1,001 research outputs found
Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings
For a poset P, a subposet A, and an order preserving map F from A into the
real numbers, the marked order polytope parametrizes the order preserving
extensions of F to P. We show that the function counting integral-valued
extensions is a piecewise polynomial in F and we prove a reciprocity statement
in terms of order-reversing maps. We apply our results to give a geometric
proof of a combinatorial reciprocity for monotone triangles due to Fischer and
Riegler (2011) and we consider the enumerative problem of counting extensions
of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3:
examples included (suggested by referee), to appear in "SIAM Journal on
Discrete Mathematics
On sets defining few ordinary lines
Let P be a set of n points in the plane, not all on a line. We show that if n
is large then there are at least n/2 ordinary lines, that is to say lines
passing through exactly two points of P. This confirms, for large n, a
conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for
this problem, as well as all sets having fewer than n - C ordinary lines for
some absolute constant C. We also solve, for large n, the "orchard-planting
problem", which asks for the maximum number of lines through exactly 3 points
of P. Underlying these results is a structure theorem which states that if P
has at most Kn ordinary lines then all but O(K) points of P lie on a cubic
curve, if n is sufficiently large depending on K.Comment: 72 pages, 16 figures. Third version prepared to take account of
suggestions made in a detailed referee repor
Blanchfield and Seifert algebra in high-dimensional boundary link theory I: Algebraic K-theory
The classification of high-dimensional mu-component boundary links motivates
decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and
the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an
arbitrary ring A, with F_mu the free group on mu generators and Sigma the set
of matrices over A[F_mu] which become invertible over A under the augmentation
A[F_mu] to A. Blanchfield A[F_mu]-modules and Seifert A-modules are abstract
algebraic analogues of the exteriors and Seifert surfaces of boundary links.
Algebraic transversality for A[F_mu]-module chain complexes is used to
establish a long exact sequence relating the algebraic K-groups of the
Blanchfield and Seifert modules, and to obtain the decompositions of
K_*(A[F_mu]) and K_*(Sigma^{-1}A[F_mu]) subject to a stable flatness condition
on Sigma^{-1}A[F_mu] for the higher K-groups.Comment: This is the version published by Geometry & Topology on 2 November
200
The characteristic polynomial and determinant are not ad hoc constructions
The typical definition of the characteristic polynomial seems totally ad hoc
to me. This note gives a canonical construction of the characteristic
polynomial as the minimal polynomial of a "generic" matrix. This approach works
not just for matrices but also for a very broad class of algebras including the
quaternions, all central simple algebras, and Jordan algebras.
The main idea of this paper dates back to the late 1800s. (In particular, it
is not due to the author.) This note is intended for a broad audience; the only
background required is one year of graduate algebra.Comment: v2 is heavily revised and somewhat expanded. The product formula for
the determinant on an algebra is prove
Finding triangular Cayley maps with graph touring
We develop a method for determining whether certain kinds of Cayley maps can exist by using multi-digraph representations of the data in the Cayley maps. Euler tours of these multi-digraphs correspond exactly to the permutations which define Cayley maps. We also begin to classify which 3-regular multi-digraphs have non-backtracking Euler tours in general
Exotic behaviour of infinite hypermaps
This is a survey of infinite hypermaps, and of how they can be constructed by using examples and techniques from combinatorial group theory, with particular emphasis on phenomena which have no analogues for finite hypermaps.<br/
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