14,324 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
(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces
We develop a kind of pregeometry consisting of a web of overlapping fuzzy
lumps which interact with each other. The individual lumps are understood as
certain closely entangled subgraphs (cliques) in a dynamically evolving network
which, in a certain approximation, can be visualized as a time-dependent random
graph. This strand of ideas is merged with another one, deriving from ideas,
developed some time ago by Menger et al, that is, the concept of probabilistic-
or random metric spaces, representing a natural extension of the metrical
continuum into a more microscopic regime. It is our general goal to find a
better adapted geometric environment for the description of microphysics. In
this sense one may it also view as a dynamical randomisation of the causal-set
framework developed by e.g. Sorkin et al. In doing this we incorporate, as a
perhaps new aspect, various concepts from fuzzy set theory.Comment: 25 pages, Latex, no figures, some references added, some minor
changes added relating to previous wor
Unusual Thermodynamics on the Fuzzy 2-Sphere
Higher spin Dirac operators on both the continuum sphere() and its fuzzy
analog() come paired with anticommuting chirality operators. A
consequence of this is seen in the fermion-like spectrum of these operators
which is especially true even for the case of integer-spin Dirac operators.
Motivated by this feature of the spectrum of a spin 1 Dirac operator on
, we assume the spin 1 particles obey Fermi-Dirac statistics. This
choice is inspite of the lack of a well defined spin-statistics relation on a
compact surface such as . The specific heats are computed in the cases of
the spin and spin 1 Dirac operators. Remarkably the specific heat
for a system of spin particles is more than that of the spin 1
case, though the number of degrees of freedom is more in the case of spin 1
particles. The reason for this is inferred through a study of the spectrums of
the Dirac operators in both the cases. The zero modes of the spin 1 Dirac
operator is studied as a function of the cut-off angular momentum and is
found to follow a simple power law. This number is such that the number of
states with positive energy for the spin 1 and spin system become
comparable. Remarks are made about the spectrums of higher spin Dirac operators
as well through a study of their zero-modes and the variation of their spectrum
with degeneracy. The mean energy as a function of temperature is studied in
both the spin and spin 1 cases. They are found to deviate from
the standard ideal gas law in 2+1 dimensions.Comment: 19 pages, 7 figures. The paper has been significantly modified. Main
results are unchange
Topology at the Planck Length
A basic arbitrariness in the determination of the topology of a manifold at
the Planck length is discussed. An explicit example is given of a `smooth'
change in topology from the 2-sphere to the 2-torus through a sequence of
noncommuting geometries. Applications are considered to the theory of D-branes
within the context of the proposed (atrix) theory.Comment: Orsay Preprint 97/34, 17 pages, Late
Fluctuating Fuzzy Funnels
It is well known that a D-string ending on a D3, D5 or D7 brane is described
in terms of a non-commutative fuzzy funnel geometry. In this article, we give a
numerical study of the fluctuations about this leading geometry. This allows us
to investigate issues related to the stability and moduli space of these
solutions. We comment on the comparison to the linearized fluctuations in
supergravity.Comment: 24 pages, 3 figures; v2 references added and correcte
On MV - topologies
En este trabajo estamos interesados en un tipo particular de topología fuzzy llamada MV-topología, la cual usa operaciones MV-algebraicas para generar abiertos fuzzy. Estos espacios topológicos fuzzy permiten generalizaciones naturales de definiciones y resultados importantes de la topología clásica. En este sentido, desarrollamos algunos conceptos y resultados centrales, con el proprósito de extender los correspondientes de la topología clásica, y al mismo tiempo siguiendo la ruta de la bien conocida teoría de espacios topológicos fuzzy. Mostramos que las MV-topologías son un tipo de topología fuzzy que goza de muy "buen comportamiento" matemático, en el sentido de que la mayoría de definiciones y resultados clásicos de topología general encuentran una extensión o adaptación natural en este marco. Entre otros resultados, también extendemos el concepto de haz para el caso en el que el espacio base es un espacio MV-topológico, y mostramos una representación por "MV-haces" para una clase de MV-álgebras.DoctoradoDOCTOR(A) EN CIENCIAS - MATEMÁTICA
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