Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions

In 1991, one of the authors showed the existence of quadratic transformations between the Painlevé VI equations with local monodromy differences (1/2, a, b, ±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformations. They are related to the quadratic transformations obtained by Manin and Ramani–Grammaticos–Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic‐pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painlevé VI functions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frequency-domain measurement of the photodegradation process of fluorescein

The frequency-domain technique is applied to measure the photodegradation rate of fluorescein in aqueous solutions. The illuminating light is modulated, and the changes in fluorescence from the illuminated region are detected synchronously. A constant flow rate is imposed on the fluorescein solution to control the mass transport of fluorescein into the illuminated region. The fluorescence response is described by a model that assumes that photodegradation occurs from the triplet excited state. The predictions of the model are consistent with the observed variations in the fluorescence response with flow rate, modulation frequency and incident power. We discuss in this article how the dependence of the model parameters on experimental conditions can be used to infer the photodegradation rate as well as some of the details of the photodegradation mechanism. The results are consistent with the known mechanism of photodegradation of fluorescein. The frequency-domain technique gives a photodegradation rate of 53 s(-1) in an air-saturated solution and 37 s(-1) in solutions purged with argon gas.     

On singular univariate specializations of bivariate hypergeometric functions

It is tempting to evaluate F₂(x,1) and similar univariate specializations of Appell's functions by evaluating the apparent power series at x=0 straight away using the Gauss formula for ₂F₁(1). But this kind of naive evaluation can lead to errors as the ₂F₁(1) coefficients might eventually diverge; then the actual power series at x=0 might involve branching terms. This paper demonstrates these complications by concrete examples.     

Symbolic evaluation of coefficients in Airy-type asymptotic expansions

Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function.     

A method for the assay of hydrolytic enzymes using dynamic light scattering

Dynamic light-scattering techniques have been successfully used for the assay of several hydrolytic enzymes. The enzymes were assayed using substrate-coated colloidal particles. Hydrolysis of the substrate coat causes destabilization of the particles followed by particle aggregation. The rate of particle aggregation can be related to the initial concentration of added enzyme.     

Additional symmetry for the electronic shell in its ground state and many-electron effects

The additional symmetry for the properties related to the ground state of the atom is considered taking into account many-electron effects. Calculations of the I _4f, I _3d,I _2p,I _3p binding energies, 4f ^N-15d - 4f^N system differences and 2p, 3p electron affinities in the second order of perturbation theory and in the configuration interaction approximation have been performed for the ground configurations with one open shell. The analysis of separate many-electron corrections for these quantities and their variation along the sequences of atoms and ions shows that the main corrections maintain the considered symmetry. Received 18 January 1999 and Received in final form 17 July 1999     

Hyperfine induced {\sf 1 s}{\sf 2 s}\ ^{\sf 1}S_{\sf 0} \rightarrow {\sf 1 s}^{\sf 2}\ ^{\sf 1}S0 M1 transition of He-like ions

Hyperfine induced $1s2s\ ^1$ S $_0 \rightarrow 1s^2\ ^1$ S₀ M1 transition probabilities of He-like ions have been calculated from relativistic configuration interaction wavefunctions including the frequency independent Breit interaction and QED effects. Present results for ¹⁵¹Eu and ¹⁵⁵Gd are in good agreement with previous calculations [L.N. Labzowsky et al., Phys. Rev. A 63, 054105 (2001)]. Electronic data are given in terms of a general scaling law in Z that, given isotopic nuclear spin and magnetic moment, allows hyperfine induced decay rates to be estimated for any isotope. The results should be helpful for future experimental investigations on QED and parity non-conservation effects.