2,862 research outputs found

    The moduli space of matroids

    Get PDF
    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 EE, the functor taking a pasture FF to the set of isomorphism classes of rank-rr FF-matroids on EE is representable by an ordered blue scheme Mat(r,E)Mat(r,E), the moduli space of rank-rr matroids on EE. In the third part, we draw conclusions on matroid theory. A classical rank-rr matroid MM on EE corresponds to a K\mathbb{K}-valued point of Mat(r,E)Mat(r,E) where K\mathbb{K} is the Krasner hyperfield. Such a point defines a residue pasture kMk_M, which we call the universal pasture of MM. We show that for every pasture FF, morphisms kM→Fk_M\to F are canonically in bijection with FF-matroid structures on MM. An analogous weak universal pasture kMwk_M^w classifies weak FF-matroid structures on MM. The unit group of kMwk_M^w can be canonically identified with the Tutte group of MM. We call the sub-pasture kMfk_M^f of kMwk_M^w generated by ``cross-ratios' the foundation of MM,. It parametrizes rescaling classes of weak FF-matroid structures on MM, and its unit group is coincides with the inner Tutte group of MM. We show that a matroid MM is regular if and only if its foundation is the regular partial field, and a non-regular matroid MM 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

    Full text link
    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

    Full text link
    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

    Full text link
    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

    Get PDF
    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

    Get PDF
    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

    Full text link
    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

    Get PDF
    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
    • …
    corecore