On the convergence of cross decomposition

Cross decomposition is a recent method for mixed integer programming problems, exploiting simultaneously both the primal and the dual structure of the problem, thus combining the advantages of Dantzig—Wolfe decomposition and Benders decomposition. Finite convergence of the algorithm equipped with some simple convergence tests has been proved. Stronger convergence tests have been proposed, but not shown to yield finite convergence.In this paper cross decomposition is generalized and applied to linear programming problems, mixed integer programming problems and nonlinear programming problems (with and without linear parts). Using the stronger convergence tests finite exact convergence is shown in the first cases. Unbounded cases are discussed and also included in the convergence tests. The behaviour of the algorithm when parts of the constraint matrix are zero is also discussed. The cross decomposition procedure is generalized (by using generalized Benders decomposition) in order to enable the solution of nonlinear programming problems.     

Spin-peierls transition in dimerized stacks of anion-radical salt (N-Me-2,5-di-Me-Pz)(TCNQ)2, (Pz is pyrazine)

An anion-radical salt (ARS) (N-Me-2,5-di-Me-Pz)(TCNQ)₂, where Pz is pyrazine, was synthesized and its crystal structure was resolved. X-ray diffraction experiments on single crystals were performed. Heat capacity was measured in the temperature range from 2 to 300 K. Magnetisation and magnetic susceptibility were measured in the temperature range from 2 to 300 K and the low-temperature part was measured in magnetic fields from 5 mT to 5 T. The experimental results were explained in terms of dimerized Heisenberg spin chain model. Numerical calculations were performed and compared with experimental data.     

Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.     

Sampling and metrology

The result of a measurement refers in principle only to the amount of substance actually contributing to the analytical signal. However, an appropriate definition of the measurand must include a specification of the system for which the result of the measurement should apply. All systems being inherently heterogeneous, representativity assumes importance for the metrological quality of a measurement, and the process needed to ascertain representativity is sampling. The contribution from this characteristic must be included when expressing the uncertainty of the reported value of the measurand. Representative sampling of systems that are infinite or non-uniform was developed by Pierre Gy in his Theory of Sampling. Finite systems can achieve uniformity by mechanical treatment and mixing; the heterogeneity of these systems can be characterized by a sampling constant, expressed in units of weight, for each particular species being determined. Examples of the contribution of sampling to the uncertainty of analytical results are discussed for some biological materials. Presented at the 2nd International Conference on Metrology – Trends and Applications in Calibration and Testing Laboratories, November 4–6, 2003, Eilat, Israel.     

Indirect determination of soluble trace elements in certified reference materials by neutron activation analysis

Since 1983 we have analyzedaqua regia extracts from environmental reference materials of 5 different soils and 4 different sludges by neutron activation analysis. Aqua regia as such is not accepted for irradiation in a nuclear reactor, and therefore the solution has to be evaporated to dryness, mixed and dried to constant weight before an aliquot can be taken and subjected to instrumental neutron activation analysis (INAA). Results for leachable Cr and Zn were found to display greater variability than results for total contents of these elements, and the reasons for this will be discussed.For the 1994 certification campaign for one soil and two sludges we have chosen to determine the total content of trace elements in these reference materials by INAA before and afteraqua regia leaching. The leachable contents are then found indirectly as a difference between the two results; in this way we eliminate the sources of uncertainty associated with the drying and handling of theaqua regia extracts, and we reduce the contributions from counting statistics.     

Quality assurance and statistical control

In scientific research laboratories it is rarely possible to use quality assurance schemes, developed for large-scale analysis. Instead methods have been developed to control the quality of modest numbers of analytical results by relying on statistical control: Analysis of precision serves to detect analytical errors by comparing thea priori precision of the analytical results with the actual variability observed among replicates or duplicates. The method relies on the chi-square distribution to detect excess variability and is quite sensitive even for 5–10 results. Interference control serves to detect analytical bias by comparing results obtained by two different analytical methods, each relying on a different detection principle and therefore exhibiting different influence from matrix elements; only 5–10 sets of results are required to establish whether a regression line passes through the origo. Calibration control is an essential link in the traceability of results. Only one or two samples of pure solid or aqueous standards with accurately known content need to be analyzed. Verification is carried out by analyzing certified reference materials from BCR, NIST, or others; their limited accuracy of 5–10% make them less suitable for calibration purposes.     

A journey through quality control

The Euroanalysis VII conference in Vienna included a two-day session: Quality Assurance in Analytical Chemistry. The contributions comprised 15 lectures devoted to: intra-laboratory quality measures, inter-laboratory control, formal aspects and accreditation and implementation. The paper presents an overview of the main items developed by the contributors.A survey on the session on Quality Assurance in Analytical Chemistry of Euroanalysis VIIThe authors thank the organizers of EUROANALYSIS VII and in particular Dr. B. Griepink and Dr. E. Maier of the Community Bureau of Reference (BCR) of the CEC for their support and considerable contributions.     

Algorithms for non-linear huber estimation

The Huber criterion for data fitting is a combination of thel ₁ and thel ₂ criteria which is robust in the sense that the influence of wild data points can be reduced. We present a trust region and a Marquardt algorithm for Huber estimation in the case where the functions used in the fit are non-linear. It is demonstrated that the algorithms converge under the usual conditions.