The Exceptional Set in Hua＇s Theorem for Three Squares of Primes

It is proved that with at most O(N^11/12 c) exceptions, all positive integers n ≤ N satisfying some necessary congruence conditions are the sum of three squares of primes. This improves substantially the previous results in this direction.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

Jackson＇s Theorem on Bounded Symmetric Domains

Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson＇s theorem to several complex variables：Eκ（f,X）≤ω（1/k,f,X）, where Eκ（f,X） is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω（1/k,f,X） is the corresponding modulus of continuity.     

Selberg＇s Normal Density Theorem for Automorphic L-Functions for GLm

Let π be an irreducible unitary cuspidal representation of GLm（AQ） with m ≥ 2, and L（s, Tr） the L-function attached to π. Under the Generalized Riemann Hypothesis for L（s,π）, we estimate the normal density of primes in short intervals for the automorphic L-function L（s, π）. Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.     

ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS

This paper discusses the accelerating of nonlinear parabolic equations. Two iterative methods for solving the implicit scheme new nonlinear iterative methods named by the implicit-explicit quasi-Newton （IEQN） method and the derivative free implicit-explicit quasi-Newton （DFIEQN） method are introduced, in which the resulting linear equations from the linearization can preserve the parabolic characteristics of the original partial differential equations. It is proved that the iterative sequence of the iteration method can converge to the solution of the implicit scheme quadratically. Moreover, compared with the Jacobian Free Newton-Krylov （JFNK） method, the DFIEQN method has some advantages, e.g., its implementation is easy, and it gives a linear algebraic system with an explicit coefficient matrix, so that the linear （inner） iteration is not restricted to the Krylov method. Computational results by the IEQN, DFIEQN, JFNK and Picard iteration methods are presented in confirmation of the theory and comparison of the performance of these methods.     

Jungck＇s Common Fixed Point Theorem and E.A Property

We prove that （E.A） property buys the required containment of range of one mapping into the range of other in common fixed point considerations up to a pair of mappings. While proving our results, we utilize the idea of implicit functions due to Popa, keeping in view their unifying power.     

A Remark on Chen’s Theorem (II)

Let p denote a prime and P2 denote an almost prime with at most two prime factors. The author proves that for sufficiently large x,∑p≤x p＋2=P2 1〉1.13Cx/log^2x,where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.     

A Note on Donaldson＇s ＂Tamed to Compatible＂, Question

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson＇s ＂tamed to compatible＂ question （Draghici T, Zhang W. A note on exact forms on almost complex manifolds, arXiv： 1111. 7287vl [math. SC]. Submitted on 30 Nov. 2011）. That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure？ They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson＇s question in dimension 4. In this note, we give another simpler proof of their result.