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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 is a
prime power.Comment: 5 pages, no figure
Analytic Continuation for Asymptotically AdS 3D Gravity
We have previously proposed that asymptotically AdS 3D wormholes and black
holes can be analytically continued to the Euclidean signature. The analytic
continuation procedure was described for non-rotating spacetimes, for which a
plane t=0 of time symmetry exists. The resulting Euclidean manifolds turned out
to be handlebodies whose boundary is the Schottky double of the geometry of the
t=0 plane. In the present paper we generalize this analytic continuation map to
the case of rotating wormholes. The Euclidean manifolds we obtain are quotients
of the hyperbolic space by a certain quasi-Fuchsian group. The group is the
Fenchel-Nielsen deformation of the group of the non-rotating spacetime. The
angular velocity of an asymptotic region is shown to be related to the
Fenchel-Nielsen twist. This solves the problem of classification of rotating
black holes and wormholes in 2+1 dimensions: the spacetimes are parametrized by
the moduli of the boundary of the corresponding Euclidean spaces. We also
comment on the thermodynamics of the wormhole spacetimes.Comment: 28 pages, 14 figure
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
Duality properties of indicatrices of knots
The bridge index and superbridge index of a knot are important invariants in
knot theory. We define the bridge map of a knot conformation, which is closely
related to these two invariants, and interpret it in terms of the tangent
indicatrix of the knot conformation. Using the concepts of dual and derivative
curves of spherical curves as introduced by Arnold, we show that the graph of
the bridge map is the union of the binormal indicatrix, its antipodal curve,
and some number of great circles. Similarly, we define the inflection map of a
knot conformation, interpret it in terms of the binormal indicatrix, and
express its graph in terms of the tangent indicatrix. This duality relationship
is also studied for another dual pair of curves, the normal and Darboux
indicatrices of a knot conformation. The analogous concepts are defined and
results are derived for stick knots.Comment: 22 pages, 9 figure
Notes about the Caratheodory number
In this paper we give sufficient conditions for a compactum in
to have Carath\'{e}odory number less than , generalizing an old result of
Fenchel. Then we prove the corresponding versions of the colorful
Carath\'{e}odory theorem and give a Tverberg type theorem for families of
convex compacta
Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2
We give a presentation by generators and relations of a certain monoid
generating a subgroup of index two in the group Aut(F_2) of automorphisms of
the rank two free group F_2 and show that it can be realized as a monoid in the
group B_4 of braids on four strings. In the second part we use Christoffel
words to construct an explicit basis of F_2 lifting any given basis of the free
abelian group Z^2. We further give an algorithm allowing to decide whether two
elements of F_2 form a basis or not. We also show that, under suitable
conditions, a basis has a unique conjugate consisting of two palindromes.Comment: 25 pages, 4 figure
On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds
Asymptotic laws for mean multiplicities of lengths of closed geodesics in
arithmetic hyperbolic three-orbifolds are derived. The sharpest results are
obtained for non-compact orbifolds associated with the Bianchi groups SL(2,o)
and some congruence subgroups. Similar results hold for cocompact arithmetic
quaternion groups, if a conjecture on the number of gaps in their length
spectra is true. The results related to the groups above give asymptotic lower
bounds for the mean multiplicities in length spectra of arbitrary arithmetic
hyperbolic three-orbifolds. The investigation of these multiplicities is
motivated by their sensitive effect on the eigenvalue spectrum of the
Laplace-Beltrami operator on a hyperbolic orbifold, which may be interpreted as
the Hamiltonian of a three-dimensional quantum system being strongly chaotic in
the classical limit.Comment: 29 pages, uuencoded ps. Revised version, to appear in NONLINEARIT
The compositional and evolutionary logic of metabolism
Metabolism displays striking and robust regularities in the forms of
modularity and hierarchy, whose composition may be compactly described. This
renders metabolic architecture comprehensible as a system, and suggests the
order in which layers of that system emerged. Metabolism also serves as the
foundation in other hierarchies, at least up to cellular integration including
bioenergetics and molecular replication, and trophic ecology. The
recapitulation of patterns first seen in metabolism, in these higher levels,
suggests metabolism as a source of causation or constraint on many forms of
organization in the biosphere.
We identify as modules widely reused subsets of chemicals, reactions, or
functions, each with a conserved internal structure. At the small molecule
substrate level, module boundaries are generally associated with the most
complex reaction mechanisms and the most conserved enzymes. Cofactors form a
structurally and functionally distinctive control layer over the small-molecule
substrate. Complex cofactors are often used at module boundaries of the
substrate level, while simpler ones participate in widely used reactions.
Cofactor functions thus act as "keys" that incorporate classes of organic
reactions within biochemistry.
The same modules that organize the compositional diversity of metabolism are
argued to have governed long-term evolution. Early evolution of core
metabolism, especially carbon-fixation, appears to have required few
innovations among a small number of conserved modules, to produce adaptations
to simple biogeochemical changes of environment. We demonstrate these features
of metabolism at several levels of hierarchy, beginning with the small-molecule
substrate and network architecture, continuing with cofactors and key conserved
reactions, and culminating in the aggregation of multiple diverse physical and
biochemical processes in cells.Comment: 56 pages, 28 figure
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