On a projection-corrected component-by-component construction

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm

Ricci-corrected derivatives and invariant differential operators

We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables us to simplify the explicit formulae for standard invariant operators given in work of Cap, Slovak and Soucek, and at the same time extend these formulae from the context of AHS structures (which include conformal and projective structures) to the more general class of all parabolic structures (including CR structures).Comment: Substantially revised, shortened and simplified, with new treatment of Weyl structures; 24 page

A fast and exact ww-stacking and ww-projection hybrid algorithm for wide-field interferometric imaging

The standard wide-field imaging technique, the $w$-projection, allows correction for wide-fields of view for non-coplanar radio interferometric arrays. However, calculating exact corrections for each measurement has not been possible due to the amount of computation required at high resolution and with the large number of visibilities from current interferometers. The required accuracy and computational cost of these corrections is one of the largest unsolved challenges facing next generation radio interferometers such as the Square Kilometre Array. We show that the same calculation can be performed with a radially symmetric $w$-projection kernel, where we use one dimensional adaptive quadrature to calculate the resulting Hankel transform, decreasing the computation required for kernel generation by several orders of magnitude, whilst preserving the accuracy. We confirm that the radial $w$-projection kernel is accurate to approximately 1% by imaging the zero-spacing with an added $w$-term. We demonstrate the potential of our radially symmetric $w$-projection kernel via sparse image reconstruction, using the software package PURIFY. We develop a distributed $w$-stacking and $w$-projection hybrid algorithm. We apply this algorithm to individually correct for non-coplanar effects in 17.5 million visibilities over a $25$ by $25$ degree field of view MWA observation for image reconstruction. Such a level of accuracy and scalability is not possible with standard $w$-projection kernel generation methods. This demonstrates that we can scale to a large number of measurements with large image sizes whilst still maintaining both speed and accuracy.Comment: 9 Figures, 19 Pages. Accepted to Ap

A hybrid 3-D reconstruction/registration algorithm for correction of head motion in emission tomography

Even with head restraint, small head movements can occur during data acquisition in emission tomography that are sufficiently large to result in detectable artifacts in the final reconstruction. Direct measurement of motion can be cumbersome and difficult to implement, whereas previous attempts to use the measured projection data for correction have been limited to simple translation orthogonal to the projection. A fully three-dimensional (3-D) algorithm is proposed that estimates the patient orientation based on the projection of motion-corrupted data, with incorporation of motion information within subsequent ordered-subset expectation-maximization subiterations. Preliminary studies have been performed using a digital version of the Hoffman brain phantom. Movement was simulated by constructing a mixed set of projections in discrete positions of the phantom. The algorithm determined the phantom orientation that best matched each constructed projection with its corresponding measured projection. In the case of a simulated single movement in 24 of 64 projections, all misaligned projections were correctly identified. Incorporating data at the determined object orientation resulted in a reduction of mean square difference (MSD) between motion-corrected and motion-free reconstructions, compared to the MSD between uncorrected and motion-free reconstructions, by a factor of 1.9

The Effect of Single-Axis Sorting on the Estimation of a Linear Regression

Microaggregation is one of the most important statistical disclosure control techniques for continuous data. The basic principle of microaggregation is to group the observations in a data set and to replace them by their corresponding group means. In this paper, we consider single-axis sorting, a frequently applied microaggregation technique where the formation of groups depends on the magnitude of a sorting variable related to the variables in the data set. The paper deals with the impact of this technique on a linear model in continuous variables. We show that parameter estimates are asymptotically biased if the sorting variable depends on the response variable of the linear model. Using this result, we develop a consistent estimator that removes the aggregation bias. Moreover, we derive the asymptotic covariance matrix of the corrected least squares estimator

Special Lagrangian fibrations, wall-crossing, and mirror symmetry

In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.Comment: 45 pages; to appear in Surveys in Differential Geometr

Interpretation of van der Waals density functionals

The nonlocal correlation energy in the van der Waals density functional (vdW-DF) method [Phys. Rev. Lett. 92, 246401 (2004); Phys. Rev. B 76, 125112 (2007); Phys. Rev. B 89, 035412 (2014)] can be interpreted in terms of a coupling of zero-point energies of characteristic modes of semilocal exchange-correlation (xc) holes. These xc holes reflect the internal functional in the framework of the vdW-DF method [Phys. Rev. B 82, 081101(2010)]. We explore the internal xc hole components, showing that they share properties with those of the generalized-gradient approximation. We use these results to illustrate the nonlocality in the vdW-DF description and analyze the vdW-DF formulation of nonlocal correlation.Comment: 13 pages, 6 figures. Submited to Physical Review