Physical Sciences and Mathematics Commons^™

Spacetime Groups, Ian M. Anderson, Charles G. Torre

All Physics Faculty Publications

A spacetime group is a connected 4-dimensional Lie group G endowed with a left invariant Lorentz metric h and such that the connected component of the isometry group of h is G itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs, (g, n), with g being a 4-dimensional Lie algebra and n being a Lorentzian inner product on g. A full analysis of the equivalence problem for spacetime Lie algebras is given which leads to a completely …

Bipartite Dot Product Graphs, Sean Bailey, David E. Brown

Mathematics and Statistics Faculty Publications

Given a bipartite graph G = (X, Y, E), the bipartite dot product representation of G is a function f : X ∪Y → ℝ^k and a positive threshold t such that for any x ∈ X and y ∈ Y , xy ∈ E if and only if f(x) · f(y) ≥ t. The minimum k such that a bipartite dot product representation exists for G is the bipartite dot product dimension of G, denoted bdp(G). We will show that such representations exist for all bipartite graphs as well as give an upper bound for the bipartite dot …

Linear Operators That Preserve Two Genera Of A Graph, Leroy B. Beasley, Kyung-Tae Kang, Seok-Zun Song

Mathematics and Statistics Faculty Publications

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of …

Characterizing The Growth Of One Student's Mathematical Understanding In A Multi-Representational Learning Environment, Hilal Gulkilik, Patricia S. Moyer-Packenham, Hasan Huseyin Ugurlu, Nejla Yuruk

Teacher Education and Leadership Faculty Publications

The purpose of this study was to characterize the growth of one student’s mathematical understanding and use of different representations about a geometric transformation, dilation. We accomplished this purpose by using the Pirie-Kieren model jointly with the Semiotic Representation Theory as a lens. Elif, a 10^th- grade student, was purposefully chosen as the case for this study because of the growth of mathematical understanding about dilation she exhibited over time. Elif participated in task-based interviews before, during and after participating in a variety of transformation lessons where she used multiple representations, including physical and virtual manipulatives. The results …

Explicit Ambient Metrics And Holonomy, Ian M. Anderson, Thomas Leistner, Pawel Nurowski

Mathematics and Statistics Faculty Publications

We present three large classes of examples of conformal structures whose Fefferman-Graham ambient metrics can be found explicitly. Our method for constructing these examples rests upon a set of sufficiency conditions under which the Fefferman-Graham equations are assured to reduce to a system of inhomogeneous linear partial differential equations. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic co-rank 3 distributions in dimensions 5 and 6.Our examples illustrate various aspects of the ambient metric construction.

The holonomy algebras of our ambient metrics are studied in detail. In particular, we exhibit a large class of …

Arbitrarily High-Order Unconditionally Energy Stable Schemes For Thermodynamically Consistent Gradient Flow Models, Yuezheng Gong, Jia Zhao, Qi Wang

Mathematics and Statistics Faculty Publications

We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization method, we formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic …