Cone types and geodesic languages for lamplighter groups and Thompson's group F

We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups $L_n$ have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F with respect to the standard finite generating set, we show there are infinitely many cone types and no regular language of geodesics with respect to the standard finite generating set. We show that the existence of families of "seesaw" elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set.Comment: 30 pages, 13 figure

From Daunting to Doable: A Practical Approach to Building Inclusive LibGuides

Universal Design for Learning (UDL) is a pedagogical approach that many universities promote to their teaching faculty to help them design courses that are accessible to all learners. After librarians at our library took a course on Universal Design for Learning, we asked, how could we apply Universal Design for Learning principles to create more inclusive LibGuides? At first, we were overwhelmed by the wide range of possible accessibility and UDL features that we could incorporate into our LibGuides. We wondered if there might be a way to identify or prioritize effective UDL elements to include in a LibGuide. We attempted to answer this question by reading the library science literature on UDL elements in LibGuides which included usability studies of various UDL features. Based on the results of these usability studies, we developed a streamlined list of research-tested, impactful UDL elements that librarians with no technical background can easily build into LibGuides to make them more inclusive and accessible

On Graphs of Sets of Reduced Words

Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or reduced words, and the collection of all such reduced words is denoted $\mathcal{R}(\sigma)$. Any reduced word of $\sigma$ can be transformed into any other by a sequence of commutation moves or long braid moves. One area of interest in these sets are the congruence classes defined by using only braid or only commutation relations. The set $\mathcal{R}(\sigma)$ can be drawn as a graph, $G(\sigma)$, where the vertices are the reduced words, and the edges denote the presence of a commutation or braid move between the words. This paper presents new work on subgraph structures in $G(\sigma)$, as well as new formulas to count the number of braid edges and commutation edges in $G(\sigma)$. We also include work on bounds for the number of braid and commutation classes in $\mathcal{R}(\sigma)$.Comment: 24 pages, 10 figure

Random subgroups of Thompson's group FF

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite and not algebraic. We then use the asymptotic growth to prove our density results.Comment: 37 pages, 11 figure

On Invariants for Spatial Graphs

We use combinatorial knot theory to construct invariants for spatial graphs. This is done by performing certain replacements at each vertex of a spatial graph diagram D , which results in a collection of knot and link diagrams in D. By applying known invariants for classical knots and links to the resulting collection, we obtain invariants for spatial graphs. We also show that for the case of undirected spatial graphs, the invariants we construct here satisfy a certain contraction-deletion recurrence relation

The Effects of a Plyometric Training Program on Jump Performance in Collegiate Figure Skaters: A Pilot Study

International Journal of Exercise Science 9(2): 175-186, 2016. Plyometric training has been implemented to increase jump height in a variety of sports, but its effects have not been researched in figure skating. The purpose of this study was to determine the effects of a plyometric training program on on-ice and off-ice jump performance. Six collegiate figure skaters (19.8±1.2 years; 164.7±4.9 cm; 60.3±11.6 kg) completed a six-week sport-specific plyometric training program, consisting of low to moderate intensity plyometric exercises, while eight collegiate figure skaters (21.1±3.9 years; 162.6±6.0 cm; 60.4±6.1 kg) served as the control group. Significant increases were found for vertical jump height, standing long jump distance, (F = 31.0, p \u3c 0.001), and flight time (F = 11.6, p = 0.007). No significant differences were found for self-reported jump evaluation (p = 0.101). Six weeks of plyometric training improved both on-ice and off-ice jump performance in collegiate figure skaters, while short-term skating training alone resulted in decreases. These results indicate that figure skaters could participate in off-ice plyometric training