Cold machining of high density tungsten and other materials

Cold machining process, which uses a sub-zero refrigerated cutting fluid, is used for machining refractory or reactive metals and alloys. Special carbide tools for turning and drilling these alloys further improve the cutting performance

Accurate pointing of tungsten welding electrodes

Thoriated-tungsten is pointed accurately and quickly by using sodium nitrite. Point produced is smooth and no effort is necessary to hold the tungsten rod concentric. The chemically produced point can be used several times longer than ground points. This method reduces time and cost of preparing tungsten electrodes

Reflections on Math Students’ Circles: Two Personal Stories from Colorado

Math Students’ Circles provide an opportunity for mathematicians to work in their communities to engage young students in mathematics as a human, aesthetic, and social endeavor. Sometimes referred to simply as Math Circles, these venues give mathematicians experience in introducing children to topics not typically seen in school curricula in an exciting, hands-on format. This article explores two Math Students’ Circles (MSCs) in the state of Colorado from the point of view of two pre-tenure faculty members. One participated in MSCs for four years while working on her Ph.D. in mathematics, the other started an MSC as an offshoot of a successful professional development program for middle school mathematics teachers. We discuss how and why MSCs have influenced our professional lives

A Topology Scavenger Hunt to Introduce Topological Data Analysis

Topology at the undergraduate level is often a theoretical mathematics course, introducing concepts from point-set topology or possibly algebraic topology. However, the last two decades have seen an explosion of growth in applied topology and topological data analysis, which are topics that can be presented in an accessible way to undergraduate students and can encourage exciting projects. For the past several years, the Topology course at Macalester College has included content from point-set and algebraic topology, as well as applied topology, culminating in a project chosen by the students. In the course, students work through a topology scavenger hunt as an activity to introduce the ideas and software behind some of the primary tools in topological data analysis, namely, persistent homology and mapper. This scavenger hunt includes a variety of point clouds of varying dimensions, such as an annulus in 2D, a bouquet of loops in 3D, a sphere in 4D, and a torus in 400D. The students' goal is to analyze each point cloud with a variety of software to infer the topological structure. After completing this activity, students are able to extend the ideas learned in the scavenger hunt to an open-ended capstone project. Examples of past projects include: using persistence to explore the relationship between country development and geography, to analyze congressional voting patterns, and to classify genres of a large corpus of texts by combining with tools from natural language processing and machine learning.Comment: To appear in "Teaching Mathematics Through Cross-Curricular Projects," volume 72 of the AMS/MAA Classroom Resource Materials serie

Exploiting geometry, topology, and optimization for knowledge discovery in big data

2013 Summer.Includes bibliographical references.In this dissertation, we consider several topics that are united by the theme of topological and geometric data analysis. First, we consider an application in landscape ecology using a well-known vector quantization algorithm to characterize and segment the color content of natural imagery. Color information in an image may be viewed naturally as clusters of pixels with similar attributes. The inherent structure and distribution of these clusters serves to quantize the information in the image and provides a basis for classification. A friendly graphical user interface called Biological Landscape Organizer and Semi-supervised Segmenting Machine (BLOSSM) was developed to aid in this classification. We consider four different choices for color space and five different metrics in which to analyze our data, and results are compared. Second, we present a novel topologically driven clustering algorithm that blends Locally Linear Embedding (LLE) and vector quantization by mapping color information to a lower dimensional space, identifying distinct color regions, and classifying pixels together based on both a proximity measure and color content. It is observed that these techniques permit a significant reduction in color resolution while maintaining the visually important features of images. Third, we develop a novel algorithm which we call Sparse LLE that leads to sparse representations in local reconstructions by using a data weighted 1-norm regularization term in the objective function of an optimization problem. It is observed that this new formulation has proven effective at automatically determining an appropriate number of nearest neighbors for each data point. We explore various optimization techniques, namely Primal Dual Interior Point algorithms, to solve this problem, comparing the computational complexity for each. Fourth, we present a novel algorithm that can be used to determine the boundary of a data set, or the vertices of a convex hull encasing a point cloud of data, in any dimension by solving a quadratic optimization problem. In this problem, each point is written as a linear combination of its nearest neighbors where the coefficients of this linear combination are penalized if they do not construct a convex combination, revealing those points that cannot be represented in this way, the vertices of the convex hull containing the data. Finally, we exploit the relatively new tool from topological data analysis, persistent homology, and consider the use of vector bundles to re-embed data in order to improve the topological signal of a data set by embedding points sampled from a projective variety into successive Grassmannians

Topological Data Analysis of Biological Aggregation Models

We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.Comment: 25 pages, 12 figures; second version contains typo corrections, minor textual additions, and a brief discussion of computational complexity; third version fixes one typo and adds small paragraph about topological stabilit