2,862 research outputs found
The moduli space of matroids
In the first part of the paper, we clarify the connections between several
algebraic objects appearing in matroid theory: both partial fields and
hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are
compatible with the respective matroid theories. Moreover, fuzzy rings are
ordered blueprints and lie in the intersection of tracts with ordered
blueprints; we call the objects of this intersection pastures.
In the second part, we construct moduli spaces for matroids over pastures. We
show that, for any non-empty finite set , the functor taking a pasture
to the set of isomorphism classes of rank- -matroids on is
representable by an ordered blue scheme , the moduli space of
rank- matroids on .
In the third part, we draw conclusions on matroid theory. A classical
rank- matroid on corresponds to a -valued point of
where is the Krasner hyperfield. Such a point defines a
residue pasture , which we call the universal pasture of . We show that
for every pasture , morphisms are canonically in bijection with
-matroid structures on .
An analogous weak universal pasture classifies weak -matroid
structures on . The unit group of can be canonically identified with
the Tutte group of . We call the sub-pasture of generated by
``cross-ratios' the foundation of ,. It parametrizes rescaling classes of
weak -matroid structures on , and its unit group is coincides with the
inner Tutte group of . We show that a matroid is regular if and only if
its foundation is the regular partial field, and a non-regular matroid is
binary if and only if its foundation is the field with two elements. This
yields a new proof of the fact that a matroid is regular if and only if it is
both binary and orientable.Comment: 83 page
Big Toy Models: Representing Physical Systems As Chu Spaces
We pursue a model-oriented rather than axiomatic approach to the foundations
of Quantum Mechanics, with the idea that new models can often suggest new
axioms. This approach has often been fruitful in Logic and Theoretical Computer
Science. Rather than seeking to construct a simplified toy model, we aim for a
`big toy model', in which both quantum and classical systems can be faithfully
represented - as well as, possibly, more exotic kinds of systems.
To this end, we show how Chu spaces can be used to represent physical systems
of various kinds. In particular, we show how quantum systems can be represented
as Chu spaces over the unit interval in such a way that the Chu morphisms
correspond exactly to the physically meaningful symmetries of the systems - the
unitaries and antiunitaries. In this way we obtain a full and faithful functor
from the groupoid of Hilbert spaces and their symmetries to Chu spaces. We also
consider whether it is possible to use a finite value set rather than the unit
interval; we show that three values suffice, while the two standard
possibilistic reductions to two values both fail to preserve fullness.Comment: 24 pages. Accepted for Synthese 16th April 2010. Published online
20th April 201
Evidence for F(uzz) Theory
We show that in the decoupling limit of an F-theory compactification, the
internal directions of the seven-branes must wrap a non-commutative four-cycle
S. We introduce a general method for obtaining fuzzy geometric spaces via toric
geometry, and develop tools for engineering four-dimensional GUT models from
this non-commutative setup. We obtain the chiral matter content and Yukawa
couplings, and show that the theory has a finite Kaluza-Klein spectrum. The
value of 1/alpha_(GUT) is predicted to be equal to the number of fuzzy points
on the internal four-cycle S. This relation puts a non-trivial restriction on
the space of gauge theories that can arise as a limit of F-theory. By viewing
the seven-brane as tiled by D3-branes sitting at the N fuzzy points of the
geometry, we argue that this theory admits a holographic dual description in
the large N limit. We also entertain the possibility of constructing string
models with large fuzzy extra dimensions, but with a high scale for quantum
gravity.Comment: v2: 66 pages, 3 figures, references and clarifications adde
Inverse limit spaces satisfying a Poincare inequality
We give conditions on Gromov-Hausdorff convergent inverse systems of metric
measure graphs (and certain higher dimensional inverse systems of metric
measure spaces) which imply that the measured Gromov-Hausdorff limit
(equivalently, the inverse limit) is a PI space, i.e. it satisfies a doubling
condition and a Poincare inequality in the sense of Heinonen-Koskela. We also
give a systematic construction of examples for which our conditions are
satisfied. Included are known examples of PI spaces, such as Laakso spaces, and
a large class of new examples. Generically our graph examples have the property
that they do not bilipschitz embed in any Banach space with Radon-Nikodym
property, but they do embed in the Banach space L_1. For Laakso spaces, these
facts were discussed in our earlier papers
The structure of classical extensions of quantum probability theory
On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed
Towards a generalisation of formal concept analysis for data mining purposes
In this paper we justify the need for a generalisation of Formal
Concept Analysis for the purpose of data mining and begin the
synthesis of such theory. For that purpose, we first review semirings and
semimodules over semirings as the appropriate objects to use in abstracting
the Boolean algebra and the notion of extents and intents, respectively.
We later bring to bear powerful theorems developed in the field
of linear algebra over idempotent semimodules to try to build a Fundamental
Theorem for K-Formal Concept Analysis, where K is a type of
idempotent semiring. Finally, we try to put Formal Concept Analysis in
new perspective by considering it as a concrete instance of the theory
developed
Convergence of Fuzzy Tori and Quantum Tori for the quantum Gromov-Hausdorff Propinquity: an explicit approach
Quantum tori are limits of finite dimensional C*-algebras for the quantum
Gromov-Hausdorff propinquity, a metric defined by the author as a strengthening
of Rieffel's quantum Gromov-Hausdorff designed to retain the C*-algebraic
structure. In this paper, we propose a proof of the continuity of the family of
quantum and fuzzy tori which relies on explicit representations of the
C*-algebras rather than on more abstract arguments, in a manner which takes
full advantage of the notion of bridge defining the quantum propinquity.Comment: 41 Pages. This paper is the second half of ArXiv:1302.4058v2. The
latter paper has been divided in two halves for publications purposes, with
the first half now the current version of 1302.4058, which has been accepted
in Trans. Amer. Math. Soc. This second half is now a stand-alone paper, with
a brief summary of 1302.4058 and a new introductio
Compact quantum metric spaces and ergodic actions of compact quantum groups
We show that for any co-amenable compact quantum group A=C(G) there exists a
unique compact Hausdorff topology on the set EA of isomorphism classes of
ergodic actions of G such that the following holds: for any continuous field of
ergodic actions of G over a locally compact Hausdorff space T the map T->EA
sending each t in T to the isomorphism class of the fibre at t is continuous if
and only if the function counting the multiplicity of gamma in each fibre is
continuous over T for every equivalence class gamma of irreducible unitary
representations of G. Generalizations for arbitrary compact quantum groups are
also obtained. In the case G is a compact group, the restriction of this
topology on the subset of isomorphism classes of ergodic actions of full
multiplicity coincides with the topology coming from the work of Landstad and
Wassermann. Podles spheres are shown to be continuous in the natural parameter
as ergodic actions of the quantum SU(2) group. When A is separable, we also
introduce a notion of regular quantum metric on G, and show how to use it to
induce a quantum metric on any ergodic action of G in the sense of Rieffel.
Furthermore, we introduce a quantum Gromov-Hausdorff distance between ergodic
actions and show that it induces the above topology.Comment: References and lemmas 5.7 and 5.8 added. To appear in JF
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