11,121 research outputs found
Examples of Berezin-Toeplitz Quantization: Finite sets and Unit Interval
We present a quantization scheme of an arbitrary measure space based on
overcomplete families of states and generalizing the Klauder and the
Berezin-Toeplitz approaches. This scheme could reveal itself as an efficient
tool for quantizing physical systems for which more traditional methods like
geometric quantization are uneasy to implement. The procedure is illustrated by
(mostly two-dimensional) elementary examples in which the measure space is a
-element set and the unit interval. Spaces of states for the -element set
and the unit interval are the 2-dimensional euclidean and hermitian
\C^2 planes
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
Intersections between some families of (U,N)- and RU-implications
(U,N)-implications and RU-implications are the generalizations of (S,N)- and R-implications to the framework of uninorms, where the t-norms and t-conorms are replaced by appropriate uninorms. In this work, we present the intersections that exist between (U,N)-implications and the different families of RU-implications obtainable from the well-established families of uninorms
Stone-type representations and dualities for varieties of bisemilattices
In this article we will focus our attention on the variety of distributive
bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and
involutive bisemilattices. After extending Balbes' representation theorem to
bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn
duality and introduce the categories of 2spaces and 2spaces. The
categories of 2spaces and 2spaces will play with respect to the
categories of distributive bisemilattices and De Morgan bisemilattices,
respectively, a role analogous to the category of Stone spaces with respect to
the category of Boolean algebras. Actually, the aim of this work is to show
that these categories are, in fact, dually equivalent
Batch and median neural gas
Neural Gas (NG) constitutes a very robust clustering algorithm given
euclidian data which does not suffer from the problem of local minima like
simple vector quantization, or topological restrictions like the
self-organizing map. Based on the cost function of NG, we introduce a batch
variant of NG which shows much faster convergence and which can be interpreted
as an optimization of the cost function by the Newton method. This formulation
has the additional benefit that, based on the notion of the generalized median
in analogy to Median SOM, a variant for non-vectorial proximity data can be
introduced. We prove convergence of batch and median versions of NG, SOM, and
k-means in a unified formulation, and we investigate the behavior of the
algorithms in several experiments.Comment: In Special Issue after WSOM 05 Conference, 5-8 september, 2005, Pari
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