Development of metrological models for internal standard single-point and multi-point calibrations for the assessment of tear gas weapons compliance with legislation

Tear gases are the most widely used non-lethal weapons, both by security forces and by the general public. The pepper spray, whose active agent is capsaicin, is the only self-defence aerosol allowed in Portugal, where capsaicin concentration must be below 5 g/100 mL. The cost-effective evaluation of the compliance of self-defence weapons with legislation involves the use of two measurement procedures with increasing quantitative capability. Samples are first assessed by preliminary measurement procedure based on single-point GC–MS calibration. Whenever the measurement uncertainty from this assessment makes evaluation inconclusive, the evaluation of sample compliance using multi-point GC–MS calibration is performed. Metrological models including sound criteria for the evaluation of sample compliance with legislation were developed for both measurement procedures. Such models include the evaluation of the impact of instrumental performance, calibration model, sample dilution and standards preparation on measurement uncertainty. The relative expanded uncertainty, in the studied range (capsaicin 3–7 g/100 mL), of measurements supported in single-point calibrations ranged from 10 to 22% and the ones supported on multi-point calibrations from 8 to 12% depending on capsaicin concentration and daily GC–MS repeatability. Measurements are fit for the intended use since they present a relative expanded uncertainty smaller than a target value of 30, or 15%, for measurements supported in single- or multi-point calibrations, respectively.     

Matrix algebras in Quasi-Newton methods for unconstrained minimization

Summary.  In this paper a new class of quasi-Newton methods, named ℒQN, is introduced in order to solve unconstrained minimization problems. The novel approach, which generalizes classical BFGS methods, is based on a Hessian updating formula involving an algebra ℒ of matrices simultaneously diagonalized by a fast unitary transform. The complexity per step of ℒQN methods is O(n log n), thereby improving considerably BFGS computational efficiency. Moreover, since ℒQN's iterative scheme utilizes single-indexed arrays, only O(n) memory allocations are required. Global convergence properties are investigated. In particular a global convergence result is obtained under suitable assumptions on f. Numerical experiences [7] confirm that ℒQN methods are particularly recommended for large scale problems. Received December 30, 2001 / Revised version received December 20, 2001 / Published online November 27, 2002 Mathematics Subject Classification (1991): 65K10 Correspondence to: P. Zellini     

Worst case uncertainty estimates for routine instrumental analysis

A methodology for the worst case measurement uncertainty estimation for analytical methods which include an instrumental quantification step, adequate for routine determinations, is presented. Although the methodology presented should be based on a careful evaluation of the analytical method, the resulting daily calculations are very simple. The methodology is based on the estimation of the maximum value for the different sources of uncertainty and requires the definition of limiting values for certain analytical parameters. The simplification of the instrumental quantification uncertainty estimation involves the use of the standard deviation obtained from control charts relating to the concentrations estimated from the calibration curves for control standards at the highest calibration level. Three levels of simplification are suggested, as alternatives to the detailed approach, which can be selected according to the proximity of the sample results to decision limits. These approaches were applied to the determination of pesticide residues in apples (CEN, EN 12393), for which the most simplified approach showed a relative expanded uncertainty of 37.2% for a confidence level of approximately 95%.     

Effect of citrate impurities on the reference pH value of potassium dihydrogen buffer solution

Solutions of potassium dihydrogen citrate, KH₂C₆H₅O₇, exhibit reference pH values close to those of the primary standard pH buffer solutions of tartrate and phthate. Nevertheless, as opposed to all other recommended primary buffer standards, the citrate reagent is not commercially available as a standard reference material (SRM).In this work, experiments have been performed on solid reagent from different sources, which revealed the presence of the various citrate species in variable amounts. This may arise from the sensitivity of the system to the purification strategy, owing to the proximity of the acidity constants of the triprotic citric acid, H₃C₆H₅O₇.An experimental procedure has been developed which leads to a satisfactory preparation of the citrate reagent, to an extent that its solutions, independently of batch on supplier, exhibit reproducible pH values, in agreement with the conventionally assigned pH (S) = 3.776 at 25 °C for 0.05 m KH₂C₆H₅O₇.Comparison of expected and experimental values is used as a quantitative test for the quality of reagent, eventually indicating the need for purification according to recommendations.     

Acidity function p(a(H) gamma(Cl)) as a step to pH assessment

The conventional assignment of pH reference buffer standards, pH(S), is achieved by means of a series of procedures that follow from measurement of Harned cell potentials for an electrolyte solution which is the buffer solution of interest. An intermediate step is assessment of the acidity function p( a(H) gamma(Cl))(0), the extrapolated value of a linear representation of the dependence of p( a(H) gamma(Cl)) on m(Cl) for at least three different molalities, m(Cl), of added alkali chloride (0.005; 0.010; 0.015 mol kg(-1) KCl). This experimental value can be compared with a theoretically expected value calculated from the dissociation constants of the buffer species. Whereas these calculations always give negative slopes for diprotic and triprotic acids and zero slope for monoprotic acids, experimental values with negative or positive slopes can be obtained for well fitting straight lines obtained for buffer solutions with ionic strengths from 0.0025 to 0.144 mol kg(-1). Such disagreement between theoretically and experimentally obtained values introduce an extra source of uncertainty in the establishment of pH(S) and on its traceability chain. In this work examples are presented and discussed for which the discrepancy between expected and experimental values leads to different intercept p( a(H) gamma(Cl))(0).     

Convergence of Newton–Kantorovich Approximations to an Approximate Zero

In this paper, we prove an a posteriori and an a priori convergence theorem for Newton–Kantorovich approximations starting from an initial point x ₀. We apply these results to operators that are analytic at interior points of a closed ball centered at x ₀ and of radius R. We obtain some theorems on approximate zeros and on approximate zeros of second kind for these operators, which improve previous results.     

$$L^{\infty }$$-norm and energy quantization for the planar Lane–Emden problem with large exponent

For any smooth bounded domain \(\Omega \subset {\mathbb {R}}^2\), we consider positive solutions to

$$\begin{aligned} \left\{ \begin{array}{lr}-\Delta u= u^p &{} \text{ in } \Omega \\ u=0 &{} \text{ on } \partial \Omega \end{array}\right. \end{aligned}$$

which satisfy the uniform energy bound

$$\begin{aligned}p\Vert \nabla u\Vert _{\infty }\le C\end{aligned}$$

for \(p>1\). We prove convergence to \(\sqrt{e}\) as \(p\rightarrow +\infty \) of the \(L^{\infty }\)-norm of any solution. We further deduce quantization of the energy to multiples of \(8\pi e\), thus completing the analysis performed in De Marchis et al. (J Fixed Point Theory Appl 19:889–916, 2017).