We study variations of tautological bundles on moduli spaces of representations of quivers with relations associated with dimer models under a change of stability parameters. We prove that if the tautological bundle induces a derived equivalence for some stability parameter, then the same holds for any generic stability parameter, and any projective crepant resolution can be obtained as the moduli space for some stability parameter. This result is used in [IU] to prove the abelian McKay correspondence without using the result of Bridgeland, King and Reid [BKR01].

This paper considers the class of abelian groups that are extensions of subgroups that are direct sums of cyclic groups by factor groups that are also of this form. This class is shown to be the projectives with respect to a natural collection of short exact sequences, and that the corresponding class of injectives consists of those groups whose first Ulm subgroup is pure-injective. This class of projectives is quite extensive, but satisfactory descriptions are given for the countable groups in the class that are either torsion-free, or else mixed groups of torsion-free rank one. Particular attention is paid to the behavior of the groups in these classes under localization at some prime.

By using the critical point theory, some new criteria are obtained for the existence and multiplicity of periodic solutions to a class of nonlinear difference equations. The proof is based on the Linking Theorem in combination with variational technique. Our results successfully generalize and improve some existing results in the literature.

Piecewise smooth systems have been consistently considered and investigated in nonlinear dynamics due to their practical applications. In this paper, we study a generic type of piecewise smooth dynamical system to deal with period-additivity and multistability in the system; an arithmetic sequence of periodic attractors appearing in the period-adding bifurcation and the coexistence of multiple attractors in the system. We state a physical observation of the phenomena and then provide rigorous mathematical arguments and numerical simulations.