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###### A sufficient condition for the existence of a k-factor excluding a given r-factor

1 Introduction For motivation and background to this work see [ 1 ]. In this paper, we consider only finite and simple graphs. Let G = ( V ( G ) , E ( G )) be a graph, where V ( G ) denotes its vertex set and E ( G ) denotes its edge set. A graph is Hamiltonian if it admits a Hamiltonian cycle. For each x ∊ V ( G ), the neighborhood N G ( x ) of x is the set of vertices of G adjacent to x , and the degree d G ( x ) of x is | N G ( x )|. For S ⊆ V ( G ), we write N G ( S ) = ∪ x∊S N G ( x ). G [ S ] denotes the subgraph of G

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###### Noether’s theorems of variable mass systems on time scales

been put forward, and a series of innovative research results have been obtained [ 22 , 23 , 24 , 25 , 26 ]. In this article, we will study the Noether theorems and its inverse problem of variable mass on time scales. In Section 2 , we review some basic definitions and properties about the calculus on time scales. In Section 3 , we obtain the Lagrange equations of systems by deriving Hamilton’s principle for variation mass systems with delta derivative. In Section 4 , based on the quasi-invariance of Hamiltonian action of the variation mass systems under the

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###### A theoretical model for the transmission dynamics of HIV/HSV-2 co-infection in the presence of poor HSV-2 treatment adherence

, we can easily verify that the objective functional is convex on the closed, convex control set U . The optimal system is bounded, which determines the compactness needed for the existence of the optimal control. In order to find an optimal solution of model system (17) , first let us define the Hamiltonian functions H for the optimal control system (17) as H ( t , X , U , λ ) = L + λ 1 d S d t + λ 2 d I 1 d t + λ 3 d I 2 d t + λ 4 d Q 1 d t + λ 5 d Q 2 d t + λ 6 d H d t λ 7 d H I 1 d t + λ 8 d H I 2 d t + λ 9 d H Q 1 d t + λ 10 d H Q 2 d t + λ 11 d A d

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