6,450 research outputs found
Specialisation and reduction of continued fractions of formal power series
We discuss and illustrate the behaviour of the continued fraction expansion
of a formal power series under specialisation of parameters or their reduction
modulo and sketch some applications of the reduction theorem here proved.Comment: 7 page
Modular embeddings of Teichmueller curves
Fuchsian groups with a modular embedding have the richest arithmetic
properties among non-arithmetic Fuchsian groups. But they are very rare, all
known examples being related either to triangle groups or to Teichmueller
curves.
In Part I of this paper we study the arithmetic properties of the modular
embedding and develop from scratch a theory of twisted modular forms for
Fuchsian groups with a modular embedding, proving dimension formulas,
coefficient growth estimates and differential equations.
In Part II we provide a modular proof for an Apery-like integrality statement
for solutions of Picard-Fuchs equations. We illustrate the theory on a worked
example, giving explicit Fourier expansions of twisted modular forms and the
equation of a Teichmueller curve in a Hilbert modular surface.
In Part III we show that genus two Teichmueller curves are cut out in Hilbert
modular surfaces by a product of theta derivatives. We rederive most of the
known properties of those Teichmueller curves from this viewpoint, without
using the theory of flat surfaces. As a consequence we give the modular
embeddings for all genus two Teichmueller curves and prove that the Fourier
developments of their twisted modular forms are algebraic up to one
transcendental scaling constant. Moreover, we prove that Bainbridge's
compactification of Hilbert modular surfaces is toroidal. The strategy to
compactify can be expressed using continued fractions and resembles
Hirzebruch's in form, but every detail is different.Comment: revision including the referee's comments, to appear in Compositio
Mat
Towards an exact adaptive algorithm for the determinant of a rational matrix
In this paper we propose several strategies for the exact computation of the
determinant of a rational matrix. First, we use the Chinese Remaindering
Theorem and the rational reconstruction to recover the rational determinant
from its modular images. Then we show a preconditioning for the determinant
which allows us to skip the rational reconstruction process and reconstruct an
integer result. We compare those approaches with matrix preconditioning which
allow us to treat integer instead of rational matrices. This allows us to
introduce integer determinant algorithms to the rational determinant problem.
In particular, we discuss the applicability of the adaptive determinant
algorithm of [9] and compare it with the integer Chinese Remaindering scheme.
We present an analysis of the complexity of the strategies and evaluate their
experimental performance on numerous examples. This experience allows us to
develop an adaptive strategy which would choose the best solution at the run
time, depending on matrix properties. All strategies have been implemented in
LinBox linear algebra library
On the Cyclotomic Quantum Algebra of Time Perception
I develop the idea that time perception is the quantum counterpart to time
measurement. Phase-locking and prime number theory were proposed as the
unifying concepts for understanding the optimal synchronization of clocks and
their 1/f frequency noise. Time perception is shown to depend on the
thermodynamics of a quantum algebra of number and phase operators already
proposed for quantum computational tasks, and to evolve according to a
Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum
model for prime numbers. The picture that emerges is a unique perception state
above a critical temperature and plenty of them allowed below, which are
parametrized by the symmetry group for the primitive roots of unity. Squeezing
of phase fluctuations close to the phase transition temperature may play a role
in memory encoding and conscious activity
Euclidean algorithms are Gaussian
This study provides new results about the probabilistic behaviour of a class
of Euclidean algorithms: the asymptotic distribution of a whole class of
cost-parameters associated to these algorithms is normal. For the cost
corresponding to the number of steps Hensley already has proved a Local Limit
Theorem; we give a new proof, and extend his result to other euclidean
algorithms and to a large class of digit costs, obtaining a faster, optimal,
rate of convergence. The paper is based on the dynamical systems methodology,
and the main tool is the transfer operator. In particular, we use recent
results of Dolgopyat.Comment: fourth revised version - 2 figures - the strict convexity condition
used has been clarifie
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