44,356 research outputs found
On Asymptotic Reducibility in SL(3,Z)
Recently we showed that Hessenberg matrices are proper to represent conjugacy
classes in SL(n,Z). In this paper we focus on the reducibility properties in
the set of Hessenberg matrices of SL(3,Z). We investigate the first interesting
open case here: the case of matrices having one real and two complex conjugate
eigenvalues.Comment: 24 pages, 7 figure
Enumerating Palindromes and Primitives in Rank Two Free Groups
Let be a rank two free group. A word in is {\sl
primitive} if it, along with another group element, generates the group. It is
a {\sl palindrome} (with respect to and ) if it reads the same forwards
and backwards. It is known that in a rank two free group any primitive element
is conjugate either to a palindrome or to the product of two palindromes, but
known iteration schemes for all primitive words give only a representative for
the conjugacy class. Here we derive a new iteration scheme that gives either
the unique palindrome in the conjugacy class or expresses the word as a unique
product of two unique palindromes. We denote these words by where
is rational number expressed in lowest terms. We prove that is
a palindrome if is even and the unique product of two unique palindromes
if is odd. We prove that the pairs generate the group
when . This improves the previously known result that held only for
and both even. The derivation of the enumeration scheme also gives a
new proof of the known results about primitives.Comment: Final revisions, to appear J Algebr
Old and new results on normality
We present a partial survey on normal numbers, including Keane's
contributions, and with recent developments in different directions.Comment: Published at http://dx.doi.org/10.1214/074921706000000248 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Continued fractions and transcendental numbers
It is widely believed that the continued fraction expansion of every
irrational algebraic number either is eventually periodic (and we know
that this is the case if and only if is a quadratic irrational), or it
contains arbitrarily large partial quotients. Apparently, this question was
first considered by Khintchine. A preliminary step towards its resolution
consists in providing explicit examples of transcendental continued fractions.
The main purpose of the present work is to present new families of
transcendental continued fractions with bounded partial quotients. Our results
are derived thanks to new combinatorial transcendence criteria recently
obtained by Adamczewski and Bugeaud
On the Maillet--Baker continued fractions
We use the Schmidt Subspace Theorem to establish the transcendence of a class
of quasi-periodic continued fractions. This improves earlier works of Maillet
and of A. Baker. We also improve an old result of Davenport and Roth on the
rate of increase of the denominators of the convergents to any real algebraic
number
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