Homogeneous Tradeoff
From right to left, k = 2, 3, 4, 5, ... .
(Note that at MSR, alpha/M = 1/k.)
From top to bottom, d - k = 0, 1, 2, 3, ... .
(beta decreases as d increases.)
Mouse instruction:
Drag to move.
Scroll at white space to zoom.
Scroll at x- or y-axis to zoom the corresponding axis,
Hover over a point to see moulin parameter.
Hover over a segment to obtain putative inequality parameter.
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1X + 1Y = 1 1X + 2Y = 1 1X + 3Y = 1 1X + 4Y = 1 1X + 5Y = 1 1X + 6Y = 1 1X + 7Y = 1 1X + 8Y = 1 1X + 9Y = 1 1X + 10Y = 1 4X + 6Y = 3 2X + 1Y = 1 7X + 17Y = 5 2X + 2Y = 1 10X + 34Y = 7 2X + 3Y = 1 13X + 57Y = 9 2X + 4Y = 1 16X + 86Y = 11 2X + 5Y = 1 19X + 121Y = 13 2X + 6Y = 1 22X + 162Y = 15 2X + 7Y = 1 25X + 209Y = 17 2X + 8Y = 1 28X + 262Y = 19 2X + 9Y = 1 31X + 321Y = 21 2X + 10Y = 1 5X + 10Y = 3 15X + 10Y = 6 3X + 1Y = 1 16X + 46Y = 9 44X + 49Y = 17 3X + 2Y = 1 11X + 42Y = 6 89X + 142Y = 34 3X + 3Y = 1 28X + 134Y = 15 150X + 313Y = 57 3X + 4Y = 1 17X + 98Y = 9 227X + 586Y = 86 3X + 5Y = 1 40X + 270Y = 21 320X + 985Y = 121 3X + 6Y = 1 23X + 178Y = 12 429X + 1534Y = 162 3X + 7Y = 1 52X + 454Y = 27 554X + 2257Y = 209 3X + 8Y = 1 29X + 282Y = 15 695X + 3178Y = 262 3X + 9Y = 1 64X + 686Y = 33 852X + 4321Y = 321 3X + 10Y = 1 2X + 5Y = 1 6X + 5Y = 2 36X + 15Y = 10 4X + 1Y = 1 15X + 50Y = 7 145X + 180Y = 46 181X + 129Y = 49 4X + 2Y = 1 20X + 85Y = 9 135X + 230Y = 42 528X + 547Y = 142 4X + 3Y = 1 25X + 130Y = 11 435X + 950Y = 134 1167X + 1593Y = 313 4X + 4Y = 1 30X + 185Y = 13 320X + 855Y = 98 2188X + 3711Y = 586 4X + 5Y = 1 7X + 50Y = 3 177X + 560Y = 54 3681X + 7465Y = 985 4X + 6Y = 1 40X + 325Y = 17 585X + 2140Y = 178 5736X + 13539Y = 1534 4X + 7Y = 1 45X + 410Y = 19 1495X + 6210Y = 454 8443X + 22737Y = 2257 4X + 8Y = 1 50X + 505Y = 21 930X + 4325Y = 282 11892X + 35983Y = 3178 4X + 9Y = 1 55X + 610Y = 23 2265X + 11660Y = 686 16173X + 54321Y = 4321 4X + 10Y = 1 7X + 21Y = 3 7X + 7Y = 2 42X + 21Y = 10 70X + 21Y = 15 5X + 1Y = 1 5X + 19Y = 2 37X + 51Y = 10 262X + 201Y = 60 614X + 321Y = 129 5X + 2Y = 1 13X + 61Y = 5 129X + 235Y = 34 2034X + 2193Y = 460 2614X + 2005Y = 547 5X + 3Y = 1 8X + 45Y = 3 100X + 229Y = 26 4224X + 5919Y = 950 7624X + 7737Y = 1593 5X + 4Y = 1 19X + 125Y = 7 287X + 795Y = 74 2542X + 4395Y = 570 17774X + 22461Y = 3711 5X + 5Y = 1 11X + 83Y = 4 39X + 127Y = 10 2502X + 5151Y = 560 35770X + 54121Y = 7465 5X + 6Y = 1 25X + 213Y = 9 509X + 1907Y = 130 19146X + 45753Y = 4280 64894X + 114381Y = 13539 5X + 7Y = 1 14X + 133Y = 5 322X + 1365Y = 82 9268X + 25221Y = 2070 109004X + 219345Y = 22737 5X + 8Y = 1 31X + 325Y = 11 795X + 3763Y = 202 38754X + 118329Y = 8650 172534X + 390277Y = 35983 5X + 9Y = 1 17X + 195Y = 6 481X + 2515Y = 122 52266X + 176955Y = 11660 260494X + 654321Y = 54321 5X + 10Y = 1 8X + 28Y = 3 12X + 14Y = 3 24X + 14Y = 5 80X + 28Y = 15 120X + 28Y = 21 6X + 1Y = 1 77X + 329Y = 27 161X + 245Y = 38 854X + 707Y = 170 1106X + 609Y = 201 1863X + 769Y = 321 6X + 2Y = 1 98X + 504Y = 33 266X + 518Y = 61 6006X + 6762Y = 1175 12166X + 9576Y = 2193 11666X + 7108Y = 2005 6X + 3Y = 1 119X + 721Y = 39 399X + 959Y = 90 11802X + 17017Y = 2290 32942X + 33985Y = 5919 45057X + 36409Y = 7737 6X + 4Y = 1 28X + 196Y = 9 112X + 322Y = 25 1372X + 2422Y = 265 24500X + 31332Y = 4395 130860X + 131836Y = 22461 6X + 5Y = 1 161X + 1281Y = 51 749X + 2513Y = 166 6594X + 13797Y = 1270 143710X + 219471Y = 25755 315395X + 380713Y = 54121 6X + 6Y = 1 182X + 1624Y = 57 966X + 3710Y = 213 49602X + 120106Y = 9535 255458X + 453712Y = 45753 666678X + 937924Y = 114381 6X + 7Y = 1 29X + 287Y = 9 173X + 749Y = 38 3386X + 9317Y = 650 20126X + 40761Y = 3603 1278621X + 2054353Y = 219345 6X + 8Y = 1 224X + 2436Y = 69 1484X + 7154Y = 325 98112X + 302442Y = 18815 661192X + 1504020Y = 118329 2275232X + 4110364Y = 390277 6X + 9Y = 1 49X + 581Y = 15 357X + 1897Y = 78 26250X + 89621Y = 5030 989030X + 2496571Y = 176955 3814815X + 7654321Y = 654321 6X + 10Y = 1 3X + 12Y = 1 9X + 12Y = 2 27X + 18Y = 5 30X + 12Y = 5 45X + 12Y = 7 189X + 36Y = 28 7X + 1Y = 1 16X + 76Y = 5 224X + 374Y = 47 992X + 886Y = 175 1888X + 1100Y = 303 12048X + 5164Y = 1827 5256X + 1793Y = 769 7X + 2Y = 1 10X + 56Y = 3 177X + 368Y = 36 1071X + 1262Y = 185 3046X + 2468Y = 483 15873X + 9844Y = 2394 48663X + 24604Y = 7108 7X + 3Y = 1 24X + 156Y = 7 516X + 1302Y = 103 4008X + 5958Y = 685 15408X + 16188Y = 2431 225768X + 184476Y = 33985 249402X + 167481Y = 36409 7X + 4Y = 1 7X + 52Y = 2 71X + 212Y = 14 677X + 1222Y = 115 16490X + 21364Y = 2595 156255X + 158692Y = 23499 903321X + 756836Y = 131836 7X + 5Y = 1 32X + 268Y = 9 936X + 3238Y = 183 10608X + 22582Y = 1795 62720X + 96748Y = 9855 1460160X + 1773676Y = 219471 2609004X + 2620201Y = 380713 7X + 6Y = 1 18X + 168Y = 5 597X + 2352Y = 116 7851X + 19278Y = 1325 54654X + 97860Y = 8579 754917X + 1067556Y = 113428 6428211X + 7526268Y = 937924 7X + 7Y = 1 40X + 412Y = 11 212X + 938Y = 41 3176X + 8842Y = 535 25456X + 51908Y = 3993 814056X + 1313668Y = 122283 14080878X + 18831569Y = 2054353 7X + 8Y = 1 11X + 124Y = 3 903X + 4436Y = 174 15189X + 47294Y = 2555 137794X + 315284Y = 21603 2503587X + 4540084Y = 376005 28174677X + 42374116Y = 4110364 7X + 9Y = 1 48X + 588Y = 13 432X + 2334Y = 83 8064X + 27774Y = 1355 408480X + 1036428Y = 64015 16625520X + 33470508Y = 2496571 52469136X + 87654321Y = 7654321 7X + 10Y = 1 10X + 45Y = 3 10X + 15Y = 2 12X + 9Y = 2 20X + 9Y = 3 50X + 15Y = 7 210X + 45Y = 28 280X + 45Y = 36 8X + 1Y = 1 39X + 204Y = 11 201X + 366Y = 38 1179X + 1134Y = 187 3078X + 1902Y = 443 14193X + 6348Y = 1925 39555X + 13878Y = 5164 14089X + 4097Y = 1793 8X + 2Y = 1 48X + 291Y = 13 153X + 339Y = 28 1188X + 1467Y = 184 4452X + 3723Y = 631 32112X + 20319Y = 4319 75681X + 38754Y = 9844 193552X + 83653Y = 24604 8X + 3Y = 1 19X + 132Y = 5 145X + 384Y = 26 1419X + 2178Y = 217 7054X + 7560Y = 993 70429X + 58260Y = 9443 473331X + 320400Y = 61492 1318003X + 757305Y = 167481 8X + 4Y = 1 66X + 519Y = 17 147X + 456Y = 26 3492X + 6453Y = 530 21780X + 28617Y = 3055 279270X + 286107Y = 37387 1222236X + 1029969Y = 158692 5957032X + 4272461Y = 756836 8X + 5Y = 1 75X + 660Y = 19 765X + 2730Y = 134 10719X + 23238Y = 1619 80670X + 125778Y = 11291 1265805X + 1547940Y = 169309 13665375X + 13786290Y = 1773676 20625685X + 17736745Y = 2620201 8X + 6Y = 1 4X + 39Y = 1 23X + 93Y = 4 372X + 927Y = 56 3284X + 5931Y = 459 61004X + 86739Y = 8155 2742285X + 3222684Y = 355852 59249344X + 59409477Y = 7526268 8X + 7Y = 1 93X + 996Y = 23 1191X + 5388Y = 206 3123X + 8802Y = 469 31662X + 65028Y = 4421 679797X + 1102044Y = 90839 70875189X + 95087196Y = 9195676 148255807X + 169826513Y = 18831569 8X + 8Y = 1 102X + 1191Y = 25 360X + 1803Y = 62 14796X + 46557Y = 2218 169476X + 390165Y = 23647 4130226X + 7519515Y = 551747 34996482X + 52778475Y = 4540084 333612088X + 429794605Y = 42374116 38263752X + 43046720Y = 4782969 37X + 468Y = 9 571X + 3138Y = 98 12993X + 45162Y = 1945 165970X + 423378Y = 23145 4526755X + 9144948Y = 604583 86008233X + 144032922Y = 11156836 690123457X + 987654320Y = 87654321 8X + 10Y = 1 11X + 55Y = 3 33X + 55Y = 6 66X + 55Y = 10 22X + 11Y = 3 165X + 55Y = 21 231X + 55Y = 28 308X + 55Y = 36 396X + 55Y = 45 9X + 1Y = 1 35X + 200Y = 9 395X + 780Y = 68 423X + 437Y = 61 1447X + 949Y = 189 18065X + 8460Y = 2219 35865X + 13000Y = 4232 120850X + 35905Y = 13878 36362X + 9217Y = 4097 9X + 2Y = 1 85X + 555Y = 21 1165X + 2745Y = 194 804X + 1041Y = 113 3812X + 3297Y = 489 71515X + 46275Y = 8687 230955X + 119985Y = 27092 1353340X + 590505Y = 155016 743036X + 280483Y = 83653 9X + 3Y = 1 25X + 185Y = 6 135X + 375Y = 22 462X + 733Y = 64 2854X + 3125Y = 363 7293X + 6115Y = 882 33177X + 22655Y = 3884 559892X + 323635Y = 64080 6728364X + 3378745Y = 757305 9X + 4Y = 1 115X + 955Y = 27 2155X + 6945Y = 346 2214X + 4193Y = 304 16990X + 22663Y = 2151 553325X + 572235Y = 66773 3261645X + 2765665Y = 381476 36010780X + 25939945Y = 4119876 37963868X + 23803711Y = 4272461 9X + 5Y = 1 13X + 120Y = 3 277X + 1020Y = 44 3333X + 7365Y = 455 30677X + 48381Y = 3873 1217515X + 1499580Y = 146741 1765623X + 1789800Y = 206392 24105854X + 20800365Y = 2757258 157615166X + 118513705Y = 17736745 9X + 6Y = 1 145X + 1475Y = 33 165X + 685Y = 26 228X + 577Y = 31 2452X + 4469Y = 309 114915X + 164335Y = 13839 6928005X + 8173475Y = 809564 112716100X + 113338835Y = 12890736 527959692X + 462945547Y = 59409477 363182464X + 282475248Y = 40353607 40X + 445Y = 9 1060X + 4905Y = 166 6624X + 18911Y = 898 11656X + 24121Y = 1467 315160X + 513375Y = 37933 22010940X + 29627405Y = 2571436 831523840X + 954785005Y = 95087196 1509264656X + 1512003792Y = 169826513 9X + 8Y = 1 175X + 2115Y = 39 5095X + 26025Y = 794 8886X + 28269Y = 1202 123422X + 285975Y = 15519 1513373X + 2766615Y = 182077 17167197X + 25963605Y = 2005204 369260156X + 476708805Y = 42222780 3819723884X + 4303999488Y = 429794605 54481008X + 54480992Y = 6053445 95X + 1240Y = 21 1005X + 5620Y = 156 3873X + 13591Y = 523 59905X + 153671Y = 7527 4106775X + 8326420Y = 493941 52202415X + 87639080Y = 6096632 1889574765X + 2709190672Y = 216049383 4388888896X + 5493827200Y = 493827161 17578124X + 19531232Y = 1953125 4X + 22Y = 1 6X + 11Y = 1 36X + 33Y = 5 40X + 22Y = 5 60X + 22Y = 7 126X + 33Y = 14 56X + 11Y = 6 144X + 22Y = 15 540X + 66Y = 55 10X + 1Y = 1 55X + 341Y = 13 253X + 539Y = 40 5896X + 6523Y = 780 3652X + 2541Y = 437 59125X + 29051Y = 6643 31405X + 11803Y = 3384 148984X + 45397Y = 15600 349888X + 90189Y = 35905 91147X + 20481Y = 9217 10X + 2Y = 1 22X + 154Y = 5 242X + 605Y = 37 7106X + 9647Y = 915 2959X + 2651Y = 347 69410X + 46024Y = 7693 57728X + 30481Y = 6170 460284X + 203005Y = 47994 5764836X + 2192212Y = 590505 2775306X + 930022Y = 280483 10X + 3Y = 1 77X + 605Y = 17 495X + 1441Y = 74 5918X + 9713Y = 750 6314X + 7073Y = 733 39699X + 33781Y = 4375 45903X + 31647Y = 4892 522148X + 303787Y = 54372 9483804X + 4784615Y = 970905 33437925X + 14913081Y = 3378745 10X + 4Y = 1 88X + 770Y = 19 1298X + 4345Y = 191 55352X + 107525Y = 6945 36322X + 49236Y = 4193 1443640X + 1508210Y = 158641 2150698X + 1835845Y = 228894 31891024X + 23078935Y = 3318798 253448536X + 159454350Y = 25939945 235595704X + 131225586Y = 23803711 10X + 5Y = 1 33X + 319Y = 7 55X + 209Y = 8 2728X + 6149Y = 340 21340X + 34067Y = 2455 1028995X + 1277089Y = 112889 1880131X + 1915573Y = 199944 6881864X + 5959547Y = 715920 203263104X + 153245477Y = 20800365 234608763X + 156728328Y = 23702741 10X + 6Y = 1 110X + 1166Y = 23 2046X + 8723Y = 295 5522X + 14201Y = 685 5027X + 9251Y = 577 285450X + 410696Y = 31283 4328940X + 5128101Y = 460138 94303748X + 95102909Y = 9808170 3323007468X + 2920218664Y = 340016505 4582376264X + 3570173280Y = 462945547 176547032X + 123582912Y = 17654703 121X + 1397Y = 25 1243X + 5885Y = 178 79354X + 229603Y = 9810 165022X + 344223Y = 18911 1541815X + 2524181Y = 168847 27056029X + 36543133Y = 2874900 683778172X + 787082725Y = 71105772 9331849516X + 9366284368Y = 954785005 935415388X + 832697728Y = 94500237 10X + 8Y = 1 44X + 550Y = 9 990X + 5159Y = 141 70356X + 226391Y = 8675 82324X + 192038Y = 9423 1219284X + 2238566Y = 133455 3472458X + 5267207Y = 368882 99881848X + 129222137Y = 10385442 7765813440X + 8764631872Y = 794514675 27737075X + 27772544Y = 2802083 17025312X + 15322752Y = 1702531 143X + 1925Y = 29 1749X + 9955Y = 248 45672X + 161865Y = 5620 118844X + 306647Y = 13591 9831855X + 20008505Y = 1075697 31358151X + 52781025Y = 3330568 1896624015X + 2724432253Y = 197187930 1655092593X + 2074759952Y = 169324417 424864968X + 472608128Y = 42920525 381470X + 381472Y = 38147 13X + 78Y = 3 13X + 26Y = 2 39X + 39Y = 5 130X + 78Y = 15 65X + 26Y = 7 273X + 78Y = 28 182X + 39Y = 18 156X + 26Y = 15 585X + 78Y = 55 715X + 78Y = 66 11X + 1Y = 1 32X + 214Y = 7 212X + 485Y = 31 2004X + 2367Y = 245 5368X + 3958Y = 593 15608X + 8054Y = 1617 53232X + 20799Y = 5282 66856X + 20965Y = 6438 218896X + 57558Y = 20635 323888X + 73758Y = 30063 223244X + 45057Y = 20481 11X + 2Y = 1 19X + 142Y = 4 99X + 262Y = 14 2313X + 3291Y = 275 8114X + 7538Y = 877 16542X + 11260Y = 1687 85203X + 45804Y = 8368 173686X + 77533Y = 16626 2945484X + 1129358Y = 276825 11824677X + 3985818Y = 1096106 10141669X + 3055786Y = 930022 11X + 3Y = 1 44X + 366Y = 9 398X + 1213Y = 55 5604X + 9519Y = 655 8264X + 9482Y = 883 44444X + 38426Y = 4501 62778X + 43737Y = 6142 180680X + 105867Y = 17262 4411824X + 2236966Y = 414255 258182820X + 115576386Y = 23923075 162645790X + 65244729Y = 14913081 11X + 4Y = 1 5X + 46Y = 1 103X + 358Y = 14 3417X + 6813Y = 395 92090X + 126974Y = 9775 155245X + 163966Y = 15666 561585X + 482814Y = 54844 2098118X + 1525883Y = 200274 22354148X + 14114382Y = 2098085 286865245X + 160217286Y = 26575725 1431274415X + 717163086Y = 131225586 537109376X + 244140640Y = 48828125 56X + 566Y = 11 216X + 847Y = 29 828X + 1905Y = 95 26440X + 42754Y = 2795 215280X + 269362Y = 21679 2380068X + 2438013Y = 232198 10951672X + 9519799Y = 1044858 86606928X + 65477614Y = 8126655 8271772920X + 5537734264Y = 766227385 8547498624X + 5137206272Y = 783641641 11X + 6Y = 1 31X + 342Y = 6 797X + 3490Y = 106 34743X + 90849Y = 3965 12246X + 22766Y = 1291 58538X + 84764Y = 5887 5361279X + 6378348Y = 522704 29327662X + 29666685Y = 2797146 1382297916X + 1217520638Y = 129685785 3940922054X + 3075841600Y = 365027333 1216948488X + 853065472Y = 111567915 21240816X + 13516928Y = 1930983 68X + 814Y = 13 962X + 4661Y = 127 23532X + 69039Y = 2675 198376X + 417262Y = 20873 2180156X + 3588110Y = 219051 32965674X + 44684127Y = 3212594 209036968X + 241235749Y = 19932618 41951509424X + 42187633816Y = 3935413625 6320363684X + 5634796736Y = 585392773 567686192X + 454696704Y = 52043609 11X + 8Y = 1 37X + 478Y = 7 381X + 2026Y = 50 20691X + 67371Y = 2345 195878X + 460154Y = 20581 608547X + 1122278Y = 61103 4177863X + 6356562Y = 407012 30134810X + 39073097Y = 2873022 103321526216X + 116805574912Y = 9691660275 8871891360X + 8894875520Y = 821684239 4733503X + 4264640Y = 433946 1316800X + 1077376Y = 119709 16X + 222Y = 3 268X + 1553Y = 35 8004X + 28659Y = 905 140200X + 363954Y = 14715 1944440X + 3972434Y = 195139 7470264X + 12607233Y = 727582 24915749100X + 35860048538Y = 2375146125 29046639268X + 36465477824Y = 2724432253 2800282912X + 3118569984Y = 259344993 80551328X + 80626688Y = 7384501 131132X + 119200Y = 11921 (2, 1)/3 (1, 1)/2 (3, 1)/5 (4, 2)/8 (4, 1)/7 (9, 3)/18 (5, 1)/9 (16, 4)/32 (6, 1)/11 (25, 5)/50 (7, 1)/13 (36, 6)/72 (8, 1)/15 (49, 7)/98 (9, 1)/17 (64, 8)/128 (10, 1)/19 (81, 9)/162 (11, 1)/21 (100, 10)/200 (3, 1)/6 (3, 2)/8 (1, 1)/3 (4, 1)/9 (7, 3)/20 (8, 4)/24 (5, 1)/12 (13, 4)/38 (27, 9)/81 (6, 1)/15 (21, 5)/62 (64, 16)/192 (7, 1)/18 (31, 6)/92 (125, 25)/375 (8, 1)/21 (43, 7)/128 (216, 36)/648 (9, 1)/24 (57, 8)/170 (343, 49)/1029 (10, 1)/27 (73, 9)/218 (512, 64)/1536 (11, 1)/30 (91, 10)/272 (729, 81)/2187 (12, 1)/33 (111, 11)/332 (1000, 100)/3000 (4, 1)/10 (6, 3)/20 (4, 3)/15 (1, 1)/4 (5, 1)/14 (11, 4)/40 (15, 7)/59 (16, 8)/64 (6, 1)/18 (18, 5)/68 (40, 13)/159 (81, 27)/324 (7, 1)/22 (27, 6)/104 (85, 21)/339 (256, 64)/1024 (8, 1)/26 (38, 7)/148 (156, 31)/623 (625, 125)/2500 (9, 1)/30 (51, 8)/200 (259, 43)/1035 (1296, 216)/5184 (10, 1)/34 (66, 9)/260 (400, 57)/1599 (2401, 343)/9604 (11, 1)/38 (83, 10)/328 (585, 73)/2339 (4096, 512)/16384 (12, 1)/42 (102, 11)/404 (820, 91)/3279 (6561, 729)/26244 (13, 1)/46 (123, 12)/488 (1111, 111)/4443 (10000, 1000)/40000 (5, 1)/15 (10, 4)/40 (10, 6)/45 (5, 4)/24 (1, 1)/5 (6, 1)/20 (16, 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