Improved bounds for Hadwiger's covering problem via thin-shell estimates

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most $N\left(n\right)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n}n\ln n$. In this note, we improve this bound by a sub-exponential factor. That is, we prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal constant $c>0$. Our approach combines ideas from previous work by Artstein-Avidan and the second named author with tools from Asymptotic Geometric Analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$; in fact, we get the same lower bound for the volume of the intersection of $K$ and $-K$ when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures. Using the same ideas, we establish an exponentially better bound for $N\left(n\right)$ when restricting our attention to convex bodies that are $\psi_{2}$. By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity

Visibility-Enhanced Bicycling Clothes with Flashing LEDs Applied for Biological Motion

Bicyclists number approximately 39 to 40 million around the world; the U.S. bicyclist population grew by 27 million in 2011 (Formosa, 2012). In 2010, roughly 52,000 bicyclists were injured, and 618 were killed in traffic accidents, which accounted for 2% of all traffic fatalities; the cause was mostly low conspicuity. Eight billion dollars are spent annually in the US to care for bicycle crash victims (NHTSA, 2012). Therefore, researchers have suggested that bicyclists wear visibility aids during at night (Kwan & Mapstone, 2004; Wood et al., 2011). Many studies have suggested using LEDs, but there is a lack of understanding about the appropriate positions for wearing these lights. Previous research (Koo & Dunne, 2011; Koo & Smith, 2010) found effectiveness for using LEDs on joints at nighttime. However, there is no research comparing the effectiveness on differing joints. Thus, the purpose of this study was to develop visibility-enhanced bicycling clothing with flashing LEDs on different configurations, focusing on body joints and evaluating the effects of this clothing design on bicyclists’ visibility. This research will provide ideas for apparel designers to develop visibility-aiding bicycling clothing as well as assistance to bicyclists in placing flashing LEDs on the most effective areas of their bodies

Neutron powder diffraction study on the iron-based nitride superconductor ThFeAsN

We report neutron diffraction and transport results on the newly discovered superconducting nitride ThFeAsN with $T_c=$ 30 K. No magnetic transition, but a weak structural distortion around 160 K, is observed cooling from 300 K to 6 K. Analysis on the resistivity, Hall transport and crystal structure suggests this material behaves as an electron optimally doped pnictide superconductors due to extra electrons from nitrogen deficiency or oxygen occupancy at the nitrogen site, which together with the low arsenic height may enhance the electron itinerancy and reduce the electron correlations, thus suppress the static magnetic order.Comment: 4 pages, 4 figures, Accepted by EP

Is Embodied Interaction Beneficial? A Study on Navigating Network Visualizations

Network visualizations are commonly used to analyze relationships in various contexts. To efficiently explore a network visualization, the user needs to quickly navigate to different parts of the network and analyze local details. Recent advancements in display and interaction technologies inspire new visions for improved visualization and interaction design. Past research into network design has identified some key benefits to visualizing networks in 3D versus 2D. However, little work has been done to study the impact of varying levels of embodied interaction on network analysis. We present a controlled user study that compared four environments featuring conditions and hardware that leveraged different amounts of embodiment and visual perception ranging from a 2D visualization desktop environment with a standard mouse to a 3D visualization virtual reality environment. We measured the accuracy, speed, perceived workload, and preferences of 20 participants as they completed three network analytic tasks, each of which required unique navigation and substantial effort. For the task that required participants to iterate over the entire visualization rather than focus on a specific area, we found that participants were more accurate using a VR and a trackball mouse than conventional desktop settings. From a workload perspective, VR was generally considered the least mentally demanding and least frustrating in two of our three tasks. It was also preferred and ranked as the most effective and visually appealing condition overall. However, using VR to compare two side-by-side networks was difficult, and it was similar to or slower than other conditions in two of the three tasks. Overall, the accuracy and workload advantages of conditions with greater embodiment in specific tasks suggest promising opportunities to create more effective environments in which to analyze network visualizations.Comment: Accepted by the Information Visualization journa

Characterization and Comparison of Mesoporous Silica Particles for Optimized Drug Delivery

In this study we have investigated the suitability of a number of different mesoporous silica nanoparticle structures for carrying a drug cargo. We have fully characterized the nanoparticles in terms of their physical parameters; size, surface area, internal pore size and structure. These data are all required if we are to make an informed judgement on the suitability of the structure for drug delivery in vivo. With these parameters in mind, we investigated the loading/unloading profile of a model therapeutic into the pore structure of the nanoparticles. We demonstrate that the release can be controlled by capping the pores on the nanoparticles to achieve temporal control of the unloading. We have also examined the rate and mechanism of the degradation of the nanoparticles over an extended period of time. The eventual dissolution of the nanoparticles after cargo release is a desirable property for a drug delivery system

Least squares estimation of spatial autoregressive models for large-scale social networks

Due to the rapid development of various social networks, the spatial autoregressive (SAR) model is becoming an important tool in social network analysis. However, major bottlenecks remain in analyzing largescale networks (e.g., Facebook has over 700 million active users), including computational scalability, estimation consistency, and proper network sampling. To address these challenges, we propose a novel least squares estimator (LSE) for analyzing large sparse networks based on the SAR model. Computationally, the LSE is linear in the network size, making it scalable to analysis of huge networks. In theory, the LSE is root n-consistent and asymptotically normal under certain regularity conditions. A new LSE-based network sampling technique is further developed, which can automatically adjust autocorrelation between sampled and unsampled units and hence guarantee valid statistical inferences. Moreover, we generalize the LSE approach for the classical SAR model to more complex networks associated with multiple sources of social interaction effect. Numerical results for simulated and real data are presented to illustrate performance of the LSE.National Natural Science Foundation of China [71532001, 11525101, 71332006, 11701560, 11401482]; Beijing Municipal Social Science Foundation [17GLC051]; Center for Applied Statistics, School of Statistics, Renmin University of China; Center of Statistical Research, Southwestern University of Finance and Economics; China's National Key Research Special Program [2016YFC0207700]; NSF [DMS-1309507, DMS-1418172]; NSFC [11571009]Open Access JournalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]