Some thoughts on a nonlinear Schrödinger equation motivated by information theory

The arguments leading to a nonlinear generalization of the Schrödinger equation in the context of the maximum uncertainty principle are reviewed. The exact and perturbative properties of that equation depend on a free regulating/interpolating parameter η, which can be fixed using energetics as is shown here. A linear theory with an external potential that reproduces some unusual exact solutions of the nonlinear equation is also discussed, together with possible symmetry enhancements in the nonlinear theory.     

Monotone periodic orbits for torus homeomorphisms

Let be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector . Then has a topologically monotone periodic orbit with the same rotation vector.

     

Zero entropy subgroups of mapping class groups

Given a group action on a surface with a finite invariant set we investigate how the algebraic properties of the induced group of permutations of that set affects the dynamical properties of the group. Our main result shows that in many circumstances if the induced permutation group is not solvable then among the homeomorphisms in the group there must be one with a pseudo-Anosov component. We formulate this in terms of the mapping class group relative to the finite set and show the stronger result that in many circumstances (e.g. if the surface has boundary) if this mapping class group has no elements with pseudo-Anosov components then it is itself solvable.     

Protective Effects of Swertiamarin against Methylglyoxal-Induced Epithelial-Mesenchymal Transition by Improving Oxidative Stress in Rat Kidney Epithelial (NRK-52E) Cells

Increased blood glucose in diabetic individuals results in the formation of advanced glycation end products (AGEs), causing various adverse effects on kidney cells, thereby leading to diabetic nephropathy (DN). In this study, the antiglycative potential of Swertiamarin (SM) isolated from the methanolic extract of E. littorale was explored. The effect of SM on protein glycation was studied by incubating bovine serum albumin with fructose at 60 °C in the presence and absence of different concentrations of swertiamarin for 24 h. For comparative analysis, metformin was also used at similar concentrations as SM. Further, to understand the role of SM in preventing DN, in vitro studies using NRK-52E cells were done by treating cells with methylglyoxal (MG) in the presence and absence of SM. SM showed better antiglycative potential as compared to metformin. In addition, SM could prevent the MG mediated pathogenesis in DN by reducing levels of argpyrimidine, oxidative stress and epithelial mesenchymal transition in kidney cells. SM also downregulated the expression of interleukin-6, tumor necrosis factor-α and interleukin-1β. This study, for the first time, reports the antiglycative potential of SM and also provides novel insights into the molecular mechanisms by which SM prevents toxicity of MG on rat kidney cells.