423 research outputs found
The 3x+1 Semigroup
The 3x+1 semigroup is the multiplicative semigroup generated by the rational
numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This
semigroup encodes backward iteration under the 3x+1 map, and the 3x+1
conjecture implies that it contains every positive integer. We prove this is
the case, and show that this semigroup consists of all positive rational
numbers a/b such that 3 does not divide b.Comment: 16 pages, latex; minor change
Calibrating an ice sheet model using high-dimensional binary spatial data
Rapid retreat of ice in the Amundsen Sea sector of West Antarctica may cause
drastic sea level rise, posing significant risks to populations in low-lying
coastal regions. Calibration of computer models representing the behavior of
the West Antarctic Ice Sheet is key for informative projections of future sea
level rise. However, both the relevant observations and the model output are
high-dimensional binary spatial data; existing computer model calibration
methods are unable to handle such data. Here we present a novel calibration
method for computer models whose output is in the form of binary spatial data.
To mitigate the computational and inferential challenges posed by our approach,
we apply a generalized principal component based dimension reduction method. To
demonstrate the utility of our method, we calibrate the PSU3D-ICE model by
comparing the output from a 499-member perturbed-parameter ensemble with
observations from the Amundsen Sea sector of the ice sheet. Our methods help
rigorously characterize the parameter uncertainty even in the presence of
systematic data-model discrepancies and dependence in the errors. Our method
also helps inform environmental risk analyses by contributing to improved
projections of sea level rise from the ice sheets
Carryless Arithmetic Mod 10
We investigate what arithmetic would look like if carry digits into other
digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 =
0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45,
47, ... .Comment: 7 pages. To the memory of Martin Gardner (October 21, 1914 -- May 22,
2010). Revised version (with a number of small improvements), July 7 201
On Asymmetric Coverings and Covering Numbers
An asymmetric covering D(n,R) is a collection of special subsets S of an
n-set such that every subset T of the n-set is contained in at least one
special S with |S| - |T| <= R. In this paper we compute the smallest size of
any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded''
versions of the problem. The latter involves the classical covering numbers
C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51,
C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also
find the number of nonisomorphic minimal covering designs in several cases.Comment: 11 page
Providing True Opportunity for Opportunity Youth: Promising Practices and Principles for Helping Youth Facing Barriers to Employment
Many "opportunity youth" -- youth who are not working or in school -- would benefit substantially from gaining work experience but need help overcoming barriers to employment and accessing the labor market.Those opportunity youth facing the most significant challenges, such as extreme poverty, homelessness, and justice system involvement, often need even more intensive assistance in entering and keeping employment, and are at risk of being left behind even by employment programs that are specifically designed to serve opportunity youth.This paper builds on the research literature with extensive interviews with employment program providers who have had success in helping the most vulnerable opportunity youth succeed in the workforce. Six principles for effectively serving these youth are identified
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