Parabolic Equations with Measurable Coefficients

We investigate the unique solvability of second order parabolic equations in non-divergence form in , p ≥ 2. The leading coefficients are only measurable in either one spatial variable or time and one spatial variable. In addition, they are VMO (vanishing mean oscillation) with respect to the remaining variables. The second author was partially supported by NSF Grant DMS-0140405.     

Oscillatory Riemann-Hilbert problems and the corona theorem

The paper is devoted to the Riemann-Hilbert problem with matrix coefficient having in Hardy spaces [H_p^±]²,1<p?∞, on half-planes . Under the assumption of existence of a non-trivial solution of corresponding homogeneous Riemann-Hilbert problem in [H_∞^±]² we study the solvability of the non-homogeneous Riemann-Hilbert problem in [H_p^±]²,1<p<∞, and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for G as well as explicit formulas for its factors in terms of solutions of two associated corona problems (in and ). A separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in H_∞⁺ are obtained. Making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2×2 matrix functions whose diagonal entries do not possess factorizations. Such matrices arise, e.g., in the theory of convolution type equations on finite intervals.     

论中值定理类命题证明中的辅助函数构造

借助实例分析的方法,讨论在证明微分与积分相结合的中值定理类命题时,关于辅助函数的构造技巧及其变形思想.     

k-素数和唯一分解*

在本文中,作者揭示了唯一k-素因数分解的更深层原因.在第二节中,首先引入S_k中的k-组合条件和费马定理;并证明了下面4论断是等价的:(1) k-组合条件成立,(2)中唯一k-素因数分解成立,(3) Sk中费马定理成立,(4)k=1或2.为了更好地理解k-素数,在第三节中作者考察了一类特殊的k-素数,即3-素数.众所周知唯一3-素因数分解一般是不成立的,那么S₃中的哪些正整数具有唯一3-素因数分解性质呢?在第三节中,作者得到一个S₃中的整数具有唯一3-素因数分解的充要条件.在第三节最后,作者引入π₃(x),它表示小于等于x的3-素数个数.由素数定理,作者得到π₃(x)的一个具体公式以及一些近似公式.     

On a new mean and functions of bounded deviation

The main purpose of the present paper is to introduce a new type of mean of a function and to study some of its special properties. In the last section we make use of this mean to show that a function of bounded deviation is not necessarily a function of bounded variation.     

Orlicz-Hardy spaces and their duals

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H(Rn)in that the class of admissible functions is largely widened.We can deal with,for example,(r)≡(rp1(log(e+1/r))q1,0r 6 1,rp2(log(e+r))q2,r1,with p1,p2∈(0,∞)and q1,q2∈(.∞,∞),where we shall establish the boundedness of the Riesz transforms on H(Rn).In particular,is neither convex nor concave when 0p11p2∞,0p21p1∞or p1=p2=1 and q1,q20.If(r)≡r(log(e+r))q,then H(Rn)=H(logH)q(Rn).We shall also establish the boundedness of the fractional integral operators I of order∈(0,∞).For example,I is shown to be bounded from H(logH)1./n(Rn)to Ln/(n.)(log L)(Rn)for 0n.     

Riesz rising sun lemma for several variables and the John-Nirenberg inequality

We obtain a multidimensional analog of the well-known Riesz rising sun lemma. We prove a more precise version of this lemma for space dimension d = 2 . We use these lemmas to establish an anisotropic analog of the John-Nirenberg inequality for functions of bounded mean oscillation with an exact constant in the exponent. Earlier, this exact constant was only known in the one-dimensional case.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 53–66.Original Russian Text Copyright © 2005 by A. A. Korenovskii.This revised version was published online in April 2005 with a corrected issue number.     

Brill-Noether theory for rank two vector bundles generated by their sections

We here study the Brill-Noether theory for rank two vector bundles generated by their sections. We generalize the vanishing theorem, the Clifford theorem and the existence theorem to such bundles.     

ON THE SENSITIVITY OF THE LDU FACTORIZATION

1　IntroductionG.W.Stewart,X.-W.Chang and A.Barralund etc.have done lot of perturbationanalyses on LU,QR and Cholesky factorizations( see[5] ,[6] ,[1 ] ,[2 ] ,[4 ] ,[3 ] ) .We givesensitivity analysisfor LDU factorization in this paper.The LDU factorization ( or LDMT) is an important method in numerical linear algebra( see [7] ) ,and can be viewed as the general form for LDLT factorization. The differencebetween LDU and LU factorizations in upper triangular matrix U,i.e. U is uni…     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

Remarks on the Defining Equations of Smooth Threefolds in P5

In this paper, we prove the vanishing of the first cohomology groups of ideal sheaves twisted by small integers for smooth threefolds in P⁵ defined over the complex number field C by using Le Potier vanishing theorem and Castelnuovo regularity for vector bundles, which contributes Peskine–Van de Ven conjecture for 2-normality.     

一个微分中值命题条件的弱化

讨论一个微分中值命题条件的弱化,将条件“f′（x）g′（x）〉0”弱化为“f（a）≠f（b）”,利用介值定理和柯西中值定理给出证明,以扩大命题的适用范围,并举出实例予以说明．     

The algebraic dimension of compact complex threefolds with vanishing second Betti number

This note investigates compact complex manifolds X of dimension 3 with second Betti number b₂(X) = 0. If X admits a non-constant meromorphic function, then we prove that either b₁(X) = 1 and b₃(X) = 0 or that b₁(X) = 0 and b₃(X) = 2. The main idea is to show that c₃(X) = 0 by means of a vanishing theorem for generic line bundles on X. As a consequence a compact complex threefold homeomorphic to the 6-sphere S⁶ cannot admit a non-constant meromorphic function. Furthermore we investigate the structure of threefolds with b₂(X) = 0 and algebraic dimension 1, in the case when the algebraic reduction X P₁ is holomorphic.