Characterization of intrinsically harmonic forms

Let $M$ be a closed oriented manifold of dimension $n$ and $\omega$ a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which $\omega$ 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 $\omega$.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

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

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

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

Learning Incoherent Subspaces: Classification via Incoherent Dictionary Learning

In this article we present the supervised iterative projections and rotations (s-ipr) algorithm, a method for learning discriminative incoherent subspaces from data. We derive s-ipr as a supervised extension of our previously proposed iterative projections and rotations (ipr) algorithm for incoherent dictionary learning, and we employ it to learn incoherent sub-spaces that model signals belonging to different classes. We test our method as a feature transform for supervised classification, first by visualising transformed features from a synthetic dataset and from the ‘iris’ dataset, then by using the resulting features in a classification experiment