Darboux dressing and undressing for the ultradiscrete KdV equation

We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We then show how to reconstruct the potential in the scattering problem at any time, using an ultradiscrete analogue of a Darboux transformation. This is achieved by obtaining data uniquely characterising the soliton content and the `background' from the initial potential by Darboux transformation.Comment: 41 pages, 5 figures // Full, unabridged version, including two appendice

Yang-Baxter maps and the discrete KP hierarchy

We present a systematic construction of the discrete KP hierarchy in terms of Sato–Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang–Baxter maps is explained in two explicit examples

Bäcklund transformations for noncommutative anti-self-dual Yang-Mills equations

We present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equations where the gauge group is G = GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants and belong to a non-commutative version of the Atiyah–Ward ansatz. In the commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard

Quasideterminant solutions of a non-Abelian Hirota-Miwa equation

A non-Abelian version of the Hirota-Miwa equation is considered. In an earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. \textbf{39}, 5053-5065] it was shown how solutions expressed as quasideterminants could be constructed for this system by means of Darboux transformations. In this paper we discuss these solutions from a different perspective and show that the solutions are quasi-Pl\"{u}cker coordinates and that the non-Abelian Hirota-Miwa equation may be written as a quasi-Pl\"{u}cker relation. The special case of the matrix Hirota-Miwa equation is also considered using a more traditional, bilinear approach and the techniques are compared

Allelic variations of the multidrug resistance gene determine susceptibility and disease behavior in ulcerative colitis

BACKGROUND AND AIMS: The MDR1 gene encodes P-glycoprotein 170, an efflux transporter that is highly expressed in intestinal epithelial cells. The MDR1 exonic single nucleotide polymorphisms (SNPs) C3435T and G2677T have been shown to correlate with activity/expression of P-glycoprotein 170.METHODS: This was a case-control analysis of MDR1 C3435T and G2677T SNPs in a large well-characterized Scottish white cohort (335 with ulcerative colitis [UC], 268 with Crohn's disease [CD], and 370 healthy controls). We conducted 2-locus haplotype and detailed univariate and multivariate genotypic-phenotypic analyses.RESULTS: The MDR1 3435 TT genotype (34.6% vs 26.5%; P = .04; odds ratio [OR], 1.60; 95% confidence interval [95% CI], 1.04-2.44) and T-allelic frequencies (58.2% vs 52.8%; P = .02; OR, 1.28; 95% CI, 1.03-1.58) were significantly higher in patients with UC compared with controls. No association was seen with CD. The association was strongest with extensive UC (TT genotype: 42.4% vs 26.5%; P = .003; OR, 2.64; 95% CI, 1.34-4.99; and T allele: 63.9% vs 52.8%; P = .009; OR, 1.70; 95% CI, 1.24-2.29), and this was also confirmed on multivariate analysis ( P = .007). The G2677T SNP was not associated with UC or CD. These 2 SNPs lie in linkage disequilibrium in our population (D', .8-.9; r 2 , .7-.8). Two-locus haplotypes showed both positive (3435T/G2677 haplotype: P = .03; OR, 1.44) and negative (C3435/2677T haplotype: P = .002; OR, .35) associations with UC. Homozygotes for the haplotype 3435T/G2677 were significantly increased in UC ( P = .017; OR, 8.88; 95% CI, 1.10-71.45).CONCLUSIONS: Allelic variations of the MDR1 gene determine disease extent as well as susceptibility to UC in the Scottish population. The present data strongly implicate the C3435T SNP, although the 2-locus haplotype data underline the need for further detailed haplotypic studies.</p

The dispersal of phytoplankton populations by enhanced turbulent mixing in a shallow coastal sea

A single tidal cycle survey in a Lagrangian reference frame was conducted in autumn 2010 to evaluate the impact of short-term, episodic and enhanced turbulent mixing on large chain-forming phytoplankton. Observations of turbulence using a free-falling microstructure profiler were undertaken, along with near-simultaneous profiles with an in-line digital holographic camera at station L4 (50° 15' N 4° 13' W, depth 50 m) in theWestern English Channel. Profiles fromeach instrumentwere collected hourly whilst following a drogued drifter. Results from an ADCP attached to the drifter showed pronounced vertical shear, indicating that the water column structure consisted of two layers, restricting interpretation of the Lagrangian experiment to the upper ̃25 m. Atmospheric conditions deteriorated during the mid-point of the survey, resulting in values of turbulent dissipation reaching a maximum of 10-4Wkg-1 toward the surface in the upper 10 m. Chain-forming phytoplankton >200 μm were counted using the data from the holographic camera for the two periods, before and after the enhanced mixing event. As mixing increased phytoplankton underwent chain breakage,were dispersed by advection through their removal from the upper to lower layer and subjected to aggregation with other suspended material. Depth averaged counts of phytoplankton were reduced from a maximum of around 2050 L-1 before the increased turbulence, to 1070 L-1 after, with each of these mechanisms contributing to this reduction. These results demonstrate the sensitivity of phytoplantkon populations to moderate increases in turbulent activity, yielding consequences for accurate forecasting of the role played by phytoplankton in climate studies and also for the ecosystem in general in their role as primary producers. © 2014 Elsevier B.V. All rights reserved

Resolving the decades-long transient FIRST J141918.9+394036: an orphan long gamma-ray burst or a young magnetar nebula?

Ofek (2017) identified FIRST J141918.9+394036 (hereafter FIRST J1419+3940) as a radio source sharing similar properties and host galaxy type to the compact, persistent radio source associated with the first known repeating fast radio burst, FRB 121102. Law et al. (2018) showed that FIRST J1419+3940 is a transient source decaying in brightness over the last few decades. One possible interpretation is that FIRST J1419+3940 is a nearby analogue to FRB 121102 and that the radio emission represents a young magnetar nebula (as several scenarios assume for FRB 121102). Another interpretation is that FIRST J1419+3940 is the afterglow of an `orphan' long gamma-ray burst (GRB). The environment is similar to where most such events are produced. To distinguish between these hypotheses, we conducted radio observations using the European VLBI Network at 1.6 GHz to spatially resolve the emission and to search for millisecond-duration radio bursts. We detect FIRST J1419+3940 as a compact radio source with a flux density of $620 \pm 20\ \mathrm{\mu Jy}$ (on 2018 September 18) and a source size of $3.9 \pm 0.7\ \mathrm{mas}$ (i.e. $1.6 \pm 0.3\ \mathrm{pc}$ given the angular diameter distance of $83\ \mathrm{Mpc}$). These results confirm that the radio emission is non-thermal and imply an average expansion velocity of $(0.10 \pm 0.02)c$. Contemporaneous high-time-resolution observations using the 100-m Effelsberg telescope detected no millisecond-duration bursts of astrophysical origin. The source properties and lack of short-duration bursts are consistent with a GRB jet expansion, whereas they disfavor a magnetar birth nebula.Comment: 8 pages, 4 figures, accepted for publication in ApJ

On a direct approach to quasideterminant solutions of a noncommutative KP equation

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the regular, commutative KP equation but, in the noncommutative case, no bilinearising transformation is available.Comment: 11 page