软代数2~X的特征

本文通过讨论有限偏序集Ｘ与基数幂２Ｘ的关系得到２Ｘ可定义成软代数的充要条件是Ｘ可定义成软偏序集．且软代数２Ｘ的型数与软偏序集Ｘ的型数相同．     

扩张原理的粗集表示

粗集、模糊集均是处理不确定信息的数据分析工具,是数据挖掘的重要方法.由Zadeh首先提出的模糊扩张原理是模糊集理论的最基本的原理之一,粗集是通过上、下近似算子来发挥作用的.本文讨论扩张原理与粗集上近似之间的关系,证明了扩张原理可以表示成粗集上近似的形式,因此,扩张原理成了粗集与模糊集之间的桥梁.此外,借助粗集上、下近似算子的公理系统解决了扩张原理的反问题.     

一类极小本原对称图的指数集

运用有向图方法完全确定出顶点带环的n阶极小本原对称有向图的本原指数集,所得的结论是:1)顶点全部自带环的n阶极小本原对称有向图所成的子图类之本原指数集E1={2,3,…,n-1};2)顶点不全带环的n阶极小本原对称有向图所成的子图类之本原指数集E2={2,3,…,2n-2}\S,其中S是{n,n+1,…,2n-2}中的所有奇数之集;3)顶点带环的n阶极小本原对称有向图所成的特殊图类之本原指数集En=E1∪E2={2,3,…,2n-2}\S.     

自反Banach空间中锥线性优化问题初始——对偶目标函数水平集的几何性质

讨论自反Banach空间中的原——对偶锥线性优化问题的目标函数水平集的几何性质．在自反Banach空间中,证明了原目标函数水平集的最大模与对偶目标函数水平集的最大内切球半径几乎是成反比例的．     

集值测度的积分表示

Ｆ．Ｈｉａｉ在集值测度有界变差的条件下，给出了Ａｕｍａｎｎ积分的Ｒ－Ｎ性质，本文在集值测度为σ－有界变差的条件下，建立了集值测度表示成Ｐｅｔｔｉｓ－Ａｕｍａｎｎ积分的充要条件。     

完备格中的成分理论

在完备格中引入了元素的成分概念．基于此,引入了元素的宽度的概念．在分配格的情形证明了元素的成分集对有限并运算封闭且有某种遗传性．证明了元素的有限宽度的成分之集是定向集．称没有非平凡成分的元素为颗粒．称每个非零元素都可表示为其颗粒成分之并的完备格为颗粒表示格．证明了拓扑空间是局部连通的充要条件是其开集格为颗粒表示格．     

一个本原矩阵类的指数集的完全刻划

本文证明了至少有一对非零对称元但非对称的ｎ（＞５）阶本原矩阵所成的类的指数集是：（１）迹非零时，无论ｎ（＞５）是奇数还是偶数，都有指数集．（２）迹为零时，（ｉ）若ｎ（＞５）是奇数，则指数集（ｉｉ）若ｎ（＞５）是偶数，则指数集．     

不允许卖空证券组合选择的有效子集

证券组合选择的有效子集是指它可取代原有的基本证券集来生成Markowits有效组合前沿.本文给出一个证券集的子集在不允许卖空的条件下是全集的有效子集的充要条件。     

内部锥-凸集值映射Henig有效性的Morea-Rockafellar定理

本文利用集值映射弱次梯度的Morea-Rockafellar定理,在内部(锥)-凸性假设下,得到了集值映射关于Henig有效性的Morea-Rockafellar定理.其结论为:在内部(锥)-凸条件下,两个集值映射和的Henig有效次梯度可以表示成它们Henig有效次梯度的和.     

启发法及其设计——综合报告(三)

4.3 确定潜在的和被考虑过的候选者所成的集当一个候选者(解状态,参考§3.2)已经选定之后,就需要去决定它的邻域(新的候选者所成的集).通常不会把单个的新的潜在候选者(邻域1)一一明确列出,它们只是被考虑的候选者所成的集的一个框架(邻域2).确定(邻域1及)邻域2的规则对于启发法来说特别重要,但很难加以规定,因为许多细节必须予以考虑.     

Determining a Rotation of a Tetrahedron from a Projection

The following problem, arising from medical imaging, is addressed: Suppose that T is a known tetrahedron in ?³ with centroid at the origin. Also known is the orthogonal projection U of the vertices of the image ?T of T under an unknown rotation ? about the origin. Under what circumstances can ? be determined from T and U?     

Symmetry-based solution of a model for a combination of a risky investment and a riskless investment

Benth and Karlsen [F.E. Benth, K.H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stoch. Anal. Appl. 23 (2005) 687-704] treated a problem of the optimisation of the selection of a portfolio based upon the Schwartz mean-reversion model. The resulting Hamilton-Jacobi-Bellman equation in 1+2 dimensions is quite nonlinear. The solution obtained by Benth and Karlsen was very ingenious. We provide a solution of the problem based on the application of the Lie theory of continuous groups to the partial differential equation and its associated boundary and terminal conditions.     

Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t_ii are arranged in increasing order. To avoid calculations using the differences t_ii ? t_jj with close (including equal) t_ii and t_jj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

     

On a mapping of a projective space into a sphere

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.     

Folding a surface to a polygon

A concept of folding for compact connected surfaces, involving the partition of the surface into combinatorially identical n-sided topological polygons, is defined. The existence of such foldings for given n and given surfaces is explored, with definitive results for the sphere and the torus. We obtain necessary conditions for the existence of such foldings in all other cases.Supported by Kuwait University Grant SM 043.     

Approaching a vertex in a shrinking domain under a nonlinear flow

We consider here the homogeneous Dirichlet problem for the equation , in a noncylindrical domain in space-time given by . By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x=0, t=T, in the three different cases p>1/2, p=1/2(vertex regular), p<1/2 (vertex irregular).