Brief Overview of Recent Developments on Negative Phase–Velocity Mediums (alias Left–Handed Materials)

This brief overview covers a rapidly occurring development in the area of microwave composite materials: isotropic dielectric–magnetic mediums wherein the phase velocity vector and the time–averaged Poynting vector are oppositely directed. Such materials have also been inappropriately designated as left–handed materials. Ramifications for aberration–free lenses, homogenization approaches, and complex mediums are also presented.     

Proof of the Mixing Theorem for Statistical Systems in Classical Physics

In a general measure-theoretic context, it is proved that the action of a stochastic map on a pair of probability distributions can be characterized by the criterion of decreasing mixing distance.     

Fluorescence correlation spectroscopic studies of diffusion within the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate

Fluorescence correlation spectroscopy (FCS) measurements have been performed in order to determine the translational diffusion coefficients of three differently charged fluorescent probes (cationic: rhodamine 6G, neutral: 4-dicyanomethylene-2-methyl-6-p-dimethylaminostyryl-4-H-pyran, DCM, and anionic: fluorescein) dissolved within the common room temperature ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate, [bmim][PF6]. These experiments demonstrate that FCS is a rapid, sensitive, precise (typical RSD from 4-8%), and low-consumption screening tool for the determination of analyte mobility within microliter ionic liquid samples.     

Scalar-type spectral operators andC(Ω)-operational calculi

It is shown that a continuous linear operatorT in a locally convex spaceX is a scalar-type spectral operator if and only if it admits aC((T))-operational calculus of a certain kind. This is a genuine extension of previous results of this type since we allow for the case when (T){} is an unbounded set in the complex planeC, a phenomenon which occurs often for continuous operatorsT defined in non- normable spacesX.     

Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides

We consider a domain Ω in ?ⁿ of the form Ω = ?^l × Ω′ with bounded Ω′ ? ?^n?l. In Ω we study the Dirichlet initial and boundary value problem for the equation ? u + [(? ? ?… ? ?)^m + (? ? ?… ? ?)^m]u = fe^?iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like t^α (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.     

Primitive lattice points in rational ellipses and related arithmetic functions

Assuming the Riemann Hypothesis to be true, an asymptotic with a sharp error term is established for the number of primitive lattice points inside a rational ellipseau ²+buv+cv ²x (a, b, c integers,b ²–4ac<0). A generalization of the result is given applying (as an example) to counting functions of Pythagorean triangles.