443 research outputs found
Sector length distributions of graph states
The sector length distribution (SLD) of a quantum state is a collection of
local unitary invariants that quantify -body correlations. We show that the
SLD of graph states can be derived by solving a graph-theoretical problem. In
this way, the mean and variance of the SLD are obtained as simple functions of
efficiently computable graph properties. Furthermore, this formulation enables
us to derive closed expressions of SLDs for some graph state families. For
cluster states, we observe that the SLD is very similar to a binomial
distribution, and we argue that this property is typical for graph states in
general. Finally, we derive an SLD-based entanglement criterion from the
majorization criterion and apply it to derive meaningful noise thresholds for
entanglement.Comment: 20+21 pages, 8+8 figure
Mathematical Knowledge for Teaching as Decision-Making for the University Mathematician Developing Coherence in Review of Discrete Mathematics
Mathematicians teaching at the university level have a deep understanding and appreciation for the mathematics that they teach. However, they rarely receive much formal training in teaching. Thus, university mathematicians must rely on their mathematical understandings and personally developed ideas of teaching to guide their decision-making. Relatively little research exists on mathematicians’ teaching practices. The purpose of this study was to examine the mathematical knowledge for teaching (MKT) of a university mathematician teaching discrete mathematics and how he leveraged his knowledge to make decisions and develop coherence among mathematical ideas during a semester review. An enactivist perspective examining a mathematician’s decision-making in planning, enacting, and reflecting upon their lessons in this study shed light on how this mathematician practically approached his teaching duties. By enacting four distinct coherence strategies, the mathematician in this case study revealed a personal standard for mathematical storytelling which guided his decision enactment. These strategies fostered rich connections among mathematical ideas and among topics from earlier in the semester meaningfully with a single culminating topic: the chromatic polynomial. Implications of this study for research include recognized advantages of graph theoretic visualizations for the analysis of teacher decisions and coherence, benefits of dual coding for the Knowledge Quartet MKT framework, and a stance on inactivism\u27s consideration of cognitive actions. For teaching, this research supports the benefits of mathematical storytelling and review units which feature a new context to reframe previously seen topics
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
- …