球面上等收敛算子的有界性

设Rⁿ 是n-维欧氏空间n≥3.用Ω_n表示Rⁿ 上的单位球面,对于函数f∈L（Ω_n）,E_N^δ（f）表示其Fourier-Laplace级数的δ阶Cesaro平均所决定的等收敛算子,其中,λ:=（n-2）/2,δ是熟知的临界指标.对于0<δ≤λ,令p₀:=(2λ)/（λ+δ）,本文主要证明了如下结果:     

Convergence rate of Cesàro means of Fourier-Laplace series

The convergence rate of Fourier-Laplace series in logarithmic subclasses of L²(Σd) defined in terms of moduli of continuity is of interest. Lin and Wang [C. Lin, K. Wang, Convergence rate of Fourier-Laplace series of L²-functions, J. Approx. Theory 128 (2004) 103-114] recently presented a characterization of those subclasses and provided the almost everywhere convergence rates of Fourier-Laplace series in those subclasses. In this note, the almost everywhere convergence rates of the Cesàro means for Fourier-Laplace series of the logarithmic subclasses are obtained. The strong approximation order of the Cesàro means and the partial summation operators are also presented.     

Growth properties of plurisubharmonic functions related to Fourier-Laplace transforms

The purpose of the paper is to study the behavior at infinity of Fourier-Laplace transforms of distributions or more generally plurisubharmonic functions u in Cⁿ with bounds of the form

PREDUAL SPACES FOR Q SPACES

To find the predual spaces Pα（R^n） of Qα（R^n） is an important motivation in the study of Q spaces. In this article, wavelet methods are used to solve this problem in a constructive way. First, an wavelet tent atomic characterization of Pα（Rn） is given, then its usual atomic characterization and Poisson extension characterization are given. Finally, the continuity on Pα of Calderon-Zygmund operators is studied, and the result can be also applied to give the Morrey characterization of Pα（Rn）.     

A TWO-SCALE HIGHER-ORDER FINITE ELEMENT DISCRETIZATION FOR SCHROEDINGER EQUATION

In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schroedinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a fine grid can be reduced to an eigenvalue problem on a much coarser grid together with some eigenvalue problems on partially fine grids. It is shown theoretically and numerically that the proposed two-scale higher-order scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.     

STABILIZED FEM FOR CONVECTION-DIFFUSION PROBLEMS ON LAYER-ADAPTED MESHES

The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent stabilization methods, especially its combination with a Galerkin method on layer-adapted meshes. Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.     

FINITE ELEMENTS WITH LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW PROBLEMS

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization （LPS）. These methods preserve the favourable stability and approximation properties of classical residual-based stabilization （RBS） techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the NavierStokes equations. Furthermore, we apply it to optimal control problems with linear（ized） flow problems, since the symmetry of the stabilization leads to the nice feature that the operations ＂discretize＂ and ＂optimize＂ commute.     

AN EFFICIENT METHOD FOR MULTIOBJECTIVE OPTIMAL CONTROL AND OPTIMAL CONTROL SUBJECT TO INTEGRAL CONSTRAINTS

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a ＂budget＂ remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian ＂marching＂ method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced ＂weighted sum＂ based algorithm for the same problem [25]. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.     

Projections,Birkhoff Orthogonality and Angles in Normed Spaces

Let X be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle Aq（x, y） between two vectors x and y in X, such that x is Birkhoff orthogonal to y if and only if Aq（x,y）=π/2. Some other properties of this angle are also discussed.     

K-controllability and Approximate K-controllability of Nonlinear Neutral Systems in Banach Spaces

In this paper, K-controllability and approximate K-controllability of non- linear neutral differential equations in Banach spaces are studied. Sufficient conditions are established for each of these types of controllability. The results are obtained by using Leray-Schauder theory.     

More about the kernel convergence and the ideal convergence

Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.     

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.     

FINITE ELEMENT APPROXIMATIONS FOR SCHROEDINGER EQUATIONS WITH APPLICATIONS TO ELECTRONIC STRUCTURE COMPUTATIONS

In this paper, both the standard finite element discretization and a two-scale finite element discretization for SchrSdinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Schroedinger equations. Very satisfying applications to electronic structure computations are provided, too.     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

Spectral gap and convergence rate for discrete-time Markov chains

Let P be a transition matrix which is symmetric with respect to a measure π.The spectral gap of P in L2(π)-space,denoted by gap(P),is defined as the distance between 1 and the rest of the spectrum of P.In this paper,we study the relationship between gap(P) and the convergence rate of Pn.When P is transient,the convergence rate of P n is equal to 1 gap(P).When P is ergodic,we give the explicit upper and lower bounds for the convergence rate of Pn in terms of gap(P).These results are extended to L∞(π)-space.     

Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L2,κ(Rd) by a subspace E2κ(σ)(SE2κ(σ)), which is a subspace of entire functions of exponential type(spherical exponential type) at most σ. Here L2,κ(Rd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight v2κ(x) =ξ∈R, which is defined by a positive+|(ξ, x)|κ(ξ)subsystem R+ of a finite root system RRdand a function κ(ξ) : R → R+ invariant under the reflection group G(R) generated by R. In the case G(R) = Zd2, we get some exact results. Moreover,the deviation of best approximation by the subspace E2κ(σ)(SE2κ(σ)) of some class of the smooth functions in the space L2,κ(Rd) is obtained.     

The product of the restrained domination numbers of a graph and its complement

Let G=(V,E) be a graph.A set S■V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S.The restrained domination number of G,denoted γr(G),is the smallest cardinality of a restrained dominating set of G.In this paper,we show that if G is a graph of order n≥4,then γr(G)γr(G)≤2n.We also characterize the graphs achieving the upper bound.     

A Quasilinear Parabolic System with Nonlocal Boundary Conditions and Localized Sources

This paper investigates the properties of solutions to a quasilinear parabolic system with nonlocal boundary conditions and localized sources. Conditions for the existence of global or blow-up solutions are given. Global blow-up property and blow- up rate estimates are also derived.     

Atomic Decompositions of Triebel-Lizorkin Spaces with Local Weights and Applications

In this paper,the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fp,q s,w(Rn)with local weight w by using the Lusin-area functions for the full ranges of the indices,and then establish their atomic decompositions for s ∈ R,p ∈(0,1] and q ∈ [p,∞).The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in(0,1].Finite atomic decompositions for smooth functions in Fp,q s,w(Rn)are also obtained,which further implies that a(sub)linear operator that maps smooth atoms of Fp,q s,w(Rn)uniformly into a bounded set of a(quasi-)Banach space is extended to a bounded operator on the whole Fp,q s,w(Rn).As an application,the boundedness of the local Riesz operator on the space Fp,q s,w(Rn)is obtained.     

A FAST SIMPLEX ALGORITHM FOR LINEAR PROGRAMMING

Recently, computational results demonstrated remarkable superiority of a so-called ＂largest-distance＂ rule and ＂nested pricing＂ rule to other major rules commonly used in practice, such as Dantzig＇s original rule, the steepest-edge rule and Devex rule. Our computational experiments show that the simplex algorithm using a combination of these rules turned out to be even more efficient.