PBW-Pairs of Varieties of Linear Algebras

The notion of a Poincaré–Birkhoff–Witt (PBW)-pair of varieties of linear algebras over a field is under consideration. Examples of PBW-pairs are given. We prove that if (𝒱, 𝒲) is a PBW-pair and the variety 𝒱 is homogeneous and Schreier, then so is 𝒲; the results similar to the Schreier property for PBW-pairs are also true for the Freiheitssatz and Word problem. In particular, it follows that the Freiheitssatz is true for the varieties of Akivis and Sabinin algebras. We give also examples of varieties that do not satisfy the Freiheitssatz. It is shown that an element u of a free algebra 𝒲[X] in a homogeneous Schreier variety of algebras 𝒲 satisfying the Freiheitssatz is a primitive element (a coordinate polynomial) if and only if the factor algebra of 𝒲[X] by the ideal generated by the element u is a free algebra in 𝒲. We consider also properties of primitive elements.     

On a Method of J. H. C. Whitehead

In this article which is partly a continuation of Goldstein (1999 Goldstein , R. Z. ( 1999 ). The length and thickness of words in a free group . Proc. Amer. Math. Soc. 127 ( 10 ): 2857 – 2863 . [CSA] [CROSSREF]  [Google Scholar]), we give some necessary and sufficient conditions for an element of a free group to be primitive. For certain words x, we also give a lower bound on the length of words obtained from x by an automorphism. In the last section we give another proof of the Primitivity Theorem recently proven by Lee, using ideas from the first part of the article.     

Maximum Principle of Optimal Control of the Primitive Equations of the Ocean With State Constraint

We investigate in this paper Pontryagin's maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) for the three-dimensional Navier–Stokes (NS) equations. The main difference between this work and Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.     

Primitive roots in quadratic fields, II

We consider an analogue of Artin's primitive root conjecture for algebraic numbers which are not units in quadratic fields. Given such an algebraic number α, for a rational prime p which is inert in the field, the maximal possible order of α modulo (p) is p²−1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. We show that for any choice of 113 algebraic numbers satisfying a certain simple restriction, at least one of the algebraic numbers has order at least for infinitely many inert primes p.     

The probability of choosing primitive sets

We generalize a theorem of Nymann that the density of points in Zd that are visible from the origin is 1/ζ(d), where ζ(a) is the Riemann zeta function . A subset S⊂Zd is called primitive if it is a Z-basis for the lattice Zd∩spanR(S), or, equivalently, if S can be completed to a Z-basis of Zd. We prove that if m points in Zd are chosen uniformly and independently at random from a large box, then as the size of the box goes to infinity, the probability that the points form a primitive set approaches 1/(ζ(d)ζ(d−1)?ζ(d−m+1)).     

基于随机模糊参数预测烧结法氧化铝生产中碱液成分

针对来自许多实际过程中数据变化的多态不确定性,提出一类新的预测方法。为了反映数据的多态不确定性,设计了一种分段推断算法来对样本数据的分布函数进行推断,在假设总体服从一类模糊随机变量的条件下,由该方法得到总体的分布函数的解析式。基于这种预测方法,对烧结法氧化铝生产过程中碱液成分进行了预测。预测结果表明新方法具有广泛的应用价值。     

一种基于梯形模糊语言变量的多属性决策方法

针对属性权重未知、属性值以梯形模糊语言变量形式给出的多属性决策问题,提出了一种扩展的VIKOR决策方法。首先介绍了梯形模糊语言变量的概念、运算法则,提出了梯形模糊语言变量值之间的距离公式,并进行了证明。在此基础上利用离差最大化方法确定了属性指标权重,并建立了基于梯形模糊语言变量的扩展VIKOR方法,给出了决策步骤。最后,通过一个算例说明该方法的有效性。     

More on the identity of Chaundy and Bullard

In what follows we present a homogeneous identity which implies a more elementary treatment of the Chaundy–Bullard identity with n variables. In a different direction we bring another ramification of the Chaundy–Bullard identity.     

The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources

We catalogue the primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources. Each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph; the primitive ideals in the Cuntz–Krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group. The Cuntz–Krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic.     

本原不可幂带号简单图的基指数集的刻划

一个本原不可幂带号有向图S的基指数l(S)是指使得在S中,从任意一点u到任意一点v都有一对长为l的SSSD途径的最小整数l.本文将完全刻划n阶本原不可幂简单图的基指数集.