CP Methods Research Articles

Comparison of different NaGdF4:Eu3+ synthesis routes and their influence on its structural and luminescent properties

Eu3+:NaGdF4 samples were obtained via co-precipitation in aqueous solution (CP), reversed micelle (RM) method, reaction between solid GdF3 and NaF solution (SR) as well as a solid-state reaction at high temperatures (SS). The synthesised materials were characterised using X-ray powder diffractometry, TEM microscopy, infrared spectroscopy and TGA analysis. For discussion of optical properties excitation and emission spectra were recorded and emission decay times were measured. The CP and RM methods allow to obtain powders with crystallite size of ∼10nm, which may be smoothly increased to about 1μm during post-fabrication heat treatment. Differences in structural and especially in optical properties of phosphors prepared by different techniques are emphasised and applicability of wet-chemistry routes for synthesis of fluoride powders is argued.

Monitoring retinal function in early age-related maculopathy: visual performance after 1 year.

To monitor visual performance in early age-related maculopathy (ARM). We measured monocular visual function-high-contrast visual acuity (HC-VA), central visual fields (mean sensitivity, MS), colour vision (desaturated Panel D-15), Pelli-Robson (P-R), and cone- and rod-mediated multifocal electroretinograms (mfERG) in 13 ARM subjects and 13 age-matched control subjects with normal fundi at baseline and after 1 year. All had visual acuity of 6/12 or better. The mfERG data were compared to templates derived from the control group at baseline. We analysed the mfERG results by averaging the central and peripheral fields and the superior and inferior fields (CP and SI methods) and by calculating the local responses. The mean rod-mediated responses were significantly delayed in the ARM group for the CP (P=0.04) and the SI methods (P=0.03) at baseline compared to the control group. This did not change significantly after 1 year, whereas the mean cone-mediated responses were within the normal range at both times. Although the local analysis revealed lower amplitudes for the cone- and rod-mediated responses at baseline this was not found after 1 year and only the local rod-mediated latencies were delayed at both times (P<0.01). HC-VA, desaturated Panel D-15 and P-R were significantly worse in the ARM group (P< or =0.01) at baseline but did not show further significant deterioration. Progressive fundus changes were found in only two subjects (18%). Although there was significant impairment of retinal function in early ARM at baseline no further deterioration was evident after 1 year.

CP methods of higher order for Sturm–Liouville and Schrödinger equations

The algorithm upon which the code SLCPM12, described in Computer Physics Communications 118 (1999) 259–277, is based, is extended to higher order. The implementation of the original algorithm, which was of order { 12 , 10 } (meaning order 12 at low energies and order 10 at high energies), was more efficient than the well-established codes SL02F, SLEDGE and SLEIGN. In the new algorithm the orders { 14 , 12 } , { 16 , 14 } and { 18 , 16 } are introduced. Besides regular Sturm–Liouville and one-dimensional Schrödinger problems also radial Schrödinger equations are considered with potentials of the form V ( r ) = S ( r ) / r + R ( r ) , where S ( r ) and R ( r ) are well behaved functions which tend to some (not necessarily equal) constants when r → 0 and r → ∞ . Numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms.

Constraint Programming Based Lagrangian Relaxation for the Automatic Recording Problem

Whereas CP methods are strong with respect to the detection of local infeasibilities, OR approaches have powerful optimization abilities that ground on tight global bounds on the objective. An integration of propagation ideas from CP and Lagrangian relaxation techniques from OR combines the merits of both approaches. We introduce a general way of how linear optimization constraints can strengthen their propagation abilities via Lagrangian relaxation. The method is evaluated on a set of benchmark problems stemming from a multimedia application. The experiments show the superiority of the combined method compared with a pure OR approach and an algorithm based on two independent optimization constraints.

CP methods for the Schrödinger equation

After a short survey over the efforts in the direction of solving the Schrödinger equation by using piecewise approximations of the potential function, the paper focuses on the piecewise perturbation methods in their CP implementation. The presentation includes a short list of problems for which CP versions are available, a sketch of the derivation of the CPM formulae, a description of various ways to construct or identify a certain version and also the main results of the error analysis. One of the most relevant results of the latter is that the energy dependence of the error is bounded, a fact which places these methods on a special position among the numerical methods for differential equations. A numerical illustration is also included in which a CPM based code for the regular Sturm–Liouville problem is compared with some other, well-established codes.

SLCPM12 — A program for solving regular Sturm—Liouville problems

The code SLCPM12 first converts the original Sturm—Liouville equation into an equation of the Schrödinger form and then it solves the latter by means of a suited highly accurate method. The conversion is done by using Liouville's transformation and the numerical method for solving the Schrödinger equation is CPM12, 10 developed by Ixaru, De Meyer and Vanden Berghe, CP Methods for the Schrödinger Equation revisited, J. Comput. Appl. Math. (1998). The new code is by far faster and more accurate than other existing codes, e.g. SLEDGE, SLEIGN and SL02F.

CP methods for the Schrödinger equation revisited

On constructing CPM propagators with an abundant number of terms by MATHEMATICA, we have shown that the CPM[ N, Q], where N is the number of polynomial terms by which the potential is approximated in each interval and Q the number of corrections introduced, is a method of order 2 N + 2 at low energies if Q ⩾ ⌊ 2 3 N⌋ + 1 and of order N at high energies if Q ⩾ 1. We have also proved that in the last case the error damps out as 1 E for both initial- and boundary-value problems. We have written a program for boundary-value problems which is of order 12, 10 at low and high energies, respectively, and have found out that it is far more efficient than the well-established codes SL02F, SLEDGE and SLEIGN.

The Little Bootstrap and Other Methods for Dimensionality Selection in Regression: X-Fixed Prediction Error

Abstract When a regression problem contains many predictor variables, it is rarely wise to try to fit the data by means of a least squares regression on all of the predictor variables. Usually, a regression equation based on a few variables will be more accurate and certainly simpler. There are various methods for picking “good” subsets of variables, and programs that do such procedures are part of every widely used statistical package. The most common methods are based on stepwise addition or deletion of variables and on “best subsets.” The latter refers to a search method that, given the number of variables to be in the equation (say, five), locates that regression equation based on five variables that has the lowest residual sum of squares among all five variable equations. All of these procedures generate a sequence of regression equations, the first based on one variable, the next based on two variables, and so on. Each member of this sequence is called a submodel and the number of variables in the equation is the dimensionality of the submodel. A complex problem is determining which submodel of the generated sequence to select. Statistical packages use various ad hoc selection methods, including F to enter, F to delete, Cp , and t-value cutoffs. Our approach to this problem is through the criterion that a good selection procedure selects dimensionality so as to give low prediction error (PE), where the PE of a regression equation is its expected squared error over the points in the X design. Because the true PE is unknown, the use of this criteria must be based on PE estimates. We introduce a method called the little bootstrap, which gives almost unbiased estimates for submodel PEs and then uses these to do submodel selection. Comparison is made to Cp and other methods by analytic examples and simulations. Little bootstrap does well—Cp and, by implication, all selection methods not based on data reuse give highly biased results and poor subset selection.

Comparison between the Self‐Consistent Continued‐Fraction and Other Continued‐Fraction Methods

AbstractAn alternative comparison of SCCF and other CP methods to that one published recently is given. It is shown that both, conceptually and on numerical merits SCCF is the best method, in existence, of calculation of the density of states in disordered materials, in principle, as it is the optimal, that is minimal yet fully tractable, approximation of the exact problem formulated.