Abstract:In this paper we propose a fractional differential equation describing the behavior of a two dimensional projectile in a resisting medium. In order to maintain the dimensionality of the physical quantities in the system, an auxiliary parameter was introduced in the derivative operator. This parameter has a dimension of inverse of seconds ( ) −1 and characterizes the existence of fractional time components in the given system. It will be shown that the trajectories of the projectile at different values of γ and different fixed values of velocity 0 and angle θ, in the fractional approach, are always less than the classical one, unlike the results obtained in other studies. All the results obtained in the ordinary case may be obtained from the fractional case when γ = 1.

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Let G be a cubic graph and Π be a polyhedral embedding of this graph. The extended graph, G e , of Π is the graph whose set of vertices is V (G e ) = V (G) and whose set of edges E(G e ) is equal to E(G) ∪ S, where S is constructed as follows: given two vertices t 0 and t 3 in V (G e ) we say [t 0 t 3 ] ∈ S, if there is a 3-path, (t 0 t 1 t 2 t 3 ) ∈ G that is a Π-facial subwalk of the embedding. We prove that there is a one to one correspondence between the set of possible extended graphs of G and polyhedral embeddings of G.

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