1,064 research outputs found
Characterization of intrinsically harmonic forms
Let be a closed oriented manifold of dimension and a closed
1-form on it. We discuss the question whether there exists a Riemannian metric
for which is co-closed. For closed 1-forms with nondegenerate zeros
the question was answered completely by Calabi in 1969. The goal of this paper
is to give an answer in the general case, i.e. not making any assumptions on
the zero set of .Comment: 8 page
Ultralocal Fields and their Relevance for Reparametrization Invariant Quantum Field Theory
Reparametrization invariant theories have a vanishing Hamiltonian and enforce
their dynamics through a constraint. We specifically choose the Dirac procedure
of quantization before the introduction of constraints. Consequently, for field
theories, and prior to the introduction of any constraints, it is argued that
the original field operator representation should be ultralocal in order to
remain totally unbiased toward those field correlations that will be imposed by
the constraints. It is shown that relativistic free and interacting theories
can be completely recovered starting from ultralocal representations followed
by a careful enforcement of the appropriate constraints. In so doing all
unnecessary features of the original ultralocal representation disappear.
The present discussion is germane to a recent theory of affine quantum
gravity in which ultralocal field representations have been invoked before the
imposition of constraints.Comment: 17 pages, LaTeX, no figure
Some Hilbert spaces related with the Dirichlet space
We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space
Embedding into the rectilinear plane in optimal O*(n^2)
We present an optimal O*(n^2) time algorithm for deciding if a metric space
(X,d) on n points can be isometrically embedded into the plane endowed with the
l_1-metric. It improves the O*(n^2 log^2 n) time algorithm of J. Edmonds
(2008). Together with some ingredients introduced by J. Edmonds, our algorithm
uses the concept of tight span and the injectivity of the l_1-plane. A
different O*(n^2) time algorithm was recently proposed by D. Eppstein (2009).Comment: 12 pages, 13 figure
Weak UCP and perturbed monopole equations
We give a simple proof of weak Unique Continuation Property for perturbed
Dirac operators, using the Carleman inequality. We apply the result to a class
of perturbations of the Seiberg-Witten monopole equations that arise in Floer
theory.Comment: 22 pages LaTeX, one .eps figur
A rescaled method for RBF approximation
In the recent paper [8], a new method to compute stable kernel-based
interpolants has been presented. This \textit{rescaled interpolation} method
combines the standard kernel interpolation with a properly defined rescaling
operation, which smooths the oscillations of the interpolant. Although
promising, this procedure lacks a systematic theoretical investigation. Through
our analysis, this novel method can be understood as standard kernel
interpolation by means of a properly rescaled kernel. This point of view allow
us to consider its error and stability properties
A rescaled method for RBF approximation
A new method to compute stable kernel-based interpolants
has been presented by the second and third authors. This rescaled interpolation method combines the
standard kernel interpolation with a properly defined rescaling operation, which
smooths the oscillations of the interpolant. Although promising, this procedure
lacks a systematic theoretical investigation.
Through our analysis, this novel method can be understood as standard
kernel interpolation by means of a properly rescaled kernel. This point of view
allow us to consider its error and stability properties.
First, we prove that the method is an instance of the Shepard\u2019s method,
when certain weight functions are used. In particular, the method can reproduce
constant functions.
Second, it is possible to define a modified set of cardinal functions strictly
related to the ones of the not-rescaled kernel. Through these functions, we
define a Lebesgue function for the rescaled interpolation process, and study its
maximum - the Lebesgue constant - in different settings.
Also, a preliminary theoretical result on the estimation of the interpolation
error is presented.
As an application, we couple our method with a partition of unity algorithm.
This setting seems to be the most promising, and we illustrate its behavior with
some experiments
Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
The fundamental solutions of the super Dirac and Laplace operators and their
natural powers are determined within the framework of Clifford analysis.Comment: 12 pages, accepted for publication in J. Math. Anal. App
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