Superlinear convergence of the affine scaling algorithm

In this paper we show that a variant of the long-step affine scaling algorithm (with variable stepsizes) is two-step superlinearly convergent when applied to general linear programming (LP) problems. Superlinear convergence of the sequence of dual estimates is also established. For homogeneous LP problems having the origin as the unique optimal solution, we also show that 2/3 is a sharp upper bound on the (fixed) stepsize that provably guarantees that the sequence of primal iterates converge to the optimal solution along a unique direction of approach. Since the point to which the sequence of dual estimates converge depend on the direction of approach of the sequence of primal iterates, this result gives a plausible (but not accurate) theoretical explanation for why 2/3 is a sharp upper bound on the (fixed) stepsize that guarantees the convergence of the dual estimates. The work of this author was based on research supported by the Overseas Research Scholars of the Ministry of Education, Science and Culture of Japan, 1992. The work of this author was based on research supported by the National Science Foundation (NSF) under grant DDM-9109404 and the Office of Naval Research (ONR) under grant N00014-93-1-0234. This work was done while the second author was a faculty member of the Systems and Industrial Engineering Department at the University of Arizona.     

Infrared study on the interaction of poly(vinyl chloride) and ketones. I. PVC-MEK film system

In a previous article we presented preliminary information on IR spectrometric evidence for the interaction between poly(vinyl chloride) and ketones by using films with residual solvent as samples for spectrometry. This article reports detailed information on PVC–MEK film systems as compared to model compounds. The results indicate that there are “specific” interactions which were observed by a frequency shift of the carbonyl absorption of the ketone and changes on intensities of the C? Cl vibrational modes of the polymer, which are similar to the effects observed for nonpolymeric systems.     

Free monadic Tarski algebras

The free monadic Tarski algebra FMT(n) with a finite set G of n free generators, was determined by A. Figallo in [7]. In this paper we indicate a formula, as a function of n, which is an easier way of calculating the number of elements of FMT(n). Received November 3, 1995; accepted in final form August 26, 1996.     

Limiting behavior of the affine scaling continuous trajectories for linear programming problems

We consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems, we show that these trajectories tend to an optimal solution which in general depends on the starting point. By considering the trajectories that arise from the Lagrangian multipliers of the above mentioned logarithmic barrier families of problems, we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem. We also present results related to asymptotic direction of the affine scaling trajectories. We briefly discuss how to apply our results to linear programs formulated in formats different from the standard form. Finally, we extend the results to the primal-dual affine scaling algorithm.     

Application of Physicochemical Treatment Allows Reutilization of <Emphasis Type="Italic">Arthrospira platensis</Emphasis> Exhausted Medium

Since cultivations of Arthrospira platensis have a high water demand, it is necessary to develop treatment methods for reusing the exhausted medium that may prevent environmental problems and obtaining useful biomass. The exhausted Schlösser medium obtained from A. platensis batch cultivation in bench-scale mini-tanks was treated by varying concentrations of different coagulants, ferric chloride (6, 10, and 14 mg L^?1) or ferric sulfate (15, 25, and 35 mg L^?1) and powdered activated carbon (PAC, 30 and 50 mg L^?1). Such treated effluent was restored with NaNO₃ and reused in new cultivations of A. platensis performed in Erlenmeyer flasks. Reusing media through the cultivation of A. platensis showed satisfactory results, particularly in the medium treated with ferric chloride and PAC. The maximum cell concentration obtained in the flasks was 1093 mg L^?1, which corresponded to the medium treated with ferric chloride (6 mg L^?1) and PAC (30 mg L^?1). This cellular growth was higher than in the medium treated with ferric sulfate and PAC, in which values of maximum cell concentration did not exceed 796 mg L^?1. The cultures in the media after treatment did not modify the biomass composition. Thus, combined coagulation/adsorption processes, commonly used in water treatment processes, can be efficient and viable for treating exhausted medium of A. platensis, allowing the production of such biomass with the reduction of production cost and saving water.     

[C6H6, C3H7 +]and [C6H7 +, C3H6] ion-neutral complexes intermediate in the reactions of protonated propylbenzenes

From the mass-analysed ion kinetic energy spectra of labelled ions, kinetic energy releases and thermodynamic data, it is proved that protonated n-propylbenzene (1) isomerizes into protonated isopropyl benzene (2). It is also shown that the dissociation of the less energetic metastable ions of (2), leading to [iso-C₃H₇]⁺ and [C₆H₇]⁺ product ions, is preceded by H exchange. This H exchange involves two interconverting ion-neutral complexes [C₆H₆, iso-C₃H₇⁺] (2π) and [C₆H₇⁺, C₃H₆] (2α).     

Aqueous room temperature metal-catalyzed living radical polymerization of vinyl chloride

This paper describes the room-temperature living radical polymerization (LRP) of vinyl chloride in H2O/THF in the presence of Cu0 or CuI salts as catalysts, tren or PEI as ligands, and iodoform as initiator. The disproportionation reaction 2CuI + L --> Cu0 + CuII(L) is the crucial step, as it continuously provides the active species for both the initiation (Cu0) and the reversible termination step (CuII). Mn was found to increase linearly with conversion and is in good agreement with Mth, with the Mw/Mn being approximately 1.5.     

Synthesis, characterization, and reactivity of Ti(IV)-monosubstituted Keggin polyoxometalates

Ti(IV)-monosubstituted Keggin-type polyoxometalates (Ti-POMs), mu-oxo dimer [Bu4N]8[(PTiW11O39)2O] (1), and three monomers [Bu4N]4[PTi(L)W11O39], where L = OH (2), OMe (3), and OAr (4, ArOH = 2,3,6-trimethylphenol (TMP)), have been prepared starting from mu-hydroxo dimer [Bu4N]7[(PTiW11O39)2OH] (5) or heteropolyacid H5PW11TiO40 or both. The compounds have been characterized by elemental analysis, IR, UV-vis, and multinuclear (31P, 1H, 183W) NMR. The interaction of 1 and 3-5 with H2O in MeCN produces 2. The hydrolysis constants, estimated from 31P and 1H NMR data, are 0.006 and 0.04 for 1 and 3, respectively. Studies by 31P NMR, IR, potentiometric titration, and cyclic voltammetry revealed that 1-3 and 5 afford the same protonated titanium peroxo complex [Bu4N]4[HPTi(O2)W11O39] (I) upon interaction with aqueous H2O2 in MeCN. The rates of formation of I correlate with the rates of hydrolysis of the Ti-POMs and follow the order of 5 > 1 > 3. A two-step mechanism of the reaction of Ti-POMs with H2O2, which involves hydrolysis of the Ti-L bonds to yield 2 followed by fast interaction of 2 with hydrogen peroxide producing I, is suggested. The equilibrium constant for the reaction of 2 with H2O2 to yield I and H2O, estimated using 31P NMR, is 10. The interaction of the Ti-POMs with TMP follows the trends similar to their interaction with H2O) and requires preliminary hydrolysis of the Ti-L bonds. All of the Ti-POMs catalyze the oxidation of TMP with H2O2 in MeCN to give 2,3,5-trimethyl-p-benzoquinone and 2,2',3,3',5,5'-hexamethyl-4,4'-biphenol. The product distribution is similar for all of the Ti-POMs. The catalytic activities of the Ti-POMs correlate with the rates of formation of I and follow the order of 2 > 5 > 1 > 3. The findings lay a basis for a better understanding of the nature of the reactivity of titanium in Ti-catalyzed oxidations.     

Interior path following primal-dual algorithms. part II: Convex quadratic programming

We describe a primal-dual interior point algorithm for convex quadratic programming problems which requires a total of number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds an approximate Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea. The total number of arithmetic operations is shown to be of the order of O(n ³ L).