Spanning trees without adjacent vertices of degree 2

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2

Long-term retrospective analysis of mackerel spawning in the North Sea: a new time series and modeling approach to CPR data

We present a unique view of mackerel (Scomber scombrus) in the North Sea based on a new time series of larvae caught by the Continuous Plankton Recorder (CPR) survey from 1948-2005, covering the period both before and after the collapse of the North Sea stock. Hydrographic backtrack modelling suggested that the effect of advection is very limited between spawning and larvae capture in the CPR survey. Using a statistical technique not previously applied to CPR data, we then generated a larval index that accounts for both catchability as well as spatial and temporal autocorrelation. The resulting time series documents the significant decrease of spawning from before 1970 to recent depleted levels. Spatial distributions of the larvae, and thus the spawning area, showed a shift from early to recent decades, suggesting that the central North Sea is no longer as important as the areas further west and south. These results provide a consistent and unique perspective on the dynamics of mackerel in this region and can potentially resolve many of the unresolved questions about this stock.lved questions about this stoc

Counting absolute number of molecules using unique molecular identifiers

Advances in molecular biology have made it easy to identify different DNA or RNA species and to copy them. Identification of nucleic acid species can be accomplished by reading the DNA sequence; currently millions of molecules can be sequenced in a single day using massively parallel sequencing. Efficient copying of DNA-molecules of arbitrary sequence was made possible by molecular cloning, and the polymerase chain reaction. Differences in the relative abundance of a large number of different sequences between two or more samples can in turn be measured using microarray hybridization and/or tag sequencing. However, determining the relative abundance of two different species and/or the absolute number of molecules present in a single sample has proven much more challenging. This is because it is hard to detect individual molecules without copying them, and even harder to make defined number of copies of molecules. We show here that this limitation can be overcome by using unique molecular identifiers (umis), which make each molecule in the sample distinct

Optimal Weak to Strong Learning

The classic algorithm AdaBoost allows to convert a weak learner, that is an algorithm that produces a hypothesis which is slightly better than chance, into a strong learner, achieving arbitrarily high accuracy when given enough training data. We present a new algorithm that constructs a strong learner from a weak learner but uses less training data than AdaBoost and all other weak to strong learners to achieve the same generalization bounds. A sample complexity lower bound shows that our new algorithm uses the minimum possible amount of training data and is thus optimal. Hence, this work settles the sample complexity of the classic problem of constructing a strong learner from a weak learner