Counting partitions of Gn,1/2G_{n,1/2} with degree congruence conditions

For $G=G_{n, 1/2}$, the Erd\H{o}s--Renyi random graph, let $X_n$ be the random variable representing the number of distinct partitions of $V(G)$ into sets $A_1, \ldots, A_q$ so that the degree of each vertex in $G[A_i]$ is divisible by $q$ for all $i\in[q]$. We prove that if $q\geq 3$ is odd then $X_n\xrightarrow{d}{\mathrm{Po}(1/q!)}$, and if $q \geq 4$ is even then $X_n\xrightarrow{d}{\mathrm{Po}(2^q/q!)}$. More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in $G[A_i]$ to be congruent to $x_i$ modulo $q$ for each $i\in[q]$, where the residues $x_i$ may be chosen freely. For $q=2$, 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