Chinese Annals of Mathematics

The author derives the same null condition as in [1] for the nonlinear elastodynamic system in a simpler way and proves the equivalence of the null conditions introduced in [1] and [7] respectively.

This paper is a continuation of the authors' previous paper [1]. In this paper the authors prove, assuming additional conditions on the initial data, some results about the existence and uniqueness of the entropy weak solutions of the Cauchy problem for the singular hyperbolic system

The authors study homogenization of some nonlinear partial differential equations of the form -div (a(hx, h²x, Du_h)) = f, where a is periodic in the first two arguments and monotone in the third. In particular the case where a satisfies degenerated structure conditions is studied. It is proved that u_h converges weakly in to the unique solution of a limit problem as h → ∞. Moreover, explicit expressions for the limit problem are obtained.

By means of the continuous Glimm functional, a proof is given on the global existence of classical solutions to Cauchy problem for general first order quasilinear hyperbolic systems with small initial total variation.

The author constructs the Casimir element of Hall algebras. By the method of Gabber-Kac theorem (see [4]), it is proved that the Serre relations are the defining relations in composition algebra.

This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Carathéodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.

A soliton hierarchy of multicomponent AKNS equations is generated from an arbitrary order matrix spectral problem, along with its bi-Hamiltonian formulation. Adjoint symmetry constraints are presented to manipulate binary nonlinearization for the associated arbitrary order matrix spectral problem. The resulting spatial and temporal constrained flows are shown to provide integrable decompositions of the multicomponent AKNS equations.

The automorphism group of the Toeplitz C*- algebra, , generated by Toeplitz operators with C¹-symbols on Dirichlet space is discussed; the K₀, K₁-groups and the first cohomology group of are computed. In addition, the author proves that the spectra of Toeplitz operators with C¹-symbols are always connected, and discusses the algebraic properties of Toeplitz operators. In particular, it is proved that there is no nontrivial selfadjoint Toeplitz operator on and if and only if T_φ is a scalar operator.

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.

Let M₁, M₂ be submodules of analytic Hilbert module X on Ω(⊂ Cⁿ) such that M₁ ⊇ M₂ and dim M₁/M₂ = k < ∞. If M₂ is an AF-cosubmodule, then the codimension dim M₁/M₂ of M₂ in M₁ equals the cardinality of zeros of M₂ related to M₁ by counting multiplicities. The codimension formula has some interesting applications. In particular, the author calculates out the dimension of Rudin quotient module, which is raised in [14].

A symplectic reduction method for symplectic G-spaces is given in this paper without using the existence of momentum mappings. By a method similar to the above one, the arthors give a symplectic reduction method for the Poisson action of Poisson Lie groups on symplectic manifolds, also without using the existence of momentum mappings. The symplectic reduction method for momentum mappings is thus a special case of the above results.