Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and figures illustrating the main results of this paper and recent results from others. Sections 0, 2, and 6 have been significantly rewritte

Fuzzy Supernova Templates II: Parameter Estimation

Wide field surveys will soon be discovering Type Ia supernovae (SNe) at rates of several thousand per year. Spectroscopic follow-up can only scratch the surface for such enormous samples, so these extensive data sets will only be useful to the extent that they can be characterized by the survey photometry alone. In a companion paper (Rodney and Tonry, 2009) we introduced the SOFT method for analyzing SNe using direct comparison to template light curves, and demonstrated its application for photometric SN classification. In this work we extend the SOFT method to derive estimates of redshift and luminosity distance for Type Ia SNe, using light curves from the SDSS and SNLS surveys as a validation set. Redshifts determined by SOFT using light curves alone are consistent with spectroscopic redshifts, showing a root-mean-square scatter in the residuals of RMS_z=0.051. SOFT can also derive simultaneous redshift and distance estimates, yielding results that are consistent with the currently favored Lambda-CDM cosmological model. When SOFT is given spectroscopic information for SN classification and redshift priors, the RMS scatter in Hubble diagram residuals is 0.18 mags for the SDSS data and 0.28 mags for the SNLS objects. Without access to any spectroscopic information, and even without any redshift priors from host galaxy photometry, SOFT can still measure reliable redshifts and distances, with an increase in the Hubble residuals to 0.37 mags for the combined SDSS and SNLS data set. Using Monte Carlo simulations we predict that SOFT will be able to improve constraints on time-variable dark energy models by a factor of 2-3 with each new generation of large-scale SN surveys.Comment: 20 pages, 7 figures, accepted to ApJ; paper 1 is arXiv:0910.370

Alignment of cryo-EM movies of individual particles by optimization of image translations

Direct detector device (DDD) cameras have revolutionized single particle electron cryomicroscopy (cryo-EM). In addition to an improved camera detective quantum efficiency, acquisition of DDD movies allows for correction of movement of the specimen, due both to instabilities in the microscope specimen stage and electron beam-induced movement. Unlike specimen stage drift, beam-induced movement is not always homogeneous within an image. Local correlation in the trajectories of nearby particles suggests that beam-induced motion is due to deformation of the ice layer. Algorithms have already been described that can correct movement for large regions of frames and for > 1 MDa protein particles. Another algorithm allows individual < 1 MDa protein particle trajectories to be estimated, but requires rolling averages to be calculated from frames and fits linear trajectories for particles. Here we describe an algorithm that allows for individual < 1 MDa particle images to be aligned without frame averaging or linear trajectories. The algorithm maximizes the overall correlation of the shifted frames with the sum of the shifted frames. The optimum in this single objective function is found efficiently by making use of analytically calculated derivatives of the function. To smooth estimates of particle trajectories, rapid changes in particle positions between frames are penalized in the objective function and weighted averaging of nearby trajectories ensures local correlation in trajectories. This individual particle motion correction, in combination with weighting of Fourier components to account for increasing radiation damage in later frames, can be used to improve 3-D maps from single particle cryo-EM.Comment: 11 pages, 4 figure

Morphogen Gradient from a Noisy Source

We investigate the effect of time-dependent noise on the shape of a morphogen gradient in a developing embryo. Perturbation theory is used to calculate the deviations from deterministic behavior in a simple reaction-diffusion model of robust gradient formation, and the results are confirmed by numerical simulation. It is shown that such deviations can disrupt robustness for sufficiently high noise levels, and the implications of these findings for more complex models of gradient-shaping pathways are discussed.Comment: Four pages, three figure

Reading and other interests of teachers

Thesis (Ed.M.)--Boston Universit

Optimal Galaxy Distance Estimators

The statistical properties of galaxy distance estimators are studied and a rigorous framework is developed for identifying and removing the effects of Malmquist bias due to obsevational selection. The prescription of Schechter (1980) for defining unbiased distance estimators is extended to more general -- and more realistic -- cases. The derivation of `optimal' unbiased distance estimators of minimum dispersion, by utilising information from additional -- suitably correlated -- observables, is discussed and the results applied to a calibrating sample from the Fornax cluster, as used in the Mathewson spiral galaxy redshift survey. The optimal distance estimator derived from I-band magnitude, diameter and 21cm line width has an intrinsic scatter which is 25 \% smaller than that of the Tully-Fisher relation quoted for this calibrating sample. (Figures are available on request).Comment: Plain Latex, 19 pages, Sussex-AST-93/9-