Chinese Annals of Mathematics

One kind of unique continuation property for a wave equation is discussed. The authors show that, if one classical solution of the wave equation vanishes in an open set on a hyperplane, then it must vanish in a larger set on this hyperplane. The result can be viewed as a localized version of Robbiano's result^[9]. The approach involves the localized Fourier-Gauss transformation and unique continuation on a line in the Laplace equation.

The deformation of a compact complex Lagrangian submanifold in a hyper-Kaehler manifold and the moduli space are studied. It is proved that the moduli space M_cl is a special Kaehler manifold, where special means that there is a real flat torsionfree symplectic connection ∇ satisfying ( is a complex structure of M_cl). Thus, following [4], one knows that T * M_cl is a hyper-Kaehler manifold and then that M_cl is a complex Lagrangian submanifold in T * M_cl.

Tomita-Takesaki theory in the real case is considered. The author introduces the conception of a nondegenerate pair of closed subspaces in a real Hilbert space. Then a satisfactory real Tomita-Takesaki theory is obtained, and it seems to be a special result of the complex case.

The author gives another linear-algebraic proof of the famous result of Zariski, Delorme, Briancon-Granger-Maisonobe about the moduli number of plane curve singularities with the same topological type as X^a + Y^b = 0 (i.e., with one characteristic pair). Since the original proof depends very much on the division theorem of Briancon, it cannot be generalized to higher dimensions. It is hopeful that the proof here will be applied to the higher dimensional cases.

The spectral flow for paths of admissible operators in arbitrary Banach space is defined, and some properties of the spectral flow is studied.

The author first proves the existences of J-holomorphic curves in the symplectizations of Legendre fibrations and then as an application confirms the Weinstein conjectures on contact manifolds of Legendre fibrations. As a corollary, a new proof on the theorem due to Hofer, Viterbo, Gluck, Ziller, Weinstein and Ljusternik-Fet Theorem is provided, which is quite different from their original proofs.

This paper investigates the maximal and minimal solutions of initial value problem for n-th order nonlinear integro-differential equations of Volterra type on an infinite interval in a Banach space by establishing a comparison result and using the monotone iterative technique.

It is proved that the occupation time of the catalytic super-Brownian motion is absolutely continuous for d = 1, and the occupation density field is jointly continuous and jointly Hölder continuous.

For a special class of quasilinear wave equations with small initial data which satisfy the nondegenerate assumption, the authors prove that the radially symmetric solution develops singularities in the second order derivatives in finite time while the first order derivatives and the solution itself remain continuous and small. More precisely, it turns out that this solution is a "geometric blowup solution of cusp type", according to the terminology posed by S. Alinhac^[2].

Let d_k(n) denote the k-fold iterated divisor function (k ≥ 2). It is proved that for sufficiently large x, d_k(n) = d_k(n + 1) holds for ≫ x(log log x)^-3 integers n ≤ x.

The authors study a 3 × 3 rate-type viscoelastic system, which is a relaxation approximation to a 2 × 2 quasi-linear hyperbolic system, including the well-known p-system. The nonlinear stability of two-mode shock waves in this relaxation approximation is proved.

The authors discuss the existence of pseudo almost periodic solutions of differential equations with piecewise constant argument by means of introducing new concept, pseudo almost periodic sequence.

When an elastic string with fixed ends is subjected to transverse vibrations, its length varies with the time: this introduces changes of the tension in the string. This induced Kirchhoff to propose a nonlinear correction of the classical D'Alembert equation. Later on, Woinowsky-Krieger (Nash & Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends.

Here a new equation for the small transverse vibrations of a simply supported beam is proposed. Such equation takes into account Kirchhoff's correction, as well as the correction for rotary inertia of the cross section of the beam and the influence of shearing strains, already present in the Timoshenko beam equation (cf. the Mindlin-Timoshenko equation for the plate). The model is inspired by a remark of Rayleigh, and by a joint paper with Panizzi & Paoli. It looks more complicated than the one proposed by Sapir & Reiss, but as a matter of fact it is easier to study, if a suitable change of variables is performed.

The author proves the local well-posedness of the initial-boundary value problem in Sobolev spaces of order ≥ 2.5. The technique is abstract, i.e. the equation is rewritten as a fourth order evolution equation in Hilbert space (thus the results could be applied also to the formally analogous equation for the plate).