On a hypergeometric function of four variables with a new aspect ofSL-symmetry

Summary The integration of the system of partial differential equations associated with the quadruple hypergeometric function K₁₆, first introduced by the author (Exton (1976), page 78), is undertaken. The interest in this function has been stimulated by the investigation of certain SL-symmetry groups by Hrabowski (1984), (1985).     

Perturbing Fredholm operators to obtain invertible operators

A condition is given on a set $\text{Ol}$ of operators on Hilbert space that guarantees it has the following property: For any Fredholm operator T of index zero there exists anA?$\text{A}$ such that T + ?A is invertible for all sufficiently small nonzero ?. As a corollary one obtains in a quite general setting the density of the invertible Toeplitz operators in the set of Fredholm Toeplitz operators of index zero.     

A note on Fermat's problem

The General Fermat Problem asks for the minimum of the weighted sum of distances fromm points inn-space. Dozens of papers have been written on variants of this problem and most of them have merely reproduced known results. This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm. Although the same algorithm has been rediscovered at least three times, there seems to be no completely correct treatment of its properties in the literature. Such a treatment, including a proof of convergence, is the sole object of this note. Other aspects of the problem are given scant attention.This paper was written while the author was Science Faculty Fellow of the National Science Foundation at the London School of Economics.     

On representing a black box as a dynamical system

Necessary conditions are derived for a black box to be representable as a dynamical system governed by ordinary differential equations. Sufficiency of these conditions depends primarily on the set of inputs. It is shown that the conditions are indeed sufficient for the set of constant inputs and under suitable hypotheses for the set of smooth inputs. These conditions cannot be verified by a finite number of input-output observations.     

Lewis-Randall to McMillan-Mayer conversion for the thermodynamic excess functions of solutions. Part II. Excess energy and volume

Derivations are given for the equations expressing the McMillan-Mayer functions in terms of Lewis-Randall functions.     

The conformation of erythronolide,the 14-membered aglycone ring of the erythromycin antibiotics

The solution conformation of erythronolide B and derivatives has been deduced from NMR and CD spectral data. The conformation is basically a “diamond lattice” type in which the ring atoms occupy cyclohexane-like positions but slightly modified to incorporate the lactone and ketone groups and to relieve an unfavorable syn-periplanar interaction. Variable temperature experiments revealed that erythronolide B and derivatives do not undergo facile ring inversion or pseudorotation and suggested the presence of a single stable conformation. Several features of the proposed conformation were demonstrated by the NMR and CD spectra of suitable derivatives. These features included the syn-periplanar relationship between the 3- and 5-OH groups, the axial orientation of the 11-OH group, and the proximity of the 6-OH group to the 9-ketone.     

A RAPID ASSAY FOR DNA PHOTOLYASE USING A MEMBRANE-BINDING TECHNIQUE*

Abstract— A rapid assay for DNA photolyase was developed that makes use of the membrane binding technique of A. D. Riggs, H. Suzuki and S. Bourgeois, J. Mol. Biol. 48 , 67–83 (1970). The complex formed between UV-irradiated bacteriophage T7 DNA-³H and the enzyme is trapped on a nitrocellulose filter after a five-minute equilibration and is counted. Using this technique, one can calculate the molar concentration of enzyme in an enzyme preparation and the saturation constant (K) for the complex formation. The latter value is 2.6 cyclobutadipyrimidines per genome (T7) for yeast DNA photolyase; thus approximately 5.2 dipyrimidines/genome are necessary for the full retention of the DNA on the membrane. The speed and reproducibility of the new method makes it a convenient one for assaying column effluents and as many as one hundred samples can be handled routinely in an afternoon.