McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions

Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N=24, 26, >= 28; 7-designs for N=24, 30, 32, 34, >= 36; 8-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >= 54; 10-designs for N=60, 62, >= 64; 11-designs for N=70, 72, >= 74; and 12-designs for N=84, >= 86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and -- although not identified as such by McLaren -- consists of the vertices of an "improved" snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5-designs exist for N=12, 16, 18, 20, >= 22. It is conjectured, albeit with decreasing confidence for t >= 9, that these lists of t-designs are complete and that no others exist. One of the constructions gives a sequence of putative spherical t-designs with N= 12m points (m >= 2) where N = t^2/2 (1+o(1)) as t -> infinity.Comment: 16 pages, 1 figur

Green Beer: Incentivizing Sustainability in California\u27s Brewing Industry

Part II of this Article examines the role of alcoholic beverages in human history, paying special attention to alcohol as a motivating factor in large-scale social change. Part III examines the prominence of California’s unique brewing industry and the economic and social ubiquity of Californian beer. As discussed in Parts IV and V, that ubiquity and prominence, as well as California’s historical leadership on environmental issues, make the state an ideal testing ground for sustainable brewing legislation. After an examination of California’s energy use in producing beer, Parts VI and VII break down the brewing process and explain a selection of opportunities to mitigate its environmental impact. These Parts discuss general and process-specific measures that either reduce energy demand or provide some other type of environmental control. Part VIII turns to various California legislative schemes that purport to achieve similar goals. It examines how various aspects of these schemes might serve as models for sustainable brewing legislation. Part IX synthesizes those models by proposing a sustainable brewing legislative scheme

Nonintersecting Subspaces Based on Finite Alphabets

Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A subseteq F? The most important case is when F is the field of complex numbers C; then M_t is the number of antennas. If A = F = GF(q) it is shown that the number of nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound can be attained if and only if m is divisible by M_t. Furthermore these subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the finite field case is essentially completely solved. In the case when F = C only the case M_t=2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2^r complex roots of unity, the number of nonintersecting planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in fact be the best that can be achieved).Comment: 14 page

The Primary Pretenders

We call a composite number q such that there exists a positive integer b with b^p == b (mod q) a prime pretender to base b. The least prime pretender to base b is the primary pretender q_b. It is shown that there are only 132 distinct primary pretenders, and that q_b is a periodic function of b whose period is the 122-digit number 19568584333460072587245340037736278982017213829337604336734362- 294738647777395483196097971852999259921329236506842360439300.Comment: 7 page