The emergence of 4-cycles in polynomial maps over the extended integers

Let $f(x) \in \mathbb{Z}[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots, f^{n_{\alpha} - 1}(\alpha), \alpha\}$ generated by the iterates of $f$ are called cycles. For $\mathbb{Z}[x]$ 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 $\mathbb{Z}$ by adjoining reciprocals of primes. Let $\mathbb{Z}[1/p_1, \ldots, 1/p_n]$ denote $\mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, \ldots, p_n$ and all their products and powers with each other and the elements of $\mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $\mathbb{Z}\left[1/p_1, \ldots, 1/p_n\right][x]$ 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 $\pm 1$ times a product of elements from the list of $n$ 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 $J$-band effective radii $R_J$ (excluding the central point source), and from 2--$4R_J$. 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 $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n(x))$ with $f_n \in\mathcal{F}_{\rm t}$, the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions ($f_n(x) = nx$ with $n > 1$ a natural number) and exponential progressions ($f_n(x) = x^n$). 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 ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands. We generalize these results to subintervals of $[F_n, F_{n+1})$ as $n \to \infty$; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $\alpha(n) \to \infty$. As $n \to \infty$, for almost all $m \in [F_n, F_{n+1})$ the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals $[m, m + F_{\alpha(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, F_{\alpha(n)})$ 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 $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ 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