Consider the graph that has as vertices all bitstrings of length 2n + 1 with exactly n or n + 1 entries equal to 1, and an edge between any two bitstrings that differ in exactly one bit. The well-known middle levels conjecture asserts that this graph has a Hamilton cycle for any n ≥ 1. In this paper we present a new proof of this conjecture, which is much shorter and more accessible than the original proof.

We studied winter climate influences on central European small herbivores by testing the prediction that direct physical effects of winter climate should be more pronounced in herbivores living above ground/snow than in subnivean/fossorial ones. Using correlation analysis and autoregressive modeling, we found that population growth rates of European hares, representing the former class of herbivores, are more efficiently predicted by the winter NAO index than those of common voles, representing the latter class of herbivores. We demonstrate that, whereas in hares the NAO index outperformed crop yield indices, used here as a proxy for plant production variability, it was crop yield indices that more effectively predicted population change in voles. These results suggest that the relative importance of direct and indirect effects of winter climate on herbivores may be related to their body size, the major determinant of above ground/snow or subnivean/fossorial mode of life.

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