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Uteshev A.Yu., Yashina M.V. On Maxwell’s Conjecture for Coulomb Potential Generated by Point Charges. Springer. Transactions on Computational Sciences XXVII. Lecture Notes in Computer Science. V.9570, 2016, pp. 68-80.


Example 1

$$ \mathcal Y(y,m_2,m_3) = $$

(2515456+206920704*m[2]^4*m[3]^2+115693056*m[2]^4*m[3]^4-852752384*m[2]^6*m[3]^4-967436288*m[2]^6*m[3]^6-4653056*m[2]^6*m[3]^10 -16348160*m[2]^8*m[3]^2+219913728*m[2]^8*m[3]^4+6492160*m[2]^8*m[3]^8-174301184*m[2]^8*m[3]^6 +110644224*m[2]^10*m[3]^2+167709696*m[2]^10*m[3]^4-7141376*m[2]^10*m[3]^6-32201728*m[2]^12*m[3]^2-892729344*m[2]^4*m[3]^6 +2523136*m[2]^4*m[3]^12-3288064*m[2]^14*m[3]^2-917504*m[2]^2*m[3]^14+5597696*m[2]^12*m[3]^4-40856576*m[2]^2*m[3]^2 +230820864*m[2]^2*m[3]^4-267411456*m[2]^2*m[3]^6+38707200*m[3]^8*m[2]^2-23134208*m[3]^12*m[2]^2+73089024*m[3]^10*m[2]^2 +281899008*m[3]^8*m[2]^4+97959936*m[3]^10*m[2]^4-36429824*m[3]^8*m[2]^6-215561216*m[2]^6*m[3]^2-14796800*m[2]^2 +73262080*m[2]^8+1124864*m[2]^16-54253568*m[2]^10+26624000*m[2]^12-7960576*m[2]^14+40456192*m[2]^4-66971648*m[2]^6 -10357760*m[3]^2+20480512*m[3]^4-25847808*m[3]^6+262144*m[3]^16+22548480*m[3]^8+6062080*m[3]^12-13959168*m[3]^10-1703936*m[3]^14)*y^34+

(-299339264-24365454336*m[2]^4*m[3]^2-12966787584*m[2]^4*m[3]^4+87566698496*m[2]^6*m[3]^4+80809603072*m[2]^6*m[3]^6+367099904*m[2]^6*m[3]^10 +3386987520*m[2]^8*m[3]^2-22368686592*m[2]^8*m[3]^4-570286080*m[2]^8*m[3]^8+12649332736*m[2]^8*m[3]^6-13944431616*m[2]^10*m[3]^2 -16643871744*m[2]^10*m[3]^4+686362624*m[2]^10*m[3]^6+4008256512*m[2]^12*m[3]^2+75329581056*m[2]^4*m[3]^6-166559744*m[2]^4*m[3]^12 +361860096*m[2]^14*m[3]^2+53477376*m[2]^2*m[3]^14-589182464*m[2]^12*m[3]^4+5073613824*m[2]^2*m[3]^2-22460445696*m[2]^2*m[3]^4 +19864879104*m[2]^2*m[3]^6+232980480*m[3]^8*m[2]^2+1069023232*m[3]^12*m[2]^2-5112938496*m[3]^10*m[2]^2-16299835392*m[3]^8*m[2]^4 -6242156544*m[3]^10*m[2]^4+4940169216*m[3]^8*m[2]^6+24524841984*m[2]^6*m[3]^2+1707550720*m[2]^2-8133524480*m[2]^8-133858816*m[2]^16 +6043482112*m[2]^10-3000524800*m[2]^12+916158464*m[2]^14-4569878528*m[2]^4+7469934592*m[2]^6+1105320960*m[3]^2-1979350528*m[3]^4 +2248445952*m[3]^6-8912896*m[3]^16-1742561280*m[3]^8-347832320*m[3]^12+944013312*m[3]^10+80216064*m[3]^14)*y^33+

(1378507415040*m[2]^4*m[3]^2+711139941120*m[2]^4*m[3]^4-4348219745280*m[2]^6*m[3]^4-3255265688576*m[2]^6*m[3]^6-13920190464*m[2]^6*m[3]^10 -265774760448*m[2]^8*m[3]^2+1119793687296*m[2]^8*m[3]^4+24446042112*m[2]^8*m[3]^8-419252056064*m[2]^8*m[3]^6+833944026624*m[2]^10*m[3]^2 +786163210752*m[2]^10*m[3]^4-32040387584*m[2]^10*m[3]^6-235785997824*m[2]^12*m[3]^2-3064371717120*m[2]^4*m[3]^6+5452251136*m[2]^4*m[3]^12 -19408836096*m[2]^14*m[3]^2-1436614656*m[2]^2*m[3]^14+30080783616*m[2]^12*m[3]^4-298272284160*m[2]^2*m[3]^2+1045159365120*m[2]^2*m[3]^4 -687439515648*m[2]^2*m[3]^6-113971888128*m[3]^8*m[2]^2-23722655744*m[3]^12*m[2]^2+169381330944*m[3]^10*m[2]^2+449041575936*m[3]^8*m[2]^4 +193259569152*m[3]^10*m[2]^4-252277407744*m[3]^8*m[2]^6-1346057046528*m[2]^6*m[3]^2-94889510912*m[2]^2+432828210688*m[2]^8+7676634368*m[2]^16 -322656843776*m[2]^10+162175290368*m[2]^12-50643453952*m[2]^14+247992749056*m[2]^4-399695083520*m[2]^6-57322655232*m[3]^2+93394150656*m[3]^4 -95752903680*m[3]^6+156499968*m[3]^16+66235219968*m[3]^8+9939632128*m[3]^12-31518867456*m[3]^10-1893138432*m[3]^14+17212007680)*y^32+

(-636993953792-49978460909568*m[2]^4*m[3]^2-25382961825792*m[2]^4*m[3]^4+138980128964608*m[2]^6*m[3]^4+84312203895808*m[2]^6*m[3]^6 +340287569920*m[2]^6*m[3]^10+12074694296576*m[2]^8*m[3]^2-36718249023488*m[2]^8*m[3]^4-677724196864*m[2]^8*m[3]^8+8149627211776*m[2]^8*m[3]^6 -31648757190656*m[2]^10*m[3]^2-23517877508096*m[2]^10*m[3]^4+968718023680*m[2]^10*m[3]^6+8762116040704*m[2]^12*m[3]^2+80271474797568*m[2]^4*m[3]^6 -115904282624*m[2]^4*m[3]^12+675229035520*m[2]^14*m[3]^2+24350228480*m[2]^2*m[3]^14-991109576704*m[2]^12*m[3]^4+11104085768192*m[2]^2*m[3]^2 -30948646914048*m[2]^2*m[3]^4+14471324233728*m[2]^2*m[3]^6+4973747552256*m[3]^8*m[2]^2+334930837504*m[3]^12*m[2]^2-3562195419136*m[3]^10*m[2]^2 -7776778498048*m[3]^8*m[2]^4-3864441880576*m[3]^10*m[2]^4+7467667521536*m[3]^8*m[2]^6+47510266047488*m[2]^6*m[3]^2+3381608587264*m[2]^2 -14700976111616*m[2]^8-282583834624*m[2]^16+10993487208448*m[2]^10-5597189718016*m[2]^12+1790350155776*m[2]^14-8609324711936*m[2]^4 +13661622378496*m[2]^6+1924687545344*m[3]^2-2863915452416*m[3]^4+2657057534976*m[3]^6-1870659584*m[3]^16-1646525239296*m[3]^8 -187480211456*m[3]^12+690295062528*m[3]^10+29580394496*m[3]^14)*y^31+

(1306425024230400*m[2]^4*m[3]^2+662171657617920*m[2]^4*m[3]^4-3212631082405888*m[2]^6*m[3]^4-1579227423148544*m[2]^6*m[3]^6 -6042264887296*m[2]^6*m[3]^10-374051880353280*m[2]^8*m[3]^2+885604808319488*m[2]^8*m[3]^4+13621854925824*m[2]^8*m[3]^8 -94734045550592*m[2]^8*m[3]^6+859021384750080*m[2]^10*m[3]^2+499872770551808*m[2]^10*m[3]^4-21314965515776*m[2]^10*m[3]^6 -231633904300032*m[2]^12*m[3]^2-1525240860446208*m[2]^4*m[3]^6+1782728728576*m[2]^4*m[3]^12-17102793337344*m[2]^14*m[3]^2 -297138192384*m[2]^2*m[3]^14+23669117395456*m[2]^12*m[3]^4-294927977121792*m[2]^2*m[3]^2+654279562291200*m[2]^2*m[3]^4 -199767525402624*m[2]^2*m[3]^6-121116844185600*m[3]^8*m[2]^2-3340263636992*m[3]^12*m[2]^2+53686649774080*m[3]^10*m[2]^2 +93127228720128*m[3]^8*m[2]^4+56032678617088*m[3]^10*m[2]^4-151073695782912*m[3]^8*m[2]^6-1212863680725504*m[2]^6*m[3]^2 -86832365662208*m[2]^2+358022280745984*m[2]^8+7503014984192*m[2]^16-268534920409088*m[2]^10+138568465817600*m[2]^12 -45468669755392*m[2]^14+214807637659648*m[2]^4-335117219028992*m[2]^6-47023361975808*m[3]^2+64102817919488*m[3]^4-53955431803392*m[3]^6 +16989028352*m[3]^16+30038261904384*m[3]^8+2613246238720*m[3]^12-11124154933248*m[3]^10-342563684352*m[3]^14+17051775648256)*y^30+

(-351777216928256-26260829142718464*m[2]^4*m[3]^2-13441742123229696*m[2]^4*m[3]^4+57236608608098304*m[2]^6*m[3]^4+22806826859788288*m[2]^6*m[3]^6 +83030601302016*m[2]^6*m[3]^10+8605168322304000*m[2]^8*m[3]^2-16726422969113088*m[2]^8*m[3]^4-211294554531840*m[2]^8*m[3]^8 +428499897278464*m[2]^8*m[3]^6-17807178598815744*m[2]^10*m[3]^2-8022387993639936*m[2]^10*m[3]^4+363327415392256*m[2]^10*m[3]^6 +4645680167651328*m[2]^12*m[3]^2+22437340423655424*m[2]^4*m[3]^6-21076674830336*m[2]^4*m[3]^12+335552871174144*m[2]^14*m[3]^2 +2806896328704*m[2]^2*m[3]^14-436413282878976*m[2]^12*m[3]^4+5962719067846656*m[2]^2*m[3]^2-10495112000769024*m[2]^2*m[3]^4 +1714767437316096*m[2]^2*m[3]^6+2041146637086720*m[3]^8*m[2]^2+24415185829888*m[3]^12*m[2]^2-619709094764544*m[3]^10*m[2]^2 -777669925337088*m[3]^8*m[2]^4-627437927817216*m[3]^10*m[2]^4+2274843374628864*m[3]^8*m[2]^6+23879551185309696*m[2]^6*m[3]^2 +1711250908426240*m[2]^2-6658696451056640*m[2]^8-153087264838144*m[2]^16+5008542328680448*m[2]^10-2620693141381120*m[2]^12 +883405895131136*m[2]^14-4102674127572992*m[2]^4+6283729069539328*m[2]^6+890651764439040*m[3]^2-1115241871314432*m[3]^4 +853748801691648*m[3]^6-124281421824*m[3]^16-427961003059200*m[3]^8-28599371816960*m[3]^12+140344453890048*m[3]^10+3127830183936*m[3]^14)*y^29+

(422765492388200448*m[2]^4*m[3]^2+220671186172532992*m[2]^4*m[3]^4-818271349009855488*m[2]^6*m[3]^4-264261651108685056*m[2]^6*m[3]^6 -917585871589376*m[2]^6*m[3]^10-154858696643230976*m[2]^8*m[3]^2+257127936667667712*m[2]^8*m[3]^4+2631049915548672*m[2]^8*m[3]^8 +7078768098003968*m[2]^8*m[3]^6+293866595927150592*m[2]^10*m[3]^2+100748101747291136*m[2]^10*m[3]^4-4985183741094656*m[2]^10*m[3]^6 -73551157067361792*m[2]^12*m[3]^2-266118067861627648*m[2]^4*m[3]^6+199336678912000*m[2]^4*m[3]^12-5297331438846720*m[2]^14*m[3]^2 -21479607828480*m[2]^2*m[3]^14+6462769696198400*m[2]^12*m[3]^4-95516653923217920*m[2]^2*m[3]^2+132395980222076928*m[2]^2*m[3]^4 -4561780688485376*m[2]^2*m[3]^6-26026198280615936*m[3]^8*m[2]^2-127322163986432*m[3]^12*m[2]^2+5710659838738432*m[3]^10*m[2]^2 +3885606327933952*m[3]^8*m[2]^4+5642858840621056*m[3]^10*m[2]^4-26844470873004032*m[3]^8*m[2]^6-377574806411799296*m[2]^6*m[3]^2 -26920661124027392*m[2]^2+98362285433723392*m[2]^8+2497028762658560*m[2]^16-74183904546163712*m[2]^10+39376142740253696*m[2]^12 -13652942011171840*m[2]^14+62381669554564096*m[2]^4-93678263151254528*m[2]^6-13604422834636032*m[3]^2+15682683385459456*m[3]^4 -10945442486930688*m[3]^6+759011737600*m[3]^16+4948938573407232*m[3]^8+255170432659456*m[3]^12-1440524777582592*m[3]^10-23413151170560*m[3]^14 +5818644341417728)*y^28+

(-79265587292767232-5604931579954876416*m[2]^4*m[3]^2-3005624814448244224*m[2]^4*m[3]^4+9654369954466277376*m[2]^6*m[3]^4 +2522282054474174208*m[2]^6*m[3]^6+8365734399856640*m[2]^6*m[3]^10+2257217996399659776*m[2]^8*m[3]^2-3300815809017463296*m[2]^8*m[3]^4 -27016828121376768*m[2]^8*m[3]^8-193109844763399168*m[2]^8*m[3]^6-3972469207683796992*m[2]^10*m[3]^2-1010903236612325888*m[2]^10*m[3]^4 +56482980734398720*m[2]^10*m[3]^6+943571266079629824*m[2]^12*m[3]^2+2613955359270753536*m[2]^4*m[3]^6-1552136720973824*m[2]^4*m[3]^12 +69013704173177088*m[2]^14*m[3]^2+137303084433408*m[2]^2*m[3]^14-78948593719084544*m[2]^12*m[3]^4+1244975151214806528*m[2]^2*m[3]^2 -1341456477239072256*m[2]^2*m[3]^4-120426914658692096*m[2]^2*m[3]^6+263804065152147456*m[3]^8*m[2]^2+378688568295424*m[3]^12*m[2]^2 -43179069926121472*m[3]^10*m[2]^2+3336718609065984*m[3]^8*m[2]^4-41824150080544768*m[3]^10*m[2]^4+256878719604129792*m[3]^8*m[2]^6 +4929599069636169984*m[2]^6*m[3]^2+347142595385110528*m[2]^2-1185051201320013824*m[2]^8-33444246987074560*m[2]^16+896007020062560256*m[2]^10 -482600628465719296*m[2]^12+172296546987069440*m[2]^14-775362522471065600*m[2]^4+1140278024101900288*m[2]^6+172101493344031488*m[3]^2 -183020324358361600*m[3]^4+116722982005033728*m[3]^6-3963155906560*m[3]^16-47676474157860864*m[3]^8-1904342497918976*m[3]^12+12344748853948416*m[3]^10 +147568710451200*m[3]^14)*y^27+

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Some check values for the zeros (in the interval $ (1; 6) $): $$ m_2=2, m_3=2 \quad \Rightarrow \quad y_1 \approx \underline{1.23443069401437}, y_2 \approx 3.06815710117825,\ y_3 \approx \underline{3.24485905846128}, y_4 \approx 3.67054641973481 ; $$ $$ m_2=2, m_3= 4 \quad \Rightarrow \quad y_1 \approx \underline{1.63887109841068},\ y_2 \approx \underline{2.27169142823895},\ y_3 \approx \underline{2.55237074510373},\ y_4 \approx 2.67254250766082,\ $$ $$ y_5 \approx \underline{2.82644406752225},\ y_6 \approx 3.20503078818970\ . $$





$$ \Psi(m_2,m_3)= $$

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Some check points: $$ m_2=2, m_3 \approx 3.3814060317111008710071911140383065346500761929048157544031201733675084287804293317288; $$ $$ m_2=2, m_3\approx 4.3529686389647017022448939375727640212433605917383268032533125912060199385192564845164; $$ For the last pair of parameter values one gets: $$ x_{\ast}\approx 2.62645919090275489414154797681759232946471,y_{\ast} \approx 1.92580865122634739337162940451665185051185 $$ is a stationary point for the Coulomb potential $ F_{} $ with $$ \det \left[ Hessian(F)\left|_{_{(x_{\ast},y_{\ast})}} \right. \right] \approx -1.0568 \times 10^{-42} \ . $$

Example 3

is HERE

matricese/optimize/coulomb_e/vspom_1.txt · Последние изменения: 2020/03/11 14:00 (внешнее изменение)