DOCSLIB.ORG

TME Volume 5, Numbers 2 and 3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
TME Volume 5, Numbers 2 and 3 The Mathematics Enthusiast Volume 5 Number 2 Numbers 2 & 3 Article 27 7-2008 TME Volume 5, Numbers 2 and 3 Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation (2008) "TME Volume 5, Numbers 2 and 3," The Mathematics Enthusiast: Vol. 5 : No. 2 , Article 27. Available at: https://scholarworks.umt.edu/tme/vol5/iss2/27 This Full Volume is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. The Montana Mathematics Enthusiast ISSN 1551-3440 VOL. 5, NOS.2&3, JULY 2008, pp.167-462 Editor-in-Chief Bharath Sriraman, The University of Montana Associate Editors: Lyn D. English, Queensland University of Technology, Australia Claus Michelsen, University of Southern Denmark, Denmark Brian Greer, Portland State University, USA Luis Moreno-Armella, University of Massachusetts-Dartmouth International Editorial Advisory Board Miriam Amit, Ben-Gurion University of the Negev, Israel. Ziya Argun, Gazi University, Turkey. Ahmet Arikan, Gazi University, Turkey. Astrid Beckmann, University of Education, Schwäbisch Gmünd, Germany. John Berry, University of Plymouth,UK. Morten Blomhøj, Roskilde University, Denmark. Robert Carson, Montana State University- Bozeman, USA. Mohan Chinnappan, University of Wollongong, Australia. Constantinos Christou, University of Cyprus, Cyprus. Bettina Dahl Søndergaard, University of Aarhus, Denmark. Helen Doerr, Syracuse University, USA. Ted Eisenberg, Ben-Gurion University of the Negev, Israel. Paul Ernest, University of Exeter, UK. Viktor Freiman, Université de Moncton, Canada. Fulvia Furinghetti, Università di Genova, Italy Eric Gutstein, University of Illinois-Chicago, USA. Marja van den Heuvel-Panhuizen, University of Utrecht, The Netherlands. Gabriele Kaiser, University of Hamburg, Germany. Libby Knott,The University of Montana, USA. Tinne Hoff Kjeldsen, Roskilde University, Denmark. Jean-Baptiste Lagrange, IUFM-Reims, France. Stephen Lerman, London South Bank University, UK. Frank Lester, Indiana University, USA. Richard Lesh, Indiana University, USA. Nicholas Mousoulides, University of Cyprus, Cyprus. Swapna Mukhopadhyay, Portland State University, USA. Norma Presmeg, Illinois State University, USA. Gudbjorg Palsdottir,Iceland University of Education, Iceland. João Pedro da Ponte, University of Lisbon, Portugal Michael Pyryt1, University of Calgary, Canada. Demetra Pitta Pantazi, University of Cyprus, Cyprus. Linda Sheffield, Northern Kentucky University, USA. Olof Bjorg Steinthorsdottir, University of North Carolina- Chapel Hill, USA. Günter Törner, University of Duisburg-Essen, Germany. Renuka Vithal, University of KwaZulu-Natal, South Africa. Dirk Wessels, UNISA, South Africa. Nurit Zehavi, The Weizmann Institute of Science, Rehovot, Israel. 1 Michael Pyryt passed away in an unfortunate manner on Jan 15, 2008. The journal expresses its condolences to the Pyryt family in Calgary, and regrets the loss of an excellent statistician, scholar and gentleman. The Montana Mathematics Enthusiast is an eclectic internationally circulated peer reviewed journal which focuses on mathematics content, mathematics education research, innovation, interdisciplinary issues and pedagogy. The journal is published by Information Age Publishing and the electronic version is hosted jointly by IAP and the Department of Mathematical Sciences- The University of Montana, on behalf of MCTM. Articles appearing in the journal address issues related to mathematical thinking, teaching and learning at all levels. The focus includes specific mathematics content and advances in that area accessible to readers, as well as political, social and cultural issues related to mathematics education. Journal articles cover a wide spectrum of topics such as mathematics content (including advanced mathematics), educational studies related to mathematics, and reports of innovative pedagogical practices with the hope of stimulating dialogue between pre-service and practicing teachers, university educators and mathematicians. The journal is interested in research based articles as well as historical, philosophical, political, cross-cultural and systems perspectives on mathematics content, its teaching and learning. The journal also includes a monograph series on special topics of interest to the community of readers The journal is accessed from 110+ countries and its readers include students of mathematics, future and practicing teachers, mathematicians, cognitive psychologists, critical theorists, mathematics/science educators, historians and philosophers of mathematics and science as well as those who pursue mathematics recreationally. The 40 member editorial board reflects this diversity. The journal exists to create a forum for argumentative and critical positions on mathematics education, and especially welcomes articles which challenge commonly held assumptions about the nature and purpose of mathematics and mathematics education. Reactions or commentaries on previously published articles are welcomed. Manuscripts are to be submitted in electronic format to the editor in APA style. The typical time period from submission to publication is 8-11 months. Please visit the journal website at http://www.montanamath.org/TMME or http://www.math.umt.edu/TMME/ Indexing Information Australian Education Index (For Australian authors); EBSCO Products (Academic Search Complete); EDNA; Cabell’s Directory of Publishing Opportunities in Educational Curriculum and Methods (2009-) Directory of Open Access Journals (DOAJ); PsycINFO (the APA Index); MathDI/MathEDUC (FiZ Karlsruhe); Journals in Higher Education (JIHE); Ulrich’s Periodicals Directory; Zentralblatt MATH THE MONTANA MATHEMATICS ENTHUSIAST ISSN 1551-3440 Vol.5, Nos.2&3, July 2008, pp.167-462 TABLE OF CONTENTS Editorial Information 0. Bharath Sriraman EDITORIAL: 2 to 3- An omnibus of features & Policy issues in mathematics education.....pp.167-168 FEATURE ARTICLES 1. The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem Jeff Babb & James Currie (Canada)................................................................................................pp.169-184 2. History of Mathematics in Mathematics Education: a Saussurean Perspective Michael Fried (Israel)........................................................................................................................pp.185-198 3. Comparison of Geometric Figures Spyros Glenis (Greece).....................................................................................................................pp.199-214 4. “Obeying a rule”: Ludwig Wittgenstein and the foundations of Set Theory Giorgio T. Bagni (Italy)....................................................................................................................pp.215-222 5. How can science history contribute to the development of new proposals in the teaching of the notion of derivatives? Arnaud Mayrargue (France).............................................................................................................pp.223-230 6. Incoherence of a concept image and erroneous conclusions in the case of differentiability Antti Viholainen (Finland)...............................................................................................................pp.231-248 7. Chess and problem solving involving patterns Dores Ferreira & Pedro Palhares (Portugal)................................................................................pp.249-256 8. Associative Operations on a Three-Element Set Friðrik Diego & Kristín Halla Jónsdóttir (Iceland)......................................................................pp.257-268 i 9. A Case Study Using Soft Systems Methodology in the Evolution of a Mathematics Module Jon Warwick (UK).............................................................................................................................pp.269-290 10. Mathematics and the World: What do Teachers Recognize as Mathematics in Real World Practice? Barbara Garii & Lillian Okumu (New York, USA)....................................................................pp.291-304 11. Mathematics for Middle School Teachers: Choices, Successes, and Challenges Linda Martin & Kristin Umland (New Mexico, USA)................................................................pp.305-314 12. Inverses – why we teach and why we need talk more about it more often! Woong Lim (Texas, USA)................................................................................................................pp.315-326 13. Magic Math Cards Steve Humble (UK)..........................................................................................................................pp.327-336 MONTANA FEATURE ARTICLES History of Logarithms and Insurance Mathematics 14. Chopping Logs: A Look at the History and Uses of Logarithms Rafael Villarreal-Calderon (Missoula, Montana, USA)................................................................pp.337-344 15. Early Insurance Mechanisms and Their Mathematical Foundations Amy Minto (Missoula, Montana, USA).........................................................................................pp.345-356 DIALOGUE ON NO CHILD LEFT BEHIND1 16. Main Points of Objections to the “No Child Left Behind” Mike O’Lear (Missoula,
Load more

Recommended publications
  • Fermat, Leibniz, Euler, and the Gang: the True History of the Concepts Of
    FERMAT, LEIBNIZ, EULER, AND THE GANG: THE TRUE HISTORY OF THE CONCEPTS OF LIMIT AND SHADOW TIZIANA BASCELLI, EMANUELE BOTTAZZI, FREDERIK HERZBERG, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TAHL NOWIK, DAVID SHERRY, AND STEVEN SHNIDER Abstract. Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding “negligible” terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern the- ories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions. Contents 1. Introduction 2 2. Methodological remarks 4 2.1. A-track and B-track 5 2.2. Formal epistemology: Easwaran on hyperreals 6 2.3. Zermelo–Fraenkel axioms and the Feferman–Levy model 8 2.4. Skolem integers and Robinson integers 9 2.5. Williamson, complexity, and other arguments 10 2.6. Infinity and infinitesimal: let both pretty severely alone 13 3. Fermat’s adequality 13 3.1. Summary of Fermat’s algorithm 14 arXiv:1407.0233v1 [math.HO] 1 Jul 2014 3.2. Tangent line and convexity of parabola 15 3.3. Fermat, Galileo, and Wallis 17 4. Leibniz’s Transcendental law of homogeneity 18 4.1. When are quantities equal? 19 4.2. Product rule 20 5. Euler’s Principle of Cancellation 20 6. What did Cauchy mean by “limit”? 22 6.1. Cauchy on Leibniz 23 6.2. Cauchy on continuity 23 7. Modern formalisations: a case study 25 8. A combinatorial approach to decreasing rearrangements 26 9.
    [Show full text]
  • Herman Heine Goldstine
    Herman Heine Goldstine Born September 13, 1913, Chicago, Ill.; Army representative to the ENIAC Project, who later worked with John von Neumann on the logical design of the JAS computer which became the prototype for many early computers-ILLIAC, JOHNNIAC, MANIAC author of The Computer from Pascal to von Neumann, one of the earliest textbooks on the history of computing. Education: BS, mathematics, University of Chicago, 1933; MS, mathematics, University of Chicago, 1934; PhD, mathematics, University of Chicago, 1936. Professional Experience: University of Chicago: research assistant, 1936-1937, instructor, 1937-1939; assistant professor, University of Michigan, 1939-1941; US Army, Ballistic Research Laboratory, Aberdeen, Md., 1941-1946; Institute for Advanced Study, Princeton University, 1946-1957; IBM: director, Mathematics Sciences Department, 1958-1965, IBM fellow, 1969. Honors and Awards: IEEE Computer Society Pioneer Award, 1980; National Medal of Science, 1985; member, Information Processing Hall of Fame, Infornart, Dallas, Texas, 1985. Herman H. Goldstine began his scientific career as a mathematician and had a life-long interest in the interaction of mathematical ideas and technology. He received his PhD in mathematics from the University of Chicago in 1936 and was an assistant professor at the University of Michigan when he entered the Army in 1941. After participating in the development of the first electronic computer (ENIAC), he left the Army in 1945, and from 1946 to 1957 he was a member of the Institute for Advanced Study (IAS), where he collaborated with John von Neumann in a series of scientific papers on subjects related to their work on the Institute computer. In 1958 he joined IBM Corporation as a member of the research planning staff.
    [Show full text]
  • Connes on the Role of Hyperreals in Mathematics
    Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V.
    [Show full text]
  • Platform Systems Vs. Step Processes—The Value of Options and the Power of Modularity by Carliss Y
    © Carliss Y. Baldwin Comments welcome. Design Rules, Volume 2: How Technology Shapes Organizations Chapter 13 Platform Systems vs. Step Processes—The Value of Options and the Power of Modularity By Carliss Y. Baldwin Note to Readers: This is a draft of Chapter 13 of Design Rules, Volume 2: How Technology Shapes Organizations. It builds on prior chapters, but I believe it is possible to read this chapter on a stand-alone basis. The chapter may be cited as: Baldwin, C. Y. (2019) “Platform Systems vs. Step Processes—The Value of Options and the Power of Modularity,” HBS Working Paper (January 2019). I would be most grateful for your comments on any aspect of this chapter! Thank you in advance, Carliss. Abstract The purpose of this chapter is to contrast the value structure of platform systems (e.g. a computer) with step processes (e.g. an assembly line). I first review the basic technical architecture of computers and argue that every computer is inherently a platform for performing computations as dictated by their programs. I state and prove five propositions about platform systems, which stand in contrast to the propositions derived for step processes in Chapter 8. The propositions suggest that platform systems and step processes call for different forms of organization. Specifically, step processes reward technical integration, unified governance, risk aversion, and the use of direct authority, while platform systems reward modularity, distributed governance, risk taking, and autonomous decision-making. Despite these differences, treating platform systems and step processes as mutually exclusive architectures sets up a false dichotomy. Creating any good requires carrying out a technical recipe, i.e., performing a series of steps.
    [Show full text]
  • Math 3010 § 1. Treibergs Final Exam Name: Practice Problems March 30
    Math 3010 x 1. Final Exam Name: Practice Problems Treibergs March 30, 2018 Here are some problems soluble by methods encountered in the course. I have tried to select problems ranging over the topics we've encountered. Admittedly, they were chosen because they're fascinating to me. As such, they may have solutions that are longer than the questions you might expect on an exam. But some of them are samples of homework problems. Here are a few of my references. References. • Carl Boyer, A History of the Calculus and its Conceptual Development, Dover Publ. Inc., New York, 1959; orig. publ. The Concepts of the Calculus, A Critical and Historical Discussion of the Derivative and the Integral, Hafner Publ. Co. Inc., 1949. • Carl Boyer, A History of Mathematics, Princeton University Press, Princeton 1985 • Lucas Bunt, Phillip Jones, Jack Bedient, The Historical Roots of Elementary mathematics, Dover, Mineola 1988; orig. publ. Prentice- Hall, Englewood Cliffs 1976 • David Burton, The History of Mathmatics, An Introduction, 7th ed., McGraw Hill, New York 2011 • Ronald Calinger, Classics of Mathematics, Prentice-Hall, Englewood Cliffs 1995; orig. publ. Moore Publ. Co. Inc., 1982 • Victor Katz, A History of Mathematics An introduction, 3rd. ed., Addison-Wesley, Boston 2009 • Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York 1972 • John Stillwell, Mathematics and its History, 3rd ed., Springer, New York, 2010. • Dirk Struik, A Concise History of Mathematics, Dover, New York 1967 • Harry N. Wright, First Course in Theory of Numbers, Dover, New York, 1971; orig. publ. John Wiley & Sons, Inc., New York, 1939.
    [Show full text]
  • Mathematical Conquerors, Unguru Polarity, and the Task of History
    Journal of Humanistic Mathematics Volume 10 | Issue 1 January 2020 Mathematical Conquerors, Unguru Polarity, and the Task of History Mikhail Katz Bar-Ilan University Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Mathematics Commons Recommended Citation Katz, M. "Mathematical Conquerors, Unguru Polarity, and the Task of History," Journal of Humanistic Mathematics, Volume 10 Issue 1 (January 2020), pages 475-515. DOI: 10.5642/jhummath.202001.27 . Available at: https://scholarship.claremont.edu/jhm/vol10/iss1/27 ©2020 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Mathematical Conquerors, Unguru Polarity, and the Task of History Mikhail G. Katz Department of Mathematics, Bar Ilan University [email protected] Synopsis I compare several approaches to the history of mathematics recently proposed by Blåsjö, Fraser–Schroter, Fried, and others. I argue that tools from both mathe- matics and history are essential for a meaningful history of the discipline. In an extension of the Unguru–Weil controversy over the concept of geometric algebra, Michael Fried presents a case against both André Weil the “privileged ob- server” and Pierre de Fermat the “mathematical conqueror.” Here I analyze Fried’s version of Unguru’s alleged polarity between a historian’s and a mathematician’s history.
    [Show full text]
  • Scientific Workplace· • Mathematical Word Processing • LATEX Typesetting Scientific Word· • Computer Algebra
    Scientific WorkPlace· • Mathematical Word Processing • LATEX Typesetting Scientific Word· • Computer Algebra (-l +lr,:znt:,-1 + 2r) ,..,_' '"""""Ke~r~UrN- r o~ r PooiliorK 1.931'J1 Po6'lf ·1.:1l26!.1 Pod:iDnZ 3.881()2 UfW'IICI(JI)( -2.801~ ""'"""U!NecteoZ l!l!iS'11 v~ 0.7815399 Animated plots ln spherical coordln1tes > To make an anlm.ted plot In spherical coordinates 1. Type an expression In thr.. variables . 2 WMh the Insertion poilt In the expression, choose Plot 3D The next exampfe shows a sphere that grows ftom radius 1 to .. Plot 3D Animated + Spherical The Gold Standard for Mathematical Publishing Scientific WorkPlace and Scientific Word Version 5.5 make writing, sharing, and doing mathematics easier. You compose and edit your documents directly on the screen, without having to think in a programming language. A click of a button allows you to typeset your documents in LAT£X. You choose to print with or without LATEX typesetting, or publish on the web. Scientific WorkPlace and Scientific Word enable both professionals and support staff to produce stunning books and articles. Also, the integrated computer algebra system in Scientific WorkPlace enables you to solve and plot equations, animate 20 and 30 plots, rotate, move, and fly through 3D plots, create 3D implicit plots, and more. MuPAD' Pro MuPAD Pro is an integrated and open mathematical problem­ solving environment for symbolic and numeric computing. Visit our website for details. cK.ichan SOFTWARE , I NC. Visit our website for free trial versions of all our products. www.mackichan.com/notices • Email: info@mac kichan.com • Toll free: 877-724-9673 It@\ A I M S \W ELEGRONIC EDITORIAL BOARD http://www.math.psu.edu/era/ Managing Editors: This electronic-only journal publishes research announcements (up to about 10 Keith Burns journal pages) of significant advances in all branches of mathematics.
    [Show full text]
  • Oral History Interview with Herman Goldstine
    An Interview with HERMAN GOLDSTINE OH 18 Conducted by Nancy Stern on 11 August 1980 Charles Babbage Institute The Center for the History of Information Processing University of Minnesota 1 Herman Goldstine Interview 11 August 1980 Abstract Goldstine, associate director of the Institute for Advanced Study (IAS) computer project from 1945 to 1956, discusses his role in the project. He describes the acquisition of funding from the Office of Naval Research, the hiring of staff, and his relationship with John von Neumann. Goldstine explains that von Neumann was responsible for convincing the Institute to sponsor the computer project. Goldstine praises von Neumann's contributions, among which he counts the first logical design of a computer and the concept of stored programming. Goldstine turns next to the relations between the project and one of its funders, the Atomic Energy Commission. He points out the conflict of interest of IAS director Robert Oppenheimer, who chaired the AEC General Advisory Committee, and von Neumann who sat on this committee, when other AEC officials discontinued funding for the project. Goldstine also recounts the problems that arose during the project over patent rights and their resolution. Goldstine concludes by discussing the many visitors to the project and the many computers (Whirlwind, ILLIAC, JOHNNIAC, IBM 70l) modeled after the IAS computer. 2 HERMAN GOLDSTINE INTERVIEW DATE: 11 August 1980 INTERVIEWER: Nancy Stern LOCATION: Princeton, NJ STERN: This is an interview with Herman Goldstine in his home; August 11, 1980. What I'd like to talk about, Herman, for the most part today, is your work with von Neumann at the Institute; that is, the computer project at the Institute; because we've spoken in the past about the Moore School work.
    [Show full text]
  • History of ENIAC
    A Short History of the Second American Revolution by Dilys Winegrad and Atsushi Akera (1) Today, the northeast corner of the old Moore School building at the University of Pennsylvania houses a bank of advanced computing workstations maintained by the professional staff of the Computing and Educational Technology Service of Penn's School of Engineering and Applied Science. There, fifty years ago, in a larger room with drab- colored walls and open rafters, stood the first general purpose electronic computer, the Electronic Numerical Integrator And Computer, or ENIAC. It spanned 150 feet in width with twenty banks of flashing lights indicating the results of its computations. ENIAC could add 5,000 numbers or do fourteen 10-digit multiplications in a second-- dead slow by present-day standards, but fast compared with the same task performed on a hand calculator. The fastest mechanical relay computers being operated experimentally at Harvard, Bell Laboratories, and elsewhere could do no more than 15 to 50 additions per second, a full two orders of magnitude slower. By showing that electronic computing circuitry could actually work, ENIAC paved the way for the modern computing industry that stands as its great legacy. ENIAC was by no means the first computer. In 1839, an Englishman Charles Babbage designed and developed the first true mechanical digital computer, which he described as a "difference engine," for solving mathematical problems including simple differential equations. He was assisted in his work by a woman mathematician, Ada Countess Lovelace, a member of the aristocracy and the daughter of Lord Byron. They worked out the mathematics of mechanical computation, which, in turn, led Babbage to design the more ambitious analytical engine.
    [Show full text]
  • Small Oscillations of the Pendulum, Euler's Method, and Adequality
    SMALL OSCILLATIONS OF THE PENDULUM, EULER’S METHOD, AND ADEQUALITY VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, AND TAHL NOWIK Abstract. Small oscillations evolved a great deal from Klein to Robinson. We propose a concept of solution of differential equa- tion based on Euler’s method with infinitesimal mesh, with well- posedness based on a relation of adequality following Fermat and Leibniz. The result is that the period of infinitesimal oscillations is independent of their amplitude. Keywords: harmonic motion; infinitesimal; pendulum; small os- cillations Contents 1. Small oscillations of a pendulum 1 2. Vector fields, walks, and integral curves 3 3. Adequality 4 4. Infinitesimal oscillations 5 5. Adjusting linear prevector field 6 6. Conclusion 7 Acknowledgments 7 References 7 arXiv:1604.06663v1 [math.HO] 13 Apr 2016 1. Small oscillations of a pendulum The breakdown of infinite divisibility at quantum scales makes irrel- evant the mathematical definitions of derivatives and integrals in terms ∆y of limits as x tends to zero. Rather, quotients like ∆x need to be taken in a certain range, or level. The work [Nowik & Katz 2015] developed a general framework for differential geometry at level λ, where λ is an infinitesimal but the formalism is a better match for a situation where infinite divisibility fails and a scale for calculations needs to be fixed ac- cordingly. In this paper we implement such an approach to give a first rigorous account “at level λ” for small oscillations of the pendulum. 1 2 VLADIMIR KANOVEI, K. KATZ, MIKHAIL KATZ, AND TAHL NOWIK In his 1908 book Elementary Mathematics from an Advanced Stand- point, Felix Klein advocated the introduction of calculus into the high- school curriculum.
    [Show full text]
  • The 1999 Notices Index, Volume 46, Number 11
    index.qxp 10/15/99 11:17 AM Page 1480 The 1999 Notices Index Index Section Page Number The AMS 1480 Meetings Information 1489 Announcements 1481 Memorial Articles 1490 Authors of Articles 1482 New Publications Offered by the AMS 1490 Reference and Book List 1420 Officers and Committee Members 1490 Deaths of Members of the Society 1483 Opinion 1490 Education 1484 Opportunities 1490 Feature Articles 1484 Prizes and Awards 1491 Forum 1485 Articles on the Profession 1492 Letters to the Editor 1486 Reviews 1492 Mathematicians 1487 Surveys 1493 Mathematics Articles 1489 Tables of Contents 1493 Mathematics History 1489 About the Cover, 26, 245, 318, 420, 534, 658, 781, 918, AMS Meeting in August 2000, 355 1073, 1208, 1367 AMS Menger Awards at the 1999 Intel-International Science and Engineering Fair, 912 Acknowledgment of Contributions, 947 AMS Officers and Committee Members, 1271 AMS Participates in Capitol Hill Exhibition, 918 The AMS AMS Participates in Faculty Preparation Program, 1072 1998 Election Results, 266 AMS Short Courses, 1171 1998 Report of the Committee on Meetings and Confer- AMS Standard Cover Sheet, 60, 270, 366, 698, 972, 1102, ences, 810 1282, 1428 1999 AMS Centennial Fellowships Awarded, 684 (The) Birth of an AMS Meeting, 1029 1999 AMS Election, 58, 267 Bylaws of the American Mathematical Society, 1252 1999 AMS-MAA-SIAM Morgan Prize, 469 Call for Proposals for 2001 Conferences, 1177, 1331 1999 Arnold Ross Lectures, 692 Conference Announced (Summer Mathematics Experience: 1999 Bôcher Prize, 463 A Working Conference on Summer
    [Show full text]
  • Document Mune
    DOCUMENT MUNE ID 175 699 SE 028 687 AUTHOR Schaaf, William L., Ed. TITLE Reprint Series: Memorable Personalities in Mathematics - Twentieth Century. RS-12. INSTITUTION Stanford Gniv.e Calif. School Mathematics Study Group. 2 SPONS AGENCT National Science Foundation, Washington, D.C. 1 PUB DATE 69 NOTE 54p.: For related documents, see SE 028 676-690 EDRS PRICE 81101/PC03 Plus Postage. DESCRIPTORS Curriculum: *Enrichsent: *History: *Instruction: *Mathematicians: Mathematics Education: *Modern Mathematics: Secondary Education: *Secondary School Sathematics: Supplementary Reading Materials IDENTIFIERS *School Mathematics study Group ABSTRACT This is one in a series of SHSG supplementary and enrichment pamphlets for high school students. This series makes available espcsitory articles which appeared in a variety of mathematical periodials. Topics covered include:(1) Srinivasa Ramanuian:12) Minkomski: (3) Stefan Banach:(4) Alfred North Whitehead: (5) Waclaw Sierptnski: and (6)J. von Neumann. (MP) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. *********************************************************************** OF NEM TN U S DEPARTMENT "PERMISSION EDUCATION II %WELFARE TO REPROnUCE OF MATERIAL THIS NATIONAL INSTITUTE HAS SEEN EDUCATION GRANTEDBY wc 0Oi ,,V11. NTHt.',Iii t iFf I ROF' all( F 0 f IA. T v A WI (I 1 Ttit P1I4S0 1.41114')WGAN,ZTIf)N01414014 AIHF0,tfvf1,14 Ts 01 ,11-A OF: Ofolv4ION.y STAll 0 04.3 NOT NI-I E55A1,0)V REPRE ,Ak NAFaINA, INSIF011 Of SF Fe 1)1. F P01,I Ot,t AT ION P00,0T,0IN OkI TO THE EDUCATIONAL INFORMATION RESOURCES CENTER(ERIC)." Financial support for the School Mathematics Study Group bas been provided by the National Science Foundation.
    [Show full text]