Discriminantal Arrangements, Fiber Polytopes and Formality

Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3–formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.     

The diffeomorphic types of the complements of arrangements in ℂℙ3 II

For any arrangement of hyperplanes in ??³, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ??³.     

静园草坪灌溉系统的改进

在本文中 ,我将讨论一个从自己的实际观察中总结出来的数学模型——静园草坪的灌溉系统问题 .文中证明了灌溉系统的最大效率是 3 32π .这个结论还可以应用于其他的一些领域 .这实际上是一个决策问题 ,它归结为用小块的图形去覆盖平面区域 .我们需要找到一种最佳的覆盖方法 ,使得重叠的部分最少 .这样 ,我们就有两个问题要解决 ,首先是如何评价覆盖方法的优劣 ,其次是如何找到最佳的覆盖方法 .文中给出了合理的评价准则 ,并且提出了一些深入研究的方向和建议 .     

The diffeomorphic types of the complements of arrangements in ??3 II

For any arrangement of hyperplanes in ℂℙ³, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ℂℙ³. This work was partially supported by NSA grant and NSF grant     

The diffeomorphic types of the complements of arrangements in CP~3Ⅱ

For any arrangement of hyperplanes in CP~3,we introduce the soul of this arrangement. The soul,which is a pseudo-complex,is determined by the combinatorics of the arrangement of hyper- planes.In this paper,we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other.In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected.This generalizes our previous result on hyperplane point arrangements in CP~3.     

An Algorithm for Reducibility of 3-arrangements

We consider a central hyperplane arrangement in a three-dimensional vector space. The definition of characteristic form to a hyperplane arrangement is given and we could make use of characteristic form to judge the reducibility of this arrangement. In addition, the relationship between the reducibility and freeness of a hyperplane arrangement is given     

Erzeugung von Permutation im Rechenautomaten mittels zyklischer Operationen

Zusammenfassung Mit Hilfe von zyklischen Vertauschungen unterschiedlicher Zykluslänge werden sämtliche Anordnungen der Elemente einer endlichen Menge aus einer beliebig vorgegebenen Anordnung der Elemente sukzessive konstruiert. Durch Einführung einer Ordnungsrelation in der Menge aller Anordnungen kann jede zu einer beliebigen Ordnungszahl gehörige Anordnung direkt aus der ersten Anordnung konstruiert werden, ohne daß man die dazwischenliegenden Anordnungen zu kennen braucht. Umgekehrt läßt sich zu einer beliebig herausgegriffenen Anordnung ihre Ordnungszahl unmittelbar bestimmen, sobald nur festgelegt ist, welche Anordnung im Sinne der Ordnungsrelation die erste sein soll.

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Summary All possible re-arrangements or permutations of elements made up from a finite number of elements can be successively constructed from any given haphazard arrangement.A schematic order of sequence of arrangements is introduced. With its help an element arrangement ranking under any desired schematic order number may be made up from the first given arrangement. There is thus no need to investigate the order number lying in between the given and the desired arrangement of elements.Conversely, the order number of any arrangement extracted at random may be determined directly if is decided which arrangement shall be first in the order of sequence.

     

Linear arrangement problems on recursively partitioned graphs

We analyze the complexity of the restrictions of linear arrangement problems that are obtained if the legal permutations of the nodes are restricted to those that can be obtained by orderings of a binary tree structuring the nodes of the graph, the so-called p-tree. These versions of the linear arrangement problems occur in several places in current circuit layout systems. There the p-tree is the result of a recursive partitioning process of the graph. We show that the MINCUT LINEAR ARRANGEMENT problem and the OPTIMAL LINEAR ARRANGEMENT problem can be solved in polynomial time, if the p-tree is balanced. All other versions of the linear arrangement problems we analyzed are NP-complete.     

Managing supply chain backorders under vendor managed inventory: An incentive approach and empirical analysis

This paper shows how a manufacturer may use an incentive contract with a distributor under a VMI arrangement to gain market share. The manufacturer promises a distributor lower inventory levels in exchange for efforts by the distributor to convert potential lost sales due to stockouts to backorders. Data gathered from a third party provider of information services are then used to illustrate that this incentive arrangement may, at least implicitly, be employed in industry. Our data estimations show that when a manufacturer and distributor are operating under a VMI arrangement, lower inventory at the distributor is associated with a higher conversion rate of lost sales stockouts to backorders.     

The double Coxeter arrangement

Let V be Euclidean space. Let be a finite irreducible reflection group. Let be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For choose such that . The arrangement is known to be free: the derivation module is a free S-module with generators of degrees equal to the exponents of W. In this paper we prove an analogous theorem for the submodule of defined by . The degrees of the basis elements are all equal to the Coxeter number. The module may be considered a deformation of the derivation module for the Shi arrangement, which is conjectured to be free. The proof is by explicit construction using a derivation introduced by K. Saito in his theory of flat generators. Received: March 13, 1997     

Optimal reinforcement of a glass-reinforced plastic plate

Relations are derived for the optimal arrangement of the reinforcement in a rectangular glass-reinforced plastic plate supported at the edges and stressed in uniaxial compression by a short-term or long-term, uniformly distributed load. (The optimal arrangement is understood to be that which maximizes the critical load.)All-Union Scientific-Research Institute of Railroad Transport, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 741–744, July–August, 1969.     

国有企业人力资本与所有权安排的实证分析

本首先对中国国有企业人力资本与所有权安排制度作一简要考察，然后对中国国有企业人力资本与所有权安排问题进行实证分析，在此基础上进一步对国有企业人力资本与所有权安排的制度进行分析。     

Reducibility of hyperplane arrangements

Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.     

Topological representations of matroids

There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the arrangement still holds. Bounded subcomplexes of an arrangement of homotopy spheres correspond to minimal cellular resolutions of the dual matroid Steiner ideal. As a result, the Betti numbers of the ideal are computed and seen to be equivalent to Stanley's formula in the special case of face ideals of independence complexes of matroids.

     

关于n类排列的最少对换次数

以排列矩阵及其等和变换为工具,给出两种计算k(Tn) 的数学方法,从而解决了如何以最少次数的对换将一个n类排列化为标准排列的问题.     

Reducibility of finite reflection groups

A finite (pseudo-)reflection group G naturally gives rise to a hyperplane arrangement,i.e.,its reflection arrangement.We show that G is reducible if and only if its reflection arrangement is reducible.     

Multivariate arrangement increasing functions with applications in probability and statistics

A real valued function of s vector arguments in Rⁿ is said to be arrangement increasing if the function increases in value as the components of the vector arguments become more similarly arranged. Various examples of arrangement increasing functions are given including many joint multivariate densities, measures of concordance between judges and the permanent of a matrix with nonnegative components. Preservation properties of the class of arrangement increasing functions are examined, and applications are given including useful probabilistic inequalities for linear combinations of exchangeable random vectors.     

The function of diorism in ancient Greek analysis

This paper is a contribution to our knowledge of Greek geometric analysis. In particular, we investigate the aspect of analysis know as diorism, which treats the conditions, arrangement, and totality of solutions to a given geometric problem, and we claim that diorism must be understood in a broader sense than historians of mathematics have generally admitted. In particular, we show that diorism was a type of mathematical investigation, not only of the limitation of a geometric solution, but also of the total number of solutions and of their arrangement. Because of the logical assumptions made in the analysis, the diorism was necessarily a separate investigation which could only be carried out after the analysis was complete.     

Free arrangements and coefficients of characteristic polynomials

Ziegler showed that the multirestriction of a free arrangement is also free. After Ziegler’s work, several results concerning the “reverse direction”, i.e., characterizing freeness of an arrangement via that of its multirestriction, have appeared. In this paper, we prove a new characterization of freeness in which the second Betti number of the arrangement plays a crucial role.