JIKUNA BIYAR, ADADIN RAWAR KAMALA MAI IYAKA

Ka yi tunanin wasu taurari ko duniyoyi kaɗan da aka jera su daidai sosai har dukan rukunin zai iya juyawa kamar abu ɗaya mai tauri, ba tare da canza siffa ba. Don hakan ya faru, jan nauyi a kan kowane jiki dole ne ya nufi daidai tsakiyar nauyi (centre of mass) na rukunin, da ƙarfin da ya yi daidai da nisansa daga gare ta. Ana kiran irin waɗannan tsare-tsare central configurations. Su ne siffofin wasu musamman mafita na matsalar jikuna n (n-body problem) waɗanda jikunan ke riƙe da tsari iri ɗaya a tsakaninsu, ciki har da relative equilibria da ke juyawa a tauye.

Idan wani tsari ya yi aiki, duk wani kwafinsa da aka juya, aka ƙara girmansa ko aka matsar ma zai yi aiki, don haka masana lissafi suna ƙidaya su “ban da kamanceceniya” (up to similarity).

Matsala ta shida ta Smale

Tambayar mai sauƙin faɗa ce: ga kowane zaɓin nauyaye masu kyau, shin adadin central configurations yana da iyaka? Ita ce matsala ta shida a jerin matsalolin da ba a warware ba da masanin lissafi Smale ya tsara.

Ga yadda al’amura suke, a cewar takardar:

Jikuna uku: biyar daidai — triangles masu daidaitattun gefe guda biyu na Lagrange da layuka madaidaita uku na Euler.
Jikuna huɗu: Hampton da Moeckel sun tabbatar da iyakar adadin, ga dukan nauyaye masu kyau.
Jikuna biyar a kan shimfiɗa: Albouy da Kaloshin sun tabbatar da iyakar a 2012, amma ga nauyaye a wajen wani saiti na musamman kawai — wani siririn iyalin haɗe-haɗen nauyi da aka ayyana ta daidaiton polynomial, inda hujjarsu ta kasa.

Rufe saitin na musamman

Tejasvi Singh Tomar ya cire wannan saiti na musamman. Babban theorem: ga kowane zaɓin nauyaye biyar masu kyau, matsalar jikuna biyar a kan shimfiɗa tana da central configurations masu iyaka ban da kamanceceniya.

Dabarar ta bi ta Albouy da Kaloshin. Idan akwai tsare-tsare marasa iyaka, za su samar da wani lanƙwasa mai ci gaba na mafita idan aka faɗaɗa daidaiton zuwa lambobin complex. Idan aka bi wannan lanƙwasa “zuwa marar iyaka”, nisan tsakanin jikunan dole ne ya yi hali ta hanyoyin da ke soke juna cikin wani daidaito na musamman. Jikunan suna kusantar ɗaya daga cikin zane-zane iyaka goma sha shida da Albouy da Kaloshin suka gano, kowanne yana sanya sharuɗɗa kan nauyayen.

Sabon aikin ya sa waɗannan sharuɗɗa su zama daidai ga kowane nauyi mai kyau, kuma ya kawar da lokuta masu wahala ɗaya bayan ɗaya:

tsare-tsaren da jikuna huɗu ke yin cunkoso mai matsewa da na biyar ke kusanta an kawar da su ta hanyar faɗaɗa makamashin wuri (potential energy) — wanda ke tsayawa ɗaya a tsawon lanƙwasar — zuwa mataki mafi girma da mafi girma; ga nauyaye masu kyau sharuɗɗan da suka fito suna saɓa wa juna;
an nuna cewa wasu zane-zane biyu ba sa taɓa faruwa;
ga duk sauran, marubucin ya bayyana dukan sharuɗɗan nauyi masu jituwa da lissafin algebra na kwamfuta na daidai — tsarin algebra 2,206 — kuma, ga kowanne, ya samar da takardar shaida (certificate) ta lambobi: jerin lambobin rational waɗanda halayensu, idan aka duba su da lissafi na daidai, suke tabbatar da cewa daidaiton ba zai yiwu ba.

Tarkon da nauyaye marasa kyau suka kafa

Akwai wata matsala da algebra zalla ba za ta iya kauce mata ba. Wani masanin lissafi, Roberts, ya sami central configurations marasa iyaka ga nauyaye (1, 1, 1, 1, −¼) — da nauyi ɗaya mara kyau (negative). Juya wasu alamomi yana mayar da shi mafita marasa iyaka na daidaiton complex ga nauyaye (4, 4, 4, 4, 1). Ba tsare-tsare na gaske ba ne, amma hujjar da ke ganin daidaiton complex kawai ba za ta iya bambance su ba.

Don haka a waɗannan nauyaye, da kuma a rabon da ke da alaƙa (1, 1, 4, 4, 4), marubucin ya ƙidaya tsare-tsaren gaske kai tsaye, da wani binciken kwamfuta da ke amfani da lissafin interval mai tabbaci:

ga nauyaye (1, 4, 4, 4, 4): tsare-tsare 306 ban da juyawa da canjin girma — 60 a layi madaidaici, 24 suna samar da pentagon mai kumburi (convex), sauran da jiki ɗaya ko biyu a ciki;
ga (1, 1, 4, 4, 4): 294.

Guda 60 na layi madaidaici sun yi daidai da ƙidayar da Moulton ya yi a baya.

Na’ura ta duba

Hujjar da ta dogara kan lissafai masu yawa haka tana buƙatar tabbatarwa mai ƙarfi. An sake duba kowace takardar shaida da wani shiri na daban da aka rubuta daga ma’anonin takardar. Fiye da duka, an tabbatar da theorem ɗin a cikin mataimakin hujja Lean 4, wani shiri da ke duba kowane mataki na hankali. Hujjar ta tsari (formal) ba ta ma ɗauki sakamakon Albouy da Kaloshin a matsayin gaskiya ba: tana sake fitar da abin da take buƙata, tana gano ta hanyar bincike zane-zane 1,376 masu lakabi da ke cika dokokinsu. Wasu masu duba hujjojin Lean guda biyu masu zaman kansu sun karɓe ta. Gina ta daga farko yana ɗaukar kusan awoyi-CPU 860.

Takardar ta ƙare da wata gajeriyar sanarwa: an yi amfani da kayan aikin AI masu samarwa (generative AI) don gano da tabbatar da sakamakon, rubuta shirye-shiryen da tsarin Lean, da kuma tsara rubutun farko. Ba a ambaci sunayen kayan aikin ba.

Abin da ta faɗa da abin da ba ta faɗa ba

Sakamakon ya shafi jikuna biyar a kan shimfiɗa. Ya tabbatar da cewa adadin tsare-tsaren yana da iyaka ga kowane saitin nauyaye masu kyau, amma yana ba da ƙidaya daidai ne kawai a ƙalilan rabe-raben nauyi da aka yi nazari.