)=x\left({n\over f_{s}}\right)
  13. n n\,
  14. T = def 1 f s T\ \stackrel{\mathrm{def}}{=}\ {1\over f_{s}}
  15. f s > R N f_{s}>R_{N}\,
  16. F T ( f ) = F 1 ( w ) FT(f)=F_{1}(w)
  17. D T F T ( f ) = F 2 ( w ) DTFT(f)=F_{2}(w)
  18. F 2 ( w ) = n = - + F 1 ( w + n f x ) F_{2}(w)=\sum_{n=-\infty}^{+\infty}F_{1}(w+nf_{x})
  19. f x f_{x}
  20. F 1 F_{1}
  21. f x f_{x}
  22. F 2 F_{2}
  23. F 1 F_{1}
  24. F 2 F_{2}
  25. F 2 F_{2}
  26. F 2 F_{2}
  27. F 2 F_{2}
  28. W B T D 1 W_{B}T_{D}\geq 1
  29. W B W_{B}
  30. T D T_{D}

Bandwidth-limited_pulse.html

  1. sech 2 \mathrm{sech^{2}}

Barbier's_theorem.html

  1. 4 π 12.566 4\pi\approx 12.566
  2. 8 π - 4 3 π 2 11.973 8\pi-\tfrac{4}{3}\pi^{2}\approx 11.973

Barry_Boehm.html

  1. P M = A * ( K S L O C ) B PM=A*(KSLOC)^{B}

Barycentric_subdivision.html

  1. n n
  2. S S
  3. v 0 , v 1 , , v n v_{0},v_{1},\dots,v_{n}
  4. p 0 , p 1 , , p n p_{0},p_{1},\dots,p_{n}
  5. S S
  6. v i v_{i}
  7. p 0 , p 1 , , p i p_{0},p_{1},\dots,p_{i}
  8. S S
  9. S S
  10. S S
  11. S S
  12. v 2 v_{2}
  13. S S
  14. v 1 v_{1}
  15. v 0 v_{0}
  16. S S
  17. S S
  18. S S
  19. n - n-
  20. n n + 1 \frac{n}{n+1}
  21. S S
  22. F 0 , F 1 , , F n F_{0},F_{1},\dots,F_{n}
  23. S S
  24. F i F_{i}
  25. F i + 1 F_{i+1}
  26. i i
  27. n - 1 n-1
  28. v i v_{i}
  29. F i F_{i}
  30. n n
  31. n n
  32. P P
  33. F 0 , F 1 , F 2 F_{0},F_{1},F_{2}
  34. P P
  35. P P
  36. P P
  37. P P
  38. F 0 , F 1 , F 2 , F 3 F_{0},F_{1},F_{2},F_{3}
  39. F 0 F_{0}
  40. F 1 F_{1}
  41. F 0 F_{0}
  42. F 2 F_{2}
  43. F 0 , F 1 F_{0},F_{1}
  44. n / ( n + 1 ) n/(n+1)
  45. ( n + 1 ) ! (n+1)!
  46. n ! n!
  47. ( n - 1 ) ! (n-1)!
  48. L K L\subset K
  49. K * K^{*}
  50. K K
  51. L L
  52. v 0 v k B ( S 1 ) B ( S l ) v_{0}\dots v_{k}B(S^{\prime}_{1})\dots B(S^{\prime}_{l})
  53. S 0 < < S k S_{0}<\cdots<S_{k}
  54. L L
  55. B ( S i ) B(S^{\prime}_{i})
  56. K L K\setminus L
  57. L L
  58. K * K^{*}
  59. L L
  60. P P
  61. P P
  62. P P
  63. n n
  64. n + 1 n+1

Baryogenesis.html

  1. B B
  2. [ B , H ] = B H - H B = 0 [B,H]=BH-HB=0
  3. η = n B - n B ¯ n γ \eta=\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}
  4. B ¯ \overline{B}
  5. n γ = 1 π 2 ( k B T c ) 3 0 x 2 exp ( x ) - 1 d x 20.3 ( T 1 K ) 3 cm - 3 n_{\gamma}=\frac{1}{\pi^{2}}{\left(\frac{k_{B}T}{\hbar c}\right)}^{3}\int_{0}^% {\infty}\frac{x^{2}}{\exp(x)-1}dx\simeq 20.3\left(\frac{T}{1\,\text{K}}\right)% ^{3}\,\text{cm}^{-3}
  6. η s = n B - n B ¯ s \eta_{s}=\frac{n_{B}-n_{\bar{B}}}{s}
  7. s = def entropy volume = p + ρ T = 2 π 2 45 g * ( T ) T 3 s\ \stackrel{\mathrm{def}}{=}\ \frac{\mathrm{entropy}}{\mathrm{volume}}=\frac{% p+\rho}{T}=\frac{2\pi^{2}}{45}g_{*}(T)T^{3}
  8. g * ( T ) = i = bosons g i ( T i T ) 3 + 7 8 j = fermions g j ( T j T ) 3 g_{*}(T)=\sum_{\mathrm{i=bosons}}g_{i}{\left(\frac{T_{i}}{T}\right)}^{3}+\frac% {7}{8}\sum_{\mathrm{j=fermions}}g_{j}{\left(\frac{T_{j}}{T}\right)}^{3}

Baryon_number.html

  1. B = 1 3 ( n q - n q ¯ ) , B=\frac{1}{3}\left(n\text{q}-n_{\bar{\,\text{q}}}\right),
  2. q ¯ \overline{q}

Basal_metabolic_rate.html

  1. P = ( 13.7516 m 1 kg + 5.0033 h 1 cm - 6.7550 a 1 year + 66.4730 ) kcal day P=\left(\frac{13.7516m}{1~{}\mbox{kg}}+\frac{5.0033h}{1~{}\mbox{cm}}-\frac{6.7% 550a}{1~{}\mbox{year}}+66.4730\right)\frac{\mbox{kcal}}{\mbox{day}}
  2. P = ( 9.5634 m 1 kg + 1.8496 h 1 cm - 4.6756 a 1 year + 655.0955 ) kcal day P=\left(\frac{9.5634m}{1~{}\mbox{kg}}+\frac{1.8496h}{1~{}\mbox{cm}}-\frac{4.67% 56a}{1~{}\mbox{year}}+655.0955\right)\frac{\mbox{kcal}}{\mbox{day}}
  3. P = ( 13.397 m 1 kg + 4.799 h 1 cm - 5.677 a 1 year + 88.362 ) kcal day P=\left(\frac{13.397m}{1~{}\mbox{kg}}+\frac{4.799h}{1~{}\mbox{cm}}-\frac{5.677% a}{1~{}\mbox{year}}+88.362\right)\frac{\mbox{kcal}}{\mbox{day}}
  4. P = ( 9.247 m 1 kg + 3.098 h 1 cm - 4.330 a 1 year + 447.593 ) kcal day P=\left(\frac{9.247m}{1~{}\mbox{kg}}+\frac{3.098h}{1~{}\mbox{cm}}-\frac{4.330a% }{1~{}\mbox{year}}+447.593\right)\frac{\mbox{kcal}}{\mbox{day}}
  5. P = ( 10.0 m 1 kg + 6.25 h 1 cm - 5.0 a 1 year + s ) kcal day P=\left(\frac{10.0m}{1~{}\mbox{kg}}+\frac{6.25h}{1~{}\mbox{cm}}-\frac{5.0a}{1~% {}\mbox{year}}+s\right)\frac{\mbox{kcal}}{\mbox{day}}
  6. P = 370 + ( 21.6 L B M ) P=370+\left({21.6\cdot LBM}\right)

Basel_problem.html

  1. n = 1 1 n 2 = lim n + ( 1 1 2 + 1 2 2 + + 1 n 2 ) . \sum_{n=1}^{\infty}\frac{1}{n^{2}}=\lim_{n\to+\infty}\left(\frac{1}{1^{2}}+% \frac{1}{2^{2}}+\cdots+\frac{1}{n^{2}}\right).
  2. π 2 6 \frac{\pi^{2}}{6}
  3. sin ( x ) = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + . \sin(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots.
  4. sin ( x ) x = 1 - x 2 3 ! + x 4 5 ! - x 6 7 ! + . \frac{\sin(x)}{x}=1-\frac{x^{2}}{3!}+\frac{x^{4}}{5!}-\frac{x^{6}}{7!}+\cdots.
  5. sin ( x ) x \displaystyle\frac{\sin(x)}{x}
  6. - ( 1 π 2 + 1 4 π 2 + 1 9 π 2 + ) = - 1 π 2 n = 1 1 n 2 . -\left(\frac{1}{\pi^{2}}+\frac{1}{4\pi^{2}}+\frac{1}{9\pi^{2}}+\cdots\right)=-% \frac{1}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}.
  7. - 1 6 = - 1 π 2 n = 1 1 n 2 . -\frac{1}{6}=-\frac{1}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}.
  8. - π 2 -\pi^{2}
  9. n = 1 1 n 2 = π 2 6 . \sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}.
  10. ζ ( s ) \zeta(s)
  11. ζ ( s ) = n = 1 1 n s . \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.
  12. ζ ( 2 ) \zeta(2)
  13. ζ ( 2 ) = n = 1 1 n 2 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + = π 2 6 1.644934. \zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{1}{1^{2}}+\frac{1}{2^{2}}+% \frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots=\frac{\pi^{2}}{6}\approx 1.644934.
  14. n = 1 N 1 n 2 < \displaystyle\sum_{n=1}^{N}\frac{1}{n^{2}}<
  15. ζ ( 2 ) < 2 \zeta(2)<2
  16. ζ ( s ) \zeta(s)
  17. s = 2 n s=2n
  18. ζ ( 2 n ) = ( 2 π ) 2 n ( - 1 ) n + 1 B 2 n 2 ( 2 n ) ! \zeta(2n)=\frac{(2\pi)^{2n}(-1)^{n+1}B_{2n}}{2\cdot(2n)!}
  19. f ( x ) = x f(x)=x
  20. π \pi
  21. π \pi
  22. f ( x ) = 2 n = 1 ( - 1 ) n + 1 n sin ( n x ) . f(x)=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\sin(nx).
  23. f ( x ) = x f(x)=x
  24. n = - | a n | 2 = 1 2 π - π π x 2 d x \sum_{n=-\infty}^{\infty}|a_{n}|^{2}=\frac{1}{2\pi}\int_{-\pi}^{\pi}x^{2}\,dx
  25. a n = 1 2 π - π π x e - i n x d x = n π cos ( n π ) - sin ( n π ) π n 2 i = cos ( n π ) n i - sin ( n π ) π n 2 i = ( - 1 ) n n i \begin{aligned}\displaystyle a_{n}&\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{% \pi}xe^{-inx}\,dx\\ &\displaystyle=\frac{n\pi\cos(n\pi)-\sin(n\pi)}{\pi n^{2}}i\\ &\displaystyle=\frac{\cos(n\pi)}{n}i-\frac{\sin(n\pi)}{\pi n^{2}}i\\ &\displaystyle=\frac{(-1)^{n}}{n}i\end{aligned}
  26. | a n | 2 = 1 n 2 |a_{n}|^{2}=\frac{1}{n^{2}}
  27. n = - | a n | 2 = 2 n = 1 1 n 2 = 1 2 π - π π x 2 d x \sum_{n=-\infty}^{\infty}|a_{n}|^{2}=2\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac% {1}{2\pi}\int_{-\pi}^{\pi}x^{2}\,dx
  28. n = 1 1 n 2 = 1 4 π - π π x 2 d x = π 2 6 \sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{1}{4\pi}\int_{-\pi}^{\pi}x^{2}\,dx=% \frac{\pi^{2}}{6}
  29. k = 1 m 1 k 2 = 1 1 2 + 1 2 2 + + 1 m 2 \sum_{k=1}^{m}\frac{1}{k^{2}}=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\cdots+\frac{1}{% m^{2}}
  30. π \pi
  31. x x
  32. 0 < x < π 2 0<x<\frac{\pi}{2}
  33. cos ( n x ) + i sin ( n x ) ( sin x ) n = ( cos x + i sin x ) n ( sin x ) n = ( cos x + i sin x sin x ) n = ( cot x + i ) n . \begin{aligned}\displaystyle\frac{\cos(nx)+i\sin(nx)}{(\sin x)^{n}}&% \displaystyle=\frac{(\cos x+i\sin x)^{n}}{(\sin x)^{n}}\\ &\displaystyle=\left(\frac{\cos x+i\sin x}{\sin x}\right)^{n}\\ &\displaystyle=(\cot x+i)^{n}.\end{aligned}
  34. ( cot x + i ) n = ( n 0 ) cot n x + ( n 1 ) ( cot n - 1 x ) i + + ( n n - 1 ) ( cot x ) i n - 1 + ( n n ) i n = [ ( n 0 ) cot n x - ( n 2 ) cot n - 2 x ± ] + i [ ( n 1 ) cot n - 1 x - ( n 3 ) cot n - 3 x ± ] . \begin{aligned}\displaystyle(\cot x+i)^{n}&\displaystyle={n\choose 0}\cot^{n}x% +{n\choose 1}(\cot^{n-1}x)i+\cdots+{n\choose{n-1}}(\cot x)i^{n-1}+{n\choose n}% i^{n}\\ &\displaystyle=\left[{n\choose 0}\cot^{n}x-{n\choose 2}\cot^{n-2}x\pm\cdots% \right]\;+\;i\left[{n\choose 1}\cot^{n-1}x-{n\choose 3}\cot^{n-3}x\pm\cdots% \right].\end{aligned}
  35. sin ( n x ) ( sin x ) n = [ ( n 1 ) cot n - 1 x - ( n 3 ) cot n - 3 x ± ] . \frac{\sin(nx)}{(\sin x)^{n}}=\left[{n\choose 1}\cot^{n-1}x-{n\choose 3}\cot^{% n-3}x\pm\cdots\right].
  36. m m
  37. n = 2 m + 1 n=2m+1\,
  38. x r = r π 2 m + 1 x_{r}=\frac{r\pi}{2m+1}
  39. r = 1 , 2 , , m r=1,2,\ldots,m
  40. n x r nx_{r}\,
  41. π \pi\,
  42. 0 = ( 2 m + 1 1 ) cot 2 m x r - ( 2 m + 1 3 ) cot 2 m - 2 x r ± + ( - 1 ) m ( 2 m + 1 2 m + 1 ) 0={{2m+1}\choose 1}\cot^{2m}x_{r}-{{2m+1}\choose 3}\cot^{2m-2}x_{r}\pm\cdots+(% -1)^{m}{{2m+1}\choose{2m+1}}
  43. r = 1 , 2 , , m r=1,2,\ldots,m
  44. x 1 , , x m x_{1},\ldots,x_{m}
  45. π \pi
  46. cot 2 x \cot^{2}x\,
  47. t r = cot 2 x r t_{r}=\cot^{2}x_{r}\,
  48. p ( t ) := ( 2 m + 1 1 ) t m - ( 2 m + 1 3 ) t m - 1 ± + ( - 1 ) m ( 2 m + 1 2 m + 1 ) . p(t):={{2m+1}\choose 1}t^{m}-{{2m+1}\choose 3}t^{m-1}\pm\cdots+(-1)^{m}{{2m+1}% \choose{2m+1}}.
  49. cot 2 x 1 + cot 2 x 2 + + cot 2 x m = ( 2 m + 1 3 ) ( 2 m + 1 1 ) = 2 m ( 2 m - 1 ) 6 . \cot^{2}x_{1}+\cot^{2}x_{2}+\cdots+\cot^{2}x_{m}=\frac{{\left({{2m+1}\atop{3}}% \right)}}{{\left({{2m+1}\atop{1}}\right)}}=\frac{2m(2m-1)}{6}.
  50. csc 2 x = cot 2 x + 1 \csc^{2}x=\cot^{2}x+1\,
  51. csc 2 x 1 + csc 2 x 2 + + csc 2 x m = 2 m ( 2 m - 1 ) 6 + m = 2 m ( 2 m + 2 ) 6 . \csc^{2}x_{1}+\csc^{2}x_{2}+\cdots+\csc^{2}x_{m}=\frac{2m(2m-1)}{6}+m=\frac{2m% (2m+2)}{6}.
  52. cot 2 x < 1 x 2 < csc 2 x \cot^{2}x<\frac{1}{x^{2}}<\csc^{2}x
  53. x r = r π 2 m + 1 x_{r}=\frac{r\pi}{2m+1}
  54. 2 m ( 2 m - 1 ) 6 < ( 2 m + 1 π ) 2 + ( 2 m + 1 2 π ) 2 + + ( 2 m + 1 m π ) 2 < 2 m ( 2 m + 2 ) 6 . \frac{2m(2m-1)}{6}<\left(\frac{2m+1}{\pi}\right)^{2}+\left(\frac{2m+1}{2\pi}% \right)^{2}+\cdots+\left(\frac{2m+1}{m\pi}\right)^{2}<\frac{2m(2m+2)}{6}.
  55. π \pi
  56. π 2 6 ( 2 m 2 m + 1 ) ( 2 m - 1 2 m + 1 ) < 1 1 2 + 1 2 2 + + 1 m 2 < π 2 6 ( 2 m 2 m + 1 ) ( 2 m + 2 2 m + 1 ) . \frac{\pi^{2}}{6}\left(\frac{2m}{2m+1}\right)\left(\frac{2m-1}{2m+1}\right)<% \frac{1}{1^{2}}+\frac{1}{2^{2}}+\cdots+\frac{1}{m^{2}}<\frac{\pi^{2}}{6}\left(% \frac{2m}{2m+1}\right)\left(\frac{2m+2}{2m+1}\right).
  57. π 2 6 \frac{\pi^{2}}{6}\,
  58. ζ ( 2 ) = k = 1 1 k 2 = lim m ( 1 1 2 + 1 2 2 + + 1 m 2 ) = π 2 6 \zeta(2)=\sum_{k=1}^{\infty}\frac{1}{k^{2}}=\lim_{m\to\infty}\left(\frac{1}{1^% {2}}+\frac{1}{2^{2}}+\cdots+\frac{1}{m^{2}}\right)=\frac{\pi^{2}}{6}
  59. 1 n \frac{1}{n}
  60. S 1 S_{1}
  61. S 1 = 1 2 + 1 3 = 5 6 S_{1}=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}
  62. S 2 S_{2}
  63. A = n = 2 1 n 2 = π 2 6 - 1 A=\sum_{n=2}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}-1
  64. S 2 = A S 1 = π 2 6 - 1 5 6 = π 2 - 6 5 = 0.77392088021 S_{2}=\frac{A}{S_{1}}=\frac{\frac{\pi^{2}}{6}-1}{\frac{5}{6}}=\frac{\pi^{2}-6}% {5}=0.77392088021\cdots

Beam_splitter.html

  1. [ E c E d ] = [ r a c t b c t a d r b d ] [ E a E b ] , \begin{bmatrix}E_{c}\\ E_{d}\end{bmatrix}=\begin{bmatrix}r_{ac}&t_{bc}\\ t_{ad}&r_{bd}\end{bmatrix}\begin{bmatrix}E_{a}\\ E_{b}\end{bmatrix},
  2. | E c | 2 + | E d | 2 = | E a | 2 + | E b | 2 . |E_{c}|^{2}+|E_{d}|^{2}=|E_{a}|^{2}+|E_{b}|^{2}.
  3. | r a c | 2 + | t a d | 2 = | r b d | 2 + | t b c | 2 = 1 |r_{ac}|^{2}+|t_{ad}|^{2}=|r_{bd}|^{2}+|t_{bc}|^{2}=1
  4. r a c t b c + t a d r b d = 0 , r_{ac}t^{\ast}_{bc}+t_{ad}r^{\ast}_{bd}=0,
  5. {}^{\ast}
  6. r a c = | r a c | e i ϕ a c r_{ac}=|r_{ac}|e^{i\phi_{ac}}
  7. | r a c | | t b c | e i ( ϕ a c - ϕ b c ) + | t a d | | r b d | e i ( ϕ a d - ϕ b d ) = 0. |r_{ac}||t_{bc}|e^{i(\phi_{ac}-\phi_{bc})}+|t_{ad}||r_{bd}|e^{i(\phi_{ad}-\phi% _{bd})}=0.
  8. | r a c | | t a d | = - | r b d | | t b c | e i ( ϕ a d - ϕ b d + ϕ b c - ϕ a c ) \frac{|r_{ac}|}{|t_{ad}|}=-\frac{|r_{bd}|}{|t_{bc}|}e^{i(\phi_{ad}-\phi_{bd}+% \phi_{bc}-\phi_{ac})}
  9. ϕ a d - ϕ b d + ϕ b c - ϕ a c = π \phi_{ad}-\phi_{bd}+\phi_{bc}-\phi_{ac}=\pi
  10. 1 - | t a d | 2 | t a d | 2 = 1 - | t b c | 2 | t b c | 2 , \frac{1-|t_{ad}|^{2}}{|t_{ad}|^{2}}=\frac{1-|t_{bc}|^{2}}{|t_{bc}|^{2}},
  11. | r a c | 2 = 1 - | t a d | 2 |r_{ac}|^{2}=1-|t_{ad}|^{2}
  12. | t a d | = | t b c | T , |t_{ad}|=|t_{bc}|\equiv T,
  13. | r a c | = | r b d | R . |r_{ac}|=|r_{bd}|\equiv R.
  14. R 2 + T 2 = 1 R^{2}+T^{2}=1
  15. [ E c E d ] = [ R e i ϕ a c T e i ϕ b c T e i ϕ a d R e i ϕ b d ] [ E a E b ] . \begin{bmatrix}E_{c}\\ E_{d}\end{bmatrix}=\begin{bmatrix}Re^{i\phi_{ac}}&Te^{i\phi_{bc}}\\ Te^{i\phi_{ad}}&Re^{i\phi_{bd}}\end{bmatrix}\begin{bmatrix}E_{a}\\ E_{b}\end{bmatrix}.
  16. a ^ 0 , a ^ 1 \hat{a}_{0},\hat{a}_{1}
  17. a ^ 2 , a ^ 3 , \hat{a}_{2},\hat{a}_{3},
  18. ( a ^ 2 a ^ 3 ) = ( t r r t ) ( a ^ 0 a ^ 1 ) . \left(\begin{matrix}\hat{a}_{2}\\ \hat{a}_{3}\end{matrix}\right)=\left(\begin{matrix}t^{\prime}&r\\ r^{\prime}&t\end{matrix}\right)\left(\begin{matrix}\hat{a}_{0}\\ \hat{a}_{1}\end{matrix}\right).
  19. [ a ^ i , a ^ j ] = δ i j [\hat{a}_{i},\hat{a}_{j}^{\dagger}]=\delta_{ij}
  20. [ a ^ i , a ^ j ] = 0. [\hat{a}_{i},\hat{a}_{j}]=0.
  21. | r | = | r | , | t | = | t | , | r | 2 + | t | 2 = 1 |r^{\prime}|=|r|,\,|t^{\prime}|=|t|,\,|r|^{2}+|t|^{2}=1
  22. r * t + r t * = r * t + r t * = 0. r^{*}t^{\prime}+r^{\prime}t^{*}=r^{*}t+r^{\prime}t^{\prime*}=0.
  23. e ± i π 2 = ± i e^{\pm i\frac{\pi}{2}}=\pm i
  24. π 2 \frac{\pi}{2}
  25. a ^ 2 = 1 2 ( a ^ 0 + i a ^ 1 ) a ^ 3 = 1 2 ( i a ^ 0 + a ^ 1 ) . \hat{a}_{2}=\frac{1}{\sqrt{2}}\left(\hat{a}_{0}+i\hat{a}_{1}\right)\,\hat{a}_{% 3}=\frac{1}{\sqrt{2}}\left(i\hat{a}_{0}+\hat{a}_{1}\right).
  26. U ^ = e i π 4 ( a ^ 0 a ^ 1 + a ^ 0 a ^ 1 ) . \hat{U}=e^{i\frac{\pi}{4}\left(\hat{a}_{0}^{\dagger}\hat{a}_{1}+\hat{a}_{0}% \hat{a}_{1}^{\dagger}\right)}.
  27. ( a ^ 2 a ^ 3 ) = U ^ ( a ^ 0 a ^ 1 ) U ^ \left(\begin{matrix}\hat{a}_{2}\\ \hat{a}_{3}\end{matrix}\right)=\hat{U}^{\dagger}\left(\begin{matrix}\hat{a}_{0% }\\ \hat{a}_{1}\end{matrix}\right)\hat{U}

Bed_load.html

  1. τ * c \tau_{*c}
  2. τ * \tau_{*}
  3. τ * = u * 2 ( s - 1 ) g d \tau_{*}=\frac{u^{2}_{*}}{(s-1)gd}
  4. u * u_{*}
  5. q s q_{s}
  6. τ * c \tau_{*c}
  7. q s q_{s}
  8. ϕ ( τ * - τ * c ) \phi(\tau_{*}-\tau_{*c})

Bell_polynomials.html

  1. B n , k ( x 1 , x 2 , , x n - k + 1 ) B_{n,k}(x_{1},x_{2},\dots,x_{n-k+1})
  2. = n ! j 1 ! j 2 ! j n - k + 1 ! ( x 1 1 ! ) j 1 ( x 2 2 ! ) j 2 ( x n - k + 1 ( n - k + 1 ) ! ) j n - k + 1 , =\sum{n!\over j_{1}!j_{2}!\cdots j_{n-k+1}!}\left({x_{1}\over 1!}\right)^{j_{1% }}\left({x_{2}\over 2!}\right)^{j_{2}}\cdots\left({x_{n-k+1}\over(n-k+1)!}% \right)^{j_{n-k+1}},
  3. j 1 + j 2 + + j n - k + 1 = k and j 1 + 2 j 2 + 3 j 3 + + ( n - k + 1 ) j n - k + 1 = n . j_{1}+j_{2}+\cdots+j_{n-k+1}=k\quad\mbox{and}~{}\quad j_{1}+2j_{2}+3j_{3}+% \cdots+(n-k+1)j_{n-k+1}=n.
  4. B n ( x 1 , , x n ) = k = 1 n B n , k ( x 1 , x 2 , , x n - k + 1 ) B_{n}(x_{1},\dots,x_{n})=\sum_{k=1}^{n}B_{n,k}(x_{1},x_{2},\dots,x_{n-k+1})
  5. B n ( x 1 , , x n ) = det [ x 1 ( n - 1 1 ) x 2 ( n - 1 2 ) x 3 ( n - 1 3 ) x 4 ( n - 1 4 ) x 5 x n - 1 x 1 ( n - 2 1 ) x 2 ( n - 2 2 ) x 3 ( n - 2 3 ) x 4 x n - 1 0 - 1 x 1 ( n - 3 1 ) x 2 ( n - 3 2 ) x 3 x n - 2 0 0 - 1 x 1 ( n - 4 1 ) x 2 x n - 3 0 0 0 - 1 x 1 x n - 4 0 0 0 0 - 1 x n - 5 0 0 0 0 0 - 1 x 1 ] . B_{n}(x_{1},\dots,x_{n})=\det\begin{bmatrix}x_{1}&{n-1\choose 1}x_{2}&{n-1% \choose 2}x_{3}&{n-1\choose 3}x_{4}&{n-1\choose 4}x_{5}&\cdots&\cdots&x_{n}\\ \\ -1&x_{1}&{n-2\choose 1}x_{2}&{n-2\choose 2}x_{3}&{n-2\choose 3}x_{4}&\cdots&% \cdots&x_{n-1}\\ \\ 0&-1&x_{1}&{n-3\choose 1}x_{2}&{n-3\choose 2}x_{3}&\cdots&\cdots&x_{n-2}\\ \\ 0&0&-1&x_{1}&{n-4\choose 1}x_{2}&\cdots&\cdots&x_{n-3}\\ \\ 0&0&0&-1&x_{1}&\cdots&\cdots&x_{n-4}\\ \\ 0&0&0&0&-1&\cdots&\cdots&x_{n-5}\\ \\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\ \\ 0&0&0&0&0&\cdots&-1&x_{1}\end{bmatrix}.
  6. B 6 , 2 ( x 1 , x 2 , x 3 , x 4 , x 5 ) = 6 x 5 x 1 + 15 x 4 x 2 + 10 x 3 2 B_{6,2}(x_{1},x_{2},x_{3},x_{4},x_{5})=6x_{5}x_{1}+15x_{4}x_{2}+10x_{3}^{2}
  7. B 6 , 3 ( x 1 , x 2 , x 3 , x 4 ) = 15 x 4 x 1 2 + 60 x 3 x 2 x 1 + 15 x 2 3 B_{6,3}(x_{1},x_{2},x_{3},x_{4})=15x_{4}x_{1}^{2}+60x_{3}x_{2}x_{1}+15x_{2}^{3}
  8. B n , k ( 1 ! , 2 ! , , ( n - k + 1 ) ! ) = ( n k ) ( n - 1 k - 1 ) ( n - k ) ! B_{n,k}(1!,2!,\dots,(n-k+1)!)={\left({{n}\atop{k}}\right)}{\left({{n-1}\atop{k% -1}}\right)}(n-k)!
  9. B n , k ( 0 ! , 1 ! , , ( n - k ) ! ) = c ( n , k ) = [ n k ] . B_{n,k}(0!,1!,\dots,(n-k)!)=c(n,k)=\left[{n\atop k}\right].
  10. B n , k ( 1 , 1 , , 1 ) = S ( n , k ) = { n k } . B_{n,k}(1,1,\dots,1)=S(n,k)=\left\{{n\atop k}\right\}.
  11. B n ( 1 , 1 , , 1 ) = k = 1 n B n , k ( 1 , 1 , , 1 ) = k = 1 n { n k } , B_{n}(1,1,\dots,1)=\sum_{k=1}^{n}B_{n,k}(1,1,\dots,1)=\sum_{k=1}^{n}\left\{{n% \atop k}\right\},
  12. T n ( x ) = k = 0 n { n k } x k T_{n}(x)=\sum_{k=0}^{n}\left\{{n\atop k}\right\}\cdot x^{k}
  13. T n ( x ) = B n ( x , x , , x ) . T_{n}(x)=B_{n}(x,x,\dots,x).
  14. ( x y ) n = j = 1 n - 1 ( n j ) x j y n - j (x\diamondsuit y)_{n}=\sum_{j=1}^{n-1}{n\choose j}x_{j}y_{n-j}
  15. x n k x_{n}^{k\diamondsuit}\,
  16. x x k factors . \displaystyle\underbrace{x\diamondsuit\cdots\diamondsuit x}_{k\ \mathrm{% factors}}.\,
  17. B n , k ( x 1 , , x n - k + 1 ) = x n k k ! . B_{n,k}(x_{1},\dots,x_{n-k+1})={x_{n}^{k\diamondsuit}\over k!}.\,
  18. B 4 , 3 ( x 1 , x 2 ) B_{4,3}(x_{1},x_{2})
  19. x = ( x 1 , x 2 , x 3 , x 4 , ) x=(x_{1}\ ,\ x_{2}\ ,\ x_{3}\ ,\ x_{4}\ ,\dots)
  20. x x = ( 0 , 2 x 1 2 , 6 x 1 x 2 , 8 x 1 x 3 + 6 x 2 2 , ) x\diamondsuit x=(0,\ 2x_{1}^{2}\ ,\ 6x_{1}x_{2}\ ,\ 8x_{1}x_{3}+6x_{2}^{2}\ ,\dots)
  21. x x x = ( 0 , 0 , 6 x 1 3 , 36 x 1 2 x 2 , ) x\diamondsuit x\diamondsuit x=(0\ ,\ 0\ ,\ 6x_{1}^{3}\ ,\ 36x_{1}^{2}x_{2}\ ,\dots)
  22. B 4 , 3 ( x 1 , x 2 ) = ( x x x ) 4 3 ! = 6 x 1 2 x 2 . B_{4,3}(x_{1},x_{2})=\frac{(x\diamondsuit x\diamondsuit x)_{4}}{3!}=6x_{1}^{2}% x_{2}.
  23. d n d x n f ( g ( x ) ) = k = 1 n f ( k ) ( g ( x ) ) B n , k ( g ( x ) , g ′′ ( x ) , , g ( n - k + 1 ) ( x ) ) . {d^{n}\over dx^{n}}f(g(x))=\sum_{k=1}^{n}f^{(k)}(g(x))B_{n,k}\left(g^{\prime}(% x),g^{\prime\prime}(x),\dots,g^{(n-k+1)}(x)\right).
  24. f ( x ) = n = 1 a n n ! x n and g ( x ) = n = 1 b n n ! x n . f(x)=\sum_{n=1}^{\infty}{a_{n}\over n!}x^{n}\qquad\mathrm{and}\qquad g(x)=\sum% _{n=1}^{\infty}{b_{n}\over n!}x^{n}.
  25. g ( f ( x ) ) = n = 1 k = 1 n b k B n , k ( a 1 , , a n - k + 1 ) n ! x n . g(f(x))=\sum_{n=1}^{\infty}{\sum_{k=1}^{n}b_{k}B_{n,k}(a_{1},\dots,a_{n-k+1})% \over n!}x^{n}.
  26. exp ( n = 1 a n n ! x n ) = n = 0 B n ( a 1 , , a n ) n ! x n . \exp\left(\sum_{n=1}^{\infty}{a_{n}\over n!}x^{n}\right)=\sum_{n=0}^{\infty}{B% _{n}(a_{1},\dots,a_{n})\over n!}x^{n}.
  27. B n ( κ 1 , , κ n ) = k = 1 n B n , k ( κ 1 , , κ n - k + 1 ) B_{n}(\kappa_{1},\dots,\kappa_{n})=\sum_{k=1}^{n}B_{n,k}(\kappa_{1},\dots,% \kappa_{n-k+1})
  28. p n ( x ) = k = 1 n B n , k ( a 1 , , a n - k + 1 ) x k . p_{n}(x)=\sum_{k=1}^{n}B_{n,k}(a_{1},\dots,a_{n-k+1})x^{k}.
  29. p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) . p_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}p_{k}(x)p_{n-k}(y).
  30. p n ( x ) p_{n}(x)
  31. h ( x ) = k = 1 a k k ! x k , h(x)=\sum_{k=1}^{\infty}{a_{k}\over k!}x^{k},
  32. h - 1 ( d d x ) p n ( x ) = n p n - 1 ( x ) . h^{-1}\left({d\over dx}\right)p_{n}(x)=np_{n-1}(x).

Benjamin_Gompertz.html

  1. N ( t ) = N ( 0 ) e - c ( e a t - 1 ) , N(t)=N(0)e^{-c(e^{at}-1)},

Beno_Gutenberg.html

  1. log E ( s ) = 11.8 + 1.5 M . \!\ \log E(s)=11.8+1.5M.
  2. E ( s ) E(s)

Berlekamp–Massey_algorithm.html

  1. S i + ν + Λ 1 S i + ν - 1 + + Λ ν - 1 S i + 1 + Λ ν S i = 0. S_{i+\nu}+\Lambda_{1}S_{i+\nu-1}+\cdots+\Lambda_{\nu-1}S_{i+1}+\Lambda_{\nu}S_% {i}=0.
  2. C ( x ) = C L x L + C L - 1 x L - 1 + + C 2 x 2 + C 1 x + 1 C(x)=C_{L}\ x^{L}+C_{L-1}\ x^{L-1}+\cdots+C_{2}\ x^{2}+C_{1}\ x+1
  3. C ( x ) = 1 + C 1 x + C 2 x 2 + + C L - 1 x L - 1 + C L x L . C(x)=1+C_{1}\ x+C_{2}\ x^{2}+\cdots+C_{L-1}\ x^{L-1}+C_{L}\ x^{L}.
  4. S n + C 1 S n - 1 + + C L S n - L = 0 S_{n}+C_{1}\ S_{n-1}+\cdots+C_{L}\ S_{n-L}=0
  5. d = S k + C 1 S k - 1 + + C L S k - L . d=S_{k}+C_{1}\ S_{k-1}+\cdots+C_{L}\ S_{k-L}.
  6. C ( x ) = C ( x ) - ( d / b ) x m B ( x ) . C(x)=C(x)\ -\ (d/b)\ x^{m}\ B(x).
  7. d = S k + C 1 S k - 1 + - ( d / b ) ( S j + B 1 S j - 1 + ) . d=S_{k}+C_{1}\ S_{k-1}+\cdots-(d/b)(S_{j}+B_{1}\ S_{j-1}+\cdots).
  8. d = d - ( d / b ) b = d - d = 0. d=d-(d/b)b=d-d=0.
  9. s 0 , s 1 , s 2 s n - 1 s_{0},s_{1},s_{2}\cdots s_{n-1}
  10. b b
  11. c c
  12. n n
  13. b 0 1 , c 0 1 b_{0}\leftarrow 1,c_{0}\leftarrow 1
  14. L 0 , m - 1 L\leftarrow 0,m\leftarrow-1
  15. N = 0 N=0
  16. N < n N<n
  17. d d
  18. s N + c 1 s N - 1 + c 2 s N - 2 + + c L s N - L s_{N}+c_{1}s_{N-1}+c_{2}s_{N-2}+\cdots+c_{L}s_{N-L}
  19. d = 0 d=0
  20. c c
  21. N - L N-L
  22. N N
  23. t t
  24. c c
  25. c N - m c N - m b 0 , c N - m + 1 c N - m + 1 b 1 , c_{N-m}\leftarrow c_{N-m}\oplus b_{0},c_{N-m+1}\leftarrow c_{N-m+1}\oplus b_{1% },\dots
  26. c n - 1 c n - 1 b n - N + m - 1 c_{n-1}\leftarrow c_{n-1}\oplus b_{n-N+m-1}
  27. \oplus
  28. L N 2 L\leq\frac{N}{2}
  29. L N + 1 - L L\leftarrow N+1-L
  30. m N m\leftarrow N
  31. b t b\leftarrow t
  32. L L
  33. m m
  34. b b
  35. L L
  36. c L s a + c L - 1 s a + 1 + c L - 2 s a + 2 + = 0 c_{L}s_{a}+c_{L-1}s_{a+1}+c_{L-2}s_{a+2}+\cdots=0
  37. a a

Bernstein_polynomial.html

  1. b ν , n ( x ) = ( n ν ) x ν ( 1 - x ) n - ν , ν = 0 , , n . b_{\nu,n}(x)={n\choose\nu}x^{\nu}\left(1-x\right)^{n-\nu},\quad\nu=0,\ldots,n.
  2. ( n ν ) {n\choose\nu}
  3. B n ( x ) = ν = 0 n β ν b ν , n ( x ) B_{n}(x)=\sum_{\nu=0}^{n}\beta_{\nu}b_{\nu,n}(x)
  4. β ν \beta_{\nu}
  5. b 0 , 0 ( x ) \displaystyle b_{0,0}(x)
  6. b ν , n ( x ) = 0 b_{\nu,n}(x)=0
  7. ν < 0 \nu<0
  8. ν > n \nu>n
  9. b ν , n ( 0 ) = δ ν , 0 b_{\nu,n}(0)=\delta_{\nu,0}
  10. b ν , n ( 1 ) = δ ν , n b_{\nu,n}(1)=\delta_{\nu,n}
  11. δ \delta
  12. b ν , n ( x ) b_{\nu,n}(x)
  13. ν \nu
  14. x = 0 x=0
  15. ν = 0 \nu=0
  16. b ν , n ( x ) b_{\nu,n}(x)
  17. ( n - ν ) \left(n-\nu\right)
  18. x = 1 x=1
  19. ν = n \nu=n
  20. b ν , n ( x ) 0 b_{\nu,n}(x)\geq 0
  21. x [ 0 , 1 ] x\in[0,\ 1]
  22. b ν , n ( 1 - x ) = b n - ν , n ( x ) b_{\nu,n}\left(1-x\right)=b_{n-\nu,n}(x)
  23. b ν , n ( x ) = n ( b ν - 1 , n - 1 ( x ) - b ν , n - 1 ( x ) ) . b^{\prime}_{\nu,n}(x)=n\left(b_{\nu-1,n-1}(x)-b_{\nu,n-1}(x)\right).
  24. n n
  25. 0 1 b ν , n ( x ) d x = 1 n + 1 ; ν = 0 , 1 n \int_{0}^{1}b_{\nu,n}(x)dx=\frac{1}{n+1};\forall\nu=0,1\dots n
  26. n 0 n\neq 0
  27. b ν , n ( x ) b_{\nu,n}(x)
  28. [ 0 , 1 ] [0,\ 1]
  29. x = ν n x=\frac{\nu}{n}
  30. ν ν n - n ( n - ν ) n - ν ( n ν ) . \nu^{\nu}n^{-n}\left(n-\nu\right)^{n-\nu}{n\choose\nu}.
  31. n n
  32. ν = 0 n b ν , n ( x ) = ν = 0 n ( n ν ) x ν ( 1 - x ) n - ν = ( x + ( 1 - x ) ) n = 1. \sum_{\nu=0}^{n}b_{\nu,n}(x)=\sum_{\nu=0}^{n}{n\choose\nu}x^{\nu}\left(1-x% \right)^{n-\nu}=\left(x+\left(1-x\right)\right)^{n}=1.
  33. ( x + y ) n (x+y)^{n}
  34. y = 1 - x y=1-x
  35. ν = 0 n ν b ν , n ( x ) = n x \sum_{\nu=0}^{n}\nu b_{\nu,n}(x)=nx
  36. ( x + y ) n (x+y)^{n}
  37. y = 1 - x y=1-x
  38. ν = 1 n ν ( ν - 1 ) b ν , n ( x ) = n ( n - 1 ) x 2 \sum_{\nu=1}^{n}\nu(\nu-1)b_{\nu,n}(x)=n(n-1)x^{2}
  39. b ν , n - 1 ( x ) = n - ν n b ν , n ( x ) + ν + 1 n b ν + 1 , n ( x ) . b_{\nu,n-1}(x)=\frac{n-\nu}{n}b_{\nu,n}(x)+\frac{\nu+1}{n}b_{\nu+1,n}(x).
  40. B n ( f ) ( x ) = ν = 0 n f ( ν n ) b ν , n ( x ) . B_{n}(f)(x)=\sum_{\nu=0}^{n}f\left(\frac{\nu}{n}\right)b_{\nu,n}(x).
  41. lim n B n ( f ) ( x ) = f ( x ) \lim_{n\to\infty}{B_{n}(f)(x)}=f(x)\,
  42. lim n sup { | f ( x ) - B n ( f ) ( x ) | : 0 x 1 } = 0. \lim_{n\to\infty}\sup\left\{\,\left|f(x)-B_{n}(f)(x)\right|\,:\,0\leq x\leq 1% \,\right\}=0.
  43. B n ( f ) ( k ) ( n ) k n k f ( k ) and f ( k ) - B n ( f ) ( k ) 0 {\left\|B_{n}(f)^{(k)}\right\|}_{\infty}\leq\frac{(n)_{k}}{n^{k}}\left\|f^{(k)% }\right\|_{\infty}\,\text{ and }\left\|f^{(k)}-B_{n}(f)^{(k)}\right\|_{\infty}\to 0
  44. ( n ) k n k = ( 1 - 0 n ) ( 1 - 1 n ) ( 1 - k - 1 n ) \frac{(n)_{k}}{n^{k}}=\left(1-\frac{0}{n}\right)\left(1-\frac{1}{n}\right)% \cdots\left(1-\frac{k-1}{n}\right)
  45. lim n P ( | K n - x | > δ ) = 0 \lim_{n\to\infty}{P\left(\left|\frac{K}{n}-x\right|>\delta\right)}=0
  46. lim n P ( | f ( K n ) - f ( x ) | > ε ) = 0 \lim_{n\to\infty}{P\left(\left|f\left(\frac{K}{n}\right)-f\left(x\right)\right% |>\varepsilon\right)}=0
  47. lim n E ( | f ( K n ) - f ( x ) | ) = 0 \lim_{n\to\infty}{E\left(\left|f\left(\frac{K}{n}\right)-f\left(x\right)\right% |\right)}=0

Berry–Esseen_theorem.html

  1. n n
  2. Y n n σ , {Y_{n}\sqrt{n}\over{\sigma}},
  3. | F n ( x ) - Φ ( x ) | C ρ σ 3 n . ( 1 ) \left|F_{n}(x)-\Phi(x)\right|\leq{C\rho\over\sigma^{3}\,\sqrt{n}}.\ \ \ \ (1)
  4. sup x | F n ( x ) - Φ ( x ) | 0.3328 ( ρ + 0.429 σ 3 ) σ 3 n , \sup_{x\in\mathbb{R}}\left|F_{n}(x)-\Phi(x)\right|\leq{0.3328(\rho+0.429\sigma% ^{3})\over\sigma^{3}\,\sqrt{n}},
  5. C 10 + 3 6 2 π 0.40973 1 2 π + 0.01079. C\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}}\approx 0.40973\approx\frac{1}{\sqrt{2\pi% }}+0.01079.
  6. σ = ( σ 1 , , σ n ) , ρ = ( ρ 1 , , ρ n ) . \vec{\sigma}=(\sigma_{1},\ldots,\sigma_{n}),\ \vec{\rho}=(\rho_{1},\ldots,\rho% _{n}).
  7. sup x | F n ( x ) - Φ ( x ) | C 1 ψ 1 , ( 2 ) \sup_{x\in\mathbb{R}}\left|F_{n}(x)-\Phi(x)\right|\leq C_{1}\cdot\psi_{1},\ \ % \ \ (2)
  8. ψ 1 = ψ 1 ( σ , ρ ) = ( i = 1 n σ i 2 ) - 1 / 2 max 1 i n ρ i σ i 2 . \psi_{1}=\psi_{1}\big(\vec{\sigma},\vec{\rho}\big)=\Big({\textstyle\sum\limits% _{i=1}^{n}\sigma_{i}^{2}}\Big)^{-1/2}\cdot\max_{1\leq i\leq n}\frac{\rho_{i}}{% \sigma_{i}^{2}}.
  9. sup x | F n ( x ) - Φ ( x ) | C 0 ψ 0 , ( 3 ) \sup_{x\in\mathbb{R}}\left|F_{n}(x)-\Phi(x)\right|\leq C_{0}\cdot\psi_{0},\ \ % \ \ (3)
  10. ψ 0 = ψ 0 ( σ , ρ ) = ( i = 1 n σ i 2 ) - 3 / 2 i = 1 n ρ i . \psi_{0}=\psi_{0}\big(\vec{\sigma},\vec{\rho}\big)=\Big({\textstyle\sum\limits% _{i=1}^{n}\sigma_{i}^{2}}\Big)^{-3/2}\cdot\sum\limits_{i=1}^{n}\rho_{i}.
  11. ψ 0 = ψ 1 = ρ 1 σ 1 3 n , \psi_{0}=\psi_{1}=\frac{\rho_{1}}{\sigma_{1}^{3}\sqrt{n}},
  12. C 0 10 + 3 6 2 π = 0.4097 . C_{0}\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}}=0.4097\ldots.

Beth_number.html

  1. \aleph
  2. \beth
  3. \aleph
  4. 0 = 0 \beth_{0}=\aleph_{0}
  5. \mathbb{N}
  6. α + 1 = 2 α , \beth_{\alpha+1}=2^{\beth_{\alpha}},
  7. α \beth_{\alpha}
  8. 0 , 1 , 2 , 3 , \beth_{0},\ \beth_{1},\ \beth_{2},\ \beth_{3},\ \dots
  9. , P ( ) , P ( P ( ) ) , P ( P ( P ( ) ) ) , . \mathbb{N},\ P(\mathbb{N}),\ P(P(\mathbb{N})),\ P(P(P(\mathbb{N}))),\ \dots.
  10. 1 \beth_{1}
  11. 𝔠 \mathfrak{c}
  12. 2 \beth_{2}
  13. λ = sup { α : α < λ } . \beth_{\lambda}=\sup\{\beth_{\alpha}:\alpha<\lambda\}.
  14. V ω + α V_{\omega+\alpha}\!
  15. α \beth_{\alpha}\!
  16. 0 \aleph_{0}
  17. 1 \aleph_{1}
  18. 1 1 . \beth_{1}\geq\aleph_{1}.
  19. α α \beth_{\alpha}\geq\aleph_{\alpha}
  20. α \alpha
  21. 1 = 1 . \beth_{1}=\aleph_{1}.
  22. α = α \beth_{\alpha}=\aleph_{\alpha}
  23. α \alpha
  24. 0 \aleph_{0}
  25. 0 \beth_{0}
  26. 1 \beth_{1}
  27. 2 \beth_{2}
  28. 2 \beth_{2}
  29. ω \beth_{\omega}
  30. α ( κ ) \beth_{\alpha}(\kappa)
  31. 0 ( κ ) = κ , \beth_{0}(\kappa)=\kappa,
  32. α + 1 ( κ ) = 2 α ( κ ) , \beth_{\alpha+1}(\kappa)=2^{\beth_{\alpha}(\kappa)},
  33. λ ( κ ) = sup { α ( κ ) : α < λ } \beth_{\lambda}(\kappa)=\sup\{\beth_{\alpha}(\kappa):\alpha<\lambda\}
  34. α = α ( 0 ) . \beth_{\alpha}=\beth_{\alpha}(\aleph_{0}).
  35. κ α ( μ ) . \kappa\leq\beth_{\alpha}(\mu).
  36. β ( α ( κ ) ) = α + β ( κ ) . \beth_{\beta}(\beth_{\alpha}(\kappa))=\beth_{\alpha+\beta}(\kappa).
  37. β ( κ ) = β ( μ ) \beth_{\beta}(\kappa)=\beth_{\beta}(\mu)

Betti_number.html

  1. H k = ker δ k / Im δ k + 1 H_{k}=\ker\delta_{k}/\mathrm{Im}\delta_{k+1}
  2. δ k s \delta_{k}s
  3. H 0 H_{0}
  4. H 1 H_{1}
  5. H 2 H_{2}
  6. 1 + x 1+x\,
  7. m - n + k . m-n+k.
  8. χ ( K ) = i = 0 ( - 1 ) i b i ( K , F ) , \chi(K)=\sum_{i=0}^{\infty}(-1)^{i}b_{i}(K,F),\,
  9. χ ( K ) \chi(K)
  10. P X × Y = P X P Y , P_{X\times Y}=P_{X}P_{Y},\,
  11. P X ( z ) = b 0 ( X ) + b 1 ( X ) z + b 2 ( X ) z 2 + , P_{X}(z)=b_{0}(X)+b_{1}(X)z+b_{2}(X)z^{2}+\cdots,\,\!
  12. b k ( X ) = b n - k ( X ) , b_{k}(X)=b_{n-k}(X),\,\!
  13. 1 + x 1+x\,
  14. ( 1 + x ) 3 = 1 + 3 x + 3 x 2 + x 3 (1+x)^{3}=1+3x+3x^{2}+x^{3}\,
  15. ( 1 + x ) n (1+x)^{n}\,
  16. 1 + x 2 + x 4 + 1+x^{2}+x^{4}+\cdots
  17. 1 1 - x 2 . \frac{1}{1-x^{2}}.
  18. a , b , c , a , b , c , , a,b,c,a,b,c,\dots,
  19. ( a + b x + c x 2 ) / ( 1 - x 3 ) (a+bx+cx^{2})/(1-x^{3})\,
  20. P S U ( n + 1 ) ( x ) = ( 1 + x 3 ) ( 1 + x 5 ) ( 1 + x 2 n + 1 ) P_{SU(n+1)}(x)=(1+x^{3})(1+x^{5})\cdots(1+x^{2n+1})
  21. P S O ( 2 n + 1 ) ( x ) = ( 1 + x 3 ) ( 1 + x 7 ) ( 1 + x 4 n - 1 ) P_{SO(2n+1)}(x)=(1+x^{3})(1+x^{7})\cdots(1+x^{4n-1})
  22. P S p ( n ) ( x ) = ( 1 + x 3 ) ( 1 + x 7 ) ( 1 + x 4 n - 1 ) P_{Sp(n)}(x)=(1+x^{3})(1+x^{7})\cdots(1+x^{4n-1})
  23. P S O ( 2 n ) ( x ) = ( 1 + x 2 n - 1 ) ( 1 + x 3 ) ( 1 + x 7 ) ( 1 + x 4 n - 5 ) P_{SO(2n)}(x)=(1+x^{2n-1})(1+x^{3})(1+x^{7})\cdots(1+x^{4n-5})
  24. P G 2 ( x ) = ( 1 + x 3 ) ( 1 + x 11 ) P_{G_{2}}(x)=(1+x^{3})(1+x^{11})
  25. P F 4 ( x ) = ( 1 + x 3 ) ( 1 + x 11 ) ( 1 + x 15 ) ( 1 + x 23 ) P_{F_{4}}(x)=(1+x^{3})(1+x^{11})(1+x^{15})(1+x^{23})
  26. P E 6 ( x ) = ( 1 + x 3 ) ( 1 + x 9 ) ( 1 + x 11 ) ( 1 + x 15 ) ( 1 + x 17 ) ( 1 + x 23 ) P_{E_{6}}(x)=(1+x^{3})(1+x^{9})(1+x^{11})(1+x^{15})(1+x^{17})(1+x^{23})
  27. P E 7 ( x ) = ( 1 + x 3 ) ( 1 + x 11 ) ( 1 + x 15 ) ( 1 + x 19 ) ( 1 + x 23 ) ( 1 + x 27 ) ( 1 + x 35 ) P_{E_{7}}(x)=(1+x^{3})(1+x^{11})(1+x^{15})(1+x^{19})(1+x^{23})(1+x^{27})(1+x^{% 35})
  28. P E 8 ( x ) = ( 1 + x 3 ) ( 1 + x 15 ) ( 1 + x 23 ) ( 1 + x 27 ) ( 1 + x 35 ) ( 1 + x 39 ) ( 1 + x 47 ) ( 1 + x 59 ) P_{E_{8}}(x)=(1+x^{3})(1+x^{15})(1+x^{23})(1+x^{27})(1+x^{35})(1+x^{39})(1+x^{% 47})(1+x^{59})
  29. X X
  30. N i N_{i}
  31. b i ( X ) - b i - 1 ( X ) + N i - N i - 1 + . b_{i}(X)-b_{i-1}(X)+\cdots\leq N_{i}-N_{i-1}+\cdots.

Bhāskara_II.html

  1. 61 x 2 + 1 = y 2 61x^{2}+1=y^{2}
  2. sin y - sin y ( y - y ) cos y \sin y^{\prime}-\sin y\approx(y^{\prime}-y)\cos y
  3. y y^{\prime}
  4. y y
  5. d d y sin y = cos y \frac{d}{dy}\sin y=\cos y
  6. sin ( a + b ) \sin\left(a+b\right)
  7. sin ( a - b ) \sin\left(a-b\right)
  8. f ( a ) = f ( b ) = 0 f\left(a\right)=f\left(b\right)=0
  9. f ( x ) = 0 f^{\prime}\left(x\right)=0
  10. x \ x
  11. a < x < b \ a<x<b
  12. x y x\approx y
  13. sin ( y ) - sin ( x ) ( y - x ) cos ( y ) \sin(y)-\sin(x)\approx(y-x)\cos(y)

Bicategory.html

  1. * : 𝐁 ( b , c ) × 𝐁 ( a , b ) 𝐁 ( a , c ) *:\mathbf{B}(b,c)\times\mathbf{B}(a,b)\to\mathbf{B}(a,c)
  2. h * ( g * f ) h*(g*f)
  3. ( h * g ) * f (h*g)*f

Bicommutant.html

  1. S ′′ S^{\prime\prime}
  2. M = M ′′ M=M^{\prime\prime}
  3. M ′′ M^{\prime\prime}
  4. S ′′′ = ( S ′′ ) S S^{\prime\prime\prime}=(S^{\prime\prime})^{\prime}\subseteq S^{\prime}
  5. S ( S ) ′′ = S ′′′ S^{\prime}\subseteq(S^{\prime})^{\prime\prime}=S^{\prime\prime\prime}
  6. S = S ′′′ S^{\prime}=S^{\prime\prime\prime}
  7. S = S ′′′ = S ′′′′′ = = S 2 n - 1 = S^{\prime}=S^{\prime\prime\prime}=S^{\prime\prime\prime\prime\prime}=\ldots=S^% {2n-1}=\ldots
  8. S S ′′ = S ′′′′ = S ′′′′′′ = = S 2 n = S\subseteq S^{\prime\prime}=S^{\prime\prime\prime\prime}=S^{\prime\prime\prime% \prime\prime\prime}=\ldots=S^{2n}=\ldots
  9. ( S 1 S 2 ) = S 1 S 2 . (S_{1}\cup S_{2})^{\prime}=S_{1}^{\prime}\cap S_{2}^{\prime}.
  10. S 1 = S 1 ′′ S_{1}=S_{1}^{\prime\prime}\,
  11. S 2 = S 2 ′′ S_{2}=S_{2}^{\prime\prime}\,
  12. ( S 1 S 2 ) ′′ = ( S 1 ′′ S 2 ′′ ) = ( S 1 S 2 ) . (S_{1}^{\prime}\cup S_{2}^{\prime})^{\prime\prime}=(S_{1}^{\prime\prime}\cap S% _{2}^{\prime\prime})^{\prime}=(S_{1}\cap S_{2})^{\prime}.