In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of such distinct closed geodesics is finite.

In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler n-dimensional real projective space (RP n , F ) when there exist only finitely many distinct non-contractible closed geodesics on (RP n , F ), where the integer n ≥ 2. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics on RP n with a bumpy and irreversible Finsler metric. Together with two previous results on bumpy and reversible Finsler metrics in [14] and [39], it yields that every RP n with a bumpy Finsler metric possesses at least two distinct non-contractible closed geodesics.

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This paper is intended to explore a new method of measuring supply chain flexibility from two dimensions (resources and time) when the supply chain is coordinated. Under the assumptions that the node enterprises' marginal costs increase with their outputs, firstly this paper analyses the coordination problem of a supply chain with buyback contract, then formulates the resource allocation with the entropy concept in the case of supply chain coordination, puts forward the concepts of resource output elasticity and time output elasticity, analyses the resources' dynamic matching problem from the two dimensions of resource and time in order to respond to the change of market demand. Finally, based on the above discussion, this paper proposes a method of measuring supply chain flexibility by integrating the resource flexibility and time flexibility. The method not only considers the resources and time required to respond the market's change, but also takes into account the coordination relationship between the node enterprises.

Let M = S n /Γ and h be a nontrivial element of finite order p in π 1 (M ), where the integer n ≥ 2, Γ is a finite group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class [h] on every Finsler compact space form (M, F ) when there exist only finitely many distinct non-contractible closed geodesics of the class [h] on (M, F ). Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class [h] on (M, F ) with a bumpy Finsler metric, which improves a result of Taimanov in [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on RP n in [25] to general compact space forms.