On the size of approximately convex sets in normed spaces

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let \Co(A) be the convex hull and \diam(A) the diameter of $A$. We prove that every $n$-dimensional normed space contains approximately convex sets $A$ with \mathcal{H}(A,\Co(A))\ge \log_2n-1 and \diam(A) \le C\sqrt n(\ln n)^2, where $\mathcal{H}$ denotes the Hausdorff distance. These estimates are reasonably sharp. For every $D>0$, we construct worst possible approximately convex sets in $C[0,1]$ such that \mathcal{H}(A,\Co(A))=\diam(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.Comment: 32 pages. See also http://www.math.sc.edu/~howard

Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$f((x+y)/2) \le (f(x)+f(y))/2 + 1.$$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function~$E$ vanishing on the vertices of a simplex. A set $A$ in a normed space is an approximately convex set iff for all $a,b\in A$ the distance of the midpoint $(a+b)/2$ to $A$ is $\le 1$. The bounds on approximately convex functions are used to show that in $\R^n$ with the Euclidean norm, for any approximately convex set $A$, any point $z$ of the convex hull of $A$ is at a distance of at most $[\log_2(n-1)]+1+(n-1)/2^{[\log_2(n-1)]}$ from $A$. Examples are given to show this is the sharp bound. Bounds for general norms on $R^n$ are also given.Comment: 39 pages. See also http://www.math.sc.edu/~howard

Nontonal floating features as grammatical morphemes

The concept of floating tones is no longer controversial in tonal analysis; important insights into the morphology of numerous tonal languages have relied on the positing of morphemes that are composed simply of prosodically unlicensed tones. Employing data from three of Africa\u27s four major language families, this paper builds on this notion by recognizing the existence of nontonal floating features -- morphemes composed solely of phonological features that have no segmental support. The first example, from Kanembu (Nilo-Sarahan, Chad), shows that the [+ATR] feature is the sole marker of incompletive aspect in the verb. Again, in Mafa and Podoko (Chadic, Cameroon), certain aspects of the verb are marked by floating labial and palatal features. One of the noun class markers in Aka (Bantu, C.A.R.) consists only of the floating feature [+voice]. And finally, the marker of the completive aspect in Mokulu (Chadic, Chad) is a morpheme that is comprised simply of the feature complex [+voice] and [+high]. In conclusion, after noting that a similar floating feature analysis may be appropriate in other languages such as Ngbaka (Ubangian, Zaire), we consider the general characteristics of floating features as full grammatical morphemes

Update of MRST parton distributions.

We discuss the latest update of the MRST parton distributions in response to the most recent data. We discuss the areas where there are hints of difficulties in the global fit, and compare to some other updated sets of parton distributions, particularly CTEQ6. We briefly discuss the issue of uncertainties associated with partons

MRST global fit update.

We discuss the impact of the most recent data on the MRST global analysis - in particular the new high-ET jet data and their implications for the gluon and the new small x structure function data. In the light of these new data we also consider the uncertainty in predictions for physical quantities depending on parton distributions, concentrating on the W cross-section at hadron colliders

Chlamydia trachomatis and the risk of spontaneous preterm birth, babies who are born small for gestational age, and stillbirth: A population-based cohort study

Background: Chlamydia trachomatis is one of the most commonly diagnosed sexually transmitted infections worldwide, but reports in the medical literature of an association between genital chlamydia infection and adverse obstetric outcomes are inconsistent. Methods: The Western Australia Data Linkage Branch created a cohort of women of reproductive age by linking records of birth registrations with the electoral roll for women in Western Australia who were born from 1974 to 1995. The cohort was then linked to both chlamydia testing records and the state perinatal registry for data on preterm births and other adverse obstetric outcomes. We determined associations between chlamydia testing, test positivity, and adverse obstetric outcomes using multivariate logistic regression analyses. Findings: From 2001 to 2012, 101558 women aged 15 to 38 years had a singleton birth. Of these women, 3921 (3·9%) had a spontaneous preterm birth, 9762 (9·6% of 101371 women with available data) had a baby who was small for gestational age, and 682 (0·7%) had a stillbirth. During their pregnancy, 21267 (20·9%) of these women had at least one chlamydia test record, and 1365 (6·4%) of those tested were positive. Before pregnancy, 19157 (18·9%) of these women were tested for chlamydia, of whom 1595 (8·3%) tested positive for chlamydia. Among all women with a test record, after adjusting for age, ethnicity, maternal smoking, and history of other infections, we found no significant association between a positive test for chlamydia and spontaneous preterm birth (adjusted odds ratio 1·08 [95% CI 0·91–1·28]; p=0·37), a baby who was small for gestational age (0·95 [0·85–1·07]; p=0·39), or stillbirth (0·93 [0·61–1·42]; p=0·74). Interpretation: A genital chlamydia infection that is diagnosed and, presumably, treated either during or before pregnancy does not substantially increase a woman’s risk of having a spontaneous preterm birth, having a baby who is small for gestational age, or having a stillbirth. Funding: Australian National Health and Medical Research Counci

29 years of vegetation community change across environmental gradients in a Mojave Desert mountain range

There is a great deal of uncertainty as to how biological communities respond to changes in land use and climate change, a situation particularly relevant in protected areas such as national parks that were designated to conserve specific biological features. Utilizing extant vegetation data sets with repeatable methodology can provide opportunities for insight into previous vegetation change and provide base line data for long-term monitoring projects useful for modeling vegetation community trajectories. We have relocated and resurveyed 106 sites from a vegetation community study initiated in 1979 in the Newberry Mountains, southern Nevada, within Lake Mead National Recreation Area managed by the National Park Service. The original methods were repeated and used to establish permanent long-term monitoring plots. All perennial plant species were measured for density, frequency and cover within each plot. In comparing 1979 and 2008 data sets we wanted to know if changes have occurred in the vegetation community and if the degree of change differs along environmental gradients and among individual species