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

Breakdown of self-similarity at the crests of large amplitude standing water waves

We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokes's sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation patterns that occur. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.Comment: 4 pages, 5 figures. Final version accepted for publicatio

Non-white frequency noise in spin torque oscillators and its effect on spectral linewidth

We measure the power spectral density of frequency fluctuations in nanocontact spin torque oscillators over time scales up to 50 ms. We use a mixer to convert oscillator signals ranging from 10 GHz to 40 GHz into a band near 70 MHz before digitizing the time domain waveform. We analyze the waveform using both zero crossing time stamps and a sliding Fourier transform, discuss the different limitations and advantages of these two methods, and combine them to obtain a frequency noise spectrum spanning more than five decades of Fourier frequency $f$. For devices having a free layer consisting of either a single Ni$_{\text{}80}$Fe$_{\text{}20}$ layer or a Co/Ni multilayer we find a frequency noise spectrum that is white at large $f$ and varies as \emph{$1/f$} at small $f$. The crossover frequency ranges from \approx\unit[10^{4}]{Hz} to \approx\unit[10^{6}]{Hz} and the $1/f$ component is stronger in the multilayer devices. Through actual and simulated spectrum analyzer measurements, we show that $1/f$ frequency noise causes both broadening and a change in shape of the oscillator's spectral line as measurement time increases. Our results indicate that the long term stability of spin torque oscillators cannot be accurately predicted from models based on thermal (white) noise sources

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

Electrical transport in the ferromagnetic state of manganites: Small-polaron metallic conduction at low temperatures

We report measurements of the resistivity in the ferromagnetic state of epitaxial thin films of La_{1-x}Ca_{x}MnO_{3} and the low temperature specific heat of a polycrystalline La_{0.8}Ca_{0.2}MnO_{3}. The resistivity below 100 K can be well fitted by \rho - \rho_{o} = E \omega_{s}/sinh^{2}(\hbar \omega_{s}/2k_{B}T) with \hbar \omega_{s}/k_{B} \simeq 100 K and E being a constant. Such behavior is consistent with small-polaron coherent motion which involves a relaxation due to a soft optical phonon mode. The specific heat data also suggest the existence of such a phonon mode. The present results thus provide evidence for small-polaron metallic conduction in the ferromagnetic state of manganites.Comment: 4 pages, 4 figures, submitted to PR

A Superbubble Feedback Model for Galaxy Simulations

We present a new stellar feedback model that reproduces superbubbles. Superbubbles from clustered young stars evolve quite differently to individual supernovae and are substantially more efficient at generating gas motions. The essential new components of the model are thermal conduction, sub-grid evaporation and a sub-grid multi-phase treatment for cases where the simulation mass resolution is insufficient to model the early stages of the superbubble. The multi-phase stage is short compared to superbubble lifetimes. Thermal conduction physically regulates the hot gas mass without requiring a free parameter. Accurately following the hot component naturally avoids overcooling. Prior approaches tend to heat too much mass, leaving the hot ISM below $10^6$ K and susceptible to rapid cooling unless ad-hoc fixes were used. The hot phase also allows feedback energy to correctly accumulate from multiple, clustered sources, including stellar winds and supernovae. We employ high-resolution simulations of a single star cluster to show the model is insensitive to numerical resolution, unresolved ISM structure and suppression of conduction by magnetic fields. We also simulate a Milky Way analog and a dwarf galaxy. Both galaxies show regulated star formation and produce strong outflows.Comment: 13 pages, 13 figures; replaced with version accepted to MNRA

Oxygen-isotope effect on the superconducting gap in the cuprate superconductor Y_{1-x}Pr_xBa_2Cu_3O_{7-\delta}

The oxygen-isotope (^{16}O/^{18}O) effect (OIE) on the zero-temperature superconducting energy gap \Delta_0 was studied for a series of Y_{1-x}Pr_xBa_2Cu_3O_{7-\delta} samples (0.0\leq x\leq0.45). The OIE on \Delta_0 was found to scale with the one on the superconducting transition temperature. These experimental results are in quantitative agreement with predictions from a polaronic model for cuprate high-temperature superconductors and rule out approaches based on purely electronic mechanisms.Comment: 5 pages, 3 figure