Averaged shelling for quasicrystals

The shelling of crystals is concerned with counting the number of atoms on spherical shells of a given radius and a fixed centre. Its straight-forward generalization to quasicrystals, the so-called central shelling, leads to non-universal answers. As one way to cope with this situation, we consider shelling averages over all quasicrystal points. We express the averaged shelling numbers in terms of the autocorrelation coefficients and give explicit results for the usual suspects, both perfect and random.Comment: 4 pages, several figures, 2 tables; updated version with minor corrections and improvements; to appear in the proceedings of ICQ

Averaged Template Matching Equations

By exploiting an analogy with averaging procedures in fluid dynamics, we present a set of averaged template matching equations. These equations are analogs of the exact template matching equations that retain all the geometric properties associated with the diffeomorphismgrou p, and which are expected to average out small scale features and so should, as in hydrodynamics, be more computationally efficient for resolving the larger scale features. Froma geometric point of view, the new equations may be viewed as coming from a change in norm that is used to measure the distance between images. The results in this paper represent first steps in a longer termpro gram: what is here is only for binary images and an algorithm for numerical computation is not yet operational. Some suggestions for further steps to develop the results given in this paper are suggested

The anisotropic averaged Euler equations

The purpose of this paper is to derive the anisotropic averaged Euler equations and to study their geometric and analytic properties. These new equations involve the evolution of a mean velocity field and an advected symmetric tensor that captures the fluctuation effects. Besides the derivation of these equations, the new results in the paper are smoothness properties of the equations in material representation, which gives well-posedness of the equations, and the derivation of a corrector to the macroscopic velocity field. The numerical implementation and physical implications of this set of equations will be explored in other publications.Comment: 24 pages, 1 figur

On Incompressible Averaged Lagrangian Hydrodynamics

This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the analytical and geometrical properties of the Lagrangian flow map. We prove existence and uniqueness of smooth-in-time solutions for initial data in $H^s$, $s > n/2 +1$ by establishing the existence of smooth geodesics of a new weak right invariant metric on new subgroups of the volume-preserving diffeomorphism group. We establish smooth limits of zero viscosity for the second-grade fluids equations even on manifolds with boundary. We prove that the weak curvature operator of the weak invariant metric is continuous in the $H^s$ topology for $s> n/2+2$, thus proving existence and uniqueness for the Jacobi equation. We show that this new metric stabilizes the Lagrangian flow of the original Euler equations by changing the sign of the sectional curvature.Comment: 35 page

Generalization bounds for averaged classifiers

We study a simple learning algorithm for binary classification. Instead of predicting with the best hypothesis in the hypothesis class, that is, the hypothesis that minimizes the training error, our algorithm predicts with a weighted average of all hypotheses, weighted exponentially with respect to their training error. We show that the prediction of this algorithm is much more stable than the prediction of an algorithm that predicts with the best hypothesis. By allowing the algorithm to abstain from predicting on some examples, we show that the predictions it makes when it does not abstain are very reliable. Finally, we show that the probability that the algorithm abstains is comparable to the generalization error of the best hypothesis in the class.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000005

Panchromatic Averaged Stellar Populations: PaasP

We study how the spectral fitting of galaxies, in terms of light fractions derived in one spectral region translates into another region, by using results from evolutionary synthesis models. In particular, we examine propagation dependencies on Evolutionary Population Synthesis (EPS, {\sc grasil}, {\sc galev}, Maraston and {\sc galaxev}) models, age, metallicity, and stellar evolution tracks over the near-UV---near infrared (NUV---NIR, 3500\AA\ to 2.5\mc) spectral region. Our main results are: as expected, young ($t \lesssim$ 400 Myr) stellar population fractions derived in the optical cannot be directly compared to those derived in the NIR, and vice versa. In contrast, intermediate to old age ($t \gtrsim$ 500 Myr) fractions are similar over the whole spectral region studied. The metallicity has a negligible effect on the propagation of the stellar population fractions derived from NUV --- NIR. The same applies to the different EPS models, but restricted to the range between 3800 \AA\ and 9000 \AA. However, a discrepancy between {\sc galev}/Maraston and {\sc grasil}/{\sc galaxev} models occurs in the NIR. Also, the initial mass function (IMF) is not important for the synthesis propagation. Compared to {\sc starlight} synthesis results, our propagation predictions agree at $\sim$95% confidence level in the optical, and $\sim$85% in the NIR. {\bf In summary, spectral fitting} performed in a restricted spectral range should not be directly propagated from the NIR to the UV/Optical, or vice versa. We provide equations and an on-line form ({\bf Pa}nchromatic {\bf A}veraged {\bf S}tellar {\bf P}opulation - \paasp) to be used for this purpose.Comment: 13 pages and 10 figures. Accepted by MNRA