A Discontinuity in the Distribution of Fixed Point Sums

The quantity $f(n,r)$, defined as the number of permutations of the set $[n]=\{1,2,... n\}$ whose fixed points sum to $r$, shows a sharp discontinuity in the neighborhood of $r=n$. We explain this discontinuity and study the possible existence of other discontinuities in $f(n,r)$ for permutations. We generalize our results to other families of structures that exhibit the same kind of discontinuities, by studying $f(n,r)$ when ``fixed points'' is replaced by ``components of size 1'' in a suitable graph of the structure. Among the objects considered are permutations, all functions and set partitions.Comment: 1 figur

Some new canonical forms for polynomials

We give some new canonical representations for forms over \cc. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can be written uniquely as a sum of the cubes of linear forms $\ell_{ij}(x_i,...,x_j)$, $1 \le i \le j \le n$. A general ternary quartic form is the sum of the square of a quadratic form and three fourth powers of linear forms. The methods are classical and elementary.Comment: I have spoken about this material under the title "steampunk canonical forms". This is the final revised version which has been accepted by the Pacific Journal of Mathematics. Apart from the usual improvements which come after a thoughtful refereeing, Theorem 1.8 is ne

The Asymptotic Number of Irreducible Partitions

A partition of [1, n] = {1,..., n} is called irreducible if no proper subinterval of [1, n] is a union of blocks. We determine the asymptotic relationship between the numbers of irreducible partitions, partitions without singleton blocks, and all partitions when the block sizes must lie in some specified set

Distribution of the Number of Encryptions in Revocation Schemes for Stateless Receivers

We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215--238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes

An observational analysis of risk factors associated with symptomatic third molar teeth

Background: Third molar teeth (wisdom teeth) are a common cause of pain and infection in young adults. The study aimed to describe the prevalence of symptomatic third molar teeth and identify factors which predispose to third molar symptoms in a birth cohort. Methods: An observational study was undertaken nested in the Avon Longitudinal Study of Parents and Children (ALSPAC), a birth cohort based in south west England. The main outcomes were self-reported third molar pain, swelling and treatment for third molar problems, taken from questionnaires completed at age 23 years. The exposures including sex, dental history, socioeconomic status, diet, and genetic factors were obtained from earlier ALSPAC data. Results: In total 4,222 ALSPAC participants responded to one or more questions about third molar teeth. The final sample included more female participants than male participants. The majority of participants (56.6%) reported at least one episode of pain associated with their third molars. Females had greater odds than males of reporting swelling (adjusted odds ratio (OR) 1.97; 95%confidence interval (CI) 1.56, 2.51), pain (adjusted OR=1.96; 95%CI 1.56, 2.51) and receiving both non-surgical and surgical treatment (adjusted OR=2.30; 95%CI 1.62, 3.35, adjusted OR=1.54; 95%CI 1.17, 2.06 respectively). Participants with previously filled teeth had greater odds of third molar extraction. There were no strong associations between index of multiple deprivation (IMD) score or sugar intake and the third molar outcomes. There was weak evidence for a genetic contribution to third molar pain. Conclusions: Symptomatic third molars are common in this age group, with over half of the participants reporting pain or other symptoms. Female participants had greater odds for third molar pain, swelling and treatment

Towards a Continuous Record of the Sky

It is currently feasible to start a continuous digital record of the entire sky sensitive to any visual magnitude brighter than 15 each night. Such a record could be created with a modest array of small telescopes, which collectively generate no more than a few Gigabytes of data daily. Alternatively, a few small telescopes could continually re-point to scan and reco rd the entire sky down to any visual magnitude brighter than 15 with a recurrence epoch of at most a few weeks, again always generating less than one Gigabyte of data each night. These estimates derive from CCD ability and budgets typical of university research projects. As a prototype, we have developed and are utilizing an inexpensive single-telescope system that obtains optical data from about 1500 square degrees. We discuss the general case of creating and storing data from a both an epochal survey, where a small number of telescopes continually scan the sky, and a continuous survey, composed of a constellation of telescopes dedicated each continually inspect a designated section of the sky. We compute specific limitations of canonical surveys in visible light, and estimate that all-sky continuous visual light surveys could be sensitive to magnitude 20 in a single night by about 2010. Possible scientific returns of continuous and epochal sky surveys include continued monitoring of most known variable stars, establishing case histories for variables of future interest, uncovering new forms of stellar variability, discovering the brightest cases of microlensing, discovering new novae and supernovae, discovering new counterparts to gamma-ray bursts, monitoring known Solar System objects, discovering new Solar System objects, and discovering objects that might strike the Earth.Comment: 38 pages, 9 postscript figures, 2 gif images. Revised and new section added. Accepted to PASP. Source code submitted to ASCL.ne