The theory of linear error-correcting codes from algebraic geometric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981–1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.

We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K 〈x, y, z〉 over a field K, i.e., automorphisms that fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K 〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and algorithms that recognize z-tame automorphisms and z-tame coordinates.

In this paper we review all the main known results about mean curvature flows with initial surfaces symplectic in a Kähler-Einstein surface, including published results and new results obtained recently. We also propose some problems that we think are very interesting.

This paper is a survey on the theory and application of some block spaces on the unit sphere introduced by Jiang and the author of this paper in the study of singular integrals and some related operators with rough kernels.

In this paper, we are concerned with a class of reflected stochastic differential equations (reflected SDEs) with non-Lipschitzian coefficients. Under the same coefficients assumptions as Fang and Zhang [Probab. Theory Relat. Fields, 2005, 132(3): 356–390] for a class of SDEs, we establish the pathwise uniqueness for the reflected SDEs. Furthermore, a strong comparison theorem is proved for the reflected SDEs in a one-dimensional case.

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow.

The problems concerned in this paper are a class of constrained min-max problems. By introducing the Lagrange multipliers to the linear constraints, such problems can be solved by some projection type prediction-correction methods. However, to obtain components of the predictor one by one, we use an alternating direction method. And then the new iterate is generated by a minor correction. Global convergence of the proposed method is proved. Finally, numerical results for a constrained single-facility location problem are provided to verify that the new method is effective for some practical problems.

Let G be a finite group, and let π_e(G) be the set of all element orders of G. In this short paper we prove that π_e(B_n(q)) ≠ π_e(C_n(q)) for all odd q.

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A₁ and finding interesting families of new noetherian rings, a class of algebras similar to U(sl₂) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.

Let G be a non-abelian group and associate a non-commuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L₂(q) (q = pⁿ, p is a prime), then G ≅ M.

This paper proposes some diagnostic tools for checking the adequacy of multivariate regression models including classical regression and time series autoregression. In statistical inference, the empirical likelihood ratio method has been well known to be a powerful tool for constructing test and confidence region. For model checking, however, the naive empirical likelihood (EL) based tests are not of Wilks’ phenomenon. Hence, we make use of bias correction to construct the EL-based score tests and derive a nonparametric version of Wilks’ theorem. Moreover, by the advantages of both the EL and score test method, the EL-based score tests share many desirable features as follows: They are self-scale invariant and can detect the alternatives that converge to the null at rate n^−1/2, the possibly fastest rate for lack-of-fit testing; they involve weight functions, which provides us with the flexibility to choose scores for improving power performance, especially under directional alternatives. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of possible alternatives. A simulation study is carried out and an application for a real dataset is analyzed.