零维多项式理想的分解算法

本文给出了一种不全用线性的变换和ｖ．ｄ．Ｗａｅｒｄｅｎ的指数方法的零维多项式理想准素分解的新方法：     

参系数零维分片代数簇的实零点

分片代数簇是一些多元样条函数的公共零点集. 文中表明: 解参系数分片代数簇问题可转化为解有限个包含严格不等式的参系数多项式系统. 利用半代数系统的正则分解和柱形代数分解方法, 提出了计算零维参系数分片代数簇无挠实零点数的上确界, 以及达到上确界时实零点在各个胞腔内的数目分布情形的算法. 该算法同时能产生达到上确界的充要条件, 以及达到上确界时实零点数在各个$n$维胞腔内 取得某种分布的充要条件. 也给出了另一算法, 用于产生零维分片代数簇在$n$维复形中的 各个$n$维胞腔内恰有指定数目的相异无挠实零点的充分必要条件.     

多项式代数与半群代数中Groebner-基的关系

本文讨论了多项式代数的理想的Groebner-基与半群代数中Groebner-基的关系，并得到一个转换定理。     

中心为4的8维幂零李代数的分类

In this paper, we give a complete classification of eight dimensional nilpotent Lie algebras with four-dimensional center by using the method of Skjelbred and Sund.     

低维幂零李代数的Yang-Baxter算子

在特征零的代数闭域F上,利用3维和4维幂零李代数的分类,通过计算刻画了3维和4维幂零李代数的Yang-Baxter算子.     

在环面同态作用下零维理想的性质

本文以矩阵变换和整理论为工具，研究了在环面同态作用下零维理想的性质，证明了零维理想的环面同态矩阵是满秩的，并且刻画了零维理想在环面作用下的对应关系．     

代数簇无穷小邻域的上同调

本文证明了无穷小邻域微分形式 Q_((A/R),n)和 Q_((A/R),n)的基本性质.建立了代数闭域上射影簇无穷小邻域对偶性定理和完备正规曲线无穷小邻域上高阶Rieman-Roch 定理.     

高维代数簇计算的一些结果

本文在［１］的基础上给出计算高维代数簇的特征值方法,进一步结果,得到了不可约集的母点和对应于极大无关变元集的多项式方程组的孤立解之间对应关系的进一步刻画．     

幂零表代数

本文研究了幂零表代数的一个有趣的性质,利用表代数的Jordan-Hlder型定理,证明了表代数满足幂零被幂零扩张仍是幂零的,但有限幂零群没有这样的扩张.     

On Homotopy Groups of Real Algebraic Varieties and their Complexifications

Let X ₀ be a topological component of a nonsingular real algebraic variety and i:X → X _C is a nonsingular projective complexification of X. In this paper, we will study the homomorphism on homotopy groups induced by the inclusion map i:X ₀ → X _C and obtain several results using rational homotopy theory and other standard tools of homotopy theory.     

The correspondence between multivariate spline ideals and piecewise algebraic varieties

As a piecewise polynomial with a certain smoothness, the spline plays an important role in computational geometry. The algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a generalization of the algebraic variety. In this paper, the correspondence between piecewise algebraic varieties and spline ideals is discussed. Furthermore, Hilbert’s Nullstellensatz for the piecewise algebraic variety is also studied.     

Algebraic Graph Theory for Sparse Flexibility Matrices

An efficient algorithm is presented for the formation of suboptimal cycle bases of graphs corresponding to sparse cycle adjacency matrices, leading to the formation of highly sparse flexibility matrices. The algorithm presented employs concepts from algebraic graph theory together with a Greedy-type algorithm to select cycles with small overlaps and uses a simple graph-theoretical method for controlling the independence of the selected cycles.     

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

K-Theory of Algebraic Tori and Toric Varieties

It is proved that under certain conditions the group K _n (X) of a smooth projective variety X over a field F is a natural direct summand of K _n (A) for some separable F-algebra A. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the groupK ₀(T) for an algebraic torus T is given.     

Puiseux Expansions and Nonisolated Points in Algebraic Varieties

We consider the problem of deciding whether a common solution to a multivariate polynomial equation system is isolated or not. We present conditions on a given truncated Puiseux series vector centered at the point ensuring that it is not isolated. In addition, in the case that the set of all common solutions of the system has dimension 1, we obtain further conditions specifying to what extent the given vector of truncated Puiseux series coincides with the initial part of a parametrization of a curve of solutions passing through the point.     

A Construction Method for Albert Algebras over Algebraic Varieties

We construct cubic Jordan algebras over an integral proper scheme X such that 2, 3 ∈ H ⁰(X, 𝒪 _X ), generalizing a construction by B. N. Allison and J. R. Faulkner. In the process, we obtain admissible cubic algebras and pseudocomposition algebras over X. Results on the structure of these algebras are obtained, as well as examples over elliptic curves.     

A Projected Algebraic Multigrid Method for Linear Complementarity Problems

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial engineering.     

On Identity Bases of Exclusion Varieties for Monoids

An exclusion variety is a variety that is maximal with respect to not containing some semigroup. It is shown that if the bases of all exclusion varieties for some periodic semigroup S are known, then the bases of exclusion varieties for the monoid S ¹ can be computed.     

一类代数系统及其应用

本文利用已有的loop代数$\widetilde{A}_{1}$构造出代数系统$X$,然后建立了一个新的等谱问题得到著名的Volterra lattice可积系,最后通过构造出的$X$的扩展代数系统$\widetilde{X}$得到已有的可积系的可积耦合系统.