In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary (null) controllability and the exact boundary observability for first order hyperbolic systems.

A bounded linear operator T acting on a Hilbert space is called Coburn operator if ker(T − λ) = {0} or ker(T − λ)* = {0} for each λ ∈ ℂ. In this paper, the authors define other Coburn type properties for Hilbert space operators and investigate the compact perturbations of operators with Coburn type properties. They characterize those operators for which has arbitrarily small compact perturbation to have some fixed Coburn property. Moreover, they study the stability of these properties under small compact perturbations.

In this paper, the authors obtain the boundedness of vector valued bilinear Calderón-Zygmund operators on products of weighted Herz-Morrey spaces with variable exponents.

For the iteration of spherical average (A ₁) ^N and the Laplace operator Δ, we consider the boundedness of the operator Δ(A ₁) ^N on the α-modulation spaces $M_{p,q}^{s,\alpha }$. The authors obtain some sufficient and necessary conditions to ensure the boundedness on the α-modulation spaces. The main theorems significantly improve some known results.

Let Z ₂ denote a cyclic group of 2 order and Z ₂ ² = Z ₂ × Z ₂ the direct product of groups. Suppose that (M, Φ) is a closed and smooth manifold M with a smooth Z ₂ ²-action whose fixed point set is the disjoint union of two real projective spaces with the same dimension. In this paper, the authors give a sufficient condition on the fixed data of the action for (M, Φ) bounding equivariantly.

Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties intersecting moving hypersurfaces targets. The main purpose of this paper is to give some interesting improvements of Ru’s second main theorem for moving hypersurfaces targets located in subgeneral position with index.

The authors give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology In addition, they give a shorter proof of a well known criterion on this subject.

Let ${\mathbb{F}_q}$ be a finite field with order q and D _2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual (LCD) codes and self-orthogonal codes in the finite dihedral group algebras ${\mathbb{F}_q}[{D_{2n}}]$. Some numerical examples are also presented to illustrate the main results.