Convergence of abstract splines

AbstractShekhtman (J. Approx. Theory, 30(1980), 237–246) gives a sufficient condition for the convergence of abstract splines. We show that his condition is not necessary and give a related condition which is both necessary and sufficient. In the process, we also give a necessary and sufficient condition for a sequence of abstract spline projectors to be bounded

On “best” interpolation

AbstractRecent interest in the problem of minimizing ∥f(k)∥∞ under the constraint that fti) = f0(ti), i = 1,…, n + k, for some given f0 and given (ti)1n+k seems to make it worthwhile to explain how Favard solved this problem in the thirties, particularly since Favard's paper on the subject is rather sketchy in places.The explanation is given in terms of a dual problem, using a technique initiated by M. Krein. In addition, the analogous problem of minimizing ∥f(k)∥p for 1 ⩽ p < ∞ under similar constraints is discussed

Resonant d-wave scattering in harmonic waveguides

We observe and analyze d-wave resonant scattering of bosons in tightly confining harmonic waveguides. It is shown that the d-wave resonance emerges in the quasi-1D regime as an imprint of a 3D d-wave shape resonance. A scaling relation for the position of the d-wave resonance is provided. By changing the trap frequency, ultracold scattering can be continuously tuned from s-wave to d-wave resonant behavior. The effect can be utilized for the realization of ultracold atomic gases interacting via higher partial waves and opens a novel possibility for studying strongly correlated atomic systems beyond s-wave physics.Comment: 6 pages, 9 figure

A comment on Ewald Quak's ``"About B-splines"

The early contributions to B-spline theory by Tiberiu Popoviciu and by Liubomir Chakalov are recalled

Illumination modelling of a mobile device environment for effective use in driving mobile apps

Copyright 2015 Society of Photo Optical Instrumentation Engineers. One print or electronic copy may be made for personal use only. Systematic electronic or print reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. This item is also available here: http://dx.doi.org/10.1117/12.2177087The present generation of Ambient Light Sensors (ALS) of a mobile handheld device suffer from two practical shortcomings. The ALSs are narrow angle, i.e. they respond effectively only within a narrow angle of operation and there is a latency of operation. As a result mobile applications that operate based on the ALS readings could perform sub-optimally especially when operated in environments with non-uniform illumination. The applications will either adopt with unacceptable levels of latency or/and may demonstrate a discrete nature of operation. In this paper we propose a framework to predict the ambient illumination of an environment in which a mobile device is present. The predictions are based on an illumination model that is developed based on a small number of readings taken during an application calibration stage. We use a machine learning based approach in developing the models. Five different regression models were developed, implemented and compared based on Polynomial, Gaussian, Sum of Sine, Fourier and Smoothing Spline functions. Approaches to remove noisy data, missing values and outliers were used prior to the modelling stage to remove their negative effects on modelling. The prediction accuracy for all models were found to be above 0.99 when measured using R-Squared test with the best performance being from Smoothing Spline. In this paper we will discuss mathematical complexity of each model and investigate how to make compromises in finding the best mode

Circular Rydberg states of atomic hydrogen in an arbitrary magnetic field

We report a theoretical scheme using a B-spline basis set to improve the poor computational accuracy of circular Rydberg states of hydrogen atoms in the intermediate magnetic field. This scheme can produce high accuracy energy levels and valid for an arbitrary magnetic field. Energy levels of hydrogen are presented for circular Rydberg states with azimuthal quantum numbers $|m|$ = 10 - 70 as a function of magnetic field strengths ranging from zero to 2.35 $\times$ 10$^9$ T. The variation of spatial distributions of electron probability densities with magnetic field strengths is discussed and competition between Coulomb and magnetic interactions is illustrated.Comment: 14 pages, 2 figure