On the optimality of gluing over scales

We show that for every $\alpha > 0$, there exist $n$-point metric spaces (X,d) where every "scale" admits a Euclidean embedding with distortion at most $\alpha$, but the whole space requires distortion at least $\Omega(\sqrt{\alpha \log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when $\alpha = \Theta(1)$ and $\alpha = \Theta(\log n)$, but nowhere in between. More specifically, we exhibit $n$-point spaces with doubling constant $\lambda$ requiring Euclidean distortion $\Omega(\sqrt{\log \lambda \log n})$, which also shows that the technique of "measured descent" [Krauthgamer, et. al., Geometric and Functional Analysis] is optimal. We extend this to obtain a similar tight result for $L_p$ spaces with $p > 1$.Comment: minor revision

Zone Diagrams in Euclidean Spaces and in Other Normed Spaces

Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.Comment: Title page + 16 pages, 20 figure

Ageism and sexuality

Sexuality remains important throughout a person’s life, but sexual behavior does not receive the same levels of acceptance at all ages. Older people are challenged by ageist attitudes and perceptions that hinder their sexual expression. They are stereotyped as non-sexual beings who should not, cannot, and do not want to have sexual relationships. Expressing sexuality or engaging in sexual activity in later life is considered by many in society as immoral or perverted. False expectations for older people also stem from ideals of beauty, centralization of the biomedical perspective on sexuality of older adults, and the association of sex with reproduction. Unfortunately, older people internalize many ageist attitudes towards sexuality in later life and become less interested in sex and less sexually active. The following chapter explores attitudes towards sexuality in later life among the media, young people, older people themselves, and care providers. In order to enable older people to express their sexuality and sexual identity freely and fully, awareness of ageist perceptions must be raised and defeated

Limitations to Frechet's Metric Embedding Method

Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is non-Frechet, and the purpose of this note is to show that this is not coincidental. Specifically, for every e>0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Frechet embedding into l_p on subsets of size at least n^{1/2 + e} is \Omega((\log n)^{1/p}).Comment: 10 pages, 1 figur

Gender differences in the association between self-rated health and hypertension in a Korean adult population

<p>Abstract</p> <p>Background</p> <p>Self-rated health (SRH) has been reported as a predictor of mortality in previous studies. This study aimed to examine whether SRH is independently associated with hypertension and if there is a gender difference in this association.</p> <p>Methods</p> <p>16,956 community dwelling adults aged 20 and over within a defined geographic area participated in this study. Data on SRH, socio-demographic factors (age, gender, marital status, education) and health behaviors (smoking status, alcohol consumption, physical activity) were collected. Body mass index and blood pressure were measured. Logistic regression models were used to determine a relationship between SRH and hypertension.</p> <p>Results</p> <p>32.5% of the participants were found to have hypertension. Women were more likely than men to rate their SRH as poor (<it>p </it>< 0.001), and the older age groups rated their SRH more negatively in both men and women (<it>p </it>< 0.001). While the multivariate-adjusted odds ratio (OR, 95% CI) of participants rating their SRH as very poor for hypertension in men was OR 1.70 (1.13-2.58), that in women was OR 2.83 (1.80-4.44). Interaction between SRH and gender was significant (<it>p </it>< 0.001).</p> <p>Conclusions</p> <p>SRH was independently associated with hypertension in a Korean adult population. This association was modified by gender.</p

Shift invariant preduals of &#8467;₁(&#8484;)

The Banach space &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) weak&lt;sup&gt;*&lt;/sup&gt;-continuous. This is equivalent to making the natural convolution multiplication on &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) separately weak*-continuous and so turning &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) into a dual Banach algebra. We call such preduals &lt;i&gt;shift-invariant&lt;/i&gt;. It is known that the only shift-invariant predual arising from the standard duality between C&lt;sub&gt;0&lt;/sub&gt;(K) (for countable locally compact K) and &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) is c&lt;sub&gt;0&lt;/sub&gt;(&#8484;). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak&lt;sup&gt;*&lt;/sup&gt;-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c&lt;sub&gt;0&lt;/sub&gt;. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of &#8484;. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c&lt;sub&gt;0&lt;/sub&gt;

Continuity properties of measurable group cohomology

A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood. This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to `regularize' measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of consequences are then derived from this argument. First, we prove that for target modules that are Fr\'echet spaces, the cohomology groups agree with those defined using continuous cocycles, and hence they vanish in positive degrees when the acting group is compact. Using this, we then show that for Fr\'echet, discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups, and also under forming direct limits of discrete target modules. Lastly, these results together enable us to establish various circumstances under which the measurable-cochains cohomology groups coincide with others defined using sheaves on a semi-simplicial space associated to the underlying group, or sheaves on a classifying space for that group. We also prove in some cases that the natural quotient topologies on the measurable-cochains cohomology groups are Hausdorff.Comment: 52 pages. [Nov 22, 2011:] Major re-write with Calvin C. Moore as new co-author. Results from previous version strengthened and several new results added. [Nov 25, 2012:] Final version now available at springerlink.co