Shellsort with three increments

A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments $(h,g,1)$. In particular, when $h=\Theta(n^{7/15})$ and $g=\Theta(h^{1/5})$, the average running time is $O(n^{23/15})$. The proof involves interesting properties of the inversions in random permutations that have been $h$-sorted and $g$-sorted

Minimal matchings of point processes

Suppose that red and blue points form independent homogeneous Poisson processes of equal intensity in R-d. For a positive (respectively, negative) parameter gamma we consider red-blue matchings that locally minimize (respectively, maximize) the sum of gamma th powers of the edge lengths, subject to locally minimizing the number of unmatched points. The parameter can be viewed as a measure of fairness. The limit gamma -> -infinity is equivalent to Gale-Shapley stable matching. We also consider limits as gamma approaches 0, 1-, 1+ and infinity. We focus on dimension d = 1. We prove that almost surely no such matching has unmatched points. (This question is open for higher d). For each gamma < 1 we establish that there is almost surely a unique such matching, and that it can be expressed as a finitary factor of the points. Moreover, its typical edge length has finite rth moment if and only if r < 1 /2. In contrast, for gamma = 1 there are uncountably many matchings, while for gamma > 1 there are countably many, but it is impossible to choose one in a translation-invariant way. We obtain existence results in higher dimensions (covering many but not all cases). We address analogous questions for one-colour matchings also

Lord Rutherford of Nelson, His 1908 Nobel Prize in Chemistry and Why He Didn't Get a Second Prize

"I have dealt with many different transformations with various periods of time, but the quickest that I have met was my own transformation in one moment from a physicist to a chemist." Ernest Rutherford (Nobel Banquet, 1908) This article is about how Ernest Rutherford (1871-1937) got the 1908 Nobel Prize in Chemistry and why he did not get a second Prize for his subsequent outstanding discoveries in physics, specially the discovery of the atomic nucleus and the proton. Who were those who nominated him and who did he nominate for the Nobel Prizes. In order to put the Prize issue into its proper context, I will briefly describe Rutherford's whereabouts. Rutherford, an exceptionally gifted scientist who revolutionized chemistry and physics, was moulded in the finest classical tradition. What were his opinions on some scientific issues such as Einstein's photon, uncertainty relations and the future prospects for atomic energy? What would he have said about the "Theory of Everything"?Comment: Extended version of an invited talk presented at the neutrino conference "Neutrino 2008", Christchurch, New Zealand, 25-31 May 200

Primer Extension Capture: Targeted Sequence Retrieval from Heavily Degraded DNA Sources

We present a method of targeted DNA sequence retrieval from DNA sources which are heavily degraded and contaminated with microbial DNA, as is typical of ancient bones. The method greatly reduces sample destruction and sequencing demands relative to direct PCR or shotgun sequencing approaches. We used this method to reconstruct the complete mitochondrial DNA (mtDNA) genomes of five Neandertals from across their geographic range. The mtDNA genetic diversity of the late Neandertals was approximately three times lower than that of contemporary modern humans. Together with analyses of mtDNA protein evolution, these data suggest that the long-term effective population size of Neandertals was smaller than that of modern humans and extant great apes

Rural-Urban Policies : Changing Conceptions of the Human-Environment Relationship

Peer reviewedPostprin