Bazı lineer kuaterniyonik ve oktonyonik denklem (sistem) lerinin incelenmeleri
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Dosyalar
Tarih
2014
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Hatay Mustafa Kemal Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Tezde, kompleks kuaterniyon cebiri H(ℂ) de AX=XA, AX=XB, AX-XA=C, AX-XB=C, AX-XA*=C, A*X-XA=C formlarındaki denklemler dikkate alındı ve reel oktonyon cebiri O(ℝ) de ise α(xα)=ρ, α(xβ)=ρ, (αx)β=ρ, αx-(αx)β=ρ, αx-α(xβ)=ρ, xα-(βx)=ρ, xα-β(xα)=ρ, (αx)β+(ϒx)δ=ρ, α(xβ)+ϒ(xδ)=ρ formlarındaki denklemler ve ax+yb=c ax+yb=c ax+by=c ax+yb=c ax+ya=c ax+yb=c xa+yb=d, ax+by=d, xa+by=d, xb+yb=d, xb+yb=d, xa+by=d formlarındaki denklem sistemleri dikkate alındı. Kompleks kuaterniyonların ve reel oktonyonların matris temsillerini kullanarak bu denklemlerin ve sistemlerin reel temsili sistemleri oluşturuldu. Temsili sistemlere dayalı bu denklem ve sistemlerin çözülebilirlikleri araştırıldı. Bu temsili sistemlerin çözümlerinin varlığı ve tekliği için gerekli ve yeterli şartlar elde edildi. Temsili sistemlerin çözümleri kapalı formlarda verildi ve buradan dikkate alınan denklem ve sistemlerin çözümlerine ulaşıldı.
In this thesis, the following linear complex quaternionic equations of the forms AX=XA, AX=XB, AX-XA=C, AX-XB=C, AX-XA*=C, A*X-XA=C are considered over the complex quaternion algebra H(ℂ), and over the real octonion algebra O(ℝ) the following linear real octonionic equations of the forms α(xα)=ρ, α(xβ)=ρ, (αx)β=ρ, αx-(αx)β=ρ, αx-α(xβ)=ρ, xα-(βx)=ρ, xα-β(xα)=ρ, (αx)β+(ϒx)δ=ρ, α(xβ)+ϒ(xδ)=ρ and the following linear real octonionic systems of the forms ax+yb=c ax+yb=c ax+by=c ax+yb=c ax+ya=c ax+yb=c xa+yb=d, ax+by=d, xa+by=d, xb+yb=d, xb+yb=d, xa+by=d are considered. Using the real matrix representation of complex quaternions and real octonions, the real representation systems for these equations and systems are established. Based on them the solvabilities of these equations and systems are investigated. Necessary and sufficient conditions for the existence of a solution or a unique solution to these representation systems are obtained. The solutions of the real representations systems are given and therefore the solutions of the considered equations and systems are reached.
In this thesis, the following linear complex quaternionic equations of the forms AX=XA, AX=XB, AX-XA=C, AX-XB=C, AX-XA*=C, A*X-XA=C are considered over the complex quaternion algebra H(ℂ), and over the real octonion algebra O(ℝ) the following linear real octonionic equations of the forms α(xα)=ρ, α(xβ)=ρ, (αx)β=ρ, αx-(αx)β=ρ, αx-α(xβ)=ρ, xα-(βx)=ρ, xα-β(xα)=ρ, (αx)β+(ϒx)δ=ρ, α(xβ)+ϒ(xδ)=ρ and the following linear real octonionic systems of the forms ax+yb=c ax+yb=c ax+by=c ax+yb=c ax+ya=c ax+yb=c xa+yb=d, ax+by=d, xa+by=d, xb+yb=d, xb+yb=d, xa+by=d are considered. Using the real matrix representation of complex quaternions and real octonions, the real representation systems for these equations and systems are established. Based on them the solvabilities of these equations and systems are investigated. Necessary and sufficient conditions for the existence of a solution or a unique solution to these representation systems are obtained. The solutions of the real representations systems are given and therefore the solutions of the considered equations and systems are reached.
Açıklama
Anahtar Kelimeler
Matematik, Mathematics
