12 research outputs found
Tensor Products of Convex Cones, Part II: Closed Cones in Finite-Dimensional Spaces
In part I, we studied tensor products of convex cones in dual pairs of real
vector spaces. This paper complements the results of the previous paper with an
overview of the most important additional properties in the finite-dimensional
case. (i) We show that the projective cone can be identified with the cone of
positive linear operators that factor through a simplex cone. (ii) We prove
that the projective tensor product of two closed convex cones is once again
closed (Tam already proved this for proper cones). (iii) We study the tensor
product of a cone with its dual, leading to another proof (and slight
extension) of a theorem of Barker and Loewy. (iv) We provide a large class of
examples where the projective and injective cones differ. As this paper was
being written, this last result was superseded by a result of Aubrun, Lami,
Palazuelos and Pl\'avala, who independently showed that the projective cone
is strictly contained in the injective cone
whenever and are closed,
proper and generating, with neither nor a simplex cone. Compared to
their result, this paper only proves a few special cases.Comment: 21 page
Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity
The tensor product of two ordered vector spaces can be ordered in more than
one way, just as the tensor product of normed spaces can be normed in multiple
ways. Two natural orderings have received considerable attention in the past,
namely the ones given by the projective and injective (or biprojective) cones.
This paper aims to show that these two cones behave similarly to their normed
counterparts, and furthermore extends the study of these two cones from the
algebraic tensor product to completed locally convex tensor products. The main
results in this paper are the following: (i) drawing parallels with the normed
theory, we show that the projective/injective cone has mapping properties
analogous to those of the projective/injective norm; (ii) we establish direct
formulas for the lineality space of the projective/injective cone, in
particular providing necessary and sufficient conditions for the cone to be
proper; (iii) we show how to construct faces of the projective/injective cone
from faces of the base cones, in particular providing a complete
characterization of the extremal rays of the projective cone; (iv) we prove
that the projective/injective tensor product of two closed proper cones is
contained in a closed proper cone (at least in the algebraic tensor product).Comment: 62 page
On the size of subsets of avoiding solutions to linear systems with repeated columns
Consider a system of balanced linear equations in variables with
coefficients in . If , then a routine application
of the slice rank method shows that there are constants
with such that, for every subset of
size at least , the system has a solution
with not all equal. Building on a
series of papers by Mimura and Tokushige and on a paper by Sauermann, this
paper investigates the problem of finding a solution of higher non-degeneracy;
that is, a solution where are pairwise distinct, or even a
solution where do not satisfy any balanced linear equation
that is not a linear combination of the equations in the system.
In this paper, we present general techniques for systems with repeated
columns. This class of linear systems is disjoint from the class covered by
Sauermann's result, and captures the systems studied by Mimura and Tokushige
into a single proof. A special case of our results shows that, if is a subset such that does not contain a non-trivial
-term arithmetic progression (where ), then must have
exponentially small density.Comment: LaTeX, 25 pages, no figure
Discrete and metric divisorial gonality can be different
This paper compares the divisorial gonality of a finite graph to the
divisorial gonality of the associated metric graph with
unit lengths. We show that is equal to the
minimal divisorial gonality of all regular subdivisions of , and we provide
a class of graphs for which this number is strictly smaller than the divisorial
gonality of . This settles a conjecture of M. Baker in the negative.Comment: 15 pages, 4 figures. Changes: improved Lemma 4.4, added Proposition
5.3, changed open question
Discrete and metric divisorial gonality can be different
This paper compares the divisorial gonality of a finite graph G to the divisorial gonality of the associated metric graph Ξ(G,1) with unit lengths. We show that dgon(Ξ(G,1)) is equal to the minimal divisorial gonality of all regular subdivisions of G, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of G. This settles a conjecture of M. Baker [3, Conjecture 3.14] in the negative
Constructing tree decompositions of graphs with bounded gonality
In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality