18,688 research outputs found
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
On perturbations of highly connected dyadic matroids
Geelen, Gerards, and Whittle [3] announced the following result: let be a prime power, and let be a proper minor-closed class of
-representable matroids, which does not contain
for sufficiently high . There exist integers
such that every vertically -connected matroid in is a
rank- perturbation of a frame matroid or the dual of a frame matroid
over . They further announced a characterization of the
perturbations through the introduction of subfield templates and frame
templates.
We show a family of dyadic matroids that form a counterexample to this
result. We offer several weaker conjectures to replace the ones in [3], discuss
consequences for some published papers, and discuss the impact of these new
conjectures on the structure of frame templates.Comment: Version 3 has a new title and a few other minor corrections; 38
pages, including a 6-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file
Koszul-Morita duality
We construct a generalization of Koszul duality in the sense of
Keller--Lef\`evre for not necessarily augmented algebras. This duality is
closely related to classical Morita duality and specializes to it in certain
cases.Comment: 12 page
Ultraviolet Limit of Open String Theory
We confirm the intuition that a string theory which is perturbatively
infrared finite is automatically perturbatively ultraviolet finite. Our
derivation based on the asymptotics of the Selberg trace formula for the Greens
function on a Riemann surface holds for both open and closed string amplitudes
and is independent of modular invariance and supersymmetry. The mass scale for
the open strings stretched between Dbranes suggests a natural world-sheet
ultraviolet regulator in the string path integral, preserving both T-duality
and open-closed string world-sheet duality. Note added (Jan 2005): Comments and
related references added.Comment: 22 pages, LaTeX. Note added (Jan 2005): comments and related ref
On the existence of asymptotically good linear codes in minor-closed classes
Let be a sequence of codes such that each
is a linear -code over some fixed finite field
, where is the length of the codewords, is the
dimension, and is the minimum distance. We say that is
asymptotically good if, for some and for all , , , and . Sequences of
asymptotically good codes exist. We prove that if is a class of
GF-linear codes (where is prime and ), closed under
puncturing and shortening, and if contains an asymptotically good
sequence, then must contain all GF-linear codes. Our proof
relies on a powerful new result from matroid structure theory
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