Mathematics
http://hdl.handle.net/10211.3/141103
Fri, 20 Oct 2017 21:47:44 GMT2017-10-20T21:47:44ZTwo problems on lattice point enumeration of rational polytopes
http://hdl.handle.net/10211.3/197219
Two problems on lattice point enumeration of rational polytopes
Vindas Melendez, Andres R.
Motivated by the generalization of Ehrhart theory with group actions, the first part
of this thesis makes progress towards obtaining the equivariant Ehrhart theory of the
permutahedron. The subset that is fixed by a group action on the permutahedron
is itself a rational polytope. We prove that these fixed polytopes are combinatorially
equivalent to lower dimensional permutahedra. Furthermore, we show that
these fixed polytopes are zonotopes, i.e., Minkowski sum of line segments. This
part is joint work with Anna Schindler. The second part of this thesis provides a
decomposition of the /i*-polynomial for rational polytopes. This decomposition is
an analogue to the decomposition proven by Ulrich Betke and Peter McMullen for
lattice polytopes.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10211.3/1972192017-01-01T00:00:00ZAlgebraic and combinatorial aspects of two symmetric polytopes
http://hdl.handle.net/10211.3/197035
Algebraic and combinatorial aspects of two symmetric polytopes
Schindler, Anna Maria
Equivariant Ehrhart theory is an extension of Ehrhart theory that considers lattice
polytopes under group actions. Ehrhart theory tells us that the lattice points in a
lattice (or rational) polytope are counted by polynomials (or quasi-polynomials). In
the equivariant analog, we consider the Ehrhart theory of the subsets of the polytope
fixed by the action. This first part of this thesis, a joint project with Andres Vindas,
focuses on the equivariant Ehrhart theory of IIn under the action of Sn. We prove
that the fixed sub-polytopes of IIn are zonotopes and are combinatorially equivalent
to permutahedra. We provide vertex and hyperplane descriptions. We also compute
the equivariant Ehrhart theory for 111, 112, 113, and II4. This thesis also includes work
in spectral graph theory. Motivated by a tropical approach to the Hodge conjecture,
we compute the spectrum of the tropical Laplacian matrix of the root polytope An,
confirming a special case of a result of Babaee and Huh.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10211.3/1970352017-01-01T00:00:00ZAnalyzing peer-assisted reflections in developmental college mathematics course
http://hdl.handle.net/10211.3/196506
Analyzing peer-assisted reflections in developmental college mathematics course
Kwon, Patrick
Formative assessment is defined as classroom activity that gives insight into studentsâ€™
understanding and learning to the teacher and students that is then used
to adapt the teaching to fit learning needs. There have been several studies conducted
that show the benefits of using formative assessment in the classroom; one
specific instance of this was seen in a study done by Reinholz [Reinholz, 2015], in
which he studied the effects of having university level calculus students partake in
peer-assisted reflections with fellow classmates. In this research study I conducted
a similar experiment to that of Reinholz in which I observed four different sections
of algebra 1 during the summer of 2016 at San Francisco State University. I
documented the ways that students seemed to be affected by participating in peerassisted
reflection, specifically studying student learning and understanding of linear
functions throughout the experimental study. This thesis discusses data and findings
based off peer-assisted reflections and their effect on student learning and other
classroom dynamics.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10211.3/1965062017-01-01T00:00:00ZThree configuration spaces in combinatorics
http://hdl.handle.net/10211.3/196244
Three configuration spaces in combinatorics
Guo, John
Roughly speaking, a reconfigurable system TZ is a discrete collection of positions of
an object, along with local reversible moves. Such a system can be encoded as a
cubical complex, which we call the configuration space S{1Z). When a configuration
space is CAT(O), there is a unique shortest path between the vertices, and there
is an efficient algorithm to compute this path. We can assess whether or not a
cubical complex is CAT(O) by determining if there exists a corresponding poset
with inconsistent pairs (PIP). In this thesis we show that the configuration spaces
of the robotic arm in a rectangular tunnel Rm,n, of the tableaux T\ of hook shape
A, and of the Dyck paths Dn of length 2n are CAT(O) cubical complexes. We do
this by identifying the associated PIPs.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10211.3/1962442017-01-01T00:00:00Z