Harmonic Functions On Manifolds Whose Large Sphere Are Small

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions

Optimal non-linear transformations for large scale structure statistics

Recently, several studies proposed non-linear transformations, such as a logarithmic or Gaussianization transformation, as efficient tools to recapture information about the (Gaussian) initial conditions. During non-linear evolution, part of the cosmologically relevant information leaks out from the second moment of the distribution. This information is accessible only through complex higher order moments or, in the worst case, becomes inaccessible to the hierarchy. The focus of this work is to investigate these transformations in the framework of Fisher information using cosmological perturbation theory of the matter field with Gaussian initial conditions. We show that at each order in perturbation theory, there is a polynomial of corresponding order exhausting the information on a given parameter. This polynomial can be interpreted as the Taylor expansion of the maximally efficient "sufficient" observable in the non-linear regime. We determine explicitly this maximally efficient observable for local transformations. Remarkably, this optimal transform is essentially the simple power transform with an exponent related to the slope of the power spectrum; when this is -1, it is indistinguishable from the logarithmic transform. This transform Gaussianizes the distribution, and recovers the linear density contrast. Thus a direct connection is revealed between undoing of the non-linear dynamics and the efficient capture of Fisher information. Our analytical results were compared with measurements from the Millennium Simulation density field. We found that our transforms remain very close to optimal even in the deeply non-linear regime with \sigma^2 \sim 10.Comment: 11 pages, matches version accepted for publication in MNRA

A New Path-Integral Representation of the TT-Matrix in Potential Scattering

We employ the method used by Barbashov and collaborators in Quantum Field Theory to derive a path-integral representation of the $T$-matrix in nonrelativistic potential scattering which is free of functional integration over fictitious variables as was necessary before. The resulting expression serves as a starting point for a variational approximation applied to high-energy scattering from a Gaussian potential. Good agreement with exact partial-wave calculations is found even at large scattering angles. A novel path-integral representation of the scattering length is obtained in the low-energy limit.Comment: 9 pages, 1 figure, Latex with amsmath, amssym; v2: some typos corrected, matches published versio

On the differential form spectrum of hyperbolic manifolds

We give a lower bound for the bottom of the $L^2$ differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodge-de Rham Laplacian and leads to applications for the (co)homology and topology of certain classes of hyperbolic manifolds