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Single-particle tracking (SPT) has been extensively used to obtain information about diffusion and directed motion in a wide range of biological applications. Recently, new methods have appeared for obtaining precise (10s of nm) spatial information in three dimensions (3D) with high temporal resolution (measurements obtained every 4ms), which promise to more accurately sense the true dynamical behavior in the natural 3D cellular environment. Despite the quantitative 3D tracking information, the range of mathematical methods for extracting information about the underlying system has been limited mostly to mean-squared displacement analysis and other techniques not accounting for complex 3D kinetic interactions. There is a great need for new analysis tools aiming to more fully extract the biological information content from in vivo SPT measurements. High-resolution SPT experimental data has enormous potential to objectively scrutinize various proposed mechanistic schemes arising from theoretical biophysics and cell biology. At the same time, methods for rigorously checking the statistical consistency of both model assumptions and estimated parameters against observed experimental data (i.e. goodness-of-fit tests) have not received great attention. We demonstrate methods enabling (1) estimation of the parameters of 3D stochastic differential equation (SDE) models of the underlying dynamics given only one trajectory; and (2) construction of hypothesis tests checking the consistency of the fitted model with the observed trajectory so that extracted parameters are not over-interpreted (the tools are applicable to linear or nonlinear SDEs calibrated from non-stationary time series data). The approach is demonstrated on high-resolution 3D trajectories of single ARG3 mRNA particles in yeast cells in order to show the power of the methods in detecting signatures of transient directed transport. The methods presented are generally relevant to a wide variety of 2D and 3D SPT tracking applications.

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We present an overview of detailed computational results for families of periodic orbits that emanate from the five libration points in the Circular Restricted 3-Body Problem, as well as for various secondary bifurcating families. Our extensive overview covers all values of the mass-ratio parameter, and includes many known families that have been studied in the past. The numerical continuation and bifurcation algorithms employed in our study are based on boundary value techniques, as implemented in the numerical continuation and bifurcation software AUTO.

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simó, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

A theory and algorithm for detecting and classifying weak, distributed patterns in network data is presented. The patterns we consider are anomalous temporal correlations between signals recorded at sensor nodes in a network. We use robust matrix completion and second order analysis to detect distributed patterns that are not discernible at the level of individual sensors. When viewed independently, the data at each node cannot provide a definitive determination of the underlying pattern, but when fused with data from across the network the relevant patterns emerge. We are specifically interested in detecting weak patterns in computer networks where the nodes (terminals, routers, servers, etc.) are sensors that provide measurements (of packet rates, user activity, central processing unit usage, etc.). The approach is applicable to many other types of sensor networks including wireless networks, mobile sensor networks, and social networks where correlated phenomena are of interest.

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