ON THE SPECTRUM OF THE TWO-PARTICLE SCHRÖDINGER OPERATOR WITH POINT POTENTIAL: ONE DIMENTIONAL CASE

In the paper a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schr\"{o}\-dinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon,$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based to the study of the operator $h_\varepsilon.$ First the essential spectrum is described. The existence of unique negative eigenvalue of the Schr\"{o}dinger operator is proved. Further, this eigenvalue and corresponding eigenfunction are found.

Guidance for use of weights: an analysis of different types of weights and their implications when using SAS PROCs

SAS and other popular statistical packages provide support for survey data with sampling weights. For example, PROC MEANS and PROC LOGISTIC in SAS have their counterparts PROC SURVEYMEANS and PROC SURVEYLOGISTIC to facilitate analysis of data from complex survey studies. On the other hand, PROC MEANS and many other classic SAS procedures also provide an option for including weights and yield identical point estimates, but different standard errors (SEs), as their corresponding survey procedures. This paper takes an in-depth look at different types of weights and provides guidance on use of different SAS procedures.

A comparison of three approaches for determining scalar illuminance from cubic illuminance data

A method for calculating scalar illuminance using cubic illuminance values in a light field has been proposed in the literature. This enables exact measurement of the illuminance vector direction and magnitude by a fixed device, as well as providing a useful basis for calculation. However, the method yields an inexact estimate of the scalar illuminance which in some cases may lead to errors. Two alternative approaches using the concept of mean spherical semi-cubic and cubic illuminances are proposed in this paper, to determine which of these approaches yields the highest accuracy, and to observe the effect of source orientation in various multiple point source configurations. Three types of test are introduced: the first involves two, three, and four identical point sources, separated by a varying angle θ; the second involves four identical point sources arranged symmetrically at varying azimuth angle ψ and incident angle α; the third involves 10,000 combinations of up to six point sources with random luminous intensities and in random positions. Comparisons between the three approaches show that the approach using mean spherical semi-cubic illuminances yields the least amount of error and thus the highest accuracy for scalar illuminance and vector/scalar illuminance ratio in the first and second test. In the third test, this approach also yields the highest accuracy, even though it tends to underestimate the scalar illuminance in scenes with more sources.

Analyzing a fractal gel of charged oblate nanoparticles in a suspension using time-resolved rheometry and DLVO theory

The disk-like nanoparticles of LAPONITE® are known to self-assemble to form a fractal gel within hours after a sufficiently large concentration of LAPONITE® is dispersed in water containing salt. The concentration of sodium counterions associated with LAPONITE® particles, however, continues to increase over a period of days, suggesting that delamination of LAPONITE® disks from stacks is sluggish and/or dissociation of counterions is slow. In either case, spontaneous self-assembly of LAPONITE® particles occurs even though delamination and/or counterion dissociation has not reached its equilibrium state. In order to determine the nature of the fractal gel as the extent of delamination and/or dissociation progresses towards equilibrium, we subject the LAPONITE® suspension to a freezing–defrosting cycle, which interestingly reinitiates the gelation process in suspension afresh. Application of time-resolved rheometry to a defrosted suspension shows that iso-frequency loss tangent curves intersect at an identical point, validating the Winter–Chambon criterion for a critical fractal gel state. Interestingly, while the time required to form a critical gel is observed to decrease with increased time elapsed since preparation, at which freezing–defrosting is carried out, the fractal dimension of the critical gel is observed to remain unaffected. We also solve DLVO theory for free energy interactions between the negatively charged LAPONITE® particle faces and analyze the observed phenomena.

Stationary states of identical point vortices and vortex foam on the sphere

Stationary configurations of identical point vortices on the sphere are investigated using a simple numerical scheme. Configurations in which the vortices are arrayed along curves on the sphere are exhibited, which approximate equilibrium configurations of vortex sheets on the sphere. Other configurations (found after starting from random initial conditions) exhibit net-like distributions of vorticity, dividing the sphere into many cells that contain no vorticity or diffuse vorticity and forming a stationary ‘vortex foam’ on the sphere. They may be viewed as intermediate-energy elements in the set of all identical point vortex equilibria on the sphere. In the continuum limit, these foam states may correspond to stationary states of multiple intersecting vortex sheets. Stationary configurations of point vortices are not found to have this character when vortices of opposite circulations are included.