166 research outputs found

    Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals

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    Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers Fn{F_n}, with initial terms F1=1,F2=2F_1 = 1, F_2 = 2. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as nn \to \infty the distribution of the number of summands in the Zeckendorf decompositions of m[Fn,Fn+1)m \in [F_n, F_{n+1}), appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in [Fn,Fn+1)[F_n, F_{n+1}) share the same potential summands. We generalize these results to subintervals of [Fn,Fn+1)[F_n, F_{n+1}) as nn \to \infty; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence α(n)\alpha(n) \to \infty. As nn \to \infty, for almost all m[Fn,Fn+1)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+Fα(n))[m, m + F_{\alpha(n)}), appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to 11, mm has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and [0,Fα(n))[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

    Benford Behavior of Zeckendorf Decompositions

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    A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers {Fi}i=1\{ F_i \}_{i = 1}^{\infty}. A set SZS \subset \mathbb{Z} is said to satisfy Benford's law if the density of the elements in SS with leading digit dd is log10(1+1d)\log_{10}{(1+\frac{1}{d})}; in other words, smaller leading digits are more likely to occur. We prove that, as nn\to\infty, for a randomly selected integer mm in [0,Fn+1)[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

    Striking the right balance in right whale conservation

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    Author Posting. © The Authors, 2009. This article is posted here by permission of NRC Research Press for personal use, not for redistribution. The definitive version was published in Canadian Journal of Fisheries and Aquatic Sciences 66 (2009): 1399-1403, doi:10.1139/F09-115.Despite many years of study and protection, the North Atlantic right whale (Eubalaena glacialis) remains on the brink of extinction. There is a crucial gap in our understanding of their habitat use in the migratory corridor along the eastern seaboard of the United States. Here, we characterize habitat suitability in migrating right whales in relation to depth, distance to shore, and the recently enacted ship speed regulations near major ports. We find that the range of suitable habitat exceeds previous estimates and that, as compared with the enacted 20 nautical mile buffer, the originally proposed 30 nautical mile buffer would protect more habitat for this critically endangered species.This work was supported in part by SERDP/DoD grant W912HQ-04-C-0011 to A.J. Read and P.N. Halpin as well as a James B. Duke Fellowship and a Harvey L. Smith Dissertation Year Fellowship to R.S. Schick

    Polarimetry and Unification of Low-Redshift Radio Galaxies

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    We have made high-quality measurements of the polarization spectra of 13 FR II radio galaxies and taken polarization images for 11 of these with the Keck telescopes. Seven of the eight narrow-line radio galaxies (NLRG) are polarized, and six of the seven show prominent broad Balmer lines in polarized light. The broad lines are also weakly visible in total flux. Some of the NLRG show bipolar regions with roughly circumferential polarization vectors, revealing a large reflection nebula illuminated by a central source. Our observations powerfully support the hidden quasar hypothesis for some NLRG. Classification as NLRG, broad-line radio galaxy (BLRG), or quasar therefore depends on orientation. However, not all objects fit into this unification scheme. Our sample is biased towards objects known in advance to be polarized, but the combination of our results with those of Hill, Goodrich and DePoy (1996) show that at least 6 out of a complete, volume and flux-limited sample of 9 FR II NLRG have broad lines, seen either in polarization or P_alpha.Comment: To appear in November 1999 Astronomical Journal. 49 pages, 13 figure

    Muscle Loss Is Associated with Overall Survival in Patients with Metastatic Colorectal Cancer Independent of Tumor Mutational Status and Weight Loss

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    Background: Survival in patients with metastatic colorectal cancer (mCRC) has been associated with tumor mutational status, muscle loss, and weight loss. We sought to explore the combined effects of these variables on overall survival. Materials and methods: We performed an observational cohort study, prospectively enrolling patients receiving chemotherapy for mCRC. We retrospectively assessed changes in muscle (using computed tomography) and weight, each dichotomized as >5% or ≤5% loss, at 3, 6, and 12 months after diagnosis of mCRC. We used regression models to assess relationships between tumor mutational status, muscle loss, weight loss, and overall survival. Additionally, we evaluated associations between muscle loss, weight loss, and tumor mutational status. Results: We included 226 patients (mean age 59 ± 13 years, 53% male). Tumor mutational status included 44% wild type, 42% RAS-mutant, and 14% BRAF-mutant. Patients with >5% muscle loss at 3 and 12 months experienced worse survival controlling for mutational status and weight (3 months hazard ratio, 2.66; p 5% muscle loss with BRAF-mutational status at 6 and 12 months. Weight loss was not associated with survival nor mutational status. Conclusion: Increased muscle loss at 3 and 12 months may identify patients with mCRC at risk for decreased overall survival, independent of tumor mutational status. Specifically, >5% muscle loss identifies patients within each category of tumor mutational status with decreased overall survival in our sample. Our findings suggest that quantifying muscle loss on serial computed tomography scans may refine survival estimates in patients with mCRC. Implications for practice: In this study of 226 patients with metastatic colorectal cancer, it was found that losing >5% skeletal muscle at 3 and 12 months after the diagnosis of metastatic disease was associated with worse overall survival, independent of tumor mutational status and weight loss. Interestingly, results did not show a significant association between weight loss and overall survival. These findings suggest that muscle quantification on serial computed tomography may refine survival estimates in patients with metastatic colorectal cancer beyond mutational status

    Catalog of Radio Galaxies with z>0.3. I:Construction of the Sample

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    The procedure of the construction of a sample of distant (z>0.3z>0.3) radio galaxies using NED, SDSS, and CATS databases for further application in statistical tests is described. The sample is assumed to be cleaned from objects with quasar properties. Primary statistical analysis of the list is performed and the regression dependence of the spectral index on redshift is found.Comment: 9 pages, 6 figures, 2 table

    Mesenchymal Stem Cell Therapy Regenerates the Native Bone-Tendon Junction after Surgical Repair in a Degenerative Rat Model

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    BACKGROUND: The enthesis, which attaches the tendon to the bone, naturally disappears with aging, thus limiting joint mobility. Surgery is frequently needed but the clinical outcome is often poor due to the decreased natural healing capacity of the elderly. This study explored the benefits of a treatment based on injecting chondrocyte and mesenchymal stem cells (MSC) in a new rat model of degenerative enthesis repair. METHODOLOGY: The Achilles' tendon was cut and the enthesis destroyed. The damage was repaired by classical surgery without cell injection (group G1, n = 52) and with chondrocyte (group G2, n = 51) or MSC injection (group G3, n = 39). The healing rate was determined macroscopically 15, 30 and 45 days later. The production and organization of a new enthesis was assessed by histological scoring of collagen II immunostaining, glycoaminoglycan production and the presence of columnar chondrocytes. The biomechanical load required to rupture the bone-tendon junction was determined. PRINCIPAL FINDINGS: The spontaneous healing rate in the G1 control group was 40%, close to those observed in humans. Cell injection significantly improved healing (69%, p = 0.0028 for G2 and p = 0.006 for G3) and the load-to-failure after 45 days (p<0.05) over controls. A new enthesis was clearly produced in cell-injected G2 and G3 rats, but not in the controls. Only the MSC-injected G3 rats had an organized enthesis with columnar chondrocytes as in a native enthesis 45 days after surgery. CONCLUSIONS: Cell therapy is an efficient procedure for reconstructing degenerative entheses. MSC treatment produced better organ regeneration than chondrocyte treatment. The morphological and biomechanical properties were similar to those of a native enthesis
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