452 research outputs found
A Discontinuity in the Distribution of Fixed Point Sums
The quantity , defined as the number of permutations of the set
whose fixed points sum to , shows a sharp discontinuity
in the neighborhood of . We explain this discontinuity and study the
possible existence of other discontinuities in for permutations. We
generalize our results to other families of structures that exhibit the same
kind of discontinuities, by studying 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
can be written uniquely as a sum of the cubes of linear forms
, . 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
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
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