Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Hyperplanes of the form xj=xi+c are called affinographic. For an affinographic hyperplane arrangement in Rn, such as the Shi arrangement, we study the function f(m) that counts integral points in n[1,m] that do not lie in any hyperplane of the arrangement. We show that f(m) is a piecewise polynomial function of positive integers m, composed of terms that appear gradually as m increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph.An application is to interval coloring in which the interval of available colors for vertex vi has the form [hi+1,m].A related problem takes colors modulo m; the number of proper modular colorations is a different piecewise polynomial that for large m becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.     

A characterization of signed graphs with generalized perfect elimination orderings

An important property of chordal graphs is that these graphs are characterized by the existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those graphs. The definition of our generalized perfect elimination orderings is motivated by a generalization of the classical result that a so-called graphic hyperplane arrangement is free if and only if the corresponding graph is chordal.     

Quadrature Frequency Conversion Scheme Using CsLiB6O10 Crystals for the Efficient Second-Harmonic Generation of High Power Nd:YAG Laser

The ability of CsLiB_6O10 (CLBO) crystals for high power second-harmonic generation (SHG) of a 1064-nm Nd:YAG laser in a quadrature arrangement was experimentally demonstrated. A 532-nm second harmonic output pulse energy of 2.25 J was obtained with 3.21 J of an input 1064-nm fundamental pulse energy at a repetition rate of 10 Hz, corresponding to a power conversion efficiency in excess of 70%.     

The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations

We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes. In terms of Aguiar-Mahajan's Lie theory of hyperplane arrangements, we show that this structure is equivalent to the action of Lie elements on faces. We use layered binary trees to encode flags of adjoint arrangement faces, allowing for the representation of certain Lie elements by antisymmetrized layered binary forests. This is dual to the well-known use of (delayered) binary trees to represent Lie elements of the braid arrangement. The discrete derivative then induces an action of layered binary forests on piecewise-constant functions, which we call the forest derivative. Our main result states that forest derivatives of functions factorize as external products of functions precisely if one restricts to functions which satisfy the Steinmann relations, which are certain four-term linear relations appearing in the foundations of axiomatic quantum field theory. We also show that the forest derivative satisfies the Lie properties of antisymmetry the Jacobi identity. It follows from these Lie properties, and also crucially factorization, that functions which satisfy the Steinmann relations form a left comodule of the Lie cooperad, with the coaction given by the forest derivative. Dually, this endows the adjoint braid arrangement modulo the Steinmann relations with the structure of a Lie algebra internal to the category of vector species. This work is a first step towards describing new connections between Hopf theory in species and quantum field theory.     

损伤自诊断自适应智能材料结构的研究

本文介绍了一种强度型损伤自诊断自适应智能复合材料结构的研究情况。这种智能材料结构基于电阻应变丝为传感元件，形状记忆合金为动作元件。为使结构具有高的灵敏度及诊断精度，文中研究了传感元件的布置方案，提出了四点布置传感元件的方法。同时采用形状记忆合金为动作元件。在自诊断出损伤后，对相应的ＳＭＡ激励，以减小损伤或防止其扩展。     

Synthesis and applications of novel fluoroalkyl end‐capped cooligomers containing diphenylacetylene segments: a new approach to the surface arrangement of diphenylacetylene segments on the traditional organic polymer

Fluoroalkyl end‐capped cooligomers containing diphenylacetylene segments [R_F‐(DPMA)_x‐(Co‐M)_y‐R_F] were prepared by reaction of fluoroalkanoyl peroxide with 4‐(phenylethynyl)phenyl methacrylate (DPMA) and radical polymerizable comonomers such as N,N‐dimethylacrylamide (DMAA) and acryloylmorpholine (ACMO) under very mild conditions. Fluorinated cooligomers containing diphenylacetylene segments thus obtained exhibited a good solubility in a variety of organic solvents. These fluorinated cooligomers were also applied to the surface modification of traditional organic polymers such as poly(methyl methacrylate) (PMMA) to exhibit not only a good surface active property imparted by fluorine but also a fluorescent characteristic related to diphenylacetylene segments on their surface. In addition, these fluorinated cooligomers could form the nanometer size‐controlled fluorinated molecular aggregates in chloroform. Interestingly, some benzenes and biphenyl (BP) derivatives could interact with these fluorinated oligomeric aggregates as guest molecules, and in particular 2‐chloro‐5‐nitrobenzotrifluoride (CNB) was most effective for enhancing the fluorescent intensity of these guest molecules. Copyright © 2009 John Wiley & Sons, Ltd.     

Determinants of the Hypergeometric Period Matrices of an Arrangement and Its Dual

We fix three natural numbers k, n, N, such that n+k+1 = N, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of N hyperplanes in a k-dimensional affine space, the other is an arrangement of N hyperplanes in an n-dimensional affine space. We assign weights α ₁, . . . , α _N to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of k-dimensional hypergeometric integrals and the second is a matrix of n-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler’s gamma function multiplied by a product of exponentials of the weights. Supported in part by NSF grant DMS-0244579.     

Homology representations arising from the half cube, II

In a previous work, we defined a family of subcomplexes of the n-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least k, and we proved that the reduced homology of such a subcomplex is concentrated in degree k−1. This homology module supports a natural action of the Coxeter group W(Dn) of type D. In this paper, we explicitly determine the characters (over C) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group Sn by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of Sn agree (over C) with the representations of Sn on the (k−2)-nd homology of the complement of the k-equal real hyperplane arrangement.     

A polyhedral study of triplet formulation for single row facility layout problem

The single row facility layout problem (SRFLP) is the problem of arranging n departments with given lengths on a straight line so as to minimize the total weighted distance between all department pairs. We present a polyhedral study of the triplet formulation of the SRFLP introduced by Amaral [A.R.S. Amaral, A new lower bound for the single row facility layout problem, Discrete Applied Mathematics 157 (1) (2009) 183-190]. For any number of departments n, we prove that the dimension of the triplet polytope is n(n−1)(n−2)/3 (this is also true for the projections of this polytope presented by Amaral). We then prove that several valid inequalities presented by Amaral for this polytope are facet-defining. These results provide theoretical support for the fact that the linear program solved over these valid inequalities gives the optimal solution for all instances studied by Amaral.