Chinese Annals of Mathematics

The author considers a linearly elastic shallow shell with variable thickness and shows that, as the thickness of the shell goes to zero, the solution of the three-dimensional equations converges to the solution of the two-dimensional shallow shell equations with variable thickness.

The authors prove well posedness in Gevrey classes of Cauchy problem for nonlinear hyperbolic equations of constant multiplicity with Hölder dependence on the time variable.

The authors define the scenery flow of the torus. The flow space is the union of all flat 2-dimensional tori of area 1 with a marketed direction (or equivalently, the union of all tori with a quadratic differential of norm 1). This is a 5-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential.

The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant 3-dimensional subset on which it is the suspension flow of that map.

The authors first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of:

• stacking and rescaling periodic tilings of the plane;

• symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the S-adic system generated by two substitutions;

• zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity;

• the natural extension of two-dimensional continued fractions;

• induction on exchanges of three intervals;

• rescaling on pairs of transverse measure foliations on the torus, or the Teichmüller flow on the twice-punctured torus.

It is well known that when the random errors are iid. with finite variance, the week and the strong consistency of LS estimate of multiple regression coefficients are equivalent. This note, by constructing a counter-example, shows that this equivalence no longer holds true in case that the random errors possess only the r-th moment with 1≤r<2.

Based on the data obtained from a survey recently made in Shanghai, this paper presents the hybrid technique for risk analysis and evaluation of some diseases.

After determination of main risk factors of these diseases by analysis of variance, the authors introduce a new concept 'Illness Fuzzy Set' and use fuzzy comprehensive evaluation to evaluate the risk of suffering from a disease for residents. Optimal technique is used to determine the weights w_i in fuzzy comprehensive evaluation, and a new method 'Improved Information Distribution' is also introduced for the treatment of small sample problem.

It is shown that the results obtained by using the hybrid technique are better than by using single fuzzy technique or single statistical method.

The boundary stabilization problem of a Timoshenko beam attached with a mass at one end is studied. First, with linear boundary force feedback and moment control simultaneously at the end attached with the load, the energy corresponding to the closed loop system is proven to be exponentially convergent to zero as time t→∞. Then, some counterexamples are given to show that, in other cases, the corresponding closed loop system is, in general, not stable asymtotically, let alone exponentially.

For the infinite delay difference equations of the general form, two new uniform asymptotic stability criteria are established in terms of the discrete Liapunov functionals.

The perturbation of semigroup by a multiplicative functional with bounded variation is investigated in the frame of weak duality. The strong continuity and Schrödinger type equation of the perturbated semigroup are discussed. A few switching identities and formulae concerning dual additive functionals and Revuz measures are given.

Let X be a complex Banach space without the analytic Radon-Nikodym property. The author shows that G={f∈H^∞(D, X) : there exists ∊>0, such that for almost all θ∈[0, 2π], is a dense open subset of H^∞(D, X). It is also shown that for every open subset B of T, there exists F∈H^∞(D, X), such that F has boundary values everywhere on B^c and F has radial limits nowhere on B. When A is a measurable subset of T with positive measure, there exists f∈H^∞(D, X), such that f has nontangential limits almost everywhere on A^c and f has radial limits almost nowhere on A.

By means of the theory of harmonic maps into the unitary group U(N), the authors study harmonic maps into the symplectic group Sp(N). The symplectic uniton and symplectic extended uniton are introduced. The method of the symplectic Bäcklund transformation and the Darboux transformation is used to construct new symplectic unitons from a known one.

This paper is concerned wih the global existence and the partial regularity for the weak solution of the Landau-Lifshitz-Maxell system in two dimensions with Neumann boundary conditions.