Experimental results: Difference between revisions
| Line 64: | Line 64: | ||
== Varying the upper and lower bounds == | == Varying the upper and lower bounds == | ||
If N(a,b) is the maximum length of a sequence | If <math>N(a,b)</math> is the maximum length of a <math>\pm 1</math> sequence with the partial sums along its HAPs bounded below by <math>-a</math> and above by <math>b</math>, then: | ||
N(a, b) = N(b, a) | <math>N(a, b) = N(b, a)</math> | ||
N(0, b) = b (everything must be +1) | <math>N(0, b) = b</math> (everything must be <math>+1</math>) | ||
N(1, 1) = 11 (there are 4 such sequences: choose | <math>N(1, 1) = 11</math> (there are <math>4</math> such sequences: choose <math>x_1</math>, and use the constraints <math>x_n +x_{2n} = 0</math> and <math>x_1 + \ldots + x_{2n} = 0</math> to determine the entries up to <math>10</math>; then choose <math>x_{11}</math>) | ||
N(1, 2) = 41 (there are 4 such sequences -- example below) | <math>N(1, 2) = 41</math> (there are <math>4</math> such sequences -- example below) | ||
0 + - - + - + + - + + - - + | 0 + - - + - + + - + + - - + | ||
| Line 80: | Line 80: | ||
+ + - + - + + - - - + - + + | + + - + - + + - - - + - + + | ||
<math>N(1, 3) = 83</math> (there are <math>216</math> such sequences -- example below) | |||
N(1, 3) = 83 (there are 216 such sequences -- example below) | |||
0 - + - + - + + - + + - - + | 0 - + - + - + + - + + - - + | ||
| Line 96: | Line 95: | ||
N(1, 4) = 131 (there are 87144 such sequences -- example below) | <math>N(1, 4) = 131</math> (there are <math>87144</math> such sequences -- example below) | ||
0 + - - + - + + - - + - | 0 + - - + - + + - - + - | ||
| Line 120: | Line 119: | ||
+ + + - + + - - + - - + | + + + - + + - - + - - + | ||
<math>N(2, 2) >= 1124</math> | |||
N(2, 2) >= 1124 | |||
--[[User:Alec|Alec]] 13:46, 9 January 2010 (UTC) | --[[User:Alec|Alec]] 13:46, 9 January 2010 (UTC) | ||
Revision as of 06:23, 9 January 2010
To return to the main Polymath5 page, click here.
Different displays of a long low-discrepancy sequence
It would be good to have some long sequences with low discrepancy displayed nicely here (or on subsidiary pages) in tabular form, together with discussions about the structures that can be found in these sequences. For now, here is the current record not displayed in a particularly helpful way. It is a sequence of discrepancy 2 and length 1124.
The raw sequence
+ - + + - - - - + + - + + + - - + - + + - - - + - + + - + - - + + - + - - + - - + + - + + - - - + + + - - + - + - - + + + + - - + - - + + - - + + + - - + + - + - - + - - + + - + - - + - + + + - - + - + - - + + - - - + + - + + + - - - + - - + - + + + + - - - - + + - + + + - - + - - + - - + - + - + + - + + + - - - - + + + + - - - + + - + - - + - + + - - + - + + - + - + - + + - - - + + - + + - - + - - + - - + + + + - + - - + - - + + - - + - + + - - - + - + + - + + + - - + + - - + - - + - + + - - + - + + - + + - - + - - + - + + - - + + - + - + - - + - + + + - - + - + + - - - - + + - - + - + + - + + - - + - + + - - + + - + - - + - + - - + + - - + - + + + - + - - + - + - - + + + + - - - + + - + - - + - + + - - + - + + - + + - - + - - + - + + - - + + - - - + + - - + - + + - + + - - + - - + + + + - - - - + + - + + - + + - - + - - + + - - - + + - + + - - + - + + - + - - - + - + + - + + - - + - - + - + + - - + + - + + + - - - + - + + - + + - - + - - - + + + - - + - + - - + + + - + - + + - + - - - + + - + - + + - - + - - + - + + - - + - + - + + - + - + - - + - + + - - + - - + - + + + - - - + + - + + + - + - - + - + + - - + - + + - + - - - + - + + - + + - - + - - - + + + - - + - + + - - + + - + - + + - + - - - + - - + - + + - - + - + + - + + - - + - + + - + + - - + - - + - + + - - + - - + - + + - + - - + + - + + - - + - - + - + + - - + - + + - + + - - + - + + - - + + - + - - + - + + - - + - - + - + + - - + - + + - + + - + + - - + - + + - - + - - + - + + - - + - + - - + + + - + - - + - + + - - + - + - - + + + - - - + + - + + - - + + - + - + + - - + - - + - - + - + + - + - - + + + - + - - + - + + - - + - + - - + + - - + - + + - + + - - + + - + - - + - - + - + + - + + - - + - + + - + + - - + - - + + + - - - + - - + - + + - - + + + + - + - - + + - - + - + + - - + - - + - + + - - + - + + - + + - - + - - + - + + - - - + + - - + + - - + - + + - + + - - + - - + - + + + - + - - + - + + - - + - + + - + - + - - + - + - + + - - + - + - - + + + - + - - - + + + - - + - - + - + + - - + + + + - + - - - + - + + - + + - - + - - + + + - - - + - - + - + + + - + - + + - + - - - + - - + + + + - - + - + + - - + - + + - + - - + - + + - - - + - - + + - + - + + - + - - - + - + + + + - + - - - - + - + + - + + - - + + - + + - + - - + - + + - + - -
The sequence, together with the corresponding integers
1+ 2- 3+ 4+ 5- 6- 7- 8- 9+ 10+ 11- 12+ 13+ 14+ 15- 16- 17+ 18- 19+ 20+ 21- 22- 23- 24+ 25- 26+ 27+ 28- 29+ 30- 31- 32+ 33+ 34- 35+ 36- 37- 38+ 39- 40- 41+ 42+ 43- 44+ 45+ 46- 47- 48- 49+ 50+ 51+ 52- 53- 54+ 55- 56+ 57- 58- 59+ 60+ 61+ 62+ 63- 64- 65+ 66- 67- 68+ 69+ 70- 71- 72+ 73+ 74+ 75- 76- 77+ 78+ 79- 80+ 81- 82- 83+ 84- 85- 86+ 87+ 88- 89+ 90- 91- 92+ 93- 94+ 95+ 96+ 97- 98- 99+ 100- 101+ 102- 103- 104+ 105+ 106- 107- 108- 109+ 110+ 111- 112+ 113+ 114+ 115- 116- 117- 118+ 119- 120- 121+ 122- 123+ 124+ 125+ 126+ 127- 128- 129- 130- 131+ 132+ 133- 134+ 135+ 136+ 137- 138- 139+ 140- 141- 142+ 143- 144- 145+ 146- 147+ 148- 149+ 150+ 151- 152+ 153+ 154+ 155- 156- 157- 158- 159+ 160+ 161+ 162+ 163- 164- 165- 166+ 167+ 168- 169+ 170- 171- 172+ 173- 174+ 175+ 176- 177- 178+ 179- 180+ 181+ 182- 183+ 184- 185+ 186- 187+ 188+ 189- 190- 191- 192+ 193+ 194- 195+ 196+ 197- 198- 199+ 200- 201- 202+ 203- 204- 205+ 206+ 207+ 208+ 209- 210+ 211- 212- 213+ 214- 215- 216+ 217+ 218- 219- 220+ 221- 222+ 223+ 224- 225- 226- 227+ 228- 229+ 230+ 231- 232+ 233+ 234+ 235- 236- 237+ 238+ 239- 240- 241+ 242- 243- 244+ 245- 246+ 247+ 248- 249- 250+ 251- 252+ 253+ 254- 255+ 256+ 257- 258- 259+ 260- 261- 262+ 263- 264+ 265+ 266- 267- 268+ 269+ 270- 271+ 272- 273+ 274- 275- 276+ 277- 278+ 279+ 280+ 281- 282- 283+ 284- 285+ 286+ 287- 288- 289- 290- 291+ 292+ 293- 294- 295+ 296- 297+ 298+ 299- 300+ 301+ 302- 303- 304+ 305- 306+ 307+ 308- 309- 310+ 311+ 312- 313+ 314- 315- 316+ 317- 318+ 319- 320- 321+ 322+ 323- 324- 325+ 326- 327+ 328+ 329+ 330- 331+ 332- 333- 334+ 335- 336+ 337- 338- 339+ 340+ 341+ 342+ 343- 344- 345- 346+ 347+ 348- 349+ 350- 351- 352+ 353- 354+ 355+ 356- 357- 358+ 359- 360+ 361+ 362- 363+ 364+ 365- 366- 367+ 368- 369- 370+ 371- 372+ 373+ 374- 375- 376+ 377+ 378- 379- 380- 381+ 382+ 383- 384- 385+ 386- 387+ 388+ 389- 390+ 391+ 392- 393- 394+ 395- 396- 397+ 398+ 399+ 400+ 401- 402- 403- 404- 405+ 406+ 407- 408+ 409+ 410- 411+ 412+ 413- 414- 415+ 416- 417- 418+ 419+ 420- 421- 422- 423+ 424+ 425- 426+ 427+ 428- 429- 430+ 431- 432+ 433+ 434- 435+ 436- 437- 438- 439+ 440- 441+ 442+ 443- 444+ 445+ 446- 447- 448+ 449- 450- 451+ 452- 453+ 454+ 455- 456- 457+ 458+ 459- 460+ 461+ 462+ 463- 464- 465- 466+ 467- 468+ 469+ 470- 471+ 472+ 473- 474- 475+ 476- 477- 478- 479+ 480+ 481+ 482- 483- 484+ 485- 486+ 487- 488- 489+ 490+ 491+ 492- 493+ 494- 495+ 496+ 497- 498+ 499- 500- 501- 502+ 503+ 504- 505+ 506- 507+ 508+ 509- 510- 511+ 512- 513- 514+ 515- 516+ 517+ 518- 519- 520+ 521- 522+ 523- 524+ 525+ 526- 527+ 528- 529+ 530- 531- 532+ 533- 534+ 535+ 536- 537- 538+ 539- 540- 541+ 542- 543+ 544+ 545+ 546- 547- 548- 549+ 550+ 551- 552+ 553+ 554+ 555- 556+ 557- 558- 559+ 560- 561+ 562+ 563- 564- 565+ 566- 567+ 568+ 569- 570+ 571- 572- 573- 574+ 575- 576+ 577+ 578- 579+ 580+ 581- 582- 583+ 584- 585- 586- 587+ 588+ 589+ 590- 591- 592+ 593- 594+ 595+ 596- 597- 598+ 599+ 600- 601+ 602- 603+ 604+ 605- 606+ 607- 608- 609- 610+ 611- 612- 613+ 614- 615+ 616+ 617- 618- 619+ 620- 621+ 622+ 623- 624+ 625+ 626- 627- 628+ 629- 630+ 631+ 632- 633+ 634+ 635- 636- 637+ 638- 639- 640+ 641- 642+ 643+ 644- 645- 646+ 647- 648- 649+ 650- 651+ 652+ 653- 654+ 655- 656- 657+ 658+ 659- 660+ 661+ 662- 663- 664+ 665- 666- 667+ 668- 669+ 670+ 671- 672- 673+ 674- 675+ 676+ 677- 678+ 679+ 680- 681- 682+ 683- 684+ 685+ 686- 687- 688+ 689+ 690- 691+ 692- 693- 694+ 695- 696+ 697+ 698- 699- 700+ 701- 702- 703+ 704- 705+ 706+ 707- 708- 709+ 710- 711+ 712+ 713- 714+ 715+ 716- 717+ 718+ 719- 720- 721+ 722- 723+ 724+ 725- 726- 727+ 728- 729- 730+ 731- 732+ 733+ 734- 735- 736+ 737- 738+ 739- 740- 741+ 742+ 743+ 744- 745+ 746- 747- 748+ 749- 750+ 751+ 752- 753- 754+ 755- 756+ 757- 758- 759+ 760+ 761+ 762- 763- 764- 765+ 766+ 767- 768+ 769+ 770- 771- 772+ 773+ 774- 775+ 776- 777+ 778+ 779- 780- 781+ 782- 783- 784+ 785- 786- 787+ 788- 789+ 790+ 791- 792+ 793- 794- 795+ 796+ 797+ 798- 799+ 800- 801- 802+ 803- 804+ 805+ 806- 807- 808+ 809- 810+ 811- 812- 813+ 814+ 815- 816- 817+ 818- 819+ 820+ 821- 822+ 823+ 824- 825- 826+ 827+ 828- 829+ 830- 831- 832+ 833- 834- 835+ 836- 837+ 838+ 839- 840+ 841+ 842- 843- 844+ 845- 846+ 847+ 848- 849+ 850+ 851- 852- 853+ 854- 855- 856+ 857+ 858+ 859- 860- 861- 862+ 863- 864- 865+ 866- 867+ 868+ 869- 870- 871+ 872+ 873+ 874+ 875- 876+ 877- 878- 879+ 880+ 881- 882- 883+ 884- 885+ 886+ 887- 888- 889+ 890- 891- 892+ 893- 894+ 895+ 896- 897- 898+ 899- 900+ 901+ 902- 903+ 904+ 905- 906- 907+ 908- 909- 910+ 911- 912+ 913+ 914- 915- 916- 917+ 918+ 919- 920- 921+ 922+ 923- 924- 925+ 926- 927+ 928+ 929- 930+ 931+ 932- 933- 934+ 935- 936- 937+ 938- 939+ 940+ 941+ 942- 943+ 944- 945- 946+ 947- 948+ 949+ 950- 951- 952+ 953- 954+ 955+ 956- 957+ 958- 959+ 960- 961- 962+ 963- 964+ 965- 966+ 967+ 968- 969- 970+ 971- 972+ 973- 974- 975+ 976+ 977+ 978- 979+ 980- 981- 982- 983+ 984+ 985+ 986- 987- 988+ 989- 990- 991+ 992- 993+ 994+ 995- 996- 997+ 998+ 999+ 1000+ 1001- 1002+ 1003- 1004- 1005- 1006+ 1007- 1008+ 1009+ 1010- 1011+ 1012+ 1013- 1014- 1015+ 1016- 1017- 1018+ 1019+ 1020+ 1021- 1022- 1023- 1024+ 1025- 1026- 1027+ 1028- 1029+ 1030+ 1031+ 1032- 1033+ 1034- 1035+ 1036+ 1037- 1038+ 1039- 1040- 1041- 1042+ 1043- 1044- 1045+ 1046+ 1047+ 1048+ 1049- 1050- 1051+ 1052- 1053+ 1054+ 1055- 1056- 1057+ 1058- 1059+ 1060+ 1061- 1062+ 1063- 1064- 1065+ 1066- 1067+ 1068+ 1069- 1070- 1071- 1072+ 1073- 1074- 1075+ 1076+ 1077- 1078+ 1079- 1080+ 1081+ 1082- 1083+ 1084- 1085- 1086- 1087+ 1088- 1089+ 1090+ 1091+ 1092+ 1093- 1094+ 1095- 1096- 1097- 1098- 1099+ 1100- 1101+ 1102+ 1103- 1104+ 1105+ 1106- 1107- 1108+ 1109+ 1110- 1111+ 1112+ 1113- 1114+ 1115- 1116- 1117+ 1118- 1119+ 1120+ 1121- 1122+ 1123- 1124-
Links to other displays and visually displayed information about the sequence
Here is an alternate formatting. Link.
Here is the sequence divided into groups of 6. Link.
Here is a Rauzy tree that conveys information about subsequences of the sequence: Link.
Multiplicativity properties of the 1124 sequence
The 1124 sequence is not weakly multiplicative, but it does appear to be close to a weakly multiplicative sequence. That is, if you are prepared to disregard a few "errors", then the number of distinct HAP-subsequences is 6. One can use this information to identify a quasi-multiplicative sequence that is in some sense close to the 1124 sequence. Some details can be found in the sequences of comments that begin here and here.
Varying the upper and lower bounds
If [math]\displaystyle{ N(a,b) }[/math] is the maximum length of a [math]\displaystyle{ \pm 1 }[/math] sequence with the partial sums along its HAPs bounded below by [math]\displaystyle{ -a }[/math] and above by [math]\displaystyle{ b }[/math], then:
[math]\displaystyle{ N(a, b) = N(b, a) }[/math]
[math]\displaystyle{ N(0, b) = b }[/math] (everything must be [math]\displaystyle{ +1 }[/math])
[math]\displaystyle{ N(1, 1) = 11 }[/math] (there are [math]\displaystyle{ 4 }[/math] such sequences: choose [math]\displaystyle{ x_1 }[/math], and use the constraints [math]\displaystyle{ x_n +x_{2n} = 0 }[/math] and [math]\displaystyle{ x_1 + \ldots + x_{2n} = 0 }[/math] to determine the entries up to [math]\displaystyle{ 10 }[/math]; then choose [math]\displaystyle{ x_{11} }[/math])
[math]\displaystyle{ N(1, 2) = 41 }[/math] (there are [math]\displaystyle{ 4 }[/math] such sequences -- example below)
0 + - - + - + + - + + - - +
- + + - + - - - + - + + - -
+ + - + - + + - - - + - + +
[math]\displaystyle{ N(1, 3) = 83 }[/math] (there are [math]\displaystyle{ 216 }[/math] such sequences -- example below)
0 - + - + - + + - + + - - +
- + + - - + - - + - + + - +
+ + - - - + + - + + - - + -
+ - - + + - - + - + + + - +
- - - + + - + - + - - + - +
+ - + + - - + - + - - - + +
[math]\displaystyle{ N(1, 4) = 131 }[/math] (there are [math]\displaystyle{ 87144 }[/math] such sequences -- example below)
0 + - - + - + + - - + -
+ + - + + + + - - - + -
- + - + + - - + + + - -
- + + - + - + - - + + -
+ + - - + - - + - + + +
+ - - + - + - - + + + -
+ - - - - + + - - - + +
- - + + + + - - - - + +
- + - + + + + - - + + -
+ - - - + + - - - + - -
+ + + - + + - - + - - +
[math]\displaystyle{ N(2, 2) \gt = 1124 }[/math]
--Alec 13:46, 9 January 2010 (UTC)
