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

The Myth of Egalitarianism in Wartime and Austerity Britain

This thesis examines the construction of a myth of egalitarianism in Britain during the Second World War and subsequent challenges to it under austerity in the immediate postwar years. It engineers the 1951 Festival of Britain as a lens through which to track a history of hope for and disillusionment with socialist reconstruction legislation, addressing the following key questions: How did expectations for postwar social harmony clash with the \u27New Britain\u27 delivered by the Labour government after 1945? In what ways did this tension motivate the organization of a festival for the nation, about the nation? Featuring the prophecies of H.G. Wells and George Orwell, baby starvation techniques, modern toilet seat aesthetics, and a particularly striking piece of steak, this thesis endeavors to question the agenda of the Festival of Britain beyond its recognized role as a tonic to the nation

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

Can PAC Learning Algorithms Tolerate Random Attribute Noise?

This paper studies the robustness of pac learning algorithms when the instances space is {0,1}n, and the examples are corrupted by purely random noise affecting only the instances (and not the labels). In the past, conflicting results on this subject have been obtained -- the best agreement rule can only tolerate small amounts of noise, yet in some cases large amounts of noise can be tolerated. We show that the truth lies somewhere in between these two alternatives. For uniform attribute noise, in which each attribute is flipped independently at random with the same probability, we present an algorithm that pac learns monomials for any (unknown) noise rate less than 1/2. Contrasting this positive result, we show that product random attribute noise, where each attribute i is flipped randomly and independently with its own probability pi, is nearly as harmful as malicious noise-- no algorithm can tolerate more than a very small amount of such noise