Chinese Annals of Mathematics

The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form

The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.

The authors obtain new characterizations of unconditional Cauchy series in terms of separation properties of subfamilies of , and a generalization of the Orlicz-Pettis Theorem is also obtained. New results on the uniform convergence on matrices and a new version of the Hahn-Schur summation theorem are proved. For matrices whose rows define unconditional Cauchy series, a better sufficient condition for the basic Matrix Theorem of Antosik and Swartz, new necessary conditions and a new proof of that theorem are given.

The authors study a generalized thin film equation. Under some assumptions on the initial value, the existence of weak solutions is established by the time-discrete method. The uniqueness and asymptotic behavior of solutions are also discussed.

An explicit construction of irreducible representations for the affine-Virasoro Lie algebra of type B_l, through the use of vertex operators and certain oscillator representations of the Virasoro algebra, is given.

In this note, the local spectral properties of unilateral operator weighted shifts are studied.

Some results of existence of positive solutions for singular boundary value problems

It is showed that if the first exit distribution leaving any ball from the center is the uniform distribution on the sphere, then the Lévy process is a scaled Brownian motion. The paper also gives a characterization of a continuous Hunt process by the first exit distribution from any ball.

Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general. This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g(t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.

In this paper the existence results of positive ω-periodic solutions are obtained for second order ordinary differential equation -u''(t)=f(t,u(t)) (t∈ℝ), and also for first order ordinary differential equation u'(t)=f(t,u(t)) (t∈ℝ), where f:ℝ×ℝ⁺→ℝ is a continuous function which is ω-periodic in t. The discussion is based on the fixed point index theory in cones.

Let X, Y be UMD-spaces that have property (α), 1<p<∞ and let ℳ be an R-bounded subset in ℒ(X, Y). It is shown that {T_(Mk)k∈ℤ : M_k, k(M_k+1-M_k)∈ℳ for k∈ℤ} is an R-bounded subset of ℒ(L^p(0, 2π; X), L^p(0, 2π; Y)), where T_(Mk)k∈ℤ denotes the L^p-multiplier given by the sequence (M_k)_k∈ℤ. This generalizes a result of Venni [10]. The author uses this result to study the strongly L^p-well-posedness of evolution equations with periodic boundary condition. Analogous results for operator-valued L^p-multipliers on ℝ are also given.