4,504 research outputs found
Light-Ray Radon Transform for Abelianin and Nonabelian Connection in 3 and 4 Dimensional Space with Minkowsky Metric
We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and
with a connection for a trivial GL(n,C) bundle over that manifold. To each
light ray on the manifold we assign the data of paralel transport along that
light ray. It turns out that these data are not enough to reconstruct the
connection, but we can add more data, which depend now not from lines but from
2-planes, and which in some sence are the data of parallel transport in the
complex light-like directions, then we can reconstruct the connection up to a
gauge transformation. There are some interesting applications of the
construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be
written as the zero curvature condition for a pair of certain first order
differential operators; one of the operators in the pair is the covariant
derivative in complex light-like direction we studied. 2) there is a relation
of this Radon transform with the supersymmetry. 3)using our Radon transform, we
can get a measure on the space of 2 dimensional planes in 4 dimensional real
space. Any such measure give rise to a Crofton 2-density. The integrals of this
2-density over surfaces in R^4 give rise to the Lagrangian for maps of real
surfaces into R^4, and therefore to some string theory. 4) there are relations
with the representation theory. In particular, a closely related transform in 3
dimensions can be used to get the Plancerel formula for representations of
SL(2,R).Comment: We add an important discussion part, establishing the relation of our
Radon transform with the self-dual Yang-Mills, string theory, and the
represntation theory of the group SL(2,R
The Associated Metric for a Particle in a Quantum Energy Level
We show that the probabilistic distribution over the space in the spectator
world, can be associated via noncommutative geometry (with some modifications)
to a metric in which the particle lives. According to this geometrical view,
the metric in the particle world is ``contracted'' or ``stretched'' in an
inverse proportion to the probability distribution.Comment: 14 pages, latex, epsf, 3 figures. Some clarifications were adde
Spatial Joint Species Distribution Modeling using Dirichlet Processes
Species distribution models usually attempt to explain presence-absence or
abundance of a species at a site in terms of the environmental features
(socalled abiotic features) present at the site. Historically, such models have
considered species individually. However, it is well-established that species
interact to influence presence-absence and abundance (envisioned as biotic
factors). As a result, there has been substantial recent interest in joint
species distribution models with various types of response, e.g.,
presence-absence, continuous and ordinal data. Such models incorporate
dependence between species response as a surrogate for interaction.
The challenge we focus on here is how to address such modeling in the context
of a large number of species (e.g., order 102) across sites numbering in the
order of 102 or 103 when, in practice, only a few species are found at any
observed site. Again, there is some recent literature to address this; we adopt
a dimension reduction approach. The novel wrinkle we add here is spatial
dependence. That is, we have a collection of sites over a relatively small
spatial region so it is anticipated that species distribution at a given site
would be similar to that at a nearby site. Specifically, we handle dimension
reduction through Dirichlet processes joined with spatial dependence through
Gaussian processes.
We use both simulated data and a plant communities dataset for the Cape
Floristic Region (CFR) of South Africa to demonstrate our approach. The latter
consists of presence-absence measurements for 639 tree species on 662
locations. Through both data examples we are able to demonstrate improved
predictive performance using the foregoing specification
Computing topological invariants with one and two-matrix models
A generalization of the Kontsevich Airy-model allows one to compute the
intersection numbers of the moduli space of p-spin curves. These models are
deduced from averages of characteristic polynomials over Gaussian ensembles of
random matrices in an external matrix source. After use of a duality, and of an
appropriate tuning of the source, we obtain in a double scaling limit these
intersection numbers as polynomials in p. One can then take the limit p to -1
which yields a matrix model for orbifold Euler characteristics. The
generalization to a time-dependent matrix model, which is equivalent to a
two-matrix model, may be treated along the same lines ; it also yields a
logarithmic potential with additional vertices for general p.Comment: 30 pages, added references, changed conten
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