18 research outputs found
Development of Light Weight Magnesium Alloys for Low Temperature Applications Annual Progress Report, 19 Jun. 1964 - 18 Jun. 1965
Lightweight, wrought magnesium alloys for high strength and ductility and low notch sensitivity applications at cryogenic temperature
Development of light weight magnesium alloys for low temperature applications Annual progress report, Jun. 19, 1965 - Mar. 20, 1966
Lightweight magnesium alloys for cryogenic application
Cyclic projectors and separation theorems in idempotent convex geometry
Semimodules over idempotent semirings like the max-plus or tropical semiring
have much in common with convex cones. This analogy is particularly apparent in
the case of subsemimodules of the n-fold cartesian product of the max-plus
semiring it is known that one can separate a vector from a closed subsemimodule
that does not contain it. We establish here a more general separation theorem,
which applies to any finite collection of closed semimodules with a trivial
intersection. In order to prove this theorem, we investigate the spectral
properties of certain nonlinear operators called here idempotent cyclic
projectors. These are idempotent analogues of the cyclic nearest-point
projections known in convex analysis. The spectrum of idempotent cyclic
projectors is characterized in terms of a suitable extension of Hilbert's
projective metric. We deduce as a corollary of our main results the idempotent
analogue of Helly's theorem.Comment: 20 pages, 1 figur
Generalizing Tanisaki's ideal via ideals of truncated symmetric functions
We define a family of ideals in the polynomial ring
that are parametrized by Hessenberg functions
(equivalently Dyck paths or ample partitions). The ideals generalize
algebraically a family of ideals called the Tanisaki ideal, which is used in a
geometric construction of permutation representations called Springer theory.
To define , we use polynomials in a proper subset of the variables
that are symmetric under the corresponding permutation
subgroup. We call these polynomials {\em truncated symmetric functions} and
show combinatorial identities relating different kinds of truncated symmetric
polynomials. We then prove several key properties of , including that if
in the natural partial order on Dyck paths then ,
and explicitly construct a Gr\"{o}bner basis for . We use a second family
of ideals for which some of the claims are easier to see, and prove that
. The ideals arise in work of Ding, Develin-Martin-Reiner, and
Gasharov-Reiner on a family of Schubert varieties called partition varieties.
Using earlier work of the first author, the current manuscript proves that the
ideals generalize the Tanisaki ideals both algebraically and
geometrically, from Springer varieties to a family of nilpotent Hessenberg
varieties.Comment: v1 had 27 pages. v2 is 29 pages and adds Appendix B, where we include
a recent proof by Federico Galetto of a conjecture given in the previous
version. We also add some connections between our work and earlier results of
Ding, Gasharov-Reiner, and Develin-Martin-Reiner. v3 corrects a typo in
Valibouze's citation in the bibliography. To appear in Journal of Algebraic
Combinatoric
Pivoting in Linear Complementarity: Two Polynomial-Time Cases
We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty’s least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris’s highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP
Carathéodory, Helly and the others in the max-plus world
International audienceCarathéodory's, Helly's and Radon's theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem-Sierksma's conjecture-although still open for the usual convexity, is shown to be true in the max-plus setting. © 2009 Springer Science+Business Media, LLC