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Yayın Epidemiyolojideki kompartman modellerinin eşlenmiş Hamilton analizi(Marmara Üniversitesi Fen Bilimleri Enstitüsü, 2021-01-13) Ateşli, Begüm; Esen, Oğul; Sütlü, SerkanEpidemiyolojideki SIR, SEIR, 2-SIR ve 2-SEIR modellerinin Hamilton formülasyonu verildi. Eşlenmiş Lie-Poisson sistemleri yazıldı. SIR ve SEIR modellerinin eşlenmiş Lie-Poisson sistemi oldukları gösterildi. Bir Lie cebiri için bükülmüş eşçevrim genişlemesi çalışıldı. Bu genişlemenin dual uzayı üzerinde eşlenmiş Lie-Poisson denklemleri elde edildi. SIR ve SEIR kompartman modellerinin iki popülasyon karşılığı olan 2-SIR ve 2-SEIR modellerinin bükülmüş eşçevrim genişlemesiyle elde edilmiş Lie-Poisson sistemi olarak ifade edilebilecekleri gösterildi.Yayın Cohomologies and generalized derivations of n-Lie algebras(Electronic Journals Project, 2022) Ateşli, Begüm; Esen, Oğul; Sütlü, SerkanA cohomology theory associated to an n-Lie algebra and a representation space of it is introduced. It is shown that this cohomology theory classifies generalized derivations of n-Lie algebras as 1-cocycles, and inner generalized derivations as 1-coboundaries.Yayın Cohomologies and generalized derivation extensions of n-Lie algebras(Cornell Univ, 2021-04-18) Ateşli, Begüm; Esen, Oğul; Sütlü, SerkanA cohomology theory, associated to a n-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for n = 3, with the known cohomology of n-Lie algebras. The abelian extensions and infinitesimal deformations of n-Lie algebras, on the other hand, are shown to be characterized by the usual cohomology of n-Lie algebras. Furthermore, the Hochschild-Serre spectral sequence of the Lie algebra cohomology is upgraded to the level of n-Lie algebras, and is applied to the cohomology of generalized derivation extensions.Yayın On matched pair Hamiltonian analysis of the compartmental models(Marmara Üniversitesi, 2020-10-30) Ateşli, Begüm; Esen, Oğul; Sütlü, Serkan; Yıldırım, KenanEpidemiological compartmental models predict the spread of an infectious disease that a specific population encounter. The population is divided into compartments representing different stages of the epidemic and the change of these compartments in time is given by nonlinear differential equations. In previous studies, the Hamiltonian analysis of these models is included. In this work, we briefly explain SIR, SEIR, 2-SIR and 2-SEIR models, and their Hamiltonian analysis. We recollect the matched pair Lie-Poisson systems and observe that SIR and SEIR models can be written as matched pair Lie-Poisson systems. We generalize the matched pair Lie-Poisson systems using the twisted cocycle extension. We attain that matched pair Lie-Poisson systems obtained by the twisted cocycle extension is convenient for 2-SIR and 2-SEIR models.
