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    Dense H-free graphs are almost (Χ(H)-1)-partite

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    By using the Szemeredi Regularity Lemma, Alon and Sudakov recently extended the classical Andrasfai-Erdos-Sos theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r+1)-partite graph H whose smallest part has t vertices, there exists a constant C such that for any given ε>0 and sufficiently large n the following is true. Whenever G is an n-vertex graph with minimum degree δ(G)≥(1 − 3/3r−1 + ε)n, either G contains H, or we can delete f(n,H)≤Cn2−1/t edges from G to obtain an r-partite graph. Further, we are able to determine the correct order of magnitude of f(n,H) in terms of the Zarankiewicz extremal function

    On the H^1-L^1 boundedness of operators

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    We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous extension to a bounded linear operator from H^1(R^n) to Y. We show that the same is true if we replace (1,q)-atoms with continuous (1,\infty)-atoms. This is known to be false for (1,\infty)-atoms.Comment: This paper will appear in Proceedings of the American Mathematical Societ

    On the Biological Standard of Living of Eighteenth-Century Americans: Taller, Richer, Healthier

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    This study analyses the physical stature of runaway apprentices and military deserters based on advertisements collected from 18th-century newspapers, in order to explore the biological welfare of colonial and early-national Americans. The results indicate that heights declined somewhat at mid-century, but increased substantially thereafter. The findings are generally in keeping with trends in mortality and in economic activity. The Americans were much taller than Europeans: by the 1780s adults were as much as 6.6 cm taller than Englishmen, and at age 16 American apprentices were some 12 cm taller than the poor children of London

    Micrometre-scale refrigerators

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    A superconductor with a gap in the density of states or a quantum dot with discrete energy levels is a central building block in realizing an electronic on-chip cooler. They can work as energy filters, allowing only hot quasiparticles to tunnel out from the electrode to be cooled. This principle has been employed experimentally since the early 1990s in investigations and demonstrations of micrometre-scale coolers at sub-kelvin temperatures. In this paper, we review the basic experimental conditions in realizing the coolers and the main practical issues that are known to limit their performance. We give an update of experiments performed on cryogenic micrometre-scale coolers in the past five years