A ring R is a QB-ring provided that aR + bR = R with a, b ∈ R implies that there exists a y ∈ R such that $a + by \in R^{{ - 1}}_{q} .$ It is said that a ring R is a JB-ring provided that R/J(R) is a QB-ring, where J(R) is the Jacobson radical of R. In this paper, various necessary and sufficient conditions, under which a ring is a JB-ring, are established. It is proved that JB-rings can be characterized by pseudo-similarity. Furthermore, the author proves that R is a JB-ring iff so is R/J(R)².

In this paper, using the Brouwer topological degree, the authors prove an existence result for finite variational inequalities. This approach is also used to obtain the existence of periodic solutions for a class of evolution variational inequalities.

The authors study the existence and long-time behavior of weak solutions to the bipolar transient quantum drift-diffusion model, a fourth order parabolic system. Using semi-discretization in time and entropy estimate, the authors get the global existence of nonnegative weak solutions to the one-dimensional model with nonnegative initial and homogenous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, it is proved that the periodic weak solution exponentially approaches its mean value as time increases to infinity.

The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices.

In this paper, the authors establish distortion theorems for various subfamilies H _k($\Bbb {B}$) of holomorphic mappings defined in the unit ball in ℂ ⁿ with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to +∞. These distortion theorems give lower bounds on | det f′(z)| and Re det f′(z). As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies β _k(M) of holomorphic mappings. Moreover, these distortion theorems are sharp. When $\Bbb {B}$ is the unit disk in ℂ, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f′(z) for locally biholomorphic mappings is also obtained.

The author explores the relationship between the cut locus of an arbitrary simply connected and compact Riemannian symmetric space and the Cartan polyhedron of corresponding restricted root system, and computes the injectivity radius and diameter for every type of irreducible ones.

The non-selfsimilar Riemann problem for two-dimensional zero-pressure flow in gas dynamics with two constant states separated by a convex curve is considered. By means of the generalized Rankine-Hugoniot relation and the generalized characteristic analysis method, the global solution involving delta shock wave and vacuum is constructed. The explicit solution for a special case is also given.

Sedimentation and erosion processes in sedimentary basins can be modeled by a parabolic equation with a limiter on the fluxes and a constraint on the time variation. This limiter happens to satisfy a stationary scalar hyperbolic inequality, within a constraint, for which the authors prove the existence and the uniqueness of the solution. Actually, this solution is shown to be the maximal element of a convenient convex set of functions. The existence proof is obtained thanks to the use of a numerical scheme.

In this paper, the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling. Here the coupling matrix can be asymmetric and reducible. Some criteria ensuring delay-independent and delay-dependent global synchronization are derived respectively. It is shown that if the coupling delay is less than a positive threshold, then the coupled network will be synchronized. On the other hand, with the increase of coupling delay, the synchronization stability of the network will be restrained, even eventually de-synchronized.