The multivariate L1-median and associated data depth

This paper gives three related results: (i) a new, simple, fast, monotonically converging algorithm for deriving the L1-median of a data cloud in ℝd, a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; (ii) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different dept functions; and (iii) a simple closed-form formula of the L1-depth function for a given data cloud in ℝd.

Integrated fossil and molecular data reconstruct bat echolocation

Molecular and morphological data have important roles in illuminating evolutionary history. DNA data often yield well resolved phylogenies for living taxa, but are generally unattainable for fossils. A distinct advantage of morphology is that some types of morphological data may be collected for extinct and extant taxa. Fossils provide a unique window on evolutionary history and may preserve combinations of primitive and derived characters that are not found in extant taxa. Given their unique character complexes, fossils are critical in documenting sequences of character transformation over geologic time and may elucidate otherwise ambiguous patterns of evolution that are not revealed by molecular data alone. Here, we employ a methodological approach that allows for the integration of molecular and paleontological data in deciphering one of the most innovative features in the evolutionary history of mammals—laryngeal echolocation in bats. Molecular data alone, including an expanded data set that includes new sequences for the A2AB gene, suggest that microbats are paraphyletic but do not resolve whether laryngeal echolocation evolved independently in different microbat lineages or evolved in the common ancestor of bats and was subsequently lost in megabats. When scaffolds from molecular phylogenies are incorporated into parsimony analyses of morphological characters, including morphological characters for the Eocene taxa Icaronycteris, Archaeonycteris, Hassianycteris, and Palaeochiropteryx, the resulting trees suggest that laryngeal echolocation evolved in the common ancestor of fossil and extant bats and was subsequently lost in megabats. Molecular dating suggests that crown-group bats last shared a common ancestor 52 to 54 million years ago.

Blood flow and oxygen delivery to human brain during functional activity: Theoretical modeling and experimental data

Coupling of cerebral blood flow (CBF) and cerebral metabolic rate for oxygen (CMRO2) in physiologically activated brain states remains the subject of debates. Recently it was suggested that CBF is tightly coupled to oxidative metabolism in a nonlinear fashion. As part of this hypothesis, mathematical models of oxygen delivery to the brain have been described in which disproportionately large increases in CBF are necessary to sustain even small increases in CMRO2 during activation. We have explored the coupling of CBF and oxygen delivery by using two complementary methods. First, a more complex mathematical model was tested that differs from those recently described in that no assumptions were made regarding tissue oxygen level. Second, [15O] water CBF positron emission tomography (PET) studies in nine healthy subjects were conducted during states of visual activation and hypoxia to examine the relationship of CBF and oxygen delivery. In contrast to previous reports, our model showed adequate tissue levels of oxygen could be maintained without the need for increased CBF or oxygen delivery. Similarly, the PET studies demonstrated that the regional increase in CBF during visual activation was not affected by hypoxia. These findings strongly indicate that the increase in CBF associated with physiological activation is regulated by factors other than local requirements in oxygen.