Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.     

Analogy between unconventional superconductivity and unconventional states of liquid crystals

An analogy is stressed between the order-parameter symmetries of the two-dimensional d-pairing wave superconductors and of liquid-crystal mesophases formed from achiral bent-shaped molecules. It leads to a definition of a class of liquid-crystal states which are the analogs of the unconventional superconducting states, and are characterized by a loss of discrete symmetry operations of the parent state.     

Wrapping Nanocellulose Nets around Graphene Oxide Sheets

Constructing advanced functional nanomaterials with pre‐designed organized morphologies from low‐dimension synthetic and biological components is challenging. Herein, we report an efficient and universal amphiphilicity‐driven assembly strategy to construct “hairy” flexible hybrid nanosheets with a 1D cellulose nanofibers (CNFs) net conformally wrapped around 2D graphene oxide (GO) monolayers. This interface‐driven bio‐synthetic assembly is facilitated by tailoring the surface chemistry of flexible GO sheets, resulting in individual sheets tightly surrounded by dense conformal nanocellulose network. The mechanical stability of the products far exceeds the compressive instability limits of both individual components. Additionally, the CNF network significantly enhances the wetting ability of initial hydrophobic reduced GO nanosheets, allowing fast water transport combined with high filtration efficiency.     

Effect of oligonucleotide length on the assembly of DNA materials: molecular dynamics simulations of layer-by-layer DNA films

DNA strand length has been found to be an important factor in many DNA-based nanoscale systems. Here, we apply molecular dynamics simulations in a synergistic effort with layer-by-layer experimental data to understand the effect of DNA strand length on the assembly of DNA films. The results indicate that short (less than 10 bases) and long (more than 30 bases) single-stranded DNAs do not exhibit optimal film growth, and this can be associated with the limited accessibility of the bases on the surface due to formation of self-protected interactions that prevent efficient hybridization. Interestingly, the presence of a duplex attached to a single strand significantly alters the persistence length of the polyT strands. Our study suggests that restrained polyT, compared to labile suspensions of free polyT, are more capable of hybridization and hence DNA-based assembly.