10,954 research outputs found
On the covering dimension of the set of solutions of some nonlinear equations
We prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose parametric equations are studied using cohomological mconclusions imply sharp, multidimensional continuation results. Applications are given to nonlinear partial differential equations
A characterization of balanced episturmian sequences
It is well known that Sturmian sequences are the aperiodic sequences that are
balanced over a 2-letter alphabet. They are also characterized by their
complexity: they have exactly factors of length . One possible
generalization of Sturmian sequences is the set of infinite sequences over a
-letter alphabet, , which are closed under reversal and have at
most one right special factor for each length. This is the set of episturmian
sequences. These are not necessarily balanced over a -letter alphabet, nor
are they necessarily aperiodic. In this paper, we characterize balanced
episturmian sequences, periodic or not, and prove Fraenkel's conjecture for the
class of episturmian sequences. This conjecture was first introduced in number
theory and has remained unsolved for more than 30 years. It states that for a
fixed , there is only one way to cover by Beatty sequences. The
problem can be translated to combinatorics on words: for a -letter alphabet,
there exists only one balanced sequence up to letter permutation that has
different letter frequencies
The thematic interpretation of plural nominalizations
Nominalizations, in German as well as in other languages, are systematically polysemous, a fact that has been widely discussed in the linguistic literature [...]. In this paper, I will discuss certain asymmetries concerning the interpretation of the postnominal genitive [...]
A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture
The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r)
is a Beatty set with density p/q. We derive a formula for the Fourier transform
\hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 \pi i j \floor{nq/p+r} / q}. A. S.
Fraenkel conjectured that there is essentially one way to partition the
integers into m>2 Beatty sets with distinct densities. We conjecture a
generalization of this, and use Fourier methods to prove several special cases
of our generalized conjecture.Comment: 24 pages, 6 figures (now with minor revisions and clarifications
Measurable realizations of abstract systems of congruences
An abstract system of congruences describes a way of partitioning a space
into finitely many pieces satisfying certain congruence relations. Examples of
abstract systems of congruences include paradoxical decompositions and
-divisibility of actions. We consider the general question of when there are
realizations of abstract systems of congruences satisfying various
measurability constraints. We completely characterize which abstract systems of
congruences can be realized by nonmeager Baire measurable pieces of the sphere
under the action of rotations on the -sphere. This answers a question of
Wagon. We also construct Borel realizations of abstract systems of congruences
for the action of on .
The combinatorial underpinnings of our proof are certain types of decomposition
of Borel graphs into paths. We also use these decompositions to obtain some
results about measurable unfriendly colorings.Comment: minor correction
Uniformly balanced words with linear complexity and prescribed letter frequencies
We consider the following problem. Let us fix a finite alphabet A; for any
given d-uple of letter frequencies, how to construct an infinite word u over
the alphabet A satisfying the following conditions: u has linear complexity
function, u is uniformly balanced, the letter frequencies in u are given by the
given d-uple. This paper investigates a construction method for such words
based on the use of mixed multidimensional continued fraction algorithms.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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