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

Automated prompting technologies in rehabilitation and at home

Purpose - The purpose of this paper is to test the efficacy of an interactive verbal prompting technology (Guide) on supporting the morning routine. Data have already established the efficacy of such prompting during procedural tasks, but the efficacy of such prompting in tasks with procedural and motivational elements remains unexamined. Such tasks, such as getting out of bed in the morning and engaging in personal care, are often the focus of rehabilitation goals. Design/methodology/approach - A single-n study with a male (age 61) who had severe cognitive impairment and was having trouble completing the morning routine. An A-B-A'-B'-A?-B? design was used, with the intervention phase occurring both in an in-patient unit (B, B') and in the participant's own home (B?). Findings - Interactive verbal prompting technology (Guide) significantly reduced support worker prompting and number of errors in the in-patient setting and in the participant's own home. Research limitations/implications - The results suggest that interactive verbal prompting can be used to support motivational tasks such as getting out of bed and the morning routine. This study used a single subject experimental design and the results need to be confirmed in a larger sample. Originality/value - This is the first report of use of interactive verbal prompting technology to support rehabilitation of a motivational task. It is also the first study to evaluate Guide in a domestic context

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

Assessing Dimensions of the Security-Liberty Trade-off in the United States

The trade-off between security and liberty has been a leading frame for understanding public opinion about domestic surveillance policies. Most of the empirical work explicitly examining whether individuals meet the trade-off framework’s core attitudinal assumptions comes from European studies. This study uses a survey of US residents to assess the veracity of the assumptions embedded in the trade-off framework, namely whether domestic counterterrorism policies are simultaneously viewed as improving security and decreasing liberty. We find that the vast majority of US respondents do not meet the basic attitudinal assumptions of the trade-off frame. Next, we evaluate the source of these attitudes with a focus on whether attitudes toward surveillance policies merely relate to core political values or whether they also depend on the messages from political leaders. We find that both political values and opinion leadership shape these attitudes. Finally, because general attitudes towards surveillance and privacy often fail to have practical implications, we assess whether these attitudes matter for understanding the structure of policy support. Our results show that heightened terrorism threat positively associates with increased support for counterterrorism policies only when people believe these policies are effective security tools

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

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

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

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

Ion Beam-Treated Space Polymers: Long-Term Stability in GEO-Simulated Environments

As part of a large program conducted between years 2010 and 2018, a number of ion beam surface treatments and proprietary coatings formed on advanced space polymers by direct ion beam deposition were developed at Integrity Testing Laboratory (ITL) Inc, in Canada. Thistechnology allowed producing surfaces with controlled surface resistivity in a wide range of charge dissipation values, with negligibly low additional RF losses and other important functional properties, that allowed using such treated materials and products in modern space antennae, solar arrays and other external applications on spacecrafts in GEO environment. This paper will present an overview of results achieved throughout the years on enhancement of radiation stability of space polymers that were ion beam treated and coated with special multifunctional thin coatings by direct ion beam deposition. The treated/coated subjects have been ground-based tested in a range of radiation conditions, simulating the GEO radiation environment, conducted in three world-recognized GEO simulation facilities