Nonholonomically Constrained Dynamics and Optimization of Rolling Isolation Systems

Rolling Isolation Systems provide a simple and effective means for protecting components from horizontal floor vibrations. In these systems a platform rolls on four steel balls which, in turn, rest within shallow bowls. The trajectories of the balls is uniquely determined by the horizontal and rotational velocity components of the rolling platform, and thus provides nonholonomic constraints. In general, the bowls are not parabolic, so the potential energy function of this system is not quadratic. This thesis presents the application of Gauss's Principle of Least Constraint to the modeling of rolling isolation platforms. The equations of motion are described in terms of a redundant set of constrained coordinates. Coordinate accelerations are uniquely determined at any point in time via Gauss's Principle by solving a linearly constrained quadratic minimization. In the absence of any modeled damping, the equations of motion conserve energy. This mathematical model is then used to find the bowl profile that minimizes response acceleration subject to displacement constraint.

Inheritance of the CENP-A chromatin domain is spatially and temporally constrained at human centromeres.

BACKGROUND: Chromatin containing the histone variant CENP-A (CEN chromatin) exists as an essential domain at every centromere and heritably marks the location of kinetochore assembly. The size of the CEN chromatin domain on alpha satellite DNA in humans has been shown to vary according to underlying array size. However, the average amount of CENP-A reported at human centromeres is largely consistent, implying the genomic extent of CENP-A chromatin domains more likely reflects variations in the number of CENP-A subdomains and/or the density of CENP-A nucleosomes within individual subdomains. Defining the organizational and spatial properties of CEN chromatin would provide insight into centromere inheritance via CENP-A loading in G1 and the dynamics of its distribution between mother and daughter strands during replication. RESULTS: Using a multi-color protein strategy to detect distinct pools of CENP-A over several cell cycles, we show that nascent CENP-A is equally distributed to sister centromeres. CENP-A distribution is independent of previous or subsequent cell cycles in that centromeres showing disproportionately distributed CENP-A in one cycle can equally divide CENP-A nucleosomes in the next cycle. Furthermore, we show using extended chromatin fibers that maintenance of the CENP-A chromatin domain is achieved by a cycle-specific oscillating pattern of new CENP-A nucleosomes next to existing CENP-A nucleosomes over multiple cell cycles. Finally, we demonstrate that the size of the CENP-A domain does not change throughout the cell cycle and is spatially fixed to a similar location within a given alpha satellite DNA array. CONCLUSIONS: We demonstrate that most human chromosomes share similar patterns of CENP-A loading and distribution and that centromere inheritance is achieved through specific placement of new CENP-A near existing CENP-A as assembly occurs each cell cycle. The loading pattern fixes the location and size of the CENP-A domain on individual chromosomes. These results suggest that spatial and temporal dynamics of CENP-A are important for maintaining centromere identity and genome stability.

Algorithms for Geometric Matching, Clustering, and Covering

With the popularization of GPS-enabled devices such as mobile phones, location data are becoming available at an unprecedented scale. The locations may be collected from many different sources such as vehicles moving around a city, user check-ins in social networks, and geo-tagged micro-blogging photos or messages. Besides the longitude and latitude, each location record may also have a timestamp and additional information such as the name of the location. Time-ordered sequences of these locations form trajectories, which together contain useful high-level information about people's movement patterns.

The first part of this thesis focuses on a few geometric problems motivated by the matching and clustering of trajectories. We first give a new algorithm for computing a matching between a pair of curves under existing models such as dynamic time warping (DTW). The algorithm is more efficient than standard dynamic programming algorithms both theoretically and practically. We then propose a new matching model for trajectories that avoids the drawbacks of existing models. For trajectory clustering, we present an algorithm that computes clusters of subtrajectories, which correspond to common movement patterns. We also consider trajectories of check-ins, and propose a statistical generative model, which identifies check-in clusters as well as the transition patterns between the clusters.

The second part of the thesis considers the problem of covering shortest paths in a road network, motivated by an EV charging station placement problem. More specifically, a subset of vertices in the road network are selected to place charging stations so that every shortest path contains enough charging stations and can be traveled by an EV without draining the battery. We first introduce a general technique for the geometric set cover problem. This technique leads to near-linear-time approximation algorithms, which are the state-of-the-art algorithms for this problem in either running time or approximation ratio. We then use this technique to develop a near-linear-time algorithm for this

shortest-path cover problem.