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The Cell Method: a Purely Algebraic Computational Method in Physics and Engineering Copyright © Momentum Press®, LLC, 2014
THE CELL METHOD THE CELL METHOD A PURELY ALGEBRAIC COMPUTATIONAL METHOD IN PHYSICS AND ENGINEERING ELENA FERRETTI MOMENTUM PRESS, LLC, NEW YORK The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering Copyright © Momentum Press®, LLC, 2014. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other—except for brief quotations, not to exceed 400 words, without the prior permission of the publisher. First published by Momentum Press®, LLC 222 East 46th Street, New York, NY 10017 www.momentumpress.net ISBN-13: 978-1-60650-604-2 (hardcover) ISBN-10: 1-60650-604-8 (hardcover) ISBN-13: 978-1-60650-606-6 (e-book) ISBN-10: 1-60650-606-4 (e-book) DOI: 10.5643/9781606506066 Cover design by Jonathan Pennell Interior design by Exeter Premedia Services Private Ltd. Chennai, India 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America CONTENTS ACKNOWLEDGMENTS vii PREFACE ix 1 A COMpaRISON BETWEEN ALGEBRAIC AND DIFFERENTIAL FORMULATIONS UNDER THE GEOMETRICAL AND TOPOLOGICAL VIEWPOINTS 1 1.1 Relationship Between How to Compute Limits and Numerical Formulations in Computational Physics 2 1.1.1 Some Basics of Calculus 2 1.1.2 The e − d Definition of a Limit 4 1.1.3 A Discussion on the Cancelation Rule for Limits 8 1.2 Field and Global Variables 15 1.3 Set Functions in Physics 20 1.4 A Comparison Between the Cell Method and the Discrete Methods 21 2 ALGEBRA AND THE GEOMETRIC INTERPRETATION -
University Microiilms, a XERQ\Company, Ann Arbor, Michigan
71-18,075 RINEHART, John McLain, 1937- IVES' COMPOSITIONAL IDIOMS: AN INVESTIGATION OF SELECTED SHORT COMPOSITIONS AS MICROCOSMS' OF HIS MUSICAL LANGUAGE. The Ohio State University, Ph.D., 1970 Music University Microiilms, A XERQ\Company, Ann Arbor, Michigan © Copyright by John McLain Rinehart 1971 tutc nTccrSTATmil HAS fiEEM MICROFILMED EXACTLY AS RECEIVED IVES' COMPOSITIONAL IDIOMS: AM IMVESTIOAT10M OF SELECTED SHORT COMPOSITIONS AS MICROCOSMS OF HIS MUSICAL LANGUAGE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy 3n the Graduate School of The Ohio State University £ JohnfRinehart, A.B., M«M. # # * -k * * # The Ohio State University 1970 Approved by .s* ' ( y ^MrrXfOor School of Music ACm.WTji.D0F,:4ENTS Grateful acknov/ledgement is made to the library of the Yale School of Music for permission to make use of manuscript materials from the Ives Collection, I further vrish to express gratitude to Professor IJoman Phelps, whose wise counsel and keen awareness of music theory have guided me in thi3 project. Finally, I wish to acknowledge my wife, Jennifer, without whose patience and expertise this project would never have come to fruition. it VITA March 17, 1937 • ••••• Dorn - Pittsburgh, Pennsylvania 1959 • • • • • .......... A#B#, Kent State University, Kent, Ohio 1960-1963 . * ........... Instructor, Cleveland Institute of Music, Cleveland, Ohio 1 9 6 1 ................ • • • M.M., Cleveland Institute of ITu3ic, Cleveland, Ohio 1963-1970 .......... • • • Associate Professor of Music, Heidelberg College, Tiffin, Ohio PUBLICATIONS Credo, for unaccompanied chorus# New York: Plymouth Music Company, 1969. FIELDS OF STUDY Major Field: Theory and Composition Studies in Theory# Professor Norman Phelps Studies in Musicology# Professors Richard Hoppin and Lee Rigsby ill TAPLE OF CC NTEKTS A C KI JO WLE DGEME MT S ............................................... -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Bestimmung Der Expositionsverteilung Von HF Feldern Im Menschlichen Körper, Unter Berücksichtigung Kleiner Strukturen Und Thermophysiologisch Relevanter Parameter
Forschungsvorhaben Bestimmung der Expositionsverteilung von HF Feldern im menschlichen Körper, unter Berücksichtigung kleiner Strukturen und thermophysiologisch relevanter Parameter Abschlussbericht Gernot Schmid, Richard Überbacher, Patrick Preiner, Theodoros Samaras, Peter Mazal, Alexandra Jappel, Wolf-Dieter Baumgartner, Manfred Tschabitscher korrigierte Ausgabe Oktober 2008 (Originalausgabe August 2006) Exemplar 1 ARC-IT-0174 Verteiler: 1-4 Bundesamt für Strahlenschutz 5 DI Lamedschwandner 6 DI Schmid 7 DI Überbacher 8 DI Preiner 9 Dr. Samaras 10 Prof. Mazal 11 Fr. Jappel 10 Prof. Baumgartner 12 Prof. Tschabitscher 13 Sekretariat ARCS/IT ARC-IT-0174 Oktober 2008 Bestimmung der Expositionsverteilung von HF Feldern im menschlichen Körper, unter Berücksichtigung kleiner Strukturen und thermophysiologisch relevanter Parameter Abschlussbericht im Auftrag des Bundesamtes für Strahlenschutz 38201 Salzgitter, Deutschland Gernot Schmid1, Richard Überbacher1, Patrick Preiner1, Theodoros Samaras2, Peter Mazal3, Alexandra Jappel4, Wolf-Dieter Baumgartner4, Manfred Tschabitscher5 1 ARC Seibersdorf research GmbH Geschäftsfeld Sichere Mobilkommunikation A-2444 Seibersdorf 2 Aristotle University of Thessaloniki, Radio Communications Laboratory, GR-54124 Thessaloniki 3 Klinisches Institut für Pathologie, Medizinische Universität Wien Währinger Gürtel 18-20 A-1090 Wien 4 Universitätsklinik für Hals- Nasen- und Ohrenkrankheiten, Medizinische Universität Wien Währinger Gürtel 18-20 A-1090 Wien 5 Zentrum für Anatomie und Zellbiologie, Medizinische Universität -
Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- Mogenization
in \Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- mogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001, Ed. L. Carbone and R. De Arcangelis, 191{211, Gakkotosho, Tokyo, 2003". On Homogenization and Γ-convergence Luc TARTAR1 In memory of Ennio DE GIORGI When in the Fall of 1976 I had chosen \Homog´en´eisationdans les ´equationsaux d´eriv´eespartielles" (Homogenization in partial differential equations) for the title of my Peccot lectures, which I gave in the beginning of 1977 at Coll`egede France in Paris, I did not know of the term Γ-convergence, which I first heard almost a year after, in a talk that Ennio DE GIORGI gave in the seminar that Jacques-Louis LIONS was organizing at Coll`egede France on Friday afternoons. I had not found the definition of Γ-convergence really new, as it was quite similar to questions that were already discussed in control theory under the name of relaxation (which was a more general question than what most people mean by that term now), and it was the convergence in the sense of Umberto MOSCO [Mos] but without the restriction to convex functionals, and it was the natural nonlinear analog of a result concerning G-convergence that Ennio DE GIORGI had obtained with Sergio SPAGNOLO [DG&Spa]; however, Ennio DE GIORGI's talk contained a quite interesting example, for which he referred to Luciano MODICA (and Stefano MORTOLA) [Mod&Mor], where functionals involving surface integrals appeared as Γ-limits of functionals involving volume integrals, and I thought that it was the interesting part of the concept, so I had found it similar to previous questions but I had felt that the point of view was slightly different. -
Modern Art Music Terms
Modern Art Music Terms Aria: A lyrical type of singing with a steady beat, accompanied by orchestra; a songful monologue or duet in an opera or other dramatic vocal work. Atonality: In modern music, the absence (intentional avoidance) of a tonal center. Avant Garde: (French for "at the forefront") Modern music that is on the cutting edge of innovation.. Counterpoint: Combining two or more independent melodies to make an intricate polyphonic texture. Form: The musical design or shape of a movement or complete work. Expressionism: A style in modern painting and music that projects the inner fear or turmoil of the artist, using abrasive colors/sounds and distortions (begun in music by Schoenberg, Webern and Berg). Impressionism: A term borrowed from 19th-century French art (Claude Monet) to loosely describe early 20th- century French music that focuses on blurred atmosphere and suggestion. Debussy "Nuages" from Trois Nocturnes (1899) Indeterminacy: (also called "Chance Music") A generic term applied to any situation where the performer is given freedom from a composer's notational prescription (when some aspect of the piece is left to chance or the choices of the performer). Metric Modulation: A technique used by Elliott Carter and others to precisely change tempo by using a note value in the original tempo as a metrical time-pivot into the new tempo. Carter String Quartet No. 5 (1995) Minimalism: An avant garde compositional approach that reiterates and slowly transforms small musical motives to create expansive and mesmerizing works. Glass Glassworks (1982); other minimalist composers are Steve Reich and John Adams. Neo-Classicism: Modern music that uses Classic gestures or forms (such as Theme and Variation Form, Rondo Form, Sonata Form, etc.) but still has modern harmonies and instrumentation. -
Mathematics to the Rescue Retiring Presidential Address
morawetz.qxp 11/13/98 11:20 AM Page 9 Mathematics to the Rescue (Retiring Presidential Address) Cathleen Synge Morawetz should like to dedicate this lecture to my tial equation by a simple difference scheme and get- teacher, Kurt Otto Friedrichs. While many ting nonsense because the scheme is unstable. people helped me on my way professionally, The answers blow up. The white knights—Courant, Friedrichs played the central role, first as my Friedrichs, and Lewy (CFL)—would then step in thesis advisor and then by feeding me early and show them that mathematics could cure the Iresearch problems. He was my friend and col- problem. That was, however, not the way. But first, league for almost forty years. As is true among what is the CFL condition? It says [1] that for sta- friends, he was sometimes exasperated with me (I ble schemes the step size in time is limited by the was too disorderly), and sometimes he exasperated step size in space. The simplest example is the dif- me (he was just too orderly). However, I think we ference scheme for a solution of the wave equa- appreciated each other’s virtues, and I learned a tion great deal from him. utt uxx =0 Sometimes we had quite different points of − view. In his old age he agreed to be interviewed, on a grid that looks like the one in Figure 1. Sec- at Jack Schwartz’s request, for archival purposes. ond derivatives are replaced by second-order I was the interviewer. At one point I asked him differences with t = x. -
Dissertation Chapter 3
CHAPTER 3: SYNTHETIC ‘KERNELS’ “…it should be borne in mind that variation is the basic creative method of folk music, and thus in Bartók’s eyes the natural mode of composition…” -Bence Szabolcsi, “Man and Nature in Bartók’s World”1 THEME… When Zoltán Székely approached Bartók with a commission for a new violin concerto in the fall of 1936, the composer’s first idea was to write a single movement in variation form.2 Székely, however, insisted on the form of a traditional concerto. Bartók obliged him with a large scale three-movement composition, but still remained faithful to his original vision: not only is the second movement a set of variations on an original theme, Bartók also derives all of the third movement from the musical material of the first movement through free variations. While the Piano Concerto No. 2 also displays thematic variation between its first and third movements, “the correspondence between the two exterior movements [in the Violin Concerto] goes much further than the simple re-use of materials: the architecture is duplicated almost section by section. Both are large sonata-form movements, with two main thematic groups and a number of pertinent 1 Bence Szabolsci, “Man and Nature in Bartók’s World,” in Bartók Studies, ed. Todd Crow (Detroit: Information Coordinators, 1976), 72. 2 Lampert cites Somfai, Tizennyolc Bartók tanulmány (1981). associates with the form.”3 In light of Bartók’s proclivity for the variation principle in his free composition,4 it is remarkable that the second movement of the Violin Concerto represents his only large-scale essay in the form of theme and variations. -
Saul Abarbanel SIAM Oral History
An interview with SAUL ABARBANEL Conducted by Philip Davis on 29 July, 2003, at the Department of Applied Mathematics, Brown University Interview conducted by the Society for Industrial and Applied Mathematics, as part of grant # DE-FG02-01ER25547 awarded by the US Department of Energy. Transcript and original tapes donated to the Computer History Museum by the Society for Industrial and Applied Mathematics © Computer History Museum Mountain View, California ABSTRACT: ABARBANEL describes his work in numerical analysis, his use of early computers, and his work with a variety of colleagues in applied mathematics. Abarbanel was born and did his early schooling in Tel Aviv, Israel, and in high school developed an interest in mathematics. After serving in the Israeli army, Abarbanel entered MIT in 1950 as an as an engineering major and took courses with Adolf Hurwitz, Francis Begnaud Hildebrand, and Philip Franklin. He found himself increasing drawn to applied mathematics, however, and by the time he began work on his Ph.D. at MIT he had switched from aeronautics to applied mathematics under the tutelage of Norman Levinson. Abarbanel recalls the frustration of dropping the punch cards for his program for the IBM 1604 that MIT was using in 1958 when he was working on his dissertation, but also notes that this work convinced him of the importance of computers. Abarbanel also relates a humorous story about Norbert Weiner, his famed linguistic aptitude, and his lesser-known interest in chess. Three years after receiving his Ph.D., Abarbanel returned to Israel, where he spent the rest of his career at Tel Aviv University. -
Glimm and Witten Receive National Medal of Science, Volume 51, Number 2
Glimm and Witten Receive National Medal of Science On October 22, 2003, President Bush named eight of the nation’s leading scientists and engineers to receive the National Medal of Science. The medal is the nation’s highest honor for achievement in sci- ence, mathematics, and engineering. The medal James G. Glimm Edward Witten also recognizes contributions to innovation, in- dustry, or education. Columbia University in 1959. He is the Distin- Among the awardees are two who work in the guished Leading Professor of Mathematics at the mathematical sciences, JAMES G. GLIMM and EDWARD State University of New York at Stony Brook. WITTEN. Edward Witten James G. Glimm Witten is a world leader in “string theory”, an attempt Glimm has made outstanding contributions to by physicists to describe in one unified way all the shock wave theory, in which mathematical models known forces of nature as well as to understand are developed to explain natural phenomena that nature at the most basic level. Witten’s contributions involve intense compression, such as air pressure while at the Institute for Advanced Study have set in sonic booms, crust displacement in earthquakes, the agenda for many developments, such as progress and density of material in volcanic eruptions and in “dualities”, which suggest that all known string other explosions. Glimm also has been a leading theories are related. theorist in operator algebras, partial differential Witten’s earliest papers produced advances in equations, mathematical physics, applied mathe- quantum chromodynamics (QCD), a theory that matics, and quantum statistical mechanics. describes the interactions among the fundamental Glimm’s work in quantum field theory and particles (quarks and gluons) that make up all statistical mechanics had a major impact on atomic nuclei. -
Literal Imagery in Music: a Thesis to Accompany Constellation Suite
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 12-2011 Literal Imagery in Music: A Thesis to Accompany Constellation Suite Evelyn Marie Pursley-Kopitzke [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Composition Commons Recommended Citation Pursley-Kopitzke, Evelyn Marie, "Literal Imagery in Music: A Thesis to Accompany Constellation Suite. " Master's Thesis, University of Tennessee, 2011. https://trace.tennessee.edu/utk_gradthes/1092 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Evelyn Marie Pursley-Kopitzke entitled "Literal Imagery in Music: A Thesis to Accompany Constellation Suite." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Music, with a major in Music. Kenneth A. Jacobs, Major Professor We have read this thesis and recommend its acceptance: Barbara A. Murphy, Donald Pederson Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Evelyn Marie Pursley-Kopitzke entitled ―Literal Imagery in Music: A Thesis to Accompany Constellation Suite.‖ I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Music, with a major in Music. -
Segmenting Musical Audio Musical Form
Analyzing structure: segmenting musical audio Musical form (1) • Can refer to the type of composition (as in multi-movement form), e.g. symphony, concerto, etc • Of more relevance to this class, it refers to the structure of a given piece (as in single-movement form), e.g. strophic, binary, sonata, fugue • We can differentiate between sectional and developmental form • “In a sectional form, the piece is built by combining small clear-cut units, sort of like stacking legos” (DeLone, 1975). This is the case with most popular music. • In developmental form the piece is built from a number of evolving presentations and combinations of small musical units (e.g. motifs, themes) • Music with continuous non-sectional, non-repetitive form is called through composed 1 Musical form (2) • Each unit can be labeled with a letter (e.g. A, B, C) or a generic name (e.g. intro, verse, chorus, bridge, interlude, coda, etc) • Strophic form: repeats the same tune in different verses, e.g. AA… • Binary form: alternates two sections, which are often repeated, e.g. ABAB or AABB • Ternary form: has three parts, third section is often a recap or a variation of the first one, e.g. AABA, AABA’, AA’BA’ • Arch form: it is a symmetric form, based on the repetition of sections around a center, e.g. ABCBA • Rondo: a main theme is alternated with sub-themes, but it always comes back (returns), e.g. ABACADA….. • Variations: theme plus variations, e.g. AAiAiiAAiii • Sonata: more complex form showing intro, exposition, development, recapitulation, coda. Musical form (3) • What is at the core of structure analysis is the idea of repetition.