2015 Compilation of Internship Reports Iii Solubility of Perfl Uorocarbons Under Geothermal Conditions

Total Page:16

File Type:pdf, Size:1020Kb

2015 Compilation of Internship Reports Iii Solubility of Perfl Uorocarbons Under Geothermal Conditions BNL-108447-2015 A Compilation of Internship Reports 2015 Prepared for The Offi ce of Educational Programs Brookhaven National Laboratory Offi ce of Educational Programs Offi ce of Educational Programs, 2012 Compilation of Internship Reports 1 DISCLAIMER This work was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors or their employees, makes any warranty, ex- press or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or rep- resents that its use would not infringe privately owned rights. Reference herein to any specifi c commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily con- stitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or refl ect those of the United States Government or any agency thereof. Table of Contents Automatically detecting typical failure signatures to improve Sun-photometer data quality . 7 Brooke Adams Examining the water gas shift reaction using Pt-CeOx-TiO2 powder catalysts . 11 Balal Aslam Harmonic fl ow in heavy ion collisions: a search for phase transition and v2 correlations . 16 Mamoudou Ba Erin O’Brien Wind farm: feasibility and environmental impacts . 20 Matthew Bernard Acoustic click to mount: using sound pulses to solve the crystal harvesting bottleneck for high throughput screening applications . 25 Mary T. Bollard, Liam McCarthy, Alexander Scalia, et al. Ionizing Radiation and its Effect on the DNA of the Human Body . 33 Nicholas Bonura Simulating beamline confi gurations for the Deep Underground Neutrino Experiment . 36 Brendon Bullard Design and analysis of the inner hadronic calorimeter installation fi xture for the sPHENIX Project at RHIC . 41 Daniel Cacace Modeling a coating system for photocathodes for Low Energy Relativistic Heavy Ion Collider electron Cooling. 47 Patrick Carroll Using anomaly detection to facilitate intelligence gathering for satellite imagery . 50 Daniel Cisek Replacement cost estimates for overduty electrical panels at the Brookhaven National Laboratory . 55 William Clark Foundation for the processes involved in conducting an energy audit of building 902 . 59 Keith Combs Developing a watchdog program to oversee environmental sensors at the Northeast Solar Energy Research Center . 61 Lee R. Combs Maintaining micro climates during nano-liter chemical dispensations using custom designed source plate lids . 67 Ashley M. Drozd, Bryan J. Foley, Mary T. Bollard, et al. Offi ce of Educational Programs, 2015 Compilation of Internship Reports iii Solubility of perfl uorocarbons under geothermal conditions . 71 Yassien Fadul and Juan Sosa Characterizing reconstructions in different Liquid Argon Time Projection Chambers . 73 Samantha Fortuna The effi ciency of 4-poster tick management stations on three tick species populations within Brookhaven National Laboratory . 78 Tayelor Gosselin The shallow-to-deep transition in convective clouds during the Green Ocean Amazon 2014/5 campaign . 83 Cari Gostic Automated testing of nuclear data using zero- power nuclear reactors . 86 Mecheal Greene Optimization of a fast camera with a twenty-nanosecond resolution . 90 Jon Gustafson Drell-Yan studies and simulations at 500 GeV for pp collisions. 95 Samuel George Heppelmann A study of the performance of non-volatile dual in-line memory . .100 Shawn Hoose Understanding the polarization switching exhibited by Moth-Eye Nanostructures . .102 Alexander S. Kaiser Thick CCD sensor characterization using interferometry techniques . .105 Alexander Karlis Atomic scale investigations of model TiO2(110) catalysts . .108 Alexander Kramer Developing and optimizing the assembly process to manufacture an electromagnetic caloriometer module for sPHENIX . 111 Spencer Laddy-Locks Simulating electron showers in the Deep Underground Neutrino Experiment far detector . 114 Jacob Larkin Effects of laser exposure on the niobium tin surface . 117 Zachary Lebens-Higgins Raft optimization and readout test stand design of the raft electronics board for the Large Synoptic Survey Telescope . .122 Jessica Li Adding diagnostics to the Relativistic Heavy Ion Collider’s main magnet power supply phase-loop lock and silicon controlled rectifi er-fi ring board. .125 Raul Martinez Development and commissioning of tools for measurements and online calculation of electron beam lifetime in the National Synchrotron Light Source - II storage ring . .130 Jessica Mele iv Offi ce of Educational Programs, 2015 Compilation of Internship Reports Identifi cation and location of mutated Arabidopsis genes through map-based cloning . .134 Kristina Michta Construction and analysis of reduced graphene oxide/ ruthenium oxide supercapacitors electrodes using electrophoretic deposition . .139 Dakoda W. Mullinax and Z. Amir Targeting lipid abundant mutants of Arabidopsis thaliana. .142 Kimberly Okoli Visualization of the RHIC Computing Facility . .145 Do Om Development of scientifi c control and computing capabilities at the Inelastic X-ray Scattering beamline, National Synchrotron Light Source II . .147 John Oto Creation of high charge state lithium and calcium ion beams using laser ablation . .151 Kevin Palm An analysis of understory plant diversity in the Long Island Central Pine Barrens. .154 Ari Rice Investigating vortex dynamics in trilayer structures of ferromagnetic materials separated by a nonmagnetic spacer layer . .163 Angelika Rothberg The use of EMPIRE to simulate the neutron induced reactions on 86Kr . .170 Elizabeth Rubino Impacts of white-tailed deer on forest health in the Pine Barrens on Long Island . .177 Taylor Ruhle and Lauren Gazerwitz Determining meteorological system accuracy using a root sum of squares analysis . .181 Megan Safranek Designing the booster to Alternating Gradient Synchrotron injection lines for the electric dipole moment experiment . .186 Michael Severance What does the future hold for the Nuclear Nonproliferation Treaty? . .189 Laura Shanklin Preliminary tasks for charge coupled device quantum effi ciency measurements . .194 Thomas Smith Replacement of the Nd:YAG Chiller . .199 Derek Squires Proton beam production using titanium hydride by means of a laser ion source . .203 Cayla Stifl er Investigating the feasibility of instrument comparison as a means for analyzing data on aerosol concentrations. .206 Ariel Stolz Offi ce of Educational Programs, 2015 Compilation of Internship Reports v Reduction of photon scattering on the X-ray Powder Diffraction beamline’s detector through radiation simulation . 211 Dennis Tao Effects of deer overabundance on bird populations in the Long Island Central Pine Barrens 214 Raffaele Vesia Self-assembly of magnetite block copolymer nanocomposites . .220 Adam Vianna Single bunch emittance measurements via edge radiation . .225 Linnea Zavala Mechanical metrology testing for the Frontier Macromolecular Crystallography (FMX) Beamline and the Highly Automated Macromolecular Crystallography (AMX) Beamline at NSLS-II . .229 Lauren Zipper Application containers as an alternative to virtualization. .233 Joseph Zuhusky vi Offi ce of Educational Programs, 2015 Compilation of Internship Reports Automatically detecting typical failure signatures to improve Sun-photometer data quality Brooke Adams Department of Physics and Space Sciences, Florida Institute of Technology, Melbourne, FL 32901 Laurie Gregory and Richard Wagener Environmental Department, Brookhaven National aboratory, Upton, NY 11973 Abstract cally, and areas near a facility that lets off a lot of smoke can ex- Cimel Sun-photometers collect data used to calculate the perience an overall cooling over time because it consistently gets aerosol optical thickness (AOT), which tells how much light less light than surrounding areas. Uneven cooling across the globe is being refl ected by particles in the air and is important for will affect future climate change, but no one is sure how, therefore climate change research. Due to the way the AOT data is nor- instruments such as the Cimel Sun-photometers have been put all malized, a cosine curve will appear in the data when some- over the world to take measurements and monitor various aspects thing is obstructing the photometer’s tubes. Using Interactive of the atmosphere, including aerosols.1,2 Data Language (IDL), the AOT data was fi t with a multi- One calculation made from the Sun-photometer data is the ple-linear regression to determine if a cosine curve was pres- aerosol optical thickness (AOT). The AOT reveals from the scat- ent. The resulting coeffi cients were plotted and a histogram tered light how thickly the aerosols have collected in a region of made to see if they can be used to fl ag problems automatically. the atmosphere, where a higher AOT corresponds to the atmo- The histogram showed the cosine coeffi cients shifting away sphere being more opaque. The instruments are placed in optimal from zero, even though it wasn’t noticeable by eye, thereby locations around the world. A problem that commonly occurs with successfully detecting a subtle change in transmission in one the Sun-photometers, however, is when an obstruction is present of the
Recommended publications
  • Generating Commit Messages from Git Diffs
    Generating Commit Messages from Git Diffs Sven van Hal Mathieu Post Kasper Wendel Delft University of Technology Delft University of Technology Delft University of Technology [email protected] [email protected] [email protected] ABSTRACT be exploited by machine learning. The hypothesis is that methods Commit messages aid developers in their understanding of a con- based on machine learning, given enough training data, are able tinuously evolving codebase. However, developers not always doc- to extract more contextual information and latent factors about ument code changes properly. Automatically generating commit the why of a change. Furthermore, Allamanis et al. [1] state that messages would relieve this burden on developers. source code is “a form of human communication [and] has similar Recently, a number of different works have demonstrated the statistical properties to natural language corpora”. Following the feasibility of using methods from neural machine translation to success of (deep) machine learning in the field of natural language generate commit messages. This work aims to reproduce a promi- processing, neural networks seem promising for automated commit nent research paper in this field, as well as attempt to improve upon message generation as well. their results by proposing a novel preprocessing technique. Jiang et al. [12] have demonstrated that generating commit mes- A reproduction of the reference neural machine translation sages with neural networks is feasible. This work aims to reproduce model was able to achieve slightly better results on the same dataset. the results from [12] on the same and a different dataset. Addition- When applying more rigorous preprocessing, however, the per- ally, efforts are made to improve upon these results by applying a formance dropped significantly.
    [Show full text]
  • Sistemas De Control De Versiones De Última Generación (DCA)
    Tema 10 - Sistemas de Control de Versiones de última generación (DCA) Antonio-M. Corbí Bellot Tema 10 - Sistemas de Control de Versiones de última generación (DCA) II HISTORIAL DE REVISIONES NÚMERO FECHA MODIFICACIONES NOMBRE Tema 10 - Sistemas de Control de Versiones de última generación (DCA) III Índice 1. ¿Qué es un Sistema de Control de Versiones (SCV)?1 2. ¿En qué consiste el control de versiones?1 3. Conceptos generales de los SCV (I) 1 4. Conceptos generales de los SCV (II) 2 5. Tipos de SCV. 2 6. Centralizados vs. Distribuidos en 90sg 2 7. ¿Qué opciones tenemos disponibles? 2 8. ¿Qué podemos hacer con un SCV? 3 9. Tipos de ramas 3 10. Formas de integrar una rama en otra (I)3 11. Formas de integrar una rama en otra (II)4 12. SCV’s con los que trabajaremos 4 13. Git (I) 5 14. Git (II) 5 15. Git (III) 5 16. Git (IV) 6 17. Git (V) 6 18. Git (VI) 7 19. Git (VII) 7 20. Git (VIII) 7 21. Git (IX) 8 22. Git (X) 8 23. Git (XI) 9 Tema 10 - Sistemas de Control de Versiones de última generación (DCA) IV 24. Git (XII) 9 25. Git (XIII) 9 26. Git (XIV) 10 27. Git (XV) 10 28. Git (XVI) 11 29. Git (XVII) 11 30. Git (XVIII) 12 31. Git (XIX) 12 32. Git. Vídeos relacionados 12 33. Mercurial (I) 12 34. Mercurial (II) 12 35. Mercurial (III) 13 36. Mercurial (IV) 13 37. Mercurial (V) 13 38. Mercurial (VI) 14 39.
    [Show full text]
  • Brno University of Technology Bulk Operation
    BRNO UNIVERSITY OF TECHNOLOGY VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ FACULTY OF INFORMATION TECHNOLOGY FAKULTA INFORMAČNÍCH TECHNOLOGIÍ DEPARTMENT OF INFORMATION SYSTEMS ÚSTAV INFORMAČNÍCH SYSTÉMŮ BULK OPERATION ORCHESTRATION IN MULTIREPO CI/CD ENVIRONMENTS HROMADNÁ ORCHESTRÁCIA V MULTIREPO CI/CD PROSTREDIACH MASTER’S THESIS DIPLOMOVÁ PRÁCE AUTHOR Bc. JAKUB VÍŠEK AUTOR PRÁCE SUPERVISOR Ing. MICHAL KOUTENSKÝ VEDOUCÍ PRÁCE BRNO 2021 Brno University of Technology Faculty of Information Technology Department of Information Systems (DIFS) Academic year 2020/2021 Master's Thesis Specification Student: Víšek Jakub, Bc. Programme: Information Technology and Artificial Intelligence Specializatio Computer Networks n: Title: Bulk Operation Orchestration in Multirepo CI/CD Environments Category: Networking Assignment: 1. Familiarize yourself with the principle of CI/CD and existing solutions. 2. Familiarize yourself with the multirepo approach to software development. 3. Analyze the shortcomings of existing CI/CD solutions in the context of multirepo development with regard to user comfort. Focus on scheduling and deploying bulk operations on multiple interdependent repositories as part of a single logical branching pipeline. 4. Propose and design a solution to these shortcomings. 5. Implement said solution. 6. Test and evaluate the solution's functionality in a production environment. Recommended literature: Humble, Jez, and David Farley. Continuous delivery. Upper Saddle River, NJ: Addison- Wesley, 2011. Forsgren, Nicole, Jez Humble, and Gene Kim. Accelerate : building and scaling high performing technology organizations. Portland, OR: IT Revolution Press, 2018. Nicolas Brousse. 2019. The issue of monorepo and polyrepo in large enterprises. In Proceedings of the Conference Companion of the 3rd International Conference on Art, Science, and Engineering of Programming (Programming '19). Association for Computing Machinery, New York, NY, USA, Article 2, 1-4.
    [Show full text]
  • Making the Most of Git and Github
    Contributing to Erlang Making the Most of Git and GitHub Tom Preston-Werner Cofounder/CTO GitHub @mojombo Quick Git Overview Git is distributed Tom PJ Chris Git is snapshot-based The Codebase 1 Snapshots have zero or more parents 1 2 Branching in Git is easy 1 2 3 4 Merging in Git is easy too 1 2 3 5 4 A branch is just a pointer to a snapshot master 1 2 3 5 4 Branches move as new snapshots are taken master 1 2 3 5 6 4 Tags are like branches that never move master 1 2 3 5 6 4 v1.0.0 Contributing to Erlang Fork, Clone, and Configure Install and Configure Git git config --global user.name "Tom Preston-Werner" git config --global user.email [email protected] Sign up on GitHub Fork github.com/erlang/otp Copy your clone URL Clone the repo locally git clone [email protected]:mojombo/otp.git replace with your username Verify the clone worked $ cd otp $ ls AUTHORS bootstrap EPLICENCE configure.in INSTALL-CROSS.md erl-build-tool-vars.sh INSTALL-WIN32.md erts INSTALL.md lib Makefile.in make View the history $ git log Add a remote for the upstream (erlang/otp) $ git remote add upstream \ git://github.com/erlang/otp.git Repositories GitHub GitHub erlang/otp mojombo/otp upstream origin Local otp Create a branch List all branches $ git branch * dev Create a branch off of “dev” and switch to it $ git checkout -b mybranch Both branches now point to the same commit dev mybranch Make Changes Each commit should: Contain a single logical change Compile cleanly Not contain any cruft Have a good commit message Review your changes $ git status $ git diff Commit your
    [Show full text]
  • Colors in Bitbucket Pull Request
    Colors In Bitbucket Pull Request Ligulate Bay blueprints his hays craving gloomily. Drearier and anaglyphic Nero license almost windingly, though Constantinos divulgating his complaints limits. Anglophilic and compartmentalized Lamar exemplified her clippings eternalised plainly or caping valorously, is Kristopher geoidal? Specifically I needed to axe at route eager to pull them a tenant ID required to hustle up. The Blue Ocean UI has a navigation bar possess the toll of its interface, Azure Repos searches the designated folders in reading order confirm, but raise some differences. Additionally for GitHub pull requests this tooltip will show assignees labels reviewers and build status. While false disables it a pull. Be objective to smell a stride, and other cases can have? Configuring project version control settings. When pulling or. This pull list is being automatically deployed with Vercel. Best practice rules to bitbucket pull harness review coverage is a vulnerability. By bitbucket request in many files in revision list. Generally speaking I rebase at lest once for every pull request I slide on GitHub It today become wildly. Disconnected from pull request commits, color coding process a remote operations. The color tags option requires all tags support. Give teams bitbucket icon now displays files from the pull request sidebar, colors in bitbucket pull request, we consider including a repo authentication failures and. Is their question about Bitbucket Cloud? Bitbucket open pull requests Bitbucket open pull requests badge bitbucketpr-rawuserrepo Bitbucket Server open pull requests Bitbucket Server open pull. Wait awhile the browser to finish rendering before scrolling. Adds syntax highlight for pull requests Double click fabric a broad to deny all occurrences.
    [Show full text]
  • Changeset-Based Topic Modeling of Software Repositories
    IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 1, NO. 1, MONTH YEAR 1 Changeset-Based Topic Modeling of Software Repositories Christopher S. Corley, Kostadin Damevski, Nicholas A. Kraft Abstract—The standard approach to applying text retrieval models to code repositories is to train models on documents representing program elements. However, code changes lead to model obsolescence and to the need to retrain the model from the latest snapshot. To address this, we previously introduced an approach that trains a model on documents representing changesets from a repository and demonstrated its feasibility for feature location. In this paper, we expand our work by investigating: a second task (developer identification), the effects of including different changeset parts in the model, the repository characteristics that affect the accuracy of our approach, and the effects of the time invariance assumption on evaluation results. Our results demonstrate that our approach is as accurate as the standard approach for projects with most changes localized to a subset of the code, but less accurate when changes are highly distributed throughout the code. Moreover, our results demonstrate that context and messages are key to the accuracy of changeset-based models and that the time invariance assumption has a statistically significant effect on evaluation results, providing overly-optimistic results. Our findings indicate that our approach is a suitable alternative to the standard approach, providing comparable accuracy while eliminating retraining costs. Index Terms—changesets; feature location; developer identification; program comprehension; mining software repositories; online topic modeling F 1 INTRODUCTION Online topic models, such as online LDA [7], natively sup- Researchers have identified numerous applications for text port the online addition of new documents, but they still retrieval (TR) models in facilitating software maintenance cannot accommodate modifications to existing documents.
    [Show full text]
  • Bitbucket Issue Pull Request
    Bitbucket Issue Pull Request Axial and follow-up Quent dolomitize her Zappa chagrin evenly or annoy uniquely, is Maximilien rheumatoid? Fusionism or streamiest, Alfredo never unionise any sandbags! Salted and Barmecidal Zechariah nod her palladium Westphalian reconsolidates and preannounces guiltlessly. The destination is the issue is old open source and bitbucket pull request will live updating in with a repo in your ability to automatically close if specified group Organizing: You can be duplicate issues, suggest the issue labels, suggest its close left open issues and ask questions on recently opened issues to stash the discussion forward. Your Bitbucket account is missing OAUTH credentials into Jira. But it is written in a way so that it is in no way limited to Jenkins. Views like a Release Hub show you sacrifice power of integrating your repository and into project manager. Sources by any new issue tracking features are two or updating in to? When there are required conditions on top of the repository administrators can perform the pull requests, changed how to pull request bitbucket. Thank you pull requests are no merge then in bitbucket issues with svn using git repositories of issue. Looking for at specific? Note before your local system admins to your pull requests for sonar stash server like user from its information and bitbucket server license key from mercurial. Explore our newest apps and recent updates. If you select multiple application links then each one will be queried until one returns issues which match the specified JQL clause. Deploying with svn using maven or restrict configuration or ssh key on them.
    [Show full text]
  • The Aliquot Constant We first Show the Following Result on the Geometric Mean for the Ordi- Nary Sum of Divisors Function
    THE ALIQUOT CONSTANT WIEB BOSMA AND BEN KANE Abstract. The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant λ when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the divisors of n less than n. Thus the geometric mean of s(n)/n, the growth factor of the function s, in the long run tends to be less than 1. This could be interpreted as probabilistic evidence that aliquot sequences tend to remain bounded. 1. Introduction This paper is concerned with the average growth of aliquot sequences. An aliquot sequence is a sequence a0, a1, a2,... of positive integers ob- tained by iteration of the sum-of-aliquot-divisors function s, which is defined for n> 1 by s(n)= d. d|n Xd<n The aliquot sequence with starting value a0 is then equal to 2 a0, a1 = s(a0), a2 = s(a1)= s (a0),... ; k we will say that the sequence terminates (at 1) if ak = s (a0)=1 for some k ≥ 0. The sequence cycles (or is said to end in a cycle) k l 0 if s (a0) = s (a0) for some k,l with 0 ≤ l<k, where s (n) = n by arXiv:0912.3660v1 [math.NT] 18 Dec 2009 definition. Note that s is related to the ordinary sum-of-divisors function σ, with σ(n)= d|n d, by s(n)= σ(n) − n for integers n> 1.
    [Show full text]
  • Mathematical Constants and Sequences
    Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) ..
    [Show full text]
  • MASON V(Irtual) Mid-Atlantic Seminar on Numbers March 27–28, 2021
    MASON V(irtual) Mid-Atlantic Seminar On Numbers March 27{28, 2021 Abstracts Amod Agashe, Florida State University A generalization of Kronecker's first limit formula with application to zeta functions of number fields The classical Kronecker's first limit formula gives the polar and constant term in the Laurent expansion of a certain two variable Eisenstein series, which in turn gives the polar and constant term in the Laurent expansion of the zeta function of a quadratic imaginary field. We will recall this formula and give its generalization to more general Eisenstein series and to zeta functions of arbitrary number fields. Max Alekseyev, George Washington University Enumeration of Payphone Permutations People's desire for privacy drives many aspects of their social behavior. One such aspect can be seen at rows of payphones, where people often pick an available payphone most distant from already occupied ones.Assuming that there are n payphones in a row and that n people pick payphones one after another as privately as possible, the resulting assignment of people to payphones defines a permutation, which we will refer to as a payphone permutation. It can be easily seen that not every permutation can be obtained this way. In the present study, we consider different variations of payphone permutations and enumerate them. Kisan Bhoi, Sambalpur University Narayana numbers as sum of two repdigits Repdigits are natural numbers formed by the repetition of a single digit. Diophantine equations involving repdigits and the terms of linear recurrence sequences have been well studied in literature. In this talk we consider Narayana's cows sequence which is a third order liner recurrence sequence originated from a herd of cows and calves problem.
    [Show full text]
  • Some Problems of Erdős on the Sum-Of-Divisors Function
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 3, Pages 1–26 (April 5, 2016) http://dx.doi.org/10.1090/btran/10 SOME PROBLEMS OF ERDOS˝ ON THE SUM-OF-DIVISORS FUNCTION PAUL POLLACK AND CARL POMERANCE For Richard Guy on his 99th birthday. May his sequence be unbounded. Abstract. Let σ(n) denote the sum of all of the positive divisors of n,and let s(n)=σ(n) − n denote the sum of the proper divisors of n. The functions σ(·)ands(·) were favorite subjects of investigation by the late Paul Erd˝os. Here we revisit three themes from Erd˝os’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n not in the range of the function s(·); these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k-sets, where a friendly σ(n) k-set is a collection of k distinct integers which share the same value of n . 1. Introduction Let σ(n) be the sum of the natural number divisors of n,sothatσ(n)= d|n d. Let s(n) be the sum of only the proper divisors of n,sothats(n)=σ(n) − n. Interest in these functions dates back to the ancient Greeks, who classified numbers as deficient, perfect,orabundant according to whether s(n) <n, s(n)=n,or s(n) >n, respectively.
    [Show full text]
  • Upper Bounds on the Sum of Principal Divisors of an Integer
    190 MATHEMATICSMAGAZINE Upper Bounds on the Sum of PrincipalDivisors of an Integer ROGER B. EGGLETON Illinois State University Normal, IL61790 [email protected] WILLIAM P. GALVIN School of Mathematical and Physical Sciences University of Newcastle Callaghan, NSW 2308 Australia A prime-poweris any integerof the form p", where p is a primeand a is a positive integer.Two prime-powersare independentif they are powers of differentprimes. The Fundamental Theorem of Arithmetic amounts to the assertion that every positive integerN is the productof a uniqueset of independentprime-powers, which we call theprincipal divisors of N. Forexample, 3 and4 arethe principaldivisors of 12, while 2, 5 and 9 are the principaldivisors of 90. The case N = 1 fits this description,using the conventionthat an emptyset has productequal to 1 (andsum equalto 0). In a recentinvited lecture, Brian Alspach noted [1]:Any odd integerN > 15 thatis not a prime-power is greater than twice the sum of its principal divisors. For instance, 21 is morethan twice 3 plus 7, and 35 is almostthree times 5 plus 7, but 15 falls just shortof twice 3 plus 5. Alspachasked for a "nice"(elegant and satisfying) proof of this observation,which he used in his lectureto provea resultabout cyclic decomposition of graphs. Respondingto the challenge,we proveAlspach's observation by a veryelementary argument.You, the reader,must be the judge of whetherour proof qualifiesas "nice." We also show how the same line of reasoningleads to several strongeryet equally elegantupper bounds on sums of principaldivisors. Perhaps surprisingly, our meth- ods will not focus on propertiesof integers.Rather, we considerproperties of finite sequencesof positivereal numbers,and use a classicalelementary inequality between the productand sum of any such sequence.But first,let us put Alspach'sobservation in its numbertheoretic context.
    [Show full text]