Numbers : Their Occult Power and Mystic Virtues
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Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Simplicius and Avicenna on the Nature of Body
Simplicius and Avicenna on the Nature of Body Abraham D. Stone August 18, 1999 1 Introduction Ibn S¯ına, known to the Latin West as Avicenna, was a medieval Aristotelian— one of the greatest of all medieval Aristotelians. He lived in Persia from 980 to 1037, and wrote mostly in Arabic. Simplicius of Cilicia was a sixth cen- tury Neoplatonist; he is known mostly for his commentaries on Aristotle. Both of these men were, broadly speaking, part of the same philosophical tradition: the tradition of Neoplatonic or Neoplatonizing Aristotelianism. There is probably no direct historical connection between them, however, and anyway I will not try to demonstrate one. In this paper I will examine their closely related, but ultimately quite different, accounts of corporeity— of what it is to be a body—and in particular of the essential relationship between corporeity and materiality.1 The problem that both Simplicius and Avicenna face in this respect is as follows. There is a certain genus of substances which forms the subject matter of the science of physics. I will refer to the members of this genus as the physical substances. On the one hand, all and only these physical substances 1A longer and more technical version of this paper will appear, under the title “Simpli- cius and Avicenna on the Essential Corporeity of Material Substance,” in R. Wisnovsky, ed., Aspects of Avicenna (= Princeton Papers: Interdisciplinary Journal of Middle Eastern Studies, vol. 9, no. 2) (Princeton: Markus Wiener, 2000). I want to emphasize at the outset that this paper is about Simplicius and Avicenna, not Aristotle. -
Std XII Computer Science
J.H. TARAPORE SCHOOL WORKSHEET NO- 5 STD XII :A/B/C COMPUTER SCIENCE (2020-2021) Topic : Java Practical Programs General Instructions: • Give the program name as “yournameQ1” eg : “SamikshaQ1,Samiksha Q2” • Execute the programs in your laptop. • Save the program and output together in a Microsoft Word File. • Submit the pendrive on the first day of the school reopening. Question 1. Write a program to check whether the given number is a Prime Adam Number or Not. Prime Number:A number which has only two factors,i.e 1 and the number itself. Adam Number : The square of a number and the square of its reverse are reverse to each other.Example,if n=31 and reverse of n=13,then 312=961 and 132= 169,which is reverse of 961.Thus ,31 is an Adam Number. So, 31 is a Prime Adam Number. Question 2. Write a program to check whether the given number is a Evil Number or Not. An Evil number is a positive whole number which has even number of 1’s in its binary equivalent. Example: Binary equivalent of 9 is 1001, which contains even number of 1’s. Thus, 9 is an Evil Number. A few Evil numbers are 3, 5, 6, 9.... Question 3. Write a program to check whether the given number is a Smith Number or Not. A Smith Number is a composite number,the sum of the digits of a number is equal to the sum of the digits of its prime factors (excluding 1). Example: 666 Prime factors of 666 : 2,3,3 and 37. -
The Fragments of the Poem of Parmenides
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by D-Scholarship@Pitt RESTORING PARMENIDES’ POEM: ESSAYS TOWARD A NEW ARRANGEMENT OF THE FRAGMENTS BASED ON A REASSESSMENT OF THE ORIGINAL SOURCES by Christopher John Kurfess B.A., St. John’s College, 1995 M.A., St. John’s College, 1996 M.A., University of Hawai‘i at Mānoa, 2000 Submitted to the Graduate Faculty of The Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2012 UNVERSITY OF PITTSBURGH The Dietrich School of Arts and Sciences This dissertation was presented by Christopher J. Kurfess It was defended on November 8, 2012 and approved by Dr. Andrew M. Miller, Professor, Department of Classics Dr. John Poulakos, Associate Professor, Department of Communication Dr. Mae J. Smethurst, Professor, Department of Classics Dissertation Supervisor: Dr. Edwin D. Floyd, Professor, Department of Classics ii Copyright © by Christopher J. Kurfess 2012 iii RESTORING PARMENIDES’ POEM Christopher J. Kurfess, Ph.D. University of Pittsburgh, 2012 The history of philosophy proper, claimed Hegel, began with the poem of the Presocratic Greek philosopher Parmenides. Today, that poem is extant only in fragmentary form, the various fragments surviving as quotations, translations or paraphrases in the works of better-preserved authors of antiquity. These range from Plato, writing within a century after Parmenides’ death, to the sixth-century C.E. commentator Simplicius of Cilicia, the latest figure known to have had access to the complete poem. Since the Renaissance, students of Parmenides have relied on collections of fragments compiled by classical scholars, and since the turn of the twentieth century, Hermann Diels’ Die Fragmente der Vorsokratiker, through a number of editions, has remained the standard collection for Presocratic material generally and for the arrangement of Parmenides’ fragments in particular. -
Eureka Issue 61
Eureka 61 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Anja Komatar © The Archimedeans (see page 94 for details) Do not copy or reprint any parts without permission. October 2011 Editorial Eureka Reinvented… efore reading any part of this issue of Eureka, you will have noticed The Team two big changes we have made: Eureka is now published in full col- our, and printed on a larger paper size than usual. We felt that, with Philipp Legner Design and Bthe internet being an increasingly large resource for mathematical articles of Illustrations all kinds, it was necessary to offer something new and exciting to keep Eu- reka as successful as it has been in the past. We moved away from the classic Anja Komatar Submissions LATEX-look, which is so common in the scientific community, to a modern, more engaging, and more entertaining design, while being conscious not to Sean Moss lose any of the mathematical clarity and rigour. Corporate Ben Millwood To make full use of the new design possibilities, many of this issue’s articles Publicity are based around mathematical images: from fractal modelling in financial Lu Zou markets (page 14) to computer rendered pictures (page 38) and mathemati- Subscriptions cal origami (page 20). The Showroom (page 46) uncovers the fundamental role pictures have in mathematics, including patterns, graphs, functions and fractals. This issue includes a wide variety of mathematical articles, problems and puzzles, diagrams, movie and book reviews. Some are more entertaining, such as Bayesian Bets (page 10), some are more technical, such as Impossible Integrals (page 80), or more philosophical, such as How to teach Physics to Mathematicians (page 42). -
Using Formal Concept Analysis in Mathematical Discovery
Using Formal Concept Analysis in Mathematical Discovery Simon Colton and Daniel Wagner sgc, [email protected] Combined Reasoning Group Department of Computing Imperial College, London http://www.doc.ic.ac.uk/crg/ Abstract. Formal concept analysis (FCA) comprises a set of powerful algorithms which can be used for data analysis and manipulation, and a set of visualisation tools which enable the discovery of meaningful re- lationships between attributes of the data. We explore the potential of combining FCA and mathematical discovery tools in order to better fa- cilitate discovery tasks. In particular, we propose a novel lookup method for the Encyclopedia of Integer Sequences, and we show how conjectures from the Graffiti discovery program can be better understood using FCA visualisation tools. We argue that, not only can FCA tools greatly en- hance the management and visualisation of mathematical knowledge, but they can also be used to drive exploratory processes. 1 Introduction Formal Concept Analysis (FCA) consists of a set of well established techniques for the analysis and manipulation of data. FCA has a strong theoretical un- derpinning, efficient implementations of fast algorithms, and useful visualisation tools. There are strong links between FCA and machine learning, and the con- nection of both fields is an active area of research [8,12,13]. We concentrate here on the combination of FCA tools with systems developed to aid mathemat- ical discovery. In particular, we are interested in addressing (i) whether FCA algorithms can be used to enhance the discovery process and (ii) whether FCA visualisation tools can enable better understanding of the discoveries made. -
Acknowledgements P. Ix Abbreviations P. X Notes on Contributors P
Acknowledgements p. ix Abbreviations p. x Notes on Contributors p. xvi Introduction p. 1 Early Developments in Reception Introduction: The Old Academy to Cicero p. 10 Speusippus and Xenocrates on the Pursuit and Ends of Philosophy p. 29 The Influence of the Platonic Dialogues on Stoic Ethics from Zeno to Panaetius of Rhodesp. 46 Plato and the Freedom of the New Academy p. 58 Return to Plato and Transition to Middle Platonism in Cicero p. 72 Early Imperial Reception of Plato Introduction: Early Imperial Reception of Plato p. 92 From Fringe Reading to Core Curriculum: Commentary, Introduction and Doctrinal p. 101 Summary Philo of Alexandria p. 115 Plutarch of Chaeronea and the Anonymous Commentator on the Theaetetus p. 130 Theon of Smyrna: Re-thinking Platonic Mathematics in Middle Platonism p. 143 Cupid's Swan from the Academy (De Plat. 1.1, 183): Apuleius' Reception of Plato p. 156 Alcinous' Reception of Plato p. 171 Numenius: Portrait of a Platonicus p. 183 Galen and Middle Platonism: The Case of the Demiurge p. 206 Variations of Receptions of Plato during the Second Sophistic p. 223 Early Christianity and Late Antique Platonism Introduction: Early Christianity and Late Antique Platonism p. 252 Origen to Evagrius p. 271 Sethian Gnostic Appropriations of Plato p. 292 Plotinus and Platonism p. 316 Porphyry p. 336 The Anonymous Commentary on the Parmenides p. 351 Iamblichus, the Commentary Tradition, and the Soul p. 366 Amelius and Theodore of Asine p. 381 Plato's Political Dialogues in the Writings of Julian the Emperor p. 400 Plato's Women Readers p. -
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number) -
Discriminators of Integer Sequences
Discriminators of Integer Sequences by Sajed Haque A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2017 c Sajed Haque 2017 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract The discriminator of an integer sequence s = (s(n))n≥0, first introduced by Arnold, Benkoski and McCabe in 1985, is the function Ds(n) that maps the integer n ≥ 1 to the smallest positive integer m such that the first n terms of s are pairwise incongruent modulo m. In this thesis, we provide a basic overview of discriminators, examining the background literature on the topic and presenting some general properties of discriminators. We also venture into various computational aspects relating to discriminators, such as providing algorithms to compute the discriminator, and establishing an upper bound on the discriminator growth rate. We provide a complete characterization of sequences whose discriminators are themselves, and also explore the problem of determining whether a given sequence is a discriminator of some other sequence with some partial results and algorithms. We briefly discuss some k-regular sequences, characterizing the discriminators for the evil and odious numbers, and show that k-regular sequences do not necessarily have k- regular discriminators. We introduce the concept of shift-invariant discriminators, i.e. -
The Cambridge History of Philosophy in Late Antiquity
THE CAMBRIDGE HISTORY OF PHILOSOPHY IN LATE ANTIQUITY The Cambridge History of Philosophy in Late Antiquity comprises over forty specially commissioned essays by experts on the philosophy of the period 200–800 ce. Designed as a successor to The Cambridge History of Later Greek and Early Medieval Philosophy (ed. A. H. Armstrong), it takes into account some forty years of schol- arship since the publication of that volume. The contributors examine philosophy as it entered literature, science and religion, and offer new and extensive assess- ments of philosophers who until recently have been mostly ignored. The volume also includes a complete digest of all philosophical works known to have been written during this period. It will be an invaluable resource for all those interested in this rich and still emerging field. lloyd p. gerson is Professor of Philosophy at the University of Toronto. He is the author of numerous books including Ancient Epistemology (Cambridge, 2009), Aristotle and Other Platonists (2005)andKnowing Persons: A Study in Plato (2004), as well as the editor of The Cambridge Companion to Plotinus (1996). The Cambridge History of Philosophy in Late Antiquity Volume II edited by LLOYD P. GERSON cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao˜ Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge cb2 8ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521876421 C Cambridge University Press 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. -
A Concise History of the Philosophy of Mathematics
A Thumbnail History of the Philosophy of Mathematics "It is beyond a doubt that all our knowledge begins with experience." - Imannuel Kant ( 1724 – 1804 ) ( However naïve realism is no substitute for truth [1] ) [1] " ... concepts have reference to sensible experience, but they are never, in a logical sense, deducible from them. For this reason I have never been able to comprehend the problem of the á priori as posed by Kant", from "The Problem of Space, Ether, and the Field in Physics" ( "Das Raum-, Äether- und Feld-Problem der Physik." ), by Albert Einstein, 1934 - source: "Beyond Geometry: Classic Papers from Riemann to Einstein", Dover Publications, by Peter Pesic, St. John's College, Sante Fe, New Mexico Mathematics does not have a universally accepted definition during any period of its development throughout the history of human thought. However for the last 2,500 years, beginning first with the pre - Hellenic Egyptians and Babylonians, mathematics encompasses possible deductive relationships concerned solely with logical truths derived by accepted philosophic methods of logic which the classical Greek thinkers of antiquity pioneered. Although it is normally associated with formulaic algorithms ( i.e., mechanical methods ), mathematics somehow arises in the human mind by the correspondence of observation and inductive experiential thinking together with its practical predictive powers in interpreting future as well as "seemingly" ephemeral phenomena. Why all of this is true in human progress, no one can answer faithfully. In other words, human experiences and intuitive thinking first suggest to the human mind the abstract symbols for which the economy of human thinking mathematics is well known; but it is those parts of mathematics most disconnected from observation and experience and therefore relying almost wholly upon its internal, self - consistent, deductive logics giving mathematics an independent reified, almost ontological, reality, that mathematics most powerfully interprets the ultimate hidden mysteries of nature. -
Dynamical Generalizations of the Prime Number Theorem And
Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions Vitaly Bergelson Florian K. Richter February 11, 2020 Abstract We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theo- rem, a theorem of Pillai-Selberg, a theorem of Erd˝os-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Hal´asz. By building on the ideas behind our ergodic results, we recast Sarnak’s M¨obius disjointness conjec- ture in a new dynamical framework. This naturally leads to an extension of Sarnak’s conjecture which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases thereof. Contents 1 Introduction 1 1.1 Dynamical generalizations of the Prime Number Theorem ................ 2 1.2 A wider framework for Sarnak’s conjecture ........................ 9 2 Distribution of orbits along Ω(n) 16 2.1 Proof of Theorem A .................................... 16 2.2 Proofs of Proposition 2.1 and Lemmas 2.2 and 2.3 .................... 19 3 Finitely generated and strongly uniquley ergodic systems 23 3.1 A counterexample ..................................... 23 3.2 Proof of Theorem B .................................... 23 3.3 Proof of Proposition 3.2 .................................. 25 4 Applications of Theorems A and B 32 5 Disjointness of additive and multiplicative semigroup actions 36 arXiv:2002.03498v1 [math.DS] 10 Feb 2020 5.1 Nilsystems, nilsequences, and a proof of Theorem C ................... 36 5.2 Horocyclic flow and a proof of Theorem D .......................