Entwining Yang-Baxter maps over Grassmann algebras

We construct novel solutions to the set-theoretical entwining Yang-Baxter equation. These solutions are birational maps involving non-commutative dynamical variables which are elements of the Grassmann algebra of order $n$. The maps arise from refactorisation problems of Lax supermatrices associated to a nonlinear Schr\"odinger equation. In this non-commutative setting, we construct a spectral curve associated to each of the obtained maps using the characteristic function of its monodromy supermatrix. We find generating functions of invariants (first integrals) for the entwining Yang-Baxter maps from the moduli of the spectral curves. Moreover, we show that a hierarchy of birational entwining Yang-Baxter maps with commutative variables can be obtained by fixing the order $n$ of the Grassmann algebra. We present the members of the hierarchy in the case $n=1$ (dual numbers) and $n=2$, and discuss their dynamical and integrability properties, such as Lax matrices, invariants, and measure preservation

Darboux–Bäcklund transformations, dressing & impurities in multi-component NLS

We consider the discrete and continuous vector non-linear Schrödinger (NLS) model. We focus on the case where space-like local discontinuities are present, and we are primarily interested in the time evolution on the defect point. This in turn yields the time part of a typical Darboux–Bäcklund transformation. Within this spirit we then explicitly work out the generic Bäcklund transformation and the dressing associated to both discrete and continuous spectrum, i.e. the Darboux transformation is expressed in the matrix and integral representation respectively

High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schroedinger equation

While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we present several high order symplectic integrators for Hamiltonian systems that can be split in exactly three integrable parts. We apply these techniques, as a practical case, for the integration of the disordered, discrete nonlinear Schroedinger equation (DDNLS) and compare their efficiencies. Three part split algorithms provide effective means to numerically study the asymptotic behavior of wave packet spreading in the DDNLS - a hotly debated subject in current scientific literature.Comment: 5 Figures, Physics Letters A (accepted

Darboux transformation with dihedral reduction group

We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix, we obtain local conservation laws of the system

Entwining Yang–Baxter maps related to NLS type equations

We construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense

Fibrinogen beta variants confer protection against coronary artery disease in a Greek case-control study

<p>Abstract</p> <p>Background</p> <p>Although plasma fibrinogen levels are related to cardiovascular risk, data regarding the role of fibrinogen genetic variation in myocardial infarction (MI) or coronary artery disease (CAD) etiology remain inconsistent. The purpose of the present study was to investigate the effect of <it>fibrinogen A (FGA)</it>, <it>fibrinogen B (FGB) </it>and <it>fibrinogen G (FGG) </it>gene SNPs and haplotypes on susceptibility to CAD in a homogeneous Greek population.</p> <p>Methods</p> <p>We genotyped for rs2070022, rs2070016, rs2070006 in <it>FGA </it>gene, the rs7673587, rs1800789, rs1800790, rs1800788, rs1800787, rs4681 and rs4220 in <it>FGB </it>gene and for the rs1118823, rs1800792 and rs2066865 SNPs in <it>FGG </it>gene applying an arrayed primer extension-based genotyping method (APEX-2) in a sample of CAD patients (n = 305) and controls (n = 305). Logistic regression analysis was used to calculate odds ratios (ORs) and 95% confidence intervals (CIs), before and after adjustment for potential confounders.</p> <p>Results</p> <p>None of the <it>FGA </it>and <it>FGG </it>SNPs and <it>FGA, FGB, FGG </it>and <it>FGA-FGG </it>haplotypes was associated with disease occurrence after adjustment. Nevertheless, rs1800787 and rs1800789 SNPs in <it>FGB </it>gene seem to decrease the risk of CAD, even after adjustment for potential confounders (OR = 0.42, 95%CI: 0.19-0.90, p = 0.026 and OR = 0.44, 95%CI:0.21-0.94, p = 0.039, respectively).</p> <p>Conclusions</p> <p><it>FGA </it>and <it>FGG </it>SNPs as well as <it>FGA, FGB, FGG </it>and <it>FGA-FGG </it>haplotypes do not seem to be important contributors to CAD occurrence in our sample. On the contrary, <it>FGB </it>rs1800787 and rs1800789 SNPs seem to confer protection to disease onset lowering the risk by about 50% in homozygotes for the minor alleles.</p