Convergence Rates for Newton’s Method at Singular Points

If Newton’s method is employed to find a root of a map from a Banach space into itself and the derivative is singular at that root, the convergence of the Newton iterates to the root is linear rather than quadratic. In this paper we give a detailed analysis of the linear convergence rates for several types of singular problems. For some of these problems we describe modifications of Newton’s method which will restore quadratic convergence

Spin-Wave Lifetimes Throughout the Brillouin Zone

We use a neutron spin-echo method with $\mu$eV resolution to determine the lifetimes of spin waves in the prototypical antiferromagnet MnF$_2$ over the entire Brillouin zone. A theory based on the interaction of magnons with longitudinal spin fluctuations provides an excellent, parameter-free description of the data, except at the lowest momenta and temperatures. This is surprising, given the prominence of alternative theories based on magnon-magnon interactions in the literature. The results and technique open up a new avenue for the investigation of fundamental concepts in magnetism. The technique also allows measurement of the lifetimes of other elementary excitations (such as lattice vibrations) throughout the Brillouin zone.Comment: 12 pages, 4 figure

Double window viewing chamber assembly

A viewing chamber which permits observation of a sample retained therein includes a pair of double window assemblies mounted in opposed openings in the walls thereof so that a light beam can directly enter and exit from the chamber. A flexible mounting arrangement for the outer windows of the window assemblies enables the windows to be brought into proper alignment. An electrical heating arrangement prevents fogging of the outer windows whereas desiccated air in the volume between the outer and inner windows prevents fogging of the latter

Recurrence spectrum in smooth dynamical systems

We prove that for conformal expanding maps the return time does have constant multifractal spectrum. This is the counterpart of the result by Feng and Wu in the symbolic setting

Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A0, the eigenpairs of A(t) ≡ (1 − t)A0 + tA1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix: • bifurcation, • ill conditioning due to nonorthogonal eigenvectors, • jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented

Momentum-resolved electron-phonon interaction in lead determined by neutron resonance spin-echo spectroscopy

Neutron resonance spin-echo spectroscopy was used to monitor the temperature evolution of the linewidths of transverse acoustic phonons in lead across the superconducting transition temperature, $T_c$, over an extended range of the Brillouin zone. For phonons with energies below the superconducting energy gap, a linewidth reduction of maximum amplitude $\sim 6 \mu$eV was observed below $T_c$. The electron-phonon contribution to the phonon lifetime extracted from these data is in satisfactory overall agreement with {\it ab-initio} lattice-dynamical calculations, but significant deviations are found

Energy Gaps and Kohn Anomalies in Elemental Superconductors

The momentum and temperature dependence of the lifetimes of acoustic phonons in the elemental superconductors Pb and Nb was determined by resonant spin-echo spectroscopy with neutrons. In both elements, the superconducting energy gap extracted from these measurements was found to converge with sharp anomalies originating from Fermi-surface nesting (Kohn anomalies) at low temperatures. The results indicate electron many-body correlations beyond the standard theoretical framework for conventional superconductivity. A possible mechanism is the interplay between superconductivity and spin- or charge-density-wave fluctuations, which may induce dynamical nesting of the Fermi surface