Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

Frequency of Sobolev and quasiconformal dimension distortion

We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in RnRn, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed m-dimensional linear subspace, whose image under f has positive HαHα measure for some fixed α>mα>m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples.

Distortion-sensitive insight into the pretransitional behavior of 4- n -octyloxy-4′-cyanobiphenyl (8OCB)

Results of studies of the static and dynamic dielectric properties in rod-like 4-n-octyloxy-4'-cyanobiphenyl (8OCB) with isotropic (I)–nematic (N)–smectic A (SmA)–crystal (Cr) mesomorphism, combined with measurements of the low-frequency nonlinear dielectric effect and heat capacity are presented. The analysis is supported by the derivative-based and distortion-sensitive transformation of experimental data. Evidence for the I–N and N–SmA pretransitional anomalies, indicating the influence of tricritical behavior, is shown. It has also been found that neither the N phase nor the SmA phase are uniform and hallmarks of fluid–fluid crossovers can be detected. The dynamics, tested via the evolution of the primary relaxation time, is clearly non-Arrhenius and described via τ(T) = τc(T−TC)−phgr. In the immediate vicinity of the I–N transition a novel anomaly has been found: Δτ ∝ 1/(T − T*), where T* is the temperature of the virtual continuous transition and Δτ is the excess over the 'background behavior'. Experimental results are confronted with the comprehensive Landau–de Gennes theory based modeling.

Distortion-free enhancement of terahertz signals measured by electro-optic sampling. I. Theory

We present three methods for the distortion-free enhancement of THz signals measured by electro-optic sampling in zinc blende-type detector crystals, e.g., ZnTe or GaP. A technique commonly used in optically heterodyne-detected optical Kerr effect spectroscopy is introduced, which is based on two measurements at opposite optical biases near the zero transmission point in a crossed polarizer detection geometry. In contrast to other techniques for an undistorted THz signal enhancement, it also works in a balanced detection scheme and does not require an elaborate procedure for the reconstruction of the true signal as the two measured waveforms are simply subtracted to remove distortions. We study three different approaches for setting an optical bias using the Jones matrix formalism and discuss them also in the framework of optical heterodyne detection. We show that there is an optimal bias point in realistic situations where a small fraction of the probe light is scattered by optical components. The experimental demonstration will be given in the second part of this two-paper series [J. Opt. Soc. Am. B, doc. ID 204877 (2014, posted online)].

Distortion-free enhancement of terahertz signals measured by electro-optic sampling. II. Experiment

Three methods for distortion-free enhancement of electro-optic sampling measurements of terahertz signals are tested. In the first part of this two-paper series [J. Opt. Soc. Am B 31, 904–910 (2014)], the theoretical framework for describing the signal enhancement was presented and discussed. As the applied optical bias is decreased, individual signal traces become enhanced but distorted. Here we experimentally show that nonlinear signal components that distort the terahertz electric field measurement can be removed by subtracting traces recorded with opposite optical bias values. In all three methods tested, we observe up to an order of magnitude increase in distortion-free signal enhancement, in agreement with the theory, making possible measurements of small terahertz-induced transient birefringence signals with increased signal-to-noise ratio.

Color polymorphism and sex ratio distortion in a cichlid fish as an incipient stage in sympatric speciation by sexual selection

We investigated a Lake Victoria cichlid with a complex colour polymorphism that apparently represents one original species and two incipient species, all of which are sympatric. In laboratory breeding experiments we observed sex ratio distortion in certain matings between original and incipient species. Mate choice experiments show that males of the incipient species exhibit mating preferences against the original species, and males and females of the original species exhibit strong mating preferences against the incipient species. Mating preferences might evolve by sex ratio selection to avoid matings with distorted progeny sex ratios. Phenotype frequencies in nature suggest that mating preferences translate into mating frequencies, thus restricting gene flow and exerting disruptive sexual selection between the original and incipient species. The incipient species do not differ in morphology or ecology from the original species, implying that colour polymorphism, associated with sex ratio distortion, can be an incipient stage in sympatric speciation, and that disruption of gene flow can precede ecological differentiation

Elicitable distortion risk measures: A concise proof

Elicitability has recently been discussed as a desirable property for risk measures. Kou and Peng (2014) showed that an elicitable distortion risk measure is either a Value-at-Risk or the mean. We give a concise alternative proof of this result, and discuss the conflict between comonotonic additivity and elicitability.