60 research outputs found
Ununfoldable Polyhedra with Convex Faces
Unfolding a convex polyhedron into a simple planar polygon is a well-studied
problem. In this paper, we study the limits of unfoldability by studying
nonconvex polyhedra with the same combinatorial structure as convex polyhedra.
In particular, we give two examples of polyhedra, one with 24 convex faces and
one with 36 triangular faces, that cannot be unfolded by cutting along edges.
We further show that such a polyhedron can indeed be unfolded if cuts are
allowed to cross faces. Finally, we prove that ``open'' polyhedra with
triangular faces may not be unfoldable no matter how they are cut.Comment: 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry:
Theory and Applications. Major revision with two new authors, solving the
open problem about triangular face
Analysis of Reaction Network Systems Using Tropical Geometry
We discuss a novel analysis method for reaction network systems with
polynomial or rational rate functions. This method is based on computing
tropical equilibrations defined by the equality of at least two dominant
monomials of opposite signs in the differential equations of each dynamic
variable. In algebraic geometry, the tropical equilibration problem is
tantamount to finding tropical prevarieties, that are finite intersections of
tropical hypersurfaces. Tropical equilibrations with the same set of dominant
monomials define a branch or equivalence class. Minimal branches are
particularly interesting as they describe the simplest states of the reaction
network. We provide a method to compute the number of minimal branches and to
find representative tropical equilibrations for each branch.Comment: Proceedings Computer Algebra in Scientific Computing CASC 201
Lattice-point generating functions for free sums of convex sets
Let \J and \K be convex sets in whose affine spans intersect at
a single rational point in \J \cap \K, and let \J \oplus \K = \conv(\J \cup
\K). We give formulas for the generating function {equation*} \sigma_{\cone(\J
\oplus \K)}(z_1,..., z_n, z_{n+1}) = \sum_{(m_1,..., m_n) \in t(\J \oplus \K)
\cap \Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points
in all integer dilates of \J \oplus \K in terms of \sigma_{\cone \J} and
\sigma_{\cone \K}, under various conditions on \J and \K. This work is
motivated by (and recovers) a product formula of B.\ Braun for the Ehrhart
series of \P \oplus \Q in the case where and \Q are lattice polytopes
containing the origin, one of which is reflexive. In particular, we find
necessary and sufficient conditions for Braun's formula and its multivariate
analogue.Comment: 17 pages, 2 figures, to appear in Journal of Combinatorial Theory
Series
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