Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems

We investigate multi-grid methods for solving linear systems arising from arc-length continuation techniques applied to nonlinear elliptic eigenvalue problems. We find that the usual multi-grid methods diverge in the neighborhood of singular points of the solution branches. As a result, the continuation method is unable to continue past a limit point in the Bratu problem. This divergence is analyzed and a modified multi-grid algorithm has been devised based on this analysis. In principle, this new multi-grid algorithm converges for elliptic systems, arbitrarily close to singularity and has been used successfully in conjunction with arc-length continuation procedures on the model problem. In the worst situation, both the storage and the computational work are only about a factor of two more than the unmodified multi-grid methods

Axiomatic Holonomy Maps and Generalized Yang-Mills Moduli Space

This article is a follow-up of ``Holonomy and Path Structures in General Relativity and Yang-Mills Theory" by Barrett, J. W. (Int.J.Theor.Phys., vol.30, No.9, 1991). Its main goal is to provide an alternative proof of this part of the reconstruction theorem which concerns the existence of a connection. A construction of connection 1-form is presented. The formula expressing the local coefficients of connection in terms of the holonomy map is obtained as an immediate consequence of that construction. Thus the derived formula coincides with that used in "On Loop Space Formulation of Gauge Theories" by Chan, H.-M., Scharbach, P. and Tsou S.T. (Ann.Phys., vol.167, 454-472, 1986). The reconstruction and representation theorems form a generalization of the fact that the pointed configuration space of the classical Yang-Mills theory is equivalent to the set of all holonomy maps. The point of this generalization is that there is a one-to-one correspondence not only between the holonomy maps and the orbits in the space of connections, but also between all maps from the loop space on $M$ to group $G$ fulfilling some axioms and all possible equivalence classes of $P(M,G)$ bundles with connection, where the equivalence relation is defined by bundle isomorphism in a natural way.Comment: amslatex, 7 pages, no figure

Stochastic blockmodel approximation of a graphon: Theory and consistent estimation

Non-parametric approaches for analyzing network data based on exchangeable graph models (ExGM) have recently gained interest. The key object that defines an ExGM is often referred to as a graphon. This non-parametric perspective on network modeling poses challenging questions on how to make inference on the graphon underlying observed network data. In this paper, we propose a computationally efficient procedure to estimate a graphon from a set of observed networks generated from it. This procedure is based on a stochastic blockmodel approximation (SBA) of the graphon. We show that, by approximating the graphon with a stochastic block model, the graphon can be consistently estimated, that is, the estimation error vanishes as the size of the graph approaches infinity.Comment: 20 pages, 4 figures, 2 algorithms. Neural Information Processing Systems (NIPS), 201

Archetypal analysis of galaxy spectra

Archetypal analysis represents each individual member of a set of data vectors as a mixture (a constrained linear combination) of the pure types or archetypes of the data set. The archetypes are themselves required to be mixtures of the data vectors. Archetypal analysis may be particularly useful in analysing data sets comprising galaxy spectra, since each spectrum is, presumably, a superposition of the emission from the various stellar populations, nebular emissions and nuclear activity making up that galaxy, and each of these emission sources corresponds to a potential archetype of the entire data set. We demonstrate archetypal analysis using sets of composite synthetic galaxy spectra, showing that the method promises to be an effective and efficient way to classify spectra. We show that archetypal analysis is robust in the presence of various types of noise.Comment: 6 pages, 5 figures, 1 style-file. Accepted for publication by MNRA

Activation barrier scaling and crossover for noise-induced switching in a micromechanical parametric oscillator

We explore fluctuation-induced switching in a parametrically-driven micromechanical torsional oscillator. The oscillator possesses one, two or three stable attractors depending on the modulation frequency. Noise induces transitions between the coexisting attractors. Near the bifurcation points, the activation barriers are found to have a power law dependence on frequency detuning with critical exponents that are in agreement with predicted universal scaling relationships. At large detuning, we observe a crossover to a different power law dependence with an exponent that is device specific.Comment: 5 pages, 5 figure

Risk and Return on Real Estate: Evidence from Equity REITs

We analyze monthly returns on an equally-weighted index of 18 to 23 equity (real property) real estate investment trusts (REITs) that were traded on major stock exchanges over the 1973-87 period. We employ a multifactor Arbitrage Pricing Model using prespecified macroeconomic factors. We also test whether equity REIT returns are related to changes in the discount on closed-end stock funds, which seems plausible given the closed-end nature of REITs. Three factors, and the percentage change in the discount on closed-end stock funds, consistently drive equity REIT returns: unexpected inflation and changes in the risk and term structures of interest rates. The impacts of these variables on equity REIT returns is around 60 percent of the impacts on corporate stock returns generally. As expected, the impacts are greater for more heavily levered REITs than for less levered REITs. Real estate, at least as measured by the return performance of equity REITs, is less risky than stocks generally, but does not offer a superior risk-adjusted return and is not a hedge against unexpected inflation.

An atlas of 1975 GEOS-3 radar altimeter data for hurricane/tropical disturbance studies, volume 1

Geographic locations of 1975 hurricanes and other tropical disturbances were correlated with the closest approaching orbits of the GEOS-3 satellite and its radar altimeter. The disturbance locations and altimeter data were gathered for a seven-month period beginning with GEOS-3 launch in mid-April 1975. Areas of coverage were the Atlantic Ocean, the Carribean, the Gulf of Mexico, the west coast of the continental United States, and the central and western Pacific Ocean. Volume 1 contains disturbance coverage data for the Atlantic Ocean, Gulf of Mexico, and Eastern Pacific Ocean. Central and Western Pacific coverage is documented in Volume II