Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold $\overline{M}$. We first assume the pull back Weitzenböck operator of $\overline{M}$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of $\overline{M}$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when $\overline{M}$ is the space form.     

Perfect matchings extend to two or more hamiltonian cycles in hypercubes

Kreweras conjectured that every perfect matching of a hypercube $Q_n$ for $n\ge 2$ can be extended to a hamiltonian cycle of $Q_n$. Fink confirmed the conjecture to be true. It is more general to ask whether every perfect matching of $Q_n$ for $n\ge 2$ can be extended to two or more hamiltonian cycles of $Q_n$. In this paper, we prove that every perfect matching of $Q_n$ for $n\ge 4$ can be extended to at least $2^{2^{n?4}}$ different hamiltonian cycles of $Q_n$.     

Revised simplex algorithm for finite Markov decision processes

We introduce a revised simplex algorithm for solving a typical type of dynamic programming equation arising from a class of finite Markov decision processes. The algorithm also applies to several types of optimal control problems with diffusion models after discretization. It is based on the regular simplex algorithm, the duality concept in linear programming, and certain special features of the dynamic programming equation itself. Convergence is established for the new algorithm. The algorithm has favorable potential applicability when the number of actions is very large or even infinite.     

A Totally(Δ+1)-colorable 1-planar Graph with Girth at Least Five

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge.In this paper,we prove that every 1-planar graph G with maximum degree Δ(G)≥12 and girth at least five is totally(Δ(G)+1)-colorable.     

Statistically and Computationally Efficient Estimating Equations for Large Spatial Datasets

For Gaussian process models, likelihood-based methods are often difficult to use with large irregularly spaced spatial datasets, because exact calculations of the likelihood for n observations require O(n³) operations and O(n²) memory. Various approximation methods have been developed to address the computational difficulties. In this article, we propose new, unbiased estimating equations (EE) based on score equation approximations that are both computationally and statistically efficient. We replace the inverse covariance matrix that appears in the score equations by a sparse matrix to approximate the quadratic forms, then set the resulting quadratic forms equal to their expected values to obtain unbiased EE. The sparse matrix is constructed by a sparse inverse Cholesky approach to approximate the inverse covariance matrix. The statistical efficiency of the resulting unbiased EE is evaluated both in theory and by numerical studies. Our methods are applied to nearly 90,000 satellite-based measurements of water vapor levels over a region in the Southeast Pacific Ocean.     

Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three-fold Zero Eigenvalue

In this paper we study the singularity at the origin with three-fold zero eigenvalue forsymmetric vector fields with nilpotent linear part and 3-jet C^∞-equivalent to y δ/δx zδ/δy ax^2yδ/δz with a≠0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularityby using the normal form and blow-up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil‘nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel‘nikov methods of a homoclinic orbit to a hyperbolic or non-hyperbolic equilibrium in a highdimensional space.     

Finite Difference Method for Reaction-Diffusion Equation with Nonlocal Boundary Conditions

In this paper, we present a numerical approach to a class of nonlinear reaction-diffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

Vacuum states on compressible Navier-Stokes equations with general density-dependent viscosity and general pressure law

In this paper,we study the one-dimensional motion of viscous gas with a general pres- sure law and a general density-dependent viscosity coefficient when the initial density connects to the vacuum state with a jump.We prove the global existence and the uniqueness of weak solutions to the compressible Navier-Stokes equations by using the line method.For this,some new a priori estimates are obtained to take care of the general viscosity coefficientμ(ρ)instead ofρ~θ.     

Stabilized finite element method based on the Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations

This paper provides an error analysis for the Crank-Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier-Stokes problem, where the finite element space pair for the approximation of the velocity and the pressure is constructed by the low-order finite element: the quadrilateral element or the triangle element with mesh size . Error estimates of the numerical solution to the exact solution with are derived.