Polynomial-Time Computation of the Degree of a Dominant Morphism in Characteristic Zero. I

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables over the field of characteristic zero. We show how to compute the degree of a dominant rational morphism from W to W′. The morphism is given by homogeneous polynomials of degree d′.This algorithms is deterministic and polynomial in (dd′)ⁿ and the size of the input. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2003, pp. 189–235.     

Efficient Smooth Stratification of an Algebraic Variety in Zero Characteristic and Its Applications

Consider a projective algebraic variety V defined as the set of common zeros of a family of homogeneous polynomials of degree less than d in variables with coefficients from a field k of zero characteristic. We prove that V can be represented as a union (respectively, a disjoint union) of at most (respectively, ) smooth quasiprojective algebraic varieties such that the degrees of these varieties are bounded from above by , where depends only on n. We propose algorithms for constructing regular sequences and sequences of local parameters for irreducible components of V and for computing the dimension of a real variety. The complexity of these algorithms is polynomial in the size of the input and in . Bibliography: 15 titles.     

零维代数簇的分解及其应用

本文利用Groebner基，给出了一种分解零维代数簇的方法，并且讨论了这种方法在理想的准素分解以及几何定理机器证明中的应用．     

On the Algorithmic Undecidability of the Embeddability Problem for Algebraic Varieties over a Field of Characteristic Zero

Mathematical Notes -     

拓扑度计算与应用

本文运用锥理论计算一类全连续场的拓扑度,极大地减弱了非线性算子下方 有界的条件, 因而本质上改进和推广了现有结果. 最后,把抽象结果应用于研究超线 性Hammerstein积分方程非平凡解的存在性.     

近似Bayes计算前沿研究进展及应用

在大数据和人工智能时代,建立能够有效处理复杂数据的模型和算法,以从数据中获取有用的信息和知识是应用数学、统计学和计算机科学面临的共同难题.为复杂数据建立生成模型并依据这些模型进行分析和推断是解决上述难题的一种有效手段.从一种宏观的视角来看,无论是应用数学中常用的微分方程和动力系统,或是统计学中表现为概率分布的统计模型,还是机器学习领域兴起的生成对抗网络和变分自编码器,都可以看作是一种广义的生成模型.随着所处理的数据规模越来越大,结构越来越复杂,在实际问题中所需要的生成模型也变得也越来越复杂,对这些生成模型的数学结构进行精确地解析刻画变得越来越困难.如何对没有精确解析形式（或其解析形式的精确计算非常困难）的生成模型进行有效的分析和推断,逐渐成为一个十分重要的问题.起源于Bayes统计推断,近似Bayes计算是一种可以免于计算似然函数的统计推断技术,近年来在复杂统计模型和生成模型的分析和推断中发挥了重要作用.该文从经典的近似Bayes计算方法出发,对近似Bayes计算方法的前沿研究进展进行了系统的综述,并对近似Bayes计算方法在复杂数据处理中的应用前景及其和前沿人工智能方法的深刻联系进行了分析和讨论.     

灰色关联度计算的改进及其应用

本文对灰色关联度提出了一种新的改进方法，改进的关联度计算不仅具有新的特性，而且能一次反映正、反相关因素，并以实例说明该方法的应用。     

Cohomology of Real Algebraic Varieties

A class of effective spaces with involution introduced by the author is studied. The cohomology ring of the fixed-point set of an effective space is completely determined by the spectral sequence of involution. Real algebraic varieties admitting a “cellular decomposition” are effective M-spaces. Under certain restrictions, one calculates the spectral sequence of involution and the total ℤ₂ Betti number of the real part for real subvarieties of real algebraic varieties that are effective GM-spaces. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2002, pp. 112–151.     

Applications of Algebraic Combinatorics to Algebraic Geometry

We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.     

On Chevalley Restriction Theorem for Semi-reductive Algebraic Groups and Its Applications

Acta Mathematica Sinica, English Series - An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive...     

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.     

Families of Algebraic Varieties and Towers of Algebraic Curves over Finite Fields

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower.     

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.     

非锥映射的不动点指数计算及其应用

本文应用锥理论获得了一类不是锥映射的非线性算子不动点指数计算的若干结果,并将所得结果应用于非线性椭圆型偏微分方程边值问题.     

层次分析法权重计算方法分析及其应用研究

介绍层次分析法的基本概念,同时也分析了层次分析法权重的计算方法及应用,层次分析法的计算方法有四种方法:几何平均法、算术平均法、特征向量法、最小二乘法,以往的文献利用层次分析法解决实际问题时,都是采用其中的某一种方法求权重向量,不同的方法会导致结果有些偏差,将对一具体实例,采用四种计算方法分别得出权重向量再进行排序和综合分析,这样可以避免采用单一方法所产生的偏差,得出的结论将更全面、更有效。     

Oriented Cohomology Theories of Algebraic Varieties

This article contains proofs of the results announced by Panin and Smirnov (http://www. math.uiuc.edu/k-theory/0459/2000) in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one correspondences between orientations, Chern structures and Thom structures on a given ring cohomology theory. The theory is illustrated by motivic cohomology, algebraic K-theory, algebraic cobordism theory and by other examples.     

Boolean Metric Spaces and Boolean Algebraic Varieties

Abstract

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps A ⁿ ?→?A ^m in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e., sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces.     

矩阵型灰色关联度的特征检验方法及其应用

针对现有灰色关联理论的不足, 本文提出灰色关联的特征检验思路, 构建了矩阵型灰色关联度的特征检验方法。首先优化矩阵型灰色关联模型, 利用行为矩阵差值定义特征差异矩阵, 采用矩阵2范数构建关联度公式。而后分析特征差异矩阵的稳定性与趋势性, 利用变异系数形式构建稳定性系数, 利用最小二乘法估计趋势性系数, 两者共同组成矩阵型灰色关联度的特征检验方法。最后, 本文模型被应用于湖北省恩施州的长期多维贫困分析, 在与现有模型的比较中, 发现关联度评估结果有效区分了恩施州8个市县的贫困情况, 特征检验方法从贫困不确定性和趋势性两方面对结果进行补充, 验证了模型的可行性与实用性。     

代数构造及在Ramsey理论中的应用

本文在Galois域上的代数构造和关于一些特定类型图的Ramseyr数之间建立了一个关系．关键问题是研究了关于Galois域上的代数构造的方程及方程组的解．我们得到了一些关于二部图的新的下界和上界．