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We use conformal maps to study a free boundary problem for a two-fluid electromechanical system, where the interface between the fluids is determined by the combined effects of electrostatic forces, gravity and surface tension. The free boundary in our system develops sharp corners/singularities in certain parameter regimes, and this is an impediment to using existing "single-scale" numerical conformal mapping methods. The difficulty is due to the phenomenon of crowding, i.e. the tendency of nodes in the preimage plane to concentrate near the sharp regions of the boundary, leaving the smooth regions of the boundary poorly resolved. A natural idea is to exploit the scale separation between the sharp regions and smooth regions to solve for each region separately, and then stitch the solutions together. However, this is not straightforward as conformal maps are rigid "global" objects, and it is not obvious how one would patch two conformal maps together to obtain a new conformal map. We develop a "multi-scale" (i.e. adaptive) conformal mapping method that allows us to carry out this program of stitching conformal maps on different scales together. We successfully apply our method to the electromechanical model problem and discuss how it generalizes to other situations., Comment: 19 pages, 12 figures, 2 Appendices, revised discussion and organization

We propose criteria that define a trend for time series with inherent multi-scale features. We call this trend the {\it tendency} of a time series. The tendency is defined empirically by a set of criteria and captures the large-scale temporal variability of the original signal as well as the most frequent events in its histogram. Among other properties, the tendency has a variance no larger than that of the original signal; the histogram of the difference between the original signal and the tendency is as symmetric as possible; and with reduced complexity, the tendency captures essential features of the signal. To find the tendency we first use the Intrinsic Time-Scale Decomposition (ITD) of the signal, introduced in 2007 by Frei and Osorio, to produce a set of candidate tendencies. We then apply the criteria to each of the candidates to single out the one that best agrees with them. While the criteria for the tendency are independent of the signal decomposition scheme, it is found that the ITD is a simple and stable methodology, well suited for multi-scale signals. The ITD is a relatively new decomposition and little is known about its outcomes. In this study we take the first steps towards a probabilistic model of the ITD analysis of random time series. This analysis yields details concerning the universality and scaling properties of the components of the decomposition., Comment: 27 Pages, 16 Figures, 1 Appendix

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to `non-tree' evolutions where cycles may develop in the network. A main part of the argument is to analyze an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, in terms of $C_0$-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003)., Comment: 25 pages

We investigate the problem of shaping radially symmetric annular beams into desired intensity patterns along the optical axis. Within the Fresnel approximation, we show that this problem can be expressed in a variational form equivalent to the one arising in phase retrieval. Using the uncertainty principle we prove rigorous lower bounds on the functional that capture how the various physical parameters in the problem determine the accuracy of the beam shaping. We also use the method of stationary phase to construct a natural ansatz for a minimizer in the short wavelength limit. We illustrate the implications of our results by applying the method of stationary phase coupled with the Gerchberg-Saxton algorithm to beam shaping problems arising in remote delivery of beams and pulses., Comment: 34 Pages, 6 figures, 1 appendix; revised figures and discussion

Wind- and current-driven flotsam, oil spills, pollutants, and nutrients, approaching the nearshore will frequently appear to park just beyond the break zone, where waves break. Moreover, the portion of these tracers that beach will do so only after a long time. Explaining why these tracers park and at what rate they reach the shore has important implications on a variety of different nearshore environmental issues, including the determination of what subscale processes are essential in computer models for the simulation of pollutant transport in the nearshore. Using a simple model we provide an explanation for the underlying mechanism responsible for the parking of tracers, the role played by the bottom topography, and the non-uniform dispersion which leads, in some circumstances, to the eventual landing of all or a portion of the tracers. We refer to the parking phenomenon in this environment as nearshore sticky waters., Comment: Revised discussion, 13 pages, 8 figures, 2 tables, Submitted Ocean Modeling

We propose a modification of a maximum likelihood procedure for tuning parameter values in models, based upon the comparison of their output to field data. Our methodology, which uses polynomial approximations of the sample space to increase the computational efficiency, differs from similar Bayesian estimation frameworks in the use of an alternative likelihood distribution, is shown to better address problems in which covariance information is lacking, than its more conventional counterpart. Lack of covariance information is a frequent challenge in large-scale geophysical estimation. This is the case in the geophysical problem considered here. We use a nearshore model for long shore currents and observational data of the same to show the contrast between both maximum likelihood methodologies. Beyond a methodological comparison, this study gives estimates of parameter values for the bottom drag and surface forcing that make the particular model most consistent with data; furthermore, we also derive sensitivity estimates that provide useful insights regarding the estimation procedure as well as of the model itself., Comment: 18 pages, 7 figures, Submitted Ocean Modeling

Microcalcifications in mammogram have been mainly targeted as a reliable earliest sign of breast cancer and their early detection is vital to improve its prognosis. Since their size is very small and may be easily overlooked by the examining radiologist, computer-based detection output can assist the radiologist to improve the diagnostic accuracy. In this paper, we have proposed an algorithm for detecting microcalcification in mammogram. The proposed microcalcification detection algorithm involves mammogram quality enhancement using multirresolution analysis based on the dyadic wavelet transform and microcalcification detection by fuzzy shell clustering. It may be possible to detect nodular components such as microcalcification accurately by introducing shape information. The effectiveness of the proposed algorithm for microcalcification detection is confirmed by experimental results., Comment: IEEE format, International Journal of Computer Science and Information Security, IJCSIS January 2010, ISSN 1947 5500, http://sites.google.com/site/ijcsis/

We investigate the electrostatic equilibria of N discrete charges of size 1/N on a two dimensional conductor (domain). We study the distribution of the charges on symmetric domains including the ellipse, the hypotrochoid and various regular polygons, with an emphasis on understanding the distributions of the charges, as the shape of the underlying conductor becomes singular. We find that there are two regimes of behavior, a symmetric regime for smooth conductors, and a symmetry broken regime for ``singular'' domains. For smooth conductors, the locations of the charges can be determined up to a certain order by an integral equation due to Pommerenke (1969). We present a derivation of a related (but different) integral equation, which has the same solutions. We also solve the equation to obtain (asymptotic) solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps. Conductors with sharp cusps and singularities show qualitatively different behavior, where the symmetry of the problem is broken, and the distribution of the discrete charges does not respect the symmetry of the underlying domain. We investigate the symmetry breaking both theoretically, and numerically, and find good agreement between our theory and the numerics. We also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime, although this distribution is very different from the one given by the integral equation., Comment: 46 pages, 46 figures, submitted to J. Stat. Phys