823 research outputs found
Computing tolerance parameters for fixturing and feeding
Fixtures and feeders are important components of automated assembly
systems: fixtures accurately hold parts and feeders move parts into alignment.
These components can fail when part shape varies. Parametric tolerance
classes specify how much variation is allowable. In this paper we consider
fixturing convex polygonal parts using right-angle brackets and feeding
polygonal parts on conveyor belts using sequences of vertical fences. For
both cases, we define new tolerance classes and give algorithms for computing
the parameter specifications such that the fixture or feeder will work for
all parts in the tolerance class. For fixturing we give an O(1) algorithm to
compute the dimensions of rectangular tolerance zones. For feeding we give
an O(n2) algorithm to compute the radius of the largest allowable tolerance
zone around each vertex. For each, we give an O(n) time algorithm for testing
if an n-sided part is in the tolerance class
A spanning tree model for the Heegaard Floer homology of a branched double-cover
Given a diagram of a link K in S^3, we write down a Heegaard diagram for the
branched-double cover Sigma(K). The generators of the associated Heegaard Floer
chain complex correspond to Kauffman states of the link diagram. Using this
model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded
group. We also conjecture the existence of a delta-grading on
\hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov
homology.Comment: 43 pages, 20 figure
When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks
Let be a supply graph and a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of and
demands to the edges of is said to satisfy the \emph{cut condition} if for
any cut in the graph, the total demand crossing the cut is no more than the
total capacity crossing it. The pair is called \emph{cut-sufficient} if
for any assignment of capacities and demands that satisfy the cut condition,
there is a multiflow routing the demands defined on within the network with
capacities defined on . We prove a previous conjecture, which states that
when the supply graph is series-parallel, the pair is
cut-sufficient if and only if does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of and delete edges
of and so that becomes the complete bipartite graph , with
odd, and is composed of a cycle connecting the vertices of
degree 2, and an edge connecting the two vertices of degree . We further
prove that if the instance is \emph{Eulerian} --- that is, the demands and
capacities are integers and the total of demands and capacities incident to
each vertex is even --- then the multiflow problem has an integral solution. We
provide a polynomial-time algorithm to find an integral solution in this case.
In order to prove these results, we formulate properties of tight cuts (cuts
for which the cut condition inequality is tight) in cut-sufficient pairs. We
believe these properties might be useful in extending our results to planar
graphs.Comment: An extended abstract of this paper will be published at the 44th
Symposium on Theory of Computing (STOC 2012
Orienting polyhedral parts by pushing
A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of an asymmetric polyhedral part by a sequence of push actions, and show that is it possible to move any such part from an unknown initial orientation into a known final orientation if these actions are performed by a jaw consisting of two orthogonal planes. We also show how to compute an orienting sequence of push actions.We propose a three-dimensional generalization of conveyor belts with fences consisting of a sequence of tilted plates with curved tips; each of the plates contains a sequence of fences. We show that it is possible to compute a set-up of plates and fences for any given asymmetric polyhedral part such that the part gets oriented on its descent along plates and fences
Sandpiles and Dominos
We consider the subgroup of the abelian sandpile group of the grid graph
consisting of configurations of sand that are symmetric with respect to central
vertical and horizontal axes. We show that the size of this group is (i) the
number of domino tilings of a corresponding weighted rectangular checkerboard;
(ii) a product of special values of Chebyshev polynomials; and (iii) a
double-product whose factors are sums of squares of values of trigonometric
functions. We provide a new derivation of the formula due to Kasteleyn and to
Temperley and Fisher for counting the number of domino tilings of a 2m x 2n
rectangular checkerboard and a new way of counting the number of domino tilings
of a 2m x 2n checkerboard on a M\"obius strip.Comment: 35 pages, 24 figure
- …