740 research outputs found
Tower-type bounds for unavoidable patterns in words
A word is said to contain the pattern if there is a way to substitute
a nonempty word for each letter in so that the resulting word is a subword
of . Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised
the patterns which are unavoidable, in the sense that any sufficiently long
word over a fixed alphabet contains . Zimin's characterisation says that a
pattern is unavoidable if and only if it is contained in a Zimin word, where
the Zimin words are defined by and . We
study the quantitative aspects of this theorem, obtaining essentially tight
tower-type bounds for the function , the least integer such that any
word of length over an alphabet of size contains . When , the first non-trivial case, we determine up to a constant factor,
showing that .Comment: 17 page
Chemoreceptor Evolution in Hymenoptera and Its Implications for the Evolution of Eusociality.
Eusocial insects, mostly Hymenoptera, have evolved unique colonial lifestyles that rely on the perception of social context mainly through pheromones, and chemoreceptors are hypothesized to have played important adaptive roles in the evolution of sociality. However, because chemoreceptor repertoires have been characterized in few social insects and their solitary relatives, a comprehensive examination of this hypothesis has not been possible. Here, we annotate ∼3,000 odorant and gustatory receptors in recently sequenced Hymenoptera genomes and systematically compare >4,000 chemoreceptors from 13 hymenopterans, representing one solitary lineage (wasps) and three independently evolved eusocial lineages (ants and two bees). We observe a strong general tendency for chemoreceptors to expand in Hymenoptera, whereas the specifics of gene gains/losses are highly diverse between lineages. We also find more frequent positive selection on chemoreceptors in a facultative eusocial bee and in the common ancestor of ants compared with solitary wasps. Our results suggest that the frequent expansions of chemoreceptors have facilitated the transition to eusociality. Divergent expression patterns of odorant receptors between honeybee and ants further indicate differential roles of chemoreceptors in parallel trajectories of social evolution
On Long Words Avoiding Zimin Patterns
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered in the n-th Zimin pattern. Given an alphabet size k, we study the minimal length f(n,k) such that every word of length f(n,k) encounters the n-th Zimin pattern. It is known that f is upper-bounded by a tower of exponentials. Our main result states that f(n,k) is lower-bounded by a tower of n-3 exponentials, even for k=2. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense
Pattern avoidance: themes and variations
AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems
Controlling instabilities along a 3DVar analysis cycle by assimilating in the unstable subspace: a comparison with the EnKF
A hybrid scheme obtained by combining 3DVar with the Assimilation in the
Unstable Subspace (3DVar-AUS) is tested in a QG model, under perfect model
conditions, with a fixed observational network, with and without observational
noise. The AUS scheme, originally formulated to assimilate adaptive
observations, is used here to assimilate the fixed observations that are found
in the region of local maxima of BDAS vectors (Bred vectors subject to
assimilation), while the remaining observations are assimilated by 3DVar.
The performance of the hybrid scheme is compared with that of 3DVar and of an
EnKF. The improvement gained by 3DVar-AUS and the EnKF with respect to 3DVar
alone is similar in the present model and observational configuration, while
3DVar-AUS outperforms the EnKF during the forecast stage. The 3DVar-AUS
algorithm is easy to implement and the results obtained in the idealized
conditions of this study encourage further investigation toward an
implementation in more realistic contexts
- …