On the generalized Busemann-Petty problem

The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ℝ ⁿ with a larger volume of all i-dimensional sections necessarily have a larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i > 3. The problem is still open for i = 2, 3. In this article we prove two specific affirmative answers to the generalized Busemann-Petty problem if the body with a smaller i-dimensional volume belongs to given classes. Our results generalize Zhang’s specific affirmative answer to the generalized Busemann-Petty problem. This work was supported, in part, by the National Natural Science Foundation of China (Grant No. 10671117)     

Busemann-Petty问题的对偶均质积分形式

本文主要建立了凸体几何中Busemann-Petty问题的一个对偶均质积分形式,并将Funk截面定理推广到了对偶均质积分形式．     

Busemann–Petty Problems for General L_p-Intersection Bodies

For 0 p 1, Haberl and Ludwig defined the notions of symmetric L p-intersection body and nonsymmetric L p-intersection body. In this paper, we introduce the general L p-intersection bodies.Furthermore, the Busemann–Petty problems for the general L p-intersection bodies are shown.     

Gaussian measure of sections of convex bodies

In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian measure of its sections. Using this formula we solve an analog of the Busemann-Petty problem for Gaussian measures.     

On an analytic generalization of the Busemann-Petty problem

In this paper, we establish an extension of the connections between an analytic generalization of the Busemann-Petty problem and the positive definite distributions. Our results show that the structure of the positive definite distributions in Rn is closely related to the analytic generalization of the Busemann-Petty problem which was posed by Koldobsky.     

Non-uniqueness of the reconstruction for connected and simply connected sets in the plane by their fixed finite projections

We discuss a problem to reconstruct the measurable sets in the plane from their fixed finite projections. In the main theorem, we construct an example of connected and simply connected polygons which are not uniquely reconstructed by their fixed finite projections. We also make a comparison between our main theorem and the known results on this problem.     

Classical Solution to the Electropainting Problem

The mathematical modelling of the electrodeposition phenomenon leads to a linear elliptic partial differential equation subject to nonlinear evolutionary mixed boundary conditions. ln this paper, the existence, uniqueness and regularity of classical solution are proved for the electropainting problem when “dissolution current” is zero.     

NON-UNIQUENESS OF THE RECONSTRUCTION FOR CONNECTED AND SIMPLY CONNECTED SETS IN THE PLANE BY THEIR FIXED FINITE PROJECTIONS

We discuss a problem to reconstruct the measurable sets in the plane from their fixed finite projections. In the main theorem, we construct an example of connected and simply connected polygons which are not uniquely reconstructed by their fixed finite projections. We also make a comparison between our main theorem and the known results on this problem.     

相同装卸工情况下装卸工问题的最优解

装卸工问题是一个新的NP困难的组合最优化问题,寻找其性能优良的近似算法是有重要的理论意义和实用价值的.相同装卸工情况下装卸工问题的系数矩阵是全么模矩阵,利用全么模矩阵的性质可以证明这种情况下的装卸工问题是多项式可解的.然而用全么模阵的性质还不能得到解的表达式.对这种情况下一辆货车的装卸工问题,用对偶单纯形法可得到最优解和最优值的解析表达式,从而可以把这个可解问题的最优值作为一般装卸工问题的近似值.这对于分析近似算法的性态是非常重要的.     

From Fibring to Cryptofibring. A Solution to the Collapsing Problem

The semantic collapse problem is perhaps the main difficulty associated to the very powerful mechanism for combining logics known as fibring. In this paper we propose cryptofibred semantics as a generalization of fibred semantics, and show that it provides a solution to the collapsing problem. In particular, given that the collapsing problem is a special case of failure of conservativeness, we formulate and prove a sufficient condition for cryptofibring to yield a conservative extension of the logics being combined. For illustration, we revisit the example of combining intuitionistic and classical propositional logics. Mathematics Subject Classification (2000): 03B22 (03B35, 03G25, 03G30)     

Regularity of the Solution to the Dirichlet Problem in Morrey Spaces

The aim of this paper is to study Morrey regularity of the solution to the Dirichlet problem for second order elliptic equation of the form Lu = -(a_{ij}u_x_i + b_j_u)_x_j + du = e + (f_j)x_j in a bounded open subset of Rn(n ≥ 3), where b_j , d, e, f_j are assumed to be in some Morrey spaces.     

Looking for Selectors of Star Bodies

Let be a family of star bodies in R ⁿ (compact bodies in R ⁿ with nonempty kernels). A function s: R ⁿ is a selector for provided that s(A) ker A for every A . To every star body A and every : [0,) [0,) we assign a function _A: ker A R defined in terms of the radial function of translates of A. We prove that if A is convex and is concave and strictly increasing, then _A has a unique maximizer, which is referred to as the radial center of A induced by (Theorem 3.1). We extend the radial center map over some family of star bodies (Theorem 4.2). Further, we define a suitable metric, _st , the star metric, on the family of all the compact star sets in R ⁿ . This new metric is topologically stronger than the Hausdorff metric (Theorem 5.7). We study the continuity of our selectors with respect to _st .     

Stable Determination of Convex Bo dies from Intersection Bo dies

In this paper, by means of the dual Brunn-Minkowski theories and methods as well as the integral transform, we have established a stability version of nonsymmetric convex bodies from intersection bodies. In addition, a stability results of symmetric convex bodies from Lp-counterparts is established.     

平衡问题解的存在性

主要研究平衡问题解的存在性.通过对目标函数和可行集合的渐近分析,给出拟单调平衡问题解集非空的条件.进而用类似的方法研究了向量平衡问题解存在的条件,并将其应用到向量优化问题上.     

凸可行问题的块迭代次梯度投影算法

本文,针对由非线性不等式系统构成的凸可行问题,提出了序列块迭代次梯度投影算法和平行块迭代次梯度投影算法.将非线性不等式系统分成若干个子系统,然后将当前迭代点在子系统各个子集上的次梯度投影的凸组合作为当前迭代点在这个子系统上的近似投影.在较弱条件下证明了两种算法的收敛性.     

求线性规划对偶问题最优解的一种方法

线性规划对偶问题的最优解有重要的经济意义,中给出了一种较为简捷的求对偶问题最优解的方法。     

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.     

凸可行问题的一种强收敛算法

无限维Hilbert空间中,解凸可行问题的平行投影算法通常是弱收敛的.本文对一般的平行投影算法进行改进,设计了一种解凸可行问题的具有强收敛性的新算法.该算法主要是在原有算法基础上引入了一个参数序列,在参数序列满足一定的控制条件下保证了算法的强收敛性.为了简单证明算法的强收敛性,我们构建了一个新的积空间,然后把原空间的这种改进平行投影算法转换为积空间中的交替投影算法.这样,改进的平行投影算法的强收敛性就可以通过交替投影算法的收敛性证明得到.     

中国邮路问题的一个解法

本给出中国邮路问题的一个解法。虽然它仍只是初始方案，但因考虑了问题的足够信息，因而这种方案至少拉近了与最优解的距离。     

不要把问题解决教学课上成问题解答介绍课——兼评分部积分法的两种不同的教学方法

结合数学分析中分部积分法这一内容的两种不同的教学方法,论述了在问题解决教学中切忌把问题解决教学课上成问题解答介绍课.