The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions A _H ↪ A, where A _H is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over A _H. Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group H_{\phi _{Z(A_H )} }^2 (H,Z(A _H)), where Z(A _H) is the center of A _H.

In this paper, the authors consider a family of smooth immersions F _t: M ⁿ → N ⁿ⁺¹ of closed hypersurfaces in Riemannian manifold N ⁿ⁺¹ with bounded geometry, moving by the H ^k mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the H ^k mean curvature flow solution with finite total mean curvature on a finite time interval [0, T _max) can be extended over T _max. This result generalizes the extension theorems in the paper of Li (see “On an extension of the H ^k mean curvature flow, Sci. China Math., 55, 2012, 99–118”).

The authors propose a dwindling filter algorithm with Zhou’s modified subproblem for nonlinear inequality constrained optimization. The feasibility restoration phase, which is always used in the traditional filter method, is not needed. Under mild conditions, global convergence and local superlinear convergence rates are obtained. Numerical results demonstrate that the new algorithm is effective.

f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970. In this paper, the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. The author proves that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map. This generalizes the well-known characterization for harmonic morphisms. Some properties and many examples as well as some non-existence of f-harmonic morphisms are given. The author also studies the f-harmonicity of conformal immersions.

In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces F _p,q ^s,w(ℝ ⁿ with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ ℝ, p ∈ (0, 1] and q ∈ [p,∞). The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in (0, 1]. Finite atomic decompositions for smooth functions in F _p,q ^s,w(ℝ ⁿ are also obtained, which further implies that a (sub)linear operator that maps smooth atoms of F _p,q ^s,w(ℝ ⁿ uniformly into a bounded set of a (quasi-)Banach space is extended to a bounded operator on the whole F _p,q ^s,w(ℝ ⁿ. As an application, the boundedness of the local Riesz operator on the space F _p,q ^s,w(ℝ ⁿ is obtained.

Associative multiplications of cubic matrices are provided. The N ³-dimensional 3-Lie algebras are realized by cubic matrices, and structures of the 3-Lie algebras are studied.

The stabilizer (additive) method and non-additive method for constructing asymmetric quantum codes have been established. In this paper, these methods are generalized to inhomogeneous quantum codes.

This paper generalizes the C*-dynamical system to the Banach algebra dynamical system (A,G, α) and define the crossed product A ⋊_α G of a given Banach algebra dynamical system (A,G, α). Then the representation of A ⋊_α G is described when A admits a bounded left approximate identity. In a natural way, the authors define the reduced crossed product A ⋊_α,r G and discuss the question when ⋊_α G coincides with ⋊_α,r G.

The authors consider the local smooth solutions to the isentropic relativistic Euler equations in (3+1)-dimensional space-time for both non-vacuum and vacuum cases. The local existence is proved by symmetrizing the system and applying the Friedrichs-Lax-Kato theory of symmetric hyperbolic systems. For the non-vacuum case, according to Godunov, firstly a strictly convex entropy function is solved out, then a suitable symmetrizer to symmetrize the system is constructed. For the vacuum case, since the coefficient matrix blows-up near the vacuum, the authors use another symmetrization which is based on the generalized Riemann invariants and the normalized velocity.