11,429 research outputs found
Epsilon-Unfolding Orthogonal Polyhedra
An unfolding of a polyhedron is produced by cutting the surface and
flattening to a single, connected, planar piece without overlap (except
possibly at boundary points). It is a long unsolved problem to determine
whether every polyhedron may be unfolded. Here we prove, via an algorithm, that
every orthogonal polyhedron (one whose faces meet at right angles) of genus
zero may be unfolded. Our cuts are not necessarily along edges of the
polyhedron, but they are always parallel to polyhedron edges. For a polyhedron
of n vertices, portions of the unfolding will be rectangular strips which, in
the worst case, may need to be as thin as epsilon = 1/2^{Omega(n)}.Comment: 23 pages, 20 figures, 7 references. Revised version improves language
and figures, updates references, and sharpens the conclusio
Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm
We show that every orthogonal polyhedron homeomorphic to a sphere can be
unfolded without overlap while using only polynomially many (orthogonal) cuts.
By contrast, the best previous such result used exponentially many cuts. More
precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts
the polyhedron only where it is met by the grid of coordinate planes passing
through the vertices, together with Theta(n^2) additional coordinate planes
between every two such grid planes.Comment: 15 pages, 10 figure
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
Higher Spin Alternating Sign Matrices
We define a higher spin alternating sign matrix to be an integer-entry square
matrix in which, for a nonnegative integer r, all complete row and column sums
are r, and all partial row and column sums extending from each end of the row
or column are nonnegative. Such matrices correspond to configurations of spin
r/2 statistical mechanical vertex models with domain-wall boundary conditions.
The case r=1 gives standard alternating sign matrices, while the case in which
all matrix entries are nonnegative gives semimagic squares. We show that the
higher spin alternating sign matrices of size n are the integer points of the
r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices
are the standard alternating sign matrices of size n. It then follows that, for
fixed n, these matrices are enumerated by an Ehrhart polynomial in r.Comment: 41 pages; v2: minor change
Quantization and Fractional Quantization of Currents in Periodically Driven Stochastic Systems I: Average Currents
This article studies Markovian stochastic motion of a particle on a graph
with finite number of nodes and periodically time-dependent transition rates
that satisfy the detailed balance condition at any time. We show that under
general conditions, the currents in the system on average become quantized or
fractionally quantized for adiabatic driving at sufficiently low temperature.
We develop the quantitative theory of this quantization and interpret it in
terms of topological invariants. By implementing the celebrated Kirchhoff
theorem we derive a general and explicit formula for the average generated
current that plays a role of an efficient tool for treating the current
quantization effects.Comment: 22 pages, 7 figure
Improved Bounds for Drawing Trees on Fixed Points with L-shaped Edges
Let be an -node tree of maximum degree 4, and let be a set of
points in the plane with no two points on the same horizontal or vertical line.
It is an open question whether always has a planar drawing on such that
each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By
giving new methods for drawing trees, we improve the bounds on the size of the
point set for which such drawings are possible to: for
maximum degree 4 trees; for maximum degree 3 (binary) trees; and
for perfect binary trees.
Drawing ordered trees with L-shaped edges is harder---we give an example that
cannot be done and a bound of points for L-shaped drawings of
ordered caterpillars, which contrasts with the known linear bound for unordered
caterpillars.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
Formation of stripes and slabs near the ferromagnetic transition
We consider Ising models in d=2 and d=3 dimensions with nearest neighbor
ferromagnetic and long-range antiferromagnetic interactions, the latter
decaying as (distance)^(-p), p>2d, at large distances. If the strength J of the
ferromagnetic interaction is larger than a critical value J_c, then the ground
state is homogeneous. It has been conjectured that when J is smaller than but
close to J_c the ground state is periodic and striped, with stripes of constant
width h=h(J), and h tends to infinity as J tends to J_c from below. (In d=3
stripes mean slabs, not columns.) Here we rigorously prove that, if we
normalize the energy in such a way that the energy of the homogeneous state is
zero, then the ratio e_0(J)/e_S(J) tends to 1 as J tends to J_c from below,
with e_S(J) being the energy per site of the optimal periodic striped/slabbed
state and e_0(J) the actual ground state energy per site of the system. Our
proof comes with explicit bounds on the difference e_0(J)-e_S(J) at small but
finite J_c-J, and also shows that in this parameter range the ground state is
striped/slabbed in a certain sense: namely, if one looks at a randomly chosen
window, of suitable size l (very large compared to the optimal stripe size
h(J)), one finds a striped/slabbed state with high probability.Comment: 23 pages, 2 figures. We discovered that our analysis works in 3D as
well as in 2D. We revised the paper and the title to reflect this. Our 2D
stripes become 3D slabs. Some references adde
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