Computation of Spiral Spectra

A computational linear stability analysis of spiral waves in a reaction-diffusion equation is performed on large disks. As the disk radius R increases, eigenvalue spectra converge to the absolute spectrum predicted by Sandstede and Scheel. The convergence rate is consistent with 1/R, except possibly near the edge of the spectrum. Eigenfunctions computed on large disks are compared with predicted exponential forms. Away from the edge of the absolute spectrum the agreement is excellent, while near the edge computed eigenfunctions deviate from predictions, probably due to finite-size effects. In addition to eigenvalues associated with the absolute spectrum, computations reveal point eigenvalues. The point eigenvalues and associated eigenfunctions responsible for both core and far-field breakup of spiral waves are shown.Comment: 20 pages, 13 figures, submitted to SIAD

Optimal Cross-Correlation Estimates from Asynchronous Tick-by-Tick Trading Data

Given two time series, A and B, sampled asynchronously at different times {t_A_i} and {t_B_j}, termed "ticks", how can one best estimate the correlation coefficient \rho between changes in A and B? We derive a natural, minimum-variance estimator that does not use any interpolation or binning, then derive from it a fast (linear time) estimator that is demonstrably nearly as good. This "fast tickwise estimator" is compared in simulation to the usual method of interpolating changes to a regular grid. Even when the grid spacing is optimized for the particular parameters (not often possible in practice), the fast tickwise estimator has generally smaller estimation errors, often by a large factor. These results are directly applicable to tick-by-tick price data of financial assets.Comment: 21 pages, 6 figures, 3 table

Magnification Ratio of the Fluctuating Light in Gravitational Lens 0957+561

Radio observations establish the B/A magnification ratio of gravitational lens 0957+561 at about 0.75. Yet, for more than 15 years, the optical magnfication ratio has been between 0.9 and 1.12. The accepted explanation is microlensing of the optical source. However, this explanation is mildly discordant with (i) the relative constancy of the optical ratio, and (ii) recent data indicating possible non-achromaticity in the ratio. To study these issues, we develop a statistical formalism for separately measuring, in a unified manner, the magnification ratio of the fluctuating and constant parts of the light curve. Applying the formalism to the published data of Kundi\'c et al. (1997), we find that the magnification ratios of fluctuating parts in both the g and r colors agrees with the magnification ratio of the constant part in g-band, and tends to disagree with the r-band value. One explanation could be about 0.1 mag of consistently unsubtracted r light from the lensing galaxy G1, which seems unlikely. Another could be that 0957+561 is approaching a caustic in the microlensing pattern.Comment: 12 pages including 1 PostScript figur

The Internet in India and China

This article compares the diffusion of the Internet in China and India. Using a six–dimension framework for characterizing the state of the Internet in a nation, we observe that, while both nations have made significant progress since our last comparison (in 1999), China enjoys a substantial lead over India. We also examine determinants of Internet diffusion. We find that the Chinese Internet has benefited from economic and trade reform begun in the late 1980s, a strong government commitment to the Internet, complementary human and capital resources, etc. The two nations have very different governments and policies, leading to differing approaches to the introduction of telecommunication competition and infrastructure development. China has pursued a strategy of competition among government–owned organizations while India has set policy via recommendations of publicly visible task forces. It remains to be seen whether India’s relatively transparent and market driven approach to Internet policy (and access) will prove effective in the long run. India and China have approximately 40 percent of the world population, and most of their inhabitants live in rural villages that lack basic telephone service. If the Internet is to succeed in raising the level of human development and curtailing migration to teeming urban centers, it must succeed in India and China. What we learn there may enable us to provide communication and information to the world\u27s 1.5 million unconnected villages

Hypergeometric Functions by Direct Path Integration

The proverb, "the longest way round is the shortest way home," dating from the 17th century, applies as well to present-day computer programming. The technique of choice in many real-life computations is not necessarily the most efficient or elegant one, but may instead be the one that is quick to program and easy to check

On formation of close binaries by two-body tidal capture

We calculate in detail the two-body tidal capture mechanism of Fabian, Pringle, and Rees: when two unbound stars have a close encounter, they may become bound by the energy that each deposits into nonradial oscillations of the other. After dimensional scalings are removed, the process depends only on a single dimensionless parameter, and on the dimensionless envelope structure of the stars. General formulae are derived; for definiteness, we apply them to the specific case of stars with an n = 3 polytropic structure. Capture cross sections as a function of velocity and capture rates for an isothermal distribution are given for the case of equal-mass stars; other cases can easily be computed from the formulae given

Fredholm and Volterra Integral Equations of the Second Kind

Integral equations are often the best way to formulate physics problems. However, the typical physics student gets almost no training in integral equations, in contrast to differential equations, for example. Many physicists thus believe that numerical solution of integral equations must be an extremely arcane topic, since it is almost never dealt with in numerical analysis textbooks