Letterplace ideals and non-commutative Gröbner bases

In this paper we propose a 1-to-1 correspondence between graded two-sided ideals of the free associative algebra and some class of ideals of the algebra of polynomials, whose variables are double-indexed commuting ones. We call these ideals the “letterplace analogues” of graded two-sided ideals. We study the behaviour of the generating sets of the ideals under this correspondence, and in particular that of the Gröbner bases. In this way, we obtain a new method for computing non-commutative homogeneous Gröbner bases via polynomials in commuting variables. Since the letterplace ideals are stable under the action of a monoid of endomorphisms of the polynomial algebra, the proposed algorithm results in an example of a Buchberger procedure “reduced by symmetry”. Owing to the portability of our algorithm to any computer algebra system able to compute commutative Gröbner bases, we present an experimental implementation of our method in Singular. By means of a representative set of examples, we show finally that our implementation is competitive with computer algebra systems that provide non-commutative Gröbner bases from classical algorithms.     

Computing generating sets of lattice ideals and Markov bases of lattices

In this article, we present an algorithm for computing generating sets of lattice ideals or equivalently for computing Markov bases of lattices. Generating sets of lattice ideals and Markov bases of lattices are essentially equivalent concepts. In contrast to other existing methods, the algorithm in this article computes with projections of lattices. This algorithm clearly outperforms other algorithms in our computational experience. Two areas of application for generating sets of lattice ideals and Markov bases lattices are algebraic statistics and integer programming.     

Simultaneous estimation of vanishing points and their converging lines using the EM algorithm

This paper introduces a new method for the simultaneous computation of sets of lines meeting at multiple vanishing points through the use of the Expectation-Maximisation (EM) algorithm. The proposed method is based on the formulation of novel error functions in the projective plane between lines and points which involve the use of non-linear optimisation procedures that allow to treat equally finite and infinite vanishing points. These functions are included into the EM framework, which handles the multi-modality of the problem naturally with mixture models. Results show that the proposed method of joint computation of vanishing points and lines converging at such points enhances the robustness and the accuracy of locating these points.     

Vanishing points and three-dimensional lines from omni-directional video

This paper describes a system for structure from motion using vanishing points and three-dimensional lines extracted from omni-directional video sequences. To track lines, we use a novel dynamic programming approach to improve ambiguity resolution, and we use delayed states to aid in the initialization of landmarks. By reobserving vanishing points we get direct measurements of the robots three-dimensional attitude that are independent of its position. Using vanishing points simplifies the representation since parallel lines share the same direction states. We show the performance of the system in various indoor and outdoor environments and include comparisons with independent two-dimensional reference maps for each experiment .     

BCI-implicative ideals of BCI-algebras

This paper introduces the notion of BCI-implicative ideals and characterizes BCI-implicative ideals and closed BCI-implicative ideals. Using these characterizations, the connections between BCI-implicative ideals and other ideals in BCI/BCK-algebras are investigated. Additionally, the extension property of BCI-implicative ideals is established. Finally, the implicative BCI-algebras are completely described using BCI-implicative ideals. The above work generalizes the corresponding results in BCK-algebras.     

Reverse search for monomial ideals

We give a set of multidegrees that support all the numerical information for a monomial ideal that can be reverse searched and hence is parallelizable and has space complexity that is polynomial in the size of the input. Our approach uses a new definition of closed sets for simplicial complexes that may be useful in other contexts.     

Flat families by strongly stable ideals and a generalization of Gröbner bases

Let J be a strongly stable monomial ideal in S=K[x₀,…,x_n] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K-vector basis of the quotient S/I. We show that an ideal I belongs to Mf(J) if and only if it is generated by a special set of polynomials, the J-marked basis of I, that in some sense generalizes the notion of reduced Gröbner basis and its constructive capabilities. Indeed, although not every J-marked basis is a Gröbner basis with respect to some term order, a sort of reduced form modulo I∈Mf(J) can be computed for every homogeneous polynomial, so that a J-marked basis can be characterized by a Buchberger-like criterion. Using J-marked bases, we prove that the family Mf(J) can be endowed, in a very natural way, with a structure of an affine scheme that turns out to be homogeneous with respect to a non-standard grading and flat in the origin (the point corresponding to J), thanks to properties ofJ-marked bases analogous to those of Gröbner bases about syzygies.     

The Gröbner basis of the ideal of vanishing polynomials

We construct an explicit minimal strong Gröbner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m≥2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Gröbner basis is independent of the monomial order and that the set of leading terms of the constructed Gröbner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Gröbner basis in Z/m[x₁,x₂,…,x_n] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.     

一种鲁棒的匹配点检测算法

控制点的快速有效检测与匹配是遥感图像应用中的一项关键技术。通过对现有的特征点检测算法的分析与比较,提出了一种控制点的检测与匹配算法。仿真结果表明,该算法不仅可以实现控制点的自动检测与匹配,而且速度快、精度高。     

Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings

The notion of intuitionistic nil radicals of intuitionistic fuzzy ideals in rings is introduced, and related properties are investigated. The notion of semiprime intuitionistic fuzzy ideals is provided, and its characterization is established. The concept of Euclidean intuitionistic fuzzy ideals is also introduced, and its characterization is established.     

A planar single facility location and border crossing problem

In this study, we tackle the problem of locating a facility in a region where a fixed line barrier divides the region into two. The resulting subregions communicate with each other through a number of passage points located on the line barrier. Our contribution is threefold. First, we formulate the problem as a Mixed Integer Nonlinear Programming (MINLP) model and provide an optimal solution methodology based on an Outer Approximation (OA) algorithm. Second, we discuss the minimax version of this problem for locating an emergency facility and use the OA algorithm to solve the problem. We then provide simple example problems and extensive computational results for both problems. Finally, we propose a one-infinity approximation approach for the latter problem which yields a linear model. Practical uses of the models have been discussed in the border crossing context.     

利用边界路由器构筑校园网络安全第一道防线

边界路由器是校园网联接外网的关键设备,不论什么原因出现死机、拒绝服务或是运行效率急剧下降,其结果都将是灾难性的,因而边界路由器自身安全是校园网络安全需考虑的重要因素之一;由于边界路由器在校园网络结构中处于关键位置,它可以在网络安全管理中担当更多角色,发挥更大作用。本文以Cisco路由器为例,介绍在配置和管理过程中采取的安全措施。     

On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings

This paper is a continuation of ideas presented by Davvaz [Roughness in rings, Inform. Sci., 164 (2004) 147-163; Roughness based on fuzzy ideals, Inform. Sci., 176 (2006) 2417-2437]. We introduce the notions of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in a ring, and give some properties of such ideals. Also, we discuss the relations between the upper and lower rough prime (primary) ideals and the upper and lower approximations of their homomorphism images.     

New length bounds for cycle bases

Based on a recent work by Abraham, Bartal and Neiman (2007), we construct a strictly fundamental cycle basis of length O(n²) for any unweighted graph, whence proving the conjecture of Deo et al. (1982).For weighted graphs, we construct cycle bases of length O(W⋅lognloglogn), where W denotes the sum of the weights of the edges. This improves the upper bound that follows from the result of Elkin et al. (2005) by a logarithmic factor and, for comparison from below, some natural classes of large girth graphs are known to exhibit minimum cycle bases of length Ω(W⋅logn).We achieve this bound for weighted graphs by not restricting ourselves to strictly fundamental cycle bases—as it is inherent to the approach of Elkin et al.—but rather also considering weakly fundamental cycle bases in our construction. This way we profit from some nice properties of Hierarchically Well-Separated Trees that were introduced by Bartal (1998).     

关于BCI-代数的余模糊理想的一些结果

目的是通过模糊集理论研究BCI-代数的余模糊理想的性质。应用余模糊理想,引入了BCI-代数的一个同余关系,进而得到了BCI-代数的一个商代数,并且证明了商代数也是一个BCI-代数。     

Approximate computation of zero-dimensional polynomial ideals

The Buchberger–Möller algorithm is a well-known efficient tool for computing the vanishing ideal of a finite set of points. If the coordinates of the points are (imprecise) measured data, the resulting Gröbner basis is numerically unstable. In this paper we introduce a numerically stable Approximate Vanishing Ideal (AVI) Algorithm which computes a set of polynomials that almost vanish at the given points and almost form a border basis. Moreover, we provide a modification of this algorithm which produces a Macaulay basis of an approximate vanishing ideal. We also generalize the Border Basis Algorithm ([Kehrein, A., Kreuzer, M., 2006. Computing border bases. J. Pure Appl. Algebra 205, 279–295]) to the approximate setting and study the approximate membership problem for zero-dimensional polynomial ideals. The algorithms are then applied to actual industrial problems.     

Mining border descriptions of emerging patterns from dataset pairs

The mining of changes or differences or other comparative patterns from a pair of datasets is an interesting problem. This paper is focused on the mining of one type of comparative pattern called emerging patterns. Emerging patterns are denoted by EPs and are defined as patterns for which support increases from one dataset to the other with a big ratio. The number of EPs is sometimes huge. To provide a good structure for and to reduce the size of mining results, we use borders to concisely describe large collections of EPs in a lossless way. Such a border consists of only the minimal (under set inclusion) and the maximal EPs in the collection. We also present an algorithm for efficiently computing the borders of some desired EPs by manipulating the input borders only. Our experience with many datasets in the UCI Repository and recent cancer diagnosis datasets demonstrated that: Both the EP pattern type and our algorithm are useful for building accurate classifiers and useful for mining multifactor interactions, for example, minimal gene groups potentially responsible for the development of cancer.