Abstract. We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T * M .

A class of distortions termed functional Bregman divergences is defined, which includes squared error and relative entropy. A functional Bregman divergence acts on functions or distributions, and generalizes the standard Bregman divergence for vectors and a previous pointwise Bregman divergence that was defined for functions. A recently published result showed that the mean minimizes the expected Bregman divergence. The new functional definition enables the extension of this result to the continuous case to show that the mean minimizes the expected functional Bregman divergence over a set of functions or distributions. It is shown how this theorem applies to the Bayesian estimation of distributions. Estimation of the uniform distribution from independent and identically drawn samples is used as a case study. OverviewBregman divergences are a useful set of distortion functions that include squared error, relative entropy, logistic loss, Mahalanobis distance, and the Itakura-Saito function. Bregman divergences are popular in statistical estimation and information theory. Analysis using the concept of Bregman divergences has played a key role in recent advances in statistical learning [1][2][3][4][5][6][7][8][9], clustering [10,11], inverse problems [12], maximum entropy estimation [13], and the applicability of the data processing theorem [14]. Recently, it was discovered that the mean is the minimizer of the expected Bregman divergence for a set of d-dimensional points [10,15].In this paper we define a functional Bregman divergence that applies to functions and distributions, and we show that this new definition is equivalent to Bregman divergence applied to vectors. The functional definition generalizes a pointwise Bregman divergence that has been previously defined for measurable functions [7,16], and thus extends the class of distortion functions that are Bregman divergences; see Section 2.1.2 for an example. Most importantly, the functional definition enables one to solve functional minimization problems using standard methods from the calculus of variations; we extend the recent result on the expectation of vector Bregman divergence [10,15] to show that the mean minimizes the expected Bregman divergence for a set of functions or distributions. We show how this theorem links to Bayesian estimation of distributions. For distributions from the exponential family distributions, many popular divergences, such as relative entropy, can be expressed as a (different) Bregman divergence on the exponential distribution parameters. The functional Bregman definition enables stronger results and a more general application.In Section 1 we state a functional definition of the Bregman divergence and give examples for total squared difference, relative entropy, and squared bias. The relationship between the functional definition and previous Bregman definitions 1 2 is established. In Section 2 we present the main theorem: that the expectation of a set of functions minimizes the expected Bregman divergence. In Section ...

The cubic nonlinear Schrödinger equation with a lattice potential is used to model a periodic dilute gas Bose-Einstein condensate. Both two-and three-dimensional condensates are considered, for atomic species with either repulsive or attractive interactions. A family of exact solutions and corresponding potential is presented in terms of elliptic functions. The dynamical stability of these exact solutions is examined using both analytical and numerical methods. For condensates with repulsive atomic interactions, all stable, trivial-phase solutions are off-set from the zero level. For condensates with attractive atomic interactions, no stable solutions are found, in contrast to the one-dimensional case [8].

The cubic nonlinear Schrödinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose-Einstein condensate in a lattice potential. A family of exact stationary solutions is presented and its stability is examined using analytical and numerical methods. All stable trivial-phase solutions are off-set from the zero level. Our results imply that a large number of condensed atoms is sufficient to form a stable, periodic condensate.