wpmath0000016_12

Quotient_type.html

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

Radiation_law_for_human_mobility.html

  1. n j o b s n_{jobs}
  2. n n
  3. z z
  4. p ( z ) p(z)
  5. n / n j o b s n/n_{jobs}
  6. p ( z ) p(z)
  7. z z
  8. < T i j T i m i n j ( m i + s i j ) ( m i + n j + s i j ) . <T_{ij}>=T_{i}\frac{m_{i}n_{j}}{(m_{i}+s_{ij})(m_{i}+n_{j}+s_{ij})}.
  9. T i T_{i}
  10. i i
  11. m i m_{i}
  12. n j n_{j}
  13. i i
  14. j j
  15. s i j s_{ij}
  16. i i
  17. j j

Radiocarbon_dating_considerations.html

  1. δ 13 C = ( ( C 13 C 12 ) sample ( C 13 C 12 ) PDB - 1 ) × 1000 o / o o \mathrm{\delta^{13}C}=\Biggl(\mathrm{\frac{\bigl(\frac{{}^{13}C}{{}^{12}C}% \bigr)_{sample}}{\bigl(\frac{{}^{13}C}{{}^{12}C}\bigr)_{PDB}}}-1\Biggr)\times 1% 000\ ^{o}\!/\!_{oo}

Rainbow_coloring.html

  1. G G
  2. G G
  3. rc ( G ) \,\text{rc}(G)
  4. G G
  5. G G
  6. src ( G ) \,\text{src}(G)
  7. G G
  8. diam ( G ) \,\text{diam}(G)
  9. diam ( G ) \,\text{diam}(G)
  10. G G
  11. m m
  12. m m
  13. G G
  14. diam ( G ) rc ( G ) src ( G ) m \,\text{diam}(G)\leq\,\text{rc}(G)\leq\,\text{src}(G)\leq m
  15. rc ( G ) = src ( G ) = 1 \,\text{rc}(G)=\,\text{src}(G)=1
  16. G G
  17. rc ( G ) = src ( G ) = m \,\text{rc}(G)=\,\text{src}(G)=m
  18. G G
  19. rc ( C n ) = src ( C n ) = n / 2 \,\text{rc}(C_{n})=\,\text{src}(C_{n})=\lceil n/2\rceil
  20. n 4 n\geq 4
  21. C n C_{n}
  22. rc ( W n ) = 3 \,\text{rc}(W_{n})=3
  23. n 7 n\geq 7
  24. src ( W n ) = n / 3 \,\text{src}(W_{n})=\lceil n/3\rceil
  25. n 3 n\geq 3
  26. W n W_{n}
  27. rc ( G ) = 2 \,\text{rc}(G)=2
  28. G G
  29. rc ( G ) = 2 \,\text{rc}(G)=2
  30. src ( G ) = 2 \,\text{src}(G)=2
  31. src ( G ) = 2 \,\text{src}(G)=2
  32. G G
  33. G G
  34. G G
  35. rvc ( G ) \,\text{rvc}(G)
  36. G G
  37. n n
  38. T T
  39. T T
  40. k k
  41. 2 k n 2\leq k\leq n
  42. G G
  43. k k
  44. S S
  45. k k
  46. G G
  47. G G
  48. S S
  49. k k
  50. rx k ( G ) \,\text{rx}_{k}(G)
  51. G G
  52. k k
  53. G G
  54. k k
  55. rx k ( G ) \,\text{rx}_{k}(G)
  56. k k
  57. rx 2 ( G ) \,\text{rx}_{2}(G)
  58. G G

Ramanujan–Sato_series.html

  1. 1 π = 2 2 99 2 k = 0 ( 4 k ) ! k ! 4 26390 k + 1103 396 4 k \frac{1}{\pi}=\frac{2\sqrt{2}}{99^{2}}\sum_{k=0}^{\infty}\frac{(4k)!}{k!^{4}}% \frac{26390k+1103}{396^{4k}}
  2. 1 π = k = 0 s ( k ) A k + B C k \frac{1}{\pi}=\sum_{k=0}^{\infty}s(k)\frac{Ak+B}{C^{k}}
  3. s ( k ) s(k)
  4. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  5. Γ 0 ( n ) \Gamma_{0}(n)
  6. q = e 2 π i τ q=e^{2\pi i\tau}
  7. j ( τ ) = ( E 4 ( τ ) η 8 ( τ ) ) 3 = 1 q + 744 + 196884 q + 21493760 q 2 + j * ( τ ) = 432 j ( τ ) + j ( τ ) - 1728 j ( τ ) - j ( τ ) - 1728 = 1 q - 120 + 10260 q - 901120 q 2 + \begin{aligned}\displaystyle j(\tau)&\displaystyle=\Big(\tfrac{E_{4}(\tau)}{% \eta^{8}(\tau)}\Big)^{3}=\tfrac{1}{q}+744+196884q+21493760q^{2}+\dots\\ \displaystyle j^{*}(\tau)&\displaystyle=432\,\frac{\sqrt{j(\tau)}+\sqrt{j(\tau% )-1728}}{\sqrt{j(\tau)}-\sqrt{j(\tau)-1728}}=\tfrac{1}{q}-120+10260q-901120q^{% 2}+\dots\end{aligned}
  8. 196883 196883
  9. s 1 A ( k ) = ( 2 k k ) ( 3 k k ) ( 6 k 3 k ) = 1 , 120 , 83160 , 81681600 , s_{1A}(k)={\textstyle\left({{2k}\atop{k}}\right)}{\textstyle\left({{3k}\atop{k% }}\right)}{\textstyle\left({{6k}\atop{3k}}\right)}=1,120,83160,81681600,\dots
  10. s 1 B ( k ) = j = 0 k ( 2 j j ) ( 3 j j ) ( 6 j 3 j ) ( k + j k - j ) ( - 432 ) k - j = 1 , - 312 , 114264 , - 44196288 , s_{1B}(k)=\sum_{j=0}^{k}{\textstyle\left({{2j}\atop{j}}\right)}{\textstyle% \left({{3j}\atop{j}}\right)}{\textstyle\left({{6j}\atop{3j}}\right)}{% \textstyle\left({{k+j}\atop{k-j}}\right)}(-432)^{k-j}=1,-312,114264,-44196288,\dots
  11. k = 0 s 1 A ( k ) 1 ( j ( τ ) ) k + 1 / 2 = ± k = 0 s 1 B ( k ) 1 ( j * ( τ ) ) k + 1 / 2 \sum_{k=0}^{\infty}s_{1A}(k)\,\frac{1}{(j(\tau))^{k+1/2}}=\pm\sum_{k=0}^{% \infty}s_{1B}(k)\,\frac{1}{(j^{*}(\tau))^{k+1/2}}
  12. 1 π = 12 s y m b o l i k = 0 s 1 A ( k ) 163 3344418 k + 13591409 ( - 640320 3 ) k + 1 / 2 , j ( 1 + - 163 2 ) = - 640320 3 \frac{1}{\pi}=12\,symbol{i}\,\sum_{k=0}^{\infty}s_{1A}(k)\,\frac{163\cdot 3344% 418k+13591409}{(-640320^{3})^{k+1/2}},\quad j\Big(\tfrac{1+\sqrt{-163}}{2}\Big% )=-640320^{3}
  13. 1 π = 24 s y m b o l i k = 0 s 1 B ( k ) - 3669 + 320 645 ( k + 1 2 ) ( - 432 U 645 3 ) k + 1 / 2 , j * ( 1 + - 43 2 ) = - 432 ( 127 + 5 645 2 ) 3 = - 432 U 645 3 \frac{1}{\pi}=24\,symbol{i}\,\sum_{k=0}^{\infty}s_{1B}(k)\,\frac{-3669+320% \sqrt{645}\,(k+\tfrac{1}{2})}{\big(-432\,U_{645}^{3}\big)^{k+1/2}},\quad j^{*}% \Big(\tfrac{1+\sqrt{-43}}{2}\Big)=-432\Big(\tfrac{127+5\sqrt{645}}{2}\Big)^{3}% =-432\,U_{645}^{3}
  14. U n U_{n}
  15. j 2 A ( τ ) = ( ( 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 + j 2 B ( τ ) = ( η ( τ ) η ( 2 τ ) ) 24 = 1 q - 24 + 276 q - 2048 q 2 + 11202 q 3 - \begin{aligned}\displaystyle j_{2A}(\tau)&\displaystyle=\Big(\big(\tfrac{\eta(% \tau)}{\eta(2\tau)}\big)^{12}+2^{6}\big(\tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{% 12}\Big)^{2}=\tfrac{1}{q}+104+4372q+96256q^{2}+1240002q^{3}+\cdots\\ \displaystyle j_{2B}(\tau)&\displaystyle=\big(\tfrac{\eta(\tau)}{\eta(2\tau)}% \big)^{24}=\tfrac{1}{q}-24+276q-2048q^{2}+11202q^{3}-\cdots\end{aligned}
  16. 4371 4371
  17. s 2 A ( k ) = ( 2 k k ) ( 2 k k ) ( 4 k 2 k ) = 1 , 24 , 2520 , 369600 , 63063000 , s_{2A}(k)={\textstyle\left({{2k}\atop{k}}\right)}{\textstyle\left({{2k}\atop{k% }}\right)}{\textstyle\left({{4k}\atop{2k}}\right)}=1,24,2520,369600,63063000,\dots
  18. s 2 B ( k ) = j = 0 k ( 2 j j ) ( 2 j j ) ( 4 j 2 j ) ( k + j k - j ) ( - 64 ) k - j = 1 , - 40 , 2008 , - 109120 , 6173656 , s_{2B}(k)=\sum_{j=0}^{k}{\textstyle\left({{2j}\atop{j}}\right)}{\textstyle% \left({{2j}\atop{j}}\right)}{\textstyle\left({{4j}\atop{2j}}\right)}{% \textstyle\left({{k+j}\atop{k-j}}\right)}(-64)^{k-j}=1,-40,2008,-109120,617365% 6,\dots
  19. k = 0 s 2 A ( k ) 1 ( j 2 A ( τ ) ) k + 1 / 2 = ± k = 0 s 2 B ( k ) 1 ( j 2 B ( τ ) ) k + 1 / 2 \sum_{k=0}^{\infty}s_{2A}(k)\,\frac{1}{(j_{2A}(\tau))^{k+1/2}}=\pm\sum_{k=0}^{% \infty}s_{2B}(k)\,\frac{1}{(j_{2B}(\tau))^{k+1/2}}
  20. 1 π = 32 2 k = 0 s 2 A ( k ) 58 455 k + 1103 ( 396 4 ) k + 1 / 2 , j 2 A ( 1 2 - 58 ) = 396 4 \frac{1}{\pi}=32\sqrt{2}\,\sum_{k=0}^{\infty}s_{2A}(k)\,\frac{58\cdot 455k+110% 3}{(396^{4})^{k+1/2}},\quad j_{2A}\Big(\tfrac{1}{2}\sqrt{-58}\Big)=396^{4}
  21. 1 π = 16 2 k = 0 s 2 B ( k ) - 24184 + 9801 29 ( k + 1 2 ) ( 64 U 29 12 ) k + 1 / 2 , j 2 B ( 1 2 - 58 ) = 64 ( 5 + 29 2 ) 12 = 64 U 29 12 \frac{1}{\pi}=16\sqrt{2}\,\sum_{k=0}^{\infty}s_{2B}(k)\,\frac{-24184+9801\sqrt% {29}\,(k+\tfrac{1}{2})}{(64\,U_{29}^{12})^{k+1/2}},\quad j_{2B}\Big(\tfrac{1}{% 2}\sqrt{-58}\Big)=64\Big(\tfrac{5+\sqrt{29}}{2}\Big)^{12}=64\,U_{29}^{12}
  22. j 3 A ( τ ) = ( ( η ( τ ) η ( 3 τ ) ) 6 + 3 3 ( η ( 3 τ ) η ( τ ) ) 6 ) 2 = 1 q + 42 + 783 q + 8672 q 2 + 65367 q 3 + j 3 B ( τ ) = ( η ( τ ) η ( 3 τ ) ) 12 = 1 q - 12 + 54 q - 76 q 2 - 243 q 3 + 1188 q 4 + \begin{aligned}\displaystyle j_{3A}(\tau)&\displaystyle=\Big(\big(\tfrac{\eta(% \tau)}{\eta(3\tau)}\big)^{6}+3^{3}\big(\tfrac{\eta(3\tau)}{\eta(\tau)}\big)^{6% }\Big)^{2}=\tfrac{1}{q}+42+783q+8672q^{2}+65367q^{3}+\dots\\ \displaystyle j_{3B}(\tau)&\displaystyle=\big(\tfrac{\eta(\tau)}{\eta(3\tau)}% \big)^{12}=\tfrac{1}{q}-12+54q-76q^{2}-243q^{3}+1188q^{4}+\dots\\ \end{aligned}
  23. 782 782
  24. s 3 A ( k ) = ( 2 k k ) ( 2 k k ) ( 3 k k ) = 1 , 12 , 540 , 33600 , 2425500 , s_{3A}(k)={\textstyle\left({{2k}\atop{k}}\right)}{\textstyle\left({{2k}\atop{k% }}\right)}{\textstyle\left({{3k}\atop{k}}\right)}=1,12,540,33600,2425500,\dots
  25. s 3 B ( k ) = j = 0 k ( 2 j j ) ( 2 j j ) ( 3 j j ) ( k + j k - j ) ( - 27 ) k - j = 1 , - 15 , 297 , - 6495 , 149481 , s_{3B}(k)=\sum_{j=0}^{k}{\textstyle\left({{2j}\atop{j}}\right)}{\textstyle% \left({{2j}\atop{j}}\right)}{\textstyle\left({{3j}\atop{j}}\right)}{\textstyle% \left({{k+j}\atop{k-j}}\right)}(-27)^{k-j}=1,-15,297,-6495,149481,\dots
  26. 1 π = 2 s y m b o l i k = 0 s 3 A ( k ) 267 53 k + 827 ( - 300 3 ) k + 1 / 2 , j 3 A ( 3 + - 267 6 ) = - 300 3 \frac{1}{\pi}=2\,symbol{i}\,\sum_{k=0}^{\infty}s_{3A}(k)\,\frac{267\cdot 53k+8% 27}{(-300^{3})^{k+1/2}},\quad j_{3A}\Big(\tfrac{3+\sqrt{-267}}{6}\Big)=-300^{3}
  27. 1 π = s y m b o l i k = 0 s 3 B ( k ) 12497 - 3000 89 ( k + 1 2 ) ( - 27 U 89 2 ) k + 1 / 2 , j 3 B ( 3 + - 267 6 ) = - 27 ( 500 + 53 89 ) 2 = - 27 U 89 2 \frac{1}{\pi}=symbol{i}\,\sum_{k=0}^{\infty}s_{3B}(k)\,\frac{12497-3000\sqrt{8% 9}\,(k+\tfrac{1}{2})}{(-27\,U_{89}^{2})^{k+1/2}},\quad j_{3B}\Big(\tfrac{3+% \sqrt{-267}}{6}\Big)=-27\,\big(500+53\sqrt{89}\big)^{2}=-27\,U_{89}^{2}
  28. j 4 A ( τ ) = ( ( η ( τ ) η ( 4 τ ) ) 4 + 4 2 ( η ( 4 τ ) η ( τ ) ) 4 ) 2 = ( η 2 ( 2 τ ) η ( τ ) η ( 4 τ ) ) 24 = 1 q + 24 + 276 q + 2048 q 2 + 11202 q 3 + j 4 C ( τ ) = ( η ( τ ) η ( 4 τ ) ) 8 = 1 q - 8 + 20 q - 62 q 3 + 216 q 5 - 641 q 7 + \begin{aligned}\displaystyle j_{4A}(\tau)&\displaystyle=\Big(\big(\tfrac{\eta(% \tau)}{\eta(4\tau)}\big)^{4}+4^{2}\big(\tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4% }\Big)^{2}=\Big(\tfrac{\eta^{2}(2\tau)}{\eta(\tau)\,\eta(4\tau)}\Big)^{24}=% \tfrac{1}{q}+24+276q+2048q^{2}+11202q^{3}+\dots\\ \displaystyle j_{4C}(\tau)&\displaystyle=\big(\tfrac{\eta(\tau)}{\eta(4\tau)}% \big)^{8}=\tfrac{1}{q}-8+20q-62q^{3}+216q^{5}-641q^{7}+\dots\\ \end{aligned}
  29. 𝔣 ( τ ) \mathfrak{f}(\tau)
  30. s 4 A ( k ) = ( 2 k k ) 3 = 1 , 8 , 216 , 8000 , 343000 , s_{4A}(k)={\textstyle\left({{2k}\atop{k}}\right)}^{3}=1,8,216,8000,343000,\dots
  31. s 4 C ( k ) = j = 0 k ( 2 j j ) 3 ( k + j k - j ) ( - 16 ) k - j = ( - 1 ) k j = 0 k ( 2 j j ) 2 ( 2 k - 2 j k - j ) 2 = 1 , - 8 , 88 , - 1088 , 14296 , s_{4C}(k)=\sum_{j=0}^{k}{\textstyle\left({{2j}\atop{j}}\right)}^{3}{\textstyle% \left({{k+j}\atop{k-j}}\right)}(-16)^{k-j}=(-1)^{k}\sum_{j=0}^{k}{\textstyle% \left({{2j}\atop{j}}\right)}^{2}{\textstyle\left({{2k-2j}\atop{k-j}}\right)}^{% 2}=1,-8,88,-1088,14296,\dots
  32. 1 π = 8 s y m b o l i k = 0 s 4 A ( k ) 6 k + 1 ( - 2 9 ) k + 1 / 2 , j 4 A ( 1 + - 4 2 ) = - 2 9 \frac{1}{\pi}=8\,symbol{i}\,\sum_{k=0}^{\infty}s_{4A}(k)\,\frac{6k+1}{(-2^{9})% ^{k+1/2}},\quad j_{4A}\Big(\tfrac{1+\sqrt{-4}}{2}\Big)=-2^{9}
  33. 1 π = 16 s y m b o l i k = 0 s 4 C ( k ) 1 - 2 2 ( k + 1 2 ) ( - 16 U 2 4 ) k + 1 / 2 , j 4 C ( 1 + - 4 2 ) = - 16 ( 1 + 2 ) 4 = - 16 U 2 4 \frac{1}{\pi}=16\,symbol{i}\,\sum_{k=0}^{\infty}s_{4C}(k)\,\frac{1-2\sqrt{2}\,% (k+\tfrac{1}{2})}{(-16\,U_{2}^{4})^{k+1/2}},\quad j_{4C}\Big(\tfrac{1+\sqrt{-4% }}{2}\Big)=-16\,\big(1+\sqrt{2}\big)^{4}=-16\,U_{2}^{4}
  34. j 5 A ( τ ) = ( η ( τ ) η ( 5 τ ) ) 6 + 5 3 ( η ( 5 τ ) η ( τ ) ) 6 + 22 = 1 q + 16 + 134 q + 760 q 2 + 3345 q 3 + j 5 B ( τ ) = ( η ( τ ) η ( 5 τ ) ) 6 = 1 q - 6 + 9 q + 10 q 2 - 30 q 3 + 6 q 4 + \begin{aligned}\displaystyle j_{5A}(\tau)&\displaystyle=\big(\tfrac{\eta(\tau)% }{\eta(5\tau)}\big)^{6}+5^{3}\big(\tfrac{\eta(5\tau)}{\eta(\tau)}\big)^{6}+22=% \tfrac{1}{q}+16+134q+760q^{2}+3345q^{3}+\dots\\ \displaystyle j_{5B}(\tau)&\displaystyle=\big(\tfrac{\eta(\tau)}{\eta(5\tau)}% \big)^{6}=\tfrac{1}{q}-6+9q+10q^{2}-30q^{3}+6q^{4}+\dots\end{aligned}
  35. s 5 A ( k ) = ( 2 k k ) j = 0 k ( k j ) 2 ( k + j j ) = 1 , 6 , 114 , 2940 , 87570 , s_{5A}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{\textstyle% \left({{k}\atop{j}}\right)}^{2}{\textstyle\left({{k+j}\atop{j}}\right)}=1,6,11% 4,2940,87570,\dots
  36. s 5 B ( k ) = j = 0 k ( - 1 ) j + k ( k j ) 3 ( 4 k - 5 j 3 k ) = 1 , - 5 , 35 , - 275 , 2275 , - 19255 , s_{5B}(k)=\sum_{j=0}^{k}(-1)^{j+k}{\textstyle\left({{k}\atop{j}}\right)}^{3}{% \textstyle\left({{4k-5j}\atop{3k}}\right)}=1,-5,35,-275,2275,-19255,\dots
  37. 1 π = 5 9 s y m b o l i k = 0 s 5 A ( k ) 682 k + 71 ( - 15228 ) k + 1 / 2 , j 5 A ( 5 + - 5 ( 47 ) 10 ) = - 15228 = - ( 18 47 ) 2 \frac{1}{\pi}=\frac{5}{9}\,symbol{i}\,\sum_{k=0}^{\infty}s_{5A}(k)\,\frac{682k% +71}{(-15228)^{k+1/2}},\quad j_{5A}\Big(\tfrac{5+\sqrt{-5(47)}}{10}\Big)=-1522% 8=-(18\sqrt{47})^{2}
  38. 1 π = 6 5 s y m b o l i k = 0 s 5 B ( k ) 25 5 - 141 ( k + 1 2 ) ( - 5 5 U 5 15 ) k + 1 / 2 , j 5 B ( 5 + - 5 ( 47 ) 10 ) = - 5 5 ( 1 + 5 2 ) 15 = - 5 5 U 5 15 \frac{1}{\pi}=\frac{6}{\sqrt{5}}\,symbol{i}\,\sum_{k=0}^{\infty}s_{5B}(k)\,% \frac{25\sqrt{5}-141(k+\tfrac{1}{2})}{(-5\sqrt{5}\,U_{5}^{15})^{k+1/2}},\quad j% _{5B}\Big(\tfrac{5+\sqrt{-5(47)}}{10}\Big)=-5\sqrt{5}\,\big(\tfrac{1+\sqrt{5}}% {2}\big)^{15}=-5\sqrt{5}\,U_{5}^{15}
  39. ζ ( 3 ) \zeta(3)
  40. j 6 A ( τ ) = j 6 B ( τ ) + 1 j 6 B ( τ ) + 2 = j 6 C ( τ ) + 64 j 6 C ( τ ) + 20 = j 6 D ( τ ) + 81 j 6 D ( τ ) + 18 = 1 q + 14 + 79 q + 352 q 2 + \begin{aligned}\displaystyle j_{6A}(\tau)&\displaystyle=j_{6B}(\tau)+\tfrac{1}% {j_{6B}(\tau)}+2=j_{6C}(\tau)+\tfrac{64}{j_{6C}(\tau)}+20=j_{6D}(\tau)+\tfrac{% 81}{j_{6D}(\tau)}+18=\tfrac{1}{q}+14+79q+352q^{2}+\dots\end{aligned}
  41. j 6 B ( τ ) = ( η ( 2 τ ) η ( 3 τ ) η ( τ ) η ( 6 τ ) ) 12 = 1 q + 12 + 78 q + 364 q 2 + 1365 q 3 + \begin{aligned}\displaystyle j_{6B}(\tau)&\displaystyle=\Big(\tfrac{\eta(2\tau% )\eta(3\tau)}{\eta(\tau)\eta(6\tau)}\Big)^{12}=\tfrac{1}{q}+12+78q+364q^{2}+13% 65q^{3}+\dots\end{aligned}
  42. j 6 C ( τ ) = ( η ( τ ) η ( 3 τ ) η ( 2 τ ) η ( 6 τ ) ) 6 = 1 q - 6 + 15 q - 32 q 2 + 87 q 3 - 192 q 4 + \begin{aligned}\displaystyle j_{6C}(\tau)&\displaystyle=\Big(\tfrac{\eta(\tau)% \eta(3\tau)}{\eta(2\tau)\eta(6\tau)}\Big)^{6}=\tfrac{1}{q}-6+15q-32q^{2}+87q^{% 3}-192q^{4}+\dots\end{aligned}
  43. j 6 D ( τ ) = ( η ( τ ) η ( 2 τ ) η ( 3 τ ) η ( 6 τ ) ) 4 = 1 q - 4 - 2 q + 28 q 2 - 27 q 3 - 52 q 4 + \begin{aligned}\displaystyle j_{6D}(\tau)&\displaystyle=\Big(\tfrac{\eta(\tau)% \eta(2\tau)}{\eta(3\tau)\eta(6\tau)}\Big)^{4}=\tfrac{1}{q}-4-2q+28q^{2}-27q^{3% }-52q^{4}+\dots\end{aligned}
  44. j 6 E ( τ ) = ( η ( 2 τ ) η 3 ( 3 τ ) η ( τ ) η 3 ( 6 τ ) ) 3 = 1 q + 3 + 6 q + 4 q 2 - 3 q 3 - 12 q 4 + \begin{aligned}\displaystyle j_{6E}(\tau)&\displaystyle=\Big(\tfrac{\eta(2\tau% )\eta^{3}(3\tau)}{\eta(\tau)\eta^{3}(6\tau)}\Big)^{3}=\tfrac{1}{q}+3+6q+4q^{2}% -3q^{3}-12q^{4}+\dots\end{aligned}
  45. T 6 A - T 6 B - T 6 C - T 6 D + 2 T 6 E = 0 T_{6A}-T_{6B}-T_{6C}-T_{6D}+2T_{6E}=0
  46. j 6 A - j 6 B - j 6 C - j 6 D + 2 j 6 E = 18 j_{6A}-j_{6B}-j_{6C}-j_{6D}+2j_{6E}=18
  47. c ( k ) = ( 2 k k ) c(k)={\textstyle\left({{2k}\atop{k}}\right)}
  48. σ 1 ( k ) = ( 2 k k ) j = 0 k ( k j ) 2 ( 2 j j ) = 1 , 6 , 90 , 1860 , 44730 , \sigma_{1}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{\textstyle% \left({{k}\atop{j}}\right)}^{2}{\textstyle\left({{2j}\atop{j}}\right)}=1,6,90,% 1860,44730,\dots
  49. σ 2 ( k ) = ( 2 k k ) j = 0 k ( k j ) ( - 8 ) k - j m = 0 j ( j m ) 3 = 1 , - 12 , 252 , - 6240 , 167580 , - 4726512 , \sigma_{2}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{\textstyle% \left({{k}\atop{j}}\right)}(-8)^{k-j}\sum_{m=0}^{j}{\textstyle\left({{j}\atop{% m}}\right)}^{3}=1,-12,252,-6240,167580,-4726512,\dots
  50. σ 3 ( k ) = ( 2 k k ) j = 0 k ( k j ) 3 = 1 , 4 , 60 , 1120 , 24220 , \sigma_{3}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{\textstyle% \left({{k}\atop{j}}\right)}^{3}=1,4,60,1120,24220,\dots
  51. s 6 B ( k ) = j = 0 k ( k j ) 2 ( k + j j ) 2 = 1 , 5 , 73 , 1445 , 33001 , s_{6B}(k)=\sum_{j=0}^{k}{\textstyle\left({{k}\atop{j}}\right)}^{2}{\textstyle% \left({{k+j}\atop{j}}\right)}^{2}=1,5,73,1445,33001,\dots
  52. s 6 C ( k ) = ( - 1 ) k j = 0 k ( k j ) 2 ( 2 ( k - j ) k - j ) ( 2 j j ) = 1 , - 4 , 28 , - 256 , 2716 , s_{6C}(k)=(-1)^{k}\sum_{j=0}^{k}{\textstyle\left({{k}\atop{j}}\right)}^{2}{% \textstyle\left({{2(k-j)}\atop{k-j}}\right)}{\textstyle\left({{2j}\atop{j}}% \right)}=1,-4,28,-256,2716,\dots
  53. s 6 D ( k ) = j = 0 k ( - 1 ) k - j 3 k - 3 j ( 3 j ) ! j ! 3 ( k 3 j ) ( k + j j ) = 1 , - 3 , 9 , - 3 , - 279 , 2997 , s_{6D}(k)=\sum_{j=0}^{k}(-1)^{k-j}\,3^{k-3j}\,\tfrac{(3j)!}{j!^{3}}{\textstyle% \left({{k}\atop{3j}}\right)}{\textstyle\left({{k+j}\atop{j}}\right)}=1,-3,9,-3% ,-279,2997,\dots
  54. P = Q = R P=Q=R
  55. P = k = 0 σ 1 ( k ) 1 ( j 6 A ( τ ) ) k + 1 / 2 = ± k = 0 s 6 B ( k ) 1 ( j 6 B ( τ ) ) k + 1 / 2 Q = k = 0 σ 2 ( k ) 1 ( j 6 A ( τ ) - 36 ) k + 1 / 2 = ± k = 0 s 6 C ( k ) 1 ( j 6 C ( τ ) ) k + 1 / 2 R = k = 0 σ 3 ( k ) 1 ( j 6 A ( τ ) - 4 ) k + 1 / 2 = ± k = 0 s 6 D ( k ) 1 ( j 6 D ( τ ) ) k + 1 / 2 \begin{aligned}\displaystyle P&\displaystyle=\sum_{k=0}^{\infty}\sigma_{1}(k)% \,\frac{1}{(j_{6A}(\tau))^{k+1/2}}=\pm\sum_{k=0}^{\infty}s_{6B}(k)\,\frac{1}{(% j_{6B}(\tau))^{k+1/2}}\\ \displaystyle Q&\displaystyle=\sum_{k=0}^{\infty}\sigma_{2}(k)\,\frac{1}{(j_{6% A}(\tau)-36)^{k+1/2}}=\pm\sum_{k=0}^{\infty}s_{6C}(k)\,\frac{1}{(j_{6C}(\tau))% ^{k+1/2}}\\ \displaystyle R&\displaystyle=\sum_{k=0}^{\infty}\sigma_{3}(k)\,\frac{1}{(j_{6% A}(\tau)-4)^{k+1/2}}=\pm\sum_{k=0}^{\infty}s_{6D}(k)\,\frac{1}{(j_{6D}(\tau))^% {k+1/2}}\end{aligned}
  56. j 6 A ( - 3 6 ) = 10 2 j_{6A}\Big(\sqrt{\tfrac{-3}{6}}\Big)=10^{2}
  57. 1 π = 3 5 2 k = 0 σ 1 ( k ) 16 k + 3 ( 10 2 ) k 1 π = 3 2 3 k = 0 σ 2 ( k ) 10 k + 3 ( 10 2 - 36 ) k 1 π = 1 3 2 k = 0 σ 3 ( k ) 5 k + 1 ( 10 2 - 4 ) k \begin{aligned}\displaystyle\frac{1}{\pi}&\displaystyle=\frac{\sqrt{3}}{5^{2}}% \,\sum_{k=0}^{\infty}\sigma_{1}(k)\,\frac{16k+3}{(10^{2})^{k}}\\ \displaystyle\frac{1}{\pi}&\displaystyle=\frac{\sqrt{3}}{2^{3}}\,\sum_{k=0}^{% \infty}\sigma_{2}(k)\,\frac{10k+3}{(10^{2}-36)^{k}}\\ \displaystyle\frac{1}{\pi}&\displaystyle=\frac{1}{3\sqrt{2}}\,\sum_{k=0}^{% \infty}\sigma_{3}(k)\,\frac{5k+1}{(10^{2}-4)^{k}}\\ \end{aligned}
  58. 1 π = 8 15 k = 0 s 6 B ( k ) ( 1 2 - 3 5 20 + k ) ( 1 ϕ 12 ) k + 1 / 2 , j 6 B ( - 5 6 ) = ( 1 + 5 2 ) 12 = ϕ 12 \frac{1}{\pi}=8\sqrt{15}\,\sum_{k=0}^{\infty}s_{6B}(k)\,\Big(\tfrac{1}{2}-% \tfrac{3\sqrt{5}}{20}+k\Big)\Big(\frac{1}{\phi^{12}}\Big)^{k+1/2},\quad j_{6B}% \Big(\sqrt{\tfrac{-5}{6}}\Big)=\Big(\tfrac{1+\sqrt{5}}{2}\Big)^{12}=\phi^{12}
  59. 1 π = 1 2 k = 0 s 6 C ( k ) 3 k + 1 32 k , j 6 C ( - 1 3 ) = 32 \frac{1}{\pi}=\frac{1}{2}\,\sum_{k=0}^{\infty}s_{6C}(k)\,\frac{3k+1}{32^{k}},% \quad j_{6C}\Big(\sqrt{\tfrac{-1}{3}}\Big)=32
  60. 1 π = 2 3 k = 0 s 6 D ( k ) 4 k + 1 81 k + 1 / 2 , j 6 D ( - 1 2 ) = 81 \frac{1}{\pi}=2\sqrt{3}\,\sum_{k=0}^{\infty}s_{6D}(k)\,\frac{4k+1}{81^{k+1/2}}% ,\quad j_{6D}\Big(\sqrt{\tfrac{-1}{2}}\Big)=81
  61. s 7 A ( k ) = j = 0 k ( k j ) 2 ( 2 j k ) ( k + j j ) = 1 , 4 , 48 , 760 , 13840 , s_{7A}(k)=\sum_{j=0}^{k}{\textstyle\left({{k}\atop{j}}\right)}^{2}{\textstyle% \left({{2j}\atop{k}}\right)}{\textstyle\left({{k+j}\atop{j}}\right)}=1,4,48,76% 0,13840,\dots
  62. j 7 A ( τ ) = ( ( η ( τ ) η ( 7 τ ) ) 2 + 7 ( η ( 7 τ ) η ( τ ) ) 2 ) 2 = 1 q + 10 + 51 q + 204 q 2 + 681 q 3 + j 7 B ( τ ) = ( η ( τ ) η ( 7 τ ) ) 4 = 1 q - 4 + 2 q + 8 q 2 - 5 q 3 - 4 q 4 - 10 q 5 + \begin{aligned}\displaystyle j_{7A}(\tau)&\displaystyle=\Big(\big(\tfrac{\eta(% \tau)}{\eta(7\tau)}\big)^{2}+7\big(\tfrac{\eta(7\tau)}{\eta(\tau)}\big)^{2}% \Big)^{2}=\tfrac{1}{q}+10+51q+204q^{2}+681q^{3}+\dots\\ \displaystyle j_{7B}(\tau)&\displaystyle=\big(\tfrac{\eta(\tau)}{\eta(7\tau)}% \big)^{4}=\tfrac{1}{q}-4+2q+8q^{2}-5q^{3}-4q^{4}-10q^{5}+\dots\end{aligned}
  63. 1 π = 3 22 3 k = 0 s 7 A ( k ) 11895 k + 1286 ( - 22 3 ) k , j 7 A ( 7 + - 427 14 ) = - 22 3 + 1 = - ( 39 7 ) 2 \frac{1}{\pi}=\frac{3}{22^{3}}\,\sum_{k=0}^{\infty}s_{7A}(k)\,\frac{11895k+128% 6}{(-22^{3})^{k}},\quad j_{7A}\Big(\tfrac{7+\sqrt{-427}}{14}\Big)=-22^{3}+1=-(% 39\sqrt{7})^{2}
  64. j 4 B ( τ ) = ( j 2 A ( 2 τ ) ) 1 / 2 = 1 q + 52 q + 834 q 3 + 4760 q 5 + 24703 q 7 + = ( ( η ( τ ) η 2 ( 4 τ ) η 2 ( 2 τ ) η ( 8 τ ) ) 4 + 4 ( η 2 ( 2 τ ) η ( 8 τ ) η ( τ ) η 2 ( 4 τ ) ) 4 ) 2 = ( ( η ( 2 τ ) η ( 4 τ ) η ( τ ) η ( 8 τ ) ) 4 - 4 ( η ( τ ) η ( 8 τ ) η ( 2 τ ) η 2 ( τ ) ) 4 ) 2 j 8 A ( τ ) = ( η ( τ ) η 2 ( 4 τ ) η 2 ( 2 τ ) η ( 8 τ ) ) 8 = 1 q - 8 + 36 q - 128 q 2 + 386 q 3 - 1024 q 4 + j 8 A ( τ ) = ( η ( 2 τ ) η ( 4 τ ) η ( τ ) η ( 8 τ ) ) 8 = 1 q + 8 + 36 q + 128 q 2 + 386 q 3 + 1024 q 4 + . \begin{aligned}\displaystyle j_{4B}(\tau)&\displaystyle=\big(j_{2A}(2\tau)\big% )^{1/2}=\tfrac{1}{q}+52q+834q^{3}+4760q^{5}+24703q^{7}+\dots\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)\,\eta^{2}(4\tau)}{\eta^{2}(2\tau)\,% \eta(8\tau)}\big)^{4}+4\big(\tfrac{\eta^{2}(2\tau)\,\eta(8\tau)}{\eta(\tau)\,% \eta^{2}(4\tau)}\big)^{4}\Big)^{2}=\Big(\big(\tfrac{\eta(2\tau)\,\eta(4\tau)}{% \eta(\tau)\,\eta(8\tau)}\big)^{4}-4\big(\tfrac{\eta(\tau)\,\eta(8\tau)}{\eta(2% \tau)\,\eta^{2}(\tau)}\big)^{4}\Big)^{2}\\ \displaystyle j_{8A^{\prime}}(\tau)&\displaystyle=\big(\tfrac{\eta(\tau)\,\eta% ^{2}(4\tau)}{\eta^{2}(2\tau)\,\eta(8\tau)}\big)^{8}=\tfrac{1}{q}-8+36q-128q^{2% }+386q^{3}-1024q^{4}+\dots\\ \displaystyle j_{8A}(\tau)&\displaystyle=\big(\tfrac{\eta(2\tau)\,\eta(4\tau)}% {\eta(\tau)\,\eta(8\tau)}\big)^{8}=\tfrac{1}{q}+8+36q+128q^{2}+386q^{3}+1024q^% {4}+\dots.\\ \end{aligned}
  65. s 4 B ( k ) = ( 2 k k ) j = 0 k 4 k - 2 j ( k 2 j ) ( 2 j j ) 2 = ( 2 k k ) j = 0 k ( k j ) ( 2 k - 2 j k - j ) ( 2 j j ) = 1 , 8 , 120 , 2240 , 47320 , s_{4B}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}4^{k-2j}{% \textstyle\left({{k}\atop{2j}}\right)}{\textstyle\left({{2j}\atop{j}}\right)}^% {2}={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{\textstyle\left({{k}% \atop{j}}\right)}{\textstyle\left({{2k-2j}\atop{k-j}}\right)}{\textstyle\left(% {{2j}\atop{j}}\right)}=1,8,120,2240,47320,\dots
  66. s 8 A ( k ) = j = 0 k ( - 1 ) k ( k j ) 2 ( 2 j k ) 2 = 1 , - 4 , 40 , - 544 , 8536 , s_{8A^{\prime}}(k)=\sum_{j=0}^{k}(-1)^{k}{\textstyle\left({{k}\atop{j}}\right)% }^{2}{\textstyle\left({{2j}\atop{k}}\right)}^{2}=1,-4,40,-544,8536,\dots
  67. 1 π = 2 2 13 k = 0 s 4 B ( k ) 70 99 k + 579 ( 16 + 396 2 ) k + 1 / 2 , j 4 B ( 1 4 - 58 ) = 396 2 \frac{1}{\pi}=\frac{2\sqrt{2}}{13}\,\sum_{k=0}^{\infty}s_{4B}(k)\,\frac{70% \cdot 99\,k+579}{(16+396^{2})^{k+1/2}},\quad j_{4B}\Big(\tfrac{1}{4}\sqrt{-58}% \Big)=396^{2}
  68. 1 π = - 2 70 k = 0 s 4 B ( k ) 58 13 99 k + 6243 ( 16 - 396 2 ) k + 1 / 2 \frac{1}{\pi}=\frac{\sqrt{-2}}{70}\,\sum_{k=0}^{\infty}s_{4B}(k)\,\frac{58% \cdot 13\cdot 99\,k+6243}{(16-396^{2})^{k+1/2}}
  69. 1 π = 2 2 k = 0 s 8 A ( k ) - 222 + 377 2 ( k + 1 2 ) ( 4 ( 1 + 2 ) 12 ) k + 1 / 2 , j 8 A ( 1 4 - 58 ) = 4 ( 1 + 2 ) 12 , j 8 A ( 1 4 - 58 ) = 4 ( 99 + 13 58 ) 2 = 4 U 58 2 \frac{1}{\pi}=2\sqrt{2}\,\sum_{k=0}^{\infty}s_{8A^{\prime}}(k)\,\frac{-222+377% \sqrt{2}\,(k+\tfrac{1}{2})}{\big(4(1+\sqrt{2})^{12}\big)^{k+1/2}},\quad j_{8A^% {\prime}}\Big(\tfrac{1}{4}\sqrt{-58}\Big)=4(1+\sqrt{2})^{12},\quad j_{8A}\Big(% \tfrac{1}{4}\sqrt{-58}\Big)=4(99+13\sqrt{58})^{2}=4U_{58}^{2}
  70. j 3 C ( τ ) = ( j ( 3 τ ) ) 1 / 3 = - 6 + ( η 2 ( 3 τ ) η ( τ ) η ( 9 τ ) ) 6 - 27 ( η ( τ ) η ( 9 τ ) η 2 ( 3 τ ) ) 6 = 1 q + 248 q 2 + 4124 q 5 + 34752 q 8 + j 9 A ( τ ) = ( η 2 ( 3 τ ) η ( τ ) η ( 9 τ ) ) 6 = 1 q + 6 + 27 q + 86 q 2 + 243 q 3 + 594 q 4 + \begin{aligned}\displaystyle j_{3C}(\tau)&\displaystyle=\big(j(3\tau))^{1/3}=-% 6+\big(\tfrac{\eta^{2}(3\tau)}{\eta(\tau)\,\eta(9\tau)}\big)^{6}-27\big(\tfrac% {\eta(\tau)\,\eta(9\tau)}{\eta^{2}(3\tau)}\big)^{6}=\tfrac{1}{q}+248q^{2}+4124% q^{5}+34752q^{8}+\dots\\ \displaystyle j_{9A}(\tau)&\displaystyle=\big(\tfrac{\eta^{2}(3\tau)}{\eta(% \tau)\,\eta(9\tau)}\big)^{6}=\tfrac{1}{q}+6+27q+86q^{2}+243q^{3}+594q^{4}+% \dots\\ \end{aligned}
  71. s 3 C ( k ) = ( 2 k k ) j = 0 k ( - 3 ) k - 3 j ( k j ) ( k - j j ) ( k - 2 j j ) = ( 2 k k ) j = 0 k ( - 3 ) k - 3 j ( k 3 j ) ( 2 j j ) ( 3 j j ) = 1 , - 6 , 54 , - 420 , 630 , s_{3C}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}(-3)^{k-3j}{% \textstyle\left({{k}\atop{j}}\right)}{\textstyle\left({{k-j}\atop{j}}\right)}{% \textstyle\left({{k-2j}\atop{j}}\right)}={\textstyle\left({{2k}\atop{k}}\right% )}\sum_{j=0}^{k}(-3)^{k-3j}{\textstyle\left({{k}\atop{3j}}\right)}{\textstyle% \left({{2j}\atop{j}}\right)}{\textstyle\left({{3j}\atop{j}}\right)}=1,-6,54,-4% 20,630,\dots
  72. s 9 A ( k ) = j = 0 k ( k j ) 2 m = 0 j ( k m ) ( j m ) ( j + m k ) = 1 , 3 , 27 , 309 , 4059 , s_{9A}(k)=\sum_{j=0}^{k}{\textstyle\left({{k}\atop{j}}\right)}^{2}\sum_{m=0}^{% j}{\textstyle\left({{k}\atop{m}}\right)}{\textstyle\left({{j}\atop{m}}\right)}% {\textstyle\left({{j+m}\atop{k}}\right)}=1,3,27,309,4059,\dots
  73. 1 π = - s y m b o l i 9 k = 0 s 3 C ( k ) 602 k + 85 ( - 960 - 12 ) k + 1 / 2 , j 3 C ( 3 + - 43 6 ) = - 960 \frac{1}{\pi}=\frac{-symbol{i}}{9}\sum_{k=0}^{\infty}s_{3C}(k)\,\frac{602k+85}% {(-960-12)^{k+1/2}},\quad j_{3C}\Big(\tfrac{3+\sqrt{-43}}{6}\Big)=-960
  74. 1 π = 6 s y m b o l i k = 0 s 9 A ( k ) 4 - 129 ( k + 1 2 ) ( - 3 3 U 129 ) k + 1 / 2 , j 9 A ( 3 + - 43 6 ) = - 3 3 ( 53 3 + 14 43 ) = - 3 3 U 129 \frac{1}{\pi}=6\,symbol{i}\,\sum_{k=0}^{\infty}s_{9A}(k)\,\frac{4-\sqrt{129}\,% (k+\tfrac{1}{2})}{\big(-3\sqrt{3U_{129}}\big)^{k+1/2}},\quad j_{9A}\Big(\tfrac% {3+\sqrt{-43}}{6}\Big)=-3\sqrt{3}\big(53\sqrt{3}+14\sqrt{43}\big)=-3\sqrt{3U_{% 129}}
  75. j 10 A ( τ ) = j 10 B ( τ ) + 16 j 10 B ( τ ) + 8 = j 10 C ( τ ) + 25 j 10 C ( τ ) + 6 = j 10 D ( τ ) + 1 j 10 D ( τ ) - 2 = 1 q + 4 + 22 q + 56 q 2 + \begin{aligned}\displaystyle j_{10A}(\tau)&\displaystyle=j_{10B}(\tau)+\tfrac{% 16}{j_{10B}(\tau)}+8=j_{10C}(\tau)+\tfrac{25}{j_{10C}(\tau)}+6=j_{10D}(\tau)+% \tfrac{1}{j_{10D}(\tau)}-2=\tfrac{1}{q}+4+22q+56q^{2}+\dots\end{aligned}
  76. j 10 B ( τ ) = ( η ( τ ) η ( 5 τ ) η ( 2 τ ) η ( 10 τ ) ) 4 = 1 q - 4 + 6 q - 8 q 2 + 17 q 3 - 32 q 4 + \begin{aligned}\displaystyle j_{10B}(\tau)&\displaystyle=\Big(\tfrac{\eta(\tau% )\eta(5\tau)}{\eta(2\tau)\eta(10\tau)}\Big)^{4}=\tfrac{1}{q}-4+6q-8q^{2}+17q^{% 3}-32q^{4}+\dots\end{aligned}
  77. j 10 C ( τ ) = ( η ( τ ) η ( 2 τ ) η ( 5 τ ) η ( 10 τ ) ) 2 = 1 q - 2 - 3 q + 6 q 2 + 2 q 3 + 2 q 4 + \begin{aligned}\displaystyle j_{10C}(\tau)&\displaystyle=\Big(\tfrac{\eta(\tau% )\eta(2\tau)}{\eta(5\tau)\eta(10\tau)}\Big)^{2}=\tfrac{1}{q}-2-3q+6q^{2}+2q^{3% }+2q^{4}+\dots\end{aligned}
  78. j 10 D ( τ ) = ( η ( 2 τ ) η ( 5 τ ) η ( τ ) η ( 10 τ ) ) 6 = 1 q + 6 + 21 q + 62 q 2 + 162 q 3 + \begin{aligned}\displaystyle j_{10D}(\tau)&\displaystyle=\Big(\tfrac{\eta(2% \tau)\eta(5\tau)}{\eta(\tau)\eta(10\tau)}\Big)^{6}=\tfrac{1}{q}+6+21q+62q^{2}+% 162q^{3}+\dots\end{aligned}
  79. j 10 E ( τ ) = ( η ( 2 τ ) η 5 ( 5 τ ) η ( τ ) η 5 ( 10 τ ) ) = 1 q + 1 + q + 2 q 2 + 2 q 3 - 2 q 4 + \begin{aligned}\displaystyle j_{10E}(\tau)&\displaystyle=\Big(\tfrac{\eta(2% \tau)\eta^{5}(5\tau)}{\eta(\tau)\eta^{5}(10\tau)}\Big)=\tfrac{1}{q}+1+q+2q^{2}% +2q^{3}-2q^{4}+\dots\end{aligned}
  80. T 10 A - T 10 B - T 10 C - T 10 D + 2 T 10 E = 0 T_{10A}-T_{10B}-T_{10C}-T_{10D}+2T_{10E}=0
  81. j 10 A - j 10 B - j 10 C - j 10 D + 2 j 10 E = 6 j_{10A}-j_{10B}-j_{10C}-j_{10D}+2j_{10E}=6
  82. v 1 ( k ) = j = 0 k ( k j ) 4 = 1 , 2 , 18 , 164 , 1810 , v_{1}(k)=\sum_{j=0}^{k}{\textstyle\left({{k}\atop{j}}\right)}^{4}=1,2,18,164,1% 810,\dots
  83. v 2 ( k ) = ( 2 k k ) j = 0 k ( 2 j j ) - 1 ( k j ) m = 0 j ( j m ) 4 = 1 , 4 , 36 , 424 , 5716 , v_{2}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{\textstyle\left% ({{2j}\atop{j}}\right)}^{-1}{\textstyle\left({{k}\atop{j}}\right)}\sum_{m=0}^{% j}{\textstyle\left({{j}\atop{m}}\right)}^{4}=1,4,36,424,5716,\dots
  84. v 3 ( k ) = ( 2 k k ) j = 0 k ( 2 j j ) - 1 ( k j ) ( - 4 ) k - j m = 0 j ( j m ) 4 = 1 , - 6 , 66 , - 876 , 12786 , v_{3}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{\textstyle\left% ({{2j}\atop{j}}\right)}^{-1}{\textstyle\left({{k}\atop{j}}\right)}(-4)^{k-j}% \sum_{m=0}^{j}{\textstyle\left({{j}\atop{m}}\right)}^{4}=1,-6,66,-876,12786,\dots
  85. v 2 ( k ) = ( 2 k k ) j = 0 k ( 2 j j ) - 1 ( k j ) ( - 1 ) k - j m = 0 j ( j m ) 4 = 1 , 0 , 12 , 24 , 564 , 2784 , v_{2}^{\prime}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{% \textstyle\left({{2j}\atop{j}}\right)}^{-1}{\textstyle\left({{k}\atop{j}}% \right)}(-1)^{k-j}\sum_{m=0}^{j}{\textstyle\left({{j}\atop{m}}\right)}^{4}=1,0% ,12,24,564,2784,\dots
  86. v 3 ( k ) = ( 2 k k ) j = 0 k ( 2 j j ) - 1 ( k j ) ( 4 ) k - j m = 0 j ( j m ) 4 = 1 , 10 , 162 , 3124 , 66994 , v_{3}^{\prime}(k)={\textstyle\left({{2k}\atop{k}}\right)}\sum_{j=0}^{k}{% \textstyle\left({{2j}\atop{j}}\right)}^{-1}{\textstyle\left({{k}\atop{j}}% \right)}(4)^{k-j}\sum_{m=0}^{j}{\textstyle\left({{j}\atop{m}}\right)}^{4}=1,10% ,162,3124,66994,\dots
  87. s 10 B ( k ) = 1 , - 2 , 10 , - 68 , 514 , - 4100 , 33940 , s_{10B}(k)=1,-2,10,-68,514,-4100,33940,\dots
  88. s 10 C ( k ) = 1 , - 1 , 1 , - 1 , 1 , 23 , - 263 , 1343 , - 2303 , s_{10C}(k)=1,-1,1,-1,1,23,-263,1343,-2303,\dots
  89. s 10 D ( k ) = 1 , 3 , 25 , 267 , 3249 , 42795 , 594145 , s_{10D}(k)=1,3,25,267,3249,42795,594145,\dots
  90. U = k = 0 v 1 ( k ) 1 ( j 10 A ( τ ) ) k + 1 / 2 = k = 0 v 2 ( k ) 1 ( j 10 A ( τ ) + 4 ) k + 1 / 2 = k = 0 v 3 ( k ) 1 ( j 10 A ( τ ) - 16 ) k + 1 / 2 U=\sum_{k=0}^{\infty}v_{1}(k)\,\frac{1}{(j_{10A}(\tau))^{k+1/2}}=\sum_{k=0}^{% \infty}v_{2}(k)\,\frac{1}{(j_{10A}(\tau)+4)^{k+1/2}}=\sum_{k=0}^{\infty}v_{3}(% k)\,\frac{1}{(j_{10A}(\tau)-16)^{k+1/2}}
  91. U = V = W U=V=W
  92. V = k = 0 v 2 ( k ) 1 ( j 10 A ( τ ) - 4 ) k + 1 / 2 = k = 0 v 3 ( k ) 1 ( j 10 A ( τ ) + 16 ) k + 1 / 2 V=\sum_{k=0}^{\infty}v_{2}^{\prime}(k)\,\frac{1}{(j_{10A}(\tau)-4)^{k+1/2}}=% \sum_{k=0}^{\infty}v_{3}^{\prime}(k)\,\frac{1}{(j_{10A}(\tau)+16)^{k+1/2}}
  93. W = k = 0 s 10 B ( k ) 1 ( j 10 B ( τ ) ) k + 1 / 2 = k = 0 s 10 C ( k ) 1 ( j 10 C ( τ ) ) k + 1 / 2 = k = 0 s 10 D ( k ) 1 ( j 10 D ( τ ) ) k + 1 / 2 W=\sum_{k=0}^{\infty}s_{10B}(k)\,\frac{1}{(j_{10B}(\tau))^{k+1/2}}=\sum_{k=0}^% {\infty}s_{10C}(k)\,\frac{1}{(j_{10C}(\tau))^{k+1/2}}=\sum_{k=0}^{\infty}s_{10% D}(k)\,\frac{1}{(j_{10D}(\tau))^{k+1/2}}
  94. j 10 A ( - 19 10 ) = 76 2 j_{10A}\Big(\sqrt{\tfrac{-19}{10}}\Big)=76^{2}
  95. 1 π = 5 95 k = 0 v 1 ( k ) 408 k + 47 ( 76 2 ) k + 1 / 2 1 π = 1 17 95 k = 0 v 2 ( k ) 19 1824 k + 3983 ( 76 2 + 4 ) k + 1 / 2 1 π = 5 481 95 k = 0 v 2 ( k ) 19 10336 k + 22675 ( 76 2 - 4 ) k + 1 / 2 1 π = 1 6 95 k = 0 v 3 ( k ) 19 646 k + 1427 ( 76 2 - 16 ) k + 1 / 2 1 π = 5 181 95 k = 0 v 3 ( k ) 19 3876 k + 8405 ( 76 2 + 16 ) k + 1 / 2 \begin{aligned}\displaystyle\frac{1}{\pi}&\displaystyle=\frac{5}{\sqrt{95}}\,% \sum_{k=0}^{\infty}v_{1}(k)\,\frac{408k+47}{(76^{2})^{k+1/2}}\\ \displaystyle\frac{1}{\pi}&\displaystyle=\frac{1}{17\sqrt{95}}\,\sum_{k=0}^{% \infty}v_{2}(k)\,\frac{19\cdot 1824k+3983}{(76^{2}+4)^{k+1/2}}\\ \displaystyle\frac{1}{\pi}&\displaystyle=\frac{5}{481\sqrt{95}}\,\sum_{k=0}^{% \infty}v_{2}^{\prime}(k)\,\frac{19\cdot 10336k+22675}{(76^{2}-4)^{k+1/2}}\\ \displaystyle\frac{1}{\pi}&\displaystyle=\frac{1}{6\sqrt{95}}\,\,\sum_{k=0}^{% \infty}v_{3}(k)\,\,\frac{19\cdot 646k+1427}{(76^{2}-16)^{k+1/2}}\\ \displaystyle\frac{1}{\pi}&\displaystyle=\frac{5}{181\sqrt{95}}\,\sum_{k=0}^{% \infty}v_{3}^{\prime}(k)\,\frac{19\cdot 3876k+8405}{(76^{2}+16)^{k+1/2}}\end{aligned}
  96. 1 π = s y m b o l i 5 k = 0 s 10 C ( k ) 10 k + 3 ( - 25 ) k + 1 / 2 , j 10 C ( 1 + s y m b o l i 2 ) = - 25 \frac{1}{\pi}=\frac{symbol{i}}{\sqrt{5}}\,\sum_{k=0}^{\infty}s_{10C}(k)\frac{1% 0k+3}{(-25)^{k+1/2}},\quad j_{10C}\Big(\tfrac{1+\,symbol{i}}{2}\Big)=-25
  97. j 11 A ( τ ) = ( 1 + 3 F ) 3 + ( 1 F + 3 F ) 2 = 1 q + 6 + 17 q + 46 q 2 + 116 q 3 + j_{11A}(\tau)=(1+3F)^{3}+(\tfrac{1}{\sqrt{F}}+3\sqrt{F})^{2}=\tfrac{1}{q}+6+17% q+46q^{2}+116q^{3}+\dots
  98. F = η ( 3 τ ) η ( 33 τ ) η ( τ ) η ( 11 τ ) F=\tfrac{\eta(3\tau)\,\eta(33\tau)}{\eta(\tau)\,\eta(11\tau)}
  99. s 11 A ( k ) = 1 , 4 , 28 , 268 , 3004 , 36784 , 476476 , s_{11A}(k)=1,\,4,\,28,\,268,\,3004,\,36784,\,476476,\dots
  100. ( k + 1 ) 3 s k + 1 = 2 ( 2 k + 1 ) ( 5 k 2 + 5 k + 2 ) s k - 8 k ( 7 k 2 + 1 ) s k - 1 + 22 k ( k - 1 ) ( 2 k - 1 ) s k - 2 (k+1)^{3}s_{k+1}=2(2k+1)(5k^{2}+5k+2)s_{k}\,-\,8k(7k^{2}+1)s_{k-1}\,+\,22k(k-1% )(2k-1)s_{k-2}
  101. 1 π = s y m b o l i 22 k = 0 s 11 A ( k ) 221 k + 67 ( - 44 ) k + 1 / 2 , j 11 A ( 1 + - 17 / 11 2 ) = - 44 \frac{1}{\pi}=\frac{symbol{i}}{22}\sum_{k=0}^{\infty}s_{11A}(k)\,\frac{221k+67% }{(-44)^{k+1/2}},\quad j_{11A}\Big(\tfrac{1+\sqrt{-17/11}}{2}\Big)=-44

Ramón_Jardí_i_Borrás.html

  1. h ( Δ T ) Δ T = i \frac{h(\Delta\mathit{T})}{\Delta\mathit{T}}=\mathit{i}\,
  2. lim Δ T 0 h ( Δ T ) Δ T = i ( t ) \lim_{\Delta\mathit{T}\rightarrow 0}\frac{h(\Delta\mathit{T})}{\Delta\mathit{T% }}=\mathit{i}(\mathit{t})

Ramsey_class.html

  1. A A
  2. B B
  3. C C
  4. k k
  5. ( B A ) {\left({{B}\atop{A}}\right)}
  6. A A^{\prime}
  7. B B
  8. A A
  9. C ( B ) k A C\rightarrow(B)^{A}_{k}
  10. X 1 X 2 X k X_{1}\cup X_{2}\cup\dots\cup X_{k}
  11. ( C A ) {\left({{C}\atop{A}}\right)}
  12. B ( C B ) B^{\prime}\in{\left({{C}\atop{B}}\right)}
  13. 1 i k 1\leq i\leq k
  14. ( B A ) X i {\left({{B^{\prime}}\atop{A}}\right)}\subseteq X_{i}
  15. K K
  16. K K
  17. k k
  18. B K B\in K
  19. C K C\in K
  20. C ( B ) k A C\rightarrow(B)^{A}_{k}
  21. K K
  22. A A
  23. A K A\in K
  24. K K

Ramsey_interferometry.html

  1. ω 0 \omega_{0}
  2. [ u b r a k e t , u k e t , u 193 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}193^{\prime}]
  3. [ u b r a k e t , u k e t , u 191 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}191^{\prime}]
  4. ω \omega
  5. Δ \Delta
  6. ω \omega
  7. ω 0 \omega_{0}
  8. ( Δ = ω - ω 0 ) (\Delta=\omega-\omega_{0})
  9. [ u b r a k e t , u k e t , u 193 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}193^{\prime}]
  10. [ u b r a k e t , u k e t , u 191 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}191^{\prime}]
  11. Δ = 0 \Delta=0
  12. P ( Δ ) P(\Delta)
  13. Δ = 0 \Delta=0
  14. Δ = 0 \Delta=0
  15. Δ = 0 \Delta=0
  16. Δ \Delta
  17. v v
  18. ω 0 \hbar\omega_{0}
  19. 𝐁 \mathbf{B}_{\|}
  20. z ^ \hat{z}
  21. ω 0 = γ | 𝐁 | \omega_{0}=\gamma|\mathbf{B}_{\|}|
  22. τ = L / v \tau=L/v
  23. 𝐁 cos ( ω t ) \mathbf{B}_{\perp}\cos(\omega t)
  24. [ u b r a k e t , u k e t , u 193 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}193^{\prime}]
  25. [ u b r a k e t , u k e t , u 191 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}191^{\prime}]
  26. Ω = γ | 𝐁 | \Omega_{\perp}=\gamma|\mathbf{B}_{\perp}|
  27. H ^ = - Δ 2 σ z ^ + Ω 2 σ x ^ \hat{H}=-\frac{\hbar\Delta}{2}\hat{\sigma_{z}}+\frac{\hbar\Omega_{\perp}}{2}% \hat{\sigma_{x}}
  28. [ u b r a k e t , u k e t , u 193 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}193^{\prime}]
  29. [ u b r a k e t , u k e t , u 191 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{2}191^{\prime}]
  30. P ( Δ , v , L , Ω ) = 1 1 + ( Δ Ω ) 2 sin 2 ( L 2 v Ω 2 + Δ 2 ) P(\Delta,v,L,\Omega_{\perp})=\frac{1}{1+\left(\frac{\Delta}{\Omega_{\perp}}% \right)^{2}}\sin^{2}\left(\frac{L}{2v}\sqrt{\Omega_{\perp}^{2}+\Delta^{2}}\right)
  31. Ω τ = π \Omega_{\perp}\tau=\pi
  32. δ \delta
  33. P ( Δ , Ω ) P(\Delta,\Omega_{\perp})
  34. Δ Ω \frac{\Delta}{\Omega_{\perp}}
  35. δ Ω π τ π v L \delta\sim\Omega_{\perp}\sim\frac{\pi}{\tau}\sim\frac{\pi v}{L}
  36. τ \tau
  37. L L
  38. Ω \Omega_{\perp}
  39. π \pi
  40. 𝐁 \mathbf{B}_{\perp}
  41. L L
  42. π 2 \frac{\pi}{2}
  43. Δ \Delta
  44. 𝐁 \mathbf{B}_{\perp}
  45. σ z ^ \hat{\sigma_{z}}
  46. π / 2 \pi/2
  47. T T
  48. π / 2 \pi/2
  49. T T
  50. | Δ | Ω |\Delta|\ll\Omega_{\perp}
  51. P ( T , Δ ) = cos 2 ( Δ T 2 ) = cos 2 ( Δ L 2 v ) P(T,\Delta)=\cos^{2}\left(\frac{\Delta T}{2}\right)=\cos^{2}\left(\frac{\Delta L% }{2v}\right)
  52. ( | Δ | Ω ) \left(|\Delta|\ll\Omega_{\perp}\right)
  53. π 2 \frac{\pi}{2}
  54. δ \delta
  55. Δ \Delta
  56. δ 1 T v L \delta\sim\frac{1}{T}\sim\frac{v}{L}
  57. T T
  58. L L
  59. ω \omega
  60. ω 0 \omega_{0}
  61. ω \omega
  62. ω 0 \omega_{0}
  63. ω = ω 0 \omega=\omega_{0}
  64. | \left|\downarrow\right\rangle
  65. | + | 2 \frac{\left|\downarrow\right\rangle+\left|\uparrow\right\rangle}{\sqrt{2}}
  66. | \left|\downarrow\right\rangle
  67. | \left|\uparrow\right\rangle
  68. | \left|\downarrow\right\rangle
  69. | \left|\downarrow\right\rangle
  70. | \left|\uparrow\right\rangle
  71. | \left|\downarrow\right\rangle
  72. Δ \Delta
  73. π 2 \frac{\pi}{2}
  74. [ u b r a k e t , u k e t , u a ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}a^{\prime}]
  75. [ u b r a k e t , u k e t , u b ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}b^{\prime}]
  76. | 𝐤 | \hbar|\mathbf{k}|
  77. 𝐤 \mathbf{k}
  78. [ u b r a k e t , u k e t , u b ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}b^{\prime}]
  79. [ u b r a k e t , u k e t , u a , 0 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}a,0^{\prime}]
  80. [ u b r a k e t , u k e t , u b , 0131 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}b,\u{2}0131^{% \prime}]
  81. [ u b r a k e t , u k e t , u a , 2 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}a,2^{\prime}]
  82. [ u b r a k e t , u k e t , u b , 1 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}b,1^{\prime}]
  83. H ^ R = - 𝛀 ( 𝐫 ^ × 𝐩 ^ ) \hat{H}_{R}=-\mathbf{\Omega}\cdot(\hat{\mathbf{r}}\times\hat{\mathbf{p}})
  84. Ω \Omega
  85. U ^ R = e x p ( i d t [ 𝛀 × 𝐫 ^ ( t ) ] [ 𝐩 𝟎 + m α 𝐤 ] ) \hat{U}_{R}=exp\left(\frac{i}{\hbar}\int dt^{\prime}[\mathbf{\Omega}\times\hat% {\mathbf{r}}(t^{\prime})]\cdot[\mathbf{p_{0}}+m_{\alpha}\hbar\mathbf{k}]\right)
  86. 𝛀 \mathbf{\Omega}
  87. e x p ( 2 i k Ω d 2 / v ) exp\left(2ik\Omega d^{2}/v\right)
  88. d d
  89. P cos [ ( Δ + 2 π Ω d λ + ϕ ) 2 d v ] P\propto\cos\left[\left(\Delta+\frac{2\pi\Omega d}{\lambda}+\phi\right)\frac{2% d}{v}\right]
  90. λ \lambda
  91. ω 0 \omega_{0}
  92. δ v = Ω d λ \delta v=\frac{\Omega d}{\lambda}
  93. Ω = π 12 h o u r s \Omega=\frac{\pi}{12hours}
  94. 2 d = 21 c m 2d=21cm
  95. λ = 6573 Å \lambda=6573\AA
  96. δ v 12 H z \delta v\approx 12Hz

Ran_space.html

  1. Ran ( X ) \operatorname{Ran}(X)
  2. { S Ran ( U 1 U m ) S U 1 , , S U m } \{S\in\operatorname{Ran}(U_{1}\cup\dots\cup U_{m})\mid S\cap U_{1}\neq% \emptyset,\dots,S\cap U_{m}\neq\emptyset\}
  3. U i X , 1 i m U_{i}\subset X,1\leq i\leq m
  4. Ran ( X ) \operatorname{Ran}(X)
  5. ( R , S , μ ) (R,S,\mu)
  6. μ : S X ( R ) \mu:S\to X(R)
  7. ( R , S , μ ) ( R , S , μ ) (R,S,\mu)\to(R^{\prime},S^{\prime},\mu^{\prime})
  8. R R R\to R^{\prime}
  9. S S S\to S^{\prime}
  10. μ \mu
  11. μ \mu^{\prime}
  12. Ran ( X ) \operatorname{Ran}(X)
  13. μ \mu
  14. Ran ( X ) \operatorname{Ran}(X)
  15. Ran ( M ) \operatorname{Ran}(M)

Random_projection.html

  1. d X N dXN
  2. O ( d k N ) O(dkN)
  3. O ( c k N ) O(ckN)
  4. S N - 1 S^{N-1}
  5. A O ( N ) A\in O(N)
  6. R i , j = 3 { + 1 w i t h p r o b a b i l i t y 1 6 0 w i t h p r o b a b i l i t y 2 3 - 1 w i t h p r o b a b i l i t y 1 6 R_{i,j}=\sqrt{3}\begin{cases}+1&with\ probability\frac{1}{6}\\ 0&with\ probability\frac{2}{3}\\ -1&with\ probability\frac{1}{6}\end{cases}

Randomness_merger.html

  1. k k
  2. X 1 , , X k X_{1},\ldots,X_{k}
  3. { 0 , 1 } n \{0,1\}^{n}
  4. n n
  5. { 0 , 1 } n \{0,1\}^{n}
  6. M : ( { 0 , 1 } n ) k × { 0 , 1 } d { 0 , 1 } n M:(\{0,1\}^{n})^{k}\times\{0,1\}^{d}\rightarrow\{0,1\}^{n}
  7. ( m , ε ) (m,\varepsilon)
  8. ( X 1 , , X k ) (X_{1},\ldots,X_{k})
  9. { 0 , 1 } n \{0,1\}^{n}
  10. Z = M ( X 1 , , X k , U d ) Z=M(X_{1},\ldots,X_{k},U_{d})
  11. H ε ( Z ) m H_{\infty}^{\varepsilon}(Z)\geq m
  12. U d U_{d}
  13. d d
  14. d d
  15. ε \varepsilon
  16. m m
  17. m m
  18. ε \varepsilon
  19. Z Z
  20. k k
  21. Z Z
  22. 2 - k 2^{-k}
  23. m m
  24. ε \varepsilon
  25. d d
  26. α > 0 \alpha>0
  27. n n
  28. k 2 o ( n ) k\leq 2^{o(n)}
  29. ( m , ε ) (m,\varepsilon)
  30. M : ( { 0 , 1 } n ) k × { 0 , 1 } d { 0 , 1 } n M:(\{0,1\}^{n})^{k}\times\{0,1\}^{d}\rightarrow\{0,1\}^{n}
  31. m = ( 1 - α ) n , m=(1-\alpha)n,
  32. d = O ( log ( n ) + log ( k ) ) , d=O(\log(n)+\log(k)),
  33. ε = O ( 1 n k ) . \varepsilon=O\left(\frac{1}{n\cdot k}\right).

Range_mode_query.html

  1. A [ 1 : n ] = [ a 1 , a 2 , , a n ] A[1:n]=[a_{1},a_{2},...,a_{n}]
  2. m o d e ( A , i : j ) mode(A,i:j)
  3. 1 i j n 1\leq i\leq j\leq n
  4. S = [ s 1 , s 2 , , s n ] S=[s_{1},s_{2},...,s_{n}]
  5. m o d e ( S ) mode(S)
  6. s i s_{i}
  7. s i s_{i}
  8. s j j { 1 , , n } s_{j}\;\forall j\in\{1,...,n\}
  9. S = [ 1 , 2 , 4 , 2 , 3 , 4 , 2 ] S=[1,2,4,2,3,4,2]
  10. m o d e ( s ) = 2 mode(s)=2
  11. A [ i : j ] = [ a i , a i + 1 , , a j ] A[i:j]=[a_{i},a_{i}+1,...,a_{j}]
  12. A A
  13. B B
  14. c c
  15. A B A\cup B
  16. c A c\notin A
  17. c c
  18. B B
  19. c A c\notin A
  20. C = A B C=A\cup B
  21. f c f_{c}
  22. C C
  23. c c
  24. B B
  25. b b
  26. f b f_{b}
  27. B B
  28. b b
  29. B B
  30. c A c\notin A
  31. f b > f c f_{b}>f_{c}
  32. b b
  33. C C
  34. O ( n ) O(n)
  35. O ( n ) O(\sqrt{n})
  36. O ( n ) O(n)
  37. O ( n / w ) O(\sqrt{n/w})
  38. w w
  39. O ( n 2 log log n / log n ) O(n^{2}\log\log n/\log n)
  40. O ( 1 ) O(1)
  41. O ( n 2 - 2 ϵ ) O(n^{2-2\epsilon})
  42. O ( n ϵ log n ) O(n^{\epsilon}\log n)
  43. 0 ϵ 1 / 2 0\leq\epsilon\leq 1/2
  44. S S
  45. w w
  46. Ω ( log n log ( S w / n ) ) \Omega\left(\frac{\log n}{\log(Sw/n)}\right)
  47. A [ i : j ] A[i:j]
  48. A [ j + i : k ] A[j+i:k]
  49. A [ i : k ] A[i:k]
  50. A [ j + i : k ] A[j+i:k]
  51. A [ j + i : k ] A[j+i:k]
  52. m o d e ( A [ i : j ] ) = a mode(A[i:j])=a
  53. f a f_{a}
  54. m o d e ( A [ j + 1 : k ] ) = b mode(A[j+1:k])=b
  55. f a f_{a}
  56. c c
  57. f a - 1 f_{a}-1
  58. A [ i : j ] A[i:j]
  59. f a - 1 f_{a}-1
  60. A [ j + 1 : k ] A[j+1:k]
  61. a c b a\not=c\not=b
  62. A [ i : k ] A[i:k]
  63. a a
  64. b b
  65. c c
  66. m o d e ( A [ i : k ] ) mode(A[i:k])
  67. a a
  68. b b
  69. O ( n + s 2 ) O(n+s^{2})
  70. O ( n / s ) O(n/s)
  71. s = n s=\sqrt{n}
  72. O ( n ) O(n)
  73. O ( n ) O(\sqrt{n})
  74. A [ 1 : n ] A[1:n]
  75. D [ 1 : Δ ] D[1:\Delta]
  76. Δ \Delta
  77. B [ 1 : n ] B[1:n]
  78. i i
  79. B [ i ] B[i]
  80. A [ i ] A[i]
  81. D D
  82. B , D B,D
  83. A A
  84. Q 1 , Q 2 , , Q Δ Q_{1},Q_{2},...,Q_{\Delta}
  85. a { 1 , , Δ } a\in\{1,...,\Delta\}
  86. Q a = { b | B [ b ] = a } Q_{a}=\{b\;|\;B[b]=a\}
  87. B [ 1 : n ] B^{\prime}[1:n]
  88. b { 1 , , n } b\in\{1,...,n\}
  89. B [ b ] B^{\prime}[b]
  90. b b
  91. Q B [ b ] Q_{B[b]}
  92. B B
  93. Q 1 , Q 2 , , Q Δ Q_{1},Q_{2},...,Q_{\Delta}
  94. B B^{\prime}
  95. B [ i ] B[i]
  96. B [ i : j ] B[i:j]
  97. q q
  98. Q B [ i ] [ B [ i ] + q - 1 ] j Q_{B[i]}[B^{\prime}[i]+q-1]\leq j
  99. s s
  100. b 1 , b 2 , , b s b_{1},b_{2},...,b_{s}
  101. t = n / s t=\lceil n/s\rceil
  102. b i b_{i}
  103. B [ i t + 1 : ( i + 1 ) t ] B[i\cdot t+1:(i+1)t]
  104. S S
  105. S S^{\prime}
  106. S [ b i , b j ] S[b_{i},b_{j}]
  107. b i b i + 1 b j b_{i}\cup b_{i+1}\cup...\cup b_{j}
  108. B [ b i t + 1 : ( b j + 1 ) t ] B[b_{i}t+1:(b_{j}+1)t]
  109. S S^{\prime}
  110. O ( s 2 ) O(s^{2})
  111. O ( s n ) O(s\cdot n)
  112. B B
  113. s s
  114. S , S S,S^{\prime}
  115. A A
  116. a , i , j a,i,j
  117. B [ a ] B[a]
  118. B [ i : j ] B[i:j]
  119. A [ a ] A[a]
  120. A [ i : j ] A[i:j]
  121. B B
  122. A A
  123. A A
  124. B B
  125. m o d e ( B , i , j ) mode(B,i,j)
  126. b i = ( i - 1 ) / t b_{i}=\lceil(i-1)/t\rceil
  127. b j = j / t - 1 b_{j}=\lfloor j/t\rfloor-1
  128. B B
  129. B [ i : m i n { b i t , j } ] B[i:min\{b_{i}t,j\}]
  130. B [ m a x { ( b j + 1 ) t + 1 , i } : j ] B[max\{(b_{j}+1)t+1,i\}:j]
  131. b j < b i b_{j}<b_{i}
  132. c c
  133. S [ b i , b j ] S[b_{i},b_{j}]
  134. f c f_{c}
  135. S [ b i , b j ] S^{\prime}[b_{i},b_{j}]
  136. f c = 0 f_{c}=0
  137. B [ i : j ] B[i:j]
  138. c c
  139. c c
  140. f c f_{c}
  141. c c
  142. B [ i : j ] B[i:j]
  143. f c f_{c}
  144. x x
  145. Q B [ x ] [ B [ x ] - 1 ] i Q_{B[x]}[B^{\prime}[x]-1]\geq i
  146. B [ i : x - 1 ] B[i:x-1]
  147. B [ x ] B[x]
  148. B [ i : j ] B[i:j]
  149. f c f_{c}
  150. B [ x : j ] B[x:j]
  151. f x f_{x}
  152. B [ x ] B[x]
  153. B [ i : j ] B[i:j]
  154. B [ x ] + f c - 1 B^{\prime}[x]+f_{c}-1
  155. Q B [ x ] Q_{B[x]}
  156. c := B [ x ] c:=B[x]
  157. f c := f x f_{c}:=f_{x}
  158. t t
  159. t t
  160. O ( t ) = O ( n / s ) O(t)=O(n/s)
  161. O ( n 2 log log n log n ) O\left(\frac{n^{2}\log{\log{n}}}{\log{n}}\right)
  162. O ( n 2 ) O(n^{2})
  163. s = n s=n
  164. A [ 1 : n ] A[1:n]
  165. A A
  166. s s
  167. b 1 , b 2 , , b s b_{1},b_{2},...,b_{s}
  168. t = n / s t=\lceil n/s\rceil
  169. S S
  170. s × s s\times s
  171. S [ i , j ] S[i,j]
  172. b i b i + 1 b j b_{i}\cup b_{i+1}\cup...\cup b_{j}
  173. O ( s 2 ) O(s^{2})
  174. m o d e ( A , i , j ) mode(A,i,j)
  175. b i b_{i^{\prime}}
  176. i i
  177. b j b_{j^{\prime}}
  178. j j
  179. A [ i : j ] A[i:j]
  180. c c
  181. S S
  182. A [ i : j ] A[i:j]
  183. A [ i : j ] A[i:j]
  184. c c
  185. 2 t 2t
  186. 0
  187. 2 t 2t
  188. [ 0 : 2 t - 1 ] [0:2t-1]
  189. 2 t 2t
  190. ( t 2 ) {\left({{t}\atop{2}}\right)}
  191. b i b_{i^{\prime}}
  192. b j b_{j^{\prime}}
  193. t 2 t^{2}
  194. ( 2 t + 1 ) t 2 (2t+1)^{t^{2}}
  195. O ( t 2 ( 2 t + 1 ) t 2 ) O(t^{2}(2t+1)^{t^{2}})
  196. S S
  197. m o d e ( A , i , j ) mode(A,i,j)
  198. i i
  199. j j
  200. O ( s t 2 ) O(st^{2})
  201. T T
  202. O ( s 2 + t 2 ( 2 t + 1 ) t 2 + s t 2 ) O(s^{2}+t^{2}(2t+1)^{t^{2}}+st^{2})
  203. O ( n 2 log log n log n ) O\left(\frac{n^{2}\log{\log{n}}}{\log{n}}\right)
  204. t = log n / log log n t=\sqrt{\log{n}/\log{\log{n}}}
  205. m o d e ( A , i , j ) mode(A,i,j)
  206. T T
  207. S S
  208. S S
  209. S [ b i , b j ] S[b_{i^{\prime}},b_{j^{\prime}}]
  210. b i , b j b_{i^{\prime}},b_{j^{\prime}}
  211. i i
  212. j j
  213. U b i , b j U_{b_{i^{\prime}},b_{j^{\prime}}}
  214. A A

Rankin–Cohen_bracket.html

  1. [ f , g ] n = r + s = n ( - 1 ) r ( k + n - 1 r ) ( h + n - 1 s ) d r f d τ r d s g d τ s . [f,g]_{n}=\sum_{r+s=n}(-1)^{r}{\left({{k+n-1}\atop{r}}\right)}{\left({{h+n-1}% \atop{s}}\right)}\frac{d^{r}f}{d\tau^{r}}\frac{d^{s}g}{d\tau^{s}}\ .

Rassias'_conjecture.html

  1. p > 2 p>2
  2. p 1 , p_{1},
  3. p 2 , p_{2},
  4. p 1 < p 2 , p_{1}<p_{2},
  5. p = p 1 + p 2 + 1 p 1 p=\frac{p_{1}+p_{2}+1}{p_{1}}
  6. p > 2 p>2
  7. p 1 , p_{1},
  8. p 2 p_{2}
  9. p 1 < p 2 , p_{1}<p_{2},
  10. ( p - 1 ) p 1 = p 2 + 1 , (p-1)p_{1}=p_{2}+1,
  11. ( p - 1 ) p 1 (p-1)p_{1}
  12. p 2 p_{2}
  13. p 2 = 2 a p 1 - 1 , p_{2}=2ap_{1}-1,
  14. 2 a + 1 2a+1
  15. p , p,
  16. 2 a p + 1 2ap+1\in\mathbb{P}
  17. \mathbb{P}
  18. s s
  19. f i ( x ) [ X ] , i = 1 , 2 , , s f_{i}(x)\in\mathbb{Z}[X],\ i=1,2,\ldots,s
  20. F ( X ) = i = 1 s f i ( x ) F(X)=\prod_{i=1}^{s}f_{i}(x)
  21. x x
  22. f i ( x ) f_{i}(x)
  23. s = 2 s=2
  24. f 1 ( x ) = x f_{1}(x)=x
  25. f 2 ( x ) = 2 a x - 1 f_{2}(x)=2ax-1
  26. p i + 1 = m p i + n , i = 1 , 2 , , k - 1 , p_{i+1}=mp_{i}+n,\ i=1,2,\ldots,k-1,
  27. m , n > 1 m,n>1
  28. 2 a , - 1 2a,-1
  29. a a
  30. 2 a - 1 = p 2a-1=p

Rational_series.html

  1. R A R\langle A\rangle
  2. w A * c ( w ) w . \sum_{w\in A^{*}}c(w)w\ .
  3. R A R\langle\langle A\rangle\rangle
  4. c + d : w c ( w ) + d ( w ) c+d:w\mapsto c(w)+d(w)
  5. c d : w u v = w c ( u ) d ( v ) . c\cdot d:w\mapsto\sum_{uv=w}c(u)\cdot d(v)\ .
  6. w L w \sum_{w\in L}w
  7. R A R\langle\langle A\rangle\rangle
  8. S * = n 0 S n S^{*}=\sum_{n\geq 0}S^{n}
  9. c * ( w ) = u 1 u 2 u n = w c ( u 1 ) c ( u 2 ) c ( u n ) . c^{*}(w)=\sum_{u_{1}u_{2}\cdots u_{n}=w}c(u_{1})c(u_{2})\cdots c(u_{n})\ .
  10. R A R\langle A\rangle

Ray_class_field.html

  1. I m / P m I^{m}/P^{m}\,
  2. U p \prod U_{p}\,
  3. U p \prod U_{p}\,

Rayleigh_dissipation_function.html

  1. N N
  2. F = 1 2 i = 1 N ( k x v i , x 2 + k y v i , y 2 + k z v i , z 2 ) . F=\frac{1}{2}\sum_{i=1}^{N}(k_{x}v_{i,x}^{2}+k_{y}v_{i,y}^{2}+k_{z}v_{i,z}^{2}).
  3. F f = - v F {F}_{f}=-\nabla_{v}F

Reactive_compatibilization.html

  1. Δ G ( m i x ) = Δ H ( m i x ) - T Δ S ( m i x ) \Delta G_{(}mix)=\Delta H_{(}mix)-T\Delta S_{(}mix)
  2. Δ S ( m i x , b l e n d ) = - R ( ϕ 1 x 1 ln ϕ 1 + ϕ 2 x 2 ln ϕ 2 ) \Delta S_{(}mix,blend)=-R\left({\phi_{1}\over x_{1}}\ln\phi_{1}+{\phi_{2}\over x% _{2}}\ln\phi_{2}\right)

Real_projective_line.html

  1. P G L ( 2 , 𝐑 ) PGL(2,\mathbf{R})
  2. V V
  3. V 0 V∖0
  4. 𝐯 𝐰 \mathbf{v}~{}\mathbf{w}
  5. t t
  6. 𝐯 = t 𝐰 \mathbf{v}=t\mathbf{w}
  7. 𝐏 ( 𝐕 ) \mathbf{P}(\mathbf{V})
  8. V V
  9. V V
  10. ( x , y ) ( w , z ) (x,y)~{}(w,z)
  11. t t
  12. ( x , y ) = ( t w , t z ) (x,y)=(tw,tz)
  13. 1 \mathbb{R}\mathbb{P}^{1}
  14. ( x , y ) (x,y)
  15. x x : y xx:y
  16. y 0 y≠0
  17. x : y x:y
  18. P P
  19. x x : y xx:y
  20. ( x , y ) (x,y)
  21. P P
  22. 𝐏 ( 𝐕 ) \mathbf{P}(\mathbf{V})
  23. V V
  24. 𝐏 ( 𝐕 ) \mathbf{P}(\mathbf{V})
  25. 𝐕 \mathbf{V}
  26. 𝐯 𝐰 \mathbf{v}~{}\mathbf{w}
  27. 𝐯 = 𝐰 \mathbf{v}=\mathbf{w}
  28. 𝐯 = 𝐰 \mathbf{v}=−\mathbf{w}
  29. y 0 , [ x : y ] x y y\neq 0,\quad[x:y]\mapsto\frac{x}{y}
  30. x 0 , [ x : y ] y x x\neq 0,\quad[x:y]\mapsto\frac{y}{x}
  31. x x
  32. y y
  33. x [ x : 1 ] . x\mapsto[x:1].
  34. 1 : 00 1:00
  35. 1 : 00 1:00
  36. x x : y xx:y
  37. x y \frac{x}{y}
  38. y 0 y≠0
  39. 0 : 11 0:11
  40. 𝐏 < s u p > 1 ( 𝐑 ) \mathbf{P}<sup>1(\mathbf{R})

Realizable_k-_ε_Model.html

  1. t ( ρ k ) + x j ( ρ k u j ) = x j [ ( μ + μ t σ k ) k x j ] + P k + P b - ρ ϵ - Y M + S k \frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{j}}(\rho ku_{j% })=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}% \right)\frac{\partial k}{\partial x_{j}}\right]+P_{k}+P_{b}-\rho\epsilon-Y_{M}% +S_{k}
  2. t ( ρ ϵ ) + x j ( ρ ϵ u j ) = x j [ ( μ + μ t σ ϵ ) ϵ x j ] + ρ C 1 S ϵ - ρ C 2 ϵ 2 k + ν ϵ + C 1 ϵ ϵ k C 3 ϵ P b + S ϵ \frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{j}}(\rho% \epsilon u_{j})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{% \sigma_{\epsilon}}\right)\frac{\partial\epsilon}{\partial x_{j}}\right]+\rho\,% C_{1}S\epsilon-\rho\,C_{2}\frac{{\epsilon}^{2}}{k+\sqrt{\nu\epsilon}}+C_{1% \epsilon}\frac{\epsilon}{k}C_{3\epsilon}P_{b}+S_{\epsilon}
  3. C 1 = max [ 0.43 , η η + 5 ] , η = S k ϵ , S = 2 S i j S i j C_{1}=\max\left[0.43,\frac{\eta}{\eta+5}\right],\;\;\;\;\;\eta=S\frac{k}{% \epsilon},\;\;\;\;\;S=\sqrt{2S_{ij}S_{ij}}
  4. P k P_{k}
  5. P b P_{b}
  6. μ t = ρ C μ k 2 ϵ \mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon}
  7. C μ = 1 A 0 + A s k U * ϵ C_{\mu}=\frac{1}{A_{0}+A_{s}\frac{kU^{*}}{\epsilon}}
  8. U * S i j S i j + Ω ~ i j Ω ~ i j U^{*}\equiv\sqrt{S_{ij}S_{ij}+\tilde{\Omega}_{ij}\tilde{\Omega}_{ij}}
  9. Ω ~ i j = Ω i j - 2 ϵ i j k ω k \tilde{\Omega}_{ij}=\Omega_{ij}-2\epsilon_{ijk}\omega_{k}
  10. Ω i j = Ω i j ¯ - ϵ i j k ω k \Omega_{ij}=\overline{\Omega_{ij}}-\epsilon_{ijk}\omega_{k}
  11. Ω i j ¯ \overline{\Omega_{ij}}
  12. ω k \omega_{k}
  13. A 0 A_{0}
  14. A s A_{s}
  15. A 0 = 4.04 , A s = 6 cos ϕ A_{0}=4.04,\;\;A_{s}=\sqrt{6}\cos\phi
  16. ϕ = 1 3 cos - 1 ( 6 W ) , W = S i j S j k S k i S ~ 3 , S ~ = S i j S i j , S i j = 1 2 ( u j x i + u i x j ) \phi=\frac{1}{3}\cos^{-1}(\sqrt{6}W),\;\;W=\frac{S_{ij}S_{jk}S_{ki}}{{\tilde{S% }}^{3}},\;\;\tilde{S}=\sqrt{S_{ij}S_{ij}},\;\;S_{ij}=\frac{1}{2}\left(\frac{% \partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right)
  17. C 1 ϵ = 1.44 , C 2 = 1.9 , σ k = 1.0 , σ ϵ = 1.2 C_{1\epsilon}=1.44,\;\;C_{2}=1.9,\;\;\sigma_{k}=1.0,\;\;\sigma_{\epsilon}=1.2

Rectangular_mask_short-time_Fourier_transform.html

  1. w ( t ) = { 1 ; | t | B 0 ; | t | > B w(t)=\begin{cases}\ 1;&|t|\leq B\\ \ 0;&|t|>B\end{cases}
  2. X ( t , f ) = t - B t + B x ( τ ) e - j 2 π f τ d τ X(t,f)=\int_{t-B}^{t+B}x(\tau)e^{-j2\pi f\tau}\,d\tau
  3. x ( t ) = - X ( t 1 , f ) e j 2 π f t d f where t - B < t 1 < t + B x(t)=\int_{-\infty}^{\infty}X(t_{1},f)e^{j2\pi ft}\,df\,\text{ where }t-B<t_{1% }<t+B
  4. - X ( t , f ) d f = t - B t + B x ( τ ) - e - j 2 π f τ d f d τ = t - B t + B x ( τ ) δ ( τ ) d τ = { x ( 0 ) ; | t | < B 0 ; otherwise \int_{-\infty}^{\infty}X(t,f)\,df=\int_{t-B}^{t+B}x(\tau)\int_{-\infty}^{% \infty}e^{-j2\pi f\tau}\,df\,d\tau=\int_{t-B}^{t+B}x(\tau)\delta(\tau)\,d\tau=% \begin{cases}\ x(0);&|t|<B\\ \ 0;&\,\text{otherwise}\end{cases}
  5. - X ( t , f ) e - j 2 π f v d f = { x ( v ) ; v - B < t < v + B 0 ; otherwise \int_{-\infty}^{\infty}X(t,f)e^{-j2\pi fv}\,df=\begin{cases}\ x(v);&v-B<t<v+B% \\ \ 0;&\,\text{otherwise}\end{cases}
  6. t - B t + B x ( τ + τ 0 ) e - j 2 π f τ d τ = X ( t + τ 0 , f ) e j 2 π f τ 0 \int_{t-B}^{t+B}x(\tau+\tau_{0})e^{-j2\pi f\tau}\,d\tau=X(t+\tau_{0},f)e^{j2% \pi f\tau_{0}}
  7. t - B t + B [ x ( τ ) e j 2 π f 0 τ ] d τ = X ( t , f - f 0 ) \int_{t-B}^{t+B}[x(\tau)e^{j2\pi f_{0}\tau}]d\tau=X(t,f-f_{0})
  8. x ( t ) = δ ( t ) , X ( t , f ) = { 1 ; | t | < B 0 ; otherwise x(t)=\delta(t),X(t,f)=\begin{cases}\ 1;&|t|<B\\ \ 0;&\,\text{otherwise}\end{cases}
  9. x ( t ) = 1 , X ( t , f ) = 2 B sinc ( 2 B f ) e j 2 π f t x(t)=1,X(t,f)=2B\operatorname{sinc}(2Bf)e^{j2\pi ft}
  10. h ( t ) = α x ( t ) + β y ( t ) h(t)=\alpha x(t)+\beta y(t)\,
  11. H ( t , f ) , X ( t , f ) , H(t,f),X(t,f),
  12. Y ( t , f ) Y(t,f)\,
  13. H ( t , f ) = α X ( t , f ) + β Y ( t , f ) . H(t,f)=\alpha X(t,f)+\beta Y(t,f).
  14. - | X ( t , f ) | 2 d f = t - B t + B | x ( τ ) | 2 d τ \int_{-\infty}^{\infty}|X(t,f)|^{2}\,df=\int_{t-B}^{t+B}|x(\tau)|^{2}\,d\tau
  15. - - | X ( t , f ) | 2 d f d t = 2 B - | x ( τ ) | 2 d τ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|X(t,f)|^{2}\,df\,dt=2B\int_{-% \infty}^{\infty}|x(\tau)|^{2}\,d\tau
  16. - X ( t , f ) Y * ( t , f ) d f = t - B t + B x ( τ ) y * ( τ ) d τ \int_{-\infty}^{\infty}X(t,f)Y^{*}(t,f)\,df=\int_{t-B}^{t+B}x(\tau)y^{*}(\tau)% \,d\tau
  17. - - X ( t , f ) Y * ( t , f ) d f d t = 2 B - x ( τ ) y * ( τ ) d τ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}X(t,f)Y^{*}(t,f)\,df\,dt=2B\int_% {-\infty}^{\infty}x(\tau)y^{*}(\tau)\,d\tau

Rectified_tesseractic_honeycomb.html

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

Recursive_neural_network.html

  1. p 1 , 2 = tanh ( W [ c 1 ; c 2 ] ) p_{1,2}=\tanh\left(W[c_{1};c_{2}]\right)
  2. n × 2 n n\times 2n

Redfield_equation.html

  1. ρ ρ
  2. t ρ ( t ) = - i [ H , ρ ( t ) ] - 1 2 m [ S m , Λ m ρ ( t ) - ρ ( t ) Λ m ] \frac{\partial}{\partial t}\rho(t)=-\frac{i}{\hbar}[H,\rho(t)]-\frac{1}{\hbar^% {2}}\sum_{m}[S_{m},\Lambda_{m}\rho(t)-\rho(t)\Lambda_{m}^{\dagger}]
  3. H H
  4. S m , Λ m S_{m},\Lambda_{m}
  5. H tot = H + H int + H env H\text{tot}=H+H\text{int}+H\text{env}
  6. H int = n S n E n H\text{int}=\sum_{n}S_{n}E_{n}
  7. S n S_{n}
  8. E n E_{n}
  9. 𝒫 \mathcal{P}
  10. 𝒬 \mathcal{Q}
  11. H int H\text{int}
  12. H 0 , S = H + H env H_{0,S}=H+H\text{env}
  13. t ρ I ( t ) = - 1 2 m , n t 0 t d t ( C m n ( t - t ) [ S m , I ( t ) , S n , I ( t ) ρ I ( t ) ] - C m n ( t - t ) [ S m , I ( t ) , ρ I ( t ) S n , I ( t ) ] ) \frac{\partial}{\partial t}\rho_{I}(t)=-\frac{1}{\hbar^{2}}\sum_{m,n}\int_{t_{% 0}}^{t}dt^{\prime}\biggl(C_{mn}(t-t^{\prime})\Bigl[S_{m,I}(t),S_{n,I}(t^{% \prime})\rho_{I}(t^{\prime})\Bigr]-C_{mn}^{\ast}(t-t^{\prime})\Bigl[S_{m,I}(t)% ,\rho_{I}(t^{\prime})S_{n,I}(t^{\prime})\Bigr]\biggr)
  14. t 0 t_{0}
  15. C m n ( t ) = tr ( E m , I ( t ) E n ρ env,eq ) C_{mn}(t)=\,\text{tr}(E_{m,I}(t)E_{n}\rho\text{env,eq})
  16. ρ env,eq \rho\text{env,eq}
  17. τ r \tau_{r}
  18. τ c \tau_{c}
  19. τ c τ r \tau_{c}\ll\tau_{r}
  20. ρ I ( t ) ρ I ( t ) \rho_{I}(t^{\prime})\approx\rho_{I}(t)
  21. t 0 - t_{0}\to-\infty
  22. t τ = t - t t^{\prime}\to\tau=t-t^{\prime}
  23. t ρ I ( t ) = - 1 2 m , n 0 d τ ( C m n ( τ ) [ S m , I ( t ) , S n , I ( t - τ ) ρ I ( t ) ] - C m n ( τ ) [ S m , I ( t ) , ρ I ( t ) S n , I ( t - τ ) ] ) \frac{\partial}{\partial t}\rho_{I}(t)=-\frac{1}{\hbar^{2}}\sum_{m,n}\int_{0}^% {\infty}d\tau\biggl(C_{mn}(\tau)\Bigl[S_{m,I}(t),S_{n,I}(t-\tau)\rho_{I}(t)% \Bigr]-C_{mn}^{\ast}(\tau)\Bigl[S_{m,I}(t),\rho_{I}(t)S_{n,I}(t-\tau)\Bigr]\biggr)
  24. Λ m = n 0 d τ C m n ( τ ) S n , I ( - τ ) \Lambda_{m}=\sum_{n}\int_{0}^{\infty}d\tau C_{mn}(\tau)S_{n,I}(-\tau)
  25. t ρ ( t ) = - i [ H , ρ ( t ) ] - 1 2 m [ S m , Λ m ρ ( t ) - ρ ( t ) Λ m ] \frac{\partial}{\partial t}\rho(t)=-\frac{i}{\hbar}[H,\rho(t)]-\frac{1}{\hbar^% {2}}\sum_{m}[S_{m},\Lambda_{m}\rho(t)-\rho(t)\Lambda_{m}^{\dagger}]

Redshift_conjecture.html

  1. K ( R ) K(R)

Redundant_proof.html

  1. ψ \psi
  2. κ \kappa
  3. ψ \psi^{\prime}
  4. κ \kappa^{\prime}
  5. κ κ \kappa^{\prime}\subseteq\kappa
  6. κ subsumes κ \kappa^{\prime}\;\,\text{subsumes}\;\kappa
  7. | ψ | < | ψ | |\psi^{\prime}|<|\psi|
  8. | φ | |\varphi|
  9. φ \varphi
  10. ( η η 1 ) ( η η 2 ) or η ( η 1 ( η η 2 ) ) (\eta\odot\eta_{1})\odot(\eta\odot\eta_{2})\,\text{ or }\eta\odot(\eta_{1}% \odot(\eta\odot\eta_{2}))
  11. η ( η 1 η 2 ) \eta\odot(\eta_{1}\odot\eta_{2})
  12. ψ [ η ] \psi\left[\eta\right]
  13. ψ [ - ] \psi\left[-\right]
  14. η \eta
  15. ψ [ ψ 1 [ η p η 1 ] ψ 2 [ η p η 2 ] ] or ψ [ ψ 1 [ η p ( η 1 ψ 2 [ η p η 2 ] ) ] ] \psi[\psi_{1}[\eta\odot_{p}\eta_{1}]\odot\psi_{2}[\eta\odot_{p}\eta_{2}]]\,% \text{ or }\psi[\psi_{1}[\eta\odot_{p}(\eta_{1}\odot\psi_{2}[\eta\odot_{p}\eta% _{2}])]]
  16. ψ [ η p ( ψ 1 [ η 1 ] ψ 2 [ η 2 ] ) ] or η p ψ [ ψ 1 [ η 1 ] ψ 2 [ η 2 ] ] or ψ [ ψ 1 [ η 1 ] ψ 2 [ η 2 ] ] . \psi[\eta\odot_{p}(\psi_{1}[\eta_{1}]\odot\psi_{2}[\eta_{2}])]\,\text{ or }% \eta\odot_{p}\psi[\psi_{1}[\eta_{1}]\odot\psi_{2}[\eta_{2}]]\text{ or }\psi[% \psi_{1}[\eta_{1}]\odot\psi_{2}[\eta_{2}]].
  17. η : p , q η 1 : ¬ p , r q , r p η 3 : ¬ q r q η η 2 : ¬ p , s , ¬ r q , s , ¬ r p η 3 s , ¬ r q ψ : s r \cfrac{\cfrac{\cfrac{\eta:\,p,q\,\,\,\,\eta_{1}:\,\neg p,r}{q,r}p\,\,\,\,\,\,% \begin{array}[]{c}\\ \eta_{3}:\,\neg q\end{array}}{r}q\,\,\,\,\,\,\,\,\,\,\,\,\,\cfrac{\cfrac{\eta% \,\,\,\,\,\,\,\,\,\,\,\,\,\eta_{2}:\,\neg p,s,\neg r}{q,s,\neg r}p\,\,\,\,% \begin{array}[]{c}\\ \eta_{3}\end{array}}{s,\neg r}q}{\psi:\,s}r
  18. ( ( η p η 1 ) η 3 ) ( ( η p η 2 ) η 3 ) . ((\eta\odot_{p}\eta_{1})\odot\eta_{3})\odot((\eta\odot_{p}\eta_{2})\odot\eta_{% 3}).
  19. ψ [ ψ 1 [ η p η 1 ] ψ 2 [ η p η 2 ] ] \psi[\psi_{1}[\eta\odot_{p}\eta_{1}]\odot\psi_{2}[\eta\odot_{p}\eta_{2}]]
  20. ψ 1 [ - ] = ψ 2 [ - ] = _ η 3 and ψ [ - ] = _ \psi_{1}[-]=\psi_{2}[-]=\_\odot\eta_{3}\,\text{ and }\psi[-]=\_
  21. ψ 1 [ η 1 ] ψ 2 [ η 2 ] \psi_{1}[\eta_{1}]\odot\psi_{2}[\eta_{2}]
  22. ψ 1 [ η 1 ] ψ 2 [ η 2 ] = ( η 1 η 3 ) ( η 2 η 3 ) \psi_{1}[\eta_{1}]\odot\psi_{2}[\eta_{2}]=(\eta_{1}\odot\eta_{3})\odot(\eta_{2% }\odot\eta_{3})
  23. η 1 \eta_{1}
  24. η 2 \eta_{2}
  25. η 3 \eta_{3}
  26. q q

Reef_knot.html

  1. 2 μ e μ π 1 2\mu e^{\mu\pi}\geq 1
  2. μ 0.24 \mu\gtrsim 0.24

Reflection_lines.html

  1. C 2 C^{2}
  2. C 1 C^{1}
  3. C 2 C^{2}
  4. p p
  5. M M
  6. n n
  7. v v
  8. r r
  9. r = ( 2 / | n | 2 ) ( ( n v ) n - v ) . r=(2/|n|^{2})((n\cdot v)n-v).
  10. a a
  11. P P
  12. d d
  13. r r
  14. P P
  15. d = r - ( r a ) a d=r-(r\cdot a)a
  16. v a v_{a}
  17. P P
  18. v a = v a ^ / | v a ^ | , v a ^ = v - ( v a ) a v_{a}=\hat{v_{a}}/|\hat{v_{a}}|,\hat{v_{a}}=v-(v\cdot a)a
  19. a a^{\perp}
  20. P P
  21. a a
  22. v a v_{a}
  23. a = a × v a a^{\perp}=a\times v_{a}
  24. θ ( p ) : M ( - π , π ] \theta(p):M\rightarrow(-\pi,\pi]
  25. v a v_{a}
  26. d d
  27. θ = arctan L ( r a , r v a ) \theta=\arctan{L(r\cdot a^{\perp},r\cdot v_{a})}
  28. a r c t a n ( y , x ) arctan(y,x)
  29. ( - π , π ] (-\pi,\pi]
  30. θ > 0 \theta>0
  31. x x
  32. θ = arctan ( n a a , n a x ) \theta=\arctan{(n_{a}\cdot a^{\perp},n_{a}\cdot x)}
  33. n a n_{a}
  34. P P
  35. n a ^ / | n a ^ | , n a ^ = n - ( n a ) a \hat{n_{a}}/|\hat{n_{a}}|,\hat{n_{a}}=n-(n\cdot a)a

Reflexive_sheaf.html

  1. F ( U ) F ( U - Y ) F(U)\to F(U-Y)

Refractive_index_and_extinction_coefficient_of_thin_film_materials.html

  1. k ( E ) = A ( E - E g ) 2 E 2 - B E + C k(E)=\frac{A(E-E_{g})^{2}}{E^{2}-BE+C}
  2. n ( E ) = n ( ) + ( B 0 E + C 0 ) E 2 - B E + C n(E)=n(\infty)+\frac{(B_{0}E+C_{0})}{E^{2}-BE+C}
  3. B 0 = A Q ( - B 2 2 + E g B - E g 2 + C ) B_{0}=\frac{A}{Q}\ \left(\frac{-B^{2}}{2}\ +E_{g}B-{E_{g}}^{2}+C\right)
  4. C 0 = A Q [ ( E g 2 + C ) B 2 - 2 E g C ] C_{0}=\frac{A}{Q}\ \left[({E_{g}}^{2}+C)\frac{B}{2}\ -2E_{g}C\right]
  5. Q = 1 2 ( 4 C - B 2 ) 1 2 Q=\frac{1}{2}\ (4C-B^{2})^{\frac{1}{2}}
  6. k ( E ) = i = 1 q [ A i ( E - E g i ) 2 E 2 - B i E + C i ] k(E)=\sum_{i=1}^{q}\left[\frac{A_{i}(E-E_{g_{i}})^{2}}{E^{2}-B_{i}E+C_{i}}\right]
  7. n ( E ) = n ( ) + i = 1 q [ B 0 i E + C 0 i E 2 - B i E + C i ] n(E)=n(\infty)+\sum_{i=1}^{q}\left[\frac{B_{0_{i}}E+C_{0_{i}}}{E^{2}-B_{i}E+C_% {i}}\right]

Regular_embedding.html

  1. i : X Y i:X\hookrightarrow Y
  2. X U X\cap U
  3. Spec B \operatorname{Spec}B
  4. I / I 2 I/I^{2}
  5. Sym ( I / I 2 ) 0 I n / I n + 1 \operatorname{Sym}(I/I^{2})\to\oplus_{0}^{\infty}I^{n}/I^{n+1}
  6. Spec ( 0 I n / I n + 1 ) \operatorname{Spec}(\oplus_{0}^{\infty}I^{n}/I^{n+1})
  7. f : X Y f:X\to Y
  8. U 𝑗 V 𝑔 Y U\overset{j}{\to}V\overset{g}{\to}Y
  9. X X × Y Y X\to X\times Y\to Y

Regularization_by_spectral_filtering.html

  1. S = { ( x 1 , y 1 ) , , ( x n , y n ) } S=\{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  2. X X
  3. n × d n\times d
  4. Y = ( y 1 , , y n ) Y=(y_{1},\dots,y_{n})
  5. k k
  6. n × n n\times n
  7. K K
  8. K i j = k ( x i , x j ) K_{ij}=k(x_{i},x_{j})
  9. \mathcal{H}
  10. k k
  11. λ \lambda
  12. g G g\in G
  13. f F f\in F
  14. G G
  15. F F
  16. L L
  17. g = L f g=Lf
  18. g g
  19. f f
  20. f f
  21. g g
  22. f f
  23. min f 1 n i = 1 n ( y i - f ( x i ) ) 2 + λ f 2 \min_{f\in\mathcal{H}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-f(x_{i}))^{2}+\lambda\|f% \|^{2}_{\mathcal{H}}
  24. f S λ ( X ) = i = 1 n c i k ( x , x i ) f_{S}^{\lambda}(X)=\sum_{i=1}^{n}c_{i}k(x,x_{i})
  25. ( K + n λ I ) c = Y (K+n\lambda I)c=Y
  26. c = ( c 1 , , c n ) c=(c_{1},\dots,c_{n})
  27. min f 1 n i = 1 n ( y i - f ( x i ) ) 2 \min_{f\in\mathcal{H}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-f(x_{i}))^{2}
  28. f S λ ( X ) = i = 1 n c i k ( x , x i ) f_{S}^{\lambda}(X)=\sum_{i=1}^{n}c_{i}k(x,x_{i})
  29. K c = Y Kc=Y
  30. { min f 1 n i = 1 n ( y i - f ( x i ) ) 2 min f 1 n i = 1 n ( y i - f ( x i ) ) 2 + λ f 2 } { K c = Y ( K + n λ I ) c = Y } . \bigg\{\min_{f\in\mathcal{H}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-f(x_{i}))^{2}% \rightarrow\min_{f\in\mathcal{H}}\frac{1}{n}\sum_{i=1}^{n}(y_{i}-f(x_{i}))^{2}% +\lambda\|f\|^{2}_{\mathcal{H}}\bigg\}\equiv\bigg\{Kc=Y\rightarrow(K+n\lambda I% )c=Y\bigg\}.
  31. K = Q Σ Q T K=Q\Sigma Q^{T}
  32. σ = diag ( σ 1 , , σ n ) , σ 1 σ 2 σ n 0 \sigma=\operatorname{diag}(\sigma_{1},\dots,\sigma_{n}),~{}\sigma_{1}\geq% \sigma_{2}\geq\cdots\geq\sigma_{n}\geq 0
  33. q 1 , , q n q_{1},\dots,q_{n}
  34. c = K - 1 Y = Q Σ - 1 Q T Y = i = 1 n 1 σ i q i , Y q i . c=K^{-1}Y=Q\Sigma^{-1}Q^{T}Y=\sum_{i=1}^{n}\frac{1}{\sigma_{i}}\langle q_{i},Y% \rangle q_{i}.
  35. G λ ( ) G_{\lambda}(\cdot)
  36. K K
  37. λ \lambda
  38. G λ ( K ) G_{\lambda}(K)
  39. f S λ ( X ) := i = 1 n c i k ( x , x i ) f_{S}^{\lambda}(X):=\sum_{i=1}^{n}c_{i}k(x,x_{i})
  40. c = G λ ( K ) Y c=G_{\lambda}(K)Y
  41. G λ ( σ ) G_{\lambda}(\sigma)
  42. G λ ( K ) = Q G λ ( Σ ) Q T , G_{\lambda}(K)=QG_{\lambda}(\Sigma)Q^{T},
  43. G λ ( K ) Y = i = 1 n G λ ( σ i ) q i , Y q i . G_{\lambda}(K)Y~{}=~{}\sum_{i=1}^{n}G_{\lambda}(\sigma_{i})\langle q_{i},Y% \rangle q_{i}.
  44. λ \lambda
  45. G λ ( σ ) 1 / σ G_{\lambda}(\sigma)~{}\rightarrow~{}1/\sigma
  46. G λ G_{\lambda}
  47. λ \lambda
  48. c = ( K + n λ I ) - 1 Y c=(K+n\lambda I)^{-1}Y
  49. c = ( K + n λ I ) - 1 Y = Q ( Σ + n λ I ) - 1 Q T Y = i = 1 n 1 σ i + n λ < q i , Y > q i . c=(K+n\lambda I)^{-1}Y=Q(\Sigma+n\lambda I)^{-1}Q^{T}Y=\sum_{i=1}^{n}\frac{1}{% \sigma_{i}+n\lambda}<q_{i},Y>q_{i}.
  50. σ λ n \sigma\gg\lambda n
  51. 1 σ i + n λ 1 σ i \frac{1}{\sigma_{i}+n\lambda}\sim\frac{1}{\sigma_{i}}
  52. σ λ n \sigma\ll\lambda n
  53. 1 σ i + n λ 1 λ n \frac{1}{\sigma_{i}+n\lambda}\sim\frac{1}{\lambda n}
  54. G λ ( σ ) = 1 σ + n λ . G_{\lambda}(\sigma)=\frac{1}{\sigma+n\lambda}.
  55. c 0 = 0 c^{0}=0\,
  56. for i = 1 , , t - 1 \,\text{for }i=1,\dots,t-1
  57. c i = c i - 1 + η ( Y - K c i - 1 ) ~{}~{}~{}~{}~{}c^{i}=c^{i-1}+\eta(Y-Kc^{i-1})
  58. end \mathrm{end}
  59. n n
  60. K K
  61. η = 2 / n \eta=2/n
  62. 1 n || Y - K c || 2 2 \frac{1}{n}||Y-Kc||_{2}^{2}
  63. t t
  64. c = η i = 0 t - 1 ( I - η K ) i Y . c=\eta\sum_{i=0}^{t-1}(I-\eta K)^{i}Y.
  65. G λ ( σ ) = η i = 0 t - 1 ( I - η σ ) i . G_{\lambda}(\sigma)=\eta\sum_{i=0}^{t-1}(I-\eta\sigma)^{i}.
  66. K - 1 K^{-1}
  67. i 0 x i = 1 / ( 1 - x ) \sum_{i\geq 0}x^{i}=1/(1-x)
  68. x x
  69. K K
  70. I - η K I-\eta K
  71. K - 1 = η i = 0 ( I - η K ) i η i = 0 t - 1 ( I - η K ) i . K^{-1}=\eta\sum_{i=0}^{\infty}(I-\eta K)^{i}\sim\eta\sum_{i=0}^{t-1}(I-\eta K)% ^{i}.
  72. t 1 / λ t\sim 1/\lambda
  73. t t
  74. t t
  75. K = Q Σ Q T K=Q\Sigma Q^{T}
  76. λ n \lambda n
  77. G λ ( σ ) = { 1 / σ , if σ λ n 0 , otherwise . G_{\lambda}(\sigma)=\left\{\begin{array}[c]{lcll}1/\sigma&,&\,\text{if }\sigma% \geq\lambda n\\ 0&,&\,\text{otherwise}\\ \end{array}\right..

Relationship_square.html

  1. j j
  2. F s F_{s}
  3. F s F_{s}
  4. s s
  5. s s
  6. F s F_{s}
  7. s s
  8. F s F_{s}
  9. j j
  10. η 2 ( j , F s ) \eta^{2}(j,F_{s})
  11. i 1 , , i 6 ) i_{1},\ldots,i_{6})
  12. ( q 1 , q 2 , q 3 ) (q_{1},q_{2},q_{3})
  13. i 1 i_{1}
  14. a a
  15. q 1 q_{1}
  16. d d
  17. q 2 q_{2}
  18. f f
  19. q 3 q_{3}
  20. q 1 q_{1}
  21. q 2 q_{2}
  22. q 3 q_{3}
  23. i 1 i_{1}
  24. q 1 q_{1}
  25. q 2 q_{2}
  26. q 3 q_{3}
  27. i 2 i_{2}
  28. q 1 q_{1}
  29. q 2 q_{2}
  30. q 3 q_{3}
  31. i 3 i_{3}
  32. q 1 q_{1}
  33. q 2 q_{2}
  34. q 3 q_{3}
  35. i 4 i_{4}
  36. q 1 q_{1}
  37. q 2 q_{2}
  38. q 3 q_{3}
  39. i 5 i_{5}
  40. q 1 q_{1}
  41. q 2 q_{2}
  42. q 3 q_{3}
  43. i 6 i_{6}
  44. q 1 q_{1}
  45. q 2 q_{2}
  46. q 3 q_{3}
  47. q 3 q_{3}
  48. q 2 q_{2}
  49. q 1 q_{1}
  50. q 3 q_{3}
  51. q 2 q_{2}

Relative_accessible_surface_area.html

  1. R e l A c c = 100 * A c c / M a x A c c RelAcc=100*Acc/MaxAcc
  2. A c c Acc
  3. M a x A c c MaxAcc
  4. Å 2 {\AA}^{2}

Relic_abundance.html

  1. Y n / T 3 , Y\equiv n/T^{3},
  2. n n
  3. n N / V n\equiv N/V
  4. Y Y_{\infty}

Relief_(feature_selection).html

  1. W i = W i - ( x i - nearHit i ) 2 + ( x i - nearMiss i ) 2 W_{i}=W_{i}-(x_{i}-\mathrm{nearHit}_{i})^{2}+(x_{i}-\mathrm{nearMiss}_{i})^{2}

Rendezvous_hashing.html

  1. O O
  2. n n
  3. n n
  4. n n
  5. n + 1 n+1
  6. n n
  7. n + 1 n+1
  8. n n
  9. n + 1 n+1
  10. n + 1 n+1
  11. O O
  12. S S
  13. S S
  14. O O
  15. w i w_{i}
  16. S i S_{i}
  17. h ( S i , O ) h(S_{i},O)
  18. O O
  19. n n
  20. S k S_{k}
  21. S S
  22. w = h ( S , O ) w=h(S,O)
  23. O O
  24. l o g ( n ) log(n)
  25. S S
  26. t i = h c ( S ^ i ) t_{i}=h_{c}(S\hat{}i)
  27. h c h_{c}
  28. h ( S , O ) h(S,O)
  29. 1 min i h c ( O ) - t i \frac{1}{\min_{i}h_{c}(O)-t_{i}}
  30. h c ( O ) - t i h_{c}(O)-t_{i}
  31. h c ( O ) h_{c}(O)
  32. t i t_{i}
  33. h c ( O ) - t i h_{c}(O)-t_{i}

Representation_on_coordinate_rings.html

  1. k [ X ] k[X]
  2. ( g f ) ( x ) = f ( g - 1 x ) (g\cdot f)(x)=f(g^{-1}x)
  3. k [ X ] ( λ ) k[X]_{(\lambda)}
  4. k [ X ] k[X]
  5. V λ V^{\lambda}
  6. λ \lambda
  7. k [ X ] k[X]
  8. k [ X ] = λ k [ X ] ( λ ) k[X]=\bigoplus_{\lambda}k[X]_{(\lambda)}
  9. V λ V^{\lambda}
  10. dim k [ X ] ( λ ) dim V λ \operatorname{dim}k[X]_{(\lambda)}\leq\operatorname{dim}V^{\lambda}
  11. G G
  12. G × G G\times G
  13. ϕ λ : V λ * ( V λ ) H k [ G / H ] ( λ ) \phi_{\lambda}:V^{{\lambda}*}\otimes(V^{\lambda})^{H}\to k[G/H]_{(\lambda)}
  14. ϕ λ ( α v ) ( g H ) = α , g v \phi_{\lambda}(\alpha\otimes v)(gH)=\langle\alpha,g\cdot v\rangle
  15. ϕ λ \phi_{\lambda}
  16. ϕ λ \phi_{\lambda}
  17. G × N G\times N
  18. k [ G / H ] = λ ϕ λ ( V λ * ( V λ ) H ) k[G/H]=\bigoplus_{\lambda}\phi_{\lambda}(V^{{\lambda}*}\otimes(V^{\lambda})^{H})
  19. G × N G\times N
  20. k [ G / H ] ( λ ) k[G/H]_{(\lambda)}
  21. V λ = W V^{\lambda}=W
  22. δ 1 \delta_{1}
  23. δ 1 ( w ) = w ( 1 ) \delta_{1}(w)=w(1)
  24. w ( g H ) = ϕ λ ( δ 1 w ) ( g H ) w(gH)=\phi_{\lambda}(\delta_{1}\otimes w)(gH)
  25. ϕ λ \phi_{\lambda}
  26. k [ G / H ] ( λ ) k[G/H]_{(\lambda)}
  27. ϕ λ \phi_{\lambda}
  28. v λ V λ v_{\lambda}\in V^{\lambda}
  29. G v λ G\cdot v_{\lambda}

Reshetnyak_gluing_theorem.html

  1. X i X_{i}
  2. κ \leq\kappa
  3. C i X i C_{i}\subset X_{i}
  4. X X
  5. X i X_{i}
  6. C i C_{i}
  7. κ \leq\kappa

Resident_Identity_Card.html

  1. a 1 , a 2 , , a 18 a_{1},a_{2},\cdots,a_{18}
  2. a 1 a_{1}
  3. W i = 2 i - 1 mod 11 W_{i}=2^{i-1}\ \bmod\ {11}
  4. S = i = 2 18 a i W i S=\sum_{i=2}^{18}a_{i}\cdot W_{i}
  5. a 1 = ( 12 - ( S mod 11 ) ) mod 11 a_{1}=(12-(S\ \bmod 11))\bmod 11

Resilience_(materials_science).html

  1. U r = σ y 2 2 E U_{r}=\frac{\sigma_{y}^{2}}{2E}

Resolution_inference.html

  1. Γ 1 { } Γ 2 { ¯ } Γ 1 Γ 2 | | \frac{\Gamma_{1}\cup\left\{\ell\right\}\,\,\,\,\Gamma_{2}\cup\left\{\overline{% \ell}\right\}}{\Gamma_{1}\cup\Gamma_{2}}|\ell|
  2. Γ 1 { } \Gamma_{1}\cup\left\{\ell\right\}
  3. Γ 2 { ¯ } \Gamma_{2}\cup\left\{\overline{\ell}\right\}
  4. Γ 1 Γ 2 \Gamma_{1}\cup\Gamma_{2}
  5. \ell
  6. ¯ \overline{\ell}
  7. | | |\ell|
  8. Γ 1 { L 1 } Γ 2 { L 2 } ( Γ 1 Γ 2 ) ϕ ϕ \frac{\Gamma_{1}\cup\left\{L_{1}\right\}\,\,\,\,\Gamma_{2}\cup\left\{L_{2}% \right\}}{(\Gamma_{1}\cup\Gamma_{2})\phi}\phi
  9. ϕ \phi
  10. L 1 L_{1}
  11. L 2 ¯ \overline{L_{2}}
  12. Γ 1 \Gamma_{1}
  13. Γ 2 \Gamma_{2}
  14. P ( x ) , Q ( x ) P(x),Q(x)
  15. ¬ P ( b ) \neg P(b)
  16. [ b / x ] [b/x]
  17. P ( x ) , Q ( x ) ¬ P ( b ) Q ( B ) [ b / x ] \frac{P(x),Q(x)\,\,\,\,\neg P(b)}{Q(B)}[b/x]
  18. P ( x ) , Q ( x ) P(x),Q(x)
  19. ¬ P ( x ) \neg P(x)
  20. Q ( b ) Q(b)
  21. P ( x ) P(x)
  22. ¬ P ( b ) \neg P(b)
  23. P P
  24. [ b / x ] [b/x]

Resonances_in_scattering_from_potentials.html

  1. ψ ( r ) = e i ( k r ) \psi(\vec{r})=e^{i(\vec{k}\cdot\vec{r})}
  2. T T
  3. T = | J trans | | J inc | T=\frac{|\vec{J}_{\mathrm{trans}}|}{|\vec{J}_{\mathrm{inc}}|}
  4. J \vec{J}
  5. σ \sigma
  6. σ l \sigma\text{l}
  7. k \vec{k}
  8. E = 2 | k | 2 2 m E=\frac{\hbar^{2}|\vec{k}|^{2}}{2m}
  9. T T
  10. σ l \sigma\text{l}
  11. V ( x ) = { V 0 , 0 < x < L , 0 , otherwise, , V(x)=\begin{cases}V_{0},&0<x<L,\\ 0,&\,\text{otherwise,}\end{cases},
  12. V 0 V_{0}
  13. E > V 0 E>V_{0}
  14. - 2 2 m d 2 ψ d x 2 + V ( x ) ψ = E ψ -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+V(x)\psi=E\psi
  15. x < 0 , 0 < x < L , x > L x<0,0<x<L,x>L
  16. ψ 1 ( x ) = { A 1 e i k 1 x + B 1 e - i k 1 x , x < 0 , A 2 e i k 2 x + B 2 e - i k 2 x , 0 < x < L , A 3 e i k 1 x + B 3 e - i k 1 x , x > L , \psi_{1}(x)=\begin{cases}A_{1}e^{ik_{1}x}+B_{1}e^{-ik_{1}x},&x<0,\\ A_{2}e^{ik_{2}x}+B_{2}e^{-ik_{2}x},&0<x<L,\\ A_{3}e^{ik_{1}x}+B_{3}e^{-ik_{1}x},&x>L,\end{cases}
  17. k 1 k_{1}
  18. k 2 k_{2}
  19. k 1 = 2 m E , k_{1}=\frac{\sqrt{2mE}}{\hbar},
  20. k 2 = 2 m ( E - V 0 ) , k_{2}=\frac{\sqrt{2m(E-V_{0})}}{\hbar},
  21. T T
  22. A 3 = 0 A_{3}=0
  23. ψ ( x ) \psi(x)
  24. d ψ d x \frac{d\psi}{dx}
  25. x = 0 x=0
  26. x = L x=L
  27. T T
  28. T = | A 3 | 2 | A 1 | 2 = 4 E ( E - V 0 ) 4 E ( E - V 0 ) + V 0 2 sin 2 [ 2 m ( E - V 0 ) L ] T=\frac{|A_{3}|^{2}}{|A_{1}|^{2}}=\frac{4E(E-V_{0})}{4E(E-V_{0})+V_{0}^{2}\sin% ^{2}[\sqrt{2m(E-V_{0})}\frac{L}{\hbar}]}
  29. T T
  30. sin 2 [ 2 m ( E - V 0 ) L ] = 0 ,or k 2 = n π L \sin^{2}[\sqrt{2m(E-V_{0})}\frac{L}{\hbar}]=0\,\text{,or }k_{2}=\frac{n\pi}{L}
  31. T T
  32. 2 L 2L
  33. λ = 2 π k \lambda=\frac{2\pi}{k}
  34. E > V 0 E>V_{0}
  35. x = 0 x=0
  36. x = 0 x=0
  37. x = L x=L
  38. x = 0 x=0
  39. L L
  40. [ 1 + V 0 2 4 E ( E - V 0 ) ] - 1 [1+\frac{V_{0}^{2}}{4E(E-V_{0})}]^{-1}
  41. π k 2 \frac{\pi}{k_{2}}
  42. E V 0 E>>V_{0}
  43. T 1 T\rightarrow 1
  44. T T
  45. T T
  46. E E
  47. 2 m V 0 2 L 2 2 \sqrt{\frac{2mV_{0}^{2}L^{2}}{\hbar^{2}}}

Resonant_ultrasound_spectroscopy.html

  1. L = V ( K E - P E ) d V ( 1 ) L=\int_{V}(KE-PE)dV(1)
  2. K E = 1 2 i ρ ω 2 u i 2 ( 2 ) KE=\frac{1}{2}\sum_{i}\rho\omega^{2}u_{i}^{2}(2)
  3. P E = 1 2 i , j , k , l c i , j , k , l d u i d x j d u k d x l ( 3 ) PE=\frac{1}{2}\sum_{i,j,k,l}c_{i,j,k,l}\frac{du_{i}}{dx_{j}}\frac{du_{k}}{dx_{% l}}(3)
  4. u i u_{i}
  5. c i , j , k , l c_{i,j,k,l}
  6. δ L = V { i [ ρ ω 2 u i - j , k , l c i , j , k , l δ 2 u k δ x j δ x l ] δ u i } d V - S { i [ j , k , l n c i , j , k , l δ u k δ x l ] d u i } d S ( 4 ) \delta L=\int_{V}\Bigl\{\sum_{i}\Bigl[\rho\omega^{2}u_{i}-\sum_{j,k,l}c_{i,j,k% ,l}\frac{\delta^{2}u_{k}}{\delta x_{j}\delta x_{l}}\Bigr]\delta u_{i}\Bigr\}dV% -\int_{S}\Bigl\{\sum_{i}\Bigl[\sum_{j,k,l}\vec{n}c_{i,j,k,l}\frac{\delta u_{k}% }{\delta x_{l}}\Bigr]du_{i}\Bigr\}dS(4)
  7. u i u_{i}
  8. n \vec{n}
  9. u i u_{i}
  10. u i u_{i}
  11. ω 2 E a = Γ a ( 5 ) \omega^{2}Ea=\Gamma a(5)
  12. f n c a l f_{n}^{cal}
  13. f n m e a f_{n}^{mea}
  14. F = n w n ( f n c a l - f n m e a ) 2 ( 6 ) F=\sum_{n}w_{n}(f_{n}^{cal}-f_{n}^{mea})^{2}(6)
  15. w n w_{n}
  16. f n f_{n}

Revised_simplex_method.html

  1. minimize s y m b o l c T s y m b o l x subject to s y m b o l A x = s y m b o l b , s y m b o l x s y m b o l 0 \begin{array}[]{rl}\,\text{minimize}&symbol{c}^{\mathrm{T}}symbol{x}\\ \,\text{subject to}&symbol{Ax}=symbol{b},symbol{x}\geq symbol{0}\end{array}
  2. 𝐀 \mathbf{A}
  3. 𝐱 𝟎 \mathbf{x}≥\mathbf{0}
  4. 𝐀𝐱 = 𝐛 \mathbf{Ax}=\mathbf{b}
  5. 𝐀 \mathbf{A}
  6. s y m b o l A x = s y m b o l b , s y m b o l A T s y m b o l λ + s y m b o l s = s y m b o l c , s y m b o l x s y m b o l 0 , s y m b o l s s y m b o l 0 , s y m b o l s T s y m b o l x = 0 \begin{aligned}\displaystyle symbol{Ax}&\displaystyle=symbol{b},\\ \displaystyle symbol{A}^{\mathrm{T}}symbol{\lambda}+symbol{s}&\displaystyle=% symbol{c},\\ \displaystyle symbol{x}&\displaystyle\geq symbol{0},\\ \displaystyle symbol{s}&\displaystyle\geq symbol{0},\\ \displaystyle symbol{s}^{\mathrm{T}}symbol{x}&\displaystyle=0\end{aligned}
  7. λ \mathbf{λ}
  8. 𝐬 \mathbf{s}
  9. 𝐀𝐱 = 𝐛 \mathbf{Ax}=\mathbf{b}
  10. 𝐱 𝟎 \mathbf{x}≥\mathbf{0}
  11. s y m b o l B T s y m b o l λ = s y m b o l c B , s y m b o l N T s y m b o l λ + s y m b o l s N = s y m b o l c N , \begin{aligned}\displaystyle symbol{B}^{\mathrm{T}}symbol{\lambda}&% \displaystyle=symbol{c_{B}},\\ \displaystyle symbol{N}^{\mathrm{T}}symbol{\lambda}+symbol{s_{N}}&% \displaystyle=symbol{c_{N}},\end{aligned}
  12. s y m b o l λ = ( s y m b o l B T ) - 1 s y m b o l c B , s y m b o l s N = s y m b o l c N - s y m b o l N T s y m b o l λ . \begin{aligned}\displaystyle symbol{\lambda}&\displaystyle=(symbol{B}^{\mathrm% {T}})^{-1}symbol{c_{B}},\\ \displaystyle symbol{s_{N}}&\displaystyle=symbol{c_{N}}-symbol{N}^{\mathrm{T}}% symbol{\lambda}.\end{aligned}
  13. 𝐱 \mathbf{x}
  14. 𝐍 \mathbf{N}
  15. 𝐁 \mathbf{B}
  16. s y m b o l B x B + s y m b o l A q x q = s y m b o l b , symbol{Bx_{B}}+symbol{A}_{q}x_{q}=symbol{b},
  17. 𝐝 𝟎 \mathbf{d}≤\mathbf{0}
  18. s y m b o l c = [ - 2 - 3 - 4 0 0 ] T , s y m b o l A = [ 3 2 1 1 0 2 5 3 0 1 ] , s y m b o l b = [ 10 15 ] . \begin{aligned}\displaystyle symbol{c}&\displaystyle=\begin{bmatrix}-2&-3&-4&0% &0\end{bmatrix}^{\mathrm{T}},\\ \displaystyle symbol{A}&\displaystyle=\begin{bmatrix}3&2&1&1&0\\ 2&5&3&0&1\end{bmatrix},\\ \displaystyle symbol{b}&\displaystyle=\begin{bmatrix}10\\ 15\end{bmatrix}.\end{aligned}
  19. s y m b o l B = [ s y m b o l A 4 s y m b o l A 5 ] , s y m b o l N = [ s y m b o l A 1 s y m b o l A 2 s y m b o l A 3 ] \begin{aligned}\displaystyle symbol{B}&\displaystyle=\begin{bmatrix}symbol{A}_% {4}&symbol{A}_{5}\end{bmatrix},\\ \displaystyle symbol{N}&\displaystyle=\begin{bmatrix}symbol{A}_{1}&symbol{A}_{% 2}&symbol{A}_{3}\end{bmatrix}\end{aligned}
  20. s y m b o l λ = [ 0 0 ] T , s y m b o l s N = [ - 2 - 3 - 4 ] T . \begin{aligned}\displaystyle symbol{\lambda}&\displaystyle=\begin{bmatrix}0&0% \end{bmatrix}^{\mathrm{T}},\\ \displaystyle symbol{s_{N}}&\displaystyle=\begin{bmatrix}-2&-3&-4\end{bmatrix}% ^{\mathrm{T}}.\end{aligned}
  21. q = 3 q=3
  22. 1 1
  23. 3 3
  24. 5 5
  25. p = 5 p=5
  26. s y m b o l B = [ s y m b o l A 3 s y m b o l A 4 ] , s y m b o l N = [ s y m b o l A 1 s y m b o l A 2 s y m b o l A 5 ] . \begin{aligned}\displaystyle symbol{B}&\displaystyle=\begin{bmatrix}symbol{A}_% {3}&symbol{A}_{4}\end{bmatrix},\\ \displaystyle symbol{N}&\displaystyle=\begin{bmatrix}symbol{A}_{1}&symbol{A}_{% 2}&symbol{A}_{5}\end{bmatrix}.\end{aligned}
  27. s y m b o l x = [ 0 0 5 5 0 ] T , s y m b o l λ = [ 0 - 4 / 3 ] T , s y m b o l s N = [ 2 / 3 11 / 3 4 ] . \begin{aligned}\displaystyle symbol{x}&\displaystyle=\begin{bmatrix}0&0&5&5&0% \end{bmatrix}^{\mathrm{T}},\\ \displaystyle symbol{\lambda}&\displaystyle=\begin{bmatrix}0&-4/3\end{bmatrix}% ^{\mathrm{T}},\\ \displaystyle symbol{s_{N}}&\displaystyle=\begin{bmatrix}2/3&11/3&4\end{% bmatrix}.\end{aligned}
  28. 𝐱 \mathbf{x}
  29. 𝐁 \mathbf{B}
  30. s y m b o l B z \displaystyle symbol{Bz}
  31. 𝐁 \mathbf{B}

Revision_theory.html

  1. G G
  2. G G
  3. G G
  4. G G
  5. = d f =_{df}
  6. G G
  7. G G
  8. G = d f G=_{df}
  9. G G
  10. G G
  11. G G
  12. L L
  13. M M
  14. D D
  15. I I
  16. 𝒟 \mathcal{D}
  17. G 1 x ¯ \displaystyle G_{1}\overline{x}
  18. A G i A_{G_{i}}
  19. G j G_{j}
  20. G i G_{i}
  21. x ¯ \overline{x}
  22. A G i A_{G_{i}}
  23. G 1 , , G n , G_{1},\ldots,G_{n},\ldots
  24. L L
  25. 𝒟 \mathcal{D}
  26. G x ¯ = D f A ( x ¯ , G ) G\overline{x}=_{Df}A(\overline{x},G)
  27. A A
  28. G G
  29. h h
  30. 𝒟 \mathcal{D}
  31. M + h M+h
  32. M M
  33. h h
  34. G i ( t ¯ ) G_{i}(\overline{t})
  35. M + h M+h
  36. M + h G i ( t ¯ ) iff I ( t ¯ ) h ( G i ) M+h\models G_{i}(\overline{t})\,\text{ iff }I(\overline{t})\in h(G_{i})
  37. 𝒟 \mathcal{D}
  38. δ M , 𝒟 \delta_{M,\mathcal{D}}
  39. G G
  40. 𝒟 \mathcal{D}
  41. M + δ M , 𝒟 ( h ) G ( t ¯ ) iff M + h A G ( t ¯ ) M+\delta_{M,\mathcal{D}}(h)\models G(\overline{t})\,\text{ iff }M+h\models A_{% G}(\overline{t})
  42. G G
  43. G G
  44. A G A_{G}
  45. A G A_{G}
  46. G G
  47. M + h M+h
  48. ω \omega
  49. 𝒮 \mathcal{S}
  50. 𝒮 α \mathcal{S}_{\alpha}
  51. α \alpha
  52. 𝒮 \mathcal{S}
  53. ω \omega
  54. 𝒮 \mathcal{S}
  55. n n
  56. 𝒮 n + 1 = δ M , 𝒟 ( 𝒮 n ) . \mathcal{S}_{n+1}=\delta_{M,\mathcal{D}}(\mathcal{S}_{n}).
  57. δ M , 𝒟 0 ( h ) = h \delta_{M,\mathcal{D}}^{0}(h)=h
  58. δ M , 𝒟 n + 1 ( h ) = δ M , 𝒟 n ( δ M , 𝒟 ( h ) ) . \delta_{M,\mathcal{D}}^{n+1}(h)=\delta_{M,\mathcal{D}}^{n}(\delta_{M,\mathcal{% D}}(h)).
  59. ω \omega
  60. h h
  61. h , δ M , 𝒟 ( h ) , δ M , 𝒟 2 ( h ) , h,\delta_{M,\mathcal{D}}(h),\delta_{M,\mathcal{D}}^{2}(h),\ldots
  62. S 0 S_{0}
  63. A A
  64. S 0 S_{0}
  65. M M
  66. D {D}
  67. n n
  68. h h
  69. m n m\geq n
  70. M + δ M , 𝒟 m ( h ) A M+\delta_{M,\mathcal{D}}^{m}(h)\models A
  71. A A
  72. D D
  73. M M
  74. S 0 S_{0}
  75. ω \omega
  76. A A
  77. α {\alpha}
  78. β α \beta\geq\alpha
  79. M + 𝒮 β A M+{\mathcal{S}_{\beta}}\models A
  80. A A
  81. α {\alpha}
  82. β α \beta\geq\alpha
  83. M + 𝒮 β ⊧̸ A M+{\mathcal{S}_{\beta}}\not\models A
  84. A A
  85. S 0 S_{0}
  86. M M
  87. A A
  88. ω \omega
  89. M M
  90. 𝒟 1 \mathcal{D}_{1}
  91. G x = D f ( x = a & G x ) ( x = b & G b ) . Gx=_{Df}(x=a\ \&\ \sim Gx)\lor(x=b\ \&\ Gb).
  92. M M
  93. I ( a ) = a I(a)=a
  94. I ( b ) = b I(b)=b
  95. M M
  96. \emptyset
  97. 𝒟 1 \mathcal{D}_{1}
  98. \emptyset
  99. \emptyset
  100. \emptyset
  101. \emptyset
  102. a a
  103. G G
  104. b b
  105. 𝒟 2 \mathcal{D}_{2}
  106. H x = D f H x H x . Hx=_{Df}Hx\lor\sim Hx.
  107. 𝒟 2 \mathcal{D}_{2}
  108. \emptyset
  109. L {L}
  110. < <
  111. k ¯ \overline{k}
  112. \mathbb{N}
  113. ω \omega
  114. I I
  115. I ( k ¯ ) = k I(\overline{k})=k
  116. I ( < ) I(<)
  117. 𝒟 3 \mathcal{D}_{3}
  118. J x = D f y ( y < x J y ) . Jx=_{Df}\forall y(y<x\supset Jy).
  119. h h
  120. \emptyset
  121. , { 0 } , { 0 , 1 } , { 0 , 1 , 2 , } , \varnothing,\ \{0\},\ \{0,1\},\ \{0,1,2,\},\ \ldots
  122. n n
  123. J n ¯ J\overline{n}
  124. \mathbb{N}
  125. x J x \forall xJx
  126. \mathbb{N}
  127. J J
  128. J J
  129. \mathbb{N}
  130. J J
  131. C 0 C_{0}
  132. A i {A}^{i}
  133. i i
  134. B i B^{i}
  135. C i C^{i}
  136. ( B & C ) i (B\&C)^{i}
  137. & \&
  138. B i B^{i}
  139. \vdots
  140. i \bot^{i}
  141. B i \sim B^{i}
  142. \sim
  143. G x ¯ = D f A G ( x ¯ ) G\overline{x}=_{Df}A_{G}(\overline{x})
  144. D D
  145. A G ( t ¯ ) i A_{G}(\overline{t})^{i}
  146. G ( t ¯ ) i + 1 G(\overline{t})^{i+1}
  147. G ( t ¯ ) i + 1 G(\overline{t})^{i+1}
  148. A G ( t ¯ ) i A_{G}(\overline{t})^{i}
  149. t ¯ \overline{t}
  150. x ¯ \overline{x}
  151. A G A_{G}
  152. B B
  153. L {L}
  154. B i B^{i}
  155. B j B^{j}
  156. i i
  157. j j
  158. C 0 C_{0}
  159. S 0 S_{0}
  160. S 0 S_{0}
  161. C 0 C_{0}
  162. S 0 S_{0}
  163. S 0 S_{0}
  164. 𝒟 3 \mathcal{D}_{3}
  165. J J
  166. x J x \forall xJx
  167. x J x \forall xJx
  168. J J
  169. S 0 S_{0}
  170. O n On
  171. O n On
  172. 𝒮 \mathcal{S}
  173. O n On
  174. d ¯ \overline{d}
  175. G G
  176. β \beta
  177. 𝒮 \mathcal{S}
  178. α β \alpha\leq\beta
  179. γ \gamma
  180. α γ < β \alpha\leq\gamma<\beta
  181. d ¯ 𝒮 γ \overline{d}\in\mathcal{S}_{\gamma}
  182. d ¯ \overline{d}
  183. G G
  184. β \beta
  185. α \alpha
  186. γ \gamma
  187. α γ < β \alpha\leq\gamma<\beta
  188. d ¯ 𝒮 γ \overline{d}\not\in\mathcal{S}_{\gamma}
  189. d ¯ \overline{d}
  190. β \beta
  191. 𝒮 \mathcal{S}
  192. h h
  193. 𝒮 \mathcal{S}
  194. β \beta
  195. d ¯ \overline{d}
  196. d ¯ \overline{d}
  197. G G
  198. β \beta
  199. 𝒮 \mathcal{S}
  200. d ¯ [ ] h ( G ) \overline{d}\in[\not\in]h(G)
  201. O n On
  202. 𝒮 \mathcal{S}
  203. α \alpha
  204. α = β + 1 \alpha=\beta+1
  205. 𝒮 α = δ M , 𝒟 ( 𝒮 β ) \mathcal{S}_{\alpha}=\delta_{M,\mathcal{D}}(\mathcal{S}_{\beta})
  206. α \alpha
  207. 𝒮 α \mathcal{S}_{\alpha}
  208. 𝒮 \mathcal{S}
  209. α \alpha
  210. ω \omega
  211. 𝒮 0 \mathcal{S}_{0}
  212. O n On
  213. S * S^{*}
  214. A A
  215. S * S^{*}
  216. M M
  217. 𝒟 \mathcal{D}
  218. O n On
  219. S {S}
  220. α \alpha
  221. A A
  222. 𝒮 \mathcal{S}
  223. α \alpha
  224. A A
  225. S * S^{*}
  226. 𝒟 \mathcal{D}
  227. M M
  228. A A
  229. S * S^{*}
  230. M M
  231. 𝒟 \mathcal{D}
  232. S # S^{\#}
  233. A {A}
  234. 𝒮 \mathcal{S}
  235. α \alpha
  236. β α \beta\geq\alpha
  237. n n
  238. m n m\geq n
  239. M + δ M , 𝒟 m ( 𝒮 β ) A . M+\delta_{M,\mathcal{D}}^{m}(\mathcal{S}_{\beta})\models A.
  240. A {A}
  241. 𝒮 \mathcal{S}
  242. α \alpha
  243. β α \beta\geq\alpha
  244. n n
  245. m n m\geq n
  246. M + δ M , 𝒟 m ( 𝒮 β ) ⊧̸ A . M+\delta_{M,\mathcal{D}}^{m}(\mathcal{S}_{\beta})\not\models A.
  247. A A
  248. S # S^{\#}
  249. M M
  250. O n On
  251. S {S}
  252. α \alpha
  253. A A
  254. 𝒮 \mathcal{S}
  255. α \alpha
  256. A A
  257. S # S^{\#}
  258. S # S^{\#}
  259. S * S^{*}
  260. S # S^{\#}
  261. 𝒟 3 \mathcal{D}_{3}
  262. x J x \forall xJx
  263. \mathbb{N}
  264. S 0 S_{0}
  265. S # S^{\#}
  266. S # S^{\#}
  267. S * S^{*}
  268. C 0 C_{0}
  269. S # S^{\#}
  270. C 0 C_{0}
  271. S 0 S_{0}
  272. S # S^{\#}
  273. S 0 S_{0}
  274. S * S^{*}
  275. C 0 C_{0}
  276. S * S^{*}
  277. S # S^{\#}
  278. S 0 S_{0}
  279. h h
  280. n > 0 n>0
  281. h = δ M , 𝒟 n ( h ) h=\delta_{M,\mathcal{D}}^{n}(h)
  282. M M
  283. h h
  284. n n
  285. δ M , 𝒟 n ( h ) \delta_{M,\mathcal{D}}^{n}(h)
  286. 𝒟 \mathcal{D}
  287. S # S^{\#}
  288. S 0 S_{0}
  289. L {L}
  290. 𝒟 \mathcal{D}
  291. 𝒟 \mathcal{D}
  292. G x ¯ = D f A ( x ¯ , G ) G\overline{x}=_{Df}A(\overline{x},G)
  293. A 0 ( x ¯ , G ) = G x ¯ A^{0}(\overline{x},G)=G\overline{x}
  294. A n + 1 ( x ¯ , G ) = A n ( x ¯ , G ) [ A ( t ¯ , G ) / G t ¯ ] A^{n+1}(\overline{x},G)=A^{n}(\overline{x},G)[A(\overline{t},G)/G\overline{t}]
  295. A n + 1 A^{n+1}
  296. G t ¯ G\overline{t}
  297. A n A^{n}
  298. A ( t ¯ , G ) A(\overline{t},G)
  299. 𝒟 \mathcal{D}
  300. 𝒟 n \mathcal{D}^{n}
  301. B B
  302. 𝒟 \mathcal{D}
  303. B n B^{n}
  304. 𝒟 n \mathcal{D}^{n}
  305. G x = D f A ( x , G ) , Gx=_{Df}A(x,G),
  306. x ( G x A ( x , G ) ) , \forall x(Gx\equiv A(x,G)),
  307. G x = D f G x Gx=_{Df}\sim Gx
  308. ¬ \lnot
  309. 𝐭 \,\textbf{t}
  310. 𝐟 \,\textbf{f}
  311. 𝐧 \,\textbf{n}
  312. 𝐟 \,\textbf{f}
  313. 𝐟 \,\textbf{f}
  314. 𝐭 \,\textbf{t}
  315. G x = D f ( x is even & G x ) ( x is odd & G x ) Gx=_{Df}(x\,\text{ is even }\&\ Gx)\vee(x\,\text{ is odd }\&\ \sim Gx)
  316. h h
  317. h = δ M , 𝒟 ( h ) h=\delta_{M,\mathcal{D}}(h)
  318. 𝒟 \mathcal{D}
  319. h ( G ) δ M , 𝒟 ( h ) ( G ) h(G)\subseteq\delta_{M,\mathcal{D}}(h)(G)
  320. G G
  321. 𝒟 \mathcal{D}
  322. Π 2 1 \Pi^{1}_{2}
  323. S # S^{\#}
  324. A A
  325. A A
  326. = D f =_{Df}
  327. b b
  328. b b
  329. b b
  330. b b
  331. S {S}
  332. S α = S α + 1 {S}_{\alpha}={S}_{\alpha+1}
  333. S # S^{\#}
  334. S * S^{*}
  335. A ( T ( A ) T ( A ) ) \forall A(T(\ulcorner\sim A\urcorner)\equiv\sim T(\ulcorner A\urcorner))
  336. A , B ( T ( A & B ) T ( A ) & T ( B ) ) \forall A,B(T(\ulcorner{A\&B}\urcorner)\equiv T(\ulcorner{A}\urcorner)\&T(% \ulcorner{B}\urcorner))
  337. A , B ( T ( A B ) T ( A ) T ( B ) ) \forall A,B(T(\ulcorner{A\lor B}\urcorner)\equiv T(\ulcorner{A}\urcorner)\lor T% (\ulcorner{B}\urcorner))
  338. A ( T ( x A ) t T ( A [ x / t ] ) ) \forall A(T(\ulcorner\forall xA\urcorner)\equiv\forall tT(\ulcorner A[x/t]% \urcorner))
  339. S # S^{\#}
  340. S * S^{*}
  341. a = T a a=\ulcorner{\sim Ta}\urcorner
  342. T T a T\ulcorner{Ta}\urcorner
  343. T T a T\ulcorner{\sim Ta}\urcorner
  344. T T a \sim T\ulcorner{Ta}\urcorner
  345. T T a T\ulcorner{\sim Ta}\urcorner
  346. S # S^{\#}
  347. ω \omega
  348. S * S^{*}
  349. ω \omega
  350. S # S^{\#}
  351. s , t ( T ( s = t ) s = t ) \forall s,t(T(\ulcorner{s=t}\urcorner)\equiv s=t)
  352. A \vdash A
  353. T ( A ) \vdash T(\ulcorner A\urcorner)
  354. T ( A ) \vdash T(\ulcorner A\urcorner)
  355. A \vdash A
  356. ω \omega
  357. x ( ( Sent ( x ) & Bew P A ( x ) ) T x ) , \forall x((\mathrm{Sent}(x)\ \&\ \mathrm{Bew}_{PA}(x))\supset Tx),
  358. Bew P A \mathrm{Bew}_{PA}
  359. Sent \mathrm{Sent}
  360. S # S^{\#}
  361. ω \omega
  362. S * S^{*}
  363. x x
  364. x n + 1 x n x 1 x . \cdots x_{n+1}\in x_{n}\in\cdots\in x_{1}\in x.
  365. x = { x } x=\{x\}
  366. S # S^{\#}
  367. S * S^{*}
  368. C 0 C_{0}
  369. x ( G x A ( x , G ) ) \forall x(Gx\equiv\Box A(x,G))

Revolving_armature_alternator.html

  1. π 2 2 1.11072073 \frac{\pi}{2\sqrt{2}}\approx 1.11072073

Reynolds_stress_equation_model.html

  1. - ρ u i u j -\rho u_{i}^{\prime}u_{j}^{\prime}
  2. μ t ( U i x j + U i x i ) - 2 3 ρ k δ i j \mu t\left(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{i}}{% \partial x_{i}}\right)-\frac{2}{3}\rho k\delta_{ij}
  3. 2 μ t E i j - 2 3 ρ k δ i j 2\mu tE_{ij}-\frac{2}{3}\rho k\delta_{ij}
  4. R i j = u i u j = - τ i j / ρ R_{ij}=u_{i}^{\prime}u_{j}^{\prime}=-\tau_{ij}/\rho
  5. D R i j D t = D i j + P i j + Π i j + Ω i j - ε i j \frac{DR_{ij}}{Dt}=D_{ij}+P_{ij}+\Pi_{ij}+\Omega_{ij}-\varepsilon_{ij}
  6. R i j R_{ij}
  7. R i j R_{ij}
  8. R i j R_{ij}
  9. R i j R_{ij}
  10. R i j R_{ij}
  11. R i j R_{ij}
  12. R i j R_{ij}
  13. P i j P_{ij}
  14. - ( R i m U j x m + R j m U i x m ) -\left(R_{im}\frac{\partial U_{j}}{\partial x_{m}}+R_{jm}\frac{\partial U_{i}}% {\partial x_{m}}\right)
  15. ϵ ij \epsilon_{\rm ij}
  16. ϵ ij \epsilon_{\rm ij}
  17. 2 / 3 ϵ δ i j 2/3\epsilon\delta_{ij}
  18. ϵ \epsilon
  19. δ i j \delta_{ij}
  20. D i j D_{ij}
  21. D i j D_{ij}
  22. D i j D_{ij}
  23. x m ( v t σ k R i j x m ) \frac{\partial}{\partial x_{m}}\left(\frac{v_{t}}{\sigma_{k}}\frac{\partial R_% {ij}}{\partial x_{m}}\right)
  24. d i v ( v t σ k ( R i j ) ) div\left(\frac{v_{t}}{\sigma_{k}}\nabla(R_{ij})\right)
  25. υ t \upsilon_{t}
  26. C μ k 2 ϵ C_{\mu}\frac{k^{2}}{\epsilon}
  27. σ k \sigma_{k}
  28. C μ C_{\mu}
  29. Π i j = - C 1 ϵ k ( R i j - 2 3 k δ i j ) - C 2 ( P i j - 2 3 P δ i j ) \Pi_{ij}=-C_{1}\frac{\epsilon}{k}\left(R_{ij}-\frac{2}{3}k\delta_{ij}\right)-C% _{2}\left(P_{ij}-\frac{2}{3}P\delta_{ij}\right)
  30. Ω i j = - 2 ω k ( R j m e i k m + R i m e j k m ) \Omega_{ij}=-2\omega_{k}\left(R_{jm}e_{ikm}+R_{im}e_{jkm}\right)
  31. ω k \omega_{k}
  32. e i j k e_{ijk}
  33. e i j k e_{ijk}
  34. e i j k e_{ijk}

Rheological_weldability.html

  1. γ S G = γ S L + γ L G cos θ c , \gamma_{SG}\ =\gamma_{SL}+\gamma_{LG}\cos{\theta_{c}},
  2. γ S G \gamma_{SG}
  3. γ S L \gamma_{SL}
  4. γ L G \gamma_{LG}
  5. θ c \theta_{c}
  6. θ C \theta_{C}
  7. γ S G \gamma_{SG}
  8. γ L G \gamma_{LG}
  9. γ S L \gamma_{SL}
  10. γ ˙ \dot{\gamma}
  11. γ ˙ \dot{\gamma}
  12. η - γ ˙ \eta-\dot{\gamma}
  13. η = C exp ( - E a R T ) , \eta=C\exp\left(\frac{-E_{a}}{RT}\right),

Ribbon_(mathematics).html

  1. ( X , U ) (X,U)
  2. X X
  3. X ( s ) X(s)
  4. s s
  5. a s b a\leq s\leq b
  6. U ( s ) U(s)
  7. X X
  8. ( X , U ) (X,U)
  9. X X
  10. U U
  11. a a
  12. b b
  13. X + ε U X+\varepsilon U
  14. X ( s ) + ε U ( s ) X(s)+\varepsilon U(s)
  15. ε \varepsilon
  16. X X
  17. L k = W r + T w , Lk=Wr+Tw\;,
  18. L k Lk
  19. W r Wr
  20. T w Tw

Ribbon_category.html

  1. 𝒞 \mathcal{C}
  2. C 1 , C 2 𝒞 C_{1},C_{2}\in\mathcal{C}
  3. C 1 C 2 𝒞 C_{1}\otimes C_{2}\in\mathcal{C}
  4. C 1 , C 2 C 1 C 2 C_{1},C_{2}\mapsto C_{1}\otimes C_{2}
  5. c C 1 , C 2 : C 1 C 2 C 2 C 1 . c_{C_{1},C_{2}}:C_{1}\otimes C_{2}\stackrel{\cong}{\rightarrow}C_{2}\otimes C_% {1}.
  6. C C
  7. C * C^{*}
  8. 1 C C * , C C * 1 1\rightarrow C\otimes C^{*},C\otimes C^{*}\rightarrow 1
  9. C * C * 1 C * ( C C * ) ( C * C ) C * 1 C * C * C^{*}\cong C^{*}\otimes 1\rightarrow C^{*}\otimes(C\otimes C^{*})\cong(C^{*}% \otimes C)\otimes C^{*}\rightarrow 1\otimes C^{*}\cong C^{*}
  10. C * C^{*}
  11. C C
  12. C 𝒞 C\in\mathcal{C}
  13. θ C : C C \theta_{C}:C\rightarrow C
  14. θ C 1 C 2 = c C 2 , C 1 c C 1 , C 2 ( θ C 1 θ C 2 ) . \theta_{C_{1}\otimes C_{2}}=c_{C_{2},C_{1}}c_{C_{1},C_{2}}(\theta_{C_{1}}% \otimes\theta_{C_{2}}).

Ricci_soliton.html

  1. g 0 g_{0}
  2. M M
  3. σ ( t ) \sigma(t)
  4. { η ( t ) } Diff ( M ) \{\eta(t)\}\subset\operatorname{Diff}(M)
  5. g ( t ) = σ ( t ) η ( t ) * g 0 g(t)=\sigma(t)\,\eta(t)^{*}g_{0}
  6. η ( t ) * g 0 \eta(t)^{*}g_{0}
  7. g 0 g_{0}
  8. η ( t ) \eta(t)
  9. g 0 g_{0}
  10. Rc ( g 0 ) = λ g 0 + X g 0 , \operatorname{Rc}(g_{0})=\lambda\,g_{0}+\mathcal{L}_{X}g_{0},
  11. Rc \operatorname{Rc}
  12. λ \lambda\in\mathbb{R}
  13. X X
  14. M M
  15. \mathcal{L}
  16. Rc ( g 0 ) = λ g 0 . \operatorname{Rc}(g_{0})=\lambda\,g_{0}.

Rich-club_coefficient.html

  1. ϕ ( k ) = 2 E > k N > k ( N > k - 1 ) \phi(k)=\frac{2E_{>k}}{N_{>k}(N_{>k}-1)}
  2. E > k E_{>k}
  3. N > k N_{>k}
  4. k m a x k_{max}
  5. ρ r a n d ( k ) = ϕ ( k ) ϕ r a n d ( k ) \rho_{rand}(k)=\frac{\phi(k)}{\phi_{rand}(k)}
  6. ϕ r a n d ( k ) \phi_{rand}(k)
  7. P ( k ) P(k)
  8. ρ r a n d ( k ) > 1 \rho_{rand}(k)>1
  9. ϕ ( r ) = 2 E > r N > r ( N > r - 1 ) \phi(r)=\frac{2E_{>r}}{N_{>r}(N_{>r}-1)}

Riffle_shuffle_permutation.html

  1. ( p + q p ) . {\left({{p+q}\atop{p}}\right)}.
  2. ( n + 1 3 ) + 1. {\left({{n+1}\atop{3}}\right)}+1.

Rigid_motion_segmentation.html

  1. ϝ : A A \digamma:A\to A
  2. [ X Y Z ] = R [ X Y Z ] + T \begin{bmatrix}X^{\prime}\\ Y^{\prime}\\ Z^{\prime}\\ \end{bmatrix}=R\cdot\begin{bmatrix}X\\ Y\\ Z\\ \end{bmatrix}+T
  3. R = [ r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 ] a n d T = [ T x T y T z ] R=\begin{bmatrix}r11&r12&r13\\ r21&r22&r23\\ r31&r32&r33\\ \end{bmatrix}and\ T=\begin{bmatrix}T_{x}\\ T_{y}\\ T_{z}\\ \end{bmatrix}
  4. X = R X + t X^{\prime}=R\cdot X+t
  5. X = [ X Y ] , X = [ X Y ] , R = [ cos ( θ ) - sin ( θ ) sin ( θ ) cos ( θ ) ] a n d t = [ t x t y ] X=\begin{bmatrix}X\\ Y\\ \end{bmatrix},\ X^{\prime}=\begin{bmatrix}X^{\prime}\\ Y^{\prime}\\ \end{bmatrix},\ R=\begin{bmatrix}\cos(\theta)&&-\sin(\theta)\\ \sin(\theta)&&\cos(\theta)\\ \end{bmatrix}and\ t=\begin{bmatrix}t_{x}\\ t_{y}\\ \end{bmatrix}

Ring_Learning_with_Errors.html

  1. a ( x ) = a 0 + a 1 x + a 2 x 2 + + a n - 2 x n - 2 + a n - 1 x n - 1 a(x)=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{n-2}x^{n-2}+a_{n-1}x^{n-1}
  2. Z / q Z = F q Z/qZ=F_{q}
  3. q q
  4. F q [ x ] F_{q}[x]
  5. F q [ x ] F_{q}[x]
  6. Φ ( x ) \Phi(x)
  7. F q [ x ] / Φ ( x ) F_{q}[x]/\Phi(x)
  8. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)
  9. Φ ( x ) \Phi(x)
  10. Φ ( x ) \Phi(x)
  11. F q F_{q}
  12. Φ ( x ) \Phi(x)
  13. || a ( x ) || = b ||a(x)||_{\infty}=b
  14. a ( x ) a(x)
  15. b b
  16. b b
  17. a ( x ) a(x)
  18. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)
  19. F q F_{q}
  20. F q F_{q}
  21. ( - ( q - 1 ) / 2 , , - 1 , 0 , 1 , , ( q - 1 ) / 2 ) (-(q-1)/2,...,-1,0,1,...,(q-1)/2)
  22. F q F_{q}
  23. a i ( x ) a_{i}(x)
  24. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)
  25. F q F_{q}
  26. e i ( x ) e_{i}(x)
  27. b b
  28. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)
  29. s ( x ) s(x)
  30. b b
  31. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)
  32. b i ( x ) = ( a i ( x ) s ( x ) ) + e i ( x ) b_{i}(x)=(a_{i}(x)\cdot s(x))+e_{i}(x)
  33. ( a i ( x ) , b i ( x ) ) (a_{i}(x),b_{i}(x))
  34. s ( x ) s(x)
  35. ( a i ( x ) , b i ( x ) ) (a_{i}(x),b_{i}(x))
  36. b i ( x ) b_{i}(x)
  37. b i ( x ) = ( a i ( x ) s ( x ) ) + e i ( x ) b_{i}(x)=(a_{i}(x)\cdot s(x))+e_{i}(x)
  38. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)
  39. F q F_{q}
  40. Φ ( x ) \Phi(x)
  41. F q F_{q}
  42. b b
  43. s ( x ) s(x)
  44. e ( x ) e(x)
  45. a ( x ) a(x)
  46. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)
  47. t ( x ) = ( a ( x ) s ( x ) ) + e ( x ) t(x)=(a(x)\cdot s(x))+e(x)
  48. a ( x ) a(x)
  49. t ( x ) t(x)
  50. s ( x ) s(x)
  51. Φ ( x ) \Phi(x)
  52. Z [ x ] / Φ ( x ) Z[x]/\Phi(x)
  53. Z [ x ] / Φ ( x ) Z[x]/\Phi(x)
  54. Z q [ x ] / Φ ( x ) Z_{q}[x]/\Phi(x)

Ring_learning_with_errors_key_exchange.html

  1. c j = 4 v j / q mod 2 c_{j}=\lfloor 4v_{j}/q\rfloor\mod 2
  2. u j = 2 v j mod 2 u_{j}=\lfloor 2v_{j}\rceil\mod 2

Ring_learning_with_errors_signature.html

  1. a ( x ) = a 0 + a 1 x + a 2 x 2 + + a n - 3 x n - 3 + a n - 2 x n - 2 + a n - 1 x n - 1 a(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n-3}x^{n-3}+a_{n-2}x^{n-2}+a_{n-1}x^{n-1}
  2. = =
  3. = =
  4. = =

Rising_film_evaporator.html

  1. 1 U = 1 h o + 1 h o d + d o ln d o d i 2 k w + d o d i 1 h i d + d o d i 1 h i \frac{1}{U}=\frac{1}{h_{o}}+\frac{1}{h_{o}d}+\frac{d_{o}\ln\frac{d_{o}}{d_{i}}% }{2k_{w}}+\frac{d_{o}}{d_{i}}\frac{1}{h_{i}d}+\frac{d_{o}}{d_{i}}\frac{1}{h_{i}}
  2. T l m = ( T h i - T c o ) - ( T h o - T c i ) ln ( T h i - T c o ) ( T h o - T c i ) T_{lm}=\left(T_{hi}-T_{co}\right)-\left(T_{ho}-T_{ci}\right)\over\ln\frac{% \left(T_{hi}-T_{co}\right)}{\left(T_{ho}-T_{ci}\right)}
  3. < v a r > T h i <var>T_{hi}

Robert_Kottwitz.html

  1. G L 3 GL_{3}

Robust_principal_component_analysis.html

  1. M = L + S M=L+S
  2. O ( m n r 2 log 1 ϵ ) O\left(mnr^{2}\log\frac{1}{\epsilon}\right)
  3. r r
  4. m × n m\times n
  5. ϵ \epsilon
  6. L ^ - L F ϵ \|\widehat{L}-L\|_{F}\leq\epsilon
  7. L L
  8. L ^ \widehat{L}
  9. 1 \ell_{1}

Robustness_of_complex_networks.html

  1. p c p_{c}
  2. < s Align g t ; <s&gt;
  3. < s > | p - p c | γ p \begin{aligned}\displaystyle<s>\sim\left|p-p_{c}\right|^{\gamma_{p}}\end{aligned}
  4. γ p \gamma_{p}
  5. p c p_{c}
  6. κ < k 2 > < k > > 2 \begin{aligned}\displaystyle\kappa\equiv\frac{<k^{2}>}{<k>}>2\end{aligned}
  7. f c = 1 - 1 < k 2 > < k > - 1 \begin{aligned}\displaystyle f_{c}=1-\frac{1}{\frac{<k^{2}>}{<k>}-1}\end{aligned}
  8. < k 2 < k > ( < k > + 1 ) <k^{2}>=<k>(<k>+1)
  9. f c E R = 1 - 1 < k > \begin{aligned}\displaystyle f_{c}^{ER}=1-\frac{1}{<k>}\end{aligned}
  10. f c = 1 - 1 κ - 1 κ = < k 2 > < k > = | 2 - γ 3 - γ | A A = K m i n , γ > 3 A = K m a x 3 - γ K m i n γ - 2 , 3 > γ > 2 A = K m a x , 2 > γ > 1 w h e r e K m a x = K m i n N 1 γ - 1 \begin{aligned}\displaystyle f_{c}&\displaystyle=1-\frac{1}{\kappa-1}\\ \displaystyle\kappa&\displaystyle=\frac{<k^{2}>}{<k>}=\left|\frac{2-\gamma}{3-% \gamma}\right|A\\ \displaystyle A&\displaystyle=K_{min},~{}\gamma>3\\ \displaystyle A&\displaystyle=K_{max}^{3-\gamma}K_{min}^{\gamma-2},~{}3>\gamma% >2\\ \displaystyle A&\displaystyle=K_{max},~{}2>\gamma>1\\ &\displaystyle where~{}K_{max}=K_{min}N^{\frac{1}{\gamma-1}}\end{aligned}
  11. κ \kappa
  12. f c 2 - γ 1 - γ = 2 + 2 - γ 3 - γ K m i n ( f c 3 - γ 1 - γ - 1 ) \begin{aligned}\displaystyle f_{c}^{\frac{2-\gamma}{1-\gamma}}=2+\frac{2-% \gamma}{3-\gamma}K_{min}(f_{c}^{\frac{3-\gamma}{1-\gamma}}-1)\end{aligned}

Rocha–Thatte_cycle_detection_algorithm.html

  1. G G
  2. G G
  3. v v
  4. ( v ) (v)
  5. v v
  6. v v
  7. v v
  8. v v
  9. ( v 1 , v 2 , , v k ) (v_{1},v_{2},\ldots,v_{k})
  10. v v
  11. v = v 1 v=v_{1}
  12. v v
  13. v = v i v=v_{i}
  14. i 2 , 3 , , k i\in{2,3,\ldots,k}
  15. v v
  16. ( v = v i , v i + 1 , , v k , v k + 1 = v ) (v=v_{i},v_{i}+1,\ldots,v_{k},v_{k}+1=v)
  17. k - i + 1 k-i+1
  18. ( v 1 , v 2 , , v k , v k + 1 = v 1 ) (v_{1},v_{2},\ldots,v_{k},v_{k}+1=v_{1})
  19. v i , i = 1 v_{i},i=1
  20. k k
  21. min { v 1 , , v k } \min\{v_{1},\ldots,v_{k}\}

Roger_Dashen.html

  1. ϕ 4 {\phi^{4}}

Rose_Peltesohn.html

  1. 1 , 2 , , v - 1 1,2,\dots,v-1
  2. mod v \bmod v
  3. a + b + c = 0 mod v a+b+c=0\bmod v
  4. mod v \bmod v
  5. a + b = c mod v a+b=c\bmod v
  6. v = 6 m + 1 v=6m+1
  7. 1 , 2 , , 3 m 1,2,\ldots,3m
  8. v = 6 m + 3 v=6m+3
  9. 1 , 2 , , 2 m , 2 m + 2 , , 3 m + 1 1,2,\ldots,2m,2m+2,\ldots,3m+1
  10. v = 13 v=13
  11. ( 1 , 3 , 4 ) (1,3,4)
  12. 1 + 3 = 4 mod 13 1+3=4\bmod 13
  13. ( 2 , 5 , 6 ) (2,5,6)
  14. 2 + 5 + 6 = 13 = 0 mod 13 2+5+6=13=0\bmod 13

Rosenau–Hyman_equation.html

  1. u t + a ( u n ) x + ( u n ) x x x = 0. u_{t}+a(u^{n})_{x}+(u^{n})_{xxx}=0.\,
  2. u ( x , t ) = ( 2 c n a ( n + 1 ) sin 2 ( n - 1 2 n a ( x - c t + b ) ) ) 1 / ( n - 1 ) , u(x,t)=\left(\frac{2cn}{a(n+1)}\sin^{2}\left(\frac{n-1}{2n}\sqrt{a}(x-ct+b)% \right)\right)^{1/(n-1)},
  3. u ( x , t ) = ( 2 c n a ( n + 1 ) sinh 2 ( n - 1 2 n - a ( x - c t + b ) ) ) 1 / ( n - 1 ) , u(x,t)=\left(\frac{2cn}{a(n+1)}\sinh^{2}\left(\frac{n-1}{2n}\sqrt{-a}(x-ct+b)% \right)\right)^{1/(n-1)},
  4. u ( x , t ) = ( 2 c n a ( n + 1 ) cosh 2 ( n - 1 2 n - a ( x - c t + b ) ) ) 1 / ( n - 1 ) . u(x,t)=\left(\frac{2cn}{a(n+1)}\cosh^{2}\left(\frac{n-1}{2n}\sqrt{-a}(x-ct+b)% \right)\right)^{1/(n-1)}.

Rosenbrock_system_matrix.html

  1. x ˙ = A x + B u , \dot{x}=Ax+Bu,
  2. y = C x + D u . y=Cx+Du.
  3. P ( s ) = ( s I - A - B C D ) . P(s)=\begin{pmatrix}sI-A&-B\\ C&D\end{pmatrix}.
  4. D D
  5. s s
  6. i i
  7. j j
  8. g i j = | s I - A - b i c j d i j | | s I - A | g_{ij}=\frac{\begin{vmatrix}sI-A&-b_{i}\\ c_{j}&d_{ij}\end{vmatrix}}{|sI-A|}
  9. b i b_{i}
  10. i i
  11. B B
  12. c j c_{j}
  13. j j
  14. C C
  15. P ( A B C D ) . P\sim\begin{pmatrix}A&B\\ C&D\end{pmatrix}.

Rossby_wave_instability_in_astrophysical_discs.html

  1. r 0 r_{0}
  2. exp ( i m ϕ ) \propto\exp({\rm i}m\phi)
  3. m = 1 , 2.. m=1,2..
  4. r 0 r_{0}
  5. L ( r ) = Σ S 2 / γ 2 ( × u ) 𝐳 ^ , {L}(r)={\Sigma~{}S^{2/\gamma}\over 2({\mathbf{\nabla}\times u})\cdot\hat{% \mathbf{z}}}~{},
  6. r 0 r_{0}
  7. Σ \Sigma
  8. 𝐮 r Ω ( r ) ϕ ^ {\mathbf{u}}\approx r\Omega(r)\hat{\phi~{}}
  9. Ω ( r ) ( G M * / r 3 ) 1 / 2 \Omega(r)\approx(GM_{*}/r^{3})^{1/2}
  10. M * M_{*}
  11. S S
  12. γ \gamma
  13. L {L}
  14. ( × u ) z / Σ ({\mathbf{\nabla}\times u})_{z}/\Sigma
  15. L ( r ) {L}(r)
  16. r ILR r_{\rm ILR}
  17. r OLR r_{\rm OLR}
  18. ω = m Ω ( r LR ) ± κ ( r LR ) \omega=m\Omega(r_{\rm LR})\pm\kappa(r_{\rm LR})
  19. κ ( r ) \kappa(r)
  20. Ω ( r ) \Omega(r)
  21. r C r_{\rm C}
  22. ω = m Ω ( r C ) \omega=m\Omega(r_{\rm C})
  23. > 300 >300
  24. f ( r ) exp ( i m ϕ - i ω t ) f(r)\exp({\rm i}m\phi-{\rm i}\omega t)
  25. m = 1 , 2 , . . m=1,2,..
  26. ω \omega
  27. ψ = δ p / ρ \psi=\delta p/\rho
  28. d 2 ψ d r 2 = V eff ( r ) ψ . {d^{2}\psi\over dr^{2}}=V_{\rm eff}(r)~{}\psi~{}.
  29. V eff ( r ) V_{\rm eff}(r)
  30. L ( r ) {L}(r)
  31. L ( r ) {L}(r)
  32. L ( r ) {L}(r)
  33. V eff V_{\rm eff}
  34. ω i = ( ω ) \omega_{i}=\Im(\omega)
  35. Δ Σ / Σ 0.2 \Delta\Sigma/\Sigma\lesssim 0.2
  36. ω i = ( 0.1 - 0.2 ) Ω ( r 0 ) \omega_{i}=(0.1-0.2)\Omega(r_{0})
  37. ω r = ( ω ) \omega_{r}=\Re(\omega)
  38. m Ω ( r 0 ) m\Omega(r_{0})
  39. V eff V_{\rm eff}
  40. r OLR r_{\rm OLR}
  41. r ILR r_{\rm ILR}
  42. S = S=
  43. V eff = 2 m Ω r ( Δ ω ) d d r [ ln ( Ω Σ κ 2 - ( Δ ω ) 2 ) ] + m 2 r 2 + κ 2 - ( Δ ω ) 2 c s 2 V_{\rm eff}={2m\Omega\over r(\Delta\omega)}{d\over dr}\left[\ln\left({\Omega% \Sigma\over\kappa^{2}-(\Delta\omega)^{2}}\right)\right]+{m^{2}\over r^{2}}+{% \kappa^{2}-(\Delta\omega)^{2}\over c_{s}^{2}}
  44. Δ ω ω - m Ω \Delta\omega\equiv\omega-m\Omega
  45. c s c_{s}
  46. κ \kappa
  47. κ 2 = r - 3 d 2 / d r \kappa^{2}=r^{-3}d\ell^{2}/dr
  48. = r u ϕ \ell=ru_{\phi}
  49. ω r < m Ω ( r ) \omega_{r}<m\Omega(r)
  50. E < 0 E<0
  51. ω r > m Ω ( r ) \omega_{r}>m\Omega(r)
  52. E > 0 E>0

Rosser's_equation.html

  1. i i\,
  2. j j\,
  3. t t\,
  4. T T\,
  5. F R A FRA\,

Rosser's_equation_(physics).html

  1. J J\,
  2. J , t J,t\,
  3. t t\,
  4. ϕ \phi\,

Roundel_(heraldry).html

  1. r 2 = a 2 * θ r^{2}=a^{2}*\theta
  2. r = a * θ r=a*\theta

Rubidium_hydrogen_sulfate.html

  1. Rb 2 S 2 O 7 + H 2 O 2 RbHSO 4 \mathrm{Rb_{2}S_{2}O_{7}+H_{2}O\longrightarrow 2\ RbHSO_{4}}
  2. H 2 SO 4 + RbCl RbHSO 4 + HCl \mathrm{H_{2}SO_{4}+\ RbCl\longrightarrow\ RbHSO_{4}+\ HCl}
  3. 2 RbHSO 4 Rb 2 S 2 O 7 + H 2 O \mathrm{2\ RbHSO_{4}\longrightarrow Rb_{2}S_{2}O_{7}+H_{2}O}

Runcicantellated_24-cell_honeycomb.html

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

Runcicantellated_tesseractic_honeycomb.html

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

Runcicantitruncated_tesseractic_honeycomb.html

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

Runcinated_16-cell_honeycomb.html

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

Runcinated_24-cell_honeycomb.html

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

Runcinated_tesseractic_honeycomb.html

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

Runcitruncated_tesseractic_honeycomb.html

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

Rüchardt_experiment.html

  1. C p C_{p}
  2. C V C_{V}
  3. γ \gamma
  4. κ \kappa
  5. κ \kappa
  6. P = P 0 + m g A P=P_{0}+\frac{mg}{A}
  7. F = A d P F=A\mathrm{d}P
  8. a = d 2 x d t 2 = A m d P a=\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=\frac{A}{m}\mathrm{d}P
  9. P V γ = constant PV^{\gamma}=\mathrm{constant}
  10. V γ d P + γ P V γ - 1 d V = 0 V^{\gamma}\mathrm{d}P+\gamma PV^{\gamma-1}\mathrm{d}V=0
  11. d x dx
  12. d V = A d x \mathrm{d}V=Adx
  13. d 2 x d t 2 + γ P A 2 m V x = 0 \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\frac{\gamma PA^{2}}{mV}x=0
  14. ω = γ P A 2 m V \omega=\sqrt{\frac{\gamma PA^{2}}{mV}}
  15. T = 2 π ω = 2 π m V γ P A 2 T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{mV}{\gamma P{A}^{2}}}
  16. γ = 4 π 2 m V A 2 P T 2 {\gamma}=\frac{4\pi^{2}mV}{A^{2}PT^{2}}
  17. γ \gamma
  18. L = 2 m g V P A 2 γ L=\frac{2mgV}{PA^{2}\gamma}
  19. γ \gamma

Rybicki_Press_algorithm.html

  1. A ( i , j ) = exp ( - a | t i - t j | ) A(i,j)=\exp(-a|t_{i}-t_{j}|)
  2. a a\in\mathbb{R}

Rytz's_construction.html

  1. d 1 d_{1}^{\prime}
  2. d 2 d_{2}^{\prime}
  3. a a
  4. b b
  5. e e
  6. e e
  7. k h k_{h}
  8. α \alpha
  9. k h k_{h}
  10. e e
  11. α \alpha
  12. k h k_{h}
  13. d 1 d_{1}^{\prime}
  14. d 2 d_{2}^{\prime}
  15. M M
  16. U U^{\prime}
  17. d 1 d_{1}^{\prime}
  18. V V^{\prime}
  19. d 2 d_{2}^{\prime}
  20. ( U M V ) \angle(U^{\prime}MV^{\prime})
  21. > 90 >90^{\circ}
  22. < 90 <90^{\circ}
  23. = 90 =90^{\circ}
  24. U U^{\prime}
  25. 90 90^{\circ}
  26. M M
  27. V V^{\prime}
  28. U r U^{\prime}_{r}
  29. U r U^{\prime}_{r}
  30. V V^{\prime}
  31. g g
  32. U r V ¯ \overline{U^{\prime}_{r}V^{\prime}}
  33. S S
  34. t t
  35. S S
  36. M M
  37. g g
  38. R R
  39. L L
  40. R R
  41. L L
  42. R R
  43. U r U^{\prime}_{r}
  44. L L
  45. V V^{\prime}
  46. S S
  47. M M
  48. R R
  49. L L
  50. M M
  51. M L ¯ \overline{ML}
  52. M R ¯ \overline{MR}
  53. V R ¯ \overline{V^{\prime}R}
  54. V L ¯ \overline{V^{\prime}L}
  55. M M
  56. a a
  57. b b
  58. S 1 S_{1}
  59. S 2 S_{2}
  60. a a
  61. M M
  62. L L
  63. S 3 S_{3}
  64. S 4 S_{4}
  65. b b
  66. M M
  67. R R

S-factor.html

  1. S ( E ) S(E)
  2. σ ( E ) σ(E)
  3. S ( E ) E exp ( - 2 π η ) σ ( E ) S(E)\equiv\frac{E}{\exp(-2\pi\eta)}\sigma(E)
  4. η η
  5. η Z 1 Z 2 e 2 4 π ϵ 0 v \eta\equiv\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}\hbar v}
  6. Z < s u b > 1 Z 2 e 2 Z<sub>1Z_{2}e^{2}

Sacks_property.html

  1. M M
  2. N N
  3. N N
  4. M M
  5. g M g\in M
  6. ω \omega
  7. ω { 0 } \omega\setminus\{0\}
  8. g g
  9. f N f\in N
  10. ω \omega
  11. ω \omega
  12. T M T\in M
  13. n n
  14. n t h n^{th}
  15. T T
  16. g ( n ) g(n)
  17. f f
  18. T T
  19. ω ω {}^{\omega}\omega

Saddlepoint_approximation_method.html

  1. M ( t ) M(t)
  2. K ( t ) = log ( M ( t ) ) K(t)=\log(M(t))
  3. f ^ ( x ) = 1 2 π K ′′ ( s ) exp ( K ( s ^ ) - s ^ x ) \hat{f}(x)=\frac{1}{\sqrt{2\pi K^{\prime\prime}(s)}}\exp(K(\hat{s})-\hat{s}x)
  4. F ^ ( x ) = { Φ ( w ^ ) + ϕ ( w ^ ) ( 1 w ^ - 1 u ^ ) for x μ 1 2 + K ′′′ ( 0 ) 6 2 π K ′′ ( 0 ) 3 / 2 for x = μ \hat{F}(x)=\begin{cases}\Phi(\hat{w})+\phi(\hat{w})(\frac{1}{\hat{w}}-\frac{1}% {\hat{u}})&\,\text{for }x\neq\mu\\ \frac{1}{2}+\frac{K^{\prime\prime\prime}(0)}{6\sqrt{2\pi}K^{\prime\prime}(0)^{% 3/2}}&\,\text{for }x=\mu\end{cases}
  5. s ^ \hat{s}
  6. K ( s ^ ) = x K^{\prime}(\hat{s})=x
  7. w ^ = sgn s ^ 2 ( s ^ x - K ( s ^ ) \hat{w}=\operatorname{sgn}{\hat{s}}\sqrt{2(\hat{s}x-K(\hat{s})}
  8. u ^ = s ^ K ′′ ( s ^ ) \hat{u}=\hat{s}\sqrt{K^{\prime\prime}(\hat{s})}

Saito–Kurokawa_lift.html

  1. ( n t / 2 t / 2 m ) = d | t , m , n d k - 1 a ( 1 t / 2 d t / 2 d n m / d 2 ) \begin{pmatrix}n&t/2\\ t/2&m\end{pmatrix}=\sum_{d|t,m,n}d^{k-1}a\begin{pmatrix}1&t/2d\\ t/2d&nm/d^{2}\end{pmatrix}

Sakai–Kasahara_scheme.html

  1. 𝕄 \mathbb{M}
  2. I U \textstyle I_{U}
  3. K U \textstyle K_{U}
  4. I U \textstyle I_{U}
  5. K U \textstyle K_{U}
  6. I U \textstyle I_{U}
  7. Z \textstyle Z
  8. E \textstyle E
  9. G \textstyle G
  10. E \textstyle E
  11. P \textstyle P
  12. Q \textstyle Q
  13. x \textstyle x
  14. [ x ] . P = Q \textstyle[x].P=Q
  15. G \textstyle G
  16. g g
  17. t t
  18. x \textstyle x
  19. g x = t \textstyle g^{x}=t
  20. e ( , ) \textstyle e(,)
  21. P \textstyle P
  22. E \textstyle E
  23. g = e ( P , P ) \textstyle g=e(P,P)
  24. G \textstyle G
  25. e ( P , [ x ] . P ) = e ( [ x ] . P , P ) = e ( P , P ) x = g x \textstyle e(P,[x].P)=e([x].P,P)=e(P,P)^{x}=g^{x}
  26. E \textstyle E
  27. E : y 2 = x 3 - 3 x \textstyle E:y^{2}=x^{3}-3x
  28. p \textstyle p
  29. P \textstyle P
  30. q \textstyle q
  31. E \textstyle E
  32. G \textstyle G
  33. P \textstyle P
  34. H 1 \textstyle H_{1}
  35. H 2 \textstyle H_{2}
  36. H 1 \textstyle H_{1}
  37. x \textstyle x
  38. 1 < x < q \textstyle 1<x<q
  39. H 2 \textstyle H_{2}
  40. n \textstyle n
  41. n \textstyle n
  42. 𝕄 \mathbb{M}
  43. z \textstyle z
  44. 1 < z < q 1<z<q
  45. Z = [ z ] . P \textstyle Z=[z].P
  46. E \textstyle E
  47. K U \textstyle K_{U}
  48. I D U \textstyle ID_{U}
  49. K U = [ 1 z + H 1 ( I D U ) ] . P \textstyle K_{U}=[\frac{1}{z+H_{1}(ID_{U})}].P
  50. 𝕄 \mathbb{M}
  51. I D U \textstyle ID_{U}
  52. Z \textstyle Z
  53. i d = H 1 ( I D U ) \textstyle id=H_{1}(ID_{U})
  54. r \textstyle r
  55. r = H 1 ( 𝕄 | | i d ) \textstyle r=H_{1}(\mathbb{M}||id)
  56. R \textstyle R
  57. E \textstyle E
  58. R = [ r ] . ( [ i d ] . P + Z ) \textstyle R=[r].([id].P+Z)
  59. S = 𝕄 H 2 ( g r ) \textstyle S=\mathbb{M}\oplus H_{2}(g^{r})
  60. ( R , S ) \textstyle(R,S)
  61. I D U \textstyle ID_{U}
  62. K U \textstyle K_{U}
  63. Z \textstyle Z
  64. i d = H 1 ( I D U ) \textstyle id=H_{1}(ID_{U})
  65. ( R , S ) \textstyle(R,S)
  66. w = e ( R , K U ) \textstyle w=e(R,K_{U})
  67. 𝕄 = S H 2 ( w ) \textstyle\mathbb{M}=S\oplus H_{2}(w)
  68. r = H 1 ( 𝕄 | | i d ) \textstyle r=H_{1}(\mathbb{M}||id)
  69. [ r ] . ( [ i d ] . P + Z ) R \textstyle[r].([id].P+Z)\equiv R
  70. w = e ( R , K U ) = e ( [ r ] . ( [ i d ] . P + Z ) , K U ) = e ( [ r ] . ( [ i d ] . P + [ z ] . P ) , K U ) = e ( [ r ( i d + z ) ] . P , K U ) \textstyle w=e(R,K_{U})=e([r].([id].P+Z),K_{U})=e([r].([id].P+[z].P),K_{U})=e(% [r(id+z)].P,K_{U})
  71. w = e ( [ r ( i d + z ) ] . P , K U ) = e ( [ r ( i d + z ) ] . P , [ 1 ( i d + z ) ] . P ) = e ( P , P ) r ( i d + z ) ( i d + z ) = g r \textstyle w=e([r(id+z)].P,K_{U})=e([r(id+z)].P,[\frac{1}{(id+z)}].P)=e(P,P)^{% \frac{r(id+z)}{(id+z)}}=g^{r}
  72. S H 2 ( w ) = ( 𝕄 H 2 ( g r ) ) H 2 ( w ) = 𝕄 \textstyle S\oplus H_{2}(w)=(\mathbb{M}\oplus H_{2}(g^{r}))\oplus H_{2}(w)=% \mathbb{M}

Salted_Challenge_Response_Authentication_Mechanism.html

  1. H H
  2. username \mathrm{username}
  3. nonce c \mathrm{nonce}_{c}
  4. nonce s \mathrm{nonce}_{s}
  5. salt \mathrm{salt}
  6. it \mathrm{it}
  7. proof c \mathrm{proof}_{c}
  8. proof s \mathrm{proof_{s}}
  9. password s \mathrm{password}_{s}
  10. password s = H i ( password , salt , it ) \mathrm{password}_{s}=H_{i}(\mathrm{password},\mathrm{salt},\mathrm{it})
  11. H i ( p , s , i ) H_{i}(p,s,i)
  12. p p
  13. s s
  14. i i
  15. H H
  16. password s \mathrm{password}_{s}
  17. salt \mathrm{salt}
  18. it \mathrm{it}
  19. Auth \mathrm{Auth}
  20. Auth = client-first , server-first , client-final-without-proof \mathrm{Auth}=\,\text{client-first},\,\text{server-first},\,\text{client-final% -without-proof}
  21. key c = H M A C ( password s , “Client Key” ) \mathrm{key}_{c}=HMAC(\mathrm{password}_{s},\,\text{``Client Key''})
  22. key s = H M A C ( password s , “Server Key” ) \mathrm{key}_{s}=HMAC(\mathrm{password}_{s},\,\text{``Server Key''})
  23. proof c = key c H M A C ( H ( key c ) , Auth ) \mathrm{proof}_{c}=\mathrm{key}_{c}\oplus HMAC(H(\mathrm{key}_{c}),\mathrm{% Auth})
  24. proof s = H M A C ( key s , Auth ) \mathrm{proof}_{s}=HMAC(\mathrm{key}_{s},\mathrm{Auth})
  25. \oplus
  26. “Client Key” \,\text{``Client Key''}
  27. “Server Key” \,\text{``Server Key''}

Sample_complexity.html

  1. S n = { ( x 1 , y 1 ) , , ( x n , y n ) } S_{n}=\{(x_{1},y_{1}),\ldots,(x_{n},y_{n})\}
  2. ρ \rho
  3. 𝒳 × 𝒴 \mathcal{X}\times\mathcal{Y}
  4. f : 𝒳 𝒴 f:\mathcal{X}\to\mathcal{Y}
  5. \mathcal{H}
  6. ρ \rho
  7. V : 𝒴 × 𝒴 + V:\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}_{+}
  8. * = inf f 𝔼 ρ [ V ( f ( x ) , y ) ] , \mathcal{R}^{*}_{\mathcal{H}}=\underset{f\in\mathcal{H}}{\inf}\mathbb{E}_{\rho% }[V(f(x),y)],
  9. f n f_{n}
  10. n n
  11. lim n S n ( 𝔼 ρ [ V ( f ( x ) , y ) ] - * > ϵ ) , \underset{n\to\infty}{\lim}\mathbb{P}_{S_{n}}(\mathbb{E}_{\rho}[V(f(x),y)]-% \mathcal{R}^{*}_{\mathcal{H}}>\epsilon),
  12. ϵ > 0 \epsilon>0
  13. S n \mathbb{P}_{S_{n}}
  14. ρ n \rho^{n}
  15. ϵ \epsilon
  16. n ( ρ , ϵ , δ ) n(\rho,\epsilon,\delta)
  17. n n ( ρ , ϵ , δ ) n\geq n(\rho,\epsilon,\delta)
  18. S n ( 𝔼 ρ [ V ( f ( x ) , y ) ] - * ϵ ) 1 - δ . \mathbb{P}_{S_{n}}(\mathbb{E}_{\rho}[V(f(x),y)]-\mathcal{R}^{*}_{\mathcal{H}}% \leq\epsilon)\geq 1-\delta.
  19. n ( ρ , ϵ , δ ) n(\rho,\epsilon,\delta)
  20. ϵ \epsilon
  21. δ \delta
  22. n ( ρ , ϵ , δ ) n(\rho,\epsilon,\delta)
  23. ϵ \epsilon
  24. 1 - δ 1-\delta
  25. ρ \rho
  26. \mathcal{H}
  27. sup 𝜌 S n ( 𝔼 ρ [ V ( f ( x ) , y ) ] - * > ϵ ) , \underset{\rho}{\sup}\ \mathbb{P}_{S_{n}}(\mathbb{E}_{\rho}[V(f(x),y)]-% \mathcal{R}^{*}_{\mathcal{H}}>\epsilon),
  28. ρ \rho
  29. \mathcal{H}
  30. \mathcal{H}
  31. * \mathcal{R}^{*}_{\mathcal{H}}
  32. * \mathcal{R}^{*}_{\mathcal{H}}

Sample_entropy.html

  1. C i m ( r ) C_{i}^{\prime m}(r)
  2. m m
  3. r r
  4. N N
  5. m m
  6. < r <r
  7. m + 1 m+1
  8. < r <r
  9. S a m p E n ( m , r , N ) SampEn(m,r,N)
  10. S a m p E n ( m , r , τ , N ) SampEn(m,r,\tau,N)
  11. τ \tau
  12. N = { x 1 , x 2 , x 3 , , x N } N={\{x_{1},x_{2},x_{3},...,x_{N}\}}
  13. τ \tau
  14. m m
  15. X m ( i ) = { x i , x i + 1 , x i + 2 , , x i + m - 1 } X_{m}(i)={\{x_{i},x_{i+1},x_{i+2},...,x_{i+m-1}\}}
  16. d [ X m ( i ) , X m ( j ) ] d[X_{m}(i),X_{m}(j)]
  17. m m
  18. m + 1 m+1
  19. d [ X m ( i ) , X m ( j ) ] < r d[X_{m}(i),X_{m}(j)]<r
  20. B B
  21. A A
  22. S a m p E n = - log A B SampEn=-\log{A\over B}
  23. A A
  24. d [ X m + 1 ( i ) , X m + 1 ( j ) ] < r d[X_{m+1}(i),X_{m+1}(j)]<r
  25. m + 1 m+1
  26. B B
  27. d [ X m ( i ) , X m ( j ) ] < r d[X_{m}(i),X_{m}(j)]<r
  28. m m
  29. A A
  30. B B
  31. S a m p E n ( m , r , τ ) SampEn(m,r,\tau)
  32. S a m p E n SampEn
  33. m m
  34. 2 2
  35. r r
  36. 0.2 × s t d 0.2\times std
  37. 0.2 × s t d 0.2\times std
  38. δ = 1 \delta=1
  39. δ \delta
  40. δ \delta
  41. X m , δ ( i ) = x i , x i + δ , x i + 2 × δ , , x i + ( m - 1 ) × δ X_{m,\delta}(i)={x_{i},x_{i+\delta},x_{i+2\times\delta},...,x_{i+(m-1)\times% \delta}}
  42. S a m p E n ( m , r , δ ) = - log A δ B δ SampEn\left(m,r,\delta\right)=-\log{A_{\delta}\over B_{\delta}}
  43. A δ A_{\delta}
  44. B δ B_{\delta}

Sandia_method.html

  1. S ( ω k ) = S ( ω = ω k ) Δ ω / 2 S(\omega_{k})=S(\omega=\omega_{k})\Delta\omega/2
  2. S ( ω k ) S(\omega_{k})
  3. ω k \omega_{k}
  4. S ( ω = ω k ) S(\omega=\omega_{k})
  5. ω = ω k \omega=\omega_{k}
  6. Δ ω \Delta\omega
  7. C o h i j k = exp - 2 C Δ r i j ω k / ( u i + u j ) Coh_{ijk}=\exp^{-2C\Delta r_{ij}\omega_{k}/(u_{i}+u_{j})}
  8. C C
  9. Δ r i j \Delta r_{ij}
  10. i i
  11. j j
  12. ω k \omega_{k}
  13. u i u_{i}
  14. u j u_{j}
  15. i i
  16. j j
  17. i i
  18. j j
  19. k k
  20. n 2 n^{2}
  21. n 2 n^{2}
  22. ω \omega
  23. ( n 4 + n 2 ) / 2 (n^{4}+n^{2})/2
  24. C o h i j k C o h i j ( ω k ) Coh_{ijk}\rightarrow Coh_{ij}(\omega_{k})
  25. S i j k S i j ( ω k ) S_{ijk}\rightarrow S_{ij}(\omega_{k})
  26. S i j k S_{ijk}
  27. ω k \omega_{k}
  28. n 2 n^{2}
  29. n 2 n^{2}
  30. n 2 n^{2}
  31. n 2 n^{2}
  32. n 2 n^{2}
  33. S i j ( ω k ) = C o h i j ( ω k ) S i i ( ω k ) S j j ( ω k ) S_{ij}(\omega_{k})=Coh_{ij}(\omega_{k})\sqrt{S_{ii}(\omega_{k})S_{jj}(\omega_{% k})}
  34. ( n 4 + n 2 ) / 2 (n^{4}+n^{2})/2
  35. S ( ω k ) S(\omega_{k})
  36. H ( ω k ) H(\omega_{k})
  37. S ( ω k ) = H ( ω k ) H ( ω k ) T S(\omega_{k})=H(\omega_{k})H(\omega_{k})^{T}
  38. H ( ω k ) H(\omega_{k})
  39. x ( t ) x(t)
  40. X ( ω ) X(\omega)
  41. S ( ω ) X ( ω ) X * ( ω ) S(\omega)\propto X(\omega)X^{*}(\omega)
  42. H ( ω k ) H(\omega_{k})
  43. H ( ω k ) H(\omega_{k})
  44. V V
  45. H ( ω k ) H(\omega_{k})
  46. n 2 n^{2}
  47. X ( ω k ) X(\omega_{k})
  48. V ( ω k ) = H ( ω k ) X ( ω k ) V(\omega_{k})=H(\omega_{k})X(\omega_{k})
  49. V i ( ω k ) V i k V_{i}(\omega_{k})\rightarrow V_{ik}

Satellite_surface_salinity.html

  1. I ν = 2 h ν 3 c 2 1 e h ν k T - 1 I_{\nu}=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT}}-1}
  2. I ν I_{\nu}
  3. ν \nu
  4. ν + d ν \nu+d\nu
  5. T T
  6. h h
  7. ν \nu
  8. c c
  9. k k
  10. T b = e T T_{b}=eT
  11. e H = 1 - [ cos θ - ( ϵ - sin 2 θ ) 1 2 cos θ + ( ϵ - sin 2 θ ) 1 2 ] 2 e_{H}=1-\left[\frac{\cos\theta-(\epsilon-\sin^{2}\theta)^{\frac{1}{2}}}{\cos% \theta+(\epsilon-\sin^{2}\theta)^{\frac{1}{2}}}\right]^{2}
  12. e V = 1 - [ ϵ cos θ - ( ϵ - sin 2 θ ) 1 2 ϵ cos θ + ( ϵ - sin 2 θ ) 1 2 ] 2 e_{V}=1-\left[\frac{\epsilon\cos\theta-(\epsilon-\sin^{2}\theta)^{\frac{1}{2}}% }{\epsilon\cos\theta+(\epsilon-\sin^{2}\theta)^{\frac{1}{2}}}\right]^{2}
  13. T b H = e H T T_{bH}=e_{H}T
  14. T b V = e V T T_{bV}=e_{V}T
  15. T b T_{b}
  16. T T

Satellite_system_(astronomy).html

  1. 25 + 621 2 \tfrac{25+\sqrt{621}}{2}

Saving-investment_balance.html

  1. Y = C + I + G + ( E X - I M ) Y=C+I+G+(EX-IM)
  2. ( Y - T - C ) + ( T - G ) - I = E X - I M (Y-T-C)+(T-G)-I=EX-IM
  3. S - I = E X - I M S-I=EX-IM

Scale_co-occurrence_matrix.html

  1. d i ( k ) = [ [ g i ] T x ] , ( i = 1 , , N ) d_{i}(k)=[[g_{i}]^{T}x],\quad(i=1,\ldots,N)

Scaling_dimension.html

  1. x λ x x\to\lambda x
  2. x λ x x\to\lambda x
  3. λ - Δ \lambda^{-\Delta}
  4. Δ \Delta
  5. O ( x ) O ( 0 ) \langle O(x)O(0)\rangle
  6. ( x 2 ) - Δ (x^{2})^{-\Delta}
  7. O 1 ( λ x 1 ) O 2 ( λ x 2 ) = λ - Δ 1 - Δ 2 - O 1 ( x 1 ) O 2 ( x 2 ) \langle O_{1}(\lambda x_{1})O_{2}(\lambda x_{2})\ldots\rangle=\lambda^{-\Delta% _{1}-\Delta_{2}-\ldots}\langle O_{1}(x_{1})O_{2}(x_{2})\ldots\rangle
  8. Δ 1 \Delta_{1}
  9. Δ 2 \Delta_{2}
  10. Δ 1 + Δ 2 \Delta_{1}+\Delta_{2}
  11. σ \sigma
  12. Δ 1 \Delta_{1}
  13. Δ 2 \Delta_{2}
  14. Δ 1 + Δ 2 \Delta_{1}+\Delta_{2}
  15. σ × σ \sigma\times\sigma
  16. ϵ \epsilon
  17. σ \sigma
  18. Δ = Δ 0 + γ ( g ) \Delta=\Delta_{0}+\gamma(g)
  19. Δ 0 \Delta_{0}
  20. γ ( g ) \gamma(g)
  21. g g
  22. γ ( g ) \gamma(g)
  23. g g
  24. γ ( g ) \gamma(g)
  25. g = g * g=g_{*}
  26. M 3 , 4 M_{3,4}
  27. σ = ϕ 1 , 2 \sigma=\phi_{1,2}
  28. ϵ = ϕ 1 , 3 \epsilon=\phi_{1,3}

Scarborough_criterion.html

  1. | a n b | | a p | { 1 , at all nodes < 1 , at one node at least \frac{\sum|a_{nb}|}{|a^{\prime}_{p}|}\begin{cases}\leq 1,&\,\text{at all nodes% }\\ <1,&\,\text{at one node at least}\end{cases}
  2. < v a r > a p > <var>a_{p}>

Schmid's_Law.html

  1. τ = σ * m \tau=\sigma*m
  2. m = cos ( ϕ ) cos ( λ ) m=\cos(\phi)\cos(\lambda)

Schröder–Bernstein_theorem.html

  1. f f
  2. g g
  3. g - 1 g^{-1}
  4. f - 1 f^{-1}
  5. f - 1 ( g - 1 ( a ) ) g - 1 ( a ) a f ( a ) g ( f ( a ) ) \cdots\rightarrow f^{-1}(g^{-1}(a))\rightarrow g^{-1}(a)\rightarrow a% \rightarrow f(a)\rightarrow g(f(a))\rightarrow\cdots
  6. f - 1 f^{-1}
  7. g - 1 g^{-1}
  8. f f
  9. g g
  10. f f
  11. g g
  12. f f
  13. g g
  14. g g

Schur_class.html

  1. γ j = f j ( 0 ) \gamma_{j}=f_{j}(0)
  2. z f j + 1 = f j ( z ) - γ j 1 - γ j ¯ f j ( z ) . zf_{j+1}=\frac{f_{j}(z)-\gamma_{j}}{1-\overline{\gamma_{j}}f_{j}(z)}.
  3. f j ( z ) = γ j + 1 - | γ j | 2 γ j ¯ + 1 z f j + 1 ( z ) f_{j}(z)=\gamma_{j}+\frac{1-|\gamma_{j}|^{2}}{\overline{\gamma_{j}}+\frac{1}{% zf_{j+1}(z)}}
  4. f 0 ( z ) = γ 0 + 1 - | γ 0 | 2 γ 0 ¯ + 1 z γ 1 + z ( 1 - | γ 1 | 2 ) γ 1 ¯ + 1 z γ 2 + . f_{0}(z)=\gamma_{0}+\frac{1-|\gamma_{0}|^{2}}{\overline{\gamma_{0}}+\frac{1}{z% \gamma_{1}+\frac{z(1-|\gamma_{1}|^{2})}{\overline{\gamma_{1}}+\frac{1}{z\gamma% _{2}+\cdots}}}}.

Schur_product_theorem.html

  1. M M
  2. N N
  3. M N M\circ N
  4. a , b a,b
  5. a T ( M N ) b = Tr ( M diag ( a ) N diag ( b ) ) a^{T}(M\circ N)b=\operatorname{Tr}(M\operatorname{diag}(a)N\operatorname{diag}% (b))
  6. Tr \operatorname{Tr}
  7. diag ( a ) \operatorname{diag}(a)
  8. a a
  9. M M
  10. N N
  11. M 1 / 2 M^{1/2}
  12. N 1 / 2 N^{1/2}
  13. Tr ( M diag ( a ) N diag ( b ) ) = Tr ( M 1 / 2 M 1 / 2 diag ( a ) N 1 / 2 N 1 / 2 diag ( b ) ) = Tr ( M 1 / 2 diag ( a ) N 1 / 2 N 1 / 2 diag ( b ) M 1 / 2 ) \operatorname{Tr}(M\operatorname{diag}(a)N\operatorname{diag}(b))=% \operatorname{Tr}(M^{1/2}M^{1/2}\operatorname{diag}(a)N^{1/2}N^{1/2}% \operatorname{diag}(b))=\operatorname{Tr}(M^{1/2}\operatorname{diag}(a)N^{1/2}% N^{1/2}\operatorname{diag}(b)M^{1/2})
  14. a = b a=b
  15. Tr ( A T A ) \operatorname{Tr}(A^{T}A)
  16. A = N 1 / 2 diag ( a ) M 1 / 2 A=N^{1/2}\operatorname{diag}(a)M^{1/2}
  17. ( M N ) (M\circ N)
  18. X X
  19. n n
  20. X i X j = M i j \langle X_{i}X_{j}\rangle=M_{ij}
  21. X i 2 X_{i}^{2}
  22. X j 2 X_{j}^{2}
  23. Cov ( X i 2 , X j 2 ) = X i 2 X j 2 - X i 2 X j 2 \operatorname{Cov}(X_{i}^{2},X_{j}^{2})=\langle X_{i}^{2}X_{j}^{2}\rangle-% \langle X_{i}^{2}\rangle\langle X_{j}^{2}\rangle
  24. X i 2 X j 2 = 2 X i X j 2 + X i 2 X j 2 \langle X_{i}^{2}X_{j}^{2}\rangle=2\langle X_{i}X_{j}\rangle^{2}+\langle X_{i}% ^{2}\rangle\langle X_{j}^{2}\rangle
  25. Cov ( X i 2 , X j 2 ) = 2 X i X j 2 = 2 M i j 2 \operatorname{Cov}(X_{i}^{2},X_{j}^{2})=2\langle X_{i}X_{j}\rangle^{2}=2M_{ij}% ^{2}
  26. M i j 2 M_{ij}^{2}
  27. X X
  28. Y Y
  29. n n
  30. X i X j = M i j \langle X_{i}X_{j}\rangle=M_{ij}
  31. Y i Y j = N i j \langle Y_{i}Y_{j}\rangle=N_{ij}
  32. X i Y j = 0 \langle X_{i}Y_{j}\rangle=0
  33. i , j i,j
  34. X i Y i X_{i}Y_{i}
  35. X j Y j X_{j}Y_{j}
  36. Cov ( X i Y i , X j Y j ) = X i Y i X j Y j - X i Y i X j Y j \operatorname{Cov}(X_{i}Y_{i},X_{j}Y_{j})=\langle X_{i}Y_{i}X_{j}Y_{j}\rangle-% \langle X_{i}Y_{i}\rangle\langle X_{j}Y_{j}\rangle
  37. X i Y i X j Y j = X i X j Y i Y j + X i Y i X j Y j + X i Y j X j Y i \langle X_{i}Y_{i}X_{j}Y_{j}\rangle=\langle X_{i}X_{j}\rangle\langle Y_{i}Y_{j% }\rangle+\langle X_{i}Y_{i}\rangle\langle X_{j}Y_{j}\rangle+\langle X_{i}Y_{j}% \rangle\langle X_{j}Y_{i}\rangle
  38. X X
  39. Y Y
  40. Cov ( X i Y i , X j Y j ) = X i X j Y i Y j = M i j N i j \operatorname{Cov}(X_{i}Y_{i},X_{j}Y_{j})=\langle X_{i}X_{j}\rangle\langle Y_{% i}Y_{j}\rangle=M_{ij}N_{ij}
  41. M i j N i j M_{ij}N_{ij}
  42. M = μ i m i m i T M=\sum\mu_{i}m_{i}m_{i}^{T}
  43. N = ν i n i n i T N=\sum\nu_{i}n_{i}n_{i}^{T}
  44. M N = i j μ i ν j ( m i m i T ) ( n j n j T ) = i j μ i ν j ( m i n j ) ( m i n j ) T M\circ N=\sum_{ij}\mu_{i}\nu_{j}(m_{i}m_{i}^{T})\circ(n_{j}n_{j}^{T})=\sum_{ij% }\mu_{i}\nu_{j}(m_{i}\circ n_{j})(m_{i}\circ n_{j})^{T}
  45. ( m i n j ) ( m i n j ) T (m_{i}\circ n_{j})(m_{i}\circ n_{j})^{T}
  46. μ i ν j > 0 \mu_{i}\nu_{j}>0
  47. M N M\circ N
  48. a 0 a\neq 0
  49. a T ( M N ) a > 0 a^{T}(M\circ N)a>0
  50. a T ( m i n j ) ( m i n j ) T a 0 a^{T}(m_{i}\circ n_{j})(m_{i}\circ n_{j})^{T}a\geq 0
  51. i i
  52. j j
  53. a T ( m i n j ) ( m i n j ) T a = ( k m i , k n j , k a k ) 2 a^{T}(m_{i}\circ n_{j})(m_{i}\circ n_{j})^{T}a=\left(\sum_{k}m_{i,k}n_{j,k}a_{% k}\right)^{2}
  54. N N
  55. j j
  56. n j , k a k n_{j,k}a_{k}
  57. k k
  58. M M
  59. i i
  60. m i , k n j , k a k m_{i,k}n_{j,k}a_{k}
  61. k k
  62. i i
  63. j j
  64. ( k m i , k n j , k a k ) 2 > 0 \left(\sum_{k}m_{i,k}n_{j,k}a_{k}\right)^{2}>0

Schwinger_magnetic_induction.html

  1. S m i = m e 2 c 2 q e 4.41401 × 10 9 tesla , S_{mi}=\frac{m_{e}^{2}c^{2}}{q_{e}\hbar}\simeq 4.41401\times 10^{9}\,\mathrm{% tesla},
  2. S m i = E s c S_{mi}=\frac{E_{\mathrm{s}}}{c}

Second_covariant_derivative.html

  1. Γ ( E ) Γ ( T * M E ) Γ ( T * M T * M E ) . \Gamma(E)\stackrel{\nabla}{\longrightarrow}\Gamma(T^{*}M\otimes E)\stackrel{% \nabla}{\longrightarrow}\Gamma(T^{*}M\otimes T^{*}M\otimes E).
  2. ( u , v 2 w ) a = u c v b c b w a (\nabla^{2}_{u,v}w)^{a}=u^{c}v^{b}\nabla_{c}\nabla_{b}w^{a}
  3. ( u v w ) a = u c c v b b w a = u c v b c b w a + ( u c c v b ) b w a = ( u , v 2 w ) a + ( u v w ) a . (\nabla_{u}\nabla_{v}w)^{a}=u^{c}\nabla_{c}v^{b}\nabla_{b}w^{a}=u^{c}v^{b}% \nabla_{c}\nabla_{b}w^{a}+(u^{c}\nabla_{c}v^{b})\nabla_{b}w^{a}=(\nabla^{2}_{u% ,v}w)^{a}+(\nabla_{\nabla_{u}v}w)^{a}.
  4. u , v 2 w = u v w - u v w . \nabla^{2}_{u,v}w=\nabla_{u}\nabla_{v}w-\nabla_{\nabla_{u}v}w.
  5. R ( u , v ) w = u , v 2 w - v , u 2 w . R(u,v)w=\nabla^{2}_{u,v}w-\nabla^{2}_{v,u}w.
  6. u , v 2 f = u c v b c b f = u v f - u v f . \nabla^{2}_{u,v}f=u^{c}v^{b}\nabla_{c}\nabla_{b}f=\nabla_{u}\nabla_{v}f-\nabla% _{\nabla_{u}v}f.
  7. u v - v u = [ u , v ] . \nabla_{u}v-\nabla_{v}u=[u,v].
  8. ( u v - v u ) ( f ) = [ u , v ] ( f ) = u ( v ( f ) ) - v ( u ( f ) ) . (\nabla_{u}v-\nabla_{v}u)(f)=[u,v](f)=u(v(f))-v(u(f)).
  9. u v f - v u f = u v f - v u f . \nabla_{\nabla_{u}v}f-\nabla_{\nabla_{v}u}f=\nabla_{u}\nabla_{v}f-\nabla_{v}% \nabla_{u}f.
  10. u , v 2 f = v , u 2 f . \nabla^{2}_{u,v}f=\nabla^{2}_{v,u}f.

Segal–Bargmann_space.html

  1. F 2 := π - n n | F ( z ) | 2 exp ( - | z | 2 ) d z < , \|F\|^{2}:=\pi^{-n}\int_{\mathbb{C}^{n}}|F(z)|^{2}\exp(-|z|^{2})\,dz<\infty,
  2. F G = π - n n F ( z ) ¯ G ( z ) exp ( - | z | 2 ) d z . \langle F\mid G\rangle=\pi^{-n}\int_{\mathbb{C}^{n}}\overline{F(z)}G(z)\exp(-|% z|^{2})\,dz.
  3. | F ( a ) | < C F . |F(a)|<C\|F\|.
  4. F ( a ) = F a F . F(a)=\langle F_{a}\mid F\rangle.
  5. F a ( z ) = exp ( a ¯ z ) F_{a}(z)=\exp(\overline{a}\cdot z)
  6. a ¯ z = j = 1 n a j ¯ z j . \overline{a}\cdot z=\sum_{j=1}^{n}\overline{a_{j}}z_{j}.
  7. κ ( a , z ) := F a ( z ) ¯ \kappa(a,z):=\overline{F_{a}(z)}
  8. F ( a ) = F a F = π - n n κ ( a , z ) F ( z ) exp ( - | z | 2 ) d z , F(a)=\langle F_{a}\mid F\rangle=\pi^{-n}\int_{\mathbb{C}^{n}}\kappa(a,z)F(z)% \exp(-|z|^{2})\,dz,
  9. F a 2 = F a F a = F a ( a ) = exp ( | a | 2 ) . \|F_{a}\|^{2}=\langle F_{a}\mid F_{a}\rangle=F_{a}(a)=\exp(|a|^{2}).
  10. | F ( a ) | F a F = exp ( | a | 2 / 2 ) F . |F(a)|\leq\|F_{a}\|\|F\|=\exp(|a|^{2}/2)\|F\|.
  11. π - n | F ( z ) | 2 exp ( - | z | 2 ) \pi^{-n}|F(z)|^{2}\exp(-|z|^{2})
  12. a j = / z j a_{j}=\partial/\partial z_{j}
  13. a j * = z j a_{j}^{*}=z_{j}
  14. [ a j , a k * ] = δ j , k [a_{j},a_{k}^{*}]=\delta_{j,k}
  15. A j = ( a j + a j * ) / 2 A_{j}=(a_{j}+a_{j}^{*})/2
  16. B j = ( a j - a j * ) / ( 2 i ) B_{j}=(a_{j}-a_{j}^{*})/(2i)
  17. B B
  18. B B
  19. ( B f ) ( z ) = n exp [ - ( z z - 2 2 z x + x x ) / 2 ] f ( x ) d x , (Bf)(z)=\int_{\mathbb{R}^{n}}\exp[-(z\cdot z-2\sqrt{2}z\cdot x+x\cdot x)/2]f(x% )\,dx~{},
  20. z z
  21. ( B f ) ( z ) (Bf)(z)
  22. f f
  23. z z
  24. f f
  25. π - n | ( B f ) ( z ) | 2 exp ( - | z | 2 ) . \pi^{-n}|(Bf)(z)|^{2}\exp(-|z|^{2})~{}.
  26. f f
  27. f f
  28. B f Bf
  29. B B
  30. B B
  31. f ( x ) = n exp [ - ( z ¯ z ¯ - 2 2 z ¯ x + x x ) / 2 ] ( B f ) ( z ) d z . f(x)=\int_{\mathbb{C}^{n}}\exp[-(\overline{z}\cdot\overline{z}-2\sqrt{2}% \overline{z}\cdot x+x\cdot x)/2](Bf)(z)\,dz~{}.
  32. B f Bf
  33. B f Bf
  34. B B
  35. f ( x ) = C exp ( - | x | 2 / 2 ) n ( B f ) ( x + i y ) exp ( - | y | 2 / 2 ) d y , f(x)=C\exp(-|x|^{2}/2)\int_{\mathbb{R}^{n}}(Bf)(x+iy)\exp(-|y|^{2}/2)\,dy,
  36. C = π - n / 4 ( 2 π ) - n / 2 C=\pi^{-n/4}(2\pi)^{-n/2}
  37. f f
  38. B f Bf

Seismic_site_effects.html

  1. h h
  2. S H SH
  3. A 2 A_{2}
  4. θ 2 \theta_{2}
  5. A 2 A_{2}^{^{\prime}}
  6. θ 2 \theta_{2}
  7. A 1 A_{1}
  8. θ 1 \theta_{1}
  9. A 1 A_{1}^{^{\prime}}
  10. θ 1 \theta_{1}
  11. A 2 A_{2}
  12. 150 c m / s 2 150~{}cm/s^{2}
  13. 18 c m / s 2 18~{}cm/s^{2}
  14. 35 c m / s 2 35~{}cm/s^{2}
  15. 170 c m / s 2 170~{}cm/s^{2}
  16. S H SH
  17. i = 1 i=1
  18. i = 2 i=2
  19. T ¯ ( ω ) \bar{T}(\omega)
  20. T ¯ ( ω ) = 2 A 1 2 A 2 = 1 cos k z 1 h + i χ ¯ sin k z 1 h \bar{T}(\omega)=\frac{2A_{1}}{2A_{2}}=\frac{1}{\cos k_{z_{1}}h+i\bar{\chi}\sin k% _{z_{1}}h}
  21. k z 1 = ω θ i V S i k_{z_{1}}=\frac{\omega\theta_{i}}{V_{S_{i}}}
  22. χ ¯ = μ 1 ρ 1 μ 2 ρ 2 cos θ 1 cos θ 2 \bar{\chi}=\sqrt{\frac{\mu_{1}\rho_{1}}{\mu_{2}\rho_{2}}}\frac{\cos\theta_{1}}% {\cos\theta_{2}}
  23. h h
  24. θ i \theta_{i}
  25. i i
  26. ρ i \rho_{i}
  27. i i
  28. μ i \mu_{i}
  29. i i
  30. k z 1 k_{z_{1}}
  31. V S i = μ i ρ i V_{S_{i}}=\sqrt{\frac{\mu_{i}}{\rho_{i}}}
  32. T ¯ \bar{T}
  33. V S 1 = 200 m / s V_{S_{1}}=200~{}m/s
  34. χ ¯ \bar{\chi}
  35. | T ¯ m a x | = 2 |\bar{T}_{max}|=2
  36. V S 2 = 800 m / s V_{S_{2}}=800~{}m/s
  37. | T ¯ m a x | 3.5 |\bar{T}_{max}|\simeq 3.5
  38. V S 2 = 2000 m / s V_{S_{2}}=2000~{}m/s
  39. | T ¯ m a x | 6 |\bar{T}_{max}|\simeq 6
  40. V S 2 = 5000 m / s V_{S_{2}}=5000~{}m/s
  41. χ ¯ 1 \bar{\chi}\gg 1
  42. f 0 = V S 1 4 h f_{0}=\frac{V_{S_{1}}}{4h}
  43. S H SH
  44. A 0 A_{0}
  45. f 0 f_{0}
  46. f 0 = 0.3 H z ; A 0 = 2.53 f_{0}=0.3Hz~{};~{}A_{0}=2.53
  47. f 0 = 0.4 H z ; A 0 = 8.83 f_{0}=0.4Hz~{};~{}A_{0}=8.83
  48. f 0 = 0.6 H z ; A 0 = 7.11 f_{0}=0.6Hz~{};~{}A_{0}=7.11

Selberg's_identity.html

  1. p < x ( log p ) 2 + p q < x log p log q = 2 x log x + O ( x ) \sum_{p<x}(\log p)^{2}+\sum_{pq<x}\log p\log q=2x\log x+O(x)

Self-assembly_of_nanoparticles.html

  1. Δ G S A = Δ H S A - T Δ S S A \Delta G_{SA}=\Delta H_{SA}-T\Delta S_{SA}\,
  2. Δ G S A \Delta G_{SA}\,
  3. Δ H S A \Delta H_{SA}\,
  4. Δ S S A \Delta S_{SA}\,
  5. T Δ S S A T\Delta S_{SA}\,
  6. Δ H S A \Delta H_{SA}\,
  7. Δ G D F = Δ H D F - T Δ S D F \Delta G_{DF}=\Delta H_{DF}-T\Delta S_{DF}\,
  8. Δ H D F \Delta H_{DF}\,
  9. Δ G D F \Delta G_{DF}\,
  10. N N 0 = e x p ( - Δ E a c t R T ) {N\over N_{0}}=exp({-\Delta E_{act}\over RT})\,
  11. Δ E a c t \Delta E_{act}\,
  12. Δ E a c t \Delta E_{act}\,
  13. Δ H D F \Delta H_{DF}\,
  14. γ \gamma
  15. E 0 - E 1 = - π r 2 γ O / W [ γ O / W - ( γ P / W - γ P / O ) ] 2 {E_{0}-E_{1}}=-{\pi r^{2}\over\gamma_{O/W}}[\gamma_{O/W}-(\gamma_{P/W}-\gamma_% {P/O})]^{2}\,

Self-healing_hydrogels.html

  1. ϕ = ζ * e κ h \phi=\zeta*e^{\kappa h}
  2. ζ \zeta
  3. κ \kappa
  4. h \ h
  5. κ = Σ ( z * e ) 2 C 0 * ϵ r ϵ 0 k B T \kappa=\sqrt{\frac{\Sigma(z*e)^{2}C_{0}^{*}}{\epsilon_{r}\epsilon_{0}k_{B}T}}
  6. z \ z
  7. e \ e
  8. C 0 * C_{0}^{*}
  9. ϵ r \epsilon_{r}
  10. ϵ 0 \epsilon_{0}
  11. k B k_{B}
  12. T \ T
  13. λ = ( γ * H 3 μ ) 1 / 4 \lambda=({\gamma*H^{3}\over\mu})^{1/4}

Semi-invariant_of_a_quiver.html

  1. Rep ( Q , 𝐝 ) := { V Rep ( Q ) : V i = 𝐝 ( i ) } \operatorname{Rep}(Q,\mathbf{d}):=\{V\in\operatorname{Rep}(Q):V_{i}=\mathbf{d}% (i)\}
  2. α Q 1 Hom k ( k 𝐝 ( s ( α ) ) , k 𝐝 ( t ( α ) ) ) \bigoplus_{\alpha\in Q_{1}}\operatorname{Hom}_{k}(k^{\mathbf{d}(s(\alpha))},k^% {\mathbf{d}(t(\alpha))})
  3. G L ( 𝐝 ) × Rep ( Q , 𝐝 ) Rep ( Q , 𝐝 ) ( ( g i ) , ( V i , V ( α ) ) ) ( V i , g t ( α ) V ( α ) g s ( α ) - 1 ) \begin{array}[]{ccc}GL(\mathbf{d})\times\operatorname{Rep}(Q,\mathbf{d})&% \longrightarrow&\operatorname{Rep}(Q,\mathbf{d})\\ \Big((g_{i}),(V_{i},V(\alpha))\Big)&\longmapsto&(V_{i},g_{t(\alpha)}\cdot V(% \alpha)\cdot g_{s(\alpha)}^{-1})\end{array}
  4. G L ( 𝐝 ) × k [ Rep ( Q , 𝐝 ) ] k Rep ( Q , 𝐝 ) ] ( g , f ) g f ( - ) := f ( g - 1 . - ) \begin{array}[]{ccc}GL(\mathbf{d})\times k[\operatorname{Rep}(Q,\mathbf{d})]&% \longrightarrow&k\operatorname{Rep}(Q,\mathbf{d})]\\ (g,f)&\longmapsto&g\cdot f(-):=f(g^{-1}.-)\end{array}
  5. I ( Q , 𝐝 ) := k [ Rep ( Q , 𝐝 ) ] G L ( 𝐝 ) I(Q,\mathbf{d}):=k[\operatorname{Rep}(Q,\mathbf{d})]^{GL(\mathbf{d})}
  6. I ( Q , 𝐝 ) = k [ c 1 , , c n ] I(Q,\mathbf{d})=k[c_{1},\ldots,c_{n}]
  7. det ( A - t 𝕀 ) = t n - c 1 ( A ) t n - 1 + + ( - 1 ) n c n ( A ) \det(A-t\mathbb{I})=t^{n}-c_{1}(A)t^{n-1}+\cdots+(-1)^{n}c_{n}(A)
  8. S I ( Q , 𝐝 ) = σ Q 0 S I ( Q , 𝐝 ) σ SI(Q,\mathbf{d})=\bigoplus_{\sigma\in\mathbb{Z}^{Q_{0}}}SI(Q,\mathbf{d})_{\sigma}
  9. S I ( Q , 𝐝 ) σ := { f k [ Rep ( Q , 𝐝 ) ] : g f = i Q 0 det ( g i ) σ i f , g G L ( 𝐝 ) } . SI(Q,\mathbf{d})_{\sigma}:=\{f\in k[\operatorname{Rep}(Q,\mathbf{d})]:g\cdot f% =\prod_{i\in Q_{0}}\det(g_{i})^{\sigma_{i}}f,\forall g\in GL(\mathbf{d})\}.
  10. 1 𝛼 2 1\xrightarrow{\ \ \alpha\ }2
  11. ( g 1 , g 2 ) det u ( B ) = det u ( g 2 - 1 B g 1 ) = det u ( g 1 ) det - u ( g 2 ) det u ( B ) (g_{1},g_{2})\cdot{\det}^{u}(B)={\det}^{u}(g_{2}^{-1}Bg_{1})={\det}^{u}(g_{1})% {\det}^{-u}(g_{2}){\det}^{u}(B)
  12. 𝖲𝖨 ( Q , 𝐝 ) = k [ det ] \mathsf{SI}(Q,\mathbf{d})=k[\det]
  13. S I ( Q , 𝐝 ) = k [ D 1 , 2 , D 3 , 4 , D 1 , 4 , D 2 , 3 , D 1 , 3 , D 2 , 4 ] D 1 , 2 D 3 , 4 + D 1 , 4 D 2 , 3 - D 1 , 3 D 2 , 4 SI(Q,\mathbf{d})=\frac{k[D_{1,2},D_{3,4},D_{1,4},D_{2,3},D_{1,3},D_{2,4}]}{D_{% 1,2}D_{3,4}+D_{1,4}D_{2,3}-D_{1,3}D_{2,4}}

Semi-simplicity.html

  1. T : V V T:V\to V
  2. M n ( D 1 ) × M n ( D 2 ) × × M n ( D r ) M_{n}(D_{1})\times M_{n}(D_{2})\times\cdots\times M_{n}(D_{r})
  3. D i D_{i}
  4. M n ( D ) M_{n}(D)
  5. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  6. M M M ′′ M\cong M^{\prime}\oplus M^{\prime\prime}
  7. F [ T ] End F ( V ) F[T]\subset\operatorname{End}_{F}(V)
  8. X α C X_{\alpha}\in C
  9. X α X_{\alpha}
  10. Mot ( k ) \operatorname{Mot}(k)_{\sim}
  11. \sim

Semi-Yao_graph.html

  1. θ \theta
  2. θ \theta
  3. θ \theta

Semicircle_law_(Quantum_Hall).html

  1. σ 1 σ 2 + ( σ h - σ h 0 ) = ( e 2 / ( 2 h ) 2 \sigma_{1}\sigma_{2}+(\sigma_{h}-\sigma^{0}_{h})=(e^{2}/(2h)^{2}
  2. σ h 0 = ( N + 1 / 2 ) e 2 / h \sigma^{0}_{h}=(N+1/2)e^{2}/h

Separable_filter.html

  1. 1 3 [ 1 1 1 ] * 1 3 [ 1 1 1 ] = 1 9 [ 1 1 1 1 1 1 1 1 1 ] \frac{1}{3}\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}*\frac{1}{3}\begin{bmatrix}1&1&1\end{bmatrix}=\frac{1}{9}\begin{% bmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{bmatrix}
  2. 1 4 [ 1 2 1 ] * 1 4 [ 1 2 1 ] = 1 16 [ 1 2 1 2 4 2 1 2 1 ] \frac{1}{4}\begin{bmatrix}1\\ 2\\ 1\end{bmatrix}*\frac{1}{4}\begin{bmatrix}1&2&1\end{bmatrix}=\frac{1}{16}\begin% {bmatrix}1&2&1\\ 2&4&2\\ 1&2&1\end{bmatrix}
  3. 𝐆 𝐱 = [ 1 0 - 1 2 0 - 2 1 0 - 1 ] * A = [ 1 2 1 ] * [ + 1 0 - 1 ] * A \mathbf{G_{x}}=\begin{bmatrix}&&\\ 1&0&-1\\ 2&0&-2\\ 1&0&-1\end{bmatrix}*A=\begin{bmatrix}1\\ 2\\ 1\end{bmatrix}*\begin{bmatrix}+1&0&-1\end{bmatrix}*A

Separative_work_units.html

  1. W SWU W_{\mathrm{SWU}}
  2. F F
  3. x f x_{f}
  4. P P
  5. x p x_{p}
  6. T T
  7. x t x_{t}
  8. W SWU = P V ( x p ) + T V ( x t ) - F V ( x f ) W_{\mathrm{SWU}}=P\cdot V\left(x_{p}\right)+T\cdot V(x_{t})-F\cdot V(x_{f})
  9. V ( x ) V\left(x\right)
  10. V ( x ) = ( 1 - 2 x ) ln ( 1 - x x ) V(x)=(1-2x)\ln\left(\frac{1-x}{x}\right)
  11. F P = x p - x t x f - x t \frac{F}{P}=\frac{x_{p}-x_{t}}{x_{f}-x_{t}}
  12. T P = x p - x f x f - x t \frac{T}{P}=\frac{x_{p}-x_{f}}{x_{f}-x_{t}}

Serre's_criterion_on_normality.html

  1. R k : A 𝔭 R_{k}:A_{\mathfrak{p}}
  2. 𝔭 \mathfrak{p}
  3. S k : depth A 𝔭 inf { k , ht ( 𝔭 ) } S_{k}:\operatorname{depth}A_{\mathfrak{p}}\geq\inf\{k,\operatorname{ht}(% \mathfrak{p})\}
  4. 𝔭 \mathfrak{p}
  5. R 0 , S 1 \Leftrightarrow R_{0},S_{1}
  6. R 1 , S 2 \Leftrightarrow R_{1},S_{2}
  7. S k \Leftrightarrow S_{k}
  8. 𝔭 i , 1 i r \mathfrak{p}_{i},\,1\leq i\leq r
  9. κ ( 𝔭 i ) = Q ( A / 𝔭 i ) \kappa(\mathfrak{p}_{i})=Q(A/\mathfrak{p}_{i})
  10. 1 = e 1 + + e r 1=e_{1}+\dots+e_{r}
  11. e i e_{i}
  12. κ ( 𝔭 i ) \kappa(\mathfrak{p}_{i})
  13. e i e j = 0 , i j e_{i}e_{j}=0,\,i\neq j
  14. e i e_{i}
  15. ( f / g ) n + a 1 ( f / g ) n - 1 + + a n = 0 (f/g)^{n}+a_{1}(f/g)^{n-1}+\dots+a_{n}=0
  16. f g A f\in gA
  17. g A gA
  18. 𝔭 \mathfrak{p}
  19. A / g A A/gA
  20. A 𝔭 A_{\mathfrak{p}}
  21. ϕ ( f ) ϕ ( g ) A 𝔭 \phi(f)\in\phi(g)A_{\mathfrak{p}}
  22. ϕ : A A 𝔭 \phi:A\to A_{\mathfrak{p}}
  23. g A = i 𝔮 i gA=\cap_{i}\mathfrak{q}_{i}
  24. 𝔮 i \mathfrak{q}_{i}
  25. 𝔭 \mathfrak{p}
  26. A / g A A/gA
  27. f ϕ - 1 ( 𝔮 i A 𝔭 ) = 𝔮 i f\in\phi^{-1}(\mathfrak{q}_{i}A_{\mathfrak{p}})=\mathfrak{q}_{i}
  28. 𝔮 i \mathfrak{q}_{i}
  29. 𝔭 \mathfrak{p}
  30. 𝔭 \mathfrak{p}
  31. A / f A A/fA
  32. 𝔭 \mathfrak{p}
  33. 𝔭 = { x A | x g 0 mod f A } \mathfrak{p}=\{x\in A|xg\equiv 0\,\text{ mod }fA\}
  34. g f A g\not\in fA
  35. y 𝔭 𝔭 y\mathfrak{p}\subset\mathfrak{p}
  36. 𝔭 \mathfrak{p}
  37. A [ y ] A[y]
  38. y y
  39. y 𝔭 = A y\mathfrak{p}=A
  40. 𝔭 = f / g A \mathfrak{p}=f/gA
  41. 𝔭 \mathfrak{p}
  42. 𝔭 \mathfrak{p}
  43. 𝔭 \mathfrak{p}
  44. 𝔭 \mathfrak{p}
  45. 𝔭 \mathfrak{p}
  46. \square