Structure of weak Descartes systems

A weak Descartes system is a basis of functions such that every ordered subset is a weak Tchebycheff system, the canonical example being the usual spline basis involving truncated power functions. By examining the intervals of degeneracy for a WD-system, we show that it is possible to produce a new basis that has a simple and convenient structure similar to the spline basis.     

Minimal condition for shortest vectors in lattices of low dimension

For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG⁺, for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum.     

欧氏空间中等角基的性质

等角基是正交基的推广,等角基具有和正交基相似的性质,因此研究等角基的性质能够为研究欧氏空间提供一种工具,加深对欧氏空间的了解.本文主要把n维欧氏空间中正交基的一些性质推广到等角基上,得到了五个关于等角基性质的定理.     

欧氏空间中等角基的性质

等角基是正交基的推广,等角基具有和正交基相似的性质,因此研究等角基的性质能够为研究欧氏空间提供一种工具,加深对欧氏空间的了解．本文主要把n维欧氏空间中正交基的一些性质推广到等角基上,得到了五个关于等角基性质的定理．     

Construction of orthonormal bases in higher symmetry classes of tensors

We present a method for constructing an orthonormal basis for a symmetry class of tensors from an orthonormal basis of the underlying vector space. The basis so obtained is not composed of decomposable symmetrized tensors. Indeed, we show that, for symmetry classes of tensors whose associated character has degree higher than one, it is impossible to construct an orthogonal basis of decomposable symmetrized tensors from any basis of the underlying vector space. We end with an open problem on the possibility of a symmetry class having an orthonormal basis of decomposable symmetrized tensors.     

关于4阶极小渐近基的一个结果

设h≥2,若h阶渐近基A的任一真子集均不是h阶渐近基,则称集合A是自然数集N的h阶极小渐近基.为进一步刻画渐近基与极小渐近基之间的关系,本文综合运用自然数的b进制表示理论及分类讨论的方法,证明了存在一个集合是4阶渐近基且其任何子集均不是4阶极小渐近基.     

Basis properties and algorithmic specializations for gub transformed networks

Applying standard transformations of generalized upper bounding (GUB) theory to a pure or generalized network basis is shown to yield a reduced working basis that is itself a basis for a reduced network. As a result, the working basis can be represented via specialized data structures for networks. The resultant GUB based specializations to the network simplex algorithm are described.     

线性规划的目标函数最速递减算法

在对偶单纯形方法的基础上，提出了线性规划的目标函数最速递减算法。它避开求初始可行基或初始基，以目标函数全局快速递减作为选基准则，将选基过程与换基迭代合二为一，从而大大减少了迭代次数。数值算例显示了该算法的有效性和优越性。     

On closed subspaces with Schauder bases in non-archimedean Fréchet spaces

The main purpose of this paper is to prove that a non-archimedean Fréchet space of countable type is normable (respectively nuclear; reflexive; a Montel space) if and only if any its closed subspace with a Schauder basis is normable (respectively nuclear; reflexive; a Montel space). It is also shown that any Schauder basis in a non-normable non-archimedean Fréchet space has a block basic sequence whose closed linear span is nuclear. It follows that any non-normable non-archimedean Fréchet space contains an infinite-dimensional nuclear closed subspace with a Schauder basis. Moreover, it is proved that a non-archimedean Fréchet space E with a Schauder basis contains an infinite-dimensional complemented nuclear closed subspace with a Schauder basis if and only if any Schauder basis in E has a subsequence whose closed linear span is nuclear.     

A Newton basis for Kernel spaces

It is well known that representations of kernel-based approximants in terms of the standard basis of translated kernels are notoriously unstable. To come up with a more useful basis, we adopt the strategy known from Newton’s interpolation formula, using generalized divided differences and a recursively computable set of basis functions vanishing at increasingly many data points. The resulting basis turns out to be orthogonal in the Hilbert space in which the kernel is reproducing, and under certain assumptions it is complete and allows convergent expansions of functions into series of interpolants. Some numerical examples show that the Newton basis is much more stable than the standard basis of kernel translates.     

Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups

Young’s orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young’s orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software.     

Minimal bases and maximal nonbases in additive number theory

An asymptotic basis $\text{A}$ of order h is minimal if no proper subset of $\text{A}$ is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal.If $\text{B}$ is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of $\text{B}$ is a basis (resp. asymptotic basis) of order h, then $\text{B}$ is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.     

Bases in semilinear spaces over zerosumfree semirings

This paper investigates the cardinality of a basis and the characterizations of a basis in semilinear space of n-dimensional vectors over zerosumfree semirings. First, it discusses the cardinality of a basis and gives some necessary and sufficient conditions that each basis has the same number of elements. Then it presents some conditions that a set of vectors is a basis and that a set of linearly independent vectors can be extended to a basis. In the end, it shows a necessary and sufficient condition that two semilinear spaces are isomorphic.     

三次数域的整基和幂元整基

如何有效地找出数域的整基，是人们长期思考的问题.本文给出了求三次数域的整基的一个简单方法.此外，人们还对数域是否存在幂元整基的问题感兴趣.二次域和分圆域均存在幂元整基，而对于三次数域情形，则尚不清楚.本文给出了三次数域存在幂元整基的充要条件，从而对于三次数域情形给出了完整的答案.     

Computing border bases

This paper presents several algorithms that compute border bases of a zero-dimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zero-dimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degree-compatible term ordering σ and hence compute a border basis that contains the reduced σ-Gröbner basis. We show an example in which this computation actually has advantages over Buchberger's algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.     

Matroid basis graphs. I

A matroid may be defined as a collection of sets, called bases, which satisfy a certain exchange axiom. The basis graph of a matroid has a vertex for each basis and an edge for each pair of bases that differ by the exchange of a single pair of elements. Two characterizations of basis graphs are obtained. The first involves certain local subgraphs and how they lie when the given graph is leveled with respect to distance from a particular vertex. The second involves the existence of a special mapping from the given graph to some “full” basis graph. It is also shown that in a natural sense all basis graphs are homotopically trivial.     

On a Question of Haskell P. Rosenthal Concerning a Characterization of c0 and lp

The following property of a normalized basis in a Banach spaceis considered: any normalized block sequence of the basis hasa subsequence equivalent to the basis. Under uniformity or othernatural assumptions, a basis with this property is equivalentto the unit vector basis of c₀ or l_p. An analogous problem concerningspreading models is also addressed. 2000 Mathematics SubjectClassification 46B20 (primary), 46B15 (secondary).     

P_n(Γ)一组Hilbert基的判定和计算

设Γ是一作用在Rn上的紧李群,Pn(Γ)是Γ不变的多项式芽环,Hilbert-Weyl定理证明了对于Pn(Γ)总存在一组由Γ不变的齐次多项式芽构成的Hilbert基.然而,如何从Γ不变的齐次多项式芽中选出一组Hilbert基?如何判定Γ不变的齐次多项式芽的一个有限集就是Pn(Γ)的一组Hilbert基?在有关的文献中,Pn(Γ)的一组Hilbert基常常是通过幂级数展开进行计算.作为一个补充,本文借助Noether环、不变积分的基本性质以及奇点理论的某些定理,证明了判定、计算Pn(Γ)的Hilbert基的有关定理和原理,这提供了计算某些Pn(Γ)一组Hilbert基的而与幂级数展开不同的方法.最后,举例加以说明.     

Properties of an h-Bernstein-like basis and curves

Blossoming is a useful technique to study bases and curve representations in computer-aided geometric design. Recently Simeonov et al. (Comput Aided Geom Des 28:549–565, 2011) have used a blossom generalization, namely the h-blossom, to derive new results about the h-Bernstein basis and h-Bézier curves that have previously been studied in approximation theory and computer-aided geometric design. This paper introduces a basis related to the h-Bernstein basis. There is a close relationship between this new basis and the h-Bernstein basis, between the new basis and the h-blossom, and between the new basis and “progressive” curves. This paper explores these relationships and uses them to derive properties both of the new basis itself, and of curves represented in terms of the new basis.     

Certificates of Primal or Dual Infeasibility in Linear Programming

In general if a linear program has an optimal solution, then a primal and dual optimal solution is a certificate of the solvable status. Furthermore, it is well known that in the solvable case, then the linear program always has an optimal basic solution. Similarly, when a linear program is primal or dual infeasible then by Farkas's Lemma a certificate of the infeasible status exists. However, in the primal or dual infeasible case then there is not an uniform definition of what a suitable basis certificate of the infeasible status is.In this work we present a definition of a basis certificate and develop a strongly polynomial algorithm which given a Farkas type certificate of infeasibility computes a basis certificate of infeasibility. This result is relevant for the recently developed interior-point methods because they do not compute a basis certificate of infeasibility in general. However, our result demonstrates that a basis certificate can be obtained at a moderate computational cost.