The combination of new insights into Ligeti by people who knew him with new analytical approaches will make this a core publication not only for Ligeti scholars, but also for readers interested in post-war music history and in Hungarian culture. Shortlisted for the RPS Music Award 2012 for Creative Communication. György Ligeti: Of Foreign Lands and Strange Sounds offers a new assessment of a composer whose constant exploration of new sound worlds- based on the musics of different cultures and ages - contributed in crucial ways to making him one of the most important musical voices of the last 50 years. The book combines texts by former students, colleagues and friends, who reflect on different and so far unknown aspects of Ligeti's persona, with new musicological interpretations of his style and several of his main works. Among the contributors are some of the most eminent Ligeti scholars, including Richard Steinitz and Paul Griffiths. Louise Duchesneau, Ligeti's assistant of over 20 years, acts not only as contributor but also as co-editor of the volume. Many of the musicological chapters are based on studies of Ligeti's sketches, which are now housed by the Paul Sacher Foundation in Basle and were made available for research only recently. Two close collaborators representing disciplines which deeply interested Ligeti - Heinz-Otto Peitgen (a mathematician who introduced Ligeti to fractal geometry, which influenced many if his works since 1985) and Simha Arom (an ethnomusicologist who acquainted Ligeti with the complex rhythmic patters of the music of Sub-saharan Africa) - also reflect on the composer for the very first time in writing. The combination of new insights into Ligeti by people who knew him with new analytical approaches will make this a core publication not only for Ligeti scholars, but also for readers interested in music of the second half of the twentieth century and in Hungarian culture. WOLFGANG MARX is Lecturer in Music, University College Dublin. LOUISE DUCHESNEAU was Ligeti's assistant for 20 years Contributors: SIMHA AROM, JONATHAN W. BERNARD, CIAR�N CRILLY, LOUISE DUCHESNEAU, BENJAMIN DWYER, TIBORC FAZEKAS, PAUL GRIFFITHS, ILDIKÓ M�NDI-FAZEKAS, WOLFGANG MARX, HEINZ-OTTO PEITGEN, FRIEDEMANN SALLIS, WOLFGANG-ANDREAS SCHULTZ, MANFRED STAHNKE, RICHARD STEINITZ

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

The fourteen chapters of this book cover the central ideas and concepts of chaos and fractals as well as many related topics including: the Mandelbrot set, Julia sets, cellular automata, L-systems, percolation and strange attractors. This new edition has been thoroughly revised throughout. The appendices of the original edition were taken out since more recent publications cover this material in more depth. Instead of the focussed computer programs in BASIC, the authors provide 10 interactive JAVA-applets for this second edition.

Now approaching its tenth year, this hugely successful book presents an unusual attempt to publicise the field of Complex Dynamics. The text was originally conceived as a supplemented catalogue to the exhibition "Frontiers of Chaos," seen in Europe and the United States, and describes the context and meaning of these fascinating images. A total of 184 illustrations - including 88 full-colour pictures of Julia sets - are suggestive of a coffee-table book.
However, the invited contributions which round off the book lend the text the required formality. Benoit Mandelbrot gives a very personal account, in his idiosyncratic self-centred style, of his discovery of the fractals named after him and Adrien Douady explains the solved and unsolved problems relating to this amusingly complex set.

This book is based on notes for the course Fractals: lntroduction, Basics and Perspectives given by MichaelF. Barnsley, RobertL. Devaney, Heinz-Otto Peit gen, Dietmar Saupe and Richard F. Voss. The course was chaired by Heinz-Otto Peitgen and was part of the SIGGRAPH '87 (Anaheim, California) course pro gram. Though the five chapters of this book have emerged from those courses we have tried to make this book a coherent and uniformly styled presentation as much as possible. It is the first book which discusses fractals solely from the point of view of computer graphics. Though fundamental concepts and algo rithms are not introduced and discussed in mathematical rigor we have made a serious attempt to justify and motivate wherever it appeared to be desirable. Ba sic algorithms are typically presented in pseudo-code or a description so close to code that a reader who is familiar with elementary computer graphics should find no problem to get started. Mandelbrot's fractal geometry provides both a description and a mathemat ical model for many of the seemingly complex forms and patterns in nature and the sciences. Fractals have blossomed enormously in the past few years and have helped reconnect pure mathematics research with both natural sciences and computing. Computer graphics has played an essential role both in its de velopment and rapidly growing popularity. Conversely, fractal geometry now plays an important role in the rendering, modelling and animation of natural phenomena and fantastic shapes in computer graphics."

ThisbookcomprisesthescientificproceedingsoftheInternationalWorkshop onDigitalMammography(IWDM),heldinBremen,Germany,fromJune22 toJune25,2002. Sinceitsinceptionin1992,IWDMhasgrownintotheIn- ternationalForumonDigitalMammography. TheuniquenessofIWDMisits attractivenesstoresearchers,medicalpractitioners,andindustrialdevelopers. F6rmerworkshopswereheldinSanJose,USA(1992),York,UK(1994), Chicago,USA(1996),Nijmegen,NL(1998),andToronto,Canada(2000). Each ofthesescientificeventshasbeencombinedwithverysuccessfulandfocused industrialandresearchexhibits,whichdemonstratedthemilestonesofdigital mammographyhard-andsoftware. Now,thatdigitalmammographyisenteringroutineclinicaluse,wewitness itsstrongimpactonscreeninganddiagnosticmammography,computeraided detection,minimallyinvasiveprocedures,andthedevelopmentofsystemicand integrateddisease-orientedbreastcare. Inviewofthis,the2002workshoppro- videdawindowtogiveusaglanceatthefuture,andtheworkwhichwas presentedintalks,postersandexhibitshasdemonstratedthatIWDM2002will takeaspecialplaceintheverysuccessfulsequenceofIWDMevents. Finally,asconferencechair,Iwouldliketothankallwhohavehelpedtopre- pareandrunIWDM2002:theScientificAdvisoryBoardforitsexcellentworkin guaranteeingscientificsignificance,theOrganizationalBoardforitshardworkto accomplishanappropriateframeworkfortheevent,andtheindustrialexhibitors andsponsorsfortheirgeneroussupport. Bremen,January2003 Heinz-Otto Peitgen ConferenceChair ProfessorofMathematicsand BiomedicalSciences VI Scientific Committee SusanAstley UniversityofManchester,UK UlrichBick CharitBerlin,Germany/UniversityofChicago,USA MichaelBrady OxfordUniversity,UK HiroshiFujita GifuUniversity,Japan MaryellenGiger UniversityofChicago,USA NicoKarssemeijer UniversityHospitalNijmegen,TheNetherlands Heinz-OttoPeitgenUniversityofBremen,Germany EttaD. Pisano UniversityofNorthCarolina,USA MartinYaffe UniversityofToronto,Canada IWDM 2002 was supported by (MeViS SIEMENS BreastCare medical I' General Electric ARC-0 Fllmle. . Perfection for DI I JMammography Table of Contents Imaging Systems and Detectors...1 DigitalMammographicApplicationofaSinglePhotonCountingPixel Detector...3 Amendolia SR, Bisogni MG, Delogu P, Fantacci ME, Novelli M, Quattrocchi M, Rosso V, Stefanini A, Zucca S BreastCancerScreeningUsingaDedicatedBreastCTScanner:A FeasibilityStudy...6 Boone JM, Lindfors KK, Seibert JA, Nelson TR PhysicalEvaluationofaPrototypefortheSectraMicrodose MammographySystem...12 Lundqvist M, Bergstrom D, Cederstrom B, Chmill V, Chuntonov A, Danielsson M, Aslund M DoseMeasurementsonaScanningMulti-slitDigitalMammography System...17 Hansson B, Cederstrom B, Danielsson M, Aslund M DigitalMammographyvs. toScreen-FilmMammography:aPhantom Study...20 Heid P, PagliuchiC, Seradour B, Motte N, Langlois L, Kurkdjian S, Piana L MammographyTaxonomyforImprovementofLesionDetectionRates...27 Howard D, Roberts SC, Tabar L AHighResolution(25 p. m)HybridSelenium/CCDSlotScanDetector forDigitalMammography...33 Hunter DM, De CrescenzoG, Mainprize JG, Mawdsley GE, SmithC, Kasap SO, Rowlands JAR, Tumer T, Yin S, Yaffe MJ Co-registeredBreastImagingwith3DX-Raysand3DUltrasound...38 Kapur A, Eberhard JW, Abdalmajeid AM, Thomenius K ImagingCharacteristicsofaDirectConversionFull-FieldDigital MammographyDetectorusingSelenium...4 3 Lee DL, Yorker lG, ling Z, leromin LS SimultaneousAcquisitionofDiffraction,ScatteringandPhase-Contrast ImagesbyMeansofaMulti-LayerEdge-OnMicrostripDetector 48 Olivo A, Arfelli P, Bergamaschi A, Dreossi D, Longo R, Menk RH, Montanari F, Pani S, Poropat P, Rigon L, Vallazza E, Castelli E VIII FullFieldDigitalMammographywithaCCDBasedSlot-Scanned Detector. PhysicalCharacteristicsMeasurements...

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

Written by the award winning authors of Chaos and Fractals (0-387-97903-4), this work introduces the reader to iterated function systems through a lively, interactive approach. This well-written, clearly illustrated book explores the history and the unlimited potential of fractals, while developing a basic mathematical understanding and appreciation for the topics.

There are many reasons for writing this first volume of strategic activities on fractals. The most pervasive is the compelling desire to provide students of mathematics with a set of accessible, hands-on experiences with fractals and their underlying mathematical principles and characteristics. Another is to show how fractals connect to many different aspects of mathematics and how the study of fractals can bring these ideas together. A third is to share the beauty of their structure and shape both through what the eye sees and what the mind visualizes. Fractals have captured the attention, enthusiasm, and interest of many people around the world. To the casual observer, their color, beauty, and geometric structure captivates the visual senses like few other things they have ever experienced in mathematics. To the computer scientist, fractals offer a rich environment in which to explore, create, and build a new visual world as an artist creating a new work. To the student, fractals bring mathematics out of past history and into the twenty-first century. To the mathematics teacher, fractals offer a unique, new opportunity to illustrate both the dynamics of mathematics and its many connecting links.

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

Fractal geometry has become popular in the last 15 years, its applications can be found in technology, science, or even arts. Fractal methods and formalism are seen today as a general, abstract, but nevertheless practical instrument for the description of nature in a wide sense. But it was Computer Graphics which made possible the increasing popularity of fractals several years ago, and long after their mathematical formulation. The two disciplines are tightly linked. The book contains the scientificcontributions presented in an international workshop in the "Computer Graphics Center" in Darmstadt, Germany. The target of the workshop was to present the wide spectrum of interrelationships and interactions between Fractal Geometry and Computer Graphics. The topics vary from fundamentals and new theoretical results to various applications and systems development. All contributions are original, unpublished papers.The presentations have been discussed in two working groups; the discussion results, together with actual trends and topics of future research, are reported in the last section. The topics of the book are divides into four sections: Fundamentals, Computer Graphics and Optical Simulation, Simulation of Natural Phenomena, Image Processing and Image Analysis.

The same factors that motivated the writing of our first volume of strategic activities on fractals continued to encourage the assembly of additional activities for this second volume. Fractals provide a setting wherein students can enjoy hands-on experiences that involve important mathematical content connected to a wide range of physical and social phenomena. The striking graphic images, unexpected geometric properties, and fascinating numerical processes offer unparalleled opportunity for enthusiastic student inquiry. Students sense the vigor present in the growing and highly integrative discipline of fractal geom etry as they are introduced to mathematical developments that have occurred during the last half of the twentieth century. Few branches of mathematics and computer science offer such a contem porary portrayal of the wonderment available in careful analysis, in the amazing dialogue between numeric and geometric processes, and in the energetic interaction between mathematics and other disciplines. Fractals continue to supply an uncommon setting for animated teaching and learn ing activities that focus upon fundamental mathematical concepts, connections, problem-solving techniques, and many other major topics of elementary and advanced mathematics. It remains our hope that, through this second volume of strategic activities, readers will find their enjoyment of mathematics heightened and their appreciation for the dynamics of the world in creased. We want experiences with fractals to enliven curiosity and to stretch the imagination."

Die Realitiit ist vielleicht das reinste Chaos. Georg Christoph Lichtenberg 1953 erkannte ich, daj3 die gerade Linie zum Untergang der Menschheitfiihrt. Aber die gerade Linie ist zur absoluten Tyrannei geworden. Die gerade Linie ist der Fluch unserer Zivilisation. Heute erleben wir den Triumph der rationalen Technik, und wiihrenddessen befinden wir uns gleichzeitig vor dem Nichts. Friedensreich Hundertwasser Dieses Buch ist weder ein typisches Mathematikbuch noch ein iibliches popularwis- senschaftliches Buch. Vielmehr war beabsichtigt, eine Art Lesebuch vorzulegen, das es auch Laien erlaubt, ohne den Ballast zu vieler technisch-mathematischer Notationen, einen soliden Einblick in die Welt der aktuellen Chaostheorie und der fraktalen Geo- metrie zu gewinnen. Dieser erste Band konzentriert sich dabei mehr auf geometrische Phanomene, wahrend der zweite Band Chaos - Bausteine der Ordnung sich vor allem auf dynamische Phanomene stiitzt. Seit Ende der siebziger Jahre lauft eine Welle durch Mathematik und Naturwissen- schaften, die in ihrer Kraft, Kreativitat und Weitraumigkeit langst ein interdisziplinares Ereignis ersten Ranges geworden ist: Chaos und Fraktale. Dies ist umso bemerkenswer- ter, als sich die Chaostheorie und die fraktale Geometrie eigentlich in keiner Hinsicht mit den groBartigen Entwiirfen dieses Jahrhunderts, wie etwa der Quantentheorie oder der Relativitatstheorie, messen k6nnen. Chaostheorie und fraktale Geometrie haben Naturwissenschaftler und Mathematiker mit einer Reihe von Dberraschungen konfrontiert, deren Konsequenzen im Verhaltnis zu den Angeboten einer sich oft omnipotent gebenden Wissenschaft und Technik zugleich emiichternd und dramatisch sind: - Zahlreiche Phanomene sind trotz strengem naturgesetzlichem Deterrninismus prin- zipiell nicht prognostizierbar.

Das vorliegende Arbeitsbuch ist Teil eines Paketes von verschiedenen Materialien, die das Ziel haben, das Thema Chaos und Fraktale in den mathematisch-naturwissenschaftlichen Unterricht einzufuhren. Ein weiteres Anliegen besteht darin, das mentale Bild von Mathematik im Bewusstsein der Schuler attraktiver zu gestalten. Mathematik ist die Antwort des Menschen auf die Komplexitat der Welt. Mathematik ist die Ordnungsmacht im Dschungel der Phanomene. Deshalb ist Mathematik le bendig, frisch und aktuell. Deshalb gibt es zwischen einzelnen Teilgebieten und Ergebnissen der Mathematik immer wieder uberraschende Querverbindungen, die oft das Verstandnis einer Sache erst wirklich erhellen. Und deshalb bietet es sich an, durch entdeckendes, explorierendes Lernen die Anziehungskraft dieser Eigenschaften der Mathematik im Unterricht auszunutzen. Chaos und Fraktale bieten hierfur eine besondere neue Chance. Beide sind jung und aktuell und belegen so ohne weiteres, dass Mathematik lebt. Fur beide gilt, dass einige ihrer schrittmachenden Entdeckungen nicht ohne Hilfe von Computern moeglich gewesen waren. Damit rucken faszinie rende Computerexperimente naturlich in den Mittelpunkt. Beide sind hochgradig interdisziplinar. Dieses heisst, dass gehaltvolle Anwendungen nicht erst muhsam konstruiert werden mussen. Beide behandeln Themen, die von sich aus wirken. Tatsachlich durchlaufen seit Ende der siebziger Jahre Mathematik und Naturwissenschaften eine Welle, die in ihrer Kraft, Kreativitat und Weitraumigkeit langst ein interdisziplinares Ereignis er sten Ranges geworden ist. Das andauernde Interesse innerhalb und ausserhalb der Wissenschaften ist in einer aufruttelnden Betroffenheit begrundet, die eine radikale Wende in dem uberkommenen naturwissenschaftlichen Weltbild und manchen uberdehnten Interpretationen ankundigt.