On kissing numbers and spherical codes in high dimensions

We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle $\theta$ in high dimensions

On the hard sphere model and sphere packings in high dimensions

We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of density $\Theta(d \cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $\Omega(d \cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_d(1)) \log(2/\sqrt{3}) d \cdot 2^{-d}$ when the ratio of the fugacity parameter to the volume covered by a single sphere is at least $3^{-d/2}$. Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh

Are the A-Team A-Okay?

This paper will look at the scene in the movie The A-Team when a tank parachutes its way to the ground with the team inside. The terminal velocity of the tank, with three parachutes and then one parachute will be calculated as per the film. The terminal velocity of the tank with three parachutes was found to reach 20.3 ms-1, and 35.1 ms-1 with only one of the parachutes. The probability of survival at these speeds was then explored, based on the amount of G-force that a person can experience before becoming fatal. The terminal velocities that were calculated were found to be 2.1 and 3.6 times larger than the fatal velocity of an impact, for three and one parachutes respectively. Therefore, it was concluded that The A-Team would not have survived the fall, as is shown in the movie

Extremes of the internal energy of the Potts model on cubic graphs

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function. Taking the zero-temperature limit gives new results in extremal combinatorics: the number of $q$-colorings of a $3$-regular graph, for any $q \ge 2$, is maximized by a union of $K_{3,3}$'s. This proves the $d=3$ case of a conjecture of Galvin and Tetali

Chloroplast DNA phylogenetics of the North American chestnuts and chinquapins (Castanea Mill., Fagaceae)

Evolutionary relationships and genetic structure of the North American Castanea were investigated using chloroplast DNA sequence data. Six plastome loci were PCR-amplified and sequenced in 77 accessions representing the three currently recognized North American Castanea species. Diagnostic morphological character states and a unique haplotype were shared among C. pumila and a plant tentatively identified as C. dentata in one sympatric site, suggesting past hybridization and chloroplast capture. Surprisingly, the cpDNA phylogeny did not agree with previous taxonomic treatments. The inability to distinguish between deep coalescence and interspecific hybridization as the causes of haplotype sharing makes phylogenetic reconstruction of the North American Castanea species difficult. Although non-D haplotypes were previously reported as diagnostic for C. pumila and hybrids, multiple non-D haplotype C. dentata were documented in the Southern Appalachians and Piedmont. The diversity of haplotypes observed in southern C. dentata populations provides further impetus to conserve C. dentata in the Southeast

Legitimate Peripheral Participation of Secondary Educators in Scientific Research Experiences: Implications for Teachers\u27 Understanding of the Nature of Science and Classroom Teaching

Both of the national reform efforts (AAAS, 1993; NRC, 1996) encouraged teachers to engage in professional development that included authentic scientific research experiences. The Department of Energy developed a program to match teachers with mentor scientists at national laboratories for three consecutive summers. Teachers produced and presented a poster summarizing their research at the conclusion of each summer. The purpose of this qualitative multiple case study was to better understand how scientific research experiences impacted teachers. Six dimensions were examined: trajectory of participation, content knowledge development, mentor relationships, beliefs about the nature of science, teacher confidence, and classroom practice. These six dimensions were integrated into three research questions which guided the research: the teachers’ ability to increase their level of participation from the first to the last summer of research, the teachers’ changes in their understanding of the nature of science (NOS), and any changes in the teachers’ classroom teaching because of their involvement in the program. In-depth interviews were triangulated with teachers’ posters to provide insights into teachers’ legitimate peripheral participation in the research laboratory. The VNOS-C (Lederman et al., 2002) was administered pre/post to the teachers. Evidence of more informed, developing, and more naive understandings of each of the tenets of NOS was collected and compared to identify changes in teachers’ beliefs. Interviews and follow-up correspondence informed the study of changes in classroom teaching. The teachers became very familiar with their mentors’ research, increased their subject content knowledge, and contributed to their mentor’s work. Mentors utilized teachers’ expertise as communicators when presenting research and hosting other student groups. The teachers’ understanding of the NOS did not change as a result of their immersion in the culture of the laboratory. The lens through which the teachers viewed science influenced how they perceived and interpreted their research experiences. Teachers who held positivist views reinforced them, while the lone teacher who held post-positivist views reinforced their positions. The teachers developed confidence in their ability to facilitate classroom inquiry, increased the number of inquiry-based in their curriculum, introduced advanced placement and scientific research courses, and rejuvenated their enthusiasm for teaching

Methodology to Analyze Tropical Cyclone Intensity from Microwave Imagery

Satellites with microwave remote sensing capabilities can be utilized to study atmospheric phenomena through high-level cloud cover (particularly cirrus), an advantage over visible and infrared bands, which only sense cloud tops. This unique capability makes microwave imagery ideal for studying the cloud structures of tropical cyclones (TCs) in detail, and relating these features to TC intensity. Techniques to estimate the intensity of TCs using infrared imagery, such as the Dvorak technique, have been used in TC forecasting for 40 years. However, due to the inherent temporal limitations of microwave imagery, no such similar technique exists for the microwave spectrum. This study utilizes pattern recognition to develop a subjective technique for estimating TC intensity using microwave imagery. The dataset includes TC composite imagery from the Special Sensor Microwave Imager (85 GHz), Advanced Microwave Scanning Radiometer-Earth Observing System (89 GHz), Advanced Microwave Scanning Radiometer 2 (89 GHz), and the Special Sensor Microwave Imager/Sounder (91 GHz) from the Atlantic basin, and aircraft reconnaissance data are used for verification. The composite imagery is binned into four categories to facilitate detection of common patterns for TCs of similar size and estimated intensity. This analysis provides the foundation for a new method to estimate TC intensity when aircraft data are unavailable. Multiple techniques are applied to explore relationships between brightness temperature values and TC intensity, and ten test cases in the Western Pacific basin are presented to validate the results. Five out of the ten TCs were classified correctly applying the microwave intensity techniques developed by this analysis

Do Ewoks pack a punch?

This paper aims to calculate the force that would be required to knock out a Stormtrooper with a rock thrown by an Ewok. In order to calculate this, the ways in which a person can get knocked out are explored, finding that the best way is to get the head to rotate in a way such that the brain hits the skull. The spin required to knock someone out was assumed, and was used to work out an angular frequency and subsequently a linear velocity. The force was then worked out, using a number of assumptions regarding the mass of a Stormtroopers head, and it was found that a force of 2797 N would be required to knock the Stormtrooper out. Therefore, the Ewok threw the rock with the same force which was 1.4 times stronger than professional featherweight boxers