From Exploration to Theory-Driven Tables (And Back Again)
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Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
Hordern House Rare Books Pty
77 vICTORIA STREET • POTTS POINT • SyDNEy NSw 2011 • AUSTRAlia • TElephONE (02) 9356 4411 • fAx (02) 9357 3635 HORDERN HOUSE RARE BOOKS PTY. LTD. A.B.N. 94 193 459 772 E-MAIL: [email protected] INTERNET: www.hordern.com DIRECTORS: ANNE McCORMICK • DEREK McDONNELL HORDERN HOUSE RARE BOOKS • MANUSCRIPTS • PAINTINGS • PRINTS • RARE BOOKS • MANUSCRIPTS • PAINTINGS • PRINTS • RARE BOOKS • MANUSCRIPTS • PAINTINGS Acquisitions • October 2015 Important Works on Longitude 2. [BOARD OF LONGITUDE]. The 3. [BUREAU DES LONGITUDES]. Nautical Almanac and Astronomical Connaissance des tems, a l’usage des Ephemeris, for the Year 1818. Astronomes et des Navigateurs pour l’an X… Octavo, very good in original polished calf, faithfully rebacked. London, John Octavo, folding world map and two Murray 1815. folding tables; an attractive copy in contemporary marbled calf, gilt, red Rare copy of the Nautical Almanac for spine label. Paris, l’Imprimerie de la 1818, a fundamental inclusion in the République, Fructidor, An VII, that is shipboard library of any Admiralty- circa August 1799. sponsored voyage. The Almanac was used for reckoning the longitude at sea A handsome copy of this rare work by the lunar method, and was closely by the French Bureau des Longitudes, studied by officers of the Royal Navy. for use by naval officers for the year The continued publication of such 1802 and 1803. The volume includes a almanacs is further proof that the handsome map of the world showing invention of the chronometer, (whilst the track of a solar eclipse that revolutionary), did not completely occurred in August of that year. Much supersede the necessity for other fail- like the British equivalent, these tables 1. -
An Introduction to Pythagorean Arithmetic
AN INTRODUCTION TO PYTHAGOREAN ARITHMETIC by JASON SCOTT NICHOLSON B.Sc. (Pure Mathematics), University of Calgary, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming tc^ the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1996 © Jason S. Nicholson, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my i department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Dale //W 39, If96. DE-6 (2/88) Abstract This thesis provides a look at some aspects of Pythagorean Arithmetic. The topic is intro• duced by looking at the historical context in which the Pythagoreans nourished, that is at the arithmetic known to the ancient Egyptians and Babylonians. The view of mathematics that the Pythagoreans held is introduced via a look at the extraordinary life of Pythagoras and a description of the mystical mathematical doctrine that he taught. His disciples, the Pythagore• ans, and their school and history are briefly mentioned. Also, the lives and works of some of the authors of the main sources for Pythagorean arithmetic and thought, namely Euclid and the Neo-Pythagoreans Nicomachus of Gerasa, Theon of Smyrna, and Proclus of Lycia, are looked i at in more detail. -
Multiplication Fact Strategies Assessment Directions and Analysis
Multiplication Fact Strategies Wichita Public Schools Curriculum and Instructional Design Mathematics Revised 2014 KCCRS version Table of Contents Introduction Page Research Connections (Strategies) 3 Making Meaning for Operations 7 Assessment 9 Tools 13 Doubles 23 Fives 31 Zeroes and Ones 35 Strategy Focus Review 41 Tens 45 Nines 48 Squared Numbers 54 Strategy Focus Review 59 Double and Double Again 64 Double and One More Set 69 Half and Then Double 74 Strategy Focus Review 80 Related Equations (fact families) 82 Practice and Review 92 Wichita Public Schools 2014 2 Research Connections Where Do Fact Strategies Fit In? Adapted from Randall Charles Fact strategies are considered a crucial second phase in a three-phase program for teaching students basic math facts. The first phase is concept learning. Here, the goal is for students to understand the meanings of multiplication and division. In this phase, students focus on actions (i.e. “groups of”, “equal parts”, “building arrays”) that relate to multiplication and division concepts. An important instructional bridge that is often neglected between concept learning and memorization is the second phase, fact strategies. There are two goals in this phase. First, students need to recognize there are clusters of multiplication and division facts that relate in certain ways. Second, students need to understand those relationships. These lessons are designed to assist with the second phase of this process. If you have students that are not ready, you will need to address the first phase of concept learning. The third phase is memorization of the basic facts. Here the goal is for students to master products and quotients so they can recall them efficiently and accurately, and retain them over time. -
WHERE WAS MEAN SOLAR TIME FIRST ADOPTED? Simone Bianchi INAF-Osservatorio Astrofisico Di Arcetri, Largo E. Fermi, 5, 50125, Flor
WHERE WAS MEAN SOLAR TIME FIRST ADOPTED? Simone Bianchi INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5, 50125, Florence, Italy [email protected] Abstract: It is usually stated in the literature that Geneva was the first city to adopt mean solar time, in 1780, followed by London (or the whole of England) in 1792, Berlin in 1810 and Paris in 1816. In this short paper I will partially revise this statement, using primary references when available, and provide dates for a few other European cities. Although no exact date was found for the first public use of mean time, the primacy seems to belong to England, followed by Geneva in 1778–1779 (for horologists), Berlin in 1810, Geneva in 1821 (for public clocks), Vienna in 1823, Paris in 1826, Rome in 1847, Turin in 1849, and Milan, Bologna and Florence in 1860. Keywords: mean solar time 1 INTRODUCTION The inclination of the Earth’s axis with respect to the orbital plane and its non-uniform revolution around the Sun are reflected in the irregularity of the length of the day, when measured from two consecutive passages of the Sun on the meridian. Though known since ancient times, the uneven length of true solar days became of practical interest only after Christiaan Huygens (1629 –1695) invented the high-accuracy pendulum clock in the 1650s. For proper registration of regularly-paced clocks, it then became necessary to convert true solar time into mean solar time, obtained from the position of a fictitious mean Sun; mean solar days all having the same duration over the course of the year. -
Le Bureau Des Longitudes: Imitation Du Board of Longitude Britannique?
Le Bureau des longitudes : imitation du Board of Longitude britannique ? Martina Schiavon To cite this version: Martina Schiavon. Le Bureau des longitudes : imitation du Board of Longitude britannique ?. 2018. hal-03218044 HAL Id: hal-03218044 https://hal.univ-lorraine.fr/hal-03218044 Submitted on 5 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - ShareAlike| 4.0 International License Le Bureau des longitudes : imitation du Board of Longitude britannique ? Martina Schiavon Figure 1 - Salle de réunion du Bureau des longitudes (Source : Bureau des longitudes) Premières réflexions après la mise en ligne des procès-verbaux du Bureau des longitudes Dans son rapport sur les besoins actuels du Bureau des longitudes du 22 septembre 1920, l’astronome et mathématicien Marie-Henri Andoyer (1862-1929) revenait ainsi sur la création du Bureau : « Le nom même de “Bureau des Longitudes” est la simple traduction du nom anglais de l’établissement analogue “Board of Longitude”, chargé de publier le Nautical Almanach pour l’usage des marins et de rechercher les meilleures méthodes pour résoudre le problème fondamental de la détermination des longitudes, soit sur mer, soit à terre. -
Siméon-Denis Poisson Mathematics in the Service of Science
S IMÉ ON-D E N I S P OISSON M ATHEMATICS I N T H E S ERVICE O F S CIENCE E XHIBITION AT THE MATHEMATICS LIBRARY U NIVE RSIT Y O F I L L I N O I S A T U RBANA - C HAMPAIGN A U G U S T 2014 Exhibition on display in the Mathematics Library of the University of Illinois at Urbana-Champaign 4 August to 14 August 2014 in association with the Poisson 2014 Conference and based on SIMEON-DENIS POISSON, LES MATHEMATIQUES AU SERVICE DE LA SCIENCE an exhibition at the Mathematics and Computer Science Research Library at the Université Pierre et Marie Curie in Paris (MIR at UPMC) 19 March to 19 June 2014 Cover Illustration: Portrait of Siméon-Denis Poisson by E. Marcellot, 1804 © Collections École Polytechnique Revised edition, February 2015 Siméon-Denis Poisson. Mathematics in the Service of Science—Exhibition at the Mathematics Library UIUC (2014) SIMÉON-DENIS POISSON (1781-1840) It is not too difficult to remember the important dates in Siméon-Denis Poisson’s life. He was seventeen in 1798 when he placed first on the entrance examination for the École Polytechnique, which the Revolution had created four years earlier. His subsequent career as a “teacher-scholar” spanned the years 1800-1840. His first publications appeared in the Journal de l’École Polytechnique in 1801, and he died in 1840. Assistant Professor at the École Polytechnique in 1802, he was named Professor in 1806, and then, in 1809, became a professor at the newly created Faculty of Sciences of the Université de Paris. -
Some Mathematics
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. C Some Mathematics C.1 Finite field theory Most linear codes are expressed in the language of Galois theory Why are Galois fields an appropriate language for linear codes? First, a defi- nition and some examples. AfieldF is a set F = {0,F} such that 1. F forms an Abelian group under an addition operation ‘+’, with + 01 · 01 0 being the identity; [Abelian means all elements commute, i.e., 0 01 0 00 satisfy a + b = b + a.] 1 10 1 01 2. F forms an Abelian group under a multiplication operation ‘·’; multiplication of any element by 0 yields 0; Table C.1. Addition and 3. these operations satisfy the distributive rule (a + b) · c = a · c + b · c. multiplication tables for GF (2). For example, the real numbers form a field, with ‘+’ and ‘·’denoting ordinary addition and multiplication. + 01AB 0 01AB AGaloisfieldGF q q ( ) is a field with a finite number of elements . 1 10BA A unique Galois field exists for any q = pm,wherep is a prime number A AB 01 and m is a positive integer; there are no other finite fields. B BA10 · 01AB GF (2). The addition and multiplication tables for GF (2) are shown in ta- 0 00 0 0 ble C.1. These are the rules of addition and multiplication modulo 2. 1 01AB A AB GF (p). -
Appendix I: on the Early History of Pi
Appendix I: On the Early History of Pi Our earliest source on 7r is a problem from the Rhind mathematical papyrus, one of the few major sources we have for the mathematics of ancient Egypt. Although the material on the papyrus in its present form comes from about 1550 B.C. during the Middle Kingdom, scholars believe that the mathematics of the document originated in the Old Kingdom, which would date it perhaps to 1900 B.C. The Egyptian source does not state an explicit value for 7r, but tells, instead, how to compute the area of a circle as the square of eight-ninths of the diameter. (One is reminded of the 'classical' problem of squaring the circle.) Quite different is the passage in I Kings 7, 23 (repeated in II Chronicles 4, 21), which clearly implies that one should multiply the diameter of a circle by 3 to find its circumference. It is worth noting, however, that the Egyptian and the Biblical texts talk about what are, a priori, different concepts: the diameter being regarded as known in both cases, the first text gives a factor used to produce the area of the circle and the other gives (by implication) a factor used to produce the circumference.1 Also from the early second millenium are Babylonian texts from Susa. In the mathematical cuneiform texts the standard factor for multiplying the diameter to produce the circumference is 3, but, according to some interpre tations, the Susa texts give a value of 31.2 Another source of ancient values of 7r is the class of Indian works known as the Sulba Sfitras, of which the four most important, and oldest are Baudhayana, .Apastamba, Katyayana and Manava (resp. -
A Reconstruction of the Mathematical Tables Project's Table Of
A reconstruction of the Mathematical Tables Project’s table of natural logarithms (4 volumes, 1941) Denis Roegel To cite this version: Denis Roegel. A reconstruction of the Mathematical Tables Project’s table of natural logarithms (4 volumes, 1941). [Research Report] LORIA, UMR 7503, Université de Lorraine, CNRS, Vandoeuvre- lès-Nancy. 2017. <hal-01615064> HAL Id: hal-01615064 https://hal.inria.fr/hal-01615064 Submitted on 13 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A reconstruction of the Mathematical Tables Project’s table of natural logarithms (1941) Denis Roegel 12 October 2017 This document is part of the LOCOMAT project: http://locomat.loria.fr “[f]or a few brief years, [the Mathematical Tables Project] was the largest computing organization in the world, and it prepared the way for the modern computing era.” D. Grier, 1997 [29] “Blanch, more than any other individual, represents that transition from hand calculation to computing machines.” D. Grier, 1997 [29] “[Gertrude Blanch] was virtually the backbone of the project, the hardest and most conscientious worker, and the one most responsible for the amount and high quality of the project’s output.” H. -
Error Correcting Codes
9 error correcting codes A main idea of this sequence of notes is that accounting is a legitimate field of academic study independent of its vocational implications and its obvious impor- tance in the economic environment. For example, the study of accounting pro- motes careful and disciplined thinking important to any intellectual pursuit. Also, studying how accounting works illuminates other fields, both academic and ap- plied. In this chapter we begin to connect the study of accounting with the study of codes. Coding theory, while theoretically rich, is also important in its applica- tions, especially with the pervasiveness of computerized information transfer and storage. Information integrity is inherently of interest to accountants, so exploring coding can be justified for both academic and applied reasons. 9.1 kinds of codes We will study three kinds of codes: error detecting, error correcting, and secret codes. The first two increase the reliability of message transmission. The third, secret codes, are designed to ensure that messages are available only to authorized users, and can’t be read by the bad guys. The sequence of events for all three types of codes is as follows. 1. determine the message to transmit. For our purposes, we will treat messages as vectors. 2. The message is encoded. The result is termed a codeword, or cyphertext, also a vector. 180 9. error correcting codes 3. The codeword is transmitted through a channel. The channel might inject noise into the codeword, or it might be vulnerable to eavesdropping. 4. The received vector is decoded. If the received vector is no longer the trans- mitted codeword, the decoding process may detect, or even correct, the er- ror. -
Document on Foundation of Bureau Des Longitudes (Pdf)
Brief History of the Bureau des Longitudes After hearing a report read by the abbé Grégoire, the Bureau des Longitudes was created by a law of the National Convention of the 7 messidor year III (June 25 1795). The purpose was to reassume "the mastery of the seas from the English", through the improvement of the determination of longitudes at sea. Charged with the compilation of Knowledge of the Times and perfecting the astronomical tables, he had responsibility for the Paris observatory, the observatory of the Military school and all the astronomy instruments that belonged to the Nation. The ten founding members had been: Lagrange, Laplace, Lalande, Delambre, Méchain, Cassini, Bougainville, Borda, Buache and Caroché. It was charged, by the decree of January 30 1854 with a larger mission bringing to it, in addition realization of the ephemerides by its "Calculations Service" created in 1802, to organize several big scientific expeditions: geodetic measurements, observation of solar eclipses, observation of the passage of Venus in front of the Sun, works that were published in the Annals of the Bureau des Longitudes. It participated equally in the foundation of several scientific organisations such as the International Office of Time (1919), the Group of Researches of Spatial Géodésie (1971) and the International Earth Rotation Service (1988). Law of the year III and Regulations FOUNDATION OF THE OFFICE OF THE LONGITUDES Report made to the National Convention in its meeting of the 7 messidor year III (June 25 1795), by the Representative of the People GRÉGOIRE, on the establishment of the Office of the Longitudes.