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

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

Head Worn Display System for Equivalent Visual Operations

Head-Worn Displays or so-called, near-to-eye displays have potentially significant advantages in terms of cost, overcoming cockpit space constraints, and for the display of spatially-integrated information. However, many technical issues need to be overcome before these technologies can be successfully introduced into commercial aircraft cockpits. The results of three activities are reported. First, the near-to-eye display design, technological, and human factors issues are described and a literature review is presented. Second, the results of a fixed-base piloted simulation, investigating the impact of near to eye displays on both operational and visual performance is reported. Straight-in approaches were flown in simulated visual and instrument conditions while using either a biocular or a monocular display placed on either the dominant or non-dominant eye. The pilot's flight performance, visual acuity, and ability to detect unsafe conditions on the runway were tested. The data generally supports a monocular design with minimal impact due to eye dominance. Finally, a method for head tracker system latency measurement is developed and used to compare two different devices

Carbon Free Boston: Buildings Technical Report

Part of a series of reports that includes: Carbon Free Boston: Summary Report; Carbon Free Boston: Social Equity Report; Carbon Free Boston: Technical Summary; Carbon Free Boston: Transportation Technical Report; Carbon Free Boston: Waste Technical Report; Carbon Free Boston: Energy Technical Report; Carbon Free Boston: Offsets Technical Report; Available at http://sites.bu.edu/cfb/OVERVIEW: Boston is known for its historic iconic buildings, from the Paul Revere House in the North End, to City Hall in Government Center, to the Old South Meeting House in Downtown Crossing, to the African Meeting House on Beacon Hill, to 200 Clarendon (the Hancock Tower) in Back Bay, to Abbotsford in Roxbury. In total, there are over 86,000 buildings that comprise more than 647 million square feet of area. Most of these buildings will still be in use in 2050. Floorspace (square footage) is almost evenly split between residential and non-residential uses, but residential buildings account for nearly 80,000 (93 percent) of the 86,000 buildings. Boston’s buildings are used for a diverse range of activities that include homes, offices, hospitals, factories, laboratories, schools, public service, retail, hotels, restaurants, and convention space. Building type strongly influences energy use; for example, restaurants, hospitals, and laboratories have high energy demands compared to other commercial uses. Boston’s building stock is characterized by thousands of turn-of-the-20th century homes and a postWorld War II building boom that expanded both residential buildings and commercial space. Boston is in the midst of another boom in building construction that is transforming neighborhoods across the city. [TRUNCATED]Published versio

When the Wheels Touch Earth and the Flight is Through, Pilots Find One Eye is Better Than Two?

This study investigated the impact of near to eye displays on both operational and visual performance by employing a human-in-the-loop simulation of straight-in ILS approaches while using a near to eye (NTE) display. The approaches were flown in simulated visual and instrument conditions while using either a biocular NTE or a monocular NTE display on either the dominant or non dominant eye. The pilot s flight performance, visual acuity, and ability to detect unsafe conditions on the runway were tested

How U.S. Ocean Policy and Market Power Can Reform the Coral Reef Wildlife Trade

As the world’s largest importer of marine ornamental species for the aquaria, curio, home décor, and jewelry industries, the United States has an opportunity to leverage its considerable market power to promote more sustainable trade and reduce the effects of ornamental trade stress on coral reefs worldwide. Evidence indicates that collection of some coral reef animals for these trades has caused virtual elimination of local populations, major changes in age structure, and promotion of collection practices that destroy reef habitats. Management and enforcement of collection activities in major source countries such as Indonesia and the Philippines remain weak. Strengthening US trade laws and enforcement capabilities combined with increasing consumer and industry demand for responsible conservation can create strong incentives for improving management in source countries. This is particularly important in light of the March 2010 failure of the parties to the Convention on International Trade in Endangered Species (CITES) to take action on key groups of corals

A tiered model of substance use severity and life complexity : potential for application to needs-based planning

Background: In order to improve long-term outcomes for individuals with substance use problems, one approach is to adopt a system planning model that considers both addiction severity and life complexities. The tiered approach has been developed and tested to describe systems-level need based on levels of risk and problem severity. Methods: An existing tiered model was modified to accommodate Australian data, incorporating substance use severity and life complexity. The hypothesis was that tiers would reflect differences in well-being amongst help seekers such that an increase in tier would be associated with a reduction in well-being, suggesting the need for more intensive (and integrated) interventions. The model was tested using 2 data sets of screening data, collected from face-to-face alcohol and other drug (AOD) service (n = 430) and online help (n = 309) seekers, drawn from a larger sample of 2,766 screens. The screen included demographic information and substance use, mental health, and quality of life measures. Results: There was a significant relationship between well-being and tier ranking, suggesting that the model adequately captured elements of severity and complexity that impact on well-being. There were notable differences between the help-seeking populations with a higher proportion of online respondents allocated to lower tiers and more face-to-face respondents allocated to higher tiers. However, there was an overlap in these populations, with more than half of online respondents classified as higher tiers and one fifth of face-to-face respondents classified as lower tiers. This suggests that the model can be used both to assess unmet need in out-of-treatment groups and demand in the absence of dependence in a subpopulation of the face-to-face treatment population. Conclusions: The tiered model provides a method to understand levels of AOD treatment need and, as part of needs-based planning, may be used to optimize treatment responses and resourcing

Calibrating Emission Lines as Quasar Bolometers

Historically, emission lines have been considered a valuable tool for estimating the bolometric thermal luminosity of the accretion flow in AGN, $L_{bol}$. We study the reliability of this method by comparing line strengths to the optical/UV continuum luminosity of SDSS DR7 radio quiet quasars with $0.4<z<0.8$. We find formulae for $L_{bol}$ as a function of single line strengths for the broad components of H$\beta$ and Mg II, as well as the narrow lines of [O III] and [O II]. We determine the standard errors of the formulae that are fitted to the data. Our new estimators are shown to be more accurate than archival line strength estimations in the literature. It is demonstrated that the broad lines are superior estimators of the continuum luminosity (and $L_{bol}$) with $H\beta$ being the most reliable. The fidelity of the each of the estimators is determined in the context of the SDSS DR7 radio loud quasars as an illustrative application of our results. In general, individual researchers can use our results as a tool to help decide if a particular line strength provides an adequate estimate of $L_{bol}$ for their purposes. Finally, it is shown that considering all four line strength, simultaneously, can yield information on both $L_{bol}$ and the radio jet power.Comment: To appear in MNRAS Letter

Intimate Partner Violence and Correlates With Risk Behaviors and HIV/STI Diagnoses Among Men Who Have Sex With Men and Men Who Have Sex With Men and Women in China: A Hidden Epidemic

BACKGROUND: Intimate partner violence (IPV) research has primarily focused on heterosexual couples but has largely ignored IPV among men who have sex with men (MSM). We examined IPV prevalence among MSM and men who have sex with men and women (MSMW) in China. METHODS: Men who have sex with men older than 16 years were recruited through 3 MSM-focused Web sites in China. An online survey containing items on sociodemographics, risk behaviors, IPV, and self-reported HIV or sexually transmitted infection diagnosis was completed. Multivariate regression was used to examine associations between IPV and risk behaviors and an HIV or sexually transmitted infection diagnosis. RESULTS: Among 610 participants, 182 (29.8%) reported experiencing at least 1 type of IPV. Men who have sex with both men and women were at significantly greater risk for IPV (adjusted odds ratio [AOR], 1.65; 95% confidence interval [CI], 1.08-2.53) compared with MSM. Men who had experienced IPV were more likely to have participated in group sex (AOR, 1.86; 95% CI, 1.08-3.21), to have had sex in exchange for gifts or money (AOR, 5.06; 95% CI, 2.47-10.35), and to report a positive HIV diagnosis (AOR, 2.59; 95% CI, 1.22-5.51). CONCLUSIONS: There is a hidden epidemic of IPV among MSM in China, especially among MSMW. The hidden nature of MSM and MSMW suggests the need for a clinical environment more conducive to disclosure. Research is needed to understand the pathways linking IPV and HIV risk among MSM to optimize the design of effective interventions