On a functional equation of Luce

Summary In a recent communication to J. Aczél, R. Duncan Luce asked about the functional equationU(x)U(G(x)F(y)) = U(G(x))U(xy) forx, y > 0, (1) which has arisen in his research on certainty equivalents of gambles. He was particularly interested in cases in which the unknowns (U, F andG) are strictly increasing functions from (0, + ) into (0, + ). In this paper we solve (1) in the case whereU, F andG are continuously differentiable with everywhere positive first derivatives. Our solution is perhaps novel in that in certain cases (1) reduces to a functional equation in a single variable and in other cases to a functional equation in several variables; see [1] for the terminology.     

Quantitative millimetre wavelength spectrometry at pressures approaching atmospheric: II. Determination of oxygen at atmospheric pressure

A millimetre wavelength (MMW) Fabry-Perot cavity spectrometer described in earlier work has been applied to the measurement of oxygen absorption at 60 GHz and atmospheric pressure in a gas matrix of nitrogen. The spectrometer has also been modified such that the MMW source is stabilised by a sub-harmonic microwave signal transmitted by an infrared carrier on a single mode telecommunications fibre optic. This is a step towards developing an instrument comprising minimal electronic components that can perform MMW spectrometry remotely. Oxygen determinations were achieved by monitoring the change in the quality factor (Q) of a resonant Fabry-Perot cavity due to the presence of an absorbing sample. The MMW absorption of the sample was determined by incrementing the frequency modulation (FM) deviation of the source frequency scanning the cavity resonance profile. The response curve of absorption signal versus fraction of oxygen in nitrogen was found to be linear throughout the working range of 1-100% O₂ (v/v) in N₂ with a slope of (1.407±0.007)×10⁻⁴ m⁻¹ (% O₂)⁻¹. The detection limit (3× standard deviation of the background) was found to be ∼0.8% (v/v). The MMW technique employed is advantageous since, unlike common MMW techniques, there is no vacuum requirement. Application of this method, to the monitoring of oxygen in gas mixtures of practical importance, is proposed. Values of the oxygen spectral absorption coefficients of lines between 55 and 60 GHz were measured at reduced pressure and found to be within ±2% of previous literature values. A pressure correction coefficient for O₂ absorption at 60 GHz in the 45-121 kPa range was obtained and found to be (1.354±0.014)×10⁻⁴ m⁻¹ kPa⁻¹.     

An algorithm for the location of branching points on reaction paths

A branching point is a point on a reaction path leading from reactants to products (via a transition state) at which it is energetically favorable for the system to break symmetry. Such a point can be defined in terms of normal modes along the reaction path and corresponds to zero curvature (a zero Hessian eigenvalue) along a symmetry-breaking mode. An effective method for the location of such points is presented and realized in an efficient, practical algorithm designed for use in the ab initio program package Gaussian 82.     

Effects of Poly(ethylene glycol) Doping on the Behavior of Pyrene, Rhodamine 6G, and Acrylodan-Labeled Bovine Serum Albumin Sequestered within Tetramethylorthosilane-Derived Sol-Gel-Processed Composites

We investigate the effects of controlled poly(ethylene glycol) (PEG) doping on the behavior of pyrene, rhodamine 6G (R6G), and acrylodan-labeled bovine serum albumin (BSA-Ac) sequestered within tetramethylorthosilicate (TMOS)-derived sol-gel-processed materials. To probe the dipolarity of the local environment within the composite we performed static fluorescence measurements on pyrene as the composites aged. We found that small levels of PEG loading effected significant enhancements in the local dipolarity surrounding the average pyrene molecule. Time-resolved fluorescence anisotropy measurements were used to follow the rotational reorientation dynamics of R6G as the composites aged. As the PEG loading increased, the R6G reorientational mobility increased. Nitrogen adsorption techniques were used to quantify the effects of PEG doping level on the surface area and final xerogel pore features. A large reduction in surface area was observed with PEG doping, but no detectable change in pore size was noted. The effects of PEG doping on a biomolecule were probed by following the time-resolved fluorescence anisotropy decay of BSA-Ac. These results showed that PEG doping resulted in increased biomolecule dynamics relative to that found for a neat, undoped TMOS-derived composites. Together these results show that PEG doping can be used to tune the sol-gel-processed composite dipolarity, alter the mobility of dopants sequestered within the composite, control analyte acessibility to the sensing chemistry, and modulate the internal dynamics within a biodopant.     

The determination of sulphuric acid in air: A specific method

Summary A new method for the measurement of sulphuric acid in airborne particulates is described. Sulphuric acid present is separated from the sample matrix by microdiffusion at 200°. The separated sulphuric acid is absorbed in sodium hydroxide, and titrated by barium perchlorate.

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Zusammenfassung Ein Verfahren zur Bestimmung von Schwefelsäure als Verunreinigung der Atmosphäre wurde beschrieben. Die Abtrennung von der Luftprobe erfolgt durch Mikrodiffusion bei 200° C. Die Schwefelsäure wird in Natronlauge absorbiert und mit Bariumperchlorat titriert.