In this paper, the author introduces some late results and puts forward a few problems on commutators of many important operators in harmonic analysis, included the Bochner-Riesz operator below the critical index, the strongly singular integral operator, the pseudo-differential operator, a class of convolution operators with oscillatory kernel, the Marcinkiewicz integral operator, and the fractional integral operator with rough kernel.

In this paper, we prove that there are no nontrivial block-transitive 6-designs for k ⩽ 100. This supports the long-standing conjecture of Cameron and Praeger that there are no nontrivial block-transitive 6-designs.

A finite group G is called an MSN-group if all maximal subgroups of the Sylow subgroups of G are subnormal in G. In this paper, we determinate the structure of non-MSN-groups in which all of whose proper subgroups are MSN-groups.

Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right H-comodules), which generalizes Majid’s transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.

We introduce an F-Willmore functional of submanifold in space forms, which generalizes the well-known Willmore functional. Its critical point is called the F-Willmore submanifold, for which the variational equation and Simons’ type integral inequality are obtained.

Let $\mathbb{Z}_{p^m } $ be the ring of integers modulo p ^m, where p is a prime and m ⩾ 1. The general linear group GL _n($\mathbb{Z}_{p^m } $) acts naturally on the polynomial algebra A _n:= $\mathbb{Z}_{p^m } $[x ₁, …, x _n]. Denote by $A_n^{GL_2 (\mathbb{Z}_{p^m } )} $ the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.

In this article, we study α-irreducible and α-strongly irreducible ideals of a commutative ring. The relations between α-strongly irreducible ideals of a ring and α-strongly irreducible ideals of localization of the ring are also studied.

In this paper, we study a reversible and non-Hamitonian system with a period annulus bounded by a hemicycle in the Poincaré disk. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. This verifies some results of the conjecture given by Gautier et al.

In this paper, the refining growth and covering theorems for f are established, where f is a quasi-convex mapping of order α and x = 0 is a zero of order k + 1 of f(x) − x. As an application, we obtain the upper and lower bounds on the distortion theorem of f(x) defined on the unit polydisc of ℂ ⁿ. The upper bound of the distortion theorem for f(x) defined on the unit ball of a complex Banach space is also given. Our results extend the growth and distortion theorems for convex functions of one complex variable to quasi-convex mappings of several complex variables.

In this paper, we will give some results on the classification of essential closed surfaces in the surface sum of product I-bundle of closed surfaces and some applications of these results.

We consider the stochastic version of the facility location problem with service installation costs. Using the primal-dual technique, we obtain a 7-approximation algorithm.

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p(ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p(ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p(ℰ(E,H) the complete random normed module generated by L ^p(ℰ, H).

This paper gives some sufficient conditions for the strongly irreducibility of operators which have the forms of upper triangular operator matrices on Banach spaces. Based on these results, strongly irreducible Cowen-Douglas operators of index n are constructed on c ₀, l _p (1 ⩽ p < ∞) for all 1 ⩽ n ⩽ ∞.

Let R = ⊕ _i=0 ^∞ R ⁱ be a connected graded commutative algebra over the field ℚ of rational numbers, and let f be a graded endomorphism of R. In this paper, we show that the Lefschetz series of f can be computed directly from the induced linear map Q(f) on the ℚ vector space of indecomposables of R. We give an explicit algorithm to compute the Lefschetz series of f from Q(f). The main tool we used is the graded algebra version of Gröbner basis theory. At the end of this paper, some examples and applications are given.

Let π be a group, and let H be a Hopf π-coalgebra. We first show that the category M ^H of right π-comodules over H is a monoidal category and there is a monoidal endofunctor (F _α, id, id) of M ^H for any α ∈ π. Then we give the definition of coquasitriangular Hopf π-coalgebras. Finally, we show that H is a coquasitriangular Hopf π-coalgebra if and only if M ^H is a braided monoidal category and (F _α, id, id) is a braided monoidal endofunctor of M ^H for any α ∈ π.