Acta Mathematica Sinica-English Series最新文献

Using the stratifications of Deligne–Mumford moduli spaces (overline{{cal{M}}}_{g,n}) indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of genus g with n external edges. By modifying the usual definition of zeta function and Möbius function of a poset, we introduce generalized (ℚ-valued) zeta function and generalized (ℚ-valued) Möbius function of the poset of stable graphs. We use them to proved a generalized Möbius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.

The elastic transmission eigenvalue problem is fundamental to the qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the elastic transmission eigenvalue problem is neither self-adjoint nor elliptic. The aim of this work is to provide a systematic spectral approximation analysis for the VEM of the elastic transmission eigenvalue problem with equal elastic tensors. Considering standard assumptions on polygonal/polyhedral meshes, we prove the stability analysis of the associated VEM bilinear forms, which shall be applied to the well-defined property of the discrete solution operator. Then the correct approximation of spectrum for the proposed VEM scheme is proven. Necessitated by supporting the convergence analysis, a series of numerical examples are reported. In addition, some negative points of the current VEM scheme are considered, including the locking phenomenon and the influence of VEM stabilization parameters. Thanks to the flexibility of construction for the VEM space, the locking-free and stabilization-free VEM approaches are utilized to tackle with these negative aspects.

In this paper, we prove that four-dimensional hypersurface M ⁴_r with proper mean curvature vector field (i.e., (Delta{vec H}) is proportional to ({vec H})) in pseudo-Riemannian space form N ⁵_s (c) has constant mean curvature, and give the value or range of this constant. As an application, we obtain that biharmonic hypersurfaces in N ⁵_s (c) are minimal in some specific case.

Zhao and Xu (2013) constructed a functor from (mathfrak{o}(n))-Mod to (mathfrak{o}(n+2))-Mod. In this paper, we use the functor successively to obtain full conformal oscillator representation of (mathfrak{o}(2n+2)) in n(n + 1) variables and determine the corresponding finite-dimensional irreducible module explicitly when the highest weight is dominant integral. We also find an equation of counting the dimension of an irreducible (mathfrak{o}(2n+2))-module in terms of certain alternating sum of the dimensions of irreducible (mathfrak{o}(2n))-modules, which leads to new combinatorial identities of classical type in the case of the Steinberg modules. One can use the results to study tensor decomposition of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch–Gordan coefficients and exact solutions of Knizhnik–Zamolodchikov equation in WZW model of conformal field theory.

In this paper, we study chaotic behavior of the (N + 1)-body planar ring problem. Firstly, based on the perturbation theory of integrable Hamiltonian systems, we utilize mass ratio as a disturbance parameter so that this ring problem is regarded as a perturbation of the two-body problem. Then, by applying the extended Melnikov method, we address that there are transversal homoclinic orbits in the ring problem. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we construct an invertible map f satisfying Conley–Moser condition and finally conclude that the ring problem possesses chaotic behavior of the Smale horseshoe type.

We introduce directional complexities for ℤ^q-measure preserving dynamical systems via a collection of new metrics along non-zero directions in ℝ^q. It turns out that a ℤ^q-measure preserving dynamical system is rigid if and only if the invariant measure has bounded directional complexity. We also obtain ergodic decomposition formula for the measure-theoretic directional entropy of a ℤ^q-measure preserving dynamical system.

For the minimal process corresponding to an explosive single-birth (or upwardly skip-free) Q-matrix on the non-negative integers, we prove the existence and uniqueness of quasi-stationary distribution, provide an explicit representation to it and show that it is the limit of quasi-stationary distributions of its truncated processes.

Let X = {X(t), t ∈ ∝^N} be a centered space-time anisotropic Gaussian random field with values in ℝ^d. Under some general conditions, the existence, joint continuity and Hölder conditions of higher-order derivative of local times of X are studied. Moreover, we obtain the uniform Hausdorff dimension of the inverse images of X. The existing results of Gaussian random fields are extended to space-time anisotropic Gaussian random fields with approximate independent components.

In this paper, we obtain the sharp Hardy space estimate of the maximal wave operator. Some applications on spherical averages and their combinations are also given.

Left-invariant Riemannian metrics on Lie groups (G_{{{mathbb R}{rm H}}^{n-1}} times {mathbb R}) and (G_{{{mathbb R}{rm H}}^{2}} times {{mathbb R}^{n-2}}) (n ≥ 3) have been classified by Hiroshi, Takahara and Tamaru. It is easily seen that (G_{{{mathbb R}{rm H}}^{n-1}} times {mathbb R}) and (G_{{{mathbb R}{rm H}}^{2}} times {{mathbb R}^{n-2}}) (n ≥ 3) have the same automorphism group, which is denoted by L_n. In this paper, we first classify n-dimensional simply connected Lie groups with automorphism group L_n, then we classify the left-invariant Riemannian metrics on such Lie groups. As an application, we get the m-quasi Einstein metrics.