4 research outputs found
Counting partitions of with degree congruence conditions
For , the Erd\H{o}s--Renyi random graph, let be the
random variable representing the number of distinct partitions of into
sets so that the degree of each vertex in is
divisible by for all . We prove that if is odd then
, and if is even then
. More generally, we show that the
distribution is still asymptotically Poisson when we require all degrees in
to be congruent to modulo for each , where the
residues may be chosen freely. For , the distribution is not
asymptotically Poisson, but it can be determined explicitly
A note on infinite antichain density
Let \scrF be an antichain of finite subsets of \BbbN . How quickly can the quantities | \scrF \cap
2
[n]
| grow as n \rightarrow \infty ? We show that for any sequence (fn)n\geq n0 \sum
of positive integers satisfying
\infty
n=n0
fn/2
n \leq 1/4 and fn \leq fn+1 \leq 2fn, there exists an infinite antichain \scrF of finite subsets of
\BbbN such that | \scrF \cap 2
[n]
| \geq fn for all n \geq n0. It follows that for any \varepsilon > 0 there exists an antichain
\scrF \subseteq 2
\BbbN such that lim infn\rightarrow \infty | \scrF \cap 2
[n]
| \cdot \bigl(
2
n
n log1+\varepsilon n
\bigr) - 1 > 0. This resolves a problem of Sudakov,
Tomon, and Wagner in a strong form and is essentially tight
Statistical evaluation of data from tractor guidance systems
Statistical tools are discussed for the analysis of data collected from tractor guidance systems. The importance of both accuracy and precision is discussed, and statistical
tools for analysis are considered which incorporate important features of the data.
In particular, accuracy is modelled using a generalized least squares model incorporating autocorrelation, and variances (inverse of precision) using a gamma generalized linear model. The methods are applied to data collected during an experiment conducted with a
Trimble receiver used with a Beeline tractor guidance system. Three different scenarios are
considered, then compared: a tractor simulating ploughing a field; the tractor pulling a
plough with the receivers on the tractor; the tractor pulling a plough with the Trimble
receiver on the plough. The change in the precision and accuracy between the scenarios is
discussed. Data were recorded over repeated swaths for each scenario. After discussing
specific statistical techniques for analysis of this type of data, the collected data are
analysed; major conclusions are: The data from the Trimble receiver showed evidence of
autocorrelation in the offsets; the plough recorded a variance about three times that
recorded by the tractor