Limit theory for point processes in manifolds

Let $Y_i,i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal {M}\subset \mathbb{R}^d$ and consider sums $\sum_{i=1}^n\xi(n^{1/m}Y_i,\{n^{1/m}Y_j\}_{j=1}^n)$, where $\xi$ is a real valued function defined on pairs $(y,\mathcal {Y})$, with $y\in \mathbb{R}^d$ and $\mathcal {Y}\subset \mathbb{R}^d$ locally finite. Subject to $\xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $\xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $\mathcal {M}$. We apply the general results to establish the limit theory of dimension and volume content estimators, R\'{e}nyi and Shannon entropy estimators and clique counts in the Vietoris-Rips complex on $\{Y_i\}_{i=1}^n$.Comment: Published in at http://dx.doi.org/10.1214/12-AAP897 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Caudal pneumaticity and pneumatic hiatuses in the sauropod dinosaurs Giraffatitan and Apatosaurus

Skeletal pneumaticity is found in the presacral vertebrae of most sauropod dinosaurs, but pneumaticity is much less common in the vertebrae of the tail. We describe previously unrecognized pneumatic fossae in the mid-caudal vertebrae of specimens of Giraffatitan and Apatosaurus. In both taxa, the most distal pneumatic vertebrae are separated from other pneumatic vertebrae by sequences of three to seven apneumatic vertebrae. Caudal pneumaticity is not prominent in most individuals of either of these taxa, and its unpredictable development means that it may be more widespread than previously recognised within Sauropoda and elsewhere in Saurischia. The erratic patterns of caudal pneumatization in Giraffatitan and Apatosaurus, including the pneumatic hiatuses, show that pneumatic diverticula were more broadly distributed in the bodies of the living animals than are their traces in the skeleton. Together with recently published evidence of cryptic diverticula--those that leave few or no skeletal traces--in basal sauropodomorphs and in pterosaurs, this is further evidence that pneumatic diverticula were widespread in ornithodirans, both across phylogeny and throughout anatomy

Why sauropods had long necks; and why giraffes have short necks

The necks of the sauropod dinosaurs reached 15 m in length: six times longer than that of the world record giraffe and five times longer than those of all other terrestrial animals. Several anatomical features enabled this extreme elongation, including: absolutely large body size and quadrupedal stance providing a stable platform for a long neck; a small, light head that did not orally process food; cervical vertebrae that were both numerous and individually elongate; an efficient air-sac-based respiratory system; and distinctive cervical architecture. Relevant features of sauropod cervical vertebrae include: pneumatic chambers that enabled the bone to be positioned in a mechanically efficient way within the envelope; and muscular attachments of varying importance to the neural spines, epipophyses and cervical ribs. Other long-necked tetrapods lacked important features of sauropods, preventing the evolution of longer necks: for example, giraffes have relatively small torsos and large, heavy heads, share the usual mammalian constraint of only seven cervical vertebrae, and lack an air-sac system and pneumatic bones. Among non-sauropods, their saurischian relatives the theropod dinosaurs seem to have been best placed to evolve long necks, and indeed they probably surpassed those of giraffes. But 150 million years of evolution did not suffice for them to exceed a relatively modest 2.5 m.Comment: 39 pages, 11 figures, 3 table