785 research outputs found
Cardinalities of topologies with small base
Let T be the family of open subsets of a topological space (not necessarily
Hausdorff or even T_0). We prove that if T has a base of cardinality <= mu,
lambda <= mu < 2^lambda, lambda strong limit of cofinality aleph_0, then T has
cardinality = 2^lambda. This is our main conclusion. First we prove
it under some set theoretic assumption, which is clear when lambda = mu ; then
we eliminate the assumption by a theorem on pcf from [Sh 460] motivated
originally by this. Next we prove that the simplest examples are the basic
ones; they occur in every example (for lambda = aleph_0 this fulfill a promise
from [Sh 454]). The main result for the case lambda = aleph_0 was proved in [Sh
454]
Technology Mapping for Circuit Optimization Using Content-Addressable Memory
The growing complexity of Field Programmable Gate Arrays (FPGA's) is leading to architectures with high input cardinality look-up tables (LUT's). This thesis describes a methodology for area-minimizing technology mapping for combinational logic, specifically designed for such FPGA architectures. This methodology, called LURU, leverages the parallel search capabilities of Content-Addressable Memories (CAM's) to outperform traditional mapping algorithms in both execution time and quality of results. The LURU algorithm is fundamentally different from other techniques for technology mapping in that LURU uses textual string representations of circuit topology in order to efficiently store and search for circuit patterns in a CAM. A circuit is mapped to the target LUT technology using both exact and inexact string matching techniques. Common subcircuit expressions (CSE's) are also identified and used for architectural optimization---a small set of CSE's is shown to effectively cover an average of 96% of the test circuits. LURU was tested with the ISCAS'85 suite of combinational benchmark circuits and compared with the mapping algorithms FlowMap and CutMap. The area reduction shown by LURU is, on average, 20% better compared to FlowMap and CutMap. The asymptotic runtime complexity of LURU is shown to be better than that of both FlowMap and CutMap
Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes
A quadrangulation is a graph embedded on the sphere such that each face is
bounded by a walk of length 4, parallel edges allowed. All quadrangulations can
be generated by a sequence of graph operations called vertex splitting,
starting from the path P_2 of length 2. We define the degree D of a splitting S
and consider restricted splittings S_{i,j} with i <= D <= j. It is known that
S_{2,3} generate all simple quadrangulations.
Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}.
First we show that the splittings S_{1,2} are exactly the monotone ones in the
sense that the resulting graph contains the original as a subgraph. Then we
show that they define a set of nontrivial ancestors beyond P_2 and each
quadrangulation has a unique ancestor.
Our results have a direct geometric interpretation in the context of
mechanical equilibria of convex bodies. The topology of the equilibria
corresponds to a 2-coloured quadrangulation with independent set sizes s, u.
The numbers s, u identify the primary equilibrium class associated with the
body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate
all primary classes from a finite set of ancestors which is closely related to
their geometric results.
If, beyond s and u, the full topology of the quadrangulation is considered,
we arrive at the more refined secondary equilibrium classes. As Domokos,
L\'angi and Szab\'o showed recently, one can create the geometric counterparts
of unrestricted splittings to generate all secondary classes. Our results show
that S_{1,2} can only generate a limited range of secondary classes from the
same ancestor. The geometric interpretation of the additional ancestors defined
by monotone splittings shows that minimal polyhedra play a key role in this
process. We also present computational results on the number of secondary
classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table
Topology Inspired Problems for Cellular Automata, and a Counterexample in Topology
We consider two relatively natural topologizations of the set of all cellular
automata on a fixed alphabet. The first turns out to be rather pathological, in
that the countable space becomes neither first-countable nor sequential. Also,
reversible automata form a closed set, while surjective ones are dense. The
second topology, which is induced by a metric, is studied in more detail.
Continuity of composition (under certain restrictions) and inversion, as well
as closedness of the set of surjective automata, are proved, and some
counterexamples are given. We then generalize this space, in the sense that
every shift-invariant measure on the configuration space induces a pseudometric
on cellular automata, and study the properties of these spaces. We also
characterize the pseudometric spaces using the Besicovitch distance, and show a
connection to the first (pathological) space.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
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