Chinese Annals of Mathematics

The authors prove the uniqueness in the inverse acoustic scattering problem within convex polygonal domains by a single incident direction in the sound-soft case and the sound-hard case, and by two incident directions in the case of the impedance boundary condition. The proof is based on analytic continuation on a straight line.

The author presents an extension of the Atiyah-Patodi-Singer invariant for unitary representations [2,3] to the non-unitary case, as well as to the case where the base manifold admits certain finer structures. In particular, when the base manifold has a fibration structure, a Riemann-Roch theorem for these invariants is established by computing the adiabatic limits of the associated η-invariants.

Consider the following Cauchy problem for the first order quasilinear strictly hyperbolic system

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. An infinite family of cubic 1-regular graphs was constructed in [7] as cyclic coverings of the three-dimensional Hypercube, and a classification of all s-regular cyclic coverings of the complete bipartite graph of order 6 was given in [8] for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. In this paper, the authors classify all s-regular dihedral coverings of the complete graph of order 4 for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed.

Here the authors are interested in the zero set of Sobolev functions and functions of bounded variation with negative power of integrability. The main result is a general Hausdorff dimension estimate on the size of zero set. The research is motivated by the model on van der waal force driven thin film, which is a singular elliptic equation. After obtaining some basic regularity result, the authors get an estimate on the size of singular set; such set corresponds to the thin film rupture set in the thin film model.

The author proves that for initial data in a set , unbounded in , the solutions of the Cauchy-Dirichlet problem for the dissipative Kirchhoff equation

The authors give two cohomology vanishing theorems for domains, which are not pseudoconvex, and characterize the holomorphy of domains with smooth boundaries in separable Hilbert spaces through cohomology vanishing.

The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general k-th Kloosterman sums, and give an interesting asymptotic formula.

This paper characterizes all connected graphs with exactly two Laplacian eigenvalues greater than two and all connected graphs with exactly one Laplacian eigenvalue greater than three.

The author establishes a deLeeuw-type theorem on maximal multiplier operators on local Hardy-Lorentz space.

The authors give an algebraic method to add uniton numbers for harmonic maps from a simply connected domain Ω⊆R²∪{∞} into the unitary group U(N) with finite uniton number. So, it is proved that any n-uniton can be obtained from a 0-uniton by purely algebraic operations and integral transforms to solve the -problem via two different ways.

This paper is motivated by S. Park [10] in which the injective cover of left R[x]-module M[x^-1] of inverse polynomials over a left R-module M was discussed. The author considers the Ω-covers of modules and shows that if η: P → M is an Ω-cover of M, then [η^S,≤] : [P^S,≤] → [M^S,≤] is an [Ω^S,≤]-cover of left [[R^S,≤]]-module [M^S,≤], where Ω is a class of left R-modules and [M^S,≤] is the left [[R^S,≤]]-module of generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[R^S,≤]]-module [M^S,≤] are discussed.

Let ξ be an irrational number with simple continued fraction expansion

Theorem. Let r > 1,R > 1 be two real numbers and

Then

(i) C_n-2 < r, C_n < R imply C_n-1 > K;

(ii) C_n-2 > r, C_n > R imply C_n-1 < K.

This theorem generalizes the main result in [1].