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Abstract. We develop persistent homology in the setting of filtrations of (Cˇech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable. 

Cell identification is an important yet difficult process in data analysis of biological images. Previously, we developed an automated cell identification method called CRF_ID and demonstrated its high performance in C. elegans whole-brain images (Chaudhary et al, 2021). However, because the method was optimized for whole-brain imaging, comparable performance could not be guaranteed for application in commonly used C. elegans multi-cell images that display a subpopulation of cells. Here, we present an advance CRF_ID 2.0 that expands the generalizability of the method to multi-cell imaging beyond whole-brain imaging. To illustrate the application of the advance, we show the characterization of CRF_ID 2.0 in multi-cell imaging and cell-specific gene expression analysis in C. elegans. This work demonstrates that high accuracy automated cell annotation in multi-cell imaging can expedite cell identification and reduce its subjectivity in C. elegans and potentially other biological images of various origins. 

Racial and ethnic representation in home ownership rates is an important public policy topic for addressing inequality within society. Although more than half of the households in the US are owned, rather than rented, the representation of home ownership is unequal among different racial and ethnic groups. Here we analyze the US Census Bureau’s American Community Survey data to conduct an exploratory and statistical analysis of home ownership in the US, and find sociodemographic factors that are associated with differences in home ownership rates. We use binomial and beta-binomial generalized linear models (GLMs) with 2020 county-level data to model the home ownership rate, and fit the beta-binomial models with Bayesian estimation. We determine that race/ethnic group, geographic region, and income all have significant associations with the home ownership rate. To make the data and results accessible to the public, we develop an Shiny web application in R with exploratory plots and model predictions. 

Contact planning is crucial to the locomotion performance of robots: to properly self-propel forward, it is not only important to determine the sequence of internal shape changes (e.g., body bending and limb shoulder joint oscillation) but also the sequence by which contact is made and broken between the mechanism and its environment. Prior work observed that properly coupling contact patterns and shape changes allows for computationally tractable gait design and efficient gait performance. The state of the art, however, made assumptions, albeit motivated by biological observation, as to how contact and shape changes can be coupled. In this paper, we extend the geometric mechanics (GM) framework to design contact patterns. Specifically, we introduce the concept of “contact space” to the GM framework. By establishing the connection between velocities in shape and position spaces, we can estimate the benefits of each contact pattern change and therefore optimize the sequence of contact patterns. In doing so, we can also analyze how a contact pattern sequence will respond to perturbations. We apply our framework to sidewinding robots and enable (1) effective locomotion direction control and (2) robust locomotion performance as the spatial resolution decreases. We also apply our framework to a hexapod robot with two back-bending joints and show that we can simplify existing hexapod gaits by properly reducing the number of contact state switches (during a gait cycle) without significant loss of locomotion speed. We test our designed gaits with robophysical experiments, and we obtain good agreement between theory and experiments. 

Information theory is used to design robots with guaranteed arrival over noisy terrain. 

The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a 1-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra-Lenstra-Lovász algorithm from computational number theory.