Physiological ecology of the ciliated protozoon Loxodes

Loxodes faces special problems in living close to the oxic-anoxic boundary. In tightly-stratified ponds like Priest Pot its optimum environment may be quite narrow and it can be displaced by the slightest turbulence. Loxodes cannot sense an O sub(2) gradient directly but its ability to perceive gravity allows it to make relatively long vertical migrations. It is also sensitive to light and oxygen and it uses these environmental cues to modulate the parameters of its random motility: in the dark, it aggregates at a low O sub(2) tension and in bright light it aggregates in anoxic water. The oxic-anoxic boundary is also a zone where O sub(2) may be a scarce and transient resource, but Loxodes) can switch to nitrate respiration and exploit the pool of nitrate that often exists close to the base of the oxycline

A unified flow approach to smooth, even LpL_p-Minkowski problems

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in $\mathbb{R}^{n+1}$ for $p>-n-1.$Comment: 21 pages. Comments are welcom

From Cosmos to Intelligent Life: The Four Ages of Astrobiology

The history of life on Earth and in other potential life-bearing planetary platforms is deeply linked to the history of the universe. Since life as we know it relies on chemical elements forged in dying heavy stars, the universe needs to be old enough for stars to form and evolve. Current cosmological theory indicates that the universe is 13.7$\pm 0.13$ billion years old and that the first stars formed hundreds of millions of years after the big bang. At least some stars formed with stable planetary systems wherein a set of biochemical reactions leading to life could have taken place. In this lecture, I argue that we can divide cosmological history into four ages, from the big bang to intelligent life. The Physical Age describes the origin of the universe, of matter, of cosmic nucleosynthesis, as well as the formation of the first stars and galaxies. The Chemical Age begun when heavy stars provided the raw ingredients for life through stellar nucleosynthesis and describes how heavier chemical elements collected in nascent planets and moons to give rise to prebiotic biomolecules. The Biological Age describes the origin of early life, its evolution through Darwinian natural selection, and the emergence of complex multicellular life forms. Finally, the Cognitive Age describes how complex life evolved into intelligent life capable of self-awareness and of developing technology through the directed manipulation of energy and materials. We conclude discussing whether we are the rule or the exception.Comment: 7 pages, Opening plenary talk delivered at the S\~ao Paulo Advanced School of Astrobiology, S\~ao Paulo, December 2011. In press, Int. J. Astrobio. Reference update

Controls on development and diversity of Early Archean stromatolites

The ≈3,450-million-year-old Strelley Pool Formation in Western Australia contains a reef-like assembly of laminated sedimentary accretion structures (stromatolites) that have macroscale characteristics suggestive of biological influence. However, direct microscale evidence of biology—namely, organic microbial remains or biosedimentary fabrics—has to date eluded discovery in the extensively-recrystallized rocks. Recently-identified outcrops with relatively good textural preservation record microscale evidence of primary sedimentary processes, including some that indicate probable microbial mat formation. Furthermore, we find relict fabrics and organic layers that covary with stromatolite morphology, linking morphologic diversity to changes in sedimentation, seafloor mineral precipitation, and inferred microbial mat development. Thus, the most direct and compelling signatures of life in the Strelley Pool Formation are those observed at the microscopic scale. By examining spatiotemporal changes in microscale characteristics it is possible not only to recognize the presence of probable microbial mats during stromatolite development, but also to infer aspects of the biological inputs to stromatolite morphogenesis. The persistence of an inferred biological signal through changing environmental circumstances and stromatolite types indicates that benthic microbial populations adapted to shifting environmental conditions in early oceans

A simple proof of the Markoff conjecture for prime powers

We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a,b,c), where a, b, and c are in increasing order, holds whenever $c$ is a prime power.Comment: 5 pages, no figure

Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard

This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace formula is compared with Gutzwiller's semiclassical periodic-orbit theory in three dimensions, and applied to a tetrahedral billiard being strongly chaotic. Geometric properties as well as the conjugacy classes of the defining group are discussed. The length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase N(E), which in turn is successfully applied in a quantization condition.Comment: 32 pages, compressed with gzip / uuencod

A new invariant on hyperbolic Dehn surgery space

In this paper we define a new invariant of the incomplete hyperbolic structures on a 1-cusped finite volume hyperbolic 3-manifold M, called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic structures and is a complete invariant of the Dehn fillings of M which admit a hyperbolic structure. We also give an explicit formula for the ortholength invariant in terms of the traces of the holonomies of certain loops in M. Conjecturally this new invariant is intimately related to the boundary of the hyperbolic Dehn surgery space of M.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-23.abs.htm

Structural results on convexity relative to cost functions

Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen's inequality for cost convex functions.Comment: 10 page