特殊曲面的规范B基表示

规范B基即最优规范的全正基,因其具有凸包性、仿射不变性、最优保形性、端点插值性及B-算法等重要性质,在CAGD中起着重要的作用.本文讨论了各种规范B基曲面,包括C-Bézier曲面、H-C-Bézier曲面、C-B-Bézier曲面和H-B-Bézier曲面,并给出了摆线柱面、旋转摆线面、悬链面等超越曲面的规范B基曲面表示.     

空间圆柱螺旋线的NURBS表示

O．引言用B样条方法或B6zier方法来表示自由曲线、曲面，是在CAD／CAM技术中广泛使用的数学手段．但是由于它们都不能精确地表示除抛物线或抛物面以外的圆锥曲线与初等二次曲面，因此近年来，另一种形式的参数样条-一参数有理多项式方法占据了主导地位．非均匀有理B样条（简称NURBS）已被国际标准组织（ISO）于1991年正式颁布为关于工业产品几何定义的STEP国际标准，将其作为定义产品形状的唯一数学方法．越来越多的CAD系统采用NURBS曲线与曲面来建立图形库，研究各种曲线与曲面的NURBS表示无疑是很有意义的．在描述圆锥曲线…     

B值鞅的一致可积条件与Banach空间的几何特性

在子σ-代数列(B_n)正规情形下,本文研究了B值鞅(f_N,B_n)的一致可积性与条件 (?)λP(S~2(f)>λ)=0之间的关系,证明了当值空间同构于g凸或p光滑空间时,两者可互推。特别当值空间同构于Hilbert空间时,两者为等价。     

周期B样条基转换矩阵的表示和计算

周期B样条基以一种简洁的形式表示闭B样条曲线.周期B样条基转换矩阵为闭B样条曲线及相关曲面的不同表示间的转换提供了一个数学模型.本文给出了周期B样条基转换矩阵的存在性条件,给出并证明了周期B样条基转换矩阵的一个简单的递归表示式.在此基础上,本文进一步给出了周期B样条基转换矩阵的计算公式和高效算法.周期B样条基转换矩阵为闭B样条曲线的节点插入、升阶、节点删除和降阶等基本运算提供了一个统一而简单的解决方法,本文给出了一些应用例子.     

B样条基转换矩阵的解析表示和计算公式

摘要B样条基的转换矩阵具有重要的理论和应用意义。本文研究其最基本的问题：存在性条件、解析表示和计算方法，利用差商展开系数得到了上述问题的有关结果，本文的结果为CAGD中B样条曲线的节点插入、节点删除、升阶、降阶、分割、组合等重要技术提供了一个统一的数学背景和实现方法。     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

几何测度空间基的研究

研究了几何测度空间中的基本对称函数μ_0,μ_1,…,μ_n和内蕴体积函数V_0,V_1,…,V_n,证明了Lⁿ上连续不变赋值函数空间中由基本对称函数构成的基{μ_0,μ_1,…,μ_n}和由内蕴体积函数构成的基{V_0,V_1,…V_n}(或均质积分构成的基{W_0,W_1,…,W_n})等价.     

双指标B值强鞅与Banach空间的几何性质

本文建立了双指标Ｂ值强鞅Ｐ均方函数的某些不等式，并且给出了Ｂａｎａｃｈ空间具有Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ性质．凸性和光滑性以及ＶＭＤ性质的等价条件．     

可数基的Banach空间上的向量值函数的一些性质

用初等的方法讨论了取值于具有可数基的Banach空间上的向量值函数的连续、可导、解析、可积及解析的充要条件等性质,得出取值于可数基的Banach空间上的向量值函数解析的两个充要条件.     

线性空间的Hamel基及维数

本文综述了域上线性空间Ｈａｍｅｌ基的存在及其维数唯一性的证明, 特别地证明了 Ｃ［０,１］空间的维数为连续统基数．     

T-B样条曲线及其应用

给出一种基于三角函数的类B样条设计方法,称其为 T B样条,它具有 B样条曲线曲面的主要优点,它还能够无需有理形式即可精确表示圆弧、椭圆弧等二次曲线弧以及球面、椭球面等二次曲面片.     

一类带有给定切线多边形的样条曲线

利用带有形状参数的基函数,构造与给定切线多边形相切的样条曲线,所构造的曲线是C2和C3连续的,且对切线多边形是保形的.曲线上的所有控制点可由多边形顶点直接计算产生,曲线具有局部修改性.最后,以实例说明算法是有效的.     

Model selection by sequentially normalized least squares

Model selection by means of the predictive least squares (PLS) principle has been thoroughly studied in the context of regression model selection and autoregressive (AR) model order estimation. We introduce a new criterion based on sequentially minimized squared deviations, which are smaller than both the usual least squares and the squared prediction errors used in PLS. We also prove that our criterion has a probabilistic interpretation as a model which is asymptotically optimal within the given class of distributions by reaching the lower bound on the logarithmic prediction errors, given by the so called stochastic complexity, and approximated by BIC. This holds when the regressor (design) matrix is non-random or determined by the observed data as in AR models. The advantages of the criterion include the fact that it can be evaluated efficiently and exactly, without asymptotic approximations, and importantly, there are no adjustable hyper-parameters, which makes it applicable to both small and large amounts of data.     

Subgroups normalized by the elementary Levi subgroup

Subgroups of the unipotent radical of a maximal parabolic subgroup of a Chevalley group over a field K, which are normalized by the commutator subgroup of the Levi subgroup, are described. It is shown that in the typical case, such subgroups are in one-to-one correspondence with the closed subsets of {1, 2, ..., n} for a natural n. In the exceptional cases the classification also involves additive subgroups of K. See the table in the paper for a detailed list of possibilities. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 199–215.     

Representation of graphs by OBDDs

Recently, it has been shown in a series of works that the representation of graphs by Ordered Binary Decision Diagrams (OBDDs) often leads to good algorithmic behavior. However, the question for which graph classes an OBDD representation is advantageous, has not been investigated, yet. In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few P₄s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous with respect to space requirements.     

Representation of automata by groups

We consider the problem of representing abstract automata by finite groups. We prove the universality of the proposed representation and study its properties.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1412–1416, October, 1992.     

Representation of integral quantales by tolerances

The central result of the paper claims that every integral quantale \(\mathbf {Q}\) has a natural embedding into the quantale of complete tolerances on the underlying lattice of \(\mathbf {Q}\). As an application, we show that the underlying lattice of any finite integral quantale is distributive in 1 and dually pseudocomplemented. Besides, we exhibit relationships between several earlier results. In particular, we give an alternative approach to Valentini’s ordered sets and show how the ordered sets are related to tolerances.     

Representation of Relations by Partial Maps

With the notions of partial morphism and relation to be understood with respect to a class M of monomorphisms in a finitely complete category C, we give sufficient conditions for the graph functor Par(C) Rel(C) to admit a right adjoint. Only under an additional condition is this right adjoint given by the naturally constructed `pierced power objects".