516 research outputs found
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
The Hot Interstellar Medium in Normal Elliptical Galaxies III: The Thermal Structure of the Gas
This is the third paper in a series analyzing X-ray emission from the hot
interstellar medium in a sample of 54 normal elliptical galaxies observed by
Chandra, focusing on 36 galaxies with sufficient signal to compute radial
temperature profiles. We distinguish four qualitatively different types of
profile: positive gradient (outwardly rising), negative gradients (falling),
quasi-isothermal (flat) and hybrid (falling at small radii, then rising). We
measure the mean logarithmic temperature gradients in two radial regions: from
0--2 -band effective radii (excluding the central point source), and
from 2--. We find the outer gradient to be uncorrelated with intrinsic
host galaxy properties, but strongly influenced by the environment: galaxies in
low-density environments tend to show negative outer gradients, while those in
high-density environments show positive outer gradients, suggesting influence
of circumgalactic hot gas. The inner temperature gradient is unaffected by the
environment but strongly correlated with intrinsic host galaxy characteristics:
negative inner gradients are more common for smaller, optically faint, low
radio-luminosity galaxies, whereas positive gradients are found in bright
galaxies with stronger radio sources. There is no evidence for bimodality in
the distribution of inner or outer gradients. We propose three scenarios to
explain the inner temperature gradients: (1) Weak AGN heat the ISM locally,
higher-luminosity AGN heat the system globally through jets inflating cavities
at larger radii; (2) The onset of negative inner gradients indicates a
declining importance of AGN heating relative to other sources, such as
compressional heating or supernovae; (3) The variety of temperature profiles
are snapshots of different stages of a time-dependent flow.Comment: 18 pages, emulateapj, 55 figures (36 online-only figures included in
astro-ph version), submitted to Ap
Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions
Two well studied Ramsey-theoretic problems consider subsets of the natural
numbers which either contain no three elements in arithmetic progression, or in
geometric progression. We study generalizations of this problem, by varying the
kinds of progressions to be avoided and the metrics used to evaluate the
density of the resulting subsets. One can view a 3-term arithmetic progression
as a sequence , where , a nonzero
integer. Thus avoiding three-term arithmetic progressions is equivalent to
containing no three elements of the form with , the set of integer translations. One can similarly
construct related progressions using different families of functions. We
investigate several such families, including geometric progressions ( with a natural number) and exponential progressions ().
Progression-free sets are often constructed "greedily," including every
number so long as it is not in progression with any of the previous elements.
Rankin characterized the greedy geometric-progression-free set in terms of the
greedy arithmetic set. We characterize the greedy exponential set and prove
that it has asymptotic density 1, and then discuss how the optimality of the
greedy set depends on the family of functions used to define progressions.
Traditionally, the size of a progression-free set is measured using the (upper)
asymptotic density, however we consider several different notions of density,
including the uniform and exponential densities.Comment: Version 1.0, 13 page
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
Revolutionary Peacemaking: Using a Critical Pedagogy Approach for Peacemaking with Terrorists
The current global political atmosphere is steeped in fear of, and intense rhetoric about, political violence and terrorism. Amidst this turbulent environment, it is clear that scholars and practitioners need to get beyond the manufactured fear and the hysterical rhetoric, peddled by what we call the corporate-state-military-media complex (or simply, the power complex ), and instead seek a deeper understanding of political groups that defend or deploy the tactics of economic sabotage (property destruction) or armed struggle in order to change repressive and violent social structures (Best and Nocella 2004; Best and Nocella 2006). Such understanding is important to slow down and reverse the current trend among legislative and policy-making bodies and political leaders who increasingly marginalize, demonize, and exclude radical opposition groups from arenas of debate
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
Observations and Scaling of Tidal Mass Transport Across the Lower Ganges-Brahmaputra Delta Plain: Implications for Delta Management and Sustainability
The landscape of southwest Bangladesh, a region constructed primarily by fluvial processes associated with the Ganges River and Brahmaputra River, is now maintained almost exclusively by tidal processes as the fluvial system has migrated east and eliminated the most direct fluvial input. In natural areas such as the Sundarbans National Forest, year-round inundation during spring high tides delivers sufficient sediment that enables vertical accretion to keep pace with relative sea-level rise. However, recent human modification of the landscape in the form of embankment construction has terminated this pathway of sediment delivery for much of the region, resulting in a startling elevation imbalance, with inhabited areas often sitting \u3e1 m below mean high water. Restoring this landscape, or preventing land loss in the natural system, requires an understanding of how rates of water and sediment flux vary across timescales ranging from hours to months. In this study, we combine time series observations of water level, salinity, and suspended sediment concentration with ship-based measurements of large tidal-channel hydrodynamics and sediment transport. To capture the greatest possible range of variability, cross-channel transects designed to encompass a 12.4 h tidal cycle were performed in both dry and wet seasons during spring and neap tides.
Regional suspended sediment concentration begins to increase in August, coincident with a decrease in local salinity, indicating the arrival of the sediment-laden, freshwater plume of the combined Ganges–Brahmaputra–Meghna rivers. We observe profound seasonality in sediment transport, despite comparatively modest seasonal variability in the magnitude of water discharge. These observations emphasize the importance of seasonal sediment delivery from the main-stem rivers to this remote tidal region. On tidal timescales, spring tides transport an order of magnitude more sediment than neap tides in both the wet and dry seasons. In aggregate, sediment transport is flood oriented, likely as a result of tidal pumping. Finally, we note that rates of sediment and water discharge in the tidal channels are of the same scale as the annually averaged values for the Ganges and Brahmaputra rivers. These observations provide context for examining the relative importance of fluvial and tidal processes in what has been defined as a quintessentially tidally influenced delta in the classification scheme of Galloway (1975). These data also inform critical questions regarding the timing and magnitude of sediment delivery to the region, which are especially important in predicting and preparing for responses of the natural system to ongoing environmental change
Restoring Lateral Incisors and Orthodontic Treatment: Perceptions among General Dentists and Othodontists
The purpose of this study was to identify and compare preferences and perceptions of orthodontists and general dentists when restoring peg-shaped lateral incisors. The investigation sought to summarize these preferences with regard to treatment planning, tooth preparation and interdisciplinary communication. A pair of mailed and electronic surveys was distributed to 1,500 general dentists and orthodontists, respectively. The results indicated that general dentists perceived that general dentists held the primary decision-making responsibility, while orthodontists disagreed (P\u3c0.0001). Orthodontists prioritized the treatment goals of Class I canine relationship and overbite/overjet more significantly than general dentists, whom valued tooth proportions more highly (P\u3c0.0001). General dentists reported receiving significantly less input than orthodontists report seeking (P\u3c0.0001).The consensus of both groups showed that the tooth should be positioned centered mesiodistally and guided by the gingival margins incisogingivally. Both groups agree that orthodontists must improve communication to improve treatment results
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