The Structure of a Bernoulli Process Variation of the Fibonacci Sequence

We consider the structure of a variation of the Fibonacci sequence which is determined by a Bernoulli process. The associated structure of all Bernoulli variations of the Fibonacci sequence can be represented by a directed binary tree, which we denote X, with vertex labels representing the specific state of the recurrence variation. Since X is a binary tree, we can consider the term of a sequence variation given by a finite traversal of X represented by a binary code t. We then prove that the traversal of X that is the reflection of the digits of t gives exactly the integer term corresponding to t. We consider how to further this result with the statement of an additional conjecture. Finally, we give connections to Fibonacci expansions, the Stern-Brocot tree, and we apply our methods to the Three Hat Problem as seen in ``Puzzle Corner'' of the ``Technology Review'' magazine.Comment: 22 pages, 1 figur

Torsion and ground state maxima: close but not the same

Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture. We construct counterexamples on the half-disk, by working with the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has merit, since the maxima of the torsion function and eigenfunction are unexpectedly close together. It is an open problem to quantify this closeness in terms of the domain and the nonlinearity

A validation framework for neuroimaging software: The case of population receptive fields

Published: June 25, 2020Neuroimaging software methods are complex, making it a near certainty that some implementations will contain errors. Modern computational techniques (i.e., public code and data repositories, continuous integration, containerization) enable the reproducibility of the analyses and reduce coding errors, but they do not guarantee the scientific validity of the results. It is difficult, nay impossible, for researchers to check the accuracy of software by reading the source code; ground truth test datasets are needed. Computational reproducibility means providing software so that for the same input anyone obtains the same result, right or wrong. Computational validity means obtaining the right result for the ground-truth test data. We describe a framework for validating and sharing software implementations, and we illustrate its usage with an example application: population receptive field (pRF) methods for functional MRI data. The framework is composed of three main components implemented with containerization methods to guarantee computational reproducibility. In our example pRF application, those components are: (1) synthesis of fMRI time series from ground-truth pRF parameters, (2) implementation of four public pRF analysis tools and standardization of inputs and outputs, and (3) report creation to compare the results with the ground truth parameters. The framework was useful in identifying realistic conditions that lead to imperfect parameter recovery in all four pRF implementations, that would remain undetected using classic validation methods. We provide means to mitigate these problems in future experiments. A computational validation framework supports scientific rigor and creativity, as opposed to the oft-repeated suggestion that investigators rely upon a few agreed upon packages. We hope that the framework will be helpful to validate other critical neuroimaging algorithms, as having a validation framework helps (1) developers to build new software, (2) research scientists to verify the software’s accuracy, and (3) reviewers to evaluate the methods used in publications and grants.Supported by a Marie Sklodowska-Curie (https://ec.europa.eu/programmes/horizon2020/ en/h2020-section/marie-sklodowska-curie-actions) grant to G.L.-U. (H2020-MSCA-IF-2017-795807- ReCiModel) and National Institutes of Health (https://www.nih.gov/) grants supporting N.C.B. and J.W. (EY027401, EY027964, MH111417). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Velocity Segregation and Systematic Biases In Velocity Dispersion Estimates With the SPT-GMOS Spectroscopic Survey

The velocity distribution of galaxies in clusters is not universal; rather, galaxies are segregated according to their spectral type and relative luminosity. We examine the velocity distributions of different populations of galaxies within 89 Sunyaev Zel'dovich (SZ) selected galaxy clusters spanning $0.28 < z < 1.08$. Our sample is primarily draw from the SPT-GMOS spectroscopic survey, supplemented by additional published spectroscopy, resulting in a final spectroscopic sample of 4148 galaxy spectra---2868 cluster members. The velocity dispersion of star-forming cluster galaxies is $17\pm4$% greater than that of passive cluster galaxies, and the velocity dispersion of bright ($m < m^{*}-0.5$) cluster galaxies is $11\pm4$% lower than the velocity dispersion of our total member population. We find good agreement with simulations regarding the shape of the relationship between the measured velocity dispersion and the fraction of passive vs. star-forming galaxies used to measure it, but we find a small offset between this relationship as measured in data and simulations in which suggests that our dispersions are systematically low by as much as 3\% relative to simulations. We argue that this offset could be interpreted as a measurement of the effective velocity bias that describes the ratio of our observed velocity dispersions and the intrinsic velocity dispersion of dark matter particles in a published simulation result. Measuring velocity bias in this way suggests that large spectroscopic surveys can improve dispersion-based mass-observable scaling relations for cosmology even in the face of velocity biases, by quantifying and ultimately calibrating them out.Comment: Accepted to ApJ; 21 pages, 11 figures, 5 table

Leatherback Turtles in the Eastern Gulf of Mexico: Foraging and Migration Behavior During the Autumn and Winter

We deployed 19 satellite tags on foraging adult leatherback turtles, including 17 females and 2 males, captured in the northeastern Gulf of Mexico in 2015, 2018, and 2019 in order to study regional distribution and movements. Prior to our study, limited data were available from leatherbacks foraging in the Gulf of Mexico. Tag deployment durations ranged from 63 to 247 days and turtles exhibited three distinct behavior types: foraging, transiting, or rapidly switching between foraging and transiting. Some females were tracked to nesting beaches in the Caribbean. Most of the leatherbacks remained on and foraged along the west Florida continental shelf whereas a few individuals foraged in waters of the central Gulf of Mexico during the autumn and winter. In addition, migration of adult females through the Yucatan Channel indicate that this is a seasonally important area for Caribbean nesting assemblages

Use of the Adult Myopathy Assessment Tool as a predictor of functional abilities in people with multiple sclerosis

Background: People with multiple sclerosis (PwMS) are at greater risk for decreased muscle performance which may lead to decreased functional abilities. The Kurtzke Expanded Disability Status Scale (EDSS) is commonly used as a disability status rating scale in PwMS. Nevertheless, the EDSS is largely comprised of neurological tests and may not best reflect functional performance. A functional battery such as The Adult Myopathy Assessment Tool (AMAT) may better reflect functional performance. The AMAT was designed to assess both functional strength and endurance in clinical settings. However, the AMAT has not been validated for the assessment of PwMS. Objective: The purpose of the study was to determine the comparative association of the AMAT and EDSS with measures of strength, fatigability, and functional performance. Methods: Twenty-nine people (mean age 48.6 ±11.2), with a history of MS (EDSS \u3c 7.0) were recruited. Participants completed functional testing (5 times sit to stand and gait speed) and an assessment of disability and functional status using the EDSS and AMAT, respectively. Muscle performance was assessed via a 60 s maximal volitional isometric contraction (MVIC) of the knee extensors using an isokinetic dynamometer, and expressed as fatigability (exhaustion time to 60% of MVIC), peak torque, and peak torque scaled to body weight. Results:The participants exhibited moderate levels of disability (EDSS, 3.6 ±1.4) and function (AMAT total score, 36.1 ±7.6; AMAT function subscale, 18.2 ±3.3). Peak force was 70.1 kg ±22.0 kg, exhaustion time was 38.4 s ±17.4 s, gait speed was 1.3 m/s ±0.3 m/s, and five time sit to stand was 11.4 s ±4.1 s. The AMAT function subscale was associated with scaled peak torque (r=0.426, p=.021), gait speed (r=0.825, p=0.00), and 5 time sit to stand (r=-0.632, p Conclusions: The AMAT was more strongly associated with scaled peak torque and functional measures in comparison to the EDSS. This may reflect the observation that relative strength is a better predictor of functional abilities than unadjusted strength measures. Whereas, the stronger association of the EDSS with fatigability may be explained by the pyramidal systems measures within the tool and the well-known association of MS-related fatigue with disability. Based on the results of the study, we suggest clinicians administer the AMAT in addition to the EDSS, to gain insight into functional impairments and assist with formulating a comprehensive plan of care

Construction and composition of the squid pen from Doryteuthis pealeii

Author Posting. © University of Chicago Press, 2019. This article is posted here by permission of University of Chicago Press for personal use, not for redistribution. The definitive version was published in Messerli, M. A., Raihan, M. J., Kobylkevich, B. M., Benson, A. C., Bruening, K. S., Shribak, M., Rosenthal, J. J. C., & Sohn, J. J. Construction and composition of the squid pen from Doryteuthis pealeii. Biological Bulletin. 237(1), (2019): 1-15, doi:10.1086/704209.The pen, or gladius, of the squid is an internalized shell. It serves as a site of attachment for important muscle groups and as a protective barrier for the visceral organs. The pen’s durability and flexibility are derived from its unique composition of chitin and protein. We report the characterization of the structure, development, and composition of pens from Doryteuthis pealeii. The nanofibrils of the polysaccharide β-chitin are arranged in an aligned configuration in only specific regions of the pen. Chitin is secreted early in development, enabling us to characterize the changes in pen morphology prior to hatching. The chitin and proteins are assembled in the shell sac surrounded by fluid that has a significantly different ionic composition from squid plasma. Two groups of proteins are associated with the pen: those on its surface and those embedded within the pen. Only 20 proteins are identified as embedded within the pen. Embedded proteins are classified into six groups, including chitin associated, protease, protease inhibitors, intracellular, extracellular matrix, and those that are unknown. The pen proteins share many conserved domains with proteins from other chitinous structures. We conclude that the pen is one of the least complex, load-bearing, chitin-rich structures currently known and is amenable to further studies to elucidate natural construction mechanisms using chitin and protein.We thank John Dowling for financial support. We thank Kasia Hammar and Louie Kerr of the Marine Biological Laboratory Central Microscopy Facility for help obtaining scanning electron micrographs. We thank Bogdan Budnik and Renee Robinson from the Mass Spectrometry and Proteomics Resource Laboratory for their help and advice with protein identification. We thank Shin-Yi Marzano and Chenchen Feng of South Dakota State University for help with rapid amplification of cDNA ends. Funding for this work was provided by the Eugene and Millicent Bell Fellowship Fund in Tissue Engineering (MAM), an Agriculture and Biological Sciences Undergraduate Research Award (KSB), National Institutes of Health grant R01 GM101701 (MS), National Science Foundation grant IOS1557748 (JJCR), and Israel-United States Binational Science Foundation 2013094 (JJCR). Literature Cited2020-07-0