#theta down columns #column labels are N # # theta N=10 11 12 13 14 15 16 17 18 19 20 0.0000000 2345 6199 16225 42811 112285 296051 777411 2049025 5384855 14190509 37313977 # slope=0/1 5.7105931 2406 6346 16607 43761 114765 302312 793739 2090830 5493752 14472712 38048175 # slope=1/10 6.3401917 2412 6362 16650 43867 115053 303009 795800 2095550 5508719 14505266 38154884 # slope=1/9 7.1250163 2421 6382 16703 44010 115401 304070 798160 2103277 5525044 14559752 38268214 # slope=1/8 8.1301024 2432 6408 16782 44184 115990 305265 802371 2111606 5554032 14617601 38464516 # slope=1/7 9.4623222 2446 6453 16877 44519 116646 307596 806922 2127264 5585502 14721875 38682874 # slope=1/6 11.3099325 2475 6506 17084 44890 118020 310164 815960 2144990 5645806 14845239 39090040 # slope=1/5 12.5288077 2491 6564 17189 45262 118733 312609 820793 2161649 5678887 14959759 39318576 # slope=2/9 15.9453959 2550 6709 17566 46260 121235 319490 837959 2208846 5797273 15282964 40136195 # slope=2/7 16.6992442 2567 6737 17680 46455 122001 320862 843020 2218556 5830555 15352389 40356407 # slope=3/10 18.4349488 2606 6802 17945 46911 123802 324049 855278 2240732 5914465 15507002 40931398 # slope=1/3 20.5560452 2642 6941 18193 47840 125503 330331 867014 2283514 5996156 15798690 41501987 # slope=3/8 21.8014095 2663 7024 18347 48391 126650 333960 875369 2307549 6055880 15959240 41924343 # slope=2/5 23.1985905 2699 7107 18588 48928 128316 337576 886778 2332719 6133410 16136592 42450936 # slope=3/7 23.9624890 2718 7156 18719 49273 129189 339979 892603 2349389 6172645 16251838 42719237 # slope=4/9 26.5650512 2787 7326 19179 50479 132302 348335 914212 2406679 6323026 16646178 43761277 # slope=1/2 29.0546041 2860 7522 19695 51796 135813 357442 938054 2469448 6487003 17076841 44895414 # slope=5/9 29.7448813 2881 7579 19843 52172 136817 360085 944825 2488199 6533163 17208389 45213984 # slope=4/7 30.9637565 2918 7680 20120 52838 138808 364632 958588 2519944 6627145 17429772 45857674 # slope=3/5 32.0053832 2949 7787 20340 53541 140384 369301 969560 2551767 6702630 17649851 46377085 # slope=5/8 33.6900675 3001 7967 20707 54714 143075 377015 988989 2603530 6839365 18004271 47324941 # slope=2/3 34.9920202 3053 8101 21041 55673 145339 383554 1004844 2647652 6951178 18303251 48107338 # slope=7/10 35.5376778 3076 8159 21185 56084 146309 386356 1011639 2666561 6999425 18431058 48449523 # slope=5/7 37.8749837 3185 8403 21876 57893 150842 399327 1042215 2757191 7210196 19054060 49921988 # slope=7/9 38.6598083 3228 8483 22142 58491 152555 403703 1053535 2788741 7286375 19277933 50442047 # slope=4/5 39.8055711 3297 8599 22592 59327 155470 409801 1072564 2832951 7412365 19595808 51286932 # slope=5/6 40.6012946 3346 8681 22926 59912 157709 413975 1087397 2862917 7510639 19810938 51941938 # slope=6/7 41.1859252 3383 8743 23176 60351 159421 417072 1099025 2884823 7588871 19966593 52466626 # slope=7/8 41.6335393 3411 8791 23371 60693 160753 419481 1108165 2901765 7651370 20085815 52891397 # slope=8/9 41.9872125 3434 8830 23527 60966 161819 421408 1115476 2915319 7701655 20180906 53236763 # slope=9/10 45.0000000 3640 9176 24928 63424 171408 438752 1181280 3037304 8154224 21036728 56354776 # slope=1/1 # c(N,theta) = a(theta) mu^N N^b # c(N,theta)/c(N,0) = a(theta)/a(0) #theta down columns #column labels are N 2345 6199 16225 42811 112285 296051 777411 2049025 5384855 14190509 37313977 2406 6346 16607 43761 114765 302312 793739 2090830 5493752 14472712 38048175 2412 6362 16650 43867 115053 303009 795800 2095550 5508719 14505266 38154884 2421 6382 16703 44010 115401 304070 798160 2103277 5525044 14559752 38268214 2432 6408 16782 44184 115990 305265 802371 2111606 5554032 14617601 38464516 2446 6453 16877 44519 116646 307596 806922 2127264 5585502 14721875 38682874 2475 6506 17084 44890 118020 310164 815960 2144990 5645806 14845239 39090040 2491 6564 17189 45262 118733 312609 820793 2161649 5678887 14959759 39318576 2550 6709 17566 46260 121235 319490 837959 2208846 5797273 15282964 40136195 2567 6737 17680 46455 122001 320862 843020 2218556 5830555 15352389 40356407 2606 6802 17945 46911 123802 324049 855278 2240732 5914465 15507002 40931398 2642 6941 18193 47840 125503 330331 867014 2283514 5996156 15798690 41501987 2663 7024 18347 48391 126650 333960 875369 2307549 6055880 15959240 41924343 2699 7107 18588 48928 128316 337576 886778 2332719 6133410 16136592 42450936 2718 7156 18719 49273 129189 339979 892603 2349389 6172645 16251838 42719237 2787 7326 19179 50479 132302 348335 914212 2406679 6323026 16646178 43761277 2860 7522 19695 51796 135813 357442 938054 2469448 6487003 17076841 44895414 2881 7579 19843 52172 136817 360085 944825 2488199 6533163 17208389 45213984 2918 7680 20120 52838 138808 364632 958588 2519944 6627145 17429772 45857674 2949 7787 20340 53541 140384 369301 969560 2551767 6702630 17649851 46377085 3001 7967 20707 54714 143075 377015 988989 2603530 6839365 18004271 47324941 3053 8101 21041 55673 145339 383554 1004844 2647652 6951178 18303251 48107338 3076 8159 21185 56084 146309 386356 1011639 2666561 6999425 18431058 48449523 3185 8403 21876 57893 150842 399327 1042215 2757191 7210196 19054060 49921988 3228 8483 22142 58491 152555 403703 1053535 2788741 7286375 19277933 50442047 3297 8599 22592 59327 155470 409801 1072564 2832951 7412365 19595808 51286932 3346 8681 22926 59912 157709 413975 1087397 2862917 7510639 19810938 51941938 3383 8743 23176 60351 159421 417072 1099025 2884823 7588871 19966593 52466626 3411 8791 23371 60693 160753 419481 1108165 2901765 7651370 20085815 52891397 3434 8830 23527 60966 161819 421408 1115476 2915319 7701655 20180906 53236763 3640 9176 24928 63424 171408 438752 1181280 3037304 8154224 21036728 56354776 #theta down columns #column labels are N # # theta N=10 11 12 13 14 15 16 17 18 19 20 0.0000000 2345 6199 16225 42811 112285 296051 777411 2049025 5384855 14190509 37313977 # slope=0/1 5.7105931 2406 6346 16607 43761 114765 302312 793739 2090830 5493752 14472712 38048175 # slope=1/10 6.3401917 2412 6362 16650 43867 115053 303009 795800 2095550 5508719 14505266 38154884 # slope=1/9 7.1250163 2421 6382 16703 44010 115401 304070 798160 2103277 5525044 14559752 38268214 # slope=1/8 8.1301024 2432 6408 16782 44184 115990 305265 802371 2111606 5554032 14617601 38464516 # slope=1/7 9.4623222 2446 6453 16877 44519 116646 307596 806922 2127264 5585502 14721875 38682874 # slope=1/6 11.3099325 2475 6506 17084 44890 118020 310164 815960 2144990 5645806 14845239 39090040 # slope=1/5 12.5288077 2491 6564 17189 45262 118733 312609 820793 2161649 5678887 14959759 39318576 # slope=2/9 15.9453959 2550 6709 17566 46260 121235 319490 837959 2208846 5797273 15282964 40136195 # slope=2/7 16.6992442 2567 6737 17680 46455 122001 320862 843020 2218556 5830555 15352389 40356407 # slope=3/10 18.4349488 2606 6802 17945 46911 123802 324049 855278 2240732 5914465 15507002 40931398 # slope=1/3 20.5560452 2642 6941 18193 47840 125503 330331 867014 2283514 5996156 15798690 41501987 # slope=3/8 21.8014095 2663 7024 18347 48391 126650 333960 875369 2307549 6055880 15959240 41924343 # slope=2/5 23.1985905 2699 7107 18588 48928 128316 337576 886778 2332719 6133410 16136592 42450936 # slope=3/7 23.9624890 2718 7156 18719 49273 129189 339979 892603 2349389 6172645 16251838 42719237 # slope=4/9 26.5650512 2787 7326 19179 50479 132302 348335 914212 2406679 6323026 16646178 43761277 # slope=1/2 29.0546041 2860 7522 19695 51796 135813 357442 938054 2469448 6487003 17076841 44895414 # slope=5/9 29.7448813 2881 7579 19843 52172 136817 360085 944825 2488199 6533163 17208389 45213984 # slope=4/7 30.9637565 2918 7680 20120 52838 138808 364632 958588 2519944 6627145 17429772 45857674 # slope=3/5 32.0053832 2949 7787 20340 53541 140384 369301 969560 2551767 6702630 17649851 46377085 # slope=5/8 33.6900675 3001 7967 20707 54714 143075 377015 988989 2603530 6839365 18004271 47324941 # slope=2/3 34.9920202 3053 8101 21041 55673 145339 383554 1004844 2647652 6951178 18303251 48107338 # slope=7/10 35.5376778 3076 8159 21185 56084 146309 386356 1011639 2666561 6999425 18431058 48449523 # slope=5/7 37.8749837 3185 8403 21876 57893 150842 399327 1042215 2757191 7210196 19054060 49921988 # slope=7/9 38.6598083 3228 8483 22142 58491 152555 403703 1053535 2788741 7286375 19277933 50442047 # slope=4/5 39.8055711 3297 8599 22592 59327 155470 409801 1072564 2832951 7412365 19595808 51286932 # slope=5/6 40.6012946 3346 8681 22926 59912 157709 413975 1087397 2862917 7510639 19810938 51941938 # slope=6/7 41.1859252 3383 8743 23176 60351 159421 417072 1099025 2884823 7588871 19966593 52466626 # slope=7/8 41.6335393 3411 8791 23371 60693 160753 419481 1108165 2901765 7651370 20085815 52891397 # slope=8/9 41.9872125 3434 8830 23527 60966 161819 421408 1115476 2915319 7701655 20180906 53236763 # slope=9/10 45.0000000 3640 9176 24928 63424 171408 438752 1181280 3037304 8154224 21036728 56354776 # slope=1/1 #theta down columns #column labels are N # # c(N,theta)/c(N,0) = a(theta)/a(0) #theta N=10 N=11 N=12 N=13 N=14 N=15 N=16 N=17 N=18 N=19 N=20 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 5.7106 1.0260 1.0237 1.0235 1.0222 1.0221 1.0211 1.0210 1.0204 1.0202 1.0199 1.0197 6.3402 1.0286 1.0263 1.0262 1.0247 1.0247 1.0235 1.0237 1.0227 1.0230 1.0222 1.0225 7.1250 1.0324 1.0295 1.0295 1.0280 1.0278 1.0271 1.0267 1.0265 1.0260 1.0260 1.0256 8.1301 1.0371 1.0337 1.0343 1.0321 1.0330 1.0311 1.0321 1.0305 1.0314 1.0301 1.0308 9.4623 1.0431 1.0410 1.0402 1.0399 1.0388 1.0390 1.0380 1.0382 1.0373 1.0374 1.0367 11.3099 1.0554 1.0495 1.0529 1.0486 1.0511 1.0477 1.0496 1.0468 1.0485 1.0461 1.0476 12.5288 1.0623 1.0589 1.0594 1.0573 1.0574 1.0559 1.0558 1.0550 1.0546 1.0542 1.0537 15.9454 1.0874 1.0823 1.0827 1.0806 1.0797 1.0792 1.0779 1.0780 1.0766 1.0770 1.0756 16.6992 1.0947 1.0868 1.0897 1.0851 1.0865 1.0838 1.0844 1.0827 1.0828 1.0819 1.0815 18.4349 1.1113 1.0973 1.1060 1.0958 1.1026 1.0946 1.1002 1.0936 1.0984 1.0928 1.0969 20.5560 1.1267 1.1197 1.1213 1.1175 1.1177 1.1158 1.1153 1.1144 1.1135 1.1133 1.1122 21.8014 1.1356 1.1331 1.1308 1.1303 1.1279 1.1280 1.1260 1.1262 1.1246 1.1246 1.1236 23.1986 1.1510 1.1465 1.1456 1.1429 1.1428 1.1403 1.1407 1.1385 1.1390 1.1371 1.1377 23.9625 1.1591 1.1544 1.1537 1.1509 1.1505 1.1484 1.1482 1.1466 1.1463 1.1453 1.1449 26.5651 1.1885 1.1818 1.1821 1.1791 1.1783 1.1766 1.1760 1.1745 1.1742 1.1731 1.1728 29.0546 1.2196 1.2134 1.2139 1.2099 1.2095 1.2074 1.2066 1.2052 1.2047 1.2034 1.2032 29.7449 1.2286 1.2226 1.2230 1.2187 1.2185 1.2163 1.2153 1.2143 1.2132 1.2127 1.2117 30.9638 1.2443 1.2389 1.2401 1.2342 1.2362 1.2317 1.2331 1.2298 1.2307 1.2283 1.2290 32.0054 1.2576 1.2562 1.2536 1.2506 1.2502 1.2474 1.2472 1.2454 1.2447 1.2438 1.2429 33.6901 1.2797 1.2852 1.2762 1.2780 1.2742 1.2735 1.2722 1.2706 1.2701 1.2688 1.2683 34.9920 1.3019 1.3068 1.2968 1.3004 1.2944 1.2956 1.2926 1.2922 1.2909 1.2898 1.2893 35.5377 1.3117 1.3162 1.3057 1.3100 1.3030 1.3050 1.3013 1.3014 1.2998 1.2988 1.2984 37.8750 1.3582 1.3555 1.3483 1.3523 1.3434 1.3488 1.3406 1.3456 1.3390 1.3427 1.3379 38.6598 1.3765 1.3684 1.3647 1.3663 1.3586 1.3636 1.3552 1.3610 1.3531 1.3585 1.3518 39.8056 1.4060 1.3872 1.3924 1.3858 1.3846 1.3842 1.3797 1.3826 1.3765 1.3809 1.3745 40.6013 1.4269 1.4004 1.4130 1.3995 1.4045 1.3983 1.3987 1.3972 1.3948 1.3961 1.3920 41.1859 1.4426 1.4104 1.4284 1.4097 1.4198 1.4088 1.4137 1.4079 1.4093 1.4070 1.4061 41.6335 1.4546 1.4181 1.4404 1.4177 1.4317 1.4169 1.4255 1.4162 1.4209 1.4154 1.4175 41.9872 1.4644 1.4244 1.4500 1.4241 1.4411 1.4234 1.4349 1.4228 1.4302 1.4221 1.4267 45.0000 1.5522 1.4802 1.5364 1.4815 1.5265 1.4820 1.5195 1.4823 1.5143 1.4825 1.5103 # Now, for each theta, estimate a(theta) as mean over a(theta, N)'s. #theta down columns #column labels are N # # c(N,theta)/c(N,0) = a(theta)/a(0) #theta a(theta)/a(0) 0.0000 1.0000 5.7106 1.0218 6.3402 1.0244 7.1250 1.0277 8.1301 1.0324 9.4623 1.0391 11.3099 1.0494 12.5288 1.0568 15.9454 1.0797 16.6992 1.0854 18.4349 1.0990 20.5560 1.1170 21.8014 1.1282 23.1986 1.1420 23.9625 1.1498 26.5651 1.1779 29.0546 1.2088 29.7449 1.2177 30.9638 1.2342 32.0054 1.2491 33.6901 1.2743 34.9920 1.2955 35.5377 1.3047 37.8750 1.3466 38.6598 1.3616 39.8056 1.3849 40.6013 1.4019 41.1859 1.4149 41.6335 1.4250 41.9872 1.4331 45.0000 1.5062 # From previous fit, estimate a(0) to be 0.1338501. # c(N,theta)/c(N,0) = a(theta)/a(0) #theta a(theta) 0.0000 0.1338 5.7106 0.1368 6.3402 0.1371 7.1250 0.1376 8.1301 0.1382 9.4623 0.1391 11.3099 0.1405 12.5288 0.1415 15.9454 0.1445 16.6992 0.1453 18.4349 0.1471 20.5560 0.1495 21.8014 0.1510 23.1986 0.1529 23.9625 0.1539 26.5651 0.1577 29.0546 0.1618 29.7449 0.1630 30.9638 0.1652 32.0054 0.1672 33.6901 0.1706 34.9920 0.1734 35.5377 0.1746 37.8750 0.1802 38.6598 0.1822 39.8056 0.1854 40.6013 0.1876 41.1859 0.1894 41.6335 0.1907 41.9872 0.1918 45.0000 0.2016 pgr \ -grey \ -xlabel '$\theta$' -ylabel '$a(\theta)$' \ -o atheta.pdf \ <<@@@ 0.0000 0.1338 5.7106 0.1368 6.3402 0.1371 7.1250 0.1376 8.1301 0.1382 9.4623 0.1391 11.3099 0.1405 12.5288 0.1415 15.9454 0.1445 16.6992 0.1453 18.4349 0.1471 20.5560 0.1495 21.8014 0.1510 23.1986 0.1529 23.9625 0.1539 26.5651 0.1577 29.0546 0.1618 29.7449 0.1630 30.9638 0.1652 32.0054 0.1672 33.6901 0.1706 34.9920 0.1734 35.5377 0.1746 37.8750 0.1802 38.6598 0.1822 39.8056 0.1854 40.6013 0.1876 41.1859 0.1894 41.6335 0.1907 41.9872 0.1918 45.0000 0.2016 @@@