Fourier transform, null variety, and Laplacian's eigenvalues

We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians.

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.

Internal nanoparticle structure of temperature-responsive self-assembled PNIPAM-b-PEG-b-PNIPAM triblock copolymers in aqueous solutions: NMR, SANS and light scattering studies

In this study we report detailed information on the internal structure of PNIPAM-b-PEG-b-PNIPAM nanoparticles formed from self-assembly in aqueous solutions upon increase in temperature. NMR spectroscopy, light scattering and small-angle neutron scattering (SANS) were used to monitor different stages of nanoparticle formation as a function of temperature, providing insight into the fundamental processes involved. The presence of PEG in a copolymer structure significantly affects the formation of nanoparticles, making their transition to occur over a broader temperature range. The crucial parameter that controls the transition is the ratio of PEG/PNIPAM. For pure PNIPAM, the transition is sharp; the higher the PEG/PNIPAM ratio results in a broader transition. This behavior is explained by different mechanisms of PNIPAM block incorporation during nanoparticle formation at different PEG/PNIPAM ratios. Contrast variation experiments using SANS show that the structure of nanoparticles above cloud point temperatures for PNIPAM-b-PEG-b-PNIPAM copolymers is drastically different from the structure of PNIPAM mesoglobules. In contrast with pure PNIPAM mesoglobules, where solid-like particles and chain network with a mesh size of 1-3 nm are present; nanoparticles formed from PNIPAM-b-PEG-b-PNIPAM copolymers have non-uniform structure with “frozen” areas interconnected by single chains in Gaussian conformation. SANS data with deuterated “invisible” PEG blocks imply that PEG is uniformly distributed inside of a nanoparticle. It is kinetically flexible PEG blocks which affect the nanoparticle formation by prevention of PNIPAM microphase separation.