Lori Ziegelmeier

Flipped Calculus: A Study of Student Performance and Perceptions

Abstract Flipping the classroom refers to moving lectures outside of the classroom to incorporate other activities into a class during its standard meeting time. This pedagogical modality has recently gained traction as a way to center the learning on students in mathematics classrooms. In an effort to better understand the efficacy of this approach, we implemented a controlled study at a small liberal arts college. We compared two sections of the entry-level course applied multivariable calculus I, with one section taught in a traditional lecture-based format and the other taught as a flipped classroom. During our study, we collected and analyzed data related to student performance, as well as perceptions of the approach and attitude toward mathematics in general. Students in both classes scored similarly on graded components of the course, and the majority of students were comfortable with the format of each section. However, some student perceptions and study habits differed.

An application of persistent homology on Grassmann manifolds for the detection of signals in hyperspectral imagery

We present an application of persistent homology to the detection of chemical plumes in hyperspectral movies. The pixels of the raw hyperspectral data cubes are mapped to the geometric framework of the real Grassmann manifold G(k, n) (whose points parameterize the k-dimensional subspaces of ℝn) where they are analyzed, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows the time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological structure, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed framework affords the processing of large data sets, such as the hyperspectral movies explored in this investigation, while retaining valuable discriminative information.

A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs

Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative description of metric graphs using topological summaries. In particular, we provide a complete characterization of the one-dimensional intrinsic Cech persistence diagrams for finite metric graphs using persistent homology. Together with complementary results by Adamaszek et al., which imply the results on intrinsic Cech persistence diagrams in all dimensions for a single cycle, our results constitute the important steps toward characterizing intrinsic Cech persistence diagrams for arbitrary finite metric graphs across all dimensions.

Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies

The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper, we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold Gk,i¾?n whose points parameterize the k-dimensional subspaces of

Stratifying High-Dimensional Data Based on Proximity to the Convex Hull Boundary

\mathbb {R}^n

Persistence Images: An Alternative Persistent Homology Representation.

Rn, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves.

Vietoris-Rips and Cech Complexes of Metric Gluings

Abstract The Locally Linear Embedding (LLE) algorithm has proven useful for determining structure preserving, dimension reducing mappings of data on manifolds. We propose a modification to the LLE optimization problem that serves to minimize the number of neighbors required for the representation of each data point. The algorithm is shown to be robust over wide ranges of the sparsity parameter producing an average number of nearest neighbors that is consistent with the best performing parameter selection for LLE. Given the number of non-zero weights may be substantially reduced in comparison to LLE, Sparse LLE can be applied to larger data sets. We provide three numerical examples including a color image, the standard swiss roll, and a gene expression data set to illustrate the behavior of the method in comparison to LLE. The resulting algorithm produces comparatively sparse representations that preserve the neighborhood geometry of the data in the spirit of LLE.

Locally Linear Embedding Clustering Algorithm for Natural Imagery

The convex hull of a set of points,