20,890 research outputs found
Outer Billiards, Arithmetic Graphs, and the Octagon
Outer Billiards is a geometrically inspired dynamical system based on a
convex shape in the plane.
When the shape is a polygon, the system has a combinatorial flavor. In the
polygonal case, there is a natural acceleration of the map, a first return map
to a certain strip in the plane. The arithmetic graph is a geometric encoding
of the symbolic dynamics of this first return map.
In the case of the regular octagon, the case we study, the arithmetic graphs
associated to periodic orbits are polygonal paths in R^8. We are interested in
the asymptotic shapes of these polygonal paths, as the period tends to
infinity. We show that the rescaled limit of essentially any sequence of these
graphs converges to a fractal curve that simultaneously projects one way onto a
variant of the Koch snowflake and another way onto a variant of the Sierpinski
carpet. In a sense, this gives a complete description of the asymptotic
behavior of the symbolic dynamics of the first return map.
What makes all our proofs work is an efficient (and basically well known)
renormalization scheme for the dynamics.Comment: 86 pages, mildly computer-aided proof. My java program
http://www.math.brown.edu/~res/Java/OctoMap2/Main.html illustrates
essentially all the ideas in the paper in an interactive and well-documented
way. This is the second version. The only difference from the first version
is that I simplified the proof of Main Theorem, Statement 2, at the end of
Ch.
2-frieze patterns and the cluster structure of the space of polygons
We study the space of 2-frieze patterns generalizing that of the classical
Coxeter-Conway frieze patterns. The geometric realization of this space is the
space of n-gons (in the projective plane and in 3-dimensional vector space)
which is a close relative of the moduli space of genus 0 curves with n marked
points. We show that the space of 2-frieze patterns is a cluster manifold and
study its algebraic and arithmetic properties.Comment: 36 pages, 4 figures, minor changes from the previous versions;
references added (v2), section 5.5 added (v3), picture added (v4
Sparse approaches for the exact distribution of patterns in long state sequences generated by a Markov source
We present two novel approaches for the computation of the exact distribution
of a pattern in a long sequence. Both approaches take into account the sparse
structure of the problem and are two-part algorithms. The first approach relies
on a partial recursion after a fast computation of the second largest
eigenvalue of the transition matrix of a Markov chain embedding. The second
approach uses fast Taylor expansions of an exact bivariate rational
reconstruction of the distribution. We illustrate the interest of both
approaches on a simple toy-example and two biological applications: the
transcription factors of the Human Chromosome 5 and the PROSITE signatures of
functional motifs in proteins. On these example our methods demonstrate their
complementarity and their hability to extend the domain of feasibility for
exact computations in pattern problems to a new level
Microbial communities and arsenic biogeochemistry at the outflow of an alkaline sulfide-rich hot spring.
Alkaline sulfide-rich hot springs provide a unique environment for microbial community and arsenic (As) biogeochemistry. In this study, a representative alkaline sulfide-rich hot spring, Zimeiquan in the Tengchong geothermal area, was chosen to study arsenic geochemistry and microbial community using Illumina MiSeq sequencing. Over 0.26 million 16S rRNA sequence reads were obtained from 5-paired parallel water and sediment samples along the hot spring's outflow channel. High ratios of As(V)/AsSum (total combined arsenate and arsenite concentrations) (0.59-0.78), coupled with high sulfide (up to 5.87 mg/L), were present in the hot spring's pools, which suggested As(III) oxidation occurred. Along the outflow channel, AsSum increased from 5.45 to 13.86 μmol/L, and the combined sulfide and sulfate concentrations increased from 292.02 to 364.28 μmol/L. These increases were primarily attributed to thioarsenic transformation. Temperature, sulfide, As and dissolved oxygen significantly shaped the microbial communities between not only the pools and downstream samples, but also water and sediment samples. Results implied that the upstream Thermocrinis was responsible for the transformation of thioarsenic to As(III) and the downstream Thermus contributed to derived As(III) oxidation. This study improves our understanding of microbially-mediated As transformation in alkaline sulfide-rich hot springs
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
Let be -bounded multiplicative
functions, and let be shifts. We consider
correlation sequences of the form where are numbers going to infinity as , and
is a generalised limit functional extending the usual limit
functional. We show a structural theorem for these sequences, namely that these
sequences are the uniform limit of periodic sequences . Furthermore,
if the multiplicative function "weakly pretends" to be a
Dirichlet character , the periodic functions can be chosen to be
-isotypic in the sense that whenever is
coprime to the periods of and , while if does not
weakly pretend to be any Dirichlet character, then must vanish identically.
As a consequence, we obtain several new cases of the logarithmically averaged
Elliott conjecture, including the logarithmically averaged Chowla conjecture
for odd order correlations. We give a number of applications of these special
cases, including the conjectured logarithmic density of all sign patterns of
the Liouville function of length up to three, and of the M\"obius function of
length up to four.Comment: 41 pages, no figures. Submitted, Duke Math. J.. Referee changes
incorporate
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