Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.Comment: 14 pages, 2 figures, 2 table

What Can We Learn about Smoking from 150 Years of Italian Data?

This paper estimates dynamic demand models for tobacco consumption in Italy from 1871 to 2010. The empirical analysis is based on an entirely new dataset. Because the tobacco sector was mostly managed by the state, rich and detailed historical documentation is available. Price elasticities are estimated both for aggregate tobacco consumption and its four major components (cigars, cigarettes, cut to- 25 bacco, and snuff) for three separate sub-periods: 1871–1913, 1919–1939, and 1946–2010. Elasticities consistently belong to a narrow set. We discuss the public policy implications of a seemingly iso-elastic tobacco demand function

Familial transitional cell carcinoma among the population of Iceland

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