Divided Differences

Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.Comment: 24 page

The Stability of One-Step Schemes for First-Order Two-Point Boundary Value Problems

The stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved. At times, this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem

Universal resonant ultracold molecular scattering

The elastic scattering amplitudes of indistinguishable, bosonic, strongly-polar molecules possess universal properties at the coldest temperatures due to wave propagation in the long-range dipole-dipole field. Universal scattering cross sections and anisotropic threshold angular distributions, independent of molecular species, result from careful tuning of the dipole moment with an applied electric field. Three distinct families of threshold resonances also occur for specific field strengths, and can be both qualitatively and quantitatively predicted using elementary adiabatic and semi-classical techniques. The temperatures and densities of heteronuclear molecular gases required to observe these univeral characteristics are predicted. PACS numbers: 34.50.Cx, 31.15.ap, 33.15.-e, 34.20.-bComment: 4 pages, 5 figure

Smoothing under Diffeomorphic Constraints with Homeomorphic Splines

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one dimensional setting incorporating diffeomorphic constraints is equivalent to impose monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to monotonize any unconstrained estimator of a regression function, and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances, and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching

Theoretical description of two ultracold atoms in finite 3D optical lattices using realistic interatomic interaction potentials

A theoretical approach is described for an exact numerical treatment of a pair of ultracold atoms interacting via a central potential that are trapped in a finite three-dimensional optical lattice. The coupling of center-of-mass and relative-motion coordinates is treated using an exact diagonalization (configuration-interaction) approach. The orthorhombic symmetry of an optical lattice with three different but orthogonal lattice vectors is explicitly considered as is the Fermionic or Bosonic symmetry in the case of indistinguishable particles.Comment: 19 pages, 5 figure

A numerical study of two-photon ionization of helium using the Pyprop framework

Few-photon induced breakup of helium is studied using a newly developed ab initio numerical framework for solving the six-dimensional time-dependent Schroedinger equation. We present details of the method and calculate (generalized) cross sections for the process of two-photon nonsequential (direct) double ionization at photon energies ranging from 39.4 to 54.4 eV, a process that has been very much debated in recent years and is not yet fully understood. In particular, we have studied the convergence property of the total cross section in the vicinity of the upper threshold (54.4 eV), versus the pulse duration of the applied laser field. We find that the cross section exhibits an increasing trend near the threshold, as has also been observed by others, and show that this rise cannot solely be attributed to an unintended inclusion of the sequential two-photon double ionization process, caused by the bandwidth of the applied field.Comment: 7 pages, 3 figure

Single Image Super-Resolution Using Multi-Scale Convolutional Neural Network

Methods based on convolutional neural network (CNN) have demonstrated tremendous improvements on single image super-resolution. However, the previous methods mainly restore images from one single area in the low resolution (LR) input, which limits the flexibility of models to infer various scales of details for high resolution (HR) output. Moreover, most of them train a specific model for each up-scale factor. In this paper, we propose a multi-scale super resolution (MSSR) network. Our network consists of multi-scale paths to make the HR inference, which can learn to synthesize features from different scales. This property helps reconstruct various kinds of regions in HR images. In addition, only one single model is needed for multiple up-scale factors, which is more efficient without loss of restoration quality. Experiments on four public datasets demonstrate that the proposed method achieved state-of-the-art performance with fast speed

Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants

We present two new adaptive quadrature routines. Both routines differ from previously published algorithms in many aspects, most significantly in how they represent the integrand, how they treat non-numerical values of the integrand, how they deal with improper divergent integrals and how they estimate the integration error. The main focus of these improvements is to increase the reliability of the algorithms without significantly impacting their efficiency. Both algorithms are implemented in Matlab and tested using both the "families" suggested by Lyness and Kaganove and the battery test used by Gander and Gautschi and Kahaner. They are shown to be more reliable, albeit in some cases less efficient, than other commonly-used adaptive integrators.Comment: 32 pages, submitted to ACM Transactions on Mathematical Softwar

Functions preserving nonnegativity of matrices

The main goal of this work is to determine which entire functions preserve nonnegativity of matrices of a fixed order $n$ -- i.e., to characterize entire functions $f$ with the property that $f(A)$ is entrywise nonnegative for every entrywise nonnegative matrix $A$ of size $n\times n$. Towards this goal, we present a complete characterization of functions preserving nonnegativity of (block) upper-triangular matrices and those preserving nonnegativity of circulant matrices. We also derive necessary conditions and sufficient conditions for entire functions that preserve nonnegativity of symmetric matrices. We also show that some of these latter conditions characterize the even or odd functions that preserve nonnegativity of symmetric matrices.Comment: 20 pages; expanded and corrected to reflect referees' remarks; to appear in SIAM J. Matrix Anal. App