Equations on partial words

It is well-known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation xMyN = zP has only periodic solutions in a free monoid, that is, if xMyN = zP holds with integers m,n,p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot of attention, was first proved by Lyndon and Schützenberger for free groups. In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of “do not know” symbols. When we speak about equations on partial words, we replace the notion of equality (=) with compatibility (↑). Among other equations, we solve xy ↑ yx, xz ↑ zy, and special cases of xmyn ↑ zp for integers m,n,p ≥ 2.

Equations on partial words

It is well known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation x m y n = z p has only periodic solutions in a free monoid, that is, if x m y n = z p holds with integers m, n, p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot of attention, was first proved by Lyndon and Schützenberger for free groups. In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of “do not know ” symbols. When we speak about equations on partial words, we replace the notion of equality (=) with compatibility (↑). Among other equations, we solve xy ↑ yx, xz ↑ zy, and special cases of x m y n ↑ z p for integers m, n, p ≥ 2

Periodicity on Partial Words

Abstract. It is well known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation x m y n = z p has only periodic solutions in a free monoid, that is, if x m y n = z p holds with integers m, n, p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot of attention, was first proved by Lyndon and Schützenberger for free groups. In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of “do not know ” symbols. When we speak about equations on partial words, we replace the notion of equality (=) with compatibility (↑). Among other equations, we solve xy ↑ yx, xz ↑ zy, and special cases of x m y n ↑ z p for integers m, n, p ≥ 2....

A draft Diabrotica virgifera virgifera genome: insights into control and host plant adaption by a major maize pest insect

Abstract Background Adaptations by arthropod pests to host plant defenses of crops determine their impacts on agricultural production. The larval host range of western corn rootworm, Diabrotica virgifera virgifera (Coleoptera: Chrysomelidae), is restricted to maize and a few grasses. Resistance of D. v. virgifera to crop rotation practices and multiple insecticides contributes to its status as the most damaging pest of cultivated maize in North America and Europe. The extent to which adaptations by this pest contributes to host plant specialization remains unknown. Results A 2.42 Gb draft D. v. virgifera genome, Dvir_v2.0, was assembled from short shotgun reads and scaffolded using long-insert mate-pair, transcriptome and linked read data. K-mer analysis predicted a repeat content of ≥ 61.5%. Ortholog assignments for Dvir_2.0 RefSeq models predict a greater number of species-specific gene duplications, including expansions in ATP binding cassette transporter and chemosensory gene families, than in other Coleoptera. A majority of annotated D. v. virgifera cytochrome P450s belong to CYP4, 6, and 9 clades. A total of 5,404 transcripts were differentially-expressed between D. v. virgifera larvae fed maize roots compared to alternative host (Miscanthus), a marginal host (Panicum virgatum), a poor host (Sorghum bicolor) and starvation treatments; Among differentially-expressed transcripts, 1,908 were shared across treatments and the least number were between Miscanthus compared to maize. Differentially-expressed transcripts were enriched for putative spliceosome, proteosome, and intracellular transport functions. General stress pathway functions were unique and enriched among up-regulated transcripts in marginal host, poor host, and starvation responses compared to responses on primary (maize) and alternate hosts. Conclusions Manual annotation of D. v. virgifera Dvir_2.0 RefSeq models predicted expansion of paralogs with gene families putatively involved in insecticide resistance and chemosensory perception. Our study also suggests that adaptations of D. v. virgifera larvae to feeding on an alternate host plant invoke fewer transcriptional changes compared to marginal or poor hosts. The shared up-regulation of stress response pathways between marginal host and poor host, and starvation treatments may reflect nutrient deprivation. This study provides insight into transcriptomic responses of larval feeding on different host plants and resources for genomic research on this economically significant pest of maize

Open data from the first and second observing runs of Advanced LIGO and Advanced Virgo

Advanced LIGO and Advanced Virgo are monitoring the sky and collecting gravitational-wave strain data with sufficient sensitivity to detect signals routinely. In this paper we describe the data recorded by these instruments during their first and second observing runs. The main data products are gravitational-wave strain time series sampled at 16384 Hz. The datasets that include this strain measurement can be freely accessed through the Gravitational Wave Open Science Center at http://gw-openscience.org, together with data-quality information essential for the analysis of LIGO and Virgo data, documentation, tutorials, and supporting software