Bounding the consistency strength of a five element linear basis

In [13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than a Mahlo cardinal, a hypothesis which can hold in the constructible universe L. A crucial notion in the proof is the saturation of an Aronszajn tree, a statement which may be of broader interest. We show that if all Aronszajn trees are saturated and PFA(ω ₁) holds, then there is a five element basis for the uncountable linear orders. We show that PFA(ω ₂) implies that all Aronszajn trees are saturated and that it is consistent to have PFA(ω ₁) plus every Aronszajn tree is saturated relative to the consistency of a reflecting Mahlo cardinal. Finally we show that a hypothesis weaker than the existence of a Mahlo cardinal is sufficient to force the existence of a five element basis for the uncountable linear orders. The first author acknowledges a fellowship granted by the French ministry of research. The research of the second author was partially supported by the Centre de Rercerca Matemàtica of the Universitat Autònoma de Barcelona, and by NSF Grant DMS-0401603. The second author would also like to thank the third and fourth authors for bringing him to Boise and Paris respectively for further discussions. The third author was supported by NSF grants DMS-0401893 and DMS-0200671. The second and fourth authors would like to thank CIRM in Luminy for hosting them during a petit group de travaille, and to thank the others participants, Ralf Schindler and Ernest Schimmerling, for discussions on this topic.     

Kinetics of Spontaneous Hydrolysis of O-(2,2-Dimethylcyclopentyl)-methylfluorophosphonate and of Hydrolysis During Constant pH Value of the Reaction Medium

Hydrolysis of O-(2,2-dimethylcyclopentyl)-methylfluorophosphonate in dilute aqueous solution was investigated. Rates of hydrolysis of O-(2,2-dimethylcyclopentyl)-methylfluorophosphonate were measured as a function of pH (in the range from 3 to 9 and without keeping the pH-value constant--in a spontaneous hydrolysis) and temperature (in the range from 25°;C to 40°;C). Objective of these investigations was to simulate real processes of degradation of highly toxic compounds in nature, getting information about rates of degradation, in order to obtain most efficient systems for their removal.     

Optimality conditions for discrete optimal control problems with equality and inequality type of constraints

We consider nonlinear discrete optimal control problems with variable endpoints and with equality and inequality type of constraints on control. We derive new nontrivial first and second-order necessary optimality conditions.      

Application for optical thermometers calibration based on virtual instruments

In this paper we present a new developed application for calibration of optical thermometers which is implemented in the Serbian Metrology Institute. The method for calibration of optical thermometer–pyrometer is based on measuring blackbody temperature via thermocouple standards and measuring voltage on pyrometer in temperature range from 550 to 1650 °C. The objective is to precisely determinate calibration curve of pyrometer—transfer function. The metrology traceability was realized through thermocouples standard which has traceability to the international standard. A developed application has a role to control and supervise calibration process. Based on virtual instruments (VI), application offers monitoring measurement parameters in real time on three charts. The main advantage of the developed VI application is reduction of measurement uncertainty of type A by increased accuracy with multi—temperature measurement up to 3 thermocouples on temperature equilibrium in the measurement system and automatic data processing by converting voltage to temperature, calculating the mean value and standard deviation of the mean value. Advantage is also reflected in permanent data logging while application is running and low cost and user friendly interface One of the goals is to improve quality of metrological services through this application and support the establishment of a traceability chain to SI.     

Parallel acquisition of two-dimensional NMR spectra of several nuclear species

We show that two or more two-dimensional NMR correlation spectra can be recorded in a single shot, using a multicoil radio-frequency probe and receiver system designed for simultaneous parallel acquisition of signals from different nuclear species such as 1H, 13C, and 15N. Dubbed PANSY (parallel acquisition NMR spectroscopy), this new technique shows promise for recording several multidimensional NMR spectra of different nuclear species in a very short time.     

First and second order necessary optimality conditions for a discrete-time optimal control problem with a vector-valued objective function

The paper deals with a nonlinear discrete-time optimal control problem with a vector-valued objective function of terminal type. For the admissible controls of the problem under consideration which satisfy the additional regularity condition of the Lyusternik type we prove the first and second necessary optimality conditions extending such the classical optimality conditions as the Euler condition and the negativity of second variation of the objective function.