Sharp ULP rounding error bound for the hypotenuse function

The hypotenuse function, , is sometimes included in math library packages. Assuming that it is being computed by a straightforward algorithm, in a binary floating point environment, with round to nearest rounding mode, a sharp roundoff error bound is derived, for arbitrary precision. For IEEE single precision, or higher, the bound implies that and . Numerical experiments indicate that this bound is sharp and cannot be improved.

     

On the role of the Coulomb interaction in the mechanism of d-wave Cooper pairing of charge carriers in high-T c superconductors

The question of the effect of the structure of the anisotropic quasi-two-dimensional electron spectrum of high-T _c superconductors on the character of the screening of the Coulomb interaction and the symmetry of the superconducting order parameter is studied. Calculations of the polarization operator of electrons are performed on the basis of the single-particle band spectrum extracted from angle-resolved photoemission spectroscopy data. It is shown that the static screened Coulomb repulsion has a minimum at small momentum transfers. This corresponds to an effective electron-electron attraction in the -wave channel of Cooper pairing of the charge carriers on account of their interaction with the long-wavelength charge-density fluctuations. This attraction together with the anisotropic electron-phonon interaction increase the critical superconducting transition temperature T _c with increasing hole density and can give quite high values of T _c while at the same time suppressing the isotope effect, in qualitative agreement with the experimental data for underdoped hole-type cuprate metal-oxide compounds. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 10, 703–710 (25 May 1999)     

The synthesis of l-α-aminoadipyl-l-cysteinyl-d-3,4-didehydrovaline,a potent inhibitor of isopenicillin synthetase

The title peptide ($\text{1}$) has been synthesized and incubated with an active cell-free extract of $\text{Cephalosporium}$$\text{acremonium}$, no conversion to active antibiotics was observed; however on co-incubation with the Arnstein tripeptide (ACV) ($\text{2}$), strong inhibition of the conversion of ACV to isopenicillin N was observed.     

m-ary closed sequences

In this paper it is shown that for any three positive integers k, n and m, there exists a closed sequence (ordered cycle) of km-ary digits 0, 1,…, m − 1 such that the k subsequences of n consecutive digits are all different.     

An intrinsic state generating the ground state rotational band of a deformed nucleus and its approximate equivalence to an algebraic-variational treatment: Illustration by an exactly soluble model

We construct a class of intrinsic or trial states to generate the ground state band of a rotating nucleus in a highly schematized shell model possessing R(5) symmetry. This model is described in the preceding paper. The exponential form of the state links this work to a line of recent developments tracing back ultimately to the work of Jancovici and Schiff on the generator coordinate method. For the present case the state is a product of BCS states and is seen to contain more general (four-particle) correlations than a single deformed BCS state. A technique for the construction of the states is described. The major result of this paper is the proof that the cranking variational principle associated with the trial state is essentially equivalent (exactly equivalent in the thermodynamic or large system limit) to the algebraic-variational method described in the preceding paper, when the latter is restricted to the oneband approximation. The existence of sharp phase transitions in the model is then investigated analytically. A spherical to deformed transition is thus confirmed, as well as an anti-pairing transition. As explained, this is unrelated to the Coriolis anti-pairing phenomena.     

Asymptotic behavior of nonexpansive mappings in normed linear spaces

LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, lim_n→x f(T ⁿ x/n)=lim_n→x‖T ⁿ x/n ‖=α, where α≡inf _y∈c ‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichT ⁿ x/n converges weakly for allx (inf_z∈f g(T ⁿ x/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT. Supported by National Science Foundation Grant MCS-79-066.     

Equations-of-motion approach to a quantum theory of large-amplitude collective motion

The equations-of-motion approach to large-amplitude collective motion is implemented both for systems of coupled bosons, also studied in a previous paper, and for systems of coupled fermions. For the fermion case, the underlying formulation is that provided by the generalized Hartree-Fock approximation (or generalized density matrix method). To obtain results valid in the semi-classical limit, as in most previous work, we compute the Wigner transform of quantum matrices in the representation in which collective coordinates are diagonal and keep only the leading contributions. Higher-order contributions can be retained, however, and, in any case, there is no ambiguity of requantization. The semi-classical limit is seen to comprise the dynamics of time-dependent Hartree-Fock theory (TDHF) and a classical canonicity condition. By utilizing a well-known parametrization of the manifold of Slater determinants in terms of classical canonical variables, we are able to derive and understand the equations of the adiabatic limit in full parallelism with the boson case. As in the previous paper, we can thus show: (i) to zero and first order in the adiabatic limit the physics is contained in Villars' equations; (ii) to second order there is consistency and no new conditions. The structure of the solution space (discussed thoroughly in the previous paper) is summarized. A discussion of associated variational principles is given. A form of the theory equivalent to self-consistent cranking is described. A method of solution is illustrated by working out several elementary examples. The relationship to previous work, especially that of Zeievinsky and Marumori and coworkers is discussed briefly. Three appendices deal respectively with the equations-of-motion method, with useful properties of Slater determinants, and with some technical details associated with the fermion equations of motion.     

Gas-chromatographic quantitation of lindane in household insecticide spray cans

Summary A method was developed for the determination of the lindane content of commercial household insecticide spray aerosols. After the propellant is removed from the chilled contents of the spray packages the hydrocarbon solution is analysed for its lindane content by electron-capture gas chromatography. Pure -isomer of hexachlorocyclohexane serves as the internal standard. Experimental error is ± 5%.

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Gas-chromatographische Bestimmung von Lindan in Haushalts-Spraydosen

Zusammenfassung Das Treibmittel wird durch Ausfrieren abgetrennt und die Insecticidlösung gas-chromatographisch mit Electron-capture Detektor analysiert. Das reine -Isomere dient als innerer Standard. Der Fehler beträgt ± 5%.