格序代数的商及其代数性质

研究了格序代数的商及其一些重要的代数性质.给出了格序代数的商的概念,定义了商f-代数、商几乎f-代数和商d-代数,并给出其等价刻画.讨论了这三类格序代数的商的半素性和交换性质,得到了这些性质的若干刻画.     

SIMPLE PROOFS OF THE CAUCHY-SCHWARTZ INEQUALITY AND THE NEGATIVE DISCRIMINANT PROPERTY IN ARCHIMEDEAN ALMOST f-ALGEBRAS

The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost f-algebras[5], based on a sequence approximation.     

STATIC REGIME IMAGING OF LOCATIONS OF CERTAIN 3D ELECTROMAGNETIC IMPERFECTIONS FROM A BOUNDARY PERTURBATION FORMULA

We are concerned, in a static regime, with a three-dimensional bounded domain of certain an imaging approach of the locations in electromagnetic imperfections. This approach is related to Electrical Impedance Tomography and makes use of a new perturbation formula in the electric fields. We present two localization procedures, from a Current Pro- jection method that deals with the single imperfection context and an inverse Fourier process that is devoted to multiple imperfections configurations. These procedures extend those that were described in our previous work, since operating for a broader class of settings. Namely, the localization is additionally performed for certain purely electric imperfections, as established from numerical simulations.     

Almost Everywhere Convergence of Sequences of Cesàro and Riesz Means of Integrable Functions with Respect to the Multidimensional Walsh System

The aim of this paper is to prove the a.e.convergence of sequences of the Cesa`ro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=(a1,...,ad):N→Nd(d∈P)such that aj(n+1)≥δsupk≤n aj(k)(j=1,...,d,n∈N)for someδ0 and a1(+∞)=···=ad(+∞)=+∞.Then,for each integrable function f∈L1(Id),we have the a.e.relation for the Ces`aro means limn→∞σαa(n)f=f and for the Riesz means limn→∞σα,γa(n)f=f for any 0αj≤1≤γj(j=1,...,d).A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Ces`aro and Riesz means of integrable functions,which was proved earlier by Weisz.     

SMALL INTO ISOMORPHISM FROM （C0） INTO C（Ω） TYPE

If T is an isomorphism of （c0） into C（Ω） （where Ω is a sequentially compact and paracompact space, or a compact metric space in particular）, which satisfies the condition ｜｜T｜｜·｜｜T^-1｜｜ ≤ 1 ＋ε for some ε ∈ （0,1/5）, then T/｜｜T｜｜ is close to an isometry with an error less than 9ε. The proof of this article is simple without using the dual space or adjoint operator.     

Pseudo Almost Automorphic Solutions for Non-autonomous Stochastic Differential Equations With Exponential Dichotomy

In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.     

ON THE ORDER OF SUMMABILITY OF THE FOURIER INVERSION FORMULA

In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.     

ON THE P1 POWELL-SABIN DIVERGENCE-FREE FINITE ELEMENT FOR THE STOKES EQUATIONS

The stability of the P1-P0 mixed-element is established on general Powell-Sabin triangular grids. The piecewise linear finite element solution approximating the velocity is divergence-free pointwise for the Stokes equations. The finite element solution approximating the pressure in the Stokes equations can be obtained as a byproduct if an iterative method is adopted for solving the discrete linear system of equations. Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the P1 Powell-Sabin divergence-free finite element method.     

拟线性系统的概周期解的存在性

<正> 考虑拟线性微分方程系 dX/dt=A(t)X十f(t)十μF(X,t,μ),(1)其中A(t)是t的n阶连续方阵,x是n向量,f(t),F(X,t,μ)是各变量的n连续向量,μ真是小参数. 当A(t)是常数方阵,f(t),F(X,t,μ)是t的一致概周期向量函数,Coddington,Levinson,等人建立了(1)的周期解的存在定理.此可参考[1]和[2].对A(t)为常数方阵,f(t),F(X,t,μ)是t的一致概周期向量函数,更进一步建立了(1)的概周期解的存在定理.     

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE WENO SCHEMES BASED ON DIFFERENT NUMERICAL FLUXES FOR THE SHALLOW WATER EQUATIONS

In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the property, and resolution of discontinuities. issues of CPU cost, accuracy, non-oscillatory     

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.     

ON THE ERROR ESTIMATES OF A NEW OPERATE SPLITTING METHOD FOR THE NAVIER-STOKES EQUATIONS

In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.     

NONLINEAR RANK-ONE MODIFICATION OF THE SYMMETRIC EIGENVALUE PROBLEM

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.     

APPLICATIONS OF THE BERNSTEIN-DURRMEYER OPERATORS IN ESTIMATING THE NORM OF MERCER KERNEL MATRICES

The paper is related to the norm estimate of Mercer kernel matrices. The lower and upper bound estimates of Rayleigh entropy numbers for some Mercer kernel matrices on [0, 1] × [0, 1] based on the Bernstein-Durrmeyer operator kernel are obtained, with which and the approximation property of the Bernstein-Durrmeyer operator the lower and upper bounds of the Rayleigh entropy number and the l2 -norm for general Mercer kernel matrices on [0, 1] x [0, 1] are provided.     

ON THE QUASI-RANDOM CHOICE METHOD FOR THE LIOUVILLE EQUATION OF GEOMETRICAL OPTICS WITH DISCONTINUOUS WAVE SPEED

We study the quasi-random choice method （QRCM） for the Liouville equation of ge- ometrical optics with discontinuous locM wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discon- tinuities, and measure-valued solutions. The so-called QRCM is a random choice method based on quasi-random sampling （a deterministic alternative to random sampling）. The method not only is viscosity-free but also provides faster convergence rate. Therefore, it is appealing for the prob!em under study which is indeed a Hamiltonian flow. Our analy- sis and computational results show that the QRCM 1） is almost first-order accurate even with the aforementioned discontinuities; 2） gives sharp resolutions for all discontinuities encountered in the problem; and 3） for measure-valued solutions, does not need the level set decomposition for finite difference/volume methods with numerical viscosities.     

ON THE SEPARABLE NONLINEAR LEAST SQUARES PROBLEMS

Separable nonlinear least squares problems are a special class of nonlinear least squares problems, where the objective functions are linear and nonlinear on different parts of variables. Such problems have broad applications in practice. Most existing algorithms for this kind of problems are derived from the variable projection method proposed by Golub and Pereyra, which utilizes the separability under a separate framework. However, the methods based on variable projection strategy would be invalid if there exist some constraints to the variables, as the real problems always do, even if the constraint is simply the ball constraint. We present a new algorithm which is based on a special approximation to the Hessian by noticing the fact that certain terms of the Hessian can be derived from the gradient. Our method maintains all the advantages of variable projection based methods, and moreover it can be combined with trust region methods easily and can be applied to general constrained separable nonlinear problems. Convergence analysis of our method is presented and numerical results are also reported.     

PRECONDITIONED HSS-LIKE ITERATIVE METHOD FOR SADDLE POINT PROBLEMS

A new HSS-like iterative method is first proposed based on HSS-like splitting of non- Hermitian （1,1） block for solving saddle point problems. The convergence analysis for the new method is given. Meanwhile, we consider the solution of saddle point systems by preconditioned Krylov subspaee method and discuss some spectral properties of the preconditioned saddle point matrices. Numerical experiments are given to validate the performances of the preconditioners.     

Three-Dimensional ℓ-Algebras

The family of all three-dimensional almost f-algebras, d-algebras, and f-algebras is constructed. It is shown that it contains all three-dimensional directly indecomposable ?-algebras. Also, a list of representatives of (algebraic) isomorphism classes of all three-dimensional algebras that can be ordered as an almost f-algebra, a d-algebra, or an f-algebra is given.     

THE GENERALIZED LOCAL HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR THE NON-HERMITIAN GENERALIZED SADDLE POINT PROBLEMS

For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 （2012） 8816-8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting （GLHSS） iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.     

GLOBAL ASYMPTOTICS TOWARD THE REREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.