392 research outputs found
Spectral Approximation for Quasiperiodic Jacobi Operators
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals
and in more general studies of structures exhibiting aperiodic order. The
spectra of these self-adjoint operators can be quite exotic, such as Cantor
sets, and their fine properties yield insight into associated dynamical
systems. Quasiperiodic operators can be approximated by periodic ones, the
spectra of which can be computed via two finite dimensional eigenvalue
problems. Since long periods are necessary to get detailed approximations, both
computational efficiency and numerical accuracy become a concern. We describe a
simple method for numerically computing the spectrum of a period- Jacobi
operator in operations, and use it to investigate the spectra of
Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse
potentials
Morphically primitive words
In the present paper, we introduce an alternative notion of the primitivity of words, that–unlike the standard understanding of this term–is not based on the power (and, hence, the concatenation) of words, but on morphisms. For any alphabet Σ, we call a word wΣ* morphically imprimitive provided that there are a shorter word v and morphisms h,h′:Σ*→Σ* satisfying h(v)=w and h′(w)=v, and we say that w is morphically primitive otherwise. We explain why this is a well-chosen terminology, we demonstrate that morphic (im-) primitivity of words is a vital attribute in many combinatorial domains based on finite words and morphisms, and we study a number of fundamental properties of the concepts under consideration
Structure theorem for U5-free tournaments
Let be the tournament with vertices , ..., such that , and if , and
. In this paper we describe the tournaments which do not have
as a subtournament. Specifically, we show that if a tournament is
"prime"---that is, if there is no subset , , such that for all , either
for all or for all ---then is
-free if and only if either is a specific tournament or
can be partitioned into sets , , such that , ,
and are transitive. From the prime -free tournaments we can
construct all the -free tournaments. We use the theorem to show that every
-free tournament with vertices has a transitive subtournament with at
least vertices, and that this bound is tight.Comment: 15 pages, 1 figure. Changes from previous version: Added a section;
added the definitions of v, A, and B to the main proof; general edit
Finding Pseudo-repetitions
Pseudo-repetitions are a natural generalization of the classical notion of repetitions in sequences. We solve fundamental algorithmic questions on pseudo-repetitions by application of insightful combinatorial results on words. More precisely, we efficiently decide whether a word is a pseudo-repetition and find all the pseudo-repetitive factors of a word
Promised streaming algorithms and finding pseudo-repetitions
As the size of data available for processing increases, new models of
computation are needed. This motivates the study of data streams, which are sequences of
information for which each element can be read only after the previous one. In
this work we study two particular types of streaming variants: promised graph streaming algorithms and
combinatorial queries on large words. We give an &omega(n) lower
bound for working memory, where n is the number of vertices of the graph, for a variety
of problems for which the graphs are promised to be forests. The crux
of the proofs is based on reductions from the field of communication complexity.
Finally, we give an upper bound for two problems related to finding
pseudo-repetitions on words via anti-/morphisms, for which we also propose streaming versions
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