TME Volume 7, Numbers 2 and 3
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Transfinite Function Interaction and Surreal Numbers
ADVANCES IN APPLIED MATHEMATICS 18, 333]350Ž. 1997 ARTICLE NO. AM960513 Transfinite Function Iteration and Surreal Numbers W. A. Beyer and J. D. Louck Theoretical Di¨ision, Los Alamos National Laboratory, Mail Stop B284, Los Alamos, New Mexico 87545 Received August 25, 1996 Louck has developed a relation between surreal numbers up to the first transfi- nite ordinal v and aspects of iterated trapezoid maps. In this paper, we present a simple connection between transfinite iterates of the inverse of the tent map and the class of all the surreal numbers. This connection extends Louck's work to all surreal numbers. In particular, one can define the arithmetic operations of addi- tion, multiplication, division, square roots, etc., of transfinite iterates by conversion of them to surreal numbers. The extension is done by transfinite induction. Inverses of other unimodal onto maps of a real interval could be considered and then the possibility exists of obtaining different structures for surreal numbers. Q 1997 Academic Press 1. INTRODUCTION In this paper, we assume the reader is familiar with the interesting topic of surreal numbers, invented by Conway and first presented in book form in Conwaywx 3 and Knuth wx 6 . In this paper we follow the development given in Gonshor's bookwx 4 which is quite different from that of Conway and Knuth. Inwx 10 , a relation was shown between surreal numbers up to the first transfinite ordinal v and aspects of iterated maps of the intervalwx 0, 2 . In this paper, we specialize the results inwx 10 to the graph of the nth iterate of the inverse tent map and extend the results ofwx 10 to all the surreal numbers. -
A Child Thinking About Infinity
A Child Thinking About Infinity David Tall Mathematics Education Research Centre University of Warwick COVENTRY CV4 7AL Young children’s thinking about infinity can be fascinating stories of extrapolation and imagination. To capture the development of an individual’s thinking requires being in the right place at the right time. When my youngest son Nic (then aged seven) spoke to me for the first time about infinity, I was fortunate to be able to tape-record the conversation for later reflection on what was happening. It proved to be a fascinating document in which he first treated infinity as a very large number and used his intuitions to think about various arithmetic operations on infinity. He also happened to know about “minus numbers” from earlier experiences with temperatures in centigrade. It was thus possible to ask him not only about arithmetic with infinity, but also about “minus infinity”. The responses were thought-provoking and amazing in their coherent relationships to his other knowledge. My research in studying infinite concepts in older students showed me that their ideas were influenced by their prior experiences. Almost always the notion of “limit” in some dynamic sense was met before the notion of one to one correspondences between infinite sets. Thus notions of “variable size” had become part of their intuition that clashed with the notion of infinite cardinals. For instance, Tall (1980) reported a student who considered that the limit of n2/n! was zero because the top is a “smaller infinity” than the bottom. It suddenly occurred to me that perhaps I could introduce Nic to the concept of cardinal infinity to see what this did to his intuitions. -
The Silk Road
The Silk Road Volume 1 Number 1 January 15, 2003 第1卷 第1號 “The Bridge between Eastern and Western Cultures” 一月十五日 In This Issue • WELCOME TO THE FIRST ISSUE WELCOME TO THE FIRST ISSUE! • [email protected]: A YE- Since the Soviet collapse, the nations of Central One reason for our distorted image of Central MENI TRADING LINK THREE THOUSAND Asia have shaken off imposed obscurity to make Asia has been the diffi culty of access for west- YEARS OLD headlines of their own. The emergence of these ern travelers, scholars, and archaeologists. new states has helped to focus attention once Russian and Chinese investigators working in • THE ORIGIN OF CHESS AND THE SILK again on their history, culture, and people. For their respective languages have done most of ROAD most of us, these were places whose names we the fi rst hand observation and reporting. The barely knew a decade ago. Collectively more experienced fi eld archaeologists • THE MONGOLS AND THE SILK ROAD they form the heart of Eurasia. Today in Russia and China—Elena Kuzmina • AGE OF MONGOLIAN EMPIRE: A BIB- they may be known as Ukraine, Armenia, from Moscow and Wang Binghua from LIOGRAPHICAL ESSAY Georgia, Azerbaijan, Turkmenistan, Urumchi, for example—have more di- Kazakhstan, Tajikistan, Uzbekistan, and rect experience with Central Asian sites Kyrghizstan, but in the more remote and materials than practically all of past, along with Afghanistan, Xinjiang, the American investigators combined. and Gansu, they evoked images of the an- Their reports and publications, in Russian and cient Silk Road—oases, caravanserai, nomads, Chinese, are available in the west to only a lim- strange empires, fantastic beasts, and exotic ited number of specialists. -
Chapter 6 the Arbelos
Chapter 6 The arbelos 6.1 Archimedes’ theorems on the arbelos Theorem 6.1 (Archimedes 1). The two circles touching CP on different sides and AC CB each touching two of the semicircles have equal diameters AB· . P W1 W2 A O1 O C O2 B A O1 O C O2 B Theorem 6.2 (Archimedes 2). The diameter of the circle tangent to all three semi- circles is AC CB AB · · . AC2 + AC CB + CB2 · We shall consider Theorem ?? in ?? later, and for now examine Archimedes’ wonderful proofs of the more remarkable§ Theorems 6.1 and 6.2. By synthetic reasoning, he computed the radii of these circles. 1Book of Lemmas, Proposition 5. 2Book of Lemmas, Proposition 6. 602 The arbelos 6.1.1 Archimedes’ proof of the twin circles theorem In the beginning of the Book of Lemmas, Archimedes has established a basic proposition 3 on parallel diameters of two tangent circles. If two circles are tangent to each other (internally or externally) at a point P , and if AB, XY are two parallel diameters of the circles, then the lines AX and BY intersect at P . D F I W1 E H W2 G A O C B Figure 6.1: Consider the circle tangent to CP at E, and to the semicircle on AC at G, to that on AB at F . If EH is the diameter through E, then AH and BE intersect at F . Also, AE and CH intersect at G. Let I be the intersection of AE with the outer semicircle. Extend AF and BI to intersect at D. -
International Journal of Division by Zero Calculus Vol. 1 (January-December, 2021)
International Journal of Division by Zero Calculus Vol. 1 (January-December, 2021) GEOMETRY AND DIVISION BY ZERO CALCULUS HIROSHI OKUMURA Abstract. We demonstrate several results in plane geometry derived from division by zero and division by zero calculus. The results show that the two new concepts open an entirely new world of mathematics. 1. Introduction The lack of division by zero has been a serious glaring omission in our mathe- matics. While recent publications [3], [7], [8], [22], [29] have offered a final solution to this [3]: z = 0 for any element z in a field: (1.1) 0 In this article we show several results in plane geometry obtained from division by zero given by (1.1) and division by zero calculus, which is a generalization of division by zero. For a meromorphic function W = f(z), we consider the Laurent expansion of f around z = a: nX=−1 X1 n n W = f(z) = Cn(z − a) + C0 + Cn(z − a) : n=−∞ n=1 Then we define f(a) = C0. This is a generalization of (1.1) called division by zero calculus [29]. Now we can consider the value f(a) = C0 at an isolated singular point a. We consider some families of circles in the plane, each of the members is rep- resented by a Cartesian equation fz(x; y) = 0 with parameter z 2 R. Here, we assume that if x and y are fixed, fz(x; y) is a meromorphic function in z. Then, for the Laurent expansion of the function fz(x; y) at z = a for fixed x; y, the corresponding coefficient Cn(a; x; y) is depending on also x and y. -
Tool 44: Prime Numbers
PART IV: Cinematography & Mathematics AGE RANGE: 13-15 1 TOOL 44: PRIME NUMBERS - THE MAN WHO KNEW INFINITY Sandgärdskolan This project has been funded with support from the European Commission. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein. Educator’s Guide Title: The man who knew infinity – prime numbers Age Range: 13-15 years old Duration: 2 hours Mathematical Concepts: Prime numbers, Infinity Artistic Concepts: Movie genres, Marvel heroes, vanishing point General Objectives: This task will make you learn more about infinity. It wants you to think for yourself about the word infinity. What does it say to you? It will also show you the connection between infinity and prime numbers. Resources: This tool provides pictures and videos for you to use in your classroom. The topics addressed in these resources will also be an inspiration for you to find other materials that you might find relevant in order to personalize and give nuance to your lesson. Tips for the educator: Learning by doing has proven to be very efficient, especially with young learners with lower attention span and learning difficulties. Don’t forget to 2 always explain what each math concept is useful for, practically. Desirable Outcomes and Competences: At the end of this tool, the student will be able to: Understand prime numbers and infinity in an improved way. Explore super hero movies. Debriefing and Evaluation: Write 3 aspects that you liked about this 1. activity: 2. 3. -
Bergson's Paradox and Cantor's
Bergson's Paradox and Cantor's Doug McLellan, July 2019 \Of a wholly new action ... which does not preexist its realization in any way, not even in the form of pure possibility, [philosophers] seem to have had no inkling. Yet that is just what free action is." Here we have what may be the last genuinely new insight to date in the eternal dispute over free will and determinism: Free will is the power, not to choose among existing possibilities, but to consciously create new realities that did not subsist beforehand even virtually or abstractly. Alas, the success of the book in which Henri Bergson introduced this insight (Creative Evolution, 1907) was not enough to win it a faithful constituency, and the insight was discarded in the 1930s as vague and unworkable.1 What prompts us to revive it now are advances in mathematics that allow the notion of \newly created possibilities" to be formalized precisely. In this form Bergson's insight promises to resolve paradoxes not only of free will and determinism, but of the foundations of mathematics and physics. For it shows that they arise from the same source: mistaken belief in a \totality of all possibilities" that is fixed once and for all, that cannot grow over time. Bergson's Paradox Although it's been recapitulated many times, many ways, let us begin by recapitulating the dilemma of free will and determinism, in a particular form that we will call \Bergson's paradox." (We so name it not because Bergson phrased it precisely the way that we will, but because he phrased it in a similar enough way, and supplied the insight that we expect will resolve it.) In this form it would be more accurate to call it a dilemma of free will and any rigorous scientific theory of the world, for although scientists today are happy to give up hard determinism for quantum randomness, the theories that result are no less vulnerable to this version of the paradox. -
Definition of Integers in Math Terms
Definition Of Integers In Math Terms whileUnpraiseworthy Ernest remains and gypsiferous impeachable Kingsley and friskier. always Unsupervised send-ups unmercifully Yuri hightails and her eluting molas his so Urtext. gently Phobic that Sancho Wynton colligated work-outs very very nautically. rippingly In order are integers of an expression is, two bases of a series that case MAT 240 Algebra I Fields Definition A motto is his set F. Story problems of math problems per worksheet. What are positive integers? Integer definition is summer of nine natural numbers the negatives of these numbers or zero. Words Describing Positive And Negative Integers Worksheets. If need add a negative integer you move very many units left another example 5 6 means they start at 5 and research move 6 units to execute left 9 5 means nothing start. Definition of Number explained with there life illustrated examples Also climb the facts to discuss understand math glossary with fun math worksheet online at. Sorry if the definition in maths study of definite value is the case of which may be taken as numbers, the principal of classes. They are integers, definition of definite results in its measurement conversions, they may not allow for instance, not obey the term applies primarily to. In terms of definition which term to this property of the definitions of a point called statistics, the opposite the most. Integers and rational numbers Algebra 1 Exploring real. Is in integer variable changes and definitions in every real numbers that term is a definition. What facility an integer Quora. Load the terms. -
Performing Place at Ancient Idalion, Cyprus: an Anthropological Perspective on the Lower City South Sanctuary Architecture Rebecca M
University of Massachusetts Amherst ScholarWorks@UMass Amherst Cultural Heritage in European Societies and Spaces CHESS Student Research Reports (CHESS) 2013 Performing Place at Ancient Idalion, Cyprus: An Anthropological Perspective on the Lower City South Sanctuary Architecture Rebecca M. Bartusewich [email protected] Follow this and additional works at: https://scholarworks.umass.edu/chess_student_research Part of the Archaeological Anthropology Commons Bartusewich, Rebecca M., "Performing Place at Ancient Idalion, Cyprus: An Anthropological Perspective on the Lower City South Sanctuary Architecture" (2013). CHESS Student Research Reports. 11. Retrieved from https://scholarworks.umass.edu/chess_student_research/11 This Article is brought to you for free and open access by the Cultural Heritage in European Societies and Spaces (CHESS) at ScholarWorks@UMass Amherst. It has been accepted for inclusion in CHESS Student Research Reports by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. Performing Place at Ancient Idalion, Cyprus: an anthropological perspective on the Lower City South sanctuary architecture Rebecca M. Bartusewich PhD Student University of Massachusetts Amherst1 Abstract: The ancient site of the Lower City South sanctuary of Idalion is a site of place making and identity formation during the 1st millennium BCE of Cyprus. This archaeological site represents repetitive building patterns and persistent cultic activity that denote a cultural tradition that withstood the changes of administrative control in the Cypro-Classical and Hellenistic periods. Certain architectural elements, like altars and water features, are characteristic of a continued tradition at the ancient site and they are evidence of a recursive building practice that falls into templates of place making and identity formation as introduced by Bourdieu and Giddens. -
Archimedean Circles Induced by Skewed Arbeloi
Journal for Geometry and Graphics Volume 16 (2012), No. 1, 13–17. Archimedean Circles Induced by Skewed Arbeloi Ryuichi¯ Nakajima1, Hiroshi Okumura2 1318-1 Goryo¯ Matsuida, Annaka Gunma 379-0302, Japan 2251 Moo 15 Ban Kesorn, Tambol Sila, Amphur Muang Khonkaen 40000, Thailand emails: [email protected], [email protected] Abstract. Infinite Archimedean circles touching one of the inner circles of the arbelos are induced by skewed arbeloi. Key Words: arbelos, skewed arbelos, infinite Archimedean circles MSC 2010: 51M04 1. Introduction On the problem to find infinite Archimedean circles of the arbelos, the following four have priority to be considered: (P1) to find infinite Archimedean circles touching the outer circle of the arbelos, (P2) to find infinite Archimedean circles passing through the tangent point of the two inner circles, (P3) to find infinite Archimedean circles touching the radical axis of the inner circles, and (P4) to find infinite Archimedean circles touching one of the inner circles. Solutions of (P1), (P2) and (P3) are given in [4], [6] and [2], respectively. But no solutions can be found for (P4). In this note we give a solution of this problem. Let O be a point on a segment AB and α, β and γ be the circles with diameters OA, OB and AB, respectively. We show that infinite Archimedean circles of the arbelos formed by the circles α, β and γ can be obtained by considering arbitrary circles touching α and β at points different from O. Let a and b be the respective radii of the circles α and β. -
SJ11-Refuchess.Pdf
POPULATION, SPACE AND PLACE Popul. Space Place 17, 140–152 (2011) Published online 7 January 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/psp.607 Refuchess: Locating Bosniac Repatriates after the War in Bosnia–Herzegovina Stef Jansen* Department of Social Anthropology, University of Manchester, Manchester, UK ABSTRACT Received 20 December 2008; revised 11 September 2009; accepted 11 November 2009 One of the central contradictions Keywords: home; ethnic cleansing; refugees; characterising the state of Bosnia–Herzegovina return; Bosnia–Herzegovina that emerged from the ashes of the 1992–1995 war concerns the territorial distribution of its population according to nationality. On the INTRODUCTION one hand, the foreign intervention, and the Dayton Peace Agreement it brokered, 1994 poster entitled REFUCHESS by the sanctioned its division into largely nationally Art Publishing Service in the then homogenised polities, consolidating military A besieged city of Sarajevo depicted a conquests. On the other hand, large foreign chessboard with two rows of 10 person icons. funds were invested in programmes of refugee Representing those displaced in the confl ict in repatriation and return to redress ethnic Bosnia–Herzegovina (BiH) as pawns, a classic cleansing, with considerable success in terms metaphor of subjection and manipulation, the of property restitution. Rather than poster implied that displacement itself had understanding these dynamics as a become a strategic parameter in the 1992–1995 resurrection of a prior situation, this article war. Wartime refuchess took many forms, of considers them as part of a set of interrelated which the military expulsion of persons of unde- transformations. Working from ethnographic sired nationality – an activity that henceforth material gathered among Bosniac repatriates came to be designated as ‘ethnic cleansing’ – was in the early phases of large-scale return, it fi rst the most documented. -
Localizing Geopolitics: Disaggregating Violence and Return in Conflict Regions
Political Geography 29 (2010) 256–265 Contents lists available at ScienceDirect Political Geography journal homepage: www.elsevier.com/locate/polgeo Localizing geopolitics: Disaggregating violence and return in conflict regions Gearo´ id O´ Tuathail School of Public and International Affairs, Virginia Tech, National Capital Region, Alexandria, VA 22314, United States abstract Keywords: Critical geopolitics began as a critique of Cold War geopolitical discourses that imposed homogenizing Critical geopolitics categories upon diverse regional conflicts and marginalized place-specific structural causes of instability Disaggregation and violence. This critique is still relevant. Implicit within it is the promise of a more geographical Localization geopolitics that, arguably, has not been realized by research. Using Bosnia–Herzegovina as an example, Fieldwork this paper examines the challenges of developing a critical geopolitics grounded in the study of contested Bosnia–Herzegovina geopolitical regions and places. Reviewing anthropological and other place-sensitive studies of violent population displacement and post-war returns in Bosnia–Herzegovina, the paper considers some conceptual dilemmas and questions raised by attempting to create a grounded critical geopolitics. Ó 2010 Elsevier Ltd. All rights reserved. In the aftermath of the 9/11 terrorist strikes the Bush adminis- known, the failures of the US and its allies in Iraq and Afghanistan tration launched a ‘global war on terror’ (GWOT) that propelled the spurred the development of a new war fighting doctrine based on United States and its allies into high stakes interventions in the ‘anthropology of counter-insurgency’ and a ‘granular knowl- Afghanistan and Iraq, territories marked by state failure, insurgent edge of the social terrains’ of conflict (Gregory, 2008; Packer, 2006).