On solution sequences of the singular Liouville equations with an integral constraint

In this paper,we analyze the blow-up behavior of sequences{uk}satisfying the following conditionsΔuk=|x|2αk V k eukinΩ,(0.1)whereΩR2,V k→V in C1,|Vk|≤A,0a≤Vk≤b,0≤αk→α∈(0,∞),andΩ|x|2αkeuk dx≤Λ1.(0.2)Furthermore,we assume that there exists someq∈(1,2)such that rq 2Br(p)|uk|qdx≤Λ2(0.3)for anyB r(p)Ω.As a result,we give a new proof of the concentration-compactness theorem for the mean feld equation.     

Weighted Estimates for the Maximal Commutator of Singular Integral Operator on Spaces of Homogeneous Type

Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associated kernel satisfies the HSlder condition on the first variable and some condition which is fairly weaker than the Holder condition on the second variable.     

NORM CALCULATIONS OF A CLASS OF INTEGRAL OPERATORS

In this paper we calculate the norm of a special class of integral operators acting on LP （C^n, dvs）, where dvs is the Gaussian measure on C^n.     

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.     

Isometric Immersions of Higher Codimension in to the Product S~κ×H~(n+p-k)

In this paper,we obtain a sufficient and necessary condition for a simply connected Riemannian manifold(M~n,g) to be isometrically immersed,as a submanifold with codimension p 1,into the product S~k×H~(n+p+k) of sphere and hyperboloid.     

Liouville type theorems for integral equations and integral systems

In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in R ^N . The main technique we use is the method of moving planes in an integral form.     

Liouville Type Theorems for Positive Solutions of Elliptic System in ℝ N

In this paper, by the Alexandrov-Serrin method of moving plane combined with integral inequalities, we prove some new Liouville type results for positive solution of semilinear elliptic system in the whole space ? ^N .     

A NON-TRIVIAL PRODUCT OF FILTRATION s＋6 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

By a method improving that of [1], the authors prove the existence of a non-trivial product of filtration, s ＋ 6, in the stable homotopy groups of sphere, πt-6S, which is represented up to non-zero scalar by β^-s＋2ho（hmbn-1 -hnbm-1） ∈ ExtA^s＋6,t＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 7, n ≥ m ＋ 2 ≥ 5, q = 2（p- 1）, 0 ≤ s 〈 p - 2, t= （s ＋ 2 ＋ （s ＋ 2）p ＋ p^m ＋ p^n）q. The advantage of this method is to extend the range of s without much complicated argument as in [1].     

A NOTE ON RICHARDSON EXTRAPOLATION OF GALERKIN METHODS FOR EIGENVALUE PROBLEMS OF FREDHOLM INTEGRAL EQUATIONS

In this paper, we introduce a new extrapolation formula by combining Richardson extrapolation and Sloan iteration algorithms. Using this extrapolation formula, we obtain some asymptotic expansions of the Galerkin finite element method for semi-simple eigenvalue problems of Fredholm integral equations of the second kind and improve the accuracy of the numerical approximations of the corresponding eigenvalues. Some numerical experiments ave carried out to demonstrate the effectiveness of our new method and to confirm our theoretical results.     

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.     

ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.     

Realization of numbers as the degrees of maps between manifolds

We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.     

SOME LIOUVILLE TYPE THEOREMS FOR THE P-SUB-LAPLACIAN ON THE GROUP OF HEISENBERG TYPE

In this paper we prove some Liouville type results for the p-sub-Laplacian on the group of Heisenberg type. A strong maximum principle and a Hopf type principle concerning p-sub-Laplacian are established.     

Symmetry and Uniqueness of Solutions of an Integral System

In this paper, we study the positive solutions for a class of integral systems and prove that all the solutions are radially symmetric and monotonically decreasing about some point. Moreover, we also obtain the uniqueness result for a special case. We use a new type of moving plane method introduced by Chen-Li-Ou [1]. Our new ingredient is the use of Hardy-Littlewood-Sobolev inequality instead of Maximum Principle.     

NON-STANDARD COMMUTATORS FOR ROUGH OSCILLATORY SINGULAR INTEGRALS

In this paper, the author studies a class of non-standard commutators with higher order remainders for oscillatory singular integral operators with phases more general than polynomials. For 1 〈 p 〈 ∞, the L^p-boundedness of such operators are obtained provided that their kernels belong to the spaces L^q（s^n-1） for some q 〉 1.     

SOME RESULTS CONCERNING GROWTH OF POLYNOMIALS

In this paper, we establish inequalities for polynomials with restricted zeros, which in particular yields interesting generalizations of some Zygmund type inequalities for polynomial.     

ON APPROXIMATION PROPERTIES OF NON-CONVOLUTION TYPE NONLINEAR INTEGRAL OPERATORS-Dedicated to Professor Paul L. Butzer on his 80th Birthday

In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]     

Liouville property for a class of quasi-harmonic sphere

In this paper we obtain a Liouville type result for a class of quasi-harmonic spheres with rotational symmetry.     

Stokes flow for the axisymmetric motion of several spherical particles perpendicular to a plane wall

The creeping flow around several spherical particles moving on a line perpendicular to a plane wall is calculated numerically using the boundary integral method. The locations of the point forces on the surfaces of the spheres are chosen so as to describe precisely the lubrication regions when the surfaces are close to one another. Earlier results are recovered for the cases of a single sphere and a wall and of two equal spheres far from a wall. New results are presented for two (equal or unequal) spheres close to a plane wall and several equal spheres far from a wall.     

Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

In this paper,we consider Li′enard systems of the form dx/dt=y,dy/dt=x+bx3-x5+ε(α+βx2+γx4)y,where b∈R,0|ε|1,(α,β,γ)∈D∈R3 and D is bounded.We prove that for |b|1(b0) the least upper bound of the number of isolated zeros of the related Abelian integrals I(h)=∮Γh(α+βx2+γx4)ydx is 2(counting the multiplicity) and this upper bound is a sharp one.