26 research outputs found
A p-adic Perron-Frobenius Theorem
We prove that if an matrix defined over (or more
generally an arbitrary complete, discretely-valued, non-Archimedean field)
satisfies a certain congruence property, then it has a strictly maximal
eigenvalue in , and that iteration of the (normalized) matrix
converges to a projection operator onto the corresponding eigenspace. This
result may be viewed as a -adic analogue of the Perron-Frobenius theorem for
positive real matrices
The emergence of 4-cycles in polynomial maps over the extended integers
Let ; for each integer it is interesting to
consider the number of iterates , if possible, needed to satisfy
. The sets generated by the iterates of are
called cycles. For it is known that cycles of length 1 and 2
occur, and no others. While much is known for extensions to number fields, we
concentrate on extending by adjoining reciprocals of primes. Let
denote extended by adding in
the reciprocals of the primes and all their products and
powers with each other and the elements of .
Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in
under the appropriate
conditions. The problem of finding criteria under which 4-cycles emerge is
equivalent to determining how often a sum of four terms is zero, where the
terms are times a product of elements from the list of primes. We
investigate conditions on sets of primes under which 4-cycles emerge. We
characterize when 4-cycles emerge if the set has one or two primes, and
(assuming a generalization of the ABC conjecture) find conditions on sets of
primes guaranteed not to cause 4-cycles to emerge.Comment: 14 pages, 1 figur
Combinatorial Aspects of the Ikeda Lift via Extended Gross-Keating Data
This thesis is motivated by the problem of studying the Ikeda lift via a converse theorem due
to R. Weissauer. We investigate a certain function; denoted by (; ), where is a positive
integer and is a symmetric positive-definite half-integral matrix; appearing in the Fourier
coefficient formulas of a linear version of the Ikeda lift due to W. Kohnen. We develop
new methods for computing (; ) via the extended Gross-Keating (EGK) datum of a
quadratic form and develop novel combinatorial interpretations for (; ) which involve
integer partitions with restrictions depending on the EGK datum attached to at each prime
Benford Behavior of Zeckendorf Decompositions
A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as the sum of non-consecutive Fibonacci numbers . A set is said to satisfy Benford's law if
the density of the elements in with leading digit is
; in other words, smaller leading digits are more
likely to occur. We prove that, as , for a randomly selected
integer in the distribution of the leading digits of the
Fibonacci summands in its Zeckendorf decomposition converge to Benford's law
almost surely. Our results hold more generally, and instead of looking at the
distribution of leading digits one obtains similar theorems concerning how
often values in sets with density are attained.Comment: Version 1.0, 12 pages, 1 figur
Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals
Zeckendorf's theorem states that every positive integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers , with initial
terms . We consider the distribution of the number of
summands involved in such decompositions. Previous work proved that as the distribution of the number of summands in the Zeckendorf
decompositions of , appropriately normalized, converges
to the standard normal. The proofs crucially used the fact that all integers in
share the same potential summands.
We generalize these results to subintervals of as ; the analysis is significantly more involved here as different integers
have different sets of potential summands. Explicitly, fix an integer sequence
. As , for almost all the distribution of the number of summands in the Zeckendorf
decompositions of integers in the subintervals ,
appropriately normalized, converges to the standard normal. The proof follows
by showing that, with probability tending to , has at least one
appropriately located large gap between indices in its decomposition. We then
use a correspondence between this interval and to obtain
the result, since the summands are known to have Gaussian behavior in the
latter interval. % We also prove the same result for more general linear
recurrences.Comment: Version 1.0, 8 page
Communication Biophysics
Contains reports on six research projects.National Institutes of Health (Grant 5 PO1 NS13126)National Institutes of Health (Grant 5 RO1 NS18682)National Institutes of Health (Grant 5 RO1 NS20322)National Institutes of Health (Grant 5 R01 NS20269)National Institutes of Health (Grant 5 T32NS 07047)Symbion, Inc.National Science Foundation (Grant BNS 83-19874)National Science Foundation (Grant BNS 83-19887)National Institutes of Health (Grant 6 RO1 NS 12846)National Institutes of Health (Grant 1 RO1 NS 21322
A planet within the debris disk around the pre-main-sequence star AU Microscopii
AU Microscopii (AU Mic) is the second closest pre main sequence star, at a
distance of 9.79 parsecs and with an age of 22 million years. AU Mic possesses
a relatively rare and spatially resolved3 edge-on debris disk extending from
about 35 to 210 astronomical units from the star, and with clumps exhibiting
non-Keplerian motion. Detection of newly formed planets around such a star is
challenged by the presence of spots, plage, flares and other manifestations of
magnetic activity on the star. Here we report observations of a planet
transiting AU Mic. The transiting planet, AU Mic b, has an orbital period of
8.46 days, an orbital distance of 0.07 astronomical units, a radius of 0.4
Jupiter radii, and a mass of less than 0.18 Jupiter masses at 3 sigma
confidence. Our observations of a planet co-existing with a debris disk offer
the opportunity to test the predictions of current models of planet formation
and evolution.Comment: Nature, published June 24th [author spelling name fix
Prognostic model to predict postoperative acute kidney injury in patients undergoing major gastrointestinal surgery based on a national prospective observational cohort study.
Background: Acute illness, existing co-morbidities and surgical stress response can all contribute to postoperative acute kidney injury (AKI) in patients undergoing major gastrointestinal surgery. The aim of this study was prospectively to develop a pragmatic prognostic model to stratify patients according to risk of developing AKI after major gastrointestinal surgery. Methods: This prospective multicentre cohort study included consecutive adults undergoing elective or emergency gastrointestinal resection, liver resection or stoma reversal in 2-week blocks over a continuous 3-month period. The primary outcome was the rate of AKI within 7 days of surgery. Bootstrap stability was used to select clinically plausible risk factors into the model. Internal model validation was carried out by bootstrap validation. Results: A total of 4544 patients were included across 173 centres in the UK and Ireland. The overall rate of AKI was 14·2 per cent (646 of 4544) and the 30-day mortality rate was 1·8 per cent (84 of 4544). Stage 1 AKI was significantly associated with 30-day mortality (unadjusted odds ratio 7·61, 95 per cent c.i. 4·49 to 12·90; P < 0·001), with increasing odds of death with each AKI stage. Six variables were selected for inclusion in the prognostic model: age, sex, ASA grade, preoperative estimated glomerular filtration rate, planned open surgery and preoperative use of either an angiotensin-converting enzyme inhibitor or an angiotensin receptor blocker. Internal validation demonstrated good model discrimination (c-statistic 0·65). Discussion: Following major gastrointestinal surgery, AKI occurred in one in seven patients. This preoperative prognostic model identified patients at high risk of postoperative AKI. Validation in an independent data set is required to ensure generalizability