Estimating the complexity of a class of path-following methods for solving linear programs by curvature integrals

In this paper we study a particular class of primal-dual path-following methods which try to follow a trajectory of interior feasible solutions in primal-dual space toward an optimal solution to the primal and dual problem. The methods investigated are so-called first-order methods: each iteration consists of a long step along the tangent of the trajectory, followed by explicit recentering steps to get close to the trajectory again. It is shown that the complexity of these methods, which can be measured by the number of points close to the trajectory which have to be computed in order to achieve a desired gain in accuracy, is bounded by an integral along the trajectory. The integrand is a suitably weighted measure of the second derivative of the trajectory with respect to a distinguished path parameter, so the integral may be loosely called a curvature integral.     

A quadratically convergent O(\sqrt n L)-iteration algorithm for linear programming

Recently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Ye's predictor—corrector interior-point algorithm for linear programming maintains the O( L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.Supported in part by NSF Grant DDM-8922636 and NSF Coop. Agr. No. CCR-8809615, the Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.Supported in part by NSF Coop. Agr. No. CCR-8809615, AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Supported in part by NSF Grant DMS-9102761 and DOE Grant DE-FG05-91ER25100.     

The theoretical basis of column chromatography in multicomponent separations. Part 2: Peak capacity. Column systems with good selectivity

The selectivity of a column system, S, defined by equation 28, includes a variation coefficient (β) of plate number with capacity ratio which has a large influence on the peak capacity, as shown in Fig. 18. Some typical chromatograms are given. In order to predict S for a column system from Kovat's Index, equations 40 and 41 are given to calculate the constant of the carbon number rule for squalane at different temperatures or for different stationary phases. The specific retention value of heptane on squalane at different temperatures can be calculated from equation 42. The nonpolarity index, defined by equation 43, was used to calculate the retention value of heptane on various stationary phases. In liquid chromatography, the order of elution may be reversed by changing the composition of the eluent on the same chemically bonded silica (manufactured in China). The linear relations between the log retention values of different kinds of solutes or of a single solute on the silicas of different surface areas when using the same eluent are given.     

On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx ₀ =x, x _j =x _j–1 –f(x _j–1)/f(x _j–1),j = 1, 2,, then |x _j –z| < (1/|x ₀ –z|)|x _j–1–z|².     

Evaluation of measurement uncertainty for analytical procedures using a linear calibration function

In the EURACHEM/CITAC draft ”Quantifying uncertainty in analytical measurement” estimations of measurement uncertainty in analytical results for linear calibration are given. In this work these estimations are compared, i.e. the uncertainty deduced from repeated observations of the sample vs. the uncertainty deduced from the standard residual deviation of the regression. As a result of this study it is shown that an uncertainty estimation based on repeated observations can give more realistic values if the condition of variance homogeneity is not correctly fulfilled in the calibration range. The complete calculation of measurement uncertainty including assessment of trueness is represented by an example concerning the determination of zinc in sediment samples using ICP-atomic emission spectrometry. Received: 9 February 2002 Accepted: 17 April 2002     

How anomalous are the anomalous properties of fluorine? Ionization energy and electron affinity revisited

It is well established that atomic fluorine has a miserly high ionization energy and avariciously high electron affinity. Yet, as was shown some 30 years ago, the latter value is anomalously low-linear extrapolation shows the value to be some 110 kJ/mol less than expected by comparison to the heavier halogens. Related literature investigation extended this anomaly to nitrogen and oxygen and their (valence) isoelectronic congeners. We extend it to related cationic species and thereby the noble gases and alkali metals, and the aforementioned species are better understood. We thus ask if the ionization energy/electron affinity anomaly of fluorine remains an anomaly.     

On linear free energy correlations in liquid-phase catalytic hydrogenation of aromatic compounds

Application of linear free energy relationships (correlation between kinetics and thermodynamics) to catalytic hydrogenation of aromatic compounds in the liquidphase over supported metal catalysts is discussed.     

Validation steps for traceability of linear calibrated chemical measurements

Based on the new draft of the EURACHEM/CITAC Guide Traceability in Chemical Measurement, this publication describes how traceability can be achieved for chemical measurements using a linear calibration function. Traceability can be accomplished without larger expenditure, if the measurement is calibrated on the basis of appropriate reference standards and the linear regression employed is selected and validated statistically in a suitable form. The determination of nickel in aqua regia eluates of sediment samples, employed for an ICP-OES measurement, is used as a practical illustration of this approach.     

Development of a miniature simultaneous MPT spectrometer

An unique miniature simultaneous microwave plasma torch (MPT) atomic emission spectrometer employing linear UV intensified charge-coupled device (ICCD) array detector has been developed and studied preliminarily. The detection limits and precisions of the spectrometer for Ag, Al, Ba, Ca, Cr, Cu, Fe, Mg, Mn, Sr and V by using different CCD exposure times have been determined. An analysis of a practical sample has been carried out. The preliminary results demonstrate that such simultaneous spectrometer has advantages of saving sample and time, especially suitable for use as detector for chromatography and in combination with flow injection systems. Taking analytical figures of merit and portability into accounts, the miniature simultaneous MPT system will have extended application areas and greater competition potential as compared with commercialized scanning MPT spectrometers.