Chebyshev constants for the unit circle

It is proven that for any system of n points z_1, ..., z_n on the (complex) unit circle, there exists another point z of norm 1, such that $$\sum 1/|z-z_k|^2 \leq n^2/4.$$ Equality holds iff the point system is a rotated copy of the nth unit roots. Two proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein's inequality.Comment: 11 page

How Should a Robot Approach a Pair of People?

This thesis experimentally investigates the comfort of pairs of seated people when they are approached by a robot from different directions. While the effect of robot approach direction on the comfort of a lone person has been investigated previously, the extension to a robot approaching pairs of people has not been explored rigorously. Three maximally-different seating configurations of paired people and eight different robot approach directions were considered. The experiment was augmented with a fourth seating configuration of a lone individual, allowing the responses of grouped and lone participants to be compared. Data obtained from the experiment were analysed using both linear and directional statistics. Results from 180 unique participants showed that the comfort of a person when a robot approached is influenced by the presence and location of a second person. Analysis of these data with directional statistics showed that participant comfort preference clusters into angular regions of ‘suitable for robot approach’ and ‘unsuitable for robot approach’. This finding shows the importance of avoiding robot approach directions of low comfort, rather than selecting a singular robot approach direction of high comfort. Rayleigh’s test of uniformity, a directional statistics method, also shows across all participant configurations that robot approach directions that minimize participant discomfort align spatially with regions that allow for good line of sight of the robot by both people, and are centred on the largest open space that a robot could approach the group from. Participants who were grouped also regarded the robot as having more social agency than did lone experimental participants. Grouped participants were less frustrated with the experimental task and also found it less physically and temporally demanding in comparison to lone experimental participants

Productivity Growth in U.S. Agriculture

Innovation and changes in technology have been a driving force for gains in productivity growth in U.S. agriculture. USDA's Economic Research Service has developed annual indexes of agricultural inputs, outputs, and total factor productivity (TFP) for 1948 through 2004. American agriculture relies almost entirely on productivity growth to raise output. By lowering the cost of agricultural commodities, productivity growth benefits not only farmers but also food manufacturers and consumers.Agriculture, productivity, productivity growth, total factor productivity, TFP, labor, farm economy, prices, agricultural research, agricultural output, technology, ERS, USDA, Production Economics, Productivity Analysis,

Entropy jumps for isotropic log-concave random vectors and spectral gap

We prove a quantitative dimension-free bound in the Shannon-Stam Entropy inequality for the convolution of two log-concave distributions in dimension d interms of the spectral gap of the density. The method relies on the analysis of the Fisher Information production, which is the second derivative of the Entropy along the (normalized) Heat semi-group. We also discuss consequences of our result in the study of the isotropic constant of log-concave distributions (slicing problem).Comment: 15 pages, The title is changed and Acknowledgement is adde