We first characterize the automorphism groups of Hodge structures of cubic threefolds and cubic fourfolds. Then we determine for some complex projective manifolds of small dimension (cubic surfaces, cubic threefolds, and non-hyperelliptic curves of genus 3 or 4), the action of their automorphism groups on Hodge structures of associated cyclic covers, and thus confirm conjectures made by Kudla and Rapoport in [KR12].

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.

No abstract

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.

In ast decades dynamic rogramming, genetic algorithms, ant colony o timization algorithms and some gradient algorithms ha e been a lied to ower o timization of gas i elines. In this a er a ower o timization model for gas i elines is de elo ed and an im ro ed article swarm o timization algorithm is a lied. ased on the testing of the arameters in ol ed in the algorithm which need to be de ned arti cially, the alues of these arameters ha e been recommended which can ma e the algorithm reach ef ciently the a ro imate o timum solution with required accuracy. Some e am les ha e shown that the relati e error of the article swarm o timization o er ant colony o timization and dynamic rogramming is less than 1% and the com utation time is much less than that of ant colony o timization and dynamic rogramming.