Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C1,1-regular; also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and such that K = G - G up to homotheties; this implies in turn that G must be C2,1- regular. Then for N = 2, we prove that G is uniformly K-dense if and only if K and G are homothetic to the same ellipse. This result was already proven by Amar, Berrone and Gianni in [3]. However, our proof removes their regularity assumptions on K and G and, more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski's inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near r = 0 for the measure of G\cap(x+rK) (needed in [3])

Cloaking due to anomalous localized resonance in plasmonic structures of confocal ellipses

If a core of dielectric material is coated by a plasmonic structure of negative dielectric material with non-zero loss parameter, then anomalous localized resonance may occur as the loss parameter tends to zero and the source outside the structure can be cloaked. It has been proved that the cloaking due to anomalous localized resonance (CALR) takes place for structures of concentric disks and the critical radius inside which the sources are cloaked has been computed. In this paper, it is proved that CALR takes place for structures of confocal ellipses and the critical elliptic radii are computed. The method of this paper uses the spectral analysis of the Neumann-Poincar\'e type operator associated with two interfaces (the boundaries of the core and the shell)

Optical Mobius Strips in Three Dimensional Ellipse Fields: Lines of Circular Polarization

The major and minor axes of the polarization ellipses that surround singular lines of circular polarization in three dimensional optical ellipse fields are shown to be organized into Mobius strips. These strips can have either one or three half-twists, and can be either right- or left-handed. The normals to the surrounding ellipses generate cone-like structures. Two special projections, one new geometrical, and seven new topological indices are developed to characterize the rather complex structures of the Mobius strips and cones. These eight indices, together with the two well-known indices used until now to characterize singular lines of circular polarization, could, if independent, generate 16,384 geometrically and topologically distinct lines. Geometric constraints and 13 selection rules are discussed that reduce the number of lines to 2,104, some 1,150 of which have been observed in practice; this number of different C lines is ~ 350 times greater than the three types of lines recognized previously. Statistical probabilities are presented for the most important index combinations in random fields. It is argued that it is presently feasible to perform experimental measurements of the Mobius strips and cones described here theoretically

Characterization of spatio-temporal epidural event-related potentials for mouse models of psychiatric disorders.

Distinctive features in sensory event-related potentials (ERPs) are endophenotypic biomarkers of psychiatric disorders, widely studied using electroencephalographic (EEG) methods in humans and model animals. Despite the popularity and unique significance of the mouse as a model species in basic research, existing EEG methods applicable to mice are far less powerful than those available for humans and large animals. We developed a new method for multi-channel epidural ERP characterization in behaving mice with high precision, reliability and convenience and report an application to time-domain ERP feature characterization of the Sp4 hypomorphic mouse model for schizophrenia. Compared to previous methods, our spatio-temporal ERP measurement robustly improved the resolving power of key signatures characteristic of the disease model. The high performance and low cost of this technique makes it suitable for high-throughput behavioral and pharmacological studies