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In a chimera state, domains composed of synchronized oscillators coexist with ones composed of desynchronized oscillators. It is a common view that, in a chimera state, oscillators in coherent domains always share the same mean phase velocity. However, recent studies have suggested that oscillators in different coherent domains may have different mean phase velocities. In this work, we study a ring of nonlocally coupled Brusselators. We find a two-frequency chimera state with mixed phase regularities in which Brusselators in adjacent coherent domains oscillate at different velocities. Moreover, Brusselators in coherent domains with higher mean phase velocity are nearly in phase. In contrast, Brusselators in coherent domains with lower mean phase velocity are randomly partitioned into two groups in antiphase. We find that the local mean fields in these two types of coherent domains perform different dynamics. Based on local mean fields, we provide an explanation for the formation of this type of chimera state. Furthermore, the stability diagrams of the two-frequency chimera state with mixed phase regularities are investigated in different parameter planes

In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber's formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation., Comment: 18 pages. arXiv admin note: text overlap with arXiv:2003.07392 by other authors

In this paper, we first introduce the notion of BiHom–Lie superalgebras, which is a generalization of both BiHom–Lie algebras and Hom–Lie superalgebras. Also, we explore some general classes of BiHom–Lie admissible superalgebras and describe all these classes via G-BiHom-associative superalgebras, where G is a subgroup of the symmetric group $$S_{3}$$ . Finally, we obtain a method to construct the solutions of the BiHom–Yang–Baxter equation from BiHom–Lie superalgebras.

The main purpose of the present paper is to develop the theory of center constructions on Hom–Hopf algebras. Let H be a Hom–Hopf algebra, we first introduce the notions of nth Yetter–Drinfeld modules and mth Drinfeld codouble for H. Also we prove that the category $${\mathcal {YD}}_H^H(n)$$ of nth Yetter–Drinfeld modules of H is a braided autonomous category. Finally, we show that $${\mathcal {YD}}_H^H(n)$$ and $$Corep^{i,j}(CD_m(H))$$ (i.e., the corepresentation category of the Drinfeld codouble of H) are braided isomorphic as the full subcategories of $$Corep^{i,j}(H)$$ .

Let H be a weak Hopf π-coalgebra and let A be a weak right π-H-comodule algebra with a total integral ϕ. This paper is mainly devoted to a Maschke type theorem for (H,A)-weak relative Hopf π-modules. As an application, we obtain a characterization for an (H,A)-weak Hopf π-module to be projective as an A-module.