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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

###### Centers: their integrability and relations with the divergence

, and we denote it by div( x , y ), as the function div ( x , y ) = ∂ X ∂ x ( x , y ) + ∂ Y ∂ y ( x , y ) . $$\begin{array}{} \displaystyle {\rm div} (x,y) \, = \, \frac{\partial X}{\partial x}(x,y) \, + \, \frac{\partial Y}{\partial y}(x,y). \end{array}$$ System 1 is said to be Hamiltonian if div( x , y ) ≡ 0. In such a case there exists a neighborhood of the origin U and an analytic function H : U ⊆ ℝ 2 → ℝ, called the Hamiltonian, such that X ( x , y ) = − ∂ H ∂ y and Y ( x , y ) = ∂ H ∂ y . $$\begin{array}{} \displaystyle X(x,y) = - \frac

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On the central configurations of the *n*-body problem

–body problem Physica D 238 2009 563 571 10.1016/j.physd.2008.12.014 [26] Llibre, J., Moeckel, R. and Simó, C., Central configurations, periodic orbits and Hamiltonian systems , Advances Courses in Math., CRM Barcelona, Birhauser, 2015. Llibre J. Moeckel R. Simó C. Central configurations, periodic orbits and Hamiltonian systems Advances Courses in Math., CRM Barcelona Birhauser 2015 [27] Long, Y. and Sun, S., Four–Body Central Configurations with some Equal Masses , Arch. Rational Mech. Anal. 162 (2002), 24–44. doi 10.1007/s002050100183 Long Y. Sun S. Four–Body Central

###### On a model for internal waves in rotating fluids

zero at infinity and smooth enough solutions. A Hamiltonian formulation is also derived. One of the relevant properties of nonlinear dispersive models for wave propagation is the existence of traveling-wave solutions of solitary type, [ 6 , 7 ] (see [ 10 , 13 ] and references therein for the case of internal waves). In this sense, and using the Concentration-Compactness theory, [ 22 ], the new model is proved to admit such solutions, under suitable conditions on the parameters. By using the Petviashvili’s iterative method, [ 28 ], to generate approximations

###### Optimal control problems for differential equations applied to tumor growth: state of the art

continuous) u : [0 ,T ] → [0 ,u max ] with A and B given by equation (6) , [ 41 , 47 ] . The necessary conditions for optimality for Problem 1 are given by the Pontryagin maximum principle [ 28 , 36 , 42 ]. The main necessary conditions for optimality is the statement that optimal controls minimize (respectively, maximize) the Hamiltonian function (9) H = H ( λ , N , u ) = q N + s u + λ ( A + u B ) N , $$\begin{equation}H=H(\lambda,N,u)=qN+su+\lambda(A+uB)N, \end{equation}$$ over the control set [0 ,u

###### 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

###### 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