Four bugs on a rectangle

The problem of four bugs in cyclic pursuit starting from a 2-by-1 rectangle is considered, and asymptotic formulas are derived to describe the motion. In contrast to the famous case of four bugs on a square, here the trajectories quickly freeze to essentially one dimension. After the first rotation about the centre point, the scale of the configuration has shrunk by a factor of 10^427907250, and this number is then exponentiated four more times with each successive cycle. Relations to Knuth’s double-arrow notation and level-index arithmetic are discussed

α\alpha-vacuum and inflationary bispectrum

In this paper, we discuss the non-Guassianity originated from the $\alpha$-vacuum on the CMB anisotropy. For $\alpha$-vacuum, there exist correlation between points in the acausal two patches of de Sitter spactime. Such kind of correlation can lead to large local form non-Guassianity in $\alpha$-vacuum. For the single field slow-roll inflationary scenario, the spacetime is in a quasi-de Sitter phase during the inflation. We will show that the $\alpha$-vacuum in this case will lead to non-Gaussianity with distinguished feature, of a large local form and a very different shape.Comment: 39 pages, 8 figures, references added, minor changes, accepted by Phys.Rev.

Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensin

Multiple roles of motor imagery during action observation

Over the last 20 years, the topics of action observation (AO) and motor imagery (MI) have been largely studied in isolation from each other, despite the early integrative account by Jeannerod (1994, 2001). Recent neuroimaging studies demonstrate enhanced cortical activity when AO and MI are performed concurrently (“AO+MI”), compared to either AO or MI performed in isolation. These results indicate the potentially beneficial effects of AO+MI, and they also demonstrate that the underlying neurocognitive processes are partly shared. We separately review the evidence for MI and AO as forms of motor simulation, and present two quantitative literature analyses that indeed indicate rather little overlap between the two bodies of research. We then propose a spectrum of concurrent AO+MI states, from congruent AO+MI where the contents of AO and MI widely overlap, over coordinative AO+MI, where observed and imagined action are different but can be coordinated with each other, to cases of conflicting AO+MI. We believe that an integrative account of AO and MI is theoretically attractive, that it should generate novel experimental approaches, and that it can also stimulate a wide range of applications in sport, occupational therapy, and neurorehabilitation