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.

0.100.150.200.250.300.350.400.450.500.550.600.650.050.100.150.200.250.300.350.400.450.501X + 1Y = 11X + 2Y = 11X + 3Y = 11X + 4Y = 11X + 5Y = 11X + 6Y = 11X + 7Y = 11X + 8Y = 11X + 9Y = 11X + 10Y = 14X + 6Y = 32X + 1Y = 17X + 17Y = 52X + 2Y = 110X + 34Y = 72X + 3Y = 113X + 57Y = 92X + 4Y = 116X + 86Y = 112X + 5Y = 119X + 121Y = 132X + 6Y = 122X + 162Y = 152X + 7Y = 125X + 209Y = 172X + 8Y = 128X + 262Y = 192X + 9Y = 131X + 321Y = 212X + 10Y = 15X + 10Y = 315X + 10Y = 63X + 1Y = 116X + 46Y = 944X + 49Y = 173X + 2Y = 111X + 42Y = 689X + 142Y = 343X + 3Y = 128X + 134Y = 15150X + 313Y = 573X + 4Y = 117X + 98Y = 9227X + 586Y = 863X + 5Y = 140X + 270Y = 21320X + 985Y = 1213X + 6Y = 123X + 178Y = 12429X + 1534Y = 1623X + 7Y = 152X + 454Y = 27554X + 2257Y = 2093X + 8Y = 129X + 282Y = 15695X + 3178Y = 2623X + 9Y = 164X + 686Y = 33852X + 4321Y = 3213X + 10Y = 12X + 5Y = 16X + 5Y = 236X + 15Y = 104X + 1Y = 115X + 50Y = 7145X + 180Y = 46181X + 129Y = 494X + 2Y = 120X + 85Y = 9135X + 230Y = 42528X + 547Y = 1424X + 3Y = 125X + 130Y = 11435X + 950Y = 1341167X + 1593Y = 3134X + 4Y = 130X + 185Y = 13320X + 855Y = 982188X + 3711Y = 5864X + 5Y = 17X + 50Y = 3177X + 560Y = 543681X + 7465Y = 9854X + 6Y = 140X + 325Y = 17585X + 2140Y = 1785736X + 13539Y = 15344X + 7Y = 145X + 410Y = 191495X + 6210Y = 4548443X + 22737Y = 22574X + 8Y = 150X + 505Y = 21930X + 4325Y = 28211892X + 35983Y = 31784X + 9Y = 155X + 610Y = 232265X + 11660Y = 68616173X + 54321Y = 43214X + 10Y = 17X + 21Y = 37X + 7Y = 242X + 21Y = 1070X + 21Y = 155X + 1Y = 15X + 19Y = 237X + 51Y = 10262X + 201Y = 60614X + 321Y = 1295X + 2Y = 113X + 61Y = 5129X + 235Y = 342034X + 2193Y = 4602614X + 2005Y = 5475X + 3Y = 18X + 45Y = 3100X + 229Y = 264224X + 5919Y = 9507624X + 7737Y = 15935X + 4Y = 119X + 125Y = 7287X + 795Y = 742542X + 4395Y = 57017774X + 22461Y = 37115X + 5Y = 111X + 83Y = 439X + 127Y = 102502X + 5151Y = 56035770X + 54121Y = 74655X + 6Y = 125X + 213Y = 9509X + 1907Y = 13019146X + 45753Y = 428064894X + 114381Y = 135395X + 7Y = 114X + 133Y = 5322X + 1365Y = 829268X + 25221Y = 2070109004X + 219345Y = 227375X + 8Y = 131X + 325Y = 11795X + 3763Y = 20238754X + 118329Y = 8650172534X + 390277Y = 359835X + 9Y = 117X + 195Y = 6481X + 2515Y = 12252266X + 176955Y = 11660260494X + 654321Y = 543215X + 10Y = 18X + 28Y = 312X + 14Y = 324X + 14Y = 580X + 28Y = 15120X + 28Y = 216X + 1Y = 177X + 329Y = 27161X + 245Y = 38854X + 707Y = 1701106X + 609Y = 2011863X + 769Y = 3216X + 2Y = 198X + 504Y = 33266X + 518Y = 616006X + 6762Y = 117512166X + 9576Y = 219311666X + 7108Y = 20056X + 3Y = 1119X + 721Y = 39399X + 959Y = 9011802X + 17017Y = 229032942X + 33985Y = 591945057X + 36409Y = 77376X + 4Y = 128X + 196Y = 9112X + 322Y = 251372X + 2422Y = 26524500X + 31332Y = 4395130860X + 131836Y = 224616X + 5Y = 1161X + 1281Y = 51749X + 2513Y = 1666594X + 13797Y = 1270143710X + 219471Y = 25755315395X + 380713Y = 541216X + 6Y = 1182X + 1624Y = 57966X + 3710Y = 21349602X + 120106Y = 9535255458X + 453712Y = 45753666678X + 937924Y = 1143816X + 7Y = 129X + 287Y = 9173X + 749Y = 383386X + 9317Y = 65020126X + 40761Y = 36031278621X + 2054353Y = 2193456X + 8Y = 1224X + 2436Y = 691484X + 7154Y = 32598112X + 302442Y = 18815661192X + 1504020Y = 1183292275232X + 4110364Y = 3902776X + 9Y = 149X + 581Y = 15357X + 1897Y = 7826250X + 89621Y = 5030989030X + 2496571Y = 1769553814815X + 7654321Y = 6543216X + 10Y = 13X + 12Y = 19X + 12Y = 227X + 18Y = 530X + 12Y = 545X + 12Y = 7189X + 36Y = 287X + 1Y = 116X + 76Y = 5224X + 374Y = 47992X + 886Y = 1751888X + 1100Y = 30312048X + 5164Y = 18275256X + 1793Y = 7697X + 2Y = 110X + 56Y = 3177X + 368Y = 361071X + 1262Y = 1853046X + 2468Y = 48315873X + 9844Y = 239448663X + 24604Y = 71087X + 3Y = 124X + 156Y = 7516X + 1302Y = 1034008X + 5958Y = 68515408X + 16188Y = 2431225768X + 184476Y = 33985249402X + 167481Y = 364097X + 4Y = 17X + 52Y = 271X + 212Y = 14677X + 1222Y = 11516490X + 21364Y = 2595156255X + 158692Y = 23499903321X + 756836Y = 1318367X + 5Y = 132X + 268Y = 9936X + 3238Y = 18310608X + 22582Y = 179562720X + 96748Y = 98551460160X + 1773676Y = 2194712609004X + 2620201Y = 3807137X + 6Y = 118X + 168Y = 5597X + 2352Y = 1167851X + 19278Y = 132554654X + 97860Y = 8579754917X + 1067556Y = 1134286428211X + 7526268Y = 9379247X + 7Y = 140X + 412Y = 11212X + 938Y = 413176X + 8842Y = 53525456X + 51908Y = 3993814056X + 1313668Y = 12228314080878X + 18831569Y = 20543537X + 8Y = 111X + 124Y = 3903X + 4436Y = 17415189X + 47294Y = 2555137794X + 315284Y = 216032503587X + 4540084Y = 37600528174677X + 42374116Y = 41103647X + 9Y = 148X + 588Y = 13432X + 2334Y = 838064X + 27774Y = 1355408480X + 1036428Y = 6401516625520X + 33470508Y = 249657152469136X + 87654321Y = 76543217X + 10Y = 110X + 45Y = 310X + 15Y = 212X + 9Y = 220X + 9Y = 350X + 15Y = 7210X + 45Y = 28280X + 45Y = 368X + 1Y = 139X + 204Y = 11201X + 366Y = 381179X + 1134Y = 1873078X + 1902Y = 44314193X + 6348Y = 192539555X + 13878Y = 516414089X + 4097Y = 17938X + 2Y = 148X + 291Y = 13153X + 339Y = 281188X + 1467Y = 1844452X + 3723Y = 63132112X + 20319Y = 431975681X + 38754Y = 9844193552X + 83653Y = 246048X + 3Y = 119X + 132Y = 5145X + 384Y = 261419X + 2178Y = 2177054X + 7560Y = 99370429X + 58260Y = 9443473331X + 320400Y = 614921318003X + 757305Y = 1674818X + 4Y = 166X + 519Y = 17147X + 456Y = 263492X + 6453Y = 53021780X + 28617Y = 3055279270X + 286107Y = 373871222236X + 1029969Y = 1586925957032X + 4272461Y = 7568368X + 5Y = 175X + 660Y = 19765X + 2730Y = 13410719X + 23238Y = 161980670X + 125778Y = 112911265805X + 1547940Y = 16930913665375X + 13786290Y = 177367620625685X + 17736745Y = 26202018X + 6Y = 14X + 39Y = 123X + 93Y = 4372X + 927Y = 563284X + 5931Y = 45961004X + 86739Y = 81552742285X + 3222684Y = 35585259249344X + 59409477Y = 75262688X + 7Y = 193X + 996Y = 231191X + 5388Y = 2063123X + 8802Y = 46931662X + 65028Y = 4421679797X + 1102044Y = 9083970875189X + 95087196Y = 9195676148255807X + 169826513Y = 188315698X + 8Y = 1102X + 1191Y = 25360X + 1803Y = 6214796X + 46557Y = 2218169476X + 390165Y = 236474130226X + 7519515Y = 55174734996482X + 52778475Y = 4540084333612088X + 429794605Y = 4237411638263752X + 43046720Y = 478296937X + 468Y = 9571X + 3138Y = 9812993X + 45162Y = 1945165970X + 423378Y = 231454526755X + 9144948Y = 60458386008233X + 144032922Y = 11156836690123457X + 987654320Y = 876543218X + 10Y = 111X + 55Y = 333X + 55Y = 666X + 55Y = 1022X + 11Y = 3165X + 55Y = 21231X + 55Y = 28308X + 55Y = 36396X + 55Y = 459X + 1Y = 135X + 200Y = 9395X + 780Y = 68423X + 437Y = 611447X + 949Y = 18918065X + 8460Y = 221935865X + 13000Y = 4232120850X + 35905Y = 1387836362X + 9217Y = 40979X + 2Y = 185X + 555Y = 211165X + 2745Y = 194804X + 1041Y = 1133812X + 3297Y = 48971515X + 46275Y = 8687230955X + 119985Y = 270921353340X + 590505Y = 155016743036X + 280483Y = 836539X + 3Y = 125X + 185Y = 6135X + 375Y = 22462X + 733Y = 642854X + 3125Y = 3637293X + 6115Y = 88233177X + 22655Y = 3884559892X + 323635Y = 640806728364X + 3378745Y = 7573059X + 4Y = 1115X + 955Y = 272155X + 6945Y = 3462214X + 4193Y = 30416990X + 22663Y = 2151553325X + 572235Y = 667733261645X + 2765665Y = 38147636010780X + 25939945Y = 411987637963868X + 23803711Y = 42724619X + 5Y = 113X + 120Y = 3277X + 1020Y = 443333X + 7365Y = 45530677X + 48381Y = 38731217515X + 1499580Y = 1467411765623X + 1789800Y = 20639224105854X + 20800365Y = 2757258157615166X + 118513705Y = 177367459X + 6Y = 1145X + 1475Y = 33165X + 685Y = 26228X + 577Y = 312452X + 4469Y = 309114915X + 164335Y = 138396928005X + 8173475Y = 809564112716100X + 113338835Y = 12890736527959692X + 462945547Y = 59409477363182464X + 282475248Y = 4035360740X + 445Y = 91060X + 4905Y = 1666624X + 18911Y = 89811656X + 24121Y = 1467315160X + 513375Y = 3793322010940X + 29627405Y = 2571436831523840X + 954785005Y = 950871961509264656X + 1512003792Y = 1698265139X + 8Y = 1175X + 2115Y = 395095X + 26025Y = 7948886X + 28269Y = 1202123422X + 285975Y = 155191513373X + 2766615Y = 18207717167197X + 25963605Y = 2005204369260156X + 476708805Y = 422227803819723884X + 4303999488Y = 42979460554481008X + 54480992Y = 605344595X + 1240Y = 211005X + 5620Y = 1563873X + 13591Y = 52359905X + 153671Y = 75274106775X + 8326420Y = 49394152202415X + 87639080Y = 60966321889574765X + 2709190672Y = 2160493834388888896X + 5493827200Y = 49382716117578124X + 19531232Y = 19531254X + 22Y = 16X + 11Y = 136X + 33Y = 540X + 22Y = 560X + 22Y = 7126X + 33Y = 1456X + 11Y = 6144X + 22Y = 15540X + 66Y = 5510X + 1Y = 155X + 341Y = 13253X + 539Y = 405896X + 6523Y = 7803652X + 2541Y = 43759125X + 29051Y = 664331405X + 11803Y = 3384148984X + 45397Y = 15600349888X + 90189Y = 3590591147X + 20481Y = 921710X + 2Y = 122X + 154Y = 5242X + 605Y = 377106X + 9647Y = 9152959X + 2651Y = 34769410X + 46024Y = 769357728X + 30481Y = 6170460284X + 203005Y = 479945764836X + 2192212Y = 5905052775306X + 930022Y = 28048310X + 3Y = 177X + 605Y = 17495X + 1441Y = 745918X + 9713Y = 7506314X + 7073Y = 73339699X + 33781Y = 437545903X + 31647Y = 4892522148X + 303787Y = 543729483804X + 4784615Y = 97090533437925X + 14913081Y = 337874510X + 4Y = 188X + 770Y = 191298X + 4345Y = 19155352X + 107525Y = 694536322X + 49236Y = 41931443640X + 1508210Y = 1586412150698X + 1835845Y = 22889431891024X + 23078935Y = 3318798253448536X + 159454350Y = 25939945235595704X + 131225586Y = 2380371110X + 5Y = 133X + 319Y = 755X + 209Y = 82728X + 6149Y = 34021340X + 34067Y = 24551028995X + 1277089Y = 1128891880131X + 1915573Y = 1999446881864X + 5959547Y = 715920203263104X + 153245477Y = 20800365234608763X + 156728328Y = 2370274110X + 6Y = 1110X + 1166Y = 232046X + 8723Y = 2955522X + 14201Y = 6855027X + 9251Y = 577285450X + 410696Y = 312834328940X + 5128101Y = 46013894303748X + 95102909Y = 98081703323007468X + 2920218664Y = 3400165054582376264X + 3570173280Y = 462945547176547032X + 123582912Y = 17654703121X + 1397Y = 251243X + 5885Y = 17879354X + 229603Y = 9810165022X + 344223Y = 189111541815X + 2524181Y = 16884727056029X + 36543133Y = 2874900683778172X + 787082725Y = 711057729331849516X + 9366284368Y = 954785005935415388X + 832697728Y = 9450023710X + 8Y = 144X + 550Y = 9990X + 5159Y = 14170356X + 226391Y = 867582324X + 192038Y = 94231219284X + 2238566Y = 1334553472458X + 5267207Y = 36888299881848X + 129222137Y = 103854427765813440X + 8764631872Y = 79451467527737075X + 27772544Y = 280208317025312X + 15322752Y = 1702531143X + 1925Y = 291749X + 9955Y = 24845672X + 161865Y = 5620118844X + 306647Y = 135919831855X + 20008505Y = 107569731358151X + 52781025Y = 33305681896624015X + 2724432253Y = 1971879301655092593X + 2074759952Y = 169324417424864968X + 472608128Y = 42920525381470X + 381472Y = 3814713X + 78Y = 313X + 26Y = 239X + 39Y = 5130X + 78Y = 1565X + 26Y = 7273X + 78Y = 28182X + 39Y = 18156X + 26Y = 15585X + 78Y = 55715X + 78Y = 6611X + 1Y = 132X + 214Y = 7212X + 485Y = 312004X + 2367Y = 2455368X + 3958Y = 59315608X + 8054Y = 161753232X + 20799Y = 528266856X + 20965Y = 6438218896X + 57558Y = 20635323888X + 73758Y = 30063223244X + 45057Y = 2048111X + 2Y = 119X + 142Y = 499X + 262Y = 142313X + 3291Y = 2758114X + 7538Y = 87716542X + 11260Y = 168785203X + 45804Y = 8368173686X + 77533Y = 166262945484X + 1129358Y = 27682511824677X + 3985818Y = 109610610141669X + 3055786Y = 93002211X + 3Y = 144X + 366Y = 9398X + 1213Y = 555604X + 9519Y = 6558264X + 9482Y = 88344444X + 38426Y = 450162778X + 43737Y = 6142180680X + 105867Y = 172624411824X + 2236966Y = 414255258182820X + 115576386Y = 23923075162645790X + 65244729Y = 1491308111X + 4Y = 15X + 46Y = 1103X + 358Y = 143417X + 6813Y = 39592090X + 126974Y = 9775155245X + 163966Y = 15666561585X + 482814Y = 548442098118X + 1525883Y = 20027422354148X + 14114382Y = 2098085286865245X + 160217286Y = 265757251431274415X + 717163086Y = 131225586537109376X + 244140640Y = 4882812556X + 566Y = 11216X + 847Y = 29828X + 1905Y = 9526440X + 42754Y = 2795215280X + 269362Y = 216792380068X + 2438013Y = 23219810951672X + 9519799Y = 104485886606928X + 65477614Y = 81266558271772920X + 5537734264Y = 7662273858547498624X + 5137206272Y = 78364164111X + 6Y = 131X + 342Y = 6797X + 3490Y = 10634743X + 90849Y = 396512246X + 22766Y = 129158538X + 84764Y = 58875361279X + 6378348Y = 52270429327662X + 29666685Y = 27971461382297916X + 1217520638Y = 1296857853940922054X + 3075841600Y = 3650273331216948488X + 853065472Y = 11156791521240816X + 13516928Y = 193098368X + 814Y = 13962X + 4661Y = 12723532X + 69039Y = 2675198376X + 417262Y = 208732180156X + 3588110Y = 21905132965674X + 44684127Y = 3212594209036968X + 241235749Y = 1993261841951509424X + 42187633816Y = 39354136256320363684X + 5634796736Y = 585392773567686192X + 454696704Y = 5204360911X + 8Y = 137X + 478Y = 7381X + 2026Y = 5020691X + 67371Y = 2345195878X + 460154Y = 20581608547X + 1122278Y = 611034177863X + 6356562Y = 40701230134810X + 39073097Y = 2873022103321526216X + 116805574912Y = 96916602758871891360X + 8894875520Y = 8216842394733503X + 4264640Y = 4339461316800X + 1077376Y = 11970916X + 222Y = 3268X + 1553Y = 358004X + 28659Y = 905140200X + 363954Y = 147151944440X + 3972434Y = 1951397470264X + 12607233Y = 72758224915749100X + 35860048538Y = 237514612529046639268X + 36465477824Y = 27244322532800282912X + 3118569984Y = 25934499380551328X + 80626688Y = 7384501131132X + 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, 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