8,601 research outputs found
Commensurable continued fractions
We compare two families of continued fractions algorithms, the symmetrized
Rosen algorithm and the Veech algorithm. Each of these algorithms expands real
numbers in terms of certain algebraic integers. We give explicit models of the
natural extension of the maps associated with these algorithms; prove that
these natural extensions are in fact conjugate to the first return map of the
geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost
every real number has an infinite number of common approximants for both
algorithms.Comment: 41 pages, 10 figure
Continuants and some decompositions into squares
In 1855 H. J. S. Smith proved Fermat's two-square using the notion of
palindromic continuants. In his paper, Smith constructed a proper
representation of a prime number as a sum of two squares, given a solution
of , and vice versa. In this paper, we extend the use of
continuants to proper representations by sums of two squares in rings of
polynomials on fields of characteristic different from 2. New deterministic
algorithms for finding the corresponding proper representations are presented.
Our approach will provide a new constructive proof of the four-square theorem
and new proofs for other representations of integers by quaternary quadratic
forms.Comment: 21 page
On the Littlewood conjecture in fields of power series
Let \k be an arbitrary field. For any fixed badly approximable power series
in \k((X^{-1})), we give an explicit construction of continuum many
badly approximable power series for which the pair
satisfies the Littlewood conjecture. We further discuss the Littlewood
conjecture for pairs of algebraic power series
Dynamic percolation theory for particle diffusion in a polymer network
Tracer-diffusion of small molecules through dense systems of chain polymers
is studied within an athermal lattice model, where hard core interactions are
taken into account by means of the site exclusion principle. An approximate
mapping of this problem onto dynamic percolation theory is proposed. This
method is shown to yield quantitative results for the tracer correlation factor
of the molecules as a function of density and chain length provided the
non-Poisson character of temporal renewals in the disorder configurations is
properly taken into account
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