10,260 research outputs found
Counting matroids in minor-closed classes
A flat cover is a collection of flats identifying the non-bases of a matroid.
We introduce the notion of cover complexity, the minimal size of such a flat
cover, as a measure for the complexity of a matroid, and present bounds on the
number of matroids on elements whose cover complexity is bounded. We apply
cover complexity to show that the class of matroids without an -minor is
asymptotically small in case is one of the sparse paving matroids
, , , , or , thus confirming a few special
cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other
hand, we show a lower bound on the number of matroids without -minor
which asymptoticaly matches the best known lower bound on the number of all
matroids, due to Knuth.Comment: 13 pages, 3 figure
On the number of matroids
We consider the problem of determining , the number of matroids on
elements. The best known lower bound on is due to Knuth (1974) who showed
that is at least . On the other hand, Piff
(1973) showed that , and it has
been conjectured since that the right answer is perhaps closer to Knuth's
bound.
We show that this is indeed the case, and prove an upper bound on that is within an additive term of Knuth's lower bound. Our proof
is based on using some structural properties of non-bases in a matroid together
with some properties of independent sets in the Johnson graph to give a
compressed representation of matroids.Comment: Final version, 17 page
An entropy argument for counting matroids
We show how a direct application of Shearers' Lemma gives an almost optimum
bound on the number of matroids on elements.Comment: Short note, 4 page
R.F. planar magnetron sputtered ZnO films II: Electrical properties
The electrical properties of r.f. planar magnetron sputtered ZnO films are studied by means of current-voltage, capacitance-voltage and Van der Pauw measurements.\ud
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These films are applied as piezoelectric transducers in micromechanical sensors and actuators. Their piezoelectric behaviour strongly depends on the electric properties.\ud
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A conduction model for the polycrystalline ZnO layers is presented. This model gives a good description of the electrical behaviour, and is useful in understanding the piezoelectric properties of the films studied
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