On the Castelnuovo-Mumford regularity of products of ideal sheaves

In this paper we give bounds on the Castelnuovo-Mumford regularity of products of ideals and ideal sheaves. In particular, we show that the regularity of a product of ideals is bounded by the sum of the regularities of its factors if the corresponding schemes intersect in a finite set of points. We also show how approximations of sheaves can be used to bound the regularity of an arrangement of two-planes in projective space.Comment: 9 pages, Minor additions to introduction, references added, typos corrected, minor change in exposition of main theore

Subspace arrangements defined by products of linear forms

We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields such generators in cases with a lot of combinatorial structure, and we present the examples that motivated our work. We give a construction which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. We also consider generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3.$Comment: 20 pages; AMSLatex; v.2: proof of Proposition 5.1.3 corrected; proof of Proposition 5.1.6 shortened; references added, v.3: minor corrections; final version; to appear in the Journal of the London Mathematical Societ

Heat sealable, flame and abrasion resistant coated fabric

Flame retardant, abrasion resistant elastomeric compositions are comprised of thermoplastic polyurethane polymer and flame retarding amounts of a filler selected from decabromodiphenyloxide and antimony oxide in a 3:1 weight ratio, and decabromodiphenyloxide, antimony oxide, and ammonium polyphosphate in a 3:1:3 weight ratio respectively. Coated fabrics employing such elastomeric compositions as coating film are flexible, lightweight, and air impermeable and can be made using heat or dielectric sealing procedures

Heat resistant protective hand covering

The heat resistant, protective glove is made up of first and second shell sections which define a palm side and a backside, respectively. The first shell section is made of a twill wave fabric of a temperature-resistant aromatic polyamide fiber. The second shell section is made of a knitted fabric of a temperature-resistant aromatic polyamide fiber. The first and second shell sections are secured to one another, e.g., by sewing, to provide the desired glove configuration and an opening for insertion of the wearer's hand. The protective glove also includes a first liner section which is secured to and overlies the inner surface of the first shell section and is made of a felt fabric of a temperature-resistant aromatic polyamide fiber and has a flame resistant, elastomenic coating on the surface facing and overlying the inner surface of the first shell section

Multigraded regularity: coarsenings and resolutions

Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to find finite subsets of Z^r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo-Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded minimal syzygies are bounded in terms of this invariant.Comment: 20 pages, 1 figure; small corrections made; final version; to appear in J. of Algebr

Hyperbanana Graphs

A bar-and-joint framework is a finite set of points together with specified distances between selected pairs. In rigidity theory we seek to understand when the remaining pairwise distances are also fixed. If there exists a pair of points which move relative to one another while maintaining the given distance constraints, the framework is flexible; otherwise, it is rigid. Counting conditions due to Maxwell give a necessary combinatorial criterion for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R^2. However, the flexible "double banana" shows that Maxwell's conditions are not sufficient to guarantee rigidity in R^3. We present a generalization of the double banana to a family of hyperbananas. In dimensions 3 and higher, these are (infinitesimally) flexible, providing counterexamples to the natural generalization of Laman's theorem

Heat resistant protective hand covering

A heat-resistant aromatic polyamide fiber is described. The outer surface of the shell is coated with a fire-resistant elastomer and liner. Generally conforming and secured to the shell and disposed inwardly of the shell, the liner is made of a felt fabric of temperature-resistant aromatic polymide fiber